id	sid	tid	token	lemma	pos
cana-1117	1	1	communications	communication	NOUN
cana-1117	1	2	on	on	ADP
cana-1117	1	3	applied	apply	VERB
cana-1117	1	4	nonlinear	nonlinear	ADJ
cana-1117	1	5	analysis	analysis	NOUN
cana-1117	1	6	issn	issn	NOUN
cana-1117	1	7	:	:	PUNCT
cana-1117	1	8	1074	1074	NUM
cana-1117	1	9	-	-	PUNCT
cana-1117	1	10	133x	133x	NUM
cana-1117	1	11	vol	vol	NOUN
cana-1117	1	12	31	31	NUM
cana-1117	1	13	no	no	NOUN
cana-1117	1	14	.	.	PUNCT
cana-1117	2	1	6s	6s	NUM
cana-1117	2	2	(	(	PUNCT
cana-1117	2	3	2024	2024	NUM
cana-1117	2	4	)	)	PUNCT
cana-1117	2	5	35	35	NUM
cana-1117	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	2	7	outcomes	outcome	NOUN
cana-1117	2	8	for	for	ADP
cana-1117	2	9	the	the	DET
cana-1117	2	10	analysis	analysis	NOUN
cana-1117	2	11	of	of	ADP
cana-1117	2	12	the	the	DET
cana-1117	2	13	existence	existence	NOUN
cana-1117	2	14	of	of	ADP
cana-1117	2	15	fuzzy	fuzzy	ADJ
cana-1117	2	16	fractional	fractional	ADJ
cana-1117	2	17	differential	differential	NOUN
cana-1117	2	18	equations	equation	NOUN
cana-1117	2	19	p.dhanalakshmi1	p.dhanalakshmi1	PROPN
cana-1117	2	20	,	,	PUNCT
cana-1117	2	21	s.padmavathi2	s.padmavathi2	NOUN
cana-1117	2	22	1department	1department	NUM
cana-1117	2	23	of	of	ADP
cana-1117	2	24	applied	apply	VERB
cana-1117	2	25	mathematics	mathematic	NOUN
cana-1117	2	26	,	,	PUNCT
cana-1117	2	27	bharathiar	bharathiar	PROPN
cana-1117	2	28	university	university	PROPN
cana-1117	2	29	,	,	PUNCT
cana-1117	2	30	coimbatore	coimbatore	PROPN
cana-1117	2	31	,	,	PUNCT
cana-1117	2	32	india	india	PROPN
cana-1117	2	33	.	.	PUNCT
cana-1117	2	34	viga_dhanasekar@yahoo.co.in	viga_dhanasekar@yahoo.co.in	PROPN
cana-1117	3	1	2department	2department	NUM
cana-1117	3	2	of	of	ADP
cana-1117	3	3	mathematics	mathematic	NOUN
cana-1117	3	4	,	,	PUNCT
cana-1117	3	5	vellalar	vellalar	ADJ
cana-1117	3	6	college	college	NOUN
cana-1117	3	7	for	for	ADP
cana-1117	3	8	women	woman	NOUN
cana-1117	3	9	,	,	PUNCT
cana-1117	3	10	erode	erode	VERB
cana-1117	3	11	,	,	PUNCT
cana-1117	3	12	india	india	PROPN
cana-1117	3	13	.	.	PUNCT
cana-1117	4	1	padmavathi201@gmail.com	padmavathi201@gmail.com	X
cana-1117	4	2	article	article	NOUN
cana-1117	4	3	history	history	NOUN
cana-1117	4	4	:	:	PUNCT
cana-1117	4	5	received	receive	VERB
cana-1117	4	6	:	:	PUNCT
cana-1117	4	7	23	23	NUM
cana-1117	4	8	-	-	SYM
cana-1117	4	9	05	05	NUM
cana-1117	4	10	-	-	PUNCT
cana-1117	4	11	2024	2024	NUM
cana-1117	4	12	revised	revise	VERB
cana-1117	4	13	:	:	PUNCT
cana-1117	4	14	26	26	NUM
cana-1117	4	15	-	-	SYM
cana-1117	4	16	06	06	NUM
cana-1117	4	17	-	-	PUNCT
cana-1117	4	18	2024	2024	NUM
cana-1117	4	19	accepted	accept	VERB
cana-1117	4	20	:	:	PUNCT
cana-1117	4	21	18	18	NUM
cana-1117	4	22	-	-	SYM
cana-1117	4	23	07	07	NUM
cana-1117	4	24	-	-	PUNCT
cana-1117	4	25	2024	2024	NUM
cana-1117	4	26	abstract	abstract	NOUN
cana-1117	4	27	:	:	PUNCT
cana-1117	4	28	in	in	ADP
cana-1117	4	29	this	this	DET
cana-1117	4	30	article	article	NOUN
cana-1117	4	31	,	,	PUNCT
cana-1117	4	32	a	a	DET
cana-1117	4	33	cauchy	cauchy	ADJ
cana-1117	4	34	problem	problem	NOUN
cana-1117	4	35	for	for	ADP
cana-1117	4	36	a	a	DET
cana-1117	4	37	fuzzy	fuzzy	ADJ
cana-1117	4	38	q	q	ADJ
cana-1117	4	39	-	-	PUNCT
cana-1117	4	40	fractional	fractional	ADJ
cana-1117	4	41	differential	differential	ADJ
cana-1117	4	42	equation	equation	NOUN
cana-1117	4	43	of	of	ADP
cana-1117	4	44	order	order	NOUN
cana-1117	4	45	α	α	PROPN
cana-1117	4	46	has	have	AUX
cana-1117	4	47	been	be	AUX
cana-1117	4	48	considered.the	considered.the	NOUN
cana-1117	4	49	results	result	NOUN
cana-1117	4	50	for	for	SCONJ
cana-1117	4	51	generalized	generalized	ADJ
cana-1117	4	52	hukuhara	hukuhara	ADJ
cana-1117	4	53	q	q	NOUN
cana-1117	4	54	-	-	NOUN
cana-1117	4	55	differentiablity	differentiablity	NOUN
cana-1117	4	56	of	of	ADP
cana-1117	4	57	a	a	DET
cana-1117	4	58	fuzzy	fuzzy	ADJ
cana-1117	4	59	function	function	NOUN
cana-1117	4	60	are	be	AUX
cana-1117	4	61	established	establish	VERB
cana-1117	4	62	.	.	PUNCT
cana-1117	5	1	this	this	DET
cana-1117	5	2	work	work	NOUN
cana-1117	5	3	has	have	AUX
cana-1117	5	4	led	lead	VERB
cana-1117	5	5	to	to	ADP
cana-1117	5	6	the	the	DET
cana-1117	5	7	study	study	NOUN
cana-1117	5	8	of	of	ADP
cana-1117	5	9	existence	existence	NOUN
cana-1117	5	10	and	and	CCONJ
cana-1117	5	11	uniqueness	uniqueness	NOUN
cana-1117	5	12	of	of	ADP
cana-1117	5	13	the	the	DET
cana-1117	5	14	fuzzy	fuzzy	ADJ
cana-1117	5	15	function	function	NOUN
cana-1117	5	16	with	with	ADP
cana-1117	5	17	caputo	caputo	PROPN
cana-1117	5	18	hukuhara	hukuhara	PROPN
cana-1117	5	19	q	q	PROPN
cana-1117	5	20	-	-	PUNCT
cana-1117	5	21	differentiability	differentiability	NOUN
cana-1117	5	22	and	and	CCONJ
cana-1117	5	23	generalized	generalized	ADJ
cana-1117	5	24	banach	banach	NOUN
cana-1117	5	25	fixed	fix	VERB
cana-1117	5	26	point	point	NOUN
cana-1117	5	27	theorem	theorem	VERB
cana-1117	5	28	.	.	PUNCT
cana-1117	6	1	a	a	DET
cana-1117	6	2	crucial	crucial	ADJ
cana-1117	6	3	qualitative	qualitative	ADJ
cana-1117	6	4	property	property	NOUN
cana-1117	6	5	of	of	ADP
cana-1117	6	6	the	the	DET
cana-1117	6	7	differential	differential	ADJ
cana-1117	6	8	equation	equation	NOUN
cana-1117	6	9	,	,	PUNCT
cana-1117	6	10	which	which	PRON
cana-1117	6	11	is	be	AUX
cana-1117	6	12	continuous	continuous	ADJ
cana-1117	6	13	dependence	dependence	NOUN
cana-1117	6	14	on	on	ADP
cana-1117	6	15	initial	initial	ADJ
cana-1117	6	16	conditions	condition	NOUN
cana-1117	6	17	and	and	CCONJ
cana-1117	6	18	the	the	DET
cana-1117	6	19	functions	function	NOUN
cana-1117	6	20	involved	involve	VERB
cana-1117	6	21	is	be	AUX
cana-1117	6	22	analysed	analyse	VERB
cana-1117	6	23	.	.	PUNCT
cana-1117	7	1	an	an	DET
cana-1117	7	2	illustrative	illustrative	ADJ
cana-1117	7	3	example	example	NOUN
cana-1117	7	4	is	be	AUX
cana-1117	7	5	given	give	VERB
cana-1117	7	6	which	which	PRON
cana-1117	7	7	ensures	ensure	VERB
cana-1117	7	8	the	the	DET
cana-1117	7	9	result	result	NOUN
cana-1117	7	10	.	.	PUNCT
cana-1117	8	1	keywords	keyword	NOUN
cana-1117	8	2	:	:	PUNCT
cana-1117	8	3	fuzzy	fuzzy	ADJ
cana-1117	8	4	fractional	fractional	ADJ
cana-1117	8	5	differential	differential	NOUN
cana-1117	8	6	equations	equation	NOUN
cana-1117	8	7	,	,	PUNCT
cana-1117	8	8	fuzzy	fuzzy	ADJ
cana-1117	8	9	q	q	ADJ
cana-1117	8	10	-	-	PUNCT
cana-1117	8	11	fractional	fractional	ADJ
cana-1117	8	12	differential	differential	NOUN
cana-1117	8	13	equation	equation	NOUN
cana-1117	8	14	,	,	PUNCT
cana-1117	8	15	existence	existence	NOUN
cana-1117	8	16	of	of	ADP
cana-1117	8	17	solution	solution	NOUN
cana-1117	8	18	,	,	PUNCT
cana-1117	8	19	fixed	fix	VERB
cana-1117	8	20	point	point	NOUN
cana-1117	8	21	theorem	theorem	ADJ
cana-1117	8	22	,	,	PUNCT
cana-1117	8	23	continuous	continuous	ADJ
cana-1117	8	24	dependence	dependence	NOUN
cana-1117	8	25	.	.	PUNCT
cana-1117	9	1	1	1	NUM
cana-1117	9	2	introduction	introduction	NOUN
cana-1117	9	3	the	the	DET
cana-1117	9	4	generalized	generalized	ADJ
cana-1117	9	5	form	form	NOUN
cana-1117	9	6	of	of	ADP
cana-1117	9	7	differential	differential	ADJ
cana-1117	9	8	equations	equation	NOUN
cana-1117	9	9	is	be	AUX
cana-1117	9	10	termed	term	VERB
cana-1117	9	11	as	as	ADP
cana-1117	9	12	fractional	fractional	ADJ
cana-1117	9	13	differential	differential	ADJ
cana-1117	9	14	equations	equation	NOUN
cana-1117	9	15	(	(	PUNCT
cana-1117	9	16	fde	fde	PROPN
cana-1117	9	17	)	)	PUNCT
cana-1117	9	18	,	,	PUNCT
cana-1117	9	19	which	which	PRON
cana-1117	9	20	emerged	emerge	VERB
cana-1117	9	21	as	as	ADP
cana-1117	9	22	an	an	DET
cana-1117	9	23	application	application	NOUN
cana-1117	9	24	of	of	ADP
cana-1117	9	25	fractional	fractional	ADJ
cana-1117	9	26	calculus	calculus	NOUN
cana-1117	9	27	.	.	PUNCT
cana-1117	10	1	fde	fde	PROPN
cana-1117	10	2	engages	engage	VERB
cana-1117	10	3	as	as	ADP
cana-1117	10	4	an	an	DET
cana-1117	10	5	important	important	ADJ
cana-1117	10	6	tool	tool	NOUN
cana-1117	10	7	to	to	PART
cana-1117	10	8	perform	perform	VERB
cana-1117	10	9	many	many	ADJ
cana-1117	10	10	physical	physical	ADJ
cana-1117	10	11	phenomena	phenomenon	NOUN
cana-1117	10	12	which	which	PRON
cana-1117	10	13	includes	include	VERB
cana-1117	10	14	viscoelasticity	viscoelasticity	NOUN
cana-1117	10	15	,	,	PUNCT
cana-1117	10	16	control	control	NOUN
cana-1117	10	17	theory	theory	NOUN
cana-1117	10	18	of	of	ADP
cana-1117	10	19	dynamical	dynamical	ADJ
cana-1117	10	20	systems	system	NOUN
cana-1117	10	21	,	,	PUNCT
cana-1117	10	22	optics	optic	NOUN
cana-1117	10	23	and	and	CCONJ
cana-1117	10	24	signal	signal	NOUN
cana-1117	10	25	processing	processing	NOUN
cana-1117	10	26	etc	etc	X
cana-1117	10	27	.	.	PUNCT
cana-1117	11	1	fde	fde	PROPN
cana-1117	11	2	attracted	attract	VERB
cana-1117	11	3	many	many	ADJ
cana-1117	11	4	scientists	scientist	NOUN
cana-1117	11	5	and	and	CCONJ
cana-1117	11	6	mathematicians	mathematician	NOUN
cana-1117	11	7	.	.	PUNCT
cana-1117	12	1	the	the	DET
cana-1117	12	2	existence	existence	NOUN
cana-1117	12	3	theory	theory	NOUN
cana-1117	12	4	of	of	ADP
cana-1117	12	5	fractional	fractional	ADJ
cana-1117	12	6	differential	differential	ADJ
cana-1117	12	7	equations	equation	NOUN
cana-1117	12	8	of	of	ADP
cana-1117	12	9	nonlinear	nonlinear	ADJ
cana-1117	12	10	type	type	NOUN
cana-1117	12	11	finds	find	VERB
cana-1117	12	12	its	its	PRON
cana-1117	12	13	consistent	consistent	ADJ
cana-1117	12	14	study	study	NOUN
cana-1117	12	15	by	by	ADP
cana-1117	12	16	many	many	ADJ
cana-1117	12	17	researchers	researcher	NOUN
cana-1117	12	18	.	.	PUNCT
cana-1117	13	1	in	in	ADP
cana-1117	13	2	recent	recent	ADJ
cana-1117	13	3	years	year	NOUN
cana-1117	13	4	,	,	PUNCT
cana-1117	13	5	cauchy	cauchy	NOUN
cana-1117	13	6	problem	problem	NOUN
cana-1117	13	7	for	for	ADP
cana-1117	13	8	nonlinear	nonlinear	ADJ
cana-1117	13	9	fde	fde	PROPN
cana-1117	13	10	has	have	AUX
cana-1117	13	11	become	become	VERB
cana-1117	13	12	the	the	DET
cana-1117	13	13	most	most	ADV
cana-1117	13	14	interesting	interesting	ADJ
cana-1117	13	15	field	field	NOUN
cana-1117	13	16	to	to	PART
cana-1117	13	17	study	study	VERB
cana-1117	13	18	the	the	DET
cana-1117	13	19	existence	existence	NOUN
cana-1117	13	20	,	,	PUNCT
cana-1117	13	21	uniqueness	uniqueness	NOUN
cana-1117	13	22	,	,	PUNCT
cana-1117	13	23	long	long	ADJ
cana-1117	13	24	time	time	NOUN
cana-1117	13	25	behavior	behavior	NOUN
cana-1117	13	26	etc	etc	X
cana-1117	13	27	.	.	X
cana-1117	13	28	fixed	fix	VERB
cana-1117	13	29	point	point	NOUN
cana-1117	13	30	theory	theory	NOUN
cana-1117	13	31	has	have	AUX
cana-1117	13	32	created	create	VERB
cana-1117	13	33	extensive	extensive	ADJ
cana-1117	13	34	interest	interest	NOUN
cana-1117	13	35	in	in	ADP
cana-1117	13	36	the	the	DET
cana-1117	13	37	researchers	researcher	NOUN
cana-1117	13	38	to	to	PART
cana-1117	13	39	utilize	utilize	VERB
cana-1117	13	40	it	it	PRON
cana-1117	13	41	for	for	ADP
cana-1117	13	42	the	the	DET
cana-1117	13	43	existence	existence	NOUN
cana-1117	13	44	of	of	ADP
cana-1117	13	45	solution	solution	NOUN
cana-1117	13	46	of	of	ADP
cana-1117	13	47	the	the	DET
cana-1117	13	48	fractional	fractional	ADJ
cana-1117	13	49	differential	differential	ADJ
cana-1117	13	50	equation	equation	NOUN
cana-1117	13	51	.	.	PUNCT
cana-1117	14	1	fractional	fractional	ADJ
cana-1117	14	2	derivative	derivative	NOUN
cana-1117	14	3	is	be	AUX
cana-1117	14	4	derived	derive	VERB
cana-1117	14	5	mostly	mostly	ADV
cana-1117	14	6	by	by	ADP
cana-1117	14	7	two	two	NUM
cana-1117	14	8	operators	operator	NOUN
cana-1117	14	9	riemann	riemann	PROPN
cana-1117	14	10	-	-	PUNCT
cana-1117	14	11	liouville	liouville	PROPN
cana-1117	14	12	and	and	CCONJ
cana-1117	14	13	caputo	caputo	PROPN
cana-1117	14	14	operators	operators	PROPN
cana-1117	14	15	.	.	PUNCT
cana-1117	15	1	but	but	CCONJ
cana-1117	15	2	the	the	DET
cana-1117	15	3	caputo	caputo	PROPN
cana-1117	15	4	operator	operator	NOUN
cana-1117	15	5	has	have	VERB
cana-1117	15	6	advantages	advantage	NOUN
cana-1117	15	7	for	for	ADP
cana-1117	15	8	initial	initial	ADJ
cana-1117	15	9	value	value	NOUN
cana-1117	15	10	problems	problem	NOUN
cana-1117	15	11	.	.	PUNCT
cana-1117	16	1	the	the	DET
cana-1117	16	2	core	core	ADJ
cana-1117	16	3	attention	attention	NOUN
cana-1117	16	4	towards	towards	ADP
cana-1117	16	5	the	the	DET
cana-1117	16	6	analysis	analysis	NOUN
cana-1117	16	7	of	of	ADP
cana-1117	16	8	existence	existence	NOUN
cana-1117	16	9	and	and	CCONJ
cana-1117	16	10	uniqueness	uniqueness	NOUN
cana-1117	16	11	of	of	ADP
cana-1117	16	12	the	the	DET
cana-1117	16	13	solution	solution	NOUN
cana-1117	16	14	of	of	ADP
cana-1117	16	15	the	the	DET
cana-1117	16	16	fdes	fde	NOUN
cana-1117	16	17	is	be	AUX
cana-1117	16	18	prolific	prolific	ADJ
cana-1117	16	19	to	to	PART
cana-1117	16	20	study	study	VERB
cana-1117	16	21	.	.	PUNCT
cana-1117	17	1	in	in	ADP
cana-1117	17	2	particular	particular	ADJ
cana-1117	17	3	,	,	PUNCT
cana-1117	17	4	the	the	DET
cana-1117	17	5	study	study	NOUN
cana-1117	17	6	of	of	ADP
cana-1117	17	7	the	the	DET
cana-1117	17	8	existence	existence	NOUN
cana-1117	17	9	of	of	ADP
cana-1117	17	10	solution	solution	NOUN
cana-1117	17	11	to	to	ADP
cana-1117	17	12	fuzzy	fuzzy	ADJ
cana-1117	17	13	fractional	fractional	ADJ
cana-1117	17	14	differential	differential	NOUN
cana-1117	17	15	equation	equation	NOUN
cana-1117	17	16	is	be	AUX
cana-1117	17	17	quite	quite	ADV
cana-1117	17	18	interesting	interesting	ADJ
cana-1117	17	19	.	.	PUNCT
cana-1117	18	1	in	in	ADP
cana-1117	18	2	addition	addition	NOUN
cana-1117	18	3	,	,	PUNCT
cana-1117	18	4	fixed	fix	VERB
cana-1117	18	5	point	point	NOUN
cana-1117	18	6	theory	theory	NOUN
cana-1117	18	7	is	be	AUX
cana-1117	18	8	an	an	DET
cana-1117	18	9	inevitable	inevitable	ADJ
cana-1117	18	10	tool	tool	NOUN
cana-1117	18	11	to	to	PART
cana-1117	18	12	study	study	VERB
cana-1117	18	13	the	the	DET
cana-1117	18	14	existence	existence	NOUN
cana-1117	18	15	and	and	CCONJ
cana-1117	18	16	uniqueness	uniqueness	NOUN
cana-1117	18	17	of	of	ADP
cana-1117	18	18	some	some	DET
cana-1117	18	19	mathematical	mathematical	ADJ
cana-1117	18	20	models	model	NOUN
cana-1117	18	21	,	,	PUNCT
cana-1117	18	22	which	which	PRON
cana-1117	18	23	has	have	AUX
cana-1117	18	24	been	be	AUX
cana-1117	18	25	studied	study	VERB
cana-1117	18	26	since	since	SCONJ
cana-1117	18	27	many	many	ADJ
cana-1117	18	28	decades	decade	NOUN
cana-1117	18	29	.	.	PUNCT
cana-1117	19	1	research	research	NOUN
cana-1117	19	2	is	be	AUX
cana-1117	19	3	progressing	progress	VERB
cana-1117	19	4	extensively	extensively	ADV
cana-1117	19	5	to	to	PART
cana-1117	19	6	study	study	VERB
cana-1117	19	7	the	the	DET
cana-1117	19	8	cauchy	cauchy	PROPN
cana-1117	19	9	problems	problem	NOUN
cana-1117	19	10	,	,	PUNCT
cana-1117	19	11	with	with	ADP
cana-1117	19	12	the	the	DET
cana-1117	19	13	utilization	utilization	NOUN
cana-1117	19	14	of	of	ADP
cana-1117	19	15	fixed	fix	VERB
cana-1117	19	16	point	point	NOUN
cana-1117	19	17	theory	theory	NOUN
cana-1117	19	18	approach	approach	NOUN
cana-1117	19	19	.	.	PUNCT
cana-1117	20	1	further	far	ADV
cana-1117	20	2	,	,	PUNCT
cana-1117	20	3	the	the	DET
cana-1117	20	4	study	study	NOUN
cana-1117	20	5	of	of	ADP
cana-1117	20	6	uncertainty	uncertainty	NOUN
cana-1117	20	7	becomes	become	VERB
cana-1117	20	8	the	the	DET
cana-1117	20	9	most	most	ADV
cana-1117	20	10	interesting	interesting	ADJ
cana-1117	20	11	study	study	NOUN
cana-1117	20	12	in	in	ADP
cana-1117	20	13	these	these	DET
cana-1117	20	14	days	day	NOUN
cana-1117	20	15	.	.	PUNCT
cana-1117	21	1	also	also	ADV
cana-1117	21	2	the	the	DET
cana-1117	21	3	dynamics	dynamic	NOUN
cana-1117	21	4	itself	itself	PRON
cana-1117	21	5	is	be	AUX
cana-1117	21	6	uncertain	uncertain	ADJ
cana-1117	21	7	due	due	ADP
cana-1117	21	8	to	to	ADP
cana-1117	21	9	its	its	PRON
cana-1117	21	10	dependence	dependence	NOUN
cana-1117	21	11	on	on	ADP
cana-1117	21	12	time	time	NOUN
cana-1117	21	13	.	.	PUNCT
cana-1117	22	1	basically	basically	ADV
cana-1117	22	2	,	,	PUNCT
cana-1117	22	3	the	the	DET
cana-1117	22	4	proposed	propose	VERB
cana-1117	22	5	ideas	idea	NOUN
cana-1117	22	6	are	be	AUX
cana-1117	22	7	a	a	DET
cana-1117	22	8	generalisation	generalisation	NOUN
cana-1117	22	9	of	of	ADP
cana-1117	22	10	the	the	DET
cana-1117	22	11	theory	theory	NOUN
cana-1117	22	12	and	and	CCONJ
cana-1117	22	13	solution	solution	NOUN
cana-1117	22	14	of	of	ADP
cana-1117	22	15	fuzzy	fuzzy	ADJ
cana-1117	22	16	differential	differential	ADJ
cana-1117	22	17	equations	equation	NOUN
cana-1117	22	18	.	.	PUNCT
cana-1117	23	1	however	however	ADV
cana-1117	23	2	,	,	PUNCT
cana-1117	23	3	the	the	DET
cana-1117	23	4	authors	author	NOUN
cana-1117	23	5	considered	consider	VERB
cana-1117	23	6	fuzzy	fuzzy	ADJ
cana-1117	23	7	fractional	fractional	ADJ
cana-1117	23	8	differential	differential	ADJ
cana-1117	23	9	equations	equation	NOUN
cana-1117	23	10	under	under	ADP
cana-1117	23	11	the	the	DET
cana-1117	23	12	riemann	riemann	PROPN
cana-1117	23	13	-	-	PUNCT
cana-1117	23	14	liouville	liouville	VERB
cana-1117	23	15	h	h	NOUN
cana-1117	23	16	-	-	PUNCT
cana-1117	23	17	derivative[4	derivative[4	X
cana-1117	23	18	]	]	PUNCT
cana-1117	23	19	.	.	PUNCT
cana-1117	24	1	again	again	ADV
cana-1117	24	2	,	,	PUNCT
cana-1117	24	3	it	it	PRON
cana-1117	24	4	requires	require	VERB
cana-1117	24	5	a	a	DET
cana-1117	24	6	communications	communication	NOUN
cana-1117	24	7	on	on	ADP
cana-1117	24	8	applied	apply	VERB
cana-1117	24	9	nonlinear	nonlinear	ADJ
cana-1117	24	10	analysis	analysis	NOUN
cana-1117	24	11	issn	issn	NOUN
cana-1117	24	12	:	:	PUNCT
cana-1117	24	13	1074	1074	NUM
cana-1117	24	14	-	-	PUNCT
cana-1117	24	15	133x	133x	NUM
cana-1117	24	16	vol	vol	NOUN
cana-1117	24	17	31	31	NUM
cana-1117	24	18	no	no	NOUN
cana-1117	24	19	.	.	PUNCT
cana-1117	25	1	6s	6s	NUM
cana-1117	25	2	(	(	PUNCT
cana-1117	25	3	2024	2024	NUM
cana-1117	25	4	)	)	PUNCT
cana-1117	25	5	36	36	NUM
cana-1117	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	25	7	quantity	quantity	NOUN
cana-1117	25	8	of	of	ADP
cana-1117	25	9	fractional	fractional	ADJ
cana-1117	25	10	h	h	NOUN
cana-1117	25	11	-	-	PUNCT
cana-1117	25	12	derivative	derivative	NOUN
cana-1117	25	13	of	of	ADP
cana-1117	25	14	an	an	DET
cana-1117	25	15	unknown	unknown	ADJ
cana-1117	25	16	solution	solution	NOUN
cana-1117	25	17	at	at	ADP
cana-1117	25	18	the	the	DET
cana-1117	25	19	fuzzy	fuzzy	ADJ
cana-1117	25	20	initial	initial	ADJ
cana-1117	25	21	point	point	NOUN
cana-1117	25	22	.	.	PUNCT
cana-1117	26	1	also	also	ADV
cana-1117	26	2	,	,	PUNCT
cana-1117	26	3	the	the	DET
cana-1117	26	4	qcalculus	qcalculus	NOUN
cana-1117	26	5	appears	appear	VERB
cana-1117	26	6	to	to	PART
cana-1117	26	7	be	be	AUX
cana-1117	26	8	the	the	DET
cana-1117	26	9	connection	connection	NOUN
cana-1117	26	10	between	between	ADP
cana-1117	26	11	mathematics	mathematic	NOUN
cana-1117	26	12	and	and	CCONJ
cana-1117	26	13	physics	physics	NOUN
cana-1117	26	14	.	.	PUNCT
cana-1117	27	1	it	it	PRON
cana-1117	27	2	has	have	VERB
cana-1117	27	3	several	several	ADJ
cana-1117	27	4	applications	application	NOUN
cana-1117	27	5	in	in	ADP
cana-1117	27	6	the	the	DET
cana-1117	27	7	areas	area	NOUN
cana-1117	27	8	like	like	ADP
cana-1117	27	9	quantum	quantum	NOUN
cana-1117	27	10	theory	theory	NOUN
cana-1117	27	11	,	,	PUNCT
cana-1117	27	12	hypo	hypo	ADJ
cana-1117	27	13	geometric	geometric	ADJ
cana-1117	27	14	functions	function	NOUN
cana-1117	27	15	and	and	CCONJ
cana-1117	27	16	electronics[1	electronics[1	NOUN
cana-1117	27	17	,	,	PUNCT
cana-1117	27	18	3	3	NUM
cana-1117	27	19	,	,	PUNCT
cana-1117	27	20	8	8	NUM
cana-1117	27	21	,	,	PUNCT
cana-1117	27	22	9	9	NUM
cana-1117	27	23	]	]	PUNCT
cana-1117	27	24	.	.	PUNCT
cana-1117	28	1	z.noeiaghdam	z.noeiaghdam	PROPN
cana-1117	28	2	et.al	et.al	PROPN
cana-1117	28	3	and	and	CCONJ
cana-1117	28	4	obaidat	obaidat	NOUN
cana-1117	28	5	et.al	et.al	PROPN
cana-1117	28	6	[	[	X
cana-1117	28	7	6	6	NUM
cana-1117	28	8	,	,	PUNCT
cana-1117	28	9	7	7	NUM
cana-1117	28	10	]	]	PUNCT
cana-1117	28	11	study	study	VERB
cana-1117	28	12	the	the	DET
cana-1117	28	13	existence	existence	NOUN
cana-1117	28	14	of	of	ADP
cana-1117	28	15	q	q	ADJ
cana-1117	28	16	-	-	PUNCT
cana-1117	28	17	fractional	fractional	ADJ
cana-1117	28	18	differential	differential	ADJ
cana-1117	28	19	equations	equation	NOUN
cana-1117	28	20	with	with	ADP
cana-1117	28	21	uncertainty	uncertainty	NOUN
cana-1117	28	22	.	.	PUNCT
cana-1117	29	1	the	the	DET
cana-1117	29	2	existence	existence	NOUN
cana-1117	29	3	of	of	ADP
cana-1117	29	4	solution	solution	NOUN
cana-1117	29	5	of	of	ADP
cana-1117	29	6	a	a	DET
cana-1117	29	7	fractional	fractional	ADJ
cana-1117	29	8	q	q	ADJ
cana-1117	29	9	-	-	PUNCT
cana-1117	29	10	integro	integro	ADJ
cana-1117	29	11	differential	differential	ADJ
cana-1117	29	12	equation	equation	NOUN
cana-1117	29	13	with	with	ADP
cana-1117	29	14	q	q	ADJ
cana-1117	29	15	-	-	ADJ
cana-1117	29	16	nonlocal	nonlocal	ADJ
cana-1117	29	17	condition	condition	NOUN
cana-1117	29	18	has	have	AUX
cana-1117	29	19	been	be	AUX
cana-1117	29	20	examined	examine	VERB
cana-1117	29	21	by	by	ADP
cana-1117	29	22	ibrahim	ibrahim	PROPN
cana-1117	29	23	et.al	et.al	PROPN
cana-1117	30	1	[	[	X
cana-1117	30	2	5	5	NUM
cana-1117	30	3	]	]	PUNCT
cana-1117	30	4	.	.	PUNCT
cana-1117	31	1	z.noeiaghdam	z.noeiaghdam	PROPN
cana-1117	31	2	et.al	et.al	PROPN
cana-1117	31	3	analysed	analyse	VERB
cana-1117	31	4	the	the	DET
cana-1117	31	5	fuzzy	fuzzy	ADJ
cana-1117	31	6	q	q	NOUN
cana-1117	31	7	-	-	ADJ
cana-1117	31	8	derivative	derivative	ADJ
cana-1117	31	9	and	and	CCONJ
cana-1117	31	10	fuzzy	fuzzy	ADJ
cana-1117	31	11	qfractional	qfractional	ADJ
cana-1117	31	12	derivative	derivative	NOUN
cana-1117	31	13	in	in	ADP
cana-1117	31	14	caputo	caputo	PROPN
cana-1117	31	15	sense	sense	NOUN
cana-1117	31	16	and	and	CCONJ
cana-1117	31	17	used	use	VERB
cana-1117	31	18	generalized	generalize	VERB
cana-1117	31	19	hukuhara	hukuhara	ADJ
cana-1117	31	20	difference	difference	NOUN
cana-1117	31	21	.	.	PUNCT
cana-1117	32	1	fuzzy	fuzzy	ADJ
cana-1117	32	2	q	q	ADJ
cana-1117	32	3	-	-	PUNCT
cana-1117	32	4	fractional	fractional	ADJ
cana-1117	32	5	differential	differential	NOUN
cana-1117	32	6	equation	equation	NOUN
cana-1117	32	7	solving	solving	NOUN
cana-1117	32	8	creates	create	VERB
cana-1117	32	9	strenuous	strenuous	ADJ
cana-1117	32	10	attention	attention	NOUN
cana-1117	32	11	to	to	ADP
cana-1117	32	12	the	the	DET
cana-1117	32	13	researchers	researcher	NOUN
cana-1117	32	14	[	[	X
cana-1117	32	15	8	8	NUM
cana-1117	32	16	]	]	PUNCT
cana-1117	32	17	.	.	PUNCT
cana-1117	33	1	motivated	motivate	VERB
cana-1117	33	2	by	by	ADP
cana-1117	33	3	those	those	PRON
cana-1117	33	4	,	,	PUNCT
cana-1117	33	5	this	this	DET
cana-1117	33	6	work	work	NOUN
cana-1117	33	7	studies	study	VERB
cana-1117	33	8	the	the	DET
cana-1117	33	9	existence	existence	NOUN
cana-1117	33	10	of	of	ADP
cana-1117	33	11	integro	integro	PROPN
cana-1117	33	12	differential	differential	ADJ
cana-1117	33	13	equations	equation	NOUN
cana-1117	33	14	𝐶𝐷𝛼(𝜙(𝜏	𝐶𝐷𝛼(𝜙(𝜏	PROPN
cana-1117	33	15	)	)	PUNCT
cana-1117	33	16	)	)	PUNCT
cana-1117	34	1	=	=	SYM
cana-1117	34	2	𝜆𝜙(𝜏	𝜆𝜙(𝜏	X
cana-1117	34	3	)	)	PUNCT
cana-1117	35	1	+	+	CCONJ
cana-1117	35	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	35	3	,	,	PUNCT
cana-1117	35	4	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	35	5	)	)	PUNCT
cana-1117	35	6	,	,	PUNCT
cana-1117	35	7	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	35	8	)	)	PUNCT
cana-1117	35	9	,	,	PUNCT
cana-1117	35	10	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	35	11	)	)	PUNCT
cana-1117	35	12	)	)	PUNCT
cana-1117	35	13	,	,	PUNCT
cana-1117	35	14	(	(	PUNCT
cana-1117	35	15	1.1	1.1	NUM
cana-1117	35	16	)	)	PUNCT
cana-1117	35	17	𝜙(𝜏0	𝜙(𝜏0	NOUN
cana-1117	35	18	)	)	PUNCT
cana-1117	35	19	=	=	SYM
cana-1117	35	20	𝜙0	𝜙0	NOUN
cana-1117	35	21	(	(	PUNCT
cana-1117	35	22	1.2	1.2	NUM
cana-1117	35	23	)	)	PUNCT
cana-1117	35	24	where	where	SCONJ
cana-1117	35	25	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	35	26	)	)	PUNCT
cana-1117	35	27	=	=	SYM
cana-1117	36	1	∫	∫	PROPN
cana-1117	36	2	𝜏	𝜏	X
cana-1117	36	3	𝜏0	𝜏0	PROPN
cana-1117	36	4	𝒦(𝜏	𝒦(𝜏	NOUN
cana-1117	36	5	,	,	PUNCT
cana-1117	36	6	𝑠)𝜙(𝑠)𝑑𝑠	𝑠)𝜙(𝑠)𝑑𝑠	NOUN
cana-1117	36	7	,	,	PUNCT
cana-1117	36	8	𝒦	𝒦	PROPN
cana-1117	36	9	∈	∈	PROPN
cana-1117	36	10	𝐶[𝒟	𝐶[𝒟	NOUN
cana-1117	36	11	,	,	PUNCT
cana-1117	36	12	ℝ+	ℝ+	ADP
cana-1117	36	13	]	]	PUNCT
cana-1117	36	14	,	,	PUNCT
cana-1117	36	15	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-1117	36	16	𝒟	𝒟	NOUN
cana-1117	36	17	=	=	SYM
cana-1117	36	18	{	{	PUNCT
cana-1117	36	19	(	(	PUNCT
cana-1117	36	20	𝜏	𝜏	NOUN
cana-1117	36	21	,	,	PUNCT
cana-1117	36	22	𝑠	𝑠	NOUN
cana-1117	36	23	)	)	PUNCT
cana-1117	36	24	∈	∈	PROPN
cana-1117	36	25	ℝ2	ℝ2	PROPN
cana-1117	36	26	:	:	PUNCT
cana-1117	36	27	0	0	NUM
