id	sid	tid	token	lemma	pos
cana-527	1	1	theorems	theorem	NOUN
cana-527	1	2	for	for	ADP
cana-527	1	3	near	near	ADJ
cana-527	1	4	stable	stable	ADJ
cana-527	1	5	points	point	NOUN
cana-527	1	6	on	on	ADP
cana-527	1	7	a	a	DET
cana-527	1	8	near	near	ADJ
cana-527	1	9	banach	banach	NOUN
cana-527	1	10	space	space	NOUN
cana-527	1	11	furnished	furnish	VERB
cana-527	1	12	with	with	ADP
cana-527	1	13	graph	graph	NOUN
cana-527	1	14	savitha.s	savitha.s	NOUN
cana-527	1	15	1	1	NUM
cana-527	1	16	,	,	PUNCT
cana-527	1	17	thirunavukarasu.p	thirunavukarasu.p	ADP
cana-527	1	18	2	2	NUM
cana-527	1	19	1research	1research	NUM
cana-527	1	20	scholar	scholar	NOUN
cana-527	1	21	(	(	PUNCT
cana-527	1	22	reg.no	reg.no	NOUN
cana-527	1	23	:	:	PUNCT
cana-527	1	24	bdu2020182778694	bdu2020182778694	NOUN
cana-527	1	25	)	)	PUNCT
cana-527	1	26	,	,	PUNCT
cana-527	1	27	pg	pg	PROPN
cana-527	1	28	&	&	CCONJ
cana-527	1	29	research	research	PROPN
cana-527	1	30	department	department	PROPN
cana-527	1	31	of	of	ADP
cana-527	1	32	mathematics	mathematics	PROPN
cana-527	1	33	(	(	PUNCT
cana-527	1	34	thanthai	thanthai	PROPN
cana-527	1	35	periyar	periyar	PROPN
cana-527	1	36	government	government	NOUN
cana-527	1	37	arts	arts	PROPN
cana-527	1	38	&	&	CCONJ
cana-527	1	39	science	science	PROPN
cana-527	1	40	college	college	PROPN
cana-527	1	41	(	(	PUNCT
cana-527	1	42	autonomous	autonomous	ADJ
cana-527	1	43	)	)	PUNCT
cana-527	1	44	,	,	PUNCT
cana-527	1	45	affiliated	affiliate	VERB
cana-527	1	46	to	to	PART
cana-527	1	47	bharathidasan	bharathidasan	VERB
cana-527	1	48	university	university	NOUN
cana-527	1	49	,	,	PUNCT
cana-527	1	50	trichy620023	trichy620023	PROPN
cana-527	1	51	,	,	PUNCT
cana-527	1	52	tamilnadu	tamilnadu	NOUN
cana-527	1	53	,	,	PUNCT
cana-527	1	54	india	india	PROPN
cana-527	1	55	)	)	PUNCT
cana-527	1	56	;	;	PUNCT
cana-527	1	57	and	and	CCONJ
cana-527	1	58	assistant	assistant	NOUN
cana-527	1	59	professor	professor	NOUN
cana-527	1	60	,	,	PUNCT
cana-527	1	61	department	department	NOUN
cana-527	1	62	of	of	ADP
cana-527	1	63	mathematics	mathematic	NOUN
cana-527	1	64	,	,	PUNCT
cana-527	1	65	kongu	kongu	PROPN
cana-527	1	66	arts	art	NOUN
cana-527	1	67	and	and	CCONJ
cana-527	1	68	science	science	PROPN
cana-527	1	69	college	college	PROPN
cana-527	1	70	(	(	PUNCT
cana-527	1	71	autonomous	autonomous	ADJ
cana-527	1	72	)	)	PUNCT
cana-527	1	73	,	,	PUNCT
cana-527	1	74	(	(	PUNCT
cana-527	1	75	affiliated	affiliate	VERB
cana-527	1	76	to	to	PART
cana-527	1	77	bharathiar	bharathiar	VERB
cana-527	1	78	university),savithamaths85@gmail.com	university),savithamaths85@gmail.com	X
cana-527	2	1	2associate	2associate	NUM
cana-527	2	2	professor	professor	NOUN
cana-527	2	3	,	,	PUNCT
cana-527	2	4	pg	pg	PROPN
cana-527	2	5	&	&	CCONJ
cana-527	2	6	research	research	PROPN
cana-527	2	7	department	department	PROPN
cana-527	2	8	of	of	ADP
cana-527	2	9	mathematics	mathematics	PROPN
cana-527	2	10	(	(	PUNCT
cana-527	2	11	thanthai	thanthai	PROPN
cana-527	2	12	periyar	periyar	PROPN
cana-527	2	13	government	government	NOUN
cana-527	2	14	arts	arts	PROPN
cana-527	2	15	&	&	CCONJ
cana-527	2	16	science	science	PROPN
cana-527	2	17	college	college	PROPN
cana-527	2	18	(	(	PUNCT
cana-527	2	19	autonomous	autonomous	ADJ
cana-527	2	20	)	)	PUNCT
cana-527	2	21	,	,	PUNCT
cana-527	2	22	affiliated	affiliate	VERB
cana-527	2	23	to	to	PART
cana-527	2	24	bharathidasan	bharathidasan	VERB
cana-527	2	25	university	university	NOUN
cana-527	2	26	,	,	PUNCT
cana-527	2	27	trichy-620023	trichy-620023	NOUN
cana-527	2	28	,	,	PUNCT
cana-527	2	29	tamilnadu	tamilnadu	NOUN
cana-527	2	30	,	,	PUNCT
cana-527	2	31	india	india	PROPN
cana-527	2	32	)	)	PUNCT
cana-527	2	33	;	;	PUNCT
cana-527	2	34	ptavinash1967@gmail.com	ptavinash1967@gmail.com	NOUN
cana-527	2	35	article	article	NOUN
cana-527	2	36	history	history	NOUN
cana-527	2	37	:	:	PUNCT
cana-527	2	38	received	receive	VERB
cana-527	2	39	:	:	PUNCT
cana-527	2	40	23	23	NUM
cana-527	2	41	-	-	SYM
cana-527	2	42	01	01	NUM
cana-527	2	43	-	-	PUNCT
cana-527	2	44	2024	2024	NUM
cana-527	2	45	revised	revise	VERB
cana-527	2	46	:	:	PUNCT
cana-527	2	47	08	08	NUM
cana-527	2	48	-	-	PUNCT
cana-527	2	49	04	04	NUM
cana-527	2	50	-	-	PUNCT
cana-527	2	51	2024	2024	NUM
cana-527	2	52	accepted	accept	VERB
cana-527	2	53	:	:	PUNCT
cana-527	2	54	26	26	NUM
cana-527	2	55	-	-	PUNCT
cana-527	2	56	04	04	NUM
cana-527	2	57	-	-	PUNCT
cana-527	2	58	2024	2024	NUM
cana-527	2	59	abstract	abstract	NOUN
cana-527	2	60	:	:	PUNCT
cana-527	2	61	the	the	DET
cana-527	2	62	paper	paper	NOUN
cana-527	2	63	introduces	introduce	VERB
cana-527	2	64	a	a	DET
cana-527	2	65	novel	novel	ADJ
cana-527	2	66	method	method	NOUN
cana-527	2	67	for	for	ADP
cana-527	2	68	defining	define	VERB
cana-527	2	69	the	the	DET
cana-527	2	70	graph	graph	NOUN
cana-527	2	71	associated	associate	VERB
cana-527	2	72	with	with	ADP
cana-527	2	73	a	a	DET
cana-527	2	74	near	near	ADJ
cana-527	2	75	banach	banach	NOUN
cana-527	2	76	space	space	NOUN
cana-527	2	77	.	.	PUNCT
cana-527	3	1	in	in	ADP
cana-527	3	2	mathematics	mathematic	NOUN
cana-527	3	3	,	,	PUNCT
cana-527	3	4	a	a	DET
cana-527	3	5	graph	graph	NOUN
cana-527	3	6	typically	typically	ADV
cana-527	3	7	represents	represent	VERB
cana-527	3	8	relationships	relationship	NOUN
cana-527	3	9	between	between	ADP
cana-527	3	10	objects	object	NOUN
cana-527	3	11	.	.	PUNCT
cana-527	4	1	here	here	ADV
cana-527	4	2	,	,	PUNCT
cana-527	4	3	it	it	PRON
cana-527	4	4	seems	seem	VERB
cana-527	4	5	the	the	DET
cana-527	4	6	graph	graph	NOUN
cana-527	4	7	is	be	AUX
cana-527	4	8	being	be	AUX
cana-527	4	9	defined	define	VERB
cana-527	4	10	in	in	ADP
cana-527	4	11	the	the	DET
cana-527	4	12	context	context	NOUN
cana-527	4	13	of	of	ADP
cana-527	4	14	a	a	DET
cana-527	4	15	near	near	ADJ
cana-527	4	16	banach	banach	NOUN
cana-527	4	17	space	space	NOUN
cana-527	4	18	,	,	PUNCT
cana-527	4	19	which	which	PRON
cana-527	4	20	is	be	AUX
cana-527	4	21	a	a	DET
cana-527	4	22	generalization	generalization	NOUN
cana-527	4	23	of	of	ADP
cana-527	4	24	banach	banach	NOUN
cana-527	4	25	spaces	space	NOUN
cana-527	4	26	allowing	allow	VERB
cana-527	4	27	the	the	DET
cana-527	4	28	norm	norm	NOUN
cana-527	4	29	to	to	PART
cana-527	4	30	take	take	VERB
cana-527	4	31	infinite	infinite	ADJ
cana-527	4	32	values	value	NOUN
cana-527	4	33	.	.	PUNCT
cana-527	5	1	an	an	DET
cana-527	5	2	iteration	iteration	NOUN
cana-527	5	3	function	function	NOUN
cana-527	5	4	is	be	AUX
cana-527	5	5	utilized	utilize	VERB
cana-527	5	6	to	to	PART
cana-527	5	7	define	define	VERB
cana-527	5	8	the	the	DET
cana-527	5	9	subgraph	subgraph	NOUN
cana-527	5	10	of	of	ADP
cana-527	5	11	the	the	DET
cana-527	5	12	graph	graph	NOUN
cana-527	5	13	associated	associate	VERB
cana-527	5	14	with	with	ADP
cana-527	5	15	the	the	DET
cana-527	5	16	near	near	ADJ
cana-527	5	17	banach	banach	NOUN
cana-527	5	18	space	space	NOUN
cana-527	5	19	.	.	PUNCT
cana-527	6	1	this	this	DET
cana-527	6	2	subgraph	subgraph	NOUN
cana-527	6	3	likely	likely	ADV
cana-527	6	4	captures	capture	VERB
cana-527	6	5	specific	specific	ADJ
cana-527	6	6	properties	property	NOUN
cana-527	6	7	or	or	CCONJ
cana-527	6	8	relationships	relationship	NOUN
cana-527	6	9	within	within	ADP
cana-527	6	10	the	the	DET
cana-527	6	11	original	original	ADJ
cana-527	6	12	graph	graph	NOUN
cana-527	6	13	.	.	PUNCT
cana-527	7	1	the	the	DET
cana-527	7	2	paper	paper	NOUN
cana-527	7	3	presents	present	VERB
cana-527	7	4	near	near	ADV
cana-527	7	5	-	-	PUNCT
cana-527	7	6	fixed	fix	VERB
cana-527	7	7	point	point	NOUN
cana-527	7	8	theorems	theorem	NOUN
cana-527	7	9	by	by	ADP
cana-527	7	10	well	well	ADV
cana-527	7	11	-	-	PUNCT
cana-527	7	12	known	know	VERB
cana-527	7	13	mathematicians	mathematician	NOUN
cana-527	7	14	such	such	ADJ
cana-527	7	15	as	as	ADP
cana-527	7	16	banach	banach	NOUN
cana-527	7	17	,	,	PUNCT
cana-527	7	18	kannan	kannan	PROPN
cana-527	7	19	,	,	PUNCT
cana-527	7	20	chatterja	chatterja	NOUN
cana-527	7	21	,	,	PUNCT
cana-527	7	22	and	and	CCONJ
cana-527	7	23	ciric	ciric	ADJ
cana-527	7	24	[	[	X
cana-527	7	25	2][18][7][9	2][18][7][9	NUM
cana-527	7	26	]	]	PUNCT
cana-527	7	27	.	.	PUNCT
cana-527	8	1	these	these	DET
cana-527	8	2	theorems	theorem	NOUN
cana-527	8	3	deal	deal	VERB
cana-527	8	4	with	with	ADP
cana-527	8	5	the	the	DET
cana-527	8	6	existence	existence	NOUN
cana-527	8	7	of	of	ADP
cana-527	8	8	points	point	NOUN
cana-527	8	9	that	that	PRON
cana-527	8	10	are	be	AUX
cana-527	8	11	approximately	approximately	ADV
cana-527	8	12	fixed	fix	VERB
cana-527	8	13	under	under	ADP
cana-527	8	14	certain	certain	ADJ
cana-527	8	15	mappings	mapping	NOUN
cana-527	8	16	or	or	CCONJ
cana-527	8	17	operations	operation	NOUN
cana-527	8	18	.	.	PUNCT
cana-527	9	1	the	the	DET
cana-527	9	2	near	near	ADV
cana-527	9	3	-	-	PUNCT
cana-527	9	4	fixed	fix	VERB
cana-527	9	5	point	point	NOUN
cana-527	9	6	theorems	theorem	NOUN
cana-527	9	7	mentioned	mention	VERB
cana-527	9	8	are	be	AUX
cana-527	9	9	obtained	obtain	VERB
cana-527	9	10	or	or	CCONJ
cana-527	9	11	derived	derive	VERB
cana-527	9	12	using	use	VERB
cana-527	9	13	the	the	DET
cana-527	9	14	new	new	ADJ
cana-527	9	15	approach	approach	NOUN
cana-527	9	16	introduced	introduce	VERB
cana-527	9	17	for	for	ADP
cana-527	9	18	defining	define	VERB
cana-527	9	19	the	the	DET
cana-527	9	20	graph	graph	NOUN
cana-527	9	21	and	and	CCONJ
cana-527	9	22	its	its	PRON
cana-527	9	23	subgraph	subgraph	NOUN
cana-527	9	24	associated	associate	VERB
cana-527	9	25	with	with	ADP
cana-527	9	26	the	the	DET
cana-527	9	27	near	near	ADJ
cana-527	9	28	banach	banach	NOUN
cana-527	9	29	space	space	NOUN
cana-527	9	30	[	[	X
cana-527	9	31	20	20	NUM
cana-527	9	32	]	]	PUNCT
cana-527	9	33	.	.	PUNCT
cana-527	10	1	this	this	PRON
cana-527	10	2	suggests	suggest	VERB
cana-527	10	3	that	that	SCONJ
cana-527	10	4	the	the	DET
cana-527	10	5	new	new	ADJ
cana-527	10	6	approach	approach	NOUN
cana-527	10	7	is	be	AUX
cana-527	10	8	effective	effective	ADJ
cana-527	10	9	in	in	ADP
cana-527	10	10	providing	provide	VERB
cana-527	10	11	a	a	DET
cana-527	10	12	framework	framework	NOUN
cana-527	10	13	for	for	ADP
cana-527	10	14	proving	prove	VERB
cana-527	10	15	these	these	DET
cana-527	10	16	theorems	theorem	NOUN
cana-527	10	17	or	or	CCONJ
cana-527	10	18	extending	extend	VERB
cana-527	10	19	their	their	PRON
cana-527	10	20	applicability	applicability	NOUN
cana-527	10	21	to	to	PART
cana-527	10	22	near	near	ADP
cana-527	10	23	banach	banach	NOUN
cana-527	10	24	spaces	space	NOUN
cana-527	10	25	.	.	PUNCT
cana-527	11	1	the	the	DET
cana-527	11	2	paper	paper	NOUN
cana-527	11	3	discusses	discuss	VERB
cana-527	11	4	a	a	DET
cana-527	11	5	fresh	fresh	ADJ
cana-527	11	6	method	method	NOUN
cana-527	11	7	for	for	ADP
cana-527	11	8	defining	define	VERB
cana-527	11	9	the	the	DET
cana-527	11	10	graph	graph	NOUN
cana-527	11	11	of	of	ADP
cana-527	11	12	a	a	DET
cana-527	11	13	near	near	ADJ
cana-527	11	14	banach	banach	NOUN
cana-527	11	15	space	space	NOUN
cana-527	11	16	,	,	PUNCT
cana-527	11	17	employs	employ	VERB
cana-527	11	18	an	an	DET
cana-527	11	19	iteration	iteration	NOUN
cana-527	11	20	function	function	NOUN
cana-527	11	21	to	to	PART
cana-527	11	22	define	define	VERB
cana-527	11	23	its	its	PRON
cana-527	11	24	subgraph	subgraph	NOUN
cana-527	11	25	,	,	PUNCT
cana-527	11	26	and	and	CCONJ
cana-527	11	27	then	then	ADV
cana-527	11	28	demonstrates	demonstrate	VERB
cana-527	11	29	the	the	DET
cana-527	11	30	utility	utility	NOUN
cana-527	11	31	of	of	ADP
cana-527	11	32	this	this	DET
cana-527	11	33	approach	approach	NOUN
cana-527	11	34	by	by	ADP
cana-527	11	35	deriving	derive	VERB
cana-527	11	36	near	near	ADV
cana-527	11	37	-	-	PUNCT
cana-527	11	38	fixed	fix	VERB
cana-527	11	39	point	point	NOUN
cana-527	11	40	theorems	theorem	NOUN
cana-527	11	41	by	by	ADP
cana-527	11	42	eminent	eminent	ADJ
cana-527	11	43	mathematicians	mathematician	NOUN
cana-527	11	44	in	in	ADP
cana-527	11	45	the	the	DET
cana-527	11	46	field	field	NOUN
cana-527	11	47	[	[	X
cana-527	11	48	14][15[16	14][15[16	NUM
cana-527	11	49	]	]	PUNCT
cana-527	11	50	.	.	PUNCT
cana-527	12	1	keywords	keyword	NOUN
cana-527	12	2	:	:	PUNCT
cana-527	12	3	cauchy	cauchy	ADJ
cana-527	12	4	sequence	sequence	NOUN
cana-527	12	5	,	,	PUNCT
cana-527	12	6	near	near	ADP
cana-527	12	7	fixed	fix	VERB
cana-527	12	8	point	point	NOUN
cana-527	12	9	,	,	PUNCT
cana-527	12	10	banach	banach	NOUN
cana-527	12	11	hyperspace	hyperspace	NOUN
cana-527	12	12	,	,	PUNCT
cana-527	12	13	iterated	iterate	VERB
cana-527	12	14	function	function	NOUN
cana-527	12	15	,	,	PUNCT
cana-527	12	16	graph	graph	NOUN
cana-527	12	17	,	,	PUNCT
cana-527	12	18	subgraph	subgraph	NOUN
cana-527	12	19	,	,	PUNCT
cana-527	12	20	wˆ	wˆ	ADP
cana-527	12	21	-sequence	-sequence	PROPN
cana-527	12	22	.	.	PUNCT
cana-527	13	1	subject	subject	ADJ
cana-527	13	2	classification	classification	NOUN
cana-527	13	3	:	:	PUNCT
cana-527	13	4	47h10	47h10	NUM
cana-527	13	5	,	,	PUNCT
cana-527	13	6	54h25	54h25	NUM
cana-527	13	7	communications	communication	NOUN
cana-527	13	8	on	on	ADP
cana-527	13	9	applied	apply	VERB
cana-527	13	10	nonlinear	nonlinear	ADJ
cana-527	13	11	analysis	analysis	NOUN
cana-527	13	12	issn	issn	NOUN
cana-527	13	13	:	:	PUNCT
cana-527	13	14	1074	1074	NUM
cana-527	13	15	-	-	PUNCT
cana-527	13	16	133x	133x	NUM
cana-527	13	17	vol	vol	NOUN
cana-527	13	18	31	31	NUM
cana-527	13	19	no	no	NOUN
cana-527	13	20	.	.	NOUN
cana-527	13	21	2	2	NUM
cana-527	13	22	(	(	PUNCT
cana-527	13	23	2024	2024	NUM
cana-527	13	24	)	)	PUNCT
cana-527	13	25	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	13	26	156	156	NUM
cana-527	13	27	mailto:sity)%2csavithamaths85@gmail.com	mailto:sity)%2csavithamaths85@gmail.com	X
cana-527	14	1	mailto:vithamaths85@gmail.com	mailto:vithamaths85@gmail.com	NOUN
cana-527	14	2	mailto:ptavinash1967@gmail.com	mailto:ptavinash1967@gmail.com	PROPN
cana-527	14	3	mailto:vinash1967@gmail.com	mailto:vinash1967@gmail.com	PROPN
cana-527	14	4	i	i	PRON
cana-527	14	5	introduction	introduction	NOUN
cana-527	14	6	jachymiski	jachymiski	PROPN
cana-527	14	7	’s	’s	PART
cana-527	14	8	[	[	X
cana-527	14	9	17	17	NUM
cana-527	14	10	]	]	X
cana-527	14	11	generalization	generalization	NOUN
cana-527	14	12	of	of	ADP
cana-527	14	13	the	the	DET
cana-527	14	14	banach	banach	NOUN
cana-527	14	15	contraction	contraction	NOUN
cana-527	14	16	principle	principle	NOUN
cana-527	14	17	by	by	ADP
cana-527	14	18	combining	combine	VERB
cana-527	14	19	fixed	fix	VERB
cana-527	14	20	point	point	NOUN
cana-527	14	21	theory	theory	NOUN
cana-527	14	22	and	and	CCONJ
cana-527	14	23	graph	graph	NOUN
cana-527	14	24	theory	theory	NOUN
cana-527	14	25	sounds	sound	VERB
cana-527	14	26	like	like	ADP
cana-527	14	27	an	an	DET
cana-527	14	28	intriguing	intriguing	ADJ
cana-527	14	29	extension	extension	NOUN
cana-527	14	30	.	.	PUNCT
cana-527	15	1	by	by	ADP
cana-527	15	2	incorpo	incorpo	NOUN
cana-527	15	3	rating	rating	NOUN
cana-527	15	4	graph	graph	NOUN
cana-527	15	5	theory	theory	NOUN
cana-527	15	6	into	into	ADP
cana-527	15	7	the	the	DET
cana-527	15	8	framework	framework	NOUN
cana-527	15	9	,	,	PUNCT
cana-527	15	10	it	it	PRON
cana-527	15	11	likely	likely	ADV
cana-527	15	12	allows	allow	VERB
cana-527	15	13	for	for	ADP
cana-527	15	14	the	the	DET
cana-527	15	15	consideration	consideration	NOUN
cana-527	15	16	of	of	ADP
cana-527	15	17	more	more	ADJ
cana-527	15	18	complex	complex	ADJ
cana-527	15	19	structures	structure	NOUN
cana-527	15	20	or	or	CCONJ
cana-527	15	21	relationships	relationship	NOUN
cana-527	15	22	between	between	ADP
cana-527	15	23	points	point	NOUN
cana-527	15	24	in	in	ADP
cana-527	15	25	the	the	DET
cana-527	15	26	space	space	NOUN
cana-527	15	27	,	,	PUNCT
cana-527	15	28	beyond	beyond	ADP
cana-527	15	29	just	just	ADV
cana-527	15	30	metric	metric	ADJ
cana-527	15	31	properties	property	NOUN
cana-527	15	32	.	.	PUNCT
cana-527	16	1	in	in	ADP
cana-527	16	2	traditional	traditional	ADJ
cana-527	16	3	fixed	fix	VERB
cana-527	16	4	-	-	PUNCT
cana-527	16	5	point	point	NOUN
cana-527	16	6	theory	theory	NOUN
cana-527	16	7	,	,	PUNCT
cana-527	16	8	the	the	DET
cana-527	16	9	banach	banach	NOUN
cana-527	16	10	contraction	contraction	NOUN
cana-527	16	11	principle	principle	NOUN
cana-527	16	12	[	[	X
cana-527	16	13	2	2	X
cana-527	16	14	]	]	PUNCT
cana-527	16	15	guarantees	guarantee	VERB
cana-527	16	16	the	the	DET
cana-527	16	17	existence	existence	NOUN
cana-527	16	18	and	and	CCONJ
cana-527	16	19	uniqueness	uniqueness	NOUN
cana-527	16	20	of	of	ADP
cana-527	16	21	fixed	fix	VERB
cana-527	16	22	points	point	NOUN
cana-527	16	23	for	for	ADP
cana-527	16	24	contraction	contraction	NOUN
cana-527	16	25	mappings	mapping	NOUN
cana-527	16	26	in	in	ADP
cana-527	16	27	complete	complete	ADJ
cana-527	16	28	metric	metric	ADJ
cana-527	16	29	spaces	space	NOUN
cana-527	16	30	.	.	PUNCT
cana-527	17	1	however	however	ADV
cana-527	17	2	,	,	PUNCT
cana-527	17	3	this	this	DET
cana-527	17	4	principle	principle	NOUN
cana-527	17	5	may	may	AUX
cana-527	17	6	not	not	PART
cana-527	17	7	directly	directly	ADV
cana-527	17	8	apply	apply	VERB
cana-527	17	9	in	in	ADP
cana-527	17	10	settings	setting	NOUN
cana-527	17	11	where	where	SCONJ
cana-527	17	12	the	the	DET
cana-527	17	13	underlying	underlie	VERB
cana-527	17	14	space	space	NOUN
cana-527	17	15	has	have	VERB
cana-527	17	16	a	a	DET
cana-527	17	17	more	more	ADV
cana-527	17	18	intricate	intricate	ADJ
cana-527	17	19	structure	structure	NOUN
cana-527	17	20	,	,	PUNCT
cana-527	17	21	such	such	ADJ
cana-527	17	22	as	as	ADP
cana-527	17	23	when	when	SCONJ
cana-527	17	24	relationships	relationship	NOUN
cana-527	17	25	between	between	ADP
cana-527	17	26	points	point	NOUN
cana-527	17	27	are	be	AUX
cana-527	17	28	de	de	ADJ
cana-527	17	29	scribed	scribe	VERB
cana-527	17	30	by	by	ADP
cana-527	17	31	a	a	DET
cana-527	17	32	graph	graph	NOUN
cana-527	17	33	.	.	PUNCT
cana-527	18	1	by	by	ADP
cana-527	18	2	leveraging	leverage	VERB
cana-527	18	3	graph	graph	NOUN
cana-527	18	4	theory	theory	NOUN
cana-527	18	5	concepts	concept	NOUN
cana-527	18	6	,	,	PUNCT
cana-527	18	7	jachymiski	jachymiski	PROPN
cana-527	18	8	’s	’s	PART
cana-527	18	9	[	[	X
cana-527	18	10	17	17	NUM
cana-527	18	11	]	]	X
cana-527	18	12	generaliza	generaliza	ADJ
cana-527	18	13	tion	tion	NOUN
cana-527	18	14	may	may	AUX
cana-527	18	15	provide	provide	VERB
cana-527	18	16	a	a	DET
cana-527	18	17	way	way	NOUN
cana-527	18	18	to	to	PART
cana-527	18	19	handle	handle	VERB
cana-527	18	20	mappings	mapping	NOUN
cana-527	18	21	that	that	PRON
cana-527	18	22	interact	interact	VERB
cana-527	18	23	with	with	ADP
cana-527	18	24	the	the	DET
cana-527	18	25	underlying	underlie	VERB
cana-527	18	26	graph	graph	NOUN
cana-527	18	27	struc	struc	PROPN
cana-527	18	28	ture	ture	NOUN
cana-527	18	29	in	in	ADP
cana-527	18	30	some	some	DET
cana-527	18	31	manner	manner	NOUN
cana-527	18	32	.	.	PUNCT
cana-527	19	1	this	this	PRON
cana-527	19	2	could	could	AUX
cana-527	19	3	involve	involve	VERB
cana-527	19	4	mappings	mapping	NOUN
cana-527	19	5	that	that	PRON
cana-527	19	6	respect	respect	VERB
cana-527	19	7	certain	certain	ADJ
cana-527	19	8	graph	graph	NOUN
cana-527	19	9	-	-	PUNCT
cana-527	19	10	theoretic	theoretic	NOUN
cana-527	19	11	properties	property	NOUN
cana-527	19	12	or	or	CCONJ
cana-527	19	13	have	have	VERB
cana-527	19	14	dependencies	dependency	NOUN
cana-527	19	15	on	on	ADP
cana-527	19	16	the	the	DET
cana-527	19	17	graph	graph	NOUN
cana-527	19	18	edges	edge	NOUN
cana-527	19	19	or	or	CCONJ
cana-527	19	20	vertices.the	vertices.the	DET
cana-527	19	21	significance	significance	NOUN
cana-527	19	22	of	of	ADP
cana-527	19	23	this	this	DET
cana-527	19	24	extension	extension	NOUN
cana-527	19	25	likely	likely	ADV
cana-527	19	26	lies	lie	VERB
cana-527	19	27	in	in	ADP
cana-527	19	28	its	its	PRON
cana-527	19	29	applicability	applicability	NOUN
cana-527	19	30	to	to	ADP
cana-527	19	31	problems	problem	NOUN
cana-527	19	32	where	where	SCONJ
cana-527	19	33	the	the	DET
cana-527	19	34	traditional	traditional	ADJ
cana-527	19	35	banach	banach	NOUN
cana-527	19	36	con	con	PROPN
cana-527	19	37	traction	traction	PROPN
cana-527	19	38	principle	principle	NOUN
cana-527	19	39	can	can	AUX
cana-527	19	40	not	not	PART
cana-527	19	41	be	be	AUX
cana-527	19	42	directly	directly	ADV
cana-527	19	43	applied	apply	VERB
cana-527	19	44	due	due	ADP
cana-527	19	45	to	to	ADP
cana-527	19	46	the	the	DET
cana-527	19	47	presence	presence	NOUN
cana-527	19	48	of	of	ADP
cana-527	19	49	a	a	DET
cana-527	19	50	graph	graph	NOUN
cana-527	19	51	structure	structure	NOUN
cana-527	19	52	.	.	PUNCT
cana-527	20	1	it	it	PRON
cana-527	20	2	opens	open	VERB
cana-527	20	3	up	up	ADP
cana-527	20	4	new	new	ADJ
cana-527	20	5	avenues	avenue	NOUN
cana-527	20	6	for	for	ADP
cana-527	20	7	studying	study	VERB
cana-527	20	8	fixed	fix	VERB
cana-527	20	9	point	point	NOUN
cana-527	20	10	properties	property	NOUN
cana-527	20	11	[	[	X
cana-527	20	12	7][9][18][20	7][9][18][20	X
cana-527	20	13	]	]	X
cana-527	20	14	in	in	ADP
cana-527	20	15	spaces	space	NOUN
cana-527	20	16	that	that	PRON
cana-527	20	17	exhibit	exhibit	VERB
cana-527	20	18	both	both	CCONJ
cana-527	20	19	metric	metric	ADJ
cana-527	20	20	and	and	CCONJ
cana-527	20	21	graph	graph	NOUN
cana-527	20	22	-	-	PUNCT
cana-527	20	23	theoretic	theoretic	NOUN
cana-527	20	24	characteristics	characteristic	NOUN
cana-527	20	25	.	.	PUNCT
cana-527	21	1	it	it	PRON
cana-527	21	2	can	can	AUX
cana-527	21	3	be	be	AUX
cana-527	21	4	speculated	speculate	VERB
cana-527	21	5	that	that	SCONJ
cana-527	21	6	jachymiski	jachymiski	PROPN
cana-527	21	7	’s	’s	PART
cana-527	21	8	[	[	X
cana-527	21	9	17	17	NUM
cana-527	21	10	]	]	X
cana-527	21	11	generalization	generalization	NOUN
cana-527	21	12	offers	offer	VERB
cana-527	21	13	a	a	DET
cana-527	21	14	way	way	NOUN
cana-527	21	15	to	to	PART
cana-527	21	16	deal	deal	VERB
cana-527	21	17	with	with	ADP
cana-527	21	18	mappings	mapping	NOUN
cana-527	21	19	that	that	PRON
cana-527	21	20	interact	interact	VERB
cana-527	21	21	with	with	ADP
cana-527	21	22	the	the	DET
cana-527	21	23	graph	graph	NOUN
cana-527	21	24	structure	structure	NOUN
cana-527	21	25	underneath	underneath	ADV
cana-527	21	26	.	.	PUNCT
cana-527	22	1	this	this	DET
cana-527	22	2	interaction	interaction	NOUN
cana-527	22	3	could	could	AUX
cana-527	22	4	take	take	VERB
cana-527	22	5	many	many	ADJ
cana-527	22	6	different	different	ADJ
cana-527	22	7	forms	form	NOUN
cana-527	22	8	,	,	PUNCT
cana-527	22	9	for	for	ADP
cana-527	22	10	example	example	NOUN
cana-527	22	11	,	,	PUNCT
cana-527	22	12	mappings	mapping	NOUN
cana-527	22	13	that	that	PRON
cana-527	22	14	depend	depend	VERB
cana-527	22	15	on	on	ADP
cana-527	22	16	the	the	DET
cana-527	22	17	graph	graph	NOUN
cana-527	22	18	’s	’s	PART
cana-527	22	19	vertices	vertex	NOUN
cana-527	22	20	and	and	CCONJ
cana-527	22	21	edges	edge	NOUN
cana-527	22	22	or	or	CCONJ
cana-527	22	23	that	that	PRON
cana-527	22	24	adhere	adhere	VERB
cana-527	22	25	to	to	ADP
cana-527	22	26	cer	cer	PROPN
cana-527	22	27	tain	tain	PROPN
cana-527	22	28	graph	graph	NOUN
cana-527	22	29	-	-	PUNCT
cana-527	22	30	theoretic	theoretic	NOUN
cana-527	22	31	features	feature	NOUN
cana-527	22	32	.	.	PUNCT
cana-527	23	1	this	this	DET
cana-527	23	2	expansion	expansion	NOUN
cana-527	23	3	provides	provide	VERB
cana-527	23	4	opportunities	opportunity	NOUN
cana-527	23	5	to	to	PART
cana-527	23	6	examine	examine	VERB
cana-527	23	7	fixed	fix	VERB
cana-527	23	8	point	point	NOUN
cana-527	23	9	attributes	attribute	VERB
cana-527	23	10	in	in	ADP
cana-527	23	11	spaces	space	NOUN
cana-527	23	12	that	that	PRON
cana-527	23	13	have	have	VERB
cana-527	23	14	both	both	CCONJ
cana-527	23	15	metric	metric	ADJ
cana-527	23	16	and	and	CCONJ
cana-527	23	17	graph	graph	NOUN
cana-527	23	18	-	-	PUNCT
cana-527	23	19	theoretic	theoretic	NOUN
cana-527	23	20	properties	property	NOUN
cana-527	23	21	;	;	PUNCT
cana-527	23	22	it	it	PRON
cana-527	23	23	also	also	ADV
cana-527	23	24	in	in	ADP
cana-527	23	25	dicates	dicate	NOUN
cana-527	23	26	a	a	DET
cana-527	23	27	more	more	ADV
cana-527	23	28	nuanced	nuanced	ADJ
cana-527	23	29	understanding	understanding	NOUN
cana-527	23	30	of	of	ADP
cana-527	23	31	the	the	DET
cana-527	23	32	interactions	interaction	NOUN
cana-527	23	33	between	between	ADP
cana-527	23	34	points	point	NOUN
cana-527	23	35	.	.	PUNCT
cana-527	24	1	the	the	DET
cana-527	24	2	possible	possible	ADJ
cana-527	24	3	applicability	applicability	NOUN
cana-527	24	4	of	of	ADP
cana-527	24	5	this	this	DET
cana-527	24	6	extension	extension	NOUN
cana-527	24	7	to	to	ADP
cana-527	24	8	problems	problem	NOUN
cana-527	24	9	where	where	SCONJ
cana-527	24	10	the	the	DET
cana-527	24	11	presence	presence	NOUN
cana-527	24	12	of	of	ADP
cana-527	24	13	a	a	DET
cana-527	24	14	graph	graph	NOUN
cana-527	24	15	structure	structure	NOUN
cana-527	24	16	poses	pose	VERB
cana-527	24	17	difficulties	difficulty	NOUN
cana-527	24	18	for	for	ADP
cana-527	24	19	the	the	DET
cana-527	24	20	classic	classic	ADJ
cana-527	24	21	banach	banach	NOUN
cana-527	24	22	contraction	contraction	NOUN
cana-527	24	23	principle	principle	NOUN
cana-527	24	24	demonstrate	demonstrate	VERB
cana-527	24	25	the	the	DET
cana-527	24	26	significance	significance	NOUN
cana-527	24	27	of	of	ADP
cana-527	24	28	this	this	DET
cana-527	24	29	work	work	NOUN
cana-527	24	30	[	[	X
cana-527	24	31	1][20	1][20	NOUN
cana-527	24	32	]	]	PUNCT
cana-527	24	33	.	.	PUNCT
cana-527	25	1	jachymiski	jachymiski	PROPN
cana-527	25	2	’s	’s	PART
cana-527	25	3	generalization	generalization	NOUN
cana-527	25	4	not	not	PART
cana-527	25	5	only	only	ADV
cana-527	25	6	broadens	broaden	VERB
cana-527	25	7	the	the	DET
cana-527	25	8	application	application	NOUN
cana-527	25	9	of	of	ADP
cana-527	25	10	the	the	DET
cana-527	25	11	banach	banach	NOUN
cana-527	25	12	con	con	NOUN
cana-527	25	13	traction	traction	PROPN
cana-527	25	14	principle	principle	NOUN
cana-527	25	15	but	but	CCONJ
cana-527	25	16	also	also	ADV
cana-527	25	17	provides	provide	VERB
cana-527	25	18	a	a	DET
cana-527	25	19	flexible	flexible	ADJ
cana-527	25	20	means	mean	NOUN
cana-527	25	21	of	of	ADP
cana-527	25	22	examining	examine	VERB
cana-527	25	23	spaces	space	NOUN
cana-527	25	24	with	with	ADP
cana-527	25	25	a	a	DET
cana-527	25	26	variety	variety	NOUN
cana-527	25	27	of	of	ADP
cana-527	25	28	structural	structural	ADJ
cana-527	25	29	features	feature	NOUN
cana-527	25	30	by	by	ADP
cana-527	25	31	bridging	bridge	VERB
cana-527	25	32	the	the	DET
cana-527	25	33	gap	gap	NOUN
cana-527	25	34	between	between	ADP
cana-527	25	35	fixed	fix	VERB
cana-527	25	36	point	point	NOUN
cana-527	25	37	theory	theory	NOUN
cana-527	25	38	and	and	CCONJ
cana-527	25	39	graph	graph	NOUN
cana-527	25	40	theory	theory	NOUN
cana-527	25	41	.	.	PUNCT
cana-527	26	1	to	to	PART
cana-527	26	2	delve	delve	VERB
cana-527	26	3	deeper	deeply	ADV
cana-527	26	4	into	into	ADP
cana-527	26	5	the	the	DET
cana-527	26	6	specifics	specific	NOUN
cana-527	26	7	of	of	ADP
cana-527	26	8	jachymiski	jachymiski	PROPN
cana-527	26	9	’s	’s	PART
cana-527	26	10	[	[	X
cana-527	26	11	17	17	NUM
cana-527	26	12	]	]	PUNCT
cana-527	26	13	generalization	generalization	NOUN
cana-527	26	14	and	and	CCONJ
cana-527	26	15	its	its	PRON
cana-527	26	16	applica	applica	PROPN
cana-527	26	17	tions	tion	NOUN
cana-527	26	18	,	,	PUNCT
cana-527	26	19	it	it	PRON
cana-527	26	20	would	would	AUX
cana-527	26	21	be	be	AUX
cana-527	26	22	necessary	necessary	ADJ
cana-527	26	23	to	to	PART
cana-527	26	24	refer	refer	VERB
cana-527	26	25	to	to	ADP
cana-527	26	26	the	the	DET
cana-527	26	27	original	original	ADJ
cana-527	26	28	paper	paper	NOUN
cana-527	26	29	[	[	X
cana-527	26	30	17	17	NUM
cana-527	26	31	]	]	PUNCT
cana-527	26	32	and	and	CCONJ
cana-527	26	33	explore	explore	VERB
cana-527	26	34	how	how	SCONJ
cana-527	26	35	the	the	DET
cana-527	26	36	combination	combination	NOUN
cana-527	26	37	of	of	ADP
cana-527	26	38	fixed	fix	VERB
cana-527	26	39	point	point	NOUN
cana-527	26	40	theory	theory	NOUN
cana-527	26	41	and	and	CCONJ
cana-527	26	42	graph	graph	NOUN
cana-527	26	43	theory	theory	NOUN
cana-527	26	44	is	be	AUX
cana-527	26	45	utilized	utilize	VERB
cana-527	26	46	to	to	PART
cana-527	26	47	establish	establish	VERB
cana-527	26	48	existence	existence	NOUN
cana-527	26	49	and	and	CCONJ
cana-527	26	50	uniqueness	uniqueness	NOUN
cana-527	26	51	results	result	NOUN
cana-527	26	52	for	for	ADP
cana-527	26	53	fixed	fix	VERB
cana-527	26	54	points	point	NOUN
cana-527	26	55	in	in	ADP
cana-527	26	56	this	this	DET
cana-527	26	57	extended	extend	VERB
cana-527	26	58	framework	framework	NOUN
cana-527	26	59	.	.	PUNCT
cana-527	27	1	the	the	DET
cana-527	27	2	graph	graph	NOUN
cana-527	27	3	was	be	AUX
cana-527	27	4	defined	define	VERB
cana-527	27	5	in	in	ADP
cana-527	27	6	previous	previous	ADJ
cana-527	27	7	studies	study	NOUN
cana-527	27	8	of	of	ADP
cana-527	27	9	fixed	fix	VERB
cana-527	27	10	point	point	NOUN
cana-527	27	11	theorems	theorem	NOUN
cana-527	27	12	on	on	ADP
cana-527	27	13	metric	metric	ADJ
cana-527	27	14	spaces	space	NOUN
cana-527	27	15	s	s	AUX
cana-527	27	16	endowed	endow	VERB
cana-527	27	17	with	with	ADP
cana-527	27	18	a	a	DET
cana-527	27	19	graph	graph	NOUN
cana-527	27	20	by	by	ADP
cana-527	27	21	considering	consider	VERB
cana-527	27	22	the	the	DET
cana-527	27	23	vertex	vertex	NOUN
cana-527	27	24	set	set	VERB
cana-527	27	25	to	to	PART
cana-527	27	26	be	be	AUX
cana-527	27	27	the	the	DET
cana-527	27	28	set	set	NOUN
cana-527	27	29	s	s	X
cana-527	27	30	and	and	CCONJ
cana-527	27	31	the	the	DET
cana-527	27	32	edge	edge	NOUN
cana-527	27	33	set	set	VERB
cana-527	27	34	communications	communication	NOUN
cana-527	27	35	on	on	ADP
cana-527	27	36	applied	apply	VERB
cana-527	27	37	nonlinear	nonlinear	ADJ
