id	sid	tid	token	lemma	pos
ejpam-4247	1	1	european	european	PROPN
ejpam-4247	1	2	journal	journal	PROPN
ejpam-4247	1	3	of	of	ADP
ejpam-4247	1	4	pure	pure	ADJ
ejpam-4247	1	5	and	and	CCONJ
ejpam-4247	1	6	applied	apply	VERB
ejpam-4247	1	7	mathematics	mathematic	NOUN
ejpam-4247	1	8	vol	vol	NOUN
ejpam-4247	1	9	.	.	PROPN
ejpam-4247	2	1	15	15	NUM
ejpam-4247	2	2	,	,	PUNCT
ejpam-4247	2	3	no	no	INTJ
ejpam-4247	2	4	.	.	NOUN
ejpam-4247	2	5	1	1	NUM
ejpam-4247	2	6	,	,	PUNCT
ejpam-4247	2	7	2022	2022	NUM
ejpam-4247	2	8	,	,	PUNCT
ejpam-4247	2	9	144	144	NUM
ejpam-4247	2	10	-	-	SYM
ejpam-4247	2	11	157	157	NUM
ejpam-4247	2	12	issn	issn	PROPN
ejpam-4247	2	13	1307	1307	NUM
ejpam-4247	2	14	-	-	SYM
ejpam-4247	2	15	5543	5543	NUM
ejpam-4247	2	16	–	–	PUNCT
ejpam-4247	2	17	ejpam.com	ejpam.com	X
ejpam-4247	2	18	published	publish	VERB
ejpam-4247	2	19	by	by	ADP
ejpam-4247	2	20	new	new	PROPN
ejpam-4247	2	21	york	york	PROPN
ejpam-4247	2	22	business	business	PROPN
ejpam-4247	2	23	global	global	ADJ
ejpam-4247	2	24	periodic	periodic	ADJ
ejpam-4247	2	25	solution	solution	NOUN
ejpam-4247	2	26	of	of	ADP
ejpam-4247	2	27	caputo	caputo	PROPN
ejpam-4247	2	28	-	-	PUNCT
ejpam-4247	2	29	fabrizio	fabrizio	PROPN
ejpam-4247	2	30	fractional	fractional	ADJ
ejpam-4247	2	31	integro	integro	PROPN
ejpam-4247	2	32	–	–	PUNCT
ejpam-4247	2	33	differential	differential	ADJ
ejpam-4247	2	34	equation	equation	NOUN
ejpam-4247	2	35	with	with	ADP
ejpam-4247	2	36	periodic	periodic	ADJ
ejpam-4247	2	37	and	and	CCONJ
ejpam-4247	2	38	integral	integral	ADJ
ejpam-4247	2	39	boundary	boundary	ADJ
ejpam-4247	2	40	conditions	condition	NOUN
ejpam-4247	2	41	ava	ava	PROPN
ejpam-4247	2	42	sh	sh	PROPN
ejpam-4247	2	43	.	.	PROPN
ejpam-4247	2	44	rafeeq	rafeeq	PROPN
ejpam-4247	2	45	department	department	PROPN
ejpam-4247	2	46	of	of	ADP
ejpam-4247	2	47	mathematics	mathematic	NOUN
ejpam-4247	2	48	,	,	PUNCT
ejpam-4247	2	49	faculty	faculty	NOUN
ejpam-4247	2	50	of	of	ADP
ejpam-4247	2	51	science	science	NOUN
ejpam-4247	2	52	,	,	PUNCT
ejpam-4247	2	53	university	university	NOUN
ejpam-4247	2	54	of	of	ADP
ejpam-4247	2	55	zakho	zakho	PROPN
ejpam-4247	2	56	,	,	PUNCT
ejpam-4247	2	57	duhok	duhok	NOUN
ejpam-4247	2	58	,	,	PUNCT
ejpam-4247	2	59	iraq	iraq	PROPN
ejpam-4247	2	60	abstract	abstract	NOUN
ejpam-4247	2	61	.	.	PUNCT
ejpam-4247	3	1	in	in	ADP
ejpam-4247	3	2	this	this	DET
ejpam-4247	3	3	paper	paper	NOUN
ejpam-4247	3	4	,	,	PUNCT
ejpam-4247	3	5	we	we	PRON
ejpam-4247	3	6	study	study	VERB
ejpam-4247	3	7	a	a	DET
ejpam-4247	3	8	new	new	ADJ
ejpam-4247	3	9	approach	approach	NOUN
ejpam-4247	3	10	of	of	ADP
ejpam-4247	3	11	investigation	investigation	NOUN
ejpam-4247	3	12	of	of	ADP
ejpam-4247	3	13	existence	existence	NOUN
ejpam-4247	3	14	,	,	PUNCT
ejpam-4247	3	15	uniqueness	uniqueness	NOUN
ejpam-4247	3	16	and	and	CCONJ
ejpam-4247	3	17	stability	stability	NOUN
ejpam-4247	3	18	of	of	ADP
ejpam-4247	3	19	the	the	DET
ejpam-4247	3	20	periodic	periodic	ADJ
ejpam-4247	3	21	solution	solution	NOUN
ejpam-4247	3	22	of	of	ADP
ejpam-4247	3	23	the	the	DET
ejpam-4247	3	24	nonlinear	nonlinear	ADJ
ejpam-4247	3	25	fractional	fractional	ADJ
ejpam-4247	3	26	integro	integro	ADJ
ejpam-4247	3	27	-	-	PUNCT
ejpam-4247	3	28	differential	differential	NOUN
ejpam-4247	3	29	equation	equation	NOUN
ejpam-4247	3	30	of	of	ADP
ejpam-4247	3	31	type	type	NOUN
ejpam-4247	3	32	caputo	caputo	PROPN
ejpam-4247	3	33	-	-	PUNCT
ejpam-4247	3	34	fabrizio	fabrizio	PROPN
ejpam-4247	3	35	fractional	fractional	ADJ
ejpam-4247	3	36	derivative	derivative	NOUN
ejpam-4247	3	37	with	with	ADP
ejpam-4247	3	38	the	the	DET
ejpam-4247	3	39	initial	initial	ADJ
ejpam-4247	3	40	condition	condition	NOUN
ejpam-4247	3	41	,	,	PUNCT
ejpam-4247	3	42	periodic	periodic	ADJ
ejpam-4247	3	43	boundary	boundary	ADJ
ejpam-4247	3	44	conditions	condition	NOUN
ejpam-4247	3	45	,	,	PUNCT
ejpam-4247	3	46	and	and	CCONJ
ejpam-4247	3	47	integral	integral	ADJ
ejpam-4247	3	48	boundary	boundary	ADJ
ejpam-4247	3	49	conditions	condition	NOUN
ejpam-4247	3	50	by	by	ADP
ejpam-4247	3	51	using	use	VERB
ejpam-4247	3	52	successive	successive	ADJ
ejpam-4247	3	53	approximations	approximation	NOUN
ejpam-4247	3	54	method	method	NOUN
ejpam-4247	3	55	and	and	CCONJ
ejpam-4247	3	56	banach	banach	ADV
ejpam-4247	3	57	fixed	fix	VERB
ejpam-4247	3	58	point	point	NOUN
ejpam-4247	3	59	theorem	theorem	VERB
ejpam-4247	3	60	.	.	PUNCT
ejpam-4247	4	1	finally	finally	ADV
ejpam-4247	4	2	,	,	PUNCT
ejpam-4247	4	3	some	some	DET
ejpam-4247	4	4	examples	example	NOUN
ejpam-4247	4	5	are	be	AUX
ejpam-4247	4	6	present	present	ADJ
ejpam-4247	4	7	to	to	PART
ejpam-4247	4	8	illustrate	illustrate	VERB
ejpam-4247	4	9	the	the	DET
ejpam-4247	4	10	theorems	theorem	NOUN
ejpam-4247	4	11	.	.	PROPN
ejpam-4247	4	12	2020	2020	NUM
ejpam-4247	4	13	mathematics	mathematic	NOUN
ejpam-4247	4	14	subject	subject	NOUN
ejpam-4247	4	15	classifications	classification	NOUN
ejpam-4247	4	16	:	:	PUNCT
ejpam-4247	4	17	34a08	34a08	NUM
ejpam-4247	4	18	,	,	PUNCT
ejpam-4247	4	19	26a33	26a33	NUM
ejpam-4247	4	20	,	,	PUNCT
ejpam-4247	4	21	34g20	34g20	NUM
ejpam-4247	4	22	,	,	PUNCT
ejpam-4247	4	23	34c25	34c25	NUM
ejpam-4247	4	24	,	,	PUNCT
ejpam-4247	4	25	45j05	45j05	ADJ
ejpam-4247	4	26	key	key	ADJ
ejpam-4247	4	27	words	word	NOUN
ejpam-4247	4	28	and	and	CCONJ
ejpam-4247	4	29	phrases	phrase	NOUN
ejpam-4247	4	30	:	:	PUNCT
ejpam-4247	4	31	caputofabrizio	caputofabrizio	PROPN
ejpam-4247	4	32	fractional	fractional	PROPN
ejpam-4247	4	33	derivative	derivative	ADJ
ejpam-4247	4	34	,	,	PUNCT
ejpam-4247	4	35	integro	integro	ADJ
ejpam-4247	4	36	-	-	PUNCT
ejpam-4247	4	37	differential	differential	NOUN
ejpam-4247	4	38	equation	equation	NOUN
ejpam-4247	4	39	,	,	PUNCT
ejpam-4247	4	40	periodic	periodic	ADJ
ejpam-4247	4	41	and	and	CCONJ
ejpam-4247	4	42	integral	integral	ADJ
ejpam-4247	4	43	boundary	boundary	ADJ
ejpam-4247	4	44	conditions	condition	NOUN
ejpam-4247	4	45	,	,	PUNCT
ejpam-4247	4	46	periodic	periodic	ADJ
ejpam-4247	4	47	solutions	solution	NOUN
ejpam-4247	4	48	,	,	PUNCT
ejpam-4247	4	49	successive	successive	ADJ
ejpam-4247	4	50	approximation	approximation	NOUN
ejpam-4247	4	51	method	method	NOUN
ejpam-4247	4	52	,	,	PUNCT
ejpam-4247	4	53	banach	banach	ADV
ejpam-4247	4	54	fixed	fix	VERB
ejpam-4247	4	55	point	point	NOUN
ejpam-4247	4	56	theorem	theorem	VERB
ejpam-4247	4	57	1	1	NUM
ejpam-4247	4	58	.	.	PUNCT
ejpam-4247	4	59	introduction	introduction	NOUN
ejpam-4247	4	60	fractional	fractional	ADJ
ejpam-4247	4	61	differential	differential	NOUN
ejpam-4247	4	62	equations	equation	NOUN
ejpam-4247	4	63	have	have	AUX
ejpam-4247	4	64	been	be	AUX
ejpam-4247	4	65	recognized	recognize	VERB
ejpam-4247	4	66	in	in	ADP
ejpam-4247	4	67	the	the	DET
ejpam-4247	4	68	last	last	ADJ
ejpam-4247	4	69	decade	decade	NOUN
ejpam-4247	4	70	as	as	ADP
ejpam-4247	4	71	important	important	ADJ
ejpam-4247	4	72	tools	tool	NOUN
ejpam-4247	4	73	to	to	PART
ejpam-4247	4	74	describe	describe	VERB
ejpam-4247	4	75	the	the	DET
ejpam-4247	4	76	mathematical	mathematical	ADJ
ejpam-4247	4	77	modeling	modeling	NOUN
ejpam-4247	4	78	of	of	ADP
ejpam-4247	4	79	processes	process	NOUN
ejpam-4247	4	80	in	in	ADP
ejpam-4247	4	81	the	the	DET
ejpam-4247	4	82	fields	field	NOUN
ejpam-4247	4	83	of	of	ADP
ejpam-4247	4	84	physics	physics	NOUN
ejpam-4247	4	85	,	,	PUNCT
ejpam-4247	4	86	chemistry	chemistry	NOUN
ejpam-4247	4	87	,	,	PUNCT
ejpam-4247	4	88	engineering	engineering	NOUN
ejpam-4247	4	89	,	,	PUNCT
ejpam-4247	4	90	statistics	statistic	NOUN
ejpam-4247	4	91	,	,	PUNCT
ejpam-4247	4	92	aerodynamics	aerodynamic	NOUN
ejpam-4247	4	93	,	,	PUNCT
ejpam-4247	4	94	control	control	NOUN
ejpam-4247	4	95	theory	theory	NOUN
ejpam-4247	4	96	,	,	PUNCT
ejpam-4247	4	97	signal	signal	NOUN
ejpam-4247	4	98	and	and	CCONJ
ejpam-4247	4	99	image	image	NOUN
ejpam-4247	4	100	processing	processing	NOUN
ejpam-4247	4	101	,	,	PUNCT
ejpam-4247	4	102	etc.[10	etc.[10	ADV
ejpam-4247	4	103	,	,	PUNCT
ejpam-4247	4	104	11	11	NUM
ejpam-4247	4	105	,	,	PUNCT
ejpam-4247	4	106	13	13	NUM
ejpam-4247	4	107	]	]	PUNCT
ejpam-4247	4	108	.	.	PUNCT
ejpam-4247	5	1	on	on	ADP
ejpam-4247	5	2	the	the	DET
ejpam-4247	5	3	other	other	ADJ
ejpam-4247	5	4	hand	hand	NOUN
ejpam-4247	5	5	,	,	PUNCT
ejpam-4247	5	6	we	we	PRON
ejpam-4247	5	7	observe	observe	VERB
ejpam-4247	5	8	periodic	periodic	ADJ
ejpam-4247	5	9	motions	motion	NOUN
ejpam-4247	5	10	in	in	ADP
ejpam-4247	5	11	every	every	DET
ejpam-4247	5	12	field	field	NOUN
ejpam-4247	5	13	of	of	ADP
ejpam-4247	5	14	science	science	NOUN
ejpam-4247	5	15	and	and	CCONJ
ejpam-4247	5	16	everywhere	everywhere	ADV
ejpam-4247	5	17	in	in	ADP
ejpam-4247	5	18	real	real	ADJ
ejpam-4247	5	19	life	life	NOUN
ejpam-4247	5	20	[	[	X
ejpam-4247	5	21	6	6	NUM
ejpam-4247	5	22	]	]	PUNCT
ejpam-4247	5	23	.	.	PUNCT
ejpam-4247	6	1	the	the	DET
ejpam-4247	6	2	theory	theory	NOUN
ejpam-4247	6	3	and	and	CCONJ
ejpam-4247	6	4	applications	application	NOUN
ejpam-4247	6	5	of	of	ADP
ejpam-4247	6	6	the	the	DET
ejpam-4247	6	7	fractional	fractional	ADJ
ejpam-4247	6	8	differential	differential	ADJ
ejpam-4247	6	9	equations	equation	NOUN
ejpam-4247	6	10	have	have	AUX
ejpam-4247	6	11	recently	recently	ADV
ejpam-4247	6	12	been	be	AUX
ejpam-4247	6	13	addressed	address	VERB
ejpam-4247	6	14	by	by	ADP
ejpam-4247	6	15	several	several	ADJ
ejpam-4247	6	16	researchers	researcher	NOUN
ejpam-4247	6	17	for	for	ADP
ejpam-4247	6	18	a	a	DET
ejpam-4247	6	19	variety	variety	NOUN
ejpam-4247	6	20	of	of	ADP
ejpam-4247	6	21	problems	problem	NOUN
ejpam-4247	6	22	,	,	PUNCT
ejpam-4247	6	23	which	which	PRON
ejpam-4247	6	24	we	we	PRON
ejpam-4247	6	25	refer	refer	VERB
ejpam-4247	6	26	the	the	DET
ejpam-4247	6	27	reader	reader	NOUN
ejpam-4247	6	28	to	to	ADP
ejpam-4247	6	29	[	[	X
ejpam-4247	6	30	1	1	NUM
ejpam-4247	6	31	,	,	PUNCT
ejpam-4247	6	32	2	2	NUM
ejpam-4247	6	33	,	,	PUNCT
ejpam-4247	6	34	4	4	NUM
ejpam-4247	6	35	]	]	PUNCT
ejpam-4247	6	36	.	.	PUNCT
ejpam-4247	7	1	we	we	PRON
ejpam-4247	7	2	mention	mention	VERB
ejpam-4247	7	3	here	here	ADV
ejpam-4247	7	4	some	some	PRON
ejpam-4247	7	5	of	of	ADP
ejpam-4247	7	6	these	these	DET
ejpam-4247	7	7	definitions	definition	NOUN
ejpam-4247	7	8	,	,	PUNCT
ejpam-4247	7	9	such	such	ADJ
ejpam-4247	7	10	as	as	ADP
ejpam-4247	7	11	riemann	riemann	PROPN
ejpam-4247	7	12	-	-	PUNCT
ejpam-4247	7	13	liouville	liouville	VERB
ejpam-4247	7	14	,	,	PUNCT
ejpam-4247	7	15	hadamard	hadamard	ADJ
ejpam-4247	7	16	,	,	PUNCT
ejpam-4247	7	17	grünwald	grünwald	NOUN
ejpam-4247	7	18	-	-	PUNCT
ejpam-4247	7	19	letnikov	letnikov	ADJ
ejpam-4247	7	20	,	,	PUNCT
ejpam-4247	7	21	weyl	weyl	VERB
ejpam-4247	7	22	,	,	PUNCT
ejpam-4247	7	23	riesz	riesz	NOUN
ejpam-4247	7	24	,	,	PUNCT
ejpam-4247	7	25	erdélyi	erdélyi	NOUN
ejpam-4247	7	26	-	-	PUNCT
ejpam-4247	7	27	kober	kober	NOUN
ejpam-4247	7	28	,	,	PUNCT
ejpam-4247	7	29	and	and	CCONJ
ejpam-4247	7	30	caputo	caputo	PROPN
ejpam-4247	7	31	.	.	PROPN
ejpam-4247	8	1	compared	compare	VERB
ejpam-4247	8	2	with	with	ADP
ejpam-4247	8	3	an	an	DET
ejpam-4247	8	4	integer	integer	NOUN
ejpam-4247	8	5	order	order	NOUN
ejpam-4247	8	6	,	,	PUNCT
ejpam-4247	8	7	a	a	DET
ejpam-4247	8	8	significant	significant	ADJ
ejpam-4247	8	9	feature	feature	NOUN
ejpam-4247	8	10	of	of	ADP
ejpam-4247	8	11	a	a	DET
ejpam-4247	8	12	fractional	fractional	ADJ
ejpam-4247	8	13	order	order	NOUN
ejpam-4247	8	14	differential	differential	NOUN
ejpam-4247	8	15	operator	operator	NOUN
ejpam-4247	8	16	appeared	appear	VERB
ejpam-4247	8	17	in	in	ADP
ejpam-4247	8	18	its	its	PRON
ejpam-4247	8	19	hereditary	hereditary	ADJ
ejpam-4247	8	20	property	property	NOUN
ejpam-4247	8	21	.	.	PUNCT
ejpam-4247	9	1	in	in	ADP
ejpam-4247	9	2	other	other	ADJ
ejpam-4247	9	3	words	word	NOUN
ejpam-4247	9	4	,	,	PUNCT
ejpam-4247	9	5	when	when	SCONJ
ejpam-4247	9	6	we	we	PRON
ejpam-4247	9	7	describe	describe	VERB
ejpam-4247	9	8	a	a	DET
ejpam-4247	9	9	process	process	NOUN
ejpam-4247	9	10	by	by	ADP
ejpam-4247	9	11	a	a	DET
ejpam-4247	9	12	fractional	fractional	ADJ
ejpam-4247	9	13	operator	operator	NOUN
ejpam-4247	9	14	,	,	PUNCT
ejpam-4247	9	15	we	we	PRON
ejpam-4247	9	16	predict	predict	VERB
ejpam-4247	9	17	the	the	DET
ejpam-4247	9	18	future	future	ADJ
ejpam-4247	9	19	state	state	NOUN
ejpam-4247	9	20	by	by	ADP
ejpam-4247	9	21	its	its	PRON
ejpam-4247	9	22	current	current	NOUN
ejpam-4247	9	23	as	as	ADV
ejpam-4247	9	24	well	well	ADV
ejpam-4247	9	25	as	as	ADP
ejpam-4247	9	26	its	its	PRON
ejpam-4247	9	27	past	past	ADJ
ejpam-4247	9	28	states	state	NOUN
ejpam-4247	9	29	[	[	X
ejpam-4247	9	30	14	14	NUM
ejpam-4247	9	31	,	,	PUNCT
ejpam-4247	9	32	16	16	NUM
ejpam-4247	9	33	]	]	PUNCT
ejpam-4247	9	34	.	.	PUNCT
ejpam-4247	10	1	however	however	ADV
ejpam-4247	10	2	,	,	PUNCT
ejpam-4247	10	3	the	the	DET
ejpam-4247	10	4	new	new	ADJ
ejpam-4247	10	5	definition	definition	NOUN
ejpam-4247	10	6	suggested	suggest	VERB
ejpam-4247	10	7	by	by	ADP
ejpam-4247	10	8	caputo	caputo	PROPN
ejpam-4247	10	9	and	and	CCONJ
ejpam-4247	10	10	fabrizio	fabrizio	PROPN
ejpam-4247	10	11	[	[	X
ejpam-4247	10	12	5	5	NUM
ejpam-4247	10	13	]	]	PUNCT
ejpam-4247	10	14	,	,	PUNCT
ejpam-4247	10	15	which	which	PRON
ejpam-4247	10	16	has	have	VERB
ejpam-4247	10	17	all	all	DET
ejpam-4247	10	18	the	the	DET
ejpam-4247	10	19	characteristics	characteristic	NOUN
ejpam-4247	10	20	of	of	ADP
ejpam-4247	10	21	the	the	DET
ejpam-4247	10	22	old	old	ADJ
ejpam-4247	10	23	definitions	definition	NOUN
ejpam-4247	10	24	,	,	PUNCT
ejpam-4247	10	25	assumes	assume	VERB
ejpam-4247	10	26	two	two	NUM
ejpam-4247	10	27	different	different	ADJ
ejpam-4247	10	28	representations	representation	NOUN
ejpam-4247	10	29	for	for	ADP
ejpam-4247	10	30	the	the	DET
ejpam-4247	10	31	temporal	temporal	ADJ
ejpam-4247	10	32	and	and	CCONJ
ejpam-4247	10	33	spatial	spatial	ADJ
ejpam-4247	10	34	variables.they	variables.they	PRON
ejpam-4247	10	35	claimed	claim	VERB
ejpam-4247	10	36	that	that	SCONJ
ejpam-4247	10	37	the	the	DET
ejpam-4247	10	38	classical	classical	ADJ
ejpam-4247	10	39	definition	definition	NOUN
ejpam-4247	10	40	given	give	VERB
ejpam-4247	10	41	by	by	ADP
ejpam-4247	10	42	caputo	caputo	PROPN
ejpam-4247	10	43	doi	doi	PROPN
ejpam-4247	10	44	:	:	PUNCT
ejpam-4247	10	45	https://doi.org/10.29020/nybg.ejpam.v15i1.4247	https://doi.org/10.29020/nybg.ejpam.v15i1.4247	ADJ
ejpam-4247	10	46	email	email	NOUN
ejpam-4247	10	47	address	address	NOUN
ejpam-4247	10	48	:	:	PUNCT
ejpam-4247	10	49	ava.rafeeq@uoz.edu.krd	ava.rafeeq@uoz.edu.krd	NOUN
ejpam-4247	10	50	(	(	PUNCT
ejpam-4247	10	51	a.	a.	PROPN
ejpam-4247	10	52	s.	s.	PROPN
ejpam-4247	10	53	rafeeq	rafeeq	PROPN
ejpam-4247	10	54	)	)	PUNCT
ejpam-4247	10	55	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-4247	11	1	144	144	NUM
ejpam-4247	12	1	©	©	PROPN
ejpam-4247	12	2	2022	2022	NUM
ejpam-4247	12	3	ejpam	ejpam	VERB
ejpam-4247	12	4	all	all	DET
ejpam-4247	12	5	rights	right	NOUN
ejpam-4247	12	6	reserved	reserve	VERB
ejpam-4247	12	7	.	.	PUNCT
ejpam-4247	13	1	a.	a.	PROPN
ejpam-4247	13	2	s.	s.	PROPN
ejpam-4247	13	3	rafeeq	rafeeq	PROPN
ejpam-4247	13	4	/	/	SYM
ejpam-4247	13	5	eur	eur	PROPN
ejpam-4247	13	6	.	.	PUNCT
ejpam-4247	14	1	j.	j.	PROPN
ejpam-4247	14	2	pure	pure	PROPN
ejpam-4247	14	3	appl	appl	PROPN
ejpam-4247	14	4	.	.	PROPN
ejpam-4247	14	5	math	math	PROPN
ejpam-4247	14	6	,	,	PUNCT
ejpam-4247	14	7	15	15	NUM
ejpam-4247	14	8	(	(	PUNCT
ejpam-4247	14	9	1	1	NUM
ejpam-4247	14	10	)	)	PUNCT
ejpam-4247	14	11	(	(	PUNCT
ejpam-4247	14	12	2022	2022	NUM
ejpam-4247	14	13	)	)	PUNCT
ejpam-4247	14	14	,	,	PUNCT
ejpam-4247	14	15	144	144	NUM
ejpam-4247	14	16	-	-	SYM
ejpam-4247	14	17	157	157	NUM
ejpam-4247	14	18	145	145	NUM
ejpam-4247	14	19	appears	appear	VERB
ejpam-4247	14	20	to	to	PART
ejpam-4247	14	21	be	be	AUX
ejpam-4247	14	22	particularly	particularly	ADV
ejpam-4247	14	23	convenient	convenient	ADJ
ejpam-4247	14	24	for	for	ADP
ejpam-4247	14	25	mechanical	mechanical	ADJ
ejpam-4247	14	26	phenomena	phenomenon	NOUN
ejpam-4247	14	27	,	,	PUNCT
ejpam-4247	14	28	related	relate	VERB
ejpam-4247	14	29	to	to	ADP
ejpam-4247	14	30	plasticity	plasticity	NOUN
ejpam-4247	14	31	,	,	PUNCT
ejpam-4247	14	32	fatigue	fatigue	NOUN
ejpam-4247	14	33	,	,	PUNCT
ejpam-4247	14	34	damage	damage	NOUN
ejpam-4247	14	35	,	,	PUNCT
ejpam-4247	14	36	and	and	CCONJ
ejpam-4247	14	37	with	with	ADP
ejpam-4247	14	38	electromagnetic	electromagnetic	ADJ
ejpam-4247	14	39	hysteresis	hysteresis	NOUN
ejpam-4247	14	40	.	.	PUNCT
ejpam-4247	15	1	the	the	DET
ejpam-4247	15	2	main	main	ADJ
ejpam-4247	15	3	advantage	advantage	NOUN
ejpam-4247	15	4	of	of	ADP
ejpam-4247	15	5	the	the	DET
ejpam-4247	15	6	caputofabrizio	caputofabrizio	NOUN
ejpam-4247	15	7	approach	approach	NOUN
ejpam-4247	15	8	is	be	AUX
ejpam-4247	15	9	that	that	SCONJ
ejpam-4247	15	10	the	the	DET
ejpam-4247	15	11	boundary	boundary	ADJ
ejpam-4247	15	12	conditions	condition	NOUN
ejpam-4247	15	13	of	of	ADP
ejpam-4247	15	14	the	the	DET
ejpam-4247	15	15	fractional	fractional	ADJ
ejpam-4247	15	16	differential	differential	ADJ
ejpam-4247	15	17	equations	equation	NOUN
ejpam-4247	15	18	with	with	ADP
ejpam-4247	15	19	caputo	caputo	PROPN
ejpam-4247	15	20	-	-	PUNCT
ejpam-4247	15	21	fabrizio	fabrizio	PROPN
ejpam-4247	15	22	derivatives	derivative	NOUN
ejpam-4247	15	23	admit	admit	VERB
ejpam-4247	15	24	the	the	DET
ejpam-4247	15	25	same	same	ADJ
ejpam-4247	15	26	form	form	NOUN
ejpam-4247	15	27	as	as	ADP
ejpam-4247	15	28	for	for	ADP
ejpam-4247	15	29	the	the	DET
ejpam-4247	15	30	integer	integer	NOUN
ejpam-4247	15	31	-	-	PUNCT
ejpam-4247	15	32	order	order	NOUN
ejpam-4247	15	33	differential	differential	ADJ
ejpam-4247	15	34	equations	equation	NOUN
ejpam-4247	15	35	.	.	PUNCT
ejpam-4247	16	1	on	on	ADP
ejpam-4247	16	2	the	the	DET
ejpam-4247	16	3	other	other	ADJ
ejpam-4247	16	4	hand	hand	NOUN
ejpam-4247	16	5	,	,	PUNCT
ejpam-4247	16	6	the	the	DET
ejpam-4247	16	7	caputo	caputo	PROPN
ejpam-4247	16	8	-	-	PUNCT
ejpam-4247	16	9	fabrizio	fabrizio	PROPN
ejpam-4247	16	10	fractional	fractional	ADJ
ejpam-4247	16	11	derivative	derivative	NOUN
ejpam-4247	16	12	has	have	VERB
ejpam-4247	16	13	many	many	ADJ
ejpam-4247	16	14	significant	significant	ADJ
ejpam-4247	16	15	properties	property	NOUN
ejpam-4247	16	16	,	,	PUNCT
ejpam-4247	16	17	such	such	ADJ
ejpam-4247	16	18	as	as	ADP
ejpam-4247	16	19	its	its	PRON
ejpam-4247	16	20	ability	ability	NOUN
ejpam-4247	16	21	in	in	ADP
ejpam-4247	16	22	describing	describe	VERB
ejpam-4247	16	23	matter	matter	NOUN
ejpam-4247	16	24	heterogeneities	heterogeneity	NOUN
ejpam-4247	16	25	and	and	CCONJ
ejpam-4247	16	26	configurations	configuration	NOUN
ejpam-4247	16	27	with	with	ADP
ejpam-4247	16	28	different	different	ADJ
ejpam-4247	16	29	scales	scale	NOUN
ejpam-4247	16	30	[	[	X
ejpam-4247	16	31	12	12	NUM
ejpam-4247	16	32	,	,	PUNCT
ejpam-4247	16	33	20	20	NUM
ejpam-4247	16	34	]	]	PUNCT
ejpam-4247	16	35	.	.	PUNCT
ejpam-4247	17	1	in	in	ADP
ejpam-4247	17	2	[	[	X
ejpam-4247	17	3	21	21	NUM
ejpam-4247	17	4	]	]	PUNCT
ejpam-4247	17	5	,	,	PUNCT
ejpam-4247	17	6	we	we	PRON
ejpam-4247	17	7	have	have	VERB
ejpam-4247	17	8	the	the	DET
ejpam-4247	17	9	analytic	analytic	ADJ
ejpam-4247	17	10	solutions	solution	NOUN
ejpam-4247	17	11	of	of	ADP
ejpam-4247	17	12	a	a	DET
ejpam-4247	17	13	viscous	viscous	ADJ
ejpam-4247	17	14	fluid	fluid	NOUN
ejpam-4247	17	15	with	with	ADP
ejpam-4247	17	16	the	the	DET
ejpam-4247	17	17	caputo	caputo	PROPN
ejpam-4247	17	18	and	and	CCONJ
ejpam-4247	17	19	caputofabrizio	caputofabrizio	PROPN
ejpam-4247	17	20	fractional	fractional	ADJ
ejpam-4247	17	21	derivatives	derivative	NOUN
ejpam-4247	17	22	.	.	PUNCT
ejpam-4247	18	1	in	in	ADP
ejpam-4247	18	2	[	[	X
ejpam-4247	18	3	8	8	NUM
ejpam-4247	18	4	]	]	PUNCT
ejpam-4247	18	5	,	,	PUNCT
ejpam-4247	18	6	the	the	DET
ejpam-4247	18	7	authors	author	NOUN
ejpam-4247	18	8	used	use	VERB
ejpam-4247	18	9	the	the	DET
ejpam-4247	18	10	fractional	fractional	ADJ
ejpam-4247	18	11	derivative	derivative	NOUN
ejpam-4247	18	12	with	with	ADP
ejpam-4247	18	13	a	a	DET
ejpam-4247	18	14	nonsingular	nonsingular	ADJ
ejpam-4247	18	15	kernel	kernel	NOUN
ejpam-4247	18	16	to	to	PART
ejpam-4247	18	17	model	model	VERB
ejpam-4247	18	18	a	a	DET
ejpam-4247	18	19	maxwell	maxwell	PROPN
ejpam-4247	18	20	fluid	fluid	NOUN
ejpam-4247	18	21	and	and	CCONJ
ejpam-4247	18	22	found	find	VERB
ejpam-4247	18	23	semianalytical	semianalytical	ADJ
ejpam-4247	18	24	solutions	solution	NOUN
ejpam-4247	18	25	.	.	PUNCT
ejpam-4247	19	1	in	in	ADP
ejpam-4247	19	2	[	[	X
ejpam-4247	19	3	22	22	NUM
ejpam-4247	19	4	]	]	PUNCT
ejpam-4247	19	5	,	,	PUNCT
ejpam-4247	19	6	we	we	PRON
ejpam-4247	19	7	found	find	VERB
ejpam-4247	19	8	a	a	DET
ejpam-4247	19	9	comparison	comparison	NOUN
ejpam-4247	19	10	approach	approach	NOUN
ejpam-4247	19	11	of	of	ADP
ejpam-4247	19	12	two	two	NUM
ejpam-4247	19	13	latest	late	ADJ
ejpam-4247	19	14	fractional	fractional	ADJ
ejpam-4247	19	15	derivatives	derivative	NOUN
ejpam-4247	19	16	models	model	NOUN
ejpam-4247	19	17	,	,	PUNCT
ejpam-4247	19	18	namely	namely	ADV
ejpam-4247	19	19	,	,	PUNCT
ejpam-4247	19	20	atangana	atangana	PROPN
ejpam-4247	19	21	-	-	PUNCT
ejpam-4247	19	22	baleanu	baleanu	PROPN
ejpam-4247	19	23	and	and	CCONJ
ejpam-4247	19	24	caputofabrizio	caputofabrizio	PROPN
ejpam-4247	19	25	,	,	PUNCT
ejpam-4247	19	26	for	for	ADP
ejpam-4247	19	27	a	a	DET
ejpam-4247	19	28	generalized	generalized	ADJ
ejpam-4247	19	29	casson	casson	NOUN
ejpam-4247	19	30	fluid	fluid	NOUN
ejpam-4247	19	31	and	and	CCONJ
ejpam-4247	19	32	obtained	obtain	VERB
ejpam-4247	19	33	exact	exact	ADJ
ejpam-4247	19	34	solutions	solution	NOUN
ejpam-4247	19	35	.	.	PUNCT
ejpam-4247	20	1	due	due	ADP
ejpam-4247	20	2	to	to	ADP
ejpam-4247	20	3	the	the	DET
ejpam-4247	20	4	abovementioned	abovementioned	ADJ
ejpam-4247	20	5	applications	application	NOUN
ejpam-4247	20	6	,	,	PUNCT
ejpam-4247	20	7	the	the	DET
ejpam-4247	20	8	existence	existence	NOUN
ejpam-4247	20	9	of	of	ADP
ejpam-4247	20	10	solutions	solution	NOUN
ejpam-4247	20	11	for	for	ADP
ejpam-4247	20	12	nonlinear	nonlinear	ADJ
ejpam-4247	20	13	differential	differential	ADJ
ejpam-4247	20	14	equations	equation	NOUN
ejpam-4247	20	15	is	be	AUX
ejpam-4247	20	16	an	an	DET
ejpam-4247	20	17	attractive	attractive	ADJ
ejpam-4247	20	18	research	research	NOUN
ejpam-4247	20	19	topic	topic	NOUN
ejpam-4247	20	20	and	and	CCONJ
ejpam-4247	20	21	has	have	AUX
ejpam-4247	20	22	been	be	AUX
ejpam-4247	20	23	studied	study	VERB
ejpam-4247	20	24	using	use	VERB
ejpam-4247	20	25	different	different	ADJ
ejpam-4247	20	26	techniques	technique	NOUN
ejpam-4247	20	27	of	of	ADP
ejpam-4247	20	28	nonlinear	nonlinear	ADJ
ejpam-4247	20	29	analysis	analysis	NOUN
ejpam-4247	20	30	[	[	X
ejpam-4247	20	31	9	9	NUM
ejpam-4247	20	32	,	,	PUNCT
ejpam-4247	20	33	18	18	NUM
ejpam-4247	20	34	]	]	PUNCT
ejpam-4247	20	35	.	.	PUNCT
ejpam-4247	21	1	one	one	NUM
ejpam-4247	21	2	of	of	ADP
ejpam-4247	21	3	the	the	DET
ejpam-4247	21	4	most	most	ADV
ejpam-4247	21	5	important	important	ADJ
ejpam-4247	21	6	theorems	theorem	NOUN
ejpam-4247	21	7	in	in	ADP
ejpam-4247	21	8	ordinary	ordinary	ADJ
ejpam-4247	21	9	differential	differential	ADJ
ejpam-4247	21	10	equations	equation	NOUN
ejpam-4247	21	11	is	be	AUX
ejpam-4247	21	12	picard	picard	NOUN
ejpam-4247	21	13	’s	’s	PART
ejpam-4247	21	14	existence	existence	NOUN
ejpam-4247	21	15	and	and	CCONJ
ejpam-4247	21	16	uniqueness	uniqueness	NOUN
ejpam-4247	21	17	theorem	theorem	VERB
ejpam-4247	21	18	.	.	PUNCT
ejpam-4247	22	1	this	this	DET
ejpam-4247	22	2	theorem	theorem	NOUN
ejpam-4247	22	3	,	,	PUNCT
ejpam-4247	22	4	which	which	PRON
ejpam-4247	22	5	is	be	AUX
ejpam-4247	22	6	applied	apply	VERB
ejpam-4247	22	7	on	on	ADP
ejpam-4247	22	8	first	first	ADJ
ejpam-4247	22	9	-	-	PUNCT
ejpam-4247	22	10	order	order	NOUN
ejpam-4247	22	11	ordinary	ordinary	ADJ
ejpam-4247	22	12	differential	differential	ADJ
ejpam-4247	22	13	equations	equation	NOUN
ejpam-4247	22	14	,	,	PUNCT
ejpam-4247	22	15	can	can	AUX
ejpam-4247	22	16	be	be	AUX
ejpam-4247	22	17	generalized	generalize	VERB
ejpam-4247	22	18	to	to	PART
ejpam-4247	22	19	establish	establish	VERB
ejpam-4247	22	20	existence	existence	NOUN
ejpam-4247	22	21	and	and	CCONJ
ejpam-4247	22	22	uniqueness	uniqueness	NOUN
ejpam-4247	22	23	results	result	NOUN
ejpam-4247	22	24	for	for	ADP
ejpam-4247	22	25	both	both	CCONJ
ejpam-4247	22	26	higher	high	ADJ
ejpam-4247	22	27	-	-	PUNCT
ejpam-4247	22	28	order	order	NOUN
ejpam-4247	22	29	ordinary	ordinary	ADJ
ejpam-4247	22	30	differential	differential	ADJ
ejpam-4247	22	31	equations	equation	NOUN
ejpam-4247	22	32	and	and	CCONJ
ejpam-4247	22	33	systems	system	NOUN
ejpam-4247	22	34	of	of	ADP
ejpam-4247	22	35	differential	differential	ADJ
ejpam-4247	22	36	equations	equation	NOUN
ejpam-4247	23	1	[	[	X
ejpam-4247	23	2	3	3	NUM
ejpam-4247	23	3	,	,	PUNCT
ejpam-4247	23	4	7	7	NUM
ejpam-4247	23	5	,	,	PUNCT
ejpam-4247	23	6	15	15	NUM
ejpam-4247	23	7	,	,	PUNCT
ejpam-4247	23	8	17	17	NUM
ejpam-4247	23	9	]	]	PUNCT
ejpam-4247	23	10	.	.	PUNCT
ejpam-4247	24	1	in	in	ADP
ejpam-4247	24	2	this	this	DET
ejpam-4247	24	3	paper	paper	NOUN
ejpam-4247	24	4	,	,	PUNCT
ejpam-4247	24	5	we	we	PRON
ejpam-4247	24	6	investigate	investigate	VERB
ejpam-4247	24	7	the	the	DET
ejpam-4247	24	8	existence	existence	NOUN
ejpam-4247	24	9	and	and	CCONJ
ejpam-4247	24	10	approximate	approximate	ADJ
ejpam-4247	24	11	periodic	periodic	ADJ
ejpam-4247	24	12	solution	solution	NOUN
ejpam-4247	24	13	of	of	ADP
ejpam-4247	24	14	the	the	DET
ejpam-4247	24	15	following	follow	VERB
ejpam-4247	24	16	nonlinear	nonlinear	ADJ
ejpam-4247	24	17	fractional	fractional	ADJ
ejpam-4247	24	18	integro	integro	ADJ
ejpam-4247	24	19	-	-	PUNCT
ejpam-4247	24	20	differential	differential	NOUN
ejpam-4247	24	21	equation	equation	NOUN
ejpam-4247	24	22	:	:	PUNCT
ejpam-4247	25	1	cf	cf	NOUN
ejpam-4247	25	2	0	0	NUM
ejpam-4247	26	1	dα	dα	ADP
ejpam-4247	26	2	t	t	PROPN
ejpam-4247	26	3	(	(	PUNCT
ejpam-4247	26	4	u(t	u(t	PROPN
ejpam-4247	26	5	)	)	PUNCT
ejpam-4247	26	6	)	)	PUNCT
ejpam-4247	27	1	=	=	SYM
ejpam-4247	27	2	h	h	NOUN
ejpam-4247	27	3	(	(	PUNCT
ejpam-4247	27	4	t	t	PROPN
ejpam-4247	27	5	,	,	PUNCT
ejpam-4247	27	6	u(t	u(t	PROPN
ejpam-4247	27	7	)	)	PUNCT
ejpam-4247	27	8	,	,	PUNCT
ejpam-4247	27	9	∫	∫	PROPN
ejpam-4247	27	10	a(t	a(t	PROPN
ejpam-4247	27	11	)	)	PUNCT
ejpam-4247	27	12	0	0	PUNCT
ejpam-4247	28	1	g(s	g(s	PROPN
ejpam-4247	28	2	,	,	PUNCT
ejpam-4247	28	3	u(s))ds	u(s))ds	PROPN
ejpam-4247	28	4	)	)	PUNCT
ejpam-4247	28	5	(	(	PUNCT
ejpam-4247	28	6	1.1	1.1	NUM
ejpam-4247	28	7	)	)	PUNCT
ejpam-4247	28	8	such	such	ADJ
ejpam-4247	28	9	that	that	SCONJ
ejpam-4247	28	10	t	t	PROPN
ejpam-4247	28	11	∈	∈	PROPN
ejpam-4247	28	12	j	j	PROPN
ejpam-4247	29	1	=	=	PUNCT
ejpam-4247	30	1	[	[	X
ejpam-4247	30	2	0	0	NUM
ejpam-4247	30	3	,	,	PUNCT
ejpam-4247	30	4	t	t	X
ejpam-4247	30	5	]	]	PUNCT
ejpam-4247	30	6	,	,	PUNCT
ejpam-4247	30	7	with	with	ADP
ejpam-4247	30	8	the	the	DET