cana-1117	36	28	≤	≤	NUM
cana-1117	36	29	𝑠	𝑠	X
cana-1117	36	30	≤	≤	NUM
cana-1117	36	31	𝜏	𝜏	PRON
cana-1117	36	32	≤	≤	NUM
cana-1117	36	33	𝕋	𝕋	NOUN
cana-1117	36	34	}	}	PUNCT
cana-1117	36	35	,	,	PUNCT
cana-1117	36	36	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	36	37	)	)	PUNCT
cana-1117	36	38	=	=	SYM
cana-1117	37	1	∫	∫	PROPN
cana-1117	38	1	𝜏	𝜏	X
cana-1117	38	2	𝜏0	𝜏0	PROPN
cana-1117	38	3	ℋ(𝜏	ℋ(𝜏	NUM
cana-1117	38	4	,	,	PUNCT
cana-1117	38	5	𝑠)𝜙(𝑠)𝑑𝑠	𝑠)𝜙(𝑠)𝑑𝑠	NOUN
cana-1117	38	6	,	,	PUNCT
cana-1117	38	7	ℋ	ℋ	PROPN
cana-1117	38	8	∈	∈	PROPN
cana-1117	38	9	𝐶[𝒟0	𝐶[𝒟0	NOUN
cana-1117	38	10	,	,	PUNCT
cana-1117	38	11	ℝ+	ℝ+	ADP
cana-1117	38	12	]	]	SYM
cana-1117	38	13	,	,	PUNCT
cana-1117	38	14	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-1117	38	15	𝒟0	𝒟0	NOUN
cana-1117	38	16	=	=	SYM
cana-1117	38	17	{	{	PUNCT
cana-1117	38	18	(	(	PUNCT
cana-1117	38	19	𝜏	𝜏	NOUN
cana-1117	38	20	,	,	PUNCT
cana-1117	38	21	𝑠	𝑠	NOUN
cana-1117	38	22	)	)	PUNCT
cana-1117	38	23	∈	∈	PROPN
cana-1117	38	24	ℝ2	ℝ2	PROPN
cana-1117	38	25	:	:	PUNCT
cana-1117	38	26	0	0	NUM
cana-1117	38	27	≤	≤	NUM
cana-1117	38	28	𝜏	𝜏	NUM
cana-1117	38	29	,	,	PUNCT
cana-1117	38	30	𝑠	𝑠	PRON
cana-1117	38	31	≤	≤	NUM
cana-1117	38	32	𝕋	𝕋	PROPN
cana-1117	38	33	}	}	PUNCT
cana-1117	38	34	where	where	SCONJ
cana-1117	38	35	0	0	NUM
cana-1117	38	36	<	<	X
cana-1117	38	37	𝛼	𝛼	PRON
cana-1117	38	38	≤	≤	NUM
cana-1117	38	39	1,𝜏	1,𝜏	NUM
cana-1117	38	40	∈	∈	PROPN
cana-1117	39	1	[	[	X
cana-1117	39	2	𝜏0	𝜏0	PROPN
cana-1117	39	3	,	,	PUNCT
cana-1117	39	4	𝕋	𝕋	PROPN
cana-1117	39	5	]	]	PUNCT
cana-1117	39	6	𝜆	𝜆	X
cana-1117	39	7	is	be	AUX
cana-1117	39	8	a	a	DET
cana-1117	39	9	constant	constant	ADJ
cana-1117	39	10	and	and	CCONJ
cana-1117	39	11	𝑓	𝑓	PRON
cana-1117	39	12	:	:	PUNCT
cana-1117	39	13	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	39	14	×	×	NOUN
cana-1117	39	15	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	39	16	×	×	PROPN
cana-1117	39	17	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	39	18	×	×	PROPN
cana-1117	39	19	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	39	20	→	→	SYM
cana-1117	39	21	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	39	22	are	be	AUX
cana-1117	39	23	continuous	continuous	ADJ
cana-1117	39	24	and	and	CCONJ
cana-1117	39	25	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	39	26	,	,	PUNCT
cana-1117	39	27	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	39	28	)	)	PUNCT
cana-1117	39	29	,	,	PUNCT
cana-1117	39	30	𝐺𝜙(𝜏	𝐺𝜙(𝜏	NOUN
cana-1117	39	31	)	)	PUNCT
cana-1117	39	32	,	,	PUNCT
cana-1117	39	33	𝑆𝜙(𝜏	𝑆𝜙(𝜏	NOUN
cana-1117	39	34	)	)	PUNCT
cana-1117	39	35	)	)	PUNCT
cana-1117	39	36	satisfies	satisfy	VERB
cana-1117	39	37	the	the	DET
cana-1117	39	38	following	follow	VERB
cana-1117	39	39	condition	condition	NOUN
cana-1117	39	40	(	(	PUNCT
cana-1117	39	41	𝐻1)‖𝑓(𝜏	𝐻1)‖𝑓(𝜏	PROPN
cana-1117	39	42	,	,	PUNCT
cana-1117	39	43	𝜙	𝜙	NOUN
cana-1117	39	44	,	,	PUNCT
cana-1117	39	45	𝒢𝜙	𝒢𝜙	PROPN
cana-1117	39	46	,	,	PUNCT
cana-1117	39	47	𝒮𝜙	𝒮𝜙	NOUN
cana-1117	39	48	)	)	PUNCT
cana-1117	39	49	−	−	NOUN
cana-1117	39	50	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	39	51	,	,	PUNCT
cana-1117	39	52	𝜓	𝜓	PROPN
cana-1117	39	53	,	,	PUNCT
cana-1117	39	54	𝒢𝜓	𝒢𝜓	PROPN
cana-1117	39	55	,	,	PUNCT
cana-1117	39	56	𝒮𝜓)‖	𝒮𝜓)‖	NOUN
cana-1117	39	57	≤	≤	NUM
cana-1117	39	58	ℒ[‖𝜙	ℒ[‖𝜙	NOUN
cana-1117	39	59	−	−	PROPN
cana-1117	39	60	𝜓‖	𝜓‖	NOUN
cana-1117	39	61	+	+	CCONJ
cana-1117	39	62	‖𝒢𝑢	‖𝒢𝑢	VERB
cana-1117	39	63	−	−	NOUN
cana-1117	39	64	𝒢𝑣‖	𝒢𝑣‖	VERB
cana-1117	39	65	+	+	CCONJ
cana-1117	39	66	‖𝒮𝑢	‖𝒮𝑢	ADJ
cana-1117	39	67	−	−	PROPN
cana-1117	39	68	𝒮𝑣‖	𝒮𝑣‖	SYM
cana-1117	39	69	]	]	PUNCT
cana-1117	39	70	with	with	ADP
cana-1117	39	71	0	0	NUM
cana-1117	39	72	<	<	X
cana-1117	39	73	ℒ	ℒ	X
cana-1117	39	74	<	<	X
cana-1117	39	75	1	1	NUM
cana-1117	39	76	.	.	SYM
cana-1117	39	77	2	2	NUM
cana-1117	39	78	preliminaries	preliminary	NOUN
cana-1117	39	79	2.1	2.1	NUM
cana-1117	39	80	basic	basic	ADJ
cana-1117	39	81	fuzzy	fuzzy	ADJ
cana-1117	39	82	concepts	concept	NOUN
cana-1117	39	83	definition	definition	NOUN
cana-1117	39	84	2.1	2.1	NUM
cana-1117	39	85	[	[	X
cana-1117	39	86	2	2	NUM
cana-1117	39	87	]	]	PUNCT
cana-1117	39	88	let	let	VERB
cana-1117	39	89	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	39	90	be	be	AUX
cana-1117	39	91	the	the	DET
cana-1117	39	92	set	set	NOUN
cana-1117	39	93	of	of	ADP
cana-1117	39	94	all	all	DET
cana-1117	39	95	fuzzy	fuzzy	ADJ
cana-1117	39	96	valued	value	VERB
cana-1117	39	97	functions	function	NOUN
cana-1117	39	98	.	.	PUNCT
cana-1117	40	1	let	let	VERB
cana-1117	40	2	𝜙	𝜙	NOUN
cana-1117	40	3	:	:	PUNCT
cana-1117	40	4	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	40	5	→	→	SYM
cana-1117	40	6	[	[	X
cana-1117	40	7	0,1	0,1	NUM
cana-1117	40	8	]	]	PUNCT
cana-1117	40	9	be	be	AUX
cana-1117	40	10	satisfying	satisfy	VERB
cana-1117	40	11	the	the	DET
cana-1117	40	12	following	following	ADJ
cana-1117	40	13	conditions	condition	NOUN
cana-1117	40	14	.	.	PUNCT
cana-1117	41	1	1	1	X
cana-1117	41	2	.	.	X
cana-1117	41	3	𝜙	𝜙	PROPN
cana-1117	41	4	is	be	AUX
cana-1117	41	5	upper	upper	ADJ
cana-1117	41	6	semi	semi	ADJ
cana-1117	41	7	-	-	ADJ
cana-1117	41	8	continuous	continuous	ADJ
cana-1117	41	9	on	on	ADP
cana-1117	41	10	ℝ	ℝ	PROPN
cana-1117	41	11	2	2	NUM
cana-1117	41	12	.	.	PUNCT
cana-1117	42	1	𝜙	𝜙	PROPN
cana-1117	42	2	is	be	AUX
cana-1117	42	3	fuzzy	fuzzy	ADJ
cana-1117	42	4	convex	convex	ADJ
cana-1117	42	5	3	3	NUM
cana-1117	42	6	.	.	PUNCT
cana-1117	43	1	𝜙	𝜙	PROPN
cana-1117	43	2	is	be	AUX
cana-1117	43	3	normal	normal	ADJ
cana-1117	43	4	4	4	NUM
cana-1117	43	5	.	.	PUNCT
cana-1117	44	1	closure	closure	NOUN
cana-1117	44	2	of	of	ADP
cana-1117	44	3	{	{	PUNCT
cana-1117	44	4	𝑡	𝑡	PROPN
cana-1117	44	5	∈	∈	PROPN
cana-1117	44	6	ℝ𝔽|𝜙(𝑡	ℝ𝔽|𝜙(𝑡	NOUN
cana-1117	44	7	)	)	PUNCT
cana-1117	44	8	>	>	X
cana-1117	44	9	0	0	NUM
cana-1117	44	10	}	}	PUNCT
cana-1117	44	11	is	be	AUX
cana-1117	44	12	compact	compact	ADJ
cana-1117	44	13	.	.	PUNCT
cana-1117	45	1	let	let	VERB
cana-1117	45	2	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	45	3	be	be	AUX
cana-1117	45	4	the	the	DET
cana-1117	45	5	space	space	NOUN
cana-1117	45	6	of	of	ADP
cana-1117	45	7	above	above	ADV
cana-1117	45	8	said	say	VERB
cana-1117	45	9	fuzzy	fuzzy	ADJ
cana-1117	45	10	numbers	number	NOUN
cana-1117	45	11	.	.	PUNCT
cana-1117	46	1	also	also	ADV
cana-1117	46	2	,	,	PUNCT
cana-1117	46	3	for	for	ADP
cana-1117	46	4	0	0	NUM
cana-1117	46	5	<	<	X
cana-1117	46	6	𝑟	𝑟	X
cana-1117	46	7	≤	≤	NUM
cana-1117	46	8	1	1	NUM
cana-1117	46	9	,	,	PUNCT
cana-1117	46	10	denote	denote	VERB
cana-1117	46	11	𝜙(𝑟	𝜙(𝑟	NOUN
cana-1117	46	12	)	)	PUNCT
cana-1117	46	13	=	=	SYM
cana-1117	46	14	{	{	PUNCT
cana-1117	46	15	𝑡	𝑡	X
cana-1117	46	16	∈	∈	PROPN
cana-1117	46	17	ℝ𝑛|𝜙(𝑡	ℝ𝑛|𝜙(𝑡	PROPN
cana-1117	46	18	)	)	PUNCT
cana-1117	46	19	>	>	X
cana-1117	47	1	𝑟	𝑟	X
cana-1117	47	2	}	}	PUNCT
cana-1117	47	3	,	,	PUNCT
cana-1117	47	4	which	which	PRON
cana-1117	47	5	is	be	AUX
cana-1117	47	6	known	know	VERB
cana-1117	47	7	as	as	ADP
cana-1117	47	8	the	the	DET
cana-1117	47	9	𝑟-level	𝑟-level	NOUN
cana-1117	47	10	set	set	NOUN
cana-1117	47	11	,	,	PUNCT
cana-1117	47	12	which	which	PRON
cana-1117	47	13	is	be	AUX
cana-1117	47	14	closed	close	VERB
cana-1117	47	15	for	for	ADP
cana-1117	47	16	all	all	DET
cana-1117	47	17	𝑟	𝑟	DET
cana-1117	47	18	∈	∈	NOUN
cana-1117	48	1	[	[	X
cana-1117	48	2	0,1	0,1	NUM
cana-1117	48	3	]	]	PUNCT
cana-1117	48	4	.	.	PUNCT
cana-1117	49	1	in	in	ADP
cana-1117	49	2	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	49	3	,	,	PUNCT
cana-1117	49	4	for	for	ADP
cana-1117	49	5	arbitrary	arbitrary	ADJ
cana-1117	49	6	𝜙	𝜙	NOUN
cana-1117	49	7	,	,	PUNCT
cana-1117	49	8	𝜓	𝜓	PROPN
cana-1117	49	9	∈	∈	PROPN
cana-1117	49	10	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	49	11	and	and	CCONJ
cana-1117	49	12	a	a	DET
cana-1117	49	13	scalar	scalar	ADJ
cana-1117	49	14	𝑘	𝑘	NOUN
cana-1117	49	15	,	,	PUNCT
cana-1117	49	16	define	define	VERB
cana-1117	49	17	the	the	DET
cana-1117	49	18	following	follow	VERB
cana-1117	49	19	binary	binary	ADJ
cana-1117	49	20	operations	operation	NOUN
cana-1117	49	21	,	,	PUNCT
cana-1117	49	22	namely	namely	ADV
cana-1117	49	23	,	,	PUNCT
cana-1117	49	24	addition	addition	NOUN
cana-1117	49	25	and	and	CCONJ
cana-1117	49	26	scalar	scalar	ADJ
cana-1117	49	27	multiplication	multiplication	NOUN
cana-1117	49	28	,	,	PUNCT
cana-1117	49	29	respectively	respectively	ADV
cana-1117	49	30	.	.	PUNCT
cana-1117	50	1	communications	communication	NOUN
cana-1117	50	2	on	on	ADP
cana-1117	50	3	applied	apply	VERB
cana-1117	50	4	nonlinear	nonlinear	ADJ
cana-1117	50	5	analysis	analysis	NOUN
cana-1117	50	6	issn	issn	NOUN
cana-1117	50	7	:	:	PUNCT
cana-1117	50	8	1074	1074	NUM
cana-1117	50	9	-	-	PUNCT
cana-1117	50	10	133x	133x	NUM
cana-1117	50	11	vol	vol	NOUN
cana-1117	50	12	31	31	NUM
cana-1117	50	13	no	no	NOUN
cana-1117	50	14	.	.	PUNCT
cana-1117	51	1	6s	6s	NUM
cana-1117	51	2	(	(	PUNCT
cana-1117	51	3	2024	2024	NUM
cana-1117	51	4	)	)	PUNCT
cana-1117	51	5	37	37	NUM
cana-1117	51	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	51	7	addition	addition	NOUN
cana-1117	51	8	:	:	PUNCT
cana-1117	51	9	(	(	PUNCT
cana-1117	51	10	𝜙	𝜙	PROPN
cana-1117	51	11	⊕	⊕	PROPN
cana-1117	51	12	𝜓)(𝑟	𝜓)(𝑟	PROPN
cana-1117	51	13	)	)	PUNCT
cana-1117	51	14	=	=	SYM
cana-1117	51	15	𝜙(𝑟	𝜙(𝑟	X
cana-1117	51	16	)	)	PUNCT
cana-1117	51	17	+	+	NUM
cana-1117	51	18	𝜓(𝑟	𝜓(𝑟	NOUN
cana-1117	51	19	)	)	PUNCT
cana-1117	51	20	scalar	scalar	ADJ
cana-1117	51	21	multiplication	multiplication	NOUN
cana-1117	51	22	:	:	PUNCT
cana-1117	51	23	(	(	PUNCT
cana-1117	51	24	𝑘	𝑘	X
cana-1117	51	25	⊙	⊙	X
cana-1117	51	26	𝜙)(𝑟	𝜙)(𝑟	PUNCT
cana-1117	51	27	)	)	PUNCT
cana-1117	51	28	=	=	SYM
cana-1117	51	29	(	(	PUNCT
cana-1117	51	30	𝑘𝜙(𝑟	𝑘𝜙(𝑟	NUM
cana-1117	51	31	)	)	PUNCT
cana-1117	51	32	,	,	PUNCT
cana-1117	51	33	𝑘𝜙(𝑟	𝑘𝜙(𝑟	NUM
cana-1117	51	34	)	)	PUNCT
cana-1117	51	35	)	)	PUNCT
cana-1117	51	36	,	,	PUNCT
cana-1117	51	37	𝑘	𝑘	DET
cana-1117	51	38	≥	≥	NOUN
cana-1117	51	39	0	0	PUNCT
cana-1117	52	1	(	(	PUNCT
cana-1117	52	2	𝑘	𝑘	PROPN
cana-1117	52	3	⊙	⊙	X
cana-1117	52	4	𝜙)(𝑟	𝜙)(𝑟	PUNCT
cana-1117	52	5	)	)	PUNCT
cana-1117	52	6	=	=	SYM
cana-1117	52	7	(	(	PUNCT
cana-1117	52	8	𝑘𝜙(𝑟	𝑘𝜙(𝑟	NUM
cana-1117	52	9	)	)	PUNCT
cana-1117	52	10	,	,	PUNCT
cana-1117	52	11	𝑘𝜙(𝑟	𝑘𝜙(𝑟	NUM
cana-1117	52	12	)	)	PUNCT
cana-1117	52	13	)	)	PUNCT
cana-1117	52	14	,	,	PUNCT
cana-1117	52	15	𝑘	𝑘	DET
cana-1117	52	16	≤	≤	NOUN
cana-1117	52	17	0	0	NUM
cana-1117	52	18	2.2	2.2	NUM
cana-1117	52	19	hukuhara	hukuhara	ADJ
cana-1117	52	20	difference	difference	NOUN
cana-1117	52	21	we	we	PRON
cana-1117	52	22	utilise	utilise	VERB
cana-1117	52	23	hukuhara	hukuhara	ADJ
cana-1117	52	24	difference	difference	NOUN
cana-1117	52	25	from	from	ADP
cana-1117	52	26	[	[	X
cana-1117	52	27	?	?	PUNCT
cana-1117	52	28	]	]	X
cana-1117	52	29	,	,	PUNCT
cana-1117	52	30	for	for	ADP
cana-1117	52	31	𝜙	𝜙	NOUN
cana-1117	52	32	,	,	PUNCT
cana-1117	52	33	𝜓	𝜓	PROPN
cana-1117	52	34	∈	∈	PROPN
cana-1117	52	35	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	52	36	,	,	PUNCT
cana-1117	52	37	as	as	SCONJ
cana-1117	52	38	follows	follow	VERB
cana-1117	52	39	.	.	PUNCT
cana-1117	53	1	i.e.	i.e.	X
cana-1117	53	2	,	,	PUNCT
cana-1117	53	3	𝜙	𝜙	PROPN
cana-1117	53	4	=	=	SYM
cana-1117	53	5	𝜓	𝜓	PROPN
cana-1117	53	6	+	+	CCONJ
cana-1117	53	7	𝜔	𝜔	PROPN
cana-1117	53	8	,	,	PUNCT
cana-1117	53	9	if	if	SCONJ
cana-1117	53	10	𝑤	𝑤	ADP
cana-1117	53	11	∈	∈	NOUN
cana-1117	53	12	ℝ𝔽.	ℝ𝔽.	NOUN
cana-1117	53	13	then	then	ADV
cana-1117	53	14	𝜔	𝜔	X
cana-1117	53	15	is	be	AUX
cana-1117	53	16	called	call	VERB
cana-1117	53	17	as	as	ADP
cana-1117	53	18	the	the	DET
cana-1117	53	19	hukuhara	hukuhara	ADJ
cana-1117	53	20	difference	difference	NOUN
cana-1117	53	21	of	of	ADP
cana-1117	53	22	𝜙	𝜙	PROPN
cana-1117	53	23	and	and	CCONJ
cana-1117	53	24	𝜓.	𝜓.	PROPN
cana-1117	53	25	also	also	ADV
cana-1117	53	26	,	,	PUNCT
cana-1117	53	27	the	the	DET
cana-1117	53	28	generalized	generalize	VERB
cana-1117	53	29	hukuhara	hukuhara	ADJ
cana-1117	53	30	difference	difference	NOUN
cana-1117	53	31	,	,	PUNCT
cana-1117	53	32	denoted	denote	VERB
cana-1117	53	33	as	as	ADP
cana-1117	53	34	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	53	35	in	in	ADP
cana-1117	53	36	short	short	ADJ
cana-1117	53	37	,	,	PUNCT
cana-1117	53	38	is	be	AUX
cana-1117	53	39	also	also	ADV
cana-1117	53	40	defined	define	VERB
cana-1117	53	41	as	as	ADP
cana-1117	53	42	,	,	PUNCT
cana-1117	53	43	𝜙	𝜙	NOUN
cana-1117	53	44	⊖	⊖	NUM
cana-1117	53	45	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	53	46	𝜓	𝜓	NOUN
cana-1117	53	47	=	=	PUNCT
cana-1117	53	48	𝜔	𝜔	X
cana-1117	53	49	⇔	⇔	X
cana-1117	53	50	{	{	PUNCT
cana-1117	53	51	(	(	PUNCT
cana-1117	53	52	𝑖)𝜙	𝑖)𝜙	PUNCT
cana-1117	53	53	=	=	SYM
cana-1117	53	54	𝜓	𝜓	PROPN
cana-1117	53	55	+	+	CCONJ
cana-1117	53	56	𝜔	𝜔	X
cana-1117	53	57	(	(	PUNCT
cana-1117	53	58	𝑖𝑖)𝜙	𝑖𝑖)𝜙	NOUN
cana-1117	53	59	=	=	SYM
cana-1117	53	60	𝜓	𝜓	PROPN
cana-1117	53	61	+	+	X
cana-1117	53	62	(	(	PUNCT
cana-1117	53	63	−1)𝜔	−1)𝜔	NOUN
cana-1117	53	64	it	it	PRON
cana-1117	53	65	is	be	AUX
cana-1117	53	66	obvious	obvious	ADJ
cana-1117	54	1	that	that	SCONJ
cana-1117	54	2	(	(	PUNCT
cana-1117	54	3	𝑖	𝑖	X
cana-1117	54	4	)	)	PUNCT
cana-1117	54	5	and	and	CCONJ
cana-1117	54	6	(	(	PUNCT
cana-1117	54	7	𝑖𝑖	𝑖𝑖	NOUN
cana-1117	54	8	)	)	PUNCT
cana-1117	54	9	are	be	AUX
cana-1117	54	10	true	true	ADJ
cana-1117	54	11	,	,	PUNCT
cana-1117	54	12	if	if	SCONJ
cana-1117	54	13	and	and	CCONJ
cana-1117	54	14	only	only	ADV
cana-1117	54	15	if	if	SCONJ
cana-1117	54	16	𝜔	𝜔	NOUN
cana-1117	54	17	is	be	AUX
cana-1117	54	18	a	a	DET
cana-1117	54	19	crisp	crisp	ADJ
cana-1117	54	20	number	number	NOUN
cana-1117	54	21	.	.	PUNCT
cana-1117	55	1	2.3	2.3	NUM
cana-1117	55	2	q	q	NOUN
cana-1117	55	3	calculus	calculus	NOUN
cana-1117	55	4	the	the	DET
cana-1117	55	5	usual	usual	ADJ
cana-1117	55	6	derivative	derivative	NOUN
cana-1117	55	7	of	of	ADP
cana-1117	55	8	a	a	DET
cana-1117	55	9	function	function	NOUN
cana-1117	55	10	′𝑓(𝜏)′	′𝑓(𝜏)′	PROPN
cana-1117	55	11	is	be	AUX
cana-1117	55	12	defined	define	VERB
cana-1117	55	13	as	as	ADP
cana-1117	55	14	𝑙𝑖𝑚⏟	𝑙𝑖𝑚⏟	PROPN
cana-1117	55	15	𝜏→𝜏0	𝜏→𝜏0	NUM
cana-1117	55	16	𝑓(𝜏)−𝑓(𝜏0	𝑓(𝜏)−𝑓(𝜏0	PROPN
cana-1117	55	17	)	)	PUNCT
cana-1117	55	18	𝜏−𝜏0	𝜏−𝜏0	NOUN
cana-1117	55	19	.	.	PUNCT
cana-1117	56	1	now	now	ADV
cana-1117	56	2	we	we	PRON
cana-1117	56	3	can	can	AUX
cana-1117	56	4	study	study	VERB
cana-1117	56	5	𝔮-derivative	𝔮-derivative	NOUN
cana-1117	56	6	of	of	ADP
cana-1117	56	7	𝑓.	𝑓.	NOUN
cana-1117	56	8	for	for	ADP
cana-1117	56	9	that	that	PRON
cana-1117	56	10	,	,	PUNCT
cana-1117	56	11	we	we	PRON
cana-1117	56	12	use	use	VERB
cana-1117	56	13	the	the	DET
cana-1117	56	14	following	follow	VERB
cana-1117	56	15	definitions	definition	NOUN
cana-1117	56	16	from	from	ADP
cana-1117	56	17	[	[	X
cana-1117	56	18	?	?	PUNCT
cana-1117	56	19	]	]	PUNCT
cana-1117	56	20	.	.	PUNCT
cana-1117	57	1	let	let	VERB
cana-1117	57	2	𝕋𝔮	𝕋𝔮	PRON
cana-1117	57	3	be	be	AUX
cana-1117	57	4	the	the	DET
cana-1117	57	5	time	time	NOUN
cana-1117	57	6	scale	scale	NOUN
cana-1117	57	7	,	,	PUNCT
cana-1117	57	8	for	for	ADP
cana-1117	57	9	0	0	NUM
cana-1117	57	10	<	<	X
cana-1117	57	11	𝔮	𝔮	X
cana-1117	57	12	<	<	X
cana-1117	57	13	1	1	NUM
cana-1117	57	14	.	.	PUNCT
cana-1117	58	1	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	58	2	=	=	SYM
cana-1117	58	3	{	{	PUNCT
cana-1117	58	4	𝔮𝑛	𝔮𝑛	NOUN
cana-1117	58	5	:	:	PUNCT
cana-1117	58	6	𝑛	𝑛	PRON
cana-1117	58	7	∈	∈	PROPN
cana-1117	58	8	ℤ	ℤ	PROPN
cana-1117	58	9	}	}	PUNCT
cana-1117	58	10	∪	∪	NOUN
cana-1117	58	11	{	{	PUNCT
cana-1117	58	12	0	0	NUM
cana-1117	58	13	}	}	PUNCT
cana-1117	58	14	𝕋𝔮	𝕋𝔮	VERB
cana-1117	58	15	𝑛	𝑛	NOUN
cana-1117	58	16	=	=	PUNCT
cana-1117	58	17	{	{	PUNCT
cana-1117	58	18	𝔮𝛼+𝑛	𝔮𝛼+𝑛	NUM
cana-1117	58	19	:	:	PUNCT
cana-1117	58	20	𝑛	𝑛	DET
cana-1117	58	21	∈	∈	PROPN
cana-1117	58	22	ℤ	ℤ	PROPN
cana-1117	58	23	}	}	PUNCT
cana-1117	58	24	∪	∪	NOUN
cana-1117	58	25	{	{	PUNCT
cana-1117	58	26	0	0	NUM
cana-1117	58	27	}	}	PUNCT
cana-1117	58	28	consider	consider	VERB
cana-1117	58	29	an	an	DET
cana-1117	58	30	arbitrary	arbitrary	ADJ
cana-1117	58	31	function	function	NOUN
cana-1117	58	32	𝑓	𝑓	NOUN
cana-1117	58	33	:	:	PUNCT
cana-1117	58	34	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	58	35	→	→	SYM
cana-1117	58	36	ℝ.	ℝ.	PROPN
cana-1117	58	37	its	its	PRON
cana-1117	58	38	𝔮-differential	𝔮-differential	NOUN
cana-1117	58	39	is	be	AUX
cana-1117	58	40	𝑑𝔮𝑓(𝜏	𝑑𝔮𝑓(𝜏	ADJ
cana-1117	58	41	)	)	PUNCT
cana-1117	58	42	=	=	SYM
cana-1117	58	43	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	58	44	)	)	PUNCT
cana-1117	59	1	−	−	PRON
cana-1117	59	2	𝑓(𝔮𝜏	𝑓(𝔮𝜏	NOUN
cana-1117	59	3	)	)	PUNCT
cana-1117	59	4	then	then	ADV
cana-1117	59	5	the	the	DET
cana-1117	59	6	𝔮-derivative	𝔮-derivative	NOUN
cana-1117	59	7	of	of	ADP
cana-1117	59	8	𝑓	𝑓	PRON
cana-1117	59	9	is	be	AUX
cana-1117	59	10	given	give	VERB
cana-1117	59	11	by	by	ADP
cana-1117	59	12	,	,	PUNCT
cana-1117	59	13	𝐷𝔮𝑓(𝜏	𝐷𝔮𝑓(𝜏	PROPN
cana-1117	59	14	)	)	PUNCT
cana-1117	59	15	=	=	PUNCT
cana-1117	59	16	𝑑𝔮𝑓(𝜏	𝑑𝔮𝑓(𝜏	ADJ
cana-1117	59	17	)	)	PUNCT
cana-1117	59	18	𝑑𝔮𝜏	𝑑𝔮𝜏	NOUN
cana-1117	59	19	=	=	SYM
cana-1117	59	20	𝑓(𝜏)−𝑓(𝔮𝜏	𝑓(𝜏)−𝑓(𝔮𝜏	PROPN
cana-1117	59	21	)	)	PUNCT
cana-1117	59	22	(	(	PUNCT
cana-1117	59	23	1−𝔮)𝜏	1−𝔮)𝜏	NUM
cana-1117	59	24	,	,	PUNCT
cana-1117	59	25	𝜏	𝜏	NOUN
cana-1117	59	26	∈	∈	NOUN
cana-1117	59	27	𝕋𝔮	𝕋𝔮	ADP
cana-1117	59	28	−	−	PROPN
cana-1117	59	29	{	{	PUNCT
cana-1117	59	30	0	0	NUM
cana-1117	59	31	}	}	PUNCT
cana-1117	59	32	the	the	DET
cana-1117	59	33	q	q	ADJ
cana-1117	59	34	-	-	PUNCT
cana-1117	59	35	gamma	gamma	NOUN
cana-1117	59	36	function	function	NOUN
cana-1117	59	37	denoted	denote	VERB
cana-1117	59	38	by	by	ADP
cana-1117	59	39	γ𝑞	γ𝑞	PROPN
cana-1117	59	40	(	(	PUNCT
cana-1117	59	41	.	.	PUNCT
cana-1117	59	42	)	)	PUNCT
cana-1117	59	43	can	can	AUX
cana-1117	59	44	be	be	AUX
cana-1117	59	45	defined	define	VERB
cana-1117	59	46	as	as	ADP
cana-1117	59	47	,	,	PUNCT
cana-1117	59	48	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	59	49	)	)	PUNCT
cana-1117	59	50	=	=	SYM
cana-1117	60	1	(	(	PUNCT
cana-1117	60	2	1−𝔮)𝔮	1−𝔮)𝔮	NUM
cana-1117	60	3	𝛼−1	𝛼−1	PROPN
cana-1117	60	4	(	(	PUNCT
cana-1117	60	5	1−𝑞)𝛼−1	1−𝑞)𝛼−1	NUM
cana-1117	60	6	,	,	PUNCT
cana-1117	60	7	𝛼	𝛼	PROPN
cana-1117	60	8	∈	∈	NOUN
cana-1117	60	9	ℝ	ℝ	NOUN
cana-1117	60	10	−	−	PROPN
cana-1117	60	11	{	{	PUNCT
cana-1117	60	12	0	0	NUM
cana-1117	60	13	}	}	PUNCT
cana-1117	60	14	∪	∪	ADP
cana-1117	60	15	ℤ	ℤ	PROPN
cana-1117	60	16	,	,	PUNCT
cana-1117	60	17	0	0	NUM
cana-1117	60	18	<	<	X
cana-1117	60	19	𝔮	𝔮	X
cana-1117	60	20	<	<	X
cana-1117	60	21	1	1	NUM
cana-1117	60	22	,	,	PUNCT
cana-1117	60	23	which	which	PRON
cana-1117	60	24	is	be	AUX
cana-1117	60	25	satisfied	satisfied	ADJ
cana-1117	60	26	in	in	ADP
cana-1117	60	27	the	the	DET
cana-1117	60	28	following	follow	VERB
cana-1117	60	29	relation	relation	NOUN
cana-1117	60	30	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	60	31	+	+	CCONJ
cana-1117	60	32	1	1	X
cana-1117	60	33	)	)	PUNCT
cana-1117	60	34	=	=	SYM
cana-1117	60	35	(	(	PUNCT
cana-1117	60	36	1−𝔮)𝛼	1−𝔮)𝛼	NUM
cana-1117	60	37	(	(	PUNCT
cana-1117	60	38	1−𝔮	1−𝔮	NUM
cana-1117	60	39	)	)	PUNCT
cana-1117	60	40	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	60	41	)	)	PUNCT
cana-1117	60	42	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	60	43	)	)	PUNCT
cana-1117	60	44	=	=	SYM
cana-1117	60	45	1	1	NUM
cana-1117	60	46	,	,	PUNCT
cana-1117	60	47	𝛼	𝛼	ADJ
cana-1117	60	48	>	>	X
cana-1117	60	49	0	0	NUM
cana-1117	60	50	communications	communication	NOUN
cana-1117	60	51	on	on	ADP
cana-1117	60	52	applied	apply	VERB
cana-1117	60	53	nonlinear	nonlinear	ADJ
cana-1117	60	54	analysis	analysis	NOUN
cana-1117	60	55	issn	issn	NOUN
cana-1117	60	56	:	:	PUNCT
cana-1117	60	57	1074	1074	NUM
cana-1117	60	58	-	-	PUNCT
cana-1117	60	59	133x	133x	NUM
cana-1117	60	60	vol	vol	NOUN
cana-1117	60	61	31	31	NUM
cana-1117	60	62	no	no	NOUN
cana-1117	60	63	.	.	PUNCT
cana-1117	61	1	6s	6s	NUM
cana-1117	61	2	(	(	PUNCT
cana-1117	61	3	2024	2024	NUM
cana-1117	61	4	)	)	PUNCT
cana-1117	61	5	38	38	NUM
cana-1117	61	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	61	7	definition	definition	NOUN
cana-1117	61	8	2.2	2.2	NUM
cana-1117	61	9	[	[	X
cana-1117	61	10	6	6	NUM
cana-1117	61	11	]	]	PUNCT
cana-1117	61	12	let	let	VERB
cana-1117	61	13	for	for	ADP
cana-1117	61	14	𝛼	𝛼	X
cana-1117	61	15	>	>	X
cana-1117	61	16	0	0	NUM
cana-1117	61	17	,	,	PUNCT
cana-1117	61	18	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	61	19	)	)	PUNCT
cana-1117	61	20	is	be	AUX
cana-1117	61	21	a	a	DET
cana-1117	61	22	𝔮-integrable	𝔮-integrable	ADJ
cana-1117	61	23	function	function	NOUN
cana-1117	61	24	,	,	PUNCT
cana-1117	61	25	the	the	DET
cana-1117	61	26	fractional	fractional	ADJ
cana-1117	61	27	𝔮-integral	𝔮-integral	NOUN
cana-1117	61	28	of	of	ADP
cana-1117	61	29	order	order	NOUN
cana-1117	61	30	𝛼	𝛼	NOUN
cana-1117	61	31	is	be	AUX
cana-1117	61	32	defined	define	VERB
cana-1117	61	33	by	by	ADP
cana-1117	61	34	,	,	PUNCT
cana-1117	61	35	𝔮𝐼𝑎	𝔮𝐼𝑎	ADJ
cana-1117	61	36	𝛼𝑓(𝜏	𝛼𝑓(𝜏	NUM
cana-1117	61	37	)	)	PUNCT
cana-1117	61	38	=	=	SYM
cana-1117	61	39	1	1	NUM
cana-1117	61	40	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	61	41	)	)	PUNCT
cana-1117	61	42	∫	∫	PROPN
cana-1117	62	1	𝜏	𝜏	X
cana-1117	62	2	𝑎	𝑎	X
cana-1117	62	3	(	(	PUNCT
cana-1117	62	4	𝜏	𝜏	PROPN
cana-1117	62	5	−	−	NUM
cana-1117	62	6	𝔮𝑠)𝔮	𝔮𝑠)𝔮	NUM
cana-1117	62	7	𝛼−1𝑓(𝑠)𝑑𝔮𝑠	𝛼−1𝑓(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	62	8	definition	definition	NOUN
cana-1117	62	9	2.3	2.3	NUM
cana-1117	63	1	[	[	X
cana-1117	63	2	6	6	NUM
cana-1117	63	3	]	]	PUNCT
cana-1117	63	4	for	for	ADP
cana-1117	63	5	an	an	DET
cana-1117	63	6	arbitrary	arbitrary	ADJ
cana-1117	63	7	fuzzy	fuzzy	ADJ
cana-1117	63	8	valued	value	VERB
cana-1117	63	9	function	function	NOUN
cana-1117	63	10	𝑓	𝑓	NOUN
cana-1117	63	11	:	:	PUNCT
cana-1117	63	12	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	63	13	→	→	SYM
cana-1117	63	14	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	63	15	,	,	PUNCT
cana-1117	63	16	the	the	DET
cana-1117	63	17	𝔮-differential	𝔮-differential	NOUN
cana-1117	63	18	by	by	ADP
cana-1117	63	19	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	63	20	difference	difference	NOUN
cana-1117	63	21	is	be	AUX
cana-1117	63	22	𝑔𝐻𝑑𝔮𝑓(𝜏	𝑔𝐻𝑑𝔮𝑓(𝜏	PROPN
cana-1117	63	23	)	)	PUNCT
cana-1117	63	24	=	=	SYM
cana-1117	63	25	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	63	26	)	)	PUNCT
cana-1117	63	27	⊖	⊖	NOUN
cana-1117	63	28	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	63	29	𝑓(𝔮𝜏	𝑓(𝔮𝜏	NOUN
cana-1117	63	30	)	)	PUNCT
cana-1117	63	31	.	.	PUNCT
cana-1117	64	1	if	if	SCONJ
cana-1117	64	2	𝑔𝐻𝑑𝔮𝑓(𝜏	𝑔𝐻𝑑𝔮𝑓(𝜏	PROPN
cana-1117	64	3	)	)	PUNCT
cana-1117	64	4	=	=	SYM
cana-1117	64	5	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	64	6	)	)	PUNCT
cana-1117	64	7	⊖	⊖	NOUN
cana-1117	64	8	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	64	9	𝑓(𝔮𝜏	𝑓(𝔮𝜏	NOUN
cana-1117	64	10	)	)	PUNCT
cana-1117	64	11	exists	exist	VERB
cana-1117	64	12	,	,	PUNCT
cana-1117	64	13	the	the	DET
cana-1117	64	14	fuzzy	fuzzy	ADJ
cana-1117	64	15	generalized	generalize	VERB
cana-1117	64	16	hukuhara	hukuhara	ADJ
cana-1117	64	17	𝔮-derivative	𝔮-derivative	NOUN
cana-1117	64	18	of	of	ADP
cana-1117	64	19	𝑓	𝑓	PRON
cana-1117	64	20	is	be	AUX
cana-1117	64	21	defined	define	VERB
cana-1117	64	22	by	by	ADP
cana-1117	64	23	,	,	PUNCT
cana-1117	64	24	𝐹𝐷𝔮𝑓(𝜏	𝐹𝐷𝔮𝑓(𝜏	NUM
cana-1117	64	25	)	)	PUNCT
cana-1117	64	26	=	=	SYM
cana-1117	64	27	𝑔𝐻𝑑𝔮𝑓(𝜏	𝑔𝐻𝑑𝔮𝑓(𝜏	PROPN
cana-1117	64	28	)	)	PUNCT
cana-1117	64	29	𝑑𝔮𝜏	𝑑𝔮𝜏	NOUN
cana-1117	64	30	=	=	SYM
cana-1117	64	31	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	64	32	)	)	PUNCT
cana-1117	64	33	⊖	⊖	NOUN
cana-1117	64	34	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	64	35	𝑓(𝔮𝜏	𝑓(𝔮𝜏	NOUN
cana-1117	64	36	)	)	PUNCT
cana-1117	64	37	(	(	PUNCT
cana-1117	64	38	1	1	NUM
cana-1117	64	39	−	−	NOUN
cana-1117	64	40	𝔮)𝜏	𝔮)𝜏	NOUN
cana-1117	64	41	,	,	PUNCT
cana-1117	64	42	𝜏	𝜏	X
cana-1117	64	43	∈	∈	NOUN
cana-1117	64	44	𝕋𝔮	𝕋𝔮	ADP
cana-1117	64	45	−	−	PROPN
cana-1117	64	46	{	{	PUNCT
cana-1117	64	47	0	0	NUM
cana-1117	64	48	}	}	PUNCT
cana-1117	64	49	definition	definition	NOUN
cana-1117	64	50	2.4	2.4	NUM
cana-1117	64	51	[	[	SYM
cana-1117	64	52	6	6	NUM
cana-1117	64	53	]	]	PUNCT
cana-1117	64	54	let	let	VERB
cana-1117	64	55	𝑓	𝑓	DET
cana-1117	64	56	:	:	PUNCT
cana-1117	64	57	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	64	58	→	→	SYM
cana-1117	64	59	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	64	60	be	be	AUX
cana-1117	64	61	a	a	DET
cana-1117	64	62	𝔮-integrable	𝔮-integrable	ADJ
cana-1117	64	63	function	function	NOUN
cana-1117	64	64	.	.	PUNCT
cana-1117	65	1	the	the	DET
cana-1117	65	2	fuzzy	fuzzy	ADJ
