id	sid	tid	token	lemma	pos
cana-3959	1	1	communications	communication	NOUN
cana-3959	1	2	on	on	ADP
cana-3959	1	3	applied	apply	VERB
cana-3959	1	4	nonlinear	nonlinear	ADJ
cana-3959	1	5	analysis	analysis	NOUN
cana-3959	1	6	issn	issn	NOUN
cana-3959	1	7	:	:	PUNCT
cana-3959	1	8	1074	1074	NUM
cana-3959	1	9	-	-	PUNCT
cana-3959	1	10	133x	133x	NUM
cana-3959	1	11	vol	vol	NOUN
cana-3959	1	12	32	32	NUM
cana-3959	1	13	no	no	NOUN
cana-3959	1	14	.	.	PUNCT
cana-3959	2	1	9s	9s	NUM
cana-3959	2	2	(	(	PUNCT
cana-3959	2	3	2025	2025	NUM
cana-3959	2	4	)	)	PUNCT
cana-3959	2	5	475	475	NUM
cana-3959	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	2	7	investigating	investigate	VERB
cana-3959	2	8	a	a	DET
cana-3959	2	9	novel	novel	ADJ
cana-3959	2	10	stability	stability	NOUN
cana-3959	2	11	results	result	NOUN
cana-3959	2	12	of	of	ADP
cana-3959	2	13	generalized	generalized	ADJ
cana-3959	2	14	alternate	alternate	ADJ
cana-3959	2	15	cubic	cubic	ADJ
cana-3959	2	16	functional	functional	ADJ
cana-3959	2	17	equations	equation	NOUN
cana-3959	2	18	:	:	PUNCT
cana-3959	2	19	classical	classical	ADJ
cana-3959	2	20	method	method	NOUN
cana-3959	2	21	for	for	ADP
cana-3959	2	22	banach	banach	NOUN
cana-3959	2	23	spaces	space	NOUN
cana-3959	2	24	and	and	CCONJ
cana-3959	2	25	direct	direct	ADJ
cana-3959	2	26	-fixed	-fixe	VERB
cana-3959	2	27	point	point	NOUN
cana-3959	2	28	approaches	approach	NOUN
cana-3959	2	29	for	for	ADP
cana-3959	2	30	fuzzy	fuzzy	ADJ
cana-3959	2	31	normed	norme	VERB
cana-3959	2	32	spaces	space	NOUN
cana-3959	2	33	p.	p.	PROPN
cana-3959	2	34	agilan	agilan	PROPN
cana-3959	3	1	𝟏∗	𝟏∗	NUM
cana-3959	3	2	,	,	PUNCT
cana-3959	3	3	v.	v.	CCONJ
cana-3959	3	4	vijayan	vijayan	PROPN
cana-3959	3	5	𝟐	𝟐	NUM
cana-3959	3	6	,	,	PUNCT
cana-3959	3	7	m.	m.	NOUN
cana-3959	3	8	sophia	sophia	PROPN
cana-3959	3	9	𝟑	𝟑	NUM
cana-3959	3	10	v.banu	v.banu	PROPN
cana-3959	3	11	priya	priya	PROPN
cana-3959	3	12	𝟒	𝟒	PROPN
cana-3959	3	13	1department	1department	NUM
cana-3959	3	14	of	of	ADP
cana-3959	3	15	mathematics	mathematic	NOUN
cana-3959	3	16	,	,	PUNCT
cana-3959	3	17	st.joseph	st.joseph	X
cana-3959	3	18	’s	’s	PART
cana-3959	3	19	college	college	NOUN
cana-3959	3	20	of	of	ADP
cana-3959	3	21	engineering	engineering	PROPN
cana-3959	3	22	,	,	PUNCT
cana-3959	3	23	omr	omr	PROPN
cana-3959	3	24	,	,	PUNCT
cana-3959	3	25	chennai	chennai	VERB
cana-3959	3	26	600	600	NUM
cana-3959	3	27	119	119	NUM
cana-3959	3	28	,	,	PUNCT
cana-3959	3	29	tamilnadu	tamilnadu	NOUN
cana-3959	3	30	,	,	PUNCT
cana-3959	3	31	india	india	PROPN
cana-3959	3	32	.	.	PUNCT
cana-3959	4	1	2department	2department	NUM
cana-3959	4	2	of	of	ADP
cana-3959	4	3	electronics	electronic	NOUN
cana-3959	4	4	and	and	CCONJ
cana-3959	4	5	instrumentation	instrumentation	NOUN
cana-3959	4	6	engineering	engineering	NOUN
cana-3959	4	7	,	,	PUNCT
cana-3959	4	8	st.joseph	st.joseph	X
cana-3959	4	9	’s	’s	PART
cana-3959	4	10	college	college	NOUN
cana-3959	4	11	of	of	ADP
cana-3959	4	12	engineering	engineering	PROPN
cana-3959	4	13	,	,	PUNCT
cana-3959	4	14	omr	omr	PROPN
cana-3959	4	15	,	,	PUNCT
cana-3959	4	16	chennai	chennai	VERB
cana-3959	4	17	600	600	NUM
cana-3959	4	18	119	119	NUM
cana-3959	4	19	,	,	PUNCT
cana-3959	4	20	tamilnadu	tamilnadu	ADJ
cana-3959	4	21	,	,	PUNCT
cana-3959	4	22	india	india	PROPN
cana-3959	4	23	.	.	PUNCT
cana-3959	5	1	e	e	X
cana-3959	5	2	-	-	NOUN
cana-3959	5	3	mail	mail	NOUN
cana-3959	5	4	:	:	PUNCT
cana-3959	5	5	vinvpn@gmail.com	vinvpn@gmail.com	PROPN
cana-3959	5	6	.	.	PROPN
cana-3959	5	7	3	3	NUM
cana-3959	5	8	department	department	NOUN
cana-3959	5	9	of	of	ADP
cana-3959	5	10	mathematics	mathematic	NOUN
cana-3959	5	11	,	,	PUNCT
cana-3959	5	12	simats	simat	NOUN
cana-3959	5	13	engineering	engineering	PROPN
cana-3959	5	14	,	,	PUNCT
cana-3959	5	15	saveetha	saveetha	PROPN
cana-3959	5	16	nagar	nagar	PROPN
cana-3959	5	17	,	,	PUNCT
cana-3959	5	18	thandalam	thandalam	PROPN
cana-3959	5	19	,	,	PUNCT
cana-3959	5	20	kanchipuram	kanchipuram	PROPN
cana-3959	5	21	-	-	PUNCT
cana-3959	5	22	chennai	chennai	PROPN
cana-3959	5	23	rd	rd	PROPN
cana-3959	5	24	,	,	PUNCT
cana-3959	5	25	chennai602105	chennai602105	PROPN
cana-3959	5	26	,	,	PUNCT
cana-3959	5	27	tamilnadu	tamilnadu	NOUN
cana-3959	5	28	,	,	PUNCT
cana-3959	5	29	india	india	PROPN
cana-3959	5	30	.	.	PUNCT
cana-3959	6	1	e	e	X
cana-3959	6	2	-	-	NOUN
cana-3959	6	3	mail	mail	NOUN
cana-3959	6	4	:	:	PUNCT
cana-3959	6	5	sophia.raj2005@gmail.com	sophia.raj2005@gmail.com	PROPN
cana-3959	6	6	.	.	PUNCT
cana-3959	7	1	4department	4department	NUM
cana-3959	7	2	of	of	ADP
cana-3959	7	3	mathematics	mathematic	NOUN
cana-3959	7	4	,	,	PUNCT
cana-3959	7	5	r.m.k	r.m.k	VERB
cana-3959	7	6	college	college	NOUN
cana-3959	7	7	of	of	ADP
cana-3959	7	8	engineering	engineering	NOUN
cana-3959	7	9	and	and	CCONJ
cana-3959	7	10	technology	technology	NOUN
cana-3959	7	11	,	,	PUNCT
cana-3959	7	12	kavaraipettai	kavaraipettai	VERB
cana-3959	7	13	601	601	NUM
cana-3959	7	14	206	206	NUM
cana-3959	7	15	,	,	PUNCT
cana-3959	7	16	tamilnadu	tamilnadu	NOUN
cana-3959	7	17	,	,	PUNCT
cana-3959	7	18	india	india	PROPN
cana-3959	7	19	.	.	PUNCT
cana-3959	8	1	e-mail:spriya.maths@gmail.com	e-mail:spriya.maths@gmail.com	PROPN
cana-3959	8	2	.	.	PUNCT
cana-3959	9	1	*	*	PUNCT
cana-3959	9	2	corresponding	correspond	VERB
cana-3959	9	3	author	author	NOUN
cana-3959	9	4	:	:	PUNCT
cana-3959	10	1	agilram@gmail.com	agilram@gmail.com	PROPN
cana-3959	10	2	.	.	PUNCT
cana-3959	10	3	article	article	PROPN
cana-3959	10	4	history	history	NOUN
cana-3959	10	5	:	:	PUNCT
cana-3959	10	6	received	receive	VERB
cana-3959	10	7	:	:	PUNCT
cana-3959	10	8	10	10	NUM
cana-3959	10	9	-	-	SYM
cana-3959	10	10	11	11	NUM
cana-3959	10	11	-	-	PUNCT
cana-3959	10	12	2024	2024	NUM
cana-3959	10	13	revised	revise	VERB
cana-3959	10	14	:	:	PUNCT
cana-3959	10	15	16	16	NUM
cana-3959	10	16	-	-	SYM
cana-3959	10	17	12	12	NUM
cana-3959	10	18	-	-	PUNCT
cana-3959	10	19	2024	2024	NUM
cana-3959	10	20	accepted	accept	VERB
cana-3959	10	21	:	:	PUNCT
cana-3959	10	22	11	11	NUM
cana-3959	10	23	-	-	SYM
cana-3959	10	24	01	01	NUM
cana-3959	10	25	-	-	PUNCT
cana-3959	10	26	2025	2025	NUM
cana-3959	10	27	abstract	abstract	NOUN
cana-3959	10	28	:	:	PUNCT
cana-3959	10	29	this	this	DET
cana-3959	10	30	paper	paper	NOUN
cana-3959	10	31	explores	explore	NOUN
cana-3959	10	32	novel	novel	ADJ
cana-3959	10	33	stability	stability	NOUN
cana-3959	10	34	results	result	NOUN
cana-3959	10	35	for	for	ADP
cana-3959	10	36	generalized	generalized	ADJ
cana-3959	10	37	alternate	alternate	ADJ
cana-3959	10	38	cubic	cubic	ADJ
cana-3959	10	39	functional	functional	ADJ
cana-3959	10	40	equation(fun	equation(fun	NOUN
cana-3959	10	41	eq	eq	ADP
cana-3959	10	42	)	)	PUNCT
cana-3959	10	43	using	use	VERB
cana-3959	10	44	two	two	NUM
cana-3959	10	45	distinct	distinct	ADJ
cana-3959	10	46	analytical	analytical	ADJ
cana-3959	10	47	frameworks	framework	NOUN
cana-3959	10	48	:	:	PUNCT
cana-3959	10	49	the	the	DET
cana-3959	10	50	classical	classical	ADJ
cana-3959	10	51	method	method	NOUN
cana-3959	10	52	for	for	ADP
cana-3959	10	53	banach	banach	NOUN
cana-3959	10	54	spaces	space	NOUN
cana-3959	10	55	and	and	CCONJ
cana-3959	10	56	the	the	DET
cana-3959	10	57	direct	direct	ADJ
cana-3959	10	58	and	and	CCONJ
cana-3959	10	59	fixed	fix	VERB
cana-3959	10	60	point	point	NOUN
cana-3959	10	61	approaches	approach	NOUN
cana-3959	10	62	for	for	ADP
cana-3959	10	63	fuzzy	fuzzy	ADJ
cana-3959	10	64	normed	normed	ADJ
cana-3959	10	65	spaces	space	NOUN
cana-3959	10	66	.	.	PUNCT
cana-3959	11	1	the	the	DET
cana-3959	11	2	study	study	NOUN
cana-3959	11	3	examines	examine	VERB
cana-3959	11	4	the	the	DET
cana-3959	11	5	stability	stability	NOUN
cana-3959	11	6	behavior	behavior	NOUN
cana-3959	11	7	of	of	ADP
cana-3959	11	8	the	the	DET
cana-3959	11	9	generalized	generalize	VERB
cana-3959	11	10	alternate	alternate	ADJ
cana-3959	11	11	cubic	cubic	ADJ
cana-3959	11	12	functional	functional	ADJ
cana-3959	11	13	equation	equation	NOUN
cana-3959	11	14	,	,	PUNCT
cana-3959	11	15	focusing	focus	VERB
cana-3959	11	16	on	on	ADP
cana-3959	11	17	how	how	SCONJ
cana-3959	11	18	small	small	ADJ
cana-3959	11	19	deviations	deviation	NOUN
cana-3959	11	20	from	from	ADP
cana-3959	11	21	exact	exact	ADJ
cana-3959	11	22	solutions	solution	NOUN
cana-3959	11	23	influence	influence	NOUN
cana-3959	11	24	the	the	DET
cana-3959	11	25	overall	overall	ADJ
cana-3959	11	26	stability	stability	NOUN
cana-3959	11	27	in	in	ADP
cana-3959	11	28	different	different	ADJ
cana-3959	11	29	normed	normed	ADJ
cana-3959	11	30	environments	environment	NOUN
cana-3959	11	31	.	.	PUNCT
cana-3959	12	1	in	in	ADP
cana-3959	12	2	banach	banach	NOUN
cana-3959	12	3	spaces	space	NOUN
cana-3959	12	4	,	,	PUNCT
cana-3959	12	5	the	the	DET
cana-3959	12	6	classical	classical	ADJ
cana-3959	12	7	approach	approach	NOUN
cana-3959	12	8	is	be	AUX
cana-3959	12	9	applied	apply	VERB
cana-3959	12	10	to	to	PART
cana-3959	12	11	derive	derive	VERB
cana-3959	12	12	conditions	condition	NOUN
cana-3959	12	13	for	for	ADP
cana-3959	12	14	hyers	hyer	NOUN
cana-3959	12	15	-	-	PUNCT
cana-3959	12	16	ulam	ulam	PROPN
cana-3959	12	17	stability	stability	NOUN
cana-3959	12	18	,	,	PUNCT
cana-3959	12	19	providing	provide	VERB
cana-3959	12	20	insight	insight	NOUN
cana-3959	12	21	into	into	ADP
cana-3959	12	22	the	the	DET
cana-3959	12	23	equation	equation	NOUN
cana-3959	12	24	’s	’s	PART
cana-3959	12	25	behavior	behavior	NOUN
cana-3959	12	26	under	under	ADP
cana-3959	12	27	small	small	ADJ
cana-3959	12	28	perturbations	perturbation	NOUN
cana-3959	12	29	.	.	PUNCT
cana-3959	13	1	for	for	ADP
cana-3959	13	2	fuzzy	fuzzy	ADJ
cana-3959	13	3	normed	normed	ADJ
cana-3959	13	4	spaces	space	NOUN
cana-3959	13	5	,	,	PUNCT
cana-3959	13	6	both	both	CCONJ
cana-3959	13	7	direct	direct	ADJ
cana-3959	13	8	and	and	CCONJ
cana-3959	13	9	fixed	fix	VERB
cana-3959	13	10	point	point	NOUN
cana-3959	13	11	methods	method	NOUN
cana-3959	13	12	are	be	AUX
cana-3959	13	13	employed	employ	VERB
cana-3959	13	14	to	to	PART
cana-3959	13	15	account	account	VERB
cana-3959	13	16	for	for	ADP
cana-3959	13	17	the	the	DET
cana-3959	13	18	inherent	inherent	ADJ
cana-3959	13	19	uncertainties	uncertainty	NOUN
cana-3959	13	20	and	and	CCONJ
cana-3959	13	21	fuzziness	fuzziness	NOUN
cana-3959	13	22	in	in	ADP
cana-3959	13	23	the	the	DET
cana-3959	13	24	normed	normed	ADJ
cana-3959	13	25	structure	structure	NOUN
cana-3959	13	26	,	,	PUNCT
cana-3959	13	27	offering	offer	VERB
cana-3959	13	28	a	a	DET
cana-3959	13	29	more	more	ADV
cana-3959	13	30	flexible	flexible	ADJ
cana-3959	13	31	stability	stability	NOUN
cana-3959	13	32	analysis	analysis	NOUN
cana-3959	13	33	.	.	PUNCT
cana-3959	14	1	the	the	DET
cana-3959	14	2	results	result	NOUN
cana-3959	14	3	obtained	obtain	VERB
cana-3959	14	4	highlight	highlight	VERB
cana-3959	14	5	the	the	DET
cana-3959	14	6	differences	difference	NOUN
cana-3959	14	7	and	and	CCONJ
cana-3959	14	8	advantages	advantage	NOUN
cana-3959	14	9	of	of	ADP
cana-3959	14	10	each	each	DET
cana-3959	14	11	approach	approach	NOUN
cana-3959	14	12	,	,	PUNCT
cana-3959	14	13	contributing	contribute	VERB
cana-3959	14	14	to	to	ADP
cana-3959	14	15	the	the	DET
cana-3959	14	16	broader	broad	ADJ
cana-3959	14	17	understanding	understanding	NOUN
cana-3959	14	18	of	of	ADP
cana-3959	14	19	functional	functional	ADJ
cana-3959	14	20	equations	equation	NOUN
cana-3959	14	21	in	in	ADP
cana-3959	14	22	both	both	CCONJ
cana-3959	14	23	deterministic	deterministic	ADJ
cana-3959	14	24	and	and	CCONJ
cana-3959	14	25	fuzzy	fuzzy	ADJ
cana-3959	14	26	frameworks	framework	NOUN
cana-3959	14	27	.	.	PUNCT
cana-3959	15	1	these	these	DET
cana-3959	15	2	findings	finding	NOUN
cana-3959	15	3	have	have	VERB
cana-3959	15	4	potential	potential	ADJ
cana-3959	15	5	applications	application	NOUN
cana-3959	15	6	in	in	ADP
cana-3959	15	7	various	various	ADJ
cana-3959	15	8	mathematical	mathematical	ADJ
cana-3959	15	9	and	and	CCONJ
cana-3959	15	10	applied	apply	VERB
cana-3959	15	11	fields	field	NOUN
cana-3959	15	12	,	,	PUNCT
cana-3959	15	13	where	where	SCONJ
cana-3959	15	14	both	both	PRON
cana-3959	15	15	precise	precise	ADJ
cana-3959	15	16	and	and	CCONJ
cana-3959	15	17	imprecise	imprecise	ADJ
cana-3959	15	18	data	datum	NOUN
cana-3959	15	19	structures	structure	NOUN
cana-3959	15	20	are	be	AUX
cana-3959	15	21	considered	consider	VERB
cana-3959	15	22	.	.	PUNCT
cana-3959	16	1	keywords	keyword	NOUN
cana-3959	16	2	:	:	PUNCT
cana-3959	16	3	banach	banach	NOUN
cana-3959	16	4	spaces	space	NOUN
cana-3959	16	5	,	,	PUNCT
cana-3959	16	6	fuzzy	fuzzy	ADJ
cana-3959	16	7	normed	normed	ADJ
cana-3959	16	8	spaces	space	NOUN
cana-3959	16	9	,	,	PUNCT
cana-3959	16	10	cubic	cubic	ADJ
cana-3959	16	11	functional	functional	ADJ
cana-3959	16	12	equations	equation	NOUN
cana-3959	16	13	,	,	PUNCT
cana-3959	16	14	ulam	ulam	PROPN
cana-3959	16	15	hyers	hyer	VERB
cana-3959	16	16	stability	stability	NOUN
cana-3959	16	17	,	,	PUNCT
cana-3959	16	18	fixed	fix	VERB
cana-3959	16	19	point	point	NOUN
cana-3959	16	20	.	.	PUNCT
cana-3959	17	1	1	1	NUM
cana-3959	17	2	introduction	introduction	NOUN
cana-3959	17	3	the	the	DET
cana-3959	17	4	ulam	ulam	PROPN
cana-3959	17	5	-	-	PUNCT
cana-3959	17	6	hyers	hyer	NOUN
cana-3959	17	7	-	-	PUNCT
cana-3959	17	8	rassias	rassias	PROPN
cana-3959	17	9	stability	stability	NOUN
cana-3959	17	10	deals	deal	NOUN
cana-3959	17	11	with	with	ADP
cana-3959	17	12	the	the	DET
cana-3959	17	13	stability	stability	NOUN
cana-3959	17	14	of	of	ADP
cana-3959	17	15	fun	fun	NOUN
cana-3959	17	16	eq	eq	NOUN
cana-3959	17	17	,	,	PUNCT
cana-3959	17	18	which	which	PRON
cana-3959	17	19	is	be	AUX
cana-3959	17	20	a	a	DET
cana-3959	17	21	branch	branch	NOUN
cana-3959	17	22	of	of	ADP
cana-3959	17	23	mathematical	mathematical	ADJ
cana-3959	17	24	analysis	analysis	NOUN
cana-3959	17	25	.	.	PUNCT
cana-3959	18	1	specifically	specifically	ADV
cana-3959	18	2	,	,	PUNCT
cana-3959	18	3	it	it	PRON
cana-3959	18	4	focuses	focus	VERB
cana-3959	18	5	on	on	ADP
cana-3959	18	6	determining	determine	VERB
cana-3959	18	7	under	under	ADP
cana-3959	18	8	what	what	DET
cana-3959	18	9	conditions	condition	NOUN
cana-3959	18	10	an	an	DET
cana-3959	18	11	approximate	approximate	ADJ
cana-3959	18	12	solution	solution	NOUN
cana-3959	18	13	of	of	ADP
cana-3959	18	14	a	a	DET
cana-3959	18	15	fun	fun	NOUN
cana-3959	18	16	eq	eq	NOUN
cana-3959	18	17	remains	remain	VERB
cana-3959	18	18	close	close	ADJ
cana-3959	18	19	to	to	ADP
cana-3959	18	20	the	the	DET
cana-3959	18	21	exact	exact	ADJ
cana-3959	18	22	solution	solution	NOUN
cana-3959	18	23	.	.	PUNCT
cana-3959	19	1	the	the	DET
cana-3959	19	2	stability	stability	NOUN
cana-3959	19	3	concept	concept	NOUN
cana-3959	19	4	was	be	AUX
cana-3959	19	5	initiated	initiate	VERB
cana-3959	19	6	by	by	ADP
cana-3959	19	7	stanislaw	stanislaw	PROPN
cana-3959	19	8	ulam	ulam	PROPN
cana-3959	19	9	in	in	ADP
cana-3959	19	10	1940	1940	NUM
cana-3959	20	1	[	[	X
cana-3959	20	2	1	1	X
cana-3959	20	3	]	]	PUNCT
cana-3959	20	4	when	when	SCONJ
cana-3959	20	5	he	he	PRON
cana-3959	20	6	asked	ask	VERB
cana-3959	20	7	whether	whether	SCONJ
cana-3959	20	8	approximate	approximate	ADJ
cana-3959	20	9	homomorphisms	homomorphism	NOUN
cana-3959	20	10	on	on	ADP
cana-3959	20	11	groups	group	NOUN
cana-3959	20	12	could	could	AUX
cana-3959	20	13	be	be	AUX
cana-3959	20	14	approximated	approximate	VERB
cana-3959	20	15	by	by	ADP
cana-3959	20	16	true	true	ADJ
cana-3959	20	17	homomorphisms	homomorphism	NOUN
cana-3959	20	18	.	.	PUNCT
cana-3959	21	1	later	later	ADV
cana-3959	21	2	,	,	PUNCT
cana-3959	21	3	in	in	ADP
cana-3959	21	4	the	the	DET
cana-3959	21	5	1940s	1940	NOUN
cana-3959	21	6	and	and	CCONJ
cana-3959	21	7	1950s	1950	NOUN
cana-3959	21	8	,	,	PUNCT
cana-3959	21	9	donald	donald	PROPN
cana-3959	21	10	h.	h.	PROPN
cana-3959	21	11	hyers	hyer	VERB
cana-3959	22	1	[	[	X
cana-3959	22	2	2	2	X
cana-3959	22	3	]	]	PUNCT
cana-3959	22	4	and	and	CCONJ
cana-3959	22	5	th.m	th.m	PROPN
cana-3959	22	6	.	.	PUNCT
cana-3959	23	1	rassias	rassias	PROPN
cana-3959	24	1	[	[	X
cana-3959	24	2	3	3	NUM
cana-3959	24	3	]	]	X
cana-3959	24	4	extended	extend	VERB
cana-3959	24	5	ulam	ulam	PROPN
cana-3959	24	6	’s	’s	PART
cana-3959	24	7	work	work	NOUN
cana-3959	24	8	to	to	PART
cana-3959	24	9	fun	fun	VERB
cana-3959	24	10	eq	eq	ADP
cana-3959	24	11	in	in	ADP
cana-3959	24	12	banach	banach	NOUN
cana-3959	24	13	spaces	space	NOUN
cana-3959	24	14	.	.	PUNCT
cana-3959	25	1	the	the	DET
cana-3959	25	2	ulam	ulam	PROPN
cana-3959	25	3	-	-	PUNCT
cana-3959	25	4	hyers	hyer	NOUN
cana-3959	25	5	-	-	PUNCT
cana-3959	25	6	rassias	rassias	PROPN
cana-3959	25	7	stability	stability	NOUN
cana-3959	25	8	theorem	theorem	VERB
cana-3959	25	9	communications	communication	NOUN
cana-3959	25	10	on	on	ADP
cana-3959	25	11	applied	apply	VERB
cana-3959	25	12	nonlinear	nonlinear	ADJ
cana-3959	25	13	analysis	analysis	NOUN
cana-3959	25	14	issn	issn	NOUN
cana-3959	25	15	:	:	PUNCT
cana-3959	25	16	1074	1074	NUM
cana-3959	25	17	-	-	PUNCT
cana-3959	25	18	133x	133x	NUM
cana-3959	25	19	vol	vol	NOUN
cana-3959	25	20	32	32	NUM
cana-3959	25	21	no	no	NOUN
cana-3959	25	22	.	.	PUNCT
cana-3959	26	1	9s	9s	NUM
cana-3959	26	2	(	(	PUNCT
cana-3959	26	3	2025	2025	NUM
cana-3959	26	4	)	)	PUNCT
cana-3959	26	5	476	476	NUM
cana-3959	26	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	26	7	provides	provide	VERB
cana-3959	26	8	conditions	condition	NOUN
cana-3959	26	9	under	under	ADP
cana-3959	26	10	which	which	PRON
cana-3959	26	11	a	a	DET
cana-3959	26	12	functional	functional	ADJ
cana-3959	26	13	equation	equation	NOUN
cana-3959	26	14	approximately	approximately	ADV
cana-3959	26	15	satisfies	satisfy	VERB
cana-3959	26	16	the	the	DET
cana-3959	26	17	equation	equation	NOUN
cana-3959	26	18	.	.	PUNCT
cana-3959	27	1	for	for	ADP
cana-3959	27	2	further	further	ADJ
cana-3959	27	3	developments	development	NOUN
cana-3959	27	4	and	and	CCONJ
cana-3959	27	5	the	the	DET
cana-3959	27	6	subsequent	subsequent	ADJ
cana-3959	27	7	contributions	contribution	NOUN
cana-3959	27	8	by	by	ADP
cana-3959	27	9	t.	t.	PROPN
cana-3959	27	10	aoki	aoki	PROPN
cana-3959	27	11	,	,	PUNCT
cana-3959	27	12	p.	p.	PROPN
cana-3959	27	13	gavruta	gavruta	PROPN
cana-3959	27	14	,	,	PUNCT
cana-3959	27	15	j.m	j.m	PROPN
cana-3959	27	16	.	.	PROPN
cana-3959	27	17	rassias	rassias	PROPN
cana-3959	27	18	,	,	PUNCT
cana-3959	27	19	isac	isac	NOUN
cana-3959	27	20	and	and	CCONJ
cana-3959	27	21	others	other	NOUN
cana-3959	28	1	[	[	X
cana-3959	28	2	4	4	NUM
cana-3959	28	3	,	,	PUNCT
cana-3959	28	4	5	5	NUM
cana-3959	28	5	,	,	PUNCT
cana-3959	28	6	6	6	NUM
cana-3959	28	7	,	,	PUNCT
cana-3959	28	8	7	7	NUM
cana-3959	28	9	,	,	PUNCT
cana-3959	28	10	8	8	NUM
cana-3959	28	11	,	,	PUNCT
cana-3959	28	12	9	9	NUM
cana-3959	28	13	,	,	PUNCT
cana-3959	28	14	10	10	NUM
cana-3959	28	15	]	]	PUNCT
cana-3959	28	16	.	.	PUNCT
cana-3959	29	1	it	it	PRON
cana-3959	29	2	’s	’	VERB
cana-3959	29	3	of	of	ADP
cana-3959	29	4	great	great	ADJ
cana-3959	29	5	significance	significance	NOUN
cana-3959	29	6	in	in	ADP
cana-3959	29	7	many	many	ADJ
cana-3959	29	8	areas	area	NOUN
cana-3959	29	9	of	of	ADP
cana-3959	29	10	mathematics	mathematic	NOUN
cana-3959	29	11	,	,	PUNCT
cana-3959	29	12	including	include	VERB
cana-3959	29	13	functional	functional	ADJ
cana-3959	29	14	analysis	analysis	NOUN
cana-3959	29	15	,	,	PUNCT
cana-3959	29	16	operator	operator	NOUN
cana-3959	29	17	theory	theory	NOUN
cana-3959	29	18	,	,	PUNCT
cana-3959	29	19	and	and	CCONJ
cana-3959	29	20	mathematical	mathematical	ADJ
cana-3959	29	21	physics	physics	NOUN
cana-3959	29	22	.	.	PUNCT
cana-3959	30	1	in	in	ADP
cana-3959	30	2	practical	practical	ADJ
cana-3959	30	3	terms	term	NOUN
cana-3959	30	4	,	,	PUNCT
cana-3959	30	5	this	this	DET
cana-3959	30	6	stability	stability	NOUN
cana-3959	30	7	theorem	theorem	NOUN
cana-3959	30	8	has	have	VERB
cana-3959	30	9	applications	application	NOUN
cana-3959	30	10	in	in	ADP
cana-3959	30	11	various	various	ADJ
cana-3959	30	12	fields	field	NOUN
cana-3959	30	13	,	,	PUNCT
cana-3959	30	14	such	such	ADJ
cana-3959	30	15	as	as	ADP
cana-3959	30	16	numerical	numerical	ADJ
cana-3959	30	17	analysis	analysis	NOUN
cana-3959	30	18	,	,	PUNCT
cana-3959	30	19	optimization	optimization	NOUN
cana-3959	30	20	,	,	PUNCT
cana-3959	30	21	control	control	NOUN
cana-3959	30	22	theory	theory	NOUN
cana-3959	30	23	,	,	PUNCT
cana-3959	30	24	and	and	CCONJ
cana-3959	30	25	signal	signal	NOUN
cana-3959	30	26	processing	processing	NOUN
cana-3959	30	27	,	,	PUNCT
cana-3959	30	28	where	where	SCONJ
cana-3959	30	29	it	it	PRON
cana-3959	30	30	’s	’	VERB
cana-3959	30	31	crucial	crucial	ADJ
cana-3959	30	32	to	to	PART
cana-3959	30	33	understand	understand	VERB
cana-3959	30	34	how	how	SCONJ
cana-3959	30	35	small	small	ADJ
cana-3959	30	36	errors	error	NOUN
cana-3959	30	37	in	in	ADP
cana-3959	30	38	input	input	NOUN
cana-3959	30	39	data	datum	NOUN
cana-3959	30	40	or	or	CCONJ
cana-3959	30	41	parameters	parameter	NOUN
cana-3959	30	42	affect	affect	VERB
cana-3959	30	43	the	the	DET
cana-3959	30	44	output	output	NOUN
cana-3959	30	45	of	of	ADP
cana-3959	30	46	a	a	DET
cana-3959	30	47	mathematical	mathematical	ADJ
cana-3959	30	48	model	model	NOUN
cana-3959	30	49	or	or	CCONJ
cana-3959	30	50	system	system	NOUN
cana-3959	30	51	.	.	PUNCT
cana-3959	31	1	the	the	DET
cana-3959	31	2	concept	concept	NOUN
cana-3959	31	3	of	of	ADP
cana-3959	31	4	hyers	hyers	PROPN
cana-3959	31	5	-	-	PUNCT
cana-3959	31	6	ulam	ulam	PROPN
cana-3959	31	7	stability	stability	NOUN
cana-3959	31	8	has	have	AUX
cana-3959	31	9	had	have	VERB
cana-3959	31	10	a	a	DET
cana-3959	31	11	significant	significant	ADJ
cana-3959	31	12	impact	impact	NOUN
cana-3959	31	13	across	across	ADP
cana-3959	31	14	various	various	ADJ
cana-3959	31	15	mathematical	mathematical	ADJ
cana-3959	31	16	domains	domain	NOUN
cana-3959	31	17	.	.	PUNCT
cana-3959	32	1	initially	initially	ADV
cana-3959	32	2	introduced	introduce	VERB
cana-3959	32	3	in	in	ADP
cana-3959	32	4	the	the	DET
cana-3959	32	5	context	context	NOUN
cana-3959	32	6	of	of	ADP
cana-3959	32	7	fun	fun	NOUN
cana-3959	32	8	eq	eq	NOUN
cana-3959	32	9	,	,	PUNCT
cana-3959	32	10	hyers	hyers	PROPN
cana-3959	32	11	-	-	PUNCT
cana-3959	32	12	ulam	ulam	PROPN
cana-3959	32	13	stability	stability	NOUN
cana-3959	32	14	focuses	focus	VERB
cana-3959	32	15	on	on	ADP
cana-3959	32	16	whether	whether	SCONJ
cana-3959	32	17	small	small	ADJ
cana-3959	32	18	deviations	deviation	NOUN
cana-3959	32	19	from	from	ADP
cana-3959	32	20	a	a	DET
cana-3959	32	21	functional	functional	ADJ
cana-3959	32	22	equation	equation	NOUN
cana-3959	32	23	still	still	ADV
cana-3959	32	24	allow	allow	VERB
cana-3959	32	25	for	for	ADP
cana-3959	32	26	an	an	DET
cana-3959	32	27	approximate	approximate	ADJ
cana-3959	32	28	solution	solution	NOUN
cana-3959	32	29	that	that	PRON
cana-3959	32	30	is	be	AUX
cana-3959	32	31	close	close	ADJ
cana-3959	32	32	to	to	ADP
cana-3959	32	33	an	an	DET
cana-3959	32	34	exact	exact	ADJ
cana-3959	32	35	solution	solution	NOUN
cana-3959	32	36	.	.	PUNCT
cana-3959	33	1	this	this	DET
cana-3959	33	2	principle	principle	NOUN
cana-3959	33	3	has	have	AUX
cana-3959	33	4	since	since	ADV
cana-3959	33	5	been	be	AUX
cana-3959	33	6	applied	apply	VERB
cana-3959	33	7	to	to	ADP
cana-3959	33	8	numerous	numerous	ADJ
cana-3959	33	9	areas	area	NOUN
cana-3959	33	10	such	such	ADJ
cana-3959	33	11	as	as	ADP
cana-3959	33	12	:	:	PUNCT
cana-3959	33	13	differential	differential	NOUN
cana-3959	33	14	equations[11	equations[11	ADV
cana-3959	33	15	,	,	PUNCT
cana-3959	33	16	12	12	NUM
cana-3959	33	17	]	]	PUNCT
cana-3959	33	18	:	:	PUNCT
cana-3959	33	19	hyers	hyers	PROPN
cana-3959	33	20	-	-	PUNCT
cana-3959	33	21	ulam	ulam	PROPN
cana-3959	33	22	stability	stability	NOUN
cana-3959	33	23	helps	help	VERB
cana-3959	33	24	assess	assess	VERB
cana-3959	33	25	the	the	DET
cana-3959	33	26	stability	stability	NOUN
cana-3959	33	27	of	of	ADP
cana-3959	33	28	differential	differential	ADJ
cana-3959	33	29	equations	equation	NOUN
cana-3959	33	30	,	,	PUNCT
cana-3959	33	31	especially	especially	ADV
cana-3959	33	32	in	in	ADP
cana-3959	33	33	determining	determine	VERB
cana-3959	33	34	whether	whether	SCONJ
cana-3959	33	35	solutions	solution	NOUN
cana-3959	33	36	to	to	PART
cana-3959	33	37	perturbed	perturb	VERB
cana-3959	33	38	equations	equation	NOUN
cana-3959	33	39	remain	remain	VERB
cana-3959	33	40	close	close	ADJ
cana-3959	33	41	to	to	ADP
cana-3959	33	42	the	the	DET
cana-3959	33	43	solutions	solution	NOUN
cana-3959	33	44	of	of	ADP
cana-3959	33	45	the	the	DET
cana-3959	33	46	original	original	ADJ
cana-3959	33	47	equation	equation	NOUN
cana-3959	33	48	.	.	PUNCT
cana-3959	34	1	integral	integral	ADJ
cana-3959	34	2	equations[13	equations[13	NOUN
cana-3959	34	3	,	,	PUNCT
cana-3959	34	4	14	14	NUM
cana-3959	34	5	]	]	X
cana-3959	34	6	:	:	PUNCT
cana-3959	34	7	in	in	ADP
cana-3959	34	8	integral	integral	ADJ
cana-3959	34	9	equations	equation	NOUN
cana-3959	34	10	,	,	PUNCT
cana-3959	34	11	the	the	DET
cana-3959	34	12	stability	stability	NOUN
cana-3959	34	13	concept	concept	NOUN
cana-3959	34	14	provides	provide	VERB
cana-3959	34	15	a	a	DET
cana-3959	34	16	framework	framework	NOUN
cana-3959	34	17	to	to	PART
cana-3959	34	18	ensure	ensure	VERB
cana-3959	34	19	that	that	SCONJ
cana-3959	34	20	approximate	approximate	ADJ
cana-3959	34	21	solutions	solution	NOUN
cana-3959	34	22	remain	remain	VERB
cana-3959	34	23	consistent	consistent	ADJ
cana-3959	34	24	even	even	ADV
cana-3959	34	25	under	under	ADP
cana-3959	34	26	perturbations	perturbation	NOUN
cana-3959	34	27	.	.	PUNCT
cana-3959	35	1	operator	operator	NOUN
cana-3959	35	2	theory[15	theory[15	PROPN
cana-3959	35	3	,	,	PUNCT
cana-3959	35	4	16	16	NUM
cana-3959	35	5	]	]	PUNCT
cana-3959	35	6	:	:	PUNCT
cana-3959	35	7	hyersulam	hyersulam	PROPN
cana-3959	35	8	stability	stability	NOUN
cana-3959	35	9	has	have	AUX
cana-3959	35	10	been	be	AUX
cana-3959	35	11	extended	extend	VERB
cana-3959	35	12	to	to	ADP
cana-3959	35	13	operator	operator	NOUN
cana-3959	35	14	equations	equation	NOUN
cana-3959	35	15	,	,	PUNCT
cana-3959	35	16	aiding	aid	VERB
cana-3959	35	17	in	in	ADP
cana-3959	35	18	the	the	DET
cana-3959	35	19	analysis	analysis	NOUN
cana-3959	35	20	of	of	ADP
cana-3959	35	21	bounded	bounded	ADJ
cana-3959	35	22	linear	linear	PROPN
cana-3959	35	23	operators	operator	NOUN
cana-3959	35	24	and	and	CCONJ
cana-3959	35	25	their	their	PRON
cana-3959	35	26	robustness	robustness	NOUN
cana-3959	35	27	under	under	ADP
cana-3959	35	28	small	small	ADJ
cana-3959	35	29	changes	change	NOUN
cana-3959	35	30	.	.	PUNCT
cana-3959	36	1	approximation	approximation	NOUN
cana-3959	36	2	theory[17	theory[17	PROPN
cana-3959	36	3	]	]	PUNCT
cana-3959	36	4	:	:	PUNCT
cana-3959	36	5	it	it	PRON
cana-3959	36	6	plays	play	VERB
cana-3959	36	7	a	a	DET
cana-3959	36	8	role	role	NOUN
cana-3959	36	9	in	in	ADP
cana-3959	36	10	approximation	approximation	NOUN
cana-3959	36	11	theory	theory	NOUN
cana-3959	36	12	by	by	ADP
cana-3959	36	13	ensuring	ensure	VERB
cana-3959	36	14	that	that	SCONJ
cana-3959	36	15	near	near	ADJ
cana-3959	36	16	solutions	solution	NOUN
cana-3959	36	17	of	of	ADP
cana-3959	36	18	approximation	approximation	NOUN
cana-3959	36	19	problems	problem	NOUN
cana-3959	36	20	can	can	AUX
cana-3959	36	21	still	still	ADV
cana-3959	36	22	yield	yield	VERB
cana-3959	36	23	good	good	ADJ
cana-3959	36	24	approximations	approximation	NOUN
cana-3959	36	25	,	,	PUNCT
cana-3959	36	26	thus	thus	ADV
cana-3959	36	27	enhancing	enhance	VERB
cana-3959	36	28	the	the	DET
cana-3959	36	29	reliability	reliability	NOUN
cana-3959	36	30	of	of	ADP
cana-3959	36	31	numerical	numerical	ADJ
cana-3959	36	32	methods	method	NOUN
cana-3959	36	33	.	.	PUNCT
cana-3959	37	1	control	control	PROPN
cana-3959	37	2	theory[18	theory[18	PROPN
cana-3959	37	3	,	,	PUNCT
cana-3959	37	4	19	19	NUM
cana-3959	37	5	]	]	PUNCT
cana-3959	37	6	:	:	PUNCT
cana-3959	37	7	in	in	ADP
cana-3959	37	8	systems	system	NOUN
cana-3959	37	9	governed	govern	VERB
cana-3959	37	10	by	by	ADP
cana-3959	37	11	control	control	NOUN
cana-3959	37	12	equations	equation	NOUN
cana-3959	37	13	,	,	PUNCT
cana-3959	37	14	hyers	hyers	PROPN
cana-3959	37	15	-	-	PUNCT
cana-3959	37	16	ulam	ulam	PROPN
cana-3959	37	17	stability	stability	NOUN
cana-3959	37	18	contributes	contribute	VERB
cana-3959	37	19	to	to	ADP
cana-3959	37	20	the	the	DET
cana-3959	37	21	robustness	robustness	NOUN
cana-3959	37	22	analysis	analysis	NOUN
cana-3959	37	23	,	,	PUNCT
cana-3959	37	24	determining	determine	VERB
cana-3959	37	25	how	how	SCONJ
cana-3959	37	26	systems	system	NOUN
cana-3959	37	27	behave	behave	VERB
cana-3959	37	28	when	when	SCONJ
cana-3959	37	29	subject	subject	ADJ
cana-3959	37	30	to	to	ADP
cana-3959	37	31	small	small	ADJ
cana-3959	37	32	external	external	ADJ
cana-3959	37	33	disturbances	disturbance	NOUN
cana-3959	37	34	.	.	PUNCT
cana-3959	38	1	overall	overall	ADV
cana-3959	38	2	,	,	PUNCT
cana-3959	38	3	hyers	hyers	PROPN
cana-3959	38	4	-	-	PUNCT
cana-3959	38	5	ulam	ulam	PROPN
cana-3959	38	6	stability	stability	NOUN
cana-3959	38	7	provides	provide	VERB
cana-3959	38	8	a	a	DET
cana-3959	38	9	foundational	foundational	ADJ
cana-3959	38	10	tool	tool	NOUN
cana-3959	38	11	to	to	PART
cana-3959	38	12	understand	understand	VERB
cana-3959	38	13	the	the	DET
cana-3959	38	14	resilience	resilience	NOUN
cana-3959	38	15	of	of	ADP
cana-3959	38	16	mathematical	mathematical	ADJ
cana-3959	38	17	models	model	NOUN
cana-3959	38	18	in	in	ADP
cana-3959	38	19	various	various	ADJ
cana-3959	38	20	applied	apply	VERB
cana-3959	38	21	and	and	CCONJ
cana-3959	38	22	theoretical	theoretical	ADJ
cana-3959	38	23	settings	setting	NOUN
cana-3959	38	24	,	,	PUNCT
cana-3959	38	25	ensuring	ensure	VERB
cana-3959	38	26	that	that	SCONJ
cana-3959	38	27	minor	minor	ADJ
cana-3959	38	28	errors	error	NOUN
cana-3959	38	29	or	or	CCONJ
cana-3959	38	30	perturbations	perturbation	NOUN
cana-3959	38	31	do	do	AUX
cana-3959	38	32	not	not	PART
cana-3959	38	33	drastically	drastically	ADV
cana-3959	38	34	alter	alter	VERB
cana-3959	38	35	solutions	solution	NOUN
cana-3959	38	36	.	.	PUNCT
cana-3959	39	1	in	in	ADP
cana-3959	39	2	fuzzy	fuzzy	ADJ
cana-3959	39	3	normed	normed	ADJ
cana-3959	39	4	spaces	space	NOUN
cana-3959	39	5	,	,	PUNCT
cana-3959	39	6	stability	stability	NOUN
cana-3959	39	7	results	result	NOUN
cana-3959	39	8	are	be	AUX
cana-3959	39	9	typically	typically	ADV
cana-3959	39	10	established	establish	VERB
cana-3959	39	11	using	use	VERB
cana-3959	39	12	fixed	fix	VERB
cana-3959	39	13	-	-	PUNCT
cana-3959	39	14	point	point	NOUN
cana-3959	39	15	methods	method	NOUN
cana-3959	39	16	or	or	CCONJ
cana-3959	39	17	direct	direct	ADJ
cana-3959	39	18	analytical	analytical	ADJ
cana-3959	39	19	approaches	approach	NOUN
cana-3959	39	20	.	.	PUNCT
cana-3959	40	1	the	the	DET
cana-3959	40	2	fuzzy	fuzzy	ADJ
cana-3959	40	3	nature	nature	NOUN
cana-3959	40	4	of	of	ADP
cana-3959	40	5	the	the	DET
cana-3959	40	6	space	space	NOUN
cana-3959	40	7	allows	allow	VERB
cana-3959	40	8	for	for	ADP
cana-3959	40	9	handling	handle	VERB
cana-3959	40	10	vagueness	vagueness	NOUN
cana-3959	40	11	or	or	CCONJ
cana-3959	40	12	uncertainty	uncertainty	NOUN
cana-3959	40	13	in	in	ADP
cana-3959	40	14	the	the	DET
cana-3959	40	15	norm	norm	NOUN
cana-3959	40	16	,	,	PUNCT
cana-3959	40	17	which	which	PRON
cana-3959	40	18	is	be	AUX
cana-3959	40	19	critical	critical	ADJ
cana-3959	40	20	for	for	ADP
cana-3959	40	21	real	real	ADJ
cana-3959	40	22	-	-	PUNCT
cana-3959	40	23	world	world	NOUN
cana-3959	40	24	applications	application	NOUN
cana-3959	40	25	where	where	SCONJ
cana-3959	40	26	data	datum	NOUN
cana-3959	40	27	may	may	AUX
cana-3959	40	28	not	not	PART
cana-3959	40	29	always	always	ADV
cana-3959	40	30	be	be	AUX
cana-3959	40	31	exact	exact	ADJ
cana-3959	40	32	.	.	PUNCT
cana-3959	41	1	the	the	DET
cana-3959	41	2	fixedpoint	fixedpoint	NOUN
cana-3959	41	3	method	method	NOUN
cana-3959	41	4	,	,	PUNCT
cana-3959	41	5	for	for	ADP
cana-3959	41	6	instance	instance	NOUN
cana-3959	41	7	,	,	PUNCT
cana-3959	41	8	is	be	AUX
cana-3959	41	9	a	a	DET
cana-3959	41	10	powerful	powerful	ADJ
cana-3959	41	11	tool	tool	NOUN
cana-3959	41	12	used	use	VERB
cana-3959	41	13	to	to	PART
cana-3959	41	14	prove	prove	VERB
cana-3959	41	15	the	the	DET
cana-3959	41	16	existence	existence	NOUN
cana-3959	41	17	of	of	ADP
cana-3959	41	18	a	a	DET
cana-3959	41	19	stable	stable	ADJ
cana-3959	41	20	cubic	cubic	ADJ
cana-3959	41	21	mapping	mapping	NOUN
cana-3959	41	22	,	,	PUNCT
cana-3959	41	23	which	which	PRON
cana-3959	41	24	satisfies	satisfy	VERB
cana-3959	41	25	the	the	DET
cana-3959	41	26	functional	functional	ADJ
cana-3959	41	27	equation	equation	NOUN
cana-3959	41	28	under	under	ADP
cana-3959	41	29	these	these	DET
cana-3959	41	30	conditions	condition	NOUN
cana-3959	41	31	[	[	X
cana-3959	41	32	20	20	NUM
cana-3959	41	33	,	,	PUNCT
cana-3959	41	34	21	21	NUM
cana-3959	41	35	,	,	PUNCT
cana-3959	41	36	22	22	NUM
cana-3959	41	37	,	,	PUNCT
cana-3959	41	38	23	23	NUM
cana-3959	41	39	,	,	PUNCT
cana-3959	41	40	24	24	NUM
cana-3959	41	41	]	]	PUNCT
cana-3959	41	42	.	.	PUNCT
cana-3959	42	1	recent	recent	ADJ
cana-3959	42	2	studies	study	NOUN
cana-3959	42	3	show	show	VERB
cana-3959	42	4	that	that	SCONJ
cana-3959	42	5	fuzzy	fuzzy	ADJ
cana-3959	42	6	normed	norme	VERB
cana-3959	42	7	spaces	space	NOUN
cana-3959	42	8	provide	provide	VERB
cana-3959	42	9	a	a	DET
cana-3959	42	10	more	more	ADV
cana-3959	42	11	flexible	flexible	ADJ
cana-3959	42	12	framework	framework	NOUN
cana-3959	42	13	for	for	ADP
cana-3959	42	14	analyzing	analyze	VERB
cana-3959	42	15	the	the	DET
cana-3959	42	16	stability	stability	NOUN
cana-3959	42	17	of	of	ADP
cana-3959	42	18	functional	functional	ADJ
cana-3959	42	19	equations	equation	NOUN
cana-3959	42	20	.	.	PUNCT
cana-3959	43	1	in	in	ADP
cana-3959	43	2	this	this	DET
cana-3959	43	3	setting	setting	NOUN
cana-3959	43	4	,	,	PUNCT
cana-3959	43	5	the	the	DET
cana-3959	43	6	stability	stability	NOUN
cana-3959	43	7	of	of	ADP
cana-3959	43	8	cubic	cubic	ADJ
cana-3959	43	9	functional	functional	ADJ
