id	sid	tid	token	lemma	pos
ejde-624	1	1	electronic	electronic	ADJ
ejde-624	1	2	journal	journal	NOUN
ejde-624	1	3	of	of	ADP
ejde-624	1	4	differential	differential	ADJ
ejde-624	1	5	equations	equation	NOUN
ejde-624	1	6	,	,	PUNCT
ejde-624	1	7	vol	vol	NOUN
ejde-624	1	8	.	.	NOUN
ejde-624	1	9	2024	2024	NUM
ejde-624	1	10	(	(	PUNCT
ejde-624	1	11	2024	2024	NUM
ejde-624	1	12	)	)	PUNCT
ejde-624	1	13	,	,	PUNCT
ejde-624	1	14	no	no	INTJ
ejde-624	1	15	.	.	NOUN
ejde-624	1	16	40	40	NUM
ejde-624	1	17	,	,	PUNCT
ejde-624	1	18	pp	pp	PROPN
ejde-624	1	19	.	.	PUNCT
ejde-624	2	1	1–16	1–16	PROPN
ejde-624	2	2	.	.	PUNCT
ejde-624	3	1	issn	issn	PROPN
ejde-624	3	2	:	:	PUNCT
ejde-624	3	3	1072	1072	NUM
ejde-624	3	4	-	-	SYM
ejde-624	3	5	6691	6691	NUM
ejde-624	3	6	.	.	PUNCT
ejde-624	4	1	url	url	PROPN
ejde-624	4	2	:	:	PUNCT
ejde-624	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-624	4	4	,	,	PUNCT
ejde-624	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-624	4	6	doi	doi	PROPN
ejde-624	4	7	:	:	PUNCT
ejde-624	4	8	10.58997	10.58997	NUM
ejde-624	4	9	/	/	SYM
ejde-624	4	10	ejde.2024.40w	ejde.2024.40w	PROPN
ejde-624	4	11	nonexistence	nonexistence	NOUN
ejde-624	4	12	results	result	NOUN
ejde-624	4	13	for	for	ADP
ejde-624	4	14	fractional	fractional	ADJ
ejde-624	4	15	differential	differential	ADJ
ejde-624	4	16	inequalities	inequality	NOUN
ejde-624	4	17	jeffrey	jeffrey	PROPN
ejde-624	4	18	r.	r.	PROPN
ejde-624	4	19	l.	l.	PROPN
ejde-624	4	20	webb	webb	PROPN
ejde-624	4	21	abstract	abstract	PROPN
ejde-624	4	22	.	.	PUNCT
ejde-624	5	1	we	we	PRON
ejde-624	5	2	prove	prove	VERB
ejde-624	5	3	nonexistence	nonexistence	NOUN
ejde-624	5	4	of	of	ADP
ejde-624	5	5	global	global	ADJ
ejde-624	5	6	solution	solution	NOUN
ejde-624	5	7	of	of	ADP
ejde-624	5	8	fractional	fractional	ADJ
ejde-624	5	9	differential	differential	ADJ
ejde-624	5	10	inequalities	inequality	NOUN
ejde-624	5	11	of	of	ADP
ejde-624	5	12	the	the	DET
ejde-624	5	13	form	form	NOUN
ejde-624	5	14	dαu(t	dαu(t	PROPN
ejde-624	5	15	)	)	PUNCT
ejde-624	5	16	≥	≥	NOUN
ejde-624	5	17	λtβ	λtβ	NOUN
ejde-624	5	18	|u(t)|p	|u(t)|p	NOUN
ejde-624	5	19	when	when	SCONJ
ejde-624	5	20	p	p	PROPN
ejde-624	5	21	>	>	X
ejde-624	5	22	1	1	NUM
ejde-624	5	23	for	for	ADP
ejde-624	5	24	each	each	PRON
ejde-624	5	25	of	of	ADP
ejde-624	5	26	the	the	DET
ejde-624	5	27	riemann	riemann	PROPN
ejde-624	5	28	-	-	PUNCT
ejde-624	5	29	liouville	liouville	PROPN
ejde-624	5	30	and	and	CCONJ
ejde-624	5	31	caputo	caputo	PROPN
ejde-624	5	32	fractional	fractional	ADJ
ejde-624	5	33	derivatives	derivative	NOUN
ejde-624	5	34	.	.	PUNCT
ejde-624	6	1	this	this	PRON
ejde-624	6	2	is	be	AUX
ejde-624	6	3	motivated	motivate	VERB
ejde-624	6	4	by	by	ADP
ejde-624	6	5	work	work	NOUN
ejde-624	6	6	of	of	ADP
ejde-624	6	7	laskri	laskri	NOUN
ejde-624	6	8	and	and	CCONJ
ejde-624	6	9	tatar	tatar	NOUN
ejde-624	6	10	(	(	PUNCT
ejde-624	6	11	comput	comput	NOUN
ejde-624	6	12	.	.	PUNCT
ejde-624	7	1	math	math	NOUN
ejde-624	7	2	.	.	PUNCT
ejde-624	8	1	appl	appl	PROPN
ejde-624	8	2	.	.	PUNCT
ejde-624	9	1	(	(	PUNCT
ejde-624	9	2	2010	2010	NUM
ejde-624	9	3	)	)	PUNCT
ejde-624	9	4	)	)	PUNCT
ejde-624	10	1	and	and	CCONJ
ejde-624	10	2	shan	shan	PROPN
ejde-624	10	3	and	and	CCONJ
ejde-624	10	4	lv	lv	PROPN
ejde-624	10	5	(	(	PUNCT
ejde-624	10	6	filomat	filomat	PROPN
ejde-624	10	7	(	(	PUNCT
ejde-624	10	8	2024	2024	NUM
ejde-624	10	9	)	)	PUNCT
ejde-624	10	10	)	)	PUNCT
ejde-624	10	11	.	.	PUNCT
ejde-624	11	1	the	the	DET
ejde-624	11	2	result	result	NOUN
ejde-624	11	3	of	of	ADP
ejde-624	11	4	laskri	laskri	NOUN
ejde-624	11	5	-	-	PUNCT
ejde-624	11	6	tatar	tatar	NOUN
ejde-624	11	7	was	be	AUX
ejde-624	11	8	claimed	claim	VERB
ejde-624	11	9	to	to	PART
ejde-624	11	10	be	be	AUX
ejde-624	11	11	false	false	ADJ
ejde-624	11	12	by	by	ADP
ejde-624	11	13	zhang	zhang	PROPN
ejde-624	11	14	,	,	PUNCT
ejde-624	11	15	liu	liu	PROPN
ejde-624	11	16	,	,	PUNCT
ejde-624	11	17	wu	wu	PROPN
ejde-624	11	18	and	and	CCONJ
ejde-624	11	19	cui	cui	PROPN
ejde-624	11	20	(	(	PUNCT
ejde-624	11	21	j.	j.	PROPN
ejde-624	11	22	funct	funct	PROPN
ejde-624	11	23	.	.	PUNCT
ejde-624	12	1	spaces	space	NOUN
ejde-624	12	2	(	(	PUNCT
ejde-624	12	3	2017	2017	NUM
ejde-624	12	4	)	)	PUNCT
ejde-624	12	5	)	)	PUNCT
ejde-624	12	6	with	with	ADP
ejde-624	12	7	a	a	DET
ejde-624	12	8	correction	correction	NOUN
ejde-624	12	9	and	and	CCONJ
ejde-624	12	10	counterexample	counterexample	NOUN
ejde-624	12	11	.	.	PUNCT
ejde-624	13	1	we	we	PRON
ejde-624	13	2	show	show	VERB
ejde-624	13	3	that	that	SCONJ
ejde-624	13	4	the	the	DET
ejde-624	13	5	counter	counter	NOUN
ejde-624	13	6	-	-	NOUN
ejde-624	13	7	example	example	NOUN
ejde-624	13	8	and	and	CCONJ
ejde-624	13	9	the	the	DET
ejde-624	13	10	claims	claim	NOUN
ejde-624	13	11	are	be	AUX
ejde-624	13	12	not	not	PART
ejde-624	13	13	accurate	accurate	ADJ
ejde-624	13	14	.	.	PUNCT
ejde-624	14	1	we	we	PRON
ejde-624	14	2	use	use	VERB
ejde-624	14	3	a	a	DET
ejde-624	14	4	different	different	ADJ
ejde-624	14	5	method	method	NOUN
ejde-624	14	6	to	to	ADP
ejde-624	14	7	that	that	PRON
ejde-624	14	8	of	of	ADP
ejde-624	14	9	laskri	laskri	NOUN
ejde-624	14	10	and	and	CCONJ
ejde-624	14	11	tatar	tatar	NOUN
ejde-624	14	12	,	,	PUNCT
ejde-624	14	13	our	our	PRON
ejde-624	14	14	result	result	NOUN
ejde-624	14	15	supports	support	VERB
ejde-624	14	16	the	the	DET
ejde-624	14	17	one	one	NUM
ejde-624	14	18	of	of	ADP
ejde-624	14	19	laskri	laskri	NOUN
ejde-624	14	20	and	and	CCONJ
ejde-624	14	21	tatar	tatar	NOUN
ejde-624	14	22	.	.	PUNCT
ejde-624	15	1	we	we	PRON
ejde-624	15	2	also	also	ADV
ejde-624	15	3	improve	improve	VERB
ejde-624	15	4	on	on	ADP
ejde-624	15	5	the	the	DET
ejde-624	15	6	result	result	NOUN
ejde-624	15	7	in	in	ADP
ejde-624	15	8	shan	shan	PROPN
ejde-624	15	9	and	and	CCONJ
ejde-624	15	10	lv	lv	PROPN
ejde-624	15	11	paper	paper	NOUN
ejde-624	15	12	by	by	ADP
ejde-624	15	13	considering	consider	VERB
ejde-624	15	14	a	a	DET
ejde-624	15	15	more	more	ADV
ejde-624	15	16	general	general	ADJ
ejde-624	15	17	problem	problem	NOUN
ejde-624	15	18	and	and	CCONJ
ejde-624	15	19	giving	give	VERB
ejde-624	15	20	a	a	DET
ejde-624	15	21	more	more	ADV
ejde-624	15	22	precise	precise	ADJ
ejde-624	15	23	conclusion	conclusion	NOUN
ejde-624	15	24	.	.	PUNCT
ejde-624	16	1	1	1	X
ejde-624	16	2	.	.	X
ejde-624	16	3	introduction	introduction	NOUN
ejde-624	16	4	some	some	DET
ejde-624	16	5	years	year	NOUN
ejde-624	16	6	ago	ago	ADV
ejde-624	16	7	laskri	laskri	NOUN
ejde-624	16	8	and	and	CCONJ
ejde-624	16	9	tatar	tatar	NOUN
ejde-624	16	10	[	[	X
ejde-624	16	11	15	15	NUM
ejde-624	16	12	]	]	PUNCT
ejde-624	16	13	proved	prove	VERB
ejde-624	16	14	that	that	SCONJ
ejde-624	16	15	global	global	ADJ
ejde-624	16	16	solutions	solution	NOUN
ejde-624	16	17	of	of	ADP
ejde-624	16	18	an	an	DET
ejde-624	16	19	inequality	inequality	NOUN
ejde-624	16	20	for	for	ADP
ejde-624	16	21	the	the	DET
ejde-624	16	22	riemann	riemann	PROPN
ejde-624	16	23	-	-	PUNCT
ejde-624	16	24	liouville	liouville	NOUN
ejde-624	16	25	(	(	PUNCT
ejde-624	16	26	r	r	NOUN
ejde-624	16	27	-	-	PUNCT
ejde-624	16	28	l	l	NOUN
ejde-624	16	29	)	)	PUNCT
ejde-624	16	30	fractional	fractional	ADJ
ejde-624	16	31	derivative	derivative	NOUN
ejde-624	16	32	do	do	AUX
ejde-624	16	33	not	not	PART
ejde-624	16	34	exist	exist	VERB
ejde-624	16	35	.	.	PUNCT
ejde-624	17	1	they	they	PRON
ejde-624	17	2	study	study	VERB
ejde-624	17	3	the	the	DET
ejde-624	17	4	inequality	inequality	NOUN
ejde-624	17	5	dαu(t	dαu(t	PROPN
ejde-624	17	6	)	)	PUNCT
ejde-624	17	7	≥	≥	NOUN
ejde-624	17	8	tβ	tβ	PROPN
ejde-624	17	9	|u(t)|p	|u(t)|p	NOUN
ejde-624	17	10	,	,	PUNCT
ejde-624	17	11	t	t	X
ejde-624	17	12	>	>	X
ejde-624	17	13	0	0	PROPN
ejde-624	17	14	,	,	PUNCT
ejde-624	17	15	where	where	SCONJ
ejde-624	17	16	p	p	X
ejde-624	17	17	>	>	X
ejde-624	17	18	1	1	NUM
ejde-624	17	19	,	,	PUNCT
ejde-624	17	20	0	0	NUM
ejde-624	17	21	<	<	X
ejde-624	17	22	α	α	X
ejde-624	17	23	<	<	X
ejde-624	17	24	1	1	NUM
ejde-624	17	25	and	and	CCONJ
ejde-624	17	26	β	β	ADJ
ejde-624	17	27	>	>	X
ejde-624	17	28	−α	−α	PROPN
ejde-624	17	29	,	,	PUNCT
ejde-624	17	30	(	(	PUNCT
ejde-624	17	31	1.1	1.1	NUM
ejde-624	17	32	)	)	PUNCT
ejde-624	17	33	with	with	ADP
ejde-624	17	34	initial	initial	ADJ
ejde-624	17	35	condition	condition	NOUN
ejde-624	17	36	(	(	PUNCT
ejde-624	17	37	ic	ic	NOUN
ejde-624	17	38	)	)	PUNCT
ejde-624	17	39	i1−αu(0	i1−αu(0	NOUN
ejde-624	17	40	)	)	PUNCT
ejde-624	18	1	=	=	SYM
ejde-624	18	2	b	b	PROPN
ejde-624	18	3	≥	≥	NOUN
ejde-624	18	4	0	0	NUM
ejde-624	18	5	,	,	PUNCT
ejde-624	18	6	where	where	SCONJ
ejde-624	18	7	i1−αu	i1−αu	NOUN
ejde-624	18	8	is	be	AUX
ejde-624	18	9	the	the	DET
ejde-624	18	10	r	r	NOUN
ejde-624	18	11	-	-	PUNCT
ejde-624	18	12	l	l	NOUN
ejde-624	18	13	fractional	fractional	ADJ
ejde-624	18	14	integral	integral	ADJ
ejde-624	18	15	and	and	CCONJ
ejde-624	18	16	dαu	dαu	ADJ
ejde-624	18	17	=	=	NOUN
ejde-624	18	18	d(i1−αu	d(i1−αu	NOUN
ejde-624	18	19	)	)	PUNCT
ejde-624	18	20	denotes	denote	VERB
ejde-624	18	21	the	the	DET
ejde-624	18	22	r	r	NOUN
ejde-624	18	23	-	-	PUNCT
ejde-624	18	24	l	l	NOUN
ejde-624	18	25	fractional	fractional	ADJ
ejde-624	18	26	derivative	derivative	NOUN
ejde-624	18	27	;	;	PUNCT
ejde-624	18	28	detailed	detailed	ADJ
ejde-624	18	29	definitions	definition	NOUN
ejde-624	18	30	are	be	AUX
ejde-624	18	31	given	give	VERB
ejde-624	18	32	later	later	ADV
ejde-624	18	33	in	in	ADP
ejde-624	18	34	the	the	DET
ejde-624	18	35	paper	paper	NOUN
ejde-624	18	36	.	.	PUNCT
ejde-624	19	1	the	the	DET
ejde-624	19	2	authors	author	NOUN
ejde-624	19	3	considered	consider	VERB
ejde-624	19	4	solutions	solution	NOUN
ejde-624	19	5	belonging	belong	VERB
ejde-624	19	6	to	to	ADP
ejde-624	19	7	a	a	DET
ejde-624	19	8	space	space	NOUN
ejde-624	19	9	they	they	PRON
ejde-624	19	10	denoted	denote	VERB
ejde-624	19	11	lα	lα	ADP
ejde-624	19	12	:	:	PUNCT
ejde-624	19	13	=	=	SYM
ejde-624	19	14	{	{	PUNCT
ejde-624	19	15	u	u	NOUN
ejde-624	19	16	∈	∈	PROPN
ejde-624	19	17	l1	l1	PROPN
ejde-624	19	18	:	:	PUNCT
ejde-624	19	19	dαu	dαu	PROPN
ejde-624	19	20	∈	∈	PROPN
ejde-624	19	21	l1	l1	PROPN
ejde-624	19	22	}	}	PUNCT
ejde-624	19	23	.	.	PUNCT
ejde-624	20	1	their	their	PRON
ejde-624	20	2	result	result	NOUN
ejde-624	20	3	is	be	AUX
ejde-624	20	4	the	the	DET
ejde-624	20	5	following	following	NOUN
ejde-624	20	6	.	.	PUNCT
ejde-624	21	1	theorem	theorem	VERB
ejde-624	21	2	1.1	1.1	NUM
ejde-624	21	3	(	(	PUNCT
ejde-624	21	4	[	[	X
ejde-624	21	5	15	15	NUM
ejde-624	21	6	,	,	PUNCT
ejde-624	21	7	theorem	theorem	VERB
ejde-624	21	8	1	1	NUM
ejde-624	21	9	]	]	PUNCT
ejde-624	21	10	)	)	PUNCT
ejde-624	21	11	.	.	PUNCT
ejde-624	22	1	assume	assume	VERB
ejde-624	22	2	that	that	SCONJ
ejde-624	22	3	β+α	β+α	PUNCT
ejde-624	22	4	>	>	X
ejde-624	22	5	0	0	PUNCT
ejde-624	22	6	and	and	CCONJ
ejde-624	22	7	1	1	NUM
ejde-624	22	8	<	<	X
ejde-624	22	9	p	p	X
ejde-624	22	10	<	<	X
ejde-624	22	11	1+β	1+β	PROPN
ejde-624	22	12	1−α	1−α	NUM
ejde-624	22	13	.	.	PUNCT
ejde-624	23	1	then	then	ADV
ejde-624	23	2	,	,	PUNCT
ejde-624	23	3	problem	problem	NOUN
ejde-624	23	4	(	(	PUNCT
ejde-624	23	5	1.1	1.1	NUM
ejde-624	23	6	)	)	PUNCT
ejde-624	23	7	does	do	AUX
ejde-624	23	8	not	not	PART
ejde-624	23	9	admit	admit	VERB
ejde-624	23	10	global	global	ADJ
ejde-624	23	11	nontrivial	nontrivial	ADJ
ejde-624	23	12	solutions	solution	NOUN
ejde-624	23	13	when	when	SCONJ
ejde-624	23	14	b	b	PROPN
ejde-624	23	15	≥	≥	NOUN
ejde-624	23	16	0	0	NUM
ejde-624	23	17	.	.	PUNCT
ejde-624	24	1	they	they	PRON
ejde-624	24	2	give	give	VERB
ejde-624	24	3	an	an	DET
ejde-624	24	4	example	example	NOUN
ejde-624	24	5	where	where	SCONJ
ejde-624	24	6	for	for	ADP
ejde-624	24	7	p	p	PRON
ejde-624	24	8	≥	≥	NOUN
ejde-624	24	9	1+β	1+β	NUM
ejde-624	24	10	1−α	1−α	NUM
ejde-624	24	11	non	non	ADJ
ejde-624	24	12	-	-	ADJ
ejde-624	24	13	zero	zero	NUM
ejde-624	24	14	solutions	solution	NOUN
ejde-624	24	15	exist	exist	VERB
ejde-624	24	16	for	for	ADP
ejde-624	24	17	all	all	DET
ejde-624	24	18	t	t	PROPN
ejde-624	24	19	,	,	PUNCT
ejde-624	24	20	so	so	ADV
ejde-624	24	21	p0	p0	NOUN
ejde-624	24	22	:	:	PUNCT
ejde-624	25	1	=	=	SYM
ejde-624	25	2	1+β	1+β	NUM
ejde-624	25	3	1−α	1−α	NUM
ejde-624	25	4	is	be	AUX
ejde-624	25	5	a	a	DET
ejde-624	25	6	critical	critical	ADJ
ejde-624	25	7	exponent	exponent	NOUN
ejde-624	25	8	.	.	PUNCT
ejde-624	26	1	we	we	PRON
ejde-624	26	2	will	will	AUX
ejde-624	26	3	show	show	VERB
ejde-624	26	4	that	that	SCONJ
ejde-624	26	5	for	for	ADP
ejde-624	26	6	p	p	PRON
ejde-624	26	7	≥	≥	NOUN
ejde-624	26	8	p0	p0	NOUN
ejde-624	26	9	and	and	CCONJ
ejde-624	26	10	b	b	NOUN
ejde-624	26	11	>	>	X
ejde-624	26	12	0	0	PUNCT
ejde-624	26	13	there	there	PRON
ejde-624	26	14	is	be	VERB
ejde-624	26	15	no	no	DET
ejde-624	26	16	solution	solution	NOUN
ejde-624	26	17	in	in	ADP
ejde-624	26	18	the	the	DET
ejde-624	26	19	space	space	NOUN
ejde-624	26	20	we	we	PRON
ejde-624	26	21	use	use	VERB
ejde-624	26	22	for	for	ADP
ejde-624	26	23	solutions	solution	NOUN
ejde-624	26	24	,	,	PUNCT
ejde-624	26	25	while	while	SCONJ
ejde-624	26	26	for	for	SCONJ
ejde-624	26	27	p	p	NOUN
ejde-624	26	28	<	<	X
ejde-624	26	29	1+β	1+β	NUM
ejde-624	26	30	1−α	1−α	NUM
ejde-624	26	31	any	any	DET
ejde-624	26	32	solution	solution	NOUN
ejde-624	26	33	must	must	AUX
ejde-624	26	34	fail	fail	VERB
ejde-624	26	35	to	to	PART
ejde-624	26	36	exist	exist	VERB
ejde-624	26	37	at	at	ADP
ejde-624	26	38	or	or	CCONJ
ejde-624	26	39	before	before	ADP
ejde-624	26	40	some	some	DET
ejde-624	26	41	finite	finite	ADJ
ejde-624	26	42	value	value	NOUN
ejde-624	26	43	t1	t1	NOUN
ejde-624	26	44	explicitly	explicitly	ADV
ejde-624	26	45	determined	determine	VERB
ejde-624	26	46	by	by	ADP
ejde-624	26	47	the	the	DET
ejde-624	26	48	data	datum	NOUN
ejde-624	26	49	and	and	CCONJ
ejde-624	26	50	parameters	parameter	NOUN
ejde-624	26	51	.	.	PUNCT
ejde-624	27	1	the	the	DET
ejde-624	27	2	reason	reason	NOUN
ejde-624	27	3	for	for	ADP
ejde-624	27	4	these	these	DET
ejde-624	27	5	differences	difference	NOUN
ejde-624	27	6	is	be	AUX
ejde-624	27	7	that	that	SCONJ
ejde-624	27	8	we	we	PRON
ejde-624	27	9	have	have	VERB
ejde-624	27	10	b	b	NUM
ejde-624	27	11	>	>	X
ejde-624	27	12	0	0	PROPN
ejde-624	27	13	,	,	PUNCT
ejde-624	27	14	the	the	DET
ejde-624	27	15	example	example	NOUN
ejde-624	27	16	has	have	VERB
ejde-624	27	17	b	b	NOUN
ejde-624	27	18	=	=	SYM
ejde-624	27	19	0	0	NUM
ejde-624	27	20	.	.	PUNCT
ejde-624	28	1	we	we	PRON
ejde-624	28	2	expect	expect	VERB
ejde-624	28	3	that	that	SCONJ
ejde-624	28	4	for	for	ADP
ejde-624	28	5	1	1	NUM
ejde-624	28	6	<	<	X
ejde-624	28	7	p	p	X
ejde-624	28	8	<	<	X
ejde-624	28	9	1+β	1+β	NUM
ejde-624	28	10	1−α	1−α	NUM
ejde-624	28	11	solutions	solution	NOUN
ejde-624	28	12	will	will	AUX
ejde-624	28	13	become	become	VERB
ejde-624	28	14	unbounded	unbounded	ADJ
ejde-624	28	15	at	at	ADP
ejde-624	28	16	some	some	DET
ejde-624	28	17	point	point	NOUN
ejde-624	28	18	t2	t2	NOUN
ejde-624	28	19	≤	≤	NUM
ejde-624	28	20	t1	t1	NOUN
ejde-624	28	21	(	(	PUNCT
ejde-624	28	22	blow	blow	NOUN
ejde-624	28	23	-	-	PUNCT
ejde-624	28	24	up	up	NOUN
ejde-624	28	25	)	)	PUNCT
ejde-624	28	26	but	but	CCONJ
ejde-624	28	27	this	this	PRON
ejde-624	28	28	requires	require	VERB
ejde-624	28	29	knowledge	knowledge	NOUN
ejde-624	28	30	of	of	ADP
ejde-624	28	31	what	what	PRON
ejde-624	28	32	occurs	occur	VERB
ejde-624	28	33	at	at	ADP
ejde-624	28	34	the	the	DET
ejde-624	28	35	2020	2020	NUM
ejde-624	28	36	mathematics	mathematic	NOUN
ejde-624	28	37	subject	subject	ADJ
ejde-624	28	38	classification	classification	NOUN
ejde-624	28	39	.	.	PUNCT
ejde-624	29	1	34a08	34a08	NUM
ejde-624	29	2	,	,	PUNCT
ejde-624	29	3	34a40	34a40	NUM
ejde-624	29	4	,	,	PUNCT
ejde-624	29	5	45d05	45d05	NUM
ejde-624	29	6	.	.	PUNCT
ejde-624	30	1	key	key	ADJ
ejde-624	30	2	words	word	NOUN
ejde-624	30	3	and	and	CCONJ
ejde-624	30	4	phrases	phrase	NOUN
ejde-624	30	5	.	.	PUNCT
ejde-624	31	1	fractional	fractional	ADJ
ejde-624	31	2	differential	differential	ADJ
ejde-624	31	3	equations	equation	NOUN
ejde-624	31	4	;	;	PUNCT
ejde-624	31	5	non	non	ADJ
ejde-624	31	6	-	-	NOUN
ejde-624	31	7	existence	existence	NOUN
ejde-624	31	8	;	;	PUNCT
ejde-624	31	9	volterra	volterra	PROPN
ejde-624	31	10	integral	integral	ADJ
ejde-624	31	11	equation	equation	NOUN
ejde-624	31	12	.	.	PUNCT
ejde-624	32	1	©	©	PROPN
ejde-624	32	2	2024	2024	NUM
ejde-624	32	3	.	.	PUNCT
ejde-624	33	1	this	this	DET
ejde-624	33	2	work	work	NOUN
ejde-624	33	3	is	be	AUX
ejde-624	33	4	licensed	license	VERB
ejde-624	33	5	under	under	ADP
ejde-624	33	6	a	a	DET
ejde-624	33	7	cc	cc	NOUN
ejde-624	33	8	by	by	ADP
ejde-624	33	9	4.0	4.0	NUM
ejde-624	33	10	license	license	NOUN
ejde-624	33	11	.	.	PUNCT
ejde-624	34	1	submitted	submit	VERB
ejde-624	34	2	march	march	PROPN
ejde-624	34	3	25	25	NUM
ejde-624	34	4	,	,	PUNCT
ejde-624	34	5	2024	2024	NUM
ejde-624	34	6	.	.	PUNCT
ejde-624	35	1	published	publish	VERB
ejde-624	35	2	july	july	PROPN
ejde-624	35	3	30	30	NUM
ejde-624	35	4	,	,	PUNCT
ejde-624	35	5	2024	2024	NUM
ejde-624	35	6	.	.	PUNCT
ejde-624	35	7	1	1	NUM
ejde-624	35	8	2	2	NUM
ejde-624	35	9	j.	j.	PROPN
ejde-624	35	10	r.	r.	PROPN
ejde-624	35	11	l.	l.	PROPN
ejde-624	35	12	webb	webb	PROPN
ejde-624	35	13	ejde-2024/40	ejde-2024/40	PROPN
ejde-624	35	14	endpoint	endpoint	NOUN
ejde-624	35	15	of	of	ADP
ejde-624	35	16	a	a	DET
ejde-624	35	17	maximal	maximal	ADJ
ejde-624	35	18	interval	interval	NOUN
ejde-624	35	19	of	of	ADP
ejde-624	35	20	existence	existence	NOUN
ejde-624	35	21	and	and	CCONJ
ejde-624	35	22	we	we	PRON
ejde-624	35	23	do	do	AUX
ejde-624	35	24	not	not	PART
ejde-624	35	25	know	know	VERB
ejde-624	35	26	of	of	ADP
ejde-624	35	27	any	any	DET
ejde-624	35	28	such	such	ADJ
ejde-624	35	29	result	result	NOUN
ejde-624	35	30	for	for	ADP
ejde-624	35	31	r	r	NOUN
ejde-624	35	32	-	-	PUNCT
ejde-624	35	33	l	l	NOUN
ejde-624	35	34	differential	differential	ADJ
ejde-624	35	35	equations	equation	NOUN
ejde-624	35	36	.	.	PUNCT
ejde-624	36	1	zhang	zhang	PROPN
ejde-624	36	2	,	,	PUNCT
ejde-624	36	3	liu	liu	PROPN
ejde-624	36	4	,	,	PUNCT
ejde-624	36	5	wu	wu	PROPN
ejde-624	36	6	,	,	PUNCT
ejde-624	36	7	and	and	CCONJ
ejde-624	36	8	cui	cui	VERB
ejde-624	36	9	[	[	X
ejde-624	36	10	25	25	NUM
ejde-624	36	11	]	]	PUNCT
ejde-624	36	12	claim	claim	NOUN
ejde-624	36	13	that	that	SCONJ
ejde-624	36	14	the	the	DET
ejde-624	36	15	proof	proof	NOUN
ejde-624	36	16	in	in	ADP
ejde-624	36	17	[	[	X
ejde-624	36	18	15	15	NUM
ejde-624	36	19	]	]	PUNCT
ejde-624	36	20	has	have	AUX
ejde-624	36	21	a	a	DET
ejde-624	36	22	flaw	flaw	NOUN
ejde-624	36	23	and	and	CCONJ
ejde-624	36	24	that	that	SCONJ
ejde-624	36	25	the	the	DET
ejde-624	36	26	result	result	NOUN
ejde-624	36	27	is	be	AUX
ejde-624	36	28	not	not	PART
ejde-624	36	29	correct	correct	ADJ
ejde-624	36	30	.	.	PUNCT
ejde-624	37	1	they	they	PRON
ejde-624	37	2	give	give	VERB
ejde-624	37	3	a	a	DET
ejde-624	37	4	‘	'	PUNCT
ejde-624	37	5	counter	counter	NOUN
ejde-624	37	6	-	-	NOUN
ejde-624	37	7	example	example	NOUN
ejde-624	37	8	’	'	PUNCT
ejde-624	37	9	and	and	CCONJ
ejde-624	37	10	a	a	DET
ejde-624	37	11	‘	'	PUNCT
ejde-624	37	12	correct	correct	ADJ
ejde-624	37	13	version	version	NOUN
ejde-624	37	14	’	'	PUNCT
ejde-624	37	15	of	of	ADP
ejde-624	37	16	theorem	theorem	NOUN
ejde-624	37	17	1.1	1.1	NUM
ejde-624	37	18	.	.	PUNCT
ejde-624	38	1	unfortunately	unfortunately	ADV
ejde-624	38	2	the	the	DET
ejde-624	38	3	correction	correction	NOUN
ejde-624	38	4	is	be	AUX
ejde-624	38	5	wrong	wrong	ADJ
ejde-624	38	6	,	,	PUNCT
ejde-624	38	7	and	and	CCONJ
ejde-624	38	8	the	the	DET
ejde-624	38	9	counter	counter	NOUN
ejde-624	38	10	-	-	NOUN
ejde-624	38	11	example	example	NOUN
ejde-624	38	12	has	have	VERB
ejde-624	38	13	an	an	DET
ejde-624	38	14	error	error	NOUN
ejde-624	38	15	,	,	PUNCT
ejde-624	38	16	which	which	PRON
ejde-624	38	17	invalidates	invalidate	VERB
ejde-624	38	18	their	their	PRON
ejde-624	38	19	claims	claim	NOUN
ejde-624	38	20	.	.	PUNCT
ejde-624	39	1	we	we	PRON
ejde-624	39	2	explain	explain	VERB
ejde-624	39	3	this	this	PRON
ejde-624	39	4	fully	fully	ADV
ejde-624	39	5	in	in	ADP
ejde-624	39	6	section	section	NOUN
ejde-624	39	7	4	4	NUM
ejde-624	39	8	.	.	PUNCT
ejde-624	40	1	one	one	NUM
ejde-624	40	2	of	of	ADP
ejde-624	40	3	the	the	DET
ejde-624	40	4	problems	problem	NOUN
ejde-624	40	5	studied	study	VERB
ejde-624	40	6	by	by	ADP
ejde-624	40	7	shan	shan	PROPN
ejde-624	40	8	and	and	CCONJ
ejde-624	40	9	lv	lv	PROPN
ejde-624	41	1	[	[	X
ejde-624	41	2	18	18	NUM
ejde-624	41	3	]	]	PUNCT
ejde-624	41	4	is	be	AUX
ejde-624	41	5	the	the	DET
ejde-624	41	6	initial	initial	ADJ
ejde-624	41	7	value	value	NOUN
ejde-624	41	8	problem	problem	NOUN
ejde-624	41	9	(	(	PUNCT
ejde-624	41	10	ivp	ivp	NOUN
ejde-624	41	11	)	)	PUNCT
ejde-624	41	12	for	for	ADP
ejde-624	41	13	the	the	DET
ejde-624	41	14	caputo	caputo	PROPN
ejde-624	41	15	fractional	fractional	PROPN
ejde-624	41	16	derivative	derivative	PROPN
ejde-624	41	17	dα	dα	PROPN
ejde-624	41	18	∗	∗	NOUN
ejde-624	41	19	u(t	u(t	NOUN
ejde-624	41	20	)	)	PUNCT
ejde-624	41	21	=	=	PUNCT
ejde-624	41	22	up(t	up(t	NOUN
ejde-624	41	23	)	)	PUNCT
ejde-624	41	24	,	,	PUNCT
ejde-624	41	25	t	t	PROPN
ejde-624	41	26	>	>	X
ejde-624	41	27	0	0	NUM
ejde-624	41	28	,	,	PUNCT
ejde-624	41	29	with	with	ADP
ejde-624	41	30	ic	ic	PROPN
ejde-624	41	31	u(0	u(0	PROPN
ejde-624	41	32	)	)	PUNCT
ejde-624	41	33	=	=	PUNCT
ejde-624	41	34	u0	u0	VERB
ejde-624	41	35	>	>	X
ejde-624	41	36	0	0	NUM
ejde-624	41	37	.	.	PUNCT
ejde-624	42	1	(	(	PUNCT
ejde-624	42	2	1.2	1.2	NUM
ejde-624	42	3	)	)	PUNCT
ejde-624	42	4	shan	shan	NOUN
ejde-624	42	5	-	-	PUNCT
ejde-624	42	6	lv	lv	PROPN
ejde-624	42	7	assert	assert	VERB
ejde-624	42	8	that	that	SCONJ
ejde-624	42	9	when	when	SCONJ
ejde-624	42	10	1	1	NUM
ejde-624	42	11	<	<	X
ejde-624	42	12	p	p	X
ejde-624	42	13	≤	≤	ADJ
ejde-624	42	14	1/(1	1/(1	NUM
ejde-624	42	15	−	−	PROPN
ejde-624	42	16	α	α	NOUN
ejde-624	42	17	)	)	PUNCT
ejde-624	42	18	every	every	DET
ejde-624	42	19	solution	solution	NOUN
ejde-624	42	20	of	of	ADP
ejde-624	42	21	(	(	PUNCT
ejde-624	42	22	1.2	1.2	NUM
ejde-624	42	23	)	)	PUNCT
ejde-624	42	24	blows	blow	NOUN
ejde-624	42	25	-	-	PUNCT
ejde-624	42	26	up	up	ADP
ejde-624	42	27	(	(	PUNCT
ejde-624	42	28	becomes	become	VERB
ejde-624	42	29	unbounded	unbounded	ADJ
ejde-624	42	30	)	)	PUNCT
ejde-624	42	31	at	at	ADP
ejde-624	42	32	some	some	DET
ejde-624	42	33	t	t	NOUN
ejde-624	42	34	∗	∗	NOUN
ejde-624	42	35	and	and	CCONJ
ejde-624	42	36	the	the	DET
ejde-624	42	37	same	same	ADJ
ejde-624	42	38	holds	hold	VERB
ejde-624	42	39	when	when	SCONJ
ejde-624	42	40	p	p	X
ejde-624	42	41	>	>	X
ejde-624	42	42	1/(1−	1/(1−	NUM
ejde-624	42	43	α	α	NOUN
ejde-624	42	44	)	)	PUNCT
ejde-624	42	45	and	and	CCONJ
ejde-624	42	46	u0	u0	PROPN
ejde-624	42	47	has	have	VERB
ejde-624	42	48	a	a	DET
ejde-624	42	49	sufficiently	sufficiently	ADV
ejde-624	42	50	large	large	ADJ
ejde-624	42	51	positive	positive	ADJ
ejde-624	42	52	lower	lower	ADV
ejde-624	42	53	bound	bind	VERB
ejde-624	42	54	.	.	PUNCT
ejde-624	43	1	they	they	PRON
ejde-624	43	2	do	do	AUX
ejde-624	43	3	not	not	PART
ejde-624	43	4	fully	fully	ADV
ejde-624	43	5	prove	prove	VERB
ejde-624	43	6	this	this	PRON
ejde-624	43	7	,	,	PUNCT
ejde-624	43	8	they	they	PRON
ejde-624	43	9	prove	prove	VERB
ejde-624	43	10	nonexistence	nonexistence	NOUN
ejde-624	43	11	but	but	CCONJ
ejde-624	43	12	do	do	AUX
ejde-624	43	13	not	not	PART
ejde-624	43	14	prove	prove	VERB
ejde-624	43	15	blow	blow	NOUN
ejde-624	43	16	-	-	PUNCT
ejde-624	43	17	up	up	NOUN
ejde-624	43	18	nor	nor	CCONJ
ejde-624	43	19	do	do	AUX
ejde-624	43	20	they	they	PRON
ejde-624	43	21	mention	mention	VERB
ejde-624	43	22	any	any	DET
ejde-624	43	23	continuation	continuation	NOUN
ejde-624	43	24	result	result	NOUN
ejde-624	43	25	that	that	PRON
ejde-624	43	26	would	would	AUX
ejde-624	43	27	give	give	VERB
ejde-624	43	28	blow	blow	NOUN
ejde-624	43	29	-	-	PUNCT
ejde-624	43	30	up	up	NOUN
ejde-624	43	31	.	.	PUNCT
ejde-624	44	1	for	for	ADP
ejde-624	44	2	this	this	DET
ejde-624	44	3	caputo	caputo	PROPN
ejde-624	44	4	derivative	derivative	ADJ
ejde-624	44	5	equation	equation	NOUN
ejde-624	44	6	there	there	PRON
ejde-624	44	7	is	be	VERB
ejde-624	44	8	a	a	DET
ejde-624	44	9	continuation	continuation	NOUN
ejde-624	44	10	result	result	NOUN
ejde-624	44	11	of	of	ADP
ejde-624	44	12	eloe	eloe	ADJ
ejde-624	44	13	and	and	CCONJ
ejde-624	44	14	masthay	masthay	ADJ
ejde-624	44	15	[	[	X
ejde-624	44	16	6	6	NUM
ejde-624	44	17	,	,	PUNCT
ejde-624	44	18	theorem	theorem	VERB
ejde-624	44	19	2.4	2.4	NUM
ejde-624	44	20	]	]	PUNCT
ejde-624	44	21	and	and	CCONJ
ejde-624	44	22	of	of	ADP
ejde-624	44	23	wu	wu	PROPN
ejde-624	44	24	and	and	CCONJ
ejde-624	44	25	liu	liu	PROPN
ejde-624	45	1	[	[	X
ejde-624	45	2	24	24	NUM
ejde-624	45	3	,	,	PUNCT
ejde-624	45	4	theorem	theorem	VERB
ejde-624	45	5	4.1	4.1	NUM
ejde-624	45	6	]	]	PUNCT
ejde-624	45	7	which	which	PRON
ejde-624	45	8	asserts	assert	VERB
ejde-624	45	9	that	that	SCONJ
ejde-624	45	10	when	when	SCONJ
ejde-624	45	11	f	f	PROPN
ejde-624	45	12	is	be	AUX
ejde-624	45	13	continuous	continuous	ADJ
ejde-624	45	14	,	,	PUNCT
ejde-624	45	15	a	a	DET
ejde-624	45	16	solution	solution	NOUN
ejde-624	45	17	u	u	NOUN
ejde-624	45	18	of	of	ADP
ejde-624	45	19	dα	dα	ADP
ejde-624	45	20	∗	∗	NOUN
ejde-624	45	21	u(t	u(t	NOUN
ejde-624	45	22	)	)	PUNCT
ejde-624	45	23	=	=	SYM
ejde-624	46	1	f(t	f(t	NOUN
ejde-624	46	2	,	,	PUNCT
ejde-624	46	3	u(t	u(t	NOUN
ejde-624	46	4	)	)	PUNCT
ejde-624	46	5	)	)	PUNCT
ejde-624	46	6	,	,	PUNCT
ejde-624	46	7	t	t	X
ejde-624	46	8	>	>	X
ejde-624	46	9	0	0	NUM
ejde-624	46	10	,	,	PUNCT
ejde-624	46	11	with	with	ADP
ejde-624	46	12	ic	ic	PROPN
ejde-624	46	13	u(0	u(0	PROPN
ejde-624	46	14	)	)	PUNCT
ejde-624	46	15	=	=	PUNCT
ejde-624	47	1	u0	u0	ADJ
ejde-624	47	2	,	,	PUNCT
ejde-624	47	3	exists	exist	VERB
ejde-624	47	4	on	on	ADP
ejde-624	47	5	a	a	DET
ejde-624	47	6	maximal	maximal	ADJ
ejde-624	47	7	interval	interval	NOUN
ejde-624	47	8	of	of	ADP
ejde-624	47	9	existence	existence	NOUN
ejde-624	47	10	[	[	X
ejde-624	47	11	0	0	NUM
ejde-624	47	12	,	,	PUNCT
ejde-624	47	13	t0	t0	PROPN
ejde-624	47	14	)	)	PUNCT
ejde-624	47	15	and	and	CCONJ
ejde-624	47	16	,	,	PUNCT
ejde-624	47	17	if	if	SCONJ
ejde-624	47	18	t0	t0	PROPN
ejde-624	47	19	is	be	AUX
ejde-624	47	20	finite	finite	ADJ
ejde-624	47	21	,	,	PUNCT
ejde-624	47	22	u(t	u(t	NOUN
ejde-624	47	23	)	)	PUNCT
ejde-624	47	24	→	→	SYM
ejde-624	47	25	∞	∞	PROPN
ejde-624	47	26	as	as	ADP
ejde-624	47	27	t	t	PROPN
ejde-624	47	28	→	→	SYM
ejde-624	47	29	t−	t−	PROPN
ejde-624	47	30	0	0	NUM
ejde-624	47	31	.	.	PUNCT
ejde-624	48	1	this	this	PRON
ejde-624	48	2	could	could	AUX
ejde-624	48	3	be	be	AUX
ejde-624	48	4	used	use	VERB
ejde-624	48	5	to	to	PART
ejde-624	48	6	justify	justify	VERB
ejde-624	48	7	the	the	DET
ejde-624	48	8	blow	blow	VERB
ejde-624	48	9	-	-	PUNCT
ejde-624	48	10	up	up	ADP
ejde-624	48	11	claims	claim	NOUN
ejde-624	48	12	of	of	ADP
ejde-624	48	13	shan	shan	NOUN
ejde-624	48	14	-	-	PUNCT
ejde-624	48	15	lv	lv	PROPN
ejde-624	48	16	.	.	PUNCT
ejde-624	49	1	we	we	PRON
ejde-624	49	2	improve	improve	VERB
ejde-624	49	3	the	the	DET
ejde-624	49	4	shan	shan	ADJ
ejde-624	49	5	-	-	PUNCT
ejde-624	49	6	lv	lv	PROPN
ejde-624	49	7	result	result	NOUN
ejde-624	49	8	by	by	ADP
ejde-624	49	9	considering	consider	VERB
ejde-624	49	10	the	the	DET
ejde-624	49	11	more	more	ADV
ejde-624	49	12	general	general	ADJ
ejde-624	49	13	inequality	inequality	NOUN
ejde-624	49	14	as	as	ADP
ejde-624	49	15	in	in	ADP
ejde-624	49	16	(	(	PUNCT
ejde-624	49	17	1.1	1.1	NUM
ejde-624	49	18	)	)	PUNCT
ejde-624	49	19	with	with	ADP
ejde-624	49	20	the	the	DET
ejde-624	49	21	possibly	possibly	ADV
ejde-624	49	22	singular	singular	ADJ
ejde-624	49	23	term	term	NOUN
ejde-624	49	24	tβ	tβ	PRON
ejde-624	49	25	.	.	PUNCT
ejde-624	50	1	we	we	PRON
ejde-624	50	2	also	also	ADV
ejde-624	50	3	prove	prove	VERB
ejde-624	50	4	,	,	PUNCT
ejde-624	50	5	by	by	ADP
ejde-624	50	6	a	a	DET
ejde-624	50	7	simple	simple	ADJ
ejde-624	50	8	method	method	NOUN
ejde-624	50	9	,	,	PUNCT
ejde-624	50	10	an	an	DET
ejde-624	50	11	improved	improved	ADJ
ejde-624	50	12	result	result	NOUN
ejde-624	50	13	,	,	PUNCT
ejde-624	50	14	namely	namely	ADV
ejde-624	50	15	that	that	SCONJ
ejde-624	50	16	,	,	PUNCT
ejde-624	50	17	for	for	ADP
ejde-624	50	18	any	any	DET
ejde-624	50	19	initial	initial	ADJ
ejde-624	50	20	value	value	NOUN
ejde-624	50	21	u0	u0	NOUN
ejde-624	50	22	>	>	X
ejde-624	50	23	0	0	PROPN
ejde-624	50	24	,	,	PUNCT
ejde-624	50	25	when	when	SCONJ
ejde-624	50	26	p	p	PROPN
ejde-624	50	27	>	>	X
ejde-624	50	28	1	1	NUM
ejde-624	50	29	global	global	ADJ
ejde-624	50	30	solutions	solution	NOUN
ejde-624	50	31	do	do	AUX
ejde-624	50	32	not	not	PART
ejde-624	50	33	exist	exist	VERB
ejde-624	50	34	.	.	PUNCT
ejde-624	51	1	since	since	SCONJ
ejde-624	51	2	we	we	PRON
ejde-624	51	3	have	have	VERB
ejde-624	51	4	a	a	DET
ejde-624	51	5	more	more	ADV
ejde-624	51	6	general	general	ADJ
ejde-624	51	7	case	case	NOUN
ejde-624	51	8	of	of	ADP
ejde-624	51	9	an	an	DET
ejde-624	51	10	inequality	inequality	NOUN
ejde-624	51	11	and	and	CCONJ
ejde-624	51	12	a	a	DET
ejde-624	51	13	singular	singular	ADJ
ejde-624	51	14	term	term	NOUN
ejde-624	51	15	,	,	PUNCT
ejde-624	51	16	we	we	PRON
ejde-624	51	17	do	do	AUX
ejde-624	51	18	not	not	PART
ejde-624	51	19	know	know	VERB
ejde-624	51	20	how	how	SCONJ
ejde-624	51	21	to	to	PART
ejde-624	51	22	prove	prove	VERB
ejde-624	51	23	that	that	SCONJ
ejde-624	51	24	this	this	PRON
ejde-624	51	25	is	be	AUX
ejde-624	51	26	blow	blow	NOUN
ejde-624	51	27	-	-	PUNCT
ejde-624	51	28	up	up	NOUN
ejde-624	51	29	.	.	PUNCT
ejde-624	52	1	there	there	PRON
ejde-624	52	2	are	be	VERB
ejde-624	52	3	a	a	DET
ejde-624	52	4	number	number	NOUN
ejde-624	52	5	of	of	ADP
ejde-624	52	6	generalizations	generalization	NOUN
ejde-624	52	7	of	of	ADP
ejde-624	52	8	the	the	DET
ejde-624	52	9	results	result	NOUN
ejde-624	52	10	in	in	ADP
ejde-624	52	11	[	[	X
ejde-624	52	12	15	15	NUM
ejde-624	52	13	]	]	PUNCT
ejde-624	52	14	,	,	PUNCT
ejde-624	52	15	for	for	ADP
ejde-624	52	16	example	example	NOUN
ejde-624	52	17	the	the	DET
ejde-624	52	18	papers	paper	NOUN
ejde-624	52	19	[	[	X
ejde-624	52	20	7	7	NUM
ejde-624	52	21	,	,	PUNCT
ejde-624	52	22	8	8	NUM
ejde-624	52	23	,	,	PUNCT
ejde-624	52	24	9	9	NUM
ejde-624	52	25	]	]	PUNCT
ejde-624	52	26	but	but	CCONJ
ejde-624	52	27	they	they	PRON
ejde-624	52	28	use	use	VERB
ejde-624	52	29	different	different	ADJ
ejde-624	52	30	methods	method	NOUN
ejde-624	52	31	and	and	CCONJ
ejde-624	52	32	have	have	VERB
ejde-624	52	33	little	little	ADJ
ejde-624	52	34	relevance	relevance	NOUN
ejde-624	52	35	to	to	ADP
ejde-624	52	36	this	this	DET
ejde-624	52	37	paper	paper	NOUN
ejde-624	52	38	so	so	SCONJ
ejde-624	52	39	we	we	PRON
ejde-624	52	40	do	do	AUX
ejde-624	52	41	not	not	PART
ejde-624	52	42	discuss	discuss	VERB
ejde-624	52	43	them	they	PRON
ejde-624	52	44	.	.	PUNCT
ejde-624	53	1	some	some	DET
ejde-624	53	2	comments	comment	NOUN
ejde-624	53	3	on	on	ADP
ejde-624	53	4	existence	existence	NOUN
ejde-624	53	5	theorems	theorem	NOUN
ejde-624	53	6	.	.	PUNCT
ejde-624	54	1	there	there	PRON
ejde-624	54	2	are	be	VERB
ejde-624	54	3	existence	existence	NOUN
ejde-624	54	4	results	result	NOUN
ejde-624	54	5	for	for	ADP
ejde-624	54	6	caputo	caputo	PROPN
ejde-624	54	7	fractional	fractional	PROPN
ejde-624	54	8	differential	differential	ADJ
ejde-624	54	9	equations	equation	NOUN
ejde-624	54	10	of	of	ADP
ejde-624	54	11	the	the	DET
ejde-624	54	12	form	form	NOUN
ejde-624	54	13	dα	dα	ADP
ejde-624	54	14	∗	∗	NOUN
ejde-624	54	15	u(t	u(t	NOUN
ejde-624	54	16	)	)	PUNCT
ejde-624	55	1	=	=	PUNCT
ejde-624	55	2	t−ηf(t	t−ηf(t	NOUN
ejde-624	55	3	,	,	PUNCT
ejde-624	55	4	u(t	u(t	NOUN
ejde-624	55	5	)	)	PUNCT
ejde-624	55	6	)	)	PUNCT
ejde-624	55	7	,	,	PUNCT
ejde-624	55	8	t	t	X
ejde-624	55	9	>	>	X
ejde-624	55	10	0	0	PROPN
ejde-624	55	11	,	,	PUNCT
ejde-624	55	12	u(0	u(0	NOUN
ejde-624	55	13	)	)	PUNCT
ejde-624	55	14	=	=	PUNCT
ejde-624	56	1	u0	u0	ADJ
ejde-624	56	2	,	,	PUNCT
ejde-624	56	3	where	where	SCONJ
ejde-624	56	4	0	0	X
ejde-624	56	5	<	<	X
ejde-624	56	6	η	η	X
ejde-624	56	7	<	<	X
ejde-624	56	8	α	α	PROPN
ejde-624	56	9	and	and	CCONJ
ejde-624	56	10	also	also	ADV
ejde-624	56	11	for	for	ADP
ejde-624	56	12	a	a	DET
ejde-624	56	13	somewhat	somewhat	ADV
ejde-624	56	14	more	more	ADV
ejde-624	56	15	general	general	ADJ
ejde-624	56	16	non	non	ADJ
ejde-624	56	17	-	-	ADJ
ejde-624	56	18	negative	negative	ADJ
ejde-624	56	19	function	function	NOUN
ejde-624	56	20	f	f	PROPN
ejde-624	56	21	.	.	PUNCT
ejde-624	57	1	for	for	ADP
ejde-624	57	2	local	local	ADJ
ejde-624	57	3	existence	existence	NOUN
ejde-624	57	4	,	,	PUNCT
ejde-624	57	5	existence	existence	NOUN
ejde-624	57	6	on	on	ADP
ejde-624	57	7	some	some	DET
ejde-624	57	8	possibly	possibly	ADV
ejde-624	57	9	short	short	ADJ
ejde-624	57	10	interval	interval	NOUN
ejde-624	57	11	,	,	PUNCT
ejde-624	57	12	continuity	continuity	NOUN
ejde-624	57	13	of	of	ADP
ejde-624	57	14	f	f	PROPN
ejde-624	57	15	is	be	AUX
ejde-624	57	16	sufficient	sufficient	ADJ
ejde-624	57	17	,	,	PUNCT
ejde-624	57	18	while	while	SCONJ
ejde-624	57	19	for	for	ADP
ejde-624	57	20	global	global	ADJ
ejde-624	57	21	existence	existence	NOUN
ejde-624	57	22	it	it	PRON
ejde-624	57	23	is	be	AUX
ejde-624	57	24	usually	usually	ADV
ejde-624	57	25	supposed	suppose	VERB
ejde-624	57	26	that	that	SCONJ
ejde-624	57	27	|f(t	|f(t	NOUN
ejde-624	57	28	,	,	PUNCT
ejde-624	57	29	u)|	u)|	NOUN
ejde-624	57	30	≤	≤	NOUN
ejde-624	57	31	c1+c2|u|p	c1+c2|u|p	NOUN
ejde-624	57	32	where	where	SCONJ
ejde-624	57	33	p	p	ADJ
ejde-624	57	34	≤	≤	NOUN
ejde-624	57	35	1	1	NUM
ejde-624	57	36	,	,	PUNCT
ejde-624	57	37	and	and	CCONJ
ejde-624	57	38	a	a	DET
ejde-624	57	39	gronwall	gronwall	ADJ
ejde-624	57	40	or	or	CCONJ
ejde-624	57	41	bihari	bihari	PROPN
ejde-624	57	42	type	type	NOUN
ejde-624	57	43	inequality	inequality	NOUN
ejde-624	57	44	is	be	AUX
ejde-624	57	45	used	use	VERB
ejde-624	57	46	to	to	PART
ejde-624	57	47	get	get	VERB
ejde-624	57	48	suitable	suitable	ADJ
ejde-624	57	49	a	a	DET
ejde-624	57	50	priori	priori	ADJ
ejde-624	57	51	bounds	bound	NOUN
ejde-624	57	52	,	,	PUNCT
ejde-624	57	53	see	see	VERB
ejde-624	57	54	for	for	ADP
ejde-624	57	55	example	example	NOUN
ejde-624	58	1	[	[	X
ejde-624	58	2	14	14	NUM
ejde-624	58	3	,	,	PUNCT
ejde-624	58	4	20	20	NUM
ejde-624	58	5	,	,	PUNCT
ejde-624	58	6	21	21	NUM
ejde-624	58	7	,	,	PUNCT
ejde-624	58	8	22	22	NUM
ejde-624	58	9	]	]	PUNCT
ejde-624	58	10	.	.	PUNCT
ejde-624	59	1	our	our	PRON
ejde-624	59	2	results	result	NOUN
ejde-624	59	3	show	show	VERB
ejde-624	59	4	that	that	SCONJ
ejde-624	59	5	for	for	ADP
ejde-624	59	6	global	global	ADJ
ejde-624	59	7	existence	existence	NOUN
ejde-624	59	8	p	p	NOUN
ejde-624	59	9	≤	≤	NOUN
ejde-624	59	10	1	1	NUM
ejde-624	59	11	can	can	AUX
ejde-624	59	12	not	not	PART
ejde-624	59	13	be	be	AUX
ejde-624	59	14	improved	improve	VERB
ejde-624	59	15	to	to	PART
ejde-624	59	16	have	have	VERB
ejde-624	59	17	p	p	X
ejde-624	59	18	>	>	X
ejde-624	59	19	1	1	NUM
ejde-624	59	20	.	.	PUNCT
ejde-624	60	1	for	for	ADP
ejde-624	60	2	the	the	DET
ejde-624	60	3	r	r	NOUN
ejde-624	60	4	-	-	PUNCT
ejde-624	60	5	l	l	NOUN
ejde-624	60	6	fractional	fractional	ADJ
ejde-624	60	7	derivative	derivative	ADJ
ejde-624	60	8	case	case	NOUN
ejde-624	60	9	dαu(t	dαu(t	PROPN
ejde-624	60	10	)	)	PUNCT
ejde-624	60	11	=	=	SYM
ejde-624	60	12	f(t	f(t	NOUN
ejde-624	60	13	,	,	PUNCT
ejde-624	60	14	u(t	u(t	NOUN
ejde-624	60	15	)	)	PUNCT
ejde-624	60	16	)	)	PUNCT
ejde-624	60	17	,	,	PUNCT
ejde-624	60	18	t	t	X
ejde-624	60	19	>	>	X
ejde-624	60	20	0	0	PROPN
ejde-624	60	21	,	,	PUNCT
ejde-624	60	22	lim	lim	PROPN
ejde-624	60	23	t→0	t→0	AUX
ejde-624	60	24	+	+	NUM
ejde-624	60	25	t1−αu(t	t1−αu(t	NOUN
ejde-624	60	26	)	)	PUNCT
ejde-624	60	27	=	=	SYM
ejde-624	60	28	u0	u0	ADJ
ejde-624	60	29	,	,	PUNCT
ejde-624	60	30	(	(	PUNCT
ejde-624	60	31	1.3	1.3	NUM
ejde-624	60	32	)	)	PUNCT
ejde-624	60	33	when	when	SCONJ
ejde-624	60	34	|f(t	|f(t	NOUN
ejde-624	60	35	,	,	PUNCT
ejde-624	60	36	u)|	u)|	NOUN
ejde-624	60	37	≤	≤	X
ejde-624	60	38	k(t	k(t	VERB
ejde-624	60	39	)	)	PUNCT
ejde-624	61	1	+	+	NUM
ejde-624	61	2	l(t)|u|	l(t)|u|	NOUN
ejde-624	61	3	,	,	PUNCT
ejde-624	61	4	under	under	ADP
ejde-624	61	5	a	a	DET
ejde-624	61	6	variety	variety	NOUN
ejde-624	61	7	of	of	ADP
ejde-624	61	8	conditions	condition	NOUN
ejde-624	61	9	on	on	ADP
ejde-624	61	10	k	k	PROPN
ejde-624	61	11	,	,	PUNCT
ejde-624	61	12	l	l	PROPN
ejde-624	61	13	,	,	PUNCT
ejde-624	61	14	which	which	PRON
ejde-624	61	15	allow	allow	VERB
ejde-624	61	16	singularities	singularity	NOUN
ejde-624	61	17	in	in	ADP
ejde-624	61	18	the	the	DET
ejde-624	61	19	t	t	PROPN
ejde-624	61	20	variable	variable	NOUN
ejde-624	61	21	,	,	PUNCT
ejde-624	61	22	zhu	zhu	PROPN
ejde-624	62	1	[	[	X
ejde-624	62	2	26	26	NUM
ejde-624	62	3	,	,	PUNCT
ejde-624	62	4	27	27	NUM
ejde-624	62	5	,	,	PUNCT
ejde-624	62	6	28	28	NUM
ejde-624	62	7	]	]	PUNCT
ejde-624	62	8	has	have	AUX
ejde-624	62	9	proved	prove	VERB
ejde-624	62	10	various	various	ADJ
ejde-624	62	11	global	global	ADJ
ejde-624	62	12	existence	existence	NOUN
ejde-624	62	13	theorems	theorem	NOUN
ejde-624	62	14	.	.	PUNCT
ejde-624	63	1	in	in	ADP
ejde-624	63	2	the	the	DET
ejde-624	63	3	papers	paper	NOUN
ejde-624	63	4	[	[	X
ejde-624	63	5	1	1	NUM
ejde-624	63	6	,	,	PUNCT
ejde-624	63	7	2	2	NUM
ejde-624	63	8	,	,	PUNCT
ejde-624	63	9	3	3	NUM
ejde-624	63	10	]	]	PUNCT
ejde-624	63	11	,	,	PUNCT
ejde-624	63	12	becker	becker	PROPN
ejde-624	63	13	-	-	PUNCT
ejde-624	63	14	burton	burton	PROPN
ejde-624	63	15	-	-	PUNCT
ejde-624	63	16	purnaras	purnaras	PROPN
ejde-624	63	17	give	give	VERB
ejde-624	63	18	interesting	interesting	ADJ
ejde-624	63	19	results	result	NOUN
ejde-624	63	20	concerning	concern	VERB
ejde-624	63	21	existence	existence	NOUN
ejde-624	63	22	theory	theory	NOUN
ejde-624	63	23	for	for	ADP
ejde-624	63	24	r	r	NOUN
ejde-624	63	25	-	-	PUNCT
ejde-624	63	26	l	l	NOUN
ejde-624	63	27	fractional	fractional	ADJ
ejde-624	63	28	differential	differential	NOUN
ejde-624	63	29	equations	equation	NOUN
ejde-624	63	30	.	.	PUNCT
ejde-624	64	1	with	with	ADP
ejde-624	64	2	a	a	DET
ejde-624	64	3	sign	sign	NOUN
ejde-624	64	4	condition	condition	NOUN
ejde-624	64	5	,	,	PUNCT
ejde-624	64	6	opposite	opposite	ADJ
ejde-624	64	7	to	to	ADP
ejde-624	64	8	the	the	DET
ejde-624	64	9	sign	sign	NOUN
ejde-624	64	10	we	we	PRON
ejde-624	64	11	have	have	VERB
ejde-624	64	12	,	,	PUNCT
ejde-624	64	13	global	global	PROPN
ejde-624	64	14	l1	l1	PROPN
ejde-624	64	15	solutions	solution	NOUN
ejde-624	64	16	are	be	AUX
ejde-624	64	17	possible	possible	ADJ
ejde-624	64	18	for	for	ADP
ejde-624	64	19	p	p	PROPN
ejde-624	64	20	>	>	X
ejde-624	64	21	1	1	NUM
ejde-624	64	22	.	.	PUNCT
ejde-624	65	1	for	for	ADP
ejde-624	65	2	ejde-2024/40	ejde-2024/40	ADJ
ejde-624	65	3	fractional	fractional	ADJ
ejde-624	65	4	differential	differential	ADJ
ejde-624	65	5	inequalities	inequality	NOUN
ejde-624	65	6	3	3	NUM
ejde-624	65	7	example	example	NOUN
ejde-624	65	8	,	,	PUNCT
ejde-624	65	9	in	in	ADP
ejde-624	65	10	[	[	PUNCT
ejde-624	65	11	1	1	NUM
ejde-624	65	12	,	,	PUNCT
ejde-624	65	13	example	example	NOUN
ejde-624	65	14	4.12	4.12	NUM
ejde-624	65	15	]	]	NOUN
ejde-624	65	16	,	,	PUNCT
ejde-624	65	17	d1/2u(t	d1/2u(t	NOUN
ejde-624	65	18	)	)	PUNCT
ejde-624	65	19	=	=	PUNCT
ejde-624	65	20	−	−	PROPN
ejde-624	66	1	√	√	NUM
ejde-624	66	2	π	π	PROPN
ejde-624	66	3	2	2	NUM
ejde-624	66	4	t3/4u3/2	t3/4u3/2	NOUN
ejde-624	66	5	with	with	ADP
ejde-624	66	6	the	the	DET
ejde-624	66	7	initial	initial	ADJ
ejde-624	66	8	condition	condition	NOUN
ejde-624	66	9	limt→0	limt→0	NOUN
ejde-624	66	10	+	+	SYM
ejde-624	66	11	t1/2u(t	t1/2u(t	NOUN
ejde-624	66	12	)	)	PUNCT
ejde-624	66	13	=	=	SYM
ejde-624	66	14	1	1	NUM
ejde-624	66	15	is	be	AUX
ejde-624	66	16	shown	show	VERB
ejde-624	66	17	to	to	PART
ejde-624	66	18	have	have	VERB
ejde-624	66	19	an	an	DET
ejde-624	66	20	explicit	explicit	ADJ
ejde-624	66	21	global	global	ADJ
ejde-624	66	22	solution	solution	NOUN
ejde-624	66	23	u(t	u(t	NOUN
ejde-624	66	24	)	)	PUNCT
ejde-624	67	1	=	=	SYM
ejde-624	67	2	1√	1√	NUM
ejde-624	67	3	t(1+t	t(1+t	NOUN
ejde-624	67	4	)	)	PUNCT
ejde-624	67	5	,	,	PUNCT
ejde-624	67	6	t	t	X
ejde-624	67	7	>	>	X
ejde-624	67	8	0	0	PROPN
ejde-624	67	9	.	.	PUNCT
ejde-624	68	1	a	a	DET
ejde-624	68	2	result	result	NOUN
ejde-624	68	3	closely	closely	ADV
ejde-624	68	4	related	relate	VERB
ejde-624	68	5	to	to	ADP
ejde-624	68	6	the	the	DET
ejde-624	68	7	problem	problem	NOUN
ejde-624	68	8	we	we	PRON
ejde-624	68	9	study	study	VERB
ejde-624	68	10	gives	give	VERB
ejde-624	68	11	local	local	ADJ
ejde-624	68	12	existence	existence	NOUN
ejde-624	68	13	.	.	PUNCT
ejde-624	69	1	theorem	theorem	VERB
ejde-624	69	2	1.2	1.2	NUM
ejde-624	69	3	(	(	PUNCT
ejde-624	69	4	[	[	X
ejde-624	69	5	2	2	NUM
ejde-624	69	6	,	,	PUNCT
ejde-624	69	7	theorem	theorem	VERB
ejde-624	69	8	3.1	3.1	NUM
ejde-624	69	9	]	]	PUNCT
ejde-624	69	10	)	)	PUNCT
ejde-624	69	11	.	.	PUNCT
ejde-624	70	1	let	let	VERB
ejde-624	70	2	0	0	PUNCT
ejde-624	70	3	<	<	X
ejde-624	70	4	α	α	X
ejde-624	70	5	<	<	X
ejde-624	70	6	1	1	NUM
ejde-624	70	7	,	,	PUNCT
