id	sid	tid	token	lemma	pos
cana-6124	1	1	communications	communication	NOUN
cana-6124	1	2	on	on	ADP
cana-6124	1	3	applied	apply	VERB
cana-6124	1	4	nonlinear	nonlinear	ADJ
cana-6124	1	5	analysis	analysis	NOUN
cana-6124	1	6	issn	issn	NOUN
cana-6124	1	7	:	:	PUNCT
cana-6124	1	8	1074	1074	NUM
cana-6124	1	9	-	-	PUNCT
cana-6124	1	10	133x	133x	NUM
cana-6124	1	11	vol	vol	VERB
cana-6124	1	12	32	32	NUM
cana-6124	1	13	no	no	NOUN
cana-6124	1	14	.	.	PUNCT
cana-6124	2	1	10s	10	NOUN
cana-6124	2	2	(	(	PUNCT
cana-6124	2	3	2025	2025	NUM
cana-6124	2	4	)	)	PUNCT
cana-6124	2	5	3426	3426	NUM
cana-6124	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	3	1	the	the	DET
cana-6124	3	2	cauchy	cauchy	PROPN
cana-6124	3	3	problems	problem	NOUN
cana-6124	3	4	for	for	ADP
cana-6124	3	5	fractional	fractional	ADJ
cana-6124	3	6	q	q	ADJ
cana-6124	3	7	-	-	PUNCT
cana-6124	3	8	difference	difference	NOUN
cana-6124	3	9	equations	equation	NOUN
cana-6124	3	10	with	with	ADP
cana-6124	3	11	integral	integral	ADJ
cana-6124	3	12	conditions	condition	NOUN
cana-6124	3	13	in	in	ADP
cana-6124	3	14	banach	banach	NOUN
cana-6124	3	15	spaces	space	NOUN
cana-6124	3	16	faouzi	faouzi	PRON
cana-6124	3	17	hireche	hireche	NOUN
cana-6124	3	18	abdelhamid	abdelhamid	PROPN
cana-6124	3	19	ibn	ibn	PROPN
cana-6124	3	20	badis	badis	PROPN
cana-6124	3	21	university	university	PROPN
cana-6124	3	22	,	,	PUNCT
cana-6124	3	23	mostaganem	mostaganem	PROPN
cana-6124	3	24	27000	27000	NUM
cana-6124	3	25	,	,	PUNCT
cana-6124	3	26	algeria	algeria	PROPN
cana-6124	3	27	.	.	PUNCT
cana-6124	3	28	faouzi.hireche.ma@gmail.com	faouzi.hireche.ma@gmail.com	PROPN
cana-6124	3	29	;	;	PUNCT
cana-6124	3	30	faouzi.hireche@univ-mosta.dz	faouzi.hireche@univ-mosta.dz	NOUN
cana-6124	3	31	article	article	NOUN
cana-6124	3	32	history	history	NOUN
cana-6124	3	33	:	:	PUNCT
cana-6124	3	34	received	receive	VERB
cana-6124	3	35	:	:	PUNCT
cana-6124	3	36	19	19	NUM
cana-6124	3	37	-	-	SYM
cana-6124	3	38	08	08	NUM
cana-6124	3	39	-	-	PUNCT
cana-6124	3	40	2025	2025	NUM
cana-6124	3	41	revised	revise	VERB
cana-6124	3	42	:	:	PUNCT
cana-6124	3	43	24	24	NUM
cana-6124	3	44	-	-	PUNCT
cana-6124	3	45	09	09	NUM
cana-6124	3	46	-	-	PUNCT
cana-6124	3	47	2025	2025	NUM
cana-6124	3	48	accepted	accept	VERB
cana-6124	3	49	:	:	PUNCT
cana-6124	3	50	15	15	NUM
cana-6124	3	51	-	-	SYM
cana-6124	3	52	10	10	NUM
cana-6124	3	53	-	-	PUNCT
cana-6124	3	54	2025	2025	NUM
cana-6124	3	55	abstract	abstract	NOUN
cana-6124	3	56	:	:	PUNCT
cana-6124	3	57	in	in	ADP
cana-6124	3	58	in	in	ADP
cana-6124	3	59	this	this	DET
cana-6124	3	60	paper	paper	NOUN
cana-6124	3	61	,	,	PUNCT
cana-6124	3	62	we	we	PRON
cana-6124	3	63	investigate	investigate	VERB
cana-6124	3	64	the	the	DET
cana-6124	3	65	existence	existence	NOUN
cana-6124	3	66	of	of	ADP
cana-6124	3	67	solutions	solution	NOUN
cana-6124	3	68	to	to	ADP
cana-6124	3	69	the	the	DET
cana-6124	3	70	fractional	fractional	ADJ
cana-6124	3	71	qdifference	qdifference	NOUN
cana-6124	3	72	equation	equation	NOUN
cana-6124	3	73	of	of	ADP
cana-6124	3	74	the	the	DET
cana-6124	3	75	type	type	NOUN
cana-6124	3	76	𝐷𝑐	𝐷𝑐	PROPN
cana-6124	3	77	𝑞	𝑞	NOUN
cana-6124	3	78	𝛼𝑢(𝑡	𝛼𝑢(𝑡	NOUN
cana-6124	3	79	)	)	PUNCT
cana-6124	3	80	=	=	SYM
cana-6124	4	1	𝐴𝑢(𝑡	𝐴𝑢(𝑡	X
cana-6124	4	2	)	)	PUNCT
cana-6124	4	3	+	+	CCONJ
cana-6124	4	4	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	4	5	,	,	PUNCT
cana-6124	4	6	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	4	7	)	)	PUNCT
cana-6124	4	8	)	)	PUNCT
cana-6124	4	9	,	,	PUNCT
cana-6124	4	10	𝑡	𝑡	PROPN
cana-6124	4	11	∈	∈	PROPN
cana-6124	4	12	𝐽	𝐽	NOUN
cana-6124	4	13	≔	≔	VERB
cana-6124	4	14	[	[	NOUN
cana-6124	4	15	0,1	0,1	NUM
cana-6124	4	16	]	]	PUNCT
cana-6124	4	17	with	with	ADP
cana-6124	4	18	𝑢(0	𝑢(0	PROPN
cana-6124	4	19	)	)	PUNCT
cana-6124	4	20	=	=	PUNCT
cana-6124	5	1	𝑎	𝑎	DET
cana-6124	5	2	∫	∫	NOUN
cana-6124	5	3	𝑢(𝑠)𝑑𝑞𝑠	𝑢(𝑠)𝑑𝑞𝑠	PROPN
cana-6124	5	4	+	+	CCONJ
cana-6124	5	5	𝑏	𝑏	NOUN
cana-6124	5	6	,	,	PUNCT
cana-6124	5	7	1	1	NUM
cana-6124	5	8	0	0	NUM
cana-6124	5	9	where	where	SCONJ
cana-6124	5	10	0	0	NUM
cana-6124	5	11	<	<	X
cana-6124	5	12	𝛼	𝛼	X
cana-6124	5	13	<	<	X
cana-6124	5	14	1	1	NUM
cana-6124	5	15	,	,	PUNCT
cana-6124	5	16	𝑓	𝑓	PRON
cana-6124	5	17	∈	∈	NOUN
cana-6124	5	18	𝐶(𝐽	𝐶(𝐽	NOUN
cana-6124	5	19	×	×	PROPN
cana-6124	5	20	𝑋	𝑋	PROPN
cana-6124	5	21	,	,	PUNCT
cana-6124	5	22	𝑋	𝑋	PROPN
cana-6124	5	23	)	)	PUNCT
cana-6124	5	24	,	,	PUNCT
cana-6124	5	25	and	and	CCONJ
cana-6124	5	26	𝐴(𝑡	𝐴(𝑡	X
cana-6124	5	27	)	)	PUNCT
cana-6124	5	28	is	be	AUX
cana-6124	5	29	a	a	DET
cana-6124	5	30	bounded	bounded	ADJ
cana-6124	5	31	linear	linear	ADJ
cana-6124	5	32	operator	operator	NOUN
cana-6124	5	33	on	on	ADP
cana-6124	5	34	a	a	DET
cana-6124	5	35	banach	banach	NOUN
cana-6124	5	36	space	space	NOUN
cana-6124	5	37	𝑋	𝑋	NOUN
cana-6124	5	38	the	the	DET
cana-6124	5	39	operator	operator	NOUN
cana-6124	6	1	𝐷𝑐	𝐷𝑐	PROPN
cana-6124	6	2	𝑞	𝑞	NOUN
cana-6124	6	3	𝛼𝑢(𝑡	𝛼𝑢(𝑡	NOUN
cana-6124	6	4	)	)	PUNCT
cana-6124	6	5	denotes	denote	VERB
cana-6124	6	6	the	the	PRON
cana-6124	6	7	caputo	caputo	PROPN
cana-6124	6	8	fractional	fractional	PROPN
cana-6124	7	1	q	q	PROPN
cana-6124	7	2	-	-	NOUN
cana-6124	7	3	derivative	derivative	NOUN
cana-6124	7	4	of	of	ADP
cana-6124	7	5	order	order	NOUN
cana-6124	7	6	𝛼.	𝛼.	VERB
cana-6124	7	7	our	our	PRON
cana-6124	7	8	existence	existence	NOUN
cana-6124	7	9	result	result	NOUN
cana-6124	7	10	are	be	AUX
cana-6124	7	11	obtained	obtain	VERB
cana-6124	7	12	by	by	ADP
cana-6124	7	13	using	use	VERB
cana-6124	7	14	the	the	DET
cana-6124	7	15	banach	banach	ADV
cana-6124	7	16	fixed	fix	VERB
cana-6124	7	17	point	point	NOUN
cana-6124	7	18	theorem	theorem	NOUN
cana-6124	7	19	and	and	CCONJ
cana-6124	7	20	the	the	DET
cana-6124	7	21	schaefer	schaefer	NOUN
cana-6124	7	22	fixed	fix	VERB
cana-6124	7	23	point	point	NOUN
cana-6124	7	24	theorem	theorem	VERB
cana-6124	7	25	.	.	PUNCT
cana-6124	8	1	keywords	keyword	NOUN
cana-6124	8	2	:	:	PUNCT
cana-6124	8	3	fixed	fixed	ADJ
cana-6124	8	4	point	point	NOUN
cana-6124	8	5	,	,	PUNCT
cana-6124	8	6	existence	existence	PROPN
cana-6124	8	7	,	,	PUNCT
cana-6124	8	8	caputo	caputo	PROPN
cana-6124	8	9	fractional	fractional	PROPN
cana-6124	8	10	q	q	ADJ
cana-6124	8	11	-	-	ADJ
cana-6124	8	12	derivative	derivative	ADJ
cana-6124	8	13	,	,	PUNCT
cana-6124	8	14	fractional	fractional	ADJ
cana-6124	8	15	qdifference	qdifference	NOUN
cana-6124	8	16	equations	equation	NOUN
cana-6124	8	17	.	.	PUNCT
cana-6124	9	1	1	1	X
cana-6124	9	2	.	.	X
cana-6124	9	3	introduction	introduction	NOUN
cana-6124	9	4	over	over	ADP
cana-6124	9	5	the	the	DET
cana-6124	9	6	past	past	ADJ
cana-6124	9	7	years	year	NOUN
cana-6124	9	8	,	,	PUNCT
cana-6124	9	9	the	the	DET
cana-6124	9	10	theory	theory	NOUN
cana-6124	9	11	of	of	ADP
cana-6124	9	12	fractional	fractional	ADJ
cana-6124	9	13	calculus	calculus	NOUN
cana-6124	9	14	has	have	AUX
cana-6124	9	15	received	receive	VERB
cana-6124	9	16	increasing	increase	VERB
cana-6124	9	17	attention	attention	NOUN
cana-6124	9	18	from	from	ADP
cana-6124	9	19	researchers	researcher	NOUN
cana-6124	9	20	and	and	CCONJ
cana-6124	9	21	has	have	AUX
cana-6124	9	22	become	become	VERB
cana-6124	9	23	one	one	NUM
cana-6124	9	24	of	of	ADP
cana-6124	9	25	the	the	DET
cana-6124	9	26	most	most	ADV
cana-6124	9	27	active	active	ADJ
cana-6124	9	28	areas	area	NOUN
cana-6124	9	29	of	of	ADP
cana-6124	9	30	research	research	NOUN
cana-6124	9	31	due	due	ADP
cana-6124	9	32	to	to	ADP
cana-6124	9	33	its	its	PRON
cana-6124	9	34	significant	significant	ADJ
cana-6124	9	35	importance	importance	NOUN
cana-6124	9	36	and	and	CCONJ
cana-6124	9	37	wide	wide	ADJ
cana-6124	9	38	applications	application	NOUN
cana-6124	9	39	on	on	ADP
cana-6124	9	40	many	many	ADJ
cana-6124	9	41	subjects	subject	NOUN
cana-6124	9	42	.	.	PUNCT
cana-6124	10	1	the	the	DET
cana-6124	10	2	importance	importance	NOUN
cana-6124	10	3	of	of	ADP
cana-6124	10	4	this	this	DET
cana-6124	10	5	theory	theory	NOUN
cana-6124	10	6	lies	lie	VERB
cana-6124	10	7	in	in	ADP
cana-6124	10	8	its	its	PRON
cana-6124	10	9	ability	ability	NOUN
cana-6124	10	10	to	to	PART
cana-6124	10	11	contribute	contribute	VERB
cana-6124	10	12	to	to	ADP
cana-6124	10	13	mathematical	mathematical	ADJ
cana-6124	10	14	modeling	modeling	NOUN
cana-6124	10	15	in	in	ADP
cana-6124	10	16	various	various	ADJ
cana-6124	10	17	fields	field	NOUN
cana-6124	10	18	such	such	ADJ
cana-6124	10	19	as	as	ADP
cana-6124	10	20	technical	technical	ADJ
cana-6124	10	21	sciences	science	NOUN
cana-6124	10	22	,	,	PUNCT
cana-6124	10	23	physics	physics	NOUN
cana-6124	10	24	,	,	PUNCT
cana-6124	10	25	engineering	engineering	NOUN
cana-6124	10	26	,	,	PUNCT
cana-6124	10	27	biophysics	biophysic	NOUN
cana-6124	10	28	and	and	CCONJ
cana-6124	10	29	biomathematics	biomathematic	NOUN
cana-6124	10	30	.	.	PUNCT
cana-6124	11	1	for	for	ADP
cana-6124	11	2	more	more	ADJ
cana-6124	11	3	details	detail	NOUN
cana-6124	11	4	,	,	PUNCT
cana-6124	11	5	see	see	VERB
cana-6124	11	6	[	[	X
cana-6124	11	7	9–11	9–11	NOUN
cana-6124	11	8	,	,	PUNCT
cana-6124	11	9	13	13	NUM
cana-6124	11	10	,	,	PUNCT
cana-6124	11	11	16	16	NUM
cana-6124	11	12	]	]	PUNCT
cana-6124	11	13	.	.	PUNCT
cana-6124	12	1	at	at	ADP
cana-6124	12	2	the	the	DET
cana-6124	12	3	beginning	beginning	NOUN
cana-6124	12	4	of	of	ADP
cana-6124	12	5	the	the	DET
cana-6124	12	6	twentieth	twentieth	ADJ
cana-6124	12	7	century	century	NOUN
cana-6124	12	8	,	,	PUNCT
cana-6124	12	9	jackson	jackson	PROPN
cana-6124	12	10	was	be	AUX
cana-6124	12	11	the	the	DET
cana-6124	12	12	first	first	ADJ
cana-6124	12	13	to	to	PART
cana-6124	12	14	develop	develop	VERB
cana-6124	12	15	quantum	quantum	ADJ
cana-6124	12	16	calculus	calculus	NOUN
cana-6124	12	17	,	,	PUNCT
cana-6124	12	18	also	also	ADV
cana-6124	12	19	known	know	VERB
cana-6124	12	20	as	as	ADP
cana-6124	12	21	q	q	ADJ
cana-6124	12	22	-	-	PUNCT
cana-6124	12	23	difference	difference	NOUN
cana-6124	12	24	calculus	calculus	NOUN
cana-6124	12	25	,	,	PUNCT
cana-6124	12	26	by	by	ADP
cana-6124	12	27	introducing	introduce	VERB
cana-6124	12	28	the	the	DET
cana-6124	12	29	concept	concept	NOUN
cana-6124	12	30	of	of	ADP
cana-6124	12	31	the	the	DET
cana-6124	12	32	qintegral	qintegral	ADJ
cana-6124	12	33	along	along	ADP
cana-6124	12	34	with	with	ADP
cana-6124	12	35	several	several	ADJ
cana-6124	12	36	other	other	ADJ
cana-6124	12	37	fundamental	fundamental	ADJ
cana-6124	12	38	notions	notion	NOUN
cana-6124	12	39	in	in	ADP
cana-6124	12	40	this	this	DET
cana-6124	12	41	theory	theory	NOUN
cana-6124	12	42	.	.	PUNCT
cana-6124	13	1	for	for	ADP
cana-6124	13	2	further	further	ADJ
cana-6124	13	3	details	detail	NOUN
cana-6124	13	4	on	on	ADP
cana-6124	13	5	this	this	DET
cana-6124	13	6	topic	topic	NOUN
cana-6124	13	7	,	,	PUNCT
cana-6124	13	8	see	see	VERB
cana-6124	13	9	references	reference	NOUN
cana-6124	13	10	[	[	X
cana-6124	13	11	8	8	NUM
cana-6124	13	12	,	,	PUNCT
cana-6124	13	13	12	12	NUM
cana-6124	13	14	]	]	PUNCT
cana-6124	13	15	.	.	PUNCT
cana-6124	14	1	in	in	ADP
cana-6124	14	2	the	the	DET
cana-6124	14	3	late	late	ADJ
cana-6124	14	4	1960s	1960	NOUN
cana-6124	14	5	,	,	PUNCT
cana-6124	14	6	a	a	DET
cana-6124	14	7	new	new	ADJ
cana-6124	14	8	branch	branch	NOUN
cana-6124	14	9	known	know	VERB
cana-6124	14	10	as	as	ADP
cana-6124	14	11	fractional	fractional	ADJ
cana-6124	14	12	q	q	ADJ
cana-6124	14	13	-	-	PUNCT
cana-6124	14	14	difference	difference	NOUN
cana-6124	14	15	calculus	calculus	NOUN
cana-6124	14	16	emerged	emerge	VERB
cana-6124	14	17	as	as	ADP
cana-6124	14	18	a	a	DET
cana-6124	14	19	generalization	generalization	NOUN
cana-6124	14	20	of	of	ADP
cana-6124	14	21	the	the	DET
cana-6124	14	22	q	q	ADJ
cana-6124	14	23	-	-	PUNCT
cana-6124	14	24	difference	difference	NOUN
cana-6124	14	25	calculus	calculus	NOUN
cana-6124	14	26	.	.	PUNCT
cana-6124	15	1	this	this	DET
cana-6124	15	2	development	development	NOUN
cana-6124	15	3	is	be	AUX
cana-6124	15	4	attributed	attribute	VERB
cana-6124	15	5	to	to	ADP
cana-6124	15	6	al	al	PROPN
cana-6124	15	7	-	-	PUNCT
cana-6124	15	8	salam	salam	PROPN
cana-6124	16	1	[	[	X
cana-6124	16	2	6	6	NUM
cana-6124	16	3	]	]	PUNCT
cana-6124	16	4	and	and	CCONJ
cana-6124	16	5	agarwal	agarwal	PROPN
cana-6124	17	1	[	[	X
cana-6124	17	2	2	2	NUM
cana-6124	17	3	]	]	PUNCT
cana-6124	17	4	.	.	PUNCT
cana-6124	18	1	this	this	DET
cana-6124	18	2	branch	branch	NOUN
cana-6124	18	3	has	have	AUX
cana-6124	18	4	received	receive	VERB
cana-6124	18	5	considerable	considerable	ADJ
cana-6124	18	6	attention	attention	NOUN
cana-6124	18	7	in	in	ADP
cana-6124	18	8	the	the	DET
cana-6124	18	9	academic	academic	ADJ
cana-6124	18	10	community	community	NOUN
cana-6124	18	11	due	due	ADP
cana-6124	18	12	to	to	ADP
cana-6124	18	13	its	its	PRON
cana-6124	18	14	wide	wide	ADJ
cana-6124	18	15	range	range	NOUN
cana-6124	18	16	of	of	ADP
cana-6124	18	17	applications	application	NOUN
cana-6124	18	18	in	in	ADP
cana-6124	18	19	modeling	model	VERB
cana-6124	18	20	mathematical	mathematical	ADJ
cana-6124	18	21	phenomena	phenomenon	NOUN
cana-6124	18	22	across	across	ADP
cana-6124	18	23	various	various	ADJ
cana-6124	18	24	scientific	scientific	ADJ
cana-6124	18	25	fields	field	NOUN
cana-6124	18	26	.	.	PUNCT
cana-6124	19	1	recently	recently	ADV
cana-6124	19	2	,	,	PUNCT
cana-6124	19	3	several	several	ADJ
cana-6124	19	4	researchers	researcher	NOUN
cana-6124	19	5	have	have	AUX
cana-6124	19	6	studied	study	VERB
cana-6124	19	7	the	the	DET
cana-6124	19	8	fractional	fractional	ADJ
cana-6124	19	9	q	q	ADJ
cana-6124	19	10	-	-	PUNCT
cana-6124	19	11	difference	difference	NOUN
cana-6124	19	12	equations	equation	NOUN
cana-6124	19	13	involving	involve	VERB
cana-6124	19	14	the	the	DET
cana-6124	19	15	caputo	caputo	PROPN
cana-6124	19	16	fractional	fractional	PROPN
cana-6124	19	17	q	q	NOUN
cana-6124	19	18	-	-	NOUN
cana-6124	19	19	derivative	derivative	NOUN
cana-6124	19	20	by	by	ADP
cana-6124	19	21	using	use	VERB
cana-6124	19	22	all	all	DET
cana-6124	19	23	kinds	kind	NOUN
cana-6124	19	24	of	of	ADP
cana-6124	19	25	fixed	fix	VERB
cana-6124	19	26	point	point	NOUN
cana-6124	19	27	theorems	theorem	NOUN
cana-6124	19	28	and	and	CCONJ
cana-6124	19	29	obtained	obtain	VERB
cana-6124	19	30	many	many	ADJ
cana-6124	19	31	interesting	interesting	ADJ
cana-6124	19	32	results	result	NOUN
cana-6124	19	33	,	,	PUNCT
cana-6124	19	34	for	for	ADP
cana-6124	19	35	example	example	NOUN
cana-6124	19	36	,	,	PUNCT
cana-6124	19	37	by	by	ADP
cana-6124	19	38	abbas	abbas	PROPN
cana-6124	19	39	et	et	PROPN
cana-6124	19	40	al	al	PROPN
cana-6124	20	1	[	[	X
cana-6124	20	2	1	1	NUM
cana-6124	20	3	]	]	PUNCT
cana-6124	20	4	,	,	PUNCT
cana-6124	20	5	ahmad	ahmad	PROPN
cana-6124	20	6	and	and	CCONJ
cana-6124	20	7	al	al	PROPN
cana-6124	20	8	.	.	PUNCT
cana-6124	21	1	[	[	X
cana-6124	21	2	5	5	NUM
cana-6124	21	3	]	]	PUNCT
cana-6124	21	4	.	.	PUNCT
cana-6124	22	1	in	in	ADP
cana-6124	22	2	[	[	X
cana-6124	22	3	3	3	NUM
cana-6124	22	4	]	]	PUNCT
cana-6124	22	5	,	,	PUNCT
cana-6124	22	6	n.	n.	PROPN
cana-6124	22	7	allouch	allouch	PROPN
cana-6124	22	8	et	et	PROPN
cana-6124	22	9	al	al	PROPN
cana-6124	22	10	studied	study	VERB
cana-6124	22	11	the	the	DET
cana-6124	22	12	existence	existence	NOUN
cana-6124	22	13	of	of	ADP
cana-6124	22	14	solutions	solution	NOUN
cana-6124	22	15	to	to	ADP
cana-6124	22	16	the	the	DET
cana-6124	22	17	following	follow	VERB
cana-6124	22	18	fractional	fractional	ADJ
cana-6124	22	19	q	q	ADJ
cana-6124	22	20	-	-	PUNCT
cana-6124	22	21	difference	difference	NOUN
cana-6124	22	22	equations	equation	NOUN
cana-6124	22	23	with	with	ADP
cana-6124	22	24	nonlinear	nonlinear	ADJ
cana-6124	22	25	integral	integral	ADJ
cana-6124	22	26	conditions	condition	NOUN
cana-6124	22	27	:	:	PUNCT
cana-6124	22	28	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	22	29	𝛼𝑐	𝛼𝑐	ADP
cana-6124	22	30	𝑢(𝑡	𝑢(𝑡	PROPN
cana-6124	22	31	)	)	PUNCT
cana-6124	22	32	=	=	PUNCT
cana-6124	23	1	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	23	2	,	,	PUNCT
cana-6124	23	3	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	23	4	)	)	PUNCT
cana-6124	23	5	)	)	PUNCT
cana-6124	23	6	,	,	PUNCT
cana-6124	23	7	𝑡	𝑡	PROPN
cana-6124	23	8	∈	∈	NOUN
cana-6124	23	9	𝐼	𝐼	ADP
cana-6124	23	10	=	=	SYM
cana-6124	24	1	[	[	X
cana-6124	24	2	0	0	NUM
cana-6124	24	3	,	,	PUNCT
cana-6124	24	4	𝑇	𝑇	PROPN
cana-6124	24	5	]	]	PUNCT
cana-6124	24	6	,	,	PUNCT
cana-6124	24	7	1	1	NUM
cana-6124	24	8	<	<	X
cana-6124	24	9	𝛼	𝛼	X
cana-6124	24	10	≤	≤	NUM
cana-6124	24	11	2	2	NUM
cana-6124	24	12	,	,	PUNCT
cana-6124	24	13	𝑢(0	𝑢(0	PROPN
cana-6124	24	14	)	)	PUNCT
cana-6124	24	15	−	−	PROPN
cana-6124	24	16	𝑢′(0	𝑢′(0	NOUN
cana-6124	24	17	)	)	PUNCT
cana-6124	24	18	=	=	SYM
cana-6124	24	19	∫	∫	PROPN
cana-6124	24	20	𝑔(𝑠	𝑔(𝑠	NOUN
cana-6124	24	21	,	,	PUNCT
cana-6124	24	22	𝑢(𝑠))𝑑𝑠	𝑢(𝑠))𝑑𝑠	PROPN
cana-6124	24	23	,	,	PUNCT
cana-6124	24	24	𝑇	𝑇	PROPN
cana-6124	24	25	0	0	NUM
cana-6124	24	26	mailto:faouzi.hireche.ma@gmail.com	mailto:faouzi.hireche.ma@gmail.com	NOUN
cana-6124	24	27	communications	communication	NOUN
cana-6124	24	28	on	on	ADP
cana-6124	24	29	applied	apply	VERB
cana-6124	24	30	nonlinear	nonlinear	ADJ
cana-6124	24	31	analysis	analysis	NOUN
cana-6124	24	32	issn	issn	NOUN
cana-6124	24	33	:	:	PUNCT
cana-6124	24	34	1074	1074	NUM
cana-6124	24	35	-	-	PUNCT
cana-6124	24	36	133x	133x	NUM
cana-6124	24	37	vol	vol	VERB
cana-6124	24	38	32	32	NUM
cana-6124	24	39	no	no	NOUN
cana-6124	24	40	.	.	PUNCT
cana-6124	25	1	10s	10	NOUN
cana-6124	25	2	(	(	PUNCT
cana-6124	25	3	2025	2025	NUM
cana-6124	25	4	)	)	PUNCT
cana-6124	25	5	3427	3427	NUM
cana-6124	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	25	7	𝑢(𝑇	𝑢(𝑇	NOUN
cana-6124	25	8	)	)	PUNCT
cana-6124	25	9	+	+	SYM
cana-6124	25	10	𝑢′(𝑇	𝑢′(𝑇	NOUN
cana-6124	25	11	)	)	PUNCT
cana-6124	25	12	=	=	SYM
cana-6124	25	13	∫	∫	PROPN
