id	sid	tid	token	lemma	pos
cana-3308	1	1	communications	communication	NOUN
cana-3308	1	2	on	on	ADP
cana-3308	1	3	applied	apply	VERB
cana-3308	1	4	nonlinear	nonlinear	ADJ
cana-3308	1	5	analysis	analysis	NOUN
cana-3308	1	6	issn	issn	NOUN
cana-3308	1	7	:	:	PUNCT
cana-3308	1	8	1074	1074	NUM
cana-3308	1	9	-	-	PUNCT
cana-3308	1	10	133x	133x	NUM
cana-3308	1	11	vol	vol	NOUN
cana-3308	1	12	32	32	NUM
cana-3308	1	13	no	no	NOUN
cana-3308	1	14	.	.	PUNCT
cana-3308	2	1	6s	6s	NUM
cana-3308	2	2	(	(	PUNCT
cana-3308	2	3	2025	2025	NUM
cana-3308	2	4	)	)	PUNCT
cana-3308	2	5	439	439	NUM
cana-3308	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	2	7	some	some	DET
cana-3308	2	8	fixed	fix	VERB
cana-3308	2	9	point	point	NOUN
cana-3308	2	10	results	result	NOUN
cana-3308	2	11	for	for	ADP
cana-3308	2	12	(	(	PUNCT
cana-3308	2	13	𝜶	𝜶	NOUN
cana-3308	2	14	,	,	PUNCT
cana-3308	2	15	𝝍	𝝍	PROPN
cana-3308	2	16	,	,	PUNCT
cana-3308	2	17	𝝋)geraghty	𝝋)geraghty	ADJ
cana-3308	2	18	contraction	contraction	NOUN
cana-3308	2	19	mappings	mapping	NOUN
cana-3308	2	20	in	in	ADP
cana-3308	2	21	bipolar	bipolar	ADJ
cana-3308	2	22	metric	metric	ADJ
cana-3308	2	23	space	space	NOUN
cana-3308	2	24	diksha1	diksha1	NOUN
cana-3308	2	25	,	,	PUNCT
cana-3308	2	26	manoj	manoj	PROPN
cana-3308	2	27	kumar2	kumar2	PROPN
cana-3308	2	28	,	,	PUNCT
cana-3308	2	29	*	*	PROPN
cana-3308	2	30	1	1	NUM
cana-3308	2	31	,	,	PUNCT
cana-3308	2	32	2department	2department	NUM
cana-3308	2	33	of	of	ADP
cana-3308	2	34	mathematics	mathematic	NOUN
cana-3308	2	35	,	,	PUNCT
cana-3308	2	36	baba	baba	PROPN
cana-3308	2	37	mastnath	mastnath	PROPN
cana-3308	2	38	university	university	PROPN
cana-3308	2	39	,	,	PUNCT
cana-3308	2	40	asthal	asthal	NOUN
cana-3308	2	41	bohar	bohar	NOUN
cana-3308	2	42	,	,	PUNCT
cana-3308	2	43	rohtak-124021	rohtak-124021	ADJ
cana-3308	2	44	,	,	PUNCT
cana-3308	2	45	haryana	haryana	PROPN
cana-3308	2	46	,	,	PUNCT
cana-3308	2	47	india	india	PROPN
cana-3308	2	48	1department	1department	NUM
cana-3308	2	49	of	of	ADP
cana-3308	2	50	mathematics	mathematic	NOUN
cana-3308	2	51	,	,	PUNCT
cana-3308	2	52	government	government	NOUN
cana-3308	2	53	college	college	NOUN
cana-3308	2	54	,	,	PUNCT
cana-3308	2	55	matanhail	matanhail	NOUN
cana-3308	2	56	,	,	PUNCT
cana-3308	2	57	jhajjar-124106	jhajjar-124106	ADV
cana-3308	2	58	,	,	PUNCT
cana-3308	2	59	haryana	haryana	PROPN
cana-3308	2	60	,	,	PUNCT
cana-3308	2	61	india	india	PROPN
cana-3308	2	62	2department	2department	PROPN
cana-3308	2	63	of	of	ADP
cana-3308	2	64	mathematics	mathematics	PROPN
cana-3308	2	65	,	,	PUNCT
cana-3308	2	66	maharishi	maharishi	PROPN
cana-3308	2	67	markandeshwar	markandeshwar	PROPN
cana-3308	2	68	(	(	PUNCT
cana-3308	2	69	deemed	deem	VERB
cana-3308	2	70	to	to	PART
cana-3308	2	71	be	be	AUX
cana-3308	2	72	university	university	NOUN
cana-3308	2	73	)	)	PUNCT
cana-3308	2	74	,	,	PUNCT
cana-3308	2	75	mullana	mullana	PROPN
cana-3308	2	76	,	,	PUNCT
cana-3308	2	77	ambala-133207	ambala-133207	NOUN
cana-3308	2	78	,	,	PUNCT
cana-3308	2	79	haryana	haryana	PROPN
cana-3308	2	80	,	,	PUNCT
cana-3308	2	81	india	india	PROPN
cana-3308	2	82	email	email	PROPN
cana-3308	2	83	addressesdikshaahlawat14@gmail.com	addressesdikshaahlawat14@gmail.com	PROPN
cana-3308	2	84	,	,	PUNCT
cana-3308	2	85	manojantil18@gmail.com	manojantil18@gmail.com	X
cana-3308	2	86	(	(	PUNCT
cana-3308	2	87	*	*	PUNCT
cana-3308	2	88	corresponding	correspond	VERB
cana-3308	2	89	author	author	NOUN
cana-3308	2	90	)	)	PUNCT
cana-3308	2	91	article	article	NOUN
cana-3308	2	92	history	history	NOUN
cana-3308	2	93	:	:	PUNCT
cana-3308	2	94	received	receive	VERB
cana-3308	2	95	:	:	PUNCT
cana-3308	2	96	22	22	NUM
cana-3308	2	97	-	-	SYM
cana-3308	2	98	10	10	NUM
cana-3308	2	99	-	-	PUNCT
cana-3308	2	100	2024	2024	NUM
cana-3308	2	101	revised	revise	VERB
cana-3308	2	102	:	:	PUNCT
cana-3308	2	103	06	06	NUM
cana-3308	2	104	-	-	SYM
cana-3308	2	105	12	12	NUM
cana-3308	2	106	-	-	PUNCT
cana-3308	2	107	2024	2024	NUM
cana-3308	2	108	accepted	accept	VERB
cana-3308	2	109	:	:	PUNCT
cana-3308	2	110	13	13	NUM
cana-3308	2	111	-	-	SYM
cana-3308	2	112	12	12	NUM
cana-3308	2	113	-	-	PUNCT
cana-3308	2	114	2024	2024	NUM
cana-3308	2	115	abstract	abstract	NOUN
cana-3308	2	116	:	:	PUNCT
cana-3308	2	117	in	in	ADP
cana-3308	2	118	this	this	DET
cana-3308	2	119	paper	paper	NOUN
cana-3308	2	120	,	,	PUNCT
cana-3308	2	121	we	we	PRON
cana-3308	2	122	explore	explore	VERB
cana-3308	2	123	a	a	DET
cana-3308	2	124	generalization	generalization	NOUN
cana-3308	2	125	of	of	ADP
cana-3308	2	126	(	(	PUNCT
cana-3308	2	127	𝛼	𝛼	PROPN
cana-3308	2	128	,	,	PUNCT
cana-3308	2	129	𝜓)geraghty	𝜓)geraghty	NOUN
cana-3308	2	130	contractions	contraction	NOUN
cana-3308	2	131	and	and	CCONJ
cana-3308	2	132	investigate	investigate	VERB
cana-3308	2	133	the	the	DET
cana-3308	2	134	existence	existence	NOUN
cana-3308	2	135	and	and	CCONJ
cana-3308	2	136	uniqueness	uniqueness	NOUN
cana-3308	2	137	of	of	ADP
cana-3308	2	138	fixed	fix	VERB
cana-3308	2	139	points	point	NOUN
cana-3308	2	140	for	for	ADP
cana-3308	2	141	mappings	mapping	NOUN
cana-3308	2	142	satisfying	satisfy	VERB
cana-3308	2	143	this	this	DET
cana-3308	2	144	condition	condition	NOUN
cana-3308	2	145	.	.	PUNCT
cana-3308	3	1	the	the	DET
cana-3308	3	2	study	study	NOUN
cana-3308	3	3	extends	extend	VERB
cana-3308	3	4	,	,	PUNCT
cana-3308	3	5	improves	improve	VERB
cana-3308	3	6	,	,	PUNCT
cana-3308	3	7	and	and	CCONJ
cana-3308	3	8	generalizes	generalize	VERB
cana-3308	3	9	some	some	DET
cana-3308	3	10	earlier	early	ADJ
cana-3308	3	11	results	result	NOUN
cana-3308	3	12	in	in	ADP
cana-3308	3	13	the	the	DET
cana-3308	3	14	literature	literature	NOUN
cana-3308	3	15	on	on	ADP
cana-3308	3	16	this	this	DET
cana-3308	3	17	topic	topic	NOUN
cana-3308	3	18	.	.	PUNCT
cana-3308	4	1	we	we	PRON
cana-3308	4	2	consider	consider	VERB
cana-3308	4	3	the	the	DET
cana-3308	4	4	interplay	interplay	NOUN
cana-3308	4	5	of	of	ADP
cana-3308	4	6	three	three	NUM
cana-3308	4	7	parameters	parameter	NOUN
cana-3308	4	8	:	:	PUNCT
cana-3308	4	9	𝛼	𝛼	X
cana-3308	4	10	,	,	PUNCT
cana-3308	4	11	𝜓	𝜓	NOUN
cana-3308	4	12	and	and	CCONJ
cana-3308	4	13	𝜑	𝜑	PROPN
cana-3308	4	14	,	,	PUNCT