ejde-624	70	8	β	β	X
ejde-624	70	9	>	>	X
ejde-624	70	10	−1	−1	NOUN
ejde-624	70	11	,	,	PUNCT
ejde-624	70	12	and	and	CCONJ
ejde-624	70	13	p	p	PRON
ejde-624	70	14	≥	≥	NOUN
ejde-624	70	15	0	0	NUM
ejde-624	70	16	satisfy	satisfy	NOUN
ejde-624	70	17	β−p+α(1+p	β−p+α(1+p	NUM
ejde-624	70	18	)	)	PUNCT
ejde-624	70	19	>	>	X
ejde-624	70	20	0	0	X
ejde-624	70	21	.	.	PUNCT
ejde-624	70	22	suppose	suppose	VERB
ejde-624	70	23	that	that	SCONJ
ejde-624	70	24	f	f	PROPN
ejde-624	70	25	:	:	PUNCT
ejde-624	70	26	(	(	PUNCT
ejde-624	70	27	0,∞)×r	0,∞)×r	X
ejde-624	70	28	→	→	SYM
ejde-624	70	29	r	r	NOUN
ejde-624	70	30	is	be	AUX
ejde-624	70	31	continuous	continuous	ADJ
ejde-624	70	32	.	.	PUNCT
ejde-624	71	1	suppose	suppose	VERB
ejde-624	71	2	there	there	PRON
ejde-624	71	3	are	be	VERB
ejde-624	71	4	nonnegative	nonnegative	ADJ
ejde-624	71	5	constants	constant	NOUN
ejde-624	71	6	k1,k2	k1,k2	PROPN
ejde-624	71	7	such	such	ADJ
ejde-624	71	8	that	that	SCONJ
ejde-624	71	9	|f(t	|f(t	NOUN
ejde-624	71	10	,	,	PUNCT
ejde-624	71	11	u)|	u)|	NOUN
ejde-624	71	12	≤	≤	NOUN
ejde-624	72	1	k1+k2	k1+k2	PROPN
ejde-624	72	2	t	t	X
ejde-624	72	3	β	β	X
ejde-624	72	4	|u|p	|u|p	NOUN
ejde-624	72	5	for	for	ADP
ejde-624	72	6	u	u	PROPN
ejde-624	72	7	∈	∈	PROPN
ejde-624	72	8	r	r	NOUN
ejde-624	72	9	and	and	CCONJ
ejde-624	72	10	0	0	NUM
ejde-624	72	11	<	<	X
ejde-624	72	12	t	t	X
ejde-624	72	13	<	<	X
ejde-624	72	14	t0	t0	PROPN
ejde-624	72	15	,	,	PUNCT
ejde-624	72	16	where	where	SCONJ
ejde-624	72	17	t0	t0	PROPN
ejde-624	72	18	∈	∈	PROPN
ejde-624	72	19	(	(	PUNCT
ejde-624	72	20	0,∞	0,∞	NOUN
ejde-624	72	21	]	]	PUNCT
ejde-624	72	22	.	.	PUNCT
ejde-624	73	1	then	then	ADV
ejde-624	73	2	,	,	PUNCT
ejde-624	73	3	for	for	ADP
ejde-624	73	4	u0	u0	ADJ
ejde-624	73	5	̸=	̸=	PROPN
ejde-624	73	6	0	0	NUM
ejde-624	73	7	,	,	PUNCT
ejde-624	73	8	the	the	DET
ejde-624	73	9	problem	problem	NOUN
ejde-624	73	10	dαu(t	dαu(t	PROPN
ejde-624	73	11	)	)	PUNCT
ejde-624	73	12	=	=	PUNCT
ejde-624	74	1	f(t	f(t	NOUN
ejde-624	74	2	,	,	PUNCT
ejde-624	74	3	u(t	u(t	NOUN
ejde-624	74	4	)	)	PUNCT
ejde-624	74	5	)	)	PUNCT
ejde-624	74	6	,	,	PUNCT
ejde-624	74	7	t	t	X
ejde-624	74	8	>	>	X
ejde-624	74	9	0	0	PROPN
ejde-624	74	10	,	,	PUNCT
ejde-624	74	11	lim	lim	PROPN
ejde-624	74	12	t→0	t→0	AUX
ejde-624	74	13	+	+	NUM
ejde-624	74	14	t1−αu(t	t1−αu(t	NOUN
ejde-624	74	15	)	)	PUNCT
ejde-624	74	16	=	=	SYM
ejde-624	75	1	u0	u0	ADJ
ejde-624	75	2	,	,	PUNCT
ejde-624	75	3	has	have	VERB
ejde-624	75	4	a	a	DET
ejde-624	75	5	solution	solution	NOUN
ejde-624	75	6	in	in	ADP
ejde-624	75	7	cα−1[0	cα−1[0	PROPN
ejde-624	75	8	,	,	PUNCT
ejde-624	75	9	t	t	X
ejde-624	75	10	]	]	PUNCT
ejde-624	75	11	for	for	ADP
ejde-624	75	12	some	some	DET
ejde-624	75	13	t	t	NOUN
ejde-624	75	14	∈	∈	PROPN
ejde-624	75	15	(	(	PUNCT
ejde-624	75	16	0	0	NUM
ejde-624	75	17	,	,	PUNCT
ejde-624	75	18	t0	t0	NOUN
ejde-624	75	19	)	)	PUNCT
ejde-624	75	20	.	.	PUNCT
ejde-624	76	1	here	here	ADV
ejde-624	76	2	u	u	PROPN
ejde-624	76	3	∈	∈	PROPN
ejde-624	76	4	cα−1	cα−1	PROPN
ejde-624	76	5	means	mean	VERB
ejde-624	76	6	that	that	SCONJ
ejde-624	76	7	u	u	PRON
ejde-624	76	8	is	be	AUX
ejde-624	76	9	continuous	continuous	ADJ
ejde-624	76	10	on	on	ADP
ejde-624	76	11	(	(	PUNCT
ejde-624	76	12	0	0	NUM
ejde-624	76	13	,	,	PUNCT
ejde-624	76	14	t	t	NOUN
ejde-624	76	15	]	]	PUNCT
ejde-624	76	16	and	and	CCONJ
ejde-624	76	17	limt→0	limt→0	PROPN
ejde-624	76	18	+	+	SYM
ejde-624	76	19	t1−αu(t	t1−αu(t	NOUN
ejde-624	76	20	)	)	PUNCT
ejde-624	76	21	exists	exist	VERB
ejde-624	76	22	.	.	PUNCT
ejde-624	77	1	note	note	VERB
ejde-624	77	2	that	that	SCONJ
ejde-624	77	3	β	β	NOUN
ejde-624	77	4	−	−	PROPN
ejde-624	77	5	p	p	X
ejde-624	78	1	+	+	CCONJ
ejde-624	78	2	α(1	α(1	PROPN
ejde-624	79	1	+	+	CCONJ
ejde-624	80	1	p	p	X
ejde-624	80	2	)	)	PUNCT
ejde-624	80	3	>	>	X
ejde-624	80	4	0	0	PUNCT
ejde-624	80	5	is	be	AUX
ejde-624	80	6	equivalent	equivalent	ADJ
ejde-624	80	7	to	to	ADP
ejde-624	80	8	p	p	X
ejde-624	80	9	<	<	X
ejde-624	80	10	α+β	α+β	NUM
ejde-624	80	11	1−α	1−α	NUM
ejde-624	81	1	so	so	SCONJ
ejde-624	81	2	it	it	PRON
ejde-624	81	3	is	be	AUX
ejde-624	81	4	implicit	implicit	ADJ
ejde-624	81	5	that	that	SCONJ
ejde-624	81	6	α+	α+	PRON
ejde-624	81	7	β	β	X
ejde-624	81	8	>	>	X
ejde-624	81	9	0	0	NUM
ejde-624	81	10	.	.	PUNCT
ejde-624	82	1	thus	thus	ADV
ejde-624	82	2	local	local	ADJ
ejde-624	82	3	existence	existence	NOUN
ejde-624	82	4	is	be	AUX
ejde-624	82	5	possible	possible	ADJ
ejde-624	82	6	in	in	ADP
ejde-624	82	7	this	this	DET
ejde-624	82	8	case	case	NOUN
ejde-624	82	9	.	.	PUNCT
ejde-624	83	1	our	our	PRON
ejde-624	83	2	theorem	theorem	ADJ
ejde-624	83	3	4.2	4.2	NUM
ejde-624	83	4	has	have	VERB
ejde-624	83	5	a	a	DET
ejde-624	83	6	similar	similar	ADJ
ejde-624	83	7	but	but	CCONJ
ejde-624	83	8	larger	large	ADJ
ejde-624	83	9	critical	critical	ADJ
ejde-624	83	10	value	value	NOUN
ejde-624	83	11	of	of	ADP
ejde-624	83	12	p.	p.	NOUN
ejde-624	83	13	we	we	PRON
ejde-624	83	14	show	show	VERB
ejde-624	83	15	that	that	SCONJ
ejde-624	83	16	when	when	SCONJ
ejde-624	83	17	1	1	NUM
ejde-624	83	18	<	<	X
ejde-624	83	19	p	p	X
ejde-624	83	20	<	<	X
ejde-624	83	21	1+β	1+β	NUM
ejde-624	83	22	1−α	1−α	NUM
ejde-624	83	23	solutions	solution	NOUN
ejde-624	83	24	can	can	AUX
ejde-624	83	25	exist	exist	VERB
ejde-624	83	26	only	only	ADV
ejde-624	83	27	on	on	ADP
ejde-624	83	28	a	a	DET
ejde-624	83	29	finite	finite	ADJ
ejde-624	83	30	interval	interval	NOUN
ejde-624	83	31	[	[	X
ejde-624	83	32	0	0	NUM
ejde-624	83	33	,	,	PUNCT
ejde-624	83	34	t	t	NOUN
ejde-624	83	35	)	)	PUNCT
ejde-624	83	36	,	,	PUNCT
ejde-624	83	37	where	where	SCONJ
ejde-624	83	38	t	t	PROPN
ejde-624	83	39	≤	≤	NUM
ejde-624	83	40	t1	t1	NOUN
ejde-624	83	41	for	for	ADP
ejde-624	83	42	some	some	DET
ejde-624	83	43	explicitly	explicitly	ADV
ejde-624	83	44	determined	determine	VERB
ejde-624	83	45	t1	t1	NOUN
ejde-624	83	46	,	,	PUNCT
ejde-624	83	47	while	while	SCONJ
ejde-624	83	48	for	for	ADP
ejde-624	83	49	p	p	PRON
ejde-624	83	50	≥	≥	NOUN
ejde-624	83	51	1+β	1+β	NUM
ejde-624	83	52	1−α	1−α	NUM
ejde-624	83	53	existence	existence	NOUN
ejde-624	83	54	in	in	ADP
ejde-624	83	55	the	the	DET
ejde-624	83	56	space	space	NOUN
ejde-624	83	57	cα−1[0	cα−1[0	PROPN
ejde-624	83	58	,	,	PUNCT
ejde-624	83	59	t	t	PROPN
ejde-624	83	60	]	]	PUNCT
ejde-624	83	61	is	be	AUX
ejde-624	83	62	impossible	impossible	ADJ
ejde-624	83	63	for	for	ADP
ejde-624	83	64	any	any	DET
ejde-624	83	65	t	t	NOUN
ejde-624	83	66	>	>	X
ejde-624	83	67	0	0	X
ejde-624	83	68	.	.	PUNCT
ejde-624	84	1	in	in	ADP
ejde-624	84	2	the	the	DET
ejde-624	84	3	paper	paper	NOUN
ejde-624	85	1	[	[	X
ejde-624	85	2	2	2	NUM
ejde-624	85	3	]	]	PUNCT
ejde-624	85	4	,	,	PUNCT
ejde-624	85	5	for	for	ADP
ejde-624	85	6	the	the	DET
ejde-624	85	7	problem	problem	NOUN
ejde-624	85	8	dαu(t	dαu(t	PROPN
ejde-624	85	9	)	)	PUNCT
ejde-624	85	10	=	=	SYM
ejde-624	85	11	un(t	un(t	NOUN
ejde-624	85	12	)	)	PUNCT
ejde-624	85	13	,	,	PUNCT
ejde-624	85	14	t	t	PROPN
ejde-624	85	15	>	>	X
ejde-624	85	16	0	0	PROPN
ejde-624	85	17	,	,	PUNCT
ejde-624	85	18	lim	lim	PROPN
ejde-624	85	19	t→0	t→0	AUX
ejde-624	85	20	+	+	NUM
ejde-624	85	21	t1−αu(t	t1−αu(t	NOUN
ejde-624	85	22	)	)	PUNCT
ejde-624	85	23	=	=	SYM
ejde-624	85	24	u0	u0	ADJ
ejde-624	85	25	,	,	PUNCT
ejde-624	85	26	(	(	PUNCT
ejde-624	85	27	1.4	1.4	NUM
ejde-624	85	28	)	)	PUNCT
ejde-624	85	29	with	with	ADP
ejde-624	85	30	n	n	PROPN
ejde-624	85	31	∈	∈	PROPN
ejde-624	85	32	n	n	CCONJ
ejde-624	85	33	,	,	PUNCT
ejde-624	85	34	the	the	DET
ejde-624	85	35	authors	author	NOUN
ejde-624	85	36	showed	show	VERB
ejde-624	85	37	that	that	SCONJ
ejde-624	85	38	whether	whether	SCONJ
ejde-624	85	39	or	or	CCONJ
ejde-624	85	40	not	not	PART
ejde-624	85	41	a	a	DET
ejde-624	85	42	solution	solution	NOUN
ejde-624	85	43	of	of	ADP
ejde-624	85	44	the	the	DET
ejde-624	85	45	initial	initial	ADJ
ejde-624	85	46	value	value	NOUN
ejde-624	85	47	problem	problem	NOUN
ejde-624	85	48	(	(	PUNCT
ejde-624	85	49	1.4	1.4	NUM
ejde-624	85	50	)	)	PUNCT
ejde-624	85	51	exists	exist	VERB
ejde-624	85	52	on	on	ADP
ejde-624	85	53	an	an	DET
ejde-624	85	54	interval	interval	NOUN
ejde-624	85	55	(	(	PUNCT
ejde-624	85	56	0	0	NUM
ejde-624	85	57	,	,	PUNCT
ejde-624	85	58	t	t	X
ejde-624	85	59	]	]	PUNCT
ejde-624	85	60	,	,	PUNCT
ejde-624	85	61	for	for	ADP
ejde-624	85	62	some	some	DET
ejde-624	85	63	t	t	PROPN
ejde-624	85	64	>	>	X
ejde-624	85	65	0	0	NUM
ejde-624	85	66	,	,	PUNCT
ejde-624	85	67	depends	depend	VERB
ejde-624	85	68	on	on	ADP
ejde-624	85	69	the	the	DET
ejde-624	85	70	value	value	NOUN
ejde-624	85	71	of	of	ADP
ejde-624	85	72	α	α	NOUN
ejde-624	85	73	.	.	PUNCT
ejde-624	86	1	one	one	NUM
ejde-624	86	2	of	of	ADP
ejde-624	86	3	their	their	PRON
ejde-624	86	4	results	result	NOUN
ejde-624	86	5	is	be	AUX
ejde-624	86	6	the	the	DET
ejde-624	86	7	following	following	NOUN
ejde-624	86	8	.	.	PUNCT
ejde-624	87	1	theorem	theorem	VERB
ejde-624	87	2	1.3	1.3	NUM
ejde-624	87	3	(	(	PUNCT
ejde-624	87	4	[	[	X
ejde-624	87	5	2	2	NUM
ejde-624	87	6	,	,	PUNCT
ejde-624	87	7	theorem	theorem	VERB
ejde-624	87	8	3.11	3.11	NUM
ejde-624	87	9	]	]	PUNCT
ejde-624	87	10	)	)	PUNCT
ejde-624	87	11	.	.	PUNCT
ejde-624	88	1	let	let	VERB
ejde-624	88	2	0	0	PUNCT
ejde-624	88	3	<	<	X
ejde-624	88	4	α	α	X
ejde-624	88	5	<	<	X
ejde-624	88	6	1	1	NUM
ejde-624	88	7	,	,	PUNCT
ejde-624	88	8	n	n	PRON
ejde-624	88	9	∈	∈	PROPN
ejde-624	88	10	n	n	CCONJ
ejde-624	88	11	and	and	CCONJ
ejde-624	88	12	u0	u0	ADJ
ejde-624	88	13	̸=	̸=	PROPN
ejde-624	88	14	0	0	NUM
ejde-624	88	15	.	.	PUNCT
ejde-624	89	1	the	the	DET
ejde-624	89	2	initial	initial	ADJ
ejde-624	89	3	value	value	NOUN
ejde-624	89	4	problem	problem	NOUN
ejde-624	89	5	(	(	PUNCT
ejde-624	89	6	1.4	1.4	NUM
ejde-624	89	7	)	)	PUNCT
ejde-624	89	8	has	have	VERB
ejde-624	89	9	a	a	DET
ejde-624	89	10	solution	solution	NOUN
ejde-624	89	11	if	if	SCONJ
ejde-624	89	12	and	and	CCONJ
ejde-624	89	13	only	only	ADV
ejde-624	89	14	if	if	SCONJ
ejde-624	89	15	α	α	PROPN
ejde-624	89	16	>	>	X
ejde-624	89	17	n−1	n−1	PROPN
ejde-624	89	18	n	n	NOUN
ejde-624	89	19	.	.	PUNCT
ejde-624	90	1	moreover	moreover	ADV
ejde-624	90	2	,	,	PUNCT
ejde-624	90	3	the	the	DET
ejde-624	90	4	solution	solution	NOUN
ejde-624	90	5	is	be	AUX
ejde-624	90	6	unique	unique	ADJ
ejde-624	90	7	.	.	PUNCT
ejde-624	91	1	for	for	ADP
ejde-624	91	2	the	the	DET
ejde-624	91	3	problem	problem	NOUN
ejde-624	91	4	(	(	PUNCT
ejde-624	91	5	1.4	1.4	NUM
ejde-624	91	6	)	)	PUNCT
ejde-624	91	7	with	with	ADP
ejde-624	91	8	n	n	CCONJ
ejde-624	91	9	replaced	replace	VERB
ejde-624	91	10	by	by	ADP
ejde-624	91	11	p	p	X
ejde-624	91	12	,	,	PUNCT
ejde-624	91	13	a	a	DET
ejde-624	91	14	special	special	ADJ
ejde-624	91	15	case	case	NOUN
ejde-624	91	16	of	of	ADP
ejde-624	91	17	our	our	PRON
ejde-624	91	18	theorem	theorem	ADJ
ejde-624	91	19	4.2	4.2	NUM
ejde-624	91	20	shows	show	VERB
ejde-624	91	21	that	that	SCONJ
ejde-624	91	22	for	for	ADP
ejde-624	91	23	u0	u0	ADJ
ejde-624	91	24	>	>	X
ejde-624	91	25	0	0	PUNCT
ejde-624	91	26	there	there	PRON
ejde-624	91	27	does	do	AUX
ejde-624	91	28	not	not	PART
ejde-624	91	29	exist	exist	VERB
ejde-624	91	30	a	a	DET
ejde-624	91	31	nontrivial	nontrivial	ADJ
ejde-624	91	32	solution	solution	NOUN
ejde-624	91	33	in	in	ADP
ejde-624	91	34	the	the	DET
ejde-624	91	35	space	space	NOUN
ejde-624	91	36	cα−1[0	cα−1[0	PROPN
ejde-624	91	37	,	,	PUNCT
ejde-624	91	38	t	t	X
ejde-624	91	39	]	]	PUNCT
ejde-624	91	40	if	if	SCONJ
ejde-624	91	41	p	p	PROPN
ejde-624	91	42	≥	≥	PUNCT
ejde-624	91	43	1	1	NUM
ejde-624	91	44	1−α	1−α	NUM
ejde-624	91	45	,	,	PUNCT
ejde-624	91	46	that	that	PRON
ejde-624	91	47	is	is	ADV
ejde-624	91	48	α	α	PRON
ejde-624	91	49	≤	≤	NUM
ejde-624	92	1	p−1	p−1	PROPN
ejde-624	92	2	p	p	PROPN
ejde-624	92	3	,	,	PUNCT
ejde-624	92	4	which	which	PRON
ejde-624	92	5	partially	partially	ADV
ejde-624	92	6	extends	extend	VERB
ejde-624	92	7	theorem	theorem	VERB
ejde-624	92	8	1.3	1.3	NUM
ejde-624	92	9	.	.	PUNCT
ejde-624	93	1	lan	lan	PROPN
ejde-624	93	2	[	[	X
ejde-624	93	3	11	11	NUM
ejde-624	93	4	,	,	PUNCT
ejde-624	93	5	12	12	NUM
ejde-624	93	6	,	,	PUNCT
ejde-624	93	7	13	13	NUM
ejde-624	93	8	]	]	PUNCT
ejde-624	93	9	has	have	AUX
ejde-624	93	10	studied	study	VERB
ejde-624	93	11	more	more	ADJ
ejde-624	93	12	general	general	ADJ
ejde-624	93	13	fractional	fractional	ADJ
ejde-624	93	14	problems	problem	NOUN
ejde-624	93	15	that	that	PRON
ejde-624	93	16	include	include	VERB
ejde-624	93	17	the	the	DET
ejde-624	93	18	caputo	caputo	PROPN
ejde-624	93	19	and	and	CCONJ
ejde-624	93	20	r	r	PROPN
ejde-624	93	21	-	-	PUNCT
ejde-624	93	22	l	l	NOUN
ejde-624	93	23	fractional	fractional	ADJ
ejde-624	93	24	equations	equation	NOUN
ejde-624	93	25	as	as	ADP
ejde-624	93	26	special	special	ADJ
ejde-624	93	27	cases	case	NOUN
ejde-624	93	28	,	,	PUNCT
ejde-624	93	29	and	and	CCONJ
ejde-624	93	30	he	he	PRON
ejde-624	93	31	has	have	AUX
ejde-624	93	32	proved	prove	VERB
ejde-624	93	33	equivalences	equivalence	NOUN
ejde-624	93	34	between	between	ADP
ejde-624	93	35	fractional	fractional	ADJ
ejde-624	93	36	differential	differential	ADJ
ejde-624	93	37	and	and	CCONJ
ejde-624	93	38	integral	integral	ADJ
ejde-624	93	39	equations	equation	NOUN
ejde-624	93	40	.	.	PUNCT
ejde-624	94	1	2	2	X
ejde-624	94	2	.	.	X
ejde-624	94	3	preliminaries	preliminary	NOUN
ejde-624	94	4	we	we	PRON
ejde-624	94	5	consider	consider	VERB
ejde-624	94	6	real	real	ADJ
ejde-624	94	7	valued	value	VERB
ejde-624	94	8	functions	function	NOUN
ejde-624	94	9	defined	define	VERB
ejde-624	94	10	on	on	ADP
ejde-624	94	11	an	an	DET
ejde-624	94	12	arbitrary	arbitrary	ADJ
ejde-624	94	13	finite	finite	ADJ
ejde-624	94	14	interval	interval	NOUN
ejde-624	94	15	[	[	X
ejde-624	94	16	0	0	NUM
ejde-624	94	17	,	,	PUNCT
ejde-624	94	18	t	t	X
ejde-624	94	19	]	]	PUNCT
ejde-624	94	20	,	,	PUNCT
ejde-624	94	21	which	which	PRON
ejde-624	94	22	is	be	AUX
ejde-624	94	23	,	,	PUNCT
ejde-624	94	24	by	by	ADP
ejde-624	94	25	a	a	DET
ejde-624	94	26	simple	simple	ADJ
ejde-624	94	27	change	change	NOUN
ejde-624	94	28	of	of	ADP
ejde-624	94	29	variable	variable	NOUN
ejde-624	94	30	,	,	PUNCT
ejde-624	94	31	equivalent	equivalent	ADJ
ejde-624	94	32	to	to	ADP
ejde-624	94	33	any	any	DET
ejde-624	94	34	finite	finite	ADJ
ejde-624	94	35	interval	interval	NOUN
ejde-624	94	36	.	.	PUNCT
ejde-624	95	1	in	in	ADP
ejde-624	95	2	this	this	DET
ejde-624	95	3	paper	paper	NOUN
ejde-624	95	4	all	all	DET
ejde-624	95	5	functions	function	NOUN
ejde-624	95	6	are	be	AUX
ejde-624	95	7	supposed	suppose	VERB
ejde-624	95	8	to	to	PART
ejde-624	95	9	be	be	AUX
ejde-624	95	10	measurable	measurable	ADJ
ejde-624	95	11	,	,	PUNCT
ejde-624	95	12	all	all	DET
ejde-624	95	13	integrals	integral	NOUN
ejde-624	95	14	are	be	AUX
ejde-624	95	15	lebesgue	lebesgue	NOUN
ejde-624	95	16	integrals	integral	NOUN
ejde-624	95	17	and	and	CCONJ
ejde-624	95	18	l1[0	l1[0	PROPN
ejde-624	95	19	,	,	PUNCT
ejde-624	95	20	t	t	PROPN
ejde-624	95	21	]	]	PUNCT
ejde-624	95	22	denotes	denote	VERB
ejde-624	95	23	the	the	DET
ejde-624	95	24	usual	usual	ADJ
ejde-624	95	25	space	space	NOUN
ejde-624	95	26	of	of	ADP
ejde-624	95	27	lebesgue	lebesgue	PROPN
ejde-624	95	28	integrable	integrable	ADJ
ejde-624	95	29	functions	function	NOUN
ejde-624	95	30	;	;	PUNCT
ejde-624	95	31	we	we	PRON
ejde-624	95	32	will	will	AUX
ejde-624	95	33	often	often	ADV
ejde-624	95	34	simply	simply	ADV
ejde-624	95	35	write	write	VERB
ejde-624	95	36	l1	l1	PROPN
ejde-624	95	37	.	.	PUNCT
ejde-624	96	1	the	the	DET
ejde-624	96	2	space	space	NOUN
ejde-624	96	3	of	of	ADP
ejde-624	96	4	functions	function	NOUN
ejde-624	96	5	that	that	PRON
ejde-624	96	6	are	be	AUX
ejde-624	96	7	continuous	continuous	ADJ
ejde-624	96	8	on	on	ADP
ejde-624	96	9	[	[	X
ejde-624	96	10	0	0	NUM
ejde-624	96	11	,	,	PUNCT
ejde-624	96	12	t	t	PROPN
ejde-624	96	13	]	]	PUNCT
ejde-624	96	14	is	be	AUX
ejde-624	96	15	denoted	denote	VERB
ejde-624	96	16	by	by	ADP
ejde-624	96	17	c[0	c[0	PROPN
ejde-624	96	18	,	,	PUNCT
ejde-624	96	19	t	t	X
ejde-624	96	20	]	]	PUNCT
ejde-624	96	21	,	,	PUNCT
ejde-624	96	22	or	or	CCONJ
ejde-624	96	23	simply	simply	ADV
ejde-624	96	24	c	c	X
ejde-624	96	25	,	,	PUNCT
ejde-624	96	26	and	and	CCONJ
ejde-624	96	27	is	be	AUX
ejde-624	96	28	endowed	endow	VERB
ejde-624	96	29	with	with	ADP
ejde-624	96	30	the	the	DET
ejde-624	96	31	supremum	supremum	ADJ
ejde-624	96	32	norm	norm	NOUN
ejde-624	96	33	∥u∥∞	∥u∥∞	X
ejde-624	96	34	:	:	PUNCT
ejde-624	97	1	=	=	NOUN
ejde-624	97	2	maxt∈[0,t	maxt∈[0,t	NOUN
ejde-624	97	3	]	]	PUNCT
ejde-624	98	1	|u(t)|	|u(t)|	NOUN
ejde-624	98	2	,	,	PUNCT
ejde-624	98	3	c1	c1	NOUN
ejde-624	98	4	=	=	SYM
ejde-624	98	5	c1[0	c1[0	PROPN
ejde-624	98	6	,	,	PUNCT
ejde-624	98	7	t	t	PROPN
ejde-624	98	8	]	]	PUNCT
ejde-624	98	9	will	will	AUX
ejde-624	98	10	denote	denote	VERB
ejde-624	98	11	the	the	DET
ejde-624	98	12	space	space	NOUN
ejde-624	98	13	of	of	ADP
ejde-624	98	14	continuously	continuously	ADV
ejde-624	98	15	differentiable	differentiable	ADJ
ejde-624	98	16	functions	function	NOUN
ejde-624	98	17	.	.	PUNCT
ejde-624	99	1	when	when	SCONJ
ejde-624	99	2	studying	study	VERB
ejde-624	99	3	fractional	fractional	ADJ
ejde-624	99	4	integrals	integral	NOUN
ejde-624	99	5	and	and	CCONJ
ejde-624	99	6	derivatives	derivative	NOUN
ejde-624	99	7	,	,	PUNCT
ejde-624	99	8	functions	function	NOUN
ejde-624	99	9	such	such	ADJ
ejde-624	99	10	as	as	ADP
ejde-624	99	11	tα−1	tα−1	NOUN
ejde-624	99	12	arise	arise	VERB
ejde-624	99	13	where	where	SCONJ
ejde-624	99	14	typically	typically	ADV
ejde-624	99	15	0	0	NUM
ejde-624	99	16	<	<	X
ejde-624	99	17	α	α	X
ejde-624	99	18	<	<	X
ejde-624	99	19	1	1	NUM
ejde-624	99	20	.	.	PUNCT
ejde-624	100	1	this	this	PRON
ejde-624	100	2	leads	lead	VERB
ejde-624	100	3	to	to	ADP
ejde-624	100	4	consideration	consideration	NOUN
ejde-624	100	5	of	of	ADP
ejde-624	100	6	a	a	DET
ejde-624	100	7	weighted	weight	VERB
ejde-624	100	8	space	space	NOUN
ejde-624	100	9	of	of	ADP
ejde-624	100	10	4	4	NUM
ejde-624	100	11	j.	j.	PROPN
ejde-624	100	12	r.	r.	PROPN
ejde-624	100	13	l.	l.	PROPN
ejde-624	100	14	webb	webb	PROPN
ejde-624	100	15	ejde-2024/40	ejde-2024/40	PROPN
ejde-624	100	16	functions	function	NOUN
ejde-624	100	17	that	that	PRON
ejde-624	100	18	are	be	AUX
ejde-624	100	19	continuous	continuous	ADJ
ejde-624	100	20	except	except	SCONJ
ejde-624	100	21	at	at	ADP
ejde-624	100	22	t	t	NOUN
ejde-624	100	23	=	=	SYM
ejde-624	100	24	0	0	PUNCT
ejde-624	100	25	and	and	CCONJ
ejde-624	100	26	have	have	VERB
ejde-624	100	27	an	an	DET
ejde-624	100	28	integrable	integrable	ADJ
ejde-624	100	29	singularity	singularity	NOUN
ejde-624	100	30	at	at	ADP
ejde-624	100	31	t	t	PROPN
ejde-624	100	32	=	=	SYM
ejde-624	100	33	0	0	NUM
ejde-624	100	34	.	.	PUNCT
ejde-624	101	1	for	for	ADP
ejde-624	101	2	γ	γ	X
ejde-624	101	3	>	>	X
ejde-624	101	4	−1	−1	NOUN
ejde-624	101	5	we	we	PRON
ejde-624	101	6	define	define	VERB
ejde-624	101	7	the	the	DET
ejde-624	101	8	space	space	NOUN
ejde-624	101	9	denoted	denote	VERB
ejde-624	101	10	cγ	cγ	X
ejde-624	101	11	=	=	SYM
ejde-624	101	12	cγ	cγ	PROPN
ejde-624	102	1	[	[	X
ejde-624	102	2	0	0	NUM
ejde-624	102	3	,	,	PUNCT
ejde-624	102	4	t	t	X
ejde-624	102	5	]	]	PUNCT
ejde-624	102	6	by	by	ADP
ejde-624	102	7	cγ	cγ	X
ejde-624	102	8	[	[	X
ejde-624	102	9	0	0	NUM
ejde-624	102	10	,	,	PUNCT
ejde-624	102	11	t	t	X
ejde-624	102	12	]	]	PUNCT
ejde-624	102	13	:	:	PUNCT
ejde-624	102	14	=	=	SYM
ejde-624	102	15	{	{	PUNCT
ejde-624	102	16	u	u	NOUN
ejde-624	102	17	∈	∈	PROPN
ejde-624	102	18	c(0	c(0	PROPN
ejde-624	102	19	,	,	PUNCT
ejde-624	102	20	t	t	X
ejde-624	102	21	]	]	PUNCT
ejde-624	102	22	such	such	ADJ
ejde-624	102	23	that	that	SCONJ
ejde-624	102	24	lim	lim	PROPN
ejde-624	102	25	t→0	t→0	AUX
ejde-624	102	26	+	+	NUM
ejde-624	102	27	t−γu(t	t−γu(t	NOUN
ejde-624	102	28	)	)	PUNCT
ejde-624	102	29	exists	exist	VERB
ejde-624	102	30	}	}	PUNCT
ejde-624	102	31	.	.	PUNCT
ejde-624	103	1	(	(	PUNCT
ejde-624	103	2	2.1	2.1	NUM
ejde-624	103	3	)	)	PUNCT
ejde-624	103	4	then	then	ADV
ejde-624	103	5	u	u	PROPN
ejde-624	103	6	∈	∈	NOUN
ejde-624	103	7	cγ	cγ	NOUN
ejde-624	103	8	if	if	SCONJ
ejde-624	103	9	and	and	CCONJ
ejde-624	103	10	only	only	ADV
ejde-624	103	11	if	if	SCONJ
ejde-624	103	12	u(t	u(t	NOUN
ejde-624	103	13	)	)	PUNCT
ejde-624	103	14	=	=	SYM
ejde-624	103	15	tγu(t	tγu(t	PROPN
ejde-624	103	16	)	)	PUNCT
ejde-624	103	17	for	for	ADP
ejde-624	103	18	some	some	DET
ejde-624	103	19	function	function	NOUN
ejde-624	103	20	u	u	PROPN
ejde-624	103	21	∈	∈	PROPN
ejde-624	103	22	c[0	c[0	PROPN
ejde-624	103	23	,	,	PUNCT
ejde-624	103	24	t	t	NOUN
ejde-624	103	25	]	]	PUNCT
ejde-624	103	26	and	and	CCONJ
ejde-624	103	27	we	we	PRON
ejde-624	103	28	define	define	VERB
ejde-624	103	29	∥u∥γ	∥u∥γ	NOUN
ejde-624	103	30	:	:	PUNCT
ejde-624	103	31	=	=	SYM
ejde-624	103	32	∥u∥∞.	∥u∥∞.	VERB
ejde-624	103	33	the	the	DET
ejde-624	103	34	spaces	space	NOUN
ejde-624	103	35	of	of	ADP
ejde-624	103	36	functions	function	NOUN
ejde-624	103	37	with	with	ADP
ejde-624	103	38	a	a	DET
ejde-624	103	39	singularity	singularity	NOUN
ejde-624	103	40	at	at	ADP
ejde-624	103	41	t	t	PROPN
ejde-624	103	42	=	=	SYM
ejde-624	103	43	0	0	NUM
ejde-624	103	44	are	be	AUX
ejde-624	103	45	c−γ	c−γ	NOUN
ejde-624	103	46	where	where	SCONJ
ejde-624	103	47	γ	γ	X
ejde-624	103	48	>	>	X
ejde-624	103	49	0	0	NUM
ejde-624	103	50	.	.	PUNCT
ejde-624	104	1	for	for	ADP
ejde-624	104	2	0	0	NUM
ejde-624	104	3	<	<	X
ejde-624	104	4	γ	γ	X
ejde-624	104	5	<	<	X
ejde-624	104	6	1	1	NUM
ejde-624	104	7	we	we	PRON
ejde-624	104	8	have	have	VERB
ejde-624	104	9	c−γ	c−γ	PROPN
ejde-624	104	10	⊂	⊂	PROPN
ejde-624	104	11	l1	l1	PROPN
ejde-624	104	12	.	.	PUNCT
ejde-624	105	1	we	we	PRON
ejde-624	105	2	also	also	ADV
ejde-624	105	3	use	use	VERB
ejde-624	105	4	the	the	DET
ejde-624	105	5	space	space	NOUN
ejde-624	105	6	of	of	ADP
ejde-624	105	7	absolutely	absolutely	ADV
ejde-624	105	8	continuous	continuous	ADJ
ejde-624	105	9	functions	function	NOUN
ejde-624	105	10	which	which	PRON
ejde-624	105	11	is	be	AUX
ejde-624	105	12	denoted	denote	VERB
ejde-624	105	13	ac	ac	PROPN
ejde-624	105	14	.	.	PUNCT
ejde-624	106	1	the	the	DET
ejde-624	106	2	space	space	NOUN
ejde-624	106	3	ac	ac	PROPN
ejde-624	106	4	is	be	AUX
ejde-624	106	5	the	the	DET
ejde-624	106	6	appropriate	appropriate	ADJ
ejde-624	106	7	space	space	NOUN
ejde-624	106	8	for	for	ADP
ejde-624	106	9	the	the	DET
ejde-624	106	10	fundamental	fundamental	ADJ
ejde-624	106	11	theorem	theorem	NOUN
ejde-624	106	12	of	of	ADP
ejde-624	106	13	the	the	DET
ejde-624	106	14	calculus	calculus	NOUN
ejde-624	106	15	for	for	ADP
ejde-624	106	16	lebesgue	lebesgue	NOUN
ejde-624	106	17	integrals	integral	NOUN
ejde-624	106	18	.	.	PUNCT
ejde-624	107	1	in	in	ADP
ejde-624	107	2	fact	fact	NOUN
ejde-624	107	3	,	,	PUNCT
ejde-624	107	4	we	we	PRON
ejde-624	107	5	have	have	VERB
ejde-624	107	6	the	the	DET
ejde-624	107	7	following	following	ADJ
ejde-624	107	8	equivalence	equivalence	NOUN
ejde-624	107	9	.	.	PUNCT
ejde-624	108	1	u	u	PROPN
ejde-624	108	2	∈	∈	PROPN
ejde-624	108	3	ac[0	ac[0	PROPN
ejde-624	108	4	,	,	PUNCT
ejde-624	108	5	t	t	X
ejde-624	108	6	]	]	PUNCT
ejde-624	108	7	if	if	SCONJ
ejde-624	108	8	and	and	CCONJ
ejde-624	108	9	only	only	ADV
ejde-624	108	10	if	if	SCONJ
ejde-624	108	11	u′	u′	X
ejde-624	108	12	∈	∈	PROPN
ejde-624	108	13	l1[0	l1[0	PROPN
ejde-624	108	14	,	,	PUNCT
ejde-624	108	15	t	t	X
ejde-624	108	16	]	]	PUNCT
ejde-624	108	17	,	,	PUNCT
ejde-624	108	18	u′(t	u′(t	NOUN
ejde-624	108	19	)	)	PUNCT
ejde-624	108	20	exists	exist	VERB
ejde-624	108	21	for	for	ADP
ejde-624	108	22	almost	almost	ADV
ejde-624	108	23	every	every	PRON
ejde-624	108	24	(	(	PUNCT
ejde-624	108	25	a.e	a.e	PROPN
ejde-624	108	26	.	.	PROPN
ejde-624	108	27	)	)	PUNCT
ejde-624	109	1	t	t	PROPN
ejde-624	109	2	∈	∈	PROPN
ejde-624	110	1	[	[	X
ejde-624	110	2	0	0	NUM
ejde-624	110	3	,	,	PUNCT
ejde-624	110	4	t	t	NOUN
ejde-624	110	5	]	]	PUNCT
ejde-624	110	6	and	and	CCONJ
ejde-624	110	7	u(t)−	u(t)−	PROPN
ejde-624	110	8	u(0	u(0	PROPN
ejde-624	110	9	)	)	PUNCT
ejde-624	110	10	=	=	SYM
ejde-624	111	1	∫	∫	PROPN
ejde-624	111	2	t	t	PROPN
ejde-624	111	3	0	0	NUM
ejde-624	111	4	u′(s	u′(s	ADV
ejde-624	111	5	)	)	PUNCT
ejde-624	111	6	ds	ds	VERB
ejde-624	111	7	for	for	ADP
ejde-624	111	8	all	all	DET
ejde-624	111	9	t	t	NOUN
ejde-624	111	10	∈	∈	PROPN
ejde-624	112	1	[	[	X
ejde-624	112	2	0	0	NUM
ejde-624	112	3	,	,	PUNCT
ejde-624	112	4	t	t	X
ejde-624	112	5	]	]	PUNCT
ejde-624	112	6	.	.	PUNCT
ejde-624	113	1	(	(	PUNCT
ejde-624	113	2	2.2	2.2	NUM
ejde-624	113	3	)	)	PUNCT
ejde-624	113	4	the	the	DET
ejde-624	113	5	gamma	gamma	NOUN
ejde-624	113	6	and	and	CCONJ
ejde-624	113	7	beta	beta	NOUN
ejde-624	113	8	functions	function	NOUN
ejde-624	113	9	frequently	frequently	ADV
ejde-624	113	10	occur	occur	VERB
ejde-624	113	11	in	in	ADP
ejde-624	113	12	fractional	fractional	ADJ
ejde-624	113	13	problems	problem	NOUN
ejde-624	113	14	.	.	PUNCT
ejde-624	114	1	the	the	DET
ejde-624	114	2	gamma	gamma	PROPN
ejde-624	114	3	function	function	NOUN
ejde-624	114	4	is	be	AUX
ejde-624	114	5	,	,	PUNCT
ejde-624	114	6	for	for	ADP
ejde-624	114	7	α	α	PROPN
ejde-624	114	8	>	>	X
ejde-624	114	9	0	0	NUM
ejde-624	114	10	,	,	PUNCT
ejde-624	114	11	given	give	VERB
ejde-624	114	12	by	by	ADP
ejde-624	114	13	γ(α	γ(α	PROPN
ejde-624	114	14	)	)	PUNCT
ejde-624	114	15	:	:	PUNCT
ejde-624	115	1	=	=	SYM
ejde-624	115	2	∫	∫	PROPN
ejde-624	115	3	∞	∞	PROPN
ejde-624	115	4	0	0	NUM
ejde-624	115	5	sα−1	sα−1	NOUN
ejde-624	115	6	exp(−s	exp(−	NOUN
ejde-624	115	7	)	)	PUNCT
ejde-624	115	8	ds	ds	PROPN
ejde-624	115	9	.	.	PUNCT
ejde-624	115	10	(	(	PUNCT
ejde-624	115	11	2.3	2.3	NUM
ejde-624	115	12	)	)	PUNCT
ejde-624	115	13	the	the	DET
ejde-624	115	14	gamma	gamma	NOUN
ejde-624	115	15	function	function	NOUN
ejde-624	115	16	has	have	VERB
ejde-624	115	17	the	the	DET
ejde-624	115	18	property	property	NOUN
ejde-624	115	19	γ(α	γ(α	NOUN
ejde-624	115	20	+	+	CCONJ
ejde-624	115	21	1	1	X
ejde-624	115	22	)	)	PUNCT
ejde-624	115	23	=	=	SYM
ejde-624	115	24	αγ(α	αγ(α	PRON
ejde-624	115	25	)	)	PUNCT
ejde-624	115	26	for	for	ADP
ejde-624	115	27	α	α	PROPN
ejde-624	115	28	>	>	X
ejde-624	115	29	0	0	PROPN
ejde-624	115	30	.	.	PUNCT
ejde-624	116	1	the	the	DET
ejde-624	116	2	beta	beta	ADJ
ejde-624	116	3	function	function	NOUN
ejde-624	116	4	is	be	AUX
ejde-624	116	5	defined	define	VERB
ejde-624	116	6	for	for	ADP
ejde-624	116	7	α	α	PROPN
ejde-624	116	8	>	>	X
ejde-624	116	9	0	0	PROPN
ejde-624	116	10	,	,	PUNCT
ejde-624	116	11	β	β	X
ejde-624	116	12	>	>	X
ejde-624	116	13	0	0	NUM
ejde-624	116	14	by	by	ADP
ejde-624	116	15	b(α	b(α	NOUN
ejde-624	116	16	,	,	PUNCT
ejde-624	116	17	β	β	NOUN
ejde-624	116	18	)	)	PUNCT
ejde-624	116	19	:	:	PUNCT
ejde-624	117	1	=	=	SYM
ejde-624	117	2	∫	∫	PROPN
ejde-624	117	3	1	1	NUM
ejde-624	117	4	0	0	NUM
ejde-624	117	5	(	(	PUNCT
ejde-624	117	6	1−	1−	NUM
ejde-624	117	7	s)α−1sβ−1	s)α−1sβ−1	NOUN
ejde-624	117	8	ds	ds	NOUN
ejde-624	117	9	.	.	PUNCT
ejde-624	117	10	(	(	PUNCT
ejde-624	117	11	2.4	2.4	NUM
ejde-624	117	12	)	)	PUNCT
ejde-624	117	13	these	these	PRON
ejde-624	117	14	are	be	AUX
ejde-624	117	15	well	well	ADV
ejde-624	117	16	defined	define	VERB
ejde-624	117	17	lebesgue	lebesgue	NOUN
ejde-624	117	18	integrals	integral	NOUN
ejde-624	117	19	and	and	CCONJ
ejde-624	117	20	it	it	PRON
ejde-624	117	21	is	be	AUX
ejde-624	117	22	well	well	ADV
ejde-624	117	23	known	know	VERB
ejde-624	117	24	that	that	SCONJ
ejde-624	117	25	b(α	b(α	NOUN
ejde-624	117	26	,	,	PUNCT
ejde-624	117	27	β	β	NOUN
ejde-624	117	28	)	)	PUNCT
ejde-624	117	29	=	=	SYM
ejde-624	117	30	γ(α)γ(β	γ(α)γ(β	NOUN
ejde-624	117	31	)	)	PUNCT
ejde-624	117	32	γ(α+	γ(α+	X
ejde-624	117	33	β	β	X
ejde-624	117	34	)	)	PUNCT
ejde-624	117	35	.	.	PUNCT
ejde-624	118	1	we	we	PRON
ejde-624	118	2	will	will	AUX
ejde-624	118	3	also	also	ADV
ejde-624	118	4	use	use	VERB
ejde-624	118	5	the	the	DET
ejde-624	118	6	following	follow	VERB
ejde-624	118	7	property	property	NOUN
ejde-624	118	8	which	which	PRON
ejde-624	118	9	is	be	AUX
ejde-624	118	10	proved	prove	VERB
ejde-624	118	11	by	by	ADP
ejde-624	118	12	the	the	DET
ejde-624	118	13	simple	simple	ADJ
ejde-624	118	14	substitution	substitution	NOUN
ejde-624	118	15	s	s	PART
ejde-624	118	16	=	=	SYM
ejde-624	118	17	tσ	tσ	PROPN
ejde-624	118	18	.	.	PUNCT
ejde-624	118	19	lemma	lemma	PROPN
ejde-624	118	20	2.1	2.1	NUM
ejde-624	118	21	.	.	PUNCT
ejde-624	119	1	let	let	VERB
ejde-624	119	2	t	t	PROPN
ejde-624	119	3	>	>	X
ejde-624	119	4	0	0	PUNCT
ejde-624	120	1	and	and	CCONJ
ejde-624	120	2	α	α	X
ejde-624	120	3	>	>	X
ejde-624	120	4	0	0	PROPN
ejde-624	120	5	,	,	PUNCT
ejde-624	120	6	β	β	X
ejde-624	120	7	>	>	X
ejde-624	120	8	0	0	X
ejde-624	120	9	.	.	PUNCT
ejde-624	121	1	then	then	ADV
ejde-624	121	2	we	we	PRON
ejde-624	121	3	have∫	have∫	VERB
ejde-624	121	4	t	t	NOUN
ejde-624	121	5	0	0	NUM
ejde-624	121	6	(	(	PUNCT
ejde-624	121	7	t−	t−	PRON
ejde-624	121	8	s)α−1sβ−1	s)α−1sβ−1	NOUN
ejde-624	121	9	ds	ds	PROPN
ejde-624	121	10	=	=	SYM
ejde-624	121	11	tα+β−1b(α	tα+β−1b(α	PROPN
ejde-624	121	12	,	,	PUNCT
ejde-624	121	13	β	β	NOUN
ejde-624	121	14	)	)	PUNCT
ejde-624	121	15	.	.	PUNCT
ejde-624	122	1	(	(	PUNCT
ejde-624	122	2	2.5	2.5	NUM
ejde-624	122	3	)	)	PUNCT
ejde-624	122	4	these	these	DET
ejde-624	122	5	properties	property	NOUN
ejde-624	122	6	will	will	AUX
ejde-624	122	7	be	be	AUX
ejde-624	122	8	used	use	VERB
ejde-624	122	9	without	without	ADP
ejde-624	122	10	further	further	ADJ
ejde-624	122	11	mention	mention	NOUN
ejde-624	122	12	.	.	PUNCT
ejde-624	123	1	we	we	PRON
ejde-624	123	2	will	will	AUX
ejde-624	123	3	use	use	VERB
ejde-624	123	4	the	the	DET
ejde-624	123	5	so	so	ADV
ejde-624	123	6	-	-	PUNCT
ejde-624	123	7	called	call	VERB
ejde-624	123	8	riemann	riemann	PROPN
ejde-624	123	9	-	-	PUNCT
ejde-624	123	10	liouville	liouville	NOUN
ejde-624	123	11	(	(	PUNCT
ejde-624	123	12	r	r	NOUN
ejde-624	123	13	-	-	PUNCT
ejde-624	123	14	l	l	NOUN
ejde-624	123	15	)	)	PUNCT
ejde-624	123	16	fractional	fractional	ADJ
ejde-624	123	17	integral	integral	ADJ
ejde-624	123	18	.	.	PUNCT
ejde-624	124	1	using	use	VERB
ejde-624	124	2	this	this	PRON
ejde-624	124	3	we	we	PRON
ejde-624	124	4	will	will	AUX
ejde-624	124	5	consider	consider	VERB
ejde-624	124	6	the	the	DET
ejde-624	124	7	two	two	NUM
ejde-624	124	8	most	most	ADV
ejde-624	124	9	often	often	ADV
ejde-624	124	10	used	use	VERB
ejde-624	124	11	fractional	fractional	ADJ
ejde-624	124	12	derivatives	derivative	NOUN
ejde-624	124	13	,	,	PUNCT
ejde-624	124	14	the	the	DET
ejde-624	124	15	r	r	NOUN
ejde-624	124	16	-	-	PUNCT
ejde-624	124	17	l	l	NOUN
ejde-624	124	18	and	and	CCONJ
ejde-624	124	19	the	the	DET
ejde-624	124	20	caputo	caputo	PROPN
ejde-624	124	21	versions	version	NOUN
ejde-624	124	22	.	.	PUNCT
ejde-624	125	1	the	the	DET
ejde-624	125	2	r	r	NOUN
ejde-624	125	3	-	-	PUNCT
ejde-624	125	4	l	l	NOUN
ejde-624	125	5	fractional	fractional	ADJ
ejde-624	125	6	integral	integral	NOUN
ejde-624	125	7	is	be	AUX
ejde-624	125	8	defined	define	VERB
ejde-624	125	9	for	for	ADP
ejde-624	125	10	l1	l1	PROPN
ejde-624	125	11	functions	function	NOUN
ejde-624	125	12	as	as	SCONJ
ejde-624	125	13	follows	follow	VERB
ejde-624	125	14	.	.	PUNCT
ejde-624	126	1	definition	definition	NOUN
ejde-624	126	2	2.2	2.2	NUM
ejde-624	126	3	.	.	PUNCT
ejde-624	127	1	the	the	DET
ejde-624	127	2	riemann	riemann	PROPN
ejde-624	127	3	-	-	PUNCT
ejde-624	127	4	liouville	liouville	NOUN
ejde-624	127	5	(	(	PUNCT
ejde-624	127	6	r	r	NOUN
ejde-624	127	7	-	-	PUNCT
ejde-624	127	8	l	l	NOUN
ejde-624	127	9	)	)	PUNCT
ejde-624	127	10	fractional	fractional	ADJ
ejde-624	127	11	integral	integral	ADJ
ejde-624	127	12	of	of	ADP
ejde-624	127	13	order	order	NOUN
ejde-624	127	14	α	α	PROPN
ejde-624	127	15	>	>	X
ejde-624	127	16	0	0	NUM
ejde-624	127	17	of	of	ADP
ejde-624	127	18	a	a	DET
ejde-624	127	19	function	function	NOUN
ejde-624	127	20	u	u	PROPN
ejde-624	127	21	∈	∈	PROPN
ejde-624	127	22	l1[0	l1[0	PROPN
ejde-624	127	23	,	,	PUNCT
ejde-624	127	24	t	t	PROPN
ejde-624	127	25	]	]	PUNCT
ejde-624	127	26	is	be	AUX
ejde-624	127	27	defined	define	VERB
ejde-624	127	28	for	for	ADP
ejde-624	127	29	a.e	a.e	PROPN
ejde-624	127	30	.	.	PROPN
ejde-624	127	31	t	t	PROPN
ejde-624	127	32	by	by	ADP
ejde-624	127	33	iαu(t	iαu(t	PROPN
ejde-624	127	34	)	)	PUNCT
ejde-624	128	1	:	:	PUNCT
ejde-624	128	2	=	=	SYM
ejde-624	128	3	1	1	NUM
ejde-624	128	4	γ(α	γ(α	NOUN
ejde-624	128	5	)	)	PUNCT
ejde-624	128	6	∫	∫	PROPN
ejde-624	128	7	t	t	PROPN
ejde-624	128	8	0	0	NUM
ejde-624	128	9	(	(	PUNCT
ejde-624	128	10	t−	t−	PROPN
ejde-624	128	11	s)α−1u(s	s)α−1u(s	ADJ
ejde-624	128	12	)	)	PUNCT
ejde-624	128	13	ds	ds	PROPN
ejde-624	128	14	.	.	PUNCT
ejde-624	128	15	(	(	PUNCT
ejde-624	128	16	2.6	2.6	NUM
ejde-624	128	17	)	)	PUNCT
ejde-624	128	18	the	the	DET
ejde-624	128	19	integral	integral	ADJ
ejde-624	128	20	iαu	iαu	NOUN
ejde-624	128	21	is	be	AUX
ejde-624	128	22	the	the	DET
ejde-624	128	23	convolution	convolution	NOUN
ejde-624	128	24	of	of	ADP
ejde-624	128	25	the	the	DET
ejde-624	128	26	l1	l1	PROPN
ejde-624	128	27	functions	functions	PROPN
ejde-624	128	28	h	h	PROPN
ejde-624	128	29	,	,	PUNCT
ejde-624	128	30	u	u	PROPN
ejde-624	128	31	where	where	SCONJ
ejde-624	128	32	h(t	h(t	X
ejde-624	128	33	)	)	PUNCT
ejde-624	129	1	=	=	PUNCT
ejde-624	129	2	tα−1	tα−1	NOUN
ejde-624	129	3	/	/	SYM
ejde-624	129	4	γ(α	γ(α	PROPN
ejde-624	129	5	)	)	PUNCT
ejde-624	130	1	,	,	PUNCT
ejde-624	130	2	so	so	CCONJ
ejde-624	130	3	by	by	ADP
ejde-624	130	4	the	the	DET
ejde-624	130	5	well	well	ADV
ejde-624	130	6	known	know	VERB
ejde-624	130	7	results	result	NOUN
ejde-624	130	8	on	on	ADP
ejde-624	130	9	convolutions	convolution	NOUN
ejde-624	130	10	iαu	iαu	NOUN
ejde-624	130	11	is	be	AUX
ejde-624	130	12	defined	define	VERB
ejde-624	130	13	as	as	ADP
ejde-624	130	14	an	an	DET
ejde-624	130	15	l1	l1	PROPN
ejde-624	130	16	function	function	NOUN
ejde-624	130	17	,	,	PUNCT
ejde-624	130	18	in	in	ADP
ejde-624	130	19	particular	particular	ADJ
ejde-624	130	20	iαu(t	iαu(t	PROPN
ejde-624	130	21	)	)	PUNCT
ejde-624	130	22	is	be	AUX
ejde-624	130	23	defined	define	VERB
ejde-624	130	24	and	and	CCONJ
ejde-624	130	25	finite	finite	VERB
ejde-624	130	26	for	for	ADP
ejde-624	130	27	a.e	a.e	PROPN
ejde-624	131	1	.	.	PUNCT
ejde-624	131	2	t.	t.	PROPN
ejde-624	132	1	if	if	SCONJ
ejde-624	132	2	α	α	NOUN
ejde-624	132	3	=	=	NOUN
ejde-624	132	4	1	1	NUM
ejde-624	132	5	this	this	PRON
ejde-624	132	6	is	be	AUX
ejde-624	132	7	the	the	DET
ejde-624	132	8	usual	usual	ADJ
ejde-624	132	9	integration	integration	NOUN
ejde-624	132	10	operator	operator	NOUN
ejde-624	132	11	which	which	PRON
ejde-624	132	12	we	we	PRON
ejde-624	132	13	denote	denote	VERB
ejde-624	132	14	i.	i.	NOUN
ejde-624	132	15	we	we	PRON
ejde-624	132	16	define	define	VERB
ejde-624	132	17	iαu(0	iαu(0	ADJ
ejde-624	132	18	)	)	PUNCT
ejde-624	133	1	:	:	PUNCT
ejde-624	133	2	=	=	PUNCT
ejde-624	133	3	limt→0	limt→0	PROPN
ejde-624	133	4	+	+	CCONJ
ejde-624	133	5	iαu(t	iαu(t	PROPN
ejde-624	133	6	)	)	PUNCT
ejde-624	133	7	if	if	SCONJ
ejde-624	133	8	this	this	DET
ejde-624	133	9	limit	limit	NOUN
ejde-624	133	10	exists	exist	VERB
ejde-624	133	11	,	,	PUNCT
ejde-624	133	12	otherwise	otherwise	ADV
ejde-624	133	13	it	it	PRON
ejde-624	133	14	is	be	AUX
ejde-624	133	15	not	not	PART
ejde-624	133	16	defined	define	VERB
ejde-624	133	17	.	.	PUNCT
ejde-624	134	1	detailed	detailed	ADJ
ejde-624	134	2	discussion	discussion	NOUN
ejde-624	134	3	of	of	ADP
ejde-624	134	4	these	these	DET
ejde-624	134	5	operators	operator	NOUN
ejde-624	134	6	can	can	AUX
ejde-624	134	7	be	be	AUX
ejde-624	134	8	found	find	VERB
ejde-624	134	9	in	in	ADP
ejde-624	134	10	the	the	DET
ejde-624	134	11	texts	text	NOUN
ejde-624	134	12	[	[	X
ejde-624	134	13	5	5	NUM
ejde-624	134	14	,	,	PUNCT
ejde-624	134	15	10	10	NUM
ejde-624	134	16	,	,	PUNCT
ejde-624	134	17	17	17	NUM
ejde-624	134	18	]	]	PUNCT
ejde-624	134	19	,	,	PUNCT
ejde-624	134	20	a	a	DET
ejde-624	134	21	survey	survey	NOUN
ejde-624	134	22	of	of	ADP
ejde-624	134	23	some	some	DET
ejde-624	134	24	important	important	ADJ
ejde-624	134	25	results	result	NOUN
ejde-624	134	26	is	be	AUX
ejde-624	134	27	given	give	VERB
ejde-624	134	28	in	in	ADP
ejde-624	134	29	the	the	DET
ejde-624	134	30	free	free	ADJ
ejde-624	134	31	to	to	PART
ejde-624	134	32	access	access	VERB
ejde-624	134	33	paper	paper	NOUN
ejde-624	134	34	[	[	X
ejde-624	134	35	21	21	NUM
ejde-624	134	36	]	]	PUNCT
ejde-624	134	37	.	.	PUNCT
ejde-624	135	1	one	one	NUM
ejde-624	135	2	useful	useful	ADJ
ejde-624	135	3	result	result	NOUN
ejde-624	135	4	is	be	AUX
ejde-624	135	5	the	the	DET
ejde-624	135	6	semigroup	semigroup	ADJ
ejde-624	135	7	property	property	NOUN
ejde-624	135	8	as	as	SCONJ
ejde-624	135	9	follows	follow	VERB
ejde-624	135	10	,	,	PUNCT
ejde-624	135	11	see	see	VERB
ejde-624	135	12	for	for	ADP
ejde-624	135	13	example	example	NOUN
ejde-624	135	14	[	[	X
ejde-624	135	15	5	5	NUM
ejde-624	135	16	,	,	PUNCT
ejde-624	135	17	theorem	theorem	VERB
ejde-624	135	18	2.2	2.2	NUM
ejde-624	135	19	]	]	PUNCT
ejde-624	135	20	,	,	PUNCT
ejde-624	135	21	[	[	X
ejde-624	135	22	17	17	NUM
ejde-624	135	23	,	,	PUNCT
ejde-624	135	24	(	(	PUNCT
ejde-624	135	25	2.21	2.21	NUM
ejde-624	135	26	)	)	PUNCT
ejde-624	135	27	]	]	PUNCT
ejde-624	135	28	,	,	PUNCT
ejde-624	135	29	[	[	X
ejde-624	135	30	21	21	NUM
ejde-624	135	31	,	,	PUNCT
ejde-624	135	32	lemma	lemma	PROPN
ejde-624	135	33	2.4	2.4	NUM
ejde-624	135	34	]	]	PUNCT
ejde-624	135	35	.	.	PUNCT
ejde-624	136	1	ejde-2024/40	ejde-2024/40	VERB
ejde-624	136	2	fractional	fractional	ADJ
ejde-624	136	3	differential	differential	ADJ
ejde-624	136	4	inequalities	inequality	NOUN
ejde-624	136	5	5	5	NUM
ejde-624	136	6	lemma	lemma	PROPN
ejde-624	136	7	2.3	2.3	NUM
ejde-624	136	8	(	(	PUNCT
ejde-624	136	9	semigroup	semigroup	ADJ
ejde-624	136	10	property	property	NOUN
ejde-624	136	11	)	)	PUNCT
ejde-624	136	12	.	.	PUNCT
ejde-624	137	1	let	let	VERB
ejde-624	137	2	α	α	PRON
ejde-624	137	3	,	,	PUNCT
ejde-624	137	4	β	β	X
ejde-624	137	5	>	>	X
ejde-624	137	6	0	0	PUNCT
ejde-624	138	1	and	and	CCONJ
ejde-624	138	2	u	u	PROPN
ejde-624	138	3	∈	∈	PROPN
ejde-624	138	4	l1[0	l1[0	PROPN
ejde-624	138	5	,	,	PUNCT
ejde-624	138	6	t	t	X
ejde-624	138	7	]	]	PUNCT
ejde-624	138	8	.	.	PUNCT
ejde-624	139	1	then	then	ADV
ejde-624	139	2	iαiβ(u	iαiβ(u	VERB
ejde-624	139	3	)	)	PUNCT
ejde-624	139	4	=	=	SYM
ejde-624	140	1	iα+β(u	iα+β(u	PROPN
ejde-624	140	2	)	)	PUNCT
ejde-624	140	3	as	as	ADP
ejde-624	140	4	l1	l1	PROPN
ejde-624	140	5	functions	function	NOUN
ejde-624	140	6	,	,	PUNCT
ejde-624	140	7	thus	thus	ADV
ejde-624	140	8	,	,	PUNCT
ejde-624	140	9	iαiβ(u)(t	iαiβ(u)(t	PROPN
ejde-624	140	10	)	)	PUNCT
ejde-624	140	11	=	=	SYM
ejde-624	140	12	iα+β(u)(t	iα+β(u)(t	PROPN
ejde-624	140	13	)	)	PUNCT
ejde-624	140	14	for	for	ADP
ejde-624	140	15	a.e	a.e	PROPN
ejde-624	140	16	.	.	PROPN
ejde-624	140	17	t	t	PROPN
ejde-624	140	18	∈	∈	PROPN
ejde-624	141	1	[	[	X
ejde-624	141	2	0	0	NUM
ejde-624	141	3	,	,	PUNCT
ejde-624	141	4	t	t	X
ejde-624	141	5	]	]	PUNCT
ejde-624	141	6	,	,	PUNCT
ejde-624	141	7	in	in	ADP
ejde-624	141	8	fact	fact	NOUN
ejde-624	141	9	for	for	ADP
ejde-624	141	10	every	every	DET
ejde-624	141	11	t	t	NOUN
ejde-624	141	12	for	for	ADP
ejde-624	141	13	which	which	PRON
ejde-624	141	14	iα+β(|u|)(t	iα+β(|u|)(t	ADJ
ejde-624	141	15	)	)	PUNCT
ejde-624	141	16	exists	exist	VERB
ejde-624	141	17	.	.	PUNCT
ejde-624	142	1	if	if	SCONJ
ejde-624	142	2	u	u	NOUN
ejde-624	142	3	is	be	AUX
ejde-624	142	4	continuous	continuous	ADJ
ejde-624	142	5	this	this	PRON
ejde-624	142	6	holds	hold	VERB
ejde-624	142	7	for	for	ADP
ejde-624	142	8	all	all	DET
ejde-624	142	9	t	t	NOUN
ejde-624	142	10	∈	∈	PROPN
ejde-624	143	1	[	[	X
ejde-624	143	2	0	0	NUM
ejde-624	143	3	,	,	PUNCT
ejde-624	143	4	t	t	X
ejde-624	143	5	]	]	PUNCT
ejde-624	143	6	.	.	PUNCT
ejde-624	144	1	if	if	SCONJ
ejde-624	144	2	u	u	PROPN
ejde-624	144	3	∈	∈	PROPN
ejde-624	144	4	l1	l1	PROPN
ejde-624	144	5	and	and	CCONJ
ejde-624	144	6	α+	α+	PUNCT
ejde-624	144	7	β	β	X
ejde-624	144	8	≥	≥	NUM
ejde-624	144	9	1	1	NUM
ejde-624	144	10	equality	equality	NOUN
ejde-624	144	11	again	again	ADV
ejde-624	144	12	holds	hold	VERB
ejde-624	144	13	for	for	ADP
ejde-624	144	14	all	all	DET
ejde-624	144	15	t	t	NOUN
ejde-624	144	16	∈	∈	PROPN
ejde-624	145	1	[	[	X
ejde-624	145	2	0	0	NUM
ejde-624	145	3	,	,	PUNCT
ejde-624	145	4	t	t	X
ejde-624	145	5	]	]	PUNCT
ejde-624	145	6	.	.	PUNCT
ejde-624	146	1	in	in	ADP
ejde-624	146	2	this	this	DET
ejde-624	146	3	paper	paper	NOUN
ejde-624	146	4	we	we	PRON
ejde-624	146	5	only	only	ADV
ejde-624	146	6	consider	consider	VERB
ejde-624	146	7	fractional	fractional	ADJ
ejde-624	146	8	derivatives	derivative	NOUN
ejde-624	146	9	of	of	ADP
ejde-624	146	10	order	order	NOUN
ejde-624	146	11	0	0	PUNCT
ejde-624	146	12	<	<	X
ejde-624	146	13	α	α	X
ejde-624	146	14	<	<	X
ejde-624	146	15	1	1	NUM
ejde-624	146	16	.	.	PUNCT
ejde-624	147	1	let	let	VERB
ejde-624	147	2	d	d	PART
ejde-624	147	3	denote	denote	VERB
ejde-624	147	4	the	the	DET
ejde-624	147	5	usual	usual	ADJ
ejde-624	147	6	differentiation	differentiation	NOUN
ejde-624	147	7	operator	operator	NOUN
ejde-624	147	8	,	,	PUNCT
ejde-624	147	9	du	du	PROPN
ejde-624	147	10	=	=	PUNCT