cana-527	27	38	analysis	analysis	NOUN
cana-527	27	39	issn	issn	NOUN
cana-527	27	40	:	:	PUNCT
cana-527	27	41	1074	1074	NUM
cana-527	27	42	-	-	PUNCT
cana-527	27	43	133x	133x	NUM
cana-527	27	44	vol	vol	NOUN
cana-527	27	45	31	31	NUM
cana-527	27	46	no	no	NOUN
cana-527	27	47	.	.	NOUN
cana-527	27	48	2	2	NUM
cana-527	27	49	(	(	PUNCT
cana-527	27	50	2024	2024	NUM
cana-527	27	51	)	)	PUNCT
cana-527	27	52	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	27	53	157	157	NUM
cana-527	27	54	to	to	PART
cana-527	27	55	be	be	AUX
cana-527	27	56	the	the	DET
cana-527	27	57	diagonal	diagonal	ADJ
cana-527	27	58	of	of	ADP
cana-527	27	59	the	the	DET
cana-527	27	60	cartesian	cartesian	ADJ
cana-527	27	61	product	product	NOUN
cana-527	27	62	s	s	PART
cana-527	27	63	xs	xs	PROPN
cana-527	27	64	.	.	PUNCT
cana-527	28	1	in	in	ADP
cana-527	28	2	another	another	DET
cana-527	28	3	way	way	NOUN
cana-527	28	4	,	,	PUNCT
cana-527	28	5	it	it	PRON
cana-527	28	6	was	be	AUX
cana-527	28	7	assumed	assume	VERB
cana-527	28	8	that	that	SCONJ
cana-527	28	9	the	the	DET
cana-527	28	10	graph	graph	NOUN
cana-527	28	11	would	would	AUX
cana-527	28	12	have	have	VERB
cana-527	28	13	loops	loop	NOUN
cana-527	28	14	at	at	ADP
cana-527	28	15	each	each	DET
cana-527	28	16	vertex	vertex	NOUN
cana-527	28	17	.	.	PUNCT
cana-527	29	1	if	if	SCONJ
cana-527	29	2	a	a	DET
cana-527	29	3	loop	loop	NOUN
cana-527	29	4	exists	exist	VERB
cana-527	29	5	at	at	ADP
cana-527	29	6	vertex	vertex	NOUN
cana-527	29	7	a	a	PRON
cana-527	29	8	in	in	ADP
cana-527	29	9	the	the	DET
cana-527	29	10	graph	graph	NOUN
cana-527	29	11	,	,	PUNCT
cana-527	29	12	then	then	ADV
cana-527	29	13	that	that	DET
cana-527	29	14	vertex	vertex	NOUN
cana-527	29	15	is	be	AUX
cana-527	29	16	the	the	DET
cana-527	29	17	fixed	fixed	ADJ
cana-527	29	18	point	point	NOUN
cana-527	29	19	of	of	ADP
cana-527	29	20	the	the	DET
cana-527	29	21	mapping	mapping	NOUN
cana-527	29	22	under	under	ADP
cana-527	29	23	consideration	consideration	NOUN
cana-527	29	24	.	.	PUNCT
cana-527	30	1	the	the	DET
cana-527	30	2	iteration	iteration	NOUN
cana-527	30	3	func	func	NOUN
cana-527	30	4	tion	tion	NOUN
cana-527	30	5	is	be	AUX
cana-527	30	6	necessary	necessary	ADJ
cana-527	30	7	to	to	PART
cana-527	30	8	take	take	VERB
cana-527	30	9	the	the	DET
cana-527	30	10	above	above	ADJ
cana-527	30	11	graph	graph	NOUN
cana-527	30	12	and	and	CCONJ
cana-527	30	13	turn	turn	VERB
cana-527	30	14	it	it	PRON
cana-527	30	15	into	into	ADP
cana-527	30	16	a	a	DET
cana-527	30	17	subgraph	subgraph	NOUN
cana-527	30	18	that	that	PRON
cana-527	30	19	shows	show	VERB
cana-527	30	20	various	various	ADJ
cana-527	30	21	contraction	contraction	NOUN
cana-527	30	22	concepts	concept	NOUN
cana-527	30	23	.	.	PUNCT
cana-527	31	1	the	the	DET
cana-527	31	2	behavior	behavior	NOUN
cana-527	31	3	of	of	ADP
cana-527	31	4	mappings	mapping	NOUN
cana-527	31	5	inside	inside	ADP
cana-527	31	6	the	the	DET
cana-527	31	7	specified	specify	VERB
cana-527	31	8	graph	graph	NOUN
cana-527	31	9	structure	structure	NOUN
cana-527	31	10	can	can	AUX
cana-527	31	11	be	be	AUX
cana-527	31	12	better	well	ADV
cana-527	31	13	seen	see	VERB
cana-527	31	14	and	and	CCONJ
cana-527	31	15	understood	understand	VERB
cana-527	31	16	thanks	thank	NOUN
cana-527	31	17	to	to	ADP
cana-527	31	18	this	this	DET
cana-527	31	19	iterative	iterative	NOUN
cana-527	31	20	procedure	procedure	NOUN
cana-527	31	21	.	.	PUNCT
cana-527	32	1	researchers	researcher	NOUN
cana-527	32	2	can	can	AUX
cana-527	32	3	investigate	investigate	VERB
cana-527	32	4	fixed	fix	VERB
cana-527	32	5	point	point	NOUN
cana-527	32	6	theorems	theorem	NOUN
cana-527	32	7	in	in	ADP
cana-527	32	8	a	a	DET
cana-527	32	9	more	more	ADV
cana-527	32	10	complex	complex	ADJ
cana-527	32	11	context	context	NOUN
cana-527	32	12	—	—	PUNCT
cana-527	32	13	one	one	NUM
cana-527	32	14	in	in	ADP
cana-527	32	15	which	which	PRON
cana-527	32	16	the	the	DET
cana-527	32	17	inter	inter	ADJ
cana-527	32	18	actions	action	NOUN
cana-527	32	19	between	between	ADP
cana-527	32	20	points	point	NOUN
cana-527	32	21	are	be	AUX
cana-527	32	22	impacted	impact	VERB
cana-527	32	23	by	by	ADP
cana-527	32	24	the	the	DET
cana-527	32	25	underlying	underlie	VERB
cana-527	32	26	graph	graph	NOUN
cana-527	32	27	structure	structure	NOUN
cana-527	32	28	in	in	ADP
cana-527	32	29	addition	addition	NOUN
cana-527	32	30	to	to	ADP
cana-527	32	31	metric	metric	ADJ
cana-527	32	32	properties	property	NOUN
cana-527	32	33	[	[	X
cana-527	32	34	22	22	NUM
cana-527	32	35	]	]	PUNCT
cana-527	32	36	—	—	PUNCT
cana-527	32	37	by	by	ADP
cana-527	32	38	combining	combine	VERB
cana-527	32	39	these	these	DET
cana-527	32	40	graph	graph	NOUN
cana-527	32	41	-	-	PUNCT
cana-527	32	42	based	base	VERB
cana-527	32	43	concerns	concern	NOUN
cana-527	32	44	.	.	PUNCT
cana-527	33	1	using	use	VERB
cana-527	33	2	the	the	DET
cana-527	33	3	iteration	iteration	NOUN
cana-527	33	4	function	function	NOUN
cana-527	33	5	,	,	PUNCT
cana-527	33	6	a	a	DET
cana-527	33	7	subgraph	subgraph	NOUN
cana-527	33	8	of	of	ADP
cana-527	33	9	the	the	DET
cana-527	33	10	aforementioned	aforementioned	ADJ
cana-527	33	11	graph	graph	NOUN
cana-527	33	12	is	be	AUX
cana-527	33	13	generated	generate	VERB
cana-527	33	14	to	to	PART
cana-527	33	15	demonstrate	demonstrate	VERB
cana-527	33	16	different	different	ADJ
cana-527	33	17	contraction	contraction	NOUN
cana-527	33	18	concepts	concept	NOUN
cana-527	33	19	.	.	PUNCT
cana-527	34	1	ii	ii	PROPN
cana-527	34	2	preliminaries	preliminary	NOUN
cana-527	34	3	a	a	DET
cana-527	34	4	mathematical	mathematical	ADJ
cana-527	34	5	framework	framework	NOUN
cana-527	34	6	known	know	VERB
cana-527	34	7	as	as	ADP
cana-527	34	8	banach	banach	ADV
cana-527	34	9	hyperspace	hyperspace	NOUN
cana-527	34	10	(	(	PUNCT
cana-527	34	11	in	in	ADP
cana-527	34	12	short	short	ADJ
cana-527	34	13	,	,	PUNCT
cana-527	34	14	bhs	bhs	PROPN
cana-527	34	15	)	)	PUNCT
cana-527	34	16	allows	allow	VERB
cana-527	34	17	one	one	NUM
cana-527	34	18	to	to	PART
cana-527	34	19	analyse	analyse	VERB
cana-527	34	20	the	the	DET
cana-527	34	21	characteristics	characteristic	NOUN
cana-527	34	22	and	and	CCONJ
cana-527	34	23	connections	connection	NOUN
cana-527	34	24	between	between	ADP
cana-527	34	25	compact	compact	ADJ
cana-527	34	26	sets	set	NOUN
cana-527	34	27	in	in	ADP
cana-527	34	28	metric	metric	ADJ
cana-527	34	29	spaces	space	NOUN
cana-527	34	30	by	by	ADP
cana-527	34	31	applying	apply	VERB
cana-527	34	32	the	the	DET
cana-527	34	33	methods	method	NOUN
cana-527	34	34	and	and	CCONJ
cana-527	34	35	structures	structure	NOUN
cana-527	34	36	of	of	ADP
cana-527	34	37	banach	banach	NOUN
cana-527	34	38	spaces	space	NOUN
cana-527	34	39	.	.	PUNCT
cana-527	35	1	these	these	DET
cana-527	35	2	spaces	space	NOUN
cana-527	35	3	find	find	VERB
cana-527	35	4	use	use	NOUN
cana-527	35	5	in	in	ADP
cana-527	35	6	many	many	ADJ
cana-527	35	7	different	different	ADJ
cana-527	35	8	areas	area	NOUN
cana-527	35	9	of	of	ADP
cana-527	35	10	mathematics	mathematic	NOUN
cana-527	35	11	,	,	PUNCT
cana-527	35	12	such	such	ADJ
cana-527	35	13	as	as	ADP
cana-527	35	14	topology	topology	NOUN
cana-527	35	15	,	,	PUNCT
cana-527	35	16	functional	functional	ADJ
cana-527	35	17	analysis	analysis	NOUN
cana-527	35	18	,	,	PUNCT
cana-527	35	19	and	and	CCONJ
cana-527	35	20	set	set	NOUN
cana-527	35	21	-	-	PUNCT
cana-527	35	22	valued	value	VERB
cana-527	35	23	analysis	analysis	NOUN
cana-527	35	24	.	.	PUNCT
cana-527	36	1	let	let	VERB
cana-527	36	2	us	we	PRON
cana-527	36	3	examine	examine	VERB
cana-527	36	4	a	a	DET
cana-527	36	5	banach	banach	NOUN
cana-527	36	6	hyperspace	hyperspace	NOUN
cana-527	36	7	,	,	PUNCT
cana-527	36	8	(	(	PUNCT
cana-527	36	9	k(s	k(s	PROPN
cana-527	36	10	)	)	PUNCT
cana-527	36	11	,	,	PUNCT
cana-527	36	12	|	|	ADV
cana-527	36	13	·	·	PUNCT
cana-527	36	14	|	|	X
cana-527	36	15	)	)	PUNCT
cana-527	36	16	,	,	PUNCT
cana-527	36	17	where	where	SCONJ
cana-527	36	18	the	the	DET
cana-527	36	19	space	space	NOUN
cana-527	36	20	k(s	k(s	PROPN
cana-527	36	21	)	)	PUNCT
cana-527	36	22	is	be	AUX
cana-527	36	23	the	the	DET
cana-527	36	24	col	col	NOUN
cana-527	36	25	lection	lection	NOUN
cana-527	36	26	of	of	ADP
cana-527	36	27	all	all	DET
cana-527	36	28	closed	closed	ADJ
cana-527	36	29	subsets	subset	NOUN
cana-527	36	30	of	of	ADP
cana-527	36	31	a	a	DET
cana-527	36	32	metric	metric	ADJ
cana-527	36	33	space	space	NOUN
cana-527	36	34	s	s	NOUN
cana-527	36	35	that	that	PRON
cana-527	36	36	are	be	AUX
cana-527	36	37	not	not	PART
cana-527	36	38	empty	empty	ADJ
cana-527	36	39	.	.	PUNCT
cana-527	37	1	the	the	DET
cana-527	37	2	norm	norm	NOUN
cana-527	37	3	of	of	ADP
cana-527	37	4	this	this	DET
cana-527	37	5	space	space	NOUN
cana-527	37	6	is	be	AUX
cana-527	37	7	|	|	ADV
cana-527	37	8	·	·	PUNCT
cana-527	38	1	|	|	ADV
cana-527	38	2	,	,	PUNCT
cana-527	38	3	and	and	CCONJ
cana-527	38	4	it	it	PRON
cana-527	38	5	is	be	AUX
cana-527	38	6	typically	typically	ADV
cana-527	38	7	defined	define	VERB
cana-527	38	8	with	with	ADP
cana-527	38	9	the	the	DET
cana-527	38	10	help	help	NOUN
cana-527	38	11	of	of	ADP
cana-527	38	12	the	the	DET
cana-527	38	13	hausdorff	hausdorff	NOUN
cana-527	38	14	metric	metric	NOUN
cana-527	38	15	.	.	PUNCT
cana-527	39	1	the	the	DET
cana-527	39	2	”	"	PUNCT
cana-527	39	3	close	close	ADJ
cana-527	39	4	ness	ness	NOUN
cana-527	39	5	”	"	PUNCT
cana-527	39	6	between	between	ADP
cana-527	39	7	two	two	NUM
cana-527	39	8	sets	set	NOUN
cana-527	39	9	can	can	AUX
cana-527	39	10	be	be	AUX
cana-527	39	11	expressed	express	VERB
cana-527	39	12	in	in	ADP
cana-527	39	13	terms	term	NOUN
cana-527	39	14	of	of	ADP
cana-527	39	15	their	their	PRON
cana-527	39	16	hausdorff	hausdorff	NOUN
cana-527	39	17	distances	distance	NOUN
cana-527	39	18	using	use	VERB
cana-527	39	19	the	the	DET
cana-527	39	20	hausdorff	hausdorff	NOUN
cana-527	39	21	metric	metric	NOUN
cana-527	39	22	.	.	PUNCT
cana-527	40	1	let	let	VERB
cana-527	40	2	ω	ω	NOUN
cana-527	40	3	now	now	ADV
cana-527	40	4	be	be	AUX
cana-527	40	5	a	a	DET
cana-527	40	6	null	null	ADJ
cana-527	40	7	set	set	NOUN
cana-527	40	8	.	.	PUNCT
cana-527	41	1	we	we	PRON
cana-527	41	2	wish	wish	VERB
cana-527	41	3	to	to	PART
cana-527	41	4	show	show	VERB
cana-527	41	5	that	that	SCONJ
cana-527	41	6	the	the	DET
cana-527	41	7	null	null	ADJ
cana-527	41	8	equality	equality	NOUN
cana-527	41	9	is	be	AUX
cana-527	41	10	satisfied	satisfied	ADJ
cana-527	41	11	by	by	ADP
cana-527	41	12	the	the	DET
cana-527	41	13	norm	norm	NOUN
cana-527	41	14	|	|	ADV
cana-527	41	15	·	·	PUNCT
cana-527	42	1	|	|	INTJ
cana-527	42	2	.	.	PUNCT
cana-527	43	1	this	this	PRON
cana-527	43	2	indicates	indicate	VERB
cana-527	43	3	,	,	PUNCT
cana-527	43	4	in	in	ADP
cana-527	43	5	mathematical	mathematical	ADJ
cana-527	43	6	words	word	NOUN
cana-527	43	7	,	,	PUNCT
cana-527	43	8	that	that	SCONJ
cana-527	43	9	for	for	ADP
cana-527	43	10	any	any	DET
cana-527	43	11	set	set	NOUN
cana-527	43	12	a	a	PRON
cana-527	43	13	that	that	PRON
cana-527	43	14	is	be	AUX
cana-527	43	15	a	a	DET
cana-527	43	16	member	member	NOUN
cana-527	43	17	of	of	ADP
cana-527	43	18	the	the	DET
cana-527	43	19	null	null	ADJ
cana-527	43	20	set	set	NOUN
cana-527	43	21	ω	ω	PROPN
cana-527	43	22	,	,	PUNCT
cana-527	43	23	|a|	|a|	PROPN
cana-527	43	24	=	=	NOUN
cana-527	43	25	0	0	PROPN
cana-527	43	26	.	.	PUNCT
cana-527	43	27	to	to	PART
cana-527	43	28	elucidate	elucidate	VERB
cana-527	43	29	,	,	PUNCT
cana-527	43	30	let	let	VERB
cana-527	43	31	us	we	PRON
cana-527	43	32	examine	examine	VERB
cana-527	43	33	many	many	ADJ
cana-527	43	34	fundamental	fundamental	ADJ
cana-527	43	35	ideas	idea	NOUN
cana-527	43	36	concerning	concern	VERB
cana-527	43	37	banach	banach	NOUN
cana-527	43	38	hyperspaces	hyperspace	NOUN
cana-527	43	39	:	:	PUNCT
cana-527	43	40	1	1	X
cana-527	43	41	.	.	X
cana-527	43	42	banach	banach	ADJ
cana-527	43	43	hyperspace	hyperspace	PROPN
cana-527	43	44	sequences	sequence	NOUN
cana-527	43	45	:	:	PUNCT
cana-527	44	1	[	[	X
cana-527	44	2	14][15][16	14][15][16	X
cana-527	44	3	]	]	X
cana-527	44	4	sequences	sequence	NOUN
cana-527	44	5	of	of	ADP
cana-527	44	6	sets	set	NOUN
cana-527	44	7	in	in	ADP
cana-527	44	8	the	the	DET
cana-527	44	9	banach	banach	NOUN
cana-527	44	10	hyperspace	hyperspace	NOUN
cana-527	44	11	can	can	AUX
cana-527	44	12	be	be	AUX
cana-527	44	13	studied	study	VERB
cana-527	44	14	,	,	PUNCT
cana-527	44	15	and	and	CCONJ
cana-527	44	16	their	their	PRON
cana-527	44	17	convergence	convergence	NOUN
cana-527	44	18	qualities	quality	NOUN
cana-527	44	19	under	under	ADP
cana-527	44	20	a	a	DET
cana-527	44	21	selected	select	VERB
cana-527	44	22	norm	norm	NOUN
cana-527	44	23	can	can	AUX
cana-527	44	24	be	be	AUX
cana-527	44	25	examined	examine	VERB
cana-527	44	26	.	.	PUNCT
cana-527	45	1	this	this	PRON
cana-527	45	2	is	be	AUX
cana-527	45	3	figuring	figure	VERB
cana-527	45	4	out	out	ADP
cana-527	45	5	when	when	SCONJ
cana-527	45	6	a	a	DET
cana-527	45	7	series	series	NOUN
cana-527	45	8	of	of	ADP
cana-527	45	9	sets	set	NOUN
cana-527	45	10	in	in	ADP
cana-527	45	11	the	the	DET
cana-527	45	12	hyperspace	hyperspace	NOUN
cana-527	45	13	converge	converge	VERB
cana-527	45	14	to	to	ADP
cana-527	45	15	a	a	DET
cana-527	45	16	limit	limit	NOUN
cana-527	45	17	set	set	VERB
cana-527	45	18	.	.	PUNCT
cana-527	46	1	2	2	X
cana-527	46	2	.	.	X
cana-527	46	3	graphs	graph	NOUN
cana-527	46	4	and	and	CCONJ
cana-527	46	5	sub	sub	NOUN
cana-527	46	6	-	-	NOUN
cana-527	46	7	graphs	graph	NOUN
cana-527	46	8	:	:	PUNCT
cana-527	46	9	graphs	graph	NOUN
cana-527	46	10	connected	connect	VERB
cana-527	46	11	to	to	ADP
cana-527	46	12	sets	set	NOUN
cana-527	46	13	in	in	ADP
cana-527	46	14	the	the	DET
cana-527	46	15	metric	metric	ADJ
cana-527	46	16	space	space	NOUN
cana-527	46	17	can	can	AUX
cana-527	46	18	be	be	AUX
cana-527	46	19	examined	examine	VERB
cana-527	46	20	thanks	thank	NOUN
cana-527	46	21	to	to	ADP
cana-527	46	22	the	the	DET
cana-527	46	23	ba	ba	PROPN
cana-527	46	24	nach	nach	PROPN
cana-527	46	25	hyperspace	hyperspace	NOUN
cana-527	46	26	.	.	PUNCT
cana-527	47	1	investigating	investigate	VERB
cana-527	47	2	sub	sub	NOUN
cana-527	47	3	-	-	NOUN
cana-527	47	4	graphs	graph	NOUN
cana-527	47	5	and	and	CCONJ
cana-527	47	6	their	their	PRON
cana-527	47	7	characteristics	characteristic	NOUN
cana-527	47	8	is	be	AUX
cana-527	47	9	one	one	NUM
cana-527	47	10	way	way	NOUN
cana-527	47	11	to	to	PART
cana-527	47	12	learn	learn	VERB
cana-527	47	13	about	about	ADP
cana-527	47	14	the	the	DET
cana-527	47	15	organisation	organisation	NOUN
cana-527	47	16	and	and	CCONJ
cana-527	47	17	connectivity	connectivity	NOUN
cana-527	47	18	of	of	ADP
cana-527	47	19	compact	compact	ADJ
cana-527	47	20	collections	collection	NOUN
cana-527	47	21	.	.	PUNCT
cana-527	48	1	communications	communication	NOUN
cana-527	48	2	on	on	ADP
cana-527	48	3	applied	apply	VERB
cana-527	48	4	nonlinear	nonlinear	ADJ
cana-527	48	5	analysis	analysis	NOUN
cana-527	48	6	issn	issn	NOUN
cana-527	48	7	:	:	PUNCT
cana-527	48	8	1074	1074	NUM
cana-527	48	9	-	-	PUNCT
cana-527	48	10	133x	133x	NUM
cana-527	48	11	vol	vol	NOUN
cana-527	48	12	31	31	NUM
cana-527	48	13	no	no	NOUN
cana-527	48	14	.	.	NOUN
cana-527	48	15	2	2	NUM
cana-527	48	16	(	(	PUNCT
cana-527	48	17	2024	2024	NUM
cana-527	48	18	)	)	PUNCT
cana-527	49	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	49	2	158	158	NUM
cana-527	49	3	3	3	NUM
cana-527	49	4	.	.	PUNCT
cana-527	49	5	near	near	ADP
cana-527	49	6	-	-	PUNCT
cana-527	49	7	static	static	ADJ
cana-527	49	8	points	point	NOUN
cana-527	49	9	:	:	PUNCT
cana-527	49	10	in	in	ADP
cana-527	49	11	order	order	NOUN
cana-527	49	12	to	to	PART
cana-527	49	13	study	study	VERB
cana-527	49	14	near	near	ADV
cana-527	49	15	-	-	PUNCT
cana-527	49	16	fixed	fix	VERB
cana-527	49	17	points	point	NOUN
cana-527	49	18	,	,	PUNCT
cana-527	49	19	one	one	PRON
cana-527	49	20	must	must	AUX
cana-527	49	21	comprehend	comprehend	VERB
cana-527	49	22	sets	set	NOUN
cana-527	49	23	that	that	SCONJ
cana-527	49	24	,	,	PUNCT
cana-527	49	25	under	under	ADP
cana-527	49	26	a	a	DET
cana-527	49	27	par	par	ADJ
cana-527	49	28	ticular	ticular	ADJ
cana-527	49	29	transformation	transformation	NOUN
cana-527	49	30	or	or	CCONJ
cana-527	49	31	mapping	mapping	NOUN
cana-527	49	32	,	,	PUNCT
cana-527	49	33	are	be	AUX
cana-527	49	34	almost	almost	ADV
cana-527	49	35	fixed	fix	VERB
cana-527	49	36	.	.	PUNCT
cana-527	50	1	the	the	DET
cana-527	50	2	stability	stability	NOUN
cana-527	50	3	of	of	ADP
cana-527	50	4	sets	set	NOUN
cana-527	50	5	under	under	ADP
cana-527	50	6	specific	specific	ADJ
cana-527	50	7	operations	operation	NOUN
cana-527	50	8	or	or	CCONJ
cana-527	50	9	mappings	mapping	NOUN
cana-527	50	10	can	can	AUX
cana-527	50	11	be	be	AUX
cana-527	50	12	studied	study	VERB
cana-527	50	13	in	in	ADP
cana-527	50	14	the	the	DET
cana-527	50	15	context	context	NOUN
cana-527	50	16	of	of	ADP
cana-527	50	17	banach	banach	NOUN
cana-527	50	18	hyper	hyper	ADJ
cana-527	50	19	spaces	space	NOUN
cana-527	50	20	.	.	PUNCT
cana-527	51	1	the	the	DET
cana-527	51	2	behaviour	behaviour	NOUN
cana-527	51	3	and	and	CCONJ
cana-527	51	4	interactions	interaction	NOUN
cana-527	51	5	of	of	ADP
cana-527	51	6	compact	compact	ADJ
cana-527	51	7	sets	set	NOUN
cana-527	51	8	in	in	ADP
cana-527	51	9	metric	metric	ADJ
cana-527	51	10	spaces	space	NOUN
cana-527	51	11	can	can	AUX
cana-527	51	12	be	be	AUX
cana-527	51	13	better	well	ADV
cana-527	51	14	understood	understand	VERB
cana-527	51	15	by	by	ADP
cana-527	51	16	mathematicians	mathematician	NOUN
cana-527	51	17	[	[	X
cana-527	51	18	12][14][15][16][22	12][14][15][16][22	X
cana-527	51	19	]	]	PUNCT
cana-527	51	20	by	by	ADP
cana-527	51	21	exploring	explore	VERB
cana-527	51	22	these	these	DET
cana-527	51	23	ideas	idea	NOUN
cana-527	51	24	within	within	ADP
cana-527	51	25	the	the	DET
cana-527	51	26	context	context	NOUN
cana-527	51	27	of	of	ADP
cana-527	51	28	the	the	DET
cana-527	51	29	banach	banach	NOUN
cana-527	51	30	hyperspace	hyperspace	NOUN
cana-527	51	31	framework	framework	NOUN
cana-527	51	32	.	.	PUNCT
cana-527	52	1	an	an	DET
cana-527	52	2	essential	essential	ADJ
cana-527	52	3	component	component	NOUN
cana-527	52	4	is	be	AUX
cana-527	52	5	the	the	DET
cana-527	52	6	proof	proof	NOUN
cana-527	52	7	of	of	ADP
cana-527	52	8	null	null	ADJ
cana-527	52	9	equality	equality	NOUN
cana-527	52	10	,	,	PUNCT
cana-527	52	11	which	which	PRON
cana-527	52	12	guarantees	guarantee	VERB
cana-527	52	13	that	that	SCONJ
cana-527	52	14	the	the	DET
cana-527	52	15	norm	norm	NOUN
cana-527	52	16	accurately	accurately	ADV
cana-527	52	17	describes	describe	VERB
cana-527	52	18	the	the	DET
cana-527	52	19	characteristics	characteristic	NOUN
cana-527	52	20	of	of	ADP
cana-527	52	21	sets	set	NOUN
cana-527	52	22	in	in	ADP
cana-527	52	23	the	the	DET
cana-527	52	24	banach	banach	ADJ
cana-527	52	25	hyperspace	hyperspace	NOUN
cana-527	52	26	.	.	PUNCT
cana-527	53	1	2.1	2.1	NUM
cana-527	53	2	sequences	sequence	NOUN
cana-527	53	3	in	in	ADP
cana-527	53	4	functional	functional	ADJ
cana-527	53	5	analysis	analysis	NOUN
cana-527	53	6	,	,	PUNCT
cana-527	53	7	sequences	sequence	NOUN
cana-527	53	8	are	be	AUX
cana-527	53	9	essential	essential	ADJ
cana-527	53	10	tools	tool	NOUN
cana-527	53	11	for	for	ADP
cana-527	53	12	examining	examine	VERB
cana-527	53	13	the	the	DET
cana-527	53	14	characteris	characteris	NOUN
cana-527	53	15	tics	tic	NOUN
cana-527	53	16	of	of	ADP
cana-527	53	17	operators	operator	NOUN
cana-527	53	18	and	and	CCONJ
cana-527	53	19	functions	function	NOUN
cana-527	53	20	defined	define	VERB
cana-527	53	21	on	on	ADP
cana-527	53	22	banach	banach	NOUN
cana-527	53	23	spaces	space	NOUN
cana-527	53	24	.	.	PUNCT
cana-527	54	1	understanding	understand	VERB
cana-527	54	2	the	the	DET
cana-527	54	3	behaviour	behaviour	NOUN
cana-527	54	4	of	of	ADP
cana-527	54	5	functions	function	NOUN
cana-527	54	6	and	and	CCONJ
cana-527	54	7	the	the	DET
cana-527	54	8	boundaries	boundary	NOUN
cana-527	54	9	of	of	ADP
cana-527	54	10	different	different	ADJ
cana-527	54	11	operations	operation	NOUN
cana-527	54	12	in	in	ADP
cana-527	54	13	these	these	DET
cana-527	54	14	spaces	space	NOUN
cana-527	54	15	requires	require	VERB
cana-527	54	16	an	an	DET
cana-527	54	17	un	un	PROPN
cana-527	54	18	derstanding	derstanding	NOUN
cana-527	54	19	of	of	ADP
cana-527	54	20	the	the	DET
cana-527	54	21	convergence	convergence	NOUN
cana-527	54	22	of	of	ADP
cana-527	54	23	sequences	sequence	NOUN
cana-527	54	24	.	.	PUNCT
cana-527	55	1	sequences	sequence	NOUN
cana-527	55	2	are	be	AUX
cana-527	55	3	important	important	ADJ
cana-527	55	4	in	in	ADP
cana-527	55	5	the	the	DET
cana-527	55	6	setting	setting	NOUN
cana-527	55	7	of	of	ADP
cana-527	55	8	banach	banach	NOUN
cana-527	55	9	spaces	space	NOUN
cana-527	55	10	,	,	PUNCT
cana-527	55	11	especially	especially	ADV
cana-527	55	12	when	when	SCONJ
cana-527	55	13	talking	talk	VERB
cana-527	55	14	about	about	ADP
cana-527	55	15	convergence	convergence	NOUN
cana-527	55	16	and	and	CCONJ
cana-527	55	17	completeness	completeness	NOUN
cana-527	55	18	.	.	PUNCT
cana-527	56	1	a	a	DET
cana-527	56	2	description	description	NOUN
cana-527	56	3	of	of	ADP
cana-527	56	4	sequences	sequence	NOUN
cana-527	56	5	in	in	ADP
cana-527	56	6	banach	banach	NOUN
cana-527	56	7	spaces	space	NOUN
cana-527	56	8	is	be	AUX
cana-527	56	9	as	as	SCONJ
cana-527	56	10	follows	follow	VERB
cana-527	56	11	:	:	PUNCT
cana-527	56	12	definition	definition	NOUN
cana-527	56	13	2.1.1	2.1.1	NUM
cana-527	56	14	:	:	PUNCT
cana-527	56	15	a	a	DET
cana-527	56	16	sequence	sequence	NOUN
cana-527	56	17	{	{	PUNCT
cana-527	56	18	yn}∞n=1	yn}∞n=1	PROPN
cana-527	56	19	is	be	AUX
cana-527	56	20	said	say	VERB
cana-527	56	21	to	to	PART
cana-527	56	22	be	be	AUX
cana-527	56	23	convergent	convergent	ADJ
cana-527	56	24	in	in	ADP
cana-527	56	25	the	the	DET
cana-527	56	26	banach	banach	ADJ
cana-527	56	27	hyperspace	hyperspace	NOUN
cana-527	56	28	(	(	PUNCT
cana-527	56	29	k(s	k(s	PROPN
cana-527	56	30	)	)	PUNCT
cana-527	56	31	,	,	PUNCT
cana-527	56	32	∥.∥	∥.∥	NUM
cana-527	56	33	)	)	PUNCT
cana-527	57	1	if	if	SCONJ
cana-527	57	2	,	,	PUNCT
cana-527	57	3	given	give	VERB
cana-527	57	4	a	a	DET
cana-527	57	5	ε	ε	PROPN
cana-527	57	6	>	>	X
cana-527	57	7	0	0	PROPN
cana-527	57	8	,	,	PUNCT
cana-527	57	9	there	there	PRON
cana-527	57	10	exists	exist	VERB
cana-527	57	11	n1	n1	PROPN
cana-527	57	12	∈	∈	PROPN
cana-527	57	13	i	i	PRON
cana-527	57	14	such	such	ADJ
cana-527	57	15	that	that	SCONJ
cana-527	57	16	∥	∥	PROPN
cana-527	58	1	yn	yn	NOUN
cana-527	58	2	−	−	NOUN
cana-527	58	3	y	y	NOUN
cana-527	58	4	∥	∥	X
cana-527	58	5	<	<	X
cana-527	58	6	ε	ε	PROPN
cana-527	58	7	,	,	PUNCT
cana-527	58	8	n	n	PRON
cana-527	58	9	≥	≥	NOUN
cana-527	58	10	n1	n1	NOUN
cana-527	58	11	.	.	PUNCT
cana-527	59	1	2.2	2.2	NUM
cana-527	59	2	graphs	graph	NOUN
cana-527	59	3	a	a	DET
cana-527	59	4	graph	graph	NOUN
cana-527	59	5	is	be	AUX
cana-527	59	6	made	make	VERB
cana-527	59	7	up	up	ADP
cana-527	59	8	of	of	ADP
cana-527	59	9	nodes	node	NOUN
cana-527	59	10	,	,	PUNCT
cana-527	59	11	or	or	CCONJ
cana-527	59	12	vertices	vertex	NOUN
cana-527	59	13	,	,	PUNCT
cana-527	59	14	and	and	CCONJ
cana-527	59	15	the	the	DET
cana-527	59	16	edges	edge	NOUN
cana-527	59	17	that	that	PRON
cana-527	59	18	join	join	VERB
cana-527	59	19	node	node	ADJ
cana-527	59	20	pairs	pair	NOUN
cana-527	59	21	.	.	PUNCT
cana-527	60	1	a	a	DET
cana-527	60	2	variety	variety	NOUN
cana-527	60	3	of	of	ADP
cana-527	60	4	ideas	idea	NOUN
cana-527	60	5	and	and	CCONJ
cana-527	60	6	structures	structure	NOUN
cana-527	60	7	are	be	AUX
cana-527	60	8	involved	involve	VERB
cana-527	60	9	in	in	ADP
cana-527	60	10	the	the	DET
cana-527	60	11	study	study	NOUN
cana-527	60	12	of	of	ADP
cana-527	60	13	graphs	graph	NOUN
cana-527	60	14	.	.	PUNCT
cana-527	61	1	analysing	analyse	VERB
cana-527	61	2	and	and	CCONJ
cana-527	61	3	com	com	NOUN
cana-527	61	4	prehending	prehending	NOUN
cana-527	61	5	graphs	graph	NOUN
cana-527	61	6	requires	require	VERB
cana-527	61	7	investigating	investigate	VERB
cana-527	61	8	attributes	attribute	NOUN
cana-527	61	9	such	such	ADJ
cana-527	61	10	as	as	ADP
cana-527	61	11	routes	route	NOUN
cana-527	61	12	,	,	PUNCT
cana-527	61	13	cycles	cycle	NOUN
cana-527	61	14	,	,	PUNCT
cana-527	61	15	connectivity	connectivity	NOUN
cana-527	61	16	,	,	PUNCT
cana-527	61	17	and	and	CCONJ
cana-527	61	18	the	the	DET
cana-527	61	19	structural	structural	ADJ
cana-527	61	20	makeup	makeup	NOUN
cana-527	61	21	of	of	ADP
cana-527	61	22	nodes	node	NOUN
cana-527	61	23	and	and	CCONJ
cana-527	61	24	edges	edge	NOUN
cana-527	61	25	.	.	PUNCT
cana-527	62	1	some	some	DET
cana-527	62	2	necessary	necessary	ADJ
cana-527	62	3	notions	notion	NOUN
cana-527	62	4	related	relate	VERB
cana-527	62	5	to	to	ADP
cana-527	62	6	graphs	graph	NOUN
cana-527	62	7	are	be	AUX
cana-527	62	8	listed	list	VERB
cana-527	62	9	below	below	ADP
cana-527	62	10	:	:	PUNCT
cana-527	62	11	definition	definition	NOUN
cana-527	62	12	2.2.1	2.2.1	NUM
cana-527	62	13	:	:	PUNCT
cana-527	62	14	an	an	DET
cana-527	62	15	ordered	order	VERB
cana-527	62	16	pair	pair	NOUN
cana-527	62	17	(	(	PUNCT
cana-527	62	18	v	v	NOUN
cana-527	62	19	,	,	PUNCT
cana-527	62	20	e	e	NOUN
cana-527	62	21	)	)	PUNCT
cana-527	62	22	,	,	PUNCT
cana-527	62	23	with	with	ADP
cana-527	62	24	v	v	NUM
cana-527	62	25	representing	represent	VERB
cana-527	62	26	the	the	DET
cana-527	62	27	set	set	NOUN
cana-527	62	28	of	of	ADP
cana-527	62	29	points	point	NOUN
cana-527	62	30	known	know	VERB
cana-527	62	31	as	as	ADP
cana-527	62	32	vertices	vertex	NOUN
cana-527	62	33	and	and	CCONJ
cana-527	62	34	e	e	NOUN
cana-527	62	35	representing	represent	VERB
cana-527	62	36	the	the	DET
cana-527	62	37	set	set	NOUN
cana-527	62	38	of	of	ADP
cana-527	62	39	lines	line	NOUN
cana-527	62	40	known	know	VERB
cana-527	62	41	as	as	ADP
cana-527	62	42	edges	edge	NOUN
cana-527	62	43	,	,	PUNCT
cana-527	62	44	constitutes	constitute	VERB
cana-527	62	45	a	a	DET
cana-527	62	46	graph	graph	NOUN
cana-527	62	47	h.	h.	NOUN
cana-527	62	48	definition	definition	NOUN
cana-527	62	49	2.2.2	2.2.2	NUM
cana-527	62	50	:	:	PUNCT
cana-527	62	51	a	a	DET
cana-527	62	52	graph	graph	NOUN
cana-527	62	53	h0	h0	NOUN
cana-527	62	54	=	=	SYM
cana-527	62	55	(	(	PUNCT
cana-527	62	56	v0	v0	PROPN
cana-527	62	57	,	,	PUNCT
cana-527	62	58	e0	e0	PROPN
cana-527	62	59	)	)	PUNCT
cana-527	62	60	is	be	AUX
cana-527	62	61	said	say	VERB
cana-527	62	62	to	to	PART
cana-527	62	63	be	be	AUX
cana-527	62	64	a	a	DET
cana-527	62	65	sub	sub	NOUN
cana-527	62	66	-	-	NOUN
cana-527	62	67	graph	graph	NOUN
cana-527	62	68	of	of	ADP
cana-527	62	69	h	h	NOUN
cana-527	62	70	=	=	SYM
cana-527	62	71	(	(	PUNCT
cana-527	62	72	v	v	NOUN
cana-527	62	73	,	,	PUNCT
cana-527	62	74	e	e	NOUN
cana-527	62	75	)	)	PUNCT
cana-527	62	76	if	if	SCONJ
cana-527	62	77	v0	v0	PROPN
cana-527	62	78	⊂	⊂	PROPN
cana-527	62	79	v	v	PROPN
cana-527	62	80	and	and	CCONJ
cana-527	62	81	e0	e0	PROPN
cana-527	63	1	⊂	⊂	PROPN
cana-527	63	2	e.	e.	PROPN
cana-527	64	1	we	we	PRON
cana-527	64	2	do	do	AUX
cana-527	64	3	not	not	PART
cana-527	64	4	like	like	VERB
cana-527	64	5	some	some	DET
cana-527	64	6	special	special	ADJ
cana-527	64	7	edge	edge	NOUN
cana-527	64	8	types	type	NOUN
cana-527	64	9	as	as	SCONJ
cana-527	64	10	they	they	PRON
cana-527	64	11	cause	cause	VERB
cana-527	64	12	complications	complication	NOUN
cana-527	64	13	in	in	ADP
cana-527	64	14	calculations	calculation	NOUN
cana-527	64	15	:	:	PUNCT
cana-527	64	16	communications	communication	NOUN
cana-527	64	17	on	on	ADP
cana-527	64	18	applied	apply	VERB
cana-527	64	19	nonlinear	nonlinear	ADJ
cana-527	64	20	analysis	analysis	NOUN
cana-527	64	21	issn	issn	NOUN
cana-527	64	22	:	:	PUNCT
cana-527	64	23	1074	1074	NUM
cana-527	64	24	-	-	PUNCT
cana-527	64	25	133x	133x	NUM
cana-527	64	26	vol	vol	NOUN
cana-527	64	27	31	31	NUM
cana-527	64	28	no	no	NOUN
cana-527	64	29	.	.	NOUN
cana-527	64	30	2	2	NUM
cana-527	64	31	(	(	PUNCT
cana-527	64	32	2024	2024	NUM
cana-527	64	33	)	)	PUNCT
cana-527	64	34	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	64	35	159	159	NUM
cana-527	64	36	definition	definition	NOUN
cana-527	64	37	2.2.3	2.2.3	NUM
cana-527	64	38	:	:	PUNCT
cana-527	64	39	if	if	SCONJ
cana-527	64	40	an	an	DET
cana-527	64	41	edge	edge	NOUN
cana-527	64	42	in	in	ADP
cana-527	64	43	a	a	DET
cana-527	64	44	graph	graph	NOUN
cana-527	64	45	h	h	NOUN
cana-527	64	46	has	have	VERB
cana-527	64	47	the	the	DET
cana-527	64	48	same	same	ADJ
cana-527	64	49	initial	initial	ADJ
cana-527	64	50	and	and	CCONJ
cana-527	64	51	terminal	terminal	ADJ
cana-527	64	52	vertices	vertex	NOUN
cana-527	64	53	,	,	PUNCT
cana-527	64	54	it	it	PRON
cana-527	64	55	is	be	AUX
cana-527	64	56	referred	refer	VERB
cana-527	64	57	to	to	ADP
cana-527	64	58	as	as	ADP
cana-527	64	59	a	a	DET
cana-527	64	60	loop	loop	NOUN
cana-527	64	61	.	.	PUNCT
cana-527	65	1	multiple	multiple	ADJ
cana-527	65	2	edges	edge	NOUN
cana-527	65	3	are	be	AUX
cana-527	65	4	those	those	PRON
cana-527	65	5	that	that	PRON
cana-527	65	6	connect	connect	VERB
cana-527	65	7	the	the	DET
cana-527	65	8	same	same	ADJ
cana-527	65	9	pair	pair	NOUN
cana-527	65	10	of	of	ADP
cana-527	65	11	vertices	vertex	NOUN
cana-527	65	12	with	with	ADP
cana-527	65	13	two	two	NUM
cana-527	65	14	or	or	CCONJ
cana-527	65	15	more	more	ADJ
cana-527	65	16	edges	edge	NOUN
cana-527	65	17	.	.	PUNCT
cana-527	66	1	a	a	DET
cana-527	66	2	simple	simple	ADJ
cana-527	66	3	graph	graph	NOUN
cana-527	66	4	is	be	AUX
cana-527	66	5	one	one	NUM
cana-527	66	6	that	that	PRON
cana-527	66	7	has	have	VERB
cana-527	66	8	neither	neither	CCONJ
cana-527	66	9	loops	loop	NOUN
cana-527	66	10	nor	nor	CCONJ
cana-527	66	11	many	many	ADJ
cana-527	66	12	edges	edge	NOUN
cana-527	66	13	.	.	PUNCT
cana-527	67	1	definition	definition	NOUN