cana-3308	4	15	which	which	PRON
cana-3308	4	16	play	play	VERB
cana-3308	4	17	a	a	DET
cana-3308	4	18	crucial	crucial	ADJ
cana-3308	4	19	role	role	NOUN
cana-3308	4	20	in	in	ADP
cana-3308	4	21	defining	define	VERB
cana-3308	4	22	the	the	DET
cana-3308	4	23	contraction	contraction	NOUN
cana-3308	4	24	properties	property	NOUN
cana-3308	4	25	.	.	PUNCT
cana-3308	5	1	our	our	PRON
cana-3308	5	2	goal	goal	NOUN
cana-3308	5	3	is	be	AUX
cana-3308	5	4	to	to	PART
cana-3308	5	5	establish	establish	VERB
cana-3308	5	6	fixed	fix	VERB
cana-3308	5	7	point	point	NOUN
cana-3308	5	8	theorems	theorem	NOUN
cana-3308	5	9	that	that	PRON
cana-3308	5	10	encompass	encompass	VERB
cana-3308	5	11	various	various	ADJ
cana-3308	5	12	scenarios	scenario	NOUN
cana-3308	5	13	within	within	ADP
cana-3308	5	14	bipolar	bipolar	ADJ
cana-3308	5	15	metric	metric	ADJ
cana-3308	5	16	spaces	space	NOUN
cana-3308	5	17	.	.	PUNCT
cana-3308	6	1	keywords	keyword	NOUN
cana-3308	6	2	:	:	PUNCT
cana-3308	6	3	fixed	fixed	ADJ
cana-3308	6	4	point	point	NOUN
cana-3308	6	5	,	,	PUNCT
cana-3308	6	6	(	(	PUNCT
cana-3308	6	7	𝛼	𝛼	X
cana-3308	6	8	,	,	PUNCT
cana-3308	6	9	𝜓	𝜓	NOUN
cana-3308	6	10	,	,	PUNCT
cana-3308	6	11	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	6	12	contraction	contraction	NOUN
cana-3308	6	13	mappings	mapping	NOUN
cana-3308	6	14	,	,	PUNCT
cana-3308	6	15	covariant	covariant	NOUN
cana-3308	6	16	and	and	CCONJ
cana-3308	6	17	contravariant	contravariant	ADJ
cana-3308	6	18	mappings	mapping	NOUN
cana-3308	6	19	,	,	PUNCT
cana-3308	6	20	bipolar	bipolar	ADJ
cana-3308	6	21	metric	metric	ADJ
cana-3308	6	22	space	space	NOUN
cana-3308	6	23	.	.	PUNCT
cana-3308	7	1	2020	2020	NUM
cana-3308	7	2	msc	msc	PROPN
cana-3308	7	3	:	:	PUNCT
cana-3308	7	4	47h10	47h10	NUM
cana-3308	7	5	,	,	PUNCT
cana-3308	7	6	54h25	54h25	NUM
cana-3308	7	7	1	1	NUM
cana-3308	7	8	.	.	PUNCT
cana-3308	7	9	introduction	introduction	NOUN
cana-3308	7	10	the	the	DET
cana-3308	7	11	concept	concept	NOUN
cana-3308	7	12	of	of	ADP
cana-3308	7	13	fixed	fix	VERB
cana-3308	7	14	points	point	NOUN
cana-3308	7	15	is	be	AUX
cana-3308	7	16	fundamental	fundamental	ADJ
cana-3308	7	17	in	in	ADP
cana-3308	7	18	mathematics	mathematic	NOUN
cana-3308	7	19	,	,	PUNCT
cana-3308	7	20	and	and	CCONJ
cana-3308	7	21	it	it	PRON
cana-3308	7	22	arises	arise	VERB
cana-3308	7	23	in	in	ADP
cana-3308	7	24	diverse	diverse	ADJ
cana-3308	7	25	fields	field	NOUN
cana-3308	7	26	such	such	ADJ
cana-3308	7	27	as	as	ADP
cana-3308	7	28	differential	differential	ADJ
cana-3308	7	29	equations	equation	NOUN
cana-3308	7	30	and	and	CCONJ
cana-3308	7	31	optimization	optimization	NOUN
cana-3308	7	32	.	.	PUNCT
cana-3308	8	1	in	in	ADP
cana-3308	8	2	1922	1922	NUM
cana-3308	8	3	,	,	PUNCT
cana-3308	8	4	banach	banach	NOUN
cana-3308	8	5	[	[	X
cana-3308	8	6	2	2	NUM
cana-3308	8	7	]	]	PUNCT
cana-3308	8	8	introduced	introduce	VERB
cana-3308	8	9	banach	banach	NOUN
cana-3308	8	10	contraction	contraction	NOUN
cana-3308	8	11	principle	principle	NOUN
cana-3308	8	12	as	as	ADP
cana-3308	8	13	the	the	DET
cana-3308	8	14	first	first	ADJ
cana-3308	8	15	constructive	constructive	ADJ
cana-3308	8	16	method	method	NOUN
cana-3308	8	17	to	to	PART
cana-3308	8	18	get	get	VERB
cana-3308	8	19	a	a	DET
cana-3308	8	20	fixed	fix	VERB
cana-3308	8	21	point	point	NOUN
cana-3308	8	22	for	for	ADP
cana-3308	8	23	a	a	DET
cana-3308	8	24	self	self	NOUN
cana-3308	8	25	map	map	NOUN
cana-3308	8	26	on	on	ADP
cana-3308	8	27	a	a	DET
cana-3308	8	28	complete	complete	ADJ
cana-3308	8	29	metric	metric	ADJ
cana-3308	8	30	space	space	NOUN
cana-3308	8	31	.	.	PUNCT
cana-3308	9	1	continuation	continuation	NOUN
cana-3308	9	2	of	of	ADP
cana-3308	9	3	this	this	PRON
cana-3308	9	4	,	,	PUNCT
cana-3308	9	5	in	in	ADP
cana-3308	9	6	1973	1973	NUM
cana-3308	9	7	,	,	PUNCT
cana-3308	9	8	geraghty	geraghty	PROPN
cana-3308	10	1	[	[	X
cana-3308	10	2	8	8	NUM
cana-3308	10	3	]	]	PUNCT
cana-3308	10	4	gives	give	VERB
cana-3308	10	5	an	an	DET
cana-3308	10	6	extension	extension	NOUN
cana-3308	10	7	of	of	ADP
cana-3308	10	8	the	the	DET
cana-3308	10	9	banach	banach	NOUN
cana-3308	10	10	contraction	contraction	NOUN
cana-3308	10	11	mapping	mapping	NOUN
cana-3308	10	12	principle	principle	NOUN
cana-3308	10	13	,	,	PUNCT
cana-3308	10	14	provides	provide	VERB
cana-3308	10	15	a	a	DET
cana-3308	10	16	powerful	powerful	ADJ
cana-3308	10	17	tool	tool	NOUN
cana-3308	10	18	for	for	ADP
cana-3308	10	19	proving	prove	VERB
cana-3308	10	20	the	the	DET
cana-3308	10	21	existence	existence	NOUN
cana-3308	10	22	of	of	ADP
cana-3308	10	23	fixed	fix	VERB
cana-3308	10	24	points	point	NOUN
cana-3308	10	25	.	.	PUNCT
cana-3308	11	1	specifically	specifically	ADV
cana-3308	11	2	,	,	PUNCT
cana-3308	11	3	geraghty	geraghty	PROPN
cana-3308	11	4	’s	’s	PART
cana-3308	11	5	result	result	NOUN
cana-3308	11	6	ensures	ensure	VERB
cana-3308	11	7	a	a	DET
cana-3308	11	8	unique	unique	ADJ
cana-3308	11	9	fixed	fix	VERB
cana-3308	11	10	point	point	NOUN
cana-3308	11	11	under	under	ADP
cana-3308	11	12	certain	certain	ADJ
cana-3308	11	13	contractive	contractive	ADJ
cana-3308	11	14	conditions	condition	NOUN
cana-3308	11	15	.	.	PUNCT
cana-3308	12	1	many	many	ADJ
cana-3308	12	2	authors	author	NOUN
cana-3308	12	3	generalized	generalize	VERB
cana-3308	12	4	his	his	PRON
cana-3308	12	5	work	work	NOUN
cana-3308	12	6	,	,	PUNCT
cana-3308	12	7	see	see	VERB
cana-3308	12	8	[	[	X
cana-3308	12	9	1,4,6,7,13	1,4,6,7,13	X
cana-3308	12	10	]	]	PUNCT
cana-3308	12	11	.	.	PUNCT
cana-3308	13	1	in	in	ADP
cana-3308	13	2	2012	2012	NUM
cana-3308	13	3	,	,	PUNCT
cana-3308	13	4	samet	samet	PROPN
cana-3308	13	5	et	et	PROPN
cana-3308	13	6	al	al	PROPN
cana-3308	13	7	.	.	PUNCT
cana-3308	14	1	[	[	X
cana-3308	14	2	14	14	NUM
cana-3308	14	3	]	]	PUNCT
cana-3308	14	4	introduced	introduce	VERB
cana-3308	14	5	the	the	DET
cana-3308	14	6	concepts	concept	NOUN
cana-3308	14	7	of	of	ADP
cana-3308	14	8	𝛼-contractive	𝛼-contractive	PROPN
cana-3308	14	9	and	and	CCONJ
cana-3308	14	10	𝛼-admissible	𝛼-admissible	ADJ
cana-3308	14	11	mappings	mapping	NOUN
cana-3308	14	12	and	and	CCONJ
cana-3308	14	13	proved	prove	VERB
cana-3308	14	14	various	various	ADJ
cana-3308	14	15	fixed	fix	VERB