ejpam-4247	30	9	initial	initial	ADJ
ejpam-4247	30	10	condition	condition	NOUN
ejpam-4247	30	11	u(0	u(0	NOUN
ejpam-4247	30	12	)	)	PUNCT
ejpam-4247	30	13	=	=	PUNCT
ejpam-4247	30	14	u0	u0	ADJ
ejpam-4247	30	15	,	,	PUNCT
ejpam-4247	30	16	where	where	SCONJ
ejpam-4247	30	17	cf	cf	NOUN
ejpam-4247	30	18	0	0	NUM
ejpam-4247	30	19	dα	dα	PROPN
ejpam-4247	30	20	t	t	PROPN
ejpam-4247	30	21	denotes	denote	VERB
ejpam-4247	30	22	the	the	DET
ejpam-4247	30	23	caputo	caputo	PROPN
ejpam-4247	30	24	-	-	PUNCT
ejpam-4247	30	25	fabrizio	fabrizio	PROPN
ejpam-4247	30	26	fractional	fractional	ADJ
ejpam-4247	30	27	derivative	derivative	NOUN
ejpam-4247	30	28	(	(	PUNCT
ejpam-4247	30	29	α	α	NOUN
ejpam-4247	30	30	∈	∈	PROPN
ejpam-4247	30	31	(	(	PUNCT
ejpam-4247	30	32	0	0	NUM
ejpam-4247	30	33	,	,	PUNCT
ejpam-4247	30	34	1	1	NUM
ejpam-4247	30	35	]	]	NUM
ejpam-4247	30	36	)	)	PUNCT
ejpam-4247	30	37	.	.	PUNCT
ejpam-4247	31	1	we	we	PRON
ejpam-4247	31	2	extend	extend	VERB
ejpam-4247	31	3	picard	picard	PROPN
ejpam-4247	31	4	’s	’s	PART
ejpam-4247	31	5	theorem	theorem	NOUN
ejpam-4247	31	6	to	to	ADP
ejpam-4247	31	7	this	this	DET
ejpam-4247	31	8	problem	problem	NOUN
ejpam-4247	31	9	,	,	PUNCT
ejpam-4247	31	10	and	and	CCONJ
ejpam-4247	31	11	by	by	ADP
ejpam-4247	31	12	the	the	DET
ejpam-4247	31	13	successive	successive	ADJ
ejpam-4247	31	14	approximation	approximation	NOUN
ejpam-4247	31	15	method	method	NOUN
ejpam-4247	31	16	,	,	PUNCT
ejpam-4247	31	17	an	an	DET
ejpam-4247	31	18	iterative	iterative	NOUN
ejpam-4247	31	19	process	process	NOUN
ejpam-4247	31	20	is	be	AUX
ejpam-4247	31	21	provided	provide	VERB
ejpam-4247	31	22	to	to	PART
ejpam-4247	31	23	obtain	obtain	VERB
ejpam-4247	31	24	the	the	DET
ejpam-4247	31	25	periodic	periodic	ADJ
ejpam-4247	31	26	solution	solution	NOUN
ejpam-4247	31	27	.	.	PUNCT
ejpam-4247	32	1	2	2	X
ejpam-4247	32	2	.	.	X
ejpam-4247	32	3	preliminaries	preliminary	NOUN
ejpam-4247	32	4	in	in	ADP
ejpam-4247	32	5	this	this	DET
ejpam-4247	32	6	section	section	NOUN
ejpam-4247	32	7	,	,	PUNCT
ejpam-4247	32	8	we	we	PRON
ejpam-4247	32	9	recall	recall	VERB
ejpam-4247	32	10	some	some	DET
ejpam-4247	32	11	notations	notation	NOUN
ejpam-4247	32	12	and	and	CCONJ
ejpam-4247	32	13	definitions	definition	NOUN
ejpam-4247	32	14	which	which	PRON
ejpam-4247	32	15	are	be	AUX
ejpam-4247	32	16	needed	need	VERB
ejpam-4247	32	17	throughout	throughout	ADP
ejpam-4247	32	18	this	this	DET
ejpam-4247	32	19	paper	paper	NOUN
ejpam-4247	32	20	.	.	PUNCT
ejpam-4247	33	1	further	far	ADV
ejpam-4247	33	2	,	,	PUNCT
ejpam-4247	33	3	some	some	DET
ejpam-4247	33	4	lemmas	lemma	NOUN
ejpam-4247	33	5	and	and	CCONJ
ejpam-4247	33	6	theorems	theorem	NOUN
ejpam-4247	33	7	are	be	AUX
ejpam-4247	33	8	stated	state	VERB
ejpam-4247	33	9	as	as	ADP
ejpam-4247	33	10	preparations	preparation	NOUN
ejpam-4247	33	11	for	for	ADP
ejpam-4247	33	12	the	the	DET
ejpam-4247	33	13	main	main	ADJ
ejpam-4247	33	14	results	result	NOUN
ejpam-4247	33	15	.	.	PUNCT
ejpam-4247	34	1	first	first	ADV
ejpam-4247	34	2	,	,	PUNCT
ejpam-4247	34	3	in	in	ADP
ejpam-4247	34	4	the	the	DET
ejpam-4247	34	5	following	following	NOUN
ejpam-4247	34	6	,	,	PUNCT
ejpam-4247	34	7	we	we	PRON
ejpam-4247	34	8	provide	provide	VERB
ejpam-4247	34	9	some	some	DET
ejpam-4247	34	10	basic	basic	ADJ
ejpam-4247	34	11	concepts	concept	NOUN
ejpam-4247	34	12	and	and	CCONJ
ejpam-4247	34	13	definitions	definition	NOUN
ejpam-4247	34	14	in	in	ADP
ejpam-4247	34	15	connection	connection	NOUN
ejpam-4247	34	16	with	with	ADP
ejpam-4247	34	17	the	the	DET
ejpam-4247	34	18	new	new	PROPN
ejpam-4247	34	19	caputo	caputo	PROPN
ejpam-4247	34	20	-	-	PUNCT
ejpam-4247	34	21	fabrizio	fabrizio	PROPN
ejpam-4247	34	22	derivative	derivative	NOUN
ejpam-4247	34	23	.	.	PUNCT
ejpam-4247	35	1	let	let	VERB
ejpam-4247	35	2	h1(a	h1(a	PRON
ejpam-4247	35	3	,	,	PUNCT
ejpam-4247	35	4	b	b	NOUN
ejpam-4247	35	5	)	)	PUNCT
ejpam-4247	35	6	=	=	SYM
ejpam-4247	35	7	{	{	PUNCT
ejpam-4247	35	8	g|g	g|g	NOUN
ejpam-4247	35	9	∈	∈	PROPN
ejpam-4247	35	10	l2(a	l2(a	PROPN
ejpam-4247	35	11	,	,	PUNCT
ejpam-4247	35	12	b	b	NOUN
ejpam-4247	35	13	)	)	PUNCT
ejpam-4247	35	14	,	,	PUNCT
ejpam-4247	35	15	g′	g′	PROPN
ejpam-4247	35	16	∈	∈	PROPN
ejpam-4247	35	17	l2(a	l2(a	PROPN
ejpam-4247	35	18	,	,	PUNCT
ejpam-4247	35	19	b	b	NOUN
ejpam-4247	35	20	)	)	PUNCT
ejpam-4247	35	21	}	}	PUNCT
ejpam-4247	35	22	,	,	PUNCT
ejpam-4247	35	23	where	where	SCONJ
ejpam-4247	35	24	l2(a	l2(a	NOUN
ejpam-4247	35	25	,	,	PUNCT
ejpam-4247	35	26	b	b	NOUN
ejpam-4247	35	27	)	)	PUNCT
ejpam-4247	35	28	is	be	AUX
ejpam-4247	35	29	the	the	DET
ejpam-4247	35	30	space	space	NOUN
ejpam-4247	35	31	of	of	ADP
ejpam-4247	35	32	square	square	ADJ
ejpam-4247	35	33	integrable	integrable	ADJ
ejpam-4247	35	34	functions	function	NOUN
ejpam-4247	35	35	on	on	ADP
ejpam-4247	35	36	the	the	DET
ejpam-4247	35	37	interval	interval	NOUN
ejpam-4247	35	38	(	(	PUNCT
ejpam-4247	35	39	a	a	DET
ejpam-4247	35	40	,	,	PUNCT
ejpam-4247	35	41	b	b	NOUN
ejpam-4247	35	42	)	)	PUNCT
ejpam-4247	35	43	.	.	PUNCT
ejpam-4247	36	1	definition	definition	NOUN
ejpam-4247	36	2	1	1	NUM
ejpam-4247	36	3	.	.	PUNCT
ejpam-4247	37	1	[	[	X
ejpam-4247	37	2	10	10	NUM
ejpam-4247	37	3	]	]	PUNCT
ejpam-4247	37	4	for	for	ADP
ejpam-4247	37	5	a	a	DET
ejpam-4247	37	6	function	function	NOUN
ejpam-4247	37	7	g	g	NOUN
ejpam-4247	37	8	:	:	PUNCT
ejpam-4247	37	9	(	(	PUNCT
ejpam-4247	37	10	0,∞	0,∞	NOUN
ejpam-4247	37	11	)	)	PUNCT
ejpam-4247	37	12	→	→	SYM
ejpam-4247	37	13	r	r	X
ejpam-4247	37	14	,	,	PUNCT
ejpam-4247	37	15	the	the	DET
ejpam-4247	37	16	caputo	caputo	PROPN
ejpam-4247	37	17	derivative	derivative	NOUN
ejpam-4247	37	18	of	of	ADP
ejpam-4247	37	19	order	order	NOUN
ejpam-4247	37	20	α	α	X
ejpam-4247	37	21	>	>	X
ejpam-4247	37	22	0	0	NUM
ejpam-4247	37	23	of	of	ADP
ejpam-4247	37	24	g	g	PROPN
ejpam-4247	37	25	is	be	AUX
ejpam-4247	37	26	defined	define	VERB
ejpam-4247	37	27	by	by	ADP
ejpam-4247	37	28	t	t	PROPN
ejpam-4247	37	29	0d	0d	X
ejpam-4247	37	30	αg(t	αg(t	PROPN
ejpam-4247	37	31	)	)	PUNCT
ejpam-4247	37	32	=	=	SYM
ejpam-4247	37	33	1	1	NUM
ejpam-4247	37	34	γ(n−	γ(n−	PROPN
ejpam-4247	37	35	α	α	NOUN
ejpam-4247	38	1	)	)	PUNCT
ejpam-4247	38	2	∫	∫	PROPN
ejpam-4247	38	3	t	t	PROPN
ejpam-4247	38	4	0	0	NUM
ejpam-4247	39	1	(	(	PUNCT
ejpam-4247	39	2	t−	t−	PROPN
ejpam-4247	39	3	s)n−α−1g(n)(s)ds	s)n−α−1g(n)(s)ds	NOUN
ejpam-4247	39	4	(	(	PUNCT
ejpam-4247	39	5	2.1	2.1	NUM
ejpam-4247	39	6	)	)	PUNCT
ejpam-4247	39	7	a.	a.	NOUN
ejpam-4247	39	8	s.	s.	PROPN
ejpam-4247	39	9	rafeeq	rafeeq	PROPN
ejpam-4247	39	10	/	/	SYM
ejpam-4247	39	11	eur	eur	PROPN
ejpam-4247	39	12	.	.	PUNCT
ejpam-4247	40	1	j.	j.	PROPN
ejpam-4247	40	2	pure	pure	PROPN
ejpam-4247	40	3	appl	appl	PROPN
ejpam-4247	40	4	.	.	PROPN
ejpam-4247	40	5	math	math	PROPN
ejpam-4247	40	6	,	,	PUNCT
ejpam-4247	40	7	15	15	NUM
ejpam-4247	40	8	(	(	PUNCT
ejpam-4247	40	9	1	1	NUM
ejpam-4247	40	10	)	)	PUNCT
ejpam-4247	40	11	(	(	PUNCT
ejpam-4247	40	12	2022	2022	NUM
ejpam-4247	40	13	)	)	PUNCT
ejpam-4247	40	14	,	,	PUNCT
ejpam-4247	40	15	144	144	NUM
ejpam-4247	40	16	-	-	SYM
ejpam-4247	40	17	157	157	NUM
ejpam-4247	40	18	146	146	NUM
ejpam-4247	41	1	where	where	SCONJ
ejpam-4247	41	2	n	n	NOUN
ejpam-4247	41	3	=	=	PUNCT
ejpam-4247	42	1	[	[	X
ejpam-4247	42	2	α	α	X
ejpam-4247	42	3	]	]	X
ejpam-4247	42	4	+	+	CCONJ
ejpam-4247	42	5	1	1	NUM
ejpam-4247	42	6	and	and	CCONJ
ejpam-4247	42	7	[	[	X
ejpam-4247	42	8	α	α	X
ejpam-4247	42	9	]	]	X
ejpam-4247	42	10	denotes	denote	VERB
ejpam-4247	42	11	the	the	DET
ejpam-4247	42	12	integer	integer	NOUN
ejpam-4247	42	13	part	part	NOUN
ejpam-4247	42	14	of	of	ADP
ejpam-4247	42	15	α	α	NOUN
ejpam-4247	42	16	,	,	PUNCT
ejpam-4247	42	17	and	and	CCONJ
ejpam-4247	42	18	γ	γ	X
ejpam-4247	42	19	(	(	PUNCT
ejpam-4247	42	20	.	.	PUNCT
ejpam-4247	42	21	)	)	PUNCT
ejpam-4247	42	22	denotes	denote	VERB
ejpam-4247	42	23	the	the	DET
ejpam-4247	42	24	gamma	gamma	PROPN
ejpam-4247	42	25	function	function	PROPN
ejpam-4247	42	26	,	,	PUNCT
ejpam-4247	42	27	i.e.	i.e.	X
ejpam-4247	42	28	,	,	PUNCT
ejpam-4247	42	29	γ(z	γ(z	ADJ
ejpam-4247	42	30	)	)	PUNCT
ejpam-4247	43	1	=	=	PUNCT
ejpam-4247	43	2	∫∞	∫∞	NOUN
ejpam-4247	43	3	0	0	PUNCT
ejpam-4247	44	1	e−ttz−1dt	e−ttz−1dt	ADJ
ejpam-4247	44	2	definition	definition	NOUN
ejpam-4247	44	3	2	2	NUM
ejpam-4247	44	4	.	.	PUNCT
ejpam-4247	45	1	[	[	X
ejpam-4247	45	2	10	10	NUM
ejpam-4247	45	3	]	]	PUNCT
ejpam-4247	45	4	let	let	VERB
ejpam-4247	45	5	g	g	PRON
ejpam-4247	45	6	be	be	AUX
ejpam-4247	45	7	a	a	DET
ejpam-4247	45	8	function	function	NOUN
ejpam-4247	45	9	which	which	PRON
ejpam-4247	45	10	is	be	AUX
ejpam-4247	45	11	defined	define	VERB
ejpam-4247	45	12	almost	almost	ADV
ejpam-4247	45	13	everywhere	everywhere	ADV
ejpam-4247	45	14	a.e	a.e	VERB
ejpam-4247	45	15	on	on	ADP
ejpam-4247	45	16	[	[	X
ejpam-4247	45	17	a	a	X
ejpam-4247	45	18	,	,	PUNCT
ejpam-4247	45	19	b	b	NOUN
ejpam-4247	45	20	]	]	X
ejpam-4247	45	21	,	,	PUNCT
ejpam-4247	45	22	for	for	ADP
ejpam-4247	45	23	α	α	PROPN
ejpam-4247	45	24	>	>	X
ejpam-4247	45	25	0	0	NUM
ejpam-4247	45	26	,	,	PUNCT
ejpam-4247	45	27	we	we	PRON
ejpam-4247	45	28	define	define	VERB
ejpam-4247	45	29	b	b	PRON
ejpam-4247	45	30	ad	ad	NOUN
ejpam-4247	45	31	−αf	−αf	X
ejpam-4247	45	32	=	=	SYM
ejpam-4247	45	33	1	1	NUM
ejpam-4247	45	34	γ(α	γ(α	NOUN
ejpam-4247	45	35	)	)	PUNCT
ejpam-4247	46	1	∫	∫	PROPN
ejpam-4247	47	1	b	b	PROPN
ejpam-4247	47	2	a	a	PRON
ejpam-4247	47	3	(	(	PUNCT
ejpam-4247	47	4	b−	b−	PROPN
ejpam-4247	47	5	t)a−1g(t)dt	t)a−1g(t)dt	PRON
ejpam-4247	47	6	(	(	PUNCT
ejpam-4247	47	7	2.2	2.2	NUM
ejpam-4247	47	8	)	)	PUNCT
ejpam-4247	47	9	provided	provide	VERB
ejpam-4247	47	10	that	that	SCONJ
ejpam-4247	47	11	the	the	DET
ejpam-4247	47	12	integral	integral	ADJ
ejpam-4247	47	13	(	(	PUNCT
ejpam-4247	47	14	lebesgue)exists	lebesgue)exists	PROPN
ejpam-4247	47	15	.	.	PUNCT
ejpam-4247	47	16	definition	definition	NOUN
ejpam-4247	47	17	3	3	NUM
ejpam-4247	47	18	.	.	PUNCT
ejpam-4247	48	1	[	[	X
ejpam-4247	48	2	5	5	X
ejpam-4247	48	3	]	]	PUNCT
ejpam-4247	48	4	let	let	VERB
ejpam-4247	48	5	g	g	PRON
ejpam-4247	48	6	be	be	AUX
ejpam-4247	48	7	a	a	DET
ejpam-4247	48	8	given	give	VERB
ejpam-4247	48	9	function	function	NOUN
ejpam-4247	48	10	in	in	ADP
ejpam-4247	48	11	h1(a	h1(a	PROPN
ejpam-4247	48	12	,	,	PUNCT
ejpam-4247	48	13	b	b	NOUN
ejpam-4247	48	14	)	)	PUNCT
ejpam-4247	48	15	.	.	PUNCT
ejpam-4247	49	1	the	the	DET
ejpam-4247	49	2	caputo	caputo	PROPN
ejpam-4247	49	3	-	-	PUNCT
ejpam-4247	49	4	fabrizio	fabrizio	PROPN
ejpam-4247	49	5	derivative	derivative	NOUN
ejpam-4247	49	6	of	of	ADP
ejpam-4247	49	7	fractional	fractional	ADJ
ejpam-4247	49	8	order	order	NOUN
ejpam-4247	49	9	α	α	X
ejpam-4247	49	10	∈	∈	PROPN
ejpam-4247	49	11	(	(	PUNCT
ejpam-4247	49	12	0	0	NUM
ejpam-4247	49	13	,	,	PUNCT
ejpam-4247	49	14	1	1	NUM
ejpam-4247	49	15	)	)	PUNCT
ejpam-4247	49	16	is	be	AUX
ejpam-4247	49	17	defined	define	VERB
ejpam-4247	49	18	as	as	ADP
ejpam-4247	49	19	cf	cf	X
ejpam-4247	49	20	a	a	DET
ejpam-4247	49	21	dα	dα	PROPN
ejpam-4247	49	22	t	t	PROPN
ejpam-4247	49	23	(	(	PUNCT
ejpam-4247	49	24	g(t	g(t	PROPN
ejpam-4247	49	25	)	)	PUNCT
ejpam-4247	49	26	)	)	PUNCT
ejpam-4247	50	1	=	=	PRON
ejpam-4247	50	2	(	(	PUNCT
ejpam-4247	50	3	n(α	n(α	PROPN
ejpam-4247	50	4	)	)	PUNCT
ejpam-4247	50	5	1−	1−	NUM
ejpam-4247	50	6	α	α	NOUN
ejpam-4247	50	7	)	)	PUNCT
ejpam-4247	50	8	∫	∫	PROPN
ejpam-4247	51	1	t	t	PROPN
ejpam-4247	51	2	a	a	DET
ejpam-4247	51	3	g′(x	g′(x	NOUN
ejpam-4247	51	4	)	)	PUNCT
ejpam-4247	51	5	exp	exp	NOUN
ejpam-4247	51	6	[	[	PUNCT
ejpam-4247	51	7	−α	−α	NOUN
ejpam-4247	51	8	t−	t−	PROPN
ejpam-4247	51	9	x	x	PROPN
ejpam-4247	51	10	1−	1−	NUM
ejpam-4247	51	11	α	α	NOUN
ejpam-4247	51	12	]	]	PUNCT
ejpam-4247	51	13	dx	dx	PROPN
ejpam-4247	51	14	(	(	PUNCT
ejpam-4247	51	15	2.3	2.3	NUM
ejpam-4247	51	16	)	)	PUNCT
ejpam-4247	51	17	where	where	SCONJ
ejpam-4247	51	18	n(α	n(α	PROPN
ejpam-4247	51	19	)	)	PUNCT
ejpam-4247	51	20	is	be	AUX
ejpam-4247	51	21	a	a	DET
ejpam-4247	51	22	normalization	normalization	NOUN
ejpam-4247	51	23	function	function	NOUN
ejpam-4247	51	24	.	.	PUNCT
ejpam-4247	52	1	also	also	ADV
ejpam-4247	52	2	,	,	PUNCT
ejpam-4247	52	3	if	if	SCONJ
ejpam-4247	52	4	a	a	DET
ejpam-4247	52	5	certain	certain	ADJ
ejpam-4247	52	6	function	function	NOUN
ejpam-4247	52	7	g	g	NOUN
ejpam-4247	52	8	does	do	AUX
ejpam-4247	52	9	not	not	PART
ejpam-4247	52	10	satisfy	satisfy	VERB
ejpam-4247	52	11	in	in	ADP
ejpam-4247	52	12	the	the	DET
ejpam-4247	52	13	restriction	restriction	NOUN
ejpam-4247	52	14	g	g	PROPN
ejpam-4247	52	15	∈	∈	PROPN
ejpam-4247	52	16	h1(a	h1(a	PROPN
ejpam-4247	52	17	,	,	PUNCT
ejpam-4247	52	18	b	b	NOUN
ejpam-4247	52	19	)	)	PUNCT
ejpam-4247	52	20	,	,	PUNCT
ejpam-4247	52	21	then	then	ADV
ejpam-4247	52	22	its	its	PRON
ejpam-4247	52	23	fractional	fractional	ADJ
ejpam-4247	52	24	derivative	derivative	NOUN
ejpam-4247	52	25	is	be	AUX
ejpam-4247	52	26	redefined	redefine	VERB
ejpam-4247	52	27	as	as	ADP
ejpam-4247	52	28	cf	cf	X
ejpam-4247	52	29	a	a	DET
ejpam-4247	52	30	dα	dα	PROPN
ejpam-4247	52	31	t	t	PROPN
ejpam-4247	52	32	(	(	PUNCT
ejpam-4247	52	33	g(t	g(t	PROPN
ejpam-4247	52	34	)	)	PUNCT
ejpam-4247	52	35	)	)	PUNCT
ejpam-4247	53	1	=	=	PUNCT
ejpam-4247	53	2	αn(α	αn(α	PROPN
ejpam-4247	53	3	)	)	PUNCT
ejpam-4247	53	4	1−	1−	NUM
ejpam-4247	54	1	α	α	NUM
ejpam-4247	54	2	∫	∫	PROPN
ejpam-4247	54	3	t	t	PROPN
ejpam-4247	55	1	a	a	PRON
ejpam-4247	55	2	(	(	PUNCT
ejpam-4247	55	3	g(t)−	g(t)−	PROPN
ejpam-4247	55	4	g(x	g(x	PROPN
ejpam-4247	55	5	)	)	PUNCT
ejpam-4247	55	6	)	)	PUNCT
ejpam-4247	55	7	exp	exp	NOUN
ejpam-4247	55	8	[	[	PUNCT
ejpam-4247	55	9	−α	−α	NOUN
ejpam-4247	55	10	t−	t−	PROPN
ejpam-4247	55	11	x	x	PROPN
ejpam-4247	55	12	1−	1−	NUM
ejpam-4247	55	13	α	α	NOUN
ejpam-4247	55	14	]	]	PUNCT
ejpam-4247	55	15	dx	dx	PROPN
ejpam-4247	55	16	(	(	PUNCT
ejpam-4247	55	17	2.4	2.4	NUM
ejpam-4247	55	18	)	)	PUNCT
ejpam-4247	55	19	clearly	clearly	ADV
ejpam-4247	55	20	,	,	PUNCT
ejpam-4247	55	21	if	if	SCONJ
ejpam-4247	55	22	one	one	NUM
ejpam-4247	55	23	sets	set	VERB
ejpam-4247	55	24	σ	σ	NOUN
ejpam-4247	55	25	=	=	SYM
ejpam-4247	55	26	(	(	PUNCT
ejpam-4247	55	27	1−	1−	NUM
ejpam-4247	55	28	α)/α	α)/α	PROPN
ejpam-4247	55	29	∈	∈	PROPN
ejpam-4247	55	30	(	(	PUNCT
ejpam-4247	55	31	0,∞	0,∞	NOUN
ejpam-4247	55	32	)	)	PUNCT
ejpam-4247	55	33	and	and	CCONJ
ejpam-4247	55	34	α	α	X
ejpam-4247	55	35	=	=	SYM
ejpam-4247	56	1	1/(1	1/(1	PROPN
ejpam-4247	56	2	+	+	NUM
ejpam-4247	56	3	σ	σ	PROPN
ejpam-4247	56	4	)	)	PUNCT
ejpam-4247	56	5	∈	∈	PROPN
ejpam-4247	56	6	(	(	PUNCT
ejpam-4247	56	7	0	0	NUM
ejpam-4247	56	8	,	,	PUNCT
ejpam-4247	56	9	1	1	NUM
ejpam-4247	56	10	)	)	PUNCT
ejpam-4247	56	11	,	,	PUNCT
ejpam-4247	56	12	then	then	ADV
ejpam-4247	56	13	the	the	DET
ejpam-4247	56	14	caputofabrizio	caputofabrizio	PROPN
ejpam-4247	56	15	definition	definition	NOUN
ejpam-4247	56	16	becomes	become	VERB
ejpam-4247	56	17	cf	cf	NOUN
ejpam-4247	56	18	a	a	DET
ejpam-4247	56	19	de	de	X
ejpam-4247	56	20	t	t	PROPN
ejpam-4247	56	21	(	(	PUNCT
ejpam-4247	56	22	g(t	g(t	PROPN
ejpam-4247	56	23	)	)	PUNCT
ejpam-4247	56	24	)	)	PUNCT
ejpam-4247	57	1	=	=	SYM
ejpam-4247	57	2	n(σ	n(σ	PROPN
ejpam-4247	57	3	)	)	PUNCT
ejpam-4247	58	1	σ	σ	PROPN
ejpam-4247	58	2	∫	∫	PROPN
ejpam-4247	58	3	t	t	PROPN
ejpam-4247	58	4	a	a	DET
ejpam-4247	58	5	g′(x	g′(x	NOUN
ejpam-4247	58	6	)	)	PUNCT
ejpam-4247	58	7	exp	exp	NOUN
ejpam-4247	58	8	[	[	PUNCT
ejpam-4247	58	9	−	−	PROPN
ejpam-4247	58	10	t−	t−	PROPN
ejpam-4247	58	11	x	x	PROPN
ejpam-4247	58	12	σ	σ	X
ejpam-4247	58	13	]	]	PUNCT
ejpam-4247	58	14	dx	dx	PROPN
ejpam-4247	58	15	(	(	PUNCT
ejpam-4247	58	16	2.5	2.5	NUM
ejpam-4247	58	17	)	)	PUNCT
ejpam-4247	58	18	where	where	SCONJ
ejpam-4247	58	19	n(0	n(0	NOUN
ejpam-4247	58	20	)	)	PUNCT
ejpam-4247	58	21	=	=	SYM
ejpam-4247	58	22	n(∞	n(∞	NOUN
ejpam-4247	58	23	)	)	PUNCT
ejpam-4247	58	24	=	=	SYM
ejpam-4247	58	25	1	1	NUM
ejpam-4247	58	26	,	,	PUNCT
ejpam-4247	58	27	and	and	CCONJ
ejpam-4247	58	28	lim	lim	PROPN
ejpam-4247	58	29	σ→0	σ→0	VERB
ejpam-4247	58	30	exp	exp	NOUN
ejpam-4247	58	31	[	[	PUNCT
ejpam-4247	58	32	−	−	PROPN
ejpam-4247	58	33	t−	t−	PROPN
ejpam-4247	58	34	x	x	SYM
ejpam-4247	58	35	σ	σ	NOUN
ejpam-4247	58	36	]	]	PUNCT
ejpam-4247	58	37	=	=	SYM
ejpam-4247	58	38	δ(x−	δ(x−	X
ejpam-4247	58	39	t	t	PROPN
ejpam-4247	58	40	)	)	PUNCT
ejpam-4247	58	41	.	.	PUNCT
ejpam-4247	59	1	(	(	PUNCT
ejpam-4247	59	2	2.6	2.6	NUM
ejpam-4247	59	3	)	)	PUNCT
ejpam-4247	59	4	also	also	ADV
ejpam-4247	59	5	,	,	PUNCT
ejpam-4247	59	6	the	the	DET
ejpam-4247	59	7	fractional	fractional	ADJ
ejpam-4247	59	8	derivative	derivative	NOUN
ejpam-4247	59	9	of	of	ADP
ejpam-4247	59	10	order	order	NOUN
ejpam-4247	59	11	(	(	PUNCT
ejpam-4247	59	12	n+α	n+α	NUM
ejpam-4247	59	13	)	)	PUNCT
ejpam-4247	59	14	when	when	SCONJ
ejpam-4247	59	15	n	n	X
ejpam-4247	59	16	≥	≥	X
ejpam-4247	59	17	1	1	NUM
ejpam-4247	59	18	and	and	CCONJ
ejpam-4247	59	19	α	α	PRON
ejpam-4247	59	20	∈	∈	PROPN
ejpam-4247	60	1	[	[	X
ejpam-4247	60	2	0	0	NUM
ejpam-4247	60	3	,	,	PUNCT
ejpam-4247	60	4	1	1	NUM
ejpam-4247	60	5	]	]	PUNCT
ejpam-4247	60	6	is	be	AUX
ejpam-4247	60	7	defined	define	VERB
ejpam-4247	60	8	by	by	ADP
ejpam-4247	60	9	the	the	DET
ejpam-4247	60	10	following	follow	VERB
ejpam-4247	60	11	cf	cf	NOUN
ejpam-4247	60	12	a	a	PRON
ejpam-4247	60	13	d	d	X
ejpam-4247	60	14	(	(	PUNCT
ejpam-4247	60	15	a+n	a+n	PROPN
ejpam-4247	60	16	)	)	PUNCT
ejpam-4247	60	17	t	t	PROPN
ejpam-4247	60	18	(	(	PUNCT
ejpam-4247	60	19	g(t	g(t	PROPN
ejpam-4247	60	20	)	)	PUNCT
ejpam-4247	60	21	)	)	PUNCT
ejpam-4247	61	1	=	=	PUNCT
ejpam-4247	61	2	αcfd	αcfd	NOUN
ejpam-4247	61	3	(	(	PUNCT
ejpam-4247	61	4	a	a	NOUN
ejpam-4247	61	5	)	)	PUNCT
ejpam-4247	61	6	t	t	NOUN
ejpam-4247	61	7	(	(	PUNCT
ejpam-4247	61	8	d	d	X
ejpam-4247	61	9	(	(	PUNCT
ejpam-4247	61	10	n	n	CCONJ
ejpam-4247	61	11	)	)	PUNCT
ejpam-4247	61	12	t	t	PROPN
ejpam-4247	61	13	g(t	g(t	PROPN
ejpam-4247	61	14	)	)	PUNCT
ejpam-4247	61	15	)	)	PUNCT
ejpam-4247	61	16	(	(	PUNCT
ejpam-4247	61	17	2.7	2.7	NUM
ejpam-4247	61	18	)	)	PUNCT
ejpam-4247	61	19	definition	definition	NOUN
ejpam-4247	61	20	4	4	NUM
ejpam-4247	61	21	.	.	PUNCT
ejpam-4247	62	1	[	[	X
ejpam-4247	62	2	5	5	X
ejpam-4247	62	3	]	]	PUNCT
ejpam-4247	62	4	let	let	VERB
ejpam-4247	62	5	g	g	PROPN
ejpam-4247	62	6	∈	∈	PROPN
ejpam-4247	62	7	h1(a	h1(a	PROPN
ejpam-4247	62	8	,	,	PUNCT
ejpam-4247	62	9	b	b	NOUN
ejpam-4247	62	10	)	)	PUNCT
ejpam-4247	62	11	,	,	PUNCT
ejpam-4247	62	12	then	then	ADV
ejpam-4247	62	13	its	its	PRON
ejpam-4247	62	14	fractional	fractional	ADJ
ejpam-4247	62	15	integral	integral	NOUN
ejpam-4247	62	16	of	of	ADP
ejpam-4247	62	17	an	an	DET
ejpam-4247	62	18	arbitrary	arbitrary	ADJ
ejpam-4247	62	19	order	order	NOUN
ejpam-4247	62	20	is	be	AUX
ejpam-4247	62	21	defined	define	VERB
ejpam-4247	62	22	as	as	SCONJ
ejpam-4247	62	23	follows	follow	VERB
ejpam-4247	62	24	:	:	PUNCT
ejpam-4247	62	25	αc	αc	NUM
ejpam-4247	62	26	x	x	SYM
ejpam-4247	62	27	t	t	PROPN
ejpam-4247	62	28	(	(	PUNCT
ejpam-4247	62	29	g(t	g(t	PROPN
ejpam-4247	62	30	)	)	PUNCT
ejpam-4247	62	31	)	)	PUNCT
ejpam-4247	62	32	=	=	PUNCT
ejpam-4247	63	1	2(1−	2(1−	NUM
ejpam-4247	63	2	α	α	X
ejpam-4247	63	3	)	)	PUNCT
ejpam-4247	63	4	(	(	PUNCT
ejpam-4247	63	5	2−	2−	NUM
ejpam-4247	63	6	α)n(α	α)n(α	NOUN
ejpam-4247	63	7	)	)	PUNCT
ejpam-4247	63	8	g(t	g(t	PROPN
ejpam-4247	63	9	)	)	PUNCT
ejpam-4247	64	1	+	+	CCONJ
ejpam-4247	64	2	2α	2α	NOUN
ejpam-4247	64	3	(	(	PUNCT
ejpam-4247	64	4	2−	2−	NUM
ejpam-4247	64	5	α)n(α	α)n(α	NOUN
ejpam-4247	64	6	)	)	PUNCT
ejpam-4247	65	1	∫	∫	PROPN
ejpam-4247	65	2	t	t	PROPN
ejpam-4247	65	3	a	a	DET
ejpam-4247	65	4	g(s)ds	g(s)ds	PROPN
ejpam-4247	65	5	,	,	PUNCT
ejpam-4247	65	6	t	t	PROPN
ejpam-4247	65	7	≥	≥	NOUN
ejpam-4247	65	8	0	0	NUM
ejpam-4247	65	9	(	(	PUNCT
ejpam-4247	65	10	2.8	2.8	NUM
ejpam-4247	65	11	)	)	PUNCT
ejpam-4247	65	12	it	it	PRON
ejpam-4247	65	13	is	be	AUX
ejpam-4247	65	14	dear	dear	ADJ
ejpam-4247	65	15	,	,	PUNCT
ejpam-4247	65	16	in	in	ADP
ejpam-4247	65	17	view	view	NOUN
ejpam-4247	65	18	of	of	ADP
ejpam-4247	65	19	the	the	DET
ejpam-4247	65	20	abowe	abowe	NOUN
ejpam-4247	65	21	definition	definition	NOUN
ejpam-4247	65	22	,	,	PUNCT
ejpam-4247	65	23	that	that	SCONJ
ejpam-4247	65	24	the	the	DET
ejpam-4247	65	25	α	α	PROPN
ejpam-4247	65	26	th	th	X
ejpam-4247	65	27	caputo	caputo	PROPN
ejpam-4247	65	28	-	-	PUNCT
ejpam-4247	65	29	fabrizio	fabrizio	PROPN
ejpam-4247	65	30	derivative	derivative	NOUN
ejpam-4247	65	31	of	of	ADP
ejpam-4247	65	32	function	function	NOUN
ejpam-4247	65	33	g	g	PROPN
ejpam-4247	65	34	is	be	AUX
ejpam-4247	65	35	average	average	ADJ
ejpam-4247	65	36	between	between	ADP
ejpam-4247	65	37	g	g	PROPN
ejpam-4247	65	38	and	and	CCONJ
ejpam-4247	65	39	its	its	PRON
ejpam-4247	65	40	first	first	ADJ
ejpam-4247	65	41	-	-	PUNCT
ejpam-4247	65	42	order	order	NOUN
ejpam-4247	65	43	integral	integral	ADJ
ejpam-4247	65	44	.	.	PUNCT
ejpam-4247	66	1	therefore	therefore	ADV
ejpam-4247	66	2	,	,	PUNCT
ejpam-4247	66	3	2(1−	2(1−	PROPN
ejpam-4247	66	4	α	α	X
ejpam-4247	66	5	)	)	PUNCT
ejpam-4247	66	6	(	(	PUNCT
ejpam-4247	66	7	2−	2−	NUM
ejpam-4247	66	8	α)n(α	α)n(α	NOUN
ejpam-4247	66	9	)	)	PUNCT
ejpam-4247	67	1	+	+	NUM
ejpam-4247	67	2	2α	2α	NOUN
ejpam-4247	67	3	(	(	PUNCT
ejpam-4247	67	4	2−	2−	NUM
ejpam-4247	67	5	α)n(α	α)n(α	NOUN
ejpam-4247	67	6	)	)	PUNCT
ejpam-4247	67	7	=	=	SYM
ejpam-4247	67	8	1	1	NUM
ejpam-4247	67	9	(	(	PUNCT
ejpam-4247	67	10	2.9	2.9	NUM
ejpam-4247	67	11	)	)	PUNCT
ejpam-4247	67	12	so	so	ADV
ejpam-4247	67	13	,	,	PUNCT
ejpam-4247	67	14	we	we	PRON
ejpam-4247	67	15	arrive	arrive	VERB
ejpam-4247	67	16	at	at	ADP
ejpam-4247	67	17	the	the	DET
ejpam-4247	67	18	following	follow	VERB
ejpam-4247	67	19	n(α	n(α	PROPN
ejpam-4247	67	20	)	)	PUNCT
ejpam-4247	67	21	=	=	PUNCT
ejpam-4247	68	1	2	2	NUM
ejpam-4247	68	2	2−	2−	NUM
ejpam-4247	68	3	α	α	NOUN
ejpam-4247	68	4	,	,	PUNCT
ejpam-4247	68	5	0	0	NUM
ejpam-4247	68	6	≤	≤	NUM
ejpam-4247	68	7	α	α	NOUN
ejpam-4247	68	8	≤	≤	NUM
ejpam-4247	68	9	1	1	NUM
ejpam-4247	68	10	(	(	PUNCT
ejpam-4247	68	11	2.10	2.10	NUM
ejpam-4247	68	12	)	)	PUNCT
ejpam-4247	68	13	a.	a.	NOUN
ejpam-4247	68	14	s.	s.	PROPN
ejpam-4247	68	15	rafeeq	rafeeq	PROPN
ejpam-4247	68	16	/	/	SYM
ejpam-4247	68	17	eur	eur	PROPN
ejpam-4247	68	18	.	.	PUNCT
ejpam-4247	69	1	j.	j.	PROPN
ejpam-4247	69	2	pure	pure	PROPN
ejpam-4247	69	3	appl	appl	PROPN
ejpam-4247	69	4	.	.	PROPN
ejpam-4247	69	5	math	math	PROPN
ejpam-4247	69	6	,	,	PUNCT
ejpam-4247	69	7	15	15	NUM
ejpam-4247	69	8	(	(	PUNCT
ejpam-4247	69	9	1	1	NUM
ejpam-4247	69	10	)	)	PUNCT
ejpam-4247	69	11	(	(	PUNCT
ejpam-4247	69	12	2022	2022	NUM
ejpam-4247	69	13	)	)	PUNCT
ejpam-4247	69	14	,	,	PUNCT
ejpam-4247	69	15	144	144	NUM
ejpam-4247	69	16	-	-	SYM
ejpam-4247	69	17	157	157	NUM
ejpam-4247	69	18	147	147	NUM
ejpam-4247	69	19	definition	definition	NOUN
ejpam-4247	69	20	5	5	NUM
ejpam-4247	69	21	.	.	PUNCT
ejpam-4247	70	1	the	the	DET
ejpam-4247	70	2	periodic	periodic	ADJ
ejpam-4247	70	3	solution	solution	NOUN
ejpam-4247	70	4	of	of	ADP
ejpam-4247	70	5	the	the	DET
ejpam-4247	70	6	fractional	fractional	ADJ
ejpam-4247	70	7	integro	integro	ADJ
ejpam-4247	70	8	-	-	PUNCT
ejpam-4247	70	9	differential	differential	NOUN
ejpam-4247	70	10	equation	equation	NOUN
ejpam-4247	70	11	(	(	PUNCT
ejpam-4247	70	12	1.1	1.1	NUM
ejpam-4247	70	13	)	)	PUNCT
ejpam-4247	70	14	,	,	PUNCT
ejpam-4247	70	15	with	with	ADP
ejpam-4247	70	16	initial	initial	ADJ
ejpam-4247	70	17	condition	condition	NOUN
ejpam-4247	70	18	u(0	u(0	NOUN
ejpam-4247	70	19	)	)	PUNCT
ejpam-4247	70	20	=	=	PUNCT
ejpam-4247	70	21	u0	u0	ADJ
ejpam-4247	70	22	and	and	CCONJ
ejpam-4247	70	23	periodic	periodic	ADJ
ejpam-4247	70	24	boundary	boundary	ADJ
ejpam-4247	70	25	condition	condition	NOUN
ejpam-4247	70	26	u(0	u(0	NOUN
ejpam-4247	70	27	)	)	PUNCT
ejpam-4247	70	28	=	=	SYM
ejpam-4247	70	29	u(t	u(t	NOUN
ejpam-4247	70	30	)	)	PUNCT
ejpam-4247	70	31	are	be	AUX
ejpam-4247	70	32	defining	define	VERB
ejpam-4247	70	33	the	the	DET
ejpam-4247	70	34	following	follow	VERB
ejpam-4247	70	35	integral	integral	ADJ
ejpam-4247	70	36	equation	equation	NOUN
ejpam-4247	70	37	u	u	PROPN
ejpam-4247	70	38	(	(	PUNCT
ejpam-4247	70	39	t	t	PROPN
ejpam-4247	70	40	,	,	PUNCT
ejpam-4247	70	41	u0	u0	ADJ
ejpam-4247	70	42	)	)	PUNCT
ejpam-4247	70	43	=	=	SYM
ejpam-4247	70	44	u0	u0	ADJ
ejpam-4247	70	45	+	+	X
ejpam-4247	70	46	2(1−α	2(1−α	NUM
ejpam-4247	70	47	)	)	PUNCT
ejpam-4247	70	48	(	(	PUNCT
ejpam-4247	70	49	2−α)n(α)h(t	2−α)n(α)h(t	NUM
ejpam-4247	70	50	,	,	PUNCT
ejpam-4247	70	51	u(t	u(t	PROPN
ejpam-4247	70	52	)	)	PUNCT
ejpam-4247	70	53	,	,	PUNCT
ejpam-4247	70	54	∫	∫	PROPN
ejpam-4247	70	55	a(t	a(t	PROPN
ejpam-4247	70	56	)	)	PUNCT
ejpam-4247	70	57	0	0	PUNCT
ejpam-4247	71	1	g(s	g(s	PROPN
ejpam-4247	71	2	,	,	PUNCT
ejpam-4247	71	3	u(s))ds	u(s))ds	PROPN
ejpam-4247	71	4	)	)	PUNCT
ejpam-4247	71	5	−	−	PROPN
ejpam-4247	71	6	(	(	PUNCT
ejpam-4247	71	7	2(1−α	2(1−α	NUM
ejpam-4247	71	8	)	)	PUNCT
ejpam-4247	71	9	(	(	PUNCT
ejpam-4247	71	10	2−α)n(α	2−α)n(α	X
ejpam-4247	71	11	)	)	PUNCT
ejpam-4247	71	12	1	1	NUM
ejpam-4247	71	13	t	t	NOUN
ejpam-4247	71	14	∫	∫	PROPN
ejpam-4247	71	15	t	t	PROPN
ejpam-4247	71	16	0	0	NUM
ejpam-4247	71	17	h(s	h(s	PROPN
ejpam-4247	71	18	,	,	PUNCT
ejpam-4247	71	19	u(s	u(s	NUM
ejpam-4247	71	20	)	)	PUNCT
ejpam-4247	71	21	,	,	PUNCT
ejpam-4247	71	22	∫	∫	PROPN
ejpam-4247	71	23	a(s	a(s	PROPN
ejpam-4247	71	24	)	)	PUNCT
ejpam-4247	71	25	0	0	NUM
ejpam-4247	71	26	g(τ	g(τ	PROPN
ejpam-4247	71	27	,	,	PUNCT
ejpam-4247	71	28	u(τ))dτ)ds+	u(τ))dτ)ds+	ADJ
ejpam-4247	71	29	2α	2α	NOUN
ejpam-4247	71	30	(	(	PUNCT
ejpam-4247	71	31	2−α)n(α)∫	2−α)n(α)∫	NUM
ejpam-4247	71	32	t	t	NOUN
ejpam-4247	71	33	0	0	NUM
ejpam-4247	71	34	(	(	PUNCT
ejpam-4247	71	35	h(s	h(s	PROPN
ejpam-4247	71	36	,	,	PUNCT
ejpam-4247	71	37	u(s	u(s	NUM
ejpam-4247	71	38	)	)	PUNCT
ejpam-4247	71	39	,	,	PUNCT
ejpam-4247	71	40	∫	∫	PROPN
ejpam-4247	71	41	a(s	a(s	PROPN
ejpam-4247	71	42	)	)	PUNCT
ejpam-4247	71	43	0	0	NUM
ejpam-4247	72	1	g(τ	g(τ	NOUN
ejpam-4247	72	2	,	,	PUNCT
ejpam-4247	72	3	u(τ))dτ)−	u(τ))dτ)−	NOUN
ejpam-4247	72	4	1	1	NUM
ejpam-4247	72	5	t	t	NOUN
ejpam-4247	72	6	∫	∫	PROPN
ejpam-4247	72	7	t	t	PROPN
ejpam-4247	72	8	0	0	NUM
ejpam-4247	72	9	h(s	h(s	PROPN
ejpam-4247	72	10	,	,	PUNCT
ejpam-4247	72	11	u(s	u(s	NUM
ejpam-4247	72	12	)	)	PUNCT
ejpam-4247	72	13	,	,	PUNCT
ejpam-4247	72	14	∫	∫	PROPN
ejpam-4247	72	15	a(s	a(s	PROPN
ejpam-4247	72	16	)	)	PUNCT
ejpam-4247	72	17	0	0	NUM
ejpam-4247	73	1	g(τ	g(τ	PROPN
ejpam-4247	73	2	,	,	PUNCT
ejpam-4247	73	3	u(τ))dτ)ds)ds	u(τ))dτ)ds)ds	PROPN
ejpam-4247	73	4	(	(	PUNCT
ejpam-4247	73	5	2.11	2.11	NUM
ejpam-4247	73	6	)	)	PUNCT
ejpam-4247	73	7	for	for	ADP
ejpam-4247	73	8	all	all	PRON
ejpam-4247	73	9	t	t	NOUN
ejpam-4247	73	10	∈	∈	PROPN
ejpam-4247	74	1	j	j	PROPN
ejpam-4247	74	2	.	.	PUNCT
ejpam-4247	75	1	lemma	lemma	PROPN
ejpam-4247	75	2	1	1	NUM
ejpam-4247	75	3	.	.	PUNCT
ejpam-4247	76	1	[	[	X
ejpam-4247	76	2	19	19	NUM
ejpam-4247	76	3	]	]	X
ejpam-4247	76	4	let	let	VERB