cana-527	67	2	2.2.4	2.2.4	NUM
cana-527	67	3	:	:	PUNCT
cana-527	67	4	the	the	DET
cana-527	67	5	term	term	NOUN
cana-527	67	6	”	"	PUNCT
cana-527	67	7	weighted	weight	VERB
cana-527	67	8	graph	graph	NOUN
cana-527	67	9	”	"	PUNCT
cana-527	67	10	refers	refer	VERB
cana-527	67	11	to	to	ADP
cana-527	67	12	a	a	DET
cana-527	67	13	basic	basic	ADJ
cana-527	67	14	graph	graph	NOUN
cana-527	67	15	in	in	ADP
cana-527	67	16	which	which	PRON
cana-527	67	17	each	each	DET
cana-527	67	18	edge	edge	NOUN
cana-527	67	19	has	have	VERB
cana-527	67	20	a	a	DET
cana-527	67	21	nu	nu	ADJ
cana-527	67	22	merical	merical	ADJ
cana-527	67	23	value	value	NOUN
cana-527	67	24	assigned	assign	VERB
cana-527	67	25	to	to	ADP
cana-527	67	26	it	it	PRON
cana-527	67	27	.	.	PUNCT
cana-527	68	1	therefore	therefore	ADV
cana-527	68	2	,	,	PUNCT
cana-527	68	3	the	the	DET
cana-527	68	4	vertex	vertex	NOUN
cana-527	68	5	set	set	NOUN
cana-527	68	6	,	,	PUNCT
cana-527	68	7	edge	edge	NOUN
cana-527	68	8	set	set	NOUN
cana-527	68	9	,	,	PUNCT
cana-527	68	10	and	and	CCONJ
cana-527	68	11	weight	weight	NOUN
cana-527	68	12	of	of	ADP
cana-527	68	13	each	each	DET
cana-527	68	14	edge	edge	NOUN
cana-527	68	15	make	make	VERB
cana-527	68	16	up	up	ADP
cana-527	68	17	a	a	DET
cana-527	68	18	weighted	weight	VERB
cana-527	68	19	graph	graph	NOUN
cana-527	68	20	.	.	PUNCT
cana-527	69	1	2.3	2.3	NUM
cana-527	69	2	near	near	ADP
cana-527	69	3	fixed	fix	VERB
cana-527	69	4	point	point	NOUN
cana-527	69	5	theorems	theorem	VERB
cana-527	69	6	the	the	DET
cana-527	69	7	classical	classical	ADJ
cana-527	69	8	fixed	fix	VERB
cana-527	69	9	point	point	NOUN
cana-527	69	10	theorems	theorem	NOUN
cana-527	69	11	are	be	AUX
cana-527	69	12	extended	extend	VERB
cana-527	69	13	to	to	ADP
cana-527	69	14	the	the	DET
cana-527	69	15	context	context	NOUN
cana-527	69	16	of	of	ADP
cana-527	69	17	banach	banach	ADV
cana-527	69	18	hyper	hyper	ADJ
cana-527	69	19	spaces	space	NOUN
cana-527	69	20	by	by	ADP
cana-527	69	21	near	near	ADP
cana-527	69	22	fixed	fix	VERB
cana-527	69	23	point	point	NOUN
cana-527	69	24	theorems	theorem	NOUN
cana-527	69	25	.	.	PUNCT
cana-527	70	1	hyperspaces	hyperspace	NOUN
cana-527	70	2	are	be	AUX
cana-527	70	3	spaces	space	NOUN
cana-527	70	4	of	of	ADP
cana-527	70	5	closed	closed	ADJ
cana-527	70	6	sets	set	NOUN
cana-527	70	7	with	with	ADP
cana-527	70	8	an	an	DET
cana-527	70	9	appropriate	appropriate	ADJ
cana-527	70	10	topology	topology	NOUN
cana-527	70	11	.	.	PUNCT
cana-527	71	1	a	a	DET
cana-527	71	2	banach	banach	NOUN
cana-527	71	3	space	space	NOUN
cana-527	71	4	with	with	ADP
cana-527	71	5	non	non	ADJ
cana-527	71	6	-	-	ADJ
cana-527	71	7	empty	empty	ADJ
cana-527	71	8	closed	closed	ADJ
cana-527	71	9	subsets	subset	NOUN
cana-527	71	10	of	of	ADP
cana-527	71	11	a	a	DET
cana-527	71	12	specified	specify	VERB
cana-527	71	13	metric	metric	ADJ
cana-527	71	14	space	space	NOUN
cana-527	71	15	,	,	PUNCT
cana-527	71	16	frequently	frequently	ADV
cana-527	71	17	furnished	furnish	VERB
cana-527	71	18	with	with	ADP
cana-527	71	19	the	the	DET
cana-527	71	20	hausdorff	hausdorff	NOUN
cana-527	71	21	metric	metric	NOUN
cana-527	71	22	,	,	PUNCT
cana-527	71	23	is	be	AUX
cana-527	71	24	called	call	VERB
cana-527	71	25	a	a	DET
cana-527	71	26	banach	banach	NOUN
cana-527	71	27	hy	hy	NOUN
cana-527	71	28	perspace	perspace	NOUN
cana-527	72	1	[	[	X
cana-527	72	2	14][15][16	14][15][16	X
cana-527	72	3	]	]	PUNCT
cana-527	72	4	.	.	PUNCT
cana-527	73	1	classical	classical	ADJ
cana-527	73	2	fixed	fix	VERB
cana-527	73	3	point	point	NOUN
cana-527	73	4	theorems	theorem	NOUN
cana-527	73	5	can	can	AUX
cana-527	73	6	be	be	AUX
cana-527	73	7	extended	extend	VERB
cana-527	73	8	to	to	PART
cana-527	73	9	include	include	VERB
cana-527	73	10	near	near	ADP
cana-527	73	11	fixed	fix	VERB
cana-527	73	12	point	point	NOUN
cana-527	73	13	theorems	theorem	NOUN
cana-527	73	14	.	.	PUNCT
cana-527	74	1	they	they	PRON
cana-527	74	2	deal	deal	VERB
cana-527	74	3	with	with	ADP
cana-527	74	4	cases	case	NOUN
cana-527	74	5	where	where	SCONJ
cana-527	74	6	a	a	DET
cana-527	74	7	mapping	mapping	NOUN
cana-527	74	8	almost	almost	ADV
cana-527	74	9	has	have	VERB
cana-527	74	10	a	a	DET
cana-527	74	11	fixed	fix	VERB
cana-527	74	12	point	point	NOUN
cana-527	74	13	,	,	PUNCT
cana-527	74	14	rather	rather	ADV
cana-527	74	15	than	than	ADP
cana-527	74	16	demanding	demand	VERB
cana-527	74	17	a	a	DET
cana-527	74	18	rigid	rigid	ADJ
cana-527	74	19	fixed	fix	VERB
cana-527	74	20	point	point	NOUN
cana-527	74	21	.	.	PUNCT
cana-527	75	1	a	a	DET
cana-527	75	2	proximity	proximity	NOUN
cana-527	75	3	or	or	CCONJ
cana-527	75	4	approximation	approximation	NOUN
cana-527	75	5	notion	notion	NOUN
cana-527	75	6	is	be	AUX
cana-527	75	7	used	use	VERB
cana-527	75	8	to	to	PART
cana-527	75	9	quantify	quantify	VERB
cana-527	75	10	”	"	PUNCT
cana-527	75	11	nearness	nearness	NOUN
cana-527	75	12	”	"	PUNCT
cana-527	75	13	.	.	PUNCT
cana-527	76	1	the	the	DET
cana-527	76	2	concept	concept	NOUN
cana-527	76	3	of	of	ADP
cana-527	76	4	”	"	PUNCT
cana-527	76	5	nearness	nearness	NOUN
cana-527	76	6	”	"	PUNCT
cana-527	76	7	in	in	ADP
cana-527	76	8	the	the	DET
cana-527	76	9	context	context	NOUN
cana-527	76	10	of	of	ADP
cana-527	76	11	mappings	mapping	NOUN
cana-527	76	12	is	be	AUX
cana-527	76	13	intro	intro	NOUN
cana-527	76	14	duced	duce	VERB
cana-527	76	15	by	by	ADP
cana-527	76	16	near	near	ADP
cana-527	76	17	fixed	fix	VERB
cana-527	76	18	point	point	NOUN
cana-527	76	19	theorems	theorem	NOUN
cana-527	76	20	in	in	ADP
cana-527	76	21	banach	banach	NOUN
cana-527	76	22	hyperspaces	hyperspace	NOUN
cana-527	76	23	,	,	PUNCT
cana-527	76	24	which	which	PRON
cana-527	76	25	expand	expand	VERB
cana-527	76	26	the	the	DET
cana-527	76	27	traditional	traditional	ADJ
cana-527	76	28	fixed	fix	VERB
cana-527	76	29	point	point	NOUN
cana-527	76	30	theory	theory	NOUN
cana-527	76	31	to	to	ADP
cana-527	76	32	the	the	DET
cana-527	76	33	space	space	NOUN
cana-527	76	34	of	of	ADP
cana-527	76	35	closed	closed	ADJ
cana-527	76	36	sets	set	NOUN
cana-527	76	37	.	.	PUNCT
cana-527	77	1	these	these	DET
cana-527	77	2	theorems	theorem	NOUN
cana-527	77	3	have	have	VERB
cana-527	77	4	applications	application	NOUN
cana-527	77	5	in	in	ADP
cana-527	77	6	many	many	ADJ
cana-527	77	7	mathematical	mathematical	ADJ
cana-527	77	8	fields	field	NOUN
cana-527	77	9	and	and	CCONJ
cana-527	77	10	are	be	AUX
cana-527	77	11	important	important	ADJ
cana-527	77	12	for	for	ADP
cana-527	77	13	understanding	understand	VERB
cana-527	77	14	the	the	DET
cana-527	77	15	dynamics	dynamic	NOUN
cana-527	77	16	of	of	ADP
cana-527	77	17	mappings	mapping	NOUN
cana-527	77	18	on	on	ADP
cana-527	77	19	closed	closed	ADJ
cana-527	77	20	sets	set	NOUN
cana-527	77	21	.	.	PUNCT
cana-527	78	1	assume	assume	VERB
cana-527	78	2	that	that	SCONJ
cana-527	78	3	there	there	PRON
cana-527	78	4	is	be	VERB
cana-527	78	5	a	a	DET
cana-527	78	6	function	function	NOUN
cana-527	78	7	s	s	PART
cana-527	78	8	:	:	PUNCT
cana-527	78	9	k(s	k(s	PROPN
cana-527	78	10	)	)	PUNCT
cana-527	78	11	→	→	SYM
cana-527	78	12	k(s	k(s	PROPN
cana-527	78	13	)	)	PUNCT
cana-527	78	14	that	that	PRON
cana-527	78	15	maps	map	VERB
cana-527	78	16	k(s	k(s	PROPN
cana-527	78	17	)	)	PUNCT
cana-527	78	18	into	into	ADP
cana-527	78	19	itself	itself	PRON
cana-527	78	20	.	.	PUNCT
cana-527	79	1	if	if	SCONJ
cana-527	79	2	and	and	CCONJ
cana-527	79	3	only	only	ADV
cana-527	79	4	if	if	SCONJ
cana-527	79	5	s(a	s(a	PROPN
cana-527	79	6	)	)	PUNCT
cana-527	79	7	=	=	SYM
cana-527	80	1	a	a	PRON
cana-527	80	2	,	,	PUNCT
cana-527	80	3	then	then	ADV
cana-527	80	4	a	a	DET
cana-527	80	5	∈	∈	PROPN
cana-527	80	6	k(s	k(s	PROPN
cana-527	80	7	)	)	PUNCT
cana-527	80	8	is	be	AUX
cana-527	80	9	a	a	DET
cana-527	80	10	fixed	fix	VERB
cana-527	80	11	point	point	NOUN
cana-527	80	12	of	of	ADP
cana-527	80	13	s.	s.	PROPN
cana-527	80	14	the	the	DET
cana-527	80	15	idea	idea	NOUN
cana-527	80	16	of	of	ADP
cana-527	80	17	a	a	DET
cana-527	80	18	fixed	fix	VERB
cana-527	80	19	point	point	NOUN
cana-527	80	20	in	in	ADP
cana-527	80	21	set	set	NOUN
cana-527	80	22	-	-	PUNCT
cana-527	80	23	valued	value	VERB
cana-527	80	24	functions	function	NOUN
cana-527	80	25	is	be	AUX
cana-527	80	26	entirely	entirely	ADV
cana-527	80	27	distinct	distinct	ADJ
cana-527	80	28	from	from	ADP
cana-527	80	29	this	this	PRON
cana-527	80	30	notion.there	notion.there	ADV
cana-527	80	31	are	be	VERB
cana-527	80	32	some	some	DET
cana-527	80	33	classical	classical	ADJ
cana-527	80	34	fixed	fix	VERB
cana-527	80	35	point	point	NOUN
cana-527	80	36	theorems	theorem	NOUN
cana-527	80	37	that	that	PRON
cana-527	80	38	depend	depend	VERB
cana-527	80	39	on	on	ADP
cana-527	80	40	the	the	DET
cana-527	80	41	normed	normed	ADJ
cana-527	80	42	space	space	NOUN
cana-527	80	43	,	,	PUNCT
cana-527	80	44	which	which	PRON
cana-527	80	45	is	be	AUX
cana-527	80	46	also	also	ADV
cana-527	80	47	a	a	DET
cana-527	80	48	vector	vector	NOUN
cana-527	80	49	space	space	NOUN
cana-527	80	50	.	.	PUNCT
cana-527	81	1	since	since	SCONJ
cana-527	81	2	(	(	PUNCT
cana-527	81	3	k(s	k(s	PROPN
cana-527	81	4	)	)	PUNCT
cana-527	81	5	,	,	PUNCT
cana-527	81	6	∥.∥	∥.∥	NUM
cana-527	81	7	)	)	PUNCT
cana-527	81	8	is	be	AUX
cana-527	81	9	not	not	PART
cana-527	81	10	a	a	DET
cana-527	81	11	vector	vector	NOUN
cana-527	81	12	space	space	NOUN
cana-527	81	13	,	,	PUNCT
cana-527	81	14	we	we	PRON
cana-527	81	15	are	be	AUX
cana-527	81	16	unable	unable	ADJ
cana-527	81	17	to	to	PART
cana-527	81	18	investigate	investigate	VERB
cana-527	81	19	the	the	DET
cana-527	81	20	related	relate	VERB
cana-527	81	21	fixed	fix	VERB
cana-527	81	22	point	point	NOUN
cana-527	81	23	theorems	theorem	NOUN
cana-527	81	24	based	base	VERB
cana-527	81	25	on	on	ADP
cana-527	81	26	(	(	PUNCT
cana-527	81	27	k(s	k(s	PROPN
cana-527	81	28	)	)	PUNCT
cana-527	81	29	,	,	PUNCT
cana-527	81	30	∥.∥	∥.∥	NUM
cana-527	81	31	)	)	PUNCT
cana-527	81	32	.	.	PUNCT
cana-527	82	1	but	but	CCONJ
cana-527	82	2	we	we	PRON
cana-527	82	3	can	can	AUX
cana-527	82	4	examine	examine	VERB
cana-527	82	5	the	the	DET
cana-527	82	6	so	so	ADV
cana-527	82	7	-	-	PUNCT
cana-527	82	8	called	call	VERB
cana-527	82	9	near	near	ADV
cana-527	82	10	fixed	fix	VERB
cana-527	82	11	point	point	NOUN
cana-527	82	12	,	,	PUNCT
cana-527	82	13	which	which	PRON
cana-527	82	14	has	have	VERB
cana-527	82	15	the	the	DET
cana-527	82	16	following	follow	VERB
cana-527	82	17	definitions	definition	NOUN
cana-527	82	18	.	.	PUNCT
cana-527	83	1	definition	definition	NOUN
cana-527	83	2	2.3.1	2.3.1	NUM
cana-527	83	3	:	:	PUNCT
cana-527	83	4	consider	consider	VERB
cana-527	83	5	a	a	DET
cana-527	83	6	function	function	NOUN
cana-527	83	7	defined	define	VERB
cana-527	83	8	on	on	ADP
cana-527	83	9	k(s	k(s	PROPN
cana-527	83	10	)	)	PUNCT
cana-527	83	11	into	into	ADP
cana-527	83	12	itself	itself	PRON
cana-527	83	13	,	,	PUNCT
cana-527	83	14	such	such	ADJ
cana-527	83	15	that	that	PRON
cana-527	83	16	s	s	VERB
cana-527	83	17	:	:	PUNCT
cana-527	83	18	k(s	k(s	PROPN
cana-527	83	19	)	)	PUNCT
cana-527	83	20	→	→	SYM
cana-527	83	21	k(s	k(s	PROPN
cana-527	83	22	)	)	PUNCT
cana-527	83	23	.	.	PUNCT
cana-527	84	1	if	if	SCONJ
cana-527	84	2	and	and	CCONJ
cana-527	84	3	only	only	ADV
cana-527	84	4	if	if	SCONJ
cana-527	84	5	s(a	s(a	PROPN
cana-527	84	6	)	)	PUNCT
cana-527	84	7	≡	≡	PROPN
cana-527	84	8	q	q	PROPN
cana-527	85	1	a	a	X
cana-527	85	2	,	,	PUNCT
cana-527	85	3	then	then	ADV
cana-527	85	4	a	a	DET
cana-527	85	5	point	point	NOUN
cana-527	85	6	a	a	DET
cana-527	85	7	∈	∈	PROPN
cana-527	85	8	k(s	k(s	PROPN
cana-527	85	9	)	)	PUNCT
cana-527	85	10	is	be	AUX
cana-527	85	11	referred	refer	VERB
cana-527	85	12	to	to	ADP
cana-527	85	13	as	as	ADP
cana-527	85	14	a	a	DET
cana-527	85	15	near	near	ADV
cana-527	85	16	fixed	fix	VERB
cana-527	85	17	point	point	NOUN
cana-527	85	18	of	of	ADP
cana-527	85	19	s.	s.	PROPN
cana-527	85	20	definition	definition	PROPN
cana-527	85	21	2.3.2	2.3.2	NUM
cana-527	85	22	:	:	PUNCT
cana-527	85	23	consider	consider	VERB
cana-527	85	24	a	a	DET
cana-527	85	25	pseudo	pseudo	NOUN
cana-527	85	26	-	-	PUNCT
cana-527	85	27	seminormed	seminorme	VERB
cana-527	85	28	hyperspace	hyperspace	NOUN
cana-527	85	29	(	(	PUNCT
cana-527	85	30	k(s	k(s	PROPN
cana-527	85	31	)	)	PUNCT
cana-527	85	32	,	,	PUNCT
cana-527	85	33	∥.∥	∥.∥	NUM
cana-527	85	34	)	)	PUNCT
cana-527	85	35	.	.	PUNCT
cana-527	86	1	if	if	SCONJ
cana-527	86	2	and	and	CCONJ
cana-527	86	3	only	only	ADV
cana-527	86	4	if	if	SCONJ
cana-527	86	5	there	there	PRON
cana-527	86	6	is	be	VERB
cana-527	86	7	a	a	DET
cana-527	86	8	real	real	ADJ
cana-527	86	9	number	number	NOUN
cana-527	86	10	0	0	NUM
cana-527	86	11	<	<	X
cana-527	86	12	β	β	X
cana-527	86	13	<	<	X
cana-527	86	14	1	1	NUM
cana-527	86	15	such	such	ADJ
cana-527	86	16	that	that	DET
cana-527	86	17	∥s(a	∥s(a	PROPN
cana-527	86	18	)	)	PUNCT
cana-527	86	19	⊖	⊖	NOUN
cana-527	86	20	s(b)∥	s(b)∥	VERB
cana-527	86	21	≤	≤	NUM
cana-527	86	22	β∥a	β∥a	PUNCT
cana-527	86	23	⊖	⊖	NOUN
cana-527	86	24	b∥	b∥	NOUN
cana-527	86	25	for	for	ADP
cana-527	86	26	any	any	DET
cana-527	86	27	a	a	PRON
cana-527	86	28	,	,	PUNCT
cana-527	86	29	b	b	PROPN
cana-527	86	30	∈	∈	PROPN
cana-527	86	31	k(s	k(s	PROPN
cana-527	86	32	)	)	PUNCT
cana-527	86	33	,	,	PUNCT
cana-527	86	34	then	then	ADV
cana-527	86	35	a	a	DET
cana-527	86	36	function	function	NOUN
cana-527	86	37	s	s	VERB
cana-527	86	38	:	:	PUNCT
cana-527	86	39	(	(	PUNCT
cana-527	86	40	k(s	k(s	PROPN
cana-527	86	41	)	)	PUNCT
cana-527	86	42	,	,	PUNCT
cana-527	86	43	∥.∥	∥.∥	NUM
cana-527	86	44	)	)	PUNCT
cana-527	87	1	→	→	PUNCT
cana-527	87	2	(	(	PUNCT
cana-527	87	3	k(s	k(s	PROPN
cana-527	87	4	)	)	PUNCT
cana-527	87	5	,	,	PUNCT
cana-527	87	6	∥.∥	∥.∥	NUM
cana-527	87	7	)	)	PUNCT
cana-527	87	8	is	be	AUX
cana-527	87	9	referred	refer	VERB
cana-527	87	10	to	to	ADP
cana-527	87	11	as	as	ADP
cana-527	87	12	a	a	DET
cana-527	87	13	contraction	contraction	NOUN
cana-527	87	14	on	on	ADP
cana-527	87	15	k(s	k(s	PROPN
cana-527	87	16	)	)	PUNCT
cana-527	87	17	.	.	PUNCT
cana-527	88	1	communications	communication	NOUN
cana-527	88	2	on	on	ADP
cana-527	88	3	applied	apply	VERB
cana-527	88	4	nonlinear	nonlinear	ADJ
cana-527	88	5	analysis	analysis	NOUN
cana-527	88	6	issn	issn	NOUN
cana-527	88	7	:	:	PUNCT
cana-527	88	8	1074	1074	NUM
cana-527	88	9	-	-	PUNCT
cana-527	88	10	133x	133x	NUM
cana-527	88	11	vol	vol	NOUN
cana-527	88	12	31	31	NUM
cana-527	88	13	no	no	NOUN
cana-527	88	14	.	.	NOUN
cana-527	88	15	2	2	NUM
cana-527	88	16	(	(	PUNCT
cana-527	88	17	2024	2024	NUM
cana-527	88	18	)	)	PUNCT
cana-527	88	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	88	20	160	160	NUM
cana-527	88	21	2	2	NUM
cana-527	88	22	2	2	NUM
cana-527	88	23	using	use	VERB
cana-527	88	24	the	the	DET
cana-527	88	25	function	function	NOUN
cana-527	88	26	s	s	PART
cana-527	88	27	,	,	PUNCT
cana-527	88	28	we	we	PRON
cana-527	88	29	define	define	VERB
cana-527	88	30	the	the	DET
cana-527	88	31	iterative	iterative	NOUN
cana-527	88	32	sequence	sequence	NOUN
cana-527	88	33	{	{	PUNCT
cana-527	88	34	an}∞n=1	an}∞n=1	X
cana-527	88	35	given	give	VERB
cana-527	88	36	any	any	DET
cana-527	88	37	initial	initial	ADJ
cana-527	88	38	element	element	NOUN
cana-527	88	39	a0	a0	PROPN
cana-527	88	40	∈	∈	PROPN
cana-527	88	41	k(s	k(s	PROPN
cana-527	88	42	)	)	PUNCT
cana-527	88	43	as	as	SCONJ
cana-527	88	44	follows	follow	VERB
cana-527	88	45	:	:	PUNCT
cana-527	88	46	a1	a1	NOUN
cana-527	88	47	=	=	PUNCT
cana-527	88	48	s(a0	s(a0	NOUN
cana-527	88	49	)	)	PUNCT
cana-527	88	50	,	,	PUNCT
cana-527	88	51	a2	a2	PROPN
cana-527	88	52	=	=	SYM
cana-527	88	53	ss(a0	ss(a0	NUM
cana-527	88	54	)	)	PUNCT
cana-527	88	55	,	,	PUNCT
cana-527	88	56	.	.	PUNCT
cana-527	88	57	.	.	PUNCT
cana-527	88	58	.	.	PUNCT
cana-527	89	1	we	we	PRON
cana-527	89	2	will	will	AUX
cana-527	89	3	demonstrate	demonstrate	VERB
cana-527	89	4	that	that	SCONJ
cana-527	89	5	the	the	DET
cana-527	89	6	series	series	NOUN
cana-527	89	7	{	{	PUNCT
cana-527	89	8	an}∞n=1	an}∞n=1	X
cana-527	89	9	can	can	AUX
cana-527	89	10	converge	converge	VERB
cana-527	89	11	to	to	ADP
cana-527	89	12	a	a	DET
cana-527	89	13	close	close	ADJ
cana-527	89	14	fixed	fix	VERB
cana-527	89	15	point	point	NOUN
cana-527	89	16	under	under	ADP
cana-527	89	17	certain	certain	ADJ
cana-527	89	18	appropriate	appropriate	ADJ
cana-527	89	19	circumstances	circumstance	NOUN
cana-527	89	20	.	.	PUNCT
cana-527	90	1	this	this	DET
cana-527	90	2	convergence	convergence	NOUN
cana-527	90	3	to	to	ADP
cana-527	90	4	a	a	DET
cana-527	90	5	fixed	fix	VERB
cana-527	90	6	point	point	NOUN
cana-527	90	7	can	can	AUX
cana-527	90	8	occur	occur	VERB
cana-527	90	9	under	under	ADP
cana-527	90	10	certain	certain	ADJ
cana-527	90	11	conditions	condition	NOUN
cana-527	90	12	,	,	PUNCT
cana-527	90	13	which	which	PRON
cana-527	90	14	we	we	PRON
cana-527	90	15	will	will	AUX
cana-527	90	16	now	now	ADV
cana-527	90	17	examine	examine	VERB
cana-527	90	18	.	.	PUNCT
cana-527	91	1	diverse	diverse	ADJ
cana-527	91	2	criteria	criterion	NOUN
cana-527	91	3	and	and	CCONJ
cana-527	91	4	theorems	theorem	NOUN
cana-527	91	5	guarantee	guarantee	VERB
cana-527	91	6	that	that	SCONJ
cana-527	91	7	a	a	DET
cana-527	91	8	sequence	sequence	NOUN
cana-527	91	9	will	will	AUX
cana-527	91	10	eventually	eventually	ADV
cana-527	91	11	converge	converge	VERB
cana-527	91	12	.	.	PUNCT
cana-527	92	1	definition	definition	NOUN
cana-527	92	2	2.3.3	2.3.3	NUM
cana-527	92	3	:	:	PUNCT
cana-527	92	4	let	let	VERB
cana-527	92	5	(	(	PUNCT
cana-527	92	6	k(s	k(s	PROPN
cana-527	92	7	)	)	PUNCT
cana-527	92	8	,	,	PUNCT
cana-527	92	9	∥.∥	∥.∥	NUM
cana-527	92	10	)	)	PUNCT
cana-527	92	11	be	be	AUX
cana-527	92	12	a	a	DET
cana-527	92	13	bhs	bhs	PROPN
cana-527	92	14	.	.	PUNCT
cana-527	93	1	a	a	DET
cana-527	93	2	mapping	mapping	NOUN
cana-527	93	3	s	s	PART
cana-527	93	4	:	:	PUNCT
cana-527	93	5	k(s	k(s	PROPN
cana-527	93	6	)	)	PUNCT
cana-527	93	7	→	→	SYM
cana-527	93	8	k(s	k(s	PROPN
cana-527	93	9	)	)	PUNCT
cana-527	93	10	is	be	AUX
cana-527	93	11	called	call	VERB
cana-527	93	12	a	a	DET
cana-527	93	13	kannan	kannan	PROPN
cana-527	93	14	mapping	mapping	NOUN
cana-527	93	15	if	if	SCONJ
cana-527	93	16	there	there	PRON
cana-527	93	17	exists	exist	VERB
cana-527	93	18	β	β	X
cana-527	93	19	∈	∈	PROPN
cana-527	93	20	(	(	PUNCT
cana-527	93	21	0	0	NUM
cana-527	93	22	,	,	PUNCT
cana-527	93	23	1	1	NUM
cana-527	93	24	)	)	PUNCT
cana-527	93	25	such	such	ADJ
cana-527	93	26	that	that	SCONJ
cana-527	93	27	,	,	PUNCT
cana-527	93	28	∥s(x	∥s(x	PROPN
cana-527	93	29	)	)	PUNCT
cana-527	93	30	⊖	⊖	NOUN
cana-527	93	31	s(y)∥	s(y)∥	VERB
cana-527	93	32	≤	≤	ADJ
cana-527	93	33	β	β	NOUN
cana-527	93	34	.	.	PUNCT
cana-527	94	1	∥x	∥x	PROPN
cana-527	94	2	⊖	⊖	VERB
cana-527	94	3	s(x)∥	s(x)∥	ADP
cana-527	94	4	⊕	⊕	PROPN
cana-527	94	5	∥y	∥y	PROPN
cana-527	94	6	⊖	⊖	AUX
cana-527	94	7	s(y)∥	s(y)∥	VERB
cana-527	94	8	.	.	PUNCT
cana-527	95	1	definition	definition	NOUN
cana-527	95	2	2.3.4	2.3.4	NUM
cana-527	95	3	:	:	PUNCT
cana-527	95	4	let	let	VERB
cana-527	95	5	(	(	PUNCT
cana-527	95	6	k(s	k(s	PROPN
cana-527	95	7	)	)	PUNCT
cana-527	95	8	,	,	PUNCT
cana-527	95	9	∥.∥	∥.∥	NUM
cana-527	95	10	)	)	PUNCT
cana-527	95	11	be	be	AUX
cana-527	95	12	a	a	DET
cana-527	95	13	bhs	bhs	PROPN
cana-527	95	14	.	.	PUNCT
cana-527	96	1	a	a	DET
cana-527	96	2	mapping	mapping	NOUN
cana-527	96	3	s	s	PART
cana-527	96	4	:	:	PUNCT
cana-527	96	5	k(s	k(s	PROPN
cana-527	96	6	)	)	PUNCT
cana-527	96	7	→	→	SYM
cana-527	96	8	k(s	k(s	PROPN
cana-527	96	9	)	)	PUNCT
cana-527	96	10	is	be	AUX
cana-527	96	11	called	call	VERB
cana-527	96	12	a	a	DET
cana-527	96	13	chatterjea	chatterjea	ADJ
cana-527	96	14	mapping	mapping	NOUN
cana-527	96	15	if	if	SCONJ
cana-527	96	16	there	there	PRON
cana-527	96	17	exists	exist	VERB
cana-527	96	18	β	β	X
cana-527	96	19	∈	∈	PROPN
cana-527	96	20	(	(	PUNCT
cana-527	96	21	0	0	NUM
cana-527	96	22	,	,	PUNCT
cana-527	96	23	1	1	NUM
cana-527	96	24	)	)	PUNCT
cana-527	96	25	such	such	ADJ
cana-527	96	26	that	that	SCONJ
cana-527	96	27	,	,	PUNCT
cana-527	96	28	∥s(x	∥s(x	PROPN
cana-527	96	29	)	)	PUNCT
cana-527	96	30	⊖	⊖	NOUN
cana-527	96	31	s(y)∥	s(y)∥	VERB
cana-527	96	32	≤	≤	ADJ
cana-527	96	33	β	β	NOUN
cana-527	96	34	.	.	PUNCT
cana-527	97	1	∥x	∥x	PROPN
cana-527	97	2	⊖	⊖	AUX
cana-527	97	3	s(y)∥	s(y)∥	VERB
cana-527	97	4	⊕	⊕	PROPN
cana-527	97	5	∥y	∥y	PROPN
cana-527	97	6	⊖	⊖	VERB
cana-527	97	7	s(x)∥	s(x)∥	ADV
cana-527	97	8	.	.	PUNCT
cana-527	98	1	definition	definition	NOUN
cana-527	98	2	2.3.5	2.3.5	NUM
cana-527	98	3	:	:	PUNCT
cana-527	98	4	let	let	VERB
cana-527	98	5	(	(	PUNCT
cana-527	98	6	k(s	k(s	PROPN
cana-527	98	7	)	)	PUNCT
cana-527	98	8	,	,	PUNCT
cana-527	98	9	∥.∥	∥.∥	NUM
cana-527	98	10	)	)	PUNCT
cana-527	98	11	be	be	AUX
cana-527	98	12	a	a	DET
cana-527	98	13	bhs	bhs	PROPN
cana-527	98	14	.	.	PUNCT
cana-527	99	1	a	a	DET
cana-527	99	2	map	map	NOUN
cana-527	99	3	s	s	VERB
cana-527	99	4	:	:	PUNCT
cana-527	99	5	k(s	k(s	PROPN
cana-527	99	6	)	)	PUNCT
cana-527	99	7	→	→	SYM
cana-527	99	8	k(s	k(s	PROPN
cana-527	99	9	)	)	PUNCT
cana-527	99	10	is	be	AUX
cana-527	99	11	λ	λ	ADJ
cana-527	99	12	-	-	ADJ
cana-527	99	13	generalized	generalized	ADJ
cana-527	99	14	contraction	contraction	NOUN
cana-527	99	15	if	if	SCONJ
cana-527	99	16	and	and	CCONJ
cana-527	99	17	only	only	ADV
cana-527	99	18	if	if	SCONJ
cana-527	99	19	for	for	ADP
cana-527	99	20	every	every	DET
cana-527	99	21	u	u	NOUN
cana-527	99	22	,	,	PUNCT
cana-527	99	23	v	v	PROPN
cana-527	99	24	∈	∈	PROPN
cana-527	99	25	k(s	k(s	PROPN
cana-527	99	26	)	)	PUNCT
cana-527	99	27	,	,	PUNCT
cana-527	99	28	there	there	PRON
cana-527	99	29	exist	exist	VERB
cana-527	99	30	non	non	ADJ
cana-527	99	31	-	-	ADJ
cana-527	99	32	negative	negative	ADJ
cana-527	99	33	numbers	number	NOUN
cana-527	99	34	q(u	q(u	ADP
cana-527	99	35	,	,	PUNCT
cana-527	99	36	v	v	NOUN
cana-527	99	37	)	)	PUNCT
cana-527	99	38	,	,	PUNCT
cana-527	99	39	r(u	r(u	PROPN
cana-527	99	40	,	,	PUNCT
cana-527	99	41	v	v	NOUN
cana-527	99	42	)	)	PUNCT
cana-527	99	43	,	,	PUNCT
cana-527	99	44	s(u	s(u	PROPN
cana-527	99	45	,	,	PUNCT
cana-527	99	46	v	v	NOUN
cana-527	99	47	)	)	PUNCT
cana-527	99	48	and	and	CCONJ
cana-527	99	49	t(u	t(u	NUM
cana-527	99	50	,	,	PUNCT
cana-527	99	51	v	v	NOUN
cana-527	99	52	)	)	PUNCT
cana-527	99	53	such	such	ADJ
cana-527	99	54	that	that	SCONJ
cana-527	99	55	,	,	PUNCT
cana-527	99	56	supu	supu	ADJ
cana-527	99	57	,	,	PUNCT
cana-527	99	58	v∈s	v∈s	NOUN
cana-527	99	59	{	{	PUNCT
cana-527	99	60	q(u	q(u	NOUN
cana-527	99	61	,	,	PUNCT
cana-527	99	62	v	v	NOUN
cana-527	99	63	)	)	PUNCT
cana-527	100	1	+	+	CCONJ
cana-527	101	1	r(u	r(u	PROPN
cana-527	101	2	,	,	PUNCT
cana-527	101	3	v	v	NOUN
cana-527	101	4	)	)	PUNCT
cana-527	101	5	+	+	CCONJ
cana-527	101	6	s(u	s(u	PROPN
cana-527	101	7	,	,	PUNCT
cana-527	101	8	v	v	NOUN
cana-527	101	9	)	)	PUNCT
cana-527	101	10	+	+	NUM
cana-527	101	11	2t(u	2t(u	NUM
cana-527	101	12	,	,	PUNCT
cana-527	101	13	v	v	NOUN
cana-527	101	14	)	)	PUNCT
cana-527	101	15	}	}	PUNCT
cana-527	101	16	=	=	PUNCT
cana-527	101	17	λ	λ	X
cana-527	101	18	<	<	X
cana-527	101	19	1	1	NUM
cana-527	101	20	and	and	CCONJ
cana-527	101	21	∥u⊖s(v)∥	∥u⊖s(v)∥	ADJ
cana-527	101	22	≤	≤	NUM
cana-527	101	23	q(u	q(u	PROPN
cana-527	101	24	,	,	PUNCT
cana-527	101	25	v)∥u⊖v∥+r(u	v)∥u⊖v∥+r(u	PROPN
cana-527	101	26	,	,	PUNCT
cana-527	101	27	v)∥u⊖s(u)∥+s(u	v)∥u⊖s(u)∥+s(u	X
cana-527	101	28	,	,	PUNCT
cana-527	101	29	v)∥v⊖s(v)∥+t(u	v)∥v⊖s(v)∥+t(u	INTJ
cana-527	101	30	,	,	PUNCT
cana-527	101	31	v	v	NOUN
cana-527	101	32	)	)	PUNCT
cana-527	102	1	[	[	X
cana-527	102	2	∥u	∥u	PROPN
cana-527	102	3	−	−	PROPN
cana-527	102	4	s(v)∥	s(v)∥	NOUN
cana-527	102	5	+	+	CCONJ
cana-527	102	6	∥v	∥v	PROPN
cana-527	102	7	−	−	PROPN
cana-527	102	8	s(u)∥	s(u)∥	VERB
cana-527	102	9	]	]	PUNCT
cana-527	102	10	holds	hold	VERB
cana-527	102	11	for	for	ADP
cana-527	102	12	every	every	DET
cana-527	102	13	u	u	NOUN
cana-527	102	14	,	,	PUNCT
cana-527	102	15	v	v	PROPN
cana-527	102	16	∈	∈	PROPN
cana-527	102	17	k(s	k(s	PROPN
cana-527	102	18	)	)	PUNCT
cana-527	102	19	.	.	PUNCT
cana-527	103	1	definition	definition	NOUN
cana-527	103	2	2.3.6	2.3.6	NUM
cana-527	103	3	:	:	PUNCT
cana-527	103	4	suppose	suppose	VERB
cana-527	103	5	there	there	PRON
cana-527	103	6	is	be	VERB
cana-527	103	7	a	a	DET
cana-527	103	8	bhs	bhs	PROPN
cana-527	103	9	(	(	PUNCT
cana-527	103	10	k(s	k(s	PROPN
cana-527	103	11	)	)	PUNCT
cana-527	103	12	,	,	PUNCT
cana-527	103	13	∥.∥	∥.∥	NUM
cana-527	103	14	)	)	PUNCT
cana-527	103	15	.	.	PUNCT
cana-527	104	1	if	if	SCONJ
cana-527	104	2	there	there	PRON
cana-527	104	3	is	be	VERB
cana-527	104	4	a	a	DET
cana-527	104	5	limit	limit	NOUN
cana-527	104	6	point	point	NOUN
cana-527	104	7	in	in	ADP
cana-527	104	8	s	s	PRON
cana-527	104	9	for	for	ADP
cana-527	104	10	every	every	DET
cana-527	104	11	cauchy	cauchy	ADJ
cana-527	104	12	sequence	sequence	NOUN
cana-527	104	13	{	{	PUNCT
cana-527	104	14	sniu	sniu	ADV
cana-527	104	15	:	:	PUNCT
cana-527	104	16	i	i	PRON
cana-527	104	17	∈	∈	PROPN
cana-527	104	18	n	n	CCONJ
cana-527	104	19	}	}	PUNCT
cana-527	104	20	,	,	PUNCT
cana-527	104	21	u	u	PROPN
cana-527	104	22	∈	∈	PROPN
cana-527	104	23	k(s	k(s	PROPN
cana-527	104	24	)	)	PUNCT
cana-527	104	25	,	,	PUNCT
cana-527	104	26	then	then	ADV
cana-527	104	27	the	the	DET
cana-527	104	28	mapping	mapping	NOUN
cana-527	104	29	s	s	PART
cana-527	104	30	:	:	PUNCT
cana-527	104	31	k(s	k(s	PROPN
cana-527	104	32	)	)	PUNCT
cana-527	105	1	→	→	SYM
cana-527	105	2	k(s	k(s	PROPN
cana-527	105	3	)	)	PUNCT
cana-527	105	4	is	be	AUX
cana-527	105	5	considered	consider	VERB
cana-527	105	6	t	t	X
cana-527	105	7	-orbitally	-orbitally	PROPN
cana-527	105	8	complete	complete	ADJ
cana-527	105	9	.	.	PUNCT
cana-527	106	1	definition	definition	NOUN
cana-527	106	2	2.3.7	2.3.7	NUM
cana-527	106	3	:	:	PUNCT
cana-527	106	4	let	let	VERB
cana-527	106	5	(	(	PUNCT
cana-527	106	6	k(s	k(s	PROPN
cana-527	106	7	)	)	PUNCT
cana-527	106	8	,	,	PUNCT
cana-527	106	9	∥.∥	∥.∥	NUM
cana-527	106	10	)	)	PUNCT
cana-527	106	11	be	be	AUX
cana-527	106	12	a	a	DET
cana-527	106	13	bhs	bhs	PROPN
cana-527	106	14	.	.	PUNCT
cana-527	107	1	a	a	DET
cana-527	107	2	mapping	mapping	NOUN
cana-527	107	3	s	s	PART
cana-527	107	4	:	:	PUNCT
cana-527	107	5	k(s	k(s	PROPN
cana-527	107	6	)	)	PUNCT
cana-527	107	7	→	→	SYM
cana-527	107	8	k(s	k(s	PROPN
cana-527	107	9	)	)	PUNCT
cana-527	107	10	is	be	AUX
cana-527	107	11	said	say	VERB
cana-527	107	12	to	to	PART
cana-527	107	13	be	be	AUX
cana-527	107	14	s	s	NOUN
cana-527	107	15	-	-	ADJ
cana-527	107	16	orbitally	orbitally	ADV
cana-527	107	17	continuous	continuous	ADJ
cana-527	107	18	if	if	SCONJ
cana-527	107	19	for	for	SCONJ
cana-527	107	20	u	u	PROPN
cana-527	107	21	∈	∈	PROPN
cana-527	107	22	s	s	PART
cana-527	107	23	then	then	ADV
cana-527	107	24	u	u	X
cana-527	107	25	=	=	PROPN
cana-527	107	26	limi→∞	limi→∞	PROPN
cana-527	107	27	sniv	sniv	ADJ
cana-527	107	28	for	for	ADP
cana-527	107	29	few	few	ADJ
cana-527	107	30	v	v	NUM
cana-527	107	31	∈	∈	PROPN
cana-527	107	32	s	s	NOUN
cana-527	107	33	,	,	PUNCT
cana-527	107	34	here	here	ADV
cana-527	107	35	su	su	PROPN
cana-527	107	36	=	=	PUNCT
cana-527	107	37	limi→∞	limi→∞	PROPN
cana-527	107	38	ssniv	ssniv	VERB
cana-527	107	39	.	.	PUNCT
cana-527	108	1	communications	communication	NOUN
cana-527	108	2	on	on	ADP
cana-527	108	3	applied	apply	VERB
cana-527	108	4	nonlinear	nonlinear	ADJ
cana-527	108	5	analysis	analysis	NOUN
cana-527	108	6	issn	issn	NOUN
cana-527	108	7	:	:	PUNCT
cana-527	108	8	1074	1074	NUM
cana-527	108	9	-	-	PUNCT
cana-527	108	10	133x	133x	NUM
cana-527	108	11	vol	vol	NOUN
cana-527	108	12	31	31	NUM
cana-527	108	13	no	no	NOUN
cana-527	108	14	.	.	NOUN
cana-527	108	15	2	2	NUM
cana-527	108	16	(	(	PUNCT
cana-527	108	17	2024	2024	NUM
cana-527	108	18	)	)	PUNCT
cana-527	108	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	108	20	161	161	NUM
cana-527	108	21	1−λ	1−λ	NUM
cana-527	108	22	theorem	theorem	VERB
cana-527	108	23	2.3.1	2.3.1	NUM
cana-527	108	24	(	(	PUNCT
cana-527	108	25	16	16	NUM
cana-527	108	26	):	):	PUNCT
cana-527	108	27	let	let	VERB
cana-527	108	28	s	s	PRON
cana-527	108	29	be	be	AUX
cana-527	108	30	a	a	DET
cana-527	108	31	λ	λ	NOUN
cana-527	108	32	-	-	ADJ
cana-527	108	33	generalized	generalized	ADJ
cana-527	108	34	contraction	contraction	NOUN
cana-527	108	35	of	of	ADP
cana-527	108	36	s	s	NOUN
cana-527	108	37	-	-	PUNCT
cana-527	108	38	orbitally	orbitally	ADV
cana-527	108	39	bhs	bhs	PROPN
cana-527	108	40	k(x	k(x	PROPN
cana-527	108	41	)	)	PUNCT
cana-527	108	42	into	into	ADP
cana-527	108	43	itself	itself	PRON
cana-527	108	44	.	.	PUNCT
cana-527	109	1	then	then	ADV
cana-527	109	2	1	1	X
cana-527	109	3	.	.	X
cana-527	110	1	there	there	PRON
cana-527	110	2	is	be	VERB
cana-527	110	3	in	in	ADP
cana-527	110	4	k(s	k(s	PROPN
cana-527	110	5	)	)	PUNCT
cana-527	110	6	a	a	DET
cana-527	110	7	unique	unique	ADJ
cana-527	110	8	near	near	ADP
cana-527	110	9	fixed	fix	VERB
cana-527	110	10	point	point	NOUN
cana-527	110	11	v	v	NOUN
cana-527	110	12	under	under	ADP
cana-527	110	13	s	s	NOUN
cana-527	110	14	,	,	PUNCT
cana-527	110	15	2	2	NUM
cana-527	110	16	.	.	PUNCT
cana-527	110	17	sn	sn	PROPN
cana-527	110	18	x	x	PUNCT
cana-527	111	1	→	→	SYM
cana-527	111	2	v	v	NOUN
cana-527	111	3	for	for	ADP
cana-527	111	4	every	every	DET
cana-527	111	5	x	x	PROPN
cana-527	111	6	∈	∈	PROPN
cana-527	111	7	k(x	k(x	PROPN
cana-527	111	8	)	)	PUNCT
cana-527	111	9	and	and	CCONJ
cana-527	111	10	3	3	X
cana-527	111	11	.	.	NOUN
cana-527	111	12	∥sn	∥sn	PROPN
cana-527	112	1	x	x	ADV
cana-527	112	2	⊖	⊖	NOUN
cana-527	112	3	v∥	v∥	NOUN
cana-527	112	4	≤	≤	NUM
cana-527	112	5	λn	λn	ADP
cana-527	112	6	∥x	∥x	PROPN
cana-527	112	7	−	−	PROPN
cana-527	112	8	s(x)∥	s(x)∥	ADV
cana-527	112	9	theorem	theorem	VERB
cana-527	112	10	2.3.2	2.3.2	NUM
cana-527	112	11	(	(	PUNCT
cana-527	112	12	16	16	NUM
cana-527	112	13	):	):	PUNCT