cana-3308	14	16	point	point	NOUN
cana-3308	14	17	theorems	theorem	NOUN
cana-3308	14	18	of	of	ADP
cana-3308	14	19	𝛼admissible	𝛼admissible	ADJ
cana-3308	14	20	contractive	contractive	ADJ
cana-3308	14	21	mappings	mapping	NOUN
cana-3308	14	22	in	in	ADP
cana-3308	14	23	complete	complete	ADJ
cana-3308	14	24	metric	metric	ADJ
cana-3308	14	25	spaces	space	NOUN
cana-3308	14	26	.	.	PUNCT
cana-3308	15	1	recently	recently	ADV
cana-3308	15	2	,	,	PUNCT
cana-3308	15	3	in	in	ADP
cana-3308	15	4	2015	2015	NUM
cana-3308	15	5	,	,	PUNCT
cana-3308	15	6	chandok	chandok	NOUN
cana-3308	15	7	[	[	X
cana-3308	15	8	4	4	X
cana-3308	15	9	]	]	PUNCT
cana-3308	15	10	introduced	introduce	VERB
cana-3308	15	11	the	the	DET
cana-3308	15	12	concept	concept	NOUN
cana-3308	15	13	of	of	ADP
cana-3308	15	14	(	(	PUNCT
cana-3308	15	15	𝛼	𝛼	PROPN
cana-3308	15	16	,	,	PUNCT
cana-3308	15	17	𝛽)admissible	𝛽)admissible	ADJ
cana-3308	15	18	geraghty	geraghty	PROPN
cana-3308	15	19	type	type	NOUN
cana-3308	15	20	contractive	contractive	ADJ
cana-3308	15	21	mappings	mapping	NOUN
cana-3308	15	22	and	and	CCONJ
cana-3308	15	23	proved	prove	VERB
cana-3308	15	24	some	some	DET
cana-3308	15	25	fixed	fix	VERB
cana-3308	15	26	point	point	NOUN
cana-3308	15	27	theorems	theorem	NOUN
cana-3308	15	28	of	of	ADP
cana-3308	15	29	such	such	ADJ
cana-3308	15	30	kind	kind	NOUN
cana-3308	15	31	of	of	ADP
cana-3308	15	32	mappings	mapping	NOUN
cana-3308	15	33	in	in	ADP
cana-3308	15	34	complete	complete	ADJ
cana-3308	15	35	metric	metric	ADJ
cana-3308	15	36	spaces	space	NOUN
cana-3308	15	37	.	.	PUNCT
cana-3308	16	1	some	some	DET
cana-3308	16	2	researcher	researcher	NOUN
cana-3308	16	3	extended	extend	VERB
cana-3308	16	4	their	their	PRON
cana-3308	16	5	work	work	NOUN
cana-3308	16	6	in	in	ADP
cana-3308	16	7	various	various	ADJ
cana-3308	16	8	spaces	space	NOUN
cana-3308	16	9	[	[	X
cana-3308	16	10	9	9	NUM
cana-3308	16	11	-	-	SYM
cana-3308	16	12	16	16	NUM
cana-3308	16	13	]	]	PUNCT
cana-3308	16	14	.	.	PUNCT
cana-3308	17	1	in	in	ADP
cana-3308	17	2	2019	2019	NUM
cana-3308	17	3	,	,	PUNCT
cana-3308	17	4	karapinar	karapinar	VERB
cana-3308	17	5	et	et	PROPN
cana-3308	17	6	al	al	PROPN
cana-3308	17	7	.	.	PUNCT
cana-3308	18	1	[	[	X
cana-3308	18	2	7	7	X
cana-3308	18	3	]	]	PUNCT
cana-3308	18	4	introduced	introduce	VERB
cana-3308	18	5	the	the	DET
cana-3308	18	6	notion	notion	NOUN
cana-3308	18	7	𝜑-geraghty	𝜑-geraghty	NUM
cana-3308	18	8	and	and	CCONJ
cana-3308	18	9	ciric	ciric	ADJ
cana-3308	18	10	type	type	NOUN
cana-3308	18	11	𝜑-geraghty	𝜑-geraghty	VERB
cana-3308	18	12	contractive	contractive	ADJ
cana-3308	18	13	mappings	mapping	NOUN
cana-3308	18	14	in	in	ADP
cana-3308	18	15	complete	complete	ADJ
cana-3308	18	16	metric	metric	ADJ
cana-3308	18	17	space	space	NOUN
cana-3308	18	18	and	and	CCONJ
cana-3308	18	19	proved	prove	VERB
cana-3308	18	20	some	some	DET
cana-3308	18	21	fixed	fix	VERB
cana-3308	18	22	points	point	NOUN
cana-3308	18	23	theorems	theorem	NOUN
cana-3308	18	24	and	and	CCONJ
cana-3308	18	25	uniqueness	uniqueness	NOUN
cana-3308	18	26	of	of	ADP
cana-3308	18	27	fixed	fix	VERB
cana-3308	18	28	points	point	NOUN
cana-3308	18	29	.	.	PUNCT
cana-3308	19	1	to	to	PART
cana-3308	19	2	get	get	VERB
cana-3308	19	3	a	a	DET
cana-3308	19	4	new	new	ADJ
cana-3308	19	5	approach	approach	NOUN
cana-3308	19	6	for	for	ADP
cana-3308	19	7	fixed	fix	VERB
cana-3308	19	8	point	point	NOUN
cana-3308	19	9	results	result	NOUN
cana-3308	19	10	in	in	ADP
cana-3308	19	11	2016	2016	NUM
cana-3308	19	12	,	,	PUNCT
cana-3308	19	13	mutlu	mutlu	PROPN
cana-3308	19	14	and	and	CCONJ
cana-3308	19	15	gürdal	gürdal	ADJ
cana-3308	19	16	[	[	X
cana-3308	19	17	10	10	NUM
cana-3308	19	18	]	]	PUNCT
cana-3308	19	19	introduced	introduce	VERB
cana-3308	19	20	the	the	DET
cana-3308	19	21	concept	concept	NOUN
cana-3308	19	22	of	of	ADP
cana-3308	19	23	bipolar	bipolar	ADJ
cana-3308	19	24	metric	metric	ADJ
cana-3308	19	25	space	space	NOUN
cana-3308	19	26	.	.	PUNCT
cana-3308	20	1	the	the	DET
cana-3308	20	2	major	major	ADJ
cana-3308	20	3	difference	difference	NOUN
cana-3308	20	4	between	between	ADP
cana-3308	20	5	the	the	DET
cana-3308	20	6	previously	previously	ADV
cana-3308	20	7	defined	define	VERB
cana-3308	20	8	spaces	space	NOUN
cana-3308	20	9	and	and	CCONJ
cana-3308	20	10	bipolar	bipolar	ADJ
cana-3308	20	11	is	be	AUX
cana-3308	20	12	of	of	ADP
cana-3308	20	13	mailto:dikshaahlawat14@gmail.com	mailto:dikshaahlawat14@gmail.com	NOUN
cana-3308	20	14	mailto:manojantil18@gmail.com	mailto:manojantil18@gmail.com	PROPN
cana-3308	20	15	communications	communication	NOUN
cana-3308	20	16	on	on	ADP
cana-3308	20	17	applied	apply	VERB
cana-3308	20	18	nonlinear	nonlinear	ADJ
cana-3308	20	19	analysis	analysis	NOUN
cana-3308	20	20	issn	issn	NOUN
cana-3308	20	21	:	:	PUNCT
cana-3308	20	22	1074	1074	NUM
cana-3308	20	23	-	-	PUNCT
cana-3308	20	24	133x	133x	NUM
cana-3308	20	25	vol	vol	NOUN
cana-3308	20	26	32	32	NUM
cana-3308	20	27	no	no	NOUN
cana-3308	20	28	.	.	PUNCT
cana-3308	21	1	6s	6s	NUM
cana-3308	21	2	(	(	PUNCT
cana-3308	21	3	2025	2025	NUM
cana-3308	21	4	)	)	PUNCT
cana-3308	21	5	440	440	NUM
cana-3308	21	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-3308	21	7	distance	distance	NOUN
cana-3308	21	8	function	function	NOUN
cana-3308	21	9	.	.	PUNCT
cana-3308	22	1	in	in	ADP
cana-3308	22	2	bipolar	bipolar	ADJ
cana-3308	22	3	metric	metric	ADJ
cana-3308	22	4	space	space	NOUN
cana-3308	22	5	,	,	PUNCT
cana-3308	22	6	the	the	DET
cana-3308	22	7	distance	distance	NOUN
cana-3308	22	8	function	function	NOUN
cana-3308	22	9	is	be	AUX
cana-3308	22	10	from	from	ADP
cana-3308	22	11	the	the	DET
cana-3308	22	12	cartesian	cartesian	ADJ
cana-3308	22	13	product	product	NOUN
cana-3308	22	14	of	of	ADP
cana-3308	22	15	two	two	NUM
cana-3308	22	16	different	different	ADJ
cana-3308	22	17	sets	set	NOUN
cana-3308	22	18	to	to	ADP
cana-3308	22	19	non	non	ADJ
cana-3308	22	20	-	-	ADJ
cana-3308	22	21	negative	negative	ADJ
cana-3308	22	22	real	real	ADJ
cana-3308	22	23	numbers	number	NOUN
cana-3308	22	24	.	.	PUNCT
cana-3308	23	1	since	since	SCONJ
cana-3308	23	2	then	then	ADV
cana-3308	23	3	,	,	PUNCT
cana-3308	23	4	many	many	ADJ
cana-3308	23	5	authors	author	NOUN