cana-6124	25	14	ℎ(𝑠	ℎ(𝑠	PROPN
cana-6124	25	15	,	,	PUNCT
cana-6124	25	16	𝑢(𝑠))𝑑𝑠	𝑢(𝑠))𝑑𝑠	PROPN
cana-6124	25	17	,	,	PUNCT
cana-6124	25	18	𝑇	𝑇	PROPN
cana-6124	25	19	0	0	NUM
cana-6124	25	20	where	where	SCONJ
cana-6124	25	21	𝑇	𝑇	PROPN
cana-6124	25	22	>	>	X
cana-6124	25	23	0	0	PROPN
cana-6124	25	24	,	,	PUNCT
cana-6124	25	25	𝑞	𝑞	X
cana-6124	25	26	∈]0,1	∈]0,1	ADV
cana-6124	25	27	[	[	X
cana-6124	25	28	,	,	PUNCT
cana-6124	25	29	𝐷𝑐	𝐷𝑐	PROPN
cana-6124	25	30	𝑞	𝑞	X
cana-6124	25	31	𝛼	𝛼	PROPN
cana-6124	25	32	denotes	denote	VERB
cana-6124	25	33	the	the	DET
cana-6124	25	34	caputo	caputo	PROPN
cana-6124	25	35	fractional	fractional	PROPN
cana-6124	25	36	q	q	ADJ
cana-6124	25	37	-	-	PUNCT
cana-6124	25	38	difference	difference	NOUN
cana-6124	25	39	derivative	derivative	NOUN
cana-6124	25	40	of	of	ADP
cana-6124	25	41	order	order	NOUN
cana-6124	26	1	1	1	NUM
cana-6124	26	2	<	<	X
cana-6124	26	3	𝛼	𝛼	X
cana-6124	26	4	≤	≤	NUM
cana-6124	26	5	2	2	NUM
cana-6124	26	6	,	,	PUNCT
cana-6124	26	7	and	and	CCONJ
cana-6124	26	8	𝑓	𝑓	X
cana-6124	26	9	,	,	PUNCT
cana-6124	26	10	𝑔	𝑔	PROPN
cana-6124	26	11	,	,	PUNCT
cana-6124	26	12	ℎ	ℎ	ADP
cana-6124	26	13	∈	∈	PROPN
cana-6124	26	14	𝐶(𝐼	𝐶(𝐼	NOUN
cana-6124	26	15	×	×	NOUN
cana-6124	26	16	𝑋,𝑋	𝑋,𝑋	NOUN
cana-6124	26	17	)	)	PUNCT
cana-6124	26	18	.	.	PUNCT
cana-6124	27	1	in	in	ADP
cana-6124	27	2	[	[	X
cana-6124	27	3	4	4	NUM
cana-6124	27	4	]	]	PUNCT
cana-6124	27	5	,	,	PUNCT
cana-6124	27	6	n.	n.	PROPN
cana-6124	27	7	allouch	allouch	PROPN
cana-6124	27	8	et	et	PROPN
cana-6124	27	9	al	al	PROPN
cana-6124	27	10	applied	apply	VERB
cana-6124	27	11	some	some	DET
cana-6124	27	12	standard	standard	ADJ
cana-6124	27	13	fixed	fix	VERB
cana-6124	27	14	point	point	NOUN
cana-6124	27	15	theorems	theorem	NOUN
cana-6124	27	16	and	and	CCONJ
cana-6124	27	17	investigated	investigate	VERB
cana-6124	27	18	the	the	DET
cana-6124	27	19	existence	existence	NOUN
cana-6124	27	20	of	of	ADP
cana-6124	27	21	solutions	solution	NOUN
cana-6124	27	22	of	of	ADP
cana-6124	27	23	fractional	fractional	ADJ
cana-6124	27	24	q	q	ADJ
cana-6124	27	25	-	-	PUNCT
cana-6124	27	26	difference	difference	NOUN
cana-6124	27	27	equations	equation	NOUN
cana-6124	27	28	of	of	ADP
cana-6124	27	29	the	the	DET
cana-6124	27	30	type	type	NOUN
cana-6124	27	31	:	:	PUNCT
cana-6124	27	32	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	27	33	𝛼𝑐	𝛼𝑐	ADP
cana-6124	27	34	𝑢(𝑡	𝑢(𝑡	PROPN
cana-6124	27	35	)	)	PUNCT
cana-6124	27	36	=	=	PUNCT
cana-6124	28	1	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	28	2	,	,	PUNCT
cana-6124	28	3	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	28	4	)	)	PUNCT
cana-6124	28	5	)	)	PUNCT
cana-6124	28	6	,	,	PUNCT
cana-6124	28	7	𝑡	𝑡	PROPN
cana-6124	28	8	∈	∈	NOUN
cana-6124	28	9	𝐼	𝐼	ADP
cana-6124	28	10	=	=	SYM
cana-6124	29	1	[	[	X
cana-6124	29	2	0	0	NUM
cana-6124	29	3	,	,	PUNCT
cana-6124	29	4	𝑇	𝑇	PROPN
cana-6124	29	5	]	]	PUNCT
cana-6124	29	6	,	,	PUNCT
cana-6124	29	7	1	1	NUM
cana-6124	29	8	<	<	X
cana-6124	29	9	𝛼	𝛼	X
cana-6124	29	10	≤	≤	NUM
cana-6124	29	11	1	1	NUM
cana-6124	29	12	,	,	PUNCT
cana-6124	29	13	𝑎𝑢(0	𝑎𝑢(0	NOUN
cana-6124	29	14	)	)	PUNCT
cana-6124	29	15	+	+	CCONJ
cana-6124	29	16	𝑏𝑢(𝑇	𝑏𝑢(𝑇	NOUN
cana-6124	29	17	)	)	PUNCT
cana-6124	29	18	=	=	SYM
cana-6124	29	19	𝑐	𝑐	NOUN
cana-6124	29	20	,	,	PUNCT
cana-6124	29	21	where	where	SCONJ
cana-6124	29	22	𝑇	𝑇	PROPN
cana-6124	29	23	>	>	X
cana-6124	29	24	0	0	PROPN
cana-6124	29	25	,	,	PUNCT
cana-6124	29	26	𝑞	𝑞	X
cana-6124	29	27	∈]0,1	∈]0,1	ADV
cana-6124	29	28	[	[	X
cana-6124	29	29	,	,	PUNCT
cana-6124	29	30	𝐷𝑐	𝐷𝑐	PROPN
cana-6124	29	31	𝑞	𝑞	X
cana-6124	29	32	𝛼	𝛼	PROPN
cana-6124	29	33	denotes	denote	VERB
cana-6124	29	34	the	the	DET
cana-6124	29	35	caputo	caputo	PROPN
cana-6124	29	36	fractional	fractional	PROPN
cana-6124	29	37	q	q	ADJ
cana-6124	29	38	-	-	PUNCT
cana-6124	29	39	difference	difference	NOUN
cana-6124	29	40	derivative	derivative	NOUN
cana-6124	29	41	of	of	ADP
cana-6124	29	42	order	order	NOUN
cana-6124	29	43	𝛼	𝛼	X
cana-6124	29	44	,	,	PUNCT
cana-6124	29	45	𝑓	𝑓	PRON
cana-6124	29	46	∈	∈	PROPN
cana-6124	30	1	𝐶(𝐼	𝐶(𝐼	NOUN
cana-6124	31	1	×	×	PROPN
cana-6124	31	2	𝑋	𝑋	PROPN
cana-6124	31	3	,	,	PUNCT
cana-6124	31	4	𝑋	𝑋	PROPN
cana-6124	31	5	)	)	PUNCT
cana-6124	31	6	an	an	DET
cana-6124	31	7	d	d	NOUN
cana-6124	31	8	𝑎	𝑎	PROPN
cana-6124	31	9	+	+	NOUN
cana-6124	31	10	𝑏	𝑏	PROPN
cana-6124	31	11	≠	≠	PROPN
cana-6124	31	12	0	0	NUM
cana-6124	31	13	.	.	PUNCT
cana-6124	32	1	in	in	ADP
cana-6124	32	2	this	this	DET
cana-6124	32	3	paper	paper	NOUN
cana-6124	32	4	,	,	PUNCT
cana-6124	32	5	we	we	PRON
cana-6124	32	6	establish	establish	VERB
cana-6124	32	7	the	the	DET
cana-6124	32	8	existence	existence	NOUN
cana-6124	32	9	of	of	ADP
cana-6124	32	10	solutions	solution	NOUN
cana-6124	32	11	to	to	ADP
cana-6124	32	12	the	the	DET
cana-6124	32	13	fractional	fractional	ADJ
cana-6124	32	14	q	q	ADJ
cana-6124	32	15	-	-	PUNCT
cana-6124	32	16	difference	difference	NOUN
cana-6124	32	17	equations	equation	NOUN
cana-6124	32	18	of	of	ADP
cana-6124	32	19	the	the	DET
cana-6124	32	20	type	type	NOUN
cana-6124	32	21	:	:	PUNCT
cana-6124	32	22	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	32	23	𝛼𝑐	𝛼𝑐	VERB
cana-6124	32	24	𝑢(𝑡	𝑢(𝑡	PROPN
cana-6124	32	25	)	)	PUNCT
cana-6124	32	26	=	=	SYM
cana-6124	32	27	𝐴(𝑡	𝐴(𝑡	PUNCT
cana-6124	32	28	)	)	PUNCT
cana-6124	32	29	+	+	CCONJ
cana-6124	32	30	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	32	31	,	,	PUNCT
cana-6124	32	32	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	32	33	)	)	PUNCT
cana-6124	32	34	)	)	PUNCT
cana-6124	32	35	,	,	PUNCT
cana-6124	32	36	𝑡	𝑡	PROPN
cana-6124	32	37	∈	∈	PROPN
cana-6124	32	38	𝐽	𝐽	NOUN
cana-6124	32	39	≔	≔	VERB
cana-6124	32	40	[	[	NOUN
cana-6124	32	41	0,1	0,1	NUM
cana-6124	32	42	]	]	PUNCT
cana-6124	32	43	,	,	PUNCT
cana-6124	32	44	(	(	PUNCT
cana-6124	32	45	1.1	1.1	NUM
cana-6124	32	46	)	)	PUNCT
cana-6124	32	47	𝑢(0	𝑢(0	PROPN
cana-6124	32	48	)	)	PUNCT
cana-6124	32	49	=	=	PUNCT
cana-6124	33	1	𝑎	𝑎	DET
cana-6124	33	2	∫	∫	NOUN
cana-6124	33	3	𝑢(𝑠)𝑑𝑞𝑠	𝑢(𝑠)𝑑𝑞𝑠	PROPN
cana-6124	33	4	+	+	CCONJ
cana-6124	33	5	𝑏	𝑏	NOUN
cana-6124	33	6	,	,	PUNCT
cana-6124	33	7	1	1	NUM
cana-6124	33	8	0	0	NUM
cana-6124	33	9	(	(	PUNCT
cana-6124	33	10	1.2	1.2	NUM
cana-6124	33	11	)	)	PUNCT
cana-6124	33	12	where	where	SCONJ
cana-6124	33	13	0	0	NUM
cana-6124	33	14	<	<	X
cana-6124	33	15	𝛼	𝛼	X
cana-6124	33	16	<	<	X
cana-6124	33	17	1	1	NUM
cana-6124	33	18	,	,	PUNCT
cana-6124	33	19	𝑓	𝑓	PRON
cana-6124	33	20	∈	∈	NOUN
cana-6124	33	21	𝐶(𝐽	𝐶(𝐽	NOUN
cana-6124	33	22	×	×	PROPN
cana-6124	33	23	𝑋	𝑋	PROPN
cana-6124	33	24	,	,	PUNCT
cana-6124	33	25	𝑋	𝑋	PROPN
cana-6124	33	26	)	)	PUNCT
cana-6124	33	27	,	,	PUNCT
cana-6124	33	28	and	and	CCONJ
cana-6124	33	29	𝐴(𝑡	𝐴(𝑡	X
cana-6124	33	30	)	)	PUNCT
cana-6124	33	31	is	be	AUX
cana-6124	33	32	a	a	DET
cana-6124	33	33	bounded	bounded	ADJ
cana-6124	33	34	linear	linear	ADJ
cana-6124	33	35	operator	operator	NOUN
cana-6124	33	36	on	on	ADP
cana-6124	33	37	a	a	DET
cana-6124	33	38	banach	banach	NOUN
cana-6124	33	39	space	space	NOUN
cana-6124	33	40	𝑋.	𝑋.	NOUN
cana-6124	33	41	the	the	DET
cana-6124	33	42	operator	operator	NOUN
cana-6124	34	1	𝐷𝑐	𝐷𝑐	PROPN
cana-6124	34	2	𝑞	𝑞	X
cana-6124	34	3	𝛼	𝛼	PROPN
cana-6124	34	4	denotes	denote	VERB
cana-6124	34	5	the	the	DET
cana-6124	34	6	caputo	caputo	PROPN
cana-6124	34	7	fractional	fractional	PROPN
cana-6124	34	8	q	q	ADJ
cana-6124	34	9	-	-	PUNCT
cana-6124	34	10	difference	difference	NOUN
cana-6124	34	11	derivative	derivative	NOUN
cana-6124	34	12	of	of	ADP
cana-6124	34	13	order	order	NOUN
cana-6124	34	14	𝛼.	𝛼.	NOUN
cana-6124	34	15	the	the	DET
cana-6124	34	16	existence	existence	NOUN
cana-6124	34	17	result	result	NOUN
cana-6124	34	18	is	be	AUX
cana-6124	34	19	based	base	VERB
cana-6124	34	20	on	on	ADP
cana-6124	34	21	the	the	DET
cana-6124	34	22	fixed	fix	VERB
cana-6124	34	23	point	point	NOUN
cana-6124	34	24	theorem	theorem	NOUN
cana-6124	34	25	and	and	CCONJ
cana-6124	34	26	the	the	DET
cana-6124	34	27	schaefer	schaefer	NOUN
cana-6124	34	28	fixed	fix	VERB
cana-6124	34	29	point	point	NOUN
cana-6124	34	30	theorem	theorem	VERB
cana-6124	34	31	.	.	PUNCT
cana-6124	35	1	the	the	DET
cana-6124	35	2	paper	paper	NOUN
cana-6124	35	3	is	be	AUX
cana-6124	35	4	structured	structure	VERB
cana-6124	35	5	as	as	SCONJ
cana-6124	35	6	follows	follow	VERB
cana-6124	35	7	.	.	PUNCT
cana-6124	36	1	in	in	ADP
cana-6124	36	2	section	section	NOUN
cana-6124	36	3	2	2	NUM
cana-6124	36	4	,	,	PUNCT
cana-6124	36	5	we	we	PRON
cana-6124	36	6	present	present	VERB
cana-6124	36	7	the	the	DET
cana-6124	36	8	notations	notation	NOUN
cana-6124	36	9	and	and	CCONJ
cana-6124	36	10	definitions	definition	NOUN
cana-6124	36	11	required	require	VERB
cana-6124	36	12	for	for	ADP
cana-6124	36	13	the	the	DET
cana-6124	36	14	study	study	NOUN
cana-6124	36	15	,	,	PUNCT
cana-6124	36	16	and	and	CCONJ
cana-6124	36	17	we	we	PRON
cana-6124	36	18	review	review	VERB
cana-6124	36	19	essential	essential	ADJ
cana-6124	36	20	preliminaries	preliminary	NOUN
cana-6124	36	21	from	from	ADP
cana-6124	36	22	fractional	fractional	ADJ
cana-6124	36	23	q	q	NOUN
cana-6124	36	24	-	-	PUNCT
cana-6124	36	25	calculus	calculus	NOUN
cana-6124	36	26	.	.	PUNCT
cana-6124	37	1	section	section	NOUN
cana-6124	37	2	3	3	NUM
cana-6124	37	3	contains	contain	VERB
cana-6124	37	4	the	the	DET
cana-6124	37	5	principal	principal	ADJ
cana-6124	37	6	results	result	NOUN
cana-6124	37	7	:	:	PUNCT
cana-6124	37	8	the	the	DET
cana-6124	37	9	first	first	ADJ
cana-6124	37	10	derived	derive	VERB
cana-6124	37	11	from	from	ADP
cana-6124	37	12	the	the	DET
cana-6124	37	13	banach	banach	ADV
cana-6124	37	14	fixed	fix	VERB
cana-6124	37	15	point	point	NOUN
cana-6124	37	16	theorem	theorem	VERB
cana-6124	37	17	,	,	PUNCT
cana-6124	37	18	and	and	CCONJ
cana-6124	37	19	the	the	DET
cana-6124	37	20	second	second	ADJ
cana-6124	37	21	from	from	ADP
cana-6124	37	22	schaefer	schaefer	PROPN
cana-6124	37	23	’s	’s	PART
cana-6124	37	24	fixed	fix	VERB
cana-6124	37	25	point	point	NOUN
cana-6124	37	26	theorem	theorem	VERB
cana-6124	37	27	.	.	PROPN
cana-6124	38	1	section	section	NOUN
cana-6124	38	2	4	4	NUM
cana-6124	38	3	is	be	AUX
cana-6124	38	4	devoted	devote	VERB
cana-6124	38	5	to	to	ADP
cana-6124	38	6	an	an	DET
cana-6124	38	7	illustrative	illustrative	ADJ
cana-6124	38	8	example	example	NOUN
cana-6124	38	9	highlighting	highlight	VERB
cana-6124	38	10	the	the	DET
cana-6124	38	11	applicability	applicability	NOUN
cana-6124	38	12	of	of	ADP
cana-6124	38	13	these	these	DET
cana-6124	38	14	results	result	NOUN
cana-6124	38	15	.	.	PUNCT
cana-6124	39	1	2	2	X
cana-6124	39	2	.	.	X
cana-6124	39	3	preliminaries	preliminary	NOUN
cana-6124	39	4	this	this	DET
cana-6124	39	5	section	section	NOUN
cana-6124	39	6	is	be	AUX
cana-6124	39	7	concerned	concern	VERB
cana-6124	39	8	with	with	ADP
cana-6124	39	9	presenting	present	VERB
cana-6124	39	10	basic	basic	ADJ
cana-6124	39	11	definitions	definition	NOUN
cana-6124	39	12	together	together	ADV
cana-6124	39	13	with	with	ADP
cana-6124	39	14	auxiliary	auxiliary	ADJ
cana-6124	39	15	results	result	NOUN
cana-6124	39	16	required	require	VERB
cana-6124	39	17	in	in	ADP
cana-6124	39	18	the	the	DET
cana-6124	39	19	later	later	ADJ
cana-6124	39	20	parts	part	NOUN
cana-6124	39	21	of	of	ADP
cana-6124	39	22	this	this	DET
cana-6124	39	23	paper	paper	NOUN
cana-6124	39	24	.	.	PUNCT
cana-6124	40	1	assume	assume	VERB
cana-6124	40	2	that	that	SCONJ
cana-6124	40	3	𝑋	𝑋	PROPN
cana-6124	40	4	is	be	AUX
cana-6124	40	5	a	a	DET
cana-6124	40	6	banach	banach	NOUN
cana-6124	40	7	space	space	NOUN
cana-6124	40	8	.	.	PUNCT
cana-6124	41	1	define	define	VERB
cana-6124	41	2	𝐽	𝐽	NOUN
cana-6124	41	3	:	:	PUNCT
cana-6124	41	4	=	=	PUNCT
cana-6124	42	1	[	[	X
cana-6124	42	2	0,1	0,1	NUM
cana-6124	42	3	]	]	PUNCT
cana-6124	42	4	and	and	CCONJ
cana-6124	42	5	let	let	VERB
cana-6124	42	6	𝐶(𝐽	𝐶(𝐽	NOUN
cana-6124	42	7	,	,	PUNCT
cana-6124	42	8	𝑋	𝑋	NOUN
cana-6124	42	9	)	)	PUNCT
cana-6124	42	10	denote	denote	VERB
cana-6124	42	11	the	the	DET
cana-6124	42	12	banach	banach	NOUN
cana-6124	42	13	space	space	NOUN
cana-6124	42	14	of	of	ADP
cana-6124	42	15	continuous	continuous	ADJ
cana-6124	42	16	functions	function	NOUN
cana-6124	42	17	𝑢	𝑢	NOUN
cana-6124	42	18	from	from	ADP
cana-6124	42	19	𝐽	𝐽	PROPN
cana-6124	42	20	into	into	ADP
cana-6124	42	21	𝑋	𝑋	PROPN
cana-6124	42	22	with	with	ADP
cana-6124	42	23	the	the	DET
cana-6124	42	24	norm	norm	NOUN
cana-6124	42	25	‖𝑢‖∞	‖𝑢‖∞	PROPN
cana-6124	42	26	=	=	SYM
cana-6124	42	27	𝑠𝑢𝑝𝑡∈𝐽|𝑢(𝑡)|	𝑠𝑢𝑝𝑡∈𝐽|𝑢(𝑡)|	PROPN
cana-6124	42	28	.	.	PUNCT
cana-6124	43	1	now	now	ADV
cana-6124	43	2	,	,	PUNCT
cana-6124	43	3	we	we	PRON
cana-6124	43	4	introduce	introduce	VERB
cana-6124	43	5	the	the	DET
cana-6124	43	6	essential	essential	ADJ
cana-6124	43	7	definitions	definition	NOUN
cana-6124	43	8	and	and	CCONJ
cana-6124	43	9	relevant	relevant	ADJ
cana-6124	43	10	properties	property	NOUN
cana-6124	43	11	of	of	ADP
cana-6124	43	12	the	the	DET
cana-6124	43	13	fractional	fractional	ADJ
cana-6124	43	14	q	q	NOUN
cana-6124	43	15	-	-	NOUN
cana-6124	43	16	calculus	calculus	NOUN
cana-6124	43	17	.	.	PUNCT
cana-6124	44	1	for	for	ADP
cana-6124	44	2	more	more	ADJ
cana-6124	44	3	details	detail	NOUN
cana-6124	44	4	,	,	PUNCT
cana-6124	44	5	see	see	VERB
cana-6124	44	6	[	[	X
cana-6124	44	7	8	8	NUM
cana-6124	44	8	,	,	PUNCT
cana-6124	44	9	12	12	NUM
cana-6124	44	10	]	]	PUNCT
cana-6124	44	11	.	.	PUNCT
cana-6124	45	1	we	we	PRON
cana-6124	45	2	assume	assume	VERB
cana-6124	45	3	that	that	SCONJ
cana-6124	45	4	𝑞	𝑞	PROPN
cana-6124	45	5	∈]0,1	∈]0,1	ADV
cana-6124	45	6	[	[	X
cana-6124	45	7	.	.	PUNCT
cana-6124	46	1	for	for	ADP
cana-6124	46	2	every	every	DET
cana-6124	46	3	𝑎	𝑎	PROPN
cana-6124	46	4	∈	∈	PROPN
cana-6124	46	5	ℝ	ℝ	PROPN
cana-6124	46	6	,	,	PUNCT
cana-6124	46	7	we	we	PRON
cana-6124	46	8	define	define	VERB
cana-6124	46	9	[	[	X
cana-6124	46	10	𝑎]𝑞	𝑎]𝑞	ADV
cana-6124	46	11	=	=	SYM
cana-6124	46	12	1	1	NUM
cana-6124	46	13	−	−	NOUN
cana-6124	46	14	𝑞𝑎	𝑞𝑎	ADP
cana-6124	46	15	1	1	NUM
cana-6124	46	16	−	−	PROPN
cana-6124	46	17	𝑞	𝑞	PROPN
cana-6124	46	18	.	.	PUNCT
cana-6124	47	1	communications	communication	NOUN
cana-6124	47	2	on	on	ADP
cana-6124	47	3	applied	apply	VERB
cana-6124	47	4	nonlinear	nonlinear	ADJ
cana-6124	47	5	analysis	analysis	NOUN
cana-6124	47	6	issn	issn	NOUN
cana-6124	47	7	:	:	PUNCT
cana-6124	47	8	1074	1074	NUM
cana-6124	47	9	-	-	PUNCT
cana-6124	47	10	133x	133x	NUM
cana-6124	47	11	vol	vol	VERB
cana-6124	47	12	32	32	NUM
cana-6124	47	13	no	no	NOUN
cana-6124	47	14	.	.	PUNCT
cana-6124	48	1	10s	10	NOUN
cana-6124	48	2	(	(	PUNCT
cana-6124	48	3	2025	2025	NUM
cana-6124	48	4	)	)	PUNCT
cana-6124	48	5	3428	3428	NUM
cana-6124	48	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	48	7	let	let	VERB
cana-6124	48	8	𝑎	𝑎	NOUN
cana-6124	48	9	,	,	PUNCT
cana-6124	48	10	𝑏	𝑏	PROPN
cana-6124	48	11	∈	∈	PROPN
cana-6124	48	12	ℝ.	ℝ.	VERB
cana-6124	48	13	the	the	DET
cana-6124	48	14	q	q	NOUN
cana-6124	48	15	-	-	PUNCT
cana-6124	48	16	analogue	analogue	NOUN
cana-6124	48	17	of	of	ADP
cana-6124	48	18	(	(	PUNCT
cana-6124	48	19	𝑎	𝑎	NOUN
cana-6124	48	20	−	−	NOUN
cana-6124	48	21	𝑏)(𝑛	𝑏)(𝑛	NOUN
cana-6124	48	22	)	)	PUNCT
cana-6124	48	23	is	be	AUX
cana-6124	48	24	given	give	VERB
cana-6124	48	25	by	by	ADP
cana-6124	48	26	:	:	PUNCT
cana-6124	48	27	(	(	PUNCT
cana-6124	48	28	𝑎	𝑎	PRON
cana-6124	48	29	−	−	NOUN
cana-6124	48	30	𝑏)(𝑛	𝑏)(𝑛	NOUN
cana-6124	48	31	)	)	PUNCT
cana-6124	48	32	=	=	PRON
cana-6124	48	33	{	{	PUNCT
cana-6124	49	1	1	1	NUM
cana-6124	49	2	𝑖𝑓	𝑖𝑓	NUM
cana-6124	49	3	𝑛	𝑛	PROPN
cana-6124	49	4	=	=	SYM
cana-6124	49	5	0	0	NUM
cana-6124	49	6	∏(𝑎	∏(𝑎	PROPN
cana-6124	49	7	−	−	PROPN
cana-6124	49	8	𝑏𝑞𝑖	𝑏𝑞𝑖	NOUN
cana-6124	49	9	𝑛−1	𝑛−1	PROPN
cana-6124	49	10	𝑖=0	𝑖=0	PUNCT
cana-6124	49	11	)	)	PUNCT
cana-6124	50	1	𝑖𝑓	𝑖𝑓	VERB
cana-6124	50	2	𝑛	𝑛	DET
cana-6124	50	3	∈	∈	PROPN
cana-6124	50	4	ℕ∗	ℕ∗	NOUN
cana-6124	50	5	for	for	ADP
cana-6124	50	6	𝛽	𝛽	PROPN
cana-6124	50	7	∈	∈	PROPN
cana-6124	50	8	ℝ	ℝ	PROPN
cana-6124	50	9	,	,	PUNCT
cana-6124	50	10	we	we	PRON
cana-6124	50	11	have	have	VERB
cana-6124	50	12	(	(	PUNCT
cana-6124	50	13	𝑎	𝑎	DET
cana-6124	50	14	−	−	NOUN
cana-6124	50	15	𝑏)(𝛽	𝑏)(𝛽	ADJ
cana-6124	50	16	)	)	PUNCT
cana-6124	50	17	=	=	PUNCT
cana-6124	51	1	𝑎𝛽∏	𝑎𝛽∏	PROPN
cana-6124	51	2	(	(	PUNCT
cana-6124	51	3	𝑎	𝑎	DET
cana-6124	51	4	−	−	NOUN
cana-6124	51	5	𝑏𝑞𝑖	𝑏𝑞𝑖	NOUN
cana-6124	51	6	𝑎	𝑎	PRON
cana-6124	51	7	−	−	PROPN
cana-6124	51	8	𝑏𝑞𝑖+𝛽	𝑏𝑞𝑖+𝛽	PROPN
cana-6124	51	9	)	)	PUNCT
cana-6124	51	10	,	,	PUNCT
cana-6124	51	11	𝑎	𝑎	X
cana-6124	51	12	,	,	PUNCT
cana-6124	51	13	𝑏	𝑏	PROPN
cana-6124	51	14	∈	∈	PROPN
cana-6124	51	15	ℝ.	ℝ.	PROPN
cana-6124	51	16	∞	∞	PROPN
cana-6124	51	17	𝑖=0	𝑖=0	PROPN
cana-6124	51	18	note	note	VERB
cana-6124	51	19	that	that	SCONJ
cana-6124	51	20	,	,	PUNCT
cana-6124	51	21	if	if	SCONJ
cana-6124	51	22	𝑏	𝑏	PROPN
cana-6124	51	23	=	=	SYM
cana-6124	51	24	0	0	NUM
cana-6124	51	25	,	,	PUNCT
cana-6124	51	26	then	then	ADV
cana-6124	51	27	𝑎(𝛽	𝑎(𝛽	NOUN
cana-6124	51	28	)	)	PUNCT
cana-6124	52	1	=	=	PRON
cana-6124	52	2	𝑎𝛽	𝑎𝛽	NOUN
cana-6124	52	3	.	.	PUNCT
cana-6124	53	1	definition	definition	NOUN
cana-6124	53	2	1	1	NUM
cana-6124	54	1	[	[	X
cana-6124	54	2	12	12	NUM
cana-6124	54	3	]	]	PUNCT
cana-6124	54	4	the	the	DET
cana-6124	54	5	q	q	ADJ
cana-6124	54	6	-	-	PUNCT
cana-6124	54	7	gamma	gamma	NOUN
cana-6124	54	8	function	function	NOUN
cana-6124	54	9	is	be	AUX
cana-6124	54	10	defined	define	VERB
cana-6124	54	11	as	as	SCONJ
cana-6124	54	12	follows	follow	VERB
cana-6124	54	13	:	:	PUNCT
cana-6124	54	14	γ𝑞(𝛽	γ𝑞(𝛽	NOUN
cana-6124	54	15	)	)	PUNCT
cana-6124	54	16	=	=	SYM
cana-6124	54	17	(	(	PUNCT
cana-6124	54	18	1	1	NUM
cana-6124	54	19	−	−	NOUN
cana-6124	54	20	𝑞)(𝛽−1	𝑞)(𝛽−1	NOUN
cana-6124	54	21	)	)	PUNCT
cana-6124	54	22	(	(	PUNCT
cana-6124	54	23	1	1	NUM
cana-6124	54	24	−	−	NOUN
