id	sid	tid	token	lemma	pos
ejde-771	1	1	electronic	electronic	ADJ
ejde-771	1	2	journal	journal	NOUN
ejde-771	1	3	of	of	ADP
ejde-771	1	4	differential	differential	ADJ
ejde-771	1	5	equations	equation	NOUN
ejde-771	1	6	,	,	PUNCT
ejde-771	1	7	vol	vol	NOUN
ejde-771	1	8	.	.	PUNCT
ejde-771	1	9	2025	2025	NUM
ejde-771	1	10	(	(	PUNCT
ejde-771	1	11	2025	2025	NUM
ejde-771	1	12	)	)	PUNCT
ejde-771	1	13	,	,	PUNCT
ejde-771	1	14	no	no	INTJ
ejde-771	1	15	.	.	NOUN
ejde-771	1	16	44	44	NUM
ejde-771	1	17	,	,	PUNCT
ejde-771	1	18	pp	pp	ADJ
ejde-771	1	19	.	.	PUNCT
ejde-771	2	1	1–15	1–15	PROPN
ejde-771	2	2	.	.	PUNCT
ejde-771	3	1	issn	issn	PROPN
ejde-771	3	2	:	:	PUNCT
ejde-771	3	3	1072	1072	NUM
ejde-771	3	4	-	-	SYM
ejde-771	3	5	6691	6691	NUM
ejde-771	3	6	.	.	PUNCT
ejde-771	4	1	url	url	PROPN
ejde-771	4	2	:	:	PUNCT
ejde-771	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-771	4	4	,	,	PUNCT
ejde-771	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-771	4	6	doi	doi	PROPN
ejde-771	4	7	:	:	PUNCT
ejde-771	4	8	10.58997	10.58997	NUM
ejde-771	4	9	/	/	SYM
ejde-771	4	10	ejde.2025.44	ejde.2025.44	NOUN
ejde-771	4	11	existence	existence	NOUN
ejde-771	4	12	of	of	ADP
ejde-771	4	13	positive	positive	ADJ
ejde-771	4	14	s	s	NOUN
ejde-771	4	15	-	-	PUNCT
ejde-771	4	16	asymptotically	asymptotically	ADV
ejde-771	4	17	ω	ω	ADJ
ejde-771	4	18	-	-	ADJ
ejde-771	4	19	periodic	periodic	ADJ
ejde-771	4	20	solutions	solution	NOUN
ejde-771	4	21	of	of	ADP
ejde-771	4	22	time	time	NOUN
ejde-771	4	23	-	-	PUNCT
ejde-771	4	24	space	space	NOUN
ejde-771	4	25	fractional	fractional	ADJ
ejde-771	4	26	nonlocal	nonlocal	ADJ
ejde-771	4	27	reaction	reaction	NOUN
ejde-771	4	28	-	-	PUNCT
ejde-771	4	29	diffusion	diffusion	NOUN
ejde-771	4	30	equations	equation	NOUN
ejde-771	4	31	xuping	xuping	PROPN
ejde-771	4	32	zhang	zhang	PROPN
ejde-771	4	33	,	,	PUNCT
ejde-771	4	34	kaibo	kaibo	VERB
ejde-771	4	35	ding	ding	NOUN
ejde-771	4	36	,	,	PUNCT
ejde-771	4	37	pengyu	pengyu	NOUN
ejde-771	4	38	chen	chen	PROPN
ejde-771	4	39	abstract	abstract	PROPN
ejde-771	4	40	.	.	PUNCT
ejde-771	5	1	this	this	DET
ejde-771	5	2	article	article	NOUN
ejde-771	5	3	studies	study	VERB
ejde-771	5	4	the	the	DET
ejde-771	5	5	asymptotically	asymptotically	ADV
ejde-771	5	6	periodic	periodic	ADJ
ejde-771	5	7	problem	problem	NOUN
ejde-771	5	8	of	of	ADP
ejde-771	5	9	time	time	NOUN
ejde-771	5	10	-	-	PUNCT
ejde-771	5	11	space	space	NOUN
ejde-771	5	12	fractional	fractional	ADJ
ejde-771	5	13	reaction	reaction	NOUN
ejde-771	5	14	-	-	PUNCT
ejde-771	5	15	diffusion	diffusion	NOUN
ejde-771	5	16	equations	equation	NOUN
ejde-771	5	17	with	with	ADP
ejde-771	5	18	nonlocal	nonlocal	ADJ
ejde-771	5	19	initial	initial	ADJ
ejde-771	5	20	conditions	condition	NOUN
ejde-771	5	21	on	on	ADP
ejde-771	5	22	infinite	infinite	ADJ
ejde-771	5	23	intervals	interval	NOUN
ejde-771	5	24	.	.	PUNCT
ejde-771	6	1	without	without	ADP
ejde-771	6	2	the	the	DET
ejde-771	6	3	assumption	assumption	NOUN
ejde-771	6	4	of	of	ADP
ejde-771	6	5	upper	upper	ADJ
ejde-771	6	6	and	and	CCONJ
ejde-771	6	7	lower	low	ADJ
ejde-771	6	8	s	s	NOUN
ejde-771	6	9	-	-	PUNCT
ejde-771	6	10	asymptotically	asymptotically	ADV
ejde-771	6	11	ω	ω	ADJ
ejde-771	6	12	-	-	ADJ
ejde-771	6	13	periodic	periodic	ADJ
ejde-771	6	14	solutions	solution	NOUN
ejde-771	6	15	,	,	PUNCT
ejde-771	6	16	the	the	DET
ejde-771	6	17	existence	existence	NOUN
ejde-771	6	18	results	result	VERB
ejde-771	6	19	of	of	ADP
ejde-771	6	20	positive	positive	ADJ
ejde-771	6	21	s	s	NOUN
ejde-771	6	22	-	-	PUNCT
ejde-771	6	23	asymptotically	asymptotically	ADV
ejde-771	6	24	ω	ω	ADJ
ejde-771	6	25	-	-	ADJ
ejde-771	6	26	periodic	periodic	ADJ
ejde-771	6	27	solutions	solution	NOUN
ejde-771	6	28	for	for	ADP
ejde-771	6	29	a	a	DET
ejde-771	6	30	class	class	NOUN
ejde-771	6	31	of	of	ADP
ejde-771	6	32	abstract	abstract	ADJ
ejde-771	6	33	time	time	NOUN
ejde-771	6	34	-	-	PUNCT
ejde-771	6	35	space	space	NOUN
ejde-771	6	36	fractional	fractional	ADJ
ejde-771	6	37	evolution	evolution	NOUN
ejde-771	6	38	equations	equation	NOUN
ejde-771	6	39	with	with	ADP
ejde-771	6	40	nonlocal	nonlocal	ADJ
ejde-771	6	41	initial	initial	ADJ
ejde-771	6	42	conditions	condition	NOUN
ejde-771	6	43	under	under	ADP
ejde-771	6	44	growth	growth	NOUN
ejde-771	6	45	and	and	CCONJ
ejde-771	6	46	order	order	NOUN
ejde-771	6	47	conditions	condition	NOUN
ejde-771	6	48	are	be	AUX
ejde-771	6	49	obtained	obtain	VERB
ejde-771	6	50	by	by	ADP
ejde-771	6	51	using	use	VERB
ejde-771	6	52	the	the	DET
ejde-771	6	53	theory	theory	NOUN
ejde-771	6	54	of	of	ADP
ejde-771	6	55	operator	operator	NOUN
ejde-771	6	56	semigroups	semigroup	NOUN
ejde-771	6	57	and	and	CCONJ
ejde-771	6	58	the	the	DET
ejde-771	6	59	method	method	NOUN
ejde-771	6	60	of	of	ADP
ejde-771	6	61	monotone	monotone	ADJ
ejde-771	6	62	iteration	iteration	NOUN
ejde-771	6	63	.	.	PUNCT
ejde-771	7	1	finally	finally	ADV
ejde-771	7	2	,	,	PUNCT
ejde-771	7	3	the	the	DET
ejde-771	7	4	abstract	abstract	ADJ
ejde-771	7	5	results	result	NOUN
ejde-771	7	6	were	be	AUX
ejde-771	7	7	applied	apply	VERB
ejde-771	7	8	to	to	ADP
ejde-771	7	9	time	time	NOUN
ejde-771	7	10	-	-	PUNCT
ejde-771	7	11	space	space	NOUN
ejde-771	7	12	fractional	fractional	ADJ
ejde-771	7	13	reaction	reaction	NOUN
ejde-771	7	14	-	-	PUNCT
ejde-771	7	15	diffusion	diffusion	NOUN
ejde-771	7	16	equations	equation	NOUN
ejde-771	7	17	with	with	ADP
ejde-771	7	18	nonlocal	nonlocal	ADJ
ejde-771	7	19	initial	initial	ADJ
ejde-771	7	20	conditions	condition	NOUN
ejde-771	7	21	and	and	CCONJ
ejde-771	7	22	some	some	DET
ejde-771	7	23	new	new	ADJ
ejde-771	7	24	results	result	NOUN
ejde-771	7	25	were	be	AUX
ejde-771	7	26	obtained	obtain	VERB
ejde-771	7	27	.	.	PUNCT
ejde-771	8	1	1	1	X
ejde-771	8	2	.	.	X
ejde-771	8	3	introduction	introduction	NOUN
ejde-771	8	4	in	in	ADP
ejde-771	8	5	this	this	DET
ejde-771	8	6	article	article	NOUN
ejde-771	8	7	,	,	PUNCT
ejde-771	8	8	we	we	PRON
ejde-771	8	9	study	study	VERB
ejde-771	8	10	the	the	DET
ejde-771	8	11	positive	positive	ADJ
ejde-771	8	12	s	s	NOUN
ejde-771	8	13	-	-	PUNCT
ejde-771	8	14	asymptotically	asymptotically	ADV
ejde-771	8	15	ω	ω	ADJ
ejde-771	8	16	-	-	ADJ
ejde-771	8	17	periodic	periodic	ADJ
ejde-771	8	18	solutions	solution	NOUN
ejde-771	8	19	for	for	ADP
ejde-771	8	20	the	the	DET
ejde-771	8	21	following	following	ADJ
ejde-771	8	22	time	time	NOUN
ejde-771	8	23	-	-	PUNCT
ejde-771	8	24	space	space	NOUN
ejde-771	8	25	fractional	fractional	ADJ
ejde-771	8	26	reaction	reaction	NOUN
ejde-771	8	27	-	-	PUNCT
ejde-771	8	28	diffusion	diffusion	NOUN
ejde-771	8	29	equation	equation	NOUN
ejde-771	8	30	with	with	ADP
ejde-771	8	31	nonlocal	nonlocal	ADJ
ejde-771	8	32	initial	initial	ADJ
ejde-771	8	33	conditions	condition	NOUN
ejde-771	8	34	cdα	cdα	NOUN
ejde-771	8	35	t	t	PROPN
ejde-771	8	36	u(t	u(t	PROPN
ejde-771	8	37	,	,	PUNCT
ejde-771	8	38	x	x	X
ejde-771	8	39	)	)	PUNCT
ejde-771	9	1	+	+	CCONJ
ejde-771	9	2	(	(	PUNCT
ejde-771	9	3	−∆)βu(t	−∆)βu(t	NOUN
ejde-771	9	4	,	,	PUNCT
ejde-771	9	5	x	x	NOUN
ejde-771	9	6	)	)	PUNCT
ejde-771	9	7	=	=	SYM
ejde-771	9	8	f	f	PROPN
ejde-771	9	9	(	(	PUNCT
ejde-771	9	10	t	t	PROPN
ejde-771	9	11	,	,	PUNCT
ejde-771	9	12	u(t	u(t	NOUN
ejde-771	9	13	,	,	PUNCT
ejde-771	9	14	x	x	NOUN
ejde-771	9	15	)	)	PUNCT
ejde-771	9	16	)	)	PUNCT
ejde-771	9	17	,	,	PUNCT
ejde-771	9	18	(	(	PUNCT
ejde-771	9	19	t	t	PROPN
ejde-771	9	20	,	,	PUNCT
ejde-771	9	21	x	x	NOUN
ejde-771	9	22	)	)	PUNCT
ejde-771	9	23	∈	∈	PROPN
ejde-771	10	1	[	[	X
ejde-771	10	2	0,+∞)×	0,+∞)×	NOUN
ejde-771	10	3	ω	ω	NUM
ejde-771	10	4	,	,	PUNCT
ejde-771	10	5	u(t	u(t	NOUN
ejde-771	10	6	,	,	PUNCT
ejde-771	10	7	x	x	NOUN
ejde-771	10	8	)	)	PUNCT
ejde-771	10	9	=	=	SYM
ejde-771	10	10	0	0	NUM
ejde-771	10	11	,	,	PUNCT
ejde-771	10	12	(	(	PUNCT
ejde-771	10	13	t	t	PROPN
ejde-771	10	14	,	,	PUNCT
ejde-771	10	15	x	x	NOUN
ejde-771	10	16	)	)	PUNCT
ejde-771	10	17	∈	∈	PROPN
ejde-771	11	1	[	[	X
ejde-771	11	2	0,+∞)×	0,+∞)×	NUM
ejde-771	11	3	∂ω	∂ω	PROPN
ejde-771	11	4	,	,	PUNCT
ejde-771	11	5	u(0	u(0	PROPN
ejde-771	11	6	,	,	PUNCT
ejde-771	11	7	x	x	NOUN
ejde-771	11	8	)	)	PUNCT
ejde-771	11	9	=	=	SYM
ejde-771	11	10	u0(x	u0(x	NOUN
ejde-771	11	11	)	)	PUNCT
ejde-771	11	12	+	+	CCONJ
ejde-771	11	13	m∑	m∑	CCONJ
ejde-771	11	14	k=1	k=1	X
ejde-771	11	15	aku(tk	aku(tk	NOUN
ejde-771	11	16	,	,	PUNCT
ejde-771	11	17	x	x	NOUN
ejde-771	11	18	)	)	PUNCT
ejde-771	11	19	,	,	PUNCT
ejde-771	11	20	x	x	PUNCT
ejde-771	11	21	∈	∈	PROPN
ejde-771	11	22	ω	ω	PROPN
ejde-771	11	23	,	,	PUNCT
ejde-771	11	24	(	(	PUNCT
ejde-771	11	25	1.1	1.1	NUM
ejde-771	11	26	)	)	PUNCT
ejde-771	11	27	where	where	SCONJ
ejde-771	11	28	cdα	cdα	NOUN
ejde-771	11	29	t	t	PROPN
ejde-771	11	30	is	be	AUX
ejde-771	11	31	the	the	DET
ejde-771	11	32	caputo	caputo	PROPN
ejde-771	11	33	fractional	fractional	PROPN
ejde-771	11	34	derivative	derivative	NOUN
ejde-771	11	35	of	of	ADP
ejde-771	11	36	order	order	NOUN
ejde-771	11	37	0	0	PUNCT
ejde-771	11	38	<	<	X
ejde-771	11	39	α	α	X
ejde-771	11	40	<	<	X
ejde-771	11	41	1	1	NUM
ejde-771	11	42	,	,	PUNCT
ejde-771	11	43	(	(	PUNCT
ejde-771	11	44	−∆)β	−∆)β	PRON
ejde-771	11	45	is	be	AUX
ejde-771	11	46	a	a	DET
ejde-771	11	47	fractional	fractional	ADJ
ejde-771	11	48	laplacian	laplacian	NOUN
ejde-771	11	49	with	with	ADP
ejde-771	11	50	0	0	NUM
ejde-771	11	51	<	<	X
ejde-771	11	52	β	β	X
ejde-771	11	53	<	<	X
ejde-771	11	54	1	1	NUM
ejde-771	11	55	,	,	PUNCT
ejde-771	11	56	ω	ω	PROPN
ejde-771	11	57	is	be	AUX
ejde-771	11	58	a	a	DET
ejde-771	11	59	bounded	bounded	ADJ
ejde-771	11	60	open	open	ADJ
ejde-771	11	61	domain	domain	NOUN
ejde-771	11	62	in	in	ADP
ejde-771	11	63	rn	rn	PROPN
ejde-771	11	64	,	,	PUNCT
ejde-771	11	65	0	0	PUNCT
ejde-771	11	66	<	<	X
ejde-771	11	67	t1	t1	NOUN
ejde-771	11	68	<	<	X
ejde-771	11	69	t2	t2	PROPN
ejde-771	11	70	<	<	X
ejde-771	11	71	·	·	PUNCT
ejde-771	11	72	·	·	PUNCT
ejde-771	11	73	·	·	PUNCT
ejde-771	11	74	<	<	X
ejde-771	12	1	tm	tm	X
ejde-771	12	2	<	<	X
ejde-771	12	3	+	+	PROPN
ejde-771	12	4	∞	∞	PROPN
ejde-771	12	5	,	,	PUNCT
ejde-771	12	6	ak	ak	PROPN
ejde-771	12	7	̸=	̸=	PROPN
ejde-771	12	8	0	0	NUM
ejde-771	12	9	are	be	AUX
ejde-771	12	10	real	real	ADJ
ejde-771	12	11	numbers	number	NOUN
ejde-771	12	12	,	,	PUNCT
ejde-771	12	13	k	k	PROPN
ejde-771	12	14	=	=	SYM
ejde-771	12	15	1	1	NUM
ejde-771	12	16	,	,	PUNCT
ejde-771	12	17	2	2	NUM
ejde-771	12	18	,	,	PUNCT
ejde-771	12	19	.	.	PUNCT
ejde-771	12	20	.	.	PUNCT
ejde-771	12	21	.	.	PUNCT
ejde-771	13	1	,	,	PUNCT
ejde-771	13	2	m	m	PROPN
ejde-771	13	3	,	,	PUNCT
ejde-771	13	4	f	f	X
ejde-771	13	5	:	:	PUNCT
ejde-771	14	1	[	[	X
ejde-771	14	2	0,+∞)×	0,+∞)×	NUM
ejde-771	14	3	r	r	NOUN
ejde-771	14	4	→	→	SYM
ejde-771	14	5	r	r	NOUN
ejde-771	14	6	is	be	AUX
ejde-771	14	7	a	a	DET
ejde-771	14	8	continuous	continuous	ADJ
ejde-771	14	9	function	function	NOUN
ejde-771	14	10	.	.	PUNCT
ejde-771	15	1	it	it	PRON
ejde-771	15	2	is	be	AUX
ejde-771	15	3	well	well	ADV
ejde-771	15	4	known	know	VERB
ejde-771	15	5	that	that	SCONJ
ejde-771	15	6	many	many	ADJ
ejde-771	15	7	realistic	realistic	ADJ
ejde-771	15	8	models	model	NOUN
ejde-771	15	9	are	be	AUX
ejde-771	15	10	not	not	PART
ejde-771	15	11	strictly	strictly	ADV
ejde-771	15	12	periodic	periodic	ADJ
ejde-771	15	13	.	.	PUNCT
ejde-771	16	1	therefore	therefore	ADV
ejde-771	16	2	,	,	PUNCT
ejde-771	16	3	since	since	SCONJ
ejde-771	16	4	the	the	DET
ejde-771	16	5	concept	concept	NOUN
ejde-771	16	6	of	of	ADP
ejde-771	16	7	s	s	NOUN
ejde-771	16	8	-	-	PUNCT
ejde-771	16	9	asymptotically	asymptotically	ADV
ejde-771	16	10	ω	ω	ADJ
ejde-771	16	11	-	-	ADJ
ejde-771	16	12	periodic	periodic	ADJ
ejde-771	16	13	function	function	NOUN
ejde-771	16	14	was	be	AUX
ejde-771	16	15	introduced	introduce	VERB
ejde-771	16	16	in	in	ADP
ejde-771	16	17	[	[	X
ejde-771	16	18	20	20	NUM
ejde-771	16	19	]	]	PUNCT
ejde-771	16	20	,	,	PUNCT
ejde-771	16	21	asymptotically	asymptotically	ADV
ejde-771	16	22	periodic	periodic	ADJ
ejde-771	16	23	problems	problem	NOUN
ejde-771	16	24	have	have	AUX
ejde-771	16	25	been	be	AUX
ejde-771	16	26	rapidly	rapidly	ADV
ejde-771	16	27	developed	develop	VERB
ejde-771	16	28	due	due	ADP
ejde-771	16	29	to	to	ADP
ejde-771	16	30	their	their	PRON
ejde-771	16	31	broad	broad	ADJ
ejde-771	16	32	physical	physical	ADJ
ejde-771	16	33	background	background	NOUN
ejde-771	16	34	and	and	CCONJ
ejde-771	16	35	realistic	realistic	ADJ
ejde-771	16	36	mathematical	mathematical	ADJ
ejde-771	16	37	models	model	NOUN
ejde-771	16	38	.	.	PUNCT
ejde-771	17	1	in	in	ADP
ejde-771	17	2	particular	particular	ADJ
ejde-771	17	3	,	,	PUNCT
ejde-771	17	4	the	the	DET
ejde-771	17	5	hereditary	hereditary	NOUN
ejde-771	17	6	and	and	CCONJ
ejde-771	17	7	memorability	memorability	NOUN
ejde-771	17	8	of	of	ADP
ejde-771	17	9	fractional	fractional	ADJ
ejde-771	17	10	derivatives	derivative	NOUN
ejde-771	17	11	provides	provide	VERB
ejde-771	17	12	an	an	DET
ejde-771	17	13	ideal	ideal	ADJ
ejde-771	17	14	tool	tool	NOUN
ejde-771	17	15	for	for	ADP
ejde-771	17	16	describing	describe	VERB
ejde-771	17	17	many	many	ADJ
ejde-771	17	18	phenomena	phenomenon	NOUN
ejde-771	17	19	and	and	CCONJ
ejde-771	17	20	processes	process	NOUN
ejde-771	17	21	.	.	PUNCT
ejde-771	18	1	it	it	PRON
ejde-771	18	2	is	be	AUX
ejde-771	18	3	worth	worth	ADJ
ejde-771	18	4	noting	note	VERB
ejde-771	18	5	that	that	SCONJ
ejde-771	18	6	there	there	PRON
ejde-771	18	7	are	be	VERB
ejde-771	18	8	many	many	ADJ
ejde-771	18	9	relevant	relevant	ADJ
ejde-771	18	10	results	result	NOUN
ejde-771	18	11	on	on	ADP
ejde-771	18	12	the	the	DET
ejde-771	18	13	existence	existence	NOUN
ejde-771	18	14	and	and	CCONJ
ejde-771	18	15	uniqueness	uniqueness	NOUN
ejde-771	18	16	of	of	ADP
ejde-771	18	17	s	s	NOUN
ejde-771	18	18	-	-	PUNCT
ejde-771	18	19	asymptotically	asymptotically	ADV
ejde-771	18	20	ω	ω	ADJ
ejde-771	18	21	-	-	ADJ
ejde-771	18	22	periodic	periodic	ADJ
ejde-771	18	23	solutions	solution	NOUN
ejde-771	18	24	to	to	ADP
ejde-771	18	25	fractional	fractional	ADJ
ejde-771	18	26	differential	differential	ADJ
ejde-771	18	27	equations	equation	NOUN
ejde-771	18	28	,	,	PUNCT
ejde-771	18	29	one	one	PRON
ejde-771	18	30	can	can	AUX
ejde-771	18	31	refer	refer	VERB
ejde-771	18	32	to	to	ADP
ejde-771	18	33	[	[	X
ejde-771	18	34	3	3	NUM
ejde-771	18	35	,	,	PUNCT
ejde-771	18	36	5	5	NUM
ejde-771	18	37	,	,	PUNCT
ejde-771	18	38	6	6	NUM
ejde-771	18	39	,	,	PUNCT
ejde-771	18	40	7	7	NUM
ejde-771	18	41	,	,	PUNCT
ejde-771	18	42	22	22	NUM
ejde-771	18	43	,	,	PUNCT
ejde-771	18	44	26	26	NUM
ejde-771	18	45	]	]	PUNCT
ejde-771	18	46	and	and	CCONJ
ejde-771	18	47	references	reference	NOUN
ejde-771	18	48	therein	therein	ADV
ejde-771	18	49	.	.	PUNCT
ejde-771	19	1	in	in	ADP
ejde-771	19	2	addition	addition	NOUN
ejde-771	19	3	,	,	PUNCT
ejde-771	19	4	in	in	ADP
ejde-771	19	5	many	many	ADJ
ejde-771	19	6	specific	specific	ADJ
ejde-771	19	7	system	system	NOUN
ejde-771	19	8	applications	application	NOUN
ejde-771	19	9	,	,	PUNCT
ejde-771	19	10	sometimes	sometimes	ADV
ejde-771	19	11	only	only	ADV
ejde-771	19	12	positive	positive	ADJ
ejde-771	19	13	solutions	solution	NOUN
ejde-771	19	14	are	be	AUX
ejde-771	19	15	significant	significant	ADJ
ejde-771	19	16	.	.	PUNCT
ejde-771	20	1	in	in	ADP
ejde-771	20	2	recent	recent	ADJ
ejde-771	20	3	years	year	NOUN
ejde-771	20	4	,	,	PUNCT
ejde-771	20	5	there	there	PRON
ejde-771	20	6	are	be	VERB
ejde-771	20	7	many	many	ADJ
ejde-771	20	8	results	result	NOUN
ejde-771	20	9	on	on	ADP
ejde-771	20	10	the	the	DET
ejde-771	20	11	existence	existence	NOUN
ejde-771	20	12	of	of	ADP
ejde-771	20	13	positive	positive	ADJ
ejde-771	20	14	solutions	solution	NOUN
ejde-771	20	15	for	for	ADP
ejde-771	20	16	fractional	fractional	ADJ
ejde-771	20	17	differential	differential	ADJ
ejde-771	20	18	equations	equation	NOUN
ejde-771	20	19	,	,	PUNCT
ejde-771	20	20	one	one	PRON
ejde-771	20	21	can	can	AUX
ejde-771	20	22	see	see	VERB
ejde-771	20	23	[	[	X
ejde-771	20	24	14	14	NUM
ejde-771	20	25	,	,	PUNCT
ejde-771	20	26	17	17	NUM
ejde-771	20	27	,	,	PUNCT
ejde-771	20	28	21	21	NUM
ejde-771	20	29	,	,	PUNCT
ejde-771	20	30	25	25	NUM
ejde-771	20	31	,	,	PUNCT
ejde-771	20	32	33	33	NUM
ejde-771	20	33	]	]	PUNCT
ejde-771	20	34	.	.	PUNCT
ejde-771	21	1	however	however	ADV
ejde-771	21	2	,	,	PUNCT
ejde-771	21	3	the	the	DET
ejde-771	21	4	existence	existence	NOUN
ejde-771	21	5	of	of	ADP
ejde-771	21	6	positive	positive	ADJ
ejde-771	21	7	s	s	NOUN
ejde-771	21	8	-	-	PUNCT
ejde-771	21	9	asymptotically	asymptotically	ADV
ejde-771	21	10	ω	ω	ADJ
ejde-771	21	11	-	-	ADJ
ejde-771	21	12	periodic	periodic	ADJ
ejde-771	21	13	solutions	solution	NOUN
ejde-771	21	14	on	on	ADP
ejde-771	21	15	infinite	infinite	ADJ
ejde-771	21	16	intervals	interval	NOUN
ejde-771	21	17	are	be	AUX
ejde-771	21	18	few	few	ADJ
ejde-771	21	19	.	.	PUNCT
ejde-771	22	1	shu	shu	PROPN
ejde-771	23	1	[	[	X
ejde-771	23	2	29	29	NUM
ejde-771	23	3	]	]	PUNCT
ejde-771	23	4	investigated	investigate	VERB
ejde-771	23	5	the	the	DET
ejde-771	23	6	existence	existence	NOUN
ejde-771	23	7	of	of	ADP
ejde-771	23	8	the	the	DET
ejde-771	23	9	positive	positive	ADJ
ejde-771	23	10	s	s	NOUN
ejde-771	23	11	-	-	PUNCT
ejde-771	23	12	asymptotically	asymptotically	ADV
ejde-771	23	13	ω	ω	ADJ
ejde-771	23	14	-	-	ADJ
ejde-771	23	15	periodic	periodic	ADJ
ejde-771	23	16	solutions	solution	NOUN
ejde-771	23	17	to	to	ADP
ejde-771	23	18	a	a	DET
ejde-771	23	19	class	class	NOUN
ejde-771	23	20	of	of	ADP
ejde-771	23	21	semilinear	semilinear	PROPN
ejde-771	23	22	neutral	neutral	PROPN
ejde-771	23	23	caputo	caputo	PROPN
ejde-771	23	24	fractional	fractional	PROPN
ejde-771	23	25	differential	differential	ADJ
ejde-771	23	26	equations	equation	NOUN
ejde-771	23	27	with	with	ADP
ejde-771	23	28	infinite	infinite	ADJ
ejde-771	23	29	delay	delay	NOUN
ejde-771	23	30	.	.	PUNCT
ejde-771	24	1	li	li	PROPN
ejde-771	24	2	et	et	PROPN
ejde-771	24	3	al	al	PROPN
ejde-771	24	4	.	.	PUNCT
ejde-771	25	1	[	[	X
ejde-771	25	2	23	23	NUM
ejde-771	25	3	]	]	PUNCT
ejde-771	25	4	discussed	discuss	VERB
ejde-771	25	5	the	the	DET
ejde-771	25	6	asymptotically	asymptotically	ADV
ejde-771	25	7	periodic	periodic	ADJ
ejde-771	25	8	problem	problem	NOUN
ejde-771	25	9	for	for	ADP
ejde-771	25	10	the	the	DET
ejde-771	25	11	abstract	abstract	ADJ
ejde-771	25	12	fractional	fractional	ADJ
ejde-771	25	13	evolution	evolution	NOUN
ejde-771	25	14	equation	equation	NOUN
ejde-771	25	15	under	under	ADP
ejde-771	25	16	order	order	NOUN
ejde-771	25	17	conditions	condition	NOUN
ejde-771	25	18	and	and	CCONJ
ejde-771	25	19	growth	growth	NOUN
ejde-771	25	20	conditions	condition	NOUN
ejde-771	25	21	as	as	ADV
ejde-771	25	22	well	well	ADV
ejde-771	25	23	as	as	ADP
ejde-771	25	24	obtained	obtain	VERB
ejde-771	25	25	some	some	DET
ejde-771	25	26	new	new	ADJ
ejde-771	25	27	results	result	NOUN
ejde-771	25	28	on	on	ADP
ejde-771	25	29	the	the	DET
ejde-771	25	30	existence	existence	NOUN
ejde-771	25	31	of	of	ADP
ejde-771	25	32	the	the	DET
ejde-771	25	33	positive	positive	ADJ
ejde-771	25	34	s	s	NOUN
ejde-771	25	35	-	-	PUNCT
ejde-771	25	36	asymptotically	asymptotically	ADV
ejde-771	25	37	2020	2020	NUM
ejde-771	25	38	mathematics	mathematic	NOUN
ejde-771	25	39	subject	subject	ADJ
ejde-771	25	40	classification	classification	NOUN
ejde-771	25	41	.	.	PUNCT
ejde-771	26	1	35r11	35r11	NUM
ejde-771	26	2	,	,	PUNCT
ejde-771	26	3	35b09	35b09	NUM
ejde-771	26	4	,	,	PUNCT
ejde-771	26	5	34g20	34g20	NUM
ejde-771	26	6	,	,	PUNCT
ejde-771	26	7	47j35	47j35	NUM
ejde-771	26	8	.	.	PUNCT
ejde-771	27	1	key	key	ADJ
ejde-771	27	2	words	word	NOUN
ejde-771	27	3	and	and	CCONJ
ejde-771	27	4	phrases	phrase	NOUN
ejde-771	27	5	.	.	PUNCT
ejde-771	28	1	time	time	NOUN
ejde-771	28	2	-	-	PUNCT
ejde-771	28	3	space	space	NOUN
ejde-771	28	4	fractional	fractional	ADJ
ejde-771	28	5	reaction	reaction	NOUN
ejde-771	28	6	-	-	PUNCT
ejde-771	28	7	diffusion	diffusion	NOUN
ejde-771	28	8	equation	equation	NOUN
ejde-771	28	9	;	;	PUNCT
ejde-771	28	10	nonlocal	nonlocal	ADJ
ejde-771	28	11	initial	initial	ADJ
ejde-771	28	12	conditions	condition	NOUN
ejde-771	28	13	;	;	PUNCT
ejde-771	28	14	positive	positive	ADJ
ejde-771	28	15	s	s	NOUN
ejde-771	28	16	-	-	PUNCT
ejde-771	28	17	asymptotically	asymptotically	ADV
ejde-771	28	18	ω	ω	ADJ
ejde-771	28	19	-	-	ADJ
ejde-771	28	20	periodic	periodic	ADJ
ejde-771	28	21	mild	mild	ADJ
ejde-771	28	22	solutions	solution	NOUN
ejde-771	28	23	;	;	PUNCT
ejde-771	28	24	monotone	monotone	ADJ
ejde-771	28	25	iterative	iterative	NOUN
ejde-771	28	26	technique	technique	NOUN
ejde-771	28	27	.	.	PUNCT
ejde-771	29	1	©	©	PROPN
ejde-771	29	2	2025	2025	NUM
ejde-771	29	3	.	.	PUNCT
ejde-771	30	1	this	this	DET
ejde-771	30	2	work	work	NOUN
ejde-771	30	3	is	be	AUX
ejde-771	30	4	licensed	license	VERB
ejde-771	30	5	under	under	ADP
ejde-771	30	6	a	a	DET
ejde-771	30	7	cc	cc	NOUN
ejde-771	30	8	by	by	ADP
ejde-771	30	9	4.0	4.0	NUM
ejde-771	30	10	license	license	NOUN
ejde-771	30	11	.	.	PUNCT
ejde-771	31	1	submitted	submit	VERB
ejde-771	31	2	july	july	PROPN
ejde-771	31	3	13	13	NUM
ejde-771	31	4	,	,	PUNCT
ejde-771	31	5	2024	2024	NUM
ejde-771	31	6	.	.	PUNCT
ejde-771	32	1	published	publish	VERB
ejde-771	32	2	april	april	PROPN
ejde-771	32	3	24	24	NUM
ejde-771	32	4	,	,	PUNCT
ejde-771	32	5	2025	2025	NUM
ejde-771	32	6	.	.	PUNCT
ejde-771	33	1	1	1	NUM
ejde-771	33	2	2	2	NUM
ejde-771	33	3	x.	x.	NOUN
ejde-771	33	4	zhang	zhang	PROPN
ejde-771	33	5	,	,	PUNCT
ejde-771	33	6	k.	k.	PROPN
ejde-771	33	7	ding	ding	PROPN
ejde-771	33	8	,	,	PUNCT
ejde-771	33	9	p.	p.	PROPN
ejde-771	33	10	chen	chen	PROPN
ejde-771	33	11	ejde-2025/44	ejde-2025/44	PROPN
ejde-771	33	12	ω	ω	PROPN
ejde-771	33	13	-	-	ADJ
ejde-771	33	14	periodic	periodic	ADJ
ejde-771	33	15	mild	mild	ADJ
ejde-771	33	16	solutions	solution	NOUN
ejde-771	33	17	.	.	PUNCT
ejde-771	34	1	gou	gou	PROPN
ejde-771	35	1	[	[	X
ejde-771	35	2	18	18	NUM
ejde-771	35	3	]	]	PUNCT
ejde-771	35	4	investigated	investigate	VERB
ejde-771	35	5	the	the	DET
ejde-771	35	6	existence	existence	NOUN
ejde-771	35	7	of	of	ADP
ejde-771	35	8	minimal	minimal	ADJ
ejde-771	35	9	positive	positive	ADJ
ejde-771	35	10	s	s	NOUN
ejde-771	35	11	-	-	PUNCT
ejde-771	35	12	asymptotically	asymptotically	ADV
ejde-771	35	13	ω	ω	ADJ
ejde-771	35	14	-	-	ADJ
ejde-771	35	15	periodic	periodic	ADJ
ejde-771	35	16	mild	mild	ADJ
ejde-771	35	17	solution	solution	NOUN
ejde-771	35	18	for	for	ADP
ejde-771	35	19	structural	structural	ADJ
ejde-771	35	20	damped	damped	ADJ
ejde-771	35	21	elastic	elastic	ADJ
ejde-771	35	22	systems	system	NOUN
ejde-771	35	23	with	with	ADP
ejde-771	35	24	delay	delay	NOUN
ejde-771	35	25	and	and	CCONJ
ejde-771	35	26	nonlocal	nonlocal	ADJ
ejde-771	35	27	conditions	condition	NOUN
ejde-771	35	28	on	on	ADP
ejde-771	35	29	infinite	infinite	ADJ
ejde-771	35	30	interval	interval	NOUN
ejde-771	35	31	.	.	PUNCT
ejde-771	36	1	besides	besides	SCONJ
ejde-771	36	2	,	,	PUNCT
ejde-771	36	3	gou	gou	X
ejde-771	37	1	[	[	X
ejde-771	37	2	19	19	NUM
ejde-771	37	3	]	]	PUNCT
ejde-771	37	4	studied	study	VERB
ejde-771	37	5	the	the	DET
ejde-771	37	6	existence	existence	NOUN
ejde-771	37	7	of	of	ADP
ejde-771	37	8	minimal	minimal	ADJ
ejde-771	37	9	positive	positive	ADJ
ejde-771	37	10	s	s	NOUN
ejde-771	37	11	-	-	PUNCT
ejde-771	37	12	asymptotically	asymptotically	ADV
ejde-771	37	13	ω	ω	ADJ
ejde-771	37	14	-	-	ADJ
ejde-771	37	15	periodic	periodic	ADJ
ejde-771	37	16	mild	mild	ADJ
ejde-771	37	17	solution	solution	NOUN
ejde-771	37	18	for	for	ADP
ejde-771	37	19	abstract	abstract	ADJ
ejde-771	37	20	evolution	evolution	NOUN
ejde-771	37	21	equation	equation	NOUN
ejde-771	37	22	with	with	ADP
ejde-771	37	23	delay	delay	NOUN
ejde-771	37	24	on	on	ADP
ejde-771	37	25	infinite	infinite	ADJ
ejde-771	37	26	interval	interval	NOUN
ejde-771	37	27	.	.	PUNCT
ejde-771	38	1	furthermore	furthermore	ADV
ejde-771	38	2	,	,	PUNCT
ejde-771	38	3	compared	compare	VERB
ejde-771	38	4	to	to	ADP
ejde-771	38	5	the	the	DET
ejde-771	38	6	classical	classical	ADJ
ejde-771	38	7	conditions	condition	NOUN
ejde-771	38	8	,	,	PUNCT
ejde-771	38	9	the	the	DET
ejde-771	38	10	nonlocal	nonlocal	ADJ
ejde-771	38	11	initial	initial	ADJ
ejde-771	38	12	conditions	condition	NOUN
ejde-771	38	13	are	be	AUX
ejde-771	38	14	more	more	ADV
ejde-771	38	15	practical	practical	ADJ
ejde-771	38	16	when	when	SCONJ
ejde-771	38	17	describe	describe	VERB
ejde-771	38	18	some	some	DET
ejde-771	38	19	physical	physical	ADJ
ejde-771	38	20	phenomena	phenomenon	NOUN
ejde-771	38	21	.	.	PUNCT
ejde-771	39	1	it	it	PRON
ejde-771	39	2	is	be	AUX
ejde-771	39	3	worth	worth	ADJ
ejde-771	39	4	noting	note	VERB
ejde-771	39	5	that	that	SCONJ
ejde-771	39	6	there	there	PRON
ejde-771	39	7	are	be	VERB
ejde-771	39	8	many	many	ADJ
ejde-771	39	9	relevant	relevant	ADJ
ejde-771	39	10	results	result	NOUN
ejde-771	39	11	on	on	ADP
ejde-771	39	12	nonlocal	nonlocal	ADJ
ejde-771	39	13	problems	problem	NOUN
ejde-771	39	14	.	.	PUNCT
ejde-771	40	1	for	for	ADP
ejde-771	40	2	more	more	ADJ
ejde-771	40	3	details	detail	NOUN
ejde-771	40	4	of	of	ADP
ejde-771	40	5	nonlocal	nonlocal	ADJ
ejde-771	40	6	conditions	condition	NOUN
ejde-771	40	7	,	,	PUNCT
ejde-771	40	8	one	one	PRON
ejde-771	40	9	can	can	AUX
ejde-771	40	10	see	see	VERB
ejde-771	40	11	[	[	X
ejde-771	40	12	9	9	NUM
ejde-771	40	13	,	,	PUNCT
ejde-771	40	14	10	10	NUM
ejde-771	40	15	,	,	PUNCT
ejde-771	40	16	11	11	NUM
ejde-771	40	17	,	,	PUNCT
ejde-771	40	18	12	12	NUM
ejde-771	40	19	,	,	PUNCT
ejde-771	40	20	34	34	NUM
ejde-771	40	21	]	]	PUNCT
ejde-771	40	22	and	and	CCONJ
ejde-771	40	23	references	reference	NOUN
ejde-771	40	24	therein	therein	ADV
ejde-771	40	25	.	.	PUNCT
ejde-771	41	1	inspired	inspire	VERB
ejde-771	41	2	by	by	ADP
ejde-771	41	3	the	the	DET
ejde-771	41	4	above	above	ADJ
ejde-771	41	5	work	work	NOUN
ejde-771	41	6	,	,	PUNCT
ejde-771	41	7	we	we	PRON
ejde-771	41	8	are	be	AUX
ejde-771	41	9	concerned	concerned	ADJ
ejde-771	41	10	about	about	ADP
ejde-771	41	11	the	the	DET
ejde-771	41	12	positive	positive	ADJ
ejde-771	41	13	s	s	NOUN
ejde-771	41	14	-	-	PUNCT
ejde-771	41	15	asymptotically	asymptotically	ADV
ejde-771	41	16	ω	ω	ADJ
ejde-771	41	17	-	-	ADJ
ejde-771	41	18	periodic	periodic	ADJ
ejde-771	41	19	solutions	solution	NOUN
ejde-771	41	20	of	of	ADP
ejde-771	41	21	nonlocal	nonlocal	ADJ
ejde-771	41	22	problem	problem	NOUN
ejde-771	41	23	(	(	PUNCT
ejde-771	41	24	1.1	1.1	NUM
ejde-771	41	25	)	)	PUNCT
ejde-771	41	26	.	.	PUNCT
ejde-771	42	1	the	the	DET
ejde-771	42	2	organization	organization	NOUN
ejde-771	42	3	of	of	ADP
ejde-771	42	4	this	this	DET
ejde-771	42	5	paper	paper	NOUN
ejde-771	42	6	can	can	AUX
ejde-771	42	7	be	be	AUX
ejde-771	42	8	described	describe	VERB
ejde-771	42	9	as	as	ADP
ejde-771	42	10	follows	follow	VERB
ejde-771	42	11	.	.	PUNCT
ejde-771	43	1	in	in	ADP
ejde-771	43	2	the	the	DET
ejde-771	43	3	section	section	NOUN
ejde-771	43	4	2	2	NUM
ejde-771	43	5	,	,	PUNCT
ejde-771	43	6	we	we	PRON
ejde-771	43	7	collect	collect	VERB
ejde-771	43	8	some	some	DET
ejde-771	43	9	necessary	necessary	ADJ
ejde-771	43	10	definitions	definition	NOUN
ejde-771	43	11	and	and	CCONJ
ejde-771	43	12	preliminary	preliminary	ADJ
ejde-771	43	13	facts	fact	NOUN
ejde-771	43	14	.	.	PUNCT
ejde-771	44	1	in	in	ADP
ejde-771	44	2	section	section	NOUN
ejde-771	44	3	3	3	NUM
ejde-771	44	4	,	,	PUNCT
ejde-771	44	5	we	we	PRON
ejde-771	44	6	present	present	VERB
ejde-771	44	7	our	our	PRON
ejde-771	44	8	abstract	abstract	ADJ
ejde-771	44	9	results	result	NOUN
ejde-771	44	10	.	.	PUNCT
ejde-771	45	1	in	in	ADP
ejde-771	45	2	the	the	DET
ejde-771	45	3	last	last	ADJ
ejde-771	45	4	section	section	NOUN
ejde-771	45	5	,	,	PUNCT
ejde-771	45	6	applying	apply	VERB
ejde-771	45	7	our	our	PRON
ejde-771	45	8	abstract	abstract	ADJ
ejde-771	45	9	results	result	NOUN
ejde-771	45	10	to	to	ADP
ejde-771	45	11	nonlocal	nonlocal	ADJ
ejde-771	45	12	problem	problem	NOUN
ejde-771	45	13	(	(	PUNCT
ejde-771	45	14	1.1	1.1	NUM
ejde-771	45	15	)	)	PUNCT
ejde-771	45	16	,	,	PUNCT
ejde-771	45	17	we	we	PRON
ejde-771	45	18	prove	prove	VERB
ejde-771	45	19	the	the	DET
ejde-771	45	20	existence	existence	NOUN
ejde-771	45	21	of	of	ADP
ejde-771	45	22	positive	positive	ADJ
ejde-771	45	23	s	s	NOUN
ejde-771	45	24	-	-	PUNCT
ejde-771	45	25	asymptotically	asymptotically	ADV
ejde-771	45	26	ω	ω	ADJ
ejde-771	45	27	-	-	ADJ
ejde-771	45	28	periodic	periodic	ADJ
ejde-771	45	29	solutions	solution	NOUN
ejde-771	45	30	.	.	PUNCT
ejde-771	46	1	2	2	X
ejde-771	46	2	.	.	X
ejde-771	46	3	preliminaries	preliminary	NOUN
ejde-771	46	4	unless	unless	SCONJ
ejde-771	46	5	stated	state	VERB
ejde-771	46	6	otherwise	otherwise	ADV
ejde-771	46	7	,	,	PUNCT
ejde-771	46	8	we	we	PRON
ejde-771	46	9	will	will	AUX
ejde-771	46	10	assume	assume	VERB
ejde-771	46	11	that	that	SCONJ
ejde-771	46	12	(	(	PUNCT
ejde-771	46	13	e	e	NOUN
ejde-771	46	14	,	,	PUNCT
ejde-771	46	15	∥	∥	X
ejde-771	46	16	·	·	PUNCT
ejde-771	46	17	∥	∥	X
ejde-771	46	18	)	)	PUNCT
ejde-771	46	19	is	be	AUX
ejde-771	46	20	an	an	DET
ejde-771	46	21	ordered	ordered	ADJ
ejde-771	46	22	banach	banach	NOUN
ejde-771	46	23	space	space	NOUN
ejde-771	46	24	with	with	ADP
ejde-771	46	25	partial	partial	ADJ
ejde-771	46	26	order	order	NOUN
ejde-771	46	27	“	"	PUNCT
ejde-771	46	28	≤	≤	NUM
ejde-771	46	29	”	"	PUNCT
ejde-771	46	30	,	,	PUNCT
ejde-771	46	31	whose	whose	DET
ejde-771	46	32	positive	positive	ADJ
ejde-771	46	33	cone	cone	NOUN
ejde-771	46	34	p	p	NOUN
ejde-771	46	35	=	=	PUNCT
ejde-771	46	36	{	{	PUNCT
ejde-771	46	37	u	u	NOUN
ejde-771	46	38	∈	∈	PROPN
ejde-771	46	39	e	e	NOUN
ejde-771	46	40	:	:	PUNCT
ejde-771	46	41	u	u	NOUN
ejde-771	46	42	≥	≥	NOUN
ejde-771	46	43	θ	θ	X
ejde-771	46	44	}	}	PUNCT
ejde-771	46	45	is	be	AUX
ejde-771	46	46	normal	normal	ADJ
ejde-771	46	47	with	with	ADP
ejde-771	46	48	normal	normal	ADJ
ejde-771	46	49	constant	constant	ADJ
ejde-771	46	50	n	n	NOUN
ejde-771	46	51	,	,	PUNCT
ejde-771	46	52	θ	θ	PROPN
ejde-771	46	53	is	be	AUX
ejde-771	46	54	the	the	DET
ejde-771	46	55	zero	zero	NUM
ejde-771	46	56	element	element	NOUN
ejde-771	46	57	of	of	ADP
ejde-771	46	58	e.	e.	PROPN
ejde-771	46	59	combining	combine	VERB
ejde-771	46	60	property	property	NOUN
ejde-771	46	61	of	of	ADP
ejde-771	46	62	exponential	exponential	ADJ
ejde-771	46	63	functions	function	NOUN
ejde-771	46	64	,	,	PUNCT
ejde-771	46	65	define	define	VERB
ejde-771	46	66	a	a	DET
ejde-771	46	67	banach	banach	NOUN
ejde-771	46	68	space	space	NOUN
ejde-771	46	69	ce([0,∞	ce([0,∞	NUM
ejde-771	46	70	)	)	PUNCT
ejde-771	46	71	,	,	PUNCT
ejde-771	47	1	e	e	X
ejde-771	47	2	)	)	PUNCT
ejde-771	47	3	=	=	SYM
ejde-771	47	4	{	{	PUNCT
ejde-771	47	5	u	u	NOUN
ejde-771	47	6	∈	∈	PROPN
ejde-771	47	7	c([0,∞	c([0,∞	PROPN
ejde-771	47	8	)	)	PUNCT
ejde-771	47	9	,	,	PUNCT
ejde-771	48	1	e	e	X
ejde-771	48	2	)	)	PUNCT
ejde-771	48	3	:	:	PUNCT
ejde-771	49	1	lim	lim	PROPN
ejde-771	49	2	t→∞	t→∞	X
ejde-771	49	3	e−t∥u(t)∥	e−t∥u(t)∥	PROPN
ejde-771	49	4	=	=	PUNCT
ejde-771	49	5	0	0	NUM
ejde-771	49	6	}	}	PUNCT
ejde-771	49	7	with	with	ADP
ejde-771	49	8	the	the	DET
ejde-771	49	9	norm	norm	NOUN
ejde-771	49	10	∥	∥	X
ejde-771	49	11	·	·	PUNCT
ejde-771	49	12	∥e	∥e	PROPN
ejde-771	49	13	=	=	SYM
ejde-771	49	14	supt∈r+	supt∈r+	NOUN
ejde-771	49	15	e−t∥u(t)∥.	e−t∥u(t)∥.	NOUN
ejde-771	49	16	we	we	PRON
ejde-771	49	17	define	define	VERB
ejde-771	49	18	a	a	DET
ejde-771	49	19	positive	positive	ADJ
ejde-771	49	20	cone	cone	NOUN
ejde-771	49	21	pe	pe	X
ejde-771	49	22	⊂	⊂	PROPN
ejde-771	49	23	ce(e	ce(e	ADV
ejde-771	49	24	)	)	PUNCT
ejde-771	49	25	by	by	ADP
ejde-771	49	26	pe	pe	PROPN
ejde-771	49	27	=	=	SYM
ejde-771	49	28	{	{	PUNCT
ejde-771	49	29	u	u	NOUN
ejde-771	49	30	∈	∈	NOUN
ejde-771	49	31	ce(e	ce(e	NOUN
ejde-771	49	32	)	)	PUNCT
ejde-771	49	33	:	:	PUNCT
ejde-771	49	34	u(t	u(t	NOUN
ejde-771	49	35	)	)	PUNCT
ejde-771	49	36	≥	≥	NUM
ejde-771	49	37	θ	θ	PROPN
ejde-771	49	38	,	,	PUNCT
ejde-771	49	39	t	t	PROPN
ejde-771	49	40	∈	∈	PROPN
ejde-771	50	1	[	[	X
ejde-771	50	2	0,∞	0,∞	NOUN
ejde-771	50	3	)	)	PUNCT
ejde-771	50	4	}	}	PUNCT
ejde-771	50	5	.	.	PUNCT
ejde-771	51	1	then	then	ADV
ejde-771	51	2	,	,	PUNCT
ejde-771	51	3	pe	pe	PROPN
ejde-771	51	4	is	be	AUX
ejde-771	51	5	normal	normal	ADJ
ejde-771	51	6	and	and	CCONJ
ejde-771	51	7	ce(e	ce(e	NUM
ejde-771	51	8	)	)	PUNCT
ejde-771	51	9	is	be	AUX
ejde-771	51	10	an	an	DET
ejde-771	51	11	ordered	ordered	ADJ
ejde-771	51	12	banach	banach	NOUN
ejde-771	51	13	space	space	NOUN
ejde-771	51	14	,	,	PUNCT
ejde-771	51	15	whose	whose	DET
ejde-771	51	16	partial	partial	ADJ
ejde-771	51	17	order	order	NOUN
ejde-771	51	18	”	"	PUNCT
ejde-771	51	19	≤	≤	NUM
ejde-771	51	20	”	"	PUNCT
ejde-771	51	21	is	be	AUX
ejde-771	51	22	induced	induce	VERB
ejde-771	51	23	by	by	ADP
ejde-771	51	24	the	the	DET
ejde-771	51	25	cone	cone	NOUN
ejde-771	51	26	pe	pe	NOUN
ejde-771	51	27	.	.	PUNCT
ejde-771	52	1	now	now	ADV
ejde-771	52	2	,	,	PUNCT
ejde-771	52	3	we	we	PRON
ejde-771	52	4	present	present	VERB
ejde-771	52	5	an	an	DET
ejde-771	52	6	important	important	ADJ
ejde-771	52	7	result	result	NOUN
ejde-771	52	8	that	that	PRON
ejde-771	52	9	will	will	AUX
ejde-771	52	10	play	play	VERB
ejde-771	52	11	an	an	DET
ejde-771	52	12	important	important	ADJ
ejde-771	52	13	role	role	NOUN
ejde-771	52	14	in	in	ADP
ejde-771	52	15	the	the	DET
ejde-771	52	16	subsequent	subsequent	ADJ
ejde-771	52	17	proof	proof	NOUN
ejde-771	52	18	.	.	PUNCT
ejde-771	53	1	lemma	lemma	PROPN
ejde-771	53	2	2.1	2.1	NUM
ejde-771	53	3	(	(	PUNCT
ejde-771	53	4	[	[	X
ejde-771	53	5	8	8	NUM
ejde-771	53	6	]	]	NUM
ejde-771	53	7	)	)	PUNCT
ejde-771	53	8	.	.	PUNCT
ejde-771	54	1	the	the	DET
ejde-771	54	2	set	set	NOUN
ejde-771	54	3	ξ	ξ	X
ejde-771	54	4	⊂	⊂	PROPN
ejde-771	54	5	ce([0,∞	ce([0,∞	NUM
ejde-771	54	6	)	)	PUNCT
ejde-771	54	7	,	,	PUNCT
ejde-771	54	8	e	e	X
ejde-771	54	9	)	)	PUNCT
ejde-771	54	10	is	be	AUX
ejde-771	54	11	relatively	relatively	ADV
ejde-771	54	12	compact	compact	ADJ
ejde-771	54	13	if	if	SCONJ
ejde-771	55	1	and	and	CCONJ
ejde-771	55	2	only	only	ADV
ejde-771	55	3	if	if	SCONJ
ejde-771	55	4	the	the	DET
ejde-771	55	5	following	follow	VERB
ejde-771	55	6	conditions	condition	NOUN
ejde-771	55	7	hold	hold	VERB
ejde-771	55	8	:	:	PUNCT
ejde-771	55	9	(	(	PUNCT
ejde-771	55	10	a	a	X
ejde-771	55	11	)	)	PUNCT
ejde-771	55	12	for	for	ADP
ejde-771	55	13	each	each	DET
ejde-771	55	14	a	a	DET
ejde-771	55	15	>	>	X
ejde-771	55	16	0	0	NUM
ejde-771	55	17	,	,	PUNCT
ejde-771	55	18	the	the	DET
ejde-771	55	19	set	set	NOUN
ejde-771	55	20	ξ	ξ	PROPN
ejde-771	55	21	is	be	AUX
ejde-771	55	22	equicontinuous	equicontinuous	ADJ
ejde-771	55	23	on	on	ADP
ejde-771	55	24	[	[	X
ejde-771	55	25	0	0	NUM
ejde-771	55	26	,	,	PUNCT
ejde-771	55	27	a	a	PRON
ejde-771	55	28	]	]	X
ejde-771	55	29	;	;	PUNCT
ejde-771	55	30	(	(	PUNCT
ejde-771	55	31	b	b	X
ejde-771	55	32	)	)	PUNCT
ejde-771	55	33	for	for	ADP
ejde-771	55	34	any	any	DET
ejde-771	55	35	t	t	NOUN
ejde-771	55	36	∈	∈	PROPN
ejde-771	56	1	[	[	X
ejde-771	56	2	0,∞	0,∞	NOUN
ejde-771	56	3	)	)	PUNCT
ejde-771	56	4	,	,	PUNCT
ejde-771	56	5	ξ(t	ξ(t	PROPN
ejde-771	56	6	)	)	PUNCT
ejde-771	56	7	=	=	SYM
ejde-771	56	8	{	{	PUNCT
ejde-771	56	9	u(t	u(t	PROPN
ejde-771	56	10	)	)	PUNCT
ejde-771	56	11	:	:	PUNCT
ejde-771	56	12	u	u	PROPN
ejde-771	56	13	∈	∈	PROPN
ejde-771	56	14	ξ	ξ	PROPN
ejde-771	56	15	}	}	PUNCT
ejde-771	56	16	is	be	AUX
ejde-771	56	17	relatively	relatively	ADV
ejde-771	56	18	compact	compact	ADJ
ejde-771	56	19	in	in	ADP
ejde-771	56	20	e	e	NOUN
ejde-771	56	21	;	;	PUNCT
ejde-771	56	22	(	(	PUNCT
ejde-771	56	23	c	c	X