cana-3308	23	6	have	have	AUX
cana-3308	23	7	proved	prove	VERB
cana-3308	23	8	several	several	ADJ
cana-3308	23	9	fixed	fix	VERB
cana-3308	23	10	point	point	NOUN
cana-3308	23	11	results	result	NOUN
cana-3308	23	12	in	in	ADP
cana-3308	23	13	bipolar	bipolar	ADJ
cana-3308	23	14	metric	metric	ADJ
cana-3308	23	15	space	space	NOUN
cana-3308	23	16	see	see	VERB
cana-3308	23	17	[	[	X
cana-3308	23	18	5	5	NUM
cana-3308	23	19	]	]	PUNCT
cana-3308	23	20	,	,	PUNCT
cana-3308	24	1	[	[	X
cana-3308	24	2	9	9	NUM
cana-3308	24	3	-	-	SYM
cana-3308	24	4	11	11	NUM
cana-3308	24	5	]	]	PUNCT
cana-3308	24	6	,	,	PUNCT
cana-3308	24	7	[	[	X
cana-3308	24	8	13	13	NUM
cana-3308	24	9	]	]	PUNCT
cana-3308	24	10	.	.	PUNCT
cana-3308	25	1	motivated	motivate	VERB
cana-3308	25	2	by	by	ADP
cana-3308	25	3	the	the	DET
cana-3308	25	4	work	work	NOUN
cana-3308	25	5	of	of	ADP
cana-3308	25	6	abduletif	abduletif	NOUN
cana-3308	25	7	et	et	NOUN
cana-3308	25	8	al	al	PROPN
cana-3308	25	9	.	.	PUNCT
cana-3308	26	1	[	[	X
cana-3308	26	2	1	1	NUM
cana-3308	26	3	]	]	PUNCT
cana-3308	26	4	,	,	PUNCT
cana-3308	26	5	the	the	DET
cana-3308	26	6	main	main	ADJ
cana-3308	26	7	objective	objective	NOUN
cana-3308	26	8	of	of	ADP
cana-3308	26	9	this	this	DET
cana-3308	26	10	manuscript	manuscript	NOUN
cana-3308	26	11	is	be	AUX
cana-3308	26	12	to	to	PART
cana-3308	26	13	prove	prove	VERB
cana-3308	26	14	some	some	DET
cana-3308	26	15	fixed	fix	VERB
cana-3308	26	16	points	point	NOUN
cana-3308	26	17	results	result	NOUN
cana-3308	26	18	and	and	CCONJ
cana-3308	26	19	their	their	PRON
cana-3308	26	20	uniqueness	uniqueness	NOUN
cana-3308	26	21	for	for	ADP
cana-3308	26	22	(	(	PUNCT
cana-3308	26	23	𝛼	𝛼	PROPN
cana-3308	26	24	,	,	PUNCT
cana-3308	26	25	𝜓	𝜓	NOUN
cana-3308	26	26	,	,	PUNCT
cana-3308	26	27	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	26	28	contraction	contraction	NOUN
cana-3308	26	29	mapping	mapping	NOUN
cana-3308	26	30	in	in	ADP
cana-3308	26	31	complete	complete	ADJ
cana-3308	26	32	bipolar	bipolar	ADJ
cana-3308	26	33	metric	metric	ADJ
cana-3308	26	34	spaces	space	NOUN
cana-3308	26	35	.	.	PUNCT
cana-3308	27	1	furthermore	furthermore	ADV
cana-3308	27	2	,	,	PUNCT
cana-3308	27	3	we	we	PRON
cana-3308	27	4	offer	offer	VERB
cana-3308	27	5	illustrations	illustration	NOUN
cana-3308	27	6	to	to	PART
cana-3308	27	7	support	support	VERB
cana-3308	27	8	our	our	PRON
cana-3308	27	9	essential	essential	ADJ
cana-3308	27	10	findings	finding	NOUN
cana-3308	27	11	.	.	PUNCT
cana-3308	28	1	2	2	X
cana-3308	28	2	.	.	X
cana-3308	28	3	preliminaries	preliminary	NOUN
cana-3308	28	4	we	we	PRON
cana-3308	28	5	need	need	VERB
cana-3308	28	6	to	to	PART
cana-3308	28	7	introduce	introduce	VERB
cana-3308	28	8	some	some	DET
cana-3308	28	9	new	new	ADJ
cana-3308	28	10	notations	notation	NOUN
cana-3308	28	11	and	and	CCONJ
cana-3308	28	12	terminology	terminology	NOUN
cana-3308	28	13	and	and	CCONJ
cana-3308	28	14	provide	provide	VERB
cana-3308	28	15	some	some	DET
cana-3308	28	16	fundamental	fundamental	ADJ
cana-3308	28	17	definitions	definition	NOUN
cana-3308	28	18	which	which	PRON
cana-3308	28	19	is	be	AUX
cana-3308	28	20	used	use	VERB
cana-3308	28	21	for	for	ADP
cana-3308	28	22	the	the	DET
cana-3308	28	23	fixed	fix	VERB
cana-3308	28	24	point	point	NOUN
cana-3308	28	25	theorems	theorem	NOUN
cana-3308	28	26	for	for	ADP
cana-3308	28	27	(	(	PUNCT
cana-3308	28	28	𝛼	𝛼	PROPN
cana-3308	28	29	,	,	PUNCT
cana-3308	28	30	𝜓	𝜓	NOUN
cana-3308	28	31	,	,	PUNCT
cana-3308	28	32	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	28	33	contraction	contraction	NOUN
cana-3308	28	34	mappings	mapping	NOUN
cana-3308	28	35	in	in	ADP
cana-3308	28	36	bipolar	bipolar	ADJ
cana-3308	28	37	metric	metric	ADJ
cana-3308	28	38	spaces	space	NOUN
cana-3308	28	39	.	.	PUNCT
cana-3308	29	1	definition	definition	NOUN
cana-3308	29	2	2.1	2.1	NUM
cana-3308	29	3	.	.	PUNCT
cana-3308	30	1	in	in	ADP
cana-3308	30	2	2016	2016	NUM
cana-3308	30	3	,	,	PUNCT
cana-3308	30	4	mutlu	mutlu	PROPN
cana-3308	30	5	and	and	CCONJ
cana-3308	30	6	gürdal	gürdal	ADJ
cana-3308	30	7	[	[	X
cana-3308	30	8	10	10	NUM
cana-3308	30	9	]	]	PUNCT
cana-3308	30	10	introduced	introduce	VERB
cana-3308	30	11	the	the	DET
cana-3308	30	12	concept	concept	NOUN
cana-3308	30	13	of	of	ADP
cana-3308	30	14	bipolar	bipolar	ADJ
cana-3308	30	15	metric	metric	ADJ
cana-3308	30	16	space	space	NOUN
cana-3308	30	17	.	.	PUNCT
cana-3308	31	1	let	let	VERB
cana-3308	31	2	𝑋	𝑋	NOUN
cana-3308	31	3	and	and	CCONJ
cana-3308	31	4	𝑌	𝑌	PROPN
cana-3308	31	5	are	be	AUX
cana-3308	31	6	two	two	NUM
cana-3308	31	7	non	non	ADJ
cana-3308	31	8	-	-	ADJ
cana-3308	31	9	empty	empty	ADJ
cana-3308	31	10	sets	set	NOUN
cana-3308	31	11	and	and	CCONJ
cana-3308	31	12	𝑑	𝑑	ADP
cana-3308	31	13	∶	∶	NOUN
cana-3308	31	14	𝑋	𝑋	NOUN
cana-3308	31	15	×	×	NOUN
cana-3308	31	16	𝑌	𝑌	PROPN
cana-3308	31	17	→	→	SYM
cana-3308	31	18	[	[	X
cana-3308	31	19	0	0	NUM
cana-3308	31	20	,	,	PUNCT
cana-3308	31	21	∞	∞	PROPN
cana-3308	31	22	)	)	PUNCT
cana-3308	31	23	be	be	VERB
cana-3308	31	24	a	a	DET
cana-3308	31	25	function	function	NOUN
cana-3308	31	26	satisfying	satisfy	VERB
cana-3308	31	27	the	the	DET
cana-3308	31	28	following	follow	VERB
cana-3308	31	29	conditions	condition	NOUN
cana-3308	31	30	:	:	PUNCT
cana-3308	31	31	(	(	PUNCT
cana-3308	31	32	bp1	bp1	PROPN
cana-3308	31	33	)	)	PUNCT
cana-3308	31	34	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	31	35	,	,	PUNCT
cana-3308	31	36	𝑦	𝑦	X
cana-3308	31	37	)	)	PUNCT
cana-3308	31	38	=	=	SYM
cana-3308	31	39	0	0	PUNCT
cana-3308	32	1	if	if	SCONJ
cana-3308	32	2	and	and	CCONJ
cana-3308	32	3	only	only	ADV
cana-3308	32	4	if	if	SCONJ
cana-3308	32	5	𝑥	𝑥	PRON
cana-3308	32	6	=	=	SYM
cana-3308	32	7	𝑦	𝑦	NOUN
cana-3308	32	8	,	,	PUNCT
cana-3308	32	9	where	where	SCONJ
cana-3308	32	10	(	(	PUNCT
cana-3308	32	11	𝑥	𝑥	NOUN
cana-3308	32	12	,	,	PUNCT
cana-3308	32	13	𝑦	𝑦	X
cana-3308	32	14	)	)	PUNCT
cana-3308	32	15	∈	∈	PROPN
cana-3308	32	16	𝑋	𝑋	PROPN
cana-3308	32	17	×	×	PROPN
cana-3308	32	18	𝑌	𝑌	PROPN
cana-3308	32	19	,	,	PUNCT
cana-3308	32	20	(	(	PUNCT
cana-3308	32	21	bp2	bp2	NOUN
cana-3308	32	22	)	)	PUNCT
cana-3308	32	23	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	32	24	,	,	PUNCT
