id	sid	tid	token	lemma	pos
cana-4127	1	1	communications	communication	NOUN
cana-4127	1	2	on	on	ADP
cana-4127	1	3	applied	apply	VERB
cana-4127	1	4	nonlinear	nonlinear	ADJ
cana-4127	1	5	analysis	analysis	NOUN
cana-4127	1	6	issn	issn	NOUN
cana-4127	1	7	:	:	PUNCT
cana-4127	1	8	1074	1074	NUM
cana-4127	1	9	-	-	PUNCT
cana-4127	1	10	133x	133x	NUM
cana-4127	1	11	vol	vol	NOUN
cana-4127	1	12	32	32	NUM
cana-4127	1	13	no	no	NOUN
cana-4127	1	14	.	.	PUNCT
cana-4127	2	1	9s	9s	NUM
cana-4127	2	2	(	(	PUNCT
cana-4127	2	3	2025	2025	NUM
cana-4127	2	4	)	)	PUNCT
cana-4127	2	5	1212	1212	NUM
cana-4127	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	2	7	analyzing	analyze	VERB
cana-4127	2	8	the	the	DET
cana-4127	2	9	stability	stability	NOUN
cana-4127	2	10	of	of	ADP
cana-4127	2	11	euler	euler	NOUN
cana-4127	2	12	-	-	PUNCT
cana-4127	2	13	lagrange	lagrange	NOUN
cana-4127	2	14	additive	additive	ADJ
cana-4127	2	15	functional	functional	ADJ
cana-4127	2	16	equations	equation	NOUN
cana-4127	2	17	in	in	ADP
cana-4127	2	18	banach	banach	NOUN
cana-4127	2	19	spaces	space	VERB
cana-4127	2	20	:	:	PUNCT
cana-4127	2	21	a	a	DET
cana-4127	2	22	fixed	fix	VERB
cana-4127	2	23	point	point	NOUN
cana-4127	2	24	and	and	CCONJ
cana-4127	2	25	direct	direct	ADJ
cana-4127	2	26	method	method	NOUN
cana-4127	2	27	perspective	perspective	NOUN
cana-4127	2	28	p.	p.	NOUN
cana-4127	2	29	agilan	agilan	PROPN
cana-4127	3	1	𝟏∗	𝟏∗	NUM
cana-4127	3	2	,	,	PUNCT
cana-4127	3	3	m.seetharaman	m.seetharaman	PROPN
cana-4127	3	4	𝟐	𝟐	NUM
cana-4127	3	5	,	,	PUNCT
cana-4127	3	6	m.	m.	NOUN
cana-4127	3	7	sophia	sophia	PROPN
cana-4127	3	8	𝟑	𝟑	NUM
cana-4127	3	9	,	,	PUNCT
cana-4127	3	10	vaddi	vaddi	VERB
cana-4127	3	11	seshagiri	seshagiri	NOUN
cana-4127	3	12	rao	rao	PROPN
cana-4127	3	13	𝟒	𝟒	NUM
cana-4127	3	14	1	1	NUM
cana-4127	3	15	department	department	NOUN
cana-4127	3	16	of	of	ADP
cana-4127	3	17	mathematics	mathematic	NOUN
cana-4127	3	18	,	,	PUNCT
cana-4127	3	19	st.joseph	st.joseph	X
cana-4127	3	20	’s	’s	PART
cana-4127	3	21	college	college	NOUN
cana-4127	3	22	of	of	ADP
cana-4127	3	23	engineering	engineering	PROPN
cana-4127	3	24	,	,	PUNCT
cana-4127	3	25	omr	omr	PROPN
cana-4127	3	26	,	,	PUNCT
cana-4127	3	27	chennai	chennai	VERB
cana-4127	3	28	600	600	NUM
cana-4127	3	29	119	119	NUM
cana-4127	3	30	,	,	PUNCT
cana-4127	3	31	tamil	tamil	PROPN
cana-4127	3	32	nadu	nadu	PROPN
cana-4127	3	33	,	,	PUNCT
cana-4127	4	1	india	india	PROPN
cana-4127	4	2	.	.	PUNCT
cana-4127	4	3	email	email	NOUN
cana-4127	4	4	:	:	PUNCT
cana-4127	5	1	agilram@gmail.com	agilram@gmail.com	X
cana-4127	6	1	2department	2department	NUM
cana-4127	6	2	of	of	ADP
cana-4127	6	3	mathematics	mathematic	NOUN
cana-4127	6	4	,	,	PUNCT
cana-4127	6	5	roever	roever	NOUN
cana-4127	6	6	engineering	engineering	NOUN
cana-4127	6	7	college	college	PROPN
cana-4127	6	8	,	,	PUNCT
cana-4127	6	9	elambalur	elambalur	NOUN
cana-4127	6	10	,	,	PUNCT
cana-4127	6	11	perambalur	perambalur	VERB
cana-4127	6	12	621	621	NUM
cana-4127	6	13	220	220	NUM
cana-4127	6	14	,	,	PUNCT
cana-4127	6	15	tamil	tamil	PROPN
cana-4127	6	16	nadu	nadu	PROPN
cana-4127	6	17	,	,	PUNCT
cana-4127	6	18	india	india	PROPN
cana-4127	6	19	.	.	PUNCT
cana-4127	6	20	email	email	NOUN
cana-4127	6	21	:	:	PUNCT
cana-4127	6	22	saram1183@gmail.com	saram1183@gmail.com	X
cana-4127	6	23	.	.	PUNCT
cana-4127	7	1	3department	3department	NUM
cana-4127	7	2	of	of	ADP
cana-4127	7	3	mathematics	mathematic	NOUN
cana-4127	7	4	,	,	PUNCT
cana-4127	7	5	simats	simat	NOUN
cana-4127	7	6	engineering	engineering	PROPN
cana-4127	7	7	,	,	PUNCT
cana-4127	7	8	saveetha	saveetha	PROPN
cana-4127	7	9	nagar	nagar	PROPN
cana-4127	7	10	,	,	PUNCT
cana-4127	7	11	thandalam	thandalam	PROPN
cana-4127	7	12	,	,	PUNCT
cana-4127	7	13	kanchipuram	kanchipuram	PROPN
cana-4127	7	14	-	-	PUNCT
cana-4127	7	15	chennai	chennai	PROPN
cana-4127	7	16	rd	rd	PROPN
cana-4127	7	17	,	,	PUNCT
cana-4127	7	18	chennai	chennai	PROPN
cana-4127	7	19	602105	602105	NUM
cana-4127	7	20	,	,	PUNCT
cana-4127	7	21	tamil	tamil	PROPN
cana-4127	7	22	nadu	nadu	PROPN
cana-4127	7	23	,	,	PUNCT
cana-4127	7	24	india	india	PROPN
cana-4127	7	25	.	.	PUNCT
cana-4127	8	1	e	e	X
cana-4127	8	2	-	-	NOUN
cana-4127	8	3	mail	mail	NOUN
cana-4127	8	4	:	:	PUNCT
cana-4127	8	5	sophia.raj2005@gmail.com	sophia.raj2005@gmail.com	PROPN
cana-4127	8	6	.	.	PROPN
cana-4127	8	7	4	4	NUM
cana-4127	8	8	department	department	NOUN
cana-4127	8	9	of	of	ADP
cana-4127	8	10	mechanical	mechanical	ADJ
cana-4127	8	11	engineering	engineering	NOUN
cana-4127	8	12	,	,	PUNCT
cana-4127	8	13	st.joseph	st.joseph	X
cana-4127	8	14	’s	’s	PART
cana-4127	8	15	college	college	NOUN
cana-4127	8	16	of	of	ADP
cana-4127	8	17	engineering	engineering	PROPN
cana-4127	8	18	,	,	PUNCT
cana-4127	8	19	omr	omr	PROPN
cana-4127	8	20	,	,	PUNCT
cana-4127	8	21	chennai	chennai	VERB
cana-4127	8	22	600	600	NUM
cana-4127	8	23	119	119	NUM
cana-4127	8	24	,	,	PUNCT
cana-4127	8	25	tamil	tamil	PROPN
cana-4127	8	26	nadu	nadu	PROPN
cana-4127	8	27	,	,	PUNCT
cana-4127	8	28	india	india	PROPN
cana-4127	8	29	.	.	PUNCT
cana-4127	9	1	e	e	X
cana-4127	9	2	-	-	NOUN
cana-4127	9	3	mail	mail	NOUN
cana-4127	9	4	:	:	PUNCT
cana-4127	10	1	raosvaddi@gmail.com	raosvaddi@gmail.com	X
cana-4127	10	2	.	.	PUNCT
cana-4127	11	1	∗corresponding	∗corresponde	VERB
cana-4127	11	2	author	author	NOUN
cana-4127	11	3	:	:	PUNCT
cana-4127	11	4	agilram@gmail.com	agilram@gmail.com	X
cana-4127	11	5	article	article	PROPN
cana-4127	11	6	history	history	NOUN
cana-4127	11	7	:	:	PUNCT
cana-4127	11	8	received	receive	VERB
cana-4127	11	9	:	:	PUNCT
cana-4127	11	10	12	12	NUM
cana-4127	11	11	-	-	SYM
cana-4127	11	12	01	01	NUM
cana-4127	11	13	-	-	PUNCT
cana-4127	11	14	2025	2025	NUM
cana-4127	11	15	revised	revise	VERB
cana-4127	11	16	:	:	PUNCT
cana-4127	11	17	15	15	NUM
cana-4127	11	18	-	-	NUM
cana-4127	11	19	02	02	NUM
cana-4127	11	20	-	-	PUNCT
cana-4127	11	21	2025	2025	NUM
cana-4127	11	22	accepted	accept	VERB
cana-4127	11	23	:	:	PUNCT
cana-4127	11	24	01	01	NUM
cana-4127	11	25	-	-	SYM
cana-4127	11	26	03	03	NUM
cana-4127	11	27	-	-	PUNCT
cana-4127	11	28	2025	2025	NUM
cana-4127	11	29	abstract	abstract	NOUN
cana-4127	11	30	:	:	PUNCT
cana-4127	11	31	this	this	DET
cana-4127	11	32	study	study	NOUN
cana-4127	11	33	explores	explore	VERB
cana-4127	11	34	the	the	DET
cana-4127	11	35	ulam	ulam	NOUN
cana-4127	11	36	-	-	PUNCT
cana-4127	11	37	hyers	hyer	NOUN
cana-4127	11	38	stability	stability	NOUN
cana-4127	11	39	of	of	ADP
cana-4127	11	40	the	the	DET
cana-4127	11	41	euler	euler	NOUN
cana-4127	11	42	-	-	PUNCT
cana-4127	11	43	lagrange	lagrange	NOUN
cana-4127	11	44	additive	additive	ADJ
cana-4127	11	45	functional	functional	ADJ
cana-4127	11	46	equation	equation	NOUN
cana-4127	11	47	in	in	ADP
cana-4127	11	48	the	the	DET
cana-4127	11	49	framework	framework	NOUN
cana-4127	11	50	of	of	ADP
cana-4127	11	51	banach	banach	NOUN
cana-4127	11	52	spaces	space	NOUN
cana-4127	11	53	.	.	PUNCT
cana-4127	12	1	by	by	ADP
cana-4127	12	2	employing	employ	VERB
cana-4127	12	3	both	both	CCONJ
cana-4127	12	4	direct	direct	ADJ
cana-4127	12	5	and	and	CCONJ
cana-4127	12	6	fixed	fix	VERB
cana-4127	12	7	point	point	NOUN
cana-4127	12	8	methods	method	NOUN
cana-4127	12	9	,	,	PUNCT
cana-4127	12	10	we	we	PRON
cana-4127	12	11	establish	establish	VERB
cana-4127	12	12	new	new	ADJ
cana-4127	12	13	stability	stability	NOUN
cana-4127	12	14	results	result	NOUN
cana-4127	12	15	that	that	PRON
cana-4127	12	16	contribute	contribute	VERB
cana-4127	12	17	to	to	ADP
cana-4127	12	18	the	the	DET
cana-4127	12	19	understanding	understanding	NOUN
cana-4127	12	20	of	of	ADP
cana-4127	12	21	functional	functional	ADJ
cana-4127	12	22	equations	equation	NOUN
cana-4127	12	23	in	in	ADP
cana-4127	12	24	normed	normed	ADJ
cana-4127	12	25	spaces	space	NOUN
cana-4127	12	26	.	.	PUNCT
cana-4127	13	1	the	the	DET
cana-4127	13	2	direct	direct	ADJ
cana-4127	13	3	method	method	NOUN
cana-4127	13	4	provides	provide	VERB
cana-4127	13	5	explicit	explicit	ADJ
cana-4127	13	6	estimates	estimate	NOUN
cana-4127	13	7	for	for	ADP
cana-4127	13	8	stability	stability	NOUN
cana-4127	13	9	bounds	bound	NOUN
cana-4127	13	10	,	,	PUNCT
cana-4127	13	11	while	while	SCONJ
cana-4127	13	12	the	the	DET
cana-4127	13	13	fixed	fix	VERB
cana-4127	13	14	point	point	NOUN
cana-4127	13	15	approach	approach	NOUN
cana-4127	13	16	ensures	ensure	VERB
cana-4127	13	17	the	the	DET
cana-4127	13	18	existence	existence	NOUN
cana-4127	13	19	and	and	CCONJ
cana-4127	13	20	uniqueness	uniqueness	NOUN
cana-4127	13	21	of	of	ADP
cana-4127	13	22	stable	stable	ADJ
cana-4127	13	23	solutions	solution	NOUN
cana-4127	13	24	under	under	ADP
cana-4127	13	25	suitable	suitable	ADJ
cana-4127	13	26	conditions	condition	NOUN
cana-4127	13	27	.	.	PUNCT
cana-4127	14	1	our	our	PRON
cana-4127	14	2	findings	finding	NOUN
cana-4127	14	3	offer	offer	VERB
cana-4127	14	4	a	a	DET
cana-4127	14	5	deeper	deep	ADJ
cana-4127	14	6	insight	insight	NOUN
cana-4127	14	7	into	into	ADP
cana-4127	14	8	the	the	DET
cana-4127	14	9	interplay	interplay	NOUN
cana-4127	14	10	between	between	ADP
cana-4127	14	11	functional	functional	ADJ
cana-4127	14	12	equations	equation	NOUN
cana-4127	14	13	and	and	CCONJ
cana-4127	14	14	stability	stability	NOUN
cana-4127	14	15	theory	theory	NOUN
cana-4127	14	16	,	,	PUNCT
cana-4127	14	17	with	with	ADP
cana-4127	14	18	potential	potential	ADJ
cana-4127	14	19	applications	application	NOUN
cana-4127	14	20	in	in	ADP
cana-4127	14	21	analysis	analysis	NOUN
cana-4127	14	22	and	and	CCONJ
cana-4127	14	23	mathematical	mathematical	ADJ
cana-4127	14	24	modeling	modeling	NOUN
cana-4127	14	25	.	.	PUNCT
cana-4127	15	1	keywords	keyword	NOUN
cana-4127	15	2	:	:	PUNCT
cana-4127	15	3	euler	euler	PROPN
cana-4127	15	4	lagrange	lagrange	PROPN
cana-4127	15	5	functional	functional	ADJ
cana-4127	15	6	equations	equation	NOUN
cana-4127	15	7	,	,	PUNCT
cana-4127	15	8	generalized	generalize	VERB
cana-4127	15	9	ulam	ulam	PROPN
cana-4127	15	10	hyers	hyer	NOUN
cana-4127	15	11	stability	stability	NOUN
cana-4127	15	12	,	,	PUNCT
cana-4127	15	13	banach	banach	NOUN
cana-4127	15	14	space	space	NOUN
cana-4127	15	15	,	,	PUNCT
cana-4127	15	16	fixed	fix	VERB
cana-4127	15	17	point	point	NOUN
cana-4127	15	18	.	.	PUNCT
cana-4127	16	1	1	1	X
cana-4127	16	2	.	.	X
cana-4127	16	3	introduction	introduction	NOUN
cana-4127	16	4	one	one	NUM
cana-4127	16	5	of	of	ADP
cana-4127	16	6	the	the	DET
cana-4127	16	7	most	most	ADV
cana-4127	16	8	influential	influential	ADJ
cana-4127	16	9	stability	stability	NOUN
cana-4127	16	10	notions	notion	NOUN
cana-4127	16	11	was	be	AUX
cana-4127	16	12	introduced	introduce	VERB
cana-4127	16	13	by	by	ADP
cana-4127	16	14	ulam	ulam	PROPN
cana-4127	16	15	in	in	ADP
cana-4127	16	16	1940	1940	NUM
cana-4127	16	17	,	,	PUNCT
cana-4127	16	18	when	when	SCONJ
cana-4127	16	19	he	he	PRON
cana-4127	16	20	posed	pose	VERB
cana-4127	16	21	a	a	DET
cana-4127	16	22	fundamental	fundamental	ADJ
cana-4127	16	23	question	question	NOUN
cana-4127	16	24	regarding	regard	VERB
cana-4127	16	25	the	the	DET
cana-4127	16	26	stability	stability	NOUN
cana-4127	16	27	of	of	ADP
cana-4127	16	28	group	group	NOUN
cana-4127	16	29	homomorphisms[1	homomorphisms[1	PROPN
cana-4127	16	30	]	]	PUNCT
cana-4127	16	31	.	.	PUNCT
cana-4127	17	1	this	this	DET
cana-4127	17	2	question	question	NOUN
cana-4127	17	3	was	be	AUX
cana-4127	17	4	later	later	ADV
cana-4127	17	5	answered	answer	VERB
cana-4127	17	6	affirmatively	affirmatively	ADV
cana-4127	17	7	by	by	ADP
cana-4127	17	8	hyers	hyer	NOUN
cana-4127	17	9	in	in	ADP
cana-4127	17	10	1941[2	1941[2	NUM
cana-4127	17	11	]	]	PUNCT
cana-4127	17	12	,	,	PUNCT
cana-4127	17	13	leading	lead	VERB
cana-4127	17	14	to	to	ADP
cana-4127	17	15	what	what	PRON
cana-4127	17	16	is	be	AUX
cana-4127	17	17	now	now	ADV
cana-4127	17	18	known	know	VERB
cana-4127	17	19	as	as	ADP
cana-4127	17	20	ulam	ulam	NOUN
cana-4127	17	21	-	-	PUNCT
cana-4127	17	22	hyers	hyer	NOUN
cana-4127	17	23	stability	stability	NOUN
cana-4127	17	24	.	.	PUNCT
cana-4127	18	1	since	since	SCONJ
cana-4127	18	2	then	then	ADV
cana-4127	18	3	,	,	PUNCT
cana-4127	18	4	this	this	DET
cana-4127	18	5	stability	stability	NOUN
cana-4127	18	6	concept	concept	NOUN
cana-4127	18	7	has	have	AUX
cana-4127	18	8	been	be	AUX
cana-4127	18	9	extensively	extensively	ADV
cana-4127	18	10	studied	study	VERB
cana-4127	18	11	and	and	CCONJ
cana-4127	18	12	extended	extend	VERB
cana-4127	18	13	to	to	ADP
cana-4127	18	14	various	various	ADJ
cana-4127	18	15	mathematical	mathematical	ADJ
cana-4127	18	16	settings	setting	NOUN
cana-4127	18	17	,	,	PUNCT
cana-4127	18	18	including	include	VERB
cana-4127	18	19	functional	functional	ADJ
cana-4127	18	20	equations	equation	NOUN
cana-4127	18	21	,	,	PUNCT
cana-4127	18	22	differential	differential	ADJ
cana-4127	18	23	equations	equation	NOUN
cana-4127	18	24	,	,	PUNCT
cana-4127	18	25	and	and	CCONJ
cana-4127	18	26	difference	difference	NOUN
cana-4127	18	27	equations	equation	NOUN
cana-4127	18	28	.	.	PUNCT
cana-4127	19	1	various	various	ADJ
cana-4127	19	2	types	type	NOUN
cana-4127	19	3	of	of	ADP
cana-4127	19	4	functional	functional	ADJ
cana-4127	19	5	equations	equation	NOUN
cana-4127	19	6	,	,	PUNCT
cana-4127	19	7	such	such	ADJ
cana-4127	19	8	as	as	ADP
cana-4127	19	9	cauchy	cauchy	PROPN
cana-4127	19	10	’s	’s	PART
cana-4127	19	11	functional	functional	ADJ
cana-4127	19	12	equation	equation	NOUN
cana-4127	19	13	,	,	PUNCT
cana-4127	19	14	jensen	jensen	PROPN
cana-4127	19	15	’s	’s	PART
cana-4127	19	16	equation	equation	NOUN
cana-4127	19	17	,	,	PUNCT
cana-4127	19	18	and	and	CCONJ
cana-4127	19	19	the	the	DET
cana-4127	19	20	eulerlagrange	eulerlagrange	ADJ
cana-4127	19	21	additive	additive	ADJ
cana-4127	19	22	functional	functional	ADJ
cana-4127	19	23	equation	equation	NOUN
cana-4127	19	24	,	,	PUNCT
cana-4127	19	25	have	have	AUX
cana-4127	19	26	been	be	AUX
cana-4127	19	27	studied	study	VERB
cana-4127	19	28	under	under	ADP
cana-4127	19	29	ulam	ulam	NOUN
cana-4127	19	30	-	-	PUNCT
cana-4127	19	31	hyers	hyer	NOUN
cana-4127	19	32	stability	stability	NOUN
cana-4127	19	33	[	[	X
cana-4127	19	34	3	3	NUM
cana-4127	19	35	,	,	PUNCT
cana-4127	19	36	4	4	NUM
cana-4127	19	37	,	,	PUNCT
cana-4127	19	38	5	5	NUM
cana-4127	19	39	]	]	PUNCT
cana-4127	19	40	.	.	PUNCT
cana-4127	20	1	in	in	ADP
cana-4127	20	2	many	many	ADJ
cana-4127	20	3	cases	case	NOUN
cana-4127	20	4	,	,	PUNCT
cana-4127	20	5	researchers	researcher	NOUN
cana-4127	20	6	employ	employ	VERB
cana-4127	20	7	direct	direct	ADJ
cana-4127	20	8	analytical	analytical	ADJ
cana-4127	20	9	methods	method	NOUN
cana-4127	20	10	,	,	PUNCT
cana-4127	20	11	fixed	fix	VERB
cana-4127	20	12	point	point	NOUN
cana-4127	20	13	theorems	theorem	NOUN
cana-4127	20	14	,	,	PUNCT
cana-4127	20	15	and	and	CCONJ
cana-4127	20	16	iterative	iterative	NOUN
cana-4127	20	17	techniques	technique	NOUN
cana-4127	20	18	to	to	PART
cana-4127	20	19	establish	establish	VERB
cana-4127	20	20	stability	stability	NOUN
cana-4127	20	21	results	result	NOUN
cana-4127	20	22	.	.	PUNCT
cana-4127	21	1	banach	banach	NOUN
cana-4127	21	2	space	space	NOUN
cana-4127	21	3	settings	setting	NOUN
cana-4127	21	4	provide	provide	VERB
cana-4127	21	5	a	a	DET
cana-4127	21	6	rich	rich	ADJ
cana-4127	21	7	framework	framework	NOUN
cana-4127	21	8	for	for	ADP
cana-4127	21	9	these	these	DET
cana-4127	21	10	studies	study	NOUN
cana-4127	21	11	,	,	PUNCT
cana-4127	21	12	as	as	SCONJ
cana-4127	21	13	normed	normed	ADJ
cana-4127	21	14	spaces	space	NOUN
cana-4127	21	15	allow	allow	VERB
cana-4127	21	16	rigorous	rigorous	ADJ
cana-4127	21	17	error	error	NOUN
cana-4127	21	18	estimations	estimation	NOUN
cana-4127	21	19	and	and	CCONJ
cana-4127	21	20	convergence	convergence	NOUN
cana-4127	21	21	analysis.[6	analysis.[6	NOUN
cana-4127	21	22	,	,	PUNCT
cana-4127	21	23	7	7	NUM
cana-4127	21	24	,	,	PUNCT
cana-4127	21	25	8	8	NUM
cana-4127	21	26	]	]	PUNCT
cana-4127	21	27	.	.	PUNCT
cana-4127	22	1	differential	differential	ADJ
cana-4127	22	2	equations	equation	NOUN
cana-4127	22	3	are	be	AUX
cana-4127	22	4	essential	essential	ADJ
cana-4127	22	5	in	in	ADP
cana-4127	22	6	modeling	model	VERB
cana-4127	22	7	physical	physical	ADJ
cana-4127	22	8	phenomena	phenomenon	NOUN
cana-4127	22	9	,	,	PUNCT
cana-4127	22	10	engineering	engineering	NOUN
cana-4127	22	11	systems	system	NOUN
cana-4127	22	12	,	,	PUNCT
cana-4127	22	13	and	and	CCONJ
cana-4127	22	14	biological	biological	ADJ
cana-4127	22	15	processes	process	NOUN
cana-4127	22	16	.	.	PUNCT
cana-4127	23	1	stability	stability	NOUN
cana-4127	23	2	analysis	analysis	NOUN
cana-4127	23	3	is	be	AUX
cana-4127	23	4	a	a	DET
cana-4127	23	5	key	key	ADJ
cana-4127	23	6	aspect	aspect	NOUN
cana-4127	23	7	of	of	ADP
cana-4127	23	8	differential	differential	ADJ
cana-4127	23	9	equation	equation	NOUN
cana-4127	23	10	theory	theory	NOUN
cana-4127	23	11	,	,	PUNCT
cana-4127	23	12	ensuring	ensure	VERB
cana-4127	23	13	that	that	SCONJ
cana-4127	23	14	communications	communication	NOUN
cana-4127	23	15	on	on	ADP
cana-4127	23	16	applied	apply	VERB
cana-4127	23	17	nonlinear	nonlinear	ADJ
cana-4127	23	18	analysis	analysis	NOUN
cana-4127	23	19	issn	issn	NOUN
cana-4127	23	20	:	:	PUNCT
cana-4127	23	21	1074	1074	NUM
cana-4127	23	22	-	-	PUNCT
cana-4127	23	23	133x	133x	NUM
cana-4127	23	24	vol	vol	NOUN
cana-4127	23	25	32	32	NUM
cana-4127	23	26	no	no	NOUN
cana-4127	23	27	.	.	PUNCT
cana-4127	24	1	9s	9s	NUM
cana-4127	24	2	(	(	PUNCT
cana-4127	24	3	2025	2025	NUM
cana-4127	24	4	)	)	PUNCT
cana-4127	24	5	1213	1213	NUM
cana-4127	24	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	24	7	small	small	ADJ
cana-4127	24	8	perturbations	perturbation	NOUN
cana-4127	24	9	in	in	ADP
cana-4127	24	10	initial	initial	ADJ
cana-4127	24	11	conditions	condition	NOUN
cana-4127	24	12	or	or	CCONJ
cana-4127	24	13	parameters	parameter	NOUN
cana-4127	24	14	do	do	AUX
cana-4127	24	15	not	not	PART
cana-4127	24	16	lead	lead	VERB
cana-4127	24	17	to	to	ADP
cana-4127	24	18	drastic	drastic	ADJ
cana-4127	24	19	changes	change	NOUN
cana-4127	24	20	in	in	ADP
cana-4127	24	21	solutions	solution	NOUN
cana-4127	24	22	[	[	X
cana-4127	24	23	9	9	NUM
cana-4127	24	24	]	]	PUNCT
cana-4127	24	25	.	.	PUNCT
cana-4127	25	1	the	the	DET
cana-4127	25	2	concept	concept	NOUN
cana-4127	25	3	of	of	ADP
cana-4127	25	4	ulam	ulam	NOUN
cana-4127	25	5	-	-	PUNCT
cana-4127	25	6	hyers	hyer	NOUN
cana-4127	25	7	stability	stability	NOUN
cana-4127	25	8	has	have	AUX
cana-4127	25	9	been	be	AUX
cana-4127	25	10	extended	extend	VERB
cana-4127	25	11	to	to	ADP
cana-4127	25	12	differential	differential	ADJ
cana-4127	25	13	equations	equation	NOUN
cana-4127	25	14	,	,	PUNCT
cana-4127	25	15	particularly	particularly	ADV
cana-4127	25	16	to	to	ADP
cana-4127	25	17	ordinary	ordinary	ADJ
cana-4127	25	18	differential	differential	ADJ
cana-4127	25	19	equations	equation	NOUN
cana-4127	25	20	(	(	PUNCT
cana-4127	25	21	odes	ode	NOUN
cana-4127	25	22	)	)	PUNCT
cana-4127	25	23	and	and	CCONJ
cana-4127	25	24	partial	partial	ADJ
cana-4127	25	25	differential	differential	ADJ
cana-4127	25	26	equations	equation	NOUN
cana-4127	25	27	(	(	PUNCT
cana-4127	25	28	pdes).for	pdes).for	ADP
cana-4127	25	29	first	first	ADJ
cana-4127	25	30	-	-	PUNCT
cana-4127	25	31	order	order	NOUN
cana-4127	25	32	and	and	CCONJ
cana-4127	25	33	higher	high	ADJ
cana-4127	25	34	-	-	PUNCT
cana-4127	25	35	order	order	NOUN
cana-4127	25	36	differential	differential	NOUN
cana-4127	25	37	equations[10	equations[10	ADV
cana-4127	25	38	,	,	PUNCT
cana-4127	25	39	11	11	NUM
cana-4127	25	40	]	]	PUNCT
cana-4127	25	41	,	,	PUNCT
cana-4127	25	42	stability	stability	NOUN
cana-4127	25	43	is	be	AUX
cana-4127	25	44	often	often	ADV
cana-4127	25	45	analyzed	analyze	VERB
cana-4127	25	46	using	use	VERB