ejde-771	56	24	)	)	PUNCT
ejde-771	56	25	limt→∞	limt→∞	ADP
ejde-771	56	26	e−t∥u(t)∥	e−t∥u(t)∥	X
ejde-771	56	27	=	=	PUNCT
ejde-771	56	28	0	0	PUNCT
ejde-771	56	29	uniformly	uniformly	ADV
ejde-771	56	30	for	for	ADP
ejde-771	56	31	u	u	PROPN
ejde-771	56	32	∈	∈	PROPN
ejde-771	56	33	ξ	ξ	PROPN
ejde-771	56	34	.	.	PUNCT
ejde-771	57	1	next	next	ADV
ejde-771	57	2	,	,	PUNCT
ejde-771	57	3	let	let	VERB
ejde-771	57	4	a	a	DET
ejde-771	57	5	:	:	PUNCT
ejde-771	57	6	d(a	d(a	PROPN
ejde-771	57	7	)	)	PUNCT
ejde-771	58	1	⊂	⊂	PROPN
ejde-771	58	2	e	e	X
ejde-771	58	3	→	→	SYM
ejde-771	58	4	e	e	X
ejde-771	58	5	and	and	CCONJ
ejde-771	58	6	−a	−a	NOUN
ejde-771	58	7	generates	generate	VERB
ejde-771	58	8	an	an	DET
ejde-771	58	9	exponentially	exponentially	ADV
ejde-771	58	10	stable	stable	ADJ
ejde-771	58	11	analytic	analytic	ADJ
ejde-771	58	12	semigroup	semigroup	NOUN
ejde-771	58	13	t	t	PROPN
ejde-771	58	14	(	(	PUNCT
ejde-771	58	15	t)(t	t)(t	ADJ
ejde-771	58	16	≥	≥	NOUN
ejde-771	58	17	0	0	NUM
ejde-771	58	18	)	)	PUNCT
ejde-771	58	19	in	in	ADP
ejde-771	58	20	e.	e.	PROPN
ejde-771	58	21	as	as	SCONJ
ejde-771	58	22	we	we	PRON
ejde-771	58	23	all	all	PRON
ejde-771	58	24	know	know	VERB
ejde-771	58	25	,	,	PUNCT
ejde-771	58	26	for	for	ADP
ejde-771	58	27	a	a	DET
ejde-771	58	28	general	general	ADJ
ejde-771	58	29	c0	c0	PROPN
ejde-771	58	30	-	-	PUNCT
ejde-771	58	31	semigroup	semigroup	PROPN
ejde-771	58	32	,	,	PUNCT
ejde-771	58	33	there	there	PRON
ejde-771	58	34	exist	exist	VERB
ejde-771	58	35	constants	constant	NOUN
ejde-771	58	36	m	m	VERB
ejde-771	58	37	≥	≥	NOUN
ejde-771	58	38	1	1	NUM
ejde-771	58	39	and	and	CCONJ
ejde-771	58	40	ν	ν	NOUN
ejde-771	58	41	∈	∈	NOUN
ejde-771	58	42	r	r	NOUN
ejde-771	59	1	such	such	ADJ
ejde-771	59	2	that	that	PRON
ejde-771	59	3	∥t	∥t	PROPN
ejde-771	59	4	(	(	PUNCT
ejde-771	59	5	t)∥	t)∥	NUM
ejde-771	59	6	≤meνt	≤meνt	ADV
ejde-771	59	7	,	,	PUNCT
ejde-771	59	8	t	t	PROPN
ejde-771	59	9	≥	≥	NUM
ejde-771	59	10	0	0	NUM
ejde-771	59	11	.	.	PUNCT
ejde-771	60	1	in	in	ADP
ejde-771	60	2	particular	particular	ADJ
ejde-771	60	3	,	,	PUNCT
ejde-771	60	4	let	let	VERB
ejde-771	60	5	growth	growth	NOUN
ejde-771	60	6	exponent	exponent	NOUN
ejde-771	60	7	ν0	ν0	PROPN
ejde-771	60	8	:	:	PUNCT
ejde-771	60	9	=	=	SYM
ejde-771	60	10	inf{ν	inf{ν	PROPN
ejde-771	60	11	∈	∈	NOUN
ejde-771	60	12	r	r	NOUN
ejde-771	60	13	:	:	PUNCT
ejde-771	60	14	∃m	∃m	PROPN
ejde-771	60	15	≥	≥	NOUN
ejde-771	60	16	1	1	NUM
ejde-771	60	17	such	such	ADJ
ejde-771	60	18	that	that	PRON
ejde-771	60	19	∥t	∥t	PROPN
ejde-771	60	20	(	(	PUNCT
ejde-771	60	21	t)∥	t)∥	NUM
ejde-771	60	22	≤meνt	≤meνt	ADV
ejde-771	60	23	,	,	PUNCT
ejde-771	60	24	t	t	PROPN
ejde-771	60	25	≥	≥	NUM
ejde-771	60	26	0	0	NUM
ejde-771	60	27	}	}	PUNCT
ejde-771	60	28	<	<	X
ejde-771	60	29	0	0	NUM
ejde-771	60	30	,	,	PUNCT
ejde-771	60	31	the	the	DET
ejde-771	60	32	semigroup	semigroup	PROPN
ejde-771	60	33	t	t	PROPN
ejde-771	60	34	(	(	PUNCT
ejde-771	60	35	t)(t	t)(t	ADJ
ejde-771	60	36	≥	≥	NOUN
ejde-771	60	37	0	0	NUM
ejde-771	60	38	)	)	PUNCT
ejde-771	60	39	is	be	AUX
ejde-771	60	40	said	say	VERB
ejde-771	60	41	to	to	PART
ejde-771	60	42	be	be	AUX
ejde-771	60	43	exponentially	exponentially	ADV
ejde-771	60	44	stable	stable	ADJ
ejde-771	60	45	.	.	PUNCT
ejde-771	61	1	it	it	PRON
ejde-771	61	2	is	be	AUX
ejde-771	61	3	well	well	ADV
ejde-771	61	4	known	known	ADJ
ejde-771	61	5	[	[	X
ejde-771	61	6	30	30	NUM
ejde-771	61	7	]	]	PUNCT
ejde-771	61	8	that	that	SCONJ
ejde-771	61	9	if	if	SCONJ
ejde-771	61	10	the	the	DET
ejde-771	61	11	semigroup	semigroup	PROPN
ejde-771	61	12	t	t	PROPN
ejde-771	61	13	(	(	PUNCT
ejde-771	61	14	t	t	PROPN
ejde-771	61	15	)	)	PUNCT
ejde-771	61	16	is	be	AUX
ejde-771	61	17	continuous	continuous	ADJ
ejde-771	61	18	in	in	ADP
ejde-771	61	19	the	the	DET
ejde-771	61	20	uniform	uniform	ADJ
ejde-771	61	21	operator	operator	NOUN
ejde-771	61	22	topology	topology	NOUN
ejde-771	61	23	for	for	ADP
ejde-771	61	24	t	t	PROPN
ejde-771	61	25	>	>	X
ejde-771	61	26	0	0	PUNCT
ejde-771	62	1	in	in	ADP
ejde-771	62	2	e	e	NOUN
ejde-771	62	3	,	,	PUNCT
ejde-771	62	4	then	then	ADV
ejde-771	62	5	ν0	ν0	PROPN
ejde-771	62	6	can	can	AUX
ejde-771	62	7	obtained	obtain	VERB
ejde-771	62	8	by	by	ADP
ejde-771	62	9	the	the	DET
ejde-771	62	10	spectrum	spectrum	NOUN
ejde-771	62	11	σ(a	σ(a	PROPN
ejde-771	62	12	)	)	PUNCT
ejde-771	62	13	of	of	ADP
ejde-771	62	14	the	the	DET
ejde-771	62	15	operator	operator	NOUN
ejde-771	62	16	a	a	DET
ejde-771	62	17	,	,	PUNCT
ejde-771	62	18	ν0	ν0	PROPN
ejde-771	62	19	=	=	SYM
ejde-771	62	20	−	−	PROPN
ejde-771	62	21	inf{reλ	inf{reλ	NOUN
ejde-771	62	22	|	|	ADV
ejde-771	62	23	λ	λ	PROPN
ejde-771	62	24	∈	∈	PROPN
ejde-771	62	25	σ(a	σ(a	PROPN
ejde-771	62	26	)	)	PUNCT
ejde-771	62	27	}	}	PUNCT
ejde-771	62	28	.	.	PUNCT
ejde-771	63	1	(	(	PUNCT
ejde-771	63	2	2.1	2.1	NUM
ejde-771	63	3	)	)	PUNCT
ejde-771	63	4	by	by	ADP
ejde-771	63	5	blakrishnan	blakrishnan	PROPN
ejde-771	63	6	’s	’s	PART
ejde-771	63	7	definition	definition	NOUN
ejde-771	63	8	[	[	X
ejde-771	63	9	4	4	NUM
ejde-771	63	10	,	,	PUNCT
ejde-771	63	11	35	35	NUM
ejde-771	63	12	]	]	PUNCT
ejde-771	63	13	,	,	PUNCT
ejde-771	63	14	the	the	DET
ejde-771	63	15	fractional	fractional	ADJ
ejde-771	63	16	power	power	NOUN
ejde-771	63	17	aβ	aβ	NOUN
ejde-771	63	18	is	be	AUX
ejde-771	63	19	well	well	ADV
ejde-771	63	20	defined	define	VERB
ejde-771	63	21	as	as	ADP
ejde-771	63	22	aβu	aβu	NOUN
ejde-771	63	23	:	:	PUNCT
ejde-771	63	24	=	=	NOUN
ejde-771	63	25	sin(βπ	sin(βπ	X
ejde-771	63	26	)	)	PUNCT
ejde-771	64	1	π	π	NOUN
ejde-771	64	2	∫	∫	PROPN
ejde-771	64	3	∞	∞	NUM
ejde-771	64	4	0	0	NUM
ejde-771	64	5	λβ−1(λ	λβ−1(λ	PROPN
ejde-771	64	6	i	i	PRON
ejde-771	64	7	+	+	NOUN
ejde-771	64	8	a)−1audλ	a)−1audλ	ADJ
ejde-771	64	9	,	,	PUNCT
ejde-771	64	10	0	0	PUNCT
ejde-771	64	11	<	<	X
ejde-771	64	12	β	β	X
ejde-771	64	13	<	<	X
ejde-771	64	14	1	1	NUM
ejde-771	64	15	,	,	PUNCT
ejde-771	64	16	u	u	PROPN
ejde-771	64	17	∈	∈	PROPN
ejde-771	64	18	d(a	d(a	PROPN
ejde-771	64	19	)	)	PUNCT
ejde-771	64	20	.	.	PUNCT
ejde-771	65	1	(	(	PUNCT
ejde-771	65	2	2.2	2.2	NUM
ejde-771	65	3	)	)	PUNCT
ejde-771	65	4	from	from	ADP
ejde-771	65	5	[	[	X
ejde-771	65	6	35	35	NUM
ejde-771	65	7	]	]	SYM
ejde-771	65	8	one	one	PRON
ejde-771	65	9	know	know	VERB
ejde-771	65	10	that	that	SCONJ
ejde-771	65	11	−aβ	−aβ	PROPN
ejde-771	65	12	is	be	AUX
ejde-771	65	13	a	a	DET
ejde-771	65	14	closed	closed	ADJ
ejde-771	65	15	densely	densely	ADV
ejde-771	65	16	defined	define	VERB
ejde-771	65	17	operator	operator	NOUN
ejde-771	65	18	,	,	PUNCT
ejde-771	65	19	which	which	PRON
ejde-771	65	20	generates	generate	VERB
ejde-771	65	21	a	a	DET
ejde-771	65	22	bounded	bounded	ADJ
ejde-771	65	23	analytic	analytic	ADJ
ejde-771	65	24	semigroup	semigroup	NOUN
ejde-771	65	25	tβ(t)(t	tβ(t)(t	X
ejde-771	65	26	≥	≥	NOUN
ejde-771	65	27	0	0	NUM
ejde-771	65	28	)	)	PUNCT
ejde-771	65	29	,	,	PUNCT
ejde-771	65	30	which	which	PRON
ejde-771	65	31	can	can	AUX
ejde-771	65	32	be	be	AUX
ejde-771	65	33	expressed	express	VERB
ejde-771	65	34	as	as	ADP
ejde-771	65	35	tβ(t	tβ(t	NOUN
ejde-771	65	36	)	)	PUNCT
ejde-771	66	1	=	=	SYM
ejde-771	66	2	∫	∫	PROPN
ejde-771	67	1	∞	∞	NOUN
ejde-771	67	2	0	0	X
ejde-771	68	1	gβ	gβ	NOUN
ejde-771	68	2	,	,	PUNCT
ejde-771	68	3	t(s)t	t(s)t	PROPN
ejde-771	68	4	(	(	PUNCT
ejde-771	68	5	s)ds	s)ds	PROPN
ejde-771	68	6	,	,	PUNCT
ejde-771	68	7	t	t	PROPN
ejde-771	68	8	>	>	X
ejde-771	68	9	0	0	PROPN
ejde-771	68	10	,	,	PUNCT
ejde-771	68	11	ejde-2025/44	ejde-2025/44	ADJ
ejde-771	68	12	time	time	NOUN
ejde-771	68	13	-	-	PUNCT
ejde-771	68	14	space	space	NOUN
ejde-771	68	15	fractional	fractional	ADJ
ejde-771	68	16	reaction	reaction	NOUN
ejde-771	68	17	-	-	PUNCT
ejde-771	68	18	diffusion	diffusion	NOUN
ejde-771	68	19	equations	equation	NOUN
ejde-771	68	20	3	3	NUM
ejde-771	68	21	where	where	SCONJ
ejde-771	68	22	gβ	gβ	PROPN
ejde-771	68	23	,	,	PUNCT
ejde-771	68	24	t	t	PROPN
ejde-771	68	25	(	(	PUNCT
ejde-771	68	26	·	·	PUNCT
ejde-771	68	27	)	)	PUNCT
ejde-771	68	28	is	be	AUX
ejde-771	68	29	defined	define	VERB
ejde-771	68	30	by	by	ADP
ejde-771	68	31	the	the	DET
ejde-771	68	32	inverse	inverse	ADJ
ejde-771	68	33	laplace	laplace	NOUN
ejde-771	68	34	integral	integral	ADJ
ejde-771	68	35	gβ	gβ	NOUN
ejde-771	68	36	,	,	PUNCT
ejde-771	68	37	t(s	t(s	PROPN
ejde-771	68	38	)	)	PUNCT
ejde-771	68	39	=	=	SYM
ejde-771	68	40	1	1	NUM
ejde-771	68	41	2πi	2πi	ADJ
ejde-771	68	42	∫	∫	PROPN
ejde-771	68	43	σ+i∞	σ+i∞	PROPN
ejde-771	68	44	σ−i∞	σ−i∞	PROPN
ejde-771	68	45	ezs−tzβ	ezs−tzβ	PROPN
ejde-771	68	46	dz	dz	PROPN
ejde-771	68	47	,	,	PUNCT
ejde-771	68	48	σ	σ	PROPN
ejde-771	68	49	>	>	X
ejde-771	68	50	0	0	NUM
ejde-771	68	51	,	,	PUNCT
ejde-771	68	52	and	and	CCONJ
ejde-771	68	53	the	the	DET
ejde-771	68	54	brach	brach	PROPN
ejde-771	68	55	of	of	ADP
ejde-771	68	56	zβ	zβ	PROPN
ejde-771	68	57	is	be	AUX
ejde-771	68	58	so	so	ADV
ejde-771	68	59	taken	take	VERB
ejde-771	68	60	that	that	DET
ejde-771	68	61	re(zβ	re(zβ	PROPN
ejde-771	68	62	)	)	PUNCT
ejde-771	68	63	>	>	X
ejde-771	68	64	0	0	PUNCT
ejde-771	68	65	for	for	ADP
ejde-771	68	66	re(z	re(z	NOUN
ejde-771	68	67	)	)	PUNCT
ejde-771	68	68	>	>	X
ejde-771	69	1	0	0	X
ejde-771	69	2	.	.	PUNCT
ejde-771	70	1	moreover	moreover	ADV
ejde-771	70	2	,	,	PUNCT
ejde-771	70	3	one	one	PRON
ejde-771	70	4	can	can	AUX
ejde-771	70	5	see	see	VERB
ejde-771	70	6	gβ	gβ	PROPN
ejde-771	70	7	,	,	PUNCT
ejde-771	70	8	t(s	t(s	PROPN
ejde-771	70	9	)	)	PUNCT
ejde-771	70	10	≥	≥	NOUN
ejde-771	70	11	0	0	NUM
ejde-771	70	12	for	for	ADP
ejde-771	70	13	s	s	PROPN
ejde-771	70	14	>	>	X
ejde-771	70	15	0	0	PROPN
ejde-771	70	16	and	and	CCONJ
ejde-771	70	17	∫∞	∫∞	NOUN
ejde-771	70	18	0	0	PUNCT
ejde-771	71	1	gβ	gβ	NOUN
ejde-771	71	2	,	,	PUNCT
ejde-771	71	3	t(s)ds	t(s)ds	NOUN
ejde-771	71	4	=	=	NOUN
ejde-771	71	5	1	1	NUM
ejde-771	71	6	in	in	ADP
ejde-771	71	7	[	[	X
ejde-771	71	8	35	35	NUM
ejde-771	71	9	]	]	PUNCT
ejde-771	71	10	.	.	PUNCT
ejde-771	72	1	it	it	PRON
ejde-771	72	2	is	be	AUX
ejde-771	72	3	valuable	valuable	ADJ
ejde-771	72	4	to	to	PART
ejde-771	72	5	note	note	VERB
ejde-771	72	6	that	that	SCONJ
ejde-771	72	7	there	there	PRON
ejde-771	72	8	is	be	VERB
ejde-771	72	9	an	an	DET
ejde-771	72	10	important	important	ADJ
ejde-771	72	11	lemma	lemma	NOUN
ejde-771	72	12	about	about	ADP
ejde-771	72	13	tβ(t	tβ(t	NUM
ejde-771	72	14	)	)	PUNCT
ejde-771	72	15	.	.	PUNCT
ejde-771	73	1	lemma	lemma	PROPN
ejde-771	73	2	2.2	2.2	NUM
ejde-771	73	3	(	(	PUNCT
ejde-771	73	4	[	[	X
ejde-771	73	5	24	24	NUM
ejde-771	73	6	]	]	PUNCT
ejde-771	73	7	)	)	PUNCT
ejde-771	73	8	.	.	PUNCT
ejde-771	74	1	if	if	SCONJ
ejde-771	74	2	the	the	DET
ejde-771	74	3	semigroup	semigroup	PROPN
ejde-771	74	4	t	t	PROPN
ejde-771	74	5	(	(	PUNCT
ejde-771	74	6	t)(t	t)(t	ADJ
ejde-771	74	7	≥	≥	NOUN
ejde-771	74	8	0	0	NUM
ejde-771	74	9	)	)	PUNCT
ejde-771	74	10	generated	generate	VERB
ejde-771	74	11	by	by	ADP
ejde-771	74	12	−a	−a	NOUN
ejde-771	74	13	is	be	AUX
ejde-771	74	14	exponentially	exponentially	ADV
ejde-771	74	15	stable	stable	ADJ
ejde-771	74	16	and	and	CCONJ
ejde-771	74	17	compact	compact	ADJ
ejde-771	74	18	,	,	PUNCT
ejde-771	74	19	then	then	ADV
ejde-771	74	20	the	the	DET
ejde-771	74	21	semigroup	semigroup	PROPN
ejde-771	74	22	tβ(t)(t	tβ(t)(t	X
ejde-771	74	23	≥	≥	NOUN
ejde-771	74	24	0	0	NUM
ejde-771	74	25	)	)	PUNCT
ejde-771	74	26	generated	generate	VERB
ejde-771	74	27	by	by	ADP
ejde-771	74	28	−aβ	−aβ	PROPN
ejde-771	74	29	is	be	AUX
ejde-771	74	30	exponentially	exponentially	ADV
ejde-771	74	31	stable	stable	ADJ
ejde-771	74	32	and	and	CCONJ
ejde-771	74	33	compact	compact	ADJ
ejde-771	74	34	.	.	PUNCT
ejde-771	75	1	in	in	ADP
ejde-771	75	2	the	the	DET
ejde-771	75	3	following	following	NOUN
ejde-771	75	4	,	,	PUNCT
ejde-771	75	5	consider	consider	VERB
ejde-771	75	6	a	a	DET
ejde-771	75	7	probability	probability	NOUN
ejde-771	75	8	density	density	NOUN
ejde-771	75	9	function	function	NOUN
ejde-771	75	10	hα(τ	hα(τ	PRON
ejde-771	75	11	)	)	PUNCT
ejde-771	75	12	defined	define	VERB
ejde-771	75	13	by	by	ADP
ejde-771	75	14	hα(τ	hα(τ	PRON
ejde-771	75	15	)	)	PUNCT
ejde-771	75	16	=	=	SYM
ejde-771	76	1	1	1	NUM
ejde-771	76	2	πα	πα	ADP
ejde-771	76	3	∞∑	∞∑	NUM
ejde-771	76	4	n=1	n=1	PROPN
ejde-771	76	5	(	(	PUNCT
ejde-771	76	6	−τ)n−1γ(nα+	−τ)n−1γ(nα+	PROPN
ejde-771	76	7	1	1	NUM
ejde-771	76	8	)	)	PUNCT
ejde-771	76	9	n	n	CCONJ
ejde-771	76	10	!	!	PUNCT
ejde-771	76	11	sin(nπα	sin(nπα	PROPN
ejde-771	76	12	)	)	PUNCT
ejde-771	76	13	,	,	PUNCT
ejde-771	76	14	τ	τ	PROPN
ejde-771	76	15	∈	∈	PROPN
ejde-771	76	16	(	(	PUNCT
ejde-771	76	17	0,∞	0,∞	NOUN
ejde-771	76	18	)	)	PUNCT
ejde-771	76	19	.	.	PUNCT
ejde-771	77	1	obviously	obviously	ADV
ejde-771	77	2	,	,	PUNCT
ejde-771	77	3	hα(τ	hα(τ	CCONJ
ejde-771	77	4	)	)	PUNCT
ejde-771	77	5	≥	≥	NOUN
ejde-771	77	6	0	0	NUM
ejde-771	77	7	,	,	PUNCT
ejde-771	77	8	∫	∫	PROPN
ejde-771	77	9	∞	∞	PROPN
ejde-771	77	10	0	0	NUM
ejde-771	78	1	hα(τ)dτ	hα(τ)dτ	ADJ
ejde-771	78	2	=	=	SYM
ejde-771	78	3	1	1	NUM
ejde-771	78	4	,	,	PUNCT
ejde-771	78	5	∫	∫	PROPN
ejde-771	78	6	∞	∞	PROPN
ejde-771	78	7	0	0	NUM
ejde-771	78	8	τhα(τ)dτ	τhα(τ)dτ	PUNCT
ejde-771	79	1	=	=	SYM
ejde-771	79	2	1	1	NUM
ejde-771	79	3	γ(1	γ(1	NOUN
ejde-771	79	4	+	+	CCONJ
ejde-771	79	5	α	α	X
ejde-771	79	6	)	)	PUNCT
ejde-771	79	7	,	,	PUNCT
ejde-771	79	8	τ	τ	PROPN
ejde-771	79	9	∈	∈	PROPN
ejde-771	79	10	(	(	PUNCT
ejde-771	79	11	0,∞	0,∞	NOUN
ejde-771	79	12	)	)	PUNCT
ejde-771	79	13	.	.	PUNCT
ejde-771	80	1	(	(	PUNCT
ejde-771	80	2	2.3	2.3	NUM
ejde-771	80	3	)	)	PUNCT
ejde-771	80	4	based	base	VERB
ejde-771	80	5	on	on	ADP
ejde-771	80	6	thesestatements	thesestatement	NOUN
ejde-771	80	7	,	,	PUNCT
ejde-771	80	8	for	for	ADP
ejde-771	80	9	t	t	PROPN
ejde-771	80	10	≥	≥	NOUN
ejde-771	80	11	0	0	NUM
ejde-771	80	12	,	,	PUNCT
ejde-771	80	13	we	we	PRON
ejde-771	80	14	define	define	VERB
ejde-771	80	15	the	the	DET
ejde-771	80	16	two	two	NUM
ejde-771	80	17	operators	operator	NOUN
ejde-771	80	18	:	:	PUNCT
ejde-771	80	19	jα	jα	NOUN
ejde-771	80	20	,	,	PUNCT
ejde-771	80	21	β(t	β(t	PROPN
ejde-771	80	22	)	)	PUNCT
ejde-771	80	23	=	=	SYM
ejde-771	81	1	∫	∫	PROPN
ejde-771	82	1	∞	∞	NUM
ejde-771	82	2	0	0	NUM
ejde-771	83	1	hα(τ)tβ(t	hα(τ)tβ(t	PROPN
ejde-771	83	2	ατ)dτ	ατ)dτ	PROPN
ejde-771	83	3	,	,	PUNCT
ejde-771	83	4	kα	kα	PROPN
ejde-771	83	5	,	,	PUNCT
ejde-771	83	6	β(t	β(t	PROPN
ejde-771	83	7	)	)	PUNCT
ejde-771	83	8	=	=	PUNCT
ejde-771	84	1	α	α	X
ejde-771	84	2	∫	∫	PROPN
ejde-771	84	3	∞	∞	PROPN
ejde-771	84	4	0	0	NUM
ejde-771	84	5	τhα(τ)tβ(t	τhα(τ)tβ(t	PRON
ejde-771	85	1	ατ)dτ	ατ)dτ	PROPN
ejde-771	85	2	.	.	PUNCT
ejde-771	86	1	similar	similar	ADJ
ejde-771	86	2	to	to	ADP
ejde-771	86	3	the	the	DET
ejde-771	86	4	proof	proof	NOUN
ejde-771	86	5	in	in	ADP
ejde-771	86	6	[	[	X
ejde-771	86	7	1	1	NUM
ejde-771	86	8	,	,	PUNCT
ejde-771	86	9	13	13	NUM
ejde-771	86	10	,	,	PUNCT
ejde-771	86	11	32	32	NUM
ejde-771	86	12	,	,	PUNCT
ejde-771	86	13	36	36	NUM
ejde-771	86	14	]	]	PUNCT
ejde-771	86	15	,	,	PUNCT
ejde-771	86	16	one	one	PRON
ejde-771	86	17	has	have	VERB
ejde-771	86	18	the	the	DET
ejde-771	86	19	following	follow	VERB
ejde-771	86	20	results	result	NOUN
ejde-771	86	21	.	.	PUNCT
ejde-771	87	1	lemma	lemma	PROPN
ejde-771	87	2	2.3	2.3	NUM
ejde-771	87	3	.	.	PUNCT
ejde-771	88	1	the	the	DET
ejde-771	88	2	operators	operator	NOUN
ejde-771	88	3	jα	jα	NOUN
ejde-771	88	4	,	,	PUNCT
ejde-771	88	5	β(t)(t	β(t)(t	PUNCT
ejde-771	88	6	≥	≥	NOUN
ejde-771	88	7	0	0	NUM
ejde-771	88	8	)	)	PUNCT
ejde-771	88	9	and	and	CCONJ
ejde-771	88	10	kα	kα	PROPN
ejde-771	88	11	,	,	PUNCT
ejde-771	88	12	β(t)(t	β(t)(t	PUNCT
ejde-771	88	13	≥	≥	NOUN
ejde-771	88	14	0	0	NUM
ejde-771	88	15	)	)	PUNCT
ejde-771	88	16	have	have	VERB
ejde-771	88	17	the	the	DET
ejde-771	88	18	following	follow	VERB
ejde-771	88	19	properties	property	NOUN
ejde-771	88	20	.	.	PUNCT
ejde-771	89	1	(	(	PUNCT
ejde-771	89	2	1	1	X
ejde-771	89	3	)	)	PUNCT
ejde-771	89	4	the	the	DET
ejde-771	89	5	operators	operator	NOUN
ejde-771	89	6	jα	jα	PROPN
ejde-771	89	7	,	,	PUNCT
ejde-771	89	8	β(t	β(t	PROPN
ejde-771	89	9	)	)	PUNCT
ejde-771	89	10	and	and	CCONJ
ejde-771	89	11	kα	kα	PROPN
ejde-771	89	12	,	,	PUNCT
ejde-771	89	13	β(t	β(t	PROPN
ejde-771	89	14	)	)	PUNCT
ejde-771	89	15	are	be	AUX
ejde-771	89	16	strongly	strongly	ADV
ejde-771	89	17	conntinuous	conntinuous	ADJ
ejde-771	89	18	operators	operator	NOUN
ejde-771	89	19	,	,	PUNCT
ejde-771	89	20	this	this	PRON
ejde-771	89	21	indicates	indicate	VERB
ejde-771	89	22	that	that	SCONJ
ejde-771	89	23	for	for	ADP
ejde-771	89	24	any	any	DET
ejde-771	89	25	x	x	SYM
ejde-771	89	26	∈	∈	PROPN
ejde-771	89	27	e	e	NOUN
ejde-771	89	28	and	and	CCONJ
ejde-771	89	29	0	0	NUM
ejde-771	89	30	≤	≤	NUM
ejde-771	89	31	t1	t1	NOUN
ejde-771	89	32	≤	≤	ADJ
ejde-771	89	33	t2	t2	NOUN
ejde-771	89	34	,	,	PUNCT
ejde-771	89	35	∥jα	∥jα	PROPN
ejde-771	89	36	,	,	PUNCT
ejde-771	89	37	β(t2)x−jα	β(t2)x−jα	PROPN
ejde-771	89	38	,	,	PUNCT
ejde-771	89	39	β(t1)x∥	β(t1)x∥	X
ejde-771	89	40	→	→	SYM
ejde-771	89	41	0	0	NUM
ejde-771	89	42	and	and	CCONJ
ejde-771	89	43	∥kα	∥kα	PROPN
ejde-771	89	44	,	,	PUNCT
ejde-771	89	45	β(t2)x−kα	β(t2)x−kα	PROPN
ejde-771	89	46	,	,	PUNCT
ejde-771	89	47	β(t1)x∥	β(t1)x∥	X
ejde-771	89	48	→	→	SYM
ejde-771	89	49	0	0	NUM
ejde-771	89	50	as	as	ADP
ejde-771	89	51	t2	t2	PROPN
ejde-771	89	52	−	−	PROPN
ejde-771	89	53	t1	t1	PROPN
ejde-771	89	54	→	→	SYM
ejde-771	89	55	0	0	NUM
ejde-771	89	56	.	.	PUNCT
ejde-771	89	57	(	(	PUNCT
ejde-771	89	58	2	2	X
ejde-771	89	59	)	)	PUNCT
ejde-771	89	60	jα	jα	NOUN
ejde-771	89	61	,	,	PUNCT
ejde-771	89	62	β(t	β(t	PROPN
ejde-771	89	63	)	)	PUNCT
ejde-771	89	64	and	and	CCONJ
ejde-771	89	65	kα	kα	PROPN
ejde-771	89	66	,	,	PUNCT
ejde-771	89	67	β(t	β(t	PROPN
ejde-771	89	68	)	)	PUNCT
ejde-771	89	69	are	be	AUX
ejde-771	89	70	linear	linear	PROPN
ejde-771	89	71	bounded	bound	VERB
ejde-771	89	72	operators	operator	NOUN
ejde-771	89	73	for	for	ADP
ejde-771	89	74	any	any	DET
ejde-771	89	75	fixed	fix	VERB
ejde-771	89	76	t	t	PROPN
ejde-771	89	77	∈	∈	PROPN
ejde-771	89	78	r+	r+	X
ejde-771	89	79	,	,	PUNCT
ejde-771	89	80	∥jα	∥jα	PROPN
ejde-771	89	81	,	,	PUNCT
ejde-771	89	82	β(t)x∥	β(t)x∥	NOUN
ejde-771	89	83	≤m∥x∥	≤m∥x∥	NOUN
ejde-771	89	84	,	,	PUNCT
ejde-771	89	85	∥kα	∥kα	PROPN
ejde-771	89	86	,	,	PUNCT
ejde-771	89	87	β(t)x∥	β(t)x∥	NOUN
ejde-771	89	88	≤	≤	NOUN
ejde-771	89	89	m	m	VERB
ejde-771	89	90	γ(α	γ(α	NOUN
ejde-771	89	91	)	)	PUNCT
ejde-771	89	92	∥x∥.	∥x∥.	INTJ
ejde-771	90	1	(	(	PUNCT
ejde-771	90	2	3	3	X
ejde-771	90	3	)	)	PUNCT
ejde-771	90	4	jα	jα	NOUN
ejde-771	90	5	,	,	PUNCT
ejde-771	90	6	β(t	β(t	PROPN
ejde-771	90	7	)	)	PUNCT
ejde-771	90	8	and	and	CCONJ
ejde-771	90	9	kα	kα	PROPN
ejde-771	90	10	,	,	PUNCT
ejde-771	90	11	β(t	β(t	PROPN
ejde-771	90	12	)	)	PUNCT
ejde-771	90	13	are	be	AUX
ejde-771	90	14	uniformly	uniformly	ADV
ejde-771	90	15	continuous	continuous	ADJ
ejde-771	90	16	for	for	ADP
ejde-771	90	17	every	every	DET
ejde-771	90	18	t	t	NOUN
ejde-771	90	19	>	>	X
ejde-771	90	20	0	0	NUM
ejde-771	90	21	.	.	PUNCT
ejde-771	91	1	(	(	PUNCT
ejde-771	91	2	4	4	X
ejde-771	91	3	)	)	PUNCT
ejde-771	91	4	if	if	SCONJ
ejde-771	91	5	semigroup	semigroup	PROPN
ejde-771	91	6	tβ(t)(t	tβ(t)(t	X
ejde-771	91	7	≥	≥	NOUN
ejde-771	91	8	0	0	NUM
ejde-771	91	9	)	)	PUNCT
ejde-771	91	10	is	be	AUX
ejde-771	91	11	compact	compact	ADJ
ejde-771	91	12	,	,	PUNCT
ejde-771	91	13	then	then	ADV
ejde-771	91	14	jα	jα	PROPN
ejde-771	91	15	,	,	PUNCT
ejde-771	91	16	β(t	β(t	PROPN
ejde-771	91	17	)	)	PUNCT
ejde-771	91	18	and	and	CCONJ
ejde-771	91	19	kα	kα	PROPN
ejde-771	91	20	,	,	PUNCT
ejde-771	91	21	β(t	β(t	PROPN
ejde-771	91	22	)	)	PUNCT
ejde-771	91	23	are	be	AUX
ejde-771	91	24	compact	compact	ADJ
ejde-771	91	25	operators	operator	NOUN
ejde-771	91	26	for	for	ADP
ejde-771	91	27	every	every	DET
ejde-771	91	28	t	t	NOUN
ejde-771	91	29	>	>	X
ejde-771	91	30	0	0	NUM
ejde-771	91	31	.	.	PUNCT
ejde-771	92	1	(	(	PUNCT
ejde-771	92	2	5	5	X
ejde-771	92	3	)	)	PUNCT
ejde-771	92	4	if	if	SCONJ
ejde-771	92	5	semigroup	semigroup	PROPN
ejde-771	92	6	tβ(t)(t	tβ(t)(t	X
ejde-771	92	7	≥	≥	NOUN
ejde-771	92	8	0	0	NUM
ejde-771	92	9	)	)	PUNCT
ejde-771	92	10	is	be	AUX
ejde-771	92	11	positive	positive	ADJ
ejde-771	92	12	,	,	PUNCT
ejde-771	92	13	then	then	ADV
ejde-771	92	14	jα	jα	PROPN
ejde-771	92	15	,	,	PUNCT
ejde-771	92	16	β(t	β(t	PROPN
ejde-771	92	17	)	)	PUNCT
ejde-771	92	18	and	and	CCONJ
ejde-771	92	19	kα	kα	PROPN
ejde-771	92	20	,	,	PUNCT
ejde-771	92	21	β(t	β(t	PROPN
ejde-771	92	22	)	)	PUNCT
ejde-771	92	23	are	be	AUX
ejde-771	92	24	positive	positive	ADJ
ejde-771	92	25	operators	operator	NOUN
ejde-771	92	26	.	.	PUNCT
ejde-771	93	1	(	(	PUNCT
ejde-771	93	2	6	6	NUM
ejde-771	93	3	)	)	PUNCT
ejde-771	93	4	if	if	SCONJ
ejde-771	93	5	semigroup	semigroup	PROPN
ejde-771	93	6	tβ(t)(t	tβ(t)(t	X
ejde-771	93	7	≥	≥	NOUN
ejde-771	93	8	0	0	NUM
ejde-771	93	9	)	)	PUNCT
ejde-771	93	10	is	be	AUX
ejde-771	93	11	exponentially	exponentially	ADV
ejde-771	93	12	stable	stable	ADJ
ejde-771	93	13	with	with	ADP
ejde-771	93	14	the	the	DET
ejde-771	93	15	growth	growth	NOUN
ejde-771	93	16	exponent	exponent	NOUN
ejde-771	93	17	−|ν0|β	−|ν0|β	PROPN
ejde-771	93	18	,	,	PUNCT
ejde-771	93	19	then	then	ADV
ejde-771	93	20	∥jα	∥jα	PROPN
ejde-771	93	21	,	,	PUNCT
ejde-771	93	22	β(t)∥	β(t)∥	X
ejde-771	93	23	≤meα(−|ν0|βtα	≤meα(−|ν0|βtα	NOUN
ejde-771	93	24	)	)	PUNCT
ejde-771	93	25	,	,	PUNCT
ejde-771	93	26	∥kα	∥kα	PROPN
ejde-771	93	27	,	,	PUNCT
ejde-771	93	28	β(t)x∥	β(t)x∥	NOUN
ejde-771	93	29	≤meα	≤meα	NUM
ejde-771	93	30	,	,	PUNCT
ejde-771	93	31	α(−|ν0|βtα	α(−|ν0|βtα	X
ejde-771	93	32	)	)	PUNCT
ejde-771	93	33	(	(	PUNCT
ejde-771	93	34	2.4	2.4	NUM
ejde-771	93	35	)	)	PUNCT
ejde-771	93	36	for	for	ADP
ejde-771	93	37	every	every	DET
ejde-771	93	38	t	t	PROPN
ejde-771	93	39	≥	≥	NOUN
ejde-771	93	40	0	0	NUM
ejde-771	93	41	,	,	PUNCT
ejde-771	93	42	where	where	SCONJ
ejde-771	93	43	eα	eα	X
ejde-771	93	44	(	(	PUNCT
ejde-771	93	45	·	·	PUNCT
ejde-771	93	46	)	)	PUNCT
ejde-771	93	47	and	and	CCONJ
ejde-771	93	48	eα	eα	PRON
ejde-771	93	49	,	,	PUNCT
ejde-771	93	50	α	α	PROPN
ejde-771	93	51	(	(	PUNCT
ejde-771	93	52	·	·	PUNCT
ejde-771	93	53	)	)	PUNCT
ejde-771	93	54	are	be	AUX
ejde-771	93	55	the	the	DET
ejde-771	93	56	mittag	mittag	ADJ
ejde-771	93	57	-	-	PUNCT
ejde-771	93	58	leffler	leffler	NOUN
ejde-771	93	59	functions	function	NOUN
ejde-771	93	60	.	.	PUNCT
ejde-771	94	1	lemma	lemma	PROPN
ejde-771	94	2	2.4	2.4	NUM
ejde-771	94	3	(	(	PUNCT
ejde-771	94	4	[	[	X
ejde-771	94	5	31	31	NUM
ejde-771	94	6	]	]	PUNCT
ejde-771	94	7	)	)	PUNCT
ejde-771	94	8	.	.	PUNCT
ejde-771	95	1	eα(−µ	eα(−µ	PROPN
ejde-771	95	2	)	)	PUNCT
ejde-771	96	1	=	=	PRON
ejde-771	96	2	∫∞	∫∞	NOUN
ejde-771	96	3	0	0	NUM
ejde-771	97	1	τhα(τ)e	τhα(τ)e	PROPN
ejde-771	97	2	−µτdτ	−µτdτ	NOUN
ejde-771	97	3	,	,	PUNCT
ejde-771	97	4	eα	eα	NOUN
ejde-771	97	5	,	,	PUNCT
ejde-771	97	6	α(−µ	α(−µ	X
ejde-771	97	7	)	)	PUNCT
ejde-771	97	8	=	=	SYM
ejde-771	97	9	α	α	PROPN
ejde-771	97	10	∫∞	∫∞	NOUN
ejde-771	97	11	0	0	NUM
ejde-771	97	12	τhα(τ)e	τhα(τ)e	PROPN
ejde-771	97	13	−µτdτ	−µτdτ	NOUN
ejde-771	97	14	.	.	PUNCT
ejde-771	98	1	now	now	ADV
ejde-771	98	2	,	,	PUNCT
ejde-771	98	3	we	we	PRON
ejde-771	98	4	provide	provide	VERB
ejde-771	98	5	a	a	DET
ejde-771	98	6	definition	definition	NOUN
ejde-771	98	7	of	of	ADP
ejde-771	98	8	s	s	NOUN
ejde-771	98	9	-	-	PUNCT
ejde-771	98	10	asymptotically	asymptotically	ADV
ejde-771	98	11	ω	ω	ADJ
ejde-771	98	12	-	-	ADJ
ejde-771	98	13	periodic	periodic	ADJ
ejde-771	98	14	function	function	NOUN
ejde-771	98	15	.	.	PUNCT
ejde-771	99	1	let	let	VERB
ejde-771	99	2	cb([0,∞	cb([0,∞	NOUN
ejde-771	99	3	)	)	PUNCT
ejde-771	99	4	,	,	PUNCT
ejde-771	99	5	e	e	X
ejde-771	99	6	)	)	PUNCT
ejde-771	99	7	denote	denote	VERB
ejde-771	99	8	the	the	DET
ejde-771	99	9	banach	banach	NOUN
ejde-771	99	10	space	space	NOUN
ejde-771	99	11	of	of	ADP
ejde-771	99	12	all	all	DET
ejde-771	99	13	bounded	bounded	ADJ
ejde-771	99	14	and	and	CCONJ
ejde-771	99	15	continuous	continuous	ADJ
ejde-771	99	16	functions	function	NOUN
ejde-771	99	17	from	from	ADP
ejde-771	99	18	[	[	X
ejde-771	99	19	0,∞	0,∞	NOUN
ejde-771	99	20	)	)	PUNCT
ejde-771	99	21	to	to	PART
ejde-771	99	22	e	e	NOUN
ejde-771	99	23	equipped	equip	VERB
ejde-771	99	24	with	with	ADP
ejde-771	99	25	the	the	DET
ejde-771	99	26	norm	norm	NOUN
ejde-771	99	27	∥u∥c	∥u∥c	NOUN
ejde-771	99	28	=	=	SYM
ejde-771	100	1	supt∈r+	supt∈r+	NOUN
ejde-771	100	2	∥u(t)∥.	∥u(t)∥.	ADJ
ejde-771	100	3	definition	definition	NOUN
ejde-771	100	4	2.5	2.5	NUM
ejde-771	100	5	(	(	PUNCT
ejde-771	100	6	[	[	X
ejde-771	100	7	20	20	NUM
ejde-771	100	8	]	]	NUM
ejde-771	100	9	)	)	PUNCT
ejde-771	100	10	.	.	PUNCT
ejde-771	101	1	a	a	DET
ejde-771	101	2	function	function	NOUN
ejde-771	101	3	f	f	PROPN
ejde-771	101	4	∈	∈	PROPN
ejde-771	101	5	cb([0,∞	cb([0,∞	NOUN
ejde-771	101	6	)	)	PUNCT
ejde-771	101	7	,	,	PUNCT
ejde-771	101	8	e)is	e)is	PROPN
ejde-771	101	9	said	say	VERB
ejde-771	101	10	to	to	PART
ejde-771	101	11	be	be	AUX
ejde-771	101	12	s	s	NOUN
ejde-771	101	13	-	-	PUNCT
ejde-771	101	14	asymptotically	asymptotically	ADV
ejde-771	101	15	ω	ω	NOUN
ejde-771	101	16	-	-	NOUN
ejde-771	101	17	periodic	periodic	ADJ
ejde-771	101	18	if	if	SCONJ
ejde-771	101	19	there	there	PRON
ejde-771	101	20	exists	exist	VERB
ejde-771	101	21	ω	ω	PROPN
ejde-771	101	22	>	>	X
ejde-771	101	23	0	0	NUM
ejde-771	101	24	such	such	ADJ
ejde-771	101	25	that	that	SCONJ
ejde-771	101	26	limt→∞	limt→∞	PROPN
ejde-771	101	27	∥f(t	∥f(t	PROPN
ejde-771	101	28	+	+	CCONJ
ejde-771	101	29	ω	ω	NUM
ejde-771	101	30	)	)	PUNCT
ejde-771	101	31	−	−	PROPN
ejde-771	101	32	f(t)∥	f(t)∥	ADV
ejde-771	101	33	=	=	SYM
ejde-771	101	34	0	0	X
ejde-771	101	35	.	.	PUNCT
ejde-771	102	1	in	in	ADP
ejde-771	102	2	this	this	DET
ejde-771	102	3	case	case	NOUN
ejde-771	102	4	we	we	PRON
ejde-771	102	5	say	say	VERB
ejde-771	102	6	that	that	SCONJ
ejde-771	102	7	ω	ω	PROPN
ejde-771	102	8	is	be	AUX
ejde-771	102	9	an	an	DET
ejde-771	102	10	asymptotic	asymptotic	ADJ
ejde-771	102	11	period	period	NOUN
ejde-771	102	12	of	of	ADP
ejde-771	102	13	f	f	PROPN
ejde-771	102	14	.	.	PUNCT
ejde-771	103	1	let	let	AUX
ejde-771	103	2	sapω(e	sapω(e	NOUN
ejde-771	103	3	)	)	PUNCT
ejde-771	103	4	represent	represent	VERB
ejde-771	103	5	the	the	DET
ejde-771	103	6	subspace	subspace	NOUN
ejde-771	103	7	of	of	ADP
ejde-771	103	8	cb([0,∞	cb([0,∞	NOUN
ejde-771	103	9	)	)	PUNCT
ejde-771	103	10	,	,	PUNCT
ejde-771	103	11	e	e	X
ejde-771	103	12	)	)	PUNCT
ejde-771	103	13	consisting	consist	VERB
ejde-771	103	14	of	of	ADP
ejde-771	103	15	all	all	DET
ejde-771	103	16	the	the	DET
ejde-771	103	17	e	e	NOUN
ejde-771	103	18	-	-	NOUN
ejde-771	103	19	value	value	ADJ
ejde-771	103	20	s	s	NOUN
ejde-771	103	21	-	-	PUNCT
ejde-771	103	22	asymptotically	asymptotically	ADV
ejde-771	103	23	ω	ω	ADJ
ejde-771	103	24	-	-	ADJ
ejde-771	103	25	periodic	periodic	ADJ
ejde-771	103	26	functions	function	NOUN
ejde-771	103	27	endowed	endow	VERB
ejde-771	103	28	with	with	ADP
ejde-771	103	29	the	the	DET
ejde-771	103	30	uniform	uniform	ADJ
ejde-771	103	31	convergence	convergence	NOUN
ejde-771	103	32	norm	norm	NOUN
ejde-771	103	33	denoted	denote	VERB
ejde-771	103	34	by	by	ADP
ejde-771	103	35	∥u∥c	∥u∥c	NOUN
ejde-771	103	36	.	.	PUNCT
ejde-771	104	1	then	then	ADV
ejde-771	104	2	sapω(e	sapω(e	NOUN
ejde-771	104	3	)	)	PUNCT
ejde-771	104	4	is	be	AUX
ejde-771	104	5	a	a	DET
ejde-771	104	6	banach	banach	NOUN
ejde-771	104	7	space	space	NOUN
ejde-771	104	8	.	.	PUNCT
ejde-771	105	1	lemma	lemma	PROPN
ejde-771	105	2	2.6	2.6	NUM
ejde-771	105	3	.	.	PUNCT
ejde-771	106	1	[	[	X
ejde-771	106	2	27	27	NUM
ejde-771	106	3	]	]	PUNCT
ejde-771	106	4	let	let	VERB
ejde-771	106	5	π	π	PRON
ejde-771	106	6	be	be	AUX
ejde-771	106	7	a	a	DET
ejde-771	106	8	convex	convex	NOUN
ejde-771	106	9	,	,	PUNCT
ejde-771	106	10	bounded	bound	VERB
ejde-771	106	11	and	and	CCONJ
ejde-771	106	12	closed	closed	ADJ
ejde-771	106	13	subset	subset	NOUN
ejde-771	106	14	of	of	ADP
ejde-771	106	15	a	a	DET
ejde-771	106	16	banach	banach	NOUN
ejde-771	106	17	space	space	NOUN
ejde-771	106	18	e.	e.	PROPN
ejde-771	106	19	if	if	SCONJ
ejde-771	106	20	θ	θ	PROPN
ejde-771	106	21	:	:	PUNCT
ejde-771	107	1	π	π	X
ejde-771	107	2	→	→	PUNCT
ejde-771	107	3	π	π	PROPN
ejde-771	107	4	is	be	AUX
ejde-771	107	5	a	a	DET
ejde-771	107	6	condensing	condense	VERB
ejde-771	107	7	map	map	NOUN
ejde-771	107	8	,	,	PUNCT
ejde-771	107	9	then	then	ADV
ejde-771	107	10	θ	θ	PROPN
ejde-771	107	11	has	have	VERB
ejde-771	107	12	a	a	DET
ejde-771	107	13	fixed	fix	VERB
ejde-771	107	14	poind	poind	NOUN
ejde-771	107	15	in	in	ADP
ejde-771	107	16	π	π	PROPN
ejde-771	107	17	.	.	PROPN
ejde-771	108	1	4	4	NUM
ejde-771	108	2	x.	x.	NOUN
ejde-771	108	3	zhang	zhang	PROPN
ejde-771	108	4	,	,	PUNCT
ejde-771	108	5	k.	k.	PROPN
ejde-771	108	6	ding	ding	PROPN
ejde-771	108	7	,	,	PUNCT
ejde-771	108	8	p.	p.	PROPN
ejde-771	108	9	chen	chen	PROPN
ejde-771	108	10	ejde-2025/44	ejde-2025/44	PROPN
ejde-771	108	11	3	3	PROPN
ejde-771	108	12	.	.	PUNCT
ejde-771	108	13	abstract	abstract	ADJ
ejde-771	108	14	results	result	NOUN
ejde-771	108	15	in	in	ADP
ejde-771	108	16	this	this	DET
ejde-771	108	17	section	section	NOUN
ejde-771	108	18	,	,	PUNCT
ejde-771	108	19	we	we	PRON
ejde-771	108	20	discuss	discuss	VERB
ejde-771	108	21	the	the	DET
ejde-771	108	22	positive	positive	ADJ
ejde-771	108	23	s	s	NOUN
ejde-771	108	24	-	-	PUNCT
ejde-771	108	25	asymptotically	asymptotically	ADV
ejde-771	108	26	ω	ω	ADJ
ejde-771	108	27	-	-	ADJ
ejde-771	108	28	periodic	periodic	ADJ
ejde-771	108	29	mild	mild	ADJ
ejde-771	108	30	solutions	solution	NOUN
ejde-771	108	31	for	for	ADP
ejde-771	108	32	the	the	DET
ejde-771	108	33	following	following	ADJ
ejde-771	108	34	abstract	abstract	ADJ
ejde-771	108	35	time	time	NOUN
ejde-771	108	36	-	-	PUNCT
ejde-771	108	37	space	space	NOUN
ejde-771	108	38	fractional	fractional	ADJ
ejde-771	108	39	evolution	evolution	NOUN
ejde-771	108	40	equations	equation	NOUN
ejde-771	108	41	with	with	ADP
ejde-771	108	42	nonlocal	nonlocal	ADJ
ejde-771	108	43	conditions	condition	NOUN
ejde-771	108	44	cdα	cdα	NOUN
ejde-771	108	45	t	t	PROPN
ejde-771	108	46	u(t	u(t	PROPN
ejde-771	108	47	)	)	PUNCT
ejde-771	109	1	+	+	NOUN
ejde-771	109	2	aβu(t	aβu(t	X
ejde-771	109	3	)	)	PUNCT
ejde-771	109	4	=	=	SYM
ejde-771	109	5	g(t	g(t	PROPN
ejde-771	109	6	,	,	PUNCT
ejde-771	109	7	u(t	u(t	NOUN
ejde-771	109	8	)	)	PUNCT
ejde-771	109	9	)	)	PUNCT
ejde-771	109	10	,	,	PUNCT
ejde-771	109	11	t	t	PROPN
ejde-771	109	12	∈	∈	PROPN
ejde-771	110	1	[	[	X
ejde-771	110	2	0,+∞	0,+∞	NUM
ejde-771	110	3	)	)	PUNCT
ejde-771	110	4	,	,	PUNCT
ejde-771	110	5	u(0	u(0	NOUN
ejde-771	110	6	)	)	PUNCT
ejde-771	110	7	=	=	PUNCT
ejde-771	110	8	u0	u0	PROPN
ejde-771	111	1	+	+	X
ejde-771	111	2	m∑	m∑	ADV
ejde-771	111	3	k=1	k=1	NOUN
ejde-771	111	4	aku(tk	aku(tk	NOUN
ejde-771	111	5	)	)	PUNCT
ejde-771	111	6	,	,	PUNCT
ejde-771	111	7	(	(	PUNCT
ejde-771	111	8	3.1	3.1	NUM
ejde-771	111	9	)	)	PUNCT
ejde-771	111	10	where	where	SCONJ
ejde-771	111	11	cdα	cdα	NOUN
ejde-771	111	12	t	t	PROPN
ejde-771	111	13	is	be	AUX
ejde-771	111	14	the	the	DET
ejde-771	111	15	caputo	caputo	PROPN
ejde-771	111	16	fractional	fractional	PROPN
ejde-771	111	17	derivative	derivative	NOUN
ejde-771	111	18	of	of	ADP
ejde-771	111	19	the	the	DET
ejde-771	111	20	order	order	NOUN
ejde-771	111	21	0	0	PUNCT
ejde-771	111	22	<	<	X
ejde-771	111	23	α	α	X
ejde-771	111	24	<	<	X
ejde-771	111	25	1	1	NUM
ejde-771	111	26	,	,	PUNCT
ejde-771	111	27	a	a	DET
ejde-771	111	28	:	:	PUNCT
ejde-771	111	29	d(a	d(a	PROPN
ejde-771	111	30	)	)	PUNCT
ejde-771	112	1	⊂	⊂	PROPN
ejde-771	112	2	e	e	X
ejde-771	112	3	→	→	PUNCT
ejde-771	112	4	e	e	X
ejde-771	112	5	is	be	AUX
ejde-771	112	6	a	a	DET
ejde-771	112	7	closed	closed	ADJ
ejde-771	112	8	linear	linear	ADJ
ejde-771	112	9	operator	operator	NOUN
ejde-771	112	10	and	and	CCONJ
ejde-771	112	11	−a	−a	NOUN
ejde-771	112	12	generates	generate	VERB
ejde-771	112	13	an	an	DET
ejde-771	112	14	exponentially	exponentially	ADV
ejde-771	112	15	stable	stable	ADJ
ejde-771	112	16	analytic	analytic	ADJ
ejde-771	112	17	semigroup	semigroup	NOUN
ejde-771	112	18	t	t	PROPN
ejde-771	112	19	(	(	PUNCT
ejde-771	112	20	t)(t	t)(t	ADJ
ejde-771	112	21	≥	≥	NOUN
ejde-771	112	22	0	0	NUM
ejde-771	112	23	)	)	PUNCT
ejde-771	112	24	in	in	ADP
ejde-771	112	25	e	e	NOUN
ejde-771	112	26	,	,	PUNCT
ejde-771	112	27	aβ	aβ	PRON
ejde-771	112	28	is	be	AUX
ejde-771	112	29	the	the	DET
ejde-771	112	30	fractional	fractional	ADJ
ejde-771	112	31	power	power	NOUN
ejde-771	112	32	operator	operator	NOUN
ejde-771	112	33	of	of	ADP
ejde-771	112	34	a	a	PRON
ejde-771	112	35	for	for	ADP
ejde-771	112	36	0	0	NUM
ejde-771	112	37	<	<	X
ejde-771	112	38	β	β	X
ejde-771	112	39	<	<	X
ejde-771	112	40	1	1	NUM
ejde-771	112	41	,	,	PUNCT
ejde-771	112	42	0	0	NUM
ejde-771	112	43	<	<	X
ejde-771	112	44	t1	t1	NOUN
ejde-771	112	45	<	<	X
ejde-771	112	46	t2	t2	PROPN
ejde-771	112	47	<	<	X
ejde-771	112	48	·	·	PUNCT
ejde-771	112	49	·	·	PUNCT
ejde-771	112	50	·	·	PUNCT
ejde-771	112	51	<	<	X
ejde-771	113	1	tm	tm	X
ejde-771	113	2	<	<	X
ejde-771	113	3	+	+	PROPN
ejde-771	113	4	∞	∞	PROPN
ejde-771	113	5	and	and	CCONJ
ejde-771	113	6	ak	ak	PROPN
ejde-771	113	7	are	be	AUX
ejde-771	113	8	real	real	ADJ
ejde-771	113	9	numbers	number	NOUN
ejde-771	113	10	,	,	PUNCT
ejde-771	113	11	g	g	NOUN
ejde-771	113	12	:	:	PUNCT
ejde-771	114	1	[	[	X
ejde-771	114	2	0,∞)×	0,∞)×	NUM
ejde-771	114	3	e	e	NOUN
ejde-771	114	4	→	→	SYM
ejde-771	114	5	e	e	X
ejde-771	114	6	is	be	AUX
ejde-771	114	7	a	a	DET
ejde-771	114	8	continuous	continuous	ADJ
ejde-771	114	9	function	function	NOUN
ejde-771	114	10	.	.	PUNCT
ejde-771	115	1	definition	definition	NOUN
ejde-771	115	2	3.1	3.1	NUM
ejde-771	115	3	.	.	PUNCT
ejde-771	116	1	a	a	DET
ejde-771	116	2	function	function	NOUN
ejde-771	116	3	u	u	NOUN
ejde-771	116	4	:	:	PUNCT
ejde-771	116	5	[	[	X
ejde-771	116	6	0,∞	0,∞	NOUN
ejde-771	116	7	)	)	PUNCT
ejde-771	116	8	→	→	PUNCT
ejde-771	116	9	e	e	NOUN
ejde-771	116	10	is	be	AUX
ejde-771	116	11	said	say	VERB
ejde-771	116	12	to	to	PART
ejde-771	116	13	be	be	AUX
ejde-771	116	14	a	a	DET
ejde-771	116	15	mild	mild	ADJ
ejde-771	116	16	solution	solution	NOUN
ejde-771	116	17	of	of	ADP
ejde-771	116	18	the	the	DET
ejde-771	116	19	nonlocal	nonlocal	ADJ
ejde-771	116	20	problem	problem	NOUN
ejde-771	116	21	(	(	PUNCT
ejde-771	116	22	3.1	3.1	NUM
ejde-771	116	23	)	)	PUNCT
ejde-771	116	24	if	if	SCONJ
ejde-771	116	25	u	u	PROPN
ejde-771	116	26	∈	∈	PROPN
ejde-771	116	27	c([0,∞	c([0,∞	PROPN
ejde-771	116	28	)	)	PUNCT
ejde-771	116	29	,	,	PUNCT
ejde-771	116	30	e	e	NOUN
ejde-771	116	31	)	)	PUNCT
ejde-771	116	32	and	and	CCONJ
ejde-771	116	33	satisfies	satisfy	VERB
ejde-771	116	34	u(t	u(t	NOUN
ejde-771	116	35	)	)	PUNCT
ejde-771	116	36	=	=	SYM
ejde-771	117	1	jα	jα	NOUN
ejde-771	117	2	,	,	PUNCT
ejde-771	117	3	β(t)λu0	β(t)λu0	ADJ
ejde-771	117	4	+	+	CCONJ
ejde-771	117	5	m∑	m∑	ADV
ejde-771	117	6	k=1	k=1	PROPN
ejde-771	117	7	akjα	akjα	NOUN
ejde-771	117	8	,	,	PUNCT
ejde-771	117	9	β(t)λ	β(t)λ	PROPN
ejde-771	117	10	∫	∫	PROPN
ejde-771	117	11	tk	tk	PROPN
ejde-771	117	12	0	0	PROPN
ejde-771	118	1	(	(	PUNCT
ejde-771	118	2	tk	tk	PROPN
ejde-771	118	3	−	−	PROPN
ejde-771	118	4	s)α−1kα	s)α−1kα	PROPN
ejde-771	118	5	,	,	PUNCT
ejde-771	118	6	β(tk	β(tk	PROPN
ejde-771	118	7	−	−	NOUN
ejde-771	118	8	s)g(s	s)g(s	NOUN
ejde-771	118	9	,	,	PUNCT
ejde-771	118	10	u(s))ds	u(s))ds	PROPN
ejde-771	119	1	+	+	CCONJ
ejde-771	119	2	∫	∫	PROPN
ejde-771	119	3	t	t	PROPN