cana-3308	32	25	𝑦	𝑦	X
cana-3308	32	26	)	)	PUNCT
cana-3308	32	27	=	=	SYM
cana-3308	32	28	𝑑(𝑦	𝑑(𝑦	NOUN
cana-3308	32	29	,	,	PUNCT
cana-3308	32	30	𝑥	𝑥	NOUN
cana-3308	32	31	)	)	PUNCT
cana-3308	32	32	for	for	ADP
cana-3308	32	33	all	all	PRON
cana-3308	32	34	𝑥	𝑥	PROPN
cana-3308	32	35	,	,	PUNCT
cana-3308	32	36	𝑦	𝑦	NOUN
cana-3308	32	37	∈	∈	NOUN
cana-3308	32	38	𝑋	𝑋	NOUN
cana-3308	32	39	∩	∩	ADJ
cana-3308	32	40	𝑌	𝑌	PROPN
cana-3308	32	41	,	,	PUNCT
cana-3308	32	42	(	(	PUNCT
cana-3308	32	43	bp3	bp3	NOUN
cana-3308	32	44	)	)	PUNCT
cana-3308	32	45	𝑑(𝑥1	𝑑(𝑥1	ADJ
cana-3308	32	46	,	,	PUNCT
cana-3308	32	47	𝑦2	𝑦2	NOUN
cana-3308	32	48	)	)	PUNCT
cana-3308	32	49	≤	≤	NOUN
cana-3308	32	50	𝑑(𝑥1	𝑑(𝑥1	ADJ
cana-3308	32	51	,	,	PUNCT
cana-3308	32	52	𝑦1	𝑦1	PROPN
cana-3308	32	53	)	)	PUNCT
cana-3308	32	54	+	+	CCONJ
cana-3308	32	55	𝑑(𝑥2	𝑑(𝑥2	NOUN
cana-3308	32	56	,	,	PUNCT
cana-3308	32	57	𝑦1	𝑦1	PROPN
cana-3308	32	58	)	)	PUNCT
cana-3308	32	59	+	+	CCONJ
cana-3308	32	60	𝑑(𝑥2	𝑑(𝑥2	NOUN
cana-3308	32	61	,	,	PUNCT
cana-3308	32	62	𝑦2	𝑦2	NOUN
cana-3308	32	63	)	)	PUNCT
cana-3308	32	64	for	for	ADP
cana-3308	32	65	all	all	DET
cana-3308	32	66	𝑥1	𝑥1	NOUN
cana-3308	32	67	,	,	PUNCT
cana-3308	32	68	𝑥2	𝑥2	NOUN
cana-3308	32	69	∈	∈	PROPN
cana-3308	32	70	𝑋	𝑋	NOUN
cana-3308	32	71	and	and	CCONJ
cana-3308	32	72	𝑦1	𝑦1	PROPN
cana-3308	32	73	,	,	PUNCT
cana-3308	32	74	𝑦2	𝑦2	PROPN
cana-3308	32	75	∈	∈	PROPN
cana-3308	32	76	𝑌.	𝑌.	PROPN
cana-3308	33	1	then	then	ADV
cana-3308	33	2	𝑑	𝑑	PROPN
cana-3308	33	3	is	be	AUX
cana-3308	33	4	called	call	VERB
cana-3308	33	5	bipolar	bipolar	ADJ
cana-3308	33	6	metric	metric	ADJ
cana-3308	33	7	and	and	CCONJ
cana-3308	33	8	(	(	PUNCT
cana-3308	33	9	𝑋	𝑋	PROPN
cana-3308	33	10	,	,	PUNCT
cana-3308	33	11	𝑌	𝑌	PROPN
cana-3308	33	12	,	,	PUNCT
cana-3308	33	13	𝑑	𝑑	NOUN
cana-3308	33	14	)	)	PUNCT
cana-3308	33	15	is	be	AUX
cana-3308	33	16	called	call	VERB
cana-3308	33	17	bipolar	bipolar	ADJ
cana-3308	33	18	metric	metric	ADJ
cana-3308	33	19	space	space	NOUN
cana-3308	33	20	.	.	PUNCT
cana-3308	34	1	if	if	SCONJ
cana-3308	34	2	𝑋	𝑋	PROPN
cana-3308	34	3	∩	∩	NOUN
cana-3308	34	4	𝑌	𝑌	PROPN
cana-3308	34	5	=	=	PUNCT
cana-3308	34	6	∅	∅	NOUN
cana-3308	34	7	,	,	PUNCT
cana-3308	34	8	then	then	ADV
cana-3308	34	9	space	space	NOUN
cana-3308	34	10	is	be	AUX
cana-3308	34	11	called	call	VERB
cana-3308	34	12	disjoint	disjoint	NOUN
cana-3308	34	13	otherwise	otherwise	ADV
cana-3308	34	14	joint	joint	ADJ
cana-3308	34	15	.	.	PUNCT
cana-3308	35	1	the	the	DET
cana-3308	35	2	set	set	NOUN
cana-3308	35	3	𝑋	𝑋	NOUN
cana-3308	35	4	is	be	AUX
cana-3308	35	5	called	call	VERB
cana-3308	35	6	left	left	ADJ
cana-3308	35	7	pole	pole	NOUN
cana-3308	35	8	and	and	CCONJ
cana-3308	35	9	𝑌	𝑌	PROPN
cana-3308	35	10	is	be	AUX
cana-3308	35	11	called	call	VERB
cana-3308	35	12	right	right	ADJ
cana-3308	35	13	pole	pole	NOUN
cana-3308	35	14	of	of	ADP
cana-3308	35	15	bipolar	bipolar	ADJ
cana-3308	35	16	metric	metric	ADJ
cana-3308	35	17	space	space	NOUN
cana-3308	35	18	(	(	PUNCT
cana-3308	35	19	𝑋	𝑋	PROPN
cana-3308	35	20	,	,	PUNCT
cana-3308	35	21	𝑌	𝑌	PROPN
cana-3308	35	22	,	,	PUNCT
cana-3308	35	23	𝑑	𝑑	NOUN
cana-3308	35	24	)	)	PUNCT
cana-3308	35	25	and	and	CCONJ
cana-3308	35	26	any	any	DET
cana-3308	35	27	element	element	NOUN
cana-3308	35	28	of	of	ADP
cana-3308	35	29	left	left	ADJ
cana-3308	35	30	pole	pole	NOUN
cana-3308	35	31	(	(	PUNCT
cana-3308	35	32	𝑋	𝑋	PROPN
cana-3308	35	33	)	)	PUNCT
cana-3308	35	34	,	,	PUNCT
cana-3308	35	35	right	right	ADJ
cana-3308	35	36	pole	pole	NOUN
cana-3308	35	37	(	(	PUNCT
cana-3308	35	38	𝑌	𝑌	PROPN
cana-3308	35	39	)	)	PUNCT
cana-3308	35	40	and	and	CCONJ
cana-3308	35	41	𝑋	𝑋	PROPN
cana-3308	35	42	∩	∩	NOUN
cana-3308	35	43	𝑌	𝑌	PROPN
cana-3308	35	44	is	be	AUX
cana-3308	35	45	called	call	VERB
cana-3308	35	46	left	left	ADJ
cana-3308	35	47	element	element	NOUN
cana-3308	35	48	,	,	PUNCT
cana-3308	35	49	right	right	ADJ
cana-3308	35	50	element	element	NOUN
cana-3308	35	51	and	and	CCONJ
cana-3308	35	52	central	central	ADJ
cana-3308	35	53	element	element	NOUN
cana-3308	35	54	respectively	respectively	ADV
cana-3308	35	55	.	.	PUNCT
cana-3308	36	1	definition	definition	NOUN
cana-3308	36	2	2.2	2.2	NUM
cana-3308	36	3	.	.	PUNCT
cana-3308	37	1	let	let	VERB
cana-3308	37	2	(	(	PUNCT
cana-3308	37	3	𝑋	𝑋	PROPN
cana-3308	37	4	,	,	PUNCT
cana-3308	37	5	𝑌	𝑌	PROPN
cana-3308	37	6	,	,	PUNCT
cana-3308	37	7	𝑑	𝑑	NOUN
cana-3308	37	8	)	)	PUNCT
cana-3308	37	9	be	be	VERB
cana-3308	37	10	a	a	DET
cana-3308	37	11	bipolar	bipolar	ADJ
cana-3308	37	12	metric	metric	ADJ
cana-3308	37	13	space	space	NOUN
cana-3308	37	14	.	.	PUNCT
cana-3308	38	1	then	then	ADV
cana-3308	38	2	any	any	DET
cana-3308	38	3	sequence	sequence	NOUN
cana-3308	38	4	(	(	PUNCT
cana-3308	38	5	𝑥𝑛	𝑥𝑛	NOUN
cana-3308	38	6	)	)	PUNCT
cana-3308	38	7	⊆	⊆	NUM
cana-3308	38	8	𝑋	𝑋	PROPN
cana-3308	38	9	is	be	AUX
cana-3308	38	10	called	call	VERB
cana-3308	38	11	left	left	ADJ
cana-3308	38	12	sequence	sequence	NOUN
cana-3308	38	13	and	and	CCONJ
cana-3308	38	14	is	be	AUX
cana-3308	38	15	said	say	VERB
cana-3308	38	16	to	to	PART
cana-3308	38	17	be	be	AUX
cana-3308	38	18	convergent	convergent	ADJ
cana-3308	38	19	to	to	ADP
cana-3308	38	20	right	right	ADJ
cana-3308	38	21	element	element	NOUN
cana-3308	38	22	say	say	VERB
cana-3308	38	23	‘	'	PUNCT
cana-3308	38	24	𝑦	𝑦	NOUN
cana-3308	38	25	’	'	PUNCT
cana-3308	38	26	if	if	SCONJ
cana-3308	38	27	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	38	28	,	,	PUNCT
cana-3308	38	29	𝑦	𝑦	NOUN
cana-3308	38	30	)	)	PUNCT
cana-3308	38	31	→	→	SYM
cana-3308	38	32	0	0	NUM
cana-3308	38	33	as	as	ADP
cana-3308	38	34	𝑛	𝑛	PROPN
cana-3308	38	35	→	→	SYM
cana-3308	38	36	∞.	∞.	PROPN
cana-3308	38	37	similarly	similarly	ADV
cana-3308	38	38	,	,	PUNCT
cana-3308	38	39	a	a	DET
cana-3308	38	40	right	right	ADJ
cana-3308	38	41	sequence	sequence	NOUN
cana-3308	38	42	(	(	PUNCT
cana-3308	38	43	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	38	44	)	)	PUNCT
cana-3308	38	45	⊆	⊆	NUM
cana-3308	38	46	𝑌	𝑌	PROPN
cana-3308	38	47	is	be	AUX
cana-3308	38	48	said	say	VERB
cana-3308	38	49	to	to	PART
cana-3308	38	50	be	be	AUX
cana-3308	38	51	convergent	convergent	ADJ
cana-3308	38	52	to	to	ADP