cana-3959	43	10	equations	equation	NOUN
cana-3959	43	11	is	be	AUX
cana-3959	43	12	guaranteed	guarantee	VERB
cana-3959	43	13	even	even	ADV
cana-3959	43	14	when	when	SCONJ
cana-3959	43	15	deviations	deviation	NOUN
cana-3959	43	16	occur	occur	VERB
cana-3959	43	17	,	,	PUNCT
cana-3959	43	18	provided	provide	VERB
cana-3959	43	19	the	the	DET
cana-3959	43	20	system	system	NOUN
cana-3959	43	21	adheres	adhere	VERB
cana-3959	43	22	to	to	ADP
cana-3959	43	23	specific	specific	ADJ
cana-3959	43	24	constraints	constraint	NOUN
cana-3959	43	25	.	.	PUNCT
cana-3959	44	1	this	this	PRON
cana-3959	44	2	makes	make	VERB
cana-3959	44	3	fuzzy	fuzzy	ADJ
cana-3959	44	4	stability	stability	NOUN
cana-3959	44	5	particularly	particularly	ADV
cana-3959	44	6	relevant	relevant	ADJ
cana-3959	44	7	in	in	ADP
cana-3959	44	8	fields	field	NOUN
cana-3959	44	9	like	like	ADP
cana-3959	44	10	applied	apply	VERB
cana-3959	44	11	mathematics	mathematic	NOUN
cana-3959	44	12	,	,	PUNCT
cana-3959	44	13	economics	economic	NOUN
cana-3959	44	14	,	,	PUNCT
cana-3959	44	15	and	and	CCONJ
cana-3959	44	16	engineering	engineering	NOUN
cana-3959	44	17	,	,	PUNCT
cana-3959	44	18	where	where	SCONJ
cana-3959	44	19	imprecision	imprecision	NOUN
cana-3959	44	20	often	often	ADV
cana-3959	44	21	exists	exist	VERB
cana-3959	44	22	.	.	PUNCT
cana-3959	45	1	by	by	ADP
cana-3959	45	2	focusing	focus	VERB
cana-3959	45	3	on	on	ADP
cana-3959	45	4	these	these	DET
cana-3959	45	5	modern	modern	ADJ
cana-3959	45	6	methods	method	NOUN
cana-3959	45	7	,	,	PUNCT
cana-3959	45	8	researchers	researcher	NOUN
cana-3959	45	9	have	have	AUX
cana-3959	45	10	successfully	successfully	ADV
cana-3959	45	11	derived	derive	VERB
cana-3959	45	12	new	new	ADJ
cana-3959	45	13	stability	stability	NOUN
cana-3959	45	14	results	result	NOUN
cana-3959	45	15	for	for	ADP
cana-3959	45	16	cubic	cubic	ADJ
cana-3959	45	17	equations	equation	NOUN
cana-3959	45	18	,	,	PUNCT
cana-3959	45	19	contributing	contribute	VERB
cana-3959	45	20	to	to	ADP
cana-3959	45	21	both	both	CCONJ
cana-3959	45	22	theoretical	theoretical	ADJ
cana-3959	45	23	mathematics	mathematic	NOUN
cana-3959	45	24	and	and	CCONJ
cana-3959	45	25	practical	practical	ADJ
cana-3959	45	26	problem	problem	NOUN
cana-3959	45	27	-	-	PUNCT
cana-3959	45	28	solving	solving	NOUN
cana-3959	45	29	in	in	ADP
cana-3959	45	30	uncertain	uncertain	ADJ
cana-3959	45	31	environments	environment	NOUN
cana-3959	45	32	the	the	DET
cana-3959	45	33	study	study	NOUN
cana-3959	45	34	focuses	focus	VERB
cana-3959	45	35	on	on	ADP
cana-3959	45	36	a	a	DET
cana-3959	45	37	generalized	generalize	VERB
cana-3959	45	38	alternate	alternate	ADJ
cana-3959	45	39	cubic	cubic	ADJ
cana-3959	45	40	functional	functional	ADJ
cana-3959	45	41	equation	equation	NOUN
cana-3959	45	42	,	,	PUNCT
cana-3959	45	43	which	which	PRON
cana-3959	45	44	is	be	AUX
cana-3959	45	45	a	a	DET
cana-3959	45	46	more	more	ADV
cana-3959	45	47	intricate	intricate	ADJ
cana-3959	45	48	form	form	NOUN
cana-3959	45	49	compared	compare	VERB
cana-3959	45	50	to	to	ADP
cana-3959	45	51	traditional	traditional	ADJ
cana-3959	45	52	cubic	cubic	ADJ
cana-3959	45	53	equations	equation	NOUN
cana-3959	45	54	.	.	PUNCT
cana-3959	46	1	exploring	explore	VERB
cana-3959	46	2	the	the	DET
cana-3959	46	3	stability	stability	NOUN
cana-3959	46	4	of	of	ADP
cana-3959	46	5	such	such	ADJ
cana-3959	46	6	equations	equation	NOUN
cana-3959	46	7	in	in	ADP
cana-3959	46	8	banach	banach	NOUN
cana-3959	46	9	and	and	CCONJ
cana-3959	46	10	fuzzy	fuzzy	ADJ
cana-3959	46	11	banach	banach	NOUN
cana-3959	46	12	spaces	space	NOUN
cana-3959	46	13	is	be	AUX
cana-3959	46	14	essential	essential	ADJ
cana-3959	46	15	because	because	SCONJ
cana-3959	46	16	these	these	DET
cana-3959	46	17	spaces	space	NOUN
cana-3959	46	18	are	be	AUX
cana-3959	46	19	widely	widely	ADV
cana-3959	46	20	used	use	VERB
cana-3959	46	21	in	in	ADP
cana-3959	46	22	various	various	ADJ
cana-3959	46	23	branches	branch	NOUN
cana-3959	46	24	of	of	ADP
cana-3959	46	25	functional	functional	ADJ
cana-3959	46	26	analysis	analysis	NOUN
cana-3959	46	27	,	,	PUNCT
cana-3959	46	28	optimization	optimization	NOUN
cana-3959	46	29	,	,	PUNCT
cana-3959	46	30	and	and	CCONJ
cana-3959	46	31	differential	differential	ADJ
cana-3959	46	32	equations	equation	NOUN
cana-3959	46	33	.	.	PUNCT
cana-3959	47	1	recently	recently	ADV
cana-3959	47	2	agilan	agilan	PROPN
cana-3959	47	3	et.al	et.al	NOUN
cana-3959	47	4	exploring	explore	VERB
cana-3959	47	5	the	the	DET
cana-3959	47	6	stability	stability	NOUN
cana-3959	47	7	results	result	VERB
cana-3959	47	8	communications	communication	NOUN
cana-3959	47	9	on	on	ADP
cana-3959	47	10	applied	apply	VERB
cana-3959	47	11	nonlinear	nonlinear	ADJ
cana-3959	47	12	analysis	analysis	NOUN
cana-3959	47	13	issn	issn	NOUN
cana-3959	47	14	:	:	PUNCT
cana-3959	47	15	1074	1074	NUM
cana-3959	47	16	-	-	PUNCT
cana-3959	47	17	133x	133x	NUM
cana-3959	47	18	vol	vol	NOUN
cana-3959	47	19	32	32	NUM
cana-3959	47	20	no	no	NOUN
cana-3959	47	21	.	.	PUNCT
cana-3959	48	1	9s	9s	NUM
cana-3959	48	2	(	(	PUNCT
cana-3959	48	3	2025	2025	NUM
cana-3959	48	4	)	)	PUNCT
cana-3959	48	5	477	477	NUM
cana-3959	48	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	48	7	in	in	ADP
cana-3959	48	8	various	various	ADJ
cana-3959	48	9	additive	additive	ADJ
cana-3959	48	10	functional	functional	ADJ
cana-3959	48	11	equation	equation	NOUN
cana-3959	48	12	through	through	ADP
cana-3959	48	13	various	various	ADJ
cana-3959	48	14	normed	norme	VERB
cana-3959	48	15	spaces	space	NOUN
cana-3959	48	16	such	such	ADJ
cana-3959	48	17	as	as	ADP
cana-3959	48	18	[	[	X
cana-3959	48	19	25	25	NUM
cana-3959	48	20	,	,	PUNCT
cana-3959	48	21	26	26	NUM
cana-3959	48	22	,	,	PUNCT
cana-3959	48	23	27	27	NUM
cana-3959	48	24	,	,	PUNCT
cana-3959	48	25	28	28	NUM
cana-3959	48	26	,	,	PUNCT
cana-3959	48	27	29	29	NUM
cana-3959	48	28	,	,	PUNCT
cana-3959	48	29	30	30	NUM
cana-3959	48	30	,	,	PUNCT
cana-3959	48	31	31	31	NUM
cana-3959	48	32	,	,	PUNCT
cana-3959	48	33	32	32	NUM
cana-3959	48	34	]	]	PUNCT
cana-3959	48	35	.	.	PUNCT
cana-3959	49	1	in	in	ADP
cana-3959	49	2	this	this	DET
cana-3959	49	3	paper	paper	NOUN
cana-3959	49	4	,	,	PUNCT
cana-3959	49	5	the	the	DET
cana-3959	49	6	authors	author	NOUN
cana-3959	49	7	investigate	investigate	VERB
cana-3959	49	8	the	the	DET
cana-3959	49	9	generalized	generalize	VERB
cana-3959	49	10	ulam	ulam	NOUN
cana-3959	49	11	-	-	PUNCT
cana-3959	49	12	hyers	hyer	NOUN
cana-3959	49	13	stability	stability	NOUN
cana-3959	49	14	of	of	ADP
cana-3959	49	15	a	a	DET
cana-3959	49	16	alternate	alternate	ADJ
cana-3959	49	17	cubic	cubic	ADJ
cana-3959	49	18	functional	functional	ADJ
cana-3959	49	19	equation	equation	NOUN
cana-3959	49	20	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	49	21	+	+	CCONJ
cana-3959	49	22	𝑤	𝑤	X
cana-3959	49	23	)	)	PUNCT
cana-3959	49	24	±ℜ	±ℜ	PUNCT
cana-3959	50	1	𝑏	𝑏	NOUN
cana-3959	50	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	50	3	−ℜ	−ℜ	PROPN
cana-3959	50	4	𝑎𝑤	𝑎𝑤	X
cana-3959	50	5	)	)	PUNCT
cana-3959	50	6	=	=	SYM
cana-3959	50	7	(	(	PUNCT
cana-3959	50	8	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	50	9	)	)	PUNCT
cana-3959	50	10	2	2	NUM
cana-3959	50	11	)	)	PUNCT
cana-3959	51	1	[	[	X
cana-3959	51	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	51	3	+	+	CCONJ
cana-3959	51	4	𝑤	𝑤	X
cana-3959	51	5	)	)	PUNCT
cana-3959	51	6	+	+	CCONJ
cana-3959	51	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	51	8	−	−	NOUN
cana-3959	51	9	𝑤	𝑤	ADP
cana-3959	51	10	)	)	PUNCT
cana-3959	51	11	]	]	PUNCT
cana-3959	52	1	+	+	CCONJ
cana-3959	52	2	(	(	PUNCT
cana-3959	52	3	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	52	4	)	)	PUNCT
cana-3959	52	5	2	2	NUM
cana-3959	52	6	)	)	PUNCT
cana-3959	53	1	[	[	X
cana-3959	53	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	53	3	+	+	NOUN
cana-3959	53	4	𝑤	𝑤	X
cana-3959	53	5	)	)	PUNCT
cana-3959	53	6	−	−	PROPN
cana-3959	53	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	53	8	−	−	NOUN
cana-3959	53	9	𝑤	𝑤	ADP
cana-3959	53	10	)	)	PUNCT
cana-3959	53	11	]	]	PUNCT
cana-3959	54	1	+	+	ADJ
cana-3959	54	2	(	(	PUNCT
cana-3959	54	3	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	54	4	−	−	PROPN
cana-3959	54	5	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	54	6	∓ℜ	∓ℜ	PROPN
cana-3959	54	7	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	54	8	)	)	PUNCT
cana-3959	54	9	∓	∓	PROPN
cana-3959	55	1	(	(	PUNCT
cana-3959	55	2	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	55	3	±	±	NUM
cana-3959	55	4	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	55	5	)	)	PUNCT
cana-3959	55	6	]	]	PUNCT
cana-3959	55	7	(	(	PUNCT
cana-3959	55	8	1	1	X
cana-3959	55	9	)	)	PUNCT
cana-3959	55	10	where	where	SCONJ
cana-3959	55	11	ℜ	ℜ	PROPN
cana-3959	55	12	,	,	PUNCT
cana-3959	55	13	𝑎	𝑎	NOUN
cana-3959	55	14	,	,	PUNCT
cana-3959	55	15	𝑏	𝑏	PROPN
cana-3959	55	16	are	be	AUX
cana-3959	55	17	integers	integer	NOUN
cana-3959	55	18	with	with	ADP
cana-3959	55	19	ℜ	ℜ	ADJ
cana-3959	55	20	≠	≠	PROPN
cana-3959	55	21	0	0	NUM
cana-3959	55	22	,	,	PUNCT
cana-3959	55	23	±1	±1	VERB
cana-3959	55	24	and	and	CCONJ
cana-3959	55	25	𝑎	𝑎	DET
cana-3959	55	26	≠	≠	PROPN
cana-3959	55	27	𝑏	𝑏	PROPN
cana-3959	55	28	≠	≠	PROPN
cana-3959	55	29	0	0	NUM
cana-3959	55	30	,	,	PUNCT
cana-3959	55	31	±1	±1	VERB
cana-3959	55	32	in	in	ADP
cana-3959	55	33	banach	banach	NOUN
cana-3959	55	34	and	and	CCONJ
cana-3959	55	35	fuzzy	fuzzy	ADJ
cana-3959	55	36	banach	banach	NOUN
cana-3959	55	37	spaces	space	VERB
cana-3959	55	38	.	.	PUNCT
cana-3959	56	1	lemma	lemma	PROPN
cana-3959	56	2	1.1	1.1	NUM
cana-3959	56	3	let	let	VERB
cana-3959	56	4	us	we	PRON
cana-3959	56	5	consider	consider	VERB
cana-3959	56	6	x	x	PRON
cana-3959	56	7	and	and	CCONJ
cana-3959	56	8	y	y	PROPN
cana-3959	56	9	be	be	AUX
cana-3959	56	10	real	real	ADJ
cana-3959	56	11	vector	vector	NOUN
cana-3959	56	12	spaces	space	NOUN
cana-3959	56	13	.	.	PUNCT
cana-3959	57	1	an	an	DET
cana-3959	57	2	odd	odd	ADJ
cana-3959	57	3	function	function	NOUN
cana-3959	57	4	satisfies	satisfy	VERB
cana-3959	57	5	the	the	DET
cana-3959	57	6	functional	functional	ADJ
cana-3959	57	7	equation	equation	NOUN
cana-3959	57	8	ℱ(𝑚𝑣	ℱ(𝑚𝑣	PROPN
cana-3959	58	1	+	+	CCONJ
cana-3959	58	2	𝑤	𝑤	X
cana-3959	58	3	)	)	PUNCT
cana-3959	58	4	+	+	CCONJ
cana-3959	58	5	ℱ(𝑚𝑣	ℱ(𝑚𝑣	NUM
cana-3959	58	6	−	−	NOUN
cana-3959	58	7	𝑤	𝑤	ADP
cana-3959	58	8	)	)	PUNCT
cana-3959	58	9	=	=	PUNCT
cana-3959	59	1	𝑚ℱ(𝑣	𝑚ℱ(𝑣	ADP
cana-3959	59	2	+	+	NUM
cana-3959	59	3	𝑤	𝑤	X
cana-3959	59	4	)	)	PUNCT
cana-3959	59	5	+	+	ADP
cana-3959	59	6	𝑚ℱ(𝑣	𝑚ℱ(𝑣	ADP
cana-3959	59	7	−	−	NOUN
cana-3959	59	8	𝑤	𝑤	SYM
cana-3959	59	9	)	)	PUNCT
cana-3959	59	10	+	+	CCONJ
cana-3959	59	11	2(𝑚3	2(𝑚3	NUM
cana-3959	59	12	−	−	ADP
cana-3959	59	13	𝑚)ℱ(𝑣	𝑚)ℱ(𝑣	PROPN
cana-3959	59	14	)	)	PUNCT
cana-3959	59	15	(	(	PUNCT
cana-3959	59	16	2	2	X
cana-3959	59	17	)	)	PUNCT
cana-3959	59	18	for	for	ADP
cana-3959	59	19	all	all	DET
cana-3959	59	20	𝑣	𝑣	NOUN
cana-3959	59	21	,	,	PUNCT
cana-3959	59	22	𝑤	𝑤	X
cana-3959	59	23	∈	∈	PROPN
cana-3959	59	24	𝑋	𝑋	NOUN
cana-3959	59	25	if	if	SCONJ
cana-3959	59	26	satisfies	satisfy	VERB
cana-3959	59	27	the	the	DET
cana-3959	59	28	fun	fun	NOUN
cana-3959	59	29	eq(1	eq(1	NOUN
cana-3959	59	30	)	)	PUNCT
cana-3959	59	31	for	for	ADP
cana-3959	59	32	all	all	DET
cana-3959	59	33	𝑣,𝑤	𝑣,𝑤	PROPN
cana-3959	59	34	∈	∈	PROPN
cana-3959	59	35	𝑋	𝑋	NOUN
cana-3959	59	36	.	.	PUNCT
cana-3959	60	1	proof	proof	NOUN
cana-3959	60	2	.	.	PUNCT
cana-3959	61	1	assume	assume	VERB
cana-3959	61	2	𝑓	𝑓	X
cana-3959	61	3	:	:	PUNCT
cana-3959	61	4	𝑋	𝑋	PROPN
cana-3959	61	5	→	→	SYM
cana-3959	61	6	𝑌	𝑌	PROPN
cana-3959	61	7	satisfies	satisfy	VERB
cana-3959	61	8	the	the	DET
cana-3959	61	9	functional	functional	ADJ
cana-3959	61	10	equation	equation	NOUN
cana-3959	61	11	(	(	PUNCT
cana-3959	61	12	2	2	NUM
cana-3959	61	13	)	)	PUNCT
cana-3959	61	14	.	.	PUNCT
cana-3959	62	1	letting	let	VERB
cana-3959	62	2	𝑣	𝑣	PRON
cana-3959	62	3	=	=	PUNCT
cana-3959	62	4	𝑤	𝑤	SYM
cana-3959	62	5	=	=	SYM
cana-3959	62	6	0	0	NUM
cana-3959	62	7	in	in	ADP
cana-3959	62	8	(	(	PUNCT
cana-3959	62	9	2	2	NUM
cana-3959	62	10	)	)	PUNCT
cana-3959	62	11	,	,	PUNCT
cana-3959	62	12	we	we	PRON
cana-3959	62	13	get	get	VERB
cana-3959	62	14	ℱ(0	ℱ(0	PRON
cana-3959	62	15	)	)	PUNCT
cana-3959	62	16	=	=	SYM
cana-3959	63	1	0	0	PUNCT
cana-3959	63	2	.	.	PUNCT
cana-3959	64	1	setting	set	VERB
cana-3959	64	2	in	in	ADP
cana-3959	64	3	(	(	PUNCT
cana-3959	64	4	2	2	NUM
cana-3959	64	5	)	)	PUNCT
cana-3959	64	6	,	,	PUNCT
cana-3959	64	7	we	we	PRON
cana-3959	64	8	have	have	VERB
cana-3959	64	9	ℱ(−𝑤	ℱ(−𝑤	NOUN
cana-3959	64	10	)	)	PUNCT
cana-3959	64	11	=	=	SYM
cana-3959	65	1	−ℱ(𝑤	−ℱ(𝑤	PROPN
cana-3959	65	2	)	)	PUNCT
cana-3959	65	3	and	and	CCONJ
cana-3959	65	4	𝑤	𝑤	X
cana-3959	65	5	=	=	SYM
cana-3959	65	6	0	0	NUM
cana-3959	65	7	we	we	PRON
cana-3959	65	8	get	get	VERB
cana-3959	65	9	ℱ(𝑚𝑣	ℱ(𝑚𝑣	PRON
cana-3959	65	10	)	)	PUNCT
cana-3959	65	11	=	=	SYM
cana-3959	66	1	𝑚3ℱ(𝑣	𝑚3ℱ(𝑣	PROPN
cana-3959	66	2	)	)	PUNCT
cana-3959	66	3	(	(	PUNCT
cana-3959	66	4	3	3	X
cana-3959	66	5	)	)	PUNCT
cana-3959	66	6	for	for	ADP
cana-3959	66	7	all	all	PRON
cana-3959	66	8	𝑣	𝑣	DET
cana-3959	66	9	∈	∈	PROPN
cana-3959	66	10	𝑋.	𝑋.	PROPN
cana-3959	66	11	in	in	ADP
cana-3959	66	12	particular	particular	ADJ
cana-3959	66	13	replace	replace	NOUN
cana-3959	66	14	𝑚	𝑚	NOUN
cana-3959	66	15	by	by	ADP
cana-3959	66	16	ℜ	ℜ	ADV
cana-3959	66	17	𝑎	𝑎	NOUN
cana-3959	66	18	in	in	ADP
cana-3959	66	19	(	(	PUNCT
cana-3959	66	20	2	2	NUM
cana-3959	66	21	)	)	PUNCT
cana-3959	66	22	,	,	PUNCT
cana-3959	66	23	we	we	PRON
cana-3959	66	24	get	get	VERB
cana-3959	66	25	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	66	26	+	+	CCONJ
cana-3959	66	27	𝑤	𝑤	X
cana-3959	66	28	)	)	PUNCT
cana-3959	67	1	+	+	CCONJ
cana-3959	67	2	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	VERB
cana-3959	67	3	−	−	NOUN
cana-3959	67	4	𝑤	𝑤	NOUN
cana-3959	67	5	)	)	PUNCT
cana-3959	67	6	=	=	PUNCT
cana-3959	67	7	ℜ	ℜ	PROPN
cana-3959	67	8	𝑎	𝑎	NOUN
cana-3959	67	9	ℱ(𝑣	ℱ(𝑣	NOUN
cana-3959	67	10	+	+	NOUN
cana-3959	67	11	𝑤	𝑤	X
cana-3959	67	12	)	)	PUNCT
cana-3959	67	13	+	+	CCONJ
cana-3959	67	14	ℜ	ℜ	ADV
cana-3959	67	15	𝑎	𝑎	PRON
cana-3959	67	16	ℱ(𝑣	ℱ(𝑣	NOUN
cana-3959	67	17	−	−	NOUN
cana-3959	67	18	𝑤	𝑤	ADP
cana-3959	67	19	)	)	PUNCT
cana-3959	67	20	+	+	CCONJ
cana-3959	67	21	2(ℜ3𝑎	2(ℜ3𝑎	NUM
cana-3959	67	22	−ℜ	−ℜ	PROPN
cana-3959	67	23	𝑎)ℱ(𝑥	𝑎)ℱ(𝑥	PROPN
cana-3959	67	24	)	)	PUNCT
cana-3959	67	25	(	(	PUNCT
cana-3959	67	26	4	4	X
cana-3959	67	27	)	)	PUNCT
cana-3959	67	28	for	for	ADP
cana-3959	67	29	all	all	DET
cana-3959	67	30	𝑣	𝑣	NOUN
cana-3959	67	31	,	,	PUNCT
cana-3959	67	32	𝑤	𝑤	ADP
cana-3959	67	33	∈	∈	PROPN
cana-3959	67	34	𝑋.	𝑋.	PROPN
cana-3959	67	35	replace	replace	NOUN
cana-3959	67	36	𝑤	𝑤	VERB
cana-3959	67	37	by	by	ADP
cana-3959	67	38	ℜ	ℜ	PROPN
cana-3959	67	39	𝑎𝑤	𝑎𝑤	VERB
cana-3959	67	40	in	in	ADP
cana-3959	67	41	(	(	PUNCT
cana-3959	67	42	4	4	NUM
cana-3959	67	43	)	)	PUNCT
cana-3959	67	44	,	,	PUNCT
cana-3959	67	45	we	we	PRON
cana-3959	67	46	obtain	obtain	VERB
cana-3959	67	47	ℱ(ℜ𝑎(𝑣	ℱ(ℜ𝑎(𝑣	NOUN
cana-3959	67	48	+	+	CCONJ
cana-3959	67	49	𝑤	𝑤	X
cana-3959	67	50	)	)	PUNCT
cana-3959	67	51	)	)	PUNCT
cana-3959	68	1	+	+	CCONJ
cana-3959	68	2	ℱ(ℜ𝑎(𝑣	ℱ(ℜ𝑎(𝑣	ADP
cana-3959	68	3	−	−	ADP
cana-3959	68	4	𝑤	𝑤	X
cana-3959	68	5	)	)	PUNCT
cana-3959	68	6	)	)	PUNCT
cana-3959	69	1	=	=	PUNCT
cana-3959	69	2	ℜ	ℜ	ADJ
cana-3959	69	3	𝑎[ℱ(𝑣	𝑎[ℱ(𝑣	ADJ
cana-3959	69	4	+	+	SYM
cana-3959	69	5	ℜ	ℜ	ADJ
cana-3959	69	6	𝑎𝑤	𝑎𝑤	VERB
cana-3959	69	7	)	)	PUNCT
cana-3959	69	8	+	+	CCONJ
cana-3959	69	9	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	69	10	−ℜ	−ℜ	PROPN
cana-3959	69	11	𝑎𝑤	𝑎𝑤	ADP
cana-3959	69	12	)	)	PUNCT
cana-3959	69	13	]	]	PUNCT
cana-3959	70	1	+	+	CCONJ
cana-3959	70	2	2(ℜ3𝑎	2(ℜ3𝑎	NUM
cana-3959	70	3	−ℜ	−ℜ	PROPN
cana-3959	70	4	𝑎)ℱ(𝑣	𝑎)ℱ(𝑣	PROPN
cana-3959	70	5	)	)	PUNCT
cana-3959	70	6	(	(	PUNCT
cana-3959	70	7	5	5	NUM
cana-3959	70	8	)	)	PUNCT
cana-3959	70	9	for	for	ADP
cana-3959	70	10	all	all	DET
cana-3959	70	11	𝑣	𝑣	NOUN
cana-3959	70	12	,	,	PUNCT
cana-3959	70	13	𝑤	𝑤	ADP
cana-3959	70	14	∈	∈	PROPN
cana-3959	70	15	𝑋.	𝑋.	PROPN
cana-3959	70	16	using	use	VERB
cana-3959	70	17	(	(	PUNCT
cana-3959	70	18	3	3	NUM
cana-3959	70	19	)	)	PUNCT
cana-3959	70	20	in	in	ADP
cana-3959	70	21	(	(	PUNCT
cana-3959	70	22	5	5	NUM
cana-3959	70	23	)	)	PUNCT
cana-3959	70	24	,	,	PUNCT
cana-3959	70	25	we	we	PRON
cana-3959	70	26	have	have	VERB
cana-3959	70	27	ℜ	ℜ	ADV
cana-3959	70	28	3𝑎[ℱ(𝑣	3𝑎[ℱ(𝑣	NUM
cana-3959	70	29	+	+	CCONJ
cana-3959	70	30	𝑤	𝑤	X
cana-3959	70	31	)	)	PUNCT
cana-3959	71	1	+	+	CCONJ
cana-3959	72	1	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	72	2	−	−	NOUN
cana-3959	72	3	𝑤	𝑤	ADP
cana-3959	72	4	)	)	PUNCT
cana-3959	72	5	]	]	PUNCT
cana-3959	73	1	=	=	PUNCT
cana-3959	73	2	ℜ	ℜ	ADJ
cana-3959	73	3	𝑎[ℱ(𝑣	𝑎[ℱ(𝑣	NOUN
cana-3959	73	4	+	+	NOUN
cana-3959	73	5	ℜ	ℜ	ADJ
cana-3959	73	6	𝑎𝑤	𝑎𝑤	VERB
cana-3959	73	7	)	)	PUNCT
cana-3959	73	8	+	+	CCONJ
cana-3959	73	9	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	73	10	−ℜ	−ℜ	PROPN
cana-3959	73	11	𝑎𝑤	𝑎𝑤	ADP
cana-3959	73	12	)	)	PUNCT
cana-3959	73	13	]	]	PUNCT
cana-3959	74	1	+	+	CCONJ
cana-3959	74	2	2(ℜ3𝑎	2(ℜ3𝑎	NUM
cana-3959	74	3	−	−	PROPN
cana-3959	74	4	ℜ	ℜ	ADJ
cana-3959	74	5	𝑎)ℱ(𝑣	𝑎)ℱ(𝑣	NOUN
cana-3959	74	6	)	)	PUNCT
cana-3959	74	7	(	(	PUNCT
cana-3959	74	8	6	6	NUM
cana-3959	74	9	)	)	PUNCT
cana-3959	74	10	for	for	ADP
cana-3959	74	11	all	all	DET
cana-3959	74	12	𝑣	𝑣	NOUN
cana-3959	74	13	,	,	PUNCT
cana-3959	74	14	𝑤	𝑤	ADP
cana-3959	74	15	∈	∈	PROPN
cana-3959	74	16	𝑋.	𝑋.	PROPN
cana-3959	74	17	divide	divide	VERB
cana-3959	74	18	the	the	DET
cana-3959	74	19	above	above	ADJ
cana-3959	74	20	equation	equation	NOUN
cana-3959	74	21	by	by	ADP
cana-3959	74	22	ℜ	ℜ	ADV
cana-3959	74	23	𝑎	𝑎	NOUN
cana-3959	74	24	,	,	PUNCT
cana-3959	74	25	we	we	PRON
cana-3959	74	26	get	get	VERB
cana-3959	74	27	ℱ(𝑣	ℱ(𝑣	PRON
cana-3959	75	1	+	+	NOUN
cana-3959	75	2	ℜ	ℜ	ADJ
cana-3959	75	3	𝑎𝑤	𝑎𝑤	VERB
cana-3959	75	4	)	)	PUNCT
cana-3959	75	5	+	+	CCONJ
cana-3959	75	6	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	75	7	−ℜ	−ℜ	PROPN
cana-3959	75	8	𝑎𝑤	𝑎𝑤	X
cana-3959	75	9	)	)	PUNCT
cana-3959	75	10	=	=	SYM
cana-3959	75	11	ℜ	ℜ	SYM
cana-3959	75	12	2𝑎[ℱ(𝑣	2𝑎[ℱ(𝑣	NUM
cana-3959	75	13	+	+	CCONJ
cana-3959	75	14	𝑤	𝑤	X
cana-3959	75	15	)	)	PUNCT
cana-3959	75	16	+	+	CCONJ
cana-3959	76	1	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	76	2	−	−	NOUN
cana-3959	76	3	𝑤	𝑤	ADP
cana-3959	76	4	)	)	PUNCT
cana-3959	76	5	]	]	PUNCT
cana-3959	77	1	−	−	PROPN
cana-3959	77	2	2(ℜ2𝑎	2(ℜ2𝑎	NUM
cana-3959	77	3	−	−	NOUN
cana-3959	77	4	1)ℱ(𝑣	1)ℱ(𝑣	NUM
cana-3959	77	5	)	)	PUNCT
cana-3959	77	6	(	(	PUNCT
cana-3959	77	7	7	7	X
cana-3959	77	8	)	)	PUNCT
cana-3959	77	9	for	for	ADP
cana-3959	77	10	all	all	DET
cana-3959	77	11	𝑣	𝑣	NOUN
cana-3959	77	12	,	,	PUNCT
cana-3959	77	13	𝑤	𝑤	ADP
cana-3959	77	14	∈	∈	PROPN
cana-3959	77	15	𝑋.	𝑋.	PROPN
cana-3959	77	16	replace	replace	VERB
cana-3959	77	17	𝑣	𝑣	PRON
cana-3959	77	18	by	by	ADP
cana-3959	77	19	𝑤	𝑤	PRON
cana-3959	77	20	and	and	CCONJ
cana-3959	77	21	𝑤	𝑤	X
cana-3959	77	22	by	by	ADP
cana-3959	77	23	𝑣	𝑣	X
cana-3959	77	24	in	in	ADP
cana-3959	77	25	(	(	PUNCT
cana-3959	77	26	7	7	NUM
cana-3959	77	27	)	)	PUNCT
cana-3959	77	28	and	and	CCONJ
cana-3959	77	29	using	use	VERB
cana-3959	77	30	oddness	oddness	NOUN
cana-3959	77	31	of	of	ADP
cana-3959	77	32	𝐶	𝐶	PROPN
cana-3959	77	33	,	,	PUNCT
cana-3959	77	34	we	we	PRON
cana-3959	77	35	obtain	obtain	VERB
cana-3959	77	36	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	PROPN
cana-3959	77	37	−	−	NOUN
cana-3959	77	38	𝑤	𝑤	ADP
cana-3959	77	39	)	)	PUNCT
cana-3959	77	40	=	=	PUNCT
cana-3959	78	1	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	78	2	+	+	CCONJ
cana-3959	78	3	𝑤	𝑤	X
cana-3959	78	4	)	)	PUNCT
cana-3959	78	5	−ℜ	−ℜ	PROPN
cana-3959	78	6	2𝑎[ℱ(𝑣	2𝑎[ℱ(𝑣	NUM
cana-3959	78	7	+	+	CCONJ
cana-3959	78	8	𝑤	𝑤	X
cana-3959	78	9	)	)	PUNCT
cana-3959	78	10	−	−	PROPN
cana-3959	79	1	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	79	2	−	−	NOUN
cana-3959	79	3	𝑤	𝑤	ADP
cana-3959	79	4	)	)	PUNCT
cana-3959	79	5	]	]	PUNCT
cana-3959	80	1	+	+	CCONJ
cana-3959	80	2	2(ℜ2𝑎	2(ℜ2𝑎	NUM
cana-3959	80	3	−	−	NOUN
cana-3959	80	4	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	80	5	)	)	PUNCT
cana-3959	80	6	(	(	PUNCT
cana-3959	80	7	8)	8)	NUM
cana-3959	80	8	for	for	ADP
cana-3959	80	9	all	all	DET
cana-3959	80	10	𝑣	𝑣	NOUN
cana-3959	80	11	,	,	PUNCT
cana-3959	80	12	𝑤	𝑤	ADP
cana-3959	80	13	∈	∈	PROPN
cana-3959	80	14	𝑋.	𝑋.	PROPN
cana-3959	80	15	substitute	substitute	PROPN
cana-3959	80	16	(	(	PUNCT
cana-3959	80	17	8)	8)	NUM
cana-3959	80	18	in	in	ADP
cana-3959	80	19	(	(	PUNCT
cana-3959	80	20	4	4	NUM
cana-3959	80	21	)	)	PUNCT
cana-3959	80	22	,	,	PUNCT
cana-3959	80	23	we	we	PRON
cana-3959	80	24	get	get	VERB
cana-3959	80	25	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	80	26	+	+	CCONJ
cana-3959	80	27	𝑤	𝑤	X
cana-3959	80	28	)	)	PUNCT
cana-3959	80	29	=	=	SYM
cana-3959	81	1	ℜ𝑎	ℜ𝑎	NOUN
cana-3959	81	2	2	2	NUM
cana-3959	82	1	[	[	X
cana-3959	82	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	82	3	+	+	CCONJ
cana-3959	82	4	𝑤	𝑤	X
cana-3959	82	5	)	)	PUNCT
cana-3959	82	6	+	+	CCONJ
cana-3959	83	1	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	83	2	−	−	NOUN
cana-3959	83	3	𝑤	𝑤	ADP
cana-3959	83	4	)	)	PUNCT
cana-3959	83	5	]	]	PUNCT
cana-3959	84	1	+	+	CCONJ
cana-3959	84	2	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	84	3	2	2	NUM
cana-3959	84	4	[	[	X
cana-3959	84	5	ℱ(𝑣	ℱ(𝑣	X
cana-3959	84	6	+	+	CCONJ
cana-3959	84	7	𝑤	𝑤	X
cana-3959	84	8	)	)	PUNCT
cana-3959	84	9	−	−	PROPN
cana-3959	84	10	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	84	11	−	−	NOUN
cana-3959	84	12	𝑤	𝑤	ADP
cana-3959	84	13	)	)	PUNCT
cana-3959	84	14	]	]	PUNCT
cana-3959	85	1	+	+	PROPN
cana-3959	85	2	(	(	PUNCT
cana-3959	85	3	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	85	4	−ℜ	−ℜ	PROPN
cana-3959	85	5	𝑎)ℱ(𝑣	𝑎)ℱ(𝑣	PROPN
cana-3959	85	6	)	)	PUNCT
cana-3959	85	7	−	−	PROPN
cana-3959	85	8	(	(	PUNCT
cana-3959	85	9	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	85	10	−	−	PROPN
cana-3959	85	11	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	85	12	)	)	PUNCT
cana-3959	85	13	(	(	PUNCT
cana-3959	85	14	9	9	NUM
cana-3959	85	15	)	)	PUNCT
cana-3959	85	16	for	for	ADP
cana-3959	85	17	all	all	DET
cana-3959	85	18	𝑣	𝑣	NOUN
cana-3959	85	19	,	,	PUNCT
cana-3959	85	20	𝑤	𝑤	ADP
cana-3959	85	21	∈	∈	PROPN
cana-3959	85	22	𝑋.	𝑋.	PROPN
cana-3959	85	23	replace	replace	VERB
cana-3959	85	24	𝑣	𝑣	X
cana-3959	85	25	by	by	ADP
cana-3959	85	26	−𝑤	−𝑤	PROPN
cana-3959	85	27	and	and	CCONJ
cana-3959	85	28	𝑤	𝑤	X
cana-3959	85	29	by	by	ADP
cana-3959	85	30	𝑣	𝑣	X
cana-3959	85	31	in	in	ADP
cana-3959	85	32	(	(	PUNCT
cana-3959	85	33	9	9	NUM
cana-3959	85	34	)	)	PUNCT
cana-3959	85	35	and	and	CCONJ
cana-3959	85	36	using	use	VERB
cana-3959	85	37	oddness	oddness	NOUN
cana-3959	85	38	of	of	ADP
cana-3959	85	39	𝐶	𝐶	PROPN
cana-3959	85	40	,	,	PUNCT
cana-3959	85	41	we	we	PRON
cana-3959	85	42	obtain	obtain	VERB
cana-3959	85	43	communications	communication	NOUN
cana-3959	85	44	on	on	ADP
cana-3959	85	45	applied	apply	VERB
cana-3959	85	46	nonlinear	nonlinear	ADJ
cana-3959	85	47	analysis	analysis	NOUN
cana-3959	85	48	issn	issn	NOUN
cana-3959	85	49	:	:	PUNCT
cana-3959	85	50	1074	1074	NUM
cana-3959	85	51	-	-	PUNCT
cana-3959	85	52	133x	133x	NUM
cana-3959	85	53	vol	vol	NOUN
cana-3959	85	54	32	32	NUM
cana-3959	85	55	no	no	NOUN
cana-3959	85	56	.	.	PUNCT
cana-3959	86	1	9s	9s	NUM
cana-3959	86	2	(	(	PUNCT
cana-3959	86	3	2025	2025	NUM
cana-3959	86	4	)	)	PUNCT
cana-3959	86	5	478	478	NUM
cana-3959	86	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	86	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	86	8	−ℜ	−ℜ	PROPN
cana-3959	86	9	𝑎𝑤	𝑎𝑤	X
cana-3959	86	10	)	)	PUNCT
cana-3959	86	11	=	=	SYM
cana-3959	87	1	ℜ𝑎	ℜ𝑎	NOUN
cana-3959	87	2	2	2	NUM
cana-3959	88	1	[	[	X
cana-3959	88	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	88	3	−	−	NOUN
cana-3959	88	4	𝑤	𝑤	ADP
cana-3959	88	5	)	)	PUNCT
cana-3959	88	6	−	−	PROPN
cana-3959	88	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	88	8	−	−	NOUN
cana-3959	88	9	𝑤	𝑤	ADP
cana-3959	88	10	)	)	PUNCT
cana-3959	88	11	]	]	PUNCT
cana-3959	89	1	+	+	CCONJ
cana-3959	89	2	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	89	3	2	2	NUM
cana-3959	90	1	[	[	X
cana-3959	90	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	90	3	−	−	NOUN
cana-3959	90	4	𝑤	𝑤	ADP
cana-3959	90	5	)	)	PUNCT
cana-3959	90	6	+	+	CCONJ
cana-3959	90	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	90	8	+	+	NUM
cana-3959	90	9	𝑤	𝑤	X
cana-3959	90	10	)	)	PUNCT
cana-3959	90	11	]	]	PUNCT
cana-3959	90	12	−(ℜ3𝑎	−(ℜ3𝑎	PROPN
cana-3959	90	13	−ℜ	−ℜ	PROPN
cana-3959	90	14	𝑎)ℱ(𝑤	𝑎)ℱ(𝑤	PROPN
cana-3959	90	15	)	)	PUNCT
cana-3959	90	16	−	−	PROPN
cana-3959	91	1	(	(	PUNCT
cana-3959	91	2	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	91	3	−	−	NUM
cana-3959	91	4	1)ℱ(𝑣	1)ℱ(𝑣	NUM
cana-3959	91	5	)	)	PUNCT
cana-3959	91	6	(	(	PUNCT
cana-3959	91	7	10	10	NUM
cana-3959	91	8	)	)	PUNCT
cana-3959	91	9	for	for	ADP
cana-3959	91	10	all	all	DET
cana-3959	91	11	𝑣	𝑣	NOUN
cana-3959	91	12	,	,	PUNCT
cana-3959	91	13	𝑤	𝑤	ADP
cana-3959	91	14	∈	∈	PROPN
cana-3959	91	15	𝑋.	𝑋.	PROPN
cana-3959	91	16	both	both	DET
cana-3959	91	17	side	side	NOUN
cana-3959	91	18	multiply	multiply	ADV
cana-3959	91	19	by	by	ADP
cana-3959	91	20	ℜ	ℜ	ADJ
cana-3959	91	21	𝑏	𝑏	PROPN
cana-3959	91	22	in	in	ADP
cana-3959	91	23	(	(	PUNCT
cana-3959	91	24	10	10	NUM
cana-3959	91	25	)	)	PUNCT
cana-3959	91	26	,	,	PUNCT
cana-3959	91	27	we	we	PRON
cana-3959	91	28	get	get	VERB
cana-3959	91	29	ℜ	ℜ	ADV
cana-3959	91	30	𝑏	𝑏	PROPN
cana-3959	91	31	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	91	32	−ℜ	−ℜ	PROPN
cana-3959	91	33	𝑎𝑤	𝑎𝑤	X
cana-3959	91	34	)	)	PUNCT
cana-3959	91	35	=	=	PUNCT
cana-3959	92	1	ℜ𝑎+𝑏	ℜ𝑎+𝑏	NUM
cana-3959	92	2	2	2	NUM
cana-3959	93	1	[	[	X
cana-3959	93	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	93	3	−	−	NOUN
cana-3959	93	4	𝑤	𝑤	ADP
cana-3959	93	5	)	)	PUNCT
cana-3959	93	6	−	−	PROPN
cana-3959	93	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	93	8	−	−	NOUN
cana-3959	93	9	𝑤	𝑤	ADP
cana-3959	93	10	)	)	PUNCT
cana-3959	93	11	]	]	PUNCT
cana-3959	94	1	+	+	CCONJ
cana-3959	94	2	ℜ2𝑎+𝑏	ℜ2𝑎+𝑏	NOUN
cana-3959	94	3	2	2	NUM
cana-3959	95	1	[	[	X
cana-3959	95	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	95	3	−	−	NOUN
cana-3959	95	4	𝑤	𝑤	ADP
cana-3959	95	5	)	)	PUNCT
cana-3959	95	6	+	+	CCONJ
cana-3959	95	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	95	8	+	+	NUM
cana-3959	95	9	𝑤	𝑤	X
cana-3959	95	10	)	)	PUNCT
cana-3959	95	11	]	]	PUNCT
cana-3959	95	12	−(ℜ2𝑎+𝑏	−(ℜ2𝑎+𝑏	ADJ
cana-3959	95	13	−	−	PROPN
cana-3959	95	14	ℜ	ℜ	PROPN
cana-3959	95	15	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	95	16	)	)	PUNCT
cana-3959	95	17	−	−	PROPN
cana-3959	96	1	(	(	PUNCT
cana-3959	96	2	ℜ3𝑎+𝑏	ℜ3𝑎+𝑏	NOUN
cana-3959	96	3	−ℜ	−ℜ	PROPN
cana-3959	96	4	𝑎+𝑏)ℱ(𝑤	𝑎+𝑏)ℱ(𝑤	NOUN
cana-3959	96	5	)	)	PUNCT
cana-3959	96	6	(	(	PUNCT
cana-3959	96	7	11	11	NUM
cana-3959	96	8	)	)	PUNCT
cana-3959	96	9	for	for	ADP
cana-3959	96	10	all	all	DET
cana-3959	96	11	𝑣	𝑣	NOUN
cana-3959	96	12	,	,	PUNCT
cana-3959	96	13	𝑤	𝑤	ADP
cana-3959	96	14	∈	∈	PROPN
cana-3959	96	15	𝑋.	𝑋.	PROPN
cana-3959	96	16	adding	add	VERB
cana-3959	96	17	(	(	PUNCT
cana-3959	96	18	9	9	NUM
cana-3959	96	19	)	)	PUNCT
cana-3959	96	20	and	and	CCONJ
cana-3959	96	21	(	(	PUNCT
cana-3959	96	22	11	11	NUM
cana-3959	96	23	)	)	PUNCT
cana-3959	96	24	,	,	PUNCT
cana-3959	96	25	we	we	PRON
cana-3959	96	26	arrive	arrive	VERB
cana-3959	96	27	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	96	28	+	+	CCONJ
cana-3959	96	29	𝑤	𝑤	X
cana-3959	96	30	)	)	PUNCT
cana-3959	97	1	+	+	NOUN
cana-3959	97	2	ℜ	ℜ	ADV
cana-3959	97	3	𝑏	𝑏	PROPN
cana-3959	97	4	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	97	5	−ℜ	−ℜ	PROPN
cana-3959	97	6	𝑎𝑤	𝑎𝑤	X
cana-3959	97	7	)	)	PUNCT
cana-3959	97	8	=	=	SYM
cana-3959	97	9	(	(	PUNCT
cana-3959	97	10	ℜ𝑎(1+ℜ𝑎+𝑏	ℜ𝑎(1+ℜ𝑎+𝑏	NOUN
cana-3959	97	11	)	)	PUNCT
cana-3959	97	12	2	2	NUM
cana-3959	97	13	)	)	PUNCT
cana-3959	98	1	[	[	X
cana-3959	98	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	98	3	+	+	CCONJ
cana-3959	98	4	𝑤	𝑤	X
cana-3959	98	5	)	)	PUNCT
cana-3959	98	6	+	+	CCONJ
cana-3959	98	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	98	8	−	−	NOUN
cana-3959	98	9	𝑤	𝑤	ADP
cana-3959	98	10	)	)	PUNCT
cana-3959	98	11	]	]	PUNCT
cana-3959	99	1	+	+	CCONJ
cana-3959	99	2	(	(	PUNCT
cana-3959	99	3	ℜ𝑎(ℜ𝑎−ℜ𝑏	ℜ𝑎(ℜ𝑎−ℜ𝑏	PROPN
cana-3959	99	4	)	)	PUNCT
cana-3959	99	5	2	2	NUM
cana-3959	99	6	)	)	PUNCT
cana-3959	100	1	[	[	X
cana-3959	100	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	100	3	+	+	NOUN
cana-3959	100	4	𝑤	𝑤	X
cana-3959	100	5	)	)	PUNCT
cana-3959	100	6	−	−	PROPN
cana-3959	100	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	100	8	−	−	NOUN
cana-3959	100	9	𝑤	𝑤	ADP
cana-3959	100	10	)	)	PUNCT
cana-3959	100	11	]	]	PUNCT
cana-3959	101	1	+	+	ADJ
cana-3959	101	2	(	(	PUNCT
cana-3959	101	3	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	101	4	−	−	NUM
cana-3959	101	5	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	101	6	−ℜ	−ℜ	PROPN
cana-3959	101	7	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	101	8	)	)	PUNCT
cana-3959	101	9	−	−	PROPN
cana-3959	102	1	(	(	PUNCT
cana-3959	102	2	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	102	3	+	+	NOUN
cana-3959	102	4	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	102	5	)	)	PUNCT
cana-3959	102	6	]	]	PUNCT
cana-3959	102	7	(	(	PUNCT
cana-3959	102	8	12	12	NUM
cana-3959	102	9	)	)	PUNCT
cana-3959	102	10	for	for	ADP
cana-3959	102	11	all	all	DET
cana-3959	102	12	𝑣	𝑣	NOUN
cana-3959	102	13	,	,	PUNCT
cana-3959	102	14	𝑤	𝑤	ADP
cana-3959	102	15	∈	∈	PROPN
cana-3959	102	16	𝑋.	𝑋.	PROPN
cana-3959	102	17	subtracting	subtracting	NOUN
cana-3959	102	18	(	(	PUNCT
cana-3959	102	19	9	9	NUM
cana-3959	102	20	)	)	PUNCT
cana-3959	102	21	and	and	CCONJ
cana-3959	102	22	(	(	PUNCT
cana-3959	102	23	11	11	NUM
cana-3959	102	24	)	)	PUNCT
cana-3959	102	25	,	,	PUNCT
cana-3959	102	26	we	we	PRON
cana-3959	102	27	arrive	arrive	VERB
cana-3959	102	28	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	102	29	+	+	CCONJ
cana-3959	102	30	𝑤	𝑤	X
cana-3959	102	31	)	)	PUNCT
cana-3959	102	32	−ℜ	−ℜ	PROPN
cana-3959	102	33	𝑏	𝑏	PROPN
cana-3959	102	34	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	102	35	−ℜ	−ℜ	PROPN
cana-3959	102	36	𝑎𝑤	𝑎𝑤	X
cana-3959	102	37	)	)	PUNCT
cana-3959	102	38	=	=	SYM
cana-3959	102	39	(	(	PUNCT
cana-3959	102	40	ℜ𝑎(1−ℜ𝑎+𝑏	ℜ𝑎(1−ℜ𝑎+𝑏	PROPN
cana-3959	102	41	)	)	PUNCT
cana-3959	102	42	2	2	NUM
cana-3959	102	43	)	)	PUNCT
cana-3959	103	1	[	[	X
cana-3959	103	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	103	3	+	+	CCONJ
cana-3959	103	4	𝑤	𝑤	X
cana-3959	103	5	)	)	PUNCT
cana-3959	103	6	+	+	CCONJ
cana-3959	103	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	103	8	−	−	NOUN
cana-3959	103	9	𝑤	𝑤	ADP
cana-3959	103	10	)	)	PUNCT
cana-3959	103	11	]	]	PUNCT
cana-3959	104	1	+	+	CCONJ
cana-3959	104	2	(	(	PUNCT
cana-3959	104	3	ℜ𝑎(ℜ𝑎+ℜ𝑏	ℜ𝑎(ℜ𝑎+ℜ𝑏	SYM
cana-3959	104	4	)	)	PUNCT
cana-3959	104	5	2	2	NUM
cana-3959	104	6	)	)	PUNCT
cana-3959	105	1	[	[	X
cana-3959	105	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	105	3	+	+	NOUN
cana-3959	105	4	𝑤	𝑤	X
cana-3959	105	5	)	)	PUNCT
cana-3959	105	6	−	−	PROPN
cana-3959	105	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	105	8	−	−	NOUN
cana-3959	105	9	𝑤	𝑤	ADP
cana-3959	105	10	)	)	PUNCT
cana-3959	105	11	]	]	PUNCT
cana-3959	106	1	+	+	ADJ
cana-3959	106	2	(	(	PUNCT
cana-3959	106	3	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	106	4	−	−	NOUN
cana-3959	106	5	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	106	6	+	+	PROPN
cana-3959	106	7	ℜ	ℜ	ADJ
cana-3959	106	8	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	NOUN
cana-3959	106	9	)	)	PUNCT
cana-3959	107	1	+	+	CCONJ
cana-3959	107	2	(	(	PUNCT
cana-3959	107	3	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	107	4	−	−	PROPN
cana-3959	107	5	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	107	6	)	)	PUNCT
cana-3959	107	7	]	]	PUNCT
cana-3959	107	8	(	(	PUNCT
cana-3959	107	9	13	13	NUM
cana-3959	107	10	)	)	PUNCT
cana-3959	107	11	for	for	ADP
cana-3959	107	12	all	all	DET
cana-3959	107	13	𝑣	𝑣	NOUN
cana-3959	107	14	,	,	PUNCT
cana-3959	107	15	𝑤	𝑤	ADP
cana-3959	107	16	∈	∈	PROPN
cana-3959	107	17	𝑋.	𝑋.	PROPN
cana-3959	107	18	combining	combine	VERB
cana-3959	107	19	both	both	PRON
cana-3959	107	20	(	(	PUNCT
cana-3959	107	21	12	12	NUM
cana-3959	107	22	)	)	PUNCT
cana-3959	107	23	and	and	CCONJ
cana-3959	107	24	(	(	PUNCT
cana-3959	107	25	13	13	NUM
cana-3959	107	26	)	)	PUNCT
cana-3959	107	27	we	we	PRON
cana-3959	107	28	arrive	arrive	VERB
cana-3959	107	29	(	(	PUNCT
cana-3959	107	30	1	1	NUM
cana-3959	107	31	)	)	PUNCT
cana-3959	107	32	.	.	PUNCT
cana-3959	108	1	2	2	NUM
cana-3959	108	2	banach	banach	NOUN
cana-3959	108	3	space	space	NOUN
cana-3959	108	4	stability	stability	NOUN
cana-3959	108	5	results	result	VERB
cana-3959	108	6	direct	direct	ADJ
cana-3959	108	7	method	method	NOUN
cana-3959	108	8	theorem	theorem	VERB
cana-3959	108	9	2.1	2.1	NUM
cana-3959	108	10	assume	assume	NOUN
cana-3959	108	11	x	x	X
cana-3959	108	12	be	be	AUX
cana-3959	108	13	normed	norme	VERB
cana-3959	108	14	linear	linear	ADJ
cana-3959	108	15	space	space	NOUN
cana-3959	108	16	and	and	CCONJ
cana-3959	108	17	y	y	PROPN
cana-3959	108	18	be	be	AUX
cana-3959	108	19	banach	banach	NOUN
cana-3959	108	20	space	space	NOUN
cana-3959	108	21	.	.	PUNCT
cana-3959	109	1	suppose	suppose	VERB
cana-3959	109	2	that	that	SCONJ
cana-3959	109	3	the	the	DET
cana-3959	109	4	function	function	NOUN
cana-3959	109	5	ℱ:x	ℱ:x	ADJ
cana-3959	109	6	→	→	PUNCT
cana-3959	109	7	y	y	NUM
cana-3959	109	8	satisfice	satisfice	NOUN
cana-3959	109	9	‖𝐷ℱ(𝑣,𝑤)‖	‖𝐷ℱ(𝑣,𝑤)‖	PROPN
cana-3959	109	10	≤	≤	PROPN
cana-3959	109	11	𝔔(𝑣	𝔔(𝑣	PROPN
cana-3959	109	12	,	,	PUNCT
cana-3959	109	13	𝑤	𝑤	ADP
cana-3959	109	14	)	)	PUNCT
cana-3959	109	15	(	(	PUNCT
cana-3959	109	16	1	1	X
cana-3959	109	17	)	)	PUNCT
cana-3959	109	18	∀𝑣,𝑤	∀𝑣,𝑤	SYM