cana-4127	25	47	fixed	fix	VERB
cana-4127	25	48	point	point	NOUN
cana-4127	25	49	methods	method	NOUN
cana-4127	25	50	,	,	PUNCT
cana-4127	25	51	integral	integral	ADJ
cana-4127	25	52	inequalities	inequality	NOUN
cana-4127	25	53	,	,	PUNCT
cana-4127	25	54	and	and	CCONJ
cana-4127	25	55	operator	operator	NOUN
cana-4127	25	56	theory	theory	NOUN
cana-4127	25	57	.	.	PUNCT
cana-4127	26	1	the	the	DET
cana-4127	26	2	stability	stability	NOUN
cana-4127	26	3	of	of	ADP
cana-4127	26	4	linear	linear	ADJ
cana-4127	26	5	and	and	CCONJ
cana-4127	26	6	nonlinear	nonlinear	ADJ
cana-4127	26	7	differential	differential	ADJ
cana-4127	26	8	equations	equation	NOUN
cana-4127	26	9	has	have	AUX
cana-4127	26	10	been	be	AUX
cana-4127	26	11	extensively	extensively	ADV
cana-4127	26	12	studied	study	VERB
cana-4127	26	13	,	,	PUNCT
cana-4127	26	14	leading	lead	VERB
cana-4127	26	15	to	to	ADP
cana-4127	26	16	significant	significant	ADJ
cana-4127	26	17	results	result	NOUN
cana-4127	26	18	that	that	PRON
cana-4127	26	19	ensure	ensure	VERB
cana-4127	26	20	the	the	DET
cana-4127	26	21	robustness	robustness	NOUN
cana-4127	26	22	of	of	ADP
cana-4127	26	23	solutions	solution	NOUN
cana-4127	26	24	under	under	ADP
cana-4127	26	25	small	small	ADJ
cana-4127	26	26	perturbations	perturbation	NOUN
cana-4127	26	27	.	.	PUNCT
cana-4127	27	1	the	the	DET
cana-4127	27	2	application	application	NOUN
cana-4127	27	3	of	of	ADP
cana-4127	27	4	fixed	fix	VERB
cana-4127	27	5	point	point	NOUN
cana-4127	27	6	theorems	theorem	NOUN
cana-4127	27	7	such	such	ADJ
cana-4127	27	8	as	as	ADP
cana-4127	27	9	banach	banach	NOUN
cana-4127	27	10	’s	’s	PART
cana-4127	27	11	contraction	contraction	NOUN
cana-4127	27	12	principle	principle	NOUN
cana-4127	27	13	has	have	AUX
cana-4127	27	14	been	be	AUX
cana-4127	27	15	particularly	particularly	ADV
cana-4127	27	16	useful	useful	ADJ
cana-4127	27	17	in	in	ADP
cana-4127	27	18	proving	prove	VERB
cana-4127	27	19	the	the	DET
cana-4127	27	20	existence	existence	NOUN
cana-4127	27	21	and	and	CCONJ
cana-4127	27	22	stability	stability	NOUN
cana-4127	27	23	of	of	ADP
cana-4127	27	24	solutions	solution	NOUN
cana-4127	27	25	.	.	PUNCT
cana-4127	28	1	in	in	ADP
cana-4127	28	2	partial	partial	ADJ
cana-4127	28	3	differential	differential	NOUN
cana-4127	28	4	equations	equation	NOUN
cana-4127	28	5	,	,	PUNCT
cana-4127	28	6	researchers	researcher	NOUN
cana-4127	28	7	have	have	AUX
cana-4127	28	8	investigated	investigate	VERB
cana-4127	28	9	ulam	ulam	NOUN
cana-4127	28	10	-	-	PUNCT
cana-4127	28	11	hyers	hyer	NOUN
cana-4127	28	12	stability	stability	NOUN
cana-4127	28	13	in	in	ADP
cana-4127	28	14	settings	setting	NOUN
cana-4127	28	15	involving	involve	VERB
cana-4127	28	16	boundary	boundary	ADJ
cana-4127	28	17	value	value	NOUN
cana-4127	28	18	problems	problem	NOUN
cana-4127	28	19	,	,	PUNCT
cana-4127	28	20	heat	heat	NOUN
cana-4127	28	21	equations	equation	NOUN
cana-4127	28	22	,	,	PUNCT
cana-4127	28	23	and	and	CCONJ
cana-4127	28	24	wave	wave	NOUN
cana-4127	28	25	equations	equation	NOUN
cana-4127	28	26	.	.	PUNCT
cana-4127	29	1	such	such	ADJ
cana-4127	29	2	studies	study	NOUN
cana-4127	29	3	are	be	AUX
cana-4127	29	4	crucial	crucial	ADJ
cana-4127	29	5	in	in	ADP
cana-4127	29	6	mathematical	mathematical	ADJ
cana-4127	29	7	physics	physics	NOUN
cana-4127	29	8	,	,	PUNCT
cana-4127	29	9	where	where	SCONJ
cana-4127	29	10	slight	slight	ADJ
cana-4127	29	11	variations	variation	NOUN
cana-4127	29	12	in	in	ADP
cana-4127	29	13	initial	initial	ADJ
cana-4127	29	14	or	or	CCONJ
cana-4127	29	15	boundary	boundary	ADJ
cana-4127	29	16	conditions	condition	NOUN
cana-4127	29	17	should	should	AUX
cana-4127	29	18	not	not	PART
cana-4127	29	19	lead	lead	VERB
cana-4127	29	20	to	to	ADP
cana-4127	29	21	unbounded	unbounded	ADJ
cana-4127	29	22	or	or	CCONJ
cana-4127	29	23	highly	highly	ADV
cana-4127	29	24	unstable	unstable	ADJ
cana-4127	29	25	behaviors	behavior	NOUN
cana-4127	29	26	.	.	PUNCT
cana-4127	30	1	difference	difference	NOUN
cana-4127	30	2	equations	equation	NOUN
cana-4127	30	3	serve	serve	VERB
cana-4127	30	4	as	as	ADP
cana-4127	30	5	discrete	discrete	ADJ
cana-4127	30	6	counterparts	counterpart	NOUN
cana-4127	30	7	to	to	PART
cana-4127	30	8	differential	differential	VERB
cana-4127	30	9	equations	equation	NOUN
cana-4127	30	10	and	and	CCONJ
cana-4127	30	11	have	have	VERB
cana-4127	30	12	applications	application	NOUN
cana-4127	30	13	in	in	ADP
cana-4127	30	14	numerical	numerical	ADJ
cana-4127	30	15	analysis	analysis	NOUN
cana-4127	30	16	,	,	PUNCT
cana-4127	30	17	dynamical	dynamical	ADJ
cana-4127	30	18	systems	system	NOUN
cana-4127	30	19	,	,	PUNCT
cana-4127	30	20	and	and	CCONJ
cana-4127	30	21	computer	computer	NOUN
cana-4127	30	22	science	science	NOUN
cana-4127	30	23	.	.	PUNCT
cana-4127	31	1	ulam	ulam	NOUN
cana-4127	31	2	-	-	PUNCT
cana-4127	31	3	hyers	hyer	NOUN
cana-4127	31	4	stability	stability	NOUN
cana-4127	31	5	in	in	ADP
cana-4127	31	6	difference	difference	NOUN
cana-4127	31	7	equations	equation	NOUN
cana-4127	31	8	ensures	ensure	VERB
cana-4127	31	9	that	that	SCONJ
cana-4127	31	10	small	small	ADJ
cana-4127	31	11	deviations	deviation	NOUN
cana-4127	31	12	in	in	ADP
cana-4127	31	13	the	the	DET
cana-4127	31	14	system	system	NOUN
cana-4127	31	15	’s	’s	PART
cana-4127	31	16	structure	structure	NOUN
cana-4127	31	17	do	do	AUX
cana-4127	31	18	not	not	PART
cana-4127	31	19	lead	lead	VERB
cana-4127	31	20	to	to	ADP
cana-4127	31	21	significant	significant	ADJ
cana-4127	31	22	instabilities	instability	NOUN
cana-4127	31	23	in	in	ADP
cana-4127	31	24	discrete	discrete	ADJ
cana-4127	31	25	-	-	PUNCT
cana-4127	31	26	time	time	NOUN
cana-4127	31	27	solutions	solution	NOUN
cana-4127	31	28	.	.	PUNCT
cana-4127	32	1	the	the	DET
cana-4127	32	2	stability	stability	NOUN
cana-4127	32	3	analysis	analysis	NOUN
cana-4127	32	4	of	of	ADP
cana-4127	32	5	difference	difference	NOUN
cana-4127	32	6	equations	equation	NOUN
cana-4127	32	7	has	have	AUX
cana-4127	32	8	been	be	AUX
cana-4127	32	9	widely	widely	ADV
cana-4127	32	10	explored	explore	VERB
cana-4127	32	11	using	use	VERB
cana-4127	32	12	discrete	discrete	ADJ
cana-4127	32	13	fixed	fix	VERB
cana-4127	32	14	point	point	NOUN
cana-4127	32	15	methods	method	NOUN
cana-4127	32	16	,	,	PUNCT
cana-4127	32	17	iterative	iterative	NOUN
cana-4127	32	18	approximation	approximation	NOUN
cana-4127	32	19	techniques	technique	NOUN
cana-4127	32	20	,	,	PUNCT
cana-4127	32	21	and	and	CCONJ
cana-4127	32	22	functional	functional	ADJ
cana-4127	32	23	inequalities	inequality	NOUN
cana-4127	32	24	.	.	PUNCT
cana-4127	33	1	linear	linear	ADJ
cana-4127	33	2	and	and	CCONJ
cana-4127	33	3	nonlinear	nonlinear	ADJ
cana-4127	33	4	difference	difference	NOUN
cana-4127	33	5	equations	equation	NOUN
cana-4127	33	6	,	,	PUNCT
cana-4127	33	7	recurrence	recurrence	NOUN
cana-4127	33	8	relations	relation	NOUN
cana-4127	33	9	,	,	PUNCT
cana-4127	33	10	and	and	CCONJ
cana-4127	33	11	fractional	fractional	ADJ
cana-4127	33	12	difference	difference	NOUN
cana-4127	33	13	equations	equation	NOUN
cana-4127	33	14	have	have	AUX
cana-4127	33	15	all	all	PRON
cana-4127	33	16	been	be	AUX
cana-4127	33	17	studied	study	VERB
cana-4127	33	18	under	under	ADP
cana-4127	33	19	the	the	DET
cana-4127	33	20	ulam	ulam	NOUN
cana-4127	33	21	-	-	PUNCT
cana-4127	33	22	hyers	hyer	NOUN
cana-4127	33	23	stability	stability	NOUN
cana-4127	33	24	framework[12	framework[12	PROPN
cana-4127	33	25	,	,	PUNCT
cana-4127	33	26	13	13	NUM
cana-4127	33	27	]	]	PUNCT
cana-4127	33	28	.	.	PUNCT
cana-4127	34	1	these	these	DET
cana-4127	34	2	results	result	NOUN
cana-4127	34	3	are	be	AUX
cana-4127	34	4	particularly	particularly	ADV
cana-4127	34	5	useful	useful	ADJ
cana-4127	34	6	in	in	ADP
cana-4127	34	7	numerical	numerical	ADJ
cana-4127	34	8	methods	method	NOUN
cana-4127	34	9	for	for	ADP
cana-4127	34	10	solving	solve	VERB
cana-4127	34	11	differential	differential	ADJ
cana-4127	34	12	equations	equation	NOUN
cana-4127	34	13	,	,	PUNCT
cana-4127	34	14	where	where	SCONJ
cana-4127	34	15	approximations	approximation	NOUN
cana-4127	34	16	and	and	CCONJ
cana-4127	34	17	discretization	discretization	NOUN
cana-4127	34	18	errors	error	NOUN
cana-4127	34	19	must	must	AUX
cana-4127	34	20	be	be	AUX
cana-4127	34	21	controlled	control	VERB
cana-4127	34	22	.	.	PUNCT
cana-4127	35	1	recently	recently	ADV
cana-4127	35	2	agilan	agilan	PROPN
cana-4127	35	3	et.al	et.al	NOUN
cana-4127	35	4	exploring	explore	VERB
cana-4127	35	5	the	the	DET
cana-4127	35	6	stability	stability	NOUN
cana-4127	35	7	results	result	NOUN
cana-4127	35	8	in	in	ADP
cana-4127	35	9	various	various	ADJ
cana-4127	35	10	additive	additive	ADJ
cana-4127	35	11	functional	functional	ADJ
cana-4127	35	12	equation	equation	NOUN
cana-4127	35	13	through	through	ADP
cana-4127	35	14	various	various	ADJ
cana-4127	35	15	normed	norme	VERB
cana-4127	35	16	spaces	space	NOUN
cana-4127	35	17	such	such	ADJ
cana-4127	35	18	as	as	ADP
cana-4127	35	19	[	[	X
cana-4127	35	20	14	14	NUM
cana-4127	35	21	,	,	PUNCT
cana-4127	35	22	15	15	NUM
cana-4127	35	23	,	,	PUNCT
cana-4127	35	24	16	16	NUM
cana-4127	35	25	,	,	PUNCT
cana-4127	35	26	17	17	NUM
cana-4127	35	27	,	,	PUNCT
cana-4127	35	28	18	18	NUM
cana-4127	35	29	,	,	PUNCT
cana-4127	35	30	19	19	NUM
cana-4127	35	31	,	,	PUNCT
cana-4127	35	32	20	20	NUM
cana-4127	35	33	,	,	PUNCT
cana-4127	35	34	21	21	NUM
cana-4127	35	35	]	]	PUNCT
cana-4127	35	36	.	.	PUNCT
cana-4127	36	1	in	in	ADP
cana-4127	36	2	this	this	DET
cana-4127	36	3	paper	paper	NOUN
cana-4127	36	4	,	,	PUNCT
cana-4127	36	5	we	we	PRON
cana-4127	36	6	introduce	introduce	VERB
cana-4127	36	7	new	new	ADJ
cana-4127	36	8	kind	kind	NOUN
cana-4127	36	9	of	of	ADP
cana-4127	36	10	functional	functional	ADJ
cana-4127	36	11	equation	equation	NOUN
cana-4127	36	12	and	and	CCONJ
cana-4127	36	13	investigate	investigate	VERB
cana-4127	36	14	the	the	DET
cana-4127	36	15	ulam	ulam	NOUN
cana-4127	36	16	-	-	PUNCT
cana-4127	36	17	hyers	hyer	NOUN
cana-4127	36	18	stability	stability	NOUN
cana-4127	36	19	of	of	ADP
cana-4127	36	20	the	the	DET
cana-4127	36	21	euler	euler	NOUN
cana-4127	36	22	-	-	PUNCT
cana-4127	36	23	lagrange	lagrange	NOUN
cana-4127	36	24	additive	additive	ADJ
cana-4127	36	25	functional	functional	ADJ
cana-4127	36	26	equation	equation	NOUN
cana-4127	36	27	in	in	ADP
cana-4127	36	28	banach	banach	NOUN
cana-4127	36	29	spaces	space	NOUN
cana-4127	36	30	using	use	VERB
cana-4127	36	31	direct	direct	ADJ
cana-4127	36	32	and	and	CCONJ
cana-4127	36	33	fixed	fix	VERB
cana-4127	36	34	point	point	NOUN
cana-4127	36	35	methods	method	NOUN
cana-4127	36	36	.	.	PUNCT
cana-4127	37	1	(	(	PUNCT
cana-4127	37	2	𝑠2	𝑠2	NOUN
cana-4127	37	3	+	+	CCONJ
cana-4127	37	4	2𝑠)ℎ(𝑝𝑥	2𝑠)ℎ(𝑝𝑥	NUM
cana-4127	37	5	+	+	CCONJ
cana-4127	37	6	𝑞𝑦	𝑞𝑦	NOUN
cana-4127	37	7	)	)	PUNCT
cana-4127	38	1	+	+	CCONJ
cana-4127	38	2	(	(	PUNCT
cana-4127	38	3	1	1	NUM
cana-4127	38	4	−	−	NUM
cana-4127	38	5	2𝑠)ℎ(𝑝𝑦	2𝑠)ℎ(𝑝𝑦	NUM
cana-4127	38	6	+	+	CCONJ
cana-4127	38	7	𝑞𝑧	𝑞𝑧	NOUN
cana-4127	38	8	)	)	PUNCT
cana-4127	39	1	+	+	CCONJ
cana-4127	39	2	2𝑠ℎ(𝑝𝑧	2𝑠ℎ(𝑝𝑧	NUM
cana-4127	39	3	+	+	NUM
cana-4127	39	4	𝑞𝑥	𝑞𝑥	PROPN
cana-4127	39	5	)	)	PUNCT
cana-4127	39	6	−	−	PROPN
cana-4127	39	7	2𝑠𝑝ℎ(𝑥	2𝑠𝑝ℎ(𝑥	NUM
cana-4127	39	8	−	−	PROPN
cana-4127	39	9	𝑦	𝑦	X
cana-4127	39	10	)	)	PUNCT
cana-4127	39	11	−2𝑠𝑞ℎ(𝑦	−2𝑠𝑞ℎ(𝑦	NOUN
cana-4127	39	12	−	−	PROPN
cana-4127	39	13	𝑧	𝑧	NOUN
cana-4127	39	14	)	)	PUNCT
cana-4127	39	15	=	=	SYM
cana-4127	39	16	(	(	PUNCT
cana-4127	39	17	𝑠2𝑝	𝑠2𝑝	PROPN
cana-4127	39	18	+	+	CCONJ
cana-4127	39	19	2𝑠𝑞)ℎ(𝑥	2𝑠𝑞)ℎ(𝑥	NUM
cana-4127	39	20	)	)	PUNCT
cana-4127	39	21	+	+	CCONJ
cana-4127	39	22	(	(	PUNCT
cana-4127	39	23	𝑝	𝑝	PROPN
cana-4127	39	24	+	+	NUM
cana-4127	39	25	𝑠2𝑞)ℎ(𝑦	𝑠2𝑞)ℎ(𝑦	NOUN
cana-4127	39	26	)	)	PUNCT
cana-4127	40	1	+	+	CCONJ
cana-4127	41	1	(	(	PUNCT
cana-4127	41	2	2𝑠𝑝	2𝑠𝑝	ADJ
cana-4127	41	3	+	+	CCONJ
cana-4127	41	4	𝑞)ℎ(𝑧	𝑞)ℎ(𝑧	NOUN
cana-4127	41	5	)	)	PUNCT
cana-4127	41	6	(	(	PUNCT
cana-4127	41	7	1	1	X
cana-4127	41	8	)	)	PUNCT
cana-4127	41	9	with	with	ADP
cana-4127	41	10	(	(	PUNCT
cana-4127	41	11	𝑠2	𝑠2	NOUN
cana-4127	41	12	+	+	CCONJ
cana-4127	41	13	2𝑠	2𝑠	NOUN
cana-4127	41	14	)	)	PUNCT
cana-4127	41	15	,	,	PUNCT
cana-4127	41	16	(	(	PUNCT
cana-4127	41	17	1	1	NUM
cana-4127	41	18	−	−	NOUN
cana-4127	41	19	2𝑠	2𝑠	NOUN
cana-4127	41	20	)	)	PUNCT
cana-4127	41	21	,	,	PUNCT
cana-4127	41	22	𝑠𝑝	𝑠𝑝	NOUN
cana-4127	41	23	,	,	PUNCT
cana-4127	41	24	𝑠𝑞	𝑠𝑞	NOUN
cana-4127	41	25	≠	≠	PROPN
cana-4127	41	26	0	0	NUM
cana-4127	41	27	and	and	CCONJ
cana-4127	41	28	𝑝	𝑝	PROPN
cana-4127	41	29	,	,	PUNCT
cana-4127	41	30	𝑞	𝑞	PROPN
cana-4127	41	31	,	,	PUNCT
cana-4127	41	32	𝑠	𝑠	PROPN
cana-4127	41	33	∈	∈	PROPN
cana-4127	41	34	ℝ	ℝ	PROPN
cana-4127	41	35	we	we	PRON
cana-4127	41	36	aim	aim	VERB
cana-4127	41	37	to:1.establish	to:1.establish	NUM
cana-4127	41	38	new	new	ADJ
cana-4127	41	39	stability	stability	NOUN
cana-4127	41	40	results	result	NOUN
cana-4127	41	41	for	for	ADP
cana-4127	41	42	the	the	DET
cana-4127	41	43	euler	euler	ADJ
cana-4127	41	44	-	-	PUNCT
cana-4127	41	45	lagrange	lagrange	NOUN
cana-4127	41	46	equation	equation	NOUN
cana-4127	41	47	under	under	ADP
cana-4127	41	48	different	different	ADJ
cana-4127	41	49	conditions	condition	NOUN
cana-4127	41	50	.	.	PUNCT
cana-4127	42	1	2.compare	2.compare	NUM
cana-4127	43	1	the	the	DET
cana-4127	43	2	effectiveness	effectiveness	NOUN
cana-4127	43	3	of	of	ADP
cana-4127	43	4	direct	direct	ADJ
cana-4127	43	5	analytical	analytical	ADJ
cana-4127	43	6	techniques	technique	NOUN
cana-4127	43	7	and	and	CCONJ
cana-4127	43	8	fixed	fix	VERB
cana-4127	43	9	point	point	NOUN
cana-4127	43	10	methods	method	NOUN
cana-4127	43	11	.	.	PUNCT
cana-4127	44	1	3.extend	3.extend	NUM
cana-4127	44	2	existing	exist	VERB
cana-4127	44	3	stability	stability	NOUN
cana-4127	44	4	results	result	NOUN
cana-4127	44	5	to	to	ADP
cana-4127	44	6	broader	broad	ADJ
cana-4127	44	7	classes	class	NOUN
cana-4127	44	8	of	of	ADP
cana-4127	44	9	functional	functional	ADJ
cana-4127	44	10	equations	equation	NOUN
cana-4127	44	11	.	.	PUNCT
cana-4127	45	1	by	by	ADP
cana-4127	45	2	doing	do	VERB
cana-4127	45	3	so	so	ADV
cana-4127	45	4	,	,	PUNCT
cana-4127	45	5	we	we	PRON
cana-4127	45	6	contribute	contribute	VERB
cana-4127	45	7	to	to	ADP
cana-4127	45	8	the	the	DET
cana-4127	45	9	ongoing	ongoing	ADJ
cana-4127	45	10	research	research	NOUN
cana-4127	45	11	in	in	ADP
cana-4127	45	12	stability	stability	NOUN
cana-4127	45	13	theory	theory	NOUN
cana-4127	45	14	,	,	PUNCT
cana-4127	45	15	offering	offer	VERB
cana-4127	45	16	insights	insight	NOUN
cana-4127	45	17	into	into	ADP
cana-4127	45	18	its	its	PRON
cana-4127	45	19	applications	application	NOUN
cana-4127	45	20	in	in	ADP
cana-4127	45	21	various	various	ADJ
cana-4127	45	22	mathematical	mathematical	ADJ
cana-4127	45	23	and	and	CCONJ
cana-4127	45	24	applied	apply	VERB
cana-4127	45	25	fields	field	NOUN
cana-4127	45	26	.	.	PUNCT
cana-4127	46	1	hereafter	hereafter	ADV
cana-4127	46	2	through	through	ADP
cana-4127	46	3	out	out	ADP
cana-4127	46	4	this	this	DET
cana-4127	46	5	paper	paper	NOUN
cana-4127	46	6	,	,	PUNCT
cana-4127	46	7	let	let	VERB
cana-4127	46	8	us	we	PRON
cana-4127	46	9	consider	consider	VERB
cana-4127	46	10	𝑋	𝑋	NOUN
cana-4127	46	11	and	and	CCONJ
cana-4127	46	12	𝑌	𝑌	PROPN
cana-4127	46	13	to	to	PART
cana-4127	46	14	be	be	AUX
cana-4127	46	15	a	a	DET
cana-4127	46	16	normed	normed	ADJ
cana-4127	46	17	linear	linear	ADJ
cana-4127	46	18	space	space	NOUN
cana-4127	46	19	and	and	CCONJ
cana-4127	46	20	a	a	DET
cana-4127	46	21	banach	banach	NOUN
cana-4127	46	22	space	space	NOUN
cana-4127	46	23	,	,	PUNCT
cana-4127	46	24	respectively	respectively	ADV
cana-4127	46	25	.	.	PUNCT
cana-4127	47	1	𝐻(𝑥	𝐻(𝑥	PROPN
cana-4127	47	2	,	,	PUNCT
cana-4127	47	3	𝑦	𝑦	NOUN
cana-4127	47	4	,	,	PUNCT
cana-4127	47	5	𝑧	𝑧	NOUN
cana-4127	47	6	)	)	PUNCT
cana-4127	47	7	=	=	SYM
cana-4127	47	8	(	(	PUNCT
cana-4127	47	9	𝑠2	𝑠2	NOUN
cana-4127	47	10	+	+	CCONJ
cana-4127	47	11	2𝑠)ℎ(𝑝𝑥	2𝑠)ℎ(𝑝𝑥	NUM
cana-4127	47	12	+	+	CCONJ
cana-4127	47	13	𝑞𝑦	𝑞𝑦	NOUN
cana-4127	47	14	)	)	PUNCT
cana-4127	48	1	+	+	CCONJ
cana-4127	48	2	(	(	PUNCT
cana-4127	48	3	1	1	NUM
cana-4127	48	4	−	−	NUM
cana-4127	48	5	2𝑠)ℎ(𝑝𝑦	2𝑠)ℎ(𝑝𝑦	NUM
cana-4127	48	6	+	+	CCONJ
cana-4127	48	7	𝑞𝑧	𝑞𝑧	NOUN
cana-4127	48	8	)	)	PUNCT
cana-4127	49	1	+	+	CCONJ
cana-4127	49	2	2𝑠ℎ(𝑝𝑧	2𝑠ℎ(𝑝𝑧	NUM
cana-4127	49	3	+	+	NUM
cana-4127	49	4	𝑞𝑥	𝑞𝑥	PROPN
cana-4127	49	5	)	)	PUNCT
cana-4127	49	6	−	−	PROPN
cana-4127	49	7	2𝑠𝑝ℎ(𝑥	2𝑠𝑝ℎ(𝑥	NUM
cana-4127	49	8	−	−	PROPN
cana-4127	49	9	𝑦	𝑦	NOUN
cana-4127	49	10	)	)	PUNCT
cana-4127	49	11	−	−	PROPN
cana-4127	50	1	2𝑠𝑞ℎ(𝑦	2𝑠𝑞ℎ(𝑦	NUM
cana-4127	50	2	−	−	PROPN
cana-4127	50	3	𝑧	𝑧	NOUN
cana-4127	50	4	)	)	PUNCT
cana-4127	50	5	−	−	PROPN
cana-4127	50	6	(	(	PUNCT
cana-4127	50	7	𝑠2𝑝	𝑠2𝑝	PROPN
cana-4127	50	8	+	+	CCONJ
cana-4127	50	9	2𝑠𝑞)ℎ(𝑥	2𝑠𝑞)ℎ(𝑥	NUM
cana-4127	50	10	)	)	PUNCT
cana-4127	50	11	−	−	PROPN
cana-4127	50	12	(	(	PUNCT
cana-4127	50	13	𝑝	𝑝	PROPN
cana-4127	50	14	+	+	NUM
cana-4127	50	15	𝑠2𝑞)ℎ(𝑦	𝑠2𝑞)ℎ(𝑦	NOUN
cana-4127	50	16	)	)	PUNCT
cana-4127	51	1	−	−	PROPN
cana-4127	51	2	(	(	PUNCT
cana-4127	51	3	2𝑠𝑝	2𝑠𝑝	ADJ
cana-4127	51	4	+	+	CCONJ
cana-4127	51	5	𝑞)ℎ(𝑧	𝑞)ℎ(𝑧	NOUN
cana-4127	51	6	)	)	PUNCT
cana-4127	51	7	communications	communication	NOUN
cana-4127	51	8	on	on	ADP
cana-4127	51	9	applied	apply	VERB
cana-4127	51	10	nonlinear	nonlinear	ADJ
cana-4127	51	11	analysis	analysis	NOUN
cana-4127	51	12	issn	issn	NOUN
cana-4127	51	13	:	:	PUNCT
cana-4127	51	14	1074	1074	NUM
cana-4127	51	15	-	-	PUNCT
cana-4127	51	16	133x	133x	NUM
cana-4127	51	17	vol	vol	NOUN
cana-4127	51	18	32	32	NUM
cana-4127	51	19	no	no	NOUN
cana-4127	51	20	.	.	PUNCT
cana-4127	52	1	9s	9s	NUM
cana-4127	52	2	(	(	PUNCT
cana-4127	52	3	2025	2025	NUM
cana-4127	52	4	)	)	PUNCT
cana-4127	52	5	1214	1214	NUM
cana-4127	52	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	52	7	where	where	SCONJ
cana-4127	52	8	(	(	PUNCT
cana-4127	52	9	𝑠2	𝑠2	NOUN
cana-4127	52	10	+	+	NOUN
cana-4127	52	11	2𝑠	2𝑠	NOUN
cana-4127	52	12	)	)	PUNCT
cana-4127	52	13	,	,	PUNCT
cana-4127	52	14	(	(	PUNCT
cana-4127	52	15	1	1	NUM
cana-4127	52	16	−	−	NOUN
cana-4127	52	17	2𝑠	2𝑠	NOUN
cana-4127	52	18	)	)	PUNCT
cana-4127	52	19	,	,	PUNCT
cana-4127	52	20	𝑠𝑝	𝑠𝑝	NOUN
cana-4127	52	21	,	,	PUNCT
cana-4127	52	22	𝑠𝑞	𝑠𝑞	NOUN
cana-4127	52	23	≠	≠	PROPN
cana-4127	52	24	0	0	NUM
cana-4127	52	25	and	and	CCONJ
cana-4127	52	26	𝑝	𝑝	PROPN
cana-4127	52	27	,	,	PUNCT
cana-4127	52	28	𝑞	𝑞	PROPN
cana-4127	52	29	,	,	PUNCT
cana-4127	52	30	𝑠	𝑠	PROPN
cana-4127	52	31	∈	∈	PROPN
cana-4127	52	32	ℝ	ℝ	PROPN
cana-4127	52	33	for	for	ADP
cana-4127	52	34	all	all	DET
cana-4127	52	35	𝑥	𝑥	PROPN
cana-4127	52	36	,	,	PUNCT
cana-4127	52	37	𝑦	𝑦	NOUN
cana-4127	52	38	,	,	PUNCT
cana-4127	52	39	𝑧	𝑧	DET
cana-4127	52	40	∈	∈	PROPN
cana-4127	52	41	𝑋.	𝑋.	PROPN
cana-4127	52	42	2	2	NUM
cana-4127	52	43	stability	stability	NOUN
cana-4127	52	44	results	result	NOUN
cana-4127	52	45	for	for	ADP
cana-4127	52	46	the	the	DET
cana-4127	52	47	functional	functional	ADJ
cana-4127	52	48	equation	equation	NOUN
cana-4127	52	49	(	(	PUNCT
cana-4127	52	50	1	1	X
cana-4127	52	51	)	)	PUNCT
cana-4127	52	52	theorem	theorem	VERB
cana-4127	52	53	2.1	2.1	NUM
cana-4127	52	54	let	let	VERB
cana-4127	52	55	j	j	PROPN
cana-4127	52	56	∈	∈	PROPN
cana-4127	52	57	{	{	PUNCT
cana-4127	52	58	−1,1	−1,1	NOUN
cana-4127	52	59	}	}	PUNCT
cana-4127	52	60	and	and	CCONJ
cana-4127	52	61	θ	θ	ADJ
cana-4127	52	62	:	:	PUNCT
cana-4127	52	63	x3	x3	ADJ
cana-4127	52	64	→	→	SYM
cana-4127	52	65	[	[	X
cana-4127	52	66	0,∞	0,∞	X
cana-4127	52	67	)	)	PUNCT
cana-4127	52	68	be	be	VERB
cana-4127	52	69	a	a	DET
cana-4127	52	70	function	function	NOUN
cana-4127	53	1	such	such	ADJ
cana-4127	53	2	that	that	SCONJ
cana-4127	53	3	lim	lim	PROPN
cana-4127	53	4	n→∞	n→∞	X