ejpam-4247	76	5	g(t	g(t	PROPN
ejpam-4247	76	6	)	)	PUNCT
ejpam-4247	76	7	be	be	AUX
ejpam-4247	76	8	a	a	DET
ejpam-4247	76	9	vector	vector	NOUN
ejpam-4247	76	10	function	function	NOUN
ejpam-4247	76	11	which	which	PRON
ejpam-4247	76	12	is	be	AUX
ejpam-4247	76	13	defined	define	VERB
ejpam-4247	76	14	in	in	ADP
ejpam-4247	76	15	the	the	DET
ejpam-4247	76	16	interval	interval	NOUN
ejpam-4247	76	17	0	0	NUM
ejpam-4247	76	18	≤	≤	NUM
ejpam-4247	76	19	t	t	PROPN
ejpam-4247	76	20	≤	≤	PROPN
ejpam-4247	76	21	t	t	NOUN
ejpam-4247	76	22	,	,	PUNCT
ejpam-4247	76	23	then	then	ADV
ejpam-4247	76	24	:	:	PUNCT
ejpam-4247	76	25	∣∣∣∣∫	∣∣∣∣∫	NOUN
ejpam-4247	76	26	t	t	NOUN
ejpam-4247	76	27	0	0	NUM
ejpam-4247	77	1	(	(	PUNCT
ejpam-4247	77	2	g(s)−	g(s)−	PROPN
ejpam-4247	77	3	1	1	NUM
ejpam-4247	77	4	t	t	NOUN
ejpam-4247	77	5	∫	∫	PROPN
ejpam-4247	77	6	t	t	PROPN
ejpam-4247	77	7	0	0	NUM
ejpam-4247	77	8	g(s)ds	g(s)ds	PROPN
ejpam-4247	77	9	)	)	PUNCT
ejpam-4247	77	10	ds	ds	PROPN
ejpam-4247	77	11	∣∣∣∣	∣∣∣∣	NOUN
ejpam-4247	77	12	≤	≤	PROPN
ejpam-4247	77	13	β(t)m	β(t)m	PROPN
ejpam-4247	77	14	,	,	PUNCT
ejpam-4247	77	15	(	(	PUNCT
ejpam-4247	77	16	2.12	2.12	NUM
ejpam-4247	77	17	)	)	PUNCT
ejpam-4247	77	18	where	where	SCONJ
ejpam-4247	77	19	m	m	NOUN
ejpam-4247	77	20	=	=	SYM
ejpam-4247	77	21	maxt∈[0,t	maxt∈[0,t	NOUN
ejpam-4247	77	22	]	]	PUNCT
ejpam-4247	78	1	|g(t)|	|g(t)|	PROPN
ejpam-4247	78	2	and	and	CCONJ
ejpam-4247	78	3	β(t	β(t	PROPN
ejpam-4247	78	4	)	)	PUNCT
ejpam-4247	78	5	=	=	PUNCT
ejpam-4247	79	1	2	2	NUM
ejpam-4247	79	2	t	t	NOUN
ejpam-4247	79	3	(	(	PUNCT
ejpam-4247	79	4	1−	1−	NUM
ejpam-4247	79	5	t	t	PROPN
ejpam-4247	79	6	t	t	PROPN
ejpam-4247	79	7	)	)	PUNCT
ejpam-4247	79	8	,	,	PUNCT
ejpam-4247	79	9	maxt∈[0,t	maxt∈[0,t	NOUN
ejpam-4247	79	10	]	]	PUNCT
ejpam-4247	80	1	|β(t)|	|β(t)|	NOUN
ejpam-4247	80	2	≤	≤	NOUN
ejpam-4247	80	3	t	t	NOUN
ejpam-4247	80	4	2	2	NUM
ejpam-4247	80	5	.	.	PUNCT
ejpam-4247	81	1	the	the	DET
ejpam-4247	81	2	proof	proof	NOUN
ejpam-4247	81	3	follows	follow	VERB
ejpam-4247	81	4	directly	directly	ADV
ejpam-4247	81	5	from	from	ADP
ejpam-4247	81	6	the	the	DET
ejpam-4247	81	7	estimate:∣∣∣∣∫	estimate:∣∣∣∣∫	PROPN
ejpam-4247	81	8	t	t	NOUN
ejpam-4247	81	9	0	0	PUNCT
ejpam-4247	82	1	(	(	PUNCT
ejpam-4247	82	2	g(s)−	g(s)−	PROPN
ejpam-4247	82	3	1	1	NUM
ejpam-4247	82	4	t	t	NOUN
ejpam-4247	82	5	∫	∫	PROPN
ejpam-4247	82	6	t	t	PROPN
ejpam-4247	82	7	0	0	NUM
ejpam-4247	82	8	g(s)ds	g(s)ds	PROPN
ejpam-4247	82	9	)	)	PUNCT
ejpam-4247	82	10	ds	ds	PROPN
ejpam-4247	82	11	∣∣∣∣	∣∣∣∣	NOUN
ejpam-4247	82	12	≤	≤	NOUN
ejpam-4247	82	13	(	(	PUNCT
ejpam-4247	82	14	1−	1−	NUM
ejpam-4247	82	15	t	t	PROPN
ejpam-4247	82	16	t	t	PROPN
ejpam-4247	82	17	)	)	PUNCT
ejpam-4247	83	1	∫	∫	PROPN
ejpam-4247	84	1	t	t	PROPN
ejpam-4247	84	2	0	0	X
ejpam-4247	84	3	|g(s)|ds+	|g(s)|ds+	PROPN
ejpam-4247	85	1	t	t	X
ejpam-4247	85	2	t	t	PROPN
ejpam-4247	85	3	∫	∫	PROPN
ejpam-4247	85	4	t	t	PROPN
ejpam-4247	85	5	t	t	PROPN
ejpam-4247	85	6	|g(s)|ds	|g(s)|ds	NOUN
ejpam-4247	85	7	≤	≤	NOUN
ejpam-4247	85	8	β(t)m	β(t)m	PUNCT
ejpam-4247	85	9	theorem	theorem	NOUN
ejpam-4247	85	10	1	1	NUM
ejpam-4247	85	11	.	.	PUNCT
ejpam-4247	86	1	[	[	X
ejpam-4247	86	2	10	10	NUM
ejpam-4247	86	3	]	]	X
ejpam-4247	86	4	(	(	PUNCT
ejpam-4247	86	5	banach	banach	ADV
ejpam-4247	86	6	fixed	fix	VERB
ejpam-4247	86	7	point	point	NOUN
ejpam-4247	86	8	theorem	theorem	ADJ
ejpam-4247	86	9	)	)	PUNCT
ejpam-4247	86	10	.	.	PUNCT
ejpam-4247	87	1	let	let	AUX
ejpam-4247	87	2	(	(	PUNCT
ejpam-4247	87	3	e	e	NOUN
ejpam-4247	87	4	,	,	PUNCT
ejpam-4247	87	5	∥.∥	∥.∥	PUNCT
ejpam-4247	87	6	)	)	PUNCT
ejpam-4247	87	7	be	be	AUX
ejpam-4247	87	8	a	a	DET
ejpam-4247	87	9	banach	banach	NOUN
ejpam-4247	87	10	space	space	NOUN
ejpam-4247	87	11	and	and	CCONJ
ejpam-4247	87	12	p	p	NOUN
ejpam-4247	87	13	:	:	PUNCT
ejpam-4247	87	14	e	e	X
ejpam-4247	87	15	→	→	PUNCT
ejpam-4247	87	16	e	e	AUX
ejpam-4247	87	17	be	be	AUX
ejpam-4247	87	18	a	a	DET
ejpam-4247	87	19	contraction	contraction	NOUN
ejpam-4247	87	20	mapping	mapping	NOUN
ejpam-4247	87	21	i.e.	i.e.	X
ejpam-4247	87	22	lipchitz	lipchitz	PROPN
ejpam-4247	87	23	continuous	continuous	ADJ
ejpam-4247	87	24	with	with	ADP
ejpam-4247	87	25	lipchitz	lipchitz	NOUN
ejpam-4247	87	26	constant	constant	ADJ
ejpam-4247	87	27	l	l	PROPN
ejpam-4247	87	28	∈	∈	PROPN
ejpam-4247	88	1	[	[	X
ejpam-4247	88	2	0	0	NUM
ejpam-4247	88	3	,	,	PUNCT
ejpam-4247	88	4	1	1	NUM
ejpam-4247	88	5	)	)	PUNCT
ejpam-4247	88	6	.	.	PUNCT
ejpam-4247	89	1	then	then	ADV
ejpam-4247	89	2	φ	φ	PROPN
ejpam-4247	89	3	∈	∈	PROPN
ejpam-4247	89	4	e	e	NOUN
ejpam-4247	89	5	has	have	VERB
ejpam-4247	89	6	a	a	DET
ejpam-4247	89	7	unique	unique	ADJ
ejpam-4247	89	8	fixed	fix	VERB
ejpam-4247	89	9	point	point	NOUN
ejpam-4247	89	10	.	.	PUNCT
ejpam-4247	90	1	3	3	X
ejpam-4247	90	2	.	.	X
ejpam-4247	90	3	conditions	condition	NOUN
ejpam-4247	90	4	for	for	ADP
ejpam-4247	90	5	convergence	convergence	NOUN
ejpam-4247	90	6	of	of	ADP
ejpam-4247	90	7	successive	successive	ADJ
ejpam-4247	90	8	approximation	approximation	NOUN
ejpam-4247	90	9	some	some	DET
ejpam-4247	90	10	conditions	condition	NOUN
ejpam-4247	90	11	are	be	AUX
ejpam-4247	90	12	needed	need	VERB
ejpam-4247	90	13	for	for	ADP
ejpam-4247	90	14	investigate	investigate	NOUN
ejpam-4247	90	15	of	of	ADP
ejpam-4247	90	16	the	the	DET
ejpam-4247	90	17	successive	successive	ADJ
ejpam-4247	90	18	approximation	approximation	NOUN
ejpam-4247	90	19	for	for	ADP
ejpam-4247	90	20	periodic	periodic	ADJ
ejpam-4247	90	21	solution	solution	NOUN
ejpam-4247	90	22	of	of	ADP
ejpam-4247	90	23	the	the	DET
ejpam-4247	90	24	problem	problem	NOUN
ejpam-4247	90	25	(	(	PUNCT
ejpam-4247	90	26	1.1	1.1	NUM
ejpam-4247	90	27	)	)	PUNCT
ejpam-4247	90	28	with	with	ADP
ejpam-4247	90	29	u(0	u(0	NOUN
ejpam-4247	90	30	)	)	PUNCT
ejpam-4247	90	31	=	=	PUNCT
ejpam-4247	90	32	u0	u0	ADJ
ejpam-4247	90	33	,	,	PUNCT
ejpam-4247	90	34	suppose	suppose	VERB
ejpam-4247	90	35	that	that	SCONJ
ejpam-4247	90	36	the	the	DET
ejpam-4247	90	37	functions	function	NOUN
ejpam-4247	90	38	h	h	NOUN
ejpam-4247	90	39	∈	∈	NOUN
ejpam-4247	90	40	c	c	X
ejpam-4247	90	41	(	(	PUNCT
ejpam-4247	90	42	[	[	X
ejpam-4247	90	43	0	0	NUM
ejpam-4247	90	44	,	,	PUNCT
ejpam-4247	90	45	t	t	X
ejpam-4247	90	46	]	]	PUNCT
ejpam-4247	90	47	×d1	×d1	ADJ
ejpam-4247	90	48	×d2,r),g	×d2,r),g	X
ejpam-4247	90	49	∈	∈	PROPN
ejpam-4247	90	50	c	c	X
ejpam-4247	90	51	(	(	PUNCT
ejpam-4247	90	52	[	[	X
ejpam-4247	90	53	0	0	NUM
ejpam-4247	90	54	,	,	PUNCT
ejpam-4247	90	55	t	t	X
ejpam-4247	90	56	]	]	PUNCT
ejpam-4247	90	57	×d1,r	×d1,r	PROPN
ejpam-4247	90	58	)	)	PUNCT
ejpam-4247	90	59	,	,	PUNCT
ejpam-4247	90	60	d1	d1	PROPN
ejpam-4247	90	61	and	and	CCONJ
ejpam-4247	90	62	d2	d2	PROPN
ejpam-4247	90	63	are	be	AUX
ejpam-4247	90	64	compact	compact	ADJ
ejpam-4247	90	65	subset	subset	NOUN
ejpam-4247	90	66	of	of	ADP
ejpam-4247	90	67	r	r	NOUN
ejpam-4247	90	68	,	,	PUNCT
ejpam-4247	90	69	a(t	a(t	NOUN
ejpam-4247	90	70	)	)	PUNCT
ejpam-4247	90	71	is	be	AUX
ejpam-4247	90	72	continuous	continuous	ADJ
ejpam-4247	90	73	functions	function	NOUN
ejpam-4247	90	74	on	on	ADP
ejpam-4247	90	75	[	[	X
ejpam-4247	90	76	0	0	NUM
ejpam-4247	90	77	,	,	PUNCT
ejpam-4247	90	78	t	t	X
ejpam-4247	90	79	]	]	PUNCT
ejpam-4247	90	80	,	,	PUNCT
ejpam-4247	90	81	moreover	moreover	ADV
ejpam-4247	90	82	define	define	VERB
ejpam-4247	90	83	|.|	|.|	NOUN
ejpam-4247	90	84	=	=	SYM
ejpam-4247	90	85	maxt∈[0,t	maxt∈[0,t	NOUN
ejpam-4247	90	86	]	]	X
ejpam-4247	90	87	|.|	|.|	NOUN
ejpam-4247	90	88	,	,	PUNCT
ejpam-4247	90	89	and	and	CCONJ
ejpam-4247	90	90	satisfies	satisfy	VERB
ejpam-4247	90	91	the	the	DET
ejpam-4247	90	92	following	follow	VERB
ejpam-4247	90	93	hypothesis	hypothesis	NOUN
ejpam-4247	90	94	.	.	PUNCT
ejpam-4247	91	1	h1	h1	VERB
ejpam-4247	91	2	there	there	PRON
ejpam-4247	91	3	exist	exist	VERB
ejpam-4247	91	4	positive	positive	ADJ
ejpam-4247	91	5	constants	constant	NOUN
ejpam-4247	91	6	m	m	NOUN
ejpam-4247	91	7	,	,	PUNCT
ejpam-4247	91	8	l	l	NOUN
ejpam-4247	91	9	,	,	PUNCT
ejpam-4247	91	10	k1	k1	NOUN
ejpam-4247	91	11	,	,	PUNCT
ejpam-4247	91	12	k2,and	k2,and	PUNCT
ejpam-4247	91	13	l1	l1	PROPN
ejpam-4247	91	14	,	,	PUNCT
ejpam-4247	91	15	such	such	ADJ
ejpam-4247	91	16	that	that	SCONJ
ejpam-4247	91	17	|h(t	|h(t	PROPN
ejpam-4247	91	18	,	,	PUNCT
ejpam-4247	91	19	u	u	NOUN
ejpam-4247	91	20	,	,	PUNCT
ejpam-4247	91	21	z)|	z)|	ADJ
ejpam-4247	91	22	≤	≤	NUM
ejpam-4247	91	23	m	m	VERB
ejpam-4247	91	24	(	(	PUNCT
ejpam-4247	91	25	3.1	3.1	NUM
ejpam-4247	91	26	)	)	PUNCT
ejpam-4247	91	27	|g(t	|g(t	NOUN
ejpam-4247	91	28	,	,	PUNCT
ejpam-4247	91	29	u)|	u)|	NOUN
ejpam-4247	91	30	≤	≤	ADJ
ejpam-4247	91	31	l	l	NOUN
ejpam-4247	91	32	(	(	PUNCT
ejpam-4247	91	33	3.2	3.2	NUM
ejpam-4247	91	34	)	)	PUNCT
ejpam-4247	91	35	|h	|h	NOUN
ejpam-4247	91	36	(	(	PUNCT
ejpam-4247	91	37	t	t	PROPN
ejpam-4247	91	38	,	,	PUNCT
ejpam-4247	91	39	u1	u1	NOUN
ejpam-4247	91	40	,	,	PUNCT
ejpam-4247	91	41	z1)−	z1)−	PROPN
ejpam-4247	91	42	h	h	NOUN
ejpam-4247	91	43	(	(	PUNCT
ejpam-4247	91	44	t	t	PROPN
ejpam-4247	91	45	,	,	PUNCT
ejpam-4247	91	46	u2	u2	PROPN
ejpam-4247	91	47	,	,	PUNCT
ejpam-4247	91	48	z2)|	z2)|	PROPN
ejpam-4247	91	49	≤	≤	NUM
ejpam-4247	91	50	k1	k1	PROPN
ejpam-4247	91	51	|u1	|u1	NOUN
ejpam-4247	92	1	−	−	PROPN
ejpam-4247	92	2	u2|+	u2|+	PROPN
ejpam-4247	92	3	k2	k2	PROPN
ejpam-4247	92	4	|z1	|z1	PROPN
ejpam-4247	92	5	−	−	PROPN
ejpam-4247	92	6	z2|	z2|	NOUN
ejpam-4247	93	1	(	(	PUNCT
ejpam-4247	93	2	3.3	3.3	NUM
ejpam-4247	93	3	)	)	PUNCT
ejpam-4247	93	4	|g	|g	NOUN
ejpam-4247	93	5	(	(	PUNCT
ejpam-4247	93	6	t	t	PROPN
ejpam-4247	93	7	,	,	PUNCT
ejpam-4247	93	8	u1)−	u1)−	PROPN
ejpam-4247	93	9	g	g	PROPN
ejpam-4247	93	10	(	(	PUNCT
ejpam-4247	93	11	t	t	PROPN
ejpam-4247	93	12	,	,	PUNCT
ejpam-4247	93	13	u2)|	u2)|	NOUN
ejpam-4247	93	14	≤	≤	PROPN
ejpam-4247	93	15	l1	l1	PROPN
ejpam-4247	93	16	|u1	|u1	PROPN
ejpam-4247	94	1	−	−	PROPN
ejpam-4247	94	2	u2|	u2|	ADJ
ejpam-4247	94	3	(	(	PUNCT
ejpam-4247	94	4	3.4	3.4	NUM
ejpam-4247	94	5	)	)	PUNCT
ejpam-4247	94	6	where	where	SCONJ
ejpam-4247	94	7	zi	zi	NOUN
ejpam-4247	94	8	=	=	SYM
ejpam-4247	94	9	∫	∫	PROPN
ejpam-4247	94	10	a(t	a(t	PROPN
ejpam-4247	94	11	)	)	PUNCT
ejpam-4247	94	12	0	0	NUM
ejpam-4247	95	1	g	g	PROPN
ejpam-4247	95	2	(	(	PUNCT
ejpam-4247	95	3	s	s	X
ejpam-4247	95	4	,	,	PUNCT
ejpam-4247	95	5	ui(s	ui(s	NUM
ejpam-4247	95	6	)	)	PUNCT
ejpam-4247	95	7	)	)	PUNCT
ejpam-4247	96	1	ds	ds	NOUN
ejpam-4247	96	2	and	and	CCONJ
ejpam-4247	96	3	for	for	ADP
ejpam-4247	96	4	all	all	DET
ejpam-4247	96	5	t	t	NOUN
ejpam-4247	96	6	∈	∈	PROPN
ejpam-4247	97	1	[	[	X
ejpam-4247	97	2	0	0	NUM
ejpam-4247	97	3	,	,	PUNCT
ejpam-4247	97	4	t	t	X
ejpam-4247	97	5	]	]	PUNCT
ejpam-4247	97	6	,	,	PUNCT
ejpam-4247	97	7	u	u	NOUN
ejpam-4247	97	8	,	,	PUNCT
ejpam-4247	97	9	u1	u1	NOUN
ejpam-4247	97	10	,	,	PUNCT
ejpam-4247	97	11	u2	u2	PROPN
ejpam-4247	97	12	∈	∈	PROPN
ejpam-4247	97	13	d1	d1	PROPN
ejpam-4247	97	14	and	and	CCONJ
ejpam-4247	97	15	zi	zi	NOUN
ejpam-4247	97	16	∈	∈	PROPN
ejpam-4247	97	17	d2	d2	PROPN
ejpam-4247	97	18	,	,	PUNCT
ejpam-4247	97	19	i	i	NOUN
ejpam-4247	97	20	=	=	NOUN
ejpam-4247	97	21	1	1	NUM
ejpam-4247	97	22	,	,	PUNCT
ejpam-4247	97	23	2	2	NUM
ejpam-4247	97	24	h2	h2	NOUN
ejpam-4247	97	25	:	:	PUNCT
ejpam-4247	97	26	there	there	PRON
ejpam-4247	97	27	exist	exist	VERB
ejpam-4247	97	28	positive	positive	ADJ
ejpam-4247	97	29	constants	constant	NOUN
ejpam-4247	97	30	at	at	ADP
ejpam-4247	97	31	,	,	PUNCT
ejpam-4247	97	32	such	such	ADJ
ejpam-4247	97	33	that	that	PRON
ejpam-4247	97	34	for	for	ADP
ejpam-4247	97	35	t	t	PROPN
ejpam-4247	97	36	∈	∈	PROPN
ejpam-4247	98	1	[	[	X
ejpam-4247	98	2	0	0	NUM
ejpam-4247	98	3	,	,	PUNCT
ejpam-4247	98	4	t	t	X
ejpam-4247	98	5	]	]	PUNCT
ejpam-4247	98	6	,	,	PUNCT
ejpam-4247	98	7	|a(t)|	|a(t)|	ADJ
ejpam-4247	98	8	≤	≤	NUM
ejpam-4247	98	9	at	at	ADP
ejpam-4247	98	10	(	(	PUNCT
ejpam-4247	98	11	3.5	3.5	NUM
ejpam-4247	98	12	)	)	PUNCT
ejpam-4247	98	13	a.	a.	NOUN
ejpam-4247	98	14	s.	s.	PROPN
ejpam-4247	98	15	rafeeq	rafeeq	PROPN
ejpam-4247	98	16	/	/	SYM
ejpam-4247	98	17	eur	eur	PROPN
ejpam-4247	98	18	.	.	PUNCT
ejpam-4247	99	1	j.	j.	PROPN
ejpam-4247	99	2	pure	pure	PROPN
ejpam-4247	99	3	appl	appl	PROPN
ejpam-4247	99	4	.	.	PROPN
ejpam-4247	99	5	math	math	PROPN
ejpam-4247	99	6	,	,	PUNCT
ejpam-4247	99	7	15	15	NUM
ejpam-4247	99	8	(	(	PUNCT
ejpam-4247	99	9	1	1	NUM
ejpam-4247	99	10	)	)	PUNCT
ejpam-4247	99	11	(	(	PUNCT
ejpam-4247	99	12	2022	2022	NUM
ejpam-4247	99	13	)	)	PUNCT
ejpam-4247	99	14	,	,	PUNCT
ejpam-4247	99	15	144	144	NUM
ejpam-4247	99	16	-	-	SYM
ejpam-4247	99	17	157	157	NUM
ejpam-4247	99	18	148	148	NUM
ejpam-4247	99	19	define	define	VERB
ejpam-4247	99	20	the	the	DET
ejpam-4247	99	21	non	non	ADJ
ejpam-4247	99	22	-	-	ADJ
ejpam-4247	99	23	empty	empty	ADJ
ejpam-4247	99	24	set	set	NOUN
ejpam-4247	99	25	dh	dh	NOUN
ejpam-4247	100	1	=	=	SYM
ejpam-4247	100	2	d1	d1	PROPN
ejpam-4247	100	3	−m1	−m1	PROPN
ejpam-4247	100	4	(	(	PUNCT
ejpam-4247	100	5	3.6	3.6	NUM
ejpam-4247	100	6	)	)	PUNCT
ejpam-4247	100	7	where	where	SCONJ
ejpam-4247	100	8	m1	m1	PROPN
ejpam-4247	100	9	=	=	PUNCT
ejpam-4247	100	10	(	(	PUNCT
ejpam-4247	100	11	2(1−	2(1−	NUM
ejpam-4247	100	12	α	α	NUM
ejpam-4247	100	13	)	)	PUNCT
ejpam-4247	101	1	+	+	CCONJ
ejpam-4247	101	2	αt	αt	NOUN
ejpam-4247	101	3	2	2	NUM
ejpam-4247	101	4	)	)	PUNCT
ejpam-4247	101	5	m	m	VERB
ejpam-4247	101	6	furthermore	furthermore	ADV
ejpam-4247	101	7	,	,	PUNCT
ejpam-4247	101	8	we	we	PRON
ejpam-4247	101	9	suppose	suppose	VERB
ejpam-4247	101	10	that	that	SCONJ
ejpam-4247	101	11	the	the	DET
ejpam-4247	101	12	following	follow	VERB
ejpam-4247	101	13	condition	condition	NOUN
ejpam-4247	101	14	is	be	AUX
ejpam-4247	101	15	valid	valid	ADJ
ejpam-4247	101	16	:	:	PUNCT
ejpam-4247	102	1	λ	λ	X
ejpam-4247	102	2	=	=	SYM
ejpam-4247	102	3	(	(	PUNCT
ejpam-4247	102	4	2(1−	2(1−	NUM
ejpam-4247	102	5	α	α	NUM
ejpam-4247	102	6	)	)	PUNCT
ejpam-4247	103	1	+	+	CCONJ
ejpam-4247	103	2	αt	αt	NOUN
ejpam-4247	103	3	2	2	NUM
ejpam-4247	103	4	)	)	PUNCT
ejpam-4247	103	5	(	(	PUNCT
ejpam-4247	103	6	k1	k1	NOUN
ejpam-4247	103	7	+	+	CCONJ
ejpam-4247	103	8	atl1k2	atl1k2	NOUN
ejpam-4247	103	9	)	)	PUNCT
ejpam-4247	103	10	<	<	X
ejpam-4247	103	11	1	1	NUM
ejpam-4247	103	12	(	(	PUNCT
ejpam-4247	103	13	3.7	3.7	NUM
ejpam-4247	103	14	)	)	PUNCT
ejpam-4247	103	15	4	4	NUM
ejpam-4247	103	16	.	.	X
ejpam-4247	103	17	main	main	ADJ
ejpam-4247	103	18	results	result	NOUN
ejpam-4247	103	19	our	our	PRON
ejpam-4247	103	20	main	main	ADJ
ejpam-4247	103	21	results	result	NOUN
ejpam-4247	103	22	separate	separate	ADJ
ejpam-4247	103	23	to	to	ADP
ejpam-4247	103	24	the	the	DET
ejpam-4247	103	25	following	follow	VERB
ejpam-4247	103	26	parts	part	NOUN
ejpam-4247	103	27	:	:	PUNCT
ejpam-4247	103	28	4.1	4.1	NUM
ejpam-4247	103	29	.	.	PUNCT
ejpam-4247	104	1	approximation	approximation	NOUN
ejpam-4247	104	2	of	of	ADP
ejpam-4247	104	3	periodic	periodic	ADJ
ejpam-4247	104	4	solution	solution	NOUN
ejpam-4247	104	5	of	of	ADP
ejpam-4247	104	6	(	(	PUNCT
ejpam-4247	104	7	1.1	1.1	NUM
ejpam-4247	104	8	)	)	PUNCT
ejpam-4247	104	9	in	in	ADP
ejpam-4247	104	10	this	this	DET
ejpam-4247	104	11	section	section	NOUN
ejpam-4247	104	12	,	,	PUNCT
ejpam-4247	104	13	we	we	PRON
ejpam-4247	104	14	study	study	VERB
ejpam-4247	104	15	the	the	DET
ejpam-4247	104	16	periodic	periodic	ADJ
ejpam-4247	104	17	approximation	approximation	NOUN
ejpam-4247	104	18	solutions	solution	NOUN
ejpam-4247	104	19	of	of	ADP
ejpam-4247	104	20	nonlinear	nonlinear	ADJ
ejpam-4247	104	21	fractional	fractional	ADJ
ejpam-4247	104	22	integro	integro	ADJ
ejpam-4247	104	23	-	-	PUNCT
ejpam-4247	104	24	differential	differential	NOUN
ejpam-4247	104	25	equations	equation	NOUN
ejpam-4247	104	26	(	(	PUNCT
ejpam-4247	104	27	1.1	1.1	NUM
ejpam-4247	104	28	)	)	PUNCT
ejpam-4247	104	29	with	with	ADP
ejpam-4247	104	30	u(0	u(0	NOUN
ejpam-4247	104	31	)	)	PUNCT
ejpam-4247	104	32	=	=	PUNCT
ejpam-4247	105	1	u0	u0	ADJ
ejpam-4247	105	2	.	.	PUNCT
ejpam-4247	106	1	in	in	ADP
ejpam-4247	106	2	the	the	DET
ejpam-4247	106	3	beginning	beginning	NOUN
ejpam-4247	106	4	,	,	PUNCT
ejpam-4247	106	5	we	we	PRON
ejpam-4247	106	6	define	define	VERB
ejpam-4247	106	7	the	the	DET
ejpam-4247	106	8	following	follow	VERB
ejpam-4247	106	9	sequence	sequence	NOUN
ejpam-4247	106	10	of	of	ADP
ejpam-4247	106	11	functions	function	NOUN
ejpam-4247	106	12	{	{	PUNCT
ejpam-4247	106	13	um+1}∞m=0	um+1}∞m=0	PROPN
ejpam-4247	106	14	given	give	VERB
ejpam-4247	106	15	by	by	ADP
ejpam-4247	106	16	the	the	DET
ejpam-4247	106	17	iterative	iterative	NOUN
ejpam-4247	106	18	formulas	formula	NOUN
ejpam-4247	106	19	um+1	um+1	PROPN
ejpam-4247	106	20	(	(	PUNCT
ejpam-4247	106	21	t	t	PROPN
ejpam-4247	106	22	,	,	PUNCT
ejpam-4247	106	23	u0	u0	ADJ
ejpam-4247	106	24	)	)	PUNCT
ejpam-4247	106	25	=	=	SYM
ejpam-4247	106	26	u0	u0	ADJ
ejpam-4247	106	27	+	+	X
ejpam-4247	106	28	2(1−α	2(1−α	NUM
ejpam-4247	106	29	)	)	PUNCT
ejpam-4247	106	30	(	(	PUNCT
ejpam-4247	106	31	2−α)n(α)h(t	2−α)n(α)h(t	NUM
ejpam-4247	106	32	,	,	PUNCT
ejpam-4247	106	33	um(t	um(t	NUM
ejpam-4247	106	34	)	)	PUNCT
ejpam-4247	106	35	,	,	PUNCT
ejpam-4247	106	36	∫	∫	PROPN
ejpam-4247	106	37	a(t	a(t	PROPN
ejpam-4247	106	38	)	)	PUNCT
ejpam-4247	106	39	0	0	NUM
ejpam-4247	107	1	g	g	PROPN
ejpam-4247	107	2	(	(	PUNCT
ejpam-4247	107	3	s	s	PROPN
ejpam-4247	107	4	,	,	PUNCT
ejpam-4247	107	5	um(s	um(s	NOUN
ejpam-4247	107	6	)	)	PUNCT
ejpam-4247	107	7	)	)	PUNCT
ejpam-4247	107	8	ds	ds	ADJ
ejpam-4247	107	9	)	)	PUNCT
ejpam-4247	107	10	−	−	NOUN
ejpam-4247	107	11	2(1−α	2(1−α	NUM
ejpam-4247	107	12	)	)	PUNCT
ejpam-4247	107	13	(	(	PUNCT
ejpam-4247	107	14	2−α)n(α	2−α)n(α	X
ejpam-4247	107	15	)	)	PUNCT
ejpam-4247	107	16	1	1	NUM
ejpam-4247	107	17	t	t	NOUN
ejpam-4247	107	18	∫	∫	PROPN
ejpam-4247	107	19	t	t	PROPN
ejpam-4247	107	20	0	0	NUM
ejpam-4247	107	21	h(s	h(s	PROPN
ejpam-4247	107	22	,	,	PUNCT
ejpam-4247	107	23	um(s	um(s	NOUN
ejpam-4247	107	24	)	)	PUNCT
ejpam-4247	107	25	,	,	PUNCT
ejpam-4247	107	26	∫	∫	PROPN
ejpam-4247	107	27	a(s	a(s	PROPN
ejpam-4247	107	28	)	)	PUNCT
ejpam-4247	107	29	0	0	NUM
ejpam-4247	108	1	g	g	NOUN
ejpam-4247	108	2	(	(	PUNCT
ejpam-4247	108	3	τ	τ	PROPN
ejpam-4247	108	4	,	,	PUNCT
ejpam-4247	108	5	um(τ))dτ	um(τ))dτ	PROPN
ejpam-4247	108	6	)	)	PUNCT
ejpam-4247	108	7	ds+	ds+	ADJ
ejpam-4247	108	8	2α	2α	NOUN
ejpam-4247	108	9	(	(	PUNCT
ejpam-4247	108	10	2−α)n(α)∫	2−α)n(α)∫	NUM
ejpam-4247	108	11	t	t	NOUN
ejpam-4247	108	12	0	0	NUM
ejpam-4247	108	13	(	(	PUNCT
ejpam-4247	108	14	h(s	h(s	PROPN
ejpam-4247	108	15	,	,	PUNCT
ejpam-4247	108	16	um(s	um(s	NOUN
ejpam-4247	108	17	)	)	PUNCT
ejpam-4247	108	18	,	,	PUNCT
ejpam-4247	108	19	∫	∫	PROPN
ejpam-4247	108	20	a(s	a(s	PROPN
ejpam-4247	108	21	)	)	PUNCT
ejpam-4247	108	22	0	0	NUM
ejpam-4247	109	1	g	g	NOUN
ejpam-4247	109	2	(	(	PUNCT
ejpam-4247	109	3	τ	τ	PROPN
ejpam-4247	109	4	,	,	PUNCT
ejpam-4247	109	5	um(τ	um(τ	NUM
ejpam-4247	109	6	)	)	PUNCT
ejpam-4247	109	7	)	)	PUNCT
ejpam-4247	110	1	dτ)−	dτ)−	NOUN
ejpam-4247	110	2	1	1	NUM
ejpam-4247	110	3	t	t	NOUN
ejpam-4247	110	4	∫	∫	PROPN
ejpam-4247	110	5	t	t	PROPN
ejpam-4247	110	6	0	0	NUM
ejpam-4247	111	1	h(s	h(s	PROPN
ejpam-4247	111	2	,	,	PUNCT
ejpam-4247	111	3	um(s	um(s	NOUN
ejpam-4247	111	4	)	)	PUNCT
ejpam-4247	111	5	,	,	PUNCT
ejpam-4247	111	6	∫	∫	PROPN
ejpam-4247	111	7	a(s	a(s	PROPN
ejpam-4247	111	8	)	)	PUNCT
ejpam-4247	111	9	0	0	NUM
ejpam-4247	112	1	g	g	NOUN
ejpam-4247	112	2	(	(	PUNCT
ejpam-4247	112	3	τ	τ	PROPN
ejpam-4247	112	4	,	,	PUNCT
ejpam-4247	112	5	um(τ	um(τ	NUM
ejpam-4247	112	6	)	)	PUNCT
ejpam-4247	112	7	)	)	PUNCT
ejpam-4247	113	1	dτ)ds)ds	dτ)ds)ds	PROPN
ejpam-4247	113	2	(	(	PUNCT
ejpam-4247	113	3	4.1	4.1	NUM
ejpam-4247	113	4	)	)	PUNCT
ejpam-4247	113	5	for	for	ADP
ejpam-4247	113	6	all	all	DET
ejpam-4247	113	7	t	t	NOUN
ejpam-4247	113	8	∈	∈	PROPN
ejpam-4247	113	9	j	j	PROPN
ejpam-4247	113	10	,	,	PUNCT
ejpam-4247	113	11	u0(t	u0(t	PROPN
ejpam-4247	113	12	)	)	PUNCT
ejpam-4247	113	13	=	=	SYM
ejpam-4247	113	14	u0,m	u0,m	PROPN
ejpam-4247	113	15	=	=	SYM
ejpam-4247	113	16	0	0	NUM
ejpam-4247	113	17	,	,	PUNCT
ejpam-4247	113	18	1	1	NUM
ejpam-4247	113	19	,	,	PUNCT
ejpam-4247	113	20	2	2	NUM
ejpam-4247	113	21	,	,	PUNCT
ejpam-4247	113	22	.	.	PUNCT
ejpam-4247	113	23	.	.	PUNCT
ejpam-4247	114	1	.	.	PUNCT
ejpam-4247	115	1	,	,	PUNCT
ejpam-4247	115	2	then	then	ADV
ejpam-4247	115	3	will	will	AUX
ejpam-4247	115	4	be	be	AUX
ejpam-4247	115	5	introduced	introduce	VERB
ejpam-4247	115	6	by	by	ADP
ejpam-4247	115	7	the	the	DET
ejpam-4247	115	8	following	follow	VERB
ejpam-4247	115	9	theorems	theorem	NOUN
ejpam-4247	115	10	.	.	PUNCT
ejpam-4247	116	1	theorem	theorem	NOUN
ejpam-4247	116	2	2	2	NUM
ejpam-4247	116	3	.	.	PUNCT
ejpam-4247	117	1	if	if	SCONJ
ejpam-4247	117	2	the	the	DET
ejpam-4247	117	3	nonlinear	nonlinear	ADJ
ejpam-4247	117	4	fractional	fractional	ADJ
ejpam-4247	117	5	integro	integro	ADJ
ejpam-4247	117	6	-	-	PUNCT
ejpam-4247	117	7	differential	differential	NOUN
ejpam-4247	117	8	equation	equation	NOUN
ejpam-4247	117	9	(	(	PUNCT
ejpam-4247	117	10	1.1	1.1	NUM
ejpam-4247	117	11	)	)	PUNCT
ejpam-4247	117	12	with	with	ADP
ejpam-4247	117	13	u(0	u(0	NOUN
ejpam-4247	117	14	)	)	PUNCT
ejpam-4247	117	15	=	=	PUNCT
ejpam-4247	117	16	u0	u0	ADV
ejpam-4247	117	17	satisfy	satisfy	VERB
ejpam-4247	117	18	the	the	DET
ejpam-4247	117	19	conditions	condition	NOUN
ejpam-4247	117	20	h1	h1	PROPN
ejpam-4247	117	21	,	,	PUNCT
ejpam-4247	117	22	and	and	CCONJ
ejpam-4247	117	23	h2	h2	NOUN
ejpam-4247	117	24	,	,	PUNCT
ejpam-4247	117	25	then	then	ADV
ejpam-4247	117	26	the	the	DET
ejpam-4247	117	27	sequence	sequence	NOUN
ejpam-4247	117	28	of	of	ADP
ejpam-4247	117	29	functions	function	NOUN
ejpam-4247	117	30	(	(	PUNCT
ejpam-4247	117	31	4.1	4.1	NUM
ejpam-4247	117	32	)	)	PUNCT
ejpam-4247	117	33	,	,	PUNCT
ejpam-4247	117	34	which	which	PRON
ejpam-4247	117	35	are	be	AUX
ejpam-4247	117	36	periodic	periodic	ADJ
ejpam-4247	117	37	in	in	ADP
ejpam-4247	117	38	t	t	PROPN
ejpam-4247	117	39	of	of	ADP
ejpam-4247	117	40	period	period	NOUN
ejpam-4247	117	41	t	t	PROPN
ejpam-4247	117	42	,	,	PUNCT
ejpam-4247	117	43	converges	converge	VERB
ejpam-4247	117	44	uniformly	uniformly	ADV
ejpam-4247	117	45	as	as	ADP
ejpam-4247	117	46	m	m	PROPN
ejpam-4247	117	47	→	→	SYM
ejpam-4247	117	48	∞	∞	NUM
ejpam-4247	117	49	on	on	ADP
ejpam-4247	117	50	the	the	DET
ejpam-4247	117	51	domain:(t	domain:(t	NOUN
ejpam-4247	117	52	,	,	PUNCT
ejpam-4247	117	53	u0	u0	ADJ
ejpam-4247	117	54	)	)	PUNCT
ejpam-4247	117	55	∈	∈	PROPN
ejpam-4247	118	1	[	[	X
ejpam-4247	118	2	0	0	NUM
ejpam-4247	118	3	,	,	PUNCT
ejpam-4247	118	4	t	t	X
ejpam-4247	118	5	]	]	PUNCT
ejpam-4247	118	6	×d1	×d1	ADJ
ejpam-4247	118	7	(	(	PUNCT
ejpam-4247	118	8	4.2	4.2	NUM
ejpam-4247	118	9	)	)	PUNCT
ejpam-4247	118	10	to	to	ADP
ejpam-4247	118	11	the	the	DET
ejpam-4247	118	12	limit	limit	NOUN
ejpam-4247	118	13	functions	function	NOUN
ejpam-4247	118	14	uθ	uθ	ADP
ejpam-4247	118	15	defined	define	VERB
ejpam-4247	118	16	on	on	ADP
ejpam-4247	118	17	the	the	DET
ejpam-4247	118	18	domain	domain	NOUN
ejpam-4247	118	19	(	(	PUNCT
ejpam-4247	118	20	4.2	4.2	NUM
ejpam-4247	118	21	)	)	PUNCT
ejpam-4247	118	22	which	which	PRON
ejpam-4247	118	23	is	be	AUX
ejpam-4247	118	24	periodic	periodic	ADJ
ejpam-4247	118	25	in	in	ADP
ejpam-4247	118	26	t	t	PROPN
ejpam-4247	118	27	of	of	ADP
ejpam-4247	118	28	period	period	NOUN
ejpam-4247	118	29	t	t	NOUN
ejpam-4247	118	30	and	and	CCONJ
ejpam-4247	118	31	satisfies	satisfy	VERB
ejpam-4247	118	32	the	the	DET
ejpam-4247	118	33	following	follow	VERB
ejpam-4247	118	34	integral	integral	ADJ
ejpam-4247	118	35	equations	equation	NOUN
ejpam-4247	118	36	:	:	PUNCT
ejpam-4247	118	37	u	u	PROPN
ejpam-4247	118	38	(	(	PUNCT
ejpam-4247	118	39	t	t	PROPN
ejpam-4247	118	40	,	,	PUNCT
ejpam-4247	118	41	u0	u0	ADJ
ejpam-4247	118	42	)	)	PUNCT
ejpam-4247	118	43	=	=	SYM
ejpam-4247	118	44	u0	u0	ADJ
ejpam-4247	118	45	+	+	X
ejpam-4247	118	46	2(1−α	2(1−α	NUM
ejpam-4247	118	47	)	)	PUNCT
ejpam-4247	118	48	(	(	PUNCT
ejpam-4247	118	49	2−α)n(α)h(t	2−α)n(α)h(t	NUM
ejpam-4247	118	50	,	,	PUNCT
ejpam-4247	118	51	u(t	u(t	PROPN
ejpam-4247	118	52	)	)	PUNCT
ejpam-4247	118	53	,	,	PUNCT
ejpam-4247	118	54	∫	∫	PROPN
ejpam-4247	118	55	a(t	a(t	PROPN
ejpam-4247	118	56	)	)	PUNCT
ejpam-4247	118	57	0	0	PUNCT
ejpam-4247	119	1	g(s	g(s	PROPN
ejpam-4247	119	2	,	,	PUNCT
ejpam-4247	119	3	u(s))ds	u(s))ds	PROPN
ejpam-4247	119	4	)	)	PUNCT
ejpam-4247	119	5	−	−	PROPN
ejpam-4247	119	6	(	(	PUNCT
ejpam-4247	119	7	2(1−α	2(1−α	NUM
ejpam-4247	119	8	)	)	PUNCT
ejpam-4247	119	9	(	(	PUNCT
ejpam-4247	119	10	2−α)n(α	2−α)n(α	X
ejpam-4247	119	11	)	)	PUNCT
ejpam-4247	119	12	1	1	NUM
ejpam-4247	119	13	t	t	NOUN
ejpam-4247	119	14	∫	∫	PROPN
ejpam-4247	119	15	t	t	PROPN
ejpam-4247	119	16	0	0	NUM
ejpam-4247	119	17	h(s	h(s	PROPN
ejpam-4247	119	18	,	,	PUNCT
ejpam-4247	119	19	u(s	u(s	NUM
ejpam-4247	119	20	)	)	PUNCT
ejpam-4247	119	21	,	,	PUNCT
ejpam-4247	119	22	∫	∫	PROPN
ejpam-4247	119	23	a(s	a(s	PROPN
ejpam-4247	119	24	)	)	PUNCT
ejpam-4247	119	25	0	0	NUM
ejpam-4247	119	26	g(τ	g(τ	PROPN
ejpam-4247	119	27	,	,	PUNCT
ejpam-4247	119	28	u(τ))dτ)ds+	u(τ))dτ)ds+	ADJ
ejpam-4247	119	29	2α	2α	NOUN
ejpam-4247	119	30	(	(	PUNCT
ejpam-4247	119	31	2−α)n(α)∫	2−α)n(α)∫	NUM
ejpam-4247	119	32	t	t	NOUN
ejpam-4247	119	33	0	0	NUM
ejpam-4247	119	34	(	(	PUNCT
ejpam-4247	119	35	h(s	h(s	PROPN
ejpam-4247	119	36	,	,	PUNCT
ejpam-4247	119	37	u(s	u(s	NUM
ejpam-4247	119	38	)	)	PUNCT
ejpam-4247	119	39	,	,	PUNCT
ejpam-4247	119	40	∫	∫	PROPN
ejpam-4247	119	41	a(s	a(s	PROPN
ejpam-4247	119	42	)	)	PUNCT
ejpam-4247	119	43	0	0	NUM
ejpam-4247	120	1	g(τ	g(τ	NOUN
ejpam-4247	120	2	,	,	PUNCT
ejpam-4247	120	3	u(τ))dτ)−	u(τ))dτ)−	NOUN
ejpam-4247	120	4	1	1	NUM
ejpam-4247	120	5	t	t	NOUN
ejpam-4247	120	6	∫	∫	PROPN
ejpam-4247	120	7	t	t	PROPN
ejpam-4247	120	8	0	0	NUM
ejpam-4247	120	9	h(s	h(s	PROPN
ejpam-4247	120	10	,	,	PUNCT
ejpam-4247	120	11	u(s	u(s	NUM
ejpam-4247	120	12	)	)	PUNCT
ejpam-4247	120	13	,	,	PUNCT
ejpam-4247	120	14	∫	∫	PROPN
ejpam-4247	120	15	a(s	a(s	PROPN
ejpam-4247	120	16	)	)	PUNCT
ejpam-4247	120	17	0	0	NUM
ejpam-4247	121	1	g(τ	g(τ	PROPN
ejpam-4247	121	2	,	,	PUNCT
ejpam-4247	121	3	u(τ))dτ)ds)ds	u(τ))dτ)ds)ds	PROPN
ejpam-4247	121	4	(	(	PUNCT
ejpam-4247	121	5	4.3	4.3	NUM
ejpam-4247	121	6	)	)	PUNCT
ejpam-4247	121	7	on	on	ADP
ejpam-4247	121	8	the	the	DET
ejpam-4247	121	9	domain	domain	NOUN
ejpam-4247	121	10	(	(	PUNCT
ejpam-4247	121	11	4.2	4.2	NUM
ejpam-4247	121	12	)	)	PUNCT
ejpam-4247	121	13	,	,	PUNCT
ejpam-4247	121	14	provided	provide	VERB
ejpam-4247	121	15	that	that	SCONJ
ejpam-4247	121	16	|u	|u	ADJ
ejpam-4247	121	17	(	(	PUNCT
ejpam-4247	121	18	t	t	PROPN
ejpam-4247	121	19	,	,	PUNCT
ejpam-4247	121	20	u0)−	u0)−	PROPN
ejpam-4247	121	21	um+1	um+1	PROPN
ejpam-4247	121	22	(	(	PUNCT
ejpam-4247	121	23	t	t	PROPN
ejpam-4247	121	24	,	,	PUNCT
ejpam-4247	121	25	u0)|	u0)|	PROPN
ejpam-4247	121	26	≤	≤	PROPN
ejpam-4247	122	1	λm(e−	λm(e−	PROPN