cana-3959	109	19	∈	∈	PROPN
cana-3959	109	20	𝑋	𝑋	NOUN
cana-3959	109	21	and	and	CCONJ
cana-3959	109	22	let	let	VERB
cana-3959	109	23	𝔔:𝑋	𝔔:𝑋	VERB
cana-3959	109	24	×	×	NOUN
cana-3959	109	25	𝑋	𝑋	NOUN
cana-3959	109	26	→	→	SYM
cana-3959	109	27	[	[	X
cana-3959	109	28	0,∞	0,∞	X
cana-3959	109	29	)	)	PUNCT
cana-3959	109	30	be	be	VERB
cana-3959	109	31	a	a	DET
cana-3959	109	32	function	function	NOUN
cana-3959	109	33	such	such	ADJ
cana-3959	109	34	that	that	SCONJ
cana-3959	109	35	lim	lim	PROPN
cana-3959	109	36	𝑛→∞	𝑛→∞	NUM
cana-3959	109	37	𝔔(ℜ𝑎𝑛𝑣,ℜ𝑎𝑛𝑤	𝔔(ℜ𝑎𝑛𝑣,ℜ𝑎𝑛𝑤	PROPN
cana-3959	109	38	)	)	PUNCT
cana-3959	109	39	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	109	40	=	=	SYM
cana-3959	109	41	0	0	PUNCT
cana-3959	110	1	(	(	PUNCT
cana-3959	110	2	2	2	NUM
cana-3959	110	3	)	)	PUNCT
cana-3959	110	4	∀𝑣,𝑤	∀𝑣,𝑤	PROPN
cana-3959	110	5	∈	∈	PROPN
cana-3959	110	6	𝑋	𝑋	PROPN
cana-3959	110	7	,	,	PUNCT
cana-3959	110	8	then	then	ADV
cana-3959	110	9	∃	∃	PROPN
cana-3959	110	10	cubic	cubic	PROPN
cana-3959	110	11	map	map	NOUN
cana-3959	110	12	ℱ	ℱ	PROPN
cana-3959	110	13	:	:	PUNCT
cana-3959	110	14	𝑋	𝑋	PROPN
cana-3959	110	15	→	→	SYM
cana-3959	110	16	𝑌	𝑌	PROPN
cana-3959	110	17	with	with	ADP
cana-3959	110	18	the	the	DET
cana-3959	110	19	the	the	DET
cana-3959	110	20	fe	fe	NOUN
cana-3959	110	21	(	(	PUNCT
cana-3959	110	22	?	?	PUNCT
cana-3959	110	23	?	?	PUNCT
cana-3959	110	24	)	)	PUNCT
cana-3959	111	1	and	and	CCONJ
cana-3959	111	2	‖ℱ(𝑣	‖ℱ(𝑣	X
cana-3959	111	3	)	)	PUNCT
cana-3959	112	1	−	−	PROPN
cana-3959	112	2	ℱ(𝑣)‖	ℱ(𝑣)‖	NOUN
cana-3959	112	3	≤	≤	NUM
cana-3959	112	4	1	1	NUM
cana-3959	112	5	ℜ3𝑎∑	ℜ3𝑎∑	NUM
cana-3959	112	6	∞	∞	NUM
cana-3959	112	7	𝑞=1	𝑞=1	VERB
cana-3959	112	8	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	PROPN
cana-3959	112	9	)	)	PUNCT
cana-3959	112	10	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	112	11	(	(	PUNCT
cana-3959	112	12	3	3	X
cana-3959	112	13	)	)	PUNCT
cana-3959	112	14	∀𝑣	∀𝑣	PROPN
cana-3959	112	15	∈	∈	PROPN
cana-3959	112	16	𝑋.	𝑋.	PROPN
cana-3959	112	17	let	let	VERB
cana-3959	112	18	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	112	19	)	)	PUNCT
cana-3959	112	20	is	be	AUX
cana-3959	112	21	defined	define	VERB
cana-3959	112	22	as	as	ADP
cana-3959	112	23	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	112	24	)	)	PUNCT
cana-3959	112	25	=	=	SYM
cana-3959	112	26	lim	lim	PROPN
cana-3959	112	27	𝑛→∞	𝑛→∞	NUM
cana-3959	112	28	ℱ(ℜ𝑎𝑛𝑢	ℱ(ℜ𝑎𝑛𝑢	NUM
cana-3959	112	29	)	)	PUNCT
cana-3959	112	30	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	112	31	(	(	PUNCT
cana-3959	112	32	4	4	X
cana-3959	112	33	)	)	PUNCT
cana-3959	112	34	∀𝑣	∀𝑣	PROPN
cana-3959	112	35	∈	∈	PROPN
cana-3959	112	36	𝑋.	𝑋.	PROPN
cana-3959	112	37	communications	communication	NOUN
cana-3959	112	38	on	on	ADP
cana-3959	112	39	applied	apply	VERB
cana-3959	112	40	nonlinear	nonlinear	ADJ
cana-3959	112	41	analysis	analysis	NOUN
cana-3959	112	42	issn	issn	NOUN
cana-3959	112	43	:	:	PUNCT
cana-3959	112	44	1074	1074	NUM
cana-3959	112	45	-	-	PUNCT
cana-3959	112	46	133x	133x	NUM
cana-3959	112	47	vol	vol	NOUN
cana-3959	112	48	32	32	NUM
cana-3959	113	1	no	no	NOUN
cana-3959	113	2	.	.	PUNCT
cana-3959	114	1	9s	9s	NUM
cana-3959	114	2	(	(	PUNCT
cana-3959	114	3	2025	2025	NUM
cana-3959	114	4	)	)	PUNCT
cana-3959	114	5	479	479	NUM
cana-3959	115	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	115	2	proof	proof	NOUN
cana-3959	115	3	.	.	PUNCT
cana-3959	116	1	considering	consider	VERB
cana-3959	116	2	(	(	PUNCT
cana-3959	116	3	𝑣,𝑤	𝑣,𝑤	ADJ
cana-3959	116	4	)	)	PUNCT
cana-3959	116	5	by	by	ADP
cana-3959	116	6	(	(	PUNCT
cana-3959	116	7	0,0	0,0	NOUN
cana-3959	116	8	)	)	PUNCT
cana-3959	116	9	in	in	ADP
cana-3959	116	10	(	(	PUNCT
cana-3959	116	11	1	1	NUM
cana-3959	116	12	)	)	PUNCT
cana-3959	116	13	,	,	PUNCT
cana-3959	116	14	then	then	ADV
cana-3959	116	15	we	we	PRON
cana-3959	116	16	have	have	VERB
cana-3959	116	17	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	116	18	)	)	PUNCT
cana-3959	116	19	=	=	SYM
cana-3959	116	20	0	0	X
cana-3959	116	21	.	.	X
cana-3959	117	1	switching	switch	VERB
cana-3959	117	2	(	(	PUNCT
cana-3959	117	3	𝑣,𝑤	𝑣,𝑤	ADJ
cana-3959	117	4	)	)	PUNCT
cana-3959	117	5	by	by	ADP
cana-3959	117	6	(	(	PUNCT
cana-3959	117	7	𝑣	𝑣	NOUN
cana-3959	117	8	,	,	PUNCT
cana-3959	117	9	0	0	NUM
cana-3959	117	10	)	)	PUNCT
cana-3959	117	11	in	in	ADP
cana-3959	117	12	(	(	PUNCT
cana-3959	117	13	1	1	NUM
cana-3959	117	14	)	)	PUNCT
cana-3959	117	15	,	,	PUNCT
cana-3959	117	16	we	we	PRON
cana-3959	117	17	get	get	AUX
cana-3959	117	18	‖ℱ(ℜ𝑎𝑣	‖ℱ(ℜ𝑎𝑣	VERB
cana-3959	117	19	)	)	PUNCT
cana-3959	117	20	−ℜ	−ℜ	PROPN
cana-3959	117	21	3𝑎	3𝑎	NUM
cana-3959	117	22	ℱ(𝑣)‖	ℱ(𝑣)‖	PROPN
cana-3959	117	23	≤	≤	PROPN
cana-3959	117	24	𝔔(𝑣	𝔔(𝑣	PROPN
cana-3959	117	25	,	,	PUNCT
cana-3959	117	26	0	0	NUM
cana-3959	117	27	)	)	PUNCT
cana-3959	117	28	(	(	PUNCT
cana-3959	117	29	5	5	X
cana-3959	117	30	)	)	PUNCT
cana-3959	117	31	∀𝑣	∀𝑣	PROPN
cana-3959	117	32	∈	∈	PROPN
cana-3959	117	33	𝑋	𝑋	PROPN
cana-3959	117	34	,	,	PUNCT
cana-3959	117	35	we	we	PRON
cana-3959	117	36	replace	replace	VERB
cana-3959	117	37	𝑣	𝑣	PRON
cana-3959	117	38	by	by	ADP
cana-3959	117	39	ℜ	ℜ	ADJ
cana-3959	117	40	𝑎(𝑞−1)𝑣	𝑎(𝑞−1)𝑣	NOUN
cana-3959	117	41	(	(	PUNCT
cana-3959	117	42	for	for	ADP
cana-3959	117	43	𝑞	𝑞	PROPN
cana-3959	117	44	∈	∈	PROPN
cana-3959	117	45	ℕ	ℕ	PROPN
cana-3959	117	46	and	and	CCONJ
cana-3959	117	47	𝑞	𝑞	X
cana-3959	117	48	≥	≥	PROPN
cana-3959	117	49	1	1	NUM
cana-3959	117	50	)	)	PUNCT
cana-3959	117	51	in	in	ADP
cana-3959	117	52	(	(	PUNCT
cana-3959	117	53	5	5	NUM
cana-3959	117	54	)	)	PUNCT
cana-3959	117	55	,	,	PUNCT
cana-3959	117	56	and	and	CCONJ
cana-3959	117	57	we	we	PRON
cana-3959	117	58	obtain	obtain	VERB
cana-3959	117	59	‖ℱ(ℜ𝑎𝑞𝑣	‖ℱ(ℜ𝑎𝑞𝑣	PUNCT
cana-3959	117	60	)	)	PUNCT
cana-3959	117	61	−ℜ	−ℜ	PROPN
cana-3959	117	62	3𝑎	3𝑎	NUM
cana-3959	117	63	ℱ(ℜ𝑎(𝑞−1)𝑣)‖	ℱ(ℜ𝑎(𝑞−1)𝑣)‖	NOUN
cana-3959	117	64	≤	≤	NUM
cana-3959	117	65	𝔔(ℜ𝑎(𝑞−1)𝑣	𝔔(ℜ𝑎(𝑞−1)𝑣	NOUN
cana-3959	117	66	,	,	PUNCT
cana-3959	117	67	0	0	NUM
cana-3959	117	68	)	)	PUNCT
cana-3959	117	69	∀𝑣	∀𝑣	PROPN
cana-3959	117	70	∈	∈	PROPN
cana-3959	117	71	𝑋.	𝑋.	PROPN
cana-3959	117	72	by	by	ADP
cana-3959	117	73	multiplying	multiply	VERB
cana-3959	117	74	both	both	DET
cana-3959	117	75	sides	side	NOUN
cana-3959	117	76	of	of	ADP
cana-3959	117	77	the	the	DET
cana-3959	117	78	aforementioned	aforementioned	ADJ
cana-3959	117	79	inequality	inequality	NOUN
cana-3959	117	80	by	by	ADP
cana-3959	117	81	1	1	NUM
cana-3959	117	82	ℜ3𝑎𝑞	ℜ3𝑎𝑞	PROPN
cana-3959	117	83	,	,	PUNCT
cana-3959	117	84	we	we	PRON
cana-3959	117	85	get	get	VERB
cana-3959	117	86	the	the	DET
cana-3959	117	87	consequence	consequence	NOUN
cana-3959	117	88	of	of	ADP
cana-3959	117	89	adding	add	VERB
cana-3959	117	90	𝑛	𝑛	DET
cana-3959	117	91	inequalities	inequality	NOUN
cana-3959	117	92	.	.	PUNCT
cana-3959	118	1	∑𝑛𝑞=1	∑𝑛𝑞=1	PROPN
cana-3959	118	2	1	1	NUM
cana-3959	118	3	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	118	4	‖ℱ(ℜ𝑎𝑞𝑣	‖ℱ(ℜ𝑎𝑞𝑣	NUM
cana-3959	118	5	)	)	PUNCT
cana-3959	118	6	−ℜ	−ℜ	PROPN
cana-3959	118	7	3𝑎𝑞	3𝑎𝑞	NOUN
cana-3959	118	8	ℱ(ℜ𝑎(𝑞−1)𝑣)‖	ℱ(ℜ𝑎(𝑞−1)𝑣)‖	NOUN
cana-3959	118	9	≤	≤	NOUN
cana-3959	119	1	∑𝑛𝑞=1	∑𝑛𝑞=1	PROPN
cana-3959	119	2	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	NOUN
cana-3959	119	3	)	)	PUNCT
cana-3959	120	1	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	120	2	making	make	VERB
cana-3959	120	3	use	use	NOUN
cana-3959	120	4	of	of	ADP
cana-3959	120	5	the	the	DET
cana-3959	120	6	triangle	triangle	NOUN
cana-3959	120	7	inequality	inequality	NOUN
cana-3959	120	8	|𝐴	|𝐴	PUNCT
cana-3959	120	9	+	+	CCONJ
cana-3959	120	10	𝐵|	𝐵|	NOUN
cana-3959	120	11	≤	≤	NOUN
cana-3959	120	12	|𝐴|	|𝐴|	VERB
cana-3959	120	13	+	+	X
cana-3959	120	14	|𝐵|	|𝐵|	NOUN
cana-3959	120	15	after	after	ADP
cana-3959	120	16	simplifying	simplify	VERB
cana-3959	120	17	,	,	PUNCT
cana-3959	120	18	we	we	PRON
cana-3959	120	19	get	get	VERB
cana-3959	120	20	at	at	ADP
cana-3959	120	21	the	the	DET
cana-3959	120	22	left	left	ADJ
cana-3959	120	23	side	side	NOUN
cana-3959	120	24	of	of	ADP
cana-3959	120	25	the	the	DET
cana-3959	120	26	inequality	inequality	NOUN
cana-3959	120	27	.	.	PUNCT
cana-3959	121	1	‖	‖	ADJ
cana-3959	121	2	1	1	NUM
cana-3959	121	3	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	121	4	ℱ(ℜ𝑎𝑛𝑣	ℱ(ℜ𝑎𝑛𝑣	NOUN
cana-3959	121	5	)	)	PUNCT
cana-3959	122	1	−	−	PROPN
cana-3959	122	2	ℱ(𝑣)‖	ℱ(𝑣)‖	PROPN
cana-3959	122	3	≤	≤	NOUN
cana-3959	123	1	∑𝑛𝑞=1	∑𝑛𝑞=1	PUNCT
cana-3959	123	2	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	NOUN
cana-3959	123	3	)	)	PUNCT
cana-3959	123	4	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	123	5	(	(	PUNCT
cana-3959	123	6	6	6	NUM
cana-3959	123	7	)	)	PUNCT
cana-3959	123	8	since	since	SCONJ
cana-3959	123	9	∑𝑛𝑞=1	∑𝑛𝑞=1	PROPN
cana-3959	123	10	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	PROPN
cana-3959	123	11	)	)	PUNCT
cana-3959	123	12	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	123	13	≤	≤	ADJ
cana-3959	123	14	∑∞	∑∞	NOUN
cana-3959	123	15	𝑞=1	𝑞=1	PUNCT
cana-3959	123	16	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	NOUN
cana-3959	123	17	)	)	PUNCT
cana-3959	123	18	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	123	19	the	the	DET
cana-3959	123	20	inequality	inequality	NOUN
cana-3959	123	21	(	(	PUNCT
cana-3959	123	22	6	6	NUM
cana-3959	123	23	)	)	PUNCT
cana-3959	123	24	yields	yield	NOUN
cana-3959	123	25	‖	‖	PROPN
cana-3959	123	26	1	1	NUM
cana-3959	123	27	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	123	28	ℱ(ℜ𝑎𝑛𝑣	ℱ(ℜ𝑎𝑛𝑣	NOUN
cana-3959	123	29	)	)	PUNCT
cana-3959	124	1	−	−	PROPN
cana-3959	125	1	ℱ(𝑣)‖	ℱ(𝑣)‖	PROPN
cana-3959	125	2	≤	≤	PROPN
cana-3959	125	3	∑∞	∑∞	NOUN
cana-3959	125	4	𝑞=1	𝑞=1	PUNCT
cana-3959	125	5	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	NOUN
cana-3959	125	6	)	)	PUNCT
cana-3959	125	7	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	125	8	∀𝑣	∀𝑣	PROPN
cana-3959	125	9	∈	∈	PROPN
cana-3959	125	10	𝑋.	𝑋.	PROPN
cana-3959	125	11	it	it	PRON
cana-3959	125	12	will	will	AUX
cana-3959	125	13	be	be	AUX
cana-3959	125	14	proven	prove	VERB
cana-3959	125	15	by	by	ADP
cana-3959	125	16	induction	induction	NOUN
cana-3959	125	17	that	that	SCONJ
cana-3959	125	18	(	(	PUNCT
cana-3959	125	19	6	6	NUM
cana-3959	125	20	)	)	PUNCT
cana-3959	125	21	exists	exist	VERB
cana-3959	125	22	∀	∀	X
cana-3959	125	23	ℕ.	ℕ.	PROPN
cana-3959	125	24	here	here	ADV
cana-3959	125	25	𝑚	𝑚	PROPN
cana-3959	125	26	>	>	X
cana-3959	125	27	𝑛	𝑛	X
cana-3959	125	28	>	>	X
cana-3959	125	29	0	0	NUM
cana-3959	125	30	,	,	PUNCT
cana-3959	125	31	then	then	ADV
cana-3959	125	32	𝑚	𝑚	ADP
cana-3959	125	33	−	−	PROPN
cana-3959	125	34	𝑛	𝑛	DET
cana-3959	125	35	∈	∈	PROPN
cana-3959	125	36	ℕ	ℕ	PROPN
cana-3959	125	37	and	and	CCONJ
cana-3959	125	38	let	let	VERB
cana-3959	125	39	𝑛	𝑛	PRON
cana-3959	125	40	by	by	ADP
cana-3959	125	41	𝑚	𝑚	ADP
cana-3959	125	42	−	−	NOUN
cana-3959	125	43	𝑛	𝑛	PROPN
cana-3959	125	44	in	in	ADP
cana-3959	125	45	(	(	PUNCT
cana-3959	125	46	6	6	NUM
cana-3959	125	47	)	)	PUNCT
cana-3959	125	48	,	,	PUNCT
cana-3959	125	49	then	then	ADV
cana-3959	125	50	‖	‖	PROPN
cana-3959	125	51	1	1	NUM
cana-3959	125	52	ℜ3𝑎(𝑚−𝑛	ℜ3𝑎(𝑚−𝑛	NUM
cana-3959	125	53	)	)	PUNCT
cana-3959	125	54	ℱ(ℜ𝑎(𝑚−𝑛)𝑣	ℱ(ℜ𝑎(𝑚−𝑛)𝑣	PROPN
cana-3959	125	55	)	)	PUNCT
cana-3959	126	1	−	−	PROPN
cana-3959	126	2	ℱ(𝑣)‖	ℱ(𝑣)‖	PROPN
cana-3959	126	3	≤	≤	PROPN
cana-3959	126	4	∑∞	∑∞	NOUN
cana-3959	126	5	𝑞=1	𝑞=1	PUNCT
cana-3959	126	6	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	PROPN
cana-3959	126	7	)	)	PUNCT
cana-3959	126	8	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	126	9	(	(	PUNCT
cana-3959	126	10	7	7	X
cana-3959	126	11	)	)	PUNCT
cana-3959	126	12	which	which	PRON
cana-3959	126	13	is	be	AUX
cana-3959	126	14	‖	‖	PROPN
cana-3959	126	15	1	1	NUM
cana-3959	126	16	ℜ3𝑎𝑚	ℜ3𝑎𝑚	PROPN
cana-3959	126	17	ℱ(ℜ𝑎(𝑚−𝑛)𝑣	ℱ(ℜ𝑎(𝑚−𝑛)𝑣	PROPN
cana-3959	126	18	)	)	PUNCT
cana-3959	126	19	−	−	NOUN
cana-3959	126	20	1	1	NUM
cana-3959	126	21	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	126	22	ℱ(𝑣)‖	ℱ(𝑣)‖	NOUN
cana-3959	126	23	≤	≤	NUM
cana-3959	126	24	1	1	NUM
cana-3959	126	25	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	126	26	∑	∑	X
cana-3959	126	27	∞	∞	NOUN
cana-3959	126	28	𝑞=1	𝑞=1	X
cana-3959	126	29	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	PROPN
cana-3959	126	30	)	)	PUNCT
cana-3959	126	31	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	126	32	(	(	PUNCT
cana-3959	126	33	8)	8)	NUM
cana-3959	126	34	∀𝑢	∀𝑢	DET
cana-3959	126	35	∈	∈	PROPN
cana-3959	126	36	𝑋.	𝑋.	PROPN
cana-3959	126	37	interchanging	interchange	VERB
cana-3959	126	38	𝑢	𝑢	PRON
cana-3959	126	39	by	by	ADP
cana-3959	126	40	ℜ	ℜ	ADJ
cana-3959	126	41	𝑎𝑛𝑣	𝑎𝑛𝑣	NOUN
cana-3959	126	42	in	in	ADP
cana-3959	126	43	(	(	PUNCT
cana-3959	126	44	8)	8)	NUM
cana-3959	126	45	,	,	PUNCT
cana-3959	126	46	we	we	PRON
cana-3959	126	47	obtain	obtain	VERB
cana-3959	126	48	‖	‖	PROPN
cana-3959	126	49	1	1	NUM
cana-3959	126	50	ℜ3𝑎𝑚	ℜ3𝑎𝑚	NOUN
cana-3959	126	51	ℱ(ℜ𝑎𝑚𝑣	ℱ(ℜ𝑎𝑚𝑣	NUM
cana-3959	126	52	)	)	PUNCT
cana-3959	127	1	−	−	ADP
cana-3959	127	2	1	1	NUM
cana-3959	127	3	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	127	4	ℱ(ℜ𝑎𝑛𝑣)‖	ℱ(ℜ𝑎𝑛𝑣)‖	NOUN
cana-3959	127	5	≤	≤	ADV
cana-3959	127	6	1	1	NUM
cana-3959	127	7	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	127	8	∑	∑	X
cana-3959	127	9	∞	∞	NUM
cana-3959	127	10	𝑞=1	𝑞=1	VERB
cana-3959	127	11	𝔔(ℜ𝑎(𝑞+𝑛−1),0	𝔔(ℜ𝑎(𝑞+𝑛−1),0	PROPN
cana-3959	127	12	)	)	PUNCT
cana-3959	127	13	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	128	1	(	(	PUNCT
cana-3959	128	2	9	9	X
cana-3959	128	3	)	)	PUNCT
cana-3959	128	4	since	since	SCONJ
cana-3959	128	5	lim	lim	PROPN
cana-3959	128	6	𝑛→∞	𝑛→∞	NUM
cana-3959	128	7	1	1	NUM
cana-3959	128	8	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	128	9	=	=	SYM
cana-3959	128	10	0	0	PUNCT
cana-3959	128	11	and	and	CCONJ
cana-3959	128	12	hence	hence	ADV
cana-3959	128	13	from	from	ADP
cana-3959	128	14	(	(	PUNCT
cana-3959	128	15	9	9	NUM
cana-3959	128	16	)	)	PUNCT
cana-3959	128	17	,	,	PUNCT
cana-3959	128	18	we	we	PRON
cana-3959	128	19	obtain	obtain	VERB
cana-3959	128	20	communications	communication	NOUN
cana-3959	128	21	on	on	ADP
cana-3959	128	22	applied	apply	VERB
cana-3959	128	23	nonlinear	nonlinear	ADJ
cana-3959	128	24	analysis	analysis	NOUN
cana-3959	128	25	issn	issn	NOUN
cana-3959	128	26	:	:	PUNCT
cana-3959	128	27	1074	1074	NUM
cana-3959	128	28	-	-	PUNCT
cana-3959	128	29	133x	133x	NUM
cana-3959	128	30	vol	vol	NOUN
cana-3959	128	31	32	32	NUM
cana-3959	128	32	no	no	NOUN
cana-3959	128	33	.	.	PUNCT
cana-3959	129	1	9s	9s	NUM
cana-3959	129	2	(	(	PUNCT
cana-3959	129	3	2025	2025	NUM
cana-3959	129	4	)	)	PUNCT
cana-3959	129	5	480	480	NUM
cana-3959	129	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	129	7	lim	lim	PROPN
cana-3959	129	8	𝑛→∞	𝑛→∞	NUM
cana-3959	129	9	‖	‖	PROPN
cana-3959	129	10	1	1	NUM
cana-3959	129	11	ℜ3𝑎𝑚	ℜ3𝑎𝑚	NOUN
cana-3959	129	12	ℱ(ℜ𝑎𝑚𝑣	ℱ(ℜ𝑎𝑚𝑣	NUM
cana-3959	129	13	)	)	PUNCT
cana-3959	130	1	−	−	ADP
cana-3959	130	2	1	1	NUM
cana-3959	130	3	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	130	4	ℱ(ℜ𝑎𝑛𝑣)‖	ℱ(ℜ𝑎𝑛𝑣)‖	NOUN
cana-3959	130	5	=	=	SYM
cana-3959	130	6	0	0	NUM
cana-3959	130	7	finally	finally	ADV
cana-3959	130	8	{	{	PUNCT
cana-3959	130	9	ℱ(ℜ𝑎𝑛𝑣	ℱ(ℜ𝑎𝑛𝑣	NUM
cana-3959	130	10	)	)	PUNCT
cana-3959	130	11	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	130	12	}	}	PUNCT
cana-3959	130	13	𝑛=1	𝑛=1	NOUN
cana-3959	130	14	∞	∞	NOUN
cana-3959	130	15	is	be	AUX
cana-3959	130	16	cauchy	cauchy	ADJ
cana-3959	130	17	sequence	sequence	NOUN
cana-3959	130	18	.	.	PUNCT
cana-3959	131	1	the	the	DET
cana-3959	131	2	sequence	sequence	NOUN
cana-3959	131	3	then	then	ADV
cana-3959	131	4	has	have	VERB
cana-3959	131	5	a	a	DET
cana-3959	131	6	limit	limit	NOUN
cana-3959	131	7	in	in	ADP
cana-3959	131	8	𝑋.	𝑋.	PROPN
cana-3959	131	9	define	define	VERB
cana-3959	131	10	𝐴(𝑣	𝐴(𝑣	NUM
cana-3959	131	11	)	)	PUNCT
cana-3959	131	12	=	=	SYM
cana-3959	131	13	lim	lim	NOUN
cana-3959	131	14	𝑛→∞	𝑛→∞	NUM
cana-3959	131	15	ℱ(ℜ𝑎𝑛𝑣	ℱ(ℜ𝑎𝑛𝑣	NUM
cana-3959	131	16	)	)	PUNCT
cana-3959	131	17	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	131	18	∀𝑢	∀𝑢	DET
cana-3959	131	19	∈	∈	PROPN
cana-3959	131	20	𝑋.	𝑋.	PROPN
cana-3959	131	21	we	we	PRON
cana-3959	131	22	prove	prove	VERB
cana-3959	131	23	𝐴	𝐴	PROPN
cana-3959	131	24	:	:	PUNCT
cana-3959	131	25	𝑋	𝑋	PROPN
cana-3959	131	26	→	→	SYM
cana-3959	131	27	𝑋	𝑋	PROPN
cana-3959	131	28	is	be	AUX
cana-3959	131	29	a	a	DET
cana-3959	131	30	linear	linear	ADJ
cana-3959	131	31	mapping	mapping	NOUN
cana-3959	131	32	.	.	PUNCT
cana-3959	132	1	‖ℱ(ℜ𝑎𝑣	‖ℱ(ℜ𝑎𝑣	VERB
cana-3959	132	2	+	+	CCONJ
cana-3959	132	3	𝑤	𝑤	X
cana-3959	132	4	)	)	PUNCT
cana-3959	132	5	±ℜ	±ℜ	PUNCT
cana-3959	133	1	𝑏	𝑏	NOUN
cana-3959	133	2	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	133	3	−ℜ	−ℜ	PROPN
cana-3959	133	4	𝑎𝑤	𝑎𝑤	X
cana-3959	133	5	)	)	PUNCT
cana-3959	133	6	−	−	PROPN
cana-3959	133	7	(	(	PUNCT
cana-3959	133	8	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	133	9	)	)	PUNCT
cana-3959	133	10	2	2	NUM
cana-3959	133	11	)	)	PUNCT
cana-3959	134	1	[	[	X
cana-3959	134	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	134	3	+	+	CCONJ
cana-3959	134	4	𝑤	𝑤	X
cana-3959	134	5	)	)	PUNCT
cana-3959	134	6	+	+	CCONJ
cana-3959	134	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	134	8	−	−	NOUN
cana-3959	134	9	𝑤	𝑤	ADP
cana-3959	134	10	)	)	PUNCT
cana-3959	134	11	]	]	PUNCT
cana-3959	134	12	−	−	PROPN
cana-3959	134	13	(	(	PUNCT
cana-3959	134	14	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	134	15	)	)	PUNCT
cana-3959	134	16	2	2	NUM
cana-3959	134	17	)	)	PUNCT
cana-3959	135	1	[	[	X
cana-3959	135	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	135	3	+	+	NOUN
cana-3959	135	4	𝑤	𝑤	X
cana-3959	135	5	)	)	PUNCT
cana-3959	135	6	−	−	PROPN
cana-3959	135	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	135	8	−	−	NOUN
cana-3959	135	9	𝑤	𝑤	PROPN
cana-3959	135	10	)	)	PUNCT
cana-3959	135	11	]	]	PUNCT
cana-3959	135	12	−(ℜ2𝑎	−(ℜ2𝑎	PROPN
cana-3959	136	1	−	−	NOUN
cana-3959	136	2	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	136	3	∓ℜ	∓ℜ	PROPN
cana-3959	136	4	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	136	5	)	)	PUNCT
cana-3959	136	6	∓	∓	PROPN
cana-3959	136	7	(	(	PUNCT
cana-3959	136	8	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	136	9	±	±	NUM
cana-3959	136	10	1)ℱ(𝑤)]‖	1)ℱ(𝑤)]‖	NUM
cana-3959	136	11	=	=	SYM
cana-3959	136	12	1	1	NUM
cana-3959	136	13	ℜ𝑎𝑛	ℜ𝑎𝑛	PROPN
cana-3959	136	14	‖ℱ(ℜ𝑎𝑣	‖ℱ(ℜ𝑎𝑣	AUX
cana-3959	136	15	+	+	CCONJ
cana-3959	136	16	𝑤	𝑤	X
cana-3959	136	17	)	)	PUNCT
cana-3959	136	18	±ℜ	±ℜ	PUNCT
cana-3959	136	19	𝑏	𝑏	PRON
cana-3959	136	20	ℱ(𝑣	ℱ(𝑣	NOUN
cana-3959	136	21	−	−	PUNCT
cana-3959	136	22	ℜ	ℜ	PROPN
cana-3959	136	23	𝑎𝑤	𝑎𝑤	VERB
cana-3959	136	24	)	)	PUNCT
cana-3959	136	25	−	−	PROPN
cana-3959	136	26	(	(	PUNCT
cana-3959	136	27	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	136	28	)	)	PUNCT
cana-3959	136	29	2	2	NUM
cana-3959	136	30	)	)	PUNCT
cana-3959	137	1	[	[	X
cana-3959	137	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	137	3	+	+	CCONJ
cana-3959	137	4	𝑤	𝑤	X
cana-3959	137	5	)	)	PUNCT
cana-3959	137	6	+	+	CCONJ
cana-3959	137	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	137	8	−	−	NOUN
cana-3959	137	9	𝑤	𝑤	ADP
cana-3959	137	10	)	)	PUNCT
cana-3959	137	11	]	]	PUNCT
cana-3959	137	12	−	−	PROPN
cana-3959	137	13	(	(	PUNCT
cana-3959	137	14	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	137	15	)	)	PUNCT
cana-3959	137	16	2	2	NUM
cana-3959	137	17	)	)	PUNCT
cana-3959	138	1	[	[	X
cana-3959	138	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	138	3	+	+	NOUN
cana-3959	138	4	𝑤	𝑤	X
cana-3959	138	5	)	)	PUNCT
cana-3959	138	6	−	−	PROPN
cana-3959	138	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	138	8	−	−	NOUN
cana-3959	138	9	𝑤	𝑤	PROPN
cana-3959	138	10	)	)	PUNCT
cana-3959	138	11	]	]	PUNCT
cana-3959	138	12	−(ℜ2𝑎	−(ℜ2𝑎	PROPN
cana-3959	139	1	−	−	NOUN
cana-3959	139	2	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	139	3	∓ℜ	∓ℜ	PROPN
cana-3959	139	4	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	139	5	)	)	PUNCT
cana-3959	139	6	∓	∓	PROPN
cana-3959	139	7	(	(	PUNCT
cana-3959	139	8	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	139	9	±	±	NUM
cana-3959	139	10	1)ℱ(𝑤)]‖	1)ℱ(𝑤)]‖	NUM
cana-3959	139	11	≤	≤	NOUN
cana-3959	139	12	lim	lim	NOUN
cana-3959	139	13	𝑛→∞	𝑛→∞	NUM
cana-3959	139	14	𝔔(ℜ𝑎𝑛𝑣,ℜ𝑎𝑛𝑤	𝔔(ℜ𝑎𝑛𝑣,ℜ𝑎𝑛𝑤	PROPN
cana-3959	139	15	)	)	PUNCT
cana-3959	139	16	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	139	17	=	=	SYM
cana-3959	139	18	0	0	PUNCT
cana-3959	139	19	hence	hence	ADV
cana-3959	139	20	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	139	21	+	+	CCONJ
cana-3959	139	22	𝑤	𝑤	X
cana-3959	139	23	)	)	PUNCT
cana-3959	139	24	±ℜ	±ℜ	PUNCT
cana-3959	140	1	𝑏	𝑏	NOUN
cana-3959	140	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	140	3	−ℜ	−ℜ	PROPN
cana-3959	140	4	𝑎𝑤	𝑎𝑤	X
cana-3959	140	5	)	)	PUNCT
cana-3959	140	6	=	=	SYM
cana-3959	140	7	(	(	PUNCT
cana-3959	140	8	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	140	9	)	)	PUNCT
cana-3959	140	10	2	2	NUM
cana-3959	140	11	)	)	PUNCT
cana-3959	141	1	[	[	X
cana-3959	141	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	141	3	+	+	CCONJ
cana-3959	141	4	𝑤	𝑤	X
cana-3959	141	5	)	)	PUNCT
cana-3959	141	6	+	+	CCONJ
cana-3959	141	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	141	8	−	−	NOUN
cana-3959	141	9	𝑤	𝑤	ADP
cana-3959	141	10	)	)	PUNCT
cana-3959	141	11	]	]	PUNCT
cana-3959	142	1	+	+	CCONJ
cana-3959	142	2	(	(	PUNCT
cana-3959	142	3	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	142	4	)	)	PUNCT
cana-3959	142	5	2	2	NUM
cana-3959	142	6	)	)	PUNCT
cana-3959	143	1	[	[	X
cana-3959	143	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	143	3	+	+	NOUN
cana-3959	143	4	𝑤	𝑤	X
cana-3959	143	5	)	)	PUNCT
cana-3959	143	6	−	−	PROPN
cana-3959	143	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	143	8	−	−	NOUN
cana-3959	143	9	𝑤	𝑤	ADP
cana-3959	143	10	)	)	PUNCT
cana-3959	143	11	]	]	PUNCT
cana-3959	144	1	+	+	ADJ
cana-3959	144	2	(	(	PUNCT
cana-3959	144	3	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	144	4	−	−	PROPN
cana-3959	144	5	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	144	6	∓ℜ	∓ℜ	PROPN
cana-3959	144	7	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	144	8	)	)	PUNCT
cana-3959	144	9	∓	∓	PROPN
cana-3959	145	1	(	(	PUNCT
cana-3959	145	2	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	145	3	±	±	NUM
cana-3959	145	4	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	145	5	)	)	PUNCT
cana-3959	145	6	]	]	PUNCT
cana-3959	146	1	∀𝑢	∀𝑢	DET
cana-3959	146	2	∈	∈	PROPN
cana-3959	146	3	𝑋.	𝑋.	PROPN
cana-3959	146	4	next	next	ADV
cana-3959	146	5	,	,	PUNCT
cana-3959	146	6	we	we	PRON
cana-3959	146	7	consider	consider	VERB
cana-3959	146	8	||𝐴(𝑣	||𝐴(𝑣	ADV
cana-3959	146	9	)	)	PUNCT
cana-3959	146	10	−	−	PROPN
cana-3959	146	11	ℱ(𝑣)||	ℱ(𝑣)||	NOUN
cana-3959	146	12	=	=	PUNCT
cana-3959	146	13	||	||	PROPN
cana-3959	147	1	lim	lim	PROPN
cana-3959	147	2	𝑛→∞	𝑛→∞	NUM
cana-3959	147	3	ℱ(ℜ𝑎𝑛𝑢	ℱ(ℜ𝑎𝑛𝑢	NUM
cana-3959	147	4	)	)	PUNCT
cana-3959	147	5	ℜ𝑎𝑛	ℜ𝑎𝑛	NOUN
cana-3959	147	6	−	−	NOUN
cana-3959	147	7	ℱ(𝑣)||	ℱ(𝑣)||	NOUN
cana-3959	147	8	=	=	PROPN
cana-3959	147	9	lim	lim	PROPN
cana-3959	147	10	𝑛→∞	𝑛→∞	NUM
cana-3959	147	11	||	||	NOUN
cana-3959	147	12	ℱ(ℜ𝑎𝑛𝑢	ℱ(ℜ𝑎𝑛𝑢	NUM
cana-3959	147	13	)	)	PUNCT
cana-3959	147	14	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	147	15	−	−	PROPN
cana-3959	147	16	ℱ(𝑣)||	ℱ(𝑣)||	ADJ
cana-3959	147	17	communications	communication	NOUN
cana-3959	147	18	on	on	ADP
cana-3959	147	19	applied	apply	VERB
cana-3959	147	20	nonlinear	nonlinear	ADJ
cana-3959	147	21	analysis	analysis	NOUN
cana-3959	147	22	issn	issn	NOUN
cana-3959	147	23	:	:	PUNCT
cana-3959	147	24	1074	1074	NUM
cana-3959	147	25	-	-	PUNCT
cana-3959	147	26	133x	133x	NUM
cana-3959	147	27	vol	vol	NOUN
cana-3959	147	28	32	32	NUM
cana-3959	147	29	no	no	NOUN
cana-3959	147	30	.	.	PUNCT
cana-3959	148	1	9s	9s	NUM
cana-3959	148	2	(	(	PUNCT
cana-3959	148	3	2025	2025	NUM
cana-3959	148	4	)	)	PUNCT
cana-3959	148	5	481	481	NUM
cana-3959	148	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	148	7	≤	≤	ADV
cana-3959	148	8	lim	lim	NOUN
cana-3959	148	9	𝑛→∞	𝑛→∞	NUM
cana-3959	148	10	1	1	NUM
cana-3959	148	11	ℜ3𝑎∑	ℜ3𝑎∑	NUM
cana-3959	148	12	∞	∞	NUM
cana-3959	148	13	𝑞=1	𝑞=1	VERB
cana-3959	148	14	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	NOUN
cana-3959	148	15	)	)	PUNCT
cana-3959	148	16	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	148	17	hence	hence	ADV
cana-3959	148	18	,	,	PUNCT
cana-3959	148	19	we	we	PRON
cana-3959	148	20	get	get	VERB
cana-3959	148	21	||𝐴(𝑣	||𝐴(𝑣	ADV
cana-3959	148	22	)	)	PUNCT
cana-3959	149	1	−	−	PROPN
cana-3959	149	2	ℱ(𝑣)||	ℱ(𝑣)||	ADJ
cana-3959	149	3	≤	≤	NOUN
cana-3959	149	4	1	1	NUM
cana-3959	149	5	ℜ3𝑎∑	ℜ3𝑎∑	NUM
cana-3959	149	6	∞	∞	NUM
cana-3959	149	7	𝑞=1	𝑞=1	VERB
cana-3959	149	8	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	PROPN
cana-3959	149	9	)	)	PUNCT
cana-3959	149	10	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	149	11	∀𝑢	∀𝑢	PROPN
cana-3959	149	12	∈	∈	PROPN
cana-3959	149	13	𝑋.	𝑋.	PROPN
cana-3959	149	14	here	here	ADV
cana-3959	149	15	to	to	PART
cana-3959	149	16	obtain	obtain	VERB
cana-3959	149	17	𝐴	𝐴	PROPN
cana-3959	149	18	is	be	AUX
cana-3959	149	19	unique	unique	ADJ
cana-3959	149	20	.	.	PUNCT
cana-3959	150	1	then	then	ADV
cana-3959	150	2	another	another	DET
cana-3959	150	3	mapping	mapping	NOUN
cana-3959	150	4	𝐵	𝐵	NOUN
cana-3959	150	5	:	:	PUNCT
cana-3959	150	6	𝑋	𝑋	PROPN
cana-3959	150	7	→	→	SYM
cana-3959	150	8	𝑌	𝑌	PROPN
cana-3959	150	9	occurs	occur	VERB
cana-3959	150	10	and	and	CCONJ
cana-3959	150	11	||𝐵(𝑣	||𝐵(𝑣	ADJ
cana-3959	150	12	)	)	PUNCT
cana-3959	151	1	−	−	NOUN
cana-3959	151	2	ℱ(𝑣)||	ℱ(𝑣)||	ADJ
cana-3959	151	3	≤	≤	NOUN
cana-3959	151	4	1	1	NUM
cana-3959	151	5	ℜ3𝑎∑	ℜ3𝑎∑	NUM
cana-3959	151	6	∞	∞	NUM
cana-3959	151	7	𝑞=1	𝑞=1	VERB
cana-3959	151	8	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	NOUN
cana-3959	151	9	)	)	PUNCT
cana-3959	151	10	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	151	11	hence	hence	ADV
cana-3959	151	12	||𝐵(𝑣	||𝐵(𝑣	ADJ
cana-3959	151	13	)	)	PUNCT
cana-3959	151	14	−	−	NOUN
cana-3959	151	15	𝐴(𝑣)||	𝐴(𝑣)||	NOUN
cana-3959	151	16	≤	≤	NUM
cana-3959	151	17	||𝐵(𝑣	||𝐵(𝑣	NOUN
cana-3959	151	18	)	)	PUNCT
cana-3959	151	19	−	−	PROPN
cana-3959	152	1	ℱ(𝑣)||	ℱ(𝑣)||	NOUN
cana-3959	152	2	+	+	CCONJ
cana-3959	152	3	||𝐴(𝑣	||𝐴(𝑣	ADV
cana-3959	152	4	)	)	PUNCT
cana-3959	152	5	−	−	PROPN
cana-3959	152	6	ℱ(𝑣)||	ℱ(𝑣)||	ADJ
cana-3959	152	7	≤	≤	NOUN
cana-3959	152	8	1	1	NUM
cana-3959	152	9	ℜ3𝑎∑	ℜ3𝑎∑	NOUN
cana-3959	152	10	𝑛	𝑛	PRON
cana-3959	152	11	𝑞=1	𝑞=1	PUNCT
cana-3959	152	12	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	NOUN
cana-3959	152	13	)	)	PUNCT
cana-3959	152	14	ℜ3𝑎𝑞	ℜ3𝑎𝑞	VERB
cana-3959	153	1	+	+	CCONJ
cana-3959	153	2	1	1	NUM
cana-3959	153	3	ℜ3𝑎∑	ℜ3𝑎∑	NUM
cana-3959	153	4	∞	∞	NUM
cana-3959	153	5	𝑞=1	𝑞=1	VERB
cana-3959	153	6	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	NOUN
cana-3959	153	7	)	)	PUNCT
cana-3959	153	8	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	153	9	=	=	NOUN
cana-3959	153	10	2	2	NUM
cana-3959	153	11	ℜ3𝑎∑	ℜ3𝑎∑	NUM
cana-3959	153	12	∞	∞	NUM
cana-3959	153	13	𝑞=1	𝑞=1	VERB
cana-3959	153	14	𝔔(ℜ𝑎(𝑞−1)𝑣,0	𝔔(ℜ𝑎(𝑞−1)𝑣,0	NOUN
cana-3959	153	15	)	)	PUNCT
cana-3959	153	16	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	153	17	because	because	SCONJ
cana-3959	153	18	the	the	DET
cana-3959	153	19	additive	additive	ADJ
cana-3959	153	20	mappings	mapping	NOUN
cana-3959	153	21	are	be	AUX
cana-3959	153	22	𝐴	𝐴	PROPN
cana-3959	153	23	and	and	CCONJ
cana-3959	153	24	𝐵	𝐵	PROPN
cana-3959	153	25	,	,	PUNCT
cana-3959	153	26	we	we	PRON
cana-3959	153	27	can	can	AUX
cana-3959	153	28	observe	observe	VERB
cana-3959	153	29	||𝐴(𝑣	||𝐴(𝑣	ADV
cana-3959	153	30	)	)	PUNCT
cana-3959	154	1	−	−	PROPN
cana-3959	154	2	𝐵(𝑣)||	𝐵(𝑣)||	NOUN
cana-3959	154	3	=	=	SYM
cana-3959	154	4	2	2	NUM
cana-3959	154	5	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	154	6	||𝐴(ℜ	||𝐴(ℜ	VERB
cana-3959	154	7	𝑎𝑛	𝑎𝑛	NOUN
cana-3959	154	8	)	)	PUNCT
cana-3959	154	9	−	−	PROPN
cana-3959	155	1	𝐵(ℜ𝑎𝑛𝑣)||	𝐵(ℜ𝑎𝑛𝑣)||	NOUN
cana-3959	155	2	≤	≤	NUM
cana-3959	155	3	2	2	NUM
cana-3959	155	4	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	155	5	∑	∑	X
cana-3959	155	6	∞	∞	NOUN
cana-3959	155	7	𝑞=1	𝑞=1	X
cana-3959	155	8	𝔔(ℜ𝑎(𝑞+𝑛−1)𝑣,0	𝔔(ℜ𝑎(𝑞+𝑛−1)𝑣,0	PROPN
cana-3959	155	9	)	)	PUNCT
cana-3959	155	10	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	155	11	(	(	PUNCT
cana-3959	155	12	10	10	NUM
cana-3959	155	13	)	)	PUNCT
cana-3959	155	14	as	as	ADP
cana-3959	155	15	a	a	DET
cana-3959	155	16	result	result	NOUN
cana-3959	155	17	(	(	PUNCT
cana-3959	155	18	10	10	NUM
cana-3959	155	19	)	)	PUNCT
cana-3959	155	20	,	,	PUNCT
cana-3959	155	21	using	use	VERB
cana-3959	155	22	the	the	DET
cana-3959	155	23	limit	limit	NOUN
cana-3959	155	24	𝑛	𝑛	ADP
cana-3959	155	25	→	→	SYM
cana-3959	155	26	∞	∞	PROPN
cana-3959	155	27	and	and	CCONJ
cana-3959	155	28	obtain	obtain	VERB
cana-3959	155	29	lim	lim	NOUN
cana-3959	155	30	𝑛→∞	𝑛→∞	PUNCT
cana-3959	155	31	||𝐴(𝑣	||𝐴(𝑣	ADV
cana-3959	155	32	)	)	PUNCT
cana-3959	155	33	−	−	PROPN
cana-3959	155	34	𝐵(𝑣)||	𝐵(𝑣)||	ADJ
cana-3959	155	35	≤	≤	PROPN
cana-3959	155	36	lim	lim	NOUN
cana-3959	155	37	𝑛→∞	𝑛→∞	NUM
cana-3959	155	38	2	2	NUM
cana-3959	155	39	ℜ3𝑎𝑛	ℜ3𝑎𝑛	NOUN
cana-3959	155	40	∑	∑	DET
cana-3959	155	41	∞	∞	NOUN
cana-3959	155	42	𝑞=1	𝑞=1	X
cana-3959	155	43	𝔔(ℜ𝑎(𝑞+𝑛−1)𝑣,0	𝔔(ℜ𝑎(𝑞+𝑛−1)𝑣,0	NOUN
cana-3959	155	44	)	)	PUNCT
cana-3959	155	45	ℜ3𝑎𝑞	ℜ3𝑎𝑞	ADP
cana-3959	155	46	hence	hence	ADV
cana-3959	155	47	||𝐴(𝑣	||𝐴(𝑣	ADV
cana-3959	155	48	)	)	PUNCT
cana-3959	155	49	−	−	PROPN
cana-3959	155	50	𝐵(𝑣)||	𝐵(𝑣)||	ADJ
cana-3959	155	51	≤	≤	NOUN
cana-3959	155	52	0	0	NUM
cana-3959	156	1	we	we	PRON
cana-3959	156	2	conclude	conclude	VERB
cana-3959	156	3	that	that	SCONJ
cana-3959	156	4	𝐴(𝑣	𝐴(𝑣	X
cana-3959	156	5	)	)	PUNCT
cana-3959	156	6	=	=	SYM
cana-3959	156	7	𝐵(𝑣	𝐵(𝑣	NOUN
cana-3959	156	8	)	)	PUNCT
cana-3959	156	9	∀	∀	X
cana-3959	157	1	𝑣	𝑣	ADP
cana-3959	157	2	∈	∈	PROPN
cana-3959	157	3	𝑋.	𝑋.	PROPN
cana-3959	157	4	at	at	ADP
cana-3959	157	5	the	the	DET
cana-3959	157	6	end	end	NOUN
cana-3959	157	7	𝐴	𝐴	PROPN
cana-3959	157	8	is	be	AUX
cana-3959	157	9	unique	unique	ADJ
cana-3959	157	10	.	.	PUNCT
cana-3959	158	1	corollary	corollary	ADJ
cana-3959	158	2	2.2	2.2	NUM
cana-3959	158	3	consider	consider	VERB
cana-3959	158	4	the	the	DET
cana-3959	158	5	map	map	NOUN
cana-3959	158	6	ℱ:x	ℱ:x	ADJ
cana-3959	158	7	→	→	ADP
cana-3959	158	8	yfulfills	yfulfill	NOUN
cana-3959	158	9	‖dℱ(v	‖dℱ(v	PROPN
cana-3959	158	10	,	,	PUNCT
cana-3959	158	11	w)‖	w)‖	NOUN
cana-3959	158	12	≤	≤	NOUN
cana-3959	158	13	{	{	PUNCT
cana-3959	158	14	𝔘	𝔘	PROPN
cana-3959	158	15	,	,	PUNCT
cana-3959	158	16	𝔘{||v||p	𝔘{||v||p	NOUN
cana-3959	158	17	+	+	CCONJ
cana-3959	158	18	||w||p	||w||p	NOUN
cana-3959	158	19	}	}	PUNCT
cana-3959	158	20	,	,	PUNCT
cana-3959	158	21	p	p	PROPN
cana-3959	158	22	≠	≠	PROPN
cana-3959	158	23	3	3	NUM
cana-3959	158	24	;	;	PUNCT
cana-3959	158	25	𝔘	𝔘	PROPN
cana-3959	158	26	{	{	PUNCT
cana-3959	158	27	||v||p||w||p	||v||p||w||p	PROPN
cana-3959	158	28	+	+	CCONJ
cana-3959	158	29	{	{	PUNCT
cana-3959	158	30	||v||2p	||v||2p	NOUN
cana-3959	158	31	+	+	CCONJ
cana-3959	158	32	||w||2p	||w||2p	ADJ
cana-3959	158	33	}	}	PUNCT
cana-3959	158	34	}	}	PUNCT
cana-3959	158	35	,	,	PUNCT
cana-3959	158	36	2p	2p	NUM
cana-3959	158	37	≠	≠	PROPN
cana-3959	158	38	3	3	NUM
cana-3959	158	39	;	;	PUNCT
cana-3959	158	40	(	(	PUNCT
cana-3959	158	41	11	11	NUM
cana-3959	158	42	)	)	PUNCT
cana-3959	158	43	and	and	CCONJ
cana-3959	158	44	the	the	DET
cana-3959	158	45	function	function	NOUN
cana-3959	158	46	𝐴	𝐴	PROPN
cana-3959	158	47	:	:	PUNCT
cana-3959	158	48	𝑋	𝑋	PROPN
cana-3959	158	49	→	→	SYM
cana-3959	158	50	𝑋	𝑋	PROPN
cana-3959	158	51	,	,	PUNCT
cana-3959	158	52	we	we	PRON
cana-3959	158	53	arrive	arrive	VERB
cana-3959	158	54	the	the	DET
cana-3959	158	55	result	result	NOUN
cana-3959	158	56	communications	communication	NOUN
cana-3959	158	57	on	on	ADP
cana-3959	158	58	applied	apply	VERB
cana-3959	158	59	nonlinear	nonlinear	ADJ
cana-3959	158	60	analysis	analysis	NOUN
cana-3959	158	61	issn	issn	NOUN
cana-3959	158	62	:	:	PUNCT
cana-3959	158	63	1074	1074	NUM
cana-3959	158	64	-	-	PUNCT
cana-3959	158	65	133x	133x	NUM
cana-3959	158	66	vol	vol	NOUN
cana-3959	158	67	32	32	NUM
cana-3959	159	1	no	no	NOUN
cana-3959	159	2	.	.	PUNCT
cana-3959	160	1	9s	9s	NUM
cana-3959	160	2	(	(	PUNCT
cana-3959	160	3	2025	2025	NUM
cana-3959	160	4	)	)	PUNCT
cana-3959	160	5	482	482	NUM
cana-3959	160	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	160	7	‖ℱ(𝑣	‖ℱ(𝑣	PROPN
cana-3959	160	8	)	)	PUNCT
cana-3959	160	9	−	−	PROPN
cana-3959	160	10	𝐴(𝑣)‖	𝐴(𝑣)‖	PROPN
cana-3959	160	11	≤	≤	PROPN
cana-3959	160	12	{	{	PUNCT
cana-3959	160	13	𝔘	𝔘	PROPN
cana-3959	160	14	|ℜ3𝑎−1|	|ℜ3𝑎−1|	PROPN
cana-3959	160	15	,	,	PUNCT
cana-3959	160	16	𝔘||𝑣||𝑝	𝔘||𝑣||𝑝	PROPN
cana-3959	160	17	|ℜ3𝑎−ℜ𝑎𝑝|	|ℜ3𝑎−ℜ𝑎𝑝|	PROPN
cana-3959	160	18	,	,	PUNCT
cana-3959	160	19	𝔘||𝑣||2𝑝	𝔘||𝑣||2𝑝	PROPN
cana-3959	160	20	|ℜ3𝑎−ℜ2𝑎𝑝|	|ℜ3𝑎−ℜ2𝑎𝑝|	PROPN
cana-3959	160	21	(	(	PUNCT
cana-3959	160	22	12	12	NUM
cana-3959	160	23	)	)	PUNCT
cana-3959	160	24	∀	∀	PUNCT
cana-3959	161	1	𝑣	𝑣	ADP
cana-3959	161	2	∈	∈	PROPN
cana-3959	161	3	𝑋.	𝑋.	PROPN
cana-3959	161	4	3	3	NUM
cana-3959	161	5	definitions	definition	NOUN
cana-3959	161	6	of	of	ADP
cana-3959	161	7	fuzzy	fuzzy	ADJ
cana-3959	161	8	normed	norme	VERB
cana-3959	161	9	spaces	space	NOUN
cana-3959	161	10	definition	definition	NOUN
cana-3959	161	11	3.1	3.1	NUM
cana-3959	161	12	let	let	VERB
cana-3959	161	13	x	x	PRON
cana-3959	161	14	be	be	AUX
cana-3959	161	15	a	a	DET
cana-3959	161	16	real	real	ADJ
cana-3959	161	17	linear	linear	ADJ
cana-3959	161	18	space	space	NOUN
cana-3959	161	19	.	.	PUNCT
cana-3959	162	1	a	a	DET
cana-3959	162	2	function	function	NOUN
cana-3959	162	3	n	n	CCONJ
cana-3959	162	4	:x	:x	PROPN
cana-3959	162	5	×	×	PROPN
cana-3959	162	6	ℝ	ℝ	PROPN
cana-3959	162	7	→	→	SYM
cana-3959	162	8	[	[	X
cana-3959	162	9	0,1](the	0,1](the	DET