cana-4127	53	5	θ(𝒬njx,𝒬njy,𝒬njz	θ(𝒬njx,𝒬njy,𝒬njz	PROPN
cana-4127	53	6	)	)	PUNCT
cana-4127	53	7	𝒬nj	𝒬nj	PROPN
cana-4127	53	8	=	=	SYM
cana-4127	53	9	0	0	PUNCT
cana-4127	53	10	(	(	PUNCT
cana-4127	53	11	1	1	NUM
cana-4127	53	12	)	)	PUNCT
cana-4127	53	13	for	for	ADP
cana-4127	53	14	all	all	DET
cana-4127	53	15	𝑥	𝑥	PROPN
cana-4127	53	16	,	,	PUNCT
cana-4127	53	17	𝑦	𝑦	NOUN
cana-4127	53	18	,	,	PUNCT
cana-4127	53	19	𝑧	𝑧	DET
cana-4127	53	20	∈	∈	PROPN
cana-4127	53	21	𝑋.	𝑋.	PROPN
cana-4127	53	22	let	let	VERB
cana-4127	53	23	ℎ	ℎ	NOUN
cana-4127	53	24	:	:	PUNCT
cana-4127	53	25	𝑋	𝑋	PROPN
cana-4127	53	26	→	→	SYM
cana-4127	53	27	𝑌	𝑌	PROPN
cana-4127	53	28	be	be	VERB
cana-4127	53	29	a	a	DET
cana-4127	53	30	function	function	NOUN
cana-4127	53	31	satisfying	satisfy	VERB
cana-4127	53	32	the	the	DET
cana-4127	53	33	inequality	inequality	NOUN
cana-4127	53	34	‖𝐻(𝑥	‖𝐻(𝑥	ADP
cana-4127	53	35	,	,	PUNCT
cana-4127	53	36	𝑦	𝑦	NOUN
cana-4127	53	37	,	,	PUNCT
cana-4127	53	38	𝑧)‖	𝑧)‖	ADJ
cana-4127	53	39	≤	≤	ADJ
cana-4127	53	40	θ(𝑥	θ(𝑥	PROPN
cana-4127	53	41	,	,	PUNCT
cana-4127	53	42	𝑦	𝑦	NOUN
cana-4127	53	43	,	,	PUNCT
cana-4127	53	44	𝑧	𝑧	NOUN
cana-4127	53	45	)	)	PUNCT
cana-4127	53	46	(	(	PUNCT
cana-4127	53	47	2	2	X
cana-4127	53	48	)	)	PUNCT
cana-4127	53	49	for	for	ADP
cana-4127	53	50	all	all	DET
cana-4127	53	51	𝑥	𝑥	PROPN
cana-4127	53	52	,	,	PUNCT
cana-4127	53	53	𝑦	𝑦	NOUN
cana-4127	53	54	,	,	PUNCT
cana-4127	53	55	𝑧	𝑧	DET
cana-4127	53	56	∈	∈	PROPN
cana-4127	53	57	𝑋.	𝑋.	PROPN
cana-4127	53	58	then	then	ADV
cana-4127	53	59	there	there	PRON
cana-4127	53	60	exists	exist	VERB
cana-4127	53	61	a	a	DET
cana-4127	53	62	unique	unique	ADJ
cana-4127	53	63	additive	additive	ADJ
cana-4127	53	64	mapping	mapping	NOUN
cana-4127	53	65	𝐴	𝐴	PROPN
cana-4127	53	66	:	:	PUNCT
cana-4127	53	67	𝑋	𝑋	PROPN
cana-4127	53	68	→	→	SYM
cana-4127	53	69	𝑌	𝑌	PROPN
cana-4127	53	70	and	and	CCONJ
cana-4127	53	71	satisfying	satisfy	VERB
cana-4127	53	72	the	the	DET
cana-4127	53	73	functional	functional	ADJ
cana-4127	53	74	equation	equation	NOUN
cana-4127	53	75	(	(	PUNCT
cana-4127	53	76	1	1	X
cana-4127	53	77	)	)	PUNCT
cana-4127	53	78	such	such	ADJ
cana-4127	53	79	that	that	SCONJ
cana-4127	53	80	‖ℎ(𝑥	‖ℎ(𝑥	X
cana-4127	53	81	)	)	PUNCT
cana-4127	53	82	−	−	PROPN
cana-4127	53	83	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	53	84	≤	≤	ADV
cana-4127	53	85	1	1	NUM
cana-4127	53	86	𝒬(𝑠2	𝒬(𝑠2	PROPN
cana-4127	53	87	+	+	NOUN
cana-4127	53	88	2𝑠+1	2𝑠+1	NUM
cana-4127	53	89	)	)	PUNCT
cana-4127	53	90	∑∞	∑∞	NOUN
cana-4127	53	91	𝑘=	𝑘=	X
cana-4127	53	92	1−𝑗	1−𝑗	NUM
cana-4127	53	93	2	2	NUM
cana-4127	53	94	θ(𝒬𝑘𝑗𝑥,𝒬𝑘𝑗𝑥,𝒬𝑘𝑗𝑥	θ(𝒬𝑘𝑗𝑥,𝒬𝑘𝑗𝑥,𝒬𝑘𝑗𝑥	NOUN
cana-4127	53	95	)	)	PUNCT
cana-4127	54	1	𝒬𝑘𝑗	𝒬𝑘𝑗	NOUN
cana-4127	54	2	(	(	PUNCT
cana-4127	54	3	3	3	NUM
cana-4127	54	4	)	)	PUNCT
cana-4127	54	5	for	for	ADP
cana-4127	54	6	all	all	DET
cana-4127	54	7	𝑥	𝑥	DET
cana-4127	54	8	∈	∈	PROPN
cana-4127	54	9	𝑋.	𝑋.	PROPN
cana-4127	54	10	the	the	DET
cana-4127	54	11	mapping	mapping	NOUN
cana-4127	54	12	𝐴(𝑥	𝐴(𝑥	NOUN
cana-4127	54	13	)	)	PUNCT
cana-4127	54	14	is	be	AUX
cana-4127	54	15	defined	define	VERB
cana-4127	54	16	by	by	ADP
cana-4127	54	17	𝐴(𝑥	𝐴(𝑥	NOUN
cana-4127	54	18	)	)	PUNCT
cana-4127	54	19	=	=	SYM
cana-4127	54	20	lim	lim	NOUN
cana-4127	54	21	𝑛→∞	𝑛→∞	NUM
cana-4127	54	22	ℎ(𝒬𝑛𝑗𝑥	ℎ(𝒬𝑛𝑗𝑥	NUM
cana-4127	54	23	)	)	PUNCT
cana-4127	54	24	𝒬𝑛𝑗	𝒬𝑛𝑗	PROPN
cana-4127	54	25	(	(	PUNCT
cana-4127	54	26	4	4	NUM
cana-4127	54	27	)	)	PUNCT
cana-4127	54	28	for	for	ADP
cana-4127	54	29	all	all	DET
cana-4127	54	30	𝑥	𝑥	DET
cana-4127	54	31	∈	∈	NOUN
cana-4127	54	32	𝑋.	𝑋.	PROPN
cana-4127	54	33	here	here	ADV
cana-4127	54	34	𝒬	𝒬	PROPN
cana-4127	54	35	=	=	SYM
cana-4127	54	36	(	(	PUNCT
cana-4127	54	37	𝑝	𝑝	PROPN
cana-4127	54	38	+	+	NUM
cana-4127	54	39	𝑞	𝑞	NOUN
cana-4127	54	40	)	)	PUNCT
cana-4127	54	41	.	.	PUNCT
cana-4127	55	1	proof	proof	NOUN
cana-4127	55	2	.	.	PUNCT
cana-4127	56	1	assume	assume	VERB
cana-4127	56	2	𝑗	𝑗	X
cana-4127	56	3	=	=	ADJ
cana-4127	56	4	1	1	X
cana-4127	56	5	.	.	X
cana-4127	57	1	replacing	replace	VERB
cana-4127	57	2	(	(	PUNCT
cana-4127	57	3	𝑥	𝑥	NOUN
cana-4127	57	4	,	,	PUNCT
cana-4127	57	5	𝑦	𝑦	NOUN
cana-4127	57	6	,	,	PUNCT
cana-4127	57	7	𝑧	𝑧	PART
cana-4127	57	8	)	)	PUNCT
cana-4127	57	9	by	by	ADP
cana-4127	57	10	(	(	PUNCT
cana-4127	57	11	𝑥	𝑥	NOUN
cana-4127	57	12	,	,	PUNCT
cana-4127	57	13	𝑥	𝑥	NOUN
cana-4127	57	14	,	,	PUNCT
cana-4127	57	15	𝑥	𝑥	NOUN
cana-4127	57	16	)	)	PUNCT
cana-4127	57	17	in	in	ADP
cana-4127	57	18	(	(	PUNCT
cana-4127	57	19	2	2	NUM
cana-4127	57	20	)	)	PUNCT
cana-4127	57	21	,	,	PUNCT
cana-4127	57	22	we	we	PRON
cana-4127	57	23	get	get	VERB
cana-4127	57	24	‖(𝑠2	‖(𝑠2	PROPN
cana-4127	57	25	+	+	NUM
cana-4127	57	26	2𝑠	2𝑠	NOUN
cana-4127	57	27	+	+	CCONJ
cana-4127	57	28	1)ℎ(𝒬𝑥	1)ℎ(𝒬𝑥	NUM
cana-4127	57	29	)	)	PUNCT
cana-4127	57	30	−	−	PROPN
cana-4127	58	1	𝒬(𝑠2	𝒬(𝑠2	X
cana-4127	58	2	+	+	NUM
cana-4127	58	3	2𝑠	2𝑠	NOUN
cana-4127	58	4	+	+	CCONJ
cana-4127	58	5	1)ℎ(𝑥)‖	1)ℎ(𝑥)‖	NUM
cana-4127	58	6	≤	≤	NUM
cana-4127	58	7	θ(𝑥	θ(𝑥	PROPN
cana-4127	58	8	,	,	PUNCT
cana-4127	58	9	𝑥	𝑥	NOUN
cana-4127	58	10	,	,	PUNCT
cana-4127	58	11	𝑥	𝑥	NOUN
cana-4127	58	12	)	)	PUNCT
cana-4127	58	13	(	(	PUNCT
cana-4127	58	14	5	5	NUM
cana-4127	58	15	)	)	PUNCT
cana-4127	58	16	for	for	ADP
cana-4127	58	17	all	all	DET
cana-4127	58	18	𝑥	𝑥	DET
cana-4127	58	19	∈	∈	PROPN
cana-4127	58	20	𝑋.	𝑋.	PROPN
cana-4127	58	21	the	the	DET
cana-4127	58	22	above	above	ADJ
cana-4127	58	23	inequality	inequality	NOUN
cana-4127	58	24	can	can	AUX
cana-4127	58	25	written	write	VERB
cana-4127	58	26	as	as	ADP
cana-4127	58	27	‖𝒬(𝑠2	‖𝒬(𝑠2	NOUN
cana-4127	58	28	+	+	X
cana-4127	58	29	2𝑠	2𝑠	NOUN
cana-4127	58	30	+	+	CCONJ
cana-4127	58	31	1)ℎ(𝑥	1)ℎ(𝑥	NUM
cana-4127	58	32	)	)	PUNCT
cana-4127	58	33	−	−	PROPN
cana-4127	58	34	(	(	PUNCT
cana-4127	58	35	𝑠2	𝑠2	NOUN
cana-4127	58	36	+	+	NOUN
cana-4127	58	37	2𝑠	2𝑠	NOUN
cana-4127	58	38	+	+	CCONJ
cana-4127	58	39	1)ℎ(𝒬𝑥)‖	1)ℎ(𝒬𝑥)‖	PROPN
cana-4127	58	40	≤	≤	NUM
cana-4127	58	41	θ(𝑥	θ(𝑥	PROPN
cana-4127	58	42	,	,	PUNCT
cana-4127	58	43	𝑥	𝑥	NOUN
cana-4127	58	44	,	,	PUNCT
cana-4127	58	45	𝑥	𝑥	NOUN
cana-4127	58	46	)	)	PUNCT
cana-4127	58	47	(	(	PUNCT
cana-4127	58	48	6	6	NUM
cana-4127	58	49	)	)	PUNCT
cana-4127	58	50	for	for	ADP
cana-4127	58	51	all	all	DET
cana-4127	58	52	𝑥	𝑥	DET
cana-4127	58	53	∈	∈	PROPN
cana-4127	58	54	𝑋.	𝑋.	PROPN
cana-4127	58	55	bothside	bothside	PROPN
cana-4127	58	56	divide	divide	NOUN
cana-4127	58	57	by	by	ADP
cana-4127	58	58	(	(	PUNCT
cana-4127	58	59	𝑠2	𝑠2	NOUN
cana-4127	58	60	+	+	CCONJ
cana-4127	58	61	2𝑠	2𝑠	NOUN
cana-4127	58	62	+	+	CCONJ
cana-4127	58	63	1)𝒬	1)𝒬	VERB
cana-4127	58	64	in	in	ADP
cana-4127	58	65	(	(	PUNCT
cana-4127	58	66	6	6	NUM
cana-4127	58	67	)	)	PUNCT
cana-4127	58	68	,	,	PUNCT
cana-4127	58	69	we	we	PRON
cana-4127	58	70	have	have	VERB
cana-4127	58	71	‖ℎ(𝑥	‖ℎ(𝑥	NOUN
cana-4127	58	72	)	)	PUNCT
cana-4127	58	73	−	−	PROPN
cana-4127	58	74	ℎ(𝒬𝑥	ℎ(𝒬𝑥	NOUN
cana-4127	58	75	)	)	PUNCT
cana-4127	58	76	𝒬	𝒬	PROPN
cana-4127	58	77	‖	‖	PROPN
cana-4127	58	78	≤	≤	PROPN
cana-4127	58	79	θ(𝑥,𝑥,𝑥	θ(𝑥,𝑥,𝑥	NOUN
cana-4127	58	80	)	)	PUNCT
cana-4127	58	81	𝒬(𝑠2	𝒬(𝑠2	PROPN
cana-4127	58	82	+	+	NOUN
cana-4127	58	83	2𝑠+1	2𝑠+1	NUM
cana-4127	58	84	)	)	PUNCT
cana-4127	58	85	(	(	PUNCT
cana-4127	58	86	7	7	X
cana-4127	58	87	)	)	PUNCT
cana-4127	58	88	for	for	ADP
cana-4127	58	89	all	all	DET
cana-4127	58	90	𝑥	𝑥	DET
cana-4127	58	91	∈	∈	PROPN
cana-4127	58	92	𝑋.	𝑋.	PROPN
cana-4127	58	93	now	now	ADV
cana-4127	58	94	replacing	replace	VERB
cana-4127	58	95	𝑥	𝑥	NOUN
cana-4127	58	96	by	by	ADP
cana-4127	58	97	𝒬𝑥	𝒬𝑥	PROPN
cana-4127	58	98	and	and	CCONJ
cana-4127	58	99	dividing	divide	VERB
cana-4127	58	100	by	by	ADP
cana-4127	58	101	𝒬	𝒬	PROPN
cana-4127	58	102	in	in	ADP
cana-4127	58	103	(	(	PUNCT
cana-4127	58	104	7	7	NUM
cana-4127	58	105	)	)	PUNCT
cana-4127	58	106	,	,	PUNCT
cana-4127	58	107	we	we	PRON
cana-4127	58	108	get	get	VERB
cana-4127	58	109	‖	‖	ADJ
cana-4127	58	110	ℎ(𝒬𝑥	ℎ(𝒬𝑥	NOUN
cana-4127	58	111	)	)	PUNCT
cana-4127	58	112	𝒬	𝒬	PROPN
cana-4127	58	113	−	−	PROPN
cana-4127	58	114	ℎ(𝒬2𝑥	ℎ(𝒬2𝑥	PROPN
cana-4127	58	115	)	)	PUNCT
cana-4127	58	116	𝒬2	𝒬2	VERB
cana-4127	59	1	‖	‖	ADJ
cana-4127	59	2	≤	≤	NOUN
cana-4127	59	3	θ(𝒬𝑥,𝒬𝑥,𝒬𝑥	θ(𝒬𝑥,𝒬𝑥,𝒬𝑥	CCONJ
cana-4127	59	4	)	)	PUNCT
cana-4127	59	5	𝒬2(𝑠2	𝒬2(𝑠2	X
cana-4127	60	1	+	+	NOUN
cana-4127	60	2	2𝑠+1	2𝑠+1	NUM
cana-4127	60	3	)	)	PUNCT
cana-4127	60	4	(	(	PUNCT
cana-4127	60	5	8)	8)	NUM
cana-4127	60	6	for	for	ADP
cana-4127	60	7	all	all	PRON
cana-4127	60	8	𝑥	𝑥	DET
cana-4127	60	9	∈	∈	PROPN
cana-4127	60	10	𝑋.	𝑋.	PROPN
cana-4127	60	11	from	from	ADP
cana-4127	60	12	(	(	PUNCT
cana-4127	60	13	7	7	NUM
cana-4127	60	14	)	)	PUNCT
cana-4127	60	15	and	and	CCONJ
cana-4127	60	16	(	(	PUNCT
cana-4127	60	17	8)	8)	NUM
cana-4127	60	18	,	,	PUNCT
cana-4127	60	19	we	we	PRON
cana-4127	60	20	obtain	obtain	VERB
cana-4127	60	21	‖ℎ(𝑥	‖ℎ(𝑥	NOUN
cana-4127	60	22	)	)	PUNCT
cana-4127	60	23	−	−	PROPN
cana-4127	60	24	ℎ(𝒬2𝑥	ℎ(𝒬2𝑥	PROPN
cana-4127	60	25	)	)	PUNCT
cana-4127	60	26	𝒬2	𝒬2	VERB
cana-4127	60	27	‖	‖	ADJ
cana-4127	60	28	≤	≤	NOUN
cana-4127	60	29	‖ℎ(𝑥	‖ℎ(𝑥	NUM
cana-4127	60	30	)	)	PUNCT
cana-4127	61	1	−	−	PROPN
cana-4127	61	2	ℎ(𝒬𝑥	ℎ(𝒬𝑥	NOUN
cana-4127	61	3	)	)	PUNCT
cana-4127	61	4	𝒬	𝒬	PROPN
cana-4127	61	5	‖	‖	PROPN
cana-4127	61	6	+	+	CCONJ
cana-4127	61	7	‖	‖	PROPN
cana-4127	61	8	ℎ(𝒬𝑥	ℎ(𝒬𝑥	NOUN
cana-4127	61	9	)	)	PUNCT
cana-4127	61	10	𝒬	𝒬	PROPN
cana-4127	61	11	−	−	PROPN
cana-4127	61	12	ℎ(𝒬2𝑥	ℎ(𝒬2𝑥	PROPN
cana-4127	61	13	)	)	PUNCT
cana-4127	61	14	𝒬2	𝒬2	VERB
cana-4127	62	1	‖	‖	ADJ
cana-4127	62	2	≤	≤	NUM
cana-4127	62	3	1	1	NUM
cana-4127	62	4	𝒬(𝑠2	𝒬(𝑠2	PROPN
cana-4127	62	5	+	+	NOUN
cana-4127	62	6	2𝑠+1	2𝑠+1	NUM
cana-4127	62	7	)	)	PUNCT
cana-4127	63	1	[	[	X
cana-4127	63	2	θ(𝑥	θ(𝑥	PROPN
cana-4127	63	3	,	,	PUNCT
cana-4127	63	4	𝑥	𝑥	NOUN
cana-4127	63	5	,	,	PUNCT
cana-4127	63	6	𝑥	𝑥	NOUN
cana-4127	63	7	)	)	PUNCT
cana-4127	63	8	+	+	CCONJ
cana-4127	63	9	θ(𝒬𝑥,𝒬𝑥,𝒬𝑥	θ(𝒬𝑥,𝒬𝑥,𝒬𝑥	X
cana-4127	63	10	)	)	PUNCT
cana-4127	63	11	𝒬	𝒬	NOUN
cana-4127	63	12	]	]	PUNCT
cana-4127	63	13	(	(	PUNCT
cana-4127	63	14	9	9	NUM
cana-4127	63	15	)	)	PUNCT
cana-4127	63	16	for	for	ADP
cana-4127	63	17	all	all	DET
cana-4127	63	18	𝑥	𝑥	DET
cana-4127	63	19	∈	∈	PROPN
cana-4127	63	20	𝑋.	𝑋.	PROPN
cana-4127	63	21	in	in	ADP
cana-4127	63	22	general	general	NOUN
cana-4127	63	23	for	for	ADP
cana-4127	63	24	any	any	DET
cana-4127	63	25	positive	positive	ADJ
cana-4127	63	26	integer	integer	NOUN
cana-4127	63	27	𝑛	𝑛	PROPN
cana-4127	63	28	,	,	PUNCT
cana-4127	63	29	we	we	PRON
cana-4127	63	30	get	get	VERB
cana-4127	63	31	‖ℎ(𝑥	‖ℎ(𝑥	NOUN
cana-4127	63	32	)	)	PUNCT
cana-4127	63	33	−	−	PROPN
cana-4127	63	34	ℎ(𝒬𝑛𝑥	ℎ(𝒬𝑛𝑥	X
cana-4127	63	35	)	)	PUNCT
cana-4127	64	1	𝒬𝑛	𝒬𝑛	PROPN
cana-4127	64	2	‖	‖	PROPN
cana-4127	64	3	≤	≤	NUM
cana-4127	64	4	1	1	NUM
cana-4127	64	5	𝒬(𝑠2	𝒬(𝑠2	PROPN
cana-4127	64	6	+	+	NOUN
cana-4127	64	7	2𝑠+1	2𝑠+1	NUM
cana-4127	64	8	)	)	PUNCT
cana-4127	64	9	∑𝑛−1𝑘=0	∑𝑛−1𝑘=0	NOUN
cana-4127	64	10	θ(𝒬𝑘𝑥,𝒬𝑘𝑥,𝒬𝑘𝑥	θ(𝒬𝑘𝑥,𝒬𝑘𝑥,𝒬𝑘𝑥	NOUN
cana-4127	64	11	)	)	PUNCT
cana-4127	65	1	𝒬𝑘	𝒬𝑘	PROPN
cana-4127	65	2	(	(	PUNCT
cana-4127	65	3	10	10	NUM
cana-4127	65	4	)	)	PUNCT
cana-4127	65	5	≤	≤	NOUN
cana-4127	65	6	1	1	NUM
cana-4127	65	7	𝒬(𝑠2	𝒬(𝑠2	PROPN
cana-4127	65	8	+	+	NOUN
cana-4127	65	9	2𝑠+1	2𝑠+1	NUM
cana-4127	65	10	)	)	PUNCT
cana-4127	65	11	∑∞𝑘=0	∑∞𝑘=0	VERB
cana-4127	65	12	θ(𝒬𝑘𝑥,𝒬𝑘𝑥,𝒬𝑘𝑥	θ(𝒬𝑘𝑥,𝒬𝑘𝑥,𝒬𝑘𝑥	NOUN
cana-4127	65	13	)	)	PUNCT
cana-4127	66	1	𝒬𝑘	𝒬𝑘	PROPN
cana-4127	66	2	communications	communication	NOUN
cana-4127	66	3	on	on	ADP
cana-4127	66	4	applied	apply	VERB
cana-4127	66	5	nonlinear	nonlinear	ADJ
cana-4127	66	6	analysis	analysis	NOUN
cana-4127	66	7	issn	issn	NOUN
cana-4127	66	8	:	:	PUNCT
cana-4127	66	9	1074	1074	NUM
cana-4127	66	10	-	-	PUNCT
cana-4127	66	11	133x	133x	NUM
cana-4127	66	12	vol	vol	NOUN
cana-4127	66	13	32	32	NUM
cana-4127	66	14	no	no	NOUN
cana-4127	66	15	.	.	PUNCT
cana-4127	67	1	9s	9s	NUM
cana-4127	67	2	(	(	PUNCT
cana-4127	67	3	2025	2025	NUM
cana-4127	67	4	)	)	PUNCT
cana-4127	67	5	1215	1215	NUM
cana-4127	67	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	67	7	for	for	ADP
cana-4127	67	8	all	all	PRON
cana-4127	67	9	𝑥	𝑥	DET
cana-4127	67	10	∈	∈	NOUN
cana-4127	67	11	𝑋.	𝑋.	PROPN
cana-4127	67	12	in	in	ADP
cana-4127	67	13	order	order	NOUN
cana-4127	67	14	to	to	PART
cana-4127	67	15	prove	prove	VERB
cana-4127	67	16	the	the	DET
cana-4127	67	17	convergence	convergence	NOUN
cana-4127	67	18	of	of	ADP
cana-4127	67	19	the	the	DET
cana-4127	67	20	sequence	sequence	NOUN
cana-4127	67	21	{	{	PUNCT
cana-4127	67	22	ℎ(𝒬𝑛𝑥	ℎ(𝒬𝑛𝑥	PROPN
cana-4127	67	23	)	)	PUNCT
cana-4127	67	24	𝒬𝑛	𝒬𝑛	PROPN
cana-4127	67	25	}	}	PUNCT
cana-4127	67	26	,	,	PUNCT
cana-4127	67	27	replace	replace	VERB
cana-4127	67	28	𝑥	𝑥	NOUN
cana-4127	67	29	by	by	ADP
cana-4127	67	30	𝒬𝑚𝑥	𝒬𝑚𝑥	PROPN
cana-4127	67	31	and	and	CCONJ
cana-4127	67	32	dividing	divide	VERB
cana-4127	67	33	by	by	ADP
cana-4127	67	34	𝒬𝑚	𝒬𝑚	PROPN
cana-4127	67	35	in	in	ADP
cana-4127	67	36	(	(	PUNCT
cana-4127	67	37	10	10	NUM
cana-4127	67	38	)	)	PUNCT
cana-4127	67	39	,	,	PUNCT
cana-4127	67	40	for	for	ADP
cana-4127	67	41	any	any	DET
cana-4127	67	42	𝑚,𝑛	𝑚,𝑛	PROPN
cana-4127	67	43	>	>	X
cana-4127	67	44	0	0	PUNCT
cana-4127	67	45	,	,	PUNCT
cana-4127	67	46	we	we	PRON
cana-4127	67	47	deduce	deduce	VERB
cana-4127	67	48	‖	‖	PROPN
cana-4127	67	49	ℎ(𝒬𝑚𝑥	ℎ(𝒬𝑚𝑥	NOUN
cana-4127	67	50	)	)	PUNCT
cana-4127	67	51	𝒬𝑚	𝒬𝑚	PROPN
cana-4127	67	52	−	−	ADP
cana-4127	67	53	ℎ(𝒬𝑛+𝑚𝑥	ℎ(𝒬𝑛+𝑚𝑥	NOUN
cana-4127	67	54	)	)	PUNCT
cana-4127	67	55	𝒬(𝑛+𝑚	𝒬(𝑛+𝑚	PUNCT
cana-4127	67	56	)	)	PUNCT
cana-4127	67	57	‖	‖	PROPN
cana-4127	68	1	=	=	NOUN
cana-4127	68	2	1	1	NUM
cana-4127	68	3	𝒬𝑚	𝒬𝑚	PROPN
cana-4127	68	4	‖ℎ(𝒬𝑚𝑥	‖ℎ(𝒬𝑚𝑥	PROPN
cana-4127	68	5	)	)	PUNCT
cana-4127	68	6	−	−	ADP
cana-4127	68	7	ℎ(𝒬𝑛⋅𝒬𝑚𝑥	ℎ(𝒬𝑛⋅𝒬𝑚𝑥	NOUN
cana-4127	68	8	)	)	PUNCT
cana-4127	68	9	𝒬𝑛	𝒬𝑛	PROPN
cana-4127	68	10	‖	‖	PROPN
cana-4127	68	11	≤	≤	ADJ
cana-4127	68	12	1	1	NUM
cana-4127	68	13	𝒬(𝑠2	𝒬(𝑠2	PROPN
cana-4127	68	14	+	+	NOUN
cana-4127	68	15	2𝑠+1	2𝑠+1	NUM
cana-4127	68	16	)	)	PUNCT
cana-4127	68	17	∑𝑛−1𝑘=0	∑𝑛−1𝑘=0	NOUN
cana-4127	68	18	θ(𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥	θ(𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥	NOUN
cana-4127	68	19	)	)	PUNCT
cana-4127	68	20	𝒬𝑘+𝑚	𝒬𝑘+𝑚	SYM
cana-4127	68	21	≤	≤	NUM
cana-4127	68	22	1	1	NUM
cana-4127	68	23	𝒬(𝑠2	𝒬(𝑠2	PROPN
cana-4127	68	24	+	+	NOUN
cana-4127	68	25	2𝑠+1	2𝑠+1	NUM
cana-4127	68	26	)	)	PUNCT
cana-4127	68	27	∑∞𝑘=0	∑∞𝑘=0	VERB
cana-4127	68	28	θ(𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥	θ(𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥,𝒬𝑘+𝑚𝑥	PROPN
cana-4127	68	29	)	)	PUNCT
cana-4127	68	30	𝒬𝑘+𝑚	𝒬𝑘+𝑚	PUNCT
cana-4127	68	31	→	→	SYM
cana-4127	68	32	0	0	NUM
cana-4127	68	33	𝑎𝑠	𝑎𝑠	ADP
cana-4127	68	34	𝑚	𝑚	PROPN
cana-4127	68	35	→	→	SYM
cana-4127	68	36	∞	∞	PROPN
cana-4127	68	37	for	for	ADP
cana-4127	68	38	all	all	DET
cana-4127	68	39	𝑥	𝑥	DET
cana-4127	68	40	∈	∈	NOUN
cana-4127	68	41	𝑋.	𝑋.	NOUN
cana-4127	68	42	hence	hence	ADV
cana-4127	68	43	the	the	DET
cana-4127	68	44	sequence	sequence	NOUN
cana-4127	68	45	{	{	PUNCT
cana-4127	68	46	ℎ(𝒬𝑛𝑥	ℎ(𝒬𝑛𝑥	PROPN
cana-4127	68	47	)	)	PUNCT
cana-4127	68	48	𝒬𝑛	𝒬𝑛	PROPN
cana-4127	68	49	}	}	PUNCT
cana-4127	68	50	is	be	AUX
cana-4127	68	51	cauchy	cauchy	ADJ
cana-4127	68	52	sequence	sequence	NOUN
cana-4127	68	53	.	.	PUNCT
cana-4127	69	1	since	since	SCONJ
cana-4127	69	2	𝑌	𝑌	PROPN
cana-4127	69	3	is	be	AUX
cana-4127	69	4	complete	complete	ADJ
cana-4127	69	5	,	,	PUNCT
cana-4127	69	6	there	there	PRON
cana-4127	69	7	exists	exist	VERB
cana-4127	69	8	a	a	DET
cana-4127	69	9	mapping	mapping	NOUN
cana-4127	69	10	𝐴:𝑋	𝐴:𝑋	NOUN
cana-4127	69	11	→	→	SYM
cana-4127	69	12	𝑌	𝑌	PROPN
cana-4127	70	1	such	such	ADJ
cana-4127	70	2	that	that	SCONJ
cana-4127	70	3	𝐴(𝑥	𝐴(𝑥	NOUN
cana-4127	70	4	)	)	PUNCT
cana-4127	70	5	=	=	SYM
cana-4127	70	6	lim	lim	PROPN
cana-4127	70	7	𝑛→∞	𝑛→∞	NUM
cana-4127	70	8	ℎ(𝒬𝑛𝑥	ℎ(𝒬𝑛𝑥	PROPN
cana-4127	70	9	)	)	PUNCT
cana-4127	70	10	𝒬𝑛	𝒬𝑛	NOUN
cana-4127	70	11	∀	∀	PUNCT
cana-4127	71	1	𝑥	𝑥	DET
cana-4127	71	2	∈	∈	PROPN
cana-4127	71	3	𝑋.	𝑋.	PROPN
cana-4127	71	4	letting	let	VERB
cana-4127	71	5	𝑛	𝑛	PRON
cana-4127	71	6	→	→	SYM
cana-4127	71	7	∞	∞	NUM
cana-4127	71	8	in	in	ADP
cana-4127	71	9	(	(	PUNCT
cana-4127	71	10	10	10	NUM
cana-4127	71	11	)	)	PUNCT
cana-4127	71	12	we	we	PRON
cana-4127	71	13	see	see	VERB
cana-4127	71	14	that	that	SCONJ
cana-4127	71	15	(	(	PUNCT
cana-4127	71	16	3	3	X
cana-4127	71	17	)	)	PUNCT
cana-4127	71	18	holds	hold	VERB
cana-4127	71	19	for	for	ADP
cana-4127	71	20	all	all	DET
cana-4127	71	21	𝑥	𝑥	DET
cana-4127	71	22	∈	∈	PROPN
cana-4127	71	23	𝑋.	𝑋.	PROPN
cana-4127	71	24	to	to	PART
cana-4127	71	25	prove	prove	VERB
cana-4127	71	26	that	that	SCONJ
cana-4127	71	27	𝐴	𝐴	PROPN
cana-4127	71	28	satisfies	satisfy	VERB
cana-4127	71	29	(	(	PUNCT
cana-4127	71	30	1	1	NUM
cana-4127	71	31	)	)	PUNCT
cana-4127	71	32	,	,	PUNCT
cana-4127	71	33	replacing	replace	VERB
cana-4127	71	34	(	(	PUNCT
cana-4127	71	35	𝑥	𝑥	NOUN
cana-4127	71	36	,	,	PUNCT
cana-4127	71	37	𝑦	𝑦	NOUN
cana-4127	71	38	,	,	PUNCT
cana-4127	71	39	𝑧	𝑧	PART
cana-4127	71	40	)	)	PUNCT
cana-4127	71	41	by	by	ADP
cana-4127	71	42	(	(	PUNCT
cana-4127	71	43	𝒬𝑛𝑥	𝒬𝑛𝑥	PROPN
cana-4127	71	44	,	,	PUNCT
cana-4127	71	45	𝒬𝑛𝑦	𝒬𝑛𝑦	PROPN
cana-4127	71	46	,	,	PUNCT
cana-4127	71	47	𝒬𝑛𝑧	𝒬𝑛𝑧	PROPN
cana-4127	71	48	)	)	PUNCT
cana-4127	71	49	and	and	CCONJ
cana-4127	71	50	dividing	divide	VERB
cana-4127	71	51	by	by	ADP
cana-4127	71	52	𝒬𝑛	𝒬𝑛	PROPN
cana-4127	71	53	in	in	ADP
cana-4127	71	54	(	(	PUNCT
cana-4127	71	55	2	2	NUM
cana-4127	71	56	)	)	PUNCT
cana-4127	71	57	,	,	PUNCT
cana-4127	71	58	we	we	PRON
cana-4127	71	59	obtain	obtain	VERB
cana-4127	71	60	1	1	NUM
cana-4127	71	61	𝒬𝑛	𝒬𝑛	PROPN
cana-4127	71	62	‖𝐻(𝒬𝑛𝑥	‖𝐻(𝒬𝑛𝑥	NOUN
cana-4127	71	63	,	,	PUNCT
cana-4127	71	64	𝒬𝑛𝑦	𝒬𝑛𝑦	PROPN
cana-4127	71	65	,	,	PUNCT
cana-4127	71	66	𝒬𝑛𝑧)‖	𝒬𝑛𝑧)‖	PUNCT
cana-4127	71	67	≤	≤	NUM
cana-4127	71	68	1	1	NUM
cana-4127	71	69	𝒬𝑛	𝒬𝑛	PROPN
cana-4127	71	70	θ(𝒬𝑛𝑥	θ(𝒬𝑛𝑥	PROPN
cana-4127	71	71	,	,	PUNCT
cana-4127	71	72	𝒬𝑛𝑦	𝒬𝑛𝑦	PROPN
cana-4127	71	73	,	,	PUNCT
cana-4127	71	74	𝒬𝑛𝑧	𝒬𝑛𝑧	PROPN
cana-4127	71	75	)	)	PUNCT
cana-4127	71	76	for	for	ADP
cana-4127	71	77	all	all	DET
cana-4127	71	78	𝑥	𝑥	PROPN
cana-4127	71	79	,	,	PUNCT
cana-4127	71	80	𝑦	𝑦	NOUN
cana-4127	71	81	,	,	PUNCT
cana-4127	71	82	𝑧	𝑧	DET
cana-4127	71	83	∈	∈	PROPN
cana-4127	71	84	𝑋.	𝑋.	PROPN
cana-4127	71	85	letting	let	VERB
cana-4127	71	86	𝑛	𝑛	PRON
cana-4127	71	87	→	→	SYM
cana-4127	71	88	∞	∞	NUM
cana-4127	71	89	in	in	ADP
cana-4127	71	90	the	the	DET
cana-4127	71	91	above	above	ADJ
cana-4127	71	92	inequality	inequality	NOUN
cana-4127	71	93	and	and	CCONJ
cana-4127	71	94	using	use	VERB
cana-4127	71	95	the	the	DET
cana-4127	71	96	definition	definition	NOUN
cana-4127	71	97	of	of	ADP
cana-4127	71	98	𝐴(𝑥	𝐴(𝑥	NOUN
cana-4127	71	99	)	)	PUNCT
cana-4127	71	100	,	,	PUNCT
cana-4127	71	101	we	we	PRON
cana-4127	71	102	see	see	VERB
cana-4127	71	103	that	that	SCONJ
cana-4127	71	104	(	(	PUNCT
cana-4127	71	105	𝑠2	𝑠2	NOUN
cana-4127	71	106	+	+	X
cana-4127	71	107	2𝑠)𝐴(𝑝𝑥	2𝑠)𝐴(𝑝𝑥	NUM
cana-4127	71	108	+	+	CCONJ
cana-4127	71	109	𝑞𝑦	𝑞𝑦	NOUN
cana-4127	71	110	)	)	PUNCT
cana-4127	71	111	+	+	CCONJ
cana-4127	71	112	(	(	PUNCT
cana-4127	71	113	1	1	NUM
cana-4127	71	114	−	−	NUM
cana-4127	71	115	2𝑠)𝐴(𝑝𝑦	2𝑠)𝐴(𝑝𝑦	PROPN
cana-4127	71	116	+	+	CCONJ
cana-4127	71	117	𝑞𝑧	𝑞𝑧	NOUN
cana-4127	71	118	)	)	PUNCT
cana-4127	71	119	+	+	CCONJ
cana-4127	71	120	2𝑠𝐴(𝑝𝑧	2𝑠𝐴(𝑝𝑧	NUM
cana-4127	71	121	+	+	NUM
cana-4127	71	122	𝑞𝑥	𝑞𝑥	NOUN
cana-4127	71	123	)	)	PUNCT
cana-4127	71	124	−	−	NOUN
cana-4127	71	125	2𝑠𝑝𝐴(𝑥	2𝑠𝑝𝐴(𝑥	NUM
cana-4127	71	126	−	−	PROPN
cana-4127	71	127	𝑦	𝑦	X
cana-4127	71	128	)	)	PUNCT
cana-4127	71	129	−2𝑠𝑞𝐴(𝑦	−2𝑠𝑞𝐴(𝑦	NOUN
cana-4127	71	130	−	−	PUNCT
cana-4127	71	131	𝑧	𝑧	X
cana-4127	71	132	)	)	PUNCT
cana-4127	71	133	=	=	SYM
cana-4127	71	134	(	(	PUNCT
cana-4127	71	135	𝑠2𝑝	𝑠2𝑝	PROPN