cana-527	112	14	let	let	VERB
cana-527	112	15	(	(	PUNCT
cana-527	112	16	k(s	k(s	PROPN
cana-527	112	17	)	)	PUNCT
cana-527	112	18	,	,	PUNCT
cana-527	112	19	∥.∥	∥.∥	NUM
cana-527	112	20	)	)	PUNCT
cana-527	112	21	be	be	AUX
cana-527	112	22	a	a	DET
cana-527	112	23	bhs	bhs	PROPN
cana-527	112	24	,	,	PUNCT
cana-527	112	25	and	and	CCONJ
cana-527	112	26	let	let	VERB
cana-527	112	27	∥.∥	∥.∥	PRON
cana-527	112	28	satisfy	satisfy	VERB
cana-527	112	29	the	the	DET
cana-527	112	30	null	null	ADJ
cana-527	112	31	equality	equality	NOUN
cana-527	112	32	,	,	PUNCT
cana-527	112	33	and	and	CCONJ
cana-527	112	34	the	the	DET
cana-527	112	35	null	null	ADJ
cana-527	112	36	set	set	NOUN
cana-527	112	37	be	be	AUX
cana-527	112	38	ω	ω	NUM
cana-527	112	39	.	.	PUNCT
cana-527	113	1	let	let	VERB
cana-527	113	2	s	s	PRON
cana-527	113	3	:	:	PUNCT
cana-527	113	4	(	(	PUNCT
cana-527	113	5	k(s	k(s	PROPN
cana-527	113	6	)	)	PUNCT
cana-527	113	7	,	,	PUNCT
cana-527	113	8	∥.∥	∥.∥	NUM
cana-527	113	9	)	)	PUNCT
cana-527	114	1	→	→	PUNCT
cana-527	114	2	(	(	PUNCT
cana-527	114	3	k(s	k(s	PROPN
cana-527	114	4	)	)	PUNCT
cana-527	114	5	,	,	PUNCT
cana-527	114	6	∥.∥	∥.∥	NUM
cana-527	114	7	)	)	PUNCT
cana-527	114	8	be	be	AUX
cana-527	114	9	a	a	DET
cana-527	114	10	contraction	contraction	NOUN
cana-527	114	11	on	on	ADP
cana-527	114	12	k(s	k(s	PROPN
cana-527	114	13	)	)	PUNCT
cana-527	114	14	.	.	PUNCT
cana-527	115	1	if	if	SCONJ
cana-527	115	2	s(b	s(b	NOUN
cana-527	115	3	)	)	PUNCT
cana-527	115	4	≏	≏	NOUN
cana-527	115	5	b	b	NOUN
cana-527	115	6	,	,	PUNCT
cana-527	115	7	then	then	ADV
cana-527	115	8	s	s	AUX
cana-527	115	9	has	have	VERB
cana-527	115	10	a	a	DET
cana-527	115	11	near	near	ADJ
cana-527	115	12	fixed	fix	VERB
cana-527	115	13	point	point	NOUN
cana-527	115	14	b	b	PROPN
cana-527	115	15	∈	∈	PROPN
cana-527	115	16	k(s	k(s	PROPN
cana-527	115	17	)	)	PUNCT
cana-527	115	18	.	.	PUNCT
cana-527	116	1	moreover	moreover	ADV
cana-527	116	2	,	,	PUNCT
cana-527	116	3	the	the	DET
cana-527	116	4	near	near	ADV
cana-527	116	5	fixed	fix	VERB
cana-527	116	6	point	point	NOUN
cana-527	116	7	b	b	PROPN
cana-527	116	8	is	be	AUX
cana-527	116	9	obtained	obtain	VERB
cana-527	116	10	by	by	ADP
cana-527	116	11	the	the	DET
cana-527	116	12	limit	limit	NOUN
cana-527	116	13	,	,	PUNCT
cana-527	116	14	∥b	∥b	PROPN
cana-527	116	15	⊖	⊖	ADJ
cana-527	116	16	bn∥	bn∥	X
cana-527	116	17	=	=	SYM
cana-527	116	18	∥bn	∥bn	NOUN
cana-527	116	19	⊖	⊖	NOUN
cana-527	116	20	b∥	b∥	NOUN
cana-527	116	21	→	→	SYM
cana-527	116	22	0	0	NUM
cana-527	116	23	as	as	ADP
cana-527	116	24	n	n	NOUN
cana-527	116	25	→	→	SYM
cana-527	116	26	∞	∞	PROPN
cana-527	116	27	in	in	ADP
cana-527	116	28	which	which	PRON
cana-527	116	29	the	the	DET
cana-527	116	30	sequence	sequence	NOUN
cana-527	116	31	{	{	PUNCT
cana-527	116	32	bn}∞n=1	bn}∞n=1	X
cana-527	116	33	is	be	AUX
cana-527	116	34	generated	generate	VERB
cana-527	116	35	according	accord	VERB
cana-527	116	36	to	to	ADP
cana-527	116	37	b1	b1	NOUN
cana-527	116	38	=	=	SYM
cana-527	116	39	s(b0	s(b0	NOUN
cana-527	116	40	)	)	PUNCT
cana-527	116	41	,	,	PUNCT
cana-527	116	42	b2	b2	NOUN
cana-527	116	43	=	=	PUNCT
cana-527	116	44	s2(b0	s2(b0	NOUN
cana-527	116	45	)	)	PUNCT
cana-527	116	46	.	.	PUNCT
cana-527	116	47	.	.	PUNCT
cana-527	117	1	.	.	PUNCT
cana-527	118	1	,	,	PUNCT
cana-527	118	2	bn	bn	NOUN
cana-527	118	3	=	=	SYM
cana-527	118	4	sn(b0	sn(b0	NOUN
cana-527	118	5	)	)	PUNCT
cana-527	118	6	.	.	PUNCT
cana-527	119	1	we	we	PRON
cana-527	119	2	also	also	ADV
cana-527	119	3	have	have	VERB
cana-527	119	4	the	the	DET
cana-527	119	5	following	follow	VERB
cana-527	119	6	properties	property	NOUN
cana-527	119	7	:	:	PUNCT
cana-527	119	8	1	1	X
cana-527	119	9	.	.	PUNCT
cana-527	119	10	since	since	SCONJ
cana-527	119	11	there	there	PRON
cana-527	119	12	is	be	VERB
cana-527	119	13	only	only	ADV
cana-527	119	14	one	one	NUM
cana-527	119	15	equivalence	equivalence	NOUN
cana-527	119	16	class	class	NOUN
cana-527	120	1	[	[	X
cana-527	120	2	b	b	X
cana-527	120	3	]	]	X
cana-527	120	4	,	,	PUNCT
cana-527	120	5	no	no	DET
cana-527	120	6	b̄	b̄	NOUN
cana-527	120	7	point	point	NOUN
cana-527	120	8	.	.	PUNCT
cana-527	121	1	this	this	PRON
cana-527	121	2	is	be	AUX
cana-527	121	3	how	how	SCONJ
cana-527	121	4	uniqueness	uniqueness	NOUN
cana-527	121	5	is	be	AUX
cana-527	121	6	defined	define	VERB
cana-527	121	7	.	.	PUNCT
cana-527	122	1	¢	¢	PRON
cana-527	123	1	[	[	X
cana-527	123	2	b	b	X
cana-527	123	3	]	]	X
cana-527	123	4	may	may	AUX
cana-527	123	5	be	be	AUX
cana-527	123	6	a	a	DET
cana-527	123	7	near	near	ADJ
cana-527	123	8	fixed	fix	VERB
cana-527	123	9	2	2	NUM
cana-527	123	10	.	.	PUNCT
cana-527	124	1	in	in	ADP
cana-527	124	2	addition	addition	NOUN
cana-527	124	3	,	,	PUNCT
cana-527	124	4	every	every	DET
cana-527	124	5	point	point	NOUN
cana-527	124	6	b̄	b̄	VERB
cana-527	124	7	fixed	fix	VERB
cana-527	124	8	point	point	NOUN
cana-527	124	9	of	of	ADP
cana-527	124	10	s.	s.	PROPN
cana-527	124	11	∈	∈	PROPN
cana-527	125	1	[	[	X
cana-527	125	2	b	b	X
cana-527	125	3	]	]	X
cana-527	125	4	that	that	PRON
cana-527	125	5	satisfies	satisfy	VERB
cana-527	125	6	s(b̄	s(b̄	NOUN
cana-527	125	7	)	)	PUNCT
cana-527	125	8	≏	≏	ADJ
cana-527	125	9	b̄	b̄	NOUN
cana-527	125	10	and	and	CCONJ
cana-527	125	11	[	[	X
cana-527	125	12	b̄	b̄	NOUN
cana-527	125	13	]	]	X
cana-527	125	14	=	=	PUNCT
cana-527	126	1	[	[	X
cana-527	126	2	b	b	X
cana-527	126	3	]	]	X
cana-527	126	4	is	be	AUX
cana-527	126	5	a	a	DET
cana-527	126	6	near	near	ADJ
cana-527	126	7	3	3	NUM
cana-527	126	8	.	.	PUNCT
cana-527	126	9	b̄	b̄	PROPN
cana-527	126	10	∈	∈	PROPN
cana-527	127	1	[	[	X
cana-527	127	2	b	b	X
cana-527	127	3	]	]	X
cana-527	127	4	,	,	PUNCT
cana-527	127	5	i.e.	i.e.	X
cana-527	127	6	[	[	X
cana-527	127	7	b̄	b̄	NOUN
cana-527	127	8	]	]	PUNCT
cana-527	127	9	=	=	PUNCT
cana-527	128	1	[	[	X
cana-527	128	2	b	b	X
cana-527	128	3	]	]	X
cana-527	128	4	,	,	PUNCT
cana-527	128	5	if	if	SCONJ
cana-527	128	6	b̄	b̄	PROPN
cana-527	128	7	is	be	AUX
cana-527	128	8	a	a	DET
cana-527	128	9	close	close	ADJ
cana-527	128	10	fixed	fix	VERB
cana-527	128	11	point	point	NOUN
cana-527	128	12	of	of	ADP
cana-527	128	13	s.	s.	PROPN
cana-527	128	14	similarly	similarly	ADV
cana-527	128	15	,	,	PUNCT
cana-527	128	16	b	b	X
cana-527	128	17	≏	≏	ADJ
cana-527	128	18	b̄	b̄	NOUN
cana-527	128	19	if	if	SCONJ
cana-527	128	20	b	b	PROPN
cana-527	128	21	and	and	CCONJ
cana-527	128	22	b̄	b̄	NOUN
cana-527	128	23	are	be	AUX
cana-527	128	24	the	the	DET
cana-527	128	25	near	near	ADJ
cana-527	128	26	fixed	fix	VERB
cana-527	128	27	points	point	NOUN
cana-527	128	28	of	of	ADP
cana-527	128	29	s.	s.	PROPN
cana-527	128	30	iii	iii	PROPN
cana-527	128	31	main	main	ADJ
cana-527	128	32	result	result	NOUN
cana-527	128	33	one	one	NUM
cana-527	128	34	method	method	NOUN
cana-527	128	35	to	to	PART
cana-527	128	36	visualise	visualise	VERB
cana-527	128	37	the	the	DET
cana-527	128	38	interactions	interaction	NOUN
cana-527	128	39	between	between	ADP
cana-527	128	40	elements	element	NOUN
cana-527	128	41	in	in	ADP
cana-527	128	42	bhs	bhs	PROPN
cana-527	128	43	is	be	AUX
cana-527	128	44	through	through	ADP
cana-527	128	45	graphs	graph	NOUN
cana-527	128	46	associated	associate	VERB
cana-527	128	47	with	with	ADP
cana-527	128	48	them	they	PRON
cana-527	128	49	,	,	PUNCT
cana-527	128	50	where	where	SCONJ
cana-527	128	51	each	each	DET
cana-527	128	52	node	node	NOUN
cana-527	128	53	of	of	ADP
cana-527	128	54	the	the	DET
cana-527	128	55	graph	graph	NOUN
cana-527	128	56	corresponds	correspond	VERB
cana-527	128	57	to	to	ADP
cana-527	128	58	a	a	DET
cana-527	128	59	compact	compact	ADJ
cana-527	128	60	set	set	NOUN
cana-527	128	61	.	.	PUNCT
cana-527	129	1	certain	certain	ADJ
cana-527	129	2	attributes	attribute	NOUN
cana-527	129	3	or	or	CCONJ
cana-527	129	4	relationships	relationship	NOUN
cana-527	129	5	between	between	ADP
cana-527	129	6	these	these	DET
cana-527	129	7	sets	set	NOUN
cana-527	129	8	are	be	AUX
cana-527	129	9	represented	represent	VERB
cana-527	129	10	by	by	ADP
cana-527	129	11	the	the	DET
cana-527	129	12	edges	edge	NOUN
cana-527	129	13	connecting	connect	VERB
cana-527	129	14	the	the	DET
cana-527	129	15	nodes	node	NOUN
cana-527	129	16	.	.	PUNCT
cana-527	130	1	the	the	DET
cana-527	130	2	way	way	NOUN
cana-527	130	3	in	in	ADP
cana-527	130	4	which	which	PRON
cana-527	130	5	these	these	DET
cana-527	130	6	graphs	graph	NOUN
cana-527	130	7	are	be	AUX
cana-527	130	8	specifically	specifically	ADV
cana-527	130	9	constructed	construct	VERB
cana-527	130	10	can	can	AUX
cana-527	130	11	change	change	VERB
cana-527	130	12	based	base	VERB
cana-527	130	13	on	on	ADP
cana-527	130	14	the	the	DET
cana-527	130	15	qualities	quality	NOUN
cana-527	130	16	of	of	ADP
cana-527	130	17	interest	interest	NOUN
cana-527	130	18	and	and	CCONJ
cana-527	130	19	the	the	DET
cana-527	130	20	environment	environment	NOUN
cana-527	130	21	.	.	PUNCT
cana-527	131	1	the	the	DET
cana-527	131	2	structure	structure	NOUN
cana-527	131	3	of	of	ADP
cana-527	131	4	banach	banach	NOUN
cana-527	131	5	hyperspaces	hyperspace	NOUN
cana-527	131	6	and	and	CCONJ
cana-527	131	7	the	the	DET
cana-527	131	8	connections	connection	NOUN
cana-527	131	9	between	between	ADP
cana-527	131	10	compact	compact	ADJ
cana-527	131	11	sets	set	NOUN
cana-527	131	12	within	within	ADP
cana-527	131	13	them	they	PRON
cana-527	131	14	can	can	AUX
cana-527	131	15	be	be	AUX
cana-527	131	16	seen	see	VERB
cana-527	131	17	and	and	CCONJ
cana-527	131	18	examined	examine	VERB
cana-527	131	19	using	use	VERB
cana-527	131	20	these	these	DET
cana-527	131	21	graphs	graph	NOUN
cana-527	131	22	.	.	PUNCT
cana-527	132	1	in	in	ADP
cana-527	132	2	this	this	DET
cana-527	132	3	final	final	ADJ
cana-527	132	4	part	part	NOUN
cana-527	132	5	of	of	ADP
cana-527	132	6	the	the	DET
cana-527	132	7	manuscript	manuscript	NOUN
cana-527	132	8	,	,	PUNCT
cana-527	132	9	we	we	PRON
cana-527	132	10	list	list	VERB
cana-527	132	11	our	our	PRON
cana-527	132	12	main	main	ADJ
cana-527	132	13	results	result	NOUN
cana-527	132	14	:	:	PUNCT
cana-527	132	15	communications	communication	NOUN
cana-527	132	16	on	on	ADP
cana-527	132	17	applied	apply	VERB
cana-527	132	18	nonlinear	nonlinear	ADJ
cana-527	132	19	analysis	analysis	NOUN
cana-527	132	20	issn	issn	NOUN
cana-527	132	21	:	:	PUNCT
cana-527	132	22	1074	1074	NUM
cana-527	132	23	-	-	PUNCT
cana-527	132	24	133x	133x	NUM
cana-527	132	25	vol	vol	NOUN
cana-527	132	26	31	31	NUM
cana-527	132	27	no	no	NOUN
cana-527	132	28	.	.	NOUN
cana-527	132	29	2	2	NUM
cana-527	132	30	(	(	PUNCT
cana-527	132	31	2024	2024	NUM
cana-527	132	32	)	)	PUNCT
cana-527	132	33	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	133	1	162	162	NUM
cana-527	133	2	1	1	NUM
cana-527	133	3	,	,	PUNCT
cana-527	133	4	x	x	PUNCT
cana-527	133	5	=	=	SYM
cana-527	133	6	3	3	NUM
cana-527	133	7	3.1	3.1	NUM
cana-527	133	8	graphs	graph	NOUN
cana-527	133	9	associate	associate	ADJ
cana-527	133	10	with	with	ADP
cana-527	133	11	banach	banach	NOUN
cana-527	133	12	hyperspaces	hyperspace	NOUN
cana-527	133	13	let	let	VERB
cana-527	133	14	(	(	PUNCT
cana-527	133	15	k(s	k(s	PROPN
cana-527	133	16	)	)	PUNCT
cana-527	133	17	,	,	PUNCT
cana-527	133	18	∥.∥	∥.∥	NUM
cana-527	133	19	)	)	PUNCT
cana-527	133	20	be	be	AUX
cana-527	133	21	a	a	DET
cana-527	133	22	bhs	bhs	PROPN
cana-527	133	23	,	,	PUNCT
cana-527	133	24	and	and	CCONJ
cana-527	133	25	let	let	VERB
cana-527	133	26	∥.∥	∥.∥	PRON
cana-527	133	27	satisfy	satisfy	VERB
cana-527	133	28	the	the	DET
cana-527	133	29	null	null	ADJ
cana-527	133	30	equality	equality	NOUN
cana-527	133	31	,	,	PUNCT
cana-527	133	32	and	and	CCONJ
cana-527	133	33	the	the	DET
cana-527	133	34	null	null	ADJ
cana-527	133	35	set	set	NOUN
cana-527	133	36	be	be	AUX
cana-527	133	37	ω	ω	NUM
cana-527	133	38	.	.	PUNCT
cana-527	134	1	s	s	PART
cana-527	134	2	:	:	PUNCT
cana-527	134	3	k(s	k(s	PROPN
cana-527	134	4	)	)	PUNCT
cana-527	134	5	→	→	SYM
cana-527	134	6	k(s	k(s	PROPN
cana-527	134	7	)	)	PUNCT
cana-527	134	8	.	.	PUNCT
cana-527	135	1	define	define	VERB
cana-527	135	2	the	the	DET
cana-527	135	3	banach	banach	NOUN
cana-527	135	4	hyperspace	hyperspace	NOUN
cana-527	135	5	-	-	PUNCT
cana-527	135	6	related	relate	VERB
cana-527	135	7	graph	graph	NOUN
cana-527	135	8	as	as	SCONJ
cana-527	135	9	follows	follow	VERB
cana-527	135	10	:	:	PUNCT
cana-527	135	11	definition	definition	NOUN
cana-527	135	12	3.1.1	3.1.1	NUM
cana-527	135	13	:	:	PUNCT
cana-527	135	14	assume	assume	VERB
cana-527	135	15	that	that	SCONJ
cana-527	135	16	the	the	DET
cana-527	135	17	bhs	bhs	PROPN
cana-527	135	18	is	be	AUX
cana-527	135	19	(	(	PUNCT
cana-527	135	20	k(s	k(s	PROPN
cana-527	135	21	)	)	PUNCT
cana-527	135	22	,	,	PUNCT
cana-527	135	23	∥.∥	∥.∥	NUM
cana-527	135	24	)	)	PUNCT
cana-527	135	25	.	.	PUNCT
cana-527	136	1	t	t	NOUN
cana-527	136	2	:	:	PUNCT
cana-527	136	3	k(s	k(s	PROPN
cana-527	136	4	)	)	PUNCT
cana-527	136	5	→	→	SYM
cana-527	136	6	k(s	k(s	PROPN
cana-527	136	7	)	)	PUNCT
cana-527	136	8	,	,	PUNCT
cana-527	136	9	let	let	VERB
cana-527	136	10	us	we	PRON
cana-527	136	11	say	say	VERB
cana-527	136	12	.	.	PUNCT
cana-527	137	1	define	define	VERB
cana-527	137	2	the	the	DET
cana-527	137	3	following	following	ADJ
cana-527	137	4	weighted	weight	VERB
cana-527	137	5	graph	graph	NOUN
cana-527	137	6	h	h	NOUN
cana-527	137	7	connected	connect	VERB
cana-527	137	8	to	to	ADP
cana-527	137	9	k(s	k(s	PROPN
cana-527	137	10	):	):	PUNCT
cana-527	137	11	let	let	VERB
cana-527	137	12	h	h	NOUN
cana-527	137	13	=	=	PUNCT
cana-527	137	14	(	(	PUNCT
cana-527	137	15	v	v	NOUN
cana-527	137	16	,	,	PUNCT
cana-527	137	17	e	e	NOUN
cana-527	137	18	)	)	PUNCT
cana-527	137	19	,	,	PUNCT
cana-527	137	20	where	where	SCONJ
cana-527	137	21	e	e	X
cana-527	137	22	=	=	PRON
cana-527	137	23	{	{	PUNCT
cana-527	137	24	(	(	PUNCT
cana-527	137	25	x	x	NOUN
cana-527	137	26	,	,	PUNCT
cana-527	137	27	s(x))/x	s(x))/x	PROPN
cana-527	137	28	∈	∈	PROPN
cana-527	137	29	k(s	k(s	PROPN
cana-527	137	30	)	)	PUNCT
cana-527	137	31	}	}	PUNCT
cana-527	137	32	and	and	CCONJ
cana-527	137	33	v	v	X
cana-527	137	34	=	=	SYM
cana-527	137	35	k(s	k(s	PROPN
cana-527	137	36	)	)	PUNCT
cana-527	137	37	.	.	PUNCT
cana-527	138	1	the	the	DET
cana-527	138	2	spacing	spacing	NOUN
cana-527	138	3	be	be	AUX
cana-527	138	4	tween	tween	VERB
cana-527	138	5	an	an	DET
cana-527	138	6	edge	edge	NOUN
cana-527	138	7	’s	’s	PART
cana-527	138	8	end	end	NOUN
cana-527	138	9	points	point	NOUN
cana-527	138	10	determines	determine	VERB
cana-527	138	11	its	its	PRON
cana-527	138	12	weight	weight	NOUN
cana-527	138	13	.	.	PUNCT
cana-527	139	1	as	as	ADP
cana-527	139	2	a	a	DET
cana-527	139	3	result	result	NOUN
cana-527	139	4	,	,	PUNCT
cana-527	139	5	(	(	PUNCT
cana-527	139	6	k(s	k(s	PROPN
cana-527	139	7	)	)	PUNCT
cana-527	139	8	,	,	PUNCT
cana-527	139	9	∥.∥	∥.∥	NUM
cana-527	139	10	)	)	PUNCT
cana-527	139	11	is	be	AUX
cana-527	139	12	transformed	transform	VERB
cana-527	139	13	into	into	ADP
cana-527	139	14	a	a	DET
cana-527	139	15	bhs	bhs	PROPN
cana-527	139	16	and	and	CCONJ
cana-527	139	17	given	give	VERB
cana-527	139	18	h.	h.	PROPN
cana-527	139	19	definition	definition	NOUN
cana-527	139	20	3.1.2	3.1.2	NUM
cana-527	139	21	:	:	PUNCT
cana-527	139	22	the	the	DET
cana-527	139	23	sub	sub	ADJ
cana-527	139	24	-	-	ADJ
cana-527	139	25	graph	graph	NOUN
cana-527	139	26	h0	h0	NOUN
cana-527	139	27	of	of	ADP
cana-527	139	28	h	h	NOUN
cana-527	139	29	is	be	AUX
cana-527	139	30	defined	define	VERB
cana-527	139	31	as	as	ADP
cana-527	139	32	,	,	PUNCT
cana-527	139	33	let	let	VERB
cana-527	139	34	y0	y0	PRON
cana-527	139	35	be	be	AUX
cana-527	139	36	any	any	DET
cana-527	139	37	random	random	ADJ
cana-527	139	38	point	point	NOUN
cana-527	139	39	of	of	ADP
cana-527	139	40	s	s	PRON
cana-527	139	41	and	and	CCONJ
cana-527	139	42	h0	h0	NOUN
cana-527	139	43	=	=	SYM
cana-527	139	44	(	(	PUNCT
cana-527	139	45	v0	v0	PROPN
cana-527	139	46	,	,	PUNCT
cana-527	139	47	e0	e0	PROPN
cana-527	139	48	)	)	PUNCT
cana-527	139	49	where	where	SCONJ
cana-527	139	50	v0	v0	NOUN
cana-527	139	51	=	=	SYM
cana-527	139	52	{	{	PUNCT
cana-527	139	53	y0	y0	PROPN
cana-527	139	54	,	,	PUNCT
cana-527	139	55	sy0	sy0	PROPN
cana-527	139	56	,	,	PUNCT
cana-527	139	57	s2y0	s2y0	PROPN
cana-527	139	58	,	,	PUNCT
cana-527	139	59	.	.	PUNCT
cana-527	139	60	.	.	PUNCT
cana-527	140	1	.	.	PUNCT
cana-527	140	2	}	}	PUNCT
cana-527	141	1	and	and	CCONJ
cana-527	141	2	let	let	VERB
cana-527	141	3	e0	e0	PROPN
cana-527	141	4	=	=	PRON
cana-527	141	5	{	{	PUNCT
cana-527	141	6	(	(	PUNCT
cana-527	141	7	y0	y0	NOUN
cana-527	141	8	,	,	PUNCT
cana-527	141	9	sy0	sy0	NOUN
cana-527	141	10	)	)	PUNCT
cana-527	141	11	,	,	PUNCT
cana-527	141	12	(	(	PUNCT
cana-527	141	13	sy0	sy0	NOUN
cana-527	141	14	,	,	PUNCT
cana-527	141	15	s2y0	s2y0	PROPN
cana-527	141	16	)	)	PUNCT
cana-527	141	17	,	,	PUNCT
cana-527	141	18	.	.	PUNCT
cana-527	141	19	.	.	PUNCT
cana-527	142	1	.	.	PUNCT
cana-527	142	2	}	}	PUNCT
cana-527	142	3	.	.	PUNCT
cana-527	143	1	then	then	ADV
cana-527	143	2	v0	v0	PROPN
cana-527	143	3	⊂	⊂	PROPN
cana-527	143	4	v	v	PROPN
cana-527	143	5	and	and	CCONJ
cana-527	143	6	e0	e0	PROPN
cana-527	143	7	⊂	⊂	PROPN
cana-527	143	8	e.	e.	PROPN
cana-527	143	9	hence	hence	PROPN
cana-527	143	10	h0	h0	PROPN
cana-527	143	11	is	be	AUX
cana-527	143	12	a	a	DET
cana-527	143	13	sub	sub	NOUN
cana-527	143	14	-	-	NOUN
cana-527	143	15	graph	graph	NOUN
cana-527	143	16	of	of	ADP
cana-527	143	17	h.	h.	PROPN
cana-527	143	18	definition	definition	NOUN
cana-527	143	19	3.1.3	3.1.3	NUM
cana-527	143	20	:	:	PUNCT
cana-527	143	21	let	let	VERB
cana-527	143	22	(	(	PUNCT
cana-527	143	23	k(s	k(s	PROPN
cana-527	143	24	)	)	PUNCT
cana-527	143	25	,	,	PUNCT
cana-527	143	26	∥.∥	∥.∥	NUM
cana-527	143	27	)	)	PUNCT
cana-527	143	28	be	be	AUX
cana-527	143	29	a	a	DET
cana-527	143	30	bhs	bhs	PROPN
cana-527	143	31	endowed	endow	VERB
cana-527	143	32	with	with	ADP
cana-527	143	33	h.	h.	PROPN
cana-527	143	34	let	let	VERB
cana-527	143	35	h0	h0	PROPN
cana-527	143	36	be	be	AUX
cana-527	143	37	the	the	DET
cana-527	143	38	sub	sub	NOUN
cana-527	143	39	-	-	NOUN
cana-527	143	40	graph	graph	NOUN
cana-527	143	41	of	of	ADP
cana-527	143	42	h	h	NOUN
cana-527	143	43	defined	define	VERB
cana-527	143	44	as	as	ADP
cana-527	143	45	in	in	ADP
cana-527	143	46	def	def	NOUN
cana-527	143	47	.	.	PUNCT
cana-527	144	1	3.1.2	3.1.2	X
cana-527	144	2	.	.	PUNCT
cana-527	144	3	let	let	VERB
cana-527	144	4	wn	wn	PROPN
cana-527	144	5	=	=	PUNCT
cana-527	144	6	∥sn−1(y0	∥sn−1(y0	PROPN
cana-527	144	7	)	)	PUNCT
cana-527	144	8	⊖	⊖	AUX
cana-527	144	9	sn(y0)∥.	sn(y0)∥.	ADJ
cana-527	144	10	then	then	ADV
cana-527	144	11	the	the	DET
cana-527	144	12	sequence	sequence	NOUN
cana-527	144	13	{	{	PUNCT
cana-527	144	14	wn}∞n=1	wn}∞n=1	X
cana-527	144	15	is	be	AUX
cana-527	144	16	called	call	VERB
cana-527	144	17	w	w	NOUN
cana-527	144	18	-	-	PUNCT
cana-527	144	19	sequence	sequence	NOUN
cana-527	144	20	of	of	ADP
cana-527	144	21	real	real	ADJ
cana-527	144	22	numbers	number	NOUN
cana-527	144	23	associated	associate	VERB
cana-527	144	24	with	with	ADP
cana-527	144	25	the	the	DET
cana-527	144	26	graph	graph	NOUN
cana-527	144	27	h0	h0	PROPN
cana-527	144	28	.	.	PROPN
cana-527	144	29	example	example	NOUN
cana-527	145	1	3.1.1	3.1.1	NUM
cana-527	145	2	:	:	PUNCT
cana-527	145	3	let	let	VERB
cana-527	145	4	k(s	k(s	PROPN
cana-527	145	5	)	)	PUNCT
cana-527	145	6	=	=	PUNCT
cana-527	145	7	{	{	PUNCT
cana-527	145	8	0	0	NUM
cana-527	145	9	,	,	PUNCT
cana-527	145	10	1	1	NUM
cana-527	145	11	,	,	PUNCT
cana-527	145	12	2	2	NUM
cana-527	145	13	,	,	PUNCT
cana-527	145	14	3	3	NUM
cana-527	145	15	}	}	PUNCT
cana-527	145	16	,	,	PUNCT
cana-527	145	17	∥x	∥x	PROPN
cana-527	145	18	−	−	PROPN
cana-527	145	19	y∥	y∥	NOUN
cana-527	145	20	=	=	PUNCT
cana-527	145	21	|x	|x	NOUN
cana-527	145	22	−	−	PROPN
cana-527	145	23	y|	y|	NOUN
cana-527	145	24	,	,	PUNCT
cana-527	145	25	x	x	PRON
cana-527	145	26	,	,	PUNCT
cana-527	145	27	y	y	PROPN
cana-527	145	28	∈	∈	PROPN
cana-527	145	29	k(s	k(s	PROPN
cana-527	145	30	)	)	PUNCT
cana-527	145	31	.	.	PUNCT
cana-527	146	1	let	let	VERB
cana-527	146	2	s	s	PRON
cana-527	146	3	:	:	PUNCT
cana-527	146	4	k(s	k(s	PROPN
cana-527	146	5	)	)	PUNCT
cana-527	147	1	→	→	SYM
cana-527	147	2	k(s	k(s	PROPN
cana-527	147	3	)	)	PUNCT
cana-527	147	4	as	as	ADP
cana-527	147	5	,	,	PUNCT
cana-527	147	6	s(x	s(x	PROPN
cana-527	147	7	)	)	PUNCT
cana-527	147	8	=	=	SYM
cana-527	147	9	0	0	NUM
cana-527	147	10	,	,	PUNCT
cana-527	147	11	x	x	X
cana-527	147	12	∈	∈	NOUN
cana-527	147	13	{	{	PUNCT
cana-527	147	14	0	0	NUM
cana-527	147	15	,	,	PUNCT
cana-527	147	16	1	1	NUM
cana-527	147	17	,	,	PUNCT
cana-527	147	18	2	2	NUM
cana-527	147	19	}	}	SYM
cana-527	147	20	v	v	NOUN
cana-527	147	21	=	=	SYM
cana-527	147	22	{	{	PUNCT
cana-527	147	23	0	0	NUM
cana-527	147	24	,	,	PUNCT
cana-527	147	25	1	1	NUM
cana-527	147	26	,	,	PUNCT
cana-527	147	27	2	2	NUM
cana-527	147	28	,	,	PUNCT
cana-527	147	29	3	3	NUM
cana-527	147	30	}	}	PUNCT
cana-527	147	31	and	and	CCONJ
cana-527	147	32	e	e	X
cana-527	147	33	=	=	PRON
cana-527	147	34	{	{	PUNCT
cana-527	147	35	(	(	PUNCT
cana-527	147	36	0	0	NUM
cana-527	147	37	,	,	PUNCT
cana-527	147	38	0	0	NUM
cana-527	147	39	)	)	PUNCT
cana-527	147	40	,	,	PUNCT
cana-527	147	41	(	(	PUNCT
cana-527	147	42	1	1	NUM
cana-527	147	43	,	,	PUNCT
cana-527	147	44	0	0	NUM
cana-527	147	45	)	)	PUNCT
cana-527	147	46	,	,	PUNCT
cana-527	147	47	(	(	PUNCT
cana-527	147	48	2	2	NUM
cana-527	147	49	,	,	PUNCT
cana-527	147	50	0	0	NUM
cana-527	147	51	)	)	PUNCT
cana-527	147	52	,	,	PUNCT
cana-527	147	53	(	(	PUNCT
cana-527	147	54	3	3	NUM
cana-527	147	55	,	,	PUNCT
cana-527	147	56	1	1	NUM
cana-527	147	57	)	)	PUNCT
cana-527	147	58	}	}	PUNCT
cana-527	147	59	figure	figure	VERB
cana-527	147	60	3.1	3.1	NUM
cana-527	147	61	:	:	PUNCT
cana-527	147	62	the	the	DET
cana-527	147	63	data	data	NOUN
cana-527	147	64	structure	structure	NOUN
cana-527	147	65	h	h	NOUN
cana-527	147	66	linked	link	VERB
cana-527	147	67	to	to	ADP
cana-527	147	68	k(s	k(s	PROPN
cana-527	147	69	)	)	PUNCT
cana-527	147	70	case	case	NOUN
cana-527	147	71	1	1	NUM
cana-527	147	72	:	:	PUNCT
cana-527	148	1	y0	y0	NOUN
cana-527	148	2	=	=	SYM
cana-527	148	3	0	0	NUM
cana-527	148	4	.	.	PUNCT
cana-527	148	5	case	case	NOUN
cana-527	148	6	2	2	NUM
cana-527	148	7	:	:	PUNCT
cana-527	148	8	y0	y0	NOUN
cana-527	148	9	=	=	SYM
cana-527	148	10	1	1	X
cana-527	148	11	.	.	PUNCT
cana-527	148	12	communications	communication	NOUN
cana-527	148	13	on	on	ADP
cana-527	148	14	applied	apply	VERB
cana-527	148	15	nonlinear	nonlinear	ADJ
cana-527	148	16	analysis	analysis	NOUN
cana-527	148	17	issn	issn	NOUN
cana-527	148	18	:	:	PUNCT
cana-527	148	19	1074	1074	NUM
cana-527	148	20	-	-	PUNCT
cana-527	148	21	133x	133x	NUM
cana-527	148	22	vol	vol	NOUN
cana-527	148	23	31	31	NUM
cana-527	148	24	no	no	NOUN
cana-527	148	25	.	.	NOUN
cana-527	148	26	2	2	NUM
cana-527	148	27	(	(	PUNCT
cana-527	148	28	2024	2024	NUM
cana-527	148	29	)	)	PUNCT
cana-527	148	30	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	148	31	163	163	NUM
cana-527	148	32	figure	figure	NOUN
cana-527	148	33	3.2	3.2	NUM
cana-527	148	34	:	:	PUNCT
cana-527	148	35	within	within	ADP
cana-527	148	36	the	the	DET
cana-527	148	37	moment	moment	NOUN
cana-527	148	38	in	in	ADP
cana-527	148	39	time	time	NOUN
cana-527	148	40	where	where	SCONJ
cana-527	148	41	by	by	ADP
cana-527	148	42	y0	y0	PROPN
cana-527	148	43	=	=	SYM
cana-527	148	44	0	0	NUM
cana-527	148	45	,	,	PUNCT
cana-527	148	46	graph	graph	NOUN
cana-527	148	47	h0	h0	NOUN
cana-527	148	48	figure	figure	NOUN
cana-527	148	49	3.3	3.3	NUM
cana-527	148	50	:	:	PUNCT
cana-527	148	51	within	within	ADP
cana-527	148	52	the	the	DET
cana-527	148	53	moment	moment	NOUN
cana-527	148	54	in	in	ADP
cana-527	148	55	time	time	NOUN
cana-527	148	56	where	where	SCONJ
cana-527	148	57	by	by	ADP
cana-527	148	58	y0	y0	PROPN
cana-527	148	59	=	=	SYM
cana-527	148	60	1	1	NUM
cana-527	148	61	,	,	PUNCT
cana-527	148	62	graph	graph	NOUN
cana-527	148	63	h0	h0	NOUN
cana-527	148	64	case	case	NOUN
cana-527	148	65	3	3	NUM
cana-527	148	66	:	:	PUNCT
cana-527	148	67	y0	y0	NOUN
cana-527	148	68	=	=	SYM
cana-527	148	69	2	2	X
cana-527	148	70	.	.	X
cana-527	148	71	figure	figure	VERB
cana-527	148	72	3.4	3.4	NUM
cana-527	148	73	:	:	PUNCT
cana-527	148	74	within	within	ADP
cana-527	148	75	the	the	DET
cana-527	148	76	moment	moment	NOUN
cana-527	148	77	in	in	ADP
cana-527	148	78	time	time	NOUN
cana-527	148	79	where	where	SCONJ
cana-527	148	80	by	by	ADP
cana-527	148	81	y0	y0	PROPN
cana-527	148	82	=	=	SYM
cana-527	148	83	2	2	NUM
cana-527	148	84	,	,	PUNCT
cana-527	148	85	graph	graph	NOUN
cana-527	148	86	h0	h0	NOUN
cana-527	148	87	case	case	NOUN
cana-527	148	88	4	4	NUM
cana-527	148	89	:	:	PUNCT
cana-527	148	90	y0	y0	NOUN
cana-527	148	91	=	=	SYM
cana-527	148	92	3	3	X
cana-527	148	93	.	.	X
cana-527	148	94	figure	figure	NOUN
cana-527	148	95	3.5	3.5	NUM
cana-527	148	96	:	:	PUNCT
cana-527	148	97	within	within	ADP
cana-527	148	98	the	the	DET
cana-527	148	99	moment	moment	NOUN
cana-527	148	100	in	in	ADP
cana-527	148	101	time	time	NOUN
cana-527	148	102	where	where	SCONJ
cana-527	148	103	by	by	ADP
cana-527	148	104	y0	y0	PROPN
cana-527	148	105	=	=	SYM
cana-527	148	106	3	3	NUM
cana-527	148	107	,	,	PUNCT
cana-527	148	108	graph	graph	NOUN
cana-527	148	109	h0	h0	NOUN
cana-527	148	110	on	on	ADP
cana-527	148	111	the	the	DET
cana-527	148	112	bhs	bhs	PROPN
cana-527	148	113	(	(	PUNCT
cana-527	148	114	k(s	k(s	PROPN
cana-527	148	115	)	)	PUNCT
cana-527	148	116	,	,	PUNCT
cana-527	148	117	∥.∥	∥.∥	NUM
cana-527	148	118	)	)	PUNCT
cana-527	149	1	equipped	equip	VERB
cana-527	149	2	with	with	ADP
cana-527	149	3	the	the	DET
cana-527	149	4	graph	graph	NOUN
cana-527	149	5	h	h	NOUN
cana-527	149	6	,	,	PUNCT
cana-527	149	7	the	the	DET
cana-527	149	8	near	near	ADJ
cana-527	149	9	fixed	fix	VERB
cana-527	149	10	point	point	NOUN
cana-527	149	11	theorems	theorem	NOUN
cana-527	149	12	are	be	AUX
cana-527	149	13	capable	capable	ADJ
cana-527	149	14	of	of	ADP
cana-527	149	15	being	be	AUX
cana-527	149	16	proved	prove	VERB
cana-527	149	17	with	with	ADP
cana-527	149	18	the	the	DET
cana-527	149	19	assistance	assistance	NOUN
cana-527	149	20	of	of	ADP
cana-527	149	21	the	the	DET
cana-527	149	22	subsequent	subsequent	ADJ
cana-527	149	23	two	two	NUM
cana-527	149	24	lemmas	lemmas	ADJ
cana-527	149	25	.	.	PUNCT
cana-527	150	1	lemma	lemma	PROPN
cana-527	150	2	3.1.1	3.1.1	NUM
cana-527	150	3	(	(	PUNCT
cana-527	150	4	16	16	NUM
cana-527	150	5	):	):	PUNCT
cana-527	150	6	given	give	VERB
cana-527	150	7	a	a	DET
cana-527	150	8	bhs	bhs	PROPN
cana-527	150	9	(	(	PUNCT
cana-527	150	10	k(s	k(s	PROPN
cana-527	150	11	)	)	PUNCT
cana-527	150	12	,	,	PUNCT
cana-527	150	13	∥.∥	∥.∥	NUM
cana-527	150	14	)	)	PUNCT
cana-527	150	15	,	,	PUNCT
cana-527	150	16	let	let	VERB
cana-527	150	17	s	s	PRON
cana-527	150	18	:	:	PUNCT
cana-527	150	19	k(s	k(s	PROPN
cana-527	150	20	)	)	PUNCT
cana-527	151	1	→	→	SYM
cana-527	151	2	k(s	k(s	PROPN
cana-527	151	3	)	)	PUNCT
cana-527	151	4	.	.	PUNCT
cana-527	152	1	assume	assume	VERB
cana-527	152	2	that	that	SCONJ
cana-527	152	3	h	h	NOUN
cana-527	152	4	is	be	AUX
cana-527	152	5	a	a	DET
cana-527	152	6	graph	graph	NOUN
cana-527	152	7	connected	connect	VERB
cana-527	152	8	to	to	ADP
cana-527	152	9	k(s	k(s	PROPN
cana-527	152	10	)	)	PUNCT
cana-527	152	11	.	.	PUNCT
cana-527	153	1	let	let	VERB
cana-527	153	2	y0	y0	PRON
cana-527	153	3	represent	represent	VERB
cana-527	153	4	any	any	DET
cana-527	153	5	random	random	ADJ
cana-527	153	6	point	point	NOUN
cana-527	153	7	in	in	ADP
cana-527	153	8	k(s	k(s	PROPN
cana-527	153	9	)	)	PUNCT
cana-527	153	10	.	.	PUNCT
cana-527	154	1	assume	assume	VERB
cana-527	154	2	that	that	SCONJ
cana-527	154	3	the	the	DET
cana-527	154	4	sub	sub	ADJ
cana-527	154	5	-	-	ADJ
cana-527	154	6	graph	graph	NOUN
cana-527	154	7	h0	h0	NOUN
cana-527	154	8	of	of	ADP
cana-527	154	9	h	h	NOUN
cana-527	154	10	is	be	AUX
cana-527	154	11	defined	define	VERB
cana-527	154	12	according	accord	VERB
cana-527	154	13	to	to	ADP
cana-527	154	14	definition	definition	NOUN
cana-527	154	15	3.1.2	3.1.2	NUM
cana-527	154	16	.	.	PUNCT
cana-527	155	1	then	then	ADV
cana-527	155	2	,	,	PUNCT
cana-527	155	3	if	if	SCONJ
cana-527	155	4	and	and	CCONJ
cana-527	155	5	only	only	ADV
cana-527	155	6	if	if	SCONJ
cana-527	155	7	the	the	DET
cana-527	155	8	w	w	NOUN
cana-527	155	9	-	-	PUNCT
cana-527	155	10	sequence	sequence	NOUN
cana-527	155	11	connected	connect	VERB
cana-527	155	12	to	to	ADP
cana-527	155	13	the	the	DET
cana-527	155	14	graph	graph	NOUN
cana-527	155	15	h0	h0	NOUN
cana-527	155	16	is	be	AUX
cana-527	155	17	non	non	ADJ
cana-527	155	18	-	-	ADJ
cana-527	155	19	increasing	increase	VERB
cana-527	155	20	,	,	PUNCT
cana-527	155	21	the	the	DET
cana-527	155	22	sequence	sequence	NOUN
cana-527	155	23	{	{	PUNCT
cana-527	155	24	s(a0	s(a0	NOUN
cana-527	155	25	)	)	PUNCT
cana-527	155	26	,	,	PUNCT
cana-527	155	27	s2(a0	s2(a0	NOUN
cana-527	155	28	)	)	PUNCT
cana-527	155	29	,	,	PUNCT
cana-527	155	30	.	.	PUNCT
cana-527	155	31	.	.	PUNCT
cana-527	156	1	.	.	PUNCT
cana-527	156	2	}	}	PUNCT
cana-527	156	3	is	be	AUX
cana-527	156	4	cauchy	cauchy	PROPN
cana-527	156	5	.	.	PUNCT
cana-527	157	1	lemma	lemma	PROPN
cana-527	157	2	3.1.2	3.1.2	NUM
cana-527	157	3	(	(	PUNCT
cana-527	157	4	16	16	NUM
cana-527	157	5	):	):	PUNCT
cana-527	157	6	let	let	VERB
cana-527	157	7	s	s	PRON
cana-527	157	8	:	:	PUNCT
cana-527	157	9	k(s	k(s	PROPN
cana-527	157	10	)	)	PUNCT
cana-527	158	1	→	→	SYM
cana-527	158	2	k(s	k(s	PROPN
cana-527	158	3	)	)	PUNCT
cana-527	158	4	be	be	AUX
cana-527	158	5	a	a	DET
cana-527	158	6	bhs	bhs	PROPN
cana-527	158	7	,	,	PUNCT
cana-527	158	8	and	and	CCONJ
cana-527	158	9	let	let	VERB
cana-527	158	10	(	(	PUNCT
cana-527	158	11	k(s	k(s	PROPN
cana-527	158	12	)	)	PUNCT
cana-527	158	13	,	,	PUNCT