ejde-771	119	4	0	0	NUM
ejde-771	119	5	(	(	PUNCT
ejde-771	119	6	t−	t−	PROPN
ejde-771	119	7	s)α−1kα	s)α−1kα	PROPN
ejde-771	119	8	,	,	PUNCT
ejde-771	119	9	β(t−	β(t−	PROPN
ejde-771	119	10	s)g(s	s)g(s	NOUN
ejde-771	119	11	,	,	PUNCT
ejde-771	119	12	u(s))ds	u(s))ds	PROPN
ejde-771	119	13	.	.	PUNCT
ejde-771	120	1	(	(	PUNCT
ejde-771	120	2	3.2	3.2	NUM
ejde-771	120	3	)	)	PUNCT
ejde-771	120	4	moreover	moreover	ADV
ejde-771	120	5	,	,	PUNCT
ejde-771	120	6	if	if	SCONJ
ejde-771	120	7	u(t	u(t	NOUN
ejde-771	120	8	)	)	PUNCT
ejde-771	120	9	≥	≥	NUM
ejde-771	120	10	θ	θ	NOUN
ejde-771	120	11	for	for	ADP
ejde-771	120	12	all	all	DET
ejde-771	120	13	t	t	PROPN
ejde-771	120	14	≥	≥	NOUN
ejde-771	120	15	0	0	NUM
ejde-771	120	16	,	,	PUNCT
ejde-771	120	17	then	then	ADV
ejde-771	120	18	it	it	PRON
ejde-771	120	19	is	be	AUX
ejde-771	120	20	said	say	VERB
ejde-771	120	21	to	to	PART
ejde-771	120	22	be	be	AUX
ejde-771	120	23	a	a	DET
ejde-771	120	24	positive	positive	ADJ
ejde-771	120	25	mild	mild	ADJ
ejde-771	120	26	solution	solution	NOUN
ejde-771	120	27	of	of	ADP
ejde-771	120	28	nonlocal	nonlocal	ADJ
ejde-771	120	29	problem	problem	NOUN
ejde-771	120	30	(	(	PUNCT
ejde-771	120	31	3.1	3.1	NUM
ejde-771	120	32	)	)	PUNCT
ejde-771	120	33	.	.	PUNCT
ejde-771	121	1	to	to	PART
ejde-771	121	2	prove	prove	VERB
ejde-771	121	3	the	the	DET
ejde-771	121	4	main	main	ADJ
ejde-771	121	5	result	result	NOUN
ejde-771	121	6	,	,	PUNCT
ejde-771	121	7	we	we	PRON
ejde-771	121	8	also	also	ADV
ejde-771	121	9	need	need	VERB
ejde-771	121	10	the	the	DET
ejde-771	121	11	following	following	ADJ
ejde-771	121	12	assumption	assumption	NOUN
ejde-771	121	13	:	:	PUNCT
ejde-771	121	14	(	(	PUNCT
ejde-771	121	15	h0	h0	NOUN
ejde-771	121	16	)	)	PUNCT
ejde-771	121	17	∑m	∑m	PROPN
ejde-771	121	18	k=1	k=1	PUNCT
ejde-771	121	19	|ak|	|ak|	PROPN
ejde-771	121	20	<	<	X
ejde-771	121	21	1	1	NUM
ejde-771	121	22	m	m	NOUN
ejde-771	121	23	.	.	PUNCT
ejde-771	122	1	it	it	PRON
ejde-771	122	2	follows	follow	VERB
ejde-771	122	3	from	from	ADP
ejde-771	122	4	lemma	lemma	PROPN
ejde-771	122	5	2.3	2.3	NUM
ejde-771	122	6	(	(	PUNCT
ejde-771	122	7	2	2	NUM
ejde-771	122	8	)	)	PUNCT
ejde-771	123	1	that	that	PRON
ejde-771	123	2	∥	∥	ADJ
ejde-771	124	1	∑m	∑m	ADJ
ejde-771	125	1	k=1	k=1	X
ejde-771	125	2	akjα	akjα	NOUN
ejde-771	125	3	,	,	PUNCT
ejde-771	125	4	β(tk)∥	β(tk)∥	NOUN
ejde-771	125	5	≤	≤	ADJ
ejde-771	125	6	m	m	VERB
ejde-771	125	7	∑m	∑m	ADJ
ejde-771	125	8	k=1	k=1	X
ejde-771	125	9	|ak|	|ak|	PROPN
ejde-771	125	10	<	<	X
ejde-771	125	11	1	1	X
ejde-771	125	12	.	.	PUNCT
ejde-771	126	1	by	by	ADP
ejde-771	126	2	the	the	DET
ejde-771	126	3	operator	operator	NOUN
ejde-771	126	4	spectral	spectral	PROPN
ejde-771	126	5	theorem	theorem	PROPN
ejde-771	126	6	,	,	PUNCT
ejde-771	126	7	(	(	PUNCT
ejde-771	126	8	h0	h0	NOUN
ejde-771	126	9	)	)	PUNCT
ejde-771	126	10	give	give	VERB
ejde-771	126	11	a	a	DET
ejde-771	126	12	sufficient	sufficient	ADJ
ejde-771	126	13	condition	condition	NOUN
ejde-771	126	14	to	to	PART
ejde-771	126	15	guarantee	guarantee	VERB
ejde-771	126	16	the	the	DET
ejde-771	126	17	operator	operator	NOUN
ejde-771	126	18	λ	λ	X
ejde-771	126	19	on	on	ADP
ejde-771	126	20	e	e	NOUN
ejde-771	126	21	given	give	VERB
ejde-771	126	22	by	by	ADP
ejde-771	126	23	λ	λ	PROPN
ejde-771	126	24	=	=	SYM
ejde-771	126	25	(	(	PUNCT
ejde-771	126	26	i	i	PRON
ejde-771	126	27	−	−	PROPN
ejde-771	126	28	m∑	m∑	INTJ
ejde-771	126	29	k=1	k=1	PROPN
ejde-771	126	30	akjα	akjα	NOUN
ejde-771	126	31	,	,	PUNCT
ejde-771	126	32	β(tk	β(tk	NOUN
ejde-771	126	33	)	)	PUNCT
ejde-771	126	34	)	)	PUNCT
ejde-771	127	1	−1	−1	NOUN
ejde-771	127	2	exists	exist	VERB
ejde-771	127	3	and	and	CCONJ
ejde-771	127	4	be	be	AUX
ejde-771	127	5	bounded	bound	VERB
ejde-771	127	6	,	,	PUNCT
ejde-771	127	7	where	where	SCONJ
ejde-771	127	8	i	i	PRON
ejde-771	127	9	is	be	AUX
ejde-771	127	10	the	the	DET
ejde-771	127	11	identity	identity	NOUN
ejde-771	127	12	operator	operator	NOUN
ejde-771	127	13	.	.	PUNCT
ejde-771	128	1	indeed	indeed	ADV
ejde-771	128	2	,	,	PUNCT
ejde-771	128	3	by	by	ADP
ejde-771	128	4	neumann	neumann	PROPN
ejde-771	128	5	formula	formula	NOUN
ejde-771	128	6	,	,	PUNCT
ejde-771	128	7	λ	λ	PROPN
ejde-771	128	8	can	can	AUX
ejde-771	128	9	be	be	AUX
ejde-771	128	10	expressed	express	VERB
ejde-771	128	11	by	by	ADP
ejde-771	128	12	λ	λ	NOUN
ejde-771	128	13	=	=	SYM
ejde-771	128	14	∞∑	∞∑	PROPN
ejde-771	128	15	n=0	n=0	NUM
ejde-771	128	16	(	(	PUNCT
ejde-771	128	17	m∑	m∑	INTJ
ejde-771	128	18	k=1	k=1	PROPN
ejde-771	128	19	akjα	akjα	NOUN
ejde-771	128	20	,	,	PUNCT
ejde-771	128	21	β(tk	β(tk	NOUN
ejde-771	128	22	)	)	PUNCT
ejde-771	128	23	)	)	PUNCT
ejde-771	129	1	n	n	X
ejde-771	129	2	.	.	PUNCT
ejde-771	130	1	therefore	therefore	ADV
ejde-771	130	2	,	,	PUNCT
ejde-771	130	3	∥λ∥	∥λ∥	ADJ
ejde-771	130	4	≤	≤	NOUN
ejde-771	130	5	∞∑	∞∑	NUM
ejde-771	130	6	n=0	n=0	NUM
ejde-771	130	7	∥	∥	PUNCT
ejde-771	130	8	m∑	m∑	VERB
ejde-771	130	9	k=1	k=1	PROPN
ejde-771	130	10	akjα	akjα	NOUN
ejde-771	130	11	,	,	PUNCT
ejde-771	130	12	β(tk)∥n	β(tk)∥n	NOUN
ejde-771	130	13	=	=	NOUN
ejde-771	130	14	1	1	NUM
ejde-771	130	15	1−	1−	NUM
ejde-771	130	16	∥	∥	NUM
ejde-771	130	17	∑m	∑m	INTJ
ejde-771	131	1	k=1	k=1	X
ejde-771	132	1	akjα	akjα	NOUN
ejde-771	132	2	,	,	PUNCT
ejde-771	132	3	β(tk)∥	β(tk)∥	NOUN
ejde-771	132	4	≤	≤	ADJ
ejde-771	132	5	1	1	NUM
ejde-771	132	6	1−m	1−m	NUM
ejde-771	132	7	∑m	∑m	PROPN
ejde-771	132	8	k=1	k=1	PROPN
ejde-771	132	9	|ak|	|ak|	PROPN
ejde-771	132	10	.	.	PUNCT
ejde-771	133	1	(	(	PUNCT
ejde-771	133	2	3.3	3.3	NUM
ejde-771	133	3	)	)	PUNCT
ejde-771	133	4	theorem	theorem	VERB
ejde-771	133	5	3.2	3.2	NUM
ejde-771	133	6	.	.	PUNCT
ejde-771	134	1	let	let	VERB
ejde-771	134	2	e	e	PRON
ejde-771	134	3	be	be	AUX
ejde-771	134	4	an	an	DET
ejde-771	134	5	ordered	ordered	ADJ
ejde-771	134	6	banach	banach	NOUN
ejde-771	134	7	space	space	NOUN
ejde-771	134	8	,	,	PUNCT
ejde-771	134	9	whose	whose	DET
ejde-771	134	10	positive	positive	ADJ
ejde-771	134	11	cone	cone	NOUN
ejde-771	134	12	p	p	NOUN
ejde-771	134	13	is	be	AUX
ejde-771	134	14	normal	normal	ADJ
ejde-771	134	15	,	,	PUNCT
ejde-771	134	16	a	a	DET
ejde-771	134	17	:	:	PUNCT
ejde-771	134	18	d(a	d(a	PROPN
ejde-771	134	19	)	)	PUNCT
ejde-771	134	20	⊂	⊂	PROPN
ejde-771	134	21	e	e	X
ejde-771	134	22	→	→	PUNCT
ejde-771	134	23	e	e	X
ejde-771	134	24	be	be	AUX
ejde-771	134	25	a	a	DET
ejde-771	134	26	closed	closed	ADJ
ejde-771	134	27	linear	linear	ADJ
ejde-771	134	28	operator	operator	NOUN
ejde-771	134	29	and	and	CCONJ
ejde-771	134	30	−a	−a	NOUN
ejde-771	134	31	generate	generate	VERB
ejde-771	134	32	an	an	DET
ejde-771	134	33	exponentially	exponentially	ADV
ejde-771	134	34	stable	stable	ADJ
ejde-771	134	35	,	,	PUNCT
ejde-771	134	36	positive	positive	ADJ
ejde-771	134	37	,	,	PUNCT
ejde-771	134	38	and	and	CCONJ
ejde-771	134	39	compact	compact	ADJ
ejde-771	134	40	analytic	analytic	ADJ
ejde-771	134	41	semigroup	semigroup	NOUN
ejde-771	134	42	t	t	PROPN
ejde-771	134	43	(	(	PUNCT
ejde-771	134	44	t)(t	t)(t	ADJ
ejde-771	134	45	≥	≥	NOUN
ejde-771	134	46	0	0	NUM
ejde-771	134	47	)	)	PUNCT
ejde-771	134	48	in	in	ADP
ejde-771	134	49	e	e	NOUN
ejde-771	134	50	,	,	PUNCT
ejde-771	134	51	whose	whose	DET
ejde-771	134	52	growth	growth	NOUN
ejde-771	134	53	exponent	exponent	NOUN
ejde-771	134	54	ν0	ν0	PROPN
ejde-771	134	55	<	<	X
ejde-771	134	56	0	0	NUM
ejde-771	134	57	,	,	PUNCT
ejde-771	134	58	the	the	DET
ejde-771	134	59	nonlinear	nonlinear	ADJ
ejde-771	134	60	function	function	NOUN
ejde-771	134	61	g	g	NOUN
ejde-771	134	62	:	:	PUNCT
ejde-771	134	63	r+	r+	NOUN
ejde-771	134	64	×	×	PROPN
ejde-771	134	65	e	e	X
ejde-771	134	66	→	→	PUNCT
ejde-771	134	67	e	e	AUX
ejde-771	134	68	be	be	AUX
ejde-771	134	69	a	a	DET
ejde-771	134	70	continuous	continuous	ADJ
ejde-771	134	71	function	function	NOUN
ejde-771	134	72	.	.	PUNCT
ejde-771	135	1	if	if	SCONJ
ejde-771	135	2	the	the	DET
ejde-771	135	3	conditions	condition	NOUN
ejde-771	135	4	(	(	PUNCT
ejde-771	135	5	h0	h0	NOUN
ejde-771	135	6	)	)	PUNCT
ejde-771	135	7	and	and	CCONJ
ejde-771	135	8	the	the	DET
ejde-771	135	9	following	follow	VERB
ejde-771	135	10	3	3	NUM
ejde-771	135	11	conditions	condition	NOUN
ejde-771	135	12	hold	hold	VERB
ejde-771	135	13	:	:	PUNCT
ejde-771	135	14	(	(	PUNCT
ejde-771	135	15	h1	h1	PROPN
ejde-771	135	16	)	)	PUNCT
ejde-771	135	17	for	for	ADP
ejde-771	135	18	t	t	PROPN
ejde-771	135	19	≥	≥	NOUN
ejde-771	135	20	0	0	PUNCT
ejde-771	135	21	and	and	CCONJ
ejde-771	135	22	x	x	SYM
ejde-771	135	23	∈	∈	PROPN
ejde-771	135	24	e	e	NOUN
ejde-771	135	25	,	,	PUNCT
ejde-771	135	26	there	there	PRON
ejde-771	135	27	exist	exist	VERB
ejde-771	135	28	positive	positive	ADJ
ejde-771	135	29	constants	constant	NOUN
ejde-771	135	30	a0	a0	PROPN
ejde-771	135	31	≥	≥	NOUN
ejde-771	135	32	0	0	NUM
ejde-771	135	33	and	and	CCONJ
ejde-771	135	34	a1	a1	PROPN
ejde-771	135	35	∈	∈	PROPN
ejde-771	135	36	(	(	PUNCT
ejde-771	135	37	0	0	NUM
ejde-771	135	38	,	,	PUNCT
ejde-771	135	39	(	(	PUNCT
ejde-771	135	40	1−m	1−m	NUM
ejde-771	135	41	∑m	∑m	NOUN
ejde-771	135	42	k=1	k=1	PUNCT
ejde-771	136	1	|ak|)|ν0|β	|ak|)|ν0|β	PROPN
ejde-771	136	2	/	/	SYM
ejde-771	136	3	m	m	VERB
ejde-771	136	4	)	)	PUNCT
ejde-771	137	1	such	such	ADJ
ejde-771	137	2	that	that	SCONJ
ejde-771	137	3	∥g(t	∥g(t	NOUN
ejde-771	137	4	,	,	PUNCT
ejde-771	137	5	etx)∥	etx)∥	PROPN
ejde-771	137	6	≤	≤	ADV
ejde-771	137	7	a1∥x∥+a0	a1∥x∥+a0	PROPN
ejde-771	137	8	,	,	PUNCT
ejde-771	137	9	(	(	PUNCT
ejde-771	137	10	h2	h2	NOUN
ejde-771	137	11	)	)	PUNCT
ejde-771	137	12	g	g	NOUN
ejde-771	137	13	is	be	AUX
ejde-771	137	14	nondecreasing	nondecrease	VERB
ejde-771	137	15	with	with	ADP
ejde-771	137	16	respect	respect	NOUN
ejde-771	137	17	to	to	ADP
ejde-771	137	18	the	the	DET
ejde-771	137	19	second	second	ADJ
ejde-771	137	20	variable	variable	NOUN
ejde-771	137	21	,	,	PUNCT
ejde-771	137	22	i.e.	i.e.	X
ejde-771	137	23	,	,	PUNCT
ejde-771	137	24	for	for	ADP
ejde-771	137	25	x2	x2	PROPN
ejde-771	137	26	≥	≥	NUM
ejde-771	137	27	x1	x1	PRON
ejde-771	137	28	≥	≥	NUM
ejde-771	137	29	θ	θ	PROPN
ejde-771	137	30	,	,	PUNCT
ejde-771	137	31	g(t	g(t	PROPN
ejde-771	137	32	,	,	PUNCT
ejde-771	137	33	x2	x2	NUM
ejde-771	137	34	)	)	PUNCT
ejde-771	137	35	≥	≥	NOUN
ejde-771	137	36	g(t	g(t	PROPN
ejde-771	137	37	,	,	PUNCT
ejde-771	137	38	x1	x1	NUM
ejde-771	137	39	)	)	PUNCT
ejde-771	137	40	≥	≥	NUM
ejde-771	137	41	θ	θ	PROPN
ejde-771	137	42	,	,	PUNCT
ejde-771	137	43	t	t	PROPN
ejde-771	137	44	≥	≥	NUM
ejde-771	137	45	0	0	NUM
ejde-771	137	46	,	,	PUNCT
ejde-771	137	47	(	(	PUNCT
ejde-771	137	48	h3	h3	NOUN
ejde-771	137	49	)	)	PUNCT
ejde-771	137	50	there	there	PRON
ejde-771	137	51	exists	exist	VERB
ejde-771	137	52	ω	ω	PROPN
ejde-771	137	53	>	>	X
ejde-771	137	54	0	0	NUM
ejde-771	137	55	,	,	PUNCT
ejde-771	137	56	for	for	ADP
ejde-771	137	57	every	every	DET
ejde-771	137	58	t	t	NOUN
ejde-771	137	59	∈	∈	PROPN
ejde-771	138	1	[	[	X
ejde-771	138	2	0,∞	0,∞	NOUN
ejde-771	138	3	)	)	PUNCT
ejde-771	138	4	,	,	PUNCT
ejde-771	138	5	x	x	PUNCT
ejde-771	138	6	∈	∈	PROPN
ejde-771	138	7	e	e	NOUN
ejde-771	138	8	,	,	PUNCT
ejde-771	138	9	lim	lim	PROPN
ejde-771	138	10	t→∞	t→∞	NUM
ejde-771	138	11	∥g(t+	∥g(t+	PROPN
ejde-771	138	12	ω	ω	PROPN
ejde-771	138	13	,	,	PUNCT
ejde-771	138	14	x)−g(t	x)−g(t	PROPN
ejde-771	138	15	,	,	PUNCT
ejde-771	138	16	x)∥	x)∥	PUNCT
ejde-771	138	17	=	=	SYM
ejde-771	138	18	0	0	NUM
ejde-771	138	19	,	,	PUNCT
ejde-771	138	20	ejde-2025/44	ejde-2025/44	ADJ
ejde-771	138	21	time	time	NOUN
ejde-771	138	22	-	-	PUNCT
ejde-771	138	23	space	space	NOUN
ejde-771	138	24	fractional	fractional	ADJ
ejde-771	138	25	reaction	reaction	NOUN
ejde-771	138	26	-	-	PUNCT
ejde-771	138	27	diffusion	diffusion	NOUN
ejde-771	138	28	equations	equation	NOUN
ejde-771	138	29	5	5	NUM
ejde-771	138	30	then	then	ADV
ejde-771	138	31	there	there	PRON
ejde-771	138	32	exist	exist	VERB
ejde-771	138	33	a	a	DET
ejde-771	138	34	minimal	minimal	ADJ
ejde-771	138	35	positive	positive	ADJ
ejde-771	138	36	s	s	NOUN
ejde-771	138	37	-	-	PUNCT
ejde-771	138	38	asymptotically	asymptotically	ADV
ejde-771	138	39	ω	ω	ADJ
ejde-771	138	40	-	-	ADJ
ejde-771	138	41	periodic	periodic	ADJ
ejde-771	138	42	mild	mild	ADJ
ejde-771	138	43	solution	solution	NOUN
ejde-771	138	44	ũ	ũ	PROPN
ejde-771	138	45	of	of	ADP
ejde-771	138	46	nonlocal	nonlocal	ADJ
ejde-771	138	47	problem	problem	NOUN
ejde-771	138	48	(	(	PUNCT
ejde-771	138	49	3.1	3.1	NUM
ejde-771	138	50	)	)	PUNCT
ejde-771	138	51	.	.	PUNCT
ejde-771	139	1	proof	proof	NOUN
ejde-771	139	2	.	.	PUNCT
ejde-771	140	1	consider	consider	VERB
ejde-771	140	2	the	the	DET
ejde-771	140	3	operator	operator	NOUN
ejde-771	140	4	θ	θ	PROPN
ejde-771	140	5	on	on	ADP
ejde-771	140	6	ce(e	ce(e	NOUN
ejde-771	140	7	)	)	PUNCT
ejde-771	140	8	defined	define	VERB
ejde-771	140	9	by	by	ADP
ejde-771	140	10	(	(	PUNCT
ejde-771	140	11	θu)(t	θu)(t	X
ejde-771	140	12	)	)	PUNCT
ejde-771	140	13	=	=	SYM
ejde-771	140	14	jα	jα	NOUN
ejde-771	140	15	,	,	PUNCT
ejde-771	140	16	β(t)λu0	β(t)λu0	ADJ
ejde-771	141	1	+	+	CCONJ
ejde-771	141	2	∫	∫	PROPN
ejde-771	141	3	t	t	NOUN
ejde-771	141	4	0	0	NUM
ejde-771	141	5	(	(	PUNCT
ejde-771	141	6	t−	t−	PROPN
ejde-771	141	7	s)α−1kα	s)α−1kα	PROPN
ejde-771	141	8	,	,	PUNCT
ejde-771	141	9	β(t−	β(t−	PROPN
ejde-771	141	10	s)g(s	s)g(s	NOUN
ejde-771	141	11	,	,	PUNCT
ejde-771	141	12	u(s))ds	u(s))ds	PROPN
ejde-771	141	13	+	+	CCONJ
ejde-771	141	14	m∑	m∑	ADV
ejde-771	141	15	k=1	k=1	PROPN
ejde-771	141	16	akjα	akjα	NOUN
ejde-771	141	17	,	,	PUNCT
ejde-771	141	18	β(t)λ	β(t)λ	PROPN
ejde-771	141	19	∫	∫	PROPN
ejde-771	141	20	tk	tk	PROPN
ejde-771	141	21	0	0	PROPN
ejde-771	142	1	(	(	PUNCT
ejde-771	142	2	tk	tk	PROPN
ejde-771	142	3	−	−	PROPN
ejde-771	142	4	s)α−1kα	s)α−1kα	PROPN
ejde-771	142	5	,	,	PUNCT
ejde-771	142	6	β(tk	β(tk	PROPN
ejde-771	142	7	−	−	NOUN
ejde-771	142	8	s)g(s	s)g(s	NOUN
ejde-771	142	9	,	,	PUNCT
ejde-771	142	10	u(s))ds	u(s))ds	PROPN
ejde-771	142	11	.	.	PUNCT
ejde-771	143	1	(	(	PUNCT
ejde-771	143	2	3.4	3.4	NUM
ejde-771	143	3	)	)	PUNCT
ejde-771	143	4	by	by	ADP
ejde-771	143	5	(	(	PUNCT
ejde-771	143	6	3.3	3.3	NUM
ejde-771	143	7	)	)	PUNCT
ejde-771	143	8	,	,	PUNCT
ejde-771	143	9	(	(	PUNCT
ejde-771	143	10	3.4	3.4	NUM
ejde-771	143	11	)	)	PUNCT
ejde-771	143	12	and	and	CCONJ
ejde-771	143	13	(	(	PUNCT
ejde-771	143	14	h1	h1	PROPN
ejde-771	143	15	)	)	PUNCT
ejde-771	143	16	,	,	PUNCT
ejde-771	143	17	one	one	PRON
ejde-771	143	18	can	can	AUX
ejde-771	143	19	conclude	conclude	VERB
ejde-771	143	20	that	that	SCONJ
ejde-771	143	21	e−t∥(θu)(t)∥	e−t∥(θu)(t)∥	PROPN
ejde-771	143	22	≤	≤	NUM
ejde-771	143	23	e−t∥jα	e−t∥jα	NOUN
ejde-771	143	24	,	,	PUNCT
ejde-771	143	25	β(t)λu0∥+	β(t)λu0∥+	NUM
ejde-771	143	26	e−t	e−t	PROPN
ejde-771	143	27	∫	∫	PROPN
ejde-771	143	28	t	t	NOUN
ejde-771	143	29	0	0	NUM
ejde-771	143	30	(	(	PUNCT
ejde-771	143	31	t−	t−	PROPN
ejde-771	143	32	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	143	33	,	,	PUNCT
ejde-771	143	34	β(t−	β(t−	PROPN
ejde-771	143	35	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	143	36	,	,	PUNCT
ejde-771	143	37	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	143	38	+	+	CCONJ
ejde-771	143	39	e−t	e−t	NOUN
ejde-771	143	40	m∑	m∑	ADP
ejde-771	143	41	k=1	k=1	PROPN
ejde-771	143	42	|ak|∥jα	|ak|∥jα	PROPN
ejde-771	143	43	,	,	PUNCT
ejde-771	143	44	β(t)∥∥λ∥	β(t)∥∥λ∥	NUM
ejde-771	144	1	∫	∫	PROPN
ejde-771	144	2	tk	tk	PROPN
ejde-771	144	3	0	0	PROPN
ejde-771	145	1	(	(	PUNCT
ejde-771	145	2	tk	tk	PROPN
ejde-771	145	3	−	−	PROPN
ejde-771	145	4	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	145	5	,	,	PUNCT
ejde-771	145	6	β(tk	β(tk	PROPN
ejde-771	145	7	−	−	NOUN
ejde-771	146	1	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	146	2	,	,	PUNCT
ejde-771	146	3	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	146	4	≤	≤	NUM
ejde-771	146	5	e−tm∥u0∥	e−tm∥u0∥	PROPN
ejde-771	146	6	1−m	1−m	NUM
ejde-771	146	7	∑m	∑m	PROPN
ejde-771	147	1	k=1	k=1	X
ejde-771	147	2	|ak|	|ak|	PROPN
ejde-771	148	1	+	+	CCONJ
ejde-771	148	2	e−tm	e−tm	PROPN
ejde-771	148	3	∑m	∑m	PROPN
ejde-771	148	4	k=1	k=1	PROPN
ejde-771	148	5	|ak|	|ak|	PROPN
ejde-771	149	1	1−m	1−m	NUM
ejde-771	149	2	∑m	∑m	PROPN
ejde-771	149	3	k=1	k=1	PUNCT
ejde-771	149	4	|ak|	|ak|	PROPN
ejde-771	150	1	×	×	NOUN
ejde-771	150	2	αm	αm	NOUN
ejde-771	150	3	∫	∫	PROPN
ejde-771	151	1	tk	tk	PROPN
ejde-771	151	2	0	0	PROPN
ejde-771	151	3	∫	∫	PROPN
ejde-771	151	4	∞	∞	PROPN
ejde-771	151	5	0	0	NUM
ejde-771	151	6	τhα(τ)(tk	τhα(τ)(tk	SYM
ejde-771	152	1	−	−	NOUN
ejde-771	152	2	s)α−1e−|ν0|β(tk−s)ατ	s)α−1e−|ν0|β(tk−s)ατ	PROPN
ejde-771	152	3	(	(	PUNCT
ejde-771	152	4	a1∥u∥e	a1∥u∥e	PUNCT
ejde-771	152	5	+	+	ADJ
ejde-771	152	6	a0)dτds	a0)dτd	NOUN
ejde-771	152	7	+	+	CCONJ
ejde-771	152	8	e−tαm	e−tαm	ADJ
ejde-771	152	9	∫	∫	PROPN
ejde-771	152	10	t	t	PROPN
ejde-771	152	11	0	0	NUM
ejde-771	152	12	∫	∫	PROPN
ejde-771	152	13	∞	∞	NUM
ejde-771	152	14	0	0	NUM
ejde-771	153	1	τhα(τ)(t−	τhα(τ)(t−	NOUN
ejde-771	154	1	s)α−1e−|ν0|β(t−s)ατ	s)α−1e−|ν0|β(t−s)ατ	NOUN
ejde-771	154	2	(	(	PUNCT
ejde-771	154	3	a1∥u∥e	a1∥u∥e	PROPN
ejde-771	154	4	+	+	ADJ
ejde-771	154	5	a0)dτds	a0)dτd	NOUN
ejde-771	154	6	≤	≤	NUM
ejde-771	154	7	e−tm∥u0∥	e−tm∥u0∥	PROPN
ejde-771	154	8	1−m	1−m	NUM
ejde-771	154	9	∑m	∑m	PROPN
ejde-771	154	10	k=1	k=1	X
ejde-771	154	11	|ak|	|ak|	PROPN
ejde-771	155	1	+	+	CCONJ
ejde-771	155	2	e−tm	e−tm	PROPN
ejde-771	155	3	∑m	∑m	PROPN
ejde-771	155	4	k=1	k=1	PROPN
ejde-771	155	5	|ak|	|ak|	PROPN
ejde-771	156	1	1−m	1−m	NUM
ejde-771	156	2	∑m	∑m	PROPN
ejde-771	156	3	k=1	k=1	PROPN
ejde-771	156	4	|ak|	|ak|	PROPN
ejde-771	156	5	m(a1∥u∥e	m(a1∥u∥e	VERB
ejde-771	156	6	+	+	NOUN
ejde-771	156	7	a0	a0	NOUN
ejde-771	156	8	)	)	PUNCT
ejde-771	156	9	∫	∫	PROPN
ejde-771	157	1	∞	∞	PROPN
ejde-771	157	2	0	0	NUM
ejde-771	158	1	hα(τ)dτ	hα(τ)dτ	ADJ
ejde-771	158	2	∫	∫	PROPN
ejde-771	158	3	∞	∞	PROPN
ejde-771	158	4	0	0	NUM
ejde-771	159	1	e−|ν0|βsds	e−|ν0|βsds	ADP
ejde-771	159	2	+	+	ADJ
ejde-771	159	3	e−tm(a1∥u∥e	e−tm(a1∥u∥e	PRON
ejde-771	159	4	+	+	ADJ
ejde-771	159	5	a0	a0	NOUN
ejde-771	159	6	)	)	PUNCT
ejde-771	159	7	∫	∫	PROPN
ejde-771	160	1	∞	∞	PROPN
ejde-771	160	2	0	0	NUM
ejde-771	161	1	hα(τ)dτ	hα(τ)dτ	ADJ
ejde-771	161	2	∫	∫	PROPN
ejde-771	161	3	∞	∞	PROPN
ejde-771	161	4	0	0	NUM
ejde-771	162	1	e−|ν0|βsds	e−|ν0|βsds	CCONJ
ejde-771	162	2	≤	≤	NUM
ejde-771	162	3	e−tm∥u0∥	e−tm∥u0∥	PROPN
ejde-771	163	1	1−m	1−m	NUM
ejde-771	163	2	∑m	∑m	PROPN
ejde-771	163	3	k=1	k=1	X
ejde-771	163	4	|ak|	|ak|	PROPN
ejde-771	164	1	+	+	CCONJ
ejde-771	164	2	e−tm	e−tm	PROPN
ejde-771	164	3	∑m	∑m	PROPN
ejde-771	164	4	k=1	k=1	PROPN
ejde-771	164	5	|ak|	|ak|	PROPN
ejde-771	165	1	1−m	1−m	NUM
ejde-771	165	2	∑m	∑m	PROPN
ejde-771	165	3	k=1	k=1	PROPN
ejde-771	165	4	|ak|	|ak|	PROPN
ejde-771	165	5	m(a1∥u∥e	m(a1∥u∥e	VERB
ejde-771	165	6	+	+	NOUN
ejde-771	165	7	a0	a0	NOUN
ejde-771	165	8	)	)	PUNCT
ejde-771	165	9	|ν0|β	|ν0|β	PUNCT
ejde-771	166	1	+	+	CCONJ
ejde-771	166	2	e−tm(a1∥u∥e	e−tm(a1∥u∥e	PRON
ejde-771	166	3	+	+	ADJ
ejde-771	166	4	a0	a0	NOUN
ejde-771	166	5	)	)	PUNCT
ejde-771	166	6	|ν0|β	|ν0|β	PUNCT
ejde-771	166	7	≤	≤	NUM
ejde-771	166	8	e−tm∥u0∥	e−tm∥u0∥	PROPN
ejde-771	166	9	1−m	1−m	NUM
ejde-771	166	10	∑m	∑m	PROPN
ejde-771	166	11	k=1	k=1	PUNCT
ejde-771	166	12	|ak|	|ak|	PROPN
ejde-771	167	1	+	+	CCONJ
ejde-771	167	2	e−t	e−t	NOUN
ejde-771	167	3	(	(	PUNCT
ejde-771	167	4	m	m	VERB
ejde-771	167	5	∑m	∑m	ADJ
ejde-771	167	6	k=1	k=1	X
ejde-771	167	7	|ak|	|ak|	PROPN
ejde-771	168	1	1−m	1−m	NUM
ejde-771	168	2	∑m	∑m	PROPN
ejde-771	168	3	k=1	k=1	PART
ejde-771	168	4	|ak|	|ak|	PROPN
ejde-771	169	1	+	+	CCONJ
ejde-771	169	2	1	1	X
ejde-771	169	3	)	)	PUNCT
ejde-771	169	4	m(a1∥u∥e	m(a1∥u∥e	NOUN
ejde-771	169	5	+	+	NOUN
ejde-771	169	6	a0	a0	NOUN
ejde-771	169	7	)	)	PUNCT
ejde-771	169	8	|ν0|β	|ν0|β	PUNCT
ejde-771	170	1	≤	≤	NUM
ejde-771	170	2	e−tm∥u0∥	e−tm∥u0∥	PROPN
ejde-771	170	3	1−m	1−m	NUM
ejde-771	170	4	∑m	∑m	PROPN
ejde-771	170	5	k=1	k=1	PUNCT
ejde-771	170	6	|ak|	|ak|	PROPN
ejde-771	171	1	+	+	PUNCT
ejde-771	171	2	e−tm(a1∥u∥e	e−tm(a1∥u∥e	PRON
ejde-771	171	3	+	+	ADJ
ejde-771	171	4	a0	a0	NOUN
ejde-771	171	5	)	)	PUNCT
ejde-771	171	6	(	(	PUNCT
ejde-771	171	7	1−m	1−m	NUM
ejde-771	171	8	∑m	∑m	PROPN
ejde-771	171	9	k=1	k=1	X
ejde-771	171	10	|ak|	|ak|	PROPN
ejde-771	171	11	)	)	PUNCT
ejde-771	172	1	|ν0|β	|ν0|β	ADV
ejde-771	172	2	.	.	PUNCT
ejde-771	173	1	(	(	PUNCT
ejde-771	173	2	3.5	3.5	NUM
ejde-771	173	3	)	)	PUNCT
ejde-771	173	4	thus	thus	ADV
ejde-771	173	5	,	,	PUNCT
ejde-771	173	6	we	we	PRON
ejde-771	173	7	can	can	AUX
ejde-771	173	8	conclude	conclude	VERB
ejde-771	173	9	that	that	SCONJ
ejde-771	173	10	∥(θu)(t)∥e	∥(θu)(t)∥e	PRON
ejde-771	173	11	≤	≤	NUM
ejde-771	173	12	m∥u0∥	m∥u0∥	PROPN
ejde-771	173	13	1−m	1−m	NUM
ejde-771	173	14	∑m	∑m	PROPN
ejde-771	174	1	k=1	k=1	PUNCT
ejde-771	174	2	|ak|	|ak|	PROPN
ejde-771	175	1	+	+	CCONJ
ejde-771	175	2	m(a1∥u∥e	m(a1∥u∥e	NOUN
ejde-771	175	3	+	+	NOUN
ejde-771	175	4	a0	a0	NOUN
ejde-771	175	5	)	)	PUNCT
ejde-771	175	6	(	(	PUNCT
ejde-771	175	7	1−m	1−m	NUM
ejde-771	175	8	∑m	∑m	PROPN
ejde-771	175	9	k=1	k=1	PUNCT
ejde-771	176	1	|ak|)|ν0|β	|ak|)|ν0|β	PROPN
ejde-771	176	2	:	:	PUNCT
ejde-771	176	3	=	=	SYM
ejde-771	176	4	φ+	φ+	NOUN
ejde-771	176	5	ψ∥u∥e	ψ∥u∥e	PROPN
ejde-771	176	6	,	,	PUNCT
ejde-771	176	7	(	(	PUNCT
ejde-771	176	8	3.6	3.6	NUM
ejde-771	176	9	)	)	PUNCT
ejde-771	177	1	where	where	SCONJ
ejde-771	177	2	φ	φ	PROPN
ejde-771	177	3	=	=	SYM
ejde-771	177	4	|ν0|βm∥u0∥+ma0	|ν0|βm∥u0∥+ma0	PROPN
ejde-771	177	5	(	(	PUNCT
ejde-771	177	6	1−m	1−m	NUM
ejde-771	177	7	∑m	∑m	PROPN
ejde-771	177	8	k=1	k=1	PUNCT
ejde-771	177	9	|ak|)|ν0|β	|ak|)|ν0|β	PROPN
ejde-771	177	10	,	,	PUNCT
ejde-771	177	11	ψ	ψ	X
ejde-771	177	12	=	=	PROPN
ejde-771	177	13	ma1	ma1	PROPN
ejde-771	177	14	(	(	PUNCT
ejde-771	177	15	1−m	1−m	NUM
ejde-771	177	16	∑m	∑m	PROPN
ejde-771	177	17	k=1	k=1	PUNCT
ejde-771	178	1	|ak|)|ν0|β	|ak|)|ν0|β	NOUN
ejde-771	178	2	are	be	AUX
ejde-771	178	3	positive	positive	ADJ
ejde-771	178	4	with	with	ADP
ejde-771	178	5	ψ	ψ	X
ejde-771	178	6	<	<	X
ejde-771	178	7	1	1	NUM
ejde-771	178	8	.	.	PUNCT
ejde-771	179	1	hence	hence	ADV
ejde-771	179	2	,	,	PUNCT
ejde-771	179	3	limt→∞	limt→∞	ADP
ejde-771	179	4	e−t∥(θu)(t)∥	e−t∥(θu)(t)∥	PROPN
ejde-771	179	5	=	=	SYM
ejde-771	179	6	0	0	PROPN
ejde-771	179	7	,	,	PUNCT
ejde-771	179	8	which	which	PRON
ejde-771	179	9	implies	imply	VERB
ejde-771	179	10	that	that	SCONJ
ejde-771	179	11	θ	θ	NOUN
ejde-771	179	12	:	:	PUNCT
ejde-771	179	13	ce(e	ce(e	X
ejde-771	179	14	)	)	PUNCT
ejde-771	179	15	→	→	SYM
ejde-771	179	16	ce(e	ce(e	NUM
ejde-771	179	17	)	)	PUNCT
ejde-771	179	18	is	be	AUX
ejde-771	179	19	well	well	ADV
ejde-771	179	20	defined	define	VERB
ejde-771	179	21	.	.	PUNCT
ejde-771	180	1	next	next	ADV
ejde-771	180	2	we	we	PRON
ejde-771	180	3	prove	prove	VERB
ejde-771	180	4	that	that	SCONJ
ejde-771	180	5	θ	θ	PROPN
ejde-771	180	6	is	be	AUX
ejde-771	180	7	continuous	continuous	ADJ
ejde-771	180	8	on	on	ADP
ejde-771	180	9	ce(e	ce(e	NUM
ejde-771	180	10	)	)	PUNCT
ejde-771	180	11	.	.	PUNCT
ejde-771	181	1	let	let	VERB
ejde-771	181	2	{	{	PUNCT
ejde-771	181	3	un	un	ADJ
ejde-771	181	4	}	}	PUNCT
ejde-771	181	5	⊂	⊂	NOUN
ejde-771	181	6	ce(e	ce(e	NOUN
ejde-771	181	7	)	)	PUNCT
ejde-771	181	8	such	such	ADJ
ejde-771	181	9	that	that	DET
ejde-771	181	10	un	un	PROPN
ejde-771	181	11	→	→	SYM
ejde-771	181	12	u	u	PROPN
ejde-771	181	13	as	as	ADP
ejde-771	181	14	n→	n→	ADV
ejde-771	181	15	∞	∞	PROPN
ejde-771	181	16	in	in	ADP
ejde-771	181	17	ce(e	ce(e	NOUN
ejde-771	181	18	)	)	PUNCT
ejde-771	181	19	.	.	PUNCT
ejde-771	182	1	from	from	ADP
ejde-771	182	2	the	the	DET
ejde-771	182	3	continuity	continuity	NOUN
ejde-771	182	4	of	of	ADP
ejde-771	182	5	g	g	NOUN
ejde-771	182	6	,	,	PUNCT
ejde-771	182	7	it	it	PRON
ejde-771	182	8	can	can	AUX
ejde-771	182	9	be	be	AUX
ejde-771	182	10	obtained	obtain	VERB
ejde-771	182	11	that	that	DET
ejde-771	182	12	sup	sup	NOUN
ejde-771	182	13	s∈[0,∞	s∈[0,∞	NOUN
ejde-771	182	14	)	)	PUNCT
ejde-771	183	1	∥g	∥g	PROPN
ejde-771	183	2	(	(	PUNCT
ejde-771	183	3	s	s	PROPN
ejde-771	183	4	,	,	PUNCT
ejde-771	183	5	un(s	un(s	NUM
ejde-771	183	6	)	)	PUNCT
ejde-771	183	7	)	)	PUNCT
ejde-771	184	1	−g	−g	NOUN
ejde-771	184	2	(	(	PUNCT
ejde-771	184	3	s	s	X
ejde-771	184	4	,	,	PUNCT
ejde-771	184	5	u(s	u(s	NUM
ejde-771	184	6	)	)	PUNCT
ejde-771	184	7	)	)	PUNCT
ejde-771	184	8	∥	∥	PUNCT
ejde-771	185	1	→	→	SYM
ejde-771	185	2	0	0	NUM
ejde-771	185	3	as	as	ADP
ejde-771	185	4	n→	n→	PROPN
ejde-771	185	5	∞.	∞.	PROPN
ejde-771	185	6	then	then	ADV
ejde-771	185	7	by	by	ADP
ejde-771	185	8	the	the	DET
ejde-771	185	9	lebesgue	lebesgue	NOUN
ejde-771	185	10	dominated	dominate	VERB
ejde-771	185	11	convergence	convergence	NOUN
ejde-771	185	12	theorem	theorem	VERB
ejde-771	185	13	,	,	PUNCT
ejde-771	185	14	∥(θun)(t)−	∥(θun)(t)−	PUNCT
ejde-771	185	15	(	(	PUNCT
ejde-771	185	16	θu)(t)∥	θu)(t)∥	PROPN
ejde-771	185	17	6	6	NUM
ejde-771	185	18	x.	x.	NOUN
ejde-771	185	19	zhang	zhang	PROPN
ejde-771	185	20	,	,	PUNCT
ejde-771	185	21	k.	k.	PROPN
ejde-771	185	22	ding	ding	PROPN
ejde-771	185	23	,	,	PUNCT
ejde-771	185	24	p.	p.	PROPN
ejde-771	185	25	chen	chen	PROPN
ejde-771	185	26	ejde-2025/44	ejde-2025/44	PROPN
ejde-771	185	27	≤m	≤m	PROPN
ejde-771	186	1	m∑	m∑	AUX
ejde-771	186	2	k=1	k=1	PROPN
ejde-771	186	3	|ak|∥λ∥	|ak|∥λ∥	VERB
ejde-771	186	4	∫	∫	PROPN
ejde-771	186	5	tk	tk	PROPN
ejde-771	186	6	0	0	PROPN
ejde-771	187	1	(	(	PUNCT
ejde-771	187	2	tk	tk	PROPN
ejde-771	187	3	−	−	PROPN
ejde-771	187	4	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	187	5	,	,	PUNCT
ejde-771	187	6	β(tk	β(tk	PROPN
ejde-771	187	7	−	−	NOUN
ejde-771	187	8	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	187	9	,	,	PUNCT
ejde-771	187	10	un(s))−g(s	un(s))−g(s	NUM
ejde-771	187	11	,	,	PUNCT
ejde-771	187	12	u(s))∥ds	u(s))∥ds	X
ejde-771	188	1	+	+	CCONJ
ejde-771	188	2	∫	∫	PROPN
ejde-771	188	3	t	t	NOUN
ejde-771	188	4	0	0	NUM
ejde-771	188	5	(	(	PUNCT
ejde-771	188	6	t−	t−	PROPN
ejde-771	188	7	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	188	8	,	,	PUNCT
ejde-771	188	9	β(tk	β(tk	PROPN
ejde-771	188	10	−	−	NOUN
ejde-771	188	11	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	188	12	,	,	PUNCT
ejde-771	188	13	un(s))−g(s	un(s))−g(s	NUM
ejde-771	188	14	,	,	PUNCT
ejde-771	188	15	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	188	16	≤	≤	PROPN
ejde-771	188	17	m	m	VERB
ejde-771	188	18	∑m	∑m	PROPN
ejde-771	188	19	i=1	i=1	PROPN
ejde-771	188	20	|ak|	|ak|	PROPN
ejde-771	188	21	1−m	1−m	NUM
ejde-771	188	22	∑m	∑m	PROPN
ejde-771	188	23	k=1	k=1	PUNCT
ejde-771	188	24	|ak|	|ak|	PROPN
ejde-771	188	25	m	m	VERB
ejde-771	188	26	∫	∫	PROPN
ejde-771	188	27	∞	∞	NUM
ejde-771	188	28	0	0	PUNCT
ejde-771	189	1	hα(τ)dτ	hα(τ)dτ	ADJ
ejde-771	189	2	∫	∫	PROPN
ejde-771	189	3	∞	∞	PROPN
ejde-771	189	4	0	0	PUNCT
ejde-771	189	5	e−|ν0|βs∥g(s	e−|ν0|βs∥g(s	NUM
ejde-771	189	6	,	,	PUNCT
ejde-771	189	7	un(s))−g(s	un(s))−g(s	NUM
ejde-771	189	8	,	,	PUNCT
ejde-771	189	9	u(s))∥ds	u(s))∥ds	X
ejde-771	189	10	+	+	NOUN
ejde-771	189	11	m	m	VERB
ejde-771	189	12	∫	∫	PROPN
ejde-771	189	13	∞	∞	PROPN
ejde-771	189	14	0	0	PUNCT
ejde-771	190	1	hα(τ)dτ	hα(τ)dτ	ADJ
ejde-771	190	2	∫	∫	PROPN
ejde-771	190	3	∞	∞	PROPN
ejde-771	190	4	0	0	PUNCT
ejde-771	190	5	e−|ν0|βs∥g(s	e−|ν0|βs∥g(s	NUM
ejde-771	190	6	,	,	PUNCT
ejde-771	190	7	un(s))−g(s	un(s))−g(s	NUM
ejde-771	190	8	,	,	PUNCT
ejde-771	190	9	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	190	10	≤	≤	PROPN
ejde-771	190	11	m	m	VERB
ejde-771	190	12	(	(	PUNCT
ejde-771	190	13	1−m	1−m	NUM
ejde-771	190	14	∑m	∑m	PROPN
ejde-771	190	15	k=1	k=1	X
ejde-771	190	16	|ak|	|ak|	PROPN
ejde-771	190	17	)	)	PUNCT
ejde-771	191	1	|ν0|β	|ν0|β	ADV
ejde-771	191	2	sup	sup	NOUN
ejde-771	191	3	s∈[0,∞	s∈[0,∞	NOUN
ejde-771	191	4	)	)	PUNCT
ejde-771	191	5	∥g(s	∥g(s	NOUN
ejde-771	191	6	,	,	PUNCT
ejde-771	191	7	un(s))−g(s	un(s))−g(s	NUM
ejde-771	191	8	,	,	PUNCT
ejde-771	191	9	u(s))∥	u(s))∥	ADP
ejde-771	191	10	→	→	SYM
ejde-771	191	11	0	0	NUM
ejde-771	191	12	as	as	ADP
ejde-771	191	13	n→	n→	PUNCT
ejde-771	191	14	∞.	∞.	PROPN
ejde-771	191	15	hence	hence	ADV
ejde-771	191	16	,	,	PUNCT
ejde-771	191	17	∥(θun)(t)−	∥(θun)(t)−	PROPN
ejde-771	191	18	(	(	PUNCT
ejde-771	191	19	θu)(t)∥e	θu)(t)∥e	NOUN
ejde-771	191	20	=	=	SYM
ejde-771	191	21	sup	sup	NOUN
ejde-771	191	22	t∈[0,∞	t∈[0,∞	NOUN
ejde-771	191	23	)	)	PUNCT
ejde-771	191	24	e−t∥(θun)(t)−	e−t∥(θun)(t)−	PROPN
ejde-771	191	25	(	(	PUNCT
ejde-771	191	26	θu)(t)∥	θu)(t)∥	PROPN
ejde-771	191	27	→	→	SYM
ejde-771	191	28	0	0	NUM
ejde-771	191	29	(	(	PUNCT
ejde-771	191	30	n→	n→	PROPN
ejde-771	191	31	∞	∞	NUM
ejde-771	191	32	)	)	PUNCT
ejde-771	191	33	,	,	PUNCT
ejde-771	191	34	which	which	PRON
ejde-771	191	35	implies	imply	VERB
ejde-771	191	36	that	that	SCONJ
ejde-771	191	37	θ	θ	NOUN
ejde-771	191	38	:	:	PUNCT
ejde-771	191	39	ce(e	ce(e	X
ejde-771	191	40	)	)	PUNCT
ejde-771	191	41	→	→	SYM
ejde-771	191	42	ce(e	ce(e	NUM
ejde-771	191	43	)	)	PUNCT
ejde-771	191	44	is	be	AUX
ejde-771	191	45	a	a	DET
ejde-771	191	46	continuous	continuous	ADJ
ejde-771	191	47	operator	operator	NOUN
ejde-771	191	48	.	.	PUNCT
ejde-771	192	1	therefore	therefore	ADV
ejde-771	192	2	,	,	PUNCT
ejde-771	192	3	one	one	PRON
ejde-771	192	4	can	can	AUX
ejde-771	192	5	deduced	deduced	VERB
ejde-771	192	6	that	that	SCONJ
ejde-771	192	7	the	the	DET
ejde-771	192	8	fixed	fix	VERB
ejde-771	192	9	points	point	NOUN
ejde-771	192	10	of	of	ADP
ejde-771	192	11	θ	θ	PROPN
ejde-771	192	12	are	be	AUX
ejde-771	192	13	mild	mild	ADJ
ejde-771	192	14	solutions	solution	NOUN
ejde-771	192	15	to	to	ADP
ejde-771	192	16	nonlocal	nonlocal	ADJ
ejde-771	192	17	problem	problem	NOUN
ejde-771	192	18	(	(	PUNCT
ejde-771	192	19	3.1	3.1	NUM
ejde-771	192	20	)	)	PUNCT
ejde-771	192	21	.	.	PUNCT
ejde-771	193	1	based	base	VERB
ejde-771	193	2	on	on	ADP
ejde-771	193	3	this	this	DET
ejde-771	193	4	fact	fact	NOUN
ejde-771	193	5	,	,	PUNCT
ejde-771	193	6	we	we	PRON
ejde-771	193	7	first	first	ADV
ejde-771	193	8	prove	prove	VERB
ejde-771	193	9	that	that	SCONJ
ejde-771	193	10	θ(sapω(e	θ(sapω(e	ADP
ejde-771	193	11	)	)	PUNCT
ejde-771	193	12	)	)	PUNCT
ejde-771	194	1	⊂	⊂	PROPN
ejde-771	194	2	sapω(e	sapω(e	NOUN
ejde-771	194	3	)	)	PUNCT
ejde-771	194	4	.	.	PUNCT
ejde-771	195	1	for	for	ADP
ejde-771	195	2	any	any	DET
ejde-771	195	3	ϵ	ϵ	X
ejde-771	195	4	>	>	X
ejde-771	195	5	0	0	PUNCT
ejde-771	195	6	and	and	CCONJ
ejde-771	195	7	u	u	PROPN
ejde-771	195	8	∈	∈	PROPN
ejde-771	195	9	sapω(e	sapω(e	NOUN
ejde-771	195	10	)	)	PUNCT
ejde-771	195	11	,	,	PUNCT
ejde-771	195	12	there	there	PRON
ejde-771	195	13	exists	exist	VERB
ejde-771	195	14	a	a	DET
ejde-771	195	15	constant	constant	ADJ
ejde-771	195	16	t1ϵ	t1ϵ	NOUN
ejde-771	195	17	>	>	X
ejde-771	195	18	0	0	NUM
ejde-771	195	19	,	,	PUNCT
ejde-771	195	20	for	for	ADP
ejde-771	195	21	t	t	PROPN
ejde-771	195	22	≥	≥	NOUN
ejde-771	195	23	t1ϵ	t1ϵ	X
ejde-771	195	24	,	,	PUNCT
ejde-771	195	25	have	have	AUX
ejde-771	195	26	∥u(t+ω)−	∥u(t+ω)−	PROPN
ejde-771	195	27	u(t)∥	u(t)∥	NOUN
ejde-771	195	28	≤	≤	NUM
ejde-771	195	29	ϵ.	ϵ.	NOUN
ejde-771	195	30	on	on	ADP
ejde-771	195	31	the	the	DET
ejde-771	195	32	one	one	NUM
ejde-771	195	33	hand	hand	NOUN
ejde-771	195	34	,	,	PUNCT
ejde-771	195	35	by	by	ADP
ejde-771	195	36	continuity	continuity	NOUN
ejde-771	195	37	of	of	ADP
ejde-771	195	38	g	g	NOUN
ejde-771	195	39	,	,	PUNCT
ejde-771	195	40	for	for	ADP
ejde-771	195	41	t	t	PROPN
ejde-771	195	42	>	>	X
ejde-771	196	1	t1ϵ	t1ϵ	X
ejde-771	196	2	,	,	PUNCT
ejde-771	196	3	∥g(t	∥g(t	ADJ
ejde-771	196	4	,	,	PUNCT
ejde-771	196	5	u(t+	u(t+	ADP
ejde-771	196	6	ω))−g(t	ω))−g(t	ADJ
ejde-771	196	7	,	,	PUNCT
ejde-771	196	8	u(t))∥	u(t))∥	VERB
ejde-771	196	9	≤	≤	NOUN
ejde-771	196	10	|ν0|β	|ν0|β	ADV
ejde-771	196	11	m	m	VERB
ejde-771	196	12	ϵ.	ϵ.	NOUN
ejde-771	196	13	(	(	PUNCT
ejde-771	196	14	3.7	3.7	NUM
ejde-771	196	15	)	)	PUNCT
ejde-771	196	16	on	on	ADP
ejde-771	196	17	the	the	DET
ejde-771	196	18	other	other	ADJ
ejde-771	196	19	hand	hand	NOUN
ejde-771	196	20	,	,	PUNCT
ejde-771	196	21	by	by	ADP
ejde-771	196	22	(	(	PUNCT
ejde-771	196	23	h3	h3	NOUN
ejde-771	196	24	)	)	PUNCT
ejde-771	196	25	,	,	PUNCT
ejde-771	196	26	there	there	PRON
ejde-771	196	27	exists	exist	VERB
ejde-771	196	28	a	a	DET
ejde-771	196	29	constant	constant	ADJ
ejde-771	196	30	t2ϵ	t2ϵ	NOUN
ejde-771	196	31	such	such	ADJ
ejde-771	196	32	that	that	PRON
ejde-771	196	33	for	for	ADP
ejde-771	196	34	t	t	PROPN
ejde-771	196	35	>	>	X
ejde-771	196	36	t2ϵ	t2ϵ	PRON
ejde-771	196	37	,	,	PUNCT
ejde-771	196	38	∥g(t+	∥g(t+	NOUN
ejde-771	196	39	ω	ω	PROPN
ejde-771	196	40	,	,	PUNCT
ejde-771	196	41	u(t+	u(t+	ADP
ejde-771	196	42	ω))−g(t	ω))−g(t	ADJ
ejde-771	196	43	,	,	PUNCT
ejde-771	196	44	u(t+	u(t+	ADJ
ejde-771	196	45	ω))∥	ω))∥	NUM
ejde-771	196	46	≤	≤	NOUN
ejde-771	196	47	|ν0|β	|ν0|β	ADV
ejde-771	196	48	m	m	VERB
ejde-771	196	49	ϵ.	ϵ.	NOUN
ejde-771	196	50	(	(	PUNCT
ejde-771	196	51	3.8	3.8	NUM
ejde-771	196	52	)	)	PUNCT
ejde-771	196	53	according	accord	VERB
ejde-771	196	54	to	to	ADP
ejde-771	196	55	(	(	PUNCT
ejde-771	196	56	2.4	2.4	NUM
ejde-771	196	57	)	)	PUNCT
ejde-771	196	58	,	,	PUNCT
ejde-771	196	59	we	we	PRON
ejde-771	196	60	let	let	VERB
ejde-771	196	61	m0	m0	PROPN
ejde-771	197	1	=	=	PROPN
ejde-771	197	2	m	m	PROPN
ejde-771	197	3	max	max	NOUN
ejde-771	197	4	{	{	PUNCT
ejde-771	197	5	sup	sup	PROPN
ejde-771	197	6	t≥0	t≥0	NOUN
ejde-771	197	7	eα(−|ν0|βtα)(1	eα(−|ν0|βtα)(1	PROPN
ejde-771	197	8	+	+	NOUN
ejde-771	197	9	t)α	t)α	NOUN
ejde-771	197	10	,	,	PUNCT
ejde-771	197	11	sup	sup	PROPN
ejde-771	197	12	t≥0	t≥0	PROPN
ejde-771	197	13	eα	eα	NOUN
ejde-771	197	14	,	,	PUNCT
ejde-771	197	15	α(−|ν0|βtα)(1	α(−|ν0|βtα)(1	NOUN
ejde-771	197	16	+	+	PUNCT
ejde-771	197	17	t)2α	t)2α	PUNCT
ejde-771	197	18	}	}	PUNCT
ejde-771	197	19	,	,	PUNCT
ejde-771	197	20	then	then	ADV
ejde-771	197	21	∥jα	∥jα	PROPN
ejde-771	197	22	,	,	PUNCT
ejde-771	197	23	β(t)∥	β(t)∥	PUNCT
ejde-771	197	24	≤	≤	NOUN
ejde-771	197	25	m0	m0	NOUN
ejde-771	197	26	(	(	PUNCT
ejde-771	197	27	1	1	NUM
ejde-771	197	28	+	+	NUM
ejde-771	197	29	t)α	t)α	NOUN
ejde-771	197	30	,	,	PUNCT
ejde-771	197	31	∥kα	∥kα	PROPN
ejde-771	197	32	,	,	PUNCT
ejde-771	197	33	β(t)∥	β(t)∥	PUNCT
ejde-771	197	34	≤	≤	NOUN
ejde-771	197	35	m0	m0	NOUN
ejde-771	197	36	(	(	PUNCT
ejde-771	197	37	1	1	NUM
ejde-771	197	38	+	+	CCONJ
ejde-771	197	39	t)2α	t)2α	NOUN
ejde-771	197	40	,	,	PUNCT
ejde-771	197	41	t	t	PROPN
ejde-771	197	42	≥	≥	NUM
ejde-771	197	43	0	0	NUM
ejde-771	197	44	.	.	PUNCT
ejde-771	197	45	(	(	PUNCT
ejde-771	197	46	3.9	3.9	NUM
ejde-771	197	47	)	)	PUNCT
ejde-771	197	48	hence	hence	ADV
ejde-771	197	49	,	,	PUNCT
ejde-771	197	50	for	for	ADP
ejde-771	197	51	t	t	PROPN
ejde-771	197	52	>	>	X
ejde-771	197	53	max{t1ϵ	max{t1ϵ	PROPN
ejde-771	197	54	,	,	PUNCT
ejde-771	197	55	t2ϵ	t2ϵ	ADV
ejde-771	197	56	}	}	PUNCT
ejde-771	197	57	,	,	PUNCT
ejde-771	197	58	it	it	PRON
ejde-771	197	59	follows	follow	VERB
ejde-771	197	60	from	from	ADP
ejde-771	197	61	(	(	PUNCT
ejde-771	197	62	3.4	3.4	NUM
ejde-771	197	63	)	)	PUNCT
ejde-771	197	64	that	that	SCONJ
ejde-771	197	65	(	(	PUNCT
ejde-771	197	66	θu)(t+	θu)(t+	PROPN
ejde-771	197	67	ω)−	ω)−	PROPN
ejde-771	197	68	(	(	PUNCT
ejde-771	197	69	θu)(t	θu)(t	PROPN
ejde-771	197	70	)	)	PUNCT
ejde-771	197	71	=	=	SYM
ejde-771	197	72	4∑	4∑	NUM
ejde-771	197	73	i=1	i=1	PROPN
ejde-771	197	74	bi(t	bi(t	NOUN
ejde-771	197	75	)	)	PUNCT
ejde-771	197	76	,	,	PUNCT
ejde-771	197	77	where	where	SCONJ
ejde-771	197	78	b1(t	b1(t	X
ejde-771	197	79	)	)	PUNCT
ejde-771	197	80	=	=	SYM