ejde-624	147	11	u′.	u′.	PROPN
ejde-624	147	12	the	the	DET
ejde-624	147	13	riemann	riemann	PROPN
ejde-624	147	14	-	-	PUNCT
ejde-624	147	15	liouville	liouville	NOUN
ejde-624	147	16	(	(	PUNCT
ejde-624	147	17	r	r	NOUN
ejde-624	147	18	-	-	PUNCT
ejde-624	147	19	l	l	NOUN
ejde-624	147	20	)	)	PUNCT
ejde-624	147	21	fractional	fractional	ADJ
ejde-624	147	22	derivative	derivative	NOUN
ejde-624	147	23	of	of	ADP
ejde-624	147	24	order	order	NOUN
ejde-624	147	25	α	α	X
ejde-624	147	26	∈	∈	PROPN
ejde-624	147	27	(	(	PUNCT
ejde-624	147	28	0	0	NUM
ejde-624	147	29	,	,	PUNCT
ejde-624	147	30	1	1	NUM
ejde-624	147	31	)	)	PUNCT
ejde-624	147	32	is	be	AUX
ejde-624	147	33	informally	informally	ADV
ejde-624	147	34	defined	define	VERB
ejde-624	147	35	by	by	ADP
ejde-624	147	36	dαu(t	dαu(t	PROPN
ejde-624	147	37	)	)	PUNCT
ejde-624	147	38	=	=	SYM
ejde-624	147	39	d(i1−αu)(t	d(i1−αu)(t	PROPN
ejde-624	147	40	)	)	PUNCT
ejde-624	147	41	.	.	PUNCT
ejde-624	148	1	for	for	SCONJ
ejde-624	148	2	d	d	PROPN
ejde-624	148	3	i1−αu(t	i1−αu(t	NOUN
ejde-624	148	4	)	)	PUNCT
ejde-624	148	5	to	to	PART
ejde-624	148	6	be	be	AUX
ejde-624	148	7	defined	define	VERB
ejde-624	148	8	at	at	ADP
ejde-624	148	9	a	a	DET
ejde-624	148	10	point	point	NOUN
ejde-624	148	11	t	t	NOUN
ejde-624	148	12	,	,	PUNCT
ejde-624	148	13	it	it	PRON
ejde-624	148	14	is	be	AUX
ejde-624	148	15	necessary	necessary	ADJ
ejde-624	148	16	that	that	SCONJ
ejde-624	148	17	i1−αu	i1−αu	NOUN
ejde-624	148	18	should	should	AUX
ejde-624	148	19	be	be	AUX
ejde-624	148	20	differentiable	differentiable	ADJ
ejde-624	148	21	at	at	ADP
ejde-624	148	22	t	t	PROPN
ejde-624	148	23	which	which	PRON
ejde-624	148	24	requires	require	VERB
ejde-624	148	25	some	some	DET
ejde-624	148	26	extra	extra	ADJ
ejde-624	148	27	condition	condition	NOUN
ejde-624	148	28	,	,	PUNCT
ejde-624	148	29	which	which	PRON
ejde-624	148	30	we	we	PRON
ejde-624	148	31	now	now	ADV
ejde-624	148	32	discuss	discuss	VERB
ejde-624	148	33	.	.	PUNCT
ejde-624	149	1	it	it	PRON
ejde-624	149	2	is	be	AUX
ejde-624	149	3	useful	useful	ADJ
ejde-624	149	4	to	to	PART
ejde-624	149	5	know	know	VERB
ejde-624	149	6	when	when	SCONJ
ejde-624	149	7	the	the	DET
ejde-624	149	8	fractional	fractional	ADJ
ejde-624	149	9	derivative	derivative	ADJ
ejde-624	149	10	and	and	CCONJ
ejde-624	149	11	fractional	fractional	ADJ
ejde-624	149	12	integral	integral	ADJ
ejde-624	149	13	are	be	AUX
ejde-624	149	14	inverse	inverse	NOUN
ejde-624	149	15	operations	operation	NOUN
ejde-624	149	16	,	,	PUNCT
ejde-624	149	17	that	that	PRON
ejde-624	149	18	is	be	AUX
ejde-624	149	19	when	when	SCONJ
ejde-624	149	20	a	a	DET
ejde-624	149	21	fractional	fractional	ADJ
ejde-624	149	22	differential	differential	ADJ
ejde-624	149	23	equation	equation	NOUN
ejde-624	149	24	(	(	PUNCT
ejde-624	149	25	fde	fde	PROPN
ejde-624	149	26	)	)	PUNCT
ejde-624	149	27	with	with	ADP
ejde-624	149	28	an	an	DET
ejde-624	149	29	initial	initial	ADJ
ejde-624	149	30	condition	condition	NOUN
ejde-624	149	31	is	be	AUX
ejde-624	149	32	equivalent	equivalent	ADJ
ejde-624	149	33	to	to	ADP
ejde-624	149	34	an	an	DET
ejde-624	149	35	integral	integral	ADJ
ejde-624	149	36	equation	equation	NOUN
ejde-624	149	37	.	.	PUNCT
ejde-624	150	1	one	one	NOUN
ejde-624	150	2	frequently	frequently	ADV
ejde-624	150	3	used	use	VERB
ejde-624	150	4	,	,	PUNCT
ejde-624	150	5	but	but	CCONJ
ejde-624	150	6	imprecise	imprecise	ADJ
ejde-624	150	7	statement	statement	NOUN
ejde-624	150	8	,	,	PUNCT
ejde-624	150	9	is	be	AUX
ejde-624	150	10	as	as	SCONJ
ejde-624	150	11	follows	follow	VERB
ejde-624	150	12	.	.	PUNCT
ejde-624	151	1	if	if	SCONJ
ejde-624	151	2	0	0	NUM
ejde-624	151	3	<	<	X
ejde-624	151	4	α	α	X
ejde-624	151	5	<	<	X
ejde-624	151	6	1	1	NUM
ejde-624	151	7	,	,	PUNCT
ejde-624	151	8	then	then	ADV
ejde-624	151	9	u	u	NOUN
ejde-624	151	10	satisfies	satisfy	VERB
ejde-624	151	11	dαu	dαu	NOUN
ejde-624	151	12	=	=	SYM
ejde-624	151	13	f	f	PROPN
ejde-624	151	14	and	and	CCONJ
ejde-624	151	15	i1−α(0	i1−α(0	PROPN
ejde-624	151	16	)	)	PUNCT
ejde-624	152	1	=	=	SYM
ejde-624	152	2	c	c	X
ejde-624	152	3	/	/	SYM
ejde-624	152	4	γ(α	γ(α	NOUN
ejde-624	152	5	)	)	PUNCT
ejde-624	153	1	if	if	SCONJ
ejde-624	153	2	and	and	CCONJ
ejde-624	153	3	only	only	ADV
ejde-624	153	4	if	if	SCONJ
ejde-624	153	5	u(t	u(t	NOUN
ejde-624	153	6	)	)	PUNCT
ejde-624	153	7	=	=	SYM
ejde-624	153	8	iαf(t	iαf(t	PROPN
ejde-624	153	9	)	)	PUNCT
ejde-624	153	10	+	+	PUNCT
ejde-624	154	1	ctα−1	ctα−1	NOUN
ejde-624	154	2	.	.	PUNCT
ejde-624	155	1	if	if	SCONJ
ejde-624	155	2	u	u	PROPN
ejde-624	155	3	∈	∈	PROPN
ejde-624	155	4	l1	l1	PROPN
ejde-624	155	5	then	then	ADV
ejde-624	155	6	i1−αu	i1−αu	VERB
ejde-624	155	7	∈	∈	PROPN
ejde-624	155	8	l1	l1	PROPN
ejde-624	155	9	but	but	CCONJ
ejde-624	155	10	need	need	AUX
ejde-624	155	11	not	not	PART
ejde-624	155	12	be	be	AUX
ejde-624	155	13	differentiable	differentiable	ADJ
ejde-624	155	14	.	.	PUNCT
ejde-624	156	1	assuming	assume	VERB
ejde-624	156	2	additionally	additionally	ADV
ejde-624	156	3	that	that	SCONJ
ejde-624	156	4	i1−αu	i1−αu	NOUN
ejde-624	156	5	is	be	AUX
ejde-624	156	6	differentiable	differentiable	ADJ
ejde-624	156	7	almost	almost	ADV
ejde-624	156	8	everywhere	everywhere	ADV
ejde-624	156	9	then	then	ADV
ejde-624	156	10	dαu(t	dαu(t	PROPN
ejde-624	156	11	)	)	PUNCT
ejde-624	157	1	=	=	SYM
ejde-624	157	2	f(t	f(t	NOUN
ejde-624	157	3	)	)	PUNCT
ejde-624	157	4	can	can	AUX
ejde-624	157	5	be	be	AUX
ejde-624	157	6	satisfied	satisfied	ADJ
ejde-624	157	7	for	for	ADP
ejde-624	157	8	a.e	a.e	PROPN
ejde-624	157	9	.	.	PROPN
ejde-624	157	10	t	t	PROPN
ejde-624	157	11	,	,	PUNCT
ejde-624	157	12	but	but	CCONJ
ejde-624	157	13	it	it	PRON
ejde-624	157	14	is	be	AUX
ejde-624	157	15	not	not	PART
ejde-624	157	16	equivalent	equivalent	ADJ
ejde-624	157	17	to	to	ADP
ejde-624	157	18	an	an	DET
ejde-624	157	19	integral	integral	ADJ
ejde-624	157	20	equation	equation	NOUN
ejde-624	157	21	,	,	PUNCT
ejde-624	157	22	it	it	PRON
ejde-624	157	23	is	be	AUX
ejde-624	157	24	necessary	necessary	ADJ
ejde-624	157	25	to	to	PART
ejde-624	157	26	always	always	ADV
ejde-624	157	27	have	have	VERB
ejde-624	157	28	i1−αu	i1−αu	VERB
ejde-624	157	29	∈	∈	PROPN
ejde-624	157	30	ac	ac	PROPN
ejde-624	157	31	.	.	PUNCT
ejde-624	158	1	this	this	PRON
ejde-624	158	2	was	be	AUX
ejde-624	158	3	noted	note	VERB
ejde-624	158	4	long	long	ADV
ejde-624	158	5	ago	ago	ADV
ejde-624	158	6	in	in	ADP
ejde-624	158	7	the	the	DET
ejde-624	158	8	monograph	monograph	NOUN
ejde-624	159	1	[	[	X
ejde-624	159	2	17	17	NUM
ejde-624	159	3	]	]	PUNCT
ejde-624	159	4	,	,	PUNCT
ejde-624	159	5	see	see	VERB
ejde-624	159	6	[	[	X
ejde-624	159	7	17	17	NUM
ejde-624	159	8	,	,	PUNCT
ejde-624	159	9	definition	definition	NOUN
ejde-624	159	10	2.4	2.4	NUM
ejde-624	159	11	]	]	PUNCT
ejde-624	159	12	and	and	CCONJ
ejde-624	159	13	the	the	DET
ejde-624	159	14	related	related	ADJ
ejde-624	159	15	comments	comment	NOUN
ejde-624	159	16	in	in	ADP
ejde-624	159	17	the	the	DET
ejde-624	159	18	‘	'	PUNCT
ejde-624	159	19	notes	note	NOUN
ejde-624	159	20	to	to	ADP
ejde-624	159	21	§	§	NOUN
ejde-624	159	22	2.6	2.6	NUM
ejde-624	159	23	’	'	PUNCT
ejde-624	159	24	.	.	PUNCT
ejde-624	160	1	it	it	PRON
ejde-624	160	2	was	be	AUX
ejde-624	160	3	recalled	recall	VERB
ejde-624	160	4	in	in	ADP
ejde-624	160	5	[	[	X
ejde-624	160	6	21	21	NUM
ejde-624	160	7	]	]	PUNCT
ejde-624	160	8	.	.	PUNCT
ejde-624	161	1	therefore	therefore	ADV
ejde-624	161	2	a	a	DET
ejde-624	161	3	suitable	suitable	ADJ
ejde-624	161	4	precise	precise	ADJ
ejde-624	161	5	definition	definition	NOUN
ejde-624	161	6	is	be	AUX
ejde-624	161	7	as	as	SCONJ
ejde-624	161	8	follows	follow	VERB
ejde-624	161	9	.	.	PUNCT
ejde-624	162	1	definition	definition	NOUN
ejde-624	162	2	2.4	2.4	NUM
ejde-624	162	3	.	.	PUNCT
ejde-624	163	1	for	for	ADP
ejde-624	163	2	α	α	DET
ejde-624	163	3	∈	∈	PROPN
ejde-624	163	4	(	(	PUNCT
ejde-624	163	5	0	0	NUM
ejde-624	163	6	,	,	PUNCT
ejde-624	163	7	1	1	NUM
ejde-624	163	8	)	)	PUNCT
ejde-624	163	9	and	and	CCONJ
ejde-624	163	10	u	u	PROPN
ejde-624	163	11	∈	∈	PROPN
ejde-624	163	12	l1	l1	PROPN
ejde-624	163	13	the	the	DET
ejde-624	163	14	r	r	NOUN
ejde-624	163	15	-	-	PUNCT
ejde-624	163	16	l	l	NOUN
ejde-624	163	17	fractional	fractional	ADJ
ejde-624	163	18	derivative	derivative	ADJ
ejde-624	163	19	dαu	dαu	NOUN
ejde-624	163	20	is	be	AUX
ejde-624	163	21	defined	define	VERB
ejde-624	163	22	when	when	SCONJ
ejde-624	163	23	i1−αu	i1−αu	VERB
ejde-624	163	24	∈	∈	PROPN
ejde-624	163	25	ac	ac	PROPN
ejde-624	163	26	as	as	ADP
ejde-624	163	27	an	an	DET
ejde-624	163	28	l1	l1	PROPN
ejde-624	163	29	function	function	NOUN
ejde-624	163	30	by	by	ADP
ejde-624	163	31	dαu(t	dαu(t	PROPN
ejde-624	163	32	)	)	PUNCT
ejde-624	163	33	:	:	PUNCT
ejde-624	164	1	=	=	SYM
ejde-624	164	2	d	d	NOUN
ejde-624	164	3	i1−αu(t	i1−αu(t	NUM
ejde-624	164	4	)	)	PUNCT
ejde-624	164	5	,	,	PUNCT
ejde-624	164	6	a.e	a.e	PROPN
ejde-624	164	7	.	.	PROPN
ejde-624	164	8	t	t	PROPN
ejde-624	164	9	∈	∈	PROPN
ejde-624	165	1	[	[	X
ejde-624	165	2	0	0	NUM
ejde-624	165	3	,	,	PUNCT
ejde-624	165	4	t	t	X
ejde-624	165	5	]	]	PUNCT
ejde-624	165	6	.	.	PUNCT
ejde-624	166	1	(	(	PUNCT
ejde-624	166	2	2.7	2.7	NUM
ejde-624	166	3	)	)	PUNCT
ejde-624	166	4	then	then	ADV
ejde-624	166	5	we	we	PRON
ejde-624	166	6	do	do	AUX
ejde-624	166	7	have	have	VERB
ejde-624	166	8	an	an	DET
ejde-624	166	9	equivalence	equivalence	NOUN
ejde-624	166	10	which	which	PRON
ejde-624	166	11	is	be	AUX
ejde-624	166	12	stated	state	VERB
ejde-624	166	13	below	below	ADP
ejde-624	166	14	in	in	ADP
ejde-624	166	15	proposition	proposition	NOUN
ejde-624	166	16	3.1	3.1	NUM
ejde-624	166	17	.	.	PUNCT
ejde-624	167	1	the	the	DET
ejde-624	167	2	caputo	caputo	PROPN
ejde-624	167	3	differential	differential	PROPN
ejde-624	167	4	operator	operator	NOUN
ejde-624	167	5	,	,	PUNCT
ejde-624	167	6	or	or	CCONJ
ejde-624	167	7	caputo	caputo	PROPN
ejde-624	167	8	fractional	fractional	PROPN
ejde-624	167	9	derivative	derivative	NOUN
ejde-624	167	10	,	,	PUNCT
ejde-624	167	11	is	be	AUX
ejde-624	167	12	usually	usually	ADV
ejde-624	167	13	used	use	VERB
ejde-624	167	14	for	for	ADP
ejde-624	167	15	continuous	continuous	ADJ
ejde-624	167	16	functions	function	NOUN
ejde-624	167	17	u	u	NOUN
ejde-624	167	18	and	and	CCONJ
ejde-624	167	19	is	be	AUX
ejde-624	167	20	defined	define	VERB
ejde-624	167	21	via	via	ADP
ejde-624	167	22	the	the	DET
ejde-624	167	23	r	r	NOUN
ejde-624	167	24	-	-	PUNCT
ejde-624	167	25	l	l	NOUN
ejde-624	167	26	derivative	derivative	NOUN
ejde-624	167	27	,	,	PUNCT
ejde-624	167	28	as	as	ADP
ejde-624	167	29	in	in	ADP
ejde-624	167	30	the	the	DET
ejde-624	167	31	texts	text	NOUN
ejde-624	168	1	[	[	X
ejde-624	168	2	5	5	NUM
ejde-624	168	3	,	,	PUNCT
ejde-624	168	4	definition	definition	NOUN
ejde-624	168	5	3.2	3.2	NUM
ejde-624	168	6	]	]	PUNCT
ejde-624	168	7	,	,	PUNCT
ejde-624	168	8	[	[	X
ejde-624	168	9	10	10	NUM
ejde-624	168	10	,	,	PUNCT
ejde-624	168	11	(	(	PUNCT
ejde-624	168	12	2.4.1	2.4.1	NUM
ejde-624	168	13	)	)	PUNCT
ejde-624	168	14	]	]	PUNCT
ejde-624	168	15	.	.	PUNCT
ejde-624	169	1	definition	definition	NOUN
ejde-624	169	2	2.5	2.5	NUM
ejde-624	169	3	.	.	PUNCT
ejde-624	170	1	for	for	ADP
ejde-624	170	2	α	α	DET
ejde-624	170	3	∈	∈	PROPN
ejde-624	170	4	(	(	PUNCT
ejde-624	170	5	0	0	NUM
ejde-624	170	6	,	,	PUNCT
ejde-624	170	7	1	1	NUM
ejde-624	170	8	)	)	PUNCT
ejde-624	170	9	,	,	PUNCT
ejde-624	170	10	u	u	PROPN
ejde-624	170	11	∈	∈	PROPN
ejde-624	170	12	c	c	NOUN
ejde-624	170	13	and	and	CCONJ
ejde-624	170	14	i1−α(u	i1−α(u	NOUN
ejde-624	170	15	−	−	PROPN
ejde-624	170	16	u(0	u(0	NOUN
ejde-624	170	17	)	)	PUNCT
ejde-624	170	18	)	)	PUNCT
ejde-624	171	1	∈	∈	PROPN
ejde-624	171	2	ac	ac	ADP
ejde-624	171	3	the	the	DET
ejde-624	171	4	caputo	caputo	PROPN
ejde-624	171	5	derivative	derivative	PROPN
ejde-624	171	6	dα	dα	PROPN
ejde-624	171	7	∗	∗	NOUN
ejde-624	171	8	u	u	NOUN
ejde-624	171	9	is	be	AUX
ejde-624	171	10	defined	define	VERB
ejde-624	171	11	by	by	ADP
ejde-624	171	12	dα	dα	PROPN
ejde-624	171	13	∗	∗	NOUN
ejde-624	171	14	u	u	NOUN
ejde-624	171	15	:	:	PUNCT
ejde-624	171	16	=	=	PUNCT
ejde-624	171	17	dα(u−	dα(u−	PROPN
ejde-624	171	18	u(0	u(0	PROPN
ejde-624	171	19	)	)	PUNCT
ejde-624	171	20	)	)	PUNCT
ejde-624	171	21	.	.	PUNCT
ejde-624	172	1	(	(	PUNCT
ejde-624	172	2	2.8	2.8	NUM
ejde-624	172	3	)	)	PUNCT
ejde-624	172	4	this	this	PRON
ejde-624	172	5	defines	define	VERB
ejde-624	172	6	dα	dα	ADP
ejde-624	172	7	∗	∗	NOUN
ejde-624	172	8	u	u	NOUN
ejde-624	172	9	as	as	ADP
ejde-624	172	10	an	an	DET
ejde-624	172	11	l1	l1	PROPN
ejde-624	172	12	function	function	NOUN
ejde-624	172	13	,	,	PUNCT
ejde-624	172	14	so	so	CCONJ
ejde-624	172	15	dα	dα	ADJ
ejde-624	172	16	∗	∗	NOUN
ejde-624	172	17	u(t	u(t	NOUN
ejde-624	172	18	)	)	PUNCT
ejde-624	172	19	is	be	AUX
ejde-624	172	20	defined	define	VERB
ejde-624	172	21	and	and	CCONJ
ejde-624	172	22	finite	finite	VERB
ejde-624	172	23	for	for	ADP
ejde-624	172	24	a.e	a.e	PROPN
ejde-624	172	25	.	.	PUNCT
ejde-624	173	1	t.	t.	PROPN
ejde-624	173	2	the	the	DET
ejde-624	173	3	caputo	caputo	PROPN
ejde-624	173	4	derivative	derivative	NOUN
ejde-624	173	5	of	of	ADP
ejde-624	173	6	a	a	DET
ejde-624	173	7	constant	constant	ADJ
ejde-624	173	8	is	be	AUX
ejde-624	173	9	0	0	NUM
ejde-624	173	10	but	but	CCONJ
ejde-624	173	11	the	the	DET
ejde-624	173	12	r	r	NOUN
ejde-624	173	13	-	-	PUNCT
ejde-624	173	14	l	l	NOUN
ejde-624	173	15	derivative	derivative	NOUN
ejde-624	173	16	is	be	AUX
ejde-624	173	17	not	not	PART
ejde-624	173	18	,	,	PUNCT
ejde-624	173	19	dαc	dαc	X
ejde-624	173	20	=	=	SYM
ejde-624	173	21	c	c	X
ejde-624	173	22	γ(1−α	γ(1−α	PROPN
ejde-624	173	23	)	)	PUNCT
ejde-624	174	1	t	t	PROPN
ejde-624	174	2	−α	−α	PROPN
ejde-624	174	3	for	for	ADP
ejde-624	174	4	t	t	PROPN
ejde-624	174	5	>	>	X
ejde-624	174	6	0	0	X
ejde-624	174	7	.	.	PUNCT
ejde-624	175	1	there	there	PRON
ejde-624	175	2	is	be	VERB
ejde-624	175	3	another	another	DET
ejde-624	175	4	commonly	commonly	ADV
ejde-624	175	5	used	use	VERB
ejde-624	175	6	definition	definition	NOUN
ejde-624	175	7	of	of	ADP
ejde-624	175	8	caputo	caputo	PROPN
ejde-624	175	9	derivative	derivative	PROPN
ejde-624	175	10	namely	namely	ADV
ejde-624	175	11	:	:	PUNCT
ejde-624	175	12	definition	definition	NOUN
ejde-624	175	13	2.6	2.6	NUM
ejde-624	175	14	.	.	PUNCT
ejde-624	176	1	for	for	ADP
ejde-624	176	2	0	0	NUM
ejde-624	176	3	<	<	X
ejde-624	176	4	α	α	X
ejde-624	176	5	<	<	X
ejde-624	176	6	1	1	NUM
ejde-624	176	7	,	,	PUNCT
ejde-624	176	8	the	the	DET
ejde-624	176	9	caputo	caputo	PROPN
ejde-624	176	10	derivative	derivative	PROPN
ejde-624	176	11	dα	dα	PROPN
ejde-624	176	12	cu	cu	PROPN
ejde-624	176	13	is	be	AUX
ejde-624	176	14	defined	define	VERB
ejde-624	176	15	for	for	ADP
ejde-624	176	16	u	u	PROPN
ejde-624	176	17	∈	∈	PROPN
ejde-624	176	18	ac	ac	PROPN
ejde-624	176	19	as	as	ADP
ejde-624	176	20	an	an	DET
ejde-624	176	21	l1	l1	PROPN
ejde-624	176	22	function	function	NOUN
ejde-624	176	23	by	by	ADP
ejde-624	176	24	dα	dα	NOUN
ejde-624	176	25	cu(t	cu(t	NOUN
ejde-624	176	26	)	)	PUNCT
ejde-624	176	27	:	:	PUNCT
ejde-624	176	28	=	=	NOUN
ejde-624	176	29	i1−αu′(t	i1−αu′(t	NOUN
ejde-624	176	30	)	)	PUNCT
ejde-624	176	31	,	,	PUNCT
ejde-624	176	32	for	for	ADP
ejde-624	176	33	a.e	a.e	PROPN
ejde-624	176	34	.	.	PROPN
ejde-624	176	35	t.	t.	PROPN
ejde-624	176	36	(	(	PUNCT
ejde-624	176	37	2.9	2.9	NUM
ejde-624	176	38	)	)	PUNCT
ejde-624	176	39	we	we	PRON
ejde-624	176	40	will	will	AUX
ejde-624	176	41	not	not	PART
ejde-624	176	42	use	use	VERB
ejde-624	176	43	the	the	DET
ejde-624	176	44	definition	definition	NOUN
ejde-624	176	45	dα	dα	ADP
ejde-624	176	46	cu	cu	PROPN
ejde-624	176	47	because	because	SCONJ
ejde-624	176	48	it	it	PRON
ejde-624	176	49	has	have	VERB
ejde-624	176	50	the	the	DET
ejde-624	176	51	severe	severe	ADJ
ejde-624	176	52	disadvantage	disadvantage	NOUN
ejde-624	176	53	that	that	SCONJ
ejde-624	176	54	for	for	ADP
ejde-624	176	55	u	u	PRON
ejde-624	176	56	continuous	continuous	VERB
ejde-624	176	57	the	the	DET
ejde-624	176	58	‘	'	PUNCT
ejde-624	176	59	equivalence	equivalence	NOUN
ejde-624	176	60	’	'	PUNCT
ejde-624	176	61	between	between	ADP
ejde-624	176	62	the	the	DET
ejde-624	176	63	fractional	fractional	ADJ
ejde-624	176	64	initial	initial	ADJ
ejde-624	176	65	value	value	NOUN
ejde-624	176	66	problem	problem	NOUN
ejde-624	176	67	(	(	PUNCT
ejde-624	176	68	ivp	ivp	NOUN
ejde-624	176	69	)	)	PUNCT
ejde-624	176	70	dαu(t	dαu(t	PROPN
ejde-624	176	71	)	)	PUNCT
ejde-624	176	72	=	=	SYM
ejde-624	176	73	f(t	f(t	NOUN
ejde-624	176	74	)	)	PUNCT
ejde-624	176	75	,	,	PUNCT
ejde-624	176	76	u(0	u(0	PROPN
ejde-624	176	77	)	)	PUNCT
ejde-624	176	78	=	=	PUNCT
ejde-624	176	79	u0	u0	ADJ
ejde-624	176	80	and	and	CCONJ
ejde-624	176	81	the	the	DET
ejde-624	176	82	volterra	volterra	NOUN
ejde-624	176	83	integral	integral	ADJ
ejde-624	176	84	equation	equation	NOUN
ejde-624	176	85	u(t	u(t	NOUN
ejde-624	176	86	)	)	PUNCT
ejde-624	176	87	=	=	PUNCT
ejde-624	177	1	u0	u0	ADJ
ejde-624	177	2	+	+	X
ejde-624	177	3	(	(	PUNCT
ejde-624	177	4	iαf)(t	iαf)(t	X
ejde-624	177	5	)	)	PUNCT
ejde-624	177	6	is	be	AUX
ejde-624	177	7	not	not	PART
ejde-624	177	8	valid	valid	ADJ
ejde-624	177	9	.	.	PUNCT
ejde-624	178	1	iα	iα	PROPN
ejde-624	178	2	maps	maps	PROPN
ejde-624	178	3	c[0	c[0	PROPN
ejde-624	178	4	,	,	PUNCT
ejde-624	178	5	t	t	X
ejde-624	178	6	]	]	PUNCT
ejde-624	178	7	into	into	ADP
ejde-624	178	8	c[0	c[0	PROPN
ejde-624	178	9	,	,	PUNCT
ejde-624	178	10	t	t	NOUN
ejde-624	178	11	]	]	PUNCT
ejde-624	178	12	but	but	CCONJ
ejde-624	178	13	not	not	PART
ejde-624	178	14	(	(	PUNCT
ejde-624	178	15	all	all	PRON
ejde-624	178	16	of	of	ADP
ejde-624	178	17	)	)	PUNCT
ejde-624	178	18	c[0	c[0	PROPN
ejde-624	178	19	,	,	PUNCT
ejde-624	178	20	t	t	NOUN
ejde-624	178	21	]	]	PUNCT
ejde-624	178	22	into	into	ADP
ejde-624	178	23	ac[0	ac[0	PROPN
ejde-624	178	24	,	,	PUNCT
ejde-624	178	25	t	t	X
ejde-624	178	26	]	]	PUNCT
ejde-624	178	27	;	;	PUNCT
ejde-624	178	28	examples	example	NOUN
ejde-624	178	29	are	be	AUX
ejde-624	178	30	in	in	ADP
ejde-624	178	31	cichon	cichon	NOUN
ejde-624	178	32	-	-	PUNCT
ejde-624	178	33	salem	salem	NOUN
ejde-624	179	1	[	[	X
ejde-624	179	2	4	4	NUM
ejde-624	179	3	,	,	PUNCT
ejde-624	179	4	counter	counter	NOUN
ejde-624	179	5	-	-	NOUN
ejde-624	179	6	example	example	NOUN
ejde-624	179	7	1	1	NUM
ejde-624	179	8	]	]	PUNCT
ejde-624	179	9	,	,	PUNCT
ejde-624	179	10	and	and	CCONJ
ejde-624	179	11	webb	webb	PROPN
ejde-624	180	1	[	[	X
ejde-624	180	2	21	21	NUM
ejde-624	180	3	,	,	PUNCT
ejde-624	180	4	addendum	addendum	PROPN
ejde-624	180	5	]	]	PUNCT
ejde-624	180	6	.	.	PUNCT
ejde-624	181	1	a	a	DET
ejde-624	181	2	detailed	detailed	ADJ
ejde-624	181	3	discussion	discussion	NOUN
ejde-624	181	4	is	be	AUX
ejde-624	181	5	given	give	VERB
ejde-624	181	6	in	in	ADP
ejde-624	181	7	[	[	X
ejde-624	181	8	14	14	NUM
ejde-624	181	9	]	]	PUNCT
ejde-624	181	10	.	.	PUNCT
ejde-624	182	1	when	when	SCONJ
ejde-624	182	2	u	u	PROPN
ejde-624	182	3	∈	∈	PROPN
ejde-624	182	4	ac	ac	PROPN
ejde-624	182	5	,	,	PUNCT
ejde-624	182	6	dα	dα	PROPN
ejde-624	182	7	∗	∗	NOUN
ejde-624	182	8	u	u	NOUN
ejde-624	182	9	=	=	PROPN
ejde-624	182	10	dα	dα	PROPN
ejde-624	182	11	cu	cu	PROPN
ejde-624	182	12	,	,	PUNCT
ejde-624	182	13	so	so	ADV
ejde-624	182	14	often	often	ADV
ejde-624	182	15	there	there	PRON
ejde-624	182	16	is	be	VERB
ejde-624	182	17	no	no	DET
ejde-624	182	18	reason	reason	NOUN
ejde-624	182	19	to	to	PART
ejde-624	182	20	use	use	VERB
ejde-624	182	21	dα	dα	ADJ
ejde-624	182	22	c	c	NOUN
ejde-624	182	23	.	.	PUNCT
ejde-624	183	1	6	6	NUM
ejde-624	183	2	j.	j.	PROPN
ejde-624	183	3	r.	r.	PROPN
ejde-624	183	4	l.	l.	PROPN
ejde-624	183	5	webb	webb	PROPN
ejde-624	183	6	ejde-2024/40	ejde-2024/40	PROPN
ejde-624	183	7	3	3	X
ejde-624	183	8	.	.	PUNCT
ejde-624	183	9	riemann	riemann	PROPN
ejde-624	183	10	-	-	PUNCT
ejde-624	183	11	liouville	liouville	NOUN
ejde-624	183	12	equivalences	equivalence	VERB
ejde-624	183	13	for	for	ADP
ejde-624	183	14	0	0	NUM
ejde-624	183	15	<	<	X
ejde-624	183	16	α	α	X
ejde-624	183	17	<	<	X
ejde-624	183	18	1	1	NUM
ejde-624	183	19	an	an	DET
ejde-624	183	20	initial	initial	ADJ
ejde-624	183	21	value	value	NOUN
ejde-624	183	22	problem	problem	NOUN
ejde-624	183	23	for	for	ADP
ejde-624	183	24	the	the	DET
ejde-624	183	25	r	r	NOUN
ejde-624	183	26	-	-	PUNCT
ejde-624	183	27	l	l	NOUN
ejde-624	183	28	fractional	fractional	ADJ
ejde-624	183	29	differential	differential	NOUN
ejde-624	183	30	equation	equation	NOUN
ejde-624	183	31	(	(	PUNCT
ejde-624	183	32	fde	fde	NOUN
ejde-624	183	33	)	)	PUNCT
ejde-624	183	34	dαu	dαu	NOUN
ejde-624	183	35	=	=	SYM
ejde-624	183	36	f	f	PROPN
ejde-624	183	37	,	,	PUNCT
ejde-624	183	38	i1−αu(0	i1−αu(0	PROPN
ejde-624	183	39	)	)	PUNCT
ejde-624	183	40	=	=	SYM
ejde-624	184	1	limt→0	limt→0	PROPN
ejde-624	184	2	+	+	CCONJ
ejde-624	184	3	iαu(t	iαu(t	PROPN
ejde-624	184	4	)	)	PUNCT
ejde-624	184	5	=	=	SYM
ejde-624	184	6	c0	c0	NOUN
ejde-624	184	7	,	,	PUNCT
ejde-624	184	8	with	with	ADP
ejde-624	184	9	f	f	PROPN
ejde-624	184	10	∈	∈	PROPN
ejde-624	184	11	l1	l1	PROPN
ejde-624	184	12	can	can	AUX
ejde-624	184	13	be	be	AUX
ejde-624	184	14	studied	study	VERB
ejde-624	184	15	in	in	ADP
ejde-624	184	16	the	the	DET
ejde-624	184	17	space	space	NOUN
ejde-624	184	18	l1	l1	PROPN
ejde-624	184	19	.	.	PUNCT
ejde-624	185	1	an	an	DET
ejde-624	185	2	equivalence	equivalence	NOUN
ejde-624	185	3	with	with	ADP
ejde-624	185	4	an	an	DET
ejde-624	185	5	integral	integral	ADJ
ejde-624	185	6	equation	equation	NOUN
ejde-624	185	7	is	be	AUX
ejde-624	185	8	given	give	VERB
ejde-624	185	9	by	by	ADP
ejde-624	185	10	the	the	DET
ejde-624	185	11	following	follow	VERB
ejde-624	185	12	result	result	NOUN
ejde-624	185	13	,	,	PUNCT
ejde-624	185	14	for	for	ADP
ejde-624	185	15	example	example	NOUN
ejde-624	185	16	[	[	X
ejde-624	185	17	10	10	NUM
ejde-624	185	18	,	,	PUNCT
ejde-624	185	19	lemma	lemma	PROPN
ejde-624	185	20	2.5(b	2.5(b	NUM
ejde-624	185	21	)	)	PUNCT
ejde-624	185	22	]	]	PUNCT
ejde-624	185	23	,	,	PUNCT
ejde-624	186	1	[	[	X
ejde-624	186	2	17	17	NUM
ejde-624	186	3	,	,	PUNCT
ejde-624	186	4	theorem	theorem	VERB
ejde-624	186	5	2.4	2.4	NUM
ejde-624	186	6	]	]	PUNCT
ejde-624	186	7	and	and	CCONJ
ejde-624	186	8	[	[	X
ejde-624	186	9	21	21	NUM
ejde-624	186	10	,	,	PUNCT
ejde-624	186	11	proposition	proposition	NOUN
ejde-624	186	12	6.1	6.1	NUM
ejde-624	186	13	]	]	PUNCT
ejde-624	186	14	.	.	PUNCT
ejde-624	187	1	proposition	proposition	NOUN
ejde-624	187	2	3.1	3.1	NUM
ejde-624	187	3	.	.	PUNCT
ejde-624	188	1	let	let	VERB
ejde-624	189	1	f	f	PROPN
ejde-624	189	2	∈	∈	PROPN
ejde-624	189	3	l1[0	l1[0	PROPN
ejde-624	189	4	,	,	PUNCT
ejde-624	189	5	t	t	X
ejde-624	189	6	]	]	PUNCT
ejde-624	189	7	and	and	CCONJ
ejde-624	189	8	c0	c0	PROPN
ejde-624	189	9	∈	∈	PROPN
ejde-624	189	10	r.	r.	PROPN
ejde-624	189	11	then	then	ADV
ejde-624	189	12	a	a	DET
ejde-624	189	13	function	function	NOUN
ejde-624	189	14	u	u	PROPN
ejde-624	189	15	∈	∈	PROPN
ejde-624	189	16	l1	l1	PROPN
ejde-624	189	17	such	such	ADJ
ejde-624	189	18	that	that	PRON
ejde-624	189	19	i1−αu	i1−αu	NUM
ejde-624	189	20	∈	∈	PROPN
ejde-624	189	21	ac	ac	PROPN
ejde-624	189	22	satisfies	satisfy	VERB
ejde-624	189	23	dαu(t	dαu(t	PROPN
ejde-624	189	24	)	)	PUNCT
ejde-624	189	25	=	=	SYM
ejde-624	189	26	f(t	f(t	NOUN
ejde-624	189	27	)	)	PUNCT
ejde-624	189	28	a.e	a.e	PROPN
ejde-624	189	29	.	.	PROPN
ejde-624	189	30	and	and	CCONJ
ejde-624	189	31	i1−αu(0	i1−αu(0	NOUN
ejde-624	189	32	)	)	PUNCT
ejde-624	190	1	=	=	SYM
ejde-624	190	2	c0	c0	NOUN
ejde-624	190	3	if	if	SCONJ
ejde-624	190	4	and	and	CCONJ
ejde-624	190	5	only	only	ADV
ejde-624	190	6	u	u	PROPN
ejde-624	190	7	∈	∈	PROPN
ejde-624	190	8	l1	l1	PROPN
ejde-624	190	9	satisfies	satisfy	VERB
ejde-624	190	10	the	the	DET
ejde-624	190	11	volterra	volterra	NOUN
ejde-624	190	12	integral	integral	ADJ
ejde-624	190	13	equation	equation	NOUN
ejde-624	190	14	u(t	u(t	NOUN
ejde-624	190	15	)	)	PUNCT
ejde-624	190	16	=	=	SYM
ejde-624	190	17	c0	c0	PROPN
ejde-624	190	18	tα−1	tα−1	PROPN
ejde-624	190	19	γ(α	γ(α	PROPN
ejde-624	190	20	)	)	PUNCT
ejde-624	191	1	+	+	CCONJ
ejde-624	191	2	1	1	NUM
ejde-624	191	3	γ(α	γ(α	NOUN
ejde-624	191	4	)	)	PUNCT
ejde-624	191	5	∫	∫	PROPN
ejde-624	192	1	t	t	PROPN
ejde-624	192	2	0	0	NUM
ejde-624	192	3	(	(	PUNCT
ejde-624	192	4	t−	t−	PROPN
ejde-624	192	5	s)α−1f(s	s)α−1f(s	ADJ
ejde-624	192	6	)	)	PUNCT
ejde-624	192	7	ds	ds	PROPN
ejde-624	192	8	,	,	PUNCT
ejde-624	192	9	a.e	a.e	PROPN
ejde-624	192	10	.	.	PROPN
ejde-624	192	11	t	t	PROPN
ejde-624	192	12	∈	∈	PROPN
ejde-624	193	1	[	[	X
ejde-624	193	2	0	0	NUM
ejde-624	193	3	,	,	PUNCT
ejde-624	193	4	t	t	X
ejde-624	193	5	]	]	PUNCT
ejde-624	193	6	.	.	PUNCT
ejde-624	194	1	(	(	PUNCT
ejde-624	194	2	3.1	3.1	NUM
ejde-624	194	3	)	)	PUNCT
ejde-624	194	4	we	we	PRON
ejde-624	194	5	illustrate	illustrate	VERB
ejde-624	194	6	part	part	NOUN
ejde-624	194	7	of	of	ADP
ejde-624	194	8	the	the	DET
ejde-624	194	9	argument	argument	NOUN
ejde-624	194	10	needed	need	VERB
ejde-624	194	11	for	for	ADP
ejde-624	194	12	the	the	DET
ejde-624	194	13	proof	proof	NOUN
ejde-624	194	14	of	of	ADP
ejde-624	194	15	proposition	proposition	NOUN
ejde-624	194	16	3.1	3.1	NUM
ejde-624	194	17	by	by	ADP
ejde-624	194	18	showing	show	VERB
ejde-624	194	19	a	a	DET
ejde-624	194	20	positivity	positivity	NOUN
ejde-624	194	21	result	result	NOUN
ejde-624	194	22	.	.	PUNCT
ejde-624	195	1	lemma	lemma	PROPN
ejde-624	195	2	3.2	3.2	NUM
ejde-624	195	3	.	.	PUNCT
ejde-624	196	1	for	for	ADP
ejde-624	196	2	0	0	NUM
ejde-624	196	3	<	<	X
ejde-624	196	4	α	α	X
ejde-624	196	5	<	<	X
ejde-624	196	6	1	1	NUM
ejde-624	196	7	suppose	suppose	VERB
ejde-624	196	8	that	that	SCONJ
ejde-624	196	9	u	u	PROPN
ejde-624	196	10	∈	∈	PROPN
ejde-624	196	11	l1	l1	PROPN
ejde-624	196	12	and	and	CCONJ
ejde-624	196	13	i1−αu	i1−αu	NOUN
ejde-624	196	14	∈	∈	PROPN
ejde-624	196	15	ac[0	ac[0	NOUN
ejde-624	196	16	,	,	PUNCT
ejde-624	196	17	t	t	X
ejde-624	196	18	]	]	PUNCT
ejde-624	196	19	and	and	CCONJ
ejde-624	196	20	that	that	SCONJ
ejde-624	196	21	dαu	dαu	VERB
ejde-624	196	22	=	=	SYM
ejde-624	196	23	f	f	X
ejde-624	196	24	where	where	SCONJ
ejde-624	196	25	f	f	PROPN
ejde-624	196	26	∈	∈	PROPN
ejde-624	196	27	l1	l1	PROPN
ejde-624	196	28	and	and	CCONJ
ejde-624	196	29	f(t	f(t	PROPN
ejde-624	196	30	)	)	PUNCT
ejde-624	196	31	≥	≥	NOUN
ejde-624	196	32	0	0	NUM
ejde-624	197	1	a.e	a.e	PROPN
ejde-624	197	2	.	.	PROPN
ejde-624	198	1	on	on	ADP
ejde-624	198	2	[	[	X
ejde-624	198	3	0	0	NUM
ejde-624	198	4	,	,	PUNCT
ejde-624	198	5	t	t	X
ejde-624	198	6	]	]	PUNCT
ejde-624	198	7	.	.	PUNCT
ejde-624	199	1	then	then	ADV
ejde-624	199	2	i1−αu(0	i1−αu(0	NOUN
ejde-624	199	3	)	)	PUNCT
ejde-624	200	1	=	=	SYM
ejde-624	200	2	c0	c0	X
ejde-624	200	3	≥	≥	NUM
ejde-624	200	4	0	0	NUM
ejde-624	200	5	implies	imply	VERB
ejde-624	200	6	that	that	SCONJ
ejde-624	200	7	u(t	u(t	NOUN
ejde-624	200	8	)	)	PUNCT
ejde-624	200	9	≥	≥	NOUN
ejde-624	200	10	c0	c0	PROPN
ejde-624	200	11	tα−1	tα−1	PROPN
ejde-624	200	12	γ(α	γ(α	PROPN
ejde-624	200	13	)	)	PUNCT
ejde-624	200	14	for	for	ADP
ejde-624	200	15	a.e	a.e	PROPN
ejde-624	200	16	.	.	PROPN
ejde-624	200	17	t	t	PROPN
ejde-624	200	18	∈	∈	PROPN
ejde-624	200	19	(	(	PUNCT
ejde-624	200	20	0	0	NUM
ejde-624	200	21	,	,	PUNCT
ejde-624	200	22	t	t	X
ejde-624	200	23	]	]	PUNCT
ejde-624	200	24	.	.	PUNCT
ejde-624	201	1	if	if	SCONJ
ejde-624	201	2	,	,	PUNCT
ejde-624	201	3	in	in	ADP
ejde-624	201	4	addition	addition	NOUN
ejde-624	201	5	,	,	PUNCT
ejde-624	201	6	u	u	PROPN
ejde-624	201	7	∈	∈	PROPN
ejde-624	201	8	cα−1	cα−1	PROPN
ejde-624	201	9	then	then	ADV
ejde-624	201	10	t1−αu(t	t1−αu(t	NOUN
ejde-624	201	11	)	)	PUNCT
ejde-624	201	12	≥	≥	PROPN
ejde-624	201	13	c0	c0	PROPN
ejde-624	201	14	γ(α	γ(α	PROPN
ejde-624	201	15	)	)	PUNCT
ejde-624	201	16	for	for	ADP
ejde-624	201	17	all	all	DET
ejde-624	201	18	t	t	NOUN
ejde-624	201	19	∈	∈	PROPN
ejde-624	201	20	(	(	PUNCT
ejde-624	201	21	0	0	NUM
ejde-624	201	22	,	,	PUNCT
ejde-624	201	23	t	t	X
ejde-624	201	24	]	]	PUNCT
ejde-624	201	25	.	.	PUNCT
ejde-624	202	1	proof	proof	NOUN
ejde-624	202	2	.	.	PUNCT
ejde-624	203	1	dαu	dαu	ADJ
ejde-624	203	2	=	=	SYM
ejde-624	203	3	f	f	X
ejde-624	203	4	means	mean	VERB
ejde-624	203	5	thatd(i1−αu	thatd(i1−αu	PUNCT
ejde-624	203	6	)	)	PUNCT
ejde-624	204	1	=	=	SYM
ejde-624	204	2	f	f	PROPN
ejde-624	204	3	.	.	PUNCT
ejde-624	205	1	since	since	SCONJ
ejde-624	205	2	i1−αu	i1−αu	NUM
ejde-624	205	3	∈	∈	PROPN
ejde-624	205	4	ac[0	ac[0	NOUN
ejde-624	205	5	,	,	PUNCT
ejde-624	205	6	t	t	X
ejde-624	205	7	]	]	PUNCT
ejde-624	205	8	and	and	CCONJ
ejde-624	205	9	i1−αu(0	i1−αu(0	NOUN
ejde-624	205	10	)	)	PUNCT
ejde-624	205	11	=	=	SYM
ejde-624	205	12	c0	c0	NOUN
ejde-624	205	13	,	,	PUNCT
ejde-624	205	14	this	this	PRON
ejde-624	205	15	can	can	AUX
ejde-624	205	16	be	be	AUX
ejde-624	205	17	integrated	integrate	VERB
ejde-624	205	18	to	to	PART
ejde-624	205	19	give	give	VERB
ejde-624	205	20	i1−αu(t	i1−αu(t	NOUN
ejde-624	205	21	)	)	PUNCT
ejde-624	206	1	=	=	SYM
ejde-624	206	2	c0	c0	NOUN
ejde-624	206	3	+	+	CCONJ
ejde-624	206	4	if(t	if(t	NOUN
ejde-624	206	5	)	)	PUNCT
ejde-624	206	6	,	,	PUNCT
ejde-624	206	7	for	for	ADP
ejde-624	206	8	all	all	DET
ejde-624	206	9	t	t	NOUN
ejde-624	206	10	∈	∈	PROPN
ejde-624	207	1	[	[	X
ejde-624	207	2	0	0	NUM
ejde-624	207	3	,	,	PUNCT
ejde-624	207	4	t	t	X
ejde-624	207	5	]	]	PUNCT
ejde-624	207	6	.	.	PUNCT
ejde-624	208	1	applying	apply	VERB
ejde-624	208	2	iα	iα	NOUN
ejde-624	208	3	and	and	CCONJ
ejde-624	208	4	using	use	VERB
ejde-624	208	5	the	the	DET
ejde-624	208	6	semigroup	semigroup	ADJ
ejde-624	208	7	property	property	NOUN
ejde-624	208	8	gives	give	VERB
ejde-624	208	9	iu	iu	ADP
ejde-624	208	10	=	=	SYM
ejde-624	208	11	c0	c0	PROPN
ejde-624	208	12	tα	tα	PROPN
ejde-624	208	13	γ(α+1	γ(α+1	NOUN
ejde-624	208	14	)	)	PUNCT
ejde-624	209	1	+	+	NOUN
ejde-624	209	2	i(iαf	i(iαf	NOUN
ejde-624	209	3	)	)	PUNCT
ejde-624	209	4	.	.	PUNCT
ejde-624	210	1	since	since	SCONJ
ejde-624	210	2	all	all	DET
ejde-624	210	3	terms	term	NOUN
ejde-624	210	4	are	be	AUX
ejde-624	210	5	ac	ac	PROPN
ejde-624	210	6	,	,	PUNCT
ejde-624	210	7	the	the	DET
ejde-624	210	8	derivatives	derivative	NOUN
ejde-624	210	9	exist	exist	VERB
ejde-624	210	10	a.e	a.e	PROPN
ejde-624	210	11	.	.	PROPN
ejde-624	210	12	,	,	PUNCT
ejde-624	210	13	which	which	PRON
ejde-624	210	14	gives	give	VERB
ejde-624	210	15	u(t	u(t	NOUN
ejde-624	210	16	)	)	PUNCT
ejde-624	211	1	=	=	SYM
ejde-624	211	2	c0	c0	PROPN
ejde-624	211	3	tα−1	tα−1	PROPN
ejde-624	211	4	γ(α	γ(α	PROPN
ejde-624	211	5	)	)	PUNCT
ejde-624	212	1	+	+	CCONJ
ejde-624	212	2	(	(	PUNCT
ejde-624	212	3	iαf)(t	iαf)(t	X
ejde-624	212	4	)	)	PUNCT
ejde-624	212	5	for	for	ADP
ejde-624	212	6	a.e	a.e	PROPN
ejde-624	212	7	.	.	PROPN
ejde-624	212	8	t	t	PROPN
ejde-624	212	9	,	,	PUNCT
ejde-624	212	10	and	and	CCONJ
ejde-624	212	11	proves	prove	VERB
ejde-624	212	12	the	the	DET
ejde-624	212	13	result	result	NOUN
ejde-624	212	14	since	since	SCONJ
ejde-624	212	15	(	(	PUNCT
ejde-624	212	16	iαf)(t	iαf)(t	ADV
ejde-624	212	17	)	)	PUNCT
ejde-624	212	18	≥	≥	X
ejde-624	212	19	0	0	NUM
ejde-624	212	20	for	for	ADP
ejde-624	212	21	a.e	a.e	PROPN
ejde-624	212	22	.	.	PROPN
ejde-624	212	23	t.	t.	PROPN
ejde-624	213	1	the	the	DET
ejde-624	213	2	last	last	ADJ
ejde-624	213	3	part	part	NOUN
ejde-624	213	4	follows	follow	VERB
ejde-624	213	5	since	since	SCONJ
ejde-624	213	6	then	then	ADV
ejde-624	213	7	both	both	DET
ejde-624	213	8	sides	side	NOUN
ejde-624	213	9	are	be	AUX
ejde-624	213	10	continuous	continuous	ADJ
ejde-624	213	11	functions	function	NOUN
ejde-624	213	12	of	of	ADP
ejde-624	213	13	t	t	PROPN
ejde-624	213	14	∈	∈	PROPN
ejde-624	213	15	(	(	PUNCT
ejde-624	213	16	0	0	NUM
ejde-624	213	17	,	,	PUNCT
ejde-624	213	18	t	t	X
ejde-624	213	19	]	]	PUNCT
ejde-624	213	20	.	.	PUNCT
ejde-624	214	1	□	□	PUNCT
ejde-624	214	2	the	the	DET
ejde-624	214	3	problem	problem	NOUN
ejde-624	214	4	(	(	PUNCT
ejde-624	214	5	3.1	3.1	NUM
ejde-624	214	6	)	)	PUNCT
ejde-624	214	7	can	can	AUX
ejde-624	214	8	also	also	ADV
ejde-624	214	9	be	be	AUX
ejde-624	214	10	studied	study	VERB
ejde-624	214	11	in	in	ADP
ejde-624	214	12	the	the	DET
ejde-624	214	13	smaller	small	ADJ
ejde-624	214	14	space	space	NOUN
ejde-624	214	15	cα−1	cα−1	PROPN
ejde-624	214	16	=	=	SYM
ejde-624	214	17	c−(1−α	c−(1−α	NOUN
ejde-624	214	18	)	)	PUNCT
ejde-624	214	19	when	when	SCONJ
ejde-624	214	20	the	the	DET
ejde-624	214	21	initial	initial	ADJ
ejde-624	214	22	condition	condition	NOUN
ejde-624	214	23	limt→0	limt→0	NOUN
ejde-624	214	24	+	+	CCONJ
ejde-624	214	25	i1−αu(t	i1−αu(t	NUM
ejde-624	214	26	)	)	PUNCT
ejde-624	215	1	=	=	SYM
ejde-624	215	2	c0	c0	NOUN
ejde-624	215	3	is	be	AUX
ejde-624	215	4	replaced	replace	VERB
ejde-624	215	5	by	by	ADP
ejde-624	215	6	limt→0	limt→0	PROPN
ejde-624	215	7	+	+	SYM
ejde-624	215	8	t1−αu(t	t1−αu(t	NOUN
ejde-624	215	9	)	)	PUNCT
ejde-624	215	10	=	=	SYM
ejde-624	215	11	c0	c0	X
ejde-624	215	12	/	/	SYM
ejde-624	215	13	γ(α	γ(α	PROPN
ejde-624	215	14	)	)	PUNCT
ejde-624	215	15	.	.	PUNCT
ejde-624	216	1	the	the	DET
ejde-624	216	2	two	two	NUM
ejde-624	216	3	limits	limit	NOUN
ejde-624	216	4	are	be	AUX
ejde-624	216	5	related	relate	VERB
ejde-624	216	6	as	as	SCONJ
ejde-624	216	7	follows	follow	VERB
ejde-624	216	8	.	.	PUNCT
ejde-624	217	1	lemma	lemma	PROPN
ejde-624	217	2	3.3	3.3	NUM
ejde-624	217	3	.	.	PUNCT
ejde-624	218	1	let	let	VERB
ejde-624	218	2	0	0	NUM
ejde-624	218	3	<	<	X
ejde-624	218	4	α	α	X
ejde-624	218	5	<	<	X
ejde-624	218	6	1	1	NUM
ejde-624	218	7	and	and	CCONJ
ejde-624	218	8	suppose	suppose	VERB
ejde-624	218	9	that	that	SCONJ
ejde-624	218	10	u	u	PROPN
ejde-624	218	11	∈	∈	PROPN
ejde-624	218	12	l1	l1	PROPN
ejde-624	218	13	.	.	PUNCT
ejde-624	219	1	then	then	ADV
ejde-624	219	2	lim	lim	PROPN
ejde-624	219	3	t→0	t→0	PROPN
ejde-624	219	4	+	+	CCONJ
ejde-624	219	5	u(t)t1−α	u(t)t1−α	NOUN
ejde-624	219	6	=	=	SYM
ejde-624	219	7	u0	u0	PROPN
ejde-624	219	8	implies	imply	VERB
ejde-624	219	9	that	that	SCONJ
ejde-624	219	10	i1−αu(0	i1−αu(0	NOUN
ejde-624	219	11	)	)	PUNCT
ejde-624	219	12	=	=	PROPN
ejde-624	219	13	lim	lim	PROPN
ejde-624	219	14	t→0	t→0	PROPN
ejde-624	219	15	+	+	CCONJ
ejde-624	219	16	i1−αu(t	i1−αu(t	NUM
ejde-624	219	17	)	)	PUNCT
ejde-624	219	18	=	=	SYM
ejde-624	219	19	u0γ(α	u0γ(α	PROPN
ejde-624	219	20	)	)	PUNCT
ejde-624	219	21	.	.	PUNCT
ejde-624	220	1	the	the	DET
ejde-624	220	2	result	result	NOUN
ejde-624	220	3	is	be	AUX
ejde-624	220	4	proved	prove	VERB
ejde-624	220	5	for	for	ADP
ejde-624	220	6	example	example	NOUN
ejde-624	220	7	in	in	ADP
ejde-624	220	8	[	[	X
ejde-624	220	9	21	21	NUM
ejde-624	220	10	,	,	PUNCT
ejde-624	220	11	lemma	lemma	PROPN
ejde-624	220	12	6.3	6.3	NUM
ejde-624	220	13	]	]	PUNCT
ejde-624	220	14	,	,	PUNCT
ejde-624	220	15	a	a	DET
ejde-624	220	16	longer	long	ADJ
ejde-624	220	17	proof	proof	NOUN
ejde-624	220	18	is	be	AUX
ejde-624	220	19	given	give	VERB
ejde-624	220	20	in	in	ADP
ejde-624	220	21	[	[	X
ejde-624	220	22	1	1	NUM
ejde-624	220	23	,	,	PUNCT
ejde-624	220	24	theorem	theorem	VERB
ejde-624	220	25	6.1	6.1	NUM
ejde-624	220	26	]	]	PUNCT
ejde-624	220	27	,	,	PUNCT
ejde-624	220	28	also	also	ADV
ejde-624	220	29	the	the	DET
ejde-624	220	30	more	more	ADV
ejde-624	220	31	general	general	ADJ
ejde-624	220	32	case	case	NOUN
ejde-624	220	33	when	when	SCONJ
ejde-624	220	34	α	α	PROPN
ejde-624	220	35	∈	∈	PROPN
ejde-624	220	36	c	c	NOUN
ejde-624	220	37	with	with	ADP
ejde-624	220	38	0	0	NUM
ejde-624	220	39	<	<	X
ejde-624	220	40	re(α	re(α	NOUN
ejde-624	220	41	)	)	PUNCT
ejde-624	220	42	<	<	X
ejde-624	220	43	1	1	NUM
ejde-624	220	44	is	be	AUX
ejde-624	220	45	proved	prove	VERB
ejde-624	220	46	in	in	ADP
ejde-624	220	47	[	[	X
ejde-624	220	48	10	10	NUM
ejde-624	220	49	,	,	PUNCT
ejde-624	220	50	lemma	lemma	PROPN
ejde-624	220	51	3.2	3.2	NUM
ejde-624	220	52	,	,	PUNCT
ejde-624	220	53	page	page	NOUN
ejde-624	220	54	151	151	NUM
ejde-624	220	55	]	]	PUNCT
ejde-624	220	56	.	.	PUNCT
ejde-624	221	1	the	the	DET
ejde-624	221	2	converse	converse	NOUN
ejde-624	221	3	of	of	ADP
ejde-624	221	4	this	this	DET
ejde-624	221	5	result	result	NOUN
ejde-624	221	6	is	be	AUX
ejde-624	221	7	false	false	ADJ
ejde-624	221	8	.	.	PUNCT
ejde-624	221	9	example	example	NOUN
ejde-624	221	10	3.4	3.4	NUM
ejde-624	221	11	.	.	PUNCT
ejde-624	222	1	let	let	VERB
ejde-624	222	2	0	0	NUM
ejde-624	222	3	<	<	X
ejde-624	222	4	γ	γ	X
ejde-624	222	5	≤	≤	PUNCT
ejde-624	222	6	α	α	NOUN
ejde-624	222	7	<	<	X
ejde-624	222	8	1	1	NUM
ejde-624	222	9	and	and	CCONJ
ejde-624	222	10	let	let	VERB
ejde-624	222	11	q	q	PRON
ejde-624	222	12	denote	denote	VERB
ejde-624	222	13	the	the	DET
ejde-624	222	14	rational	rational	ADJ
ejde-624	222	15	numbers	number	NOUN
ejde-624	222	16	.	.	PUNCT
ejde-624	223	1	let	let	VERB
ejde-624	223	2	u(t	u(t	NOUN
ejde-624	223	3	)	)	PUNCT
ejde-624	224	1	:	:	PUNCT
ejde-624	224	2	=	=	X
ejde-624	224	3	{	{	PUNCT
ejde-624	224	4	tγ−1	tγ−1	NOUN
ejde-624	224	5	,	,	PUNCT
ejde-624	224	6	for	for	ADP
ejde-624	224	7	t	t	PROPN
ejde-624	224	8	∈	∈	PROPN
ejde-624	224	9	(	(	PUNCT
ejde-624	224	10	0	0	NUM
ejde-624	224	11	,	,	PUNCT
ejde-624	224	12	t	t	X
ejde-624	224	13	]	]	PUNCT
ejde-624	224	14	∩q	∩q	PROPN
ejde-624	224	15	,	,	PUNCT
ejde-624	224	16	0	0	NUM
ejde-624	224	17	,	,	PUNCT
ejde-624	224	18	otherwise	otherwise	ADV
ejde-624	224	19	.	.	PUNCT
ejde-624	225	1	then	then	ADV
ejde-624	225	2	u	u	PROPN
ejde-624	225	3	∈	∈	PROPN
ejde-624	225	4	l1[0	l1[0	PROPN
ejde-624	225	5	,	,	PUNCT
ejde-624	225	6	t	t	X
ejde-624	225	7	]	]	PUNCT
ejde-624	225	8	,	,	PUNCT
ejde-624	225	9	limt→0	limt→0	PROPN
ejde-624	225	10	+	+	CCONJ
ejde-624	225	11	i1−αu(t	i1−αu(t	NUM
ejde-624	225	12	)	)	PUNCT
ejde-624	225	13	=	=	SYM
ejde-624	225	14	0	0	NUM
ejde-624	226	1	but	but	CCONJ
ejde-624	226	2	limt→0	limt→0	PROPN
ejde-624	226	3	+	+	SYM
ejde-624	226	4	t1−αu(t	t1−αu(t	NOUN
ejde-624	226	5	)	)	PUNCT
ejde-624	226	6	does	do	AUX
ejde-624	226	7	not	not	PART
ejde-624	226	8	exist	exist	VERB
ejde-624	226	9	.	.	PUNCT
ejde-624	227	1	proof	proof	NOUN
ejde-624	227	2	.	.	PUNCT
ejde-624	228	1	since	since	SCONJ
ejde-624	228	2	u(t	u(t	NOUN
ejde-624	228	3	)	)	PUNCT
ejde-624	228	4	=	=	SYM
ejde-624	228	5	0	0	NUM
ejde-624	228	6	a.e	a.e	PROPN
ejde-624	228	7	.	.	PROPN
ejde-624	229	1	on	on	ADP
ejde-624	229	2	[	[	X
ejde-624	229	3	0	0	NUM
ejde-624	229	4	,	,	PUNCT
ejde-624	229	5	t	t	X
ejde-624	229	6	]	]	PUNCT
ejde-624	229	7	,	,	PUNCT
ejde-624	229	8	(	(	PUNCT
ejde-624	229	9	i1−αu)(t	i1−αu)(t	ADJ
ejde-624	229	10	)	)	PUNCT
ejde-624	229	11	=	=	SYM
ejde-624	229	12	0	0	NUM
ejde-624	229	13	for	for	ADP
ejde-624	229	14	every	every	DET
ejde-624	229	15	t	t	NOUN
ejde-624	229	16	>	>	X
ejde-624	229	17	0	0	PUNCT
ejde-624	230	1	so	so	ADV
ejde-624	230	2	limt→0	limt→0	NOUN
ejde-624	230	3	+	+	CCONJ
ejde-624	230	4	i1−αu(t	i1−αu(t	NUM
ejde-624	230	5	)	)	PUNCT
ejde-624	230	6	=	=	SYM
ejde-624	230	7	0	0	X
ejde-624	230	8	.	.	PUNCT
ejde-624	231	1	also	also	ADV
ejde-624	231	2	we	we	PRON
ejde-624	231	3	have	have	VERB
ejde-624	231	4	t1−αu(t	t1−αu(t	NOUN
ejde-624	231	5	)	)	PUNCT
ejde-624	231	6	:	:	PUNCT
ejde-624	232	1	=	=	PRON
ejde-624	232	2	{	{	PUNCT
ejde-624	232	3	tγ−α	tγ−α	NOUN
ejde-624	232	4	,	,	PUNCT
ejde-624	232	5	for	for	ADP
ejde-624	232	6	t	t	PROPN
ejde-624	232	7	∈	∈	PROPN
ejde-624	232	8	(	(	PUNCT
ejde-624	232	9	0	0	NUM
ejde-624	232	10	,	,	PUNCT
ejde-624	232	11	t	t	X
ejde-624	232	12	]	]	PUNCT
ejde-624	232	13	∩q	∩q	PROPN
ejde-624	232	14	,	,	PUNCT
ejde-624	232	15	0	0	NUM
ejde-624	232	16	,	,	PUNCT
ejde-624	232	17	otherwise	otherwise	ADV
ejde-624	232	18	,	,	PUNCT
ejde-624	232	19	and	and	CCONJ
ejde-624	232	20	for	for	ADP
ejde-624	232	21	α	α	PROPN
ejde-624	232	22	≥	≥	X
ejde-624	232	23	γ	γ	NOUN
ejde-624	232	24	,	,	PUNCT
ejde-624	232	25	limt→0	limt→0	PROPN
ejde-624	232	26	+	+	SYM
ejde-624	232	27	t1−αu(t	t1−αu(t	NOUN
ejde-624	232	28	)	)	PUNCT
ejde-624	232	29	does	do	AUX
ejde-624	232	30	not	not	PART
ejde-624	232	31	exist	exist	VERB
ejde-624	232	32	.	.	PUNCT
ejde-624	233	1	□	□	PUNCT
ejde-624	233	2	ejde-2024/40	ejde-2024/40	NUM
ejde-624	233	3	fractional	fractional	ADJ
ejde-624	233	4	differential	differential	ADJ
ejde-624	233	5	inequalities	inequality	NOUN
ejde-624	233	6	7	7	NUM
ejde-624	233	7	the	the	DET
ejde-624	233	8	following	follow	VERB
ejde-624	233	9	proposition	proposition	NOUN
ejde-624	233	10	[	[	X
ejde-624	233	11	1	1	NUM
ejde-624	233	12	,	,	PUNCT
ejde-624	233	13	theorem	theorem	VERB
ejde-624	233	14	6.2	6.2	NUM
ejde-624	233	15	]	]	PUNCT
ejde-624	233	16	proves	prove	VERB
ejde-624	233	17	the	the	DET
ejde-624	233	18	equivalence	equivalence	NOUN
ejde-624	233	19	between	between	ADP
ejde-624	233	20	the	the	DET
ejde-624	233	21	fde	fde	NOUN
ejde-624	233	22	and	and	CCONJ
ejde-624	233	23	the	the	DET
ejde-624	233	24	volterra	volterra	NOUN
ejde-624	233	25	integral	integral	ADJ
ejde-624	233	26	equation	equation	NOUN
ejde-624	233	27	in	in	ADP
ejde-624	233	28	the	the	DET
ejde-624	233	29	space	space	NOUN
ejde-624	233	30	cα−1	cα−1	PROPN
ejde-624	233	31	.	.	PUNCT
ejde-624	234	1	proposition	proposition	NOUN
ejde-624	234	2	3.5	3.5	NUM
ejde-624	234	3	.	.	PUNCT
ejde-624	235	1	let	let	VERB
ejde-624	235	2	0	0	NUM
ejde-624	235	3	<	<	X