cana-4127	71	136	+	+	CCONJ
cana-4127	71	137	2𝑠𝑞)𝐴(𝑥	2𝑠𝑞)𝐴(𝑥	NUM
cana-4127	71	138	)	)	PUNCT
cana-4127	71	139	+	+	CCONJ
cana-4127	71	140	(	(	PUNCT
cana-4127	71	141	𝑝	𝑝	PROPN
cana-4127	71	142	+	+	NUM
cana-4127	71	143	𝑠2𝑞)𝐴(𝑦	𝑠2𝑞)𝐴(𝑦	NOUN
cana-4127	71	144	)	)	PUNCT
cana-4127	72	1	+	+	CCONJ
cana-4127	72	2	(	(	PUNCT
cana-4127	72	3	2𝑠𝑝	2𝑠𝑝	ADJ
cana-4127	72	4	+	+	CCONJ
cana-4127	72	5	𝑞)𝐴(𝑧	𝑞)𝐴(𝑧	NOUN
cana-4127	72	6	)	)	PUNCT
cana-4127	72	7	hence	hence	ADV
cana-4127	72	8	𝐴	𝐴	PROPN
cana-4127	72	9	satisfies	satisfie	NOUN
cana-4127	72	10	(	(	PUNCT
cana-4127	72	11	1	1	NUM
cana-4127	72	12	)	)	PUNCT
cana-4127	72	13	for	for	ADP
cana-4127	72	14	all	all	DET
cana-4127	72	15	𝑥	𝑥	PROPN
cana-4127	72	16	,	,	PUNCT
cana-4127	72	17	𝑦	𝑦	NOUN
cana-4127	72	18	,	,	PUNCT
cana-4127	72	19	𝑧	𝑧	DET
cana-4127	72	20	∈	∈	PROPN
cana-4127	72	21	𝑋.	𝑋.	NOUN
cana-4127	72	22	to	to	PART
cana-4127	72	23	prove	prove	VERB
cana-4127	72	24	𝐴	𝐴	PROPN
cana-4127	72	25	is	be	AUX
cana-4127	72	26	unique	unique	ADJ
cana-4127	72	27	,	,	PUNCT
cana-4127	72	28	we	we	PRON
cana-4127	72	29	let	let	VERB
cana-4127	72	30	𝐵(𝑥	𝐵(𝑥	PRON
cana-4127	72	31	)	)	PUNCT
cana-4127	72	32	be	be	AUX
cana-4127	72	33	another	another	DET
cana-4127	72	34	mapping	mapping	NOUN
cana-4127	72	35	satisfying	satisfy	VERB
cana-4127	72	36	(	(	PUNCT
cana-4127	72	37	1	1	NUM
cana-4127	72	38	)	)	PUNCT
cana-4127	72	39	and	and	CCONJ
cana-4127	72	40	(	(	PUNCT
cana-4127	72	41	3	3	NUM
cana-4127	72	42	)	)	PUNCT
cana-4127	72	43	,	,	PUNCT
cana-4127	72	44	then	then	ADV
cana-4127	72	45	‖𝐴(𝑥	‖𝐴(𝑥	NOUN
cana-4127	72	46	)	)	PUNCT
cana-4127	72	47	−	−	PROPN
cana-4127	72	48	𝐵(𝑥)‖	𝐵(𝑥)‖	ADJ
cana-4127	72	49	=	=	SYM
cana-4127	72	50	1	1	NUM
cana-4127	72	51	𝒬𝑛	𝒬𝑛	PROPN
cana-4127	72	52	‖𝐴(𝒬𝑛𝑥	‖𝐴(𝒬𝑛𝑥	NOUN
cana-4127	72	53	)	)	PUNCT
cana-4127	73	1	−	−	ADP
cana-4127	73	2	𝐵(𝒬𝑛𝑥)‖	𝐵(𝒬𝑛𝑥)‖	ADP
cana-4127	74	1	≤	≤	ADV
cana-4127	74	2	1	1	NUM
cana-4127	74	3	𝒬𝑛	𝒬𝑛	PROPN
cana-4127	74	4	{	{	PUNCT
cana-4127	74	5	‖𝐴(𝒬𝑛𝑥	‖𝐴(𝒬𝑛𝑥	NOUN
cana-4127	74	6	)	)	PUNCT
cana-4127	74	7	−	−	ADP
cana-4127	74	8	ℎ(𝒬𝑛𝑥)‖	ℎ(𝒬𝑛𝑥)‖	X
cana-4127	74	9	+	+	CCONJ
cana-4127	74	10	‖ℎ(𝒬𝑛𝑥	‖ℎ(𝒬𝑛𝑥	NOUN
cana-4127	74	11	)	)	PUNCT
cana-4127	74	12	−	−	ADP
cana-4127	74	13	𝐵(𝒬𝑛𝑥)‖	𝐵(𝒬𝑛𝑥)‖	ADP
cana-4127	74	14	}	}	PUNCT
cana-4127	74	15	≤	≤	NUM
cana-4127	74	16	2	2	NUM
cana-4127	74	17	(	(	PUNCT
cana-4127	74	18	𝑠2	𝑠2	NOUN
cana-4127	74	19	+	+	NOUN
cana-4127	74	20	2𝑠+1	2𝑠+1	NUM
cana-4127	74	21	)	)	PUNCT
cana-4127	74	22	∑∞𝑘=0	∑∞𝑘=0	VERB
cana-4127	74	23	θ(𝒬𝑘+𝑛𝑥,𝒬𝑘+𝑛𝑥,𝒬𝑘+𝑛𝑥	θ(𝒬𝑘+𝑛𝑥,𝒬𝑘+𝑛𝑥,𝒬𝑘+𝑛𝑥	NOUN
cana-4127	74	24	)	)	PUNCT
cana-4127	74	25	𝒬(𝑘+𝑛	𝒬(𝑘+𝑛	NOUN
cana-4127	74	26	)	)	PUNCT
cana-4127	74	27	→	→	SYM
cana-4127	74	28	0	0	NUM
cana-4127	74	29	𝑎𝑠	𝑎𝑠	PROPN
cana-4127	74	30	𝑛	𝑛	PROPN
cana-4127	74	31	→	→	SYM
cana-4127	74	32	∞	∞	PROPN
cana-4127	74	33	for	for	ADP
cana-4127	74	34	all	all	DET
cana-4127	74	35	𝑥	𝑥	DET
cana-4127	74	36	∈	∈	NOUN
cana-4127	74	37	𝑋.	𝑋.	PROPN
cana-4127	74	38	hence	hence	ADV
cana-4127	74	39	𝐴	𝐴	PROPN
cana-4127	74	40	is	be	AUX
cana-4127	74	41	unique	unique	ADJ
cana-4127	74	42	.	.	PUNCT
cana-4127	75	1	corollary	corollary	ADJ
cana-4127	75	2	2.2	2.2	NUM
cana-4127	75	3	let	let	VERB
cana-4127	75	4	θ	θ	PROPN
cana-4127	75	5	and	and	CCONJ
cana-4127	75	6	s	s	AUX
cana-4127	75	7	be	be	AUX
cana-4127	75	8	nonnegative	nonnegative	ADJ
cana-4127	75	9	real	real	ADJ
cana-4127	75	10	numbers	number	NOUN
cana-4127	75	11	.	.	PUNCT
cana-4127	76	1	let	let	VERB
cana-4127	76	2	a	a	DET
cana-4127	76	3	function	function	NOUN
cana-4127	76	4	h	h	NOUN
cana-4127	76	5	:	:	PUNCT
cana-4127	76	6	x	x	X
cana-4127	76	7	→	→	SYM
cana-4127	76	8	y	y	PROPN
cana-4127	76	9	satisfies	satisfy	VERB
cana-4127	76	10	the	the	DET
cana-4127	76	11	inequality	inequality	NOUN
cana-4127	76	12	‖𝐻(𝑥	‖𝐻(𝑥	ADP
cana-4127	76	13	,	,	PUNCT
cana-4127	76	14	𝑦	𝑦	NOUN
cana-4127	76	15	,	,	PUNCT
cana-4127	76	16	𝑧)‖	𝑧)‖	ADJ
cana-4127	76	17	≤	≤	NOUN
cana-4127	76	18	{	{	PUNCT
cana-4127	76	19	θ	θ	PROPN
cana-4127	76	20	,	,	PUNCT
cana-4127	76	21	θ{||𝑥||𝑠	θ{||𝑥||𝑠	NOUN
cana-4127	76	22	+	+	CCONJ
cana-4127	76	23	||𝑦||𝑠	||𝑦||𝑠	PROPN
cana-4127	76	24	+	+	X
cana-4127	76	25	||𝑧||𝑠	||𝑧||𝑠	NOUN
cana-4127	76	26	}	}	PUNCT
cana-4127	76	27	,	,	PUNCT
cana-4127	76	28	𝑠	𝑠	X
cana-4127	76	29	≠	≠	PROPN
cana-4127	76	30	1	1	NUM
cana-4127	76	31	;	;	PUNCT
cana-4127	76	32	θ||𝑥||𝑠||𝑦||𝑠||𝑧||𝑠	θ||𝑥||𝑠||𝑦||𝑠||𝑧||𝑠	PROPN
cana-4127	76	33	,	,	PUNCT
cana-4127	76	34	3𝑠	3𝑠	NUM
cana-4127	76	35	≠	≠	PROPN
cana-4127	76	36	1	1	NUM
cana-4127	76	37	;	;	PUNCT
cana-4127	76	38	θ{||𝑥||𝑠||𝑦||𝑠||𝑧||𝑠	θ{||𝑥||𝑠||𝑦||𝑠||𝑧||𝑠	PROPN
cana-4127	76	39	+	+	CCONJ
cana-4127	76	40	{	{	PUNCT
cana-4127	76	41	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	76	42	+	+	CCONJ
cana-4127	76	43	||𝑦||3𝑠	||𝑦||3𝑠	NOUN
cana-4127	77	1	+	+	X
cana-4127	77	2	||𝑧||3𝑠	||𝑧||3𝑠	NOUN
cana-4127	77	3	}	}	PUNCT
cana-4127	77	4	}	}	PUNCT
cana-4127	77	5	,	,	PUNCT
cana-4127	77	6	3𝑠	3𝑠	NUM
cana-4127	77	7	≠	≠	PROPN
cana-4127	77	8	1	1	NUM
cana-4127	77	9	;	;	PUNCT
cana-4127	77	10	(	(	PUNCT
cana-4127	77	11	11	11	X
cana-4127	77	12	)	)	PUNCT
cana-4127	77	13	communications	communication	NOUN
cana-4127	77	14	on	on	ADP
cana-4127	77	15	applied	apply	VERB
cana-4127	77	16	nonlinear	nonlinear	ADJ
cana-4127	77	17	analysis	analysis	NOUN
cana-4127	77	18	issn	issn	NOUN
cana-4127	77	19	:	:	PUNCT
cana-4127	77	20	1074	1074	NUM
cana-4127	77	21	-	-	PUNCT
cana-4127	77	22	133x	133x	NUM
cana-4127	77	23	vol	vol	NOUN
cana-4127	77	24	32	32	NUM
cana-4127	77	25	no	no	NOUN
cana-4127	77	26	.	.	PUNCT
cana-4127	78	1	9s	9s	NUM
cana-4127	78	2	(	(	PUNCT
cana-4127	78	3	2025	2025	NUM
cana-4127	78	4	)	)	PUNCT
cana-4127	78	5	1216	1216	NUM
cana-4127	78	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	78	7	for	for	ADP
cana-4127	78	8	all	all	DET
cana-4127	78	9	𝑥	𝑥	PROPN
cana-4127	78	10	,	,	PUNCT
cana-4127	78	11	𝑦	𝑦	NOUN
cana-4127	78	12	,	,	PUNCT
cana-4127	78	13	𝑧	𝑧	DET
cana-4127	78	14	∈	∈	PROPN
cana-4127	78	15	𝑋.	𝑋.	PROPN
cana-4127	78	16	then	then	ADV
cana-4127	78	17	there	there	PRON
cana-4127	78	18	exists	exist	VERB
cana-4127	78	19	a	a	DET
cana-4127	78	20	unique	unique	ADJ
cana-4127	78	21	additive	additive	ADJ
cana-4127	78	22	function	function	NOUN
cana-4127	78	23	𝐴	𝐴	PROPN
cana-4127	78	24	:	:	PUNCT
cana-4127	78	25	𝑋	𝑋	PROPN
cana-4127	78	26	→	→	SYM
cana-4127	78	27	𝑌	𝑌	PROPN
cana-4127	79	1	such	such	ADJ
cana-4127	79	2	that	that	SCONJ
cana-4127	79	3	‖ℎ(𝑥	‖ℎ(𝑥	X
cana-4127	79	4	)	)	PUNCT
cana-4127	79	5	−	−	ADP
cana-4127	80	1	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	80	2	≤	≤	X
cana-4127	80	3	{	{	PUNCT
cana-4127	80	4	θ	θ	NOUN
cana-4127	80	5	(	(	PUNCT
cana-4127	80	6	𝑠2	𝑠2	PROPN
cana-4127	80	7	+	+	PROPN
cana-4127	80	8	2𝑠+1)|𝒬−1|	2𝑠+1)|𝒬−1|	NUM
cana-4127	80	9	,	,	PUNCT
cana-4127	80	10	3	3	NUM
cana-4127	80	11	θ||𝑥||𝑠	θ||𝑥||𝑠	NOUN
cana-4127	80	12	(	(	PUNCT
cana-4127	80	13	𝑠2	𝑠2	PROPN
cana-4127	80	14	+	+	PROPN
cana-4127	80	15	2𝑠+1)|𝒬−𝒬𝑠|	2𝑠+1)|𝒬−𝒬𝑠|	NUM
cana-4127	80	16	,	,	PUNCT
cana-4127	80	17	θ||𝑥||3𝑠	θ||𝑥||3𝑠	PROPN
cana-4127	80	18	(	(	PUNCT
cana-4127	80	19	𝑠2	𝑠2	PROPN
cana-4127	80	20	+	+	PROPN
cana-4127	80	21	2𝑠+1)|𝒬−𝒬3𝑠|	2𝑠+1)|𝒬−𝒬3𝑠|	NUM
cana-4127	80	22	,	,	PUNCT
cana-4127	80	23	4	4	NUM
cana-4127	80	24	θ||𝑥||3𝑠	θ||𝑥||3𝑠	PROPN
cana-4127	80	25	(	(	PUNCT
cana-4127	80	26	𝑠2	𝑠2	PROPN
cana-4127	80	27	+	+	PROPN
cana-4127	80	28	2𝑠+1)|𝒬−𝒬3𝑠|	2𝑠+1)|𝒬−𝒬3𝑠|	NUM
cana-4127	80	29	(	(	PUNCT
cana-4127	80	30	12	12	NUM
cana-4127	80	31	)	)	PUNCT
cana-4127	80	32	for	for	ADP
cana-4127	80	33	all	all	DET
cana-4127	80	34	𝑥	𝑥	DET
cana-4127	80	35	∈	∈	PROPN
cana-4127	80	36	𝑋.	𝑋.	PROPN
cana-4127	80	37	3	3	NUM
cana-4127	80	38	stability	stability	NOUN
cana-4127	80	39	results	result	NOUN
cana-4127	80	40	:	:	PUNCT
cana-4127	80	41	fixed	fixed	ADJ
cana-4127	80	42	point	point	NOUN
cana-4127	80	43	method	method	NOUN
cana-4127	80	44	theorem	theorem	VERB
cana-4127	80	45	3.1	3.1	NUM
cana-4127	81	1	[	[	X
cana-4127	81	2	22](the	22](the	NUM
cana-4127	81	3	alternative	alternative	NOUN
cana-4127	81	4	of	of	ADP
cana-4127	81	5	fixed	fix	VERB
cana-4127	81	6	point	point	NOUN
cana-4127	81	7	)	)	PUNCT
cana-4127	81	8	suppose	suppose	VERB
cana-4127	82	1	that	that	SCONJ
cana-4127	82	2	for	for	ADP
cana-4127	82	3	a	a	DET
cana-4127	82	4	complete	complete	ADJ
cana-4127	82	5	generalized	generalize	VERB
cana-4127	82	6	metric	metric	ADJ
cana-4127	82	7	space	space	NOUN
cana-4127	82	8	(	(	PUNCT
cana-4127	82	9	x	x	X
cana-4127	82	10	,	,	PUNCT
cana-4127	82	11	d	d	NOUN
cana-4127	82	12	)	)	PUNCT
cana-4127	82	13	and	and	CCONJ
cana-4127	82	14	a	a	DET
cana-4127	82	15	strictly	strictly	ADV
cana-4127	82	16	contractive	contractive	ADJ
cana-4127	82	17	mapping	mapping	NOUN
cana-4127	82	18	t	t	PROPN
cana-4127	82	19	:	:	PUNCT
cana-4127	82	20	x	x	SYM
cana-4127	82	21	→	→	SYM
cana-4127	82	22	x	x	PUNCT
cana-4127	82	23	with	with	ADP
cana-4127	82	24	lipschitz	lipschitz	NOUN
cana-4127	82	25	constant	constant	ADJ
cana-4127	82	26	l.	l.	NOUN
cana-4127	82	27	then	then	ADV
cana-4127	82	28	,	,	PUNCT
cana-4127	82	29	for	for	ADP
cana-4127	82	30	each	each	DET
cana-4127	82	31	given	give	VERB
cana-4127	82	32	element	element	NOUN
cana-4127	82	33	x	x	SYM
cana-4127	82	34	∈	∈	PROPN
cana-4127	82	35	x	x	NOUN
cana-4127	82	36	,	,	PUNCT
cana-4127	82	37	either	either	CCONJ
cana-4127	82	38	(	(	PUNCT
cana-4127	82	39	b1	b1	PROPN
cana-4127	82	40	)	)	PUNCT
cana-4127	82	41	d(t	d(t	PROPN
cana-4127	82	42	nx	nx	PROPN
cana-4127	82	43	,	,	PUNCT
cana-4127	82	44	tn+1x	tn+1x	ADV
cana-4127	82	45	)	)	PUNCT
cana-4127	82	46	=	=	SYM
cana-4127	82	47	∞	∞	NUM
cana-4127	82	48	∀	∀	X
cana-4127	82	49	n	n	PRON
cana-4127	82	50	≥	≥	NOUN
cana-4127	82	51	0	0	NUM
cana-4127	82	52	,	,	PUNCT
cana-4127	82	53	(	(	PUNCT
cana-4127	82	54	b2	b2	NOUN
cana-4127	82	55	)	)	PUNCT
cana-4127	82	56	there	there	PRON
cana-4127	82	57	exists	exist	VERB
cana-4127	82	58	a	a	DET
cana-4127	82	59	natural	natural	ADJ
cana-4127	82	60	number	number	NOUN
cana-4127	82	61	n0	n0	NOUN
cana-4127	82	62	such	such	ADJ
cana-4127	82	63	that	that	PRON
cana-4127	82	64	:	:	PUNCT
cana-4127	82	65	(	(	PUNCT
cana-4127	82	66	i	i	NOUN
cana-4127	82	67	)	)	PUNCT
cana-4127	82	68	d(tnx	d(tnx	PROPN
cana-4127	82	69	,	,	PUNCT
cana-4127	82	70	tn+1x	tn+1x	ADV
cana-4127	82	71	)	)	PUNCT
cana-4127	82	72	<	<	X
cana-4127	82	73	∞	∞	PROPN
cana-4127	82	74	for	for	ADP
cana-4127	82	75	all	all	DET
cana-4127	82	76	n	n	DET
cana-4127	82	77	≥	≥	NOUN
cana-4127	82	78	n0	n0	NUM
cana-4127	82	79	;	;	PUNCT
cana-4127	82	80	(	(	PUNCT
cana-4127	82	81	ii)the	ii)the	DET
cana-4127	82	82	sequence	sequence	NOUN
cana-4127	82	83	(	(	PUNCT
cana-4127	82	84	tnx	tnx	NOUN
cana-4127	82	85	)	)	PUNCT
cana-4127	82	86	is	be	AUX
cana-4127	82	87	convergent	convergent	ADJ
cana-4127	82	88	to	to	ADP
cana-4127	82	89	a	a	DET
cana-4127	82	90	fixed	fix	VERB
cana-4127	82	91	point	point	NOUN
cana-4127	82	92	y∗	y∗	PROPN
cana-4127	82	93	of	of	ADP
cana-4127	82	94	t	t	PROPN
cana-4127	82	95	;	;	PUNCT
cana-4127	82	96	(	(	PUNCT
cana-4127	82	97	iii	iii	X
cana-4127	82	98	)	)	PUNCT
cana-4127	82	99	y∗	y∗	ADV
cana-4127	82	100	is	be	AUX
cana-4127	82	101	the	the	DET
cana-4127	82	102	unique	unique	ADJ
cana-4127	82	103	fixed	fix	VERB
cana-4127	82	104	point	point	NOUN
cana-4127	82	105	of	of	ADP
cana-4127	82	106	t	t	PROPN
cana-4127	82	107	in	in	ADP
cana-4127	82	108	the	the	DET
cana-4127	82	109	set	set	NOUN
cana-4127	82	110	y	y	PROPN
cana-4127	82	111	=	=	PRON
cana-4127	82	112	{	{	PUNCT
cana-4127	82	113	y	y	PROPN
cana-4127	82	114	∈	∈	PROPN
cana-4127	82	115	x	x	X
cana-4127	82	116	:	:	PUNCT
cana-4127	82	117	d(tn0x	d(tn0x	ADJ
cana-4127	82	118	,	,	PUNCT
cana-4127	82	119	y	y	NOUN
cana-4127	82	120	)	)	PUNCT
cana-4127	82	121	<	<	X
cana-4127	82	122	∞	∞	PROPN
cana-4127	82	123	}	}	PUNCT
cana-4127	82	124	;	;	PUNCT
cana-4127	82	125	(	(	PUNCT
cana-4127	82	126	iv	iv	X
cana-4127	82	127	)	)	PUNCT
cana-4127	82	128	d(y∗	d(y∗	NOUN
cana-4127	82	129	,	,	PUNCT
cana-4127	82	130	y	y	NOUN
cana-4127	82	131	)	)	PUNCT
cana-4127	82	132	≤	≤	NUM
cana-4127	82	133	1	1	NUM
cana-4127	82	134	1−l	1−l	NUM
cana-4127	82	135	d(y	d(y	NOUN
cana-4127	82	136	,	,	PUNCT
cana-4127	82	137	ty	ty	INTJ
cana-4127	82	138	)	)	PUNCT
cana-4127	82	139	for	for	ADP
cana-4127	82	140	all	all	PRON
cana-4127	82	141	y	y	PROPN
cana-4127	82	142	∈	∈	PROPN
cana-4127	82	143	y.	y.	PROPN
cana-4127	82	144	theorem	theorem	VERB
cana-4127	82	145	3.2	3.2	NUM
cana-4127	82	146	let	let	VERB
cana-4127	82	147	h	h	NOUN
cana-4127	82	148	:	:	PUNCT
cana-4127	82	149	v	v	PROPN
cana-4127	83	1	→	→	SYM
cana-4127	83	2	b	b	X
cana-4127	83	3	be	be	AUX
cana-4127	83	4	a	a	DET
cana-4127	83	5	mapping	mapping	NOUN
cana-4127	83	6	for	for	ADP
cana-4127	83	7	which	which	PRON
cana-4127	83	8	there	there	PRON
cana-4127	83	9	exists	exist	VERB
cana-4127	83	10	functions	function	NOUN
cana-4127	83	11	α	α	NOUN
cana-4127	83	12	,	,	PUNCT
cana-4127	83	13	β	β	X
cana-4127	83	14	,	,	PUNCT
cana-4127	83	15	γ	γ	PROPN
cana-4127	83	16	:	:	PUNCT
cana-4127	83	17	v3	v3	PROPN
cana-4127	83	18	→	→	PUNCT
cana-4127	84	1	[	[	X
cana-4127	84	2	0,∞	0,∞	NOUN
cana-4127	84	3	)	)	PUNCT
cana-4127	84	4	with	with	ADP
cana-4127	84	5	the	the	DET
cana-4127	84	6	condition	condition	NOUN
cana-4127	84	7	lim	lim	PROPN
cana-4127	84	8	k→∞	k→∞	PROPN
cana-4127	85	1	α(μi	α(μi	PROPN
cana-4127	85	2	kx	kx	PROPN
cana-4127	85	3	,	,	PUNCT
cana-4127	85	4	μi	μi	PROPN
cana-4127	85	5	ky	ky	PROPN
cana-4127	85	6	,	,	PUNCT
cana-4127	85	7	μi	μi	PROPN
cana-4127	85	8	kz	kz	PROPN
cana-4127	85	9	)	)	PUNCT
cana-4127	85	10	μi	μi	VERB
cana-4127	86	1	k	k	NOUN
cana-4127	87	1	=	=	SYM
cana-4127	88	1	0	0	PROPN
cana-4127	89	1	,	,	PUNCT
cana-4127	89	2	(	(	PUNCT
cana-4127	89	3	13	13	NUM
cana-4127	89	4	)	)	PUNCT
cana-4127	89	5	where	where	SCONJ
cana-4127	89	6	μi	μi	ADV
cana-4127	89	7	=	=	SYM
cana-4127	89	8	{	{	PUNCT
cana-4127	89	9	𝒬	𝒬	PROPN
cana-4127	89	10	,	,	PUNCT
cana-4127	89	11	i	i	NOUN
cana-4127	89	12	=	=	NOUN
cana-4127	89	13	0	0	NUM
cana-4127	89	14	,	,	PUNCT
cana-4127	89	15	1	1	NUM
cana-4127	89	16	𝒬	𝒬	NOUN
cana-4127	89	17	,	,	PUNCT
cana-4127	89	18	i	i	PRON
cana-4127	89	19	=	=	NOUN
cana-4127	89	20	1	1	NUM
cana-4127	89	21	satisfying	satisfy	VERB
cana-4127	89	22	the	the	DET
cana-4127	89	23	functional	functional	ADJ
cana-4127	89	24	inequality	inequality	NOUN
cana-4127	89	25	‖h(x	‖h(x	PROPN
cana-4127	89	26	,	,	PUNCT
cana-4127	89	27	y	y	PROPN
cana-4127	89	28	,	,	PUNCT
cana-4127	89	29	z)‖	z)‖	ADJ
cana-4127	89	30	≤	≤	PROPN
cana-4127	89	31	α(x	α(x	PROPN
cana-4127	89	32	,	,	PUNCT
cana-4127	89	33	y	y	PROPN
cana-4127	89	34	,	,	PUNCT
cana-4127	89	35	z	z	NOUN
cana-4127	89	36	)	)	PUNCT
cana-4127	89	37	(	(	PUNCT
cana-4127	89	38	14	14	NUM
cana-4127	89	39	)	)	PUNCT
cana-4127	89	40	for	for	ADP
cana-4127	89	41	all	all	DET
cana-4127	89	42	x	x	NOUN
cana-4127	89	43	,	,	PUNCT
cana-4127	89	44	y	y	PROPN
cana-4127	89	45	,	,	PUNCT
cana-4127	89	46	z	z	NOUN
cana-4127	89	47	∈	∈	PROPN
cana-4127	89	48	v.	v.	CCONJ
cana-4127	89	49	if	if	SCONJ
cana-4127	89	50	there	there	PRON
cana-4127	89	51	exists	exist	VERB
cana-4127	89	52	an	an	DET
cana-4127	89	53	l	l	NOUN
cana-4127	89	54	=	=	PUNCT
cana-4127	89	55	l(i	l(i	PROPN
cana-4127	89	56	)	)	PUNCT
cana-4127	89	57	<	<	X
cana-4127	89	58	1	1	NUM
cana-4127	89	59	such	such	ADJ
cana-4127	89	60	that	that	SCONJ
cana-4127	89	61	the	the	DET
cana-4127	89	62	function	function	NOUN
cana-4127	89	63	x	x	INTJ
cana-4127	89	64	→	→	SYM
cana-4127	89	65	γ(x	γ(x	NOUN
cana-4127	89	66	)	)	PUNCT
cana-4127	89	67	=	=	SYM
cana-4127	89	68	1	1	X
cana-4127	89	69	(	(	PUNCT
cana-4127	89	70	s2	s2	PROPN
cana-4127	89	71	+	+	NOUN
cana-4127	89	72	2s+1	2s+1	NOUN
cana-4127	89	73	)	)	PUNCT
cana-4127	89	74	θ	θ	NOUN
cana-4127	89	75	(	(	PUNCT
cana-4127	89	76	x	x	SYM
cana-4127	89	77	𝒬	𝒬	PROPN
cana-4127	89	78	)	)	PUNCT
cana-4127	89	79	,	,	PUNCT
cana-4127	89	80	one	one	PRON
cana-4127	89	81	has	have	VERB
cana-4127	89	82	the	the	DET
cana-4127	89	83	property	property	NOUN
cana-4127	89	84	γ(x	γ(x	NOUN
cana-4127	89	85	)	)	PUNCT
cana-4127	90	1	=	=	SYM
cana-4127	90	2	l	l	NOUN
cana-4127	90	3	μi	μi	NOUN
cana-4127	90	4	γ	γ	X
cana-4127	90	5	(	(	PUNCT
cana-4127	90	6	x	x	SYM
cana-4127	90	7	μi	μi	PROPN
cana-4127	90	8	)	)	PUNCT
cana-4127	90	9	(	(	PUNCT
cana-4127	90	10	15	15	X
cana-4127	90	11	)	)	PUNCT
cana-4127	90	12	communications	communication	NOUN
cana-4127	90	13	on	on	ADP
cana-4127	90	14	applied	apply	VERB
cana-4127	90	15	nonlinear	nonlinear	ADJ
cana-4127	90	16	analysis	analysis	NOUN
cana-4127	90	17	issn	issn	NOUN
cana-4127	90	18	:	:	PUNCT
cana-4127	90	19	1074	1074	NUM
cana-4127	90	20	-	-	PUNCT
cana-4127	90	21	133x	133x	NUM
cana-4127	90	22	vol	vol	NOUN
cana-4127	90	23	32	32	NUM
cana-4127	90	24	no	no	NOUN
cana-4127	90	25	.	.	PUNCT
cana-4127	91	1	9s	9s	NUM
cana-4127	91	2	(	(	PUNCT
cana-4127	91	3	2025	2025	NUM
cana-4127	91	4	)	)	PUNCT
cana-4127	91	5	1217	1217	NUM
cana-4127	91	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	91	7	for	for	ADP
cana-4127	91	8	all	all	DET
cana-4127	91	9	x	x	SYM
cana-4127	91	10	∈	∈	PROPN
cana-4127	92	1	v.	v.	CCONJ
cana-4127	92	2	then	then	ADV
cana-4127	92	3	there	there	PRON
cana-4127	92	4	exists	exist	VERB
cana-4127	92	5	a	a	DET
cana-4127	92	6	unique	unique	ADJ
cana-4127	92	7	additive	additive	ADJ
cana-4127	92	8	function	function	NOUN
cana-4127	92	9	a	a	DET
cana-4127	92	10	:	:	PUNCT
cana-4127	92	11	v	v	NOUN
cana-4127	92	12	→	→	SYM
cana-4127	92	13	b	b	X
cana-4127	92	14	satisfying	satisfy	VERB
cana-4127	92	15	the	the	DET
cana-4127	92	16	functional	functional	ADJ
cana-4127	92	17	equation	equation	NOUN
cana-4127	92	18	(	(	PUNCT
cana-4127	92	19	1	1	NUM
cana-4127	92	20	)	)	PUNCT
cana-4127	92	21	and	and	CCONJ
cana-4127	92	22	‖h(x	‖h(x	NUM
cana-4127	92	23	)	)	PUNCT
cana-4127	92	24	−	−	PROPN
cana-4127	93	1	a(x)‖	a(x)‖	CCONJ
cana-4127	93	2	≤	≤	PUNCT
cana-4127	93	3	l1−i	l1−i	PROPN
cana-4127	93	4	1−l	1−l	NUM
cana-4127	93	5	γ(x	γ(x	NOUN
cana-4127	93	6	)	)	PUNCT
cana-4127	93	7	(	(	PUNCT
cana-4127	93	8	16	16	NUM
cana-4127	93	9	)	)	PUNCT
cana-4127	93	10	holds	hold	VERB
cana-4127	93	11	for	for	ADP
cana-4127	93	12	all	all	DET
cana-4127	93	13	x	x	SYM
cana-4127	93	14	∈	∈	ADJ
cana-4127	93	15	x.	x.	NOUN
cana-4127	93	16	proof	proof	NOUN
cana-4127	93	17	.	.	PUNCT
cana-4127	94	1	consider	consider	VERB
cana-4127	94	2	the	the	DET
cana-4127	94	3	set	set	NOUN
cana-4127	94	4	𝑋	𝑋	NOUN
cana-4127	94	5	=	=	SYM
cana-4127	94	6	{	{	PUNCT
cana-4127	94	7	𝑝/𝑝	𝑝/𝑝	PROPN
cana-4127	94	8	:	:	PUNCT
cana-4127	94	9	𝑉	𝑉	PROPN
cana-4127	94	10	→	→	SYM
cana-4127	94	11	𝐵	𝐵	PROPN
cana-4127	94	12	,	,	PUNCT
cana-4127	94	13	𝑝(0	𝑝(0	PROPN
cana-4127	94	14	)	)	PUNCT
cana-4127	94	15	=	=	SYM
cana-4127	94	16	0	0	PUNCT
cana-4127	94	17	}	}	PUNCT
cana-4127	94	18	and	and	CCONJ
cana-4127	94	19	introduce	introduce	VERB
cana-4127	94	20	the	the	DET
cana-4127	94	21	generalized	generalize	VERB
cana-4127	94	22	metric	metric	NOUN
cana-4127	94	23	on	on	ADP
cana-4127	94	24	𝑋	𝑋	PROPN
cana-4127	94	25	,	,	PUNCT
cana-4127	94	26	𝑑(𝑝	𝑑(𝑝	PROPN
cana-4127	94	27	,	,	PUNCT
cana-4127	94	28	𝑞	𝑞	X
cana-4127	94	29	)	)	PUNCT
cana-4127	94	30	=	=	SYM
cana-4127	94	31	inf{𝐾	inf{𝐾	PROPN
cana-4127	94	32	∈	∈	PROPN
cana-4127	94	33	(	(	PUNCT
cana-4127	94	34	0,∞	0,∞	NOUN
cana-4127	94	35	):	):	PUNCT
cana-4127	94	36	∥	∥	X
cana-4127	94	37	𝑝(𝑥	𝑝(𝑥	NOUN
cana-4127	94	38	)	)	PUNCT
cana-4127	94	39	−	−	ADP
cana-4127	94	40	𝑞(𝑥	𝑞(𝑥	PROPN
cana-4127	94	41	)	)	PUNCT
cana-4127	94	42	∥≤	∥≤	PROPN
cana-4127	94	43	𝐾𝛾(𝑥	𝐾𝛾(𝑥	NOUN
cana-4127	94	44	)	)	PUNCT
cana-4127	94	45	,	,	PUNCT
cana-4127	94	46	𝑥	𝑥	PROPN
cana-4127	94	47	∈	∈	PROPN
cana-4127	94	48	𝑉	𝑉	PROPN
cana-4127	94	49	}	}	PUNCT
cana-4127	94	50	.	.	PUNCT
cana-4127	95	1	it	it	PRON
cana-4127	95	2	is	be	AUX
cana-4127	95	3	easy	easy	ADJ
cana-4127	95	4	to	to	PART
cana-4127	95	5	see	see	VERB
cana-4127	95	6	that	that	PRON
cana-4127	95	7	(	(	PUNCT
cana-4127	95	8	𝑋	𝑋	PROPN