cana-527	158	14	∥.∥	∥.∥	NUM
cana-527	158	15	)	)	PUNCT
cana-527	158	16	be	be	AUX
cana-527	158	17	its	its	PRON
cana-527	158	18	boundary	boundary	NOUN
cana-527	158	19	.	.	PUNCT
cana-527	159	1	assume	assume	VERB
cana-527	159	2	that	that	SCONJ
cana-527	159	3	h	h	NOUN
cana-527	159	4	is	be	AUX
cana-527	159	5	a	a	DET
cana-527	159	6	graph	graph	NOUN
cana-527	159	7	connected	connect	VERB
cana-527	159	8	to	to	ADP
cana-527	159	9	k(s	k(s	PROPN
cana-527	159	10	)	)	PUNCT
cana-527	159	11	.	.	PUNCT
cana-527	160	1	if	if	SCONJ
cana-527	160	2	the	the	DET
cana-527	160	3	graph	graph	NOUN
cana-527	160	4	h	h	NOUN
cana-527	160	5	includes	include	VERB
cana-527	160	6	a	a	DET
cana-527	160	7	loop	loop	NOUN
cana-527	160	8	at	at	ADP
cana-527	160	9	y∗	y∗	PROPN
cana-527	160	10	,	,	PUNCT
cana-527	160	11	then	then	ADV
cana-527	160	12	the	the	DET
cana-527	160	13	point	point	NOUN
cana-527	160	14	y∗	y∗	ADV
cana-527	160	15	of	of	ADP
cana-527	160	16	k(s	k(s	PROPN
cana-527	160	17	)	)	PUNCT
cana-527	160	18	is	be	AUX
cana-527	160	19	a	a	DET
cana-527	160	20	near	near	ADJ
cana-527	160	21	fixed	fix	VERB
cana-527	160	22	point	point	NOUN
cana-527	160	23	of	of	ADP
cana-527	160	24	s.	s.	PROPN
cana-527	160	25	communications	communications	PROPN
cana-527	160	26	on	on	ADP
cana-527	160	27	applied	apply	VERB
cana-527	160	28	nonlinear	nonlinear	ADJ
cana-527	160	29	analysis	analysis	NOUN
cana-527	160	30	issn	issn	NOUN
cana-527	160	31	:	:	PUNCT
cana-527	160	32	1074	1074	NUM
cana-527	160	33	-	-	PUNCT
cana-527	160	34	133x	133x	NUM
cana-527	160	35	vol	vol	NOUN
cana-527	160	36	31	31	NUM
cana-527	160	37	no	no	NOUN
cana-527	160	38	.	.	NOUN
cana-527	160	39	2	2	NUM
cana-527	160	40	(	(	PUNCT
cana-527	160	41	2024	2024	NUM
cana-527	160	42	)	)	PUNCT
cana-527	160	43	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	160	44	164	164	NUM
cana-527	160	45	theorem	theorem	VERB
cana-527	160	46	3.1.1	3.1.1	NOUN
cana-527	160	47	:	:	PUNCT
cana-527	160	48	let	let	VERB
cana-527	160	49	(	(	PUNCT
cana-527	160	50	k(s	k(s	PROPN
cana-527	160	51	)	)	PUNCT
cana-527	160	52	,	,	PUNCT
cana-527	160	53	∥.∥	∥.∥	NUM
cana-527	160	54	)	)	PUNCT
cana-527	160	55	be	be	AUX
cana-527	160	56	a	a	DET
cana-527	160	57	bhs	bhs	PROPN
cana-527	160	58	,	,	PUNCT
cana-527	160	59	and	and	CCONJ
cana-527	160	60	let	let	VERB
cana-527	160	61	∥.∥	∥.∥	PRON
cana-527	160	62	satisfy	satisfy	VERB
cana-527	160	63	the	the	DET
cana-527	160	64	null	null	ADJ
cana-527	160	65	equality	equality	NOUN
cana-527	160	66	,	,	PUNCT
cana-527	160	67	and	and	CCONJ
cana-527	160	68	the	the	DET
cana-527	160	69	null	null	ADJ
cana-527	160	70	set	set	NOUN
cana-527	160	71	be	be	AUX
cana-527	160	72	ω	ω	PROPN
cana-527	160	73	.	.	PUNCT
cana-527	161	1	assume	assume	VERB
cana-527	161	2	we	we	PRON
cana-527	161	3	have	have	VERB
cana-527	161	4	a	a	DET
cana-527	161	5	contraction	contraction	NOUN
cana-527	161	6	function	function	NOUN
cana-527	161	7	s	s	PART
cana-527	161	8	:	:	PUNCT
cana-527	161	9	(	(	PUNCT
cana-527	161	10	k(s	k(s	PROPN
cana-527	161	11	)	)	PUNCT
cana-527	161	12	,	,	PUNCT
cana-527	161	13	∥.∥	∥.∥	NUM
cana-527	161	14	)	)	PUNCT
cana-527	162	1	→	→	PUNCT
cana-527	162	2	(	(	PUNCT
cana-527	162	3	k(s	k(s	PROPN
cana-527	162	4	)	)	PUNCT
cana-527	162	5	,	,	PUNCT
cana-527	162	6	∥.∥	∥.∥	NUM
cana-527	162	7	)	)	PUNCT
cana-527	163	1	on	on	ADP
cana-527	163	2	k(s	k(s	PROPN
cana-527	163	3	)	)	PUNCT
cana-527	163	4	.	.	PUNCT
cana-527	163	5	assume	assume	VERB
cana-527	163	6	that	that	SCONJ
cana-527	163	7	g	g	PROPN
cana-527	163	8	is	be	AUX
cana-527	163	9	the	the	DET
cana-527	163	10	graph	graph	NOUN
cana-527	163	11	connected	connect	VERB
cana-527	163	12	to	to	ADP
cana-527	163	13	k(s	k(s	PROPN
cana-527	163	14	)	)	PUNCT
cana-527	163	15	.	.	PUNCT
cana-527	164	1	then	then	ADV
cana-527	164	2	,	,	PUNCT
cana-527	164	3	y∗	y∗	PROPN
cana-527	164	4	∈	∈	PROPN
cana-527	164	5	k(s	k(s	PROPN
cana-527	164	6	)	)	PUNCT
cana-527	164	7	is	be	AUX
cana-527	164	8	the	the	DET
cana-527	164	9	unique	unique	ADJ
cana-527	164	10	near	near	ADP
cana-527	164	11	fixed	fix	VERB
cana-527	164	12	point	point	NOUN
cana-527	164	13	of	of	ADP
cana-527	164	14	s.	s.	PROPN
cana-527	164	15	proof	proof	PROPN
cana-527	164	16	:	:	PUNCT
cana-527	164	17	considering	consider	VERB
cana-527	164	18	any	any	DET
cana-527	164	19	starting	starting	NOUN
cana-527	164	20	element	element	NOUN
cana-527	164	21	y0	y0	PROPN
cana-527	164	22	∈	∈	PROPN
cana-527	164	23	k(s	k(s	PROPN
cana-527	164	24	)	)	PUNCT
cana-527	164	25	.	.	PUNCT
cana-527	165	1	the	the	DET
cana-527	165	2	definitions	definition	NOUN
cana-527	165	3	of	of	ADP
cana-527	165	4	the	the	DET
cana-527	165	5	graph	graph	NOUN
cana-527	165	6	h	h	NOUN
cana-527	165	7	and	and	CCONJ
cana-527	165	8	sub	sub	NOUN
cana-527	165	9	graph	graph	NOUN
cana-527	165	10	h0	h0	PROPN
cana-527	165	11	are	be	AUX
cana-527	165	12	,	,	PUNCT
cana-527	165	13	”	"	PUNCT
cana-527	165	14	let	let	VERB
cana-527	165	15	the	the	DET
cana-527	165	16	banach	banach	NOUN
cana-527	165	17	hyperspace	hyperspace	NOUN
cana-527	165	18	(	(	PUNCT
cana-527	165	19	k(s	k(s	PROPN
cana-527	165	20	)	)	PUNCT
cana-527	165	21	,	,	PUNCT
cana-527	165	22	∥.∥	∥.∥	NUM
cana-527	165	23	)	)	PUNCT
cana-527	165	24	be	be	AUX
cana-527	165	25	described	describe	VERB
cana-527	165	26	.	.	PUNCT
cana-527	166	1	assume	assume	VERB
cana-527	166	2	that	that	SCONJ
cana-527	166	3	s	s	VERB
cana-527	166	4	:	:	PUNCT
cana-527	166	5	k(s	k(s	PROPN
cana-527	166	6	)	)	PUNCT
cana-527	167	1	→	→	SYM
cana-527	167	2	k(s	k(s	PROPN
cana-527	167	3	)	)	PUNCT
cana-527	167	4	.	.	PUNCT
cana-527	168	1	as	as	SCONJ
cana-527	168	2	shown	show	VERB
cana-527	168	3	below	below	ADV
cana-527	168	4	,	,	PUNCT
cana-527	168	5	define	define	VERB
cana-527	168	6	a	a	DET
cana-527	168	7	weighted	weight	VERB
cana-527	168	8	graph	graph	NOUN
cana-527	168	9	g	g	PROPN
cana-527	168	10	connected	connect	VERB
cana-527	168	11	to	to	ADP
cana-527	168	12	k(s	k(s	PROPN
cana-527	168	13	)	)	PUNCT
cana-527	168	14	.	.	PUNCT
cana-527	169	1	with	with	ADP
cana-527	169	2	v	v	NOUN
cana-527	169	3	=	=	SYM
cana-527	169	4	k(s	k(s	PROPN
cana-527	169	5	)	)	PUNCT
cana-527	169	6	and	and	CCONJ
cana-527	169	7	e	e	X
cana-527	169	8	=	=	PUNCT
cana-527	169	9	{	{	PUNCT
cana-527	169	10	x	x	PROPN
cana-527	169	11	,	,	PUNCT
cana-527	169	12	s(x)/x	s(x)/x	PROPN
cana-527	169	13	∈	∈	PROPN
cana-527	169	14	k(s	k(s	PROPN
cana-527	169	15	)	)	PUNCT
cana-527	169	16	}	}	PUNCT
cana-527	169	17	,	,	PUNCT
cana-527	169	18	let	let	VERB
cana-527	169	19	h	h	NOUN
cana-527	169	20	=	=	PUNCT
cana-527	169	21	(	(	PUNCT
cana-527	169	22	v	v	NOUN
cana-527	169	23	,	,	PUNCT
cana-527	169	24	e	e	NOUN
cana-527	169	25	)	)	PUNCT
cana-527	169	26	.	.	PUNCT
cana-527	170	1	the	the	DET
cana-527	170	2	spacing	spacing	NOUN
cana-527	170	3	between	between	ADP
cana-527	170	4	an	an	DET
cana-527	170	5	edge	edge	NOUN
cana-527	170	6	’s	’s	PART
cana-527	170	7	endpoints	endpoint	NOUN
cana-527	170	8	determines	determine	VERB
cana-527	170	9	its	its	PRON
cana-527	170	10	weight	weight	NOUN
cana-527	170	11	.	.	PUNCT
cana-527	171	1	at	at	ADP
cana-527	171	2	this	this	DET
cana-527	171	3	point	point	NOUN
cana-527	171	4	,	,	PUNCT
cana-527	171	5	(	(	PUNCT
cana-527	171	6	k(s	k(s	PROPN
cana-527	171	7	)	)	PUNCT
cana-527	171	8	,	,	PUNCT
cana-527	171	9	∥.∥	∥.∥	NUM
cana-527	171	10	)	)	PUNCT
cana-527	171	11	is	be	AUX
cana-527	171	12	transformed	transform	VERB
cana-527	171	13	into	into	ADP
cana-527	171	14	a	a	DET
cana-527	171	15	bhs	bhs	PROPN
cana-527	171	16	and	and	CCONJ
cana-527	171	17	given	give	VERB
cana-527	171	18	the	the	DET
cana-527	171	19	graph	graph	NOUN
cana-527	171	20	h.	h.	NOUN
cana-527	172	1	this	this	PRON
cana-527	172	2	is	be	AUX
cana-527	172	3	the	the	DET
cana-527	172	4	definition	definition	NOUN
cana-527	172	5	of	of	ADP
cana-527	172	6	h0	h0	PROPN
cana-527	172	7	,	,	PUNCT
cana-527	172	8	the	the	DET
cana-527	172	9	sub	sub	NOUN
cana-527	172	10	-	-	NOUN
cana-527	172	11	graph	graph	NOUN
cana-527	172	12	of	of	ADP
cana-527	172	13	h.	h.	NOUN
cana-527	172	14	let	let	VERB
cana-527	172	15	h0	h0	PROPN
cana-527	172	16	=	=	SYM
cana-527	172	17	(	(	PUNCT
cana-527	172	18	v0	v0	PROPN
cana-527	172	19	,	,	PUNCT
cana-527	172	20	e0	e0	PROPN
cana-527	172	21	)	)	PUNCT
cana-527	172	22	,	,	PUNCT
cana-527	172	23	where	where	SCONJ
cana-527	172	24	v0	v0	NOUN
cana-527	172	25	=	=	SYM
cana-527	172	26	{	{	PUNCT
cana-527	172	27	sy0	sy0	NOUN
cana-527	172	28	,	,	PUNCT
cana-527	172	29	s2y0	s2y0	PROPN
cana-527	172	30	,	,	PUNCT
cana-527	172	31	s3y0	s3y0	ADV
cana-527	172	32	,	,	PUNCT
cana-527	172	33	.	.	PUNCT
cana-527	172	34	.	.	PUNCT
cana-527	172	35	.	.	PUNCT
cana-527	173	1	sny0	sny0	PROPN
cana-527	173	2	}	}	PUNCT
cana-527	173	3	and	and	CCONJ
cana-527	173	4	let	let	VERB
cana-527	173	5	e0	e0	PROPN
cana-527	173	6	=	=	PRON
cana-527	173	7	{	{	PUNCT
cana-527	173	8	(	(	PUNCT
cana-527	173	9	sy0	sy0	PROPN
cana-527	173	10	,	,	PUNCT
cana-527	173	11	s2y0	s2y0	PROPN
cana-527	173	12	)	)	PUNCT
cana-527	173	13	,	,	PUNCT
cana-527	173	14	(	(	PUNCT
cana-527	173	15	s2y0	s2y0	ADP
cana-527	173	16	,	,	PUNCT
cana-527	173	17	s3y0	s3y0	PROPN
cana-527	173	18	)	)	PUNCT
cana-527	173	19	,	,	PUNCT
cana-527	173	20	.	.	PUNCT
cana-527	173	21	.	.	PUNCT
cana-527	173	22	.	.	PUNCT
cana-527	173	23	}	}	PUNCT
cana-527	173	24	,	,	PUNCT
cana-527	173	25	e0	e0	PROPN
cana-527	173	26	⊂	⊂	PROPN
cana-527	173	27	e	e	PROPN
cana-527	173	28	and	and	CCONJ
cana-527	173	29	v0	v0	PROPN
cana-527	173	30	⊂	⊂	PROPN
cana-527	173	31	v	v	PART
cana-527	173	32	follow	follow	VERB
cana-527	173	33	.	.	PUNCT
cana-527	174	1	therefore	therefore	ADV
cana-527	174	2	a	a	DET
cana-527	174	3	sub	sub	NOUN
cana-527	174	4	graph	graph	NOUN
cana-527	174	5	of	of	ADP
cana-527	174	6	h	h	NOUN
cana-527	174	7	is	be	AUX
cana-527	174	8	h0	h0	PROPN
cana-527	174	9	.	.	PUNCT
cana-527	175	1	consider	consider	VERB
cana-527	175	2	,	,	PUNCT
cana-527	175	3	the	the	DET
cana-527	175	4	iterated	iterated	ADJ
cana-527	175	5	sequence	sequence	NOUN
cana-527	175	6	given	give	VERB
cana-527	175	7	any	any	DET
cana-527	175	8	initial	initial	ADJ
cana-527	175	9	element	element	NOUN
cana-527	175	10	y0	y0	PROPN
cana-527	175	11	∈	∈	PROPN
cana-527	175	12	k(s	k(s	PROPN
cana-527	175	13	)	)	PUNCT
cana-527	175	14	,	,	PUNCT
cana-527	175	15	{	{	PUNCT
cana-527	175	16	sy0	sy0	NOUN
cana-527	175	17	,	,	PUNCT
cana-527	175	18	s2y0	s2y0	PROPN
cana-527	175	19	,	,	PUNCT
cana-527	175	20	s3y0	s3y0	ADP
cana-527	175	21	,	,	PUNCT
cana-527	175	22	.	.	PUNCT
cana-527	175	23	.	.	PUNCT
cana-527	176	1	.	.	PUNCT
cana-527	176	2	}	}	PUNCT
cana-527	177	1	∈	∈	PROPN
cana-527	177	2	k(s	k(s	PROPN
cana-527	177	3	)	)	PUNCT
cana-527	177	4	.	.	PUNCT
cana-527	178	1	y1	y1	NOUN
cana-527	178	2	=	=	SYM
cana-527	178	3	sy0	sy0	PROPN
cana-527	178	4	,	,	PUNCT
cana-527	178	5	y2	y2	PROPN
cana-527	178	6	=	=	SYM
cana-527	178	7	s(y1	s(y1	NOUN
cana-527	178	8	)	)	PUNCT
cana-527	178	9	=	=	SYM
cana-527	178	10	s(sy0	s(sy0	PROPN
cana-527	178	11	)	)	PUNCT
cana-527	178	12	=	=	SYM
cana-527	178	13	s2y0	s2y0	PROPN
cana-527	178	14	,	,	PUNCT
cana-527	178	15	.	.	PUNCT
cana-527	178	16	.	.	PUNCT
cana-527	178	17	.	.	PUNCT
cana-527	179	1	yn	yn	PRON
cana-527	179	2	=	=	PUNCT
cana-527	179	3	sn(y0	sn(y0	PROPN
cana-527	179	4	)	)	PUNCT
cana-527	179	5	from	from	ADP
cana-527	179	6	lemma	lemma	PROPN
cana-527	179	7	3.1	3.1	NUM
cana-527	179	8	of	of	ADP
cana-527	179	9	[	[	X
cana-527	179	10	20	20	NUM
cana-527	179	11	]	]	PUNCT
cana-527	179	12	to	to	PART
cana-527	179	13	say	say	VERB
cana-527	179	14	that	that	SCONJ
cana-527	179	15	this	this	DET
cana-527	179	16	sequence	sequence	NOUN
cana-527	179	17	is	be	AUX
cana-527	179	18	cauchy	cauchy	NOUN
cana-527	179	19	.	.	PUNCT
cana-527	180	1	it	it	PRON
cana-527	180	2	is	be	AUX
cana-527	180	3	sufficient	sufficient	ADJ
cana-527	180	4	to	to	PART
cana-527	180	5	show	show	VERB
cana-527	180	6	that	that	SCONJ
cana-527	180	7	w	w	ADJ
cana-527	180	8	-	-	PUNCT
cana-527	180	9	sequence	sequence	NOUN
cana-527	180	10	associates	associate	NOUN
cana-527	180	11	with	with	ADP
cana-527	180	12	graph	graph	NOUN
cana-527	180	13	g	g	PROPN
cana-527	180	14	is	be	AUX
cana-527	180	15	non	non	ADJ
cana-527	180	16	-	-	ADJ
cana-527	180	17	increasing	increase	VERB
cana-527	180	18	.	.	PUNCT
cana-527	181	1	since	since	SCONJ
cana-527	181	2	s	s	PROPN
cana-527	181	3	is	be	AUX
cana-527	181	4	a	a	DET
cana-527	181	5	contraction	contraction	NOUN
cana-527	181	6	on	on	ADP
cana-527	181	7	k(x	k(x	PROPN
cana-527	181	8	)	)	PUNCT
cana-527	181	9	,	,	PUNCT
cana-527	181	10	we	we	PRON
cana-527	181	11	have	have	AUX
cana-527	181	12	wn+1	wn+1	VERB
cana-527	181	13	=	=	SYM
cana-527	181	14	∥sny0	∥sny0	NOUN
cana-527	181	15	⊖	⊖	NOUN
cana-527	181	16	sn+1y0∥	sn+1y0∥	VERB
cana-527	181	17	≤	≤	NUM
cana-527	181	18	β∥sn−1y0	β∥sn−1y0	NOUN
cana-527	181	19	⊖	⊖	NOUN
cana-527	181	20	sny0∥	sny0∥	NOUN
cana-527	181	21	=	=	SYM
cana-527	181	22	βwn	βwn	NOUN
cana-527	181	23	(	(	PUNCT
cana-527	181	24	i.e	i.e	X
cana-527	181	25	)	)	PUNCT
cana-527	181	26	wn+1	wn+1	VERB
cana-527	181	27	<	<	X
cana-527	181	28	wn	wn	PROPN
cana-527	181	29	since	since	SCONJ
cana-527	181	30	0	0	NUM
cana-527	181	31	≤	≤	NUM
cana-527	181	32	β	β	X
cana-527	181	33	<	<	X
cana-527	181	34	1	1	NUM
cana-527	181	35	,	,	PUNCT
cana-527	181	36	n	n	PRON
cana-527	181	37	∈	∈	NOUN
cana-527	182	1	i	i	PRON
cana-527	182	2	hence	hence	ADV
cana-527	182	3	the	the	DET
cana-527	182	4	w	w	NOUN
cana-527	182	5	-	-	PUNCT
cana-527	182	6	sequence	sequence	NOUN
cana-527	182	7	associated	associate	VERB
cana-527	182	8	with	with	ADP
cana-527	182	9	h0	h0	PROPN
cana-527	182	10	is	be	AUX
cana-527	182	11	non	non	ADJ
cana-527	182	12	-	-	ADJ
cana-527	182	13	increasing	increase	VERB
cana-527	182	14	.	.	PUNCT
cana-527	183	1	by	by	ADP
cana-527	183	2	lemma	lemma	PROPN
cana-527	183	3	3.1	3.1	NUM
cana-527	183	4	of	of	ADP
cana-527	183	5	[	[	X
cana-527	183	6	20	20	NUM
cana-527	183	7	]	]	PUNCT
cana-527	183	8	,	,	PUNCT
cana-527	183	9	{	{	PUNCT
cana-527	183	10	sy0	sy0	NOUN
cana-527	183	11	,	,	PUNCT
cana-527	183	12	s2y0	s2y0	PROPN
cana-527	183	13	,	,	PUNCT
cana-527	183	14	.	.	PUNCT
cana-527	183	15	.	.	PUNCT
cana-527	183	16	.	.	PUNCT
cana-527	183	17	}	}	PUNCT
cana-527	183	18	is	be	AUX
cana-527	183	19	an	an	DET
cana-527	183	20	iterated	iterated	ADJ
cana-527	183	21	sequence	sequence	NOUN
cana-527	183	22	that	that	PRON
cana-527	183	23	is	be	AUX
cana-527	183	24	cauchy	cauchy	PROPN
cana-527	183	25	.	.	PUNCT
cana-527	184	1	however	however	ADV
cana-527	184	2	,	,	PUNCT
cana-527	184	3	k(s	k(s	PROPN
cana-527	184	4	)	)	PUNCT
cana-527	184	5	is	be	AUX
cana-527	184	6	complete	complete	ADJ
cana-527	184	7	.	.	PUNCT
cana-527	185	1	we	we	PRON
cana-527	185	2	can	can	AUX
cana-527	185	3	see	see	VERB
cana-527	185	4	that	that	SCONJ
cana-527	185	5	,	,	PUNCT
cana-527	185	6	the	the	DET
cana-527	185	7	series	series	NOUN
cana-527	185	8	converges	converge	VERB
cana-527	185	9	to	to	PART
cana-527	185	10	,	,	PUNCT
cana-527	185	11	say	say	INTJ
cana-527	185	12	,	,	PUNCT
cana-527	185	13	v∗	v∗	PROPN
cana-527	185	14	∈	∈	PROPN
cana-527	185	15	k(s	k(s	PROPN
cana-527	185	16	)	)	PUNCT
cana-527	185	17	as	as	ADP
cana-527	185	18	a	a	DET
cana-527	185	19	result	result	NOUN
cana-527	185	20	.	.	PUNCT
cana-527	186	1	∴	∴	NOUN
cana-527	186	2	the	the	DET
cana-527	186	3	mapping	mapping	NOUN
cana-527	186	4	τ	τ	PROPN
cana-527	186	5	,	,	PUNCT
cana-527	186	6	which	which	PRON
cana-527	186	7	is	be	AUX
cana-527	186	8	a	a	DET
cana-527	186	9	contraction	contraction	NOUN
cana-527	186	10	,	,	PUNCT
cana-527	186	11	is	be	AUX
cana-527	186	12	continuous	continuous	ADJ
cana-527	186	13	,	,	PUNCT
cana-527	186	14	which	which	PRON
cana-527	186	15	is	be	AUX
cana-527	186	16	clearly	clearly	ADV
cana-527	186	17	visible	visible	ADJ
cana-527	186	18	from	from	ADP
cana-527	186	19	the	the	DET
cana-527	186	20	above	above	ADJ
cana-527	186	21	result	result	NOUN
cana-527	186	22	.	.	PUNCT
cana-527	187	1	thus	thus	ADV
cana-527	187	2	,	,	PUNCT
cana-527	187	3	τv∗	τv∗	ADJ
cana-527	187	4	is	be	AUX
cana-527	187	5	the	the	DET
cana-527	187	6	convergence	convergence	NOUN
cana-527	187	7	point	point	NOUN
cana-527	187	8	of	of	ADP
cana-527	187	9	the	the	DET
cana-527	187	10	series	series	NOUN
cana-527	187	11	[	[	X
cana-527	187	12	ssn−1y0]∞	ssn−1y0]∞	PROPN
cana-527	187	13	n=1	n=1	PROPN
cana-527	187	14	.	.	PUNCT
cana-527	188	1	but	but	CCONJ
cana-527	188	2	the	the	DET
cana-527	188	3	sequence	sequence	NOUN
cana-527	189	1	[	[	X
cana-527	189	2	ssny0]n	ssny0]n	ADJ
cana-527	189	3	∞	∞	NUM
cana-527	189	4	=	=	SYM
cana-527	189	5	1	1	NUM
cana-527	189	6	is	be	AUX
cana-527	189	7	a	a	DET
cana-527	189	8	sub	sub	NOUN
cana-527	189	9	-	-	NOUN
cana-527	189	10	sequence	sequence	NOUN
cana-527	189	11	of	of	ADP
cana-527	189	12	the	the	DET
cana-527	189	13	sequence	sequence	NOUN
cana-527	189	14	[	[	X
cana-527	189	15	sn−1y0]∞	sn−1y0]∞	NOUN
cana-527	189	16	n=1	n=1	PROPN
cana-527	189	17	.	.	PUNCT
cana-527	190	1	hence	hence	ADV
cana-527	190	2	the	the	DET
cana-527	190	3	sub	sub	NOUN
cana-527	190	4	-	-	NOUN
cana-527	190	5	sequence	sequence	NOUN
cana-527	190	6	must	must	AUX
cana-527	190	7	have	have	VERB
cana-527	190	8	the	the	DET
cana-527	190	9	same	same	ADJ
cana-527	190	10	limit	limit	NOUN
cana-527	190	11	as	as	ADP
cana-527	190	12	the	the	DET
cana-527	190	13	parent	parent	NOUN
cana-527	190	14	sequence	sequence	NOUN
cana-527	190	15	.	.	PUNCT
cana-527	191	1	but	but	CCONJ
cana-527	191	2	the	the	DET
cana-527	191	3	limit	limit	NOUN
cana-527	191	4	communications	communication	NOUN
cana-527	191	5	on	on	ADP
cana-527	191	6	applied	apply	VERB
cana-527	191	7	nonlinear	nonlinear	ADJ
cana-527	191	8	analysis	analysis	NOUN
cana-527	191	9	issn	issn	NOUN
cana-527	191	10	:	:	PUNCT
cana-527	191	11	1074	1074	NUM
cana-527	191	12	-	-	PUNCT
cana-527	191	13	133x	133x	NUM
cana-527	191	14	vol	vol	NOUN
cana-527	191	15	31	31	NUM
cana-527	191	16	no	no	NOUN
cana-527	191	17	.	.	NOUN
cana-527	191	18	2	2	NUM
cana-527	191	19	(	(	PUNCT
cana-527	191	20	2024	2024	NUM
cana-527	191	21	)	)	PUNCT
cana-527	192	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	192	2	165	165	NUM
cana-527	192	3	2	2	NUM
cana-527	192	4	.	.	PUNCT
cana-527	192	5	.	.	PUNCT
cana-527	193	1	of	of	ADP
cana-527	193	2	a	a	DET
cana-527	193	3	sequence	sequence	NOUN
cana-527	193	4	is	be	AUX
cana-527	193	5	unique	unique	ADJ
cana-527	193	6	.	.	PUNCT
cana-527	194	1	hence	hence	ADV
cana-527	194	2	we	we	PRON
cana-527	194	3	must	must	AUX
cana-527	194	4	have	have	VERB
cana-527	194	5	,	,	PUNCT
cana-527	194	6	sv∗	sv∗	NOUN
cana-527	194	7	=	=	SYM
cana-527	194	8	v∗	v∗	PROPN
cana-527	194	9	=	=	NOUN
cana-527	194	10	⇒	⇒	NOUN
cana-527	194	11	(	(	PUNCT
cana-527	194	12	v∗	v∗	ADJ
cana-527	194	13	,	,	PUNCT
cana-527	194	14	v∗	v∗	ADJ
cana-527	194	15	)	)	PUNCT
cana-527	194	16	∈	∈	PROPN
cana-527	194	17	g	g	PROPN
cana-527	194	18	(	(	PUNCT
cana-527	194	19	i.e	i.e	PROPN
cana-527	194	20	)	)	PUNCT
cana-527	194	21	,	,	PUNCT
cana-527	194	22	h	h	PROPN
cana-527	194	23	has	have	VERB
cana-527	194	24	a	a	DET
cana-527	194	25	loop	loop	NOUN
cana-527	194	26	at	at	ADP
cana-527	194	27	v∗.	v∗.	PROPN
cana-527	194	28	by	by	ADP
cana-527	194	29	lemma	lemma	PROPN
cana-527	194	30	3.2	3.2	NUM
cana-527	194	31	of	of	ADP
cana-527	194	32	[	[	X
cana-527	194	33	20	20	NUM
cana-527	194	34	]	]	PUNCT
cana-527	194	35	,	,	PUNCT
cana-527	194	36	v∗	v∗	PROPN
cana-527	194	37	is	be	AUX
cana-527	194	38	a	a	DET
cana-527	194	39	fixed	fix	VERB
cana-527	194	40	point	point	NOUN
cana-527	194	41	of	of	ADP
cana-527	194	42	s.	s.	PROPN
cana-527	194	43	to	to	PART
cana-527	194	44	prove	prove	VERB
cana-527	194	45	uniqueness	uniqueness	NOUN
cana-527	194	46	:	:	PUNCT
cana-527	194	47	let	let	VERB
cana-527	194	48	if	if	SCONJ
cana-527	194	49	possible	possible	ADJ
cana-527	194	50	w∗	w∗	NOUN
cana-527	194	51	be	be	AUX
cana-527	194	52	any	any	DET
cana-527	194	53	other	other	ADJ
cana-527	194	54	near	near	ADP
cana-527	194	55	fixed	fix	VERB
cana-527	194	56	point	point	NOUN
cana-527	194	57	of	of	ADP
cana-527	194	58	s.	s.	PROPN
cana-527	194	59	then	then	ADV
cana-527	194	60	sw∗	sw∗	PROPN
cana-527	194	61	=	=	SYM
cana-527	194	62	w∗.	w∗.	PROPN
cana-527	194	63	since	since	SCONJ
cana-527	194	64	s	s	PROPN
cana-527	194	65	is	be	AUX
cana-527	194	66	a	a	DET
cana-527	194	67	contraction	contraction	NOUN
cana-527	194	68	on	on	ADP
cana-527	194	69	k(s	k(s	PROPN
cana-527	194	70	)	)	PUNCT
cana-527	194	71	we	we	PRON
cana-527	194	72	have	have	AUX
cana-527	194	73	,	,	PUNCT
cana-527	194	74	∥sv∗	∥sv∗	NOUN
cana-527	194	75	⊖	⊖	NOUN
cana-527	194	76	sw∗∥	sw∗∥	VERB
cana-527	194	77	≤	≤	NOUN
cana-527	194	78	∥v∗	∥v∗	PUNCT
cana-527	194	79	⊖	⊖	SYM
cana-527	194	80	w∗∥	w∗∥	PROPN
cana-527	194	81	,	,	PUNCT
cana-527	194	82	0	0	PUNCT
cana-527	194	83	<	<	X
cana-527	194	84	α	α	X
cana-527	194	85	<	<	X
cana-527	194	86	1	1	NUM
cana-527	194	87	,	,	PUNCT
cana-527	194	88	∥v∗	∥v∗	PROPN
cana-527	194	89	⊖	⊖	AUX
cana-527	194	90	w∗∥	w∗∥	PROPN
cana-527	194	91	<	<	X
cana-527	194	92	∥v∗	∥v∗	PROPN
cana-527	194	93	⊖	⊖	AUX
cana-527	194	94	w∗	w∗	NOUN
cana-527	194	95	it	it	PRON
cana-527	194	96	is	be	AUX
cana-527	194	97	a	a	DET
cana-527	194	98	contradiction	contradiction	NOUN
cana-527	194	99	.	.	PUNCT
cana-527	195	1	hence	hence	ADV
cana-527	195	2	the	the	DET
cana-527	195	3	near	near	ADJ
cana-527	195	4	fixed	fix	VERB
cana-527	195	5	point	point	NOUN
cana-527	195	6	of	of	ADP
cana-527	195	7	s	s	NOUN
cana-527	195	8	is	be	AUX
cana-527	195	9	unique	unique	ADJ
cana-527	195	10	.	.	PUNCT
cana-527	196	1	theorem	theorem	ADJ
cana-527	196	2	3.1.2	3.1.2	NOUN
cana-527	196	3	:	:	PUNCT
cana-527	196	4	given	give	VERB
cana-527	196	5	a	a	DET
cana-527	196	6	bhs	bhs	PROPN
cana-527	196	7	(	(	PUNCT
cana-527	196	8	k(s	k(s	PROPN
cana-527	196	9	)	)	PUNCT
cana-527	196	10	,	,	PUNCT
cana-527	196	11	∥.∥	∥.∥	NUM
cana-527	196	12	)	)	PUNCT
cana-527	196	13	,	,	PUNCT
cana-527	196	14	its	its	PRON
cana-527	196	15	null	null	ADJ
cana-527	196	16	set	set	NOUN
cana-527	196	17	is	be	AUX
cana-527	196	18	ω	ω	PROPN
cana-527	196	19	.	.	PUNCT
cana-527	197	1	let	let	VERB
cana-527	197	2	h	h	NOUN
cana-527	197	3	be	be	AUX
cana-527	197	4	connected	connect	VERB
cana-527	197	5	to	to	ADP
cana-527	197	6	k(s	k(s	PROPN
cana-527	197	7	)	)	PUNCT
cana-527	197	8	,	,	PUNCT
cana-527	197	9	and	and	CCONJ
cana-527	197	10	allow	allow	VERB
cana-527	197	11	s	s	PRON
cana-527	197	12	:	:	PUNCT
cana-527	197	13	k(s	k(s	PROPN
cana-527	197	14	)	)	PUNCT
cana-527	197	15	→	→	SYM
cana-527	197	16	k(s	k(s	PROPN
cana-527	197	17	)	)	PUNCT
cana-527	197	18	.	.	PUNCT
cana-527	198	1	if	if	SCONJ
cana-527	198	2	s	s	PRON
cana-527	198	3	satisfy	satisfy	VERB
cana-527	198	4	,	,	PUNCT
cana-527	198	5	∥s(x	∥s(x	PROPN
cana-527	198	6	)	)	PUNCT
cana-527	198	7	⊖	⊖	NOUN
cana-527	198	8	s(y)∥	s(y)∥	VERB
cana-527	198	9	≤	≤	NUM
cana-527	198	10	β(∥x	β(∥x	PROPN
cana-527	198	11	⊖	⊖	X
cana-527	198	12	s(x)∥	s(x)∥	ADP
cana-527	198	13	⊕	⊕	PROPN
cana-527	198	14	∥y	∥y	PROPN
cana-527	198	15	⊖	⊖	VERB
cana-527	198	16	s(y)∥	s(y)∥	NOUN
cana-527	198	17	)	)	PUNCT
cana-527	198	18	(	(	PUNCT
cana-527	198	19	1	1	X
cana-527	198	20	)	)	PUNCT
cana-527	198	21	for	for	ADP
cana-527	198	22	all	all	DET
cana-527	198	23	x	x	NOUN
cana-527	198	24	,	,	PUNCT
cana-527	198	25	y	y	PROPN
cana-527	198	26	∈	∈	PROPN
cana-527	198	27	k(s	k(s	PROPN
cana-527	198	28	)	)	PUNCT
cana-527	198	29	,	,	PUNCT
cana-527	198	30	where	where	SCONJ
cana-527	198	31	β	β	X
cana-527	198	32	∈	∈	PROPN
cana-527	198	33	(	(	PUNCT
cana-527	198	34	0	0	NUM
cana-527	198	35	,	,	PUNCT
cana-527	198	36	1	1	NUM
cana-527	198	37	)	)	PUNCT
cana-527	198	38	,	,	PUNCT
cana-527	198	39	then	then	ADV
cana-527	198	40	s	s	VERB
cana-527	198	41	has	have	AUX
cana-527	198	42	unique	unique	ADJ
cana-527	198	43	near	near	ADP
cana-527	198	44	fixed	fix	VERB
cana-527	198	45	point	point	NOUN
cana-527	198	46	.	.	PUNCT
cana-527	199	1	proof	proof	NOUN
cana-527	199	2	:	:	PUNCT
cana-527	199	3	given	give	VERB
cana-527	199	4	that	that	SCONJ
cana-527	199	5	any	any	DET
cana-527	199	6	initial	initial	ADJ
cana-527	199	7	element	element	NOUN
cana-527	199	8	y0	y0	PROPN
cana-527	199	9	∈	∈	PROPN
cana-527	199	10	k(s	k(s	PROPN
cana-527	199	11	)	)	PUNCT
cana-527	199	12	.	.	PUNCT
cana-527	200	1	the	the	DET
cana-527	200	2	h	h	NOUN
cana-527	200	3	and	and	CCONJ
cana-527	200	4	h0	h0	PROPN
cana-527	200	5	are	be	AUX
cana-527	200	6	defined	define	VERB
cana-527	200	7	as	as	ADP
cana-527	200	8	in	in	ADP
cana-527	200	9	definition	definition	NOUN
cana-527	200	10	3.1.1	3.1.1	NUM
cana-527	200	11	and	and	CCONJ
cana-527	200	12	3.1.2	3.1.2	NUM
cana-527	200	13	and	and	CCONJ
cana-527	200	14	refer	refer	VERB
cana-527	200	15	[	[	X
cana-527	200	16	20	20	NUM
cana-527	200	17	]	]	PUNCT
cana-527	200	18	given	give	VERB
cana-527	200	19	any	any	DET
cana-527	200	20	initial	initial	ADJ
cana-527	200	21	element	element	NOUN
cana-527	200	22	y0	y0	PROPN
cana-527	200	23	∈	∈	PROPN
cana-527	200	24	k(s	k(s	PROPN
cana-527	200	25	)	)	PUNCT
cana-527	200	26	,	,	PUNCT
cana-527	200	27	we	we	PRON
cana-527	200	28	define	define	VERB
cana-527	200	29	the	the	DET
cana-527	200	30	iterative	iterative	NOUN
cana-527	200	31	sequence	sequence	NOUN
cana-527	200	32	,	,	PUNCT
cana-527	200	33	y1	y1	NOUN
cana-527	200	34	=	=	SYM
cana-527	200	35	s(y0	s(y0	NOUN
cana-527	200	36	)	)	PUNCT
cana-527	200	37	,	,	PUNCT
cana-527	200	38	y2	y2	NOUN
cana-527	200	39	=	=	SYM
cana-527	200	40	s(y1	s(y1	NOUN
cana-527	200	41	)	)	PUNCT
cana-527	200	42	=	=	SYM
cana-527	200	43	s(sy0	s(sy0	PROPN
cana-527	200	44	)	)	PUNCT
cana-527	200	45	=	=	SYM
cana-527	200	46	s2y0	s2y0	PROPN
cana-527	200	47	.	.	PUNCT
cana-527	200	48	.	.	PUNCT
cana-527	200	49	.	.	PUNCT
cana-527	201	1	yn	yn	PRON
cana-527	201	2	=	=	SYM
cana-527	201	3	sny0	sny0	PROPN
cana-527	201	4	(	(	PUNCT
cana-527	201	5	i.e	i.e	PROPN
cana-527	201	6	)	)	PUNCT
cana-527	201	7	{	{	PUNCT
cana-527	201	8	s(y0	s(y0	NOUN
cana-527	201	9	)	)	PUNCT
cana-527	201	10	,	,	PUNCT
cana-527	201	11	s2(y0	s2(y0	NOUN
cana-527	201	12	)	)	PUNCT
cana-527	201	13	,	,	PUNCT
cana-527	201	14	s3(y0	s3(y0	NOUN
cana-527	201	15	)	)	PUNCT
cana-527	201	16	,	,	PUNCT
cana-527	201	17	.	.	PUNCT
cana-527	201	18	.	.	PUNCT
cana-527	201	19	.	.	PUNCT
cana-527	201	20	}	}	PUNCT
cana-527	202	1	∈	∈	PROPN
cana-527	202	2	k(x	k(x	PROPN
cana-527	202	3	)	)	PUNCT
cana-527	202	4	lemma	lemma	PROPN
cana-527	202	5	3.1	3.1	NUM
cana-527	202	6	of	of	ADP
cana-527	202	7	[	[	X
cana-527	202	8	20	20	NUM
cana-527	202	9	]	]	PUNCT
cana-527	202	10	states	state	VERB
cana-527	202	11	that	that	SCONJ
cana-527	202	12	demonstrating	demonstrate	VERB
cana-527	202	13	the	the	DET
cana-527	202	14	non	non	ADJ
cana-527	202	15	-	-	ADJ
cana-527	202	16	increasing	increasing	ADJ
cana-527	202	17	nature	nature	NOUN
cana-527	202	18	of	of	ADP
cana-527	202	19	the	the	DET
cana-527	202	20	w	w	NOUN
cana-527	202	21	-	-	PUNCT
cana-527	202	22	sequence	sequence	NOUN
cana-527	202	23	connected	connect	VERB
cana-527	202	24	to	to	ADP
cana-527	202	25	the	the	DET
cana-527	202	26	graph	graph	NOUN
cana-527	202	27	h0	h0	NOUN
cana-527	202	28	is	be	AUX
cana-527	202	29	sufficient	sufficient	ADJ
cana-527	202	30	to	to	PART
cana-527	202	31	establish	establish	VERB
cana-527	202	32	the	the	DET
cana-527	202	33	cauchy	cauchy	ADJ
cana-527	202	34	nature	nature	NOUN
cana-527	202	35	of	of	ADP
cana-527	202	36	this	this	DET
cana-527	202	37	sequence	sequence	NOUN
cana-527	202	38	.	.	PUNCT
cana-527	203	1	from	from	ADP
cana-527	203	2	eqn	eqn	PROPN
cana-527	203	3	1	1	NUM
cana-527	203	4	,	,	PUNCT
cana-527	203	5	we	we	PRON
cana-527	203	6	have	have	AUX
cana-527	203	7	,	,	PUNCT
cana-527	203	8	wn+1	wn+1	VERB
cana-527	203	9	=	=	SYM
cana-527	203	10	∥sny0	∥sny0	NOUN
cana-527	203	11	⊖	⊖	NOUN
cana-527	203	12	sn+1y0∥	sn+1y0∥	VERB
cana-527	203	13	≤	≤	NOUN
cana-527	203	14	β	β	NOUN
cana-527	203	15	∥sn−1y0	∥sn−1y0	X
cana-527	203	16	⊖	⊖	X
cana-527	203	17	sny0∥	sny0∥	PROPN
cana-527	203	18	⊕	⊕	PROPN
cana-527	203	19	∥sny0	∥sny0	NOUN
cana-527	203	20	⊖	⊖	NOUN
cana-527	203	21	sn+1y0∥	sn+1y0∥	VERB
cana-527	203	22	wn+1	wn+1	VERB
cana-527	203	23	≤	≤	NOUN
cana-527	203	24	β[wn	β[wn	ADJ
cana-527	203	25	+	+	CCONJ
cana-527	203	26	wn+1	wn+1	X
cana-527	203	27	]	]	X
cana-527	203	28	β	β	X
cana-527	203	29	1	1	NUM
cana-527	203	30	wn	wn	PROPN
cana-527	203	31	1	1	NUM
cana-527	203	32	≤	≤	NUM
cana-527	203	33	wn	wn	NOUN
cana-527	203	34	<	<	X
cana-527	203	35	wn	wn	PROPN
cana-527	203	36	since	since	SCONJ
cana-527	203	37	0	0	NUM
cana-527	203	38	≤	≤	NUM
cana-527	203	39	β	β	X
cana-527	203	40	<	<	X
cana-527	203	41	1	1	NUM
cana-527	203	42	−	−	NOUN
cana-527	203	43	β	β	NOUN
cana-527	203	44	2	2	NUM
cana-527	203	45	+	+	NUM
cana-527	203	46	communications	communication	NOUN
cana-527	203	47	on	on	ADP
cana-527	203	48	applied	apply	VERB
cana-527	203	49	nonlinear	nonlinear	ADJ
cana-527	203	50	analysis	analysis	NOUN
cana-527	203	51	issn	issn	NOUN
cana-527	203	52	:	:	PUNCT
cana-527	203	53	1074	1074	NUM
cana-527	203	54	-	-	PUNCT
cana-527	203	55	133x	133x	NUM
cana-527	203	56	vol	vol	NOUN
cana-527	203	57	31	31	NUM
cana-527	203	58	no	no	NOUN
cana-527	203	59	.	.	NOUN
cana-527	203	60	2	2	NUM
cana-527	203	61	(	(	PUNCT
cana-527	203	62	2024	2024	NUM
cana-527	203	63	)	)	PUNCT
cana-527	203	64	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	203	65	166	166	NUM
cana-527	203	66	.	.	PUNCT
cana-527	203	67	.	.	PUNCT
cana-527	204	1	2	2	X
cana-527	204	2	.	.	PUNCT
cana-527	204	3	.	.	PUNCT
cana-527	205	1	2	2	NUM
cana-527	205	2	hence	hence	ADV
cana-527	205	3	the	the	DET
cana-527	205	4	w	w	NOUN
cana-527	205	5	-	-	PUNCT
cana-527	205	6	sequence	sequence	NOUN
cana-527	205	7	associated	associate	VERB
cana-527	205	8	with	with	ADP
cana-527	205	9	h0	h0	PROPN
cana-527	205	10	is	be	AUX
cana-527	205	11	non	non	ADJ
cana-527	205	12	-	-	ADJ
cana-527	205	13	increasing	increase	VERB
cana-527	205	14	.	.	PUNCT
cana-527	206	1	from	from	ADP