ejpam-4247	122	2	λ)−1m1	λ)−1m1	PROPN
ejpam-4247	122	3	(	(	PUNCT
ejpam-4247	122	4	4.4	4.4	NUM
ejpam-4247	122	5	)	)	PUNCT
ejpam-4247	122	6	a.	a.	NOUN
ejpam-4247	122	7	s.	s.	PROPN
ejpam-4247	122	8	rafeeq	rafeeq	PROPN
ejpam-4247	122	9	/	/	SYM
ejpam-4247	122	10	eur	eur	PROPN
ejpam-4247	122	11	.	.	PUNCT
ejpam-4247	123	1	j.	j.	PROPN
ejpam-4247	123	2	pure	pure	PROPN
ejpam-4247	123	3	appl	appl	PROPN
ejpam-4247	123	4	.	.	PROPN
ejpam-4247	123	5	math	math	PROPN
ejpam-4247	123	6	,	,	PUNCT
ejpam-4247	123	7	15	15	NUM
ejpam-4247	123	8	(	(	PUNCT
ejpam-4247	123	9	1	1	NUM
ejpam-4247	123	10	)	)	PUNCT
ejpam-4247	123	11	(	(	PUNCT
ejpam-4247	123	12	2022	2022	NUM
ejpam-4247	123	13	)	)	PUNCT
ejpam-4247	123	14	,	,	PUNCT
ejpam-4247	123	15	144	144	NUM
ejpam-4247	123	16	-	-	SYM
ejpam-4247	123	17	157	157	NUM
ejpam-4247	123	18	149	149	NUM
ejpam-4247	123	19	for	for	ADP
ejpam-4247	123	20	all	all	DET
ejpam-4247	123	21	m	m	PROPN
ejpam-4247	123	22	≥	≥	NOUN
ejpam-4247	123	23	0	0	NUM
ejpam-4247	123	24	,	,	PUNCT
ejpam-4247	123	25	u0	u0	PROPN
ejpam-4247	123	26	∈	∈	PROPN
ejpam-4247	123	27	d	d	NOUN
ejpam-4247	123	28	,	,	PUNCT
ejpam-4247	123	29	and	and	CCONJ
ejpam-4247	123	30	t	t	PROPN
ejpam-4247	123	31	∈	∈	PROPN
ejpam-4247	123	32	j	j	PROPN
ejpam-4247	123	33	proof	proof	NOUN
ejpam-4247	123	34	.	.	PUNCT
ejpam-4247	124	1	setting	set	VERB
ejpam-4247	124	2	m	m	NOUN
ejpam-4247	124	3	=	=	X
ejpam-4247	124	4	0	0	NUM
ejpam-4247	124	5	in	in	ADP
ejpam-4247	124	6	the	the	DET
ejpam-4247	124	7	sequence	sequence	NOUN
ejpam-4247	124	8	of	of	ADP
ejpam-4247	124	9	functions	function	NOUN
ejpam-4247	124	10	(	(	PUNCT
ejpam-4247	124	11	4.1	4.1	NUM
ejpam-4247	124	12	)	)	PUNCT
ejpam-4247	124	13	and	and	CCONJ
ejpam-4247	124	14	by	by	ADP
ejpam-4247	124	15	using	use	VERB
ejpam-4247	124	16	lemma	lemma	PROPN
ejpam-4247	124	17	1	1	NUM
ejpam-4247	124	18	,	,	PUNCT
ejpam-4247	124	19	we	we	PRON
ejpam-4247	124	20	have	have	VERB
ejpam-4247	124	21	|u1	|u1	PRON
ejpam-4247	124	22	(	(	PUNCT
ejpam-4247	124	23	t	t	PROPN
ejpam-4247	124	24	,	,	PUNCT
ejpam-4247	124	25	u0)−	u0)−	PROPN
ejpam-4247	124	26	u0|	u0|	PROPN
ejpam-4247	124	27	≤	≤	NUM
ejpam-4247	124	28	(	(	PUNCT
ejpam-4247	124	29	4(1−	4(1−	NUM
ejpam-4247	124	30	α	α	NOUN
ejpam-4247	124	31	)	)	PUNCT
ejpam-4247	124	32	(	(	PUNCT
ejpam-4247	124	33	2−	2−	NUM
ejpam-4247	124	34	α)n(α	α)n(α	NOUN
ejpam-4247	124	35	)	)	PUNCT
ejpam-4247	125	1	+	+	NUM
ejpam-4247	125	2	2α	2α	NOUN
ejpam-4247	125	3	(	(	PUNCT
ejpam-4247	125	4	2−	2−	NUM
ejpam-4247	125	5	α)n(α	α)n(α	NOUN
ejpam-4247	125	6	)	)	PUNCT
ejpam-4247	125	7	β(t	β(t	PROPN
ejpam-4247	125	8	)	)	PUNCT
ejpam-4247	125	9	)	)	PUNCT
ejpam-4247	126	1	m	m	VERB
ejpam-4247	126	2	≤	≤	NOUN
ejpam-4247	126	3	(	(	PUNCT
ejpam-4247	126	4	2(1−	2(1−	NUM
ejpam-4247	126	5	α	α	NOUN
ejpam-4247	126	6	)	)	PUNCT
ejpam-4247	127	1	+	+	CCONJ
ejpam-4247	127	2	αt	αt	NOUN
ejpam-4247	127	3	2	2	NUM
ejpam-4247	127	4	)	)	PUNCT
ejpam-4247	127	5	m	m	PROPN
ejpam-4247	127	6	=	=	NOUN
ejpam-4247	127	7	m1	m1	PROPN
ejpam-4247	127	8	for	for	ADP
ejpam-4247	127	9	all	all	DET
ejpam-4247	127	10	t	t	NOUN
ejpam-4247	127	11	∈	∈	PROPN
ejpam-4247	128	1	[	[	X
ejpam-4247	128	2	0	0	NUM
ejpam-4247	128	3	,	,	PUNCT
ejpam-4247	128	4	t	t	X
ejpam-4247	128	5	]	]	PUNCT
ejpam-4247	128	6	,	,	PUNCT
ejpam-4247	128	7	u0	u0	PROPN
ejpam-4247	128	8	∈	∈	PROPN
ejpam-4247	128	9	dh	dh	NOUN
ejpam-4247	128	10	we	we	PRON
ejpam-4247	128	11	get	get	VERB
ejpam-4247	128	12	u1	u1	NOUN
ejpam-4247	128	13	(	(	PUNCT
ejpam-4247	128	14	t	t	PROPN
ejpam-4247	128	15	,	,	PUNCT
ejpam-4247	128	16	u0	u0	ADJ
ejpam-4247	128	17	)	)	PUNCT
ejpam-4247	128	18	∈	∈	NOUN
ejpam-4247	128	19	d1	d1	NOUN
ejpam-4247	128	20	.	.	PUNCT
ejpam-4247	129	1	thus	thus	ADV
ejpam-4247	129	2	by	by	ADP
ejpam-4247	129	3	mathematical	mathematical	ADJ
ejpam-4247	129	4	induction	induction	NOUN
ejpam-4247	129	5	,	,	PUNCT
ejpam-4247	129	6	we	we	PRON
ejpam-4247	129	7	find	find	VERB
ejpam-4247	129	8	that	that	SCONJ
ejpam-4247	129	9	|um	|um	PRON
ejpam-4247	129	10	(	(	PUNCT
ejpam-4247	129	11	t	t	PROPN
ejpam-4247	129	12	,	,	PUNCT
ejpam-4247	129	13	u0)−	u0)−	PROPN
ejpam-4247	129	14	u0|	u0|	PROPN
ejpam-4247	129	15	≤	≤	ADJ
ejpam-4247	129	16	m1	m1	NOUN
ejpam-4247	129	17	(	(	PUNCT
ejpam-4247	129	18	4.5	4.5	NUM
ejpam-4247	129	19	)	)	PUNCT
ejpam-4247	129	20	mean	mean	VERB
ejpam-4247	129	21	that	that	SCONJ
ejpam-4247	129	22	for	for	ADP
ejpam-4247	129	23	all	all	DET
ejpam-4247	129	24	t	t	NOUN
ejpam-4247	129	25	∈	∈	PROPN
ejpam-4247	130	1	[	[	X
ejpam-4247	130	2	0	0	NUM
ejpam-4247	130	3	,	,	PUNCT
ejpam-4247	130	4	t	t	X
ejpam-4247	130	5	]	]	PUNCT
ejpam-4247	130	6	,	,	PUNCT
ejpam-4247	130	7	u0	u0	PROPN
ejpam-4247	130	8	∈	∈	PROPN
ejpam-4247	130	9	dh	dh	NOUN
ejpam-4247	130	10	we	we	PRON
ejpam-4247	130	11	get	get	VERB
ejpam-4247	130	12	um	um	INTJ
ejpam-4247	131	1	(	(	PUNCT
ejpam-4247	131	2	t	t	PROPN
ejpam-4247	131	3	,	,	PUNCT
ejpam-4247	131	4	u0	u0	ADJ
ejpam-4247	131	5	)	)	PUNCT
ejpam-4247	131	6	∈	∈	PROPN
ejpam-4247	131	7	d1,m	d1,m	PROPN
ejpam-4247	131	8	=	=	SYM
ejpam-4247	131	9	0	0	NUM
ejpam-4247	131	10	,	,	PUNCT
ejpam-4247	131	11	1	1	NUM
ejpam-4247	131	12	,	,	PUNCT
ejpam-4247	131	13	2	2	NUM
ejpam-4247	131	14	,	,	PUNCT
ejpam-4247	131	15	...	...	PUNCT
ejpam-4247	131	16	now	now	ADV
ejpam-4247	131	17	,	,	PUNCT
ejpam-4247	131	18	we	we	PRON
ejpam-4247	131	19	claim	claim	VERB
ejpam-4247	131	20	that	that	SCONJ
ejpam-4247	131	21	the	the	DET
ejpam-4247	131	22	sequences	sequence	NOUN
ejpam-4247	131	23	of	of	ADP
ejpam-4247	131	24	functions	function	NOUN
ejpam-4247	131	25	(	(	PUNCT
ejpam-4247	131	26	4.1	4.1	NUM
ejpam-4247	131	27	)	)	PUNCT
ejpam-4247	131	28	are	be	AUX
ejpam-4247	131	29	uniformly	uniformly	ADV
ejpam-4247	131	30	convergent	convergent	NOUN
ejpam-4247	131	31	on	on	ADP
ejpam-4247	131	32	the	the	DET
ejpam-4247	131	33	domain	domain	NOUN
ejpam-4247	131	34	(	(	PUNCT
ejpam-4247	131	35	4.2	4.2	NUM
ejpam-4247	131	36	)	)	PUNCT
ejpam-4247	131	37	.	.	PUNCT
ejpam-4247	132	1	by	by	ADP
ejpam-4247	132	2	the	the	DET
ejpam-4247	132	3	inequalities	inequality	NOUN
ejpam-4247	132	4	(	(	PUNCT
ejpam-4247	132	5	3.3)-(3.5	3.3)-(3.5	NUM
ejpam-4247	132	6	)	)	PUNCT
ejpam-4247	132	7	,	,	PUNCT
ejpam-4247	132	8	we	we	PRON
ejpam-4247	132	9	obtain	obtain	VERB
ejpam-4247	132	10	|um+1	|um+1	PROPN
ejpam-4247	132	11	(	(	PUNCT
ejpam-4247	132	12	t	t	PROPN
ejpam-4247	132	13	,	,	PUNCT
ejpam-4247	132	14	u0)−	u0)−	PROPN
ejpam-4247	132	15	um	um	INTJ
ejpam-4247	132	16	(	(	PUNCT
ejpam-4247	132	17	t	t	PROPN
ejpam-4247	132	18	,	,	PUNCT
ejpam-4247	132	19	u0)|	u0)|	NOUN
ejpam-4247	132	20	≤	≤	PROPN
ejpam-4247	132	21	(	(	PUNCT
ejpam-4247	132	22	2(1−	2(1−	NUM
ejpam-4247	132	23	α	α	NOUN
ejpam-4247	132	24	)	)	PUNCT
ejpam-4247	133	1	+	+	CCONJ
ejpam-4247	133	2	αt	αt	NOUN
ejpam-4247	133	3	2	2	NUM
ejpam-4247	133	4	)	)	PUNCT
ejpam-4247	133	5	(	(	PUNCT
ejpam-4247	133	6	k1	k1	NOUN
ejpam-4247	133	7	+	+	CCONJ
ejpam-4247	133	8	atl1k2	atl1k2	NOUN
ejpam-4247	133	9	)	)	PUNCT
ejpam-4247	133	10	|um	|um	PRON
ejpam-4247	133	11	(	(	PUNCT
ejpam-4247	133	12	t	t	PROPN
ejpam-4247	133	13	,	,	PUNCT
ejpam-4247	133	14	,	,	PUNCT
ejpam-4247	133	15	u0)−	u0)−	PROPN
ejpam-4247	133	16	um−1	um−1	PROPN
ejpam-4247	133	17	(	(	PUNCT
ejpam-4247	133	18	t	t	PROPN
ejpam-4247	133	19	,	,	PUNCT
ejpam-4247	133	20	,	,	PUNCT
ejpam-4247	133	21	u0)|	u0)|	NOUN
ejpam-4247	133	22	=	=	SYM
ejpam-4247	133	23	λ	λ	X
ejpam-4247	133	24	|um	|um	X
ejpam-4247	133	25	(	(	PUNCT
ejpam-4247	133	26	t	t	PROPN
ejpam-4247	133	27	,	,	PUNCT
ejpam-4247	133	28	,	,	PUNCT
ejpam-4247	133	29	u0)−	u0)−	PROPN
ejpam-4247	133	30	um−1	um−1	PROPN
ejpam-4247	133	31	(	(	PUNCT
ejpam-4247	133	32	tr	tr	VERB
ejpam-4247	133	33	,	,	PUNCT
ejpam-4247	133	34	u0)|	u0)|	NOUN
ejpam-4247	133	35	(	(	PUNCT
ejpam-4247	133	36	4.6	4.6	NUM
ejpam-4247	133	37	)	)	PUNCT
ejpam-4247	133	38	by	by	ADP
ejpam-4247	133	39	mathematical	mathematical	ADJ
ejpam-4247	133	40	induction	induction	NOUN
ejpam-4247	133	41	,	,	PUNCT
ejpam-4247	133	42	we	we	PRON
ejpam-4247	133	43	obtain	obtain	VERB
ejpam-4247	133	44	that	that	SCONJ
ejpam-4247	133	45	|um+1	|um+1	PROPN
ejpam-4247	133	46	(	(	PUNCT
ejpam-4247	133	47	t	t	PROPN
ejpam-4247	133	48	,	,	PUNCT
ejpam-4247	133	49	u0)−	u0)−	PROPN
ejpam-4247	133	50	um	um	INTJ
ejpam-4247	133	51	(	(	PUNCT
ejpam-4247	133	52	t	t	PROPN
ejpam-4247	133	53	,	,	PUNCT
ejpam-4247	133	54	u0)|	u0)|	PROPN
ejpam-4247	133	55	≤	≤	PROPN
ejpam-4247	133	56	λm	λm	ADP
ejpam-4247	133	57	|u1	|u1	PROPN
ejpam-4247	133	58	(	(	PUNCT
ejpam-4247	133	59	t	t	PROPN
ejpam-4247	133	60	,	,	PUNCT
ejpam-4247	133	61	u0)−	u0)−	PROPN
ejpam-4247	133	62	u0|	u0|	PROPN
ejpam-4247	133	63	(	(	PUNCT
ejpam-4247	133	64	4.7	4.7	NUM
ejpam-4247	133	65	)	)	PUNCT
ejpam-4247	133	66	now	now	ADV
ejpam-4247	133	67	from	from	ADP
ejpam-4247	133	68	m	m	PROPN
ejpam-4247	133	69	=	=	SYM
ejpam-4247	133	70	1	1	NUM
ejpam-4247	133	71	,	,	PUNCT
ejpam-4247	133	72	2	2	NUM
ejpam-4247	133	73	,	,	PUNCT
ejpam-4247	133	74	.	.	PUNCT
ejpam-4247	133	75	.	.	PUNCT
ejpam-4247	134	1	.	.	PUNCT
ejpam-4247	135	1	and	and	CCONJ
ejpam-4247	135	2	p	p	PRON
ejpam-4247	135	3	≥	≥	NUM
ejpam-4247	135	4	1	1	NUM
ejpam-4247	135	5	,	,	PUNCT
ejpam-4247	135	6	we	we	PRON
ejpam-4247	135	7	find	find	VERB
ejpam-4247	135	8	that	that	SCONJ
ejpam-4247	135	9	|um+p	|um+p	X
ejpam-4247	135	10	(	(	PUNCT
ejpam-4247	135	11	t	t	PROPN
ejpam-4247	135	12	,	,	PUNCT
ejpam-4247	135	13	u0)−	u0)−	PROPN
ejpam-4247	135	14	um	um	INTJ
ejpam-4247	135	15	(	(	PUNCT
ejpam-4247	135	16	t	t	PROPN
ejpam-4247	135	17	,	,	PUNCT
ejpam-4247	135	18	u0)|	u0)|	NOUN
ejpam-4247	135	19	≤	≤	PUNCT
ejpam-4247	136	1	λm(1−	λm(1−	ADJ
ejpam-4247	136	2	λ)−1	λ)−1	NOUN
ejpam-4247	136	3	(	(	PUNCT
ejpam-4247	136	4	(	(	PUNCT
ejpam-4247	136	5	2(1−	2(1−	NUM
ejpam-4247	136	6	α	α	NOUN
ejpam-4247	136	7	)	)	PUNCT
ejpam-4247	136	8	+	+	CCONJ
ejpam-4247	136	9	αt	αt	NOUN
ejpam-4247	136	10	2	2	NUM
ejpam-4247	136	11	)	)	PUNCT
ejpam-4247	136	12	m	m	VERB
ejpam-4247	136	13	)	)	PUNCT
ejpam-4247	136	14	≤	≤	PUNCT
ejpam-4247	137	1	λm(1−	λm(1−	PROPN
ejpam-4247	137	2	λ)−1m1	λ)−1m1	PROPN
ejpam-4247	137	3	,	,	PUNCT
ejpam-4247	137	4	(	(	PUNCT
ejpam-4247	137	5	4.8	4.8	NUM
ejpam-4247	137	6	)	)	PUNCT
ejpam-4247	137	7	for	for	ADP
ejpam-4247	137	8	all	all	DET
ejpam-4247	137	9	t	t	NOUN
ejpam-4247	137	10	∈	∈	PROPN
ejpam-4247	138	1	[	[	X
ejpam-4247	138	2	0	0	NUM
ejpam-4247	138	3	,	,	PUNCT
ejpam-4247	138	4	t	t	PROPN
ejpam-4247	138	5	]	]	PUNCT
ejpam-4247	138	6	,	,	PUNCT
ejpam-4247	138	7	u0	u0	PROPN
ejpam-4247	138	8	∈	∈	PROPN
ejpam-4247	138	9	dh	dh	NOUN
ejpam-4247	138	10	.	.	PUNCT
ejpam-4247	139	1	since	since	SCONJ
ejpam-4247	139	2	λ	λ	X
ejpam-4247	139	3	=	=	PRON
ejpam-4247	139	4	(	(	PUNCT
ejpam-4247	139	5	2(1−	2(1−	NUM
ejpam-4247	139	6	α	α	NUM
ejpam-4247	139	7	)	)	PUNCT
ejpam-4247	139	8	+	+	CCONJ
ejpam-4247	139	9	αt	αt	NOUN
ejpam-4247	139	10	2	2	NUM
ejpam-4247	139	11	)	)	PUNCT
ejpam-4247	139	12	(	(	PUNCT
ejpam-4247	139	13	k1	k1	NOUN
ejpam-4247	139	14	+	+	CCONJ
ejpam-4247	139	15	atl1k2	atl1k2	NOUN
ejpam-4247	139	16	)	)	PUNCT
ejpam-4247	139	17	<	<	X
ejpam-4247	139	18	1	1	NUM
ejpam-4247	139	19	and	and	CCONJ
ejpam-4247	139	20	limm→∞	limm→∞	PROPN
ejpam-4247	139	21	λm	λm	ADP
ejpam-4247	139	22	=	=	SYM
ejpam-4247	139	23	0	0	NUM
ejpam-4247	139	24	,	,	PUNCT
ejpam-4247	139	25	so	so	SCONJ
ejpam-4247	139	26	that	that	SCONJ
ejpam-4247	139	27	the	the	DET
ejpam-4247	139	28	right	right	ADJ
ejpam-4247	139	29	side	side	NOUN
ejpam-4247	139	30	of	of	ADP
ejpam-4247	139	31	(	(	PUNCT
ejpam-4247	139	32	4.8	4.8	NUM
ejpam-4247	139	33	)	)	PUNCT
ejpam-4247	139	34	tends	tend	VERB
ejpam-4247	139	35	to	to	ADP
ejpam-4247	139	36	zero	zero	NUM
ejpam-4247	139	37	.	.	PUNCT
ejpam-4247	140	1	therefore	therefore	ADV
ejpam-4247	140	2	the	the	DET
ejpam-4247	140	3	sequence	sequence	NOUN
ejpam-4247	140	4	of	of	ADP
ejpam-4247	140	5	functions	function	NOUN
ejpam-4247	140	6	um	um	INTJ
ejpam-4247	140	7	(	(	PUNCT
ejpam-4247	140	8	t	t	PROPN
ejpam-4247	140	9	,	,	PUNCT
ejpam-4247	140	10	u0	u0	ADJ
ejpam-4247	140	11	)	)	PUNCT
ejpam-4247	140	12	,	,	PUNCT
ejpam-4247	140	13	m	m	VERB
ejpam-4247	140	14	=	=	NOUN
ejpam-4247	140	15	1	1	NUM
ejpam-4247	140	16	,	,	PUNCT
ejpam-4247	140	17	2	2	NUM
ejpam-4247	140	18	,	,	PUNCT
ejpam-4247	140	19	3	3	NUM
ejpam-4247	140	20	,	,	PUNCT
ejpam-4247	140	21	.	.	PUNCT
ejpam-4247	140	22	.	.	PUNCT
ejpam-4247	141	1	.	.	PUNCT
ejpam-4247	142	1	is	be	AUX
ejpam-4247	142	2	converges	converge	VERB
ejpam-4247	142	3	uniformly	uniformly	ADV
ejpam-4247	142	4	on	on	ADP
ejpam-4247	142	5	the	the	DET
ejpam-4247	142	6	domain	domain	NOUN
ejpam-4247	142	7	(	(	PUNCT
ejpam-4247	142	8	4.2	4.2	NUM
ejpam-4247	142	9	)	)	PUNCT
ejpam-4247	142	10	to	to	ADP
ejpam-4247	142	11	the	the	DET
ejpam-4247	142	12	limit	limit	NOUN
ejpam-4247	142	13	function	function	NOUN
ejpam-4247	142	14	u	u	PROPN
ejpam-4247	142	15	(	(	PUNCT
ejpam-4247	142	16	t	t	PROPN
ejpam-4247	142	17	,	,	PUNCT
ejpam-4247	142	18	u0	u0	PROPN
ejpam-4247	142	19	)	)	PUNCT
ejpam-4247	142	20	which	which	PRON
ejpam-4247	142	21	is	be	AUX
ejpam-4247	142	22	defined	define	VERB
ejpam-4247	142	23	on	on	ADP
ejpam-4247	142	24	the	the	DET
ejpam-4247	142	25	same	same	ADJ
ejpam-4247	142	26	domain	domain	NOUN
ejpam-4247	142	27	.	.	PUNCT
ejpam-4247	143	1	let	let	VERB
ejpam-4247	143	2	lim	lim	PROPN
ejpam-4247	143	3	m→∞	m→∞	PROPN
ejpam-4247	143	4	um	um	INTJ
ejpam-4247	143	5	(	(	PUNCT
ejpam-4247	143	6	t	t	PROPN
ejpam-4247	143	7	,	,	PUNCT
ejpam-4247	143	8	u0	u0	ADJ
ejpam-4247	143	9	)	)	PUNCT
ejpam-4247	143	10	=	=	SYM
ejpam-4247	144	1	uθ	uθ	PROPN
ejpam-4247	144	2	(	(	PUNCT
ejpam-4247	144	3	t	t	PROPN
ejpam-4247	144	4	,	,	PUNCT
ejpam-4247	144	5	u0	u0	ADJ
ejpam-4247	144	6	)	)	PUNCT
ejpam-4247	144	7	(	(	PUNCT
ejpam-4247	144	8	4.9	4.9	NUM
ejpam-4247	144	9	)	)	PUNCT
ejpam-4247	144	10	since	since	SCONJ
ejpam-4247	144	11	the	the	DET
ejpam-4247	144	12	sequence	sequence	NOUN
ejpam-4247	144	13	of	of	ADP
ejpam-4247	144	14	functions	function	NOUN
ejpam-4247	144	15	(	(	PUNCT
ejpam-4247	144	16	4.1	4.1	NUM
ejpam-4247	144	17	)	)	PUNCT
ejpam-4247	144	18	are	be	AUX
ejpam-4247	144	19	periodic	periodic	ADJ
ejpam-4247	144	20	in	in	ADP
ejpam-4247	144	21	t	t	PROPN
ejpam-4247	144	22	of	of	ADP
ejpam-4247	144	23	period	period	NOUN
ejpam-4247	144	24	t	t	PROPN
ejpam-4247	144	25	,	,	PUNCT
ejpam-4247	144	26	then	then	ADV
ejpam-4247	144	27	the	the	DET
ejpam-4247	144	28	limiting	limit	VERB
ejpam-4247	144	29	function	function	NOUN
ejpam-4247	144	30	uθ	uθ	ADP
ejpam-4247	144	31	(	(	PUNCT
ejpam-4247	144	32	t	t	PROPN
ejpam-4247	144	33	,	,	PUNCT
ejpam-4247	144	34	u0	u0	ADJ
ejpam-4247	144	35	)	)	PUNCT
ejpam-4247	144	36	is	be	AUX
ejpam-4247	144	37	also	also	ADV
ejpam-4247	144	38	periodic	periodic	ADJ
ejpam-4247	144	39	in	in	ADP
ejpam-4247	144	40	t	t	PROPN
ejpam-4247	144	41	of	of	ADP
ejpam-4247	144	42	period	period	NOUN
ejpam-4247	144	43	t.	t.	NOUN
ejpam-4247	144	44	by	by	ADP
ejpam-4247	144	45	using	use	VERB
ejpam-4247	144	46	the	the	DET
ejpam-4247	144	47	relation	relation	NOUN
ejpam-4247	144	48	(	(	PUNCT
ejpam-4247	144	49	4.9	4.9	NUM
ejpam-4247	144	50	)	)	PUNCT
ejpam-4247	144	51	and	and	CCONJ
ejpam-4247	144	52	proceeding	proceed	VERB
ejpam-4247	144	53	in	in	ADP
ejpam-4247	144	54	(	(	PUNCT
ejpam-4247	144	55	4.1	4.1	NUM
ejpam-4247	144	56	)	)	PUNCT
ejpam-4247	144	57	to	to	PART
ejpam-4247	144	58	limit	limit	VERB
ejpam-4247	144	59	,	,	PUNCT
ejpam-4247	144	60	when	when	SCONJ
ejpam-4247	144	61	m	m	PROPN
ejpam-4247	144	62	→	→	SYM
ejpam-4247	144	63	∞	∞	PROPN
ejpam-4247	144	64	,	,	PUNCT
ejpam-4247	144	65	it	it	PRON
ejpam-4247	144	66	is	be	AUX
ejpam-4247	144	67	converging	converge	VERB
ejpam-4247	144	68	that	that	SCONJ
ejpam-4247	144	69	the	the	DET
ejpam-4247	144	70	limiting	limit	VERB
ejpam-4247	144	71	function	function	NOUN
ejpam-4247	144	72	u	u	PROPN
ejpam-4247	144	73	(	(	PUNCT
ejpam-4247	144	74	t	t	PROPN
ejpam-4247	144	75	,	,	PUNCT
ejpam-4247	144	76	u0	u0	ADJ
ejpam-4247	144	77	)	)	PUNCT
ejpam-4247	144	78	is	be	AUX
ejpam-4247	144	79	the	the	DET
ejpam-4247	144	80	periodic	periodic	ADJ
ejpam-4247	144	81	solution	solution	NOUN
ejpam-4247	144	82	of	of	ADP
ejpam-4247	144	83	the	the	DET
ejpam-4247	144	84	integral	integral	ADJ
ejpam-4247	144	85	equation	equation	NOUN
ejpam-4247	144	86	(	(	PUNCT
ejpam-4247	144	87	4.3	4.3	NUM
ejpam-4247	144	88	)	)	PUNCT
ejpam-4247	144	89	.	.	PUNCT
ejpam-4247	145	1	theorem	theorem	NOUN
ejpam-4247	145	2	3	3	X
ejpam-4247	145	3	.	.	PUNCT
ejpam-4247	146	1	if	if	SCONJ
ejpam-4247	146	2	all	all	DET
ejpam-4247	146	3	assumptions	assumption	NOUN
ejpam-4247	146	4	of	of	ADP
ejpam-4247	146	5	the	the	DET
ejpam-4247	146	6	theorem	theorem	ADJ
ejpam-4247	146	7	2	2	NUM
ejpam-4247	146	8	are	be	AUX
ejpam-4247	146	9	satisfy	satisfy	ADJ
ejpam-4247	146	10	,	,	PUNCT
ejpam-4247	146	11	then	then	ADV
ejpam-4247	146	12	u	u	X
ejpam-4247	146	13	(	(	PUNCT
ejpam-4247	146	14	t	t	PROPN
ejpam-4247	146	15	,	,	PUNCT
ejpam-4247	146	16	u0	u0	ADJ
ejpam-4247	146	17	)	)	PUNCT
ejpam-4247	146	18	is	be	AUX
ejpam-4247	146	19	a	a	DET
ejpam-4247	146	20	unique	unique	ADJ
ejpam-4247	146	21	solution	solution	NOUN
ejpam-4247	146	22	of	of	ADP
ejpam-4247	146	23	the	the	DET
ejpam-4247	146	24	problem	problem	NOUN
ejpam-4247	146	25	(	(	PUNCT
ejpam-4247	146	26	1.1	1.1	NUM
ejpam-4247	146	27	)	)	PUNCT
ejpam-4247	146	28	with	with	ADP
ejpam-4247	146	29	u(0	u(0	NOUN
ejpam-4247	146	30	)	)	PUNCT
ejpam-4247	146	31	=	=	PUNCT
ejpam-4247	147	1	u0	u0	PROPN
ejpam-4247	147	2	.	.	PUNCT
ejpam-4247	147	3	a.	a.	PROPN
ejpam-4247	147	4	s.	s.	PROPN
ejpam-4247	147	5	rafeeq	rafeeq	PROPN
ejpam-4247	147	6	/	/	SYM
ejpam-4247	147	7	eur	eur	PROPN
ejpam-4247	147	8	.	.	PUNCT
ejpam-4247	148	1	j.	j.	PROPN
ejpam-4247	148	2	pure	pure	PROPN
ejpam-4247	148	3	appl	appl	PROPN
ejpam-4247	148	4	.	.	PROPN
ejpam-4247	148	5	math	math	PROPN
ejpam-4247	148	6	,	,	PUNCT
ejpam-4247	148	7	15	15	NUM
ejpam-4247	148	8	(	(	PUNCT
ejpam-4247	148	9	1	1	NUM
ejpam-4247	148	10	)	)	PUNCT
ejpam-4247	148	11	(	(	PUNCT
ejpam-4247	148	12	2022	2022	NUM
ejpam-4247	148	13	)	)	PUNCT
ejpam-4247	148	14	,	,	PUNCT
ejpam-4247	148	15	144	144	NUM
ejpam-4247	148	16	-	-	SYM
ejpam-4247	148	17	157	157	NUM
ejpam-4247	148	18	150	150	NUM
ejpam-4247	148	19	proof	proof	NOUN
ejpam-4247	148	20	.	.	PUNCT
ejpam-4247	149	1	assume	assume	VERB
ejpam-4247	149	2	that	that	SCONJ
ejpam-4247	149	3	û	û	PROPN
ejpam-4247	149	4	(	(	PUNCT
ejpam-4247	149	5	t	t	PROPN
ejpam-4247	149	6	,	,	PUNCT
ejpam-4247	149	7	u0	u0	ADJ
ejpam-4247	149	8	)	)	PUNCT
ejpam-4247	149	9	is	be	AUX
ejpam-4247	149	10	another	another	DET
ejpam-4247	149	11	solution	solution	NOUN
ejpam-4247	149	12	of	of	ADP
ejpam-4247	149	13	the	the	DET
ejpam-4247	149	14	problem	problem	NOUN
ejpam-4247	149	15	(	(	PUNCT
ejpam-4247	149	16	1.1	1.1	NUM
ejpam-4247	149	17	)	)	PUNCT
ejpam-4247	149	18	with	with	ADP
ejpam-4247	149	19	u(0	u(0	NOUN
ejpam-4247	149	20	)	)	PUNCT
ejpam-4247	149	21	=	=	PUNCT
ejpam-4247	150	1	u0	u0	ADJ
ejpam-4247	150	2	,	,	PUNCT
ejpam-4247	150	3	as	as	SCONJ
ejpam-4247	150	4	follows	follow	VERB
ejpam-4247	150	5	û	û	NUM
ejpam-4247	150	6	(	(	PUNCT
ejpam-4247	150	7	t	t	PROPN
ejpam-4247	150	8	,	,	PUNCT
ejpam-4247	150	9	u0	u0	ADJ
ejpam-4247	150	10	)	)	PUNCT
ejpam-4247	150	11	=	=	SYM
ejpam-4247	150	12	u0	u0	ADJ
ejpam-4247	151	1	+	+	X
ejpam-4247	151	2	2(1−α	2(1−α	NUM
ejpam-4247	151	3	)	)	PUNCT
ejpam-4247	151	4	(	(	PUNCT
ejpam-4247	151	5	2−α)n(α)h(t	2−α)n(α)h(t	NUM
ejpam-4247	151	6	,	,	PUNCT
ejpam-4247	151	7	û(t	û(t	NOUN
ejpam-4247	151	8	)	)	PUNCT
ejpam-4247	151	9	,	,	PUNCT
ejpam-4247	151	10	∫	∫	PROPN
ejpam-4247	151	11	a(t	a(t	PROPN
ejpam-4247	151	12	)	)	PUNCT
ejpam-4247	151	13	0	0	PUNCT
ejpam-4247	152	1	g(s	g(s	NOUN
ejpam-4247	152	2	,	,	PUNCT
ejpam-4247	152	3	û(s))ds	û(s))ds	NOUN
ejpam-4247	152	4	)	)	PUNCT
ejpam-4247	152	5	−	−	PROPN
ejpam-4247	152	6	(	(	PUNCT
ejpam-4247	152	7	2(1−α	2(1−α	NUM
ejpam-4247	152	8	)	)	PUNCT
ejpam-4247	152	9	(	(	PUNCT
ejpam-4247	152	10	2−α)n(α	2−α)n(α	X
ejpam-4247	152	11	)	)	PUNCT
ejpam-4247	152	12	1	1	NUM
ejpam-4247	152	13	t	t	NOUN
ejpam-4247	152	14	∫	∫	PROPN
ejpam-4247	152	15	t	t	PROPN
ejpam-4247	152	16	0	0	NUM
ejpam-4247	152	17	h(s	h(s	PROPN
ejpam-4247	152	18	,	,	PUNCT
ejpam-4247	152	19	û(s	û(s	ADJ
ejpam-4247	152	20	)	)	PUNCT
ejpam-4247	152	21	,	,	PUNCT
ejpam-4247	152	22	∫	∫	PROPN
ejpam-4247	152	23	a(s	a(s	PROPN
ejpam-4247	152	24	)	)	PUNCT
ejpam-4247	152	25	0	0	PUNCT
ejpam-4247	153	1	g(τ	g(τ	PROPN
ejpam-4247	153	2	,	,	PUNCT
ejpam-4247	153	3	û(τ))dτ)ds+	û(τ))dτ)ds+	ADJ
ejpam-4247	153	4	2α	2α	NOUN
ejpam-4247	153	5	(	(	PUNCT
ejpam-4247	153	6	2−α)n(α)∫	2−α)n(α)∫	NUM
ejpam-4247	153	7	t	t	NOUN
ejpam-4247	153	8	0	0	NUM
ejpam-4247	153	9	(	(	PUNCT
ejpam-4247	153	10	h(s	h(s	PROPN
ejpam-4247	153	11	,	,	PUNCT
ejpam-4247	153	12	û(s	û(s	ADJ
ejpam-4247	153	13	)	)	PUNCT
ejpam-4247	153	14	,	,	PUNCT
ejpam-4247	153	15	∫	∫	PROPN
ejpam-4247	153	16	a(s	a(s	PROPN
ejpam-4247	153	17	)	)	PUNCT
ejpam-4247	153	18	0	0	NUM
ejpam-4247	154	1	g(τ	g(τ	PROPN
ejpam-4247	154	2	,	,	PUNCT
ejpam-4247	154	3	û(τ))dτ)−	û(τ))dτ)−	NOUN
ejpam-4247	154	4	1	1	NUM
ejpam-4247	154	5	t	t	NOUN
ejpam-4247	154	6	∫	∫	PROPN
ejpam-4247	154	7	t	t	PROPN
ejpam-4247	154	8	0	0	NUM
ejpam-4247	154	9	h(s	h(s	PROPN
ejpam-4247	154	10	,	,	PUNCT
ejpam-4247	154	11	û(s	û(s	ADJ
ejpam-4247	154	12	)	)	PUNCT
ejpam-4247	154	13	,	,	PUNCT
ejpam-4247	154	14	∫	∫	PROPN
ejpam-4247	154	15	a(s	a(s	PROPN
ejpam-4247	154	16	)	)	PUNCT
ejpam-4247	154	17	0	0	NUM
ejpam-4247	155	1	g(τ	g(τ	PROPN
ejpam-4247	155	2	,	,	PUNCT
ejpam-4247	155	3	û(τ))dτ)ds)ds	û(τ))dτ)ds)ds	X
ejpam-4247	155	4	(	(	PUNCT
ejpam-4247	155	5	4.10	4.10	NUM
ejpam-4247	155	6	)	)	PUNCT
ejpam-4247	155	7	now	now	ADV
ejpam-4247	155	8	,	,	PUNCT
ejpam-4247	155	9	the	the	DET
ejpam-4247	155	10	difference	difference	NOUN
ejpam-4247	155	11	between	between	ADP
ejpam-4247	155	12	the	the	DET
ejpam-4247	155	13	two	two	NUM
ejpam-4247	155	14	solutions	solution	NOUN
ejpam-4247	155	15	u	u	NOUN
ejpam-4247	155	16	(	(	PUNCT
ejpam-4247	155	17	t	t	PROPN
ejpam-4247	155	18	,	,	PUNCT
ejpam-4247	155	19	u0	u0	ADJ
ejpam-4247	155	20	)	)	PUNCT
ejpam-4247	155	21	and	and	CCONJ
ejpam-4247	155	22	û	û	NUM
ejpam-4247	155	23	(	(	PUNCT
ejpam-4247	155	24	t	t	PROPN
ejpam-4247	155	25	,	,	PUNCT
ejpam-4247	155	26	u0),for	u0),for	ADJ
ejpam-4247	155	27	all	all	DET
ejpam-4247	155	28	t	t	NOUN
ejpam-4247	155	29	∈	∈	PROPN
ejpam-4247	156	1	[	[	X
ejpam-4247	156	2	0	0	NUM
ejpam-4247	156	3	,	,	PUNCT
ejpam-4247	156	4	t	t	PROPN
ejpam-4247	156	5	]	]	PUNCT
ejpam-4247	156	6	and	and	CCONJ
ejpam-4247	156	7	u0	u0	PROPN
ejpam-4247	156	8	∈	∈	PROPN
ejpam-4247	156	9	dh	dh	NOUN
ejpam-4247	156	10	,	,	PUNCT
ejpam-4247	156	11	hence	hence	ADV
ejpam-4247	156	12	,	,	PUNCT
ejpam-4247	156	13	by	by	ADP
ejpam-4247	156	14	the	the	DET
ejpam-4247	156	15	inequalities	inequality	NOUN
ejpam-4247	156	16	(	(	PUNCT
ejpam-4247	156	17	3	3	NUM
ejpam-4247	156	18	,	,	PUNCT
ejpam-4247	156	19	3)−	3)−	NUM
ejpam-4247	156	20	(	(	PUNCT
ejpam-4247	156	21	3.5	3.5	NUM
ejpam-4247	156	22	)	)	PUNCT
ejpam-4247	156	23	,	,	PUNCT
ejpam-4247	156	24	we	we	PRON
ejpam-4247	156	25	get	get	VERB
ejpam-4247	156	26	|u	|u	ADJ
ejpam-4247	156	27	(	(	PUNCT
ejpam-4247	156	28	t	t	PROPN
ejpam-4247	156	29	,	,	PUNCT
ejpam-4247	156	30	u0)−	u0)−	PROPN
ejpam-4247	156	31	û	û	X
ejpam-4247	156	32	(	(	PUNCT
ejpam-4247	156	33	t	t	PROPN
ejpam-4247	156	34	,	,	PUNCT
ejpam-4247	156	35	u0)|	u0)|	NOUN
ejpam-4247	156	36	≤	≤	PROPN
ejpam-4247	156	37	(	(	PUNCT
ejpam-4247	156	38	2(1−	2(1−	NUM
ejpam-4247	156	39	α	α	NOUN
ejpam-4247	156	40	)	)	PUNCT
ejpam-4247	157	1	+	+	CCONJ
ejpam-4247	157	2	αt	αt	NOUN
ejpam-4247	157	3	2	2	NUM
ejpam-4247	157	4	)	)	PUNCT
ejpam-4247	157	5	(	(	PUNCT
ejpam-4247	157	6	k1	k1	NOUN
ejpam-4247	157	7	+	+	CCONJ
ejpam-4247	157	8	ay	ay	X
ejpam-4247	157	9	l1k2	l1k2	NOUN
ejpam-4247	157	10	)	)	PUNCT
ejpam-4247	157	11	|u	|u	ADJ
ejpam-4247	157	12	(	(	PUNCT
ejpam-4247	157	13	t	t	PROPN
ejpam-4247	157	14	,	,	PUNCT
ejpam-4247	157	15	u0)−	u0)−	PROPN
ejpam-4247	157	16	û	û	X
ejpam-4247	157	17	(	(	PUNCT
ejpam-4247	157	18	t	t	PROPN
ejpam-4247	157	19	,	,	PUNCT
ejpam-4247	157	20	u0)|	u0)|	ADJ
ejpam-4247	157	21	≤	≤	PROPN
ejpam-4247	157	22	λ	λ	PROPN
ejpam-4247	157	23	|u	|u	ADJ
ejpam-4247	157	24	(	(	PUNCT
ejpam-4247	157	25	tru0)−	tru0)−	INTJ
ejpam-4247	157	26	û	û	PROPN
ejpam-4247	157	27	(	(	PUNCT
ejpam-4247	157	28	t	t	PROPN
ejpam-4247	157	29	,	,	PUNCT
ejpam-4247	157	30	u0)|	u0)|	PROPN
ejpam-4247	157	31	(	(	PUNCT
ejpam-4247	157	32	4.11	4.11	NUM
ejpam-4247	157	33	)	)	PUNCT
ejpam-4247	157	34	by	by	ADP
ejpam-4247	157	35	mathematical	mathematical	ADJ
ejpam-4247	157	36	induction	induction	NOUN
ejpam-4247	157	37	,	,	PUNCT
ejpam-4247	157	38	we	we	PRON
ejpam-4247	157	39	find	find	VERB
ejpam-4247	157	40	that	that	SCONJ
ejpam-4247	157	41	|u	|u	ADJ
ejpam-4247	157	42	(	(	PUNCT
ejpam-4247	157	43	t	t	PROPN
ejpam-4247	157	44	,	,	PUNCT
ejpam-4247	157	45	u0)−	u0)−	PROPN
ejpam-4247	157	46	û	û	X
ejpam-4247	157	47	(	(	PUNCT
ejpam-4247	157	48	t	t	PROPN
ejpam-4247	157	49	,	,	PUNCT
ejpam-4247	157	50	u0)|	u0)|	PROPN
ejpam-4247	157	51	≤	≤	NOUN
ejpam-4247	157	52	λm	λm	ADP
ejpam-4247	157	53	|u	|u	ADJ
ejpam-4247	157	54	(	(	PUNCT
ejpam-4247	157	55	t	t	PROPN
ejpam-4247	157	56	,	,	PUNCT
ejpam-4247	157	57	u0)−	u0)−	PROPN
ejpam-4247	157	58	û	û	X
ejpam-4247	157	59	(	(	PUNCT
ejpam-4247	157	60	t	t	PROPN
ejpam-4247	157	61	,	,	PUNCT
ejpam-4247	157	62	u0)|	u0)|	PROPN
ejpam-4247	157	63	(	(	PUNCT
ejpam-4247	157	64	4.12	4.12	NUM
ejpam-4247	157	65	)	)	PUNCT
ejpam-4247	157	66	from	from	ADP
ejpam-4247	157	67	the	the	DET
ejpam-4247	157	68	condition	condition	NOUN
ejpam-4247	157	69	(	(	PUNCT
ejpam-4247	157	70	3.7	3.7	NUM
ejpam-4247	157	71	)	)	PUNCT
ejpam-4247	157	72	,	,	PUNCT
ejpam-4247	157	73	shows	show	VERB
ejpam-4247	157	74	that	that	SCONJ
ejpam-4247	157	75	the	the	DET
ejpam-4247	157	76	solution	solution	NOUN
ejpam-4247	157	77	u	u	NOUN
ejpam-4247	157	78	(	(	PUNCT
ejpam-4247	157	79	t	t	PROPN
ejpam-4247	157	80	,	,	PUNCT
ejpam-4247	157	81	u0	u0	ADJ
ejpam-4247	157	82	)	)	PUNCT
ejpam-4247	157	83	=	=	SYM
ejpam-4247	157	84	û	û	X
ejpam-4247	157	85	(	(	PUNCT
ejpam-4247	157	86	t	t	PROPN
ejpam-4247	157	87	,	,	PUNCT
ejpam-4247	157	88	u0	u0	NOUN
ejpam-4247	157	89	)	)	PUNCT
ejpam-4247	157	90	,	,	PUNCT
ejpam-4247	157	91	thus	thus	ADV
ejpam-4247	157	92	u	u	X
ejpam-4247	157	93	(	(	PUNCT
ejpam-4247	157	94	t	t	PROPN
ejpam-4247	157	95	,	,	PUNCT
ejpam-4247	157	96	u0	u0	ADJ
ejpam-4247	157	97	)	)	PUNCT
ejpam-4247	157	98	is	be	AUX
ejpam-4247	157	99	a	a	DET
ejpam-4247	157	100	unique	unique	ADJ
ejpam-4247	157	101	periodic	periodic	ADJ
ejpam-4247	157	102	solution	solution	NOUN
ejpam-4247	157	103	on	on	ADP
ejpam-4247	157	104	the	the	DET
ejpam-4247	157	105	domain	domain	NOUN
ejpam-4247	157	106	(	(	PUNCT
ejpam-4247	157	107	4.2	4.2	NUM
ejpam-4247	157	108	)	)	PUNCT
ejpam-4247	157	109	.	.	PUNCT
ejpam-4247	158	1	4.2	4.2	NUM
ejpam-4247	158	2	.	.	PUNCT