cana-3308	38	53	a	a	DET
cana-3308	38	54	left	left	ADJ
cana-3308	38	55	element	element	NOUN
cana-3308	38	56	say	say	VERB
cana-3308	38	57	‘	'	PUNCT
cana-3308	38	58	𝑥	𝑥	NOUN
cana-3308	38	59	’	'	PUNCT
cana-3308	38	60	if	if	SCONJ
cana-3308	38	61	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	38	62	,	,	PUNCT
cana-3308	38	63	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	38	64	)	)	PUNCT
cana-3308	38	65	→	→	SYM
cana-3308	38	66	0	0	NUM
cana-3308	38	67	as	as	ADP
cana-3308	38	68	𝑛	𝑛	PROPN
cana-3308	38	69	→	→	SYM
cana-3308	38	70	∞.	∞.	PROPN
cana-3308	38	71	definition	definition	NOUN
cana-3308	38	72	2.3	2.3	NUM
cana-3308	38	73	.	.	PUNCT
cana-3308	39	1	let	let	VERB
cana-3308	39	2	(	(	PUNCT
cana-3308	39	3	𝑋1	𝑋1	PROPN
cana-3308	39	4	,	,	PUNCT
cana-3308	39	5	𝑌1	𝑌1	NOUN
cana-3308	39	6	,	,	PUNCT
cana-3308	39	7	𝑑1	𝑑1	NOUN
cana-3308	39	8	)	)	PUNCT
cana-3308	39	9	and	and	CCONJ
cana-3308	39	10	(	(	PUNCT
cana-3308	39	11	𝑋2	𝑋2	PROPN
cana-3308	39	12	,	,	PUNCT
cana-3308	39	13	𝑌2	𝑌2	NOUN
cana-3308	39	14	,	,	PUNCT
cana-3308	39	15	𝑑2	𝑑2	NOUN
cana-3308	39	16	)	)	PUNCT
cana-3308	39	17	be	be	AUX
cana-3308	39	18	two	two	NUM
cana-3308	39	19	bipolar	bipolar	ADJ
cana-3308	39	20	metric	metric	ADJ
cana-3308	39	21	spaces	space	NOUN
cana-3308	39	22	.	.	PUNCT
cana-3308	40	1	let	let	VERB
cana-3308	40	2	𝑇	𝑇	PROPN
cana-3308	40	3	∶	∶	VERB
cana-3308	40	4	𝑋1	𝑋1	NOUN
cana-3308	40	5	∪	∪	ADV
cana-3308	40	6	𝑌1	𝑌1	PROPN
cana-3308	40	7	→	→	SYM
cana-3308	40	8	𝑋2	𝑋2	VERB
cana-3308	40	9	∪	∪	ADJ
cana-3308	40	10	𝑌2	𝑌2	NOUN
cana-3308	40	11	be	be	VERB
cana-3308	40	12	a	a	DET
cana-3308	40	13	function	function	NOUN
cana-3308	40	14	such	such	ADJ
cana-3308	40	15	that	that	SCONJ
cana-3308	40	16	(	(	PUNCT
cana-3308	40	17	i)if	i)if	PROPN
cana-3308	40	18	𝑇(𝑋1	𝑇(𝑋1	ADJ
cana-3308	40	19	)	)	PUNCT
cana-3308	40	20	⊆	⊆	NUM
cana-3308	40	21	𝑋2	𝑋2	ADJ
cana-3308	40	22	and	and	CCONJ
cana-3308	40	23	𝑇(𝑌1	𝑇(𝑌1	NOUN
cana-3308	40	24	)	)	PUNCT
cana-3308	40	25	⊆	⊆	NUM
cana-3308	40	26	𝑌2	𝑌2	NOUN
cana-3308	40	27	,	,	PUNCT
cana-3308	40	28	then	then	ADV
cana-3308	40	29	𝑇	𝑇	PROPN
cana-3308	40	30	is	be	AUX
cana-3308	40	31	called	call	VERB
cana-3308	40	32	covariant	covariant	ADJ
cana-3308	40	33	map	map	NOUN
cana-3308	40	34	and	and	CCONJ
cana-3308	40	35	is	be	AUX
cana-3308	40	36	denoted	denote	VERB
cana-3308	40	37	by	by	ADP
cana-3308	40	38	𝑇	𝑇	PROPN
cana-3308	40	39	∶	∶	PROPN
cana-3308	40	40	(	(	PUNCT
cana-3308	40	41	𝑋1	𝑋1	PROPN
cana-3308	40	42	,	,	PUNCT
cana-3308	40	43	𝑌1	𝑌1	NOUN
cana-3308	40	44	,	,	PUNCT
cana-3308	40	45	𝑑1	𝑑1	NOUN
cana-3308	40	46	)	)	PUNCT
cana-3308	40	47	⇉	⇉	PUNCT
cana-3308	41	1	(	(	PUNCT
cana-3308	41	2	𝑋2	𝑋2	ADJ
cana-3308	41	3	,	,	PUNCT
cana-3308	41	4	𝑌2	𝑌2	NOUN
cana-3308	41	5	,	,	PUNCT
cana-3308	41	6	𝑑2	𝑑2	NOUN
cana-3308	41	7	)	)	PUNCT
cana-3308	41	8	.	.	PUNCT
cana-3308	42	1	(	(	PUNCT
cana-3308	42	2	ii)if	ii)if	PROPN
cana-3308	42	3	𝑇(𝑋1	𝑇(𝑋1	ADJ
cana-3308	42	4	)	)	PUNCT
cana-3308	42	5	⊆	⊆	NUM
cana-3308	42	6	𝑌2	𝑌2	NOUN
cana-3308	42	7	and	and	CCONJ
cana-3308	42	8	𝑇(𝑌1	𝑇(𝑌1	NOUN
cana-3308	42	9	)	)	PUNCT
cana-3308	42	10	⊆	⊆	NUM
cana-3308	42	11	𝑋2	𝑋2	ADJ
cana-3308	42	12	,	,	PUNCT
cana-3308	42	13	then	then	ADV
cana-3308	42	14	𝑇	𝑇	PROPN
cana-3308	42	15	is	be	AUX
cana-3308	42	16	called	call	VERB
cana-3308	42	17	contravariant	contravariant	ADJ
cana-3308	42	18	map	map	NOUN
cana-3308	42	19	and	and	CCONJ
cana-3308	42	20	is	be	AUX
cana-3308	42	21	denoted	denote	VERB
cana-3308	42	22	by	by	ADP
cana-3308	42	23	𝑇	𝑇	PROPN
cana-3308	42	24	∶	∶	PROPN
cana-3308	42	25	(	(	PUNCT
cana-3308	42	26	𝑋1	𝑋1	PROPN
cana-3308	42	27	,	,	PUNCT
cana-3308	42	28	𝑌1	𝑌1	NOUN
cana-3308	42	29	,	,	PUNCT
cana-3308	42	30	𝑑1	𝑑1	NOUN
cana-3308	42	31	)	)	PUNCT
cana-3308	42	32	⤨	⤨	PROPN
cana-3308	42	33	(	(	PUNCT
cana-3308	42	34	𝑋2	𝑋2	PROPN
cana-3308	42	35	,	,	PUNCT
cana-3308	42	36	𝑌2	𝑌2	NOUN
cana-3308	42	37	,	,	PUNCT
cana-3308	42	38	𝑑2	𝑑2	NOUN
cana-3308	42	39	)	)	PUNCT
cana-3308	42	40	.	.	PUNCT
cana-3308	43	1	definition	definition	NOUN
cana-3308	43	2	2.4	2.4	NUM
cana-3308	43	3	.	.	PUNCT
cana-3308	44	1	let	let	VERB
cana-3308	44	2	(	(	PUNCT
cana-3308	44	3	𝑋1	𝑋1	PROPN
cana-3308	44	4	,	,	PUNCT
cana-3308	44	5	𝑌1	𝑌1	NOUN
cana-3308	44	6	,	,	PUNCT
cana-3308	44	7	𝑑1	𝑑1	NOUN
cana-3308	44	8	)	)	PUNCT
cana-3308	44	9	and	and	CCONJ
cana-3308	44	10	(	(	PUNCT
cana-3308	44	11	𝑋2	𝑋2	PROPN
cana-3308	44	12	,	,	PUNCT
cana-3308	44	13	𝑌2	𝑌2	NOUN
cana-3308	44	14	,	,	PUNCT
cana-3308	44	15	𝑑2	𝑑2	NOUN
cana-3308	44	16	)	)	PUNCT
cana-3308	44	17	be	be	AUX
cana-3308	44	18	two	two	NUM
cana-3308	44	19	bipolar	bipolar	ADJ
cana-3308	44	20	metric	metric	ADJ
cana-3308	44	21	spaces	space	NOUN
cana-3308	44	22	.	.	PUNCT
cana-3308	45	1	(	(	PUNCT
cana-3308	45	2	i)a	i)a	X
cana-3308	45	3	map	map	VERB
cana-3308	45	4	𝑇	𝑇	PROPN
cana-3308	45	5	∶	∶	PROPN
cana-3308	45	6	(	(	PUNCT
cana-3308	45	7	𝑋1	𝑋1	PROPN
cana-3308	45	8	,	,	PUNCT
cana-3308	45	9	𝑌1	𝑌1	NOUN
cana-3308	45	10	,	,	PUNCT
cana-3308	45	11	𝑑1	𝑑1	NOUN
cana-3308	45	12	)	)	PUNCT
cana-3308	45	13	⇉	⇉	PUNCT
cana-3308	46	1	(	(	PUNCT
cana-3308	46	2	𝑋2	𝑋2	ADJ
cana-3308	46	3	,	,	PUNCT
cana-3308	46	4	𝑌2	𝑌2	NOUN
cana-3308	46	5	,	,	PUNCT
cana-3308	46	6	𝑑2	𝑑2	NOUN
cana-3308	46	7	)	)	PUNCT
cana-3308	46	8	is	be	AUX
cana-3308	46	9	called	call	VERB
cana-3308	46	10	left	leave	VERB
cana-3308	46	11	continuous	continuous	ADJ
cana-3308	46	12	at	at	ADP
cana-3308	46	13	a	a	DET
cana-3308	46	14	point	point	NOUN
cana-3308	46	15	𝑥0	𝑥0	NOUN
cana-3308	46	16	∈	∈	PROPN
cana-3308	46	17	𝑋1	𝑋1	NOUN
cana-3308	46	18	if	if	SCONJ
cana-3308	46	19	for	for	ADP
cana-3308	46	20	every	every	DET
cana-3308	46	21	𝜖	𝜖	X
cana-3308	46	22	>	>	X
cana-3308	46	23	0	0	PUNCT
cana-3308	47	1	there	there	PRON
cana-3308	47	2	exists	exist	VERB
cana-3308	47	3	𝛿	𝛿	PROPN
cana-3308	47	4	>	>	X
cana-3308	47	5	0	0	NUM
cana-3308	47	6	such	such	ADJ
cana-3308	47	7	that	that	SCONJ
cana-3308	47	8	𝑑2(𝑇𝑥0	𝑑2(𝑇𝑥0	NOUN
cana-3308	47	9	,	,	PUNCT
cana-3308	47	10	𝑇𝑦	𝑇𝑦	NOUN
cana-3308	47	11	)	)	PUNCT
cana-3308	47	12	<	<	X
cana-3308	48	1	휀	휀	X
cana-3308	48	2	whenever	whenever	SCONJ
cana-3308	48	3	𝑑1(𝑥0	𝑑1(𝑥0	ADP
cana-3308	48	4	,	,	PUNCT
cana-3308	48	5	𝑦	𝑦	NOUN
cana-3308	48	6	)	)	PUNCT
cana-3308	48	7	<	<	X