cana-527	206	2	lemma	lemma	PROPN
cana-527	206	3	3.1	3.1	NUM
cana-527	206	4	of	of	ADP
cana-527	206	5	[	[	X
cana-527	206	6	20	20	NUM
cana-527	206	7	]	]	PUNCT
cana-527	206	8	,	,	PUNCT
cana-527	206	9	the	the	DET
cana-527	206	10	iterated	iterated	ADJ
cana-527	206	11	sequence	sequence	NOUN
cana-527	206	12	{	{	PUNCT
cana-527	206	13	s(y0	s(y0	NOUN
cana-527	206	14	)	)	PUNCT
cana-527	206	15	,	,	PUNCT
cana-527	206	16	s2(y0	s2(y0	NOUN
cana-527	206	17	)	)	PUNCT
cana-527	206	18	,	,	PUNCT
cana-527	206	19	.	.	PUNCT
cana-527	206	20	.	.	PUNCT
cana-527	207	1	.	.	PUNCT
cana-527	207	2	}	}	PUNCT
cana-527	207	3	is	be	AUX
cana-527	207	4	a	a	DET
cana-527	207	5	cauchy	cauchy	ADJ
cana-527	207	6	sequence	sequence	NOUN
cana-527	207	7	in	in	ADP
cana-527	207	8	k(s	k(s	PROPN
cana-527	207	9	)	)	PUNCT
cana-527	207	10	.	.	PUNCT
cana-527	208	1	but	but	CCONJ
cana-527	208	2	k(s	k(s	PROPN
cana-527	208	3	)	)	PUNCT
cana-527	208	4	is	be	AUX
cana-527	208	5	complete	complete	ADJ
cana-527	208	6	.	.	PUNCT
cana-527	209	1	therefore	therefore	ADV
cana-527	209	2	the	the	DET
cana-527	209	3	sequence	sequence	NOUN
cana-527	209	4	converges	converge	VERB
cana-527	209	5	in	in	ADP
cana-527	209	6	k(s	k(s	PROPN
cana-527	209	7	)	)	PUNCT
cana-527	209	8	.	.	PUNCT
cana-527	210	1	let	let	VERB
cana-527	210	2	y∗	y∗	ADV
cana-527	210	3	=	=	SYM
cana-527	210	4	limn→∞	limn→∞	PROPN
cana-527	210	5	sny0	sny0	PROPN
cana-527	210	6	∥y∗	∥y∗	PROPN
cana-527	210	7	⊖	⊖	NOUN
cana-527	210	8	s(y∗)∥	s(y∗)∥	VERB
cana-527	210	9	≤	≤	NOUN
cana-527	210	10	∥y∗	∥y∗	PROPN
cana-527	210	11	⊖	⊖	ADJ
cana-527	210	12	sn(y0)∥	sn(y0)∥	NOUN
cana-527	210	13	+	+	SYM
cana-527	210	14	∥sn(y0	∥sn(y0	NOUN
cana-527	210	15	)	)	PUNCT
cana-527	210	16	⊖	⊖	NOUN
cana-527	210	17	s(y∗)∥	s(y∗)∥	VERB
cana-527	210	18	≤	≤	NOUN
cana-527	210	19	∥y∗	∥y∗	PROPN
cana-527	210	20	⊖	⊖	NOUN
cana-527	210	21	sn(y0)∥	sn(y0)∥	NOUN
cana-527	211	1	+	+	X
cana-527	211	2	β	β	X
cana-527	211	3	∥sn−1(y0	∥sn−1(y0	PROPN
cana-527	211	4	)	)	PUNCT
cana-527	211	5	⊖	⊖	VERB
cana-527	211	6	sn(y0)∥	sn(y0)∥	PROPN
cana-527	212	1	+	+	CCONJ
cana-527	212	2	∥y∗	∥y∗	PROPN
cana-527	212	3	⊖	⊖	NOUN
cana-527	212	4	s(y∗)∥	s(y∗)∥	VERB
cana-527	212	5	∴	∴	PROPN
cana-527	212	6	(	(	PUNCT
cana-527	212	7	1	1	NUM
cana-527	212	8	−	−	NOUN
cana-527	212	9	β)∥y∗	β)∥y∗	NOUN
cana-527	212	10	⊖	⊖	NOUN
cana-527	212	11	s(y∗)∥	s(y∗)∥	VERB
cana-527	212	12	≤	≤	NOUN
cana-527	212	13	∥y∗	∥y∗	PROPN
cana-527	212	14	⊖	⊖	NOUN
cana-527	212	15	sn(y0)∥	sn(y0)∥	NOUN
cana-527	213	1	+	+	CCONJ
cana-527	213	2	α∥sn−1(y0	α∥sn−1(y0	PROPN
cana-527	213	3	)	)	PUNCT
cana-527	213	4	⊖	⊖	AUX
cana-527	213	5	sn(y0)∥	sn(y0)∥	PROPN
cana-527	213	6	allow	allow	VERB
cana-527	213	7	,	,	PUNCT
cana-527	213	8	n	n	PROPN
cana-527	213	9	→	→	SYM
cana-527	213	10	∞	∞	NUM
cana-527	213	11	on	on	ADP
cana-527	213	12	both	both	DET
cana-527	213	13	sides	side	NOUN
cana-527	213	14	,	,	PUNCT
cana-527	213	15	then	then	ADV
cana-527	213	16	we	we	PRON
cana-527	213	17	have	have	VERB
cana-527	213	18	,	,	PUNCT
cana-527	213	19	(	(	PUNCT
cana-527	213	20	1	1	NUM
cana-527	213	21	−	−	NOUN
cana-527	213	22	β)∥y∗	β)∥y∗	NOUN
cana-527	213	23	⊖	⊖	NOUN
cana-527	213	24	s(y∗)∥	s(y∗)∥	VERB
cana-527	213	25	≤	≤	NOUN
cana-527	213	26	∥y∗	∥y∗	PROPN
cana-527	213	27	⊖	⊖	NOUN
cana-527	213	28	y∗∥	y∗∥	ADV
cana-527	214	1	+	+	CCONJ
cana-527	215	1	β∥y∗	β∥y∗	PROPN
cana-527	215	2	⊖	⊖	NOUN
cana-527	215	3	y∗∥	y∗∥	ADV
cana-527	215	4	hence	hence	ADV
cana-527	215	5	∥y∗	∥y∗	PROPN
cana-527	215	6	−	−	PROPN
cana-527	215	7	s(y∗)∥	s(y∗)∥	NOUN
cana-527	215	8	=	=	NOUN
cana-527	215	9	0	0	PUNCT
cana-527	216	1	=	=	NOUN
cana-527	216	2	⇒	⇒	NOUN
cana-527	216	3	s(y∗	s(y∗	NUM
cana-527	216	4	)	)	PUNCT
cana-527	217	1	=	=	SYM
cana-527	217	2	y∗	y∗	PROPN
cana-527	217	3	(	(	PUNCT
cana-527	217	4	i.e	i.e	PROPN
cana-527	217	5	)	)	PUNCT
cana-527	217	6	,	,	PUNCT
cana-527	217	7	y∗	y∗	PROPN
cana-527	217	8	is	be	AUX
cana-527	217	9	a	a	DET
cana-527	217	10	fixed	fix	VERB
cana-527	217	11	point	point	NOUN
cana-527	217	12	of	of	ADP
cana-527	217	13	s.	s.	PROPN
cana-527	217	14	to	to	PART
cana-527	217	15	prove	prove	VERB
cana-527	217	16	uniqueness	uniqueness	NOUN
cana-527	217	17	:	:	PUNCT
cana-527	217	18	let	let	VERB
cana-527	217	19	if	if	SCONJ
cana-527	217	20	possible	possible	ADJ
cana-527	217	21	,	,	PUNCT
cana-527	217	22	z∗	z∗	X
cana-527	217	23	be	be	AUX
cana-527	217	24	any	any	DET
cana-527	217	25	other	other	ADJ
cana-527	217	26	near	near	ADP
cana-527	217	27	fixed	fix	VERB
cana-527	217	28	point	point	NOUN
cana-527	217	29	of	of	ADP
cana-527	217	30	s.	s.	PROPN
cana-527	217	31	then	then	ADV
cana-527	217	32	s(z∗	s(z∗	X
cana-527	217	33	)	)	PUNCT
cana-527	217	34	=	=	SYM
cana-527	217	35	z∗.	z∗.	PUNCT
cana-527	217	36	from	from	ADP
cana-527	217	37	eqn	eqn	PROPN
cana-527	217	38	1	1	NUM
cana-527	217	39	,	,	PUNCT
cana-527	217	40	we	we	PRON
cana-527	217	41	have	have	VERB
cana-527	217	42	where	where	SCONJ
cana-527	217	43	,	,	PUNCT
cana-527	217	44	0	0	NUM
cana-527	217	45	≤	≤	NUM
cana-527	218	1	β	β	X
cana-527	218	2	<	<	X
cana-527	218	3	1	1	NUM
cana-527	218	4	∥s(y∗	∥s(y∗	NUM
cana-527	218	5	)	)	PUNCT
cana-527	218	6	⊖	⊖	PROPN
cana-527	218	7	s(z∗)∥	s(z∗)∥	VERB
cana-527	218	8	≤	≤	ADJ
cana-527	218	9	β	β	X
cana-527	218	10	.	.	PUNCT
cana-527	219	1	∥y∗	∥y∗	PROPN
cana-527	219	2	⊖	⊖	NOUN
cana-527	219	3	s(y∗)∥	s(y∗)∥	VERB
cana-527	219	4	+	+	CCONJ
cana-527	219	5	∥z∗	∥z∗	PROPN
cana-527	219	6	⊖	⊖	NOUN
cana-527	219	7	s(z∗)∥	s(z∗)∥	VERB
cana-527	219	8	.	.	PUNCT
cana-527	220	1	∥y∗	∥y∗	PROPN
cana-527	220	2	⊖	⊖	ADJ
cana-527	220	3	z∗∥	z∗∥	X
cana-527	220	4	≤	≤	NUM
cana-527	220	5	β	β	X
cana-527	220	6	∥y∗	∥y∗	PROPN
cana-527	220	7	⊖	⊖	X
cana-527	220	8	y∗∥	y∗∥	PROPN
cana-527	220	9	+	+	CCONJ
cana-527	220	10	∥z∗	∥z∗	PROPN
cana-527	220	11	⊖	⊖	ADJ
cana-527	220	12	z∗∥	z∗∥	X
cana-527	220	13	=	=	NOUN
cana-527	220	14	⇒	⇒	NOUN
cana-527	220	15	∥y∗	∥y∗	PROPN
cana-527	220	16	⊖	⊖	SYM
cana-527	220	17	z∗∥	z∗∥	X
cana-527	220	18	=	=	SYM
cana-527	220	19	0	0	PUNCT
cana-527	221	1	=	=	NOUN
cana-527	221	2	⇒	⇒	VERB
cana-527	221	3	y∗	y∗	ADV
cana-527	221	4	=	=	PUNCT
cana-527	221	5	z∗	z∗	NOUN
cana-527	221	6	hence	hence	ADV
cana-527	221	7	the	the	DET
cana-527	221	8	near	near	ADJ
cana-527	221	9	fixed	fix	VERB
cana-527	221	10	point	point	NOUN
cana-527	221	11	of	of	ADP
cana-527	221	12	s	s	NOUN
cana-527	221	13	is	be	AUX
cana-527	221	14	unique	unique	ADJ
cana-527	221	15	.	.	PUNCT
cana-527	222	1	theorem	theorem	NOUN
cana-527	222	2	3.1.3	3.1.3	NUM
cana-527	222	3	:	:	PUNCT
cana-527	222	4	suppose	suppose	VERB
cana-527	222	5	there	there	PRON
cana-527	222	6	is	be	VERB
cana-527	222	7	a	a	DET
cana-527	222	8	bhs	bhs	PROPN
cana-527	222	9	(	(	PUNCT
cana-527	222	10	k(s	k(s	PROPN
cana-527	222	11	)	)	PUNCT
cana-527	222	12	,	,	PUNCT
cana-527	222	13	∥.∥	∥.∥	NUM
cana-527	222	14	)	)	PUNCT
cana-527	222	15	.	.	PUNCT
cana-527	223	1	let	let	VERB
cana-527	223	2	h	h	PRON
cana-527	223	3	be	be	AUX
cana-527	223	4	the	the	DET
cana-527	223	5	graph	graph	NOUN
cana-527	223	6	connected	connect	VERB
cana-527	223	7	to	to	ADP
cana-527	223	8	k(s	k(s	PROPN
cana-527	223	9	)	)	PUNCT
cana-527	223	10	,	,	PUNCT
cana-527	223	11	and	and	CCONJ
cana-527	223	12	allow	allow	VERB
cana-527	223	13	s	s	PRON
cana-527	223	14	:	:	PUNCT
cana-527	223	15	k(s	k(s	PROPN
cana-527	223	16	)	)	PUNCT
cana-527	223	17	→	→	SYM
cana-527	223	18	k(s	k(s	PROPN
cana-527	223	19	)	)	PUNCT
cana-527	223	20	.	.	PUNCT
cana-527	224	1	then	then	ADV
cana-527	224	2	s	s	VERB
cana-527	224	3	satisfies	satisfie	NOUN
cana-527	224	4	,	,	PUNCT
cana-527	224	5	∥s(x	∥s(x	PROPN
cana-527	224	6	)	)	PUNCT
cana-527	224	7	⊖	⊖	NOUN
cana-527	224	8	s(y)∥	s(y)∥	VERB
cana-527	224	9	≤	≤	NOUN
cana-527	224	10	β∥x	β∥x	PUNCT
cana-527	224	11	⊖	⊖	ADJ
cana-527	224	12	s(y)∥	s(y)∥	VERB
cana-527	225	1	+	+	CCONJ
cana-527	226	1	∥y	∥y	PROPN
cana-527	226	2	⊖	⊖	AUX
cana-527	226	3	s(x)∥	s(x)∥	X
cana-527	226	4	(	(	PUNCT
cana-527	226	5	2	2	X
cana-527	226	6	)	)	PUNCT
cana-527	226	7	for	for	ADP
cana-527	226	8	all	all	DET
cana-527	226	9	x	x	NOUN
cana-527	226	10	,	,	PUNCT
cana-527	226	11	y	y	PROPN
cana-527	226	12	∈	∈	PROPN
cana-527	226	13	k(s	k(s	PROPN
cana-527	226	14	)	)	PUNCT
cana-527	226	15	where	where	SCONJ
cana-527	226	16	,	,	PUNCT
cana-527	226	17	β	β	X
cana-527	226	18	∈	∈	PROPN
cana-527	227	1	[	[	X
cana-527	227	2	0	0	NUM
cana-527	227	3	,	,	PUNCT
cana-527	227	4	1	1	NUM
cana-527	227	5	)	)	PUNCT
cana-527	227	6	.	.	PUNCT
cana-527	228	1	then	then	ADV
cana-527	228	2	s	s	AUX
cana-527	228	3	has	have	VERB
cana-527	228	4	a	a	DET
cana-527	228	5	unique	unique	ADJ
cana-527	228	6	fixed	fix	VERB
cana-527	228	7	point	point	NOUN
cana-527	228	8	.	.	PUNCT
cana-527	229	1	proof	proof	NOUN
cana-527	229	2	:	:	PUNCT
cana-527	229	3	let	let	VERB
cana-527	229	4	y0	y0	PRON
cana-527	229	5	be	be	AUX
cana-527	229	6	a	a	DET
cana-527	229	7	random	random	ADJ
cana-527	229	8	point	point	NOUN
cana-527	229	9	in	in	ADP
cana-527	229	10	k(s	k(s	PROPN
cana-527	229	11	)	)	PUNCT
cana-527	229	12	.	.	PUNCT
cana-527	230	1	these	these	PRON
cana-527	230	2	apply	apply	VERB
cana-527	230	3	to	to	ADP
cana-527	230	4	h	h	NOUN
cana-527	230	5	and	and	CCONJ
cana-527	230	6	h0	h0	VERB
cana-527	230	7	as	as	ADP
cana-527	230	8	in	in	ADP
cana-527	230	9	definitions	definition	NOUN
cana-527	230	10	3.1.1	3.1.1	NUM
cana-527	230	11	and	and	CCONJ
cana-527	230	12	communications	communication	NOUN
cana-527	230	13	on	on	ADP
cana-527	230	14	applied	apply	VERB
cana-527	230	15	nonlinear	nonlinear	ADJ
cana-527	230	16	analysis	analysis	NOUN
cana-527	230	17	issn	issn	NOUN
cana-527	230	18	:	:	PUNCT
cana-527	230	19	1074	1074	NUM
cana-527	230	20	-	-	PUNCT
cana-527	230	21	133x	133x	NUM
cana-527	230	22	vol	vol	NOUN
cana-527	230	23	31	31	NUM
cana-527	230	24	no	no	NOUN
cana-527	230	25	.	.	NOUN
cana-527	230	26	2	2	NUM
cana-527	230	27	(	(	PUNCT
cana-527	230	28	2024	2024	NUM
cana-527	230	29	)	)	PUNCT
cana-527	230	30	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	230	31	167	167	NUM
cana-527	230	32	.	.	PUNCT
cana-527	230	33	.	.	PUNCT
cana-527	230	34	.	.	PUNCT
cana-527	230	35	.	.	PUNCT
cana-527	230	36	.	.	PUNCT
cana-527	231	1	.	.	PUNCT
cana-527	232	1	3.1.2	3.1.2	NUM
cana-527	232	2	and	and	CCONJ
cana-527	232	3	refer	refer	VERB
cana-527	232	4	[	[	X
cana-527	232	5	20	20	NUM
cana-527	232	6	]	]	PUNCT
cana-527	232	7	.	.	PUNCT
cana-527	233	1	consider	consider	VERB
cana-527	233	2	,	,	PUNCT
cana-527	233	3	the	the	DET
cana-527	233	4	iterated	iterated	ADJ
cana-527	233	5	sequence	sequence	NOUN
cana-527	233	6	,	,	PUNCT
cana-527	233	7	{	{	PUNCT
cana-527	233	8	s(y0	s(y0	NOUN
cana-527	233	9	)	)	PUNCT
cana-527	233	10	,	,	PUNCT
cana-527	233	11	s2(y0	s2(y0	NOUN
cana-527	233	12	)	)	PUNCT
cana-527	233	13	,	,	PUNCT
cana-527	233	14	s3(y0	s3(y0	NOUN
cana-527	233	15	)	)	PUNCT
cana-527	233	16	,	,	PUNCT
cana-527	233	17	.	.	PUNCT
cana-527	233	18	.	.	PUNCT
cana-527	234	1	.	.	PUNCT
cana-527	234	2	}	}	PUNCT
cana-527	235	1	∈	∈	PROPN
cana-527	235	2	k(s	k(s	PROPN
cana-527	235	3	)	)	PUNCT
cana-527	235	4	according	accord	VERB
cana-527	235	5	to	to	ADP
cana-527	235	6	lemma	lemma	PROPN
cana-527	235	7	3.1	3.1	NUM
cana-527	235	8	of	of	ADP
cana-527	235	9	[	[	X
cana-527	235	10	20	20	NUM
cana-527	235	11	]	]	PUNCT
cana-527	235	12	,	,	PUNCT
cana-527	235	13	to	to	PART
cana-527	235	14	demonstrate	demonstrate	VERB
cana-527	235	15	the	the	DET
cana-527	235	16	cauchy	cauchy	ADJ
cana-527	235	17	nature	nature	NOUN
cana-527	235	18	of	of	ADP
cana-527	235	19	this	this	DET
cana-527	235	20	sequence	sequence	NOUN
cana-527	235	21	.	.	PUNCT
cana-527	236	1	it	it	PRON
cana-527	236	2	suffices	suffice	VERB
cana-527	236	3	to	to	PART
cana-527	236	4	demonstrate	demonstrate	VERB
cana-527	236	5	that	that	SCONJ
cana-527	236	6	the	the	DET
cana-527	236	7	graph	graph	NOUN
cana-527	236	8	h0	h0	PROPN
cana-527	236	9	’s	’s	PART
cana-527	236	10	w	w	NOUN
cana-527	236	11	-	-	PUNCT
cana-527	236	12	sequence	sequence	NOUN
cana-527	236	13	is	be	AUX
cana-527	236	14	non	non	ADJ
cana-527	236	15	-	-	ADJ
cana-527	236	16	increasing	increase	VERB
cana-527	236	17	.	.	PUNCT
cana-527	237	1	from	from	ADP
cana-527	237	2	eqn	eqn	NOUN
cana-527	237	3	2	2	NUM
cana-527	237	4	,	,	PUNCT
cana-527	237	5	we	we	PRON
cana-527	237	6	have	have	AUX
cana-527	237	7	,	,	PUNCT
cana-527	237	8	wn+1	wn+1	VERB
cana-527	237	9	=	=	SYM
cana-527	237	10	∥sny0	∥sny0	NOUN
cana-527	237	11	⊖	⊖	NOUN
cana-527	237	12	sn+1y0∥	sn+1y0∥	VERB
cana-527	237	13	≤	≤	NOUN
cana-527	237	14	β	β	NOUN
cana-527	237	15	∥sn−1y0	∥sn−1y0	VERB
cana-527	237	16	⊖	⊖	PROPN
cana-527	237	17	sn+1y0∥	sn+1y0∥	VERB
cana-527	237	18	⊕	⊕	PROPN
cana-527	237	19	∥sny0	∥sny0	NOUN
cana-527	237	20	⊖	⊖	VERB
cana-527	237	21	sny0∥	sny0∥	NOUN
cana-527	237	22	wn+1	wn+1	VERB
cana-527	237	23	≤	≤	NOUN
cana-527	237	24	β	β	X
cana-527	237	25	∥sn−1y0	∥sn−1y0	X
cana-527	237	26	⊖	⊖	X
cana-527	237	27	sny0∥	sny0∥	PROPN
cana-527	237	28	⊕	⊕	PROPN
cana-527	237	29	∥sny0	∥sny0	NOUN
cana-527	237	30	⊖	⊖	NOUN
cana-527	237	31	sn+1y0∥	sn+1y0∥	VERB
cana-527	237	32	wn+1	wn+1	VERB
cana-527	237	33	≤	≤	NOUN
cana-527	237	34	β[wn	β[wn	ADJ
cana-527	237	35	+	+	CCONJ
cana-527	237	36	wn+1	wn+1	X
cana-527	237	37	]	]	X
cana-527	237	38	β	β	X
cana-527	237	39	1	1	NUM
cana-527	237	40	wn	wn	PROPN
cana-527	237	41	1	1	NUM
cana-527	237	42	≤	≤	NUM
cana-527	237	43	wn	wn	NOUN
cana-527	237	44	<	<	X
cana-527	237	45	wn	wn	PROPN
cana-527	237	46	since	since	SCONJ
cana-527	237	47	0	0	NUM
cana-527	237	48	≤	≤	NUM
cana-527	237	49	β	β	X
cana-527	237	50	<	<	X
cana-527	237	51	1	1	NUM
cana-527	237	52	−	−	NOUN
cana-527	237	53	β	β	NOUN
cana-527	237	54	2	2	NUM
cana-527	237	55	hence	hence	ADV
cana-527	237	56	the	the	DET
cana-527	237	57	w	w	NOUN
cana-527	237	58	-	-	PUNCT
cana-527	237	59	sequence	sequence	NOUN
cana-527	237	60	associated	associate	VERB
cana-527	237	61	with	with	ADP
cana-527	237	62	h0	h0	PROPN
cana-527	237	63	is	be	AUX
cana-527	237	64	non	non	ADJ
cana-527	237	65	-	-	ADJ
cana-527	237	66	increasing	increase	VERB
cana-527	237	67	.	.	PUNCT
cana-527	238	1	from	from	ADP
cana-527	238	2	lemma	lemma	PROPN
cana-527	238	3	3.1	3.1	NUM
cana-527	238	4	of	of	ADP
cana-527	238	5	[	[	X
cana-527	238	6	20	20	NUM
cana-527	238	7	]	]	PUNCT
cana-527	238	8	,	,	PUNCT
cana-527	238	9	the	the	DET
cana-527	238	10	iterated	iterated	ADJ
cana-527	238	11	sequence	sequence	NOUN
cana-527	238	12	is	be	AUX
cana-527	238	13	cauchy	cauchy	ADJ
cana-527	238	14	sequence	sequence	NOUN
cana-527	238	15	in	in	ADP
cana-527	238	16	k(s	k(s	PROPN
cana-527	238	17	)	)	PUNCT
cana-527	238	18	.	.	PUNCT
cana-527	239	1	but	but	CCONJ
cana-527	239	2	k(s	k(s	PROPN
cana-527	239	3	)	)	PUNCT
cana-527	239	4	is	be	AUX
cana-527	239	5	complete	complete	ADJ
cana-527	239	6	.	.	PUNCT
cana-527	240	1	∴	∴	NOUN
cana-527	240	2	the	the	DET
cana-527	240	3	sequence	sequence	NOUN
cana-527	240	4	converges	converge	VERB
cana-527	240	5	in	in	ADP
cana-527	240	6	k(s	k(s	PROPN
cana-527	240	7	)	)	PUNCT
cana-527	240	8	.	.	PUNCT
cana-527	241	1	let	let	VERB
cana-527	241	2	y∗	y∗	ADV
cana-527	241	3	=	=	SYM
cana-527	241	4	limn→∞	limn→∞	ADJ
cana-527	241	5	sn(y0	sn(y0	PRON
cana-527	241	6	)	)	PUNCT
cana-527	241	7	consider	consider	NOUN
cana-527	241	8	,	,	PUNCT
cana-527	241	9	∥y∗	∥y∗	PROPN
cana-527	241	10	⊖	⊖	NOUN
cana-527	241	11	s(y∗)∥	s(y∗)∥	VERB
cana-527	241	12	≤	≤	NOUN
cana-527	241	13	∥y∗	∥y∗	PROPN
cana-527	241	14	⊖	⊖	ADJ
cana-527	241	15	sn(y0)∥	sn(y0)∥	NOUN
cana-527	241	16	+	+	SYM
cana-527	241	17	∥sn(y0	∥sn(y0	NOUN
cana-527	241	18	)	)	PUNCT
cana-527	241	19	−	−	NOUN
cana-527	241	20	s(y∗)∥	s(y∗)∥	VERB
cana-527	241	21	≤	≤	ADJ
cana-527	241	22	β	β	X
cana-527	241	23	.	.	PUNCT
cana-527	242	1	∥y∗	∥y∗	PROPN
cana-527	243	1	−	−	NUM
cana-527	243	2	sn(y0)∥	sn(y0)∥	NOUN
cana-527	243	3	+	+	CCONJ
cana-527	243	4	β∥sn−1(y0	β∥sn−1(y0	NOUN
cana-527	243	5	)	)	PUNCT
cana-527	244	1	−	−	NOUN
cana-527	244	2	s(y∗)∥	s(y∗)∥	VERB
cana-527	244	3	+	+	CCONJ
cana-527	244	4	∥y∗	∥y∗	PROPN
cana-527	244	5	⊖	⊖	PROPN
cana-527	244	6	sn(y0)∥	sn(y0)∥	PROPN
cana-527	244	7	.	.	PUNCT
cana-527	245	1	allow	allow	VERB
cana-527	245	2	n	n	PRON
cana-527	245	3	→	→	SYM
cana-527	245	4	∞	∞	NUM
cana-527	245	5	on	on	ADP
cana-527	245	6	both	both	DET
cana-527	245	7	sides	side	NOUN
cana-527	245	8	.	.	PUNCT
cana-527	246	1	then	then	ADV
cana-527	246	2	we	we	PRON
cana-527	246	3	have	have	VERB
cana-527	246	4	,	,	PUNCT
cana-527	246	5	∥y∗	∥y∗	PROPN
cana-527	246	6	⊖	⊖	NOUN
cana-527	246	7	s(y∗)∥	s(y∗)∥	VERB
cana-527	246	8	≤	≤	NOUN
cana-527	246	9	∥y∗	∥y∗	PROPN
cana-527	246	10	−	−	PROPN
cana-527	247	1	y∗∥	y∗∥	PROPN
cana-527	247	2	+	+	NUM
cana-527	247	3	β	β	AUX
cana-527	247	4	∥y∗	∥y∗	NOUN
cana-527	247	5	−	−	NOUN
cana-527	247	6	s(y∗)∥	s(y∗)∥	NOUN
cana-527	247	7	+	+	CCONJ
cana-527	247	8	∥y∗	∥y∗	PROPN
cana-527	247	9	−	−	PROPN
cana-527	247	10	y∗∥	y∗∥	CCONJ
cana-527	247	11	(	(	PUNCT
cana-527	247	12	1	1	NUM
cana-527	247	13	−	−	PROPN
cana-527	247	14	β)∥y∗	β)∥y∗	NOUN
cana-527	247	15	−	−	NOUN
cana-527	247	16	s(y∗)∥	s(y∗)∥	VERB
cana-527	247	17	≤	≤	NOUN
cana-527	247	18	0	0	PUNCT
cana-527	248	1	hence	hence	ADV
cana-527	248	2	∥y∗	∥y∗	PROPN
cana-527	248	3	−	−	PROPN
cana-527	248	4	s(y∗)∥	s(y∗)∥	NOUN
cana-527	248	5	=	=	NOUN
cana-527	248	6	0	0	PUNCT
cana-527	249	1	=	=	NOUN
cana-527	249	2	⇒	⇒	NOUN
cana-527	249	3	s(y∗	s(y∗	NUM
cana-527	249	4	)	)	PUNCT
cana-527	250	1	=	=	SYM
cana-527	250	2	y∗	y∗	PROPN
cana-527	250	3	∴	∴	PROPN
cana-527	250	4	y∗	y∗	PROPN
cana-527	250	5	is	be	AUX
cana-527	250	6	a	a	DET
cana-527	250	7	fixed	fix	VERB
cana-527	250	8	point	point	NOUN
cana-527	250	9	of	of	ADP
cana-527	250	10	s.	s.	PROPN
cana-527	250	11	to	to	PART
cana-527	250	12	prove	prove	VERB
cana-527	250	13	uniqueness	uniqueness	NOUN
cana-527	250	14	:	:	PUNCT
cana-527	250	15	let	let	VERB
cana-527	250	16	if	if	SCONJ
cana-527	250	17	possible	possible	ADJ
cana-527	250	18	,	,	PUNCT
cana-527	250	19	z∗	z∗	X
cana-527	250	20	be	be	AUX
cana-527	250	21	any	any	DET
cana-527	250	22	other	other	ADJ
cana-527	250	23	fixed	fix	VERB
cana-527	250	24	point	point	NOUN
cana-527	250	25	of	of	ADP
cana-527	250	26	s.	s.	PROPN
cana-527	250	27	then	then	ADV
cana-527	250	28	s(z∗	s(z∗	X
cana-527	250	29	)	)	PUNCT
cana-527	250	30	=	=	SYM
cana-527	250	31	z∗.	z∗.	PUNCT
cana-527	250	32	from	from	ADP
cana-527	250	33	eqn	eqn	PROPN
cana-527	250	34	2	2	NUM
cana-527	250	35	,	,	PUNCT
cana-527	250	36	we	we	PRON
cana-527	250	37	have	have	VERB
cana-527	250	38	∥s(y∗	∥s(y∗	NUM
cana-527	250	39	)	)	PUNCT
cana-527	250	40	⊖	⊖	NOUN
cana-527	250	41	s(z∗)∥	s(z∗)∥	VERB
cana-527	250	42	≤	≤	ADJ
cana-527	250	43	β	β	X
cana-527	250	44	.	.	PUNCT
cana-527	251	1	∥y∗	∥y∗	PROPN
cana-527	251	2	⊖	⊖	PUNCT
cana-527	251	3	s(z∗)∥	s(z∗)∥	VERB
cana-527	251	4	+	+	CCONJ
cana-527	251	5	∥z∗	∥z∗	PROPN
cana-527	251	6	⊖	⊖	NOUN
cana-527	251	7	s(y∗)∥	s(y∗)∥	VERB
cana-527	251	8	.	.	PUNCT
cana-527	252	1	+	+	CCONJ
cana-527	252	2	communications	communication	NOUN
cana-527	252	3	on	on	ADP
cana-527	252	4	applied	apply	VERB
cana-527	252	5	nonlinear	nonlinear	ADJ
cana-527	252	6	analysis	analysis	NOUN
cana-527	252	7	issn	issn	NOUN
cana-527	252	8	:	:	PUNCT
cana-527	252	9	1074	1074	NUM
cana-527	252	10	-	-	PUNCT
cana-527	252	11	133x	133x	NUM
cana-527	252	12	vol	vol	NOUN
cana-527	252	13	31	31	NUM
cana-527	252	14	no	no	NOUN
cana-527	252	15	.	.	NOUN
cana-527	252	16	2	2	NUM
cana-527	252	17	(	(	PUNCT
cana-527	252	18	2024	2024	NUM
cana-527	252	19	)	)	PUNCT
cana-527	252	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	252	21	168	168	NUM
cana-527	252	22	2	2	NUM
cana-527	252	23	∥	∥	NUM
cana-527	252	24	∗	∗	NOUN
cana-527	252	25	∗	∗	NOUN
cana-527	252	26	∗	∗	NOUN
cana-527	252	27	∗y	∗y	PROPN
cana-527	252	28	−	−	PROPN
cana-527	252	29	z	z	NOUN
cana-527	252	30	∥	∥	PUNCT
cana-527	252	31	≤	≤	NOUN
cana-527	253	1	2β∥y	2β∥y	NUM
cana-527	253	2	−	−	PROPN
cana-527	253	3	z	z	NOUN
cana-527	253	4	∥	∥	PUNCT
cana-527	253	5	⊂	⊂	PUNCT
cana-527	253	6	∈	∈	PROPN
cana-527	253	7	n	n	CCONJ
cana-527	253	8	where	where	SCONJ
cana-527	253	9	,	,	PUNCT
cana-527	253	10	0	0	NUM
cana-527	253	11	≤	≤	NUM
cana-527	253	12	α	α	PRON
cana-527	253	13	<	<	X
cana-527	253	14	1	1	NUM
cana-527	253	15	∥y∗	∥y∗	PROPN
cana-527	253	16	⊖	⊖	NOUN
cana-527	253	17	z∗∥	z∗∥	X
cana-527	253	18	≤	≤	NUM
cana-527	253	19	β	β	X
cana-527	253	20	.	.	PUNCT
cana-527	254	1	∥y∗	∥y∗	PROPN
cana-527	254	2	⊖	⊖	ADJ
cana-527	254	3	z∗∥	z∗∥	X
cana-527	254	4	+	+	CCONJ
cana-527	254	5	∥z∗	∥z∗	PROPN
cana-527	254	6	⊖	⊖	X
cana-527	254	7	y∗∥	y∗∥	ADV
cana-527	254	8	.	.	PUNCT
cana-527	255	1	1	1	NUM
cana-527	255	2	=	=	X
cana-527	255	3	⇒	⇒	NOUN
cana-527	255	4	β	β	X
cana-527	255	5	≥	≥	NUM
cana-527	255	6	2	2	NUM
cana-527	255	7	this	this	PRON
cana-527	255	8	is	be	AUX
cana-527	255	9	a	a	DET
cana-527	255	10	contradiction	contradiction	NOUN
cana-527	255	11	.	.	PUNCT
cana-527	256	1	hence	hence	ADV
cana-527	256	2	y∗	y∗	ADV
cana-527	256	3	=	=	SYM
cana-527	256	4	z∗.	z∗.	X
cana-527	256	5	(	(	PUNCT
cana-527	256	6	i.e	i.e	PROPN
cana-527	256	7	)	)	PUNCT
cana-527	256	8	,	,	PUNCT
cana-527	256	9	the	the	DET
cana-527	256	10	near	near	ADV
cana-527	256	11	fixed	fix	VERB
cana-527	256	12	point	point	NOUN
cana-527	256	13	of	of	ADP
cana-527	256	14	t	t	PROPN
cana-527	256	15	is	be	AUX
cana-527	256	16	unique	unique	ADJ
cana-527	256	17	.	.	PUNCT
cana-527	257	1	definition	definition	NOUN
cana-527	257	2	3.1.4	3.1.4	NUM
cana-527	257	3	:	:	PUNCT
cana-527	257	4	let	let	VERB
cana-527	257	5	the	the	DET
cana-527	257	6	mapping	mapping	NOUN
cana-527	257	7	f	f	NOUN
cana-527	257	8	:	:	PUNCT
cana-527	257	9	t+	t+	PROPN
cana-527	257	10	→	→	SYM
cana-527	257	11	t	t	PROPN
cana-527	257	12	satisfies	satisfy	VERB
cana-527	257	13	the	the	DET
cana-527	257	14	following	follow	VERB
cana-527	257	15	conditions	condition	NOUN
cana-527	257	16	,	,	PUNCT
cana-527	257	17	1	1	X
cana-527	257	18	.	.	X
cana-527	257	19	f	f	PROPN
cana-527	257	20	is	be	AUX
cana-527	257	21	strictly	strictly	ADV
cana-527	257	22	increasing	increase	VERB
cana-527	257	23	2	2	NUM
cana-527	257	24	.	.	PUNCT
cana-527	258	1	for	for	ADP
cana-527	258	2	each	each	DET
cana-527	258	3	sequence	sequence	NOUN
cana-527	258	4	sn	sn	PROPN
cana-527	258	5	t+	t+	PROPN
cana-527	258	6	,	,	PUNCT
cana-527	258	7	lim	lim	PROPN
cana-527	258	8	n→+∞	n→+∞	VERB
cana-527	258	9	sn	sn	PROPN
cana-527	258	10	=	=	SYM
cana-527	258	11	0	0	PROPN
cana-527	258	12	iff	iff	PROPN
cana-527	258	13	lim	lim	PROPN
cana-527	258	14	n→	n→	PUNCT
cana-527	258	15	+	+	PROPN
cana-527	258	16	∞	∞	PROPN
cana-527	258	17	f(sn	f(sn	NUM
cana-527	258	18	)	)	PUNCT
cana-527	258	19	=	=	PUNCT
cana-527	258	20	−∞	−∞	ADP
cana-527	258	21	3	3	NUM
cana-527	258	22	.	.	PUNCT
cana-527	259	1	there	there	PRON
cana-527	259	2	exists	exist	VERB
cana-527	259	3	m	m	VERB
cana-527	259	4	(	(	PUNCT
cana-527	259	5	0	0	NUM
cana-527	259	6	,	,	PUNCT
cana-527	259	7	1	1	NUM
cana-527	259	8	)	)	PUNCT
cana-527	259	9	provided	provide	VERB
cana-527	259	10	that	that	SCONJ
cana-527	259	11	lim	lim	PROPN
cana-527	259	12	λmf(λ	λmf(λ	PROPN
cana-527	259	13	)	)	PUNCT
cana-527	259	14	=	=	PUNCT
cana-527	260	1	0	0	X
cana-527	260	2	.	.	PUNCT
cana-527	261	1	the	the	DET
cana-527	261	2	collection	collection	NOUN
cana-527	261	3	of	of	ADP
cana-527	261	4	all	all	DET
cana-527	261	5	such	such	ADJ
cana-527	261	6	λ→0	λ→0	ADJ
cana-527	261	7	+	+	NUM
cana-527	261	8	mappings	mapping	NOUN
cana-527	261	9	is	be	AUX
cana-527	261	10	denoted	denote	VERB
cana-527	261	11	by	by	ADP
cana-527	261	12	ω	ω	PROPN
cana-527	261	13	.	.	PUNCT
cana-527	262	1	definition	definition	NOUN
cana-527	262	2	3.1.5	3.1.5	NUM
cana-527	262	3	:	:	PUNCT
cana-527	262	4	let	let	VERB
cana-527	262	5	(	(	PUNCT
cana-527	262	6	k(s	k(s	PROPN
cana-527	262	7	)	)	PUNCT
cana-527	262	8	,	,	PUNCT
cana-527	262	9	∥.∥	∥.∥	NUM
cana-527	262	10	)	)	PUNCT
cana-527	262	11	be	be	AUX
cana-527	262	12	a	a	DET
cana-527	262	13	bhs	bhs	PROPN
cana-527	262	14	.	.	PUNCT
cana-527	263	1	a	a	DET
cana-527	263	2	map	map	NOUN
cana-527	263	3	s	s	VERB
cana-527	263	4	:	:	PUNCT
cana-527	263	5	k(s	k(s	PROPN
cana-527	263	6	)	)	PUNCT
cana-527	263	7	→	→	SYM
cana-527	263	8	k(s	k(s	PROPN
cana-527	263	9	)	)	PUNCT
cana-527	263	10	is	be	AUX
cana-527	263	11	f−contraction	f−contraction	NUM
cana-527	263	12	if	if	SCONJ
cana-527	263	13	there	there	PRON
cana-527	263	14	exist	exist	VERB
cana-527	263	15	f	f	PROPN
cana-527	263	16	∈	∈	PROPN
cana-527	263	17	ω	ω	PROPN
cana-527	263	18	and	and	CCONJ
cana-527	263	19	τ	τ	PROPN
cana-527	263	20	>	>	X
cana-527	263	21	0	0	PUNCT
cana-527	263	22	provided	provide	VERB
cana-527	263	23	that	that	SCONJ
cana-527	263	24	||γl⊖γn||	||γl⊖γn||	NUM
cana-527	263	25	>	>	SYM
cana-527	263	26	0	0	PUNCT
cana-527	263	27	⇒	⇒	NOUN
cana-527	263	28	τ⊕f(||γl⊕γn||	τ⊕f(||γl⊕γn||	NUM
cana-527	263	29	)	)	PUNCT
cana-527	263	30	≤	≤	NUM
cana-527	263	31	f(||l⊕n||	f(||l⊕n||	PROPN
cana-527	263	32	)	)	PUNCT
cana-527	263	33	.	.	PUNCT
cana-527	263	34	.	.	PUNCT
cana-527	263	35	.	.	PUNCT
cana-527	264	1	(	(	PUNCT
cana-527	264	2	1	1	NUM
cana-527	264	3	)	)	PUNCT
cana-527	264	4	,	,	PUNCT
cana-527	264	5	∀	∀	X
cana-527	264	6	l	l	NOUN
cana-527	264	7	,	,	PUNCT
cana-527	264	8	n	n	PROPN
cana-527	264	9	∈	∈	PROPN
cana-527	264	10	k(s	k(s	PROPN
cana-527	264	11	)	)	PUNCT
cana-527	264	12	.	.	PUNCT
cana-527	265	1	example	example	NOUN
cana-527	266	1	3.1.2	3.1.2	NUM
cana-527	266	2	:	:	PUNCT
cana-527	266	3	let	let	VERB
cana-527	266	4	f	f	PROPN
cana-527	266	5	∈	∈	PROPN
cana-527	266	6	ω	ω	PROPN
cana-527	266	7	be	be	AUX
cana-527	266	8	f(β	f(β	NOUN
cana-527	266	9	)	)	PUNCT
cana-527	267	1	=	=	NOUN
cana-527	267	2	inβ	inβ	NOUN
cana-527	267	3	for	for	ADP
cana-527	267	4	any	any	DET
cana-527	267	5	m	m	NOUN
cana-527	267	6	∈	∈	NOUN
cana-527	267	7	(	(	PUNCT
cana-527	267	8	0	0	NUM
cana-527	267	9	,	,	PUNCT
cana-527	267	10	1	1	NUM
cana-527	267	11	)	)	PUNCT
cana-527	267	12	here	here	ADV
cana-527	267	13	,	,	PUNCT
cana-527	267	14	every	every	DET
cana-527	267	15	map	map	NOUN
cana-527	267	16	s	s	VERB
cana-527	267	17	:	:	PUNCT
cana-527	267	18	k(s	k(s	PROPN
cana-527	267	19	)	)	PUNCT
cana-527	267	20	→	→	SYM
cana-527	267	21	k(s	k(s	PROPN
cana-527	267	22	)	)	PUNCT
cana-527	267	23	satisfying	satisfy	VERB
cana-527	267	24	(	(	PUNCT
cana-527	267	25	1	1	NUM
cana-527	267	26	)	)	PUNCT
cana-527	267	27	is	be	AUX
cana-527	267	28	an	an	DET
cana-527	267	29	f	f	PROPN
cana-527	267	30	contraction	contraction	NOUN
cana-527	267	31	such	such	DET
cana-527	267	32	that	that	DET
cana-527	267	33	||γl	||γl	NOUN
cana-527	267	34	⊕	⊕	PROPN
cana-527	267	35	γn||	γn||	PUNCT
cana-527	267	36	≤	≤	NUM
cana-527	267	37	e−τ||l	e−τ||l	PROPN
cana-527	267	38	⊕	⊕	PROPN
cana-527	267	39	n||	n||	PROPN
cana-527	267	40	,	,	PUNCT
cana-527	267	41	for	for	ADP
cana-527	267	42	every	every	DET
cana-527	267	43	l	l	NOUN
cana-527	267	44	,	,	PUNCT
cana-527	267	45	n	n	PROPN
cana-527	267	46	∈	∈	PROPN
cana-527	267	47	k(s	k(s	PROPN
cana-527	267	48	)	)	PUNCT
cana-527	267	49	,	,	PUNCT
cana-527	267	50	γl	γl	PROPN
cana-527	267	51	≠	≠	PROPN
cana-527	267	52	γn	γn	ADP
cana-527	267	53	example	example	NOUN
cana-527	267	54	3.1.3	3.1.3	NUM
cana-527	267	55	:	:	PUNCT
cana-527	267	56	consider	consider	VERB
cana-527	267	57	f	f	PROPN
cana-527	267	58	∈	∈	PROPN
cana-527	267	59	ω	ω	PROPN
cana-527	267	60	be	be	AUX
cana-527	267	61	f(β	f(β	NOUN
cana-527	267	62	)	)	PUNCT
cana-527	267	63	=	=	SYM
cana-527	267	64	−√1	−√1	NOUN
cana-527	267	65	,	,	PUNCT
cana-527	267	66	β	β	X
cana-527	267	67	>	>	X
cana-527	267	68	0	0	X
cana-527	267	69	.	.	PUNCT
cana-527	268	1	in	in	ADP
cana-527	268	2	this	this	DET
cana-527	268	3	case	case	NOUN
cana-527	268	4	,	,	PUNCT
cana-527	268	5	for	for	ADP
cana-527	268	6	any	any	DET
cana-527	268	7	m	m	NOUN
cana-527	268	8	∈	∈	NOUN
cana-527	268	9	(	(	PUNCT
cana-527	268	10	1	1	NUM
cana-527	268	11	,	,	PUNCT
cana-527	268	12	1	1	NUM
cana-527	268	13	)	)	PUNCT
cana-527	268	14	every	every	DET
cana-527	268	15	f	f	NOUN
cana-527	268	16	-	-	PUNCT
cana-527	268	17	contraction	contraction	NOUN
cana-527	268	18	γ	γ	NOUN
cana-527	268	19	satisfies	satisfie	NOUN
cana-527	268	20	,	,	PUNCT
cana-527	268	21	||γl	||γl	NOUN
cana-527	268	22	⊖	⊖	NOUN
cana-527	268	23	γn||	γn||	PUNCT
cana-527	268	24	≤	≤	ADJ
cana-527	268	25	1	1	NUM
cana-527	268	26	theorem	theorem	VERB
cana-527	268	27	3.1.4	3.1.4	NUM
cana-527	268	28	:	:	PUNCT
cana-527	268	29	β	β	X
cana-527	268	30	1	1	NUM
cana-527	268	31	⊕	⊕	NUM
cana-527	268	32	τ	τ	NOUN
cana-527	268	33	√	√	PROPN
cana-527	268	34	||l	||l	NOUN
cana-527	268	35	⊖	⊖	VERB
cana-527	268	36	2	2	NUM
cana-527	268	37	2	2	NUM
cana-527	268	38	||l	||l	NOUN
cana-527	268	39	⊖	⊖	X
cana-527	268	40	n||	n||	PROPN
cana-527	268	41	,	,	PUNCT
cana-527	268	42	∀	∀	X
cana-527	268	43	l	l	NOUN
cana-527	268	44	,	,	PUNCT
cana-527	268	45	n	n	PROPN
cana-527	268	46	∈	∈	PROPN
cana-527	268	47	k(x	k(x	PROPN
cana-527	268	48	)	)	PUNCT
cana-527	268	49	,	,	PUNCT
cana-527	269	1	γl	γl	PROPN
cana-527	269	2	≠	≠	PROPN
cana-527	269	3	γn	γn	ADV
cana-527	269	4	||	||	ADV
cana-527	269	5	let	let	VERB
cana-527	269	6	s	s	PRON
cana-527	269	7	:	:	PUNCT
cana-527	269	8	k(s	k(s	PROPN
cana-527	269	9	)	)	PUNCT
cana-527	269	10	→	→	SYM
cana-527	269	11	k(x	k(x	PROPN
cana-527	269	12	)	)	PUNCT
cana-527	269	13	be	be	AUX
cana-527	269	14	an	an	DET
cana-527	269	15	f	f	NOUN
cana-527	269	16	-	-	PUNCT
cana-527	269	17	contraction	contraction	NOUN
cana-527	269	18	and	and	CCONJ
cana-527	269	19	(	(	PUNCT
cana-527	269	20	k(s	k(s	PROPN
cana-527	269	21	)	)	PUNCT
cana-527	269	22	,	,	PUNCT
cana-527	269	23	∥.∥	∥.∥	NUM
cana-527	269	24	)	)	PUNCT
cana-527	269	25	be	be	AUX
cana-527	269	26	a	a	DET
cana-527	269	27	banach	banach	NOUN
cana-527	269	28	hyperspace	hyperspace	NOUN
cana-527	269	29	.	.	PUNCT
cana-527	270	1	then	then	ADV
cana-527	270	2	s	s	AUX
cana-527	270	3	has	have	VERB
cana-527	270	4	a	a	DET
cana-527	270	5	unique	unique	ADJ
cana-527	270	6	near	near	ADP
cana-527	270	7	fixed	fix	VERB
cana-527	270	8	point	point	NOUN
cana-527	270	9	l∗	l∗	PROPN
cana-527	270	10	∈	∈	PROPN