ejde-771	197	81	(	(	PUNCT
ejde-771	197	82	jα	jα	NOUN
ejde-771	197	83	,	,	PUNCT
ejde-771	197	84	β(t+	β(t+	NOUN
ejde-771	197	85	ω)−	ω)−	PROPN
ejde-771	197	86	jα	jα	NOUN
ejde-771	197	87	,	,	PUNCT
ejde-771	197	88	β(t	β(t	PROPN
ejde-771	197	89	)	)	PUNCT
ejde-771	197	90	)	)	PUNCT
ejde-771	198	1	(	(	PUNCT
ejde-771	198	2	λu0	λu0	X
ejde-771	198	3	+	+	CCONJ
ejde-771	198	4	m∑	m∑	ADV
ejde-771	198	5	k=1	k=1	PROPN
ejde-771	198	6	akλ	akλ	PROPN
ejde-771	198	7	∫	∫	PROPN
ejde-771	198	8	tk	tk	PROPN
ejde-771	198	9	0	0	PROPN
ejde-771	199	1	(	(	PUNCT
ejde-771	199	2	tk	tk	PROPN
ejde-771	199	3	−	−	PROPN
ejde-771	199	4	s)α−1kα	s)α−1kα	PROPN
ejde-771	199	5	,	,	PUNCT
ejde-771	199	6	β(tk	β(tk	PROPN
ejde-771	199	7	−	−	NOUN
ejde-771	199	8	s)g(s	s)g(s	ADJ
ejde-771	199	9	,	,	PUNCT
ejde-771	199	10	u(s))ds	u(s))ds	PROPN
ejde-771	199	11	)	)	PUNCT
ejde-771	199	12	,	,	PUNCT
ejde-771	199	13	b2(t	b2(t	PROPN
ejde-771	199	14	)	)	PUNCT
ejde-771	199	15	=	=	SYM
ejde-771	200	1	∫	∫	PROPN
ejde-771	200	2	ω	ω	NUM
ejde-771	200	3	0	0	NUM
ejde-771	200	4	(	(	PUNCT
ejde-771	200	5	t+	t+	NOUN
ejde-771	200	6	ω	ω	PROPN
ejde-771	200	7	−	−	X
ejde-771	200	8	s)α−1kα	s)α−1kα	PROPN
ejde-771	200	9	,	,	PUNCT
ejde-771	200	10	β(t+	β(t+	NOUN
ejde-771	200	11	ω	ω	NUM
ejde-771	200	12	−	−	NOUN
ejde-771	200	13	s)g(s	s)g(s	NOUN
ejde-771	200	14	,	,	PUNCT
ejde-771	200	15	u(s))ds	u(s))ds	PROPN
ejde-771	200	16	,	,	PUNCT
ejde-771	200	17	b3(t	b3(t	PROPN
ejde-771	200	18	)	)	PUNCT
ejde-771	200	19	=	=	SYM
ejde-771	201	1	∫	∫	PROPN
ejde-771	201	2	t	t	PROPN
ejde-771	201	3	0	0	NUM
ejde-771	201	4	(	(	PUNCT
ejde-771	201	5	t−	t−	PROPN
ejde-771	201	6	s)α−1kα	s)α−1kα	PROPN
ejde-771	201	7	,	,	PUNCT
ejde-771	201	8	β(t−	β(t−	PROPN
ejde-771	201	9	s	s	X
ejde-771	201	10	)	)	PUNCT
ejde-771	201	11	(	(	PUNCT
ejde-771	201	12	g(s	g(s	PROPN
ejde-771	201	13	,	,	PUNCT
ejde-771	201	14	u(s+	u(s+	ADJ
ejde-771	201	15	ω))−g(s	ω))−g(s	NOUN
ejde-771	201	16	,	,	PUNCT
ejde-771	201	17	u(s	u(s	NUM
ejde-771	201	18	)	)	PUNCT
ejde-771	201	19	)	)	PUNCT
ejde-771	201	20	)	)	PUNCT
ejde-771	202	1	ds	ds	PROPN
ejde-771	202	2	,	,	PUNCT
ejde-771	202	3	b4(t	b4(t	NUM
ejde-771	202	4	)	)	PUNCT
ejde-771	202	5	=	=	SYM
ejde-771	202	6	∫	∫	PROPN
ejde-771	202	7	t	t	PROPN
ejde-771	202	8	0	0	NUM
ejde-771	202	9	(	(	PUNCT
ejde-771	202	10	t−	t−	PROPN
ejde-771	202	11	s)α−1kα	s)α−1kα	PROPN
ejde-771	202	12	,	,	PUNCT
ejde-771	202	13	β(t−	β(t−	PROPN
ejde-771	202	14	s	s	X
ejde-771	202	15	)	)	PUNCT
ejde-771	202	16	(	(	PUNCT
ejde-771	202	17	g(s+	g(s+	NUM
ejde-771	202	18	ω	ω	NOUN
ejde-771	202	19	,	,	PUNCT
ejde-771	202	20	u(s+	u(s+	ADJ
ejde-771	202	21	ω))−g(s	ω))−g(s	NOUN
ejde-771	202	22	,	,	PUNCT
ejde-771	202	23	u(s+	u(s+	PROPN
ejde-771	202	24	ω	ω	NUM
ejde-771	202	25	)	)	PUNCT
ejde-771	202	26	)	)	PUNCT
ejde-771	202	27	)	)	PUNCT
ejde-771	203	1	ds	ds	PROPN
ejde-771	203	2	.	.	PROPN
ejde-771	203	3	ejde-2025/44	ejde-2025/44	PROPN
ejde-771	203	4	time	time	NOUN
ejde-771	203	5	-	-	PUNCT
ejde-771	203	6	space	space	NOUN
ejde-771	203	7	fractional	fractional	ADJ
ejde-771	203	8	reaction	reaction	NOUN
ejde-771	203	9	-	-	PUNCT
ejde-771	203	10	diffusion	diffusion	NOUN
ejde-771	203	11	equations	equation	NOUN
ejde-771	203	12	7	7	NUM
ejde-771	203	13	this	this	PRON
ejde-771	203	14	implies	imply	VERB
ejde-771	203	15	that	that	SCONJ
ejde-771	203	16	∥(θu)(t+	∥(θu)(t+	PROPN
ejde-771	203	17	ω)−	ω)−	PROPN
ejde-771	203	18	(	(	PUNCT
ejde-771	203	19	θu)(t)∥	θu)(t)∥	PROPN
ejde-771	203	20	≤	≤	ADV
ejde-771	203	21	4∑	4∑	NUM
ejde-771	203	22	i=1	i=1	PROPN
ejde-771	203	23	∥bi(t)∥.	∥bi(t)∥.	NOUN
ejde-771	203	24	let	let	VERB
ejde-771	203	25	us	we	PRON
ejde-771	203	26	start	start	VERB
ejde-771	203	27	with	with	ADP
ejde-771	203	28	estimations	estimation	NOUN
ejde-771	203	29	of	of	ADP
ejde-771	203	30	∥b1(t)∥	∥b1(t)∥	NOUN
ejde-771	203	31	and	and	CCONJ
ejde-771	203	32	∥b2(t)∥.	∥b2(t)∥.	ADJ
ejde-771	203	33	by	by	ADP
ejde-771	203	34	(	(	PUNCT
ejde-771	203	35	3.9	3.9	NUM
ejde-771	203	36	)	)	PUNCT
ejde-771	203	37	,	,	PUNCT
ejde-771	203	38	one	one	PRON
ejde-771	203	39	can	can	AUX
ejde-771	203	40	see	see	VERB
ejde-771	203	41	that	that	SCONJ
ejde-771	203	42	∥b1(t)∥	∥b1(t)∥	NOUN
ejde-771	203	43	=	=	SYM
ejde-771	203	44	∥	∥	X
ejde-771	203	45	(	(	PUNCT
ejde-771	203	46	jα	jα	NOUN
ejde-771	203	47	,	,	PUNCT
ejde-771	203	48	β(t+	β(t+	NOUN
ejde-771	203	49	ω)−	ω)−	PROPN
ejde-771	203	50	jα	jα	NOUN
ejde-771	203	51	,	,	PUNCT
ejde-771	203	52	β(t	β(t	PROPN
ejde-771	203	53	)	)	PUNCT
ejde-771	203	54	)	)	PUNCT
ejde-771	203	55	∥∥λu0	∥∥λu0	VERB
ejde-771	204	1	+	+	CCONJ
ejde-771	204	2	m∑	m∑	PROPN
ejde-771	204	3	k=1	k=1	PROPN
ejde-771	204	4	|ak|λ	|ak|λ	PROPN
ejde-771	204	5	∫	∫	PROPN
ejde-771	204	6	tk	tk	PROPN
ejde-771	204	7	0	0	PROPN
ejde-771	204	8	(	(	PUNCT
ejde-771	204	9	tk	tk	PROPN
ejde-771	204	10	−	−	PROPN
ejde-771	204	11	s)α−1kα	s)α−1kα	PROPN
ejde-771	204	12	,	,	PUNCT
ejde-771	204	13	β(tk	β(tk	PROPN
ejde-771	204	14	−	−	NOUN
ejde-771	204	15	s)g(s	s)g(s	NOUN
ejde-771	204	16	,	,	PUNCT
ejde-771	204	17	u(s))ds∥	u(s))ds∥	ADV
ejde-771	204	18	≤	≤	ADJ
ejde-771	204	19	2m0	2m0	NUM
ejde-771	204	20	(	(	PUNCT
ejde-771	204	21	1	1	NUM
ejde-771	204	22	+	+	NUM
ejde-771	204	23	t)α	t)α	AUX
ejde-771	204	24	(	(	PUNCT
ejde-771	204	25	∥λu0∥+	∥λu0∥+	PROPN
ejde-771	204	26	∥λ∥	∥λ∥	VERB
ejde-771	204	27	m∑	m∑	VERB
ejde-771	204	28	k=1	k=1	PROPN
ejde-771	204	29	|ak|	|ak|	PROPN
ejde-771	204	30	∫	∫	PROPN
ejde-771	204	31	tk	tk	PROPN
ejde-771	204	32	0	0	PROPN
ejde-771	205	1	(	(	PUNCT
ejde-771	205	2	tk	tk	PROPN
ejde-771	205	3	−	−	PROPN
ejde-771	205	4	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	205	5	,	,	PUNCT
ejde-771	205	6	β(tk	β(tk	PROPN
ejde-771	205	7	−	−	NOUN
ejde-771	206	1	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	206	2	,	,	PUNCT
ejde-771	206	3	u(s))∥ds	u(s))∥ds	X
ejde-771	206	4	)	)	PUNCT
ejde-771	206	5	and	and	CCONJ
ejde-771	206	6	∥b2(t)∥	∥b2(t)∥	NOUN
ejde-771	206	7	=	=	SYM
ejde-771	207	1	∥	∥	NUM
ejde-771	207	2	∫	∫	PROPN
ejde-771	207	3	ω	ω	NOUN
ejde-771	207	4	0	0	NUM
ejde-771	207	5	(	(	PUNCT
ejde-771	207	6	t+	t+	NOUN
ejde-771	207	7	ω	ω	PROPN
ejde-771	207	8	−	−	X
ejde-771	207	9	s)α−1kα	s)α−1kα	PROPN
ejde-771	207	10	,	,	PUNCT
ejde-771	207	11	β(t+	β(t+	NOUN
ejde-771	207	12	ω	ω	NUM
ejde-771	208	1	−	−	NOUN
ejde-771	208	2	s)g(s	s)g(s	NOUN
ejde-771	208	3	,	,	PUNCT
ejde-771	208	4	u(s))ds∥	u(s))ds∥	ADP
ejde-771	208	5	≤	≤	NUM
ejde-771	208	6	∫	∫	PROPN
ejde-771	208	7	ω	ω	NOUN
ejde-771	208	8	0	0	NUM
ejde-771	208	9	(	(	PUNCT
ejde-771	208	10	t+	t+	X
ejde-771	208	11	ω	ω	NUM
ejde-771	208	12	−	−	PROPN
ejde-771	208	13	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	208	14	,	,	PUNCT
ejde-771	208	15	β(t+	β(t+	NOUN
ejde-771	208	16	ω	ω	NUM
ejde-771	208	17	−	−	NOUN
ejde-771	208	18	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	208	19	,	,	PUNCT
ejde-771	208	20	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	208	21	≤	≤	NUM
ejde-771	208	22	∫	∫	PROPN
ejde-771	208	23	ω	ω	NOUN
ejde-771	208	24	0	0	NUM
ejde-771	208	25	(	(	PUNCT
ejde-771	208	26	t+	t+	NOUN
ejde-771	208	27	ω	ω	NUM
ejde-771	208	28	−	−	NOUN
ejde-771	208	29	s)α−1	s)α−1	NOUN
ejde-771	208	30	(	(	PUNCT
ejde-771	208	31	a1∥u(s)∥+a0)m0	a1∥u(s)∥+a0)m0	X
ejde-771	208	32	(	(	PUNCT
ejde-771	208	33	1	1	NUM
ejde-771	208	34	+	+	NUM
ejde-771	208	35	t+	t+	NOUN
ejde-771	208	36	ω	ω	NUM
ejde-771	208	37	−	−	PROPN
ejde-771	208	38	s)2α	s)2α	NOUN
ejde-771	208	39	ds	ds	PROPN
ejde-771	208	40	≤	≤	NOUN
ejde-771	208	41	(	(	PUNCT
ejde-771	208	42	a1∥u∥c	a1∥u∥c	PROPN
ejde-771	208	43	+	+	NUM
ejde-771	208	44	a0	a0	NOUN
ejde-771	208	45	)	)	PUNCT
ejde-771	208	46	m0((t+	m0((t+	NOUN
ejde-771	208	47	ω)α	ω)α	NOUN
ejde-771	209	1	−	−	PROPN
ejde-771	209	2	tα	tα	PROPN
ejde-771	209	3	)	)	PUNCT
ejde-771	209	4	α(1	α(1	PROPN
ejde-771	210	1	+	+	PUNCT
ejde-771	210	2	t)2α	t)2α	ADJ
ejde-771	210	3	≤	≤	NOUN
ejde-771	210	4	(	(	PUNCT
ejde-771	210	5	a1∥u∥c	a1∥u∥c	PROPN
ejde-771	210	6	+	+	PROPN
ejde-771	210	7	a0	a0	PROPN
ejde-771	210	8	)	)	PUNCT
ejde-771	210	9	m0ω	m0ω	ADP
ejde-771	210	10	α	α	PROPN
ejde-771	210	11	α(1	α(1	PROPN
ejde-771	210	12	+	+	X
ejde-771	210	13	t)2α	t)2α	PUNCT
ejde-771	210	14	.	.	PUNCT
ejde-771	211	1	by	by	ADP
ejde-771	211	2	(	(	PUNCT
ejde-771	211	3	h1	h1	PROPN
ejde-771	211	4	)	)	PUNCT
ejde-771	211	5	,	,	PUNCT
ejde-771	211	6	(	(	PUNCT
ejde-771	211	7	3.7	3.7	NUM
ejde-771	211	8	)	)	PUNCT
ejde-771	211	9	and	and	CCONJ
ejde-771	211	10	(	(	PUNCT
ejde-771	211	11	3.9	3.9	NUM
ejde-771	211	12	)	)	PUNCT
ejde-771	211	13	,	,	PUNCT
ejde-771	211	14	one	one	PRON
ejde-771	211	15	can	can	AUX
ejde-771	211	16	obtain	obtain	VERB
ejde-771	211	17	that	that	DET
ejde-771	211	18	∥b3(t)∥	∥b3(t)∥	PROPN
ejde-771	211	19	=	=	PUNCT
ejde-771	212	1	∥	∥	NUM
ejde-771	212	2	∫	∫	NOUN
ejde-771	212	3	tϵ	tϵ	NOUN
ejde-771	212	4	0	0	NUM
ejde-771	212	5	(	(	PUNCT
ejde-771	212	6	t−	t−	PROPN
ejde-771	212	7	s)α−1kα	s)α−1kα	PROPN
ejde-771	212	8	,	,	PUNCT
ejde-771	212	9	β(t−	β(t−	PROPN
ejde-771	212	10	s	s	X
ejde-771	212	11	)	)	PUNCT
ejde-771	212	12	(	(	PUNCT
ejde-771	212	13	g(s	g(s	PROPN
ejde-771	212	14	,	,	PUNCT
ejde-771	212	15	u(s+	u(s+	ADJ
ejde-771	212	16	ω))−g(s	ω))−g(s	NOUN
ejde-771	212	17	,	,	PUNCT
ejde-771	212	18	u(s	u(s	NUM
ejde-771	212	19	)	)	PUNCT
ejde-771	212	20	)	)	PUNCT
ejde-771	212	21	)	)	PUNCT
ejde-771	213	1	ds∥	ds∥	NOUN
ejde-771	213	2	+	+	CCONJ
ejde-771	213	3	∥	∥	NUM
ejde-771	213	4	∫	∫	PROPN
ejde-771	213	5	t	t	PROPN
ejde-771	213	6	tϵ	tϵ	INTJ
ejde-771	213	7	(	(	PUNCT
ejde-771	213	8	t−	t−	PROPN
ejde-771	213	9	s)α−1kα	s)α−1kα	PROPN
ejde-771	213	10	,	,	PUNCT
ejde-771	213	11	β(t−	β(t−	PROPN
ejde-771	213	12	s	s	X
ejde-771	213	13	)	)	PUNCT
ejde-771	213	14	(	(	PUNCT
ejde-771	213	15	g(s	g(s	PROPN
ejde-771	213	16	,	,	PUNCT
ejde-771	213	17	u(s+	u(s+	ADJ
ejde-771	213	18	ω))−g(s	ω))−g(s	NOUN
ejde-771	213	19	,	,	PUNCT
ejde-771	213	20	u(s	u(s	NUM
ejde-771	213	21	)	)	PUNCT
ejde-771	213	22	)	)	PUNCT
ejde-771	213	23	)	)	PUNCT
ejde-771	214	1	ds∥	ds∥	VERB
ejde-771	214	2	≤	≤	NUM
ejde-771	214	3	∫	∫	PROPN
ejde-771	214	4	tϵ	tϵ	NOUN
ejde-771	214	5	0	0	NUM
ejde-771	215	1	(	(	PUNCT
ejde-771	215	2	t−	t−	PROPN
ejde-771	215	3	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	215	4	,	,	PUNCT
ejde-771	215	5	β(t−	β(t−	PROPN
ejde-771	215	6	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	215	7	,	,	PUNCT
ejde-771	215	8	u(s+	u(s+	ADJ
ejde-771	215	9	ω))−g(s	ω))−g(s	NOUN
ejde-771	215	10	,	,	PUNCT
ejde-771	215	11	u(s))∥ds	u(s))∥ds	X
ejde-771	215	12	+	+	CCONJ
ejde-771	216	1	∫	∫	PROPN
ejde-771	217	1	t	t	PROPN
ejde-771	218	1	tϵ	tϵ	INTJ
ejde-771	219	1	(	(	PUNCT
ejde-771	219	2	t−	t−	PROPN
ejde-771	219	3	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	219	4	,	,	PUNCT
ejde-771	219	5	β(t−	β(t−	PROPN
ejde-771	219	6	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	219	7	,	,	PUNCT
ejde-771	219	8	u(s+	u(s+	ADJ
ejde-771	219	9	ω))−g(s	ω))−g(s	NOUN
ejde-771	219	10	,	,	PUNCT
ejde-771	219	11	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	219	12	≤	≤	NOUN
ejde-771	219	13	2m0	2m0	NUM
ejde-771	220	1	∫	∫	NOUN
ejde-771	220	2	tϵ	tϵ	NOUN
ejde-771	220	3	0	0	NUM
ejde-771	220	4	(	(	PUNCT
ejde-771	220	5	t−	t−	PROPN
ejde-771	220	6	s)α−1	s)α−1	NOUN
ejde-771	220	7	(	(	PUNCT
ejde-771	220	8	1	1	NUM
ejde-771	220	9	+	+	ADP
ejde-771	220	10	t−	t−	PRON
ejde-771	220	11	s)2α	s)2α	NOUN
ejde-771	220	12	(	(	PUNCT
ejde-771	220	13	a1∥u(s)∥+a0)ds	a1∥u(s)∥+a0)ds	PUNCT
ejde-771	221	1	+	+	NUM
ejde-771	221	2	∫	∫	PROPN
ejde-771	221	3	t	t	PROPN
ejde-771	221	4	0	0	NUM
ejde-771	221	5	(	(	PUNCT
ejde-771	221	6	t−	t−	PROPN
ejde-771	221	7	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	221	8	,	,	PUNCT
ejde-771	221	9	β(t−	β(t−	PROPN
ejde-771	222	1	s)∥ds	s)∥ds	PROPN
ejde-771	222	2	|ν0|	|ν0|	NOUN
ejde-771	222	3	β	β	X
ejde-771	222	4	m	m	VERB
ejde-771	222	5	ϵ	ϵ	PRON
ejde-771	222	6	≤	≤	NUM
ejde-771	222	7	2m0(a1∥u∥c	2m0(a1∥u∥c	PROPN
ejde-771	223	1	+	+	NOUN
ejde-771	223	2	a0	a0	NOUN
ejde-771	223	3	)	)	PUNCT
ejde-771	223	4	(	(	PUNCT
ejde-771	223	5	t−	t−	PROPN
ejde-771	223	6	tϵ	tϵ	NOUN
ejde-771	223	7	)	)	PUNCT
ejde-771	223	8	−α	−α	NOUN
ejde-771	223	9	−	−	PROPN
ejde-771	223	10	t−α	t−α	PROPN
ejde-771	224	1	α	α	NOUN
ejde-771	224	2	+	+	NOUN
ejde-771	224	3	mα	mα	PROPN
ejde-771	224	4	∫	∫	PROPN
ejde-771	224	5	t	t	PROPN
ejde-771	224	6	0	0	NUM
ejde-771	224	7	(	(	PUNCT
ejde-771	224	8	(	(	PUNCT
ejde-771	224	9	t−	t−	PROPN
ejde-771	224	10	s)α−1	s)α−1	PROPN
ejde-771	224	11	∫	∫	PROPN
ejde-771	224	12	∞	∞	PROPN
ejde-771	224	13	0	0	NUM
ejde-771	224	14	τhα(τ)e	τhα(τ)e	NOUN
ejde-771	224	15	−|ν0|β(t−s)ατdτ)ds	−|ν0|β(t−s)ατdτ)ds	PROPN
ejde-771	224	16	|ν0|β	|ν0|β	ADV
ejde-771	224	17	m	m	VERB
ejde-771	224	18	ϵ	ϵ	PRON
ejde-771	224	19	≤	≤	NUM
ejde-771	224	20	2m0(a1∥u∥c	2m0(a1∥u∥c	PROPN
ejde-771	224	21	+	+	NOUN
ejde-771	224	22	a0	a0	NOUN
ejde-771	224	23	)	)	PUNCT
ejde-771	224	24	(	(	PUNCT
ejde-771	224	25	t−	t−	PROPN
ejde-771	224	26	tϵ	tϵ	NOUN
ejde-771	224	27	)	)	PUNCT
ejde-771	224	28	−α	−α	NOUN
ejde-771	224	29	−	−	PROPN
ejde-771	224	30	t−α	t−α	PROPN
ejde-771	225	1	α	α	NOUN
ejde-771	226	1	+	+	X
ejde-771	226	2	ϵ	ϵ	X
ejde-771	226	3	,	,	PUNCT
ejde-771	226	4	which	which	PRON
ejde-771	226	5	implies	imply	VERB
ejde-771	226	6	that	that	SCONJ
ejde-771	226	7	∥b3(t)∥	∥b3(t)∥	NOUN
ejde-771	226	8	tend	tend	VERB
ejde-771	226	9	to	to	ADP
ejde-771	226	10	0	0	NUM
ejde-771	226	11	as	as	SCONJ
ejde-771	226	12	t→	t→	DET
ejde-771	226	13	∞.	∞.	PROPN
ejde-771	226	14	similarly	similarly	ADV
ejde-771	226	15	,	,	PUNCT
ejde-771	226	16	by	by	ADP
ejde-771	226	17	(	(	PUNCT
ejde-771	226	18	h1	h1	PROPN
ejde-771	226	19	)	)	PUNCT
ejde-771	226	20	,	,	PUNCT
ejde-771	226	21	(	(	PUNCT
ejde-771	226	22	3.8	3.8	NUM
ejde-771	226	23	)	)	PUNCT
ejde-771	226	24	and	and	CCONJ
ejde-771	226	25	(	(	PUNCT
ejde-771	226	26	3.9	3.9	NUM
ejde-771	226	27	)	)	PUNCT
ejde-771	226	28	,	,	PUNCT
ejde-771	226	29	we	we	PRON
ejde-771	226	30	can	can	AUX
ejde-771	226	31	get	get	VERB
ejde-771	226	32	that	that	SCONJ
ejde-771	226	33	∥b4(t)∥	∥b4(t)∥	PROPN
ejde-771	226	34	tend	tend	VERB
ejde-771	226	35	to	to	ADP
ejde-771	226	36	0	0	NUM
ejde-771	226	37	as	as	ADP
ejde-771	226	38	t→	t→	DET
ejde-771	226	39	∞.	∞.	PROPN
ejde-771	226	40	summing	sum	VERB
ejde-771	226	41	up	up	ADP
ejde-771	226	42	,	,	PUNCT
ejde-771	226	43	it	it	PRON
ejde-771	226	44	follows	follow	VERB
ejde-771	226	45	from	from	ADP
ejde-771	226	46	above	above	ADP
ejde-771	226	47	results	result	NOUN
ejde-771	226	48	for	for	ADP
ejde-771	226	49	∥bi(t)∥(i	∥bi(t)∥(i	ADJ
ejde-771	226	50	=	=	SYM
ejde-771	226	51	1	1	NUM
ejde-771	226	52	,	,	PUNCT
ejde-771	226	53	2	2	NUM
ejde-771	226	54	,	,	PUNCT
ejde-771	226	55	3	3	NUM
ejde-771	226	56	,	,	PUNCT
ejde-771	226	57	4	4	NUM
ejde-771	226	58	)	)	PUNCT
ejde-771	226	59	that	that	PRON
ejde-771	226	60	θu	θu	VERB
ejde-771	226	61	∈	∈	PROPN
ejde-771	226	62	sapω(e	sapω(e	NOUN
ejde-771	226	63	)	)	PUNCT
ejde-771	226	64	,	,	PUNCT
ejde-771	226	65	which	which	PRON
ejde-771	226	66	justifies	justify	VERB
ejde-771	226	67	the	the	DET
ejde-771	226	68	following	follow	VERB
ejde-771	226	69	inclusion	inclusion	NOUN
ejde-771	226	70	,	,	PUNCT
ejde-771	226	71	that	that	PRON
ejde-771	226	72	is	be	AUX
ejde-771	226	73	θ(sapω(e	θ(sapω(e	ADP
ejde-771	226	74	)	)	PUNCT
ejde-771	226	75	)	)	PUNCT
ejde-771	227	1	⊂	⊂	PROPN
ejde-771	227	2	sapω(e	sapω(e	NOUN
ejde-771	227	3	)	)	PUNCT
ejde-771	227	4	.	.	PUNCT
ejde-771	228	1	8	8	NUM
ejde-771	228	2	x.	x.	NOUN
ejde-771	228	3	zhang	zhang	PROPN
ejde-771	228	4	,	,	PUNCT
ejde-771	228	5	k.	k.	PROPN
ejde-771	228	6	ding	ding	PROPN
ejde-771	228	7	,	,	PUNCT
ejde-771	228	8	p.	p.	PROPN
ejde-771	228	9	chen	chen	PROPN
ejde-771	228	10	ejde-2025/44	ejde-2025/44	PROPN
ejde-771	228	11	in	in	ADP
ejde-771	228	12	what	what	PRON
ejde-771	228	13	follows	follow	VERB
ejde-771	228	14	,	,	PUNCT
ejde-771	228	15	we	we	PRON
ejde-771	228	16	prove	prove	VERB
ejde-771	228	17	the	the	DET
ejde-771	228	18	existence	existence	NOUN
ejde-771	228	19	of	of	ADP
ejde-771	228	20	positive	positive	ADJ
ejde-771	228	21	solutions	solution	NOUN
ejde-771	228	22	by	by	ADP
ejde-771	228	23	a	a	DET
ejde-771	228	24	monotone	monotone	ADJ
ejde-771	228	25	iterative	iterative	NOUN
ejde-771	228	26	technique	technique	NOUN
ejde-771	228	27	.	.	PUNCT
ejde-771	229	1	for	for	ADP
ejde-771	229	2	any	any	DET
ejde-771	229	3	u	u	NOUN
ejde-771	229	4	,	,	PUNCT
ejde-771	229	5	v	v	PROPN
ejde-771	229	6	∈	∈	NOUN
ejde-771	229	7	pe	pe	NOUN
ejde-771	229	8	with	with	ADP
ejde-771	229	9	u	u	NOUN
ejde-771	229	10	≤	≤	NUM
ejde-771	229	11	v	v	NOUN
ejde-771	229	12	,	,	PUNCT
ejde-771	229	13	by	by	ADP
ejde-771	229	14	(	(	PUNCT
ejde-771	229	15	h2	h2	NOUN
ejde-771	229	16	)	)	PUNCT
ejde-771	229	17	,	,	PUNCT
ejde-771	229	18	(	(	PUNCT
ejde-771	229	19	3.4	3.4	NUM
ejde-771	229	20	)	)	PUNCT
ejde-771	229	21	,	,	PUNCT
ejde-771	229	22	u0	u0	ADJ
ejde-771	229	23	≥	≥	NUM
ejde-771	229	24	θ	θ	PROPN
ejde-771	229	25	,	,	PUNCT
ejde-771	229	26	the	the	DET
ejde-771	229	27	positivity	positivity	NOUN
ejde-771	229	28	of	of	ADP
ejde-771	229	29	jα	jα	PROPN
ejde-771	229	30	,	,	PUNCT
ejde-771	229	31	β(t	β(t	PROPN
ejde-771	229	32	)	)	PUNCT
ejde-771	229	33	and	and	CCONJ
ejde-771	229	34	kα	kα	PROPN
ejde-771	229	35	,	,	PUNCT
ejde-771	229	36	β(t	β(t	PROPN
ejde-771	229	37	)	)	PUNCT
ejde-771	229	38	,	,	PUNCT
ejde-771	229	39	one	one	PRON
ejde-771	229	40	can	can	AUX
ejde-771	229	41	find	find	VERB
ejde-771	229	42	that	that	SCONJ
ejde-771	229	43	for	for	ADP
ejde-771	229	44	all	all	DET
ejde-771	229	45	t	t	NOUN
ejde-771	229	46	∈	∈	PROPN
ejde-771	230	1	[	[	X
ejde-771	230	2	0,∞	0,∞	NOUN
ejde-771	230	3	)	)	PUNCT
ejde-771	230	4	,	,	PUNCT
ejde-771	230	5	θ	θ	PROPN
ejde-771	230	6	≤	≤	NOUN
ejde-771	230	7	(	(	PUNCT
ejde-771	230	8	θu)(t	θu)(t	PROPN
ejde-771	230	9	)	)	PUNCT
ejde-771	230	10	≤	≤	NOUN
ejde-771	230	11	(	(	PUNCT
ejde-771	230	12	θv)(t	θv)(t	NUM
ejde-771	230	13	)	)	PUNCT
ejde-771	230	14	.	.	PUNCT
ejde-771	231	1	thus	thus	ADV
ejde-771	231	2	θ	θ	PROPN
ejde-771	231	3	is	be	AUX
ejde-771	231	4	a	a	DET
ejde-771	231	5	monotonically	monotonically	ADV
ejde-771	231	6	increasing	increase	VERB
ejde-771	231	7	operator	operator	NOUN
ejde-771	231	8	.	.	PUNCT
ejde-771	232	1	let	let	VERB
ejde-771	232	2	v0	v0	NOUN
ejde-771	232	3	=	=	SYM
ejde-771	232	4	θ	θ	PROPN
ejde-771	232	5	∈	∈	PROPN
ejde-771	232	6	pe	pe	PROPN
ejde-771	232	7	∩	∩	PROPN
ejde-771	232	8	sapω(e	sapω(e	NOUN
ejde-771	232	9	)	)	PUNCT
ejde-771	232	10	and	and	CCONJ
ejde-771	232	11	define	define	VERB
ejde-771	232	12	a	a	DET
ejde-771	232	13	sequence	sequence	NOUN
ejde-771	232	14	{	{	PUNCT
ejde-771	232	15	vn	vn	NOUN
ejde-771	232	16	}	}	PUNCT
ejde-771	232	17	by	by	ADP
ejde-771	232	18	vn	vn	NOUN
ejde-771	232	19	=	=	SYM
ejde-771	232	20	θvn−1	θvn−1	PROPN
ejde-771	232	21	,	,	PUNCT
ejde-771	232	22	n	n	NOUN
ejde-771	232	23	=	=	SYM
ejde-771	232	24	1	1	NUM
ejde-771	232	25	,	,	PUNCT
ejde-771	232	26	2	2	NUM
ejde-771	232	27	,	,	PUNCT
ejde-771	232	28	.	.	PUNCT
ejde-771	232	29	.	.	PUNCT
ejde-771	232	30	.	.	PUNCT
ejde-771	233	1	.	.	PUNCT
ejde-771	234	1	(	(	PUNCT
ejde-771	234	2	3.10	3.10	NUM
ejde-771	234	3	)	)	PUNCT
ejde-771	234	4	it	it	PRON
ejde-771	234	5	follows	follow	VERB
ejde-771	234	6	from	from	ADP
ejde-771	234	7	the	the	DET
ejde-771	234	8	monotonicity	monotonicity	NOUN
ejde-771	234	9	of	of	ADP
ejde-771	234	10	θ	θ	PROPN
ejde-771	234	11	,	,	PUNCT
ejde-771	234	12	(	(	PUNCT
ejde-771	234	13	3.6	3.6	NUM
ejde-771	234	14	)	)	PUNCT
ejde-771	234	15	and	and	CCONJ
ejde-771	234	16	(	(	PUNCT
ejde-771	234	17	3.10	3.10	NUM
ejde-771	234	18	)	)	PUNCT
ejde-771	234	19	that	that	SCONJ
ejde-771	234	20	{	{	PUNCT
ejde-771	234	21	vn	vn	NOUN
ejde-771	234	22	}	}	PUNCT
ejde-771	234	23	⊂	⊂	PROPN
ejde-771	234	24	pe	pe	PROPN
ejde-771	234	25	∩	∩	ADJ
ejde-771	234	26	sapω(e	sapω(e	NOUN
ejde-771	234	27	)	)	PUNCT
ejde-771	234	28	and	and	CCONJ
ejde-771	234	29	v0	v0	PROPN
ejde-771	234	30	≤	≤	NUM
ejde-771	234	31	v1	v1	PROPN
ejde-771	234	32	≤	≤	NUM
ejde-771	234	33	·	·	PUNCT
ejde-771	234	34	·	·	PUNCT
ejde-771	234	35	·	·	PUNCT
ejde-771	235	1	≤	≤	NUM
ejde-771	235	2	vn	vn	VERB
ejde-771	235	3	≤	≤	NUM
ejde-771	235	4	.	.	PUNCT
ejde-771	235	5	.	.	PUNCT
ejde-771	235	6	.	.	PUNCT
ejde-771	236	1	,	,	PUNCT
ejde-771	236	2	(	(	PUNCT
ejde-771	236	3	3.11	3.11	NUM
ejde-771	236	4	)	)	PUNCT
ejde-771	236	5	∥vn∥e	∥vn∥e	NOUN
ejde-771	236	6	≤	≤	PROPN
ejde-771	236	7	φ+	φ+	NOUN
ejde-771	236	8	ψ∥vn−1∥e	ψ∥vn−1∥e	NOUN
ejde-771	236	9	.	.	PUNCT
ejde-771	237	1	(	(	PUNCT
ejde-771	237	2	3.12	3.12	NUM
ejde-771	237	3	)	)	PUNCT
ejde-771	237	4	since	since	SCONJ
ejde-771	237	5	∥v0∥e	∥v0∥e	PROPN
ejde-771	237	6	≡	≡	PROPN
ejde-771	237	7	0	0	NUM
ejde-771	237	8	,	,	PUNCT
ejde-771	237	9	by	by	ADP
ejde-771	237	10	(	(	PUNCT
ejde-771	237	11	3.12	3.12	NUM
ejde-771	237	12	)	)	PUNCT
ejde-771	237	13	,	,	PUNCT
ejde-771	237	14	one	one	PRON
ejde-771	237	15	can	can	AUX
ejde-771	237	16	find	find	VERB
ejde-771	237	17	that	that	DET
ejde-771	237	18	∥vn∥e	∥vn∥e	PROPN
ejde-771	237	19	≤	≤	PROPN
ejde-771	237	20	φ+	φ+	X
ejde-771	237	21	φψ	φψ	X
ejde-771	237	22	+	+	CCONJ
ejde-771	237	23	φψ2	φψ2	VERB
ejde-771	237	24	+	+	X
ejde-771	237	25	·	·	PUNCT
ejde-771	237	26	·	·	PUNCT
ejde-771	237	27	·	·	PUNCT
ejde-771	238	1	+	+	NUM
ejde-771	238	2	φψn−1	φψn−1	PROPN
ejde-771	238	3	=	=	SYM
ejde-771	238	4	φ	φ	PROPN
ejde-771	238	5	1−	1−	NUM
ejde-771	238	6	ψn	ψn	INTJ
ejde-771	238	7	1−	1−	NUM
ejde-771	238	8	ψ	ψ	NOUN
ejde-771	238	9	≤	≤	NUM
ejde-771	238	10	φ	φ	NUM
ejde-771	238	11	1−	1−	NUM
ejde-771	238	12	ψ	ψ	NOUN
ejde-771	238	13	,	,	PUNCT
ejde-771	238	14	(	(	PUNCT
ejde-771	238	15	3.13	3.13	NUM
ejde-771	238	16	)	)	PUNCT
ejde-771	238	17	which	which	PRON
ejde-771	238	18	implies	imply	VERB
ejde-771	238	19	that	that	SCONJ
ejde-771	238	20	the	the	DET
ejde-771	238	21	sequence	sequence	NOUN
ejde-771	238	22	{	{	PUNCT
ejde-771	238	23	vn	vn	NOUN
ejde-771	238	24	}	}	PUNCT
ejde-771	238	25	is	be	AUX
ejde-771	238	26	uniformly	uniformly	ADV
ejde-771	238	27	bounded	bound	VERB
ejde-771	238	28	.	.	PUNCT
ejde-771	239	1	at	at	ADP
ejde-771	239	2	this	this	DET
ejde-771	239	3	level	level	NOUN
ejde-771	239	4	,	,	PUNCT
ejde-771	239	5	we	we	PRON
ejde-771	239	6	verify	verify	VERB
ejde-771	239	7	that	that	SCONJ
ejde-771	239	8	the	the	DET
ejde-771	239	9	sequence	sequence	NOUN
ejde-771	239	10	{	{	PUNCT
ejde-771	239	11	vn	vn	NOUN
ejde-771	239	12	}	}	PUNCT
ejde-771	239	13	is	be	AUX
ejde-771	239	14	uniformly	uniformly	ADV
ejde-771	239	15	convergent	convergent	NOUN
ejde-771	239	16	.	.	PUNCT
ejde-771	240	1	next	next	ADV
ejde-771	240	2	,	,	PUNCT
ejde-771	240	3	suppose	suppose	VERB
ejde-771	240	4	that	that	SCONJ
ejde-771	240	5	0	0	PUNCT
ejde-771	240	6	<	<	X
ejde-771	240	7	a	a	PRON
ejde-771	240	8	<	<	X
ejde-771	240	9	+	+	NOUN
ejde-771	240	10	∞	∞	PROPN
ejde-771	240	11	is	be	AUX
ejde-771	240	12	an	an	DET
ejde-771	240	13	arbitrary	arbitrary	ADJ
ejde-771	240	14	constant	constant	ADJ
ejde-771	240	15	,	,	PUNCT
ejde-771	240	16	we	we	PRON
ejde-771	240	17	need	need	VERB
ejde-771	240	18	to	to	PART
ejde-771	240	19	verify	verify	VERB
ejde-771	240	20	{	{	PUNCT
ejde-771	240	21	vn	vn	NOUN
ejde-771	240	22	}	}	PUNCT
ejde-771	240	23	⊂	⊂	PUNCT
ejde-771	240	24	pe∩sapω(e	pe∩sapω(e	NOUN
ejde-771	240	25	)	)	PUNCT
ejde-771	240	26	is	be	AUX
ejde-771	240	27	locally	locally	ADV
ejde-771	240	28	equicontinuous	equicontinuous	ADJ
ejde-771	240	29	in	in	ADP
ejde-771	240	30	[	[	X
ejde-771	240	31	0	0	NUM
ejde-771	240	32	,	,	PUNCT
ejde-771	240	33	a	a	DET
ejde-771	240	34	]	]	X
ejde-771	240	35	.	.	PUNCT
ejde-771	241	1	for	for	ADP
ejde-771	241	2	any	any	DET
ejde-771	241	3	u	u	PROPN
ejde-771	241	4	∈	∈	PROPN
ejde-771	241	5	{	{	PUNCT
ejde-771	241	6	vn	vn	NOUN
ejde-771	241	7	}	}	PUNCT
ejde-771	241	8	and	and	CCONJ
ejde-771	241	9	0	0	NUM
ejde-771	241	10	≤	≤	NUM
ejde-771	241	11	t1	t1	NOUN
ejde-771	241	12	≤	≤	PUNCT
ejde-771	241	13	t2	t2	PROPN
ejde-771	241	14	≤	≤	PROPN
ejde-771	241	15	a	a	PRON
ejde-771	241	16	,	,	PUNCT
ejde-771	241	17	a	a	DET
ejde-771	241	18	direct	direct	ADJ
ejde-771	241	19	computation	computation	NOUN
ejde-771	241	20	allows	allow	VERB
ejde-771	241	21	us	we	PRON
ejde-771	241	22	to	to	PART
ejde-771	241	23	obtain	obtain	VERB
ejde-771	241	24	∥(θu)(t2)−	∥(θu)(t2)−	X
ejde-771	242	1	(	(	PUNCT
ejde-771	242	2	θu)(t1)∥	θu)(t1)∥	NOUN
ejde-771	242	3	≤	≤	NOUN
ejde-771	242	4	5∑	5∑	PROPN
ejde-771	242	5	i=1	i=1	PROPN
ejde-771	242	6	di	di	PROPN
ejde-771	242	7	,	,	PUNCT
ejde-771	242	8	where	where	SCONJ
ejde-771	242	9	d1	d1	PROPN
ejde-771	242	10	=	=	SYM
ejde-771	242	11	∥jα	∥jα	PROPN
ejde-771	242	12	,	,	PUNCT
ejde-771	242	13	β(t2)λu0	β(t2)λu0	NOUN
ejde-771	242	14	−	−	PROPN
ejde-771	242	15	jα	jα	NOUN
ejde-771	242	16	,	,	PUNCT
ejde-771	242	17	β(t1)λu0∥	β(t1)λu0∥	PROPN
ejde-771	242	18	,	,	PUNCT
ejde-771	242	19	d2	d2	PROPN
ejde-771	242	20	=	=	SYM
ejde-771	242	21	∥jα	∥jα	PROPN
ejde-771	242	22	,	,	PUNCT
ejde-771	242	23	β(t2)−	β(t2)−	PROPN
ejde-771	242	24	jα	jα	PROPN
ejde-771	242	25	,	,	PUNCT
ejde-771	242	26	β(t1)∥	β(t1)∥	VERB
ejde-771	242	27	m∑	m∑	CCONJ
ejde-771	242	28	k=1	k=1	PROPN
ejde-771	242	29	|ak|∥λ∥	|ak|∥λ∥	VERB
ejde-771	242	30	∫	∫	PROPN
ejde-771	242	31	tk	tk	PROPN
ejde-771	242	32	0	0	PROPN
ejde-771	243	1	(	(	PUNCT
ejde-771	243	2	tk	tk	PROPN
ejde-771	243	3	−	−	PROPN
ejde-771	243	4	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	243	5	,	,	PUNCT
ejde-771	243	6	β(tk	β(tk	PROPN
ejde-771	243	7	−	−	NOUN
ejde-771	243	8	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	243	9	,	,	PUNCT
ejde-771	243	10	u(s))∥ds	u(s))∥ds	PROPN
ejde-771	243	11	,	,	PUNCT
ejde-771	243	12	d3	d3	PROPN
ejde-771	243	13	=	=	SYM
ejde-771	243	14	∫	∫	PROPN
ejde-771	243	15	t1	t1	NOUN
ejde-771	243	16	0	0	NUM
ejde-771	244	1	(	(	PUNCT
ejde-771	244	2	(	(	PUNCT
ejde-771	244	3	t2	t2	PROPN
ejde-771	244	4	−	−	PROPN
ejde-771	244	5	s)α−1	s)α−1	NOUN
ejde-771	244	6	−	−	PROPN
ejde-771	244	7	(	(	PUNCT
ejde-771	244	8	t1	t1	NOUN
ejde-771	244	9	−	−	PROPN
ejde-771	244	10	s)α−1)∥kα	s)α−1)∥kα	NOUN
ejde-771	244	11	,	,	PUNCT
ejde-771	244	12	β(t2	β(t2	VERB
ejde-771	244	13	−	−	ADP
ejde-771	244	14	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	244	15	,	,	PUNCT
ejde-771	244	16	u(s))∥ds	u(s))∥ds	PROPN
ejde-771	244	17	,	,	PUNCT
ejde-771	244	18	d4	d4	PROPN
ejde-771	244	19	=	=	SYM
ejde-771	244	20	∫	∫	PROPN
ejde-771	245	1	t1	t1	NOUN
ejde-771	245	2	0	0	NUM
ejde-771	246	1	(	(	PUNCT
ejde-771	246	2	t1	t1	NOUN
ejde-771	246	3	−	−	PROPN
ejde-771	246	4	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	246	5	,	,	PUNCT
ejde-771	246	6	β(t2	β(t2	VERB
ejde-771	246	7	−	−	PROPN
ejde-771	246	8	s)−	s)−	PROPN
ejde-771	246	9	kα	kα	PROPN
ejde-771	246	10	,	,	PUNCT
ejde-771	246	11	β(t1	β(t1	VERB
ejde-771	247	1	−	−	PROPN
ejde-771	247	2	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	247	3	,	,	PUNCT
ejde-771	247	4	u(s))∥ds	u(s))∥ds	NOUN
ejde-771	247	5	,	,	PUNCT
ejde-771	247	6	d5	d5	NOUN
ejde-771	247	7	=	=	SYM
ejde-771	247	8	∫	∫	PROPN
ejde-771	247	9	t2	t2	PROPN
ejde-771	247	10	t1	t1	PROPN
ejde-771	247	11	(	(	PUNCT
ejde-771	247	12	t2	t2	PROPN
ejde-771	247	13	−	−	PROPN
ejde-771	247	14	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	247	15	,	,	PUNCT
ejde-771	247	16	β(t2	β(t2	VERB
ejde-771	247	17	−	−	ADP
ejde-771	247	18	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	247	19	,	,	PUNCT
ejde-771	247	20	u(s))∥ds	u(s))∥ds	PROPN
ejde-771	247	21	.	.	PUNCT
ejde-771	248	1	we	we	PRON
ejde-771	248	2	just	just	ADV
ejde-771	248	3	need	need	VERB
ejde-771	248	4	to	to	PART
ejde-771	248	5	examine	examine	VERB
ejde-771	248	6	that	that	PRON
ejde-771	248	7	di	di	NOUN
ejde-771	248	8	tend	tend	VERB
ejde-771	248	9	to	to	PART
ejde-771	248	10	0	0	NUM
ejde-771	248	11	independently	independently	ADV
ejde-771	248	12	of	of	ADP
ejde-771	248	13	u	u	PROPN
ejde-771	248	14	∈	∈	PROPN
ejde-771	248	15	{	{	PUNCT
ejde-771	248	16	vn	vn	NOUN
ejde-771	248	17	}	}	PUNCT
ejde-771	248	18	as	as	ADP
ejde-771	248	19	t2	t2	PROPN
ejde-771	248	20	−	−	PROPN
ejde-771	248	21	t1	t1	PROPN
ejde-771	248	22	→	→	SYM
ejde-771	248	23	0	0	NUM
ejde-771	248	24	for	for	ADP
ejde-771	248	25	i	i	PRON
ejde-771	248	26	=	=	NOUN
ejde-771	248	27	1	1	NUM
ejde-771	248	28	,	,	PUNCT
ejde-771	248	29	2	2	NUM
ejde-771	248	30	,	,	PUNCT
ejde-771	248	31	3	3	NUM
ejde-771	248	32	,	,	PUNCT
ejde-771	248	33	4	4	NUM
ejde-771	248	34	,	,	PUNCT
ejde-771	248	35	5	5	NUM
ejde-771	248	36	.	.	PUNCT
ejde-771	248	37	thus	thus	ADV
ejde-771	248	38	,	,	PUNCT
ejde-771	248	39	by	by	ADP
ejde-771	248	40	lemma	lemma	PROPN
ejde-771	248	41	2.3	2.3	NUM
ejde-771	248	42	,	,	PUNCT
ejde-771	248	43	we	we	PRON
ejde-771	248	44	obtain	obtain	VERB
ejde-771	248	45	d1	d1	NOUN
ejde-771	249	1	=	=	SYM
ejde-771	249	2	∥jα	∥jα	PROPN
ejde-771	249	3	,	,	PUNCT
ejde-771	249	4	β(t2)λu0	β(t2)λu0	NOUN
ejde-771	249	5	−	−	PROPN
ejde-771	249	6	jα	jα	NOUN
ejde-771	249	7	,	,	PUNCT
ejde-771	249	8	β(t1)λu0∥	β(t1)λu0∥	NOUN
ejde-771	249	9	≤	≤	NUM
ejde-771	249	10	∥jα	∥jα	NOUN
ejde-771	249	11	,	,	PUNCT
ejde-771	249	12	β(t2)−	β(t2)−	PROPN
ejde-771	249	13	jα	jα	NOUN
ejde-771	249	14	,	,	PUNCT
ejde-771	249	15	β(t1)∥∥λ∥∥u0∥	β(t1)∥∥λ∥∥u0∥	PUNCT
ejde-771	249	16	→	→	SYM
ejde-771	249	17	0	0	PUNCT
ejde-771	249	18	as	as	ADP
ejde-771	249	19	t2	t2	PROPN
ejde-771	249	20	−	−	PROPN
ejde-771	249	21	t1	t1	PROPN
ejde-771	249	22	→	→	SYM
ejde-771	249	23	0	0	NUM
ejde-771	249	24	.	.	PUNCT
ejde-771	250	1	similarly	similarly	ADV
ejde-771	250	2	,	,	PUNCT
ejde-771	250	3	d2	d2	PROPN
ejde-771	250	4	≤	≤	PROPN
ejde-771	250	5	∥jα	∥jα	PROPN
ejde-771	250	6	,	,	PUNCT
ejde-771	250	7	β(t2)−	β(t2)−	PROPN
ejde-771	250	8	jα	jα	PROPN
ejde-771	250	9	,	,	PUNCT
ejde-771	250	10	β(t1)∥	β(t1)∥	VERB
ejde-771	250	11	m∑	m∑	CCONJ
ejde-771	250	12	k=1	k=1	PROPN
ejde-771	250	13	|ak|∥λ∥	|ak|∥λ∥	VERB
ejde-771	250	14	∫	∫	PROPN
ejde-771	250	15	tk	tk	PROPN
ejde-771	250	16	0	0	PROPN
ejde-771	250	17	(	(	PUNCT
ejde-771	250	18	tk	tk	PROPN
ejde-771	250	19	−	−	PROPN
ejde-771	250	20	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	250	21	,	,	PUNCT
ejde-771	250	22	β(tk	β(tk	PROPN
ejde-771	250	23	−	−	NOUN
ejde-771	250	24	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	250	25	,	,	PUNCT
ejde-771	250	26	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	250	27	→	→	SYM
ejde-771	250	28	0	0	NUM
ejde-771	250	29	ast2	ast2	NOUN
ejde-771	250	30	−	−	PROPN
ejde-771	250	31	t1	t1	PROPN
ejde-771	250	32	→	→	SYM
ejde-771	250	33	0	0	NUM
ejde-771	250	34	.	.	X
ejde-771	251	1	for	for	ADP
ejde-771	251	2	d3	d3	PROPN
ejde-771	251	3	,	,	PUNCT
ejde-771	251	4	it	it	PRON
ejde-771	251	5	follows	follow	VERB
ejde-771	251	6	from	from	ADP
ejde-771	251	7	(	(	PUNCT
ejde-771	251	8	h1	h1	PROPN
ejde-771	251	9	)	)	PUNCT
ejde-771	251	10	and	and	CCONJ
ejde-771	251	11	(	(	PUNCT
ejde-771	251	12	3.13	3.13	NUM
ejde-771	251	13	)	)	PUNCT
ejde-771	251	14	that	that	PRON
ejde-771	251	15	d3	d3	PROPN
ejde-771	251	16	=	=	SYM
ejde-771	251	17	∫	∫	PROPN
ejde-771	251	18	t1	t1	NOUN
ejde-771	251	19	0	0	NUM
ejde-771	251	20	(	(	PUNCT
ejde-771	251	21	(	(	PUNCT
ejde-771	251	22	t2	t2	PROPN
ejde-771	251	23	−	−	PROPN
ejde-771	251	24	s)α−1	s)α−1	NOUN
ejde-771	251	25	−	−	PROPN
ejde-771	251	26	(	(	PUNCT
ejde-771	251	27	t1	t1	NOUN
ejde-771	251	28	−	−	PROPN
ejde-771	251	29	s)α−1)∥kα	s)α−1)∥kα	NOUN
ejde-771	251	30	,	,	PUNCT
ejde-771	251	31	β(t2	β(t2	VERB
ejde-771	251	32	−	−	ADP
ejde-771	251	33	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	251	34	,	,	PUNCT
ejde-771	251	35	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	251	36	≤	≤	NUM
ejde-771	251	37	m	m	VERB
ejde-771	251	38	γ(α+	γ(α+	DET
ejde-771	251	39	1	1	NUM
ejde-771	251	40	)	)	PUNCT
ejde-771	252	1	(	(	PUNCT
ejde-771	252	2	a1	a1	PROPN
ejde-771	252	3	φ	φ	X
ejde-771	252	4	1−	1−	NUM
ejde-771	252	5	ψ	ψ	X
ejde-771	252	6	+	+	NOUN
ejde-771	252	7	a0	a0	NOUN
ejde-771	252	8	)	)	PUNCT
ejde-771	252	9	(	(	PUNCT
ejde-771	252	10	tα1	tα1	NOUN
ejde-771	252	11	−	−	NOUN
ejde-771	252	12	tα2	tα2	NOUN
ejde-771	252	13	+	+	CCONJ
ejde-771	252	14	(	(	PUNCT
ejde-771	252	15	t2	t2	PROPN
ejde-771	252	16	−	−	PROPN
ejde-771	252	17	t1	t1	PROPN
ejde-771	252	18	)	)	PUNCT
ejde-771	252	19	α	α	PROPN
ejde-771	252	20	)	)	PUNCT
ejde-771	252	21	ejde-2025/44	ejde-2025/44	NOUN
ejde-771	252	22	time	time	NOUN
ejde-771	252	23	-	-	PUNCT
ejde-771	252	24	space	space	NOUN
ejde-771	252	25	fractional	fractional	ADJ
ejde-771	252	26	reaction	reaction	NOUN
ejde-771	252	27	-	-	PUNCT
ejde-771	252	28	diffusion	diffusion	NOUN
ejde-771	252	29	equations	equation	NOUN
ejde-771	252	30	9	9	NUM
ejde-771	252	31	≤	≤	NUM
ejde-771	252	32	m	m	VERB
ejde-771	252	33	γ(α+	γ(α+	X
ejde-771	252	34	1	1	NUM
ejde-771	252	35	)	)	PUNCT
ejde-771	252	36	(	(	PUNCT
ejde-771	252	37	a1	a1	PROPN
ejde-771	252	38	φ	φ	X
ejde-771	252	39	1−	1−	NUM
ejde-771	252	40	ψ	ψ	X
ejde-771	252	41	+	+	NOUN
ejde-771	252	42	a0	a0	NOUN
ejde-771	252	43	)	)	PUNCT
ejde-771	252	44	(	(	PUNCT
ejde-771	252	45	t2	t2	PROPN
ejde-771	252	46	−	−	PROPN
ejde-771	252	47	t1	t1	PROPN
ejde-771	252	48	)	)	PUNCT
ejde-771	252	49	α	α	PROPN
ejde-771	252	50	→	→	SYM
ejde-771	252	51	0	0	NUM
ejde-771	252	52	a	a	DET
ejde-771	252	53	st2	st2	NOUN
ejde-771	252	54	−	−	PROPN
ejde-771	252	55	t1	t1	NOUN
ejde-771	252	56	→	→	SYM
ejde-771	252	57	0	0	NUM
ejde-771	252	58	.	.	PUNCT
ejde-771	253	1	for	for	ADP
ejde-771	253	2	t1	t1	NOUN
ejde-771	253	3	=	=	SYM
ejde-771	253	4	0	0	NUM
ejde-771	253	5	and	and	CCONJ
ejde-771	253	6	t2	t2	PROPN
ejde-771	253	7	>	>	X
ejde-771	253	8	0	0	NUM
ejde-771	253	9	,	,	PUNCT
ejde-771	253	10	it	it	PRON
ejde-771	253	11	is	be	AUX
ejde-771	253	12	conspicuous	conspicuous	ADJ
ejde-771	253	13	that	that	SCONJ
ejde-771	253	14	d4	d4	PROPN
ejde-771	253	15	=	=	SYM
ejde-771	253	16	0	0	X
ejde-771	253	17	.	.	PUNCT
ejde-771	254	1	now	now	ADV
ejde-771	254	2	,	,	PUNCT
ejde-771	254	3	for	for	ADP
ejde-771	254	4	t1	t1	PROPN
ejde-771	254	5	>	>	X
ejde-771	254	6	0	0	PUNCT
ejde-771	254	7	and	and	CCONJ
ejde-771	254	8	ϵ	ϵ	X
ejde-771	254	9	>	>	X
ejde-771	254	10	0	0	PUNCT
ejde-771	254	11	small	small	ADJ
ejde-771	254	12	enough	enough	ADV
ejde-771	254	13	,	,	PUNCT
ejde-771	254	14	by	by	ADP
ejde-771	254	15	(	(	PUNCT
ejde-771	254	16	h1	h1	PROPN
ejde-771	254	17	)	)	PUNCT
ejde-771	254	18	,	,	PUNCT
ejde-771	254	19	(	(	PUNCT
ejde-771	254	20	3.13	3.13	NUM
ejde-771	254	21	)	)	PUNCT
ejde-771	254	22	and	and	CCONJ
ejde-771	254	23	lemma	lemma	PROPN
ejde-771	254	24	2.3(2	2.3(2	NUM
ejde-771	254	25	)	)	PUNCT
ejde-771	254	26	,	,	PUNCT
ejde-771	254	27	we	we	PRON
ejde-771	254	28	obtain	obtain	VERB
ejde-771	254	29	d4	d4	PROPN
ejde-771	254	30	≤	≤	NUM
ejde-771	254	31	∫	∫	PROPN
ejde-771	255	1	t1−ε	t1−ε	PROPN
ejde-771	255	2	0	0	NUM
ejde-771	256	1	(	(	PUNCT
ejde-771	256	2	t1	t1	NOUN
ejde-771	256	3	−	−	PROPN
ejde-771	256	4	s)α−1∥(kα	s)α−1∥(kα	ADV
ejde-771	256	5	,	,	PUNCT
ejde-771	256	6	β(t2	β(t2	VERB
ejde-771	256	7	−	−	PROPN
ejde-771	256	8	s)−	s)−	PROPN
ejde-771	256	9	kα	kα	PROPN
ejde-771	256	10	,	,	PUNCT
ejde-771	256	11	β(t1	β(t1	VERB
ejde-771	257	1	−	−	NOUN
ejde-771	257	2	s))∥∥g(s	s))∥∥g(s	NOUN
ejde-771	257	3	,	,	PUNCT
ejde-771	257	4	u(s))∥ds	u(s))∥ds	X
ejde-771	258	1	+	+	CCONJ
ejde-771	258	2	∫	∫	PROPN
ejde-771	258	3	t1	t1	PROPN
ejde-771	258	4	t1−ε	t1−ε	PROPN
ejde-771	258	5	(	(	PUNCT
ejde-771	258	6	t1	t1	NOUN
ejde-771	258	7	−	−	PROPN
ejde-771	258	8	s)α−1∥(kα	s)α−1∥(kα	ADV
ejde-771	258	9	,	,	PUNCT
ejde-771	258	10	β(t2	β(t2	VERB
ejde-771	258	11	−	−	PROPN
ejde-771	258	12	s)−	s)−	PROPN
ejde-771	258	13	kα	kα	PROPN
ejde-771	258	14	,	,	PUNCT
ejde-771	258	15	β(t1	β(t1	VERB