cana-3959	162	10	so	so	ADV
cana-3959	162	11	-	-	PUNCT
cana-3959	162	12	called	call	VERB
cana-3959	162	13	fuzzy	fuzzy	ADJ
cana-3959	162	14	subset	subset	NOUN
cana-3959	162	15	)	)	PUNCT
cana-3959	162	16	is	be	AUX
cana-3959	162	17	said	say	VERB
cana-3959	162	18	to	to	PART
cana-3959	162	19	be	be	AUX
cana-3959	162	20	a	a	DET
cana-3959	162	21	fuzzy	fuzzy	ADJ
cana-3959	162	22	norm	norm	NOUN
cana-3959	162	23	on	on	ADP
cana-3959	162	24	x	x	SYM
cana-3959	162	25	if	if	SCONJ
cana-3959	162	26	for	for	ADP
cana-3959	162	27	all	all	DET
cana-3959	162	28	v	v	NOUN
cana-3959	162	29	,	,	PUNCT
cana-3959	162	30	w	w	PROPN
cana-3959	162	31	∈	∈	PROPN
cana-3959	162	32	x	x	X
cana-3959	162	33	and	and	CCONJ
cana-3959	162	34	all	all	PRON
cana-3959	162	35	s	s	PROPN
cana-3959	162	36	,	,	PUNCT
cana-3959	162	37	t	t	PROPN
cana-3959	162	38	∈	∈	PROPN
cana-3959	162	39	ℝ	ℝ	PROPN
cana-3959	162	40	,	,	PUNCT
cana-3959	162	41	(	(	PUNCT
cana-3959	162	42	𝐹1	𝐹1	NOUN
cana-3959	162	43	)	)	PUNCT
cana-3959	162	44	𝒩(𝑣	𝒩(𝑣	PROPN
cana-3959	162	45	,	,	PUNCT
cana-3959	162	46	𝑐	𝑐	NOUN
cana-3959	162	47	)	)	PUNCT
cana-3959	162	48	=	=	SYM
cana-3959	162	49	0	0	NUM
cana-3959	162	50	for	for	ADP
cana-3959	162	51	𝑐	𝑐	NOUN
cana-3959	162	52	≤	≤	NOUN
cana-3959	162	53	0	0	NUM
cana-3959	162	54	;	;	PUNCT
cana-3959	162	55	(	(	PUNCT
cana-3959	162	56	𝐹2	𝐹2	NOUN
cana-3959	162	57	)	)	PUNCT
cana-3959	162	58	𝑣	𝑣	NOUN
cana-3959	163	1	=	=	SYM
cana-3959	163	2	0	0	PUNCT
cana-3959	164	1	if	if	SCONJ
cana-3959	164	2	and	and	CCONJ
cana-3959	164	3	only	only	ADV
cana-3959	164	4	if	if	SCONJ
cana-3959	164	5	𝒩(𝑣	𝒩(𝑣	VERB
cana-3959	164	6	,	,	PUNCT
cana-3959	164	7	𝑐	𝑐	NOUN
cana-3959	164	8	)	)	PUNCT
cana-3959	164	9	=	=	SYM
cana-3959	164	10	1	1	NUM
cana-3959	164	11	for	for	ADP
cana-3959	164	12	all	all	DET
cana-3959	164	13	𝑐	𝑐	PROPN
cana-3959	164	14	>	>	X
cana-3959	164	15	0	0	NUM
cana-3959	164	16	;	;	PUNCT
cana-3959	164	17	(	(	PUNCT
cana-3959	164	18	𝐹3	𝐹3	PROPN
cana-3959	164	19	)	)	PUNCT
cana-3959	164	20	𝒩(𝑐𝑣	𝒩(𝑐𝑣	X
cana-3959	164	21	,	,	PUNCT
cana-3959	164	22	𝑡	𝑡	NOUN
cana-3959	164	23	)	)	PUNCT
cana-3959	164	24	=	=	SYM
cana-3959	164	25	𝒩	𝒩	PROPN
cana-3959	164	26	(	(	PUNCT
cana-3959	164	27	𝑣	𝑣	NOUN
cana-3959	164	28	,	,	PUNCT
cana-3959	164	29	𝑡	𝑡	X
cana-3959	164	30	|𝑐|	|𝑐|	ADV
cana-3959	164	31	)	)	PUNCT
cana-3959	164	32	if	if	SCONJ
cana-3959	164	33	𝑐	𝑐	PROPN
cana-3959	164	34	≠	≠	PROPN
cana-3959	164	35	0	0	NUM
cana-3959	164	36	;	;	PUNCT
cana-3959	164	37	(	(	PUNCT
cana-3959	164	38	𝐹4	𝐹4	NOUN
cana-3959	164	39	)	)	PUNCT
cana-3959	164	40	𝒩(𝑣	𝒩(𝑣	PUNCT
cana-3959	164	41	+	+	CCONJ
cana-3959	164	42	𝑤	𝑤	X
cana-3959	164	43	,	,	PUNCT
cana-3959	164	44	𝑠	𝑠	PROPN
cana-3959	164	45	+	+	CCONJ
cana-3959	164	46	𝑡	𝑡	PROPN
cana-3959	164	47	)	)	PUNCT
cana-3959	164	48	≥	≥	NOUN
cana-3959	164	49	𝑚𝑖𝑛{𝒩(𝑣	𝑚𝑖𝑛{𝒩(𝑣	PROPN
cana-3959	164	50	,	,	PUNCT
cana-3959	164	51	𝑠),𝒩(𝑤	𝑠),𝒩(𝑤	NOUN
cana-3959	164	52	,	,	PUNCT
cana-3959	164	53	𝑡	𝑡	PROPN
cana-3959	164	54	)	)	PUNCT
cana-3959	164	55	}	}	PUNCT
cana-3959	164	56	;	;	PUNCT
cana-3959	164	57	(	(	PUNCT
cana-3959	164	58	𝐹5	𝐹5	NOUN
cana-3959	164	59	)	)	PUNCT
cana-3959	164	60	𝒩(𝑣,⋅	𝒩(𝑣,⋅	NOUN
cana-3959	164	61	)	)	PUNCT
cana-3959	164	62	is	be	AUX
cana-3959	164	63	a	a	DET
cana-3959	164	64	non	non	ADJ
cana-3959	164	65	-	-	ADJ
cana-3959	164	66	decreasing	decrease	VERB
cana-3959	164	67	function	function	NOUN
cana-3959	164	68	on	on	ADP
cana-3959	164	69	ℝ	ℝ	PROPN
cana-3959	164	70	and	and	CCONJ
cana-3959	164	71	𝑙𝑖𝑚𝑡→∞𝒩(𝑣	𝑙𝑖𝑚𝑡→∞𝒩(𝑣	PROPN
cana-3959	164	72	,	,	PUNCT
cana-3959	164	73	𝑡	𝑡	NOUN
cana-3959	164	74	)	)	PUNCT
cana-3959	164	75	=	=	SYM
cana-3959	164	76	1	1	NUM
cana-3959	164	77	;	;	PUNCT
cana-3959	164	78	(	(	PUNCT
cana-3959	164	79	𝐹6	𝐹6	NOUN
cana-3959	164	80	)	)	PUNCT
cana-3959	164	81	for	for	ADP
cana-3959	164	82	𝑣	𝑣	DET
cana-3959	164	83	≠	≠	PROPN
cana-3959	164	84	0,𝒩(𝑣,⋅	0,𝒩(𝑣,⋅	NUM
cana-3959	164	85	)	)	PUNCT
cana-3959	164	86	is	be	AUX
cana-3959	164	87	(	(	PUNCT
cana-3959	164	88	upper	upper	ADJ
cana-3959	164	89	semi	semi	NOUN
cana-3959	164	90	)	)	PUNCT
cana-3959	164	91	continuous	continuous	ADJ
cana-3959	164	92	on	on	ADP
cana-3959	164	93	ℝ.	ℝ.	PROPN
cana-3959	164	94	the	the	DET
cana-3959	164	95	pair	pair	NOUN
cana-3959	164	96	(	(	PUNCT
cana-3959	164	97	𝑋,𝑁	𝑋,𝑁	NOUN
cana-3959	164	98	)	)	PUNCT
cana-3959	164	99	is	be	AUX
cana-3959	164	100	called	call	VERB
cana-3959	164	101	a	a	DET
cana-3959	164	102	fuzzy	fuzzy	ADJ
cana-3959	164	103	normed	norme	VERB
cana-3959	164	104	linear	linear	ADJ
cana-3959	164	105	space	space	NOUN
cana-3959	164	106	.	.	PUNCT
cana-3959	165	1	one	one	PRON
cana-3959	165	2	may	may	AUX
cana-3959	165	3	regard	regard	VERB
cana-3959	165	4	𝒩(𝑋	𝒩(𝑋	PROPN
cana-3959	165	5	,	,	PUNCT
cana-3959	165	6	𝑡	𝑡	PROPN
cana-3959	165	7	)	)	PUNCT
cana-3959	165	8	as	as	ADP
cana-3959	165	9	the	the	DET
cana-3959	165	10	truth	truth	NOUN
cana-3959	165	11	-	-	PUNCT
cana-3959	165	12	value	value	NOUN
cana-3959	165	13	of	of	ADP
cana-3959	165	14	the	the	DET
cana-3959	165	15	statement	statement	NOUN
cana-3959	165	16	the	the	DET
cana-3959	165	17	norm	norm	NOUN
cana-3959	165	18	of	of	ADP
cana-3959	165	19	𝑣	𝑣	PRON
cana-3959	165	20	is	be	AUX
cana-3959	165	21	less	less	ADJ
cana-3959	165	22	than	than	ADP
cana-3959	165	23	or	or	CCONJ
cana-3959	165	24	equal	equal	ADJ
cana-3959	165	25	to	to	ADP
cana-3959	165	26	the	the	DET
cana-3959	165	27	real	real	ADJ
cana-3959	165	28	number	number	NOUN
cana-3959	165	29	𝑡	𝑡	NOUN
cana-3959	165	30	’	'	PUNCT
cana-3959	165	31	.	.	PUNCT
cana-3959	165	32	example	example	NOUN
cana-3959	165	33	3.2	3.2	NUM
cana-3959	165	34	let	let	VERB
cana-3959	165	35	(	(	PUNCT
cana-3959	165	36	x	x	NOUN
cana-3959	165	37	,	,	PUNCT
cana-3959	165	38	||	||	PROPN
cana-3959	166	1	⋅	⋅	PROPN
cana-3959	166	2	||	||	NUM
cana-3959	166	3	)	)	PUNCT
cana-3959	166	4	be	be	AUX
cana-3959	166	5	a	a	DET
cana-3959	166	6	normed	normed	ADJ
cana-3959	166	7	linear	linear	ADJ
cana-3959	166	8	space	space	NOUN
cana-3959	166	9	.	.	PUNCT
cana-3959	167	1	then	then	ADV
cana-3959	167	2	𝒩(𝑣	𝒩(𝑣	VERB
cana-3959	167	3	,	,	PUNCT
cana-3959	167	4	𝑡	𝑡	PROPN
cana-3959	167	5	)	)	PUNCT
cana-3959	167	6	=	=	SYM
cana-3959	167	7	{	{	PUNCT
cana-3959	167	8	𝑡	𝑡	X
cana-3959	167	9	𝑡+‖𝑣‖	𝑡+‖𝑣‖	PROPN
cana-3959	167	10	,	,	PUNCT
cana-3959	167	11	𝑡	𝑡	X
cana-3959	167	12	>	>	X
cana-3959	167	13	0	0	NUM
cana-3959	167	14	,	,	PUNCT
cana-3959	167	15	𝑣	𝑣	DET
cana-3959	167	16	∈	∈	PROPN
cana-3959	167	17	𝑋	𝑋	PROPN
cana-3959	167	18	,	,	PUNCT
cana-3959	167	19	0	0	NUM
cana-3959	167	20	,	,	PUNCT
cana-3959	167	21	𝑡	𝑡	VERB
cana-3959	167	22	≤	≤	ADV
cana-3959	167	23	0	0	NUM
cana-3959	167	24	,	,	PUNCT
cana-3959	167	25	𝑣	𝑣	DET
cana-3959	167	26	∈	∈	NOUN
cana-3959	167	27	𝑋	𝑋	NOUN
cana-3959	167	28	is	be	AUX
cana-3959	167	29	a	a	DET
cana-3959	167	30	fuzzy	fuzzy	ADJ
cana-3959	167	31	norm	norm	NOUN
cana-3959	167	32	on	on	ADP
cana-3959	167	33	𝑋.	𝑋.	PROPN
cana-3959	167	34	4	4	NUM
cana-3959	167	35	direct	direct	ADJ
cana-3959	167	36	method	method	NOUN
cana-3959	167	37	of	of	ADP
cana-3959	167	38	fuzzy	fuzzy	ADJ
cana-3959	167	39	stability	stability	NOUN
cana-3959	167	40	result	result	NOUN
cana-3959	167	41	𝐷	𝐷	PROPN
cana-3959	167	42	ℱ(𝑣,𝑤	ℱ(𝑣,𝑤	PUNCT
cana-3959	167	43	)	)	PUNCT
cana-3959	168	1	=	=	PUNCT
cana-3959	169	1	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	169	2	+	+	CCONJ
cana-3959	169	3	𝑤	𝑤	X
cana-3959	169	4	)	)	PUNCT
cana-3959	169	5	±ℜ	±ℜ	PUNCT
cana-3959	170	1	𝑏	𝑏	NOUN
cana-3959	170	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	170	3	−ℜ	−ℜ	PROPN
cana-3959	170	4	𝑎𝑤	𝑎𝑤	X
cana-3959	170	5	)	)	PUNCT
cana-3959	170	6	=	=	SYM
cana-3959	170	7	(	(	PUNCT
cana-3959	170	8	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	170	9	)	)	PUNCT
cana-3959	170	10	2	2	NUM
cana-3959	170	11	)	)	PUNCT
cana-3959	171	1	[	[	X
cana-3959	171	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	171	3	+	+	CCONJ
cana-3959	171	4	𝑤	𝑤	X
cana-3959	171	5	)	)	PUNCT
cana-3959	171	6	+	+	CCONJ
cana-3959	171	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	171	8	−	−	NOUN
cana-3959	171	9	𝑤	𝑤	ADP
cana-3959	171	10	)	)	PUNCT
cana-3959	171	11	]	]	PUNCT
cana-3959	172	1	+	+	CCONJ
cana-3959	172	2	(	(	PUNCT
cana-3959	172	3	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	172	4	−	−	PROPN
cana-3959	172	5	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	172	6	∓ℜ	∓ℜ	PROPN
cana-3959	172	7	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	172	8	)	)	PUNCT
cana-3959	172	9	∓	∓	PROPN
cana-3959	172	10	(	(	PUNCT
cana-3959	172	11	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	172	12	±	±	NUM
cana-3959	172	13	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	172	14	)	)	PUNCT
cana-3959	172	15	]	]	PUNCT
cana-3959	172	16	theorem	theorem	VERB
cana-3959	172	17	4.1	4.1	NUM
cana-3959	172	18	assume	assume	VERB
cana-3959	172	19	that	that	SCONJ
cana-3959	172	20	x	x	SYM
cana-3959	172	21	linear	linear	ADJ
cana-3959	172	22	space	space	NOUN
cana-3959	172	23	,	,	PUNCT
cana-3959	172	24	(	(	PUNCT
cana-3959	172	25	z	z	NOUN
cana-3959	172	26	,	,	PUNCT
cana-3959	172	27	n′	n′	ADJ
cana-3959	172	28	)	)	PUNCT
cana-3959	172	29	fuzzy	fuzzy	ADJ
cana-3959	172	30	normed	normed	ADJ
cana-3959	172	31	space	space	NOUN
cana-3959	172	32	and	and	CCONJ
cana-3959	172	33	(	(	PUNCT
cana-3959	172	34	y	y	NOUN
cana-3959	172	35	,	,	PUNCT
cana-3959	172	36	n′)fuzzy	n′)fuzzy	X
cana-3959	172	37	banach	banach	NOUN
cana-3959	172	38	space	space	NOUN
cana-3959	172	39	.	.	PUNCT
cana-3959	173	1	let	let	VERB
cana-3959	173	2	β	β	X
cana-3959	173	3	∈	∈	PROPN
cana-3959	173	4	{	{	PUNCT
cana-3959	173	5	−1,1	−1,1	NOUN
cana-3959	173	6	}	}	PUNCT
cana-3959	173	7	be	be	AUX
cana-3959	173	8	fixed	fix	VERB
cana-3959	173	9	and	and	CCONJ
cana-3959	173	10	let	let	VERB
cana-3959	173	11	𝔔:x2	𝔔:x2	NOUN
cana-3959	173	12	→	→	PUNCT
cana-3959	173	13	z	z	NOUN
cana-3959	173	14	be	be	AUX
cana-3959	173	15	a	a	DET
cana-3959	173	16	mapping	mapping	NOUN
cana-3959	173	17	such	such	ADJ
cana-3959	173	18	that	that	PRON
cana-3959	173	19	for	for	ADP
cana-3959	173	20	some	some	DET
cana-3959	173	21	d	d	NOUN
cana-3959	173	22	with	with	ADP
cana-3959	173	23	0	0	NUM
cana-3959	173	24	<	<	X
cana-3959	173	25	(	(	PUNCT
cana-3959	173	26	d	d	PROPN
cana-3959	173	27	ℜ3a	ℜ3a	PROPN
cana-3959	173	28	)	)	PUNCT
cana-3959	173	29	β	β	X
cana-3959	173	30	<	<	X
cana-3959	173	31	1	1	NUM
cana-3959	173	32	𝑁′(𝔔(ℜ𝑎𝛽𝑣,ℜ𝑎𝛽𝑤	𝑁′(𝔔(ℜ𝑎𝛽𝑣,ℜ𝑎𝛽𝑤	PROPN
cana-3959	173	33	)	)	PUNCT
cana-3959	173	34	,	,	PUNCT
cana-3959	173	35	𝑟	𝑟	X
cana-3959	173	36	)	)	PUNCT
cana-3959	173	37	≥	≥	NOUN
cana-3959	173	38	𝑁′(𝑑𝑎𝛽𝔔(𝑣,𝑤	𝑁′(𝑑𝑎𝛽𝔔(𝑣,𝑤	NOUN
cana-3959	173	39	)	)	PUNCT
cana-3959	173	40	,	,	PUNCT
cana-3959	173	41	𝑟	𝑟	X
cana-3959	173	42	)	)	PUNCT
cana-3959	173	43	(	(	PUNCT
cana-3959	173	44	1	1	X
cana-3959	173	45	)	)	PUNCT
cana-3959	173	46	for	for	ADP
cana-3959	173	47	all	all	DET
cana-3959	173	48	𝑣	𝑣	DET
cana-3959	173	49	∈	∈	NOUN
cana-3959	173	50	𝑋	𝑋	NOUN
cana-3959	173	51	and	and	CCONJ
cana-3959	173	52	all	all	DET
cana-3959	173	53	𝑟	𝑟	NOUN
cana-3959	173	54	>	>	X
cana-3959	173	55	0	0	NUM
cana-3959	173	56	,	,	PUNCT
cana-3959	173	57	𝑑	𝑑	PROPN
cana-3959	173	58	>	>	X
cana-3959	173	59	0	0	NUM
cana-3959	173	60	,	,	PUNCT
cana-3959	173	61	and	and	CCONJ
cana-3959	173	62	communications	communication	NOUN
cana-3959	173	63	on	on	ADP
cana-3959	173	64	applied	apply	VERB
cana-3959	173	65	nonlinear	nonlinear	ADJ
cana-3959	173	66	analysis	analysis	NOUN
cana-3959	173	67	issn	issn	NOUN
cana-3959	173	68	:	:	PUNCT
cana-3959	173	69	1074	1074	NUM
cana-3959	173	70	-	-	PUNCT
cana-3959	173	71	133x	133x	NUM
cana-3959	173	72	vol	vol	NOUN
cana-3959	173	73	32	32	NUM
cana-3959	173	74	no	no	NOUN
cana-3959	173	75	.	.	PUNCT
cana-3959	174	1	9s	9s	NUM
cana-3959	174	2	(	(	PUNCT
cana-3959	174	3	2025	2025	NUM
cana-3959	174	4	)	)	PUNCT
cana-3959	174	5	483	483	NUM
cana-3959	174	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	174	7	lim	lim	PROPN
cana-3959	174	8	𝑘→∞	𝑘→∞	PUNCT
cana-3959	174	9	𝑁′(𝔔(ℜ𝑎𝛽𝑘𝑣,ℜ𝑎𝛽𝑘𝑤),ℜ𝑎𝛽3𝑘𝑟	𝑁′(𝔔(ℜ𝑎𝛽𝑘𝑣,ℜ𝑎𝛽𝑘𝑤),ℜ𝑎𝛽3𝑘𝑟	PROPN
cana-3959	174	10	)	)	PUNCT
cana-3959	174	11	=	=	SYM
cana-3959	174	12	1	1	NUM
cana-3959	174	13	(	(	PUNCT
cana-3959	174	14	2	2	NUM
cana-3959	174	15	)	)	PUNCT
cana-3959	174	16	for	for	ADP
cana-3959	174	17	all	all	DET
cana-3959	174	18	𝑣	𝑣	NOUN
cana-3959	174	19	,	,	PUNCT
cana-3959	174	20	𝑤	𝑤	ADP
cana-3959	174	21	∈	∈	PROPN
cana-3959	174	22	𝑋	𝑋	NOUN
cana-3959	174	23	and	and	CCONJ
cana-3959	174	24	all	all	PRON
cana-3959	174	25	𝑟	𝑟	NOUN
cana-3959	174	26	>	>	X
cana-3959	174	27	0	0	X
cana-3959	174	28	.	.	PUNCT
cana-3959	174	29	suppose	suppose	VERB
cana-3959	174	30	that	that	SCONJ
cana-3959	174	31	a	a	DET
cana-3959	174	32	function	function	NOUN
cana-3959	174	33	𝑓	𝑓	NOUN
cana-3959	174	34	:	:	PUNCT
cana-3959	174	35	𝑋	𝑋	PROPN
cana-3959	174	36	→	→	SYM
cana-3959	174	37	𝑌	𝑌	PROPN
cana-3959	174	38	satisfies	satisfy	VERB
cana-3959	174	39	the	the	DET
cana-3959	174	40	inequality	inequality	NOUN
cana-3959	174	41	𝒩(𝐷ℱ(𝑣	𝒩(𝐷ℱ(𝑣	NOUN
cana-3959	174	42	,	,	PUNCT
cana-3959	174	43	𝑤	𝑤	X
cana-3959	174	44	)	)	PUNCT
cana-3959	174	45	,	,	PUNCT
cana-3959	174	46	𝑟	𝑟	X
cana-3959	174	47	)	)	PUNCT
cana-3959	174	48	≥	≥	NOUN
cana-3959	174	49	𝑁′(𝔔(𝑣,𝑤	𝑁′(𝔔(𝑣,𝑤	NUM
cana-3959	174	50	)	)	PUNCT
cana-3959	174	51	,	,	PUNCT
cana-3959	174	52	𝑟	𝑟	X
cana-3959	174	53	)	)	PUNCT
cana-3959	174	54	(	(	PUNCT
cana-3959	174	55	3	3	X
cana-3959	174	56	)	)	PUNCT
cana-3959	174	57	for	for	ADP
cana-3959	174	58	all	all	DET
cana-3959	174	59	𝑟	𝑟	NOUN
cana-3959	174	60	>	>	PUNCT
cana-3959	174	61	0	0	PUNCT
cana-3959	174	62	and	and	CCONJ
cana-3959	174	63	all	all	DET
cana-3959	174	64	𝑣,𝑤	𝑣,𝑤	PROPN
cana-3959	174	65	∈	∈	PROPN
cana-3959	174	66	𝑋.	𝑋.	PROPN
cana-3959	174	67	then	then	ADV
cana-3959	174	68	the	the	DET
cana-3959	174	69	limit	limit	NOUN
cana-3959	174	70	𝒞(𝑣	𝒞(𝑣	X
cana-3959	174	71	)	)	PUNCT
cana-3959	174	72	=	=	SYM
cana-3959	174	73	𝑁	𝑁	PROPN
cana-3959	174	74	−	−	PROPN
cana-3959	174	75	lim	lim	PROPN
cana-3959	174	76	𝑘→∞	𝑘→∞	PUNCT
cana-3959	174	77	ℱ(ℜ𝑎𝛽𝑘𝑣	ℱ(ℜ𝑎𝛽𝑘𝑣	PROPN
cana-3959	174	78	)	)	PUNCT
cana-3959	174	79	ℜ𝑎𝛽3𝑘	ℜ𝑎𝛽3𝑘	X
cana-3959	174	80	(	(	PUNCT
cana-3959	174	81	4	4	X
cana-3959	174	82	)	)	PUNCT
cana-3959	174	83	exists	exist	VERB
cana-3959	174	84	for	for	ADP
cana-3959	174	85	all	all	DET
cana-3959	174	86	𝑣	𝑣	DET
cana-3959	174	87	∈	∈	NOUN
cana-3959	174	88	𝑋	𝑋	NOUN
cana-3959	174	89	and	and	CCONJ
cana-3959	174	90	the	the	DET
cana-3959	174	91	mapping	mapping	NOUN
cana-3959	174	92	𝐶	𝐶	PROPN
cana-3959	174	93	:	:	PUNCT
cana-3959	174	94	𝑋	𝑋	PROPN
cana-3959	174	95	→	→	SYM
cana-3959	174	96	𝑌	𝑌	PROPN
cana-3959	174	97	is	be	AUX
cana-3959	174	98	a	a	DET
cana-3959	174	99	unique	unique	ADJ
cana-3959	174	100	cubic	cubic	ADJ
cana-3959	174	101	mapping	mapping	NOUN
cana-3959	174	102	such	such	ADJ
cana-3959	174	103	that	that	DET
cana-3959	174	104	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	174	105	)	)	PUNCT
cana-3959	174	106	−	−	PROPN
cana-3959	174	107	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	174	108	)	)	PUNCT
cana-3959	174	109	,	,	PUNCT
cana-3959	174	110	𝑟	𝑟	X
cana-3959	174	111	)	)	PUNCT
cana-3959	174	112	≥	≥	NOUN
cana-3959	174	113	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	VERB
cana-3959	174	114	,	,	PUNCT
cana-3959	174	115	0	0	NUM
cana-3959	174	116	)	)	PUNCT
cana-3959	174	117	,	,	PUNCT
cana-3959	174	118	|ℜ3𝑎	|ℜ3𝑎	PROPN
cana-3959	174	119	−	−	PROPN
cana-3959	174	120	𝑑𝑎|𝑟	𝑑𝑎|𝑟	NOUN
cana-3959	174	121	)	)	PUNCT
cana-3959	174	122	(	(	PUNCT
cana-3959	174	123	5	5	X
cana-3959	174	124	)	)	PUNCT
cana-3959	174	125	for	for	ADP
cana-3959	174	126	all	all	DET
cana-3959	174	127	𝑣	𝑣	DET
cana-3959	174	128	∈	∈	NOUN
cana-3959	174	129	𝑋	𝑋	NOUN
cana-3959	174	130	and	and	CCONJ
cana-3959	174	131	all	all	DET
cana-3959	174	132	𝑟	𝑟	NOUN
cana-3959	174	133	>	>	X
cana-3959	174	134	0	0	X
cana-3959	174	135	.	.	PUNCT
cana-3959	175	1	proof	proof	NOUN
cana-3959	175	2	.	.	PUNCT
cana-3959	176	1	first	first	ADV
cana-3959	176	2	assume	assume	VERB
cana-3959	176	3	𝛽	𝛽	NOUN
cana-3959	176	4	=	=	NOUN
cana-3959	176	5	1	1	X
cana-3959	176	6	.	.	X
cana-3959	177	1	replacing	replace	VERB
cana-3959	177	2	(	(	PUNCT
cana-3959	177	3	𝑣	𝑣	NOUN
cana-3959	177	4	,	,	PUNCT
cana-3959	177	5	𝑤	𝑤	ADP
cana-3959	177	6	)	)	PUNCT
cana-3959	177	7	by	by	ADP
cana-3959	177	8	(	(	PUNCT
cana-3959	177	9	𝑣	𝑣	NOUN
cana-3959	177	10	,	,	PUNCT
cana-3959	177	11	0	0	NUM
cana-3959	177	12	)	)	PUNCT
cana-3959	177	13	in	in	ADP
cana-3959	177	14	(	(	PUNCT
cana-3959	177	15	3	3	NUM
cana-3959	177	16	)	)	PUNCT
cana-3959	177	17	,	,	PUNCT
cana-3959	177	18	we	we	PRON
cana-3959	177	19	get	get	VERB
cana-3959	177	20	𝒩(ℱ(ℜ𝑎𝑣	𝒩(ℱ(ℜ𝑎𝑣	VERB
cana-3959	177	21	)	)	PUNCT
cana-3959	178	1	−ℜ	−ℜ	PROPN
cana-3959	178	2	3𝑎	3𝑎	NUM
cana-3959	178	3	ℱ(𝑣	ℱ(𝑣	NOUN
cana-3959	178	4	)	)	PUNCT
cana-3959	178	5	,	,	PUNCT
cana-3959	178	6	𝑟	𝑟	X
cana-3959	178	7	)	)	PUNCT
cana-3959	178	8	≥	≥	NOUN
cana-3959	178	9	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	VERB
cana-3959	178	10	,	,	PUNCT
cana-3959	178	11	0	0	NUM
cana-3959	178	12	)	)	PUNCT
cana-3959	178	13	,	,	PUNCT
cana-3959	178	14	𝑟	𝑟	X
cana-3959	178	15	)	)	PUNCT
cana-3959	178	16	(	(	PUNCT
cana-3959	178	17	6	6	NUM
cana-3959	178	18	)	)	PUNCT
cana-3959	178	19	for	for	ADP
cana-3959	178	20	all	all	DET
cana-3959	178	21	𝑣	𝑣	DET
cana-3959	178	22	∈	∈	NOUN
cana-3959	178	23	𝑋	𝑋	NOUN
cana-3959	178	24	and	and	CCONJ
cana-3959	178	25	all	all	PRON
cana-3959	178	26	𝑟	𝑟	NOUN
cana-3959	178	27	>	>	X
cana-3959	178	28	0	0	X
cana-3959	178	29	.	.	PUNCT
cana-3959	179	1	replacing	replace	VERB
cana-3959	179	2	𝑣	𝑣	PRON
cana-3959	179	3	by	by	ADP
cana-3959	179	4	ℜ	ℜ	ADJ
cana-3959	179	5	𝑎𝑘𝑣	𝑎𝑘𝑣	X
cana-3959	179	6	in	in	ADP
cana-3959	179	7	(	(	PUNCT
cana-3959	179	8	6	6	NUM
cana-3959	179	9	)	)	PUNCT
cana-3959	179	10	,	,	PUNCT
cana-3959	179	11	we	we	PRON
cana-3959	179	12	obtain	obtain	VERB
cana-3959	179	13	𝒩	𝒩	PROPN
cana-3959	179	14	(	(	PUNCT
cana-3959	179	15	ℱ(ℜ𝑎(𝑘+1)𝑣	ℱ(ℜ𝑎(𝑘+1)𝑣	NOUN
cana-3959	179	16	)	)	PUNCT
cana-3959	179	17	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	179	18	−	−	PROPN
cana-3959	179	19	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	179	20	)	)	PUNCT
cana-3959	179	21	,	,	PUNCT
cana-3959	179	22	𝑟	𝑟	PRON
cana-3959	179	23	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	179	24	)	)	PUNCT
cana-3959	179	25	≥	≥	NOUN
cana-3959	179	26	𝑁′(𝔔(ℜ𝑎𝑘𝑣	𝑁′(𝔔(ℜ𝑎𝑘𝑣	PROPN
cana-3959	179	27	,	,	PUNCT
cana-3959	179	28	0	0	NUM
cana-3959	179	29	)	)	PUNCT
cana-3959	179	30	,	,	PUNCT
cana-3959	179	31	𝑟	𝑟	X
cana-3959	179	32	)	)	PUNCT
cana-3959	179	33	(	(	PUNCT
cana-3959	179	34	7	7	X
cana-3959	179	35	)	)	PUNCT
cana-3959	179	36	for	for	ADP
cana-3959	179	37	all	all	DET
cana-3959	179	38	𝑣	𝑣	DET
cana-3959	179	39	∈	∈	NOUN
cana-3959	179	40	𝑋	𝑋	NOUN
cana-3959	179	41	and	and	CCONJ
cana-3959	179	42	all	all	PRON
cana-3959	179	43	𝑟	𝑟	NOUN
cana-3959	179	44	>	>	X
cana-3959	179	45	0	0	X
cana-3959	179	46	.	.	PUNCT
cana-3959	180	1	using	use	VERB
cana-3959	180	2	(	(	PUNCT
cana-3959	180	3	1	1	NUM
cana-3959	180	4	)	)	PUNCT
cana-3959	180	5	,	,	PUNCT
cana-3959	180	6	(	(	PUNCT
cana-3959	180	7	𝐹3	𝐹3	PROPN
cana-3959	180	8	)	)	PUNCT
cana-3959	180	9	in	in	ADP
cana-3959	180	10	(	(	PUNCT
cana-3959	180	11	7	7	NUM
cana-3959	180	12	)	)	PUNCT
cana-3959	180	13	,	,	PUNCT
cana-3959	180	14	we	we	PRON
cana-3959	180	15	arrive	arrive	VERB
cana-3959	180	16	𝒩	𝒩	PROPN
cana-3959	180	17	(	(	PUNCT
cana-3959	180	18	ℱ(ℜ𝑎(𝑘+1)𝑣	ℱ(ℜ𝑎(𝑘+1)𝑣	NOUN
cana-3959	180	19	)	)	PUNCT
cana-3959	180	20	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	180	21	−	−	PROPN
cana-3959	180	22	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	180	23	)	)	PUNCT
cana-3959	180	24	,	,	PUNCT
cana-3959	180	25	𝑟	𝑟	PRON
cana-3959	180	26	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	180	27	)	)	PUNCT
cana-3959	180	28	≥	≥	PROPN
cana-3959	180	29	𝑁′	𝑁′	X
cana-3959	180	30	(	(	PUNCT
cana-3959	180	31	𝔔(𝑣	𝔔(𝑣	PROPN
cana-3959	180	32	,	,	PUNCT
cana-3959	180	33	0	0	NUM
cana-3959	180	34	)	)	PUNCT
cana-3959	180	35	,	,	PUNCT
cana-3959	180	36	𝑟	𝑟	X
cana-3959	180	37	𝑑𝑎𝑘	𝑑𝑎𝑘	NOUN
cana-3959	180	38	)	)	PUNCT
cana-3959	180	39	(	(	PUNCT
cana-3959	180	40	8)	8)	NUM
cana-3959	180	41	for	for	ADP
cana-3959	180	42	all	all	DET
cana-3959	180	43	𝑣	𝑣	DET
cana-3959	180	44	∈	∈	NOUN
cana-3959	180	45	𝑋	𝑋	NOUN
cana-3959	180	46	and	and	CCONJ
cana-3959	180	47	all	all	PRON
cana-3959	180	48	𝑟	𝑟	NOUN
cana-3959	180	49	>	>	X
cana-3959	180	50	0	0	X
cana-3959	180	51	.	.	PUNCT
cana-3959	181	1	it	it	PRON
cana-3959	181	2	is	be	AUX
cana-3959	181	3	easy	easy	ADJ
cana-3959	181	4	to	to	PART
cana-3959	181	5	verify	verify	VERB
cana-3959	181	6	from	from	ADP
cana-3959	181	7	(	(	PUNCT
cana-3959	181	8	8)	8)	NUM
cana-3959	181	9	,	,	PUNCT
cana-3959	181	10	that	that	SCONJ
cana-3959	181	11	𝒩	𝒩	PROPN
cana-3959	181	12	(	(	PUNCT
cana-3959	181	13	ℱ(ℜ𝑎(𝑘+1)𝑣	ℱ(ℜ𝑎(𝑘+1)𝑣	NOUN
cana-3959	181	14	)	)	PUNCT
cana-3959	181	15	ℜ3𝑎(𝑘+1	ℜ3𝑎(𝑘+1	NOUN
cana-3959	181	16	)	)	PUNCT
cana-3959	181	17	−	−	NUM
cana-3959	181	18	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	181	19	)	)	PUNCT
cana-3959	182	1	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	182	2	,	,	PUNCT
cana-3959	182	3	𝑟	𝑟	X
cana-3959	182	4	ℜ3𝑎⋅ℜ3𝑎𝑘	ℜ3𝑎⋅ℜ3𝑎𝑘	PROPN
cana-3959	182	5	)	)	PUNCT
cana-3959	182	6	≥	≥	NOUN
cana-3959	182	7	𝑁′	𝑁′	X
cana-3959	182	8	(	(	PUNCT
cana-3959	182	9	𝔔(𝑣	𝔔(𝑣	PROPN
cana-3959	182	10	,	,	PUNCT
cana-3959	182	11	0	0	NUM
cana-3959	182	12	)	)	PUNCT
cana-3959	182	13	,	,	PUNCT
cana-3959	182	14	𝑟	𝑟	X
cana-3959	182	15	𝑑𝑎𝑘	𝑑𝑎𝑘	NOUN
cana-3959	182	16	)	)	PUNCT
cana-3959	182	17	(	(	PUNCT
cana-3959	182	18	9	9	X
cana-3959	182	19	)	)	PUNCT
cana-3959	182	20	holds	hold	VERB
cana-3959	182	21	for	for	ADP
cana-3959	182	22	all	all	DET
cana-3959	182	23	𝑣	𝑣	DET
cana-3959	182	24	∈	∈	NOUN
cana-3959	182	25	𝑋	𝑋	NOUN
cana-3959	182	26	and	and	CCONJ
cana-3959	182	27	all	all	DET
cana-3959	182	28	𝑟	𝑟	NOUN
cana-3959	182	29	>	>	X
cana-3959	182	30	0	0	X
cana-3959	182	31	.	.	PUNCT
cana-3959	182	32	replacing	replace	VERB
cana-3959	182	33	𝑟	𝑟	NOUN
cana-3959	182	34	by	by	ADP
cana-3959	182	35	ℜ	ℜ	ADJ
cana-3959	182	36	𝑎𝑘𝑟	𝑎𝑘𝑟	NOUN
cana-3959	182	37	in	in	ADP
cana-3959	182	38	(	(	PUNCT
cana-3959	182	39	9	9	NUM
cana-3959	182	40	)	)	PUNCT
cana-3959	182	41	,	,	PUNCT
cana-3959	182	42	we	we	PRON
cana-3959	182	43	get	get	VERB
cana-3959	182	44	𝒩	𝒩	PROPN
cana-3959	182	45	(	(	PUNCT
cana-3959	182	46	ℱ(ℜ𝑎(𝑘+1)𝑣	ℱ(ℜ𝑎(𝑘+1)𝑣	NOUN
cana-3959	182	47	)	)	PUNCT
cana-3959	182	48	ℜ3𝑎(𝑘+1	ℜ3𝑎(𝑘+1	NOUN
cana-3959	182	49	)	)	PUNCT
cana-3959	182	50	−	−	NUM
cana-3959	183	1	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	183	2	)	)	PUNCT
cana-3959	184	1	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	184	2	,	,	PUNCT
cana-3959	184	3	𝑑𝑎𝑘	𝑑𝑎𝑘	NOUN
cana-3959	184	4	𝑟	𝑟	PRON
cana-3959	184	5	ℜ3𝑎⋅ℜ3𝑎𝑘	ℜ3𝑎⋅ℜ3𝑎𝑘	PROPN
cana-3959	184	6	)	)	PUNCT
cana-3959	184	7	≥	≥	NOUN
cana-3959	184	8	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	VERB
cana-3959	184	9	,	,	PUNCT
cana-3959	184	10	0	0	NUM
cana-3959	184	11	)	)	PUNCT
cana-3959	184	12	,	,	PUNCT
cana-3959	184	13	𝑟	𝑟	X
cana-3959	184	14	)	)	PUNCT
cana-3959	184	15	(	(	PUNCT
cana-3959	184	16	10	10	NUM
cana-3959	184	17	)	)	PUNCT
cana-3959	184	18	for	for	ADP
cana-3959	184	19	all	all	DET
cana-3959	184	20	𝑣	𝑣	DET
cana-3959	184	21	∈	∈	NOUN
cana-3959	184	22	𝑋	𝑋	NOUN
cana-3959	184	23	and	and	CCONJ
cana-3959	184	24	all	all	PRON
cana-3959	184	25	𝑟	𝑟	NOUN
cana-3959	184	26	>	>	X
cana-3959	184	27	0	0	X
cana-3959	184	28	.	.	PUNCT
cana-3959	185	1	it	it	PRON
cana-3959	185	2	is	be	AUX
cana-3959	185	3	easy	easy	ADJ
cana-3959	185	4	to	to	PART
cana-3959	185	5	see	see	VERB
cana-3959	185	6	that	that	PRON
cana-3959	185	7	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	PUNCT
cana-3959	185	8	)	)	PUNCT
cana-3959	186	1	ℜ3𝑎𝑘	ℜ3𝑎𝑘	PRON
cana-3959	186	2	−	−	NOUN
cana-3959	186	3	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	186	4	)	)	PUNCT
cana-3959	186	5	=	=	PUNCT
cana-3959	186	6	∑𝑘−1	∑𝑘−1	X
cana-3959	186	7	𝑖=0	𝑖=0	PROPN
cana-3959	186	8	[	[	PUNCT
cana-3959	186	9	ℱ(ℜ𝑎(𝑖+1)𝑥	ℱ(ℜ𝑎(𝑖+1)𝑥	NOUN
cana-3959	186	10	)	)	PUNCT
cana-3959	186	11	ℜ3𝑎(𝑖+1	ℜ3𝑎(𝑖+1	PROPN
cana-3959	186	12	)	)	PUNCT
cana-3959	186	13	−	−	NOUN
cana-3959	186	14	ℱ(ℜ𝑎𝑖𝑣	ℱ(ℜ𝑎𝑖𝑣	NUM
cana-3959	186	15	)	)	PUNCT
cana-3959	186	16	ℜ3𝑎𝑖	ℜ3𝑎𝑖	NOUN
cana-3959	186	17	]	]	PUNCT
cana-3959	187	1	(	(	PUNCT
cana-3959	187	2	11	11	NUM
cana-3959	187	3	)	)	PUNCT
cana-3959	187	4	for	for	ADP
cana-3959	187	5	all	all	PRON
cana-3959	187	6	𝑣	𝑣	DET
cana-3959	187	7	∈	∈	PROPN
cana-3959	187	8	𝑋.	𝑋.	PROPN
cana-3959	187	9	from	from	ADP
cana-3959	187	10	equations	equation	NOUN
cana-3959	187	11	(	(	PUNCT
cana-3959	187	12	10	10	NUM
cana-3959	187	13	)	)	PUNCT
cana-3959	187	14	and	and	CCONJ
cana-3959	187	15	(	(	PUNCT
cana-3959	187	16	11	11	NUM
cana-3959	187	17	)	)	PUNCT
cana-3959	187	18	,	,	PUNCT
cana-3959	187	19	we	we	PRON
cana-3959	187	20	have	have	VERB
cana-3959	187	21	𝒩	𝒩	PROPN
cana-3959	187	22	(	(	PUNCT
cana-3959	187	23	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	187	24	)	)	PUNCT
cana-3959	188	1	ℜ3𝑎𝑘	ℜ3𝑎𝑘	PRON
cana-3959	188	2	−	−	PROPN
cana-3959	188	3	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	188	4	)	)	PUNCT
cana-3959	188	5	,	,	PUNCT
cana-3959	188	6	∑𝑘−1	∑𝑘−1	PROPN
cana-3959	188	7	𝑖=0	𝑖=0	PROPN
cana-3959	188	8	𝑑𝑖	𝑑𝑖	PART
cana-3959	188	9	𝑟	𝑟	DET
cana-3959	188	10	ℜ3𝑎⋅ℜ3𝑎𝑖	ℜ3𝑎⋅ℜ3𝑎𝑖	PROPN
cana-3959	188	11	)	)	PUNCT
cana-3959	188	12	≥	≥	NOUN
cana-3959	188	13	𝑚𝑖𝑛⋃𝑘−1	𝑚𝑖𝑛⋃𝑘−1	X
cana-3959	188	14	𝑖=0	𝑖=0	PROPN
cana-3959	188	15	{	{	PUNCT
cana-3959	188	16	ℱ(ℜ𝑎(𝑖+1)𝑣	ℱ(ℜ𝑎(𝑖+1)𝑣	NOUN
cana-3959	188	17	)	)	PUNCT
cana-3959	188	18	ℜ3𝑎(𝑖+1	ℜ3𝑎(𝑖+1	PROPN
cana-3959	188	19	)	)	PUNCT
cana-3959	188	20	−	−	ADP
cana-3959	188	21	ℱ(ℜ𝑖𝑎𝑣	ℱ(ℜ𝑖𝑎𝑣	NUM
cana-3959	188	22	)	)	PUNCT
cana-3959	188	23	ℜ3𝑎𝑖	ℜ3𝑎𝑖	NOUN
cana-3959	188	24	,	,	PUNCT
cana-3959	188	25	𝑑𝑖	𝑑𝑖	VERB
cana-3959	188	26	𝑟	𝑟	PRON
cana-3959	188	27	ℜ3𝑎⋅ℜ3𝑎𝑖	ℜ3𝑎⋅ℜ3𝑎𝑖	PROPN
cana-3959	188	28	}	}	PUNCT
cana-3959	188	29	≥	≥	NOUN
cana-3959	188	30	𝑚𝑖𝑛⋃𝑘−1	𝑚𝑖𝑛⋃𝑘−1	X
cana-3959	188	31	𝑖=0	𝑖=0	SYM
cana-3959	188	32	{	{	PUNCT
cana-3959	188	33	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	NOUN
cana-3959	188	34	,	,	PUNCT
cana-3959	188	35	0	0	NUM
cana-3959	188	36	)	)	PUNCT
cana-3959	188	37	,	,	PUNCT
cana-3959	188	38	𝑟	𝑟	X
cana-3959	188	39	)	)	PUNCT
cana-3959	188	40	}	}	PUNCT
cana-3959	188	41	≥	≥	NOUN
cana-3959	188	42	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	VERB
cana-3959	188	43	,	,	PUNCT
cana-3959	188	44	0	0	NUM
cana-3959	188	45	)	)	PUNCT
cana-3959	188	46	,	,	PUNCT
cana-3959	188	47	𝑟	𝑟	X
cana-3959	188	48	)	)	PUNCT
cana-3959	188	49	(	(	PUNCT
cana-3959	188	50	12	12	NUM
cana-3959	188	51	)	)	PUNCT
cana-3959	188	52	communications	communication	NOUN
cana-3959	188	53	on	on	ADP
cana-3959	188	54	applied	apply	VERB
cana-3959	188	55	nonlinear	nonlinear	ADJ
cana-3959	188	56	analysis	analysis	NOUN
cana-3959	188	57	issn	issn	NOUN
cana-3959	188	58	:	:	PUNCT
cana-3959	188	59	1074	1074	NUM
cana-3959	188	60	-	-	PUNCT
cana-3959	188	61	133x	133x	NUM
cana-3959	188	62	vol	vol	NOUN
cana-3959	188	63	32	32	NUM
cana-3959	188	64	no	no	NOUN
cana-3959	188	65	.	.	PUNCT
cana-3959	189	1	9s	9s	NUM
cana-3959	189	2	(	(	PUNCT
cana-3959	189	3	2025	2025	NUM
cana-3959	189	4	)	)	PUNCT
cana-3959	189	5	484	484	NUM
cana-3959	189	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	189	7	for	for	ADP
cana-3959	189	8	all	all	DET
cana-3959	189	9	𝑣	𝑣	DET
cana-3959	189	10	∈	∈	NOUN
cana-3959	189	11	𝑋	𝑋	NOUN
cana-3959	189	12	and	and	CCONJ
cana-3959	189	13	all	all	PRON
cana-3959	189	14	𝑟	𝑟	NOUN
cana-3959	189	15	>	>	X
cana-3959	189	16	0	0	X
cana-3959	189	17	.	.	PUNCT
cana-3959	190	1	replacing	replace	VERB
cana-3959	190	2	𝑣	𝑣	PRON
cana-3959	190	3	by	by	ADP
cana-3959	190	4	ℜ	ℜ	PROPN
cana-3959	190	5	𝑚𝑎𝑣	𝑚𝑎𝑣	NOUN
cana-3959	190	6	in	in	ADP
cana-3959	190	7	(	(	PUNCT
cana-3959	190	8	12	12	NUM
cana-3959	190	9	)	)	PUNCT
cana-3959	190	10	and	and	CCONJ
cana-3959	190	11	using	use	VERB
cana-3959	190	12	(	(	PUNCT
cana-3959	190	13	1	1	NUM
cana-3959	190	14	)	)	PUNCT
cana-3959	190	15	,	,	PUNCT
cana-3959	190	16	(	(	PUNCT
cana-3959	190	17	𝐹3	𝐹3	PROPN
cana-3959	190	18	)	)	PUNCT
cana-3959	190	19	,	,	PUNCT
cana-3959	190	20	we	we	PRON
cana-3959	190	21	obtain	obtain	VERB
cana-3959	190	22	𝒩	𝒩	PROPN
cana-3959	190	23	(	(	PUNCT
cana-3959	190	24	ℱ(ℜ𝑎(𝑘+𝑚)𝑣	ℱ(ℜ𝑎(𝑘+𝑚)𝑣	NOUN
cana-3959	190	25	)	)	PUNCT
cana-3959	190	26	ℜ3𝑎(𝑘+𝑚	ℜ3𝑎(𝑘+𝑚	PROPN
cana-3959	190	27	)	)	PUNCT
cana-3959	190	28	−	−	NOUN
cana-3959	190	29	ℱ(ℜ𝑚𝑎𝑣	ℱ(ℜ𝑚𝑎𝑣	NOUN
cana-3959	190	30	)	)	PUNCT
cana-3959	191	1	ℜ3𝑎𝑚	ℜ3𝑎𝑚	NOUN
cana-3959	191	2	,	,	PUNCT
cana-3959	191	3	∑𝑘−1	∑𝑘−1	PROPN
cana-3959	191	4	𝑖=0	𝑖=0	PROPN
cana-3959	191	5	𝑑𝑖	𝑑𝑖	PART
cana-3959	191	6	𝑟	𝑟	DET
cana-3959	191	7	ℜ3𝑎⋅ℜ3𝑎(𝑖+𝑚	ℜ3𝑎⋅ℜ3𝑎(𝑖+𝑚	PROPN
cana-3959	191	8	)	)	PUNCT
cana-3959	191	9	)	)	PUNCT
cana-3959	191	10	≥	≥	PROPN
cana-3959	191	11	𝑁′	𝑁′	X
cana-3959	191	12	(	(	PUNCT
cana-3959	191	13	𝔔(𝑣	𝔔(𝑣	PROPN
cana-3959	191	14	,	,	PUNCT
cana-3959	191	15	0	0	NUM
cana-3959	191	16	)	)	PUNCT
cana-3959	191	17	,	,	PUNCT
cana-3959	191	18	𝑟	𝑟	X
cana-3959	191	19	𝑑𝑎𝑚	𝑑𝑎𝑚	NOUN
cana-3959	191	20	)	)	PUNCT
cana-3959	191	21	(	(	PUNCT
cana-3959	191	22	13	13	NUM
cana-3959	191	23	)	)	PUNCT
cana-3959	191	24	for	for	ADP
cana-3959	191	25	all	all	DET
cana-3959	191	26	𝑣	𝑣	DET
cana-3959	191	27	∈	∈	NOUN
cana-3959	191	28	𝑋	𝑋	NOUN
cana-3959	191	29	and	and	CCONJ
cana-3959	191	30	all	all	DET
cana-3959	191	31	𝑟	𝑟	NOUN
cana-3959	191	32	>	>	PUNCT
cana-3959	191	33	0	0	PUNCT
cana-3959	191	34	and	and	CCONJ
cana-3959	191	35	all	all	DET
cana-3959	191	36	𝑚	𝑚	NOUN
cana-3959	191	37	,	,	PUNCT
cana-3959	191	38	𝑘	𝑘	DET
cana-3959	191	39	≥	≥	NOUN
cana-3959	191	40	0	0	NUM
cana-3959	191	41	.	.	PUNCT
cana-3959	192	1	replacing	replace	VERB
cana-3959	192	2	𝑟	𝑟	NOUN
cana-3959	192	3	by	by	ADP
cana-3959	192	4	𝑑𝑎𝑚𝑟	𝑑𝑎𝑚𝑟	NOUN
cana-3959	192	5	in	in	ADP
cana-3959	192	6	(	(	PUNCT
cana-3959	192	7	13	13	NUM
cana-3959	192	8	)	)	PUNCT
cana-3959	192	9	,	,	PUNCT
cana-3959	192	10	we	we	PRON
cana-3959	192	11	get	get	VERB
cana-3959	192	12	𝒩	𝒩	PROPN
cana-3959	192	13	(	(	PUNCT
cana-3959	192	14	ℱ(ℜ𝑎(𝑘+𝑚)𝑣	ℱ(ℜ𝑎(𝑘+𝑚)𝑣	ADJ
cana-3959	192	15	)	)	PUNCT
cana-3959	192	16	ℜ3𝑎(𝑘+𝑚	ℜ3𝑎(𝑘+𝑚	PROPN
cana-3959	192	17	)	)	PUNCT
cana-3959	192	18	−	−	PROPN
cana-3959	192	19	ℱ(ℜ𝑎𝑚𝑣	ℱ(ℜ𝑎𝑚𝑣	NUM
cana-3959	192	20	)	)	PUNCT
cana-3959	193	1	ℜ3𝑎𝑚	ℜ3𝑎𝑚	PROPN
cana-3959	193	2	,	,	PUNCT
cana-3959	193	3	∑𝑚+𝑘−1	∑𝑚+𝑘−1	PROPN
cana-3959	193	4	𝑖=𝑚	𝑖=𝑚	PROPN
cana-3959	193	5	𝑑𝑎𝑖	𝑑𝑎𝑖	VERB
cana-3959	193	6	𝑟	𝑟	DET
cana-3959	193	7	ℜ3𝑎⋅ℜ3𝑎𝑖	ℜ3𝑎⋅ℜ3𝑎𝑖	PROPN
cana-3959	193	8	)	)	PUNCT
cana-3959	193	9	≥	≥	NOUN
cana-3959	193	10	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	VERB
cana-3959	193	11	,	,	PUNCT
cana-3959	193	12	0	0	NUM
cana-3959	193	13	)	)	PUNCT
cana-3959	193	14	,	,	PUNCT
cana-3959	193	15	𝑟	𝑟	X
cana-3959	193	16	)	)	PUNCT
cana-3959	193	17	(	(	PUNCT
cana-3959	193	18	14	14	NUM
cana-3959	193	19	)	)	PUNCT
cana-3959	193	20	for	for	ADP
cana-3959	193	21	all	all	DET
cana-3959	193	22	𝑣	𝑣	DET
cana-3959	193	23	∈	∈	NOUN
cana-3959	193	24	𝑋	𝑋	NOUN
cana-3959	193	25	and	and	CCONJ
cana-3959	193	26	all	all	DET
cana-3959	193	27	𝑟	𝑟	NOUN
cana-3959	193	28	>	>	PUNCT
cana-3959	193	29	0	0	PUNCT
cana-3959	193	30	and	and	CCONJ
cana-3959	193	31	all	all	DET
cana-3959	193	32	𝑚	𝑚	NOUN
cana-3959	193	33	,	,	PUNCT
cana-3959	193	34	𝑘	𝑘	DET
cana-3959	193	35	≥	≥	NOUN
cana-3959	193	36	0	0	NUM
cana-3959	193	37	.	.	PUNCT
cana-3959	194	1	using	use	VERB
cana-3959	194	2	(	(	PUNCT
cana-3959	194	3	𝐹3	𝐹3	PROPN
cana-3959	194	4	)	)	PUNCT
cana-3959	194	5	in	in	ADP
cana-3959	194	6	(	(	PUNCT
cana-3959	194	7	14	14	NUM
cana-3959	194	8	)	)	PUNCT
cana-3959	194	9	,	,	PUNCT
cana-3959	194	10	we	we	PRON
cana-3959	194	11	obtain	obtain	VERB
cana-3959	194	12	𝒩	𝒩	PROPN
cana-3959	194	13	(	(	PUNCT
cana-3959	194	14	ℱ(ℜ𝑎(𝑘+𝑚)𝑣	ℱ(ℜ𝑎(𝑘+𝑚)𝑣	NOUN
cana-3959	194	15	)	)	PUNCT
cana-3959	194	16	ℜ3𝑎(𝑘+𝑚	ℜ3𝑎(𝑘+𝑚	PROPN
cana-3959	194	17	)	)	PUNCT
cana-3959	194	18	−	−	PROPN
cana-3959	194	19	ℱ(ℜ𝑎𝑚𝑣	ℱ(ℜ𝑎𝑚𝑣	NUM
cana-3959	194	20	)	)	PUNCT
cana-3959	195	1	ℜ3𝑎𝑚	ℜ3𝑎𝑚	NOUN
cana-3959	195	2	,	,	PUNCT
cana-3959	195	3	𝑟	𝑟	X
cana-3959	195	4	)	)	PUNCT
cana-3959	195	5	≥	≥	NOUN
cana-3959	195	6	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	VERB
cana-3959	195	7	,	,	PUNCT
cana-3959	195	8	0	0	NUM
cana-3959	195	9	)	)	PUNCT
cana-3959	195	10	,	,	PUNCT
cana-3959	195	11	𝑟	𝑟	X
cana-3959	195	12	∑𝑚+𝑘−1	∑𝑚+𝑘−1	SYM
cana-3959	195	13	𝑖=𝑚	𝑖=𝑚	PROPN
cana-3959	195	14	𝑑𝑎𝑖	𝑑𝑎𝑖	VERB
cana-3959	195	15	ℜ	ℜ	ADJ
cana-3959	195	16	3𝑎⋅ℜ3𝑎𝑖	3𝑎⋅ℜ3𝑎𝑖	NUM
cana-3959	195	17	)	)	PUNCT
cana-3959	195	18	(	(	PUNCT
cana-3959	195	19	15	15	NUM
cana-3959	195	20	)	)	PUNCT
cana-3959	195	21	for	for	ADP
cana-3959	195	22	all	all	DET
cana-3959	195	23	𝑣	𝑣	DET
cana-3959	195	24	∈	∈	NOUN
cana-3959	195	25	𝑋	𝑋	NOUN
cana-3959	195	26	and	and	CCONJ
cana-3959	195	27	all	all	DET
cana-3959	195	28	𝑟	𝑟	NOUN
cana-3959	195	29	>	>	PUNCT
cana-3959	195	30	0	0	PUNCT
cana-3959	196	1	and	and	CCONJ
cana-3959	196	2	all	all	DET
cana-3959	196	3	𝑚	𝑚	NOUN
cana-3959	196	4	,	,	PUNCT
cana-3959	196	5	𝑘	𝑘	DET
cana-3959	196	6	≥	≥	NOUN
cana-3959	196	7	0	0	NUM
cana-3959	196	8	.	.	PUNCT
cana-3959	197	1	since	since	SCONJ
cana-3959	197	2	0	0	NUM
cana-3959	197	3	<	<	X
cana-3959	197	4	𝑑	𝑑	X
cana-3959	197	5	<	<	X
cana-3959	197	6	ℜ	ℜ	ADJ
cana-3959	197	7	3𝑎	3𝑎	NOUN
cana-3959	197	8	and	and	CCONJ
cana-3959	197	9	∑𝑘𝑖=0	∑𝑘𝑖=0	PROPN
cana-3959	197	10	(	(	PUNCT
cana-3959	197	11	𝑑	𝑑	PROPN
cana-3959	197	12	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	197	13	)	)	PUNCT
cana-3959	197	14	𝑖	𝑖	X
cana-3959	197	15	<	<	X
cana-3959	197	16	∞	∞	PROPN
cana-3959	197	17	,	,	PUNCT
cana-3959	197	18	the	the	DET
cana-3959	197	19	cauchy	cauchy	ADJ
cana-3959	197	20	criterion	criterion	NOUN
cana-3959	197	21	for	for	ADP
cana-3959	197	22	convergence	convergence	NOUN
cana-3959	197	23	and	and	CCONJ
cana-3959	197	24	(	(	PUNCT
cana-3959	197	25	𝐹5	𝐹5	NOUN
cana-3959	197	26	)	)	PUNCT
cana-3959	197	27	implies	imply	VERB
cana-3959	197	28	that	that	SCONJ
cana-3959	197	29	{	{	PUNCT
cana-3959	197	30	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	197	31	)	)	PUNCT
cana-3959	197	32	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	197	33	}	}	PUNCT
cana-3959	197	34	is	be	AUX