cana-527	270	11	k(s	k(s	PROPN
cana-527	270	12	)	)	PUNCT
cana-527	270	13	and	and	CCONJ
cana-527	270	14	for	for	ADP
cana-527	270	15	every	every	DET
cana-527	270	16	l	l	NOUN
cana-527	270	17	∈	∈	PROPN
cana-527	270	18	k(s	k(s	PROPN
cana-527	270	19	)	)	PUNCT
cana-527	270	20	the	the	DET
cana-527	270	21	sequence	sequence	NOUN
cana-527	270	22	(	(	PUNCT
cana-527	270	23	snl	snl	PROPN
cana-527	270	24	)	)	PUNCT
cana-527	270	25	,	,	PUNCT
cana-527	270	26	n	n	PRON
cana-527	270	27	∈	∈	PROPN
cana-527	270	28	n	n	PRON
cana-527	270	29	converges	converge	VERB
cana-527	270	30	to	to	PART
cana-527	270	31	l∗.	l∗.	NOUN
cana-527	270	32	remark	remark	NOUN
cana-527	270	33	:	:	PUNCT
cana-527	270	34	every	every	DET
cana-527	270	35	f	f	NOUN
cana-527	270	36	-	-	PUNCT
cana-527	270	37	contraction	contraction	NOUN
cana-527	270	38	γ	γ	NOUN
cana-527	270	39	is	be	AUX
cana-527	270	40	a	a	DET
cana-527	270	41	contractive	contractive	ADJ
cana-527	270	42	map	map	NOUN
cana-527	270	43	.	.	PUNCT
cana-527	271	1	(	(	PUNCT
cana-527	271	2	i.e	i.e	X
cana-527	271	3	)	)	PUNCT
cana-527	271	4	||γl	||γl	NOUN
cana-527	271	5	⊖	⊖	NOUN
cana-527	271	6	γn||	γn||	PUNCT
cana-527	271	7	≤	≤	ADJ
cana-527	271	8	||l	||l	NOUN
cana-527	271	9	⊖	⊖	X
cana-527	271	10	n||	n||	PROPN
cana-527	271	11	,	,	PUNCT
cana-527	271	12	∀l	∀l	NOUN
cana-527	271	13	,	,	PUNCT
cana-527	271	14	n	n	PROPN
cana-527	271	15	∈	∈	PROPN
cana-527	271	16	k(s	k(s	PROPN
cana-527	271	17	)	)	PUNCT
cana-527	271	18	and	and	CCONJ
cana-527	271	19	γl	γl	PROPN
cana-527	271	20	≠	≠	PROPN
cana-527	271	21	γn	γn	NUM
cana-527	271	22	.	.	PUNCT
cana-527	272	1	thus	thus	ADV
cana-527	272	2	every	every	DET
cana-527	272	3	f	f	X
cana-527	272	4	-	-	PUNCT
cana-527	272	5	contraction	contraction	NOUN
cana-527	272	6	is	be	AUX
cana-527	272	7	continuous	continuous	ADJ
cana-527	272	8	map	map	NOUN
cana-527	272	9	.	.	PUNCT
cana-527	273	1	communications	communication	NOUN
cana-527	273	2	on	on	ADP
cana-527	273	3	applied	apply	VERB
cana-527	273	4	nonlinear	nonlinear	ADJ
cana-527	273	5	analysis	analysis	NOUN
cana-527	273	6	issn	issn	NOUN
cana-527	273	7	:	:	PUNCT
cana-527	273	8	1074	1074	NUM
cana-527	273	9	-	-	PUNCT
cana-527	273	10	133x	133x	NUM
cana-527	273	11	vol	vol	NOUN
cana-527	273	12	31	31	NUM
cana-527	273	13	no	no	NOUN
cana-527	273	14	.	.	NOUN
cana-527	273	15	2	2	NUM
cana-527	273	16	(	(	PUNCT
cana-527	273	17	2024	2024	NUM
cana-527	273	18	)	)	PUNCT
cana-527	273	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	273	20	169	169	NUM
cana-527	273	21	n→∞	n→∞	NUM
cana-527	273	22	n	n	CCONJ
cana-527	273	23	n	n	CCONJ
cana-527	273	24	≤	≤	X
cana-527	273	25	∀	∀	X
cana-527	273	26	≥	≥	NOUN
cana-527	273	27	n	n	CCONJ
cana-527	273	28	n	n	CCONJ
cana-527	273	29	n	n	ADV
cana-527	273	30	3.2	3.2	NUM
cana-527	273	31	near	near	ADP
cana-527	273	32	fixed	fix	VERB
cana-527	273	33	point	point	NOUN
cana-527	273	34	theorems	theorem	NOUN
cana-527	273	35	in	in	ADP
cana-527	273	36	bhs	bhs	PROPN
cana-527	273	37	endowed	endow	VERB
cana-527	273	38	with	with	ADP
cana-527	273	39	a	a	DET
cana-527	273	40	graph	graph	NOUN
cana-527	273	41	we	we	PRON
cana-527	273	42	provide	provide	VERB
cana-527	273	43	fixed	fix	VERB
cana-527	273	44	point	point	NOUN
cana-527	273	45	theorems	theorem	NOUN
cana-527	273	46	for	for	ADP
cana-527	273	47	mappings	mapping	NOUN
cana-527	273	48	in	in	ADP
cana-527	273	49	banach	banach	NOUN
cana-527	273	50	hyperspace	hyperspace	NOUN
cana-527	273	51	endowed	endow	VERB
cana-527	273	52	with	with	ADP
cana-527	273	53	a	a	DET
cana-527	273	54	graph	graph	NOUN
cana-527	273	55	by	by	ADP
cana-527	273	56	utilising	utilise	VERB
cana-527	273	57	the	the	DET
cana-527	273	58	idea	idea	NOUN
cana-527	273	59	of	of	ADP
cana-527	273	60	f	f	NOUN
cana-527	273	61	-	-	PUNCT
cana-527	273	62	contraction	contraction	NOUN
cana-527	274	1	[	[	X
cana-527	274	2	3][4][5][8][13][19][21	3][4][5][8][13][19][21	X
cana-527	274	3	]	]	X
cana-527	274	4	.	.	PUNCT
cana-527	275	1	a	a	DET
cana-527	275	2	particular	particular	ADJ
cana-527	275	3	kind	kind	NOUN
cana-527	275	4	of	of	ADP
cana-527	275	5	contraction	contraction	NOUN
cana-527	275	6	mapping	mapping	NOUN
cana-527	275	7	that	that	PRON
cana-527	275	8	is	be	AUX
cana-527	275	9	defined	define	VERB
cana-527	275	10	in	in	ADP
cana-527	275	11	the	the	DET
cana-527	275	12	context	context	NOUN
cana-527	275	13	of	of	ADP
cana-527	275	14	the	the	DET
cana-527	275	15	investigation	investigation	NOUN
cana-527	275	16	is	be	AUX
cana-527	275	17	referred	refer	VERB
cana-527	275	18	to	to	ADP
cana-527	275	19	as	as	ADP
cana-527	275	20	a	a	DET
cana-527	275	21	f	f	NOUN
cana-527	275	22	-	-	PUNCT
cana-527	275	23	contraction	contraction	NOUN
cana-527	275	24	.	.	PUNCT
cana-527	276	1	regarding	regard	VERB
cana-527	276	2	a	a	DET
cana-527	276	3	specific	specific	ADJ
cana-527	276	4	set	set	NOUN
cana-527	276	5	or	or	CCONJ
cana-527	276	6	function	function	NOUN
cana-527	276	7	class	class	NOUN
cana-527	276	8	indicated	indicate	VERB
cana-527	276	9	by	by	ADP
cana-527	276	10	f	f	PROPN
cana-527	276	11	,	,	PUNCT
cana-527	276	12	it	it	PRON
cana-527	276	13	suggests	suggest	VERB
cana-527	276	14	a	a	DET
cana-527	276	15	contractive	contractive	ADJ
cana-527	276	16	feature	feature	NOUN
cana-527	276	17	.	.	PUNCT
cana-527	277	1	a	a	DET
cana-527	277	2	measure	measure	NOUN
cana-527	277	3	of	of	ADP
cana-527	277	4	the	the	DET
cana-527	277	5	”	"	PUNCT
cana-527	277	6	nearness	nearness	NOUN
cana-527	277	7	”	"	PUNCT
cana-527	277	8	between	between	ADP
cana-527	277	9	the	the	DET
cana-527	277	10	pictures	picture	NOUN
cana-527	277	11	of	of	ADP
cana-527	277	12	distinct	distinct	ADJ
cana-527	277	13	locations	location	NOUN
cana-527	277	14	is	be	AUX
cana-527	277	15	provided	provide	VERB
cana-527	277	16	by	by	ADP
cana-527	277	17	the	the	DET
cana-527	277	18	contraction	contraction	NOUN
cana-527	277	19	property	property	NOUN
cana-527	277	20	that	that	PRON
cana-527	277	21	the	the	DET
cana-527	277	22	mappings	mapping	NOUN
cana-527	277	23	under	under	ADP
cana-527	277	24	consideration	consideration	NOUN
cana-527	277	25	under	under	ADP
cana-527	277	26	f	f	NOUN
cana-527	277	27	-	-	PUNCT
cana-527	277	28	contractions	contraction	NOUN
cana-527	277	29	display	display	NOUN
cana-527	277	30	inside	inside	ADP
cana-527	277	31	the	the	DET
cana-527	277	32	designated	designate	VERB
cana-527	277	33	function	function	NOUN
cana-527	277	34	class	class	NOUN
cana-527	277	35	.	.	PUNCT
cana-527	278	1	a	a	DET
cana-527	278	2	graph	graph	NOUN
cana-527	278	3	is	be	AUX
cana-527	278	4	present	present	ADJ
cana-527	278	5	in	in	ADP
cana-527	278	6	the	the	DET
cana-527	278	7	banach	banach	ADJ
cana-527	278	8	hyperspace	hyperspace	NOUN
cana-527	278	9	,	,	PUNCT
cana-527	278	10	indicating	indicate	VERB
cana-527	278	11	a	a	DET
cana-527	278	12	visual	visual	ADJ
cana-527	278	13	depiction	depiction	NOUN
cana-527	278	14	of	of	ADP
cana-527	278	15	connections	connection	NOUN
cana-527	278	16	among	among	ADP
cana-527	278	17	compact	compact	ADJ
cana-527	278	18	sets	set	NOUN
cana-527	278	19	.	.	PUNCT
cana-527	279	1	this	this	DET
cana-527	279	2	incorporation	incorporation	NOUN
cana-527	279	3	of	of	ADP
cana-527	279	4	graph	graph	NOUN
cana-527	279	5	theory	theory	NOUN
cana-527	279	6	into	into	ADP
cana-527	279	7	the	the	DET
cana-527	279	8	context	context	NOUN
cana-527	279	9	of	of	ADP
cana-527	279	10	banach	banach	ADJ
cana-527	279	11	hyperspace	hyperspace	NOUN
cana-527	279	12	probably	probably	ADV
cana-527	279	13	offers	offer	VERB
cana-527	279	14	a	a	DET
cana-527	279	15	more	more	ADV
cana-527	279	16	illustrative	illustrative	ADJ
cana-527	279	17	and	and	CCONJ
cana-527	279	18	possibly	possibly	ADV
cana-527	279	19	enlightening	enlighten	VERB
cana-527	279	20	viewpoint	viewpoint	NOUN
cana-527	279	21	[	[	X
cana-527	279	22	6	6	NUM
cana-527	279	23	]	]	PUNCT
cana-527	279	24	.	.	PUNCT
cana-527	280	1	theorem	theorem	ADJ
cana-527	280	2	3.2.1	3.2.1	NUM
cana-527	280	3	:	:	PUNCT
cana-527	280	4	suppose	suppose	VERB
cana-527	280	5	(	(	PUNCT
cana-527	280	6	k(s	k(s	PROPN
cana-527	280	7	)	)	PUNCT
cana-527	280	8	,	,	PUNCT
cana-527	280	9	d	d	X
cana-527	280	10	,	,	PUNCT
cana-527	280	11	h	h	NOUN
cana-527	280	12	)	)	PUNCT
cana-527	280	13	be	be	VERB
cana-527	280	14	a	a	DET
cana-527	280	15	bhs	bhs	PROPN
cana-527	280	16	with	with	ADP
cana-527	280	17	a	a	DET
cana-527	280	18	weakly	weakly	ADV
cana-527	280	19	connected	connected	ADJ
cana-527	280	20	and	and	CCONJ
cana-527	280	21	directed	direct	VERB
cana-527	280	22	graph	graph	NOUN
cana-527	280	23	h	h	NOUN
cana-527	280	24	holds	hold	VERB
cana-527	280	25	the	the	DET
cana-527	280	26	following	follow	VERB
cana-527	280	27	property	property	NOUN
cana-527	280	28	,	,	PUNCT
cana-527	280	29	for	for	ADP
cana-527	280	30	any	any	DET
cana-527	280	31	sequence	sequence	NOUN
cana-527	280	32	{	{	PUNCT
cana-527	280	33	sn}∞n=1	sn}∞n=1	X
cana-527	280	34	⊂	⊂	X
cana-527	280	35	k(x	k(x	PROPN
cana-527	280	36	)	)	PUNCT
cana-527	280	37	with	with	ADP
cana-527	280	38	sn	sn	PROPN
cana-527	280	39	→	→	SYM
cana-527	280	40	s	s	NOUN
cana-527	280	41	as	as	ADP
cana-527	280	42	n	n	PROPN
cana-527	280	43	→	→	SYM
cana-527	280	44	∞	∞	NUM
cana-527	280	45	and	and	CCONJ
cana-527	280	46	(	(	PUNCT
cana-527	280	47	sn	sn	PROPN
cana-527	280	48	,	,	PUNCT
cana-527	280	49	sn+1	sn+1	X
cana-527	280	50	)	)	PUNCT
cana-527	280	51	∈	∈	PROPN
cana-527	280	52	e(h	e(h	PROPN
cana-527	280	53	)	)	PUNCT
cana-527	280	54	,	,	PUNCT
cana-527	280	55	∀n	∀n	NUM
cana-527	280	56	∈	∈	PROPN
cana-527	280	57	n	n	CCONJ
cana-527	280	58	,	,	PUNCT
cana-527	280	59	there	there	PRON
cana-527	280	60	exist	exist	VERB
cana-527	280	61	a	a	DET
cana-527	280	62	subsequence	subsequence	NOUN
cana-527	280	63	{	{	PUNCT
cana-527	280	64	s\	s\	NOUN
cana-527	280	65	}	}	PUNCT
cana-527	280	66	n	n	NOUN
cana-527	280	67	∞	∞	NUM
cana-527	280	68	=	=	SYM
cana-527	280	69	1	1	NUM
cana-527	280	70	,	,	PUNCT
cana-527	280	71	satisfying	satisfy	VERB
cana-527	280	72	(	(	PUNCT
cana-527	280	73	smn	smn	PROPN
cana-527	280	74	,	,	PUNCT
cana-527	280	75	s	s	NOUN
cana-527	280	76	)	)	PUNCT
cana-527	280	77	∈	∈	PROPN
cana-527	280	78	e(g	e(g	PROPN
cana-527	280	79	)	)	PUNCT
cana-527	280	80	)	)	PUNCT
cana-527	280	81	,	,	PUNCT
cana-527	280	82	∀n	∀n	X
cana-527	280	83	∈	∈	PROPN
cana-527	280	84	n	n	ADV
cana-527	280	85	let	let	VERB
cana-527	280	86	s	s	PRON
cana-527	280	87	:	:	PUNCT
cana-527	280	88	k(s	k(s	PROPN
cana-527	280	89	)	)	PUNCT
cana-527	281	1	→	→	SYM
cana-527	281	2	k(s	k(s	PROPN
cana-527	281	3	)	)	PUNCT
cana-527	281	4	be	be	AUX
cana-527	281	5	a	a	DET
cana-527	281	6	h	h	NOUN
cana-527	281	7	f	f	NOUN
cana-527	281	8	-	-	PUNCT
cana-527	281	9	contraction	contraction	NOUN
cana-527	281	10	,	,	PUNCT
cana-527	281	11	if	if	SCONJ
cana-527	281	12	the	the	DET
cana-527	281	13	set	set	NOUN
cana-527	281	14	sγ	sγ	X
cana-527	281	15	=	=	PUNCT
cana-527	281	16	{	{	PUNCT
cana-527	281	17	s	s	NOUN
cana-527	281	18	∈	∈	X
cana-527	281	19	k(s	k(s	PROPN
cana-527	281	20	)	)	PUNCT
cana-527	281	21	;	;	PUNCT
cana-527	281	22	(	(	PUNCT
cana-527	281	23	s	s	X
cana-527	281	24	,	,	PUNCT
cana-527	281	25	γ	γ	NOUN
cana-527	281	26	γl	γl	PROPN
cana-527	281	27	)	)	PUNCT
cana-527	281	28	∈	∈	PROPN
cana-527	281	29	e(h	e(h	PROPN
cana-527	281	30	)	)	PUNCT
cana-527	281	31	}	}	PUNCT
cana-527	281	32	is	be	AUX
cana-527	281	33	non	non	ADJ
cana-527	281	34	-	-	ADJ
cana-527	281	35	empty	empty	ADJ
cana-527	281	36	,	,	PUNCT
cana-527	281	37	then	then	ADV
cana-527	281	38	s	s	VERB
cana-527	281	39	has	have	VERB
cana-527	281	40	a	a	DET
cana-527	281	41	unique	unique	ADJ
cana-527	281	42	near	near	ADP
cana-527	281	43	fixed	fix	VERB
cana-527	281	44	point	point	NOUN
cana-527	281	45	in	in	ADP
cana-527	281	46	k(s	k(s	PROPN
cana-527	281	47	)	)	PUNCT
cana-527	281	48	.	.	PUNCT
cana-527	282	1	proof	proof	NOUN
cana-527	282	2	:	:	PUNCT
cana-527	282	3	let	let	VERB
cana-527	282	4	sy0	sy0	PROPN
cana-527	282	5	∈	∈	PROPN
cana-527	282	6	sγ	sγ	PROPN
cana-527	282	7	,	,	PUNCT
cana-527	282	8	therefore	therefore	ADV
cana-527	282	9	(	(	PUNCT
cana-527	282	10	sy0	sy0	PROPN
cana-527	282	11	,	,	PUNCT
cana-527	282	12	γsy0	γsy0	PROPN
cana-527	282	13	)	)	PUNCT
cana-527	282	14	∈	∈	PROPN
cana-527	282	15	e(h	e(h	PROPN
cana-527	282	16	)	)	PUNCT
cana-527	282	17	,	,	PUNCT
cana-527	282	18	we	we	PRON
cana-527	282	19	get	get	VERB
cana-527	282	20	(	(	PUNCT
cana-527	282	21	γnsy0	γnsy0	NOUN
cana-527	282	22	,	,	PUNCT
cana-527	282	23	γn+1sy0	γn+1sy0	NUM
cana-527	282	24	)	)	PUNCT
cana-527	282	25	∈	∈	PROPN
cana-527	282	26	e(h	e(h	PROPN
cana-527	282	27	)	)	PUNCT
cana-527	282	28	,	,	PUNCT
cana-527	282	29	∀n	∀n	NUM
cana-527	282	30	∈	∈	PROPN
cana-527	282	31	n	n	PRON
cana-527	282	32	denote	denote	VERB
cana-527	282	33	sn	sn	PROPN
cana-527	282	34	=	=	SYM
cana-527	282	35	γnsy0	γnsy0	NOUN
cana-527	282	36	,	,	PUNCT
cana-527	282	37	∀n	∀n	X
cana-527	282	38	∈	∈	PROPN
cana-527	282	39	n.	n.	NOUN
cana-527	282	40	by	by	ADP
cana-527	282	41	the	the	DET
cana-527	282	42	fact	fact	NOUN
cana-527	282	43	that	that	SCONJ
cana-527	282	44	γ	γ	PROPN
cana-527	282	45	is	be	AUX
cana-527	282	46	a	a	DET
cana-527	282	47	h	h	NOUN
cana-527	282	48	f	f	NOUN
cana-527	282	49	-	-	PUNCT
cana-527	282	50	contraction	contraction	NOUN
cana-527	282	51	and	and	CCONJ
cana-527	282	52	using	use	VERB
cana-527	282	53	self	self	NOUN
cana-527	282	54	fcontraction	fcontraction	NOUN
cana-527	282	55	case	case	NOUN
cana-527	282	56	,	,	PUNCT
cana-527	282	57	we	we	PRON
cana-527	282	58	get	get	VERB
cana-527	282	59	,	,	PUNCT
cana-527	282	60	f||sn	f||sn	NOUN
cana-527	282	61	⊖	⊖	NOUN
cana-527	282	62	sn+1||	sn+1||	VERB
cana-527	282	63	≤	≤	ADJ
cana-527	282	64	f||sn−1	f||sn−1	PROPN
cana-527	282	65	,	,	PUNCT
cana-527	282	66	sn||	sn||	ADV
cana-527	282	67	−	−	PROPN
cana-527	282	68	τ	τ	NOUN
cana-527	282	69	,	,	PUNCT
cana-527	282	70	∀	∀	X
cana-527	282	71	n	n	PRON
cana-527	282	72	∈	∈	NOUN
cana-527	282	73	n	n	PRON
cana-527	282	74	denote	denote	VERB
cana-527	282	75	βn	βn	NOUN
cana-527	282	76	=	=	SYM
cana-527	282	77	||sn	||sn	PROPN
cana-527	282	78	⊖	⊖	NUM
cana-527	282	79	sn+1||	sn+1||	NOUN
cana-527	282	80	,	,	PUNCT
cana-527	282	81	n	n	NOUN
cana-527	282	82	=	=	SYM
cana-527	282	83	0	0	NUM
cana-527	282	84	,	,	PUNCT
cana-527	282	85	1	1	NUM
cana-527	282	86	,	,	PUNCT
cana-527	282	87	2	2	NUM
cana-527	282	88	,	,	PUNCT
cana-527	282	89	.......	.......	PUNCT
cana-527	283	1	take	take	VERB
cana-527	283	2	sn+1	sn+1	VERB
cana-527	283	3	≠	≠	PROPN
cana-527	283	4	sn	sn	PROPN
cana-527	283	5	,	,	PUNCT
cana-527	283	6	∀n	∀n	X
cana-527	283	7	∈	∈	PROPN
cana-527	283	8	n	n	NOUN
cana-527	283	9	∪	∪	VERB
cana-527	283	10	0	0	NUM
cana-527	283	11	.	.	PUNCT
cana-527	284	1	then	then	ADV
cana-527	284	2	βn	βn	VERB
cana-527	284	3	>	>	X
cana-527	284	4	0	0	NUM
cana-527	284	5	,	,	PUNCT
cana-527	284	6	∀n	∀n	NUM
cana-527	284	7	∈	∈	NOUN
cana-527	284	8	n	n	NOUN
cana-527	284	9	∪	∪	VERB
cana-527	284	10	0	0	NUM
cana-527	284	11	and	and	CCONJ
cana-527	284	12	by	by	ADP
cana-527	284	13	using	use	VERB
cana-527	284	14	the	the	DET
cana-527	284	15	known	know	VERB
cana-527	284	16	result	result	NOUN
cana-527	284	17	,	,	PUNCT
cana-527	284	18	we	we	PRON
cana-527	284	19	get	get	VERB
cana-527	284	20	f(βn	f(βn	ADJ
cana-527	284	21	)	)	PUNCT
cana-527	284	22	=	=	SYM
cana-527	284	23	f(βn−1	f(βn−1	PROPN
cana-527	284	24	)	)	PUNCT
cana-527	284	25	⊖	⊖	NOUN
cana-527	284	26	τ	τ	PROPN
cana-527	284	27	≤	≤	NUM
cana-527	284	28	f(βn−2	f(βn−2	NOUN
cana-527	284	29	)	)	PUNCT
cana-527	284	30	−	−	PROPN
cana-527	284	31	2τ	2τ	NOUN
cana-527	284	32	≤	≤	NOUN
cana-527	284	33	......	......	PUNCT
cana-527	284	34	≤	≤	NUM
cana-527	284	35	f(β0	f(β0	NOUN
cana-527	284	36	)	)	PUNCT
cana-527	284	37	−	−	PROPN
cana-527	285	1	nτ	nτ	PROPN
cana-527	285	2	∴	∴	PROPN
cana-527	285	3	lim	lim	PROPN
cana-527	285	4	f(βn	f(βn	PROPN
cana-527	285	5	)	)	PUNCT
cana-527	285	6	=	=	SYM
cana-527	286	1	−∞	−∞	NOUN
cana-527	286	2	,	,	PUNCT
cana-527	286	3	obtain	obtain	VERB
cana-527	286	4	βn	βn	NOUN
cana-527	286	5	→	→	SYM
cana-527	286	6	0as	0as	NOUN
cana-527	286	7	n	n	NOUN
cana-527	286	8	→	→	SYM
cana-527	286	9	∞	∞	NUM
cana-527	286	10	there	there	PRON
cana-527	286	11	exist	exist	VERB
cana-527	286	12	m	m	VERB
cana-527	286	13	∈	∈	NOUN
cana-527	286	14	(	(	PUNCT
cana-527	286	15	0	0	NUM
cana-527	286	16	,	,	PUNCT
cana-527	286	17	1	1	NUM
cana-527	286	18	)	)	PUNCT
cana-527	286	19	such	such	ADJ
cana-527	286	20	that	that	SCONJ
cana-527	286	21	lim	lim	PROPN
cana-527	286	22	βkf(βn	βkf(βn	ADJ
cana-527	286	23	)	)	PUNCT
cana-527	286	24	=	=	SYM
cana-527	287	1	0	0	X
cana-527	287	2	.	.	NOUN
cana-527	287	3	n→∞	n→∞	PRON
cana-527	287	4	n	n	CCONJ
cana-527	287	5	βk	βk	ADP
cana-527	287	6	f(βn	f(βn	ADJ
cana-527	287	7	)	)	PUNCT
cana-527	288	1	⊖	⊖	NOUN
cana-527	288	2	βk	βk	ADP
cana-527	288	3	f(β0	f(β0	NOUN
cana-527	288	4	)	)	PUNCT
cana-527	288	5	≤	≤	NUM
cana-527	288	6	βk(f(β0	βk(f(β0	PROPN
cana-527	288	7	)	)	PUNCT
cana-527	288	8	⊖	⊖	AUX
cana-527	288	9	nτ	nτ	NOUN
cana-527	288	10	)	)	PUNCT
cana-527	288	11	⊖	⊖	NOUN
cana-527	288	12	βk	βk	ADP
cana-527	288	13	f(β0	f(β0	NOUN
cana-527	288	14	)	)	PUNCT
cana-527	289	1	=	=	SYM
cana-527	289	2	−βk	−βk	PROPN
cana-527	289	3	nτ	nτ	PROPN
cana-527	289	4	n	n	CCONJ
cana-527	289	5	n	n	ADV
cana-527	289	6	holds	hold	VERB
cana-527	289	7	for	for	ADP
cana-527	289	8	all	all	PRON
cana-527	289	9	n	n	PRON
cana-527	289	10	∈	∈	PROPN
cana-527	289	11	n.	n.	NOUN
cana-527	289	12	take	take	VERB
cana-527	289	13	n	n	NOUN
cana-527	289	14	→	→	SYM
cana-527	289	15	∞	∞	PROPN
cana-527	289	16	,	,	PUNCT
cana-527	289	17	lim	lim	PROPN
cana-527	289	18	nβk	nβk	ADV
cana-527	289	19	=	=	PROPN
cana-527	290	1	0	0	X
cana-527	290	2	.	.	NOUN
cana-527	290	3	n→∞	n→∞	PRON
cana-527	291	1	n	n	PRON
cana-527	291	2	observe	observe	VERB
cana-527	291	3	that	that	SCONJ
cana-527	291	4	there	there	PRON
cana-527	291	5	exist	exist	VERB
cana-527	291	6	n′	n′	PRON
cana-527	291	7	∈	∈	PROPN
cana-527	291	8	n	n	PRON
cana-527	291	9	such	such	ADJ
cana-527	291	10	that	that	SCONJ
cana-527	291	11	nβk	nβk	ADV
cana-527	291	12	≤	≤	ADJ
cana-527	291	13	1	1	NUM
cana-527	291	14	,	,	PUNCT
cana-527	291	15	∀n	∀n	NUM
cana-527	291	16	≥	≥	NOUN
cana-527	291	17	n′	n′	PROPN
cana-527	291	18	we	we	PRON
cana-527	291	19	have	have	VERB
cana-527	291	20	,	,	PUNCT
cana-527	291	21	β	β	X
cana-527	291	22	1	1	NUM
cana-527	291	23	,	,	PUNCT
cana-527	291	24	n	n	PRON
cana-527	291	25	n′	n′	PROPN
cana-527	291	26	n1	n1	PROPN
cana-527	291	27	/	/	SYM
cana-527	291	28	k	k	PROPN
cana-527	291	29	communications	communication	NOUN
cana-527	291	30	on	on	ADP
cana-527	291	31	applied	apply	VERB
cana-527	291	32	nonlinear	nonlinear	ADJ
cana-527	291	33	analysis	analysis	NOUN
cana-527	291	34	issn	issn	NOUN
cana-527	291	35	:	:	PUNCT
cana-527	291	36	1074	1074	NUM
cana-527	291	37	-	-	PUNCT
cana-527	291	38	133x	133x	NUM
cana-527	291	39	vol	vol	NOUN
cana-527	291	40	31	31	NUM
cana-527	291	41	no	no	NOUN
cana-527	291	42	.	.	NOUN
cana-527	291	43	2	2	NUM
cana-527	291	44	(	(	PUNCT
cana-527	291	45	2024	2024	NUM
cana-527	291	46	)	)	PUNCT
cana-527	291	47	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	291	48	170	170	NUM
cana-527	291	49	−	−	PROPN
cana-527	291	50	j1	j1	PROPN
cana-527	291	51	/	/	SYM
cana-527	291	52	k	k	PROPN
cana-527	291	53	choose	choose	PROPN
cana-527	291	54	l	l	PROPN
cana-527	291	55	,	,	PUNCT
cana-527	291	56	n	n	PROPN
cana-527	291	57	∈	∈	PROPN
cana-527	291	58	n	n	CCONJ
cana-527	291	59	,	,	PUNCT
cana-527	291	60	such	such	ADJ
cana-527	291	61	that	that	SCONJ
cana-527	291	62	l	l	PROPN
cana-527	291	63	≥	≥	X
cana-527	291	64	n	n	PRON
cana-527	291	65	≥	≥	NOUN
cana-527	291	66	n′	n′	PROPN
cana-527	291	67	,	,	PUNCT
cana-527	291	68	we	we	PRON
cana-527	291	69	get	get	VERB
cana-527	291	70	∞	∞	PROPN
cana-527	291	71	∞	∞	PRON
cana-527	291	72	||sl	||sl	PROPN
cana-527	291	73	⊖	⊖	NOUN
cana-527	291	74	sn||	sn||	ADV
cana-527	291	75	≤	≤	NUM
cana-527	291	76	βl	βl	ADP
cana-527	291	77	1	1	NUM
cana-527	291	78	+	+	CCONJ
cana-527	291	79	.	.	PUNCT
cana-527	291	80	.	.	PUNCT
cana-527	291	81	.	.	PUNCT
cana-527	292	1	βn	βn	PROPN
cana-527	293	1	x	x	SYM
cana-527	293	2	βn	βn	VERB
cana-527	293	3	≤	≤	NUM
cana-527	293	4	x	x	SYM
cana-527	293	5	1	1	NUM
cana-527	293	6	the	the	DET
cana-527	293	7	convergence	convergence	NOUN
cana-527	293	8	of	of	ADP
cana-527	293	9	the	the	DET
cana-527	293	10	above	above	ADJ
cana-527	293	11	series	series	NOUN
cana-527	293	12	that	that	SCONJ
cana-527	293	13	{	{	PUNCT
cana-527	293	14	sn	sn	NOUN
cana-527	293	15	}	}	PUNCT
cana-527	293	16	is	be	AUX
cana-527	293	17	a	a	DET
cana-527	293	18	cauchy	cauchy	ADJ
cana-527	293	19	sequence	sequence	NOUN
cana-527	293	20	,	,	PUNCT
cana-527	293	21	it	it	PRON
cana-527	293	22	is	be	AUX
cana-527	293	23	convergent	convergent	NOUN
cana-527	293	24	in	in	ADP
cana-527	293	25	(	(	PUNCT
cana-527	293	26	k(s	k(s	PROPN
cana-527	293	27	)	)	PUNCT
cana-527	293	28	,	,	PUNCT
cana-527	293	29	d	d	X
cana-527	293	30	,	,	PUNCT
cana-527	293	31	h	h	NOUN
cana-527	293	32	)	)	PUNCT
cana-527	293	33	.	.	PUNCT
cana-527	294	1	∴	∴	PROPN
cana-527	294	2	lim	lim	PROPN
cana-527	294	3	sn	sn	PROPN
cana-527	294	4	=	=	PRON
cana-527	294	5	s∗	s∗	VERB
cana-527	294	6	n→∞	n→∞	X
cana-527	294	7	the	the	DET
cana-527	294	8	subsequence	subsequence	NOUN
cana-527	294	9	{	{	PUNCT
cana-527	294	10	smn	smn	NOUN
cana-527	294	11	}	}	PUNCT
cana-527	294	12	satisfying	satisfy	VERB
cana-527	294	13	(	(	PUNCT
cana-527	294	14	smn	smn	PROPN
cana-527	294	15	,	,	PUNCT
cana-527	294	16	s∗	s∗	PROPN
cana-527	294	17	)	)	PUNCT
cana-527	294	18	∈	∈	PROPN
cana-527	294	19	e(h	e(h	PROPN
cana-527	294	20	)	)	PUNCT
cana-527	294	21	,	,	PUNCT
cana-527	294	22	∀n	∀n	NUM
cana-527	294	23	∈	∈	PROPN
cana-527	294	24	n	n	CCONJ
cana-527	294	25	,	,	PUNCT
cana-527	294	26	we	we	PRON
cana-527	294	27	get	get	VERB
cana-527	294	28	f||γsmn	f||γsmn	NOUN
cana-527	294	29	⊖	⊖	NUM
cana-527	294	30	γs∗||	γs∗||	NOUN
cana-527	294	31	≤	≤	PROPN
cana-527	294	32	f||smn	f||smn	PROPN
cana-527	294	33	⊖	⊖	NOUN
cana-527	294	34	s∗||	s∗||	PUNCT
cana-527	294	35	−	−	PROPN
cana-527	295	1	τ	τ	X
cana-527	295	2	<	<	X
cana-527	295	3	f||γmn	f||γmn	PROPN
cana-527	295	4	⊖	⊖	NOUN
cana-527	295	5	s∗||	s∗||	PROPN
cana-527	295	6	||γsmn	||γsmn	VERB
cana-527	295	7	⊖	⊖	NUM
cana-527	295	8	γs∗||	γs∗||	NOUN
cana-527	295	9	≤	≤	NUM
cana-527	295	10	||smn	||smn	ADJ
cana-527	295	11	⊖	⊖	X
cana-527	295	12	s∗||	s∗||	NUM
cana-527	295	13	by	by	ADP
cana-527	295	14	triangle	triangle	NOUN
cana-527	295	15	inequality	inequality	NOUN
cana-527	295	16	,	,	PUNCT
cana-527	295	17	we	we	PRON
cana-527	295	18	have	have	VERB
cana-527	295	19	||s∗	||s∗	NOUN
cana-527	295	20	⊖	⊖	NOUN
cana-527	295	21	γs∗||	γs∗||	X
cana-527	295	22	=	=	X
cana-527	295	23	||s∗	||s∗	NOUN
cana-527	295	24	⊖	⊖	AUX
cana-527	295	25	smn	smn	PROPN
cana-527	296	1	||	||	PROPN
cana-527	296	2	⊕	⊕	PROPN
cana-527	296	3	||γsmn	||γsmn	PROPN
cana-527	296	4	⊖	⊖	X
cana-527	296	5	γs∗||	γs∗||	NOUN
cana-527	296	6	,	,	PUNCT
cana-527	296	7	∀	∀	X
cana-527	296	8	n	n	PRON
cana-527	296	9	≥	≥	NOUN
cana-527	296	10	1	1	NUM
cana-527	296	11	assuming	assume	VERB
cana-527	296	12	n	n	X
cana-527	296	13	→	→	SYM
cana-527	296	14	∞	∞	NUM
cana-527	296	15	and	and	CCONJ
cana-527	296	16	using	use	VERB
cana-527	296	17	the	the	DET
cana-527	296	18	above	above	ADJ
cana-527	296	19	results	result	NOUN
cana-527	296	20	we	we	PRON
cana-527	296	21	get	get	VERB
cana-527	296	22	,	,	PUNCT
cana-527	296	23	||s∗	||s∗	NOUN
cana-527	296	24	,	,	PUNCT
cana-527	296	25	γs∗||	γs∗||	X
cana-527	296	26	=	=	NOUN
cana-527	296	27	0	0	X
cana-527	296	28	.	.	PUNCT
cana-527	297	1	⇒	⇒	PROPN
cana-527	297	2	s∗	s∗	PROPN
cana-527	297	3	=	=	PUNCT
cana-527	297	4	γs∗	γs∗	ADV
cana-527	297	5	⇒	⇒	PROPN
cana-527	297	6	s∗	s∗	PROPN
cana-527	297	7	is	be	AUX
cana-527	297	8	a	a	DET
cana-527	297	9	fixed	fix	VERB
cana-527	297	10	point	point	NOUN
cana-527	297	11	of	of	ADP
cana-527	297	12	γ	γ	NOUN
cana-527	297	13	we	we	PRON
cana-527	297	14	can	can	AUX
cana-527	297	15	extend	extend	VERB
cana-527	297	16	this	this	PRON
cana-527	297	17	to	to	ADP
cana-527	297	18	non	non	ADJ
cana-527	297	19	-	-	ADJ
cana-527	297	20	self	self	NOUN
cana-527	297	21	-	-	PUNCT
cana-527	297	22	f	f	NOUN
cana-527	297	23	-	-	PUNCT
cana-527	297	24	contraction	contraction	NOUN
cana-527	297	25	also	also	ADV
cana-527	297	26	.	.	PUNCT
cana-527	298	1	iv	iv	X
cana-527	298	2	conclusion	conclusion	NOUN
cana-527	298	3	the	the	DET
cana-527	298	4	detailed	detailed	ADJ
cana-527	298	5	research	research	NOUN
cana-527	298	6	seems	seem	VERB
cana-527	298	7	to	to	PART
cana-527	298	8	present	present	VERB
cana-527	298	9	a	a	DET
cana-527	298	10	novel	novel	ADJ
cana-527	298	11	way	way	NOUN
cana-527	298	12	of	of	ADP
cana-527	298	13	conceptualising	conceptualise	VERB
cana-527	298	14	banach	banach	NOUN
cana-527	298	15	hy	hy	NOUN
cana-527	298	16	perspace	perspace	NOUN
cana-527	298	17	through	through	ADP
cana-527	298	18	graph	graph	NOUN
cana-527	298	19	associations	association	NOUN
cana-527	298	20	.	.	PUNCT
cana-527	299	1	furthermore	furthermore	ADV
cana-527	299	2	,	,	PUNCT
cana-527	299	3	the	the	DET
cana-527	299	4	development	development	NOUN
cana-527	299	5	of	of	ADP
cana-527	299	6	the	the	DET
cana-527	299	7	w	w	NOUN
cana-527	299	8	-	-	PUNCT
cana-527	299	9	sequence	sequence	NOUN
cana-527	299	10	is	be	AUX
cana-527	299	11	discussed	discuss	VERB
cana-527	299	12	,	,	PUNCT
cana-527	299	13	which	which	PRON
cana-527	299	14	quantifies	quantify	VERB
cana-527	299	15	the	the	DET
cana-527	299	16	edge	edge	NOUN
cana-527	299	17	intensities	intensity	NOUN
cana-527	299	18	in	in	ADP
cana-527	299	19	the	the	DET
cana-527	299	20	graph	graph	NOUN
cana-527	299	21	.	.	PUNCT
cana-527	300	1	it	it	PRON
cana-527	300	2	appears	appear	VERB
cana-527	300	3	that	that	SCONJ
cana-527	300	4	this	this	DET
cana-527	300	5	se	se	PROPN
cana-527	300	6	quence	quence	NOUN
cana-527	300	7	is	be	AUX
cana-527	300	8	essential	essential	ADJ
cana-527	300	9	to	to	ADP
cana-527	300	10	showing	show	VERB
cana-527	300	11	how	how	SCONJ
cana-527	300	12	a	a	DET
cana-527	300	13	set	set	NOUN
cana-527	300	14	of	of	ADP
cana-527	300	15	iterated	iterated	ADJ
cana-527	300	16	functions	function	NOUN
cana-527	300	17	converges	converge	NOUN
cana-527	300	18	to	to	ADP
cana-527	300	19	a	a	DET
cana-527	300	20	cauchy	cauchy	ADJ
cana-527	300	21	sequence	sequence	NOUN
cana-527	300	22	.	.	PUNCT
cana-527	301	1	moreover	moreover	ADV
cana-527	301	2	,	,	PUNCT
cana-527	301	3	the	the	DET
cana-527	301	4	approach	approach	NOUN
cana-527	301	5	is	be	AUX
cana-527	301	6	said	say	VERB
cana-527	301	7	to	to	PART
cana-527	301	8	be	be	AUX
cana-527	301	9	useful	useful	ADJ
cana-527	301	10	in	in	ADP
cana-527	301	11	illustrating	illustrate	VERB
cana-527	301	12	different	different	ADJ
cana-527	301	13	contrac	contrac	ADJ
cana-527	301	14	tion	tion	NOUN
cana-527	301	15	concepts	concept	NOUN
cana-527	301	16	.	.	PUNCT
cana-527	302	1	below	below	ADV
cana-527	302	2	is	be	AUX
cana-527	302	3	a	a	DET
cana-527	302	4	summary	summary	NOUN
cana-527	302	5	of	of	ADP
cana-527	302	6	the	the	DET
cana-527	302	7	essential	essential	ADJ
cana-527	302	8	components	component	NOUN
cana-527	302	9	:	:	PUNCT
cana-527	302	10	banach	banach	NOUN
cana-527	302	11	hyperspace	hyperspace	NOUN
cana-527	302	12	graphs	graph	NOUN
cana-527	302	13	:	:	PUNCT
cana-527	302	14	graphs	graph	NOUN
cana-527	302	15	are	be	AUX
cana-527	302	16	associated	associate	VERB
cana-527	302	17	with	with	ADP
cana-527	302	18	the	the	DET
cana-527	302	19	banach	banach	ADJ
cana-527	302	20	hyperspace	hyperspace	NOUN
cana-527	302	21	,	,	PUNCT
cana-527	302	22	indicating	indicate	VERB
cana-527	302	23	that	that	SCONJ
cana-527	302	24	the	the	DET
cana-527	302	25	structure	structure	NOUN
cana-527	302	26	and	and	CCONJ
cana-527	302	27	connections	connection	NOUN
cana-527	302	28	among	among	ADP
cana-527	302	29	compact	compact	ADJ
cana-527	302	30	sets	set	NOUN
cana-527	302	31	are	be	AUX
cana-527	302	32	being	be	AUX
cana-527	302	33	portrayed	portray	VERB
cana-527	302	34	visually	visually	ADV
cana-527	302	35	.	.	PUNCT
cana-527	303	1	the	the	DET
cana-527	303	2	inquiry	inquiry	NOUN
cana-527	303	3	may	may	AUX
cana-527	303	4	re	re	AUX
cana-527	303	5	volve	volve	VERB
cana-527	303	6	around	around	ADP
cana-527	303	7	the	the	DET
cana-527	303	8	nature	nature	NOUN
cana-527	303	9	of	of	ADP
cana-527	303	10	these	these	DET
cana-527	303	11	graphs	graph	NOUN
cana-527	303	12	and	and	CCONJ
cana-527	303	13	how	how	SCONJ
cana-527	303	14	they	they	PRON
cana-527	303	15	represent	represent	VERB
cana-527	303	16	the	the	DET
cana-527	303	17	banach	banach	ADJ
cana-527	303	18	hyperspace	hyperspace	NOUN
cana-527	303	19	.	.	PUNCT
cana-527	304	1	w	w	NOUN
cana-527	304	2	-	-	PUNCT
cana-527	304	3	sequence	sequence	NOUN
cana-527	304	4	:	:	PUNCT
cana-527	304	5	the	the	DET
cana-527	304	6	w	w	NOUN
cana-527	304	7	-	-	PUNCT
cana-527	304	8	sequence	sequence	NOUN
cana-527	304	9	appears	appear	VERB
cana-527	304	10	to	to	PART
cana-527	304	11	be	be	AUX
cana-527	304	12	a	a	DET
cana-527	304	13	novel	novel	ADJ
cana-527	304	14	concept	concept	NOUN
cana-527	304	15	introduced	introduce	VERB
cana-527	304	16	to	to	ADP
cana-527	304	17	mea	mea	VERB
cana-527	304	18	sure	sure	ADJ
cana-527	304	19	the	the	DET
cana-527	304	20	intensities	intensity	NOUN
cana-527	304	21	of	of	ADP
cana-527	304	22	the	the	DET
cana-527	304	23	edges	edge	NOUN
cana-527	304	24	in	in	ADP
cana-527	304	25	the	the	DET
cana-527	304	26	associated	associated	ADJ
cana-527	304	27	graph	graph	NOUN
cana-527	304	28	.	.	PUNCT
cana-527	305	1	it	it	PRON
cana-527	305	2	’s	’	VERB
cana-527	305	3	likely	likely	ADJ
cana-527	305	4	that	that	SCONJ
cana-527	305	5	the	the	DET
cana-527	305	6	w	w	NOUN
cana-527	305	7	-	-	PUNCT
cana-527	305	8	sequence	sequence	NOUN
cana-527	305	9	plays	play	VERB
cana-527	305	10	a	a	DET
cana-527	305	11	significant	significant	ADJ