ejde-771	258	16	−	−	NOUN
ejde-771	258	17	s))∥∥g(s	s))∥∥g(s	NOUN
ejde-771	258	18	,	,	PUNCT
ejde-771	258	19	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	258	20	≤	≤	X
ejde-771	258	21	(	(	PUNCT
ejde-771	258	22	a1	a1	PROPN
ejde-771	258	23	φ	φ	X
ejde-771	258	24	1−	1−	NUM
ejde-771	258	25	ψ	ψ	SYM
ejde-771	258	26	+	+	NOUN
ejde-771	258	27	a0	a0	NOUN
ejde-771	258	28	)	)	PUNCT
ejde-771	258	29	sup	sup	NOUN
ejde-771	258	30	s∈[0,t1−ε	s∈[0,t1−ε	PROPN
ejde-771	258	31	]	]	PUNCT
ejde-771	258	32	∥(kα	∥(kα	NOUN
ejde-771	258	33	,	,	PUNCT
ejde-771	258	34	β(t2	β(t2	VERB
ejde-771	258	35	−	−	PROPN
ejde-771	258	36	s)−	s)−	PROPN
ejde-771	258	37	kα	kα	PROPN
ejde-771	258	38	,	,	PUNCT
ejde-771	258	39	β(t1	β(t1	VERB
ejde-771	258	40	−	−	PROPN
ejde-771	258	41	s))∥	s))∥	ADJ
ejde-771	258	42	∫	∫	PROPN
ejde-771	258	43	t1−ε	t1−ε	PROPN
ejde-771	258	44	0	0	NUM
ejde-771	259	1	(	(	PUNCT
ejde-771	259	2	t1	t1	NOUN
ejde-771	259	3	−	−	PROPN
ejde-771	259	4	s)α−1ds	s)α−1ds	NOUN
ejde-771	259	5	+	+	CCONJ
ejde-771	259	6	2	2	NUM
ejde-771	259	7	m	m	NOUN
ejde-771	259	8	γ(α	γ(α	NOUN
ejde-771	259	9	)	)	PUNCT
ejde-771	260	1	(	(	PUNCT
ejde-771	260	2	a1	a1	PROPN
ejde-771	260	3	φ	φ	X
ejde-771	260	4	1−	1−	NUM
ejde-771	260	5	ψ	ψ	X
ejde-771	260	6	+	+	PROPN
ejde-771	260	7	a0	a0	NOUN
ejde-771	260	8	)	)	PUNCT
ejde-771	260	9	∫	∫	PROPN
ejde-771	260	10	t1	t1	PROPN
ejde-771	260	11	t1−ε	t1−ε	PROPN
ejde-771	260	12	(	(	PUNCT
ejde-771	260	13	t1	t1	NOUN
ejde-771	260	14	−	−	PROPN
ejde-771	260	15	s)α−1ds	s)α−1ds	NOUN
ejde-771	260	16	≤	≤	NOUN
ejde-771	260	17	(	(	PUNCT
ejde-771	260	18	a1	a1	PROPN
ejde-771	260	19	φ	φ	X
ejde-771	260	20	1−	1−	NUM
ejde-771	260	21	ψ	ψ	X
ejde-771	260	22	+	+	PROPN
ejde-771	260	23	a0	a0	NOUN
ejde-771	260	24	)	)	PUNCT
ejde-771	260	25	(	(	PUNCT
ejde-771	260	26	sup	sup	NOUN
ejde-771	260	27	s∈[0,t1−ε	s∈[0,t1−ε	PROPN
ejde-771	260	28	]	]	PUNCT
ejde-771	260	29	∥(kα	∥(kα	NOUN
ejde-771	260	30	,	,	PUNCT
ejde-771	260	31	β(t2	β(t2	VERB
ejde-771	260	32	−	−	PROPN
ejde-771	260	33	s)−	s)−	PROPN
ejde-771	260	34	kα	kα	PROPN
ejde-771	260	35	,	,	PUNCT
ejde-771	260	36	β(t1	β(t1	VERB
ejde-771	260	37	−	−	NOUN
ejde-771	260	38	s))∥	s))∥	ADJ
ejde-771	260	39	t	t	PROPN
ejde-771	260	40	α	α	NOUN
ejde-771	260	41	1	1	NUM
ejde-771	260	42	−	−	NOUN
ejde-771	260	43	εα	εα	PROPN
ejde-771	260	44	α	α	NOUN
ejde-771	261	1	+	+	CCONJ
ejde-771	261	2	2	2	NUM
ejde-771	261	3	m	m	NOUN
ejde-771	261	4	γ(α+	γ(α+	ADJ
ejde-771	261	5	1	1	NUM
ejde-771	261	6	)	)	PUNCT
ejde-771	261	7	εα	εα	ADJ
ejde-771	261	8	)	)	PUNCT
ejde-771	261	9	→	→	SYM
ejde-771	261	10	0	0	NUM
ejde-771	261	11	as	as	ADP
ejde-771	261	12	t2	t2	PROPN
ejde-771	261	13	−	−	PROPN
ejde-771	261	14	t1	t1	PROPN
ejde-771	261	15	→	→	SYM
ejde-771	261	16	0	0	NUM
ejde-771	261	17	,	,	PUNCT
ejde-771	261	18	ϵ→	ϵ→	PROPN
ejde-771	261	19	0	0	X
ejde-771	261	20	.	.	PUNCT
ejde-771	262	1	for	for	ADP
ejde-771	262	2	d5	d5	NOUN
ejde-771	262	3	,	,	PUNCT
ejde-771	262	4	we	we	PRON
ejde-771	262	5	observe	observe	VERB
ejde-771	262	6	that	that	SCONJ
ejde-771	262	7	d5	d5	NOUN
ejde-771	262	8	≤	≤	PROPN
ejde-771	262	9	∫	∫	PROPN
ejde-771	262	10	t2	t2	PROPN
ejde-771	262	11	t1	t1	PROPN
ejde-771	262	12	(	(	PUNCT
ejde-771	262	13	t2	t2	PROPN
ejde-771	262	14	−	−	PROPN
ejde-771	262	15	s)α−1∥kα	s)α−1∥kα	PROPN
ejde-771	262	16	,	,	PUNCT
ejde-771	262	17	β(t2	β(t2	VERB
ejde-771	262	18	−	−	ADP
ejde-771	262	19	s)∥∥g(s	s)∥∥g(s	NOUN
ejde-771	262	20	,	,	PUNCT
ejde-771	262	21	u(s))∥ds	u(s))∥ds	ADJ
ejde-771	262	22	≤	≤	NUM
ejde-771	262	23	m	m	VERB
ejde-771	262	24	γ(α+	γ(α+	X
ejde-771	262	25	1	1	NUM
ejde-771	262	26	)	)	PUNCT
ejde-771	262	27	(	(	PUNCT
ejde-771	262	28	a1	a1	PROPN
ejde-771	262	29	φ	φ	X
ejde-771	262	30	1−	1−	NUM
ejde-771	262	31	ψ	ψ	X
ejde-771	262	32	+	+	NOUN
ejde-771	262	33	a0)(t2	a0)(t2	NOUN
ejde-771	262	34	−	−	PROPN
ejde-771	262	35	t1	t1	NOUN
ejde-771	262	36	)	)	PUNCT
ejde-771	262	37	α	α	PROPN
ejde-771	262	38	→	→	SYM
ejde-771	262	39	0	0	NUM
ejde-771	262	40	as	as	ADP
ejde-771	262	41	t2	t2	PROPN
ejde-771	262	42	−	−	PROPN
ejde-771	262	43	t1	t1	PROPN
ejde-771	262	44	→	→	SYM
ejde-771	262	45	0	0	X
ejde-771	262	46	.	.	PUNCT
ejde-771	263	1	combining	combine	VERB
ejde-771	263	2	all	all	DET
ejde-771	263	3	the	the	DET
ejde-771	263	4	above	above	ADJ
ejde-771	263	5	arguments	argument	NOUN
ejde-771	263	6	,	,	PUNCT
ejde-771	263	7	one	one	PRON
ejde-771	263	8	can	can	AUX
ejde-771	263	9	deduced	deduced	VERB
ejde-771	263	10	that	that	SCONJ
ejde-771	263	11	∥(θu)(t2)−	∥(θu)(t2)−	PROPN
ejde-771	263	12	(	(	PUNCT
ejde-771	263	13	θu)(t1)∥	θu)(t1)∥	NOUN
ejde-771	263	14	→	→	SYM
ejde-771	263	15	0	0	PUNCT
ejde-771	263	16	as	as	ADP
ejde-771	263	17	t2	t2	PROPN
ejde-771	263	18	−	−	PROPN
ejde-771	263	19	t1	t1	PROPN
ejde-771	263	20	→	→	SYM
ejde-771	263	21	0	0	NUM
ejde-771	263	22	,	,	PUNCT
ejde-771	263	23	which	which	PRON
ejde-771	263	24	means	mean	VERB
ejde-771	263	25	that	that	SCONJ
ejde-771	263	26	the	the	DET
ejde-771	263	27	operator	operator	NOUN
ejde-771	263	28	θ	θ	PROPN
ejde-771	263	29	is	be	AUX
ejde-771	263	30	locally	locally	ADV
ejde-771	263	31	equicontinuous	equicontinuous	ADJ
ejde-771	263	32	in	in	ADP
ejde-771	263	33	[	[	X
ejde-771	263	34	0	0	NUM
ejde-771	263	35	,	,	PUNCT
ejde-771	263	36	a	a	PRON
ejde-771	263	37	]	]	X
ejde-771	263	38	for	for	ADP
ejde-771	263	39	arbitrary	arbitrary	ADJ
ejde-771	263	40	constant	constant	ADJ
ejde-771	263	41	0	0	NUM
ejde-771	263	42	<	<	X
ejde-771	263	43	a	a	PRON
ejde-771	263	44	<	<	X
ejde-771	263	45	+	+	NOUN
ejde-771	263	46	∞.	∞.	PROPN
ejde-771	263	47	subsequently	subsequently	ADV
ejde-771	263	48	,	,	PUNCT
ejde-771	263	49	we	we	PRON
ejde-771	263	50	need	need	VERB
ejde-771	263	51	to	to	PART
ejde-771	263	52	prove	prove	VERB
ejde-771	263	53	{	{	PUNCT
ejde-771	263	54	vn(t	vn(t	NUM
ejde-771	263	55	)	)	PUNCT
ejde-771	263	56	}	}	PUNCT
ejde-771	263	57	is	be	AUX
ejde-771	263	58	relatively	relatively	ADV
ejde-771	263	59	compact	compact	ADJ
ejde-771	263	60	on	on	ADP
ejde-771	263	61	e	e	NOUN
ejde-771	263	62	for	for	ADP
ejde-771	263	63	t	t	PROPN
ejde-771	263	64	∈	∈	PROPN
ejde-771	264	1	[	[	X
ejde-771	264	2	0,∞	0,∞	NOUN
ejde-771	264	3	)	)	PUNCT
ejde-771	264	4	.	.	PUNCT
ejde-771	265	1	let	let	VERB
ejde-771	265	2	v	v	VERB
ejde-771	265	3	=	=	PUNCT
ejde-771	265	4	{	{	PUNCT
ejde-771	265	5	vn	vn	NOUN
ejde-771	265	6	}	}	PUNCT
ejde-771	265	7	and	and	CCONJ
ejde-771	265	8	v0	v0	PROPN
ejde-771	265	9	=	=	SYM
ejde-771	265	10	v	v	NOUN
ejde-771	265	11	∪	∪	X
ejde-771	265	12	{	{	PUNCT
ejde-771	265	13	v0	v0	NOUN
ejde-771	265	14	}	}	PUNCT
ejde-771	265	15	.	.	PUNCT
ejde-771	266	1	obviously	obviously	ADV
ejde-771	266	2	,	,	PUNCT
ejde-771	266	3	v(t	v(t	NOUN
ejde-771	266	4	)	)	PUNCT
ejde-771	266	5	=	=	SYM
ejde-771	266	6	(	(	PUNCT
ejde-771	266	7	θv0)(t	θv0)(t	PROPN
ejde-771	266	8	)	)	PUNCT
ejde-771	266	9	for	for	ADP
ejde-771	266	10	t	t	PROPN
ejde-771	266	11	∈	∈	PROPN
ejde-771	267	1	[	[	X
ejde-771	267	2	0,∞	0,∞	NOUN
ejde-771	267	3	)	)	PUNCT
ejde-771	267	4	.	.	PUNCT
ejde-771	268	1	it	it	PRON
ejde-771	268	2	is	be	AUX
ejde-771	268	3	easy	easy	ADJ
ejde-771	268	4	to	to	PART
ejde-771	268	5	prove	prove	VERB
ejde-771	268	6	that	that	SCONJ
ejde-771	268	7	{	{	PUNCT
ejde-771	268	8	vn(0	vn(0	NOUN
ejde-771	268	9	)	)	PUNCT
ejde-771	268	10	}	}	PUNCT
ejde-771	268	11	is	be	AUX
ejde-771	268	12	relatively	relatively	ADV
ejde-771	268	13	compact	compact	ADJ
ejde-771	268	14	on	on	ADP
ejde-771	268	15	e.	e.	PROPN
ejde-771	268	16	we	we	PRON
ejde-771	268	17	only	only	ADV
ejde-771	268	18	consider	consider	VERB
ejde-771	268	19	the	the	DET
ejde-771	268	20	case	case	NOUN
ejde-771	268	21	t	t	X
ejde-771	268	22	>	>	X
ejde-771	268	23	0	0	NUM
ejde-771	268	24	,	,	PUNCT
ejde-771	268	25	for	for	ADP
ejde-771	268	26	all	all	DET
ejde-771	268	27	∀ϵ	∀ϵ	NOUN
ejde-771	268	28	∈	∈	PROPN
ejde-771	268	29	(	(	PUNCT
ejde-771	268	30	0	0	NUM
ejde-771	268	31	,	,	PUNCT
ejde-771	268	32	t	t	PROPN
ejde-771	268	33	)	)	PUNCT
ejde-771	268	34	and	and	CCONJ
ejde-771	268	35	δ	δ	PROPN
ejde-771	268	36	>	>	X
ejde-771	268	37	0	0	PROPN
ejde-771	268	38	,	,	PUNCT
ejde-771	268	39	define	define	VERB
ejde-771	268	40	θϵ,δvn	θϵ,δvn	X
ejde-771	268	41	by	by	ADP
ejde-771	268	42	(	(	PUNCT
ejde-771	268	43	θϵ,δvn)(t	θϵ,δvn)(t	PROPN
ejde-771	268	44	)	)	PUNCT
ejde-771	268	45	=	=	SYM
ejde-771	268	46	jα	jα	NOUN
ejde-771	268	47	,	,	PUNCT
ejde-771	268	48	β(t)λvn−1(0	β(t)λvn−1(0	PUNCT
ejde-771	268	49	)	)	PUNCT
ejde-771	269	1	+	+	CCONJ
ejde-771	269	2	α	α	PRON
ejde-771	269	3	m∑	m∑	VERB
ejde-771	269	4	k=1	k=1	X
ejde-771	269	5	akλjα	akλjα	ADV
ejde-771	269	6	,	,	PUNCT
ejde-771	269	7	β(t	β(t	PROPN
ejde-771	269	8	)	)	PUNCT
ejde-771	270	1	×	×	NOUN
ejde-771	270	2	∫	∫	PROPN
ejde-771	270	3	tk	tk	PROPN
ejde-771	270	4	0	0	PROPN
ejde-771	270	5	∫	∫	PROPN
ejde-771	270	6	∞	∞	PROPN
ejde-771	270	7	0	0	NUM
ejde-771	271	1	(	(	PUNCT
ejde-771	271	2	tk	tk	PROPN
ejde-771	271	3	−	−	PROPN
ejde-771	271	4	s)α−1τhα(τ)tβ((tk	s)α−1τhα(τ)tβ((tk	NOUN
ejde-771	271	5	−	−	PROPN
ejde-771	271	6	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	271	7	,	,	PUNCT
ejde-771	271	8	vn−1(s))dτds	vn−1(s))dτd	VERB
ejde-771	271	9	+	+	CCONJ
ejde-771	271	10	α	α	DET
ejde-771	271	11	∫	∫	NOUN
ejde-771	271	12	t−ϵ	t−ϵ	ADV
ejde-771	271	13	0	0	NUM
ejde-771	271	14	∫	∫	PROPN
ejde-771	271	15	∞	∞	PROPN
ejde-771	271	16	δ	δ	PROPN
ejde-771	271	17	(	(	PUNCT
ejde-771	271	18	t−	t−	PROPN
ejde-771	271	19	s)α−1τhα(τ)tβ((t−	s)α−1τhα(τ)tβ((t−	PROPN
ejde-771	271	20	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	271	21	,	,	PUNCT
ejde-771	271	22	vn−1(s))dτds	vn−1(s))dτd	VERB
ejde-771	271	23	=	=	SYM
ejde-771	271	24	jα	jα	NOUN
ejde-771	271	25	,	,	PUNCT
ejde-771	271	26	β(t)λvn−1(0	β(t)λvn−1(0	PUNCT
ejde-771	271	27	)	)	PUNCT
ejde-771	272	1	+	+	CCONJ
ejde-771	272	2	α	α	PRON
ejde-771	272	3	m∑	m∑	VERB
ejde-771	272	4	k=1	k=1	X
ejde-771	272	5	akλjα	akλjα	ADV
ejde-771	272	6	,	,	PUNCT
ejde-771	272	7	β(t	β(t	PROPN
ejde-771	272	8	)	)	PUNCT
ejde-771	273	1	×	×	NOUN
ejde-771	273	2	∫	∫	PROPN
ejde-771	273	3	tk	tk	PROPN
ejde-771	273	4	0	0	PROPN
ejde-771	273	5	∫	∫	PROPN
ejde-771	273	6	∞	∞	PROPN
ejde-771	273	7	0	0	NUM
ejde-771	274	1	(	(	PUNCT
ejde-771	274	2	tk	tk	PROPN
ejde-771	274	3	−	−	PROPN
ejde-771	274	4	s)α−1τhα(τ)tβ((tk	s)α−1τhα(τ)tβ((tk	NOUN
ejde-771	274	5	−	−	PROPN
ejde-771	274	6	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	274	7	,	,	PUNCT
ejde-771	274	8	vn−1(s))dτds	vn−1(s))dτd	VERB
ejde-771	274	9	+	+	NUM
ejde-771	274	10	αtβ(ϵ	αtβ(ϵ	NUM
ejde-771	274	11	αδ	αδ	NOUN
ejde-771	274	12	)	)	PUNCT
ejde-771	274	13	∫	∫	PROPN
ejde-771	274	14	t−ϵ	t−ϵ	ADV
ejde-771	274	15	0	0	NUM
ejde-771	274	16	∫	∫	PROPN
ejde-771	274	17	∞	∞	PROPN
ejde-771	274	18	δ	δ	PROPN
ejde-771	274	19	(	(	PUNCT
ejde-771	274	20	t−	t−	PROPN
ejde-771	274	21	s)α−1τhα(τ)tβ((t−	s)α−1τhα(τ)tβ((t−	PROPN
ejde-771	275	1	s)ατ	s)ατ	PROPN
ejde-771	275	2	−	−	PROPN
ejde-771	276	1	ϵαδ)g(s	ϵαδ)g(s	PROPN
ejde-771	276	2	,	,	PUNCT
ejde-771	276	3	vn−1(s))dτds	vn−1(s))dτd	VERB
ejde-771	276	4	.	.	PUNCT
ejde-771	276	5	10	10	NUM
ejde-771	276	6	x.	x.	NOUN
ejde-771	276	7	zhang	zhang	PROPN
ejde-771	276	8	,	,	PUNCT
ejde-771	276	9	k.	k.	PROPN
ejde-771	276	10	ding	ding	PROPN
ejde-771	276	11	,	,	PUNCT
ejde-771	276	12	p.	p.	PROPN
ejde-771	276	13	chen	chen	PROPN
ejde-771	276	14	ejde-2025/44	ejde-2025/44	VERB
ejde-771	276	15	the	the	DET
ejde-771	276	16	compactness	compactness	NOUN
ejde-771	276	17	of	of	ADP
ejde-771	276	18	jα	jα	PROPN
ejde-771	276	19	,	,	PUNCT
ejde-771	276	20	β(t	β(t	PROPN
ejde-771	276	21	)	)	PUNCT
ejde-771	276	22	and	and	CCONJ
ejde-771	276	23	tβ(ϵ	tβ(ϵ	NOUN
ejde-771	276	24	αδ	αδ	CCONJ
ejde-771	276	25	)	)	PUNCT
ejde-771	276	26	implies	imply	VERB
ejde-771	276	27	that	that	SCONJ
ejde-771	276	28	the	the	DET
ejde-771	276	29	set	set	NOUN
ejde-771	276	30	(	(	PUNCT
ejde-771	276	31	θϵ,δv0)(t	θϵ,δv0)(t	PROPN
ejde-771	276	32	)	)	PUNCT
ejde-771	276	33	is	be	AUX
ejde-771	276	34	relatively	relatively	ADV
ejde-771	276	35	compact	compact	ADJ
ejde-771	276	36	in	in	ADP
ejde-771	276	37	e.	e.	PROPN
ejde-771	276	38	moreover	moreover	ADV
ejde-771	276	39	,	,	PUNCT
ejde-771	276	40	for	for	ADP
ejde-771	276	41	∀vn	∀vn	PROPN
ejde-771	276	42	∈	∈	PROPN
ejde-771	276	43	v0	v0	NOUN
ejde-771	276	44	and	and	CCONJ
ejde-771	276	45	t	t	NOUN
ejde-771	276	46	∈	∈	PROPN
ejde-771	276	47	(	(	PUNCT
ejde-771	276	48	0,∞	0,∞	NOUN
ejde-771	276	49	)	)	PUNCT
ejde-771	276	50	,	,	PUNCT
ejde-771	276	51	one	one	PRON
ejde-771	276	52	can	can	AUX
ejde-771	276	53	obtain	obtain	VERB
ejde-771	276	54	that	that	DET
ejde-771	276	55	∥(θvn)(t)−	∥(θvn)(t)−	PROPN
ejde-771	276	56	(	(	PUNCT
ejde-771	276	57	θϵ,δvn)(t)∥	θϵ,δvn)(t)∥	NOUN
ejde-771	276	58	=	=	PUNCT
ejde-771	277	1	∥α	∥α	PROPN
ejde-771	277	2	∫	∫	PROPN
ejde-771	277	3	t	t	PROPN
ejde-771	277	4	0	0	NUM
ejde-771	277	5	∫	∫	PROPN
ejde-771	277	6	δ	δ	PROPN
ejde-771	277	7	0	0	NUM
ejde-771	278	1	(	(	PUNCT
ejde-771	278	2	t−	t−	PROPN
ejde-771	278	3	s)α−1τhα(τ)tβ((t−	s)α−1τhα(τ)tβ((t−	PROPN
ejde-771	278	4	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	278	5	,	,	PUNCT
ejde-771	278	6	vn−1(s))dτds∥	vn−1(s))dτds∥	NOUN
ejde-771	279	1	+	+	CCONJ
ejde-771	279	2	∥α	∥α	NOUN
ejde-771	279	3	∫	∫	PROPN
ejde-771	279	4	t	t	PROPN
ejde-771	279	5	t−ϵ	t−ϵ	ADV
ejde-771	279	6	∫	∫	PROPN
ejde-771	279	7	∞	∞	PROPN
ejde-771	279	8	δ	δ	PROPN
ejde-771	279	9	(	(	PUNCT
ejde-771	279	10	t−	t−	PROPN
ejde-771	279	11	s)α−1τhα(τ)tβ((t−	s)α−1τhα(τ)tβ((t−	PROPN
ejde-771	279	12	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	279	13	,	,	PUNCT
ejde-771	279	14	vn−1(s))dτds∥	vn−1(s))dτds∥	VERB
ejde-771	279	15	≤	≤	NUM
ejde-771	279	16	(	(	PUNCT
ejde-771	279	17	a1	a1	PROPN
ejde-771	279	18	φ	φ	X
ejde-771	279	19	1−	1−	NUM
ejde-771	279	20	ψ	ψ	X
ejde-771	279	21	+	+	PROPN
ejde-771	279	22	a0	a0	NOUN
ejde-771	279	23	)	)	PUNCT
ejde-771	280	1	α	α	PROPN
ejde-771	280	2	∫	∫	PROPN
ejde-771	280	3	t	t	PROPN
ejde-771	280	4	0	0	NUM
ejde-771	280	5	∫	∫	PROPN
ejde-771	280	6	δ	δ	PROPN
ejde-771	280	7	0	0	NUM
ejde-771	281	1	(	(	PUNCT
ejde-771	281	2	t−	t−	X
ejde-771	281	3	s)α−1τhα(τ)∥tβ((t−	s)α−1τhα(τ)∥tβ((t−	NOUN
ejde-771	281	4	s)ατ)∥dτds	s)ατ)∥dτd	NOUN
ejde-771	281	5	+	+	CCONJ
ejde-771	281	6	(	(	PUNCT
ejde-771	281	7	a1	a1	PROPN
ejde-771	281	8	φ	φ	NOUN
ejde-771	281	9	1−	1−	NUM
ejde-771	281	10	ψ	ψ	X
ejde-771	281	11	+	+	PROPN
ejde-771	281	12	a0	a0	NOUN
ejde-771	281	13	)	)	PUNCT
ejde-771	281	14	α	α	PROPN
ejde-771	281	15	∫	∫	PROPN
ejde-771	281	16	t	t	PROPN
ejde-771	281	17	t−ϵ	t−ϵ	ADV
ejde-771	281	18	∫	∫	PROPN
ejde-771	281	19	∞	∞	PROPN
ejde-771	281	20	δ	δ	PROPN
ejde-771	281	21	(	(	PUNCT
ejde-771	281	22	t−	t−	X
ejde-771	281	23	s)α−1τhα(τ)∥tβ((t−	s)α−1τhα(τ)∥tβ((t−	PROPN
ejde-771	281	24	s)ατ)∥dτds	s)ατ)∥dτds	PROPN
ejde-771	281	25	≤m(a1	≤m(a1	PROPN
ejde-771	281	26	φ	φ	X
ejde-771	281	27	1−	1−	NUM
ejde-771	281	28	ψ	ψ	SYM
ejde-771	281	29	+	+	PROPN
ejde-771	281	30	a0	a0	NOUN
ejde-771	281	31	)	)	PUNCT
ejde-771	281	32	·	·	PUNCT
ejde-771	281	33	(	(	PUNCT
ejde-771	281	34	∫	∫	PROPN
ejde-771	281	35	t	t	PROPN
ejde-771	281	36	0	0	NUM
ejde-771	281	37	(	(	PUNCT
ejde-771	281	38	t−	t−	PROPN
ejde-771	281	39	s)α−1ds	s)α−1ds	NOUN
ejde-771	281	40	∫	∫	PROPN
ejde-771	281	41	δ	δ	PROPN
ejde-771	281	42	0	0	PROPN
ejde-771	281	43	τhα(τ)dτ	τhα(τ)dτ	PUNCT
ejde-771	281	44	+	+	NUM
ejde-771	281	45	∫	∫	PROPN
ejde-771	281	46	t	t	NOUN
ejde-771	281	47	t−ϵ	t−ϵ	ADV
ejde-771	281	48	(	(	PUNCT
ejde-771	281	49	t−	t−	PROPN
ejde-771	281	50	s)α−1ds	s)α−1ds	NOUN
ejde-771	281	51	∫	∫	PROPN
ejde-771	281	52	∞	∞	PROPN
ejde-771	281	53	δ	δ	PROPN
ejde-771	281	54	τhα(τ)dτ	τhα(τ)dτ	PUNCT
ejde-771	281	55	)	)	PUNCT
ejde-771	281	56	→	→	SYM
ejde-771	281	57	0	0	PUNCT
ejde-771	281	58	as	as	ADP
ejde-771	281	59	ϵ→	ϵ→	PROPN
ejde-771	281	60	0	0	NUM
ejde-771	281	61	,	,	PUNCT
ejde-771	281	62	δ	δ	PROPN
ejde-771	281	63	→	→	SYM
ejde-771	281	64	0	0	X
ejde-771	281	65	.	.	PUNCT
ejde-771	282	1	we	we	PRON
ejde-771	282	2	conclude	conclude	VERB
ejde-771	282	3	that	that	SCONJ
ejde-771	282	4	there	there	PRON
ejde-771	282	5	is	be	VERB
ejde-771	282	6	a	a	DET
ejde-771	282	7	relatively	relatively	ADV
ejde-771	282	8	compact	compact	ADJ
ejde-771	282	9	set	set	NOUN
ejde-771	282	10	(	(	PUNCT
ejde-771	282	11	θϵ,δv0)(t	θϵ,δv0)(t	NOUN
ejde-771	282	12	)	)	PUNCT
ejde-771	282	13	arbitrarily	arbitrarily	ADV
ejde-771	282	14	close	close	ADJ
ejde-771	282	15	to	to	ADP
ejde-771	282	16	the	the	DET
ejde-771	282	17	set	set	NOUN
ejde-771	282	18	(	(	PUNCT
ejde-771	282	19	θv0)(t	θv0)(t	PROPN
ejde-771	282	20	)	)	PUNCT
ejde-771	282	21	on	on	ADP
ejde-771	282	22	e	e	PROPN
ejde-771	282	23	for	for	ADP
ejde-771	282	24	t	t	PROPN
ejde-771	282	25	∈	∈	PROPN
ejde-771	282	26	(	(	PUNCT
ejde-771	282	27	0,∞	0,∞	NOUN
ejde-771	282	28	)	)	PUNCT
ejde-771	282	29	.	.	PUNCT
ejde-771	283	1	consequently	consequently	ADV
ejde-771	283	2	,	,	PUNCT
ejde-771	283	3	we	we	PRON
ejde-771	283	4	can	can	AUX
ejde-771	283	5	obtain	obtain	VERB
ejde-771	283	6	that	that	PRON
ejde-771	283	7	{	{	PUNCT
ejde-771	283	8	vn(t	vn(t	NUM
ejde-771	283	9	)	)	PUNCT
ejde-771	283	10	}	}	PUNCT
ejde-771	283	11	is	be	AUX
ejde-771	283	12	relatively	relatively	ADV
ejde-771	283	13	compact	compact	ADJ
ejde-771	283	14	on	on	ADP
ejde-771	283	15	e	e	NOUN
ejde-771	283	16	for	for	ADP
ejde-771	283	17	t	t	PROPN
ejde-771	283	18	∈	∈	PROPN
ejde-771	284	1	[	[	X
ejde-771	284	2	0,∞	0,∞	NOUN
ejde-771	284	3	)	)	PUNCT
ejde-771	284	4	.	.	PUNCT
ejde-771	285	1	further	far	ADV
ejde-771	285	2	,	,	PUNCT
ejde-771	285	3	for	for	ADP
ejde-771	285	4	any	any	DET
ejde-771	285	5	u	u	PROPN
ejde-771	285	6	∈	∈	PROPN
ejde-771	285	7	{	{	PUNCT
ejde-771	285	8	vn	vn	NOUN
ejde-771	285	9	}	}	PUNCT
ejde-771	285	10	,	,	PUNCT
ejde-771	285	11	by	by	ADP
ejde-771	285	12	(	(	PUNCT
ejde-771	285	13	3.5	3.5	NUM
ejde-771	285	14	)	)	PUNCT
ejde-771	285	15	and	and	CCONJ
ejde-771	285	16	(	(	PUNCT
ejde-771	285	17	3.13	3.13	NUM
ejde-771	285	18	)	)	PUNCT
ejde-771	285	19	,	,	PUNCT
ejde-771	285	20	we	we	PRON
ejde-771	285	21	can	can	AUX
ejde-771	285	22	easily	easily	ADV
ejde-771	285	23	get	get	VERB
ejde-771	285	24	that	that	DET
ejde-771	285	25	lim	lim	PROPN
ejde-771	285	26	t→∞	t→∞	X
ejde-771	285	27	e−t∥(θu)(t)∥	e−t∥(θu)(t)∥	PROPN
ejde-771	285	28	=	=	SYM
ejde-771	285	29	0	0	NUM
ejde-771	285	30	.	.	PUNCT
ejde-771	286	1	hence	hence	ADV
ejde-771	286	2	,	,	PUNCT
ejde-771	286	3	it	it	PRON
ejde-771	286	4	follows	follow	VERB
ejde-771	286	5	from	from	ADP
ejde-771	286	6	lemma	lemma	PROPN
ejde-771	286	7	2.1	2.1	NUM
ejde-771	286	8	that	that	PRON
ejde-771	286	9	{	{	PUNCT
ejde-771	286	10	vn	vn	NOUN
ejde-771	286	11	}	}	PUNCT
ejde-771	286	12	is	be	AUX
ejde-771	286	13	relatively	relatively	ADV
ejde-771	286	14	compact	compact	ADJ
ejde-771	286	15	in	in	ADP
ejde-771	286	16	pe	pe	X
ejde-771	286	17	∩	∩	NOUN
ejde-771	286	18	sapω(e	sapω(e	NOUN
ejde-771	286	19	)	)	PUNCT
ejde-771	286	20	.	.	PUNCT
ejde-771	287	1	hence	hence	ADV
ejde-771	287	2	,	,	PUNCT
ejde-771	287	3	there	there	PRON
ejde-771	287	4	exist	exist	VERB
ejde-771	287	5	convergent	convergent	ADJ
ejde-771	287	6	subsequence	subsequence	NOUN
ejde-771	287	7	in	in	ADP
ejde-771	287	8	{	{	PUNCT
ejde-771	287	9	vn	vn	NOUN
ejde-771	287	10	}	}	PUNCT
ejde-771	287	11	.	.	PUNCT
ejde-771	288	1	as	as	ADP
ejde-771	288	2	a	a	DET
ejde-771	288	3	result	result	NOUN
ejde-771	288	4	,	,	PUNCT
ejde-771	288	5	one	one	PRON
ejde-771	288	6	can	can	AUX
ejde-771	288	7	obtain	obtain	VERB
ejde-771	288	8	that	that	DET
ejde-771	288	9	{	{	PUNCT
ejde-771	288	10	vn	vn	NOUN
ejde-771	288	11	}	}	PUNCT
ejde-771	288	12	itself	itself	PRON
ejde-771	288	13	is	be	AUX
ejde-771	288	14	uniformly	uniformly	ADV
ejde-771	288	15	convergent	convergent	ADJ
ejde-771	288	16	through	through	ADP
ejde-771	288	17	the	the	DET
ejde-771	288	18	monotonicity	monotonicity	NOUN
ejde-771	288	19	of	of	ADP
ejde-771	288	20	sequence	sequence	NOUN
ejde-771	288	21	and	and	CCONJ
ejde-771	288	22	the	the	DET
ejde-771	288	23	normality	normality	NOUN
ejde-771	288	24	of	of	ADP
ejde-771	288	25	cone	cone	NOUN
ejde-771	288	26	,	,	PUNCT
ejde-771	288	27	which	which	PRON
ejde-771	288	28	means	mean	VERB
ejde-771	288	29	that	that	SCONJ
ejde-771	288	30	there	there	PRON
ejde-771	288	31	exist	exist	VERB
ejde-771	288	32	ũ	ũ	PROPN
ejde-771	288	33	∈	∈	PROPN
ejde-771	288	34	pe	pe	PROPN
ejde-771	288	35	∩	∩	ADJ
ejde-771	288	36	sapω(e	sapω(e	NOUN
ejde-771	288	37	)	)	PUNCT
ejde-771	288	38	such	such	ADJ
ejde-771	288	39	that	that	SCONJ
ejde-771	288	40	limn→∞	limn→∞	PROPN
ejde-771	288	41	vn	vn	X
ejde-771	288	42	=	=	SYM
ejde-771	288	43	ũ.	ũ.	PROPN
ejde-771	288	44	moreover	moreover	ADV
ejde-771	288	45	,	,	PUNCT
ejde-771	288	46	taking	take	VERB
ejde-771	288	47	the	the	DET
ejde-771	288	48	limit	limit	NOUN
ejde-771	288	49	in	in	ADP
ejde-771	288	50	(	(	PUNCT
ejde-771	288	51	3.10	3.10	NUM
ejde-771	288	52	)	)	PUNCT
ejde-771	288	53	,	,	PUNCT
ejde-771	288	54	we	we	PRON
ejde-771	288	55	can	can	AUX
ejde-771	288	56	obtain	obtain	VERB
ejde-771	288	57	ũ	ũ	PROPN
ejde-771	288	58	=	=	PUNCT
ejde-771	288	59	θũ.	θũ.	PROPN
ejde-771	288	60	therefore	therefore	ADV
ejde-771	288	61	,	,	PUNCT
ejde-771	288	62	ũ	ũ	PROPN
ejde-771	288	63	∈	∈	PROPN
ejde-771	288	64	pe	pe	PROPN
ejde-771	288	65	∩	∩	PROPN
ejde-771	288	66	sapω(e	sapω(e	NOUN
ejde-771	288	67	)	)	PUNCT
ejde-771	288	68	is	be	AUX
ejde-771	288	69	fixed	fix	VERB
ejde-771	288	70	point	point	NOUN
ejde-771	288	71	of	of	ADP
ejde-771	288	72	θ	θ	PROPN
ejde-771	288	73	,	,	PUNCT
ejde-771	288	74	which	which	PRON
ejde-771	288	75	is	be	AUX
ejde-771	288	76	a	a	DET
ejde-771	288	77	positive	positive	ADJ
ejde-771	288	78	s	s	NOUN
ejde-771	288	79	-	-	PUNCT
ejde-771	288	80	asymptotically	asymptotically	ADV
ejde-771	288	81	ω	ω	ADJ
ejde-771	288	82	-	-	ADJ
ejde-771	288	83	periodic	periodic	ADJ
ejde-771	288	84	mild	mild	ADJ
ejde-771	288	85	solution	solution	NOUN
ejde-771	288	86	of	of	ADP
ejde-771	288	87	nonlocal	nonlocal	ADJ
ejde-771	288	88	problem	problem	NOUN
ejde-771	288	89	(	(	PUNCT
ejde-771	288	90	3.1	3.1	NUM
ejde-771	288	91	)	)	PUNCT
ejde-771	288	92	.	.	PUNCT
ejde-771	289	1	we	we	PRON
ejde-771	289	2	need	need	VERB
ejde-771	289	3	to	to	PART
ejde-771	289	4	verify	verify	VERB
ejde-771	289	5	that	that	SCONJ
ejde-771	289	6	ũ	ũ	PROPN
ejde-771	289	7	is	be	AUX
ejde-771	289	8	the	the	DET
ejde-771	289	9	minimal	minimal	ADJ
ejde-771	289	10	positive	positive	ADJ
ejde-771	289	11	s	s	NOUN
ejde-771	289	12	-	-	PUNCT
ejde-771	289	13	asymptotically	asymptotically	ADV
ejde-771	289	14	ω	ω	ADJ
ejde-771	289	15	-	-	ADJ
ejde-771	289	16	periodic	periodic	ADJ
ejde-771	289	17	mild	mild	ADJ
ejde-771	289	18	solution	solution	NOUN
ejde-771	289	19	.	.	PUNCT
ejde-771	290	1	let	let	VERB
ejde-771	290	2	û	û	NUM
ejde-771	290	3	∈	∈	NOUN
ejde-771	290	4	pe	pe	X
ejde-771	290	5	∩	∩	PROPN
ejde-771	290	6	sapω(e	sapω(e	NOUN
ejde-771	290	7	)	)	PUNCT
ejde-771	290	8	be	be	AUX
ejde-771	290	9	a	a	DET
ejde-771	290	10	positive	positive	ADJ
ejde-771	290	11	s	s	NOUN
ejde-771	290	12	-	-	PUNCT
ejde-771	290	13	asymptotically	asymptotically	ADV
ejde-771	290	14	ω	ω	ADJ
ejde-771	290	15	-	-	ADJ
ejde-771	290	16	periodic	periodic	ADJ
ejde-771	290	17	mild	mild	ADJ
ejde-771	290	18	solution	solution	NOUN
ejde-771	290	19	of	of	ADP
ejde-771	290	20	nonlocal	nonlocal	ADJ
ejde-771	290	21	problem	problem	NOUN
ejde-771	290	22	(	(	PUNCT
ejde-771	290	23	3.1	3.1	NUM
ejde-771	290	24	)	)	PUNCT
ejde-771	290	25	,	,	PUNCT
ejde-771	290	26	which	which	PRON
ejde-771	290	27	means	mean	VERB
ejde-771	290	28	that	that	SCONJ
ejde-771	290	29	û(t	û(t	SYM
ejde-771	290	30	)	)	PUNCT
ejde-771	290	31	=	=	SYM
ejde-771	290	32	θû(t	θû(t	PROPN
ejde-771	290	33	)	)	PUNCT
ejde-771	290	34	for	for	ADP
ejde-771	290	35	every	every	DET
ejde-771	290	36	t	t	NOUN
ejde-771	290	37	∈	∈	PROPN
ejde-771	291	1	[	[	X
ejde-771	291	2	0,∞	0,∞	NOUN
ejde-771	291	3	)	)	PUNCT
ejde-771	291	4	.	.	PUNCT
ejde-771	292	1	obviously	obviously	ADV
ejde-771	292	2	,	,	PUNCT
ejde-771	292	3	û(t	û(t	ADJ
ejde-771	292	4	)	)	PUNCT
ejde-771	292	5	≥	≥	NOUN
ejde-771	292	6	v0	v0	NOUN
ejde-771	292	7	=	=	SYM
ejde-771	292	8	0	0	X
ejde-771	292	9	.	.	X
ejde-771	293	1	taking	take	VERB
ejde-771	293	2	into	into	ADP
ejde-771	293	3	account	account	NOUN
ejde-771	293	4	the	the	DET
ejde-771	293	5	monotonicity	monotonicity	NOUN
ejde-771	293	6	of	of	ADP
ejde-771	293	7	θ	θ	PROPN
ejde-771	293	8	,	,	PUNCT
ejde-771	293	9	one	one	PRON
ejde-771	293	10	can	can	AUX
ejde-771	293	11	deduced	deduced	VERB
ejde-771	293	12	that	that	SCONJ
ejde-771	293	13	û(t	û(t	SYM
ejde-771	293	14	)	)	PUNCT
ejde-771	293	15	=	=	SYM
ejde-771	293	16	(	(	PUNCT
ejde-771	293	17	θû)(t	θû)(t	PROPN
ejde-771	293	18	)	)	PUNCT
ejde-771	293	19	≥	≥	NOUN
ejde-771	293	20	(	(	PUNCT
ejde-771	293	21	θv0)(t	θv0)(t	PROPN
ejde-771	293	22	)	)	PUNCT
ejde-771	293	23	=	=	PUNCT
ejde-771	294	1	v1(t	v1(t	PROPN
ejde-771	294	2	)	)	PUNCT
ejde-771	294	3	,	,	PUNCT
ejde-771	294	4	(	(	PUNCT
ejde-771	294	5	3.14	3.14	X
ejde-771	294	6	)	)	PUNCT
ejde-771	294	7	it	it	PRON
ejde-771	294	8	follows	follow	VERB
ejde-771	294	9	that	that	SCONJ
ejde-771	294	10	û	û	NUM
ejde-771	294	11	>	>	SYM
ejde-771	294	12	v1	v1	NOUN
ejde-771	294	13	.	.	PUNCT
ejde-771	295	1	repeat	repeat	VERB
ejde-771	295	2	this	this	DET
ejde-771	295	3	process	process	NOUN
ejde-771	295	4	,	,	PUNCT
ejde-771	295	5	one	one	PRON
ejde-771	295	6	can	can	AUX
ejde-771	295	7	see	see	VERB
ejde-771	295	8	û	û	NUM
ejde-771	295	9	>	>	X
ejde-771	295	10	vn	vn	X
ejde-771	295	11	,	,	PUNCT
ejde-771	295	12	n	n	NOUN
ejde-771	295	13	=	=	SYM
ejde-771	295	14	1	1	NUM
ejde-771	295	15	,	,	PUNCT
ejde-771	295	16	2	2	NUM
ejde-771	295	17	,	,	PUNCT
ejde-771	295	18	.	.	PUNCT
ejde-771	295	19	.	.	PUNCT
ejde-771	295	20	.	.	PUNCT
ejde-771	296	1	.	.	PUNCT
ejde-771	297	1	it	it	PRON
ejde-771	297	2	’s	’	VERB
ejde-771	297	3	worth	worth	ADJ
ejde-771	297	4	noting	note	VERB
ejde-771	297	5	that	that	SCONJ
ejde-771	297	6	one	one	PRON
ejde-771	297	7	can	can	AUX
ejde-771	297	8	obtain	obtain	VERB
ejde-771	297	9	û	û	NUM
ejde-771	297	10	>	>	X
ejde-771	297	11	ũ	ũ	PROPN
ejde-771	297	12	through	through	ADP
ejde-771	297	13	taking	take	VERB
ejde-771	297	14	the	the	DET
ejde-771	297	15	limit	limit	NOUN
ejde-771	297	16	in	in	ADP
ejde-771	297	17	(	(	PUNCT
ejde-771	297	18	3.14	3.14	NUM
ejde-771	297	19	)	)	PUNCT
ejde-771	297	20	as	as	ADP
ejde-771	297	21	n	n	PROPN
ejde-771	297	22	→	→	SYM
ejde-771	297	23	∞	∞	PROPN
ejde-771	297	24	,	,	PUNCT
ejde-771	297	25	which	which	PRON
ejde-771	297	26	means	mean	VERB
ejde-771	297	27	that	that	SCONJ
ejde-771	297	28	ũ	ũ	PROPN
ejde-771	297	29	is	be	AUX
ejde-771	297	30	the	the	DET
ejde-771	297	31	minimal	minimal	ADJ
ejde-771	297	32	positive	positive	ADJ
ejde-771	297	33	s	s	NOUN
ejde-771	297	34	-	-	PUNCT
ejde-771	297	35	asymptotically	asymptotically	ADV
ejde-771	297	36	ω	ω	ADJ
ejde-771	297	37	-	-	ADJ
ejde-771	297	38	periodic	periodic	ADJ
ejde-771	297	39	mild	mild	ADJ
ejde-771	297	40	solution	solution	NOUN
ejde-771	297	41	of	of	ADP
ejde-771	297	42	nonlocal	nonlocal	ADJ
ejde-771	297	43	problem	problem	NOUN
ejde-771	297	44	(	(	PUNCT
ejde-771	297	45	3.1	3.1	NUM
ejde-771	297	46	)	)	PUNCT
ejde-771	297	47	.	.	PUNCT
ejde-771	298	1	this	this	PRON
ejde-771	298	2	completes	complete	VERB
ejde-771	298	3	the	the	DET
ejde-771	298	4	proof	proof	NOUN
ejde-771	298	5	of	of	ADP
ejde-771	298	6	theorem	theorem	ADJ
ejde-771	298	7	3.2	3.2	NUM
ejde-771	298	8	.	.	PUNCT
ejde-771	299	1	□	□	PUNCT
ejde-771	299	2	now	now	ADV
ejde-771	299	3	,	,	PUNCT
ejde-771	299	4	we	we	PRON
ejde-771	299	5	assume	assume	VERB
ejde-771	299	6	that	that	SCONJ
ejde-771	299	7	the	the	DET
ejde-771	299	8	cone	cone	NOUN
ejde-771	299	9	p	p	NOUN
ejde-771	299	10	is	be	AUX
ejde-771	299	11	a	a	DET
ejde-771	299	12	regeneration	regeneration	NOUN
ejde-771	299	13	cone	cone	NOUN
ejde-771	299	14	on	on	ADP
ejde-771	299	15	e	e	PROPN
ejde-771	299	16	and	and	CCONJ
ejde-771	299	17	t	t	PROPN
ejde-771	299	18	(	(	PUNCT
ejde-771	299	19	t)(t	t)(t	ADJ
ejde-771	299	20	≥	≥	NOUN
ejde-771	299	21	0	0	NUM
ejde-771	299	22	)	)	PUNCT
ejde-771	299	23	generated	generate	VERB
ejde-771	299	24	by	by	ADP
ejde-771	299	25	−a	−a	NOUN
ejde-771	299	26	is	be	AUX
ejde-771	299	27	a	a	DET
ejde-771	299	28	positive	positive	ADJ
ejde-771	299	29	semigroup	semigroup	NOUN
ejde-771	299	30	,	,	PUNCT
ejde-771	299	31	it	it	PRON
ejde-771	299	32	follows	follow	VERB
ejde-771	299	33	that	that	SCONJ
ejde-771	299	34	λ0i+a	λ0i+a	PROPN
ejde-771	299	35	has	have	VERB
ejde-771	299	36	positive	positive	ADJ
ejde-771	299	37	bounded	bounded	ADJ
ejde-771	299	38	inverse	inverse	NOUN
ejde-771	299	39	operator	operator	NOUN
ejde-771	299	40	(	(	PUNCT
ejde-771	299	41	λ0i+a	λ0i+a	NOUN
ejde-771	299	42	)	)	PUNCT
ejde-771	299	43	−1	−1	NOUN
ejde-771	299	44	if	if	SCONJ
ejde-771	299	45	λ0	λ0	NOUN
ejde-771	299	46	>	>	X
ejde-771	299	47	−	−	PROPN
ejde-771	299	48	inf{reλ	inf{reλ	NOUN
ejde-771	300	1	|	|	ADV
ejde-771	300	2	λ	λ	PROPN
ejde-771	300	3	∈	∈	PROPN
ejde-771	300	4	σ(a	σ(a	PROPN
ejde-771	300	5	)	)	PUNCT
ejde-771	300	6	}	}	PUNCT
ejde-771	300	7	is	be	AUX
ejde-771	300	8	sufficiently	sufficiently	ADV
ejde-771	300	9	large	large	ADJ
ejde-771	300	10	through	through	ADP
ejde-771	300	11	the	the	DET
ejde-771	300	12	characteristic	characteristic	NOUN
ejde-771	300	13	of	of	ADP
ejde-771	300	14	positive	positive	ADJ
ejde-771	300	15	semigroups	semigroup	NOUN
ejde-771	300	16	.	.	PUNCT
ejde-771	301	1	since	since	SCONJ
ejde-771	301	2	σ(a	σ(a	PROPN
ejde-771	301	3	)	)	PUNCT
ejde-771	301	4	̸=	̸=	PROPN
ejde-771	301	5	∅	∅	NOUN
ejde-771	301	6	,	,	PUNCT
ejde-771	301	7	the	the	DET
ejde-771	301	8	spectral	spectral	ADJ
ejde-771	301	9	radius	radius	NOUN
ejde-771	301	10	r((λ0i	r((λ0i	NOUN
ejde-771	301	11	+	+	NOUN
ejde-771	301	12	a)−1	a)−1	NOUN
ejde-771	301	13	)	)	PUNCT
ejde-771	301	14	=	=	SYM
ejde-771	301	15	1	1	NUM
ejde-771	301	16	dist(−λ0	dist(−λ0	NOUN
ejde-771	301	17	,	,	PUNCT
ejde-771	301	18	σ(a	σ(a	PROPN
ejde-771	301	19	)	)	PUNCT
ejde-771	301	20	)	)	PUNCT
ejde-771	301	21	>	>	X
ejde-771	302	1	0	0	X
ejde-771	302	2	.	.	PUNCT
ejde-771	302	3	based	base	VERB
ejde-771	302	4	on	on	ADP
ejde-771	302	5	the	the	DET
ejde-771	302	6	famous	famous	ADJ
ejde-771	302	7	krein	krein	PROPN
ejde-771	302	8	-	-	PUNCT
ejde-771	302	9	rutman	rutman	NOUN
ejde-771	302	10	theorem(see	theorem(see	NOUN
ejde-771	302	11	[	[	X
ejde-771	302	12	15	15	NUM
ejde-771	302	13	,	,	PUNCT
ejde-771	302	14	16	16	NUM
ejde-771	302	15	]	]	PUNCT
ejde-771	302	16	)	)	PUNCT
ejde-771	302	17	,	,	PUNCT
ejde-771	302	18	a	a	PRON
ejde-771	302	19	has	have	AUX
ejde-771	302	20	the	the	DET
ejde-771	302	21	first	first	ADJ
ejde-771	302	22	eigenvalue	eigenvalue	PROPN
ejde-771	302	23	λ1	λ1	PROPN
ejde-771	302	24	>	>	SYM
ejde-771	302	25	0	0	PROPN
ejde-771	302	26	,	,	PUNCT
ejde-771	302	27	associated	associate	VERB
ejde-771	302	28	a	a	DET
ejde-771	302	29	positive	positive	ADJ
ejde-771	302	30	eigenfunction	eigenfunction	NOUN
ejde-771	302	31	e1	e1	NOUN
ejde-771	302	32	,	,	PUNCT
ejde-771	302	33	and	and	CCONJ
ejde-771	302	34	λ1	λ1	ADJ
ejde-771	302	35	=	=	SYM
ejde-771	302	36	inf{reλ	inf{reλ	NOUN
ejde-771	302	37	|	|	ADV
ejde-771	302	38	λ	λ	PROPN
ejde-771	302	39	∈	∈	PROPN
ejde-771	302	40	σ(a	σ(a	PROPN
ejde-771	302	41	)	)	PUNCT
ejde-771	302	42	}	}	PUNCT
ejde-771	302	43	.	.	PUNCT
ejde-771	303	1	therefore	therefore	ADV
ejde-771	303	2	,	,	PUNCT
ejde-771	303	3	it	it	PRON
ejde-771	303	4	follows	follow	VERB
ejde-771	303	5	from	from	ADP
ejde-771	303	6	(	(	PUNCT
ejde-771	303	7	2.1	2.1	NUM
ejde-771	303	8	)	)	PUNCT
ejde-771	303	9	that	that	SCONJ
ejde-771	303	10	ν0	ν0	PROPN
ejde-771	303	11	=	=	SYM
ejde-771	303	12	−λ1	−λ1	PROPN
ejde-771	303	13	.	.	PUNCT
ejde-771	304	1	by	by	ADP
ejde-771	304	2	theorem	theorem	NOUN
ejde-771	304	3	(	(	PUNCT
ejde-771	304	4	3.2	3.2	NUM
ejde-771	304	5	)	)	PUNCT
ejde-771	304	6	,	,	PUNCT
ejde-771	304	7	we	we	PRON
ejde-771	304	8	have	have	VERB
ejde-771	304	9	the	the	DET
ejde-771	304	10	following	follow	VERB
ejde-771	304	11	results	result	NOUN
ejde-771	304	12	.	.	PUNCT
ejde-771	305	1	ejde-2025/44	ejde-2025/44	ADJ
ejde-771	305	2	time	time	NOUN
ejde-771	305	3	-	-	PUNCT
ejde-771	305	4	space	space	NOUN
ejde-771	305	5	fractional	fractional	ADJ
ejde-771	305	6	reaction	reaction	NOUN
ejde-771	305	7	-	-	PUNCT
ejde-771	305	8	diffusion	diffusion	NOUN
ejde-771	305	9	equations	equation	NOUN
ejde-771	305	10	11	11	NUM
ejde-771	305	11	corollary	corollary	ADJ
ejde-771	305	12	3.3	3.3	NUM
ejde-771	305	13	.	.	PUNCT
ejde-771	306	1	let	let	AUX
ejde-771	306	2	e	e	PRON
ejde-771	306	3	be	be	AUX
ejde-771	306	4	an	an	DET
ejde-771	306	5	ordered	ordered	ADJ
ejde-771	306	6	banach	banach	NOUN
ejde-771	306	7	space	space	NOUN
ejde-771	306	8	,	,	PUNCT
ejde-771	306	9	whose	whose	DET
ejde-771	306	10	positive	positive	ADJ
ejde-771	306	11	cone	cone	NOUN
ejde-771	306	12	p	p	NOUN
ejde-771	306	13	is	be	AUX
ejde-771	306	14	a	a	DET
ejde-771	306	15	regeneration	regeneration	NOUN
ejde-771	306	16	cone	cone	NOUN
ejde-771	306	17	,	,	PUNCT
ejde-771	306	18	let	let	VERB
ejde-771	306	19	a	a	DET
ejde-771	306	20	:	:	PUNCT
ejde-771	306	21	d(a	d(a	PROPN
ejde-771	306	22	)	)	PUNCT
ejde-771	306	23	⊂	⊂	PROPN
ejde-771	306	24	e	e	X
ejde-771	306	25	→	→	PUNCT
ejde-771	306	26	e	e	X
ejde-771	306	27	be	be	AUX
ejde-771	306	28	a	a	DET
ejde-771	306	29	closed	closed	ADJ
ejde-771	306	30	linear	linear	ADJ
ejde-771	306	31	operator	operator	NOUN
ejde-771	306	32	and	and	CCONJ
ejde-771	306	33	−a	−a	NOUN
ejde-771	306	34	generate	generate	VERB
ejde-771	306	35	an	an	DET
ejde-771	306	36	exponentially	exponentially	ADV
ejde-771	306	37	stable	stable	ADJ
ejde-771	306	38	,	,	PUNCT
ejde-771	306	39	positive	positive	ADJ
ejde-771	306	40	,	,	PUNCT
ejde-771	306	41	and	and	CCONJ
ejde-771	306	42	compact	compact	ADJ
ejde-771	306	43	semigroup	semigroup	PROPN
ejde-771	306	44	t	t	PROPN
ejde-771	306	45	(	(	PUNCT
ejde-771	306	46	t)(t	t)(t	ADJ
ejde-771	306	47	≥	≥	NOUN
ejde-771	306	48	0	0	NUM
ejde-771	306	49	)	)	PUNCT
ejde-771	306	50	in	in	ADP
ejde-771	306	51	e	e	NOUN
ejde-771	306	52	,	,	PUNCT
ejde-771	306	53	u0	u0	ADJ
ejde-771	306	54	≥	≥	NUM
ejde-771	306	55	θ	θ	PROPN
ejde-771	306	56	.	.	PROPN
ejde-771	306	57	assume	assume	VERB
ejde-771	306	58	that	that	SCONJ
ejde-771	306	59	g	g	NOUN
ejde-771	306	60	:	:	PUNCT
ejde-771	307	1	[	[	X
ejde-771	307	2	0,∞)×	0,∞)×	NUM
ejde-771	307	3	e	e	NOUN
ejde-771	307	4	→	→	SYM
ejde-771	307	5	e	e	X
ejde-771	307	6	is	be	AUX
ejde-771	307	7	a	a	DET
ejde-771	307	8	continuous	continuous	ADJ
ejde-771	307	9	function	function	NOUN
ejde-771	307	10	,	,	PUNCT
ejde-771	307	11	and	and	CCONJ
ejde-771	307	12	let	let	VERB
ejde-771	307	13	conditions	condition	NOUN
ejde-771	307	14	(	(	PUNCT
ejde-771	307	15	h0	h0	NOUN
ejde-771	307	16	)	)	PUNCT
ejde-771	307	17	,	,	PUNCT
ejde-771	307	18	(	(	PUNCT
ejde-771	307	19	h2	h2	NOUN
ejde-771	307	20	)	)	PUNCT
ejde-771	307	21	,	,	PUNCT
ejde-771	307	22	(	(	PUNCT
ejde-771	307	23	h3	h3	NOUN
ejde-771	307	24	)	)	PUNCT
ejde-771	307	25	,	,	PUNCT
ejde-771	307	26	and	and	CCONJ
ejde-771	307	27	(	(	PUNCT
ejde-771	307	28	h4	h4	PROPN
ejde-771	307	29	)	)	PUNCT
ejde-771	307	30	for	for	ADP
ejde-771	307	31	t	t	PROPN
ejde-771	307	32	≥	≥	NOUN
ejde-771	307	33	0	0	PUNCT
ejde-771	307	34	and	and	CCONJ
ejde-771	307	35	x	x	SYM
ejde-771	307	36	∈	∈	PROPN
ejde-771	307	37	e	e	NOUN
ejde-771	307	38	,	,	PUNCT
ejde-771	307	39	there	there	PRON
ejde-771	307	40	exist	exist	VERB
ejde-771	307	41	positive	positive	ADJ
ejde-771	307	42	constants	constant	NOUN
ejde-771	307	43	a0	a0	PROPN
ejde-771	307	44	≥	≥	NOUN
ejde-771	307	45	0	0	NUM