cana-4127	95	9	,	,	PUNCT
cana-4127	95	10	𝑑	𝑑	NOUN
cana-4127	95	11	)	)	PUNCT
cana-4127	95	12	is	be	AUX
cana-4127	95	13	complete	complete	ADJ
cana-4127	95	14	.	.	PUNCT
cana-4127	96	1	define	define	VERB
cana-4127	96	2	𝑇	𝑇	PROPN
cana-4127	96	3	:	:	PUNCT
cana-4127	96	4	𝑋	𝑋	PROPN
cana-4127	96	5	→	→	SYM
cana-4127	96	6	𝑋	𝑋	NOUN
cana-4127	96	7	by	by	ADP
cana-4127	96	8	𝑇𝑝(𝑥	𝑇𝑝(𝑥	NOUN
cana-4127	96	9	)	)	PUNCT
cana-4127	96	10	=	=	PUNCT
cana-4127	97	1	1	1	NUM
cana-4127	97	2	𝜇𝑖	𝜇𝑖	ADP
cana-4127	97	3	𝑝(𝜇𝑖𝑥	𝑝(𝜇𝑖𝑥	PROPN
cana-4127	97	4	)	)	PUNCT
cana-4127	97	5	,	,	PUNCT
cana-4127	97	6	∀	∀	PUNCT
cana-4127	97	7	𝑥	𝑥	DET
cana-4127	97	8	∈	∈	PROPN
cana-4127	97	9	𝑉.	𝑉.	PROPN
cana-4127	97	10	now	now	ADV
cana-4127	97	11	𝑝	𝑝	VERB
cana-4127	97	12	,	,	PUNCT
cana-4127	97	13	𝑞	𝑞	PROPN
cana-4127	97	14	∈	∈	PROPN
cana-4127	97	15	𝑋	𝑋	PROPN
cana-4127	97	16	,	,	PUNCT
cana-4127	97	17	𝑑(𝑝	𝑑(𝑝	PROPN
cana-4127	97	18	,	,	PUNCT
cana-4127	97	19	𝑞	𝑞	NOUN
cana-4127	97	20	)	)	PUNCT
cana-4127	97	21	≤	≤	NOUN
cana-4127	98	1	𝐾	𝐾	PROPN
cana-4127	98	2	⇒∥	⇒∥	ADJ
cana-4127	98	3	𝑝(𝑥	𝑝(𝑥	PROPN
cana-4127	98	4	)	)	PUNCT
cana-4127	98	5	−	−	ADP
cana-4127	98	6	𝑞(𝑥	𝑞(𝑥	PROPN
cana-4127	98	7	)	)	PUNCT
cana-4127	98	8	∥≤	∥≤	PROPN
cana-4127	98	9	𝐾𝛾(𝑥	𝐾𝛾(𝑥	NOUN
cana-4127	98	10	)	)	PUNCT
cana-4127	98	11	,	,	PUNCT
cana-4127	98	12	𝑥	𝑥	PROPN
cana-4127	98	13	∈	∈	PROPN
cana-4127	98	14	𝑉.	𝑉.	PROPN
cana-4127	98	15	⇒	⇒	NOUN
cana-4127	98	16	‖	‖	PROPN
cana-4127	98	17	1	1	NUM
cana-4127	98	18	𝜇𝑖	𝜇𝑖	ADP
cana-4127	98	19	𝑝(𝜇𝑖𝑥	𝑝(𝜇𝑖𝑥	PROPN
cana-4127	98	20	)	)	PUNCT
cana-4127	99	1	−	−	PROPN
cana-4127	99	2	1	1	NUM
cana-4127	99	3	𝜇𝑖	𝜇𝑖	ADP
cana-4127	99	4	𝑞(𝜇𝑖𝑥)‖	𝑞(𝜇𝑖𝑥)‖	NOUN
cana-4127	99	5	≤	≤	ADV
cana-4127	99	6	1	1	NUM
cana-4127	99	7	𝜇𝑖	𝜇𝑖	ADP
cana-4127	99	8	𝐾𝛾(𝜇𝑖𝑥	𝐾𝛾(𝜇𝑖𝑥	NOUN
cana-4127	99	9	)	)	PUNCT
cana-4127	99	10	,	,	PUNCT
cana-4127	99	11	𝑥	𝑥	DET
cana-4127	99	12	∈	∈	PROPN
cana-4127	99	13	𝑉	𝑉	PROPN
cana-4127	99	14	,	,	PUNCT
cana-4127	99	15	⇒	⇒	NOUN
cana-4127	99	16	‖	‖	PROPN
cana-4127	99	17	1	1	NUM
cana-4127	99	18	𝜇𝑖	𝜇𝑖	ADP
cana-4127	99	19	𝑝(𝜇𝑖𝑥	𝑝(𝜇𝑖𝑥	PROPN
cana-4127	99	20	)	)	PUNCT
cana-4127	100	1	−	−	PROPN
cana-4127	100	2	1	1	NUM
cana-4127	100	3	𝜇𝑖	𝜇𝑖	ADP
cana-4127	100	4	𝑞(𝜇𝑖𝑥)‖	𝑞(𝜇𝑖𝑥)‖	NOUN
cana-4127	100	5	≤	≤	PROPN
cana-4127	100	6	𝐿𝐾𝛾(𝑥	𝐿𝐾𝛾(𝑥	NOUN
cana-4127	100	7	)	)	PUNCT
cana-4127	100	8	,	,	PUNCT
cana-4127	100	9	𝑥	𝑥	PRON
cana-4127	100	10	∈	∈	PROPN
cana-4127	100	11	𝑉	𝑉	PROPN
cana-4127	100	12	,	,	PUNCT
cana-4127	100	13	⇒∥	⇒∥	ADJ
cana-4127	100	14	𝑇𝑝(𝑥	𝑇𝑝(𝑥	NOUN
cana-4127	100	15	)	)	PUNCT
cana-4127	100	16	−	−	ADP
cana-4127	100	17	𝑇𝑞(𝑥	𝑇𝑞(𝑥	NOUN
cana-4127	100	18	)	)	PUNCT
cana-4127	100	19	∥≤	∥≤	PROPN
cana-4127	100	20	𝐿𝐾𝛾(𝑥	𝐿𝐾𝛾(𝑥	PROPN
cana-4127	100	21	)	)	PUNCT
cana-4127	100	22	,	,	PUNCT
cana-4127	100	23	𝑥	𝑥	PROPN
cana-4127	100	24	∈	∈	PROPN
cana-4127	100	25	𝑉	𝑉	PROPN
cana-4127	100	26	,	,	PUNCT
cana-4127	100	27	⇒	⇒	NOUN
cana-4127	100	28	𝑑(𝑇𝑝	𝑑(𝑇𝑝	NUM
cana-4127	100	29	,	,	PUNCT
cana-4127	100	30	𝑇𝑞	𝑇𝑞	PROPN
cana-4127	100	31	)	)	PUNCT
cana-4127	100	32	≤	≤	NUM
cana-4127	100	33	𝐿𝐾.	𝐿𝐾.	VERB
cana-4127	100	34	this	this	PRON
cana-4127	100	35	implies	imply	VERB
cana-4127	100	36	𝑑(𝑇𝑝	𝑑(𝑇𝑝	NOUN
cana-4127	100	37	,	,	PUNCT
cana-4127	100	38	𝑇𝑞	𝑇𝑞	PROPN
cana-4127	100	39	)	)	PUNCT
cana-4127	100	40	≤	≤	NOUN
cana-4127	101	1	𝐿𝑑(𝑝	𝐿𝑑(𝑝	NOUN
cana-4127	101	2	,	,	PUNCT
cana-4127	101	3	𝑞	𝑞	NOUN
cana-4127	101	4	)	)	PUNCT
cana-4127	101	5	,	,	PUNCT
cana-4127	101	6	for	for	ADP
cana-4127	101	7	all	all	DET
cana-4127	101	8	𝑝	𝑝	NOUN
cana-4127	101	9	,	,	PUNCT
cana-4127	101	10	𝑞	𝑞	PROPN
cana-4127	101	11	∈	∈	PROPN
cana-4127	101	12	𝑋	𝑋	PROPN
cana-4127	101	13	.	.	PUNCT
cana-4127	102	1	i.e.	i.e.	X
cana-4127	102	2	,	,	PUNCT
cana-4127	102	3	𝑇	𝑇	PROPN
cana-4127	102	4	is	be	AUX
cana-4127	102	5	a	a	DET
cana-4127	102	6	strictly	strictly	ADV
cana-4127	102	7	contractive	contractive	ADJ
cana-4127	102	8	mapping	mapping	NOUN
cana-4127	102	9	on	on	ADP
cana-4127	102	10	𝑋	𝑋	PROPN
cana-4127	102	11	with	with	ADP
cana-4127	102	12	lipschitz	lipschitz	NOUN
cana-4127	102	13	constant	constant	ADJ
cana-4127	102	14	𝐿.	𝐿.	VERB
cana-4127	102	15	from	from	ADP
cana-4127	102	16	(	(	PUNCT
cana-4127	102	17	7	7	NUM
cana-4127	102	18	)	)	PUNCT
cana-4127	102	19	,	,	PUNCT
cana-4127	102	20	we	we	PRON
cana-4127	102	21	have	have	VERB
cana-4127	102	22	‖ℎ(𝑥	‖ℎ(𝑥	NOUN
cana-4127	102	23	)	)	PUNCT
cana-4127	102	24	−	−	PROPN
cana-4127	102	25	ℎ(𝒬𝑥	ℎ(𝒬𝑥	NOUN
cana-4127	102	26	)	)	PUNCT
cana-4127	102	27	𝒬	𝒬	PROPN
cana-4127	102	28	‖	‖	PROPN
cana-4127	102	29	≤	≤	PROPN
cana-4127	102	30	θ(𝑥,𝑥,𝑥	θ(𝑥,𝑥,𝑥	NOUN
cana-4127	102	31	)	)	PUNCT
cana-4127	102	32	𝒬(𝑠2	𝒬(𝑠2	PROPN
cana-4127	102	33	+	+	NOUN
cana-4127	102	34	2𝑠+1	2𝑠+1	NUM
cana-4127	102	35	)	)	PUNCT
cana-4127	102	36	(	(	PUNCT
cana-4127	102	37	17	17	NUM
cana-4127	102	38	)	)	PUNCT
cana-4127	102	39	where	where	SCONJ
cana-4127	102	40	𝛽(𝑥	𝛽(𝑥	NOUN
cana-4127	102	41	)	)	PUNCT
cana-4127	103	1	=	=	SYM
cana-4127	103	2	θ(𝑥,𝑥,𝑥	θ(𝑥,𝑥,𝑥	NOUN
cana-4127	103	3	)	)	PUNCT
cana-4127	103	4	𝒬(𝑠2	𝒬(𝑠2	PROPN
cana-4127	103	5	+	+	NOUN
cana-4127	103	6	2𝑠+1	2𝑠+1	NUM
cana-4127	103	7	)	)	PUNCT
cana-4127	103	8	for	for	ADP
cana-4127	103	9	all	all	DET
cana-4127	103	10	𝑥	𝑥	DET
cana-4127	103	11	∈	∈	PROPN
cana-4127	103	12	𝑉.	𝑉.	NOUN
cana-4127	103	13	using	use	VERB
cana-4127	103	14	(	(	PUNCT
cana-4127	103	15	15	15	NUM
cana-4127	103	16	)	)	PUNCT
cana-4127	103	17	for	for	ADP
cana-4127	103	18	the	the	DET
cana-4127	103	19	case	case	NOUN
cana-4127	103	20	𝑖	𝑖	X
cana-4127	103	21	=	=	SYM
cana-4127	103	22	0	0	NUM
cana-4127	103	23	,	,	PUNCT
cana-4127	103	24	it	it	PRON
cana-4127	103	25	reduces	reduce	VERB
cana-4127	103	26	to	to	ADP
cana-4127	103	27	‖	‖	PROPN
cana-4127	103	28	1	1	NUM
cana-4127	103	29	𝒬	𝒬	PROPN
cana-4127	103	30	ℎ(𝒬𝑥	ℎ(𝒬𝑥	NOUN
cana-4127	103	31	)	)	PUNCT
cana-4127	103	32	−	−	ADP
cana-4127	103	33	ℎ(𝑥)‖	ℎ(𝑥)‖	NOUN
cana-4127	103	34	≤	≤	NUM
cana-4127	103	35	1	1	NUM
cana-4127	103	36	𝒬	𝒬	PROPN
cana-4127	103	37	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	103	38	)	)	PUNCT
cana-4127	103	39	for	for	ADP
cana-4127	103	40	all	all	DET
cana-4127	103	41	𝑥	𝑥	DET
cana-4127	103	42	∈	∈	PROPN
cana-4127	103	43	𝑉.	𝑉.	NOUN
cana-4127	103	44	𝑖.	𝑖.	NOUN
cana-4127	103	45	𝑒.	𝑒.	PROPN
cana-4127	103	46	,	,	PUNCT
cana-4127	103	47	𝑑(𝑇ℎ	𝑑(𝑇ℎ	INTJ
cana-4127	103	48	,	,	PUNCT
cana-4127	103	49	ℎ	ℎ	PROPN
cana-4127	103	50	)	)	PUNCT
cana-4127	103	51	≤	≤	NUM
cana-4127	103	52	1	1	NUM
cana-4127	103	53	𝒬	𝒬	NOUN
cana-4127	103	54	=	=	SYM
cana-4127	103	55	𝐿	𝐿	NOUN
cana-4127	103	56	=	=	SYM
cana-4127	103	57	𝐿1−0	𝐿1−0	PROPN
cana-4127	103	58	=	=	SYM
cana-4127	103	59	𝐿1−𝑖	𝐿1−𝑖	NOUN
cana-4127	103	60	<	<	X
cana-4127	103	61	∞.	∞.	PROPN
cana-4127	103	62	again	again	ADV
cana-4127	103	63	replacing	replace	VERB
cana-4127	103	64	𝑥	𝑥	NOUN
cana-4127	103	65	=	=	PUNCT
cana-4127	103	66	𝑥	𝑥	DET
cana-4127	103	67	𝒬	𝒬	PROPN
cana-4127	103	68	in	in	ADP
cana-4127	103	69	(	(	PUNCT
cana-4127	103	70	17	17	NUM
cana-4127	103	71	)	)	PUNCT
cana-4127	103	72	,	,	PUNCT
cana-4127	103	73	we	we	PRON
cana-4127	103	74	get	get	VERB
cana-4127	103	75	communications	communication	NOUN
cana-4127	103	76	on	on	ADP
cana-4127	103	77	applied	apply	VERB
cana-4127	103	78	nonlinear	nonlinear	ADJ
cana-4127	103	79	analysis	analysis	NOUN
cana-4127	103	80	issn	issn	NOUN
cana-4127	103	81	:	:	PUNCT
cana-4127	103	82	1074	1074	NUM
cana-4127	103	83	-	-	PUNCT
cana-4127	103	84	133x	133x	NUM
cana-4127	103	85	vol	vol	NOUN
cana-4127	103	86	32	32	NUM
cana-4127	103	87	no	no	NOUN
cana-4127	103	88	.	.	PUNCT
cana-4127	104	1	9s	9s	NUM
cana-4127	104	2	(	(	PUNCT
cana-4127	104	3	2025	2025	NUM
cana-4127	104	4	)	)	PUNCT
cana-4127	104	5	1218	1218	NUM
cana-4127	104	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	104	7	‖ℎ(𝑥	‖ℎ(𝑥	X
cana-4127	104	8	)	)	PUNCT
cana-4127	104	9	−	−	PROPN
cana-4127	105	1	𝒬𝑓	𝒬𝑓	NOUN
cana-4127	105	2	(	(	PUNCT
cana-4127	105	3	𝑥	𝑥	PROPN
cana-4127	105	4	𝒬	𝒬	PROPN
cana-4127	105	5	)	)	PUNCT
cana-4127	105	6	‖	‖	PROPN
cana-4127	105	7	≤	≤	NUM
cana-4127	105	8	1	1	NUM
cana-4127	105	9	(	(	PUNCT
cana-4127	105	10	𝑠2	𝑠2	NOUN
cana-4127	105	11	+	+	NOUN
cana-4127	105	12	2𝑠+1	2𝑠+1	NUM
cana-4127	105	13	)	)	PUNCT
cana-4127	105	14	θ	θ	PROPN
cana-4127	105	15	(	(	PUNCT
cana-4127	105	16	𝑥	𝑥	PROPN
cana-4127	105	17	𝒬	𝒬	PROPN
cana-4127	105	18	)	)	PUNCT
cana-4127	105	19	.	.	PUNCT
cana-4127	106	1	for	for	ADP
cana-4127	106	2	all	all	DET
cana-4127	106	3	𝑥	𝑥	DET
cana-4127	106	4	∈	∈	PROPN
cana-4127	106	5	𝑉.	𝑉.	NOUN
cana-4127	106	6	using	use	VERB
cana-4127	106	7	(	(	PUNCT
cana-4127	106	8	15	15	NUM
cana-4127	106	9	)	)	PUNCT
cana-4127	106	10	for	for	ADP
cana-4127	106	11	the	the	DET
cana-4127	106	12	case	case	NOUN
cana-4127	106	13	𝑖	𝑖	X
cana-4127	106	14	=	=	SYM
cana-4127	106	15	1	1	NUM
cana-4127	106	16	,	,	PUNCT
cana-4127	106	17	it	it	PRON
cana-4127	106	18	reduces	reduce	VERB
cana-4127	106	19	to	to	ADP
cana-4127	106	20	‖ℎ(𝑥	‖ℎ(𝑥	NUM
cana-4127	106	21	)	)	PUNCT
cana-4127	106	22	−	−	PROPN
cana-4127	107	1	𝒬𝑓	𝒬𝑓	NOUN
cana-4127	107	2	(	(	PUNCT
cana-4127	107	3	𝑥	𝑥	PROPN
cana-4127	107	4	𝒬	𝒬	PROPN
cana-4127	107	5	)	)	PUNCT
cana-4127	107	6	‖	‖	PROPN
cana-4127	107	7	≤	≤	X
cana-4127	107	8	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	107	9	)	)	PUNCT
cana-4127	107	10	for	for	ADP
cana-4127	107	11	all	all	DET
cana-4127	107	12	𝑥	𝑥	DET
cana-4127	107	13	∈	∈	PROPN
cana-4127	107	14	𝑉.	𝑉.	NOUN
cana-4127	107	15	𝑖.	𝑖.	NOUN
cana-4127	107	16	𝑒.	𝑒.	PROPN
cana-4127	107	17	,	,	PUNCT
cana-4127	107	18	𝑑(ℎ	𝑑(ℎ	PROPN
cana-4127	107	19	,	,	PUNCT
cana-4127	107	20	𝑇ℎ	𝑇ℎ	ADJ
cana-4127	107	21	)	)	PUNCT
cana-4127	107	22	≤	≤	NUM
cana-4127	107	23	1	1	NUM
cana-4127	107	24	=	=	NUM
cana-4127	107	25	𝐿0	𝐿0	ADJ
cana-4127	107	26	=	=	PUNCT
cana-4127	107	27	𝐿1−1	𝐿1−1	PROPN
cana-4127	107	28	=	=	SYM
cana-4127	107	29	𝐿1−𝑖	𝐿1−𝑖	PROPN
cana-4127	107	30	<	<	X
cana-4127	107	31	∞.	∞.	PROPN
cana-4127	107	32	in	in	ADP
cana-4127	107	33	the	the	DET
cana-4127	107	34	above	above	ADJ
cana-4127	107	35	cases	case	NOUN
cana-4127	107	36	,	,	PUNCT
cana-4127	107	37	we	we	PRON
cana-4127	107	38	arrive	arrive	VERB
cana-4127	107	39	𝑑(ℎ	𝑑(ℎ	PROPN
cana-4127	107	40	,	,	PUNCT
cana-4127	107	41	𝑇ℎ	𝑇ℎ	PROPN
cana-4127	107	42	)	)	PUNCT
cana-4127	107	43	≤	≤	NUM
cana-4127	107	44	𝐿1−𝑖	𝐿1−𝑖	NOUN
cana-4127	107	45	.	.	PUNCT
cana-4127	108	1	therefore	therefore	ADV
cana-4127	108	2	(	(	PUNCT
cana-4127	108	3	𝐵2(𝑖	𝐵2(𝑖	PROPN
cana-4127	108	4	)	)	PUNCT
cana-4127	108	5	)	)	PUNCT
cana-4127	108	6	holds	hold	VERB
cana-4127	108	7	.	.	PUNCT
cana-4127	109	1	by	by	ADP
cana-4127	109	2	(	(	PUNCT
cana-4127	109	3	𝐵2(𝑖𝑖	𝐵2(𝑖𝑖	PROPN
cana-4127	109	4	)	)	PUNCT
cana-4127	109	5	)	)	PUNCT
cana-4127	109	6	,	,	PUNCT
cana-4127	109	7	it	it	PRON
cana-4127	109	8	follows	follow	VERB
cana-4127	109	9	that	that	SCONJ
cana-4127	109	10	there	there	PRON
cana-4127	109	11	exists	exist	VERB
cana-4127	109	12	a	a	DET
cana-4127	109	13	fixed	fixed	ADJ
cana-4127	109	14	point	point	NOUN
cana-4127	109	15	𝐴	𝐴	PROPN
cana-4127	109	16	of	of	ADP
cana-4127	109	17	𝑇	𝑇	PROPN
cana-4127	109	18	in	in	ADP
cana-4127	109	19	𝑋	𝑋	PROPN
cana-4127	109	20	such	such	ADJ
cana-4127	109	21	that	that	SCONJ
cana-4127	109	22	𝐴(𝑥	𝐴(𝑥	NOUN
cana-4127	109	23	)	)	PUNCT
cana-4127	109	24	=	=	SYM
cana-4127	109	25	lim	lim	PROPN
cana-4127	109	26	𝑘→∞	𝑘→∞	NUM
cana-4127	109	27	ℎ(𝜇𝑖	ℎ(𝜇𝑖	PROPN
cana-4127	109	28	𝑘𝑥	𝑘𝑥	PROPN
cana-4127	109	29	)	)	PUNCT
cana-4127	109	30	𝜇𝑖	𝜇𝑖	ADP
cana-4127	109	31	𝑘	𝑘	PRON
cana-4127	109	32	,	,	PUNCT
cana-4127	109	33	∀	∀	VERB
cana-4127	109	34	𝑥	𝑥	DET
cana-4127	109	35	∈	∈	PROPN
cana-4127	109	36	𝑉.	𝑉.	NOUN
cana-4127	109	37	(	(	PUNCT
cana-4127	109	38	18	18	NUM
cana-4127	109	39	)	)	PUNCT
cana-4127	109	40	claim	claim	NOUN
cana-4127	109	41	that	that	SCONJ
cana-4127	109	42	𝐴	𝐴	PROPN
cana-4127	109	43	:	:	PUNCT
cana-4127	109	44	𝑉	𝑉	PROPN
cana-4127	109	45	→	→	PUNCT
cana-4127	109	46	𝐵	𝐵	NOUN
cana-4127	109	47	is	be	AUX
cana-4127	109	48	additive	additive	ADJ
cana-4127	109	49	.	.	PUNCT
cana-4127	110	1	replacing	replace	VERB
cana-4127	110	2	(	(	PUNCT
cana-4127	110	3	𝑥	𝑥	NOUN
cana-4127	110	4	,	,	PUNCT
cana-4127	110	5	𝑦	𝑦	NOUN
cana-4127	110	6	,	,	PUNCT
cana-4127	110	7	𝑧	𝑧	PART
cana-4127	110	8	)	)	PUNCT
cana-4127	110	9	by	by	ADP
cana-4127	110	10	(	(	PUNCT
cana-4127	110	11	𝜇𝑖	𝜇𝑖	ADP
cana-4127	110	12	𝑘𝑥	𝑘𝑥	NOUN
cana-4127	110	13	,	,	PUNCT
cana-4127	110	14	𝜇𝑖	𝜇𝑖	ADP
cana-4127	110	15	𝑘𝑦	𝑘𝑦	PROPN
cana-4127	110	16	,	,	PUNCT
cana-4127	110	17	𝜇𝑖	𝜇𝑖	ADP
cana-4127	110	18	𝑘𝑧	𝑘𝑧	PRON
cana-4127	110	19	)	)	PUNCT
cana-4127	110	20	in	in	ADP
cana-4127	110	21	(	(	PUNCT
cana-4127	110	22	14	14	NUM
cana-4127	110	23	)	)	PUNCT
cana-4127	110	24	and	and	CCONJ
cana-4127	110	25	dividing	divide	VERB
cana-4127	110	26	by	by	ADP
cana-4127	110	27	𝜇𝑖	𝜇𝑖	ADP
cana-4127	110	28	𝑘	𝑘	PRON
cana-4127	110	29	,	,	PUNCT
cana-4127	110	30	it	it	PRON
cana-4127	110	31	follows	follow	VERB
cana-4127	110	32	from	from	ADP
cana-4127	110	33	(	(	PUNCT
cana-4127	110	34	13	13	NUM
cana-4127	110	35	)	)	PUNCT
cana-4127	110	36	and	and	CCONJ
cana-4127	110	37	(	(	PUNCT
cana-4127	110	38	18	18	NUM
cana-4127	110	39	)	)	PUNCT
cana-4127	110	40	,	,	PUNCT
cana-4127	110	41	𝐴	𝐴	PROPN
cana-4127	110	42	satisfies	satisfy	VERB
cana-4127	110	43	(	(	PUNCT
cana-4127	110	44	1	1	NUM
cana-4127	110	45	)	)	PUNCT
cana-4127	110	46	for	for	ADP
cana-4127	110	47	all	all	DET
cana-4127	110	48	𝑥	𝑥	PROPN
cana-4127	110	49	,	,	PUNCT
cana-4127	110	50	𝑦	𝑦	NOUN
cana-4127	110	51	,	,	PUNCT
cana-4127	110	52	𝑧	𝑧	DET
cana-4127	110	53	∈	∈	PROPN
cana-4127	110	54	𝑋.	𝑋.	PROPN
cana-4127	110	55	by	by	ADP
cana-4127	110	56	(	(	PUNCT
cana-4127	110	57	𝐵2(𝑖𝑖𝑖	𝐵2(𝑖𝑖𝑖	ADJ
cana-4127	110	58	)	)	PUNCT
cana-4127	110	59	)	)	PUNCT
cana-4127	110	60	,	,	PUNCT
cana-4127	110	61	𝐴	𝐴	PROPN
cana-4127	110	62	is	be	AUX
cana-4127	110	63	the	the	DET
cana-4127	110	64	unique	unique	ADJ
cana-4127	110	65	fixed	fix	VERB
cana-4127	110	66	point	point	NOUN
cana-4127	110	67	of	of	ADP
cana-4127	110	68	𝑇	𝑇	PROPN
cana-4127	110	69	in	in	ADP
cana-4127	110	70	the	the	DET
cana-4127	110	71	set	set	NOUN
cana-4127	110	72	𝑌	𝑌	PROPN
cana-4127	110	73	=	=	PUNCT
cana-4127	110	74	{	{	PUNCT
cana-4127	110	75	ℎ	ℎ	PROPN
cana-4127	110	76	∈	∈	PROPN
cana-4127	110	77	𝑋	𝑋	NOUN
cana-4127	110	78	:	:	PUNCT
cana-4127	110	79	𝑑(𝑇ℎ	𝑑(𝑇ℎ	ADJ
cana-4127	110	80	,	,	PUNCT
cana-4127	110	81	𝐴	𝐴	PROPN
cana-4127	110	82	)	)	PUNCT
cana-4127	110	83	<	<	X
cana-4127	110	84	∞	∞	PROPN
cana-4127	110	85	}	}	PUNCT
cana-4127	110	86	,	,	PUNCT
cana-4127	110	87	using	use	VERB
cana-4127	110	88	the	the	DET
cana-4127	110	89	fixed	fix	VERB
cana-4127	110	90	point	point	NOUN
cana-4127	110	91	alternative	alternative	ADJ
cana-4127	110	92	result	result	NOUN
cana-4127	110	93	𝐴	𝐴	PROPN
cana-4127	110	94	is	be	AUX
cana-4127	110	95	the	the	DET
cana-4127	110	96	unique	unique	ADJ
cana-4127	110	97	function	function	NOUN
cana-4127	110	98	such	such	ADJ
cana-4127	110	99	that	that	SCONJ
cana-4127	110	100	‖ℎ(𝑥	‖ℎ(𝑥	X
cana-4127	110	101	)	)	PUNCT
cana-4127	110	102	−	−	PROPN
cana-4127	110	103	𝐴(𝑥)‖	𝐴(𝑥)‖	ADJ
cana-4127	110	104	≤	≤	X
cana-4127	110	105	𝐾𝛾(𝑥	𝐾𝛾(𝑥	NOUN
cana-4127	110	106	)	)	PUNCT
cana-4127	110	107	for	for	ADP
cana-4127	110	108	all	all	PRON
cana-4127	110	109	𝑥	𝑥	DET
cana-4127	110	110	∈	∈	PROPN
cana-4127	110	111	𝑉	𝑉	PROPN
cana-4127	110	112	and	and	CCONJ
cana-4127	110	113	𝐾	𝐾	PROPN
cana-4127	110	114	>	>	X
cana-4127	110	115	0	0	X
cana-4127	110	116	.	.	PUNCT
cana-4127	110	117	finally	finally	ADV
cana-4127	110	118	by	by	ADP
cana-4127	110	119	(	(	PUNCT
cana-4127	110	120	𝐵2(𝑖𝑣	𝐵2(𝑖𝑣	PROPN
cana-4127	110	121	)	)	PUNCT
cana-4127	110	122	)	)	PUNCT
cana-4127	110	123	,	,	PUNCT
cana-4127	110	124	we	we	PRON
cana-4127	110	125	obtain	obtain	VERB
cana-4127	110	126	𝑑(ℎ	𝑑(ℎ	PROPN
cana-4127	110	127	,	,	PUNCT
cana-4127	110	128	𝐴	𝐴	PROPN
cana-4127	110	129	)	)	PUNCT
cana-4127	110	130	≤	≤	NUM
cana-4127	110	131	1	1	NUM
cana-4127	110	132	1−𝐿	1−𝐿	NUM
cana-4127	110	133	𝑑(𝑔	𝑑(𝑔	PROPN
cana-4127	110	134	,	,	PUNCT
cana-4127	110	135	𝑇𝑔	𝑇𝑔	PROPN
cana-4127	110	136	)	)	PUNCT
cana-4127	110	137	implying	imply	VERB
cana-4127	110	138	𝑑(ℎ	𝑑(ℎ	PROPN
cana-4127	110	139	,	,	PUNCT
cana-4127	110	140	𝐴	𝐴	PROPN
cana-4127	110	141	)	)	PUNCT
cana-4127	110	142	≤	≤	NUM
cana-4127	110	143	𝐿1−𝑖	𝐿1−𝑖	PUNCT
cana-4127	110	144	1−𝐿	1−𝐿	NUM
cana-4127	110	145	.	.	PUNCT
cana-4127	111	1	hence	hence	ADV
cana-4127	111	2	we	we	PRON
cana-4127	111	3	conclude	conclude	VERB
cana-4127	111	4	that	that	SCONJ
cana-4127	111	5	∥	∥	X
cana-4127	111	6	ℎ(𝑥	ℎ(𝑥	NOUN
cana-4127	111	7	)	)	PUNCT
cana-4127	111	8	−	−	ADP
cana-4127	111	9	𝐴(𝑥	𝐴(𝑥	NOUN
cana-4127	111	10	)	)	PUNCT
cana-4127	111	11	∥≤	∥≤	PROPN
cana-4127	111	12	𝐿1−𝑖	𝐿1−𝑖	NOUN
cana-4127	111	13	1−𝐿	1−𝐿	NUM
cana-4127	111	14	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	111	15	)	)	PUNCT
cana-4127	111	16	.	.	PUNCT
cana-4127	112	1	for	for	ADP
cana-4127	112	2	all	all	PRON
cana-4127	112	3	𝑥	𝑥	DET
cana-4127	112	4	∈	∈	PROPN
cana-4127	112	5	𝑉.	𝑉.	NOUN
cana-4127	112	6	corollary	corollary	NOUN
cana-4127	112	7	3.3	3.3	NUM
cana-4127	112	8	let	let	VERB
cana-4127	112	9	h	h	NOUN
cana-4127	112	10	:	:	PUNCT
cana-4127	112	11	v	v	PROPN
cana-4127	112	12	→	→	SYM
cana-4127	112	13	b	b	X
cana-4127	112	14	be	be	AUX
cana-4127	112	15	a	a	DET
cana-4127	112	16	mapping	mapping	NOUN
cana-4127	112	17	and	and	CCONJ
cana-4127	112	18	there	there	PRON
cana-4127	112	19	exist	exist	VERB
cana-4127	112	20	real	real	ADJ
cana-4127	112	21	numbers	number	NOUN
cana-4127	112	22	θ	θ	PROPN
cana-4127	112	23	and	and	CCONJ
cana-4127	112	24	s	s	VERB
cana-4127	112	25	such	such	ADJ
cana-4127	112	26	that	that	SCONJ
cana-4127	112	27	‖h(x	‖h(x	PROPN
cana-4127	112	28	,	,	PUNCT
cana-4127	112	29	y	y	PROPN
cana-4127	112	30	,	,	PUNCT
cana-4127	112	31	z)‖	z)‖	ADJ
cana-4127	112	32	≤	≤	PROPN
cana-4127	112	33	{	{	PUNCT
cana-4127	112	34	θ	θ	NOUN
cana-4127	112	35	,	,	PUNCT
cana-4127	112	36	θ{||x||s	θ{||x||s	NOUN
cana-4127	112	37	+	+	CCONJ
cana-4127	112	38	||y||s	||y||s	PROPN
cana-4127	112	39	+	+	NUM
cana-4127	112	40	||z||s	||z||s	PROPN
cana-4127	112	41	}	}	PUNCT
cana-4127	112	42	,	,	PUNCT
cana-4127	112	43	s	s	VERB
cana-4127	112	44	≠	≠	PROPN
cana-4127	112	45	1	1	NUM
cana-4127	112	46	;	;	PUNCT
cana-4127	112	47	θ||x||s||y||s||z||s	θ||x||s||y||s||z||s	PROPN
cana-4127	112	48	,	,	PUNCT
cana-4127	112	49	3s	3s	NUM
cana-4127	112	50	≠	≠	PROPN
cana-4127	112	51	1	1	NUM
cana-4127	112	52	;	;	PUNCT
cana-4127	112	53	θ{||x||s||y||s||z||s	θ{||x||s||y||s||z||s	X
cana-4127	112	54	+	+	CCONJ
cana-4127	112	55	{	{	PUNCT
cana-4127	112	56	||x||3s	||x||3s	NOUN
cana-4127	112	57	+	+	CCONJ
cana-4127	112	58	||y||3s	||y||3s	PROPN
cana-4127	112	59	+	+	CCONJ
cana-4127	112	60	||z||3s	||z||3s	PROPN
cana-4127	112	61	}	}	PUNCT
cana-4127	112	62	}	}	PUNCT
cana-4127	112	63	,	,	PUNCT
cana-4127	112	64	3s	3s	NUM
cana-4127	112	65	≠	≠	PROPN
cana-4127	112	66	1	1	NUM
cana-4127	112	67	;	;	PUNCT
cana-4127	112	68	(	(	PUNCT
cana-4127	112	69	19	19	NUM
cana-4127	112	70	)	)	PUNCT
cana-4127	112	71	communications	communication	NOUN
cana-4127	112	72	on	on	ADP
cana-4127	112	73	applied	apply	VERB
cana-4127	112	74	nonlinear	nonlinear	ADJ
cana-4127	112	75	analysis	analysis	NOUN
cana-4127	112	76	issn	issn	NOUN
cana-4127	112	77	:	:	PUNCT
cana-4127	112	78	1074	1074	NUM
cana-4127	112	79	-	-	PUNCT
cana-4127	112	80	133x	133x	NUM
cana-4127	112	81	vol	vol	NOUN
cana-4127	112	82	32	32	NUM
cana-4127	113	1	no	no	NOUN
cana-4127	113	2	.	.	PUNCT
cana-4127	114	1	9s	9s	NUM
cana-4127	114	2	(	(	PUNCT
cana-4127	114	3	2025	2025	NUM
cana-4127	114	4	)	)	PUNCT
cana-4127	114	5	1219	1219	NUM
cana-4127	114	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	114	7	for	for	ADP
cana-4127	114	8	all	all	DET
cana-4127	114	9	x	x	NOUN
cana-4127	114	10	,	,	PUNCT
cana-4127	114	11	y	y	PROPN
cana-4127	114	12	,	,	PUNCT
cana-4127	114	13	z	z	PROPN
cana-4127	114	14	∈	∈	PROPN
cana-4127	115	1	x	x	X
cana-4127	115	2	,	,	PUNCT