cana-3959	197	35	a	a	DET
cana-3959	197	36	cauchy	cauchy	ADJ
cana-3959	197	37	sequence	sequence	NOUN
cana-3959	197	38	in	in	ADP
cana-3959	197	39	(	(	PUNCT
cana-3959	197	40	𝑌,𝑁	𝑌,𝑁	ADJ
cana-3959	197	41	)	)	PUNCT
cana-3959	197	42	.	.	PUNCT
cana-3959	198	1	since	since	SCONJ
cana-3959	198	2	(	(	PUNCT
cana-3959	198	3	𝑌	𝑌	PROPN
cana-3959	198	4	,	,	PUNCT
cana-3959	198	5	𝑁	𝑁	PROPN
cana-3959	198	6	)	)	PUNCT
cana-3959	198	7	is	be	AUX
cana-3959	198	8	a	a	DET
cana-3959	198	9	fuzzy	fuzzy	ADJ
cana-3959	198	10	banach	banach	NOUN
cana-3959	198	11	space	space	NOUN
cana-3959	198	12	,	,	PUNCT
cana-3959	198	13	this	this	DET
cana-3959	198	14	sequence	sequence	NOUN
cana-3959	198	15	converges	converge	VERB
cana-3959	198	16	to	to	ADP
cana-3959	198	17	some	some	DET
cana-3959	198	18	point	point	NOUN
cana-3959	198	19	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	198	20	)	)	PUNCT
cana-3959	198	21	∈	∈	PROPN
cana-3959	198	22	𝑌.	𝑌.	PROPN
cana-3959	199	1	so	so	ADV
cana-3959	199	2	one	one	NOUN
cana-3959	199	3	can	can	AUX
cana-3959	199	4	define	define	VERB
cana-3959	199	5	the	the	DET
cana-3959	199	6	mapping	mapping	NOUN
cana-3959	199	7	𝐶	𝐶	PROPN
cana-3959	199	8	:	:	PUNCT
cana-3959	199	9	𝑋	𝑋	PROPN
cana-3959	199	10	→	→	SYM
cana-3959	199	11	𝑌	𝑌	PROPN
cana-3959	199	12	by	by	ADP
cana-3959	199	13	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	199	14	)	)	PUNCT
cana-3959	199	15	=	=	SYM
cana-3959	200	1	𝑁	𝑁	PROPN
cana-3959	200	2	−	−	PROPN
cana-3959	200	3	lim	lim	PROPN
cana-3959	200	4	𝑘→∞	𝑘→∞	NUM
cana-3959	200	5	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	200	6	)	)	PUNCT
cana-3959	200	7	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	200	8	for	for	ADP
cana-3959	200	9	all	all	DET
cana-3959	200	10	𝑣	𝑣	DET
cana-3959	200	11	∈	∈	PROPN
cana-3959	200	12	𝑋.	𝑋.	PROPN
cana-3959	200	13	letting	let	VERB
cana-3959	200	14	𝑚	𝑚	X
cana-3959	200	15	=	=	SYM
cana-3959	200	16	0	0	NUM
cana-3959	200	17	in	in	ADP
cana-3959	200	18	(	(	PUNCT
cana-3959	200	19	15	15	NUM
cana-3959	200	20	)	)	PUNCT
cana-3959	200	21	,	,	PUNCT
cana-3959	200	22	we	we	PRON
cana-3959	200	23	get	get	VERB
cana-3959	200	24	𝒩	𝒩	PROPN
cana-3959	200	25	(	(	PUNCT
cana-3959	200	26	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	200	27	)	)	PUNCT
cana-3959	201	1	ℜ3𝑎𝑘	ℜ3𝑎𝑘	PRON
cana-3959	201	2	−	−	PROPN
cana-3959	201	3	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	201	4	)	)	PUNCT
cana-3959	201	5	,	,	PUNCT
cana-3959	201	6	𝑟	𝑟	X
cana-3959	201	7	)	)	PUNCT
cana-3959	201	8	≥	≥	NOUN
cana-3959	201	9	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	VERB
cana-3959	201	10	,	,	PUNCT
cana-3959	201	11	0	0	NUM
cana-3959	201	12	)	)	PUNCT
cana-3959	201	13	,	,	PUNCT
cana-3959	201	14	𝑟	𝑟	X
cana-3959	201	15	∑𝑘−1	∑𝑘−1	X
cana-3959	201	16	𝑖=0	𝑖=0	PROPN
cana-3959	201	17	𝑑𝑎𝑖	𝑑𝑎𝑖	VERB
cana-3959	201	18	ℜ	ℜ	SYM
cana-3959	201	19	3𝑖⋅ℜ3𝑎𝑖	3𝑖⋅ℜ3𝑎𝑖	NUM
cana-3959	201	20	)	)	PUNCT
cana-3959	201	21	(	(	PUNCT
cana-3959	201	22	16	16	NUM
cana-3959	201	23	)	)	PUNCT
cana-3959	201	24	for	for	ADP
cana-3959	201	25	all	all	DET
cana-3959	201	26	𝑣	𝑣	DET
cana-3959	201	27	∈	∈	NOUN
cana-3959	201	28	𝑋	𝑋	NOUN
cana-3959	201	29	and	and	CCONJ
cana-3959	201	30	all	all	DET
cana-3959	201	31	𝑟	𝑟	NOUN
cana-3959	201	32	>	>	X
cana-3959	201	33	0	0	X
cana-3959	201	34	.	.	PUNCT
cana-3959	202	1	letting	let	VERB
cana-3959	202	2	𝑘	𝑘	X
cana-3959	202	3	→	→	SYM
cana-3959	202	4	∞	∞	NUM
cana-3959	202	5	in	in	ADP
cana-3959	202	6	(	(	PUNCT
cana-3959	202	7	16	16	NUM
cana-3959	202	8	)	)	PUNCT
cana-3959	202	9	and	and	CCONJ
cana-3959	202	10	using	use	VERB
cana-3959	202	11	(	(	PUNCT
cana-3959	202	12	𝐹6	𝐹6	NOUN
cana-3959	202	13	)	)	PUNCT
cana-3959	202	14	,	,	PUNCT
cana-3959	202	15	we	we	PRON
cana-3959	202	16	arrive	arrive	VERB
cana-3959	202	17	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	202	18	)	)	PUNCT
cana-3959	202	19	−	−	PROPN
cana-3959	202	20	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	202	21	)	)	PUNCT
cana-3959	202	22	,	,	PUNCT
cana-3959	202	23	𝑟	𝑟	X
cana-3959	202	24	)	)	PUNCT
cana-3959	202	25	≥	≥	NOUN
cana-3959	202	26	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	VERB
cana-3959	202	27	,	,	PUNCT
cana-3959	202	28	0	0	NUM
cana-3959	202	29	)	)	PUNCT
cana-3959	202	30	,	,	PUNCT
cana-3959	202	31	𝑟(ℜ3𝑎	𝑟(ℜ3𝑎	PROPN
cana-3959	202	32	−	−	PROPN
cana-3959	202	33	𝑑	𝑑	NOUN
cana-3959	202	34	)	)	PUNCT
cana-3959	202	35	)	)	PUNCT
cana-3959	202	36	for	for	ADP
cana-3959	202	37	all	all	DET
cana-3959	202	38	𝑣	𝑣	DET
cana-3959	202	39	∈	∈	NOUN
cana-3959	202	40	𝑋	𝑋	NOUN
cana-3959	202	41	and	and	CCONJ
cana-3959	202	42	all	all	DET
cana-3959	202	43	𝑟	𝑟	NOUN
cana-3959	202	44	>	>	X
cana-3959	202	45	0	0	X
cana-3959	202	46	.	.	PUNCT
cana-3959	202	47	to	to	PART
cana-3959	202	48	prove	prove	VERB
cana-3959	202	49	𝐶	𝐶	PROPN
cana-3959	202	50	satisfies	satisfy	VERB
cana-3959	202	51	the	the	DET
cana-3959	202	52	(	(	PUNCT
cana-3959	202	53	1	1	NUM
cana-3959	202	54	)	)	PUNCT
cana-3959	202	55	,	,	PUNCT
cana-3959	202	56	replacing	replace	VERB
cana-3959	202	57	(	(	PUNCT
cana-3959	202	58	𝑣	𝑣	NOUN
cana-3959	202	59	,	,	PUNCT
cana-3959	202	60	𝑤	𝑤	ADP
cana-3959	202	61	)	)	PUNCT
cana-3959	202	62	by	by	ADP
cana-3959	202	63	(	(	PUNCT
cana-3959	202	64	ℜ𝑎𝑘𝑣,ℜ𝑎𝑘𝑤	ℜ𝑎𝑘𝑣,ℜ𝑎𝑘𝑤	NUM
cana-3959	202	65	)	)	PUNCT
cana-3959	202	66	in	in	ADP
cana-3959	202	67	(	(	PUNCT
cana-3959	202	68	3	3	NUM
cana-3959	202	69	)	)	PUNCT
cana-3959	202	70	,	,	PUNCT
cana-3959	202	71	respectively	respectively	ADV
cana-3959	202	72	,	,	PUNCT
cana-3959	202	73	we	we	PRON
cana-3959	202	74	obtain	obtain	VERB
cana-3959	202	75	𝒩	𝒩	PROPN
cana-3959	202	76	(	(	PUNCT
cana-3959	202	77	1	1	NUM
cana-3959	202	78	ℜ3𝑎𝑘𝐷ℱ(ℜ𝑎𝑘𝑣,ℜ𝑎𝑘𝑤	ℜ3𝑎𝑘𝐷ℱ(ℜ𝑎𝑘𝑣,ℜ𝑎𝑘𝑤	NOUN
cana-3959	202	79	)	)	PUNCT
cana-3959	202	80	,	,	PUNCT
cana-3959	202	81	𝑟	𝑟	X
cana-3959	202	82	)	)	PUNCT
cana-3959	202	83	≥	≥	NOUN
cana-3959	202	84	𝑁′(𝔔(ℜ𝑎𝑘𝑣,ℜ𝑎𝑘𝑤),ℜ3𝑎𝑘𝑟	𝑁′(𝔔(ℜ𝑎𝑘𝑣,ℜ𝑎𝑘𝑤),ℜ3𝑎𝑘𝑟	PROPN
cana-3959	202	85	)	)	PUNCT
cana-3959	202	86	(	(	PUNCT
cana-3959	202	87	17	17	NUM
cana-3959	202	88	)	)	PUNCT
cana-3959	202	89	for	for	ADP
cana-3959	202	90	all	all	DET
cana-3959	202	91	𝑟	𝑟	NOUN
cana-3959	202	92	>	>	PUNCT
cana-3959	202	93	0	0	PUNCT
cana-3959	202	94	and	and	CCONJ
cana-3959	202	95	all	all	DET
cana-3959	202	96	𝑣,𝑤	𝑣,𝑤	PROPN
cana-3959	202	97	∈	∈	PROPN
cana-3959	202	98	𝑋.	𝑋.	PROPN
cana-3959	202	99	now	now	ADV
cana-3959	202	100	,	,	PUNCT
cana-3959	202	101	𝒩(ℱ(ℜ𝑎𝑣	𝒩(ℱ(ℜ𝑎𝑣	X
cana-3959	202	102	+	+	SYM
cana-3959	202	103	𝑤	𝑤	X
cana-3959	202	104	)	)	PUNCT
cana-3959	202	105	±ℜ	±ℜ	PUNCT
cana-3959	203	1	𝑏	𝑏	NOUN
cana-3959	203	2	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	203	3	−ℜ	−ℜ	PROPN
cana-3959	203	4	𝑎𝑤	𝑎𝑤	X
cana-3959	203	5	)	)	PUNCT
cana-3959	203	6	−	−	PROPN
cana-3959	203	7	(	(	PUNCT
cana-3959	203	8	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	203	9	)	)	PUNCT
cana-3959	203	10	2	2	NUM
cana-3959	203	11	)	)	PUNCT
cana-3959	204	1	[	[	X
cana-3959	204	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	204	3	+	+	CCONJ
cana-3959	204	4	𝑤	𝑤	X
cana-3959	204	5	)	)	PUNCT
cana-3959	204	6	+	+	CCONJ
cana-3959	204	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	204	8	−	−	NOUN
cana-3959	204	9	𝑤	𝑤	ADP
cana-3959	204	10	)	)	PUNCT
cana-3959	204	11	]	]	PUNCT
cana-3959	204	12	−	−	PROPN
cana-3959	204	13	(	(	PUNCT
cana-3959	204	14	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	204	15	)	)	PUNCT
cana-3959	204	16	2	2	NUM
cana-3959	204	17	)	)	PUNCT
cana-3959	205	1	[	[	X
cana-3959	205	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	205	3	+	+	NOUN
cana-3959	205	4	𝑤	𝑤	X
cana-3959	205	5	)	)	PUNCT
cana-3959	205	6	−	−	PROPN
cana-3959	205	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	205	8	−	−	NOUN
cana-3959	205	9	𝑤	𝑤	PROPN
cana-3959	205	10	)	)	PUNCT
cana-3959	205	11	]	]	PUNCT
cana-3959	205	12	−(ℜ2𝑎	−(ℜ2𝑎	PROPN
cana-3959	206	1	−	−	NOUN
cana-3959	206	2	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	206	3	∓	∓	PROPN
cana-3959	206	4	ℜ	ℜ	PROPN
cana-3959	206	5	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	206	6	)	)	PUNCT
cana-3959	206	7	∓	∓	PROPN
cana-3959	206	8	(	(	PUNCT
cana-3959	206	9	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	206	10	±	±	NUM
cana-3959	206	11	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	206	12	)	)	PUNCT
cana-3959	206	13	]	]	PUNCT
cana-3959	206	14	,	,	PUNCT
cana-3959	206	15	𝑟	𝑟	PRON
cana-3959	206	16	6	6	NUM
cana-3959	206	17	)	)	PUNCT
cana-3959	206	18	≥	≥	NOUN
cana-3959	206	19	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-3959	206	20	{	{	PUNCT
cana-3959	206	21	𝒩	𝒩	PROPN
cana-3959	206	22	(	(	PUNCT
cana-3959	206	23	𝒞(ℜ𝑎𝑣	𝒞(ℜ𝑎𝑣	NOUN
cana-3959	206	24	+	+	NOUN
cana-3959	206	25	𝑤	𝑤	X
cana-3959	206	26	)	)	PUNCT
cana-3959	206	27	−	−	PROPN
cana-3959	207	1	1	1	NUM
cana-3959	207	2	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	207	3	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	207	4	+	+	CCONJ
cana-3959	207	5	𝑤	𝑤	X
cana-3959	207	6	)	)	PUNCT
cana-3959	207	7	,	,	PUNCT
cana-3959	207	8	𝑟	𝑟	PRON
cana-3959	207	9	6	6	NUM
cana-3959	207	10	)	)	PUNCT
cana-3959	207	11	,	,	PUNCT
cana-3959	207	12	communications	communication	NOUN
cana-3959	207	13	on	on	ADP
cana-3959	207	14	applied	apply	VERB
cana-3959	207	15	nonlinear	nonlinear	ADJ
cana-3959	207	16	analysis	analysis	NOUN
cana-3959	207	17	issn	issn	NOUN
cana-3959	207	18	:	:	PUNCT
cana-3959	207	19	1074	1074	NUM
cana-3959	207	20	-	-	PUNCT
cana-3959	207	21	133x	133x	NUM
cana-3959	207	22	vol	vol	NOUN
cana-3959	207	23	32	32	NUM
cana-3959	207	24	no	no	NOUN
cana-3959	207	25	.	.	PUNCT
cana-3959	208	1	9s	9s	NUM
cana-3959	208	2	(	(	PUNCT
cana-3959	208	3	2025	2025	NUM
cana-3959	208	4	)	)	PUNCT
cana-3959	208	5	485	485	NUM
cana-3959	208	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	208	7	𝒩	𝒩	PROPN
cana-3959	208	8	(	(	PUNCT
cana-3959	208	9	±ℜ	±ℜ	VERB
cana-3959	208	10	𝑏𝒞(𝑣	𝑏𝒞(𝑣	PROPN
cana-3959	208	11	−	−	PROPN
cana-3959	208	12	ℜ	ℜ	PROPN
cana-3959	208	13	𝑎𝑤	𝑎𝑤	ADP
cana-3959	208	14	)	)	PUNCT
cana-3959	208	15	±	±	NUM
cana-3959	209	1	1	1	NUM
cana-3959	209	2	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	209	3	ℜ	ℜ	NOUN
cana-3959	209	4	𝑏	𝑏	PROPN
cana-3959	209	5	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	209	6	−ℜ	−ℜ	PROPN
cana-3959	209	7	𝑎𝑤	𝑎𝑤	ADP
cana-3959	209	8	)	)	PUNCT
cana-3959	209	9	,	,	PUNCT
cana-3959	209	10	𝑟	𝑟	PRON
cana-3959	209	11	6	6	NUM
cana-3959	209	12	)	)	PUNCT
cana-3959	209	13	,	,	PUNCT
cana-3959	209	14	𝒩	𝒩	PROPN
cana-3959	209	15	(	(	PUNCT
cana-3959	209	16	−	−	PROPN
cana-3959	209	17	(	(	PUNCT
cana-3959	209	18	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	209	19	)	)	SYM
cana-3959	209	20	2	2	NUM
cana-3959	209	21	)	)	PUNCT
cana-3959	210	1	[	[	X
cana-3959	210	2	𝒞(𝑣	𝒞(𝑣	X
cana-3959	210	3	+	+	CCONJ
cana-3959	210	4	𝑤	𝑤	X
cana-3959	210	5	)	)	PUNCT
cana-3959	211	1	+	+	PROPN
cana-3959	211	2	𝒞(𝑣	𝒞(𝑣	CCONJ
cana-3959	211	3	−	−	PROPN
cana-3959	211	4	𝑤	𝑤	ADP
cana-3959	211	5	)	)	PUNCT
cana-3959	211	6	]	]	PUNCT
cana-3959	212	1	−	−	PROPN
cana-3959	212	2	1	1	NUM
cana-3959	212	3	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	212	4	(	(	PUNCT
cana-3959	212	5	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	212	6	)	)	PUNCT
cana-3959	212	7	2	2	NUM
cana-3959	212	8	)	)	PUNCT
cana-3959	213	1	[	[	X
cana-3959	213	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	213	3	+	+	CCONJ
cana-3959	213	4	𝑤	𝑤	X
cana-3959	213	5	)	)	PUNCT
cana-3959	213	6	+	+	CCONJ
cana-3959	214	1	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	214	2	−	−	NOUN
cana-3959	214	3	𝑤	𝑤	ADP
cana-3959	214	4	)	)	PUNCT
cana-3959	214	5	]	]	PUNCT
cana-3959	214	6	,	,	PUNCT
cana-3959	214	7	𝑟	𝑟	PRON
cana-3959	214	8	6	6	NUM
cana-3959	214	9	)	)	PUNCT
cana-3959	214	10	,	,	PUNCT
cana-3959	214	11	𝒩	𝒩	PROPN
cana-3959	214	12	(	(	PUNCT
cana-3959	214	13	−	−	PROPN
cana-3959	214	14	(	(	PUNCT
cana-3959	214	15	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	214	16	)	)	PUNCT
cana-3959	214	17	2	2	NUM
cana-3959	214	18	)	)	PUNCT
cana-3959	215	1	[	[	X
cana-3959	215	2	𝒞(𝑣	𝒞(𝑣	X
cana-3959	215	3	+	+	CCONJ
cana-3959	215	4	𝑤	𝑤	X
cana-3959	215	5	)	)	PUNCT
cana-3959	215	6	−	−	PROPN
cana-3959	216	1	𝒞(𝑣	𝒞(𝑣	CCONJ
cana-3959	216	2	−	−	PROPN
cana-3959	216	3	𝑤	𝑤	ADP
cana-3959	216	4	)	)	PUNCT
cana-3959	216	5	]	]	PUNCT
cana-3959	217	1	−	−	PROPN
cana-3959	217	2	1	1	NUM
cana-3959	217	3	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	217	4	(	(	PUNCT
cana-3959	217	5	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	217	6	)	)	PUNCT
cana-3959	217	7	2	2	NUM
cana-3959	217	8	)	)	PUNCT
cana-3959	218	1	[	[	X
cana-3959	218	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	218	3	+	+	NOUN
cana-3959	218	4	𝑤	𝑤	X
cana-3959	218	5	)	)	PUNCT
cana-3959	218	6	−	−	PROPN
cana-3959	218	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	218	8	−	−	NOUN
cana-3959	218	9	𝑤	𝑤	ADP
cana-3959	218	10	)	)	PUNCT
cana-3959	218	11	]	]	PUNCT
cana-3959	218	12	,	,	PUNCT
cana-3959	218	13	𝑟	𝑟	PRON
cana-3959	218	14	6	6	NUM
cana-3959	218	15	)	)	PUNCT
cana-3959	218	16	,	,	PUNCT
cana-3959	218	17	𝒩(−(ℜ2𝑎	𝒩(−(ℜ2𝑎	PROPN
cana-3959	218	18	−	−	NOUN
cana-3959	218	19	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	218	20	∓	∓	PROPN
cana-3959	218	21	ℜ	ℜ	ADJ
cana-3959	218	22	𝑏)𝒞(𝑣	𝑏)𝒞(𝑣	PROPN
cana-3959	218	23	)	)	PUNCT
cana-3959	218	24	∓	∓	PROPN
cana-3959	219	1	(	(	PUNCT
cana-3959	219	2	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	219	3	±	±	NUM
cana-3959	219	4	1)𝒞(𝑤	1)𝒞(𝑤	NUM
cana-3959	219	5	)	)	PUNCT
cana-3959	219	6	]	]	PUNCT
cana-3959	220	1	−	−	PROPN
cana-3959	220	2	1	1	NUM
cana-3959	220	3	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	220	4	(	(	PUNCT
cana-3959	220	5	ℜ	ℜ	ADV
cana-3959	220	6	2𝑎	2𝑎	NUM
cana-3959	220	7	−	−	NOUN
cana-3959	220	8	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	220	9	∓ℜ	∓ℜ	PROPN
cana-3959	220	10	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	220	11	)	)	PUNCT
cana-3959	220	12	∓	∓	PROPN
cana-3959	220	13	(	(	PUNCT
cana-3959	220	14	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	220	15	±	±	NUM
cana-3959	220	16	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	220	17	)	)	PUNCT
cana-3959	220	18	]	]	PUNCT
cana-3959	220	19	,	,	PUNCT
cana-3959	220	20	𝑟	𝑟	PRON
cana-3959	220	21	6	6	NUM
cana-3959	220	22	)	)	PUNCT
cana-3959	220	23	,	,	PUNCT
cana-3959	220	24	𝒩	𝒩	PROPN
cana-3959	220	25	(	(	PUNCT
cana-3959	220	26	1	1	NUM
cana-3959	220	27	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	220	28	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	X
cana-3959	220	29	+	+	CCONJ
cana-3959	220	30	𝑤	𝑤	X
cana-3959	220	31	)	)	PUNCT
cana-3959	220	32	±	±	NUM
cana-3959	220	33	1	1	NUM
cana-3959	221	1	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	221	2	ℜ	ℜ	NOUN
cana-3959	221	3	𝑏	𝑏	PROPN
cana-3959	221	4	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	221	5	−ℜ	−ℜ	PROPN
cana-3959	221	6	𝑎𝑤	𝑎𝑤	X
cana-3959	221	7	)	)	PUNCT
cana-3959	221	8	−	−	PROPN
cana-3959	221	9	1	1	NUM
cana-3959	221	10	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	221	11	(	(	PUNCT
cana-3959	221	12	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	221	13	)	)	PUNCT
cana-3959	221	14	2	2	NUM
cana-3959	221	15	)	)	PUNCT
cana-3959	222	1	[	[	X
cana-3959	222	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	222	3	+	+	CCONJ
cana-3959	222	4	𝑤	𝑤	X
cana-3959	222	5	)	)	PUNCT
cana-3959	222	6	+	+	CCONJ
cana-3959	222	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	222	8	−	−	NOUN
cana-3959	222	9	𝑤	𝑤	ADP
cana-3959	222	10	)	)	PUNCT
cana-3959	222	11	]	]	PUNCT
cana-3959	223	1	−	−	PROPN
cana-3959	223	2	1	1	NUM
cana-3959	223	3	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	223	4	(	(	PUNCT
cana-3959	223	5	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	223	6	)	)	PUNCT
cana-3959	223	7	2	2	NUM
cana-3959	223	8	)	)	PUNCT
cana-3959	224	1	[	[	X
cana-3959	224	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	224	3	+	+	NOUN
cana-3959	224	4	𝑤	𝑤	X
cana-3959	224	5	)	)	PUNCT
cana-3959	224	6	−	−	PROPN
cana-3959	224	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	224	8	−	−	NOUN
cana-3959	224	9	𝑤	𝑤	ADP
cana-3959	224	10	)	)	PUNCT
cana-3959	224	11	]	]	PUNCT
cana-3959	225	1	−	−	PROPN
cana-3959	225	2	1	1	NUM
cana-3959	225	3	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	225	4	(	(	PUNCT
cana-3959	225	5	ℜ	ℜ	ADV
cana-3959	225	6	2𝑎	2𝑎	NUM
cana-3959	225	7	−	−	NOUN
cana-3959	225	8	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	225	9	∓ℜ	∓ℜ	PROPN
cana-3959	225	10	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	225	11	)	)	PUNCT
cana-3959	225	12	∓	∓	PROPN
cana-3959	225	13	(	(	PUNCT
cana-3959	225	14	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	225	15	±	±	NUM
cana-3959	225	16	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	225	17	)	)	PUNCT
cana-3959	225	18	]	]	PUNCT
cana-3959	225	19	,	,	PUNCT
cana-3959	225	20	𝑟	𝑟	PRON
cana-3959	225	21	6	6	NUM
cana-3959	225	22	)	)	PUNCT
cana-3959	225	23	}	}	PUNCT
cana-3959	225	24	for	for	ADP
cana-3959	225	25	all	all	DET
cana-3959	225	26	𝑣	𝑣	NOUN
cana-3959	225	27	,	,	PUNCT
cana-3959	225	28	𝑤	𝑤	ADP
cana-3959	225	29	∈	∈	PROPN
cana-3959	225	30	𝑋	𝑋	NOUN
cana-3959	225	31	and	and	CCONJ
cana-3959	225	32	all	all	DET
cana-3959	225	33	𝑟	𝑟	NOUN
cana-3959	225	34	>	>	X
cana-3959	225	35	0	0	X
cana-3959	225	36	.	.	PUNCT
cana-3959	226	1	𝒩(ℱ(ℜ𝑎𝑣	𝒩(ℱ(ℜ𝑎𝑣	AUX
cana-3959	226	2	+	+	CCONJ
cana-3959	226	3	𝑤	𝑤	X
cana-3959	226	4	)	)	PUNCT
cana-3959	226	5	±ℜ	±ℜ	PUNCT
cana-3959	227	1	𝑏	𝑏	NOUN
cana-3959	227	2	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	227	3	−ℜ	−ℜ	PROPN
cana-3959	227	4	𝑎𝑤	𝑎𝑤	X
cana-3959	227	5	)	)	PUNCT
cana-3959	227	6	−	−	PROPN
cana-3959	227	7	(	(	PUNCT
cana-3959	227	8	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	227	9	)	)	PUNCT
cana-3959	227	10	2	2	NUM
cana-3959	227	11	)	)	PUNCT
cana-3959	228	1	[	[	X
cana-3959	228	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	228	3	+	+	CCONJ
cana-3959	228	4	𝑤	𝑤	X
cana-3959	228	5	)	)	PUNCT
cana-3959	228	6	+	+	CCONJ
cana-3959	228	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	228	8	−	−	NOUN
cana-3959	228	9	𝑤	𝑤	ADP
cana-3959	228	10	)	)	PUNCT
cana-3959	228	11	]	]	PUNCT
cana-3959	228	12	−	−	PROPN
cana-3959	228	13	(	(	PUNCT
cana-3959	228	14	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	228	15	)	)	PUNCT
cana-3959	228	16	2	2	NUM
cana-3959	228	17	)	)	PUNCT
cana-3959	229	1	[	[	X
cana-3959	229	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	229	3	+	+	NOUN
cana-3959	229	4	𝑤	𝑤	X
cana-3959	229	5	)	)	PUNCT
cana-3959	229	6	−	−	PROPN
cana-3959	229	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	229	8	−	−	NOUN
cana-3959	229	9	𝑤	𝑤	PROPN
cana-3959	229	10	)	)	PUNCT
cana-3959	229	11	]	]	PUNCT
cana-3959	229	12	−(ℜ2𝑎	−(ℜ2𝑎	PROPN
cana-3959	230	1	−	−	NOUN
cana-3959	230	2	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	230	3	∓	∓	PROPN
cana-3959	230	4	ℜ	ℜ	PROPN
cana-3959	230	5	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	230	6	)	)	PUNCT
cana-3959	230	7	∓	∓	PROPN
cana-3959	230	8	(	(	PUNCT
cana-3959	230	9	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	230	10	±	±	NUM
cana-3959	230	11	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	230	12	)	)	PUNCT
cana-3959	230	13	]	]	PUNCT
cana-3959	230	14	,	,	PUNCT
cana-3959	230	15	𝑟	𝑟	PRON
cana-3959	230	16	6	6	NUM
cana-3959	230	17	)	)	PUNCT
cana-3959	230	18	≥	≥	NOUN
cana-3959	230	19	𝑚𝑖𝑛{1,1,1,1,1,1,1𝑁′(𝔔(ℜ𝑎𝑘𝑣	𝑚𝑖𝑛{1,1,1,1,1,1,1𝑁′(𝔔(ℜ𝑎𝑘𝑣	NUM
cana-3959	230	20	,	,	PUNCT
cana-3959	230	21	0),ℜ3𝑎𝑘𝑟	0),ℜ3𝑎𝑘𝑟	NUM
cana-3959	230	22	)	)	PUNCT
cana-3959	230	23	}	}	PUNCT
cana-3959	230	24	≥	≥	PROPN
cana-3959	230	25	𝑁′(𝔔(ℜ𝑎𝑘𝑣	𝑁′(𝔔(ℜ𝑎𝑘𝑣	NOUN
cana-3959	230	26	,	,	PUNCT
cana-3959	230	27	0),ℜ3𝑎𝑘𝑟	0),ℜ3𝑎𝑘𝑟	NUM
cana-3959	230	28	)	)	PUNCT
cana-3959	230	29	(	(	PUNCT
cana-3959	230	30	18	18	NUM
cana-3959	230	31	)	)	PUNCT
cana-3959	230	32	for	for	ADP
cana-3959	230	33	all	all	DET
cana-3959	230	34	𝑣	𝑣	NOUN
cana-3959	230	35	,	,	PUNCT
cana-3959	230	36	𝑤	𝑤	ADP
cana-3959	230	37	∈	∈	PROPN
cana-3959	230	38	𝑋	𝑋	NOUN
cana-3959	230	39	and	and	CCONJ
cana-3959	230	40	all	all	PRON
cana-3959	230	41	𝑟	𝑟	NOUN
cana-3959	230	42	>	>	X
cana-3959	230	43	0	0	X
cana-3959	230	44	.	.	PUNCT
cana-3959	231	1	letting	let	VERB
cana-3959	231	2	𝑘	𝑘	X
cana-3959	231	3	→	→	SYM
cana-3959	231	4	∞	∞	NUM
cana-3959	231	5	in	in	ADP
cana-3959	231	6	(	(	PUNCT
cana-3959	231	7	18	18	NUM
cana-3959	231	8	)	)	PUNCT
cana-3959	231	9	and	and	CCONJ
cana-3959	231	10	using	use	VERB
cana-3959	231	11	(	(	PUNCT
cana-3959	231	12	2	2	NUM
cana-3959	231	13	)	)	PUNCT
cana-3959	231	14	,	,	PUNCT
cana-3959	231	15	we	we	PRON
cana-3959	231	16	see	see	VERB
cana-3959	231	17	that	that	PRON
cana-3959	231	18	𝒩(ℱ(ℜ𝑎𝑣	𝒩(ℱ(ℜ𝑎𝑣	AUX
cana-3959	231	19	+	+	SYM
cana-3959	231	20	𝑤	𝑤	X
cana-3959	231	21	)	)	PUNCT
cana-3959	231	22	±ℜ	±ℜ	PUNCT
cana-3959	231	23	𝑏	𝑏	NOUN
cana-3959	231	24	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	231	25	−ℜ	−ℜ	PROPN
cana-3959	231	26	𝑎𝑤	𝑎𝑤	X
cana-3959	231	27	)	)	PUNCT
cana-3959	231	28	−	−	PROPN
cana-3959	231	29	(	(	PUNCT
cana-3959	231	30	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	231	31	)	)	PUNCT
cana-3959	231	32	2	2	NUM
cana-3959	231	33	)	)	PUNCT
cana-3959	232	1	[	[	X
cana-3959	232	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	232	3	+	+	CCONJ
cana-3959	232	4	𝑤	𝑤	X
cana-3959	232	5	)	)	PUNCT
cana-3959	232	6	+	+	CCONJ
cana-3959	232	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	232	8	−	−	NOUN
cana-3959	232	9	𝑤	𝑤	ADP
cana-3959	232	10	)	)	PUNCT
cana-3959	232	11	]	]	PUNCT
cana-3959	232	12	−	−	PROPN
cana-3959	232	13	(	(	PUNCT
cana-3959	232	14	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	232	15	)	)	PUNCT
cana-3959	232	16	2	2	NUM
cana-3959	232	17	)	)	PUNCT
cana-3959	233	1	[	[	X
cana-3959	233	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	233	3	+	+	NOUN
cana-3959	233	4	𝑤	𝑤	X
cana-3959	233	5	)	)	PUNCT
cana-3959	233	6	−	−	PROPN
cana-3959	233	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	233	8	−	−	NOUN
cana-3959	233	9	𝑤	𝑤	PROPN
cana-3959	233	10	)	)	PUNCT
cana-3959	233	11	]	]	PUNCT
cana-3959	234	1	communications	communication	NOUN
cana-3959	234	2	on	on	ADP
cana-3959	234	3	applied	apply	VERB
cana-3959	234	4	nonlinear	nonlinear	ADJ
cana-3959	234	5	analysis	analysis	NOUN
cana-3959	234	6	issn	issn	NOUN
cana-3959	234	7	:	:	PUNCT
cana-3959	234	8	1074	1074	NUM
cana-3959	234	9	-	-	PUNCT
cana-3959	234	10	133x	133x	NUM
cana-3959	234	11	vol	vol	NOUN
cana-3959	234	12	32	32	NUM
cana-3959	234	13	no	no	NOUN
cana-3959	234	14	.	.	PUNCT
cana-3959	235	1	9s	9s	NUM
cana-3959	235	2	(	(	PUNCT
cana-3959	235	3	2025	2025	NUM
cana-3959	235	4	)	)	PUNCT
cana-3959	235	5	486	486	NUM
cana-3959	235	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	235	7	−(ℜ2𝑎	−(ℜ2𝑎	NOUN
cana-3959	235	8	−	−	NOUN
cana-3959	235	9	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	235	10	∓	∓	PROPN
cana-3959	235	11	ℜ	ℜ	PROPN
cana-3959	235	12	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	235	13	)	)	PUNCT
cana-3959	235	14	∓	∓	PROPN
cana-3959	236	1	(	(	PUNCT
cana-3959	236	2	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	236	3	±	±	NUM
cana-3959	236	4	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	236	5	)	)	PUNCT
cana-3959	236	6	]	]	PUNCT
cana-3959	236	7	,	,	PUNCT
cana-3959	236	8	𝑟	𝑟	PRON
cana-3959	236	9	6	6	NUM
cana-3959	236	10	)	)	PUNCT
cana-3959	236	11	=	=	SYM
cana-3959	236	12	1	1	NUM
cana-3959	236	13	(	(	PUNCT
cana-3959	236	14	19	19	NUM
cana-3959	236	15	)	)	PUNCT
cana-3959	236	16	for	for	ADP
cana-3959	236	17	all	all	DET
cana-3959	236	18	𝑣	𝑣	NOUN
cana-3959	236	19	,	,	PUNCT
cana-3959	236	20	𝑤	𝑤	ADP
cana-3959	236	21	∈	∈	PROPN
cana-3959	236	22	𝑋	𝑋	NOUN
cana-3959	236	23	and	and	CCONJ
cana-3959	236	24	all	all	PRON
cana-3959	236	25	𝑟	𝑟	NOUN
cana-3959	236	26	>	>	X
cana-3959	236	27	0	0	X
cana-3959	236	28	.	.	PUNCT
cana-3959	237	1	using	use	VERB
cana-3959	237	2	(	(	PUNCT
cana-3959	237	3	𝐹2	𝐹2	NOUN
cana-3959	237	4	)	)	PUNCT
cana-3959	237	5	in	in	ADP
cana-3959	237	6	the	the	DET
cana-3959	237	7	above	above	ADJ
cana-3959	237	8	inequality	inequality	NOUN
cana-3959	237	9	gives	give	VERB
cana-3959	237	10	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	PROPN
cana-3959	237	11	+	+	CCONJ
cana-3959	237	12	𝑤	𝑤	X
cana-3959	237	13	)	)	PUNCT
cana-3959	237	14	±ℜ	±ℜ	PUNCT
cana-3959	238	1	𝑏	𝑏	NOUN
cana-3959	238	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	238	3	−ℜ	−ℜ	PROPN
cana-3959	238	4	𝑎𝑤	𝑎𝑤	X
cana-3959	238	5	)	)	PUNCT
cana-3959	238	6	=	=	SYM
cana-3959	238	7	(	(	PUNCT
cana-3959	238	8	ℜ𝑎(1±ℜ𝑎+𝑏	ℜ𝑎(1±ℜ𝑎+𝑏	NOUN
cana-3959	238	9	)	)	PUNCT
cana-3959	238	10	2	2	NUM
cana-3959	238	11	)	)	PUNCT
cana-3959	239	1	[	[	X
cana-3959	239	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	239	3	+	+	CCONJ
cana-3959	239	4	𝑤	𝑤	X
cana-3959	239	5	)	)	PUNCT
cana-3959	239	6	+	+	CCONJ
cana-3959	239	7	ℱ(𝑣	ℱ(𝑣	ADP
cana-3959	239	8	−	−	NOUN
cana-3959	239	9	𝑤	𝑤	ADP
cana-3959	239	10	)	)	PUNCT
cana-3959	239	11	]	]	PUNCT
cana-3959	240	1	+	+	CCONJ
cana-3959	240	2	(	(	PUNCT
cana-3959	240	3	ℜ𝑎(ℜ𝑎∓ℜ𝑏	ℜ𝑎(ℜ𝑎∓ℜ𝑏	PROPN
cana-3959	240	4	)	)	PUNCT
cana-3959	240	5	2	2	NUM
cana-3959	240	6	)	)	PUNCT
cana-3959	241	1	[	[	X
cana-3959	241	2	ℱ(𝑣	ℱ(𝑣	X
cana-3959	241	3	+	+	NOUN
cana-3959	241	4	𝑤	𝑤	X
cana-3959	241	5	)	)	PUNCT
cana-3959	241	6	−	−	PROPN
cana-3959	241	7	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	241	8	−	−	NOUN
cana-3959	241	9	𝑤	𝑤	ADP
cana-3959	241	10	)	)	PUNCT
cana-3959	241	11	]	]	PUNCT
cana-3959	242	1	+	+	ADJ
cana-3959	242	2	(	(	PUNCT
cana-3959	242	3	ℜ2𝑎	ℜ2𝑎	PROPN
cana-3959	242	4	−	−	NOUN
cana-3959	242	5	1)[(ℜ𝑎	1)[(ℜ𝑎	NUM
cana-3959	242	6	∓	∓	PROPN
cana-3959	242	7	ℜ	ℜ	PROPN
cana-3959	242	8	𝑏)ℱ(𝑣	𝑏)ℱ(𝑣	PROPN
cana-3959	242	9	)	)	PUNCT
cana-3959	242	10	∓	∓	PROPN
cana-3959	243	1	(	(	PUNCT
cana-3959	243	2	ℜ𝑎+𝑏	ℜ𝑎+𝑏	PROPN
cana-3959	243	3	±	±	NUM
cana-3959	243	4	1)ℱ(𝑤	1)ℱ(𝑤	NUM
cana-3959	243	5	)	)	PUNCT
cana-3959	243	6	]	]	PUNCT
cana-3959	243	7	for	for	ADP
cana-3959	243	8	all	all	DET
cana-3959	243	9	𝑣	𝑣	NOUN
cana-3959	243	10	,	,	PUNCT
cana-3959	243	11	𝑤	𝑤	ADP
cana-3959	243	12	∈	∈	PROPN
cana-3959	243	13	𝑋.	𝑋.	PROPN
cana-3959	243	14	hence	hence	ADV
cana-3959	243	15	𝐶	𝐶	PROPN
cana-3959	243	16	satisfies	satisfy	VERB
cana-3959	243	17	the	the	DET
cana-3959	243	18	cubic	cubic	ADJ
cana-3959	243	19	functional	functional	ADJ
cana-3959	243	20	equation	equation	NOUN
cana-3959	243	21	(	(	PUNCT
cana-3959	243	22	1	1	NUM
cana-3959	243	23	)	)	PUNCT
cana-3959	243	24	.	.	PUNCT
cana-3959	244	1	in	in	ADP
cana-3959	244	2	order	order	NOUN
cana-3959	244	3	to	to	PART
cana-3959	244	4	prove	prove	VERB
cana-3959	244	5	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	244	6	)	)	PUNCT
cana-3959	244	7	is	be	AUX
cana-3959	244	8	unique	unique	ADJ
cana-3959	244	9	,	,	PUNCT
cana-3959	244	10	let	let	VERB
cana-3959	244	11	ℱ′(𝑣	ℱ′(𝑣	PRON
cana-3959	244	12	)	)	PUNCT
cana-3959	244	13	be	be	AUX
cana-3959	244	14	another	another	DET
cana-3959	244	15	cubic	cubic	ADJ
cana-3959	244	16	functional	functional	ADJ
cana-3959	244	17	equation	equation	NOUN
cana-3959	244	18	satisfying	satisfy	VERB
cana-3959	244	19	(	(	PUNCT
cana-3959	244	20	1	1	NUM
cana-3959	244	21	)	)	PUNCT
cana-3959	244	22	and	and	CCONJ
cana-3959	244	23	(	(	PUNCT
cana-3959	244	24	5	5	NUM
cana-3959	244	25	)	)	PUNCT
cana-3959	244	26	.	.	PUNCT
cana-3959	245	1	hence	hence	ADV
cana-3959	245	2	,	,	PUNCT
cana-3959	245	3	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	245	4	)	)	PUNCT
cana-3959	245	5	−	−	NUM
cana-3959	246	1	ℱ′(𝑣	ℱ′(𝑣	PROPN
cana-3959	246	2	)	)	PUNCT
cana-3959	246	3	,	,	PUNCT
cana-3959	246	4	𝑟	𝑟	X
cana-3959	246	5	)	)	PUNCT
cana-3959	246	6	=	=	SYM
cana-3959	246	7	𝒩	𝒩	PROPN
cana-3959	246	8	(	(	PUNCT
cana-3959	246	9	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	246	10	)	)	PUNCT
cana-3959	247	1	ℜ3𝑎𝑘	ℜ3𝑎𝑘	PRON
cana-3959	247	2	−	−	PROPN
cana-3959	247	3	ℱ′(ℜ𝑎𝑘𝑣	ℱ′(ℜ𝑎𝑘𝑣	PROPN
cana-3959	247	4	)	)	PUNCT
cana-3959	248	1	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	248	2	,	,	PUNCT
cana-3959	248	3	𝑟	𝑟	X
cana-3959	248	4	)	)	PUNCT
cana-3959	248	5	≥	≥	NOUN
cana-3959	248	6	𝑚𝑖𝑛{𝒩	𝑚𝑖𝑛{𝒩	VERB
cana-3959	248	7	(	(	PUNCT
cana-3959	248	8	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	248	9	)	)	PUNCT
cana-3959	248	10	ℜ3𝑎𝑘	ℜ3𝑎𝑘	PRON
cana-3959	248	11	−	−	PROPN
cana-3959	248	12	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	248	13	)	)	PUNCT
cana-3959	248	14	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	248	15	,	,	PUNCT
cana-3959	248	16	𝑟	𝑟	NOUN
cana-3959	248	17	2	2	NUM
cana-3959	248	18	)	)	PUNCT
cana-3959	248	19	,	,	PUNCT
cana-3959	248	20	𝒩	𝒩	PROPN
cana-3959	248	21	(	(	PUNCT
cana-3959	248	22	ℱ(ℜ𝑎𝑘𝑣	ℱ(ℜ𝑎𝑘𝑣	NUM
cana-3959	248	23	)	)	PUNCT
cana-3959	248	24	ℜ3𝑎𝑘	ℜ3𝑎𝑘	PRON
cana-3959	248	25	−	−	PROPN
cana-3959	248	26	ℱ′(ℜ𝑎𝑘𝑣	ℱ′(ℜ𝑎𝑘𝑣	PROPN
cana-3959	248	27	)	)	PUNCT
cana-3959	249	1	ℜ3𝑎𝑘	ℜ3𝑎𝑘	NOUN
cana-3959	249	2	,	,	PUNCT
cana-3959	249	3	𝑟	𝑟	NOUN
cana-3959	249	4	2	2	NUM
cana-3959	249	5	)	)	PUNCT
cana-3959	249	6	}	}	PUNCT
cana-3959	249	7	≥	≥	NOUN
cana-3959	249	8	𝑁′	𝑁′	X
cana-3959	249	9	(	(	PUNCT
cana-3959	249	10	𝔔(ℜ𝑎𝑘𝑢	𝔔(ℜ𝑎𝑘𝑢	PROPN
cana-3959	249	11	,	,	PUNCT
cana-3959	249	12	0	0	NUM
cana-3959	249	13	)	)	PUNCT
cana-3959	249	14	,	,	PUNCT
cana-3959	249	15	(	(	PUNCT
cana-3959	249	16	ℜ3𝑎𝑘−𝑑𝑎)𝑟	ℜ3𝑎𝑘−𝑑𝑎)𝑟	NOUN
cana-3959	249	17	2	2	NUM
cana-3959	249	18	)	)	PUNCT
cana-3959	249	19	≥	≥	NOUN
cana-3959	249	20	𝑁′	𝑁′	X
cana-3959	249	21	(	(	PUNCT
cana-3959	249	22	𝔔(𝑣	𝔔(𝑣	PROPN
cana-3959	249	23	,	,	PUNCT
cana-3959	249	24	0	0	NUM
cana-3959	249	25	)	)	PUNCT
cana-3959	249	26	,	,	PUNCT
cana-3959	249	27	(	(	PUNCT
cana-3959	249	28	ℜ3𝑎𝑘−𝑑𝑎)𝑟	ℜ3𝑎𝑘−𝑑𝑎)𝑟	NOUN
cana-3959	249	29	2𝑑𝑘	2𝑑𝑘	ADJ
cana-3959	249	30	)	)	PUNCT
cana-3959	249	31	for	for	ADP
cana-3959	249	32	all	all	DET
cana-3959	249	33	𝑢	𝑢	PRON
cana-3959	249	34	∈	∈	PROPN
cana-3959	249	35	𝑋	𝑋	NOUN
cana-3959	249	36	and	and	CCONJ
cana-3959	249	37	all	all	DET
cana-3959	249	38	𝑟	𝑟	NOUN
cana-3959	249	39	>	>	X
cana-3959	249	40	0	0	X
cana-3959	249	41	.	.	PUNCT
cana-3959	250	1	since	since	SCONJ
cana-3959	250	2	lim	lim	PROPN
cana-3959	250	3	𝑘→∞	𝑘→∞	NUM
cana-3959	250	4	(	(	PUNCT
cana-3959	250	5	ℜ3𝑎𝑘−𝑑𝑎)𝑟	ℜ3𝑎𝑘−𝑑𝑎)𝑟	NOUN
cana-3959	250	6	2𝑑𝑘	2𝑑𝑘	NOUN
cana-3959	250	7	=	=	SYM
cana-3959	250	8	∞	∞	PROPN
cana-3959	250	9	,	,	PUNCT
cana-3959	250	10	we	we	PRON
cana-3959	250	11	obtain	obtain	VERB
cana-3959	250	12	lim	lim	PROPN
cana-3959	250	13	𝑘→∞	𝑘→∞	PUNCT
cana-3959	250	14	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	PROPN
cana-3959	250	15	,	,	PUNCT
cana-3959	250	16	0	0	NUM
cana-3959	250	17	)	)	PUNCT
cana-3959	250	18	,	,	PUNCT
cana-3959	250	19	(	(	PUNCT
cana-3959	250	20	ℜ3𝑎𝑘−𝑑𝑎)𝑟	ℜ3𝑎𝑘−𝑑𝑎)𝑟	NOUN
cana-3959	250	21	2𝑑𝑘	2𝑑𝑘	ADJ
cana-3959	250	22	)	)	PUNCT
cana-3959	251	1	=	=	SYM
cana-3959	251	2	1	1	X
cana-3959	251	3	.	.	PUNCT
cana-3959	251	4	thus	thus	ADV
cana-3959	251	5	𝒩(ℱ(𝑣	𝒩(ℱ(𝑣	NOUN
cana-3959	251	6	)	)	PUNCT
cana-3959	251	7	−	−	PROPN
cana-3959	251	8	ℱ′(𝑣	ℱ′(𝑣	PROPN
cana-3959	251	9	)	)	PUNCT
cana-3959	251	10	,	,	PUNCT
cana-3959	251	11	𝑟	𝑟	X
cana-3959	251	12	)	)	PUNCT
cana-3959	251	13	=	=	SYM
cana-3959	251	14	1	1	NUM
cana-3959	251	15	for	for	ADP
cana-3959	251	16	all	all	DET
cana-3959	251	17	𝑣	𝑣	DET
cana-3959	251	18	∈	∈	NOUN
cana-3959	251	19	𝑋	𝑋	NOUN
cana-3959	251	20	and	and	CCONJ
cana-3959	251	21	all	all	DET
cana-3959	251	22	𝑟	𝑟	NOUN
cana-3959	251	23	>	>	X
cana-3959	251	24	0	0	NUM
cana-3959	251	25	,	,	PUNCT
cana-3959	251	26	hence	hence	ADV
cana-3959	251	27	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	251	28	)	)	PUNCT
cana-3959	251	29	=	=	SYM
cana-3959	251	30	ℱ′(𝑣	ℱ′(𝑣	PROPN
cana-3959	251	31	)	)	PUNCT
cana-3959	251	32	.	.	PUNCT
cana-3959	252	1	therefore	therefore	ADV
cana-3959	252	2	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	252	3	)	)	PUNCT
cana-3959	252	4	is	be	AUX
cana-3959	252	5	unique	unique	ADJ
cana-3959	252	6	.	.	PUNCT
cana-3959	253	1	corollary	corollary	ADJ
cana-3959	253	2	4.2	4.2	NUM
cana-3959	253	3	suppose	suppose	VERB
cana-3959	253	4	that	that	SCONJ
cana-3959	253	5	a	a	DET
cana-3959	253	6	function	function	NOUN
cana-3959	253	7	f	f	X
cana-3959	253	8	:x	:x	PROPN
cana-3959	253	9	→	→	SYM
cana-3959	253	10	y	y	PROPN
cana-3959	253	11	satisfies	satisfy	VERB
cana-3959	253	12	the	the	DET
cana-3959	253	13	inequality	inequality	NOUN
cana-3959	253	14	𝑁(𝐷ℱ(𝑣	𝑁(𝐷ℱ(𝑣	PROPN
cana-3959	253	15	,	,	PUNCT
cana-3959	253	16	𝑤	𝑤	ADP
cana-3959	253	17	)	)	PUNCT
cana-3959	253	18	,	,	PUNCT
cana-3959	253	19	𝑟	𝑟	X
cana-3959	253	20	)	)	PUNCT
cana-3959	253	21	≥	≥	NOUN
cana-3959	253	22	{	{	PUNCT
cana-3959	253	23	𝑁′(𝜖	𝑁′(𝜖	PROPN
cana-3959	253	24	,	,	PUNCT
cana-3959	253	25	𝑟	𝑟	NOUN
cana-3959	253	26	)	)	PUNCT
cana-3959	253	27	,	,	PUNCT
cana-3959	253	28	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	X
cana-3959	253	29	+	+	CCONJ
cana-3959	253	30	||𝑤||𝑠	||𝑤||𝑠	PROPN
cana-3959	253	31	,	,	PUNCT
cana-3959	253	32	𝑟	𝑟	NOUN
cana-3959	253	33	)	)	PUNCT
cana-3959	253	34	,	,	PUNCT
cana-3959	253	35	𝑠	𝑠	PROPN
cana-3959	253	36	≠	≠	PROPN
cana-3959	253	37	3	3	NUM
cana-3959	253	38	;	;	PUNCT
cana-3959	253	39	𝑁′(𝜖(||𝑣||𝑠||𝑤||𝑠	𝑁′(𝜖(||𝑣||𝑠||𝑤||𝑠	PROPN
cana-3959	253	40	+	+	CCONJ
cana-3959	253	41	{	{	PUNCT
cana-3959	253	42	||𝑣||2𝑠	||𝑣||2𝑠	ADJ
cana-3959	253	43	+	+	CCONJ
cana-3959	253	44	||𝑤||2𝑠	||𝑤||2𝑠	PROPN
cana-3959	253	45	}	}	PUNCT
cana-3959	253	46	)	)	PUNCT
cana-3959	253	47	,	,	PUNCT
cana-3959	253	48	𝑟	𝑟	X
cana-3959	253	49	)	)	PUNCT
cana-3959	253	50	,	,	PUNCT
cana-3959	253	51	𝑠	𝑠	PROPN
cana-3959	253	52	≠	≠	PROPN
cana-3959	253	53	3	3	NUM
cana-3959	253	54	2	2	NUM
cana-3959	253	55	;	;	PUNCT
cana-3959	253	56	(	(	PUNCT
cana-3959	253	57	20	20	NUM
cana-3959	253	58	)	)	PUNCT
cana-3959	253	59	for	for	ADP
cana-3959	253	60	all	all	DET
cana-3959	253	61	𝑣	𝑣	NOUN
cana-3959	253	62	,	,	PUNCT
cana-3959	253	63	𝑤	𝑤	ADP
cana-3959	253	64	∈	∈	PROPN
cana-3959	253	65	𝑋	𝑋	NOUN
cana-3959	253	66	and	and	CCONJ
cana-3959	253	67	all	all	DET
cana-3959	253	68	𝑟	𝑟	NOUN
cana-3959	253	69	>	>	X
cana-3959	253	70	0	0	NUM
cana-3959	253	71	,	,	PUNCT
cana-3959	253	72	where	where	SCONJ
cana-3959	253	73	𝜖	𝜖	X
cana-3959	253	74	,	,	PUNCT
cana-3959	253	75	𝑠	𝑠	PROPN
cana-3959	253	76	are	be	AUX