ejpam-4247	158	3	existence	existence	NOUN
ejpam-4247	158	4	of	of	ADP
ejpam-4247	158	5	periodic	periodic	ADJ
ejpam-4247	158	6	solutions	solution	NOUN
ejpam-4247	158	7	of	of	ADP
ejpam-4247	158	8	(	(	PUNCT
ejpam-4247	158	9	1.1	1.1	NUM
ejpam-4247	158	10	)	)	PUNCT
ejpam-4247	158	11	the	the	DET
ejpam-4247	158	12	problem	problem	NOUN
ejpam-4247	158	13	of	of	ADP
ejpam-4247	158	14	the	the	DET
ejpam-4247	158	15	existence	existence	NOUN
ejpam-4247	158	16	of	of	ADP
ejpam-4247	158	17	the	the	DET
ejpam-4247	158	18	periodic	periodic	ADJ
ejpam-4247	158	19	solution	solution	NOUN
ejpam-4247	158	20	for	for	ADP
ejpam-4247	158	21	the	the	DET
ejpam-4247	158	22	problem	problem	NOUN
ejpam-4247	158	23	(	(	PUNCT
ejpam-4247	158	24	1.1	1.1	NUM
ejpam-4247	158	25	)	)	PUNCT
ejpam-4247	158	26	with	with	ADP
ejpam-4247	158	27	u(0	u(0	NOUN
ejpam-4247	158	28	)	)	PUNCT
ejpam-4247	158	29	=	=	PRON
ejpam-4247	159	1	u0	u0	PROPN
ejpam-4247	159	2	is	be	AUX
ejpam-4247	159	3	uniquely	uniquely	ADV
ejpam-4247	159	4	connected	connect	VERB
ejpam-4247	159	5	with	with	ADP
ejpam-4247	159	6	the	the	DET
ejpam-4247	159	7	existence	existence	NOUN
ejpam-4247	159	8	of	of	ADP
ejpam-4247	159	9	the	the	DET
ejpam-4247	159	10	zeros	zero	NOUN
ejpam-4247	159	11	of	of	ADP
ejpam-4247	159	12	the	the	DET
ejpam-4247	159	13	functions:µ	functions:µ	NOUN
ejpam-4247	159	14	(	(	PUNCT
ejpam-4247	159	15	0,u0	0,u0	NOUN
ejpam-4247	159	16	)	)	PUNCT
ejpam-4247	160	1	=	=	SYM
ejpam-4247	161	1	1	1	NUM
ejpam-4247	161	2	t	t	NOUN
ejpam-4247	161	3	∫	∫	PROPN
ejpam-4247	161	4	t	t	PROPN
ejpam-4247	161	5	0	0	NUM
ejpam-4247	161	6	h	h	PROPN
ejpam-4247	161	7	(	(	PUNCT
ejpam-4247	161	8	s	s	PROPN
ejpam-4247	161	9	,	,	PUNCT
ejpam-4247	161	10	u(s	u(s	PROPN
ejpam-4247	161	11	)	)	PUNCT
ejpam-4247	161	12	,	,	PUNCT
ejpam-4247	161	13	∫	∫	PROPN
ejpam-4247	161	14	a(s	a(s	PROPN
ejpam-4247	161	15	)	)	PUNCT
ejpam-4247	161	16	0	0	PUNCT
ejpam-4247	162	1	g(τ	g(τ	PROPN
ejpam-4247	162	2	,	,	PUNCT
ejpam-4247	162	3	u(τ))dτ	u(τ))dτ	NOUN
ejpam-4247	162	4	)	)	PUNCT
ejpam-4247	162	5	ds	ds	NOUN
ejpam-4247	162	6	(	(	PUNCT
ejpam-4247	162	7	4.13	4.13	NUM
ejpam-4247	162	8	)	)	PUNCT
ejpam-4247	162	9	also	also	ADV
ejpam-4247	162	10	,	,	PUNCT
ejpam-4247	162	11	we	we	PRON
ejpam-4247	162	12	define	define	VERB
ejpam-4247	162	13	the	the	DET
ejpam-4247	162	14	sequences	sequence	NOUN
ejpam-4247	162	15	of	of	ADP
ejpam-4247	162	16	functions	function	NOUN
ejpam-4247	162	17	µm	µm	X
ejpam-4247	162	18	(	(	PUNCT
ejpam-4247	162	19	0	0	NUM
ejpam-4247	162	20	,	,	PUNCT
ejpam-4247	162	21	u0	u0	ADJ
ejpam-4247	162	22	)	)	PUNCT
ejpam-4247	162	23	are	be	AUX
ejpam-4247	162	24	approximately	approximately	ADV
ejpam-4247	162	25	determined	determine	VERB
ejpam-4247	162	26	by	by	ADP
ejpam-4247	162	27	the	the	DET
ejpam-4247	162	28	following	following	NOUN
ejpam-4247	162	29	:	:	PUNCT
ejpam-4247	162	30	µm	µm	X
ejpam-4247	162	31	(	(	PUNCT
ejpam-4247	162	32	0,u0	0,u0	NOUN
ejpam-4247	162	33	)	)	PUNCT
ejpam-4247	162	34	=	=	SYM
ejpam-4247	163	1	1	1	NUM
ejpam-4247	163	2	t	t	NOUN
ejpam-4247	163	3	∫	∫	PROPN
ejpam-4247	163	4	t	t	PROPN
ejpam-4247	163	5	0	0	NUM
ejpam-4247	163	6	h	h	PROPN
ejpam-4247	163	7	(	(	PUNCT
ejpam-4247	163	8	s	s	PROPN
ejpam-4247	163	9	,	,	PUNCT
ejpam-4247	163	10	um(s	um(s	NOUN
ejpam-4247	163	11	)	)	PUNCT
ejpam-4247	163	12	,	,	PUNCT
ejpam-4247	163	13	∫	∫	PROPN
ejpam-4247	163	14	a(s	a(s	PROPN
ejpam-4247	163	15	)	)	PUNCT
ejpam-4247	163	16	0	0	NUM
ejpam-4247	164	1	g	g	NOUN
ejpam-4247	164	2	(	(	PUNCT
ejpam-4247	164	3	τ	τ	PROPN
ejpam-4247	164	4	,	,	PUNCT
ejpam-4247	164	5	um(τ	um(τ	NUM
ejpam-4247	164	6	)	)	PUNCT
ejpam-4247	164	7	)	)	PUNCT
ejpam-4247	164	8	dτ	dτ	NOUN
ejpam-4247	164	9	)	)	PUNCT
ejpam-4247	164	10	ds	ds	PROPN
ejpam-4247	164	11	(	(	PUNCT
ejpam-4247	164	12	4.14	4.14	NUM
ejpam-4247	164	13	)	)	PUNCT
ejpam-4247	164	14	theorem	theorem	NOUN
ejpam-4247	164	15	4	4	NUM
ejpam-4247	164	16	.	.	PUNCT
ejpam-4247	165	1	if	if	SCONJ
ejpam-4247	165	2	the	the	DET
ejpam-4247	165	3	hypotheses	hypothesis	NOUN
ejpam-4247	165	4	and	and	CCONJ
ejpam-4247	165	5	all	all	DET
ejpam-4247	165	6	the	the	DET
ejpam-4247	165	7	conditions	condition	NOUN
ejpam-4247	165	8	of	of	ADP
ejpam-4247	165	9	the	the	DET
ejpam-4247	165	10	theorem	theorem	ADJ
ejpam-4247	165	11	2	2	NUM
ejpam-4247	165	12	are	be	AUX
ejpam-4247	165	13	given	give	VERB
ejpam-4247	165	14	,	,	PUNCT
ejpam-4247	165	15	the	the	DET
ejpam-4247	165	16	following	follow	VERB
ejpam-4247	165	17	inequalities	inequality	NOUN
ejpam-4247	165	18	are	be	AUX
ejpam-4247	165	19	satisfied:|µ	satisfied:|µ	PROPN
ejpam-4247	165	20	(	(	PUNCT
ejpam-4247	165	21	0	0	PROPN
ejpam-4247	165	22	,	,	PUNCT
ejpam-4247	165	23	u0)−	u0)−	PROPN
ejpam-4247	165	24	µm	µm	X
ejpam-4247	165	25	(	(	PUNCT
ejpam-4247	165	26	0,u0)|	0,u0)|	NOUN
ejpam-4247	165	27	≤	≤	NUM
ejpam-4247	165	28	(	(	PUNCT
ejpam-4247	165	29	k1	k1	NOUN
ejpam-4247	165	30	+	+	CCONJ
ejpam-4247	165	31	atl1k2	atl1k2	NOUN
ejpam-4247	165	32	)	)	PUNCT
ejpam-4247	165	33	λ	λ	PROPN
ejpam-4247	165	34	m(1−	m(1−	PROPN
ejpam-4247	165	35	λ)−1m1	λ)−1m1	PROPN
ejpam-4247	165	36	(	(	PUNCT
ejpam-4247	165	37	4.15	4.15	NUM
ejpam-4247	165	38	)	)	PUNCT
ejpam-4247	165	39	holds	hold	VERB
ejpam-4247	165	40	for	for	ADP
ejpam-4247	165	41	all	all	DET
ejpam-4247	165	42	m	m	PROPN
ejpam-4247	165	43	≥	≥	NOUN
ejpam-4247	165	44	0	0	NUM
ejpam-4247	165	45	proof	proof	NOUN
ejpam-4247	165	46	.	.	PUNCT
ejpam-4247	166	1	from	from	ADP
ejpam-4247	166	2	equations	equation	NOUN
ejpam-4247	166	3	(	(	PUNCT
ejpam-4247	166	4	4.13	4.13	NUM
ejpam-4247	166	5	)	)	PUNCT
ejpam-4247	166	6	to	to	ADP
ejpam-4247	166	7	(	(	PUNCT
ejpam-4247	166	8	4.14	4.14	NUM
ejpam-4247	166	9	)	)	PUNCT
ejpam-4247	166	10	,	,	PUNCT
ejpam-4247	166	11	we	we	PRON
ejpam-4247	166	12	obtain	obtain	VERB
ejpam-4247	166	13	that	that	DET
ejpam-4247	166	14	|µ	|µ	NOUN
ejpam-4247	166	15	(	(	PUNCT
ejpam-4247	166	16	0	0	NUM
ejpam-4247	166	17	,	,	PUNCT
ejpam-4247	166	18	u0)−	u0)−	PROPN
ejpam-4247	166	19	µm	µm	X
ejpam-4247	166	20	(	(	PUNCT
ejpam-4247	166	21	0,u0)|	0,u0)|	NOUN
ejpam-4247	166	22	≤	≤	NUM
ejpam-4247	166	23	(	(	PUNCT
ejpam-4247	166	24	k1	k1	NOUN
ejpam-4247	166	25	+	+	CCONJ
ejpam-4247	166	26	atl1k2	atl1k2	NOUN
ejpam-4247	166	27	)	)	PUNCT
ejpam-4247	166	28	|u	|u	ADJ
ejpam-4247	166	29	(	(	PUNCT
ejpam-4247	166	30	t	t	PROPN
ejpam-4247	166	31	,	,	PUNCT
ejpam-4247	166	32	u0)−	u0)−	PROPN
ejpam-4247	166	33	um	um	INTJ
ejpam-4247	166	34	(	(	PUNCT
ejpam-4247	166	35	t	t	PROPN
ejpam-4247	166	36	,	,	PUNCT
ejpam-4247	166	37	u0)|	u0)|	PROPN
ejpam-4247	166	38	≤	≤	NOUN
ejpam-4247	166	39	(	(	PUNCT
ejpam-4247	166	40	k1	k1	NOUN
ejpam-4247	166	41	+	+	CCONJ
ejpam-4247	166	42	arl1k2	arl1k2	NOUN
ejpam-4247	166	43	)	)	PUNCT
ejpam-4247	166	44	λ	λ	PROPN
ejpam-4247	166	45	m(1−	m(1−	PROPN
ejpam-4247	166	46	λ)−1m1	λ)−1m1	PROPN
ejpam-4247	166	47	(	(	PUNCT
ejpam-4247	166	48	4.16	4.16	NUM
ejpam-4247	166	49	)	)	PUNCT
ejpam-4247	166	50	a.	a.	NOUN
ejpam-4247	166	51	s.	s.	PROPN
ejpam-4247	166	52	rafeeq	rafeeq	PROPN
ejpam-4247	166	53	/	/	SYM
ejpam-4247	166	54	eur	eur	PROPN
ejpam-4247	166	55	.	.	PUNCT
ejpam-4247	167	1	j.	j.	PROPN
ejpam-4247	167	2	pure	pure	PROPN
ejpam-4247	167	3	appl	appl	PROPN
ejpam-4247	167	4	.	.	PROPN
ejpam-4247	167	5	math	math	PROPN
ejpam-4247	167	6	,	,	PUNCT
ejpam-4247	167	7	15	15	NUM
ejpam-4247	167	8	(	(	PUNCT
ejpam-4247	167	9	1	1	NUM
ejpam-4247	167	10	)	)	PUNCT
ejpam-4247	167	11	(	(	PUNCT
ejpam-4247	167	12	2022	2022	NUM
ejpam-4247	167	13	)	)	PUNCT
ejpam-4247	167	14	,	,	PUNCT
ejpam-4247	167	15	144	144	NUM
ejpam-4247	167	16	-	-	SYM
ejpam-4247	167	17	157	157	NUM
ejpam-4247	167	18	151	151	NUM
ejpam-4247	167	19	the	the	DET
ejpam-4247	167	20	inequality	inequality	NOUN
ejpam-4247	167	21	(	(	PUNCT
ejpam-4247	167	22	4.15	4.15	NUM
ejpam-4247	167	23	)	)	PUNCT
ejpam-4247	167	24	is	be	AUX
ejpam-4247	167	25	hold	hold	NOUN
ejpam-4247	167	26	for	for	ADP
ejpam-4247	167	27	all	all	DET
ejpam-4247	167	28	m	m	PROPN
ejpam-4247	167	29	≥	≥	NOUN
ejpam-4247	167	30	0	0	NUM
ejpam-4247	167	31	.	.	PUNCT
ejpam-4247	168	1	theorem	theorem	NOUN
ejpam-4247	168	2	5	5	NUM
ejpam-4247	168	3	.	.	PUNCT
ejpam-4247	169	1	let	let	VERB
ejpam-4247	169	2	the	the	DET
ejpam-4247	169	3	function	function	NOUN
ejpam-4247	169	4	h(s	h(s	PROPN
ejpam-4247	169	5	,	,	PUNCT
ejpam-4247	169	6	u(s	u(s	NUM
ejpam-4247	169	7	)	)	PUNCT
ejpam-4247	169	8	,	,	PUNCT
ejpam-4247	169	9	z(t	z(t	PROPN
ejpam-4247	169	10	)	)	PUNCT
ejpam-4247	169	11	)	)	PUNCT
ejpam-4247	169	12	be	be	AUX
ejpam-4247	169	13	defined	define	VERB
ejpam-4247	169	14	on	on	ADP
ejpam-4247	169	15	the	the	DET
ejpam-4247	169	16	intervals	interval	NOUN
ejpam-4247	169	17	[	[	X
ejpam-4247	169	18	c	c	X
ejpam-4247	169	19	,	,	PUNCT
ejpam-4247	169	20	d	d	X
ejpam-4247	169	21	]	]	X
ejpam-4247	169	22	on	on	ADP
ejpam-4247	169	23	r	r	NOUN
ejpam-4247	169	24	and	and	CCONJ
ejpam-4247	169	25	periodic	periodic	NOUN
ejpam-4247	169	26	in	in	ADP
ejpam-4247	169	27	t	t	PROPN
ejpam-4247	169	28	of	of	ADP
ejpam-4247	169	29	period	period	NOUN
ejpam-4247	169	30	t	t	PROPN
ejpam-4247	169	31	,	,	PUNCT
ejpam-4247	169	32	suppose	suppose	VERB
ejpam-4247	169	33	that	that	SCONJ
ejpam-4247	169	34	for	for	ADP
ejpam-4247	169	35	all	all	DET
ejpam-4247	169	36	m	m	PROPN
ejpam-4247	169	37	≥	≥	NOUN
ejpam-4247	169	38	0	0	NUM
ejpam-4247	169	39	,	,	PUNCT
ejpam-4247	169	40	then	then	ADV
ejpam-4247	169	41	the	the	DET
ejpam-4247	169	42	sequences	sequence	NOUN
ejpam-4247	169	43	of	of	ADP
ejpam-4247	169	44	the	the	DET
ejpam-4247	169	45	functions	function	NOUN
ejpam-4247	169	46	µm	µm	X
ejpam-4247	169	47	(	(	PUNCT
ejpam-4247	169	48	0	0	NUM
ejpam-4247	169	49	,	,	PUNCT
ejpam-4247	169	50	u0	u0	PROPN
ejpam-4247	169	51	)	)	PUNCT
ejpam-4247	169	52	which	which	PRON
ejpam-4247	169	53	are	be	AUX
ejpam-4247	169	54	defined	define	VERB
ejpam-4247	169	55	in	in	ADP
ejpam-4247	169	56	(	(	PUNCT
ejpam-4247	169	57	4.14	4.14	NUM
ejpam-4247	169	58	)	)	PUNCT
ejpam-4247	169	59	satisfy	satisfy	VERB
ejpam-4247	169	60	the	the	DET
ejpam-4247	169	61	inequalities	inequality	NOUN
ejpam-4247	169	62	:	:	PUNCT
ejpam-4247	169	63	minu0∈[c	minu0∈[c	ADJ
ejpam-4247	169	64	,	,	PUNCT
ejpam-4247	169	65	d	d	X
ejpam-4247	169	66	]	]	X
ejpam-4247	169	67	µm	µm	X
ejpam-4247	169	68	(	(	PUNCT
ejpam-4247	169	69	0	0	NUM
ejpam-4247	169	70	,	,	PUNCT
ejpam-4247	169	71	u0	u0	ADJ
ejpam-4247	169	72	)	)	PUNCT
ejpam-4247	169	73	≤	≤	NUM
ejpam-4247	170	1	−	−	PROPN
ejpam-4247	171	1	(	(	PUNCT
ejpam-4247	171	2	k1	k1	NOUN
ejpam-4247	171	3	+	+	CCONJ
ejpam-4247	171	4	aτl1k2	aτl1k2	PROPN
ejpam-4247	171	5	)	)	PUNCT
ejpam-4247	171	6	λ	λ	PROPN
ejpam-4247	171	7	m(1−	m(1−	PROPN
ejpam-4247	171	8	λ)−1m1	λ)−1m1	PROPN
ejpam-4247	171	9	maxu0∈[c	maxu0∈[c	PROPN
ejpam-4247	171	10	,	,	PUNCT
ejpam-4247	171	11	d	d	X
ejpam-4247	171	12	]	]	X
ejpam-4247	171	13	µm	µm	X
ejpam-4247	171	14	(	(	PUNCT
ejpam-4247	171	15	0,u0	0,u0	NOUN
ejpam-4247	171	16	)	)	PUNCT
ejpam-4247	171	17	≥	≥	PROPN
ejpam-4247	171	18	(	(	PUNCT
ejpam-4247	171	19	k1	k1	X
ejpam-4247	171	20	+	+	CCONJ
ejpam-4247	171	21	arl1k2	arl1k2	NOUN
ejpam-4247	171	22	)	)	PUNCT
ejpam-4247	171	23	λ	λ	PROPN
ejpam-4247	171	24	m(1−	m(1−	PROPN
ejpam-4247	171	25	λ)−1m1	λ)−1m1	PROPN
ejpam-4247	171	26	}	}	PUNCT
ejpam-4247	171	27	(	(	PUNCT
ejpam-4247	171	28	4.17	4.17	NUM
ejpam-4247	171	29	)	)	PUNCT
ejpam-4247	171	30	then	then	ADV
ejpam-4247	171	31	the	the	DET
ejpam-4247	171	32	problem	problem	NOUN
ejpam-4247	171	33	(	(	PUNCT
ejpam-4247	171	34	1.1	1.1	NUM
ejpam-4247	171	35	)	)	PUNCT
ejpam-4247	171	36	has	have	VERB
ejpam-4247	171	37	a	a	DET
ejpam-4247	171	38	periodic	periodic	ADJ
ejpam-4247	171	39	solution	solution	NOUN
ejpam-4247	171	40	u	u	PROPN
ejpam-4247	171	41	(	(	PUNCT
ejpam-4247	171	42	t	t	PROPN
ejpam-4247	171	43	,	,	PUNCT
ejpam-4247	171	44	u0	u0	ADJ
ejpam-4247	171	45	)	)	PUNCT
ejpam-4247	171	46	such	such	ADJ
ejpam-4247	171	47	that	that	DET
ejpam-4247	171	48	u0	u0	PROPN
ejpam-4247	171	49	∈	∈	PROPN
ejpam-4247	172	1	[	[	X
ejpam-4247	172	2	c	c	X
ejpam-4247	172	3	,	,	PUNCT
ejpam-4247	172	4	d	d	X
ejpam-4247	172	5	]	]	X
ejpam-4247	172	6	=	=	PUNCT
ejpam-4247	173	1	[	[	X
ejpam-4247	173	2	c	c	X
ejpam-4247	173	3	+	+	NOUN
ejpam-4247	173	4	m1	m1	NOUN
ejpam-4247	173	5	,	,	PUNCT
ejpam-4247	173	6	d−m1	d−m1	NOUN
ejpam-4247	173	7	]	]	X
ejpam-4247	173	8	proof	proof	NOUN
ejpam-4247	173	9	.	.	PUNCT
ejpam-4247	174	1	let	let	VERB
ejpam-4247	174	2	u1	u1	NOUN
ejpam-4247	174	3	and	and	CCONJ
ejpam-4247	174	4	u2	u2	NOUN
ejpam-4247	174	5	be	be	VERB
ejpam-4247	174	6	any	any	DET
ejpam-4247	174	7	points	point	NOUN
ejpam-4247	174	8	belonging	belong	VERB
ejpam-4247	174	9	to	to	ADP
ejpam-4247	174	10	the	the	DET
ejpam-4247	174	11	intervals	interval	NOUN
ejpam-4247	174	12	[	[	X
ejpam-4247	174	13	c	c	X
ejpam-4247	174	14	,	,	PUNCT
ejpam-4247	174	15	d	d	X
ejpam-4247	174	16	]	]	X
ejpam-4247	174	17	,	,	PUNCT
ejpam-4247	174	18	such	such	ADJ
ejpam-4247	174	19	that	that	PRON
ejpam-4247	174	20	µm	µm	NOUN
ejpam-4247	174	21	(	(	PUNCT
ejpam-4247	174	22	0,u1	0,u1	ADJ
ejpam-4247	174	23	)	)	PUNCT
ejpam-4247	174	24	=	=	SYM
ejpam-4247	174	25	minu0∈[c	minu0∈[c	NOUN
ejpam-4247	174	26	,	,	PUNCT
ejpam-4247	174	27	d	d	X
ejpam-4247	174	28	]	]	X
ejpam-4247	174	29	µm	µm	X
ejpam-4247	174	30	(	(	PUNCT
ejpam-4247	174	31	0	0	NUM
ejpam-4247	174	32	,	,	PUNCT
ejpam-4247	174	33	u0	u0	ADJ
ejpam-4247	174	34	)	)	PUNCT
ejpam-4247	174	35	µm	µm	ADP
ejpam-4247	174	36	(	(	PUNCT
ejpam-4247	174	37	0,u2	0,u2	NUM
ejpam-4247	174	38	)	)	PUNCT
ejpam-4247	175	1	=	=	SYM
ejpam-4247	175	2	maxu0∈[c	maxu0∈[c	PROPN
ejpam-4247	175	3	,	,	PUNCT
ejpam-4247	175	4	d	d	X
ejpam-4247	175	5	]	]	X
ejpam-4247	175	6	µm	µm	X
ejpam-4247	175	7	(	(	PUNCT
ejpam-4247	175	8	0	0	NUM
ejpam-4247	175	9	,	,	PUNCT
ejpam-4247	175	10	u0	u0	ADJ
ejpam-4247	175	11	)	)	PUNCT
ejpam-4247	175	12	}	}	PUNCT
ejpam-4247	175	13	(	(	PUNCT
ejpam-4247	175	14	4.18	4.18	NUM
ejpam-4247	175	15	)	)	PUNCT
ejpam-4247	175	16	by	by	ADP
ejpam-4247	175	17	using	use	VERB
ejpam-4247	175	18	inequalities	inequality	NOUN
ejpam-4247	175	19	(	(	PUNCT
ejpam-4247	175	20	4.15	4.15	NUM
ejpam-4247	175	21	)	)	PUNCT
ejpam-4247	175	22	to	to	ADP
ejpam-4247	175	23	(	(	PUNCT
ejpam-4247	175	24	4.18	4.18	NUM
ejpam-4247	175	25	)	)	PUNCT
ejpam-4247	175	26	,	,	PUNCT
ejpam-4247	175	27	,	,	PUNCT
ejpam-4247	175	28	the	the	DET
ejpam-4247	175	29	following	follow	VERB
ejpam-4247	175	30	are	be	AUX
ejpam-4247	175	31	obtained:µ	obtained:µ	NOUN
ejpam-4247	175	32	(	(	PUNCT
ejpam-4247	175	33	0	0	NUM
ejpam-4247	175	34	,	,	PUNCT
ejpam-4247	175	35	u1	u1	NOUN
ejpam-4247	175	36	)	)	PUNCT
ejpam-4247	175	37	=	=	SYM
ejpam-4247	175	38	µm	µm	X
ejpam-4247	175	39	(	(	PUNCT
ejpam-4247	175	40	0	0	NUM
ejpam-4247	175	41	,	,	PUNCT
ejpam-4247	175	42	u1	u1	NOUN
ejpam-4247	175	43	)	)	PUNCT
ejpam-4247	176	1	+	+	CCONJ
ejpam-4247	176	2	(	(	PUNCT
ejpam-4247	176	3	µ	µ	X
ejpam-4247	176	4	(	(	PUNCT
ejpam-4247	176	5	0	0	NUM
ejpam-4247	176	6	,	,	PUNCT
ejpam-4247	176	7	u1)−	u1)−	ADJ
ejpam-4247	176	8	µm	µm	X
ejpam-4247	176	9	(	(	PUNCT
ejpam-4247	176	10	0,u1	0,u1	NUM
ejpam-4247	176	11	)	)	PUNCT
ejpam-4247	176	12	)	)	PUNCT
ejpam-4247	177	1	<	<	X
ejpam-4247	177	2	0	0	NUM
ejpam-4247	177	3	µ	µ	X
ejpam-4247	177	4	(	(	PUNCT
ejpam-4247	177	5	0	0	NUM
ejpam-4247	177	6	,	,	PUNCT
ejpam-4247	177	7	u2	u2	NOUN
ejpam-4247	177	8	)	)	PUNCT
ejpam-4247	177	9	=	=	SYM
ejpam-4247	177	10	µm	µm	X
ejpam-4247	177	11	(	(	PUNCT
ejpam-4247	177	12	0	0	NUM
ejpam-4247	177	13	,	,	PUNCT
ejpam-4247	177	14	u2	u2	NOUN
ejpam-4247	177	15	)	)	PUNCT
ejpam-4247	177	16	+	+	CCONJ
ejpam-4247	177	17	(	(	PUNCT
ejpam-4247	177	18	µ	µ	X
ejpam-4247	177	19	(	(	PUNCT
ejpam-4247	177	20	0,u2)−	0,u2)−	NUM
ejpam-4247	177	21	µm	µm	ADP
ejpam-4247	177	22	(	(	PUNCT
ejpam-4247	177	23	0	0	NUM
ejpam-4247	177	24	,	,	PUNCT
ejpam-4247	177	25	u2	u2	NOUN
ejpam-4247	177	26	)	)	PUNCT
ejpam-4247	177	27	)	)	PUNCT
ejpam-4247	177	28	>	>	X
ejpam-4247	177	29	0	0	PUNCT
ejpam-4247	177	30	}	}	PUNCT
ejpam-4247	177	31	(	(	PUNCT
ejpam-4247	177	32	4.19	4.19	NUM
ejpam-4247	177	33	)	)	PUNCT
ejpam-4247	177	34	and	and	CCONJ
ejpam-4247	177	35	from	from	ADP
ejpam-4247	177	36	the	the	DET
ejpam-4247	177	37	continuity	continuity	NOUN
ejpam-4247	177	38	of	of	ADP
ejpam-4247	177	39	the	the	DET
ejpam-4247	177	40	functions	function	NOUN
ejpam-4247	177	41	µ	µ	X
ejpam-4247	177	42	(	(	PUNCT
ejpam-4247	177	43	0	0	NUM
ejpam-4247	177	44	,	,	PUNCT
ejpam-4247	177	45	u1	u1	NOUN
ejpam-4247	177	46	)	)	PUNCT
ejpam-4247	177	47	,	,	PUNCT
ejpam-4247	177	48	µ	µ	X
ejpam-4247	177	49	(	(	PUNCT
ejpam-4247	177	50	0	0	NUM
ejpam-4247	177	51	,	,	PUNCT
ejpam-4247	177	52	u2	u2	NOUN
ejpam-4247	177	53	)	)	PUNCT
ejpam-4247	177	54	and	and	CCONJ
ejpam-4247	177	55	the	the	DET
ejpam-4247	177	56	inequalities	inequality	NOUN
ejpam-4247	177	57	(	(	PUNCT
ejpam-4247	177	58	4.19	4.19	NUM
ejpam-4247	177	59	)	)	PUNCT
ejpam-4247	177	60	,	,	PUNCT
ejpam-4247	177	61	then	then	ADV
ejpam-4247	177	62	the	the	DET
ejpam-4247	177	63	isolated	isolated	ADJ
ejpam-4247	177	64	singular	singular	PROPN
ejpam-4247	177	65	points	point	NOUN
ejpam-4247	177	66	u0	u0	PROPN
ejpam-4247	177	67	∈	∈	PROPN
ejpam-4247	178	1	[	[	X
ejpam-4247	178	2	c	c	X
ejpam-4247	178	3	,	,	PUNCT
ejpam-4247	178	4	d	d	X
ejpam-4247	178	5	]	]	PUNCT
ejpam-4247	178	6	exist	exist	VERB
ejpam-4247	178	7	such	such	ADJ
ejpam-4247	178	8	that	that	SCONJ
ejpam-4247	178	9	µ	µ	X
ejpam-4247	178	10	(	(	PUNCT
ejpam-4247	178	11	0	0	NUM
ejpam-4247	178	12	,	,	PUNCT
ejpam-4247	178	13	u0	u0	ADJ
ejpam-4247	178	14	)	)	PUNCT
ejpam-4247	179	1	=	=	SYM
ejpam-4247	179	2	0	0	X
ejpam-4247	179	3	.	.	PUNCT
ejpam-4247	180	1	this	this	PRON
ejpam-4247	180	2	means	mean	VERB
ejpam-4247	180	3	that	that	SCONJ
ejpam-4247	180	4	(	(	PUNCT
ejpam-4247	180	5	1.1	1.1	NUM
ejpam-4247	180	6	)	)	PUNCT
ejpam-4247	180	7	has	have	VERB
ejpam-4247	180	8	a	a	DET
ejpam-4247	180	9	periodic	periodic	ADJ
ejpam-4247	180	10	solution	solution	NOUN
ejpam-4247	180	11	u	u	NOUN
ejpam-4247	180	12	(	(	PUNCT
ejpam-4247	180	13	t	t	PROPN
ejpam-4247	180	14	,	,	PUNCT
ejpam-4247	180	15	u0	u0	ADJ
ejpam-4247	180	16	)	)	PUNCT
ejpam-4247	180	17	.	.	PUNCT
ejpam-4247	181	1	4.3	4.3	NUM
ejpam-4247	181	2	.	.	PUNCT
ejpam-4247	181	3	stability	stability	NOUN
ejpam-4247	181	4	of	of	ADP
ejpam-4247	181	5	periodic	periodic	ADJ
ejpam-4247	181	6	solution	solution	NOUN
ejpam-4247	181	7	of	of	ADP
ejpam-4247	181	8	(	(	PUNCT
ejpam-4247	181	9	1.1	1.1	NUM
ejpam-4247	181	10	)	)	PUNCT
ejpam-4247	181	11	in	in	ADP
ejpam-4247	181	12	this	this	DET
ejpam-4247	181	13	section	section	NOUN
ejpam-4247	181	14	,	,	PUNCT
ejpam-4247	181	15	we	we	PRON
ejpam-4247	181	16	investigate	investigate	VERB
ejpam-4247	181	17	the	the	DET
ejpam-4247	181	18	stability	stability	NOUN
ejpam-4247	181	19	or	or	CCONJ
ejpam-4247	181	20	periodic	periodic	ADJ
ejpam-4247	181	21	solution	solution	NOUN
ejpam-4247	181	22	of	of	ADP
ejpam-4247	181	23	(	(	PUNCT
ejpam-4247	181	24	1.1	1.1	NUM
ejpam-4247	181	25	)	)	PUNCT
ejpam-4247	181	26	.	.	PUNCT
ejpam-4247	182	1	theorem	theorem	VERB
ejpam-4247	182	2	6	6	NUM
ejpam-4247	182	3	.	.	PUNCT
ejpam-4247	183	1	let	let	VERB
ejpam-4247	183	2	the	the	DET
ejpam-4247	183	3	function	function	NOUN
ejpam-4247	183	4	µ	µ	X
ejpam-4247	183	5	(	(	PUNCT
ejpam-4247	183	6	0	0	NUM
ejpam-4247	183	7	,	,	PUNCT
ejpam-4247	183	8	u0	u0	ADJ
ejpam-4247	183	9	)	)	PUNCT
ejpam-4247	183	10	be	be	AUX
ejpam-4247	183	11	defined	define	VERB
ejpam-4247	183	12	by	by	ADP
ejpam-4247	183	13	the	the	DET
ejpam-4247	183	14	equation	equation	NOUN
ejpam-4247	183	15	(	(	PUNCT
ejpam-4247	183	16	4.13	4.13	NUM
ejpam-4247	183	17	)	)	PUNCT
ejpam-4247	183	18	where	where	SCONJ
ejpam-4247	183	19	u	u	PROPN
ejpam-4247	183	20	(	(	PUNCT
ejpam-4247	183	21	t	t	PROPN
ejpam-4247	183	22	,	,	PUNCT
ejpam-4247	183	23	u0	u0	ADJ
ejpam-4247	183	24	)	)	PUNCT
ejpam-4247	183	25	is	be	AUX
ejpam-4247	183	26	a	a	DET
ejpam-4247	183	27	limit	limit	NOUN
ejpam-4247	183	28	of	of	ADP
ejpam-4247	183	29	the	the	DET
ejpam-4247	183	30	sequence	sequence	NOUN
ejpam-4247	183	31	of	of	ADP
ejpam-4247	183	32	the	the	DET
ejpam-4247	183	33	function	function	NOUN
ejpam-4247	183	34	(	(	PUNCT
ejpam-4247	183	35	4.1	4.1	NUM
ejpam-4247	183	36	)	)	PUNCT
ejpam-4247	183	37	,	,	PUNCT
ejpam-4247	183	38	then	then	ADV
ejpam-4247	183	39	the	the	DET
ejpam-4247	183	40	following	follow	VERB
ejpam-4247	183	41	inequalities	inequality	NOUN
ejpam-4247	183	42	yield:|µ	yield:|µ	PROPN
ejpam-4247	183	43	(	(	PUNCT
ejpam-4247	183	44	0	0	NUM
ejpam-4247	183	45	,	,	PUNCT
ejpam-4247	183	46	u0)|	u0)|	PROPN
ejpam-4247	183	47	≤	≤	PUNCT
ejpam-4247	183	48	m	m	VERB
ejpam-4247	183	49	(	(	PUNCT
ejpam-4247	183	50	4.20	4.20	NUM
ejpam-4247	183	51	)	)	PUNCT
ejpam-4247	183	52	and	and	CCONJ
ejpam-4247	183	53	∣∣µ	∣∣µ	PROPN
ejpam-4247	183	54	(	(	PUNCT
ejpam-4247	183	55	0,u10)−	0,u10)−	NOUN
ejpam-4247	183	56	µ	µ	X
ejpam-4247	183	57	(	(	PUNCT
ejpam-4247	183	58	0	0	NUM
ejpam-4247	183	59	,	,	PUNCT
ejpam-4247	183	60	u20	u20	NUM
ejpam-4247	183	61	)	)	PUNCT
ejpam-4247	184	1	∣∣	∣∣	X
ejpam-4247	184	2	≤	≤	NOUN
ejpam-4247	184	3	f2f3	f2f3	ADP
ejpam-4247	184	4	∣∣u10	∣∣u10	PROPN
ejpam-4247	184	5	−	−	PROPN
ejpam-4247	184	6	u20	u20	PROPN
ejpam-4247	184	7	∣∣	∣∣	NUM
ejpam-4247	184	8	(	(	PUNCT
ejpam-4247	184	9	4.21	4.21	NUM
ejpam-4247	184	10	)	)	PUNCT
ejpam-4247	184	11	where	where	SCONJ
ejpam-4247	184	12	f1	f1	NOUN
ejpam-4247	184	13	=	=	SYM
ejpam-4247	184	14	2(1−	2(1−	NUM
ejpam-4247	184	15	α	α	NUM
ejpam-4247	184	16	)	)	PUNCT
ejpam-4247	184	17	+	+	CCONJ
ejpam-4247	184	18	αt	αt	NOUN
ejpam-4247	184	19	2	2	NUM
ejpam-4247	184	20	,	,	PUNCT
ejpam-4247	184	21	f2	f2	PROPN
ejpam-4247	184	22	=	=	SYM
ejpam-4247	184	23	k1	k1	PROPN
ejpam-4247	184	24	+	+	CCONJ
ejpam-4247	184	25	atl1k2	atl1k2	NOUN
ejpam-4247	184	26	,	,	PUNCT
ejpam-4247	184	27	f3	f3	NOUN
ejpam-4247	184	28	=	=	SYM
ejpam-4247	184	29	(	(	PUNCT
ejpam-4247	184	30	1−	1−	NUM
ejpam-4247	184	31	f1f2	f1f2	NOUN
ejpam-4247	184	32	)	)	PUNCT
ejpam-4247	184	33	−1	−1	NOUN
ejpam-4247	184	34	proof	proof	NOUN
ejpam-4247	184	35	from	from	ADP
ejpam-4247	184	36	the	the	DET
ejpam-4247	184	37	properties	property	NOUN
ejpam-4247	184	38	of	of	ADP
ejpam-4247	184	39	the	the	DET
ejpam-4247	184	40	function	function	NOUN
ejpam-4247	184	41	u	u	PROPN
ejpam-4247	184	42	(	(	PUNCT
ejpam-4247	184	43	t	t	PROPN
ejpam-4247	184	44	,	,	PUNCT
ejpam-4247	184	45	u0	u0	ADJ
ejpam-4247	184	46	)	)	PUNCT
ejpam-4247	184	47	as	as	ADP
ejpam-4247	184	48	in	in	ADP
ejpam-4247	184	49	the	the	DET
ejpam-4247	184	50	theorem	theorem	NOUN
ejpam-4247	184	51	2	2	NUM
ejpam-4247	184	52	,	,	PUNCT
ejpam-4247	184	53	the	the	DET
ejpam-4247	184	54	function	function	NOUN
ejpam-4247	184	55	µ	µ	X
ejpam-4247	184	56	(	(	PUNCT
ejpam-4247	184	57	0	0	NUM
ejpam-4247	184	58	,	,	PUNCT
ejpam-4247	184	59	u0	u0	ADJ
ejpam-4247	184	60	)	)	PUNCT
ejpam-4247	184	61	,	,	PUNCT
ejpam-4247	184	62	u0	u0	PROPN
ejpam-4247	184	63	∈	∈	PROPN
ejpam-4247	185	1	d	d	NOUN
ejpam-4247	185	2	is	be	AUX
ejpam-4247	185	3	continuous	continuous	ADJ
ejpam-4247	185	4	and	and	CCONJ
ejpam-4247	185	5	bounded	bound	VERB
ejpam-4247	185	6	by	by	ADP
ejpam-4247	185	7	1−α	1−α	NUM
ejpam-4247	186	1	αt	αt	NOUN
ejpam-4247	186	2	gt	gt	PROPN
ejpam-4247	187	1	+	+	NUM
ejpam-4247	187	2	m	m	VERB
ejpam-4247	187	3	in	in	ADP
ejpam-4247	187	4	the	the	DET
ejpam-4247	187	5	domain	domain	NOUN
ejpam-4247	188	1	(	(	PUNCT
ejpam-4247	188	2	4.2	4.2	NUM
ejpam-4247	188	3	)	)	PUNCT
ejpam-4247	188	4	.	.	PUNCT
ejpam-4247	189	1	from	from	ADP
ejpam-4247	189	2	(	(	PUNCT
ejpam-4247	189	3	4.13	4.13	NUM
ejpam-4247	189	4	)	)	PUNCT
ejpam-4247	189	5	,	,	PUNCT
ejpam-4247	189	6	we	we	PRON
ejpam-4247	189	7	obtained	obtain	VERB
ejpam-4247	189	8	that	that	DET
ejpam-4247	189	9	|µ	|µ	NOUN
ejpam-4247	189	10	(	(	PUNCT
ejpam-4247	189	11	0	0	NUM
ejpam-4247	189	12	,	,	PUNCT
ejpam-4247	189	13	u0)|	u0)|	NOUN
ejpam-4247	189	14	≤	≤	NOUN
ejpam-4247	189	15	1	1	NUM
ejpam-4247	190	1	t	t	NOUN
ejpam-4247	190	2	∫	∫	PROPN
ejpam-4247	190	3	t	t	PROPN
ejpam-4247	190	4	0	0	NUM
ejpam-4247	190	5	∣∣∣∣∣h	∣∣∣∣∣h	PROPN
ejpam-4247	190	6	(	(	PUNCT
ejpam-4247	190	7	s	s	X
ejpam-4247	190	8	,	,	PUNCT
ejpam-4247	190	9	u(s	u(s	PROPN
ejpam-4247	190	10	)	)	PUNCT
ejpam-4247	190	11	,	,	PUNCT
ejpam-4247	190	12	∫	∫	PROPN
ejpam-4247	190	13	a(s	a(s	PROPN
ejpam-4247	190	14	)	)	PUNCT
ejpam-4247	190	15	0	0	PUNCT
ejpam-4247	191	1	g(τ	g(τ	PROPN
ejpam-4247	191	2	,	,	PUNCT
ejpam-4247	191	3	u(τ))dτ	u(τ))dτ	NOUN
ejpam-4247	191	4	)	)	PUNCT
ejpam-4247	192	1	∣∣∣∣∣	∣∣∣∣∣	ADP
ejpam-4247	192	2	ds	ds	PROPN
ejpam-4247	192	3	≤	≤	NUM
ejpam-4247	192	4	m	m	VERB
ejpam-4247	192	5	(	(	PUNCT
ejpam-4247	192	6	4.22	4.22	NUM
ejpam-4247	192	7	)	)	PUNCT
ejpam-4247	192	8	a.	a.	NOUN
ejpam-4247	192	9	s.	s.	PROPN
ejpam-4247	192	10	rafeeq	rafeeq	PROPN
ejpam-4247	192	11	/	/	SYM
ejpam-4247	192	12	eur	eur	PROPN
ejpam-4247	192	13	.	.	PUNCT
ejpam-4247	193	1	j.	j.	PROPN
ejpam-4247	193	2	pure	pure	PROPN
ejpam-4247	193	3	appl	appl	PROPN
ejpam-4247	193	4	.	.	PROPN
ejpam-4247	193	5	math	math	PROPN
ejpam-4247	193	6	,	,	PUNCT
ejpam-4247	193	7	15	15	NUM
ejpam-4247	193	8	(	(	PUNCT
ejpam-4247	193	9	1	1	NUM
ejpam-4247	193	10	)	)	PUNCT
ejpam-4247	193	11	(	(	PUNCT
ejpam-4247	193	12	2022	2022	NUM
ejpam-4247	193	13	)	)	PUNCT
ejpam-4247	193	14	,	,	PUNCT
ejpam-4247	193	15	144	144	NUM
ejpam-4247	193	16	-	-	SYM
ejpam-4247	193	17	157	157	NUM
ejpam-4247	193	18	152	152	NUM
ejpam-4247	193	19	next	next	ADV
ejpam-4247	193	20	,	,	PUNCT
ejpam-4247	193	21	from	from	ADP
ejpam-4247	193	22	inequality	inequality	NOUN
ejpam-4247	193	23	(	(	PUNCT
ejpam-4247	193	24	4.13	4.13	NUM
ejpam-4247	193	25	)	)	PUNCT
ejpam-4247	193	26	,	,	PUNCT
ejpam-4247	193	27	we	we	PRON
ejpam-4247	193	28	get∣∣µ	get∣∣µ	VERB
ejpam-4247	193	29	(	(	PUNCT
ejpam-4247	193	30	0,u10)−	0,u10)−	PROPN
ejpam-4247	193	31	µ	µ	X
ejpam-4247	193	32	(	(	PUNCT
ejpam-4247	193	33	0,u20	0,u20	NUM
ejpam-4247	193	34	)	)	PUNCT
ejpam-4247	193	35	∣∣	∣∣	PROPN
ejpam-4247	193	36	≤	≤	NUM
ejpam-4247	193	37	(	(	PUNCT
ejpam-4247	193	38	k1	k1	NOUN
ejpam-4247	193	39	+	+	CCONJ
ejpam-4247	193	40	atl1k2	atl1k2	NOUN
ejpam-4247	193	41	)	)	PUNCT
ejpam-4247	193	42	∣∣u	∣∣u	NOUN
ejpam-4247	193	43	(	(	PUNCT
ejpam-4247	193	44	t	t	PROPN
ejpam-4247	193	45	,	,	PUNCT
ejpam-4247	193	46	u10)−	u10)−	NOUN
ejpam-4247	193	47	u	u	PROPN
ejpam-4247	193	48	(	(	PUNCT
ejpam-4247	193	49	t	t	PROPN
ejpam-4247	193	50	,	,	PUNCT
ejpam-4247	193	51	u20	u20	PROPN
ejpam-4247	193	52	)	)	PUNCT
ejpam-4247	193	53	∣∣	∣∣	PROPN
ejpam-4247	193	54	≤	≤	NOUN
ejpam-4247	193	55	f2	f2	ADV
ejpam-4247	193	56	∣∣u	∣∣u	PROPN
ejpam-4247	193	57	(	(	PUNCT
ejpam-4247	193	58	t	t	PROPN
ejpam-4247	193	59	,	,	PUNCT
ejpam-4247	193	60	u10)−	u10)−	NOUN
ejpam-4247	194	1	u	u	PROPN
ejpam-4247	194	2	(	(	PUNCT
ejpam-4247	194	3	t	t	PROPN
ejpam-4247	194	4	,	,	PUNCT
ejpam-4247	194	5	u20	u20	NUM
ejpam-4247	194	6	)	)	PUNCT
ejpam-4247	194	7	∣∣	∣∣	X
ejpam-4247	194	8	(	(	PUNCT
ejpam-4247	194	9	4.23	4.23	NUM