cana-6124	54	25	𝑞)𝛽−1	𝑞)𝛽−1	NOUN
cana-6124	54	26	,	,	PUNCT
cana-6124	54	27	𝛽	𝛽	PROPN
cana-6124	54	28	>	>	X
cana-6124	54	29	0	0	X
cana-6124	54	30	.	.	PUNCT
cana-6124	54	31	observe	observe	VERB
cana-6124	54	32	that	that	SCONJ
cana-6124	54	33	the	the	DET
cana-6124	54	34	q	q	ADJ
cana-6124	54	35	-	-	PUNCT
cana-6124	54	36	gamma	gamma	NOUN
cana-6124	54	37	function	function	NOUN
cana-6124	54	38	verifies	verifie	NOUN
cana-6124	54	39	γ𝑞(𝛽	γ𝑞(𝛽	PUNCT
cana-6124	54	40	+	+	NOUN
cana-6124	54	41	1	1	X
cana-6124	54	42	)	)	PUNCT
cana-6124	54	43	=	=	PUNCT
cana-6124	55	1	[	[	X
cana-6124	55	2	𝛽]𝑞γ𝑞(𝛽	𝛽]𝑞γ𝑞(𝛽	NOUN
cana-6124	55	3	)	)	PUNCT
cana-6124	55	4	.	.	PUNCT
cana-6124	56	1	definition	definition	NOUN
cana-6124	56	2	2	2	NUM
cana-6124	57	1	[	[	X
cana-6124	57	2	12	12	NUM
cana-6124	57	3	]	]	PUNCT
cana-6124	57	4	let	let	VERB
cana-6124	57	5	𝑓	𝑓	X
cana-6124	57	6	:	:	PUNCT
cana-6124	57	7	𝐽	𝐽	PROPN
cana-6124	57	8	→	→	PUNCT
cana-6124	57	9	ℝ.	ℝ.	PROPN
cana-6124	57	10	the	the	DET
cana-6124	57	11	q	q	NOUN
cana-6124	57	12	-	-	NOUN
cana-6124	57	13	derivative	derivative	NOUN
cana-6124	57	14	of	of	ADP
cana-6124	57	15	order	order	NOUN
cana-6124	57	16	𝑛	𝑛	DET
cana-6124	57	17	∈	∈	PROPN
cana-6124	57	18	ℕ	ℕ	PROPN
cana-6124	57	19	is	be	AUX
cana-6124	57	20	given	give	VERB
cana-6124	57	21	by	by	ADP
cana-6124	57	22	:	:	PUNCT
cana-6124	57	23	(	(	PUNCT
cana-6124	57	24	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	57	25	0𝑓)(𝑡	0𝑓)(𝑡	NUM
cana-6124	57	26	)	)	PUNCT
cana-6124	58	1	=	=	PUNCT
cana-6124	58	2	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	58	3	)	)	PUNCT
cana-6124	58	4	,	,	PUNCT
cana-6124	58	5	(	(	PUNCT
cana-6124	58	6	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	58	7	1𝑓)(𝑡	1𝑓)(𝑡	NUM
cana-6124	58	8	)	)	PUNCT
cana-6124	58	9	=	=	SYM
cana-6124	58	10	𝑓(𝑡	𝑓(𝑡	VERB
cana-6124	58	11	)	)	PUNCT
cana-6124	58	12	−	−	ADP
cana-6124	58	13	𝑓(𝑞𝑡	𝑓(𝑞𝑡	NOUN
cana-6124	58	14	)	)	PUNCT
cana-6124	58	15	(	(	PUNCT
cana-6124	58	16	1	1	NUM
cana-6124	58	17	−	−	NOUN
cana-6124	58	18	𝑞)𝑡	𝑞)𝑡	NOUN
cana-6124	58	19	,	,	PUNCT
cana-6124	58	20	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-6124	58	21	(	(	PUNCT
cana-6124	58	22	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	58	23	𝑛𝑓)(𝑡	𝑛𝑓)(𝑡	PROPN
cana-6124	58	24	)	)	PUNCT
cana-6124	58	25	=	=	NOUN
cana-6124	58	26	(	(	PUNCT
cana-6124	58	27	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	58	28	1𝐷𝑞	1𝐷𝑞	PROPN
cana-6124	58	29	𝑛−1𝑓)(𝑡	𝑛−1𝑓)(𝑡	PROPN
cana-6124	58	30	)	)	PUNCT
cana-6124	58	31	,	,	PUNCT
cana-6124	58	32	𝑛	𝑛	DET
cana-6124	58	33	∈	∈	PROPN
cana-6124	58	34	ℕ∗.	ℕ∗.	PROPN
cana-6124	58	35	definition	definition	NOUN
cana-6124	58	36	3	3	NUM
cana-6124	58	37	[	[	X
cana-6124	58	38	12	12	NUM
cana-6124	58	39	]	]	PUNCT
cana-6124	58	40	let	let	VERB
cana-6124	58	41	𝐽𝑡	𝐽𝑡	VERB
cana-6124	58	42	=	=	PUNCT
cana-6124	58	43	{	{	PUNCT
cana-6124	58	44	𝑡𝑞	𝑡𝑞	INTJ
cana-6124	58	45	𝑛	𝑛	PROPN
cana-6124	58	46	:	:	PUNCT
cana-6124	58	47	𝑛	𝑛	PRON
cana-6124	58	48	∈	∈	PROPN
cana-6124	58	49	ℕ}⋃{0	ℕ}⋃{0	PROPN
cana-6124	58	50	}	}	PUNCT
cana-6124	58	51	.	.	PUNCT
cana-6124	59	1	the	the	DET
cana-6124	59	2	q	q	NOUN
cana-6124	59	3	-	-	ADJ
cana-6124	59	4	integral	integral	ADJ
cana-6124	59	5	of	of	ADP
cana-6124	59	6	a	a	DET
cana-6124	59	7	function	function	NOUN
cana-6124	59	8	𝑓	𝑓	NOUN
cana-6124	59	9	:	:	PUNCT
cana-6124	59	10	𝐽𝑡	𝐽𝑡	PROPN
cana-6124	59	11	→	→	SYM
cana-6124	59	12	ℝ	ℝ	PROPN
cana-6124	59	13	is	be	AUX
cana-6124	59	14	defined	define	VERB
cana-6124	59	15	by	by	ADP
cana-6124	59	16	:	:	PUNCT
cana-6124	59	17	(	(	PUNCT
cana-6124	59	18	𝐼𝑞𝑓)(𝑡	𝐼𝑞𝑓)(𝑡	PROPN
cana-6124	59	19	)	)	PUNCT
cana-6124	59	20	=	=	SYM
cana-6124	59	21	∫	∫	PROPN
cana-6124	60	1	𝑓(𝑠)𝑑𝑞𝑠	𝑓(𝑠)𝑑𝑞𝑠	PROPN
cana-6124	60	2	=	=	PUNCT
cana-6124	60	3	∑𝑡(1	∑𝑡(1	ADP
cana-6124	60	4	−	−	PROPN
cana-6124	60	5	𝑞)𝑞𝑛𝑓(𝑡𝑞𝑛	𝑞)𝑞𝑛𝑓(𝑡𝑞𝑛	PROPN
cana-6124	60	6	)	)	PUNCT
cana-6124	60	7	,	,	PUNCT
cana-6124	60	8	∝	∝	PROPN
cana-6124	60	9	𝑛=0	𝑛=0	VERB
cana-6124	60	10	1	1	NUM
cana-6124	60	11	0	0	NUM
cana-6124	60	12	under	under	ADP
cana-6124	60	13	the	the	DET
cana-6124	60	14	assumption	assumption	NOUN
cana-6124	60	15	that	that	SCONJ
cana-6124	60	16	the	the	DET
cana-6124	60	17	series	series	NOUN
cana-6124	60	18	converges	converge	VERB
cana-6124	60	19	.	.	PUNCT
cana-6124	61	1	note	note	VERB
cana-6124	61	2	that	that	SCONJ
cana-6124	61	3	(	(	PUNCT
cana-6124	61	4	𝐷𝑞𝐼𝑞𝑓)(𝑡	𝐷𝑞𝐼𝑞𝑓)(𝑡	PROPN
cana-6124	61	5	)	)	PUNCT
cana-6124	61	6	=	=	PUNCT
cana-6124	62	1	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	62	2	)	)	PUNCT
cana-6124	62	3	,	,	PUNCT
cana-6124	62	4	furthermore	furthermore	ADV
cana-6124	62	5	,	,	PUNCT
cana-6124	62	6	if	if	SCONJ
cana-6124	62	7	𝑓	𝑓	PRON
cana-6124	62	8	is	be	AUX
cana-6124	62	9	continuous	continuous	ADJ
cana-6124	62	10	at	at	ADP
cana-6124	62	11	0	0	NUM
cana-6124	62	12	,	,	PUNCT
cana-6124	62	13	then	then	ADV
cana-6124	62	14	(	(	PUNCT
cana-6124	62	15	𝐼𝑞𝐷𝑞𝑓)(𝑡	𝐼𝑞𝐷𝑞𝑓)(𝑡	PROPN
cana-6124	62	16	)	)	PUNCT
cana-6124	62	17	=	=	SYM
cana-6124	62	18	𝑓(𝑡	𝑓(𝑡	PROPN
cana-6124	62	19	)	)	PUNCT
cana-6124	62	20	−	−	PROPN
cana-6124	63	1	𝑓(0	𝑓(0	NOUN
cana-6124	63	2	)	)	PUNCT
cana-6124	63	3	.	.	PUNCT
cana-6124	64	1	definition	definition	NOUN
cana-6124	64	2	4	4	NUM
cana-6124	64	3	[	[	X
cana-6124	64	4	2	2	NUM
cana-6124	64	5	]	]	PUNCT
cana-6124	64	6	let	let	VERB
cana-6124	64	7	𝑓	𝑓	X
cana-6124	64	8	:	:	PUNCT
cana-6124	64	9	𝐽	𝐽	PROPN
cana-6124	64	10	→	→	PUNCT
cana-6124	64	11	ℝ.	ℝ.	PROPN
cana-6124	64	12	the	the	DET
cana-6124	64	13	riemann	riemann	PROPN
cana-6124	64	14	-	-	PUNCT
cana-6124	64	15	liouville	liouville	VERB
cana-6124	64	16	fractional	fractional	ADJ
cana-6124	64	17	q	q	NOUN
cana-6124	64	18	-	-	ADJ
cana-6124	64	19	integral	integral	ADJ
cana-6124	64	20	of	of	ADP
cana-6124	64	21	order	order	NOUN
cana-6124	64	22	𝛼	𝛼	PRON
cana-6124	64	23	≥	≥	NOUN
cana-6124	64	24	0	0	NUM
cana-6124	64	25	is	be	AUX
cana-6124	64	26	defined	define	VERB
cana-6124	64	27	as	as	ADP
cana-6124	64	28	:	:	PUNCT
cana-6124	64	29	(	(	PUNCT
cana-6124	64	30	𝐼𝑞	𝐼𝑞	PROPN
cana-6124	64	31	𝛼𝑓)(𝑡	𝛼𝑓)(𝑡	PROPN
cana-6124	64	32	)	)	PUNCT
cana-6124	65	1	=	=	PRON
cana-6124	65	2	{	{	PUNCT
cana-6124	65	3	𝑓(𝑡	𝑓(𝑡	PROPN
cana-6124	65	4	)	)	PUNCT
cana-6124	65	5	𝑖𝑓	𝑖𝑓	ADP
cana-6124	65	6	𝛼	𝛼	NOUN
cana-6124	65	7	=	=	SYM
cana-6124	65	8	0	0	NUM
cana-6124	65	9	,	,	PUNCT
cana-6124	65	10	∫	∫	PROPN
cana-6124	65	11	(	(	PUNCT
cana-6124	65	12	𝑡	𝑡	PROPN
cana-6124	65	13	−	−	NOUN
cana-6124	65	14	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	65	15	)	)	PUNCT
cana-6124	65	16	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	65	17	)	)	PUNCT
cana-6124	66	1	𝑓(𝑠)𝑑𝑞𝑠	𝑓(𝑠)𝑑𝑞𝑠	PROPN
cana-6124	66	2	𝑡	𝑡	NOUN
cana-6124	66	3	0	0	NUM
cana-6124	66	4	𝑖𝑓	𝑖𝑓	ADP
cana-6124	66	5	𝛼	𝛼	PROPN
cana-6124	66	6	>	>	X
cana-6124	66	7	0	0	X
cana-6124	66	8	.	.	PUNCT
cana-6124	66	9	observe	observe	VERB
cana-6124	66	10	that	that	SCONJ
cana-6124	66	11	when	when	SCONJ
cana-6124	66	12	𝛼	𝛼	X
cana-6124	66	13	=	=	SYM
cana-6124	66	14	1	1	NUM
cana-6124	66	15	,	,	PUNCT
cana-6124	66	16	we	we	PRON
cana-6124	66	17	have	have	VERB
cana-6124	66	18	(	(	PUNCT
cana-6124	66	19	𝐼𝑞	𝐼𝑞	PROPN
cana-6124	66	20	1𝑓)(𝑡	1𝑓)(𝑡	NUM
cana-6124	66	21	)	)	PUNCT
cana-6124	66	22	=	=	PUNCT
cana-6124	66	23	(	(	PUNCT
cana-6124	66	24	𝐼𝑞𝑓)(𝑡	𝐼𝑞𝑓)(𝑡	PROPN
cana-6124	66	25	)	)	PUNCT
cana-6124	66	26	.	.	PUNCT
cana-6124	67	1	communications	communication	NOUN
cana-6124	67	2	on	on	ADP
cana-6124	67	3	applied	apply	VERB
cana-6124	67	4	nonlinear	nonlinear	ADJ
cana-6124	67	5	analysis	analysis	NOUN
cana-6124	67	6	issn	issn	NOUN
cana-6124	67	7	:	:	PUNCT
cana-6124	67	8	1074	1074	NUM
cana-6124	67	9	-	-	PUNCT
cana-6124	67	10	133x	133x	NUM
cana-6124	67	11	vol	vol	VERB
cana-6124	67	12	32	32	NUM
cana-6124	67	13	no	no	NOUN
cana-6124	67	14	.	.	PUNCT
cana-6124	68	1	10s	10	NOUN
cana-6124	68	2	(	(	PUNCT
cana-6124	68	3	2025	2025	NUM
cana-6124	68	4	)	)	PUNCT
cana-6124	68	5	3429	3429	NUM
cana-6124	68	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	69	1	lemma	lemma	PROPN
cana-6124	69	2	5	5	NUM
cana-6124	70	1	[	[	X
cana-6124	70	2	15	15	NUM
cana-6124	70	3	]	]	PUNCT
cana-6124	70	4	for	for	ADP
cana-6124	70	5	all	all	DET
cana-6124	70	6	𝛼	𝛼	PRON
cana-6124	70	7	≥	≥	NOUN
cana-6124	70	8	0	0	NUM
cana-6124	70	9	and	and	CCONJ
cana-6124	70	10	𝛽	𝛽	ADP
cana-6124	70	11	∈	∈	PROPN
cana-6124	70	12	]	]	PUNCT
cana-6124	70	13	−	−	PROPN
cana-6124	71	1	1,+∞	1,+∞	NUM
cana-6124	71	2	[	[	X
cana-6124	71	3	,	,	PUNCT
cana-6124	71	4	we	we	PRON
cana-6124	71	5	have	have	VERB
cana-6124	71	6	(	(	PUNCT
cana-6124	71	7	𝐼𝑞	𝐼𝑞	PROPN
cana-6124	71	8	𝛼(𝑡	𝛼(𝑡	PROPN
cana-6124	71	9	−	−	NOUN
cana-6124	71	10	𝑎)(𝛽))(𝑡	𝑎)(𝛽))(𝑡	NOUN
cana-6124	71	11	)	)	PUNCT
cana-6124	71	12	=	=	SYM
cana-6124	71	13	γ𝑞(𝛽	γ𝑞(𝛽	PUNCT
cana-6124	72	1	+	+	NOUN
cana-6124	72	2	1	1	NUM
cana-6124	72	3	)	)	PUNCT
cana-6124	72	4	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	73	1	+	+	CCONJ
cana-6124	73	2	𝛽	𝛽	NOUN
cana-6124	73	3	+	+	NOUN
cana-6124	73	4	1	1	NUM
cana-6124	73	5	)	)	PUNCT
cana-6124	73	6	(	(	PUNCT
cana-6124	73	7	𝑡	𝑡	PROPN
cana-6124	73	8	−	−	NOUN
cana-6124	73	9	𝑎)(𝛼+𝛽	𝑎)(𝛼+𝛽	ADJ
cana-6124	73	10	)	)	PUNCT
cana-6124	73	11	,	,	PUNCT
cana-6124	73	12	0	0	PUNCT
cana-6124	73	13	<	<	X
cana-6124	73	14	𝛼	𝛼	X
cana-6124	73	15	<	<	X
cana-6124	73	16	𝑡	𝑡	X
cana-6124	73	17	<	<	X
cana-6124	73	18	1	1	NUM
cana-6124	73	19	.	.	PUNCT
cana-6124	74	1	in	in	ADP
cana-6124	74	2	particular	particular	ADJ
cana-6124	74	3	,	,	PUNCT
cana-6124	74	4	(	(	PUNCT
cana-6124	74	5	𝐼𝑞	𝐼𝑞	PROPN
cana-6124	74	6	𝛼1)(𝑡	𝛼1)(𝑡	PROPN
cana-6124	74	7	)	)	PUNCT
cana-6124	74	8	=	=	SYM
cana-6124	74	9	1	1	NUM
cana-6124	74	10	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	74	11	+	+	NOUN
cana-6124	74	12	1	1	X
cana-6124	74	13	)	)	PUNCT
cana-6124	74	14	𝑡(𝛼	𝑡(𝛼	PROPN
cana-6124	74	15	)	)	PUNCT
cana-6124	74	16	.	.	PUNCT
cana-6124	75	1	definition	definition	NOUN
cana-6124	75	2	6	6	NUM
cana-6124	75	3	[	[	X
cana-6124	75	4	14	14	NUM
cana-6124	75	5	]	]	PUNCT
cana-6124	75	6	the	the	DET
cana-6124	75	7	riemann	riemann	PROPN
cana-6124	75	8	-	-	PUNCT
cana-6124	75	9	liouville	liouville	VERB
cana-6124	75	10	fractional	fractional	ADJ
cana-6124	75	11	q	q	NOUN
cana-6124	75	12	-	-	NOUN
cana-6124	75	13	derivative	derivative	NOUN
cana-6124	75	14	of	of	ADP
cana-6124	75	15	order	order	NOUN
cana-6124	75	16	𝛼	𝛼	NOUN
cana-6124	75	17	≥	≥	NOUN
cana-6124	75	18	0	0	NUM
cana-6124	75	19	for	for	ADP
cana-6124	75	20	a	a	DET
cana-6124	75	21	function	function	NOUN
cana-6124	75	22	𝑓	𝑓	PROPN
cana-6124	75	23	:	:	PUNCT
cana-6124	75	24	𝐽	𝐽	PROPN
cana-6124	75	25	→	→	PUNCT
cana-6124	75	26	ℝ	ℝ	PROPN
cana-6124	75	27	is	be	AUX
cana-6124	75	28	defined	define	VERB
cana-6124	75	29	as	as	SCONJ
cana-6124	75	30	follows	follow	VERB
cana-6124	75	31	:	:	PUNCT
cana-6124	75	32	(	(	PUNCT
cana-6124	75	33	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	75	34	0𝑓)(𝑡	0𝑓)(𝑡	NUM
cana-6124	75	35	)	)	PUNCT
cana-6124	75	36	=	=	VERB
cana-6124	76	1	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	76	2	)	)	PUNCT
cana-6124	76	3	and	and	CCONJ
cana-6124	76	4	(	(	PUNCT
cana-6124	76	5	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	76	6	𝛼𝑓)(𝑡	𝛼𝑓)(𝑡	PROPN
cana-6124	76	7	)	)	PUNCT
cana-6124	76	8	=	=	PUNCT
cana-6124	77	1	(	(	PUNCT
cana-6124	77	2	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	77	3	[	[	X
cana-6124	77	4	𝛼]𝐼𝑞	𝛼]𝐼𝑞	PROPN
cana-6124	77	5	[	[	NOUN
cana-6124	77	6	𝛼]−𝛼𝑓	𝛼]−𝛼𝑓	NOUN
cana-6124	77	7	)	)	PUNCT
cana-6124	77	8	(	(	PUNCT
cana-6124	77	9	𝑡	𝑡	NOUN
cana-6124	77	10	)	)	PUNCT
cana-6124	77	11	,	,	PUNCT
cana-6124	77	12	𝑡	𝑡	PROPN
cana-6124	77	13	∈	∈	PROPN
cana-6124	77	14	𝐽	𝐽	PROPN
cana-6124	77	15	,	,	PUNCT
cana-6124	77	16	where	where	SCONJ
cana-6124	77	17	[	[	X
cana-6124	77	18	𝛼	𝛼	X
cana-6124	77	19	]	]	X
cana-6124	77	20	is	be	AUX
cana-6124	77	21	the	the	DET
cana-6124	77	22	integer	integer	ADJ
cana-6124	77	23	part	part	NOUN
cana-6124	77	24	of	of	ADP
cana-6124	77	25	𝛼.	𝛼.	ADJ
cana-6124	77	26	definition	definition	NOUN
cana-6124	77	27	7	7	NUM
cana-6124	77	28	[	[	X
cana-6124	77	29	14	14	NUM
cana-6124	77	30	]	]	PUNCT
cana-6124	77	31	consider	consider	VERB
cana-6124	77	32	a	a	DET
cana-6124	77	33	function	function	NOUN
cana-6124	77	34	𝑓	𝑓	NOUN
cana-6124	77	35	:	:	PUNCT
cana-6124	77	36	𝐽	𝐽	PROPN
cana-6124	77	37	→	→	PUNCT
cana-6124	77	38	ℝ	ℝ	PROPN
cana-6124	77	39	and	and	CCONJ
cana-6124	77	40	let	let	VERB
cana-6124	77	41	𝛼	𝛼	PRON
cana-6124	77	42	≥	≥	VERB
cana-6124	77	43	0	0	NUM
cana-6124	77	44	.	.	PUNCT
cana-6124	78	1	the	the	DET
cana-6124	78	2	caputo	caputo	PROPN
cana-6124	78	3	fractional	fractional	PROPN
cana-6124	78	4	qderivative	qderivative	ADJ
cana-6124	78	5	of	of	ADP
cana-6124	78	6	order	order	NOUN
cana-6124	78	7	𝛼	𝛼	NOUN
cana-6124	78	8	is	be	AUX
cana-6124	78	9	defined	define	VERB
cana-6124	78	10	as	as	SCONJ
cana-6124	78	11	follows	follow	VERB
cana-6124	78	12	:	:	PUNCT
cana-6124	78	13	(	(	PUNCT
cana-6124	78	14	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	78	15	0𝑓)(𝑡	0𝑓)(𝑡	NUM
cana-6124	78	16	)	)	PUNCT
cana-6124	78	17	=	=	VERB
cana-6124	79	1	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	79	2	)	)	PUNCT
cana-6124	79	3	and	and	CCONJ
cana-6124	79	4	(	(	PUNCT
cana-6124	79	5	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	79	6	𝛼𝑐	𝛼𝑐	NOUN
cana-6124	79	7	𝑓)(𝑡	𝑓)(𝑡	NOUN
cana-6124	79	8	)	)	PUNCT
cana-6124	79	9	=	=	PUNCT
cana-6124	79	10	(	(	PUNCT
cana-6124	79	11	𝐼𝑞	𝐼𝑞	PROPN
cana-6124	79	12	[	[	X
cana-6124	79	13	𝛼]−𝛼	𝛼]−𝛼	ADJ
cana-6124	79	14	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	79	15	[	[	X
cana-6124	79	16	𝛼	𝛼	X
cana-6124	79	17	]	]	X
cana-6124	79	18	𝑓	𝑓	X
cana-6124	79	19	)	)	PUNCT
cana-6124	79	20	(	(	PUNCT
cana-6124	79	21	𝑡	𝑡	NOUN
cana-6124	79	22	)	)	PUNCT
cana-6124	79	23	,	,	PUNCT
cana-6124	79	24	𝑡	𝑡	PROPN
cana-6124	79	25	∈	∈	PROPN
cana-6124	79	26	𝐽	𝐽	PROPN
cana-6124	79	27	,	,	PUNCT
cana-6124	79	28	where	where	SCONJ
cana-6124	79	29	[	[	X
cana-6124	79	30	𝛼	𝛼	X
cana-6124	79	31	]	]	X
cana-6124	79	32	is	be	AUX
cana-6124	79	33	the	the	DET
cana-6124	79	34	integer	integer	ADJ
cana-6124	79	35	part	part	NOUN
cana-6124	79	36	of	of	ADP
cana-6124	79	37	𝛼.	𝛼.	PROPN
cana-6124	79	38	lemma	lemma	PROPN
cana-6124	79	39	8	8	NUM
cana-6124	80	1	[	[	X
cana-6124	80	2	14	14	NUM
cana-6124	80	3	]	]	PUNCT
cana-6124	80	4	suppose	suppose	VERB
cana-6124	80	5	that	that	SCONJ
cana-6124	80	6	𝛼,𝛽	𝛼,𝛽	VERB
cana-6124	80	7	≥	≥	NOUN
cana-6124	80	8	0	0	NUM
cana-6124	80	9	,	,	PUNCT
cana-6124	80	10	and	and	CCONJ
cana-6124	80	11	let	let	VERB
cana-6124	80	12	𝑓	𝑓	X
cana-6124	80	13	:	:	PUNCT
cana-6124	80	14	𝐽	𝐽	PROPN
cana-6124	80	15	→	→	PUNCT
cana-6124	80	16	ℝ	ℝ	PROPN
cana-6124	80	17	be	be	AUX
cana-6124	80	18	a	a	DET
cana-6124	80	19	given	give	VERB
cana-6124	80	20	function	function	NOUN
cana-6124	80	21	.	.	PUNCT
cana-6124	81	1	then	then	ADV
cana-6124	81	2	,	,	PUNCT
cana-6124	81	3	the	the	DET
cana-6124	81	4	following	follow	VERB
cana-6124	81	5	identities	identity	NOUN
cana-6124	81	6	hold	hold	VERB
cana-6124	81	7	:	:	PUNCT
cana-6124	81	8	(	(	PUNCT
cana-6124	81	9	i	i	NOUN
cana-6124	81	10	)	)	PUNCT
cana-6124	81	11	(	(	PUNCT
cana-6124	81	12	𝐼𝑞	𝐼𝑞	PROPN
cana-6124	81	13	𝛼𝐼𝑞	𝛼𝐼𝑞	NOUN
cana-6124	81	14	𝛽	𝛽	NOUN
cana-6124	81	15	𝑓)(𝑡	𝑓)(𝑡	NOUN
cana-6124	81	16	)	)	PUNCT
cana-6124	81	17	=	=	SYM
cana-6124	81	18	(	(	PUNCT
cana-6124	81	19	𝐼𝑞	𝐼𝑞	PROPN
cana-6124	81	20	𝛼+𝛽	𝛼+𝛽	NUM
cana-6124	81	21	𝑓)(𝑡	𝑓)(𝑡	NUM
cana-6124	81	22	)	)	PUNCT
cana-6124	81	23	,	,	PUNCT
cana-6124	81	24	(	(	PUNCT
cana-6124	81	25	ii	ii	NOUN
cana-6124	81	26	)	)	PUNCT
cana-6124	81	27	(	(	PUNCT
cana-6124	81	28	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	81	29	𝛼𝐼𝑞	𝛼𝐼𝑞	NOUN
cana-6124	81	30	𝛽	𝛽	NOUN
cana-6124	81	31	𝑓)(𝑡	𝑓)(𝑡	NOUN
cana-6124	81	32	)	)	PUNCT
cana-6124	81	33	=	=	PUNCT
cana-6124	81	34	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	81	35	)	)	PUNCT
cana-6124	81	36	.	.	PUNCT
cana-6124	82	1	lemma	lemma	PROPN
cana-6124	82	2	9	9	NUM
cana-6124	83	1	[	[	X
cana-6124	83	2	14	14	NUM
cana-6124	83	3	]	]	PUNCT
cana-6124	83	4	assume	assume	VERB
cana-6124	83	5	𝛼	𝛼	X
cana-6124	83	6	≥	≥	NOUN
cana-6124	83	7	0	0	NUM
cana-6124	83	8	,	,	PUNCT
cana-6124	83	9	and	and	CCONJ
cana-6124	83	10	let	let	VERB
cana-6124	83	11	𝑓	𝑓	PRON
cana-6124	83	12	be	be	AUX
cana-6124	83	13	a	a	DET
cana-6124	83	14	function	function	NOUN
cana-6124	83	15	defined	define	VERB
cana-6124	83	16	on	on	ADP
cana-6124	83	17	the	the	DET
cana-6124	83	18	interval	interval	NOUN
cana-6124	83	19	𝐽.	𝐽.	PROPN
cana-6124	84	1	the	the	DET
cana-6124	84	2	following	follow	VERB
cana-6124	84	3	identity	identity	NOUN
cana-6124	84	4	holds	hold	VERB
cana-6124	84	5	(	(	PUNCT
cana-6124	84	6	𝐼𝑞	𝐼𝑞	PROPN
cana-6124	84	7	𝛼	𝛼	PRON
cana-6124	84	8	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	84	9	𝛼𝑐	𝛼𝑐	NOUN
cana-6124	84	10	𝑓)(𝑡	𝑓)(𝑡	NOUN
cana-6124	84	11	)	)	PUNCT
cana-6124	84	12	=	=	SYM
cana-6124	84	13	𝑓(𝑡	𝑓(𝑡	VERB
cana-6124	84	14	)	)	PUNCT
cana-6124	84	15	−	−	PROPN
cana-6124	85	1	∑	∑	ADV
cana-6124	85	2	𝑡𝑘	𝑡𝑘	ADV
cana-6124	85	3	γ𝑞(𝑘	γ𝑞(𝑘	PUNCT
cana-6124	86	1	+	+	CCONJ
cana-6124	86	2	1	1	X
cana-6124	86	3	)	)	PUNCT
cana-6124	86	4	(	(	PUNCT
cana-6124	86	5	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	86	6	𝛼𝑓)(0	𝛼𝑓)(0	NUM
cana-6124	86	7	)	)	PUNCT
cana-6124	86	8	.	.	PUNCT
cana-6124	87	1	[	[	X
cana-6124	87	2	𝛼]−1	𝛼]−1	NUM
cana-6124	87	3	𝑘=0	𝑘=0	VERB
cana-6124	87	4	if	if	SCONJ
cana-6124	87	5	𝛼	𝛼	PRON
cana-6124	87	6	∈]0,1	∈]0,1	NOUN
cana-6124	87	7	[	[	X
cana-6124	87	8	,	,	PUNCT
cana-6124	87	9	we	we	PRON
cana-6124	87	10	have	have	VERB
cana-6124	87	11	(	(	PUNCT
cana-6124	87	12	𝐼𝑞	𝐼𝑞	PROPN
cana-6124	87	13	𝛼	𝛼	PRON
cana-6124	87	14	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	87	15	𝛼𝑐	𝛼𝑐	NOUN
cana-6124	87	16	𝑓)(𝑡	𝑓)(𝑡	NOUN
cana-6124	87	17	)	)	PUNCT
cana-6124	87	18	=	=	SYM
cana-6124	87	19	𝑓(𝑡	𝑓(𝑡	PROPN
cana-6124	87	20	)	)	PUNCT