cana-3308	48	8	𝛿.	𝛿.	ADJ
cana-3308	48	9	(	(	PUNCT
cana-3308	48	10	ii)a	ii)a	NOUN
cana-3308	48	11	map	map	NOUN
cana-3308	48	12	𝑇	𝑇	PROPN
cana-3308	48	13	∶	∶	NOUN
cana-3308	48	14	(	(	PUNCT
cana-3308	48	15	𝑋1	𝑋1	PROPN
cana-3308	48	16	,	,	PUNCT
cana-3308	48	17	𝑌1	𝑌1	NOUN
cana-3308	48	18	,	,	PUNCT
cana-3308	48	19	𝑑1	𝑑1	NOUN
cana-3308	48	20	)	)	PUNCT
cana-3308	48	21	⇉	⇉	PUNCT
cana-3308	49	1	(	(	PUNCT
cana-3308	49	2	𝑋2	𝑋2	ADJ
cana-3308	49	3	,	,	PUNCT
cana-3308	49	4	𝑌2	𝑌2	NOUN
cana-3308	49	5	,	,	PUNCT
cana-3308	49	6	𝑑2	𝑑2	NOUN
cana-3308	49	7	)	)	PUNCT
cana-3308	49	8	is	be	AUX
cana-3308	49	9	called	call	VERB
cana-3308	49	10	right	right	ADV
cana-3308	49	11	continuous	continuous	ADJ
cana-3308	49	12	at	at	ADP
cana-3308	49	13	a	a	DET
cana-3308	49	14	point	point	NOUN
cana-3308	49	15	𝑦0	𝑦0	NOUN
cana-3308	49	16	∈	∈	NOUN
cana-3308	49	17	𝑌1	𝑌1	NOUN
cana-3308	49	18	if	if	SCONJ
cana-3308	49	19	for	for	ADP
cana-3308	49	20	every	every	DET
cana-3308	49	21	𝜖	𝜖	X
cana-3308	49	22	>	>	X
cana-3308	49	23	0	0	PUNCT
cana-3308	50	1	there	there	PRON
cana-3308	50	2	exists	exist	VERB
cana-3308	50	3	𝛿	𝛿	PROPN
cana-3308	50	4	>	>	X
cana-3308	50	5	0	0	NUM
cana-3308	50	6	such	such	ADJ
cana-3308	50	7	that	that	PRON
cana-3308	50	8	𝑑2(𝑇𝑥	𝑑2(𝑇𝑥	ADP
cana-3308	50	9	,	,	PUNCT
cana-3308	50	10	𝑇𝑦0	𝑇𝑦0	NOUN
cana-3308	50	11	)	)	PUNCT
cana-3308	51	1	<	<	X
cana-3308	51	2	휀	휀	X
cana-3308	51	3	whenever	whenever	SCONJ
cana-3308	51	4	𝑑1(𝑥	𝑑1(𝑥	NUM
cana-3308	51	5	,	,	PUNCT
cana-3308	51	6	𝑦0	𝑦0	NOUN
cana-3308	51	7	)	)	PUNCT
cana-3308	51	8	<	<	X
cana-3308	51	9	𝛿.	𝛿.	ADJ
cana-3308	51	10	communications	communication	NOUN
cana-3308	51	11	on	on	ADP
cana-3308	51	12	applied	apply	VERB
cana-3308	51	13	nonlinear	nonlinear	ADJ
cana-3308	51	14	analysis	analysis	NOUN
cana-3308	51	15	issn	issn	NOUN
cana-3308	51	16	:	:	PUNCT
cana-3308	51	17	1074	1074	NUM
cana-3308	51	18	-	-	PUNCT
cana-3308	51	19	133x	133x	NUM
cana-3308	51	20	vol	vol	NOUN
cana-3308	51	21	32	32	NUM
cana-3308	51	22	no	no	NOUN
cana-3308	51	23	.	.	PUNCT
cana-3308	52	1	6s	6s	NUM
cana-3308	52	2	(	(	PUNCT
cana-3308	52	3	2025	2025	NUM
cana-3308	52	4	)	)	PUNCT
cana-3308	52	5	441	441	NUM
cana-3308	52	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	52	7	(	(	PUNCT
cana-3308	52	8	iii)a	iii)a	NOUN
cana-3308	52	9	map	map	NOUN
cana-3308	52	10	𝑇	𝑇	PROPN
cana-3308	52	11	is	be	AUX
cana-3308	52	12	called	call	VERB
cana-3308	52	13	continuous	continuous	ADJ
cana-3308	52	14	,	,	PUNCT
cana-3308	52	15	if	if	SCONJ
cana-3308	52	16	it	it	PRON
cana-3308	52	17	is	be	AUX
cana-3308	52	18	left	leave	VERB
cana-3308	52	19	continuous	continuous	ADJ
cana-3308	52	20	at	at	ADP
cana-3308	52	21	each	each	DET
cana-3308	52	22	𝑥0	𝑥0	PROPN
cana-3308	52	23	∈	∈	PROPN
cana-3308	52	24	𝑋1	𝑋1	NOUN
cana-3308	52	25	and	and	CCONJ
cana-3308	52	26	right	right	ADV
cana-3308	52	27	continuous	continuous	ADJ
cana-3308	52	28	at	at	ADP
cana-3308	52	29	each	each	DET
cana-3308	52	30	𝑦0	𝑦0	NOUN
cana-3308	52	31	∈	∈	PROPN
cana-3308	52	32	𝑌1	𝑌1	PROPN
cana-3308	52	33	.	.	PUNCT
cana-3308	53	1	(	(	PUNCT
cana-3308	53	2	iv)a	iv)a	PROPN
cana-3308	53	3	contravariant	contravariant	PROPN
cana-3308	53	4	map	map	NOUN
cana-3308	53	5	𝑇	𝑇	PROPN
cana-3308	53	6	∶	∶	PROPN
cana-3308	53	7	(	(	PUNCT
cana-3308	53	8	𝑋1	𝑋1	PROPN
cana-3308	53	9	,	,	PUNCT
cana-3308	53	10	𝑌1	𝑌1	NOUN
cana-3308	53	11	,	,	PUNCT
cana-3308	53	12	𝑑1	𝑑1	NOUN
cana-3308	53	13	)	)	PUNCT
cana-3308	53	14	⤨	⤨	PROPN
cana-3308	53	15	(	(	PUNCT
cana-3308	53	16	𝑋2	𝑋2	PROPN
cana-3308	53	17	,	,	PUNCT
cana-3308	53	18	𝑌2	𝑌2	NOUN
cana-3308	53	19	,	,	PUNCT
cana-3308	53	20	𝑑2	𝑑2	NOUN
cana-3308	53	21	)	)	PUNCT
cana-3308	53	22	is	be	AUX
cana-3308	53	23	continuous	continuous	ADJ
cana-3308	53	24	if	if	SCONJ
cana-3308	53	25	and	and	CCONJ
cana-3308	53	26	only	only	ADV
cana-3308	53	27	if	if	SCONJ
cana-3308	53	28	it	it	PRON
cana-3308	53	29	is	be	AUX
cana-3308	53	30	continuous	continuous	ADJ
cana-3308	53	31	as	as	ADP
cana-3308	53	32	a	a	DET
cana-3308	53	33	covariant	covariant	ADJ
cana-3308	53	34	map	map	NOUN
cana-3308	53	35	𝑇	𝑇	PROPN
cana-3308	53	36	∶	∶	PROPN
cana-3308	53	37	(	(	PUNCT
cana-3308	53	38	𝑋1	𝑋1	PROPN
cana-3308	53	39	,	,	PUNCT
cana-3308	53	40	𝑌1	𝑌1	NOUN
cana-3308	53	41	,	,	PUNCT
cana-3308	53	42	𝑑1	𝑑1	NOUN
cana-3308	53	43	)	)	PUNCT
cana-3308	53	44	⇉	⇉	PUNCT
cana-3308	54	1	(	(	PUNCT
cana-3308	54	2	𝑋2	𝑋2	ADJ
cana-3308	54	3	,	,	PUNCT
cana-3308	54	4	𝑌2	𝑌2	NOUN
cana-3308	54	5	,	,	PUNCT
cana-3308	54	6	𝑑2	𝑑2	NOUN
cana-3308	54	7	)	)	PUNCT
cana-3308	54	8	.	.	PUNCT
cana-3308	55	1	definition	definition	NOUN
cana-3308	55	2	2.5	2.5	NUM
cana-3308	55	3	.	.	PUNCT
cana-3308	56	1	let	let	VERB
cana-3308	56	2	(	(	PUNCT
cana-3308	56	3	𝑋	𝑋	PROPN
cana-3308	56	4	,	,	PUNCT
cana-3308	56	5	𝑌	𝑌	PROPN
cana-3308	56	6	,	,	PUNCT
cana-3308	56	7	𝑑	𝑑	NOUN
cana-3308	56	8	)	)	PUNCT
cana-3308	56	9	be	be	VERB
cana-3308	56	10	a	a	DET
cana-3308	56	11	bipolar	bipolar	ADJ
cana-3308	56	12	metric	metric	ADJ
cana-3308	56	13	space	space	NOUN
cana-3308	56	14	.	.	PUNCT
cana-3308	57	1	(	(	PUNCT
cana-3308	57	2	i)a	i)a	X
cana-3308	57	3	sequence	sequence	NOUN
cana-3308	57	4	{	{	PUNCT
cana-3308	57	5	(	(	PUNCT
cana-3308	57	6	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	57	7	,	,	PUNCT
cana-3308	57	8	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	57	9	)	)	PUNCT
cana-3308	57	10	}	}	PUNCT
cana-3308	57	11	on	on	ADP
cana-3308	57	12	the	the	DET
cana-3308	57	13	set	set	NOUN
cana-3308	57	14	𝑋	𝑋	PROPN
cana-3308	57	15	×	×	PROPN
cana-3308	57	16	𝑌	𝑌	PROPN
cana-3308	57	17	is	be	AUX
cana-3308	57	18	called	call	VERB
cana-3308	57	19	a	a	DET
cana-3308	57	20	bisequence	bisequence	NOUN
cana-3308	57	21	on	on	ADP
cana-3308	57	22	(	(	PUNCT
cana-3308	57	23	𝑋	𝑋	PROPN
cana-3308	57	24	,	,	PUNCT
cana-3308	57	25	𝑌	𝑌	PROPN
cana-3308	57	26	,	,	PUNCT
cana-3308	57	27	𝑑	𝑑	NOUN
cana-3308	57	28	)	)	PUNCT
cana-3308	57	29	.	.	PUNCT
cana-3308	58	1	(	(	PUNCT
cana-3308	58	2	ii)if	ii)if	ADP
cana-3308	58	3	both	both	DET
cana-3308	58	4	the	the	DET
cana-3308	58	5	sequences	sequence	NOUN
cana-3308	58	6	(	(	PUNCT
cana-3308	58	7	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	58	8	)	)	PUNCT
cana-3308	58	9	and	and	CCONJ
cana-3308	58	10	(	(	PUNCT