cana-1117	65	3	𝔮-fractional	𝔮-fractional	ADJ
cana-1117	65	4	integral	integral	ADJ
cana-1117	65	5	of	of	ADP
cana-1117	65	6	order	order	NOUN
cana-1117	65	7	𝛼	𝛼	NOUN
cana-1117	65	8	for	for	ADP
cana-1117	65	9	𝑓	𝑓	PROPN
cana-1117	65	10	is	be	AUX
cana-1117	65	11	defined	define	VERB
cana-1117	65	12	by	by	ADP
cana-1117	65	13	,	,	PUNCT
cana-1117	65	14	𝐹𝐼𝑎	𝐹𝐼𝑎	NOUN
cana-1117	65	15	𝛼𝑓(𝜏	𝛼𝑓(𝜏	NUM
cana-1117	65	16	)	)	PUNCT
cana-1117	65	17	=	=	SYM
cana-1117	65	18	1	1	NUM
cana-1117	65	19	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	65	20	)	)	PUNCT
cana-1117	66	1	⊙	⊙	PROPN
cana-1117	66	2	∫	∫	PROPN
cana-1117	67	1	𝜏	𝜏	X
cana-1117	67	2	𝑎	𝑎	X
cana-1117	67	3	(	(	PUNCT
cana-1117	67	4	𝜏	𝜏	PROPN
cana-1117	67	5	−	−	PROPN
cana-1117	67	6	𝔮𝑠)𝔮	𝔮𝑠)𝔮	PROPN
cana-1117	68	1	𝛼−1	𝛼−1	DET
cana-1117	68	2	⊙	⊙	NOUN
cana-1117	68	3	𝑓(𝑠)𝑑𝔮𝑠	𝑓(𝑠)𝑑𝔮𝑠	PROPN
cana-1117	68	4	definition	definition	NOUN
cana-1117	68	5	2.5	2.5	NUM
cana-1117	68	6	[	[	SYM
cana-1117	68	7	6	6	NUM
cana-1117	68	8	]	]	PUNCT
cana-1117	68	9	let	let	VERB
cana-1117	68	10	∀𝑚	∀𝑚	NUM
cana-1117	68	11	,	,	PUNCT
cana-1117	68	12	𝔮	𝔮	X
cana-1117	68	13	𝐹𝐷𝑚𝑓	𝐹𝐷𝑚𝑓	NOUN
cana-1117	68	14	be	be	AUX
cana-1117	68	15	continuous	continuous	ADJ
cana-1117	68	16	and	and	CCONJ
cana-1117	68	17	𝔮-integrable	𝔮-integrable	ADJ
cana-1117	68	18	functions	function	NOUN
cana-1117	68	19	in	in	ADP
cana-1117	68	20	𝕋𝔮.	𝕋𝔮.	PROPN
cana-1117	68	21	the	the	DET
cana-1117	68	22	fuzzy	fuzzy	ADJ
cana-1117	68	23	caputo	caputo	PROPN
cana-1117	68	24	q	q	PROPN
cana-1117	68	25	-	-	PUNCT
cana-1117	68	26	fractional	fractional	ADJ
cana-1117	68	27	derivative	derivative	NOUN
cana-1117	68	28	of	of	ADP
cana-1117	68	29	order	order	NOUN
cana-1117	68	30	𝛼	𝛼	NOUN
cana-1117	68	31	of	of	ADP
cana-1117	68	32	fuzzy	fuzzy	ADJ
cana-1117	68	33	valued	value	VERB
cana-1117	68	34	function	function	NOUN
cana-1117	68	35	is	be	AUX
cana-1117	68	36	defined	define	VERB
cana-1117	68	37	,	,	PUNCT
cana-1117	68	38	𝔮	𝔮	NOUN
cana-1117	68	39	𝐹𝐶𝐷𝛼𝑓(𝑡	𝐹𝐶𝐷𝛼𝑓(𝑡	NOUN
cana-1117	68	40	)	)	PUNCT
cana-1117	69	1	=	=	NOUN
cana-1117	69	2	𝑎	𝑎	DET
cana-1117	69	3	𝐹	𝐹	PROPN
cana-1117	69	4	𝐼𝑎	𝐼𝑎	PROPN
cana-1117	69	5	𝑚−𝛼(𝐹𝐷𝔮	𝑚−𝛼(𝐹𝐷𝔮	PROPN
cana-1117	69	6	𝑚𝑓)(𝜏	𝑚𝑓)(𝜏	NOUN
cana-1117	69	7	)	)	PUNCT
cana-1117	70	1	=	=	SYM
cana-1117	70	2	1	1	NUM
cana-1117	70	3	γ𝔮(𝑚	γ𝔮(𝑚	NUM
cana-1117	70	4	−	−	PROPN
cana-1117	70	5	𝛼	𝛼	X
cana-1117	70	6	)	)	PUNCT
cana-1117	70	7	⊙	⊙	PROPN
cana-1117	70	8	∫	∫	PROPN
cana-1117	71	1	𝜏	𝜏	X
cana-1117	71	2	𝑎	𝑎	X
cana-1117	71	3	(	(	PUNCT
cana-1117	71	4	𝜏	𝜏	PROPN
cana-1117	71	5	−	−	PROPN
cana-1117	71	6	𝔮𝑠)𝔮	𝔮𝑠)𝔮	NUM
cana-1117	71	7	𝑚−𝛼−1	𝑚−𝛼−1	PROPN
cana-1117	71	8	⊙𝔮	⊙𝔮	PROPN
cana-1117	71	9	𝐹	𝐹	PROPN
cana-1117	71	10	𝐷𝑚𝑓(𝑠)𝑑𝔮𝑠	𝐷𝑚𝑓(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	71	11	where	where	SCONJ
cana-1117	71	12	𝑚	𝑚	PROPN
cana-1117	71	13	−	−	PROPN
cana-1117	71	14	1	1	NUM
cana-1117	71	15	<	<	X
cana-1117	71	16	𝛼	𝛼	X
cana-1117	71	17	≤	≤	NUM
cana-1117	71	18	𝑚	𝑚	ADP
cana-1117	71	19	,	,	PUNCT
cana-1117	71	20	𝑚	𝑚	PROPN
cana-1117	71	21	∈	∈	PROPN
cana-1117	71	22	ℕ	ℕ	PROPN
cana-1117	71	23	,	,	PUNCT
cana-1117	71	24	𝑡	𝑡	PROPN
cana-1117	71	25	>	>	X
cana-1117	71	26	𝑎.	𝑎.	PROPN
cana-1117	71	27	definition	definition	NOUN
cana-1117	71	28	2.6	2.6	NUM
cana-1117	72	1	[	[	X
cana-1117	72	2	6	6	NUM
cana-1117	72	3	]	]	PUNCT
cana-1117	72	4	for	for	ADP
cana-1117	72	5	0	0	NUM
cana-1117	72	6	<	<	X
cana-1117	72	7	𝛼	𝛼	PROPN
cana-1117	72	8	≤	≤	NUM
cana-1117	72	9	1	1	NUM
cana-1117	72	10	,	,	PUNCT
cana-1117	72	11	the	the	DET
cana-1117	72	12	fuzzy	fuzzy	ADJ
cana-1117	72	13	caputo	caputo	PROPN
cana-1117	72	14	𝔮-fractional	𝔮-fractional	PROPN
cana-1117	72	15	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	72	16	derivative	derivative	NOUN
cana-1117	72	17	will	will	AUX
cana-1117	72	18	be	be	AUX
cana-1117	72	19	,	,	PUNCT
cana-1117	72	20	𝔮	𝔮	X
cana-1117	72	21	𝐹𝐶𝐷𝛼𝑓(𝜏	𝐹𝐶𝐷𝛼𝑓(𝜏	NOUN
cana-1117	72	22	)	)	PUNCT
cana-1117	73	1	=	=	NOUN
cana-1117	73	2	𝑎	𝑎	X
cana-1117	73	3	𝐹	𝐹	PROPN
cana-1117	73	4	𝐼𝑎	𝐼𝑎	PROPN
cana-1117	73	5	1−𝛼(𝐹𝐷𝔮𝑓)(𝜏	1−𝛼(𝐹𝐷𝔮𝑓)(𝜏	NUM
cana-1117	73	6	)	)	PUNCT
cana-1117	73	7	=	=	SYM
cana-1117	73	8	1	1	NUM
cana-1117	73	9	γ𝔮(1	γ𝔮(1	PROPN
cana-1117	73	10	−	−	PROPN
cana-1117	73	11	𝛼	𝛼	NOUN
cana-1117	73	12	)	)	PUNCT
cana-1117	73	13	⊙	⊙	PROPN
cana-1117	74	1	∫	∫	PROPN
cana-1117	75	1	𝜏	𝜏	X
cana-1117	75	2	𝑎	𝑎	X
cana-1117	75	3	(	(	PUNCT
cana-1117	75	4	𝜏	𝜏	PROPN
cana-1117	75	5	−	−	PROPN
cana-1117	75	6	𝔮𝑠)𝔮	𝔮𝑠)𝔮	NUM
cana-1117	75	7	−𝛼	−𝛼	PROPN
cana-1117	75	8	⊙𝔮	⊙𝔮	PROPN
cana-1117	75	9	𝐹	𝐹	PROPN
cana-1117	75	10	𝐷𝑓(𝑠)𝑑𝔮	𝐷𝑓(𝑠)𝑑𝔮	PROPN
cana-1117	75	11	for	for	ADP
cana-1117	75	12	𝜏	𝜏	PROPN
cana-1117	75	13	>	>	PUNCT
cana-1117	75	14	𝑎.	𝑎.	NOUN
cana-1117	75	15	definition	definition	NOUN
cana-1117	75	16	2.7	2.7	NUM
cana-1117	75	17	[	[	X
cana-1117	75	18	7	7	NUM
cana-1117	75	19	]	]	PUNCT
cana-1117	75	20	for	for	ADP
cana-1117	75	21	a	a	DET
cana-1117	75	22	fuzzy	fuzzy	ADJ
cana-1117	75	23	valued	value	VERB
cana-1117	75	24	function	function	NOUN
cana-1117	75	25	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	75	26	)	)	PUNCT
cana-1117	75	27	,	,	PUNCT
cana-1117	75	28	where	where	SCONJ
cana-1117	75	29	𝜏	𝜏	X
cana-1117	75	30	∈	∈	PROPN
cana-1117	75	31	𝑇𝑞	𝑇𝑞	PROPN
cana-1117	75	32	,	,	PUNCT
cana-1117	75	33	the	the	DET
cana-1117	75	34	caputo	caputo	PROPN
cana-1117	75	35	𝔮-integral	𝔮-integral	PROPN
cana-1117	75	36	of	of	ADP
cana-1117	75	37	order	order	NOUN
cana-1117	75	38	𝛼	𝛼	NOUN
cana-1117	75	39	is	be	AUX
cana-1117	75	40	𝔮𝐼𝑎	𝔮𝐼𝑎	ADP
cana-1117	75	41	𝛼𝑓(𝜏	𝛼𝑓(𝜏	NUM
cana-1117	75	42	)	)	PUNCT
cana-1117	75	43	=	=	SYM
cana-1117	75	44	1	1	NUM
cana-1117	75	45	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	75	46	)	)	PUNCT
cana-1117	75	47	∫	∫	PROPN
cana-1117	76	1	𝜏	𝜏	X
cana-1117	76	2	𝑎	𝑎	X
cana-1117	76	3	(	(	PUNCT
cana-1117	76	4	𝜏	𝜏	PROPN
cana-1117	76	5	−	−	NUM
cana-1117	76	6	𝔮𝑠)𝔮	𝔮𝑠)𝔮	NUM
cana-1117	76	7	𝛼−1𝑓(𝑠)𝑑𝔮𝑠	𝛼−1𝑓(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	76	8	definition	definition	NOUN
cana-1117	76	9	2.8	2.8	NUM
cana-1117	77	1	[	[	X
cana-1117	77	2	6	6	NUM
cana-1117	77	3	]	]	PUNCT
cana-1117	77	4	for	for	ADP
cana-1117	77	5	a	a	DET
cana-1117	77	6	fuzzy	fuzzy	ADJ
cana-1117	77	7	valued	value	VERB
cana-1117	77	8	function	function	NOUN
cana-1117	77	9	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	77	10	)	)	PUNCT
cana-1117	77	11	,	,	PUNCT
cana-1117	77	12	where	where	SCONJ
cana-1117	77	13	𝜏	𝜏	X
cana-1117	77	14	∈	∈	PROPN
cana-1117	77	15	𝑇𝑞	𝑇𝑞	PROPN
cana-1117	77	16	,	,	PUNCT
cana-1117	77	17	𝔮𝐼𝑎	𝔮𝐼𝑎	ADV
cana-1117	77	18	𝛼(𝔮	𝛼(𝔮	PROPN
cana-1117	77	19	𝐶𝐷𝛼)(𝑓(𝜏	𝐶𝐷𝛼)(𝑓(𝜏	PROPN
cana-1117	77	20	)	)	PUNCT
cana-1117	77	21	)	)	PUNCT
cana-1117	78	1	=	=	SYM
cana-1117	78	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	78	3	)	)	PUNCT
cana-1117	78	4	−	−	NOUN
cana-1117	78	5	∑	∑	PUNCT
cana-1117	78	6	𝑚−1	𝑚−1	PROPN
cana-1117	78	7	𝑘=0	𝑘=0	VERB
cana-1117	78	8	(	(	PUNCT
cana-1117	78	9	𝜏	𝜏	NOUN
cana-1117	78	10	−	−	NOUN
cana-1117	78	11	𝑎)𝔮	𝑎)𝔮	NOUN
cana-1117	78	12	𝑘	𝑘	ADP
cana-1117	78	13	γ𝔮(𝑘	γ𝔮(𝑘	PUNCT
cana-1117	78	14	+	+	NOUN
cana-1117	78	15	1	1	X
cana-1117	78	16	)	)	PUNCT
cana-1117	78	17	𝐷𝔮	𝐷𝔮	PROPN
cana-1117	78	18	𝑘𝑓(𝑎	𝑘𝑓(𝑎	NOUN
cana-1117	78	19	)	)	PUNCT
cana-1117	78	20	where	where	SCONJ
cana-1117	78	21	𝑚	𝑚	PROPN
cana-1117	78	22	−	−	PROPN
cana-1117	78	23	1	1	NUM
cana-1117	78	24	<	<	X
cana-1117	78	25	𝛼	𝛼	X
cana-1117	78	26	≤	≤	NUM
cana-1117	78	27	𝑚	𝑚	NOUN
cana-1117	78	28	,	,	PUNCT
cana-1117	78	29	𝑚	𝑚	PROPN
cana-1117	78	30	∈	∈	PROPN
cana-1117	78	31	ℕ	ℕ	PROPN
cana-1117	78	32	and	and	CCONJ
cana-1117	78	33	for	for	ADP
cana-1117	78	34	the	the	DET
cana-1117	78	35	special	special	ADJ
cana-1117	78	36	case	case	NOUN
cana-1117	78	37	0	0	PUNCT
cana-1117	78	38	<	<	X
cana-1117	78	39	𝛼	𝛼	X
cana-1117	78	40	≤	≤	NUM
cana-1117	78	41	1	1	NUM
cana-1117	78	42	,	,	PUNCT
cana-1117	78	43	𝔮𝐼𝑎	𝔮𝐼𝑎	ADV
cana-1117	78	44	𝛼(𝔮	𝛼(𝔮	ADJ
cana-1117	78	45	𝐶𝐷)(𝑓(𝜏	𝐶𝐷)(𝑓(𝜏	NOUN
cana-1117	78	46	)	)	PUNCT
cana-1117	78	47	)	)	PUNCT
cana-1117	79	1	=	=	SYM
cana-1117	79	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	79	3	)	)	PUNCT
cana-1117	79	4	−	−	PRON
cana-1117	79	5	𝑓(𝑎	𝑓(𝑎	NOUN
cana-1117	79	6	)	)	PUNCT
cana-1117	79	7	communications	communication	NOUN
cana-1117	79	8	on	on	ADP
cana-1117	79	9	applied	apply	VERB
cana-1117	79	10	nonlinear	nonlinear	ADJ
cana-1117	79	11	analysis	analysis	NOUN
cana-1117	79	12	issn	issn	NOUN
cana-1117	79	13	:	:	PUNCT
cana-1117	79	14	1074	1074	NUM
cana-1117	79	15	-	-	PUNCT
cana-1117	79	16	133x	133x	NUM
cana-1117	79	17	vol	vol	NOUN
cana-1117	79	18	31	31	NUM
cana-1117	79	19	no	no	NOUN
cana-1117	79	20	.	.	PUNCT
cana-1117	80	1	6s	6s	NUM
cana-1117	80	2	(	(	PUNCT
cana-1117	80	3	2024	2024	NUM
cana-1117	80	4	)	)	PUNCT
cana-1117	80	5	39	39	NUM
cana-1117	80	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	80	7	definition	definition	NOUN
cana-1117	80	8	2.9	2.9	NUM
cana-1117	81	1	[	[	X
cana-1117	81	2	6	6	NUM
cana-1117	81	3	]	]	PUNCT
cana-1117	81	4	let	let	VERB
cana-1117	81	5	𝑓	𝑓	PRON
cana-1117	81	6	:	:	PUNCT
cana-1117	81	7	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	81	8	→	→	SYM
cana-1117	81	9	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	81	10	be	be	PROPN
cana-1117	81	11	caputo	caputo	PROPN
cana-1117	81	12	differentiable	differentiable	NOUN
cana-1117	81	13	at	at	ADP
cana-1117	81	14	𝜙0	𝜙0	NOUN
cana-1117	81	15	∈	∈	PROPN
cana-1117	81	16	𝕋𝔮.	𝕋𝔮.	PROPN
cana-1117	82	1	we	we	PRON
cana-1117	82	2	say	say	VERB
cana-1117	82	3	that	that	SCONJ
cana-1117	82	4	𝑓	𝑓	PROPN
cana-1117	82	5	is	be	AUX
cana-1117	82	6	caputo	caputo	PROPN
cana-1117	82	7	[	[	X
cana-1117	82	8	(	(	PUNCT
cana-1117	82	9	𝑖)−𝑔𝐻	𝑖)−𝑔𝐻	NOUN
cana-1117	82	10	]	]	X
cana-1117	82	11	differentiable	differentiable	NOUN
cana-1117	82	12	at	at	ADP
cana-1117	82	13	𝜙0	𝜙0	NOUN
cana-1117	82	14	,	,	PUNCT
cana-1117	82	15	if	if	SCONJ
cana-1117	82	16	(	(	PUNCT
cana-1117	82	17	𝑖	𝑖	X
cana-1117	82	18	)	)	PUNCT
cana-1117	82	19	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	82	20	𝐶	𝐶	PROPN
cana-1117	82	21	𝐷𝛼𝑓(𝜙0	𝐷𝛼𝑓(𝜙0	NOUN
cana-1117	82	22	;	;	PUNCT
cana-1117	82	23	𝑟	𝑟	X
cana-1117	82	24	)	)	PUNCT
cana-1117	82	25	=	=	SYM
cana-1117	82	26	[	[	PUNCT
cana-1117	82	27	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	82	28	𝐶	𝐶	PROPN
cana-1117	82	29	𝐷𝛼𝑓(𝜙0	𝐷𝛼𝑓(𝜙0	NOUN
cana-1117	82	30	;	;	PUNCT
cana-1117	82	31	𝑟	𝑟	X
cana-1117	82	32	)	)	PUNCT
cana-1117	82	33	,	,	PUNCT
cana-1117	82	34	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	82	35	𝐶	𝐶	PROPN
cana-1117	82	36	𝐷𝛼𝑓(𝜙0	𝐷𝛼𝑓(𝜙0	NOUN
cana-1117	82	37	;	;	PUNCT
cana-1117	82	38	𝑟	𝑟	X
cana-1117	82	39	)	)	PUNCT
cana-1117	82	40	]	]	PUNCT
cana-1117	82	41	and	and	CCONJ
cana-1117	82	42	𝑓	𝑓	PROPN
cana-1117	82	43	is	be	AUX
cana-1117	82	44	caputo	caputo	PROPN
cana-1117	82	45	[	[	X
cana-1117	82	46	(	(	PUNCT
cana-1117	82	47	𝑖𝑖)−𝑔𝐻	𝑖𝑖)−𝑔𝐻	NOUN
cana-1117	82	48	]	]	X
cana-1117	82	49	differentiable	differentiable	NOUN
cana-1117	82	50	at	at	ADP
cana-1117	82	51	𝜙0	𝜙0	NOUN
cana-1117	82	52	,	,	PUNCT
cana-1117	82	53	if	if	SCONJ
cana-1117	82	54	(	(	PUNCT
cana-1117	82	55	𝑖𝑖	𝑖𝑖	ADJ
cana-1117	82	56	)	)	PUNCT
cana-1117	82	57	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	82	58	𝐶	𝐶	PROPN
cana-1117	82	59	𝐷𝛼𝑓(𝜙0	𝐷𝛼𝑓(𝜙0	NOUN
cana-1117	82	60	;	;	PUNCT
cana-1117	82	61	𝑟	𝑟	X
cana-1117	82	62	)	)	PUNCT
cana-1117	83	1	=	=	SYM
cana-1117	83	2	[	[	PUNCT
cana-1117	83	3	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	83	4	𝐶	𝐶	PROPN
cana-1117	83	5	𝐷𝛼𝑓(𝜙0	𝐷𝛼𝑓(𝜙0	NOUN
cana-1117	83	6	;	;	PUNCT
cana-1117	83	7	𝑟	𝑟	X
cana-1117	83	8	)	)	PUNCT
cana-1117	83	9	,	,	PUNCT
cana-1117	83	10	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	83	11	𝐶	𝐶	PROPN
cana-1117	83	12	𝐷𝛼𝑓(𝜙0	𝐷𝛼𝑓(𝜙0	NOUN
cana-1117	83	13	;	;	PUNCT
cana-1117	83	14	𝑟	𝑟	X
cana-1117	83	15	)	)	PUNCT
cana-1117	83	16	]	]	PUNCT
cana-1117	83	17	,	,	PUNCT
cana-1117	83	18	for	for	ADP
cana-1117	83	19	0	0	NUM
cana-1117	83	20	≤	≤	NUM
cana-1117	83	21	𝑟	𝑟	NOUN
cana-1117	83	22	≤	≤	NUM
cana-1117	83	23	1	1	NUM
cana-1117	83	24	definition	definition	NOUN
cana-1117	83	25	2.10	2.10	NUM
cana-1117	83	26	the	the	DET
cana-1117	83	27	function	function	NOUN
cana-1117	83	28	𝐸𝛼(𝛼	𝐸𝛼(𝛼	PROPN
cana-1117	83	29	>	>	X
cana-1117	83	30	0	0	X
cana-1117	83	31	)	)	PUNCT
cana-1117	83	32	defined	define	VERB
cana-1117	83	33	by	by	ADP
cana-1117	83	34	𝐸𝛼(𝑧	𝐸𝛼(𝑧	NOUN
cana-1117	83	35	)	)	PUNCT
cana-1117	83	36	=	=	PUNCT
cana-1117	83	37	∑	∑	PUNCT
cana-1117	83	38	∞	∞	PROPN
cana-1117	83	39	𝑘=0	𝑘=0	ADP
cana-1117	83	40	𝑧𝑘	𝑧𝑘	ADP
cana-1117	83	41	γ(𝑘𝛼	γ(𝑘𝛼	NUM
cana-1117	83	42	+	+	NOUN
cana-1117	83	43	1	1	NUM
cana-1117	83	44	)	)	PUNCT
cana-1117	83	45	is	be	AUX
cana-1117	83	46	called	call	VERB
cana-1117	83	47	mittag	mittag	ADJ
cana-1117	83	48	-	-	PUNCT
cana-1117	83	49	leffler	leffler	NOUN
cana-1117	83	50	function	function	NOUN
cana-1117	83	51	of	of	ADP
cana-1117	83	52	order	order	NOUN
cana-1117	83	53	𝛼.	𝛼.	PROPN
cana-1117	83	54	lemma	lemma	PROPN
cana-1117	83	55	2.1	2.1	NUM
cana-1117	83	56	suppose	suppose	VERB
cana-1117	83	57	that	that	SCONJ
cana-1117	83	58	𝑓	𝑓	X
cana-1117	83	59	:	:	PUNCT
cana-1117	83	60	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	83	61	→	→	SYM
cana-1117	83	62	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	83	63	be	be	AUX
cana-1117	83	64	a	a	DET
cana-1117	83	65	fuzzy	fuzzy	ADJ
cana-1117	83	66	valued	value	VERB
cana-1117	83	67	function	function	NOUN
cana-1117	83	68	and	and	CCONJ
cana-1117	83	69	𝑓	𝑓	PRON
cana-1117	83	70	is	be	AUX
cana-1117	83	71	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	83	72	𝔮-differentiable	𝔮-differentiable	ADJ
cana-1117	83	73	and	and	CCONJ
cana-1117	83	74	𝔮-integrable	𝔮-integrable	ADJ
cana-1117	83	75	in	in	ADP
cana-1117	83	76	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	83	77	,	,	PUNCT
cana-1117	83	78	then	then	ADV
cana-1117	83	79	for	for	ADP
cana-1117	83	80	0	0	NUM
cana-1117	83	81	<	<	X
cana-1117	83	82	𝛼	𝛼	X
cana-1117	83	83	≤	≤	NUM
cana-1117	83	84	1	1	NUM
cana-1117	83	85	,	,	PUNCT
cana-1117	83	86	then	then	ADV
cana-1117	83	87	𝔮	𝔮	X
cana-1117	83	88	𝐹𝐼𝑎	𝐹𝐼𝑎	X
cana-1117	83	89	𝛼(𝔮	𝛼(𝔮	X
cana-1117	83	90	𝐹𝐶𝐷𝛼𝑓(𝜏	𝐹𝐶𝐷𝛼𝑓(𝜏	NOUN
cana-1117	83	91	)	)	PUNCT
cana-1117	83	92	)	)	PUNCT
cana-1117	84	1	=	=	SYM
cana-1117	84	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	84	3	)	)	PUNCT
cana-1117	84	4	⊖	⊖	NOUN
cana-1117	84	5	𝑔𝐻	𝑔𝐻	PUNCT
cana-1117	84	6	𝑓(𝑎	𝑓(𝑎	NOUN
cana-1117	84	7	)	)	PUNCT
cana-1117	84	8	lemma	lemma	PROPN
cana-1117	84	9	2.2	2.2	NUM
cana-1117	84	10	let	let	VERB
cana-1117	84	11	𝑣	𝑣	NOUN
cana-1117	84	12	:	:	PUNCT
cana-1117	85	1	[	[	X
cana-1117	85	2	𝜏0	𝜏0	PROPN
cana-1117	85	3	,	,	PUNCT
cana-1117	85	4	𝑇	𝑇	PROPN
cana-1117	85	5	]	]	PUNCT
cana-1117	85	6	→	→	PUNCT
cana-1117	85	7	[	[	X
cana-1117	85	8	0	0	NUM
cana-1117	85	9	,	,	PUNCT
cana-1117	85	10	+	+	NOUN
cana-1117	85	11	∞	∞	NOUN
cana-1117	85	12	)	)	PUNCT
cana-1117	85	13	be	be	VERB
cana-1117	85	14	a	a	DET
cana-1117	85	15	real	real	ADJ
cana-1117	85	16	function	function	NOUN
cana-1117	85	17	and	and	CCONJ
cana-1117	85	18	𝑤	𝑤	X
cana-1117	85	19	(	(	PUNCT
cana-1117	85	20	.	.	PUNCT
cana-1117	85	21	)	)	PUNCT
cana-1117	86	1	is	be	AUX
cana-1117	86	2	a	a	DET
cana-1117	86	3	non	non	ADJ
cana-1117	86	4	negative	negative	ADJ
cana-1117	86	5	,	,	PUNCT
cana-1117	86	6	locally	locally	ADV
cana-1117	86	7	integrable	integrable	ADJ
cana-1117	86	8	function	function	NOUN
cana-1117	86	9	on	on	ADP
cana-1117	86	10	[	[	X
cana-1117	86	11	𝜏0	𝜏0	PROPN
cana-1117	86	12	,	,	PUNCT
cana-1117	86	13	𝑇	𝑇	PROPN
cana-1117	86	14	]	]	PUNCT
cana-1117	86	15	.	.	PUNCT
cana-1117	87	1	assume	assume	VERB
cana-1117	87	2	that	that	SCONJ
cana-1117	87	3	there	there	PRON
cana-1117	87	4	is	be	VERB
cana-1117	87	5	a	a	DET
cana-1117	87	6	constant	constant	ADJ
cana-1117	87	7	𝛼	𝛼	NOUN
cana-1117	87	8	such	such	ADJ
cana-1117	87	9	that	that	SCONJ
cana-1117	87	10	0	0	NUM
cana-1117	87	11	<	<	X
cana-1117	87	12	𝛼	𝛼	X
cana-1117	87	13	≤	≤	NUM
cana-1117	87	14	1	1	NUM
cana-1117	87	15	.	.	PUNCT
cana-1117	88	1	𝑣(𝜏	𝑣(𝜏	NOUN
cana-1117	88	2	)	)	PUNCT
cana-1117	88	3	≤	≤	NOUN
cana-1117	88	4	𝑤(𝜏	𝑤(𝜏	PROPN
cana-1117	88	5	)	)	PUNCT
cana-1117	89	1	+	+	CCONJ
cana-1117	90	1	𝑎	𝑎	PRON
cana-1117	90	2	∫	∫	NOUN
cana-1117	90	3	𝜏	𝜏	X
cana-1117	90	4	0	0	NUM
cana-1117	90	5	(	(	PUNCT
cana-1117	90	6	𝜏	𝜏	NOUN
cana-1117	90	7	−	−	NOUN
cana-1117	90	8	𝑠)−𝛼𝑣(𝑠)𝑑𝑠.	𝑠)−𝛼𝑣(𝑠)𝑑𝑠.	NOUN
cana-1117	90	9	then	then	ADV
cana-1117	90	10	,	,	PUNCT
cana-1117	90	11	there	there	PRON
cana-1117	90	12	exists	exist	VERB
cana-1117	90	13	a	a	DET
cana-1117	90	14	constant	constant	ADJ
cana-1117	90	15	𝐾	𝐾	NOUN
cana-1117	90	16	=	=	SYM
cana-1117	90	17	𝐾(𝛼	𝐾(𝛼	PROPN
cana-1117	90	18	)	)	PUNCT
cana-1117	90	19	such	such	ADJ
cana-1117	90	20	that	that	SCONJ
cana-1117	90	21	𝑣(𝜏	𝑣(𝜏	NOUN
cana-1117	90	22	)	)	PUNCT
cana-1117	90	23	≤	≤	NOUN
cana-1117	90	24	𝑤(𝜏	𝑤(𝜏	PROPN
cana-1117	90	25	)	)	PUNCT
cana-1117	91	1	+	+	CCONJ
cana-1117	91	2	𝐾𝑎	𝐾𝑎	PROPN
cana-1117	91	3	∫	∫	PROPN
cana-1117	91	4	𝜏	𝜏	X
cana-1117	91	5	0	0	NUM
cana-1117	91	6	(	(	PUNCT
cana-1117	91	7	𝜏	𝜏	NOUN
cana-1117	91	8	−	−	X
cana-1117	91	9	𝑠)−𝛼𝑤(𝑠)𝑑𝑠	𝑠)−𝛼𝑤(𝑠)𝑑𝑠	PROPN
cana-1117	91	10	for	for	ADP
cana-1117	91	11	every	every	DET
cana-1117	91	12	𝜏	𝜏	PROPN
cana-1117	91	13	∈	∈	PROPN
cana-1117	91	14	[	[	X
cana-1117	91	15	𝜏0	𝜏0	PROPN
cana-1117	91	16	,	,	PUNCT
cana-1117	91	17	𝑇	𝑇	PROPN
cana-1117	91	18	]	]	PUNCT
cana-1117	91	19	.	.	PUNCT
cana-1117	92	1	we	we	PRON
cana-1117	92	2	consider	consider	VERB
cana-1117	92	3	here	here	ADV
cana-1117	92	4	the	the	DET
cana-1117	92	5	generalized	generalized	ADJ
cana-1117	92	6	banach	banach	ADV
cana-1117	92	7	fixed	fix	VERB
cana-1117	92	8	point	point	NOUN
cana-1117	92	9	theorem	theorem	VERB
cana-1117	92	10	,	,	PUNCT
cana-1117	92	11	which	which	PRON
cana-1117	92	12	is	be	AUX
cana-1117	92	13	used	use	VERB
cana-1117	92	14	to	to	PART
cana-1117	92	15	prove	prove	VERB
cana-1117	92	16	the	the	DET
cana-1117	92	17	existence	existence	NOUN
cana-1117	92	18	results	result	NOUN
cana-1117	92	19	.	.	PUNCT
cana-1117	93	1	theorem	theorem	VERB
cana-1117	93	2	2.3	2.3	NUM
cana-1117	93	3	let	let	VERB
cana-1117	93	4	𝑈	𝑈	PROPN
cana-1117	93	5	be	be	AUX
cana-1117	93	6	a	a	DET
cana-1117	93	7	nonempty	nonempty	ADJ
cana-1117	93	8	closed	close	VERB
cana-1117	93	9	subset	subset	NOUN
cana-1117	93	10	of	of	ADP
cana-1117	93	11	a	a	DET
cana-1117	93	12	banach	banach	NOUN
cana-1117	93	13	space	space	NOUN
cana-1117	93	14	𝐵	𝐵	NOUN
cana-1117	93	15	,	,	PUNCT
cana-1117	93	16	and	and	CCONJ
cana-1117	93	17	let	let	VERB
cana-1117	93	18	𝛼𝑛	𝛼𝑛	PROPN
cana-1117	93	19	≥	≥	PRON
cana-1117	93	20	0	0	NUM
cana-1117	93	21	,	,	PUNCT
cana-1117	93	22	𝑛	𝑛	DET
cana-1117	93	23	∈	∈	PROPN
cana-1117	93	24	ℕ	ℕ	PROPN
cana-1117	93	25	∪	∪	X
cana-1117	93	26	{	{	PUNCT
cana-1117	93	27	0	0	NUM
cana-1117	93	28	}	}	PUNCT
cana-1117	93	29	,	,	PUNCT
cana-1117	93	30	be	be	AUX
cana-1117	93	31	a	a	DET
cana-1117	93	32	sequence	sequence	NOUN
cana-1117	93	33	such	such	ADJ
cana-1117	93	34	that	that	DET
cana-1117	93	35	∑∞	∑∞	NOUN
cana-1117	93	36	𝑛=0	𝑛=0	PROPN
cana-1117	93	37	𝛼𝑛	𝛼𝑛	PROPN
cana-1117	93	38	converges	converge	NOUN
cana-1117	93	39	.	.	PUNCT
cana-1117	94	1	moreover	moreover	ADV
cana-1117	94	2	,	,	PUNCT
cana-1117	94	3	let	let	VERB
cana-1117	94	4	the	the	DET
cana-1117	94	5	mapping	mapping	NOUN
cana-1117	94	6	𝐹	𝐹	PROPN
cana-1117	94	7	:	:	PUNCT
cana-1117	94	8	𝑈	𝑈	PROPN
cana-1117	94	9	→	→	SYM
cana-1117	94	10	𝑈	𝑈	PROPN
cana-1117	94	11	satisfy	satisfy	VERB
cana-1117	94	12	the	the	DET
cana-1117	94	13	inequality	inequality	NOUN
cana-1117	94	14	‖𝐹𝑛𝑢	‖𝐹𝑛𝑢	PROPN
cana-1117	94	15	−	−	PROPN
cana-1117	94	16	𝐹𝑛𝑣‖	𝐹𝑛𝑣‖	PROPN
cana-1117	94	17	≤	≤	ADJ
cana-1117	94	18	𝛼𝑛‖𝑢	𝛼𝑛‖𝑢	NOUN
cana-1117	94	19	−	−	NOUN
cana-1117	94	20	𝑣‖	𝑣‖	NOUN
cana-1117	94	21	∀	∀	NOUN
cana-1117	94	22	𝑛	𝑛	DET
cana-1117	94	23	∈	∈	PROPN
cana-1117	94	24	ℕ	ℕ	PROPN
cana-1117	94	25	∪	∪	X
cana-1117	94	26	{	{	PUNCT
cana-1117	94	27	0	0	NUM
cana-1117	94	28	}	}	PUNCT
cana-1117	94	29	,	,	PUNCT
cana-1117	94	30	and	and	CCONJ
cana-1117	94	31	for	for	ADP
cana-1117	94	32	every	every	DET
cana-1117	94	33	𝑢	𝑢	PROPN
cana-1117	94	34	,	,	PUNCT
cana-1117	94	35	𝑣	𝑣	DET
cana-1117	94	36	∈	∈	PROPN
cana-1117	94	37	𝑈.	𝑈.	PROPN
cana-1117	94	38	then	then	ADV
cana-1117	94	39	𝐹	𝐹	PROPN
cana-1117	94	40	has	have	VERB
cana-1117	94	41	a	a	DET
cana-1117	94	42	uniquely	uniquely	ADV
cana-1117	94	43	defined	define	VERB
cana-1117	94	44	fixed	fix	VERB
cana-1117	94	45	point	point	NOUN
cana-1117	94	46	𝑢∗.furthermore	𝑢∗.furthermore	PROPN
cana-1117	94	47	,	,	PUNCT
cana-1117	94	48	the	the	DET
cana-1117	94	49	sequence	sequence	NOUN
cana-1117	94	50	{	{	PUNCT
cana-1117	94	51	𝐹𝑛𝑢0}𝑛=1	𝐹𝑛𝑢0}𝑛=1	NUM
cana-1117	94	52	∞	∞	NUM
cana-1117	94	53	converges	converge	NOUN
cana-1117	94	54	to	to	ADP
cana-1117	94	55	the	the	DET
cana-1117	94	56	fixed	fix	VERB
cana-1117	94	57	point	point	NOUN
cana-1117	94	58	𝑢∗	𝑢∗	NOUN
cana-1117	94	59	for	for	ADP
cana-1117	94	60	every	every	DET
cana-1117	94	61	𝑢0	𝑢0	PROPN
cana-1117	94	62	∈	∈	PROPN
cana-1117	94	63	𝑈.	𝑈.	PROPN
cana-1117	94	64	3	3	NUM
cana-1117	94	65	results	result	NOUN
cana-1117	94	66	on	on	ADP
cana-1117	94	67	fuzzy	fuzzy	ADJ
cana-1117	94	68	q	q	ADJ
cana-1117	94	69	-	-	PUNCT
cana-1117	94	70	fractional	fractional	ADJ
cana-1117	94	71	derivative	derivative	NOUN
cana-1117	94	72	by	by	ADP
cana-1117	94	73	𝒈𝑯	𝒈𝑯	ADJ
cana-1117	94	74	difference	difference	NOUN
cana-1117	94	75	let	let	VERB
cana-1117	94	76	𝑋	𝑋	NOUN
cana-1117	94	77	be	be	AUX
cana-1117	94	78	a	a	DET
cana-1117	94	79	nonempty	nonempty	ADJ
cana-1117	94	80	closed	close	VERB
cana-1117	94	81	subset	subset	NOUN
cana-1117	94	82	of	of	ADP
cana-1117	94	83	a	a	DET
cana-1117	94	84	ℝ.	ℝ.	PROPN
cana-1117	94	85	consider	consider	VERB
cana-1117	94	86	an	an	DET
cana-1117	94	87	arbitrary	arbitrary	ADJ
cana-1117	94	88	fuzzy	fuzzy	ADJ
cana-1117	94	89	valued	value	VERB
cana-1117	94	90	function	function	NOUN
cana-1117	94	91	ℎ	ℎ	PROPN
cana-1117	94	92	:	:	PUNCT
cana-1117	94	93	𝑋	𝑋	PROPN
cana-1117	94	94	→	→	SYM
cana-1117	94	95	ℝ𝐹	ℝ𝐹	PROPN
cana-1117	94	96	,	,	PUNCT
cana-1117	94	97	where	where	SCONJ
cana-1117	94	98	ℝ	ℝ	PROPN
cana-1117	94	99	is	be	AUX
cana-1117	94	100	the	the	DET
cana-1117	94	101	set	set	NOUN
cana-1117	94	102	of	of	ADP
cana-1117	94	103	all	all	DET
cana-1117	94	104	real	real	ADJ
cana-1117	94	105	numbers	number	NOUN
cana-1117	94	106	,	,	PUNCT
cana-1117	94	107	and	and	CCONJ
cana-1117	94	108	[	[	X
cana-1117	94	109	ℎ(𝜏)]𝑟	ℎ(𝜏)]𝑟	NOUN
cana-1117	94	110	=	=	PUNCT
cana-1117	95	1	[	[	X
cana-1117	95	2	ℎ(𝜏	ℎ(𝜏	NOUN
cana-1117	95	3	;	;	PUNCT
cana-1117	95	4	𝑟	𝑟	NOUN
cana-1117	95	5	)	)	PUNCT
cana-1117	95	6	,	,	PUNCT
cana-1117	95	7	ℎ(𝜏	ℎ(𝜏	PROPN
cana-1117	95	8	;	;	PUNCT
cana-1117	95	9	𝑟	𝑟	NOUN
cana-1117	95	10	)	)	PUNCT
cana-1117	95	11	]	]	PUNCT
cana-1117	95	12	is	be	AUX
cana-1117	95	13	called	call	VERB
cana-1117	95	14	as	as	ADP
cana-1117	95	15	𝑟	𝑟	NOUN
cana-1117	95	16	-cut	-cut	ADJ
cana-1117	95	17	or	or	CCONJ
cana-1117	95	18	parametric	parametric	ADJ
cana-1117	95	19	form	form	NOUN
cana-1117	95	20	of	of	ADP
cana-1117	95	21	the	the	DET
cana-1117	95	22	fuzzy	fuzzy	ADJ
cana-1117	95	23	valued	value	VERB
cana-1117	95	24	function	function	NOUN
cana-1117	95	25	𝑓.	𝑓.	NOUN
cana-1117	95	26	communications	communication	NOUN
cana-1117	95	27	on	on	ADP
cana-1117	95	28	applied	apply	VERB
cana-1117	95	29	nonlinear	nonlinear	ADJ
cana-1117	95	30	analysis	analysis	NOUN
cana-1117	95	31	issn	issn	NOUN
cana-1117	95	32	:	:	PUNCT
cana-1117	95	33	1074	1074	NUM
cana-1117	95	34	-	-	PUNCT
cana-1117	95	35	133x	133x	NUM
cana-1117	95	36	vol	vol	NOUN
cana-1117	95	37	31	31	NUM
cana-1117	95	38	no	no	NOUN
cana-1117	95	39	.	.	PUNCT
cana-1117	96	1	6s	6s	NUM
cana-1117	96	2	(	(	PUNCT
cana-1117	96	3	2024	2024	NUM
cana-1117	96	4	)	)	PUNCT
cana-1117	96	5	40	40	NUM
cana-1117	96	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	97	1	the	the	DET
cana-1117	97	2	results	result	NOUN
cana-1117	97	3	on	on	ADP
cana-1117	97	4	fuzzy	fuzzy	ADJ
cana-1117	97	5	𝔮	𝔮	X
cana-1117	97	6	-fractional	-fractional	ADJ
cana-1117	97	7	derivative	derivative	NOUN
cana-1117	97	8	by	by	ADP
cana-1117	97	9	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	97	10	difference	difference	NOUN
cana-1117	97	11	of	of	ADP
cana-1117	97	12	the	the	DET
cana-1117	97	13	fuzzy	fuzzy	ADJ
cana-1117	97	14	function	function	NOUN
cana-1117	97	15	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	97	16	,	,	PUNCT
cana-1117	97	17	𝑢(𝜏	𝑢(𝜏	PROPN
cana-1117	97	18	)	)	PUNCT
cana-1117	97	19	,	,	PUNCT
cana-1117	97	20	𝒢𝑢(𝜏	𝒢𝑢(𝜏	PROPN
cana-1117	97	21	)	)	PUNCT
cana-1117	97	22	,	,	PUNCT
cana-1117	97	23	𝒮𝑢(𝜏	𝒮𝑢(𝜏	PROPN
cana-1117	97	24	)	)	PUNCT
cana-1117	97	25	)	)	PUNCT
cana-1117	97	26	has	have	AUX
cana-1117	97	27	been	be	AUX
cana-1117	97	28	studied	study	VERB
cana-1117	97	29	by	by	ADP
cana-1117	97	30	z.noeiaghdam	z.noeiaghdam	PROPN
cana-1117	97	31	et.al	et.al	PROPN
cana-1117	97	32	in	in	ADP
cana-1117	97	33	[	[	X
cana-1117	97	34	?	?	PUNCT