cana-3959	253	77	constants	constant	NOUN
cana-3959	253	78	with	with	ADP
cana-3959	253	79	𝜖	𝜖	PROPN
cana-3959	253	80	>	>	X
cana-3959	253	81	0	0	NUM
cana-3959	253	82	.	.	PUNCT
cana-3959	254	1	then	then	ADV
cana-3959	254	2	there	there	PRON
cana-3959	254	3	exists	exist	VERB
cana-3959	254	4	a	a	DET
cana-3959	254	5	unique	unique	ADJ
cana-3959	254	6	cubic	cubic	ADJ
cana-3959	254	7	mapping	mapping	NOUN
cana-3959	254	8	𝐶	𝐶	PROPN
cana-3959	254	9	:	:	PUNCT
cana-3959	254	10	𝑋	𝑋	PROPN
cana-3959	254	11	→	→	SYM
cana-3959	254	12	𝑌	𝑌	PROPN
cana-3959	254	13	such	such	ADJ
cana-3959	254	14	that	that	SCONJ
cana-3959	254	15	communications	communication	NOUN
cana-3959	254	16	on	on	ADP
cana-3959	254	17	applied	apply	VERB
cana-3959	254	18	nonlinear	nonlinear	ADJ
cana-3959	254	19	analysis	analysis	NOUN
cana-3959	254	20	issn	issn	NOUN
cana-3959	254	21	:	:	PUNCT
cana-3959	254	22	1074	1074	NUM
cana-3959	254	23	-	-	PUNCT
cana-3959	254	24	133x	133x	NUM
cana-3959	254	25	vol	vol	NOUN
cana-3959	254	26	32	32	NUM
cana-3959	254	27	no	no	NOUN
cana-3959	254	28	.	.	PUNCT
cana-3959	255	1	9s	9s	NUM
cana-3959	255	2	(	(	PUNCT
cana-3959	255	3	2025	2025	NUM
cana-3959	255	4	)	)	PUNCT
cana-3959	255	5	487	487	NUM
cana-3959	255	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	255	7	𝑁(𝒞(𝑣	𝑁(𝒞(𝑣	NOUN
cana-3959	255	8	)	)	PUNCT
cana-3959	255	9	−	−	ADP
cana-3959	256	1	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	256	2	)	)	PUNCT
cana-3959	256	3	,	,	PUNCT
cana-3959	256	4	𝑟	𝑟	X
cana-3959	256	5	)	)	PUNCT
cana-3959	256	6	≥	≥	NOUN
cana-3959	256	7	{	{	PUNCT
cana-3959	256	8	𝑁′(𝜖	𝑁′(𝜖	PROPN
cana-3959	256	9	,	,	PUNCT
cana-3959	256	10	|ℜ3𝑎	|ℜ3𝑎	PROPN
cana-3959	256	11	−	−	PROPN
cana-3959	256	12	1|𝑟	1|𝑟	NUM
cana-3959	256	13	)	)	PUNCT
cana-3959	256	14	,	,	PUNCT
cana-3959	256	15	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	PROPN
cana-3959	256	16	,	,	PUNCT
cana-3959	256	17	|ℜ3𝑎	|ℜ3𝑎	PROPN
cana-3959	256	18	−ℜ	−ℜ	PROPN
cana-3959	256	19	𝑎𝑠|𝑟	𝑎𝑠|𝑟	NOUN
cana-3959	256	20	)	)	PUNCT
cana-3959	256	21	,	,	PUNCT
cana-3959	256	22	𝑁′(𝜖||𝑣||2𝑠	𝑁′(𝜖||𝑣||2𝑠	NOUN
cana-3959	256	23	,	,	PUNCT
cana-3959	256	24	|ℜ3𝑎	|ℜ3𝑎	PROPN
cana-3959	256	25	−ℜ	−ℜ	PROPN
cana-3959	256	26	3𝑎𝑠|𝑟	3𝑎𝑠|𝑟	NUM
cana-3959	256	27	)	)	PUNCT
cana-3959	256	28	(	(	PUNCT
cana-3959	256	29	21	21	NUM
cana-3959	256	30	)	)	PUNCT
cana-3959	256	31	for	for	ADP
cana-3959	256	32	all	all	DET
cana-3959	256	33	𝑣	𝑣	DET
cana-3959	256	34	∈	∈	NOUN
cana-3959	256	35	𝑋	𝑋	NOUN
cana-3959	256	36	and	and	CCONJ
cana-3959	256	37	all	all	PRON
cana-3959	256	38	𝑟	𝑟	NOUN
cana-3959	256	39	>	>	X
cana-3959	256	40	0	0	X
cana-3959	256	41	.	.	NOUN
cana-3959	256	42	5	5	NUM
cana-3959	256	43	fixed	fix	VERB
cana-3959	256	44	point	point	NOUN
cana-3959	256	45	method	method	NOUN
cana-3959	256	46	of	of	ADP
cana-3959	256	47	fuzzy	fuzzy	ADJ
cana-3959	256	48	stability	stability	NOUN
cana-3959	256	49	results	result	VERB
cana-3959	256	50	for	for	ADP
cana-3959	256	51	to	to	PART
cana-3959	256	52	prove	prove	VERB
cana-3959	256	53	the	the	DET
cana-3959	256	54	stability	stability	NOUN
cana-3959	256	55	result	result	VERB
cana-3959	256	56	we	we	PRON
cana-3959	256	57	define	define	VERB
cana-3959	256	58	the	the	DET
cana-3959	256	59	following	following	NOUN
cana-3959	256	60	:	:	PUNCT
cana-3959	256	61	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	256	62	is	be	AUX
cana-3959	256	63	a	a	DET
cana-3959	256	64	constant	constant	ADJ
cana-3959	256	65	such	such	ADJ
cana-3959	256	66	that	that	PRON
cana-3959	256	67	𝛿𝑖	𝛿𝑖	NOUN
cana-3959	256	68	=	=	SYM
cana-3959	256	69	{	{	PUNCT
cana-3959	256	70	ℜ	ℜ	PROPN
cana-3959	256	71	𝑎	𝑎	NOUN
cana-3959	256	72	𝑖𝑓	𝑖𝑓	NOUN
cana-3959	256	73	𝑖	𝑖	SYM
cana-3959	256	74	=	=	SYM
cana-3959	256	75	0	0	NUM
cana-3959	256	76	,	,	PUNCT
cana-3959	256	77	1	1	NUM
cana-3959	256	78	ℜ𝑎	ℜ𝑎	NOUN
cana-3959	256	79	𝑖𝑓	𝑖𝑓	NOUN
cana-3959	256	80	𝑖	𝑖	SYM
cana-3959	257	1	=	=	SYM
cana-3959	257	2	1	1	NUM
cana-3959	257	3	and	and	CCONJ
cana-3959	257	4	ω	ω	PROPN
cana-3959	257	5	is	be	AUX
cana-3959	257	6	the	the	DET
cana-3959	257	7	set	set	NOUN
cana-3959	257	8	such	such	ADJ
cana-3959	257	9	that	that	SCONJ
cana-3959	257	10	ω	ω	PROPN
cana-3959	257	11	=	=	PRON
cana-3959	257	12	{	{	PUNCT
cana-3959	257	13	𝑔	𝑔	PROPN
cana-3959	257	14	|	|	ADV
cana-3959	257	15	𝑔	𝑔	NOUN
cana-3959	257	16	:	:	PUNCT
cana-3959	257	17	𝑋	𝑋	PROPN
cana-3959	257	18	→	→	SYM
cana-3959	257	19	𝑌	𝑌	PROPN
cana-3959	257	20	,	,	PUNCT
cana-3959	257	21	𝑔(0	𝑔(0	PROPN
cana-3959	257	22	)	)	PUNCT
cana-3959	257	23	=	=	NOUN
cana-3959	257	24	0	0	NUM
cana-3959	257	25	}	}	PUNCT
cana-3959	257	26	.	.	PUNCT
cana-3959	258	1	theorem	theorem	VERB
cana-3959	258	2	5.1	5.1	NUM
cana-3959	258	3	let	let	VERB
cana-3959	258	4	f	f	NOUN
cana-3959	258	5	:x	:x	PROPN
cana-3959	258	6	→	→	SYM
cana-3959	258	7	y	y	PROPN
cana-3959	258	8	be	be	AUX
cana-3959	258	9	a	a	DET
cana-3959	258	10	mapping	mapping	NOUN
cana-3959	258	11	for	for	ADP
cana-3959	258	12	which	which	PRON
cana-3959	258	13	there	there	PRON
cana-3959	258	14	exist	exist	VERB
cana-3959	258	15	a	a	DET
cana-3959	258	16	function	function	NOUN
cana-3959	258	17	𝔔:x2	𝔔:x2	PUNCT
cana-3959	258	18	→	→	SYM
cana-3959	258	19	z	z	NOUN
cana-3959	258	20	with	with	ADP
cana-3959	258	21	the	the	DET
cana-3959	258	22	condition	condition	NOUN
cana-3959	258	23	lim	lim	NOUN
cana-3959	258	24	𝑘→∞	𝑘→∞	NUM
cana-3959	258	25	𝑁′(𝔔(𝛿𝑖	𝑁′(𝔔(𝛿𝑖	PROPN
cana-3959	258	26	𝑘𝑣	𝑘𝑣	PROPN
cana-3959	258	27	,	,	PUNCT
cana-3959	258	28	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	258	29	𝑘𝑤	𝑘𝑤	PROPN
cana-3959	258	30	)	)	PUNCT
cana-3959	258	31	,	,	PUNCT
cana-3959	258	32	𝛿𝑖	𝛿𝑖	ADP
cana-3959	258	33	3𝑘𝑟	3𝑘𝑟	ADJ
cana-3959	258	34	)	)	PUNCT
cana-3959	259	1	=	=	SYM
cana-3959	259	2	1	1	NUM
cana-3959	259	3	∀	∀	NOUN
cana-3959	259	4	𝑣	𝑣	NOUN
cana-3959	259	5	,	,	PUNCT
cana-3959	259	6	𝑤	𝑤	ADP
cana-3959	259	7	∈	∈	PROPN
cana-3959	259	8	𝑋	𝑋	PROPN
cana-3959	259	9	,	,	PUNCT
cana-3959	259	10	𝑟	𝑟	X
cana-3959	259	11	>	>	X
cana-3959	259	12	0	0	PUNCT
cana-3959	259	13	(	(	PUNCT
cana-3959	259	14	22	22	NUM
cana-3959	259	15	)	)	PUNCT
cana-3959	259	16	and	and	CCONJ
cana-3959	259	17	satisfying	satisfy	VERB
cana-3959	259	18	the	the	DET
cana-3959	259	19	functional	functional	ADJ
cana-3959	259	20	inequality	inequality	NOUN
cana-3959	259	21	𝒩(𝐷	𝒩(𝐷	ADP
cana-3959	259	22	ℱ(𝑣	ℱ(𝑣	NOUN
cana-3959	259	23	,	,	PUNCT
cana-3959	259	24	𝑤	𝑤	ADP
cana-3959	259	25	)	)	PUNCT
cana-3959	259	26	,	,	PUNCT
cana-3959	259	27	𝑟	𝑟	X
cana-3959	259	28	)	)	PUNCT
cana-3959	259	29	≥	≥	NOUN
cana-3959	259	30	𝑁′(𝔔(𝑣,𝑤	𝑁′(𝔔(𝑣,𝑤	NUM
cana-3959	259	31	)	)	PUNCT
cana-3959	259	32	,	,	PUNCT
cana-3959	259	33	𝑟	𝑟	NOUN
cana-3959	259	34	)	)	PUNCT
cana-3959	259	35	∀	∀	X
cana-3959	260	1	𝑣	𝑣	NOUN
cana-3959	260	2	,	,	PUNCT
cana-3959	260	3	𝑤	𝑤	ADP
cana-3959	260	4	∈	∈	PROPN
cana-3959	260	5	𝑋	𝑋	PROPN
cana-3959	260	6	,	,	PUNCT
cana-3959	260	7	𝑟	𝑟	X
cana-3959	260	8	>	>	X
cana-3959	260	9	0	0	NUM
cana-3959	260	10	.	.	PUNCT
cana-3959	261	1	(	(	PUNCT
cana-3959	261	2	23	23	NUM
cana-3959	261	3	)	)	PUNCT
cana-3959	261	4	if	if	SCONJ
cana-3959	261	5	there	there	PRON
cana-3959	261	6	exists	exist	VERB
cana-3959	261	7	𝐿	𝐿	PROPN
cana-3959	261	8	=	=	SYM
cana-3959	261	9	𝐿(𝑖	𝐿(𝑖	NUM
cana-3959	261	10	)	)	PUNCT
cana-3959	261	11	such	such	ADJ
cana-3959	261	12	that	that	SCONJ
cana-3959	261	13	the	the	DET
cana-3959	261	14	function	function	NOUN
cana-3959	261	15	𝑣	𝑣	X
cana-3959	261	16	→	→	SYM
cana-3959	261	17	𝛽(𝑣	𝛽(𝑣	ADJ
cana-3959	261	18	)	)	PUNCT
cana-3959	261	19	=	=	SYM
cana-3959	261	20	𝔔	𝔔	PROPN
cana-3959	261	21	(	(	PUNCT
cana-3959	261	22	𝑣	𝑣	ADP
cana-3959	261	23	ℜ𝑎	ℜ𝑎	PROPN
cana-3959	261	24	,	,	PUNCT
cana-3959	261	25	0	0	NUM
cana-3959	261	26	)	)	PUNCT
cana-3959	261	27	,	,	PUNCT
cana-3959	261	28	has	have	VERB
cana-3959	261	29	the	the	DET
cana-3959	261	30	property	property	NOUN
cana-3959	261	31	𝑁′	𝑁′	ADJ
cana-3959	261	32	(	(	PUNCT
cana-3959	261	33	𝐿	𝐿	PROPN
cana-3959	261	34	1	1	NUM
cana-3959	261	35	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	261	36	3𝛽(𝛿𝑖𝑣	3𝛽(𝛿𝑖𝑣	NUM
cana-3959	261	37	)	)	PUNCT
cana-3959	261	38	,	,	PUNCT
cana-3959	261	39	𝑟	𝑟	X
cana-3959	261	40	)	)	PUNCT
cana-3959	261	41	=	=	SYM
cana-3959	261	42	𝑁′(𝛽(𝑣	𝑁′(𝛽(𝑣	NOUN
cana-3959	261	43	)	)	PUNCT
cana-3959	261	44	,	,	PUNCT
cana-3959	261	45	𝑟	𝑟	NOUN
cana-3959	261	46	)	)	PUNCT
cana-3959	261	47	,	,	PUNCT
cana-3959	261	48	∀	∀	PUNCT
cana-3959	262	1	𝑣	𝑣	ADP
cana-3959	262	2	∈	∈	PROPN
cana-3959	262	3	𝑋	𝑋	PROPN
cana-3959	262	4	,	,	PUNCT
cana-3959	262	5	𝑟	𝑟	X
cana-3959	262	6	>	>	X
cana-3959	262	7	0	0	NUM
cana-3959	262	8	.	.	PUNCT
cana-3959	262	9	(	(	PUNCT
cana-3959	262	10	24	24	NUM
cana-3959	262	11	)	)	PUNCT
cana-3959	262	12	then	then	ADV
cana-3959	262	13	there	there	PRON
cana-3959	262	14	exists	exist	VERB
cana-3959	262	15	unique	unique	ADJ
cana-3959	262	16	cubic	cubic	ADJ
cana-3959	262	17	function	function	NOUN
cana-3959	262	18	𝐶	𝐶	PROPN
cana-3959	262	19	:	:	PUNCT
cana-3959	262	20	𝑋	𝑋	PROPN
cana-3959	262	21	→	→	SYM
cana-3959	262	22	𝑌	𝑌	PROPN
cana-3959	262	23	satisfying	satisfy	VERB
cana-3959	262	24	the	the	DET
cana-3959	262	25	functional	functional	ADJ
cana-3959	262	26	equation	equation	NOUN
cana-3959	262	27	(	(	PUNCT
cana-3959	262	28	1	1	NUM
cana-3959	262	29	)	)	PUNCT
cana-3959	262	30	and	and	CCONJ
cana-3959	262	31	𝒩(ℱ(𝑣	𝒩(ℱ(𝑣	NOUN
cana-3959	262	32	)	)	PUNCT
cana-3959	262	33	−	−	NOUN
cana-3959	263	1	𝒞(𝑣	𝒞(𝑣	CCONJ
cana-3959	263	2	)	)	PUNCT
cana-3959	263	3	,	,	PUNCT
cana-3959	263	4	𝑟	𝑟	X
cana-3959	263	5	)	)	PUNCT
cana-3959	263	6	≥	≥	NOUN
cana-3959	263	7	𝑁′	𝑁′	ADJ
cana-3959	263	8	(	(	PUNCT
cana-3959	263	9	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-3959	263	10	1−𝐿	1−𝐿	NUM
cana-3959	263	11	𝛽(𝑥	𝛽(𝑥	NOUN
cana-3959	263	12	)	)	PUNCT
cana-3959	263	13	,	,	PUNCT
cana-3959	263	14	𝑟	𝑟	NOUN
cana-3959	263	15	)	)	PUNCT
cana-3959	263	16	,	,	PUNCT
cana-3959	263	17	∀	∀	PUNCT
cana-3959	264	1	𝑣	𝑣	ADP
cana-3959	264	2	∈	∈	PROPN
cana-3959	264	3	𝑋	𝑋	PROPN
cana-3959	264	4	,	,	PUNCT
cana-3959	264	5	𝑟	𝑟	X
cana-3959	264	6	>	>	X
cana-3959	264	7	0	0	NUM
cana-3959	264	8	.	.	PUNCT
cana-3959	264	9	(	(	PUNCT
cana-3959	264	10	25	25	NUM
cana-3959	264	11	)	)	PUNCT
cana-3959	264	12	proof	proof	NOUN
cana-3959	264	13	.	.	PUNCT
cana-3959	265	1	let	let	VERB
cana-3959	265	2	𝑑	𝑑	PRON
cana-3959	265	3	be	be	AUX
cana-3959	265	4	a	a	DET
cana-3959	265	5	general	general	ADJ
cana-3959	265	6	metric	metric	NOUN
cana-3959	265	7	on	on	ADP
cana-3959	265	8	ω	ω	PROPN
cana-3959	265	9	,	,	PUNCT
cana-3959	265	10	such	such	ADJ
cana-3959	265	11	that	that	SCONJ
cana-3959	265	12	𝑑(𝑔	𝑑(𝑔	PROPN
cana-3959	265	13	,	,	PUNCT
cana-3959	265	14	ℎ	ℎ	PROPN
cana-3959	265	15	)	)	PUNCT
cana-3959	265	16	=	=	SYM
cana-3959	265	17	𝑖𝑛𝑓{𝐾𝒩(0,∞)|𝒩(𝑔(𝑣	𝑖𝑛𝑓{𝐾𝒩(0,∞)|𝒩(𝑔(𝑣	NOUN
cana-3959	265	18	)	)	PUNCT
cana-3959	265	19	−	−	PROPN
cana-3959	265	20	ℎ(𝑣	ℎ(𝑣	NOUN
cana-3959	265	21	)	)	PUNCT
cana-3959	265	22	,	,	PUNCT
cana-3959	265	23	𝑟	𝑟	X
cana-3959	265	24	)	)	PUNCT
cana-3959	265	25	≥	≥	NOUN
cana-3959	265	26	𝑁′(𝐾𝛽(𝑣	𝑁′(𝐾𝛽(𝑣	PROPN
cana-3959	265	27	)	)	PUNCT
cana-3959	265	28	,	,	PUNCT
cana-3959	265	29	𝑟	𝑟	NOUN
cana-3959	265	30	)	)	PUNCT
cana-3959	265	31	,	,	PUNCT
cana-3959	265	32	𝑣	𝑣	PROPN
cana-3959	265	33	∈	∈	PROPN
cana-3959	265	34	𝑋	𝑋	PROPN
cana-3959	265	35	,	,	PUNCT
cana-3959	265	36	𝑟	𝑟	X
cana-3959	265	37	>	>	X
cana-3959	265	38	0	0	NUM
cana-3959	265	39	}	}	PUNCT
cana-3959	265	40	.	.	PUNCT
cana-3959	266	1	it	it	PRON
cana-3959	266	2	is	be	AUX
cana-3959	266	3	easy	easy	ADJ
cana-3959	266	4	to	to	PART
cana-3959	266	5	see	see	VERB
cana-3959	266	6	that	that	PRON
cana-3959	266	7	(	(	PUNCT
cana-3959	266	8	ω	ω	NOUN
cana-3959	266	9	,	,	PUNCT
cana-3959	266	10	𝑑	𝑑	NOUN
cana-3959	266	11	)	)	PUNCT
cana-3959	266	12	is	be	AUX
cana-3959	266	13	complete	complete	ADJ
cana-3959	266	14	.	.	PUNCT
cana-3959	267	1	define	define	VERB
cana-3959	267	2	𝑇:ω	𝑇:ω	NOUN
cana-3959	267	3	→	→	SYM
cana-3959	267	4	ω	ω	NUM
cana-3959	267	5	by	by	ADP
cana-3959	267	6	𝑇𝑔(𝑥	𝑇𝑔(𝑥	PROPN
cana-3959	267	7	)	)	PUNCT
cana-3959	267	8	=	=	SYM
cana-3959	268	1	1	1	NUM
cana-3959	268	2	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	268	3	3𝑔(𝛿𝑖𝑣	3𝑔(𝛿𝑖𝑣	PROPN
cana-3959	268	4	)	)	PUNCT
cana-3959	268	5	,	,	PUNCT
cana-3959	268	6	for	for	ADP
cana-3959	268	7	all	all	PRON
cana-3959	268	8	𝑣	𝑣	DET
cana-3959	268	9	∈	∈	PROPN
cana-3959	268	10	𝑋.	𝑋.	PROPN
cana-3959	268	11	for	for	ADP
cana-3959	268	12	𝑔	𝑔	PROPN
cana-3959	268	13	,	,	PUNCT
cana-3959	268	14	ℎ	ℎ	PROPN
cana-3959	268	15	∈	∈	PROPN
cana-3959	268	16	ω	ω	PROPN
cana-3959	268	17	,	,	PUNCT
cana-3959	268	18	we	we	PRON
cana-3959	268	19	have	have	VERB
cana-3959	268	20	𝑑(𝑔	𝑑(𝑔	PROPN
cana-3959	268	21	,	,	PUNCT
cana-3959	268	22	ℎ	ℎ	PROPN
cana-3959	268	23	)	)	PUNCT
cana-3959	268	24	≤	≤	NOUN
cana-3959	268	25	𝐾	𝐾	PROPN
cana-3959	268	26	⇒	⇒	NOUN
cana-3959	268	27	𝒩(𝑔(𝑣	𝒩(𝑔(𝑣	PUNCT
cana-3959	268	28	)	)	PUNCT
cana-3959	268	29	−	−	PROPN
cana-3959	268	30	ℎ(𝑣	ℎ(𝑣	NOUN
cana-3959	268	31	)	)	PUNCT
cana-3959	268	32	,	,	PUNCT
cana-3959	268	33	𝑟	𝑟	X
cana-3959	268	34	)	)	PUNCT
cana-3959	268	35	≥	≥	NOUN
cana-3959	268	36	𝑁′(𝐾𝛽(𝑣	𝑁′(𝐾𝛽(𝑣	PROPN
cana-3959	268	37	)	)	PUNCT
cana-3959	268	38	,	,	PUNCT
cana-3959	268	39	𝑟	𝑟	X
cana-3959	268	40	)	)	PUNCT
cana-3959	268	41	⇒	⇒	PROPN
cana-3959	268	42	𝒩	𝒩	PROPN
cana-3959	268	43	(	(	PUNCT
cana-3959	268	44	𝑔(𝛿𝑖𝑣	𝑔(𝛿𝑖𝑣	PROPN
cana-3959	268	45	)	)	PUNCT
cana-3959	268	46	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	268	47	3	3	NUM
cana-3959	268	48	−	−	PROPN
cana-3959	268	49	ℎ(𝛿𝑖𝑣	ℎ(𝛿𝑖𝑣	PROPN
cana-3959	268	50	)	)	PUNCT
cana-3959	268	51	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	268	52	3	3	NUM
cana-3959	268	53	,	,	PUNCT
cana-3959	268	54	𝑟	𝑟	NOUN
cana-3959	268	55	)	)	PUNCT
cana-3959	268	56	≥	≥	NOUN
cana-3959	268	57	𝑁′	𝑁′	ADJ
cana-3959	268	58	(	(	PUNCT
cana-3959	268	59	𝐾	𝐾	PROPN
cana-3959	268	60	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	268	61	3𝛽(𝛿𝑖𝑣	3𝛽(𝛿𝑖𝑣	PROPN
cana-3959	268	62	)	)	PUNCT
cana-3959	268	63	,	,	PUNCT
cana-3959	268	64	𝑟	𝑟	X
cana-3959	268	65	)	)	PUNCT
cana-3959	268	66	communications	communication	NOUN
cana-3959	268	67	on	on	ADP
cana-3959	268	68	applied	apply	VERB
cana-3959	268	69	nonlinear	nonlinear	ADJ
cana-3959	268	70	analysis	analysis	NOUN
cana-3959	268	71	issn	issn	NOUN
cana-3959	268	72	:	:	PUNCT
cana-3959	268	73	1074	1074	NUM
cana-3959	268	74	-	-	PUNCT
cana-3959	268	75	133x	133x	NUM
cana-3959	268	76	vol	vol	NOUN
cana-3959	268	77	32	32	NUM
cana-3959	268	78	no	no	NOUN
cana-3959	268	79	.	.	PUNCT
cana-3959	269	1	9s	9s	NUM
cana-3959	269	2	(	(	PUNCT
cana-3959	269	3	2025	2025	NUM
cana-3959	269	4	)	)	PUNCT
cana-3959	270	1	488	488	NUM
cana-3959	270	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	270	3	⇒	⇒	PROPN
cana-3959	270	4	𝒩(𝑇𝑔(𝑣	𝒩(𝑇𝑔(𝑣	NOUN
cana-3959	270	5	)	)	PUNCT
cana-3959	270	6	−	−	PROPN
cana-3959	270	7	𝑇ℎ(𝑣	𝑇ℎ(𝑣	NOUN
cana-3959	270	8	)	)	PUNCT
cana-3959	270	9	,	,	PUNCT
cana-3959	270	10	𝑟	𝑟	X
cana-3959	270	11	)	)	PUNCT
cana-3959	270	12	≥	≥	PROPN
cana-3959	270	13	𝑁′(𝐾𝐿𝛽(𝑣	𝑁′(𝐾𝐿𝛽(𝑣	NOUN
cana-3959	270	14	)	)	PUNCT
cana-3959	270	15	,	,	PUNCT
cana-3959	270	16	𝑟	𝑟	X
cana-3959	270	17	)	)	PUNCT
cana-3959	270	18	⇒	⇒	NOUN
cana-3959	270	19	𝑑(𝑇𝑔(𝑣	𝑑(𝑇𝑔(𝑣	PROPN
cana-3959	270	20	)	)	PUNCT
cana-3959	270	21	,	,	PUNCT
cana-3959	270	22	𝑇ℎ(𝑣	𝑇ℎ(𝑣	NUM
cana-3959	270	23	)	)	PUNCT
cana-3959	270	24	)	)	PUNCT
cana-3959	270	25	≤	≤	PUNCT
cana-3959	271	1	𝐾𝐿	𝐾𝐿	PROPN
cana-3959	271	2	⇒	⇒	VERB
cana-3959	271	3	𝑑(𝑇𝑔	𝑑(𝑇𝑔	PROPN
cana-3959	271	4	,	,	PUNCT
cana-3959	271	5	𝑇ℎ	𝑇ℎ	PROPN
cana-3959	271	6	)	)	PUNCT
cana-3959	271	7	≤	≤	NOUN
cana-3959	271	8	𝐿𝑑(𝑔	𝐿𝑑(𝑔	PROPN
cana-3959	271	9	,	,	PUNCT
cana-3959	271	10	ℎ	ℎ	PROPN
cana-3959	271	11	)	)	PUNCT
cana-3959	271	12	(	(	PUNCT
cana-3959	271	13	26	26	NUM
cana-3959	271	14	)	)	PUNCT
cana-3959	271	15	for	for	ADP
cana-3959	271	16	all	all	DET
cana-3959	271	17	𝑔	𝑔	NOUN
cana-3959	271	18	,	,	PUNCT
cana-3959	271	19	ℎ	ℎ	PROPN
cana-3959	271	20	∈	∈	PROPN
cana-3959	271	21	ω	ω	NOUN
cana-3959	271	22	.	.	PUNCT
cana-3959	272	1	there	there	ADV
cana-3959	272	2	fore	fore	NOUN
cana-3959	272	3	𝑇	𝑇	PROPN
cana-3959	272	4	is	be	AUX
cana-3959	272	5	strictly	strictly	ADV
cana-3959	272	6	contractive	contractive	ADJ
cana-3959	272	7	mapping	mapping	NOUN
cana-3959	272	8	on	on	ADP
cana-3959	272	9	ω	ω	PROPN
cana-3959	272	10	with	with	ADP
cana-3959	272	11	lipschitz	lipschitz	NOUN
cana-3959	272	12	constant	constant	ADJ
cana-3959	272	13	𝐿.	𝐿.	NOUN
cana-3959	272	14	replacing	replacing	NOUN
cana-3959	272	15	(	(	PUNCT
cana-3959	272	16	𝑣	𝑣	NOUN
cana-3959	272	17	,	,	PUNCT
cana-3959	272	18	𝑤	𝑤	ADP
cana-3959	272	19	)	)	PUNCT
cana-3959	272	20	by	by	ADP
cana-3959	272	21	(	(	PUNCT
cana-3959	272	22	𝑣	𝑣	NOUN
cana-3959	272	23	,	,	PUNCT
cana-3959	272	24	0	0	NUM
cana-3959	272	25	)	)	PUNCT
cana-3959	272	26	in	in	ADP
cana-3959	272	27	(	(	PUNCT
cana-3959	272	28	23	23	NUM
cana-3959	272	29	)	)	PUNCT
cana-3959	272	30	,	,	PUNCT
cana-3959	272	31	we	we	PRON
cana-3959	272	32	get	get	AUX
cana-3959	272	33	𝒩(ℱ(ℜ𝑎𝑣	𝒩(ℱ(ℜ𝑎𝑣	VERB
cana-3959	272	34	)	)	PUNCT
cana-3959	273	1	−ℜ	−ℜ	PROPN
cana-3959	273	2	3𝑎	3𝑎	NUM
cana-3959	273	3	ℱ(𝑣	ℱ(𝑣	NOUN
cana-3959	273	4	)	)	PUNCT
cana-3959	273	5	,	,	PUNCT
cana-3959	273	6	𝑟	𝑟	X
cana-3959	273	7	)	)	PUNCT
cana-3959	273	8	≥	≥	NOUN
cana-3959	273	9	𝑁′(𝔔(𝑣	𝑁′(𝔔(𝑣	VERB
cana-3959	273	10	,	,	PUNCT
cana-3959	273	11	0	0	NUM
cana-3959	273	12	)	)	PUNCT
cana-3959	273	13	,	,	PUNCT
cana-3959	273	14	𝑟	𝑟	NOUN
cana-3959	273	15	)	)	PUNCT
cana-3959	273	16	.	.	PUNCT
cana-3959	274	1	(	(	PUNCT
cana-3959	274	2	27	27	NUM
cana-3959	274	3	)	)	PUNCT
cana-3959	274	4	for	for	ADP
cana-3959	274	5	all	all	PRON
cana-3959	274	6	𝑣	𝑣	DET
cana-3959	274	7	∈	∈	PROPN
cana-3959	274	8	𝑋	𝑋	PROPN
cana-3959	274	9	,	,	PUNCT
cana-3959	274	10	𝑟	𝑟	X
cana-3959	274	11	>	>	X
cana-3959	274	12	0	0	X
cana-3959	274	13	.	.	PUNCT
cana-3959	275	1	using	use	VERB
cana-3959	275	2	(	(	PUNCT
cana-3959	275	3	f3	f3	ADJ
cana-3959	275	4	)	)	PUNCT
cana-3959	275	5	in	in	ADP
cana-3959	275	6	(	(	PUNCT
cana-3959	275	7	27	27	NUM
cana-3959	275	8	)	)	PUNCT
cana-3959	275	9	,	,	PUNCT
cana-3959	275	10	we	we	PRON
cana-3959	275	11	arrive	arrive	VERB
cana-3959	275	12	𝒩	𝒩	PROPN
cana-3959	275	13	(	(	PUNCT
cana-3959	275	14	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	PROPN
cana-3959	275	15	)	)	PUNCT
cana-3959	275	16	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	275	17	−	−	PROPN
cana-3959	275	18	ℱ(𝑣	ℱ(𝑣	NOUN
cana-3959	275	19	)	)	PUNCT
cana-3959	275	20	,	,	PUNCT
cana-3959	275	21	𝑟	𝑟	X
cana-3959	275	22	)	)	PUNCT
cana-3959	275	23	≥	≥	NOUN
cana-3959	275	24	𝑁′	𝑁′	X
cana-3959	275	25	(	(	PUNCT
cana-3959	275	26	1	1	NUM
cana-3959	275	27	ℜ3𝑎𝔔(𝑣	ℜ3𝑎𝔔(𝑣	ADJ
cana-3959	275	28	,	,	PUNCT
cana-3959	275	29	0	0	NUM
cana-3959	275	30	)	)	PUNCT
cana-3959	275	31	,	,	PUNCT
cana-3959	275	32	𝑟	𝑟	X
cana-3959	275	33	)	)	PUNCT
cana-3959	275	34	(	(	PUNCT
cana-3959	275	35	28	28	NUM
cana-3959	275	36	)	)	PUNCT
cana-3959	275	37	for	for	ADP
cana-3959	275	38	all	all	PRON
cana-3959	275	39	𝑣	𝑣	DET
cana-3959	275	40	∈	∈	PROPN
cana-3959	275	41	𝑋	𝑋	PROPN
cana-3959	275	42	,	,	PUNCT
cana-3959	275	43	𝑟	𝑟	X
cana-3959	275	44	>	>	X
cana-3959	275	45	0	0	PUNCT
cana-3959	275	46	with	with	ADP
cana-3959	275	47	the	the	DET
cana-3959	275	48	help	help	NOUN
cana-3959	275	49	of	of	ADP
cana-3959	275	50	(	(	PUNCT
cana-3959	275	51	24	24	NUM
cana-3959	275	52	)	)	PUNCT
cana-3959	275	53	when	when	SCONJ
cana-3959	275	54	𝑖	𝑖	X
cana-3959	275	55	=	=	NOUN
cana-3959	275	56	0	0	NUM
cana-3959	275	57	,	,	PUNCT
cana-3959	275	58	it	it	PRON
cana-3959	275	59	follows	follow	VERB
cana-3959	275	60	from	from	ADP
cana-3959	275	61	(	(	PUNCT
cana-3959	275	62	28	28	NUM
cana-3959	275	63	)	)	PUNCT
cana-3959	275	64	,	,	PUNCT
cana-3959	275	65	we	we	PRON
cana-3959	275	66	get	get	VERB
cana-3959	275	67	⇒	⇒	NOUN
cana-3959	275	68	𝒩	𝒩	PROPN
cana-3959	275	69	(	(	PUNCT
cana-3959	275	70	ℱ(ℜ𝑎𝑣	ℱ(ℜ𝑎𝑣	PROPN
cana-3959	275	71	)	)	PUNCT
cana-3959	275	72	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	275	73	−	−	PROPN
cana-3959	275	74	ℱ(𝑣	ℱ(𝑣	NOUN
cana-3959	275	75	)	)	PUNCT
cana-3959	275	76	,	,	PUNCT
cana-3959	275	77	𝑟	𝑟	X
cana-3959	275	78	)	)	PUNCT
cana-3959	275	79	≥	≥	PROPN
cana-3959	275	80	𝑁′(𝐿𝛽(𝑣	𝑁′(𝐿𝛽(𝑣	PROPN
cana-3959	275	81	)	)	PUNCT
cana-3959	275	82	,	,	PUNCT
cana-3959	275	83	𝑟	𝑟	X
cana-3959	275	84	)	)	PUNCT
cana-3959	275	85	⇒	⇒	PROPN
cana-3959	275	86	𝑑(𝑇𝐶	𝑑(𝑇𝐶	PROPN
cana-3959	275	87	,	,	PUNCT
cana-3959	275	88	𝐶	𝐶	PROPN
cana-3959	275	89	)	)	PUNCT
cana-3959	275	90	≤	≤	NUM
cana-3959	276	1	𝐿	𝐿	PROPN
cana-3959	276	2	=	=	SYM
cana-3959	276	3	𝐿1	𝐿1	PROPN
cana-3959	276	4	=	=	SYM
cana-3959	276	5	𝐿1−𝑖	𝐿1−𝑖	PROPN
cana-3959	276	6	(	(	PUNCT
cana-3959	276	7	29	29	NUM
cana-3959	276	8	)	)	PUNCT
cana-3959	276	9	replacing	replace	VERB
cana-3959	276	10	𝑣	𝑣	PRON
cana-3959	276	11	by	by	ADP
cana-3959	276	12	𝑣	𝑣	PRON
cana-3959	276	13	ℜ𝑎	ℜ𝑎	PROPN
cana-3959	276	14	in	in	ADP
cana-3959	276	15	(	(	PUNCT
cana-3959	276	16	27	27	NUM
cana-3959	276	17	)	)	PUNCT
cana-3959	276	18	,	,	PUNCT
cana-3959	276	19	we	we	PRON
cana-3959	276	20	obtain	obtain	VERB
cana-3959	276	21	𝒩(ℱ(𝑣	𝒩(ℱ(𝑣	NOUN
cana-3959	276	22	)	)	PUNCT
cana-3959	277	1	−ℜ	−ℜ	PROPN
cana-3959	277	2	3𝑎	3𝑎	NUM
cana-3959	277	3	ℱ	ℱ	PROPN
cana-3959	277	4	(	(	PUNCT
cana-3959	277	5	𝑣	𝑣	ADP
cana-3959	277	6	ℜ𝑎	ℜ𝑎	PROPN
cana-3959	277	7	)	)	PUNCT
cana-3959	277	8	,	,	PUNCT
cana-3959	277	9	𝑟	𝑟	X
cana-3959	277	10	)	)	PUNCT
cana-3959	277	11	≥	≥	PROPN
cana-3959	277	12	𝑁′	𝑁′	X
cana-3959	277	13	(	(	PUNCT
cana-3959	277	14	𝔔	𝔔	PROPN
cana-3959	277	15	(	(	PUNCT
cana-3959	277	16	𝑣	𝑣	ADP
cana-3959	277	17	ℜ𝑎	ℜ𝑎	PROPN
cana-3959	277	18	,	,	PUNCT
cana-3959	277	19	0	0	NUM
cana-3959	277	20	)	)	PUNCT
cana-3959	277	21	,	,	PUNCT
cana-3959	277	22	𝑟	𝑟	X
cana-3959	277	23	)	)	PUNCT
cana-3959	277	24	(	(	PUNCT
cana-3959	277	25	30	30	NUM
cana-3959	277	26	)	)	PUNCT
cana-3959	277	27	for	for	ADP
cana-3959	277	28	all	all	PRON
cana-3959	277	29	𝑣	𝑣	DET
cana-3959	277	30	∈	∈	PROPN
cana-3959	277	31	𝑋	𝑋	PROPN
cana-3959	277	32	,	,	PUNCT
cana-3959	277	33	𝑟	𝑟	X
cana-3959	277	34	>	>	X
cana-3959	277	35	0	0	PUNCT
cana-3959	277	36	with	with	ADP
cana-3959	277	37	the	the	DET
cana-3959	277	38	help	help	NOUN
cana-3959	277	39	of	of	ADP
cana-3959	277	40	(	(	PUNCT
cana-3959	277	41	24	24	NUM
cana-3959	277	42	)	)	PUNCT
cana-3959	277	43	when	when	SCONJ
cana-3959	277	44	𝑖	𝑖	X
cana-3959	277	45	=	=	SYM
cana-3959	277	46	1	1	NUM
cana-3959	277	47	,	,	PUNCT
cana-3959	277	48	it	it	PRON
cana-3959	277	49	follows	follow	VERB
cana-3959	277	50	from	from	ADP
cana-3959	277	51	(	(	PUNCT
cana-3959	277	52	30	30	NUM
cana-3959	277	53	)	)	PUNCT
cana-3959	277	54	we	we	PRON
cana-3959	277	55	get	get	VERB
cana-3959	277	56	⇒	⇒	NOUN
cana-3959	277	57	𝒩	𝒩	PROPN
cana-3959	277	58	(	(	PUNCT
cana-3959	277	59	ℱ(𝑣	ℱ(𝑣	NOUN
cana-3959	277	60	)	)	PUNCT
cana-3959	277	61	−ℜ	−ℜ	PROPN
cana-3959	277	62	3𝑎	3𝑎	PROPN
cana-3959	277	63	ℱ	ℱ	PROPN
cana-3959	277	64	(	(	PUNCT
cana-3959	277	65	𝑣	𝑣	ADP
cana-3959	277	66	ℜ𝑎	ℜ𝑎	PROPN
cana-3959	277	67	)	)	PUNCT
cana-3959	277	68	,	,	PUNCT
cana-3959	277	69	𝑟	𝑟	X
cana-3959	277	70	)	)	PUNCT
cana-3959	277	71	≥	≥	NOUN
cana-3959	277	72	𝑁′(𝛽(𝑣	𝑁′(𝛽(𝑣	NUM
cana-3959	277	73	)	)	PUNCT
cana-3959	277	74	,	,	PUNCT
cana-3959	277	75	𝑟	𝑟	X
cana-3959	277	76	)	)	PUNCT
cana-3959	277	77	⇒	⇒	NOUN
cana-3959	277	78	𝑑(𝐶	𝑑(𝐶	PUNCT
cana-3959	277	79	,	,	PUNCT
cana-3959	277	80	𝑇𝐶	𝑇𝐶	PROPN
cana-3959	277	81	)	)	PUNCT
cana-3959	277	82	≤	≤	NOUN
cana-3959	277	83	1	1	NUM
cana-3959	277	84	=	=	NUM
cana-3959	277	85	𝐿0	𝐿0	ADJ
cana-3959	277	86	=	=	PUNCT
cana-3959	277	87	𝐿1−𝑖	𝐿1−𝑖	X
cana-3959	277	88	(	(	PUNCT
cana-3959	277	89	31	31	NUM
cana-3959	277	90	)	)	PUNCT
cana-3959	277	91	then	then	ADV
cana-3959	277	92	from	from	ADP
cana-3959	277	93	(	(	PUNCT
cana-3959	277	94	29	29	NUM
cana-3959	277	95	)	)	PUNCT
cana-3959	277	96	and	and	CCONJ
cana-3959	277	97	(	(	PUNCT
cana-3959	277	98	31	31	NUM
cana-3959	277	99	)	)	PUNCT
cana-3959	277	100	we	we	PRON
cana-3959	277	101	can	can	AUX
cana-3959	277	102	conclude	conclude	VERB
cana-3959	277	103	,	,	PUNCT
cana-3959	277	104	𝑑(𝐶	𝑑(𝐶	ADJ
cana-3959	277	105	,	,	PUNCT
cana-3959	277	106	𝑇𝐶	𝑇𝐶	ADJ
cana-3959	277	107	)	)	PUNCT
cana-3959	277	108	≤	≤	NUM
cana-3959	278	1	𝐿1−𝑖	𝐿1−𝑖	ADP
cana-3959	278	2	<	<	X
cana-3959	278	3	∞	∞	NUM
cana-3959	278	4	now	now	ADV
cana-3959	278	5	from	from	ADP
cana-3959	278	6	the	the	DET
cana-3959	278	7	fixed	fix	VERB
cana-3959	278	8	point	point	NOUN
cana-3959	278	9	alternative	alternative	NOUN
cana-3959	278	10	in	in	ADP
cana-3959	278	11	both	both	DET
cana-3959	278	12	cases	case	NOUN
cana-3959	278	13	,	,	PUNCT
cana-3959	278	14	it	it	PRON
cana-3959	278	15	follows	follow	VERB
cana-3959	278	16	that	that	SCONJ
cana-3959	278	17	there	there	PRON
cana-3959	278	18	exists	exist	VERB
cana-3959	278	19	a	a	DET
cana-3959	278	20	fixed	fix	VERB
cana-3959	278	21	point	point	NOUN
cana-3959	278	22	𝐶	𝐶	PROPN
cana-3959	278	23	of	of	ADP
cana-3959	278	24	𝑇	𝑇	PROPN
cana-3959	278	25	in	in	ADP
cana-3959	278	26	ω	ω	NUM
cana-3959	278	27	such	such	ADJ
cana-3959	278	28	that	that	SCONJ
cana-3959	278	29	𝒞(𝑣	𝒞(𝑣	NOUN
cana-3959	278	30	)	)	PUNCT
cana-3959	279	1	=	=	SYM
cana-3959	279	2	𝑁	𝑁	PROPN
cana-3959	279	3	−	−	PROPN
cana-3959	279	4	lim	lim	NOUN
cana-3959	279	5	𝑘→∞	𝑘→∞	NUM
cana-3959	279	6	ℱ(𝑛𝑘𝑣	ℱ(𝑛𝑘𝑣	NOUN
cana-3959	279	7	)	)	PUNCT
cana-3959	279	8	𝑛3𝑘	𝑛3𝑘	NOUN
cana-3959	279	9	,	,	PUNCT
cana-3959	279	10	∀𝑣	∀𝑣	PROPN
cana-3959	279	11	∈	∈	PROPN
cana-3959	279	12	𝑋	𝑋	PROPN
cana-3959	279	13	,	,	PUNCT
cana-3959	279	14	𝑟	𝑟	X
cana-3959	279	15	>	>	X
cana-3959	279	16	0	0	NUM
cana-3959	279	17	.	.	PUNCT
cana-3959	280	1	(	(	PUNCT
cana-3959	280	2	32	32	NUM
cana-3959	280	3	)	)	PUNCT
cana-3959	280	4	replacing	replace	VERB
cana-3959	280	5	(	(	PUNCT
cana-3959	280	6	𝑣,𝑤	𝑣,𝑤	ADJ
cana-3959	280	7	)	)	PUNCT
cana-3959	280	8	by	by	ADP
cana-3959	280	9	(	(	PUNCT
cana-3959	280	10	𝛿𝑖𝑣	𝛿𝑖𝑣	NOUN
cana-3959	280	11	,	,	PUNCT
cana-3959	280	12	𝛿𝑖𝑤	𝛿𝑖𝑤	NOUN
cana-3959	280	13	)	)	PUNCT
cana-3959	280	14	in	in	ADP
cana-3959	280	15	(	(	PUNCT
cana-3959	280	16	23	23	NUM
cana-3959	280	17	)	)	PUNCT
cana-3959	280	18	,	,	PUNCT
cana-3959	280	19	we	we	PRON
cana-3959	280	20	arrive	arrive	VERB
cana-3959	280	21	𝒩	𝒩	PROPN
cana-3959	280	22	(	(	PUNCT
cana-3959	280	23	1	1	NUM
cana-3959	280	24	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	280	25	3𝑘𝐷ℱ(𝛿𝑖𝑣	3𝑘𝐷ℱ(𝛿𝑖𝑣	NUM
cana-3959	280	26	,	,	PUNCT
cana-3959	280	27	𝛿𝑖𝑤	𝛿𝑖𝑤	PROPN
cana-3959	280	28	)	)	PUNCT
cana-3959	280	29	,	,	PUNCT
cana-3959	280	30	𝑟	𝑟	X
cana-3959	280	31	)	)	PUNCT
cana-3959	280	32	≥	≥	NOUN
cana-3959	280	33	𝑁′(𝔔(𝛿𝑖𝑣	𝑁′(𝔔(𝛿𝑖𝑣	PROPN
cana-3959	280	34	,	,	PUNCT
cana-3959	280	35	𝛿𝑖𝑤	𝛿𝑖𝑤	PROPN
cana-3959	280	36	)	)	PUNCT
cana-3959	280	37	,	,	PUNCT
cana-3959	280	38	𝛿𝑖	𝛿𝑖	ADP
cana-3959	280	39	3𝑘𝑟	3𝑘𝑟	ADJ
cana-3959	280	40	)	)	PUNCT
cana-3959	280	41	(	(	PUNCT
cana-3959	280	42	33	33	NUM
cana-3959	280	43	)	)	PUNCT
cana-3959	280	44	for	for	ADP
cana-3959	280	45	all	all	DET
cana-3959	280	46	𝑟	𝑟	NOUN
cana-3959	280	47	>	>	PUNCT
cana-3959	280	48	0	0	PUNCT
cana-3959	280	49	and	and	CCONJ
cana-3959	280	50	all	all	DET
cana-3959	280	51	𝑣,𝑤	𝑣,𝑤	PROPN
cana-3959	280	52	∈	∈	PROPN
cana-3959	280	53	𝑋	𝑋	PROPN
cana-3959	280	54	,	,	PUNCT
cana-3959	280	55	we	we	PRON
cana-3959	280	56	can	can	AUX
cana-3959	280	57	prove	prove	VERB
cana-3959	280	58	the	the	DET
cana-3959	280	59	function	function	NOUN
cana-3959	280	60	,	,	PUNCT
cana-3959	280	61	𝐶	𝐶	PROPN
cana-3959	280	62	:	:	PUNCT
cana-3959	280	63	𝑋	𝑋	PROPN
cana-3959	280	64	→	→	SYM
cana-3959	280	65	𝑌	𝑌	PROPN
cana-3959	280	66	satisfies	satisfy	VERB
cana-3959	280	67	the	the	DET
cana-3959	280	68	functional	functional	ADJ
cana-3959	280	69	equation	equation	NOUN
cana-3959	280	70	(	(	PUNCT
cana-3959	280	71	1	1	NUM
cana-3959	280	72	)	)	PUNCT
cana-3959	280	73	.	.	PUNCT
cana-3959	281	1	by	by	ADP
cana-3959	281	2	fixed	fix	VERB
cana-3959	281	3	point	point	NOUN
cana-3959	281	4	alternative	alternative	NOUN
cana-3959	281	5	,	,	PUNCT
cana-3959	281	6	since	since	SCONJ
cana-3959	281	7	𝐶	𝐶	PROPN
cana-3959	281	8	is	be	AUX
cana-3959	281	9	unique	unique	ADJ
cana-3959	281	10	fixed	fix	VERB
cana-3959	281	11	point	point	NOUN
cana-3959	281	12	of	of	ADP
cana-3959	281	13	𝑇	𝑇	PROPN
cana-3959	281	14	in	in	ADP
cana-3959	281	15	the	the	DET
cana-3959	281	16	set	set	NOUN
cana-3959	281	17	δ	δ	NOUN
cana-3959	281	18	=	=	PRON
cana-3959	281	19	{	{	PUNCT
cana-3959	281	20	𝑓	𝑓	PROPN
cana-3959	281	21	∈	∈	PROPN
cana-3959	281	22	ω|𝑑(𝑓	ω|𝑑(𝑓	PROPN
cana-3959	281	23	,	,	PUNCT
cana-3959	281	24	𝐶	𝐶	PROPN
cana-3959	281	25	)	)	PUNCT
cana-3959	281	26	<	<	X
cana-3959	281	27	∞	∞	PROPN
cana-3959	281	28	}	}	PUNCT
cana-3959	281	29	,	,	PUNCT
cana-3959	281	30	therefore	therefore	ADV
cana-3959	281	31	𝐶	𝐶	PROPN
cana-3959	281	32	is	be	AUX
cana-3959	281	33	a	a	DET
cana-3959	281	34	uniqe	uniqe	ADJ
cana-3959	281	35	function	function	NOUN
cana-3959	281	36	such	such	ADJ
cana-3959	281	37	that	that	SCONJ
cana-3959	281	38	communications	communication	NOUN
cana-3959	281	39	on	on	ADP
cana-3959	281	40	applied	apply	VERB
cana-3959	281	41	nonlinear	nonlinear	ADJ
cana-3959	281	42	analysis	analysis	NOUN
cana-3959	281	43	issn	issn	NOUN
cana-3959	281	44	:	:	PUNCT
cana-3959	281	45	1074	1074	NUM
cana-3959	281	46	-	-	PUNCT
cana-3959	281	47	133x	133x	NUM
cana-3959	281	48	vol	vol	NOUN
cana-3959	281	49	32	32	NUM
cana-3959	281	50	no	no	NOUN
cana-3959	281	51	.	.	PUNCT
cana-3959	282	1	9s	9s	NUM
cana-3959	282	2	(	(	PUNCT
cana-3959	282	3	2025	2025	NUM
cana-3959	282	4	)	)	PUNCT
cana-3959	282	5	489	489	NUM
cana-3959	282	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	282	7	𝒩(ℱ(𝑣	𝒩(ℱ(𝑣	NOUN
cana-3959	282	8	)	)	PUNCT
cana-3959	282	9	−	−	PROPN
cana-3959	283	1	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	283	2	)	)	PUNCT
cana-3959	283	3	,	,	PUNCT
cana-3959	283	4	𝑟	𝑟	X
cana-3959	283	5	)	)	PUNCT
cana-3959	283	6	≥	≥	NOUN
cana-3959	283	7	𝑁′(𝐾𝛽(𝑣	𝑁′(𝐾𝛽(𝑣	PROPN
cana-3959	283	8	)	)	PUNCT
cana-3959	283	9	,	,	PUNCT
cana-3959	284	1	𝑟	𝑟	X
cana-3959	284	2	)	)	PUNCT
cana-3959	284	3	(	(	PUNCT
cana-3959	284	4	34	34	NUM
cana-3959	284	5	)	)	PUNCT
cana-3959	284	6	for	for	ADP
cana-3959	284	7	all	all	PRON
cana-3959	284	8	𝑣	𝑣	DET
cana-3959	284	9	∈	∈	PROPN
cana-3959	284	10	𝑋	𝑋	PROPN
cana-3959	284	11	,	,	PUNCT
cana-3959	284	12	𝑟	𝑟	X
cana-3959	284	13	>	>	PUNCT
cana-3959	284	14	0	0	PUNCT
cana-3959	285	1	and	and	CCONJ
cana-3959	285	2	𝐾	𝐾	PROPN
cana-3959	285	3	>	>	X
cana-3959	285	4	0	0	X
cana-3959	285	5	.	.	PUNCT
cana-3959	285	6	again	again	ADV
cana-3959	285	7	using	use	VERB
cana-3959	285	8	the	the	DET
cana-3959	285	9	fixed	fix	VERB
cana-3959	285	10	point	point	NOUN
cana-3959	285	11	alternative	alternative	NOUN
cana-3959	285	12	,	,	PUNCT
cana-3959	285	13	we	we	PRON
cana-3959	285	14	obtain	obtain	VERB
cana-3959	285	15	𝑑(𝐶	𝑑(𝐶	ADJ
cana-3959	285	16	,	,	PUNCT
cana-3959	285	17	𝒞	𝒞	PROPN
cana-3959	285	18	)	)	PUNCT
cana-3959	285	19	≤	≤	NUM
cana-3959	285	20	1	1	NUM
cana-3959	285	21	1−𝐿	1−𝐿	NUM
cana-3959	285	22	𝑑(𝐶	𝑑(𝐶	ADJ
cana-3959	285	23	,	,	PUNCT
cana-3959	285	24	𝑇𝐶	𝑇𝐶	ADJ
cana-3959	285	25	)	)	PUNCT
cana-3959	285	26	⇒	⇒	NOUN
cana-3959	285	27	𝑑(𝐶	𝑑(𝐶	PUNCT
cana-3959	285	28	,	,	PUNCT
cana-3959	285	29	𝒞	𝒞	PROPN
cana-3959	285	30	)	)	PUNCT
cana-3959	285	31	≤	≤	NUM
cana-3959	285	32	𝐿1−𝑖	𝐿1−𝑖	NOUN
cana-3959	285	33	1−𝐿	1−𝐿	NUM
cana-3959	285	34	⇒	⇒	NOUN
cana-3959	285	35	𝒩(ℱ(𝑣	𝒩(ℱ(𝑣	NOUN
cana-3959	285	36	)	)	PUNCT
cana-3959	285	37	−	−	PROPN
cana-3959	286	1	𝒞(𝑣	𝒞(𝑣	CCONJ
cana-3959	286	2	)	)	PUNCT
cana-3959	286	3	,	,	PUNCT
cana-3959	286	4	𝑟	𝑟	X
cana-3959	286	5	)	)	PUNCT
cana-3959	286	6	≥	≥	NOUN
cana-3959	286	7	𝑁′	𝑁′	ADJ
cana-3959	286	8	(	(	PUNCT
cana-3959	286	9	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-3959	286	10	1−𝐿	1−𝐿	NUM
cana-3959	286	11	𝛽(𝑣	𝛽(𝑣	NOUN
cana-3959	286	12	)	)	PUNCT
cana-3959	286	13	,	,	PUNCT
cana-3959	286	14	𝑟	𝑟	NOUN
cana-3959	286	15	)	)	PUNCT
cana-3959	286	16	,	,	PUNCT
cana-3959	286	17	(	(	PUNCT
cana-3959	286	18	35	35	NUM
cana-3959	286	19	)	)	PUNCT
cana-3959	286	20	for	for	ADP
cana-3959	286	21	all	all	DET
cana-3959	286	22	𝑣	𝑣	DET
cana-3959	286	23	∈	∈	NOUN
cana-3959	286	24	𝑋	𝑋	NOUN
cana-3959	286	25	and	and	CCONJ
cana-3959	286	26	𝑟	𝑟	NOUN
cana-3959	286	27	>	>	X
cana-3959	286	28	0	0	X
cana-3959	286	29	.	.	PUNCT
cana-3959	287	1	corollary	corollary	ADJ
cana-3959	287	2	5.2	5.2	NUM
cana-3959	287	3	suppose	suppose	VERB
cana-3959	287	4	that	that	SCONJ
cana-3959	287	5	a	a	DET
cana-3959	287	6	function	function	NOUN
cana-3959	287	7	f	f	X
cana-3959	287	8	:x	:x	PROPN
cana-3959	287	9	→	→	SYM
cana-3959	287	10	y	y	PROPN
cana-3959	287	11	satisfies	satisfy	VERB
cana-3959	287	12	theinequality	theinequality	NOUN
cana-3959	287	13	𝑁(𝐷ℱ(𝑣	𝑁(𝐷ℱ(𝑣	PROPN
cana-3959	287	14	,	,	PUNCT
cana-3959	287	15	𝑤	𝑤	ADP
cana-3959	287	16	)	)	PUNCT
cana-3959	287	17	,	,	PUNCT
cana-3959	287	18	𝑟	𝑟	X
cana-3959	287	19	)	)	PUNCT
cana-3959	287	20	≥	≥	NOUN
cana-3959	287	21	{	{	PUNCT
cana-3959	287	22	𝑁′(𝜖	𝑁′(𝜖	PROPN
cana-3959	287	23	,	,	PUNCT
cana-3959	287	24	𝑟	𝑟	NOUN
cana-3959	287	25	)	)	PUNCT
cana-3959	287	26	,	,	PUNCT
cana-3959	287	27	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	X
cana-3959	287	28	+	+	CCONJ
cana-3959	287	29	||𝑤||𝑠	||𝑤||𝑠	PROPN
cana-3959	287	30	,	,	PUNCT
cana-3959	287	31	𝑟	𝑟	NOUN
cana-3959	287	32	)	)	PUNCT
cana-3959	287	33	,	,	PUNCT
cana-3959	287	34	𝑠	𝑠	PROPN
cana-3959	287	35	≠	≠	PROPN
cana-3959	287	36	3	3	NUM
cana-3959	287	37	;	;	PUNCT
cana-3959	287	38	𝑁′(𝜖(||𝑣||𝑠||𝑤||𝑠	𝑁′(𝜖(||𝑣||𝑠||𝑤||𝑠	PROPN
cana-3959	287	39	+	+	CCONJ
cana-3959	287	40	{	{	PUNCT
cana-3959	287	41	||𝑣||2𝑠	||𝑣||2𝑠	ADJ
cana-3959	287	42	+	+	CCONJ
cana-3959	287	43	||𝑤||2𝑠	||𝑤||2𝑠	PROPN
cana-3959	287	44	}	}	PUNCT
cana-3959	287	45	)	)	PUNCT
cana-3959	287	46	,	,	PUNCT
cana-3959	287	47	𝑟	𝑟	X
cana-3959	287	48	)	)	PUNCT
cana-3959	287	49	,	,	PUNCT
cana-3959	287	50	𝑠	𝑠	PROPN
cana-3959	287	51	≠	≠	PROPN
cana-3959	287	52	3	3	NUM
cana-3959	287	53	2	2	NUM
cana-3959	287	54	;	;	PUNCT
cana-3959	287	55	(	(	PUNCT
cana-3959	287	56	36	36	NUM
cana-3959	287	57	)	)	PUNCT
cana-3959	287	58	for	for	ADP
cana-3959	287	59	all	all	DET
cana-3959	287	60	𝑣	𝑣	NOUN
cana-3959	287	61	,	,	PUNCT
cana-3959	287	62	𝑤	𝑤	ADP
cana-3959	287	63	∈	∈	PROPN
cana-3959	287	64	𝑋	𝑋	NOUN
cana-3959	287	65	and	and	CCONJ
cana-3959	287	66	𝑟	𝑟	NOUN
cana-3959	287	67	>	>	X