ejde-624	235	4	α	α	X
ejde-624	235	5	<	<	X
ejde-624	235	6	1	1	NUM
ejde-624	235	7	,	,	PUNCT
ejde-624	235	8	and	and	CCONJ
ejde-624	235	9	let	let	VERB
ejde-624	235	10	f	f	PRON
ejde-624	235	11	be	be	AUX
ejde-624	235	12	continuous	continuous	ADJ
ejde-624	235	13	on	on	ADP
ejde-624	235	14	(	(	PUNCT
ejde-624	235	15	0	0	NUM
ejde-624	235	16	,	,	PUNCT
ejde-624	235	17	t	t	X
ejde-624	235	18	]	]	PUNCT
ejde-624	235	19	×	×	PROPN
ejde-624	235	20	j	j	PROPN
ejde-624	235	21	where	where	SCONJ
ejde-624	235	22	j	j	PROPN
ejde-624	235	23	⊂	⊂	PROPN
ejde-624	235	24	r	r	PROPN
ejde-624	235	25	is	be	AUX
ejde-624	235	26	an	an	DET
ejde-624	235	27	unbounded	unbounded	ADJ
ejde-624	235	28	interval	interval	NOUN
ejde-624	235	29	.	.	PUNCT
ejde-624	236	1	if	if	SCONJ
ejde-624	236	2	u	u	PRON
ejde-624	236	3	:	:	PUNCT
ejde-624	236	4	(	(	PUNCT
ejde-624	236	5	0	0	NUM
ejde-624	236	6	,	,	PUNCT
ejde-624	236	7	t	t	X
ejde-624	236	8	]	]	PUNCT
ejde-624	236	9	→	→	PUNCT
ejde-624	236	10	j	j	PROPN
ejde-624	236	11	is	be	AUX
ejde-624	236	12	continuous	continuous	ADJ
ejde-624	236	13	,	,	PUNCT
ejde-624	236	14	u	u	PROPN
ejde-624	236	15	∈	∈	PROPN
ejde-624	236	16	l1[0	l1[0	PROPN
ejde-624	236	17	,	,	PUNCT
ejde-624	236	18	t	t	X
ejde-624	236	19	]	]	PUNCT
ejde-624	236	20	and	and	CCONJ
ejde-624	236	21	t	t	PROPN
ejde-624	236	22	7→	7→	NUM
ejde-624	236	23	f(t	f(t	NOUN
ejde-624	236	24	,	,	PUNCT
ejde-624	236	25	u(t	u(t	NOUN
ejde-624	236	26	)	)	PUNCT
ejde-624	236	27	)	)	PUNCT
ejde-624	236	28	belongs	belong	VERB
ejde-624	236	29	to	to	ADP
ejde-624	236	30	l1[0	l1[0	PROPN
ejde-624	236	31	,	,	PUNCT
ejde-624	236	32	t	t	X
ejde-624	236	33	]	]	PUNCT
ejde-624	236	34	,	,	PUNCT
ejde-624	236	35	then	then	ADV
ejde-624	236	36	u	u	PRON
ejde-624	236	37	satisfies	satisfy	VERB
ejde-624	236	38	the	the	DET
ejde-624	236	39	initial	initial	ADJ
ejde-624	236	40	value	value	NOUN
ejde-624	236	41	problem	problem	NOUN
ejde-624	236	42	,	,	PUNCT
ejde-624	236	43	dαu(t	dαu(t	PROPN
ejde-624	236	44	)	)	PUNCT
ejde-624	236	45	=	=	SYM
ejde-624	236	46	f(t	f(t	NOUN
ejde-624	236	47	,	,	PUNCT
ejde-624	236	48	u(t	u(t	NOUN
ejde-624	236	49	)	)	PUNCT
ejde-624	236	50	)	)	PUNCT
ejde-624	236	51	,	,	PUNCT
ejde-624	236	52	t	t	PROPN
ejde-624	236	53	∈	∈	PROPN
ejde-624	236	54	(	(	PUNCT
ejde-624	236	55	0	0	NUM
ejde-624	236	56	,	,	PUNCT
ejde-624	236	57	t	t	X
ejde-624	236	58	]	]	PUNCT
ejde-624	236	59	,	,	PUNCT
ejde-624	236	60	lim	lim	PROPN
ejde-624	236	61	t→0	t→0	AUX
ejde-624	236	62	+	+	NUM
ejde-624	236	63	t1−αu(t	t1−αu(t	NOUN
ejde-624	236	64	)	)	PUNCT
ejde-624	236	65	=	=	SYM
ejde-624	237	1	u0	u0	ADJ
ejde-624	237	2	,	,	PUNCT
ejde-624	237	3	(	(	PUNCT
ejde-624	237	4	3.2	3.2	NUM
ejde-624	237	5	)	)	PUNCT
ejde-624	237	6	if	if	SCONJ
ejde-624	238	1	and	and	CCONJ
ejde-624	238	2	only	only	ADV
ejde-624	238	3	if	if	SCONJ
ejde-624	238	4	it	it	PRON
ejde-624	238	5	satisfies	satisfy	VERB
ejde-624	238	6	the	the	DET
ejde-624	238	7	volterra	volterra	NOUN
ejde-624	238	8	integral	integral	ADJ
ejde-624	238	9	equation	equation	NOUN
ejde-624	238	10	u(t	u(t	NOUN
ejde-624	238	11	)	)	PUNCT
ejde-624	238	12	=	=	PUNCT
ejde-624	239	1	u0tα−1	u0tα−1	NOUN
ejde-624	239	2	+	+	CCONJ
ejde-624	239	3	1	1	NUM
ejde-624	239	4	γ(α	γ(α	NOUN
ejde-624	239	5	)	)	PUNCT
ejde-624	240	1	∫	∫	PROPN
ejde-624	240	2	t	t	PROPN
ejde-624	240	3	0	0	NUM
ejde-624	240	4	(	(	PUNCT
ejde-624	240	5	t−	t−	PROPN
ejde-624	240	6	s)α−1f(s	s)α−1f(s	ADJ
ejde-624	240	7	,	,	PUNCT
ejde-624	240	8	u(s	u(s	ADJ
ejde-624	240	9	)	)	PUNCT
ejde-624	240	10	)	)	PUNCT
ejde-624	241	1	ds	ds	PROPN
ejde-624	241	2	,	,	PUNCT
ejde-624	241	3	t	t	PROPN
ejde-624	241	4	∈	∈	PROPN
ejde-624	241	5	(	(	PUNCT
ejde-624	241	6	0	0	NUM
ejde-624	241	7	,	,	PUNCT
ejde-624	241	8	t	t	X
ejde-624	241	9	]	]	PUNCT
ejde-624	241	10	.	.	PUNCT
ejde-624	242	1	(	(	PUNCT
ejde-624	242	2	3.3	3.3	NUM
ejde-624	242	3	)	)	PUNCT
ejde-624	242	4	the	the	DET
ejde-624	242	5	difference	difference	NOUN
ejde-624	242	6	between	between	ADP
ejde-624	242	7	these	these	DET
ejde-624	242	8	equivalence	equivalence	NOUN
ejde-624	242	9	results	result	NOUN
ejde-624	242	10	is	be	AUX
ejde-624	242	11	that	that	SCONJ
ejde-624	242	12	proposition	proposition	NOUN
ejde-624	242	13	3.5	3.5	NUM
ejde-624	242	14	assumes	assume	VERB
ejde-624	242	15	thatdαu(t	thatdαu(t	NOUN
ejde-624	242	16	)	)	PUNCT
ejde-624	242	17	exists	exist	VERB
ejde-624	242	18	for	for	ADP
ejde-624	242	19	every	every	DET
ejde-624	242	20	t	t	NOUN
ejde-624	242	21	∈	∈	PROPN
ejde-624	242	22	(	(	PUNCT
ejde-624	242	23	0	0	NUM
ejde-624	242	24	,	,	PUNCT
ejde-624	242	25	t	t	NOUN
ejde-624	242	26	]	]	PUNCT
ejde-624	242	27	and	and	CCONJ
ejde-624	242	28	functions	function	NOUN
ejde-624	242	29	that	that	PRON
ejde-624	242	30	are	be	AUX
ejde-624	242	31	continuous	continuous	ADJ
ejde-624	242	32	on	on	ADP
ejde-624	242	33	(	(	PUNCT
ejde-624	242	34	0	0	NUM
ejde-624	242	35	,	,	PUNCT
ejde-624	242	36	t	t	PROPN
ejde-624	242	37	]	]	PUNCT
ejde-624	242	38	are	be	AUX
ejde-624	242	39	considered	consider	VERB
ejde-624	242	40	,	,	PUNCT
ejde-624	242	41	as	as	SCONJ
ejde-624	242	42	opposed	oppose	VERB
ejde-624	242	43	to	to	ADP
ejde-624	242	44	supposing	suppose	VERB
ejde-624	242	45	that	that	SCONJ
ejde-624	242	46	functions	function	NOUN
ejde-624	242	47	are	be	AUX
ejde-624	242	48	in	in	ADP
ejde-624	242	49	l1	l1	PROPN
ejde-624	242	50	and	and	CCONJ
ejde-624	242	51	dαu(t	dαu(t	PROPN
ejde-624	242	52	)	)	PUNCT
ejde-624	242	53	exists	exist	VERB
ejde-624	242	54	only	only	ADV
ejde-624	242	55	a.e	a.e	PROPN
ejde-624	242	56	.	.	PROPN
ejde-624	243	1	in	in	ADP
ejde-624	243	2	proposition	proposition	NOUN
ejde-624	243	3	3.1	3.1	NUM
ejde-624	243	4	.	.	PUNCT
ejde-624	244	1	the	the	DET
ejde-624	244	2	conditions	condition	NOUN
ejde-624	244	3	in	in	ADP
ejde-624	244	4	proposition	proposition	NOUN
ejde-624	244	5	3.5	3.5	NUM
ejde-624	244	6	imply	imply	NOUN
ejde-624	244	7	that	that	PRON
ejde-624	244	8	i1−αu	i1−αu	VERB
ejde-624	244	9	∈	∈	PROPN
ejde-624	244	10	ac	ac	PROPN
ejde-624	244	11	as	as	SCONJ
ejde-624	244	12	shown	show	VERB
ejde-624	244	13	in	in	ADP
ejde-624	244	14	[	[	X
ejde-624	244	15	21	21	NUM
ejde-624	244	16	,	,	PUNCT
ejde-624	244	17	remark	remark	VERB
ejde-624	244	18	6.6	6.6	NUM
ejde-624	244	19	]	]	PUNCT
ejde-624	244	20	.	.	PUNCT
ejde-624	245	1	4	4	X
ejde-624	245	2	.	.	X
ejde-624	245	3	non	non	ADJ
ejde-624	246	1	-	-	NOUN
ejde-624	246	2	existence	existence	NOUN
ejde-624	246	3	for	for	ADP
ejde-624	246	4	r	r	NOUN
ejde-624	246	5	-	-	PUNCT
ejde-624	246	6	l	l	NOUN
ejde-624	246	7	inequalities	inequality	NOUN
ejde-624	246	8	henceforth	henceforth	ADV
ejde-624	246	9	for	for	ADP
ejde-624	246	10	non	non	ADJ
ejde-624	246	11	-	-	ADJ
ejde-624	246	12	existence	existence	ADJ
ejde-624	246	13	results	result	NOUN
ejde-624	246	14	we	we	PRON
ejde-624	246	15	consider	consider	VERB
ejde-624	246	16	p	p	PRON
ejde-624	246	17	>	>	X
ejde-624	246	18	1	1	NUM
ejde-624	246	19	since	since	SCONJ
ejde-624	246	20	there	there	PRON
ejde-624	246	21	are	be	VERB
ejde-624	246	22	theorems	theorem	NOUN
ejde-624	246	23	of	of	ADP
ejde-624	246	24	existence	existence	NOUN
ejde-624	246	25	on	on	ADP
ejde-624	246	26	an	an	DET
ejde-624	246	27	arbitrary	arbitrary	ADJ
ejde-624	246	28	interval	interval	NOUN
ejde-624	246	29	[	[	X
ejde-624	246	30	0	0	NUM
ejde-624	246	31	,	,	PUNCT
ejde-624	246	32	t	t	NOUN
ejde-624	246	33	]	]	PUNCT
ejde-624	246	34	when	when	SCONJ
ejde-624	246	35	p	p	PRON
ejde-624	246	36	≤	≤	NOUN
ejde-624	246	37	1	1	NUM
ejde-624	246	38	,	,	PUNCT
ejde-624	246	39	for	for	ADP
ejde-624	246	40	example	example	NOUN
ejde-624	247	1	zhu	zhu	PROPN
ejde-624	248	1	[	[	X
ejde-624	248	2	26	26	NUM
ejde-624	248	3	,	,	PUNCT
ejde-624	248	4	27	27	NUM
ejde-624	248	5	]	]	PUNCT
ejde-624	248	6	for	for	ADP
ejde-624	248	7	the	the	DET
ejde-624	248	8	r	r	NOUN
ejde-624	248	9	-	-	PUNCT
ejde-624	248	10	l	l	NOUN
ejde-624	248	11	case	case	NOUN
ejde-624	248	12	,	,	PUNCT
ejde-624	248	13	and	and	CCONJ
ejde-624	248	14	[	[	X
ejde-624	248	15	14	14	NUM
ejde-624	248	16	,	,	PUNCT
ejde-624	248	17	23	23	NUM
ejde-624	248	18	]	]	PUNCT
ejde-624	248	19	for	for	ADP
ejde-624	248	20	the	the	DET
ejde-624	248	21	caputo	caputo	PROPN
ejde-624	248	22	case	case	NOUN
ejde-624	248	23	.	.	PUNCT
ejde-624	249	1	in	in	ADP
ejde-624	249	2	the	the	DET
ejde-624	249	3	paper	paper	NOUN
ejde-624	249	4	[	[	X
ejde-624	249	5	15	15	NUM
ejde-624	249	6	]	]	X
ejde-624	249	7	laskri	laskri	NOUN
ejde-624	249	8	and	and	CCONJ
ejde-624	249	9	tatar	tatar	NOUN
ejde-624	249	10	study	study	VERB
ejde-624	249	11	the	the	DET
ejde-624	249	12	inequality	inequality	NOUN
ejde-624	249	13	dαu(t	dαu(t	PROPN
ejde-624	249	14	)	)	PUNCT
ejde-624	249	15	≥	≥	NOUN
ejde-624	249	16	tβ	tβ	PROPN
ejde-624	249	17	|u(t)|p	|u(t)|p	NOUN
ejde-624	249	18	,	,	PUNCT
ejde-624	249	19	t	t	X
ejde-624	249	20	>	>	X
ejde-624	249	21	0	0	PROPN
ejde-624	249	22	,	,	PUNCT
ejde-624	249	23	where	where	SCONJ
ejde-624	249	24	p	p	X
ejde-624	249	25	>	>	X
ejde-624	249	26	1	1	NUM
ejde-624	249	27	,	,	PUNCT
ejde-624	249	28	with	with	ADP
ejde-624	249	29	ic	ic	PROPN
ejde-624	249	30	i1−αu(0	i1−αu(0	PROPN
ejde-624	249	31	)	)	PUNCT
ejde-624	249	32	=	=	SYM
ejde-624	249	33	b.	b.	PROPN
ejde-624	249	34	(	(	PUNCT
ejde-624	249	35	4.1	4.1	NUM
ejde-624	249	36	)	)	PUNCT
ejde-624	249	37	they	they	PRON
ejde-624	249	38	consider	consider	VERB
ejde-624	249	39	u	u	PRON
ejde-624	249	40	∈	∈	NOUN
ejde-624	249	41	lα	lα	ADP
ejde-624	249	42	where	where	SCONJ
ejde-624	249	43	lα	lα	NOUN
ejde-624	249	44	:	:	PUNCT
ejde-624	249	45	=	=	SYM
ejde-624	249	46	{	{	PUNCT
ejde-624	249	47	u	u	NOUN
ejde-624	249	48	∈	∈	PROPN
ejde-624	249	49	l1	l1	PROPN
ejde-624	249	50	:	:	PUNCT
ejde-624	249	51	dαu	dαu	PROPN
ejde-624	249	52	∈	∈	PROPN
ejde-624	249	53	l1	l1	PROPN
ejde-624	249	54	}	}	PUNCT
ejde-624	249	55	.	.	PUNCT
ejde-624	250	1	their	their	PRON
ejde-624	250	2	result	result	NOUN
ejde-624	250	3	is	be	AUX
ejde-624	250	4	as	as	SCONJ
ejde-624	250	5	follows	follow	NOUN
ejde-624	250	6	.	.	PUNCT
ejde-624	251	1	theorem	theorem	VERB
ejde-624	251	2	4.1	4.1	NUM
ejde-624	251	3	(	(	PUNCT
ejde-624	251	4	[	[	X
ejde-624	251	5	15	15	NUM
ejde-624	251	6	,	,	PUNCT
ejde-624	251	7	theorem	theorem	VERB
ejde-624	251	8	1	1	NUM
ejde-624	251	9	]	]	PUNCT
ejde-624	251	10	)	)	PUNCT
ejde-624	251	11	.	.	PUNCT
ejde-624	252	1	assume	assume	VERB
ejde-624	252	2	that	that	SCONJ
ejde-624	252	3	β+α	β+α	PUNCT
ejde-624	252	4	>	>	X
ejde-624	252	5	0	0	PUNCT
ejde-624	252	6	and	and	CCONJ
ejde-624	252	7	1	1	NUM
ejde-624	252	8	<	<	X
ejde-624	252	9	p	p	X
ejde-624	252	10	<	<	X
ejde-624	252	11	1+β	1+β	PROPN
ejde-624	252	12	1−α	1−α	NUM
ejde-624	252	13	.	.	PUNCT
ejde-624	253	1	then	then	ADV
ejde-624	253	2	,	,	PUNCT
ejde-624	253	3	problem	problem	NOUN
ejde-624	253	4	(	(	PUNCT
ejde-624	253	5	4.1	4.1	NUM
ejde-624	253	6	)	)	PUNCT
ejde-624	253	7	does	do	AUX
ejde-624	253	8	not	not	PART
ejde-624	253	9	admit	admit	VERB
ejde-624	253	10	global	global	ADJ
ejde-624	253	11	nontrivial	nontrivial	ADJ
ejde-624	253	12	solutions	solution	NOUN
ejde-624	253	13	when	when	SCONJ
ejde-624	253	14	b	b	PROPN
ejde-624	253	15	≥	≥	NOUN
ejde-624	253	16	0	0	NUM
ejde-624	253	17	.	.	PUNCT
ejde-624	254	1	from	from	ADP
ejde-624	254	2	lemma	lemma	PROPN
ejde-624	254	3	3.2	3.2	NUM
ejde-624	254	4	we	we	PRON
ejde-624	254	5	see	see	VERB
ejde-624	254	6	that	that	PRON
ejde-624	254	7	for	for	ADP
ejde-624	254	8	b	b	PROPN
ejde-624	254	9	>	>	X
ejde-624	254	10	0	0	PUNCT
ejde-624	255	1	any	any	DET
ejde-624	255	2	solution	solution	NOUN
ejde-624	255	3	must	must	AUX
ejde-624	255	4	be	be	AUX
ejde-624	255	5	positive	positive	ADJ
ejde-624	255	6	a.e	a.e	PROPN
ejde-624	255	7	..	..	PROPN
ejde-624	255	8	when	when	SCONJ
ejde-624	255	9	b	b	X
ejde-624	256	1	=	=	NOUN
ejde-624	256	2	0	0	PUNCT
ejde-624	256	3	there	there	PRON
ejde-624	256	4	is	be	VERB
ejde-624	256	5	the	the	DET
ejde-624	256	6	trivial	trivial	ADJ
ejde-624	256	7	solution	solution	NOUN
ejde-624	256	8	u	u	NOUN
ejde-624	256	9	=	=	NOUN
ejde-624	256	10	0	0	NUM
ejde-624	257	1	but	but	CCONJ
ejde-624	257	2	nontrivial	nontrivial	ADJ
ejde-624	257	3	solutions	solution	NOUN
ejde-624	257	4	are	be	AUX
ejde-624	257	5	possible	possible	ADJ
ejde-624	257	6	,	,	PUNCT
ejde-624	257	7	see	see	VERB
ejde-624	257	8	remark	remark	NOUN
ejde-624	257	9	4.5	4.5	NUM
ejde-624	257	10	below	below	ADV
ejde-624	257	11	.	.	PUNCT
ejde-624	258	1	laskri	laskri	NOUN
ejde-624	258	2	and	and	CCONJ
ejde-624	258	3	tatar	tatar	NOUN
ejde-624	258	4	use	use	VERB
ejde-624	258	5	a	a	DET
ejde-624	258	6	test	test	NOUN
ejde-624	258	7	function	function	NOUN
ejde-624	258	8	method	method	NOUN
ejde-624	258	9	due	due	ADP
ejde-624	258	10	to	to	ADP
ejde-624	258	11	mitidieri	mitidieri	VERB
ejde-624	258	12	and	and	CCONJ
ejde-624	258	13	pokhozhaev	pokhozhaev	VERB
ejde-624	259	1	[	[	X
ejde-624	259	2	16	16	NUM
ejde-624	259	3	]	]	PUNCT
ejde-624	259	4	.	.	PUNCT
ejde-624	260	1	zhang	zhang	PROPN
ejde-624	260	2	,	,	PUNCT
ejde-624	260	3	liu	liu	PROPN
ejde-624	260	4	,	,	PUNCT
ejde-624	260	5	wu	wu	PROPN
ejde-624	260	6	,	,	PUNCT
ejde-624	260	7	and	and	CCONJ
ejde-624	260	8	cui	cui	VERB
ejde-624	260	9	[	[	X
ejde-624	260	10	25	25	NUM
ejde-624	260	11	]	]	PUNCT
ejde-624	260	12	claim	claim	NOUN
ejde-624	260	13	that	that	SCONJ
ejde-624	260	14	the	the	DET
ejde-624	260	15	proof	proof	NOUN
ejde-624	260	16	in	in	ADP
ejde-624	260	17	[	[	X
ejde-624	260	18	15	15	NUM
ejde-624	260	19	]	]	PUNCT
ejde-624	260	20	has	have	AUX
ejde-624	260	21	some	some	DET
ejde-624	260	22	flaws	flaw	NOUN
ejde-624	260	23	and	and	CCONJ
ejde-624	260	24	that	that	SCONJ
ejde-624	260	25	the	the	DET
ejde-624	260	26	result	result	NOUN
ejde-624	260	27	is	be	AUX
ejde-624	260	28	not	not	PART
ejde-624	260	29	correct	correct	ADJ
ejde-624	260	30	.	.	PUNCT
ejde-624	261	1	they	they	PRON
ejde-624	261	2	give	give	VERB
ejde-624	261	3	a	a	DET
ejde-624	261	4	‘	'	PUNCT
ejde-624	261	5	counter	counter	NOUN
ejde-624	261	6	-	-	NOUN
ejde-624	261	7	example	example	NOUN
ejde-624	261	8	’	'	PUNCT
ejde-624	261	9	and	and	CCONJ
ejde-624	261	10	a	a	DET
ejde-624	261	11	‘	'	PUNCT
ejde-624	261	12	correct	correct	ADJ
ejde-624	261	13	version	version	NOUN
ejde-624	261	14	’	'	PUNCT
ejde-624	261	15	of	of	ADP
ejde-624	261	16	theorem	theorem	NOUN
ejde-624	261	17	4.1	4.1	NUM
ejde-624	261	18	.	.	PUNCT
ejde-624	262	1	unfortunately	unfortunately	ADV
ejde-624	262	2	both	both	CCONJ
ejde-624	262	3	the	the	DET
ejde-624	262	4	correction	correction	NOUN
ejde-624	262	5	and	and	CCONJ
ejde-624	262	6	the	the	DET
ejde-624	262	7	counter	counter	NOUN
ejde-624	262	8	-	-	NOUN
ejde-624	262	9	example	example	NOUN
ejde-624	262	10	are	be	AUX
ejde-624	262	11	wrong	wrong	ADJ
ejde-624	262	12	.	.	PUNCT
ejde-624	263	1	we	we	PRON
ejde-624	263	2	now	now	ADV
ejde-624	263	3	explain	explain	VERB
ejde-624	263	4	these	these	DET
ejde-624	263	5	points	point	NOUN
ejde-624	263	6	.	.	PUNCT
ejde-624	264	1	laskri	laskri	NOUN
ejde-624	264	2	-	-	PUNCT
ejde-624	264	3	tatar	tatar	NOUN
ejde-624	264	4	[	[	X
ejde-624	264	5	15	15	NUM
ejde-624	264	6	]	]	PUNCT
ejde-624	264	7	consider	consider	VERB
ejde-624	264	8	a	a	DET
ejde-624	264	9	nonincreasing	nonincrease	VERB
ejde-624	264	10	c1[0,∞	c1[0,∞	PROPN
ejde-624	264	11	)	)	PUNCT
ejde-624	264	12	test	test	NOUN
ejde-624	264	13	function	function	NOUN
ejde-624	264	14	φ	φ	PROPN
ejde-624	264	15	≥	≥	PROPN
ejde-624	264	16	0	0	NUM
ejde-624	264	17	such	such	ADJ
ejde-624	264	18	that	that	SCONJ
ejde-624	264	19	,	,	PUNCT
ejde-624	264	20	for	for	ADP
ejde-624	264	21	some	some	DET
ejde-624	264	22	τ	τ	PROPN
ejde-624	264	23	>	>	X
ejde-624	264	24	0	0	PROPN
ejde-624	264	25	,	,	PUNCT
ejde-624	264	26	φ(t	φ(t	PROPN
ejde-624	264	27	)	)	PUNCT
ejde-624	264	28	=	=	PRON
ejde-624	264	29	{	{	PUNCT
ejde-624	265	1	1	1	NUM
ejde-624	265	2	,	,	PUNCT
ejde-624	265	3	if	if	SCONJ
ejde-624	265	4	t	t	PROPN
ejde-624	265	5	≤	≤	NOUN
ejde-624	265	6	τ/2	τ/2	PROPN
ejde-624	265	7	,	,	PUNCT
ejde-624	265	8	0	0	NUM
ejde-624	265	9	,	,	PUNCT
ejde-624	265	10	if	if	SCONJ
ejde-624	265	11	t	t	PROPN
ejde-624	265	12	≥	≥	PROPN
ejde-624	265	13	τ	τ	X
ejde-624	265	14	.	.	PUNCT
ejde-624	266	1	then	then	ADV
ejde-624	266	2	for	for	ADP
ejde-624	266	3	a	a	DET
ejde-624	266	4	supposed	suppose	VERB
ejde-624	266	5	positive	positive	ADJ
ejde-624	266	6	solution	solution	NOUN
ejde-624	266	7	u	u	NOUN
ejde-624	266	8	of	of	ADP
ejde-624	266	9	(	(	PUNCT
ejde-624	266	10	4.1	4.1	NUM
ejde-624	266	11	)	)	PUNCT
ejde-624	266	12	,	,	PUNCT
ejde-624	266	13	the	the	DET
ejde-624	266	14	integral	integral	ADJ
ejde-624	266	15	∫	∫	PROPN
ejde-624	266	16	τ	τ	PROPN
ejde-624	266	17	0	0	NUM
ejde-624	266	18	φ′(t)i1−αu(t	φ′(t)i1−αu(t	PROPN
ejde-624	266	19	)	)	PUNCT
ejde-624	266	20	dt	dt	PUNCT
ejde-624	266	21	is	be	AUX
ejde-624	266	22	estimated	estimate	VERB
ejde-624	266	23	from	from	ADP
ejde-624	266	24	above	above	ADV
ejde-624	266	25	.	.	PUNCT
ejde-624	267	1	the	the	DET
ejde-624	267	2	integrand	integrand	NOUN
ejde-624	267	3	in	in	ADP
ejde-624	267	4	this	this	DET
ejde-624	267	5	integral	integral	NOUN
ejde-624	267	6	is	be	AUX
ejde-624	267	7	non	non	ADJ
ejde-624	267	8	-	-	ADJ
ejde-624	267	9	positive	positive	ADJ
ejde-624	267	10	and	and	CCONJ
ejde-624	267	11	is	be	AUX
ejde-624	267	12	negative	negative	ADJ
ejde-624	267	13	on	on	ADP
ejde-624	267	14	an	an	DET
ejde-624	267	15	interval	interval	NOUN
ejde-624	267	16	.	.	PUNCT
ejde-624	268	1	the	the	DET
ejde-624	268	2	mistake	mistake	NOUN
ejde-624	268	3	in	in	ADP
ejde-624	268	4	[	[	X
ejde-624	268	5	15	15	NUM
ejde-624	268	6	]	]	PUNCT
ejde-624	268	7	is	be	AUX
ejde-624	268	8	that	that	SCONJ
ejde-624	268	9	absolute	absolute	ADJ
ejde-624	268	10	value	value	NOUN
ejde-624	268	11	signs	sign	NOUN
ejde-624	268	12	have	have	AUX
ejde-624	268	13	been	be	AUX
ejde-624	268	14	omitted	omit	VERB
ejde-624	268	15	but	but	CCONJ
ejde-624	268	16	,	,	PUNCT
ejde-624	268	17	in	in	ADP
ejde-624	268	18	fact	fact	NOUN
ejde-624	268	19	,	,	PUNCT
ejde-624	268	20	it	it	PRON
ejde-624	268	21	is	be	AUX
ejde-624	268	22	the	the	DET
ejde-624	268	23	absolute	absolute	ADJ
ejde-624	268	24	value	value	NOUN
ejde-624	268	25	of	of	ADP
ejde-624	268	26	the	the	DET
ejde-624	268	27	integral	integral	NOUN
ejde-624	268	28	that	that	PRON
ejde-624	268	29	is	be	AUX
ejde-624	268	30	estimated	estimate	VERB
ejde-624	268	31	.	.	PUNCT
ejde-624	269	1	when	when	SCONJ
ejde-624	269	2	the	the	DET
ejde-624	269	3	absolute	absolute	ADJ
ejde-624	269	4	value	value	NOUN
ejde-624	269	5	signs	sign	NOUN
ejde-624	269	6	are	be	AUX
ejde-624	269	7	added	add	VERB
ejde-624	269	8	in	in	ADP
ejde-624	269	9	[	[	X
ejde-624	269	10	15	15	NUM
ejde-624	269	11	]	]	PUNCT
ejde-624	269	12	the	the	DET
ejde-624	269	13	flaw	flaw	NOUN
ejde-624	269	14	claimed	claim	VERB
ejde-624	269	15	in	in	ADP
ejde-624	269	16	[	[	X
ejde-624	269	17	25	25	NUM
ejde-624	269	18	]	]	PUNCT
ejde-624	269	19	disappears	disappear	VERB
ejde-624	269	20	.	.	PUNCT
ejde-624	270	1	the	the	DET
ejde-624	270	2	paper	paper	NOUN
ejde-624	270	3	[	[	X
ejde-624	270	4	25	25	NUM
ejde-624	270	5	]	]	PUNCT
ejde-624	270	6	claims	claim	NOUN
ejde-624	270	7	,	,	PUNCT
ejde-624	270	8	that	that	SCONJ
ejde-624	270	9	for	for	ADP
ejde-624	270	10	1	1	NUM
ejde-624	270	11	<	<	X
ejde-624	270	12	p	p	X
ejde-624	270	13	<	<	X
ejde-624	270	14	1+β	1+β	NUM
ejde-624	270	15	1−α	1−α	NUM
ejde-624	271	1	and	and	CCONJ
ejde-624	271	2	b	b	NOUN
ejde-624	271	3	≥	≥	NOUN
ejde-624	271	4	0	0	NUM
ejde-624	271	5	,	,	PUNCT
ejde-624	271	6	based	base	VERB
ejde-624	271	7	on	on	ADP
ejde-624	271	8	their	their	PRON
ejde-624	271	9	‘	'	PUNCT
ejde-624	271	10	counterexamples	counterexample	NOUN
ejde-624	271	11	’	'	PUNCT
ejde-624	271	12	,	,	PUNCT
ejde-624	271	13	the	the	DET
ejde-624	271	14	problem	problem	NOUN
ejde-624	271	15	has	have	VERB
ejde-624	271	16	infinitely	infinitely	ADV
ejde-624	271	17	many	many	ADJ
ejde-624	271	18	global	global	ADJ
ejde-624	271	19	nontrivial	nontrivial	ADJ
ejde-624	271	20	positive	positive	ADJ
ejde-624	271	21	solutions	solution	NOUN
ejde-624	271	22	.	.	PUNCT
ejde-624	272	1	also	also	ADV
ejde-624	272	2	8	8	NUM
ejde-624	272	3	j.	j.	PROPN
ejde-624	272	4	r.	r.	PROPN
ejde-624	272	5	l.	l.	PROPN
ejde-624	272	6	webb	webb	PROPN
ejde-624	272	7	ejde-2024/40	ejde-2024/40	PROPN
ejde-624	273	1	it	it	PRON
ejde-624	273	2	is	be	AUX
ejde-624	273	3	claimed	claim	VERB
ejde-624	273	4	that	that	SCONJ
ejde-624	273	5	the	the	DET
ejde-624	273	6	problem	problem	NOUN
ejde-624	273	7	does	do	AUX
ejde-624	273	8	not	not	PART
ejde-624	273	9	have	have	VERB
ejde-624	273	10	any	any	DET
ejde-624	273	11	global	global	ADJ
ejde-624	273	12	nontrivial	nontrivial	ADJ
ejde-624	273	13	negative	negative	ADJ
ejde-624	273	14	solutions	solution	NOUN
ejde-624	273	15	.	.	PUNCT
ejde-624	274	1	the	the	DET
ejde-624	274	2	last	last	ADJ
ejde-624	274	3	point	point	NOUN
ejde-624	274	4	actually	actually	ADV
ejde-624	274	5	follows	follow	VERB
ejde-624	274	6	immediately	immediately	ADV
ejde-624	274	7	from	from	ADP
ejde-624	274	8	lemma	lemma	PROPN
ejde-624	274	9	3.2	3.2	NUM
ejde-624	274	10	,	,	PUNCT
ejde-624	274	11	not	not	PART
ejde-624	274	12	from	from	ADP
ejde-624	274	13	a	a	DET
ejde-624	274	14	correction	correction	NOUN
ejde-624	274	15	of	of	ADP
ejde-624	274	16	the	the	DET
ejde-624	274	17	proof	proof	NOUN
ejde-624	274	18	of	of	ADP
ejde-624	274	19	[	[	X
ejde-624	274	20	15	15	NUM
ejde-624	274	21	]	]	PUNCT
ejde-624	274	22	.	.	PUNCT
ejde-624	275	1	the	the	DET
ejde-624	275	2	‘	'	PUNCT
ejde-624	275	3	counter	counter	NOUN
ejde-624	275	4	-	-	NOUN
ejde-624	275	5	example	example	NOUN
ejde-624	275	6	’	'	PUNCT
ejde-624	275	7	claimed	claim	VERB
ejde-624	275	8	in	in	ADP
ejde-624	275	9	[	[	X
ejde-624	275	10	25	25	NUM
ejde-624	275	11	]	]	PUNCT
ejde-624	275	12	is	be	AUX
ejde-624	275	13	for	for	ADP
ejde-624	275	14	the	the	DET
ejde-624	275	15	case	case	NOUN
ejde-624	275	16	α	α	X
ejde-624	275	17	=	=	SYM
ejde-624	275	18	1/2	1/2	NUM
ejde-624	275	19	,	,	PUNCT
ejde-624	275	20	β	β	NOUN
ejde-624	275	21	=	=	SYM
ejde-624	275	22	−1/6	−1/6	NOUN
ejde-624	275	23	,	,	PUNCT
ejde-624	275	24	p	p	NOUN
ejde-624	275	25	=	=	NOUN
ejde-624	275	26	3/2	3/2	NUM
ejde-624	275	27	.	.	PUNCT
ejde-624	276	1	it	it	PRON
ejde-624	276	2	is	be	AUX
ejde-624	276	3	stated	state	VERB
ejde-624	276	4	that	that	SCONJ
ejde-624	276	5	,	,	PUNCT
ejde-624	276	6	for	for	ADP
ejde-624	276	7	c	c	NOUN
ejde-624	276	8	=	=	SYM
ejde-624	276	9	γ(3/2	γ(3/2	PROPN
ejde-624	276	10	)	)	PUNCT
ejde-624	276	11	,	,	PUNCT
ejde-624	276	12	u1(t	u1(t	X
ejde-624	276	13	)	)	PUNCT
ejde-624	276	14	=	=	PRON
ejde-624	276	15	{	{	PUNCT
ejde-624	276	16	c2t1/2	c2t1/2	NOUN
ejde-624	276	17	,	,	PUNCT
ejde-624	276	18	if	if	SCONJ
ejde-624	276	19	t	t	NOUN
ejde-624	276	20	≤	≤	NOUN
ejde-624	276	21	1	1	NUM
ejde-624	276	22	,	,	PUNCT
ejde-624	276	23	c2t−4/5	c2t−4/5	NOUN
ejde-624	276	24	,	,	PUNCT
ejde-624	276	25	if	if	SCONJ
ejde-624	276	26	t	t	PROPN
ejde-624	276	27	>	>	X
ejde-624	276	28	1	1	NUM
ejde-624	276	29	,	,	PUNCT
ejde-624	276	30	is	be	AUX
ejde-624	276	31	a	a	DET
ejde-624	276	32	global	global	ADJ
ejde-624	276	33	solution	solution	NOUN
ejde-624	276	34	of	of	ADP
ejde-624	276	35	(	(	PUNCT
ejde-624	276	36	4.1	4.1	NUM
ejde-624	276	37	)	)	PUNCT
ejde-624	276	38	with	with	ADP
ejde-624	276	39	b	b	NOUN
ejde-624	276	40	=	=	SYM
ejde-624	276	41	0	0	PROPN
ejde-624	276	42	.	.	PUNCT
ejde-624	277	1	for	for	ADP
ejde-624	277	2	t	t	PROPN
ejde-624	277	3	≤	≤	NUM
ejde-624	277	4	1	1	NUM
ejde-624	277	5	,	,	PUNCT
ejde-624	277	6	d1/2u1(t	d1/2u1(t	NUM
ejde-624	277	7	)	)	PUNCT
ejde-624	277	8	=	=	SYM
ejde-624	277	9	d(i1/2u1)(t	d(i1/2u1)(t	X
ejde-624	277	10	)	)	PUNCT
ejde-624	277	11	=	=	SYM
ejde-624	277	12	c3	c3	PROPN
ejde-624	277	13	and	and	CCONJ
ejde-624	277	14	the	the	DET
ejde-624	277	15	inequality	inequality	NOUN
ejde-624	277	16	(	(	PUNCT
ejde-624	277	17	4.1	4.1	NUM
ejde-624	277	18	)	)	PUNCT
ejde-624	277	19	holds	hold	VERB
ejde-624	277	20	.	.	PUNCT
ejde-624	278	1	for	for	ADP
ejde-624	278	2	t	t	PROPN
ejde-624	278	3	>	>	X
ejde-624	278	4	1	1	NUM
ejde-624	278	5	the	the	DET
ejde-624	278	6	authors	author	NOUN
ejde-624	278	7	give	give	VERB
ejde-624	278	8	d1/2u1(t	d1/2u1(t	PUNCT
ejde-624	278	9	)	)	PUNCT
ejde-624	279	1	=	=	SYM
ejde-624	279	2	c3t−13/10	c3t−13/10	PROPN
ejde-624	279	3	.	.	PUNCT
ejde-624	280	1	this	this	PRON
ejde-624	280	2	is	be	AUX
ejde-624	280	3	not	not	PART
ejde-624	280	4	correct	correct	ADJ
ejde-624	280	5	.	.	PUNCT
ejde-624	281	1	it	it	PRON
ejde-624	281	2	seems	seem	VERB
ejde-624	281	3	that	that	SCONJ
ejde-624	281	4	they	they	PRON
ejde-624	281	5	apply	apply	VERB
ejde-624	281	6	a	a	DET
ejde-624	281	7	known	know	VERB
ejde-624	281	8	formula	formula	NOUN
ejde-624	281	9	,	,	PUNCT
ejde-624	281	10	but	but	CCONJ
ejde-624	281	11	that	that	DET
ejde-624	281	12	formula	formula	NOUN
ejde-624	281	13	is	be	AUX
ejde-624	281	14	for	for	ADP
ejde-624	281	15	a	a	DET
ejde-624	281	16	function	function	NOUN
ejde-624	281	17	equal	equal	ADJ
ejde-624	281	18	to	to	ADP
ejde-624	281	19	a	a	DET
ejde-624	281	20	single	single	ADJ
ejde-624	281	21	power	power	NOUN
ejde-624	281	22	of	of	ADP
ejde-624	281	23	t	t	PROPN
ejde-624	281	24	for	for	ADP
ejde-624	281	25	all	all	DET
ejde-624	281	26	t	t	PROPN
ejde-624	281	27	>	>	X
ejde-624	281	28	0	0	X
ejde-624	281	29	.	.	PUNCT
ejde-624	282	1	for	for	ADP
ejde-624	282	2	h(t	h(t	NUM
ejde-624	282	3	)	)	PUNCT
ejde-624	282	4	=	=	PUNCT
ejde-624	282	5	c2t−4/5	c2t−4/5	NOUN
ejde-624	282	6	for	for	ADP
ejde-624	282	7	t	t	PROPN
ejde-624	282	8	>	>	X
ejde-624	282	9	0	0	PROPN
ejde-624	282	10	,	,	PUNCT
ejde-624	282	11	the	the	DET
ejde-624	282	12	formula	formula	NOUN
ejde-624	282	13	gives	give	VERB
ejde-624	282	14	d1/2h(t	d1/2h(t	NOUN
ejde-624	282	15	)	)	PUNCT
ejde-624	282	16	=	=	PUNCT
ejde-624	283	1	c2c1	c2c1	NOUN
ejde-624	283	2	t	t	NOUN
ejde-624	284	1	−13/10	−13/10	VERB
ejde-624	284	2	where	where	SCONJ
ejde-624	284	3	the	the	DET
ejde-624	284	4	constant	constant	ADJ
ejde-624	284	5	c1	c1	NOUN
ejde-624	284	6	is	be	AUX
ejde-624	284	7	negative	negative	ADJ
ejde-624	284	8	,	,	PUNCT
ejde-624	284	9	so	so	SCONJ
ejde-624	284	10	it	it	PRON
ejde-624	284	11	is	be	AUX
ejde-624	284	12	not	not	PART
ejde-624	284	13	equal	equal	ADJ
ejde-624	284	14	to	to	ADP
ejde-624	284	15	c3t−13/10	c3t−13/10	NOUN
ejde-624	284	16	.	.	PUNCT
ejde-624	285	1	the	the	DET
ejde-624	285	2	correct	correct	ADJ
ejde-624	285	3	calculation	calculation	NOUN
ejde-624	285	4	of	of	ADP
ejde-624	285	5	d1/2u1(t	d1/2u1(t	PROPN
ejde-624	285	6	)	)	PUNCT
ejde-624	285	7	=	=	SYM
ejde-624	285	8	d(i1/2u1)(t	d(i1/2u1)(t	X
ejde-624	285	9	)	)	PUNCT
ejde-624	285	10	for	for	ADP
ejde-624	285	11	t	t	PROPN
ejde-624	285	12	>	>	SYM
ejde-624	285	13	1	1	NUM
ejde-624	285	14	starts	start	NOUN
ejde-624	285	15	as	as	SCONJ
ejde-624	285	16	follows	follow	VERB
ejde-624	285	17	.	.	PUNCT
ejde-624	286	1	i1/2u1(t	i1/2u1(t	PUNCT
ejde-624	286	2	)	)	PUNCT
ejde-624	287	1	=	=	SYM
ejde-624	287	2	1	1	NUM
ejde-624	287	3	γ(1/2	γ(1/2	NOUN
ejde-624	287	4	)	)	PUNCT
ejde-624	287	5	∫	∫	PROPN
ejde-624	287	6	t	t	PROPN
ejde-624	287	7	0	0	NUM
ejde-624	288	1	(	(	PUNCT
ejde-624	288	2	t−	t−	PROPN
ejde-624	288	3	s)−1/2u1(s	s)−1/2u1(s	NOUN
ejde-624	288	4	)	)	PUNCT
ejde-624	288	5	ds	ds	PROPN
ejde-624	288	6	=	=	PUNCT
ejde-624	288	7	c2	c2	PROPN
ejde-624	288	8	1	1	NUM
ejde-624	288	9	γ(1/2	γ(1/2	NOUN
ejde-624	288	10	)	)	PUNCT
ejde-624	288	11	(	(	PUNCT
ejde-624	288	12	∫	∫	PROPN
ejde-624	288	13	1	1	NUM
ejde-624	288	14	0	0	NUM
ejde-624	288	15	(	(	PUNCT
ejde-624	288	16	t−	t−	PROPN
ejde-624	288	17	s)−1/2s1/2	s)−1/2s1/2	PROPN
ejde-624	288	18	ds+	ds+	PROPN
ejde-624	288	19	∫	∫	PROPN
ejde-624	288	20	t	t	PROPN
ejde-624	288	21	1	1	NUM
ejde-624	288	22	(	(	PUNCT
ejde-624	288	23	t−	t−	PROPN
ejde-624	288	24	s)−1/2s−4/5	s)−1/2s−4/5	NOUN
ejde-624	288	25	ds	ds	ADJ
ejde-624	288	26	)	)	PUNCT
ejde-624	288	27	.	.	PUNCT
ejde-624	289	1	the	the	DET
ejde-624	289	2	first	first	ADJ
ejde-624	289	3	integral	integral	NOUN
ejde-624	289	4	is	be	AUX
ejde-624	289	5	a	a	DET
ejde-624	289	6	decreasing	decrease	VERB
ejde-624	289	7	function	function	NOUN
ejde-624	289	8	of	of	ADP
ejde-624	289	9	t	t	PROPN
ejde-624	289	10	so	so	SCONJ
ejde-624	289	11	it	it	PRON
ejde-624	289	12	contributes	contribute	VERB
ejde-624	289	13	a	a	DET
ejde-624	289	14	negative	negative	ADJ
ejde-624	289	15	amount	amount	NOUN
ejde-624	289	16	to	to	ADP
ejde-624	289	17	the	the	DET
ejde-624	289	18	fractional	fractional	ADJ
ejde-624	289	19	derivative	derivative	NOUN
ejde-624	289	20	,	,	PUNCT
ejde-624	289	21	the	the	DET
ejde-624	289	22	other	other	ADJ
ejde-624	289	23	term	term	NOUN
ejde-624	289	24	is	be	AUX
ejde-624	289	25	not	not	PART
ejde-624	289	26	known	know	VERB
ejde-624	289	27	in	in	ADP
ejde-624	289	28	terms	term	NOUN
ejde-624	289	29	of	of	ADP
ejde-624	289	30	elementary	elementary	ADJ
ejde-624	289	31	functions	function	NOUN
ejde-624	289	32	and	and	CCONJ
ejde-624	289	33	its	its	PRON
ejde-624	289	34	monotonicity	monotonicity	NOUN
ejde-624	289	35	properties	property	NOUN
ejde-624	289	36	are	be	AUX
ejde-624	289	37	not	not	PART
ejde-624	289	38	easy	easy	ADJ
ejde-624	289	39	to	to	PART
ejde-624	289	40	prove	prove	VERB
ejde-624	289	41	,	,	PUNCT
ejde-624	289	42	but	but	CCONJ
ejde-624	289	43	a	a	DET
ejde-624	289	44	maple	maple	NOUN
ejde-624	289	45	calculation	calculation	NOUN
ejde-624	289	46	and	and	CCONJ
ejde-624	289	47	graph	graph	NOUN
ejde-624	289	48	suggest	suggest	VERB
ejde-624	289	49	that	that	SCONJ
ejde-624	289	50	it	it	PRON
ejde-624	289	51	first	first	ADV
ejde-624	289	52	increases	increase	VERB
ejde-624	289	53	then	then	ADV
ejde-624	289	54	decreases	decrease	VERB
ejde-624	289	55	,	,	PUNCT
ejde-624	289	56	so	so	SCONJ
ejde-624	289	57	it	it	PRON
ejde-624	289	58	would	would	AUX
ejde-624	289	59	not	not	PART
ejde-624	289	60	be	be	AUX
ejde-624	289	61	a	a	DET
ejde-624	289	62	counter	counter	NOUN
ejde-624	289	63	-	-	NOUN
ejde-624	289	64	example	example	NOUN
ejde-624	289	65	.	.	PUNCT
ejde-624	290	1	however	however	ADV
ejde-624	290	2	,	,	PUNCT
ejde-624	290	3	the	the	DET
ejde-624	290	4	authors	author	NOUN
ejde-624	290	5	of	of	ADP
ejde-624	290	6	[	[	X
ejde-624	290	7	25	25	NUM
ejde-624	290	8	]	]	PUNCT
ejde-624	290	9	claim	claim	NOUN
ejde-624	290	10	that	that	SCONJ
ejde-624	290	11	,	,	PUNCT
ejde-624	290	12	by	by	ADP
ejde-624	290	13	“	"	PUNCT
ejde-624	290	14	the	the	DET
ejde-624	290	15	proof	proof	NOUN
ejde-624	290	16	of	of	ADP
ejde-624	290	17	the	the	DET
ejde-624	290	18	above	above	ADJ
ejde-624	290	19	example	example	NOUN
ejde-624	290	20	”	"	PUNCT
ejde-624	290	21	,	,	PUNCT
ejde-624	290	22	a	a	DET
ejde-624	290	23	more	more	ADV
ejde-624	290	24	general	general	ADJ
ejde-624	290	25	case	case	NOUN
ejde-624	290	26	also	also	ADV
ejde-624	290	27	holds	hold	VERB
ejde-624	290	28	,	,	PUNCT
ejde-624	290	29	namely	namely	ADV
ejde-624	290	30	for	for	ADP
ejde-624	290	31	u2(t	u2(t	NOUN
ejde-624	290	32	)	)	PUNCT
ejde-624	290	33	=	=	PRON
ejde-624	290	34	{	{	PUNCT
ejde-624	290	35	ktα	ktα	PROPN
ejde-624	290	36	,	,	PUNCT
ejde-624	290	37	if	if	SCONJ
ejde-624	290	38	t	t	PROPN
ejde-624	290	39	≤	≤	NOUN
ejde-624	290	40	1	1	NUM
ejde-624	290	41	,	,	PUNCT
ejde-624	290	42	kt−µ−(α+β)/(p−1	kt−µ−(α+β)/(p−1	NOUN
ejde-624	290	43	)	)	PUNCT
ejde-624	290	44	,	,	PUNCT
ejde-624	290	45	if	if	SCONJ
ejde-624	290	46	t	t	PROPN
ejde-624	290	47	>	>	X
ejde-624	290	48	1	1	NUM
ejde-624	290	49	,	,	PUNCT
ejde-624	290	50	where	where	SCONJ
ejde-624	290	51	p	p	NOUN
ejde-624	290	52	>	>	X
ejde-624	290	53	1	1	NUM
ejde-624	290	54	and	and	CCONJ
ejde-624	290	55	k	k	PROPN
ejde-624	290	56	=	=	PROPN
ejde-624	290	57	γ(α)1/(p−1	γ(α)1/(p−1	PROPN
ejde-624	290	58	)	)	PUNCT
ejde-624	290	59	,	,	PUNCT
ejde-624	290	60	µ	µ	X
ejde-624	290	61	≥	≥	NOUN
ejde-624	290	62	1	1	NUM
ejde-624	290	63	,	,	PUNCT
ejde-624	290	64	α+β	α+β	PROPN
ejde-624	290	65	>	>	SYM
ejde-624	290	66	0	0	NUM
ejde-624	290	67	,	,	PUNCT
ejde-624	290	68	it	it	PRON
ejde-624	290	69	is	be	AUX
ejde-624	290	70	claimed	claim	VERB
ejde-624	290	71	that	that	SCONJ
ejde-624	290	72	u2	u2	PROPN
ejde-624	290	73	is	be	AUX
ejde-624	290	74	a	a	DET
ejde-624	290	75	global	global	ADJ
ejde-624	290	76	solution	solution	NOUN
ejde-624	290	77	of	of	ADP
ejde-624	290	78	the	the	DET
ejde-624	290	79	inequality	inequality	NOUN
ejde-624	290	80	(	(	PUNCT
ejde-624	290	81	4.1	4.1	NUM
ejde-624	290	82	)	)	PUNCT
ejde-624	290	83	.	.	PUNCT
ejde-624	291	1	for	for	ADP
ejde-624	291	2	this	this	DET
ejde-624	291	3	case	case	NOUN
ejde-624	291	4	we	we	PRON
ejde-624	291	5	can	can	AUX
ejde-624	291	6	prove	prove	VERB
ejde-624	291	7	this	this	PRON
ejde-624	291	8	is	be	AUX
ejde-624	291	9	false	false	ADJ
ejde-624	291	10	by	by	ADP
ejde-624	291	11	giving	give	VERB
ejde-624	291	12	a	a	DET
ejde-624	291	13	counter	counter	NOUN
ejde-624	291	14	-	-	NOUN
ejde-624	291	15	example	example	NOUN
ejde-624	291	16	that	that	PRON
ejde-624	291	17	can	can	AUX
ejde-624	291	18	be	be	AUX
ejde-624	291	19	readily	readily	ADV
ejde-624	291	20	checked	check	VERB
ejde-624	291	21	.	.	PUNCT
ejde-624	292	1	let	let	VERB
ejde-624	292	2	α	α	NOUN
ejde-624	292	3	=	=	SYM
ejde-624	292	4	1/2	1/2	NUM
ejde-624	292	5	,	,	PUNCT
ejde-624	292	6	β	β	X
ejde-624	292	7	=	=	SYM
ejde-624	292	8	1	1	NUM
ejde-624	292	9	,	,	PUNCT
ejde-624	292	10	µ	µ	NOUN
ejde-624	292	11	=	=	SYM
ejde-624	292	12	1	1	NUM
ejde-624	292	13	and	and	CCONJ
ejde-624	292	14	p	p	X
ejde-624	292	15	=	=	ADJ
ejde-624	292	16	2	2	X
ejde-624	292	17	.	.	PUNCT
ejde-624	293	1	the	the	DET
ejde-624	293	2	corresponding	corresponding	ADJ
ejde-624	293	3	fractional	fractional	ADJ
ejde-624	293	4	integral	integral	ADJ
ejde-624	293	5	is	be	AUX
ejde-624	293	6	as	as	SCONJ
ejde-624	293	7	follows	follow	VERB
ejde-624	293	8	.	.	PUNCT
ejde-624	294	1	for	for	ADP
ejde-624	294	2	t	t	PROPN
ejde-624	294	3	≤	≤	NUM
ejde-624	294	4	1	1	NUM
ejde-624	294	5	,	,	PUNCT
ejde-624	294	6	i1/2u2(t	i1/2u2(t	NOUN
ejde-624	294	7	)	)	PUNCT
ejde-624	294	8	=	=	SYM
ejde-624	294	9	1	1	NUM
ejde-624	294	10	γ(1/2	γ(1/2	NOUN
ejde-624	294	11	)	)	PUNCT
ejde-624	294	12	∫	∫	PROPN
ejde-624	294	13	t	t	PROPN
ejde-624	294	14	0	0	NUM
ejde-624	294	15	(	(	PUNCT
ejde-624	294	16	t−	t−	PRON
ejde-624	294	17	s)−1/2ks1/2	s)−1/2ks1/2	ADJ
ejde-624	294	18	ds	ds	X
ejde-624	294	19	=	=	SYM
ejde-624	294	20	ktγ(3/2	ktγ(3/2	PROPN
ejde-624	294	21	)	)	PUNCT
ejde-624	294	22	.	.	PUNCT
ejde-624	295	1	the	the	DET
ejde-624	295	2	fractional	fractional	ADJ
ejde-624	295	3	derivative	derivative	ADJ
ejde-624	295	4	d1/2u2(t	d1/2u2(t	NOUN
ejde-624	295	5	)	)	PUNCT
ejde-624	295	6	=	=	SYM
ejde-624	295	7	d(i1/2u2)(t	d(i1/2u2)(t	X
ejde-624	295	8	)	)	PUNCT
ejde-624	295	9	=	=	SYM
ejde-624	295	10	kγ(3/2	kγ(3/2	PROPN
ejde-624	295	11	)	)	PUNCT
ejde-624	295	12	is	be	AUX
ejde-624	295	13	constant	constant	ADJ
ejde-624	295	14	for	for	ADP
ejde-624	295	15	t	t	NOUN
ejde-624	295	16	≤	≤	NUM
ejde-624	295	17	1	1	NUM
ejde-624	295	18	.	.	PUNCT
ejde-624	296	1	for	for	ADP
ejde-624	296	2	t	t	PROPN
ejde-624	296	3	>	>	X
ejde-624	296	4	1	1	NUM
ejde-624	296	5	we	we	PRON
ejde-624	296	6	have	have	VERB
ejde-624	296	7	i1/2u2(t	i1/2u2(t	PRON
ejde-624	296	8	)	)	PUNCT
ejde-624	297	1	=	=	SYM
ejde-624	297	2	1	1	NUM
ejde-624	297	3	γ(1/2	γ(1/2	NOUN
ejde-624	297	4	)	)	PUNCT
ejde-624	297	5	∫	∫	PROPN
ejde-624	297	6	t	t	PROPN
ejde-624	297	7	0	0	NUM
ejde-624	298	1	(	(	PUNCT
ejde-624	298	2	t−	t−	PROPN
ejde-624	298	3	s)−1/2u2(s	s)−1/2u2(s	NOUN
ejde-624	298	4	)	)	PUNCT
ejde-624	298	5	ds	ds	PROPN
ejde-624	298	6	=	=	SYM
ejde-624	298	7	1	1	NUM
ejde-624	298	8	γ(1/2	γ(1/2	NOUN
ejde-624	298	9	)	)	PUNCT
ejde-624	298	10	(	(	PUNCT
ejde-624	298	11	∫	∫	PROPN
ejde-624	298	12	1	1	NUM
ejde-624	298	13	0	0	NUM
ejde-624	298	14	(	(	PUNCT
ejde-624	298	15	t−	t−	PROPN
ejde-624	298	16	s)−1/2ks1/2	s)−1/2ks1/2	PROPN
ejde-624	298	17	ds+	ds+	PROPN
ejde-624	298	18	∫	∫	PROPN
ejde-624	298	19	t	t	PROPN
ejde-624	298	20	1	1	NUM
ejde-624	298	21	(	(	PUNCT
ejde-624	298	22	t−	t−	PROPN
ejde-624	298	23	s)−1/2ks−5/2	s)−1/2ks−5/2	ADJ
ejde-624	298	24	ds	ds	NOUN
ejde-624	298	25	)	)	PUNCT
ejde-624	298	26	.	.	PUNCT
ejde-624	299	1	the	the	DET
ejde-624	299	2	second	second	ADJ
ejde-624	299	3	integral	integral	NOUN
ejde-624	299	4	can	can	AUX
ejde-624	299	5	be	be	AUX
ejde-624	299	6	evaluated	evaluate	VERB
ejde-624	299	7	and	and	CCONJ
ejde-624	299	8	we	we	PRON
ejde-624	299	9	have	have	VERB
ejde-624	299	10	,	,	PUNCT
ejde-624	299	11	for	for	ADP
ejde-624	299	12	t	t	PROPN
ejde-624	299	13	>	>	X
ejde-624	299	14	1,∫	1,∫	NUM
ejde-624	299	15	t	t	NOUN
ejde-624	299	16	1	1	NUM
ejde-624	300	1	(	(	PUNCT
ejde-624	300	2	t−	t−	PROPN
ejde-624	300	3	s)−1/2ks−5/2	s)−1/2ks−5/2	ADJ
ejde-624	300	4	ds	ds	NOUN
ejde-624	300	5	=	=	SYM
ejde-624	300	6	k	k	PROPN
ejde-624	300	7	2	2	NUM
ejde-624	300	8	√	√	NOUN
ejde-624	300	9	t−	t−	PROPN
ejde-624	300	10	1(t−	1(t−	NUM
ejde-624	300	11	2	2	NUM
ejde-624	300	12	)	)	PUNCT
ejde-624	300	13	3t2	3t2	NUM
ejde-624	300	14	.	.	PUNCT
ejde-624	301	1	ejde-2024/40	ejde-2024/40	VERB
ejde-624	301	2	fractional	fractional	ADJ
ejde-624	301	3	differential	differential	ADJ
ejde-624	301	4	inequalities	inequality	NOUN
ejde-624	301	5	9	9	NUM
ejde-624	301	6	this	this	DET
ejde-624	301	7	function	function	NOUN
ejde-624	301	8	of	of	ADP
ejde-624	301	9	t	t	PROPN
ejde-624	301	10	first	first	ADV
ejde-624	301	11	decreases	decrease	VERB
ejde-624	301	12	for	for	ADP
ejde-624	301	13	t	t	PROPN
ejde-624	301	14	∈	∈	PROPN
ejde-624	301	15	(	(	PUNCT
ejde-624	301	16	1	1	NUM
ejde-624	301	17	,	,	PUNCT
ejde-624	301	18	t1	t1	NOUN
ejde-624	301	19	]	]	PUNCT
ejde-624	302	1	where	where	SCONJ
ejde-624	302	2	t1	t1	NOUN
ejde-624	302	3	=	=	PUNCT
ejde-624	302	4	4−	4−	NUM
ejde-624	302	5	2	2	NUM
ejde-624	302	6	√	√	NOUN
ejde-624	302	7	2	2	NUM
ejde-624	302	8	≈	≈	PROPN
ejde-624	302	9	1.17157	1.17157	NUM
ejde-624	302	10	,	,	PUNCT
ejde-624	302	11	then	then	ADV
ejde-624	302	12	increases	increase	VERB
ejde-624	302	13	to	to	ADP
ejde-624	302	14	a	a	DET
ejde-624	302	15	maximum	maximum	NOUN
ejde-624	302	16	at	at	ADP
ejde-624	302	17	t2	t2	NOUN
ejde-624	302	18	=	=	PUNCT
ejde-624	302	19	4	4	NUM
ejde-624	302	20	+	+	NOUN
ejde-624	302	21	2	2	NUM
ejde-624	302	22	√	√	ADP
ejde-624	302	23	2	2	NUM
ejde-624	302	24	≈	≈	PROPN
ejde-624	302	25	6.82843	6.82843	NUM
ejde-624	302	26	and	and	CCONJ
ejde-624	302	27	then	then	ADV
ejde-624	302	28	decreases	decrease	VERB
ejde-624	302	29	again	again	ADV
ejde-624	302	30	.	.	PUNCT
ejde-624	303	1	since	since	SCONJ
ejde-624	303	2	the	the	DET
ejde-624	303	3	first	first	ADJ
ejde-624	303	4	part	part	NOUN
ejde-624	303	5	of	of	ADP
ejde-624	303	6	i1/2u2(t	i1/2u2(t	NOUN
ejde-624	303	7	)	)	PUNCT
ejde-624	303	8	is	be	AUX
ejde-624	303	9	a	a	DET
ejde-624	303	10	decreasing	decrease	VERB
ejde-624	303	11	function	function	NOUN
ejde-624	303	12	of	of	ADP
ejde-624	303	13	t	t	NOUN
ejde-624	303	14	we	we	PRON
ejde-624	303	15	see	see	VERB
ejde-624	303	16	that	that	SCONJ
ejde-624	303	17	the	the	DET
ejde-624	303	18	fractional	fractional	ADJ
ejde-624	303	19	derivative	derivative	NOUN
ejde-624	303	20	is	be	AUX
ejde-624	303	21	certainly	certainly	ADV
ejde-624	303	22	negative	negative	ADJ
ejde-624	303	23	except	except	SCONJ
ejde-624	303	24	possibly	possibly	ADV
ejde-624	303	25	for	for	ADP
ejde-624	303	26	t	t	PROPN
ejde-624	303	27	in	in	ADP
ejde-624	303	28	part	part	NOUN
ejde-624	303	29	of	of	ADP
ejde-624	303	30	the	the	DET
ejde-624	303	31	interval	interval	NOUN
ejde-624	303	32	(	(	PUNCT
ejde-624	303	33	t1	t1	NOUN
ejde-624	303	34	,	,	PUNCT
ejde-624	303	35	t2	t2	NOUN
ejde-624	303	36	)	)	PUNCT
ejde-624	303	37	,	,	PUNCT
ejde-624	303	38	so	so	ADV
ejde-624	303	39	can	can	AUX
ejde-624	303	40	never	never	ADV
ejde-624	303	41	be	be	AUX
ejde-624	303	42	a	a	DET
ejde-624	303	43	global	global	ADJ
ejde-624	303	44	solution	solution	NOUN
ejde-624	303	45	of	of	ADP
ejde-624	303	46	(	(	PUNCT
ejde-624	303	47	4.1	4.1	NUM
ejde-624	303	48	)	)	PUNCT
ejde-624	303	49	since	since	SCONJ
ejde-624	303	50	the	the	DET
ejde-624	303	51	right	right	ADJ
ejde-624	303	52	side	side	NOUN
ejde-624	303	53	is	be	AUX
ejde-624	303	54	always	always	ADV
ejde-624	303	55	non	non	ADJ
ejde-624	303	56	-	-	ADJ
ejde-624	303	57	negative	negative	ADJ
ejde-624	303	58	.	.	PUNCT
ejde-624	304	1	for	for	ADP
ejde-624	304	2	the	the	DET
ejde-624	304	3	case	case	NOUN
ejde-624	304	4	b	b	NOUN
ejde-624	304	5	=	=	SYM
ejde-624	304	6	0	0	NUM
ejde-624	305	1	it	it	PRON
ejde-624	305	2	is	be	AUX
ejde-624	305	3	possible	possible	ADJ
ejde-624	305	4	to	to	PART
ejde-624	305	5	have	have	VERB
ejde-624	305	6	explicit	explicit	ADJ
ejde-624	305	7	solutions	solution	NOUN
ejde-624	305	8	of	of	ADP
ejde-624	305	9	the	the	DET
ejde-624	305	10	corresponding	correspond	VERB
ejde-624	305	11	rl	rl	PROPN
ejde-624	305	12	equation	equation	NOUN
ejde-624	305	13	dαu(t	dαu(t	PROPN
ejde-624	305	14	)	)	PUNCT
ejde-624	305	15	=	=	SYM
ejde-624	305	16	tβ	tβ	PROPN
ejde-624	305	17	|u(t)|p	|u(t)|p	NOUN
ejde-624	305	18	when	when	SCONJ
ejde-624	305	19	p	p	PROPN
ejde-624	305	20	>	>	X
ejde-624	305	21	1	1	NUM
ejde-624	305	22	for	for	ADP
ejde-624	305	23	some	some	DET
ejde-624	305	24	values	value	NOUN
ejde-624	305	25	of	of	ADP
ejde-624	305	26	α	α	PROPN
ejde-624	305	27	,	,	PUNCT
ejde-624	305	28	β	β	X
ejde-624	305	29	,	,	PUNCT