cana-1117	97	35	]	]	X
cana-1117	97	36	.	.	PUNCT
cana-1117	98	1	for	for	ADP
cana-1117	98	2	an	an	DET
cana-1117	98	3	arbitrary	arbitrary	ADJ
cana-1117	98	4	fuzzy	fuzzy	ADJ
cana-1117	98	5	valued	value	VERB
cana-1117	98	6	function	function	NOUN
cana-1117	98	7	𝑓	𝑓	NOUN
cana-1117	98	8	:	:	PUNCT
cana-1117	98	9	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	98	10	→	→	SYM
cana-1117	98	11	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	98	12	,	,	PUNCT
cana-1117	98	13	q	q	NOUN
cana-1117	98	14	-	-	NOUN
cana-1117	98	15	differential	differential	NOUN
cana-1117	98	16	by	by	ADP
cana-1117	98	17	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	98	18	difference	difference	NOUN
cana-1117	98	19	is	be	AUX
cana-1117	98	20	𝑑𝑞,𝑔𝐻	𝑑𝑞,𝑔𝐻	NOUN
cana-1117	98	21	=	=	SYM
cana-1117	98	22	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	98	23	)	)	PUNCT
cana-1117	98	24	⊖	⊖	NOUN
cana-1117	98	25	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	98	26	𝑓(𝔮𝜏	𝑓(𝔮𝜏	NOUN
cana-1117	98	27	)	)	PUNCT
cana-1117	98	28	if	if	SCONJ
cana-1117	98	29	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	98	30	)	)	PUNCT
cana-1117	98	31	⊖	⊖	NOUN
cana-1117	98	32	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	98	33	𝑓(𝔮𝜏	𝑓(𝔮𝜏	NOUN
cana-1117	98	34	)	)	PUNCT
cana-1117	98	35	exists	exist	VERB
cana-1117	98	36	.	.	PUNCT
cana-1117	99	1	then	then	ADV
cana-1117	99	2	the	the	DET
cana-1117	99	3	fuzzy	fuzzy	ADJ
cana-1117	99	4	generalized	generalize	VERB
cana-1117	99	5	hukuhara	hukuhara	ADV
cana-1117	99	6	𝔮	𝔮	X
cana-1117	99	7	derivative	derivative	NOUN
cana-1117	99	8	of	of	ADP
cana-1117	99	9	𝑓	𝑓	PROPN
cana-1117	99	10	is	be	AUX
cana-1117	99	11	defined	define	VERB
cana-1117	99	12	by	by	ADP
cana-1117	99	13	𝐷𝐹,𝑞𝑓(𝜏	𝐷𝐹,𝑞𝑓(𝜏	PROPN
cana-1117	99	14	)	)	PUNCT
cana-1117	100	1	=	=	X
cana-1117	100	2	𝑑𝑞,𝑔𝐻	𝑑𝑞,𝑔𝐻	NOUN
cana-1117	100	3	𝑑𝑞𝜏	𝑑𝑞𝜏	NOUN
cana-1117	100	4	=	=	SYM
cana-1117	100	5	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	100	6	)	)	PUNCT
cana-1117	100	7	⊖	⊖	NOUN
cana-1117	100	8	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	100	9	𝑓(𝔮𝜏	𝑓(𝔮𝜏	NOUN
cana-1117	100	10	)	)	PUNCT
cana-1117	100	11	(	(	PUNCT
cana-1117	100	12	1	1	NUM
cana-1117	100	13	−	−	NOUN
cana-1117	100	14	𝔮)𝜏	𝔮)𝜏	NOUN
cana-1117	100	15	,	,	PUNCT
cana-1117	100	16	𝜏	𝜏	X
cana-1117	100	17	∈	∈	NOUN
cana-1117	100	18	𝕋𝔮	𝕋𝔮	VERB
cana-1117	100	19	−	−	PROPN
cana-1117	100	20	0	0	SYM
cana-1117	100	21	lemma	lemma	PROPN
cana-1117	100	22	3.1	3.1	NUM
cana-1117	100	23	the	the	DET
cana-1117	100	24	function	function	NOUN
cana-1117	100	25	𝑓	𝑓	NOUN
cana-1117	100	26	is	be	AUX
cana-1117	100	27	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	100	28	differentiable	differentiable	ADJ
cana-1117	100	29	if	if	SCONJ
cana-1117	100	30	and	and	CCONJ
cana-1117	100	31	only	only	ADV
cana-1117	100	32	if	if	SCONJ
cana-1117	100	33	,	,	PUNCT
cana-1117	100	34	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	100	35	,	,	PUNCT
cana-1117	100	36	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	100	37	)	)	PUNCT
cana-1117	100	38	,	,	PUNCT
cana-1117	100	39	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	100	40	)	)	PUNCT
cana-1117	100	41	,	,	PUNCT
cana-1117	100	42	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	100	43	)	)	PUNCT
cana-1117	100	44	;	;	PUNCT
cana-1117	100	45	𝑟	𝑟	X
cana-1117	100	46	)	)	PUNCT
cana-1117	100	47	and	and	CCONJ
cana-1117	100	48	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	100	49	,	,	PUNCT
cana-1117	100	50	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	100	51	)	)	PUNCT
cana-1117	100	52	,	,	PUNCT
cana-1117	100	53	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	100	54	)	)	PUNCT
cana-1117	100	55	,	,	PUNCT
cana-1117	100	56	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	100	57	)	)	PUNCT
cana-1117	100	58	)	)	PUNCT
cana-1117	100	59	are	be	AUX
cana-1117	100	60	differentiable	differentiable	ADJ
cana-1117	100	61	with	with	ADP
cana-1117	100	62	respect	respect	NOUN
cana-1117	100	63	to	to	ADP
cana-1117	100	64	𝜏	𝜏	PRON
cana-1117	100	65	for	for	ADP
cana-1117	100	66	all	all	DET
cana-1117	100	67	𝑟	𝑟	PRON
cana-1117	100	68	∈	∈	NOUN
cana-1117	101	1	[	[	X
cana-1117	101	2	0,1	0,1	NUM
cana-1117	101	3	]	]	PUNCT
cana-1117	101	4	and	and	CCONJ
cana-1117	101	5	𝑓′	𝑓′	NOUN
cana-1117	101	6	𝑔𝐻	𝑔𝐻	X
cana-1117	101	7	(	(	PUNCT
cana-1117	101	8	𝜏	𝜏	NOUN
cana-1117	101	9	,	,	PUNCT
cana-1117	101	10	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	101	11	)	)	PUNCT
cana-1117	101	12	,	,	PUNCT
cana-1117	101	13	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	101	14	)	)	PUNCT
cana-1117	101	15	,	,	PUNCT
cana-1117	101	16	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	101	17	)	)	PUNCT
cana-1117	101	18	;	;	PUNCT
cana-1117	101	19	𝑟	𝑟	X
cana-1117	101	20	)	)	PUNCT
cana-1117	101	21	=	=	PUNCT
cana-1117	102	1	[	[	X
cana-1117	102	2	min	min	NOUN
cana-1117	102	3	(	(	PUNCT
cana-1117	102	4	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	102	5	,	,	PUNCT
cana-1117	102	6	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	102	7	)	)	PUNCT
cana-1117	102	8	,	,	PUNCT
cana-1117	102	9	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	102	10	)	)	PUNCT
cana-1117	102	11	,	,	PUNCT
cana-1117	102	12	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	102	13	)	)	PUNCT
cana-1117	102	14	;	;	PUNCT
cana-1117	102	15	𝑟	𝑟	X
cana-1117	102	16	)	)	PUNCT
cana-1117	102	17	,	,	PUNCT
cana-1117	102	18	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	102	19	,	,	PUNCT
cana-1117	102	20	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	102	21	)	)	PUNCT
cana-1117	102	22	,	,	PUNCT
cana-1117	102	23	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	102	24	)	)	PUNCT
cana-1117	102	25	,	,	PUNCT
cana-1117	102	26	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	102	27	)	)	PUNCT
cana-1117	102	28	;	;	PUNCT
cana-1117	102	29	𝑟	𝑟	X
cana-1117	102	30	)	)	PUNCT
cana-1117	102	31	)	)	PUNCT
cana-1117	102	32	,	,	PUNCT
cana-1117	102	33	max	max	PROPN
cana-1117	102	34	(	(	PUNCT
cana-1117	102	35	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	102	36	,	,	PUNCT
cana-1117	102	37	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	102	38	)	)	PUNCT
cana-1117	102	39	,	,	PUNCT
cana-1117	102	40	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	102	41	)	)	PUNCT
cana-1117	102	42	,	,	PUNCT
cana-1117	102	43	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	102	44	)	)	PUNCT
cana-1117	102	45	;	;	PUNCT
cana-1117	102	46	𝑟	𝑟	X
cana-1117	102	47	)	)	PUNCT
cana-1117	102	48	,	,	PUNCT
cana-1117	102	49	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	102	50	,	,	PUNCT
cana-1117	102	51	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	102	52	)	)	PUNCT
cana-1117	102	53	,	,	PUNCT
cana-1117	102	54	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	102	55	)	)	PUNCT
cana-1117	102	56	,	,	PUNCT
cana-1117	102	57	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	102	58	)	)	PUNCT
cana-1117	102	59	;	;	PUNCT
cana-1117	102	60	𝑟	𝑟	X
cana-1117	102	61	)	)	PUNCT
cana-1117	102	62	)	)	PUNCT
cana-1117	102	63	]	]	PUNCT
cana-1117	102	64	theorem	theorem	VERB
cana-1117	102	65	3.2	3.2	NUM
cana-1117	102	66	let	let	VERB
cana-1117	102	67	𝑓	𝑓	PRON
cana-1117	102	68	:	:	PUNCT
cana-1117	102	69	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	102	70	×	×	PROPN
cana-1117	102	71	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	102	72	×	×	PROPN
cana-1117	102	73	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	102	74	×	×	PROPN
cana-1117	102	75	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	102	76	→	→	SYM
cana-1117	102	77	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	102	78	be	be	AUX
cana-1117	102	79	a	a	DET
cana-1117	102	80	fuzzy	fuzzy	ADJ
cana-1117	102	81	valued	value	VERB
cana-1117	102	82	function	function	NOUN
cana-1117	102	83	on	on	ADP
cana-1117	102	84	𝕁.	𝕁.	PROPN
cana-1117	102	85	(	(	PUNCT
cana-1117	102	86	a	a	NOUN
cana-1117	102	87	)	)	PUNCT
cana-1117	102	88	if	if	SCONJ
cana-1117	102	89	𝑓	𝑓	PRON
cana-1117	102	90	is	be	AUX
cana-1117	102	91	[	[	PUNCT
cana-1117	102	92	(	(	PUNCT
cana-1117	102	93	𝑖)−	𝑖)−	PROPN
cana-1117	102	94	𝑔𝐻	𝑔𝐻	NOUN
cana-1117	102	95	]	]	X
cana-1117	102	96	differentiable	differentiable	NOUN
cana-1117	102	97	at	at	ADP
cana-1117	102	98	𝜙0	𝜙0	NOUN
cana-1117	102	99	∈	∈	PROPN
cana-1117	102	100	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	102	101	,	,	PUNCT
cana-1117	102	102	then	then	ADV
cana-1117	102	103	𝑓	𝑓	PROPN
cana-1117	102	104	is	be	AUX
cana-1117	102	105	caputo	caputo	PROPN
cana-1117	102	106	[	[	X
cana-1117	102	107	(	(	PUNCT
cana-1117	102	108	𝑖)−	𝑖)−	PROPN
cana-1117	102	109	𝑔𝐻	𝑔𝐻	NOUN
cana-1117	102	110	]	]	X
cana-1117	102	111	differentiable	differentiable	NOUN
cana-1117	102	112	at	at	ADP
cana-1117	102	113	𝜙0	𝜙0	PROPN
cana-1117	102	114	(	(	PUNCT
cana-1117	102	115	b	b	NOUN
cana-1117	102	116	)	)	PUNCT
cana-1117	102	117	if	if	SCONJ
cana-1117	102	118	𝑓	𝑓	PRON
cana-1117	102	119	is	be	AUX
cana-1117	102	120	[	[	X
cana-1117	102	121	(	(	PUNCT
cana-1117	102	122	𝑖𝑖)−	𝑖𝑖)−	PUNCT
cana-1117	102	123	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	102	124	]	]	X
cana-1117	102	125	differentiable	differentiable	NOUN
cana-1117	102	126	at	at	ADP
cana-1117	102	127	𝜙0	𝜙0	NOUN
cana-1117	102	128	∈	∈	PROPN
cana-1117	102	129	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	102	130	,	,	PUNCT
cana-1117	102	131	then	then	ADV
cana-1117	102	132	𝑓	𝑓	PROPN
cana-1117	102	133	is	be	AUX
cana-1117	102	134	caputo	caputo	PROPN
cana-1117	102	135	[	[	X
cana-1117	102	136	(	(	PUNCT
cana-1117	102	137	𝑖𝑖)−	𝑖𝑖)−	PUNCT
cana-1117	102	138	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	102	139	]	]	X
cana-1117	102	140	differentiable	differentiable	NOUN
cana-1117	102	141	at	at	ADP
cana-1117	102	142	𝜙0	𝜙0	NOUN
cana-1117	102	143	proof	proof	NOUN
cana-1117	102	144	.	.	PUNCT
cana-1117	103	1	c	c	NOUN
cana-1117	103	2	𝐷𝛼(𝑓(𝜏0	𝐷𝛼(𝑓(𝜏0	NOUN
cana-1117	103	3	,	,	PUNCT
cana-1117	103	4	𝜙(𝜏0	𝜙(𝜏0	NOUN
cana-1117	103	5	)	)	PUNCT
cana-1117	103	6	,	,	PUNCT
cana-1117	103	7	𝒢𝜙(𝜏0	𝒢𝜙(𝜏0	PROPN
cana-1117	103	8	)	)	PUNCT
cana-1117	103	9	,	,	PUNCT
cana-1117	103	10	𝒮𝜙(𝜏0	𝒮𝜙(𝜏0	PROPN
cana-1117	103	11	)	)	PUNCT
cana-1117	103	12	)	)	PUNCT
cana-1117	103	13	;	;	PUNCT
cana-1117	103	14	𝑟	𝑟	X
cana-1117	103	15	)	)	PUNCT
cana-1117	103	16	=	=	SYM
cana-1117	104	1	𝐼1−𝛼(𝑓′)(𝜏0	𝐼1−𝛼(𝑓′)(𝜏0	PROPN
cana-1117	104	2	,	,	PUNCT
cana-1117	104	3	𝜙(𝜏0	𝜙(𝜏0	NOUN
cana-1117	104	4	)	)	PUNCT
cana-1117	104	5	,	,	PUNCT
cana-1117	104	6	𝒢𝜙(𝜏0	𝒢𝜙(𝜏0	PROPN
cana-1117	104	7	)	)	PUNCT
cana-1117	104	8	,	,	PUNCT
cana-1117	104	9	𝒮𝜙(𝜏0	𝒮𝜙(𝜏0	PROPN
cana-1117	104	10	)	)	PUNCT
cana-1117	104	11	)	)	PUNCT
cana-1117	104	12	;	;	PUNCT
cana-1117	104	13	𝑟	𝑟	X
cana-1117	104	14	)	)	PUNCT
cana-1117	104	15	=	=	PUNCT
cana-1117	105	1	[	[	PUNCT
cana-1117	105	2	1	1	NUM
cana-1117	105	3	γ(1−𝛼	γ(1−𝛼	NOUN
cana-1117	105	4	)	)	PUNCT
cana-1117	105	5	∫	∫	PROPN
cana-1117	106	1	𝑓′(𝜏0,𝜙(𝜏0),𝒢𝜙(𝜏0),𝒮𝜙(𝜏0));𝑟	𝑓′(𝜏0,𝜙(𝜏0),𝒢𝜙(𝜏0),𝒮𝜙(𝜏0));𝑟	PROPN
cana-1117	106	2	)	)	PUNCT
cana-1117	106	3	(	(	PUNCT
cana-1117	106	4	𝜏0−𝑠)𝛼	𝜏0−𝑠)𝛼	NOUN
cana-1117	106	5	𝑑𝑠	𝑑𝑠	NOUN
cana-1117	106	6	,	,	PUNCT
cana-1117	106	7	1	1	NUM
cana-1117	106	8	γ(1−𝛼	γ(1−𝛼	NOUN
cana-1117	106	9	)	)	PUNCT
cana-1117	106	10	∫	∫	PROPN
cana-1117	107	1	𝑓′(𝜏0,𝜙(𝜏0),𝒢𝜙(𝜏0),𝒮𝜙(𝜏0));𝑟	𝑓′(𝜏0,𝜙(𝜏0),𝒢𝜙(𝜏0),𝒮𝜙(𝜏0));𝑟	PROPN
cana-1117	107	2	)	)	PUNCT
cana-1117	107	3	(	(	PUNCT
cana-1117	107	4	𝜏0−𝑠)𝛼	𝜏0−𝑠)𝛼	NOUN
cana-1117	107	5	𝑑𝑠	𝑑𝑠	X
cana-1117	107	6	]	]	X
cana-1117	107	7	=	=	PUNCT
cana-1117	107	8	[	[	PUNCT
cana-1117	107	9	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	107	10	𝐶	𝐶	PROPN
cana-1117	107	11	𝐷𝛼𝑓(𝜏0	𝐷𝛼𝑓(𝜏0	NOUN
cana-1117	107	12	,	,	PUNCT
cana-1117	107	13	𝜙(𝜏0	𝜙(𝜏0	NOUN
cana-1117	107	14	)	)	PUNCT
cana-1117	107	15	,	,	PUNCT
cana-1117	107	16	𝒢𝜙(𝜏0	𝒢𝜙(𝜏0	PROPN
cana-1117	107	17	)	)	PUNCT
cana-1117	107	18	,	,	PUNCT
cana-1117	107	19	𝒮𝜙(𝜏0	𝒮𝜙(𝜏0	PROPN
cana-1117	107	20	)	)	PUNCT
cana-1117	107	21	)	)	PUNCT
cana-1117	107	22	,	,	PUNCT
cana-1117	107	23	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	107	24	𝐶	𝐶	PROPN
cana-1117	107	25	𝐷𝛼𝑓(𝜏0	𝐷𝛼𝑓(𝜏0	NOUN
cana-1117	107	26	,	,	PUNCT
cana-1117	107	27	𝜙(𝜏0	𝜙(𝜏0	NOUN
cana-1117	107	28	)	)	PUNCT
cana-1117	107	29	,	,	PUNCT
cana-1117	107	30	𝒢	𝒢	PROPN
cana-1117	107	31	𝜙(𝜏0	𝜙(𝜏0	NOUN
cana-1117	107	32	)	)	PUNCT
cana-1117	107	33	,	,	PUNCT
cana-1117	107	34	𝒮𝜙(𝜏0	𝒮𝜙(𝜏0	PROPN
cana-1117	107	35	)	)	PUNCT
cana-1117	107	36	)	)	PUNCT
cana-1117	107	37	]	]	PUNCT
cana-1117	107	38	,	,	PUNCT
cana-1117	107	39	0	0	NUM
cana-1117	107	40	≤	≤	NUM
cana-1117	107	41	𝑟	𝑟	X
cana-1117	107	42	≤	≤	NUM
cana-1117	107	43	1	1	NUM
cana-1117	107	44	therefore	therefore	ADV
cana-1117	107	45	,	,	PUNCT
cana-1117	107	46	𝑓	𝑓	PRON
cana-1117	107	47	is	be	AUX
cana-1117	107	48	[	[	PUNCT
cana-1117	107	49	(	(	PUNCT
cana-1117	107	50	𝑖)−	𝑖)−	PROPN
cana-1117	107	51	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	107	52	]	]	X
cana-1117	107	53	caputo	caputo	PROPN
cana-1117	107	54	differentiable	differentiable	PROPN
cana-1117	107	55	.	.	PUNCT
cana-1117	108	1	similarly	similarly	ADV
cana-1117	108	2	we	we	PRON
cana-1117	108	3	can	can	AUX
cana-1117	108	4	prove	prove	VERB
cana-1117	108	5	that	that	SCONJ
cana-1117	108	6	𝑓	𝑓	PRON
cana-1117	108	7	is	be	AUX
cana-1117	108	8	[	[	X
cana-1117	108	9	(	(	PUNCT
cana-1117	108	10	𝑖𝑖)−	𝑖𝑖)−	PUNCT
cana-1117	108	11	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	108	12	]	]	X
cana-1117	108	13	caputo	caputo	PROPN
cana-1117	108	14	differentiable	differentiable	PROPN
cana-1117	108	15	.	.	PUNCT
cana-1117	109	1	4	4	NUM
cana-1117	109	2	existence	existence	NOUN
cana-1117	109	3	result	result	VERB
cana-1117	109	4	for	for	ADP
cana-1117	109	5	fuzzy	fuzzy	ADJ
cana-1117	109	6	𝖖-fractional	𝖖-fractional	ADJ
cana-1117	109	7	differential	differential	ADJ
cana-1117	109	8	equations	equation	NOUN
cana-1117	109	9	theorem	theorem	VERB
cana-1117	109	10	4.1	4.1	NUM
cana-1117	109	11	if	if	SCONJ
cana-1117	109	12	the	the	DET
cana-1117	109	13	function	function	NOUN
cana-1117	109	14	𝑓	𝑓	X
cana-1117	109	15	:	:	PUNCT
cana-1117	109	16	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	109	17	×	×	PROPN
cana-1117	109	18	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	109	19	×	×	PROPN
cana-1117	109	20	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	109	21	×	×	PROPN
cana-1117	109	22	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	109	23	→	→	SYM
cana-1117	109	24	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	109	25	is	be	AUX
cana-1117	109	26	a	a	DET
cana-1117	109	27	continuous	continuous	ADJ
cana-1117	109	28	fuzzy	fuzzy	ADJ
cana-1117	109	29	valued	value	VERB
cana-1117	109	30	function	function	NOUN
cana-1117	109	31	,	,	PUNCT
cana-1117	109	32	then	then	ADV
cana-1117	109	33	the	the	DET
cana-1117	109	34	given	give	VERB
cana-1117	109	35	equation	equation	NOUN
cana-1117	109	36	(	(	PUNCT
cana-1117	109	37	1.1	1.1	NUM
cana-1117	109	38	)	)	PUNCT
cana-1117	109	39	with	with	ADP
cana-1117	109	40	the	the	DET
cana-1117	109	41	initial	initial	ADJ
cana-1117	109	42	condition	condition	NOUN
cana-1117	109	43	(	(	PUNCT
cana-1117	109	44	1.2	1.2	NUM
cana-1117	109	45	)	)	PUNCT
cana-1117	109	46	is	be	AUX
cana-1117	109	47	equivalent	equivalent	ADJ
cana-1117	109	48	to	to	ADP
cana-1117	109	49	the	the	DET
cana-1117	109	50	following	follow	VERB
cana-1117	109	51	integral	integral	ADJ
cana-1117	109	52	equation	equation	NOUN
cana-1117	109	53	communications	communication	NOUN
cana-1117	109	54	on	on	ADP
cana-1117	109	55	applied	apply	VERB
cana-1117	109	56	nonlinear	nonlinear	ADJ
cana-1117	109	57	analysis	analysis	NOUN
cana-1117	109	58	issn	issn	NOUN
cana-1117	109	59	:	:	PUNCT
cana-1117	109	60	1074	1074	NUM
cana-1117	109	61	-	-	PUNCT
cana-1117	109	62	133x	133x	NUM
cana-1117	109	63	vol	vol	NOUN
cana-1117	109	64	31	31	NUM
cana-1117	109	65	no	no	NOUN
cana-1117	109	66	.	.	PUNCT
cana-1117	110	1	6s	6s	NUM
cana-1117	110	2	(	(	PUNCT
cana-1117	110	3	2024	2024	NUM
cana-1117	110	4	)	)	PUNCT
cana-1117	110	5	41	41	NUM
cana-1117	110	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	110	7	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	110	8	)	)	PUNCT
cana-1117	110	9	⊖	⊖	NOUN
cana-1117	111	1	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	111	2	𝜙0	𝜙0	NOUN
cana-1117	111	3	=	=	SYM
cana-1117	111	4	𝜆	𝜆	PRON
cana-1117	111	5	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	111	6	)	)	PUNCT
cana-1117	111	7	⊙	⊙	PROPN
cana-1117	111	8	∫	∫	PROPN
cana-1117	112	1	𝜏	𝜏	PROPN
cana-1117	112	2	𝜏0	𝜏0	PROPN
cana-1117	112	3	(	(	PUNCT
cana-1117	112	4	𝜏	𝜏	NOUN
cana-1117	112	5	−	−	NOUN
cana-1117	112	6	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	112	7	+	+	CCONJ
cana-1117	112	8	1	1	NUM
cana-1117	112	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	112	10	)	)	PUNCT
cana-1117	112	11	⊙	⊙	PROPN
cana-1117	112	12	∫	∫	PROPN
cana-1117	113	1	𝜏	𝜏	PROPN
cana-1117	113	2	𝜏0	𝜏0	PROPN
cana-1117	113	3	(	(	PUNCT
cana-1117	113	4	𝜏	𝜏	NOUN
cana-1117	113	5	−	−	NOUN
cana-1117	113	6	𝔮𝑠)𝛼−1𝑓(𝑠	𝔮𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	113	7	,	,	PUNCT
cana-1117	113	8	𝜙(𝑠	𝜙(𝑠	PROPN
cana-1117	113	9	)	)	PUNCT
cana-1117	113	10	,	,	PUNCT
cana-1117	113	11	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	113	12	)	)	PUNCT
cana-1117	113	13	,	,	PUNCT
cana-1117	113	14	𝒮𝜙(𝑠))𝑑𝔮𝑠	𝒮𝜙(𝑠))𝑑𝔮𝑠	NOUN
cana-1117	113	15	(	(	PUNCT
cana-1117	113	16	4.1	4.1	NUM
cana-1117	113	17	)	)	PUNCT
cana-1117	113	18	proof	proof	NOUN
cana-1117	113	19	.	.	PUNCT
cana-1117	114	1	let	let	AUX
cana-1117	114	2	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	114	3	)	)	PUNCT
cana-1117	114	4	be	be	AUX
cana-1117	114	5	a	a	DET
cana-1117	114	6	solution	solution	NOUN
cana-1117	114	7	of	of	ADP
cana-1117	114	8	(	(	PUNCT
cana-1117	114	9	1.1	1.1	NUM
cana-1117	114	10	)	)	PUNCT
cana-1117	114	11	.	.	PUNCT
cana-1117	115	1	we	we	PRON
cana-1117	115	2	claim	claim	VERB
cana-1117	115	3	that	that	SCONJ
cana-1117	115	4	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	115	5	)	)	PUNCT
cana-1117	115	6	is	be	AUX
cana-1117	115	7	also	also	ADV
cana-1117	115	8	a	a	DET
cana-1117	115	9	solution	solution	NOUN
cana-1117	115	10	of	of	ADP
cana-1117	115	11	(	(	PUNCT
cana-1117	115	12	4.1	4.1	NUM
cana-1117	115	13	)	)	PUNCT
cana-1117	115	14	.	.	PUNCT
cana-1117	116	1	to	to	PART
cana-1117	116	2	assure	assure	VERB
cana-1117	116	3	that	that	SCONJ
cana-1117	116	4	,	,	PUNCT
cana-1117	116	5	let	let	VERB
cana-1117	116	6	us	we	PRON
cana-1117	116	7	consider	consider	VERB
cana-1117	116	8	,	,	PUNCT
cana-1117	116	9	𝑟(𝜏	𝑟(𝜏	PROPN
cana-1117	116	10	)	)	PUNCT
cana-1117	116	11	=	=	SYM
cana-1117	116	12	𝜆𝜙(𝜏	𝜆𝜙(𝜏	X
cana-1117	116	13	)	)	PUNCT
cana-1117	117	1	+	+	CCONJ
cana-1117	117	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	117	3	,	,	PUNCT
cana-1117	117	4	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	117	5	)	)	PUNCT
cana-1117	117	6	,	,	PUNCT
cana-1117	117	7	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	117	8	)	)	PUNCT
cana-1117	117	9	,	,	PUNCT
cana-1117	117	10	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	117	11	)	)	PUNCT
cana-1117	117	12	)	)	PUNCT
cana-1117	117	13	.	.	PUNCT
cana-1117	118	1	then	then	ADV
cana-1117	118	2	we	we	PRON
cana-1117	118	3	have	have	VERB
cana-1117	118	4	,	,	PUNCT
cana-1117	118	5	𝑟(𝜏	𝑟(𝜏	NOUN
cana-1117	118	6	)	)	PUNCT
cana-1117	119	1	=	=	PROPN
cana-1117	119	2	𝐶	𝐶	PROPN
cana-1117	119	3	𝐷𝛼𝜙(𝜏	𝐷𝛼𝜙(𝜏	NOUN
cana-1117	119	4	)	)	PUNCT
cana-1117	119	5	.	.	PUNCT
cana-1117	120	1	assume	assume	VERB
cana-1117	120	2	that	that	SCONJ
cana-1117	120	3	here	here	ADV
cana-1117	120	4	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	120	5	)	)	PUNCT
cana-1117	120	6	is	be	AUX
cana-1117	120	7	a	a	DET
cana-1117	120	8	monotone	monotone	ADJ
cana-1117	120	9	function	function	NOUN
cana-1117	120	10	for	for	ADP
cana-1117	120	11	𝜏	𝜏	PROPN
cana-1117	120	12	∈	∈	PROPN
cana-1117	120	13	𝕋𝔮.	𝕋𝔮.	PROPN
cana-1117	120	14	using	use	VERB
cana-1117	120	15	lemma	lemma	PROPN
cana-1117	120	16	(	(	PUNCT
cana-1117	120	17	2.1	2.1	NUM
cana-1117	120	18	)	)	PUNCT
cana-1117	120	19	,	,	PUNCT
cana-1117	120	20	we	we	PRON
cana-1117	120	21	have	have	VERB
cana-1117	120	22	𝐼𝛼(𝐶𝐷𝛼)(𝜙(𝜏	𝐼𝛼(𝐶𝐷𝛼)(𝜙(𝜏	NOUN
cana-1117	120	23	)	)	PUNCT
cana-1117	120	24	)	)	PUNCT
cana-1117	121	1	=	=	SYM
cana-1117	121	2	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	121	3	)	)	PUNCT
cana-1117	121	4	⊖	⊖	NOUN
cana-1117	121	5	𝑔𝐻	𝑔𝐻	ADJ
cana-1117	121	6	𝜙(𝜏0	𝜙(𝜏0	NOUN
cana-1117	121	7	)	)	PUNCT
cana-1117	121	8	=	=	SYM
cana-1117	121	9	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	121	10	)	)	PUNCT
cana-1117	121	11	⊖	⊖	NOUN
cana-1117	121	12	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	121	13	𝜙0	𝜙0	NOUN
cana-1117	121	14	hence	hence	ADV
cana-1117	121	15	𝐼𝛼𝑟(𝜏	𝐼𝛼𝑟(𝜏	PROPN
cana-1117	121	16	)	)	PUNCT
cana-1117	121	17	=	=	SYM
cana-1117	121	18	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	121	19	)	)	PUNCT
cana-1117	121	20	⊖	⊖	NOUN
cana-1117	121	21	𝑔𝐻	𝑔𝐻	PUNCT
cana-1117	121	22	𝜙0	𝜙0	NOUN
cana-1117	121	23	i.e.	i.e.	X
cana-1117	121	24	,	,	PUNCT
cana-1117	121	25	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	121	26	)	)	PUNCT
cana-1117	121	27	⊖	⊖	NOUN
cana-1117	121	28	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	121	29	𝜙0	𝜙0	NOUN
cana-1117	121	30	=	=	SYM
cana-1117	121	31	𝜆	𝜆	NOUN
cana-1117	121	32	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-1117	121	33	)	)	PUNCT
cana-1117	121	34	⊙	⊙	PROPN
cana-1117	121	35	∫	∫	PROPN
cana-1117	122	1	𝜏	𝜏	PROPN
cana-1117	122	2	𝜏0	𝜏0	PROPN
cana-1117	122	3	(	(	PUNCT
cana-1117	122	4	𝜏	𝜏	NOUN
cana-1117	122	5	−	−	NOUN
cana-1117	122	6	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	122	7	+	+	CCONJ
cana-1117	122	8	1	1	NUM
cana-1117	122	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	122	10	)	)	PUNCT
cana-1117	122	11	⊙	⊙	PROPN
cana-1117	122	12	∫	∫	PROPN
cana-1117	123	1	𝜏	𝜏	PROPN
cana-1117	123	2	𝜏0	𝜏0	PROPN
cana-1117	123	3	(	(	PUNCT
cana-1117	123	4	𝜏	𝜏	NOUN
cana-1117	123	5	−	−	NOUN
cana-1117	123	6	𝔮𝑠)𝛼−1𝑓(𝑠	𝔮𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	123	7	,	,	PUNCT
cana-1117	123	8	𝜙(𝑠	𝜙(𝑠	PROPN
cana-1117	123	9	)	)	PUNCT
cana-1117	123	10	,	,	PUNCT
cana-1117	123	11	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	123	12	)	)	PUNCT
cana-1117	123	13	,	,	PUNCT
cana-1117	123	14	𝒮𝜙(𝑠))𝑑𝔮𝑠	𝒮𝜙(𝑠))𝑑𝔮𝑠	NOUN
cana-1117	123	15	the	the	DET
cana-1117	123	16	necessary	necessary	ADJ
cana-1117	123	17	condition	condition	NOUN
cana-1117	123	18	is	be	AUX
cana-1117	123	19	satisfied	satisfied	ADJ
cana-1117	123	20	.	.	PUNCT
cana-1117	124	1	next	next	ADV
cana-1117	124	2	,	,	PUNCT
cana-1117	124	3	let	let	VERB
cana-1117	124	4	us	we	PRON
cana-1117	124	5	consider	consider	VERB
cana-1117	124	6	that	that	DET
cana-1117	124	7	𝜙(𝜏	𝜙(𝜏	NOUN
cana-1117	124	8	)	)	PUNCT
cana-1117	124	9	is	be	AUX
cana-1117	124	10	a	a	DET
cana-1117	124	11	monotone	monotone	ADJ
cana-1117	124	12	function	function	NOUN
cana-1117	124	13	for	for	ADP
cana-1117	124	14	𝜏	𝜏	PRON
cana-1117	124	15	∈	∈	PRON
cana-1117	124	16	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	124	17	,	,	PUNCT
cana-1117	124	18	such	such	ADJ
cana-1117	124	19	that	that	SCONJ
cana-1117	124	20	(	(	PUNCT
cana-1117	124	21	4.1	4.1	NUM
cana-1117	124	22	)	)	PUNCT
cana-1117	124	23	is	be	AUX
cana-1117	124	24	satisfied	satisfied	ADJ
cana-1117	124	25	.	.	PUNCT
cana-1117	125	1	operating	operate	VERB
cana-1117	125	2	fuzzy	fuzzy	ADJ
cana-1117	125	3	caputo	caputo	PROPN
cana-1117	125	4	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	125	5	derivative	derivative	NOUN
cana-1117	125	6	of	of	ADP
cana-1117	125	7	order	order	NOUN
cana-1117	125	8	𝛼	𝛼	X
cana-1117	125	9	,	,	PUNCT
cana-1117	125	10	by	by	ADP
cana-1117	125	11	employing	employ	VERB
cana-1117	125	12	definition	definition	NOUN
cana-1117	125	13	(	(	PUNCT
cana-1117	125	14	2.5	2.5	NUM
cana-1117	125	15	)	)	PUNCT
cana-1117	125	16	,	,	PUNCT
cana-1117	125	17	on	on	ADP
cana-1117	125	18	both	both	CCONJ
cana-1117	125	19	the	the	DET
cana-1117	125	20	sides	side	NOUN
cana-1117	125	21	of	of	ADP
cana-1117	125	22	equation	equation	NOUN
cana-1117	125	23	(	(	PUNCT
cana-1117	125	24	4.1	4.1	NUM
cana-1117	125	25	)	)	PUNCT
cana-1117	125	26	,	,	PUNCT
cana-1117	125	27	we	we	PRON
cana-1117	125	28	get	get	VERB
cana-1117	125	29	𝐶𝐷𝛼(𝜙(𝜏	𝐶𝐷𝛼(𝜙(𝜏	ADP
cana-1117	125	30	)	)	PUNCT
cana-1117	125	31	)	)	PUNCT
cana-1117	126	1	=	=	SYM
cana-1117	126	2	𝜆𝜙(𝜏	𝜆𝜙(𝜏	X
cana-1117	126	3	)	)	PUNCT
cana-1117	127	1	+	+	CCONJ
cana-1117	127	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	127	3	,	,	PUNCT
cana-1117	127	4	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	127	5	)	)	PUNCT
cana-1117	127	6	,	,	PUNCT
cana-1117	127	7	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	127	8	)	)	PUNCT
cana-1117	127	9	,	,	PUNCT