cana-3959	287	68	0	0	NUM
cana-3959	287	69	,	,	PUNCT
cana-3959	287	70	where	where	SCONJ
cana-3959	287	71	𝜖	𝜖	X
cana-3959	287	72	,	,	PUNCT
cana-3959	287	73	𝑠	𝑠	PROPN
cana-3959	287	74	are	be	AUX
cana-3959	287	75	constants	constant	NOUN
cana-3959	287	76	with	with	ADP
cana-3959	287	77	𝜖	𝜖	PROPN
cana-3959	287	78	>	>	X
cana-3959	287	79	0	0	NUM
cana-3959	287	80	.	.	PUNCT
cana-3959	288	1	then	then	ADV
cana-3959	288	2	there	there	PRON
cana-3959	288	3	exists	exist	VERB
cana-3959	288	4	a	a	DET
cana-3959	288	5	unique	unique	ADJ
cana-3959	288	6	cubic	cubic	ADJ
cana-3959	288	7	mapping	mapping	NOUN
cana-3959	288	8	𝐶	𝐶	PROPN
cana-3959	288	9	:	:	PUNCT
cana-3959	288	10	𝑋	𝑋	PROPN
cana-3959	288	11	→	→	SYM
cana-3959	288	12	𝑌	𝑌	PROPN
cana-3959	288	13	such	such	ADJ
cana-3959	288	14	that	that	DET
cana-3959	288	15	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	288	16	)	)	PUNCT
cana-3959	288	17	−	−	PROPN
cana-3959	288	18	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	288	19	)	)	PUNCT
cana-3959	288	20	,	,	PUNCT
cana-3959	288	21	𝑟	𝑟	X
cana-3959	288	22	)	)	PUNCT
cana-3959	288	23	≥	≥	NOUN
cana-3959	288	24	{	{	PUNCT
cana-3959	288	25	𝑁′(𝜖	𝑁′(𝜖	PROPN
cana-3959	288	26	,	,	PUNCT
cana-3959	288	27	|ℜ3𝑎	|ℜ3𝑎	PROPN
cana-3959	288	28	−	−	PROPN
cana-3959	288	29	1|𝑟	1|𝑟	NUM
cana-3959	288	30	)	)	PUNCT
cana-3959	288	31	,	,	PUNCT
cana-3959	288	32	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	PROPN
cana-3959	288	33	,	,	PUNCT
cana-3959	288	34	|ℜ3𝑎	|ℜ3𝑎	PROPN
cana-3959	288	35	−ℜ	−ℜ	PROPN
cana-3959	288	36	𝑎𝑠|𝑟	𝑎𝑠|𝑟	NOUN
cana-3959	288	37	)	)	PUNCT
cana-3959	288	38	,	,	PUNCT
cana-3959	288	39	𝑁′(𝜖||𝑣||2𝑠	𝑁′(𝜖||𝑣||2𝑠	NOUN
cana-3959	288	40	,	,	PUNCT
cana-3959	288	41	|ℜ3𝑎	|ℜ3𝑎	PROPN
cana-3959	288	42	−ℜ	−ℜ	PROPN
cana-3959	288	43	3𝑎𝑠|𝑟	3𝑎𝑠|𝑟	NUM
cana-3959	288	44	)	)	PUNCT
cana-3959	288	45	,	,	PUNCT
cana-3959	288	46	(	(	PUNCT
cana-3959	288	47	37	37	NUM
cana-3959	288	48	)	)	PUNCT
cana-3959	288	49	for	for	ADP
cana-3959	288	50	all	all	DET
cana-3959	288	51	𝑣	𝑣	DET
cana-3959	288	52	∈	∈	NOUN
cana-3959	288	53	𝑋	𝑋	NOUN
cana-3959	288	54	and	and	CCONJ
cana-3959	288	55	all	all	DET
cana-3959	288	56	𝑟	𝑟	NOUN
cana-3959	288	57	>	>	X
cana-3959	288	58	0	0	X
cana-3959	288	59	.	.	PUNCT
cana-3959	289	1	proof	proof	NOUN
cana-3959	289	2	.	.	PUNCT
cana-3959	290	1	setting	set	VERB
cana-3959	290	2	𝔔(𝑣,𝑤	𝔔(𝑣,𝑤	PUNCT
cana-3959	290	3	)	)	PUNCT
cana-3959	291	1	=	=	SYM
cana-3959	291	2	{	{	PUNCT
cana-3959	291	3	𝑁′(𝜖	𝑁′(𝜖	PROPN
cana-3959	291	4	,	,	PUNCT
cana-3959	291	5	𝑟	𝑟	NOUN
cana-3959	291	6	)	)	PUNCT
cana-3959	291	7	,	,	PUNCT
cana-3959	291	8	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	X
cana-3959	291	9	+	+	CCONJ
cana-3959	291	10	||𝑤||𝑠	||𝑤||𝑠	PROPN
cana-3959	291	11	,	,	PUNCT
cana-3959	291	12	𝑟	𝑟	NOUN
cana-3959	291	13	)	)	PUNCT
cana-3959	291	14	,	,	PUNCT
cana-3959	291	15	𝑁′(𝜖(||𝑣||𝑠||𝑤||𝑠	𝑁′(𝜖(||𝑣||𝑠||𝑤||𝑠	PROPN
cana-3959	291	16	+	+	CCONJ
cana-3959	291	17	{	{	PUNCT
cana-3959	291	18	||𝑣||2𝑠	||𝑣||2𝑠	ADJ
cana-3959	291	19	+	+	CCONJ
cana-3959	291	20	||𝑤||2𝑠	||𝑤||2𝑠	PROPN
cana-3959	291	21	}	}	PUNCT
cana-3959	291	22	)	)	PUNCT
cana-3959	291	23	,	,	PUNCT
cana-3959	291	24	𝑟	𝑟	X
cana-3959	291	25	)	)	PUNCT
cana-3959	291	26	for	for	ADP
cana-3959	291	27	all	all	DET
cana-3959	291	28	𝑣	𝑣	NOUN
cana-3959	291	29	,	,	PUNCT
cana-3959	291	30	𝑤	𝑤	X
cana-3959	291	31	∈	∈	PROPN
cana-3959	291	32	𝑋.	𝑋.	PROPN
cana-3959	291	33	then	then	ADV
cana-3959	291	34	,	,	PUNCT
cana-3959	291	35	𝑁′(𝔔(𝛿𝑖	𝑁′(𝔔(𝛿𝑖	PROPN
cana-3959	291	36	𝑘𝑣	𝑘𝑣	PROPN
cana-3959	291	37	,	,	PUNCT
cana-3959	291	38	0	0	NUM
cana-3959	291	39	)	)	PUNCT
cana-3959	291	40	,	,	PUNCT
cana-3959	291	41	𝛿𝑖	𝛿𝑖	ADP
cana-3959	291	42	3𝑘𝑟	3𝑘𝑟	ADJ
cana-3959	291	43	)	)	PUNCT
cana-3959	292	1	=	=	SYM
cana-3959	292	2	{	{	PUNCT
cana-3959	292	3	𝑁′(𝜖	𝑁′(𝜖	PROPN
cana-3959	292	4	,	,	PUNCT
cana-3959	292	5	𝛿𝑖	𝛿𝑖	ADP
cana-3959	292	6	3𝑘𝑟	3𝑘𝑟	ADJ
cana-3959	292	7	)	)	PUNCT
cana-3959	293	1	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	PROPN
cana-3959	293	2	,	,	PUNCT
cana-3959	293	3	𝛿𝑖	𝛿𝑖	X
cana-3959	293	4	(	(	PUNCT
cana-3959	293	5	3−𝑠)𝑘	3−𝑠)𝑘	PROPN
cana-3959	293	6	𝑟	𝑟	NOUN
cana-3959	293	7	)	)	PUNCT
cana-3959	293	8	𝑁′(𝜖||𝑣||2𝑠	𝑁′(𝜖||𝑣||2𝑠	NOUN
cana-3959	293	9	,	,	PUNCT
cana-3959	293	10	𝛿𝑖	𝛿𝑖	X
cana-3959	293	11	(	(	PUNCT
cana-3959	293	12	3−2𝑠)𝑘	3−2𝑠)𝑘	NUM
cana-3959	293	13	𝑟	𝑟	NOUN
cana-3959	293	14	)	)	PUNCT
cana-3959	293	15	=	=	PRON
cana-3959	293	16	{	{	PUNCT
cana-3959	293	17	→	→	SYM
cana-3959	293	18	1	1	NUM
cana-3959	293	19	𝑎𝑠	𝑎𝑠	PROPN
cana-3959	293	20	𝑘	𝑘	PROPN
cana-3959	293	21	→	→	SYM
cana-3959	293	22	∞	∞	PROPN
cana-3959	293	23	,	,	PUNCT
cana-3959	293	24	→	→	SYM
cana-3959	293	25	1	1	NUM
cana-3959	293	26	𝑎𝑠	𝑎𝑠	PROPN
cana-3959	293	27	𝑘	𝑘	PROPN
cana-3959	293	28	→	→	SYM
cana-3959	293	29	∞	∞	PROPN
cana-3959	293	30	,	,	PUNCT
cana-3959	293	31	→	→	SYM
cana-3959	293	32	1	1	NUM
cana-3959	293	33	𝑎𝑠	𝑎𝑠	PROPN
cana-3959	293	34	𝑘	𝑘	PROPN
cana-3959	293	35	→	→	SYM
cana-3959	293	36	∞.	∞.	PROPN
cana-3959	293	37	thus	thus	ADV
cana-3959	293	38	,	,	PUNCT
cana-3959	293	39	(	(	PUNCT
cana-3959	293	40	22	22	NUM
cana-3959	293	41	)	)	PUNCT
cana-3959	293	42	is	be	AUX
cana-3959	293	43	holds	hold	NOUN
cana-3959	293	44	.	.	PUNCT
cana-3959	294	1	but	but	CCONJ
cana-3959	294	2	we	we	PRON
cana-3959	294	3	have	have	VERB
cana-3959	294	4	𝛽(𝑣	𝛽(𝑣	ADJ
cana-3959	294	5	)	)	PUNCT
cana-3959	295	1	=	=	SYM
cana-3959	295	2	𝔔	𝔔	PROPN
cana-3959	295	3	(	(	PUNCT
cana-3959	295	4	𝑣	𝑣	ADP
cana-3959	295	5	ℜ𝑎	ℜ𝑎	PROPN
cana-3959	295	6	,	,	PUNCT
cana-3959	295	7	0	0	NUM
cana-3959	295	8	)	)	PUNCT
cana-3959	295	9	has	have	VERB
cana-3959	295	10	the	the	DET
cana-3959	295	11	property	property	NOUN
cana-3959	295	12	𝑁′	𝑁′	ADJ
cana-3959	295	13	(	(	PUNCT
cana-3959	295	14	𝐿	𝐿	PROPN
cana-3959	295	15	1	1	NUM
cana-3959	295	16	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	295	17	3𝛽(𝛿𝑖𝑣	3𝛽(𝛿𝑖𝑣	NUM
cana-3959	295	18	)	)	PUNCT
cana-3959	295	19	,	,	PUNCT
cana-3959	295	20	𝑟	𝑟	X
cana-3959	295	21	)	)	PUNCT
cana-3959	295	22	≥	≥	NOUN
cana-3959	295	23	𝑁′(𝛽(𝑣	𝑁′(𝛽(𝑣	NUM
cana-3959	295	24	)	)	PUNCT
cana-3959	295	25	,	,	PUNCT
cana-3959	295	26	𝑟	𝑟	NOUN
cana-3959	295	27	)	)	PUNCT
cana-3959	295	28	∀	∀	X
cana-3959	296	1	𝑣	𝑣	ADP
cana-3959	296	2	∈	∈	PROPN
cana-3959	296	3	𝑋	𝑋	PROPN
cana-3959	296	4	,	,	PUNCT
cana-3959	296	5	𝑟	𝑟	X
cana-3959	296	6	>	>	X
cana-3959	296	7	0	0	X
cana-3959	296	8	.	.	PUNCT
cana-3959	297	1	communications	communication	NOUN
cana-3959	297	2	on	on	ADP
cana-3959	297	3	applied	apply	VERB
cana-3959	297	4	nonlinear	nonlinear	ADJ
cana-3959	297	5	analysis	analysis	NOUN
cana-3959	297	6	issn	issn	NOUN
cana-3959	297	7	:	:	PUNCT
cana-3959	297	8	1074	1074	NUM
cana-3959	297	9	-	-	PUNCT
cana-3959	297	10	133x	133x	NUM
cana-3959	297	11	vol	vol	NOUN
cana-3959	297	12	32	32	NUM
cana-3959	297	13	no	no	NOUN
cana-3959	297	14	.	.	PUNCT
cana-3959	298	1	9s	9s	NUM
cana-3959	298	2	(	(	PUNCT
cana-3959	298	3	2025	2025	NUM
cana-3959	298	4	)	)	PUNCT
cana-3959	298	5	490	490	NUM
cana-3959	298	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	298	7	hence	hence	ADV
cana-3959	298	8	𝑁′(𝛽(𝑣	𝑁′(𝛽(𝑣	NUM
cana-3959	298	9	)	)	PUNCT
cana-3959	298	10	,	,	PUNCT
cana-3959	298	11	𝑟	𝑟	X
cana-3959	298	12	)	)	PUNCT
cana-3959	298	13	=	=	SYM
cana-3959	299	1	𝑁′	𝑁′	NOUN
cana-3959	299	2	(	(	PUNCT
cana-3959	299	3	𝔔	𝔔	PROPN
cana-3959	299	4	(	(	PUNCT
cana-3959	299	5	𝑥	𝑥	PROPN
cana-3959	299	6	ℜ𝑎	ℜ𝑎	PROPN
cana-3959	299	7	,	,	PUNCT
cana-3959	299	8	0	0	NUM
cana-3959	299	9	)	)	PUNCT
cana-3959	299	10	,	,	PUNCT
cana-3959	299	11	𝑟	𝑟	X
cana-3959	299	12	)	)	PUNCT
cana-3959	299	13	=	=	SYM
cana-3959	299	14	{	{	PUNCT
cana-3959	299	15	𝑁′(𝜖	𝑁′(𝜖	PROPN
cana-3959	299	16	,	,	PUNCT
cana-3959	299	17	𝑟	𝑟	NOUN
cana-3959	299	18	)	)	PUNCT
cana-3959	299	19	,	,	PUNCT
cana-3959	299	20	𝑁′	𝑁′	X
cana-3959	299	21	(	(	PUNCT
cana-3959	299	22	𝜖	𝜖	PROPN
cana-3959	299	23	ℜ𝑎𝑠	ℜ𝑎𝑠	PROPN
cana-3959	299	24	||𝑣||	||𝑣||	PROPN
cana-3959	299	25	𝑠	𝑠	PROPN
cana-3959	299	26	,	,	PUNCT
cana-3959	299	27	𝑟	𝑟	NOUN
cana-3959	299	28	)	)	PUNCT
cana-3959	299	29	,	,	PUNCT
cana-3959	299	30	𝑁′	𝑁′	X
cana-3959	299	31	(	(	PUNCT
cana-3959	299	32	𝜖	𝜖	PROPN
cana-3959	299	33	ℜ2𝑎𝑠	ℜ2𝑎𝑠	NOUN
cana-3959	299	34	||𝑣||	||𝑣||	PROPN
cana-3959	299	35	2𝑠	2𝑠	NOUN
cana-3959	299	36	,	,	PUNCT
cana-3959	299	37	𝑟	𝑟	NOUN
cana-3959	299	38	)	)	PUNCT
cana-3959	299	39	.	.	PUNCT
cana-3959	300	1	now	now	ADV
cana-3959	300	2	,	,	PUNCT
cana-3959	300	3	𝑁′	𝑁′	X
cana-3959	300	4	(	(	PUNCT
cana-3959	300	5	1	1	NUM
cana-3959	300	6	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	300	7	3𝛽(𝛿𝑖𝑣	3𝛽(𝛿𝑖𝑣	NUM
cana-3959	300	8	)	)	PUNCT
cana-3959	300	9	,	,	PUNCT
cana-3959	300	10	𝑟	𝑟	X
cana-3959	300	11	)	)	PUNCT
cana-3959	300	12	=	=	SYM
cana-3959	300	13	{	{	PUNCT
cana-3959	300	14	𝑁′	𝑁′	X
cana-3959	300	15	(	(	PUNCT
cana-3959	300	16	𝜖	𝜖	X
cana-3959	300	17	𝛿𝑖	𝛿𝑖	PROPN
cana-3959	300	18	3	3	NUM
cana-3959	300	19	,	,	PUNCT
cana-3959	300	20	𝑟	𝑟	NOUN
cana-3959	300	21	)	)	PUNCT
cana-3959	300	22	,	,	PUNCT
cana-3959	300	23	𝑁′	𝑁′	X
cana-3959	300	24	(	(	PUNCT
cana-3959	300	25	𝜖	𝜖	X
cana-3959	300	26	𝛿𝑖	𝛿𝑖	ADP
cana-3959	300	27	3	3	NUM
cana-3959	300	28	(	(	PUNCT
cana-3959	300	29	1	1	NUM
cana-3959	300	30	ℜ𝑎𝑠	ℜ𝑎𝑠	NOUN
cana-3959	300	31	)	)	PUNCT
cana-3959	300	32	||𝛿𝑖𝑣||	||𝛿𝑖𝑣||	NUM
cana-3959	300	33	𝑠	𝑠	NUM
cana-3959	300	34	,	,	PUNCT
cana-3959	300	35	𝑟	𝑟	NOUN
cana-3959	300	36	)	)	PUNCT
cana-3959	300	37	,	,	PUNCT
cana-3959	300	38	𝑁′	𝑁′	X
cana-3959	300	39	(	(	PUNCT
cana-3959	300	40	𝜖	𝜖	X
cana-3959	300	41	𝛿𝑖	𝛿𝑖	ADP
cana-3959	300	42	3	3	NUM
cana-3959	300	43	(	(	PUNCT
cana-3959	300	44	1	1	NUM
cana-3959	300	45	ℜ2𝑎𝑠	ℜ2𝑎𝑠	NOUN
cana-3959	300	46	)	)	PUNCT
cana-3959	300	47	||𝛿𝑖𝑣||	||𝛿𝑖𝑣||	NUM
cana-3959	300	48	2𝑠	2𝑠	NOUN
cana-3959	300	49	,	,	PUNCT
cana-3959	300	50	𝑟	𝑟	NOUN
cana-3959	300	51	)	)	PUNCT
cana-3959	300	52	=	=	SYM
cana-3959	300	53	{	{	PUNCT
cana-3959	300	54	𝑁′(𝛿𝑖	𝑁′(𝛿𝑖	NUM
cana-3959	300	55	−3𝛽(𝑣	−3𝛽(𝑣	PROPN
cana-3959	300	56	)	)	PUNCT
cana-3959	300	57	,	,	PUNCT
cana-3959	300	58	𝑟	𝑟	NOUN
cana-3959	300	59	)	)	PUNCT
cana-3959	300	60	,	,	PUNCT
cana-3959	300	61	𝑁′(𝛿𝑖	𝑁′(𝛿𝑖	X
cana-3959	300	62	𝑠−3𝛽(𝑣	𝑠−3𝛽(𝑣	PROPN
cana-3959	300	63	)	)	PUNCT
cana-3959	300	64	,	,	PUNCT
cana-3959	300	65	𝑟	𝑟	X
cana-3959	300	66	)	)	PUNCT
cana-3959	300	67	,	,	PUNCT
cana-3959	300	68	𝑁′(𝛿𝑖	𝑁′(𝛿𝑖	NUM
cana-3959	300	69	2𝑠−3𝛽(𝑣	2𝑠−3𝛽(𝑣	NOUN
cana-3959	300	70	)	)	PUNCT
cana-3959	300	71	,	,	PUNCT
cana-3959	300	72	𝑟	𝑟	NOUN
cana-3959	300	73	)	)	PUNCT
cana-3959	300	74	.	.	PUNCT
cana-3959	301	1	now	now	ADV
cana-3959	301	2	from	from	ADP
cana-3959	301	3	(	(	PUNCT
cana-3959	301	4	25	25	NUM
cana-3959	301	5	)	)	PUNCT
cana-3959	301	6	,	,	PUNCT
cana-3959	301	7	we	we	PRON
cana-3959	301	8	prove	prove	VERB
cana-3959	301	9	the	the	DET
cana-3959	301	10	following	follow	VERB
cana-3959	301	11	cases	case	NOUN
cana-3959	301	12	for	for	ADP
cana-3959	301	13	conditions	condition	NOUN
cana-3959	301	14	(	(	PUNCT
cana-3959	301	15	𝑖	𝑖	X
cana-3959	301	16	)	)	PUNCT
cana-3959	301	17	and	and	CCONJ
cana-3959	301	18	(	(	PUNCT
cana-3959	301	19	𝑖𝑖	𝑖𝑖	NOUN
cana-3959	301	20	)	)	PUNCT
cana-3959	301	21	.	.	PUNCT
cana-3959	302	1	case:1	case:1	PROPN
cana-3959	302	2	𝐿	𝐿	PROPN
cana-3959	302	3	=	=	PROPN
cana-3959	302	4	ℜ	ℜ	PROPN
cana-3959	302	5	−3𝑎	−3𝑎	PROPN
cana-3959	302	6	for	for	ADP
cana-3959	302	7	𝑠	𝑠	PROPN
cana-3959	302	8	=	=	SYM
cana-3959	302	9	0	0	PUNCT
cana-3959	303	1	if	if	SCONJ
cana-3959	303	2	𝑖	𝑖	PRON
cana-3959	303	3	=	=	NOUN
cana-3959	303	4	0	0	NUM
cana-3959	303	5	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	303	6	)	)	PUNCT
cana-3959	303	7	−	−	PROPN
cana-3959	304	1	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	304	2	)	)	PUNCT
cana-3959	304	3	,	,	PUNCT
cana-3959	304	4	𝑟	𝑟	X
cana-3959	304	5	)	)	PUNCT
cana-3959	304	6	≥	≥	NOUN
cana-3959	305	1	𝑁′	𝑁′	ADJ
cana-3959	305	2	(	(	PUNCT
cana-3959	305	3	ℜ−3𝑎	ℜ−3𝑎	PROPN
cana-3959	305	4	1−ℜ−3𝑎	1−ℜ−3𝑎	NUM
cana-3959	305	5	𝛽(𝑣	𝛽(𝑣	NOUN
cana-3959	305	6	)	)	PUNCT
cana-3959	305	7	,	,	PUNCT
cana-3959	305	8	𝑟	𝑟	X
cana-3959	305	9	)	)	PUNCT
cana-3959	305	10	=	=	SYM
cana-3959	305	11	𝑁′	𝑁′	NOUN
cana-3959	305	12	(	(	PUNCT
cana-3959	305	13	𝜖	𝜖	X
cana-3959	305	14	(	(	PUNCT
cana-3959	305	15	ℜ3𝑎−1	ℜ3𝑎−1	NOUN
cana-3959	305	16	)	)	PUNCT
cana-3959	305	17	,	,	PUNCT
cana-3959	305	18	𝑟	𝑟	X
cana-3959	305	19	)	)	PUNCT
cana-3959	305	20	=	=	SYM
cana-3959	305	21	𝑁′(𝜖	𝑁′(𝜖	PROPN
cana-3959	305	22	,	,	PUNCT
cana-3959	305	23	(	(	PUNCT
cana-3959	305	24	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	305	25	−	−	PROPN
cana-3959	305	26	1)𝑟	1)𝑟	NUM
cana-3959	305	27	)	)	PUNCT
cana-3959	305	28	.	.	PUNCT
cana-3959	306	1	case:2	case:2	PROPN
cana-3959	306	2	𝐿	𝐿	PROPN
cana-3959	306	3	=	=	SYM
cana-3959	306	4	ℜ	ℜ	PROPN
cana-3959	306	5	3𝑎	3𝑎	NOUN
cana-3959	306	6	for	for	ADP
cana-3959	306	7	𝑠	𝑠	PROPN
cana-3959	306	8	=	=	SYM
cana-3959	306	9	0	0	PUNCT
cana-3959	307	1	if	if	SCONJ
cana-3959	307	2	𝑖	𝑖	PRON
cana-3959	307	3	=	=	SYM
cana-3959	307	4	1	1	NUM
cana-3959	307	5	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	307	6	)	)	PUNCT
cana-3959	307	7	−	−	PROPN
cana-3959	308	1	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	308	2	)	)	PUNCT
cana-3959	308	3	,	,	PUNCT
cana-3959	308	4	𝑟	𝑟	X
cana-3959	308	5	)	)	PUNCT
cana-3959	308	6	≥	≥	NOUN
cana-3959	308	7	𝑁′	𝑁′	X
cana-3959	308	8	(	(	PUNCT
cana-3959	308	9	1	1	NUM
cana-3959	308	10	1−ℜ3𝑎𝛽(𝑣	1−ℜ3𝑎𝛽(𝑣	NUM
cana-3959	308	11	)	)	PUNCT
cana-3959	308	12	,	,	PUNCT
cana-3959	308	13	𝑟	𝑟	X
cana-3959	308	14	)	)	PUNCT
cana-3959	308	15	=	=	SYM
cana-3959	309	1	𝑁′	𝑁′	NOUN
cana-3959	309	2	(	(	PUNCT
cana-3959	309	3	𝜖	𝜖	X
cana-3959	309	4	(	(	PUNCT
cana-3959	309	5	1−ℜ3𝑎	1−ℜ3𝑎	NUM
cana-3959	309	6	)	)	PUNCT
cana-3959	309	7	,	,	PUNCT
cana-3959	309	8	𝑟	𝑟	X
cana-3959	309	9	)	)	PUNCT
cana-3959	309	10	=	=	SYM
cana-3959	309	11	𝑁′(𝜖	𝑁′(𝜖	PROPN
cana-3959	309	12	,	,	PUNCT
cana-3959	309	13	(	(	PUNCT
cana-3959	309	14	1−ℜ	1−ℜ	NUM
cana-3959	309	15	3𝑎)𝑟	3𝑎)𝑟	NUM
cana-3959	309	16	)	)	PUNCT
cana-3959	309	17	.	.	PUNCT
cana-3959	310	1	case:3	case:3	PROPN
cana-3959	310	2	𝐿	𝐿	PROPN
cana-3959	310	3	=	=	SYM
cana-3959	310	4	ℜ	ℜ	PROPN
cana-3959	310	5	𝑎(𝑠−3	𝑎(𝑠−3	PROPN
cana-3959	310	6	)	)	PUNCT
cana-3959	310	7	for	for	ADP
cana-3959	310	8	𝑠	𝑠	PROPN
cana-3959	310	9	>	>	ADP
cana-3959	310	10	3	3	NUM
cana-3959	311	1	if	if	SCONJ
cana-3959	311	2	𝑖	𝑖	ADP
cana-3959	311	3	=	=	NOUN
cana-3959	311	4	0	0	NUM
cana-3959	311	5	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	311	6	)	)	PUNCT
cana-3959	311	7	−	−	PROPN
cana-3959	311	8	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	311	9	)	)	PUNCT
cana-3959	311	10	,	,	PUNCT
cana-3959	311	11	𝑟	𝑟	X
cana-3959	311	12	)	)	PUNCT
cana-3959	311	13	≥	≥	NOUN
cana-3959	311	14	𝑁′	𝑁′	X
cana-3959	311	15	(	(	PUNCT
cana-3959	311	16	ℜ𝑎(𝑠−3	ℜ𝑎(𝑠−3	PROPN
cana-3959	311	17	)	)	PUNCT
cana-3959	311	18	1−ℜ𝑎(𝑠−3)𝛽(𝑣	1−ℜ𝑎(𝑠−3)𝛽(𝑣	NUM
cana-3959	311	19	)	)	PUNCT
cana-3959	311	20	,	,	PUNCT
cana-3959	311	21	𝑟	𝑟	X
cana-3959	311	22	)	)	PUNCT
cana-3959	311	23	=	=	SYM
cana-3959	311	24	𝑁′	𝑁′	NOUN
cana-3959	311	25	(	(	PUNCT
cana-3959	311	26	𝜖	𝜖	X
cana-3959	311	27	(	(	PUNCT
cana-3959	311	28	ℜ3𝑎−ℜ𝑎𝑠	ℜ3𝑎−ℜ𝑎𝑠	NOUN
cana-3959	311	29	)	)	PUNCT
cana-3959	311	30	||𝑣||𝑠	||𝑣||𝑠	PROPN
cana-3959	311	31	,	,	PUNCT
cana-3959	311	32	𝑟	𝑟	NOUN
cana-3959	311	33	)	)	PUNCT
cana-3959	311	34	=	=	SYM
cana-3959	311	35	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	NOUN
cana-3959	311	36	,	,	PUNCT
cana-3959	311	37	(	(	PUNCT
cana-3959	311	38	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	311	39	−ℜ	−ℜ	PROPN
cana-3959	311	40	𝑎𝑠)𝑟	𝑎𝑠)𝑟	PROPN
cana-3959	311	41	)	)	PUNCT
cana-3959	311	42	.	.	PUNCT
cana-3959	312	1	case:4	case:4	PRON
cana-3959	312	2	𝐿	𝐿	PROPN
cana-3959	312	3	=	=	PUNCT
cana-3959	312	4	ℜ	ℜ	PROPN
cana-3959	312	5	𝑎(3−𝑠	𝑎(3−𝑠	NOUN
cana-3959	312	6	)	)	PUNCT
cana-3959	312	7	for	for	ADP
cana-3959	312	8	𝑠	𝑠	PROPN
cana-3959	312	9	<	<	X
cana-3959	312	10	3	3	NUM
cana-3959	312	11	if	if	SCONJ
cana-3959	312	12	𝑖	𝑖	PRON
cana-3959	312	13	=	=	SYM
cana-3959	312	14	1	1	NUM
cana-3959	312	15	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	312	16	)	)	PUNCT
cana-3959	312	17	−	−	PROPN
cana-3959	312	18	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	312	19	)	)	PUNCT
cana-3959	312	20	,	,	PUNCT
cana-3959	312	21	𝑟	𝑟	X
cana-3959	312	22	)	)	PUNCT
cana-3959	312	23	≥	≥	NOUN
cana-3959	312	24	𝑁′	𝑁′	X
cana-3959	312	25	(	(	PUNCT
cana-3959	312	26	1	1	NUM
cana-3959	312	27	1−ℜ𝑎(3−𝑠	1−ℜ𝑎(3−𝑠	NUM
cana-3959	312	28	)	)	PUNCT
cana-3959	312	29	𝛽(𝑣	𝛽(𝑣	NOUN
cana-3959	312	30	)	)	PUNCT
cana-3959	312	31	,	,	PUNCT
cana-3959	312	32	𝑟	𝑟	X
cana-3959	312	33	)	)	PUNCT
cana-3959	312	34	=	=	SYM
cana-3959	312	35	𝑁′	𝑁′	NOUN
cana-3959	312	36	(	(	PUNCT
cana-3959	312	37	𝜖	𝜖	X
cana-3959	312	38	(	(	PUNCT
cana-3959	312	39	ℜ𝑎𝑠−ℜ3𝑎	ℜ𝑎𝑠−ℜ3𝑎	PROPN
cana-3959	312	40	)	)	PUNCT
cana-3959	312	41	||𝑣||𝑠	||𝑣||𝑠	PROPN
cana-3959	312	42	,	,	PUNCT
cana-3959	312	43	𝑟	𝑟	NOUN
cana-3959	312	44	)	)	PUNCT
cana-3959	313	1	=	=	SYM
cana-3959	313	2	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	PROPN
cana-3959	313	3	,	,	PUNCT
cana-3959	313	4	(	(	PUNCT
cana-3959	313	5	ℜ𝑎𝑠	ℜ𝑎𝑠	PROPN
cana-3959	313	6	−ℜ	−ℜ	PROPN
cana-3959	313	7	3𝑎)𝑟	3𝑎)𝑟	NUM
cana-3959	313	8	)	)	PUNCT
cana-3959	313	9	.	.	PUNCT
cana-3959	314	1	case:5	case:5	VERB
cana-3959	314	2	𝐿	𝐿	PROPN
cana-3959	314	3	=	=	SYM
cana-3959	314	4	ℜ	ℜ	PROPN
cana-3959	314	5	𝑎(2𝑠−3	𝑎(2𝑠−3	NOUN
cana-3959	314	6	)	)	PUNCT
cana-3959	314	7	for	for	ADP
cana-3959	314	8	𝑠	𝑠	PROPN
cana-3959	314	9	>	>	SYM
cana-3959	314	10	3	3	NUM
cana-3959	314	11	2	2	NUM
cana-3959	314	12	if	if	SCONJ
cana-3959	314	13	𝑖	𝑖	ADP
cana-3959	314	14	=	=	NOUN
cana-3959	314	15	0	0	NUM
cana-3959	314	16	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	314	17	)	)	PUNCT
cana-3959	314	18	−	−	PROPN
cana-3959	315	1	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	315	2	)	)	PUNCT
cana-3959	315	3	,	,	PUNCT
cana-3959	315	4	𝑟	𝑟	X
cana-3959	315	5	)	)	PUNCT
cana-3959	315	6	≥	≥	NOUN
cana-3959	315	7	𝑁′	𝑁′	X
cana-3959	315	8	(	(	PUNCT
cana-3959	315	9	ℜ𝑎(2𝑠−3	ℜ𝑎(2𝑠−3	PROPN
cana-3959	315	10	)	)	PUNCT
cana-3959	315	11	1−ℜ𝑎(2𝑠−3)𝛽(𝑣	1−ℜ𝑎(2𝑠−3)𝛽(𝑣	NUM
cana-3959	315	12	)	)	PUNCT
cana-3959	315	13	,	,	PUNCT
cana-3959	316	1	𝑟	𝑟	X
cana-3959	316	2	)	)	PUNCT
cana-3959	316	3	=	=	SYM
cana-3959	316	4	𝑁′	𝑁′	NOUN
cana-3959	316	5	(	(	PUNCT
cana-3959	316	6	𝜖	𝜖	X
cana-3959	316	7	(	(	PUNCT
cana-3959	316	8	ℜ3𝑎−ℜ2𝑎𝑠	ℜ3𝑎−ℜ2𝑎𝑠	NOUN
cana-3959	316	9	)	)	PUNCT
cana-3959	316	10	||𝑣||𝑠	||𝑣||𝑠	PROPN
cana-3959	316	11	,	,	PUNCT
cana-3959	316	12	𝑟	𝑟	NOUN
cana-3959	316	13	)	)	PUNCT
cana-3959	316	14	=	=	SYM
cana-3959	316	15	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	NOUN
cana-3959	316	16	,	,	PUNCT
cana-3959	316	17	(	(	PUNCT
cana-3959	316	18	ℜ3𝑎	ℜ3𝑎	PROPN
cana-3959	316	19	−ℜ	−ℜ	PROPN
cana-3959	316	20	2𝑎𝑠)𝑟	2𝑎𝑠)𝑟	PROPN
cana-3959	316	21	)	)	PUNCT
cana-3959	316	22	.	.	PUNCT
cana-3959	317	1	communications	communication	NOUN
cana-3959	317	2	on	on	ADP
cana-3959	317	3	applied	apply	VERB
cana-3959	317	4	nonlinear	nonlinear	ADJ
cana-3959	317	5	analysis	analysis	NOUN
cana-3959	317	6	issn	issn	NOUN
cana-3959	317	7	:	:	PUNCT
cana-3959	317	8	1074	1074	NUM
cana-3959	317	9	-	-	PUNCT
cana-3959	317	10	133x	133x	NUM
cana-3959	317	11	vol	vol	NOUN
cana-3959	317	12	32	32	NUM
cana-3959	317	13	no	no	NOUN
cana-3959	317	14	.	.	PUNCT
cana-3959	318	1	9s	9s	NUM
cana-3959	318	2	(	(	PUNCT
cana-3959	318	3	2025	2025	NUM
cana-3959	318	4	)	)	PUNCT
cana-3959	318	5	491	491	NUM
cana-3959	319	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	319	2	case:6	case:6	NOUN
cana-3959	319	3	𝐿	𝐿	PROPN
cana-3959	319	4	=	=	PROPN
cana-3959	319	5	ℜ	ℜ	PROPN
cana-3959	319	6	𝑎(3−2𝑠	𝑎(3−2𝑠	NOUN
cana-3959	319	7	)	)	PUNCT
cana-3959	319	8	for	for	ADP
cana-3959	319	9	𝑠	𝑠	PROPN
cana-3959	319	10	<	<	X
cana-3959	319	11	3	3	NUM
cana-3959	319	12	2	2	NUM
cana-3959	319	13	if	if	SCONJ
cana-3959	319	14	𝑖	𝑖	PRON
cana-3959	319	15	=	=	SYM
cana-3959	319	16	1	1	NUM
cana-3959	319	17	𝒩(𝒞(𝑣	𝒩(𝒞(𝑣	NOUN
cana-3959	319	18	)	)	PUNCT
cana-3959	319	19	−	−	PROPN
cana-3959	319	20	ℱ(𝑣	ℱ(𝑣	NUM
cana-3959	319	21	)	)	PUNCT
cana-3959	319	22	,	,	PUNCT
cana-3959	319	23	𝑟	𝑟	X
cana-3959	319	24	)	)	PUNCT
cana-3959	319	25	≥	≥	NOUN
cana-3959	319	26	𝑁′	𝑁′	X
cana-3959	319	27	(	(	PUNCT
cana-3959	319	28	1	1	NUM
cana-3959	319	29	1−ℜ𝑎(3−2𝑠)𝛽(𝑣	1−ℜ𝑎(3−2𝑠)𝛽(𝑣	NUM
cana-3959	319	30	)	)	PUNCT
cana-3959	319	31	,	,	PUNCT
cana-3959	319	32	𝑟	𝑟	X
cana-3959	319	33	)	)	PUNCT
cana-3959	319	34	=	=	SYM
cana-3959	320	1	𝑁′	𝑁′	NOUN
cana-3959	320	2	(	(	PUNCT
cana-3959	320	3	𝜖	𝜖	X
cana-3959	320	4	(	(	PUNCT
cana-3959	320	5	ℜ2𝑎𝑠−ℜ3𝑎	ℜ2𝑎𝑠−ℜ3𝑎	NOUN
cana-3959	320	6	)	)	PUNCT
cana-3959	320	7	||𝑣||𝑠	||𝑣||𝑠	PROPN
cana-3959	320	8	,	,	PUNCT
cana-3959	320	9	𝑟	𝑟	NOUN
cana-3959	320	10	)	)	PUNCT
cana-3959	320	11	=	=	SYM
cana-3959	320	12	𝑁′(𝜖||𝑣||𝑠	𝑁′(𝜖||𝑣||𝑠	NOUN
cana-3959	320	13	,	,	PUNCT
cana-3959	320	14	(	(	PUNCT
cana-3959	320	15	ℜ2𝑎𝑠	ℜ2𝑎𝑠	NOUN
cana-3959	320	16	−ℜ	−ℜ	PROPN
cana-3959	320	17	3𝑎)𝑟	3𝑎)𝑟	NUM
cana-3959	320	18	)	)	PUNCT
cana-3959	320	19	.	.	PUNCT
cana-3959	321	1	6	6	NUM
cana-3959	321	2	conclusion	conclusion	NOUN
cana-3959	321	3	in	in	ADP
cana-3959	321	4	this	this	DET
cana-3959	321	5	paper	paper	NOUN
cana-3959	321	6	,	,	PUNCT
cana-3959	321	7	we	we	PRON
cana-3959	321	8	have	have	AUX
cana-3959	321	9	established	establish	VERB
cana-3959	321	10	novel	novel	ADJ
cana-3959	321	11	stability	stability	NOUN
cana-3959	321	12	results	result	NOUN
cana-3959	321	13	for	for	ADP
cana-3959	321	14	generalized	generalized	ADJ
cana-3959	321	15	alternate	alternate	ADJ
cana-3959	321	16	cubic	cubic	ADJ
cana-3959	321	17	functional	functional	ADJ
cana-3959	321	18	equations	equation	NOUN
cana-3959	321	19	using	use	VERB
cana-3959	321	20	the	the	DET
cana-3959	321	21	classical	classical	ADJ
cana-3959	321	22	method	method	NOUN
cana-3959	321	23	for	for	ADP
cana-3959	321	24	banach	banach	NOUN
cana-3959	321	25	spaces	space	NOUN
cana-3959	321	26	and	and	CCONJ
cana-3959	321	27	both	both	CCONJ
cana-3959	321	28	direct	direct	ADJ
cana-3959	321	29	and	and	CCONJ
cana-3959	321	30	fixed	fix	VERB
cana-3959	321	31	point	point	NOUN
cana-3959	321	32	approaches	approach	NOUN
cana-3959	321	33	for	for	ADP
cana-3959	321	34	fuzzy	fuzzy	ADJ
cana-3959	321	35	normed	normed	ADJ
cana-3959	321	36	spaces	space	NOUN
cana-3959	321	37	.	.	PUNCT
cana-3959	322	1	the	the	DET
cana-3959	322	2	classical	classical	ADJ
cana-3959	322	3	method	method	NOUN
cana-3959	322	4	provided	provide	VERB
cana-3959	322	5	a	a	DET
cana-3959	322	6	clear	clear	ADJ
cana-3959	322	7	pathway	pathway	NOUN
cana-3959	322	8	to	to	PART
cana-3959	322	9	demonstrate	demonstrate	VERB
cana-3959	322	10	hyers	hyers	PROPN
cana-3959	322	11	-	-	PUNCT
cana-3959	322	12	ulam	ulam	PROPN
cana-3959	322	13	stability	stability	NOUN
cana-3959	322	14	in	in	ADP
cana-3959	322	15	banach	banach	NOUN
cana-3959	322	16	spaces	space	NOUN
cana-3959	322	17	,	,	PUNCT
cana-3959	322	18	revealing	reveal	VERB
cana-3959	322	19	how	how	SCONJ
cana-3959	322	20	small	small	ADJ
cana-3959	322	21	deviations	deviation	NOUN
cana-3959	322	22	affect	affect	VERB
cana-3959	322	23	the	the	DET
cana-3959	322	24	functional	functional	ADJ
cana-3959	322	25	equation	equation	NOUN
cana-3959	322	26	’s	’s	PART
cana-3959	322	27	solutions	solution	NOUN
cana-3959	322	28	.	.	PUNCT
cana-3959	323	1	in	in	ADP
cana-3959	323	2	contrast	contrast	NOUN
cana-3959	323	3	,	,	PUNCT
cana-3959	323	4	the	the	DET
cana-3959	323	5	fuzzy	fuzzy	ADJ
cana-3959	323	6	normed	normed	ADJ
cana-3959	323	7	space	space	NOUN
cana-3959	323	8	framework	framework	NOUN
cana-3959	323	9	,	,	PUNCT
cana-3959	323	10	enriched	enrich	VERB
cana-3959	323	11	by	by	ADP
cana-3959	323	12	direct	direct	ADJ
cana-3959	323	13	and	and	CCONJ
cana-3959	323	14	fixed	fix	VERB
cana-3959	323	15	point	point	NOUN
cana-3959	323	16	methods	method	NOUN
cana-3959	323	17	,	,	PUNCT
cana-3959	323	18	allowed	allow	VERB
cana-3959	323	19	for	for	ADP
cana-3959	323	20	a	a	DET
cana-3959	323	21	more	more	ADV
cana-3959	323	22	nuanced	nuanced	ADJ
cana-3959	323	23	stability	stability	NOUN
cana-3959	323	24	analysis	analysis	NOUN
cana-3959	323	25	,	,	PUNCT
cana-3959	323	26	accommodating	accommodate	VERB
cana-3959	323	27	uncertainties	uncertainty	NOUN
cana-3959	323	28	and	and	CCONJ
cana-3959	323	29	imprecisions	imprecision	NOUN
cana-3959	323	30	intrinsic	intrinsic	ADJ
cana-3959	323	31	to	to	ADP
cana-3959	323	32	fuzzy	fuzzy	ADJ
cana-3959	323	33	systems	system	NOUN
cana-3959	323	34	.	.	PUNCT
cana-3959	324	1	our	our	PRON
cana-3959	324	2	results	result	NOUN
cana-3959	324	3	highlight	highlight	VERB
cana-3959	324	4	the	the	DET
cana-3959	324	5	effectiveness	effectiveness	NOUN
cana-3959	324	6	of	of	ADP
cana-3959	324	7	combining	combine	VERB
cana-3959	324	8	classical	classical	ADJ
cana-3959	324	9	techniques	technique	NOUN
cana-3959	324	10	with	with	ADP
cana-3959	324	11	fixed	fix	VERB
cana-3959	324	12	point	point	NOUN
cana-3959	324	13	theory	theory	NOUN
cana-3959	324	14	in	in	ADP
cana-3959	324	15	analyzing	analyze	VERB
cana-3959	324	16	functional	functional	ADJ
cana-3959	324	17	equations	equation	NOUN
cana-3959	324	18	under	under	ADP
cana-3959	324	19	different	different	ADJ
cana-3959	324	20	normed	normed	ADJ
cana-3959	324	21	environments	environment	NOUN
cana-3959	324	22	.	.	PUNCT
cana-3959	325	1	the	the	DET
cana-3959	325	2	comparative	comparative	ADJ
cana-3959	325	3	analysis	analysis	NOUN
cana-3959	325	4	between	between	ADP
cana-3959	325	5	deterministic	deterministic	ADJ
cana-3959	325	6	banach	banach	NOUN
cana-3959	325	7	spaces	space	NOUN
cana-3959	325	8	and	and	CCONJ
cana-3959	325	9	the	the	DET
cana-3959	325	10	more	more	ADV
cana-3959	325	11	flexible	flexible	ADJ
cana-3959	325	12	fuzzy	fuzzy	ADJ
cana-3959	325	13	normed	normed	ADJ
cana-3959	325	14	spaces	space	NOUN
cana-3959	325	15	underscores	underscore	VERB
cana-3959	325	16	the	the	DET
cana-3959	325	17	adaptability	adaptability	NOUN
cana-3959	325	18	of	of	ADP
cana-3959	325	19	the	the	DET
cana-3959	325	20	generalized	generalize	VERB
cana-3959	325	21	alternate	alternate	ADJ
cana-3959	325	22	cubic	cubic	ADJ
cana-3959	325	23	functional	functional	ADJ
cana-3959	325	24	equation	equation	NOUN
cana-3959	325	25	across	across	ADP
cana-3959	325	26	various	various	ADJ
cana-3959	325	27	mathematical	mathematical	ADJ
cana-3959	325	28	contexts	contexts	NOUN
cana-3959	325	29	.	.	PUNCT
cana-3959	326	1	these	these	DET
cana-3959	326	2	findings	finding	NOUN
cana-3959	326	3	offer	offer	VERB
cana-3959	326	4	significant	significant	ADJ
cana-3959	326	5	contributions	contribution	NOUN
cana-3959	326	6	to	to	ADP
cana-3959	326	7	the	the	DET
cana-3959	326	8	stability	stability	NOUN
cana-3959	326	9	theory	theory	NOUN
cana-3959	326	10	of	of	ADP
cana-3959	326	11	functional	functional	ADJ
cana-3959	326	12	equations	equation	NOUN
cana-3959	326	13	and	and	CCONJ
cana-3959	326	14	lay	lie	VERB
cana-3959	326	15	the	the	DET
cana-3959	326	16	groundwork	groundwork	NOUN
cana-3959	326	17	for	for	ADP
cana-3959	326	18	future	future	ADJ
cana-3959	326	19	applications	application	NOUN
cana-3959	326	20	in	in	ADP
cana-3959	326	21	both	both	CCONJ
cana-3959	326	22	pure	pure	ADJ
cana-3959	326	23	and	and	CCONJ
cana-3959	326	24	applied	applied	ADJ
cana-3959	326	25	mathematical	mathematical	ADJ
cana-3959	326	26	fields	field	NOUN
cana-3959	326	27	,	,	PUNCT
cana-3959	326	28	particularly	particularly	ADV
cana-3959	326	29	in	in	ADP
cana-3959	326	30	scenarios	scenario	NOUN
cana-3959	326	31	involving	involve	VERB
cana-3959	326	32	uncertain	uncertain	ADJ
cana-3959	326	33	or	or	CCONJ
cana-3959	326	34	fuzzy	fuzzy	ADJ
cana-3959	326	35	data	datum	NOUN
cana-3959	326	36	.	.	PUNCT
cana-3959	327	1	conflict	conflict	NOUN
cana-3959	327	2	of	of	ADP
cana-3959	327	3	interest	interest	NOUN
cana-3959	327	4	.	.	PUNCT
cana-3959	328	1	the	the	DET
cana-3959	328	2	authors	author	NOUN
cana-3959	328	3	declare	declare	VERB
cana-3959	328	4	that	that	SCONJ
cana-3959	328	5	they	they	PRON
cana-3959	328	6	have	have	VERB
cana-3959	328	7	no	no	DET
cana-3959	328	8	competing	compete	VERB
cana-3959	328	9	interests	interest	NOUN
cana-3959	328	10	.	.	PUNCT
cana-3959	329	1	references	reference	NOUN
cana-3959	329	2	[	[	X
cana-3959	329	3	1	1	NUM
cana-3959	329	4	]	]	X
cana-3959	329	5	s.m	s.m	PROPN
cana-3959	329	6	.	.	PROPN
cana-3959	329	7	ulam	ulam	PROPN
cana-3959	329	8	,	,	PUNCT
cana-3959	329	9	problems	problem	NOUN
cana-3959	329	10	in	in	ADP
cana-3959	329	11	modern	modern	ADJ
cana-3959	329	12	mathematics	mathematic	NOUN
cana-3959	329	13	,	,	PUNCT
cana-3959	329	14	science	science	NOUN
cana-3959	329	15	editions	edition	NOUN
cana-3959	329	16	,	,	PUNCT
cana-3959	329	17	wiley	wiley	NOUN
cana-3959	329	18	,	,	PUNCT
cana-3959	329	19	newyork	newyork	PROPN
cana-3959	329	20	,	,	PUNCT
cana-3959	329	21	1964	1964	NUM
cana-3959	329	22	.	.	PUNCT
cana-3959	330	1	[	[	X
cana-3959	330	2	2	2	NUM
cana-3959	330	3	]	]	X
cana-3959	330	4	d.h	d.h	PROPN
cana-3959	330	5	.	.	PROPN
cana-3959	330	6	hyers	hyer	NOUN
cana-3959	330	7	,	,	PUNCT
cana-3959	330	8	on	on	ADP
cana-3959	330	9	the	the	DET
cana-3959	330	10	stability	stability	NOUN
cana-3959	330	11	of	of	ADP
cana-3959	330	12	the	the	DET
cana-3959	330	13	linear	linear	ADJ
cana-3959	330	14	functional	functional	ADJ
cana-3959	330	15	equation	equation	NOUN
cana-3959	330	16	,	,	PUNCT
cana-3959	330	17	proc.nat	proc.nat	PROPN
cana-3959	330	18	.	.	PUNCT
cana-3959	331	1	acad.sci	acad.sci	X
cana-3959	331	2	.	.	PUNCT
cana-3959	331	3	,u.s.a	,u.s.a	PROPN
cana-3959	331	4	.	.	PUNCT
cana-3959	332	1	,27	,27	PROPN
cana-3959	332	2	,	,	PUNCT
cana-3959	332	3	1941	1941	NUM
cana-3959	332	4	,	,	PUNCT
cana-3959	332	5	222	222	NUM
cana-3959	332	6	-	-	SYM
cana-3959	332	7	224	224	NUM
cana-3959	332	8	.	.	PUNCT
cana-3959	333	1	[	[	X
cana-3959	333	2	3	3	NUM
cana-3959	333	3	]	]	PUNCT
cana-3959	333	4	th.m	th.m	PROPN
cana-3959	333	5	.	.	PUNCT
cana-3959	334	1	rassias	rassias	PROPN
cana-3959	334	2	,	,	PUNCT
cana-3959	334	3	on	on	ADP
cana-3959	334	4	the	the	DET
cana-3959	334	5	stability	stability	NOUN
cana-3959	334	6	of	of	ADP
cana-3959	334	7	the	the	DET
cana-3959	334	8	linear	linear	ADJ
cana-3959	334	9	mapping	mapping	NOUN
cana-3959	334	10	in	in	ADP
cana-3959	334	11	banach	banach	NOUN
cana-3959	334	12	spaces	space	NOUN
cana-3959	334	13	,	,	PUNCT
cana-3959	334	14	proc.amer.math.soc	proc.amer.math.soc	PROPN
cana-3959	334	15	.	.	PROPN
cana-3959	334	16	,	,	PUNCT
cana-3959	334	17	72	72	NUM
cana-3959	334	18	,	,	PUNCT
cana-3959	334	19	1978	1978	NUM
cana-3959	334	20	,	,	PUNCT
cana-3959	334	21	297	297	NUM
cana-3959	334	22	-	-	SYM
cana-3959	334	23	300	300	NUM
cana-3959	334	24	.	.	PUNCT
cana-3959	335	1	[	[	X
cana-3959	335	2	4	4	X
cana-3959	335	3	]	]	PUNCT
cana-3959	335	4	t.	t.	PROPN
cana-3959	335	5	aoki	aoki	PROPN
cana-3959	335	6	,	,	PUNCT
cana-3959	335	7	on	on	ADP
cana-3959	335	8	the	the	DET
cana-3959	335	9	stability	stability	NOUN
cana-3959	335	10	of	of	ADP
cana-3959	335	11	the	the	DET
cana-3959	335	12	linear	linear	ADJ
cana-3959	335	13	transformation	transformation	NOUN
cana-3959	335	14	in	in	ADP
cana-3959	335	15	banach	banach	NOUN
cana-3959	335	16	spaces	space	NOUN
cana-3959	335	17	,	,	PUNCT
cana-3959	335	18	j.	j.	PROPN
cana-3959	335	19	math	math	PROPN
cana-3959	335	20	.	.	PUNCT
cana-3959	336	1	soc	soc	PROPN
cana-3959	336	2	.	.	PUNCT
cana-3959	337	1	japan	japan	PROPN
cana-3959	337	2	,	,	PUNCT
cana-3959	337	3	2	2	NUM
cana-3959	337	4	,	,	PUNCT
cana-3959	337	5	1950	1950	NUM
cana-3959	337	6	,	,	PUNCT
cana-3959	337	7	64	64	NUM
cana-3959	337	8	-	-	SYM
cana-3959	337	9	66	66	NUM
cana-3959	337	10	.	.	PUNCT
cana-3959	338	1	[	[	X
cana-3959	338	2	5	5	X
cana-3959	338	3	]	]	PUNCT
cana-3959	338	4	p.	p.	NOUN
cana-3959	338	5	gavruta	gavruta	PROPN
cana-3959	338	6	,	,	PUNCT
cana-3959	338	7	a	a	DET
cana-3959	338	8	generalization	generalization	NOUN
cana-3959	338	9	of	of	ADP
cana-3959	338	10	the	the	DET
cana-3959	338	11	hyers	hyers	PROPN
cana-3959	338	12	-	-	PUNCT
cana-3959	338	13	ulam	ulam	ADJ
cana-3959	338	14	-	-	PUNCT
cana-3959	338	15	rassias	rassias	PROPN
cana-3959	338	16	stability	stability	NOUN
cana-3959	338	17	of	of	ADP
cana-3959	338	18	approximately	approximately	ADV
cana-3959	338	19	additive	additive	ADJ
cana-3959	338	20	mappings	mapping	NOUN
cana-3959	338	21	,	,	PUNCT
cana-3959	338	22	j.	j.	PROPN
cana-3959	338	23	math	math	PROPN
cana-3959	338	24	.	.	PUNCT
cana-3959	339	1	anal	anal	PROPN
cana-3959	339	2	.	.	PUNCT
cana-3959	339	3	appl	appl	PROPN
cana-3959	339	4	.	.	PROPN
cana-3959	340	1	,	,	PUNCT
cana-3959	341	1	184	184	NUM
cana-3959	341	2	1994	1994	NUM
cana-3959	341	3	,	,	PUNCT
cana-3959	341	4	431	431	NUM
cana-3959	341	5	-	-	SYM
cana-3959	341	6	436	436	NUM
cana-3959	341	7	.	.	PUNCT
cana-3959	342	1	[	[	X
cana-3959	342	2	6	6	NUM
cana-3959	342	3	]	]	X
cana-3959	342	4	j.m	j.m	PROPN
cana-3959	342	5	.	.	PROPN
cana-3959	342	6	rassias	rassias	PROPN
cana-3959	342	7	,	,	PUNCT
cana-3959	342	8	on	on	ADP
cana-3959	342	9	approximately	approximately	ADV
cana-3959	342	10	of	of	ADP
cana-3959	342	11	approximately	approximately	ADV
cana-3959	342	12	linear	linear	ADJ
cana-3959	342	13	mappings	mapping	NOUN
cana-3959	342	14	by	by	ADP
cana-3959	342	15	linear	linear	PROPN
cana-3959	342	16	mappings	mapping	NOUN
cana-3959	342	17	,	,	PUNCT
cana-3959	342	18	j.	j.	PROPN
cana-3959	342	19	funct	funct	PROPN
cana-3959	342	20	.	.	PUNCT
cana-3959	343	1	anal	anal	PROPN
cana-3959	343	2	.	.	PUNCT
cana-3959	344	1	usa	usa	PROPN
cana-3959	344	2	,	,	PUNCT
cana-3959	344	3	46	46	NUM
cana-3959	344	4	,	,	PUNCT
cana-3959	344	5	1982	1982	NUM
cana-3959	344	6	,	,	PUNCT
cana-3959	344	7	126	126	NUM
cana-3959	344	8	-	-	SYM
cana-3959	344	9	130	130	NUM
cana-3959	344	10	.	.	PUNCT
cana-3959	345	1	[	[	X
cana-3959	345	2	7	7	X
cana-3959	345	3	]	]	X
cana-3959	345	4	d.h	d.h	PROPN
cana-3959	345	5	.	.	PROPN
cana-3959	345	6	hyers	hyers	PROPN
cana-3959	345	7	,	,	PUNCT
cana-3959	345	8	g.	g.	PROPN
cana-3959	345	9	isac	isac	PROPN
cana-3959	345	10	,	,	PUNCT
cana-3959	345	11	th.m	th.m	PROPN
cana-3959	345	12	.	.	PUNCT
cana-3959	346	1	rassias	rassias	PROPN
cana-3959	346	2	,	,	PUNCT
cana-3959	346	3	stability	stability	NOUN
cana-3959	346	4	of	of	ADP
cana-3959	346	5	fun	fun	NOUN
cana-3959	346	6	eq	eq	NOUN