cana-4127	115	3	then	then	ADV
cana-4127	115	4	there	there	PRON
cana-4127	115	5	exists	exist	VERB
cana-4127	115	6	a	a	DET
cana-4127	115	7	unique	unique	ADJ
cana-4127	115	8	additive	additive	ADJ
cana-4127	115	9	function	function	NOUN
cana-4127	115	10	a	a	DET
cana-4127	115	11	:	:	PUNCT
cana-4127	115	12	v	v	NOUN
cana-4127	115	13	→	→	SYM
cana-4127	115	14	b	b	X
cana-4127	115	15	such	such	ADJ
cana-4127	115	16	that	that	PRON
cana-4127	115	17	‖h(x	‖h(x	PUNCT
cana-4127	115	18	)	)	PUNCT
cana-4127	115	19	−	−	PROPN
cana-4127	115	20	a(x)‖	a(x)‖	PROPN
cana-4127	115	21	≤	≤	PROPN
cana-4127	115	22	{	{	PUNCT
cana-4127	115	23	θ	θ	NOUN
cana-4127	115	24	(	(	PUNCT
cana-4127	115	25	s2	s2	VERB
cana-4127	115	26	+	+	NOUN
cana-4127	115	27	2s+1)|𝒬−1|	2s+1)|𝒬−1|	NUM
cana-4127	115	28	,	,	PUNCT
cana-4127	115	29	3	3	NUM
cana-4127	115	30	θ||x||s	θ||x||s	NOUN
cana-4127	115	31	(	(	PUNCT
cana-4127	115	32	s2	s2	PROPN
cana-4127	115	33	+	+	PROPN
cana-4127	115	34	2s+1)|𝒬−𝒬s|	2s+1)|𝒬−𝒬s|	NOUN
cana-4127	115	35	,	,	PUNCT
cana-4127	115	36	θ||x||3s	θ||x||3s	PROPN
cana-4127	115	37	(	(	PUNCT
cana-4127	115	38	s2	s2	VERB
cana-4127	115	39	+	+	PROPN
cana-4127	115	40	2s+1)|𝒬−𝒬3s|	2s+1)|𝒬−𝒬3s|	NUM
cana-4127	115	41	,	,	PUNCT
cana-4127	115	42	4	4	NUM
cana-4127	115	43	θ||x||3s	θ||x||3s	PROPN
cana-4127	115	44	(	(	PUNCT
cana-4127	115	45	s2	s2	VERB
cana-4127	115	46	+	+	PROPN
cana-4127	115	47	2s+1)|𝒬−𝒬3s|	2s+1)|𝒬−𝒬3s|	NUM
cana-4127	115	48	(	(	PUNCT
cana-4127	115	49	20	20	NUM
cana-4127	115	50	)	)	PUNCT
cana-4127	115	51	for	for	ADP
cana-4127	115	52	all	all	DET
cana-4127	115	53	x	x	SYM
cana-4127	115	54	∈	∈	ADJ
cana-4127	115	55	x.	x.	NOUN
cana-4127	115	56	proof	proof	NOUN
cana-4127	115	57	.	.	PUNCT
cana-4127	116	1	let	let	VERB
cana-4127	116	2	us	we	PRON
cana-4127	116	3	set	set	VERB
cana-4127	116	4	𝛼(𝑥	𝛼(𝑥	PROPN
cana-4127	116	5	,	,	PUNCT
cana-4127	116	6	𝑦	𝑦	NOUN
cana-4127	116	7	,	,	PUNCT
cana-4127	116	8	𝑧	𝑧	NOUN
cana-4127	116	9	)	)	PUNCT
cana-4127	116	10	=	=	SYM
cana-4127	116	11	{	{	PUNCT
cana-4127	116	12	θ	θ	PROPN
cana-4127	116	13	,	,	PUNCT
cana-4127	116	14	θ{||𝑥||𝑠	θ{||𝑥||𝑠	NOUN
cana-4127	116	15	+	+	CCONJ
cana-4127	116	16	||𝑦||𝑠	||𝑦||𝑠	PROPN
cana-4127	116	17	+	+	X
cana-4127	116	18	||𝑧||𝑠	||𝑧||𝑠	NOUN
cana-4127	116	19	}	}	PUNCT
cana-4127	116	20	,	,	PUNCT
cana-4127	117	1	θ	θ	PROPN
cana-4127	117	2	||𝑥||𝑠	||𝑥||𝑠	PROPN
cana-4127	117	3	||𝑦||𝑠	||𝑦||𝑠	PROPN
cana-4127	117	4	||𝑧||𝑠	||𝑧||𝑠	PROPN
cana-4127	117	5	,	,	PUNCT
cana-4127	117	6	θ{||𝑥||𝑠||𝑦||𝑠||𝑧||𝑠	θ{||𝑥||𝑠||𝑦||𝑠||𝑧||𝑠	PROPN
cana-4127	117	7	+	+	CCONJ
cana-4127	117	8	(	(	PUNCT
cana-4127	117	9	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	117	10	+	+	CCONJ
cana-4127	117	11	||𝑦||3𝑠	||𝑦||3𝑠	PROPN
cana-4127	117	12	+	+	X
cana-4127	117	13	||𝑧||3𝑠	||𝑧||3𝑠	NOUN
cana-4127	117	14	)	)	PUNCT
cana-4127	117	15	}	}	PUNCT
cana-4127	117	16	for	for	ADP
cana-4127	117	17	all	all	DET
cana-4127	117	18	𝑥	𝑥	PROPN
cana-4127	117	19	,	,	PUNCT
cana-4127	117	20	𝑦	𝑦	NOUN
cana-4127	117	21	,	,	PUNCT
cana-4127	117	22	𝑧	𝑧	DET
cana-4127	117	23	∈	∈	PROPN
cana-4127	117	24	𝑋.	𝑋.	PROPN
cana-4127	117	25	now	now	ADV
cana-4127	117	26	𝛼(𝜇𝑖	𝛼(𝜇𝑖	NUM
cana-4127	117	27	𝑘𝑥,𝜇𝑖	𝑘𝑥,𝜇𝑖	VERB
cana-4127	117	28	𝑘𝑦,𝜇𝑖	𝑘𝑦,𝜇𝑖	PROPN
cana-4127	117	29	𝑘𝑧	𝑘𝑧	PRON
cana-4127	117	30	)	)	PUNCT
cana-4127	117	31	𝜇𝑖	𝜇𝑖	ADP
cana-4127	117	32	𝑘	𝑘	PROPN
cana-4127	117	33	=	=	PUNCT
cana-4127	117	34	{	{	PUNCT
cana-4127	117	35	θ	θ	NOUN
cana-4127	117	36	𝜇𝑖	𝜇𝑖	ADP
cana-4127	117	37	𝑘	𝑘	PRON
cana-4127	117	38	,	,	PUNCT
cana-4127	117	39	θ	θ	PROPN
cana-4127	117	40	𝜇𝑖	𝜇𝑖	ADP
cana-4127	117	41	𝑘	𝑘	PROPN
cana-4127	117	42	{	{	PUNCT
cana-4127	117	43	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	117	44	𝑘𝑥||𝑠	𝑘𝑥||𝑠	NOUN
cana-4127	117	45	+	+	SYM
cana-4127	117	46	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	117	47	𝑘𝑦||𝑠	𝑘𝑦||𝑠	PROPN
cana-4127	117	48	+	+	CCONJ
cana-4127	117	49	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	117	50	𝑘𝑧||𝑠	𝑘𝑧||𝑠	NOUN
cana-4127	117	51	}	}	PUNCT
cana-4127	117	52	,	,	PUNCT
cana-4127	117	53	θ	θ	PROPN
cana-4127	117	54	𝜇𝑖	𝜇𝑖	ADP
cana-4127	117	55	𝑛	𝑛	PROPN
cana-4127	117	56	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	117	57	𝑛𝑥||𝑠	𝑛𝑥||𝑠	PROPN
cana-4127	117	58	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	117	59	𝑛𝑦||𝑠	𝑛𝑦||𝑠	NOUN
cana-4127	117	60	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	117	61	𝑛𝑧||𝑠	𝑛𝑧||𝑠	NOUN
cana-4127	117	62	,	,	PUNCT
cana-4127	117	63	θ	θ	PROPN
cana-4127	117	64	𝜇𝑖	𝜇𝑖	ADP
cana-4127	117	65	𝑘	𝑘	PROPN
cana-4127	117	66	{	{	PUNCT
cana-4127	117	67	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	117	68	𝑘𝑥||𝑠	𝑘𝑥||𝑠	NOUN
cana-4127	117	69	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	117	70	𝑘𝑦||𝑠	𝑘𝑦||𝑠	PROPN
cana-4127	117	71	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	117	72	𝑘𝑧||𝑠{||𝜇𝑖	𝑘𝑧||𝑠{||𝜇𝑖	VERB
cana-4127	117	73	𝑘𝑥||3𝑠	𝑘𝑥||3𝑠	PUNCT
cana-4127	118	1	+	+	CCONJ
cana-4127	118	2	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	118	3	𝑘𝑦||3𝑠	𝑘𝑦||3𝑠	PUNCT
cana-4127	118	4	+	+	NOUN
cana-4127	118	5	||𝜇𝑖	||𝜇𝑖	NOUN
cana-4127	118	6	𝑘𝑧||3𝑠	𝑘𝑧||3𝑠	VERB
cana-4127	118	7	}	}	PUNCT
cana-4127	118	8	}	}	PUNCT
cana-4127	118	9	=	=	SYM
cana-4127	118	10	{	{	PUNCT
cana-4127	118	11	→	→	SYM
cana-4127	118	12	0	0	NUM
cana-4127	118	13	𝑎𝑠	𝑎𝑠	PROPN
cana-4127	118	14	𝑘	𝑘	PROPN
cana-4127	118	15	→	→	SYM
cana-4127	118	16	∞	∞	PROPN
cana-4127	118	17	,	,	PUNCT
cana-4127	118	18	→	→	SYM
cana-4127	118	19	0	0	NUM
cana-4127	118	20	𝑎𝑠	𝑎𝑠	PROPN
cana-4127	118	21	𝑘	𝑘	PROPN
cana-4127	118	22	→	→	SYM
cana-4127	118	23	∞	∞	PROPN
cana-4127	118	24	,	,	PUNCT
cana-4127	118	25	→	→	SYM
cana-4127	118	26	0	0	NUM
cana-4127	118	27	𝑎𝑠	𝑎𝑠	PROPN
cana-4127	118	28	𝑘	𝑘	PROPN
cana-4127	118	29	→	→	SYM
cana-4127	118	30	∞	∞	PROPN
cana-4127	118	31	,	,	PUNCT
cana-4127	118	32	→	→	SYM
cana-4127	118	33	0	0	NUM
cana-4127	118	34	𝑎𝑠	𝑎𝑠	PROPN
cana-4127	118	35	𝑘	𝑘	PROPN
cana-4127	118	36	→	→	SYM
cana-4127	118	37	∞.	∞.	PROPN
cana-4127	118	38	i.e.	i.e.	X
cana-4127	118	39	,	,	PUNCT
cana-4127	118	40	(	(	PUNCT
cana-4127	118	41	13	13	NUM
cana-4127	118	42	)	)	PUNCT
cana-4127	118	43	is	be	AUX
cana-4127	118	44	holds	hold	NOUN
cana-4127	118	45	.	.	PUNCT
cana-4127	119	1	but	but	CCONJ
cana-4127	119	2	,	,	PUNCT
cana-4127	119	3	we	we	PRON
cana-4127	119	4	have	have	VERB
cana-4127	119	5	𝛾(𝑥	𝛾(𝑥	NOUN
cana-4127	119	6	)	)	PUNCT
cana-4127	119	7	=	=	SYM
cana-4127	120	1	1	1	NUM
cana-4127	120	2	(	(	PUNCT
cana-4127	120	3	𝑠2	𝑠2	NOUN
cana-4127	120	4	+	+	NOUN
cana-4127	120	5	2𝑠+1	2𝑠+1	NUM
cana-4127	120	6	)	)	PUNCT
cana-4127	121	1	[	[	X
cana-4127	121	2	θ	θ	X
cana-4127	121	3	(	(	PUNCT
cana-4127	121	4	𝑥	𝑥	PROPN
cana-4127	121	5	𝒬	𝒬	PROPN
cana-4127	121	6	,	,	PUNCT
cana-4127	121	7	𝑥	𝑥	PROPN
cana-4127	121	8	𝒬	𝒬	PROPN
cana-4127	121	9	,	,	PUNCT
cana-4127	121	10	𝑥	𝑥	PROPN
cana-4127	121	11	𝒬	𝒬	PROPN
cana-4127	121	12	)	)	PUNCT
cana-4127	121	13	]	]	PUNCT
cana-4127	121	14	.	.	PUNCT
cana-4127	122	1	hence	hence	ADV
cana-4127	122	2	𝛾(𝑥	𝛾(𝑥	NOUN
cana-4127	122	3	)	)	PUNCT
cana-4127	122	4	=	=	SYM
cana-4127	123	1	1	1	NUM
cana-4127	123	2	(	(	PUNCT
cana-4127	123	3	𝑠2	𝑠2	NOUN
cana-4127	123	4	+	+	NOUN
cana-4127	123	5	2𝑠+1	2𝑠+1	NUM
cana-4127	123	6	)	)	PUNCT
cana-4127	124	1	[	[	X
cana-4127	124	2	θ	θ	X
cana-4127	124	3	(	(	PUNCT
cana-4127	124	4	𝑥	𝑥	PROPN
cana-4127	124	5	𝒬	𝒬	PROPN
cana-4127	124	6	,	,	PUNCT
cana-4127	124	7	𝑥	𝑥	PROPN
cana-4127	124	8	𝒬	𝒬	PROPN
cana-4127	124	9	,	,	PUNCT
cana-4127	124	10	𝑥	𝑥	PROPN
cana-4127	124	11	𝒬	𝒬	PROPN
cana-4127	124	12	)	)	PUNCT
cana-4127	124	13	]	]	PUNCT
cana-4127	125	1	=	=	PRON
cana-4127	125	2	{	{	PUNCT
cana-4127	125	3	θ	θ	PROPN
cana-4127	125	4	(	(	PUNCT
cana-4127	125	5	𝑠2	𝑠2	PROPN
cana-4127	125	6	+	+	PROPN
cana-4127	125	7	2𝑠+1)𝒬	2𝑠+1)𝒬	NUM
cana-4127	125	8	,	,	PUNCT
cana-4127	125	9	3θ	3θ	NUM
cana-4127	125	10	(	(	PUNCT
cana-4127	125	11	𝑠2	𝑠2	PROPN
cana-4127	125	12	+	+	PROPN
cana-4127	125	13	2𝑠+1)𝒬𝑠	2𝑠+1)𝒬𝑠	NUM
cana-4127	125	14	||𝑥||𝑠	||𝑥||𝑠	NOUN
cana-4127	125	15	,	,	PUNCT
cana-4127	125	16	θ	θ	PROPN
cana-4127	125	17	(	(	PUNCT
cana-4127	125	18	𝑠2	𝑠2	PROPN
cana-4127	125	19	+	+	PROPN
cana-4127	125	20	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	PROPN
cana-4127	125	21	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	125	22	,	,	PUNCT
cana-4127	125	23	4θ	4θ	NOUN
cana-4127	125	24	(	(	PUNCT
cana-4127	125	25	𝑠2	𝑠2	PROPN
cana-4127	125	26	+	+	PROPN
cana-4127	125	27	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	125	28	||𝑥||3𝑠.	||𝑥||3𝑠.	NOUN
cana-4127	125	29	also	also	ADV
cana-4127	125	30	,	,	PUNCT
cana-4127	125	31	communications	communication	NOUN
cana-4127	125	32	on	on	ADP
cana-4127	125	33	applied	apply	VERB
cana-4127	125	34	nonlinear	nonlinear	ADJ
cana-4127	125	35	analysis	analysis	NOUN
cana-4127	125	36	issn	issn	NOUN
cana-4127	125	37	:	:	PUNCT
cana-4127	125	38	1074	1074	NUM
cana-4127	125	39	-	-	PUNCT
cana-4127	125	40	133x	133x	NUM
cana-4127	125	41	vol	vol	NOUN
cana-4127	125	42	32	32	NUM
cana-4127	125	43	no	no	NOUN
cana-4127	125	44	.	.	PUNCT
cana-4127	126	1	9s	9s	NUM
cana-4127	126	2	(	(	PUNCT
cana-4127	126	3	2025	2025	NUM
cana-4127	126	4	)	)	PUNCT
cana-4127	126	5	1220	1220	NUM
cana-4127	126	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	126	7	1	1	NUM
cana-4127	126	8	𝜇𝑖	𝜇𝑖	ADP
cana-4127	126	9	𝛾(𝜇𝑖𝑥	𝛾(𝜇𝑖𝑥	NOUN
cana-4127	126	10	)	)	PUNCT
cana-4127	127	1	=	=	PRON
cana-4127	127	2	{	{	PUNCT
cana-4127	127	3	θ	θ	PROPN
cana-4127	127	4	𝜇𝑖⋅(𝑠	𝜇𝑖⋅(𝑠	PROPN
cana-4127	127	5	2	2	NUM
cana-4127	127	6	+	+	PROPN
cana-4127	127	7	2𝑠+1)𝒬	2𝑠+1)𝒬	NUM
cana-4127	127	8	,	,	PUNCT
cana-4127	127	9	3θ	3θ	PROPN
cana-4127	127	10	𝜇𝑖⋅(𝑠	𝜇𝑖⋅(𝑠	PROPN
cana-4127	127	11	2	2	NUM
cana-4127	127	12	+	+	PROPN
cana-4127	127	13	2𝑠+1)𝒬𝑠	2𝑠+1)𝒬𝑠	NUM
cana-4127	127	14	||𝜇𝑖𝑥||	||𝜇𝑖𝑥||	X
cana-4127	127	15	𝑠	𝑠	INTJ
cana-4127	127	16	,	,	PUNCT
cana-4127	127	17	θ	θ	PROPN
cana-4127	127	18	𝜇𝑖⋅(𝑠	𝜇𝑖⋅(𝑠	PROPN
cana-4127	127	19	2	2	NUM
cana-4127	127	20	+	+	PROPN
cana-4127	127	21	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	127	22	||𝜇𝑖𝑥||	||𝜇𝑖𝑥||	X
cana-4127	127	23	3𝑠	3𝑠	NOUN
cana-4127	127	24	,	,	PUNCT
cana-4127	127	25	θ	θ	PROPN
cana-4127	127	26	𝜇𝑖⋅(𝑠	𝜇𝑖⋅(𝑠	PROPN
cana-4127	127	27	2	2	NUM
cana-4127	127	28	+	+	PROPN
cana-4127	127	29	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	127	30	||𝜇𝑖𝑥||	||𝜇𝑖𝑥||	X
cana-4127	127	31	3𝑠.	3𝑠.	NUM
cana-4127	128	1	=	=	NOUN
cana-4127	128	2	{	{	PUNCT
cana-4127	128	3	𝜇𝑖	𝜇𝑖	ADP
cana-4127	128	4	−1	−1	NOUN
cana-4127	128	5	θ	θ	NOUN
cana-4127	128	6	(	(	PUNCT
cana-4127	128	7	𝑠2	𝑠2	PROPN
cana-4127	128	8	+	+	NOUN
cana-4127	128	9	2𝑠+1	2𝑠+1	NUM
cana-4127	128	10	)	)	PUNCT
cana-4127	128	11	,	,	PUNCT
cana-4127	128	12	𝜇𝑖	𝜇𝑖	ADP
cana-4127	128	13	𝑠−1	𝑠−1	PROPN
cana-4127	128	14	3θ	3θ	NUM
cana-4127	128	15	(	(	PUNCT
cana-4127	128	16	𝑠2	𝑠2	PROPN
cana-4127	128	17	+	+	PROPN
cana-4127	128	18	2𝑠+1)𝒬𝑠	2𝑠+1)𝒬𝑠	NUM
cana-4127	128	19	||𝑥||𝑠	||𝑥||𝑠	NOUN
cana-4127	128	20	,	,	PUNCT
cana-4127	128	21	𝜇𝑖	𝜇𝑖	ADP
cana-4127	128	22	3𝑠−1	3𝑠−1	NUM
cana-4127	128	23	θ	θ	NOUN
cana-4127	128	24	(	(	PUNCT
cana-4127	128	25	𝑠2	𝑠2	PROPN
cana-4127	128	26	+	+	PROPN
cana-4127	128	27	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	PROPN
cana-4127	128	28	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	128	29	,	,	PUNCT
cana-4127	128	30	𝜇𝑖	𝜇𝑖	ADP
cana-4127	128	31	3𝑠−1	3𝑠−1	ADJ
cana-4127	128	32	4θ	4θ	NOUN
cana-4127	128	33	(	(	PUNCT
cana-4127	128	34	𝑠2	𝑠2	PROPN
cana-4127	128	35	+	+	PROPN
cana-4127	128	36	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	128	37	||𝑥||3𝑠.	||𝑥||3𝑠.	NOUN
cana-4127	128	38	=	=	PUNCT
cana-4127	128	39	{	{	PUNCT
cana-4127	128	40	𝜇𝑖	𝜇𝑖	ADP
cana-4127	128	41	−1𝛾(𝑥	−1𝛾(𝑥	NOUN
cana-4127	128	42	)	)	PUNCT
cana-4127	128	43	,	,	PUNCT
cana-4127	128	44	𝜇𝑖	𝜇𝑖	ADP
cana-4127	128	45	𝑠−1𝛾(𝑥	𝑠−1𝛾(𝑥	NOUN
cana-4127	128	46	)	)	PUNCT
cana-4127	128	47	,	,	PUNCT
cana-4127	128	48	𝜇𝑖	𝜇𝑖	ADP
cana-4127	128	49	3𝑠−1𝛾(𝑥	3𝑠−1𝛾(𝑥	NOUN
cana-4127	128	50	)	)	PUNCT
cana-4127	128	51	,	,	PUNCT
cana-4127	128	52	𝜇𝑖	𝜇𝑖	ADP
cana-4127	128	53	3𝑠−1𝛾(𝑥	3𝑠−1𝛾(𝑥	NUM
cana-4127	128	54	)	)	PUNCT
cana-4127	128	55	.	.	PUNCT
cana-4127	129	1	we	we	PRON
cana-4127	129	2	prove	prove	VERB
cana-4127	129	3	the	the	DET
cana-4127	129	4	following	follow	VERB
cana-4127	129	5	cases	case	NOUN
cana-4127	129	6	for	for	ADP
cana-4127	129	7	conditions	condition	NOUN
cana-4127	129	8	using	use	VERB
cana-4127	129	9	(	(	PUNCT
cana-4127	129	10	16	16	NUM
cana-4127	129	11	)	)	PUNCT
cana-4127	129	12	case:1	case:1	PROPN
cana-4127	129	13	𝐿	𝐿	NOUN
cana-4127	129	14	=	=	NOUN
cana-4127	129	15	𝒬−1	𝒬−1	NOUN
cana-4127	129	16	if	if	SCONJ
cana-4127	129	17	𝑖	𝑖	PRON
cana-4127	129	18	=	=	NOUN
cana-4127	129	19	0	0	NUM
cana-4127	129	20	‖ℎ(𝑥	‖ℎ(𝑥	NOUN
cana-4127	129	21	)	)	PUNCT
cana-4127	129	22	−	−	PROPN
cana-4127	129	23	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	129	24	≤	≤	NUM
cana-4127	129	25	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-4127	129	26	1−𝐿	1−𝐿	NUM
cana-4127	129	27	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	129	28	)	)	PUNCT
cana-4127	129	29	=	=	PUNCT
cana-4127	130	1	(	(	PUNCT
cana-4127	130	2	𝒬−1	𝒬−1	NOUN
cana-4127	130	3	)	)	PUNCT
cana-4127	130	4	1−0	1−0	NUM
cana-4127	130	5	1−(𝒬)−1	1−(𝒬)−1	NUM
cana-4127	130	6	⋅	⋅	PROPN
cana-4127	130	7	θ	θ	PROPN
cana-4127	130	8	(	(	PUNCT
cana-4127	130	9	𝑠2	𝑠2	PROPN
cana-4127	130	10	+	+	PROPN
cana-4127	130	11	2𝑠+1)𝒬	2𝑠+1)𝒬	VERB
cana-4127	130	12	=	=	SYM
cana-4127	130	13	θ	θ	PROPN
cana-4127	130	14	(	(	PUNCT
cana-4127	130	15	𝑠2	𝑠2	PROPN
cana-4127	130	16	+	+	NOUN
cana-4127	130	17	2𝑠+1)(𝒬−1	2𝑠+1)(𝒬−1	NUM
cana-4127	130	18	)	)	PUNCT
cana-4127	130	19	.	.	PUNCT
cana-4127	131	1	case:2	case:2	X
cana-4127	131	2	𝐿	𝐿	PROPN
cana-4127	131	3	=	=	PROPN
cana-4127	131	4	𝒬	𝒬	PROPN
cana-4127	131	5	if	if	SCONJ
cana-4127	131	6	𝑖	𝑖	NOUN
cana-4127	131	7	=	=	NOUN
cana-4127	131	8	1	1	NUM
cana-4127	131	9	‖ℎ(𝑥	‖ℎ(𝑥	NUM
cana-4127	131	10	)	)	PUNCT
cana-4127	132	1	−	−	PROPN
cana-4127	132	2	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	132	3	≤	≤	NUM
cana-4127	132	4	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-4127	132	5	1−𝐿	1−𝐿	NUM
cana-4127	132	6	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	132	7	)	)	PUNCT
cana-4127	132	8	=	=	PUNCT
cana-4127	133	1	(	(	PUNCT
cana-4127	133	2	𝒬)1−1	𝒬)1−1	PROPN
cana-4127	133	3	1−𝒬	1−𝒬	ADJ
cana-4127	133	4	⋅	⋅	PROPN
cana-4127	133	5	θ	θ	PROPN
cana-4127	133	6	(	(	PUNCT
cana-4127	133	7	𝑠2	𝑠2	PROPN
cana-4127	133	8	+	+	PROPN
cana-4127	133	9	2𝑠+1)𝒬	2𝑠+1)𝒬	VERB
cana-4127	133	10	=	=	SYM
cana-4127	133	11	θ	θ	PROPN
cana-4127	133	12	(	(	PUNCT
cana-4127	133	13	𝑠2	𝑠2	PROPN
cana-4127	133	14	+	+	NOUN
cana-4127	133	15	2𝑠+1)(1−𝒬	2𝑠+1)(1−𝒬	NUM
cana-4127	133	16	)	)	PUNCT
cana-4127	133	17	.	.	PUNCT
cana-4127	134	1	case:1	case:1	PROPN
cana-4127	135	1	𝐿	𝐿	PROPN
cana-4127	135	2	=	=	NOUN
cana-4127	135	3	𝒬𝑠−1	𝒬𝑠−1	PROPN
cana-4127	136	1	if	if	SCONJ
cana-4127	136	2	𝑖	𝑖	ADP
cana-4127	136	3	=	=	NOUN
cana-4127	136	4	0	0	NUM
cana-4127	136	5	‖ℎ(𝑥	‖ℎ(𝑥	NOUN
cana-4127	136	6	)	)	PUNCT
cana-4127	136	7	−	−	PROPN
cana-4127	136	8	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	136	9	≤	≤	NUM
cana-4127	136	10	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-4127	136	11	1−𝐿	1−𝐿	NUM
cana-4127	136	12	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	136	13	)	)	PUNCT
cana-4127	136	14	=	=	PUNCT
cana-4127	137	1	(	(	PUNCT
cana-4127	137	2	𝒬𝑠−1	𝒬𝑠−1	NOUN
cana-4127	137	3	)	)	PUNCT
cana-4127	137	4	1−0	1−0	NUM
cana-4127	137	5	1−𝒬𝑠−1	1−𝒬𝑠−1	NUM
cana-4127	137	6	3θ	3θ	NUM
cana-4127	137	7	(	(	PUNCT
cana-4127	137	8	𝑠2	𝑠2	PROPN
cana-4127	137	9	+	+	PROPN
cana-4127	137	10	2𝑠+1)𝒬𝑠	2𝑠+1)𝒬𝑠	NUM
cana-4127	137	11	||𝑥||𝑠	||𝑥||𝑠	NOUN
cana-4127	137	12	=	=	SYM
cana-4127	138	1	𝒬𝑠	𝒬𝑠	PROPN
cana-4127	138	2	𝒬−𝒬𝑠	𝒬−𝒬𝑠	PROPN
cana-4127	138	3	3θ	3θ	NUM
cana-4127	138	4	(	(	PUNCT
cana-4127	138	5	𝑠2	𝑠2	PROPN
cana-4127	138	6	+	+	PROPN
cana-4127	138	7	2𝑠+1)𝒬𝑠	2𝑠+1)𝒬𝑠	NOUN
cana-4127	138	8	||𝑥||𝑠	||𝑥||𝑠	NOUN
cana-4127	138	9	=	=	SYM
cana-4127	138	10	3θ||𝑥||𝑠	3θ||𝑥||𝑠	NUM
cana-4127	138	11	(	(	PUNCT
cana-4127	138	12	𝑠2	𝑠2	NOUN
cana-4127	138	13	+	+	NOUN
cana-4127	138	14	2𝑠+1)(𝒬−𝒬𝑠	2𝑠+1)(𝒬−𝒬𝑠	NOUN
cana-4127	138	15	)	)	PUNCT
cana-4127	138	16	.	.	PUNCT
cana-4127	139	1	case:2	case:2	X
cana-4127	139	2	𝐿	𝐿	NOUN
cana-4127	139	3	=	=	PROPN
cana-4127	139	4	1	1	NUM
cana-4127	139	5	𝒬𝑠−1	𝒬𝑠−1	NOUN
cana-4127	139	6	if	if	SCONJ
cana-4127	139	7	𝑖	𝑖	SYM
cana-4127	139	8	=	=	NOUN
cana-4127	139	9	1	1	NUM
cana-4127	139	10	‖ℎ(𝑥	‖ℎ(𝑥	NUM
cana-4127	139	11	)	)	PUNCT
cana-4127	139	12	−	−	PROPN
cana-4127	139	13	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	139	14	≤	≤	NUM
cana-4127	139	15	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-4127	139	16	1−𝐿	1−𝐿	NUM
cana-4127	139	17	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	139	18	)	)	PUNCT
cana-4127	139	19	=	=	PUNCT
cana-4127	139	20	(	(	PUNCT
cana-4127	139	21	1	1	NUM
cana-4127	139	22	𝒬𝑠−1	𝒬𝑠−1	NOUN
cana-4127	139	23	)	)	PUNCT
cana-4127	139	24	1−1	1−1	NUM
cana-4127	139	25	1−	1−	NUM
cana-4127	139	26	1	1	NUM
cana-4127	139	27	(	(	PUNCT
cana-4127	139	28	𝑠2	𝑠2	NOUN
cana-4127	139	29	+	+	PROPN
cana-4127	139	30	2𝑠+1)𝒬𝑠−1	2𝑠+1)𝒬𝑠−1	NUM
cana-4127	139	31	3θ	3θ	NUM
cana-4127	139	32	(	(	PUNCT
cana-4127	139	33	𝑠2	𝑠2	PROPN
cana-4127	139	34	+	+	PROPN
cana-4127	139	35	2𝑠+1)𝒬𝑠	2𝑠+1)𝒬𝑠	NUM
cana-4127	139	36	||𝑥||𝑠	||𝑥||𝑠	NOUN
cana-4127	139	37	=	=	SYM
cana-4127	140	1	𝒬𝑠	𝒬𝑠	PROPN
cana-4127	140	2	𝒬𝑠−𝒬	𝒬𝑠−𝒬	NOUN
cana-4127	140	3	3θ	3θ	NUM
cana-4127	140	4	(	(	PUNCT
cana-4127	140	5	𝑠2	𝑠2	PROPN
cana-4127	140	6	+	+	PROPN
cana-4127	140	7	2𝑠+1)𝒬𝑠	2𝑠+1)𝒬𝑠	NOUN
cana-4127	140	8	||𝑥||𝑠	||𝑥||𝑠	NOUN
cana-4127	140	9	=	=	SYM
cana-4127	140	10	3θ||𝑥||𝑠	3θ||𝑥||𝑠	NUM
cana-4127	140	11	(	(	PUNCT
cana-4127	140	12	𝑠2	𝑠2	PROPN
cana-4127	140	13	+	+	NOUN
cana-4127	140	14	2𝑠+1)(𝒬𝑠−𝒬	2𝑠+1)(𝒬𝑠−𝒬	NUM
cana-4127	140	15	)	)	PUNCT
cana-4127	140	16	.	.	PUNCT
cana-4127	141	1	communications	communication	NOUN
cana-4127	141	2	on	on	ADP
cana-4127	141	3	applied	apply	VERB
cana-4127	141	4	nonlinear	nonlinear	ADJ
cana-4127	141	5	analysis	analysis	NOUN
cana-4127	141	6	issn	issn	NOUN
cana-4127	141	7	:	:	PUNCT
cana-4127	141	8	1074	1074	NUM
cana-4127	141	9	-	-	PUNCT
cana-4127	141	10	133x	133x	NUM
cana-4127	141	11	vol	vol	NOUN
cana-4127	141	12	32	32	NUM
cana-4127	141	13	no	no	NOUN
cana-4127	141	14	.	.	PUNCT
cana-4127	142	1	9s	9s	NUM
cana-4127	142	2	(	(	PUNCT
cana-4127	142	3	2025	2025	NUM
cana-4127	142	4	)	)	PUNCT
cana-4127	142	5	1221	1221	NUM
cana-4127	142	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	143	1	case:1	case:1	PROPN
cana-4127	143	2	𝐿	𝐿	PROPN
cana-4127	143	3	=	=	NOUN
cana-4127	143	4	𝒬3𝑠−1	𝒬3𝑠−1	PROPN
cana-4127	144	1	if	if	SCONJ
cana-4127	144	2	𝑖	𝑖	ADP
cana-4127	144	3	=	=	NOUN
cana-4127	144	4	0	0	NUM
cana-4127	144	5	‖ℎ(𝑥	‖ℎ(𝑥	NOUN
cana-4127	144	6	)	)	PUNCT
cana-4127	144	7	−	−	PROPN
cana-4127	144	8	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	144	9	≤	≤	NUM
cana-4127	144	10	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-4127	144	11	1−𝐿	1−𝐿	NUM
cana-4127	144	12	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	144	13	)	)	PUNCT
cana-4127	144	14	=	=	SYM
cana-4127	144	15	(	(	PUNCT
cana-4127	144	16	𝒬3𝑠−1	𝒬3𝑠−1	NUM
cana-4127	144	17	)	)	PUNCT
cana-4127	144	18	1−0	1−0	NUM
cana-4127	144	19	1−𝒬3𝑠−1	1−𝒬3𝑠−1	NUM
cana-4127	144	20	θ	θ	PROPN
cana-4127	144	21	(	(	PUNCT
cana-4127	144	22	𝑠2	𝑠2	PROPN
cana-4127	144	23	+	+	PROPN
cana-4127	144	24	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	144	25	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	145	1	=	=	SYM
cana-4127	146	1	𝒬3𝑠	𝒬3𝑠	PROPN
cana-4127	146	2	𝒬−𝒬3𝑠	𝒬−𝒬3𝑠	NUM
cana-4127	146	3	θ	θ	X
cana-4127	146	4	(	(	PUNCT
cana-4127	146	5	𝑠2	𝑠2	PROPN
cana-4127	146	6	+	+	PROPN
cana-4127	146	7	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	146	8	||𝑥||3𝑠	||𝑥||3𝑠	PROPN
cana-4127	146	9	=	=	SYM
cana-4127	146	10	θ||𝑥||3𝑠	θ||𝑥||3𝑠	PROPN
cana-4127	146	11	(	(	PUNCT