ejde-624	305	30	see	see	VERB
ejde-624	305	31	the	the	DET
ejde-624	305	32	details	detail	NOUN
ejde-624	305	33	in	in	ADP
ejde-624	305	34	example	example	NOUN
ejde-624	305	35	4.6	4.6	NUM
ejde-624	305	36	below	below	ADV
ejde-624	305	37	.	.	PUNCT
ejde-624	306	1	the	the	DET
ejde-624	306	2	values	value	NOUN
ejde-624	306	3	used	use	VERB
ejde-624	306	4	in	in	ADP
ejde-624	306	5	the	the	DET
ejde-624	306	6	above	above	ADJ
ejde-624	306	7	discussion	discussion	NOUN
ejde-624	306	8	do	do	AUX
ejde-624	306	9	not	not	PART
ejde-624	306	10	fit	fit	VERB
ejde-624	306	11	the	the	DET
ejde-624	306	12	example	example	NOUN
ejde-624	306	13	.	.	PUNCT
ejde-624	307	1	we	we	PRON
ejde-624	307	2	will	will	AUX
ejde-624	307	3	prove	prove	VERB
ejde-624	307	4	non	non	ADJ
ejde-624	307	5	-	-	ADJ
ejde-624	307	6	existence	existence	ADJ
ejde-624	307	7	results	result	NOUN
ejde-624	307	8	by	by	ADP
ejde-624	307	9	a	a	DET
ejde-624	307	10	different	different	ADJ
ejde-624	307	11	method	method	NOUN
ejde-624	307	12	to	to	ADP
ejde-624	307	13	that	that	PRON
ejde-624	307	14	used	use	VERB
ejde-624	307	15	by	by	ADP
ejde-624	307	16	laskritatar	laskritatar	NOUN
ejde-624	307	17	[	[	X
ejde-624	307	18	15	15	NUM
ejde-624	307	19	]	]	PUNCT
ejde-624	307	20	which	which	PRON
ejde-624	307	21	supports	support	VERB
ejde-624	307	22	their	their	PRON
ejde-624	307	23	conclusion	conclusion	NOUN
ejde-624	307	24	.	.	PUNCT
ejde-624	308	1	we	we	PRON
ejde-624	308	2	do	do	AUX
ejde-624	308	3	not	not	PART
ejde-624	308	4	believe	believe	VERB
ejde-624	308	5	the	the	DET
ejde-624	308	6	space	space	NOUN
ejde-624	308	7	lα	lα	NOUN
ejde-624	308	8	is	be	AUX
ejde-624	308	9	an	an	DET
ejde-624	308	10	adequate	adequate	ADJ
ejde-624	308	11	space	space	NOUN
ejde-624	308	12	to	to	PART
ejde-624	308	13	consider	consider	VERB
ejde-624	308	14	the	the	DET
ejde-624	308	15	inequality	inequality	NOUN
ejde-624	308	16	(	(	PUNCT
ejde-624	308	17	4.1	4.1	NUM
ejde-624	308	18	)	)	PUNCT
ejde-624	308	19	.	.	PUNCT
ejde-624	309	1	for	for	ADP
ejde-624	309	2	,	,	PUNCT
ejde-624	309	3	in	in	ADP
ejde-624	309	4	the	the	DET
ejde-624	309	5	proof	proof	NOUN
ejde-624	309	6	of	of	ADP
ejde-624	309	7	[	[	X
ejde-624	309	8	15	15	NUM
ejde-624	309	9	,	,	PUNCT
ejde-624	309	10	theorem	theorem	VERB
ejde-624	309	11	1	1	NUM
ejde-624	309	12	]	]	PUNCT
ejde-624	309	13	,	,	PUNCT
ejde-624	309	14	the	the	DET
ejde-624	309	15	integration	integration	NOUN
ejde-624	309	16	by	by	ADP
ejde-624	309	17	parts	part	NOUN
ejde-624	309	18	requires	require	VERB
ejde-624	309	19	i1−α	i1−α	PROPN
ejde-624	309	20	∈	∈	PROPN
ejde-624	309	21	ac	ac	NOUN
ejde-624	309	22	,	,	PUNCT
ejde-624	309	23	thus	thus	ADV
ejde-624	309	24	a	a	DET
ejde-624	309	25	better	well	ADJ
ejde-624	309	26	definition	definition	NOUN
ejde-624	309	27	of	of	ADP
ejde-624	309	28	lα	lα	NOUN
ejde-624	309	29	is	be	AUX
ejde-624	309	30	lα	lα	ADJ
ejde-624	309	31	:	:	PUNCT
ejde-624	309	32	=	=	SYM
ejde-624	309	33	{	{	PUNCT
ejde-624	309	34	u	u	NOUN
ejde-624	309	35	∈	∈	PROPN
ejde-624	309	36	l1	l1	PROPN
ejde-624	309	37	:	:	PUNCT
ejde-624	309	38	i1−αu	i1−αu	VERB
ejde-624	309	39	∈	∈	PROPN
ejde-624	309	40	ac	ac	PROPN
ejde-624	309	41	}	}	PUNCT
ejde-624	309	42	.	.	PUNCT
ejde-624	310	1	we	we	PRON
ejde-624	310	2	will	will	AUX
ejde-624	310	3	use	use	VERB
ejde-624	310	4	a	a	DET
ejde-624	310	5	smaller	small	ADJ
ejde-624	310	6	space	space	NOUN
ejde-624	310	7	and	and	CCONJ
ejde-624	310	8	have	have	VERB
ejde-624	310	9	u	u	PROPN
ejde-624	310	10	∈	∈	PROPN
ejde-624	310	11	cα−1	cα−1	PROPN
ejde-624	310	12	so	so	SCONJ
ejde-624	310	13	that	that	SCONJ
ejde-624	310	14	u	u	PROPN
ejde-624	310	15	∈	∈	PROPN
ejde-624	310	16	l1	l1	PROPN
ejde-624	310	17	and	and	CCONJ
ejde-624	310	18	is	be	AUX
ejde-624	310	19	continuous	continuous	ADJ
ejde-624	310	20	on	on	ADP
ejde-624	310	21	(	(	PUNCT
ejde-624	310	22	0	0	NUM
ejde-624	310	23	,	,	PUNCT
ejde-624	310	24	t	t	X
ejde-624	310	25	]	]	PUNCT
ejde-624	310	26	.	.	PUNCT
ejde-624	311	1	for	for	ADP
ejde-624	311	2	0	0	NUM
ejde-624	311	3	<	<	X
ejde-624	311	4	α	α	X
ejde-624	311	5	<	<	X
ejde-624	311	6	1	1	NUM
ejde-624	311	7	,	,	PUNCT
ejde-624	311	8	α+	α+	X
ejde-624	311	9	β	β	X
ejde-624	311	10	>	>	X
ejde-624	311	11	0	0	PUNCT
ejde-624	312	1	and	and	CCONJ
ejde-624	312	2	p	p	X
ejde-624	312	3	>	>	X
ejde-624	312	4	1	1	NUM
ejde-624	312	5	we	we	PRON
ejde-624	312	6	will	will	AUX
ejde-624	312	7	study	study	VERB
ejde-624	312	8	the	the	DET
ejde-624	312	9	following	follow	VERB
ejde-624	312	10	problem	problem	NOUN
ejde-624	312	11	.	.	PUNCT
ejde-624	313	1	dαu(t	dαu(t	NUM
ejde-624	313	2	)	)	PUNCT
ejde-624	313	3	≥	≥	NOUN
ejde-624	313	4	tβ	tβ	PROPN
ejde-624	313	5	|u(t)|p	|u(t)|p	NOUN
ejde-624	313	6	,	,	PUNCT
ejde-624	313	7	t	t	X
ejde-624	313	8	>	>	X
ejde-624	313	9	0	0	NUM
ejde-624	313	10	,	,	PUNCT
ejde-624	313	11	with	with	ADP
ejde-624	313	12	ic	ic	PROPN
ejde-624	313	13	lim	lim	PROPN
ejde-624	313	14	t→0	t→0	PROPN
ejde-624	313	15	+	+	NUM
ejde-624	313	16	t1−αu(t	t1−αu(t	NOUN
ejde-624	313	17	)	)	PUNCT
ejde-624	313	18	=	=	SYM
ejde-624	314	1	u0	u0	ADJ
ejde-624	314	2	.	.	PUNCT
ejde-624	315	1	(	(	PUNCT
ejde-624	315	2	4.2	4.2	NUM
ejde-624	315	3	)	)	PUNCT
ejde-624	315	4	by	by	ADP
ejde-624	315	5	a	a	DET
ejde-624	315	6	solution	solution	NOUN
ejde-624	315	7	u	u	NOUN
ejde-624	315	8	of	of	ADP
ejde-624	315	9	(	(	PUNCT
ejde-624	315	10	4.2	4.2	NUM
ejde-624	315	11	)	)	PUNCT
ejde-624	315	12	on	on	ADP
ejde-624	315	13	an	an	DET
ejde-624	315	14	interval	interval	NOUN
ejde-624	315	15	[	[	X
ejde-624	315	16	0	0	NUM
ejde-624	315	17	,	,	PUNCT
ejde-624	315	18	t	t	X
ejde-624	315	19	]	]	PUNCT
ejde-624	315	20	we	we	PRON
ejde-624	315	21	will	will	AUX
ejde-624	315	22	mean	mean	VERB
ejde-624	315	23	that	that	SCONJ
ejde-624	315	24	u	u	PROPN
ejde-624	315	25	∈	∈	PROPN
ejde-624	315	26	cα−1[0	cα−1[0	PROPN
ejde-624	315	27	,	,	PUNCT
ejde-624	315	28	t	t	X
ejde-624	315	29	]	]	PUNCT
ejde-624	315	30	,	,	PUNCT
ejde-624	315	31	i1−αu	i1−αu	NOUN
ejde-624	315	32	∈	∈	PROPN
ejde-624	315	33	ac[0	ac[0	NOUN
ejde-624	315	34	,	,	PUNCT
ejde-624	315	35	t	t	X
ejde-624	315	36	]	]	PUNCT
ejde-624	315	37	,	,	PUNCT
ejde-624	315	38	dαu	dαu	PROPN
ejde-624	315	39	is	be	AUX
ejde-624	315	40	continuous	continuous	ADJ
ejde-624	315	41	on	on	ADP
ejde-624	315	42	(	(	PUNCT
ejde-624	315	43	0	0	NUM
ejde-624	315	44	,	,	PUNCT
ejde-624	315	45	t	t	X
ejde-624	315	46	]	]	PUNCT
ejde-624	315	47	,	,	PUNCT
ejde-624	315	48	tβ	tβ	PROPN
ejde-624	315	49	|u(t)|p	|u(t)|p	NOUN
ejde-624	315	50	∈	∈	PROPN
ejde-624	316	1	l1[0	l1[0	PROPN
ejde-624	316	2	,	,	PUNCT
ejde-624	316	3	t	t	X
ejde-624	316	4	]	]	PUNCT
ejde-624	316	5	,	,	PUNCT
ejde-624	316	6	the	the	DET
ejde-624	316	7	inequality	inequality	NOUN
ejde-624	316	8	is	be	AUX
ejde-624	316	9	satisfied	satisfied	ADJ
ejde-624	316	10	for	for	ADP
ejde-624	316	11	all	all	DET
ejde-624	316	12	t	t	NOUN
ejde-624	316	13	∈	∈	PROPN
ejde-624	316	14	(	(	PUNCT
ejde-624	316	15	0	0	NUM
ejde-624	316	16	,	,	PUNCT
ejde-624	316	17	t	t	NOUN
ejde-624	316	18	]	]	PUNCT
ejde-624	316	19	and	and	CCONJ
ejde-624	316	20	the	the	DET
ejde-624	316	21	ic	ic	PROPN
ejde-624	316	22	is	be	AUX
ejde-624	316	23	satisfied	satisfied	ADJ
ejde-624	316	24	.	.	PUNCT
ejde-624	317	1	a	a	DET
ejde-624	317	2	global	global	ADJ
ejde-624	317	3	solution	solution	NOUN
ejde-624	317	4	would	would	AUX
ejde-624	317	5	be	be	AUX
ejde-624	317	6	a	a	DET
ejde-624	317	7	solution	solution	NOUN
ejde-624	317	8	u(t	u(t	NOUN
ejde-624	317	9	)	)	PUNCT
ejde-624	317	10	which	which	PRON
ejde-624	317	11	exists	exist	VERB
ejde-624	317	12	for	for	ADP
ejde-624	317	13	all	all	DET
ejde-624	317	14	finite	finite	PROPN
ejde-624	317	15	t	t	PROPN
ejde-624	317	16	>	>	X
ejde-624	317	17	0	0	X
ejde-624	317	18	.	.	PUNCT
ejde-624	318	1	we	we	PRON
ejde-624	318	2	will	will	AUX
ejde-624	318	3	prove	prove	VERB
ejde-624	318	4	that	that	SCONJ
ejde-624	318	5	no	no	DET
ejde-624	318	6	global	global	ADJ
ejde-624	318	7	solution	solution	NOUN
ejde-624	318	8	exists	exist	VERB
ejde-624	318	9	,	,	PUNCT
ejde-624	318	10	a	a	DET
ejde-624	318	11	precise	precise	ADJ
ejde-624	318	12	statement	statement	NOUN
ejde-624	318	13	is	be	AUX
ejde-624	318	14	given	give	VERB
ejde-624	318	15	in	in	ADP
ejde-624	318	16	the	the	DET
ejde-624	318	17	following	following	NOUN
ejde-624	318	18	theorem	theorem	VERB
ejde-624	318	19	.	.	PUNCT
ejde-624	319	1	firstly	firstly	ADV
ejde-624	319	2	we	we	PRON
ejde-624	319	3	prove	prove	VERB
ejde-624	319	4	a	a	DET
ejde-624	319	5	result	result	NOUN
ejde-624	319	6	for	for	ADP
ejde-624	319	7	an	an	DET
ejde-624	319	8	equation	equation	NOUN
ejde-624	319	9	.	.	PUNCT
ejde-624	320	1	theorem	theorem	VERB
ejde-624	320	2	4.2	4.2	NUM
ejde-624	320	3	.	.	PUNCT
ejde-624	321	1	for	for	ADP
ejde-624	321	2	p	p	PROPN
ejde-624	321	3	>	>	SYM
ejde-624	321	4	1	1	NUM
ejde-624	321	5	and	and	CCONJ
ejde-624	321	6	u0	u0	ADJ
ejde-624	321	7	>	>	X
ejde-624	321	8	0	0	PUNCT
ejde-624	321	9	let	let	VERB
ejde-624	321	10	f	f	PROPN
ejde-624	321	11	∈	∈	PROPN
ejde-624	321	12	l1	l1	PROPN
ejde-624	321	13	with	with	ADP
ejde-624	321	14	f	f	PROPN
ejde-624	321	15	continuous	continuous	ADJ
ejde-624	321	16	on	on	ADP
ejde-624	321	17	(	(	PUNCT
ejde-624	321	18	0	0	NUM
ejde-624	321	19	,	,	PUNCT
ejde-624	321	20	t	t	NOUN
ejde-624	321	21	]	]	PUNCT
ejde-624	321	22	and	and	CCONJ
ejde-624	321	23	f(t	f(t	NOUN
ejde-624	321	24	)	)	PUNCT
ejde-624	321	25	≥	≥	NOUN
ejde-624	321	26	0	0	NUM
ejde-624	321	27	for	for	ADP
ejde-624	321	28	t	t	PROPN
ejde-624	321	29	>	>	X
ejde-624	321	30	0	0	PROPN
ejde-624	321	31	.	.	PUNCT
ejde-624	322	1	then	then	ADV
ejde-624	322	2	for	for	ADP
ejde-624	322	3	λ	λ	PROPN
ejde-624	322	4	>	>	X
ejde-624	322	5	0	0	PUNCT
ejde-624	322	6	and	and	CCONJ
ejde-624	322	7	β	β	X
ejde-624	322	8	+	+	X
ejde-624	322	9	α	α	PROPN
ejde-624	322	10	>	>	X
ejde-624	322	11	0	0	PROPN
ejde-624	322	12	,	,	PUNCT
ejde-624	322	13	the	the	DET
ejde-624	322	14	equation	equation	NOUN
ejde-624	322	15	dαu(t	dαu(t	PROPN
ejde-624	322	16	)	)	PUNCT
ejde-624	322	17	=	=	PUNCT
ejde-624	322	18	λtβ	λtβ	NOUN
ejde-624	322	19	|u(t)|p	|u(t)|p	NOUN
ejde-624	322	20	+	+	CCONJ
ejde-624	322	21	f(t	f(t	PROPN
ejde-624	322	22	)	)	PUNCT
ejde-624	322	23	,	,	PUNCT
ejde-624	322	24	t	t	X
ejde-624	322	25	>	>	X
ejde-624	322	26	0	0	PUNCT
ejde-624	323	1	with	with	ADP
ejde-624	323	2	limt→0	limt→0	PROPN
ejde-624	323	3	t	t	NOUN
ejde-624	323	4	1−αu(t	1−αu(t	NUM
ejde-624	323	5	)	)	PUNCT
ejde-624	323	6	=	=	PRON
ejde-624	323	7	u0	u0	PROPN
ejde-624	323	8	does	do	AUX
ejde-624	323	9	not	not	PART
ejde-624	323	10	have	have	VERB
ejde-624	323	11	a	a	DET
ejde-624	323	12	global	global	ADJ
ejde-624	323	13	solution	solution	NOUN
ejde-624	323	14	u	u	PROPN
ejde-624	323	15	∈	∈	PROPN
ejde-624	323	16	cα−1	cα−1	PROPN
ejde-624	323	17	.	.	PUNCT
ejde-624	324	1	in	in	ADP
ejde-624	324	2	fact	fact	NOUN
ejde-624	324	3	,	,	PUNCT
ejde-624	324	4	for	for	ADP
ejde-624	324	5	p	p	PRON
ejde-624	324	6	≥	≥	NOUN
ejde-624	324	7	1+β	1+β	NUM
ejde-624	324	8	1−α	1−α	NUM
ejde-624	324	9	there	there	PRON
ejde-624	324	10	is	be	VERB
ejde-624	324	11	no	no	DET
ejde-624	324	12	solution	solution	NOUN
ejde-624	324	13	in	in	ADP
ejde-624	324	14	the	the	DET
ejde-624	324	15	space	space	NOUN
ejde-624	324	16	cα−1[0	cα−1[0	PROPN
ejde-624	324	17	,	,	PUNCT
ejde-624	324	18	t	t	X
ejde-624	324	19	]	]	PUNCT
ejde-624	324	20	for	for	ADP
ejde-624	324	21	any	any	DET
ejde-624	324	22	t	t	PROPN
ejde-624	324	23	>	>	X
ejde-624	324	24	0	0	NUM
ejde-624	324	25	,	,	PUNCT
ejde-624	324	26	while	while	SCONJ
ejde-624	324	27	for	for	ADP
ejde-624	324	28	1	1	NUM
ejde-624	324	29	<	<	X
ejde-624	324	30	p	p	X
ejde-624	324	31	<	<	X
ejde-624	324	32	1+β	1+β	NUM
ejde-624	324	33	1−α	1−α	NUM
ejde-624	324	34	there	there	PRON
ejde-624	324	35	exists	exist	VERB
ejde-624	324	36	t1	t1	NOUN
ejde-624	324	37	,	,	PUNCT
ejde-624	324	38	explicitly	explicitly	ADV
ejde-624	324	39	determined	determine	VERB
ejde-624	324	40	by	by	ADP
ejde-624	324	41	the	the	DET
ejde-624	324	42	given	give	VERB
ejde-624	324	43	data	datum	NOUN
ejde-624	324	44	and	and	CCONJ
ejde-624	324	45	parameters	parameter	NOUN
ejde-624	324	46	,	,	PUNCT
ejde-624	324	47	such	such	ADJ
ejde-624	324	48	that	that	SCONJ
ejde-624	324	49	a	a	DET
ejde-624	324	50	solution	solution	NOUN
ejde-624	324	51	can	can	AUX
ejde-624	324	52	exist	exist	VERB
ejde-624	324	53	on	on	ADP
ejde-624	324	54	an	an	DET
ejde-624	324	55	interval	interval	NOUN
ejde-624	324	56	[	[	X
ejde-624	324	57	0	0	NUM
ejde-624	324	58	,	,	PUNCT
ejde-624	324	59	t	t	X
ejde-624	324	60	]	]	PUNCT
ejde-624	324	61	only	only	ADV
ejde-624	324	62	if	if	SCONJ
ejde-624	324	63	t	t	PROPN
ejde-624	324	64	<	<	X
ejde-624	324	65	t1	t1	PROPN
ejde-624	324	66	.	.	PUNCT
ejde-624	325	1	note	note	VERB
ejde-624	325	2	that	that	SCONJ
ejde-624	325	3	β	β	NOUN
ejde-624	325	4	can	can	AUX
ejde-624	325	5	be	be	AUX
ejde-624	325	6	negative	negative	ADJ
ejde-624	325	7	.	.	PUNCT
ejde-624	326	1	the	the	DET
ejde-624	326	2	hypothesis	hypothesis	NOUN
ejde-624	326	3	β	β	NOUN
ejde-624	326	4	+	+	X
ejde-624	326	5	α	α	PROPN
ejde-624	326	6	>	>	X
ejde-624	326	7	0	0	NUM
ejde-624	326	8	ensures	ensure	VERB
ejde-624	326	9	that	that	SCONJ
ejde-624	326	10	1+β	1+β	NUM
ejde-624	326	11	1−α	1−α	NUM
ejde-624	326	12	>	>	X
ejde-624	326	13	1	1	X
ejde-624	326	14	.	.	PUNCT
ejde-624	327	1	proof	proof	NOUN
ejde-624	327	2	.	.	PUNCT
ejde-624	328	1	from	from	ADP
ejde-624	328	2	lemma	lemma	PROPN
ejde-624	328	3	3.2	3.2	NUM
ejde-624	328	4	we	we	PRON
ejde-624	328	5	note	note	VERB
ejde-624	328	6	that	that	SCONJ
ejde-624	328	7	,	,	PUNCT
ejde-624	328	8	if	if	SCONJ
ejde-624	328	9	a	a	DET
ejde-624	328	10	solution	solution	NOUN
ejde-624	328	11	u	u	NOUN
ejde-624	328	12	exists	exist	VERB
ejde-624	328	13	,	,	PUNCT
ejde-624	328	14	then	then	ADV
ejde-624	328	15	t1−αu(t	t1−αu(t	NUM
ejde-624	328	16	)	)	PUNCT
ejde-624	328	17	≥	≥	NOUN
ejde-624	328	18	u0	u0	VERB
ejde-624	328	19	>	>	X
ejde-624	328	20	0	0	PUNCT
ejde-624	328	21	for	for	ADP
ejde-624	328	22	all	all	DET
ejde-624	328	23	t	t	NOUN
ejde-624	328	24	in	in	ADP
ejde-624	328	25	its	its	PRON
ejde-624	328	26	interval	interval	NOUN
ejde-624	328	27	of	of	ADP
ejde-624	328	28	existence	existence	NOUN
ejde-624	328	29	,	,	PUNCT
ejde-624	328	30	so	so	ADV
ejde-624	328	31	u(t	u(t	NOUN
ejde-624	328	32	)	)	PUNCT
ejde-624	328	33	>	>	X
ejde-624	328	34	0	0	PUNCT
ejde-624	329	1	for	for	ADP
ejde-624	329	2	t	t	PROPN
ejde-624	329	3	>	>	X
ejde-624	329	4	0	0	PUNCT
ejde-624	329	5	in	in	ADP
ejde-624	329	6	its	its	PRON
ejde-624	329	7	interval	interval	NOUN
ejde-624	329	8	of	of	ADP
ejde-624	329	9	existence	existence	NOUN
ejde-624	329	10	.	.	PUNCT
ejde-624	330	1	since	since	SCONJ
ejde-624	330	2	dαu	dαu	PROPN
ejde-624	330	3	∈	∈	PROPN
ejde-624	330	4	l1	l1	PROPN
ejde-624	330	5	,	,	PUNCT
ejde-624	330	6	it	it	PRON
ejde-624	330	7	is	be	AUX
ejde-624	330	8	necessary	necessary	ADJ
ejde-624	330	9	that	that	SCONJ
ejde-624	330	10	tβup(t	tβup(t	NOUN
ejde-624	330	11	)	)	PUNCT
ejde-624	330	12	∈	∈	PROPN
ejde-624	330	13	l1	l1	PROPN
ejde-624	330	14	,	,	PUNCT
ejde-624	330	15	otherwise	otherwise	ADV
ejde-624	330	16	no	no	DET
ejde-624	330	17	solution	solution	NOUN
ejde-624	330	18	can	can	AUX
ejde-624	330	19	exist	exist	VERB
ejde-624	330	20	.	.	PUNCT
ejde-624	331	1	however	however	ADV
ejde-624	331	2	,	,	PUNCT
ejde-624	331	3	when	when	SCONJ
ejde-624	331	4	t1−αu(t	t1−αu(t	NOUN
ejde-624	331	5	)	)	PUNCT
ejde-624	331	6	=	=	SYM
ejde-624	331	7	u(t	u(t	NOUN
ejde-624	331	8	)	)	PUNCT
ejde-624	331	9	≥	≥	NOUN
ejde-624	331	10	u0	u0	VERB
ejde-624	331	11	>	>	X
ejde-624	331	12	0	0	NUM
ejde-624	331	13	,	,	PUNCT
ejde-624	331	14	with	with	ADP
ejde-624	331	15	u	u	NOUN
ejde-624	331	16	continuous	continuous	ADJ
ejde-624	331	17	on	on	ADP
ejde-624	331	18	[	[	X
ejde-624	331	19	0	0	NUM
ejde-624	331	20	,	,	PUNCT
ejde-624	331	21	t	t	X
ejde-624	331	22	]	]	PUNCT
ejde-624	331	23	,	,	PUNCT
ejde-624	331	24	tβup(t	tβup(t	NOUN
ejde-624	331	25	)	)	PUNCT
ejde-624	331	26	=	=	SYM
ejde-624	331	27	tβ−p(1−α)up(t	tβ−p(1−α)up(t	PROPN
ejde-624	331	28	)	)	PUNCT
ejde-624	331	29	is	be	AUX
ejde-624	331	30	in	in	ADP
ejde-624	331	31	l1[0	l1[0	PROPN
ejde-624	331	32	,	,	PUNCT
ejde-624	331	33	t	t	X
ejde-624	331	34	]	]	PUNCT
ejde-624	331	35	for	for	ADP
ejde-624	331	36	some	some	DET
ejde-624	331	37	t	t	NOUN
ejde-624	331	38	>	>	X
ejde-624	331	39	0	0	PUNCT
ejde-624	332	1	if	if	SCONJ
ejde-624	332	2	and	and	CCONJ
ejde-624	332	3	only	only	ADV
ejde-624	332	4	if	if	SCONJ
ejde-624	332	5	β−p(1−α	β−p(1−α	NOUN
ejde-624	332	6	)	)	PUNCT
ejde-624	332	7	>	>	X
ejde-624	333	1	−1	−1	NOUN
ejde-624	333	2	,	,	PUNCT
ejde-624	333	3	that	that	PRON
ejde-624	333	4	is	be	AUX
ejde-624	333	5	p	p	X
ejde-624	333	6	<	<	X
ejde-624	333	7	1+β	1+β	NUM
ejde-624	333	8	1−α	1−α	NUM
ejde-624	333	9	,	,	PUNCT
ejde-624	333	10	so	so	CCONJ
ejde-624	333	11	no	no	DET
ejde-624	333	12	solution	solution	NOUN
ejde-624	333	13	can	can	AUX
ejde-624	333	14	exist	exist	VERB
ejde-624	333	15	when	when	SCONJ
ejde-624	333	16	p	p	PRON
ejde-624	333	17	≥	≥	NOUN
ejde-624	333	18	1+β	1+β	NUM
ejde-624	333	19	1−α	1−α	NUM
ejde-624	333	20	.	.	PUNCT
ejde-624	334	1	we	we	PRON
ejde-624	334	2	recover	recover	VERB
ejde-624	334	3	this	this	PRON
ejde-624	334	4	again	again	ADV
ejde-624	334	5	below	below	ADV
ejde-624	334	6	but	but	CCONJ
ejde-624	334	7	for	for	ADP
ejde-624	334	8	now	now	ADV
ejde-624	334	9	we	we	PRON
ejde-624	334	10	suppose	suppose	VERB
ejde-624	334	11	a	a	DET
ejde-624	334	12	solution	solution	NOUN
ejde-624	334	13	exists	exist	VERB
ejde-624	334	14	on	on	ADP
ejde-624	334	15	an	an	DET
ejde-624	334	16	interval	interval	NOUN
ejde-624	334	17	[	[	X
ejde-624	334	18	0	0	NUM
ejde-624	334	19	,	,	PUNCT
ejde-624	334	20	t	t	X
ejde-624	334	21	]	]	PUNCT
ejde-624	334	22	with	with	ADP
ejde-624	334	23	t	t	PROPN
ejde-624	334	24	>	>	X
ejde-624	334	25	0	0	X
ejde-624	334	26	.	.	PUNCT
ejde-624	335	1	let	let	VERB
ejde-624	335	2	v	v	NOUN
ejde-624	335	3	=	=	SYM
ejde-624	335	4	cu	cu	PROPN
ejde-624	335	5	where	where	SCONJ
ejde-624	335	6	cp−1	cp−1	ADJ
ejde-624	335	7	=	=	SYM
ejde-624	335	8	λ	λ	PROPN
ejde-624	335	9	,	,	PUNCT
ejde-624	335	10	then	then	ADV
ejde-624	335	11	v	v	NOUN
ejde-624	335	12	∈	∈	PROPN
ejde-624	335	13	cα−1	cα−1	PROPN
ejde-624	335	14	satisfies	satisfie	NOUN
ejde-624	335	15	dαv(t	dαv(t	NUM
ejde-624	335	16	)	)	PUNCT
ejde-624	335	17	=	=	SYM
ejde-624	335	18	tβvp(t	tβvp(t	PROPN
ejde-624	335	19	)	)	PUNCT
ejde-624	336	1	+	+	NUM
ejde-624	336	2	cf(t	cf(t	NOUN
ejde-624	336	3	)	)	PUNCT
ejde-624	336	4	,	,	PUNCT
ejde-624	336	5	t	t	PROPN
ejde-624	336	6	∈	∈	PROPN
ejde-624	336	7	(	(	PUNCT
ejde-624	336	8	0	0	NUM
ejde-624	336	9	,	,	PUNCT
ejde-624	336	10	t	t	X
ejde-624	336	11	]	]	PUNCT
ejde-624	336	12	,	,	PUNCT
ejde-624	336	13	with	with	ADP
ejde-624	336	14	ic	ic	PROPN
ejde-624	336	15	lim	lim	PROPN
ejde-624	336	16	t→0	t→0	PUNCT
ejde-624	336	17	t1−αv(t	t1−αv(t	PROPN
ejde-624	336	18	)	)	PUNCT
ejde-624	337	1	=	=	PUNCT
ejde-624	337	2	c	c	NOUN
ejde-624	337	3	u0	u0	X
ejde-624	337	4	.	.	PUNCT
ejde-624	338	1	by	by	ADP
ejde-624	338	2	proposition	proposition	NOUN
ejde-624	338	3	3.5	3.5	NUM
ejde-624	338	4	,	,	PUNCT
ejde-624	338	5	v	v	X
ejde-624	338	6	satisfies	satisfie	NOUN
ejde-624	338	7	v(t	v(t	NUM
ejde-624	338	8	)	)	PUNCT
ejde-624	338	9	=	=	PUNCT
ejde-624	339	1	c	c	NOUN
ejde-624	339	2	u0tα−1	u0tα−1	NOUN
ejde-624	339	3	+	+	CCONJ
ejde-624	339	4	iα(sβvp(s))(t	iα(sβvp(s))(t	ADJ
ejde-624	339	5	)	)	PUNCT
ejde-624	339	6	+	+	NUM
ejde-624	339	7	ciαf(t	ciαf(t	PROPN
ejde-624	339	8	)	)	PUNCT
ejde-624	339	9	,	,	PUNCT
ejde-624	339	10	t	t	PROPN
ejde-624	339	11	∈	∈	PROPN
ejde-624	339	12	(	(	PUNCT
ejde-624	339	13	0	0	NUM
ejde-624	339	14	,	,	PUNCT
ejde-624	339	15	t	t	X
ejde-624	339	16	]	]	PUNCT
ejde-624	339	17	.	.	PUNCT
ejde-624	340	1	10	10	NUM
ejde-624	340	2	j.	j.	PROPN
ejde-624	340	3	r.	r.	PROPN
ejde-624	340	4	l.	l.	PROPN
ejde-624	340	5	webb	webb	PROPN
ejde-624	340	6	ejde-2024/40	ejde-2024/40	PROPN
ejde-624	340	7	then	then	ADV
ejde-624	340	8	w(t	w(t	PROPN
ejde-624	340	9	)	)	PUNCT
ejde-624	341	1	=	=	SYM
ejde-624	341	2	t1−αv(t	t1−αv(t	NOUN
ejde-624	341	3	)	)	PUNCT
ejde-624	341	4	is	be	AUX
ejde-624	341	5	continuous	continuous	ADJ
ejde-624	341	6	for	for	ADP
ejde-624	341	7	t	t	PROPN
ejde-624	341	8	∈	∈	PROPN
ejde-624	342	1	[	[	X
ejde-624	342	2	0	0	NUM
ejde-624	342	3	,	,	PUNCT
ejde-624	342	4	t	t	X
ejde-624	342	5	]	]	PUNCT
ejde-624	342	6	.	.	PUNCT
ejde-624	343	1	discarding	discard	VERB
ejde-624	343	2	the	the	DET
ejde-624	343	3	last	last	ADJ
ejde-624	343	4	non	non	ADJ
ejde-624	343	5	-	-	ADJ
ejde-624	343	6	negative	negative	ADJ
ejde-624	343	7	term	term	NOUN
ejde-624	343	8	,	,	PUNCT
ejde-624	343	9	we	we	PRON
ejde-624	343	10	have	have	VERB
ejde-624	343	11	w	w	ADJ
ejde-624	343	12	satisfies	satisfie	NOUN
ejde-624	343	13	the	the	DET
ejde-624	343	14	inequality	inequality	NOUN
ejde-624	343	15	w(t	w(t	PROPN
ejde-624	343	16	)	)	PUNCT
ejde-624	343	17	≥	≥	NOUN
ejde-624	344	1	c	c	NOUN
ejde-624	344	2	u0	u0	NOUN
ejde-624	344	3	+	+	CCONJ
ejde-624	344	4	t1−α	t1−α	PROPN
ejde-624	344	5	1	1	NUM
ejde-624	344	6	γ(α	γ(α	NOUN
ejde-624	344	7	)	)	PUNCT
ejde-624	345	1	∫	∫	PROPN
ejde-624	345	2	t	t	PROPN
ejde-624	345	3	0	0	NUM
ejde-624	346	1	(	(	PUNCT
ejde-624	346	2	t−	t−	PROPN
ejde-624	346	3	s)α−1sβvp(s	s)α−1sβvp(s	NOUN
ejde-624	346	4	)	)	PUNCT
ejde-624	346	5	ds	ds	NOUN
ejde-624	346	6	=	=	PUNCT
ejde-624	346	7	c	c	X
ejde-624	346	8	u0	u0	ADJ
ejde-624	346	9	+	+	PROPN
ejde-624	346	10	1	1	NUM
ejde-624	346	11	γ(α	γ(α	NOUN
ejde-624	346	12	)	)	PUNCT
ejde-624	346	13	∫	∫	PROPN
ejde-624	347	1	t	t	PROPN
ejde-624	347	2	0	0	NUM
ejde-624	348	1	(	(	PUNCT
ejde-624	348	2	1−	1−	NUM
ejde-624	348	3	s	s	PROPN
ejde-624	348	4	/	/	SYM
ejde-624	348	5	t)α−1sβ−p(1−α)wp(s	t)α−1sβ−p(1−α)wp(s	NOUN
ejde-624	348	6	)	)	PUNCT
ejde-624	348	7	ds	ds	ADJ
ejde-624	348	8	≥	≥	NOUN
ejde-624	348	9	c	c	X
ejde-624	348	10	u0	u0	ADJ
ejde-624	348	11	+	+	PROPN
ejde-624	348	12	1	1	NUM
ejde-624	348	13	γ(α	γ(α	NOUN
ejde-624	348	14	)	)	PUNCT
ejde-624	349	1	∫	∫	PROPN
ejde-624	349	2	t	t	NOUN
ejde-624	349	3	0	0	NUM
ejde-624	349	4	sβ−p(1−α)wp(s	sβ−p(1−α)wp(s	ADJ
ejde-624	349	5	)	)	PUNCT
ejde-624	349	6	ds	ds	NOUN
ejde-624	349	7	,	,	PUNCT
ejde-624	349	8	where	where	SCONJ
ejde-624	349	9	we	we	PRON
ejde-624	349	10	used	use	VERB
ejde-624	349	11	(	(	PUNCT
ejde-624	349	12	1	1	NUM
ejde-624	349	13	−	−	PROPN
ejde-624	349	14	s	s	NOUN
ejde-624	349	15	/	/	SYM
ejde-624	349	16	t)α−1	t)α−1	NOUN
ejde-624	349	17	≥	≥	NOUN
ejde-624	349	18	1	1	NUM
ejde-624	349	19	.	.	PUNCT
ejde-624	349	20	note	note	VERB
ejde-624	349	21	that	that	SCONJ
ejde-624	349	22	w(t	w(t	PROPN
ejde-624	349	23	)	)	PUNCT
ejde-624	349	24	≥	≥	NOUN
ejde-624	349	25	c	c	X
ejde-624	349	26	u0	u0	PROPN
ejde-624	349	27	for	for	ADP
ejde-624	349	28	all	all	DET
ejde-624	349	29	t	t	NOUN
ejde-624	349	30	∈	∈	PROPN
ejde-624	350	1	[	[	X
ejde-624	350	2	0	0	NUM
ejde-624	350	3	,	,	PUNCT
ejde-624	350	4	t	t	X
ejde-624	350	5	]	]	PUNCT
ejde-624	350	6	.	.	PUNCT
ejde-624	351	1	if	if	SCONJ
ejde-624	351	2	β	β	X
ejde-624	351	3	−	−	PROPN
ejde-624	351	4	p(1−	p(1−	PROPN
ejde-624	351	5	α	α	NOUN
ejde-624	351	6	)	)	PUNCT
ejde-624	351	7	≤	≤	NOUN
ejde-624	351	8	−1	−1	NOUN
ejde-624	351	9	,	,	PUNCT
ejde-624	351	10	that	that	ADV
ejde-624	351	11	is	is	ADV
ejde-624	351	12	,	,	PUNCT
ejde-624	351	13	p	p	PRON
ejde-624	351	14	≥	≥	NOUN
ejde-624	351	15	1+β	1+β	NUM
ejde-624	351	16	1−α	1−α	NUM
ejde-624	351	17	,	,	PUNCT
ejde-624	352	1	then∫	then∫	NOUN
ejde-624	352	2	t	t	NOUN
ejde-624	352	3	0	0	PUNCT
ejde-624	352	4	sβ−p(1−α)wp(s	sβ−p(1−α)wp(s	ADJ
ejde-624	352	5	)	)	PUNCT
ejde-624	352	6	ds	ds	ADJ
ejde-624	352	7	≥	≥	NUM
ejde-624	352	8	∫	∫	PROPN
ejde-624	352	9	t	t	PROPN
ejde-624	352	10	0	0	NUM
ejde-624	352	11	sβ−p(1−α)(c	sβ−p(1−α)(c	PROPN
ejde-624	352	12	u0)p	u0)p	PROPN
ejde-624	352	13	ds	ds	PROPN
ejde-624	352	14	,	,	PUNCT
ejde-624	352	15	and	and	CCONJ
ejde-624	352	16	the	the	DET
ejde-624	352	17	last	last	ADJ
ejde-624	352	18	integral	integral	NOUN
ejde-624	352	19	does	do	AUX
ejde-624	352	20	not	not	PART
ejde-624	352	21	exist	exist	VERB
ejde-624	352	22	for	for	ADP
ejde-624	352	23	any	any	DET
ejde-624	352	24	t	t	NOUN
ejde-624	352	25	>	>	X
ejde-624	352	26	0	0	NUM
ejde-624	352	27	,	,	PUNCT
ejde-624	352	28	so	so	SCONJ
ejde-624	352	29	there	there	PRON
ejde-624	352	30	is	be	VERB
ejde-624	352	31	no	no	DET
ejde-624	352	32	solution	solution	NOUN
ejde-624	352	33	in	in	ADP
ejde-624	352	34	the	the	DET
ejde-624	352	35	space	space	NOUN
ejde-624	352	36	cα−1[0	cα−1[0	PROPN
ejde-624	352	37	,	,	PUNCT
ejde-624	352	38	t	t	X
ejde-624	352	39	]	]	PUNCT
ejde-624	352	40	for	for	ADP
ejde-624	352	41	any	any	DET
ejde-624	352	42	t	t	PROPN
ejde-624	352	43	>	>	X
ejde-624	352	44	0	0	X
ejde-624	352	45	.	.	PUNCT
ejde-624	353	1	this	this	PRON
ejde-624	353	2	can	can	AUX
ejde-624	353	3	be	be	AUX
ejde-624	353	4	thought	think	VERB
ejde-624	353	5	of	of	ADP
ejde-624	353	6	as	as	ADP
ejde-624	353	7	instantaneous	instantaneous	ADJ
ejde-624	353	8	blow	blow	NOUN
ejde-624	353	9	-	-	PUNCT
ejde-624	353	10	up	up	NOUN
ejde-624	353	11	,	,	PUNCT
ejde-624	353	12	or	or	CCONJ
ejde-624	353	13	blow	blow	NOUN
ejde-624	353	14	-	-	PUNCT
ejde-624	353	15	up	up	NOUN
ejde-624	353	16	at	at	ADP
ejde-624	353	17	0	0	NUM
ejde-624	353	18	.	.	PUNCT
ejde-624	354	1	for	for	ADP
ejde-624	354	2	β	β	X
ejde-624	354	3	−	−	PROPN
ejde-624	354	4	p(1	p(1	PROPN
ejde-624	354	5	−	−	PROPN
ejde-624	354	6	α	α	NOUN
ejde-624	354	7	)	)	PUNCT
ejde-624	354	8	>	>	X
ejde-624	354	9	−1	−1	NOUN
ejde-624	354	10	,	,	PUNCT
ejde-624	354	11	that	that	ADV
ejde-624	354	12	is	is	ADV
ejde-624	354	13	,	,	PUNCT
ejde-624	354	14	for	for	ADP
ejde-624	354	15	1	1	NUM
ejde-624	354	16	<	<	X
ejde-624	354	17	p	p	X
ejde-624	354	18	<	<	X
ejde-624	354	19	1+β	1+β	NUM
ejde-624	354	20	1−α	1−α	NUM
ejde-624	354	21	,	,	PUNCT
ejde-624	354	22	let	let	VERB
ejde-624	354	23	γ	γ	X
ejde-624	354	24	=	=	VERB
ejde-624	354	25	p(1	p(1	PROPN
ejde-624	354	26	−	−	NOUN
ejde-624	354	27	α	α	NOUN
ejde-624	354	28	)	)	PUNCT
ejde-624	354	29	−	−	ADP
ejde-624	354	30	β	β	NOUN
ejde-624	354	31	,	,	PUNCT
ejde-624	354	32	then	then	ADV
ejde-624	354	33	γ	γ	X
ejde-624	354	34	<	<	X
ejde-624	354	35	1	1	NUM
ejde-624	354	36	.	.	PUNCT
ejde-624	354	37	by	by	ADP
ejde-624	354	38	theorem	theorem	ADJ
ejde-624	354	39	1.2	1.2	NUM
ejde-624	354	40	,	,	PUNCT
ejde-624	354	41	local	local	ADJ
ejde-624	354	42	solutions	solution	NOUN
ejde-624	354	43	can	can	AUX
ejde-624	354	44	certainly	certainly	ADV
ejde-624	354	45	exist	exist	VERB
ejde-624	354	46	in	in	ADP
ejde-624	354	47	some	some	DET
ejde-624	354	48	cases	case	NOUN
ejde-624	354	49	.	.	PUNCT
ejde-624	355	1	we	we	PRON
ejde-624	355	2	suppose	suppose	VERB
ejde-624	355	3	that	that	SCONJ
ejde-624	355	4	u	u	PROPN
ejde-624	355	5	and	and	CCONJ
ejde-624	355	6	hence	hence	ADV
ejde-624	355	7	also	also	ADV
ejde-624	355	8	w	w	NOUN
ejde-624	355	9	exist	exist	VERB
ejde-624	355	10	on	on	ADP
ejde-624	355	11	an	an	DET
ejde-624	355	12	interval	interval	NOUN
ejde-624	355	13	(	(	PUNCT
ejde-624	355	14	0	0	NUM
ejde-624	355	15	,	,	PUNCT
ejde-624	355	16	t	t	X
ejde-624	355	17	]	]	PUNCT
ejde-624	355	18	with	with	ADP
ejde-624	355	19	t	t	PROPN
ejde-624	355	20	>	>	X
ejde-624	355	21	0	0	PROPN
ejde-624	355	22	.	.	PUNCT
ejde-624	356	1	thus	thus	ADV
ejde-624	356	2	we	we	PRON
ejde-624	356	3	have	have	AUX
ejde-624	356	4	w(t	w(t	PROPN
ejde-624	356	5	)	)	PUNCT
ejde-624	356	6	≥	≥	NOUN
ejde-624	356	7	c	c	NOUN
ejde-624	356	8	u0	u0	ADJ
ejde-624	356	9	+	+	PROPN
ejde-624	356	10	1	1	NUM
ejde-624	356	11	γ(α	γ(α	NOUN
ejde-624	356	12	)	)	PUNCT
ejde-624	357	1	∫	∫	PROPN
ejde-624	357	2	t	t	PROPN
ejde-624	357	3	0	0	NUM
ejde-624	357	4	s−γwp(s	s−γwp(s	NUM
ejde-624	357	5	)	)	PUNCT
ejde-624	357	6	ds	ds	NOUN
ejde-624	357	7	for	for	ADP
ejde-624	357	8	all	all	DET
ejde-624	357	9	t	t	NOUN
ejde-624	357	10	∈	∈	PROPN
ejde-624	358	1	[	[	X
ejde-624	358	2	0	0	NUM
ejde-624	358	3	,	,	PUNCT
ejde-624	358	4	t	t	X
ejde-624	358	5	]	]	PUNCT
ejde-624	358	6	since	since	SCONJ
ejde-624	358	7	terms	term	NOUN
ejde-624	358	8	are	be	AUX
ejde-624	358	9	continuous	continuous	ADJ
ejde-624	358	10	.	.	PUNCT
ejde-624	359	1	let	let	VERB
ejde-624	359	2	g(t	g(t	PROPN
ejde-624	359	3	)	)	PUNCT
ejde-624	360	1	:	:	PUNCT
ejde-624	360	2	=	=	PUNCT
ejde-624	360	3	c	c	PRON
ejde-624	360	4	u0	u0	ADJ
ejde-624	360	5	+	+	PROPN
ejde-624	360	6	1	1	NUM
ejde-624	360	7	γ(α	γ(α	NOUN
ejde-624	360	8	)	)	PUNCT
ejde-624	361	1	∫	∫	PROPN
ejde-624	361	2	t	t	PROPN
ejde-624	361	3	0	0	NUM
ejde-624	361	4	s−γwp(s	s−γwp(s	NUM
ejde-624	361	5	)	)	PUNCT
ejde-624	361	6	ds	ds	NOUN
ejde-624	361	7	,	,	PUNCT
ejde-624	361	8	then	then	ADV
ejde-624	361	9	g	g	PROPN
ejde-624	361	10	∈	∈	PROPN
ejde-624	361	11	ac	ac	PROPN
ejde-624	361	12	,	,	PUNCT
ejde-624	361	13	g	g	PROPN
ejde-624	361	14	is	be	AUX
ejde-624	361	15	strictly	strictly	ADV
ejde-624	361	16	increasing	increase	VERB
ejde-624	361	17	,	,	PUNCT
ejde-624	361	18	g(t	g(t	PROPN
ejde-624	361	19	)	)	PUNCT
ejde-624	361	20	≥	≥	NOUN
ejde-624	362	1	g(0	g(0	NOUN
ejde-624	362	2	)	)	PUNCT
ejde-624	362	3	=	=	PUNCT
ejde-624	363	1	c	c	X
ejde-624	363	2	u0	u0	X
ejde-624	363	3	>	>	X
ejde-624	363	4	0	0	NUM
ejde-624	363	5	,	,	PUNCT
ejde-624	363	6	and	and	CCONJ
ejde-624	363	7	g′(t	g′(t	PROPN
ejde-624	363	8	)	)	PUNCT
ejde-624	363	9	=	=	SYM
ejde-624	363	10	t−γwp(t	t−γwp(t	NUM
ejde-624	363	11	)	)	PUNCT
ejde-624	363	12	≥	≥	NOUN
ejde-624	363	13	1	1	NUM
ejde-624	363	14	γ(α	γ(α	NOUN
ejde-624	363	15	)	)	PUNCT
ejde-624	363	16	t	t	PROPN
ejde-624	363	17	−γgp(t	−γgp(t	PROPN
ejde-624	363	18	)	)	PUNCT
ejde-624	363	19	.	.	PUNCT
ejde-624	364	1	since	since	SCONJ
ejde-624	364	2	gp	gp	PROPN
ejde-624	364	3	∈	∈	PROPN
ejde-624	364	4	ac	ac	PROPN
ejde-624	364	5	is	be	AUX
ejde-624	364	6	positive	positive	ADJ
ejde-624	364	7	,	,	PUNCT
ejde-624	364	8	integrating	integrate	VERB
ejde-624	364	9	g′	g′	NOUN
ejde-624	364	10	gp	gp	NOUN
ejde-624	364	11	≥	≥	PROPN
ejde-624	364	12	1	1	NUM
ejde-624	364	13	γ(α	γ(α	NOUN
ejde-624	364	14	)	)	PUNCT
ejde-624	364	15	t	t	NOUN
ejde-624	364	16	−γ	−γ	NOUN
ejde-624	364	17	gives	give	VERB
ejde-624	364	18	g1−p(t)−	g1−p(t)−	PROPN
ejde-624	364	19	g1−p(0	g1−p(0	NOUN
ejde-624	364	20	)	)	PUNCT
ejde-624	364	21	1−	1−	NUM
ejde-624	365	1	p	p	NOUN
ejde-624	365	2	≥	≥	NUM
ejde-624	365	3	1	1	NUM
ejde-624	365	4	γ(α	γ(α	NOUN
ejde-624	365	5	)	)	PUNCT
ejde-624	366	1	t1−γ	t1−γ	PROPN
ejde-624	366	2	1−	1−	NUM
ejde-624	366	3	γ	γ	X
ejde-624	366	4	,	,	PUNCT
ejde-624	366	5	that	that	PRON
ejde-624	366	6	is	be	AUX
ejde-624	366	7	g1−p(t	g1−p(t	NOUN
ejde-624	366	8	)	)	PUNCT
ejde-624	366	9	≤	≤	NOUN
ejde-624	366	10	(	(	PUNCT
ejde-624	366	11	cu0)1−p	cu0)1−p	VERB
ejde-624	366	12	−	−	PROPN
ejde-624	366	13	(	(	PUNCT
ejde-624	366	14	p−	p−	NOUN
ejde-624	366	15	1	1	NUM
ejde-624	366	16	)	)	SYM
ejde-624	366	17	1	1	NUM
ejde-624	366	18	γ(α	γ(α	NOUN
ejde-624	366	19	)	)	PUNCT
ejde-624	367	1	t1−γ	t1−γ	PROPN
ejde-624	367	2	1−	1−	NUM
ejde-624	367	3	γ	γ	X
ejde-624	367	4	,	,	PUNCT
ejde-624	367	5	for	for	ADP
ejde-624	367	6	all	all	DET
ejde-624	367	7	t	t	NOUN
ejde-624	367	8	∈	∈	PROPN
ejde-624	368	1	[	[	X
ejde-624	368	2	0	0	NUM
ejde-624	368	3	,	,	PUNCT
ejde-624	368	4	t	t	X
ejde-624	368	5	]	]	PUNCT
ejde-624	368	6	.	.	PUNCT
ejde-624	369	1	let	let	VERB
ejde-624	369	2	t1	t1	NOUN
ejde-624	369	3	=	=	PUNCT
ejde-624	369	4	[	[	PUNCT
ejde-624	369	5	(	(	PUNCT
ejde-624	369	6	1−γ)γ(α	1−γ)γ(α	INTJ
ejde-624	369	7	)	)	PUNCT
ejde-624	369	8	(	(	PUNCT
ejde-624	369	9	p−1)(cu0)p−1	p−1)(cu0)p−1	NOUN
ejde-624	369	10	]	]	PUNCT
ejde-624	369	11	1	1	NUM
ejde-624	369	12	1−γ	1−γ	NUM
ejde-624	369	13	.	.	PUNCT
ejde-624	370	1	then	then	ADV
ejde-624	370	2	we	we	PRON
ejde-624	370	3	have	have	AUX
ejde-624	370	4	(	(	PUNCT
ejde-624	370	5	cu0)1−p	cu0)1−p	VERB
ejde-624	370	6	−	−	PROPN
ejde-624	371	1	(	(	PUNCT
ejde-624	371	2	p	p	NOUN
ejde-624	371	3	−	−	PROPN
ejde-624	371	4	1	1	NUM
ejde-624	371	5	)	)	SYM
ejde-624	371	6	1	1	NUM
ejde-624	371	7	γ(α	γ(α	NOUN
ejde-624	371	8	)	)	PUNCT
ejde-624	371	9	t	t	NOUN
ejde-624	371	10	1−γ	1−γ	NUM
ejde-624	372	1	1	1	NUM
ejde-624	372	2	1−γ	1−γ	NUM
ejde-624	372	3	≤	≤	NUM
ejde-624	372	4	0	0	NUM
ejde-624	372	5	,	,	PUNCT
ejde-624	372	6	but	but	CCONJ
ejde-624	372	7	since	since	SCONJ
ejde-624	372	8	g(t	g(t	PROPN
ejde-624	372	9	)	)	PUNCT
ejde-624	372	10	is	be	AUX
ejde-624	372	11	positive	positive	ADJ
ejde-624	372	12	on	on	ADP
ejde-624	372	13	its	its	PRON
ejde-624	372	14	interval	interval	NOUN
ejde-624	372	15	of	of	ADP
ejde-624	372	16	existence	existence	NOUN
ejde-624	372	17	,	,	PUNCT
ejde-624	372	18	g1−p(t	g1−p(t	NOUN
ejde-624	372	19	)	)	PUNCT
ejde-624	372	20	can	can	AUX
ejde-624	372	21	not	not	PART
ejde-624	372	22	exist	exist	VERB
ejde-624	372	23	at	at	ADP
ejde-624	372	24	t	t	PROPN
ejde-624	372	25	=	=	SYM
ejde-624	372	26	t1	t1	PROPN
ejde-624	372	27	.	.	PUNCT
ejde-624	373	1	thus	thus	ADV
ejde-624	373	2	w	w	NOUN
ejde-624	373	3	,	,	PUNCT
ejde-624	373	4	and	and	CCONJ
ejde-624	373	5	hence	hence	ADV
ejde-624	373	6	u	u	NOUN
ejde-624	373	7	,	,	PUNCT
ejde-624	373	8	can	can	AUX
ejde-624	373	9	exist	exist	VERB
ejde-624	373	10	on	on	ADP
ejde-624	373	11	an	an	DET
ejde-624	373	12	interval	interval	NOUN
ejde-624	373	13	[	[	X
ejde-624	373	14	0	0	NUM
ejde-624	373	15	,	,	PUNCT
ejde-624	373	16	t	t	X
ejde-624	373	17	]	]	PUNCT
ejde-624	373	18	only	only	ADV
ejde-624	373	19	for	for	ADP
ejde-624	373	20	t	t	PROPN
ejde-624	373	21	<	<	X
ejde-624	373	22	t1	t1	NOUN
ejde-624	373	23	.	.	PUNCT
ejde-624	374	1	□	□	PUNCT
ejde-624	374	2	remark	remark	NOUN
ejde-624	374	3	4.3	4.3	NUM
ejde-624	374	4	.	.	PUNCT
ejde-624	375	1	we	we	PRON
ejde-624	375	2	expect	expect	VERB
ejde-624	375	3	that	that	SCONJ
ejde-624	375	4	the	the	DET
ejde-624	375	5	solution	solution	NOUN
ejde-624	375	6	u	u	NOUN
ejde-624	375	7	exists	exist	VERB
ejde-624	375	8	on	on	ADP
ejde-624	375	9	an	an	DET
ejde-624	375	10	interval	interval	NOUN
ejde-624	375	11	(	(	PUNCT
ejde-624	375	12	0	0	NUM
ejde-624	375	13	,	,	PUNCT
ejde-624	375	14	t2	t2	NOUN
ejde-624	375	15	)	)	PUNCT
ejde-624	375	16	for	for	ADP
ejde-624	375	17	some	some	DET
ejde-624	375	18	t2	t2	PROPN
ejde-624	375	19	≤	≤	PUNCT
ejde-624	375	20	t1	t1	NOUN
ejde-624	375	21	and	and	CCONJ
ejde-624	375	22	blows	blow	VERB
ejde-624	375	23	up	up	ADP
ejde-624	375	24	at	at	ADP
ejde-624	375	25	t2	t2	NOUN
ejde-624	375	26	but	but	CCONJ
ejde-624	375	27	we	we	PRON
ejde-624	375	28	do	do	AUX
ejde-624	375	29	not	not	PART
ejde-624	375	30	know	know	VERB
ejde-624	375	31	a	a	DET
ejde-624	375	32	proof	proof	NOUN
ejde-624	375	33	of	of	ADP
ejde-624	375	34	this	this	PRON
ejde-624	375	35	.	.	PUNCT
ejde-624	376	1	if	if	SCONJ
ejde-624	376	2	by	by	ADP
ejde-624	376	3	some	some	DET
ejde-624	376	4	means	mean	NOUN
ejde-624	376	5	we	we	PRON
ejde-624	376	6	knew	know	VERB
ejde-624	376	7	that	that	DET
ejde-624	376	8	t2	t2	NOUN
ejde-624	376	9	=	=	PROPN
ejde-624	376	10	t1	t1	NOUN
ejde-624	376	11	then	then	ADV
ejde-624	376	12	we	we	PRON
ejde-624	376	13	would	would	AUX
ejde-624	376	14	have	have	VERB
ejde-624	376	15	blow	blow	NOUN
ejde-624	376	16	-	-	PUNCT
ejde-624	376	17	up	up	NOUN
ejde-624	376	18	.	.	PUNCT
ejde-624	377	1	laskri	laskri	NOUN
ejde-624	377	2	-	-	PUNCT
ejde-624	377	3	tatar	tatar	NOUN
ejde-624	377	4	[	[	X
ejde-624	377	5	15	15	NUM
ejde-624	377	6	]	]	PUNCT
ejde-624	377	7	prove	prove	VERB
ejde-624	377	8	non	non	ADJ
ejde-624	377	9	-	-	NOUN
ejde-624	377	10	existence	existence	NOUN
ejde-624	377	11	of	of	ADP
ejde-624	377	12	a	a	DET
ejde-624	377	13	global	global	ADJ
ejde-624	377	14	solution	solution	NOUN
ejde-624	377	15	but	but	CCONJ
ejde-624	377	16	do	do	AUX
ejde-624	377	17	not	not	PART
ejde-624	377	18	have	have	VERB
ejde-624	377	19	any	any	DET
ejde-624	377	20	estimate	estimate	NOUN
ejde-624	377	21	of	of	ADP
ejde-624	377	22	the	the	DET
ejde-624	377	23	interval	interval	NOUN
ejde-624	377	24	of	of	ADP
ejde-624	377	25	existence	existence	NOUN
ejde-624	377	26	(	(	PUNCT
ejde-624	377	27	0	0	NUM
ejde-624	377	28	,	,	PUNCT
ejde-624	377	29	t2	t2	NOUN
ejde-624	377	30	)	)	PUNCT
ejde-624	377	31	.	.	PUNCT
ejde-624	378	1	theorem	theorem	VERB
ejde-624	378	2	4.4	4.4	NUM
ejde-624	378	3	.	.	PUNCT
ejde-624	379	1	let	let	VERB
ejde-624	379	2	p	p	PRON
ejde-624	379	3	>	>	X
ejde-624	379	4	1	1	NUM
ejde-624	379	5	and	and	CCONJ
ejde-624	379	6	λ	λ	X
ejde-624	379	7	>	>	X
ejde-624	379	8	0	0	NUM
ejde-624	379	9	.	.	PUNCT
ejde-624	380	1	the	the	DET
ejde-624	380	2	fractional	fractional	ADJ
ejde-624	380	3	inequality	inequality	NOUN
ejde-624	380	4	dαu(t	dαu(t	PROPN
ejde-624	380	5	)	)	PUNCT
ejde-624	380	6	≥	≥	NOUN
ejde-624	380	7	λtβup(t	λtβup(t	NOUN
ejde-624	380	8	)	)	PUNCT
ejde-624	380	9	for	for	ADP
ejde-624	380	10	t	t	PROPN
ejde-624	380	11	>	>	X
ejde-624	380	12	0	0	PROPN
ejde-624	380	13	,	,	PUNCT
ejde-624	380	14	together	together	ADV
ejde-624	380	15	with	with	ADP
ejde-624	380	16	limt→0	limt→0	PROPN
ejde-624	380	17	t	t	NOUN
ejde-624	380	18	1−αu(t	1−αu(t	NUM
ejde-624	380	19	)	)	PUNCT
ejde-624	381	1	=	=	PUNCT
ejde-624	381	2	u0	u0	VERB
ejde-624	381	3	>	>	X
ejde-624	381	4	0	0	PROPN
ejde-624	381	5	,	,	PUNCT
ejde-624	381	6	does	do	AUX
ejde-624	381	7	not	not	PART
ejde-624	381	8	have	have	VERB
ejde-624	381	9	a	a	DET
ejde-624	381	10	global	global	ADJ
ejde-624	381	11	solution	solution	NOUN
ejde-624	381	12	u	u	PROPN
ejde-624	381	13	∈	∈	PROPN
ejde-624	381	14	cα−1	cα−1	PROPN
ejde-624	381	15	.	.	PUNCT
ejde-624	382	1	in	in	ADP
ejde-624	382	2	fact	fact	NOUN
ejde-624	382	3	,	,	PUNCT
ejde-624	382	4	for	for	ADP
ejde-624	382	5	p	p	PRON
ejde-624	382	6	≥	≥	NOUN
ejde-624	382	7	1+β	1+β	NUM
ejde-624	382	8	1−α	1−α	NUM
ejde-624	382	9	there	there	PRON
ejde-624	382	10	is	be	VERB
ejde-624	382	11	no	no	DET
ejde-624	382	12	solution	solution	NOUN
ejde-624	382	13	in	in	ADP
ejde-624	382	14	cα−1[0	cα−1[0	PROPN
ejde-624	382	15	,	,	PUNCT
ejde-624	382	16	t	t	X
ejde-624	382	17	]	]	PUNCT
ejde-624	382	18	for	for	ADP
ejde-624	382	19	any	any	DET
ejde-624	382	20	t	t	PROPN
ejde-624	382	21	>	>	X
ejde-624	382	22	0	0	NUM
ejde-624	382	23	,	,	PUNCT
ejde-624	382	24	while	while	SCONJ
ejde-624	382	25	for	for	SCONJ
ejde-624	382	26	p	p	NOUN
ejde-624	382	27	<	<	X
ejde-624	382	28	1+β	1+β	NUM
ejde-624	382	29	1−α	1−α	NUM
ejde-624	382	30	any	any	DET
ejde-624	382	31	solution	solution	NOUN
ejde-624	382	32	can	can	AUX
ejde-624	382	33	only	only	ADV
ejde-624	382	34	exist	exist	VERB
ejde-624	382	35	on	on	ADP
ejde-624	382	36	an	an	DET
ejde-624	382	37	interval	interval	NOUN
ejde-624	382	38	[	[	X
ejde-624	382	39	0	0	NUM
ejde-624	382	40	,	,	PUNCT
ejde-624	382	41	t	t	X
ejde-624	382	42	]	]	PUNCT
ejde-624	382	43	for	for	ADP
ejde-624	382	44	t	t	PROPN
ejde-624	382	45	<	<	X
ejde-624	382	46	t1	t1	PROPN
ejde-624	382	47	,	,	PUNCT
ejde-624	382	48	with	with	ADP
ejde-624	382	49	t1	t1	NOUN
ejde-624	382	50	as	as	SCONJ
ejde-624	382	51	given	give	VERB
ejde-624	382	52	in	in	ADP
ejde-624	382	53	theorem	theorem	ADJ
ejde-624	382	54	4.2	4.2	NUM
ejde-624	382	55	.	.	PUNCT
ejde-624	383	1	ejde-2024/40	ejde-2024/40	VERB
ejde-624	383	2	fractional	fractional	ADJ
ejde-624	383	3	differential	differential	ADJ
ejde-624	383	4	inequalities	inequality	NOUN
ejde-624	383	5	11	11	NUM
ejde-624	383	6	proof	proof	NOUN
ejde-624	383	7	.	.	PUNCT
ejde-624	384	1	let	let	VERB
ejde-624	384	2	f(t	f(t	NOUN
ejde-624	384	3	)	)	PUNCT
ejde-624	384	4	=	=	SYM
ejde-624	384	5	dαu(t)−λtβup(t	dαu(t)−λtβup(t	PROPN
ejde-624	384	6	)	)	PUNCT
ejde-624	384	7	.	.	PUNCT
ejde-624	385	1	for	for	ADP
ejde-624	385	2	p	p	PRON
ejde-624	385	3	≥	≥	NOUN
ejde-624	385	4	1+β	1+β	NUM
ejde-624	385	5	1−α	1−α	NUM
ejde-624	385	6	and	and	CCONJ
ejde-624	385	7	u	u	PROPN
ejde-624	385	8	∈	∈	PROPN
ejde-624	385	9	cα−1	cα−1	PROPN
ejde-624	385	10	the	the	DET
ejde-624	385	11	term	term	NOUN
ejde-624	385	12	λtβup(t	λtβup(t	NOUN
ejde-624	385	13	)	)	PUNCT
ejde-624	385	14	is	be	AUX
ejde-624	385	15	not	not	PART
ejde-624	385	16	integrable	integrable	ADJ
ejde-624	385	17	and	and	CCONJ
ejde-624	385	18	no	no	DET
ejde-624	385	19	solution	solution	NOUN
ejde-624	385	20	exists	exist	VERB
ejde-624	385	21	.	.	PUNCT
ejde-624	386	1	otherwise	otherwise	ADV
ejde-624	386	2	we	we	PRON
ejde-624	386	3	can	can	AUX
ejde-624	386	4	apply	apply	VERB
ejde-624	386	5	theorem	theorem	ADJ
ejde-624	386	6	4.2	4.2	NUM
ejde-624	386	7	noting	note	VERB
ejde-624	386	8	that	that	SCONJ
ejde-624	386	9	t1	t1	NOUN
ejde-624	386	10	does	do	AUX
ejde-624	386	11	not	not	PART
ejde-624	386	12	depend	depend	VERB
ejde-624	386	13	on	on	ADP
ejde-624	386	14	f	f	PROPN
ejde-624	386	15	.	.	PUNCT
ejde-624	387	1	□	□	PUNCT
ejde-624	387	2	remark	remark	NOUN
ejde-624	387	3	4.5	4.5	NUM
ejde-624	387	4	.	.	PUNCT
ejde-624	388	1	laskri	laskri	NOUN
ejde-624	388	2	-	-	PUNCT
ejde-624	388	3	tatar	tatar	NOUN
ejde-624	388	4	[	[	X
ejde-624	388	5	15	15	NUM
ejde-624	388	6	]	]	X
ejde-624	388	7	state	state	NOUN
ejde-624	389	1	that	that	SCONJ
ejde-624	389	2	p0	p0	NOUN
ejde-624	389	3	=	=	PUNCT
ejde-624	390	1	1+β	1+β	NUM
ejde-624	390	2	1−α	1−α	NUM
ejde-624	390	3	is	be	AUX
ejde-624	390	4	a	a	DET
ejde-624	390	5	critical	critical	ADJ
ejde-624	390	6	exponent	exponent	NOUN
ejde-624	390	7	,	,	PUNCT
ejde-624	390	8	arguing	argue	VERB
ejde-624	390	9	that	that	SCONJ