cana-6124	87	21	−	−	PROPN
cana-6124	87	22	𝑓(0	𝑓(0	NOUN
cana-6124	87	23	)	)	PUNCT
cana-6124	87	24	.	.	PUNCT
cana-6124	88	1	theorem	theorem	ADJ
cana-6124	88	2	10	10	NUM
cana-6124	88	3	(	(	PUNCT
cana-6124	88	4	banach	banach	NOUN
cana-6124	88	5	contraction	contraction	NOUN
cana-6124	88	6	principle	principle	NOUN
cana-6124	88	7	)	)	PUNCT
cana-6124	89	1	[	[	X
cana-6124	89	2	7	7	X
cana-6124	89	3	]	]	PUNCT
cana-6124	89	4	suppose	suppose	VERB
cana-6124	89	5	that	that	SCONJ
cana-6124	89	6	𝐶	𝐶	PROPN
cana-6124	89	7	is	be	AUX
cana-6124	89	8	a	a	DET
cana-6124	89	9	non	non	ADJ
cana-6124	89	10	-	-	ADJ
cana-6124	89	11	empty	empty	ADJ
cana-6124	89	12	closed	closed	ADJ
cana-6124	89	13	subset	subset	NOUN
cana-6124	89	14	of	of	ADP
cana-6124	89	15	a	a	DET
cana-6124	89	16	banach	banach	NOUN
cana-6124	89	17	space	space	NOUN
cana-6124	89	18	𝑋.	𝑋.	NOUN
cana-6124	89	19	if	if	SCONJ
cana-6124	89	20	𝐻	𝐻	PROPN
cana-6124	89	21	:	:	PUNCT
cana-6124	89	22	𝐶	𝐶	PROPN
cana-6124	89	23	→	→	SYM
cana-6124	89	24	𝐶	𝐶	PROPN
cana-6124	89	25	is	be	AUX
cana-6124	89	26	a	a	DET
cana-6124	89	27	contraction	contraction	NOUN
cana-6124	89	28	,	,	PUNCT
cana-6124	89	29	then	then	ADV
cana-6124	89	30	𝐻	𝐻	PROPN
cana-6124	89	31	admits	admit	VERB
cana-6124	89	32	a	a	DET
cana-6124	89	33	unique	unique	ADJ
cana-6124	89	34	fixed	fix	VERB
cana-6124	89	35	point	point	NOUN
cana-6124	89	36	in	in	ADP
cana-6124	89	37	𝐶.	𝐶.	PROPN
cana-6124	89	38	theorem	theorem	NOUN
cana-6124	89	39	11	11	NUM
cana-6124	89	40	(	(	PUNCT
cana-6124	89	41	schaefer	schaefer	NOUN
cana-6124	89	42	)	)	PUNCT
cana-6124	90	1	[	[	X
cana-6124	90	2	17	17	NUM
cana-6124	90	3	]	]	PUNCT
cana-6124	90	4	let	let	VERB
cana-6124	90	5	𝑋	𝑋	NOUN
cana-6124	90	6	be	be	AUX
cana-6124	90	7	a	a	DET
cana-6124	90	8	banach	banach	NOUN
cana-6124	90	9	space	space	NOUN
cana-6124	90	10	and	and	CCONJ
cana-6124	90	11	let	let	VERB
cana-6124	90	12	𝐻:𝑋	𝐻:𝑋	PROPN
cana-6124	90	13	→	→	SYM
cana-6124	90	14	𝑋	𝑋	NOUN
cana-6124	90	15	be	be	VERB
cana-6124	90	16	a	a	DET
cana-6124	90	17	completely	completely	ADV
cana-6124	90	18	continuous	continuous	ADJ
cana-6124	90	19	operator	operator	NOUN
cana-6124	90	20	.	.	PUNCT
cana-6124	91	1	assume	assume	VERB
cana-6124	91	2	that	that	SCONJ
cana-6124	91	3	the	the	DET
cana-6124	91	4	set	set	NOUN
cana-6124	91	5	ℰ	ℰ	NOUN
cana-6124	91	6	≔	≔	NOUN
cana-6124	91	7	{	{	PUNCT
cana-6124	91	8	𝑢	𝑢	NOUN
cana-6124	91	9	∈	∈	PROPN
cana-6124	91	10	𝑋	𝑋	NOUN
cana-6124	91	11	|𝑢	|𝑢	PROPN
cana-6124	91	12	=	=	SYM
cana-6124	91	13	𝜆𝐻(𝑢	𝜆𝐻(𝑢	NUM
cana-6124	91	14	)	)	PUNCT
cana-6124	91	15	,	,	PUNCT
cana-6124	91	16	𝜆	𝜆	SCONJ
cana-6124	91	17	∈]0,1	∈]0,1	PRON
cana-6124	91	18	[	[	X
cana-6124	91	19	}	}	PUNCT
cana-6124	91	20	is	be	AUX
cana-6124	91	21	bounded	bound	VERB
cana-6124	91	22	.	.	PUNCT
cana-6124	92	1	then	then	ADV
cana-6124	92	2	𝐻	𝐻	PROPN
cana-6124	92	3	admits	admit	VERB
cana-6124	92	4	a	a	DET
cana-6124	92	5	fixed	fixed	ADJ
cana-6124	92	6	point	point	NOUN
cana-6124	92	7	in	in	ADP
cana-6124	92	8	𝑋.	𝑋.	PROPN
cana-6124	92	9	communications	communication	NOUN
cana-6124	92	10	on	on	ADP
cana-6124	92	11	applied	apply	VERB
cana-6124	92	12	nonlinear	nonlinear	ADJ
cana-6124	92	13	analysis	analysis	NOUN
cana-6124	92	14	issn	issn	NOUN
cana-6124	92	15	:	:	PUNCT
cana-6124	92	16	1074	1074	NUM
cana-6124	92	17	-	-	PUNCT
cana-6124	92	18	133x	133x	NUM
cana-6124	92	19	vol	vol	VERB
cana-6124	92	20	32	32	NUM
cana-6124	92	21	no	no	NOUN
cana-6124	92	22	.	.	PUNCT
cana-6124	93	1	10s	10	NOUN
cana-6124	93	2	(	(	PUNCT
cana-6124	93	3	2025	2025	NUM
cana-6124	93	4	)	)	PUNCT
cana-6124	93	5	3430	3430	NUM
cana-6124	93	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	93	7	3	3	X
cana-6124	93	8	.	.	NOUN
cana-6124	93	9	existence	existence	NOUN
cana-6124	93	10	in	in	ADP
cana-6124	93	11	this	this	DET
cana-6124	93	12	section	section	NOUN
cana-6124	93	13	,	,	PUNCT
cana-6124	93	14	we	we	PRON
cana-6124	93	15	investigate	investigate	VERB
cana-6124	93	16	the	the	DET
cana-6124	93	17	existence	existence	NOUN
cana-6124	93	18	of	of	ADP
cana-6124	93	19	solutions	solution	NOUN
cana-6124	93	20	for	for	ADP
cana-6124	93	21	the	the	DET
cana-6124	93	22	fractional	fractional	ADJ
cana-6124	93	23	qqq	qqq	NOUN
cana-6124	93	24	-	-	PUNCT
cana-6124	93	25	difference	difference	NOUN
cana-6124	93	26	problem	problem	NOUN
cana-6124	93	27	given	give	VERB
cana-6124	93	28	by	by	ADP
cana-6124	93	29	(	(	PUNCT
cana-6124	93	30	1.1)-(1.2	1.1)-(1.2	NUM
cana-6124	93	31	)	)	PUNCT
cana-6124	93	32	.	.	PUNCT
cana-6124	94	1	definition	definition	NOUN
cana-6124	94	2	12	12	NUM
cana-6124	94	3	a	a	DET
cana-6124	94	4	function	function	NOUN
cana-6124	94	5	𝑢	𝑢	ADP
cana-6124	94	6	∈	∈	PROPN
cana-6124	94	7	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	94	8	]	]	X
cana-6124	94	9	,	,	PUNCT
cana-6124	94	10	𝑋	𝑋	PROPN
cana-6124	94	11	)	)	PUNCT
cana-6124	94	12	is	be	AUX
cana-6124	94	13	called	call	VERB
cana-6124	94	14	a	a	DET
cana-6124	94	15	solution	solution	NOUN
cana-6124	94	16	of	of	ADP
cana-6124	94	17	the	the	DET
cana-6124	94	18	fractional	fractional	ADJ
cana-6124	94	19	q	q	ADJ
cana-6124	94	20	-	-	PUNCT
cana-6124	94	21	difference	difference	NOUN
cana-6124	94	22	problem	problem	NOUN
cana-6124	94	23	(	(	PUNCT
cana-6124	94	24	1.1)-(1.2	1.1)-(1.2	NUM
cana-6124	94	25	)	)	PUNCT
cana-6124	94	26	if	if	SCONJ
cana-6124	94	27	𝑢	𝑢	NOUN
cana-6124	94	28	satisfies	satisfy	VERB
cana-6124	94	29	the	the	DET
cana-6124	94	30	equation	equation	NOUN
cana-6124	94	31	𝐷𝑐	𝐷𝑐	PROPN
cana-6124	94	32	𝑞	𝑞	NOUN
cana-6124	94	33	𝛼𝑢(𝑡	𝛼𝑢(𝑡	NOUN
cana-6124	94	34	)	)	PUNCT
cana-6124	94	35	=	=	SYM
cana-6124	94	36	𝐴𝑢(𝑡	𝐴𝑢(𝑡	X
cana-6124	94	37	)	)	PUNCT
cana-6124	95	1	+	+	CCONJ
cana-6124	95	2	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	95	3	,	,	PUNCT
cana-6124	95	4	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	95	5	)	)	PUNCT
cana-6124	95	6	)	)	PUNCT
cana-6124	95	7	on	on	ADP
cana-6124	95	8	𝐽	𝐽	PROPN
cana-6124	95	9	=	=	PUNCT
cana-6124	96	1	[	[	X
cana-6124	96	2	0,1	0,1	NUM
cana-6124	96	3	]	]	PUNCT
cana-6124	96	4	,	,	PUNCT
cana-6124	96	5	and	and	CCONJ
cana-6124	96	6	the	the	DET
cana-6124	96	7	condition	condition	NOUN
cana-6124	96	8	𝑢(0	𝑢(0	PROPN
cana-6124	96	9	)	)	PUNCT
cana-6124	96	10	=	=	PUNCT
cana-6124	97	1	𝑎	𝑎	DET
cana-6124	97	2	∫	∫	NOUN
cana-6124	97	3	𝑢(𝑠)𝑑𝑞𝑠	𝑢(𝑠)𝑑𝑞𝑠	PROPN
cana-6124	97	4	+	+	CCONJ
cana-6124	97	5	𝑏.	𝑏.	VERB
cana-6124	97	6	1	1	NUM
cana-6124	97	7	0	0	NUM
cana-6124	97	8	to	to	PART
cana-6124	97	9	establish	establish	VERB
cana-6124	97	10	the	the	DET
cana-6124	97	11	existence	existence	NOUN
cana-6124	97	12	of	of	ADP
cana-6124	97	13	solutions	solution	NOUN
cana-6124	97	14	for	for	ADP
cana-6124	97	15	the	the	DET
cana-6124	97	16	fractional	fractional	ADJ
cana-6124	97	17	problem	problem	NOUN
cana-6124	97	18	(	(	PUNCT
cana-6124	97	19	1.1)-(1.2	1.1)-(1.2	NUM
cana-6124	97	20	)	)	PUNCT
cana-6124	97	21	,	,	PUNCT
cana-6124	97	22	we	we	PRON
cana-6124	97	23	require	require	VERB
cana-6124	97	24	the	the	DET
cana-6124	97	25	following	follow	VERB
cana-6124	97	26	lemma	lemma	PROPN
cana-6124	97	27	:	:	PUNCT
cana-6124	97	28	lemma	lemma	PROPN
cana-6124	97	29	13	13	NUM
cana-6124	97	30	let	let	VERB
cana-6124	97	31	𝛼	𝛼	PRON
cana-6124	97	32	∈]0,1	∈]0,1	VERB
cana-6124	97	33	[	[	PUNCT
cana-6124	97	34	and	and	CCONJ
cana-6124	97	35	let	let	VERB
cana-6124	97	36	ℎ	ℎ	NOUN
cana-6124	97	37	:	:	PUNCT
cana-6124	97	38	[	[	X
cana-6124	97	39	0,1	0,1	NUM
cana-6124	97	40	]	]	X
cana-6124	97	41	×	×	NOUN
cana-6124	97	42	𝑋	𝑋	PROPN
cana-6124	97	43	→	→	SYM
cana-6124	97	44	𝑋	𝑋	PROPN
cana-6124	97	45	be	be	VERB
cana-6124	97	46	continuous	continuous	ADJ
cana-6124	97	47	function	function	NOUN
cana-6124	97	48	.	.	PUNCT
cana-6124	98	1	the	the	DET
cana-6124	98	2	solution	solution	NOUN
cana-6124	98	3	of	of	ADP
cana-6124	98	4	the	the	DET
cana-6124	98	5	fractional	fractional	ADJ
cana-6124	98	6	q	q	ADJ
cana-6124	98	7	-	-	PUNCT
cana-6124	98	8	difference	difference	NOUN
cana-6124	98	9	problem	problem	NOUN
cana-6124	98	10	𝐷𝑞	𝐷𝑞	PROPN
cana-6124	98	11	𝛼𝑐	𝛼𝑐	VERB
cana-6124	98	12	𝑢(𝑡	𝑢(𝑡	PROPN
cana-6124	98	13	)	)	PUNCT
cana-6124	98	14	=	=	SYM
cana-6124	98	15	𝐴(𝑡	𝐴(𝑡	PUNCT
cana-6124	98	16	)	)	PUNCT
cana-6124	98	17	+	+	CCONJ
cana-6124	98	18	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	98	19	,	,	PUNCT
cana-6124	98	20	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	98	21	)	)	PUNCT
cana-6124	98	22	)	)	PUNCT
cana-6124	98	23	,	,	PUNCT
cana-6124	98	24	𝑡	𝑡	PROPN
cana-6124	98	25	∈	∈	PROPN
cana-6124	98	26	𝐽	𝐽	NOUN
cana-6124	98	27	=	=	PUNCT
cana-6124	99	1	[	[	X
cana-6124	99	2	0,1	0,1	NUM
cana-6124	99	3	]	]	PUNCT
cana-6124	99	4	,	,	PUNCT
cana-6124	99	5	𝑢(0	𝑢(0	PROPN
cana-6124	99	6	)	)	PUNCT
cana-6124	99	7	=	=	PUNCT
cana-6124	99	8	𝑎	𝑎	DET
cana-6124	99	9	∫	∫	NOUN
cana-6124	99	10	𝑢(𝑠)𝑑𝑞𝑠	𝑢(𝑠)𝑑𝑞𝑠	PROPN
cana-6124	99	11	+	+	CCONJ
cana-6124	99	12	𝑏	𝑏	NOUN
cana-6124	99	13	,	,	PUNCT
cana-6124	99	14	1	1	NUM
cana-6124	99	15	0	0	NUM
cana-6124	99	16	is	be	AUX
cana-6124	99	17	given	give	VERB
cana-6124	99	18	by	by	ADP
cana-6124	99	19	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	99	20	)	)	PUNCT
cana-6124	99	21	=	=	SYM
cana-6124	99	22	𝑎∫	𝑎∫	ADJ
cana-6124	99	23	𝑢(𝑠)𝑑𝑞𝑠	𝑢(𝑠)𝑑𝑞𝑠	PROPN
cana-6124	99	24	+	+	CCONJ
cana-6124	99	25	𝑏	𝑏	PROPN
cana-6124	99	26	+	+	CCONJ
cana-6124	99	27	1	1	NUM
cana-6124	99	28	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	99	29	)	)	PUNCT
cana-6124	99	30	∫	∫	PROPN
cana-6124	99	31	(	(	PUNCT
cana-6124	99	32	𝑡	𝑡	PROPN
cana-6124	99	33	−	−	NOUN
cana-6124	99	34	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	99	35	)	)	PUNCT
cana-6124	99	36	𝑡	𝑡	PROPN
cana-6124	99	37	0	0	NUM
cana-6124	99	38	𝑓(𝑠	𝑓(𝑠	NOUN
cana-6124	99	39	,	,	PUNCT
cana-6124	99	40	𝑢(𝑠))𝑑𝑞𝑠	𝑢(𝑠))𝑑𝑞𝑠	VERB
cana-6124	99	41	1	1	NUM
cana-6124	99	42	0	0	NUM
cana-6124	99	43	+	+	CCONJ
cana-6124	99	44	1	1	NUM
cana-6124	99	45	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	99	46	)	)	PUNCT
cana-6124	99	47	∫	∫	PROPN
cana-6124	99	48	(	(	PUNCT
cana-6124	99	49	𝑡	𝑡	X
cana-6124	99	50	−	−	PROPN
cana-6124	99	51	𝑞𝑠)(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠.	𝑞𝑠)(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠.	NOUN
cana-6124	100	1	𝑡	𝑡	PROPN
cana-6124	100	2	0	0	NOUN
cana-6124	100	3	the	the	DET
cana-6124	100	4	first	first	ADJ
cana-6124	100	5	result	result	NOUN
cana-6124	100	6	is	be	AUX
cana-6124	100	7	obtained	obtain	VERB
cana-6124	100	8	by	by	ADP
cana-6124	100	9	applying	apply	VERB
cana-6124	100	10	the	the	DET
cana-6124	100	11	banach	banach	ADV
cana-6124	100	12	fixed	fix	VERB
cana-6124	100	13	point	point	NOUN
cana-6124	100	14	theorem	theorem	VERB
cana-6124	100	15	.	.	PUNCT
cana-6124	101	1	theorem	theorem	PROPN
cana-6124	101	2	14	14	NUM
cana-6124	101	3	suppose	suppose	VERB
cana-6124	101	4	that	that	SCONJ
cana-6124	101	5	:	:	PUNCT
cana-6124	101	6	(	(	PUNCT
cana-6124	101	7	𝐻1	𝐻1	PROPN
cana-6124	101	8	)	)	PUNCT
cana-6124	101	9	there	there	PRON
cana-6124	101	10	exist	exist	VERB
cana-6124	101	11	𝑘	𝑘	ADP
cana-6124	101	12	>	>	X
cana-6124	101	13	0	0	NUM
cana-6124	101	14	such	such	ADJ
cana-6124	101	15	that	that	SCONJ
cana-6124	101	16	∀𝑡	∀𝑡	PROPN
cana-6124	101	17	∈	∈	PROPN
cana-6124	101	18	𝐽	𝐽	PROPN
cana-6124	101	19	,	,	PUNCT
cana-6124	101	20	∀𝑢	∀𝑢	PROPN
cana-6124	101	21	,	,	PUNCT
cana-6124	101	22	𝑣	𝑣	PROPN
cana-6124	101	23	∈	∈	PROPN
cana-6124	101	24	𝑋	𝑋	PROPN
cana-6124	101	25	,	,	PUNCT
cana-6124	101	26	|𝑓(𝑡	|𝑓(𝑡	PROPN
cana-6124	101	27	,	,	PUNCT
cana-6124	101	28	𝑢	𝑢	NOUN
cana-6124	101	29	)	)	PUNCT
cana-6124	101	30	−	−	PROPN
cana-6124	102	1	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	102	2	,	,	PUNCT
cana-6124	102	3	𝑣)|	𝑣)|	NOUN
cana-6124	102	4	≤	≤	NUM
cana-6124	102	5	𝑘|𝑢	𝑘|𝑢	NOUN
cana-6124	102	6	−	−	PROPN
cana-6124	103	1	𝑣|	𝑣|	PROPN
cana-6124	103	2	.	.	PUNCT
cana-6124	104	1	if	if	SCONJ
cana-6124	104	2	|𝛼|	|𝛼|	PRON
cana-6124	104	3	+	+	CCONJ
cana-6124	104	4	𝑘	𝑘	PROPN
cana-6124	104	5	+	+	NUM
cana-6124	104	6	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	NOUN
cana-6124	104	7	)	)	PUNCT
cana-6124	104	8	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	105	1	+	+	CCONJ
cana-6124	105	2	1	1	X
cana-6124	105	3	)	)	PUNCT
cana-6124	105	4	<	<	X
cana-6124	105	5	1	1	X
cana-6124	105	6	.	.	PUNCT
cana-6124	105	7	(	(	PUNCT
cana-6124	105	8	3.1	3.1	NUM
cana-6124	105	9	)	)	PUNCT
cana-6124	105	10	then	then	ADV
cana-6124	105	11	,	,	PUNCT
cana-6124	105	12	the	the	DET
cana-6124	105	13	fractional	fractional	ADJ
cana-6124	105	14	q	q	ADJ
cana-6124	105	15	-	-	PUNCT
cana-6124	105	16	difference	difference	NOUN
cana-6124	105	17	problem	problem	NOUN
cana-6124	105	18	(	(	PUNCT
cana-6124	105	19	1.1)-(1.2	1.1)-(1.2	NUM
cana-6124	105	20	)	)	PUNCT
cana-6124	105	21	admits	admit	VERB
cana-6124	105	22	a	a	DET
cana-6124	105	23	unique	unique	ADJ
cana-6124	105	24	solution	solution	NOUN
cana-6124	105	25	on	on	ADP
cana-6124	105	26	[	[	X
cana-6124	105	27	0,1	0,1	NUM
cana-6124	105	28	]	]	PUNCT
cana-6124	105	29	.	.	PUNCT
cana-6124	106	1	proof	proof	NOUN
cana-6124	106	2	15	15	NUM
cana-6124	106	3	the	the	DET
cana-6124	106	4	fractional	fractional	ADJ
cana-6124	106	5	q	q	ADJ
cana-6124	106	6	-	-	PUNCT
cana-6124	106	7	difference	difference	NOUN
cana-6124	106	8	problem	problem	NOUN
cana-6124	106	9	(	(	PUNCT
cana-6124	106	10	1.1)-(1.2	1.1)-(1.2	NUM
cana-6124	106	11	)	)	PUNCT
cana-6124	106	12	can	can	AUX
cana-6124	106	13	be	be	AUX
cana-6124	106	14	reformulated	reformulate	VERB
cana-6124	106	15	in	in	ADP
cana-6124	106	16	terms	term	NOUN
cana-6124	106	17	of	of	ADP
cana-6124	106	18	a	a	DET
cana-6124	106	19	fixed	fix	VERB
cana-6124	106	20	point	point	NOUN
cana-6124	106	21	problem	problem	NOUN
cana-6124	106	22	.	.	PUNCT
cana-6124	107	1	for	for	ADP
cana-6124	107	2	this	this	DET
cana-6124	107	3	purpose	purpose	NOUN
cana-6124	107	4	,	,	PUNCT
cana-6124	107	5	we	we	PRON
cana-6124	107	6	define	define	VERB
cana-6124	107	7	the	the	DET
cana-6124	107	8	operator	operator	NOUN
cana-6124	107	9	𝐹	𝐹	PROPN
cana-6124	107	10	:	:	PUNCT
cana-6124	107	11	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	107	12	]	]	X
cana-6124	107	13	,	,	PUNCT
cana-6124	107	14	𝑋	𝑋	PROPN
cana-6124	107	15	)	)	PUNCT
cana-6124	107	16	→	→	SYM
cana-6124	107	17	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	107	18	]	]	X
cana-6124	107	19	,	,	PUNCT
cana-6124	107	20	𝑋	𝑋	PROPN
cana-6124	107	21	)	)	PUNCT
cana-6124	107	22	,	,	PUNCT
cana-6124	107	23	where	where	SCONJ
cana-6124	107	24	communications	communication	NOUN
cana-6124	107	25	on	on	ADP
cana-6124	107	26	applied	apply	VERB
cana-6124	107	27	nonlinear	nonlinear	ADJ
cana-6124	107	28	analysis	analysis	NOUN
cana-6124	107	29	issn	issn	NOUN
cana-6124	107	30	:	:	PUNCT
cana-6124	107	31	1074	1074	NUM
cana-6124	107	32	-	-	PUNCT
cana-6124	107	33	133x	133x	NUM
cana-6124	107	34	vol	vol	VERB
cana-6124	107	35	32	32	NUM
cana-6124	107	36	no	no	NOUN
cana-6124	107	37	.	.	PUNCT
cana-6124	108	1	10s	10	NOUN
cana-6124	108	2	(	(	PUNCT
cana-6124	108	3	2025	2025	NUM
cana-6124	108	4	)	)	PUNCT
cana-6124	108	5	3431	3431	NUM
cana-6124	108	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	108	7	𝐹(𝑢(𝑡	𝐹(𝑢(𝑡	NOUN
cana-6124	108	8	)	)	PUNCT
cana-6124	108	9	)	)	PUNCT
cana-6124	108	10	≔	≔	VERB
cana-6124	108	11	𝑎∫	𝑎∫	ADJ
cana-6124	108	12	𝑢(𝑠)𝑑𝑞𝑠	𝑢(𝑠)𝑑𝑞𝑠	PROPN
cana-6124	108	13	+	+	CCONJ
cana-6124	108	14	𝑏	𝑏	PROPN
cana-6124	108	15	+	+	CCONJ
cana-6124	108	16	1	1	NUM
cana-6124	108	17	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	108	18	)	)	PUNCT
cana-6124	108	19	∫	∫	PROPN
cana-6124	108	20	(	(	PUNCT
cana-6124	108	21	𝑡	𝑡	PROPN
cana-6124	108	22	−	−	NOUN
cana-6124	108	23	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	108	24	)	)	PUNCT
cana-6124	108	25	𝑡	𝑡	PROPN
cana-6124	108	26	0	0	NUM
cana-6124	108	27	𝑓(𝑠	𝑓(𝑠	NOUN
cana-6124	108	28	,	,	PUNCT
cana-6124	108	29	𝑢(𝑠))𝑑𝑞𝑠	𝑢(𝑠))𝑑𝑞𝑠	VERB
cana-6124	108	30	1	1	NUM
cana-6124	108	31	0	0	NUM
cana-6124	108	32	(	(	PUNCT
cana-6124	108	33	3.2	3.2	NUM
cana-6124	108	34	)	)	PUNCT
cana-6124	108	35	+	+	CCONJ
cana-6124	108	36	1	1	NUM
cana-6124	108	37	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	108	38	)	)	PUNCT
cana-6124	108	39	∫	∫	PROPN
cana-6124	109	1	(	(	PUNCT
cana-6124	109	2	𝑡	𝑡	X
cana-6124	109	3	−	−	PROPN
cana-6124	109	4	𝑞𝑠)(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠.	𝑞𝑠)(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠.	NOUN
cana-6124	110	1	𝑡	𝑡	NOUN
cana-6124	110	2	0	0	NUM
cana-6124	111	1	it	it	PRON
cana-6124	111	2	is	be	AUX
cana-6124	111	3	evident	evident	ADJ
cana-6124	111	4	that	that	SCONJ
cana-6124	111	5	the	the	DET
cana-6124	111	6	fixed	fix	VERB
cana-6124	111	7	points	point	NOUN
cana-6124	111	8	of	of	ADP
cana-6124	111	9	the	the	DET
cana-6124	111	10	operator	operator	NOUN
cana-6124	111	11	𝐹	𝐹	PRON
cana-6124	111	12	correspond	correspond	VERB
cana-6124	111	13	to	to	ADP
cana-6124	111	14	solutions	solution	NOUN
cana-6124	111	15	of	of	ADP
cana-6124	111	16	the	the	DET
cana-6124	111	17	fractional	fractional	ADJ
cana-6124	111	18	qdifference	qdifference	NOUN
cana-6124	111	19	problem	problem	NOUN
cana-6124	111	20	(	(	PUNCT
cana-6124	111	21	1.1)-(1.2	1.1)-(1.2	NUM
cana-6124	111	22	)	)	PUNCT
cana-6124	111	23	.	.	PUNCT
cana-6124	112	1	we	we	PRON
cana-6124	112	2	shall	shall	AUX
cana-6124	112	3	use	use	VERB
cana-6124	112	4	the	the	DET
cana-6124	112	5	banach	banach	NOUN
cana-6124	112	6	contraction	contraction	NOUN
cana-6124	112	7	mapping	mapping	NOUN
cana-6124	112	8	principle	principle	NOUN
cana-6124	112	9	to	to	PART
cana-6124	112	10	demonstrate	demonstrate	VERB
cana-6124	112	11	that	that	SCONJ
cana-6124	112	12	𝐹	𝐹	PROPN
cana-6124	112	13	defined	define	VERB
cana-6124	112	14	by	by	ADP
cana-6124	112	15	(	(	PUNCT
cana-6124	112	16	3.2	3.2	NUM
cana-6124	112	17	)	)	PUNCT
cana-6124	112	18	has	have	VERB
cana-6124	112	19	a	a	DET
cana-6124	112	20	fixed	fix	VERB