ejpam-4247	194	10	)	)	PUNCT
ejpam-4247	194	11	where	where	SCONJ
ejpam-4247	194	12	the	the	DET
ejpam-4247	194	13	functions	function	NOUN
ejpam-4247	194	14	u	u	X
ejpam-4247	194	15	(	(	PUNCT
ejpam-4247	194	16	t	t	PROPN
ejpam-4247	194	17	,	,	PUNCT
ejpam-4247	194	18	u10	u10	PROPN
ejpam-4247	194	19	)	)	PUNCT
ejpam-4247	194	20	and	and	CCONJ
ejpam-4247	194	21	u	u	PROPN
ejpam-4247	194	22	(	(	PUNCT
ejpam-4247	194	23	t	t	PROPN
ejpam-4247	194	24	,	,	PUNCT
ejpam-4247	194	25	u20	u20	PROPN
ejpam-4247	194	26	)	)	PUNCT
ejpam-4247	194	27	are	be	AUX
ejpam-4247	194	28	solutions	solution	NOUN
ejpam-4247	194	29	of	of	ADP
ejpam-4247	194	30	the	the	DET
ejpam-4247	194	31	integral	integral	ADJ
ejpam-4247	194	32	equation	equation	NOUN
ejpam-4247	194	33	:	:	PUNCT
ejpam-4247	194	34	u	u	NOUN
ejpam-4247	194	35	(	(	PUNCT
ejpam-4247	194	36	t	t	PROPN
ejpam-4247	194	37	,	,	PUNCT
ejpam-4247	194	38	uk0	uk0	NOUN
ejpam-4247	194	39	)	)	PUNCT
ejpam-4247	194	40	=	=	SYM
ejpam-4247	195	1	uk0	uk0	ADJ
ejpam-4247	195	2	+	+	CCONJ
ejpam-4247	195	3	2(1−α	2(1−α	NUM
ejpam-4247	195	4	)	)	PUNCT
ejpam-4247	195	5	(	(	PUNCT
ejpam-4247	195	6	2−α)n(α)h	2−α)n(α)h	NUM
ejpam-4247	195	7	(	(	PUNCT
ejpam-4247	195	8	t	t	PROPN
ejpam-4247	195	9	,	,	PUNCT
ejpam-4247	195	10	u	u	PROPN
ejpam-4247	195	11	(	(	PUNCT
ejpam-4247	195	12	t	t	PROPN
ejpam-4247	195	13	,	,	PUNCT
ejpam-4247	195	14	uk0	uk0	NOUN
ejpam-4247	195	15	)	)	PUNCT
ejpam-4247	195	16	,	,	PUNCT
ejpam-4247	195	17	∫	∫	PROPN
ejpam-4247	195	18	a(t	a(t	PROPN
ejpam-4247	195	19	)	)	PUNCT
ejpam-4247	195	20	0	0	NUM
ejpam-4247	196	1	g	g	PROPN
ejpam-4247	196	2	(	(	PUNCT
ejpam-4247	196	3	s	s	PROPN
ejpam-4247	196	4	,	,	PUNCT
ejpam-4247	196	5	u	u	NOUN
ejpam-4247	196	6	(	(	PUNCT
ejpam-4247	196	7	s	s	PROPN
ejpam-4247	196	8	,	,	PUNCT
ejpam-4247	196	9	uk0	uk0	ADJ
ejpam-4247	196	10	)	)	PUNCT
ejpam-4247	196	11	)	)	PUNCT
ejpam-4247	196	12	ds	ds	X
ejpam-4247	196	13	)	)	PUNCT
ejpam-4247	196	14	−	−	NOUN
ejpam-4247	196	15	2(1−α	2(1−α	NUM
ejpam-4247	196	16	)	)	PUNCT
ejpam-4247	196	17	(	(	PUNCT
ejpam-4247	196	18	2−α)n(α	2−α)n(α	X
ejpam-4247	196	19	)	)	PUNCT
ejpam-4247	196	20	1	1	NUM
ejpam-4247	196	21	t	t	NOUN
ejpam-4247	196	22	∫	∫	PROPN
ejpam-4247	196	23	t	t	PROPN
ejpam-4247	196	24	0	0	NUM
ejpam-4247	196	25	h	h	PROPN
ejpam-4247	196	26	(	(	PUNCT
ejpam-4247	196	27	s	s	PROPN
ejpam-4247	196	28	,	,	PUNCT
ejpam-4247	196	29	u	u	NOUN
ejpam-4247	196	30	(	(	PUNCT
ejpam-4247	196	31	s	s	PROPN
ejpam-4247	196	32	,	,	PUNCT
ejpam-4247	196	33	uk0	uk0	ADJ
ejpam-4247	196	34	)	)	PUNCT
ejpam-4247	196	35	,	,	PUNCT
ejpam-4247	196	36	∫	∫	PROPN
ejpam-4247	196	37	a(s	a(s	PROPN
ejpam-4247	196	38	)	)	PUNCT
ejpam-4247	196	39	0	0	NUM
ejpam-4247	197	1	g	g	NOUN
ejpam-4247	197	2	(	(	PUNCT
ejpam-4247	197	3	τ	τ	PROPN
ejpam-4247	197	4	,	,	PUNCT
ejpam-4247	197	5	u	u	PROPN
ejpam-4247	197	6	(	(	PUNCT
ejpam-4247	197	7	τ	τ	PROPN
ejpam-4247	197	8	,	,	PUNCT
ejpam-4247	197	9	uk0	uk0	NOUN
ejpam-4247	197	10	)	)	PUNCT
ejpam-4247	197	11	)	)	PUNCT
ejpam-4247	197	12	dτ	dτ	NOUN
ejpam-4247	197	13	)	)	PUNCT
ejpam-4247	197	14	ds+	ds+	PROPN
ejpam-4247	197	15	2α	2α	NOUN
ejpam-4247	197	16	(	(	PUNCT
ejpam-4247	197	17	2−α)n(α	2−α)n(α	NUM
ejpam-4247	197	18	)	)	PUNCT
ejpam-4247	197	19	∫	∫	PROPN
ejpam-4247	198	1	t	t	PROPN
ejpam-4247	198	2	0	0	NUM
ejpam-4247	198	3	(	(	PUNCT
ejpam-4247	198	4	h(s	h(s	PROPN
ejpam-4247	198	5	,	,	PUNCT
ejpam-4247	198	6	u	u	NOUN
ejpam-4247	198	7	(	(	PUNCT
ejpam-4247	198	8	s	s	PROPN
ejpam-4247	198	9	,	,	PUNCT
ejpam-4247	198	10	uk0	uk0	ADJ
ejpam-4247	198	11	)	)	PUNCT
ejpam-4247	198	12	,	,	PUNCT
ejpam-4247	198	13	∫	∫	PROPN
ejpam-4247	198	14	a(s	a(s	PROPN
ejpam-4247	198	15	)	)	PUNCT
ejpam-4247	198	16	0	0	NUM
ejpam-4247	199	1	g	g	NOUN
ejpam-4247	199	2	(	(	PUNCT
ejpam-4247	199	3	τ	τ	PROPN
ejpam-4247	199	4	,	,	PUNCT
ejpam-4247	199	5	u	u	PROPN
ejpam-4247	199	6	(	(	PUNCT
ejpam-4247	199	7	τ	τ	PROPN
ejpam-4247	199	8	,	,	PUNCT
ejpam-4247	199	9	uk0	uk0	NOUN
ejpam-4247	199	10	)	)	PUNCT
ejpam-4247	199	11	)	)	PUNCT
ejpam-4247	200	1	dτ)−	dτ)−	NOUN
ejpam-4247	200	2	1	1	NUM
ejpam-4247	200	3	t	t	NOUN
ejpam-4247	200	4	∫	∫	PROPN
ejpam-4247	200	5	t	t	PROPN
ejpam-4247	200	6	0	0	NUM
ejpam-4247	200	7	h(s	h(s	PROPN
ejpam-4247	200	8	,	,	PUNCT
ejpam-4247	200	9	u	u	NOUN
ejpam-4247	200	10	(	(	PUNCT
ejpam-4247	200	11	s	s	PROPN
ejpam-4247	200	12	,	,	PUNCT
ejpam-4247	200	13	uk0	uk0	ADJ
ejpam-4247	200	14	)	)	PUNCT
ejpam-4247	200	15	,	,	PUNCT
ejpam-4247	200	16	∫	∫	PROPN
ejpam-4247	200	17	a(s	a(s	PROPN
ejpam-4247	200	18	)	)	PUNCT
ejpam-4247	200	19	0	0	NUM
ejpam-4247	201	1	g	g	NOUN
ejpam-4247	201	2	(	(	PUNCT
ejpam-4247	201	3	τ	τ	PROPN
ejpam-4247	201	4	,	,	PUNCT
ejpam-4247	201	5	u	u	PROPN
ejpam-4247	201	6	(	(	PUNCT
ejpam-4247	201	7	τ	τ	PROPN
ejpam-4247	201	8	,	,	PUNCT
ejpam-4247	201	9	uk0	uk0	ADJ
ejpam-4247	201	10	)	)	PUNCT
ejpam-4247	201	11	)	)	PUNCT
ejpam-4247	202	1	dτ)ds)ds	dτ)ds)ds	PROPN
ejpam-4247	202	2	(	(	PUNCT
ejpam-4247	202	3	4.24	4.24	NUM
ejpam-4247	202	4	)	)	PUNCT
ejpam-4247	202	5	where	where	SCONJ
ejpam-4247	202	6	k	k	NOUN
ejpam-4247	202	7	=	=	SYM
ejpam-4247	202	8	1	1	NUM
ejpam-4247	202	9	,	,	PUNCT
ejpam-4247	202	10	2	2	NUM
ejpam-4247	202	11	,	,	PUNCT
ejpam-4247	202	12	from	from	ADP
ejpam-4247	202	13	(	(	PUNCT
ejpam-4247	202	14	4.24	4.24	NUM
ejpam-4247	202	15	)	)	PUNCT
ejpam-4247	202	16	,	,	PUNCT
ejpam-4247	202	17	we	we	PRON
ejpam-4247	202	18	get∣∣u	get∣∣u	PROPN
ejpam-4247	202	19	(	(	PUNCT
ejpam-4247	202	20	t	t	PROPN
ejpam-4247	202	21	,	,	PUNCT
ejpam-4247	202	22	u10)−	u10)−	NOUN
ejpam-4247	203	1	u	u	PROPN
ejpam-4247	203	2	(	(	PUNCT
ejpam-4247	203	3	t	t	PROPN
ejpam-4247	203	4	,	,	PUNCT
ejpam-4247	203	5	u20	u20	NUM
ejpam-4247	203	6	)	)	PUNCT
ejpam-4247	203	7	∣∣	∣∣	PROPN
ejpam-4247	203	8	≤|	≤|	X
ejpam-4247	203	9	u10	u10	PROPN
ejpam-4247	203	10	−	−	PROPN
ejpam-4247	203	11	u20	u20	PROPN
ejpam-4247	203	12	]	]	PUNCT
ejpam-4247	204	1	+	+	CCONJ
ejpam-4247	204	2	4(1−α	4(1−α	NUM
ejpam-4247	204	3	)	)	PUNCT
ejpam-4247	204	4	(	(	PUNCT
ejpam-4247	204	5	2−α)n(a	2−α)n(a	NOUN
ejpam-4247	204	6	)	)	PUNCT
ejpam-4247	204	7	(	(	PUNCT
ejpam-4247	204	8	k1	k1	NOUN
ejpam-4247	204	9	+	+	CCONJ
ejpam-4247	204	10	atl1k2	atl1k2	NOUN
ejpam-4247	204	11	)	)	PUNCT
ejpam-4247	204	12	∣∣u	∣∣u	NOUN
ejpam-4247	204	13	(	(	PUNCT
ejpam-4247	204	14	t	t	PROPN
ejpam-4247	204	15	,	,	PUNCT
ejpam-4247	204	16	u10)−	u10)−	NOUN
ejpam-4247	204	17	u	u	PROPN
ejpam-4247	204	18	(	(	PUNCT
ejpam-4247	204	19	t	t	PROPN
ejpam-4247	204	20	,	,	PUNCT
ejpam-4247	204	21	u20	u20	PROPN
ejpam-4247	204	22	)	)	PUNCT
ejpam-4247	204	23	∣∣+	∣∣+	PROPN
ejpam-4247	205	1	αt	αt	PROPN
ejpam-4247	205	2	(	(	PUNCT
ejpam-4247	205	3	2−	2−	NUM
ejpam-4247	205	4	α)n(α	α)n(α	NUM
ejpam-4247	205	5	)	)	PUNCT
ejpam-4247	205	6	(	(	PUNCT
ejpam-4247	205	7	k1	k1	NOUN
ejpam-4247	205	8	+	+	CCONJ
ejpam-4247	205	9	atl1k2	atl1k2	NOUN
ejpam-4247	205	10	)	)	PUNCT
ejpam-4247	205	11	∣∣u	∣∣u	NOUN
ejpam-4247	205	12	(	(	PUNCT
ejpam-4247	205	13	t	t	PROPN
ejpam-4247	205	14	,	,	PUNCT
ejpam-4247	205	15	u10)−	u10)−	NOUN
ejpam-4247	205	16	u	u	PROPN
ejpam-4247	205	17	(	(	PUNCT
ejpam-4247	205	18	t	t	PROPN
ejpam-4247	205	19	,	,	PUNCT
ejpam-4247	205	20	u20	u20	NUM
ejpam-4247	205	21	)	)	PUNCT
ejpam-4247	205	22	∣∣	∣∣	X
ejpam-4247	205	23	(	(	PUNCT
ejpam-4247	205	24	4.25	4.25	NUM
ejpam-4247	205	25	)	)	PUNCT
ejpam-4247	205	26	therefore	therefore	ADV
ejpam-4247	205	27	,	,	PUNCT
ejpam-4247	205	28	we	we	PRON
ejpam-4247	205	29	obtain	obtain	VERB
ejpam-4247	205	30	that∣∣u	that∣∣u	NOUN
ejpam-4247	205	31	(	(	PUNCT
ejpam-4247	205	32	t	t	PROPN
ejpam-4247	205	33	,	,	PUNCT
ejpam-4247	205	34	u10)−	u10)−	NOUN
ejpam-4247	205	35	u	u	PROPN
ejpam-4247	205	36	(	(	PUNCT
ejpam-4247	205	37	t	t	PROPN
ejpam-4247	205	38	,	,	PUNCT
ejpam-4247	205	39	u20	u20	PROPN
ejpam-4247	205	40	)	)	PUNCT
ejpam-4247	205	41	∣∣	∣∣	X
ejpam-4247	205	42	≤	≤	ADJ
ejpam-4247	205	43	∣∣u10	∣∣u10	PROPN
ejpam-4247	205	44	−	−	PROPN
ejpam-4247	205	45	u20	u20	PROPN
ejpam-4247	205	46	∣∣+	∣∣+	X
ejpam-4247	205	47	(	(	PUNCT
ejpam-4247	205	48	2(1−	2(1−	NUM
ejpam-4247	205	49	α	α	NUM
ejpam-4247	205	50	)	)	PUNCT
ejpam-4247	206	1	+	+	CCONJ
ejpam-4247	206	2	αt	αt	NOUN
ejpam-4247	206	3	2	2	NUM
ejpam-4247	206	4	)	)	PUNCT
ejpam-4247	206	5	(	(	PUNCT
ejpam-4247	206	6	k1	k1	NOUN
ejpam-4247	206	7	+	+	CCONJ
ejpam-4247	206	8	atl1k2	atl1k2	NOUN
ejpam-4247	206	9	)	)	PUNCT
ejpam-4247	206	10	∣∣u	∣∣u	NOUN
ejpam-4247	206	11	(	(	PUNCT
ejpam-4247	206	12	t	t	PROPN
ejpam-4247	206	13	,	,	PUNCT
ejpam-4247	206	14	u10)−	u10)−	NOUN
ejpam-4247	206	15	u	u	PROPN
ejpam-4247	206	16	(	(	PUNCT
ejpam-4247	206	17	t	t	PROPN
ejpam-4247	206	18	,	,	PUNCT
ejpam-4247	206	19	u20	u20	PROPN
ejpam-4247	206	20	)	)	PUNCT
ejpam-4247	207	1	∣∣	∣∣	X
ejpam-4247	207	2	≤	≤	ADJ
ejpam-4247	207	3	∣∣u10	∣∣u10	PROPN
ejpam-4247	207	4	−	−	PROPN
ejpam-4247	207	5	u20	u20	PROPN
ejpam-4247	207	6	∣∣+	∣∣+	X
ejpam-4247	207	7	f1f2	f1f2	X
ejpam-4247	207	8	∣∣u	∣∣u	X
ejpam-4247	207	9	(	(	PUNCT
ejpam-4247	207	10	t	t	PROPN
ejpam-4247	207	11	,	,	PUNCT
ejpam-4247	207	12	u10)−	u10)−	NOUN
ejpam-4247	207	13	u	u	PROPN
ejpam-4247	207	14	(	(	PUNCT
ejpam-4247	207	15	t	t	PROPN
ejpam-4247	207	16	,	,	PUNCT
ejpam-4247	207	17	u20	u20	NUM
ejpam-4247	207	18	)	)	PUNCT
ejpam-4247	207	19	∣∣	∣∣	X
ejpam-4247	207	20	(	(	PUNCT
ejpam-4247	207	21	4.26	4.26	NUM
ejpam-4247	207	22	)	)	PUNCT
ejpam-4247	207	23	from	from	ADP
ejpam-4247	207	24	equations	equation	NOUN
ejpam-4247	207	25	(	(	PUNCT
ejpam-4247	207	26	4.26	4.26	NUM
ejpam-4247	207	27	)	)	PUNCT
ejpam-4247	207	28	,	,	PUNCT
ejpam-4247	207	29	we	we	PRON
ejpam-4247	207	30	have∣∣u	have∣∣u	ADV
ejpam-4247	207	31	(	(	PUNCT
ejpam-4247	207	32	t	t	PROPN
ejpam-4247	207	33	,	,	PUNCT
ejpam-4247	207	34	u10)−	u10)−	NOUN
ejpam-4247	207	35	u	u	PROPN
ejpam-4247	207	36	(	(	PUNCT
ejpam-4247	207	37	t	t	PROPN
ejpam-4247	207	38	,	,	PUNCT
ejpam-4247	207	39	u20	u20	PROPN
ejpam-4247	207	40	)	)	PUNCT
ejpam-4247	207	41	∣∣	∣∣	PROPN
ejpam-4247	207	42	≤	≤	NUM
ejpam-4247	207	43	f3	f3	VERB
ejpam-4247	207	44	∣∣u10	∣∣u10	PROPN
ejpam-4247	207	45	−	−	PROPN
ejpam-4247	207	46	u20	u20	PROPN
ejpam-4247	207	47	∣∣	∣∣	NUM
ejpam-4247	207	48	(	(	PUNCT
ejpam-4247	207	49	4.27	4.27	NUM
ejpam-4247	207	50	)	)	PUNCT
ejpam-4247	207	51	substitutes	substitute	NOUN
ejpam-4247	207	52	(	(	PUNCT
ejpam-4247	207	53	4.27	4.27	NUM
ejpam-4247	207	54	)	)	PUNCT
ejpam-4247	207	55	in	in	ADP
ejpam-4247	207	56	(	(	PUNCT
ejpam-4247	207	57	4.23	4.23	NUM
ejpam-4247	207	58	)	)	PUNCT
ejpam-4247	207	59	,	,	PUNCT
ejpam-4247	207	60	we	we	PRON
ejpam-4247	207	61	get	get	VERB
ejpam-4247	207	62	that	that	DET
ejpam-4247	207	63	(	(	PUNCT
ejpam-4247	207	64	4.21	4.21	NUM
ejpam-4247	207	65	)	)	PUNCT
ejpam-4247	207	66	remark	remark	NOUN
ejpam-4247	207	67	1	1	NUM
ejpam-4247	207	68	.	.	PUNCT
ejpam-4247	208	1	[	[	X
ejpam-4247	208	2	22	22	NUM
ejpam-4247	208	3	]	]	PUNCT
ejpam-4247	208	4	.	.	PUNCT
ejpam-4247	209	1	theorem	theorem	NOUN
ejpam-4247	209	2	6	6	NUM
ejpam-4247	209	3	confirms	confirm	VERB
ejpam-4247	209	4	the	the	DET
ejpam-4247	209	5	stability	stability	NOUN
ejpam-4247	209	6	of	of	ADP
ejpam-4247	209	7	the	the	DET
ejpam-4247	209	8	solution	solution	NOUN
ejpam-4247	209	9	of	of	ADP
ejpam-4247	209	10	the	the	DET
ejpam-4247	209	11	problem	problem	NOUN
ejpam-4247	209	12	(	(	PUNCT
ejpam-4247	209	13	1.1	1.1	NUM
ejpam-4247	209	14	)	)	PUNCT
ejpam-4247	209	15	,	,	PUNCT
ejpam-4247	209	16	when	when	SCONJ
ejpam-4247	209	17	a	a	DET
ejpam-4247	209	18	slight	slight	ADJ
ejpam-4247	209	19	change	change	NOUN
ejpam-4247	209	20	happens	happen	VERB
ejpam-4247	209	21	in	in	ADP
ejpam-4247	209	22	the	the	DET
ejpam-4247	209	23	points	point	NOUN
ejpam-4247	209	24	u0	u0	ADJ
ejpam-4247	209	25	,	,	PUNCT
ejpam-4247	209	26	then	then	ADV
ejpam-4247	209	27	a	a	DET
ejpam-4247	209	28	slight	slight	ADJ
ejpam-4247	209	29	change	change	NOUN
ejpam-4247	209	30	will	will	AUX
ejpam-4247	209	31	happen	happen	VERB
ejpam-4247	209	32	in	in	ADP
ejpam-4247	209	33	the	the	DET
ejpam-4247	209	34	function	function	NOUN
ejpam-4247	209	35	µ	µ	X
ejpam-4247	209	36	(	(	PUNCT
ejpam-4247	209	37	0	0	NUM
ejpam-4247	209	38	,	,	PUNCT
ejpam-4247	209	39	u0	u0	ADJ
ejpam-4247	209	40	)	)	PUNCT
ejpam-4247	209	41	.	.	PUNCT
ejpam-4247	210	1	4.4	4.4	NUM
ejpam-4247	210	2	.	.	PUNCT
ejpam-4247	210	3	existence	existence	NOUN
ejpam-4247	210	4	and	and	CCONJ
ejpam-4247	210	5	uniqueness	uniqueness	NOUN
ejpam-4247	210	6	of	of	ADP
ejpam-4247	210	7	periodic	periodic	ADJ
ejpam-4247	210	8	solution	solution	NOUN
ejpam-4247	210	9	of	of	ADP
ejpam-4247	210	10	(	(	PUNCT
ejpam-4247	210	11	1.1	1.1	NUM
ejpam-4247	210	12	)	)	PUNCT
ejpam-4247	210	13	with	with	ADP
ejpam-4247	210	14	integral	integral	ADJ
ejpam-4247	210	15	boundary	boundary	ADJ
ejpam-4247	210	16	condition	condition	NOUN
ejpam-4247	210	17	in	in	ADP
ejpam-4247	210	18	this	this	DET
ejpam-4247	210	19	section	section	NOUN
ejpam-4247	210	20	,	,	PUNCT
ejpam-4247	210	21	we	we	PRON
ejpam-4247	210	22	investigate	investigate	VERB
ejpam-4247	210	23	the	the	DET
ejpam-4247	210	24	periodic	periodic	ADJ
ejpam-4247	210	25	solution	solution	NOUN
ejpam-4247	210	26	of	of	ADP
ejpam-4247	210	27	the	the	DET
ejpam-4247	210	28	problem	problem	NOUN
ejpam-4247	210	29	(	(	PUNCT
ejpam-4247	210	30	1.1	1.1	NUM
ejpam-4247	210	31	)	)	PUNCT
ejpam-4247	210	32	with	with	ADP
ejpam-4247	210	33	integral	integral	ADJ
ejpam-4247	210	34	boundary	boundary	ADJ
ejpam-4247	210	35	conditions	condition	NOUN
ejpam-4247	210	36	:	:	PUNCT
ejpam-4247	210	37	u(0)−	u(0)−	PROPN
ejpam-4247	210	38	u(t	u(t	PROPN
ejpam-4247	210	39	)	)	PUNCT
ejpam-4247	211	1	=	=	SYM
ejpam-4247	212	1	∫	∫	PROPN
ejpam-4247	212	2	t	t	PROPN
ejpam-4247	212	3	0	0	NUM
ejpam-4247	212	4	h(u(s))ds	h(u(s))d	NOUN
ejpam-4247	212	5	(	(	PUNCT
ejpam-4247	212	6	4.28	4.28	NUM
ejpam-4247	212	7	)	)	PUNCT
ejpam-4247	212	8	where	where	SCONJ
ejpam-4247	212	9	the	the	DET
ejpam-4247	212	10	function	function	NOUN
ejpam-4247	212	11	h(u(s	h(u(s	PROPN
ejpam-4247	212	12	)	)	PUNCT
ejpam-4247	212	13	)	)	PUNCT
ejpam-4247	212	14	defined	define	VERB
ejpam-4247	212	15	and	and	CCONJ
ejpam-4247	212	16	continuous	continuous	ADJ
ejpam-4247	212	17	on	on	ADP
ejpam-4247	212	18	are	be	AUX
ejpam-4247	212	19	compact	compact	ADJ
ejpam-4247	212	20	subset	subset	NOUN
ejpam-4247	212	21	of	of	ADP
ejpam-4247	212	22	r	r	NOUN
ejpam-4247	212	23	and	and	CCONJ
ejpam-4247	212	24	periodic	periodic	NOUN
ejpam-4247	212	25	on	on	ADP
ejpam-4247	212	26	t	t	PROPN
ejpam-4247	212	27	of	of	ADP
ejpam-4247	212	28	periodic	periodic	ADJ
ejpam-4247	212	29	t.	t.	PROPN
ejpam-4247	212	30	a.	a.	PROPN
ejpam-4247	212	31	s.	s.	PROPN
ejpam-4247	212	32	rafeeq	rafeeq	PROPN
ejpam-4247	212	33	/	/	SYM
ejpam-4247	212	34	eur	eur	PROPN
ejpam-4247	212	35	.	.	PUNCT
ejpam-4247	213	1	j.	j.	PROPN
ejpam-4247	213	2	pure	pure	PROPN
ejpam-4247	213	3	appl	appl	PROPN
ejpam-4247	213	4	.	.	PROPN
ejpam-4247	213	5	math	math	PROPN
ejpam-4247	213	6	,	,	PUNCT
ejpam-4247	213	7	15	15	NUM
ejpam-4247	213	8	(	(	PUNCT
ejpam-4247	213	9	1	1	NUM
ejpam-4247	213	10	)	)	PUNCT
ejpam-4247	213	11	(	(	PUNCT
ejpam-4247	213	12	2022	2022	NUM
ejpam-4247	213	13	)	)	PUNCT
ejpam-4247	213	14	,	,	PUNCT
ejpam-4247	213	15	144	144	NUM
ejpam-4247	213	16	-	-	SYM
ejpam-4247	213	17	157	157	NUM
ejpam-4247	213	18	153	153	NUM
ejpam-4247	213	19	theorem	theorem	NOUN
ejpam-4247	213	20	7	7	NUM
ejpam-4247	213	21	.	.	PUNCT
ejpam-4247	214	1	all	all	DET
ejpam-4247	214	2	assumptions	assumption	NOUN
ejpam-4247	214	3	of	of	ADP
ejpam-4247	214	4	the	the	DET
ejpam-4247	214	5	theorem	theorem	ADJ
ejpam-4247	214	6	2	2	NUM
ejpam-4247	214	7	are	be	AUX
ejpam-4247	214	8	satisfy	satisfy	ADJ
ejpam-4247	214	9	,	,	PUNCT
ejpam-4247	214	10	and	and	CCONJ
ejpam-4247	214	11	the	the	DET
ejpam-4247	214	12	function	function	NOUN
ejpam-4247	214	13	h(u(s	h(u(s	PROPN
ejpam-4247	214	14	)	)	PUNCT
ejpam-4247	214	15	)	)	PUNCT
ejpam-4247	215	1	satisfies	satisfy	VERB
ejpam-4247	215	2	|h	|h	X
ejpam-4247	215	3	(	(	PUNCT
ejpam-4247	215	4	u1)−h	u1)−h	PROPN
ejpam-4247	215	5	(	(	PUNCT
ejpam-4247	215	6	u2)|	u2)|	NOUN
ejpam-4247	215	7	≤	≤	NOUN
ejpam-4247	215	8	l2	l2	NOUN
ejpam-4247	215	9	|u1	|u1	NOUN
ejpam-4247	216	1	−	−	NOUN
ejpam-4247	216	2	u2|	u2|	ADJ
ejpam-4247	216	3	(	(	PUNCT
ejpam-4247	216	4	4.29	4.29	NUM
ejpam-4247	216	5	)	)	PUNCT
ejpam-4247	216	6	then	then	ADV
ejpam-4247	216	7	the	the	DET
ejpam-4247	216	8	problem	problem	NOUN
ejpam-4247	216	9	(	(	PUNCT
ejpam-4247	216	10	1.1	1.1	NUM
ejpam-4247	216	11	)	)	PUNCT
ejpam-4247	216	12	and	and	CCONJ
ejpam-4247	216	13	integral	integral	ADJ
ejpam-4247	216	14	boundary	boundary	ADJ
ejpam-4247	216	15	condition(4.28	condition(4.28	NOUN
ejpam-4247	216	16	)	)	PUNCT
ejpam-4247	216	17	has	have	VERB
ejpam-4247	216	18	unique	unique	ADJ
ejpam-4247	216	19	solution	solution	NOUN
ejpam-4247	216	20	if	if	SCONJ
ejpam-4247	216	21	q	q	NOUN
ejpam-4247	216	22	=	=	X
ejpam-4247	216	23	(	(	PUNCT
ejpam-4247	216	24	2(1−	2(1−	NUM
ejpam-4247	216	25	α	α	NUM
ejpam-4247	216	26	)	)	PUNCT
ejpam-4247	217	1	+	+	CCONJ
ejpam-4247	217	2	αt	αt	NOUN
ejpam-4247	217	3	2	2	NUM
ejpam-4247	217	4	)	)	PUNCT
ejpam-4247	217	5	(	(	PUNCT
ejpam-4247	217	6	k1	k1	NOUN
ejpam-4247	217	7	+	+	CCONJ
ejpam-4247	217	8	atl1k2	atl1k2	NOUN
ejpam-4247	217	9	)	)	PUNCT
ejpam-4247	217	10	+	+	CCONJ
ejpam-4247	217	11	(	(	PUNCT
ejpam-4247	217	12	1−	1−	NUM
ejpam-4247	217	13	α+	α+	X
ejpam-4247	217	14	αt	αt	NOUN
ejpam-4247	217	15	)	)	PUNCT
ejpam-4247	217	16	α	α	PRON
ejpam-4247	217	17	l2	l2	NOUN
ejpam-4247	217	18	<	<	X
ejpam-4247	217	19	1	1	NUM
ejpam-4247	217	20	(	(	PUNCT
ejpam-4247	217	21	4.30	4.30	NUM
ejpam-4247	217	22	)	)	PUNCT
ejpam-4247	217	23	proof	proof	NOUN
ejpam-4247	217	24	.	.	PUNCT
ejpam-4247	218	1	we	we	PRON
ejpam-4247	218	2	define	define	VERB
ejpam-4247	218	3	an	an	DET
ejpam-4247	218	4	operator	operator	NOUN
ejpam-4247	218	5	p	p	NOUN
ejpam-4247	218	6	:	:	PUNCT
ejpam-4247	218	7	c[0	c[0	PROPN
ejpam-4247	218	8	,	,	PUNCT
ejpam-4247	218	9	t	t	X
ejpam-4247	218	10	]	]	PUNCT
ejpam-4247	218	11	→	→	SYM
ejpam-4247	218	12	c[0	c[0	PROPN
ejpam-4247	218	13	,	,	PUNCT
ejpam-4247	218	14	t	t	X
ejpam-4247	218	15	]	]	X
ejpam-4247	218	16	p	p	X
ejpam-4247	218	17	(	(	PUNCT
ejpam-4247	218	18	u(t	u(t	NOUN
ejpam-4247	218	19	)	)	PUNCT
ejpam-4247	218	20	)	)	PUNCT
ejpam-4247	219	1	=	=	PUNCT
ejpam-4247	219	2	u0	u0	ADJ
ejpam-4247	219	3	−	−	PROPN
ejpam-4247	219	4	(	(	PUNCT
ejpam-4247	219	5	1−α+αt	1−α+αt	NUM
ejpam-4247	219	6	)	)	PUNCT
ejpam-4247	219	7	αt	αt	NOUN
ejpam-4247	219	8	∫	∫	PROPN
ejpam-4247	219	9	t	t	PROPN
ejpam-4247	219	10	0	0	NUM
ejpam-4247	219	11	h(u(s))dt+	h(u(s))dt+	NOUN
ejpam-4247	219	12	2(1−α	2(1−α	NUM
ejpam-4247	219	13	)	)	PUNCT
ejpam-4247	219	14	(	(	PUNCT
ejpam-4247	219	15	2−α)n(a	2−α)n(a	X
ejpam-4247	219	16	)	)	PUNCT
ejpam-4247	219	17	[	[	X
ejpam-4247	219	18	h(t	h(t	ADJ
ejpam-4247	219	19	,	,	PUNCT
ejpam-4247	219	20	u(t	u(t	NOUN
ejpam-4247	219	21	)	)	PUNCT
ejpam-4247	219	22	,	,	PUNCT
ejpam-4247	219	23	∫	∫	PROPN
ejpam-4247	219	24	a(t	a(t	PROPN
ejpam-4247	219	25	)	)	PUNCT
ejpam-4247	219	26	0	0	PUNCT
ejpam-4247	220	1	g(s	g(s	PROPN
ejpam-4247	220	2	,	,	PUNCT
ejpam-4247	220	3	u(s))ds)−	u(s))ds)−	ADP
ejpam-4247	220	4	1	1	NUM
ejpam-4247	220	5	t	t	NOUN
ejpam-4247	220	6	∫	∫	PROPN
ejpam-4247	220	7	t	t	PROPN
ejpam-4247	220	8	0	0	NUM
ejpam-4247	220	9	h(s	h(s	PROPN
ejpam-4247	220	10	,	,	PUNCT
ejpam-4247	220	11	u(s	u(s	NUM
ejpam-4247	220	12	)	)	PUNCT
ejpam-4247	220	13	,	,	PUNCT
ejpam-4247	220	14	∫	∫	PROPN
ejpam-4247	220	15	a(s	a(s	PROPN
ejpam-4247	220	16	)	)	PUNCT
ejpam-4247	220	17	0	0	NUM
ejpam-4247	221	1	g(τ	g(τ	PROPN
ejpam-4247	221	2	,	,	PUNCT
ejpam-4247	221	3	u(τ))dτ)ds]+	u(τ))dτ)ds]+	PROPN
ejpam-4247	221	4	2α	2α	NOUN
ejpam-4247	221	5	(	(	PUNCT
ejpam-4247	221	6	2−α)n(α	2−α)n(α	NUM
ejpam-4247	221	7	)	)	PUNCT
ejpam-4247	221	8	∫	∫	PROPN
ejpam-4247	221	9	t	t	PROPN
ejpam-4247	221	10	0	0	NUM
ejpam-4247	221	11	(	(	PUNCT
ejpam-4247	221	12	h(s	h(s	PROPN
ejpam-4247	221	13	,	,	PUNCT
ejpam-4247	221	14	u(s	u(s	NUM
ejpam-4247	221	15	)	)	PUNCT
ejpam-4247	221	16	,	,	PUNCT
ejpam-4247	221	17	∫	∫	PROPN
ejpam-4247	221	18	a(s	a(s	PROPN
ejpam-4247	221	19	)	)	PUNCT
ejpam-4247	221	20	0	0	NUM
ejpam-4247	222	1	g(τ	g(τ	NOUN
ejpam-4247	222	2	,	,	PUNCT
ejpam-4247	222	3	u(τ))dτ)−	u(τ))dτ)−	NOUN
ejpam-4247	222	4	1	1	NUM
ejpam-4247	222	5	t	t	NOUN
ejpam-4247	222	6	∫	∫	PROPN
ejpam-4247	222	7	t	t	PROPN
ejpam-4247	222	8	0	0	NUM
ejpam-4247	222	9	h(s	h(s	PROPN
ejpam-4247	222	10	,	,	PUNCT
ejpam-4247	222	11	u(s	u(s	NUM
ejpam-4247	222	12	)	)	PUNCT
ejpam-4247	222	13	,	,	PUNCT
ejpam-4247	222	14	∫	∫	PROPN
ejpam-4247	222	15	a(s	a(s	PROPN
ejpam-4247	222	16	)	)	PUNCT
ejpam-4247	222	17	0	0	NUM
ejpam-4247	223	1	g(τ	g(τ	PROPN
ejpam-4247	223	2	,	,	PUNCT
ejpam-4247	223	3	u(τ))dτ)ds)ds	u(τ))dτ)ds)ds	PROPN
ejpam-4247	223	4	therefore	therefore	ADV
ejpam-4247	223	5	,	,	PUNCT
ejpam-4247	223	6	we	we	PRON
ejpam-4247	223	7	get	get	VERB
ejpam-4247	223	8	|p	|p	PRON
ejpam-4247	223	9	(	(	PUNCT
ejpam-4247	223	10	u(t))−	u(t))−	ADP
ejpam-4247	223	11	p	p	X
ejpam-4247	223	12	(	(	PUNCT
ejpam-4247	223	13	w(t))|	w(t))|	NOUN
ejpam-4247	223	14	=	=	SYM
ejpam-4247	223	15	(	(	PUNCT
ejpam-4247	223	16	(	(	PUNCT
ejpam-4247	223	17	2(1−	2(1−	NUM
ejpam-4247	223	18	α	α	NUM
ejpam-4247	223	19	)	)	PUNCT
ejpam-4247	224	1	+	+	CCONJ
ejpam-4247	224	2	αt	αt	NOUN
ejpam-4247	224	3	2	2	NUM
ejpam-4247	224	4	)	)	PUNCT
ejpam-4247	224	5	(	(	PUNCT
ejpam-4247	224	6	k1	k1	NOUN
ejpam-4247	224	7	+	+	CCONJ
ejpam-4247	224	8	atl1k2	atl1k2	NOUN
ejpam-4247	224	9	)	)	PUNCT
ejpam-4247	224	10	+	+	CCONJ
ejpam-4247	224	11	(	(	PUNCT
ejpam-4247	224	12	1−	1−	NUM
ejpam-4247	224	13	α+	α+	X
ejpam-4247	224	14	αt	αt	NOUN
ejpam-4247	224	15	)	)	PUNCT
ejpam-4247	225	1	α	α	PROPN
ejpam-4247	225	2	l2)|u(t)−	l2)|u(t)−	PROPN
ejpam-4247	226	1	w(t)|	w(t)|	PROPN
ejpam-4247	226	2	from	from	ADP
ejpam-4247	226	3	(	(	PUNCT
ejpam-4247	226	4	4.30	4.30	NUM
ejpam-4247	226	5	)	)	PUNCT
ejpam-4247	226	6	,	,	PUNCT
ejpam-4247	226	7	the	the	DET
ejpam-4247	226	8	operator	operator	NOUN
ejpam-4247	226	9	p	p	NOUN
ejpam-4247	226	10	satisfies	satisfie	NOUN
ejpam-4247	226	11	contraction	contraction	NOUN
ejpam-4247	226	12	mapping	mapping	NOUN
ejpam-4247	226	13	,	,	PUNCT
ejpam-4247	226	14	hence	hence	ADV
ejpam-4247	226	15	the	the	DET
ejpam-4247	226	16	problem	problem	NOUN
ejpam-4247	226	17	(	(	PUNCT
ejpam-4247	226	18	1.1	1.1	NUM
ejpam-4247	226	19	)	)	PUNCT
ejpam-4247	226	20	and	and	CCONJ
ejpam-4247	226	21	(	(	PUNCT
ejpam-4247	226	22	4.28	4.28	NUM
ejpam-4247	226	23	)	)	PUNCT
ejpam-4247	226	24	has	have	VERB
ejpam-4247	226	25	unique	unique	ADJ
ejpam-4247	226	26	solution	solution	NOUN
ejpam-4247	226	27	.	.	PUNCT
ejpam-4247	227	1	theorem	theorem	VERB
ejpam-4247	227	2	8	8	NUM
ejpam-4247	227	3	.	.	PUNCT
ejpam-4247	228	1	if	if	SCONJ
ejpam-4247	228	2	the	the	DET
ejpam-4247	228	3	hypotheses	hypothesis	NOUN
ejpam-4247	228	4	and	and	CCONJ
ejpam-4247	228	5	all	all	DET
ejpam-4247	228	6	the	the	DET
ejpam-4247	228	7	conditions	condition	NOUN
ejpam-4247	228	8	of	of	ADP
ejpam-4247	228	9	the	the	DET
ejpam-4247	228	10	theorem	theorem	NOUN
ejpam-4247	228	11	2	2	NUM
ejpam-4247	228	12	and	and	CCONJ
ejpam-4247	228	13	the	the	DET
ejpam-4247	228	14	inequality	inequality	NOUN
ejpam-4247	228	15	(	(	PUNCT
ejpam-4247	228	16	4.29	4.29	NUM
ejpam-4247	228	17	)	)	PUNCT
ejpam-4247	228	18	are	be	AUX
ejpam-4247	228	19	given	give	VERB
ejpam-4247	228	20	,	,	PUNCT
ejpam-4247	228	21	the	the	DET
ejpam-4247	228	22	following	follow	VERB
ejpam-4247	228	23	inequalities	inequality	NOUN
ejpam-4247	228	24	are	be	AUX
ejpam-4247	228	25	satisfied:|σ	satisfied:|σ	NOUN
ejpam-4247	228	26	(	(	PUNCT
ejpam-4247	228	27	0,u0)−	0,u0)−	NUM
ejpam-4247	228	28	σm	σm	X
ejpam-4247	228	29	(	(	PUNCT
ejpam-4247	228	30	0,u0)|	0,u0)|	NUM
ejpam-4247	228	31	≤	≤	NUM
ejpam-4247	228	32	(	(	PUNCT
ejpam-4247	228	33	k1	k1	NOUN
ejpam-4247	228	34	+	+	CCONJ
ejpam-4247	228	35	aυl1k2	aυl1k2	PROPN
ejpam-4247	228	36	+	+	CCONJ
ejpam-4247	228	37	l1	l1	PROPN
ejpam-4247	228	38	α	α	PROPN
ejpam-4247	228	39	)	)	PUNCT
ejpam-4247	228	40	qm(1−q)−1m3	qm(1−q)−1m3	PROPN
ejpam-4247	228	41	(	(	PUNCT
ejpam-4247	228	42	4.31	4.31	NUM
ejpam-4247	228	43	)	)	PUNCT
ejpam-4247	229	1	where	where	SCONJ
ejpam-4247	229	2	σm	σm	INTJ
ejpam-4247	229	3	(	(	PUNCT
ejpam-4247	229	4	0	0	NUM
ejpam-4247	229	5	,	,	PUNCT
ejpam-4247	229	6	u0	u0	ADJ
ejpam-4247	229	7	)	)	PUNCT
ejpam-4247	229	8	=	=	SYM
ejpam-4247	229	9	1	1	NUM
ejpam-4247	229	10	t	t	NOUN
ejpam-4247	229	11	∫	∫	PROPN
ejpam-4247	229	12	t	t	PROPN
ejpam-4247	229	13	0	0	NUM
ejpam-4247	229	14	h	h	PROPN
ejpam-4247	229	15	(	(	PUNCT
ejpam-4247	229	16	s	s	PROPN
ejpam-4247	229	17	,	,	PUNCT
ejpam-4247	229	18	um(s	um(s	NOUN
ejpam-4247	229	19	)	)	PUNCT
ejpam-4247	229	20	,	,	PUNCT
ejpam-4247	229	21	∫	∫	PROPN
ejpam-4247	229	22	a(s	a(s	PROPN
ejpam-4247	229	23	)	)	PUNCT
ejpam-4247	229	24	0	0	NUM
ejpam-4247	230	1	g	g	NOUN
ejpam-4247	230	2	(	(	PUNCT
ejpam-4247	230	3	τ	τ	PROPN
ejpam-4247	230	4	,	,	PUNCT
ejpam-4247	230	5	um(τ	um(τ	NUM
ejpam-4247	230	6	)	)	PUNCT
ejpam-4247	230	7	)	)	PUNCT
ejpam-4247	230	8	dτ	dτ	NOUN
ejpam-4247	230	9	)	)	PUNCT
ejpam-4247	230	10	ds	ds	PROPN
ejpam-4247	230	11	+	+	CCONJ
ejpam-4247	230	12	(	(	PUNCT
ejpam-4247	230	13	2−	2−	NUM
ejpam-4247	230	14	α)n(α	α)n(α	NOUN
ejpam-4247	230	15	)	)	PUNCT
ejpam-4247	231	1	2αt	2αt	NOUN
ejpam-4247	231	2	∫	∫	PROPN
ejpam-4247	231	3	t	t	PROPN
ejpam-4247	231	4	0	0	NUM
ejpam-4247	231	5	h	h	NOUN
ejpam-4247	231	6	(	(	PUNCT
ejpam-4247	231	7	um(s	um(s	NOUN
ejpam-4247	231	8	)	)	PUNCT
ejpam-4247	231	9	)	)	PUNCT
ejpam-4247	232	1	ds	ds	NOUN
ejpam-4247	232	2	(	(	PUNCT
ejpam-4247	232	3	4.32	4.32	NUM
ejpam-4247	232	4	)	)	PUNCT
ejpam-4247	232	5	holds	hold	VERB
ejpam-4247	232	6	for	for	ADP
ejpam-4247	232	7	all	all	DET
ejpam-4247	232	8	m	m	PROPN
ejpam-4247	232	9	≥	≥	NOUN
ejpam-4247	232	10	0	0	NUM
ejpam-4247	232	11	,	,	PUNCT
ejpam-4247	232	12	here	here	ADV
ejpam-4247	232	13	m3	m3	PROPN
ejpam-4247	232	14	=	=	PROPN
ejpam-4247	232	15	m1	m1	PROPN
ejpam-4247	232	16	+	+	CCONJ
ejpam-4247	232	17	(	(	PUNCT
ejpam-4247	232	18	1−	1−	NUM
ejpam-4247	232	19	α+	α+	X
ejpam-4247	232	20	αt	αt	PROPN
ejpam-4247	232	21	)	)	PUNCT
ejpam-4247	232	22	a	a	DET
ejpam-4247	232	23	m2	m2	PROPN
ejpam-4247	232	24	(	(	PUNCT
ejpam-4247	232	25	4.33	4.33	NUM
ejpam-4247	232	26	)	)	PUNCT
ejpam-4247	232	27	and	and	CCONJ
ejpam-4247	232	28	m2	m2	PROPN
ejpam-4247	232	29	≥	≥	PROPN
ejpam-4247	232	30	|h(u(t))|	|h(u(t))|	PROPN
ejpam-4247	232	31	(	(	PUNCT
ejpam-4247	232	32	4.34	4.34	NUM
ejpam-4247	232	33	)	)	PUNCT
ejpam-4247	232	34	the	the	DET
ejpam-4247	232	35	proof	proof	NOUN
ejpam-4247	232	36	of	of	ADP
ejpam-4247	232	37	this	this	DET
ejpam-4247	232	38	theorem	theorem	NOUN
ejpam-4247	232	39	is	be	AUX
ejpam-4247	232	40	direct	direct	ADJ
ejpam-4247	232	41	.	.	PUNCT
ejpam-4247	233	1	a.	a.	PROPN
ejpam-4247	233	2	s.	s.	PROPN
ejpam-4247	233	3	rafeeq	rafeeq	PROPN
ejpam-4247	233	4	/	/	SYM
ejpam-4247	233	5	eur	eur	PROPN