ejde-624	390	10	no	no	DET
ejde-624	390	11	global	global	ADJ
ejde-624	390	12	solution	solution	NOUN
ejde-624	390	13	exists	exist	VERB
ejde-624	390	14	for	for	ADP
ejde-624	390	15	1	1	NUM
ejde-624	390	16	<	<	X
ejde-624	390	17	p	p	X
ejde-624	390	18	<	<	X
ejde-624	390	19	p0	p0	NOUN
ejde-624	390	20	,	,	PUNCT
ejde-624	390	21	while	while	SCONJ
ejde-624	390	22	for	for	SCONJ
ejde-624	390	23	p	p	PROPN
ejde-624	390	24	≥	≥	NOUN
ejde-624	390	25	p0	p0	NOUN
ejde-624	390	26	a	a	DET
ejde-624	390	27	global	global	ADJ
ejde-624	390	28	solution	solution	NOUN
ejde-624	390	29	exists	exist	VERB
ejde-624	390	30	by	by	ADP
ejde-624	390	31	quoting	quote	VERB
ejde-624	390	32	an	an	DET
ejde-624	390	33	explicit	explicit	ADJ
ejde-624	390	34	example	example	NOUN
ejde-624	390	35	given	give	VERB
ejde-624	390	36	in	in	ADP
ejde-624	390	37	[	[	X
ejde-624	390	38	10	10	NUM
ejde-624	390	39	,	,	PUNCT
ejde-624	390	40	example	example	NOUN
ejde-624	390	41	3.3	3.3	NUM
ejde-624	390	42	]	]	PUNCT
ejde-624	390	43	.	.	PUNCT
ejde-624	391	1	this	this	PRON
ejde-624	391	2	appears	appear	VERB
ejde-624	391	3	to	to	PART
ejde-624	391	4	contradict	contradict	VERB
ejde-624	391	5	our	our	PRON
ejde-624	391	6	result	result	NOUN
ejde-624	391	7	in	in	ADP
ejde-624	391	8	theorem	theorem	ADJ
ejde-624	391	9	4.2	4.2	NUM
ejde-624	391	10	,	,	PUNCT
ejde-624	391	11	but	but	CCONJ
ejde-624	391	12	the	the	DET
ejde-624	391	13	real	real	ADJ
ejde-624	391	14	reason	reason	NOUN
ejde-624	391	15	is	be	AUX
ejde-624	391	16	that	that	SCONJ
ejde-624	391	17	the	the	DET
ejde-624	391	18	example	example	NOUN
ejde-624	391	19	has	have	VERB
ejde-624	391	20	u0	u0	ADJ
ejde-624	391	21	=	=	SYM
ejde-624	391	22	0	0	PUNCT
ejde-624	392	1	while	while	SCONJ
ejde-624	392	2	we	we	PRON
ejde-624	392	3	have	have	VERB
ejde-624	392	4	u0	u0	ADJ
ejde-624	392	5	>	>	X
ejde-624	392	6	0	0	X
ejde-624	392	7	.	.	PUNCT
ejde-624	393	1	the	the	DET
ejde-624	393	2	case	case	NOUN
ejde-624	393	3	p	p	X
ejde-624	393	4	≥	≥	NOUN
ejde-624	393	5	p0	p0	NOUN
ejde-624	393	6	is	be	AUX
ejde-624	393	7	not	not	PART
ejde-624	393	8	discussed	discuss	VERB
ejde-624	393	9	in	in	ADP
ejde-624	393	10	[	[	X
ejde-624	393	11	15	15	NUM
ejde-624	393	12	]	]	PUNCT
ejde-624	393	13	when	when	SCONJ
ejde-624	393	14	u0	u0	VERB
ejde-624	393	15	>	>	X
ejde-624	393	16	0	0	X
ejde-624	393	17	.	.	PUNCT
ejde-624	394	1	we	we	PRON
ejde-624	394	2	now	now	ADV
ejde-624	394	3	give	give	VERB
ejde-624	394	4	extra	extra	ADJ
ejde-624	394	5	details	detail	NOUN
ejde-624	394	6	of	of	ADP
ejde-624	394	7	this	this	DET
ejde-624	394	8	example	example	NOUN
ejde-624	394	9	since	since	SCONJ
ejde-624	394	10	it	it	PRON
ejde-624	394	11	is	be	AUX
ejde-624	394	12	only	only	ADV
ejde-624	394	13	stated	state	VERB
ejde-624	394	14	in	in	ADP
ejde-624	394	15	[	[	X
ejde-624	394	16	15	15	NUM
ejde-624	394	17	]	]	PUNCT
ejde-624	394	18	.	.	PUNCT
ejde-624	395	1	they	they	PRON
ejde-624	395	2	cite	cite	VERB
ejde-624	395	3	the	the	DET
ejde-624	395	4	monograph	monograph	NOUN
ejde-624	396	1	[	[	X
ejde-624	396	2	10	10	NUM
ejde-624	396	3	,	,	PUNCT
ejde-624	396	4	example	example	NOUN
ejde-624	396	5	3.3	3.3	NUM
ejde-624	396	6	]	]	PUNCT
ejde-624	396	7	,	,	PUNCT
ejde-624	396	8	but	but	CCONJ
ejde-624	396	9	there	there	PRON
ejde-624	396	10	is	be	VERB
ejde-624	396	11	a	a	DET
ejde-624	396	12	missing	miss	VERB
ejde-624	396	13	minus	minus	NOUN
ejde-624	396	14	sign	sign	NOUN
ejde-624	396	15	in	in	ADP
ejde-624	396	16	the	the	DET
ejde-624	396	17	constant	constant	ADJ
ejde-624	396	18	term	term	NOUN
ejde-624	396	19	in	in	ADP
ejde-624	396	20	the	the	DET
ejde-624	396	21	formula	formula	NOUN
ejde-624	396	22	in	in	ADP
ejde-624	396	23	[	[	X
ejde-624	396	24	10	10	NUM
ejde-624	396	25	,	,	PUNCT
ejde-624	396	26	page	page	NOUN
ejde-624	396	27	177	177	NUM
ejde-624	396	28	]	]	PUNCT
ejde-624	396	29	)	)	PUNCT
ejde-624	396	30	which	which	DET
ejde-624	396	31	misprint	misprint	NOUN
ejde-624	396	32	is	be	AUX
ejde-624	396	33	copied	copy	VERB
ejde-624	396	34	in	in	ADP
ejde-624	396	35	[	[	X
ejde-624	396	36	15	15	NUM
ejde-624	396	37	]	]	PUNCT
ejde-624	396	38	.	.	PUNCT
ejde-624	397	1	example	example	NOUN
ejde-624	397	2	4.6	4.6	NUM
ejde-624	397	3	.	.	PUNCT
ejde-624	398	1	consider	consider	VERB
ejde-624	398	2	the	the	DET
ejde-624	398	3	equation	equation	NOUN
ejde-624	398	4	dαu	dαu	NOUN
ejde-624	398	5	=	=	SYM
ejde-624	398	6	λtβ	λtβ	NOUN
ejde-624	398	7	|u(t)|p	|u(t)|p	NOUN
ejde-624	398	8	,	,	PUNCT
ejde-624	398	9	t	t	X
ejde-624	398	10	>	>	X
ejde-624	398	11	0	0	NUM
ejde-624	398	12	,	,	PUNCT
ejde-624	398	13	with	with	ADP
ejde-624	398	14	lim	lim	PROPN
ejde-624	398	15	t→0	t→0	PUNCT
ejde-624	398	16	t1−αu(t	t1−αu(t	PROPN
ejde-624	398	17	)	)	PUNCT
ejde-624	398	18	=	=	SYM
ejde-624	398	19	0	0	NUM
ejde-624	398	20	and	and	CCONJ
ejde-624	398	21	λ	λ	X
ejde-624	398	22	>	>	X
ejde-624	398	23	0	0	PROPN
ejde-624	398	24	,	,	PUNCT
ejde-624	398	25	p	p	X
ejde-624	398	26	>	>	X
ejde-624	398	27	0	0	NUM
ejde-624	398	28	.	.	PUNCT
ejde-624	399	1	(	(	PUNCT
ejde-624	399	2	4.3	4.3	NUM
ejde-624	399	3	)	)	PUNCT
ejde-624	399	4	the	the	DET
ejde-624	399	5	zero	zero	NUM
ejde-624	399	6	function	function	NOUN
ejde-624	399	7	is	be	AUX
ejde-624	399	8	a	a	DET
ejde-624	399	9	solution	solution	NOUN
ejde-624	399	10	.	.	PUNCT
ejde-624	400	1	we	we	PRON
ejde-624	400	2	show	show	VERB
ejde-624	400	3	that	that	SCONJ
ejde-624	400	4	a	a	DET
ejde-624	400	5	positive	positive	ADJ
ejde-624	400	6	solution	solution	NOUN
ejde-624	400	7	of	of	ADP
ejde-624	400	8	the	the	DET
ejde-624	400	9	form	form	NOUN
ejde-624	400	10	u(t	u(t	NOUN
ejde-624	400	11	)	)	PUNCT
ejde-624	400	12	=	=	SYM
ejde-624	400	13	ctr	ctr	PROPN
ejde-624	400	14	,	,	PUNCT
ejde-624	400	15	c	c	X
ejde-624	400	16	>	>	X
ejde-624	400	17	0	0	NUM
ejde-624	400	18	can	can	AUX
ejde-624	400	19	exist	exist	VERB
ejde-624	400	20	.	.	PUNCT
ejde-624	401	1	let	let	VERB
ejde-624	401	2	p0	p0	NOUN
ejde-624	401	3	=	=	PUNCT
ejde-624	401	4	1+β	1+β	NUM
ejde-624	401	5	1−α	1−α	NUM
ejde-624	401	6	,	,	PUNCT
ejde-624	401	7	where	where	SCONJ
ejde-624	401	8	0	0	X
ejde-624	401	9	<	<	X
ejde-624	401	10	α	α	X
ejde-624	401	11	<	<	X
ejde-624	401	12	1	1	NUM
ejde-624	401	13	,	,	PUNCT
ejde-624	401	14	but	but	CCONJ
ejde-624	401	15	β	β	X
ejde-624	401	16	can	can	AUX
ejde-624	401	17	be	be	AUX
ejde-624	401	18	of	of	ADP
ejde-624	401	19	either	either	DET
ejde-624	401	20	sign	sign	NOUN
ejde-624	401	21	.	.	PUNCT
ejde-624	402	1	we	we	PRON
ejde-624	402	2	have	have	VERB
ejde-624	402	3	by	by	ADP
ejde-624	402	4	direct	direct	ADJ
ejde-624	402	5	calculation	calculation	NOUN
ejde-624	402	6	,	,	PUNCT
ejde-624	402	7	i1−αu	i1−αu	NOUN
ejde-624	402	8	=	=	SYM
ejde-624	402	9	1	1	NUM
ejde-624	402	10	γ(1−	γ(1−	NOUN
ejde-624	402	11	α	α	NUM
ejde-624	402	12	)	)	PUNCT
ejde-624	402	13	∫	∫	PROPN
ejde-624	402	14	t	t	PROPN
ejde-624	402	15	0	0	NUM
ejde-624	402	16	(	(	PUNCT
ejde-624	402	17	t−	t−	PROPN
ejde-624	402	18	s)−αcsrds	s)−αcsrds	PROPN
ejde-624	402	19	=	=	SYM
ejde-624	402	20	ct1−α+r	ct1−α+r	PROPN
ejde-624	403	1	γ(1	γ(1	PROPN
ejde-624	403	2	+	+	CCONJ
ejde-624	403	3	r	r	NOUN
ejde-624	403	4	)	)	PUNCT
ejde-624	403	5	γ(2	γ(2	NOUN
ejde-624	404	1	+	+	PUNCT
ejde-624	404	2	r	r	NOUN
ejde-624	404	3	−	−	NOUN
ejde-624	404	4	α	α	NOUN
ejde-624	404	5	)	)	PUNCT
ejde-624	404	6	,	,	PUNCT
ejde-624	404	7	where	where	SCONJ
ejde-624	404	8	we	we	PRON
ejde-624	404	9	impose	impose	VERB
ejde-624	404	10	1	1	NUM
ejde-624	404	11	+	+	CCONJ
ejde-624	404	12	r−α	r−α	VERB
ejde-624	404	13	>	>	X
ejde-624	404	14	0	0	PUNCT
ejde-624	405	1	for	for	SCONJ
ejde-624	405	2	this	this	PRON
ejde-624	405	3	to	to	PART
ejde-624	405	4	be	be	AUX
ejde-624	405	5	ac	ac	PROPN
ejde-624	405	6	and	and	CCONJ
ejde-624	405	7	to	to	PART
ejde-624	405	8	have	have	VERB
ejde-624	405	9	a	a	DET
ejde-624	405	10	nonzero	nonzero	ADJ
ejde-624	405	11	derivative	derivative	NOUN
ejde-624	405	12	.	.	PUNCT
ejde-624	406	1	therefore	therefore	ADV
ejde-624	406	2	dαu(t	dαu(t	PROPN
ejde-624	406	3	)	)	PUNCT
ejde-624	406	4	=	=	PUNCT
ejde-624	406	5	c	c	NOUN
ejde-624	406	6	γ(1+r	γ(1+r	PROPN
ejde-624	406	7	)	)	PUNCT
ejde-624	406	8	γ(1+r−α	γ(1+r−α	PROPN
ejde-624	406	9	)	)	PUNCT
ejde-624	406	10	t	t	PROPN
ejde-624	406	11	r−α	r−α	VERB
ejde-624	406	12	for	for	ADP
ejde-624	406	13	1	1	NUM
ejde-624	406	14	+	+	NOUN
ejde-624	406	15	r	r	NOUN
ejde-624	406	16	−	−	NOUN
ejde-624	406	17	α	α	NOUN
ejde-624	406	18	>	>	X
ejde-624	406	19	0	0	NUM
ejde-624	406	20	.	.	PUNCT
ejde-624	407	1	hence	hence	ADV
ejde-624	407	2	,	,	PUNCT
ejde-624	407	3	for	for	SCONJ
ejde-624	407	4	u	u	PROPN
ejde-624	407	5	=	=	PROPN
ejde-624	407	6	ctr	ctr	PROPN
ejde-624	407	7	to	to	PART
ejde-624	407	8	be	be	AUX
ejde-624	407	9	a	a	DET
ejde-624	407	10	solution	solution	NOUN
ejde-624	407	11	,	,	PUNCT
ejde-624	407	12	we	we	PRON
ejde-624	407	13	require	require	VERB
ejde-624	407	14	that	that	SCONJ
ejde-624	408	1	r	r	NOUN
ejde-624	408	2	−	−	NOUN
ejde-624	408	3	α	α	NOUN
ejde-624	408	4	=	=	PUNCT
ejde-624	408	5	β	β	X
ejde-624	408	6	+	+	CCONJ
ejde-624	408	7	pr	pr	NOUN
ejde-624	408	8	and	and	CCONJ
ejde-624	408	9	cp−1	cp−1	NOUN
ejde-624	408	10	=	=	SYM
ejde-624	409	1	γ(1	γ(1	PROPN
ejde-624	409	2	+	+	NUM
ejde-624	409	3	r	r	X
ejde-624	409	4	)	)	PUNCT
ejde-624	410	1	λγ(1	λγ(1	X
ejde-624	410	2	+	+	NOUN
ejde-624	411	1	r	r	NOUN
ejde-624	411	2	−	−	NOUN
ejde-624	411	3	α	α	NOUN
ejde-624	411	4	)	)	PUNCT
ejde-624	411	5	.	.	PUNCT
ejde-624	412	1	(	(	PUNCT
ejde-624	412	2	4.4	4.4	NUM
ejde-624	412	3	)	)	PUNCT
ejde-624	412	4	the	the	DET
ejde-624	412	5	value	value	NOUN
ejde-624	412	6	p	p	NOUN
ejde-624	412	7	=	=	SYM
ejde-624	412	8	1	1	NUM
ejde-624	412	9	is	be	AUX
ejde-624	412	10	possible	possible	ADJ
ejde-624	412	11	only	only	ADV
ejde-624	412	12	if	if	SCONJ
ejde-624	412	13	α+	α+	X
ejde-624	412	14	β	β	X
ejde-624	412	15	=	=	SYM
ejde-624	412	16	0	0	NUM
ejde-624	412	17	and	and	CCONJ
ejde-624	412	18	λ	λ	PROPN
ejde-624	412	19	,	,	PUNCT
ejde-624	412	20	r	r	NOUN
ejde-624	412	21	satisfy	satisfy	NOUN
ejde-624	412	22	γ(1+r	γ(1+r	NOUN
ejde-624	412	23	)	)	PUNCT
ejde-624	412	24	γ(1+r−α	γ(1+r−α	PROPN
ejde-624	412	25	)	)	PUNCT
ejde-624	413	1	=	=	SYM
ejde-624	413	2	λ	λ	X
ejde-624	413	3	.	.	PUNCT
ejde-624	414	1	if	if	SCONJ
ejde-624	414	2	α	α	PROPN
ejde-624	414	3	+	+	X
ejde-624	414	4	β	β	X
ejde-624	414	5	=	=	SYM
ejde-624	414	6	0	0	NUM
ejde-624	414	7	and	and	CCONJ
ejde-624	414	8	p	p	PROPN
ejde-624	414	9	̸=	̸=	PROPN
ejde-624	414	10	1	1	NUM
ejde-624	414	11	then	then	ADV
ejde-624	414	12	r	r	NOUN
ejde-624	414	13	=	=	SYM
ejde-624	414	14	0	0	NUM
ejde-624	414	15	which	which	PRON
ejde-624	414	16	gives	give	VERB
ejde-624	414	17	a	a	DET
ejde-624	414	18	constant	constant	ADJ
ejde-624	414	19	solution	solution	NOUN
ejde-624	414	20	with	with	ADP
ejde-624	414	21	cp−1	cp−1	NOUN
ejde-624	414	22	=	=	SYM
ejde-624	414	23	1/(λγ(1−	1/(λγ(1−	NUM
ejde-624	414	24	α	α	NOUN
ejde-624	414	25	)	)	PUNCT
ejde-624	414	26	)	)	PUNCT
ejde-624	414	27	.	.	PUNCT
ejde-624	415	1	we	we	PRON
ejde-624	415	2	now	now	ADV
ejde-624	415	3	suppose	suppose	VERB
ejde-624	415	4	that	that	SCONJ
ejde-624	415	5	α	α	PROPN
ejde-624	415	6	+	+	X
ejde-624	415	7	β	β	X
ejde-624	415	8	̸=	̸=	PROPN
ejde-624	415	9	0	0	NUM
ejde-624	415	10	.	.	PUNCT
ejde-624	416	1	for	for	ADP
ejde-624	416	2	p	p	PROPN
ejde-624	416	3	>	>	X
ejde-624	416	4	1	1	NUM
ejde-624	416	5	,	,	PUNCT
ejde-624	416	6	we	we	PRON
ejde-624	416	7	have	have	VERB
ejde-624	416	8	r	r	NOUN
ejde-624	416	9	=	=	SYM
ejde-624	416	10	−(α+β	−(α+β	NOUN
ejde-624	416	11	)	)	PUNCT
ejde-624	416	12	p−1	p−1	PROPN
ejde-624	416	13	.	.	PUNCT
ejde-624	417	1	it	it	PRON
ejde-624	417	2	is	be	AUX
ejde-624	417	3	readily	readily	ADV
ejde-624	417	4	verified	verify	VERB
ejde-624	417	5	that	that	SCONJ
ejde-624	417	6	1−	1−	NUM
ejde-624	417	7	α+	α+	PRON
ejde-624	417	8	r	r	NOUN
ejde-624	417	9	>	>	X
ejde-624	417	10	0	0	PUNCT
ejde-624	418	1	if	if	SCONJ
ejde-624	418	2	p(1−	p(1−	PROPN
ejde-624	418	3	α	α	NOUN
ejde-624	418	4	)	)	PUNCT
ejde-624	418	5	>	>	X
ejde-624	418	6	1	1	NUM
ejde-624	419	1	+	+	CCONJ
ejde-624	419	2	β	β	NOUN
ejde-624	419	3	,	,	PUNCT
ejde-624	419	4	that	that	PRON
ejde-624	419	5	is	be	AUX
ejde-624	419	6	p	p	X
ejde-624	419	7	>	>	X
ejde-624	419	8	p0	p0	NOUN
ejde-624	419	9	.	.	PUNCT
ejde-624	420	1	also	also	ADV
ejde-624	420	2	it	it	PRON
ejde-624	420	3	then	then	ADV
ejde-624	420	4	follows	follow	VERB
ejde-624	420	5	that	that	SCONJ
ejde-624	420	6	u	u	PROPN
ejde-624	420	7	∈	∈	PROPN
ejde-624	420	8	cα−1[0	cα−1[0	PROPN
ejde-624	420	9	,	,	PUNCT
ejde-624	420	10	t	t	X
ejde-624	420	11	]	]	PUNCT
ejde-624	420	12	and	and	CCONJ
ejde-624	420	13	λtβup	λtβup	NOUN
ejde-624	420	14	∈	∈	PROPN
ejde-624	420	15	l1[0	l1[0	PROPN
ejde-624	420	16	,	,	PUNCT
ejde-624	420	17	t	t	X
ejde-624	420	18	]	]	PUNCT
ejde-624	420	19	for	for	ADP
ejde-624	420	20	every	every	DET
ejde-624	420	21	t	t	PROPN
ejde-624	420	22	>	>	X
ejde-624	420	23	0	0	PROPN
ejde-624	420	24	,	,	PUNCT
ejde-624	420	25	thus	thus	ADV
ejde-624	420	26	u	u	NOUN
ejde-624	420	27	is	be	AUX
ejde-624	420	28	a	a	DET
ejde-624	420	29	solution	solution	NOUN
ejde-624	420	30	when	when	SCONJ
ejde-624	420	31	p	p	PROPN
ejde-624	420	32	>	>	X
ejde-624	420	33	1	1	NUM
ejde-624	420	34	and	and	CCONJ
ejde-624	420	35	p	p	X
ejde-624	420	36	>	>	X
ejde-624	420	37	p0	p0	NOUN
ejde-624	420	38	.	.	PUNCT
ejde-624	421	1	note	note	VERB
ejde-624	421	2	that	that	SCONJ
ejde-624	421	3	p0	p0	NOUN
ejde-624	421	4	>	>	X
ejde-624	421	5	1	1	NUM
ejde-624	421	6	if	if	SCONJ
ejde-624	421	7	and	and	CCONJ
ejde-624	421	8	only	only	ADV
ejde-624	421	9	if	if	SCONJ
ejde-624	421	10	α	α	PROPN
ejde-624	421	11	+	+	X
ejde-624	421	12	β	β	X
ejde-624	421	13	>	>	X
ejde-624	421	14	0	0	X
ejde-624	421	15	.	.	PUNCT
ejde-624	422	1	therefore	therefore	ADV
ejde-624	422	2	if	if	SCONJ
ejde-624	422	3	α	α	PROPN
ejde-624	422	4	+	+	X
ejde-624	422	5	β	β	X
ejde-624	422	6	>	>	X
ejde-624	422	7	0	0	NUM
ejde-624	422	8	we	we	PRON
ejde-624	422	9	must	must	AUX
ejde-624	422	10	have	have	VERB
ejde-624	422	11	p	p	X
ejde-624	422	12	>	>	X
ejde-624	422	13	p0	p0	NOUN
ejde-624	422	14	>	>	X
ejde-624	422	15	1	1	NUM
ejde-624	422	16	,	,	PUNCT
ejde-624	422	17	whereas	whereas	SCONJ
ejde-624	422	18	if	if	SCONJ
ejde-624	422	19	α	α	PROPN
ejde-624	422	20	+	+	X
ejde-624	422	21	β	β	X
ejde-624	422	22	<	<	X
ejde-624	422	23	0	0	NUM
ejde-624	423	1	any	any	DET
ejde-624	423	2	p	p	X
ejde-624	423	3	>	>	X
ejde-624	423	4	1	1	NUM
ejde-624	423	5	is	be	AUX
ejde-624	423	6	allowed	allow	VERB
ejde-624	423	7	.	.	PUNCT
ejde-624	424	1	when	when	SCONJ
ejde-624	424	2	α	α	PROPN
ejde-624	424	3	+	+	X
ejde-624	424	4	β	β	X
ejde-624	424	5	<	<	X
ejde-624	424	6	0	0	NUM
ejde-624	424	7	,	,	PUNCT
ejde-624	424	8	we	we	PRON
ejde-624	424	9	have	have	VERB
ejde-624	424	10	r	r	NOUN
ejde-624	424	11	>	>	X
ejde-624	424	12	0	0	PUNCT
ejde-624	424	13	and	and	CCONJ
ejde-624	424	14	u	u	NOUN
ejde-624	424	15	is	be	AUX
ejde-624	424	16	continuous	continuous	ADJ
ejde-624	424	17	;	;	PUNCT
ejde-624	424	18	this	this	PRON
ejde-624	424	19	also	also	ADV
ejde-624	424	20	gives	give	VERB
ejde-624	424	21	an	an	DET
ejde-624	424	22	explicit	explicit	ADJ
ejde-624	424	23	solution	solution	NOUN
ejde-624	424	24	for	for	ADP
ejde-624	424	25	the	the	DET
ejde-624	424	26	caputo	caputo	PROPN
ejde-624	424	27	derivative	derivative	NOUN
ejde-624	424	28	case	case	NOUN
ejde-624	424	29	when	when	SCONJ
ejde-624	424	30	u(0	u(0	NOUN
ejde-624	424	31	)	)	PUNCT
ejde-624	424	32	=	=	SYM
ejde-624	425	1	0	0	X
ejde-624	425	2	.	.	PUNCT
ejde-624	426	1	for	for	ADP
ejde-624	426	2	0	0	NUM
ejde-624	426	3	<	<	X
ejde-624	426	4	p	p	X
ejde-624	426	5	<	<	X
ejde-624	426	6	1	1	NUM
ejde-624	426	7	,	,	PUNCT
ejde-624	426	8	we	we	PRON
ejde-624	426	9	have	have	VERB
ejde-624	426	10	r	r	NOUN
ejde-624	426	11	=	=	SYM
ejde-624	426	12	α+β	α+β	NUM
ejde-624	426	13	1−p	1−p	NUM
ejde-624	426	14	and	and	CCONJ
ejde-624	426	15	1	1	NUM
ejde-624	426	16	+	+	CCONJ
ejde-624	426	17	r	r	NOUN
ejde-624	426	18	−	−	NOUN
ejde-624	426	19	α	α	NOUN
ejde-624	426	20	>	>	X
ejde-624	426	21	0	0	NUM
ejde-624	426	22	requires	require	VERB
ejde-624	426	23	p	p	X
ejde-624	426	24	<	<	X
ejde-624	426	25	p0	p0	NOUN
ejde-624	426	26	.	.	PUNCT
ejde-624	427	1	if	if	SCONJ
ejde-624	427	2	p0	p0	NOUN
ejde-624	427	3	≥	≥	NOUN
ejde-624	427	4	1	1	NUM
ejde-624	427	5	,	,	PUNCT
ejde-624	427	6	that	that	PRON
ejde-624	427	7	is	be	AUX
ejde-624	427	8	α	α	PRON
ejde-624	428	1	+	+	X
ejde-624	428	2	β	β	X
ejde-624	428	3	>	>	X
ejde-624	428	4	0	0	PROPN
ejde-624	428	5	,	,	PUNCT
ejde-624	428	6	any	any	DET
ejde-624	428	7	p	p	X
ejde-624	428	8	<	<	X
ejde-624	428	9	1	1	NUM
ejde-624	428	10	is	be	AUX
ejde-624	428	11	allowed	allow	VERB
ejde-624	428	12	,	,	PUNCT
ejde-624	428	13	while	while	SCONJ
ejde-624	428	14	if	if	SCONJ
ejde-624	428	15	p0	p0	NOUN
ejde-624	428	16	≤	≤	NOUN
ejde-624	428	17	1	1	NUM
ejde-624	428	18	,	,	PUNCT
ejde-624	428	19	that	that	PRON
ejde-624	428	20	is	be	AUX
ejde-624	428	21	α	α	PRON
ejde-624	428	22	+	+	X
ejde-624	428	23	β	β	X
ejde-624	428	24	≤	≤	NUM
ejde-624	428	25	0	0	NUM
ejde-624	428	26	,	,	PUNCT
ejde-624	428	27	it	it	PRON
ejde-624	428	28	must	must	AUX
ejde-624	428	29	be	be	AUX
ejde-624	428	30	0	0	NUM
ejde-624	428	31	<	<	X
ejde-624	428	32	p	p	X
ejde-624	428	33	<	<	X
ejde-624	428	34	p0	p0	NOUN
ejde-624	428	35	<	<	X
ejde-624	428	36	1	1	NUM
ejde-624	428	37	.	.	PUNCT
ejde-624	428	38	remark	remark	NOUN
ejde-624	428	39	4.7	4.7	NUM
ejde-624	428	40	.	.	PUNCT
ejde-624	429	1	our	our	PRON
ejde-624	429	2	theorem	theorem	ADJ
ejde-624	429	3	4.2	4.2	NUM
ejde-624	429	4	shows	show	VERB
ejde-624	429	5	that	that	SCONJ
ejde-624	429	6	,	,	PUNCT
ejde-624	429	7	when	when	SCONJ
ejde-624	429	8	α	α	PROPN
ejde-624	429	9	+	+	X
ejde-624	429	10	β	β	X
ejde-624	429	11	>	>	X
ejde-624	429	12	0	0	PROPN
ejde-624	429	13	,	,	PUNCT
ejde-624	429	14	p0	p0	NOUN
ejde-624	429	15	is	be	AUX
ejde-624	429	16	critical	critical	ADJ
ejde-624	429	17	for	for	ADP
ejde-624	429	18	u0	u0	PROPN
ejde-624	429	19	>	>	X
ejde-624	429	20	0	0	PUNCT
ejde-624	430	1	because	because	SCONJ
ejde-624	430	2	when	when	SCONJ
ejde-624	430	3	1	1	NUM
ejde-624	430	4	<	<	X
ejde-624	430	5	p	p	X
ejde-624	430	6	<	<	X
ejde-624	430	7	p0	p0	NOUN
ejde-624	430	8	solutions	solution	NOUN
ejde-624	430	9	can	can	AUX
ejde-624	430	10	exist	exist	VERB
ejde-624	430	11	only	only	ADV
ejde-624	430	12	for	for	ADP
ejde-624	430	13	0	0	NUM
ejde-624	430	14	<	<	X
ejde-624	430	15	t	t	PROPN
ejde-624	430	16	≤	≤	X
ejde-624	430	17	t	t	PROPN
ejde-624	430	18	<	<	X
ejde-624	430	19	t1	t1	NOUN
ejde-624	430	20	for	for	ADP
ejde-624	430	21	an	an	DET
ejde-624	430	22	explicit	explicit	ADJ
ejde-624	430	23	t1	t1	NOUN
ejde-624	430	24	,	,	PUNCT
ejde-624	430	25	but	but	CCONJ
ejde-624	430	26	solutions	solution	NOUN
ejde-624	430	27	do	do	AUX
ejde-624	430	28	not	not	PART
ejde-624	430	29	exist	exist	VERB
ejde-624	430	30	in	in	ADP
ejde-624	430	31	the	the	DET
ejde-624	430	32	space	space	NOUN
ejde-624	430	33	cα−1[0	cα−1[0	PROPN
ejde-624	430	34	,	,	PUNCT
ejde-624	430	35	t	t	X
ejde-624	430	36	]	]	PUNCT
ejde-624	430	37	for	for	ADP
ejde-624	430	38	any	any	DET
ejde-624	430	39	t	t	NOUN
ejde-624	430	40	>	>	X
ejde-624	430	41	0	0	PUNCT
ejde-624	430	42	when	when	SCONJ
ejde-624	430	43	p	p	PRON
ejde-624	430	44	≥	≥	NOUN
ejde-624	430	45	p0	p0	VERB
ejde-624	430	46	>	>	X
ejde-624	430	47	1	1	NUM
ejde-624	430	48	.	.	PUNCT
ejde-624	430	49	thus	thus	ADV
ejde-624	430	50	u0	u0	ADJ
ejde-624	430	51	=	=	SYM
ejde-624	430	52	0	0	NUM
ejde-624	430	53	is	be	AUX
ejde-624	430	54	an	an	DET
ejde-624	430	55	exceptional	exceptional	ADJ
ejde-624	430	56	case	case	NOUN
ejde-624	430	57	.	.	PUNCT
ejde-624	431	1	12	12	NUM
ejde-624	431	2	j.	j.	PROPN
ejde-624	431	3	r.	r.	PROPN
ejde-624	431	4	l.	l.	PROPN
ejde-624	431	5	webb	webb	PROPN
ejde-624	431	6	ejde-2024/40	ejde-2024/40	PROPN
ejde-624	431	7	5	5	X
ejde-624	431	8	.	.	PUNCT
ejde-624	431	9	blow	blow	NOUN
ejde-624	431	10	-	-	PUNCT
ejde-624	431	11	up	up	NOUN
ejde-624	431	12	for	for	ADP
ejde-624	431	13	caputo	caputo	PROPN
ejde-624	431	14	derivative	derivative	ADJ
ejde-624	431	15	inequalities	inequality	NOUN
ejde-624	431	16	for	for	ADP
ejde-624	431	17	0	0	NUM
ejde-624	431	18	<	<	X
ejde-624	431	19	α	α	X
ejde-624	431	20	<	<	X
ejde-624	431	21	1	1	NUM
ejde-624	431	22	,	,	PUNCT
ejde-624	431	23	α	α	PROPN
ejde-624	431	24	+	+	X
ejde-624	431	25	β	β	X
ejde-624	431	26	≥	≥	NOUN
ejde-624	431	27	0	0	NUM
ejde-624	431	28	,	,	PUNCT
ejde-624	431	29	λ	λ	X
ejde-624	431	30	>	>	X
ejde-624	431	31	0	0	PUNCT
ejde-624	432	1	and	and	CCONJ
ejde-624	432	2	p	p	X
ejde-624	432	3	>	>	X
ejde-624	432	4	1	1	NUM
ejde-624	432	5	,	,	PUNCT
ejde-624	432	6	we	we	PRON
ejde-624	432	7	will	will	AUX
ejde-624	432	8	investigate	investigate	VERB
ejde-624	432	9	continuous	continuous	ADJ
ejde-624	432	10	functions	function	NOUN
ejde-624	432	11	u	u	NOUN
ejde-624	432	12	that	that	PRON
ejde-624	432	13	satisfy	satisfy	VERB
ejde-624	432	14	the	the	DET
ejde-624	432	15	inequality	inequality	NOUN
ejde-624	432	16	dα	dα	ADP
ejde-624	432	17	∗	∗	NOUN
ejde-624	432	18	u(t	u(t	PROPN
ejde-624	432	19	)	)	PUNCT
ejde-624	432	20	≥	≥	NOUN
ejde-624	432	21	λtβ	λtβ	NOUN
ejde-624	432	22	|u(t)|p	|u(t)|p	NOUN
ejde-624	432	23	with	with	ADP
ejde-624	432	24	u(0	u(0	PROPN
ejde-624	432	25	)	)	PUNCT
ejde-624	432	26	=	=	PUNCT
ejde-624	432	27	u0	u0	VERB
ejde-624	432	28	>	>	X
ejde-624	432	29	0	0	NUM
ejde-624	432	30	.	.	PUNCT
ejde-624	433	1	(	(	PUNCT
ejde-624	433	2	5.1	5.1	NUM
ejde-624	433	3	)	)	PUNCT
ejde-624	433	4	our	our	PRON
ejde-624	433	5	aim	aim	NOUN
ejde-624	433	6	is	be	AUX
ejde-624	433	7	to	to	PART
ejde-624	433	8	prove	prove	VERB
ejde-624	433	9	nonexistence	nonexistence	NOUN
ejde-624	433	10	of	of	ADP
ejde-624	433	11	nontrivial	nontrivial	ADJ
ejde-624	433	12	global	global	ADJ
ejde-624	433	13	solutions	solution	NOUN
ejde-624	433	14	.	.	PUNCT
ejde-624	434	1	of	of	ADP
ejde-624	434	2	course	course	ADV
ejde-624	434	3	,	,	PUNCT
ejde-624	434	4	for	for	ADP
ejde-624	434	5	u0	u0	ADJ
ejde-624	434	6	=	=	NOUN
ejde-624	434	7	0	0	PROPN
ejde-624	434	8	the	the	DET
ejde-624	434	9	trivial	trivial	ADJ
ejde-624	434	10	solution	solution	NOUN
ejde-624	434	11	u	u	NOUN
ejde-624	434	12	=	=	SYM
ejde-624	434	13	0	0	NUM
ejde-624	434	14	exists	exist	VERB
ejde-624	434	15	for	for	ADP
ejde-624	434	16	all	all	DET
ejde-624	434	17	t.	t.	NOUN
ejde-624	434	18	since	since	SCONJ
ejde-624	434	19	the	the	DET
ejde-624	434	20	caputo	caputo	PROPN
ejde-624	434	21	derivative	derivative	NOUN
ejde-624	434	22	is	be	AUX
ejde-624	434	23	defined	define	VERB
ejde-624	434	24	in	in	ADP
ejde-624	434	25	terms	term	NOUN
ejde-624	434	26	of	of	ADP
ejde-624	434	27	the	the	DET
ejde-624	434	28	r	r	NOUN
ejde-624	434	29	-	-	PUNCT
ejde-624	434	30	l	l	NOUN
ejde-624	434	31	derivative	derivative	NOUN
ejde-624	434	32	it	it	PRON
ejde-624	434	33	should	should	AUX
ejde-624	434	34	be	be	AUX
ejde-624	434	35	no	no	DET
ejde-624	434	36	surprise	surprise	NOUN
ejde-624	434	37	that	that	SCONJ
ejde-624	434	38	a	a	DET
ejde-624	434	39	similar	similar	ADJ
ejde-624	434	40	result	result	NOUN
ejde-624	434	41	to	to	PART
ejde-624	434	42	theorem	theorem	VERB
ejde-624	434	43	4.2	4.2	NUM
ejde-624	434	44	holds	hold	NOUN
ejde-624	434	45	.	.	PUNCT
ejde-624	435	1	however	however	ADV
ejde-624	435	2	there	there	PRON
ejde-624	435	3	are	be	VERB
ejde-624	435	4	some	some	DET
ejde-624	435	5	differences	difference	NOUN
ejde-624	435	6	.	.	PUNCT
ejde-624	436	1	in	in	ADP
ejde-624	436	2	the	the	DET
ejde-624	436	3	caputo	caputo	PROPN
ejde-624	436	4	case	case	NOUN
ejde-624	436	5	it	it	PRON
ejde-624	436	6	is	be	AUX
ejde-624	436	7	supposed	suppose	VERB
ejde-624	436	8	that	that	SCONJ
ejde-624	436	9	u	u	PRON
ejde-624	436	10	is	be	AUX
ejde-624	436	11	continuous	continuous	ADJ
ejde-624	436	12	but	but	CCONJ
ejde-624	436	13	dα	dα	ADJ
ejde-624	436	14	∗	∗	NOUN
ejde-624	436	15	u	u	PRON
ejde-624	436	16	need	need	AUX
ejde-624	436	17	not	not	PART
ejde-624	436	18	be	be	AUX
ejde-624	436	19	continuous	continuous	ADJ
ejde-624	436	20	.	.	PUNCT
ejde-624	437	1	by	by	ADP
ejde-624	437	2	a	a	DET
ejde-624	437	3	solution	solution	NOUN
ejde-624	437	4	u	u	NOUN
ejde-624	437	5	of	of	ADP
ejde-624	437	6	the	the	DET
ejde-624	437	7	problem	problem	NOUN
ejde-624	437	8	(	(	PUNCT
ejde-624	437	9	5.1	5.1	NUM
ejde-624	437	10	)	)	PUNCT
ejde-624	437	11	on	on	ADP
ejde-624	437	12	an	an	DET
ejde-624	437	13	interval	interval	NOUN
ejde-624	437	14	[	[	X
ejde-624	437	15	0	0	NUM
ejde-624	437	16	,	,	PUNCT
ejde-624	437	17	t	t	X
ejde-624	437	18	]	]	PUNCT
ejde-624	437	19	we	we	PRON
ejde-624	437	20	will	will	AUX
ejde-624	437	21	mean	mean	VERB
ejde-624	437	22	that	that	SCONJ
ejde-624	437	23	u	u	PROPN
ejde-624	437	24	∈	∈	PROPN
ejde-624	437	25	c[0	c[0	PROPN
ejde-624	437	26	,	,	PUNCT
ejde-624	437	27	t	t	X
ejde-624	437	28	]	]	PUNCT
ejde-624	437	29	,	,	PUNCT
ejde-624	437	30	i1−α	i1−α	PROPN
ejde-624	437	31	∈	∈	PROPN
ejde-624	437	32	ac[0	ac[0	NOUN
ejde-624	437	33	,	,	PUNCT
ejde-624	437	34	t	t	X
ejde-624	437	35	]	]	PUNCT
ejde-624	437	36	,	,	PUNCT
ejde-624	437	37	and	and	CCONJ
ejde-624	438	1	tβ	tβ	PROPN
ejde-624	438	2	|u(t)|p	|u(t)|p	NOUN
ejde-624	438	3	∈	∈	PROPN
ejde-624	438	4	l1[0	l1[0	PROPN
ejde-624	438	5	,	,	PUNCT
ejde-624	438	6	t	t	X
ejde-624	438	7	]	]	PUNCT
ejde-624	438	8	,	,	PUNCT
ejde-624	438	9	the	the	DET
ejde-624	438	10	inequality	inequality	NOUN
ejde-624	438	11	is	be	AUX
ejde-624	438	12	satisfied	satisfied	ADJ
ejde-624	438	13	for	for	ADP
ejde-624	438	14	t	t	PROPN
ejde-624	438	15	∈	∈	PROPN
ejde-624	438	16	(	(	PUNCT
ejde-624	438	17	0	0	NUM
ejde-624	438	18	,	,	PUNCT
ejde-624	438	19	t	t	NOUN
ejde-624	438	20	]	]	PUNCT
ejde-624	438	21	and	and	CCONJ
ejde-624	438	22	the	the	DET
ejde-624	438	23	ic	ic	PROPN
ejde-624	438	24	is	be	AUX
ejde-624	438	25	satisfied	satisfied	ADJ
ejde-624	438	26	.	.	PUNCT
ejde-624	439	1	by	by	ADP
ejde-624	439	2	a	a	DET
ejde-624	439	3	global	global	ADJ
ejde-624	439	4	solution	solution	NOUN
ejde-624	439	5	we	we	PRON
ejde-624	439	6	mean	mean	VERB
ejde-624	439	7	u(t	u(t	PROPN
ejde-624	439	8	)	)	PUNCT
ejde-624	439	9	is	be	AUX
ejde-624	439	10	a	a	DET
ejde-624	439	11	solution	solution	NOUN
ejde-624	439	12	for	for	ADP
ejde-624	439	13	all	all	DET
ejde-624	439	14	t	t	PROPN
ejde-624	439	15	>	>	X
ejde-624	439	16	0	0	X
ejde-624	439	17	.	.	PUNCT
ejde-624	440	1	we	we	PRON
ejde-624	440	2	first	first	ADV
ejde-624	440	3	give	give	VERB
ejde-624	440	4	a	a	DET
ejde-624	440	5	result	result	NOUN
ejde-624	440	6	which	which	PRON
ejde-624	440	7	will	will	AUX
ejde-624	440	8	prove	prove	VERB
ejde-624	440	9	useful	useful	ADJ
ejde-624	440	10	.	.	PUNCT
ejde-624	441	1	it	it	PRON
ejde-624	441	2	can	can	AUX
ejde-624	441	3	be	be	AUX
ejde-624	441	4	deduced	deduce	VERB
ejde-624	441	5	from	from	ADP
ejde-624	441	6	more	more	ADJ
ejde-624	441	7	general	general	ADJ
ejde-624	441	8	known	know	VERB
ejde-624	441	9	results	result	NOUN
ejde-624	441	10	,	,	PUNCT
ejde-624	441	11	for	for	ADP
ejde-624	441	12	example	example	NOUN
ejde-624	441	13	[	[	X
ejde-624	441	14	11	11	NUM
ejde-624	441	15	,	,	PUNCT
ejde-624	441	16	theorem	theorem	VERB
ejde-624	441	17	3.2	3.2	NUM
ejde-624	441	18	]	]	PUNCT
ejde-624	441	19	,	,	PUNCT
ejde-624	441	20	[	[	X
ejde-624	441	21	14	14	NUM
ejde-624	441	22	,	,	PUNCT
ejde-624	441	23	lemma	lemma	PROPN
ejde-624	441	24	4	4	NUM
ejde-624	441	25	]	]	PUNCT
ejde-624	441	26	,	,	PUNCT
ejde-624	441	27	[	[	X
ejde-624	441	28	21	21	NUM
ejde-624	441	29	,	,	PUNCT
ejde-624	441	30	theorem	theorem	VERB
ejde-624	441	31	5.1	5.1	NUM
ejde-624	441	32	]	]	PUNCT
ejde-624	441	33	.	.	PUNCT
ejde-624	442	1	for	for	ADP
ejde-624	442	2	completeness	completeness	NOUN
ejde-624	442	3	we	we	PRON
ejde-624	442	4	give	give	VERB
ejde-624	442	5	the	the	DET
ejde-624	442	6	simple	simple	ADJ
ejde-624	442	7	proof	proof	NOUN
ejde-624	442	8	.	.	PUNCT
ejde-624	443	1	lemma	lemma	PROPN
ejde-624	443	2	5.1	5.1	NUM
ejde-624	443	3	.	.	PUNCT
ejde-624	444	1	let	let	VERB
ejde-624	444	2	0	0	NUM
ejde-624	444	3	<	<	X
ejde-624	444	4	α	α	X
ejde-624	444	5	<	<	X
ejde-624	444	6	1	1	NUM
ejde-624	444	7	,	,	PUNCT
ejde-624	444	8	and	and	CCONJ
ejde-624	444	9	for	for	ADP
ejde-624	444	10	f	f	PROPN
ejde-624	444	11	∈	∈	PROPN
ejde-624	444	12	l1	l1	PROPN
ejde-624	444	13	suppose	suppose	VERB
ejde-624	444	14	that	that	SCONJ
ejde-624	444	15	u	u	PROPN
ejde-624	444	16	∈	∈	PROPN
ejde-624	444	17	c[0	c[0	PROPN
ejde-624	444	18	,	,	PUNCT
ejde-624	444	19	t	t	X
ejde-624	444	20	]	]	PUNCT
ejde-624	444	21	and	and	CCONJ
ejde-624	444	22	i1−α(u−	i1−α(u−	PUNCT
ejde-624	444	23	u0	u0	ADJ
ejde-624	444	24	)	)	PUNCT
ejde-624	444	25	∈	∈	PROPN
ejde-624	444	26	ac[0	ac[0	PROPN
ejde-624	444	27	,	,	PUNCT
ejde-624	444	28	t	t	X
ejde-624	444	29	]	]	PUNCT
ejde-624	444	30	satisfies	satisfy	VERB
ejde-624	444	31	dα	dα	ADP
ejde-624	444	32	∗	∗	NOUN
ejde-624	444	33	u(t	u(t	NOUN
ejde-624	444	34	)	)	PUNCT
ejde-624	444	35	=	=	SYM
ejde-624	445	1	f(t	f(t	NOUN
ejde-624	445	2	)	)	PUNCT
ejde-624	445	3	,	,	PUNCT
ejde-624	445	4	for	for	ADP
ejde-624	445	5	a.e	a.e	PROPN
ejde-624	445	6	.	.	PROPN
ejde-624	445	7	t	t	PROPN
ejde-624	445	8	>	>	X
ejde-624	445	9	0	0	NUM
ejde-624	445	10	,	,	PUNCT
ejde-624	445	11	with	with	ADP
ejde-624	445	12	ic	ic	PROPN
ejde-624	445	13	u(0	u(0	PROPN
ejde-624	445	14	)	)	PUNCT
ejde-624	446	1	=	=	PUNCT
ejde-624	446	2	u0	u0	ADJ
ejde-624	446	3	,	,	PUNCT
ejde-624	446	4	(	(	PUNCT
ejde-624	446	5	5.2	5.2	NUM
ejde-624	446	6	)	)	PUNCT
ejde-624	446	7	then	then	ADV
ejde-624	446	8	u	u	PRON
ejde-624	446	9	satisfies	satisfy	VERB
ejde-624	446	10	the	the	DET
ejde-624	446	11	volterra	volterra	NOUN
ejde-624	446	12	integral	integral	ADJ
ejde-624	446	13	equation	equation	NOUN
ejde-624	446	14	u(t	u(t	NOUN
ejde-624	446	15	)	)	PUNCT
ejde-624	446	16	=	=	PUNCT
ejde-624	447	1	u0	u0	ADJ
ejde-624	447	2	+	+	ADJ
ejde-624	447	3	1	1	NUM
ejde-624	447	4	γ(α	γ(α	NOUN
ejde-624	447	5	)	)	PUNCT
ejde-624	448	1	∫	∫	PROPN
ejde-624	448	2	t	t	PROPN
ejde-624	448	3	0	0	NUM
ejde-624	448	4	(	(	PUNCT
ejde-624	448	5	t−	t−	PROPN
ejde-624	448	6	s)α−1f(s	s)α−1f(s	ADJ
ejde-624	448	7	)	)	PUNCT
ejde-624	448	8	ds	ds	PROPN
ejde-624	448	9	,	,	PUNCT
ejde-624	448	10	a.e	a.e	PROPN
ejde-624	448	11	.	.	PROPN
ejde-624	448	12	t	t	PROPN
ejde-624	448	13	∈	∈	PROPN
ejde-624	449	1	[	[	X
ejde-624	449	2	0	0	NUM
ejde-624	449	3	,	,	PUNCT
ejde-624	449	4	t	t	X
ejde-624	449	5	]	]	PUNCT
ejde-624	449	6	.	.	PUNCT
ejde-624	450	1	(	(	PUNCT
ejde-624	450	2	5.3	5.3	NUM
ejde-624	450	3	)	)	PUNCT
ejde-624	450	4	proof	proof	NOUN
ejde-624	450	5	.	.	PUNCT
ejde-624	451	1	since	since	SCONJ
ejde-624	451	2	u	u	NOUN
ejde-624	451	3	is	be	AUX
ejde-624	451	4	continuous	continuous	ADJ
ejde-624	451	5	,	,	PUNCT
ejde-624	451	6	for	for	ADP
ejde-624	451	7	m	m	PROPN
ejde-624	451	8	>	>	X
ejde-624	451	9	0	0	PUNCT
ejde-624	451	10	there	there	PRON
ejde-624	451	11	exists	exist	VERB
ejde-624	451	12	δ	δ	PROPN
ejde-624	451	13	>	>	X
ejde-624	451	14	0	0	NUM
ejde-624	452	1	such	such	ADJ
ejde-624	452	2	that	that	SCONJ
ejde-624	452	3	|u(s)−u0|	|u(s)−u0|	PROPN
ejde-624	452	4	<	<	X
ejde-624	452	5	m	m	PROPN
ejde-624	452	6	for	for	ADP
ejde-624	452	7	0	0	NUM
ejde-624	452	8	≤	≤	NUM
ejde-624	452	9	s	s	PART
ejde-624	452	10	<	<	X
ejde-624	452	11	δ	δ	PROPN
ejde-624	452	12	.	.	PUNCT
ejde-624	453	1	then	then	ADV
ejde-624	453	2	we	we	PRON
ejde-624	453	3	have	have	AUX
ejde-624	453	4	,	,	PUNCT
ejde-624	453	5	for	for	ADP
ejde-624	453	6	0	0	NUM
ejde-624	453	7	<	<	X
ejde-624	453	8	t	t	X
ejde-624	453	9	<	<	X
ejde-624	453	10	δ	δ	PROPN
ejde-624	453	11	,	,	PUNCT
ejde-624	453	12	|i1−α(u−	|i1−α(u−	PROPN
ejde-624	453	13	u0)(t)|	u0)(t)|	VERB
ejde-624	453	14	≤	≤	NUM
ejde-624	453	15	1	1	NUM
ejde-624	453	16	γ(1−	γ(1−	NOUN
ejde-624	453	17	α	α	NUM
ejde-624	453	18	)	)	PUNCT
ejde-624	453	19	∫	∫	PROPN
ejde-624	453	20	t	t	PROPN
ejde-624	453	21	0	0	NUM
ejde-624	454	1	(	(	PUNCT
ejde-624	454	2	t−	t−	PROPN
ejde-624	454	3	s)−α|u(s)−	s)−α|u(s)−	ADP
ejde-624	454	4	u0|	u0|	VERB
ejde-624	454	5	ds	ds	PRON
ejde-624	454	6	≤	≤	NUM
ejde-624	454	7	m	m	VERB
ejde-624	454	8	γ(2−	γ(2−	NOUN
ejde-624	454	9	α	α	NOUN
ejde-624	454	10	)	)	PUNCT
ejde-624	454	11	t1−α	t1−α	PROPN
ejde-624	454	12	,	,	PUNCT
ejde-624	454	13	thus	thus	ADV
ejde-624	454	14	i1−α(u−u0)(0	i1−α(u−u0)(0	NUM
ejde-624	454	15	)	)	PUNCT
ejde-624	454	16	=	=	SYM
ejde-624	455	1	0	0	X
ejde-624	455	2	.	.	PUNCT
ejde-624	456	1	by	by	ADP
ejde-624	456	2	definition	definition	NOUN
ejde-624	456	3	,	,	PUNCT
ejde-624	456	4	dα	dα	PROPN
ejde-624	456	5	∗	∗	NOUN
ejde-624	456	6	u	u	NOUN
ejde-624	456	7	=	=	PROPN
ejde-624	456	8	f	f	PROPN
ejde-624	456	9	means	mean	VERB
ejde-624	456	10	that	that	SCONJ
ejde-624	456	11	d(i1−α(u−u0	d(i1−α(u−u0	NOUN
ejde-624	456	12	)	)	PUNCT
ejde-624	456	13	)	)	PUNCT
ejde-624	457	1	=	=	SYM
ejde-624	457	2	f	f	X
ejde-624	457	3	.	.	PUNCT
ejde-624	458	1	since	since	SCONJ
ejde-624	458	2	i1−α(u−u0	i1−α(u−u0	PROPN
ejde-624	458	3	)	)	PUNCT
ejde-624	458	4	)	)	PUNCT
ejde-624	459	1	∈	∈	PROPN
ejde-624	459	2	ac[0	ac[0	PROPN
ejde-624	459	3	,	,	PUNCT
ejde-624	459	4	t	t	X
ejde-624	459	5	]	]	PUNCT
ejde-624	459	6	,	,	PUNCT
ejde-624	459	7	by	by	ADP
ejde-624	459	8	integration	integration	NOUN
ejde-624	459	9	and	and	CCONJ
ejde-624	459	10	the	the	DET
ejde-624	459	11	above	above	ADJ
ejde-624	459	12	calculation	calculation	NOUN
ejde-624	459	13	we	we	PRON
ejde-624	459	14	obtain	obtain	VERB
ejde-624	459	15	i1−α(u	i1−α(u	NOUN
ejde-624	459	16	−	−	NOUN
ejde-624	459	17	u0))(t	u0))(t	SYM
ejde-624	459	18	)	)	PUNCT
ejde-624	459	19	=	=	NOUN
ejde-624	459	20	if(t	if(t	NOUN
ejde-624	459	21	)	)	PUNCT
ejde-624	459	22	for	for	ADP
ejde-624	459	23	all	all	DET
ejde-624	459	24	t.	t.	NOUN
ejde-624	459	25	then	then	ADV
ejde-624	459	26	,	,	PUNCT
ejde-624	459	27	applying	apply	VERB
ejde-624	459	28	iα	iα	NOUN
ejde-624	459	29	and	and	CCONJ
ejde-624	459	30	using	use	VERB
ejde-624	459	31	the	the	DET
ejde-624	459	32	semigroup	semigroup	ADJ
ejde-624	459	33	property	property	NOUN
ejde-624	459	34	gives	give	VERB
ejde-624	459	35	i(u−u0)(t	i(u−u0)(t	ADJ
ejde-624	459	36	)	)	PUNCT
ejde-624	459	37	=	=	SYM
ejde-624	459	38	i(iαf)(t	i(iαf)(t	NOUN
ejde-624	459	39	)	)	PUNCT
ejde-624	459	40	.	.	PUNCT
ejde-624	460	1	the	the	DET
ejde-624	460	2	functions	function	NOUN
ejde-624	460	3	on	on	ADP
ejde-624	460	4	both	both	DET
ejde-624	460	5	sides	side	NOUN
ejde-624	460	6	of	of	ADP
ejde-624	460	7	this	this	DET
ejde-624	460	8	equation	equation	NOUN
ejde-624	460	9	are	be	AUX
ejde-624	460	10	absolutely	absolutely	ADV
ejde-624	460	11	continuous	continuous	ADJ
ejde-624	460	12	,	,	PUNCT
ejde-624	460	13	so	so	ADV
ejde-624	460	14	are	be	AUX
ejde-624	460	15	differentiable	differentiable	ADJ
ejde-624	460	16	almost	almost	ADV
ejde-624	460	17	everywhere	everywhere	ADV
ejde-624	460	18	,	,	PUNCT
ejde-624	460	19	and	and	CCONJ
ejde-624	460	20	we	we	PRON
ejde-624	460	21	get	get	VERB
ejde-624	460	22	u(t)−	u(t)−	PROPN
ejde-624	460	23	u0	u0	NOUN
ejde-624	460	24	=	=	PUNCT
ejde-624	460	25	iαf(t	iαf(t	PROPN
ejde-624	460	26	)	)	PUNCT
ejde-624	460	27	for	for	ADP
ejde-624	460	28	a.e	a.e	PROPN
ejde-624	460	29	.	.	PUNCT
ejde-624	460	30	t.	t.	PROPN
ejde-624	460	31	□	□	PUNCT
ejde-624	460	32	remark	remark	NOUN
ejde-624	460	33	5.2	5.2	NUM
ejde-624	460	34	.	.	PUNCT
ejde-624	461	1	the	the	DET
ejde-624	461	2	converse	converse	NOUN
ejde-624	461	3	needs	need	VERB
ejde-624	461	4	more	more	ADJ
ejde-624	461	5	condition	condition	NOUN
ejde-624	461	6	since	since	SCONJ
ejde-624	461	7	,	,	PUNCT
ejde-624	461	8	for	for	ADP
ejde-624	461	9	f	f	PROPN
ejde-624	461	10	∈	∈	PROPN
ejde-624	461	11	l1	l1	PROPN
ejde-624	461	12	,	,	PUNCT
ejde-624	461	13	we	we	PRON
ejde-624	461	14	only	only	ADV
ejde-624	461	15	have	have	VERB
ejde-624	461	16	iαf	iαf	NOUN
ejde-624	461	17	∈	∈	PROPN
ejde-624	461	18	l1	l1	PROPN
ejde-624	461	19	,	,	PUNCT
ejde-624	461	20	and	and	CCONJ
ejde-624	461	21	u(t	u(t	NOUN
ejde-624	461	22	)	)	PUNCT
ejde-624	461	23	−	−	NOUN
ejde-624	461	24	u0	u0	ADJ
ejde-624	461	25	=	=	PUNCT
ejde-624	461	26	iαf(t	iαf(t	PROPN
ejde-624	461	27	)	)	PUNCT
ejde-624	461	28	for	for	ADP
ejde-624	461	29	a.e	a.e	PROPN
ejde-624	461	30	.	.	PROPN
ejde-624	461	31	t	t	PROPN
ejde-624	461	32	does	do	AUX
ejde-624	461	33	not	not	PART
ejde-624	461	34	imply	imply	VERB
ejde-624	461	35	u(0	u(0	NOUN
ejde-624	461	36	)	)	PUNCT
ejde-624	462	1	=	=	PUNCT
ejde-624	462	2	u0	u0	ADJ
ejde-624	462	3	.	.	PUNCT
ejde-624	463	1	there	there	PRON
ejde-624	463	2	is	be	VERB
ejde-624	463	3	an	an	DET
ejde-624	463	4	equivalence	equivalence	NOUN
ejde-624	463	5	when	when	SCONJ
ejde-624	463	6	f	f	PROPN
ejde-624	463	7	is	be	AUX
ejde-624	463	8	continuous	continuous	ADJ
ejde-624	463	9	as	as	SCONJ
ejde-624	463	10	is	be	AUX
ejde-624	463	11	proved	prove	VERB
ejde-624	463	12	in	in	ADP
ejde-624	463	13	diethelm	diethelm	NOUN
ejde-624	463	14	[	[	X
ejde-624	463	15	5	5	NUM
ejde-624	463	16	,	,	PUNCT
ejde-624	463	17	lemma	lemma	PROPN
ejde-624	463	18	6.2	6.2	NUM
ejde-624	463	19	]	]	PUNCT
ejde-624	463	20	,	,	PUNCT
ejde-624	463	21	and	and	CCONJ
ejde-624	463	22	there	there	PRON
ejde-624	463	23	are	be	VERB
ejde-624	463	24	equivalences	equivalence	NOUN
ejde-624	463	25	under	under	ADP
ejde-624	463	26	some	some	DET
ejde-624	463	27	conditions	condition	NOUN
ejde-624	463	28	on	on	ADP
ejde-624	463	29	f	f	X
ejde-624	463	30	weaker	weak	ADJ
ejde-624	463	31	than	than	ADP
ejde-624	463	32	continuity	continuity	NOUN
ejde-624	463	33	,	,	PUNCT
ejde-624	463	34	for	for	ADP
ejde-624	463	35	example	example	NOUN
ejde-624	463	36	[	[	X
ejde-624	463	37	11	11	NUM
ejde-624	463	38	,	,	PUNCT
ejde-624	463	39	theorem	theorem	VERB
ejde-624	463	40	3.2	3.2	NUM
ejde-624	463	41	]	]	PUNCT
ejde-624	463	42	,	,	PUNCT
ejde-624	463	43	[	[	X
ejde-624	463	44	14	14	NUM
ejde-624	463	45	,	,	PUNCT
ejde-624	463	46	lemma	lemma	PROPN
ejde-624	463	47	4	4	NUM
ejde-624	463	48	]	]	PUNCT
ejde-624	463	49	and	and	CCONJ
ejde-624	463	50	[	[	X
ejde-624	463	51	21	21	NUM
ejde-624	463	52	,	,	PUNCT
ejde-624	463	53	theorem	theorem	VERB
ejde-624	463	54	4.6	4.6	NUM
ejde-624	463	55	]	]	PUNCT
ejde-624	463	56	.	.	PUNCT
ejde-624	464	1	shan	shan	PROPN
ejde-624	464	2	-	-	PUNCT
ejde-624	464	3	lv	lv	PROPN
ejde-624	465	1	[	[	X
ejde-624	465	2	18	18	NUM
ejde-624	465	3	]	]	PUNCT
ejde-624	465	4	studied	study	VERB
ejde-624	465	5	the	the	DET
ejde-624	465	6	caputo	caputo	PROPN
ejde-624	465	7	ivp	ivp	PROPN
ejde-624	465	8	dα	dα	PROPN
ejde-624	465	9	∗	∗	NOUN
ejde-624	465	10	u(t	u(t	NOUN
ejde-624	465	11	)	)	PUNCT
ejde-624	465	12	=	=	PUNCT
ejde-624	465	13	up(t	up(t	NOUN
ejde-624	465	14	)	)	PUNCT
ejde-624	465	15	,	,	PUNCT
ejde-624	465	16	t	t	PROPN
ejde-624	465	17	>	>	X
ejde-624	465	18	0	0	NUM
ejde-624	465	19	,	,	PUNCT
ejde-624	465	20	with	with	ADP
ejde-624	465	21	ic	ic	PROPN
ejde-624	465	22	u(0	u(0	PROPN
ejde-624	465	23	)	)	PUNCT
ejde-624	466	1	=	=	PUNCT
ejde-624	466	2	u0	u0	ADJ
ejde-624	466	3	.	.	PUNCT
ejde-624	467	1	(	(	PUNCT
ejde-624	467	2	5.4	5.4	NUM
ejde-624	467	3	)	)	PUNCT
ejde-624	467	4	this	this	PRON
ejde-624	467	5	is	be	AUX
ejde-624	467	6	a	a	DET
ejde-624	467	7	special	special	ADJ
ejde-624	467	8	case	case	NOUN
ejde-624	467	9	of	of	ADP
ejde-624	467	10	the	the	DET
ejde-624	467	11	problem	problem	NOUN
ejde-624	467	12	we	we	PRON
ejde-624	467	13	study	study	VERB
ejde-624	467	14	with	with	ADP
ejde-624	467	15	β	β	X
ejde-624	467	16	=	=	SYM
ejde-624	467	17	0	0	X
ejde-624	467	18	.	.	PUNCT
ejde-624	468	1	they	they	PRON
ejde-624	468	2	asserted	assert	VERB
ejde-624	468	3	that	that	SCONJ
ejde-624	468	4	if	if	SCONJ
ejde-624	468	5	u0	u0	ADJ
ejde-624	468	6	>	>	X
ejde-624	468	7	0	0	PUNCT
ejde-624	469	1	and	and	CCONJ
ejde-624	469	2	1	1	NUM
ejde-624	469	3	<	<	X
ejde-624	469	4	p	p	X
ejde-624	469	5	≤	≤	ADJ
ejde-624	469	6	1/(1	1/(1	NUM
ejde-624	469	7	−	−	PROPN
ejde-624	469	8	α	α	NOUN
ejde-624	469	9	)	)	PUNCT
ejde-624	469	10	then	then	ADV
ejde-624	469	11	every	every	DET
ejde-624	469	12	solution	solution	NOUN
ejde-624	469	13	of	of	ADP
ejde-624	469	14	(	(	PUNCT
ejde-624	469	15	5.4	5.4	NUM
ejde-624	469	16	)	)	PUNCT
ejde-624	469	17	blows	blow	NOUN
ejde-624	469	18	-	-	PUNCT
ejde-624	469	19	up	up	NOUN
ejde-624	469	20	in	in	ADP
ejde-624	469	21	finite	finite	ADJ
ejde-624	469	22	time	time	NOUN
ejde-624	469	23	,	,	PUNCT
ejde-624	469	24	[	[	X
ejde-624	469	25	18	18	NUM
ejde-624	469	26	,	,	PUNCT
ejde-624	469	27	theorem	theorem	VERB
ejde-624	469	28	2.2	2.2	NUM
ejde-624	469	29	]	]	PUNCT
ejde-624	469	30	.	.	PUNCT
ejde-624	470	1	they	they	PRON
ejde-624	470	2	also	also	ADV
ejde-624	470	3	asserted	assert	VERB
ejde-624	470	4	that	that	SCONJ
ejde-624	470	5	when	when	SCONJ
ejde-624	470	6	p	p	X
ejde-624	470	7	>	>	X
ejde-624	470	8	1/(1	1/(1	NUM
ejde-624	470	9	−	−	PROPN
ejde-624	470	10	α	α	NOUN
ejde-624	470	11	)	)	PUNCT
ejde-624	470	12	and	and	CCONJ
ejde-624	470	13	u0	u0	PROPN
ejde-624	470	14	has	have	AUX
ejde-624	470	15	ejde-2024/40	ejde-2024/40	VERB
ejde-624	470	16	fractional	fractional	ADJ
ejde-624	470	17	differential	differential	ADJ
ejde-624	470	18	inequalities	inequality	NOUN
ejde-624	470	19	13	13	NUM
ejde-624	470	20	an	an	DET
ejde-624	470	21	explicit	explicit	ADJ
ejde-624	470	22	positive	positive	ADJ
ejde-624	470	23	lower	lower	ADV
ejde-624	470	24	bound	bind	VERB
ejde-624	470	25	,	,	PUNCT