cana-6124	112	21	point	point	NOUN
cana-6124	112	22	.	.	PUNCT
cana-6124	113	1	we	we	PRON
cana-6124	113	2	shall	shall	AUX
cana-6124	113	3	demonstrate	demonstrate	VERB
cana-6124	113	4	that	that	SCONJ
cana-6124	113	5	𝐹	𝐹	PROPN
cana-6124	113	6	is	be	AUX
cana-6124	113	7	a	a	DET
cana-6124	113	8	contraction	contraction	NOUN
cana-6124	113	9	.	.	PUNCT
cana-6124	114	1	let	let	VERB
cana-6124	114	2	𝑢	𝑢	NOUN
cana-6124	114	3	,	,	PUNCT
cana-6124	114	4	𝑣	𝑣	PRON
cana-6124	114	5	∈	∈	PROPN
cana-6124	114	6	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	114	7	]	]	X
cana-6124	114	8	,	,	PUNCT
cana-6124	114	9	𝑋	𝑋	PROPN
cana-6124	114	10	)	)	PUNCT
cana-6124	114	11	and	and	CCONJ
cana-6124	114	12	𝑡	𝑡	PROPN
cana-6124	114	13	∈	∈	PROPN
cana-6124	114	14	[	[	X
cana-6124	114	15	0,1	0,1	NUM
cana-6124	114	16	]	]	PUNCT
cana-6124	114	17	,	,	PUNCT
cana-6124	114	18	we	we	PRON
cana-6124	114	19	obtain	obtain	VERB
cana-6124	114	20	|𝐹(𝑢)(𝑡	|𝐹(𝑢)(𝑡	NOUN
cana-6124	114	21	)	)	PUNCT
cana-6124	114	22	−	−	PROPN
cana-6124	114	23	𝐹(𝑣)(𝑡)|	𝐹(𝑣)(𝑡)|	PROPN
cana-6124	114	24	≤	≤	PROPN
cana-6124	114	25	|𝑎|∫	|𝑎|∫	PROPN
cana-6124	114	26	|𝑢(𝑠	|𝑢(𝑠	PROPN
cana-6124	114	27	)	)	PUNCT
cana-6124	114	28	−	−	PROPN
cana-6124	115	1	𝑣(𝑠)|𝑑𝑞𝑠	𝑣(𝑠)|𝑑𝑞𝑠	ADP
cana-6124	115	2	+	+	NOUN
cana-6124	115	3	1	1	NUM
cana-6124	115	4	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	115	5	)	)	PUNCT
cana-6124	115	6	∫	∫	PROPN
cana-6124	115	7	(	(	PUNCT
cana-6124	115	8	𝑡	𝑡	PROPN
cana-6124	115	9	−	−	NOUN
cana-6124	115	10	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	115	11	)	)	PUNCT
cana-6124	115	12	𝑡	𝑡	PROPN
cana-6124	115	13	0	0	PROPN
cana-6124	115	14	|𝑓(𝑠	|𝑓(𝑠	PROPN
cana-6124	115	15	,	,	PUNCT
cana-6124	115	16	𝑢(𝑠	𝑢(𝑠	NOUN
cana-6124	115	17	)	)	PUNCT
cana-6124	115	18	)	)	PUNCT
cana-6124	115	19	−	−	PROPN
cana-6124	115	20	𝑓(𝑠	𝑓(𝑠	NOUN
cana-6124	115	21	,	,	PUNCT
cana-6124	115	22	𝑣(𝑠))|𝑑𝑞𝑠	𝑣(𝑠))|𝑑𝑞𝑠	PROPN
cana-6124	115	23	1	1	NUM
cana-6124	115	24	0	0	NUM
cana-6124	115	25	+	+	CCONJ
cana-6124	115	26	1	1	NUM
cana-6124	115	27	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	115	28	)	)	PUNCT
cana-6124	115	29	∫	∫	PROPN
cana-6124	115	30	(	(	PUNCT
cana-6124	115	31	𝑡	𝑡	PROPN
cana-6124	115	32	−	−	PROPN
cana-6124	115	33	𝑞𝑠)(𝛼−1)‖𝐴‖ℒ(𝑋)|𝑢(𝑠	𝑞𝑠)(𝛼−1)‖𝐴‖ℒ(𝑋)|𝑢(𝑠	NOUN
cana-6124	115	34	)	)	PUNCT
cana-6124	115	35	𝑡	𝑡	PROPN
cana-6124	115	36	0	0	NUM
cana-6124	115	37	−	−	NOUN
cana-6124	115	38	𝑣(𝑠)|𝑑𝑞𝑠	𝑣(𝑠)|𝑑𝑞𝑠	ADP
cana-6124	115	39	≤	≤	NUM
cana-6124	115	40	|𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠	|𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠	PROPN
cana-6124	115	41	)	)	PUNCT
cana-6124	115	42	−	−	ADP
cana-6124	116	1	𝑣(𝑠)|	𝑣(𝑠)|	NOUN
cana-6124	116	2	+	+	CCONJ
cana-6124	116	3	𝑘	𝑘	PRON
cana-6124	116	4	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	116	5	)	)	PUNCT
cana-6124	116	6	𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠	𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠	NOUN
cana-6124	116	7	)	)	PUNCT
cana-6124	116	8	−	−	PROPN
cana-6124	116	9	𝑣(𝑠)|∫	𝑣(𝑠)|∫	PROPN
cana-6124	116	10	(	(	PUNCT
cana-6124	116	11	𝑡	𝑡	PROPN
cana-6124	116	12	−	−	PROPN
cana-6124	116	13	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	116	14	)	)	PUNCT
cana-6124	116	15	(	(	PUNCT
cana-6124	116	16	𝛼−1)𝑑𝑞𝑠	𝛼−1)𝑑𝑞𝑠	NUM
cana-6124	116	17	𝑡	𝑡	NOUN
cana-6124	116	18	0	0	PUNCT
cana-6124	116	19	+	+	NUM
cana-6124	116	20	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	PROPN
cana-6124	116	21	)	)	PUNCT
cana-6124	116	22	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-6124	116	23	)	)	PUNCT
cana-6124	116	24	𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠	𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠	NOUN
cana-6124	116	25	)	)	PUNCT
cana-6124	116	26	−	−	PROPN
cana-6124	116	27	𝑣(𝑠)|∫	𝑣(𝑠)|∫	PROPN
cana-6124	116	28	(	(	PUNCT
cana-6124	116	29	𝑡	𝑡	PROPN
cana-6124	116	30	−	−	PROPN
cana-6124	116	31	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	116	32	)	)	PUNCT
cana-6124	116	33	(	(	PUNCT
cana-6124	116	34	𝛼−1)𝑑𝑞𝑠	𝛼−1)𝑑𝑞𝑠	NUM
cana-6124	116	35	𝑡	𝑡	NOUN
cana-6124	116	36	0	0	NUM
cana-6124	116	37	≤	≤	NUM
cana-6124	116	38	|𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠	|𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠	PROPN
cana-6124	116	39	)	)	PUNCT
cana-6124	116	40	−	−	ADP
cana-6124	117	1	𝑣(𝑠)|	𝑣(𝑠)|	NOUN
cana-6124	117	2	+	+	NUM
cana-6124	117	3	𝑘	𝑘	DET
cana-6124	117	4	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	117	5	+	+	CCONJ
cana-6124	117	6	1	1	X
cana-6124	117	7	)	)	PUNCT
cana-6124	117	8	𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠	𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠	NOUN
cana-6124	117	9	)	)	PUNCT
cana-6124	117	10	−	−	ADP
cana-6124	118	1	𝑣(𝑠)|	𝑣(𝑠)|	NOUN
cana-6124	118	2	+	+	NUM
cana-6124	118	3	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	PROPN
cana-6124	118	4	)	)	PUNCT
cana-6124	118	5	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	119	1	+	+	CCONJ
cana-6124	119	2	1	1	X
cana-6124	119	3	)	)	PUNCT
cana-6124	119	4	𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠	𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠	NOUN
cana-6124	119	5	)	)	PUNCT
cana-6124	119	6	−	−	ADP
cana-6124	119	7	𝑣(𝑠)|	𝑣(𝑠)|	NOUN
cana-6124	119	8	=	=	PUNCT
cana-6124	119	9	(	(	PUNCT
cana-6124	119	10	|𝑎|	|𝑎|	PROPN
cana-6124	119	11	+	+	CCONJ
cana-6124	119	12	𝑘	𝑘	PROPN
cana-6124	119	13	+	+	NUM
cana-6124	119	14	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	NOUN
cana-6124	119	15	)	)	PUNCT
cana-6124	119	16	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	120	1	+	+	CCONJ
cana-6124	120	2	1	1	NUM
cana-6124	120	3	)	)	PUNCT
cana-6124	120	4	)	)	PUNCT
cana-6124	120	5	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠	NOUN
cana-6124	120	6	)	)	PUNCT
cana-6124	120	7	−	−	ADP
cana-6124	120	8	𝑣(𝑠)|	𝑣(𝑠)|	NOUN
cana-6124	120	9	.	.	PUNCT
cana-6124	121	1	hence	hence	ADV
cana-6124	121	2	,	,	PUNCT
cana-6124	121	3	‖𝐹(𝑢	‖𝐹(𝑢	PROPN
cana-6124	121	4	)	)	PUNCT
cana-6124	121	5	−	−	NOUN
cana-6124	121	6	𝐹(𝑣)‖∞	𝐹(𝑣)‖∞	NOUN
cana-6124	121	7	≤	≤	NOUN
cana-6124	122	1	[	[	X
cana-6124	122	2	|𝑎|	|𝑎|	PROPN
cana-6124	122	3	+	+	CCONJ
cana-6124	122	4	𝑘	𝑘	PROPN
cana-6124	122	5	+	+	NUM
cana-6124	122	6	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	NOUN
cana-6124	122	7	)	)	PUNCT
cana-6124	122	8	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	123	1	+	+	CCONJ
cana-6124	123	2	1	1	NUM
cana-6124	123	3	)	)	PUNCT
cana-6124	123	4	]	]	PUNCT
cana-6124	124	1	‖𝑢	‖𝑢	DET
cana-6124	124	2	−	−	PROPN
cana-6124	124	3	𝑣‖∞.	𝑣‖∞.	PROPN
cana-6124	124	4	according	accord	VERB
cana-6124	124	5	to	to	ADP
cana-6124	124	6	(	(	PUNCT
cana-6124	124	7	3.2	3.2	NUM
cana-6124	124	8	)	)	PUNCT
cana-6124	124	9	,	,	PUNCT
cana-6124	124	10	the	the	DET
cana-6124	124	11	operator	operator	NOUN
cana-6124	124	12	𝐹	𝐹	PROPN
cana-6124	124	13	satisfies	satisfy	VERB
cana-6124	124	14	the	the	DET
cana-6124	124	15	contraction	contraction	NOUN
cana-6124	124	16	condition	condition	NOUN
cana-6124	124	17	.	.	PUNCT
cana-6124	125	1	hence	hence	ADV
cana-6124	125	2	,	,	PUNCT
cana-6124	125	3	by	by	ADP
cana-6124	125	4	the	the	DET
cana-6124	125	5	banach	banach	ADV
cana-6124	125	6	fixed	fix	VERB
cana-6124	125	7	point	point	NOUN
cana-6124	125	8	theorem	theorem	VERB
cana-6124	125	9	,	,	PUNCT
cana-6124	125	10	𝐹	𝐹	PROPN
cana-6124	125	11	admits	admit	VERB
cana-6124	125	12	a	a	DET
cana-6124	125	13	fixed	fixed	ADJ
cana-6124	125	14	point	point	NOUN
cana-6124	125	15	,	,	PUNCT
cana-6124	125	16	representing	represent	VERB
cana-6124	125	17	the	the	DET
cana-6124	125	18	solution	solution	NOUN
cana-6124	125	19	of	of	ADP
cana-6124	125	20	the	the	DET
cana-6124	125	21	fractional	fractional	ADJ
cana-6124	125	22	q	q	ADJ
cana-6124	125	23	-	-	PUNCT
cana-6124	125	24	difference	difference	NOUN
cana-6124	125	25	problem	problem	NOUN
cana-6124	125	26	(	(	PUNCT
cana-6124	125	27	1.1)-(1.2	1.1)-(1.2	NUM
cana-6124	125	28	)	)	PUNCT
cana-6124	125	29	.	.	PUNCT
cana-6124	126	1	theorem	theorem	ADJ
cana-6124	126	2	16	16	NUM
cana-6124	126	3	suppose	suppose	VERB
cana-6124	126	4	that	that	SCONJ
cana-6124	126	5	:	:	PUNCT
cana-6124	126	6	(	(	PUNCT
cana-6124	126	7	𝐻2	𝐻2	PROPN
cana-6124	126	8	)	)	PUNCT
cana-6124	126	9	the	the	DET
cana-6124	126	10	function	function	NOUN
cana-6124	126	11	𝑓	𝑓	X
cana-6124	126	12	:	:	PUNCT
cana-6124	126	13	[	[	X
cana-6124	126	14	0,1	0,1	NUM
cana-6124	126	15	]	]	X
cana-6124	126	16	×	×	NOUN
cana-6124	126	17	𝑋	𝑋	PROPN
cana-6124	126	18	→	→	SYM
cana-6124	126	19	𝑋	𝑋	PROPN
cana-6124	126	20	is	be	AUX
cana-6124	126	21	continuous	continuous	ADJ
cana-6124	126	22	.	.	PUNCT
cana-6124	127	1	(	(	PUNCT
cana-6124	127	2	𝐻3	𝐻3	PROPN
cana-6124	127	3	)	)	PUNCT
cana-6124	128	1	∃𝑀	∃𝑀	NOUN
cana-6124	128	2	>	>	PUNCT
cana-6124	128	3	0	0	NUM
cana-6124	128	4	,	,	PUNCT
cana-6124	128	5	∀𝑡	∀𝑡	PROPN
cana-6124	128	6	∈	∈	PROPN
cana-6124	128	7	𝐽	𝐽	PROPN
cana-6124	128	8	,	,	PUNCT
cana-6124	128	9	∀𝑢	∀𝑢	PRON
cana-6124	128	10	∈	∈	PROPN
cana-6124	128	11	𝑋	𝑋	PROPN
cana-6124	128	12	,	,	PUNCT
cana-6124	128	13	|𝑓(𝑡	|𝑓(𝑡	PROPN
cana-6124	128	14	,	,	PUNCT
cana-6124	128	15	𝑢)|	𝑢)|	VERB
cana-6124	128	16	≤	≤	NUM
cana-6124	128	17	𝑀.	𝑀.	PROPN
cana-6124	128	18	hence	hence	ADV
cana-6124	128	19	,	,	PUNCT
cana-6124	128	20	the	the	DET
cana-6124	128	21	fractional	fractional	ADJ
cana-6124	128	22	q	q	ADJ
cana-6124	128	23	-	-	PUNCT
cana-6124	128	24	difference	difference	NOUN
cana-6124	128	25	problem	problem	NOUN
cana-6124	128	26	(	(	PUNCT
cana-6124	128	27	1.1)-(1.2	1.1)-(1.2	NUM
cana-6124	128	28	)	)	PUNCT
cana-6124	128	29	admits	admit	VERB
cana-6124	128	30	at	at	ADP
cana-6124	128	31	least	least	ADV
cana-6124	128	32	one	one	NUM
cana-6124	128	33	solution	solution	NOUN
cana-6124	128	34	on	on	ADP
cana-6124	128	35	[	[	X
cana-6124	128	36	0,1	0,1	NUM
cana-6124	128	37	]	]	PUNCT
cana-6124	128	38	.	.	PUNCT
cana-6124	129	1	communications	communication	NOUN
cana-6124	129	2	on	on	ADP
cana-6124	129	3	applied	apply	VERB
cana-6124	129	4	nonlinear	nonlinear	ADJ
cana-6124	129	5	analysis	analysis	NOUN
cana-6124	129	6	issn	issn	NOUN
cana-6124	129	7	:	:	PUNCT
cana-6124	129	8	1074	1074	NUM
cana-6124	129	9	-	-	PUNCT
cana-6124	129	10	133x	133x	NUM
cana-6124	129	11	vol	vol	VERB
cana-6124	129	12	32	32	NUM
cana-6124	129	13	no	no	NOUN
cana-6124	129	14	.	.	PUNCT
cana-6124	130	1	10s	10	NOUN
cana-6124	130	2	(	(	PUNCT
cana-6124	130	3	2025	2025	NUM
cana-6124	130	4	)	)	PUNCT
cana-6124	130	5	3432	3432	NUM
cana-6124	131	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	131	2	proof	proof	NOUN
cana-6124	131	3	17	17	NUM
cana-6124	131	4	we	we	PRON
cana-6124	131	5	apply	apply	VERB
cana-6124	131	6	schaefer	schaefer	PROPN
cana-6124	131	7	’s	’s	PART
cana-6124	131	8	fixed	fix	VERB
cana-6124	131	9	point	point	NOUN
cana-6124	131	10	theorem	theorem	VERB
cana-6124	131	11	to	to	PART
cana-6124	131	12	verify	verify	VERB
cana-6124	131	13	that	that	SCONJ
cana-6124	131	14	the	the	DET
cana-6124	131	15	operator	operator	NOUN
cana-6124	131	16	𝐹	𝐹	PROPN
cana-6124	131	17	,	,	PUNCT
cana-6124	131	18	defined	define	VERB
cana-6124	131	19	in	in	ADP
cana-6124	131	20	(	(	PUNCT
cana-6124	131	21	3.2	3.2	NUM
cana-6124	131	22	)	)	PUNCT
cana-6124	131	23	,	,	PUNCT
cana-6124	131	24	admits	admit	VERB
cana-6124	131	25	a	a	DET
cana-6124	131	26	fixed	fixed	ADJ
cana-6124	131	27	point	point	NOUN
cana-6124	131	28	.	.	PUNCT
cana-6124	132	1	the	the	DET
cana-6124	132	2	proof	proof	NOUN
cana-6124	132	3	is	be	AUX
cana-6124	132	4	organized	organize	VERB
cana-6124	132	5	in	in	ADP
cana-6124	132	6	four	four	NUM
cana-6124	132	7	steps	step	NOUN
cana-6124	132	8	.	.	PUNCT
cana-6124	133	1	step	step	NOUN
cana-6124	133	2	1	1	NUM
cana-6124	133	3	:	:	PUNCT
cana-6124	133	4	𝐹	𝐹	PROPN
cana-6124	133	5	is	be	AUX
cana-6124	133	6	continuous	continuous	ADJ
cana-6124	133	7	operator	operator	NOUN
cana-6124	133	8	.	.	PUNCT
cana-6124	134	1	consider	consider	VERB
cana-6124	134	2	a	a	DET
cana-6124	134	3	sequence	sequence	NOUN
cana-6124	134	4	{	{	PUNCT
cana-6124	134	5	𝑢𝑛	𝑢𝑛	X
cana-6124	134	6	}	}	PUNCT
cana-6124	134	7	such	such	ADJ
cana-6124	134	8	that	that	SCONJ
cana-6124	134	9	𝑢𝑛	𝑢𝑛	NOUN
cana-6124	134	10	→	→	SYM
cana-6124	134	11	𝑢	𝑢	X
cana-6124	134	12	in	in	ADP
cana-6124	134	13	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	134	14	]	]	PUNCT
cana-6124	134	15	,	,	PUNCT
cana-6124	134	16	𝑋	𝑋	PROPN
cana-6124	134	17	)	)	PUNCT
cana-6124	134	18	.	.	PUNCT
cana-6124	135	1	then	then	ADV
cana-6124	135	2	for	for	ADP
cana-6124	135	3	all	all	DET
cana-6124	135	4	𝑡	𝑡	ADP
cana-6124	135	5	∈	∈	PROPN
cana-6124	135	6	[	[	X
cana-6124	135	7	0,1	0,1	NUM
cana-6124	135	8	]	]	PUNCT
cana-6124	135	9	:	:	PUNCT
cana-6124	135	10	|𝐹(𝑢𝑛)(𝑡	|𝐹(𝑢𝑛)(𝑡	X
cana-6124	135	11	)	)	PUNCT
cana-6124	136	1	−	−	PROPN
cana-6124	136	2	𝐹(𝑢)(𝑡)|	𝐹(𝑢)(𝑡)|	PROPN
cana-6124	136	3	≤	≤	PROPN
cana-6124	136	4	|𝑎|∫	|𝑎|∫	PROPN
cana-6124	137	1	|𝑢𝑛(𝑠	|𝑢𝑛(𝑠	PROPN
cana-6124	137	2	)	)	PUNCT
cana-6124	138	1	−	−	NOUN
cana-6124	138	2	𝑢(𝑠)|𝑑𝑞𝑠	𝑢(𝑠)|𝑑𝑞𝑠	NOUN
cana-6124	138	3	1	1	NUM
cana-6124	138	4	0	0	NUM
cana-6124	139	1	+	+	CCONJ
cana-6124	139	2	1	1	NUM
cana-6124	139	3	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	139	4	)	)	PUNCT
cana-6124	139	5	∫	∫	PROPN
cana-6124	139	6	(	(	PUNCT
cana-6124	139	7	𝑡	𝑡	PROPN
cana-6124	139	8	−	−	PROPN
cana-6124	139	9	𝑞𝑠)(𝛼−1)|𝑓(𝑠	𝑞𝑠)(𝛼−1)|𝑓(𝑠	PROPN
cana-6124	139	10	,	,	PUNCT
cana-6124	139	11	𝑢𝑛(𝑠	𝑢𝑛(𝑠	PROPN
cana-6124	139	12	)	)	PUNCT
cana-6124	139	13	)	)	PUNCT
cana-6124	140	1	−	−	PROPN
cana-6124	140	2	𝑓(𝑠	𝑓(𝑠	NOUN
cana-6124	140	3	,	,	PUNCT
cana-6124	140	4	𝑢(𝑠))|𝑑𝑞𝑠	𝑢(𝑠))|𝑑𝑞𝑠	PROPN
cana-6124	140	5	1	1	NUM
cana-6124	140	6	0	0	NUM
cana-6124	140	7	+	+	CCONJ
cana-6124	140	8	1	1	NUM
cana-6124	140	9	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	140	10	)	)	PUNCT
cana-6124	140	11	∫	∫	PROPN
cana-6124	140	12	(	(	PUNCT
cana-6124	140	13	𝑡	𝑡	PROPN
cana-6124	140	14	−	−	PROPN
cana-6124	140	15	𝑞𝑠)(𝛼−1)‖𝐴‖ℒ(𝑋)|𝑢𝑛(𝑠	𝑞𝑠)(𝛼−1)‖𝐴‖ℒ(𝑋)|𝑢𝑛(𝑠	NOUN
cana-6124	140	16	)	)	PUNCT
cana-6124	140	17	−	−	PROPN
cana-6124	141	1	𝑢(𝑠)|𝑑𝑞𝑠	𝑢(𝑠)|𝑑𝑞𝑠	NOUN
cana-6124	141	2	𝑡	𝑡	NOUN
cana-6124	141	3	0	0	NUM
cana-6124	141	4	≤	≤	NUM
cana-6124	141	5	|𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	|𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	PROPN
cana-6124	141	6	)	)	PUNCT
cana-6124	141	7	−	−	NOUN
cana-6124	141	8	𝑢(𝑠)|	𝑢(𝑠)|	NOUN
cana-6124	141	9	+	+	CCONJ
cana-6124	141	10	1	1	NUM
cana-6124	141	11	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	141	12	)	)	PUNCT
cana-6124	141	13	𝑠𝑢𝑝𝑠∈[0,1]|𝑓(𝑠	𝑠𝑢𝑝𝑠∈[0,1]|𝑓(𝑠	NOUN
cana-6124	141	14	,	,	PUNCT
cana-6124	141	15	𝑢𝑛(𝑠	𝑢𝑛(𝑠	PROPN
cana-6124	141	16	)	)	PUNCT
cana-6124	141	17	)	)	PUNCT
cana-6124	142	1	−	−	PROPN
cana-6124	142	2	𝑓(𝑠	𝑓(𝑠	PROPN
cana-6124	142	3	,	,	PUNCT
cana-6124	142	4	𝑢(𝑠))|∫	𝑢(𝑠))|∫	PROPN
cana-6124	142	5	(	(	PUNCT
cana-6124	142	6	𝑡	𝑡	PROPN
cana-6124	142	7	−	−	NOUN
cana-6124	142	8	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	142	9	)	)	PUNCT
cana-6124	142	10	(	(	PUNCT
cana-6124	142	11	𝛼−1	𝛼−1	NOUN
cana-6124	142	12	)	)	PUNCT
cana-6124	142	13	𝑡	𝑡	PROPN
cana-6124	142	14	0	0	NUM
cana-6124	142	15	𝑑𝑞𝑠	𝑑𝑞𝑠	NOUN
cana-6124	142	16	+	+	NUM
cana-6124	142	17	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	PROPN
cana-6124	142	18	)	)	PUNCT
cana-6124	142	19	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	142	20	)	)	PUNCT
cana-6124	142	21	𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	NOUN
cana-6124	142	22	)	)	PUNCT
cana-6124	142	23	−	−	PROPN
cana-6124	143	1	𝑢(𝑠)|∫	𝑢(𝑠)|∫	PROPN
cana-6124	143	2	(	(	PUNCT
cana-6124	143	3	𝑡	𝑡	PROPN
cana-6124	143	4	−	−	NOUN
cana-6124	143	5	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	143	6	)	)	PUNCT
cana-6124	143	7	(	(	PUNCT
cana-6124	143	8	𝛼−1)𝑑𝑞𝑠	𝛼−1)𝑑𝑞𝑠	NUM
cana-6124	143	9	𝑡	𝑡	NOUN
cana-6124	143	10	0	0	NUM
cana-6124	143	11	≤	≤	NUM
cana-6124	143	12	|𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	|𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	PROPN
cana-6124	143	13	)	)	PUNCT
cana-6124	143	14	−	−	NOUN
cana-6124	143	15	𝑢(𝑠)|	𝑢(𝑠)|	NOUN
cana-6124	143	16	+	+	CCONJ
cana-6124	143	17	1	1	NUM
cana-6124	143	18	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	143	19	+	+	CCONJ
cana-6124	143	20	1	1	X
cana-6124	143	21	)	)	PUNCT
cana-6124	143	22	𝑠𝑢𝑝𝑠∈[0,1]|𝑓(𝑠	𝑠𝑢𝑝𝑠∈[0,1]|𝑓(𝑠	NOUN
cana-6124	143	23	,	,	PUNCT
cana-6124	143	24	𝑢𝑛(𝑠	𝑢𝑛(𝑠	PROPN
cana-6124	143	25	)	)	PUNCT
cana-6124	143	26	)	)	PUNCT
cana-6124	144	1	−	−	PROPN
cana-6124	144	2	𝑓(𝑠	𝑓(𝑠	PROPN
cana-6124	144	3	,	,	PUNCT
cana-6124	144	4	𝑢(𝑠))|	𝑢(𝑠))|	PROPN
cana-6124	144	5	+	+	PROPN
cana-6124	144	6	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	PROPN
cana-6124	144	7	)	)	PUNCT
cana-6124	144	8	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	145	1	+	+	CCONJ
cana-6124	145	2	1	1	X
cana-6124	145	3	)	)	PUNCT
cana-6124	145	4	𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	NOUN
cana-6124	145	5	)	)	PUNCT
cana-6124	145	6	−	−	NOUN
cana-6124	145	7	𝑢(𝑠)|	𝑢(𝑠)|	NOUN
cana-6124	145	8	=	=	PUNCT
cana-6124	145	9	(	(	PUNCT
cana-6124	145	10	|𝑎|	|𝑎|	PROPN
cana-6124	145	11	+	+	CCONJ
cana-6124	145	12	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	PROPN
cana-6124	145	13	)	)	PUNCT
cana-6124	145	14	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	146	1	+	+	CCONJ
cana-6124	146	2	1	1	NUM
cana-6124	146	3	)	)	PUNCT
cana-6124	146	4	)	)	PUNCT
cana-6124	147	1	𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠	NOUN
cana-6124	147	2	)	)	PUNCT
cana-6124	147	3	−	−	NOUN
cana-6124	147	4	𝑢(𝑠)|	𝑢(𝑠)|	NOUN
cana-6124	147	5	+	+	CCONJ
cana-6124	147	6	1	1	NUM
cana-6124	147	7	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	147	8	+	+	SYM
cana-6124	147	9	1	1	X
cana-6124	147	10	)	)	PUNCT
cana-6124	147	11	𝑠𝑢𝑝𝑠∈[0.1]|𝑓(𝑠	𝑠𝑢𝑝𝑠∈[0.1]|𝑓(𝑠	PROPN
cana-6124	147	12	,	,	PUNCT
cana-6124	147	13	𝑢𝑛(𝑠	𝑢𝑛(𝑠	PROPN