cana-1117	127	10	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	127	11	)	)	PUNCT
cana-1117	127	12	)	)	PUNCT
cana-1117	127	13	for	for	ADP
cana-1117	127	14	𝜏	𝜏	PROPN
cana-1117	127	15	∈	∈	PROPN
cana-1117	127	16	𝕋𝔮.	𝕋𝔮.	PROPN
cana-1117	127	17	now	now	ADV
cana-1117	127	18	,	,	PUNCT
cana-1117	127	19	this	this	DET
cana-1117	127	20	part	part	NOUN
cana-1117	127	21	is	be	AUX
cana-1117	127	22	sufficient	sufficient	ADJ
cana-1117	127	23	to	to	PART
cana-1117	127	24	prove	prove	VERB
cana-1117	127	25	the	the	DET
cana-1117	127	26	theorem	theorem	PROPN
cana-1117	127	27	.	.	PUNCT
cana-1117	127	28	theorem	theorem	VERB
cana-1117	127	29	4.2	4.2	NUM
cana-1117	127	30	let	let	VERB
cana-1117	127	31	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	127	32	≥	≥	PRON
cana-1117	127	33	0	0	PUNCT
cana-1117	127	34	and	and	CCONJ
cana-1117	127	35	let	let	VERB
cana-1117	127	36	𝜖	𝜖	PROPN
cana-1117	127	37	≥	≥	PRON
cana-1117	127	38	0	0	NUM
cana-1117	127	39	be	be	AUX
cana-1117	127	40	a	a	DET
cana-1117	127	41	constant	constant	ADJ
cana-1117	127	42	such	such	ADJ
cana-1117	127	43	that	that	DET
cana-1117	127	44	0	0	NUM
cana-1117	127	45	∈	∈	PROPN
cana-1117	128	1	[	[	X
cana-1117	128	2	𝜙0	𝜙0	NOUN
cana-1117	128	3	−	−	NOUN
cana-1117	128	4	𝜖	𝜖	PROPN
cana-1117	128	5	,	,	PUNCT
cana-1117	128	6	𝜙0	𝜙0	NOUN
cana-1117	128	7	+	+	SYM
cana-1117	128	8	𝜖	𝜖	PROPN
cana-1117	128	9	]	]	X
cana-1117	128	10	.	.	PUNCT
cana-1117	129	1	assume	assume	VERB
cana-1117	129	2	that	that	SCONJ
cana-1117	129	3	𝑓	𝑓	X
cana-1117	129	4	:	:	PUNCT
cana-1117	129	5	𝕋𝔮	𝕋𝔮	ADP
cana-1117	129	6	×	×	NOUN
cana-1117	130	1	[	[	X
cana-1117	130	2	𝜙0	𝜙0	NOUN
cana-1117	130	3	−	−	PROPN
cana-1117	130	4	𝜖	𝜖	PROPN
cana-1117	130	5	,	,	PUNCT
cana-1117	130	6	𝜙0	𝜙0	NOUN
cana-1117	130	7	+	+	SYM
cana-1117	130	8	𝜖	𝜖	PROPN
cana-1117	130	9	]	]	X
cana-1117	130	10	×	×	NOUN
cana-1117	130	11	[	[	X
cana-1117	130	12	𝜙0	𝜙0	NOUN
cana-1117	130	13	−	−	PROPN
cana-1117	130	14	𝜖	𝜖	PROPN
cana-1117	130	15	,	,	PUNCT
cana-1117	130	16	𝜙0	𝜙0	NOUN
cana-1117	130	17	+	+	SYM
cana-1117	130	18	𝜖	𝜖	PROPN
cana-1117	130	19	]	]	X
cana-1117	130	20	×	×	NOUN
cana-1117	130	21	[	[	X
cana-1117	130	22	𝜙0	𝜙0	NOUN
cana-1117	130	23	−	−	PROPN
cana-1117	130	24	𝜖	𝜖	PROPN
cana-1117	130	25	,	,	PUNCT
cana-1117	130	26	𝜙0	𝜙0	NOUN
cana-1117	130	27	+	+	CCONJ
cana-1117	130	28	𝜖	𝜖	X
cana-1117	130	29	]	]	X
cana-1117	130	30	→	→	X
cana-1117	130	31	[	[	X
cana-1117	130	32	𝜙0	𝜙0	NOUN
cana-1117	130	33	−	−	PROPN
cana-1117	130	34	𝜖	𝜖	PROPN
cana-1117	130	35	,	,	PUNCT
cana-1117	130	36	𝜙0	𝜙0	NOUN
cana-1117	130	37	+	+	SYM
cana-1117	130	38	𝜖	𝜖	X
cana-1117	130	39	]	]	X
cana-1117	130	40	satisfies	satisfy	VERB
cana-1117	130	41	the	the	DET
cana-1117	130	42	condition	condition	NOUN
cana-1117	130	43	(	(	PUNCT
cana-1117	130	44	𝐻1	𝐻1	PROPN
cana-1117	130	45	)	)	PUNCT
cana-1117	130	46	.	.	PUNCT
cana-1117	131	1	𝜅	𝜅	X
cana-1117	131	2	=	=	SYM
cana-1117	131	3	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-1117	131	4	{	{	PUNCT
cana-1117	131	5	𝕋𝔮	𝕋𝔮	PROPN
cana-1117	131	6	,	,	PUNCT
cana-1117	131	7	[	[	PUNCT
cana-1117	131	8	γ𝔮(𝛼	γ𝔮(𝛼	X
cana-1117	131	9	+	+	CCONJ
cana-1117	131	10	1)𝜖	1)𝜖	NOUN
cana-1117	131	11	(	(	PUNCT
cana-1117	131	12	𝜖	𝜖	X
cana-1117	131	13	+	+	NUM
cana-1117	131	14	‖𝜙0‖)(𝜆	‖𝜙0‖)(𝜆	NOUN
cana-1117	131	15	+	+	CCONJ
cana-1117	131	16	𝐿(1	𝐿(1	X
cana-1117	132	1	+	+	NUM
cana-1117	132	2	𝐾∗	𝐾∗	NUM
cana-1117	132	3	+	+	CCONJ
cana-1117	132	4	𝐻∗	𝐻∗	NUM
cana-1117	132	5	)	)	PUNCT
cana-1117	133	1	+	+	NUM
cana-1117	133	2	𝑀	𝑀	NOUN
cana-1117	133	3	]	]	PUNCT
cana-1117	133	4	1	1	NUM
cana-1117	133	5	𝛼	𝛼	NOUN
cana-1117	133	6	}	}	PUNCT
cana-1117	133	7	,	,	PUNCT
cana-1117	133	8	where	where	SCONJ
cana-1117	133	9	𝑀	𝑀	PROPN
cana-1117	133	10	=	=	AUX
cana-1117	133	11	𝑆𝑢𝑝𝜏∈[𝜏0,𝑇]|𝑓(𝜏	𝑆𝑢𝑝𝜏∈[𝜏0,𝑇]|𝑓(𝜏	PROPN
cana-1117	133	12	,	,	PUNCT
cana-1117	133	13	0,0,0)|	0,0,0)|	PROPN
cana-1117	133	14	.	.	PUNCT
cana-1117	134	1	then	then	ADV
cana-1117	134	2	the	the	DET
cana-1117	134	3	cauchy	cauchy	PROPN
cana-1117	134	4	problem	problem	NOUN
cana-1117	134	5	for	for	ADP
cana-1117	134	6	(	(	PUNCT
cana-1117	134	7	1.1	1.1	NUM
cana-1117	134	8	)	)	PUNCT
cana-1117	134	9	has	have	VERB
cana-1117	134	10	a	a	DET
cana-1117	134	11	unique	unique	ADJ
cana-1117	134	12	solution	solution	NOUN
cana-1117	134	13	𝑢	𝑢	ADP
cana-1117	134	14	:	:	PUNCT
cana-1117	134	15	[	[	X
cana-1117	134	16	0	0	NUM
cana-1117	134	17	,	,	PUNCT
cana-1117	134	18	𝜅	𝜅	X
cana-1117	134	19	]	]	PUNCT
cana-1117	134	20	→	→	SYM
cana-1117	134	21	ℝ𝔽.	ℝ𝔽.	X
cana-1117	134	22	communications	communication	NOUN
cana-1117	134	23	on	on	ADP
cana-1117	134	24	applied	apply	VERB
cana-1117	134	25	nonlinear	nonlinear	ADJ
cana-1117	134	26	analysis	analysis	NOUN
cana-1117	134	27	issn	issn	NOUN
cana-1117	134	28	:	:	PUNCT
cana-1117	134	29	1074	1074	NUM
cana-1117	134	30	-	-	PUNCT
cana-1117	134	31	133x	133x	NUM
cana-1117	134	32	vol	vol	NOUN
cana-1117	134	33	31	31	NUM
cana-1117	134	34	no	no	NOUN
cana-1117	134	35	.	.	PUNCT
cana-1117	135	1	6s	6s	NUM
cana-1117	135	2	(	(	PUNCT
cana-1117	135	3	2024	2024	NUM
cana-1117	135	4	)	)	PUNCT
cana-1117	135	5	42	42	NUM
cana-1117	135	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	135	7	proof	proof	NOUN
cana-1117	135	8	.	.	PUNCT
cana-1117	136	1	define	define	VERB
cana-1117	136	2	the	the	DET
cana-1117	136	3	set	set	NOUN
cana-1117	136	4	𝔹	𝔹	NOUN
cana-1117	136	5	=	=	PUNCT
cana-1117	136	6	{	{	PUNCT
cana-1117	136	7	𝜙	𝜙	NOUN
cana-1117	136	8	∈	∈	PROPN
cana-1117	136	9	𝐶([𝜏0	𝐶([𝜏0	NOUN
cana-1117	136	10	,	,	PUNCT
cana-1117	136	11	𝜅	𝜅	PROPN
cana-1117	136	12	]	]	X
cana-1117	136	13	,	,	PUNCT
cana-1117	136	14	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	136	15	):	):	PUNCT
cana-1117	136	16	𝜙(𝜏0	𝜙(𝜏0	NOUN
cana-1117	136	17	)	)	PUNCT
cana-1117	136	18	=	=	SYM
cana-1117	136	19	𝜙0	𝜙0	NOUN
cana-1117	136	20	,	,	PUNCT
cana-1117	136	21	‖𝜙	‖𝜙	VERB
cana-1117	136	22	−	−	PROPN
cana-1117	136	23	𝜙0‖	𝜙0‖	PROPN
cana-1117	136	24	≤	≤	PUNCT
cana-1117	136	25	𝜖	𝜖	PART
cana-1117	136	26	}	}	PUNCT
cana-1117	136	27	.	.	PUNCT
cana-1117	137	1	since	since	SCONJ
cana-1117	137	2	𝜙0	𝜙0	NOUN
cana-1117	137	3	∈	∈	PROPN
cana-1117	137	4	𝔹	𝔹	PROPN
cana-1117	137	5	,	,	PUNCT
cana-1117	137	6	𝔹	𝔹	PROPN
cana-1117	137	7	is	be	AUX
cana-1117	137	8	nonempty	nonempty	X
cana-1117	137	9	.	.	PUNCT
cana-1117	138	1	also	also	ADV
cana-1117	138	2	,	,	PUNCT
cana-1117	138	3	𝔹	𝔹	PROPN
cana-1117	138	4	is	be	AUX
cana-1117	138	5	a	a	DET
cana-1117	138	6	closed	closed	ADJ
cana-1117	138	7	,	,	PUNCT
cana-1117	138	8	bounded	bound	VERB
cana-1117	138	9	and	and	CCONJ
cana-1117	138	10	convex	convex	PROPN
cana-1117	138	11	subset	subset	NOUN
cana-1117	138	12	of	of	ADP
cana-1117	138	13	banach	banach	NOUN
cana-1117	138	14	space	space	NOUN
cana-1117	138	15	𝐶([0	𝐶([0	PROPN
cana-1117	138	16	,	,	PUNCT
cana-1117	138	17	𝜅	𝜅	ADP
cana-1117	138	18	]	]	X
cana-1117	138	19	,	,	PUNCT
cana-1117	138	20	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	138	21	)	)	PUNCT
cana-1117	138	22	.	.	PUNCT
cana-1117	139	1	on	on	ADP
cana-1117	139	2	𝔹	𝔹	PROPN
cana-1117	139	3	,	,	PUNCT
cana-1117	139	4	define	define	VERB
cana-1117	139	5	an	an	DET
cana-1117	139	6	operator	operator	NOUN
cana-1117	139	7	𝒦	𝒦	NOUN
cana-1117	139	8	by	by	ADP
cana-1117	139	9	𝒦𝜙(𝜏	𝒦𝜙(𝜏	NOUN
cana-1117	139	10	)	)	PUNCT
cana-1117	139	11	⊖	⊖	NOUN
cana-1117	139	12	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	139	13	𝜙0	𝜙0	NOUN
cana-1117	139	14	=	=	SYM
cana-1117	139	15	𝜆	𝜆	NOUN
cana-1117	139	16	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-1117	139	17	)	)	PUNCT
cana-1117	139	18	⊙	⊙	PROPN
cana-1117	139	19	∫	∫	PROPN
cana-1117	140	1	𝜏	𝜏	PROPN
cana-1117	140	2	𝜏0	𝜏0	PROPN
cana-1117	140	3	(	(	PUNCT
cana-1117	140	4	𝜏	𝜏	NOUN
cana-1117	140	5	−	−	NOUN
cana-1117	140	6	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	140	7	+	+	CCONJ
cana-1117	140	8	1	1	NUM
cana-1117	140	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	140	10	)	)	PUNCT
cana-1117	140	11	⊙	⊙	PROPN
cana-1117	140	12	∫	∫	PROPN
cana-1117	141	1	𝜏	𝜏	PROPN
cana-1117	141	2	𝜏0	𝜏0	PROPN
cana-1117	141	3	(	(	PUNCT
cana-1117	141	4	𝜏	𝜏	NOUN
cana-1117	141	5	−	−	NOUN
cana-1117	141	6	𝔮𝑠)𝛼−1𝑓(𝑠	𝔮𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	141	7	,	,	PUNCT
cana-1117	141	8	𝑢(𝑠	𝑢(𝑠	NOUN
cana-1117	141	9	)	)	PUNCT
cana-1117	141	10	,	,	PUNCT
cana-1117	141	11	𝒢𝑢(𝑠	𝒢𝑢(𝑠	NOUN
cana-1117	141	12	)	)	PUNCT
cana-1117	141	13	,	,	PUNCT
cana-1117	141	14	𝒮𝑢(𝑠))𝑑𝔮𝑠	𝒮𝑢(𝑠))𝑑𝔮𝑠	ADV
cana-1117	141	15	now	now	ADV
cana-1117	141	16	,	,	PUNCT
cana-1117	141	17	it	it	PRON
cana-1117	141	18	needs	need	VERB
cana-1117	141	19	to	to	PART
cana-1117	141	20	prove	prove	VERB
cana-1117	141	21	that	that	SCONJ
cana-1117	141	22	𝒦	𝒦	PROPN
cana-1117	141	23	maps	map	NOUN
cana-1117	141	24	𝔹	𝔹	VERB
cana-1117	141	25	to	to	ADP
cana-1117	141	26	itself	itself	PRON
cana-1117	141	27	.	.	PUNCT
cana-1117	142	1	consider	consider	VERB
cana-1117	142	2	any	any	DET
cana-1117	142	3	𝜙	𝜙	PRON
cana-1117	142	4	∈	∈	PROPN
cana-1117	142	5	𝔹	𝔹	NOUN
cana-1117	142	6	and	and	CCONJ
cana-1117	142	7	𝜏	𝜏	NOUN
cana-1117	142	8	∈	∈	NOUN
cana-1117	143	1	[	[	X
cana-1117	143	2	0	0	NUM
cana-1117	143	3	,	,	PUNCT
cana-1117	143	4	𝜅	𝜅	NOUN
cana-1117	143	5	]	]	PUNCT
cana-1117	143	6	.	.	PUNCT
cana-1117	143	7	‖𝒦𝜙(𝜏	‖𝒦𝜙(𝜏	NOUN
cana-1117	143	8	)	)	PUNCT
cana-1117	143	9	⊖	⊖	PROPN
cana-1117	143	10	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	143	11	𝜙0‖	𝜙0‖	PROPN
cana-1117	143	12	≤	≤	NUM
cana-1117	143	13	𝜆	𝜆	ADP
cana-1117	143	14	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-1117	143	15	)	)	PUNCT
cana-1117	143	16	⊙	⊙	PROPN
cana-1117	143	17	∫	∫	PROPN
cana-1117	144	1	𝜏	𝜏	PROPN
cana-1117	144	2	𝜏0	𝜏0	PROPN
cana-1117	144	3	(	(	PUNCT
cana-1117	144	4	𝜏	𝜏	NOUN
cana-1117	144	5	−	−	NUM
cana-1117	144	6	𝔮𝑠)𝛼−1‖𝜙(𝑠)‖𝑑𝔮𝑠	𝔮𝑠)𝛼−1‖𝜙(𝑠)‖𝑑𝔮𝑠	NOUN
cana-1117	144	7	+	+	SYM
cana-1117	144	8	1	1	NUM
cana-1117	144	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	144	10	)	)	PUNCT
cana-1117	144	11	⊙	⊙	PROPN
cana-1117	144	12	∫	∫	PROPN
cana-1117	145	1	𝜏	𝜏	PROPN
cana-1117	145	2	𝜏0	𝜏0	PROPN
cana-1117	145	3	(	(	PUNCT
cana-1117	145	4	𝜏	𝜏	PROPN
cana-1117	145	5	−	−	PROPN
cana-1117	145	6	𝔮𝑠)𝛼−1‖𝑓(𝑠	𝔮𝑠)𝛼−1‖𝑓(𝑠	PROPN
cana-1117	145	7	,	,	PUNCT
cana-1117	145	8	𝜙(𝑠	𝜙(𝑠	NOUN
cana-1117	145	9	)	)	PUNCT
cana-1117	145	10	,	,	PUNCT
cana-1117	145	11	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	145	12	)	)	PUNCT
cana-1117	145	13	,	,	PUNCT
cana-1117	145	14	𝒮𝜙(𝑠))‖𝑑𝔮𝑠	𝒮𝜙(𝑠))‖𝑑𝔮𝑠	ADV
cana-1117	145	15	≤	≤	ADJ
cana-1117	145	16	𝜆	𝜆	DET
cana-1117	145	17	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	145	18	)	)	PUNCT
cana-1117	145	19	⊙	⊙	PROPN
cana-1117	145	20	∫	∫	PROPN
cana-1117	146	1	𝜏	𝜏	PROPN
cana-1117	146	2	𝜏0	𝜏0	PROPN
cana-1117	146	3	(	(	PUNCT
cana-1117	146	4	𝜏	𝜏	NOUN
cana-1117	146	5	−	−	NOUN
cana-1117	146	6	𝔮𝑠)𝛼−1(‖𝜙(𝑠	𝔮𝑠)𝛼−1(‖𝜙(𝑠	NOUN
cana-1117	146	7	)	)	PUNCT
cana-1117	146	8	−	−	PROPN
cana-1117	146	9	𝜙0‖	𝜙0‖	PROPN
cana-1117	146	10	+	+	CCONJ
cana-1117	146	11	‖𝜙0‖)𝑑𝔮𝑠	‖𝜙0‖)𝑑𝔮𝑠	ADJ
cana-1117	146	12	+	+	CCONJ
cana-1117	146	13	1	1	NUM
cana-1117	146	14	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	146	15	)	)	PUNCT
cana-1117	146	16	⊙	⊙	PROPN
cana-1117	146	17	∫	∫	PROPN
cana-1117	147	1	𝜏	𝜏	PROPN
cana-1117	147	2	𝜏0	𝜏0	PROPN
cana-1117	147	3	(	(	PUNCT
cana-1117	147	4	𝜏	𝜏	PROPN
cana-1117	147	5	−	−	NOUN
cana-1117	147	6	𝔮𝑠)𝛼−1(‖𝑓(𝑠	𝔮𝑠)𝛼−1(‖𝑓(𝑠	NOUN
cana-1117	147	7	,	,	PUNCT
cana-1117	147	8	𝜙(𝑠	𝜙(𝑠	NOUN
cana-1117	147	9	)	)	PUNCT
cana-1117	147	10	,	,	PUNCT
cana-1117	147	11	𝒢𝑢(𝑠	𝒢𝑢(𝑠	NOUN
cana-1117	147	12	)	)	PUNCT
cana-1117	147	13	,	,	PUNCT
cana-1117	147	14	𝒮𝑢(𝑠	𝒮𝑢(𝑠	NOUN
cana-1117	147	15	)	)	PUNCT
cana-1117	147	16	)	)	PUNCT
cana-1117	147	17	−	−	PROPN
cana-1117	147	18	𝑓(𝑠	𝑓(𝑠	NOUN
cana-1117	147	19	,	,	PUNCT
cana-1117	147	20	0,0,0)‖)𝑑𝔮𝑠	0,0,0)‖)𝑑𝔮𝑠	NUM
cana-1117	148	1	+	+	CCONJ
cana-1117	148	2	1	1	NUM
cana-1117	148	3	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	148	4	)	)	PUNCT
cana-1117	148	5	⊙	⊙	PROPN
cana-1117	148	6	∫	∫	PROPN
cana-1117	149	1	𝜏	𝜏	PROPN
cana-1117	149	2	𝜏0	𝜏0	PROPN
cana-1117	149	3	(	(	PUNCT
cana-1117	149	4	𝜏	𝜏	PROPN
cana-1117	149	5	−	−	PROPN
cana-1117	149	6	𝔮𝑠)𝛼−1‖𝑓(𝑠	𝔮𝑠)𝛼−1‖𝑓(𝑠	PROPN
cana-1117	149	7	,	,	PUNCT
cana-1117	149	8	0,0,0)‖𝑑𝔮𝑠	0,0,0)‖𝑑𝔮𝑠	NOUN
cana-1117	149	9	by	by	ADP
cana-1117	149	10	the	the	DET
cana-1117	149	11	assumption	assumption	NOUN
cana-1117	149	12	(	(	PUNCT
cana-1117	149	13	𝐻1	𝐻1	PROPN
cana-1117	149	14	)	)	PUNCT
cana-1117	149	15	and	and	CCONJ
cana-1117	149	16	the	the	DET
cana-1117	149	17	definition	definition	NOUN
cana-1117	149	18	of	of	ADP
cana-1117	149	19	𝔹	𝔹	PROPN
cana-1117	149	20	for	for	ADP
cana-1117	149	21	any	any	DET
cana-1117	149	22	𝜏	𝜏	PROPN
cana-1117	149	23	∈	∈	NOUN
cana-1117	149	24	[	[	X
cana-1117	149	25	0	0	NUM
cana-1117	149	26	,	,	PUNCT
cana-1117	149	27	𝜅	𝜅	ADP
cana-1117	149	28	]	]	X
cana-1117	149	29	,	,	PUNCT
cana-1117	149	30	‖𝜙(𝜏)‖	‖𝜙(𝜏)‖	PUNCT
cana-1117	149	31	≤	≤	NUM
cana-1117	149	32	‖𝜙(𝜏	‖𝜙(𝜏	NOUN
cana-1117	149	33	)	)	PUNCT
cana-1117	149	34	−	−	PROPN
cana-1117	149	35	𝜙0‖	𝜙0‖	PROPN
cana-1117	149	36	+	+	CCONJ
cana-1117	149	37	‖𝜙0‖	‖𝜙0‖	NOUN
cana-1117	149	38	≤	≤	X
cana-1117	149	39	𝜖	𝜖	X
cana-1117	150	1	+	+	CCONJ
cana-1117	150	2	‖𝜙0‖	‖𝜙0‖	NOUN
cana-1117	150	3	(	(	PUNCT
cana-1117	150	4	4.2	4.2	NUM
cana-1117	150	5	)	)	PUNCT
cana-1117	150	6	consider	consider	VERB
cana-1117	150	7	,	,	PUNCT
cana-1117	150	8	1	1	NUM
cana-1117	150	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	150	10	)	)	PUNCT
cana-1117	151	1	⊙	⊙	PROPN
cana-1117	151	2	∫	∫	PROPN
cana-1117	152	1	𝜏	𝜏	PROPN
cana-1117	152	2	𝜏0	𝜏0	PROPN
cana-1117	152	3	(	(	PUNCT
cana-1117	152	4	𝜏	𝜏	PROPN
cana-1117	152	5	−	−	NOUN
cana-1117	152	6	𝔮𝑠)𝛼−1(‖𝑓(𝑠	𝔮𝑠)𝛼−1(‖𝑓(𝑠	NOUN
cana-1117	152	7	,	,	PUNCT
cana-1117	152	8	𝜙(𝑠	𝜙(𝑠	NOUN
cana-1117	152	9	)	)	PUNCT
cana-1117	152	10	,	,	PUNCT
cana-1117	152	11	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	152	12	)	)	PUNCT
cana-1117	152	13	,	,	PUNCT
cana-1117	152	14	𝒮𝜙(𝑠	𝒮𝜙(𝑠	NOUN
cana-1117	152	15	)	)	PUNCT
cana-1117	152	16	)	)	PUNCT
cana-1117	152	17	−	−	PROPN
cana-1117	152	18	𝑓(𝑠	𝑓(𝑠	PROPN
cana-1117	152	19	,	,	PUNCT
cana-1117	152	20	0,0,0)‖)𝑑𝔮𝑠	0,0,0)‖)𝑑𝔮𝑠	NUM
cana-1117	152	21	≤	≤	NUM
cana-1117	152	22	1	1	NUM
cana-1117	152	23	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	152	24	)	)	PUNCT
cana-1117	152	25	⊙	⊙	PROPN
cana-1117	152	26	∫	∫	PROPN
cana-1117	153	1	𝜏	𝜏	PROPN
cana-1117	153	2	𝜏0	𝜏0	PROPN
cana-1117	153	3	(	(	PUNCT
cana-1117	153	4	𝜏	𝜏	PROPN
cana-1117	153	5	−	−	PROPN
cana-1117	153	6	𝔮𝑠)𝛼−1ℒ[‖𝜙‖	𝔮𝑠)𝛼−1ℒ[‖𝜙‖	PROPN
cana-1117	154	1	+	+	CCONJ
cana-1117	154	2	‖𝒢𝜙‖	‖𝒢𝜙‖	NUM
cana-1117	155	1	+	+	NUM
cana-1117	155	2	‖𝒮𝜙‖]𝑑𝔮𝑠	‖𝒮𝜙‖]𝑑𝔮𝑠	ADJ
cana-1117	155	3	ℒ	ℒ	ADJ
cana-1117	155	4	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	155	5	)	)	PUNCT
cana-1117	155	6	⊙	⊙	PROPN
cana-1117	155	7	∫	∫	PROPN
cana-1117	156	1	𝜏	𝜏	PROPN
cana-1117	156	2	𝜏0	𝜏0	PROPN
cana-1117	156	3	(	(	PUNCT
cana-1117	156	4	𝜏	𝜏	NOUN
cana-1117	156	5	−	−	NOUN
cana-1117	156	6	𝔮𝑠)𝛼−1‖𝜙‖(1	𝔮𝑠)𝛼−1‖𝜙‖(1	NOUN
cana-1117	157	1	+	+	CCONJ
cana-1117	157	2	𝐾∗	𝐾∗	PUNCT
cana-1117	158	1	+	+	CCONJ
cana-1117	159	1	𝐻∗)𝑑𝔮𝑠	𝐻∗)𝑑𝔮𝑠	ADJ
cana-1117	159	2	where	where	SCONJ
cana-1117	159	3	𝑚𝑎𝑥‖𝒢𝜙‖	𝑚𝑎𝑥‖𝒢𝜙‖	NUM
cana-1117	159	4	=	=	NOUN
cana-1117	159	5	𝐾∗	𝐾∗	NUM
cana-1117	159	6	and	and	CCONJ
cana-1117	159	7	𝑚𝑎𝑥‖𝒮𝜙‖	𝑚𝑎𝑥‖𝒮𝜙‖	NUM
cana-1117	159	8	=	=	SYM
cana-1117	159	9	𝐻∗	𝐻∗	PROPN
cana-1117	159	10	therefore	therefore	ADV
cana-1117	159	11	,	,	PUNCT
cana-1117	159	12	‖𝒦𝜙(𝜏	‖𝒦𝜙(𝜏	NOUN
cana-1117	159	13	)	)	PUNCT
cana-1117	159	14	⊖	⊖	NOUN
cana-1117	159	15	𝑔𝐻	𝑔𝐻	PROPN
cana-1117	159	16	𝜙0‖	𝜙0‖	PROPN
cana-1117	159	17	≤	≤	NUM
cana-1117	159	18	𝜆	𝜆	DET
cana-1117	159	19	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	159	20	)	)	PUNCT
cana-1117	159	21	⊙	⊙	PROPN
cana-1117	159	22	∫	∫	PROPN
cana-1117	159	23	𝜏	𝜏	PROPN
cana-1117	159	24	𝜏0	𝜏0	PROPN
cana-1117	159	25	(	(	PUNCT
cana-1117	159	26	𝜏	𝜏	PROPN
cana-1117	159	27	−	−	NUM
cana-1117	159	28	𝔮𝑠)𝛼−1(𝜖	𝔮𝑠)𝛼−1(𝜖	NUM
cana-1117	159	29	+	+	CCONJ
cana-1117	159	30	‖𝜙0‖)𝑑𝔮𝑠	‖𝜙0‖)𝑑𝔮𝑠	NOUN
cana-1117	159	31	+	+	CCONJ
cana-1117	159	32	ℒ	ℒ	ADJ
cana-1117	159	33	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	159	34	)	)	PUNCT
cana-1117	159	35	⊙	⊙	PROPN
cana-1117	159	36	∫	∫	PROPN
cana-1117	159	37	𝜏	𝜏	PROPN
cana-1117	159	38	𝜏0	𝜏0	PROPN
cana-1117	159	39	(	(	PUNCT
cana-1117	159	40	𝜏	𝜏	PROPN
cana-1117	159	41	−	−	PROPN
cana-1117	159	42	𝔮𝑠)𝛼−1(𝜖	𝔮𝑠)𝛼−1(𝜖	NUM
cana-1117	159	43	+	+	CCONJ
cana-1117	159	44	‖𝜙0‖	‖𝜙0‖	NOUN
cana-1117	159	45	)	)	PUNCT
cana-1117	159	46	+	+	CCONJ
cana-1117	159	47	𝑀	𝑀	PROPN
cana-1117	159	48	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	159	49	)	)	PUNCT
cana-1117	159	50	⊙	⊙	PROPN
cana-1117	159	51	∫	∫	PROPN
cana-1117	160	1	𝜏	𝜏	PROPN
cana-1117	160	2	𝜏0	𝜏0	PROPN
cana-1117	160	3	(	(	PUNCT
cana-1117	160	4	𝜏	𝜏	NOUN
cana-1117	160	5	−	−	NOUN
cana-1117	160	6	𝔮𝑠)𝛼−1𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝑑𝔮𝑠	ADJ
cana-1117	160	7	communications	communication	NOUN
cana-1117	160	8	on	on	ADP
cana-1117	160	9	applied	apply	VERB
cana-1117	160	10	nonlinear	nonlinear	ADJ
cana-1117	160	11	analysis	analysis	NOUN
cana-1117	160	12	issn	issn	NOUN
cana-1117	160	13	:	:	PUNCT
cana-1117	160	14	1074	1074	NUM
cana-1117	160	15	-	-	PUNCT
cana-1117	160	16	133x	133x	NUM
cana-1117	160	17	vol	vol	NOUN
cana-1117	160	18	31	31	NUM
cana-1117	160	19	no	no	NOUN
cana-1117	160	20	.	.	PUNCT
cana-1117	161	1	6s	6s	NUM
cana-1117	161	2	(	(	PUNCT
cana-1117	161	3	2024	2024	NUM
cana-1117	161	4	)	)	PUNCT
cana-1117	161	5	43	43	NUM
cana-1117	162	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	162	2	=	=	SYM
cana-1117	162	3	(	(	PUNCT
cana-1117	162	4	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	NUM
cana-1117	162	5	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	162	6	)	)	PUNCT
cana-1117	162	7	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	162	8	)	)	PUNCT
cana-1117	162	9	γ𝔮(𝛼+1	γ𝔮(𝛼+1	NUM
cana-1117	162	10	)	)	PUNCT
cana-1117	162	11	(	(	PUNCT
cana-1117	162	12	𝜏	𝜏	X
cana-1117	162	13	−	−	NOUN
cana-1117	162	14	𝜏0)𝛼	𝜏0)𝛼	ADJ
cana-1117	162	15	≤	≤	X
cana-1117	162	16	(	(	PUNCT
cana-1117	162	17	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	NUM
cana-1117	162	18	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	162	19	)	)	PUNCT
cana-1117	162	20	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	162	21	)	)	PUNCT
cana-1117	162	22	γ𝔮(𝛼+1	γ𝔮(𝛼+1	NUM
cana-1117	162	23	)	)	PUNCT
cana-1117	162	24	(	(	PUNCT
cana-1117	162	25	𝜅)𝛼	𝜅)𝛼	X
cana-1117	162	26	=	=	SYM
cana-1117	162	27	𝜖	𝜖	X
cana-1117	162	28	hence	hence	ADV
cana-1117	162	29	𝒦	𝒦	PROPN
cana-1117	162	30	maps	map	NOUN
cana-1117	162	31	𝔹	𝔹	VERB
cana-1117	162	32	to	to	ADP
cana-1117	162	33	itself	itself	PRON
cana-1117	162	34	.	.	PUNCT
cana-1117	163	1	next	next	ADV
cana-1117	163	2	we	we	PRON
cana-1117	163	3	consider	consider	VERB
cana-1117	163	4	that	that	PRON
cana-1117	163	5	,	,	PUNCT
cana-1117	163	6	for	for	ADP
cana-1117	163	7	0	0	NUM
cana-1117	163	8	≤	≤	NUM
cana-1117	163	9	𝜏1	𝜏1	NOUN
cana-1117	163	10	≤	≤	PUNCT
cana-1117	163	11	𝜏2	𝜏2	PROPN
cana-1117	163	12	≤	≤	NUM
cana-1117	163	13	𝜅	𝜅	NUM
cana-1117	163	14	,	,	PUNCT
cana-1117	163	15	‖𝒦𝜙(𝜏1	‖𝒦𝜙(𝜏1	NUM
cana-1117	163	16	)	)	PUNCT
cana-1117	163	17	−	−	ADP
cana-1117	163	18	𝒦𝜙(𝜏2)‖	𝒦𝜙(𝜏2)‖	NOUN
cana-1117	163	19	≤	≤	NUM
cana-1117	163	20	𝜆	𝜆	DET
cana-1117	163	21	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	163	22	)	)	PUNCT
cana-1117	164	1	⊙	⊙	PROPN
cana-1117	164	2	‖∫	‖∫	PROPN
cana-1117	164	3	𝜏1	𝜏1	PROPN
cana-1117	164	4	𝜏0	𝜏0	PROPN
cana-1117	164	5	(	(	PUNCT
cana-1117	164	6	𝜏1	𝜏1	NOUN
cana-1117	164	7	−	−	PROPN
cana-1117	164	8	𝔮𝑠)𝛼−1𝜙(𝑠	𝔮𝑠)𝛼−1𝜙(𝑠	NOUN
cana-1117	164	9	)	)	PUNCT
cana-1117	164	10	𝑑𝔮𝑠	𝑑𝔮𝑠	NOUN
cana-1117	164	11	−	−	PROPN
cana-1117	164	12	∫	∫	PROPN
cana-1117	164	13	𝜏2	𝜏2	PROPN
cana-1117	164	14	𝜏0	𝜏0	PROPN
cana-1117	164	15	(	(	PUNCT
cana-1117	164	16	𝜏2	𝜏2	PROPN
cana-1117	164	17	−	−	PROPN
cana-1117	164	18	𝔮𝑠)𝛼−1𝜙(𝑠	𝔮𝑠)𝛼−1𝜙(𝑠	NOUN
cana-1117	164	19	)	)	PUNCT
cana-1117	164	20	𝑑𝔮𝑠‖	𝑑𝔮𝑠‖	PUNCT
cana-1117	165	1	+	+	SYM
cana-1117	165	2	1	1	NUM
cana-1117	165	3	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	165	4	)	)	PUNCT
cana-1117	165	5	⊙	⊙	PROPN
cana-1117	165	6	‖∫	‖∫	PROPN
cana-1117	165	7	𝜏1	𝜏1	PROPN
cana-1117	165	8	𝜏0	𝜏0	PROPN
cana-1117	165	9	(	(	PUNCT
cana-1117	165	10	𝜏1	𝜏1	NOUN
cana-1117	165	11	−	−	PROPN
cana-1117	165	12	𝔮𝑠)𝛼−1𝑓(𝑠	𝔮𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	165	13	,	,	PUNCT
cana-1117	165	14	𝜙(𝑠	𝜙(𝑠	NOUN
cana-1117	165	15	)	)	PUNCT
cana-1117	165	16	,	,	PUNCT
cana-1117	165	17	𝒢𝑢(𝑠	𝒢𝑢(𝑠	NOUN
cana-1117	165	18	)	)	PUNCT
cana-1117	165	19	,	,	PUNCT
cana-1117	165	20	𝒮𝑢(𝑠	𝒮𝑢(𝑠	NOUN
cana-1117	165	21	)	)	PUNCT
cana-1117	165	22	)	)	PUNCT
cana-1117	165	23	𝑑𝔮𝑠	𝑑𝔮𝑠	NOUN
cana-1117	165	24	−	−	PROPN
cana-1117	165	25	∫	∫	PROPN
cana-1117	165	26	𝜏2	𝜏2	PROPN
cana-1117	165	27	𝜏0	𝜏0	PROPN
cana-1117	165	28	(	(	PUNCT
cana-1117	165	29	𝜏2	𝜏2	PROPN
cana-1117	165	30	−	−	PROPN
cana-1117	165	31	𝔮𝑠)𝛼−1𝑓(𝑠	𝔮𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	165	32	,	,	PUNCT
cana-1117	165	33	𝜙(𝑠	𝜙(𝑠	PROPN
cana-1117	165	34	)	)	PUNCT
cana-1117	165	35	,	,	PUNCT
cana-1117	165	36	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	165	37	)	)	PUNCT
cana-1117	165	38	,	,	PUNCT
cana-1117	165	39	𝒮𝜙(𝑠	𝒮𝜙(𝑠	NOUN
cana-1117	165	40	)	)	PUNCT
cana-1117	165	41	)	)	PUNCT
cana-1117	165	42	𝑑𝔮𝑠‖	𝑑𝔮𝑠‖	PUNCT
cana-1117	165	43	consider	consider	VERB
cana-1117	165	44	𝜆	𝜆	NOUN
cana-1117	165	45	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	165	46	)	)	PUNCT
cana-1117	165	47	⊙	⊙	PROPN
cana-1117	165	48	‖∫	‖∫	PROPN
cana-1117	165	49	𝜏1	𝜏1	PROPN
cana-1117	165	50	𝜏0	𝜏0	PROPN
cana-1117	165	51	(	(	PUNCT
cana-1117	165	52	𝜏1	𝜏1	NOUN
cana-1117	165	53	−	−	NOUN
cana-1117	165	54	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	165	55	−	−	PROPN
cana-1117	165	56	∫	∫	PROPN
cana-1117	165	57	𝜏2	𝜏2	PROPN
cana-1117	165	58	𝜏0	𝜏0	PROPN
cana-1117	165	59	(	(	PUNCT
cana-1117	165	60	𝜏2	𝜏2	PROPN
cana-1117	165	61	−	−	PROPN
cana-1117	165	62	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠‖	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝔮𝑠‖	PROPN
cana-1117	165	63	≤	≤	PROPN
cana-1117	165	64	𝜆	𝜆	DET
cana-1117	165	65	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	165	66	)	)	PUNCT
cana-1117	166	1	⊙	⊙	PROPN
cana-1117	166	2	∫	∫	PROPN
cana-1117	166	3	𝜏1	𝜏1	PROPN
cana-1117	166	4	𝜏0	𝜏0	PROPN
cana-1117	166	5	(	(	PUNCT
cana-1117	166	6	(	(	PUNCT
cana-1117	166	7	𝜏1	𝜏1	NOUN
cana-1117	166	8	−	−	NOUN
cana-1117	166	9	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	166	10	−	−	PROPN
cana-1117	166	11	(	(	PUNCT
cana-1117	166	12	𝜏2	𝜏2	PROPN
cana-1117	166	13	−	−	PROPN
cana-1117	166	14	𝔮𝑠)𝛼−1)‖𝜙(𝑠)‖𝑑𝔮	𝔮𝑠)𝛼−1)‖𝜙(𝑠)‖𝑑𝔮	PROPN
cana-1117	166	15	+	+	CCONJ
cana-1117	166	16	𝜆	𝜆	DET
cana-1117	166	17	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	166	18	)	)	PUNCT
cana-1117	166	19	⊙	⊙	PROPN
cana-1117	166	20	∫	∫	PROPN
cana-1117	166	21	𝜏2	𝜏2	PROPN
cana-1117	166	22	𝜏0	𝜏0	PROPN
cana-1117	166	23	(	(	PUNCT
cana-1117	166	24	(	(	PUNCT
cana-1117	166	25	𝜏2	𝜏2	PROPN
cana-1117	166	26	−	−	PROPN
cana-1117	166	27	𝔮𝑠)𝛼−1)‖𝜙(𝑠)‖𝑑𝔮𝑠	𝔮𝑠)𝛼−1)‖𝜙(𝑠)‖𝑑𝔮𝑠	NOUN
cana-1117	166	28	≤	≤	NOUN
cana-1117	166	29	𝜆	𝜆	PRON
cana-1117	166	30	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	166	31	)	)	PUNCT
cana-1117	166	32	⊙	⊙	PROPN
cana-1117	166	33	∫	∫	PROPN
cana-1117	166	34	𝜏1	𝜏1	PROPN
cana-1117	166	35	𝜏0	𝜏0	PROPN
cana-1117	166	36	(	(	PUNCT
cana-1117	166	37	(	(	PUNCT
cana-1117	166	38	𝜏1	𝜏1	NOUN
cana-1117	166	39	−	−	NOUN
cana-1117	166	40	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	166	41	−	−	PROPN
cana-1117	166	42	(	(	PUNCT
cana-1117	166	43	𝜏2	𝜏2	PROPN
cana-1117	166	44	−	−	PROPN
cana-1117	166	45	𝔮𝑠)𝛼−1)(‖𝜙(𝑠	𝔮𝑠)𝛼−1)(‖𝜙(𝑠	NOUN
cana-1117	166	46	)	)	PUNCT
cana-1117	166	47	−	−	PROPN
cana-1117	166	48	𝜙0‖	𝜙0‖	PROPN
cana-1117	166	49	+	+	CCONJ
cana-1117	166	50	‖𝜙0‖)𝑑𝔮𝑠	‖𝜙0‖)𝑑𝔮𝑠	NOUN
cana-1117	166	51	+	+	CCONJ
cana-1117	166	52	𝜆	𝜆	NOUN
cana-1117	166	53	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	166	54	)	)	PUNCT
cana-1117	166	55	⊙	⊙	PROPN
cana-1117	166	56	∫	∫	PROPN
cana-1117	166	57	𝜏2	𝜏2	PROPN
cana-1117	166	58	𝜏0	𝜏0	PROPN
cana-1117	166	59	(	(	PUNCT
cana-1117	166	60	(	(	PUNCT
cana-1117	166	61	𝜏2	𝜏2	PROPN
cana-1117	166	62	−	−	PROPN
cana-1117	166	63	𝔮𝑠)𝛼−1)(‖𝜙(𝑠	𝔮𝑠)𝛼−1)(‖𝜙(𝑠	NOUN
cana-1117	166	64	)	)	PUNCT
cana-1117	166	65	−	−	PROPN
cana-1117	166	66	𝜙0‖	𝜙0‖	PROPN
cana-1117	166	67	+	+	CCONJ
cana-1117	166	68	‖𝜙0‖)𝑑𝔮𝑠	‖𝜙0‖)𝑑𝔮𝑠	PROPN
cana-1117	166	69	≤	≤	NOUN
cana-1117	166	70	𝜆	𝜆	PRON
cana-1117	166	71	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	166	72	)	)	PUNCT
cana-1117	166	73	⊙	⊙	PROPN
cana-1117	166	74	∫	∫	PROPN
cana-1117	166	75	𝜏1	𝜏1	PROPN
cana-1117	166	76	𝜏0	𝜏0	PROPN
cana-1117	166	77	(	(	PUNCT
cana-1117	166	78	(	(	PUNCT
cana-1117	166	79	𝜏1	𝜏1	NOUN
cana-1117	166	80	−	−	NOUN
cana-1117	166	81	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	166	82	−	−	PROPN