cana-4127	146	12	𝑠2	𝑠2	PROPN
cana-4127	146	13	+	+	PROPN
cana-4127	146	14	2𝑠+1)(𝒬−𝒬3𝑠	2𝑠+1)(𝒬−𝒬3𝑠	NUM
cana-4127	146	15	)	)	PUNCT
cana-4127	146	16	.	.	PUNCT
cana-4127	147	1	case:2	case:2	X
cana-4127	147	2	𝐿	𝐿	NOUN
cana-4127	147	3	=	=	SYM
cana-4127	147	4	1	1	NUM
cana-4127	147	5	𝒬3𝑠−1	𝒬3𝑠−1	NOUN
cana-4127	147	6	if	if	SCONJ
cana-4127	147	7	𝑖	𝑖	PRON
cana-4127	147	8	=	=	NOUN
cana-4127	147	9	1	1	NUM
cana-4127	147	10	‖ℎ(𝑥	‖ℎ(𝑥	NUM
cana-4127	147	11	)	)	PUNCT
cana-4127	147	12	−	−	PROPN
cana-4127	147	13	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	147	14	≤	≤	NUM
cana-4127	147	15	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-4127	147	16	1−𝐿	1−𝐿	NUM
cana-4127	147	17	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	147	18	)	)	PUNCT
cana-4127	147	19	=	=	PUNCT
cana-4127	147	20	(	(	PUNCT
cana-4127	147	21	1	1	NUM
cana-4127	147	22	𝒬3𝑠−1	𝒬3𝑠−1	NOUN
cana-4127	147	23	)	)	PUNCT
cana-4127	148	1	1−1	1−1	NUM
cana-4127	148	2	1−	1−	NUM
cana-4127	148	3	1	1	NUM
cana-4127	148	4	𝒬3𝑠−1	𝒬3𝑠−1	PROPN
cana-4127	148	5	θ	θ	PROPN
cana-4127	148	6	(	(	PUNCT
cana-4127	148	7	𝑠2	𝑠2	PROPN
cana-4127	148	8	+	+	PROPN
cana-4127	148	9	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	148	10	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	148	11	=	=	SYM
cana-4127	149	1	𝒬3𝑠	𝒬3𝑠	CCONJ
cana-4127	149	2	𝒬3𝑠−𝒬	𝒬3𝑠−𝒬	NOUN
cana-4127	149	3	θ	θ	PROPN
cana-4127	149	4	(	(	PUNCT
cana-4127	149	5	𝑠2	𝑠2	PROPN
cana-4127	149	6	+	+	PROPN
cana-4127	149	7	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	149	8	||𝑥||3𝑠	||𝑥||3𝑠	PROPN
cana-4127	149	9	=	=	SYM
cana-4127	149	10	θ||𝑥||3𝑠	θ||𝑥||3𝑠	PROPN
cana-4127	149	11	(	(	PUNCT
cana-4127	149	12	𝑠2	𝑠2	PROPN
cana-4127	149	13	+	+	PROPN
cana-4127	149	14	2𝑠+1)(𝒬3𝑠−𝒬	2𝑠+1)(𝒬3𝑠−𝒬	NUM
cana-4127	149	15	)	)	PUNCT
cana-4127	149	16	.	.	PUNCT
cana-4127	150	1	case:1	case:1	PROPN
cana-4127	150	2	𝐿	𝐿	PROPN
cana-4127	150	3	=	=	NOUN
cana-4127	150	4	𝒬3𝑠−1	𝒬3𝑠−1	PROPN
cana-4127	151	1	if	if	SCONJ
cana-4127	151	2	𝑖	𝑖	ADP
cana-4127	151	3	=	=	NOUN
cana-4127	151	4	0	0	NUM
cana-4127	151	5	‖ℎ(𝑥	‖ℎ(𝑥	NOUN
cana-4127	151	6	)	)	PUNCT
cana-4127	151	7	−	−	PROPN
cana-4127	151	8	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	151	9	≤	≤	NUM
cana-4127	151	10	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-4127	151	11	1−𝐿	1−𝐿	NUM
cana-4127	151	12	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	151	13	)	)	PUNCT
cana-4127	151	14	=	=	SYM
cana-4127	151	15	(	(	PUNCT
cana-4127	151	16	𝒬3𝑠−1	𝒬3𝑠−1	NUM
cana-4127	151	17	)	)	PUNCT
cana-4127	151	18	1−0	1−0	NUM
cana-4127	151	19	1−𝒬3𝑠−1	1−𝒬3𝑠−1	NUM
cana-4127	151	20	4θ	4θ	NOUN
cana-4127	151	21	(	(	PUNCT
cana-4127	151	22	𝑠2	𝑠2	PROPN
cana-4127	151	23	+	+	PROPN
cana-4127	151	24	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	151	25	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	151	26	=	=	SYM
cana-4127	152	1	𝒬3𝑠	𝒬3𝑠	PROPN
cana-4127	152	2	𝒬−𝒬3𝑠	𝒬−𝒬3𝑠	X
cana-4127	152	3	4θ	4θ	NOUN
cana-4127	152	4	(	(	PUNCT
cana-4127	152	5	𝑠2	𝑠2	PROPN
cana-4127	152	6	+	+	PROPN
cana-4127	152	7	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	152	8	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	152	9	=	=	SYM
cana-4127	152	10	4θ||𝑥||3𝑠	4θ||𝑥||3𝑠	PROPN
cana-4127	152	11	(	(	PUNCT
cana-4127	152	12	𝑠2	𝑠2	PROPN
cana-4127	152	13	+	+	NOUN
cana-4127	152	14	2𝑠+1)(𝒬−𝒬3𝑠	2𝑠+1)(𝒬−𝒬3𝑠	NUM
cana-4127	152	15	)	)	PUNCT
cana-4127	152	16	.	.	PUNCT
cana-4127	153	1	case:2	case:2	X
cana-4127	153	2	𝐿	𝐿	NOUN
cana-4127	153	3	=	=	SYM
cana-4127	153	4	1	1	NUM
cana-4127	153	5	𝒬3𝑠−1	𝒬3𝑠−1	NOUN
cana-4127	153	6	if	if	SCONJ
cana-4127	153	7	𝑖	𝑖	PRON
cana-4127	153	8	=	=	NOUN
cana-4127	153	9	1	1	NUM
cana-4127	153	10	‖ℎ(𝑥	‖ℎ(𝑥	NUM
cana-4127	153	11	)	)	PUNCT
cana-4127	153	12	−	−	PROPN
cana-4127	153	13	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-4127	153	14	≤	≤	NUM
cana-4127	153	15	𝐿1−𝑖	𝐿1−𝑖	NUM
cana-4127	153	16	1−𝐿	1−𝐿	NUM
cana-4127	153	17	𝛾(𝑥	𝛾(𝑥	PROPN
cana-4127	153	18	)	)	PUNCT
cana-4127	153	19	=	=	PUNCT
cana-4127	153	20	(	(	PUNCT
cana-4127	153	21	1	1	NUM
cana-4127	153	22	𝒬3𝑠−1	𝒬3𝑠−1	NOUN
cana-4127	153	23	)	)	PUNCT
cana-4127	154	1	1−1	1−1	NUM
cana-4127	154	2	1−	1−	NUM
cana-4127	154	3	1	1	NUM
cana-4127	154	4	𝒬3𝑠−1	𝒬3𝑠−1	PRON
cana-4127	154	5	4θ	4θ	NOUN
cana-4127	154	6	(	(	PUNCT
cana-4127	154	7	𝑠2	𝑠2	PROPN
cana-4127	154	8	+	+	PROPN
cana-4127	154	9	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	154	10	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	154	11	=	=	SYM
cana-4127	155	1	𝒬3𝑠	𝒬3𝑠	CCONJ
cana-4127	155	2	𝒬3𝑠−𝒬	𝒬3𝑠−𝒬	NOUN
cana-4127	155	3	4θ	4θ	NOUN
cana-4127	155	4	(	(	PUNCT
cana-4127	155	5	𝑠2	𝑠2	PROPN
cana-4127	155	6	+	+	PROPN
cana-4127	155	7	2𝑠+1)𝒬3𝑠	2𝑠+1)𝒬3𝑠	NOUN
cana-4127	155	8	||𝑥||3𝑠	||𝑥||3𝑠	NOUN
cana-4127	155	9	=	=	SYM
cana-4127	155	10	4θ||𝑥||3𝑠	4θ||𝑥||3𝑠	PROPN
cana-4127	155	11	(	(	PUNCT
cana-4127	155	12	𝑠2	𝑠2	PROPN
cana-4127	155	13	+	+	PROPN
cana-4127	155	14	2𝑠+1)(𝒬3𝑠−𝒬	2𝑠+1)(𝒬3𝑠−𝒬	NUM
cana-4127	155	15	)	)	PUNCT
cana-4127	155	16	.	.	PUNCT
cana-4127	156	1	4	4	NUM
cana-4127	156	2	conclusion	conclusion	NOUN
cana-4127	156	3	the	the	DET
cana-4127	156	4	research	research	NOUN
cana-4127	156	5	article	article	NOUN
cana-4127	156	6	titled	title	VERB
cana-4127	156	7	"	"	PUNCT
cana-4127	156	8	analyzing	analyze	VERB
cana-4127	156	9	the	the	DET
cana-4127	156	10	stability	stability	NOUN
cana-4127	156	11	of	of	ADP
cana-4127	156	12	euler	euler	NOUN
cana-4127	156	13	-	-	PUNCT
cana-4127	156	14	lagrange	lagrange	NOUN
cana-4127	156	15	additive	additive	ADJ
cana-4127	156	16	functional	functional	ADJ
cana-4127	156	17	equations	equation	NOUN
cana-4127	156	18	in	in	ADP
cana-4127	156	19	banach	banach	NOUN
cana-4127	156	20	spaces	space	VERB
cana-4127	156	21	:	:	PUNCT
cana-4127	156	22	a	a	DET
cana-4127	156	23	fixed	fix	VERB
cana-4127	156	24	point	point	NOUN
cana-4127	156	25	and	and	CCONJ
cana-4127	156	26	direct	direct	ADJ
cana-4127	156	27	method	method	NOUN
cana-4127	156	28	perspective	perspective	NOUN
cana-4127	156	29	"	"	PUNCT
cana-4127	156	30	investigates	investigate	VERB
cana-4127	156	31	the	the	DET
cana-4127	156	32	stability	stability	NOUN
cana-4127	156	33	of	of	ADP
cana-4127	156	34	specific	specific	ADJ
cana-4127	156	35	functional	functional	ADJ
cana-4127	156	36	equations	equation	NOUN
cana-4127	156	37	within	within	ADP
cana-4127	156	38	banach	banach	NOUN
cana-4127	156	39	spaces	space	NOUN
cana-4127	156	40	.	.	PUNCT
cana-4127	157	1	the	the	DET
cana-4127	157	2	study	study	NOUN
cana-4127	157	3	employs	employ	VERB
cana-4127	157	4	both	both	PRON
cana-4127	157	5	fixed	fix	VERB
cana-4127	157	6	point	point	NOUN
cana-4127	157	7	and	and	CCONJ
cana-4127	157	8	direct	direct	ADJ
cana-4127	157	9	methods	method	NOUN
cana-4127	157	10	to	to	PART
cana-4127	157	11	establish	establish	VERB
cana-4127	157	12	conditions	condition	NOUN
cana-4127	157	13	under	under	ADP
cana-4127	157	14	which	which	PRON
cana-4127	157	15	these	these	DET
cana-4127	157	16	equations	equation	NOUN
cana-4127	157	17	exhibit	exhibit	VERB
cana-4127	157	18	stability	stability	NOUN
cana-4127	157	19	.	.	PUNCT
cana-4127	158	1	the	the	DET
cana-4127	158	2	findings	finding	NOUN
cana-4127	158	3	contribute	contribute	VERB
cana-4127	158	4	to	to	ADP
cana-4127	158	5	a	a	DET
cana-4127	158	6	deeper	deep	ADJ
cana-4127	158	7	understanding	understanding	NOUN
cana-4127	158	8	of	of	ADP
cana-4127	158	9	the	the	DET
cana-4127	158	10	behavior	behavior	NOUN
cana-4127	158	11	of	of	ADP
cana-4127	158	12	euler	euler	NOUN
cana-4127	158	13	-	-	PUNCT
cana-4127	158	14	lagrange	lagrange	NOUN
cana-4127	158	15	additive	additive	ADJ
cana-4127	158	16	functional	functional	ADJ
cana-4127	158	17	equations	equation	NOUN
cana-4127	158	18	in	in	ADP
cana-4127	158	19	the	the	DET
cana-4127	158	20	context	context	NOUN
cana-4127	158	21	of	of	ADP
cana-4127	158	22	banach	banach	NOUN
cana-4127	158	23	spaces	space	NOUN
cana-4127	158	24	,	,	PUNCT
cana-4127	158	25	offering	offer	VERB
cana-4127	158	26	valuable	valuable	ADJ
cana-4127	158	27	insights	insight	NOUN
cana-4127	158	28	for	for	ADP
cana-4127	158	29	further	further	ADJ
cana-4127	158	30	research	research	NOUN
cana-4127	158	31	in	in	ADP
cana-4127	158	32	this	this	DET
cana-4127	158	33	area	area	NOUN
cana-4127	158	34	.	.	PUNCT
cana-4127	159	1	conflict	conflict	NOUN
cana-4127	159	2	of	of	ADP
cana-4127	159	3	interest	interest	NOUN
cana-4127	159	4	.	.	PUNCT
cana-4127	160	1	the	the	DET
cana-4127	160	2	authors	author	NOUN
cana-4127	160	3	declare	declare	VERB
cana-4127	160	4	that	that	SCONJ
cana-4127	160	5	they	they	PRON
cana-4127	160	6	have	have	VERB
cana-4127	160	7	no	no	DET
cana-4127	160	8	competing	compete	VERB
cana-4127	160	9	interests	interest	NOUN
cana-4127	160	10	.	.	PUNCT
cana-4127	161	1	communications	communication	NOUN
cana-4127	161	2	on	on	ADP
cana-4127	161	3	applied	apply	VERB
cana-4127	161	4	nonlinear	nonlinear	ADJ
cana-4127	161	5	analysis	analysis	NOUN
cana-4127	161	6	issn	issn	NOUN
cana-4127	161	7	:	:	PUNCT
cana-4127	161	8	1074	1074	NUM
cana-4127	161	9	-	-	PUNCT
cana-4127	161	10	133x	133x	NUM
cana-4127	161	11	vol	vol	NOUN
cana-4127	161	12	32	32	NUM
cana-4127	161	13	no	no	NOUN
cana-4127	161	14	.	.	PUNCT
cana-4127	162	1	9s	9s	NUM
cana-4127	162	2	(	(	PUNCT
cana-4127	162	3	2025	2025	NUM
cana-4127	162	4	)	)	PUNCT
cana-4127	162	5	1222	1222	NUM
cana-4127	162	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	162	7	references	reference	NOUN
cana-4127	162	8	[	[	X
cana-4127	162	9	1	1	NUM
cana-4127	162	10	]	]	X
cana-4127	162	11	s.m	s.m	PROPN
cana-4127	162	12	.	.	PROPN
cana-4127	162	13	ulam	ulam	PROPN
cana-4127	162	14	,	,	PUNCT
cana-4127	162	15	problems	problem	NOUN
cana-4127	162	16	in	in	ADP
cana-4127	162	17	modern	modern	ADJ
cana-4127	162	18	mathematics	mathematic	NOUN
cana-4127	162	19	,	,	PUNCT
cana-4127	162	20	science	science	NOUN
cana-4127	162	21	editions	edition	NOUN
cana-4127	162	22	,	,	PUNCT
cana-4127	162	23	wiley	wiley	NOUN
cana-4127	162	24	,	,	PUNCT
cana-4127	162	25	newyork	newyork	PROPN
cana-4127	162	26	,	,	PUNCT
cana-4127	162	27	1964	1964	NUM
cana-4127	162	28	(	(	PUNCT
cana-4127	162	29	chapter	chapter	NOUN
cana-4127	162	30	vi	vi	PROPN
cana-4127	162	31	,	,	PUNCT
cana-4127	162	32	some	some	DET
cana-4127	162	33	questions	question	NOUN
cana-4127	162	34	in	in	ADP
cana-4127	162	35	analysis	analysis	NOUN
cana-4127	162	36	:	:	PUNCT
cana-4127	162	37	1	1	NUM
cana-4127	162	38	,	,	PUNCT
cana-4127	162	39	stability	stability	NOUN
cana-4127	162	40	)	)	PUNCT
cana-4127	162	41	.	.	PUNCT
cana-4127	163	1	[	[	X
cana-4127	163	2	2	2	NUM
cana-4127	163	3	]	]	X
cana-4127	163	4	d.h	d.h	PROPN
cana-4127	163	5	.	.	PROPN
cana-4127	163	6	hyers	hyer	NOUN
cana-4127	163	7	,	,	PUNCT
cana-4127	163	8	on	on	ADP
cana-4127	163	9	the	the	DET
cana-4127	163	10	stability	stability	NOUN
cana-4127	163	11	of	of	ADP
cana-4127	163	12	the	the	DET
cana-4127	163	13	linear	linear	ADJ
cana-4127	163	14	functional	functional	ADJ
cana-4127	163	15	equation	equation	NOUN
cana-4127	163	16	,	,	PUNCT
cana-4127	163	17	proc.nat	proc.nat	PROPN
cana-4127	163	18	.	.	PUNCT
cana-4127	164	1	acad.sci	acad.sci	X
cana-4127	164	2	.	.	PUNCT
cana-4127	164	3	,u.s.a	,u.s.a	PROPN
cana-4127	164	4	.	.	PUNCT
cana-4127	164	5	,27	,27	PUNCT
cana-4127	164	6	(	(	PUNCT
cana-4127	164	7	1941	1941	NUM
cana-4127	164	8	)	)	PUNCT
cana-4127	164	9	222	222	NUM
cana-4127	164	10	-	-	SYM
cana-4127	164	11	224	224	NUM
cana-4127	164	12	.	.	PUNCT
cana-4127	165	1	[	[	X
cana-4127	165	2	3	3	X
cana-4127	165	3	]	]	PUNCT
cana-4127	165	4	t.	t.	PROPN
cana-4127	165	5	aoki	aoki	PROPN
cana-4127	165	6	,	,	PUNCT
cana-4127	165	7	on	on	ADP
cana-4127	165	8	the	the	DET
cana-4127	165	9	stability	stability	NOUN
cana-4127	165	10	of	of	ADP
cana-4127	165	11	the	the	DET
cana-4127	165	12	linear	linear	ADJ
cana-4127	165	13	transformation	transformation	NOUN
cana-4127	165	14	in	in	ADP
cana-4127	165	15	banach	banach	NOUN
cana-4127	165	16	spaces	space	NOUN
cana-4127	165	17	,	,	PUNCT
cana-4127	165	18	j.	j.	PROPN
cana-4127	165	19	math	math	PROPN
cana-4127	165	20	.	.	PUNCT
cana-4127	166	1	soc	soc	PROPN
cana-4127	166	2	.	.	PUNCT
cana-4127	167	1	japan	japan	PROPN
cana-4127	167	2	,	,	PUNCT
cana-4127	167	3	2	2	NUM
cana-4127	167	4	(	(	PUNCT
cana-4127	167	5	1950	1950	NUM
cana-4127	167	6	)	)	PUNCT
cana-4127	167	7	,	,	PUNCT
cana-4127	167	8	64	64	NUM
cana-4127	167	9	-	-	SYM
cana-4127	167	10	66	66	NUM
cana-4127	167	11	.	.	PUNCT
cana-4127	168	1	[	[	X
cana-4127	168	2	4	4	NUM
cana-4127	168	3	]	]	X
cana-4127	168	4	p.	p.	NOUN
cana-4127	168	5	găvrută	găvrută	NOUN
cana-4127	168	6	,	,	PUNCT
cana-4127	168	7	an	an	DET
cana-4127	168	8	answer	answer	NOUN
cana-4127	168	9	to	to	ADP
cana-4127	168	10	a	a	DET
cana-4127	168	11	question	question	NOUN
cana-4127	168	12	of	of	ADP
cana-4127	168	13	j.m.rassias	j.m.rassia	NOUN
cana-4127	168	14	concerning	concern	VERB
cana-4127	168	15	the	the	DET
cana-4127	168	16	stability	stability	NOUN
cana-4127	168	17	of	of	ADP
cana-4127	168	18	cauchy	cauchy	ADJ
cana-4127	168	19	functional	functional	ADJ
cana-4127	168	20	equation	equation	NOUN
cana-4127	168	21	,	,	PUNCT
cana-4127	168	22	advances	advance	NOUN
cana-4127	168	23	in	in	ADP
cana-4127	168	24	equations	equation	NOUN
cana-4127	168	25	and	and	CCONJ
cana-4127	168	26	inequalities	inequality	NOUN
cana-4127	168	27	,	,	PUNCT
cana-4127	168	28	hadronic	hadronic	ADJ
cana-4127	168	29	math	math	NOUN
cana-4127	168	30	.	.	PUNCT
cana-4127	169	1	ser	ser	PROPN
cana-4127	169	2	.	.	PROPN
cana-4127	169	3	,	,	PUNCT
cana-4127	169	4	(	(	PUNCT
cana-4127	169	5	1999	1999	NUM
cana-4127	169	6	)	)	PUNCT
cana-4127	169	7	,	,	PUNCT
cana-4127	169	8	67	67	NUM
cana-4127	169	9	-	-	SYM
cana-4127	169	10	71	71	NUM
cana-4127	169	11	.	.	PUNCT
cana-4127	170	1	[	[	X
cana-4127	170	2	5	5	NUM
cana-4127	170	3	]	]	PUNCT
cana-4127	170	4	p.	p.	NOUN
cana-4127	170	5	găvrută	găvrută	NOUN
cana-4127	170	6	,	,	PUNCT
cana-4127	170	7	on	on	ADP
cana-4127	170	8	a	a	DET
cana-4127	170	9	problem	problem	NOUN
cana-4127	170	10	of	of	ADP
cana-4127	170	11	g.	g.	PROPN
cana-4127	170	12	isac	isac	PROPN
cana-4127	170	13	and	and	CCONJ
cana-4127	170	14	th	th	PROPN
cana-4127	170	15	.	.	PUNCT
cana-4127	171	1	m.	m.	NOUN
cana-4127	171	2	rassias	rassias	PROPN
cana-4127	171	3	concerning	concern	VERB
cana-4127	171	4	the	the	DET
cana-4127	171	5	stability	stability	NOUN
cana-4127	171	6	of	of	ADP
cana-4127	171	7	mappings	mapping	NOUN
cana-4127	171	8	,	,	PUNCT
cana-4127	171	9	j.	j.	PROPN
cana-4127	171	10	math	math	PROPN
cana-4127	171	11	.	.	PUNCT
cana-4127	172	1	anal	anal	PROPN
cana-4127	172	2	.	.	PUNCT
cana-4127	173	1	appl	appl	PROPN
cana-4127	173	2	.	.	PUNCT
cana-4127	174	1	261	261	NUM
cana-4127	174	2	(	(	PUNCT
cana-4127	174	3	2001	2001	NUM
cana-4127	174	4	)	)	PUNCT
cana-4127	174	5	,	,	PUNCT
cana-4127	174	6	543	543	NUM
cana-4127	174	7	-	-	SYM
cana-4127	174	8	553	553	NUM
cana-4127	174	9	.	.	PUNCT
cana-4127	175	1	[	[	X
cana-4127	175	2	6	6	NUM
cana-4127	175	3	]	]	X
cana-4127	175	4	j.m	j.m	PROPN
cana-4127	175	5	.	.	PROPN
cana-4127	175	6	rassias	rassias	PROPN
cana-4127	175	7	,	,	PUNCT
cana-4127	175	8	on	on	ADP
cana-4127	175	9	approximately	approximately	ADV
cana-4127	175	10	of	of	ADP
cana-4127	175	11	approximately	approximately	ADV
cana-4127	175	12	linear	linear	ADJ
cana-4127	175	13	mappings	mapping	NOUN
cana-4127	175	14	by	by	ADP
cana-4127	175	15	linear	linear	PROPN
cana-4127	175	16	mappings	mapping	NOUN
cana-4127	175	17	,	,	PUNCT
cana-4127	175	18	j.	j.	PROPN
cana-4127	175	19	funct	funct	PROPN
cana-4127	175	20	.	.	PUNCT
cana-4127	176	1	anal	anal	PROPN
cana-4127	176	2	.	.	PUNCT
cana-4127	177	1	usa	usa	PROPN
cana-4127	177	2	,	,	PUNCT
cana-4127	177	3	46	46	NUM
cana-4127	177	4	,	,	PUNCT
cana-4127	177	5	(	(	PUNCT
cana-4127	177	6	1982	1982	NUM
cana-4127	177	7	)	)	PUNCT
cana-4127	177	8	126	126	NUM
cana-4127	177	9	-	-	SYM
cana-4127	177	10	130	130	NUM
cana-4127	177	11	.	.	PUNCT
cana-4127	178	1	[	[	X
cana-4127	178	2	7	7	X
cana-4127	178	3	]	]	X
cana-4127	178	4	j.m	j.m	PROPN
cana-4127	178	5	.	.	PROPN
cana-4127	178	6	rassias	rassias	PROPN
cana-4127	178	7	,	,	PUNCT
cana-4127	178	8	on	on	ADP
cana-4127	178	9	approximately	approximately	ADV
cana-4127	178	10	of	of	ADP
cana-4127	178	11	approximately	approximately	ADV
cana-4127	178	12	linear	linear	ADJ
cana-4127	178	13	mappings	mapping	NOUN
cana-4127	178	14	by	by	ADP
cana-4127	178	15	linear	linear	ADJ
cana-4127	178	16	mappings	mapping	NOUN
cana-4127	178	17	,	,	PUNCT
cana-4127	178	18	bull	bull	NOUN
cana-4127	178	19	.	.	PUNCT
cana-4127	179	1	sc	sc	PROPN
cana-4127	179	2	.	.	PROPN
cana-4127	179	3	math	math	PROPN
cana-4127	179	4	,	,	PUNCT
cana-4127	179	5	108	108	NUM
cana-4127	179	6	,	,	PUNCT
cana-4127	179	7	(	(	PUNCT
cana-4127	179	8	1984	1984	NUM
cana-4127	179	9	)	)	PUNCT
cana-4127	179	10	445	445	NUM
cana-4127	179	11	-	-	SYM
cana-4127	179	12	446	446	NUM
cana-4127	179	13	.	.	PUNCT
cana-4127	180	1	[	[	X
cana-4127	180	2	8	8	NUM
cana-4127	180	3	]	]	X
cana-4127	180	4	j.m	j.m	PROPN
cana-4127	180	5	.	.	PROPN
cana-4127	180	6	rassias	rassias	PROPN
cana-4127	180	7	,	,	PUNCT
cana-4127	180	8	k.w	k.w	PROPN
cana-4127	180	9	.	.	PROPN
cana-4127	180	10	jun	jun	PROPN
cana-4127	180	11	,	,	PUNCT
cana-4127	180	12	h.m	h.m	PROPN
cana-4127	180	13	.	.	PROPN
cana-4127	180	14	kim	kim	PROPN
cana-4127	180	15	,	,	PUNCT
cana-4127	180	16	approximate	approximate	ADJ
cana-4127	180	17	(	(	PUNCT
cana-4127	180	18	m	m	PROPN
cana-4127	180	19	,	,	PUNCT
cana-4127	180	20	n	n	CCONJ
cana-4127	180	21	)	)	PUNCT
cana-4127	180	22	−cauchy	−cauchy	ADJ
cana-4127	180	23	jensen	jensen	PROPN
cana-4127	180	24	additive	additive	ADJ
cana-4127	180	25	mappings	mapping	NOUN
cana-4127	180	26	in	in	ADP
cana-4127	180	27	c	c	PROPN
cana-4127	180	28	∗-algebras	∗-algebra	NOUN
cana-4127	180	29	,	,	PUNCT
cana-4127	180	30	acta	acta	PROPN
cana-4127	180	31	mathematica	mathematica	PROPN
cana-4127	180	32	sinica	sinica	PROPN
cana-4127	180	33	,	,	PUNCT
cana-4127	180	34	english	english	ADJ
cana-4127	180	35	series	series	NOUN
cana-4127	180	36	,	,	PUNCT
cana-4127	180	37	vol	vol	NOUN
cana-4127	180	38	.	.	PROPN
cana-4127	180	39	27	27	NUM
cana-4127	180	40	,	,	PUNCT
cana-4127	180	41	no	no	INTJ
cana-4127	180	42	.	.	NOUN
cana-4127	180	43	10	10	NUM
cana-4127	180	44	,	,	PUNCT
cana-4127	180	45	(	(	PUNCT
cana-4127	180	46	2011	2011	NUM
cana-4127	180	47	)	)	PUNCT
cana-4127	180	48	,	,	PUNCT
cana-4127	180	49	1907	1907	NUM
cana-4127	180	50	-	-	SYM
cana-4127	180	51	1922	1922	NUM
cana-4127	180	52	.	.	PUNCT
cana-4127	181	1	[	[	X
cana-4127	181	2	9	9	NUM
cana-4127	181	3	]	]	SYM
cana-4127	181	4	rus	rus	NOUN
cana-4127	181	5	,	,	PUNCT
cana-4127	181	6	ioan	ioan	PROPN
cana-4127	181	7	a.	a.	PROPN
cana-4127	181	8	"	"	PUNCT
cana-4127	181	9	ulam	ulam	X
cana-4127	181	10	stability	stability	NOUN
cana-4127	181	11	of	of	ADP
cana-4127	181	12	ordinary	ordinary	ADJ
cana-4127	181	13	differential	differential	ADJ
cana-4127	181	14	equation	equation	NOUN
cana-4127	181	15	.	.	PUNCT
cana-4127	181	16	"	"	PUNCT
cana-4127	182	1	studia	studia	PROPN
cana-4127	182	2	universitatis	universitatis	PROPN
cana-4127	182	3	babesbolyai	babesbolyai	PROPN
cana-4127	182	4	,	,	PUNCT
cana-4127	182	5	mathematica	mathematica	PROPN
cana-4127	182	6	4	4	NUM
cana-4127	182	7	(	(	PUNCT
cana-4127	182	8	2009	2009	NUM
cana-4127	182	9	)	)	PUNCT
cana-4127	182	10	.	.	PUNCT
cana-4127	183	1	[	[	X
cana-4127	183	2	10	10	NUM
cana-4127	183	3	]	]	PUNCT
cana-4127	183	4	kumama	kumama	NOUN
cana-4127	183	5	,	,	PUNCT
cana-4127	183	6	poom	poom	NOUN
cana-4127	183	7	,	,	PUNCT
cana-4127	183	8	amjad	amjad	PROPN
cana-4127	183	9	ali	ali	PROPN
cana-4127	183	10	,	,	PUNCT
cana-4127	183	11	kamal	kamal	PROPN
cana-4127	183	12	shah	shah	PROPN
cana-4127	183	13	,	,	PUNCT
cana-4127	183	14	and	and	CCONJ
cana-4127	183	15	rahmat	rahmat	PROPN