cana-6124	147	14	)	)	PUNCT
cana-6124	147	15	)	)	PUNCT
cana-6124	148	1	−	−	PROPN
cana-6124	148	2	𝑓(𝑠	𝑓(𝑠	PROPN
cana-6124	148	3	,	,	PUNCT
cana-6124	148	4	𝑢(𝑠))|	𝑢(𝑠))|	PROPN
cana-6124	148	5	.	.	PUNCT
cana-6124	149	1	since	since	SCONJ
cana-6124	149	2	the	the	DET
cana-6124	149	3	function	function	NOUN
cana-6124	149	4	𝑓	𝑓	PRON
cana-6124	149	5	is	be	AUX
cana-6124	149	6	a	a	DET
cana-6124	149	7	continuous	continuous	ADJ
cana-6124	149	8	,	,	PUNCT
cana-6124	149	9	we	we	PRON
cana-6124	149	10	have	have	VERB
cana-6124	149	11	‖𝐹(𝑢𝑛	‖𝐹(𝑢𝑛	NUM
cana-6124	149	12	)	)	PUNCT
cana-6124	149	13	−	−	NOUN
cana-6124	149	14	𝐹(𝑢)‖∞	𝐹(𝑢)‖∞	ADJ
cana-6124	149	15	≤	≤	NOUN
cana-6124	149	16	(	(	PUNCT
cana-6124	149	17	|𝑎|	|𝑎|	PROPN
cana-6124	149	18	+	+	CCONJ
cana-6124	149	19	‖𝐴‖𝓛(𝑋	‖𝐴‖𝓛(𝑋	NUM
cana-6124	149	20	)	)	PUNCT
cana-6124	149	21	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	150	1	+	+	CCONJ
cana-6124	150	2	1	1	NUM
cana-6124	150	3	)	)	PUNCT
cana-6124	150	4	)	)	PUNCT
cana-6124	151	1	‖𝑢𝑛(∙	‖𝑢𝑛(∙	X
cana-6124	151	2	)	)	PUNCT
cana-6124	151	3	−	−	ADP
cana-6124	151	4	𝑢(∙)‖∞	𝑢(∙)‖∞	NOUN
cana-6124	151	5	+	+	CCONJ
cana-6124	151	6	1	1	NUM
cana-6124	151	7	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	151	8	+	+	NOUN
cana-6124	151	9	1	1	X
cana-6124	151	10	)	)	PUNCT
cana-6124	151	11	‖𝑓(∙	‖𝑓(∙	NOUN
cana-6124	151	12	,	,	PUNCT
cana-6124	151	13	𝑢𝑛(∙	𝑢𝑛(∙	NOUN
cana-6124	151	14	)	)	PUNCT
cana-6124	151	15	)	)	PUNCT
cana-6124	151	16	−	−	PROPN
cana-6124	151	17	𝑓(∙	𝑓(∙	NOUN
cana-6124	151	18	,	,	PUNCT
cana-6124	151	19	𝑢(∙))‖∞	𝑢(∙))‖∞	NOUN
cana-6124	151	20	𝑛→+∞	𝑛→+∞	PROPN
cana-6124	151	21	→	→	SYM
cana-6124	151	22	0	0	X
cana-6124	151	23	.	.	X
cana-6124	151	24	step	step	NOUN
cana-6124	151	25	2	2	NUM
cana-6124	151	26	:	:	PUNCT
cana-6124	151	27	𝐹	𝐹	PROPN
cana-6124	151	28	maps	map	VERB
cana-6124	151	29	bounded	bound	VERB
cana-6124	151	30	sets	set	NOUN
cana-6124	151	31	into	into	ADP
cana-6124	151	32	bounded	bounded	ADJ
cana-6124	151	33	sets	set	NOUN
cana-6124	151	34	in	in	ADP
cana-6124	151	35	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	151	36	]	]	PUNCT
cana-6124	151	37	,	,	PUNCT
cana-6124	151	38	𝑋	𝑋	PROPN
cana-6124	151	39	)	)	PUNCT
cana-6124	151	40	.	.	PUNCT
cana-6124	152	1	indeed	indeed	ADV
cana-6124	152	2	,	,	PUNCT
cana-6124	152	3	it	it	PRON
cana-6124	152	4	is	be	AUX
cana-6124	152	5	enough	enough	ADJ
cana-6124	152	6	to	to	PART
cana-6124	152	7	prove	prove	VERB
cana-6124	152	8	that	that	SCONJ
cana-6124	152	9	for	for	ADP
cana-6124	152	10	every	every	DET
cana-6124	152	11	𝜂∗	𝜂∗	PROPN
cana-6124	152	12	>	>	X
cana-6124	152	13	0	0	PROPN
cana-6124	152	14	,	,	PUNCT
cana-6124	152	15	there	there	PRON
cana-6124	152	16	exists	exist	VERB
cana-6124	152	17	a	a	DET
cana-6124	152	18	constant	constant	ADJ
cana-6124	152	19	ℓ	ℓ	NOUN
cana-6124	152	20	>	>	X
cana-6124	152	21	0	0	NUM
cana-6124	152	22	such	such	ADJ
cana-6124	152	23	that	that	PRON
cana-6124	152	24	for	for	ADP
cana-6124	152	25	all	all	DET
cana-6124	152	26	𝑢	𝑢	PROPN
cana-6124	152	27	∈	∈	NOUN
cana-6124	152	28	𝐵𝜂∗	𝐵𝜂∗	NOUN
cana-6124	152	29	=	=	SYM
cana-6124	152	30	{	{	PUNCT
cana-6124	152	31	𝑢	𝑢	X
cana-6124	152	32	∈	∈	PROPN
cana-6124	152	33	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	152	34	]	]	X
cana-6124	152	35	,	,	PUNCT
cana-6124	152	36	ℝ	ℝ	PROPN
cana-6124	152	37	):	):	PUNCT
cana-6124	152	38	‖𝑢‖∞	‖𝑢‖∞	PROPN
cana-6124	152	39	≤	≤	NOUN
cana-6124	152	40	𝜂	𝜂	NOUN
cana-6124	152	41	∗	∗	NOUN
cana-6124	152	42	}	}	PUNCT
cana-6124	152	43	,	,	PUNCT
cana-6124	152	44	we	we	PRON
cana-6124	152	45	have	have	VERB
cana-6124	152	46	‖𝐹(𝑢)‖∞	‖𝐹(𝑢)‖∞	ADJ
cana-6124	152	47	≤	≤	PROPN
cana-6124	152	48	ℓ.	ℓ.	NOUN
cana-6124	152	49	by	by	ADP
cana-6124	152	50	(	(	PUNCT
cana-6124	152	51	𝐻3	𝐻3	PROPN
cana-6124	152	52	)	)	PUNCT
cana-6124	152	53	we	we	PRON
cana-6124	152	54	have	have	VERB
cana-6124	152	55	for	for	ADP
cana-6124	152	56	every	every	DET
cana-6124	152	57	𝑡	𝑡	PROPN
cana-6124	152	58	∈	∈	PROPN
cana-6124	153	1	[	[	X
cana-6124	153	2	0,1	0,1	NUM
cana-6124	153	3	]	]	PUNCT
cana-6124	153	4	:	:	PUNCT
cana-6124	153	5	communications	communication	NOUN
cana-6124	153	6	on	on	ADP
cana-6124	153	7	applied	apply	VERB
cana-6124	153	8	nonlinear	nonlinear	ADJ
cana-6124	153	9	analysis	analysis	NOUN
cana-6124	153	10	issn	issn	NOUN
cana-6124	153	11	:	:	PUNCT
cana-6124	153	12	1074	1074	NUM
cana-6124	153	13	-	-	PUNCT
cana-6124	153	14	133x	133x	NUM
cana-6124	153	15	vol	vol	VERB
cana-6124	153	16	32	32	NUM
cana-6124	153	17	no	no	NOUN
cana-6124	153	18	.	.	PUNCT
cana-6124	154	1	10s	10	NOUN
cana-6124	154	2	(	(	PUNCT
cana-6124	154	3	2025	2025	NUM
cana-6124	154	4	)	)	PUNCT
cana-6124	154	5	3433	3433	NUM
cana-6124	154	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	154	7	|𝐹(𝑢)(𝑡)|	|𝐹(𝑢)(𝑡)|	PROPN
cana-6124	154	8	≤	≤	X
cana-6124	154	9	|𝑎|	|𝑎|	NUM
cana-6124	154	10	∫	∫	PROPN
cana-6124	154	11	|𝑢(𝑠)|𝑑𝑞𝑠	|𝑢(𝑠)|𝑑𝑞𝑠	PUNCT
cana-6124	154	12	1	1	NUM
cana-6124	154	13	0	0	NUM
cana-6124	154	14	+	+	CCONJ
cana-6124	154	15	|𝑏|	|𝑏|	PROPN
cana-6124	154	16	+	+	CCONJ
cana-6124	154	17	1	1	NUM
cana-6124	154	18	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	154	19	)	)	PUNCT
cana-6124	154	20	∫	∫	PROPN
cana-6124	154	21	(	(	PUNCT
cana-6124	154	22	𝑡	𝑡	PROPN
cana-6124	154	23	−	−	NOUN
cana-6124	154	24	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	154	25	)	)	PUNCT
cana-6124	154	26	𝑡	𝑡	PROPN
cana-6124	154	27	0	0	PROPN
cana-6124	154	28	|𝑓(𝑠	|𝑓(𝑠	PROPN
cana-6124	154	29	,	,	PUNCT
cana-6124	154	30	𝑢(𝑠))|𝑑𝑞𝑠	𝑢(𝑠))|𝑑𝑞𝑠	PROPN
cana-6124	154	31	+	+	CCONJ
cana-6124	154	32	1	1	NUM
cana-6124	154	33	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	154	34	)	)	PUNCT
cana-6124	154	35	∫	∫	PROPN
cana-6124	155	1	(	(	PUNCT
cana-6124	155	2	𝑡	𝑡	PROPN
cana-6124	155	3	−	−	NOUN
cana-6124	155	4	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	155	5	)	)	PUNCT
cana-6124	155	6	𝑡	𝑡	PROPN
cana-6124	155	7	0	0	NUM
cana-6124	155	8	‖𝐴‖ℒ(𝑋)|𝑢(𝑠)|𝑑𝑞𝑠	‖𝐴‖ℒ(𝑋)|𝑢(𝑠)|𝑑𝑞𝑠	ADJ
cana-6124	155	9	≤	≤	NUM
cana-6124	155	10	|𝑎|𝜂∗	|𝑎|𝜂∗	PROPN
cana-6124	155	11	+	+	CCONJ
cana-6124	155	12	|𝑏|	|𝑏|	PROPN
cana-6124	155	13	+	+	CCONJ
cana-6124	155	14	𝑀	𝑀	PROPN
cana-6124	155	15	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-6124	155	16	)	)	PUNCT
cana-6124	155	17	∫	∫	PROPN
cana-6124	155	18	(	(	PUNCT
cana-6124	155	19	𝑡	𝑡	PROPN
cana-6124	155	20	−	−	PROPN
cana-6124	155	21	𝑞𝑠)(𝛼−1)𝑑𝑞𝑠	𝑞𝑠)(𝛼−1)𝑑𝑞𝑠	PROPN
cana-6124	155	22	𝑡	𝑡	PROPN
cana-6124	155	23	0	0	PUNCT
cana-6124	156	1	+	+	NUM
cana-6124	156	2	‖𝐴‖ℒ(𝑋)𝜂	‖𝐴‖ℒ(𝑋)𝜂	NOUN
cana-6124	156	3	∗	∗	NOUN
cana-6124	156	4	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	156	5	)	)	PUNCT
cana-6124	156	6	∫	∫	PROPN
cana-6124	156	7	(	(	PUNCT
cana-6124	156	8	𝑡	𝑡	PROPN
cana-6124	156	9	−	−	NOUN
cana-6124	156	10	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	156	11	)	)	PUNCT
cana-6124	156	12	𝑡	𝑡	PROPN
cana-6124	156	13	0	0	PROPN
cana-6124	156	14	𝑑𝑞𝑠	𝑑𝑞𝑠	PROPN
cana-6124	156	15	≤	≤	NUM
cana-6124	156	16	|𝑎|𝜂∗	|𝑎|𝜂∗	NOUN
cana-6124	156	17	+	+	CCONJ
cana-6124	156	18	|𝑏|	|𝑏|	PROPN
cana-6124	156	19	+	+	CCONJ
cana-6124	156	20	𝑀	𝑀	PROPN
cana-6124	156	21	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	157	1	+	+	CCONJ
cana-6124	157	2	1	1	X
cana-6124	157	3	)	)	PUNCT
cana-6124	157	4	+	+	NUM
cana-6124	157	5	‖𝐴‖ℒ(𝑋)𝜂	‖𝐴‖ℒ(𝑋)𝜂	NOUN
cana-6124	157	6	∗	∗	NOUN
cana-6124	157	7	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	158	1	+	+	CCONJ
cana-6124	158	2	1	1	NUM
cana-6124	158	3	)	)	PUNCT
cana-6124	158	4	.	.	PUNCT
cana-6124	159	1	hence	hence	ADV
cana-6124	159	2	‖𝐹(𝑢)‖∞	‖𝐹(𝑢)‖∞	ADJ
cana-6124	159	3	≤	≤	PUNCT
cana-6124	159	4	|𝑎|𝜂	|𝑎|𝜂	PROPN
cana-6124	159	5	∗	∗	NOUN
cana-6124	159	6	+	+	CCONJ
cana-6124	159	7	|𝑏|	|𝑏|	PROPN
cana-6124	159	8	+	+	CCONJ
cana-6124	159	9	𝑀	𝑀	PROPN
cana-6124	159	10	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	159	11	+	+	CCONJ
cana-6124	159	12	1	1	X
cana-6124	159	13	)	)	PUNCT
cana-6124	159	14	+	+	NUM
cana-6124	159	15	‖𝐴‖ℒ(𝑋)𝜂	‖𝐴‖ℒ(𝑋)𝜂	NOUN
cana-6124	159	16	∗	∗	NOUN
cana-6124	159	17	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	160	1	+	+	CCONJ
cana-6124	160	2	1	1	X
cana-6124	160	3	)	)	PUNCT
cana-6124	160	4	≔	≔	NOUN
cana-6124	160	5	ℓ.	ℓ.	NOUN
cana-6124	160	6	step	step	NOUN
cana-6124	160	7	3	3	NUM
cana-6124	160	8	:	:	PUNCT
cana-6124	160	9	the	the	DET
cana-6124	160	10	operator	operator	NOUN
cana-6124	160	11	𝐹	𝐹	PROPN
cana-6124	160	12	maps	map	VERB
cana-6124	160	13	bounded	bound	VERB
cana-6124	160	14	sets	set	NOUN
cana-6124	160	15	into	into	ADP
cana-6124	160	16	equicontinuous	equicontinuous	ADJ
cana-6124	160	17	sets	set	NOUN
cana-6124	160	18	of	of	ADP
cana-6124	160	19	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	160	20	]	]	X
cana-6124	160	21	,	,	PUNCT
cana-6124	160	22	𝑋	𝑋	PROPN
cana-6124	160	23	)	)	PUNCT
cana-6124	160	24	.	.	PUNCT
cana-6124	161	1	for	for	ADP
cana-6124	161	2	𝑡1	𝑡1	NOUN
cana-6124	161	3	,	,	PUNCT
cana-6124	161	4	𝑡2	𝑡2	NOUN
cana-6124	161	5	∈	∈	PROPN
cana-6124	161	6	[	[	X
cana-6124	161	7	0,1	0,1	NUM
cana-6124	161	8	]	]	PUNCT
cana-6124	161	9	with	with	ADP
cana-6124	161	10	𝑡1	𝑡1	NOUN
cana-6124	161	11	<	<	X
cana-6124	161	12	𝑡2	𝑡2	PROPN
cana-6124	161	13	,	,	PUNCT
cana-6124	161	14	and	and	CCONJ
cana-6124	161	15	for	for	ADP
cana-6124	161	16	𝑢	𝑢	PROPN
cana-6124	161	17	∈	∈	PROPN
cana-6124	161	18	𝐵𝜂∗	𝐵𝜂∗	NOUN
cana-6124	161	19	,	,	PUNCT
cana-6124	161	20	where	where	SCONJ
cana-6124	161	21	𝐵𝜂∗	𝐵𝜂∗	NOUN
cana-6124	161	22	is	be	AUX
cana-6124	161	23	the	the	DET
cana-6124	161	24	bounded	bounded	ADJ
cana-6124	161	25	subset	subset	NOUN
cana-6124	161	26	of	of	ADP
cana-6124	161	27	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	161	28	]	]	X
cana-6124	161	29	,	,	PUNCT
cana-6124	161	30	𝑋	𝑋	PROPN
cana-6124	161	31	)	)	PUNCT
cana-6124	161	32	defined	define	VERB
cana-6124	161	33	in	in	ADP
cana-6124	161	34	step	step	NOUN
cana-6124	161	35	2	2	NUM
cana-6124	161	36	,	,	PUNCT
cana-6124	161	37	we	we	PRON
cana-6124	161	38	have	have	VERB
cana-6124	161	39	|𝐹(𝑢)(𝑡2	|𝐹(𝑢)(𝑡2	NOUN
cana-6124	161	40	)	)	PUNCT
cana-6124	162	1	−	−	PROPN
cana-6124	163	1	𝐹(𝑢)(𝑡1)|	𝐹(𝑢)(𝑡1)|	NUM
cana-6124	163	2	=	=	SYM
cana-6124	164	1	|	|	ADV
cana-6124	164	2	1	1	NUM
cana-6124	164	3	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	164	4	)	)	PUNCT
cana-6124	164	5	∫	∫	PROPN
cana-6124	165	1	[	[	X
cana-6124	165	2	(	(	PUNCT
cana-6124	165	3	𝑡2	𝑡2	ADJ
cana-6124	165	4	−	−	PROPN
cana-6124	165	5	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	165	6	)	)	PUNCT
cana-6124	165	7	(	(	PUNCT
cana-6124	165	8	𝛼−1	𝛼−1	NOUN
cana-6124	165	9	)	)	PUNCT
cana-6124	165	10	−	−	PROPN
cana-6124	165	11	(	(	PUNCT
cana-6124	165	12	𝑡1	𝑡1	NOUN
cana-6124	165	13	−	−	PROPN
cana-6124	165	14	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	165	15	)	)	PUNCT
cana-6124	165	16	(	(	PUNCT
cana-6124	165	17	𝛼−1)]𝑓(𝑠	𝛼−1)]𝑓(𝑠	PROPN
cana-6124	165	18	,	,	PUNCT
cana-6124	165	19	𝑢(𝑠))𝑑𝑞𝑠	𝑢(𝑠))𝑑𝑞𝑠	VERB
cana-6124	165	20	𝑡1	𝑡1	NOUN
cana-6124	165	21	0	0	PUNCT
cana-6124	166	1	+	+	CCONJ
cana-6124	166	2	1	1	NUM
cana-6124	166	3	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	166	4	)	)	PUNCT
cana-6124	166	5	∫	∫	PROPN
cana-6124	167	1	[	[	X
cana-6124	167	2	(	(	PUNCT
cana-6124	167	3	𝑡2	𝑡2	ADJ
cana-6124	167	4	−	−	PROPN
cana-6124	167	5	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	167	6	)	)	PUNCT
cana-6124	167	7	(	(	PUNCT
cana-6124	167	8	𝛼−1	𝛼−1	NOUN
cana-6124	167	9	)	)	PUNCT
cana-6124	167	10	−	−	PROPN
cana-6124	167	11	(	(	PUNCT
cana-6124	167	12	𝑡1	𝑡1	NOUN
cana-6124	167	13	−	−	PROPN
cana-6124	167	14	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	167	15	)	)	PUNCT
cana-6124	167	16	(	(	PUNCT
cana-6124	167	17	𝛼−1)]𝐴𝑢(𝑠)𝑑𝑞𝑠	𝛼−1)]𝐴𝑢(𝑠)𝑑𝑞𝑠	ADJ
cana-6124	167	18	𝑡1	𝑡1	NOUN
cana-6124	167	19	0	0	NUM
cana-6124	168	1	+	+	CCONJ
cana-6124	168	2	1	1	NUM
cana-6124	168	3	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	168	4	)	)	PUNCT
cana-6124	168	5	∫	∫	PROPN
cana-6124	169	1	(	(	PUNCT
cana-6124	169	2	𝑡2	𝑡2	NOUN
cana-6124	169	3	−	−	PROPN
cana-6124	169	4	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	169	5	)	)	PUNCT
cana-6124	169	6	(	(	PUNCT
cana-6124	169	7	𝛼−1)𝑓(𝑠	𝛼−1)𝑓(𝑠	NUM
cana-6124	169	8	,	,	PUNCT
cana-6124	169	9	𝑢(𝑠))𝑑𝑞𝑠	𝑢(𝑠))𝑑𝑞𝑠	VERB
cana-6124	169	10	+	+	X
cana-6124	169	11	1	1	NUM
cana-6124	169	12	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	169	13	)	)	PUNCT
cana-6124	169	14	∫	∫	PROPN
cana-6124	170	1	(	(	PUNCT
cana-6124	170	2	𝑡2	𝑡2	NOUN
cana-6124	170	3	−	−	PROPN
cana-6124	170	4	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	170	5	)	)	PUNCT
cana-6124	170	6	(	(	PUNCT
cana-6124	170	7	𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠	𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠	NUM
cana-6124	170	8	𝑡2	𝑡2	PROPN
cana-6124	170	9	𝑡1	𝑡1	NOUN
cana-6124	170	10	𝑡2	𝑡2	PROPN
cana-6124	170	11	𝑡1	𝑡1	NOUN
cana-6124	170	12	|	|	ADV
cana-6124	170	13	≤	≤	NUM
cana-6124	170	14	𝑀	𝑀	PROPN
cana-6124	170	15	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-6124	170	16	)	)	PUNCT
cana-6124	170	17	∫	∫	PROPN
cana-6124	171	1	[	[	X
cana-6124	171	2	(	(	PUNCT
cana-6124	171	3	𝑡1	𝑡1	NOUN
cana-6124	171	4	−	−	NOUN
cana-6124	171	5	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	171	6	)	)	PUNCT
cana-6124	171	7	(	(	PUNCT
cana-6124	171	8	𝛼−1	𝛼−1	NOUN
cana-6124	171	9	)	)	PUNCT
cana-6124	171	10	−	−	PROPN
cana-6124	172	1	(	(	PUNCT
cana-6124	172	2	𝑡2	𝑡2	NOUN
cana-6124	172	3	−	−	PROPN
cana-6124	172	4	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	172	5	)	)	PUNCT
cana-6124	172	6	(	(	PUNCT
cana-6124	172	7	𝛼−1)]𝑓(𝑠	𝛼−1)]𝑓(𝑠	PROPN
cana-6124	172	8	,	,	PUNCT
cana-6124	172	9	𝑢(𝑠))𝑑𝑞𝑠	𝑢(𝑠))𝑑𝑞𝑠	VERB
cana-6124	172	10	𝑡1	𝑡1	NOUN
cana-6124	172	11	0	0	NUM
cana-6124	173	1	+	+	CCONJ
cana-6124	173	2	𝜂∗‖𝐴‖ℒ(𝑋	𝜂∗‖𝐴‖ℒ(𝑋	PROPN
cana-6124	173	3	)	)	PUNCT
cana-6124	173	4	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	173	5	)	)	PUNCT
cana-6124	174	1	∫	∫	PROPN
cana-6124	175	1	[	[	X
cana-6124	175	2	(	(	PUNCT
cana-6124	175	3	𝑡1	𝑡1	NOUN
cana-6124	175	4	−	−	NOUN
cana-6124	175	5	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	175	6	)	)	PUNCT
cana-6124	175	7	(	(	PUNCT
cana-6124	175	8	𝛼−1	𝛼−1	NOUN
cana-6124	175	9	)	)	PUNCT
cana-6124	175	10	−	−	PROPN
cana-6124	176	1	(	(	PUNCT
cana-6124	176	2	𝑡2	𝑡2	NOUN
cana-6124	176	3	−	−	PROPN
cana-6124	176	4	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	176	5	)	)	PUNCT
cana-6124	176	6	(	(	PUNCT
cana-6124	176	7	𝛼−1)]𝑑𝑞𝑠	𝛼−1)]𝑑𝑞𝑠	CCONJ
cana-6124	176	8	𝑡1	𝑡1	NOUN
cana-6124	176	9	0	0	PUNCT
cana-6124	177	1	+	+	CCONJ
cana-6124	177	2	𝑀	𝑀	PROPN
cana-6124	177	3	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-6124	177	4	)	)	PUNCT
cana-6124	177	5	∫	∫	PROPN
cana-6124	178	1	(	(	PUNCT
cana-6124	178	2	𝑡2	𝑡2	NOUN
cana-6124	178	3	−	−	PROPN
cana-6124	178	4	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	178	5	)	)	PUNCT
cana-6124	178	6	(	(	PUNCT
cana-6124	178	7	𝛼−1)𝑑𝑞𝑠	𝛼−1)𝑑𝑞𝑠	NUM
cana-6124	178	8	+	+	NUM
cana-6124	178	9	𝜂∗‖𝐴‖ℒ(𝑋	𝜂∗‖𝐴‖ℒ(𝑋	NOUN
cana-6124	178	10	)	)	PUNCT
cana-6124	178	11	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	178	12	)	)	PUNCT
cana-6124	178	13	∫	∫	PROPN
cana-6124	179	1	(	(	PUNCT
cana-6124	179	2	𝑡2	𝑡2	NOUN
cana-6124	179	3	−	−	PROPN
cana-6124	179	4	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	179	5	)	)	PUNCT
cana-6124	179	6	(	(	PUNCT
cana-6124	179	7	𝛼−1)𝑑𝑞𝑠	𝛼−1)𝑑𝑞𝑠	NUM
cana-6124	179	8	𝑡2	𝑡2	PROPN
cana-6124	179	9	𝑡1	𝑡1	NOUN
cana-6124	179	10	𝑡2	𝑡2	PROPN
cana-6124	179	11	𝑡1	𝑡1	NOUN
cana-6124	179	12	≤	≤	PROPN
cana-6124	179	13	𝑀	𝑀	PROPN
cana-6124	179	14	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-6124	180	1	+	+	CCONJ
cana-6124	180	2	1	1	X
cana-6124	180	3	)	)	PUNCT
cana-6124	180	4	[	[	X
cana-6124	180	5	(	(	PUNCT
cana-6124	180	6	𝑡2	𝑡2	ADJ
cana-6124	180	7	−	−	PROPN
cana-6124	180	8	𝑡1	𝑡1	NOUN
cana-6124	180	9	)	)	PUNCT
cana-6124	180	10	(	(	PUNCT
cana-6124	180	11	𝛼	𝛼	X
cana-6124	180	12	)	)	PUNCT
cana-6124	180	13	+	+	CCONJ
cana-6124	180	14	𝑡1	𝑡1	NOUN
cana-6124	180	15	(	(	PUNCT
cana-6124	180	16	𝛼	𝛼	NOUN
cana-6124	180	17	)	)	PUNCT
cana-6124	180	18	−	−	PROPN
cana-6124	180	19	𝑡2	𝑡2	NOUN
cana-6124	180	20	(	(	PUNCT
cana-6124	180	21	𝛼	𝛼	NOUN
cana-6124	180	22	)	)	PUNCT
cana-6124	180	23	]	]	PUNCT
cana-6124	181	1	+	+	PUNCT
cana-6124	181	2	𝜂∗‖𝐴‖ℒ(𝑋	𝜂∗‖𝐴‖ℒ(𝑋	X
cana-6124	181	3	)	)	PUNCT
cana-6124	181	4	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	182	1	+	+	CCONJ
cana-6124	182	2	1	1	X
cana-6124	182	3	)	)	PUNCT
cana-6124	182	4	[	[	X
cana-6124	182	5	(	(	PUNCT
cana-6124	182	6	𝑡2	𝑡2	ADJ
cana-6124	182	7	−	−	PROPN
cana-6124	182	8	𝑡1	𝑡1	NOUN
cana-6124	182	9	)	)	PUNCT
cana-6124	182	10	(	(	PUNCT
cana-6124	182	11	𝛼	𝛼	X
cana-6124	182	12	)	)	PUNCT
cana-6124	182	13	+	+	CCONJ
cana-6124	182	14	𝑡1	𝑡1	NOUN
cana-6124	182	15	(	(	PUNCT
cana-6124	182	16	𝛼	𝛼	NOUN
cana-6124	182	17	)	)	PUNCT
cana-6124	182	18	−	−	PROPN
cana-6124	182	19	𝑡2	𝑡2	NOUN
cana-6124	182	20	(	(	PUNCT
cana-6124	182	21	𝛼	𝛼	NOUN
cana-6124	182	22	)	)	PUNCT
cana-6124	182	23	]	]	PUNCT
cana-6124	183	1	+	+	CCONJ
cana-6124	183	2	𝑀	𝑀	PROPN
cana-6124	183	3	γ𝑞(𝛼	γ𝑞(𝛼	AUX
cana-6124	183	4	+	+	CCONJ
cana-6124	183	5	1	1	X
cana-6124	183	6	)	)	PUNCT
cana-6124	183	7	(	(	PUNCT
cana-6124	183	8	𝑡2	𝑡2	ADJ
cana-6124	183	9	−	−	PROPN
cana-6124	183	10	𝑡1	𝑡1	NOUN
cana-6124	183	11	)	)	PUNCT
cana-6124	183	12	(	(	PUNCT
cana-6124	183	13	𝛼	𝛼	X
cana-6124	183	14	)	)	PUNCT
cana-6124	183	15	+	+	NUM
cana-6124	183	16	𝜂∗‖𝐴‖ℒ(𝑋	𝜂∗‖𝐴‖ℒ(𝑋	X
cana-6124	183	17	)	)	PUNCT
cana-6124	183	18	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	184	1	+	+	CCONJ
cana-6124	184	2	1	1	X
cana-6124	184	3	)	)	PUNCT
cana-6124	184	4	(	(	PUNCT
cana-6124	184	5	𝑡2	𝑡2	ADJ
cana-6124	184	6	−	−	PROPN
cana-6124	184	7	𝑡1	𝑡1	NOUN
cana-6124	184	8	)	)	PUNCT
cana-6124	184	9	(	(	PUNCT
cana-6124	184	10	𝛼	𝛼	NOUN
cana-6124	184	11	)	)	PUNCT
cana-6124	184	12	≤	≤	NOUN
cana-6124	184	13	[	[	PUNCT