cana-1117	166	83	(	(	PUNCT
cana-1117	166	84	𝜏2	𝜏2	PROPN
cana-1117	166	85	−	−	PROPN
cana-1117	166	86	𝔮𝑠)𝛼−1)(𝜖	𝔮𝑠)𝛼−1)(𝜖	NOUN
cana-1117	167	1	+	+	CCONJ
cana-1117	167	2	‖𝜙0‖)𝑑𝔮𝑠	‖𝜙0‖)𝑑𝔮𝑠	ADJ
cana-1117	167	3	+	+	CCONJ
cana-1117	167	4	𝜆	𝜆	NOUN
cana-1117	167	5	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	167	6	)	)	PUNCT
cana-1117	167	7	⊙	⊙	PROPN
cana-1117	167	8	∫	∫	PROPN
cana-1117	167	9	𝜏2	𝜏2	PROPN
cana-1117	167	10	𝜏0	𝜏0	PROPN
cana-1117	167	11	(	(	PUNCT
cana-1117	167	12	(	(	PUNCT
cana-1117	167	13	𝜏2	𝜏2	PROPN
cana-1117	167	14	−	−	PROPN
cana-1117	167	15	𝔮𝑠)𝛼−1)(𝜖	𝔮𝑠)𝛼−1)(𝜖	NOUN
cana-1117	167	16	+	+	CCONJ
cana-1117	167	17	‖𝜙0‖)𝑑𝔮𝑠	‖𝜙0‖)𝑑𝔮𝑠	ADJ
cana-1117	167	18	≤	≤	NUM
cana-1117	167	19	𝜆(𝜖+‖𝜙0‖	𝜆(𝜖+‖𝜙0‖	NOUN
cana-1117	167	20	)	)	PUNCT
cana-1117	167	21	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	167	22	)	)	PUNCT
cana-1117	167	23	consider	consider	VERB
cana-1117	167	24	1	1	NUM
cana-1117	167	25	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	167	26	)	)	PUNCT
cana-1117	167	27	⊙	⊙	PROPN
cana-1117	167	28	‖	‖	PROPN
cana-1117	167	29	∫	∫	PROPN
cana-1117	167	30	𝜏1	𝜏1	PROPN
cana-1117	167	31	𝜏0	𝜏0	PROPN
cana-1117	167	32	(	(	PUNCT
cana-1117	167	33	𝜏1	𝜏1	NOUN
cana-1117	167	34	−	−	PROPN
cana-1117	167	35	𝔮𝑠)𝛼−1𝑓(𝑠	𝔮𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	167	36	,	,	PUNCT
cana-1117	167	37	𝜙(𝑠	𝜙(𝑠	PROPN
cana-1117	167	38	)	)	PUNCT
cana-1117	167	39	,	,	PUNCT
cana-1117	167	40	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	167	41	)	)	PUNCT
cana-1117	167	42	,	,	PUNCT
cana-1117	167	43	𝒮𝜙(𝑠))𝑑𝔮𝑠	𝒮𝜙(𝑠))𝑑𝔮𝑠	ADJ
cana-1117	167	44	−	−	PROPN
cana-1117	167	45	∫	∫	PROPN
cana-1117	167	46	𝜏2	𝜏2	PROPN
cana-1117	167	47	𝜏0	𝜏0	PROPN
cana-1117	167	48	(	(	PUNCT
cana-1117	167	49	𝜏2	𝜏2	PROPN
cana-1117	167	50	−	−	PROPN
cana-1117	167	51	𝔮𝑠)𝛼−1𝑓(𝑠	𝔮𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	167	52	,	,	PUNCT
cana-1117	167	53	𝜙(𝑠	𝜙(𝑠	PROPN
cana-1117	167	54	)	)	PUNCT
cana-1117	167	55	,	,	PUNCT
cana-1117	167	56	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	167	57	)	)	PUNCT
cana-1117	167	58	,	,	PUNCT
cana-1117	167	59	𝒮𝜙(𝑠))𝑑𝔮𝑠	𝒮𝜙(𝑠))𝑑𝔮𝑠	NOUN
cana-1117	167	60	‖	‖	PROPN
cana-1117	167	61	≤	≤	ADV
cana-1117	167	62	1	1	NUM
cana-1117	167	63	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	167	64	)	)	PUNCT
cana-1117	167	65	⊙	⊙	PROPN
cana-1117	167	66	∫	∫	PROPN
cana-1117	167	67	𝜏1	𝜏1	PROPN
cana-1117	167	68	𝜏0	𝜏0	PROPN
cana-1117	167	69	(	(	PUNCT
cana-1117	167	70	(	(	PUNCT
cana-1117	167	71	𝜏1	𝜏1	NOUN
cana-1117	167	72	−	−	NOUN
cana-1117	167	73	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	167	74	−	−	PROPN
cana-1117	167	75	(	(	PUNCT
cana-1117	167	76	𝜏2	𝜏2	PROPN
cana-1117	167	77	−	−	PROPN
cana-1117	167	78	𝔮𝑠)𝛼−1)‖𝑓(𝑠	𝔮𝑠)𝛼−1)‖𝑓(𝑠	NOUN
cana-1117	167	79	,	,	PUNCT
cana-1117	167	80	𝜙(𝑠	𝜙(𝑠	PROPN
cana-1117	167	81	)	)	PUNCT
cana-1117	167	82	,	,	PUNCT
cana-1117	167	83	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	167	84	)	)	PUNCT
cana-1117	167	85	,	,	PUNCT
cana-1117	167	86	𝒮𝜙(𝑠	𝒮𝜙(𝑠	NOUN
cana-1117	167	87	)	)	PUNCT
cana-1117	167	88	)	)	PUNCT
cana-1117	167	89	−	−	PROPN
cana-1117	167	90	𝑓(𝑠	𝑓(𝑠	NOUN
cana-1117	167	91	,	,	PUNCT
cana-1117	167	92	0,0,0)‖𝑑𝔮𝑠	0,0,0)‖𝑑𝔮𝑠	NOUN
cana-1117	167	93	+	+	CCONJ
cana-1117	167	94	1	1	NUM
cana-1117	167	95	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	167	96	)	)	PUNCT
cana-1117	167	97	⊙	⊙	PROPN
cana-1117	167	98	∫	∫	PROPN
cana-1117	167	99	𝜏1	𝜏1	PROPN
cana-1117	167	100	𝜏0	𝜏0	PROPN
cana-1117	167	101	(	(	PUNCT
cana-1117	167	102	𝜏2	𝜏2	PROPN
cana-1117	167	103	−	−	PROPN
cana-1117	167	104	𝔮𝑠)𝛼−1‖𝑓(𝑠	𝔮𝑠)𝛼−1‖𝑓(𝑠	PROPN
cana-1117	167	105	,	,	PUNCT
cana-1117	167	106	0,0,0)‖𝑑𝔮𝑠	0,0,0)‖𝑑𝔮𝑠	NOUN
cana-1117	167	107	communications	communication	NOUN
cana-1117	167	108	on	on	ADP
cana-1117	167	109	applied	apply	VERB
cana-1117	167	110	nonlinear	nonlinear	ADJ
cana-1117	167	111	analysis	analysis	NOUN
cana-1117	167	112	issn	issn	NOUN
cana-1117	167	113	:	:	PUNCT
cana-1117	167	114	1074	1074	NUM
cana-1117	167	115	-	-	PUNCT
cana-1117	167	116	133x	133x	NUM
cana-1117	167	117	vol	vol	NOUN
cana-1117	167	118	31	31	NUM
cana-1117	167	119	no	no	NOUN
cana-1117	167	120	.	.	PUNCT
cana-1117	168	1	6s	6s	NUM
cana-1117	168	2	(	(	PUNCT
cana-1117	168	3	2024	2024	NUM
cana-1117	168	4	)	)	PUNCT
cana-1117	168	5	44	44	NUM
cana-1117	168	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	168	7	+	+	CCONJ
cana-1117	168	8	1	1	NUM
cana-1117	168	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	168	10	)	)	PUNCT
cana-1117	169	1	⊙	⊙	PROPN
cana-1117	169	2	∫	∫	PROPN
cana-1117	170	1	𝜏2	𝜏2	PROPN
cana-1117	170	2	𝜏1	𝜏1	NOUN
cana-1117	170	3	(	(	PUNCT
cana-1117	170	4	(	(	PUNCT
cana-1117	170	5	𝜏1	𝜏1	NOUN
cana-1117	170	6	−	−	NOUN
cana-1117	170	7	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	170	8	−	−	PROPN
cana-1117	170	9	(	(	PUNCT
cana-1117	170	10	𝜏2	𝜏2	PROPN
cana-1117	170	11	−	−	PROPN
cana-1117	170	12	𝔮𝑠)𝛼−1)‖𝑓(𝑠	𝔮𝑠)𝛼−1)‖𝑓(𝑠	NOUN
cana-1117	170	13	,	,	PUNCT
cana-1117	170	14	𝜙(𝑠	𝜙(𝑠	PROPN
cana-1117	170	15	)	)	PUNCT
cana-1117	170	16	,	,	PUNCT
cana-1117	170	17	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	170	18	)	)	PUNCT
cana-1117	170	19	,	,	PUNCT
cana-1117	170	20	𝒮𝜙(𝑠	𝒮𝜙(𝑠	NOUN
cana-1117	170	21	)	)	PUNCT
cana-1117	170	22	)	)	PUNCT
cana-1117	171	1	−	−	PROPN
cana-1117	171	2	𝑓(𝑠	𝑓(𝑠	NOUN
cana-1117	171	3	,	,	PUNCT
cana-1117	171	4	0,0,0)‖𝑑𝔮𝑠	0,0,0)‖𝑑𝔮𝑠	NOUN
cana-1117	171	5	1	1	NUM
cana-1117	171	6	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	171	7	)	)	PUNCT
cana-1117	171	8	⊙	⊙	PROPN
cana-1117	171	9	∫	∫	PROPN
cana-1117	171	10	𝜏2	𝜏2	PROPN
cana-1117	171	11	𝜏1	𝜏1	PROPN
cana-1117	171	12	(	(	PUNCT
cana-1117	171	13	𝜏2	𝜏2	PROPN
cana-1117	171	14	−	−	PROPN
cana-1117	171	15	𝔮𝑠)𝛼−1‖𝑓(𝑠	𝔮𝑠)𝛼−1‖𝑓(𝑠	PROPN
cana-1117	171	16	,	,	PUNCT
cana-1117	171	17	0,0,0)‖𝑑𝔮𝑠	0,0,0)‖𝑑𝔮𝑠	NOUN
cana-1117	171	18	≤	≤	NUM
cana-1117	171	19	ℒ(1+𝐾∗+𝐻∗	ℒ(1+𝐾∗+𝐻∗	PROPN
cana-1117	171	20	)	)	PUNCT
cana-1117	171	21	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	171	22	)	)	PUNCT
cana-1117	171	23	⊙	⊙	PROPN
cana-1117	171	24	∫	∫	PROPN
cana-1117	171	25	𝜏1	𝜏1	PROPN
cana-1117	171	26	𝜏0	𝜏0	PROPN
cana-1117	171	27	(	(	PUNCT
cana-1117	171	28	(	(	PUNCT
cana-1117	171	29	𝜏1	𝜏1	NOUN
cana-1117	171	30	−	−	NOUN
cana-1117	171	31	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	171	32	−	−	PROPN
cana-1117	171	33	(	(	PUNCT
cana-1117	171	34	𝜏2	𝜏2	PROPN
cana-1117	171	35	−	−	PROPN
cana-1117	171	36	𝔮𝑠)𝛼−1)(𝜖	𝔮𝑠)𝛼−1)(𝜖	NOUN
cana-1117	171	37	+	+	CCONJ
cana-1117	171	38	‖𝜙0‖)𝑑𝔮𝑠	‖𝜙0‖)𝑑𝔮𝑠	PROPN
cana-1117	171	39	+	+	NUM
cana-1117	171	40	𝑀	𝑀	PROPN
cana-1117	171	41	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-1117	171	42	)	)	PUNCT
cana-1117	171	43	⊙	⊙	PROPN
cana-1117	171	44	∫	∫	PROPN
cana-1117	171	45	𝜏1	𝜏1	PROPN
cana-1117	171	46	𝜏0	𝜏0	PROPN
cana-1117	171	47	(	(	PUNCT
cana-1117	171	48	(	(	PUNCT
cana-1117	171	49	𝜏1	𝜏1	NOUN
cana-1117	171	50	−	−	NOUN
cana-1117	171	51	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	171	52	−	−	PROPN
cana-1117	171	53	(	(	PUNCT
cana-1117	171	54	𝜏2	𝜏2	PROPN
cana-1117	171	55	−	−	PROPN
cana-1117	171	56	𝔮𝑠)𝛼−1)𝑑𝔮𝑠	𝔮𝑠)𝛼−1)𝑑𝔮𝑠	NUM
cana-1117	171	57	+	+	CCONJ
cana-1117	171	58	ℒ(1+𝐾∗+𝐻∗	ℒ(1+𝐾∗+𝐻∗	NOUN
cana-1117	171	59	)	)	PUNCT
cana-1117	171	60	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	171	61	)	)	PUNCT
cana-1117	171	62	⊙	⊙	PROPN
cana-1117	171	63	∫	∫	PROPN
cana-1117	171	64	𝜏2	𝜏2	PROPN
cana-1117	171	65	𝜏1	𝜏1	NOUN
cana-1117	171	66	(	(	PUNCT
cana-1117	171	67	(	(	PUNCT
cana-1117	171	68	𝜏1	𝜏1	NOUN
cana-1117	171	69	−	−	NOUN
cana-1117	171	70	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	171	71	−	−	PROPN
cana-1117	171	72	(	(	PUNCT
cana-1117	171	73	𝜏2	𝜏2	PROPN
cana-1117	171	74	−	−	PROPN
cana-1117	171	75	𝔮𝑠)𝛼−1)(𝜖	𝔮𝑠)𝛼−1)(𝜖	NOUN
cana-1117	171	76	+	+	CCONJ
cana-1117	171	77	‖𝜙0‖)𝑑𝔮𝑠	‖𝜙0‖)𝑑𝔮𝑠	PROPN
cana-1117	171	78	+	+	NUM
cana-1117	171	79	𝑀	𝑀	PROPN
cana-1117	171	80	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	171	81	)	)	PUNCT
cana-1117	171	82	⊙	⊙	PROPN
cana-1117	171	83	∫	∫	PROPN
cana-1117	171	84	𝜏2	𝜏2	PROPN
cana-1117	171	85	𝜏1	𝜏1	NOUN
cana-1117	171	86	(	(	PUNCT
cana-1117	171	87	(	(	PUNCT
cana-1117	171	88	𝜏1	𝜏1	NOUN
cana-1117	171	89	−	−	NOUN
cana-1117	171	90	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	171	91	−	−	PROPN
cana-1117	171	92	(	(	PUNCT
cana-1117	171	93	𝜏2	𝜏2	PROPN
cana-1117	171	94	−	−	PROPN
cana-1117	171	95	𝔮𝑠)𝛼−1)𝑑𝔮𝑠	𝔮𝑠)𝛼−1)𝑑𝔮𝑠	NUM
cana-1117	171	96	then	then	ADV
cana-1117	171	97	‖𝒦𝜙(𝜏1	‖𝒦𝜙(𝜏1	NUM
cana-1117	171	98	)	)	PUNCT
cana-1117	172	1	−	−	ADP
cana-1117	172	2	𝒦𝜙(𝜏2)‖	𝒦𝜙(𝜏2)‖	NOUN
cana-1117	172	3	≤	≤	NOUN
cana-1117	172	4	(	(	PUNCT
cana-1117	172	5	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	X
cana-1117	172	6	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-1117	172	7	)	)	PUNCT
cana-1117	172	8	⊙	⊙	PROPN
cana-1117	172	9	∫	∫	PROPN
cana-1117	172	10	𝜏1	𝜏1	PROPN
cana-1117	172	11	𝜏0	𝜏0	PROPN
cana-1117	172	12	(	(	PUNCT
cana-1117	172	13	(	(	PUNCT
cana-1117	172	14	𝜏1	𝜏1	NOUN
cana-1117	172	15	−	−	NOUN
cana-1117	172	16	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	172	17	−(𝜏2	−(𝜏2	NUM
cana-1117	172	18	−	−	NOUN
cana-1117	172	19	𝔮𝑠)𝛼−1	𝔮𝑠)𝛼−1	NUM
cana-1117	172	20	)	)	PUNCT
cana-1117	172	21	𝑑𝔮𝑠	𝑑𝔮𝑠	NOUN
cana-1117	172	22	+	+	CCONJ
cana-1117	172	23	(	(	PUNCT
cana-1117	172	24	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	NUM
cana-1117	172	25	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	172	26	)	)	PUNCT
cana-1117	172	27	⊙	⊙	PROPN
cana-1117	172	28	∫	∫	PROPN
cana-1117	172	29	𝜏2	𝜏2	PROPN
cana-1117	172	30	𝜏1	𝜏1	PROPN
cana-1117	172	31	(	(	PUNCT
cana-1117	172	32	𝜏2	𝜏2	PROPN
cana-1117	172	33	−	−	PROPN
cana-1117	172	34	𝔮𝑠)𝛼−1𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝑑𝔮𝑠	NOUN
cana-1117	172	35	=	=	SYM
cana-1117	172	36	(	(	PUNCT
cana-1117	172	37	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	𝜆+ℒ(1+𝐾∗+𝐻∗))(𝜖+𝜙0)+𝑀	X
cana-1117	172	38	γ𝑞(𝛼	γ𝑞(𝛼	X
cana-1117	172	39	)	)	PUNCT
cana-1117	172	40	⊙	⊙	NOUN
cana-1117	172	41	(	(	PUNCT
cana-1117	172	42	2(𝜏2	2(𝜏2	PROPN
cana-1117	172	43	−	−	NOUN
cana-1117	172	44	𝜏1)𝛼	𝜏1)𝛼	NOUN
cana-1117	172	45	+	+	CCONJ
cana-1117	172	46	𝜏1	𝜏1	NOUN
cana-1117	172	47	𝛼	𝛼	NOUN
cana-1117	172	48	−	−	PROPN
cana-1117	172	49	𝜏2	𝜏2	PROPN
cana-1117	172	50	𝛼	𝛼	NOUN
cana-1117	172	51	)	)	PUNCT
cana-1117	172	52	hence	hence	ADV
cana-1117	172	53	𝒦𝜙	𝒦𝜙	PROPN
cana-1117	172	54	is	be	AUX
cana-1117	172	55	continuous	continuous	ADJ
cana-1117	172	56	.	.	PUNCT
cana-1117	173	1	hence	hence	ADV
cana-1117	173	2	for	for	ADP
cana-1117	173	3	any	any	DET
cana-1117	173	4	𝜙	𝜙	PRON
cana-1117	173	5	∈	∈	PROPN
cana-1117	173	6	𝔹	𝔹	PROPN
cana-1117	173	7	,	,	PUNCT
cana-1117	173	8	we	we	PRON
cana-1117	173	9	have	have	VERB
cana-1117	173	10	𝒦𝜙	𝒦𝜙	PROPN
cana-1117	173	11	∈	∈	PROPN
cana-1117	173	12	𝐶([0	𝐶([0	PROPN
cana-1117	173	13	,	,	PUNCT
cana-1117	173	14	𝜅	𝜅	ADP
cana-1117	173	15	]	]	X
cana-1117	173	16	,	,	PUNCT
cana-1117	173	17	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	173	18	)	)	PUNCT
cana-1117	173	19	i.e.	i.e.	X
cana-1117	173	20	𝒦𝜙(𝜏0	𝒦𝜙(𝜏0	PROPN
cana-1117	173	21	)	)	PUNCT
cana-1117	174	1	=	=	NOUN
cana-1117	174	2	𝜙0	𝜙0	NOUN
cana-1117	174	3	and	and	CCONJ
cana-1117	174	4	‖𝒦𝜙	‖𝒦𝜙	ADJ
cana-1117	174	5	−	−	PROPN
cana-1117	174	6	𝜙0‖	𝜙0‖	PROPN
cana-1117	174	7	≤	≤	NUM
cana-1117	174	8	𝜖.	𝜖.	NOUN
cana-1117	175	1	this	this	PRON
cana-1117	175	2	concludes	conclude	VERB
cana-1117	175	3	that	that	SCONJ
cana-1117	175	4	𝒦𝜙	𝒦𝜙	PROPN
cana-1117	175	5	∈	∈	PROPN
cana-1117	175	6	𝔹	𝔹	PROPN
cana-1117	175	7	whenever	whenever	SCONJ
cana-1117	175	8	𝑢	𝑢	X
cana-1117	175	9	∈	∈	PROPN
cana-1117	175	10	𝔹.	𝔹.	NOUN
cana-1117	175	11	i.e.	i.e.	X
cana-1117	175	12	𝒦𝜙	𝒦𝜙	PROPN
cana-1117	175	13	maps	map	NOUN
cana-1117	175	14	𝔹	𝔹	VERB
cana-1117	175	15	into	into	ADP
cana-1117	175	16	itself	itself	PRON
cana-1117	175	17	.	.	PUNCT
cana-1117	176	1	the	the	DET
cana-1117	176	2	next	next	ADJ
cana-1117	176	3	step	step	NOUN
cana-1117	176	4	is	be	AUX
cana-1117	176	5	to	to	PART
cana-1117	176	6	prove	prove	VERB
cana-1117	176	7	that	that	SCONJ
cana-1117	176	8	,	,	PUNCT
cana-1117	176	9	for	for	ADP
cana-1117	176	10	every	every	DET
cana-1117	176	11	𝑛	𝑛	DET
cana-1117	176	12	∈	∈	PROPN
cana-1117	176	13	ℕ	ℕ	PROPN
cana-1117	176	14	∪	∪	X
cana-1117	176	15	{	{	PUNCT
cana-1117	176	16	0	0	NUM
cana-1117	176	17	}	}	PUNCT
cana-1117	176	18	,	,	PUNCT
cana-1117	176	19	and	and	CCONJ
cana-1117	176	20	every	every	DET
cana-1117	176	21	𝜙	𝜙	NOUN
cana-1117	176	22	,	,	PUNCT
cana-1117	176	23	𝜓	𝜓	PROPN
cana-1117	176	24	∈	∈	PROPN
cana-1117	176	25	𝔹	𝔹	PROPN
cana-1117	176	26	,	,	PUNCT
cana-1117	176	27	we	we	PRON
cana-1117	176	28	have	have	VERB
cana-1117	176	29	‖𝒦𝑛𝜙	‖𝒦𝑛𝜙	PROPN
cana-1117	176	30	−	−	PROPN
cana-1117	176	31	𝒦𝑛𝜓‖	𝒦𝑛𝜓‖	PROPN
cana-1117	176	32	≤	≤	NOUN
cana-1117	177	1	[	[	X
cana-1117	177	2	𝜆+ℒ(1+𝐾∗+𝐻∗)]𝑛	𝜆+ℒ(1+𝐾∗+𝐻∗)]𝑛	NOUN
cana-1117	177	3	γ𝑞(𝑛𝛼+1	γ𝑞(𝑛𝛼+1	NOUN
cana-1117	177	4	)	)	PUNCT
cana-1117	177	5	‖𝜙	‖𝜙	NOUN
cana-1117	177	6	−	−	NOUN
cana-1117	177	7	𝜓‖	𝜓‖	NOUN
cana-1117	177	8	,	,	PUNCT
cana-1117	177	9	𝜏	𝜏	PRON
cana-1117	177	10	∈	∈	NOUN
cana-1117	178	1	[	[	X
cana-1117	178	2	0	0	NUM
cana-1117	178	3	,	,	PUNCT
cana-1117	178	4	𝜅	𝜅	X
cana-1117	178	5	]	]	X
cana-1117	178	6	(	(	PUNCT
cana-1117	178	7	4.3	4.3	NUM
cana-1117	178	8	)	)	PUNCT
cana-1117	178	9	this	this	PRON
cana-1117	178	10	can	can	AUX
cana-1117	178	11	be	be	AUX
cana-1117	178	12	seen	see	VERB
cana-1117	178	13	by	by	ADP
cana-1117	178	14	induction	induction	NOUN
cana-1117	178	15	.	.	PUNCT
cana-1117	179	1	for	for	ADP
cana-1117	179	2	𝑛	𝑛	PROPN
cana-1117	179	3	=	=	SYM
cana-1117	179	4	0	0	NUM
cana-1117	179	5	,	,	PUNCT
cana-1117	179	6	the	the	DET
cana-1117	179	7	inequality	inequality	NOUN
cana-1117	179	8	(	(	PUNCT
cana-1117	179	9	4.4	4.4	NUM
cana-1117	179	10	)	)	PUNCT
cana-1117	179	11	is	be	AUX
cana-1117	179	12	trivially	trivially	ADV
cana-1117	179	13	true	true	ADJ
cana-1117	179	14	.	.	PUNCT
cana-1117	180	1	we	we	PRON
cana-1117	180	2	assume	assume	VERB
cana-1117	180	3	that	that	SCONJ
cana-1117	180	4	(	(	PUNCT
cana-1117	180	5	4.4	4.4	NUM
cana-1117	180	6	)	)	PUNCT
cana-1117	180	7	is	be	AUX
cana-1117	180	8	true	true	ADJ
cana-1117	180	9	for	for	ADP
cana-1117	180	10	𝑛	𝑛	PROPN
cana-1117	180	11	=	=	SYM
cana-1117	180	12	𝑚	𝑚	PROPN
cana-1117	180	13	−	−	NUM
cana-1117	180	14	1	1	NUM
cana-1117	180	15	and	and	CCONJ
cana-1117	180	16	prove	prove	VERB
cana-1117	180	17	it	it	PRON
cana-1117	180	18	for	for	ADP
cana-1117	180	19	𝑛	𝑛	NOUN
cana-1117	180	20	=	=	PUNCT
cana-1117	180	21	𝑚.	𝑚.	ADV
cana-1117	180	22	by	by	ADP
cana-1117	180	23	using	use	VERB
cana-1117	180	24	definition	definition	NOUN
cana-1117	180	25	of	of	ADP
cana-1117	180	26	operator	operator	NOUN
cana-1117	180	27	𝒦	𝒦	PROPN
cana-1117	180	28	,	,	PUNCT
cana-1117	180	29	we	we	PRON
cana-1117	180	30	have	have	AUX
cana-1117	180	31	‖𝒦𝑚𝜙(𝜏	‖𝒦𝑚𝜙(𝜏	NOUN
cana-1117	180	32	)	)	PUNCT
cana-1117	181	1	−	−	PROPN
cana-1117	181	2	𝒦𝑚𝜓(𝜏)‖	𝒦𝑚𝜓(𝜏)‖	VERB
cana-1117	181	3	≤	≤	NOUN
cana-1117	181	4	𝜆	𝜆	DET
cana-1117	181	5	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	181	6	)	)	PUNCT
cana-1117	181	7	⊙	⊙	PROPN
cana-1117	181	8	∫	∫	PROPN
cana-1117	182	1	𝜏	𝜏	PROPN
cana-1117	182	2	𝜏0	𝜏0	PROPN
cana-1117	182	3	(	(	PUNCT
cana-1117	182	4	𝜏	𝜏	NOUN
cana-1117	182	5	−	−	NOUN
cana-1117	182	6	𝑠)𝛼−1‖𝒦𝑚−1𝜙(𝑠	𝑠)𝛼−1‖𝒦𝑚−1𝜙(𝑠	ADJ
cana-1117	182	7	)	)	PUNCT
cana-1117	182	8	−	−	PROPN
cana-1117	182	9	𝒦𝑚−1𝜓(𝑠)‖𝑑𝔮𝑠	𝒦𝑚−1𝜓(𝑠)‖𝑑𝔮𝑠	ADJ
cana-1117	182	10	+	+	CCONJ
cana-1117	182	11	1	1	NUM
cana-1117	182	12	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	182	13	)	)	PUNCT
cana-1117	182	14	⊙	⊙	PROPN
cana-1117	182	15	∫	∫	PROPN
cana-1117	183	1	𝜏	𝜏	PROPN
cana-1117	183	2	𝜏0	𝜏0	PROPN
cana-1117	183	3	(	(	PUNCT
cana-1117	183	4	𝜏	𝜏	NOUN
cana-1117	183	5	−	−	NOUN
cana-1117	183	6	𝑠)𝛼−1	𝑠)𝛼−1	NOUN
cana-1117	183	7	‖	‖	ADJ
cana-1117	183	8	𝑓(𝑠	𝑓(𝑠	NOUN
cana-1117	183	9	,	,	PUNCT
cana-1117	183	10	𝒦𝑚−1𝜙(𝑠	𝒦𝑚−1𝜙(𝑠	NOUN
cana-1117	183	11	)	)	PUNCT
cana-1117	183	12	,	,	PUNCT
cana-1117	183	13	𝒦𝑚−1𝒢𝜙(𝑠	𝒦𝑚−1𝒢𝜙(𝑠	NOUN
cana-1117	183	14	)	)	PUNCT
cana-1117	183	15	,	,	PUNCT
cana-1117	183	16	𝒦𝑚−1𝒮𝜙(𝑠	𝒦𝑚−1𝒮𝜙(𝑠	NOUN
cana-1117	183	17	)	)	PUNCT
cana-1117	183	18	)	)	PUNCT
cana-1117	183	19	−𝑓(𝑠	−𝑓(𝑠	NOUN
cana-1117	183	20	,	,	PUNCT
cana-1117	183	21	𝒦𝑚−1𝜓(𝑠	𝒦𝑚−1𝜓(𝑠	NOUN
cana-1117	183	22	)	)	PUNCT
cana-1117	183	23	,	,	PUNCT
cana-1117	183	24	𝒦𝑚−1𝒢𝜓(𝑠	𝒦𝑚−1𝒢𝜓(𝑠	PROPN
cana-1117	183	25	)	)	PUNCT
cana-1117	183	26	,	,	PUNCT
cana-1117	183	27	𝒦𝑚−1𝒮𝜓(𝑠	𝒦𝑚−1𝒮𝜓(𝑠	PROPN
cana-1117	183	28	)	)	PUNCT
cana-1117	183	29	)	)	PUNCT
cana-1117	184	1	‖	‖	PROPN
cana-1117	184	2	𝑑𝑞𝑠	𝑑𝑞𝑠	VERB
cana-1117	184	3	for	for	ADP
cana-1117	184	4	𝑛	𝑛	PROPN
cana-1117	184	5	=	=	SYM
cana-1117	184	6	𝑚	𝑚	PROPN
cana-1117	184	7	−	−	NUM
cana-1117	184	8	1	1	NUM
cana-1117	184	9	,	,	PUNCT
cana-1117	184	10	we	we	PRON
cana-1117	184	11	get	get	VERB
cana-1117	184	12	‖𝒦𝑚𝜙(𝜏	‖𝒦𝑚𝜙(𝜏	NOUN
cana-1117	184	13	)	)	PUNCT
cana-1117	185	1	−	−	PROPN
cana-1117	185	2	𝒦𝑚𝜓(𝜏)‖	𝒦𝑚𝜓(𝜏)‖	VERB
cana-1117	185	3	≤	≤	NUM
cana-1117	185	4	𝜆	𝜆	ADP
cana-1117	185	5	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-1117	185	6	)	)	PUNCT
cana-1117	185	7	(	(	PUNCT
cana-1117	185	8	𝜆+ℒ(1+𝐾∗+𝐻∗))𝑚−1	𝜆+ℒ(1+𝐾∗+𝐻∗))𝑚−1	PROPN
cana-1117	185	9	γ𝑞((𝑚−1)𝛼+1	γ𝑞((𝑚−1)𝛼+1	NUM
cana-1117	185	10	)	)	PUNCT
cana-1117	185	11	‖𝜙	‖𝜙	NOUN
cana-1117	185	12	−	−	NOUN
cana-1117	185	13	𝜓‖	𝜓‖	NOUN
cana-1117	185	14	⊙	⊙	PROPN
cana-1117	185	15	∫	∫	PROPN
cana-1117	185	16	𝜏	𝜏	PROPN
cana-1117	185	17	𝜏0	𝜏0	PROPN
cana-1117	185	18	(	(	PUNCT
cana-1117	185	19	𝜏	𝜏	NOUN
cana-1117	185	20	−	−	PROPN
cana-1117	185	21	𝑠)𝛼−1𝑠𝑚𝛼−𝛼𝑑𝔮𝑠	𝑠)𝛼−1𝑠𝑚𝛼−𝛼𝑑𝔮𝑠	X
cana-1117	185	22	+	+	CCONJ
cana-1117	185	23	ℒ(1+𝐾∗+𝐻∗	ℒ(1+𝐾∗+𝐻∗	NOUN
cana-1117	185	24	)	)	PUNCT
cana-1117	185	25	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	185	26	)	)	PUNCT
cana-1117	185	27	⊙	⊙	PROPN
cana-1117	185	28	∫	∫	PROPN
cana-1117	186	1	𝜏	𝜏	PROPN
cana-1117	186	2	𝜏0	𝜏0	PROPN
cana-1117	186	3	(	(	PUNCT
cana-1117	186	4	𝜏	𝜏	NOUN
cana-1117	186	5	−	−	NOUN
cana-1117	186	6	𝑠)𝛼−1‖𝒦𝑚−1𝜙(𝑠	𝑠)𝛼−1‖𝒦𝑚−1𝜙(𝑠	ADJ
cana-1117	186	7	)	)	PUNCT
cana-1117	186	8	−	−	PROPN
cana-1117	186	9	𝒦𝑚−1𝜓(𝑠)‖𝑑𝔮𝑠	𝒦𝑚−1𝜓(𝑠)‖𝑑𝔮𝑠	ADJ
cana-1117	186	10	≤	≤	NOUN
cana-1117	186	11	(	(	PUNCT
cana-1117	186	12	𝜆+ℒ	𝜆+ℒ	NUM
cana-1117	186	13	)	)	PUNCT
cana-1117	186	14	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	186	15	)	)	PUNCT
cana-1117	186	16	(	(	PUNCT
cana-1117	186	17	𝜆+ℒ(1+𝐾∗+𝐻∗))𝑚−1	𝜆+ℒ(1+𝐾∗+𝐻∗))𝑚−1	NOUN
cana-1117	186	18	γ𝔮((𝑚−1)𝛼+1	γ𝔮((𝑚−1)𝛼+1	NUM
cana-1117	186	19	)	)	PUNCT
cana-1117	186	20	‖𝜙	‖𝜙	NOUN
cana-1117	186	21	−	−	NOUN
cana-1117	186	22	𝜓‖	𝜓‖	NOUN
cana-1117	186	23	⊙	⊙	PROPN
cana-1117	186	24	∫	∫	PROPN
cana-1117	187	1	𝜏	𝜏	PROPN
cana-1117	187	2	𝜏0	𝜏0	PROPN
cana-1117	187	3	(	(	PUNCT
cana-1117	187	4	𝜏	𝜏	NOUN
cana-1117	187	5	−	−	PRON
cana-1117	187	6	𝑠)𝛼−1𝑠𝑚𝛼−𝛼𝑑𝔮𝑠	𝑠)𝛼−1𝑠𝑚𝛼−𝛼𝑑𝔮𝑠	NOUN
cana-1117	187	7	communications	communication	NOUN
cana-1117	187	8	on	on	ADP
cana-1117	187	9	applied	apply	VERB
cana-1117	187	10	nonlinear	nonlinear	ADJ
cana-1117	187	11	analysis	analysis	NOUN
cana-1117	187	12	issn	issn	NOUN
cana-1117	187	13	:	:	PUNCT
cana-1117	187	14	1074	1074	NUM
cana-1117	187	15	-	-	PUNCT
cana-1117	187	16	133x	133x	NUM
cana-1117	187	17	vol	vol	NOUN
cana-1117	187	18	31	31	NUM
cana-1117	187	19	no	no	NOUN
cana-1117	187	20	.	.	PUNCT
cana-1117	188	1	6s	6s	NUM
cana-1117	188	2	(	(	PUNCT
cana-1117	188	3	2024	2024	NUM
cana-1117	188	4	)	)	PUNCT
cana-1117	188	5	45	45	NUM
cana-1117	188	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	188	7	=	=	PUNCT
cana-1117	189	1	[	[	X
cana-1117	189	2	(	(	PUNCT
cana-1117	189	3	𝜆+ℒ(1+𝐾∗+𝐻∗))𝜏𝛼]𝑚	𝜆+ℒ(1+𝐾∗+𝐻∗))𝜏𝛼]𝑚	NUM
cana-1117	189	4	γ𝔮(𝑚𝛼+1	γ𝔮(𝑚𝛼+1	NOUN
cana-1117	189	5	)	)	PUNCT
cana-1117	189	6	‖𝜙	‖𝜙	NOUN
cana-1117	189	7	−	−	ADP
cana-1117	189	8	𝜓‖	𝜓‖	NOUN
cana-1117	189	9	which	which	PRON
cana-1117	189	10	is	be	AUX
cana-1117	189	11	our	our	PRON
cana-1117	189	12	desired	desire	VERB
cana-1117	189	13	inequality	inequality	NOUN
cana-1117	189	14	(	(	PUNCT
cana-1117	189	15	4.4	4.4	NUM
cana-1117	189	16	)	)	PUNCT
cana-1117	189	17	.	.	PUNCT
cana-1117	190	1	hence	hence	ADV
cana-1117	190	2	we	we	PRON
cana-1117	190	3	have	have	VERB
cana-1117	190	4	‖𝒦𝑛𝜙	‖𝒦𝑛𝜙	PROPN
cana-1117	190	5	−	−	PROPN
cana-1117	190	6	𝒦𝑛𝜓‖	𝒦𝑛𝜓‖	PROPN
cana-1117	190	7	≤	≤	PUNCT
cana-1117	191	1	[	[	X
cana-1117	191	2	(	(	PUNCT
cana-1117	191	3	𝜆+ℒ(1+𝐾∗+𝐻∗))𝜅𝛼]𝑛	𝜆+ℒ(1+𝐾∗+𝐻∗))𝜅𝛼]𝑛	NUM
cana-1117	191	4	γ𝔮(𝑚𝛼)+1	γ𝔮(𝑚𝛼)+1	PROPN
cana-1117	191	5	‖𝜙	‖𝜙	NOUN
cana-1117	191	6	−	−	NOUN
cana-1117	191	7	𝜓‖	𝜓‖	NOUN
cana-1117	191	8	by	by	ADP
cana-1117	191	9	definition(2.10	definition(2.10	NUM
cana-1117	191	10	)	)	PUNCT
cana-1117	191	11	,	,	PUNCT
cana-1117	191	12	we	we	PRON
cana-1117	191	13	have	have	VERB
cana-1117	191	14	∑∞	∑∞	VERB
cana-1117	191	15	𝑛=0	𝑛=0	X
cana-1117	192	1	[	[	X
cana-1117	192	2	(	(	PUNCT
cana-1117	192	3	𝜆+ℒ(1+𝐾∗+𝐻∗))𝜅𝛼]𝑛	𝜆+ℒ(1+𝐾∗+𝐻∗))𝜅𝛼]𝑛	PUNCT
cana-1117	192	4	γ𝑞(𝑛𝛼+1	γ𝑞(𝑛𝛼+1	NOUN
cana-1117	192	5	)	)	PUNCT
cana-1117	192	6	=	=	PUNCT
cana-1117	193	1	𝐸(𝜆	𝐸(𝜆	X
cana-1117	193	2	+	+	CCONJ
cana-1117	193	3	ℒ(1	ℒ(1	PUNCT
cana-1117	193	4	+	+	NUM
cana-1117	193	5	𝐾∗	𝐾∗	PUNCT
cana-1117	194	1	+	+	CCONJ
cana-1117	194	2	𝐻∗))𝜅𝛼	𝐻∗))𝜅𝛼	NOUN
cana-1117	194	3	we	we	PRON
cana-1117	194	4	have	have	AUX
cana-1117	194	5	proved	prove	VERB
cana-1117	194	6	that	that	SCONJ
cana-1117	194	7	the	the	DET
cana-1117	194	8	operator	operator	NOUN
cana-1117	194	9	𝒦	𝒦	PROPN
cana-1117	194	10	satisfies	satisfy	VERB
cana-1117	194	11	all	all	DET
cana-1117	194	12	the	the	DET
cana-1117	194	13	conditions	condition	NOUN
cana-1117	194	14	of	of	ADP
cana-1117	194	15	theorem	theorem	NOUN
cana-1117	194	16	(	(	PUNCT
cana-1117	194	17	2.3	2.3	NUM
cana-1117	194	18	)	)	PUNCT
cana-1117	194	19	and	and	CCONJ
cana-1117	194	20	hence	hence	ADV
cana-1117	194	21	ℒ	ℒ	PROPN
cana-1117	194	22	has	have	VERB
cana-1117	194	23	a	a	DET
cana-1117	194	24	unique	unique	ADJ
cana-1117	194	25	fixed	fix	VERB
cana-1117	194	26	point	point	NOUN
cana-1117	194	27	𝜙	𝜙	NOUN
cana-1117	194	28	:	:	PUNCT
cana-1117	194	29	[	[	X
cana-1117	194	30	0	0	NUM
cana-1117	194	31	,	,	PUNCT
cana-1117	194	32	𝜅	𝜅	ADP
cana-1117	194	33	]	]	X
cana-1117	194	34	→	→	SYM
cana-1117	194	35	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	194	36	which	which	PRON
cana-1117	194	37	is	be	AUX
cana-1117	194	38	the	the	DET
cana-1117	194	39	solution	solution	NOUN
cana-1117	194	40	of	of	ADP
cana-1117	194	41	(	(	PUNCT
cana-1117	194	42	1.1	1.1	NUM
cana-1117	194	43	)	)	PUNCT
cana-1117	194	44	.	.	PUNCT
cana-1117	195	1	5	5	NUM
cana-1117	195	2	continuous	continuous	ADJ
cana-1117	195	3	dependence	dependence	NOUN
cana-1117	195	4	and	and	CCONJ
cana-1117	195	5	uniqueness	uniqueness	NOUN
cana-1117	195	6	of	of	ADP
cana-1117	195	7	solutions	solution	NOUN
cana-1117	195	8	theorem	theorem	VERB
cana-1117	195	9	5.1	5.1	NUM
cana-1117	195	10	suppose	suppose	VERB
cana-1117	195	11	that	that	SCONJ
cana-1117	195	12	the	the	DET
cana-1117	195	13	function	function	NOUN
cana-1117	195	14	𝑓	𝑓	NOUN
cana-1117	195	15	:	:	PUNCT
cana-1117	195	16	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	195	17	×	×	NOUN
cana-1117	195	18	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	195	19	×	×	PROPN
cana-1117	195	20	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	195	21	×	×	PROPN
cana-1117	195	22	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	195	23	→	→	SYM
cana-1117	195	24	ℝ𝔽.	ℝ𝔽.	X
cana-1117	195	25	let	let	VERB
cana-1117	195	26	𝜙1(𝜏	𝜙1(𝜏	NUM
cana-1117	195	27	)	)	PUNCT
cana-1117	195	28	and	and	CCONJ
cana-1117	195	29	𝜙2(𝑡	𝜙2(𝑡	NUM
cana-1117	195	30	)	)	PUNCT
cana-1117	195	31	be	be	VERB
cana-1117	195	32	the	the	DET
cana-1117	195	33	solutions	solution	NOUN
cana-1117	195	34	of	of	ADP
cana-1117	195	35	equation	equation	NOUN
cana-1117	195	36	,	,	PUNCT
cana-1117	195	37	𝐶𝐷𝛼𝜙(𝜏	𝐶𝐷𝛼𝜙(𝜏	PROPN
cana-1117	195	38	)	)	PUNCT
cana-1117	195	39	=	=	SYM
cana-1117	195	40	𝜆𝜙(𝜏	𝜆𝜙(𝜏	X
cana-1117	195	41	)	)	PUNCT
cana-1117	196	1	+	+	CCONJ
cana-1117	196	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	196	3	,	,	PUNCT
cana-1117	196	4	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	196	5	)	)	PUNCT
cana-1117	196	6	,	,	PUNCT
cana-1117	196	7	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	196	8	)	)	PUNCT
cana-1117	196	9	,	,	PUNCT
cana-1117	196	10	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	196	11	)	)	PUNCT
cana-1117	196	12	)	)	PUNCT
cana-1117	196	13	,	,	PUNCT
cana-1117	196	14	𝜏	𝜏	PROPN
cana-1117	196	15	∈	∈	PROPN
cana-1117	196	16	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	196	17	(	(	PUNCT
cana-1117	196	18	5.1	5.1	NUM
cana-1117	196	19	)	)	PUNCT
cana-1117	196	20	corresponding	correspond	VERB
cana-1117	196	21	to	to	ADP
cana-1117	196	22	𝜙1(0	𝜙1(0	PROPN
cana-1117	196	23	)	)	PUNCT
cana-1117	196	24	=	=	SYM
cana-1117	196	25	𝜙0	𝜙0	NOUN
cana-1117	196	26	and	and	CCONJ
cana-1117	196	27	𝜙2(0	𝜙2(0	PROPN
cana-1117	196	28	)	)	PUNCT