ejde-771	308	1	and	and	CCONJ
ejde-771	308	2	a1	a1	PROPN
ejde-771	308	3	∈	∈	PROPN
ejde-771	308	4	(	(	PUNCT
ejde-771	308	5	0	0	NUM
ejde-771	308	6	,	,	PUNCT
ejde-771	308	7	(	(	PUNCT
ejde-771	308	8	1−m	1−m	NUM
ejde-771	308	9	∑m	∑m	NOUN
ejde-771	308	10	k=1	k=1	X
ejde-771	308	11	|ak|)λ	|ak|)λ	X
ejde-771	308	12	β	β	X
ejde-771	308	13	1	1	NUM
ejde-771	308	14	/	/	SYM
ejde-771	308	15	m	m	NOUN
ejde-771	308	16	)	)	PUNCT
ejde-771	308	17	such	such	ADJ
ejde-771	308	18	that	that	SCONJ
ejde-771	308	19	∥g(t	∥g(t	NOUN
ejde-771	308	20	,	,	PUNCT
ejde-771	308	21	etx)∥	etx)∥	PROPN
ejde-771	308	22	≤	≤	ADV
ejde-771	308	23	a1∥x∥+a0	a1∥x∥+a0	PROPN
ejde-771	308	24	,	,	PUNCT
ejde-771	308	25	hold	hold	VERB
ejde-771	308	26	,	,	PUNCT
ejde-771	308	27	then	then	ADV
ejde-771	308	28	there	there	PRON
ejde-771	308	29	exist	exist	VERB
ejde-771	308	30	a	a	DET
ejde-771	308	31	minimal	minimal	ADJ
ejde-771	308	32	positive	positive	ADJ
ejde-771	308	33	s	s	NOUN
ejde-771	308	34	-	-	PUNCT
ejde-771	308	35	asymptotically	asymptotically	ADV
ejde-771	308	36	ω	ω	ADJ
ejde-771	308	37	-	-	ADJ
ejde-771	308	38	periodic	periodic	ADJ
ejde-771	308	39	mild	mild	ADJ
ejde-771	308	40	solution	solution	NOUN
ejde-771	308	41	ũ	ũ	PROPN
ejde-771	308	42	of	of	ADP
ejde-771	308	43	nonlocal	nonlocal	ADJ
ejde-771	308	44	problem	problem	NOUN
ejde-771	308	45	(	(	PUNCT
ejde-771	308	46	3.1	3.1	NUM
ejde-771	308	47	)	)	PUNCT
ejde-771	308	48	.	.	PUNCT
ejde-771	309	1	theorem	theorem	VERB
ejde-771	309	2	3.4	3.4	NUM
ejde-771	309	3	.	.	PUNCT
ejde-771	310	1	let	let	VERB
ejde-771	310	2	e	e	PRON
ejde-771	310	3	be	be	AUX
ejde-771	310	4	an	an	DET
ejde-771	310	5	ordered	ordered	ADJ
ejde-771	310	6	banach	banach	NOUN
ejde-771	310	7	space	space	NOUN
ejde-771	310	8	,	,	PUNCT
ejde-771	310	9	whose	whose	DET
ejde-771	310	10	positive	positive	ADJ
ejde-771	310	11	cone	cone	NOUN
ejde-771	310	12	p	p	NOUN
ejde-771	310	13	is	be	AUX
ejde-771	310	14	normal	normal	ADJ
ejde-771	310	15	,	,	PUNCT
ejde-771	310	16	a	a	DET
ejde-771	310	17	:	:	PUNCT
ejde-771	310	18	d(a	d(a	PROPN
ejde-771	310	19	)	)	PUNCT
ejde-771	310	20	⊂	⊂	PROPN
ejde-771	310	21	e	e	X
ejde-771	310	22	→	→	PUNCT
ejde-771	310	23	e	e	X
ejde-771	310	24	be	be	AUX
ejde-771	310	25	a	a	DET
ejde-771	310	26	closed	closed	ADJ
ejde-771	310	27	linear	linear	ADJ
ejde-771	310	28	operator	operator	NOUN
ejde-771	310	29	and	and	CCONJ
ejde-771	310	30	−a	−a	NOUN
ejde-771	310	31	generate	generate	VERB
ejde-771	310	32	an	an	DET
ejde-771	310	33	exponentially	exponentially	ADV
ejde-771	310	34	stable	stable	ADJ
ejde-771	310	35	,	,	PUNCT
ejde-771	310	36	positive	positive	ADJ
ejde-771	310	37	and	and	CCONJ
ejde-771	310	38	compact	compact	ADJ
ejde-771	310	39	analytic	analytic	ADJ
ejde-771	310	40	semigroup	semigroup	NOUN
ejde-771	310	41	t	t	PROPN
ejde-771	310	42	(	(	PUNCT
ejde-771	310	43	t)(t	t)(t	ADJ
ejde-771	310	44	≥	≥	NOUN
ejde-771	310	45	0	0	NUM
ejde-771	310	46	)	)	PUNCT
ejde-771	310	47	in	in	ADP
ejde-771	310	48	e	e	NOUN
ejde-771	310	49	,	,	PUNCT
ejde-771	310	50	whose	whose	DET
ejde-771	310	51	growth	growth	NOUN
ejde-771	310	52	exponent	exponent	NOUN
ejde-771	310	53	ν0	ν0	PROPN
ejde-771	310	54	<	<	X
ejde-771	310	55	0	0	NUM
ejde-771	310	56	,	,	PUNCT
ejde-771	310	57	the	the	DET
ejde-771	310	58	nonlinear	nonlinear	ADJ
ejde-771	310	59	function	function	NOUN
ejde-771	310	60	g	g	NOUN
ejde-771	310	61	:	:	PUNCT
ejde-771	310	62	r+	r+	NOUN
ejde-771	310	63	×	×	PROPN
ejde-771	310	64	e	e	X
ejde-771	310	65	→	→	PUNCT
ejde-771	310	66	e	e	AUX
ejde-771	310	67	be	be	AUX
ejde-771	310	68	a	a	DET
ejde-771	310	69	continuous	continuous	ADJ
ejde-771	310	70	mapping	mapping	NOUN
ejde-771	310	71	.	.	PUNCT
ejde-771	311	1	if	if	SCONJ
ejde-771	311	2	the	the	DET
ejde-771	311	3	conditions	condition	NOUN
ejde-771	311	4	(	(	PUNCT
ejde-771	311	5	h0),(h3),(h4)and	h0),(h3),(h4)and	NOUN
ejde-771	311	6	(	(	PUNCT
ejde-771	311	7	h5	h5	PROPN
ejde-771	311	8	)	)	PUNCT
ejde-771	311	9	for	for	ADP
ejde-771	311	10	each	each	DET
ejde-771	311	11	u	u	NOUN
ejde-771	311	12	∈	∈	NOUN
ejde-771	311	13	ce(e	ce(e	NOUN
ejde-771	311	14	)	)	PUNCT
ejde-771	311	15	with	with	ADP
ejde-771	311	16	u(t	u(t	NOUN
ejde-771	311	17	)	)	PUNCT
ejde-771	311	18	≥	≥	NOUN
ejde-771	311	19	ςe1	ςe1	NOUN
ejde-771	311	20	,	,	PUNCT
ejde-771	311	21	there	there	PRON
ejde-771	311	22	is	be	VERB
ejde-771	311	23	a	a	DET
ejde-771	311	24	constant	constant	ADJ
ejde-771	311	25	ς	ς	X
ejde-771	311	26	>	>	X
ejde-771	311	27	0	0	NUM
ejde-771	311	28	such	such	ADJ
ejde-771	311	29	that	that	SCONJ
ejde-771	311	30	g(t	g(t	PROPN
ejde-771	311	31	,	,	PUNCT
ejde-771	311	32	u(t	u(t	NOUN
ejde-771	311	33	)	)	PUNCT
ejde-771	311	34	)	)	PUNCT
ejde-771	311	35	≥	≥	PROPN
ejde-771	312	1	g(t	g(t	PROPN
ejde-771	312	2	,	,	PUNCT
ejde-771	312	3	ςe1	ςe1	PROPN
ejde-771	312	4	)	)	PUNCT
ejde-771	312	5	≥	≥	PROPN
ejde-771	312	6	λβ1	λβ1	PROPN
ejde-771	312	7	ςe1	ςe1	PROPN
ejde-771	312	8	,	,	PUNCT
ejde-771	312	9	hold	hold	NOUN
ejde-771	312	10	and	and	CCONJ
ejde-771	312	11	u(0	u(0	PROPN
ejde-771	312	12	)	)	PUNCT
ejde-771	312	13	≥	≥	NOUN
ejde-771	312	14	ςe1	ςe1	NOUN
ejde-771	312	15	,	,	PUNCT
ejde-771	312	16	then	then	ADV
ejde-771	312	17	the	the	DET
ejde-771	312	18	nonlocal	nonlocal	ADJ
ejde-771	312	19	problem	problem	NOUN
ejde-771	312	20	(	(	PUNCT
ejde-771	312	21	3.1	3.1	NUM
ejde-771	312	22	)	)	PUNCT
ejde-771	312	23	has	have	VERB
ejde-771	312	24	at	at	ADV
ejde-771	312	25	least	least	ADV
ejde-771	312	26	one	one	NUM
ejde-771	312	27	positive	positive	ADJ
ejde-771	312	28	s	s	NOUN
ejde-771	312	29	-	-	PUNCT
ejde-771	312	30	asymptotically	asymptotically	ADV
ejde-771	312	31	ω	ω	ADJ
ejde-771	312	32	-	-	ADJ
ejde-771	312	33	periodic	periodic	ADJ
ejde-771	312	34	mild	mild	ADJ
ejde-771	312	35	solution	solution	NOUN
ejde-771	312	36	.	.	PUNCT
ejde-771	313	1	proof	proof	NOUN
ejde-771	313	2	.	.	PUNCT
ejde-771	314	1	let	let	VERB
ejde-771	314	2	θ	θ	NOUN
ejde-771	314	3	be	be	AUX
ejde-771	314	4	defined	define	VERB
ejde-771	314	5	by	by	ADP
ejde-771	314	6	(	(	PUNCT
ejde-771	314	7	3.4	3.4	NUM
ejde-771	314	8	)	)	PUNCT
ejde-771	314	9	,	,	PUNCT
ejde-771	314	10	it	it	PRON
ejde-771	314	11	follows	follow	VERB
ejde-771	314	12	from	from	ADP
ejde-771	314	13	the	the	DET
ejde-771	314	14	proof	proof	NOUN
ejde-771	314	15	of	of	ADP
ejde-771	314	16	theorem	theorem	NOUN
ejde-771	314	17	(	(	PUNCT
ejde-771	314	18	3.2	3.2	NUM
ejde-771	314	19	)	)	PUNCT
ejde-771	314	20	that	that	SCONJ
ejde-771	314	21	θ(sapω(e	θ(sapω(e	ADP
ejde-771	314	22	)	)	PUNCT
ejde-771	314	23	)	)	PUNCT
ejde-771	315	1	⊂	⊂	PROPN
ejde-771	315	2	sapω(e	sapω(e	NOUN
ejde-771	315	3	)	)	PUNCT
ejde-771	315	4	.	.	PUNCT
ejde-771	316	1	we	we	PRON
ejde-771	316	2	denote	denote	VERB
ejde-771	316	3	br0	br0	VERB
ejde-771	316	4	:	:	PUNCT
ejde-771	316	5	=	=	SYM
ejde-771	316	6	{	{	PUNCT
ejde-771	316	7	u	u	NOUN
ejde-771	316	8	∈	∈	NOUN
ejde-771	316	9	ce(e	ce(e	NOUN
ejde-771	316	10	)	)	PUNCT
ejde-771	316	11	|	|	ADV
ejde-771	316	12	∥u∥e	∥u∥e	ADJ
ejde-771	316	13	≤	≤	NUM
ejde-771	316	14	r0	r0	NOUN
ejde-771	316	15	,	,	PUNCT
ejde-771	316	16	u(t	u(t	PROPN
ejde-771	316	17	)	)	PUNCT
ejde-771	316	18	≥	≥	NOUN
ejde-771	316	19	ςe1	ςe1	PROPN
ejde-771	316	20	,	,	PUNCT
ejde-771	316	21	t	t	PROPN
ejde-771	316	22	≥	≥	NUM
ejde-771	316	23	0	0	NUM
ejde-771	316	24	}	}	PUNCT
ejde-771	316	25	(	(	PUNCT
ejde-771	316	26	3.15	3.15	NUM
ejde-771	316	27	)	)	PUNCT
ejde-771	316	28	which	which	PRON
ejde-771	316	29	is	be	AUX
ejde-771	316	30	a	a	DET
ejde-771	316	31	nonempty	nonempty	ADV
ejde-771	316	32	bounded	bound	VERB
ejde-771	316	33	convex	convex	NOUN
ejde-771	316	34	closed	close	VERB
ejde-771	316	35	set	set	VERB
ejde-771	316	36	for	for	ADP
ejde-771	316	37	r0	r0	NOUN
ejde-771	316	38	≥	≥	NOUN
ejde-771	316	39	m(λβ1∥u0∥+a0	m(λβ1∥u0∥+a0	NOUN
ejde-771	316	40	)	)	PUNCT
ejde-771	316	41	(	(	PUNCT
ejde-771	316	42	1−m	1−m	NUM
ejde-771	316	43	∑m	∑m	PROPN
ejde-771	316	44	k=1	k=1	PROPN
ejde-771	316	45	|ak|	|ak|	PROPN
ejde-771	316	46	)	)	PUNCT
ejde-771	316	47	λβ1	λβ1	PROPN
ejde-771	316	48	−ma1	−ma1	PROPN
ejde-771	316	49	.	.	PUNCT
ejde-771	317	1	hence	hence	ADV
ejde-771	317	2	,	,	PUNCT
ejde-771	317	3	for	for	SCONJ
ejde-771	317	4	any	any	DET
ejde-771	317	5	u	u	PROPN
ejde-771	317	6	∈	∈	PROPN
ejde-771	317	7	br0	br0	NOUN
ejde-771	317	8	and	and	CCONJ
ejde-771	317	9	t	t	PROPN
ejde-771	317	10	≥	≥	PROPN
ejde-771	317	11	0	0	NUM
ejde-771	317	12	,	,	PUNCT
ejde-771	317	13	exploiting	exploit	VERB
ejde-771	317	14	(	(	PUNCT
ejde-771	317	15	h4	h4	NOUN
ejde-771	317	16	)	)	PUNCT
ejde-771	317	17	,	,	PUNCT
ejde-771	317	18	according	accord	VERB
ejde-771	317	19	to	to	ADP
ejde-771	317	20	e−t	e−t	NOUN
ejde-771	317	21	≤	≤	NUM
ejde-771	317	22	1	1	NUM
ejde-771	317	23	,	,	PUNCT
ejde-771	317	24	one	one	PRON
ejde-771	317	25	can	can	AUX
ejde-771	317	26	obtain	obtain	VERB
ejde-771	317	27	∥(θu)(t)∥e	∥(θu)(t)∥e	NOUN
ejde-771	317	28	=	=	NOUN
ejde-771	317	29	sup	sup	NOUN
ejde-771	317	30	t∈r+	t∈r+	NOUN
ejde-771	317	31	e−t∥(θu)(t)∥	e−t∥(θu)(t)∥	PROPN
ejde-771	317	32	≤	≤	PROPN
ejde-771	317	33	∥(θu)(t)∥	∥(θu)(t)∥	NUM
ejde-771	317	34	≤	≤	NUM
ejde-771	317	35	m∥u0∥	m∥u0∥	PROPN
ejde-771	317	36	1−m	1−m	NUM
ejde-771	317	37	∑m	∑m	PROPN
ejde-771	318	1	k=1	k=1	PUNCT
ejde-771	318	2	|ak|	|ak|	PROPN
ejde-771	319	1	+	+	CCONJ
ejde-771	319	2	m(a1∥u∥e	m(a1∥u∥e	NOUN
ejde-771	319	3	+	+	NOUN
ejde-771	319	4	a0	a0	NOUN
ejde-771	319	5	)	)	PUNCT
ejde-771	319	6	(	(	PUNCT
ejde-771	319	7	1−m	1−m	NUM
ejde-771	319	8	∑m	∑m	NOUN
ejde-771	319	9	k=1	k=1	X
ejde-771	319	10	|ak|)λ	|ak|)λ	X
ejde-771	319	11	β	β	X
ejde-771	319	12	1	1	NUM
ejde-771	319	13	≤	≤	NOUN
ejde-771	319	14	r0	r0	NOUN
ejde-771	319	15	.	.	PUNCT
ejde-771	320	1	let	let	VERB
ejde-771	320	2	w0	w0	PROPN
ejde-771	320	3	=	=	PUNCT
ejde-771	320	4	ςe1	ςe1	PROPN
ejde-771	320	5	.	.	PUNCT
ejde-771	321	1	then	then	ADV
ejde-771	321	2	w0(t	w0(t	NUM
ejde-771	321	3	)	)	PUNCT
ejde-771	321	4	=	=	SYM
ejde-771	321	5	ςe1	ςe1	NOUN
ejde-771	321	6	for	for	ADP
ejde-771	321	7	any	any	DET
ejde-771	321	8	t	t	PROPN
ejde-771	321	9	≥	≥	NOUN
ejde-771	321	10	0	0	NUM
ejde-771	321	11	,	,	PUNCT
ejde-771	321	12	and	and	CCONJ
ejde-771	321	13	η(t	η(t	NOUN
ejde-771	321	14	)	)	PUNCT
ejde-771	321	15	:	:	PUNCT
ejde-771	322	1	=	=	NOUN
ejde-771	322	2	c	c	X
ejde-771	322	3	dα	dα	ADP
ejde-771	322	4	t	t	PROPN
ejde-771	322	5	w0(t	w0(t	PROPN
ejde-771	322	6	)	)	PUNCT
ejde-771	322	7	+	+	NOUN
ejde-771	322	8	aβw0(t	aβw0(t	NOUN
ejde-771	322	9	)	)	PUNCT
ejde-771	322	10	=	=	SYM
ejde-771	322	11	λβ1	λβ1	NOUN
ejde-771	322	12	ςe1	ςe1	NOUN
ejde-771	322	13	≤	≤	PROPN
ejde-771	322	14	g(t	g(t	PROPN
ejde-771	322	15	,	,	PUNCT
ejde-771	322	16	ςe1	ςe1	PROPN
ejde-771	322	17	)	)	PUNCT
ejde-771	322	18	,	,	PUNCT
ejde-771	322	19	t	t	PROPN
ejde-771	322	20	≥	≥	NUM
ejde-771	322	21	0	0	NUM
ejde-771	322	22	.	.	PUNCT
ejde-771	323	1	by	by	ADP
ejde-771	323	2	the	the	DET
ejde-771	323	3	positivity	positivity	NOUN
ejde-771	323	4	of	of	ADP
ejde-771	323	5	semigroup	semigroup	PROPN
ejde-771	323	6	tβ(t)(t	tβ(t)(t	X
ejde-771	323	7	≥	≥	NOUN
ejde-771	323	8	0	0	NUM
ejde-771	323	9	)	)	PUNCT
ejde-771	323	10	,	,	PUNCT
ejde-771	323	11	condition	condition	NOUN
ejde-771	323	12	(	(	PUNCT
ejde-771	323	13	h5	h5	PROPN
ejde-771	323	14	)	)	PUNCT
ejde-771	323	15	and	and	CCONJ
ejde-771	323	16	(	(	PUNCT
ejde-771	323	17	3.4	3.4	NUM
ejde-771	323	18	)	)	PUNCT
ejde-771	323	19	,	,	PUNCT
ejde-771	323	20	for	for	SCONJ
ejde-771	323	21	any	any	DET
ejde-771	323	22	u	u	PROPN
ejde-771	323	23	∈	∈	PROPN
ejde-771	323	24	br0	br0	NOUN
ejde-771	323	25	and	and	CCONJ
ejde-771	323	26	t	t	PROPN
ejde-771	323	27	≥	≥	NUM
ejde-771	323	28	0	0	NUM
ejde-771	323	29	,	,	PUNCT
ejde-771	323	30	one	one	PRON
ejde-771	323	31	can	can	AUX
ejde-771	323	32	see	see	VERB
ejde-771	323	33	that	that	DET
ejde-771	323	34	ςe1	ςe1	NOUN
ejde-771	323	35	=	=	SYM
ejde-771	323	36	w0(t	w0(t	PROPN
ejde-771	323	37	)	)	PUNCT
ejde-771	323	38	=	=	SYM
ejde-771	323	39	jα	jα	NOUN
ejde-771	323	40	,	,	PUNCT
ejde-771	323	41	β(t)λw0(0	β(t)λw0(0	NUM
ejde-771	323	42	)	)	PUNCT
ejde-771	324	1	+	+	CCONJ
ejde-771	324	2	m∑	m∑	ADV
ejde-771	324	3	k=1	k=1	PRON
ejde-771	324	4	akjα	akjα	NOUN
ejde-771	324	5	,	,	PUNCT
ejde-771	324	6	β(t)λ	β(t)λ	PROPN
ejde-771	324	7	∫	∫	PROPN
ejde-771	324	8	tk	tk	PROPN
ejde-771	324	9	0	0	PROPN
ejde-771	325	1	(	(	PUNCT
ejde-771	325	2	tk	tk	PROPN
ejde-771	325	3	−	−	PROPN
ejde-771	325	4	s)α−1kα	s)α−1kα	PROPN
ejde-771	325	5	,	,	PUNCT
ejde-771	325	6	β(tk	β(tk	PROPN
ejde-771	325	7	−	−	NOUN
ejde-771	326	1	s)η(s)ds	s)η(s)ds	PROPN
ejde-771	327	1	+	+	CCONJ
ejde-771	327	2	∫	∫	PROPN
ejde-771	327	3	t	t	PROPN
ejde-771	327	4	0	0	NUM
ejde-771	327	5	(	(	PUNCT
ejde-771	327	6	t−	t−	PROPN
ejde-771	327	7	s)α−1kα	s)α−1kα	PROPN
ejde-771	327	8	,	,	PUNCT
ejde-771	327	9	β(t−	β(t−	PROPN
ejde-771	327	10	s)η(s)ds	s)η(s)ds	PROPN
ejde-771	327	11	≤	≤	NUM
ejde-771	327	12	jα	jα	NOUN
ejde-771	327	13	,	,	PUNCT
ejde-771	327	14	β(t)λςe1	β(t)λςe1	NOUN
ejde-771	328	1	+	+	CCONJ
ejde-771	328	2	m∑	m∑	CCONJ
ejde-771	328	3	k=1	k=1	PROPN
ejde-771	328	4	akjα	akjα	NOUN
ejde-771	328	5	,	,	PUNCT
ejde-771	328	6	β(t)λ	β(t)λ	PROPN
ejde-771	328	7	∫	∫	PROPN
ejde-771	328	8	tk	tk	PROPN
ejde-771	328	9	0	0	PROPN
ejde-771	329	1	(	(	PUNCT
ejde-771	329	2	tk	tk	PROPN
ejde-771	329	3	−	−	PROPN
ejde-771	329	4	s)α−1kα	s)α−1kα	PROPN
ejde-771	329	5	,	,	PUNCT
ejde-771	329	6	β(tk	β(tk	PROPN
ejde-771	329	7	−	−	NOUN
ejde-771	329	8	s)g(s	s)g(s	NOUN
ejde-771	329	9	,	,	PUNCT
ejde-771	329	10	ςe1)ds	ςe1)ds	NOUN
ejde-771	330	1	+	+	CCONJ
ejde-771	330	2	∫	∫	PROPN
ejde-771	330	3	t	t	PROPN
ejde-771	330	4	0	0	NUM
ejde-771	330	5	(	(	PUNCT
ejde-771	330	6	t−	t−	PROPN
ejde-771	330	7	s)α−1kα	s)α−1kα	PROPN
ejde-771	330	8	,	,	PUNCT
ejde-771	330	9	β(t−	β(t−	PROPN
ejde-771	330	10	s)g(s	s)g(s	NOUN
ejde-771	330	11	,	,	PUNCT
ejde-771	330	12	ςe1)ds	ςe1)ds	NUM
ejde-771	330	13	12	12	NUM
ejde-771	330	14	x.	x.	NOUN
ejde-771	330	15	zhang	zhang	PROPN
ejde-771	330	16	,	,	PUNCT
ejde-771	330	17	k.	k.	PROPN
ejde-771	330	18	ding	ding	PROPN
ejde-771	330	19	,	,	PUNCT
ejde-771	330	20	p.	p.	PROPN
ejde-771	330	21	chen	chen	PROPN
ejde-771	331	1	ejde-2025/44	ejde-2025/44	PROPN
ejde-771	331	2	≤	≤	PROPN
ejde-771	331	3	jα	jα	NOUN
ejde-771	331	4	,	,	PUNCT
ejde-771	331	5	β(t)λu0	β(t)λu0	ADJ
ejde-771	331	6	+	+	CCONJ
ejde-771	331	7	m∑	m∑	ADV
ejde-771	331	8	k=1	k=1	PROPN
ejde-771	331	9	akjα	akjα	NOUN
ejde-771	331	10	,	,	PUNCT
ejde-771	331	11	β(t)λ	β(t)λ	PROPN
ejde-771	331	12	∫	∫	PROPN
ejde-771	331	13	tk	tk	PROPN
ejde-771	331	14	0	0	PROPN
ejde-771	332	1	(	(	PUNCT
ejde-771	332	2	tk	tk	PROPN
ejde-771	332	3	−	−	PROPN
ejde-771	332	4	s)α−1kα	s)α−1kα	PROPN
ejde-771	332	5	,	,	PUNCT
ejde-771	332	6	β(tk	β(tk	PROPN
ejde-771	332	7	−	−	NOUN
ejde-771	332	8	s)g(s	s)g(s	NOUN
ejde-771	332	9	,	,	PUNCT
ejde-771	332	10	u(s))ds	u(s))ds	PROPN
ejde-771	333	1	+	+	CCONJ
ejde-771	333	2	∫	∫	PROPN
ejde-771	333	3	t	t	PROPN
ejde-771	333	4	0	0	NUM
ejde-771	333	5	(	(	PUNCT
ejde-771	333	6	t−	t−	PROPN
ejde-771	333	7	s)α−1kα	s)α−1kα	PROPN
ejde-771	333	8	,	,	PUNCT
ejde-771	333	9	β(t−	β(t−	PROPN
ejde-771	333	10	s)g(s	s)g(s	NOUN
ejde-771	333	11	,	,	PUNCT
ejde-771	333	12	u(s))ds	u(s))ds	PROPN
ejde-771	333	13	=	=	SYM
ejde-771	333	14	(	(	PUNCT
ejde-771	333	15	θu)(t	θu)(t	PROPN
ejde-771	333	16	)	)	PUNCT
ejde-771	333	17	.	.	PUNCT
ejde-771	334	1	thus	thus	ADV
ejde-771	334	2	,	,	PUNCT
ejde-771	334	3	θ(br0	θ(br0	NOUN
ejde-771	334	4	)	)	PUNCT
ejde-771	334	5	⊂	⊂	PROPN
ejde-771	334	6	br0	br0	VERB
ejde-771	334	7	and	and	CCONJ
ejde-771	334	8	(	(	PUNCT
ejde-771	334	9	θu)(t	θu)(t	PROPN
ejde-771	334	10	)	)	PUNCT
ejde-771	334	11	≥	≥	NOUN
ejde-771	334	12	ςe1	ςe1	NOUN
ejde-771	334	13	for	for	SCONJ
ejde-771	334	14	any	any	DET
ejde-771	334	15	u	u	PROPN
ejde-771	334	16	∈	∈	PROPN
ejde-771	334	17	br0	br0	NOUN
ejde-771	334	18	and	and	CCONJ
ejde-771	334	19	t	t	PROPN
ejde-771	334	20	≥	≥	NUM
ejde-771	334	21	0	0	NUM
ejde-771	334	22	.	.	PUNCT
ejde-771	335	1	next	next	ADV
ejde-771	335	2	,	,	PUNCT
ejde-771	335	3	we	we	PRON
ejde-771	335	4	prove	prove	VERB
ejde-771	335	5	that	that	SCONJ
ejde-771	335	6	θ	θ	NOUN
ejde-771	335	7	:	:	PUNCT
ejde-771	335	8	br0	br0	VERB
ejde-771	335	9	→	→	PUNCT
ejde-771	335	10	br0	br0	VERB
ejde-771	335	11	is	be	AUX
ejde-771	335	12	a	a	DET
ejde-771	335	13	completely	completely	ADV
ejde-771	335	14	continuous	continuous	ADJ
ejde-771	335	15	operator	operator	NOUN
ejde-771	335	16	.	.	PUNCT
ejde-771	336	1	from	from	ADP
ejde-771	336	2	assumptions	assumption	NOUN
ejde-771	336	3	(	(	PUNCT
ejde-771	336	4	h3	h3	NOUN
ejde-771	336	5	)	)	PUNCT
ejde-771	336	6	and	and	CCONJ
ejde-771	336	7	(	(	PUNCT
ejde-771	336	8	h4	h4	PROPN
ejde-771	336	9	)	)	PUNCT
ejde-771	336	10	,	,	PUNCT
ejde-771	336	11	there	there	PRON
ejde-771	336	12	is	be	VERB
ejde-771	336	13	a	a	DET
ejde-771	336	14	constant	constant	ADJ
ejde-771	336	15	w	w	ADP
ejde-771	336	16	such	such	ADJ
ejde-771	336	17	that	that	PRON
ejde-771	336	18	for	for	SCONJ
ejde-771	336	19	all	all	PRON
ejde-771	336	20	u	u	PROPN
ejde-771	336	21	∈	∈	PROPN
ejde-771	336	22	br0	br0	VERB
ejde-771	336	23	,	,	PUNCT
ejde-771	336	24	sup	sup	NOUN
ejde-771	336	25	t∈[0,∞	t∈[0,∞	NOUN
ejde-771	336	26	)	)	PUNCT
ejde-771	337	1	∥g(t	∥g(t	NOUN
ejde-771	337	2	,	,	PUNCT
ejde-771	337	3	u(t))∥	u(t))∥	VERB
ejde-771	337	4	≤	≤	PUNCT
ejde-771	337	5	w.	w.	NOUN
ejde-771	337	6	(	(	PUNCT
ejde-771	337	7	3.16	3.16	NUM
ejde-771	337	8	)	)	PUNCT
ejde-771	337	9	it	it	PRON
ejde-771	337	10	should	should	AUX
ejde-771	337	11	be	be	AUX
ejde-771	337	12	noted	note	VERB
ejde-771	337	13	that	that	SCONJ
ejde-771	337	14	the	the	DET
ejde-771	337	15	set	set	NOUN
ejde-771	337	16	θ(br0	θ(br0	NOUN
ejde-771	337	17	)	)	PUNCT
ejde-771	337	18	is	be	AUX
ejde-771	337	19	locally	locally	ADV
ejde-771	337	20	equicontinuous	equicontinuous	ADJ
ejde-771	337	21	on	on	ADP
ejde-771	337	22	e	e	NOUN
ejde-771	337	23	by	by	ADP
ejde-771	337	24	using	use	VERB
ejde-771	337	25	the	the	DET
ejde-771	337	26	method	method	NOUN
ejde-771	337	27	similar	similar	ADJ
ejde-771	337	28	to	to	ADP
ejde-771	337	29	theorem	theorem	NOUN
ejde-771	337	30	(	(	PUNCT
ejde-771	337	31	3.2	3.2	NUM
ejde-771	337	32	)	)	PUNCT
ejde-771	337	33	and	and	CCONJ
ejde-771	337	34	for	for	SCONJ
ejde-771	337	35	any	any	DET
ejde-771	337	36	u	u	PROPN
ejde-771	337	37	∈	∈	PROPN
ejde-771	337	38	br0	br0	NOUN
ejde-771	337	39	,	,	PUNCT
ejde-771	337	40	lim	lim	PROPN
ejde-771	337	41	t→∞	t→∞	X
ejde-771	337	42	e−t∥(θu)(t)∥	e−t∥(θu)(t)∥	PROPN
ejde-771	337	43	=	=	SYM
ejde-771	337	44	0	0	X
ejde-771	337	45	.	.	PUNCT
ejde-771	338	1	so	so	ADV
ejde-771	338	2	we	we	PRON
ejde-771	338	3	only	only	ADV
ejde-771	338	4	need	need	VERB
ejde-771	338	5	to	to	PART
ejde-771	338	6	show	show	VERB
ejde-771	338	7	that	that	SCONJ
ejde-771	338	8	for	for	ADP
ejde-771	338	9	any	any	DET
ejde-771	338	10	t	t	NOUN
ejde-771	338	11	∈	∈	PROPN
ejde-771	339	1	[	[	X
ejde-771	339	2	0,∞	0,∞	NOUN
ejde-771	339	3	)	)	PUNCT
ejde-771	339	4	,	,	PUNCT
ejde-771	339	5	{	{	PUNCT
ejde-771	339	6	(	(	PUNCT
ejde-771	339	7	θu)(t	θu)(t	X
ejde-771	339	8	)	)	PUNCT
ejde-771	340	1	|	|	CCONJ
ejde-771	340	2	u	u	PROPN
ejde-771	340	3	∈	∈	PROPN
ejde-771	340	4	br0	br0	VERB
ejde-771	340	5	}	}	PUNCT
ejde-771	340	6	is	be	AUX
ejde-771	340	7	relatively	relatively	ADV
ejde-771	340	8	compact	compact	ADJ
ejde-771	340	9	in	in	ADP
ejde-771	340	10	e.	e.	PROPN
ejde-771	340	11	obviously	obviously	ADV
ejde-771	340	12	,	,	PUNCT
ejde-771	340	13	{	{	PUNCT
ejde-771	340	14	(	(	PUNCT
ejde-771	340	15	θu)(0	θu)(0	NUM
ejde-771	340	16	)	)	PUNCT
ejde-771	340	17	:	:	PUNCT
ejde-771	340	18	u	u	PROPN
ejde-771	340	19	∈	∈	PROPN
ejde-771	340	20	br0	br0	VERB
ejde-771	340	21	}	}	PUNCT
ejde-771	340	22	is	be	AUX
ejde-771	340	23	relatively	relatively	ADV
ejde-771	340	24	compact	compact	ADJ
ejde-771	340	25	in	in	ADP
ejde-771	340	26	e.	e.	PROPN
ejde-771	340	27	we	we	PRON
ejde-771	340	28	only	only	ADV
ejde-771	340	29	consider	consider	VERB
ejde-771	340	30	the	the	DET
ejde-771	340	31	case	case	NOUN
ejde-771	340	32	t	t	X
ejde-771	340	33	>	>	X
ejde-771	340	34	0	0	NUM
ejde-771	340	35	,	,	PUNCT
ejde-771	340	36	for	for	ADP
ejde-771	340	37	all	all	DET
ejde-771	340	38	δ	δ	PROPN
ejde-771	340	39	>	>	X
ejde-771	340	40	0	0	PUNCT
ejde-771	341	1	and	and	CCONJ
ejde-771	341	2	ϵ	ϵ	PRON
ejde-771	341	3	∈	∈	PROPN
ejde-771	341	4	(	(	PUNCT
ejde-771	341	5	0	0	NUM
ejde-771	341	6	,	,	PUNCT
ejde-771	341	7	t	t	PROPN
ejde-771	341	8	)	)	PUNCT
ejde-771	341	9	,	,	PUNCT
ejde-771	341	10	define	define	VERB
ejde-771	341	11	(	(	PUNCT
ejde-771	341	12	θϵ,δu	θϵ,δu	NOUN
ejde-771	341	13	)	)	PUNCT
ejde-771	341	14	by	by	ADP
ejde-771	341	15	(	(	PUNCT
ejde-771	341	16	θϵ,δu)(t	θϵ,δu)(t	PROPN
ejde-771	341	17	)	)	PUNCT
ejde-771	341	18	=	=	SYM
ejde-771	341	19	jα	jα	NOUN
ejde-771	341	20	,	,	PUNCT
ejde-771	341	21	β(t)λu0	β(t)λu0	ADJ
ejde-771	341	22	+	+	CCONJ
ejde-771	341	23	α	α	X
ejde-771	341	24	m∑	m∑	VERB
ejde-771	341	25	k=1	k=1	X
ejde-771	341	26	akλjα	akλjα	ADV
ejde-771	341	27	,	,	PUNCT
ejde-771	341	28	β(t	β(t	PROPN
ejde-771	341	29	)	)	PUNCT
ejde-771	342	1	×	×	NOUN
ejde-771	342	2	∫	∫	PROPN
ejde-771	342	3	tk	tk	PROPN
ejde-771	342	4	0	0	PROPN
ejde-771	342	5	∫	∫	PROPN
ejde-771	342	6	∞	∞	PROPN
ejde-771	342	7	0	0	NUM
ejde-771	343	1	(	(	PUNCT
ejde-771	343	2	tk	tk	PROPN
ejde-771	343	3	−	−	PROPN
ejde-771	343	4	s)α−1τhα(τ)tβ((tk	s)α−1τhα(τ)tβ((tk	NOUN
ejde-771	343	5	−	−	PROPN
ejde-771	343	6	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	343	7	,	,	PUNCT
ejde-771	343	8	u(s))dτds	u(s))dτds	PROPN
ejde-771	343	9	+	+	NUM
ejde-771	343	10	α	α	NOUN
ejde-771	343	11	∫	∫	NOUN
ejde-771	343	12	t−ϵ	t−ϵ	ADV
ejde-771	343	13	0	0	NUM
ejde-771	343	14	∫	∫	PROPN
ejde-771	343	15	∞	∞	PROPN
ejde-771	343	16	δ	δ	PROPN
ejde-771	343	17	(	(	PUNCT
ejde-771	343	18	t−	t−	PROPN
ejde-771	343	19	s)α−1τhα(τ)tβ((t−	s)α−1τhα(τ)tβ((t−	PROPN
ejde-771	343	20	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	343	21	,	,	PUNCT
ejde-771	343	22	u(s))dτds	u(s))dτds	PROPN
ejde-771	343	23	=	=	SYM
ejde-771	343	24	jα	jα	NOUN
ejde-771	343	25	,	,	PUNCT
ejde-771	343	26	β(t)λu0	β(t)λu0	ADJ
ejde-771	343	27	+	+	CCONJ
ejde-771	344	1	α	α	X
ejde-771	344	2	m∑	m∑	VERB
ejde-771	344	3	k=1	k=1	X
ejde-771	344	4	akλjα	akλjα	ADV
ejde-771	344	5	,	,	PUNCT
ejde-771	344	6	β(t	β(t	PROPN
ejde-771	344	7	)	)	PUNCT
ejde-771	345	1	×	×	NOUN
ejde-771	345	2	∫	∫	PROPN
ejde-771	345	3	tk	tk	PROPN
ejde-771	345	4	0	0	PROPN
ejde-771	345	5	∫	∫	PROPN
ejde-771	345	6	∞	∞	PROPN
ejde-771	345	7	0	0	NUM
ejde-771	346	1	(	(	PUNCT
ejde-771	346	2	tk	tk	PROPN
ejde-771	346	3	−	−	PROPN
ejde-771	346	4	s)α−1τhα(τ)tβ((tk	s)α−1τhα(τ)tβ((tk	NOUN
ejde-771	346	5	−	−	PROPN
ejde-771	346	6	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	346	7	,	,	PUNCT
ejde-771	346	8	u(s))dτds	u(s))dτds	PROPN
ejde-771	346	9	+	+	NUM
ejde-771	346	10	αtβ(ϵ	αtβ(ϵ	PROPN
ejde-771	346	11	αδ	αδ	NOUN
ejde-771	346	12	)	)	PUNCT
ejde-771	346	13	∫	∫	PROPN
ejde-771	346	14	t−ϵ	t−ϵ	ADV
ejde-771	346	15	0	0	NUM
ejde-771	346	16	∫	∫	PROPN
ejde-771	346	17	∞	∞	PROPN
ejde-771	346	18	δ	δ	PROPN
ejde-771	346	19	(	(	PUNCT
ejde-771	346	20	t−	t−	PROPN
ejde-771	346	21	s)α−1τhα(τ)tβ((t−	s)α−1τhα(τ)tβ((t−	PROPN
ejde-771	347	1	s)ατ	s)ατ	PROPN
ejde-771	347	2	−	−	PROPN
ejde-771	348	1	ϵαδ)g(s	ϵαδ)g(s	PROPN
ejde-771	348	2	,	,	PUNCT
ejde-771	348	3	u(s))dτds	u(s))dτds	PROPN
ejde-771	348	4	.	.	PROPN
ejde-771	349	1	from	from	ADP
ejde-771	349	2	the	the	DET
ejde-771	349	3	compactness	compactness	NOUN
ejde-771	349	4	of	of	ADP
ejde-771	349	5	jα	jα	PROPN
ejde-771	349	6	,	,	PUNCT
ejde-771	349	7	β(t	β(t	PROPN
ejde-771	349	8	)	)	PUNCT
ejde-771	349	9	and	and	CCONJ
ejde-771	349	10	tβ(ϵ	tβ(ϵ	NOUN
ejde-771	349	11	αδ	αδ	ADP
ejde-771	349	12	)	)	PUNCT
ejde-771	349	13	,	,	PUNCT
ejde-771	349	14	one	one	PRON
ejde-771	349	15	gets	get	VERB
ejde-771	349	16	that	that	PRON
ejde-771	349	17	{	{	PUNCT
ejde-771	349	18	(	(	PUNCT
ejde-771	349	19	θϵ,δu)(t	θϵ,δu)(t	PROPN
ejde-771	349	20	)	)	PUNCT
ejde-771	349	21	|	|	CCONJ
ejde-771	349	22	u	u	NOUN
ejde-771	349	23	∈	∈	PROPN
ejde-771	349	24	br0	br0	VERB
ejde-771	349	25	}	}	PUNCT
ejde-771	349	26	is	be	AUX
ejde-771	349	27	relatively	relatively	ADV
ejde-771	349	28	compact	compact	ADJ
ejde-771	349	29	in	in	ADP
ejde-771	349	30	e.	e.	PROPN
ejde-771	349	31	thus	thus	ADV
ejde-771	349	32	,	,	PUNCT
ejde-771	349	33	for	for	SCONJ
ejde-771	349	34	every	every	DET
ejde-771	349	35	u	u	PROPN
ejde-771	349	36	∈	∈	PROPN
ejde-771	349	37	br0	br0	VERB
ejde-771	349	38	,	,	PUNCT
ejde-771	349	39	it	it	PRON
ejde-771	349	40	follows	follow	VERB
ejde-771	349	41	from	from	ADP
ejde-771	349	42	(	(	PUNCT
ejde-771	349	43	3.16	3.16	NUM
ejde-771	349	44	)	)	PUNCT
ejde-771	349	45	that	that	PRON
ejde-771	349	46	∥(θu)(t)−	∥(θu)(t)−	PROPN
ejde-771	349	47	(	(	PUNCT
ejde-771	349	48	θϵ,δu)(t)∥	θϵ,δu)(t)∥	X
ejde-771	349	49	=	=	SYM
ejde-771	350	1	∥α	∥α	NOUN
ejde-771	350	2	∫	∫	PROPN
ejde-771	350	3	t	t	PROPN
ejde-771	350	4	0	0	NUM
ejde-771	350	5	∫	∫	PROPN
ejde-771	350	6	δ	δ	PROPN
ejde-771	350	7	0	0	NUM
ejde-771	351	1	(	(	PUNCT
ejde-771	351	2	t−	t−	PROPN
ejde-771	351	3	s)α−1τhα(τ)tβ((t−	s)α−1τhα(τ)tβ((t−	PROPN
ejde-771	351	4	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	351	5	,	,	PUNCT
ejde-771	351	6	u(s))dτds∥	u(s))dτds∥	NOUN
ejde-771	352	1	+	+	CCONJ
ejde-771	352	2	∥α	∥α	NOUN
ejde-771	352	3	∫	∫	PROPN
ejde-771	352	4	t	t	PROPN
ejde-771	352	5	t−ϵ	t−ϵ	ADV
ejde-771	352	6	∫	∫	PROPN
ejde-771	352	7	∞	∞	PROPN
ejde-771	352	8	δ	δ	PROPN
ejde-771	352	9	(	(	PUNCT
ejde-771	352	10	t−	t−	PROPN
ejde-771	352	11	s)α−1τhα(τ)tβ((t−	s)α−1τhα(τ)tβ((t−	PROPN
ejde-771	352	12	s)ατ)g(s	s)ατ)g(s	PROPN
ejde-771	352	13	,	,	PUNCT
ejde-771	352	14	u(s))dτds∥	u(s))dτds∥	VERB
ejde-771	352	15	≤	≤	NUM
ejde-771	353	1	w	w	PROPN
ejde-771	353	2	∫	∫	PROPN
ejde-771	353	3	t	t	PROPN
ejde-771	353	4	0	0	NUM
ejde-771	353	5	∫	∫	PROPN
ejde-771	353	6	δ	δ	PROPN
ejde-771	353	7	0	0	NUM
ejde-771	354	1	(	(	PUNCT
ejde-771	354	2	t−	t−	X
ejde-771	354	3	s)α−1τhα(τ)∥tβ((t−	s)α−1τhα(τ)∥tβ((t−	ADJ
ejde-771	354	4	s)ατ)∥dτds	s)ατ)∥dτd	NOUN
ejde-771	354	5	+	+	PROPN
ejde-771	354	6	w	w	PROPN
ejde-771	354	7	∫	∫	PROPN
ejde-771	354	8	t	t	PROPN
ejde-771	354	9	t−ϵ	t−ϵ	ADV
ejde-771	354	10	∫	∫	PROPN
ejde-771	354	11	∞	∞	PROPN
ejde-771	354	12	δ	δ	PROPN
ejde-771	354	13	(	(	PUNCT
ejde-771	355	1	t−	t−	X
ejde-771	355	2	s)α−1τhα(τ)∥tβ((t−	s)α−1τhα(τ)∥tβ((t−	PROPN
ejde-771	355	3	s)ατ)∥dτds	s)ατ)∥dτds	PROPN
ejde-771	355	4	≤mw	≤mw	PROPN
ejde-771	355	5	∫	∫	PROPN
ejde-771	355	6	t	t	PROPN
ejde-771	355	7	0	0	NUM
ejde-771	356	1	(	(	PUNCT
ejde-771	356	2	t−	t−	PROPN
ejde-771	356	3	s)α−1ds	s)α−1ds	NOUN
ejde-771	356	4	∫	∫	PROPN
ejde-771	356	5	δ	δ	PROPN
ejde-771	356	6	0	0	NUM
ejde-771	356	7	τhα(τ)dτ	τhα(τ)dτ	PUNCT
ejde-771	357	1	+	+	X
ejde-771	357	2	mw	mw	PROPN
ejde-771	357	3	∫	∫	PROPN
ejde-771	357	4	t	t	PROPN
ejde-771	357	5	t−ϵ	t−ϵ	ADV
ejde-771	357	6	(	(	PUNCT
ejde-771	357	7	t−	t−	PROPN
ejde-771	357	8	s)α−1ds	s)α−1ds	NOUN
ejde-771	357	9	∫	∫	PROPN
ejde-771	357	10	∞	∞	PROPN
ejde-771	357	11	δ	δ	PROPN
ejde-771	357	12	τhα(τ)dτ	τhα(τ)dτ	PUNCT
ejde-771	357	13	→	→	SYM
ejde-771	357	14	0	0	PUNCT
ejde-771	357	15	as	as	ADP
ejde-771	357	16	ϵ→	ϵ→	PROPN
ejde-771	357	17	0	0	NUM
ejde-771	357	18	,	,	PUNCT
ejde-771	357	19	δ	δ	PROPN
ejde-771	357	20	→	→	SYM
ejde-771	357	21	0	0	NUM
ejde-771	357	22	,	,	PUNCT
ejde-771	357	23	which	which	PRON
ejde-771	357	24	implies	imply	VERB
ejde-771	357	25	that	that	SCONJ
ejde-771	357	26	there	there	PRON
ejde-771	357	27	is	be	VERB
ejde-771	357	28	a	a	DET
ejde-771	357	29	relatively	relatively	ADV
ejde-771	357	30	compact	compact	ADJ
ejde-771	357	31	set	set	NOUN
ejde-771	357	32	{	{	PUNCT
ejde-771	357	33	(	(	PUNCT
ejde-771	357	34	θϵ,δu)(t	θϵ,δu)(t	PROPN
ejde-771	357	35	)	)	PUNCT
ejde-771	358	1	|	|	ADV
ejde-771	358	2	u	u	PROPN
ejde-771	358	3	∈	∈	PROPN
ejde-771	358	4	br0	br0	VERB
ejde-771	358	5	}	}	PUNCT
ejde-771	358	6	arbitrarily	arbitrarily	ADV
ejde-771	358	7	close	close	ADJ
ejde-771	358	8	to	to	ADP
ejde-771	358	9	the	the	DET
ejde-771	358	10	set	set	NOUN
ejde-771	358	11	{	{	PUNCT
ejde-771	358	12	(	(	PUNCT
ejde-771	358	13	θu)(t	θu)(t	X
ejde-771	358	14	)	)	PUNCT
ejde-771	358	15	|	|	CCONJ
ejde-771	358	16	u	u	PROPN
ejde-771	358	17	∈	∈	PROPN
ejde-771	358	18	br0	br0	VERB
ejde-771	358	19	}	}	PUNCT
ejde-771	358	20	in	in	ADP
ejde-771	358	21	e	e	PROPN
ejde-771	358	22	for	for	ADP
ejde-771	358	23	t	t	PROPN
ejde-771	358	24	∈	∈	PROPN
ejde-771	358	25	(	(	PUNCT
ejde-771	358	26	0,∞	0,∞	NOUN
ejde-771	358	27	)	)	PUNCT
ejde-771	358	28	.	.	PUNCT
ejde-771	359	1	therefore	therefore	ADV
ejde-771	359	2	,	,	PUNCT
ejde-771	359	3	the	the	DET
ejde-771	359	4	set	set	NOUN
ejde-771	359	5	{	{	PUNCT
ejde-771	359	6	(	(	PUNCT
ejde-771	359	7	θu)(t	θu)(t	X
ejde-771	359	8	)	)	PUNCT
ejde-771	359	9	|	|	CCONJ
ejde-771	359	10	u	u	PROPN
ejde-771	359	11	∈	∈	PROPN
ejde-771	359	12	br0	br0	VERB
ejde-771	359	13	}	}	PUNCT
ejde-771	359	14	is	be	AUX
ejde-771	359	15	relatively	relatively	ADV
ejde-771	359	16	ejde-2025/44	ejde-2025/44	ADJ
ejde-771	359	17	time	time	NOUN
ejde-771	359	18	-	-	PUNCT
ejde-771	359	19	space	space	NOUN
ejde-771	359	20	fractional	fractional	ADJ
ejde-771	359	21	reaction	reaction	NOUN
ejde-771	359	22	-	-	PUNCT
ejde-771	359	23	diffusion	diffusion	NOUN
ejde-771	359	24	equations	equation	NOUN
ejde-771	359	25	13	13	NUM
ejde-771	359	26	compact	compact	ADJ
ejde-771	359	27	on	on	ADP
ejde-771	359	28	e	e	PROPN
ejde-771	359	29	for	for	ADP
ejde-771	359	30	t	t	PROPN
ejde-771	359	31	∈	∈	PROPN
ejde-771	360	1	[	[	X
ejde-771	360	2	0,∞	0,∞	NOUN
ejde-771	360	3	)	)	PUNCT
ejde-771	360	4	.	.	PUNCT
ejde-771	361	1	moreover	moreover	ADV
ejde-771	361	2	,	,	PUNCT
ejde-771	361	3	it	it	PRON
ejde-771	361	4	follows	follow	VERB
ejde-771	361	5	from	from	ADP
ejde-771	361	6	lemma	lemma	PROPN
ejde-771	361	7	2.1	2.1	NUM
ejde-771	361	8	that	that	PRON
ejde-771	361	9	θ(br0	θ(br0	NOUN
ejde-771	361	10	)	)	PUNCT
ejde-771	361	11	is	be	AUX
ejde-771	361	12	relatively	relatively	ADV
ejde-771	361	13	compact	compact	ADJ
ejde-771	361	14	in	in	ADP
ejde-771	361	15	ce(e	ce(e	NUM
ejde-771	361	16	)	)	PUNCT
ejde-771	361	17	.	.	PUNCT
ejde-771	362	1	based	base	VERB
ejde-771	362	2	on	on	ADP
ejde-771	362	3	above	above	ADP
ejde-771	362	4	results	result	NOUN
ejde-771	362	5	,	,	PUNCT
ejde-771	362	6	one	one	PRON
ejde-771	362	7	can	can	AUX
ejde-771	362	8	find	find	VERB
ejde-771	362	9	that	that	SCONJ
ejde-771	362	10	θ	θ	NOUN
ejde-771	362	11	:	:	PUNCT
ejde-771	362	12	br0	br0	VERB
ejde-771	362	13	∩	∩	ADJ
ejde-771	362	14	sapω(e	sapω(e	NOUN
ejde-771	362	15	)	)	PUNCT
ejde-771	362	16	→	→	PUNCT
ejde-771	362	17	br0	br0	VERB
ejde-771	362	18	∩	∩	NOUN
ejde-771	362	19	sapω(e	sapω(e	NOUN
ejde-771	362	20	)	)	PUNCT
ejde-771	362	21	is	be	AUX
ejde-771	362	22	a	a	DET
ejde-771	362	23	completely	completely	ADV
ejde-771	362	24	continuous	continuous	ADJ
ejde-771	362	25	operator	operator	NOUN
ejde-771	362	26	,	,	PUNCT
ejde-771	362	27	which	which	PRON
ejde-771	362	28	implies	imply	VERB
ejde-771	362	29	that	that	SCONJ
ejde-771	362	30	θ	θ	PROPN
ejde-771	362	31	is	be	AUX
ejde-771	362	32	a	a	DET
ejde-771	362	33	condensing	condense	VERB
ejde-771	362	34	mapping	mapping	NOUN
ejde-771	362	35	from	from	ADP
ejde-771	362	36	br0	br0	NOUN
ejde-771	362	37	∩	∩	NOUN
ejde-771	362	38	sapω(e	sapω(e	NOUN
ejde-771	362	39	)	)	PUNCT
ejde-771	362	40	into	into	ADP
ejde-771	362	41	br0	br0	NOUN
ejde-771	362	42	∩	∩	ADJ
ejde-771	362	43	sapω(e	sapω(e	NOUN
ejde-771	362	44	)	)	PUNCT
ejde-771	362	45	.	.	PUNCT
ejde-771	363	1	therefore	therefore	ADV
ejde-771	363	2	,	,	PUNCT
ejde-771	363	3	lemma	lemma	PROPN
ejde-771	363	4	2.6	2.6	NUM
ejde-771	363	5	implies	imply	VERB
ejde-771	363	6	that	that	SCONJ
ejde-771	363	7	θ	θ	PROPN
ejde-771	363	8	has	have	VERB
ejde-771	363	9	a	a	DET
ejde-771	363	10	fixed	fix	VERB
ejde-771	363	11	point	point	NOUN
ejde-771	363	12	ũ	ũ	PROPN
ejde-771	363	13	∈	∈	PROPN
ejde-771	363	14	br0	br0	VERB
ejde-771	363	15	∩	∩	NOUN
ejde-771	363	16	sapω(e	sapω(e	NOUN
ejde-771	363	17	)	)	PUNCT
ejde-771	363	18	.	.	PUNCT
ejde-771	364	1	we	we	PRON
ejde-771	364	2	need	need	VERB
ejde-771	364	3	to	to	PART
ejde-771	364	4	verify	verify	VERB
ejde-771	364	5	that	that	DET
ejde-771	364	6	ũ	ũ	PROPN
ejde-771	364	7	∈	∈	PROPN
ejde-771	364	8	sapω(e	sapω(e	NOUN
ejde-771	364	9	)	)	PUNCT
ejde-771	364	10	.	.	PUNCT
ejde-771	365	1	let	let	VERB
ejde-771	365	2	{	{	PUNCT
ejde-771	365	3	un	un	ADJ
ejde-771	365	4	}	}	PUNCT
ejde-771	365	5	⊂	⊂	PROPN
ejde-771	365	6	br0	br0	VERB
ejde-771	365	7	∩	∩	ADJ
ejde-771	365	8	sapω(e	sapω(e	NOUN
ejde-771	365	9	)	)	PUNCT
ejde-771	365	10	converge	converge	VERB
ejde-771	365	11	to	to	ADP
ejde-771	365	12	ũ	ũ	PROPN
ejde-771	365	13	,	,	PUNCT
ejde-771	365	14	it	it	PRON
ejde-771	365	15	follows	follow	VERB
ejde-771	365	16	from	from	ADP
ejde-771	365	17	the	the	DET
ejde-771	365	18	continuity	continuity	NOUN
ejde-771	365	19	of	of	ADP
ejde-771	365	20	θ	θ	PROPN
ejde-771	365	21	and	and	CCONJ
ejde-771	365	22	(	(	PUNCT
ejde-771	365	23	3.15	3.15	NUM
ejde-771	365	24	)	)	PUNCT
ejde-771	365	25	that	that	SCONJ
ejde-771	365	26	{	{	PUNCT
ejde-771	365	27	θun	θun	NOUN
ejde-771	365	28	}	}	PUNCT
ejde-771	365	29	converges	converge	VERB
ejde-771	365	30	to	to	ADP
ejde-771	365	31	θũ	θũ	SYM
ejde-771	365	32	=	=	PUNCT
ejde-771	365	33	ũ	ũ	PROPN
ejde-771	365	34	uniformly	uniformly	ADV
ejde-771	365	35	in	in	ADP
ejde-771	365	36	[	[	X
ejde-771	365	37	0,∞	0,∞	NOUN
ejde-771	365	38	)	)	PUNCT
ejde-771	365	39	and	and	CCONJ
ejde-771	365	40	ũ	ũ	PROPN
ejde-771	365	41	≥	≥	NUM
ejde-771	365	42	ςe1	ςe1	NOUN
ejde-771	365	43	,	,	PUNCT
ejde-771	365	44	which	which	PRON
ejde-771	365	45	implies	imply	VERB
ejde-771	365	46	that	that	SCONJ
ejde-771	365	47	ũ	ũ	PROPN
ejde-771	365	48	∈	∈	PROPN
ejde-771	365	49	sapω(e	sapω(e	NOUN
ejde-771	365	50	)	)	PUNCT
ejde-771	365	51	is	be	AUX
ejde-771	365	52	a	a	DET
ejde-771	365	53	positive	positive	ADJ
ejde-771	365	54	s	s	NOUN
ejde-771	365	55	-	-	PUNCT
ejde-771	365	56	asymptotically	asymptotically	ADV
ejde-771	365	57	ω	ω	ADJ
ejde-771	365	58	-	-	ADJ
ejde-771	365	59	periodic	periodic	ADJ
ejde-771	365	60	mild	mild	ADJ
ejde-771	365	61	solution	solution	NOUN
ejde-771	365	62	of	of	ADP
ejde-771	365	63	nonlocal	nonlocal	ADJ
ejde-771	365	64	problem	problem	NOUN
ejde-771	365	65	(	(	PUNCT
ejde-771	365	66	3.1	3.1	NUM
ejde-771	365	67	)	)	PUNCT
ejde-771	365	68	.	.	PUNCT
ejde-771	366	1	this	this	PRON
ejde-771	366	2	completes	complete	VERB
ejde-771	366	3	the	the	DET
ejde-771	366	4	proof	proof	NOUN
ejde-771	366	5	of	of	ADP
ejde-771	366	6	theorem	theorem	ADJ
ejde-771	366	7	3.4	3.4	NUM
ejde-771	366	8	.	.	PUNCT
ejde-771	367	1	□	□	SYM
ejde-771	367	2	4	4	X
ejde-771	367	3	.	.	X
ejde-771	367	4	application	application	NOUN
ejde-771	367	5	to	to	ADP
ejde-771	367	6	nonlocal	nonlocal	ADJ
ejde-771	367	7	problem	problem	NOUN
ejde-771	367	8	(	(	PUNCT
ejde-771	367	9	1.1	1.1	NUM
ejde-771	367	10	)	)	PUNCT
ejde-771	367	11	let	let	VERB
ejde-771	367	12	e	e	NOUN
ejde-771	367	13	=	=	PUNCT
ejde-771	367	14	l2(ω	l2(ω	PROPN
ejde-771	367	15	)	)	PUNCT
ejde-771	367	16	with	with	ADP
ejde-771	367	17	the	the	DET
ejde-771	367	18	l2	l2	NOUN
ejde-771	367	19	-	-	PUNCT
ejde-771	367	20	norm	norm	NOUN
ejde-771	367	21	∥	∥	X
ejde-771	367	22	·	·	PUNCT
ejde-771	367	23	∥2	∥2	NOUN
ejde-771	367	24	and	and	CCONJ
ejde-771	367	25	partial	partial	ADJ
ejde-771	367	26	order	order	NOUN
ejde-771	367	27	≤	≤	NOUN
ejde-771	367	28	,	,	PUNCT
ejde-771	367	29	p	p	X
ejde-771	367	30	=	=	X
ejde-771	367	31	{	{	PUNCT
ejde-771	367	32	u	u	NOUN
ejde-771	367	33	∈	∈	PROPN