cana-4127	183	16	ali	ali	PROPN
cana-4127	183	17	khan	khan	PROPN
cana-4127	183	18	.	.	PUNCT
cana-4127	184	1	"	"	PUNCT
cana-4127	184	2	existence	existence	NOUN
cana-4127	184	3	results	result	NOUN
cana-4127	184	4	and	and	CCONJ
cana-4127	184	5	hyers	hyer	NOUN
cana-4127	184	6	–	–	PUNCT
cana-4127	184	7	ulam	ulam	PROPN
cana-4127	184	8	stability	stability	NOUN
cana-4127	184	9	to	to	ADP
cana-4127	184	10	a	a	DET
cana-4127	184	11	class	class	NOUN
cana-4127	184	12	of	of	ADP
cana-4127	184	13	nonlinear	nonlinear	ADJ
cana-4127	184	14	arbitrary	arbitrary	ADJ
cana-4127	184	15	order	order	NOUN
cana-4127	184	16	differential	differential	NOUN
cana-4127	184	17	equations	equation	NOUN
cana-4127	184	18	.	.	PUNCT
cana-4127	184	19	"	"	PUNCT
cana-4127	185	1	j.	j.	PROPN
cana-4127	185	2	nonlinear	nonlinear	PROPN
cana-4127	185	3	sci	sci	PROPN
cana-4127	185	4	.	.	PUNCT
cana-4127	185	5	appl	appl	PROPN
cana-4127	185	6	10	10	NUM
cana-4127	185	7	,	,	PUNCT
cana-4127	185	8	no	no	INTJ
cana-4127	185	9	.	.	NOUN
cana-4127	185	10	6	6	NUM
cana-4127	185	11	(	(	PUNCT
cana-4127	185	12	2017	2017	NUM
cana-4127	185	13	):	):	PUNCT
cana-4127	185	14	2986	2986	NUM
cana-4127	185	15	-	-	SYM
cana-4127	185	16	2997	2997	NUM
cana-4127	185	17	.	.	PUNCT
cana-4127	186	1	[	[	X
cana-4127	186	2	11	11	NUM
cana-4127	186	3	]	]	X
cana-4127	186	4	agarwal	agarwal	PROPN
cana-4127	186	5	,	,	PUNCT
cana-4127	186	6	ravi	ravi	PROPN
cana-4127	186	7	p.	p.	PROPN
cana-4127	186	8	,	,	PUNCT
cana-4127	186	9	snezhana	snezhana	PROPN
cana-4127	186	10	hristova	hristova	PROPN
cana-4127	186	11	,	,	PUNCT
cana-4127	186	12	and	and	CCONJ
cana-4127	186	13	donal	donal	PROPN
cana-4127	186	14	o’regan	o’regan	PROPN
cana-4127	186	15	.	.	PUNCT
cana-4127	187	1	"	"	PUNCT
cana-4127	187	2	ulam	ulam	PROPN
cana-4127	187	3	stability	stability	NOUN
cana-4127	187	4	for	for	ADP
cana-4127	187	5	boundary	boundary	ADJ
cana-4127	187	6	value	value	NOUN
cana-4127	187	7	problems	problem	NOUN
cana-4127	187	8	of	of	ADP
cana-4127	187	9	differential	differential	ADJ
cana-4127	187	10	equations	equation	NOUN
cana-4127	187	11	—	—	PUNCT
cana-4127	187	12	main	main	ADJ
cana-4127	187	13	misunderstandings	misunderstanding	NOUN
cana-4127	187	14	and	and	CCONJ
cana-4127	187	15	how	how	SCONJ
cana-4127	187	16	to	to	PART
cana-4127	187	17	avoid	avoid	VERB
cana-4127	187	18	them	they	PRON
cana-4127	187	19	.	.	PUNCT
cana-4127	187	20	"	"	PUNCT
cana-4127	188	1	mathematics	mathematic	NOUN
cana-4127	188	2	12	12	NUM
cana-4127	188	3	,	,	PUNCT
cana-4127	188	4	no	no	INTJ
cana-4127	188	5	.	.	NOUN
cana-4127	188	6	11	11	NUM
cana-4127	188	7	(	(	PUNCT
cana-4127	188	8	2024	2024	NUM
cana-4127	188	9	):	):	PUNCT
cana-4127	188	10	1626	1626	NUM
cana-4127	188	11	.	.	PUNCT
cana-4127	189	1	[	[	X
cana-4127	189	2	12	12	NUM
cana-4127	189	3	]	]	PUNCT
cana-4127	189	4	baias	baia	NOUN
cana-4127	189	5	,	,	PUNCT
cana-4127	189	6	alina	alina	PROPN
cana-4127	189	7	ramona	ramona	PROPN
cana-4127	189	8	,	,	PUNCT
cana-4127	189	9	and	and	CCONJ
cana-4127	189	10	dorian	dorian	PROPN
cana-4127	189	11	popa	popa	NOUN
cana-4127	189	12	.	.	PUNCT
cana-4127	190	1	"	"	PUNCT
cana-4127	190	2	on	on	ADP
cana-4127	190	3	ulam	ulam	PROPN
cana-4127	190	4	stability	stability	NOUN
cana-4127	190	5	of	of	ADP
cana-4127	190	6	a	a	DET
cana-4127	190	7	linear	linear	ADJ
cana-4127	190	8	difference	difference	NOUN
cana-4127	190	9	equation	equation	NOUN
cana-4127	190	10	in	in	ADP
cana-4127	190	11	banach	banach	NOUN
cana-4127	190	12	spaces	space	NOUN
cana-4127	190	13	.	.	PUNCT
cana-4127	190	14	"	"	PUNCT
cana-4127	190	15	bulletin	bulletin	NOUN
cana-4127	190	16	of	of	ADP
cana-4127	190	17	the	the	DET
cana-4127	190	18	malaysian	malaysian	PROPN
cana-4127	190	19	mathematical	mathematical	PROPN
cana-4127	190	20	sciences	sciences	PROPN
cana-4127	190	21	society	society	NOUN
cana-4127	190	22	43	43	NUM
cana-4127	190	23	,	,	PUNCT
cana-4127	190	24	no	no	INTJ
cana-4127	190	25	.	.	NOUN
cana-4127	190	26	2	2	NUM
cana-4127	190	27	(	(	PUNCT
cana-4127	190	28	2020	2020	NUM
cana-4127	190	29	):	):	PUNCT
cana-4127	190	30	13571371	13571371	NUM
cana-4127	190	31	.	.	PUNCT
cana-4127	191	1	[	[	X
cana-4127	191	2	13	13	NUM
cana-4127	191	3	]	]	X
cana-4127	191	4	tripathy	tripathy	ADJ
cana-4127	191	5	,	,	PUNCT
cana-4127	191	6	a.k	a.k	PROPN
cana-4127	191	7	.	.	PROPN
cana-4127	191	8	,	,	PUNCT
cana-4127	191	9	2021	2021	NUM
cana-4127	191	10	.	.	PUNCT
cana-4127	192	1	hyers	hyer	NOUN
cana-4127	192	2	-	-	PUNCT
cana-4127	192	3	ulam	ulam	PROPN
cana-4127	192	4	stability	stability	NOUN
cana-4127	192	5	of	of	ADP
cana-4127	192	6	ordinary	ordinary	ADJ
cana-4127	192	7	differential	differential	ADJ
cana-4127	192	8	equations	equation	NOUN
cana-4127	192	9	.	.	PUNCT
cana-4127	193	1	chapman	chapman	NOUN
cana-4127	193	2	and	and	CCONJ
cana-4127	193	3	hall	hall	PROPN
cana-4127	193	4	/	/	SYM
cana-4127	193	5	crc	crc	NOUN
cana-4127	193	6	.	.	PUNCT
cana-4127	194	1	[	[	X
cana-4127	194	2	14	14	NUM
cana-4127	194	3	]	]	X
cana-4127	194	4	pasupathi	pasupathi	NOUN
cana-4127	194	5	,	,	PUNCT
cana-4127	194	6	a.	a.	NOUN
cana-4127	194	7	;	;	PUNCT
cana-4127	194	8	konsalraj	konsalraj	PROPN
cana-4127	194	9	,	,	PUNCT
cana-4127	194	10	j.	j.	PROPN
cana-4127	194	11	;	;	PUNCT
cana-4127	194	12	fatima	fatima	PROPN
cana-4127	194	13	,	,	PUNCT
cana-4127	194	14	n.	n.	NOUN
cana-4127	194	15	;	;	PUNCT
cana-4127	194	16	velusamy	velusamy	PROPN
cana-4127	194	17	,	,	PUNCT
cana-4127	194	18	v.	v.	PROPN
cana-4127	194	19	;	;	PUNCT
cana-4127	194	20	mlaiki	mlaiki	PROPN
cana-4127	194	21	,	,	PUNCT
cana-4127	194	22	n.	n.	NOUN
cana-4127	194	23	;	;	PUNCT
cana-4127	194	24	souayah	souayah	NOUN
cana-4127	194	25	,	,	PUNCT
cana-4127	194	26	n.	n.	NOUN
cana-4127	194	27	direct	direct	ADJ
cana-4127	194	28	and	and	CCONJ
cana-4127	194	29	fixedpoint	fixedpoint	NOUN
cana-4127	194	30	stability	stability	NOUN
cana-4127	194	31	–	–	PUNCT
cana-4127	194	32	instability	instability	NOUN
cana-4127	194	33	of	of	ADP
cana-4127	194	34	additive	additive	ADJ
cana-4127	194	35	functional	functional	ADJ
cana-4127	194	36	equation	equation	NOUN
cana-4127	194	37	in	in	ADP
cana-4127	194	38	banach	banach	NOUN
cana-4127	194	39	and	and	CCONJ
cana-4127	194	40	quasi	quasi	ADJ
cana-4127	194	41	-	-	ADJ
cana-4127	194	42	beta	beta	ADJ
cana-4127	194	43	normed	norme	VERB
cana-4127	194	44	spaces	space	NOUN
cana-4127	194	45	.	.	PUNCT
cana-4127	195	1	symmetry	symmetry	NOUN
cana-4127	195	2	2022	2022	NUM
cana-4127	195	3	,	,	PUNCT
cana-4127	195	4	14	14	NUM
cana-4127	195	5	,	,	PUNCT
cana-4127	195	6	1700	1700	NUM
cana-4127	195	7	.	.	PUNCT
cana-4127	196	1	[	[	X
cana-4127	196	2	15	15	NUM
cana-4127	196	3	]	]	X
cana-4127	196	4	agilan	agilan	ADJ
cana-4127	196	5	,	,	PUNCT
cana-4127	196	6	p.	p.	NOUN
cana-4127	196	7	;	;	PUNCT
cana-4127	196	8	julietraja	julietraja	PROPN
cana-4127	196	9	,	,	PUNCT
cana-4127	196	10	k.	k.	PROPN
cana-4127	196	11	;	;	PUNCT
cana-4127	196	12	mlaiki	mlaiki	PROPN
cana-4127	196	13	,	,	PUNCT
cana-4127	196	14	n.	n.	NOUN
cana-4127	196	15	;	;	PUNCT
cana-4127	196	16	mukheimer	mukheimer	NOUN
cana-4127	196	17	,	,	PUNCT
cana-4127	196	18	a.	a.	NOUN
cana-4127	196	19	intuitionistic	intuitionistic	ADJ
cana-4127	196	20	fuzzy	fuzzy	ADJ
cana-4127	196	21	stability	stability	NOUN
cana-4127	196	22	of	of	ADP
cana-4127	196	23	an	an	DET
cana-4127	196	24	euler	euler	NOUN
cana-4127	196	25	–	–	PUNCT
cana-4127	196	26	lagrange	lagrange	NOUN
cana-4127	196	27	symmetry	symmetry	NOUN
cana-4127	196	28	additive	additive	ADJ
cana-4127	196	29	functional	functional	ADJ
cana-4127	196	30	equation	equation	NOUN
cana-4127	196	31	via	via	ADP
cana-4127	196	32	direct	direct	ADJ
cana-4127	196	33	and	and	CCONJ
cana-4127	196	34	fixed	fix	VERB
cana-4127	196	35	point	point	NOUN
cana-4127	196	36	technique	technique	NOUN
cana-4127	196	37	(	(	PUNCT
cana-4127	196	38	fpt	fpt	PROPN
cana-4127	196	39	)	)	PUNCT
cana-4127	196	40	.	.	PUNCT
cana-4127	197	1	symmetry	symmetry	PROPN
cana-4127	197	2	2022	2022	NUM
cana-4127	197	3	,	,	PUNCT
cana-4127	197	4	14	14	NUM
cana-4127	197	5	,	,	PUNCT
cana-4127	197	6	2454	2454	NUM
cana-4127	197	7	.	.	PUNCT
cana-4127	198	1	communications	communication	NOUN
cana-4127	198	2	on	on	ADP
cana-4127	198	3	applied	apply	VERB
cana-4127	198	4	nonlinear	nonlinear	ADJ
cana-4127	198	5	analysis	analysis	NOUN
cana-4127	198	6	issn	issn	NOUN
cana-4127	198	7	:	:	PUNCT
cana-4127	198	8	1074	1074	NUM
cana-4127	198	9	-	-	PUNCT
cana-4127	198	10	133x	133x	NUM
cana-4127	198	11	vol	vol	NOUN
cana-4127	198	12	32	32	NUM
cana-4127	198	13	no	no	NOUN
cana-4127	198	14	.	.	PUNCT
cana-4127	199	1	9s	9s	NUM
cana-4127	199	2	(	(	PUNCT
cana-4127	199	3	2025	2025	NUM
cana-4127	199	4	)	)	PUNCT
cana-4127	199	5	1223	1223	NUM
cana-4127	199	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4127	200	1	[	[	X
cana-4127	200	2	16	16	NUM
cana-4127	200	3	]	]	X
cana-4127	200	4	agilan	agilan	ADJ
cana-4127	200	5	,	,	PUNCT
cana-4127	200	6	p.	p.	NOUN
cana-4127	200	7	;	;	PUNCT
cana-4127	200	8	almazah	almazah	PROPN
cana-4127	200	9	,	,	PUNCT
cana-4127	200	10	m.a.a	m.a.a	PROPN
cana-4127	200	11	.	.	PUNCT
cana-4127	200	12	;	;	PUNCT
cana-4127	200	13	julietraja	julietraja	PROPN
cana-4127	200	14	,	,	PUNCT
cana-4127	200	15	k.	k.	PROPN
cana-4127	200	16	;	;	PUNCT
cana-4127	200	17	alsinai	alsinai	PROPN
cana-4127	200	18	,	,	PUNCT
cana-4127	200	19	a.	a.	NOUN
cana-4127	200	20	classical	classical	NOUN
cana-4127	200	21	and	and	CCONJ
cana-4127	200	22	fixed	fix	VERB
cana-4127	200	23	point	point	NOUN
cana-4127	200	24	approach	approach	NOUN
cana-4127	200	25	to	to	ADP
cana-4127	200	26	the	the	DET
cana-4127	200	27	stability	stability	NOUN
cana-4127	200	28	analysis	analysis	NOUN
cana-4127	200	29	of	of	ADP
cana-4127	200	30	a	a	DET
cana-4127	200	31	bilateral	bilateral	ADJ
cana-4127	200	32	symmetric	symmetric	ADJ
cana-4127	200	33	additive	additive	ADJ
cana-4127	200	34	functional	functional	ADJ
cana-4127	200	35	equation	equation	NOUN
cana-4127	200	36	in	in	ADP
cana-4127	200	37	fuzzy	fuzzy	ADJ
cana-4127	200	38	and	and	CCONJ
cana-4127	200	39	random	random	ADJ
cana-4127	200	40	normed	normed	ADJ
cana-4127	200	41	spaces	space	NOUN
cana-4127	200	42	.	.	PUNCT
cana-4127	201	1	mathematics	mathematic	NOUN
cana-4127	201	2	2023	2023	NUM
cana-4127	201	3	,	,	PUNCT
cana-4127	201	4	11	11	NUM
cana-4127	201	5	,	,	PUNCT
cana-4127	201	6	681	681	NUM
cana-4127	201	7	.	.	PUNCT
cana-4127	202	1	[	[	X
cana-4127	202	2	17	17	NUM
cana-4127	202	3	]	]	SYM
cana-4127	202	4	agilan	agilan	ADJ
cana-4127	202	5	,	,	PUNCT
cana-4127	202	6	p.	p.	NOUN
cana-4127	202	7	;	;	PUNCT
cana-4127	202	8	julietraja	julietraja	PROPN
cana-4127	202	9	.	.	PUNCT
cana-4127	202	10	;	;	PUNCT
cana-4127	202	11	k	k	PROPN
cana-4127	202	12	almazah	almazah	PROPN
cana-4127	202	13	,	,	PUNCT
cana-4127	202	14	m.a.a	m.a.a	PROPN
cana-4127	202	15	.	.	PUNCT
cana-4127	202	16	;	;	PUNCT
cana-4127	203	1	alsinai	alsinai	PROPN
cana-4127	203	2	,	,	PUNCT
cana-4127	203	3	a.	a.	NOUN
cana-4127	203	4	stability	stability	NOUN
cana-4127	203	5	analysis	analysis	NOUN
cana-4127	203	6	of	of	ADP
cana-4127	203	7	a	a	DET
cana-4127	203	8	new	new	ADJ
cana-4127	203	9	class	class	NOUN
cana-4127	203	10	of	of	ADP
cana-4127	203	11	series	series	NOUN
cana-4127	203	12	type	type	NOUN
cana-4127	203	13	additive	additive	ADJ
cana-4127	203	14	functional	functional	ADJ
cana-4127	203	15	equation	equation	NOUN
cana-4127	203	16	in	in	ADP
cana-4127	203	17	banach	banach	NOUN
cana-4127	203	18	spaces	space	NOUN
cana-4127	203	19	:	:	PUNCT
cana-4127	203	20	direct	direct	ADJ
cana-4127	203	21	and	and	CCONJ
cana-4127	203	22	fixed	fix	VERB
cana-4127	203	23	point	point	NOUN
cana-4127	203	24	techniques	technique	NOUN
cana-4127	203	25	mathematics	mathematic	NOUN
cana-4127	203	26	2023	2023	NUM
cana-4127	203	27	,	,	PUNCT
cana-4127	203	28	11	11	NUM
cana-4127	203	29	,	,	PUNCT
cana-4127	203	30	887	887	NUM
cana-4127	203	31	.	.	PUNCT
cana-4127	203	32	doi.org/10.3390/math11040887	doi.org/10.3390/math11040887	VERB
cana-4127	203	33	.	.	PUNCT
cana-4127	204	1	[	[	X
cana-4127	204	2	18	18	NUM
cana-4127	204	3	]	]	PUNCT
cana-4127	204	4	aloqaily	aloqaily	ADV
cana-4127	204	5	,	,	PUNCT
cana-4127	204	6	ahmad	ahmad	PROPN
cana-4127	204	7	,	,	PUNCT
cana-4127	204	8	p.	p.	PROPN
cana-4127	204	9	agilan	agilan	PROPN
cana-4127	204	10	,	,	PUNCT
cana-4127	204	11	k.	k.	PROPN
cana-4127	204	12	julietraja	julietraja	PROPN
cana-4127	204	13	,	,	PUNCT
cana-4127	204	14	s.	s.	PROPN
cana-4127	204	15	annadurai	annadurai	PROPN
cana-4127	204	16	,	,	PUNCT
cana-4127	204	17	and	and	CCONJ
cana-4127	204	18	nabil	nabil	PROPN
cana-4127	204	19	mlaiki	mlaiki	PROPN
cana-4127	204	20	.	.	PUNCT
cana-4127	205	1	a	a	DET
cana-4127	205	2	novel	novel	ADJ
cana-4127	205	3	stability	stability	NOUN
cana-4127	205	4	analysis	analysis	NOUN
cana-4127	205	5	of	of	ADP
cana-4127	205	6	functional	functional	ADJ
cana-4127	205	7	equation	equation	NOUN
cana-4127	205	8	in	in	ADP
cana-4127	205	9	neutrosophic	neutrosophic	ADJ
cana-4127	205	10	normed	norme	VERB
cana-4127	205	11	spaces	space	NOUN
cana-4127	205	12	.	.	PUNCT
cana-4127	206	1	boundary	boundary	ADJ
cana-4127	206	2	value	value	NOUN
cana-4127	206	3	problems	problem	NOUN
cana-4127	206	4	,	,	PUNCT
cana-4127	206	5	2024	2024	NUM
cana-4127	206	6	,	,	PUNCT
cana-4127	206	7	no	no	INTJ
cana-4127	206	8	.	.	NOUN
cana-4127	206	9	1	1	NUM
cana-4127	206	10	(	(	PUNCT
cana-4127	206	11	2024	2024	NUM
cana-4127	206	12	)	)	PUNCT
cana-4127	206	13	,	,	PUNCT
cana-4127	206	14	47	47	NUM
cana-4127	206	15	.	.	PUNCT
cana-4127	207	1	[	[	X
cana-4127	207	2	19	19	NUM
cana-4127	207	3	]	]	SYM
cana-4127	207	4	agilan	agilan	ADJ
cana-4127	207	5	,	,	PUNCT
cana-4127	207	6	p.	p.	PROPN
cana-4127	207	7	,	,	PUNCT
cana-4127	207	8	julietraja	julietraja	PROPN
cana-4127	207	9	,	,	PUNCT
cana-4127	207	10	k.	k.	PROPN
cana-4127	207	11	,	,	PUNCT
cana-4127	207	12	kanimozhi	kanimozhi	PROPN
cana-4127	207	13	,	,	PUNCT
cana-4127	207	14	b.	b.	PROPN
cana-4127	207	15	and	and	CCONJ
cana-4127	207	16	alsinai	alsinai	PROPN
cana-4127	207	17	,	,	PUNCT
cana-4127	207	18	a.	a.	PROPN
cana-4127	207	19	,	,	PUNCT
cana-4127	207	20	hyers	hyer	NOUN
cana-4127	207	21	stability	stability	NOUN
cana-4127	207	22	of	of	ADP
cana-4127	207	23	aqc	aqc	PROPN
cana-4127	207	24	functional	functional	ADJ
cana-4127	207	25	equation	equation	NOUN
cana-4127	207	26	.	.	PUNCT
cana-4127	208	1	dynamics	dynamic	NOUN
cana-4127	208	2	of	of	ADP
cana-4127	208	3	continuous	continuous	ADJ
cana-4127	208	4	,	,	PUNCT
cana-4127	208	5	discrete	discrete	ADJ
cana-4127	208	6	and	and	CCONJ
cana-4127	208	7	impulsive	impulsive	ADJ
cana-4127	208	8	systems	system	NOUN
cana-4127	208	9	series	series	NOUN
cana-4127	208	10	b	b	NOUN
cana-4127	208	11	:	:	PUNCT
cana-4127	208	12	applications	application	NOUN
cana-4127	208	13	and	and	CCONJ
cana-4127	208	14	algorithms	algorithm	NOUN
cana-4127	208	15	,	,	PUNCT
cana-4127	208	16	2024	2024	NUM
cana-4127	208	17	,	,	PUNCT
cana-4127	208	18	31	31	NUM
cana-4127	208	19	,	,	PUNCT
cana-4127	208	20	63	63	NUM
cana-4127	208	21	-	-	SYM
cana-4127	208	22	75	75	NUM
cana-4127	208	23	.	.	PUNCT
cana-4127	209	1	[	[	X
cana-4127	209	2	20	20	NUM
cana-4127	209	3	]	]	SYM
cana-4127	209	4	agilan	agilan	ADJ
cana-4127	209	5	,	,	PUNCT
cana-4127	209	6	p.	p.	PROPN
cana-4127	209	7	,	,	PUNCT
cana-4127	209	8	julietraja	julietraja	PROPN
cana-4127	209	9	,	,	PUNCT
cana-4127	209	10	k	k	PROPN
cana-4127	209	11	,	,	PUNCT
cana-4127	209	12	sarah	sarah	PROPN
cana-4127	209	13	aljohani	aljohani	PROPN
cana-4127	209	14	,	,	PUNCT
cana-4127	209	15	nabil	nabil	PROPN
cana-4127	209	16	mlaiki	mlaiki	PROPN
cana-4127	209	17	,	,	PUNCT
cana-4127	209	18	generalised	generalise	VERB
cana-4127	209	19	ulam	ulam	PROPN
cana-4127	209	20	-	-	PUNCT
cana-4127	209	21	hyers	hyer	NOUN
cana-4127	209	22	stability	stability	NOUN
cana-4127	209	23	analysis	analysis	NOUN
cana-4127	209	24	for	for	ADP
cana-4127	209	25	system	system	NOUN
cana-4127	209	26	of	of	ADP
cana-4127	209	27	additive	additive	ADJ
cana-4127	209	28	functional	functional	ADJ
cana-4127	209	29	equation	equation	NOUN
cana-4127	209	30	in	in	ADP
cana-4127	209	31	fuzzy	fuzzy	ADJ
cana-4127	209	32	and	and	CCONJ
cana-4127	209	33	random	random	ADJ
cana-4127	209	34	normed	normed	ADJ
cana-4127	209	35	spaces	space	NOUN
cana-4127	209	36	:	:	PUNCT
cana-4127	209	37	direct	direct	ADJ
cana-4127	209	38	and	and	CCONJ
cana-4127	209	39	fixed	fix	VERB
cana-4127	209	40	point	point	NOUN
cana-4127	209	41	approach	approach	NOUN
cana-4127	209	42	.	.	PUNCT
cana-4127	210	1	int	int	NOUN
cana-4127	210	2	.	.	PUNCT
cana-4127	211	1	j.	j.	PROPN
cana-4127	211	2	anal	anal	PROPN
cana-4127	211	3	.	.	PUNCT
cana-4127	212	1	appl	appl	PROPN
cana-4127	212	2	.	.	PROPN
cana-4127	212	3	,	,	PUNCT
cana-4127	212	4	22	22	NUM
cana-4127	212	5	2024	2024	NUM
cana-4127	212	6	,	,	PUNCT
cana-4127	212	7	201	201	NUM
cana-4127	212	8	.	.	PUNCT
cana-4127	213	1	[	[	X
cana-4127	213	2	21	21	NUM
cana-4127	213	3	]	]	SYM
cana-4127	213	4	agilan.p	agilan.p	PROPN
cana-4127	213	5	,	,	PUNCT
cana-4127	213	6	vijayan.v	vijayan.v	PROPN
cana-4127	213	7	,	,	PUNCT
cana-4127	213	8	sophia.m	sophia.m	NUM
cana-4127	213	9	,	,	PUNCT
cana-4127	213	10	ganapathy.g	ganapathy.g	PROPN
cana-4127	213	11	.	.	PUNCT
cana-4127	213	12	,	,	PUNCT
cana-4127	213	13	exploring	explore	VERB
cana-4127	213	14	advanced	advanced	ADJ
cana-4127	213	15	stability	stability	NOUN
cana-4127	213	16	of	of	ADP
cana-4127	213	17	higher	high	ADJ
cana-4127	213	18	-	-	PUNCT
cana-4127	213	19	order	order	NOUN
cana-4127	213	20	functional	functional	ADJ
cana-4127	213	21	equations	equation	NOUN
cana-4127	213	22	in	in	ADP
cana-4127	213	23	neutrosophic	neutrosophic	ADJ
cana-4127	213	24	normed	norme	VERB
cana-4127	213	25	spaces	space	NOUN
cana-4127	213	26	via	via	ADP
cana-4127	213	27	hyers	hyer	NOUN
cana-4127	213	28	-	-	PUNCT
cana-4127	213	29	ulam	ulam	PROPN
cana-4127	213	30	methodologies	methodology	NOUN
cana-4127	213	31	.	.	PUNCT
cana-4127	214	1	communications	communication	NOUN
cana-4127	214	2	on	on	ADP
cana-4127	214	3	applied	apply	VERB
cana-4127	214	4	nonlinear	nonlinear	ADJ
cana-4127	214	5	analysis	analysis	NOUN
cana-4127	214	6	,	,	PUNCT
cana-4127	214	7	2025	2025	NUM
cana-4127	214	8	,	,	PUNCT
cana-4127	214	9	vol	vol	NOUN
cana-4127	214	10	32	32	NUM
cana-4127	214	11	no	no	NOUN
cana-4127	214	12	.	.	PUNCT
cana-4127	215	1	7s	7	NOUN
cana-4127	215	2	(	(	PUNCT
cana-4127	215	3	2025	2025	NUM
cana-4127	215	4	)	)	PUNCT
cana-4127	215	5	,	,	PUNCT
cana-4127	215	6	806	806	NUM
cana-4127	215	7	-	-	SYM
cana-4127	215	8	822	822	NUM
cana-4127	215	9	.	.	PUNCT
cana-4127	216	1	[	[	X
cana-4127	216	2	22	22	NUM
cana-4127	216	3	]	]	PUNCT
cana-4127	216	4	b.margoils	b.margoil	NOUN
cana-4127	216	5	,	,	PUNCT
cana-4127	216	6	j.b.diaz	j.b.diaz	PROPN
cana-4127	216	7	,	,	PUNCT
cana-4127	216	8	a	a	DET
cana-4127	216	9	fixed	fix	VERB
cana-4127	216	10	point	point	NOUN
cana-4127	216	11	theorem	theorem	NOUN
cana-4127	216	12	of	of	ADP
cana-4127	216	13	the	the	DET
cana-4127	216	14	alternative	alternative	NOUN
cana-4127	216	15	for	for	ADP
cana-4127	216	16	contractions	contraction	NOUN
cana-4127	216	17	on	on	ADP
cana-4127	216	18	a	a	DET
cana-4127	216	19	generalized	generalized	ADJ
cana-4127	216	20	complete	complete	ADJ
cana-4127	216	21	metric	metric	ADJ
cana-4127	216	22	space	space	NOUN
cana-4127	216	23	,	,	PUNCT
cana-4127	216	24	bull.amer	bull.amer	NOUN
cana-4127	216	25	.	.	PUNCT
cana-4127	216	26	math	math	NOUN
cana-4127	216	27	.	.	PUNCT
cana-4127	217	1	soc	soc	PROPN
cana-4127	217	2	.	.	PUNCT
cana-4127	218	1	126	126	NUM
cana-4127	218	2	74	74	NUM
cana-4127	218	3	(	(	PUNCT
cana-4127	218	4	1968	1968	NUM
cana-4127	218	5	)	)	PUNCT
cana-4127	218	6	,	,	PUNCT
cana-4127	218	7	305	305	NUM
cana-4127	218	8	-	-	SYM
cana-4127	218	9	309	309	NUM
cana-4127	218	10	.	.	PUNCT