cana-1117	196	29	=	=	SYM
cana-1117	196	30	𝜙0	𝜙0	NOUN
cana-1117	196	31	∗	∗	NOUN
cana-1117	196	32	respectively	respectively	ADV
cana-1117	196	33	.	.	PUNCT
cana-1117	197	1	then	then	ADV
cana-1117	197	2	‖𝜙1	‖𝜙1	PROPN
cana-1117	197	3	−	−	PROPN
cana-1117	197	4	𝜙2‖	𝜙2‖	PROPN
cana-1117	197	5	≤	≤	PROPN
cana-1117	197	6	{	{	PUNCT
cana-1117	197	7	1	1	NUM
cana-1117	197	8	+	+	PROPN
cana-1117	197	9	𝐾(𝜆+ℒ(1+𝐾∗+𝐻∗	𝐾(𝜆+ℒ(1+𝐾∗+𝐻∗	ADJ
cana-1117	197	10	)	)	PUNCT
cana-1117	197	11	)	)	PUNCT
cana-1117	197	12	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	197	13	)	)	PUNCT
cana-1117	198	1	𝑇𝛼	𝑇𝛼	NOUN
cana-1117	198	2	}	}	PUNCT
cana-1117	198	3	‖𝜙0	‖𝜙0	NUM
cana-1117	198	4	−	−	PROPN
cana-1117	198	5	𝜙0	𝜙0	NOUN
cana-1117	198	6	∗‖	∗‖	PROPN
cana-1117	198	7	,	,	PUNCT
cana-1117	198	8	𝜏	𝜏	PROPN
cana-1117	198	9	∈	∈	PROPN
cana-1117	198	10	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	198	11	let	let	VERB
cana-1117	198	12	𝜙1(𝜏	𝜙1(𝜏	PRON
cana-1117	198	13	)	)	PUNCT
cana-1117	198	14	and	and	CCONJ
cana-1117	198	15	𝜙2(𝜏	𝜙2(𝜏	PROPN
cana-1117	198	16	)	)	PUNCT
cana-1117	198	17	be	be	VERB
cana-1117	198	18	the	the	DET
cana-1117	198	19	solutions	solution	NOUN
cana-1117	198	20	of	of	ADP
cana-1117	198	21	equation	equation	NOUN
cana-1117	198	22	(	(	PUNCT
cana-1117	198	23	5.1	5.1	NUM
cana-1117	198	24	)	)	PUNCT
cana-1117	198	25	corresponding	correspond	VERB
cana-1117	198	26	to	to	ADP
cana-1117	198	27	𝜙1(0	𝜙1(0	PROPN
cana-1117	198	28	)	)	PUNCT
cana-1117	199	1	=	=	SYM
cana-1117	199	2	𝜙0	𝜙0	NOUN
cana-1117	199	3	and	and	CCONJ
cana-1117	199	4	𝜙2(0	𝜙2(0	PROPN
cana-1117	199	5	)	)	PUNCT
cana-1117	199	6	=	=	SYM
cana-1117	200	1	𝜙0	𝜙0	NOUN
cana-1117	200	2	∗	∗	NOUN
cana-1117	200	3	respectively	respectively	ADV
cana-1117	200	4	.	.	PUNCT
cana-1117	201	1	𝐶𝐷𝛼𝜙1(𝜏	𝐶𝐷𝛼𝜙1(𝜏	NOUN
cana-1117	201	2	)	)	PUNCT
cana-1117	201	3	=	=	SYM
cana-1117	201	4	𝜆𝜙1(𝜏	𝜆𝜙1(𝜏	NOUN
cana-1117	201	5	)	)	PUNCT
cana-1117	202	1	+	+	CCONJ
cana-1117	202	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	202	3	,	,	PUNCT
cana-1117	202	4	𝜙1(𝜏	𝜙1(𝜏	NOUN
cana-1117	202	5	)	)	PUNCT
cana-1117	202	6	,	,	PUNCT
cana-1117	202	7	𝒢𝜙1(𝜏	𝒢𝜙1(𝜏	PROPN
cana-1117	202	8	)	)	PUNCT
cana-1117	202	9	,	,	PUNCT
cana-1117	202	10	𝒮𝜙1(𝜏	𝒮𝜙1(𝜏	NOUN
cana-1117	202	11	)	)	PUNCT
cana-1117	202	12	)	)	PUNCT
cana-1117	202	13	,	,	PUNCT
cana-1117	202	14	𝜙1(0	𝜙1(0	NOUN
cana-1117	202	15	)	)	PUNCT
cana-1117	202	16	=	=	SYM
cana-1117	202	17	𝜙0	𝜙0	NOUN
cana-1117	202	18	,	,	PUNCT
cana-1117	202	19	𝐶𝐷𝛼𝜙2(𝜏	𝐶𝐷𝛼𝜙2(𝜏	ADV
cana-1117	202	20	)	)	PUNCT
cana-1117	202	21	=	=	SYM
cana-1117	202	22	𝜆𝜙2(𝜏	𝜆𝜙2(𝜏	PROPN
cana-1117	202	23	)	)	PUNCT
cana-1117	203	1	+	+	CCONJ
cana-1117	203	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	203	3	,	,	PUNCT
cana-1117	203	4	𝜙2(𝜏	𝜙2(𝜏	PROPN
cana-1117	203	5	)	)	PUNCT
cana-1117	203	6	,	,	PUNCT
cana-1117	203	7	𝒢𝜙2(𝜏	𝒢𝜙2(𝜏	PROPN
cana-1117	203	8	)	)	PUNCT
cana-1117	203	9	,	,	PUNCT
cana-1117	203	10	𝒮𝜙2(𝜏	𝒮𝜙2(𝜏	NOUN
cana-1117	203	11	)	)	PUNCT
cana-1117	203	12	)	)	PUNCT
cana-1117	203	13	,	,	PUNCT
cana-1117	203	14	𝜏2(0	𝜏2(0	PROPN
cana-1117	203	15	)	)	PUNCT
cana-1117	204	1	=	=	SYM
cana-1117	204	2	𝜙0	𝜙0	NOUN
cana-1117	204	3	∗	∗	NOUN
cana-1117	204	4	this	this	PRON
cana-1117	204	5	implies	imply	VERB
cana-1117	204	6	that	that	SCONJ
cana-1117	204	7	,	,	PUNCT
cana-1117	204	8	𝜙1(𝜏	𝜙1(𝜏	NOUN
cana-1117	204	9	)	)	PUNCT
cana-1117	204	10	=	=	SYM
cana-1117	204	11	𝜙0	𝜙0	NOUN
cana-1117	204	12	+	+	CCONJ
cana-1117	204	13	𝜆	𝜆	PRON
cana-1117	204	14	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	204	15	)	)	PUNCT
cana-1117	204	16	⊙	⊙	PROPN
cana-1117	204	17	∫	∫	PROPN
cana-1117	205	1	𝜏	𝜏	X
cana-1117	205	2	0	0	NUM
cana-1117	205	3	(	(	PUNCT
cana-1117	205	4	𝜏	𝜏	NOUN
cana-1117	205	5	−	−	NOUN
cana-1117	205	6	𝔮𝑠)𝛼−1𝜙1(𝑠)𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝜙1(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	205	7	+	+	CCONJ
cana-1117	205	8	1	1	NUM
cana-1117	205	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	205	10	)	)	PUNCT
cana-1117	205	11	⊙	⊙	PROPN
cana-1117	205	12	∫	∫	PROPN
cana-1117	206	1	𝜏	𝜏	X
cana-1117	206	2	0	0	NUM
cana-1117	206	3	(	(	PUNCT
cana-1117	206	4	𝜏	𝜏	NOUN
cana-1117	206	5	−	−	NOUN
cana-1117	206	6	𝔮𝑠)𝛼−1𝑓(𝑠	𝔮𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	206	7	,	,	PUNCT
cana-1117	206	8	𝜙1(𝑠	𝜙1(𝑠	NUM
cana-1117	206	9	)	)	PUNCT
cana-1117	206	10	,	,	PUNCT
cana-1117	206	11	𝒢𝜙1(𝑠	𝒢𝜙1(𝑠	PROPN
cana-1117	206	12	)	)	PUNCT
cana-1117	206	13	,	,	PUNCT
cana-1117	206	14	𝒮𝜙1(𝑠))𝑑𝔮𝑠	𝒮𝜙1(𝑠))𝑑𝔮𝑠	ADV
cana-1117	206	15	and	and	CCONJ
cana-1117	206	16	𝜙2(𝜏	𝜙2(𝜏	PRON
cana-1117	206	17	)	)	PUNCT
cana-1117	206	18	=	=	SYM
cana-1117	206	19	𝜙0	𝜙0	NOUN
cana-1117	206	20	∗	∗	NOUN
cana-1117	206	21	+	+	CCONJ
cana-1117	206	22	𝜆	𝜆	DET
cana-1117	206	23	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	206	24	)	)	PUNCT
cana-1117	206	25	⊙	⊙	PROPN
cana-1117	206	26	∫	∫	PROPN
cana-1117	207	1	𝜏	𝜏	X
cana-1117	207	2	0	0	NUM
cana-1117	207	3	(	(	PUNCT
cana-1117	207	4	𝜏	𝜏	NOUN
cana-1117	207	5	−	−	NOUN
cana-1117	207	6	𝔮𝑠)𝛼−1𝜙2(𝑠)𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝜙2(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	207	7	+	+	CCONJ
cana-1117	207	8	1	1	NUM
cana-1117	207	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	207	10	)	)	PUNCT
cana-1117	207	11	⊙	⊙	PROPN
cana-1117	207	12	∫	∫	PROPN
cana-1117	208	1	𝜏	𝜏	X
cana-1117	208	2	0	0	NUM
cana-1117	208	3	(	(	PUNCT
cana-1117	208	4	𝜏	𝜏	NOUN
cana-1117	208	5	−	−	NOUN
cana-1117	208	6	𝑠)𝛼−1𝑓(𝑠	𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	208	7	,	,	PUNCT
cana-1117	208	8	𝜙2(𝑠	𝜙2(𝑠	PROPN
cana-1117	208	9	)	)	PUNCT
cana-1117	208	10	,	,	PUNCT
cana-1117	208	11	𝒢𝜙2(𝑠	𝒢𝜙2(𝑠	PROPN
cana-1117	208	12	)	)	PUNCT
cana-1117	208	13	,	,	PUNCT
cana-1117	208	14	𝒮𝜙2(𝑠))𝑑𝔮𝑠	𝒮𝜙2(𝑠))𝑑𝔮𝑠	ADV
cana-1117	208	15	communications	communication	NOUN
cana-1117	208	16	on	on	ADP
cana-1117	208	17	applied	apply	VERB
cana-1117	208	18	nonlinear	nonlinear	ADJ
cana-1117	208	19	analysis	analysis	NOUN
cana-1117	208	20	issn	issn	NOUN
cana-1117	208	21	:	:	PUNCT
cana-1117	208	22	1074	1074	NUM
cana-1117	208	23	-	-	PUNCT
cana-1117	208	24	133x	133x	NUM
cana-1117	208	25	vol	vol	NOUN
cana-1117	208	26	31	31	NUM
cana-1117	208	27	no	no	NOUN
cana-1117	208	28	.	.	PUNCT
cana-1117	209	1	6s	6s	NUM
cana-1117	209	2	(	(	PUNCT
cana-1117	209	3	2024	2024	NUM
cana-1117	209	4	)	)	PUNCT
cana-1117	209	5	46	46	NUM
cana-1117	209	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	209	7	using	use	VERB
cana-1117	209	8	the	the	DET
cana-1117	209	9	hypothesis	hypothesis	NOUN
cana-1117	209	10	(	(	PUNCT
cana-1117	209	11	𝐻1	𝐻1	PROPN
cana-1117	209	12	)	)	PUNCT
cana-1117	209	13	,	,	PUNCT
cana-1117	209	14	for	for	ADP
cana-1117	209	15	any	any	DET
cana-1117	209	16	𝜏	𝜏	PROPN
cana-1117	209	17	∈	∈	PROPN
cana-1117	209	18	[	[	X
cana-1117	209	19	𝜏0	𝜏0	PROPN
cana-1117	209	20	,	,	PUNCT
cana-1117	209	21	𝑇	𝑇	PROPN
cana-1117	209	22	]	]	PUNCT
cana-1117	209	23	,	,	PUNCT
cana-1117	209	24	we	we	PRON
cana-1117	209	25	obtain	obtain	VERB
cana-1117	209	26	‖𝜙1(𝜏	‖𝜙1(𝜏	NOUN
cana-1117	209	27	)	)	PUNCT
cana-1117	210	1	−	−	ADP
cana-1117	210	2	𝜙2(𝜏)‖	𝜙2(𝜏)‖	PROPN
cana-1117	210	3	≤	≤	NUM
cana-1117	210	4	‖𝜙0	‖𝜙0	PUNCT
cana-1117	210	5	−	−	PROPN
cana-1117	210	6	𝜙0	𝜙0	NOUN
cana-1117	210	7	∗‖	∗‖	PROPN
cana-1117	210	8	+	+	CCONJ
cana-1117	210	9	𝜆	𝜆	DET
cana-1117	210	10	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	210	11	)	)	PUNCT
cana-1117	210	12	⊙	⊙	PROPN
cana-1117	210	13	∫	∫	PROPN
cana-1117	211	1	𝜏	𝜏	PROPN
cana-1117	211	2	𝜏0	𝜏0	PROPN
cana-1117	211	3	(	(	PUNCT
cana-1117	211	4	𝜏	𝜏	PROPN
cana-1117	211	5	−	−	PROPN
cana-1117	211	6	𝔮𝑠)𝛼−1‖𝜙1(𝑠	𝔮𝑠)𝛼−1‖𝜙1(𝑠	NUM
cana-1117	211	7	)	)	PUNCT
cana-1117	211	8	−	−	ADP
cana-1117	211	9	𝜙2(𝑠)‖𝑑𝔮𝑠	𝜙2(𝑠)‖𝑑𝔮𝑠	PROPN
cana-1117	211	10	+	+	CCONJ
cana-1117	211	11	1	1	NUM
cana-1117	211	12	γ(𝛼	γ(𝛼	NUM
cana-1117	211	13	)	)	PUNCT
cana-1117	211	14	⊙	⊙	PROPN
cana-1117	211	15	∫	∫	PROPN
cana-1117	212	1	𝜏	𝜏	X
cana-1117	212	2	0	0	NUM
cana-1117	212	3	(	(	PUNCT
cana-1117	212	4	𝜏	𝜏	NOUN
cana-1117	212	5	−	−	PROPN
cana-1117	212	6	𝔮𝑠)𝛼−1‖𝑓(𝑠	𝔮𝑠)𝛼−1‖𝑓(𝑠	PROPN
cana-1117	212	7	,	,	PUNCT
cana-1117	212	8	𝜙1(𝑠	𝜙1(𝑠	NUM
cana-1117	212	9	)	)	PUNCT
cana-1117	212	10	,	,	PUNCT
cana-1117	212	11	𝒢𝜙1(𝑠	𝒢𝜙1(𝑠	PROPN
cana-1117	212	12	)	)	PUNCT
cana-1117	212	13	,	,	PUNCT
cana-1117	212	14	𝒮𝜙1(𝑠	𝒮𝜙1(𝑠	PROPN
cana-1117	212	15	)	)	PUNCT
cana-1117	212	16	)	)	PUNCT
cana-1117	213	1	−	−	PROPN
cana-1117	213	2	𝑓(𝑠	𝑓(𝑠	PROPN
cana-1117	213	3	,	,	PUNCT
cana-1117	213	4	𝜙2(𝑠	𝜙2(𝑠	PROPN
cana-1117	213	5	)	)	PUNCT
cana-1117	213	6	,	,	PUNCT
cana-1117	213	7	𝒢𝜙2(𝑠	𝒢𝜙2(𝑠	PROPN
cana-1117	213	8	)	)	PUNCT
cana-1117	213	9	,	,	PUNCT
cana-1117	213	10	𝒮𝑢2(𝑠))‖𝑑𝔮𝑠	𝒮𝑢2(𝑠))‖𝑑𝔮𝑠	ADJ
cana-1117	213	11	≤	≤	ADJ
cana-1117	213	12	‖𝜙0	‖𝜙0	PUNCT
cana-1117	213	13	−	−	PROPN
cana-1117	213	14	𝜙0	𝜙0	NOUN
cana-1117	213	15	∗‖	∗‖	PROPN
cana-1117	213	16	+	+	CCONJ
cana-1117	213	17	𝐾(𝜆	𝐾(𝜆	SYM
cana-1117	213	18	+	+	CCONJ
cana-1117	213	19	ℒ(1	ℒ(1	PUNCT
cana-1117	213	20	+	+	NUM
cana-1117	213	21	𝐾∗	𝐾∗	PUNCT
cana-1117	213	22	+	+	CCONJ
cana-1117	213	23	𝐻∗	𝐻∗	NUM
cana-1117	213	24	)	)	PUNCT
cana-1117	213	25	)	)	PUNCT
cana-1117	214	1	⊙	⊙	PROPN
cana-1117	214	2	∫	∫	PROPN
cana-1117	215	1	𝜏	𝜏	PROPN
cana-1117	215	2	𝜏0	𝜏0	PROPN
cana-1117	215	3	(	(	PUNCT
cana-1117	215	4	𝜏	𝜏	PROPN
cana-1117	215	5	−	−	NOUN
cana-1117	215	6	𝔮𝑠)𝛼−1‖𝜙0	𝔮𝑠)𝛼−1‖𝜙0	NOUN
cana-1117	215	7	−	−	PROPN
cana-1117	215	8	𝜙0	𝜙0	NOUN
cana-1117	215	9	∗‖𝑑𝔮𝑠	∗‖𝑑𝔮𝑠	NOUN
cana-1117	215	10	=	=	SYM
cana-1117	215	11	‖𝜙0	‖𝜙0	NUM
cana-1117	215	12	−	−	PROPN
cana-1117	215	13	𝜙0	𝜙0	NOUN
cana-1117	215	14	∗‖	∗‖	PROPN
cana-1117	215	15	{	{	PUNCT
cana-1117	215	16	1	1	NUM
cana-1117	215	17	+	+	NUM
cana-1117	215	18	𝐾(𝜆+ℒ(1+𝐾∗+𝐻)∗	𝐾(𝜆+ℒ(1+𝐾∗+𝐻)∗	NOUN
cana-1117	215	19	)	)	PUNCT
cana-1117	215	20	)	)	PUNCT
cana-1117	215	21	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	215	22	)	)	PUNCT
cana-1117	215	23	⊙	⊙	PROPN
cana-1117	215	24	∫	∫	PROPN
cana-1117	216	1	𝜏	𝜏	PROPN
cana-1117	216	2	𝜏0	𝜏0	PROPN
cana-1117	216	3	(	(	PUNCT
cana-1117	216	4	𝜏	𝜏	PROPN
cana-1117	216	5	−	−	NOUN
cana-1117	216	6	𝑠)𝛼−1𝑑𝔮𝑠	𝑠)𝛼−1𝑑𝔮𝑠	NOUN
cana-1117	216	7	}	}	PUNCT
cana-1117	216	8	≤	≤	NUM
cana-1117	216	9	‖𝜙0	‖𝜙0	PUNCT
cana-1117	216	10	−	−	PROPN
cana-1117	216	11	𝜙0	𝜙0	NOUN
cana-1117	216	12	∗‖	∗‖	PROPN
cana-1117	216	13	{	{	PUNCT
cana-1117	216	14	1	1	NUM
cana-1117	216	15	+	+	NUM
cana-1117	216	16	𝐾(𝜆+ℒ(1+𝐾∗+𝐻)∗	𝐾(𝜆+ℒ(1+𝐾∗+𝐻)∗	NOUN
cana-1117	216	17	)	)	PUNCT
cana-1117	216	18	)	)	PUNCT
cana-1117	216	19	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	216	20	)	)	PUNCT
cana-1117	216	21	𝑇𝛼	𝑇𝛼	NOUN
cana-1117	216	22	}	}	PUNCT
cana-1117	216	23	,	,	PUNCT
cana-1117	216	24	𝜏	𝜏	PROPN
cana-1117	216	25	∈	∈	PROPN
cana-1117	216	26	𝕋𝑞.	𝕋𝑞.	PROPN
cana-1117	216	27	this	this	PRON
cana-1117	216	28	proves	prove	VERB
cana-1117	216	29	the	the	DET
cana-1117	216	30	uniqueness	uniqueness	NOUN
cana-1117	216	31	of	of	ADP
cana-1117	216	32	the	the	DET
cana-1117	216	33	solution	solution	NOUN
cana-1117	216	34	of	of	ADP
cana-1117	216	35	(	(	PUNCT
cana-1117	216	36	1.1	1.1	NUM
cana-1117	216	37	)	)	PUNCT
cana-1117	216	38	.	.	PUNCT
cana-1117	217	1	6	6	NUM
cana-1117	217	2	continuous	continuous	ADJ
cana-1117	217	3	dependence	dependence	NOUN
cana-1117	217	4	on	on	ADP
cana-1117	217	5	functions	function	NOUN
cana-1117	217	6	involved	involve	VERB
cana-1117	217	7	and	and	CCONJ
cana-1117	217	8	parameters	parameter	NOUN
cana-1117	217	9	consider	consider	VERB
cana-1117	217	10	(	(	PUNCT
cana-1117	217	11	1.1	1.1	NUM
cana-1117	217	12	)	)	PUNCT
cana-1117	217	13	and	and	CCONJ
cana-1117	217	14	𝐶𝐷𝛼𝜒(𝜏	𝐶𝐷𝛼𝜒(𝜏	NOUN
cana-1117	217	15	)	)	PUNCT
cana-1117	217	16	=	=	PUNCT
cana-1117	217	17	𝜆𝜒(𝜏	𝜆𝜒(𝜏	X
cana-1117	217	18	)	)	PUNCT
cana-1117	218	1	+	+	CCONJ
cana-1117	219	1	𝑓∗(𝜏	𝑓∗(𝜏	INTJ
cana-1117	219	2	,	,	PUNCT
cana-1117	219	3	𝜒(𝜏	𝜒(𝜏	PROPN
cana-1117	219	4	)	)	PUNCT
cana-1117	219	5	,	,	PUNCT
cana-1117	219	6	𝒢𝜒(𝜏	𝒢𝜒(𝜏	NOUN
cana-1117	219	7	)	)	PUNCT
cana-1117	219	8	,	,	PUNCT
cana-1117	219	9	𝒮𝜒(𝜏	𝒮𝜒(𝜏	NOUN
cana-1117	219	10	)	)	PUNCT
cana-1117	219	11	)	)	PUNCT
cana-1117	219	12	(	(	PUNCT
cana-1117	219	13	6.1	6.1	NUM
cana-1117	219	14	)	)	PUNCT
cana-1117	219	15	𝜒(0	𝜒(0	NOUN
cana-1117	219	16	)	)	PUNCT
cana-1117	219	17	=	=	SYM
cana-1117	219	18	𝜒0	𝜒0	NOUN
cana-1117	219	19	,	,	PUNCT
cana-1117	219	20	where	where	SCONJ
cana-1117	219	21	𝑓	𝑓	X
cana-1117	219	22	:	:	PUNCT
cana-1117	219	23	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	219	24	×	×	NOUN
cana-1117	219	25	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	219	26	×	×	PROPN
cana-1117	219	27	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	219	28	×	×	PROPN
cana-1117	219	29	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	219	30	→	→	SYM
cana-1117	219	31	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	219	32	theorem	theorem	ADJ
cana-1117	219	33	6.1	6.1	NUM
cana-1117	219	34	suppose	suppose	VERB
cana-1117	219	35	that	that	SCONJ
cana-1117	219	36	𝑓	𝑓	X
cana-1117	219	37	in	in	ADP
cana-1117	219	38	(	(	PUNCT
cana-1117	219	39	1.1	1.1	NUM
cana-1117	219	40	)	)	PUNCT
cana-1117	219	41	satisfies	satisfy	VERB
cana-1117	219	42	the	the	DET
cana-1117	219	43	hypothesis	hypothesis	NOUN
cana-1117	219	44	(	(	PUNCT
cana-1117	219	45	𝐻1	𝐻1	PROPN
cana-1117	219	46	)	)	PUNCT
cana-1117	219	47	.	.	PUNCT
cana-1117	220	1	let	let	VERB
cana-1117	220	2	𝜒(𝑡	𝜒(𝑡	PRON
cana-1117	220	3	)	)	PUNCT
cana-1117	220	4	be	be	AUX
cana-1117	220	5	a	a	DET
cana-1117	220	6	solution	solution	NOUN
cana-1117	220	7	of	of	ADP
cana-1117	220	8	(	(	PUNCT
cana-1117	220	9	6.1	6.1	NUM
cana-1117	220	10	)	)	PUNCT
cana-1117	220	11	and	and	CCONJ
cana-1117	220	12	suppose	suppose	VERB
cana-1117	220	13	that	that	SCONJ
cana-1117	220	14	‖𝑓(𝜏	‖𝑓(𝜏	NOUN
cana-1117	220	15	,	,	PUNCT
cana-1117	220	16	𝜒(𝜏	𝜒(𝜏	PROPN
cana-1117	220	17	)	)	PUNCT
cana-1117	220	18	,	,	PUNCT
cana-1117	220	19	𝒢𝜒(𝜏	𝒢𝜒(𝜏	NOUN
cana-1117	220	20	)	)	PUNCT
cana-1117	220	21	,	,	PUNCT
cana-1117	220	22	𝒮𝜒(𝜏	𝒮𝜒(𝜏	NOUN
cana-1117	220	23	)	)	PUNCT
cana-1117	220	24	)	)	PUNCT
cana-1117	221	1	−	−	PROPN
cana-1117	222	1	𝑓∗(𝜏	𝑓∗(𝜏	INTJ
cana-1117	222	2	,	,	PUNCT
cana-1117	222	3	𝜒(𝜏	𝜒(𝜏	PROPN
cana-1117	222	4	)	)	PUNCT
cana-1117	222	5	,	,	PUNCT
cana-1117	222	6	𝒢𝜒(𝑡	𝒢𝜒(𝑡	PROPN
cana-1117	222	7	)	)	PUNCT
cana-1117	222	8	,	,	PUNCT
cana-1117	222	9	𝒮𝜒(𝑡))‖	𝒮𝜒(𝑡))‖	VERB
cana-1117	222	10	≤	≤	X
cana-1117	222	11	𝜖	𝜖	PROPN
cana-1117	222	12	,	,	PUNCT
cana-1117	222	13	𝜏	𝜏	PROPN
cana-1117	222	14	∈	∈	PROPN
cana-1117	222	15	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	222	16	and	and	CCONJ
cana-1117	222	17	‖𝜙0	‖𝜙0	NOUN
cana-1117	222	18	−	−	PROPN
cana-1117	222	19	𝜓0‖	𝜓0‖	NOUN
cana-1117	222	20	<	<	X
cana-1117	222	21	𝛿	𝛿	PROPN
cana-1117	222	22	where	where	SCONJ
cana-1117	222	23	𝜖	𝜖	X
cana-1117	222	24	,	,	PUNCT
cana-1117	222	25	𝛿	𝛿	PROPN
cana-1117	222	26	>	>	X
cana-1117	222	27	0	0	NUM
cana-1117	222	28	are	be	AUX
cana-1117	222	29	arbitrary	arbitrary	ADJ
cana-1117	222	30	small	small	ADJ
cana-1117	222	31	constants	constant	NOUN
cana-1117	222	32	.	.	PUNCT
cana-1117	223	1	then	then	ADV
cana-1117	223	2	the	the	DET
cana-1117	223	3	solution	solution	NOUN
cana-1117	223	4	of	of	ADP
cana-1117	223	5	𝜙(𝜏	𝜙(𝜏	NOUN
cana-1117	223	6	)	)	PUNCT
cana-1117	223	7	of	of	ADP
cana-1117	223	8	(	(	PUNCT
cana-1117	223	9	1.1	1.1	NUM
cana-1117	223	10	)	)	PUNCT
cana-1117	223	11	depends	depend	VERB
cana-1117	223	12	continuously	continuously	ADV
cana-1117	223	13	on	on	ADP
cana-1117	223	14	the	the	DET
cana-1117	223	15	functions	function	NOUN
cana-1117	223	16	involved	involve	VERB
cana-1117	223	17	therein	therein	ADV
cana-1117	223	18	.	.	PUNCT
cana-1117	224	1	proof	proof	NOUN
cana-1117	224	2	.	.	PUNCT
cana-1117	225	1	let	let	AUX
cana-1117	225	2	𝜙(𝜏	𝜙(𝜏	NOUN
cana-1117	225	3	)	)	PUNCT
cana-1117	225	4	and	and	CCONJ
cana-1117	225	5	𝜓(𝜏	𝜓(𝜏	NOUN
cana-1117	225	6	)	)	PUNCT
cana-1117	225	7	be	be	VERB
cana-1117	225	8	solutions	solution	NOUN
cana-1117	225	9	of	of	ADP
cana-1117	225	10	(	(	PUNCT
cana-1117	225	11	1.1	1.1	NUM
cana-1117	225	12	)	)	PUNCT
cana-1117	225	13	and	and	CCONJ
cana-1117	225	14	(	(	PUNCT
cana-1117	225	15	6.1	6.1	NUM
cana-1117	225	16	)	)	PUNCT
cana-1117	225	17	respectively	respectively	ADV
cana-1117	225	18	.	.	PUNCT
cana-1117	226	1	𝐶𝐷𝛼𝜙(𝜏	𝐶𝐷𝛼𝜙(𝜏	NOUN
cana-1117	226	2	)	)	PUNCT
cana-1117	226	3	=	=	SYM
cana-1117	226	4	𝜆𝜙(𝜏	𝜆𝜙(𝜏	X
cana-1117	226	5	)	)	PUNCT
cana-1117	227	1	+	+	CCONJ
cana-1117	227	2	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	227	3	,	,	PUNCT
cana-1117	227	4	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	227	5	)	)	PUNCT
cana-1117	227	6	,	,	PUNCT
cana-1117	227	7	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	227	8	)	)	PUNCT
cana-1117	227	9	,	,	PUNCT
cana-1117	227	10	𝒮𝜙(𝑡	𝒮𝜙(𝑡	NOUN
cana-1117	227	11	)	)	PUNCT
cana-1117	227	12	)	)	PUNCT
cana-1117	227	13	,	,	PUNCT
cana-1117	227	14	𝜙(0	𝜙(0	PROPN
cana-1117	227	15	)	)	PUNCT
cana-1117	227	16	=	=	SYM
cana-1117	227	17	𝜙0	𝜙0	NOUN
cana-1117	227	18	𝐶𝐷𝛼𝜓(𝜏	𝐶𝐷𝛼𝜓(𝜏	NOUN
cana-1117	227	19	)	)	PUNCT
cana-1117	228	1	=	=	SYM
cana-1117	228	2	𝜆𝜓(𝜏	𝜆𝜓(𝜏	PUNCT
cana-1117	228	3	)	)	PUNCT
cana-1117	229	1	+	+	CCONJ
cana-1117	229	2	𝑓∗(𝜏	𝑓∗(𝜏	INTJ
cana-1117	229	3	,	,	PUNCT
cana-1117	229	4	𝜓(𝜏	𝜓(𝜏	PROPN
cana-1117	229	5	)	)	PUNCT
cana-1117	229	6	,	,	PUNCT
cana-1117	229	7	𝒢𝜓(𝜏	𝒢𝜓(𝜏	NOUN
cana-1117	229	8	)	)	PUNCT
cana-1117	229	9	,	,	PUNCT
cana-1117	229	10	𝒮𝜓(𝜏	𝒮𝜓(𝜏	NOUN
cana-1117	229	11	)	)	PUNCT
cana-1117	229	12	)	)	PUNCT
cana-1117	229	13	,	,	PUNCT
cana-1117	229	14	𝜓(0	𝜓(0	PROPN
cana-1117	229	15	)	)	PUNCT
cana-1117	229	16	=	=	PUNCT
cana-1117	229	17	𝜓0	𝜓0	PROPN
cana-1117	229	18	this	this	PRON
cana-1117	229	19	implies	imply	VERB
cana-1117	229	20	that	that	SCONJ
cana-1117	229	21	,	,	PUNCT
cana-1117	229	22	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	229	23	)	)	PUNCT
cana-1117	230	1	=	=	SYM
cana-1117	230	2	𝜙0	𝜙0	NOUN
cana-1117	230	3	+	+	CCONJ
cana-1117	230	4	𝜆	𝜆	PRON
cana-1117	230	5	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	230	6	)	)	PUNCT
cana-1117	230	7	⊙	⊙	PROPN
cana-1117	230	8	∫	∫	PROPN
cana-1117	231	1	𝜏	𝜏	PROPN
cana-1117	231	2	𝜏0	𝜏0	PROPN
cana-1117	231	3	(	(	PUNCT
cana-1117	231	4	𝜏	𝜏	NOUN
cana-1117	231	5	−	−	PROPN
cana-1117	231	6	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝑠	𝔮𝑠)𝛼−1𝜙(𝑠)𝑑𝑠	NOUN
cana-1117	231	7	+	+	CCONJ
cana-1117	231	8	1	1	NUM
cana-1117	231	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	231	10	)	)	PUNCT
cana-1117	231	11	⊙	⊙	PROPN
cana-1117	231	12	∫	∫	PROPN
cana-1117	232	1	𝜏	𝜏	PROPN
cana-1117	232	2	𝜏0	𝜏0	PROPN
cana-1117	232	3	(	(	PUNCT
cana-1117	232	4	𝜏	𝜏	NOUN
cana-1117	232	5	−	−	NOUN
cana-1117	232	6	𝔮𝑠)𝛼−1𝑓(𝑠	𝔮𝑠)𝛼−1𝑓(𝑠	NOUN
cana-1117	232	7	,	,	PUNCT
cana-1117	232	8	𝜙(𝑠	𝜙(𝑠	PROPN
cana-1117	232	9	)	)	PUNCT
cana-1117	232	10	,	,	PUNCT
cana-1117	232	11	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	232	12	)	)	PUNCT
cana-1117	232	13	,	,	PUNCT
cana-1117	232	14	𝒮𝜙(𝑠))𝑑𝔮𝑠	𝒮𝜙(𝑠))𝑑𝔮𝑠	ADJ
cana-1117	232	15	communications	communication	NOUN
cana-1117	232	16	on	on	ADP
cana-1117	232	17	applied	apply	VERB
cana-1117	232	18	nonlinear	nonlinear	ADJ
cana-1117	232	19	analysis	analysis	NOUN
cana-1117	232	20	issn	issn	NOUN
cana-1117	232	21	:	:	PUNCT
cana-1117	232	22	1074	1074	NUM
cana-1117	232	23	-	-	PUNCT
cana-1117	232	24	133x	133x	NUM
cana-1117	232	25	vol	vol	NOUN
cana-1117	232	26	31	31	NUM
cana-1117	232	27	no	no	NOUN
cana-1117	232	28	.	.	PUNCT
cana-1117	233	1	6s	6s	NUM
cana-1117	233	2	(	(	PUNCT
cana-1117	233	3	2024	2024	NUM
cana-1117	233	4	)	)	PUNCT
cana-1117	233	5	47	47	NUM
cana-1117	233	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1117	233	7	and	and	CCONJ
cana-1117	233	8	𝜓(𝑡	𝜓(𝑡	PROPN
cana-1117	233	9	)	)	PUNCT
cana-1117	234	1	=	=	PUNCT
cana-1117	234	2	𝜓0	𝜓0	PROPN
cana-1117	234	3	+	+	CCONJ
cana-1117	234	4	𝜆	𝜆	DET
cana-1117	234	5	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	234	6	)	)	PUNCT
cana-1117	234	7	⊙	⊙	PROPN
cana-1117	234	8	∫	∫	PROPN
cana-1117	235	1	𝜏	𝜏	PROPN
cana-1117	235	2	𝜏0	𝜏0	PROPN
cana-1117	235	3	(	(	PUNCT
cana-1117	235	4	𝜏	𝜏	NOUN
cana-1117	235	5	−	−	NUM
cana-1117	235	6	𝔮𝑠)𝛼−1𝜓(𝑠)𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝜓(𝑠)𝑑𝔮𝑠	NOUN
cana-1117	235	7	+	+	CCONJ
cana-1117	235	8	1	1	NUM
cana-1117	235	9	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	235	10	)	)	PUNCT
cana-1117	235	11	⊙	⊙	PROPN
cana-1117	235	12	∫	∫	PROPN
cana-1117	236	1	𝜏	𝜏	PROPN
cana-1117	236	2	𝜏0	𝜏0	PROPN
cana-1117	236	3	(	(	PUNCT
cana-1117	236	4	𝜏	𝜏	NOUN
cana-1117	236	5	−	−	PROPN
cana-1117	236	6	𝔮𝑠)𝛼−1𝑓∗(𝑠	𝔮𝑠)𝛼−1𝑓∗(𝑠	PROPN
cana-1117	236	7	,	,	PUNCT
cana-1117	236	8	𝜓(𝑠	𝜓(𝑠	PROPN
cana-1117	236	9	)	)	PUNCT
cana-1117	236	10	,	,	PUNCT
cana-1117	236	11	𝒢𝜓(𝑠	𝒢𝜓(𝑠	PROPN
cana-1117	236	12	)	)	PUNCT
cana-1117	236	13	,	,	PUNCT
cana-1117	236	14	𝒮𝜓(𝑠))𝑑𝔮𝑠	𝒮𝜓(𝑠))𝑑𝔮𝑠	PROPN
cana-1117	236	15	‖𝜙(𝜏	‖𝜙(𝜏	NOUN
cana-1117	236	16	)	)	PUNCT
cana-1117	236	17	−	−	PROPN
cana-1117	236	18	𝜓(𝜏)‖	𝜓(𝜏)‖	NOUN
cana-1117	236	19	≤	≤	NUM
cana-1117	236	20	‖𝜙0	‖𝜙0	PUNCT
cana-1117	236	21	−	−	PROPN
cana-1117	236	22	𝜓0‖	𝜓0‖	NOUN
cana-1117	236	23	+	+	CCONJ
cana-1117	236	24	𝜆	𝜆	PRON
cana-1117	236	25	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	236	26	)	)	PUNCT
cana-1117	236	27	⊙	⊙	PROPN
cana-1117	236	28	∫	∫	PROPN
cana-1117	237	1	𝜏	𝜏	PROPN
cana-1117	237	2	𝜏0	𝜏0	PROPN
cana-1117	237	3	(	(	PUNCT
cana-1117	237	4	𝜏	𝜏	NOUN
cana-1117	237	5	−	−	PROPN
cana-1117	237	6	𝔮𝑠)𝛼−1‖𝜙(𝑠	𝔮𝑠)𝛼−1‖𝜙(𝑠	NOUN
cana-1117	237	7	)	)	PUNCT
cana-1117	237	8	−	−	ADP
cana-1117	238	1	𝜓(𝑠)‖𝑑𝔮𝑠	𝜓(𝑠)‖𝑑𝔮𝑠	PROPN
cana-1117	238	2	+	+	CCONJ
cana-1117	238	3	1	1	NUM
cana-1117	238	4	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	238	5	)	)	PUNCT
cana-1117	238	6	⊙	⊙	PROPN
cana-1117	238	7	∫	∫	PROPN
cana-1117	239	1	𝜏	𝜏	PROPN
cana-1117	239	2	𝜏0	𝜏0	PROPN
cana-1117	239	3	(	(	PUNCT
cana-1117	239	4	𝜏	𝜏	PROPN
cana-1117	239	5	−	−	PROPN
cana-1117	239	6	𝔮𝑠)𝛼−1‖𝑓(𝑠	𝔮𝑠)𝛼−1‖𝑓(𝑠	PROPN
cana-1117	239	7	,	,	PUNCT
cana-1117	239	8	𝜙(𝑠	𝜙(𝑠	NOUN
cana-1117	239	9	)	)	PUNCT
cana-1117	239	10	,	,	PUNCT
cana-1117	239	11	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	239	12	)	)	PUNCT
cana-1117	239	13	,	,	PUNCT
cana-1117	239	14	𝒮𝜙(𝑠	𝒮𝜙(𝑠	NOUN
cana-1117	239	15	)	)	PUNCT
cana-1117	239	16	)	)	PUNCT
cana-1117	240	1	−	−	PROPN
cana-1117	241	1	𝑓∗(𝑠	𝑓∗(𝑠	PROPN
cana-1117	241	2	,	,	PUNCT
cana-1117	241	3	𝜙(𝑠	𝜙(𝑠	PROPN
cana-1117	241	4	)	)	PUNCT
cana-1117	241	5	,	,	PUNCT
cana-1117	241	6	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	241	7	)	)	PUNCT
cana-1117	241	8	,	,	PUNCT
cana-1117	241	9	𝒮𝜙(𝑠))‖𝑑𝔮𝑠	𝒮𝜙(𝑠))‖𝑑𝔮𝑠	ADV
cana-1117	241	10	≤	≤	ADJ
cana-1117	241	11	‖𝜙0	‖𝜙0	PUNCT
cana-1117	241	12	−	−	PROPN
cana-1117	241	13	𝜓0‖	𝜓0‖	NOUN
cana-1117	241	14	+	+	CCONJ
cana-1117	241	15	𝜆	𝜆	PRON
cana-1117	241	16	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	241	17	)	)	PUNCT
cana-1117	241	18	⊙	⊙	PROPN
cana-1117	241	19	∫	∫	PROPN
cana-1117	242	1	𝜏	𝜏	PROPN
cana-1117	242	2	𝜏0	𝜏0	PROPN
cana-1117	242	3	(	(	PUNCT
cana-1117	242	4	𝜏	𝜏	NOUN
cana-1117	242	5	−	−	PROPN
cana-1117	242	6	𝔮𝑠)𝛼−1‖𝜙(𝑠	𝔮𝑠)𝛼−1‖𝜙(𝑠	NOUN
cana-1117	242	7	)	)	PUNCT
cana-1117	242	8	−	−	ADP
cana-1117	243	1	𝜓(𝑠)‖𝑑𝔮𝑠	𝜓(𝑠)‖𝑑𝔮𝑠	PROPN
cana-1117	243	2	+	+	CCONJ
cana-1117	243	3	1	1	NUM
cana-1117	243	4	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-1117	243	5	)	)	PUNCT
cana-1117	243	6	⊙	⊙	PROPN
cana-1117	243	7	∫	∫	PROPN
cana-1117	244	1	𝜏	𝜏	PROPN
cana-1117	244	2	𝜏0	𝜏0	PROPN
cana-1117	244	3	(	(	PUNCT
cana-1117	244	4	𝜏	𝜏	PROPN
cana-1117	244	5	−	−	PROPN
cana-1117	244	6	𝔮𝑠)𝛼−1‖𝑓(𝑠	𝔮𝑠)𝛼−1‖𝑓(𝑠	PROPN
cana-1117	244	7	,	,	PUNCT
cana-1117	244	8	𝜙(𝑠	𝜙(𝑠	NOUN
cana-1117	244	9	)	)	PUNCT
cana-1117	244	10	,	,	PUNCT
cana-1117	244	11	𝒢𝜙(𝑠	𝒢𝜙(𝑠	ADJ
cana-1117	244	12	)	)	PUNCT
cana-1117	244	13	,	,	PUNCT
cana-1117	244	14	𝒮𝜙(𝑠	𝒮𝜙(𝑠	NOUN
cana-1117	244	15	)	)	PUNCT
cana-1117	244	16	)	)	PUNCT
cana-1117	244	17	−	−	PROPN
cana-1117	244	18	𝑓(𝑠	𝑓(𝑠	PROPN
cana-1117	244	19	,	,	PUNCT
cana-1117	244	20	𝜓(𝑠	𝜓(𝑠	PROPN
cana-1117	244	21	)	)	PUNCT
cana-1117	244	22	,	,	PUNCT
cana-1117	244	23	𝒢𝜓(𝑠	𝒢𝜓(𝑠	PROPN
cana-1117	244	24	)	)	PUNCT
cana-1117	244	25	,	,	PUNCT
cana-1117	244	26	𝒮𝜓(𝑠))‖𝑑𝔮𝑠	𝒮𝜓(𝑠))‖𝑑𝔮𝑠	PROPN
cana-1117	244	27	+	+	CCONJ
cana-1117	244	28	1	1	NUM
cana-1117	244	29	γ𝑞(𝛼	γ𝑞(𝛼	NUM
cana-1117	244	30	)	)	PUNCT
cana-1117	244	31	⊙	⊙	PROPN
cana-1117	244	32	∫	∫	PROPN
cana-1117	245	1	𝜏	𝜏	PROPN
cana-1117	245	2	𝜏0	𝜏0	PROPN
cana-1117	245	3	(	(	PUNCT
cana-1117	245	4	𝜏	𝜏	PROPN
cana-1117	245	5	−	−	PROPN
cana-1117	245	6	𝔮𝑠)𝛼−1‖𝑓(𝑠	𝔮𝑠)𝛼−1‖𝑓(𝑠	PROPN
cana-1117	245	7	,	,	PUNCT
cana-1117	245	8	𝜓(𝑠	𝜓(𝑠	PROPN
cana-1117	245	9	)	)	PUNCT
cana-1117	245	10	,	,	PUNCT
cana-1117	245	11	𝒢𝜓(𝑠	𝒢𝜓(𝑠	PROPN
cana-1117	245	12	)	)	PUNCT
cana-1117	245	13	,	,	PUNCT
cana-1117	245	14	𝒮𝜓(𝑠	𝒮𝜓(𝑠	NOUN
cana-1117	245	15	)	)	PUNCT
cana-1117	245	16	)	)	PUNCT
cana-1117	246	1	−	−	PROPN
cana-1117	247	1	𝑓∗(𝑠	𝑓∗(𝑠	PROPN
cana-1117	247	2	,	,	PUNCT
cana-1117	247	3	𝜓(𝑠	𝜓(𝑠	PROPN
cana-1117	247	4	)	)	PUNCT
cana-1117	247	5	,	,	PUNCT
cana-1117	247	6	𝒢𝜓(𝑠	𝒢𝜓(𝑠	PROPN