ejde-771	367	34	l2(ω	l2(ω	NOUN
ejde-771	367	35	)	)	PUNCT
ejde-771	367	36	|	|	ADV
ejde-771	367	37	u(x	u(x	VERB
ejde-771	367	38	)	)	PUNCT
ejde-771	367	39	≥	≥	NOUN
ejde-771	367	40	0	0	NUM
ejde-771	367	41	,	,	PUNCT
ejde-771	367	42	a.e.x	a.e.x	PROPN
ejde-771	367	43	∈	∈	PROPN
ejde-771	367	44	ω	ω	PROPN
ejde-771	367	45	}	}	PUNCT
ejde-771	367	46	is	be	AUX
ejde-771	367	47	a	a	DET
ejde-771	367	48	normal	normal	ADJ
ejde-771	367	49	cone	cone	NOUN
ejde-771	367	50	in	in	ADP
ejde-771	367	51	l2(ω	l2(ω	PROPN
ejde-771	367	52	)	)	PUNCT
ejde-771	367	53	,	,	PUNCT
ejde-771	367	54	then	then	ADV
ejde-771	367	55	p	p	NOUN
ejde-771	367	56	is	be	AUX
ejde-771	367	57	a	a	DET
ejde-771	367	58	regular	regular	ADJ
ejde-771	367	59	cone	cone	NOUN
ejde-771	367	60	of	of	ADP
ejde-771	367	61	e.	e.	PROPN
ejde-771	367	62	we	we	PRON
ejde-771	367	63	define	define	VERB
ejde-771	367	64	the	the	DET
ejde-771	367	65	operator	operator	NOUN
ejde-771	367	66	a	a	DET
ejde-771	367	67	:	:	PUNCT
ejde-771	367	68	d(a	d(a	PROPN
ejde-771	367	69	)	)	PUNCT
ejde-771	367	70	⊂	⊂	PROPN
ejde-771	368	1	e	e	X
ejde-771	368	2	→	→	SYM
ejde-771	368	3	e	e	PROPN
ejde-771	368	4	as	as	SCONJ
ejde-771	368	5	follows	follow	VERB
ejde-771	368	6	:	:	PUNCT
ejde-771	368	7	d(a	d(a	PROPN
ejde-771	368	8	)	)	PUNCT
ejde-771	369	1	=	=	SYM
ejde-771	369	2	w	w	PROPN
ejde-771	369	3	2,2(ω	2,2(ω	NUM
ejde-771	369	4	)	)	PUNCT
ejde-771	369	5	∩w	∩w	ADJ
ejde-771	369	6	1,2	1,2	NUM
ejde-771	369	7	0	0	NUM
ejde-771	369	8	(	(	PUNCT
ejde-771	369	9	ω	ω	NOUN
ejde-771	369	10	)	)	PUNCT
ejde-771	369	11	,	,	PUNCT
ejde-771	369	12	au	au	ADP
ejde-771	369	13	=	=	SYM
ejde-771	369	14	−∆u	−∆u	X
ejde-771	369	15	.	.	PUNCT
ejde-771	370	1	(	(	PUNCT
ejde-771	370	2	4.1	4.1	NUM
ejde-771	370	3	)	)	PUNCT
ejde-771	370	4	let	let	VERB
ejde-771	370	5	u(t	u(t	NOUN
ejde-771	370	6	,	,	PUNCT
ejde-771	370	7	x	x	NOUN
ejde-771	370	8	)	)	PUNCT
ejde-771	370	9	=	=	SYM
ejde-771	371	1	u(t)(x	u(t)(x	NUM
ejde-771	371	2	)	)	PUNCT
ejde-771	372	1	and	and	CCONJ
ejde-771	372	2	f	f	PROPN
ejde-771	372	3	(	(	PUNCT
ejde-771	372	4	t	t	PROPN
ejde-771	372	5	,	,	PUNCT
ejde-771	372	6	u(t	u(t	NOUN
ejde-771	372	7	,	,	PUNCT
ejde-771	372	8	x	x	NOUN
ejde-771	372	9	)	)	PUNCT
ejde-771	372	10	)	)	PUNCT
ejde-771	373	1	=	=	SYM
ejde-771	373	2	g(t	g(t	PROPN
ejde-771	373	3	,	,	PUNCT
ejde-771	373	4	u(t))(x	u(t))(x	PROPN
ejde-771	373	5	)	)	PUNCT
ejde-771	373	6	,	,	PUNCT
ejde-771	373	7	u0	u0	PROPN
ejde-771	373	8	+	+	X
ejde-771	373	9	m∑	m∑	ADV
ejde-771	373	10	k=1	k=1	X
ejde-771	373	11	aku(tk	aku(tk	NOUN
ejde-771	373	12	,	,	PUNCT
ejde-771	373	13	x	x	PRON
ejde-771	373	14	)	)	PUNCT
ejde-771	373	15	=	=	SYM
ejde-771	373	16	u0	u0	ADJ
ejde-771	374	1	+	+	X
ejde-771	374	2	m∑	m∑	INTJ
ejde-771	374	3	k=1	k=1	PROPN
ejde-771	374	4	aku(tk)(x	aku(tk)(x	PROPN
ejde-771	374	5	)	)	PUNCT
ejde-771	374	6	.	.	PUNCT
ejde-771	375	1	(	(	PUNCT
ejde-771	375	2	4.2	4.2	NUM
ejde-771	375	3	)	)	PUNCT
ejde-771	375	4	then	then	ADV
ejde-771	375	5	the	the	DET
ejde-771	375	6	nonlocal	nonlocal	ADJ
ejde-771	375	7	problem	problem	NOUN
ejde-771	375	8	(	(	PUNCT
ejde-771	375	9	1.1	1.1	NUM
ejde-771	375	10	)	)	PUNCT
ejde-771	375	11	can	can	AUX
ejde-771	375	12	be	be	AUX
ejde-771	375	13	rewritten	rewrite	VERB
ejde-771	375	14	as	as	ADP
ejde-771	375	15	an	an	DET
ejde-771	375	16	abstract	abstract	ADJ
ejde-771	375	17	evolution	evolution	NOUN
ejde-771	375	18	equation	equation	NOUN
ejde-771	375	19	with	with	ADP
ejde-771	375	20	nonlocal	nonlocal	ADJ
ejde-771	375	21	conditions	condition	NOUN
ejde-771	375	22	(	(	PUNCT
ejde-771	375	23	3.1	3.1	NUM
ejde-771	375	24	)	)	PUNCT
ejde-771	375	25	in	in	ADP
ejde-771	375	26	l2(ω	l2(ω	NOUN
ejde-771	375	27	)	)	PUNCT
ejde-771	375	28	.	.	PUNCT
ejde-771	376	1	according	accord	VERB
ejde-771	376	2	to	to	ADP
ejde-771	376	3	(	(	PUNCT
ejde-771	376	4	2.2	2.2	NUM
ejde-771	376	5	)	)	PUNCT
ejde-771	376	6	,	,	PUNCT
ejde-771	376	7	the	the	DET
ejde-771	376	8	fractional	fractional	ADJ
ejde-771	376	9	laplacian	laplacian	NOUN
ejde-771	376	10	is	be	AUX
ejde-771	376	11	well	well	ADV
ejde-771	376	12	defined	define	VERB
ejde-771	376	13	.	.	PUNCT
ejde-771	377	1	besides	besides	SCONJ
ejde-771	377	2	,	,	PUNCT
ejde-771	377	3	if	if	SCONJ
ejde-771	377	4	λn(n	λn(n	NUM
ejde-771	377	5	=	=	SYM
ejde-771	377	6	1	1	NUM
ejde-771	377	7	,	,	PUNCT
ejde-771	377	8	2	2	NUM
ejde-771	377	9	.	.	PUNCT
ejde-771	377	10	.	.	PUNCT
ejde-771	377	11	.	.	PUNCT
ejde-771	377	12	)	)	PUNCT
ejde-771	378	1	are	be	AUX
ejde-771	378	2	the	the	DET
ejde-771	378	3	eigenvalues	eigenvalue	NOUN
ejde-771	378	4	of	of	ADP
ejde-771	378	5	−∆	−∆	NOUN
ejde-771	378	6	with	with	ADP
ejde-771	378	7	homogeneous	homogeneous	ADJ
ejde-771	378	8	dirichlet	dirichlet	PROPN
ejde-771	378	9	boundary	boundary	PROPN
ejde-771	378	10	conditions	condition	NOUN
ejde-771	378	11	cnsidered	cnsidere	VERB
ejde-771	378	12	in	in	ADP
ejde-771	378	13	l2(ω	l2(ω	PROPN
ejde-771	378	14	)	)	PUNCT
ejde-771	378	15	and	and	CCONJ
ejde-771	378	16	en	en	ADV
ejde-771	378	17	as	as	ADP
ejde-771	378	18	its	its	PRON
ejde-771	378	19	corresponding	corresponding	ADJ
ejde-771	378	20	eigenfuction	eigenfuction	NOUN
ejde-771	378	21	,	,	PUNCT
ejde-771	378	22	it	it	PRON
ejde-771	378	23	follows	follow	VERB
ejde-771	378	24	that	that	SCONJ
ejde-771	378	25	(	(	PUNCT
ejde-771	378	26	−∆)βen	−∆)βen	PROPN
ejde-771	378	27	=	=	SYM
ejde-771	378	28	λβnen	λβnen	X
ejde-771	378	29	,	,	PUNCT
ejde-771	378	30	x	x	X
ejde-771	378	31	∈	∈	PROPN
ejde-771	378	32	ω	ω	PROPN
ejde-771	378	33	,	,	PUNCT
ejde-771	378	34	e	e	X
ejde-771	378	35	∣∣	∣∣	X
ejde-771	378	36	∂ω	∂ω	ADJ
ejde-771	378	37	=	=	SYM
ejde-771	378	38	0	0	NUM
ejde-771	378	39	,	,	PUNCT
ejde-771	378	40	which	which	PRON
ejde-771	378	41	λn	λn	NOUN
ejde-771	378	42	=	=	SYM
ejde-771	378	43	n2π2	n2π2	ADJ
ejde-771	378	44	and	and	CCONJ
ejde-771	378	45	corresponding	correspond	VERB
ejde-771	378	46	eigenfunctions	eigenfunction	NOUN
ejde-771	378	47	en(x	en(x	NOUN
ejde-771	378	48	)	)	PUNCT
ejde-771	378	49	=	=	SYM
ejde-771	378	50	√	√	ADP
ejde-771	378	51	2	2	NUM
ejde-771	378	52	sin(nπx	sin(nπx	NOUN
ejde-771	378	53	)	)	PUNCT
ejde-771	378	54	,	,	PUNCT
ejde-771	378	55	n	n	NOUN
ejde-771	378	56	=	=	SYM
ejde-771	378	57	1	1	NUM
ejde-771	378	58	,	,	PUNCT
ejde-771	378	59	2	2	NUM
ejde-771	378	60	.	.	PUNCT
ejde-771	378	61	.	.	PUNCT
ejde-771	378	62	.	.	PUNCT
ejde-771	378	63	.	.	PUNCT
ejde-771	379	1	hence	hence	ADV
ejde-771	379	2	,	,	PUNCT
ejde-771	379	3	based	base	VERB
ejde-771	379	4	on	on	ADP
ejde-771	379	5	corollary	corollary	ADJ
ejde-771	379	6	3.3	3.3	NUM
ejde-771	379	7	and	and	CCONJ
ejde-771	379	8	theorem	theorem	VERB
ejde-771	379	9	3.4	3.4	NUM
ejde-771	379	10	,	,	PUNCT
ejde-771	379	11	we	we	PRON
ejde-771	379	12	can	can	AUX
ejde-771	379	13	establish	establish	VERB
ejde-771	379	14	the	the	DET
ejde-771	379	15	following	follow	VERB
ejde-771	379	16	results	result	NOUN
ejde-771	379	17	.	.	PUNCT
ejde-771	380	1	theorem	theorem	VERB
ejde-771	380	2	4.1	4.1	NUM
ejde-771	380	3	.	.	PUNCT
ejde-771	381	1	let	let	VERB
ejde-771	381	2	nonlinear	nonlinear	ADJ
ejde-771	381	3	function	function	NOUN
ejde-771	381	4	f	f	NOUN
ejde-771	381	5	:	:	PUNCT
ejde-771	382	1	[	[	X
ejde-771	382	2	0,∞	0,∞	NUM
ejde-771	382	3	)	)	PUNCT
ejde-771	382	4	×	×	NOUN
ejde-771	382	5	p	p	NOUN
ejde-771	382	6	→	→	PUNCT
ejde-771	382	7	p	p	X
ejde-771	382	8	be	be	AUX
ejde-771	382	9	a	a	DET
ejde-771	382	10	continuous	continuous	ADJ
ejde-771	382	11	mapping	mapping	NOUN
ejde-771	382	12	.	.	PUNCT
ejde-771	383	1	if	if	SCONJ
ejde-771	383	2	the	the	DET
ejde-771	383	3	followinf	followinf	NOUN
ejde-771	383	4	4	4	NUM
ejde-771	383	5	conditions	condition	NOUN
ejde-771	383	6	hold	hold	VERB
ejde-771	383	7	:	:	PUNCT
ejde-771	383	8	(	(	PUNCT
ejde-771	383	9	k0	k0	PROPN
ejde-771	383	10	)	)	PUNCT
ejde-771	383	11	∑m	∑m	PROPN
ejde-771	383	12	k=1	k=1	PUNCT
ejde-771	383	13	|ak|	|ak|	PROPN
ejde-771	383	14	<	<	X
ejde-771	383	15	1	1	NUM
ejde-771	383	16	,	,	PUNCT
ejde-771	383	17	(	(	PUNCT
ejde-771	383	18	k1	k1	NOUN
ejde-771	383	19	)	)	PUNCT
ejde-771	383	20	there	there	PRON
ejde-771	383	21	are	be	VERB
ejde-771	383	22	nonnegative	nonnegative	ADJ
ejde-771	383	23	constants	constant	NOUN
ejde-771	383	24	a1	a1	NOUN
ejde-771	383	25	∈	∈	PROPN
ejde-771	383	26	(	(	PUNCT
ejde-771	383	27	0	0	NUM
ejde-771	383	28	,	,	PUNCT
ejde-771	383	29	(	(	PUNCT
ejde-771	383	30	1−	1−	NUM
ejde-771	383	31	∑m	∑m	INTJ
ejde-771	383	32	k=1	k=1	PROPN
ejde-771	383	33	|ak|)π2β	|ak|)π2β	PROPN
ejde-771	383	34	)	)	PUNCT
ejde-771	383	35	,	,	PUNCT
ejde-771	383	36	a0	a0	PROPN
ejde-771	383	37	≥	≥	NOUN
ejde-771	383	38	0	0	NUM
ejde-771	383	39	and	and	CCONJ
ejde-771	383	40	a	a	DET
ejde-771	383	41	nondecreasing	nondecrease	VERB
ejde-771	383	42	function	function	NOUN
ejde-771	383	43	et	et	NOUN
ejde-771	383	44	∈	∈	NOUN
ejde-771	383	45	c(r+	c(r+	NOUN
ejde-771	383	46	,	,	PUNCT
ejde-771	383	47	[	[	X
ejde-771	383	48	1,∞	1,∞	NUM
ejde-771	383	49	)	)	PUNCT
ejde-771	383	50	)	)	PUNCT
ejde-771	383	51	with	with	ADP
ejde-771	383	52	limt→∞	limt→∞	ADP
ejde-771	383	53	et	et	NOUN
ejde-771	383	54	=	=	PUNCT
ejde-771	384	1	+	+	NUM
ejde-771	384	2	∞	∞	NUM
ejde-771	384	3	such	such	ADJ
ejde-771	384	4	that	that	SCONJ
ejde-771	384	5	∥f	∥f	PROPN
ejde-771	384	6	(	(	PUNCT
ejde-771	384	7	t	t	PROPN
ejde-771	384	8	,	,	PUNCT
ejde-771	384	9	etξ)∥2	etξ)∥2	ADJ
ejde-771	384	10	≤	≤	ADJ
ejde-771	384	11	a1∥ξ∥2	a1∥ξ∥2	NOUN
ejde-771	384	12	+	+	NOUN
ejde-771	384	13	a0	a0	NOUN
ejde-771	384	14	,	,	PUNCT
ejde-771	384	15	t	t	PROPN
ejde-771	384	16	≥	≥	NUM
ejde-771	384	17	0	0	NUM
ejde-771	384	18	,	,	PUNCT
ejde-771	384	19	ξ	ξ	X
ejde-771	384	20	∈	∈	PROPN
ejde-771	384	21	e	e	NOUN
ejde-771	384	22	,	,	PUNCT
ejde-771	384	23	(	(	PUNCT
ejde-771	384	24	k2	k2	NOUN
ejde-771	384	25	)	)	PUNCT
ejde-771	384	26	for	for	ADP
ejde-771	384	27	any	any	DET
ejde-771	384	28	ξ1	ξ1	NOUN
ejde-771	384	29	,	,	PUNCT
ejde-771	384	30	ξ2	ξ2	NOUN
ejde-771	384	31	∈	∈	PROPN
ejde-771	384	32	e	e	NOUN
ejde-771	384	33	with	with	ADP
ejde-771	384	34	ξ2	ξ2	PROPN
ejde-771	384	35	≥	≥	NUM
ejde-771	384	36	ξ1	ξ1	PROPN
ejde-771	384	37	≥	≥	NUM
ejde-771	384	38	θ	θ	PROPN
ejde-771	384	39	,	,	PUNCT
ejde-771	384	40	f	f	PROPN
ejde-771	384	41	(	(	PUNCT
ejde-771	384	42	t	t	PROPN
ejde-771	384	43	,	,	PUNCT
ejde-771	384	44	ξ2	ξ2	PROPN
ejde-771	384	45	)	)	PUNCT
ejde-771	384	46	≥	≥	PROPN
ejde-771	384	47	f	f	X
ejde-771	384	48	(	(	PUNCT
ejde-771	384	49	t	t	PROPN
ejde-771	384	50	,	,	PUNCT
ejde-771	384	51	ξ1	ξ1	PROPN
ejde-771	384	52	)	)	PUNCT
ejde-771	384	53	≥	≥	NUM
ejde-771	384	54	θ	θ	PROPN
ejde-771	384	55	,	,	PUNCT
ejde-771	384	56	t	t	PROPN
ejde-771	384	57	≥	≥	NUM
ejde-771	384	58	0	0	NUM
ejde-771	384	59	,	,	PUNCT
ejde-771	384	60	(	(	PUNCT
ejde-771	384	61	k3	k3	PROPN
ejde-771	384	62	)	)	PUNCT
ejde-771	384	63	there	there	PRON
ejde-771	384	64	exist	exist	VERB
ejde-771	384	65	ω	ω	PROPN
ejde-771	384	66	>	>	X
ejde-771	384	67	0	0	NUM
ejde-771	385	1	such	such	ADJ
ejde-771	385	2	that	that	SCONJ
ejde-771	385	3	lim	lim	PROPN
ejde-771	385	4	t→∞	t→∞	X
ejde-771	385	5	∥f	∥f	PROPN
ejde-771	385	6	(	(	PUNCT
ejde-771	385	7	t+	t+	NOUN
ejde-771	385	8	ω	ω	NOUN
ejde-771	385	9	,	,	PUNCT
ejde-771	385	10	ξ)−	ξ)−	PROPN
ejde-771	385	11	f	f	X
ejde-771	385	12	(	(	PUNCT
ejde-771	385	13	t	t	PROPN
ejde-771	385	14	,	,	PUNCT
ejde-771	385	15	ξ)∥2	ξ)∥2	PROPN
ejde-771	385	16	=	=	SYM
ejde-771	385	17	0	0	NUM
ejde-771	385	18	,	,	PUNCT
ejde-771	385	19	ξ	ξ	PROPN
ejde-771	385	20	∈	∈	PROPN
ejde-771	385	21	e	e	PROPN
ejde-771	385	22	,	,	PUNCT
ejde-771	385	23	t	t	PROPN
ejde-771	385	24	≥	≥	NUM
ejde-771	385	25	0	0	NUM
ejde-771	385	26	,	,	PUNCT
ejde-771	385	27	then	then	ADV
ejde-771	385	28	nonlocal	nonlocal	ADJ
ejde-771	385	29	problem	problem	NOUN
ejde-771	385	30	(	(	PUNCT
ejde-771	385	31	1.1	1.1	NUM
ejde-771	385	32	)	)	PUNCT
ejde-771	385	33	exist	exist	VERB
ejde-771	385	34	a	a	DET
ejde-771	385	35	minimal	minimal	ADJ
ejde-771	385	36	positive	positive	ADJ
ejde-771	385	37	s	s	NOUN
ejde-771	385	38	-	-	PUNCT
ejde-771	385	39	asymptotically	asymptotically	ADV
ejde-771	385	40	ω	ω	ADJ
ejde-771	385	41	-	-	ADJ
ejde-771	385	42	periodic	periodic	ADJ
ejde-771	385	43	solution	solution	NOUN
ejde-771	385	44	.	.	PUNCT
ejde-771	386	1	proof	proof	NOUN
ejde-771	386	2	.	.	PUNCT
ejde-771	387	1	from	from	ADP
ejde-771	387	2	[	[	X
ejde-771	387	3	2	2	X
ejde-771	387	4	]	]	PUNCT
ejde-771	387	5	one	one	PRON
ejde-771	387	6	can	can	AUX
ejde-771	387	7	see	see	VERB
ejde-771	387	8	−a	−a	NOUN
ejde-771	387	9	generates	generate	VERB
ejde-771	387	10	a	a	DET
ejde-771	387	11	uniformly	uniformly	ADV
ejde-771	387	12	bounded	bound	VERB
ejde-771	387	13	analytic	analytic	ADJ
ejde-771	387	14	semigroup	semigroup	PROPN
ejde-771	387	15	t	t	PROPN
ejde-771	387	16	(	(	PUNCT
ejde-771	387	17	t)(t	t)(t	ADJ
ejde-771	387	18	≥	≥	NOUN
ejde-771	387	19	0	0	NUM
ejde-771	387	20	)	)	PUNCT
ejde-771	387	21	in	in	ADP
ejde-771	387	22	e	e	NOUN
ejde-771	387	23	,	,	PUNCT
ejde-771	387	24	and	and	CCONJ
ejde-771	387	25	t	t	PROPN
ejde-771	387	26	(	(	PUNCT
ejde-771	387	27	t)(t	t)(t	ADJ
ejde-771	387	28	≥	≥	NOUN
ejde-771	387	29	0	0	NUM
ejde-771	387	30	)	)	PUNCT
ejde-771	387	31	is	be	AUX
ejde-771	387	32	contractive	contractive	ADJ
ejde-771	387	33	in	in	ADP
ejde-771	387	34	e	e	PROPN
ejde-771	387	35	means	mean	VERB
ejde-771	387	36	that	that	SCONJ
ejde-771	388	1	∥t	∥t	PROPN
ejde-771	388	2	(	(	PUNCT
ejde-771	388	3	t)∥	t)∥	NUM
ejde-771	388	4	≤	≤	NUM
ejde-771	388	5	1	1	NUM
ejde-771	388	6	for	for	ADP
ejde-771	388	7	t	t	PROPN
ejde-771	388	8	≥	≥	NOUN
ejde-771	388	9	0	0	NUM
ejde-771	388	10	.	.	PUNCT
ejde-771	389	1	in	in	ADP
ejde-771	389	2	addition	addition	NOUN
ejde-771	389	3	,	,	PUNCT
ejde-771	389	4	from	from	ADP
ejde-771	389	5	[	[	X
ejde-771	389	6	28	28	NUM
ejde-771	389	7	]	]	PUNCT
ejde-771	389	8	the	the	DET
ejde-771	389	9	operator	operator	NOUN
ejde-771	389	10	a	a	PRON
ejde-771	389	11	has	have	AUX
ejde-771	389	12	compact	compact	ADJ
ejde-771	389	13	resolvent	resolvent	NOUN
ejde-771	389	14	in	in	ADP
ejde-771	389	15	l2(ω	l2(ω	NOUN
ejde-771	389	16	)	)	PUNCT
ejde-771	389	17	implies	imply	VERB
ejde-771	389	18	that	that	SCONJ
ejde-771	389	19	the	the	DET
ejde-771	389	20	semigroup	semigroup	PROPN
ejde-771	389	21	t	t	PROPN
ejde-771	389	22	(	(	PUNCT
ejde-771	389	23	t)(t	t)(t	ADJ
ejde-771	389	24	≥	≥	NOUN
ejde-771	389	25	0	0	NUM
ejde-771	389	26	)	)	PUNCT
ejde-771	389	27	is	be	AUX
ejde-771	389	28	compact	compact	ADJ
ejde-771	389	29	.	.	PUNCT
ejde-771	390	1	besides	besides	SCONJ
ejde-771	390	2	,	,	PUNCT
ejde-771	390	3	λi	λi	X
ejde-771	390	4	+	+	X
ejde-771	390	5	a	a	PRON
ejde-771	390	6	has	have	AUX
ejde-771	390	7	a	a	DET
ejde-771	390	8	positive	positive	ADJ
ejde-771	390	9	bounded	bounded	ADJ
ejde-771	390	10	inverse	inverse	NOUN
ejde-771	390	11	operator	operator	NOUN
ejde-771	390	12	(	(	PUNCT
ejde-771	390	13	λi	λi	ADP
ejde-771	390	14	+	+	NUM
ejde-771	390	15	a)−1	a)−1	NOUN
ejde-771	390	16	for	for	ADP
ejde-771	390	17	λ	λ	PROPN
ejde-771	390	18	>	>	X
ejde-771	390	19	0	0	NUM
ejde-771	390	20	implies	imply	VERB
ejde-771	390	21	that	that	SCONJ
ejde-771	390	22	t	t	PROPN
ejde-771	390	23	(	(	PUNCT
ejde-771	390	24	t)(t	t)(t	ADJ
ejde-771	390	25	≥	≥	NOUN
ejde-771	390	26	0	0	NUM
ejde-771	390	27	)	)	PUNCT
ejde-771	390	28	is	be	AUX
ejde-771	390	29	a	a	DET
ejde-771	390	30	positive	positive	ADJ
ejde-771	390	31	semigroup	semigroup	NOUN
ejde-771	390	32	.	.	PUNCT
ejde-771	391	1	therefore	therefore	ADV
ejde-771	391	2	,	,	PUNCT
ejde-771	391	3	based	base	VERB
ejde-771	391	4	on	on	ADP
ejde-771	391	5	the	the	DET
ejde-771	391	6	argument	argument	NOUN
ejde-771	391	7	in	in	ADP
ejde-771	391	8	preliminaries	preliminary	NOUN
ejde-771	391	9	and	and	CCONJ
ejde-771	391	10	the	the	DET
ejde-771	391	11	properties	property	NOUN
ejde-771	391	12	of	of	ADP
ejde-771	391	13	the	the	DET
ejde-771	391	14	semigroup	semigroup	PROPN
ejde-771	391	15	t	t	PROPN
ejde-771	391	16	(	(	PUNCT
ejde-771	391	17	t)(t	t)(t	ADJ
ejde-771	391	18	≥	≥	NOUN
ejde-771	391	19	0	0	NUM
ejde-771	391	20	)	)	PUNCT
ejde-771	391	21	generated	generate	VERB
ejde-771	391	22	by	by	ADP
ejde-771	391	23	−a	−a	ADV
ejde-771	391	24	,	,	PUNCT
ejde-771	391	25	one	one	PRON
ejde-771	391	26	can	can	AUX
ejde-771	391	27	deduce	deduce	VERB
ejde-771	391	28	that	that	SCONJ
ejde-771	391	29	the	the	DET
ejde-771	391	30	analytic	analytic	ADJ
ejde-771	391	31	semigroup	semigroup	NOUN
ejde-771	391	32	tβ(t)(t	tβ(t)(t	X
ejde-771	391	33	≥	≥	NOUN
ejde-771	391	34	0	0	NUM
ejde-771	391	35	)	)	PUNCT
ejde-771	391	36	generated	generate	VERB
ejde-771	391	37	by	by	ADP
ejde-771	391	38	−aβ	−aβ	PROPN
ejde-771	391	39	is	be	AUX
ejde-771	391	40	compact	compact	ADJ
ejde-771	391	41	,	,	PUNCT
ejde-771	391	42	positive	positive	ADJ
ejde-771	391	43	and	and	CCONJ
ejde-771	391	44	exponentially	exponentially	ADV
ejde-771	391	45	stable	stable	ADJ
ejde-771	391	46	on	on	ADP
ejde-771	391	47	e	e	X
ejde-771	391	48	as	as	ADV
ejde-771	391	49	well	well	ADV
ejde-771	391	50	as	as	ADP
ejde-771	391	51	∥tβ(t)∥	∥tβ(t)∥	PROPN
ejde-771	391	52	≤	≤	NUM
ejde-771	391	53	1	1	NUM
ejde-771	391	54	for	for	ADP
ejde-771	391	55	all	all	DET
ejde-771	391	56	t	t	PROPN
ejde-771	391	57	≥	≥	NOUN
ejde-771	391	58	0	0	NUM
ejde-771	391	59	.	.	PUNCT
ejde-771	392	1	let	let	VERB
ejde-771	392	2	m	m	VERB
ejde-771	392	3	=	=	SYM
ejde-771	392	4	1	1	NUM
ejde-771	392	5	and	and	CCONJ
ejde-771	392	6	ν0	ν0	PROPN
ejde-771	393	1	=	=	SYM
ejde-771	393	2	−λ1	−λ1	NOUN
ejde-771	393	3	=	=	SYM
ejde-771	393	4	−π2	−π2	NOUN
ejde-771	393	5	,	,	PUNCT
ejde-771	393	6	by	by	ADP
ejde-771	393	7	conditions	condition	NOUN
ejde-771	393	8	(	(	PUNCT
ejde-771	393	9	k0	k0	PROPN
ejde-771	393	10	)	)	PUNCT
ejde-771	393	11	and	and	CCONJ
ejde-771	393	12	(	(	PUNCT
ejde-771	393	13	k1	k1	NOUN
ejde-771	393	14	)	)	PUNCT
ejde-771	393	15	,	,	PUNCT
ejde-771	393	16	we	we	PRON
ejde-771	393	17	can	can	AUX
ejde-771	393	18	deduced	deduce	VERB
ejde-771	393	19	that	that	SCONJ
ejde-771	393	20	conditions	condition	NOUN
ejde-771	393	21	(	(	PUNCT
ejde-771	393	22	h0	h0	NOUN
ejde-771	393	23	)	)	PUNCT
ejde-771	393	24	and	and	CCONJ
ejde-771	393	25	(	(	PUNCT
ejde-771	393	26	h4	h4	NOUN
ejde-771	393	27	)	)	PUNCT
ejde-771	393	28	hold	hold	VERB
ejde-771	393	29	.	.	PUNCT
ejde-771	394	1	from	from	ADP
ejde-771	394	2	the	the	DET
ejde-771	394	3	conditions	condition	NOUN
ejde-771	394	4	(	(	PUNCT
ejde-771	394	5	k2	k2	NOUN
ejde-771	394	6	)	)	PUNCT
ejde-771	394	7	and	and	CCONJ
ejde-771	394	8	(	(	PUNCT
ejde-771	394	9	k3	k3	PROPN
ejde-771	394	10	)	)	PUNCT
ejde-771	394	11	,	,	PUNCT
ejde-771	394	12	we	we	PRON
ejde-771	394	13	can	can	AUX
ejde-771	394	14	deduced	deduce	VERB
ejde-771	394	15	that	that	SCONJ
ejde-771	394	16	conditions	condition	NOUN
ejde-771	394	17	(	(	PUNCT
ejde-771	394	18	h2	h2	NOUN
ejde-771	394	19	)	)	PUNCT
ejde-771	394	20	and	and	CCONJ
ejde-771	394	21	(	(	PUNCT
ejde-771	394	22	h3	h3	NOUN
ejde-771	394	23	)	)	PUNCT
ejde-771	394	24	hold	hold	NOUN
ejde-771	394	25	.	.	PUNCT
ejde-771	395	1	thus	thus	ADV
ejde-771	395	2	,	,	PUNCT
ejde-771	395	3	by	by	ADP
ejde-771	395	4	corrollary	corrollary	ADJ
ejde-771	395	5	3.3	3.3	NUM
ejde-771	395	6	one	one	NOUN
ejde-771	395	7	can	can	AUX
ejde-771	395	8	deduced	deduced	VERB
ejde-771	395	9	that	that	DET
ejde-771	395	10	nonlocal	nonlocal	ADJ
ejde-771	395	11	problem	problem	NOUN
ejde-771	395	12	(	(	PUNCT
ejde-771	395	13	1.1	1.1	NUM
ejde-771	395	14	)	)	PUNCT
ejde-771	395	15	exist	exist	VERB
ejde-771	395	16	a	a	DET
ejde-771	395	17	minimal	minimal	ADJ
ejde-771	395	18	positive	positive	ADJ
ejde-771	395	19	s	s	NOUN
ejde-771	395	20	-	-	PUNCT
ejde-771	395	21	asymptotically	asymptotically	ADV
ejde-771	395	22	ω	ω	ADJ
ejde-771	395	23	-	-	ADJ
ejde-771	395	24	periodic	periodic	ADJ
ejde-771	395	25	solution	solution	NOUN
ejde-771	395	26	.	.	PUNCT
ejde-771	396	1	□	□	PUNCT
ejde-771	396	2	14	14	NUM
ejde-771	396	3	x.	x.	NOUN
ejde-771	396	4	zhang	zhang	PROPN
ejde-771	396	5	,	,	PUNCT
ejde-771	396	6	k.	k.	PROPN
ejde-771	396	7	ding	ding	PROPN
ejde-771	396	8	,	,	PUNCT
ejde-771	396	9	p.	p.	PROPN
ejde-771	396	10	chen	chen	PROPN
ejde-771	396	11	ejde-2025/44	ejde-2025/44	PROPN
ejde-771	396	12	based	base	VERB
ejde-771	396	13	on	on	ADP
ejde-771	396	14	the	the	DET
ejde-771	396	15	proof	proof	NOUN
ejde-771	396	16	of	of	ADP
ejde-771	396	17	this	this	DET
ejde-771	396	18	theorem	theorem	NOUN
ejde-771	396	19	,	,	PUNCT
ejde-771	396	20	it	it	PRON
ejde-771	396	21	is	be	AUX
ejde-771	396	22	not	not	PART
ejde-771	396	23	difficult	difficult	ADJ
ejde-771	396	24	to	to	PART
ejde-771	396	25	obtain	obtain	VERB
ejde-771	396	26	the	the	DET
ejde-771	396	27	following	follow	VERB
ejde-771	396	28	result	result	NOUN
ejde-771	396	29	.	.	PUNCT
ejde-771	397	1	theorem	theorem	VERB
ejde-771	397	2	4.2	4.2	NUM
ejde-771	397	3	.	.	PUNCT
ejde-771	398	1	let	let	VERB
ejde-771	398	2	nonlinear	nonlinear	ADJ
ejde-771	398	3	function	function	NOUN
ejde-771	398	4	f	f	NOUN
ejde-771	398	5	:	:	PUNCT
ejde-771	399	1	[	[	X
ejde-771	399	2	0,∞	0,∞	NUM
ejde-771	399	3	)	)	PUNCT
ejde-771	399	4	×	×	NOUN
ejde-771	399	5	p	p	NOUN
ejde-771	399	6	→	→	PUNCT
ejde-771	399	7	p	p	X
ejde-771	399	8	be	be	AUX
ejde-771	399	9	a	a	DET
ejde-771	399	10	continuous	continuous	ADJ
ejde-771	399	11	mapping	mapping	NOUN
ejde-771	399	12	.	.	PUNCT
ejde-771	400	1	if	if	SCONJ
ejde-771	400	2	the	the	DET
ejde-771	400	3	conditions	condition	NOUN
ejde-771	400	4	(	(	PUNCT
ejde-771	400	5	k0)–(k4	k0)–(k4	NOUN
ejde-771	400	6	)	)	PUNCT
ejde-771	400	7	hold	hold	VERB
ejde-771	400	8	for	for	ADP
ejde-771	400	9	any	any	DET
ejde-771	400	10	ξ	ξ	PROPN
ejde-771	400	11	∈	∈	PROPN
ejde-771	400	12	e	e	X
ejde-771	400	13	with	with	ADP
ejde-771	400	14	ξ	ξ	PROPN
ejde-771	400	15	≥	≥	NUM
ejde-771	400	16	ς	ς	NOUN
ejde-771	400	17	√	√	ADV
ejde-771	400	18	2	2	NUM
ejde-771	400	19	sin(πx	sin(πx	NOUN
ejde-771	400	20	)	)	PUNCT
ejde-771	400	21	,	,	PUNCT
ejde-771	400	22	there	there	PRON
ejde-771	400	23	is	be	VERB
ejde-771	400	24	a	a	DET
ejde-771	400	25	constant	constant	ADJ
ejde-771	400	26	ς	ς	X
ejde-771	400	27	>	>	X
ejde-771	400	28	0	0	NUM
ejde-771	400	29	such	such	ADJ
ejde-771	400	30	that	that	SCONJ
ejde-771	400	31	f	f	PROPN
ejde-771	400	32	(	(	PUNCT
ejde-771	400	33	t	t	PROPN
ejde-771	400	34	,	,	PUNCT
ejde-771	400	35	ξ	ξ	PROPN
ejde-771	400	36	)	)	PUNCT
ejde-771	400	37	≥	≥	NOUN
ejde-771	400	38	f	f	X
ejde-771	400	39	(	(	PUNCT
ejde-771	400	40	t	t	PROPN
ejde-771	400	41	,	,	PUNCT
ejde-771	400	42	ς	ς	PROPN
ejde-771	400	43	√	√	ADJ
ejde-771	400	44	2	2	NUM
ejde-771	400	45	sin(πx	sin(πx	NOUN
ejde-771	400	46	)	)	PUNCT
ejde-771	400	47	)	)	PUNCT
ejde-771	400	48	≥	≥	NOUN
ejde-771	400	49	π2βς	π2βς	PUNCT
ejde-771	400	50	√	√	NUM
ejde-771	400	51	2	2	NUM
ejde-771	400	52	sin(πx	sin(πx	NOUN
ejde-771	400	53	)	)	PUNCT
ejde-771	400	54	,	,	PUNCT
ejde-771	400	55	hold	hold	VERB
ejde-771	400	56	,	,	PUNCT
ejde-771	400	57	and	and	CCONJ
ejde-771	400	58	u0(x	u0(x	X
ejde-771	400	59	)	)	PUNCT
ejde-771	400	60	≥	≥	NOUN
ejde-771	400	61	ς	ς	NOUN
ejde-771	400	62	√	√	ADV
ejde-771	400	63	2	2	NUM
ejde-771	400	64	sin(πx	sin(πx	NOUN
ejde-771	400	65	)	)	PUNCT
ejde-771	400	66	,	,	PUNCT
ejde-771	400	67	then	then	ADV
ejde-771	400	68	nonlocal	nonlocal	ADJ
ejde-771	400	69	problem	problem	NOUN
ejde-771	400	70	(	(	PUNCT
ejde-771	400	71	1.1	1.1	NUM
ejde-771	400	72	)	)	PUNCT
ejde-771	400	73	has	have	AUX
ejde-771	400	74	at	at	ADV
ejde-771	400	75	least	least	ADV
ejde-771	400	76	one	one	NUM
ejde-771	400	77	positive	positive	ADJ
ejde-771	400	78	s	s	NOUN
ejde-771	400	79	-	-	PUNCT
ejde-771	400	80	asymptotically	asymptotically	ADV
ejde-771	400	81	ω	ω	ADJ
ejde-771	400	82	-	-	ADJ
ejde-771	400	83	periodic	periodic	ADJ
ejde-771	400	84	solution	solution	NOUN
ejde-771	400	85	.	.	PUNCT
ejde-771	401	1	acknowledgments	acknowledgment	NOUN
ejde-771	401	2	.	.	PUNCT
ejde-771	402	1	this	this	DET
ejde-771	402	2	work	work	NOUN
ejde-771	402	3	was	be	AUX
ejde-771	402	4	supported	support	VERB
ejde-771	402	5	by	by	ADP
ejde-771	402	6	the	the	DET
ejde-771	402	7	the	the	DET
ejde-771	402	8	national	national	ADJ
ejde-771	402	9	natural	natural	PROPN
ejde-771	402	10	science	science	PROPN
ejde-771	402	11	foundation	foundation	PROPN
ejde-771	402	12	of	of	ADP
ejde-771	402	13	china	china	PROPN
ejde-771	402	14	(	(	PUNCT
ejde-771	402	15	no	no	INTJ
ejde-771	402	16	.	.	NOUN
ejde-771	402	17	12061063	12061063	NUM
ejde-771	402	18	)	)	PUNCT
ejde-771	402	19	,	,	PUNCT
ejde-771	402	20	by	by	ADP
ejde-771	402	21	the	the	DET
ejde-771	402	22	outstanding	outstanding	ADJ
ejde-771	402	23	youth	youth	NOUN
ejde-771	402	24	science	science	NOUN
ejde-771	402	25	fund	fund	NOUN
ejde-771	402	26	of	of	ADP
ejde-771	402	27	gansu	gansu	PROPN
ejde-771	402	28	province	province	PROPN
ejde-771	402	29	(	(	PUNCT
ejde-771	402	30	no	no	INTJ
ejde-771	402	31	.	.	PUNCT
ejde-771	403	1	24jrra122	24jrra122	NUM
ejde-771	403	2	)	)	PUNCT
ejde-771	403	3	,	,	PUNCT
ejde-771	403	4	by	by	ADP
ejde-771	403	5	the	the	DET
ejde-771	403	6	young	young	ADJ
ejde-771	403	7	doctor	doctor	NOUN
ejde-771	403	8	fund	fund	NOUN
ejde-771	403	9	project	project	NOUN
ejde-771	403	10	of	of	ADP
ejde-771	403	11	gansu	gansu	PROPN
ejde-771	403	12	provincial	provincial	PROPN
ejde-771	403	13	department	department	PROPN
ejde-771	403	14	of	of	ADP
ejde-771	403	15	education	education	PROPN
ejde-771	403	16	(	(	PUNCT
ejde-771	403	17	no	no	INTJ
ejde-771	403	18	.	.	NOUN
ejde-771	403	19	2023qb-111	2023qb-111	NUM
ejde-771	403	20	)	)	PUNCT
ejde-771	403	21	,	,	PUNCT
ejde-771	403	22	by	by	ADP
ejde-771	403	23	the	the	DET
ejde-771	403	24	funds	fund	NOUN
ejde-771	403	25	for	for	ADP
ejde-771	403	26	innovative	innovative	ADJ
ejde-771	403	27	fundamental	fundamental	ADJ
ejde-771	403	28	research	research	NOUN
ejde-771	403	29	group	group	NOUN
ejde-771	403	30	project	project	NOUN
ejde-771	403	31	of	of	ADP
ejde-771	403	32	gansu	gansu	PROPN
ejde-771	403	33	province	province	PROPN
ejde-771	403	34	(	(	PUNCT
ejde-771	403	35	no	no	INTJ
ejde-771	403	36	.	.	PUNCT
ejde-771	404	1	23jrra684	23jrra684	NUM
ejde-771	404	2	)	)	PUNCT
ejde-771	404	3	,	,	PUNCT
ejde-771	404	4	and	and	CCONJ
ejde-771	404	5	by	by	ADP
ejde-771	404	6	the	the	DET
ejde-771	404	7	natural	natural	ADJ
ejde-771	404	8	science	science	PROPN
ejde-771	404	9	foundation	foundation	NOUN
ejde-771	404	10	of	of	ADP
ejde-771	404	11	gansu	gansu	PROPN
ejde-771	404	12	province	province	PROPN
ejde-771	404	13	(	(	PUNCT
ejde-771	404	14	no	no	INTJ
ejde-771	404	15	.	.	PUNCT
ejde-771	405	1	24jrra780	24jrra780	NUM
ejde-771	405	2	)	)	PUNCT
ejde-771	406	1	and	and	CCONJ
ejde-771	406	2	project	project	NOUN
ejde-771	406	3	2024kglx01017	2024kglx01017	NUM
ejde-771	406	4	.	.	PUNCT
ejde-771	407	1	references	reference	NOUN
ejde-771	407	2	[	[	X
ejde-771	407	3	1	1	NUM
ejde-771	407	4	]	]	PUNCT
ejde-771	407	5	c.	c.	PROPN
ejde-771	407	6	t.	t.	PROPN
ejde-771	407	7	anh	anh	PROPN
ejde-771	407	8	,	,	PUNCT
ejde-771	407	9	t.	t.	PROPN
ejde-771	407	10	d.	d.	PROPN
ejde-771	407	11	ke	ke	PROPN
ejde-771	407	12	;	;	PUNCT
ejde-771	407	13	on	on	ADP
ejde-771	407	14	nonlocal	nonlocal	ADJ
ejde-771	407	15	problems	problem	NOUN
ejde-771	407	16	for	for	ADP
ejde-771	407	17	retarded	retarded	ADJ
ejde-771	407	18	fractional	fractional	ADJ
ejde-771	407	19	differential	differential	ADJ
ejde-771	407	20	equations	equation	NOUN
ejde-771	407	21	in	in	ADP
ejde-771	407	22	banach	banach	NOUN
ejde-771	407	23	spaces	space	NOUN
ejde-771	407	24	.	.	PUNCT
ejde-771	408	1	fixed	fix	VERB
ejde-771	408	2	point	point	NOUN
ejde-771	408	3	theory	theory	NOUN
ejde-771	408	4	,	,	PUNCT
ejde-771	408	5	15.2(2014	15.2(2014	NUM
ejde-771	408	6	)	)	PUNCT
ejde-771	408	7	,	,	PUNCT
ejde-771	408	8	373	373	NUM
ejde-771	408	9	-	-	SYM
ejde-771	408	10	392	392	NUM
ejde-771	408	11	.	.	PUNCT
ejde-771	409	1	[	[	X
ejde-771	409	2	2	2	X
ejde-771	409	3	]	]	PUNCT
ejde-771	409	4	h.	h.	PROPN
ejde-771	409	5	amann	amann	PROPN
ejde-771	409	6	;	;	PUNCT
ejde-771	409	7	periodic	periodic	ADJ
ejde-771	409	8	solutions	solution	NOUN
ejde-771	409	9	of	of	ADP
ejde-771	409	10	semilinear	semilinear	PROPN
ejde-771	409	11	parabolic	parabolic	PROPN
ejde-771	409	12	equations	equation	NOUN
ejde-771	409	13	,	,	PUNCT
ejde-771	409	14	in	in	ADP
ejde-771	409	15	nonlinear	nonlinear	ADJ
ejde-771	409	16	analysis	analysis	NOUN
ejde-771	409	17	,	,	PUNCT
ejde-771	409	18	collection	collection	NOUN
ejde-771	409	19	of	of	ADP
ejde-771	409	20	papers	paper	NOUN
ejde-771	409	21	in	in	ADP
ejde-771	409	22	honor	honor	NOUN
ejde-771	409	23	of	of	ADP
ejde-771	409	24	erich	erich	PROPN
ejde-771	409	25	h.	h.	PROPN
ejde-771	409	26	rothe	rothe	PROPN
ejde-771	409	27	.	.	PUNCT
ejde-771	410	1	academic	academic	ADJ
ejde-771	410	2	press	press	PROPN
ejde-771	410	3	,	,	PUNCT
ejde-771	410	4	new	new	PROPN
ejde-771	410	5	york	york	PROPN
ejde-771	410	6	,	,	PUNCT
ejde-771	410	7	(	(	PUNCT
ejde-771	410	8	1978	1978	NUM
ejde-771	410	9	)	)	PUNCT
ejde-771	410	10	,	,	PUNCT
ejde-771	410	11	1	1	NUM
ejde-771	410	12	-	-	SYM
ejde-771	410	13	29	29	NUM
ejde-771	410	14	.	.	PUNCT
ejde-771	411	1	[	[	X
ejde-771	411	2	3	3	X
ejde-771	411	3	]	]	PUNCT
ejde-771	411	4	z.	z.	PROPN
ejde-771	411	5	alsheekhhussain	alsheekhhussain	PROPN
ejde-771	411	6	,	,	PUNCT
ejde-771	411	7	a.	a.	NOUN
ejde-771	411	8	g.	g.	PROPN
ejde-771	411	9	ibrahim	ibrahim	PROPN
ejde-771	411	10	,	,	PUNCT
ejde-771	411	11	r.	r.	PROPN
ejde-771	411	12	a.	a.	PROPN
ejde-771	411	13	ramadan	ramadan	PROPN
ejde-771	411	14	;	;	PUNCT
ejde-771	411	15	existence	existence	NOUN
ejde-771	411	16	of	of	ADP
ejde-771	411	17	s	s	NOUN
ejde-771	411	18	-	-	PUNCT
ejde-771	411	19	asymptotically	asymptotically	ADV
ejde-771	411	20	ω	ω	ADJ
ejde-771	411	21	-	-	ADJ
ejde-771	411	22	periodic	periodic	ADJ
ejde-771	411	23	solutions	solution	NOUN
ejde-771	411	24	for	for	ADP
ejde-771	411	25	non	non	ADJ
ejde-771	411	26	-	-	ADJ
ejde-771	411	27	instantaneous	instantaneous	ADJ
ejde-771	411	28	impulsive	impulsive	ADJ
ejde-771	411	29	semilinear	semilinear	ADJ
ejde-771	411	30	differential	differential	NOUN
ejde-771	411	31	equations	equation	NOUN
ejde-771	411	32	and	and	CCONJ
ejde-771	411	33	inclusions	inclusion	NOUN
ejde-771	411	34	of	of	ADP
ejde-771	411	35	fractional	fractional	ADJ
ejde-771	411	36	order	order	NOUN
ejde-771	411	37	1	1	NUM
ejde-771	411	38	<	<	X
ejde-771	411	39	α	α	X
ejde-771	411	40	<	<	X
ejde-771	411	41	2	2	NUM
ejde-771	411	42	.	.	PUNCT
ejde-771	411	43	aims	aim	VERB
ejde-771	411	44	math	math	NOUN
ejde-771	411	45	.	.	PUNCT
ejde-771	411	46	,	,	PUNCT
ejde-771	411	47	8.1(2023	8.1(2023	NUM
ejde-771	411	48	)	)	PUNCT
ejde-771	411	49	,	,	PUNCT
ejde-771	411	50	76	76	NUM
ejde-771	411	51	-	-	SYM
ejde-771	411	52	101	101	NUM
ejde-771	411	53	.	.	PUNCT
ejde-771	412	1	[	[	X
ejde-771	412	2	4	4	NUM
ejde-771	412	3	]	]	PUNCT
ejde-771	412	4	a.	a.	NOUN
ejde-771	412	5	v.	v.	PROPN
ejde-771	412	6	balakrishnan	balakrishnan	PROPN
ejde-771	412	7	;	;	PUNCT
ejde-771	412	8	fractional	fractional	ADJ
ejde-771	412	9	powers	power	NOUN
ejde-771	412	10	of	of	ADP
ejde-771	412	11	closed	closed	ADJ
ejde-771	412	12	operators	operator	NOUN
ejde-771	412	13	and	and	CCONJ
ejde-771	412	14	the	the	DET
ejde-771	412	15	semigroups	semigroup	NOUN
ejde-771	412	16	generated	generate	VERB
ejde-771	412	17	by	by	ADP
ejde-771	412	18	them	they	PRON
ejde-771	412	19	.	.	PUNCT
ejde-771	413	1	pacific	pacific	PROPN
ejde-771	413	2	j	j	PROPN
ejde-771	413	3	math	math	PROPN
ejde-771	413	4	.	.	PUNCT
ejde-771	413	5	,	,	PUNCT
ejde-771	413	6	10(1961	10(1961	NUM
ejde-771	413	7	)	)	PUNCT
ejde-771	413	8	,	,	PUNCT
ejde-771	413	9	419	419	NUM
ejde-771	413	10	-	-	SYM
ejde-771	413	11	437	437	NUM
ejde-771	413	12	.	.	PUNCT
ejde-771	414	1	[	[	X
ejde-771	414	2	5	5	X
ejde-771	414	3	]	]	PUNCT
ejde-771	414	4	d.	d.	PROPN
ejde-771	414	5	brindle	brindle	PROPN
ejde-771	414	6	,	,	PUNCT
ejde-771	414	7	g.	g.	PROPN
ejde-771	414	8	m.	m.	PROPN
ejde-771	414	9	n’guerekata	n’guerekata	PROPN
ejde-771	414	10	;	;	PUNCT
ejde-771	414	11	s	s	X
ejde-771	414	12	-	-	PUNCT
ejde-771	414	13	asymptotically	asymptotically	ADV
ejde-771	414	14	ω	ω	ADJ
ejde-771	414	15	-	-	ADJ
ejde-771	414	16	periodic	periodic	ADJ
ejde-771	414	17	mild	mild	ADJ
ejde-771	414	18	solutions	solution	NOUN
ejde-771	414	19	to	to	ADP
ejde-771	414	20	fractional	fractional	ADJ
ejde-771	414	21	differential	differential	ADJ
ejde-771	414	22	equations	equation	NOUN
ejde-771	414	23	.	.	PUNCT
ejde-771	415	1	electron	electron	PROPN
ejde-771	415	2	.	.	PUNCT
ejde-771	416	1	j.	j.	PROPN
ejde-771	416	2	differential	differential	PROPN
ejde-771	416	3	equations	equations	PROPN
ejde-771	416	4	,	,	PUNCT
ejde-771	416	5	30(2020	30(2020	NUM
ejde-771	416	6	)	)	PUNCT
ejde-771	416	7	,	,	PUNCT
ejde-771	416	8	1	1	NUM
ejde-771	416	9	-	-	SYM
ejde-771	416	10	12	12	NUM
ejde-771	416	11	.	.	PUNCT
ejde-771	417	1	[	[	X
ejde-771	417	2	6	6	NUM
ejde-771	417	3	]	]	PUNCT
ejde-771	417	4	p.	p.	PROPN
ejde-771	417	5	bedi	bedi	PROPN
ejde-771	417	6	,	,	PUNCT
ejde-771	417	7	a.	a.	PROPN
ejde-771	417	8	kumar	kumar	PROPN
ejde-771	417	9	,	,	PUNCT
ejde-771	417	10	t.	t.	PROPN
ejde-771	417	11	abdeljawad	abdeljawad	PROPN
ejde-771	417	12	,	,	PUNCT
ejde-771	417	13	a.	a.	PROPN
ejde-771	417	14	khan	khan	PROPN
ejde-771	417	15	;	;	PUNCT
ejde-771	417	16	s	s	X
ejde-771	417	17	-	-	PUNCT
ejde-771	417	18	asymptotically	asymptotically	ADV
ejde-771	417	19	ω	ω	ADJ
ejde-771	417	20	-	-	ADJ
ejde-771	417	21	periodic	periodic	ADJ
ejde-771	417	22	mild	mild	ADJ
ejde-771	417	23	solutions	solution	NOUN
ejde-771	417	24	and	and	CCONJ
ejde-771	417	25	stability	stability	NOUN
ejde-771	417	26	analysis	analysis	NOUN
ejde-771	417	27	of	of	ADP
ejde-771	417	28	hilfer	hilfer	NOUN
ejde-771	417	29	fractional	fractional	ADJ
ejde-771	417	30	evolution	evolution	NOUN
ejde-771	417	31	equations	equation	NOUN
ejde-771	417	32	.	.	PUNCT
ejde-771	418	1	evol	evol	NOUN
ejde-771	418	2	.	.	PUNCT
ejde-771	419	1	equ	equ	PROPN
ejde-771	419	2	.	.	PUNCT
ejde-771	419	3	control	control	PROPN
ejde-771	419	4	theory	theory	PROPN
ejde-771	419	5	,	,	PUNCT
ejde-771	419	6	10	10	NUM
ejde-771	419	7	(	(	PUNCT
ejde-771	419	8	2021	2021	NUM
ejde-771	419	9	)	)	PUNCT
ejde-771	419	10	,	,	PUNCT
ejde-771	419	11	733	733	NUM
ejde-771	419	12	-	-	SYM
ejde-771	419	13	748	748	NUM
ejde-771	419	14	.	.	PUNCT
ejde-771	420	1	[	[	X
ejde-771	420	2	7	7	X
ejde-771	420	3	]	]	X
ejde-771	420	4	j.	j.	PROPN
ejde-771	420	5	cao	cao	PROPN
ejde-771	420	6	,	,	PUNCT
ejde-771	420	7	z.	z.	PROPN
ejde-771	420	8	huang	huang	PROPN
ejde-771	420	9	;	;	PUNCT
ejde-771	420	10	existence	existence	NOUN
ejde-771	420	11	of	of	ADP
ejde-771	420	12	asymptotically	asymptotically	ADV
ejde-771	420	13	periodic	periodic	ADJ
ejde-771	420	14	solutions	solution	NOUN
ejde-771	420	15	for	for	ADP
ejde-771	420	16	semilinear	semilinear	ADJ
ejde-771	420	17	evolution	evolution	NOUN
ejde-771	420	18	equations	equation	NOUN
ejde-771	420	19	with	with	ADP
ejde-771	420	20	nonlocal	nonlocal	ADJ
ejde-771	420	21	initial	initial	ADJ
ejde-771	420	22	conditions	condition	NOUN
ejde-771	420	23	.	.	PUNCT
ejde-771	421	1	open	open	ADJ
ejde-771	421	2	math	math	NOUN
ejde-771	421	3	.	.	PUNCT
ejde-771	421	4	,	,	PUNCT
ejde-771	421	5	16(2018	16(2018	NUM
ejde-771	421	6	)	)	PUNCT
ejde-771	421	7	,	,	PUNCT
ejde-771	421	8	792	792	NUM
ejde-771	421	9	-	-	SYM
ejde-771	421	10	805	805	NUM
ejde-771	421	11	.	.	PUNCT
ejde-771	422	1	[	[	X
ejde-771	422	2	8	8	NUM
ejde-771	422	3	]	]	PUNCT
ejde-771	422	4	p.	p.	NOUN
ejde-771	422	5	chen	chen	PROPN
ejde-771	422	6	,	,	PUNCT
ejde-771	422	7	a.	a.	PROPN
ejde-771	422	8	abdelmonem	abdelmonem	PROPN