cana-3959	346	7	in	in	ADP
cana-3959	346	8	several	several	ADJ
cana-3959	346	9	variables	variable	NOUN
cana-3959	346	10	,	,	PUNCT
cana-3959	346	11	birkhauser	birkhauser	NOUN
cana-3959	346	12	,	,	PUNCT
cana-3959	346	13	basel	basel	PROPN
cana-3959	346	14	,	,	PUNCT
cana-3959	346	15	1998	1998	NUM
cana-3959	346	16	.	.	PUNCT
cana-3959	347	1	[	[	X
cana-3959	347	2	8	8	X
cana-3959	347	3	]	]	PUNCT
cana-3959	347	4	j.	j.	PROPN
cana-3959	347	5	aczel	aczel	PROPN
cana-3959	347	6	and	and	CCONJ
cana-3959	347	7	j.	j.	PROPN
cana-3959	347	8	dhombres	dhombres	PROPN
cana-3959	347	9	,	,	PUNCT
cana-3959	347	10	fun	fun	NOUN
cana-3959	347	11	eq	eq	NOUN
cana-3959	347	12	in	in	ADP
cana-3959	347	13	several	several	ADJ
cana-3959	347	14	variables	variable	NOUN
cana-3959	347	15	,	,	PUNCT
cana-3959	347	16	cambridge	cambridge	PROPN
cana-3959	347	17	univ	univ	PROPN
cana-3959	347	18	,	,	PUNCT
cana-3959	347	19	press	press	NOUN
cana-3959	347	20	,	,	PUNCT
cana-3959	347	21	1989	1989	NUM
cana-3959	347	22	.	.	PUNCT
cana-3959	348	1	[	[	X
cana-3959	348	2	9	9	NUM
cana-3959	348	3	]	]	X
cana-3959	348	4	s.	s.	PROPN
cana-3959	348	5	czerwik	czerwik	PROPN
cana-3959	348	6	,	,	PUNCT
cana-3959	348	7	fun	fun	NOUN
cana-3959	348	8	eq	eq	NOUN
cana-3959	348	9	and	and	CCONJ
cana-3959	348	10	inequalities	inequality	NOUN
cana-3959	348	11	in	in	ADP
cana-3959	348	12	several	several	ADJ
cana-3959	348	13	variables	variable	NOUN
cana-3959	348	14	,	,	PUNCT
cana-3959	348	15	world	world	NOUN
cana-3959	348	16	scientific	scientific	ADJ
cana-3959	348	17	,	,	PUNCT
cana-3959	348	18	river	river	NOUN
cana-3959	348	19	edge	edge	NOUN
cana-3959	348	20	,	,	PUNCT
cana-3959	348	21	nj	nj	PROPN
cana-3959	348	22	,	,	PUNCT
cana-3959	348	23	2002	2002	NUM
cana-3959	348	24	.	.	PUNCT
cana-3959	349	1	[	[	X
cana-3959	349	2	10	10	NUM
cana-3959	349	3	]	]	X
cana-3959	349	4	s.m	s.m	PROPN
cana-3959	349	5	.	.	PROPN
cana-3959	349	6	jung	jung	PROPN
cana-3959	349	7	,	,	PUNCT
cana-3959	349	8	hyers	hyers	PROPN
cana-3959	349	9	-	-	PUNCT
cana-3959	349	10	ulam	ulam	ADJ
cana-3959	349	11	-	-	PUNCT
cana-3959	349	12	rassias	rassias	PROPN
cana-3959	349	13	stability	stability	NOUN
cana-3959	349	14	of	of	ADP
cana-3959	349	15	fun	fun	NOUN
cana-3959	349	16	eq	eq	NOUN
cana-3959	349	17	in	in	ADP
cana-3959	349	18	mathematical	mathematical	ADJ
cana-3959	349	19	analysis	analysis	NOUN
cana-3959	349	20	,	,	PUNCT
cana-3959	349	21	hadronic	hadronic	ADJ
cana-3959	349	22	press	press	NOUN
cana-3959	349	23	,	,	PUNCT
cana-3959	349	24	palm	palm	NOUN
cana-3959	349	25	harbor,2001	harbor,2001	NOUN
cana-3959	349	26	.	.	PUNCT
cana-3959	350	1	[	[	X
cana-3959	350	2	11	11	NUM
cana-3959	350	3	]	]	PUNCT
cana-3959	350	4	tunç	tunç	PROPN
cana-3959	350	5	,	,	PUNCT
cana-3959	350	6	osman	osman	PROPN
cana-3959	350	7	.	.	PUNCT
cana-3959	351	1	new	new	ADJ
cana-3959	351	2	results	result	NOUN
cana-3959	351	3	on	on	ADP
cana-3959	351	4	the	the	DET
cana-3959	351	5	ulam	ulam	NOUN
cana-3959	351	6	–	–	PUNCT
cana-3959	351	7	hyers	hyer	NOUN
cana-3959	351	8	–	–	PUNCT
cana-3959	351	9	mittag	mittag	ADJ
cana-3959	351	10	–	–	PUNCT
cana-3959	351	11	leffler	leffler	NOUN
cana-3959	351	12	stability	stability	NOUN
cana-3959	351	13	of	of	ADP
cana-3959	351	14	caputo	caputo	PROPN
cana-3959	351	15	fractional	fractional	ADJ
cana-3959	351	16	-	-	PUNCT
cana-3959	351	17	order	order	NOUN
cana-3959	351	18	delay	delay	NOUN
cana-3959	351	19	differential	differential	ADJ
cana-3959	351	20	equations	equation	NOUN
cana-3959	351	21	.	.	PUNCT
cana-3959	351	22	"	"	PUNCT
cana-3959	352	1	mathematics	mathematic	NOUN
cana-3959	352	2	12	12	NUM
cana-3959	352	3	,	,	PUNCT
cana-3959	352	4	no	no	INTJ
cana-3959	352	5	.	.	NOUN
cana-3959	352	6	9	9	NUM
cana-3959	352	7	2024	2024	NUM
cana-3959	352	8	:	:	PUNCT
cana-3959	352	9	1342	1342	NUM
cana-3959	352	10	.	.	PUNCT
cana-3959	353	1	[	[	X
cana-3959	353	2	12	12	NUM
cana-3959	353	3	]	]	X
cana-3959	353	4	selvam	selvam	PROPN
cana-3959	353	5	,	,	PUNCT
cana-3959	353	6	a.	a.	PROPN
cana-3959	353	7	,	,	PUNCT
cana-3959	353	8	sabarinathan	sabarinathan	PROPN
cana-3959	353	9	,	,	PUNCT
cana-3959	353	10	s.	s.	PROPN
cana-3959	353	11	,	,	PUNCT
cana-3959	353	12	sooppy	sooppy	ADJ
cana-3959	353	13	nisar	nisar	PROPN
cana-3959	353	14	,	,	PUNCT
cana-3959	353	15	k.	k.	PROPN
cana-3959	353	16	,	,	PUNCT
cana-3959	353	17	ravichandran	ravichandran	NOUN
cana-3959	353	18	,	,	PUNCT
cana-3959	353	19	c.	c.	PROPN
cana-3959	353	20	and	and	CCONJ
cana-3959	353	21	senthil	senthil	PROPN
cana-3959	353	22	kumar	kumar	PROPN
cana-3959	353	23	,	,	PUNCT
cana-3959	353	24	b.v	b.v	PROPN
cana-3959	353	25	.	.	PROPN
cana-3959	353	26	results	result	NOUN
cana-3959	353	27	on	on	ADP
cana-3959	353	28	ulam‐type	ulam‐type	ADJ
cana-3959	353	29	stability	stability	NOUN
cana-3959	353	30	of	of	ADP
cana-3959	353	31	linear	linear	PROPN
cana-3959	353	32	differential	differential	ADJ
cana-3959	353	33	equation	equation	NOUN
cana-3959	353	34	with	with	ADP
cana-3959	353	35	integral	integral	ADJ
cana-3959	353	36	transform	transform	NOUN
cana-3959	353	37	.	.	PUNCT
cana-3959	354	1	mathematical	mathematical	ADJ
cana-3959	354	2	methods	method	NOUN
cana-3959	354	3	in	in	ADP
cana-3959	354	4	the	the	DET
cana-3959	354	5	applied	apply	VERB
cana-3959	354	6	sciences	science	NOUN
cana-3959	354	7	,	,	PUNCT
cana-3959	354	8	47(4	47(4	NUM
cana-3959	354	9	)	)	PUNCT
cana-3959	354	10	,	,	PUNCT
cana-3959	354	11	2024.pp.2311	2024.pp.2311	NUM
cana-3959	354	12	-	-	SYM
cana-3959	354	13	2323	2323	NUM
cana-3959	354	14	.	.	PUNCT
cana-3959	355	1	[	[	X
cana-3959	355	2	13	13	NUM
cana-3959	355	3	]	]	PUNCT
cana-3959	355	4	tunç	tunç	PROPN
cana-3959	355	5	,	,	PUNCT
cana-3959	355	6	osman	osman	PROPN
cana-3959	355	7	,	,	PUNCT
cana-3959	355	8	cemil	cemil	PROPN
cana-3959	355	9	tunç	tunç	PROPN
cana-3959	355	10	,	,	PUNCT
cana-3959	355	11	and	and	CCONJ
cana-3959	355	12	jen	jen	PROPN
cana-3959	355	13	-	-	PUNCT
cana-3959	355	14	chih	chih	PROPN
cana-3959	355	15	yao	yao	PROPN
cana-3959	355	16	.	.	PUNCT
cana-3959	356	1	new	new	ADJ
cana-3959	356	2	results	result	NOUN
cana-3959	356	3	on	on	ADP
cana-3959	356	4	ulam	ulam	PROPN
cana-3959	356	5	stabilities	stability	NOUN
cana-3959	356	6	of	of	ADP
cana-3959	356	7	nlinear	nlinear	ADJ
cana-3959	356	8	integral	integral	ADJ
cana-3959	356	9	equations	equation	NOUN
cana-3959	356	10	.	.	PUNCT
cana-3959	356	11	"	"	PUNCT
cana-3959	357	1	mathematics	mathematic	NOUN
cana-3959	357	2	12	12	NUM
cana-3959	357	3	,	,	PUNCT
cana-3959	357	4	no	no	INTJ
cana-3959	357	5	.	.	NOUN
cana-3959	357	6	5	5	NUM
cana-3959	357	7	2024	2024	NUM
cana-3959	357	8	:	:	PUNCT
cana-3959	357	9	682	682	NUM
cana-3959	357	10	.	.	PUNCT
cana-3959	358	1	communications	communication	NOUN
cana-3959	358	2	on	on	ADP
cana-3959	358	3	applied	apply	VERB
cana-3959	358	4	nonlinear	nonlinear	ADJ
cana-3959	358	5	analysis	analysis	NOUN
cana-3959	358	6	issn	issn	NOUN
cana-3959	358	7	:	:	PUNCT
cana-3959	358	8	1074	1074	NUM
cana-3959	358	9	-	-	PUNCT
cana-3959	358	10	133x	133x	NUM
cana-3959	358	11	vol	vol	NOUN
cana-3959	358	12	32	32	NUM
cana-3959	358	13	no	no	NOUN
cana-3959	358	14	.	.	PUNCT
cana-3959	359	1	9s	9s	NUM
cana-3959	359	2	(	(	PUNCT
cana-3959	359	3	2025	2025	NUM
cana-3959	359	4	)	)	PUNCT
cana-3959	359	5	492	492	NUM
cana-3959	359	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3959	360	1	[	[	X
cana-3959	360	2	14	14	NUM
cana-3959	360	3	]	]	X
cana-3959	360	4	tunç	tunç	PROPN
cana-3959	360	5	,	,	PUNCT
cana-3959	360	6	osman	osman	PROPN
cana-3959	360	7	,	,	PUNCT
cana-3959	360	8	and	and	CCONJ
cana-3959	360	9	cemil	cemil	PROPN
cana-3959	360	10	tunç	tunç	PROPN
cana-3959	360	11	.	.	PUNCT
cana-3959	361	1	on	on	ADP
cana-3959	361	2	ulam	ulam	PROPN
cana-3959	361	3	stabilities	stability	NOUN
cana-3959	361	4	of	of	ADP
cana-3959	361	5	delay	delay	PROPN
cana-3959	361	6	hammerstein	hammerstein	PROPN
cana-3959	361	7	integral	integral	ADJ
cana-3959	361	8	equation	equation	NOUN
cana-3959	361	9	symmetry	symmetry	NOUN
cana-3959	361	10	15	15	NUM
cana-3959	361	11	,	,	PUNCT
cana-3959	361	12	no	no	INTJ
cana-3959	361	13	.	.	NOUN
cana-3959	361	14	9	9	NUM
cana-3959	361	15	2023	2023	NUM
cana-3959	361	16	:	:	PUNCT
cana-3959	361	17	1736	1736	NUM
cana-3959	361	18	.	.	PUNCT
cana-3959	362	1	[	[	X
cana-3959	362	2	15	15	NUM
cana-3959	362	3	]	]	X
cana-3959	362	4	novac	novac	PROPN
cana-3959	362	5	,	,	PUNCT
cana-3959	362	6	adela	adela	PROPN
cana-3959	362	7	,	,	PUNCT
cana-3959	362	8	diana	diana	PROPN
cana-3959	362	9	otrocol	otrocol	ADV
cana-3959	362	10	,	,	PUNCT
cana-3959	362	11	and	and	CCONJ
cana-3959	362	12	dorian	dorian	PROPN
cana-3959	362	13	popa	popa	NOUN
cana-3959	362	14	.	.	PUNCT
cana-3959	363	1	on	on	ADP
cana-3959	363	2	ulam	ulam	PROPN
cana-3959	363	3	stability	stability	NOUN
cana-3959	363	4	of	of	ADP
cana-3959	363	5	a	a	DET
cana-3959	363	6	partial	partial	ADJ
cana-3959	363	7	differential	differential	NOUN
cana-3959	363	8	operator	operator	NOUN
cana-3959	363	9	in	in	ADP
cana-3959	363	10	banach	banach	NOUN
cana-3959	363	11	spaces	space	NOUN
cana-3959	363	12	.	.	PUNCT
cana-3959	363	13	"	"	PUNCT
cana-3959	364	1	mathematics	mathematic	NOUN
cana-3959	364	2	11	11	NUM
cana-3959	364	3	,	,	PUNCT
cana-3959	364	4	no	no	INTJ
cana-3959	364	5	.	.	NOUN
cana-3959	364	6	11	11	NUM
cana-3959	364	7	2023	2023	NUM
cana-3959	364	8	:	:	PUNCT
cana-3959	364	9	2488	2488	NUM
cana-3959	364	10	.	.	PUNCT
cana-3959	365	1	[	[	X
cana-3959	365	2	16	16	NUM
cana-3959	365	3	]	]	X
cana-3959	365	4	garcía	garcía	ADJ
cana-3959	365	5	,	,	PUNCT
cana-3959	365	6	gonzalo	gonzalo	PROPN
cana-3959	365	7	,	,	PUNCT
cana-3959	365	8	and	and	CCONJ
cana-3959	365	9	gaspar	gaspar	PROPN
cana-3959	365	10	mora	mora	PROPN
cana-3959	365	11	.	.	PUNCT
cana-3959	366	1	the	the	DET
cana-3959	366	2	degree	degree	NOUN
cana-3959	366	3	of	of	ADP
cana-3959	366	4	nondensifiability	nondensifiability	NOUN
cana-3959	366	5	of	of	ADP
cana-3959	366	6	linear	linear	PROPN
cana-3959	366	7	bounded	bounded	PROPN
cana-3959	366	8	operators	operator	NOUN
cana-3959	366	9	and	and	CCONJ
cana-3959	366	10	its	its	PRON
cana-3959	366	11	applications	application	NOUN
cana-3959	366	12	.	.	PUNCT
cana-3959	366	13	"	"	PUNCT
cana-3959	366	14	applied	apply	VERB
cana-3959	366	15	general	general	ADJ
cana-3959	366	16	topology	topology	NOUN
cana-3959	366	17	25	25	NUM
cana-3959	366	18	,	,	PUNCT
cana-3959	366	19	no	no	INTJ
cana-3959	366	20	.	.	NOUN
cana-3959	366	21	1	1	NUM
cana-3959	366	22	2024	2024	NUM
cana-3959	366	23	:	:	PUNCT
cana-3959	366	24	213	213	NUM
cana-3959	366	25	-	-	SYM
cana-3959	366	26	228	228	NUM
cana-3959	366	27	.	.	PUNCT
cana-3959	367	1	[	[	X
cana-3959	367	2	17	17	NUM
cana-3959	367	3	]	]	X
cana-3959	367	4	zada	zada	PROPN
cana-3959	367	5	,	,	PUNCT
cana-3959	367	6	akbar	akbar	NOUN
cana-3959	367	7	,	,	PUNCT
cana-3959	367	8	peiguang	peiguang	PROPN
cana-3959	367	9	wang	wang	PROPN
cana-3959	367	10	,	,	PUNCT
cana-3959	367	11	dhaou	dhaou	NOUN
cana-3959	367	12	lassoued	lassoue	VERB
cana-3959	367	13	,	,	PUNCT
cana-3959	367	14	and	and	CCONJ
cana-3959	367	15	tongxing	tongxe	VERB
cana-3959	367	16	li	li	PROPN
cana-3959	367	17	.	.	PUNCT
cana-3959	367	18	connections	connection	NOUN
cana-3959	367	19	between	between	ADP
cana-3959	367	20	hyers	hyer	NOUN
cana-3959	367	21	-	-	PUNCT
cana-3959	367	22	ulam	ulam	PROPN
cana-3959	367	23	stability	stability	PROPN
cana-3959	367	24	and	and	CCONJ
cana-3959	367	25	uniform	uniform	ADJ
cana-3959	367	26	exponential	exponential	ADJ
cana-3959	367	27	stability	stability	NOUN
cana-3959	367	28	of	of	ADP
cana-3959	367	29	2	2	NUM
cana-3959	367	30	-	-	PUNCT
cana-3959	367	31	periodic	periodic	ADJ
cana-3959	367	32	linear	linear	ADJ
cana-3959	367	33	nonautonomoussystems	nonautonomoussystem	NOUN
cana-3959	367	34	advances	advance	VERB
cana-3959	367	35	in	in	ADP
cana-3959	367	36	difference	difference	NOUN
cana-3959	367	37	equations	equation	NOUN
cana-3959	367	38	2017	2017	NUM
cana-3959	367	39	(	(	PUNCT
cana-3959	367	40	2017	2017	NUM
cana-3959	367	41	):	):	PUNCT
cana-3959	367	42	1	1	NUM
cana-3959	367	43	-	-	SYM
cana-3959	367	44	7	7	NUM
cana-3959	367	45	.	.	PUNCT
cana-3959	368	1	[	[	X
cana-3959	368	2	18	18	NUM
cana-3959	368	3	]	]	X
cana-3959	368	4	kumar	kumar	PROPN
cana-3959	368	5	,	,	PUNCT
cana-3959	368	6	bhim	bhim	PROPN
cana-3959	368	7	,	,	PUNCT
cana-3959	368	8	and	and	CCONJ
cana-3959	368	9	muslim	muslim	PROPN
cana-3959	368	10	malik	malik	PROPN
cana-3959	368	11	.	.	PUNCT
cana-3959	369	1	existence	existence	PROPN
cana-3959	369	2	,	,	PUNCT
cana-3959	369	3	controllability	controllability	NOUN
cana-3959	369	4	and	and	CCONJ
cana-3959	369	5	hyers	hyer	NOUN
cana-3959	369	6	–	–	PUNCT
cana-3959	369	7	ulam	ulam	X
cana-3959	369	8	stability	stability	NOUN
cana-3959	369	9	of	of	ADP
cana-3959	369	10	a	a	DET
cana-3959	369	11	hybrid	hybrid	ADJ
cana-3959	369	12	neutral	neutral	ADJ
cana-3959	369	13	switched	switch	VERB
cana-3959	369	14	system	system	NOUN
cana-3959	369	15	with	with	ADP
cana-3959	369	16	impulsive	impulsive	ADJ
cana-3959	369	17	effects	effect	NOUN
cana-3959	369	18	.	.	PUNCT
cana-3959	369	19	"	"	PUNCT
cana-3959	370	1	international	international	ADJ
cana-3959	370	2	journal	journal	NOUN
cana-3959	370	3	of	of	ADP
cana-3959	370	4	systems	system	NOUN
cana-3959	370	5	science	science	NOUN
cana-3959	370	6	55	55	NUM
cana-3959	370	7	,	,	PUNCT
cana-3959	370	8	no	no	INTJ
cana-3959	370	9	.	.	NOUN
cana-3959	370	10	3	3	NUM
cana-3959	370	11	2024	2024	NUM
cana-3959	370	12	:	:	PUNCT
cana-3959	370	13	517	517	NUM
cana-3959	370	14	-	-	SYM
cana-3959	370	15	534	534	NUM
cana-3959	370	16	.	.	PUNCT
cana-3959	371	1	[	[	X
cana-3959	371	2	19	19	NUM
cana-3959	371	3	]	]	X
cana-3959	371	4	almarri	almarri	NOUN
cana-3959	371	5	,	,	PUNCT
cana-3959	371	6	barakah	barakah	PROPN
cana-3959	371	7	,	,	PUNCT
cana-3959	371	8	xingtao	xingtao	PROPN
cana-3959	371	9	wang	wang	PROPN
cana-3959	371	10	,	,	PUNCT
cana-3959	371	11	and	and	CCONJ
cana-3959	371	12	ahmed	ahmed	PROPN
cana-3959	371	13	m.	m.	PROPN
cana-3959	371	14	elshenhab	elshenhab	PROPN
cana-3959	371	15	.	.	PUNCT
cana-3959	372	1	controllability	controllability	NOUN
cana-3959	372	2	and	and	CCONJ
cana-3959	372	3	hyers	hyer	NOUN
cana-3959	372	4	–	–	PUNCT
cana-3959	372	5	ulam	ulam	PROPN
cana-3959	372	6	stability	stability	NOUN
cana-3959	372	7	of	of	ADP
cana-3959	372	8	fractional	fractional	ADJ
cana-3959	372	9	systems	system	NOUN
cana-3959	372	10	with	with	ADP
cana-3959	372	11	pure	pure	ADJ
cana-3959	372	12	delay	delay	NOUN
cana-3959	372	13	.	.	PUNCT
cana-3959	372	14	"	"	PUNCT
cana-3959	373	1	fractal	fractal	ADJ
cana-3959	373	2	and	and	CCONJ
cana-3959	373	3	fractional	fractional	ADJ
cana-3959	373	4	6	6	NUM
cana-3959	373	5	,	,	PUNCT
cana-3959	373	6	no	no	INTJ
cana-3959	373	7	.	.	NOUN
cana-3959	373	8	10	10	NUM
cana-3959	373	9	,	,	PUNCT
cana-3959	373	10	2022	2022	NUM
cana-3959	373	11	:	:	PUNCT
cana-3959	373	12	611	611	NUM
cana-3959	373	13	.	.	PUNCT
cana-3959	374	1	[	[	X
cana-3959	374	2	20	20	NUM
cana-3959	374	3	]	]	PUNCT
cana-3959	374	4	agarwal	agarwal	PROPN
cana-3959	374	5	,	,	PUNCT
cana-3959	374	6	r.	r.	PROPN
cana-3959	374	7	p.	p.	PROPN
cana-3959	374	8	,	,	PUNCT
cana-3959	374	9	cho	cho	PROPN
cana-3959	374	10	,	,	PUNCT
cana-3959	374	11	y.	y.	PROPN
cana-3959	374	12	j.	j.	PROPN
cana-3959	374	13	,	,	PUNCT
cana-3959	374	14	saadati	saadati	PROPN
cana-3959	374	15	,	,	PUNCT
cana-3959	374	16	r.	r.	PROPN
cana-3959	374	17	,	,	PUNCT
cana-3959	374	18	wang	wang	PROPN
cana-3959	374	19	,	,	PUNCT
cana-3959	374	20	s	s	PROPN
cana-3959	374	21	,	,	PUNCT
cana-3959	374	22	nonlinear	nonlinear	ADJ
cana-3959	374	23	fuzzy	fuzzy	ADJ
cana-3959	374	24	stability	stability	NOUN
cana-3959	374	25	of	of	ADP
cana-3959	374	26	cubic	cubic	ADJ
cana-3959	374	27	functional	functional	ADJ
cana-3959	374	28	equations	equation	NOUN
cana-3959	374	29	.	.	PUNCT
cana-3959	375	1	journal	journal	PROPN
cana-3959	375	2	of	of	ADP
cana-3959	375	3	inequalities	inequality	NOUN
cana-3959	375	4	and	and	CCONJ
cana-3959	375	5	applications	application	NOUN
cana-3959	375	6	,	,	PUNCT
cana-3959	375	7	2012	2012	NUM
cana-3959	375	8	,	,	PUNCT
cana-3959	375	9	1	1	NUM
cana-3959	375	10	-	-	SYM
cana-3959	375	11	19	19	NUM
cana-3959	375	12	.	.	PUNCT
cana-3959	376	1	[	[	X
cana-3959	376	2	21	21	NUM
cana-3959	376	3	]	]	X
cana-3959	376	4	saadati	saadati	PROPN
cana-3959	376	5	,	,	PUNCT
cana-3959	376	6	r.	r.	PROPN
cana-3959	376	7	,	,	PUNCT
cana-3959	376	8	cho	cho	PROPN
cana-3959	376	9	,	,	PUNCT
cana-3959	376	10	y.	y.	PROPN
cana-3959	376	11	j.	j.	PROPN
cana-3959	376	12	,	,	PUNCT
cana-3959	376	13	rassias	rassias	PROPN
cana-3959	376	14	,	,	PUNCT
cana-3959	376	15	j.	j.	PROPN
cana-3959	376	16	m	m	PROPN
cana-3959	376	17	,	,	PUNCT
cana-3959	376	18	nonlinear	nonlinear	ADJ
cana-3959	376	19	l	l	ADJ
cana-3959	376	20	-	-	ADJ
cana-3959	376	21	fuzzy	fuzzy	ADJ
cana-3959	376	22	stability	stability	NOUN
cana-3959	376	23	of	of	ADP
cana-3959	376	24	k	k	ADJ
cana-3959	376	25	-	-	ADJ
cana-3959	376	26	cubic	cubic	ADJ
cana-3959	376	27	functional	functional	ADJ
cana-3959	376	28	equation	equation	NOUN
cana-3959	376	29	.	.	PUNCT
cana-3959	377	1	filomat	filomat	NOUN
cana-3959	377	2	,	,	PUNCT
cana-3959	377	3	29(5	29(5	NUM
cana-3959	377	4	)	)	PUNCT
cana-3959	377	5	,	,	PUNCT
cana-3959	377	6	(	(	PUNCT
cana-3959	377	7	2015	2015	NUM
cana-3959	377	8	)	)	PUNCT
cana-3959	377	9	1137	1137	NUM
cana-3959	377	10	-	-	SYM
cana-3959	377	11	1148	1148	NUM
cana-3959	377	12	.	.	PUNCT
cana-3959	378	1	[	[	X
cana-3959	378	2	22	22	NUM
cana-3959	378	3	]	]	X
cana-3959	378	4	mohiuddine	mohiuddine	NOUN
cana-3959	378	5	,	,	PUNCT
cana-3959	378	6	s.a	s.a	PROPN
cana-3959	378	7	.	.	PROPN
cana-3959	378	8	and	and	CCONJ
cana-3959	378	9	alotaibi	alotaibi	PROPN
cana-3959	378	10	,	,	PUNCT
cana-3959	378	11	a.	a.	NOUN
cana-3959	378	12	,	,	PUNCT
cana-3959	378	13	fuzzy	fuzzy	ADJ
cana-3959	378	14	stability	stability	NOUN
cana-3959	378	15	of	of	ADP
cana-3959	378	16	a	a	DET
cana-3959	378	17	cubic	cubic	ADJ
cana-3959	378	18	functional	functional	ADJ
cana-3959	378	19	equation	equation	NOUN
cana-3959	378	20	via	via	ADP
cana-3959	378	21	fixed	fix	VERB
cana-3959	378	22	point	point	NOUN
cana-3959	378	23	technique	technique	NOUN
cana-3959	378	24	.	.	PUNCT
cana-3959	379	1	advances	advance	NOUN
cana-3959	379	2	in	in	ADP
cana-3959	379	3	difference	difference	NOUN
cana-3959	379	4	equations	equation	NOUN
cana-3959	379	5	,	,	PUNCT
cana-3959	379	6	2012	2012	NUM
cana-3959	379	7	,	,	PUNCT
cana-3959	379	8	pp.1	pp.1	NOUN
cana-3959	379	9	-	-	PUNCT
cana-3959	379	10	8	8	NUM
cana-3959	379	11	.	.	PUNCT
cana-3959	380	1	[	[	X
cana-3959	380	2	23	23	NUM
cana-3959	380	3	]	]	X
cana-3959	380	4	lee	lee	PROPN
cana-3959	380	5	,	,	PUNCT
cana-3959	380	6	yang	yang	PROPN
cana-3959	380	7	-	-	PUNCT
cana-3959	380	8	hi	hi	PROPN
cana-3959	380	9	,	,	PUNCT
cana-3959	380	10	and	and	CCONJ
cana-3959	380	11	soon	soon	ADV
cana-3959	380	12	-	-	PUNCT
cana-3959	380	13	mo	mo	PROPN
cana-3959	380	14	jung	jung	PROPN
cana-3959	380	15	.	.	PUNCT
cana-3959	381	1	fuzzy	fuzzy	ADJ
cana-3959	381	2	stability	stability	NOUN
cana-3959	381	3	of	of	ADP
cana-3959	381	4	the	the	DET
cana-3959	381	5	cubic	cubic	ADJ
cana-3959	381	6	and	and	CCONJ
cana-3959	381	7	quadratic	quadratic	ADJ
cana-3959	381	8	functional	functional	ADJ
cana-3959	381	9	equation	equation	NOUN
cana-3959	381	10	.	.	PUNCT
cana-3959	382	1	appl	appl	PROPN
cana-3959	382	2	.	.	PROPN
cana-3959	382	3	math	math	PROPN
cana-3959	382	4	.	.	PUNCT
cana-3959	383	1	sci.(ruse	sci.(ruse	CCONJ
cana-3959	383	2	)	)	PUNCT
cana-3959	383	3	10	10	NUM
cana-3959	383	4	(	(	PUNCT
cana-3959	383	5	2016	2016	NUM
cana-3959	383	6	):	):	PUNCT
cana-3959	383	7	2671	2671	NUM
cana-3959	383	8	-	-	SYM
cana-3959	383	9	2686	2686	NUM
cana-3959	383	10	.	.	PUNCT
cana-3959	384	1	[	[	X
cana-3959	384	2	24	24	NUM
cana-3959	384	3	]	]	X
cana-3959	384	4	javadi	javadi	PROPN
cana-3959	384	5	,	,	PUNCT
cana-3959	384	6	s.	s.	PROPN
cana-3959	384	7	and	and	CCONJ
cana-3959	384	8	rassias	rassias	PROPN
cana-3959	384	9	,	,	PUNCT
cana-3959	384	10	j.m	j.m	PROPN
cana-3959	384	11	.	.	PROPN
cana-3959	384	12	,	,	PUNCT
cana-3959	384	13	stability	stability	NOUN
cana-3959	384	14	of	of	ADP
cana-3959	384	15	general	general	ADJ
cana-3959	384	16	cubic	cubic	ADJ
cana-3959	384	17	mapping	mapping	NOUN
cana-3959	384	18	in	in	ADP
cana-3959	384	19	fuzzy	fuzzy	ADJ
cana-3959	384	20	normed	normed	ADJ
cana-3959	384	21	spaces	space	NOUN
cana-3959	384	22	.	.	PUNCT
cana-3959	385	1	analele	analele	ADP
cana-3959	385	2	ştiinţifice	ştiinţifice	PROPN
cana-3959	385	3	ale	ale	NOUN
cana-3959	385	4	universităţii	universităţii	PROPN
cana-3959	385	5	"	"	PUNCT
cana-3959	385	6	ovidius	ovidius	NOUN
cana-3959	385	7	"	"	PUNCT
cana-3959	385	8	constanţa	constanţa	NOUN
cana-3959	385	9	.	.	PUNCT
cana-3959	386	1	seria	seria	PROPN
cana-3959	386	2	matematică	matematică	PROPN
cana-3959	386	3	,	,	PUNCT
cana-3959	386	4	2012,20(1	2012,20(1	NOUN
cana-3959	386	5	)	)	PUNCT
cana-3959	386	6	,	,	PUNCT
cana-3959	386	7	pp.129	pp.129	PROPN
cana-3959	386	8	-	-	PUNCT
cana-3959	386	9	150	150	NUM
cana-3959	386	10	.	.	PUNCT
cana-3959	387	1	[	[	X
cana-3959	387	2	25	25	NUM
cana-3959	387	3	]	]	X
cana-3959	387	4	pasupathi	pasupathi	NOUN
cana-3959	387	5	,	,	PUNCT
cana-3959	387	6	a.	a.	NOUN
cana-3959	387	7	;	;	PUNCT
cana-3959	387	8	konsalraj	konsalraj	PROPN
cana-3959	387	9	,	,	PUNCT
cana-3959	387	10	j.	j.	PROPN
cana-3959	387	11	;	;	PUNCT
cana-3959	387	12	fatima	fatima	PROPN
cana-3959	387	13	,	,	PUNCT
cana-3959	387	14	n.	n.	NOUN
cana-3959	387	15	;	;	PUNCT
cana-3959	387	16	velusamy	velusamy	PROPN
cana-3959	387	17	,	,	PUNCT
cana-3959	387	18	v.	v.	PROPN
cana-3959	387	19	;	;	PUNCT
cana-3959	387	20	mlaiki	mlaiki	PROPN
cana-3959	387	21	,	,	PUNCT
cana-3959	387	22	n.	n.	NOUN
cana-3959	387	23	;	;	PUNCT
cana-3959	387	24	souayah	souayah	NOUN
cana-3959	387	25	,	,	PUNCT
cana-3959	387	26	n.	n.	NOUN
cana-3959	387	27	direct	direct	ADJ
cana-3959	387	28	and	and	CCONJ
cana-3959	387	29	fixed	fix	VERB
cana-3959	387	30	-	-	PUNCT
cana-3959	387	31	point	point	NOUN
cana-3959	387	32	stability	stability	NOUN
cana-3959	387	33	–	–	PUNCT
cana-3959	387	34	instability	instability	NOUN
cana-3959	387	35	of	of	ADP
cana-3959	387	36	additive	additive	ADJ
cana-3959	387	37	functional	functional	ADJ
cana-3959	387	38	equation	equation	NOUN
cana-3959	387	39	in	in	ADP
cana-3959	387	40	banach	banach	NOUN
cana-3959	387	41	and	and	CCONJ
cana-3959	387	42	quasi	quasi	ADJ
cana-3959	387	43	-	-	ADJ
cana-3959	387	44	beta	beta	ADJ
cana-3959	387	45	normed	norme	VERB
cana-3959	387	46	spaces	space	NOUN
cana-3959	387	47	.	.	PUNCT
cana-3959	388	1	symmetry	symmetry	NOUN
cana-3959	388	2	2022	2022	NUM
cana-3959	388	3	,	,	PUNCT
cana-3959	388	4	14	14	NUM
cana-3959	388	5	,	,	PUNCT
cana-3959	388	6	1700	1700	NUM
cana-3959	388	7	.	.	PUNCT
cana-3959	389	1	[	[	X
cana-3959	389	2	26	26	NUM
cana-3959	389	3	]	]	SYM
cana-3959	389	4	agilan	agilan	ADJ
cana-3959	389	5	,	,	PUNCT
cana-3959	389	6	p.	p.	NOUN
cana-3959	389	7	;	;	PUNCT
cana-3959	389	8	julietraja	julietraja	PROPN
cana-3959	389	9	,	,	PUNCT
cana-3959	389	10	k.	k.	PROPN
cana-3959	389	11	;	;	PUNCT
cana-3959	389	12	mlaiki	mlaiki	PROPN
cana-3959	389	13	,	,	PUNCT
cana-3959	389	14	n.	n.	NOUN
cana-3959	389	15	;	;	PUNCT
cana-3959	389	16	mukheimer	mukheimer	NOUN
cana-3959	389	17	,	,	PUNCT
cana-3959	389	18	a.	a.	NOUN
cana-3959	389	19	intuitionistic	intuitionistic	ADJ
cana-3959	389	20	fuzzy	fuzzy	ADJ
cana-3959	389	21	stability	stability	NOUN
cana-3959	389	22	of	of	ADP
cana-3959	389	23	an	an	DET
cana-3959	389	24	euler	euler	ADJ
cana-3959	389	25	–	–	PUNCT
cana-3959	389	26	lagrange	lagrange	ADJ
cana-3959	389	27	symmetry	symmetry	NOUN
cana-3959	389	28	additive	additive	ADJ
cana-3959	389	29	functional	functional	ADJ
cana-3959	389	30	equation	equation	NOUN
cana-3959	389	31	via	via	ADP
cana-3959	389	32	direct	direct	ADJ
cana-3959	389	33	and	and	CCONJ
cana-3959	389	34	fixed	fix	VERB
cana-3959	389	35	point	point	NOUN
cana-3959	389	36	technique	technique	NOUN
cana-3959	389	37	(	(	PUNCT
cana-3959	389	38	fpt	fpt	PROPN
cana-3959	389	39	)	)	PUNCT
cana-3959	389	40	.	.	PUNCT
cana-3959	390	1	symmetry	symmetry	PROPN
cana-3959	390	2	2022	2022	NUM
cana-3959	390	3	,	,	PUNCT
cana-3959	390	4	14	14	NUM
cana-3959	390	5	,	,	PUNCT
cana-3959	390	6	2454	2454	NUM
cana-3959	390	7	.	.	PUNCT
cana-3959	391	1	[	[	X
cana-3959	391	2	27	27	NUM
cana-3959	391	3	]	]	SYM
cana-3959	391	4	agilan	agilan	ADJ
cana-3959	391	5	,	,	PUNCT
cana-3959	391	6	p.	p.	NOUN
cana-3959	391	7	;	;	PUNCT
cana-3959	391	8	almazah	almazah	PROPN
cana-3959	391	9	,	,	PUNCT
cana-3959	391	10	m.a.a	m.a.a	PROPN
cana-3959	391	11	.	.	PUNCT
cana-3959	391	12	;	;	PUNCT
cana-3959	391	13	julietraja	julietraja	PROPN
cana-3959	391	14	,	,	PUNCT
cana-3959	391	15	k.	k.	PROPN
cana-3959	391	16	;	;	PUNCT
cana-3959	391	17	alsinai	alsinai	PROPN
cana-3959	391	18	,	,	PUNCT
cana-3959	391	19	a.	a.	NOUN
cana-3959	391	20	classical	classical	NOUN
cana-3959	391	21	and	and	CCONJ
cana-3959	391	22	fixed	fix	VERB
cana-3959	391	23	point	point	NOUN
cana-3959	391	24	approach	approach	NOUN
cana-3959	391	25	to	to	ADP
cana-3959	391	26	the	the	DET
cana-3959	391	27	stability	stability	NOUN
cana-3959	391	28	analysis	analysis	NOUN
cana-3959	391	29	of	of	ADP
cana-3959	391	30	a	a	DET
cana-3959	391	31	bilateral	bilateral	ADJ
cana-3959	391	32	symmetric	symmetric	ADJ
cana-3959	391	33	additive	additive	ADJ
cana-3959	391	34	functional	functional	ADJ
cana-3959	391	35	equation	equation	NOUN
cana-3959	391	36	in	in	ADP
cana-3959	391	37	fuzzy	fuzzy	ADJ
cana-3959	391	38	and	and	CCONJ
cana-3959	391	39	random	random	ADJ
cana-3959	391	40	normed	normed	ADJ
cana-3959	391	41	spaces	space	NOUN
cana-3959	391	42	.	.	PUNCT
cana-3959	392	1	mathematics	mathematic	NOUN
cana-3959	392	2	2023	2023	NUM
cana-3959	392	3	,	,	PUNCT
cana-3959	392	4	11	11	NUM
cana-3959	392	5	,	,	PUNCT
cana-3959	392	6	681	681	NUM
cana-3959	392	7	.	.	PUNCT
cana-3959	393	1	[	[	X
cana-3959	393	2	28	28	NUM
cana-3959	393	3	]	]	X
cana-3959	393	4	agilan	agilan	ADJ
cana-3959	393	5	,	,	PUNCT
cana-3959	393	6	p.	p.	NOUN
cana-3959	393	7	;	;	PUNCT
cana-3959	393	8	julietraja	julietraja	PROPN
cana-3959	393	9	.	.	PUNCT
cana-3959	393	10	;	;	PUNCT
cana-3959	393	11	k	k	PROPN
cana-3959	393	12	almazah	almazah	PROPN
cana-3959	393	13	,	,	PUNCT
cana-3959	393	14	m.a.a	m.a.a	PROPN
cana-3959	393	15	.	.	PUNCT
cana-3959	393	16	;	;	PUNCT
cana-3959	394	1	alsinai	alsinai	PROPN
cana-3959	394	2	,	,	PUNCT
cana-3959	394	3	a.	a.	NOUN
cana-3959	394	4	stability	stability	NOUN
cana-3959	394	5	analysis	analysis	NOUN
cana-3959	394	6	of	of	ADP
cana-3959	394	7	a	a	DET
cana-3959	394	8	new	new	ADJ
cana-3959	394	9	class	class	NOUN
cana-3959	394	10	of	of	ADP
cana-3959	394	11	series	series	NOUN
cana-3959	394	12	type	type	NOUN
cana-3959	394	13	additive	additive	ADJ
cana-3959	394	14	functional	functional	ADJ
cana-3959	394	15	equation	equation	NOUN
cana-3959	394	16	in	in	ADP
cana-3959	394	17	banach	banach	NOUN
cana-3959	394	18	spaces	space	NOUN
cana-3959	394	19	:	:	PUNCT
cana-3959	394	20	direct	direct	ADJ
cana-3959	394	21	and	and	CCONJ
cana-3959	394	22	fixed	fix	VERB
cana-3959	394	23	point	point	NOUN
cana-3959	394	24	techniques	technique	NOUN
cana-3959	394	25	mathematics	mathematic	NOUN
cana-3959	394	26	2023	2023	NUM
cana-3959	394	27	,	,	PUNCT
cana-3959	394	28	11	11	NUM
cana-3959	394	29	,	,	PUNCT
cana-3959	394	30	887	887	NUM
cana-3959	394	31	.	.	PUNCT
cana-3959	394	32	doi.org/10.3390/math11040887	doi.org/10.3390/math11040887	VERB
cana-3959	394	33	.	.	PUNCT
cana-3959	395	1	[	[	X
cana-3959	395	2	29	29	NUM
cana-3959	395	3	]	]	PUNCT
cana-3959	395	4	aloqaily	aloqaily	ADV
cana-3959	395	5	,	,	PUNCT
cana-3959	395	6	ahmad	ahmad	PROPN
cana-3959	395	7	,	,	PUNCT
cana-3959	395	8	p.	p.	PROPN
cana-3959	395	9	agilan	agilan	PROPN
cana-3959	395	10	,	,	PUNCT
cana-3959	395	11	k.	k.	PROPN
cana-3959	395	12	julietraja	julietraja	PROPN
cana-3959	395	13	,	,	PUNCT
cana-3959	395	14	s.	s.	PROPN
cana-3959	395	15	annadurai	annadurai	PROPN
cana-3959	395	16	,	,	PUNCT
cana-3959	395	17	and	and	CCONJ
cana-3959	395	18	nabil	nabil	PROPN
cana-3959	395	19	mlaiki	mlaiki	PROPN
cana-3959	395	20	.	.	PUNCT
cana-3959	396	1	a	a	DET
cana-3959	396	2	novel	novel	ADJ
cana-3959	396	3	stability	stability	NOUN
cana-3959	396	4	analysis	analysis	NOUN
cana-3959	396	5	of	of	ADP
cana-3959	396	6	functional	functional	ADJ
cana-3959	396	7	equation	equation	NOUN
cana-3959	396	8	in	in	ADP
cana-3959	396	9	neutrosophic	neutrosophic	ADJ
cana-3959	396	10	normed	norme	VERB
cana-3959	396	11	spaces	space	NOUN
cana-3959	396	12	.	.	PUNCT
cana-3959	397	1	boundary	boundary	ADJ
cana-3959	397	2	value	value	NOUN
cana-3959	397	3	problems	problem	NOUN
cana-3959	397	4	,	,	PUNCT
cana-3959	397	5	2024	2024	NUM
cana-3959	397	6	,	,	PUNCT
cana-3959	397	7	no	no	INTJ
cana-3959	397	8	.	.	NOUN
cana-3959	397	9	1	1	NUM
cana-3959	397	10	(	(	PUNCT
cana-3959	397	11	2024	2024	NUM
cana-3959	397	12	)	)	PUNCT
cana-3959	397	13	,	,	PUNCT
cana-3959	397	14	47	47	NUM
cana-3959	397	15	.	.	PUNCT
cana-3959	398	1	[	[	X
cana-3959	398	2	30	30	NUM
cana-3959	398	3	]	]	X
cana-3959	398	4	agilan	agilan	ADJ
cana-3959	398	5	,	,	PUNCT
cana-3959	398	6	p.	p.	PROPN
cana-3959	398	7	,	,	PUNCT
cana-3959	398	8	julietraja	julietraja	PROPN
cana-3959	398	9	,	,	PUNCT
cana-3959	398	10	k.	k.	PROPN
cana-3959	398	11	,	,	PUNCT
cana-3959	398	12	kanimozhi	kanimozhi	PROPN
cana-3959	398	13	,	,	PUNCT
cana-3959	398	14	b.	b.	PROPN
cana-3959	398	15	and	and	CCONJ
cana-3959	398	16	alsinai	alsinai	PROPN
cana-3959	398	17	,	,	PUNCT
cana-3959	398	18	a.	a.	PROPN
cana-3959	398	19	,	,	PUNCT
cana-3959	398	20	hyers	hyer	NOUN
cana-3959	398	21	stability	stability	NOUN
cana-3959	398	22	of	of	ADP
cana-3959	398	23	aqc	aqc	PROPN
cana-3959	398	24	functional	functional	ADJ
cana-3959	398	25	equation	equation	NOUN
cana-3959	398	26	.	.	PUNCT
cana-3959	399	1	dynamics	dynamic	NOUN
cana-3959	399	2	of	of	ADP
cana-3959	399	3	continuous	continuous	ADJ
cana-3959	399	4	,	,	PUNCT
cana-3959	399	5	discrete	discrete	ADJ
cana-3959	399	6	and	and	CCONJ
cana-3959	399	7	impulsive	impulsive	ADJ
cana-3959	399	8	systems	system	NOUN
cana-3959	399	9	series	series	NOUN
cana-3959	399	10	b	b	NOUN
cana-3959	399	11	:	:	PUNCT
cana-3959	399	12	applications	application	NOUN
cana-3959	399	13	and	and	CCONJ
cana-3959	399	14	algorithms	algorithm	NOUN
cana-3959	399	15	,	,	PUNCT
cana-3959	399	16	2024	2024	NUM
cana-3959	399	17	,	,	PUNCT
cana-3959	399	18	31	31	NUM
cana-3959	399	19	,	,	PUNCT
cana-3959	399	20	63	63	NUM
cana-3959	399	21	-	-	SYM
cana-3959	399	22	75	75	NUM
cana-3959	399	23	.	.	PUNCT
cana-3959	400	1	[	[	X
cana-3959	400	2	31	31	NUM
cana-3959	400	3	]	]	SYM
cana-3959	400	4	agilan	agilan	ADJ
cana-3959	400	5	,	,	PUNCT
cana-3959	400	6	p.	p.	PROPN
cana-3959	400	7	,	,	PUNCT
cana-3959	400	8	julietraja	julietraja	PROPN
cana-3959	400	9	,	,	PUNCT
cana-3959	400	10	k	k	PROPN
cana-3959	400	11	,	,	PUNCT
cana-3959	400	12	sarah	sarah	PROPN
cana-3959	400	13	aljohani	aljohani	PROPN
cana-3959	400	14	,	,	PUNCT
cana-3959	400	15	nabil	nabil	PROPN
cana-3959	400	16	mlaiki	mlaiki	PROPN
cana-3959	400	17	,	,	PUNCT
cana-3959	400	18	generalised	generalise	VERB
cana-3959	400	19	ulam	ulam	PROPN
cana-3959	400	20	-	-	PUNCT
cana-3959	400	21	hyers	hyer	NOUN
cana-3959	400	22	stability	stability	NOUN
cana-3959	400	23	analysis	analysis	NOUN
cana-3959	400	24	for	for	ADP
cana-3959	400	25	system	system	NOUN
cana-3959	400	26	of	of	ADP
cana-3959	400	27	additive	additive	ADJ
cana-3959	400	28	functional	functional	ADJ
cana-3959	400	29	equation	equation	NOUN
cana-3959	400	30	in	in	ADP
cana-3959	400	31	fuzzy	fuzzy	ADJ
cana-3959	400	32	and	and	CCONJ
cana-3959	400	33	random	random	ADJ
cana-3959	400	34	normed	normed	ADJ
cana-3959	400	35	spaces	space	NOUN
cana-3959	400	36	:	:	PUNCT
cana-3959	400	37	direct	direct	ADJ
cana-3959	400	38	and	and	CCONJ
cana-3959	400	39	fixed	fix	VERB
cana-3959	400	40	point	point	NOUN
cana-3959	400	41	approach	approach	NOUN
cana-3959	400	42	.	.	PUNCT
cana-3959	401	1	int	int	NOUN
cana-3959	401	2	.	.	PUNCT
cana-3959	402	1	j.	j.	PROPN
cana-3959	402	2	anal	anal	PROPN
cana-3959	402	3	.	.	PUNCT
cana-3959	403	1	appl	appl	PROPN
cana-3959	403	2	.	.	PROPN
cana-3959	403	3	,	,	PUNCT
cana-3959	403	4	22	22	NUM
cana-3959	403	5	2024	2024	NUM
cana-3959	403	6	,	,	PUNCT
cana-3959	403	7	201	201	NUM
cana-3959	403	8	.	.	PUNCT
cana-3959	404	1	[	[	X
cana-3959	404	2	32	32	NUM
cana-3959	404	3	]	]	SYM
cana-3959	404	4	agilan.p	agilan.p	PROPN
cana-3959	404	5	,	,	PUNCT
cana-3959	404	6	vijayan.v	vijayan.v	PROPN
cana-3959	404	7	,	,	PUNCT
cana-3959	404	8	sophia.m	sophia.m	NUM
cana-3959	404	9	,	,	PUNCT
cana-3959	404	10	ganapathy.g	ganapathy.g	PROPN
cana-3959	404	11	.	.	PUNCT
cana-3959	404	12	,	,	PUNCT
cana-3959	404	13	exploring	explore	VERB
cana-3959	404	14	advanced	advanced	ADJ
cana-3959	404	15	stability	stability	NOUN
cana-3959	404	16	of	of	ADP
cana-3959	404	17	higher	high	ADJ
cana-3959	404	18	-	-	PUNCT
cana-3959	404	19	order	order	NOUN
cana-3959	404	20	functional	functional	ADJ
cana-3959	404	21	equations	equation	NOUN
cana-3959	404	22	in	in	ADP
cana-3959	404	23	neutrosophic	neutrosophic	ADJ
cana-3959	404	24	normed	norme	VERB
cana-3959	404	25	spaces	space	NOUN
cana-3959	404	26	via	via	ADP
cana-3959	404	27	hyers	hyer	NOUN
cana-3959	404	28	-	-	PUNCT
cana-3959	404	29	ulam	ulam	PROPN
cana-3959	404	30	methodologies	methodology	NOUN
cana-3959	404	31	,	,	PUNCT
cana-3959	404	32	communications	communication	NOUN
cana-3959	404	33	on	on	ADP
cana-3959	404	34	applied	apply	VERB
cana-3959	404	35	nonlinear	nonlinear	ADJ
cana-3959	404	36	analysis	analysis	NOUN
cana-3959	404	37	,	,	PUNCT
cana-3959	404	38	2025	2025	NUM
cana-3959	404	39	,	,	PUNCT
cana-3959	404	40	vol	vol	NOUN
cana-3959	404	41	32	32	NUM
cana-3959	404	42	no	no	NOUN
cana-3959	404	43	.	.	PUNCT
cana-3959	405	1	7s	7	NOUN
cana-3959	405	2	(	(	PUNCT
cana-3959	405	3	2025	2025	NUM
cana-3959	405	4	)	)	PUNCT
cana-3959	405	5	,	,	PUNCT
cana-3959	405	6	806	806	NUM
cana-3959	405	7	-	-	SYM
cana-3959	405	8	822	822	NUM
cana-3959	405	9	.	.	PUNCT