cana-527	305	12	role	role	NOUN
cana-527	305	13	in	in	ADP
cana-527	305	14	quantifying	quantify	VERB
cana-527	305	15	the	the	DET
cana-527	305	16	relationships	relationship	NOUN
cana-527	305	17	or	or	CCONJ
cana-527	305	18	properties	property	NOUN
cana-527	305	19	of	of	ADP
cana-527	305	20	the	the	DET
cana-527	305	21	sets	set	NOUN
cana-527	305	22	in	in	ADP
cana-527	305	23	the	the	DET
cana-527	305	24	hyperspace	hyperspace	NOUN
cana-527	305	25	.	.	PUNCT
cana-527	306	1	convergence	convergence	NOUN
cana-527	306	2	of	of	ADP
cana-527	306	3	iterated	iterated	ADJ
cana-527	306	4	functions	function	NOUN
cana-527	306	5	:	:	PUNCT
cana-527	306	6	the	the	DET
cana-527	306	7	investigation	investigation	NOUN
cana-527	306	8	demonstrates	demonstrate	VERB
cana-527	306	9	that	that	SCONJ
cana-527	306	10	a	a	DET
cana-527	306	11	series	series	NOUN
cana-527	306	12	of	of	ADP
cana-527	306	13	iterated	iterated	ADJ
cana-527	306	14	functions	function	NOUN
cana-527	306	15	form	form	VERB
cana-527	306	16	a	a	DET
cana-527	306	17	cauchy	cauchy	ADJ
cana-527	306	18	sequence	sequence	NOUN
cana-527	306	19	,	,	PUNCT
cana-527	306	20	and	and	CCONJ
cana-527	306	21	this	this	PRON
cana-527	306	22	is	be	AUX
cana-527	306	23	done	do	VERB
cana-527	306	24	using	use	VERB
cana-527	306	25	the	the	DET
cana-527	306	26	w	w	NOUN
cana-527	306	27	-	-	NOUN
cana-527	306	28	sequence	sequence	NOUN
cana-527	306	29	.	.	PUNCT
cana-527	307	1	j	j	X
cana-527	308	1	=	=	PROPN
cana-527	308	2	n	n	PROPN
cana-527	308	3	j	j	NOUN
cana-527	308	4	=	=	PROPN
cana-527	308	5	n	n	PROPN
cana-527	308	6	communications	communication	NOUN
cana-527	308	7	on	on	ADP
cana-527	308	8	applied	apply	VERB
cana-527	308	9	nonlinear	nonlinear	ADJ
cana-527	308	10	analysis	analysis	NOUN
cana-527	308	11	issn	issn	NOUN
cana-527	308	12	:	:	PUNCT
cana-527	308	13	1074	1074	NUM
cana-527	308	14	-	-	PUNCT
cana-527	308	15	133x	133x	NUM
cana-527	308	16	vol	vol	NOUN
cana-527	308	17	31	31	NUM
cana-527	308	18	no	no	NOUN
cana-527	308	19	.	.	NOUN
cana-527	308	20	2	2	NUM
cana-527	308	21	(	(	PUNCT
cana-527	308	22	2024	2024	NUM
cana-527	308	23	)	)	PUNCT
cana-527	309	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	309	2	171	171	NUM
cana-527	309	3	.	.	PUNCT
cana-527	310	1	this	this	PRON
cana-527	310	2	suggests	suggest	VERB
cana-527	310	3	a	a	DET
cana-527	310	4	connection	connection	NOUN
cana-527	310	5	between	between	ADP
cana-527	310	6	the	the	DET
cana-527	310	7	properties	property	NOUN
cana-527	310	8	of	of	ADP
cana-527	310	9	the	the	DET
cana-527	310	10	associated	associated	ADJ
cana-527	310	11	graph	graph	NOUN
cana-527	310	12	and	and	CCONJ
cana-527	310	13	the	the	DET
cana-527	310	14	con	con	PROPN
cana-527	310	15	vergence	vergence	PROPN
cana-527	310	16	behavior	behavior	NOUN
cana-527	310	17	of	of	ADP
cana-527	310	18	the	the	DET
cana-527	310	19	iterated	iterated	ADJ
cana-527	310	20	functions	function	NOUN
cana-527	310	21	.	.	PUNCT
cana-527	311	1	application	application	NOUN
cana-527	311	2	to	to	ADP
cana-527	311	3	contraction	contraction	NOUN
cana-527	311	4	principles	principle	NOUN
cana-527	311	5	:	:	PUNCT
cana-527	311	6	the	the	DET
cana-527	311	7	method	method	NOUN
cana-527	311	8	is	be	AUX
cana-527	311	9	claimed	claim	VERB
cana-527	311	10	to	to	PART
cana-527	311	11	be	be	AUX
cana-527	311	12	applicable	applicable	ADJ
cana-527	311	13	in	in	ADP
cana-527	311	14	demonstrating	demonstrate	VERB
cana-527	311	15	various	various	ADJ
cana-527	311	16	contraction	contraction	NOUN
cana-527	311	17	principles	principle	NOUN
cana-527	311	18	.	.	PUNCT
cana-527	312	1	this	this	PRON
cana-527	312	2	indicates	indicate	VERB
cana-527	312	3	that	that	SCONJ
cana-527	312	4	the	the	DET
cana-527	312	5	insights	insight	NOUN
cana-527	312	6	gained	gain	VERB
cana-527	312	7	from	from	ADP
cana-527	312	8	the	the	DET
cana-527	312	9	w	w	NOUN
cana-527	312	10	-	-	PUNCT
cana-527	312	11	sequence	sequence	NOUN
cana-527	312	12	and	and	CCONJ
cana-527	312	13	the	the	DET
cana-527	312	14	associated	associated	ADJ
cana-527	312	15	graph	graph	NOUN
cana-527	312	16	are	be	AUX
cana-527	312	17	useful	useful	ADJ
cana-527	312	18	in	in	ADP
cana-527	312	19	proving	prove	VERB
cana-527	312	20	contraction	contraction	NOUN
cana-527	312	21	proper	proper	ADJ
cana-527	312	22	ties	tie	NOUN
cana-527	312	23	,	,	PUNCT
cana-527	312	24	potentially	potentially	ADV
cana-527	312	25	extending	extend	VERB
cana-527	312	26	the	the	DET
cana-527	312	27	classical	classical	ADJ
cana-527	312	28	banach	banach	NOUN
cana-527	312	29	contraction	contraction	NOUN
cana-527	312	30	principle	principle	NOUN
cana-527	312	31	.	.	PUNCT
cana-527	313	1	overall	overall	ADV
cana-527	313	2	,	,	PUNCT
cana-527	313	3	the	the	DET
cana-527	313	4	investigation	investigation	NOUN
cana-527	313	5	seems	seem	VERB
cana-527	313	6	to	to	PART
cana-527	313	7	introduce	introduce	VERB
cana-527	313	8	a	a	DET
cana-527	313	9	new	new	ADJ
cana-527	313	10	perspective	perspective	NOUN
cana-527	313	11	by	by	ADP
cana-527	313	12	incorporating	incorporate	VERB
cana-527	313	13	graph	graph	NOUN
cana-527	313	14	theory	theory	NOUN
cana-527	313	15	concepts	concept	NOUN
cana-527	313	16	,	,	PUNCT
cana-527	313	17	w	w	NOUN
cana-527	313	18	-	-	PUNCT
cana-527	313	19	sequences	sequence	NOUN
cana-527	313	20	,	,	PUNCT
cana-527	313	21	and	and	CCONJ
cana-527	313	22	iterated	iterated	ADJ
cana-527	313	23	functions	function	NOUN
cana-527	313	24	to	to	PART
cana-527	313	25	study	study	VERB
cana-527	313	26	the	the	DET
cana-527	313	27	properties	property	NOUN
cana-527	313	28	of	of	ADP
cana-527	313	29	banach	banach	ADJ
cana-527	313	30	hyperspace	hyperspace	NOUN
cana-527	313	31	.	.	PUNCT
cana-527	314	1	it	it	PRON
cana-527	314	2	’s	’	VERB
cana-527	314	3	an	an	DET
cana-527	314	4	interesting	interesting	ADJ
cana-527	314	5	integration	integration	NOUN
cana-527	314	6	of	of	ADP
cana-527	314	7	different	different	ADJ
cana-527	314	8	mathematical	mathematical	ADJ
cana-527	314	9	ideas	idea	NOUN
cana-527	314	10	to	to	ADP
cana-527	314	11	ex	ex	PROPN
cana-527	314	12	plore	plore	NOUN
cana-527	314	13	the	the	DET
cana-527	314	14	convergence	convergence	NOUN
cana-527	314	15	and	and	CCONJ
cana-527	314	16	contraction	contraction	NOUN
cana-527	314	17	properties	property	NOUN
cana-527	314	18	of	of	ADP
cana-527	314	19	mappings	mapping	NOUN
cana-527	314	20	in	in	ADP
cana-527	314	21	this	this	DET
cana-527	314	22	particular	particular	ADJ
cana-527	314	23	setting	setting	NOUN
cana-527	314	24	.	.	PUNCT
cana-527	315	1	finding	find	VERB
cana-527	315	2	:	:	PUNCT
cana-527	315	3	no	no	DET
cana-527	315	4	external	external	ADJ
cana-527	315	5	findings	finding	NOUN
cana-527	315	6	were	be	AUX
cana-527	315	7	obtained	obtain	VERB
cana-527	315	8	for	for	ADP
cana-527	315	9	this	this	DET
cana-527	315	10	study	study	NOUN
cana-527	315	11	.	.	PUNCT
cana-527	316	1	author	author	NOUN
cana-527	316	2	contributions	contribution	NOUN
cana-527	316	3	:	:	PUNCT
cana-527	316	4	this	this	DET
cana-527	316	5	paper	paper	NOUN
cana-527	316	6	’s	’s	PART
cana-527	316	7	entire	entire	ADJ
cana-527	316	8	work	work	NOUN
cana-527	316	9	is	be	AUX
cana-527	316	10	the	the	DET
cana-527	316	11	responsibility	responsibility	NOUN
cana-527	316	12	of	of	ADP
cana-527	316	13	its	its	PRON
cana-527	316	14	two	two	NUM
cana-527	316	15	authors	author	NOUN
cana-527	316	16	.	.	PUNCT
cana-527	317	1	conflicts	conflict	NOUN
cana-527	317	2	of	of	ADP
cana-527	317	3	interest	interest	NOUN
cana-527	317	4	:	:	PUNCT
cana-527	317	5	there	there	PRON
cana-527	317	6	is	be	VERB
cana-527	317	7	certainly	certainly	ADV
cana-527	317	8	no	no	DET
cana-527	317	9	conflict	conflict	NOUN
cana-527	317	10	of	of	ADP
cana-527	317	11	interest	interest	NOUN
cana-527	317	12	revealed	reveal	VERB
cana-527	317	13	by	by	ADP
cana-527	317	14	the	the	DET
cana-527	317	15	author	author	NOUN
cana-527	317	16	.	.	PUNCT
cana-527	318	1	references	reference	NOUN
cana-527	318	2	[	[	X
cana-527	318	3	1	1	NUM
cana-527	318	4	]	]	PUNCT
cana-527	318	5	balog.l	balog.l	NOUN
cana-527	318	6	and	and	CCONJ
cana-527	318	7	berinde.v	berinde.v	PROPN
cana-527	318	8	.	.	PUNCT
cana-527	319	1	fixed	fix	VERB
cana-527	319	2	point	point	NOUN
cana-527	319	3	theorems	theorem	NOUN
cana-527	319	4	for	for	ADP
cana-527	319	5	non	non	ADJ
cana-527	319	6	-	-	ADJ
cana-527	319	7	self	self	ADJ
cana-527	319	8	kannan	kannan	PROPN
cana-527	319	9	type	type	NOUN
cana-527	319	10	contraction	contraction	NOUN
cana-527	319	11	in	in	ADP
cana-527	319	12	banach	banach	NOUN
cana-527	319	13	spaces	space	NOUN
cana-527	319	14	endowed	endow	VERB
cana-527	319	15	with	with	ADP
cana-527	319	16	graph	graph	NOUN
cana-527	319	17	.	.	PUNCT
cana-527	320	1	carpathian	carpathian	ADJ
cana-527	320	2	j.math	j.math	NOUN
cana-527	320	3	,	,	PUNCT
cana-527	320	4	32:293	32:293	NUM
cana-527	320	5	-	-	SYM
cana-527	320	6	302	302	NUM
cana-527	320	7	,	,	PUNCT
cana-527	320	8	2016	2016	NUM
cana-527	320	9	.	.	PUNCT
cana-527	321	1	[	[	X
cana-527	321	2	2	2	X
cana-527	321	3	]	]	PUNCT
cana-527	321	4	banahc.s	banahc.	VERB
cana-527	321	5	surles	surle	NOUN
cana-527	321	6	operations	operation	NOUN
cana-527	321	7	dans	dan	NOUN
cana-527	321	8	les	le	NOUN
cana-527	321	9	ensembles	ensemble	NOUN
cana-527	321	10	abstraits	abstrait	NOUN
cana-527	321	11	et	et	PROPN
cana-527	321	12	leurs	leurs	PROPN
cana-527	321	13	applications	application	NOUN
cana-527	321	14	.	.	PUNCT
cana-527	322	1	fundam.math	fundam.math	ADJ
cana-527	322	2	,	,	PUNCT
cana-527	322	3	3:133	3:133	NUM
cana-527	322	4	-	-	SYM
cana-527	322	5	181	181	NUM
cana-527	322	6	,	,	PUNCT
cana-527	322	7	1922	1922	NUM
cana-527	322	8	.	.	PUNCT
cana-527	323	1	[	[	X
cana-527	323	2	3	3	X
cana-527	323	3	]	]	SYM
cana-527	323	4	began.i	began.i	NUM
cana-527	323	5	butt	butt	PROPN
cana-527	323	6	a.r	a.r	PROPN
cana-527	323	7	and	and	CCONJ
cana-527	323	8	radojeric.s	radojeric.s	PROPN
cana-527	323	9	”	"	PUNCT
cana-527	323	10	the	the	DET
cana-527	323	11	contraction	contraction	NOUN
cana-527	323	12	principle	principle	NOUN
cana-527	323	13	for	for	ADP
cana-527	323	14	set	set	VERB
cana-527	323	15	valued	value	VERB
cana-527	323	16	mappings	mapping	NOUN
cana-527	323	17	on	on	ADP
cana-527	323	18	a	a	DET
cana-527	323	19	metric	metric	ADJ
cana-527	323	20	space	space	NOUN
cana-527	323	21	with	with	ADP
cana-527	323	22	a	a	DET
cana-527	323	23	graph	graph	NOUN
cana-527	323	24	”	"	PUNCT
cana-527	323	25	comput.math.appl,60	comput.math.appl,60	NUM
cana-527	323	26	,	,	PUNCT
cana-527	323	27	1214	1214	NUM
cana-527	323	28	-	-	SYM
cana-527	323	29	1219	1219	NUM
cana-527	323	30	,	,	PUNCT
cana-527	323	31	2010	2010	NUM
cana-527	323	32	.	.	PUNCT
cana-527	324	1	[	[	X
cana-527	324	2	4	4	NUM
cana-527	324	3	]	]	PUNCT
cana-527	324	4	berinde.v	berinde.v	VERB
cana-527	325	1	and	and	CCONJ
cana-527	325	2	pacurar.m	pacurar.m	PRON
cana-527	325	3	”	"	PUNCT
cana-527	325	4	the	the	DET
cana-527	325	5	contraction	contraction	NOUN
cana-527	325	6	principle	principle	NOUN
cana-527	325	7	for	for	ADP
cana-527	325	8	non	non	ADJ
cana-527	325	9	-	-	ADJ
cana-527	325	10	self	self	ADJ
cana-527	325	11	mappings	mapping	NOUN
cana-527	325	12	on	on	ADP
cana-527	325	13	banach	banach	NOUN
cana-527	325	14	spaces	space	NOUN
cana-527	325	15	endowed	endow	VERB
cana-527	325	16	with	with	ADP
cana-527	325	17	a	a	DET
cana-527	325	18	graph	graph	NOUN
cana-527	325	19	”	"	PUNCT
cana-527	325	20	j	j	PROPN
cana-527	325	21	non	non	ADJ
cana-527	325	22	-	-	ADJ
cana-527	325	23	linear	linear	ADJ
cana-527	325	24	convex.appl	convex.appl	PROPN
cana-527	325	25	,	,	PUNCT
cana-527	325	26	16	16	NUM
cana-527	325	27	,	,	PUNCT
cana-527	325	28	1925	1925	NUM
cana-527	325	29	-	-	SYM
cana-527	325	30	1936	1936	NUM
cana-527	325	31	,	,	PUNCT
cana-527	325	32	2015	2015	NUM
cana-527	325	33	.	.	PUNCT
cana-527	326	1	[	[	X
cana-527	326	2	5	5	NUM
cana-527	326	3	]	]	PUNCT
cana-527	326	4	bajor.f	bajor.f	PROPN
cana-527	326	5	”	"	PUNCT
cana-527	326	6	fixed	fix	VERB
cana-527	326	7	point	point	NOUN
cana-527	326	8	of	of	ADP
cana-527	326	9	kannan	kannan	PROPN
cana-527	326	10	mappings	mapping	NOUN
cana-527	326	11	in	in	ADP
cana-527	326	12	metric	metric	ADJ
cana-527	326	13	spaces	space	NOUN
cana-527	326	14	endowed	endow	VERB
cana-527	326	15	with	with	ADP
cana-527	326	16	a	a	DET
cana-527	326	17	graph	graph	NOUN
cana-527	326	18	”	"	PUNCT
cana-527	326	19	.	.	PUNCT
cana-527	327	1	an	an	DET
cana-527	327	2	stiint	stiint	PROPN
cana-527	327	3	univ	univ	PROPN
cana-527	327	4	.	.	PUNCT
cana-527	327	5	”	"	PUNCT
cana-527	327	6	ovidius	ovidius	PROPN
cana-527	327	7	”	"	PUNCT
cana-527	327	8	constanta	constanta	PROPN
cana-527	327	9	set.mat	set.mat	PROPN
cana-527	327	10	20(1	20(1	PROPN
cana-527	327	11	)	)	PUNCT
cana-527	327	12	,	,	PUNCT
cana-527	327	13	31	31	NUM
cana-527	327	14	-	-	SYM
cana-527	327	15	40	40	NUM
cana-527	327	16	,	,	PUNCT
cana-527	327	17	2012	2012	NUM
cana-527	327	18	.	.	PUNCT
cana-527	328	1	[	[	X
cana-527	328	2	6	6	NUM
cana-527	328	3	]	]	PUNCT
cana-527	328	4	bondy.j.a	bondy.j.a	PROPN
cana-527	328	5	and	and	CCONJ
cana-527	328	6	murthy	murthy	ADJ
cana-527	328	7	u.s.r	u.s.r	PROPN
cana-527	328	8	.	.	PROPN
cana-527	328	9	graph	graph	NOUN
cana-527	328	10	theory	theory	NOUN
cana-527	328	11	.	.	PUNCT
cana-527	329	1	springer	springer	NOUN
cana-527	329	2	,	,	PUNCT
cana-527	329	3	new	new	PROPN
cana-527	329	4	york	york	PROPN
cana-527	329	5	,	,	PUNCT
cana-527	329	6	2008	2008	NUM
cana-527	329	7	.	.	PUNCT
cana-527	330	1	[	[	X
cana-527	330	2	7	7	NUM
cana-527	330	3	]	]	SYM
cana-527	330	4	chatterjea.s.k	chatterjea.s.k	NOUN
cana-527	330	5	.	.	PUNCT
cana-527	330	6	fixed	fix	VERB
cana-527	330	7	point	point	NOUN
cana-527	330	8	theorems	theorem	NOUN
cana-527	330	9	.	.	PUNCT
cana-527	331	1	c.r.acad.bulgare.sci	c.r.acad.bulgare.sci	PROPN
cana-527	331	2	,	,	PUNCT
cana-527	331	3	25:727–730	25:727–730	NUM
cana-527	331	4	,	,	PUNCT
cana-527	331	5	1972	1972	NUM
cana-527	331	6	.	.	PUNCT
cana-527	332	1	[	[	X
cana-527	332	2	8	8	NUM
cana-527	332	3	]	]	SYM
cana-527	332	4	chifu.c.i	chifu.c.i	NOUN
cana-527	332	5	and	and	CCONJ
cana-527	332	6	petrusal	petrusal	PROPN
cana-527	332	7	g.r	g.r	PROPN
cana-527	332	8	”	"	PUNCT
cana-527	332	9	generalization	generalization	NOUN
cana-527	332	10	contractions	contraction	NOUN
cana-527	332	11	in	in	ADP
cana-527	332	12	metric	metric	ADJ
cana-527	332	13	spaces	space	NOUN
cana-527	332	14	endowed	endow	VERB
cana-527	332	15	with	with	ADP
cana-527	332	16	a	a	DET
cana-527	332	17	graph	graph	NOUN
cana-527	332	18	”	"	PUNCT
cana-527	332	19	fixed	fix	VERB
cana-527	332	20	point	point	NOUN
cana-527	332	21	theory	theory	NOUN
cana-527	332	22	and	and	CCONJ
cana-527	332	23	applications	application	NOUN
cana-527	332	24	1	1	NUM
cana-527	332	25	,	,	PUNCT
cana-527	332	26	1	1	NUM
cana-527	332	27	-	-	SYM
cana-527	332	28	9	9	NUM
cana-527	332	29	,	,	PUNCT
cana-527	332	30	2012	2012	NUM
cana-527	332	31	.	.	PUNCT
cana-527	333	1	[	[	X
cana-527	333	2	9	9	NUM
cana-527	333	3	]	]	SYM
cana-527	333	4	circ.l.j	circ.l.j	NOUN
cana-527	333	5	.	.	PUNCT
cana-527	333	6	generalized	generalized	ADJ
cana-527	333	7	contractions	contraction	NOUN
cana-527	333	8	and	and	CCONJ
cana-527	333	9	fixed	fix	VERB
cana-527	333	10	point	point	NOUN
cana-527	333	11	theorems	theorem	NOUN
cana-527	333	12	.	.	PUNCT
cana-527	334	1	publications	publication	NOUN
cana-527	334	2	de.l	de.l	PROPN
cana-527	334	3	institut	institut	PROPN
cana-527	334	4	mathematique	mathematique	PROPN
cana-527	334	5	,	,	PUNCT
cana-527	334	6	26:19–26	26:19–26	NUM
cana-527	334	7	,	,	PUNCT
cana-527	334	8	1971	1971	NUM
cana-527	334	9	.	.	PUNCT
cana-527	335	1	[	[	X
cana-527	335	2	10	10	NUM
cana-527	335	3	]	]	X
cana-527	335	4	e.m	e.m	PROPN
cana-527	335	5	.	.	PROPN
cana-527	335	6	delstein	delstein	PROPN
cana-527	335	7	.	.	PUNCT
cana-527	336	1	an	an	DET
cana-527	336	2	extension	extension	NOUN
cana-527	336	3	of	of	ADP
cana-527	336	4	banach	banach	NOUN
cana-527	336	5	contraction	contraction	NOUN
cana-527	336	6	principle	principle	NOUN
cana-527	336	7	.	.	PUNCT
cana-527	337	1	proc.am.math.soc	proc.am.math.soc	PROPN
cana-527	337	2	.	.	PROPN
cana-527	337	3	,	,	PUNCT
cana-527	337	4	12:7–10	12:7–10	NUM
cana-527	337	5	,	,	PUNCT
cana-527	337	6	1961	1961	NUM
cana-527	337	7	.	.	PUNCT
cana-527	338	1	[	[	X
cana-527	338	2	11	11	NUM
cana-527	338	3	]	]	PUNCT
cana-527	338	4	eswari.p.k	eswari.p.k	NOUN
cana-527	338	5	thirunavukarasu.p	thirunavukarasu.p	PUNCT
cana-527	338	6	and	and	CCONJ
cana-527	338	7	manjula.r	manjula.r	PROPN
cana-527	338	8	.	.	PUNCT
cana-527	338	9	fixed	fix	VERB
cana-527	338	10	points	point	NOUN
cana-527	338	11	for	for	ADP
cana-527	338	12	cyclic	cyclic	ADJ
cana-527	338	13	contractions	contraction	NOUN
cana-527	338	14	in	in	ADP
cana-527	338	15	symmetric	symmetric	ADJ
cana-527	338	16	spaces	space	NOUN
cana-527	338	17	and	and	CCONJ
cana-527	338	18	partial	partial	ADJ
cana-527	338	19	symmetric	symmetric	ADJ
cana-527	338	20	spaces	space	NOUN
cana-527	338	21	.	.	PUNCT
cana-527	339	1	cikitusi	cikitusi	PROPN
cana-527	339	2	journal	journal	PROPN
cana-527	339	3	for	for	ADP
cana-527	339	4	multidiciplinary	multidiciplinary	ADJ
cana-527	339	5	research	research	NOUN
cana-527	339	6	,	,	PUNCT
cana-527	339	7	6(5	6(5	NUM
cana-527	339	8	)	)	PUNCT
cana-527	339	9	,	,	PUNCT
cana-527	339	10	2019	2019	NUM
cana-527	339	11	.	.	PUNCT
cana-527	340	1	[	[	X
cana-527	340	2	12	12	NUM
cana-527	340	3	]	]	PUNCT
cana-527	340	4	espinola.r	espinola.r	X
cana-527	340	5	and	and	CCONJ
cana-527	340	6	kirk.w.a	kirk.w.a	PROPN
cana-527	340	7	”	"	PUNCT
cana-527	340	8	fixed	fix	VERB
cana-527	340	9	point	point	NOUN
cana-527	340	10	theorems	theorem	NOUN
cana-527	340	11	in	in	ADP
cana-527	340	12	r	r	NOUN
cana-527	340	13	-	-	PUNCT
cana-527	340	14	trees	tree	NOUN
cana-527	340	15	with	with	ADP
cana-527	340	16	applications	application	NOUN
cana-527	340	17	to	to	PART
cana-527	340	18	graph	graph	VERB
cana-527	340	19	theory	theory	NOUN
cana-527	340	20	”	"	PUNCT
cana-527	340	21	top.appl.153	top.appl.153	NOUN
cana-527	340	22	,	,	PUNCT
cana-527	340	23	1046	1046	NUM
cana-527	340	24	-	-	SYM
cana-527	340	25	1055	1055	NUM
cana-527	340	26	,	,	PUNCT
cana-527	340	27	2006	2006	NUM
cana-527	340	28	.	.	PUNCT
cana-527	341	1	[	[	X
cana-527	341	2	13	13	NUM
cana-527	341	3	]	]	PUNCT
cana-527	341	4	fallahik	fallahik	NOUN
cana-527	341	5	and	and	CCONJ
cana-527	341	6	aghanianas.a	aghanianas.a	NOUN
cana-527	341	7	”	"	PUNCT
cana-527	341	8	on	on	ADP
cana-527	341	9	quais	quais	PROPN
cana-527	341	10	-	-	PUNCT
cana-527	341	11	contractions	contraction	NOUN
cana-527	341	12	in	in	ADP
cana-527	341	13	metric	metric	ADJ
cana-527	341	14	spaces	space	NOUN
cana-527	341	15	with	with	ADP
cana-527	341	16	a	a	DET
cana-527	341	17	graph	graph	NOUN
cana-527	341	18	”	"	PUNCT
cana-527	341	19	hacettepe	hacettepe	ADJ
cana-527	341	20	j.math	j.math	NOUN
cana-527	341	21	and	and	CCONJ
cana-527	341	22	statistics	statistic	NOUN
cana-527	341	23	45(4),1033	45(4),1033	NUM
cana-527	341	24	-	-	SYM
cana-527	341	25	1047,2016	1047,2016	NUM
cana-527	341	26	.	.	PUNCT
cana-527	342	1	[	[	X
cana-527	342	2	14	14	NUM
cana-527	342	3	]	]	X
cana-527	342	4	hsien	hsien	PROPN
cana-527	342	5	-	-	PUNCT
cana-527	342	6	chung	chung	PROPN
cana-527	342	7	wu	wu	PROPN
cana-527	342	8	.	.	PUNCT
cana-527	343	1	near	near	ADP
cana-527	343	2	fixed	fix	VERB
cana-527	343	3	point	point	NOUN
cana-527	343	4	theorems	theorem	NOUN
cana-527	343	5	in	in	ADP
cana-527	343	6	hyperspace	hyperspace	NOUN
cana-527	343	7	.	.	PUNCT
cana-527	344	1	pages	page	NOUN
cana-527	344	2	1269	1269	NUM
cana-527	344	3	–	–	PUNCT
cana-527	344	4	1303	1303	NUM
cana-527	344	5	,	,	PUNCT
cana-527	344	6	1968	1968	NUM
cana-527	344	7	.	.	PUNCT
cana-527	345	1	[	[	X
cana-527	345	2	15	15	NUM
cana-527	345	3	]	]	X
cana-527	345	4	hsien	hsien	PROPN
cana-527	345	5	-	-	PUNCT
cana-527	345	6	chung	chung	PROPN
cana-527	345	7	wu	wu	PROPN
cana-527	345	8	.	.	PUNCT
cana-527	346	1	near	near	ADP
cana-527	346	2	fixed	fix	VERB
cana-527	346	3	point	point	NOUN
cana-527	346	4	theorems	theorem	NOUN
cana-527	346	5	in	in	ADP
cana-527	346	6	hyperspace	hyperspace	NOUN
cana-527	346	7	.	.	PUNCT
cana-527	347	1	mathematics	mathematic	NOUN
cana-527	347	2	mdpi	mdpi	PROPN
cana-527	347	3	,	,	PUNCT
cana-527	347	4	pages	page	NOUN
cana-527	347	5	1–15	1–15	PROPN
cana-527	347	6	,	,	PUNCT
cana-527	347	7	2018	2018	NUM
cana-527	347	8	.	.	PUNCT
cana-527	348	1	communications	communication	NOUN
cana-527	348	2	on	on	ADP
cana-527	348	3	applied	apply	VERB
cana-527	348	4	nonlinear	nonlinear	ADJ
cana-527	348	5	analysis	analysis	NOUN
cana-527	348	6	issn	issn	NOUN
cana-527	348	7	:	:	PUNCT
cana-527	348	8	1074	1074	NUM
cana-527	348	9	-	-	PUNCT
cana-527	348	10	133x	133x	NUM
cana-527	348	11	vol	vol	NOUN
cana-527	348	12	31	31	NUM
cana-527	348	13	no	no	NOUN
cana-527	348	14	.	.	NOUN
cana-527	348	15	2	2	NUM
cana-527	348	16	(	(	PUNCT
cana-527	348	17	2024	2024	NUM
cana-527	348	18	)	)	PUNCT
cana-527	348	19	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	348	20	172	172	NUM
cana-527	349	1	[	[	X
cana-527	349	2	16	16	NUM
cana-527	349	3	]	]	X
cana-527	349	4	hsien	hsien	PROPN
cana-527	349	5	-	-	PUNCT
cana-527	349	6	chung	chung	PROPN
cana-527	349	7	,	,	PUNCT
cana-527	349	8	wu	wu	PROPN
cana-527	349	9	,	,	PUNCT
cana-527	349	10	near	near	ADP
cana-527	349	11	fixed	fix	VERB
cana-527	349	12	point	point	NOUN
cana-527	349	13	theorems	theorem	NOUN
cana-527	349	14	in	in	ADP
cana-527	349	15	near	near	ADP
cana-527	349	16	banach	banach	NOUN
cana-527	349	17	spaces	space	NOUN
cana-527	349	18	,	,	PUNCT
cana-527	349	19	aims	aim	VERB
cana-527	349	20	mathemativs,8(1):12269	mathemativs,8(1):12269	NOUN
cana-527	349	21	-	-	PUNCT
cana-527	349	22	1303,2022	1303,2022	NOUN
cana-527	349	23	.	.	PUNCT
cana-527	350	1	[	[	X
cana-527	350	2	17	17	NUM
cana-527	350	3	]	]	PUNCT
cana-527	350	4	jachymski.j	jachymski.j	PROPN
cana-527	350	5	.	.	PUNCT
cana-527	351	1	the	the	DET
cana-527	351	2	centraction	centraction	NOUN
cana-527	351	3	principle	principle	NOUN
cana-527	351	4	for	for	ADP
cana-527	351	5	mappings	mapping	NOUN
cana-527	351	6	on	on	ADP
cana-527	351	7	a	a	DET
cana-527	351	8	metric	metric	ADJ
cana-527	351	9	space	space	NOUN
cana-527	351	10	with	with	ADP
cana-527	351	11	a	a	DET
cana-527	351	12	graph	graph	NOUN
cana-527	351	13	.	.	PUNCT
cana-527	352	1	proc.am.math.soc	proc.am.math.soc	NOUN
cana-527	352	2	,	,	PUNCT
cana-527	352	3	1(136):1359–1373	1(136):1359–1373	NOUN
cana-527	352	4	,	,	PUNCT
cana-527	352	5	2008	2008	NUM
cana-527	352	6	.	.	PUNCT
cana-527	353	1	[	[	X
cana-527	353	2	18	18	NUM
cana-527	353	3	]	]	X
cana-527	353	4	kannan.r	kannan.r	PROPN
cana-527	353	5	.	.	PUNCT
cana-527	354	1	some	some	DET
cana-527	354	2	results	result	NOUN
cana-527	354	3	on	on	ADP
cana-527	354	4	fixed	fix	VERB
cana-527	354	5	points	point	NOUN
cana-527	354	6	,	,	PUNCT
cana-527	354	7	bull.calcutta.math.soc	bull.calcutta.math.soc	PROPN
cana-527	354	8	.	.	PROPN
cana-527	354	9	10:71–76	10:71–76	NUM
cana-527	354	10	,	,	PUNCT
cana-527	354	11	1968	1968	NUM
cana-527	354	12	.	.	PUNCT
cana-527	355	1	[	[	X
cana-527	355	2	19	19	NUM
cana-527	355	3	]	]	SYM
cana-527	355	4	nicolae.a	nicolae.a	PROPN
cana-527	355	5	,	,	PUNCT
cana-527	355	6	o	o	NOUN
cana-527	355	7	’	'	PUNCT
cana-527	355	8	ragan.d	ragan.d	ADJ
cana-527	355	9	and	and	CCONJ
cana-527	355	10	petrusal.a	petrusal.a	NOUN
cana-527	355	11	”	"	PUNCT
cana-527	355	12	fixed	fix	VERB
cana-527	355	13	point	point	NOUN
cana-527	355	14	theorems	theorem	NOUN
cana-527	355	15	for	for	ADP
cana-527	355	16	single	single	ADJ
cana-527	355	17	valued	value	VERB
cana-527	355	18	and	and	CCONJ
cana-527	355	19	multivalued	multivalued	ADJ
cana-527	355	20	generalized	generalized	ADJ
cana-527	355	21	contractions	contraction	NOUN
cana-527	355	22	in	in	ADP
cana-527	355	23	metric	metric	ADJ
cana-527	355	24	space	space	NOUN
cana-527	355	25	endowed	endow	VERB
cana-527	355	26	with	with	ADP
cana-527	355	27	a	a	DET
cana-527	355	28	graph	graph	NOUN
cana-527	355	29	”	"	PUNCT
cana-527	355	30	georgian	georgian	NOUN
cana-527	355	31	math.j	math.j	NOUN
cana-527	355	32	,	,	PUNCT
cana-527	355	33	2	2	NUM
cana-527	355	34	,	,	PUNCT
cana-527	355	35	307	307	NUM
cana-527	355	36	-	-	SYM
cana-527	355	37	327	327	NUM
cana-527	355	38	,	,	PUNCT
cana-527	355	39	2011	2011	NUM
cana-527	355	40	.	.	PUNCT
cana-527	356	1	[	[	X
cana-527	356	2	20	20	NUM
cana-527	356	3	]	]	X
cana-527	356	4	om	om	PROPN
cana-527	356	5	gayathri.r	gayathri.r	PROPN
cana-527	356	6	hemavathy.r	hemavathy.r	PROPN
cana-527	356	7	.	.	PUNCT
cana-527	357	1	a	a	DET
cana-527	357	2	new	new	ADJ
cana-527	357	3	approach	approach	NOUN
cana-527	357	4	to	to	ADP
cana-527	357	5	fixed	fix	VERB
cana-527	357	6	point	point	NOUN
cana-527	357	7	theorems	theorem	NOUN
cana-527	357	8	on	on	ADP
cana-527	357	9	a	a	DET
cana-527	357	10	metric	metric	ADJ
cana-527	357	11	space	space	NOUN
cana-527	357	12	endowed	endow	VERB
cana-527	357	13	with	with	ADP
cana-527	357	14	graph	graph	NOUN
cana-527	357	15	.	.	PUNCT
cana-527	358	1	american	american	PROPN
cana-527	358	2	journal	journal	PROPN
cana-527	358	3	of	of	ADP
cana-527	358	4	applied	apply	VERB
cana-527	358	5	mathematics	mathematic	NOUN
cana-527	358	6	and	and	CCONJ
cana-527	358	7	statistics	statistic	NOUN
cana-527	358	8	,	,	PUNCT
cana-527	358	9	3:69–75	3:69–75	NUM
cana-527	358	10	,	,	PUNCT
cana-527	358	11	2022	2022	NUM
cana-527	358	12	.	.	PUNCT
cana-527	359	1	[	[	X
cana-527	359	2	21	21	NUM
cana-527	359	3	]	]	PUNCT
cana-527	359	4	samreen.m	samreen.m	PROPN
cana-527	359	5	,	,	PUNCT
cana-527	359	6	kannan.t	kannan.t	PROPN
cana-527	359	7	and	and	CCONJ
cana-527	359	8	shahzad	shahzad	PROPN
cana-527	359	9	,	,	PUNCT
cana-527	359	10	”	"	PUNCT
cana-527	359	11	some	some	DET
cana-527	359	12	fixed	fix	VERB
cana-527	359	13	point	point	NOUN
cana-527	359	14	theorems	theorem	NOUN
cana-527	359	15	in	in	ADP
cana-527	359	16	b	b	NOUN
cana-527	359	17	-	-	PUNCT
cana-527	359	18	metric	metric	ADJ
cana-527	359	19	space	space	NOUN
cana-527	359	20	endowed	endow	VERB
cana-527	359	21	with	with	ADP
cana-527	359	22	a	a	DET
cana-527	359	23	graph	graph	NOUN
cana-527	359	24	”	"	PUNCT
cana-527	359	25	abstra.app.anas.article	abstra.app.anas.article	NOUN
cana-527	359	26	i	i	PROPN
cana-527	359	27	d	d	PROPN
cana-527	359	28	967132	967132	NUM
cana-527	359	29	,	,	PUNCT
cana-527	359	30	2013	2013	NUM
cana-527	359	31	.	.	PUNCT
cana-527	360	1	[	[	X
cana-527	360	2	22	22	NUM
cana-527	360	3	]	]	PUNCT
cana-527	360	4	shukla.s	shukla.s	PROPN
cana-527	360	5	,	,	PUNCT
cana-527	360	6	radenoric.s	radenoric.s	PUNCT
cana-527	360	7	and	and	CCONJ
cana-527	360	8	vetro.c	vetro.c	NUM
cana-527	360	9	”	"	PUNCT
cana-527	360	10	graphical	graphical	ADJ
cana-527	360	11	metric	metric	ADJ
cana-527	360	12	space	space	NOUN
cana-527	360	13	a	a	DET
cana-527	360	14	generalized	generalized	ADJ
cana-527	360	15	setting	setting	NOUN
cana-527	360	16	in	in	ADP
cana-527	360	17	fixed	fix	VERB
cana-527	360	18	point	point	NOUN
cana-527	360	19	theory	theory	NOUN
cana-527	360	20	”	"	PUNCT
cana-527	360	21	rev.real	rev.real	NOUN
cana-527	360	22	acad.cienc	acad.cienc	PROPN
cana-527	360	23	ser.a.mat	ser.a.mat	PROPN
cana-527	360	24	.	.	PROPN
cana-527	360	25	,	,	PUNCT
cana-527	360	26	111,641	111,641	NUM
cana-527	360	27	-	-	SYM
cana-527	360	28	655	655	NUM
cana-527	360	29	,	,	PUNCT
cana-527	360	30	2017	2017	NUM
cana-527	360	31	.	.	PUNCT
cana-527	361	1	[	[	X
cana-527	361	2	23	23	NUM
cana-527	361	3	]	]	PUNCT
cana-527	361	4	thirunavukarasu.p	thirunavukarasu.p	ADV
cana-527	361	5	and	and	CCONJ
cana-527	361	6	savitha.s	savitha.s	PROPN
cana-527	361	7	.	.	PUNCT
cana-527	362	1	cyclic	cyclic	ADJ
cana-527	362	2	contractions	contraction	NOUN
cana-527	362	3	and	and	CCONJ
cana-527	362	4	fixed	fix	VERB
cana-527	362	5	point	point	NOUN
cana-527	362	6	theorems	theorem	NOUN
cana-527	362	7	in	in	ADP
cana-527	362	8	banach	banach	NOUN
cana-527	362	9	spaces	space	NOUN
cana-527	362	10	.	.	PUNCT
cana-527	363	1	ijas	ijas	PROPN
cana-527	363	2	,	,	PUNCT
cana-527	363	3	2:112–118	2:112–118	NUM
cana-527	363	4	,	,	PUNCT
cana-527	363	5	2021	2021	NUM
cana-527	363	6	.	.	PUNCT
cana-527	364	1	[	[	X
cana-527	364	2	24	24	NUM
cana-527	364	3	]	]	PUNCT
cana-527	364	4	tibebu	tibebu	NOUN
cana-527	364	5	worku	worku	PROPN
cana-527	364	6	hunde	hunde	PROPN
cana-527	364	7	.	.	PUNCT
cana-527	365	1	approximation	approximation	NOUN
cana-527	365	2	of	of	ADP
cana-527	365	3	a	a	DET
cana-527	365	4	common	common	ADJ
cana-527	365	5	fixed	fix	VERB
cana-527	365	6	point	point	NOUN
cana-527	365	7	of	of	ADP
cana-527	365	8	a	a	DET
cana-527	365	9	family	family	NOUN
cana-527	365	10	of	of	ADP
cana-527	365	11	gnonexpansive	gnonexpansive	ADJ
cana-527	365	12	mappings	mapping	NOUN
cana-527	365	13	in	in	ADP
cana-527	365	14	banach	banach	NOUN
cana-527	365	15	spaces	space	NOUN
cana-527	365	16	with	with	ADP
cana-527	365	17	a	a	DET
cana-527	365	18	graph	graph	NOUN
cana-527	365	19	.	.	PUNCT
cana-527	366	1	int.jour.advance	int.jour.advance	NOUN
cana-527	366	2	in	in	ADP
cana-527	366	3	mathematics	mathematic	NOUN
cana-527	366	4	,	,	PUNCT
cana-527	366	5	6:137–152	6:137–152	NUM
cana-527	366	6	,	,	PUNCT
cana-527	366	7	2017	2017	NUM
cana-527	366	8	.	.	PUNCT
cana-527	367	1	communications	communication	NOUN
cana-527	367	2	on	on	ADP
cana-527	367	3	applied	apply	VERB
cana-527	367	4	nonlinear	nonlinear	ADJ
cana-527	367	5	analysis	analysis	NOUN
cana-527	367	6	issn	issn	NOUN
cana-527	367	7	:	:	PUNCT
cana-527	367	8	1074	1074	NUM
cana-527	367	9	-	-	PUNCT
cana-527	367	10	133x	133x	NUM
cana-527	367	11	vol	vol	NOUN
cana-527	367	12	31	31	NUM
cana-527	367	13	no	no	NOUN
cana-527	367	14	.	.	NOUN
cana-527	367	15	2	2	NUM
cana-527	367	16	(	(	PUNCT
cana-527	367	17	2024	2024	NUM
cana-527	367	18	)	)	PUNCT
cana-527	368	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-527	368	2	173	173	NUM