ejde-624	470	26	then	then	ADV
ejde-624	470	27	solutions	solution	VERB
ejde-624	470	28	blow	blow	NOUN
ejde-624	470	29	-	-	PUNCT
ejde-624	470	30	up	up	NOUN
ejde-624	470	31	in	in	ADP
ejde-624	470	32	finite	finite	ADJ
ejde-624	470	33	time	time	NOUN
ejde-624	470	34	.	.	PUNCT
ejde-624	471	1	in	in	ADP
ejde-624	471	2	fact	fact	NOUN
ejde-624	471	3	they	they	PRON
ejde-624	471	4	proved	prove	VERB
ejde-624	471	5	non	non	ADJ
ejde-624	471	6	-	-	NOUN
ejde-624	471	7	existence	existence	NOUN
ejde-624	471	8	of	of	ADP
ejde-624	471	9	global	global	ADJ
ejde-624	471	10	solutions	solution	NOUN
ejde-624	471	11	but	but	CCONJ
ejde-624	471	12	did	do	AUX
ejde-624	471	13	not	not	PART
ejde-624	471	14	prove	prove	VERB
ejde-624	471	15	that	that	SCONJ
ejde-624	471	16	this	this	PRON
ejde-624	471	17	is	be	AUX
ejde-624	471	18	blow	blow	NOUN
ejde-624	471	19	-	-	PUNCT
ejde-624	471	20	up	up	NOUN
ejde-624	471	21	.	.	PUNCT
ejde-624	472	1	we	we	PRON
ejde-624	472	2	improve	improve	VERB
ejde-624	472	3	their	their	PRON
ejde-624	472	4	result	result	NOUN
ejde-624	472	5	by	by	ADP
ejde-624	472	6	considering	consider	VERB
ejde-624	472	7	the	the	DET
ejde-624	472	8	more	more	ADV
ejde-624	472	9	general	general	ADJ
ejde-624	472	10	inequality	inequality	NOUN
ejde-624	472	11	with	with	ADP
ejde-624	472	12	the	the	DET
ejde-624	472	13	possibly	possibly	ADV
ejde-624	472	14	singular	singular	ADJ
ejde-624	472	15	term	term	NOUN
ejde-624	472	16	tβ	tβ	PRON
ejde-624	472	17	.	.	PUNCT
ejde-624	473	1	we	we	PRON
ejde-624	473	2	prove	prove	VERB
ejde-624	473	3	that	that	SCONJ
ejde-624	473	4	,	,	PUNCT
ejde-624	473	5	for	for	ADP
ejde-624	473	6	for	for	ADP
ejde-624	473	7	every	every	DET
ejde-624	473	8	p	p	NOUN
ejde-624	473	9	>	>	X
ejde-624	473	10	1	1	NUM
ejde-624	473	11	and	and	CCONJ
ejde-624	473	12	any	any	DET
ejde-624	473	13	initial	initial	ADJ
ejde-624	473	14	value	value	NOUN
ejde-624	473	15	u0	u0	NOUN
ejde-624	473	16	>	>	X
ejde-624	473	17	0	0	PROPN
ejde-624	473	18	,	,	PUNCT
ejde-624	473	19	solutions	solution	NOUN
ejde-624	473	20	can	can	AUX
ejde-624	473	21	only	only	ADV
ejde-624	473	22	exist	exist	VERB
ejde-624	473	23	on	on	ADP
ejde-624	473	24	a	a	DET
ejde-624	473	25	finite	finite	ADJ
ejde-624	473	26	interval	interval	NOUN
ejde-624	473	27	[	[	X
ejde-624	473	28	0	0	NUM
ejde-624	473	29	,	,	PUNCT
ejde-624	473	30	t	t	X
ejde-624	473	31	]	]	PUNCT
ejde-624	473	32	with	with	ADP
ejde-624	473	33	an	an	DET
ejde-624	473	34	explicit	explicit	ADJ
ejde-624	473	35	upper	upper	ADJ
ejde-624	473	36	bound	bind	VERB
ejde-624	473	37	t	t	PROPN
ejde-624	473	38	<	<	X
ejde-624	473	39	t1	t1	PROPN
ejde-624	473	40	.	.	PUNCT
ejde-624	474	1	we	we	PRON
ejde-624	474	2	start	start	VERB
ejde-624	474	3	with	with	ADP
ejde-624	474	4	the	the	DET
ejde-624	474	5	ivp	ivp	NOUN
ejde-624	474	6	,	,	PUNCT
ejde-624	474	7	for	for	ADP
ejde-624	474	8	α+	α+	PRON
ejde-624	474	9	β	β	X
ejde-624	474	10	>	>	X
ejde-624	474	11	0	0	PROPN
ejde-624	474	12	,	,	PUNCT
ejde-624	474	13	f	f	PROPN
ejde-624	474	14	∈	∈	PROPN
ejde-624	474	15	l1	l1	PROPN
ejde-624	474	16	,	,	PUNCT
ejde-624	474	17	f(t	f(t	PROPN
ejde-624	474	18	)	)	PUNCT
ejde-624	474	19	≥	≥	NOUN
ejde-624	474	20	0	0	NUM
ejde-624	474	21	for	for	ADP
ejde-624	474	22	t	t	PROPN
ejde-624	474	23	>	>	X
ejde-624	474	24	0	0	X
ejde-624	474	25	.	.	PUNCT
ejde-624	475	1	dα	dα	PROPN
ejde-624	475	2	∗	∗	NOUN
ejde-624	475	3	u(t	u(t	PROPN
ejde-624	475	4	)	)	PUNCT
ejde-624	475	5	=	=	SYM
ejde-624	475	6	λtβup(t	λtβup(t	NOUN
ejde-624	475	7	)	)	PUNCT
ejde-624	475	8	+	+	CCONJ
ejde-624	475	9	f(t	f(t	NOUN
ejde-624	475	10	)	)	PUNCT
ejde-624	475	11	with	with	ADP
ejde-624	475	12	u(0	u(0	NOUN
ejde-624	475	13	)	)	PUNCT
ejde-624	475	14	=	=	PUNCT
ejde-624	475	15	u0	u0	VERB
ejde-624	475	16	>	>	X
ejde-624	475	17	0	0	NUM
ejde-624	475	18	.	.	PUNCT
ejde-624	476	1	(	(	PUNCT
ejde-624	476	2	5.5	5.5	NUM
ejde-624	476	3	)	)	PUNCT
ejde-624	476	4	a	a	DET
ejde-624	476	5	solution	solution	NOUN
ejde-624	476	6	u	u	PROPN
ejde-624	476	7	∈	∈	PROPN
ejde-624	476	8	c[0	c[0	PROPN
ejde-624	476	9	,	,	PUNCT
ejde-624	476	10	t	t	PROPN
ejde-624	476	11	]	]	PUNCT
ejde-624	476	12	of	of	ADP
ejde-624	476	13	(	(	PUNCT
ejde-624	476	14	5.5	5.5	NUM
ejde-624	476	15	)	)	PUNCT
ejde-624	476	16	satisfies	satisfy	VERB
ejde-624	476	17	u(t	u(t	NOUN
ejde-624	476	18	)	)	PUNCT
ejde-624	476	19	=	=	PUNCT
ejde-624	476	20	u0	u0	ADJ
ejde-624	476	21	+	+	CCONJ
ejde-624	476	22	λ	λ	PROPN
ejde-624	476	23	γ(α	γ(α	NOUN
ejde-624	476	24	)	)	PUNCT
ejde-624	477	1	∫	∫	PROPN
ejde-624	477	2	t	t	PROPN
ejde-624	477	3	0	0	NUM
ejde-624	477	4	(	(	PUNCT
ejde-624	477	5	t−	t−	NOUN
ejde-624	477	6	s)α−1(sβup(s	s)α−1(sβup(s	NUM
ejde-624	477	7	)	)	PUNCT
ejde-624	477	8	+	+	NUM
ejde-624	477	9	f(s	f(	NOUN
ejde-624	477	10	)	)	PUNCT
ejde-624	477	11	)	)	PUNCT
ejde-624	478	1	ds	ds	PROPN
ejde-624	478	2	,	,	PUNCT
ejde-624	478	3	a.e	a.e	PROPN
ejde-624	478	4	.	.	PROPN
ejde-624	478	5	t	t	PROPN
ejde-624	478	6	∈	∈	PROPN
ejde-624	479	1	[	[	X
ejde-624	479	2	0	0	NUM
ejde-624	479	3	,	,	PUNCT
ejde-624	479	4	t	t	X
ejde-624	479	5	]	]	PUNCT
ejde-624	479	6	.	.	PUNCT
ejde-624	480	1	(	(	PUNCT
ejde-624	480	2	5.6	5.6	NUM
ejde-624	480	3	)	)	PUNCT
ejde-624	480	4	since	since	SCONJ
ejde-624	480	5	p	p	NOUN
ejde-624	480	6	is	be	AUX
ejde-624	480	7	a	a	DET
ejde-624	480	8	real	real	ADJ
ejde-624	480	9	number	number	NOUN
ejde-624	480	10	,	,	PUNCT
ejde-624	480	11	it	it	PRON
ejde-624	480	12	is	be	AUX
ejde-624	480	13	implicit	implicit	ADJ
ejde-624	480	14	that	that	SCONJ
ejde-624	480	15	solutions	solution	NOUN
ejde-624	480	16	of	of	ADP
ejde-624	480	17	(	(	PUNCT
ejde-624	480	18	5.5	5.5	NUM
ejde-624	480	19	)	)	PUNCT
ejde-624	480	20	are	be	AUX
ejde-624	480	21	positive	positive	ADJ
ejde-624	480	22	.	.	PUNCT
ejde-624	481	1	in	in	ADP
ejde-624	481	2	fact	fact	NOUN
ejde-624	481	3	,	,	PUNCT
ejde-624	481	4	u0	u0	VERB
ejde-624	481	5	>	>	X
ejde-624	481	6	0	0	PUNCT
ejde-624	481	7	and	and	CCONJ
ejde-624	481	8	continuity	continuity	NOUN
ejde-624	481	9	imply	imply	VERB
ejde-624	481	10	that	that	SCONJ
ejde-624	481	11	a	a	DET
ejde-624	481	12	solution	solution	NOUN
ejde-624	481	13	u(t	u(t	NOUN
ejde-624	481	14	)	)	PUNCT
ejde-624	481	15	will	will	AUX
ejde-624	481	16	be	be	AUX
ejde-624	481	17	positive	positive	ADJ
ejde-624	481	18	for	for	ADP
ejde-624	481	19	small	small	ADJ
ejde-624	481	20	t	t	PROPN
ejde-624	481	21	>	>	X
ejde-624	481	22	0	0	PUNCT
ejde-624	482	1	and	and	CCONJ
ejde-624	482	2	then	then	ADV
ejde-624	482	3	from	from	ADP
ejde-624	482	4	(	(	PUNCT
ejde-624	482	5	5.6	5.6	NUM
ejde-624	482	6	)	)	PUNCT
ejde-624	482	7	it	it	PRON
ejde-624	482	8	follows	follow	VERB
ejde-624	482	9	that	that	SCONJ
ejde-624	482	10	u(t	u(t	NOUN
ejde-624	482	11	)	)	PUNCT
ejde-624	482	12	>	>	X
ejde-624	482	13	u0	u0	PROPN
ejde-624	482	14	for	for	ADP
ejde-624	482	15	t	t	PROPN
ejde-624	482	16	>	>	X
ejde-624	482	17	0	0	PUNCT
ejde-624	483	1	on	on	ADP
ejde-624	483	2	its	its	PRON
ejde-624	483	3	interval	interval	NOUN
ejde-624	483	4	of	of	ADP
ejde-624	483	5	existence	existence	NOUN
ejde-624	483	6	.	.	PUNCT
ejde-624	484	1	we	we	PRON
ejde-624	484	2	first	first	ADV
ejde-624	484	3	consider	consider	VERB
ejde-624	484	4	the	the	DET
ejde-624	484	5	case	case	NOUN
ejde-624	484	6	1	1	NUM
ejde-624	484	7	<	<	X
ejde-624	484	8	p	p	X
ejde-624	484	9	≤	≤	NOUN
ejde-624	484	10	1+β	1+β	NUM
ejde-624	484	11	1−α	1−α	NUM
ejde-624	484	12	.	.	PUNCT
ejde-624	485	1	theorem	theorem	VERB
ejde-624	485	2	5.3	5.3	NUM
ejde-624	485	3	.	.	PUNCT
ejde-624	486	1	let	let	VERB
ejde-624	486	2	f	f	PROPN
ejde-624	486	3	∈	∈	PROPN
ejde-624	486	4	l1	l1	PROPN
ejde-624	486	5	be	be	VERB
ejde-624	486	6	non	non	ADJ
ejde-624	486	7	-	-	ADJ
ejde-624	486	8	negative	negative	ADJ
ejde-624	486	9	and	and	CCONJ
ejde-624	486	10	let	let	VERB
ejde-624	486	11	λ	λ	PRON
ejde-624	486	12	>	>	X
ejde-624	486	13	0	0	PUNCT
ejde-624	486	14	and	and	CCONJ
ejde-624	486	15	α+	α+	PUNCT
ejde-624	486	16	β	β	X
ejde-624	486	17	>	>	X
ejde-624	486	18	0	0	X
ejde-624	486	19	.	.	PUNCT
ejde-624	487	1	consider	consider	VERB
ejde-624	487	2	the	the	DET
ejde-624	487	3	problem	problem	NOUN
ejde-624	487	4	dα	dα	ADP
ejde-624	487	5	∗	∗	NOUN
ejde-624	487	6	u(t	u(t	PROPN
ejde-624	487	7	)	)	PUNCT
ejde-624	488	1	=	=	SYM
ejde-624	488	2	λtβup(t	λtβup(t	NOUN
ejde-624	488	3	)	)	PUNCT
ejde-624	488	4	+	+	CCONJ
ejde-624	488	5	f(t	f(t	NOUN
ejde-624	488	6	)	)	PUNCT
ejde-624	488	7	,	,	PUNCT
ejde-624	488	8	a.e	a.e	PROPN
ejde-624	488	9	.	.	PROPN
ejde-624	488	10	t	t	PROPN
ejde-624	488	11	>	>	X
ejde-624	488	12	0	0	NUM
ejde-624	488	13	,	,	PUNCT
ejde-624	488	14	with	with	ADP
ejde-624	488	15	ic	ic	PROPN
ejde-624	488	16	u(0	u(0	PROPN
ejde-624	488	17	)	)	PUNCT
ejde-624	488	18	=	=	PUNCT
ejde-624	489	1	u0	u0	ADJ
ejde-624	489	2	.	.	PUNCT
ejde-624	490	1	(	(	PUNCT
ejde-624	490	2	5.7	5.7	NUM
ejde-624	490	3	)	)	PUNCT
ejde-624	490	4	for	for	ADP
ejde-624	490	5	any	any	DET
ejde-624	490	6	u0	u0	NOUN
ejde-624	490	7	>	>	X
ejde-624	490	8	0	0	PUNCT
ejde-624	490	9	and	and	CCONJ
ejde-624	490	10	for	for	ADP
ejde-624	490	11	1	1	NUM
ejde-624	490	12	<	<	X
ejde-624	490	13	p	p	X
ejde-624	490	14	≤	≤	NOUN
ejde-624	490	15	1+β	1+β	NUM
ejde-624	490	16	1−α	1−α	NUM
ejde-624	490	17	there	there	PRON
ejde-624	490	18	does	do	AUX
ejde-624	490	19	not	not	PART
ejde-624	490	20	exist	exist	VERB
ejde-624	490	21	a	a	DET
ejde-624	490	22	global	global	ADJ
ejde-624	490	23	solution	solution	NOUN
ejde-624	490	24	u.	u.	VERB
ejde-624	490	25	more	more	ADV
ejde-624	490	26	precisely	precisely	ADV
ejde-624	490	27	,	,	PUNCT
ejde-624	490	28	there	there	PRON
ejde-624	490	29	exists	exist	VERB
ejde-624	490	30	t1	t1	NOUN
ejde-624	490	31	>	>	X
ejde-624	490	32	0	0	PROPN
ejde-624	490	33	,	,	PUNCT
ejde-624	490	34	explicitly	explicitly	ADV
ejde-624	490	35	determined	determine	VERB
ejde-624	490	36	by	by	ADP
ejde-624	490	37	the	the	DET
ejde-624	490	38	parameters	parameter	NOUN
ejde-624	490	39	of	of	ADP
ejde-624	490	40	the	the	DET
ejde-624	490	41	problem	problem	NOUN
ejde-624	490	42	,	,	PUNCT
ejde-624	490	43	such	such	ADJ
ejde-624	490	44	that	that	SCONJ
ejde-624	490	45	a	a	DET
ejde-624	490	46	solution	solution	NOUN
ejde-624	490	47	u	u	NOUN
ejde-624	490	48	can	can	AUX
ejde-624	490	49	only	only	ADV
ejde-624	490	50	exist	exist	VERB
ejde-624	490	51	on	on	ADP
ejde-624	490	52	an	an	DET
ejde-624	490	53	interval	interval	NOUN
ejde-624	490	54	[	[	X
ejde-624	490	55	0	0	NUM
ejde-624	490	56	,	,	PUNCT
ejde-624	490	57	t	t	NOUN
ejde-624	490	58	]	]	PUNCT
ejde-624	490	59	where	where	SCONJ
ejde-624	490	60	t	t	PROPN
ejde-624	490	61	<	<	X
ejde-624	490	62	t1	t1	PROPN
ejde-624	490	63	.	.	PUNCT
ejde-624	491	1	proof	proof	NOUN
ejde-624	491	2	.	.	PUNCT
ejde-624	492	1	suppose	suppose	VERB
ejde-624	492	2	a	a	DET
ejde-624	492	3	solution	solution	NOUN
ejde-624	492	4	u	u	NOUN
ejde-624	492	5	exists	exist	VERB
ejde-624	492	6	on	on	ADP
ejde-624	492	7	an	an	DET
ejde-624	492	8	interval	interval	NOUN
ejde-624	492	9	[	[	X
ejde-624	492	10	0	0	NUM
ejde-624	492	11	,	,	PUNCT
ejde-624	492	12	t	t	NOUN
ejde-624	492	13	]	]	PUNCT
ejde-624	492	14	where	where	SCONJ
ejde-624	492	15	t	t	PROPN
ejde-624	492	16	>	>	X
ejde-624	492	17	1	1	NUM
ejde-624	492	18	;	;	PUNCT
ejde-624	492	19	otherwise	otherwise	ADV
ejde-624	492	20	we	we	PRON
ejde-624	492	21	can	can	AUX
ejde-624	492	22	take	take	VERB
ejde-624	492	23	any	any	DET
ejde-624	492	24	t1	t1	NOUN
ejde-624	492	25	>	>	X
ejde-624	492	26	1	1	X
ejde-624	492	27	.	.	PUNCT
ejde-624	492	28	define	define	VERB
ejde-624	492	29	c	c	PROPN
ejde-624	492	30	>	>	X
ejde-624	492	31	0	0	NUM
ejde-624	492	32	by	by	ADP
ejde-624	492	33	cp−1	cp−1	PROPN
ejde-624	492	34	=	=	SYM
ejde-624	492	35	λ	λ	PROPN
ejde-624	492	36	.	.	PUNCT
ejde-624	493	1	then	then	ADV
ejde-624	493	2	v	v	X
ejde-624	493	3	=	=	SYM
ejde-624	493	4	cu	cu	PROPN
ejde-624	493	5	is	be	AUX
ejde-624	493	6	a	a	DET
ejde-624	493	7	solution	solution	NOUN
ejde-624	493	8	of	of	ADP
ejde-624	493	9	dα	dα	ADP
ejde-624	493	10	∗	∗	NOUN
ejde-624	493	11	v(t	v(t	NOUN
ejde-624	493	12	)	)	PUNCT
ejde-624	493	13	=	=	SYM
ejde-624	493	14	tβvp(t	tβvp(t	PROPN
ejde-624	493	15	)	)	PUNCT
ejde-624	493	16	+	+	NUM
ejde-624	493	17	cf(t	cf(t	NOUN
ejde-624	493	18	)	)	PUNCT
ejde-624	493	19	,	,	PUNCT
ejde-624	493	20	a.e	a.e	PROPN
ejde-624	493	21	.	.	PROPN
ejde-624	493	22	t	t	PROPN
ejde-624	493	23	∈	∈	PROPN
ejde-624	493	24	(	(	PUNCT
ejde-624	493	25	0	0	NUM
ejde-624	493	26	,	,	PUNCT
ejde-624	493	27	t	t	NOUN
ejde-624	493	28	)	)	PUNCT
ejde-624	493	29	,	,	PUNCT
ejde-624	493	30	with	with	ADP
ejde-624	493	31	ic	ic	PROPN
ejde-624	493	32	v(0	v(0	PROPN
ejde-624	493	33	)	)	PUNCT
ejde-624	493	34	=	=	SYM
ejde-624	493	35	v0	v0	NOUN
ejde-624	493	36	=	=	SYM
ejde-624	493	37	cu0	cu0	PROPN
ejde-624	493	38	>	>	X
ejde-624	493	39	0	0	PROPN
ejde-624	493	40	,	,	PUNCT
ejde-624	493	41	by	by	ADP
ejde-624	493	42	lemma	lemma	PROPN
ejde-624	493	43	5.1	5.1	NUM
ejde-624	493	44	,	,	PUNCT
ejde-624	493	45	for	for	ADP
ejde-624	493	46	a.e	a.e	PROPN
ejde-624	493	47	.	.	PROPN
ejde-624	493	48	t	t	PROPN
ejde-624	493	49	∈	∈	PROPN
ejde-624	493	50	(	(	PUNCT
ejde-624	493	51	0	0	NUM
ejde-624	493	52	,	,	PUNCT
ejde-624	493	53	t	t	X
ejde-624	493	54	]	]	PUNCT
ejde-624	493	55	,	,	PUNCT
ejde-624	493	56	v	v	X
ejde-624	493	57	satisfies	satisfy	VERB
ejde-624	493	58	the	the	DET
ejde-624	493	59	equation	equation	NOUN
ejde-624	493	60	v(t	v(t	NOUN
ejde-624	493	61	)	)	PUNCT
ejde-624	493	62	=	=	SYM
ejde-624	493	63	v0	v0	NOUN
ejde-624	493	64	+	+	CCONJ
ejde-624	493	65	1	1	NUM
ejde-624	493	66	γ(α	γ(α	NOUN
ejde-624	493	67	)	)	PUNCT
ejde-624	494	1	∫	∫	PROPN
ejde-624	494	2	t	t	PROPN
ejde-624	494	3	0	0	NUM
ejde-624	495	1	(	(	PUNCT
ejde-624	495	2	t−	t−	PROPN
ejde-624	495	3	s)α−1sβvp(s	s)α−1sβvp(s	NOUN
ejde-624	495	4	)	)	PUNCT
ejde-624	495	5	ds+	ds+	NOUN
ejde-624	495	6	1	1	NUM
ejde-624	495	7	γ(α	γ(α	NOUN
ejde-624	495	8	)	)	PUNCT
ejde-624	496	1	∫	∫	PROPN
ejde-624	496	2	t	t	PROPN
ejde-624	496	3	0	0	NUM
ejde-624	497	1	(	(	PUNCT
ejde-624	497	2	t−	t−	PROPN
ejde-624	497	3	s)α−1cf(s	s)α−1cf(s	PROPN
ejde-624	497	4	)	)	PUNCT
ejde-624	497	5	ds	ds	NOUN
ejde-624	497	6	,	,	PUNCT
ejde-624	497	7	=	=	SYM
ejde-624	497	8	v0	v0	NOUN
ejde-624	497	9	+	+	CCONJ
ejde-624	497	10	1	1	NUM
ejde-624	497	11	γ(α	γ(α	NOUN
ejde-624	497	12	)	)	PUNCT
ejde-624	498	1	tα−1	tα−1	NOUN
ejde-624	498	2	∫	∫	PROPN
ejde-624	498	3	t	t	PROPN
ejde-624	498	4	0	0	NUM
ejde-624	499	1	(	(	PUNCT
ejde-624	499	2	1−	1−	NUM
ejde-624	499	3	s	s	PROPN
ejde-624	499	4	/	/	SYM
ejde-624	499	5	t)α−1sβvp(s	t)α−1sβvp(s	NOUN
ejde-624	499	6	)	)	PUNCT
ejde-624	499	7	ds	ds	ADJ
ejde-624	499	8	+	+	CCONJ
ejde-624	499	9	1	1	NUM
ejde-624	499	10	γ(α	γ(α	NOUN
ejde-624	499	11	)	)	PUNCT
ejde-624	499	12	∫	∫	PROPN
ejde-624	499	13	t	t	PROPN
ejde-624	499	14	0	0	NUM
ejde-624	500	1	(	(	PUNCT
ejde-624	500	2	t−	t−	PROPN
ejde-624	500	3	s)α−1cf(s	s)α−1cf(s	PROPN
ejde-624	500	4	)	)	PUNCT
ejde-624	500	5	ds	ds	NOUN
ejde-624	500	6	.	.	PUNCT
ejde-624	500	7	(	(	PUNCT
ejde-624	500	8	5.8	5.8	NUM
ejde-624	500	9	)	)	PUNCT
ejde-624	500	10	the	the	DET
ejde-624	500	11	last	last	ADJ
ejde-624	500	12	term	term	NOUN
ejde-624	500	13	can	can	AUX
ejde-624	500	14	be	be	AUX
ejde-624	500	15	discarded	discard	VERB
ejde-624	500	16	and	and	CCONJ
ejde-624	500	17	we	we	PRON
ejde-624	500	18	obtain	obtain	VERB
ejde-624	500	19	t1−αv(t	t1−αv(t	NUM
ejde-624	500	20	)	)	PUNCT
ejde-624	500	21	≥	≥	NOUN
ejde-624	501	1	t1−αv0	t1−αv0	ADJ
ejde-624	501	2	+	+	CCONJ
ejde-624	501	3	1	1	NUM
ejde-624	501	4	γ(α	γ(α	NOUN
ejde-624	501	5	)	)	PUNCT
ejde-624	501	6	∫	∫	PROPN
ejde-624	502	1	t	t	NOUN
ejde-624	502	2	0	0	NUM
ejde-624	503	1	sβ−p(1−α)(s1−αv(s))p	sβ−p(1−α)(s1−αv(s))p	ADJ
ejde-624	503	2	ds	ds	PROPN
ejde-624	503	3	,	,	PUNCT
ejde-624	503	4	a.e	a.e	PROPN
ejde-624	503	5	.	.	PROPN
ejde-624	503	6	t.	t.	PROPN
ejde-624	503	7	let	let	VERB
ejde-624	503	8	w(t	w(t	PROPN
ejde-624	503	9	)	)	PUNCT
ejde-624	503	10	:	:	PUNCT
ejde-624	504	1	=	=	SYM
ejde-624	504	2	t1−αv(t	t1−αv(t	NOUN
ejde-624	504	3	)	)	PUNCT
ejde-624	504	4	.	.	PUNCT
ejde-624	505	1	then	then	ADV
ejde-624	505	2	w	w	PROPN
ejde-624	505	3	is	be	AUX
ejde-624	505	4	continuous	continuous	ADJ
ejde-624	505	5	and	and	CCONJ
ejde-624	505	6	satisfies	satisfie	NOUN
ejde-624	505	7	w(t	w(t	PROPN
ejde-624	505	8	)	)	PUNCT
ejde-624	505	9	≥	≥	NOUN
ejde-624	506	1	t1−αv0	t1−αv0	ADJ
ejde-624	506	2	+	+	CCONJ
ejde-624	506	3	1	1	NUM
ejde-624	506	4	γ(α	γ(α	NOUN
ejde-624	506	5	)	)	PUNCT
ejde-624	506	6	∫	∫	PROPN
ejde-624	506	7	t	t	NOUN
ejde-624	506	8	0	0	NUM
ejde-624	506	9	sβ−p(1−α)wp(s	sβ−p(1−α)wp(s	ADJ
ejde-624	506	10	)	)	PUNCT
ejde-624	506	11	ds	ds	NOUN
ejde-624	506	12	.	.	PUNCT
ejde-624	506	13	(	(	PUNCT
ejde-624	506	14	5.9	5.9	NUM
ejde-624	506	15	)	)	PUNCT
ejde-624	506	16	for	for	ADP
ejde-624	506	17	t	t	PROPN
ejde-624	506	18	∈	∈	PROPN
ejde-624	507	1	[	[	X
ejde-624	507	2	1	1	NUM
ejde-624	507	3	,	,	PUNCT
ejde-624	507	4	t	t	X
ejde-624	507	5	]	]	PUNCT
ejde-624	507	6	we	we	PRON
ejde-624	507	7	have	have	AUX
ejde-624	507	8	w(t	w(t	PROPN
ejde-624	507	9	)	)	PUNCT
ejde-624	507	10	≥	≥	NOUN
ejde-624	507	11	v0	v0	NOUN
ejde-624	507	12	+	+	CCONJ
ejde-624	507	13	1	1	NUM
ejde-624	507	14	γ(α	γ(α	NOUN
ejde-624	507	15	)	)	PUNCT
ejde-624	507	16	∫	∫	PROPN
ejde-624	507	17	t	t	PROPN
ejde-624	507	18	1	1	NUM
ejde-624	507	19	sβ−p(1−α)wp(s	sβ−p(1−α)wp(s	NOUN
ejde-624	507	20	)	)	PUNCT
ejde-624	507	21	ds	ds	NOUN
ejde-624	507	22	.	.	PUNCT
ejde-624	507	23	(	(	PUNCT
ejde-624	507	24	5.10	5.10	NUM
ejde-624	507	25	)	)	PUNCT
ejde-624	507	26	14	14	NUM
ejde-624	507	27	j.	j.	PROPN
ejde-624	507	28	r.	r.	PROPN
ejde-624	507	29	l.	l.	PROPN
ejde-624	507	30	webb	webb	PROPN
ejde-624	507	31	ejde-2024/40	ejde-2024/40	PROPN
ejde-624	507	32	let	let	VERB
ejde-624	507	33	γ	γ	X
ejde-624	507	34	=	=	VERB
ejde-624	507	35	p(1	p(1	PROPN
ejde-624	507	36	−	−	NOUN
ejde-624	507	37	α	α	NOUN
ejde-624	507	38	)	)	PUNCT
ejde-624	507	39	−	−	ADP
ejde-624	507	40	β	β	NOUN
ejde-624	507	41	,	,	PUNCT
ejde-624	507	42	then	then	ADV
ejde-624	507	43	1	1	NUM
ejde-624	507	44	<	<	X
ejde-624	507	45	p	p	X
ejde-624	507	46	≤	≤	NOUN
ejde-624	507	47	1+β	1+β	NUM
ejde-624	507	48	1−α	1−α	NUM
ejde-624	507	49	implies	imply	VERB
ejde-624	507	50	that	that	SCONJ
ejde-624	507	51	γ	γ	PROPN
ejde-624	507	52	≤	≤	NOUN
ejde-624	507	53	1	1	NUM
ejde-624	507	54	and	and	CCONJ
ejde-624	507	55	(	(	PUNCT
ejde-624	507	56	5.10	5.10	NUM
ejde-624	507	57	)	)	PUNCT
ejde-624	507	58	can	can	AUX
ejde-624	507	59	be	be	AUX
ejde-624	507	60	written	write	VERB
ejde-624	507	61	w(t	w(t	PROPN
ejde-624	507	62	)	)	PUNCT
ejde-624	507	63	≥	≥	NOUN
ejde-624	507	64	v0	v0	NOUN
ejde-624	507	65	+	+	CCONJ
ejde-624	507	66	1	1	NUM
ejde-624	507	67	γ(α	γ(α	NOUN
ejde-624	507	68	)	)	PUNCT
ejde-624	508	1	∫	∫	PROPN
ejde-624	508	2	t	t	PROPN
ejde-624	509	1	1	1	NUM
ejde-624	509	2	s−γwp(s	s−γwp(s	NUM
ejde-624	509	3	)	)	PUNCT
ejde-624	509	4	ds	ds	PROPN
ejde-624	509	5	.	.	NOUN
ejde-624	509	6	let	let	VERB
ejde-624	509	7	g(t	g(t	PROPN
ejde-624	509	8	)	)	PUNCT
ejde-624	510	1	=	=	SYM
ejde-624	510	2	v0	v0	NOUN
ejde-624	510	3	+	+	CCONJ
ejde-624	510	4	1	1	NUM
ejde-624	510	5	γ(α	γ(α	NOUN
ejde-624	510	6	)	)	PUNCT
ejde-624	510	7	∫	∫	PROPN
ejde-624	510	8	t	t	PROPN
ejde-624	510	9	1	1	NUM
ejde-624	510	10	s−γwp(s	s−γwp(s	NUM
ejde-624	510	11	)	)	PUNCT
ejde-624	510	12	ds	ds	NOUN
ejde-624	510	13	for	for	ADP
ejde-624	510	14	t	t	PROPN
ejde-624	510	15	∈	∈	PROPN
ejde-624	511	1	[	[	X
ejde-624	511	2	1	1	NUM
ejde-624	511	3	,	,	PUNCT
ejde-624	511	4	t	t	X
ejde-624	511	5	]	]	PUNCT
ejde-624	511	6	.	.	PUNCT
ejde-624	512	1	now	now	ADV
ejde-624	512	2	terms	term	NOUN
ejde-624	512	3	are	be	AUX
ejde-624	512	4	continuous	continuous	ADJ
ejde-624	512	5	so	so	SCONJ
ejde-624	512	6	g	g	PROPN
ejde-624	512	7	∈	∈	PROPN
ejde-624	512	8	ac	ac	PROPN
ejde-624	512	9	,	,	PUNCT
ejde-624	512	10	g(1	g(1	NOUN
ejde-624	512	11	)	)	PUNCT
ejde-624	512	12	=	=	SYM
ejde-624	512	13	v0	v0	PROPN
ejde-624	512	14	,	,	PUNCT
ejde-624	512	15	g(t	g(t	PROPN
ejde-624	512	16	)	)	PUNCT
ejde-624	512	17	≥	≥	PROPN
ejde-624	512	18	v0	v0	X
ejde-624	512	19	>	>	X
ejde-624	512	20	0	0	PUNCT
ejde-624	513	1	for	for	ADP
ejde-624	513	2	all	all	DET
ejde-624	513	3	t	t	PROPN
ejde-624	513	4	≥	≥	NOUN
ejde-624	513	5	1	1	NUM
ejde-624	513	6	and	and	CCONJ
ejde-624	513	7	g′(t	g′(t	PROPN
ejde-624	513	8	)	)	PUNCT
ejde-624	513	9	=	=	SYM
ejde-624	513	10	1	1	NUM
ejde-624	513	11	γ(α	γ(α	NOUN
ejde-624	513	12	)	)	PUNCT
ejde-624	513	13	t−γwp(t	t−γwp(t	NUM
ejde-624	513	14	)	)	PUNCT
ejde-624	513	15	≥	≥	NOUN
ejde-624	513	16	1	1	NUM
ejde-624	513	17	γ(α	γ(α	NOUN
ejde-624	513	18	)	)	PUNCT
ejde-624	513	19	t−γgp(t	t−γgp(t	NUM
ejde-624	513	20	)	)	PUNCT
ejde-624	513	21	,	,	PUNCT
ejde-624	513	22	so	so	ADV
ejde-624	513	23	g′	g′	NOUN
ejde-624	513	24	gp	gp	PROPN
ejde-624	513	25	≥	≥	PROPN
ejde-624	513	26	t−γ	t−γ	PROPN
ejde-624	513	27	γ(α	γ(α	PROPN
ejde-624	513	28	)	)	PUNCT
ejde-624	513	29	,	,	PUNCT
ejde-624	513	30	t	t	PROPN
ejde-624	513	31	∈	∈	PROPN
ejde-624	514	1	[	[	X
ejde-624	514	2	1	1	NUM
ejde-624	514	3	,	,	PUNCT
ejde-624	514	4	t	t	X
ejde-624	514	5	]	]	PUNCT
ejde-624	514	6	.	.	PUNCT
ejde-624	515	1	we	we	PRON
ejde-624	515	2	can	can	AUX
ejde-624	515	3	integrate	integrate	VERB
ejde-624	515	4	from	from	ADP
ejde-624	515	5	1	1	NUM
ejde-624	515	6	to	to	ADP
ejde-624	515	7	t	t	PROPN
ejde-624	515	8	≤	≤	X
ejde-624	515	9	t	t	PROPN
ejde-624	515	10	to	to	PART
ejde-624	515	11	obtain	obtain	VERB
ejde-624	515	12	g1−p(t	g1−p(t	NOUN
ejde-624	515	13	)	)	PUNCT
ejde-624	515	14	≤	≤	PUNCT
ejde-624	516	1	v1−p	v1−p	PROPN
ejde-624	516	2	0	0	NUM
ejde-624	517	1	−	−	PROPN
ejde-624	517	2	(	(	PUNCT
ejde-624	517	3	p−	p−	NOUN
ejde-624	517	4	1	1	NUM
ejde-624	517	5	)	)	PUNCT
ejde-624	517	6	γ(α	γ(α	PROPN
ejde-624	517	7	)	)	PUNCT
ejde-624	518	1	(	(	PUNCT
ejde-624	518	2	t1−γ	t1−γ	PROPN
ejde-624	518	3	−	−	PROPN
ejde-624	518	4	1	1	NUM
ejde-624	518	5	)	)	PUNCT
ejde-624	518	6	1−	1−	NUM
ejde-624	518	7	γ	γ	NOUN
ejde-624	518	8	,	,	PUNCT
ejde-624	518	9	for	for	ADP
ejde-624	518	10	γ	γ	X
ejde-624	518	11	<	<	X
ejde-624	518	12	1	1	NUM
ejde-624	518	13	,	,	PUNCT
ejde-624	518	14	g1−p(t	g1−p(t	NOUN
ejde-624	518	15	)	)	PUNCT
ejde-624	518	16	≤	≤	PUNCT
ejde-624	518	17	v1−p	v1−p	VERB
ejde-624	518	18	0	0	NUM
ejde-624	519	1	−	−	PROPN
ejde-624	520	1	(	(	PUNCT
ejde-624	520	2	p−	p−	NOUN
ejde-624	520	3	1	1	NUM
ejde-624	520	4	)	)	PUNCT
ejde-624	520	5	γ(α	γ(α	PROPN
ejde-624	520	6	)	)	PUNCT
ejde-624	520	7	ln	ln	PROPN
ejde-624	520	8	t	t	PROPN
ejde-624	520	9	,	,	PUNCT
ejde-624	520	10	for	for	ADP
ejde-624	520	11	γ	γ	X
ejde-624	520	12	=	=	SYM
ejde-624	520	13	1	1	NUM
ejde-624	520	14	.	.	PUNCT
ejde-624	521	1	(	(	PUNCT
ejde-624	521	2	5.11	5.11	NUM
ejde-624	521	3	)	)	PUNCT
ejde-624	521	4	it	it	PRON
ejde-624	521	5	is	be	AUX
ejde-624	521	6	clear	clear	ADJ
ejde-624	521	7	that	that	SCONJ
ejde-624	521	8	there	there	PRON
ejde-624	521	9	exists	exist	VERB
ejde-624	521	10	t1	t1	NOUN
ejde-624	521	11	>	>	X
ejde-624	521	12	1	1	NUM
ejde-624	521	13	(	(	PUNCT
ejde-624	521	14	it	it	PRON
ejde-624	521	15	can	can	AUX
ejde-624	521	16	be	be	AUX
ejde-624	521	17	written	write	VERB
ejde-624	521	18	explicitly	explicitly	ADV
ejde-624	521	19	)	)	PUNCT
ejde-624	521	20	such	such	ADJ
ejde-624	521	21	that	that	SCONJ
ejde-624	521	22	the	the	DET
ejde-624	521	23	terms	term	NOUN
ejde-624	521	24	on	on	ADP
ejde-624	521	25	the	the	DET
ejde-624	521	26	right	right	NOUN
ejde-624	521	27	become	become	VERB
ejde-624	521	28	zero	zero	NUM
ejde-624	521	29	,	,	PUNCT
ejde-624	521	30	hence	hence	ADV
ejde-624	521	31	g1−p(t1	g1−p(t1	PROPN
ejde-624	521	32	)	)	PUNCT
ejde-624	521	33	does	do	AUX
ejde-624	521	34	not	not	PART
ejde-624	521	35	exist	exist	VERB
ejde-624	521	36	,	,	PUNCT
ejde-624	521	37	thus	thus	ADV
ejde-624	521	38	u	u	PRON
ejde-624	521	39	can	can	AUX
ejde-624	521	40	only	only	ADV
ejde-624	521	41	exist	exist	VERB
ejde-624	521	42	on	on	ADP
ejde-624	521	43	an	an	DET
ejde-624	521	44	interval	interval	NOUN
ejde-624	521	45	[	[	X
ejde-624	521	46	0	0	NUM
ejde-624	521	47	,	,	PUNCT
ejde-624	521	48	t	t	X
ejde-624	521	49	]	]	PUNCT
ejde-624	521	50	with	with	ADP
ejde-624	521	51	t	t	PROPN
ejde-624	521	52	<	<	X
ejde-624	521	53	t1	t1	NOUN
ejde-624	521	54	.	.	PUNCT
ejde-624	522	1	□	□	PUNCT
ejde-624	522	2	corollary	corollary	ADJ
ejde-624	522	3	5.4	5.4	NUM
ejde-624	522	4	.	.	PUNCT
ejde-624	523	1	let	let	VERB
ejde-624	523	2	1	1	NUM
ejde-624	523	3	<	<	X
ejde-624	523	4	p	p	X
ejde-624	523	5	≤	≤	NOUN
ejde-624	523	6	1+β	1+β	NUM
ejde-624	523	7	1−α	1−α	NUM
ejde-624	523	8	,	,	PUNCT
ejde-624	523	9	λ	λ	INTJ
ejde-624	523	10	>	>	X
ejde-624	523	11	0	0	PUNCT
ejde-624	524	1	and	and	CCONJ
ejde-624	524	2	α	α	PRON
ejde-624	525	1	+	+	X
ejde-624	525	2	β	β	X
ejde-624	525	3	>	>	X
ejde-624	525	4	0	0	X
ejde-624	525	5	.	.	PUNCT
ejde-624	526	1	the	the	DET
ejde-624	526	2	problem	problem	NOUN
ejde-624	526	3	dα	dα	ADP
ejde-624	526	4	∗	∗	NOUN
ejde-624	526	5	u(t	u(t	PROPN
ejde-624	526	6	)	)	PUNCT
ejde-624	526	7	≥	≥	NOUN
ejde-624	526	8	λtβup(t	λtβup(t	NOUN
ejde-624	526	9	)	)	PUNCT
ejde-624	526	10	,	,	PUNCT
ejde-624	526	11	t	t	X
ejde-624	526	12	>	>	X
ejde-624	526	13	0	0	NUM
ejde-624	526	14	,	,	PUNCT
ejde-624	526	15	with	with	ADP
ejde-624	526	16	u(0	u(0	NOUN
ejde-624	526	17	)	)	PUNCT
ejde-624	526	18	=	=	PUNCT
ejde-624	526	19	u0	u0	VERB
ejde-624	526	20	>	>	X
ejde-624	526	21	0	0	PROPN
ejde-624	526	22	,	,	PUNCT
ejde-624	526	23	does	do	AUX
ejde-624	526	24	not	not	PART
ejde-624	526	25	have	have	VERB
ejde-624	526	26	a	a	DET
ejde-624	526	27	global	global	ADJ
ejde-624	526	28	solution	solution	NOUN
ejde-624	526	29	.	.	PUNCT
ejde-624	527	1	proof	proof	NOUN
ejde-624	527	2	.	.	PUNCT
ejde-624	528	1	the	the	DET
ejde-624	528	2	proof	proof	NOUN
ejde-624	528	3	of	of	ADP
ejde-624	528	4	theorem	theorem	ADJ
ejde-624	528	5	5.3	5.3	NUM
ejde-624	528	6	applies	apply	VERB
ejde-624	528	7	since	since	SCONJ
ejde-624	528	8	for	for	ADP
ejde-624	528	9	f(t	f(t	NOUN
ejde-624	528	10	)	)	PUNCT
ejde-624	529	1	=	=	PUNCT
ejde-624	529	2	dα	dα	ADP
ejde-624	529	3	∗	∗	NOUN
ejde-624	529	4	u(t)−λtβup(t	u(t)−λtβup(t	NOUN
ejde-624	529	5	)	)	PUNCT
ejde-624	529	6	,	,	PUNCT
ejde-624	529	7	f	f	PROPN
ejde-624	529	8	∈	∈	PROPN
ejde-624	529	9	l1	l1	PROPN
ejde-624	529	10	and	and	CCONJ
ejde-624	529	11	f(t	f(t	PROPN
ejde-624	529	12	)	)	PUNCT
ejde-624	529	13	≥	≥	NOUN
ejde-624	529	14	0	0	NUM
ejde-624	529	15	for	for	ADP
ejde-624	529	16	t	t	PROPN
ejde-624	529	17	>	>	X
ejde-624	529	18	0	0	NUM
ejde-624	529	19	.	.	PUNCT
ejde-624	530	1	□	□	PUNCT
ejde-624	530	2	we	we	PRON
ejde-624	530	3	now	now	ADV
ejde-624	530	4	can	can	AUX
ejde-624	530	5	deal	deal	VERB
ejde-624	530	6	with	with	ADP
ejde-624	530	7	the	the	DET
ejde-624	530	8	case	case	NOUN
ejde-624	530	9	p	p	X
ejde-624	530	10	>	>	X
ejde-624	530	11	1+β	1+β	NUM
ejde-624	530	12	1−α	1−α	NUM
ejde-624	530	13	very	very	ADV
ejde-624	530	14	simply	simply	ADV
ejde-624	530	15	and	and	CCONJ
ejde-624	530	16	it	it	PRON
ejde-624	530	17	gives	give	VERB
ejde-624	530	18	the	the	DET
ejde-624	530	19	following	follow	VERB
ejde-624	530	20	result	result	NOUN
ejde-624	530	21	.	.	PUNCT
ejde-624	531	1	theorem	theorem	VERB
ejde-624	531	2	5.5	5.5	NUM
ejde-624	531	3	.	.	PUNCT
ejde-624	532	1	let	let	VERB
ejde-624	532	2	λ	λ	PRON
ejde-624	532	3	>	>	X
ejde-624	532	4	0	0	PUNCT
ejde-624	532	5	and	and	CCONJ
ejde-624	532	6	α+β	α+β	NUM
ejde-624	532	7	>	>	X
ejde-624	532	8	0	0	X
ejde-624	532	9	.	.	PUNCT
ejde-624	533	1	for	for	ADP
ejde-624	533	2	any	any	DET
ejde-624	533	3	p	p	NOUN
ejde-624	533	4	>	>	X
ejde-624	533	5	1	1	NUM
ejde-624	533	6	and	and	CCONJ
ejde-624	533	7	any	any	DET
ejde-624	533	8	u0	u0	NOUN
ejde-624	533	9	>	>	X
ejde-624	533	10	0	0	PUNCT
ejde-624	534	1	there	there	PRON
ejde-624	534	2	does	do	AUX
ejde-624	534	3	not	not	PART
ejde-624	534	4	exist	exist	VERB
ejde-624	534	5	a	a	DET
ejde-624	534	6	global	global	ADJ
ejde-624	534	7	solution	solution	NOUN
ejde-624	534	8	u	u	NOUN
ejde-624	534	9	of	of	ADP
ejde-624	534	10	the	the	DET
ejde-624	534	11	problem	problem	NOUN
ejde-624	534	12	dα	dα	ADP
ejde-624	534	13	∗	∗	NOUN
ejde-624	534	14	u(t	u(t	PROPN
ejde-624	534	15	)	)	PUNCT
ejde-624	534	16	≥	≥	NOUN
ejde-624	534	17	λtβup(t	λtβup(t	NOUN
ejde-624	534	18	)	)	PUNCT
ejde-624	534	19	,	,	PUNCT
ejde-624	534	20	t	t	X
ejde-624	534	21	>	>	X
ejde-624	534	22	0,with	0,with	ADP
ejde-624	534	23	u(0	u(0	NOUN
ejde-624	534	24	)	)	PUNCT
ejde-624	534	25	=	=	PUNCT
ejde-624	534	26	u0	u0	VERB
ejde-624	534	27	>	>	X
ejde-624	534	28	0	0	NUM
ejde-624	534	29	.	.	PUNCT
ejde-624	535	1	(	(	PUNCT
ejde-624	535	2	5.12	5.12	NUM
ejde-624	535	3	)	)	PUNCT
ejde-624	535	4	proof	proof	NOUN
ejde-624	535	5	.	.	PUNCT
ejde-624	536	1	a	a	DET
ejde-624	536	2	solution	solution	NOUN
ejde-624	536	3	u	u	NOUN
ejde-624	536	4	of	of	ADP
ejde-624	536	5	(	(	PUNCT
ejde-624	536	6	5.12	5.12	NUM
ejde-624	536	7	)	)	PUNCT
ejde-624	536	8	will	will	AUX
ejde-624	536	9	satisfy	satisfy	VERB
ejde-624	536	10	u(t	u(t	PROPN
ejde-624	536	11	)	)	PUNCT
ejde-624	536	12	≥	≥	NOUN
ejde-624	536	13	u0	u0	VERB
ejde-624	536	14	>	>	X
ejde-624	536	15	0	0	PUNCT
ejde-624	537	1	on	on	ADP
ejde-624	537	2	its	its	PRON
ejde-624	537	3	interval	interval	NOUN
ejde-624	537	4	of	of	ADP
ejde-624	537	5	existence	existence	NOUN
ejde-624	537	6	.	.	PUNCT
ejde-624	538	1	for	for	ADP
ejde-624	538	2	1	1	NUM
ejde-624	538	3	<	<	X
ejde-624	538	4	p	p	X
ejde-624	538	5	≤	≤	NOUN
ejde-624	538	6	1+β	1+β	NUM
ejde-624	538	7	1−α	1−α	NUM
ejde-624	538	8	the	the	DET
ejde-624	538	9	result	result	NOUN
ejde-624	538	10	is	be	AUX
ejde-624	538	11	shown	show	VERB
ejde-624	538	12	in	in	ADP
ejde-624	538	13	corollary	corollary	ADJ
ejde-624	538	14	5.4	5.4	NUM
ejde-624	538	15	,	,	PUNCT
ejde-624	538	16	so	so	ADV
ejde-624	538	17	suppose	suppose	VERB
ejde-624	538	18	that	that	SCONJ
ejde-624	538	19	p	p	PROPN
ejde-624	538	20	>	>	X
ejde-624	538	21	1+β	1+β	NUM
ejde-624	538	22	1−α	1−α	NUM
ejde-624	538	23	.	.	PUNCT
ejde-624	539	1	write	write	VERB
ejde-624	539	2	p	p	NOUN
ejde-624	539	3	=	=	PROPN
ejde-624	539	4	p0	p0	NOUN
ejde-624	539	5	+	+	CCONJ
ejde-624	539	6	p1	p1	PROPN
ejde-624	539	7	where	where	SCONJ
ejde-624	539	8	p0	p0	NOUN
ejde-624	539	9	=	=	PUNCT
ejde-624	539	10	1+β	1+β	PROPN
ejde-624	539	11	1−α	1−α	NUM
ejde-624	539	12	.	.	PUNCT
ejde-624	540	1	then	then	ADV
ejde-624	540	2	up	up	ADV
ejde-624	540	3	=	=	PUNCT
ejde-624	540	4	up1u0	up1u0	PROPN
ejde-624	540	5	≥	≥	PUNCT
ejde-624	540	6	up1	up1	X
ejde-624	540	7	0	0	PUNCT
ejde-624	540	8	up0	up0	NUM
ejde-624	540	9	and	and	CCONJ
ejde-624	540	10	from	from	ADP
ejde-624	540	11	(	(	PUNCT
ejde-624	540	12	5.12	5.12	NUM
ejde-624	540	13	)	)	PUNCT
ejde-624	540	14	we	we	PRON
ejde-624	540	15	obtain	obtain	VERB
ejde-624	540	16	dα	dα	ADP
ejde-624	540	17	∗	∗	NOUN
ejde-624	540	18	u(t	u(t	NOUN
ejde-624	540	19	)	)	PUNCT
ejde-624	540	20	≥	≥	NOUN
ejde-624	540	21	(	(	PUNCT
ejde-624	540	22	up1	up1	PROPN
ejde-624	540	23	0	0	NUM
ejde-624	540	24	λ)tβup0(t	λ)tβup0(t	PROPN
ejde-624	540	25	)	)	PUNCT
ejde-624	540	26	,	,	PUNCT
ejde-624	540	27	with	with	ADP
ejde-624	540	28	ic	ic	PROPN
ejde-624	540	29	u(0	u(0	PROPN
ejde-624	540	30	)	)	PUNCT
ejde-624	540	31	=	=	PUNCT
ejde-624	540	32	u0	u0	VERB
ejde-624	540	33	>	>	X
ejde-624	540	34	0	0	X
ejde-624	540	35	.	.	PUNCT
ejde-624	541	1	by	by	ADP
ejde-624	541	2	corollary	corollary	ADJ
ejde-624	541	3	5.4	5.4	NUM
ejde-624	541	4	,	,	PUNCT
ejde-624	541	5	u	u	NOUN
ejde-624	541	6	can	can	AUX
ejde-624	541	7	only	only	ADV
ejde-624	541	8	exist	exist	VERB
ejde-624	541	9	on	on	ADP
ejde-624	541	10	some	some	DET
ejde-624	541	11	interval	interval	NOUN
ejde-624	541	12	[	[	X
ejde-624	541	13	0	0	NUM
ejde-624	541	14	,	,	PUNCT
ejde-624	541	15	t	t	NOUN
ejde-624	541	16	]	]	PUNCT
ejde-624	541	17	where	where	SCONJ
ejde-624	541	18	t	t	NOUN
ejde-624	541	19	<	<	X
ejde-624	542	1	t̂1	t̂1	PROPN
ejde-624	543	1	and	and	CCONJ
ejde-624	543	2	t̂1	t̂1	PROPN
ejde-624	543	3	is	be	AUX
ejde-624	543	4	determined	determine	VERB
ejde-624	543	5	by	by	ADP
ejde-624	543	6	the	the	DET
ejde-624	543	7	parameters	parameter	NOUN
ejde-624	543	8	of	of	ADP
ejde-624	543	9	the	the	DET
ejde-624	543	10	problem	problem	NOUN
ejde-624	543	11	.	.	PUNCT
ejde-624	544	1	□	□	PUNCT
ejde-624	544	2	remark	remark	NOUN
ejde-624	544	3	5.6	5.6	NUM
ejde-624	544	4	.	.	PUNCT
ejde-624	545	1	when	when	SCONJ
ejde-624	545	2	α	α	PROPN
ejde-624	545	3	+	+	X
ejde-624	545	4	β	β	X
ejde-624	545	5	<	<	X
ejde-624	545	6	0	0	PROPN
ejde-624	545	7	,	,	PUNCT
ejde-624	545	8	the	the	DET
ejde-624	545	9	opposite	opposite	ADJ
ejde-624	545	10	sign	sign	NOUN
ejde-624	545	11	to	to	ADP
ejde-624	545	12	the	the	DET
ejde-624	545	13	one	one	NOUN
ejde-624	545	14	we	we	PRON
ejde-624	545	15	have	have	AUX
ejde-624	545	16	considered	consider	VERB
ejde-624	545	17	,	,	PUNCT
ejde-624	545	18	and	and	CCONJ
ejde-624	545	19	p	p	X
ejde-624	545	20	>	>	X
ejde-624	545	21	1	1	NUM
ejde-624	545	22	,	,	PUNCT
ejde-624	545	23	there	there	PRON
ejde-624	545	24	exists	exist	VERB
ejde-624	545	25	a	a	DET
ejde-624	545	26	continuous	continuous	ADJ
ejde-624	545	27	solution	solution	NOUN
ejde-624	545	28	of	of	ADP
ejde-624	545	29	the	the	DET
ejde-624	545	30	form	form	NOUN
ejde-624	545	31	ctr	ctr	PROPN
ejde-624	545	32	with	with	ADP
ejde-624	545	33	c	c	PROPN
ejde-624	545	34	>	>	PUNCT
ejde-624	545	35	0	0	NUM
ejde-624	545	36	for	for	ADP
ejde-624	545	37	the	the	DET
ejde-624	545	38	caputo	caputo	PROPN
ejde-624	545	39	problem	problem	NOUN
ejde-624	545	40	with	with	ADP
ejde-624	545	41	initial	initial	ADJ
ejde-624	545	42	data	datum	NOUN
ejde-624	545	43	0	0	NUM
ejde-624	545	44	dα	dα	PROPN
ejde-624	545	45	∗	∗	NOUN
ejde-624	545	46	u(t	u(t	NOUN
ejde-624	545	47	)	)	PUNCT
ejde-624	546	1	=	=	SYM
ejde-624	546	2	λtβup(t	λtβup(t	NOUN
ejde-624	546	3	)	)	PUNCT
ejde-624	546	4	,	,	PUNCT
ejde-624	546	5	u(0	u(0	PROPN
ejde-624	546	6	)	)	PUNCT
ejde-624	546	7	=	=	PUNCT
ejde-624	546	8	u0	u0	ADJ
ejde-624	546	9	=	=	NOUN
ejde-624	546	10	0	0	PROPN
ejde-624	546	11	.	.	PUNCT
ejde-624	547	1	the	the	DET
ejde-624	547	2	calculation	calculation	NOUN
ejde-624	547	3	is	be	AUX
ejde-624	547	4	given	give	VERB
ejde-624	547	5	above	above	ADV
ejde-624	547	6	in	in	ADP
ejde-624	547	7	example	example	NOUN
ejde-624	547	8	4.6	4.6	NUM
ejde-624	547	9	.	.	PUNCT
ejde-624	548	1	the	the	DET
ejde-624	548	2	solution	solution	NOUN
ejde-624	548	3	is	be	AUX
ejde-624	548	4	ctr	ctr	PROPN
ejde-624	548	5	for	for	ADP
ejde-624	548	6	r	r	NOUN
ejde-624	548	7	=	=	PUNCT
ejde-624	548	8	−α−β	−α−β	PROPN
ejde-624	548	9	p−1	p−1	PROPN
ejde-624	548	10	.	.	PUNCT
ejde-624	549	1	this	this	DET
ejde-624	549	2	nontrivial	nontrivial	ADJ
ejde-624	549	3	continuous	continuous	ADJ
ejde-624	549	4	solution	solution	NOUN
ejde-624	549	5	exists	exist	VERB
ejde-624	549	6	if	if	SCONJ
ejde-624	549	7	α+	α+	ADP
ejde-624	549	8	β	β	X
ejde-624	549	9	<	<	X
ejde-624	549	10	0	0	NUM
ejde-624	549	11	for	for	ADP
ejde-624	549	12	any	any	DET
ejde-624	549	13	p	p	NOUN
ejde-624	549	14	>	>	X
ejde-624	549	15	1	1	NUM
ejde-624	549	16	.	.	PUNCT
ejde-624	549	17	ejde-2024/40	ejde-2024/40	VERB
ejde-624	549	18	fractional	fractional	ADJ
ejde-624	549	19	differential	differential	ADJ
ejde-624	549	20	inequalities	inequality	NOUN
ejde-624	549	21	15	15	NUM
ejde-624	549	22	remark	remark	NOUN
ejde-624	549	23	5.7	5.7	NUM
ejde-624	549	24	.	.	PUNCT
ejde-624	550	1	in	in	ADP
ejde-624	550	2	the	the	DET
ejde-624	550	3	paper	paper	NOUN
ejde-624	550	4	by	by	ADP
ejde-624	550	5	shan	shan	PROPN
ejde-624	550	6	and	and	CCONJ
ejde-624	550	7	lv	lv	PROPN
ejde-624	550	8	[	[	X
ejde-624	550	9	18	18	NUM
ejde-624	550	10	]	]	PUNCT
ejde-624	550	11	,	,	PUNCT
ejde-624	550	12	who	who	PRON
ejde-624	550	13	have	have	AUX
ejde-624	550	14	the	the	DET
ejde-624	550	15	special	special	ADJ
ejde-624	550	16	case	case	NOUN
ejde-624	550	17	β	β	X
ejde-624	550	18	=	=	SYM
ejde-624	550	19	0	0	NUM
ejde-624	550	20	,	,	PUNCT
ejde-624	550	21	at	at	ADP
ejde-624	550	22	the	the	DET
ejde-624	550	23	corresponding	corresponding	ADJ
ejde-624	550	24	stage	stage	NOUN
ejde-624	550	25	(	(	PUNCT
ejde-624	550	26	5.8	5.8	NUM
ejde-624	550	27	)	)	PUNCT
ejde-624	550	28	of	of	ADP
ejde-624	550	29	our	our	PRON
ejde-624	550	30	proof	proof	NOUN
ejde-624	550	31	of	of	ADP
ejde-624	550	32	theorem	theorem	ADJ
ejde-624	550	33	5.3	5.3	NUM
ejde-624	550	34	,	,	PUNCT
ejde-624	550	35	the	the	DET
ejde-624	550	36	authors	author	NOUN
ejde-624	550	37	use	use	VERB
ejde-624	550	38	the	the	DET
ejde-624	550	39	inequality	inequality	NOUN
ejde-624	550	40	(	(	PUNCT
ejde-624	550	41	t	t	PROPN
ejde-624	550	42	−	−	PROPN
ejde-624	550	43	s)α−1	s)α−1	PROPN
ejde-624	550	44	≥	≥	X
ejde-624	550	45	(	(	PUNCT
ejde-624	550	46	t	t	NOUN
ejde-624	550	47	+	+	CCONJ
ejde-624	550	48	1)α−1	1)α−1	NUM
ejde-624	550	49	together	together	ADV
ejde-624	550	50	with	with	ADP
ejde-624	550	51	comparison	comparison	NOUN
ejde-624	550	52	principles	principle	NOUN
ejde-624	550	53	.	.	PUNCT
ejde-624	551	1	we	we	PRON
ejde-624	551	2	discuss	discuss	VERB
ejde-624	551	3	the	the	DET
ejde-624	551	4	more	more	ADV
ejde-624	551	5	general	general	ADJ
ejde-624	551	6	result	result	NOUN
ejde-624	551	7	using	use	VERB
ejde-624	551	8	a	a	DET
ejde-624	551	9	different	different	ADJ
ejde-624	551	10	inequality	inequality	NOUN
ejde-624	551	11	at	at	ADP
ejde-624	551	12	that	that	DET
ejde-624	551	13	point	point	NOUN
ejde-624	551	14	and	and	CCONJ
ejde-624	551	15	simple	simple	ADJ
ejde-624	551	16	comparisons	comparison	NOUN
ejde-624	551	17	.	.	PUNCT
ejde-624	552	1	the	the	DET
ejde-624	552	2	special	special	ADJ
ejde-624	552	3	case	case	NOUN
ejde-624	552	4	of	of	ADP
ejde-624	552	5	β	β	X
ejde-624	552	6	=	=	SYM
ejde-624	552	7	0	0	PUNCT
ejde-624	552	8	in	in	ADP
ejde-624	552	9	theorem	theorem	ADJ
ejde-624	552	10	5.3	5.3	NUM
ejde-624	552	11	gives	give	VERB
ejde-624	552	12	a	a	DET
ejde-624	552	13	similar	similar	ADJ
ejde-624	552	14	conclusion	conclusion	NOUN
ejde-624	552	15	to	to	ADP
ejde-624	552	16	[	[	X
ejde-624	552	17	18	18	NUM
ejde-624	552	18	,	,	PUNCT
ejde-624	552	19	theorem	theorem	VERB
ejde-624	552	20	2.2	2.2	NUM
ejde-624	552	21	(	(	PUNCT
ejde-624	552	22	1	1	NUM
ejde-624	552	23	)	)	PUNCT
ejde-624	552	24	]	]	PUNCT
ejde-624	552	25	.	.	PUNCT
ejde-624	553	1	theorem	theorem	VERB
ejde-624	553	2	5.5	5.5	NUM
ejde-624	553	3	improves	improve	VERB
ejde-624	553	4	[	[	X
ejde-624	553	5	18	18	NUM
ejde-624	553	6	,	,	PUNCT
ejde-624	553	7	theorem	theorem	VERB
ejde-624	553	8	2.2	2.2	NUM
ejde-624	553	9	(	(	PUNCT
ejde-624	553	10	2	2	NUM
ejde-624	553	11	)	)	PUNCT
ejde-624	553	12	]	]	PUNCT
ejde-624	553	13	,	,	PUNCT
ejde-624	553	14	which	which	PRON
ejde-624	553	15	uses	use	VERB
ejde-624	553	16	an	an	DET
ejde-624	553	17	inequality	inequality	NOUN
ejde-624	553	18	that	that	PRON
ejde-624	553	19	requires	require	VERB
ejde-624	553	20	the	the	DET
ejde-624	553	21	initial	initial	ADJ
ejde-624	553	22	condition	condition	NOUN
ejde-624	553	23	u0	u0	ADJ
ejde-624	553	24	to	to	PART
ejde-624	553	25	be	be	AUX
ejde-624	553	26	bounded	bound	VERB
ejde-624	553	27	below	below	ADV
ejde-624	553	28	by	by	ADP
ejde-624	553	29	a	a	DET
ejde-624	553	30	sufficiently	sufficiently	ADV
ejde-624	553	31	large	large	ADJ
ejde-624	553	32	explicit	explicit	ADJ
ejde-624	553	33	constant	constant	ADJ
ejde-624	553	34	.	.	PUNCT
ejde-624	554	1	they	they	PRON
ejde-624	554	2	write	write	VERB
ejde-624	554	3	that	that	SCONJ
ejde-624	554	4	the	the	DET
ejde-624	554	5	solution	solution	NOUN
ejde-624	554	6	blows	blow	VERB
ejde-624	554	7	-	-	PUNCT
ejde-624	554	8	up	up	NOUN
ejde-624	554	9	but	but	CCONJ
ejde-624	554	10	do	do	AUX
ejde-624	554	11	not	not	PART
ejde-624	554	12	mention	mention	VERB
ejde-624	554	13	any	any	DET
ejde-624	554	14	continuation	continuation	NOUN
ejde-624	554	15	theorem	theorem	NOUN
ejde-624	554	16	that	that	PRON
ejde-624	554	17	would	would	AUX
ejde-624	554	18	give	give	VERB
ejde-624	554	19	the	the	DET
ejde-624	554	20	proof	proof	NOUN
ejde-624	554	21	of	of	ADP
ejde-624	554	22	this	this	PRON
ejde-624	554	23	.	.	PUNCT
ejde-624	555	1	by	by	ADP
ejde-624	555	2	using	use	VERB
ejde-624	555	3	wu	wu	PROPN
ejde-624	555	4	and	and	CCONJ
ejde-624	555	5	liu	liu	PROPN
ejde-624	556	1	[	[	X
ejde-624	556	2	24	24	NUM
ejde-624	556	3	,	,	PUNCT
ejde-624	556	4	theorem	theorem	VERB
ejde-624	556	5	4.1	4.1	NUM
ejde-624	556	6	]	]	PUNCT
ejde-624	556	7	,	,	PUNCT
ejde-624	556	8	or	or	CCONJ
ejde-624	556	9	eloe	eloe	ADJ
ejde-624	556	10	-	-	PUNCT
ejde-624	556	11	masthay	masthay	NOUN
ejde-624	557	1	[	[	X
ejde-624	557	2	6	6	NUM
ejde-624	557	3	,	,	PUNCT
ejde-624	557	4	theorem	theorem	VERB
ejde-624	557	5	2.4	2.4	NUM
ejde-624	557	6	]	]	PUNCT
ejde-624	557	7	,	,	PUNCT
ejde-624	557	8	the	the	DET
ejde-624	557	9	blow	blow	NOUN
ejde-624	557	10	-	-	PUNCT
ejde-624	557	11	up	up	NOUN
ejde-624	557	12	at	at	ADP
ejde-624	557	13	some	some	DET
ejde-624	557	14	t2	t2	NOUN
ejde-624	557	15	<	<	AUX
ejde-624	557	16	t	t	PROPN
ejde-624	557	17	∗	∗	NOUN
ejde-624	557	18	can	can	AUX
ejde-624	557	19	be	be	AUX
ejde-624	557	20	justified	justify	VERB
ejde-624	557	21	for	for	ADP
ejde-624	557	22	the	the	DET
ejde-624	557	23	case	case	NOUN
ejde-624	557	24	of	of	ADP
ejde-624	557	25	an	an	DET
ejde-624	557	26	equation	equation	NOUN
ejde-624	557	27	.	.	PUNCT
ejde-624	558	1	systems	system	NOUN