cana-1117	247	7	)	)	PUNCT
cana-1117	247	8	,	,	PUNCT
cana-1117	247	9	𝒮𝜓(𝑠))‖𝑑𝔮𝑠	𝒮𝜓(𝑠))‖𝑑𝔮𝑠	PROPN
cana-1117	247	10	≤	≤	PROPN
cana-1117	247	11	‖𝜙0	‖𝜙0	PUNCT
cana-1117	247	12	−	−	PROPN
cana-1117	247	13	𝜓0‖	𝜓0‖	NOUN
cana-1117	247	14	+	+	CCONJ
cana-1117	247	15	𝜆	𝜆	PRON
cana-1117	247	16	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	247	17	)	)	PUNCT
cana-1117	247	18	⊙	⊙	PROPN
cana-1117	247	19	∫	∫	PROPN
cana-1117	248	1	𝜏	𝜏	PROPN
cana-1117	248	2	𝜏0	𝜏0	PROPN
cana-1117	248	3	(	(	PUNCT
cana-1117	248	4	𝜏	𝜏	NOUN
cana-1117	248	5	−	−	PROPN
cana-1117	248	6	𝔮𝑠)𝛼−1‖𝜙(𝑠	𝔮𝑠)𝛼−1‖𝜙(𝑠	NOUN
cana-1117	248	7	)	)	PUNCT
cana-1117	248	8	−	−	ADP
cana-1117	249	1	𝜓(𝑠)‖𝑑𝔮𝑠	𝜓(𝑠)‖𝑑𝔮𝑠	PROPN
cana-1117	249	2	+	+	CCONJ
cana-1117	249	3	1	1	NUM
cana-1117	249	4	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	249	5	)	)	PUNCT
cana-1117	249	6	⊙	⊙	PROPN
cana-1117	249	7	∫	∫	PROPN
cana-1117	250	1	𝜏	𝜏	PROPN
cana-1117	250	2	𝜏0	𝜏0	PROPN
cana-1117	250	3	(	(	PUNCT
cana-1117	250	4	𝜏	𝜏	PROPN
cana-1117	250	5	−	−	PROPN
cana-1117	250	6	𝔮𝑠)𝛼−1ℒ[‖𝜙	𝔮𝑠)𝛼−1ℒ[‖𝜙	NOUN
cana-1117	250	7	−	−	PROPN
cana-1117	250	8	𝜓‖	𝜓‖	NOUN
cana-1117	250	9	+	+	CCONJ
cana-1117	250	10	‖𝒢𝜙	‖𝒢𝜙	NOUN
cana-1117	250	11	−	−	ADP
cana-1117	250	12	𝒢𝜓‖	𝒢𝜓‖	PROPN
cana-1117	250	13	+	+	CCONJ
cana-1117	250	14	‖𝒮𝜙	‖𝒮𝜙	PROPN
cana-1117	250	15	−	−	NOUN
cana-1117	250	16	𝒮𝜓‖]𝑑𝔮𝑠	𝒮𝜓‖]𝑑𝔮𝑠	X
cana-1117	251	1	+	+	CCONJ
cana-1117	251	2	𝜖	𝜖	X
cana-1117	251	3	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	251	4	)	)	PUNCT
cana-1117	251	5	⊙	⊙	PROPN
cana-1117	251	6	∫	∫	PROPN
cana-1117	252	1	𝜏	𝜏	PROPN
cana-1117	252	2	𝜏0	𝜏0	PROPN
cana-1117	252	3	(	(	PUNCT
cana-1117	252	4	𝜏	𝜏	NOUN
cana-1117	252	5	−	−	NOUN
cana-1117	252	6	𝔮𝑠)𝛼−1𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝑑𝔮𝑠	ADJ
cana-1117	252	7	≤	≤	ADJ
cana-1117	252	8	‖𝜙0	‖𝜙0	PUNCT
cana-1117	252	9	−	−	PROPN
cana-1117	252	10	𝜓0‖	𝜓0‖	NOUN
cana-1117	252	11	+	+	CCONJ
cana-1117	252	12	𝜆	𝜆	PRON
cana-1117	252	13	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	252	14	)	)	PUNCT
cana-1117	252	15	⊙	⊙	PROPN
cana-1117	252	16	∫	∫	PROPN
cana-1117	253	1	𝜏	𝜏	PROPN
cana-1117	253	2	𝜏0	𝜏0	PROPN
cana-1117	253	3	(	(	PUNCT
cana-1117	253	4	𝜏	𝜏	NOUN
cana-1117	253	5	−	−	PROPN
cana-1117	253	6	𝔮𝑠)𝛼−1‖𝜙	𝔮𝑠)𝛼−1‖𝜙	ADJ
cana-1117	253	7	−	−	NOUN
cana-1117	253	8	𝜓‖𝑑𝔮𝑠	𝜓‖𝑑𝔮𝑠	ADJ
cana-1117	253	9	+	+	X
cana-1117	253	10	1	1	NUM
cana-1117	253	11	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	253	12	)	)	PUNCT
cana-1117	254	1	⊙	⊙	PROPN
cana-1117	254	2	∫	∫	PROPN
cana-1117	255	1	𝜏	𝜏	PROPN
cana-1117	255	2	𝜏0	𝜏0	PROPN
cana-1117	255	3	(	(	PUNCT
cana-1117	255	4	𝜏	𝜏	PROPN
cana-1117	255	5	−	−	PROPN
cana-1117	255	6	𝔮𝑠)𝛼−1ℒ(1	𝔮𝑠)𝛼−1ℒ(1	NUM
cana-1117	255	7	+	+	NUM
cana-1117	255	8	𝐾∗	𝐾∗	PUNCT
cana-1117	256	1	+	+	CCONJ
cana-1117	256	2	𝐻∗)‖𝜙	𝐻∗)‖𝜙	NOUN
cana-1117	256	3	−	−	NOUN
cana-1117	256	4	𝜓‖𝑑𝔮𝑠	𝜓‖𝑑𝔮𝑠	ADJ
cana-1117	256	5	+	+	CCONJ
cana-1117	256	6	𝜖	𝜖	X
cana-1117	256	7	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	256	8	)	)	PUNCT
cana-1117	256	9	⊙	⊙	PROPN
cana-1117	256	10	∫	∫	PROPN
cana-1117	257	1	𝜏	𝜏	PROPN
cana-1117	257	2	𝜏0	𝜏0	PROPN
cana-1117	257	3	(	(	PUNCT
cana-1117	257	4	𝜏	𝜏	NOUN
cana-1117	257	5	−	−	PROPN
cana-1117	257	6	𝔮𝑠)𝛼−1𝑑𝔮𝑠	𝔮𝑠)𝛼−1𝑑𝔮𝑠	NOUN
cana-1117	257	7	≤	≤	NOUN
cana-1117	258	1	𝛿	𝛿	PROPN
cana-1117	258	2	+	+	NUM
cana-1117	258	3	𝜖	𝜖	PROPN
cana-1117	258	4	γ𝔮(𝛼+1	γ𝔮(𝛼+1	NOUN
cana-1117	258	5	)	)	PUNCT
cana-1117	258	6	𝜏𝛼	𝜏𝛼	ADP
cana-1117	258	7	+	+	CCONJ
cana-1117	258	8	(	(	PUNCT
cana-1117	258	9	𝜆+ℒ(1+𝐾∗+𝐻∗	𝜆+ℒ(1+𝐾∗+𝐻∗	PROPN
cana-1117	258	10	)	)	PUNCT
cana-1117	258	11	γ𝔮(𝛼	γ𝔮(𝛼	NOUN
cana-1117	258	12	)	)	PUNCT
cana-1117	258	13	)	)	PUNCT
cana-1117	259	1	⊙	⊙	PROPN
cana-1117	259	2	∫	∫	PROPN
cana-1117	260	1	𝜏	𝜏	PROPN
cana-1117	260	2	𝜏0	𝜏0	PROPN
cana-1117	260	3	(	(	PUNCT
cana-1117	260	4	𝜏	𝜏	NOUN
cana-1117	260	5	−	−	PROPN
cana-1117	260	6	𝔮𝑠)𝛼−1‖𝜙	𝔮𝑠)𝛼−1‖𝜙	ADJ
cana-1117	260	7	−	−	NOUN
cana-1117	260	8	𝜓‖𝑑𝔮𝑠	𝜓‖𝑑𝔮𝑠	VERB
cana-1117	260	9	by	by	ADP
cana-1117	260	10	lemma	lemma	PROPN
cana-1117	260	11	(	(	PUNCT
cana-1117	260	12	2.2	2.2	NUM
cana-1117	260	13	)	)	PUNCT
cana-1117	260	14	,	,	PUNCT
cana-1117	260	15	‖𝜙(𝑡	‖𝜙(𝑡	X
cana-1117	260	16	)	)	PUNCT
cana-1117	260	17	−	−	PROPN
cana-1117	260	18	𝜓(𝑡)‖	𝜓(𝑡)‖	ADP
cana-1117	260	19	≤	≤	NOUN
cana-1117	260	20	𝛿	𝛿	PROPN
cana-1117	260	21	+	+	NUM
cana-1117	260	22	𝜖	𝜖	PROPN
cana-1117	260	23	γ𝔮(𝛼+1	γ𝔮(𝛼+1	NOUN
cana-1117	260	24	)	)	PUNCT
cana-1117	260	25	𝜏𝛼	𝜏𝛼	ADP
cana-1117	260	26	+	+	CCONJ
cana-1117	260	27	𝐾(𝜆+ℒ(1+𝐾∗+𝐻∗))𝛿	𝐾(𝜆+ℒ(1+𝐾∗+𝐻∗))𝛿	PROPN
cana-1117	260	28	γ𝔮(𝛼+1	γ𝔮(𝛼+1	NUM
cana-1117	260	29	)	)	PUNCT
cana-1117	260	30	𝜏𝛼	𝜏𝛼	ADP
cana-1117	260	31	+	+	PROPN
cana-1117	260	32	𝐾	𝐾	PROPN
cana-1117	260	33	𝜆+𝐿(1+𝐾∗+𝐻∗	𝜆+𝐿(1+𝐾∗+𝐻∗	PROPN
cana-1117	260	34	)	)	PUNCT
cana-1117	260	35	γ𝑞(2𝛼+1	γ𝑞(2𝛼+1	NOUN
cana-1117	260	36	)	)	PUNCT
cana-1117	260	37	𝜏2𝛼	𝜏2𝛼	X
cana-1117	260	38	,	,	PUNCT
cana-1117	260	39	𝜏	𝜏	PROPN
cana-1117	260	40	∈	∈	PROPN
cana-1117	260	41	𝕋𝑞.	𝕋𝑞.	PROPN
cana-1117	260	42	from	from	ADP
cana-1117	260	43	the	the	DET
cana-1117	260	44	above	above	ADJ
cana-1117	260	45	inequality	inequality	NOUN
cana-1117	260	46	it	it	PRON
cana-1117	260	47	comes	come	VERB
cana-1117	260	48	to	to	PART
cana-1117	260	49	know	know	VERB
cana-1117	260	50	that	that	SCONJ
cana-1117	260	51	the	the	DET
cana-1117	260	52	solution	solution	NOUN
cana-1117	260	53	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	260	54	)	)	PUNCT
cana-1117	260	55	depends	depend	VERB
cana-1117	260	56	continuously	continuously	ADV
cana-1117	260	57	on	on	ADP
cana-1117	260	58	the	the	DET
cana-1117	260	59	functions	function	NOUN
cana-1117	260	60	involved	involve	VERB
cana-1117	260	61	in	in	ADP
cana-1117	260	62	the	the	DET
cana-1117	260	63	given	give	VERB
cana-1117	260	64	fde	fde	PROPN
cana-1117	260	65	(	(	PUNCT
cana-1117	260	66	1.1	1.1	NUM
cana-1117	260	67	)	)	PUNCT
cana-1117	260	68	.	.	PUNCT
cana-1117	261	1	if	if	SCONJ
cana-1117	261	2	𝜖	𝜖	PROPN
cana-1117	261	3	=	=	SYM
cana-1117	261	4	0	0	PROPN
cana-1117	261	5	,	,	PUNCT
cana-1117	261	6	the	the	DET
cana-1117	261	7	continuous	continuous	ADJ
cana-1117	261	8	dependence	dependence	NOUN
cana-1117	261	9	of	of	ADP
cana-1117	261	10	solutions	solution	NOUN
cana-1117	261	11	of	of	ADP
cana-1117	261	12	the	the	DET
cana-1117	261	13	inequality	inequality	NOUN
cana-1117	261	14	is	be	AUX
cana-1117	261	15	on	on	ADP
cana-1117	261	16	the	the	DET
cana-1117	261	17	initial	initial	ADJ
cana-1117	261	18	conditions	condition	NOUN
cana-1117	261	19	.	.	PUNCT
cana-1117	262	1	here	here	ADV
cana-1117	262	2	it	it	PRON
cana-1117	262	3	is	be	AUX
cana-1117	262	4	noted	note	VERB
cana-1117	262	5	that	that	SCONJ
cana-1117	262	6	as	as	ADP
cana-1117	262	7	𝜖	𝜖	PROPN
cana-1117	262	8	,	,	PUNCT
cana-1117	262	9	𝛿	𝛿	PROPN
cana-1117	262	10	>	>	X
cana-1117	262	11	0	0	NUM
cana-1117	262	12	were	be	AUX
cana-1117	262	13	arbitrary	arbitrary	ADJ
cana-1117	262	14	,	,	PUNCT
cana-1117	262	15	by	by	ADP
cana-1117	262	16	taking	take	VERB
cana-1117	262	17	𝜖	𝜖	PROPN
cana-1117	262	18	,	,	PUNCT
cana-1117	262	19	𝛿	𝛿	PROPN
cana-1117	262	20	>	>	X
cana-1117	262	21	0	0	NUM
cana-1117	262	22	,	,	PUNCT
cana-1117	262	23	we	we	PRON
cana-1117	262	24	have	have	VERB
cana-1117	262	25	𝜙	𝜙	PROPN
cana-1117	262	26	→	→	SYM
cana-1117	262	27	𝜓	𝜓	PROPN
cana-1117	262	28	,	,	PUNCT
cana-1117	262	29	where	where	SCONJ
cana-1117	262	30	𝜙	𝜙	NOUN
cana-1117	262	31	:	:	PUNCT
cana-1117	262	32	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	262	33	→	→	SYM
cana-1117	262	34	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	262	35	and	and	CCONJ
cana-1117	262	36	𝜓	𝜓	NOUN
cana-1117	262	37	:	:	PUNCT
cana-1117	262	38	𝕋𝑞	𝕋𝑞	PROPN
cana-1117	262	39	→	→	SYM
cana-1117	262	40	ℝ𝔽	ℝ𝔽	PROPN
cana-1117	262	41	are	be	AUX
cana-1117	262	42	the	the	DET
cana-1117	262	43	solutions	solution	NOUN
cana-1117	262	44	of	of	ADP
cana-1117	262	45	(	(	PUNCT
cana-1117	262	46	1.1	1.1	NUM
cana-1117	262	47	)	)	PUNCT
cana-1117	262	48	and	and	CCONJ
cana-1117	262	49	(	(	PUNCT
cana-1117	262	50	6.1	6.1	NUM
cana-1117	262	51	)	)	PUNCT
cana-1117	262	52	respectively	respectively	ADV
cana-1117	262	53	.	.	PUNCT
cana-1117	263	1	communications	communication	NOUN
cana-1117	263	2	on	on	ADP
cana-1117	263	3	applied	apply	VERB
cana-1117	263	4	nonlinear	nonlinear	ADJ
cana-1117	263	5	analysis	analysis	NOUN
cana-1117	263	6	issn	issn	NOUN
cana-1117	263	7	:	:	PUNCT
cana-1117	263	8	1074	1074	NUM
cana-1117	263	9	-	-	PUNCT
cana-1117	263	10	133x	133x	NUM
cana-1117	263	11	vol	vol	NOUN
cana-1117	263	12	31	31	NUM
cana-1117	263	13	no	no	NOUN
cana-1117	263	14	.	.	PUNCT
cana-1117	264	1	6s	6s	NUM
cana-1117	264	2	(	(	PUNCT
cana-1117	264	3	2024	2024	NUM
cana-1117	264	4	)	)	PUNCT
cana-1117	264	5	48	48	NUM
cana-1117	264	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-1117	264	7	7	7	NUM
cana-1117	264	8	illustrative	illustrative	ADJ
cana-1117	264	9	example	example	NOUN
cana-1117	264	10	the	the	DET
cana-1117	264	11	incorporation	incorporation	NOUN
cana-1117	264	12	of	of	ADP
cana-1117	264	13	an	an	DET
cana-1117	264	14	illustrative	illustrative	ADJ
cana-1117	264	15	example	example	NOUN
cana-1117	264	16	can	can	AUX
cana-1117	264	17	serve	serve	VERB
cana-1117	264	18	as	as	ADP
cana-1117	264	19	a	a	DET
cana-1117	264	20	powerful	powerful	ADJ
cana-1117	264	21	tool	tool	NOUN
cana-1117	264	22	to	to	PART
cana-1117	264	23	amplify	amplify	VERB
cana-1117	264	24	the	the	DET
cana-1117	264	25	overall	overall	ADJ
cana-1117	264	26	impact	impact	NOUN
cana-1117	264	27	of	of	ADP
cana-1117	264	28	the	the	DET
cana-1117	264	29	presented	present	VERB
cana-1117	264	30	idea	idea	NOUN
cana-1117	264	31	in	in	ADP
cana-1117	264	32	the	the	DET
cana-1117	264	33	article	article	NOUN
cana-1117	264	34	.	.	PUNCT
cana-1117	265	1	to	to	PART
cana-1117	265	2	accomplish	accomplish	VERB
cana-1117	265	3	the	the	DET
cana-1117	265	4	same	same	ADJ
cana-1117	265	5	,	,	PUNCT
cana-1117	265	6	we	we	PRON
cana-1117	265	7	provide	provide	VERB
cana-1117	265	8	an	an	DET
cana-1117	265	9	example	example	NOUN
cana-1117	265	10	to	to	PART
cana-1117	265	11	illustrate	illustrate	VERB
cana-1117	265	12	the	the	DET
cana-1117	265	13	execution	execution	NOUN
cana-1117	265	14	of	of	ADP
cana-1117	265	15	our	our	PRON
cana-1117	265	16	main	main	ADJ
cana-1117	265	17	result	result	NOUN
cana-1117	265	18	.	.	PUNCT
cana-1117	266	1	we	we	PRON
cana-1117	266	2	consider	consider	VERB
cana-1117	266	3	𝐶𝐷𝛼[𝜙(𝜏	𝐶𝐷𝛼[𝜙(𝜏	NOUN
cana-1117	266	4	)	)	PUNCT
cana-1117	266	5	]	]	PUNCT
cana-1117	267	1	=	=	SYM
cana-1117	267	2	𝜏+1	𝜏+1	NUM
cana-1117	267	3	11	11	NUM
cana-1117	267	4	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	267	5	)	)	PUNCT
cana-1117	267	6	+	+	CCONJ
cana-1117	267	7	𝜏2	𝜏2	PROPN
cana-1117	267	8	2	2	NUM
cana-1117	267	9	+	+	NUM
cana-1117	267	10	∫	∫	PROPN
cana-1117	267	11	1	1	NUM
cana-1117	267	12	0	0	NUM
cana-1117	267	13	1	1	NUM
cana-1117	267	14	(	(	PUNCT
cana-1117	267	15	𝑠+𝜏+2)2	𝑠+𝜏+2)2	X
cana-1117	267	16	𝜙(𝑠)𝑑𝑠	𝜙(𝑠)𝑑𝑠	NUM
cana-1117	267	17	+	+	NUM
cana-1117	267	18	∫	∫	PROPN
cana-1117	267	19	1	1	NUM
cana-1117	267	20	0	0	NUM
cana-1117	267	21	𝜏	𝜏	PROPN
cana-1117	267	22	𝑠+1	𝑠+1	NUM
cana-1117	267	23	𝜙(𝑠)𝑑𝑠	𝜙(𝑠)𝑑𝑠	NUM
cana-1117	267	24	,	,	PUNCT
cana-1117	267	25	𝜏	𝜏	PROPN
cana-1117	267	26	∈	∈	PROPN
cana-1117	267	27	[	[	X
cana-1117	267	28	𝜏0	𝜏0	PROPN
cana-1117	267	29	,	,	PUNCT
cana-1117	267	30	𝑇	𝑇	PROPN
cana-1117	267	31	]	]	PUNCT
cana-1117	267	32	(	(	PUNCT
cana-1117	267	33	7.1	7.1	NUM
cana-1117	267	34	)	)	PUNCT
cana-1117	267	35	𝜙(0	𝜙(0	NOUN
cana-1117	267	36	)	)	PUNCT
cana-1117	267	37	=	=	SYM
cana-1117	267	38	0	0	NUM
cana-1117	267	39	define	define	VERB
cana-1117	267	40	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	267	41	,	,	PUNCT
cana-1117	267	42	𝜙	𝜙	NOUN
cana-1117	267	43	,	,	PUNCT
cana-1117	267	44	𝒢𝜙	𝒢𝜙	PROPN
cana-1117	267	45	,	,	PUNCT
cana-1117	267	46	𝒮𝜙	𝒮𝜙	PROPN
cana-1117	267	47	)	)	PUNCT
cana-1117	267	48	=	=	SYM
cana-1117	267	49	𝜏+1	𝜏+1	NUM
cana-1117	267	50	11	11	NUM
cana-1117	267	51	𝑢(𝜏	𝑢(𝜏	PROPN
cana-1117	267	52	)	)	PUNCT
cana-1117	267	53	+	+	CCONJ
cana-1117	267	54	𝜏2	𝜏2	PROPN
cana-1117	267	55	2	2	NUM
cana-1117	267	56	+	+	NUM
cana-1117	267	57	∫	∫	PROPN
cana-1117	267	58	1	1	NUM
cana-1117	267	59	0	0	NUM
cana-1117	267	60	1	1	NUM
cana-1117	267	61	(	(	PUNCT
cana-1117	267	62	𝑠+𝜏+2)2	𝑠+𝜏+2)2	X
cana-1117	267	63	𝜙(𝑠)𝑑𝑠	𝜙(𝑠)𝑑𝑠	NUM
cana-1117	267	64	+	+	NUM
cana-1117	267	65	∫	∫	PROPN
cana-1117	267	66	1	1	NUM
cana-1117	267	67	0	0	NUM
cana-1117	267	68	𝜏	𝜏	PROPN
cana-1117	267	69	𝑠+1	𝑠+1	PROPN
cana-1117	267	70	𝜙(𝑠)𝑑𝑠	𝜙(𝑠)𝑑𝑠	NUM
cana-1117	267	71	clearly	clearly	ADV
cana-1117	267	72	the	the	DET
cana-1117	267	73	function	function	NOUN
cana-1117	267	74	𝑓	𝑓	ADP
cana-1117	267	75	and	and	CCONJ
cana-1117	267	76	𝑔	𝑔	PROPN
cana-1117	267	77	continuous	continuous	ADJ
cana-1117	267	78	,	,	PUNCT
cana-1117	267	79	since	since	SCONJ
cana-1117	267	80	‖𝑓(𝜏	‖𝑓(𝜏	NOUN
cana-1117	267	81	,	,	PUNCT
cana-1117	267	82	𝜙(𝜏	𝜙(𝜏	PROPN
cana-1117	267	83	)	)	PUNCT
cana-1117	267	84	,	,	PUNCT
cana-1117	267	85	𝒢𝜙(𝜏	𝒢𝜙(𝜏	NOUN
cana-1117	267	86	)	)	PUNCT
cana-1117	267	87	,	,	PUNCT
cana-1117	267	88	𝒮𝜙(𝜏	𝒮𝜙(𝜏	NOUN
cana-1117	267	89	)	)	PUNCT
cana-1117	267	90	)	)	PUNCT
cana-1117	267	91	−	−	PROPN
cana-1117	268	1	𝑓(𝜏	𝑓(𝜏	PROPN
cana-1117	268	2	,	,	PUNCT
cana-1117	268	3	𝜓(𝜏	𝜓(𝜏	PROPN
cana-1117	268	4	)	)	PUNCT
cana-1117	268	5	,	,	PUNCT
cana-1117	268	6	𝒢𝜓(𝜏	𝒢𝜓(𝜏	NOUN
cana-1117	268	7	)	)	PUNCT
cana-1117	268	8	,	,	PUNCT
cana-1117	268	9	𝒮𝜓(𝜏))‖	𝒮𝜓(𝜏))‖	NOUN
cana-1117	268	10	≤	≤	NUM
cana-1117	268	11	169	169	NUM
cana-1117	268	12	264	264	NUM
cana-1117	268	13	‖𝜙	‖𝜙	NOUN
cana-1117	268	14	−	−	NOUN
cana-1117	268	15	𝜓‖	𝜓‖	NOUN
cana-1117	268	16	here	here	ADV
cana-1117	268	17	by	by	ADP
cana-1117	268	18	(	(	PUNCT
cana-1117	268	19	1.2	1.2	NUM
cana-1117	268	20	)	)	PUNCT
cana-1117	268	21	,	,	PUNCT
cana-1117	268	22	we	we	PRON
cana-1117	268	23	get	get	VERB
cana-1117	268	24	ℒ	ℒ	NOUN
cana-1117	268	25	=	=	SYM
cana-1117	268	26	169	169	NUM
cana-1117	268	27	264	264	NUM
cana-1117	268	28	≤	≤	NUM
cana-1117	268	29	1	1	NUM
cana-1117	268	30	.	.	PUNCT
cana-1117	269	1	therefore	therefore	ADV
cana-1117	269	2	by	by	ADP
cana-1117	269	3	theorem(3.2	theorem(3.2	PROPN
cana-1117	269	4	)	)	PUNCT
cana-1117	269	5	this	this	DET
cana-1117	269	6	problem	problem	NOUN
cana-1117	269	7	has	have	VERB
cana-1117	269	8	a	a	DET
cana-1117	269	9	unique	unique	ADJ
cana-1117	269	10	solution	solution	NOUN
cana-1117	269	11	.	.	PUNCT
cana-1117	270	1	references	reference	NOUN
cana-1117	270	2	[	[	X
cana-1117	270	3	1	1	NUM
cana-1117	270	4	]	]	PUNCT
cana-1117	270	5	r.	r.	PROPN
cana-1117	270	6	p.	p.	PROPN
cana-1117	270	7	agarwal	agarwal	PROPN
cana-1117	270	8	.	.	PUNCT
cana-1117	271	1	certain	certain	ADJ
cana-1117	271	2	fractional	fractional	ADJ
cana-1117	271	3	q	q	NOUN
cana-1117	271	4	-	-	PUNCT
cana-1117	271	5	integrals	integral	NOUN
cana-1117	271	6	and	and	CCONJ
cana-1117	271	7	q	q	NOUN
cana-1117	271	8	-	-	NOUN
cana-1117	271	9	derivatives	derivative	NOUN
cana-1117	271	10	.	.	PUNCT
cana-1117	272	1	mathematical	mathematical	ADJ
cana-1117	272	2	proceedings	proceeding	NOUN
cana-1117	272	3	of	of	ADP
cana-1117	272	4	the	the	DET
cana-1117	272	5	cambridge	cambridge	PROPN
cana-1117	272	6	philosophical	philosophical	ADJ
cana-1117	272	7	society	society	NOUN
cana-1117	272	8	,	,	PUNCT
cana-1117	272	9	66(2):365–370	66(2):365–370	PROPN
cana-1117	272	10	,	,	PUNCT
cana-1117	272	11	1969	1969	NUM
cana-1117	272	12	.	.	PUNCT
cana-1117	273	1	[	[	X
cana-1117	273	2	2	2	NUM
cana-1117	273	3	]	]	PUNCT
cana-1117	273	4	tofigh	tofigh	NOUN
cana-1117	273	5	allahviranloo	allahviranloo	NOUN
cana-1117	273	6	,	,	PUNCT
cana-1117	273	7	atefeh	atefeh	PROPN
cana-1117	273	8	armand	armand	PROPN
cana-1117	273	9	,	,	PUNCT
cana-1117	273	10	and	and	CCONJ
cana-1117	273	11	zienab	zienab	ADV
cana-1117	273	12	gouyandeh.fuzzy	gouyandeh.fuzzy	INTJ
cana-1117	273	13	fractional	fractional	ADJ
cana-1117	273	14	differential	differential	ADJ
cana-1117	273	15	equations	equation	NOUN
cana-1117	273	16	under	under	ADP
cana-1117	273	17	generalized	generalized	ADJ
cana-1117	273	18	fuzzy	fuzzy	ADJ
cana-1117	273	19	caputo	caputo	PROPN
cana-1117	273	20	derivative	derivative	PROPN
cana-1117	273	21	.	.	PUNCT
cana-1117	274	1	journal	journal	PROPN
cana-1117	274	2	of	of	ADP
cana-1117	274	3	intelligent	intelligent	ADJ
cana-1117	274	4	&	&	CCONJ
cana-1117	274	5	fuzzy	fuzzy	ADJ
cana-1117	274	6	systems	system	NOUN
cana-1117	274	7	,	,	PUNCT
cana-1117	274	8	26(3):1481–1490	26(3):1481–1490	NUM
cana-1117	274	9	,	,	PUNCT
cana-1117	274	10	2014	2014	NUM
cana-1117	274	11	.	.	PUNCT
cana-1117	275	1	[	[	X
cana-1117	275	2	3	3	X
cana-1117	275	3	]	]	X
cana-1117	275	4	jehad	jehad	PROPN
cana-1117	275	5	alzabut	alzabut	PROPN
cana-1117	275	6	,	,	PUNCT
cana-1117	275	7	mohamed	mohamed	PROPN
cana-1117	275	8	houas	houas	PROPN
cana-1117	275	9	,	,	PUNCT
cana-1117	275	10	and	and	CCONJ
cana-1117	275	11	mohamed	mohamed	PROPN
cana-1117	275	12	i.	i.	PROPN
cana-1117	275	13	abbas	abbas	PROPN
cana-1117	275	14	.	.	PUNCT
cana-1117	276	1	application	application	NOUN
cana-1117	276	2	of	of	ADP
cana-1117	276	3	fractional	fractional	ADJ
cana-1117	276	4	quantum	quantum	NOUN
cana-1117	276	5	calculus	calculus	NOUN
cana-1117	276	6	on	on	ADP
cana-1117	276	7	coupled	couple	VERB
cana-1117	276	8	hybrid	hybrid	ADJ
cana-1117	276	9	differential	differential	NOUN
cana-1117	276	10	systems	system	NOUN
cana-1117	276	11	within	within	ADP
cana-1117	276	12	the	the	DET
cana-1117	276	13	sequential	sequential	ADJ
cana-1117	276	14	caputo	caputo	PROPN
cana-1117	276	15	fractional	fractional	PROPN
cana-1117	276	16	q	q	NOUN
cana-1117	276	17	-	-	PUNCT
cana-1117	276	18	derivatives	derivative	NOUN
cana-1117	276	19	.	.	PUNCT
cana-1117	277	1	demonstratio	demonstratio	PROPN
cana-1117	277	2	mathematica	mathematica	PROPN
cana-1117	277	3	,	,	PUNCT
cana-1117	277	4	56(1):20220205	56(1):20220205	NUM
cana-1117	277	5	,	,	PUNCT
cana-1117	277	6	2023	2023	NUM
cana-1117	277	7	.	.	PUNCT
cana-1117	278	1	[	[	X
cana-1117	278	2	4	4	X
cana-1117	278	3	]	]	PUNCT
cana-1117	278	4	a	a	DET
cana-1117	278	5	harir	harir	NOUN
cana-1117	278	6	,	,	PUNCT
cana-1117	278	7	s	s	NOUN
cana-1117	278	8	melliani	melliani	NOUN
cana-1117	278	9	,	,	PUNCT
cana-1117	278	10	and	and	CCONJ
cana-1117	278	11	ls	ls	PRON
cana-1117	278	12	chadli.fuzzy	chadli.fuzzy	NUM
cana-1117	278	13	fractional	fractional	ADJ
cana-1117	278	14	hybrid	hybrid	ADJ
cana-1117	278	15	differential	differential	NOUN
cana-1117	278	16	equations	equation	NOUN
cana-1117	278	17	.	.	PUNCT
cana-1117	279	1	carpathian	carpathian	ADJ
cana-1117	279	2	mathematical	mathematical	PROPN
cana-1117	279	3	publications,14(2):332–344,2022	publications,14(2):332–344,2022	NOUN
cana-1117	279	4	.	.	PUNCT
cana-1117	280	1	[	[	X
cana-1117	280	2	5	5	NUM
cana-1117	280	3	]	]	PUNCT
cana-1117	280	4	amira	amira	PROPN
cana-1117	280	5	abd	abd	PROPN
cana-1117	280	6	-	-	PUNCT
cana-1117	280	7	elall	elall	PROPN
cana-1117	280	8	ibrahim	ibrahim	PROPN
cana-1117	280	9	,	,	PUNCT
cana-1117	280	10	afaf	afaf	VERB
cana-1117	280	11	as	as	ADP
cana-1117	280	12	zaghrout	zaghrout	PROPN
cana-1117	280	13	,	,	PUNCT
cana-1117	280	14	kr	kr	PROPN
cana-1117	280	15	raslan	raslan	NOUN
cana-1117	280	16	,	,	PUNCT
cana-1117	280	17	and	and	CCONJ
cana-1117	280	18	khalid	khalid	PROPN
cana-1117	280	19	k	k	PROPN
cana-1117	280	20	ali.on	ali.on	PROPN
cana-1117	280	21	the	the	DET
cana-1117	280	22	analytical	analytical	ADJ
cana-1117	280	23	and	and	CCONJ
cana-1117	280	24	numerical	numerical	ADJ
cana-1117	280	25	study	study	NOUN
cana-1117	280	26	for	for	ADP
cana-1117	280	27	fractional	fractional	ADJ
cana-1117	280	28	q	q	ADJ
cana-1117	280	29	-	-	PUNCT
cana-1117	280	30	integrodifferential	integrodifferential	ADJ
cana-1117	280	31	equations.boundary	equations.boundary	ADJ
cana-1117	280	32	value	value	NOUN
cana-1117	280	33	problems,2022(1):98,2022	problems,2022(1):98,2022	NOUN
cana-1117	280	34	.	.	PUNCT
cana-1117	281	1	[	[	X
cana-1117	281	2	6	6	NUM
cana-1117	281	3	]	]	PUNCT
cana-1117	281	4	z	z	NOUN
cana-1117	281	5	noeiaghdam	noeiaghdam	NOUN
cana-1117	281	6	,	,	PUNCT
cana-1117	281	7	tofigh	tofigh	ADJ
cana-1117	281	8	allahviranloo	allahviranloo	NOUN
cana-1117	281	9	,	,	PUNCT
cana-1117	281	10	and	and	CCONJ
cana-1117	281	11	juanjnieto	juanjnieto	NOUN
cana-1117	281	12	.	.	PUNCT
cana-1117	282	1	q	q	X
cana-1117	282	2	-	-	PUNCT
cana-1117	282	3	fractional	fractional	ADJ
cana-1117	282	4	differential	differential	ADJ
cana-1117	282	5	equations	equation	NOUN
cana-1117	282	6	with	with	ADP
cana-1117	282	7	uncertainty	uncertainty	NOUN
cana-1117	282	8	.	.	PUNCT
cana-1117	283	1	soft	soft	ADJ
cana-1117	283	2	computing	computing	NOUN
cana-1117	283	3	,	,	PUNCT
cana-1117	283	4	23:9507–9524	23:9507–9524	NUM
cana-1117	283	5	,	,	PUNCT
cana-1117	283	6	2019	2019	NUM
cana-1117	283	7	.	.	PUNCT
cana-1117	284	1	[	[	X
cana-1117	284	2	7	7	NUM
cana-1117	284	3	]	]	X
cana-1117	284	4	o	o	NOUN
cana-1117	284	5	obaidat	obaidat	NOUN
cana-1117	284	6	,	,	PUNCT
cana-1117	284	7	s	s	PART
cana-1117	284	8	al	al	PROPN
cana-1117	284	9	-	-	PUNCT
cana-1117	284	10	omari	omari	PROPN
cana-1117	284	11	,	,	PUNCT
cana-1117	284	12	m	m	PROPN
cana-1117	284	13	alabedalhadi	alabedalhadi	NOUN
cana-1117	284	14	,	,	PUNCT
cana-1117	284	15	s	s	NOUN
cana-1117	284	16	momani	momani	NOUN
cana-1117	284	17	,	,	PUNCT
cana-1117	284	18	m	m	PROPN
cana-1117	284	19	al	al	PROPN
cana-1117	284	20	-	-	PUNCT
cana-1117	284	21	smadi	smadi	NOUN
cana-1117	284	22	,	,	PUNCT
cana-1117	284	23	and	and	CCONJ
cana-1117	284	24	malaroud	malaroud	NOUN
cana-1117	284	25	.	.	PUNCT
cana-1117	285	1	certain	certain	ADJ
cana-1117	285	2	results	result	NOUN
cana-1117	285	3	associated	associate	VERB
cana-1117	285	4	with	with	ADP
cana-1117	285	5	q	q	ADJ
cana-1117	285	6	-	-	PUNCT
cana-1117	285	7	fractional	fractional	ADJ
cana-1117	285	8	integrals	integral	NOUN
cana-1117	285	9	and	and	CCONJ
cana-1117	285	10	some	some	DET
cana-1117	285	11	application	application	NOUN
cana-1117	285	12	.	.	PUNCT
cana-1117	286	1	in	in	ADP
cana-1117	286	2	2023	2023	NUM
cana-1117	286	3	international	international	ADJ
cana-1117	286	4	conference	conference	NOUN
cana-1117	286	5	on	on	ADP
cana-1117	286	6	fractional	fractional	ADJ
cana-1117	286	7	differentiation	differentiation	NOUN
cana-1117	286	8	and	and	CCONJ
cana-1117	286	9	its	its	PRON
cana-1117	286	10	applications	application	NOUN
cana-1117	286	11	(	(	PUNCT
cana-1117	286	12	icfda	icfda	PROPN
cana-1117	286	13	)	)	PUNCT
cana-1117	286	14	,	,	PUNCT
cana-1117	286	15	pages	page	NOUN
cana-1117	286	16	1–6	1–6	NUM
cana-1117	286	17	.	.	PUNCT
cana-1117	286	18	ieee	ieee	NOUN
cana-1117	286	19	,	,	PUNCT
cana-1117	286	20	2023	2023	NUM
cana-1117	286	21	.	.	PUNCT
cana-1117	287	1	[	[	X
cana-1117	287	2	8	8	NUM
cana-1117	287	3	]	]	PUNCT
cana-1117	287	4	mawia	mawia	NOUN
cana-1117	287	5	osman	osman	NOUN
cana-1117	287	6	and	and	CCONJ
cana-1117	287	7	yonghui	yonghui	PROPN
cana-1117	287	8	xia	xia	PROPN
cana-1117	287	9	.	.	PUNCT
cana-1117	288	1	solving	solve	VERB
cana-1117	288	2	fuzzy	fuzzy	ADJ
cana-1117	288	3	fractional	fractional	ADJ
cana-1117	288	4	q	q	ADJ
cana-1117	288	5	-	-	PUNCT
cana-1117	288	6	differential	differential	ADJ
cana-1117	288	7	equations	equation	NOUN
cana-1117	288	8	via	via	ADP
cana-1117	288	9	fuzzy	fuzzy	ADJ
cana-1117	288	10	q	q	ADJ
cana-1117	288	11	-	-	PUNCT
cana-1117	288	12	differential	differential	ADJ
cana-1117	288	13	transform	transform	NOUN
cana-1117	288	14	.	.	PUNCT
cana-1117	289	1	journal	journal	NOUN
cana-1117	289	2	of	of	ADP
cana-1117	289	3	intelligent	intelligent	ADJ
cana-1117	289	4	&	&	CCONJ
cana-1117	289	5	fuzzy	fuzzy	ADJ
cana-1117	289	6	systems	system	NOUN
cana-1117	289	7	,	,	PUNCT
cana-1117	289	8	44(2):2791–2846	44(2):2791–2846	PROPN
cana-1117	289	9	,	,	PUNCT
cana-1117	289	10	2023	2023	NUM
cana-1117	289	11	.	.	PUNCT
cana-1117	290	1	[	[	X
cana-1117	290	2	9	9	NUM
cana-1117	290	3	]	]	X
cana-1117	290	4	p	p	PROPN
cana-1117	290	5	njionou	njionou	NOUN
cana-1117	290	6	sadjang	sadjang	NOUN
cana-1117	290	7	and	and	CCONJ
cana-1117	290	8	s	s	VERB
cana-1117	290	9	mboutngam	mboutngam	NOUN
cana-1117	290	10	.	.	PUNCT
cana-1117	291	1	a	a	DET
cana-1117	291	2	fractional	fractional	ADJ
cana-1117	291	3	g	g	NOUN
cana-1117	291	4	-	-	PUNCT
cana-1117	291	5	derivative	derivative	ADJ
cana-1117	291	6	operator	operator	NOUN
cana-1117	291	7	and	and	CCONJ
cana-1117	291	8	fractional	fractional	ADJ
cana-1117	291	9	extensions	extension	NOUN
cana-1117	291	10	of	of	ADP
cana-1117	291	11	some	some	DET
cana-1117	291	12	gorthogonal	gorthogonal	ADJ
cana-1117	291	13	polynomials	polynomial	NOUN
cana-1117	291	14	.	.	PUNCT
cana-1117	292	1	ramanujan	ramanujan	PROPN
cana-1117	292	2	journal	journal	PROPN
cana-1117	292	3	,	,	PUNCT
cana-1117	292	4	54(1):29–41	54(1):29–41	NUM
cana-1117	292	5	,	,	PUNCT
cana-1117	292	6	2021	2021	NUM
cana-1117	292	7	.	.	PUNCT