cana-6124	184	14	2𝑀	2𝑀	NOUN
cana-6124	184	15	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	184	16	+	+	CCONJ
cana-6124	184	17	1	1	X
cana-6124	184	18	)	)	PUNCT
cana-6124	184	19	+	+	CCONJ
cana-6124	184	20	2𝜂∗‖𝐴‖ℒ(𝑋	2𝜂∗‖𝐴‖ℒ(𝑋	NUM
cana-6124	184	21	)	)	PUNCT
cana-6124	184	22	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	185	1	+	+	CCONJ
cana-6124	185	2	1	1	NUM
cana-6124	185	3	)	)	PUNCT
cana-6124	185	4	]	]	PUNCT
cana-6124	185	5	(	(	PUNCT
cana-6124	185	6	𝑡2	𝑡2	ADJ
cana-6124	185	7	−	−	PROPN
cana-6124	185	8	𝑡1	𝑡1	NOUN
cana-6124	185	9	)	)	PUNCT
cana-6124	185	10	(	(	PUNCT
cana-6124	185	11	𝛼	𝛼	X
cana-6124	185	12	)	)	PUNCT
cana-6124	186	1	+	+	CCONJ
cana-6124	186	2	[	[	PUNCT
cana-6124	186	3	𝑀	𝑀	NOUN
cana-6124	186	4	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	186	5	+	+	CCONJ
cana-6124	186	6	1	1	X
cana-6124	186	7	)	)	PUNCT
cana-6124	186	8	+	+	NUM
cana-6124	186	9	𝜂∗‖𝐴‖ℒ(𝑋	𝜂∗‖𝐴‖ℒ(𝑋	X
cana-6124	186	10	)	)	PUNCT
cana-6124	186	11	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	187	1	+	+	CCONJ
cana-6124	187	2	1	1	NUM
cana-6124	187	3	)	)	PUNCT
cana-6124	187	4	]	]	PUNCT
cana-6124	187	5	(	(	PUNCT
cana-6124	187	6	𝑡2	𝑡2	ADJ
cana-6124	187	7	−	−	PROPN
cana-6124	187	8	𝑡1	𝑡1	NOUN
cana-6124	187	9	)	)	PUNCT
cana-6124	187	10	(	(	PUNCT
cana-6124	187	11	𝛼	𝛼	NOUN
cana-6124	187	12	)	)	PUNCT
cana-6124	187	13	.	.	PUNCT
cana-6124	188	1	as	as	ADP
cana-6124	188	2	𝑡1	𝑡1	PROPN
cana-6124	188	3	→	→	SYM
cana-6124	188	4	𝑡2	𝑡2	PROPN
cana-6124	188	5	,	,	PUNCT
cana-6124	188	6	the	the	DET
cana-6124	188	7	right	right	ADJ
cana-6124	188	8	-	-	PUNCT
cana-6124	188	9	hand	hand	NOUN
cana-6124	188	10	side	side	NOUN
cana-6124	188	11	of	of	ADP
cana-6124	188	12	the	the	DET
cana-6124	188	13	preceding	precede	VERB
cana-6124	188	14	inequality	inequality	NOUN
cana-6124	188	15	tends	tend	VERB
cana-6124	188	16	to	to	ADP
cana-6124	188	17	zero	zero	NUM
cana-6124	188	18	.	.	PUNCT
cana-6124	189	1	as	as	ADP
cana-6124	189	2	a	a	DET
cana-6124	189	3	consequence	consequence	NOUN
cana-6124	189	4	of	of	ADP
cana-6124	189	5	step	step	NOUN
cana-6124	189	6	1	1	NUM
cana-6124	189	7	to	to	ADP
cana-6124	189	8	3	3	NUM
cana-6124	189	9	tougher	tough	ADJ
cana-6124	189	10	with	with	ADP
cana-6124	189	11	the	the	DET
cana-6124	189	12	arzelà	arzelà	PROPN
cana-6124	189	13	-	-	PUNCT
cana-6124	189	14	ascoli	ascoli	NOUN
cana-6124	189	15	theorem	theorem	PROPN
cana-6124	189	16	,	,	PUNCT
cana-6124	189	17	we	we	PRON
cana-6124	189	18	can	can	AUX
cana-6124	189	19	conclude	conclude	VERB
cana-6124	189	20	that	that	SCONJ
cana-6124	189	21	the	the	DET
cana-6124	189	22	operator	operator	NOUN
cana-6124	189	23	𝐹	𝐹	PROPN
cana-6124	189	24	:	:	PUNCT
cana-6124	189	25	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	189	26	]	]	X
cana-6124	189	27	,	,	PUNCT
cana-6124	189	28	𝑋	𝑋	PROPN
cana-6124	189	29	)	)	PUNCT
cana-6124	189	30	→	→	SYM
cana-6124	189	31	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	189	32	]	]	PUNCT
cana-6124	189	33	,	,	PUNCT
cana-6124	189	34	𝑋	𝑋	PROPN
cana-6124	189	35	)	)	PUNCT
cana-6124	189	36	is	be	AUX
cana-6124	189	37	continuous	continuous	ADJ
cana-6124	189	38	and	and	CCONJ
cana-6124	189	39	completely	completely	ADV
cana-6124	189	40	continuous	continuous	ADJ
cana-6124	189	41	.	.	PUNCT
cana-6124	190	1	communications	communication	NOUN
cana-6124	190	2	on	on	ADP
cana-6124	190	3	applied	apply	VERB
cana-6124	190	4	nonlinear	nonlinear	ADJ
cana-6124	190	5	analysis	analysis	NOUN
cana-6124	190	6	issn	issn	NOUN
cana-6124	190	7	:	:	PUNCT
cana-6124	190	8	1074	1074	NUM
cana-6124	190	9	-	-	PUNCT
cana-6124	190	10	133x	133x	NUM
cana-6124	190	11	vol	vol	VERB
cana-6124	190	12	32	32	NUM
cana-6124	190	13	no	no	NOUN
cana-6124	190	14	.	.	PUNCT
cana-6124	191	1	10s	10	NOUN
cana-6124	191	2	(	(	PUNCT
cana-6124	191	3	2025	2025	NUM
cana-6124	191	4	)	)	PUNCT
cana-6124	191	5	3434	3434	NUM
cana-6124	191	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	191	7	step	step	NOUN
cana-6124	191	8	4	4	NUM
cana-6124	191	9	:	:	PUNCT
cana-6124	191	10	a	a	DET
cana-6124	191	11	priori	priori	ADJ
cana-6124	191	12	bounds	bound	NOUN
cana-6124	191	13	.	.	PUNCT
cana-6124	192	1	by	by	ADP
cana-6124	192	2	schauder	schauder	NOUN
cana-6124	192	3	’s	’s	PART
cana-6124	192	4	fixed	fix	VERB
cana-6124	192	5	point	point	NOUN
cana-6124	192	6	theorem	theorem	VERB
cana-6124	192	7	,	,	PUNCT
cana-6124	192	8	it	it	PRON
cana-6124	192	9	is	be	AUX
cana-6124	192	10	sufficient	sufficient	ADJ
cana-6124	192	11	to	to	PART
cana-6124	192	12	show	show	VERB
cana-6124	192	13	that	that	SCONJ
cana-6124	192	14	the	the	DET
cana-6124	192	15	set	set	NOUN
cana-6124	192	16	:	:	PUNCT
cana-6124	192	17	ℰ	ℰ	PROPN
cana-6124	192	18	=	=	PUNCT
cana-6124	192	19	{	{	PUNCT
cana-6124	192	20	𝑢	𝑢	PROPN
cana-6124	192	21	∈	∈	PROPN
cana-6124	192	22	𝐶([0,1	𝐶([0,1	NOUN
cana-6124	192	23	]	]	X
cana-6124	192	24	,	,	PUNCT
cana-6124	192	25	𝑋	𝑋	PROPN
cana-6124	192	26	)	)	PUNCT
cana-6124	192	27	|𝑢	|𝑢	PROPN
cana-6124	192	28	=	=	SYM
cana-6124	192	29	𝜆𝐹(𝑢	𝜆𝐹(𝑢	PROPN
cana-6124	192	30	)	)	PUNCT
cana-6124	192	31	,	,	PUNCT
cana-6124	192	32	𝜆	𝜆	ADP
cana-6124	192	33	∈]0,1	∈]0,1	PRON
cana-6124	192	34	[	[	X
cana-6124	192	35	}	}	PUNCT
cana-6124	192	36	is	be	AUX
cana-6124	192	37	bounded	bound	VERB
cana-6124	192	38	.	.	PUNCT
cana-6124	193	1	let	let	VERB
cana-6124	193	2	𝑢	𝑢	PRON
cana-6124	193	3	∈	∈	PROPN
cana-6124	193	4	ℰ.	ℰ.	NOUN
cana-6124	193	5	then	then	ADV
cana-6124	193	6	there	there	PRON
cana-6124	193	7	exists	exist	VERB
cana-6124	193	8	𝜆	𝜆	DET
cana-6124	193	9	∈]0,1	∈]0,1	NOUN
cana-6124	193	10	[	[	PUNCT
cana-6124	193	11	such	such	ADJ
cana-6124	193	12	that	that	SCONJ
cana-6124	193	13	𝑢	𝑢	X
cana-6124	193	14	=	=	SYM
cana-6124	193	15	𝜆𝐹(𝑢	𝜆𝐹(𝑢	NOUN
cana-6124	193	16	)	)	PUNCT
cana-6124	193	17	.	.	PUNCT
cana-6124	194	1	hence	hence	ADV
cana-6124	194	2	,	,	PUNCT
cana-6124	194	3	for	for	ADP
cana-6124	194	4	every	every	DET
cana-6124	194	5	𝑡	𝑡	PROPN
cana-6124	194	6	∈	∈	PROPN
cana-6124	194	7	[	[	X
cana-6124	194	8	0,1	0,1	NUM
cana-6124	194	9	]	]	PUNCT
cana-6124	194	10	,	,	PUNCT
cana-6124	194	11	we	we	PRON
cana-6124	194	12	have	have	VERB
cana-6124	194	13	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	194	14	)	)	PUNCT
cana-6124	194	15	=	=	SYM
cana-6124	195	1	𝜆	𝜆	X
cana-6124	196	1	[	[	X
cana-6124	196	2	𝑎∫	𝑎∫	ADJ
cana-6124	196	3	𝑢(𝑠)𝑑𝑞𝑠	𝑢(𝑠)𝑑𝑞𝑠	PROPN
cana-6124	196	4	+	+	CCONJ
cana-6124	196	5	𝑏	𝑏	PROPN
cana-6124	196	6	+	+	CCONJ
cana-6124	196	7	1	1	NUM
cana-6124	196	8	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	196	9	)	)	PUNCT
cana-6124	196	10	∫	∫	PROPN
cana-6124	196	11	(	(	PUNCT
cana-6124	196	12	𝑡	𝑡	PROPN
cana-6124	196	13	−	−	NOUN
cana-6124	196	14	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	196	15	)	)	PUNCT
cana-6124	196	16	𝑡	𝑡	PROPN
cana-6124	196	17	0	0	NUM
cana-6124	196	18	𝑓(𝑠	𝑓(𝑠	NOUN
cana-6124	196	19	,	,	PUNCT
cana-6124	196	20	𝑢(𝑠))𝑑𝑞𝑠	𝑢(𝑠))𝑑𝑞𝑠	VERB
cana-6124	196	21	1	1	NUM
cana-6124	196	22	0	0	NUM
cana-6124	196	23	+	+	CCONJ
cana-6124	196	24	1	1	NUM
cana-6124	196	25	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	196	26	)	)	PUNCT
cana-6124	196	27	∫	∫	PROPN
cana-6124	196	28	(	(	PUNCT
cana-6124	196	29	𝑡	𝑡	PROPN
cana-6124	196	30	−	−	NOUN
cana-6124	196	31	𝑞𝑠)(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠	𝑞𝑠)(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠	PUNCT
cana-6124	196	32	𝑡	𝑡	PROPN
cana-6124	196	33	0	0	NUM
cana-6124	196	34	]	]	PUNCT
cana-6124	196	35	.	.	PUNCT
cana-6124	197	1	under	under	ADP
cana-6124	197	2	hypothesis	hypothesis	NOUN
cana-6124	197	3	(	(	PUNCT
cana-6124	197	4	𝐻3	𝐻3	PROPN
cana-6124	197	5	)	)	PUNCT
cana-6124	197	6	,	,	PUNCT
cana-6124	197	7	it	it	PRON
cana-6124	197	8	follows	follow	VERB
cana-6124	197	9	that	that	SCONJ
cana-6124	197	10	for	for	ADP
cana-6124	197	11	every	every	DET
cana-6124	197	12	𝑡	𝑡	PROPN
cana-6124	197	13	∈	∈	PROPN
cana-6124	198	1	[	[	X
cana-6124	198	2	0,1	0,1	NUM
cana-6124	198	3	]	]	PUNCT
cana-6124	198	4	:	:	PUNCT
cana-6124	198	5	|𝐹(𝑢)(𝑡)|	|𝐹(𝑢)(𝑡)|	X
cana-6124	198	6	≤	≤	PUNCT
cana-6124	198	7	|𝑎|∫	|𝑎|∫	NOUN
cana-6124	198	8	|𝑢(𝑠)|𝑑𝑞𝑠	|𝑢(𝑠)|𝑑𝑞𝑠	PUNCT
cana-6124	198	9	1	1	NUM
cana-6124	198	10	0	0	NUM
cana-6124	198	11	+	+	CCONJ
cana-6124	198	12	|𝑏|	|𝑏|	PROPN
cana-6124	198	13	+	+	CCONJ
cana-6124	198	14	1	1	NUM
cana-6124	198	15	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	198	16	)	)	PUNCT
cana-6124	198	17	∫	∫	PROPN
cana-6124	198	18	(	(	PUNCT
cana-6124	198	19	𝑡	𝑡	PROPN
cana-6124	198	20	−	−	NOUN
cana-6124	198	21	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	198	22	)	)	PUNCT
cana-6124	198	23	𝑡	𝑡	PROPN
cana-6124	198	24	0	0	PROPN
cana-6124	198	25	|𝑓(𝑠	|𝑓(𝑠	PROPN
cana-6124	198	26	,	,	PUNCT
cana-6124	198	27	𝑢(𝑠))|𝑑𝑞𝑠	𝑢(𝑠))|𝑑𝑞𝑠	PROPN
cana-6124	198	28	+	+	CCONJ
cana-6124	198	29	1	1	NUM
cana-6124	198	30	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-6124	198	31	)	)	PUNCT
cana-6124	198	32	∫	∫	PROPN
cana-6124	198	33	(	(	PUNCT
cana-6124	198	34	𝑡	𝑡	PROPN
cana-6124	198	35	−	−	NOUN
cana-6124	198	36	𝑞𝑠)(𝛼−1	𝑞𝑠)(𝛼−1	NOUN
cana-6124	198	37	)	)	PUNCT
cana-6124	198	38	𝑡	𝑡	PROPN
cana-6124	198	39	0	0	NUM
cana-6124	198	40	‖𝐴‖ℒ(𝑋)|𝑢(𝑠)|𝑑𝑞𝑠	‖𝐴‖ℒ(𝑋)|𝑢(𝑠)|𝑑𝑞𝑠	NOUN
cana-6124	198	41	≤	≤	VERB
cana-6124	199	1	|𝑎|	|𝑎|	ADP
cana-6124	199	2	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	NOUN
cana-6124	199	3	+	+	CCONJ
cana-6124	199	4	|𝑏|	|𝑏|	PROPN
cana-6124	199	5	+	+	CCONJ
cana-6124	199	6	𝑀	𝑀	PROPN
cana-6124	199	7	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-6124	199	8	)	)	PUNCT
cana-6124	199	9	∫	∫	PROPN
cana-6124	199	10	(	(	PUNCT
cana-6124	199	11	𝑡	𝑡	PROPN
cana-6124	199	12	−	−	PROPN
cana-6124	199	13	𝑞𝑠)(𝛼−1)𝑑𝑞𝑠	𝑞𝑠)(𝛼−1)𝑑𝑞𝑠	PROPN
cana-6124	199	14	𝑡	𝑡	PROPN
cana-6124	199	15	0	0	PUNCT
cana-6124	199	16	+	+	NUM
cana-6124	199	17	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	PROPN
cana-6124	199	18	)	)	PUNCT
cana-6124	199	19	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-6124	199	20	)	)	PUNCT
cana-6124	199	21	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|∫	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|∫	PROPN
cana-6124	199	22	(	(	PUNCT
cana-6124	199	23	𝑡	𝑡	NOUN
cana-6124	199	24	−	−	NOUN
cana-6124	199	25	𝑞𝑠	𝑞𝑠	PROPN
cana-6124	199	26	)	)	PUNCT
cana-6124	199	27	(	(	PUNCT
cana-6124	199	28	𝛼−1	𝛼−1	NOUN
cana-6124	199	29	)	)	PUNCT
cana-6124	199	30	𝑡	𝑡	PROPN
cana-6124	199	31	0	0	PROPN
cana-6124	199	32	𝑑𝑞𝑠	𝑑𝑞𝑠	PROPN
cana-6124	199	33	≤	≤	NOUN
cana-6124	199	34	|𝑎|	|𝑎|	ADP
cana-6124	199	35	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	NOUN
cana-6124	199	36	+	+	CCONJ
cana-6124	199	37	|𝑏|	|𝑏|	PROPN
cana-6124	199	38	+	+	CCONJ
cana-6124	199	39	𝑀	𝑀	PROPN
cana-6124	199	40	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	200	1	+	+	CCONJ
cana-6124	200	2	1	1	X
cana-6124	200	3	)	)	PUNCT
cana-6124	200	4	+	+	NUM
cana-6124	200	5	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	PROPN
cana-6124	200	6	)	)	PUNCT
cana-6124	200	7	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	201	1	+	+	CCONJ
cana-6124	201	2	1	1	X
cana-6124	201	3	)	)	PUNCT
cana-6124	201	4	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	NOUN
cana-6124	201	5	.	.	PUNCT
cana-6124	202	1	then	then	ADV
cana-6124	202	2	,	,	PUNCT
cana-6124	202	3	for	for	ADP
cana-6124	202	4	every	every	DET
cana-6124	202	5	𝑡	𝑡	PROPN
cana-6124	202	6	∈	∈	PROPN
cana-6124	202	7	[	[	X
cana-6124	202	8	0,1	0,1	NUM
cana-6124	202	9	]	]	PUNCT
cana-6124	202	10	,	,	PUNCT
cana-6124	202	11	we	we	PRON
cana-6124	202	12	have	have	VERB
cana-6124	202	13	‖𝐹(𝑢)‖∞	‖𝐹(𝑢)‖∞	ADJ
cana-6124	202	14	≤	≤	NOUN
cana-6124	202	15	|𝑎|	|𝑎|	ADP
cana-6124	202	16	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	NOUN
cana-6124	202	17	+	+	CCONJ
cana-6124	202	18	|𝑏|	|𝑏|	PROPN
cana-6124	202	19	+	+	CCONJ
cana-6124	202	20	𝑀	𝑀	PROPN
cana-6124	202	21	γ𝑞(𝛼	γ𝑞(𝛼	NOUN
cana-6124	203	1	+	+	CCONJ
cana-6124	203	2	1	1	X
cana-6124	203	3	)	)	PUNCT
cana-6124	203	4	+	+	NUM
cana-6124	203	5	‖𝐴‖ℒ(𝑋	‖𝐴‖ℒ(𝑋	PROPN
cana-6124	203	6	)	)	PUNCT
cana-6124	203	7	γ𝑞(𝛼	γ𝑞(𝛼	PUNCT
cana-6124	204	1	+	+	CCONJ
cana-6124	204	2	1	1	X
cana-6124	204	3	)	)	PUNCT
cana-6124	204	4	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|	NOUN
cana-6124	204	5	≔	≔	NOUN
cana-6124	204	6	𝑅.	𝑅.	NOUN
cana-6124	204	7	this	this	PRON
cana-6124	204	8	shows	show	VERB
cana-6124	204	9	that	that	SCONJ
cana-6124	204	10	the	the	DET
cana-6124	204	11	set	set	NOUN
cana-6124	204	12	ℰ	ℰ	PROPN
cana-6124	204	13	is	be	AUX
cana-6124	204	14	bounded	bound	VERB
cana-6124	204	15	.	.	PUNCT
cana-6124	205	1	consequently	consequently	ADV
cana-6124	205	2	,	,	PUNCT
cana-6124	205	3	by	by	ADP
cana-6124	205	4	schaefer	schaefer	NOUN
cana-6124	205	5	fixed	fix	VERB
cana-6124	205	6	point	point	NOUN
cana-6124	205	7	theorem	theorem	VERB
cana-6124	205	8	,	,	PUNCT
cana-6124	205	9	we	we	PRON
cana-6124	205	10	conclude	conclude	VERB
cana-6124	205	11	that	that	SCONJ
cana-6124	205	12	𝐹	𝐹	PROPN
cana-6124	205	13	admits	admit	VERB
cana-6124	205	14	a	a	DET
cana-6124	205	15	fixed	fixed	ADJ
cana-6124	205	16	point	point	NOUN
cana-6124	205	17	which	which	PRON
cana-6124	205	18	is	be	AUX
cana-6124	205	19	a	a	DET
cana-6124	205	20	solution	solution	NOUN
cana-6124	205	21	of	of	ADP
cana-6124	205	22	the	the	DET
cana-6124	205	23	problem	problem	NOUN
cana-6124	205	24	(	(	PUNCT
cana-6124	205	25	1.1)-(1.2	1.1)-(1.2	NUM
cana-6124	205	26	)	)	PUNCT
cana-6124	205	27	.	.	PUNCT
cana-6124	206	1	4	4	X
cana-6124	206	2	.	.	X
cana-6124	206	3	application	application	NOUN
cana-6124	206	4	in	in	ADP
cana-6124	206	5	this	this	DET
cana-6124	206	6	section	section	NOUN
cana-6124	206	7	,	,	PUNCT
cana-6124	206	8	we	we	PRON
cana-6124	206	9	provide	provide	VERB
cana-6124	206	10	an	an	DET
cana-6124	206	11	illustrative	illustrative	ADJ
cana-6124	206	12	example	example	NOUN
cana-6124	206	13	to	to	PART
cana-6124	206	14	demonstrate	demonstrate	VERB
cana-6124	206	15	the	the	DET
cana-6124	206	16	applicability	applicability	NOUN
cana-6124	206	17	of	of	ADP
cana-6124	206	18	our	our	PRON
cana-6124	206	19	result	result	NOUN
cana-6124	206	20	.	.	PUNCT
cana-6124	207	1	consider	consider	VERB
cana-6124	207	2	the	the	DET
cana-6124	207	3	following	follow	VERB
cana-6124	207	4	fractional	fractional	ADJ
cana-6124	207	5	q	q	ADJ
cana-6124	207	6	-	-	PUNCT
cana-6124	207	7	difference	difference	NOUN
cana-6124	207	8	problem	problem	NOUN
cana-6124	207	9	with	with	ADP
cana-6124	207	10	an	an	DET
cana-6124	207	11	integral	integral	ADJ
cana-6124	207	12	boundary	boundary	ADJ
cana-6124	207	13	condition	condition	NOUN
cana-6124	207	14	:	:	PUNCT
cana-6124	207	15	𝐷1	𝐷1	NOUN
cana-6124	207	16	3	3	NUM
cana-6124	207	17	1	1	NUM
cana-6124	207	18	2𝑐	2𝑐	NUM
cana-6124	207	19	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	207	20	)	)	PUNCT
cana-6124	207	21	=	=	SYM
cana-6124	207	22	𝑡	𝑡	NOUN
cana-6124	207	23	100	100	NUM
cana-6124	207	24	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	207	25	)	)	PUNCT
cana-6124	208	1	+	+	CCONJ
cana-6124	208	2	𝑢(𝑡	𝑢(𝑡	NOUN
cana-6124	208	3	)	)	PUNCT
cana-6124	208	4	(	(	PUNCT
cana-6124	208	5	1+𝑐𝑜𝑠(𝑢(𝑡	1+𝑐𝑜𝑠(𝑢(𝑡	NOUN
cana-6124	208	6	)	)	PUNCT
cana-6124	208	7	)	)	PUNCT
cana-6124	208	8	)	)	PUNCT
cana-6124	208	9	,	,	PUNCT
cana-6124	208	10	𝑡	𝑡	PROPN
cana-6124	208	11	∈	∈	PROPN
cana-6124	209	1	[	[	X
cana-6124	209	2	0,1	0,1	NUM
cana-6124	209	3	]	]	PUNCT
cana-6124	209	4	,	,	PUNCT
cana-6124	209	5	(	(	PUNCT
cana-6124	209	6	4.1	4.1	NUM
cana-6124	209	7	)	)	PUNCT
cana-6124	209	8	𝑢(0	𝑢(0	PROPN
cana-6124	209	9	)	)	PUNCT
cana-6124	209	10	=	=	SYM
cana-6124	209	11	2	2	NUM
cana-6124	209	12	3	3	NUM
cana-6124	209	13	∫	∫	NOUN
cana-6124	209	14	𝑢(𝑠)𝑑1	𝑢(𝑠)𝑑1	PROPN
cana-6124	209	15	3	3	NUM
cana-6124	209	16	𝑠	𝑠	PROPN
cana-6124	209	17	+	+	NOUN
cana-6124	209	18	1	1	NUM
cana-6124	209	19	2	2	NUM
cana-6124	209	20	.	.	PUNCT
cana-6124	209	21	1	1	NUM
cana-6124	209	22	0	0	NUM
cana-6124	209	23	(	(	PUNCT
cana-6124	209	24	4.2	4.2	NUM
cana-6124	209	25	)	)	PUNCT
cana-6124	209	26	communications	communication	NOUN
cana-6124	209	27	on	on	ADP
cana-6124	209	28	applied	apply	VERB
cana-6124	209	29	nonlinear	nonlinear	ADJ
cana-6124	209	30	analysis	analysis	NOUN
cana-6124	209	31	issn	issn	NOUN
cana-6124	209	32	:	:	PUNCT
cana-6124	209	33	1074	1074	NUM
cana-6124	209	34	-	-	PUNCT
cana-6124	209	35	133x	133x	NUM
cana-6124	209	36	vol	vol	VERB
cana-6124	209	37	32	32	NUM
cana-6124	209	38	no	no	NOUN
cana-6124	209	39	.	.	PUNCT
cana-6124	210	1	10s	10	NOUN
cana-6124	210	2	(	(	PUNCT
cana-6124	210	3	2025	2025	NUM
cana-6124	210	4	)	)	PUNCT
cana-6124	210	5	3435	3435	NUM
cana-6124	210	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	210	7	let	let	VERB
cana-6124	210	8	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	210	9	,	,	PUNCT
cana-6124	210	10	𝑢	𝑢	NOUN
cana-6124	210	11	)	)	PUNCT
cana-6124	210	12	=	=	SYM
cana-6124	210	13	𝑢	𝑢	NOUN
cana-6124	210	14	(	(	PUNCT
cana-6124	210	15	1	1	NUM
cana-6124	210	16	+	+	NUM
cana-6124	210	17	𝑐𝑜𝑠(𝑢	𝑐𝑜𝑠(𝑢	NOUN
cana-6124	210	18	)	)	PUNCT
cana-6124	210	19	)	)	PUNCT
cana-6124	210	20	,	,	PUNCT
cana-6124	210	21	(	(	PUNCT
cana-6124	210	22	𝑡	𝑡	X
cana-6124	210	23	,	,	PUNCT
cana-6124	210	24	𝑢	𝑢	PART
cana-6124	210	25	)	)	PUNCT
cana-6124	210	26	∈	∈	NOUN
cana-6124	211	1	[	[	X
cana-6124	211	2	0,1	0,1	NUM
cana-6124	211	3	]	]	X
cana-6124	211	4	×]0,+∞	×]0,+∞	NOUN
cana-6124	211	5	[	[	PUNCT
cana-6124	211	6	and	and	CCONJ
cana-6124	211	7	𝐴(𝑡	𝐴(𝑡	PROPN
cana-6124	211	8	)	)	PUNCT
cana-6124	211	9	=	=	SYM
cana-6124	211	10	𝑡	𝑡	PROPN
cana-6124	211	11	100	100	NUM
cana-6124	211	12	.	.	PUNCT
cana-6124	212	1	for	for	ADP
cana-6124	212	2	all	all	DET
cana-6124	212	3	𝑢	𝑢	NOUN
cana-6124	212	4	,	,	PUNCT
cana-6124	212	5	𝑣	𝑣	PRON
cana-6124	212	6	∈]0,+∞	∈]0,+∞	PUNCT
cana-6124	212	7	[	[	PUNCT
cana-6124	212	8	and	and	CCONJ
cana-6124	212	9	𝑡	𝑡	PROPN
cana-6124	212	10	∈	∈	PROPN
cana-6124	212	11	[	[	X
cana-6124	212	12	0,1	0,1	NUM