ejpam-4247	233	6	.	.	PUNCT
ejpam-4247	234	1	j.	j.	PROPN
ejpam-4247	234	2	pure	pure	PROPN
ejpam-4247	234	3	appl	appl	PROPN
ejpam-4247	234	4	.	.	PROPN
ejpam-4247	234	5	math	math	PROPN
ejpam-4247	234	6	,	,	PUNCT
ejpam-4247	234	7	15	15	NUM
ejpam-4247	234	8	(	(	PUNCT
ejpam-4247	234	9	1	1	NUM
ejpam-4247	234	10	)	)	PUNCT
ejpam-4247	234	11	(	(	PUNCT
ejpam-4247	234	12	2022	2022	NUM
ejpam-4247	234	13	)	)	PUNCT
ejpam-4247	234	14	,	,	PUNCT
ejpam-4247	234	15	144	144	NUM
ejpam-4247	234	16	-	-	SYM
ejpam-4247	234	17	157	157	NUM
ejpam-4247	234	18	154	154	NUM
ejpam-4247	234	19	theorem	theorem	NOUN
ejpam-4247	234	20	9	9	NUM
ejpam-4247	234	21	.	.	PUNCT
ejpam-4247	235	1	let	let	VERB
ejpam-4247	235	2	the	the	DET
ejpam-4247	235	3	functions	function	NOUN
ejpam-4247	235	4	h(s	h(s	PROPN
ejpam-4247	235	5	,	,	PUNCT
ejpam-4247	235	6	u(s	u(s	NUM
ejpam-4247	235	7	)	)	PUNCT
ejpam-4247	235	8	,	,	PUNCT
ejpam-4247	235	9	z(t	z(t	NOUN
ejpam-4247	235	10	)	)	PUNCT
ejpam-4247	235	11	)	)	PUNCT
ejpam-4247	235	12	and	and	CCONJ
ejpam-4247	235	13	h(u(t	h(u(t	PROPN
ejpam-4247	235	14	)	)	PUNCT
ejpam-4247	235	15	)	)	PUNCT
ejpam-4247	235	16	be	be	AUX
ejpam-4247	235	17	defined	define	VERB
ejpam-4247	235	18	on	on	ADP
ejpam-4247	235	19	the	the	DET
ejpam-4247	235	20	intervals	interval	NOUN
ejpam-4247	235	21	[	[	X
ejpam-4247	235	22	c1	c1	NOUN
ejpam-4247	235	23	,	,	PUNCT
ejpam-4247	235	24	d1	d1	PROPN
ejpam-4247	235	25	]	]	PUNCT
ejpam-4247	235	26	on	on	ADP
ejpam-4247	235	27	r	r	NOUN
ejpam-4247	235	28	and	and	CCONJ
ejpam-4247	235	29	periodic	periodic	NOUN
ejpam-4247	235	30	in	in	ADP
ejpam-4247	235	31	t	t	PROPN
ejpam-4247	235	32	of	of	ADP
ejpam-4247	235	33	period	period	NOUN
ejpam-4247	235	34	t	t	PROPN
ejpam-4247	235	35	,	,	PUNCT
ejpam-4247	235	36	suppose	suppose	VERB
ejpam-4247	235	37	that	that	SCONJ
ejpam-4247	235	38	for	for	ADP
ejpam-4247	235	39	all	all	DET
ejpam-4247	235	40	m	m	PROPN
ejpam-4247	235	41	≥	≥	NOUN
ejpam-4247	235	42	0	0	NUM
ejpam-4247	235	43	,	,	PUNCT
ejpam-4247	235	44	then	then	ADV
ejpam-4247	235	45	the	the	DET
ejpam-4247	235	46	sequences	sequence	NOUN
ejpam-4247	235	47	of	of	ADP
ejpam-4247	235	48	the	the	DET
ejpam-4247	235	49	functions	function	NOUN
ejpam-4247	235	50	σm	σm	X
ejpam-4247	235	51	(	(	PUNCT
ejpam-4247	235	52	0,u0	0,u0	NOUN
ejpam-4247	235	53	)	)	PUNCT
ejpam-4247	235	54	which	which	PRON
ejpam-4247	235	55	are	be	AUX
ejpam-4247	235	56	defined	define	VERB
ejpam-4247	235	57	in	in	ADP
ejpam-4247	235	58	(	(	PUNCT
ejpam-4247	235	59	4.32	4.32	NUM
ejpam-4247	235	60	)	)	PUNCT
ejpam-4247	235	61	satisfy	satisfy	VERB
ejpam-4247	235	62	the	the	DET
ejpam-4247	235	63	inequalities	inequality	NOUN
ejpam-4247	235	64	:	:	PUNCT
ejpam-4247	235	65	minu0∈[c1,d1	minu0∈[c1,d1	NOUN
ejpam-4247	235	66	]	]	PUNCT
ejpam-4247	235	67	σm	σm	X
ejpam-4247	235	68	(	(	PUNCT
ejpam-4247	235	69	0	0	NUM
ejpam-4247	235	70	,	,	PUNCT
ejpam-4247	235	71	u0	u0	ADJ
ejpam-4247	235	72	)	)	PUNCT
ejpam-4247	235	73	≤	≤	NUM
ejpam-4247	236	1	−	−	PROPN
ejpam-4247	237	1	(	(	PUNCT
ejpam-4247	237	2	k1	k1	X
ejpam-4247	237	3	+	+	CCONJ
ejpam-4247	237	4	atl1k2	atl1k2	NOUN
ejpam-4247	237	5	+	+	CCONJ
ejpam-4247	237	6	l2	l2	NOUN
ejpam-4247	237	7	α	α	NOUN
ejpam-4247	237	8	)	)	PUNCT
ejpam-4247	237	9	qm(1−q)−1m3	qm(1−q)−1m3	ADJ
ejpam-4247	237	10	maxu0∈[c1,d11	maxu0∈[c1,d11	NOUN
ejpam-4247	237	11	σm	σm	X
ejpam-4247	237	12	(	(	PUNCT
ejpam-4247	237	13	0	0	NUM
ejpam-4247	237	14	,	,	PUNCT
ejpam-4247	237	15	u0	u0	ADJ
ejpam-4247	237	16	)	)	PUNCT
ejpam-4247	237	17	≥	≥	PROPN
ejpam-4247	237	18	(	(	PUNCT
ejpam-4247	237	19	k1	k1	X
ejpam-4247	237	20	+	+	CCONJ
ejpam-4247	237	21	atl1k2	atl1k2	NOUN
ejpam-4247	237	22	+	+	CCONJ
ejpam-4247	237	23	l2	l2	NOUN
ejpam-4247	237	24	α	α	NOUN
ejpam-4247	237	25	)	)	PUNCT
ejpam-4247	237	26	qm(1−q)−1m3	qm(1−q)−1m3	ADJ
ejpam-4247	237	27	}	}	PUNCT
ejpam-4247	237	28	(	(	PUNCT
ejpam-4247	237	29	4.35	4.35	NUM
ejpam-4247	237	30	)	)	PUNCT
ejpam-4247	237	31	then	then	ADV
ejpam-4247	237	32	the	the	DET
ejpam-4247	237	33	problem	problem	NOUN
ejpam-4247	237	34	(	(	PUNCT
ejpam-4247	237	35	1.1	1.1	NUM
ejpam-4247	237	36	)	)	PUNCT
ejpam-4247	237	37	with	with	ADP
ejpam-4247	237	38	(	(	PUNCT
ejpam-4247	237	39	4.28	4.28	NUM
ejpam-4247	237	40	)	)	PUNCT
ejpam-4247	237	41	has	have	VERB
ejpam-4247	237	42	a	a	DET
ejpam-4247	237	43	periodic	periodic	ADJ
ejpam-4247	237	44	solution	solution	NOUN
ejpam-4247	237	45	such	such	ADJ
ejpam-4247	237	46	that	that	DET
ejpam-4247	237	47	u0	u0	PROPN
ejpam-4247	237	48	∈	∈	PROPN
ejpam-4247	238	1	[	[	X
ejpam-4247	238	2	c1	c1	PROPN
ejpam-4247	238	3	+	+	NOUN
ejpam-4247	238	4	m3	m3	PROPN
ejpam-4247	238	5	,	,	PUNCT
ejpam-4247	238	6	d1	d1	PROPN
ejpam-4247	238	7	−m3	−m3	NOUN
ejpam-4247	238	8	]	]	PUNCT
ejpam-4247	238	9	where	where	SCONJ
ejpam-4247	238	10	m3	m3	PROPN
ejpam-4247	238	11	defined	define	VERB
ejpam-4247	238	12	in	in	ADP
ejpam-4247	238	13	(	(	PUNCT
ejpam-4247	238	14	4.33	4.33	NUM
ejpam-4247	238	15	)	)	PUNCT
ejpam-4247	238	16	.	.	PUNCT
ejpam-4247	239	1	this	this	DET
ejpam-4247	239	2	theorem	theorem	VERB
ejpam-4247	239	3	’s	’s	PART
ejpam-4247	239	4	proof	proof	NOUN
ejpam-4247	239	5	was	be	AUX
ejpam-4247	239	6	similar	similar	ADJ
ejpam-4247	239	7	to	to	ADP
ejpam-4247	239	8	that	that	PRON
ejpam-4247	239	9	of	of	ADP
ejpam-4247	239	10	theorem	theorem	ADJ
ejpam-4247	239	11	5	5	NUM
ejpam-4247	239	12	.	.	PUNCT
ejpam-4247	239	13	theorem	theorem	NOUN
ejpam-4247	239	14	10	10	NUM
ejpam-4247	239	15	.	.	PUNCT
ejpam-4247	240	1	let	let	VERB
ejpam-4247	240	2	the	the	DET
ejpam-4247	240	3	function	function	NOUN
ejpam-4247	240	4	σ	σ	PROPN
ejpam-4247	240	5	(	(	PUNCT
ejpam-4247	240	6	0,u0	0,u0	NOUN
ejpam-4247	240	7	)	)	PUNCT
ejpam-4247	240	8	be	be	AUX
ejpam-4247	240	9	defined	define	VERB
ejpam-4247	240	10	by	by	ADP
ejpam-4247	240	11	the	the	DET
ejpam-4247	240	12	equations	equation	NOUN
ejpam-4247	240	13	(	(	PUNCT
ejpam-4247	240	14	4.32	4.32	NUM
ejpam-4247	240	15	)	)	PUNCT
ejpam-4247	241	1	,	,	PUNCT
ejpam-4247	241	2	then	then	ADV
ejpam-4247	241	3	the	the	DET
ejpam-4247	241	4	following	follow	VERB
ejpam-4247	241	5	inequalities	inequality	NOUN
ejpam-4247	241	6	yield:|σ	yield:|σ	PROPN
ejpam-4247	241	7	(	(	PUNCT
ejpam-4247	241	8	0	0	NUM
ejpam-4247	241	9	,	,	PUNCT
ejpam-4247	241	10	u0)|	u0)|	PROPN
ejpam-4247	241	11	≤	≤	PUNCT
ejpam-4247	241	12	m	m	VERB
ejpam-4247	241	13	+	+	NUM
ejpam-4247	241	14	m2	m2	PROPN
ejpam-4247	241	15	α	α	PROPN
ejpam-4247	241	16	(	(	PUNCT
ejpam-4247	241	17	4.36	4.36	NUM
ejpam-4247	241	18	)	)	PUNCT
ejpam-4247	241	19	and	and	CCONJ
ejpam-4247	241	20	∣∣σ	∣∣σ	PROPN
ejpam-4247	241	21	(	(	PUNCT
ejpam-4247	241	22	0,u10)−	0,u10)−	PROPN
ejpam-4247	241	23	σ	σ	PROPN
ejpam-4247	241	24	(	(	PUNCT
ejpam-4247	241	25	0	0	NUM
ejpam-4247	241	26	,	,	PUNCT
ejpam-4247	241	27	u20	u20	NUM
ejpam-4247	241	28	)	)	PUNCT
ejpam-4247	241	29	∣∣	∣∣	PROPN
ejpam-4247	241	30	≤	≤	X
ejpam-4247	241	31	e2e3	e2e3	ADP
ejpam-4247	241	32	∣∣u10	∣∣u10	PROPN
ejpam-4247	241	33	−	−	PROPN
ejpam-4247	241	34	u20	u20	PROPN
ejpam-4247	241	35	∣∣	∣∣	NUM
ejpam-4247	241	36	(	(	PUNCT
ejpam-4247	241	37	4.37	4.37	NUM
ejpam-4247	241	38	)	)	PUNCT
ejpam-4247	241	39	where	where	SCONJ
ejpam-4247	241	40	e1	e1	NOUN
ejpam-4247	241	41	=	=	SYM
ejpam-4247	241	42	2(1−	2(1−	NUM
ejpam-4247	241	43	α	α	X
ejpam-4247	241	44	)	)	PUNCT
ejpam-4247	242	1	+	+	CCONJ
ejpam-4247	242	2	αt	αt	NOUN
ejpam-4247	242	3	2	2	NUM
ejpam-4247	242	4	,	,	PUNCT
ejpam-4247	242	5	e2	e2	NOUN
ejpam-4247	242	6	=	=	SYM
ejpam-4247	242	7	k1	k1	PROPN
ejpam-4247	242	8	+	+	CCONJ
ejpam-4247	242	9	atl1k2	atl1k2	NOUN
ejpam-4247	242	10	+	+	CCONJ
ejpam-4247	242	11	l1	l1	PROPN
ejpam-4247	242	12	α	α	PROPN
ejpam-4247	242	13	,	,	PUNCT
ejpam-4247	242	14	e3	e3	NOUN
ejpam-4247	242	15	=	=	SYM
ejpam-4247	242	16	(	(	PUNCT
ejpam-4247	242	17	1−	1−	NUM
ejpam-4247	242	18	e1e2	e1e2	NOUN
ejpam-4247	242	19	)	)	PUNCT
ejpam-4247	242	20	−1	−1	NOUN
ejpam-4247	242	21	the	the	DET
ejpam-4247	242	22	proof	proof	NOUN
ejpam-4247	242	23	of	of	ADP
ejpam-4247	242	24	this	this	DET
ejpam-4247	242	25	theorem	theorem	NOUN
ejpam-4247	242	26	was	be	AUX
ejpam-4247	242	27	similar	similar	ADJ
ejpam-4247	242	28	to	to	ADP
ejpam-4247	242	29	the	the	DET
ejpam-4247	242	30	proof	proof	NOUN
ejpam-4247	242	31	of	of	ADP
ejpam-4247	242	32	theorem	theorem	ADJ
ejpam-4247	242	33	6	6	NUM
ejpam-4247	242	34	.	.	NOUN
ejpam-4247	242	35	5	5	NUM
ejpam-4247	242	36	.	.	PUNCT
ejpam-4247	242	37	examples	example	NOUN
ejpam-4247	242	38	in	in	ADP
ejpam-4247	242	39	this	this	DET
ejpam-4247	242	40	section	section	NOUN
ejpam-4247	242	41	contains	contain	VERB
ejpam-4247	242	42	two	two	NUM
ejpam-4247	242	43	example	example	NOUN
ejpam-4247	242	44	to	to	PART
ejpam-4247	242	45	illustrate	illustrate	VERB
ejpam-4247	242	46	the	the	DET
ejpam-4247	242	47	previous	previous	ADJ
ejpam-4247	242	48	theorems	theorem	NOUN
ejpam-4247	242	49	.	.	PUNCT
ejpam-4247	242	50	example	example	NOUN
ejpam-4247	242	51	5.1	5.1	NUM
ejpam-4247	242	52	.	.	PUNCT
ejpam-4247	243	1	consider	consider	VERB
ejpam-4247	243	2	the	the	DET
ejpam-4247	243	3	following	follow	VERB
ejpam-4247	243	4	fractional	fractional	ADJ
ejpam-4247	243	5	integro	integro	ADJ
ejpam-4247	243	6	-	-	PUNCT
ejpam-4247	243	7	differential	differential	NOUN
ejpam-4247	243	8	equation	equation	NOUN
ejpam-4247	243	9	cf	cf	NOUN
ejpam-4247	243	10	0	0	NUM
ejpam-4247	243	11	d0.7	d0.7	PROPN
ejpam-4247	243	12	t	t	PROPN
ejpam-4247	243	13	(	(	PUNCT
ejpam-4247	243	14	u(t	u(t	PROPN
ejpam-4247	243	15	)	)	PUNCT
ejpam-4247	243	16	)	)	PUNCT
ejpam-4247	244	1	=	=	SYM
ejpam-4247	244	2	1	1	NUM
ejpam-4247	244	3	et	et	NOUN
ejpam-4247	244	4	+	+	CCONJ
ejpam-4247	244	5	5	5	NUM
ejpam-4247	244	6	u(t	u(t	NOUN
ejpam-4247	244	7	)	)	PUNCT
ejpam-4247	245	1	+	+	NUM
ejpam-4247	245	2	∫	∫	PROPN
ejpam-4247	245	3	t2	t2	NOUN
ejpam-4247	245	4	0	0	NUM
ejpam-4247	245	5	1	1	NUM
ejpam-4247	245	6	2(s+	2(s+	NUM
ejpam-4247	245	7	2)3	2)3	NUM
ejpam-4247	245	8	sin(u(s))ds	sin(u(s))ds	NUM
ejpam-4247	245	9	(	(	PUNCT
ejpam-4247	245	10	5.1	5.1	NUM
ejpam-4247	245	11	)	)	PUNCT
ejpam-4247	245	12	such	such	ADJ
ejpam-4247	245	13	that	that	SCONJ
ejpam-4247	245	14	t	t	PROPN
ejpam-4247	245	15	∈	∈	PROPN
ejpam-4247	245	16	j	j	PROPN
ejpam-4247	246	1	=	=	PUNCT
ejpam-4247	247	1	[	[	X
ejpam-4247	247	2	0	0	NUM
ejpam-4247	247	3	,	,	PUNCT
ejpam-4247	247	4	2	2	NUM
ejpam-4247	247	5	]	]	PUNCT
ejpam-4247	247	6	,	,	PUNCT
ejpam-4247	247	7	with	with	ADP
ejpam-4247	247	8	the	the	DET
ejpam-4247	247	9	initial	initial	ADJ
ejpam-4247	247	10	condition	condition	NOUN
ejpam-4247	247	11	u(0	u(0	NOUN
ejpam-4247	247	12	)	)	PUNCT
ejpam-4247	247	13	=	=	SYM
ejpam-4247	247	14	1	1	NUM
ejpam-4247	247	15	,	,	PUNCT
ejpam-4247	247	16	where	where	SCONJ
ejpam-4247	247	17	cf	cf	NOUN
ejpam-4247	247	18	0	0	NUM
ejpam-4247	247	19	dα	dα	PROPN
ejpam-4247	247	20	t	t	PROPN
ejpam-4247	247	21	denotes	denote	VERB
ejpam-4247	247	22	the	the	DET
ejpam-4247	247	23	fractional	fractional	PROPN
ejpam-4247	247	24	caputo	caputo	PROPN
ejpam-4247	247	25	-	-	PUNCT
ejpam-4247	247	26	fabrizio	fabrizio	PROPN
ejpam-4247	247	27	derivative	derivative	NOUN
ejpam-4247	247	28	(	(	PUNCT
ejpam-4247	247	29	a	a	PRON
ejpam-4247	247	30	=	=	SYM
ejpam-4247	247	31	0.7	0.7	NUM
ejpam-4247	247	32	∈	∈	NOUN
ejpam-4247	247	33	(	(	PUNCT
ejpam-4247	247	34	0	0	NUM
ejpam-4247	247	35	,	,	PUNCT
ejpam-4247	247	36	1	1	NUM
ejpam-4247	247	37	]	]	NUM
ejpam-4247	247	38	)	)	PUNCT
ejpam-4247	247	39	.	.	PUNCT
ejpam-4247	248	1	here	here	ADV
ejpam-4247	248	2	t	t	PROPN
ejpam-4247	248	3	=	=	SYM
ejpam-4247	248	4	2	2	NUM
ejpam-4247	248	5	,	,	PUNCT
ejpam-4247	248	6	,	,	PUNCT
ejpam-4247	248	7	a(t	a(t	NOUN
ejpam-4247	248	8	)	)	PUNCT
ejpam-4247	248	9	=	=	SYM
ejpam-4247	248	10	t2	t2	NOUN
ejpam-4247	248	11	,	,	PUNCT
ejpam-4247	248	12	h(t	h(t	PROPN
ejpam-4247	248	13	,	,	PUNCT
ejpam-4247	248	14	u(t	u(t	NOUN
ejpam-4247	248	15	)	)	PUNCT
ejpam-4247	248	16	,	,	PUNCT
ejpam-4247	248	17	z(t	z(t	NOUN
ejpam-4247	248	18	)	)	PUNCT
ejpam-4247	248	19	)	)	PUNCT
ejpam-4247	249	1	=	=	SYM
ejpam-4247	249	2	1	1	NUM
ejpam-4247	249	3	et	et	NOUN
ejpam-4247	249	4	+	+	CCONJ
ejpam-4247	249	5	5	5	NUM
ejpam-4247	249	6	u(t	u(t	NOUN
ejpam-4247	249	7	)	)	PUNCT
ejpam-4247	250	1	+	+	NUM
ejpam-4247	250	2	∫	∫	PROPN
ejpam-4247	250	3	t2	t2	NOUN
ejpam-4247	250	4	0	0	NUM
ejpam-4247	250	5	1	1	NUM
ejpam-4247	250	6	2(s+	2(s+	NUM
ejpam-4247	250	7	2)3	2)3	NUM
ejpam-4247	250	8	sin(u(s))ds	sin(u(s))ds	NUM
ejpam-4247	250	9	g(t	g(t	PROPN
ejpam-4247	250	10	,	,	PUNCT
ejpam-4247	250	11	u(t	u(t	NOUN
ejpam-4247	250	12	)	)	PUNCT
ejpam-4247	250	13	)	)	PUNCT
ejpam-4247	251	1	=	=	SYM
ejpam-4247	251	2	1	1	NUM
ejpam-4247	251	3	2(s+	2(s+	NUM
ejpam-4247	251	4	2)3	2)3	NUM
ejpam-4247	251	5	sin(u(s	sin(u(s	NOUN
ejpam-4247	251	6	)	)	PUNCT
ejpam-4247	251	7	)	)	PUNCT
ejpam-4247	252	1	we	we	PRON
ejpam-4247	252	2	obtain	obtain	VERB
ejpam-4247	252	3	that	that	DET
ejpam-4247	252	4	k1	k1	NOUN
ejpam-4247	252	5	=	=	NUM
ejpam-4247	252	6	0.2	0.2	NUM
ejpam-4247	252	7	,	,	PUNCT
ejpam-4247	252	8	k2	k2	NOUN
ejpam-4247	252	9	=	=	SYM
ejpam-4247	252	10	1	1	NUM
ejpam-4247	252	11	,	,	PUNCT
ejpam-4247	252	12	at	at	ADP
ejpam-4247	252	13	=	=	NOUN
ejpam-4247	252	14	4	4	NUM
ejpam-4247	252	15	,	,	PUNCT
ejpam-4247	252	16	l1	l1	PROPN
ejpam-4247	252	17	=	=	PROPN
ejpam-4247	252	18	0.0625	0.0625	NUM
ejpam-4247	252	19	,	,	PUNCT
ejpam-4247	252	20	so	so	SCONJ
ejpam-4247	252	21	that	that	SCONJ
ejpam-4247	252	22	λ	λ	X
ejpam-4247	252	23	=	=	PRON
ejpam-4247	252	24	(	(	PUNCT
ejpam-4247	252	25	2(1−	2(1−	NUM
ejpam-4247	252	26	α	α	X
ejpam-4247	252	27	)	)	PUNCT
ejpam-4247	252	28	+	+	CCONJ
ejpam-4247	252	29	αr	αr	NUM
ejpam-4247	252	30	2	2	NUM
ejpam-4247	252	31	)	)	PUNCT
ejpam-4247	252	32	(	(	PUNCT
ejpam-4247	252	33	k1	k1	NOUN
ejpam-4247	252	34	+	+	CCONJ
ejpam-4247	252	35	atl1k2	atl1k2	NOUN
ejpam-4247	252	36	)	)	PUNCT
ejpam-4247	252	37	=	=	PUNCT
ejpam-4247	252	38	0.585	0.585	NUM
ejpam-4247	252	39	<	<	X
ejpam-4247	252	40	1	1	NUM
ejpam-4247	252	41	.	.	PUNCT
ejpam-4247	252	42	therefore	therefore	ADV
ejpam-4247	252	43	,	,	PUNCT
ejpam-4247	252	44	by	by	ADP
ejpam-4247	252	45	theorem	theorem	NOUN
ejpam-4247	252	46	2	2	NUM
ejpam-4247	252	47	and	and	CCONJ
ejpam-4247	252	48	theorem	theorem	VERB
ejpam-4247	252	49	3	3	NUM
ejpam-4247	252	50	,	,	PUNCT
ejpam-4247	252	51	the	the	DET
ejpam-4247	252	52	fractional	fractional	ADJ
ejpam-4247	252	53	differential	differential	ADJ
ejpam-4247	252	54	equation	equation	NOUN
ejpam-4247	252	55	(	(	PUNCT
ejpam-4247	252	56	5.1	5.1	NUM
ejpam-4247	252	57	)	)	PUNCT
ejpam-4247	252	58	has	have	VERB
ejpam-4247	252	59	references	reference	NOUN
ejpam-4247	252	60	155	155	NUM
ejpam-4247	252	61	exactly	exactly	ADV
ejpam-4247	252	62	one	one	NUM
ejpam-4247	252	63	periodic	periodic	ADJ
ejpam-4247	252	64	solution	solution	NOUN
ejpam-4247	252	65	.	.	PUNCT
ejpam-4247	253	1	example	example	NOUN
ejpam-4247	254	1	5.2	5.2	NUM
ejpam-4247	254	2	.	.	PUNCT
ejpam-4247	255	1	consider	consider	VERB
ejpam-4247	255	2	the	the	DET
ejpam-4247	255	3	fractional	fractional	ADJ
ejpam-4247	255	4	integro	integro	ADJ
ejpam-4247	255	5	-	-	PUNCT
ejpam-4247	255	6	differential	differential	NOUN
ejpam-4247	255	7	equation	equation	NOUN
ejpam-4247	255	8	(	(	PUNCT
ejpam-4247	255	9	5.1	5.1	NUM
ejpam-4247	255	10	)	)	PUNCT
ejpam-4247	255	11	with	with	ADP
ejpam-4247	255	12	integral	integral	ADJ
ejpam-4247	255	13	boundary	boundary	ADJ
ejpam-4247	255	14	conditions	condition	NOUN
ejpam-4247	255	15	u(0)−	u(0)−	PROPN
ejpam-4247	255	16	u(1	u(1	PROPN
ejpam-4247	255	17	)	)	PUNCT
ejpam-4247	255	18	=	=	SYM
ejpam-4247	256	1	∫	∫	PROPN
ejpam-4247	256	2	1	1	NUM
ejpam-4247	256	3	0	0	NUM
ejpam-4247	256	4	1	1	NUM
ejpam-4247	256	5	2	2	NUM
ejpam-4247	256	6	cos(u(t))dt	cos(u(t))dt	NUM
ejpam-4247	256	7	(	(	PUNCT
ejpam-4247	256	8	5.2	5.2	NUM
ejpam-4247	256	9	)	)	PUNCT
ejpam-4247	256	10	such	such	ADJ
ejpam-4247	256	11	that	that	SCONJ
ejpam-4247	256	12	t	t	PROPN
ejpam-4247	256	13	∈	∈	PROPN
ejpam-4247	256	14	(	(	PUNCT
ejpam-4247	256	15	0	0	NUM
ejpam-4247	256	16	,	,	PUNCT
ejpam-4247	256	17	1	1	NUM
ejpam-4247	256	18	]	]	PUNCT
ejpam-4247	256	19	,	,	PUNCT
ejpam-4247	256	20	where	where	SCONJ
ejpam-4247	256	21	cf	cf	NOUN
ejpam-4247	256	22	0	0	NUM
ejpam-4247	256	23	dα	dα	PROPN
ejpam-4247	256	24	t	t	PROPN
ejpam-4247	256	25	denotes	denote	VERB
ejpam-4247	256	26	the	the	DET
ejpam-4247	256	27	fractional	fractional	PROPN
ejpam-4247	256	28	caputo	caputo	PROPN
ejpam-4247	256	29	-	-	PUNCT
ejpam-4247	256	30	fabrizio	fabrizio	PROPN
ejpam-4247	256	31	derivative	derivative	NOUN
ejpam-4247	256	32	(	(	PUNCT
ejpam-4247	256	33	α	α	NOUN
ejpam-4247	256	34	=	=	NOUN
ejpam-4247	256	35	0.7	0.7	NUM
ejpam-4247	256	36	∈	∈	NOUN
ejpam-4247	257	1	[	[	X
ejpam-4247	257	2	0	0	NUM
ejpam-4247	257	3	,	,	PUNCT
ejpam-4247	257	4	1	1	NUM
ejpam-4247	257	5	]	]	NUM
ejpam-4247	257	6	)	)	PUNCT
ejpam-4247	257	7	.	.	PUNCT
ejpam-4247	258	1	here	here	ADV
ejpam-4247	258	2	t	t	PROPN
ejpam-4247	258	3	=	=	SYM
ejpam-4247	258	4	1	1	NUM
ejpam-4247	258	5	,	,	PUNCT
ejpam-4247	258	6	,	,	PUNCT
ejpam-4247	258	7	a(t	a(t	PROPN
ejpam-4247	258	8	)	)	PUNCT
ejpam-4247	258	9	,	,	PUNCT
ejpam-4247	258	10	h(t	h(t	PROPN
ejpam-4247	258	11	,	,	PUNCT
ejpam-4247	258	12	u(t	u(t	NOUN
ejpam-4247	258	13	)	)	PUNCT
ejpam-4247	258	14	,	,	PUNCT
ejpam-4247	258	15	z(t	z(t	NOUN
ejpam-4247	258	16	)	)	PUNCT
ejpam-4247	258	17	)	)	PUNCT
ejpam-4247	258	18	,	,	PUNCT
ejpam-4247	258	19	g(t	g(t	PROPN
ejpam-4247	258	20	,	,	PUNCT
ejpam-4247	258	21	u(t	u(t	NOUN
ejpam-4247	258	22	)	)	PUNCT
ejpam-4247	258	23	)	)	PUNCT
ejpam-4247	258	24	are	be	AUX
ejpam-4247	258	25	defined	define	VERB
ejpam-4247	258	26	in	in	ADP
ejpam-4247	258	27	previous	previous	ADJ
ejpam-4247	258	28	example	example	NOUN
ejpam-4247	258	29	and	and	CCONJ
ejpam-4247	258	30	h(u(t	h(u(t	PROPN
ejpam-4247	258	31	)	)	PUNCT
ejpam-4247	258	32	)	)	PUNCT
ejpam-4247	259	1	=	=	SYM
ejpam-4247	259	2	1	1	NUM
ejpam-4247	259	3	2	2	NUM
ejpam-4247	259	4	cos(u(t	cos(u(t	NOUN
ejpam-4247	259	5	)	)	PUNCT
ejpam-4247	259	6	)	)	PUNCT
ejpam-4247	260	1	we	we	PRON
ejpam-4247	260	2	obtain	obtain	VERB
ejpam-4247	260	3	that	that	DET
ejpam-4247	260	4	k1	k1	NOUN
ejpam-4247	260	5	=	=	NUM
ejpam-4247	260	6	0.2	0.2	NUM
ejpam-4247	260	7	,	,	PUNCT
ejpam-4247	260	8	k2	k2	NOUN
ejpam-4247	260	9	=	=	SYM
ejpam-4247	260	10	1	1	NUM
ejpam-4247	260	11	,	,	PUNCT
ejpam-4247	260	12	at	at	ADP
ejpam-4247	260	13	=	=	NOUN
ejpam-4247	260	14	1	1	NUM
ejpam-4247	260	15	,	,	PUNCT
ejpam-4247	260	16	l1	l1	PROPN
ejpam-4247	260	17	=	=	PROPN
ejpam-4247	260	18	0.0625	0.0625	PROPN
ejpam-4247	260	19	,	,	PUNCT
ejpam-4247	260	20	and	and	CCONJ
ejpam-4247	260	21	l2	l2	NOUN
ejpam-4247	260	22	=	=	SYM
ejpam-4247	260	23	0.5	0.5	NUM
ejpam-4247	260	24	,	,	PUNCT
ejpam-4247	260	25	so	so	SCONJ
ejpam-4247	260	26	that	that	SCONJ
ejpam-4247	260	27	q	q	NOUN
ejpam-4247	260	28	=	=	X
ejpam-4247	260	29	(	(	PUNCT
ejpam-4247	260	30	2(1−	2(1−	NUM
ejpam-4247	260	31	α	α	NUM
ejpam-4247	260	32	)	)	PUNCT
ejpam-4247	260	33	+	+	CCONJ
ejpam-4247	260	34	at	at	ADP
ejpam-4247	260	35	2	2	NUM
ejpam-4247	260	36	)	)	PUNCT
ejpam-4247	260	37	(	(	PUNCT
ejpam-4247	260	38	k1	k1	NOUN
ejpam-4247	260	39	+	+	CCONJ
ejpam-4247	260	40	atl1k2	atl1k2	NOUN
ejpam-4247	260	41	)	)	PUNCT
ejpam-4247	260	42	+	+	CCONJ
ejpam-4247	260	43	(	(	PUNCT
ejpam-4247	260	44	1−	1−	NUM
ejpam-4247	260	45	α+	α+	X
ejpam-4247	260	46	αt	αt	NOUN
ejpam-4247	260	47	)	)	PUNCT
ejpam-4247	260	48	a	a	DET
ejpam-4247	260	49	l2	l2	NOUN
ejpam-4247	260	50	=	=	SYM
ejpam-4247	260	51	0.9637	0.9637	NUM
ejpam-4247	260	52	<	<	X
ejpam-4247	260	53	1	1	NUM
ejpam-4247	260	54	therefore	therefore	ADV
ejpam-4247	260	55	,	,	PUNCT
ejpam-4247	260	56	by	by	ADP
ejpam-4247	260	57	theorem	theorem	NOUN
ejpam-4247	260	58	7	7	NUM
ejpam-4247	260	59	,	,	PUNCT
ejpam-4247	260	60	the	the	DET
ejpam-4247	260	61	boundary	boundary	ADJ
ejpam-4247	260	62	value	value	NOUN
ejpam-4247	260	63	problem	problem	NOUN
ejpam-4247	260	64	(	(	PUNCT
ejpam-4247	260	65	5.1	5.1	NUM
ejpam-4247	260	66	)	)	PUNCT
ejpam-4247	260	67	and	and	CCONJ
ejpam-4247	260	68	(	(	PUNCT
ejpam-4247	260	69	5.2	5.2	NUM
ejpam-4247	260	70	)	)	PUNCT
ejpam-4247	260	71	has	have	VERB
ejpam-4247	260	72	exactly	exactly	ADV
ejpam-4247	260	73	one	one	NUM
ejpam-4247	260	74	periodic	periodic	ADJ
ejpam-4247	260	75	solution	solution	NOUN
ejpam-4247	260	76	.	.	PUNCT
ejpam-4247	261	1	6	6	X
ejpam-4247	261	2	.	.	X
ejpam-4247	261	3	conclusion	conclusion	NOUN
ejpam-4247	261	4	in	in	ADP
ejpam-4247	261	5	this	this	DET
ejpam-4247	261	6	paper	paper	NOUN
ejpam-4247	261	7	,	,	PUNCT
ejpam-4247	261	8	we	we	PRON
ejpam-4247	261	9	studied	study	VERB
ejpam-4247	261	10	the	the	DET
ejpam-4247	261	11	existence	existence	NOUN
ejpam-4247	261	12	,	,	PUNCT
ejpam-4247	261	13	uniqueness	uniqueness	NOUN
ejpam-4247	261	14	,	,	PUNCT
ejpam-4247	261	15	and	and	CCONJ
ejpam-4247	261	16	stability	stability	NOUN
ejpam-4247	261	17	of	of	ADP
ejpam-4247	261	18	periodic	periodic	ADJ
ejpam-4247	261	19	solutions	solution	NOUN
ejpam-4247	261	20	of	of	ADP
ejpam-4247	261	21	nonlinear	nonlinear	ADJ
ejpam-4247	261	22	fractional	fractional	ADJ
ejpam-4247	261	23	integro	integro	ADJ
ejpam-4247	261	24	-	-	PUNCT
ejpam-4247	261	25	differential	differential	NOUN
ejpam-4247	261	26	equation	equation	NOUN
ejpam-4247	261	27	(	(	PUNCT
ejpam-4247	261	28	1.1	1.1	NUM
ejpam-4247	261	29	)	)	PUNCT
ejpam-4247	261	30	where	where	SCONJ
ejpam-4247	261	31	cf	cf	NOUN
ejpam-4247	261	32	0	0	NUM
ejpam-4247	261	33	dα	dα	PROPN
ejpam-4247	261	34	t	t	PROPN
ejpam-4247	261	35	denotes	denote	VERB
ejpam-4247	261	36	the	the	DET
ejpam-4247	261	37	fractional	fractional	PROPN
ejpam-4247	261	38	caputo	caputo	PROPN
ejpam-4247	261	39	-	-	PUNCT
ejpam-4247	261	40	fabrizio	fabrizio	PROPN
ejpam-4247	261	41	derivative	derivative	NOUN
ejpam-4247	261	42	with	with	ADP
ejpam-4247	261	43	the	the	DET
ejpam-4247	261	44	initial	initial	ADJ
ejpam-4247	261	45	condition	condition	NOUN
ejpam-4247	261	46	,	,	PUNCT
ejpam-4247	261	47	periodic	periodic	ADJ
ejpam-4247	261	48	boundary	boundary	ADJ
ejpam-4247	261	49	conditions	condition	NOUN
ejpam-4247	261	50	,	,	PUNCT
ejpam-4247	261	51	and	and	CCONJ
ejpam-4247	261	52	integral	integral	ADJ
ejpam-4247	261	53	boundary	boundary	ADJ
ejpam-4247	261	54	conditions	condition	NOUN
ejpam-4247	261	55	by	by	ADP
ejpam-4247	261	56	using	use	VERB
ejpam-4247	261	57	technique	technique	NOUN
ejpam-4247	261	58	successive	successive	ADJ
ejpam-4247	261	59	approximations	approximation	NOUN
ejpam-4247	261	60	method	method	NOUN
ejpam-4247	261	61	and	and	CCONJ
ejpam-4247	261	62	banach	banach	ADV
ejpam-4247	261	63	fixed	fix	VERB
ejpam-4247	261	64	point	point	NOUN
ejpam-4247	261	65	theorem	theorem	VERB
ejpam-4247	261	66	.	.	PUNCT
ejpam-4247	262	1	here	here	ADV
ejpam-4247	262	2	conclude	conclude	VERB
ejpam-4247	262	3	that	that	SCONJ
ejpam-4247	262	4	we	we	PRON
ejpam-4247	262	5	could	could	AUX
ejpam-4247	262	6	investigate	investigate	VERB
ejpam-4247	262	7	the	the	DET
ejpam-4247	262	8	existence	existence	NOUN
ejpam-4247	262	9	,	,	PUNCT
ejpam-4247	262	10	uniqueness	uniqueness	NOUN
ejpam-4247	262	11	,	,	PUNCT
ejpam-4247	262	12	and	and	CCONJ
ejpam-4247	262	13	stability	stability	NOUN
ejpam-4247	262	14	of	of	ADP
ejpam-4247	262	15	periodic	periodic	ADJ
ejpam-4247	262	16	solution	solution	NOUN
ejpam-4247	262	17	of	of	ADP
ejpam-4247	262	18	caputo	caputo	PROPN
ejpam-4247	262	19	-	-	PUNCT
ejpam-4247	262	20	fabrizio	fabrizio	PROPN
ejpam-4247	262	21	fractional	fractional	ADJ
ejpam-4247	262	22	differential	differential	NOUN
ejpam-4247	262	23	equation	equation	NOUN
ejpam-4247	262	24	with	with	ADP
ejpam-4247	262	25	integral	integral	ADJ
ejpam-4247	262	26	boundary	boundary	ADJ
ejpam-4247	262	27	condition	condition	NOUN
ejpam-4247	262	28	au(0)−bu(t	au(0)−bu(t	PROPN
ejpam-4247	262	29	)	)	PUNCT
ejpam-4247	263	1	=	=	PUNCT
ejpam-4247	263	2	m∑	m∑	INTJ
ejpam-4247	263	3	i=1	i=1	PROPN
ejpam-4247	263	4	ci	ci	PROPN
ejpam-4247	263	5	∫	∫	PROPN
ejpam-4247	263	6	t	t	PROPN
ejpam-4247	263	7	0	0	NUM
ejpam-4247	263	8	hi(u(s))ds	hi(u(s))ds	PROPN
ejpam-4247	263	9	,	,	PUNCT
ejpam-4247	263	10	where	where	SCONJ
ejpam-4247	263	11	a	a	DET
ejpam-4247	263	12	,	,	PUNCT
ejpam-4247	263	13	b	b	NOUN
ejpam-4247	263	14	and	and	CCONJ
ejpam-4247	263	15	ci	ci	NOUN
ejpam-4247	263	16	,	,	PUNCT
ejpam-4247	263	17	i	i	NOUN
ejpam-4247	263	18	=	=	NOUN
ejpam-4247	263	19	1	1	NUM
ejpam-4247	263	20	,	,	PUNCT
ejpam-4247	263	21	2	2	NUM
ejpam-4247	263	22	,	,	PUNCT
ejpam-4247	263	23	...	...	PUNCT
ejpam-4247	263	24	,	,	PUNCT
ejpam-4247	263	25	m	m	PROPN
ejpam-4247	263	26	are	be	AUX
ejpam-4247	263	27	constants	constant	NOUN
ejpam-4247	263	28	,	,	PUNCT
ejpam-4247	263	29	and	and	CCONJ
ejpam-4247	263	30	hi	hi	INTJ
ejpam-4247	263	31	,	,	PUNCT
ejpam-4247	263	32	i	i	PRON
ejpam-4247	263	33	=	=	NOUN
ejpam-4247	263	34	1	1	NUM
ejpam-4247	263	35	,	,	PUNCT
ejpam-4247	263	36	2	2	NUM
ejpam-4247	263	37	,	,	PUNCT
ejpam-4247	263	38	...	...	PUNCT
ejpam-4247	263	39	,	,	PUNCT
ejpam-4247	263	40	m	m	VERB
ejpam-4247	263	41	are	be	AUX
ejpam-4247	263	42	defined	define	VERB
ejpam-4247	263	43	and	and	CCONJ
ejpam-4247	263	44	continuous	continuous	ADJ
ejpam-4247	263	45	functions	function	NOUN
ejpam-4247	263	46	on	on	ADP
ejpam-4247	263	47	[	[	X
ejpam-4247	263	48	0	0	NUM
ejpam-4247	263	49	,	,	PUNCT
ejpam-4247	263	50	t	t	X
ejpam-4247	263	51	]	]	PUNCT
ejpam-4247	263	52	.	.	PUNCT
ejpam-4247	264	1	finally	finally	ADV
ejpam-4247	264	2	,	,	PUNCT
ejpam-4247	264	3	some	some	DET
ejpam-4247	264	4	examples	example	NOUN
ejpam-4247	264	5	are	be	AUX
ejpam-4247	264	6	presented	present	VERB
ejpam-4247	264	7	to	to	PART
ejpam-4247	264	8	illustrate	illustrate	VERB
ejpam-4247	264	9	the	the	DET
ejpam-4247	264	10	previous	previous	ADJ
ejpam-4247	264	11	theorems	theorem	NOUN
ejpam-4247	264	12	.	.	PUNCT
ejpam-4247	265	1	references	reference	NOUN
ejpam-4247	265	2	[	[	X
ejpam-4247	265	3	1	1	NUM
ejpam-4247	265	4	]	]	PUNCT
ejpam-4247	265	5	m.	m.	NOUN
ejpam-4247	265	6	belmekki	belmekki	PROPN
ejpam-4247	265	7	,	,	PUNCT
ejpam-4247	265	8	j.	j.	PROPN
ejpam-4247	265	9	j.	j.	PROPN
ejpam-4247	265	10	nieto	nieto	PROPN
ejpam-4247	265	11	,	,	PUNCT
ejpam-4247	265	12	and	and	CCONJ
ejpam-4247	265	13	r.	r.	PROPN
ejpam-4247	265	14	r.	r.	PROPN
ejpam-4247	265	15	lopez	lopez	PROPN
ejpam-4247	265	16	.	.	PUNCT
ejpam-4247	266	1	existence	existence	NOUN
ejpam-4247	266	2	of	of	ADP
ejpam-4247	266	3	periodic	periodic	ADJ
ejpam-4247	266	4	solution	solution	NOUN
ejpam-4247	266	5	for	for	ADP
ejpam-4247	266	6	a	a	DET
ejpam-4247	266	7	nonlinear	nonlinear	ADJ
ejpam-4247	266	8	fractional	fractional	ADJ
ejpam-4247	266	9	differential	differential	NOUN
ejpam-4247	266	10	equation	equation	NOUN
ejpam-4247	266	11	.	.	PUNCT
ejpam-4247	267	1	boundary	boundary	ADJ
ejpam-4247	267	2	value	value	NOUN
ejpam-4247	267	3	problems	problem	NOUN
ejpam-4247	267	4	,	,	PUNCT
ejpam-4247	267	5	2009:1–18	2009:1–18	NUM
ejpam-4247	267	6	,	,	PUNCT