ejde-771	422	9	,	,	PUNCT
ejde-771	422	10	y.	y.	PROPN
ejde-771	422	11	li	li	PROPN
ejde-771	422	12	;	;	PUNCT
ejde-771	422	13	global	global	ADJ
ejde-771	422	14	existence	existence	NOUN
ejde-771	422	15	and	and	CCONJ
ejde-771	422	16	asymptotic	asymptotic	ADJ
ejde-771	422	17	stability	stability	NOUN
ejde-771	422	18	of	of	ADP
ejde-771	422	19	mild	mild	ADJ
ejde-771	422	20	solutions	solution	NOUN
ejde-771	422	21	for	for	ADP
ejde-771	422	22	stochastic	stochastic	ADJ
ejde-771	422	23	evolution	evolution	NOUN
ejde-771	422	24	equations	equation	NOUN
ejde-771	422	25	with	with	ADP
ejde-771	422	26	nonlocal	nonlocal	ADJ
ejde-771	422	27	initial	initial	ADJ
ejde-771	422	28	conditions	condition	NOUN
ejde-771	422	29	.	.	PUNCT
ejde-771	423	1	j.	j.	PROPN
ejde-771	423	2	integral	integral	PROPN
ejde-771	423	3	equations	equation	NOUN
ejde-771	423	4	appl	appl	PROPN
ejde-771	423	5	.	.	PROPN
ejde-771	423	6	,	,	PUNCT
ejde-771	423	7	29.2(2017	29.2(2017	NUM
ejde-771	423	8	)	)	PUNCT
ejde-771	423	9	,	,	PUNCT
ejde-771	423	10	325	325	NUM
ejde-771	423	11	-	-	SYM
ejde-771	423	12	348	348	NUM
ejde-771	423	13	.	.	PUNCT
ejde-771	424	1	[	[	X
ejde-771	424	2	9	9	NUM
ejde-771	424	3	]	]	PUNCT
ejde-771	424	4	p.	p.	NOUN
ejde-771	424	5	chen	chen	PROPN
ejde-771	424	6	,	,	PUNCT
ejde-771	424	7	x.	x.	PROPN
ejde-771	424	8	zhang	zhang	PROPN
ejde-771	424	9	;	;	PUNCT
ejde-771	424	10	approximate	approximate	ADJ
ejde-771	424	11	controllability	controllability	NOUN
ejde-771	424	12	of	of	ADP
ejde-771	424	13	nonlocal	nonlocal	ADJ
ejde-771	424	14	problem	problem	NOUN
ejde-771	424	15	for	for	ADP
ejde-771	424	16	non	non	ADJ
ejde-771	424	17	-	-	ADJ
ejde-771	424	18	autonomous	autonomous	ADJ
ejde-771	424	19	stochastic	stochastic	ADJ
ejde-771	424	20	evolution	evolution	NOUN
ejde-771	424	21	equations	equation	NOUN
ejde-771	424	22	.	.	PUNCT
ejde-771	425	1	evol	evol	NOUN
ejde-771	425	2	.	.	PUNCT
ejde-771	426	1	equ	equ	PROPN
ejde-771	426	2	.	.	PUNCT
ejde-771	426	3	control	control	PROPN
ejde-771	426	4	theory	theory	PROPN
ejde-771	426	5	,	,	PUNCT
ejde-771	426	6	10.3(2021	10.3(2021	NUM
ejde-771	426	7	)	)	PUNCT
ejde-771	426	8	,	,	PUNCT
ejde-771	426	9	471	471	NUM
ejde-771	426	10	-	-	SYM
ejde-771	426	11	489	489	NUM
ejde-771	426	12	.	.	PUNCT
ejde-771	427	1	[	[	X
ejde-771	427	2	10	10	NUM
ejde-771	427	3	]	]	PUNCT
ejde-771	427	4	p.	p.	NOUN
ejde-771	427	5	chen	chen	PROPN
ejde-771	427	6	,	,	PUNCT
ejde-771	427	7	x.	x.	PROPN
ejde-771	427	8	zhang	zhang	PROPN
ejde-771	427	9	;	;	PUNCT
ejde-771	427	10	non	non	ADJ
ejde-771	427	11	-	-	ADJ
ejde-771	427	12	autonomous	autonomous	ADJ
ejde-771	427	13	stochastic	stochastic	ADJ
ejde-771	427	14	evolution	evolution	NOUN
ejde-771	427	15	equations	equation	NOUN
ejde-771	427	16	of	of	ADP
ejde-771	427	17	parabolic	parabolic	ADJ
ejde-771	427	18	type	type	NOUN
ejde-771	427	19	with	with	ADP
ejde-771	427	20	nonlocal	nonlocal	ADJ
ejde-771	427	21	initial	initial	ADJ
ejde-771	427	22	conditions	condition	NOUN
ejde-771	427	23	.	.	PUNCT
ejde-771	428	1	discrete	discrete	ADJ
ejde-771	428	2	contin	contin	NOUN
ejde-771	428	3	.	.	PUNCT
ejde-771	429	1	dyn	dyn	NOUN
ejde-771	429	2	.	.	PUNCT
ejde-771	430	1	syst	syst	PROPN
ejde-771	430	2	.	.	PUNCT
ejde-771	431	1	ser	ser	PROPN
ejde-771	431	2	.	.	PUNCT
ejde-771	432	1	b	b	NUM
ejde-771	432	2	,	,	PUNCT
ejde-771	432	3	26.9(2021	26.9(2021	NUM
ejde-771	432	4	)	)	PUNCT
ejde-771	432	5	,	,	PUNCT
ejde-771	432	6	4681	4681	NUM
ejde-771	432	7	-	-	SYM
ejde-771	432	8	4695	4695	NUM
ejde-771	432	9	.	.	PUNCT
ejde-771	433	1	[	[	X
ejde-771	433	2	11	11	NUM
ejde-771	433	3	]	]	PUNCT
ejde-771	433	4	p.	p.	NOUN
ejde-771	433	5	chen	chen	PROPN
ejde-771	433	6	,	,	PUNCT
ejde-771	433	7	x.	x.	PROPN
ejde-771	433	8	zhang	zhang	PROPN
ejde-771	433	9	,	,	PUNCT
ejde-771	433	10	y.	y.	PROPN
ejde-771	433	11	li	li	PROPN
ejde-771	433	12	;	;	PUNCT
ejde-771	433	13	existence	existence	NOUN
ejde-771	433	14	and	and	CCONJ
ejde-771	433	15	approximate	approximate	ADJ
ejde-771	433	16	controllability	controllability	NOUN
ejde-771	433	17	of	of	ADP
ejde-771	433	18	fractional	fractional	ADJ
ejde-771	433	19	evolution	evolution	NOUN
ejde-771	433	20	equations	equation	NOUN
ejde-771	433	21	with	with	ADP
ejde-771	433	22	nonlocal	nonlocal	ADJ
ejde-771	433	23	conditions	condition	NOUN
ejde-771	433	24	via	via	ADP
ejde-771	433	25	resolvent	resolvent	ADJ
ejde-771	433	26	operators	operator	NOUN
ejde-771	433	27	.	.	PUNCT
ejde-771	434	1	fract	fract	PROPN
ejde-771	434	2	.	.	PUNCT
ejde-771	435	1	calc	calc	PROPN
ejde-771	435	2	.	.	PUNCT
ejde-771	436	1	appl	appl	PROPN
ejde-771	436	2	.	.	PUNCT
ejde-771	437	1	anal	anal	PROPN
ejde-771	437	2	.	.	PROPN
ejde-771	437	3	,	,	PUNCT
ejde-771	437	4	23.1(2020	23.1(2020	NUM
ejde-771	437	5	)	)	PUNCT
ejde-771	437	6	,	,	PUNCT
ejde-771	437	7	268	268	NUM
ejde-771	437	8	-	-	SYM
ejde-771	437	9	291	291	NUM
ejde-771	437	10	.	.	PUNCT
ejde-771	438	1	[	[	X
ejde-771	438	2	12	12	NUM
ejde-771	438	3	]	]	PUNCT
ejde-771	438	4	p.	p.	NOUN
ejde-771	438	5	chen	chen	PROPN
ejde-771	438	6	,	,	PUNCT
ejde-771	438	7	x.	x.	PROPN
ejde-771	438	8	zhang	zhang	PROPN
ejde-771	438	9	,	,	PUNCT
ejde-771	438	10	y.	y.	PROPN
ejde-771	438	11	li	li	PROPN
ejde-771	438	12	;	;	PUNCT
ejde-771	438	13	approximate	approximate	ADJ
ejde-771	438	14	controllability	controllability	NOUN
ejde-771	438	15	of	of	ADP
ejde-771	438	16	non	non	ADJ
ejde-771	438	17	-	-	ADJ
ejde-771	438	18	autonomous	autonomous	ADJ
ejde-771	438	19	evolution	evolution	NOUN
ejde-771	438	20	system	system	NOUN
ejde-771	438	21	with	with	ADP
ejde-771	438	22	nonlocal	nonlocal	ADJ
ejde-771	438	23	conditions	condition	NOUN
ejde-771	438	24	.	.	PUNCT
ejde-771	439	1	j.	j.	PROPN
ejde-771	439	2	dyn	dyn	PROPN
ejde-771	439	3	.	.	PUNCT
ejde-771	440	1	control	control	PROPN
ejde-771	440	2	syst	syst	PROPN
ejde-771	440	3	.	.	PUNCT
ejde-771	440	4	,	,	PUNCT
ejde-771	440	5	26.1(2020	26.1(2020	NUM
ejde-771	440	6	)	)	PUNCT
ejde-771	440	7	,	,	PUNCT
ejde-771	440	8	1	1	NUM
ejde-771	440	9	-	-	SYM
ejde-771	440	10	16	16	NUM
ejde-771	440	11	.	.	PUNCT
ejde-771	441	1	[	[	X
ejde-771	441	2	13	13	NUM
ejde-771	441	3	]	]	PUNCT
ejde-771	441	4	p.	p.	NOUN
ejde-771	441	5	chen	chen	PROPN
ejde-771	441	6	,	,	PUNCT
ejde-771	441	7	y.	y.	PROPN
ejde-771	441	8	li	li	PROPN
ejde-771	441	9	;	;	PUNCT
ejde-771	441	10	existence	existence	NOUN
ejde-771	441	11	of	of	ADP
ejde-771	441	12	mild	mild	ADJ
ejde-771	441	13	solutions	solution	NOUN
ejde-771	441	14	for	for	ADP
ejde-771	441	15	fractional	fractional	ADJ
ejde-771	441	16	evolution	evolution	NOUN
ejde-771	441	17	equations	equation	NOUN
ejde-771	441	18	with	with	ADP
ejde-771	441	19	mixed	mixed	ADJ
ejde-771	441	20	monotone	monotone	ADJ
ejde-771	441	21	nonlocal	nonlocal	ADJ
ejde-771	441	22	conditions	condition	NOUN
ejde-771	441	23	.	.	PUNCT
ejde-771	442	1	z.	z.	PROPN
ejde-771	442	2	angew	angew	PROPN
ejde-771	442	3	.	.	PUNCT
ejde-771	443	1	math	math	NOUN
ejde-771	443	2	.	.	PUNCT
ejde-771	444	1	phys	phy	NOUN
ejde-771	444	2	.	.	PUNCT
ejde-771	444	3	,	,	PUNCT
ejde-771	444	4	65(2014	65(2014	NUM
ejde-771	444	5	)	)	PUNCT
ejde-771	444	6	,	,	PUNCT
ejde-771	444	7	711	711	NUM
ejde-771	444	8	-	-	SYM
ejde-771	444	9	728	728	NUM
ejde-771	444	10	.	.	PUNCT
ejde-771	445	1	[	[	X
ejde-771	445	2	14	14	NUM
ejde-771	445	3	]	]	X
ejde-771	445	4	y.	y.	PROPN
ejde-771	445	5	chen	chen	PROPN
ejde-771	445	6	,	,	PUNCT
ejde-771	445	7	z.	z.	PROPN
ejde-771	445	8	lv	lv	PROPN
ejde-771	445	9	,	,	PUNCT
ejde-771	445	10	l.	l.	PROPN
ejde-771	445	11	zhang	zhang	PROPN
ejde-771	445	12	;	;	PUNCT
ejde-771	445	13	existence	existence	NOUN
ejde-771	445	14	and	and	CCONJ
ejde-771	445	15	uniqueness	uniqueness	NOUN
ejde-771	445	16	of	of	ADP
ejde-771	445	17	positive	positive	ADJ
ejde-771	445	18	mild	mild	ADJ
ejde-771	445	19	solutions	solution	NOUN
ejde-771	445	20	for	for	ADP
ejde-771	445	21	a	a	DET
ejde-771	445	22	class	class	NOUN
ejde-771	445	23	of	of	ADP
ejde-771	445	24	fractional	fractional	ADJ
ejde-771	445	25	evolution	evolution	NOUN
ejde-771	445	26	equations	equation	NOUN
ejde-771	445	27	on	on	ADP
ejde-771	445	28	infinite	infinite	ADJ
ejde-771	445	29	interval	interval	NOUN
ejde-771	445	30	.	.	PUNCT
ejde-771	446	1	bound	bind	VERB
ejde-771	446	2	.	.	PUNCT
ejde-771	446	3	value	value	PROPN
ejde-771	446	4	probl	probl	PROPN
ejde-771	446	5	.	.	PUNCT
ejde-771	446	6	,	,	PUNCT
ejde-771	446	7	2017(2017	2017(2017	NUM
ejde-771	446	8	)	)	PUNCT
ejde-771	446	9	,	,	PUNCT
ejde-771	446	10	1	1	NUM
ejde-771	446	11	-	-	SYM
ejde-771	446	12	15	15	NUM
ejde-771	446	13	.	.	PUNCT
ejde-771	447	1	[	[	X
ejde-771	447	2	15	15	NUM
ejde-771	447	3	]	]	X
ejde-771	447	4	k.	k.	PROPN
ejde-771	447	5	deimling	deimling	PROPN
ejde-771	447	6	;	;	PUNCT
ejde-771	447	7	nonlinear	nonlinear	ADJ
ejde-771	447	8	functional	functional	ADJ
ejde-771	447	9	analysis	analysis	NOUN
ejde-771	447	10	.	.	PUNCT
ejde-771	448	1	springer	springer	NOUN
ejde-771	448	2	,	,	PUNCT
ejde-771	448	3	new	new	PROPN
ejde-771	448	4	york	york	PROPN
ejde-771	448	5	,	,	PUNCT
ejde-771	448	6	1985	1985	NUM
ejde-771	448	7	.	.	PUNCT
ejde-771	449	1	[	[	X
ejde-771	449	2	16	16	NUM
ejde-771	449	3	]	]	X
ejde-771	449	4	d.	d.	PROPN
ejde-771	449	5	guo	guo	PROPN
ejde-771	449	6	,	,	PUNCT
ejde-771	449	7	v.	v.	ADP
ejde-771	449	8	lakshmikantham	lakshmikantham	ADV
ejde-771	449	9	;	;	PUNCT
ejde-771	449	10	nonlinear	nonlinear	ADJ
ejde-771	449	11	problems	problem	NOUN
ejde-771	449	12	in	in	ADP
ejde-771	449	13	abstract	abstract	ADJ
ejde-771	449	14	cone	cone	NOUN
ejde-771	449	15	.	.	PUNCT
ejde-771	450	1	academic	academic	ADJ
ejde-771	450	2	press	press	PROPN
ejde-771	450	3	,	,	PUNCT
ejde-771	450	4	orlando	orlando	PROPN
ejde-771	450	5	,	,	PUNCT
ejde-771	450	6	1988	1988	NUM
ejde-771	450	7	.	.	PUNCT
ejde-771	451	1	[	[	X
ejde-771	451	2	17	17	NUM
ejde-771	451	3	]	]	X
ejde-771	451	4	h.	h.	PROPN
ejde-771	451	5	gou	gou	PROPN
ejde-771	451	6	;	;	PUNCT
ejde-771	451	7	positive	positive	ADJ
ejde-771	451	8	solutions	solution	NOUN
ejde-771	451	9	for	for	ADP
ejde-771	451	10	a	a	DET
ejde-771	451	11	class	class	NOUN
ejde-771	451	12	of	of	ADP
ejde-771	451	13	nonlinear	nonlinear	ADJ
ejde-771	451	14	fractional	fractional	ADJ
ejde-771	451	15	differential	differential	ADJ
ejde-771	451	16	equations	equation	NOUN
ejde-771	451	17	with	with	ADP
ejde-771	451	18	derivative	derivative	ADJ
ejde-771	451	19	terms	term	NOUN
ejde-771	451	20	.	.	PUNCT
ejde-771	452	1	rocky	rocky	ADJ
ejde-771	452	2	mountain	mountain	PROPN
ejde-771	452	3	j.	j.	PROPN
ejde-771	452	4	math	math	PROPN
ejde-771	452	5	.	.	PUNCT
ejde-771	452	6	,	,	PUNCT
ejde-771	452	7	52.5(2022	52.5(2022	PROPN
ejde-771	452	8	)	)	PUNCT
ejde-771	452	9	,	,	PUNCT
ejde-771	452	10	1619	1619	NUM
ejde-771	452	11	-	-	SYM
ejde-771	452	12	1641	1641	NUM
ejde-771	452	13	.	.	PUNCT
ejde-771	453	1	[	[	X
ejde-771	453	2	18	18	NUM
ejde-771	453	3	]	]	X
ejde-771	453	4	h.	h.	PROPN
ejde-771	453	5	gou	gou	PROPN
ejde-771	453	6	;	;	PUNCT
ejde-771	453	7	a	a	DET
ejde-771	453	8	study	study	NOUN
ejde-771	453	9	on	on	ADP
ejde-771	453	10	s	s	NOUN
ejde-771	453	11	-	-	PUNCT
ejde-771	453	12	asymptotically	asymptotically	ADV
ejde-771	453	13	ω	ω	ADJ
ejde-771	453	14	-	-	ADJ
ejde-771	453	15	periodic	periodic	ADJ
ejde-771	453	16	positive	positive	ADJ
ejde-771	453	17	mild	mild	ADJ
ejde-771	453	18	solutions	solution	NOUN
ejde-771	453	19	for	for	ADP
ejde-771	453	20	damped	damped	ADJ
ejde-771	453	21	elastic	elastic	ADJ
ejde-771	453	22	systems	system	NOUN
ejde-771	453	23	.	.	PUNCT
ejde-771	454	1	bull	bull	NOUN
ejde-771	454	2	.	.	PUNCT
ejde-771	455	1	sci	sci	PROPN
ejde-771	455	2	.	.	PUNCT
ejde-771	455	3	math	math	PROPN
ejde-771	455	4	.	.	PUNCT
ejde-771	455	5	,	,	PUNCT
ejde-771	455	6	187(2023	187(2023	NUM
ejde-771	455	7	)	)	PUNCT
ejde-771	455	8	,	,	PUNCT
ejde-771	455	9	38pp	38pp	NOUN
ejde-771	455	10	.	.	PUNCT
ejde-771	456	1	[	[	X
ejde-771	456	2	19	19	NUM
ejde-771	456	3	]	]	X
ejde-771	456	4	h.	h.	PROPN
ejde-771	456	5	gou	gou	PROPN
ejde-771	456	6	,	,	PUNCT
ejde-771	456	7	y.	y.	PROPN
ejde-771	456	8	li	li	PROPN
ejde-771	456	9	;	;	PUNCT
ejde-771	456	10	a	a	DET
ejde-771	456	11	study	study	NOUN
ejde-771	456	12	on	on	ADP
ejde-771	456	13	asymptotically	asymptotically	ADV
ejde-771	456	14	periodic	periodic	ADJ
ejde-771	456	15	behavior	behavior	NOUN
ejde-771	456	16	for	for	ADP
ejde-771	456	17	evolution	evolution	NOUN
ejde-771	456	18	equations	equation	NOUN
ejde-771	456	19	with	with	ADP
ejde-771	456	20	delay	delay	NOUN
ejde-771	456	21	in	in	ADP
ejde-771	456	22	banach	banach	NOUN
ejde-771	456	23	spaces	space	NOUN
ejde-771	456	24	.	.	PUNCT
ejde-771	457	1	qual.theory	qual.theory	ADJ
ejde-771	457	2	dyn.syst	dyn.syst	NOUN
ejde-771	457	3	.	.	PUNCT
ejde-771	457	4	,	,	PUNCT
ejde-771	457	5	23.1(2024	23.1(2024	NUM
ejde-771	457	6	)	)	PUNCT
ejde-771	457	7	,	,	PUNCT
ejde-771	457	8	1	1	NUM
ejde-771	457	9	-	-	SYM
ejde-771	457	10	27	27	NUM
ejde-771	457	11	.	.	PUNCT
ejde-771	458	1	[	[	X
ejde-771	458	2	20	20	NUM
ejde-771	458	3	]	]	X
ejde-771	458	4	h.r	h.r	PROPN
ejde-771	458	5	.	.	PROPN
ejde-771	458	6	henŕıquez	henŕıquez	PROPN
ejde-771	458	7	,	,	PUNCT
ejde-771	458	8	m.	m.	NOUN
ejde-771	458	9	pierri	pierri	PROPN
ejde-771	458	10	,	,	PUNCT
ejde-771	458	11	p.	p.	NOUN
ejde-771	458	12	táboas	táboas	PROPN
ejde-771	458	13	;	;	PUNCT
ejde-771	458	14	on	on	ADP
ejde-771	458	15	s	s	NOUN
ejde-771	458	16	-	-	PUNCT
ejde-771	458	17	asymptotically	asymptotically	ADV
ejde-771	458	18	ω	ω	ADJ
ejde-771	458	19	-	-	ADJ
ejde-771	458	20	periodic	periodic	ADJ
ejde-771	458	21	functions	function	NOUN
ejde-771	458	22	on	on	ADP
ejde-771	458	23	banach	banach	NOUN
ejde-771	458	24	spaces	space	NOUN
ejde-771	458	25	and	and	CCONJ
ejde-771	458	26	applications	application	NOUN
ejde-771	458	27	.	.	PUNCT
ejde-771	459	1	j.	j.	PROPN
ejde-771	459	2	math	math	PROPN
ejde-771	459	3	.	.	PUNCT
ejde-771	460	1	anal	anal	PROPN
ejde-771	460	2	.	.	PUNCT
ejde-771	461	1	appl	appl	PROPN
ejde-771	461	2	.	.	PROPN
ejde-771	461	3	,	,	PUNCT
ejde-771	461	4	343(2008	343(2008	NOUN
ejde-771	461	5	)	)	PUNCT
ejde-771	461	6	,	,	PUNCT
ejde-771	461	7	1119	1119	NUM
ejde-771	461	8	-	-	SYM
ejde-771	461	9	1130	1130	NUM
ejde-771	461	10	.	.	PUNCT
ejde-771	462	1	[	[	X
ejde-771	462	2	21	21	NUM
ejde-771	462	3	]	]	PUNCT
ejde-771	462	4	s.	s.	PROPN
ejde-771	462	5	hussain	hussain	PROPN
ejde-771	462	6	,	,	PUNCT
ejde-771	462	7	m.	m.	NOUN
ejde-771	462	8	sarwar	sarwar	PROPN
ejde-771	462	9	,	,	PUNCT
ejde-771	462	10	k.	k.	PROPN
ejde-771	462	11	s.	s.	PROPN
ejde-771	462	12	nisar	nisar	PROPN
ejde-771	462	13	,	,	PUNCT
ejde-771	462	14	k.	k.	PROPN
ejde-771	462	15	shah	shah	PROPN
ejde-771	462	16	;	;	PUNCT
ejde-771	462	17	controllability	controllability	NOUN
ejde-771	462	18	of	of	ADP
ejde-771	462	19	fractional	fractional	ADJ
ejde-771	462	20	differential	differential	ADJ
ejde-771	462	21	evolution	evolution	NOUN
ejde-771	462	22	equation	equation	NOUN
ejde-771	462	23	of	of	ADP
ejde-771	462	24	order	order	NOUN
ejde-771	462	25	γ	γ	X
ejde-771	462	26	∈	∈	X
ejde-771	462	27	(	(	PUNCT
ejde-771	462	28	1	1	NUM
ejde-771	462	29	,	,	PUNCT
ejde-771	462	30	2	2	NUM
ejde-771	462	31	)	)	PUNCT
ejde-771	462	32	with	with	ADP
ejde-771	462	33	nonlocal	nonlocal	ADJ
ejde-771	462	34	conditions	condition	NOUN
ejde-771	462	35	.	.	PUNCT
ejde-771	462	36	aims	aim	VERB
ejde-771	462	37	math	math	NOUN
ejde-771	462	38	.	.	PUNCT
ejde-771	462	39	,	,	PUNCT
ejde-771	462	40	8.6(2023	8.6(2023	NUM
ejde-771	462	41	)	)	PUNCT
ejde-771	462	42	,	,	PUNCT
ejde-771	462	43	14188	14188	NUM
ejde-771	462	44	-	-	SYM
ejde-771	462	45	14206	14206	NUM
ejde-771	462	46	.	.	PUNCT
ejde-771	463	1	[	[	X
ejde-771	463	2	22	22	NUM
ejde-771	463	3	]	]	X
ejde-771	463	4	l.	l.	PROPN
ejde-771	463	5	m.	m.	PROPN
ejde-771	463	6	issaka	issaka	PROPN
ejde-771	463	7	,	,	PUNCT
ejde-771	463	8	a.	a.	PROPN
ejde-771	463	9	diop	diop	PROPN
ejde-771	463	10	,	,	PUNCT
ejde-771	463	11	m.	m.	NOUN
ejde-771	463	12	niang	niang	PROPN
ejde-771	463	13	,	,	PUNCT
ejde-771	463	14	m.	m.	NOUN
ejde-771	463	15	a.	a.	PROPN
ejde-771	463	16	diop	diop	PROPN
ejde-771	463	17	;	;	PUNCT
ejde-771	463	18	on	on	ADP
ejde-771	463	19	s	s	NOUN
ejde-771	463	20	-	-	PUNCT
ejde-771	463	21	asymptotically	asymptotically	ADV
ejde-771	463	22	ω	ω	ADJ
ejde-771	463	23	-	-	ADJ
ejde-771	463	24	periodic	periodic	ADJ
ejde-771	463	25	mild	mild	ADJ
ejde-771	463	26	solutions	solution	NOUN
ejde-771	463	27	of	of	ADP
ejde-771	463	28	some	some	DET
ejde-771	463	29	integrodifferential	integrodifferential	ADJ
ejde-771	463	30	inclusions	inclusion	NOUN
ejde-771	463	31	of	of	ADP
ejde-771	463	32	volterra	volterra	NOUN
ejde-771	463	33	-	-	PUNCT
ejde-771	463	34	type	type	NOUN
ejde-771	463	35	.	.	PUNCT
ejde-771	464	1	j.	j.	PROPN
ejde-771	464	2	anal	anal	PROPN
ejde-771	464	3	.	.	PROPN
ejde-771	464	4	,	,	PUNCT
ejde-771	464	5	31.4(2023	31.4(2023	NUM
ejde-771	464	6	)	)	PUNCT
ejde-771	464	7	,	,	PUNCT
ejde-771	464	8	2943	2943	NUM
ejde-771	464	9	-	-	SYM
ejde-771	464	10	2972	2972	NUM
ejde-771	464	11	.	.	PUNCT
ejde-771	465	1	[	[	X
ejde-771	465	2	23	23	NUM
ejde-771	465	3	]	]	X
ejde-771	465	4	q.	q.	PROPN
ejde-771	465	5	li	li	PROPN
ejde-771	465	6	,	,	PUNCT
ejde-771	465	7	l.	l.	PROPN
ejde-771	465	8	liu	liu	PROPN
ejde-771	465	9	,	,	PUNCT
ejde-771	465	10	m.	m.	PROPN
ejde-771	465	11	wei	wei	PROPN
ejde-771	465	12	;	;	PUNCT
ejde-771	465	13	existence	existence	NOUN
ejde-771	465	14	of	of	ADP
ejde-771	465	15	positive	positive	ADJ
ejde-771	465	16	s	s	NOUN
ejde-771	465	17	-	-	PUNCT
ejde-771	465	18	asymptotically	asymptotically	ADV
ejde-771	465	19	periodic	periodic	ADJ
ejde-771	465	20	solutions	solution	NOUN
ejde-771	465	21	of	of	ADP
ejde-771	465	22	the	the	DET
ejde-771	465	23	fractional	fractional	ADJ
ejde-771	465	24	evolution	evolution	NOUN
ejde-771	465	25	equations	equation	NOUN
ejde-771	465	26	in	in	ADP
ejde-771	465	27	ordered	order	VERB
ejde-771	465	28	banach	banach	NOUN
ejde-771	465	29	spaces	space	VERB
ejde-771	465	30	.	.	PUNCT
ejde-771	466	1	nonlinear	nonlinear	ADJ
ejde-771	466	2	anal	anal	PROPN
ejde-771	466	3	.	.	PUNCT
ejde-771	467	1	model	model	PROPN
ejde-771	467	2	.	.	PUNCT
ejde-771	468	1	control	control	PROPN
ejde-771	468	2	.	.	PUNCT
ejde-771	468	3	,	,	PUNCT
ejde-771	468	4	26.5(2021	26.5(2021	NUM
ejde-771	468	5	)	)	PUNCT
ejde-771	468	6	,	,	PUNCT
ejde-771	468	7	928	928	NUM
ejde-771	468	8	-	-	SYM
ejde-771	468	9	946	946	NUM
ejde-771	468	10	.	.	PUNCT
ejde-771	469	1	ejde-2025/44	ejde-2025/44	ADJ
ejde-771	469	2	time	time	NOUN
ejde-771	469	3	-	-	PUNCT
ejde-771	469	4	space	space	NOUN
ejde-771	469	5	fractional	fractional	ADJ
ejde-771	469	6	reaction	reaction	NOUN
ejde-771	469	7	-	-	PUNCT
ejde-771	469	8	diffusion	diffusion	NOUN
ejde-771	469	9	equations	equation	NOUN
ejde-771	469	10	15	15	NUM
ejde-771	470	1	[	[	X
ejde-771	470	2	24	24	NUM
ejde-771	470	3	]	]	PUNCT
ejde-771	470	4	q.	q.	PROPN
ejde-771	470	5	li	li	PROPN
ejde-771	470	6	,	,	PUNCT
ejde-771	470	7	l.	l.	PROPN
ejde-771	470	8	liu	liu	PROPN
ejde-771	470	9	,	,	PUNCT
ejde-771	470	10	m.	m.	PROPN
ejde-771	470	11	wei	wei	PROPN
ejde-771	470	12	;	;	PUNCT
ejde-771	470	13	s	s	X
ejde-771	470	14	-	-	PUNCT
ejde-771	470	15	asymptotically	asymptotically	ADV
ejde-771	470	16	periodic	periodic	ADJ
ejde-771	470	17	solutions	solution	NOUN
ejde-771	470	18	for	for	ADP
ejde-771	470	19	time	time	NOUN
ejde-771	470	20	-	-	PUNCT
ejde-771	470	21	space	space	NOUN
ejde-771	470	22	fractional	fractional	ADJ
ejde-771	470	23	evolution	evolution	NOUN
ejde-771	470	24	equation	equation	NOUN
ejde-771	470	25	.	.	PUNCT
ejde-771	471	1	mediterr	mediterr	PROPN
ejde-771	471	2	.	.	PUNCT
ejde-771	472	1	j.	j.	PROPN
ejde-771	472	2	math	math	PROPN
ejde-771	472	3	.	.	PROPN
ejde-771	472	4	,	,	PUNCT
ejde-771	472	5	18(2021	18(2021	NUM
ejde-771	472	6	)	)	PUNCT
ejde-771	472	7	,	,	PUNCT
ejde-771	472	8	21	21	NUM
ejde-771	472	9	pp	pp	NOUN
ejde-771	472	10	.	.	PUNCT
ejde-771	473	1	[	[	X
ejde-771	473	2	25	25	NUM
ejde-771	473	3	]	]	X
ejde-771	473	4	s.	s.	PROPN
ejde-771	473	5	li	li	PROPN
ejde-771	473	6	,	,	PUNCT
ejde-771	473	7	c.	c.	PROPN
ejde-771	473	8	zhai	zhai	PROPN
ejde-771	473	9	;	;	PUNCT
ejde-771	473	10	positive	positive	ADJ
ejde-771	473	11	solutions	solution	NOUN
ejde-771	473	12	for	for	ADP
ejde-771	473	13	a	a	DET
ejde-771	473	14	new	new	ADJ
ejde-771	473	15	class	class	NOUN
ejde-771	473	16	of	of	ADP
ejde-771	473	17	hadamard	hadamard	ADJ
ejde-771	473	18	fractional	fractional	ADJ
ejde-771	473	19	differential	differential	ADJ
ejde-771	473	20	equations	equation	NOUN
ejde-771	473	21	on	on	ADP
ejde-771	473	22	infinite	infinite	ADJ
ejde-771	473	23	intervals	interval	NOUN
ejde-771	473	24	.	.	PUNCT
ejde-771	474	1	j.	j.	PROPN
ejde-771	474	2	inequal	inequal	PROPN
ejde-771	474	3	.	.	PUNCT
ejde-771	475	1	appl	appl	PROPN
ejde-771	475	2	.	.	PROPN
ejde-771	475	3	,	,	PUNCT
ejde-771	475	4	2019(2019	2019(2019	NUM
ejde-771	475	5	)	)	PUNCT
ejde-771	475	6	,	,	PUNCT
ejde-771	475	7	1	1	NUM
ejde-771	475	8	-	-	SYM
ejde-771	475	9	9	9	NUM
ejde-771	475	10	.	.	PUNCT
ejde-771	476	1	[	[	X
ejde-771	476	2	26	26	NUM
ejde-771	476	3	]	]	PUNCT
ejde-771	476	4	j.	j.	PROPN
ejde-771	476	5	mu	mu	PROPN
ejde-771	476	6	,	,	PUNCT
ejde-771	476	7	j.	j.	PROPN
ejde-771	476	8	nan	nan	PROPN
ejde-771	476	9	,	,	PUNCT
ejde-771	476	10	y.	y.	PROPN
ejde-771	476	11	zhou	zhou	PROPN
ejde-771	476	12	,	,	PUNCT
ejde-771	476	13	existence	existence	NOUN
ejde-771	476	14	of	of	ADP
ejde-771	476	15	periodic	periodic	ADJ
ejde-771	476	16	and	and	CCONJ
ejde-771	476	17	s	s	NOUN
ejde-771	476	18	-	-	PUNCT
ejde-771	476	19	asymptotically	asymptotically	ADV
ejde-771	476	20	periodic	periodic	ADJ
ejde-771	476	21	solutions	solution	NOUN
ejde-771	476	22	to	to	ADP
ejde-771	476	23	fractional	fractional	ADJ
ejde-771	476	24	diffusion	diffusion	NOUN
ejde-771	476	25	equations	equation	NOUN
ejde-771	476	26	with	with	ADP
ejde-771	476	27	analytic	analytic	ADJ
ejde-771	476	28	semigroups	semigroup	NOUN
ejde-771	476	29	.	.	PUNCT
ejde-771	476	30	math	math	NOUN
ejde-771	476	31	.	.	PUNCT
ejde-771	477	1	methods	method	NOUN
ejde-771	477	2	appl	appl	PROPN
ejde-771	477	3	.	.	PUNCT
ejde-771	478	1	sci	sci	PROPN
ejde-771	478	2	.	.	PROPN
ejde-771	478	3	,	,	PUNCT
ejde-771	478	4	44.3(2021	44.3(2021	NUM
ejde-771	478	5	)	)	PUNCT
ejde-771	478	6	,	,	PUNCT
ejde-771	478	7	2393	2393	NUM
ejde-771	478	8	-	-	SYM
ejde-771	478	9	2404	2404	NUM
ejde-771	478	10	.	.	PUNCT
ejde-771	479	1	[	[	X
ejde-771	479	2	27	27	NUM
ejde-771	479	3	]	]	X
ejde-771	479	4	r.h	r.h	PROPN
ejde-771	479	5	.	.	PROPN
ejde-771	479	6	martin	martin	PROPN
ejde-771	479	7	jr	jr	PROPN
ejde-771	479	8	.	.	PROPN
ejde-771	479	9	;	;	PUNCT
ejde-771	479	10	nonlinear	nonlinear	ADJ
ejde-771	479	11	operators	operator	NOUN
ejde-771	479	12	and	and	CCONJ
ejde-771	479	13	differential	differential	ADJ
ejde-771	479	14	equations	equation	NOUN
ejde-771	479	15	in	in	ADP
ejde-771	479	16	banach	banach	NOUN
ejde-771	479	17	spaces	space	NOUN
ejde-771	479	18	.	.	PUNCT
ejde-771	480	1	krieger	krieger	PROPN
ejde-771	480	2	,	,	PUNCT
ejde-771	480	3	malabar	malabar	PROPN
ejde-771	480	4	,	,	PUNCT
ejde-771	480	5	fl	fl	PROPN
ejde-771	480	6	,	,	PUNCT
ejde-771	480	7	1986	1986	NUM
ejde-771	480	8	.	.	PUNCT
ejde-771	481	1	[	[	X
ejde-771	481	2	28	28	NUM
ejde-771	481	3	]	]	X
ejde-771	481	4	a.	a.	NOUN
ejde-771	481	5	pazy	pazy	NOUN
ejde-771	481	6	;	;	PUNCT
ejde-771	481	7	semigroup	semigroup	NOUN
ejde-771	481	8	of	of	ADP
ejde-771	481	9	linear	linear	PROPN
ejde-771	481	10	operators	operator	NOUN
ejde-771	481	11	and	and	CCONJ
ejde-771	481	12	applications	application	NOUN
ejde-771	481	13	to	to	ADP
ejde-771	481	14	partial	partial	ADJ
ejde-771	481	15	differential	differential	NOUN
ejde-771	481	16	equations	equation	NOUN
ejde-771	481	17	.	.	PUNCT
ejde-771	482	1	springer	springer	NOUN
ejde-771	482	2	,	,	PUNCT
ejde-771	482	3	new	new	PROPN
ejde-771	482	4	york	york	PROPN
ejde-771	482	5	,	,	PUNCT
ejde-771	482	6	1993	1993	NUM
ejde-771	482	7	.	.	PUNCT
ejde-771	483	1	[	[	X
ejde-771	483	2	29	29	NUM
ejde-771	483	3	]	]	PUNCT
ejde-771	483	4	x.	x.	PROPN
ejde-771	483	5	shu	shu	PROPN
ejde-771	483	6	,	,	PUNCT
ejde-771	483	7	f.	f.	PROPN
ejde-771	483	8	xu	xu	PROPN
ejde-771	483	9	,	,	PUNCT
ejde-771	483	10	y.	y.	PROPN
ejde-771	483	11	shi	shi	PROPN
ejde-771	483	12	;	;	PUNCT
ejde-771	483	13	s	s	X
ejde-771	483	14	-	-	PUNCT
ejde-771	483	15	asymptotically	asymptotically	ADV
ejde-771	483	16	ω	ω	ADJ
ejde-771	483	17	-	-	ADJ
ejde-771	483	18	positive	positive	ADJ
ejde-771	483	19	periodic	periodic	ADJ
ejde-771	483	20	solutions	solution	NOUN
ejde-771	483	21	for	for	ADP
ejde-771	483	22	a	a	DET
ejde-771	483	23	class	class	NOUN
ejde-771	483	24	of	of	ADP
ejde-771	483	25	neutral	neutral	ADJ
ejde-771	483	26	fractional	fractional	ADJ
ejde-771	483	27	differential	differential	NOUN
ejde-771	483	28	equations	equation	NOUN
ejde-771	483	29	.	.	PUNCT
ejde-771	484	1	appl	appl	PROPN
ejde-771	484	2	.	.	PROPN
ejde-771	484	3	math	math	PROPN
ejde-771	484	4	.	.	PUNCT
ejde-771	485	1	comput	comput	NOUN
ejde-771	485	2	.	.	PUNCT
ejde-771	485	3	,	,	PUNCT
ejde-771	485	4	270(2015	270(2015	NUM
ejde-771	485	5	)	)	PUNCT
ejde-771	485	6	,	,	PUNCT
ejde-771	485	7	768	768	NUM
ejde-771	485	8	-	-	SYM
ejde-771	485	9	776	776	NUM
ejde-771	485	10	.	.	PUNCT
ejde-771	486	1	[	[	X
ejde-771	486	2	30	30	NUM
ejde-771	486	3	]	]	X
ejde-771	486	4	r.	r.	NOUN
ejde-771	486	5	triggiani	triggiani	PROPN
ejde-771	486	6	;	;	PUNCT
ejde-771	486	7	on	on	ADP
ejde-771	486	8	the	the	DET
ejde-771	486	9	stabilizability	stabilizability	NOUN
ejde-771	486	10	problem	problem	NOUN
ejde-771	486	11	in	in	ADP
ejde-771	486	12	banach	banach	NOUN
ejde-771	486	13	spaces	space	NOUN
ejde-771	486	14	.	.	PUNCT
ejde-771	487	1	j.	j.	PROPN
ejde-771	487	2	math	math	PROPN
ejde-771	487	3	.	.	PUNCT
ejde-771	488	1	anal	anal	PROPN
ejde-771	488	2	.	.	PUNCT
ejde-771	489	1	appl	appl	PROPN
ejde-771	489	2	.	.	PROPN
ejde-771	489	3	,	,	PUNCT
ejde-771	489	4	52(1975	52(1975	NUM
ejde-771	489	5	)	)	PUNCT
ejde-771	489	6	,	,	PUNCT
ejde-771	489	7	383	383	NUM
ejde-771	489	8	-	-	SYM
ejde-771	489	9	403	403	NUM
ejde-771	489	10	.	.	PUNCT
ejde-771	490	1	[	[	X
ejde-771	490	2	31	31	NUM
ejde-771	490	3	]	]	PUNCT
ejde-771	490	4	r.	r.	PROPN
ejde-771	490	5	wang	wang	PROPN
ejde-771	490	6	,	,	PUNCT
ejde-771	490	7	d.	d.	PROPN
ejde-771	490	8	chen	chen	PROPN
ejde-771	490	9	,	,	PUNCT
ejde-771	490	10	t.	t.	PROPN
ejde-771	490	11	xiao	xiao	PROPN
ejde-771	490	12	;	;	PUNCT
ejde-771	490	13	abstract	abstract	ADJ
ejde-771	490	14	fractional	fractional	ADJ
ejde-771	490	15	cauchy	cauchy	NOUN
ejde-771	490	16	problems	problem	NOUN
ejde-771	490	17	with	with	ADP
ejde-771	490	18	almost	almost	ADV
ejde-771	490	19	sectorial	sectorial	ADJ
ejde-771	490	20	operators	operator	NOUN
ejde-771	490	21	.	.	PUNCT
ejde-771	491	1	j.	j.	PROPN
ejde-771	491	2	differential	differential	PROPN
ejde-771	491	3	equations	equations	PROPN
ejde-771	491	4	,	,	PUNCT
ejde-771	491	5	252(2012	252(2012	NUM
ejde-771	491	6	)	)	PUNCT
ejde-771	491	7	,	,	PUNCT
ejde-771	491	8	202	202	NUM
ejde-771	491	9	-	-	SYM
ejde-771	491	10	235	235	NUM
ejde-771	491	11	.	.	PUNCT
ejde-771	492	1	[	[	X
ejde-771	492	2	32	32	NUM
ejde-771	492	3	]	]	PUNCT
ejde-771	492	4	j.	j.	PROPN
ejde-771	492	5	wang	wang	PROPN
ejde-771	492	6	,	,	PUNCT
ejde-771	492	7	y.	y.	PROPN
ejde-771	492	8	zhou	zhou	PROPN
ejde-771	492	9	;	;	PUNCT
ejde-771	492	10	a	a	DET
ejde-771	492	11	class	class	NOUN
ejde-771	492	12	of	of	ADP
ejde-771	492	13	fractional	fractional	ADJ
ejde-771	492	14	evolution	evolution	NOUN
ejde-771	492	15	equations	equation	NOUN
ejde-771	492	16	and	and	CCONJ
ejde-771	492	17	optimal	optimal	ADJ
ejde-771	492	18	controls	control	NOUN
ejde-771	492	19	.	.	PUNCT
ejde-771	493	1	nonlinear	nonlinear	ADJ
ejde-771	493	2	anal	anal	PROPN
ejde-771	493	3	.	.	PUNCT
ejde-771	494	1	real	real	ADJ
ejde-771	494	2	world	world	NOUN
ejde-771	494	3	appl	appl	PROPN
ejde-771	494	4	.	.	PROPN
ejde-771	494	5	,	,	PUNCT
ejde-771	494	6	12(2011	12(2011	NUM
ejde-771	494	7	)	)	PUNCT
ejde-771	494	8	,	,	PUNCT
ejde-771	494	9	262	262	NUM
ejde-771	494	10	-	-	SYM
ejde-771	494	11	272	272	NUM
ejde-771	494	12	.	.	PUNCT
ejde-771	495	1	[	[	X
ejde-771	495	2	33	33	NUM
ejde-771	495	3	]	]	PUNCT
ejde-771	495	4	m.	m.	PROPN
ejde-771	495	5	wei	wei	PROPN
ejde-771	495	6	,	,	PUNCT
ejde-771	495	7	y.	y.	PROPN
ejde-771	495	8	li	li	PROPN
ejde-771	495	9	,	,	PUNCT
ejde-771	495	10	q.	q.	PROPN
ejde-771	495	11	li	li	PROPN
ejde-771	495	12	;	;	PUNCT
ejde-771	495	13	positive	positive	ADJ
ejde-771	495	14	mild	mild	ADJ
ejde-771	495	15	solutions	solution	NOUN
ejde-771	495	16	for	for	ADP
ejde-771	495	17	damped	damped	ADJ
ejde-771	495	18	elastic	elastic	ADJ
ejde-771	495	19	systems	system	NOUN
ejde-771	495	20	with	with	ADP
ejde-771	495	21	delay	delay	NOUN
ejde-771	495	22	and	and	CCONJ
ejde-771	495	23	nonlocal	nonlocal	ADJ
ejde-771	495	24	conditions	condition	NOUN
ejde-771	495	25	in	in	ADP
ejde-771	495	26	ordered	order	VERB
ejde-771	495	27	banach	banach	NOUN
ejde-771	495	28	space	space	NOUN
ejde-771	495	29	.	.	PUNCT
ejde-771	496	1	qual	qual	X
ejde-771	496	2	.	.	PUNCT
ejde-771	496	3	theory	theory	NOUN
ejde-771	496	4	dyn	dyn	PROPN
ejde-771	496	5	.	.	PUNCT
ejde-771	497	1	syst	syst	PROPN
ejde-771	497	2	.	.	PROPN
ejde-771	497	3	,	,	PUNCT
ejde-771	497	4	21.4(2022	21.4(2022	NUM
ejde-771	497	5	)	)	PUNCT
ejde-771	497	6	,	,	PUNCT
ejde-771	497	7	22	22	NUM
ejde-771	497	8	pp	pp	NOUN
ejde-771	497	9	.	.	PUNCT
ejde-771	498	1	[	[	X
ejde-771	498	2	34	34	NUM
ejde-771	498	3	]	]	X
ejde-771	498	4	h.	h.	PROPN
ejde-771	498	5	yang	yang	PROPN
ejde-771	498	6	,	,	PUNCT
ejde-771	498	7	y.	y.	PROPN
ejde-771	498	8	zhao	zhao	PROPN
ejde-771	498	9	;	;	PUNCT
ejde-771	498	10	existence	existence	NOUN
ejde-771	498	11	and	and	CCONJ
ejde-771	498	12	optimal	optimal	ADJ
ejde-771	498	13	controls	control	NOUN
ejde-771	498	14	of	of	ADP
ejde-771	498	15	non	non	ADJ
ejde-771	498	16	-	-	ADJ
ejde-771	498	17	autonomous	autonomous	ADJ
ejde-771	498	18	impulsive	impulsive	ADJ
ejde-771	498	19	integro	integro	ADJ
ejde-771	498	20	-	-	PUNCT
ejde-771	498	21	differential	differential	NOUN
ejde-771	498	22	evolution	evolution	NOUN
ejde-771	498	23	equation	equation	NOUN
ejde-771	498	24	with	with	ADP
ejde-771	498	25	nonlocal	nonlocal	ADJ
ejde-771	498	26	conditions	condition	NOUN
ejde-771	498	27	.	.	PUNCT
ejde-771	499	1	chaos	chaos	NOUN
ejde-771	499	2	solitons	soliton	NOUN
ejde-771	499	3	fractals	fractal	NOUN
ejde-771	499	4	,	,	PUNCT
ejde-771	499	5	148(2021	148(2021	NUM
ejde-771	499	6	)	)	PUNCT
ejde-771	499	7	,	,	PUNCT
ejde-771	499	8	9pp	9pp	NOUN
ejde-771	499	9	.	.	PUNCT
ejde-771	500	1	[	[	X
ejde-771	500	2	35	35	NUM
ejde-771	500	3	]	]	PUNCT
ejde-771	500	4	k.	k.	PROPN
ejde-771	500	5	yosida	yosida	PROPN
ejde-771	500	6	;	;	PUNCT
ejde-771	500	7	functional	functional	ADJ
ejde-771	500	8	analysis	analysis	NOUN
ejde-771	500	9	.	.	PUNCT
ejde-771	501	1	berlin	berlin	ADJ
ejde-771	501	2	:	:	PUNCT
ejde-771	501	3	springer	springer	NOUN
ejde-771	501	4	-	-	PUNCT
ejde-771	501	5	verlag	verlag	PROPN
ejde-771	501	6	,	,	PUNCT
ejde-771	501	7	1965	1965	NUM
ejde-771	501	8	.	.	PUNCT
ejde-771	502	1	[	[	X
ejde-771	502	2	36	36	NUM
ejde-771	502	3	]	]	X
ejde-771	502	4	y.	y.	PROPN
ejde-771	502	5	zhou	zhou	PROPN
ejde-771	502	6	,	,	PUNCT
ejde-771	502	7	f.	f.	PROPN
ejde-771	502	8	jiao	jiao	PROPN
ejde-771	502	9	;	;	PUNCT
ejde-771	502	10	existence	existence	NOUN
ejde-771	502	11	of	of	ADP
ejde-771	502	12	mild	mild	ADJ
ejde-771	502	13	solutions	solution	NOUN
ejde-771	502	14	for	for	ADP
ejde-771	502	15	fractional	fractional	ADJ
ejde-771	502	16	neutral	neutral	ADJ
ejde-771	502	17	evolution	evolution	NOUN
ejde-771	502	18	equations	equation	NOUN
ejde-771	502	19	.	.	PUNCT
ejde-771	503	1	comput	comput	NOUN
ejde-771	503	2	.	.	PUNCT
ejde-771	504	1	math	math	NOUN
ejde-771	504	2	.	.	PUNCT
ejde-771	505	1	appl	appl	PROPN
ejde-771	505	2	.	.	PROPN
ejde-771	505	3	,	,	PUNCT
ejde-771	505	4	59(2010	59(2010	NOUN
ejde-771	505	5	)	)	PUNCT
ejde-771	505	6	,	,	PUNCT
ejde-771	505	7	1063	1063	NUM
ejde-771	505	8	-	-	SYM
ejde-771	505	9	1077	1077	NUM
ejde-771	505	10	.	.	PUNCT
ejde-771	506	1	xuping	xupe	VERB
ejde-771	506	2	zhang	zhang	PROPN
ejde-771	506	3	(	(	PUNCT
ejde-771	506	4	corresponding	correspond	VERB
ejde-771	506	5	author	author	NOUN
ejde-771	506	6	)	)	PUNCT
ejde-771	506	7	department	department	NOUN
ejde-771	506	8	of	of	ADP
ejde-771	506	9	mathematics	mathematic	NOUN
ejde-771	506	10	,	,	PUNCT
ejde-771	506	11	northwest	northwest	PROPN
ejde-771	506	12	normal	normal	ADJ
ejde-771	506	13	university	university	NOUN
ejde-771	506	14	,	,	PUNCT
ejde-771	506	15	lanzhou	lanzhou	PROPN
ejde-771	506	16	730070	730070	NUM
ejde-771	506	17	,	,	PUNCT
ejde-771	506	18	china	china	PROPN
ejde-771	506	19	email	email	NOUN
ejde-771	506	20	address	address	NOUN
ejde-771	506	21	:	:	PUNCT
ejde-771	506	22	lanyu9986@126.com	lanyu9986@126.com	PROPN
ejde-771	506	23	kaibo	kaibo	PROPN
ejde-771	506	24	ding	ding	PROPN
ejde-771	506	25	department	department	PROPN
ejde-771	506	26	of	of	ADP
ejde-771	506	27	mathematics	mathematic	NOUN
ejde-771	506	28	,	,	PUNCT
ejde-771	506	29	northwest	northwest	PROPN
ejde-771	506	30	normal	normal	ADJ
ejde-771	506	31	university	university	NOUN
ejde-771	506	32	,	,	PUNCT
ejde-771	506	33	lanzhou	lanzhou	PROPN
ejde-771	506	34	730070	730070	NUM
ejde-771	506	35	,	,	PUNCT
ejde-771	506	36	china	china	PROPN
ejde-771	506	37	email	email	NOUN
ejde-771	506	38	address	address	NOUN
ejde-771	506	39	:	:	PUNCT
ejde-771	507	1	dingkb583x@163.com	dingkb583x@163.com	X
ejde-771	507	2	pengyu	pengyu	PROPN
ejde-771	507	3	chen	chen	PROPN
ejde-771	507	4	department	department	PROPN
ejde-771	507	5	of	of	ADP
ejde-771	507	6	mathematics	mathematics	PROPN
ejde-771	507	7	,	,	PUNCT
ejde-771	507	8	northwest	northwest	PROPN
ejde-771	507	9	normal	normal	ADJ
ejde-771	507	10	university	university	NOUN
ejde-771	507	11	,	,	PUNCT
ejde-771	507	12	lanzhou	lanzhou	PROPN
ejde-771	507	13	730070	730070	NUM
ejde-771	507	14	,	,	PUNCT
ejde-771	507	15	china	china	PROPN
ejde-771	507	16	.	.	PUNCT
ejde-771	508	1	gansu	gansu	PROPN
ejde-771	508	2	provincial	provincial	ADJ
ejde-771	508	3	research	research	NOUN
ejde-771	508	4	center	center	NOUN
ejde-771	508	5	for	for	ADP
ejde-771	508	6	basic	basic	ADJ
ejde-771	508	7	disciplines	discipline	NOUN
ejde-771	508	8	of	of	ADP
ejde-771	508	9	mathematics	mathematic	NOUN
ejde-771	508	10	and	and	CCONJ
ejde-771	508	11	statistics	statistic	NOUN
ejde-771	508	12	,	,	PUNCT
ejde-771	508	13	lanzhou	lanzhou	PROPN
ejde-771	508	14	730070	730070	NUM
ejde-771	508	15	,	,	PUNCT
ejde-771	508	16	china	china	PROPN
ejde-771	508	17	email	email	NOUN
ejde-771	508	18	address	address	PROPN
ejde-771	508	19	:	:	PUNCT
ejde-771	508	20	chpengyu123@163.com	chpengyu123@163.com	X
ejde-771	508	21	1	1	X
ejde-771	508	22	.	.	PUNCT
ejde-771	508	23	introduction	introduction	NOUN
ejde-771	508	24	2	2	NUM
ejde-771	508	25	.	.	PUNCT
ejde-771	508	26	preliminaries	preliminary	NOUN
ejde-771	508	27	3	3	NUM
ejde-771	508	28	.	.	PUNCT
ejde-771	509	1	abstract	abstract	ADJ
ejde-771	509	2	results	result	VERB
ejde-771	509	3	4	4	NUM
ejde-771	509	4	.	.	X
ejde-771	510	1	application	application	NOUN
ejde-771	510	2	to	to	ADP
ejde-771	510	3	nonlocal	nonlocal	ADJ
ejde-771	510	4	problem	problem	NOUN
ejde-771	510	5	(	(	PUNCT
ejde-771	510	6	1.1	1.1	NUM
ejde-771	510	7	)	)	PUNCT
ejde-771	510	8	acknowledgments	acknowledgment	NOUN
ejde-771	510	9	references	reference	NOUN