cana-6124	212	13	]	]	PUNCT
cana-6124	212	14	.	.	PUNCT
cana-6124	213	1	we	we	PRON
cana-6124	213	2	have	have	VERB
cana-6124	213	3	|𝑓(𝑡	|𝑓(𝑡	PROPN
cana-6124	213	4	,	,	PUNCT
cana-6124	213	5	𝑢	𝑢	NOUN
cana-6124	213	6	)	)	PUNCT
cana-6124	213	7	−	−	PROPN
cana-6124	214	1	𝑓(𝑡	𝑓(𝑡	NOUN
cana-6124	214	2	,	,	PUNCT
cana-6124	214	3	𝑣)|	𝑣)|	NOUN
cana-6124	214	4	≤	≤	NOUN
cana-6124	214	5	|	|	ADV
cana-6124	214	6	𝑢	𝑢	ADP
cana-6124	214	7	−	−	PROPN
cana-6124	214	8	𝑣	𝑣	X
cana-6124	214	9	(	(	PUNCT
cana-6124	214	10	1	1	NUM
cana-6124	214	11	+	+	NUM
cana-6124	214	12	cos(𝑢))(1	cos(𝑢))(1	NOUN
cana-6124	214	13	+	+	CCONJ
cana-6124	214	14	cos(𝑣	cos(𝑣	PROPN
cana-6124	214	15	)	)	PUNCT
cana-6124	214	16	)	)	PUNCT
cana-6124	215	1	|	|	ADV
cana-6124	215	2	≤	≤	NUM
cana-6124	215	3	1	1	NUM
cana-6124	215	4	4	4	NUM
cana-6124	215	5	|𝑢	|𝑢	ADJ
cana-6124	215	6	−	−	PROPN
cana-6124	215	7	𝑣|	𝑣|	PROPN
cana-6124	215	8	.	.	PUNCT
cana-6124	216	1	we	we	PRON
cana-6124	216	2	see	see	VERB
cana-6124	216	3	that	that	SCONJ
cana-6124	216	4	the	the	DET
cana-6124	216	5	condition	condition	NOUN
cana-6124	216	6	|𝑎|	|𝑎|	ADP
cana-6124	216	7	+	+	ADJ
cana-6124	216	8	𝑘+‖𝐴‖	𝑘+‖𝐴‖	NUM
cana-6124	216	9	γ𝑞(𝛼+1	γ𝑞(𝛼+1	PROPN
cana-6124	216	10	)	)	PUNCT
cana-6124	217	1	≈	≈	PROPN
cana-6124	217	2	0.93	0.93	NUM
cana-6124	217	3	<	<	SYM
cana-6124	217	4	1	1	NUM
cana-6124	217	5	holds	hold	VERB
cana-6124	217	6	with	with	ADP
cana-6124	217	7	𝛼	𝛼	NOUN
cana-6124	217	8	=	=	SYM
cana-6124	217	9	1	1	NUM
cana-6124	217	10	2	2	NUM
cana-6124	217	11	,	,	PUNCT
cana-6124	217	12	𝑘	𝑘	NOUN
cana-6124	217	13	=	=	NOUN
cana-6124	217	14	1	1	NUM
cana-6124	217	15	4	4	NUM
cana-6124	217	16	,	,	PUNCT
cana-6124	217	17	𝑞	𝑞	X
cana-6124	217	18	=	=	NOUN
cana-6124	217	19	1	1	NUM
cana-6124	217	20	3	3	NUM
cana-6124	217	21	,	,	PUNCT
cana-6124	217	22	‖𝐴‖	‖𝐴‖	PROPN
cana-6124	217	23	=	=	SYM
cana-6124	217	24	1	1	NUM
cana-6124	217	25	100	100	NUM
cana-6124	217	26	and	and	CCONJ
cana-6124	217	27	γ1	γ1	PROPN
cana-6124	217	28	3	3	NUM
cana-6124	217	29	(	(	PUNCT
cana-6124	217	30	3	3	NUM
cana-6124	217	31	2	2	NUM
cana-6124	217	32	)	)	PUNCT
cana-6124	217	33	≈	≈	PROPN
cana-6124	217	34	0.9376	0.9376	NUM
cana-6124	217	35	.	.	PUNCT
cana-6124	218	1	consequently	consequently	ADV
cana-6124	218	2	,	,	PUNCT
cana-6124	218	3	by	by	ADP
cana-6124	218	4	theorem	theorem	NOUN
cana-6124	218	5	14	14	NUM
cana-6124	218	6	,	,	PUNCT
cana-6124	218	7	the	the	DET
cana-6124	218	8	fractional	fractional	ADJ
cana-6124	218	9	q	q	ADJ
cana-6124	218	10	-	-	PUNCT
cana-6124	218	11	difference	difference	NOUN
cana-6124	218	12	problem	problem	NOUN
cana-6124	218	13	(	(	PUNCT
cana-6124	218	14	4.1)(4.2	4.1)(4.2	NOUN
cana-6124	218	15	)	)	PUNCT
cana-6124	218	16	admits	admit	VERB
cana-6124	218	17	a	a	DET
cana-6124	218	18	unique	unique	ADJ
cana-6124	218	19	solution	solution	NOUN
cana-6124	218	20	on	on	ADP
cana-6124	218	21	the	the	DET
cana-6124	218	22	interval	interval	NOUN
cana-6124	218	23	[	[	X
cana-6124	218	24	0,1	0,1	NUM
cana-6124	218	25	]	]	PUNCT
cana-6124	218	26	.	.	PUNCT
cana-6124	219	1	5	5	X
cana-6124	219	2	.	.	X
cana-6124	219	3	conclusion	conclusion	NOUN
cana-6124	219	4	in	in	ADP
cana-6124	219	5	this	this	DET
cana-6124	219	6	study	study	NOUN
cana-6124	219	7	,	,	PUNCT
cana-6124	219	8	we	we	PRON
cana-6124	219	9	establish	establish	VERB
cana-6124	219	10	the	the	DET
cana-6124	219	11	existence	existence	NOUN
cana-6124	219	12	of	of	ADP
cana-6124	219	13	solutions	solution	NOUN
cana-6124	219	14	to	to	PART
cana-6124	219	15	cauchy	cauchy	VERB
cana-6124	219	16	problems	problem	NOUN
cana-6124	219	17	for	for	ADP
cana-6124	219	18	fractional	fractional	ADJ
cana-6124	219	19	qdifference	qdifference	NOUN
cana-6124	219	20	equations	equation	NOUN
cana-6124	219	21	with	with	ADP
cana-6124	219	22	integral	integral	ADJ
cana-6124	219	23	conditions	condition	NOUN
cana-6124	219	24	in	in	ADP
cana-6124	219	25	banach	banach	NOUN
cana-6124	219	26	spaces	space	NOUN
cana-6124	219	27	.	.	PUNCT
cana-6124	220	1	the	the	DET
cana-6124	220	2	analysis	analysis	NOUN
cana-6124	220	3	is	be	AUX
cana-6124	220	4	conducted	conduct	VERB
cana-6124	220	5	employing	employ	VERB
cana-6124	220	6	the	the	DET
cana-6124	220	7	banach	banach	ADV
cana-6124	220	8	fixed	fix	VERB
cana-6124	220	9	point	point	NOUN
cana-6124	220	10	theorem	theorem	NOUN
cana-6124	220	11	and	and	CCONJ
cana-6124	220	12	the	the	DET
cana-6124	220	13	schaefer	schaefer	NOUN
cana-6124	220	14	fixed	fix	VERB
cana-6124	220	15	point	point	NOUN
cana-6124	220	16	theorem	theorem	VERB
cana-6124	220	17	.	.	PUNCT
cana-6124	221	1	furthermore	furthermore	ADV
cana-6124	221	2	,	,	PUNCT
cana-6124	221	3	an	an	DET
cana-6124	221	4	illustrative	illustrative	ADJ
cana-6124	221	5	example	example	NOUN
cana-6124	221	6	is	be	AUX
cana-6124	221	7	provided	provide	VERB
cana-6124	221	8	to	to	PART
cana-6124	221	9	demonstrate	demonstrate	VERB
cana-6124	221	10	the	the	DET
cana-6124	221	11	applicability	applicability	NOUN
cana-6124	221	12	and	and	CCONJ
cana-6124	221	13	effectiveness	effectiveness	NOUN
cana-6124	221	14	of	of	ADP
cana-6124	221	15	the	the	DET
cana-6124	221	16	obtained	obtain	VERB
cana-6124	221	17	results	result	NOUN
cana-6124	221	18	.	.	PUNCT
cana-6124	222	1	references	reference	NOUN
cana-6124	222	2	[	[	X
cana-6124	222	3	1	1	X
cana-6124	222	4	]	]	PUNCT
cana-6124	222	5	s.	s.	PROPN
cana-6124	222	6	abbas	abbas	PROPN
cana-6124	222	7	,	,	PUNCT
cana-6124	222	8	m.	m.	NOUN
cana-6124	222	9	benchohra	benchohra	NOUN
cana-6124	222	10	,	,	PUNCT
cana-6124	222	11	j.	j.	PROPN
cana-6124	222	12	henderson	henderson	PROPN
cana-6124	222	13	,	,	PUNCT
cana-6124	222	14	existence	existence	NOUN
cana-6124	222	15	and	and	CCONJ
cana-6124	222	16	oscillation	oscillation	NOUN
cana-6124	222	17	for	for	ADP
cana-6124	222	18	coupled	couple	VERB
cana-6124	222	19	fractional	fractional	ADJ
cana-6124	222	20	q	q	NOUN
cana-6124	222	21	difference	difference	NOUN
cana-6124	222	22	systems	system	NOUN
cana-6124	222	23	,	,	PUNCT
cana-6124	222	24	fract	fract	NOUN
cana-6124	222	25	.	.	PUNCT
cana-6124	223	1	calc	calc	PROPN
cana-6124	223	2	.	.	PUNCT
cana-6124	224	1	appl	appl	PROPN
cana-6124	224	2	.	.	PUNCT
cana-6124	225	1	anal	anal	PROPN
cana-6124	225	2	.	.	PROPN
cana-6124	225	3	,	,	PUNCT
cana-6124	225	4	12	12	NUM
cana-6124	225	5	(	(	PUNCT
cana-6124	225	6	2021	2021	NUM
cana-6124	225	7	)	)	PUNCT
cana-6124	225	8	,	,	PUNCT
cana-6124	225	9	143	143	NUM
cana-6124	225	10	-	-	SYM
cana-6124	225	11	155	155	NUM
cana-6124	225	12	.	.	PUNCT
cana-6124	226	1	[	[	X
cana-6124	226	2	2	2	NUM
cana-6124	226	3	]	]	X
cana-6124	226	4	r.	r.	PROPN
cana-6124	226	5	agarwal	agarwal	PROPN
cana-6124	226	6	,	,	PUNCT
cana-6124	226	7	certain	certain	ADJ
cana-6124	226	8	fractional	fractional	ADJ
cana-6124	226	9	q	q	NOUN
cana-6124	226	10	-	-	PUNCT
cana-6124	226	11	integrals	integral	NOUN
cana-6124	226	12	and	and	CCONJ
cana-6124	226	13	q	q	NOUN
cana-6124	226	14	-	-	PUNCT
cana-6124	226	15	derivatives	derivative	NOUN
cana-6124	226	16	,	,	PUNCT
cana-6124	226	17	proc	proc	NOUN
cana-6124	226	18	.	.	PUNCT
cana-6124	227	1	camb	camb	PROPN
cana-6124	227	2	.	.	PUNCT
cana-6124	228	1	philos	philos	PROPN
cana-6124	228	2	.	.	PUNCT
cana-6124	229	1	soc	soc	PROPN
cana-6124	229	2	.	.	PUNCT
cana-6124	229	3	,	,	PUNCT
cana-6124	229	4	66(1969	66(1969	PROPN
cana-6124	229	5	)	)	PUNCT
cana-6124	229	6	,	,	PUNCT
cana-6124	229	7	365	365	NUM
cana-6124	229	8	-	-	SYM
cana-6124	229	9	370	370	NUM
cana-6124	229	10	.	.	PUNCT
cana-6124	230	1	[	[	X
cana-6124	230	2	3	3	NUM
cana-6124	230	3	]	]	X
cana-6124	230	4	n.	n.	NOUN
cana-6124	230	5	allouch	allouch	PROPN
cana-6124	230	6	,	,	PUNCT
cana-6124	230	7	j.r	j.r	PROPN
cana-6124	230	8	.	.	PROPN
cana-6124	230	9	graef	graef	PROPN
cana-6124	230	10	,	,	PUNCT
cana-6124	230	11	s.	s.	PROPN
cana-6124	230	12	hamani	hamani	PROPN
cana-6124	230	13	,	,	PUNCT
cana-6124	230	14	boundary	boundary	ADJ
cana-6124	230	15	value	value	NOUN
cana-6124	230	16	problem	problem	NOUN
cana-6124	230	17	for	for	ADP
cana-6124	230	18	fractional	fractional	ADJ
cana-6124	230	19	q	q	ADJ
cana-6124	230	20	-	-	PUNCT
cana-6124	230	21	difference	difference	NOUN
cana-6124	230	22	equations	equation	NOUN
cana-6124	230	23	with	with	ADP
cana-6124	230	24	integral	integral	ADJ
cana-6124	230	25	conditions	condition	NOUN
cana-6124	230	26	in	in	ADP
cana-6124	230	27	banach	banach	NOUN
cana-6124	230	28	space	space	NOUN
cana-6124	230	29	,	,	PUNCT
cana-6124	230	30	fractal	fractal	ADJ
cana-6124	230	31	fract	fract	NOUN
cana-6124	230	32	.	.	PUNCT
cana-6124	230	33	,	,	PUNCT
cana-6124	230	34	6	6	NUM
cana-6124	230	35	(	(	PUNCT
cana-6124	230	36	2022	2022	NUM
cana-6124	230	37	)	)	PUNCT
cana-6124	230	38	,	,	PUNCT
cana-6124	230	39	1	1	NUM
cana-6124	230	40	-	-	SYM
cana-6124	230	41	11	11	NUM
cana-6124	230	42	.	.	PUNCT
cana-6124	231	1	[	[	X
cana-6124	231	2	4	4	NUM
cana-6124	231	3	]	]	X
cana-6124	231	4	n.	n.	NOUN
cana-6124	231	5	allouch	allouch	PROPN
cana-6124	231	6	,	,	PUNCT
cana-6124	231	7	s.	s.	PROPN
cana-6124	231	8	hamani	hamani	PROPN
cana-6124	231	9	,	,	PUNCT
cana-6124	231	10	j.	j.	PROPN
cana-6124	231	11	henderson	henderson	PROPN
cana-6124	231	12	,	,	PUNCT
cana-6124	231	13	boundary	boundary	ADJ
cana-6124	231	14	value	value	NOUN
cana-6124	231	15	problem	problem	NOUN
cana-6124	231	16	for	for	ADP
cana-6124	231	17	fractional	fractional	ADJ
cana-6124	231	18	q	q	ADJ
cana-6124	231	19	-	-	PUNCT
cana-6124	231	20	difference	difference	NOUN
cana-6124	231	21	equations	equation	NOUN
cana-6124	231	22	,	,	PUNCT
cana-6124	231	23	nonlinear	nonlinear	ADJ
cana-6124	231	24	dyn	dyn	NOUN
cana-6124	231	25	.	.	PUNCT
cana-6124	232	1	syst	syst	PROPN
cana-6124	232	2	.	.	PUNCT
cana-6124	233	1	theory	theory	NOUN
cana-6124	233	2	,	,	PUNCT
cana-6124	233	3	24	24	NUM
cana-6124	233	4	(	(	PUNCT
cana-6124	233	5	2024	2024	NUM
cana-6124	233	6	)	)	PUNCT
cana-6124	233	7	,	,	PUNCT
cana-6124	233	8	111	111	NUM
cana-6124	233	9	-	-	SYM
cana-6124	233	10	122	122	NUM
cana-6124	233	11	.	.	PUNCT
cana-6124	234	1	[	[	X
cana-6124	234	2	5	5	NUM
cana-6124	234	3	]	]	PUNCT
cana-6124	234	4	b.	b.	PROPN
cana-6124	234	5	ahmad	ahmad	PROPN
cana-6124	234	6	,	,	PUNCT
cana-6124	234	7	s.k	s.k	PROPN
cana-6124	234	8	.	.	PROPN
cana-6124	234	9	ntouyas	ntouyas	PROPN
cana-6124	234	10	,	,	PUNCT
cana-6124	234	11	i.k	i.k	PROPN
cana-6124	234	12	.	.	PROPN
cana-6124	234	13	purnaras	purnaras	PROPN
cana-6124	234	14	,	,	PUNCT
cana-6124	234	15	existence	existence	NOUN
cana-6124	234	16	results	result	VERB
cana-6124	234	17	for	for	ADP
cana-6124	234	18	nonlocal	nonlocal	ADJ
cana-6124	234	19	boundary	boundary	ADJ
cana-6124	234	20	value	value	NOUN
cana-6124	234	21	problems	problem	NOUN
cana-6124	234	22	of	of	ADP
cana-6124	234	23	nonlinear	nonlinear	ADJ
cana-6124	234	24	fractional	fractional	ADJ
cana-6124	234	25	q	q	ADJ
cana-6124	234	26	-	-	PUNCT
cana-6124	234	27	difference	difference	NOUN
cana-6124	234	28	equations	equation	NOUN
cana-6124	234	29	,	,	PUNCT
cana-6124	234	30	adv	adv	PROPN
cana-6124	234	31	,	,	PUNCT
cana-6124	234	32	differ	differ	VERB
cana-6124	234	33	.	.	PUNCT
cana-6124	235	1	equ	equ	PROPN
cana-6124	235	2	.	.	PROPN
cana-6124	235	3	,	,	PUNCT
cana-6124	235	4	140	140	NUM
cana-6124	235	5	(	(	PUNCT
cana-6124	235	6	2012	2012	NUM
cana-6124	235	7	)	)	PUNCT
cana-6124	235	8	.	.	PUNCT
cana-6124	236	1	[	[	X
cana-6124	236	2	6	6	NUM
cana-6124	236	3	]	]	PUNCT
cana-6124	236	4	w.	w.	PROPN
cana-6124	236	5	al	al	PROPN
cana-6124	236	6	-	-	PUNCT
cana-6124	236	7	salam	salam	PROPN
cana-6124	236	8	,	,	PUNCT
cana-6124	236	9	some	some	DET
cana-6124	236	10	fractional	fractional	ADJ
cana-6124	236	11	q	q	ADJ
cana-6124	236	12	-	-	ADJ
cana-6124	236	13	integral	integral	ADJ
cana-6124	236	14	and	and	CCONJ
cana-6124	236	15	q	q	NOUN
cana-6124	236	16	-	-	PUNCT
cana-6124	236	17	derivatives	derivative	NOUN
cana-6124	236	18	,	,	PUNCT
cana-6124	236	19	proc	proc	NOUN
cana-6124	236	20	.	.	PUNCT
cana-6124	237	1	edinb	edinb	PROPN
cana-6124	237	2	.	.	PUNCT
cana-6124	238	1	math	math	NOUN
cana-6124	238	2	.	.	PUNCT
cana-6124	239	1	soc	soc	PROPN
cana-6124	239	2	.	.	PUNCT
cana-6124	239	3	,	,	PUNCT
cana-6124	239	4	15	15	NUM
cana-6124	239	5	(	(	PUNCT
cana-6124	239	6	1967	1967	NUM
cana-6124	239	7	)	)	PUNCT
cana-6124	239	8	,	,	PUNCT
cana-6124	239	9	135	135	NUM
cana-6124	239	10	-	-	SYM
cana-6124	239	11	140	140	NUM
cana-6124	239	12	.	.	PUNCT
cana-6124	240	1	[	[	X
cana-6124	240	2	7	7	NUM
cana-6124	240	3	]	]	PUNCT
cana-6124	240	4	a.	a.	NOUN
cana-6124	240	5	granas	granas	PROPN
cana-6124	240	6	,	,	PUNCT
cana-6124	240	7	j.	j.	PROPN
cana-6124	240	8	dugundji	dugundji	PROPN
cana-6124	240	9	,	,	PUNCT
cana-6124	240	10	fixed	fix	VERB
cana-6124	240	11	point	point	NOUN
cana-6124	240	12	theory	theory	NOUN
cana-6124	240	13	,	,	PUNCT
cana-6124	240	14	springer	springer	NOUN
cana-6124	240	15	,	,	PUNCT
cana-6124	240	16	new	new	PROPN
cana-6124	240	17	york	york	PROPN
cana-6124	240	18	,	,	PUNCT
cana-6124	240	19	2003	2003	NUM
cana-6124	240	20	.	.	PUNCT
cana-6124	241	1	[	[	X
cana-6124	241	2	8	8	NUM
cana-6124	241	3	]	]	X
cana-6124	241	4	g.	g.	PROPN
cana-6124	241	5	gasper	gasper	PROPN
cana-6124	241	6	,	,	PUNCT
cana-6124	241	7	m.	m.	PROPN
cana-6124	241	8	rahman	rahman	PROPN
cana-6124	241	9	,	,	PUNCT
cana-6124	241	10	basic	basic	ADJ
cana-6124	241	11	hypergeometric	hypergeometric	ADJ
cana-6124	241	12	series	series	NOUN
cana-6124	241	13	,	,	PUNCT
cana-6124	241	14	encyclopedia	encyclopedia	NOUN
cana-6124	241	15	math	math	NOUN
cana-6124	241	16	.	.	PUNCT
cana-6124	242	1	appl	appl	PROPN
cana-6124	242	2	.	.	PROPN
cana-6124	242	3	,	,	PUNCT
cana-6124	242	4	vol	vol	NOUN
cana-6124	242	5	.	.	PROPN
cana-6124	242	6	96	96	NUM
cana-6124	242	7	,	,	PUNCT
cana-6124	242	8	cambridge	cambridge	PROPN
cana-6124	242	9	univ	univ	PROPN
cana-6124	242	10	.	.	PUNCT
cana-6124	243	1	press	press	PROPN
cana-6124	243	2	,	,	PUNCT
cana-6124	243	3	cambridge	cambridge	PROPN
cana-6124	243	4	,	,	PUNCT
cana-6124	243	5	1990	1990	NUM
cana-6124	243	6	.	.	PUNCT
cana-6124	244	1	communications	communication	NOUN
cana-6124	244	2	on	on	ADP
cana-6124	244	3	applied	apply	VERB
cana-6124	244	4	nonlinear	nonlinear	ADJ
cana-6124	244	5	analysis	analysis	NOUN
cana-6124	244	6	issn	issn	NOUN
cana-6124	244	7	:	:	PUNCT
cana-6124	244	8	1074	1074	NUM
cana-6124	244	9	-	-	PUNCT
cana-6124	244	10	133x	133x	NUM
cana-6124	244	11	vol	vol	VERB
cana-6124	244	12	32	32	NUM
cana-6124	244	13	no	no	NOUN
cana-6124	244	14	.	.	PUNCT
cana-6124	245	1	10s	10	NOUN
cana-6124	245	2	(	(	PUNCT
cana-6124	245	3	2025	2025	NUM
cana-6124	245	4	)	)	PUNCT
cana-6124	245	5	3436	3436	NUM
cana-6124	245	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6124	246	1	[	[	X
cana-6124	246	2	9	9	NUM
cana-6124	246	3	]	]	X
cana-6124	246	4	r.	r.	NOUN
cana-6124	246	5	hilfer	hilfer	PROPN
cana-6124	246	6	,	,	PUNCT
cana-6124	246	7	applications	application	NOUN
cana-6124	246	8	of	of	ADP
cana-6124	246	9	fractional	fractional	ADJ
cana-6124	246	10	calculus	calculus	NOUN
cana-6124	246	11	in	in	ADP
cana-6124	246	12	physics	physics	PROPN
cana-6124	246	13	,	,	PUNCT
cana-6124	246	14	world	world	NOUN
cana-6124	246	15	scientific	scientific	PROPN
cana-6124	246	16	,	,	PUNCT
cana-6124	246	17	singapore	singapore	PROPN
cana-6124	246	18	,	,	PUNCT
cana-6124	246	19	2000	2000	NUM
cana-6124	246	20	.	.	PUNCT
cana-6124	247	1	[	[	X
cana-6124	247	2	10	10	NUM
cana-6124	247	3	]	]	X
cana-6124	247	4	k.s	k.s	PROPN
cana-6124	247	5	.	.	PROPN
cana-6124	247	6	miller	miller	PROPN
cana-6124	247	7	,	,	PUNCT
cana-6124	247	8	b.	b.	PROPN
cana-6124	247	9	ross	ross	PROPN
cana-6124	247	10	,	,	PUNCT
cana-6124	247	11	an	an	DET
cana-6124	247	12	introduction	introduction	NOUN
cana-6124	247	13	to	to	ADP
cana-6124	247	14	fractional	fractional	ADJ
cana-6124	247	15	calculus	calculus	NOUN
cana-6124	247	16	and	and	CCONJ
cana-6124	247	17	differential	differential	ADJ
cana-6124	247	18	equations	equation	NOUN
cana-6124	247	19	,	,	PUNCT
cana-6124	247	20	wiley	wiley	NOUN
cana-6124	247	21	,	,	PUNCT
cana-6124	247	22	new	new	PROPN
cana-6124	247	23	york	york	PROPN
cana-6124	247	24	,	,	PUNCT
cana-6124	247	25	1993	1993	NUM
cana-6124	247	26	.	.	PUNCT
cana-6124	248	1	[	[	X
cana-6124	248	2	11	11	NUM
cana-6124	248	3	]	]	PUNCT
cana-6124	248	4	i.	i.	NOUN
cana-6124	248	5	podlubny	podlubny	PROPN
cana-6124	248	6	,	,	PUNCT
cana-6124	248	7	fractional	fractional	ADJ
cana-6124	248	8	differential	differential	NOUN
cana-6124	248	9	equations	equation	NOUN
cana-6124	248	10	,	,	PUNCT
cana-6124	248	11	academic	academic	ADJ
cana-6124	248	12	press	press	NOUN
cana-6124	248	13	,	,	PUNCT
cana-6124	248	14	san	san	PROPN
cana-6124	248	15	diego	diego	PROPN
cana-6124	248	16	,	,	PUNCT
cana-6124	248	17	1998	1998	NUM
cana-6124	248	18	.	.	PUNCT
cana-6124	249	1	[	[	X
cana-6124	249	2	12	12	NUM
cana-6124	249	3	]	]	X
cana-6124	249	4	v.	v.	PROPN
cana-6124	249	5	kac	kac	PROPN
cana-6124	249	6	,	,	PUNCT
cana-6124	249	7	p.	p.	PROPN
cana-6124	249	8	cheung	cheung	PROPN
cana-6124	249	9	,	,	PUNCT
cana-6124	249	10	quantum	quantum	NOUN
cana-6124	249	11	calculus	calculus	NOUN
cana-6124	249	12	,	,	PUNCT
cana-6124	249	13	springer	springer	NOUN
cana-6124	249	14	,	,	PUNCT
cana-6124	249	15	new	new	PROPN
cana-6124	249	16	york	york	PROPN
cana-6124	249	17	,	,	PUNCT
cana-6124	249	18	2002	2002	NUM
cana-6124	249	19	.	.	PUNCT
cana-6124	250	1	[	[	X
cana-6124	250	2	13	13	NUM
cana-6124	250	3	]	]	SYM
cana-6124	250	4	a.a	a.a	PROPN
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cana-6124	253	2	,	,	PUNCT
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cana-6124	257	2	.	.	PUNCT
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cana-6124	263	9	.	.	PROPN
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cana-6124	265	10	theorems	theorem	NOUN
cana-6124	265	11	,	,	PUNCT
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