ejpam-4247	267	7	2009	2009	NUM
ejpam-4247	267	8	.	.	PUNCT
ejpam-4247	268	1	[	[	X
ejpam-4247	268	2	2	2	NUM
ejpam-4247	268	3	]	]	PUNCT
ejpam-4247	268	4	f.	f.	PROPN
ejpam-4247	268	5	e.	e.	PROPN
ejpam-4247	268	6	bouzenna	bouzenna	PROPN
ejpam-4247	268	7	,	,	PUNCT
ejpam-4247	268	8	m.	m.	NOUN
ejpam-4247	268	9	t.	t.	NOUN
ejpam-4247	268	10	meftah	meftah	NOUN
ejpam-4247	268	11	,	,	PUNCT
ejpam-4247	268	12	and	and	CCONJ
ejpam-4247	268	13	m.	m.	NOUN
ejpam-4247	268	14	difallah	difallah	NOUN
ejpam-4247	268	15	.	.	PUNCT
ejpam-4247	269	1	application	application	NOUN
ejpam-4247	269	2	of	of	ADP
ejpam-4247	269	3	the	the	DET
ejpam-4247	269	4	caputo	caputo	PROPN
ejpam-4247	269	5	–	–	PUNCT
ejpam-4247	269	6	fabrizio	fabrizio	PROPN
ejpam-4247	269	7	derivative	derivative	NOUN
ejpam-4247	269	8	without	without	ADP
ejpam-4247	269	9	singular	singular	ADJ
ejpam-4247	269	10	kernel	kernel	NOUN
ejpam-4247	269	11	to	to	PART
ejpam-4247	269	12	fractional	fractional	VERB
ejpam-4247	269	13	schrodinger	schrodinger	PROPN
ejpam-4247	269	14	equations	equation	NOUN
ejpam-4247	269	15	.	.	PUNCT
ejpam-4247	270	1	pramana	pramana	PROPN
ejpam-4247	270	2	,	,	PUNCT
ejpam-4247	270	3	94	94	NUM
ejpam-4247	270	4	,	,	PUNCT
ejpam-4247	270	5	2020	2020	NUM
ejpam-4247	270	6	.	.	PUNCT
ejpam-4247	271	1	references	reference	NOUN
ejpam-4247	271	2	156	156	NUM
ejpam-4247	272	1	[	[	X
ejpam-4247	272	2	3	3	NUM
ejpam-4247	272	3	]	]	X
ejpam-4247	272	4	r.	r.	PROPN
ejpam-4247	272	5	n.	n.	PROPN
ejpam-4247	272	6	butris	butris	PROPN
ejpam-4247	272	7	and	and	CCONJ
ejpam-4247	272	8	a.	a.	NOUN
ejpam-4247	272	9	sh	sh	PROPN
ejpam-4247	272	10	.	.	PROPN
ejpam-4247	272	11	rafeeq	rafeeq	PROPN
ejpam-4247	272	12	.	.	PUNCT
ejpam-4247	273	1	solutions	solution	NOUN
ejpam-4247	273	2	for	for	ADP
ejpam-4247	273	3	nonlinear	nonlinear	ADJ
ejpam-4247	273	4	system	system	NOUN
ejpam-4247	273	5	of	of	ADP
ejpam-4247	273	6	fractional	fractional	ADJ
ejpam-4247	273	7	integro	integro	ADJ
ejpam-4247	273	8	–	–	PUNCT
ejpam-4247	273	9	differential	differential	ADJ
ejpam-4247	273	10	equations	equation	NOUN
ejpam-4247	273	11	with	with	ADP
ejpam-4247	273	12	non	non	ADJ
ejpam-4247	273	13	-	-	ADJ
ejpam-4247	273	14	separated	separated	ADJ
ejpam-4247	273	15	integral	integral	ADJ
ejpam-4247	273	16	coupled	couple	VERB
ejpam-4247	273	17	boundary	boundary	ADJ
ejpam-4247	273	18	conditions	condition	NOUN
ejpam-4247	273	19	.	.	PUNCT
ejpam-4247	274	1	international	international	ADJ
ejpam-4247	274	2	journal	journal	NOUN
ejpam-4247	274	3	of	of	ADP
ejpam-4247	274	4	advanced	advanced	ADJ
ejpam-4247	274	5	trends	trend	NOUN
ejpam-4247	274	6	in	in	ADP
ejpam-4247	274	7	computer	computer	NOUN
ejpam-4247	274	8	science	science	NOUN
ejpam-4247	274	9	and	and	CCONJ
ejpam-4247	274	10	engineering	engineering	NOUN
ejpam-4247	274	11	,	,	PUNCT
ejpam-4247	274	12	9	9	NUM
ejpam-4247	274	13	,	,	PUNCT
ejpam-4247	274	14	2020	2020	NUM
ejpam-4247	274	15	.	.	PUNCT
ejpam-4247	275	1	[	[	X
ejpam-4247	275	2	4	4	NUM
ejpam-4247	275	3	]	]	PUNCT
ejpam-4247	275	4	a.	a.	NOUN
ejpam-4247	275	5	cabada	cabada	PROPN
ejpam-4247	275	6	and	and	CCONJ
ejpam-4247	275	7	t.	t.	PROPN
ejpam-4247	275	8	kisela	kisela	PROPN
ejpam-4247	275	9	.	.	PUNCT
ejpam-4247	276	1	existence	existence	NOUN
ejpam-4247	276	2	of	of	ADP
ejpam-4247	276	3	positive	positive	ADJ
ejpam-4247	276	4	periodic	periodic	ADJ
ejpam-4247	276	5	solutions	solution	NOUN
ejpam-4247	276	6	of	of	ADP
ejpam-4247	276	7	some	some	DET
ejpam-4247	276	8	nonlinear	nonlinear	ADJ
ejpam-4247	276	9	fractional	fractional	ADJ
ejpam-4247	276	10	differential	differential	ADJ
ejpam-4247	276	11	equations	equation	NOUN
ejpam-4247	276	12	.	.	PUNCT
ejpam-4247	277	1	communications	communication	NOUN
ejpam-4247	277	2	in	in	ADP
ejpam-4247	277	3	nonlinear	nonlinear	ADJ
ejpam-4247	277	4	science	science	NOUN
ejpam-4247	277	5	and	and	CCONJ
ejpam-4247	277	6	numerical	numerical	PROPN
ejpam-4247	277	7	simulation	simulation	PROPN
ejpam-4247	277	8	,	,	PUNCT
ejpam-4247	277	9	50:51–67	50:51–67	PROPN
ejpam-4247	277	10	,	,	PUNCT
ejpam-4247	277	11	2017	2017	NUM
ejpam-4247	277	12	.	.	PUNCT
ejpam-4247	278	1	[	[	X
ejpam-4247	278	2	5	5	X
ejpam-4247	278	3	]	]	PUNCT
ejpam-4247	278	4	m.	m.	NOUN
ejpam-4247	278	5	caputo	caputo	PROPN
ejpam-4247	278	6	and	and	CCONJ
ejpam-4247	278	7	fabrizio	fabrizio	PROPN
ejpam-4247	278	8	.	.	PUNCT
ejpam-4247	279	1	a	a	DET
ejpam-4247	279	2	new	new	ADJ
ejpam-4247	279	3	definition	definition	NOUN
ejpam-4247	279	4	of	of	ADP
ejpam-4247	279	5	fractional	fractional	ADJ
ejpam-4247	279	6	derivative	derivative	NOUN
ejpam-4247	279	7	without	without	ADP
ejpam-4247	279	8	singular	singular	ADJ
ejpam-4247	279	9	kernel	kernel	PROPN
ejpam-4247	279	10	.	.	PUNCT
ejpam-4247	280	1	progress	progress	NOUN
ejpam-4247	280	2	in	in	ADP
ejpam-4247	280	3	fractional	fractional	ADJ
ejpam-4247	280	4	differentiation	differentiation	NOUN
ejpam-4247	280	5	and	and	CCONJ
ejpam-4247	280	6	applications	application	NOUN
ejpam-4247	280	7	,	,	PUNCT
ejpam-4247	280	8	2:73–85	2:73–85	NUM
ejpam-4247	280	9	,	,	PUNCT
ejpam-4247	280	10	2015	2015	NUM
ejpam-4247	280	11	.	.	PUNCT
ejpam-4247	281	1	[	[	X
ejpam-4247	281	2	6	6	NUM
ejpam-4247	281	3	]	]	PUNCT
ejpam-4247	281	4	m.	m.	NOUN
ejpam-4247	281	5	farkas	farkas	PROPN
ejpam-4247	281	6	.	.	PUNCT
ejpam-4247	282	1	periodic	periodic	ADJ
ejpam-4247	282	2	motions	motion	NOUN
ejpam-4247	282	3	.	.	PUNCT
ejpam-4247	283	1	springer	springer	NOUN
ejpam-4247	283	2	-	-	PUNCT
ejpam-4247	283	3	verlag	verlag	PROPN
ejpam-4247	283	4	,	,	PUNCT
ejpam-4247	283	5	new	new	PROPN
ejpam-4247	283	6	york	york	PROPN
ejpam-4247	283	7	,	,	PUNCT
ejpam-4247	283	8	1994	1994	NUM
ejpam-4247	283	9	.	.	PUNCT
ejpam-4247	284	1	[	[	X
ejpam-4247	284	2	7	7	X
ejpam-4247	284	3	]	]	PUNCT
ejpam-4247	284	4	m.	m.	NOUN
ejpam-4247	284	5	feckan	feckan	PROPN
ejpam-4247	284	6	and	and	CCONJ
ejpam-4247	284	7	k.	k.	PROPN
ejpam-4247	284	8	marynets	marynets	PROPN
ejpam-4247	284	9	.	.	PUNCT
ejpam-4247	285	1	approximation	approximation	NOUN
ejpam-4247	285	2	approach	approach	NOUN
ejpam-4247	285	3	to	to	ADP
ejpam-4247	285	4	periodic	periodic	ADJ
ejpam-4247	285	5	bvp	bvp	NOUN
ejpam-4247	285	6	for	for	ADP
ejpam-4247	285	7	mixed	mixed	ADJ
ejpam-4247	285	8	fractional	fractional	ADJ
ejpam-4247	285	9	differential	differential	NOUN
ejpam-4247	285	10	systems	system	NOUN
ejpam-4247	285	11	.	.	PUNCT
ejpam-4247	286	1	j.	j.	PROPN
ejpam-4247	286	2	comput	comput	PROPN
ejpam-4247	286	3	.	.	PUNCT
ejpam-4247	287	1	appl	appl	PROPN
ejpam-4247	287	2	.	.	PROPN
ejpam-4247	287	3	math	math	PROPN
ejpam-4247	287	4	.	.	PUNCT
ejpam-4247	287	5	,	,	PUNCT
ejpam-4247	287	6	339:208–217	339:208–217	NUM
ejpam-4247	287	7	,	,	PUNCT
ejpam-4247	287	8	2018	2018	NUM
ejpam-4247	287	9	.	.	PUNCT
ejpam-4247	288	1	[	[	X
ejpam-4247	288	2	8	8	NUM
ejpam-4247	288	3	]	]	PUNCT
ejpam-4247	288	4	m.	m.	NOUN
ejpam-4247	288	5	a.	a.	PROPN
ejpam-4247	288	6	imran	imran	PROPN
ejpam-4247	288	7	,	,	PUNCT
ejpam-4247	288	8	m.	m.	PROPN
ejpam-4247	288	9	b.	b.	PROPN
ejpam-4247	288	10	riaz	riaz	PROPN
ejpam-4247	288	11	,	,	PUNCT
ejpam-4247	288	12	n.	n.	PROPN
ejpam-4247	288	13	a.	a.	NOUN
ejpam-4247	288	14	shah	shah	PROPN
ejpam-4247	288	15	,	,	PUNCT
ejpam-4247	288	16	and	and	CCONJ
ejpam-4247	288	17	a.	a.	NOUN
ejpam-4247	288	18	a.	a.	PROPN
ejpam-4247	288	19	zafar	zafar	PROPN
ejpam-4247	288	20	.	.	PUNCT
ejpam-4247	289	1	boundary	boundary	ADJ
ejpam-4247	289	2	layer	layer	NOUN
ejpam-4247	289	3	flow	flow	NOUN
ejpam-4247	289	4	of	of	ADP
ejpam-4247	289	5	mhd	mhd	PROPN
ejpam-4247	289	6	generalized	generalized	ADJ
ejpam-4247	289	7	maxwell	maxwell	PROPN
ejpam-4247	289	8	fluid	fluid	NOUN
ejpam-4247	289	9	over	over	ADP
ejpam-4247	289	10	an	an	DET
ejpam-4247	289	11	exponentially	exponentially	ADV
ejpam-4247	289	12	accelerated	accelerate	VERB
ejpam-4247	289	13	infinite	infinite	ADJ
ejpam-4247	289	14	vertical	vertical	ADJ
ejpam-4247	289	15	surface	surface	NOUN
ejpam-4247	289	16	with	with	ADP
ejpam-4247	289	17	slip	slip	NOUN
ejpam-4247	289	18	and	and	CCONJ
ejpam-4247	289	19	newtonian	newtonian	ADJ
ejpam-4247	289	20	heating	heating	NOUN
ejpam-4247	289	21	at	at	ADP
ejpam-4247	289	22	the	the	DET
ejpam-4247	289	23	boundary	boundary	NOUN
ejpam-4247	289	24	.	.	PUNCT
ejpam-4247	290	1	results	result	NOUN
ejpam-4247	290	2	in	in	ADP
ejpam-4247	290	3	physics	physics	NOUN
ejpam-4247	290	4	,	,	PUNCT
ejpam-4247	290	5	8:1061–1067	8:1061–1067	NUM
ejpam-4247	290	6	,	,	PUNCT
ejpam-4247	290	7	2018	2018	NUM
ejpam-4247	290	8	.	.	PUNCT
ejpam-4247	291	1	[	[	X
ejpam-4247	291	2	9	9	NUM
ejpam-4247	291	3	]	]	PUNCT
ejpam-4247	291	4	k.	k.	PROPN
ejpam-4247	291	5	jothimani	jothimani	PROPN
ejpam-4247	291	6	,	,	PUNCT
ejpam-4247	291	7	k.	k.	PROPN
ejpam-4247	291	8	kaliraj	kaliraj	PROPN
ejpam-4247	291	9	,	,	PUNCT
ejpam-4247	291	10	z.	z.	PROPN
ejpam-4247	291	11	hammouch	hammouch	PROPN
ejpam-4247	291	12	,	,	PUNCT
ejpam-4247	291	13	and	and	CCONJ
ejpam-4247	291	14	c.	c.	PROPN
ejpam-4247	291	15	ravichandran	ravichandran	NOUN
ejpam-4247	291	16	.	.	PUNCT
ejpam-4247	292	1	new	new	ADJ
ejpam-4247	292	2	results	result	NOUN
ejpam-4247	292	3	on	on	ADP
ejpam-4247	292	4	controllability	controllability	NOUN
ejpam-4247	292	5	in	in	ADP
ejpam-4247	292	6	the	the	DET
ejpam-4247	292	7	framework	framework	NOUN
ejpam-4247	292	8	of	of	ADP
ejpam-4247	292	9	fractional	fractional	ADJ
ejpam-4247	292	10	integrodifferential	integrodifferential	ADJ
ejpam-4247	292	11	equations	equation	NOUN
ejpam-4247	292	12	with	with	ADP
ejpam-4247	292	13	nondense	nondense	NOUN
ejpam-4247	292	14	domain	domain	NOUN
ejpam-4247	292	15	.	.	PUNCT
ejpam-4247	293	1	the	the	DET
ejpam-4247	293	2	european	european	PROPN
ejpam-4247	293	3	physical	physical	PROPN
ejpam-4247	293	4	journal	journal	PROPN
ejpam-4247	293	5	plus	plus	CCONJ
ejpam-4247	293	6	,	,	PUNCT
ejpam-4247	293	7	134	134	NUM
ejpam-4247	293	8	,	,	PUNCT
ejpam-4247	293	9	2019	2019	NUM
ejpam-4247	293	10	.	.	PUNCT
ejpam-4247	294	1	[	[	X
ejpam-4247	294	2	10	10	NUM
ejpam-4247	294	3	]	]	PUNCT
ejpam-4247	294	4	a.	a.	NOUN
ejpam-4247	294	5	a.	a.	NOUN
ejpam-4247	294	6	kilbas	kilbas	PROPN
ejpam-4247	294	7	,	,	PUNCT
ejpam-4247	294	8	h.	h.	PROPN
ejpam-4247	294	9	m	m	PROPN
ejpam-4247	294	10	srivastava	srivastava	PROPN
ejpam-4247	294	11	,	,	PUNCT
ejpam-4247	294	12	and	and	CCONJ
ejpam-4247	294	13	j.	j.	PROPN
ejpam-4247	294	14	j.	j.	PROPN
ejpam-4247	294	15	trujillo	trujillo	PROPN
ejpam-4247	294	16	.	.	PUNCT
ejpam-4247	294	17	theory	theory	NOUN
ejpam-4247	294	18	and	and	CCONJ
ejpam-4247	294	19	applications	application	NOUN
ejpam-4247	294	20	of	of	ADP
ejpam-4247	294	21	fractional	fractional	ADJ
ejpam-4247	294	22	differential	differential	ADJ
ejpam-4247	294	23	equations	equation	NOUN
ejpam-4247	294	24	.	.	PUNCT
ejpam-4247	295	1	north	north	NOUN
ejpam-4247	295	2	-	-	PUNCT
ejpam-4247	295	3	holland	holland	PROPN
ejpam-4247	295	4	mathematics	mathematics	PROPN
ejpam-4247	295	5	studies	study	NOUN
ejpam-4247	295	6	,	,	PUNCT
ejpam-4247	295	7	vol	vol	NOUN
ejpam-4247	295	8	.	.	PUNCT
ejpam-4247	296	1	204	204	NUM
ejpam-4247	296	2	.	.	PUNCT
ejpam-4247	297	1	elsevier	elsevier	NOUN
ejpam-4247	297	2	,	,	PUNCT
ejpam-4247	297	3	amsterdam	amsterdam	PROPN
ejpam-4247	297	4	,	,	PUNCT
ejpam-4247	297	5	2006	2006	NUM
ejpam-4247	297	6	.	.	PUNCT
ejpam-4247	298	1	[	[	X
ejpam-4247	298	2	11	11	NUM
ejpam-4247	298	3	]	]	X
ejpam-4247	298	4	v.	v.	CCONJ
ejpam-4247	298	5	lakshmikantham	lakshmikantham	PROPN
ejpam-4247	298	6	,	,	PUNCT
ejpam-4247	298	7	s.	s.	PROPN
ejpam-4247	298	8	leela	leela	PROPN
ejpam-4247	298	9	,	,	PUNCT
ejpam-4247	298	10	and	and	CCONJ
ejpam-4247	298	11	v.	v.	ADP
ejpam-4247	298	12	devi	devi	PROPN
ejpam-4247	298	13	.	.	PUNCT
ejpam-4247	299	1	theory	theory	NOUN
ejpam-4247	299	2	of	of	ADP
ejpam-4247	299	3	fractional	fractional	ADJ
ejpam-4247	299	4	dynamic	dynamic	ADJ
ejpam-4247	299	5	systems	system	NOUN
ejpam-4247	299	6	.	.	PUNCT
ejpam-4247	300	1	cambridge	cambridge	PROPN
ejpam-4247	300	2	academic	academic	ADJ
ejpam-4247	300	3	publishers	publisher	NOUN
ejpam-4247	300	4	,	,	PUNCT
ejpam-4247	300	5	cambridge	cambridge	PROPN
ejpam-4247	300	6	,	,	PUNCT
ejpam-4247	300	7	2009	2009	NUM
ejpam-4247	300	8	.	.	PUNCT
ejpam-4247	301	1	[	[	X
ejpam-4247	301	2	12	12	NUM
ejpam-4247	301	3	]	]	X
ejpam-4247	301	4	j.	j.	PROPN
ejpam-4247	301	5	losada	losada	PROPN
ejpam-4247	301	6	and	and	CCONJ
ejpam-4247	301	7	j.	j.	PROPN
ejpam-4247	301	8	j.	j.	PROPN
ejpam-4247	301	9	nieto	nieto	PROPN
ejpam-4247	301	10	.	.	PUNCT
ejpam-4247	302	1	properties	property	NOUN
ejpam-4247	302	2	of	of	ADP
ejpam-4247	302	3	the	the	DET
ejpam-4247	302	4	new	new	ADJ
ejpam-4247	302	5	fractional	fractional	ADJ
ejpam-4247	302	6	derivative	derivative	NOUN
ejpam-4247	302	7	without	without	ADP
ejpam-4247	302	8	singular	singular	ADJ
ejpam-4247	302	9	kernel	kernel	PROPN
ejpam-4247	302	10	.	.	PUNCT
ejpam-4247	303	1	progress	progress	NOUN
ejpam-4247	303	2	in	in	ADP
ejpam-4247	303	3	fractional	fractional	ADJ
ejpam-4247	303	4	differentiation	differentiation	NOUN
ejpam-4247	303	5	and	and	CCONJ
ejpam-4247	303	6	applications	application	NOUN
ejpam-4247	303	7	,	,	PUNCT
ejpam-4247	303	8	1:87–92	1:87–92	NUM
ejpam-4247	303	9	,	,	PUNCT
ejpam-4247	303	10	2015	2015	NUM
ejpam-4247	303	11	.	.	PUNCT
ejpam-4247	304	1	[	[	X
ejpam-4247	304	2	13	13	NUM
ejpam-4247	304	3	]	]	PUNCT
ejpam-4247	304	4	k.	k.	PROPN
ejpam-4247	304	5	s.	s.	PROPN
ejpam-4247	304	6	miller	miller	PROPN
ejpam-4247	304	7	and	and	CCONJ
ejpam-4247	304	8	b.	b.	PROPN
ejpam-4247	304	9	ross	ross	PROPN
ejpam-4247	304	10	.	.	PUNCT
ejpam-4247	305	1	an	an	DET
ejpam-4247	305	2	introduction	introduction	NOUN
ejpam-4247	305	3	to	to	ADP
ejpam-4247	305	4	the	the	DET
ejpam-4247	305	5	fractional	fractional	ADJ
ejpam-4247	305	6	calculus	calculus	NOUN
ejpam-4247	305	7	and	and	CCONJ
ejpam-4247	305	8	fractional	fractional	ADJ
ejpam-4247	305	9	differential	differential	ADJ
ejpam-4247	305	10	equations	equation	NOUN
ejpam-4247	305	11	.	.	PUNCT
ejpam-4247	306	1	wiley	wiley	PROPN
ejpam-4247	306	2	,	,	PUNCT
ejpam-4247	306	3	new	new	PROPN
ejpam-4247	306	4	york	york	PROPN
ejpam-4247	306	5	,	,	PUNCT
ejpam-4247	306	6	1993	1993	NUM
ejpam-4247	306	7	.	.	PUNCT
ejpam-4247	307	1	[	[	X
ejpam-4247	307	2	14	14	NUM
ejpam-4247	307	3	]	]	X
ejpam-4247	307	4	sh	sh	PROPN
ejpam-4247	307	5	.	.	PROPN
ejpam-4247	307	6	a.	a.	PROPN
ejpam-4247	307	7	murad	murad	PROPN
ejpam-4247	307	8	and	and	CCONJ
ejpam-4247	307	9	s.	s.	PROPN
ejpam-4247	307	10	b.	b.	PROPN
ejpam-4247	307	11	hadid	hadid	PROPN
ejpam-4247	307	12	.	.	PUNCT
ejpam-4247	308	1	existence	existence	NOUN
ejpam-4247	308	2	and	and	CCONJ
ejpam-4247	308	3	uniquness	uniquness	NOUN
ejpam-4247	308	4	theorem	theorem	NOUN
ejpam-4247	308	5	for	for	ADP
ejpam-4247	308	6	fractional	fractional	ADJ
ejpam-4247	308	7	differential	differential	ADJ
ejpam-4247	308	8	equation	equation	NOUN
ejpam-4247	308	9	with	with	ADP
ejpam-4247	308	10	integral	integral	ADJ
ejpam-4247	308	11	boundary	boundary	ADJ
ejpam-4247	308	12	condition	condition	NOUN
ejpam-4247	308	13	.	.	PUNCT
ejpam-4247	309	1	journal	journal	NOUN
ejpam-4247	309	2	of	of	ADP
ejpam-4247	309	3	fractional	fractional	ADJ
ejpam-4247	309	4	calculus	calculus	NOUN
ejpam-4247	309	5	and	and	CCONJ
ejpam-4247	309	6	applications	application	NOUN
ejpam-4247	309	7	,	,	PUNCT
ejpam-4247	309	8	,	,	PUNCT
ejpam-4247	309	9	3:1–9	3:1–9	NUM
ejpam-4247	309	10	,	,	PUNCT
ejpam-4247	309	11	2012	2012	NUM
ejpam-4247	309	12	.	.	PUNCT
ejpam-4247	310	1	[	[	X
ejpam-4247	310	2	15	15	NUM
ejpam-4247	310	3	]	]	X
ejpam-4247	310	4	sh	sh	PROPN
ejpam-4247	310	5	.	.	PROPN
ejpam-4247	310	6	a.	a.	PROPN
ejpam-4247	310	7	murad	murad	PROPN
ejpam-4247	310	8	and	and	CCONJ
ejpam-4247	310	9	a.	a.	NOUN
ejpam-4247	310	10	sh	sh	PROPN
ejpam-4247	310	11	.	.	PROPN
ejpam-4247	310	12	rafeeq	rafeeq	PROPN
ejpam-4247	310	13	.	.	PUNCT
ejpam-4247	311	1	existence	existence	NOUN
ejpam-4247	311	2	of	of	ADP
ejpam-4247	311	3	solutions	solution	NOUN
ejpam-4247	311	4	of	of	ADP
ejpam-4247	311	5	integro	integro	ADJ
ejpam-4247	311	6	-	-	PUNCT
ejpam-4247	311	7	fractional	fractional	ADJ
ejpam-4247	311	8	differential	differential	ADJ
ejpam-4247	311	9	equation	equation	NOUN
ejpam-4247	311	10	when	when	SCONJ
ejpam-4247	311	11	α	α	X
ejpam-4247	311	12	∈	∈	PROPN
ejpam-4247	311	13	(	(	PUNCT
ejpam-4247	311	14	2	2	NUM
ejpam-4247	311	15	,	,	PUNCT
ejpam-4247	311	16	3	3	NUM
ejpam-4247	311	17	]	]	PUNCT
ejpam-4247	311	18	through	through	ADP
ejpam-4247	311	19	fixed	fix	VERB
ejpam-4247	311	20	point	point	NOUN
ejpam-4247	311	21	theorem	theorem	VERB
ejpam-4247	311	22	.	.	PUNCT
ejpam-4247	312	1	j.	j.	PROPN
ejpam-4247	312	2	math	math	PROPN
ejpam-4247	312	3	.	.	PUNCT
ejpam-4247	313	1	comput	comput	NOUN
ejpam-4247	313	2	.	.	PUNCT
ejpam-4247	314	1	sci	sci	PROPN
ejpam-4247	314	2	.	.	PROPN
ejpam-4247	314	3	,	,	PUNCT
ejpam-4247	314	4	11:6392–6402	11:6392–6402	NUM
ejpam-4247	314	5	,	,	PUNCT
ejpam-4247	314	6	2021	2021	NUM
ejpam-4247	314	7	.	.	PUNCT
ejpam-4247	315	1	references	reference	NOUN
ejpam-4247	315	2	157	157	NUM
ejpam-4247	315	3	[	[	X
ejpam-4247	315	4	16	16	NUM
ejpam-4247	315	5	]	]	X
ejpam-4247	315	6	sh	sh	PROPN
ejpam-4247	315	7	.	.	PUNCT
ejpam-4247	315	8	a.	a.	PROPN
ejpam-4247	315	9	murad	murad	PROPN
ejpam-4247	315	10	,	,	PUNCT
ejpam-4247	315	11	h.	h.	PROPN
ejpam-4247	315	12	j.	j.	PROPN
ejpam-4247	315	13	zekri	zekri	PROPN
ejpam-4247	315	14	,	,	PUNCT
ejpam-4247	315	15	and	and	CCONJ
ejpam-4247	315	16	s.	s.	PROPN
ejpam-4247	315	17	hadid	hadid	PROPN
ejpam-4247	315	18	.	.	PUNCT
ejpam-4247	316	1	existence	existence	NOUN
ejpam-4247	316	2	and	and	CCONJ
ejpam-4247	316	3	uniqueness	uniqueness	NOUN
ejpam-4247	316	4	theorem	theorem	NOUN
ejpam-4247	316	5	of	of	ADP
ejpam-4247	316	6	fractional	fractional	ADJ
ejpam-4247	316	7	mixed	mixed	ADJ
ejpam-4247	316	8	volterra	volterra	NOUN
ejpam-4247	316	9	-	-	PUNCT
ejpam-4247	316	10	fredholm	fredholm	NOUN
ejpam-4247	316	11	integrodifferential	integrodifferential	ADJ
ejpam-4247	316	12	equation	equation	NOUN
ejpam-4247	316	13	with	with	ADP
ejpam-4247	316	14	integral	integral	ADJ
ejpam-4247	316	15	boundary	boundary	ADJ
ejpam-4247	316	16	conditions	condition	NOUN
ejpam-4247	316	17	.	.	PUNCT
ejpam-4247	317	1	international	international	ADJ
ejpam-4247	317	2	journal	journal	PROPN
ejpam-4247	317	3	of	of	ADP
ejpam-4247	317	4	differential	differential	ADJ
ejpam-4247	317	5	equations	equation	NOUN
ejpam-4247	317	6	,	,	PUNCT
ejpam-4247	317	7	2011:1–16	2011:1–16	NUM
ejpam-4247	317	8	,	,	PUNCT
ejpam-4247	317	9	2011	2011	NUM
ejpam-4247	317	10	.	.	PUNCT
ejpam-4247	318	1	[	[	X
ejpam-4247	318	2	17	17	NUM
ejpam-4247	318	3	]	]	X
ejpam-4247	318	4	a.	a.	NOUN
ejpam-4247	318	5	sh	sh	PROPN
ejpam-4247	318	6	.	.	PROPN
ejpam-4247	318	7	rafeeq	rafeeq	PROPN
ejpam-4247	318	8	.	.	PUNCT
ejpam-4247	319	1	on	on	ADP
ejpam-4247	319	2	the	the	DET
ejpam-4247	319	3	parametrization	parametrization	NOUN
ejpam-4247	319	4	of	of	ADP
ejpam-4247	319	5	nonlinear	nonlinear	ADJ
ejpam-4247	319	6	impulsive	impulsive	ADJ
ejpam-4247	319	7	fractional	fractional	ADJ
ejpam-4247	319	8	integro	integro	ADJ
ejpam-4247	319	9	–	–	PUNCT
ejpam-4247	319	10	differential	differential	ADJ
ejpam-4247	319	11	system	system	NOUN
ejpam-4247	319	12	with	with	ADP
ejpam-4247	319	13	non	non	ADJ
ejpam-4247	319	14	-	-	ADJ
ejpam-4247	319	15	separated	separated	ADJ
ejpam-4247	319	16	integral	integral	ADJ
ejpam-4247	319	17	coupled	couple	VERB
ejpam-4247	319	18	boundary	boundary	ADJ
ejpam-4247	319	19	conditions	condition	NOUN
ejpam-4247	319	20	.	.	PUNCT
ejpam-4247	320	1	science	science	NOUN
ejpam-4247	320	2	journal	journal	PROPN
ejpam-4247	320	3	of	of	ADP
ejpam-4247	320	4	university	university	PROPN
ejpam-4247	320	5	of	of	ADP
ejpam-4247	320	6	zakho	zakho	PROPN
ejpam-4247	320	7	,	,	PUNCT
ejpam-4247	320	8	8:160–168	8:160–168	NUM
ejpam-4247	320	9	,	,	PUNCT
ejpam-4247	320	10	2020	2020	NUM
ejpam-4247	320	11	.	.	PUNCT
ejpam-4247	321	1	[	[	X
ejpam-4247	321	2	18	18	NUM
ejpam-4247	321	3	]	]	X
ejpam-4247	321	4	c.	c.	NOUN
ejpam-4247	321	5	ravichandran	ravichandran	PROPN
ejpam-4247	321	6	,	,	PUNCT
ejpam-4247	321	7	k.	k.	PROPN
ejpam-4247	321	8	logeswari	logeswari	PROPN
ejpam-4247	321	9	,	,	PUNCT
ejpam-4247	321	10	and	and	CCONJ
ejpam-4247	321	11	f.	f.	PROPN
ejpam-4247	321	12	jarad	jarad	PROPN
ejpam-4247	321	13	.	.	PUNCT
ejpam-4247	322	1	new	new	ADJ
ejpam-4247	322	2	results	result	NOUN
ejpam-4247	322	3	on	on	ADP
ejpam-4247	322	4	existence	existence	NOUN
ejpam-4247	322	5	in	in	ADP
ejpam-4247	322	6	the	the	DET
ejpam-4247	322	7	framework	framework	NOUN
ejpam-4247	322	8	of	of	ADP
ejpam-4247	322	9	atangana	atangana	PROPN
ejpam-4247	322	10	–	–	PUNCT
ejpam-4247	322	11	baleanu	baleanu	ADJ
ejpam-4247	322	12	derivative	derivative	NOUN
ejpam-4247	322	13	for	for	ADP
ejpam-4247	322	14	fractional	fractional	ADJ
ejpam-4247	322	15	integro	integro	ADJ
ejpam-4247	322	16	-	-	PUNCT
ejpam-4247	322	17	differential	differential	NOUN
ejpam-4247	322	18	equations	equation	NOUN
ejpam-4247	322	19	.	.	PUNCT
ejpam-4247	323	1	chaos	chaos	NOUN
ejpam-4247	323	2	,	,	PUNCT
ejpam-4247	323	3	solitons	soliton	NOUN
ejpam-4247	323	4	and	and	CCONJ
ejpam-4247	323	5	fractals	fractal	NOUN
ejpam-4247	323	6	,	,	PUNCT
ejpam-4247	323	7	125:194–200	125:194–200	NUM
ejpam-4247	323	8	,	,	PUNCT
ejpam-4247	323	9	2019	2019	NUM
ejpam-4247	323	10	.	.	PUNCT
ejpam-4247	324	1	[	[	X
ejpam-4247	324	2	19	19	NUM
ejpam-4247	324	3	]	]	PUNCT
ejpam-4247	324	4	m.	m.	NOUN
ejpam-4247	324	5	ronto	ronto	NOUN
ejpam-4247	324	6	and	and	CCONJ
ejpam-4247	324	7	a.	a.	NOUN
ejpam-4247	324	8	m.	m.	PROPN
ejpam-4247	324	9	samoilenko	samoilenko	PROPN
ejpam-4247	324	10	.	.	PUNCT
ejpam-4247	325	1	numerical	numerical	ADJ
ejpam-4247	325	2	-	-	PUNCT
ejpam-4247	325	3	analytic	analytic	ADJ
ejpam-4247	325	4	methods	method	NOUN
ejpam-4247	325	5	in	in	ADP
ejpam-4247	325	6	the	the	DET
ejpam-4247	325	7	theory	theory	NOUN
ejpam-4247	325	8	of	of	ADP
ejpam-4247	325	9	boundary	boundary	ADJ
ejpam-4247	325	10	-	-	PUNCT
ejpam-4247	325	11	value	value	NOUN
ejpam-4247	325	12	problems	problem	NOUN
ejpam-4247	325	13	.	.	PUNCT
ejpam-4247	326	1	world	world	PROPN
ejpam-4247	326	2	scientific	scientific	PROPN
ejpam-4247	326	3	,	,	PUNCT
ejpam-4247	326	4	singapore	singapore	PROPN
ejpam-4247	326	5	,	,	PUNCT
ejpam-4247	326	6	2000	2000	NUM
ejpam-4247	326	7	.	.	PUNCT
ejpam-4247	327	1	[	[	X
ejpam-4247	327	2	20	20	NUM
ejpam-4247	327	3	]	]	PUNCT
ejpam-4247	327	4	k.	k.	PROPN
ejpam-4247	327	5	salim	salim	PROPN
ejpam-4247	327	6	and	and	CCONJ
ejpam-4247	327	7	s.	s.	PROPN
ejpam-4247	327	8	abbas	abbas	PROPN
ejpam-4247	327	9	.	.	PUNCT
ejpam-4247	328	1	boundary	boundary	ADJ
ejpam-4247	328	2	value	value	NOUN
ejpam-4247	328	3	problem	problem	NOUN
ejpam-4247	328	4	for	for	ADP
ejpam-4247	328	5	implicit	implicit	ADJ
ejpam-4247	328	6	caputo	caputo	PROPN
ejpam-4247	328	7	–	–	PUNCT
ejpam-4247	328	8	fabrizio	fabrizio	PROPN
ejpam-4247	328	9	fractional	fractional	ADJ
ejpam-4247	328	10	differential	differential	NOUN
ejpam-4247	328	11	equations	equation	NOUN
ejpam-4247	328	12	.	.	PUNCT
ejpam-4247	329	1	international	international	ADJ
ejpam-4247	329	2	journal	journal	PROPN
ejpam-4247	329	3	of	of	ADP
ejpam-4247	329	4	difference	difference	NOUN
ejpam-4247	329	5	equations	equation	NOUN
ejpam-4247	329	6	,	,	PUNCT
ejpam-4247	329	7	15:493–510	15:493–510	NUM
ejpam-4247	329	8	,	,	PUNCT
ejpam-4247	329	9	2020	2020	NUM
ejpam-4247	329	10	.	.	PUNCT
ejpam-4247	330	1	[	[	X
ejpam-4247	330	2	21	21	NUM
ejpam-4247	330	3	]	]	X
ejpam-4247	330	4	n.	n.	NOUN
ejpam-4247	330	5	a.	a.	NOUN
ejpam-4247	330	6	shah	shah	PROPN
ejpam-4247	330	7	,	,	PUNCT
ejpam-4247	330	8	m.	m.	NOUN
ejpam-4247	330	9	a.	a.	PROPN
ejpam-4247	330	10	imran	imran	PROPN
ejpam-4247	330	11	,	,	PUNCT
ejpam-4247	330	12	and	and	CCONJ
ejpam-4247	330	13	f.	f.	PROPN
ejpam-4247	330	14	miraj	miraj	PROPN
ejpam-4247	330	15	.	.	PUNCT
ejpam-4247	331	1	exact	exact	ADJ
ejpam-4247	331	2	solutions	solution	NOUN
ejpam-4247	331	3	of	of	ADP
ejpam-4247	331	4	time	time	NOUN
ejpam-4247	331	5	fractional	fractional	ADJ
ejpam-4247	331	6	free	free	ADJ
ejpam-4247	331	7	convection	convection	NOUN
ejpam-4247	331	8	flows	flow	NOUN
ejpam-4247	331	9	of	of	ADP
ejpam-4247	331	10	viscous	viscous	ADJ
ejpam-4247	331	11	fluid	fluid	NOUN
ejpam-4247	331	12	over	over	ADP
ejpam-4247	331	13	an	an	DET
ejpam-4247	331	14	isothermal	isothermal	ADJ
ejpam-4247	331	15	vertical	vertical	ADJ
ejpam-4247	331	16	plate	plate	NOUN
ejpam-4247	331	17	with	with	ADP
ejpam-4247	331	18	caputo	caputo	PROPN
ejpam-4247	331	19	and	and	CCONJ
ejpam-4247	331	20	caputo	caputo	PROPN
ejpam-4247	331	21	-	-	PUNCT
ejpam-4247	331	22	fabrizio	fabrizio	PROPN
ejpam-4247	331	23	derivatives	derivative	NOUN
ejpam-4247	331	24	.	.	PUNCT
ejpam-4247	332	1	journal	journal	NOUN
ejpam-4247	332	2	of	of	ADP
ejpam-4247	332	3	prime	prime	ADJ
ejpam-4247	332	4	research	research	NOUN
ejpam-4247	332	5	in	in	ADP
ejpam-4247	332	6	mathematics	mathematic	NOUN
ejpam-4247	332	7	,	,	PUNCT
ejpam-4247	332	8	13:56–74	13:56–74	NUM
ejpam-4247	332	9	,	,	PUNCT
ejpam-4247	332	10	2017	2017	NUM
ejpam-4247	332	11	.	.	PUNCT
ejpam-4247	333	1	[	[	X
ejpam-4247	333	2	22	22	NUM
ejpam-4247	333	3	]	]	X
ejpam-4247	333	4	n.	n.	PROPN
ejpam-4247	333	5	a.	a.	NOUN
ejpam-4247	333	6	sheikh	sheikh	PROPN
ejpam-4247	333	7	,	,	PUNCT
ejpam-4247	333	8	f.	f.	PROPN
ejpam-4247	333	9	ali	ali	PROPN
ejpam-4247	333	10	,	,	PUNCT
ejpam-4247	333	11	m.	m.	NOUN
ejpam-4247	333	12	saqib	saqib	PROPN
ejpam-4247	333	13	,	,	PUNCT
ejpam-4247	333	14	i.	i.	PROPN
ejpam-4247	333	15	khan	khan	PROPN
ejpam-4247	333	16	,	,	PUNCT
ejpam-4247	333	17	and	and	CCONJ
ejpam-4247	333	18	s.	s.	PROPN
ejpam-4247	333	19	a.	a.	PROPN
ejpam-4247	333	20	a.	a.	PROPN
ejpam-4247	333	21	jan	jan	PROPN
ejpam-4247	333	22	.	.	PROPN
ejpam-4247	334	1	a	a	DET
ejpam-4247	334	2	comparative	comparative	ADJ
ejpam-4247	334	3	study	study	NOUN
ejpam-4247	334	4	of	of	ADP
ejpam-4247	334	5	atangana	atangana	PROPN
ejpam-4247	334	6	-	-	PUNCT
ejpam-4247	334	7	baleanu	baleanu	PROPN
ejpam-4247	334	8	and	and	CCONJ
ejpam-4247	334	9	caputofabrizio	caputofabrizio	VERB
ejpam-4247	334	10	fractional	fractional	ADJ
ejpam-4247	334	11	derivatives	derivative	NOUN
ejpam-4247	334	12	to	to	ADP
ejpam-4247	334	13	the	the	DET
ejpam-4247	334	14	convective	convective	ADJ
ejpam-4247	334	15	flow	flow	NOUN
ejpam-4247	334	16	of	of	ADP
ejpam-4247	334	17	a	a	DET
ejpam-4247	334	18	generalized	generalized	ADJ
ejpam-4247	334	19	casson	casson	NOUN
ejpam-4247	334	20	fluid	fluid	NOUN
ejpam-4247	334	21	.	.	PUNCT
ejpam-4247	335	1	the	the	DET
ejpam-4247	335	2	european	european	PROPN
ejpam-4247	335	3	physical	physical	PROPN
ejpam-4247	335	4	journal	journal	PROPN
ejpam-4247	335	5	plus	plus	CCONJ
ejpam-4247	335	6	,	,	PUNCT
ejpam-4247	335	7	132	132	NUM
ejpam-4247	335	8	,	,	PUNCT
ejpam-4247	335	9	2017	2017	NUM
ejpam-4247	335	10	.	.	PUNCT