ejde-624	558	2	of	of	ADP
ejde-624	558	3	caputo	caputo	PROPN
ejde-624	558	4	inequalities	inequality	NOUN
ejde-624	558	5	are	be	AUX
ejde-624	558	6	studied	study	VERB
ejde-624	558	7	in	in	ADP
ejde-624	558	8	[	[	X
ejde-624	558	9	19	19	NUM
ejde-624	558	10	]	]	PUNCT
ejde-624	558	11	.	.	PUNCT
ejde-624	559	1	a	a	DET
ejde-624	559	2	nonexistence	nonexistence	NOUN
ejde-624	559	3	result	result	NOUN
ejde-624	559	4	for	for	SCONJ
ejde-624	559	5	the	the	DET
ejde-624	559	6	inequality	inequality	NOUN
ejde-624	559	7	dα	dα	NOUN
ejde-624	559	8	cu(t	cu(t	PUNCT
ejde-624	559	9	)	)	PUNCT
ejde-624	559	10	≥	≥	NOUN
ejde-624	559	11	λtβup	λtβup	NOUN
ejde-624	559	12	for	for	ADP
ejde-624	559	13	β	β	PROPN
ejde-624	559	14	≥	≥	NOUN
ejde-624	559	15	0	0	NUM
ejde-624	559	16	and	and	CCONJ
ejde-624	559	17	p	p	X
ejde-624	559	18	>	>	X
ejde-624	559	19	1	1	NUM
ejde-624	559	20	is	be	AUX
ejde-624	559	21	given	give	VERB
ejde-624	559	22	in	in	ADP
ejde-624	559	23	[	[	X
ejde-624	559	24	19	19	NUM
ejde-624	559	25	,	,	PUNCT
ejde-624	559	26	proposition	proposition	NOUN
ejde-624	559	27	3.1	3.1	NUM
ejde-624	559	28	]	]	PUNCT
ejde-624	559	29	,	,	PUNCT
ejde-624	559	30	using	use	VERB
ejde-624	559	31	a	a	DET
ejde-624	559	32	test	test	NOUN
ejde-624	559	33	function	function	NOUN
ejde-624	559	34	and	and	CCONJ
ejde-624	559	35	capacity	capacity	NOUN
ejde-624	559	36	method	method	NOUN
ejde-624	559	37	.	.	PUNCT
ejde-624	560	1	the	the	DET
ejde-624	560	2	given	give	VERB
ejde-624	560	3	condition	condition	NOUN
ejde-624	560	4	is	be	AUX
ejde-624	560	5	β	β	NOUN
ejde-624	560	6	+	+	PROPN
ejde-624	560	7	1	1	NUM
ejde-624	560	8	≥	≥	NUM
ejde-624	560	9	βp′	βp′	NUM
ejde-624	561	1	where	where	SCONJ
ejde-624	561	2	1	1	X
ejde-624	561	3	/	/	SYM
ejde-624	561	4	p	p	NOUN
ejde-624	561	5	+	+	NOUN
ejde-624	561	6	1	1	NUM
ejde-624	561	7	/	/	SYM
ejde-624	561	8	p′	p′	NOUN
ejde-624	561	9	=	=	SYM
ejde-624	561	10	1	1	X
ejde-624	561	11	.	.	PUNCT
ejde-624	562	1	this	this	PRON
ejde-624	562	2	is	be	AUX
ejde-624	562	3	equivalent	equivalent	ADJ
ejde-624	562	4	to	to	ADP
ejde-624	562	5	β	β	X
ejde-624	562	6	<	<	X
ejde-624	562	7	p	p	X
ejde-624	562	8	−	−	PROPN
ejde-624	562	9	1	1	NUM
ejde-624	562	10	so	so	SCONJ
ejde-624	562	11	it	it	PRON
ejde-624	562	12	can	can	AUX
ejde-624	562	13	not	not	PART
ejde-624	562	14	be	be	AUX
ejde-624	562	15	a	a	DET
ejde-624	562	16	sharp	sharp	ADJ
ejde-624	562	17	estimate	estimate	NOUN
ejde-624	562	18	;	;	PUNCT
ejde-624	562	19	perhaps	perhaps	ADV
ejde-624	562	20	there	there	PRON
ejde-624	562	21	is	be	VERB
ejde-624	562	22	a	a	DET
ejde-624	562	23	typo	typo	NOUN
ejde-624	562	24	.	.	PUNCT
ejde-624	563	1	blow	blow	VERB
ejde-624	563	2	-	-	PUNCT
ejde-624	563	3	up	up	NOUN
ejde-624	563	4	is	be	AUX
ejde-624	563	5	claimed	claim	VERB
ejde-624	563	6	but	but	CCONJ
ejde-624	563	7	it	it	PRON
ejde-624	563	8	seems	seem	VERB
ejde-624	563	9	to	to	PART
ejde-624	563	10	need	need	VERB
ejde-624	563	11	further	further	ADJ
ejde-624	563	12	explanation	explanation	NOUN
ejde-624	563	13	for	for	ADP
ejde-624	563	14	fractional	fractional	ADJ
ejde-624	563	15	inequalities	inequality	NOUN
ejde-624	563	16	.	.	PUNCT
ejde-624	564	1	references	reference	NOUN
ejde-624	564	2	[	[	X
ejde-624	564	3	1	1	NUM
ejde-624	564	4	]	]	PUNCT
ejde-624	564	5	l.	l.	PROPN
ejde-624	564	6	c.	c.	PROPN
ejde-624	564	7	becker	becker	PROPN
ejde-624	564	8	,	,	PUNCT
ejde-624	564	9	t.	t.	PROPN
ejde-624	564	10	a.	a.	PROPN
ejde-624	564	11	burton	burton	PROPN
ejde-624	564	12	,	,	PUNCT
ejde-624	564	13	i.	i.	PROPN
ejde-624	564	14	k.	k.	PROPN
ejde-624	564	15	purnaras	purnaras	PROPN
ejde-624	564	16	;	;	PUNCT
ejde-624	564	17	complementary	complementary	ADJ
ejde-624	564	18	equations	equation	NOUN
ejde-624	564	19	:	:	PUNCT
ejde-624	564	20	a	a	DET
ejde-624	564	21	fractional	fractional	ADJ
ejde-624	564	22	differential	differential	ADJ
ejde-624	564	23	equation	equation	NOUN
ejde-624	564	24	and	and	CCONJ
ejde-624	564	25	a	a	DET
ejde-624	564	26	volterra	volterra	NOUN
ejde-624	564	27	integral	integral	ADJ
ejde-624	564	28	equation	equation	NOUN
ejde-624	564	29	.	.	PUNCT
ejde-624	565	1	electron	electron	PROPN
ejde-624	565	2	.	.	PUNCT
ejde-624	566	1	j.	j.	PROPN
ejde-624	566	2	qual	qual	PROPN
ejde-624	566	3	.	.	PROPN
ejde-624	566	4	theory	theory	NOUN
ejde-624	566	5	differ	differ	VERB
ejde-624	566	6	.	.	PUNCT
ejde-624	567	1	equ	equ	PROPN
ejde-624	567	2	.	.	PROPN
ejde-624	567	3	2015	2015	NUM
ejde-624	567	4	,	,	PUNCT
ejde-624	567	5	no	no	INTJ
ejde-624	567	6	.	.	NOUN
ejde-624	567	7	12	12	NUM
ejde-624	567	8	,	,	PUNCT
ejde-624	567	9	24	24	NUM
ejde-624	567	10	pp	pp	NOUN
ejde-624	567	11	.	.	PUNCT
ejde-624	568	1	[	[	X
ejde-624	568	2	2	2	NUM
ejde-624	568	3	]	]	PUNCT
ejde-624	568	4	l.	l.	PROPN
ejde-624	568	5	c.	c.	PROPN
ejde-624	568	6	becker	becker	PROPN
ejde-624	568	7	,	,	PUNCT
ejde-624	568	8	t.	t.	PROPN
ejde-624	568	9	a.	a.	PROPN
ejde-624	568	10	burton	burton	PROPN
ejde-624	568	11	,	,	PUNCT
ejde-624	568	12	i.	i.	PROPN
ejde-624	568	13	k.	k.	PROPN
ejde-624	568	14	purnaras	purnaras	PROPN
ejde-624	568	15	;	;	PUNCT
ejde-624	568	16	existence	existence	NOUN
ejde-624	568	17	of	of	ADP
ejde-624	568	18	solutions	solution	NOUN
ejde-624	568	19	of	of	ADP
ejde-624	568	20	nonlinear	nonlinear	ADJ
ejde-624	568	21	fractional	fractional	ADJ
ejde-624	568	22	differential	differential	ADJ
ejde-624	568	23	equations	equation	NOUN
ejde-624	568	24	of	of	ADP
ejde-624	568	25	riemann	riemann	PROPN
ejde-624	568	26	-	-	PUNCT
ejde-624	568	27	liouville	liouville	NOUN
ejde-624	568	28	type	type	NOUN
ejde-624	568	29	.	.	PUNCT
ejde-624	569	1	j.	j.	PROPN
ejde-624	569	2	fract	fract	PROPN
ejde-624	569	3	.	.	PUNCT
ejde-624	570	1	calc	calc	PROPN
ejde-624	570	2	.	.	PUNCT
ejde-624	571	1	appl	appl	PROPN
ejde-624	571	2	.	.	PROPN
ejde-624	572	1	7	7	NUM
ejde-624	572	2	(	(	PUNCT
ejde-624	572	3	2016	2016	NUM
ejde-624	572	4	)	)	PUNCT
ejde-624	572	5	,	,	PUNCT
ejde-624	573	1	no	no	INTJ
ejde-624	573	2	.	.	NOUN
ejde-624	573	3	2	2	NUM
ejde-624	573	4	,	,	PUNCT
ejde-624	573	5	20–39	20–39	NUM
ejde-624	573	6	.	.	PUNCT
ejde-624	574	1	[	[	X
ejde-624	574	2	3	3	NUM
ejde-624	574	3	]	]	X
ejde-624	574	4	l.	l.	PROPN
ejde-624	574	5	c.	c.	PROPN
ejde-624	574	6	becker	becker	PROPN
ejde-624	574	7	,	,	PUNCT
ejde-624	574	8	t.	t.	PROPN
ejde-624	574	9	a.	a.	PROPN
ejde-624	574	10	burton	burton	PROPN
ejde-624	574	11	,	,	PUNCT
ejde-624	574	12	i.	i.	PROPN
ejde-624	574	13	k.	k.	PROPN
ejde-624	574	14	purnaras	purnaras	PROPN
ejde-624	574	15	;	;	PUNCT
ejde-624	574	16	integral	integral	ADJ
ejde-624	574	17	and	and	CCONJ
ejde-624	574	18	fractional	fractional	ADJ
ejde-624	574	19	equations	equation	NOUN
ejde-624	574	20	,	,	PUNCT
ejde-624	574	21	positive	positive	ADJ
ejde-624	574	22	solutions	solution	NOUN
ejde-624	574	23	,	,	PUNCT
ejde-624	574	24	and	and	CCONJ
ejde-624	574	25	schaefer	schaefer	PROPN
ejde-624	574	26	’s	’s	PART
ejde-624	574	27	fixed	fix	VERB
ejde-624	574	28	point	point	NOUN
ejde-624	574	29	theorem	theorem	VERB
ejde-624	574	30	.	.	PROPN
ejde-624	574	31	opuscula	opuscula	PROPN
ejde-624	574	32	math	math	PROPN
ejde-624	574	33	.	.	PUNCT
ejde-624	575	1	36	36	NUM
ejde-624	575	2	(	(	PUNCT
ejde-624	575	3	2016	2016	NUM
ejde-624	575	4	)	)	PUNCT
ejde-624	575	5	,	,	PUNCT
ejde-624	575	6	no	no	INTJ
ejde-624	575	7	.	.	NOUN
ejde-624	575	8	4	4	NUM
ejde-624	575	9	,	,	PUNCT
ejde-624	575	10	431–458	431–458	NUM
ejde-624	575	11	.	.	PUNCT
ejde-624	576	1	[	[	X
ejde-624	576	2	4	4	NUM
ejde-624	576	3	]	]	PUNCT
ejde-624	576	4	m.	m.	NOUN
ejde-624	576	5	cichon	cichon	PROPN
ejde-624	576	6	,	,	PUNCT
ejde-624	576	7	h.	h.	PROPN
ejde-624	576	8	a.	a.	PROPN
ejde-624	576	9	h.	h.	PROPN
ejde-624	576	10	salem	salem	PROPN
ejde-624	576	11	;	;	PUNCT
ejde-624	576	12	on	on	ADP
ejde-624	576	13	the	the	DET
ejde-624	576	14	lack	lack	NOUN
ejde-624	576	15	of	of	ADP
ejde-624	576	16	equivalence	equivalence	NOUN
ejde-624	576	17	between	between	ADP
ejde-624	576	18	differential	differential	ADJ
ejde-624	576	19	and	and	CCONJ
ejde-624	576	20	integral	integral	ADJ
ejde-624	576	21	forms	form	NOUN
ejde-624	576	22	of	of	ADP
ejde-624	576	23	the	the	DET
ejde-624	576	24	caputo	caputo	NOUN
ejde-624	576	25	-	-	PUNCT
ejde-624	576	26	type	type	NOUN
ejde-624	576	27	fractional	fractional	ADJ
ejde-624	576	28	problems	problem	NOUN
ejde-624	576	29	.	.	PUNCT
ejde-624	577	1	j.	j.	PROPN
ejde-624	577	2	pseudo	pseudo	PROPN
ejde-624	577	3	-	-	PUNCT
ejde-624	577	4	differ	differ	VERB
ejde-624	577	5	.	.	PUNCT
ejde-624	578	1	oper	oper	PROPN
ejde-624	578	2	.	.	PUNCT
ejde-624	578	3	appl	appl	PROPN
ejde-624	578	4	.	.	PROPN
ejde-624	579	1	11	11	NUM
ejde-624	579	2	(	(	PUNCT
ejde-624	579	3	2020	2020	NUM
ejde-624	579	4	)	)	PUNCT
ejde-624	579	5	,	,	PUNCT
ejde-624	579	6	1869–1895	1869–1895	NUM
ejde-624	579	7	.	.	PUNCT
ejde-624	580	1	[	[	X
ejde-624	580	2	5	5	NUM
ejde-624	580	3	]	]	PUNCT
ejde-624	580	4	k.	k.	PROPN
ejde-624	580	5	diethelm	diethelm	PROPN
ejde-624	580	6	;	;	PUNCT
ejde-624	580	7	the	the	DET
ejde-624	580	8	analysis	analysis	NOUN
ejde-624	580	9	of	of	ADP
ejde-624	580	10	fractional	fractional	ADJ
ejde-624	580	11	differential	differential	ADJ
ejde-624	580	12	equations	equation	NOUN
ejde-624	580	13	.	.	PUNCT
ejde-624	581	1	an	an	DET
ejde-624	581	2	application	application	NOUN
ejde-624	581	3	-	-	PUNCT
ejde-624	581	4	oriented	orient	VERB
ejde-624	581	5	exposition	exposition	NOUN
ejde-624	581	6	using	use	VERB
ejde-624	581	7	differential	differential	ADJ
ejde-624	581	8	operators	operator	NOUN
ejde-624	581	9	of	of	ADP
ejde-624	581	10	caputo	caputo	PROPN
ejde-624	581	11	type	type	PROPN
ejde-624	581	12	.	.	PUNCT
ejde-624	582	1	lecture	lecture	NOUN
ejde-624	582	2	notes	note	NOUN
ejde-624	582	3	in	in	ADP
ejde-624	582	4	mathematics	mathematics	PROPN
ejde-624	582	5	no	no	INTJ
ejde-624	582	6	.	.	PUNCT
ejde-624	582	7	2004	2004	NUM
ejde-624	582	8	.	.	PUNCT
ejde-624	583	1	springer	springer	NOUN
ejde-624	583	2	-	-	PUNCT
ejde-624	583	3	verlag	verlag	PROPN
ejde-624	583	4	,	,	PUNCT
ejde-624	583	5	berlin	berlin	PROPN
ejde-624	583	6	,	,	PUNCT
ejde-624	583	7	2010	2010	NUM
ejde-624	583	8	.	.	PUNCT
ejde-624	584	1	[	[	X
ejde-624	584	2	6	6	NUM
ejde-624	584	3	]	]	PUNCT
ejde-624	584	4	p.	p.	NOUN
ejde-624	584	5	w.	w.	PROPN
ejde-624	584	6	eloe	eloe	PROPN
ejde-624	584	7	,	,	PUNCT
ejde-624	584	8	t.	t.	PROPN
ejde-624	584	9	masthay	masthay	PROPN
ejde-624	584	10	;	;	PUNCT
ejde-624	584	11	initial	initial	ADJ
ejde-624	584	12	value	value	NOUN
ejde-624	584	13	problems	problem	NOUN
ejde-624	584	14	for	for	ADP
ejde-624	584	15	caputo	caputo	PROPN
ejde-624	584	16	fractional	fractional	PROPN
ejde-624	584	17	differential	differential	PROPN
ejde-624	584	18	equations	equation	NOUN
ejde-624	584	19	,	,	PUNCT
ejde-624	584	20	j.	j.	PROPN
ejde-624	584	21	fract	fract	PROPN
ejde-624	584	22	.	.	PUNCT
ejde-624	585	1	calc	calc	PROPN
ejde-624	585	2	.	.	PUNCT
ejde-624	586	1	appl	appl	PROPN
ejde-624	586	2	.	.	PROPN
ejde-624	586	3	,	,	PUNCT
ejde-624	586	4	9	9	NUM
ejde-624	586	5	(	(	PUNCT
ejde-624	586	6	2018	2018	NUM
ejde-624	586	7	)	)	PUNCT
ejde-624	586	8	,	,	PUNCT
ejde-624	586	9	178–195	178–195	NUM
ejde-624	586	10	.	.	PUNCT
ejde-624	587	1	[	[	X
ejde-624	587	2	7	7	X
ejde-624	587	3	]	]	PUNCT
ejde-624	587	4	m.	m.	NOUN
ejde-624	587	5	d.	d.	PROPN
ejde-624	587	6	kassim	kassim	PROPN
ejde-624	587	7	,	,	PUNCT
ejde-624	587	8	k.	k.	PROPN
ejde-624	587	9	m.	m.	PROPN
ejde-624	587	10	furati	furati	PROPN
ejde-624	587	11	,	,	PUNCT
ejde-624	587	12	n	n	CCONJ
ejde-624	587	13	-	-	PUNCT
ejde-624	587	14	e	e	NOUN
ejde-624	587	15	tatar	tatar	NOUN
ejde-624	587	16	;	;	PUNCT
ejde-624	587	17	non	non	ADJ
ejde-624	587	18	-	-	NOUN
ejde-624	587	19	existence	existence	NOUN
ejde-624	587	20	for	for	ADP
ejde-624	587	21	fractionally	fractionally	ADV
ejde-624	587	22	damped	damp	VERB
ejde-624	587	23	fractional	fractional	ADJ
ejde-624	587	24	differential	differential	NOUN
ejde-624	587	25	problems	problem	NOUN
ejde-624	587	26	.	.	PUNCT
ejde-624	588	1	acta	acta	PROPN
ejde-624	588	2	math	math	PROPN
ejde-624	588	3	.	.	PUNCT
ejde-624	589	1	sci	sci	PROPN
ejde-624	589	2	.	.	PUNCT
ejde-624	589	3	ser	ser	PROPN
ejde-624	589	4	.	.	PUNCT
ejde-624	590	1	b	b	PROPN
ejde-624	590	2	(	(	PUNCT
ejde-624	590	3	engl	engl	PROPN
ejde-624	590	4	.	.	PUNCT
ejde-624	591	1	ed	ed	NOUN
ejde-624	591	2	.	.	PUNCT
ejde-624	591	3	)	)	PUNCT
ejde-624	592	1	37	37	NUM
ejde-624	592	2	(	(	PUNCT
ejde-624	592	3	2017	2017	NUM
ejde-624	592	4	)	)	PUNCT
ejde-624	592	5	,	,	PUNCT
ejde-624	592	6	no	no	INTJ
ejde-624	592	7	.	.	NOUN
ejde-624	592	8	1	1	NUM
ejde-624	592	9	,	,	PUNCT
ejde-624	592	10	119–130	119–130	NUM
ejde-624	592	11	.	.	PUNCT
ejde-624	593	1	[	[	X
ejde-624	593	2	8	8	NUM
ejde-624	593	3	]	]	PUNCT
ejde-624	593	4	m.	m.	NOUN
ejde-624	593	5	d.	d.	PROPN
ejde-624	593	6	kassim	kassim	PROPN
ejde-624	593	7	,	,	PUNCT
ejde-624	593	8	k.	k.	PROPN
ejde-624	593	9	m.	m.	PROPN
ejde-624	593	10	furati	furati	PROPN
ejde-624	593	11	,	,	PUNCT
ejde-624	593	12	n	n	CCONJ
ejde-624	593	13	-	-	PUNCT
ejde-624	593	14	e	e	NOUN
ejde-624	593	15	tatar	tatar	NOUN
ejde-624	593	16	;	;	PUNCT
ejde-624	593	17	nonexistence	nonexistence	NOUN
ejde-624	593	18	of	of	ADP
ejde-624	593	19	global	global	ADJ
ejde-624	593	20	solutions	solution	NOUN
ejde-624	593	21	for	for	ADP
ejde-624	593	22	a	a	DET
ejde-624	593	23	fractional	fractional	ADJ
ejde-624	593	24	differential	differential	NOUN
ejde-624	593	25	problem	problem	NOUN
ejde-624	593	26	.	.	PUNCT
ejde-624	594	1	j.	j.	PROPN
ejde-624	594	2	comput	comput	PROPN
ejde-624	594	3	.	.	PUNCT
ejde-624	595	1	appl	appl	PROPN
ejde-624	595	2	.	.	PROPN
ejde-624	595	3	math	math	PROPN
ejde-624	595	4	.	.	PUNCT
ejde-624	596	1	314	314	NUM
ejde-624	596	2	(	(	PUNCT
ejde-624	596	3	2017	2017	NUM
ejde-624	596	4	)	)	PUNCT
ejde-624	596	5	,	,	PUNCT
ejde-624	596	6	61–68	61–68	NUM
ejde-624	596	7	.	.	PUNCT
ejde-624	597	1	[	[	X
ejde-624	597	2	9	9	NUM
ejde-624	597	3	]	]	PUNCT
ejde-624	597	4	m.	m.	NOUN
ejde-624	597	5	d.	d.	PROPN
ejde-624	597	6	kassim	kassim	PROPN
ejde-624	597	7	,	,	PUNCT
ejde-624	597	8	m.	m.	PROPN
ejde-624	597	9	alqahtani	alqahtani	PROPN
ejde-624	597	10	,	,	PUNCT
ejde-624	597	11	n.-e	n.-e	PROPN
ejde-624	597	12	.	.	PUNCT
ejde-624	598	1	tatar	tatar	NOUN
ejde-624	598	2	,	,	PUNCT
ejde-624	598	3	a.	a.	NOUN
ejde-624	598	4	laadhari	laadhari	PROPN
ejde-624	598	5	;	;	PUNCT
ejde-624	598	6	nonexistence	nonexistence	NOUN
ejde-624	598	7	results	result	VERB
ejde-624	598	8	for	for	ADP
ejde-624	598	9	a	a	DET
ejde-624	598	10	sequential	sequential	ADJ
ejde-624	598	11	fractional	fractional	ADJ
ejde-624	598	12	differential	differential	NOUN
ejde-624	598	13	problem	problem	NOUN
ejde-624	598	14	,	,	PUNCT
ejde-624	598	15	math	math	NOUN
ejde-624	598	16	.	.	PUNCT
ejde-624	599	1	meth	meth	NOUN
ejde-624	599	2	.	.	PUNCT
ejde-624	600	1	appl	appl	PROPN
ejde-624	600	2	.	.	PUNCT
ejde-624	601	1	sci	sci	PROPN
ejde-624	601	2	.	.	PROPN
ejde-624	601	3	46	46	NUM
ejde-624	601	4	(	(	PUNCT
ejde-624	601	5	2023	2023	NUM
ejde-624	601	6	)	)	PUNCT
ejde-624	601	7	,	,	PUNCT
ejde-624	601	8	16305–16317	16305–16317	NUM
ejde-624	601	9	.	.	PUNCT
ejde-624	602	1	[	[	X
ejde-624	602	2	10	10	NUM
ejde-624	602	3	]	]	PUNCT
ejde-624	602	4	a.	a.	NOUN
ejde-624	602	5	a.	a.	NOUN
ejde-624	602	6	kilbas	kilbas	PROPN
ejde-624	602	7	,	,	PUNCT
ejde-624	602	8	h.	h.	PROPN
ejde-624	602	9	m.	m.	PROPN
ejde-624	602	10	srivastava	srivastava	PROPN
ejde-624	602	11	,	,	PUNCT
ejde-624	602	12	j.	j.	PROPN
ejde-624	602	13	j.	j.	PROPN
ejde-624	602	14	trujillo	trujillo	PROPN
ejde-624	602	15	;	;	PUNCT
ejde-624	602	16	theory	theory	NOUN
ejde-624	602	17	and	and	CCONJ
ejde-624	602	18	applications	application	NOUN
ejde-624	602	19	of	of	ADP
ejde-624	602	20	fractional	fractional	ADJ
ejde-624	602	21	differential	differential	ADJ
ejde-624	602	22	equations	equation	NOUN
ejde-624	602	23	.	.	PUNCT
ejde-624	603	1	north	north	NOUN
ejde-624	603	2	-	-	PUNCT
ejde-624	603	3	holland	holland	PROPN
ejde-624	603	4	mathematics	mathematics	PROPN
ejde-624	603	5	studies	study	NOUN
ejde-624	603	6	204	204	NUM
ejde-624	603	7	,	,	PUNCT
ejde-624	603	8	elsevier	elsevier	PROPN
ejde-624	603	9	science	science	NOUN
ejde-624	603	10	:	:	PUNCT
ejde-624	604	1	b.v	b.v	PROPN
ejde-624	604	2	.	.	PROPN
ejde-624	604	3	amsterdam	amsterdam	PROPN
ejde-624	604	4	,	,	PUNCT
ejde-624	604	5	2006	2006	NUM
ejde-624	604	6	.	.	PUNCT
ejde-624	605	1	[	[	X
ejde-624	605	2	11	11	NUM
ejde-624	605	3	]	]	PUNCT
ejde-624	605	4	k.	k.	PROPN
ejde-624	605	5	q.	q.	PROPN
ejde-624	605	6	lan	lan	PROPN
ejde-624	605	7	;	;	PUNCT
ejde-624	605	8	linear	linear	PROPN
ejde-624	605	9	first	first	ADJ
ejde-624	605	10	order	order	NOUN
ejde-624	605	11	riemann	riemann	NOUN
ejde-624	605	12	-	-	PUNCT
ejde-624	605	13	liouville	liouville	VERB
ejde-624	605	14	fractional	fractional	ADJ
ejde-624	605	15	differential	differential	NOUN
ejde-624	605	16	and	and	CCONJ
ejde-624	605	17	perturbed	perturb	VERB
ejde-624	605	18	abel	abel	PROPN
ejde-624	605	19	’s	’s	PART
ejde-624	605	20	integral	integral	ADJ
ejde-624	605	21	equations	equation	NOUN
ejde-624	605	22	.	.	PUNCT
ejde-624	606	1	j.	j.	PROPN
ejde-624	606	2	differential	differential	PROPN
ejde-624	606	3	equations	equation	NOUN
ejde-624	606	4	306	306	NUM
ejde-624	606	5	(	(	PUNCT
ejde-624	606	6	2022	2022	NUM
ejde-624	606	7	)	)	PUNCT
ejde-624	606	8	,	,	PUNCT
ejde-624	606	9	28–59	28–59	NUM
ejde-624	606	10	.	.	PUNCT
ejde-624	607	1	[	[	X
ejde-624	607	2	12	12	NUM
ejde-624	607	3	]	]	PUNCT
ejde-624	607	4	k.	k.	PROPN
ejde-624	607	5	q.	q.	PROPN
ejde-624	607	6	lan	lan	PROPN
ejde-624	607	7	;	;	PUNCT
ejde-624	607	8	corrigendum	corrigendum	VERB
ejde-624	607	9	to	to	PART
ejde-624	607	10	“	"	PUNCT
ejde-624	607	11	linear	linear	VERB
ejde-624	607	12	first	first	ADJ
ejde-624	607	13	order	order	NOUN
ejde-624	607	14	riemann	riemann	NOUN
ejde-624	607	15	-	-	PUNCT
ejde-624	607	16	liouville	liouville	VERB
ejde-624	607	17	fractional	fractional	ADJ
ejde-624	607	18	differential	differential	NOUN
ejde-624	607	19	and	and	CCONJ
ejde-624	607	20	perturbed	perturb	VERB
ejde-624	607	21	abel	abel	PROPN
ejde-624	607	22	’s	’s	PART
ejde-624	607	23	integral	integral	ADJ
ejde-624	607	24	equations	equation	NOUN
ejde-624	607	25	”	"	PUNCT
ejde-624	608	1	[	[	X
ejde-624	608	2	j.	j.	PROPN
ejde-624	608	3	differ	differ	VERB
ejde-624	608	4	.	.	PUNCT
ejde-624	609	1	equ	equ	PROPN
ejde-624	609	2	.	.	PROPN
ejde-624	609	3	306	306	NUM
ejde-624	609	4	(	(	PUNCT
ejde-624	609	5	2022	2022	NUM
ejde-624	609	6	)	)	PUNCT
ejde-624	609	7	28	28	NUM
ejde-624	609	8	-	-	SYM
ejde-624	609	9	59	59	NUM
ejde-624	609	10	]	]	PUNCT
ejde-624	609	11	.	.	PUNCT
ejde-624	610	1	j.	j.	PROPN
ejde-624	610	2	differential	differential	PROPN
ejde-624	610	3	equations	equation	NOUN
ejde-624	610	4	345	345	NUM
ejde-624	610	5	(	(	PUNCT
ejde-624	610	6	2023	2023	NUM
ejde-624	610	7	)	)	PUNCT
ejde-624	610	8	,	,	PUNCT
ejde-624	610	9	519	519	NUM
ejde-624	610	10	-	-	SYM
ejde-624	610	11	520	520	NUM
ejde-624	610	12	.	.	PUNCT
ejde-624	611	1	[	[	X
ejde-624	611	2	13	13	NUM
ejde-624	611	3	]	]	PUNCT
ejde-624	611	4	k.	k.	PROPN
ejde-624	611	5	q.	q.	PROPN
ejde-624	611	6	lan	lan	PROPN
ejde-624	611	7	;	;	PUNCT
ejde-624	611	8	linear	linear	ADJ
ejde-624	611	9	higher	high	ADJ
ejde-624	611	10	-	-	PUNCT
ejde-624	611	11	order	order	NOUN
ejde-624	611	12	fractional	fractional	ADJ
ejde-624	611	13	differential	differential	NOUN
ejde-624	611	14	and	and	CCONJ
ejde-624	611	15	integral	integral	ADJ
ejde-624	611	16	equations	equation	NOUN
ejde-624	611	17	.	.	PUNCT
ejde-624	612	1	electron	electron	PROPN
ejde-624	612	2	.	.	PUNCT
ejde-624	613	1	j.	j.	PROPN
ejde-624	613	2	differential	differential	PROPN
ejde-624	613	3	equations	equation	NOUN
ejde-624	613	4	2023	2023	NUM
ejde-624	613	5	,	,	PUNCT
ejde-624	613	6	paper	paper	NOUN
ejde-624	613	7	no	no	NOUN
ejde-624	613	8	.	.	PROPN
ejde-624	613	9	01	01	NUM
ejde-624	613	10	,	,	PUNCT
ejde-624	613	11	20	20	NUM
ejde-624	613	12	pp	pp	NOUN
ejde-624	613	13	.	.	PUNCT
ejde-624	614	1	[	[	X
ejde-624	614	2	14	14	NUM
ejde-624	614	3	]	]	X
ejde-624	614	4	k.q	k.q	PROPN
ejde-624	614	5	.	.	PROPN
ejde-624	614	6	lan	lan	PROPN
ejde-624	614	7	,	,	PUNCT
ejde-624	614	8	j.	j.	PROPN
ejde-624	614	9	r.	r.	PROPN
ejde-624	614	10	l.	l.	PROPN
ejde-624	614	11	webb	webb	PROPN
ejde-624	614	12	;	;	PUNCT
ejde-624	614	13	a	a	DET
ejde-624	614	14	new	new	ADJ
ejde-624	614	15	bihari	bihari	PROPN
ejde-624	614	16	inequality	inequality	NOUN
ejde-624	614	17	and	and	CCONJ
ejde-624	614	18	initial	initial	ADJ
ejde-624	614	19	value	value	NOUN
ejde-624	614	20	problems	problem	NOUN
ejde-624	614	21	of	of	ADP
ejde-624	614	22	first	first	ADJ
ejde-624	614	23	order	order	NOUN
ejde-624	614	24	fractional	fractional	ADJ
ejde-624	614	25	differential	differential	ADJ
ejde-624	614	26	equations	equation	NOUN
ejde-624	614	27	.	.	PUNCT
ejde-624	615	1	fract	fract	PROPN
ejde-624	615	2	.	.	PUNCT
ejde-624	616	1	calc	calc	PROPN
ejde-624	616	2	.	.	PUNCT
ejde-624	617	1	appl	appl	PROPN
ejde-624	617	2	.	.	PUNCT
ejde-624	618	1	anal	anal	PROPN
ejde-624	618	2	.	.	PUNCT
ejde-624	619	1	26	26	NUM
ejde-624	619	2	(	(	PUNCT
ejde-624	619	3	2023	2023	NUM
ejde-624	619	4	)	)	PUNCT
ejde-624	619	5	,	,	PUNCT
ejde-624	619	6	no	no	INTJ
ejde-624	619	7	.	.	NOUN
ejde-624	619	8	3	3	NUM
ejde-624	619	9	,	,	PUNCT
ejde-624	619	10	962	962	NUM
ejde-624	619	11	-	-	SYM
ejde-624	619	12	988	988	NUM
ejde-624	619	13	.	.	PUNCT
ejde-624	620	1	[	[	X
ejde-624	620	2	15	15	NUM
ejde-624	620	3	]	]	X
ejde-624	620	4	y.	y.	PROPN
ejde-624	620	5	laskri	laskri	PROPN
ejde-624	620	6	,	,	PUNCT
ejde-624	620	7	n	n	CCONJ
ejde-624	620	8	-	-	PUNCT
ejde-624	620	9	e	e	NOUN
ejde-624	620	10	tatar	tatar	NOUN
ejde-624	620	11	;	;	PUNCT
ejde-624	620	12	the	the	DET
ejde-624	620	13	critical	critical	ADJ
ejde-624	620	14	exponent	exponent	NOUN
ejde-624	620	15	for	for	ADP
ejde-624	620	16	an	an	DET
ejde-624	620	17	ordinary	ordinary	ADJ
ejde-624	620	18	fractional	fractional	ADJ
ejde-624	620	19	differential	differential	NOUN
ejde-624	620	20	problem	problem	NOUN
ejde-624	620	21	.	.	PUNCT
ejde-624	621	1	comput	comput	NOUN
ejde-624	621	2	.	.	PUNCT
ejde-624	622	1	math	math	NOUN
ejde-624	622	2	.	.	PUNCT
ejde-624	623	1	appl	appl	PROPN
ejde-624	623	2	.	.	PUNCT
ejde-624	624	1	59	59	NUM
ejde-624	624	2	(	(	PUNCT
ejde-624	624	3	2010	2010	NUM
ejde-624	624	4	)	)	PUNCT
ejde-624	624	5	,	,	PUNCT
ejde-624	624	6	no	no	INTJ
ejde-624	624	7	.	.	NOUN
ejde-624	624	8	3	3	NUM
ejde-624	624	9	,	,	PUNCT
ejde-624	624	10	1266–1270	1266–1270	NUM
ejde-624	624	11	.	.	PUNCT
ejde-624	625	1	16	16	NUM
ejde-624	625	2	j.	j.	PROPN
ejde-624	625	3	r.	r.	PROPN
ejde-624	625	4	l.	l.	PROPN
ejde-624	625	5	webb	webb	PROPN
ejde-624	625	6	ejde-2024/40	ejde-2024/40	PROPN
ejde-624	626	1	[	[	X
ejde-624	626	2	16	16	NUM
ejde-624	626	3	]	]	X
ejde-624	626	4	e.	e.	PROPN
ejde-624	626	5	mitidieri	mitidieri	PROPN
ejde-624	626	6	,	,	PUNCT
ejde-624	626	7	s.	s.	PROPN
ejde-624	626	8	i.	i.	PROPN
ejde-624	626	9	pokhozhaev	pokhozhaev	PROPN
ejde-624	626	10	;	;	PUNCT
ejde-624	626	11	a	a	DET
ejde-624	626	12	priori	priori	ADJ
ejde-624	626	13	estimates	estimate	NOUN
ejde-624	626	14	and	and	CCONJ
ejde-624	626	15	the	the	DET
ejde-624	626	16	absence	absence	NOUN
ejde-624	626	17	of	of	ADP
ejde-624	626	18	solutions	solution	NOUN
ejde-624	626	19	of	of	ADP
ejde-624	626	20	nonlinear	nonlinear	ADJ
ejde-624	626	21	partial	partial	ADJ
ejde-624	626	22	differential	differential	ADJ
ejde-624	626	23	equations	equation	NOUN
ejde-624	626	24	and	and	CCONJ
ejde-624	626	25	inequalities	inequality	NOUN
ejde-624	626	26	.	.	PUNCT
ejde-624	627	1	(	(	PUNCT
ejde-624	627	2	russian	russian	PROPN
ejde-624	627	3	)	)	PUNCT
ejde-624	627	4	tr	tr	VERB
ejde-624	627	5	.	.	PROPN
ejde-624	627	6	mat	mat	PROPN
ejde-624	627	7	.	.	PROPN
ejde-624	627	8	inst	inst	PROPN
ejde-624	627	9	.	.	PUNCT
ejde-624	628	1	steklova	steklova	PROPN
ejde-624	628	2	234	234	NUM
ejde-624	628	3	(	(	PUNCT
ejde-624	628	4	2001	2001	NUM
ejde-624	628	5	)	)	PUNCT
ejde-624	628	6	,	,	PUNCT
ejde-624	628	7	1	1	NUM
ejde-624	628	8	-	-	SYM
ejde-624	628	9	384	384	NUM
ejde-624	628	10	;	;	PUNCT
ejde-624	628	11	translation	translation	NOUN
ejde-624	628	12	in	in	ADP
ejde-624	628	13	proc	proc	PROPN
ejde-624	628	14	.	.	PUNCT
ejde-624	629	1	steklov	steklov	PROPN
ejde-624	629	2	inst	inst	PROPN
ejde-624	629	3	.	.	PUNCT
ejde-624	630	1	math	math	NOUN
ejde-624	630	2	.	.	PUNCT
ejde-624	631	1	2001	2001	NUM
ejde-624	631	2	,	,	PUNCT
ejde-624	631	3	no	no	INTJ
ejde-624	631	4	.	.	NOUN
ejde-624	631	5	3	3	NUM
ejde-624	631	6	(	(	PUNCT
ejde-624	631	7	234	234	NUM
ejde-624	631	8	)	)	PUNCT
ejde-624	631	9	,	,	PUNCT
ejde-624	631	10	1–362	1–362	NUM
ejde-624	631	11	.	.	PUNCT
ejde-624	632	1	[	[	X
ejde-624	632	2	17	17	NUM
ejde-624	632	3	]	]	PUNCT
ejde-624	632	4	s.	s.	PROPN
ejde-624	632	5	g.	g.	PROPN
ejde-624	632	6	samko	samko	PROPN
ejde-624	632	7	,	,	PUNCT
ejde-624	632	8	a.	a.	NOUN
ejde-624	632	9	a.	a.	NOUN
ejde-624	632	10	kilbas	kilbas	PROPN
ejde-624	632	11	,	,	PUNCT
ejde-624	632	12	o.i	o.i	PROPN
ejde-624	632	13	.	.	PROPN
ejde-624	632	14	marichev	marichev	PROPN
ejde-624	632	15	;	;	PUNCT
ejde-624	632	16	fractional	fractional	ADJ
ejde-624	632	17	integrals	integral	NOUN
ejde-624	632	18	and	and	CCONJ
ejde-624	632	19	derivatives	derivative	NOUN
ejde-624	632	20	:	:	PUNCT
ejde-624	632	21	theory	theory	NOUN
ejde-624	632	22	and	and	CCONJ
ejde-624	632	23	applications	application	NOUN
ejde-624	632	24	.	.	PUNCT
ejde-624	633	1	gordon	gordon	PROPN
ejde-624	633	2	and	and	CCONJ
ejde-624	633	3	breach	breach	VERB
ejde-624	633	4	science	science	NOUN
ejde-624	633	5	publishers	publisher	NOUN
ejde-624	633	6	,	,	PUNCT
ejde-624	633	7	yverdon	yverdon	PROPN
ejde-624	633	8	,	,	PUNCT
ejde-624	633	9	1993	1993	NUM
ejde-624	633	10	.	.	PUNCT
ejde-624	634	1	[	[	X
ejde-624	634	2	18	18	NUM
ejde-624	634	3	]	]	X
ejde-624	634	4	y.	y.	PROPN
ejde-624	634	5	shan	shan	PROPN
ejde-624	634	6	,	,	PUNCT
ejde-624	634	7	g.	g.	PROPN
ejde-624	634	8	lv	lv	PROPN
ejde-624	634	9	;	;	PUNCT
ejde-624	634	10	new	new	ADJ
ejde-624	634	11	criteria	criterion	NOUN
ejde-624	634	12	for	for	ADP
ejde-624	634	13	blow	blow	NOUN
ejde-624	634	14	-	-	PUNCT
ejde-624	634	15	up	up	NOUN
ejde-624	634	16	of	of	ADP
ejde-624	634	17	fractional	fractional	ADJ
ejde-624	634	18	differential	differential	ADJ
ejde-624	634	19	equations	equation	NOUN
ejde-624	634	20	,	,	PUNCT
ejde-624	634	21	filomat	filomat	PROPN
ejde-624	634	22	38:4	38:4	NUM
ejde-624	634	23	(	(	PUNCT
ejde-624	634	24	2024	2024	NUM
ejde-624	634	25	)	)	PUNCT
ejde-624	634	26	,	,	PUNCT
ejde-624	634	27	1305–1315	1305–1315	NUM
ejde-624	634	28	.	.	PUNCT
ejde-624	635	1	[	[	X
ejde-624	635	2	19	19	NUM
ejde-624	635	3	]	]	X
ejde-624	635	4	j.	j.	PROPN
ejde-624	635	5	villa	villa	PROPN
ejde-624	635	6	-	-	PUNCT
ejde-624	635	7	morales	morale	NOUN
ejde-624	635	8	;	;	PUNCT
ejde-624	635	9	upper	upper	ADJ
ejde-624	635	10	bounds	bound	NOUN
ejde-624	635	11	for	for	ADP
ejde-624	635	12	the	the	DET
ejde-624	635	13	blow	blow	NOUN
ejde-624	635	14	-	-	PUNCT
ejde-624	635	15	up	up	ADP
ejde-624	635	16	time	time	NOUN
ejde-624	635	17	of	of	ADP
ejde-624	635	18	a	a	DET
ejde-624	635	19	system	system	NOUN
ejde-624	635	20	of	of	ADP
ejde-624	635	21	fractional	fractional	ADJ
ejde-624	635	22	differential	differential	ADJ
ejde-624	635	23	equations	equation	NOUN
ejde-624	635	24	with	with	ADP
ejde-624	635	25	caputo	caputo	PROPN
ejde-624	635	26	derivatives	derivative	NOUN
ejde-624	635	27	,	,	PUNCT
ejde-624	635	28	results	result	VERB
ejde-624	635	29	appl	appl	PROPN
ejde-624	635	30	.	.	PROPN
ejde-624	635	31	math	math	NOUN
ejde-624	635	32	.	.	PUNCT
ejde-624	636	1	20	20	NUM
ejde-624	636	2	(	(	PUNCT
ejde-624	636	3	2023	2023	NUM
ejde-624	636	4	)	)	PUNCT
ejde-624	637	1	,	,	PUNCT
ejde-624	637	2	paper	paper	NOUN
ejde-624	637	3	no	no	NOUN
ejde-624	637	4	.	.	PROPN
ejde-624	637	5	100408	100408	NUM
ejde-624	637	6	,	,	PUNCT
ejde-624	637	7	7	7	NUM
ejde-624	637	8	pp	pp	NOUN
ejde-624	637	9	.	.	PUNCT
ejde-624	638	1	[	[	X
ejde-624	638	2	20	20	NUM
ejde-624	638	3	]	]	PUNCT
ejde-624	638	4	j.	j.	PROPN
ejde-624	638	5	r.	r.	PROPN
ejde-624	638	6	l.	l.	PROPN
ejde-624	638	7	webb	webb	PROPN
ejde-624	638	8	;	;	PUNCT
ejde-624	638	9	weakly	weakly	ADV
ejde-624	638	10	singular	singular	ADJ
ejde-624	638	11	gronwall	gronwall	ADJ
ejde-624	638	12	inequalities	inequality	NOUN
ejde-624	638	13	and	and	CCONJ
ejde-624	638	14	applications	application	NOUN
ejde-624	638	15	to	to	ADP
ejde-624	638	16	fractional	fractional	ADJ
ejde-624	638	17	differential	differential	ADJ
ejde-624	638	18	equations	equation	NOUN
ejde-624	638	19	.	.	PUNCT
ejde-624	639	1	j.	j.	PROPN
ejde-624	639	2	math	math	PROPN
ejde-624	639	3	.	.	PUNCT
ejde-624	640	1	anal	anal	PROPN
ejde-624	640	2	.	.	PUNCT
ejde-624	640	3	appl	appl	PROPN
ejde-624	640	4	.	.	PUNCT
ejde-624	641	1	471	471	NUM
ejde-624	641	2	(	(	PUNCT
ejde-624	641	3	2019	2019	NUM
ejde-624	641	4	)	)	PUNCT
ejde-624	641	5	,	,	PUNCT
ejde-624	641	6	no	no	INTJ
ejde-624	641	7	.	.	NOUN
ejde-624	641	8	1	1	NUM
ejde-624	641	9	-	-	SYM
ejde-624	641	10	2	2	NUM
ejde-624	641	11	,	,	PUNCT
ejde-624	641	12	692–711	692–711	NUM
ejde-624	641	13	.	.	PUNCT
ejde-624	642	1	[	[	X
ejde-624	642	2	21	21	NUM
ejde-624	642	3	]	]	X
ejde-624	642	4	j.	j.	PROPN
ejde-624	642	5	r.	r.	PROPN
ejde-624	642	6	l.	l.	PROPN
ejde-624	642	7	webb	webb	PROPN
ejde-624	642	8	;	;	PUNCT
ejde-624	642	9	initial	initial	ADJ
ejde-624	642	10	value	value	NOUN
ejde-624	642	11	problems	problem	NOUN
ejde-624	642	12	for	for	ADP
ejde-624	642	13	caputo	caputo	PROPN
ejde-624	642	14	fractional	fractional	PROPN
ejde-624	642	15	equations	equation	NOUN
ejde-624	642	16	with	with	ADP
ejde-624	642	17	singular	singular	ADJ
ejde-624	642	18	nonlinearities	nonlinearitie	NOUN
ejde-624	642	19	,	,	PUNCT
ejde-624	642	20	electron	electron	NOUN
ejde-624	642	21	.	.	PUNCT
ejde-624	643	1	j.	j.	PROPN
ejde-624	643	2	differential	differential	PROPN
ejde-624	643	3	equations	equation	NOUN
ejde-624	643	4	2019	2019	NUM
ejde-624	643	5	,	,	PUNCT
ejde-624	643	6	no	no	INTJ
ejde-624	643	7	.	.	NOUN
ejde-624	643	8	117	117	NUM
ejde-624	643	9	,	,	PUNCT
ejde-624	643	10	32	32	NUM
ejde-624	643	11	pp	pp	NOUN
ejde-624	643	12	.	.	PUNCT
ejde-624	644	1	[	[	X
ejde-624	644	2	22	22	NUM
ejde-624	644	3	]	]	PUNCT
ejde-624	644	4	j.	j.	PROPN
ejde-624	644	5	r.	r.	PROPN
ejde-624	644	6	l.	l.	PROPN
ejde-624	644	7	webb	webb	PROPN
ejde-624	644	8	;	;	PUNCT
ejde-624	644	9	a	a	DET
ejde-624	644	10	fractional	fractional	ADJ
ejde-624	644	11	gronwall	gronwall	ADJ
ejde-624	644	12	inequality	inequality	NOUN
ejde-624	644	13	and	and	CCONJ
ejde-624	644	14	the	the	DET
ejde-624	644	15	asymptotic	asymptotic	ADJ
ejde-624	644	16	behaviour	behaviour	NOUN
ejde-624	644	17	of	of	ADP
ejde-624	644	18	global	global	ADJ
ejde-624	644	19	solutions	solution	NOUN
ejde-624	644	20	of	of	ADP
ejde-624	644	21	caputo	caputo	PROPN
ejde-624	644	22	fractional	fractional	PROPN
ejde-624	644	23	problems	problem	NOUN
ejde-624	644	24	,	,	PUNCT
ejde-624	644	25	electron	electron	PROPN
ejde-624	644	26	.	.	PUNCT
ejde-624	645	1	j.	j.	PROPN
ejde-624	645	2	differential	differential	PROPN
ejde-624	645	3	equations	equation	NOUN
ejde-624	645	4	2021	2021	NUM
ejde-624	645	5	,	,	PUNCT
ejde-624	645	6	paper	paper	NOUN
ejde-624	645	7	no	no	NOUN
ejde-624	645	8	.	.	PROPN
ejde-624	645	9	80	80	NUM
ejde-624	645	10	,	,	PUNCT
ejde-624	645	11	22	22	NUM
ejde-624	645	12	pp	pp	NOUN
ejde-624	645	13	.	.	PUNCT
ejde-624	646	1	[	[	X
ejde-624	646	2	23	23	NUM
ejde-624	646	3	]	]	PUNCT
ejde-624	646	4	j.	j.	PROPN
ejde-624	646	5	r.	r.	PROPN
ejde-624	646	6	l.	l.	PROPN
ejde-624	646	7	webb	webb	PROPN
ejde-624	646	8	;	;	PUNCT
ejde-624	646	9	compactness	compactness	NOUN
ejde-624	646	10	of	of	ADP
ejde-624	646	11	nonlinear	nonlinear	ADJ
ejde-624	646	12	integral	integral	ADJ
ejde-624	646	13	operators	operator	NOUN
ejde-624	646	14	with	with	ADP
ejde-624	646	15	discontinuous	discontinuous	ADJ
ejde-624	646	16	and	and	CCONJ
ejde-624	646	17	with	with	ADP
ejde-624	646	18	singular	singular	ADJ
ejde-624	646	19	kernels	kernel	NOUN
ejde-624	646	20	.	.	PUNCT
ejde-624	647	1	j.	j.	PROPN
ejde-624	647	2	math	math	PROPN
ejde-624	647	3	.	.	PUNCT
ejde-624	648	1	anal	anal	PROPN
ejde-624	648	2	.	.	PUNCT
ejde-624	648	3	appl	appl	PROPN
ejde-624	648	4	.	.	PUNCT
ejde-624	649	1	509	509	NUM
ejde-624	649	2	(	(	PUNCT
ejde-624	649	3	2022	2022	NUM
ejde-624	649	4	)	)	PUNCT
ejde-624	649	5	,	,	PUNCT
ejde-624	649	6	no	no	INTJ
ejde-624	649	7	.	.	NOUN
ejde-624	649	8	2	2	NUM
ejde-624	649	9	,	,	PUNCT
ejde-624	649	10	paper	paper	NOUN
ejde-624	649	11	no	no	NOUN
ejde-624	649	12	.	.	PROPN
ejde-624	649	13	126000	126000	NUM
ejde-624	649	14	,	,	PUNCT
ejde-624	649	15	17	17	NUM
ejde-624	649	16	pp	pp	NOUN
ejde-624	649	17	.	.	PUNCT
ejde-624	650	1	[	[	X
ejde-624	650	2	24	24	NUM
ejde-624	650	3	]	]	PUNCT
ejde-624	650	4	c.	c.	PROPN
ejde-624	650	5	wu	wu	PROPN
ejde-624	650	6	,	,	PUNCT
ejde-624	650	7	x.	x.	PROPN
ejde-624	650	8	liu	liu	PROPN
ejde-624	650	9	;	;	PUNCT
ejde-624	650	10	the	the	DET
ejde-624	650	11	continuation	continuation	NOUN
ejde-624	650	12	of	of	ADP
ejde-624	650	13	solutions	solution	NOUN
ejde-624	650	14	to	to	ADP
ejde-624	650	15	systems	system	NOUN
ejde-624	650	16	of	of	ADP
ejde-624	650	17	caputo	caputo	PROPN
ejde-624	650	18	fractional	fractional	PROPN
ejde-624	650	19	order	order	NOUN
ejde-624	650	20	differential	differential	NOUN
ejde-624	650	21	equations	equation	NOUN
ejde-624	650	22	,	,	PUNCT
ejde-624	650	23	fract	fract	NOUN
ejde-624	650	24	.	.	PUNCT
ejde-624	651	1	calc	calc	PROPN
ejde-624	651	2	.	.	PUNCT
ejde-624	652	1	appl	appl	PROPN
ejde-624	652	2	.	.	PUNCT
ejde-624	653	1	anal	anal	PROPN
ejde-624	653	2	.	.	PUNCT
ejde-624	654	1	23	23	NUM
ejde-624	654	2	(	(	PUNCT
ejde-624	654	3	2020	2020	NUM
ejde-624	654	4	)	)	PUNCT
ejde-624	654	5	,	,	PUNCT
ejde-624	654	6	no	no	INTJ
ejde-624	654	7	.	.	NOUN
ejde-624	654	8	2	2	NUM
ejde-624	654	9	,	,	PUNCT
ejde-624	654	10	591	591	NUM
ejde-624	654	11	-	-	SYM
ejde-624	654	12	599	599	NUM
ejde-624	654	13	.	.	PUNCT
ejde-624	655	1	[	[	X
ejde-624	655	2	25	25	NUM
ejde-624	655	3	]	]	PUNCT
ejde-624	655	4	x.	x.	PROPN
ejde-624	655	5	zhang	zhang	PROPN
ejde-624	655	6	,	,	PUNCT
ejde-624	655	7	l.	l.	PROPN
ejde-624	655	8	liu	liu	PROPN
ejde-624	655	9	,	,	PUNCT
ejde-624	655	10	y.	y.	PROPN
ejde-624	655	11	wu	wu	PROPN
ejde-624	655	12	,	,	PUNCT
ejde-624	655	13	y.	y.	PROPN
ejde-624	655	14	cui	cui	PROPN
ejde-624	655	15	;	;	PUNCT
ejde-624	655	16	new	new	ADJ
ejde-624	655	17	result	result	NOUN
ejde-624	655	18	on	on	ADP
ejde-624	655	19	the	the	DET
ejde-624	655	20	critical	critical	ADJ
ejde-624	655	21	exponent	exponent	NOUN
ejde-624	655	22	for	for	ADP
ejde-624	655	23	solution	solution	NOUN
ejde-624	655	24	of	of	ADP
ejde-624	655	25	an	an	DET
ejde-624	655	26	ordinary	ordinary	ADJ
ejde-624	655	27	fractional	fractional	ADJ
ejde-624	655	28	differential	differential	NOUN
ejde-624	655	29	problem	problem	NOUN
ejde-624	655	30	.	.	PUNCT
ejde-624	656	1	j.	j.	PROPN
ejde-624	656	2	funct	funct	PROPN
ejde-624	656	3	.	.	PUNCT
ejde-624	657	1	spaces	space	VERB
ejde-624	657	2	2017	2017	NUM
ejde-624	657	3	,	,	PUNCT
ejde-624	657	4	art	art	NOUN
ejde-624	657	5	.	.	PUNCT
ejde-624	658	1	i	i	PRON
ejde-624	658	2	d	d	PROPN
ejde-624	658	3	3976469	3976469	NUM
ejde-624	658	4	,	,	PUNCT
ejde-624	658	5	4	4	NUM
ejde-624	658	6	pp	pp	NOUN
ejde-624	658	7	.	.	PUNCT
ejde-624	659	1	[	[	X
ejde-624	659	2	26	26	NUM
ejde-624	659	3	]	]	PUNCT
ejde-624	659	4	t.	t.	PROPN
ejde-624	659	5	zhu	zhu	PROPN
ejde-624	659	6	;	;	PUNCT
ejde-624	659	7	fractional	fractional	ADJ
ejde-624	659	8	integral	integral	ADJ
ejde-624	659	9	inequalities	inequality	NOUN
ejde-624	659	10	and	and	CCONJ
ejde-624	659	11	global	global	ADJ
ejde-624	659	12	solutions	solution	NOUN
ejde-624	659	13	of	of	ADP
ejde-624	659	14	fractional	fractional	ADJ
ejde-624	659	15	differential	differential	ADJ
ejde-624	659	16	equations	equation	NOUN
ejde-624	659	17	.	.	PUNCT
ejde-624	660	1	electron	electron	PROPN
ejde-624	660	2	.	.	PUNCT
ejde-624	661	1	j.	j.	PROPN
ejde-624	661	2	qual	qual	PROPN
ejde-624	661	3	.	.	PROPN
ejde-624	661	4	theory	theory	NOUN
ejde-624	661	5	differ	differ	VERB
ejde-624	661	6	.	.	PUNCT
ejde-624	662	1	equ	equ	PROPN
ejde-624	662	2	.	.	PROPN
ejde-624	662	3	2020	2020	NUM
ejde-624	662	4	,	,	PUNCT
ejde-624	662	5	paper	paper	NOUN
ejde-624	662	6	no	no	NOUN
ejde-624	662	7	.	.	PROPN
ejde-624	662	8	5	5	NUM
ejde-624	662	9	,	,	PUNCT
ejde-624	662	10	16	16	NUM
ejde-624	662	11	pp	pp	NOUN
ejde-624	662	12	.	.	PUNCT
ejde-624	663	1	[	[	X
ejde-624	663	2	27	27	NUM
ejde-624	663	3	]	]	PUNCT
ejde-624	663	4	t.	t.	PROPN
ejde-624	663	5	zhu	zhu	PROPN
ejde-624	663	6	;	;	PUNCT
ejde-624	663	7	weakly	weakly	ADJ
ejde-624	663	8	singular	singular	ADJ
ejde-624	663	9	integral	integral	ADJ
ejde-624	663	10	inequalities	inequality	NOUN
ejde-624	663	11	and	and	CCONJ
ejde-624	663	12	global	global	ADJ
ejde-624	663	13	solutions	solution	NOUN
ejde-624	663	14	for	for	ADP
ejde-624	663	15	fractional	fractional	ADJ
ejde-624	663	16	differential	differential	ADJ
ejde-624	663	17	equations	equation	NOUN
ejde-624	663	18	of	of	ADP
ejde-624	663	19	riemann	riemann	PROPN
ejde-624	663	20	-	-	PUNCT
ejde-624	663	21	liouville	liouville	NOUN
ejde-624	663	22	type	type	NOUN
ejde-624	663	23	.	.	PUNCT
ejde-624	664	1	mediterr	mediterr	PROPN
ejde-624	664	2	.	.	PUNCT
ejde-624	665	1	j.	j.	PROPN
ejde-624	665	2	math	math	PROPN
ejde-624	665	3	.	.	PUNCT
ejde-624	666	1	18	18	NUM
ejde-624	666	2	(	(	PUNCT
ejde-624	666	3	2021	2021	NUM
ejde-624	666	4	)	)	PUNCT
ejde-624	666	5	,	,	PUNCT
ejde-624	666	6	no	no	INTJ
ejde-624	666	7	.	.	NOUN
ejde-624	666	8	5	5	NUM
ejde-624	666	9	,	,	PUNCT
ejde-624	666	10	paper	paper	NOUN
ejde-624	666	11	no	no	NOUN
ejde-624	666	12	.	.	PROPN
ejde-624	667	1	184	184	NUM
ejde-624	667	2	,	,	PUNCT
ejde-624	667	3	17	17	NUM
ejde-624	667	4	pp	pp	NOUN
ejde-624	667	5	.	.	PUNCT
ejde-624	668	1	[	[	X
ejde-624	668	2	28	28	NUM
ejde-624	668	3	]	]	PUNCT
ejde-624	668	4	t.	t.	PROPN
ejde-624	668	5	zhu	zhu	PROPN
ejde-624	668	6	;	;	PUNCT
ejde-624	668	7	attractivity	attractivity	NOUN
ejde-624	668	8	of	of	ADP
ejde-624	668	9	solutions	solution	NOUN
ejde-624	668	10	of	of	ADP
ejde-624	668	11	riemann	riemann	PROPN
ejde-624	668	12	-	-	PUNCT
ejde-624	668	13	liouville	liouville	VERB
ejde-624	668	14	fractional	fractional	ADJ
ejde-624	668	15	differential	differential	NOUN
ejde-624	668	16	equations	equation	NOUN
ejde-624	668	17	.	.	PUNCT
ejde-624	669	1	electron	electron	PROPN
ejde-624	669	2	.	.	PUNCT
ejde-624	670	1	j.	j.	PROPN
ejde-624	670	2	qual	qual	PROPN
ejde-624	670	3	.	.	PUNCT
ejde-624	671	1	theory	theory	NOUN
ejde-624	671	2	.	.	PUNCT
ejde-624	672	1	differ	differ	VERB
ejde-624	672	2	.	.	PUNCT
ejde-624	673	1	equ	equ	PROPN
ejde-624	673	2	.	.	PUNCT
ejde-624	674	1	no	no	INTJ
ejde-624	674	2	.	.	NOUN
ejde-624	674	3	52	52	NUM
ejde-624	674	4	,	,	PUNCT
ejde-624	674	5	1	1	NUM
ejde-624	674	6	-	-	SYM
ejde-624	674	7	12	12	NUM
ejde-624	674	8	(	(	PUNCT
ejde-624	674	9	2022	2022	NUM
ejde-624	674	10	)	)	PUNCT
ejde-624	674	11	.	.	PUNCT
ejde-624	675	1	jeffrey	jeffrey	PROPN
ejde-624	675	2	r.	r.	PROPN
ejde-624	675	3	l.	l.	PROPN
ejde-624	675	4	webb	webb	PROPN
ejde-624	675	5	school	school	PROPN
ejde-624	675	6	of	of	ADP
ejde-624	675	7	mathematics	mathematic	NOUN
ejde-624	675	8	and	and	CCONJ
ejde-624	675	9	statistics	statistic	NOUN
ejde-624	675	10	,	,	PUNCT
ejde-624	675	11	university	university	NOUN
ejde-624	675	12	of	of	ADP
ejde-624	675	13	glasgow	glasgow	PROPN
ejde-624	675	14	,	,	PUNCT
ejde-624	675	15	glasgow	glasgow	PROPN
ejde-624	675	16	g12	g12	PROPN
ejde-624	675	17	8sq	8sq	PROPN
ejde-624	675	18	,	,	PUNCT
ejde-624	675	19	uk	uk	PROPN
ejde-624	675	20	email	email	NOUN
ejde-624	675	21	address	address	NOUN
ejde-624	675	22	:	:	PUNCT
ejde-624	675	23	jeffrey.webb@glasgow.ac.uk	jeffrey.webb@glasgow.ac.uk	NUM
ejde-624	675	24	1	1	X
ejde-624	675	25	.	.	X
ejde-624	675	26	introduction	introduction	NOUN
ejde-624	675	27	some	some	DET
ejde-624	675	28	comments	comment	NOUN
ejde-624	675	29	on	on	ADP
ejde-624	675	30	existence	existence	NOUN
ejde-624	675	31	theorems	theorem	VERB
ejde-624	675	32	2	2	X
ejde-624	675	33	.	.	NOUN
ejde-624	675	34	preliminaries	preliminary	NOUN
ejde-624	675	35	3	3	NUM
ejde-624	675	36	.	.	PUNCT
ejde-624	675	37	riemann	riemann	PROPN
ejde-624	675	38	-	-	PUNCT
ejde-624	675	39	liouville	liouville	NOUN
ejde-624	675	40	equivalences	equivalence	VERB
ejde-624	675	41	4	4	NUM
ejde-624	675	42	.	.	PUNCT
ejde-624	676	1	non	non	ADJ
ejde-624	677	1	-	-	NOUN
ejde-624	678	1	existence	existence	NOUN
ejde-624	678	2	for	for	ADP
ejde-624	678	3	r	r	NOUN
ejde-624	678	4	-	-	PUNCT
ejde-624	678	5	l	l	NOUN
ejde-624	678	6	inequalities	inequality	NOUN
ejde-624	678	7	5	5	NUM
ejde-624	678	8	.	.	PUNCT
ejde-624	679	1	blow	blow	NOUN
ejde-624	679	2	-	-	PUNCT
ejde-624	679	3	up	up	NOUN
ejde-624	679	4	for	for	ADP
ejde-624	679	5	caputo	caputo	PROPN
ejde-624	679	6	derivative	derivative	ADJ
ejde-624	679	7	inequalities	inequality	NOUN
ejde-624	679	8	references	reference	NOUN