cana-3308	58	11	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	58	12	)	)	PUNCT
cana-3308	58	13	converge	converge	NOUN
cana-3308	58	14	,	,	PUNCT
cana-3308	58	15	then	then	ADV
cana-3308	58	16	bisequence	bisequence	NOUN
cana-3308	58	17	{	{	PUNCT
cana-3308	58	18	(	(	PUNCT
cana-3308	58	19	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	58	20	,	,	PUNCT
cana-3308	58	21	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	58	22	)	)	PUNCT
cana-3308	58	23	}	}	PUNCT
cana-3308	58	24	is	be	AUX
cana-3308	58	25	said	say	VERB
cana-3308	58	26	to	to	PART
cana-3308	58	27	be	be	AUX
cana-3308	58	28	convergent	convergent	ADJ
cana-3308	58	29	.	.	PUNCT
cana-3308	59	1	if	if	SCONJ
cana-3308	59	2	both	both	PRON
cana-3308	59	3	the	the	DET
cana-3308	59	4	sequences	sequence	NOUN
cana-3308	59	5	(	(	PUNCT
cana-3308	59	6	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	59	7	)	)	PUNCT
cana-3308	59	8	and	and	CCONJ
cana-3308	59	9	(	(	PUNCT
cana-3308	59	10	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	59	11	)	)	PUNCT
cana-3308	59	12	converge	converge	VERB
cana-3308	59	13	to	to	ADP
cana-3308	59	14	same	same	ADJ
cana-3308	59	15	point	point	NOUN
cana-3308	59	16	𝑣	𝑣	ADP
cana-3308	59	17	and	and	CCONJ
cana-3308	59	18	𝑣	𝑣	PRON
cana-3308	59	19	∈	∈	NOUN
cana-3308	59	20	𝑋	𝑋	PROPN
cana-3308	59	21	∩	∩	ADJ
cana-3308	59	22	𝑌	𝑌	PROPN
cana-3308	59	23	,	,	PUNCT
cana-3308	59	24	then	then	ADV
cana-3308	59	25	this	this	DET
cana-3308	59	26	bisequence	bisequence	NOUN
cana-3308	59	27	is	be	AUX
cana-3308	59	28	said	say	VERB
cana-3308	59	29	to	to	PART
cana-3308	59	30	be	be	AUX
cana-3308	59	31	biconvergent	biconvergent	NOUN
cana-3308	59	32	.	.	PUNCT
cana-3308	60	1	(	(	PUNCT
cana-3308	60	2	iii)a	iii)a	PROPN
cana-3308	60	3	bisequence	bisequence	NOUN
cana-3308	60	4	{	{	PUNCT
cana-3308	60	5	(	(	PUNCT
cana-3308	60	6	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	60	7	,	,	PUNCT
cana-3308	60	8	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	60	9	)	)	PUNCT
cana-3308	60	10	}	}	PUNCT
cana-3308	60	11	on	on	ADP
cana-3308	60	12	(	(	PUNCT
cana-3308	60	13	𝑋	𝑋	PROPN
cana-3308	60	14	,	,	PUNCT
cana-3308	60	15	𝑌	𝑌	PROPN
cana-3308	60	16	,	,	PUNCT
cana-3308	60	17	𝑑	𝑑	NOUN
cana-3308	60	18	)	)	PUNCT
cana-3308	60	19	is	be	AUX
cana-3308	60	20	said	say	VERB
cana-3308	60	21	to	to	PART
cana-3308	60	22	be	be	AUX
cana-3308	60	23	cauchy	cauchy	ADJ
cana-3308	60	24	bisequence	bisequence	NOUN
cana-3308	60	25	,	,	PUNCT
cana-3308	60	26	if	if	SCONJ
cana-3308	60	27	for	for	ADP
cana-3308	60	28	each	each	PRON
cana-3308	60	29	𝜖	𝜖	X
cana-3308	60	30	>	>	X
cana-3308	60	31	0	0	PUNCT
cana-3308	60	32	there	there	PRON
cana-3308	60	33	exists	exist	VERB
cana-3308	60	34	a	a	DET
cana-3308	60	35	positive	positive	ADJ
cana-3308	60	36	integer	integer	NOUN
cana-3308	60	37	𝑁	𝑁	PROPN
cana-3308	60	38	∈	∈	PROPN
cana-3308	60	39	ℕ	ℕ	NOUN
cana-3308	60	40	such	such	ADJ
cana-3308	60	41	that	that	SCONJ
cana-3308	60	42	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	60	43	,	,	PUNCT
cana-3308	60	44	𝑦𝑚	𝑦𝑚	NOUN
cana-3308	60	45	)	)	PUNCT
cana-3308	60	46	<	<	X
cana-3308	60	47	𝜖	𝜖	PROPN
cana-3308	60	48	for	for	ADP
cana-3308	60	49	all	all	DET
cana-3308	60	50	𝑛	𝑛	PROPN
cana-3308	60	51	,	,	PUNCT
cana-3308	60	52	𝑚	𝑚	X
cana-3308	60	53	≥	≥	PRON
cana-3308	60	54	𝑁.	𝑁.	PROPN
cana-3308	60	55	(	(	PUNCT
cana-3308	60	56	iv)a	iv)a	X
cana-3308	60	57	bipolar	bipolar	ADJ
cana-3308	60	58	metric	metric	ADJ
cana-3308	60	59	space	space	NOUN
cana-3308	60	60	is	be	AUX
cana-3308	60	61	said	say	VERB
cana-3308	60	62	to	to	PART
cana-3308	60	63	be	be	AUX
cana-3308	60	64	complete	complete	ADJ
cana-3308	60	65	if	if	SCONJ
cana-3308	60	66	every	every	DET
cana-3308	60	67	cauchy	cauchy	ADJ
cana-3308	60	68	bisequence	bisequence	NOUN
cana-3308	60	69	is	be	AUX
cana-3308	60	70	convergent	convergent	ADJ
cana-3308	60	71	in	in	ADP
cana-3308	60	72	this	this	DET
cana-3308	60	73	space	space	NOUN
cana-3308	60	74	.	.	PUNCT
cana-3308	61	1	definition	definition	NOUN
cana-3308	61	2	2.6	2.6	NUM
cana-3308	61	3	.	.	PUNCT
cana-3308	62	1	[	[	X
cana-3308	62	2	11	11	NUM
cana-3308	62	3	]	]	PUNCT
cana-3308	62	4	let	let	VERB
cana-3308	62	5	𝑋	𝑋	NOUN
cana-3308	62	6	and	and	CCONJ
cana-3308	62	7	𝑌	𝑌	PROPN
cana-3308	62	8	be	be	VERB
cana-3308	62	9	two	two	NUM
cana-3308	62	10	non	non	ADJ
cana-3308	62	11	-	-	ADJ
cana-3308	62	12	empty	empty	ADJ
cana-3308	62	13	sets	set	NOUN
cana-3308	62	14	.	.	PUNCT
cana-3308	63	1	let	let	VERB
cana-3308	63	2	𝑇	𝑇	PROPN
cana-3308	63	3	∶	∶	NOUN
cana-3308	63	4	(	(	PUNCT
cana-3308	63	5	𝑋	𝑋	PROPN
cana-3308	63	6	,	,	PUNCT
cana-3308	63	7	𝑌	𝑌	PROPN
cana-3308	63	8	)	)	PUNCT
cana-3308	63	9	⇉	⇉	PUNCT
cana-3308	64	1	(	(	PUNCT
cana-3308	64	2	𝑋	𝑋	PROPN
cana-3308	64	3	,	,	PUNCT
cana-3308	64	4	𝑌	𝑌	PROPN
cana-3308	64	5	)	)	PUNCT
cana-3308	64	6	and	and	CCONJ
cana-3308	64	7	𝛼	𝛼	X
cana-3308	64	8	∶	∶	NOUN
cana-3308	64	9	𝑋	𝑋	NOUN
cana-3308	64	10	×	×	NOUN
cana-3308	64	11	𝑌	𝑌	PROPN
cana-3308	64	12	→	→	SYM
cana-3308	65	1	[	[	X
cana-3308	65	2	0	0	NUM
cana-3308	65	3	,	,	PUNCT
cana-3308	65	4	+	+	NOUN
cana-3308	65	5	∞	∞	NOUN
cana-3308	65	6	)	)	PUNCT
cana-3308	65	7	.	.	PUNCT
cana-3308	66	1	then	then	ADV
cana-3308	66	2	𝑇	𝑇	PROPN
cana-3308	66	3	is	be	AUX
cana-3308	66	4	called	call	VERB
cana-3308	66	5	𝛼-admissible	𝛼-admissible	ADJ
cana-3308	66	6	(	(	PUNCT
cana-3308	66	7	covariant	covariant	NOUN
cana-3308	66	8	)	)	PUNCT
cana-3308	66	9	if	if	SCONJ
cana-3308	66	10	𝛼(𝑥	𝛼(𝑥	PROPN
cana-3308	66	11	,	,	PUNCT
cana-3308	66	12	𝑦	𝑦	NOUN
cana-3308	66	13	)	)	PUNCT
cana-3308	66	14	≥	≥	NOUN
cana-3308	66	15	1	1	NUM
cana-3308	66	16	⇒	⇒	NOUN
cana-3308	66	17	𝛼(𝑇𝑥	𝛼(𝑇𝑥	NOUN
cana-3308	66	18	,	,	PUNCT
cana-3308	66	19	𝑇𝑦	𝑇𝑦	PROPN
cana-3308	66	20	)	)	PUNCT
cana-3308	66	21	≥	≥	NOUN
cana-3308	66	22	1	1	NUM
cana-3308	66	23	for	for	ADP
cana-3308	66	24	all	all	PRON
cana-3308	66	25	𝑥	𝑥	DET
cana-3308	66	26	∈	∈	PROPN
cana-3308	66	27	𝑋	𝑋	NOUN
cana-3308	66	28	and	and	CCONJ
cana-3308	66	29	𝑦	𝑦	NOUN
cana-3308	66	30	∈	∈	NOUN
cana-3308	66	31	𝑌.	𝑌.	ADJ
cana-3308	66	32	definition	definition	NOUN
cana-3308	66	33	2.7	2.7	NUM
cana-3308	66	34	.	.	PUNCT
cana-3308	67	1	[	[	X
cana-3308	67	2	11	11	NUM
cana-3308	67	3	]	]	PUNCT
cana-3308	67	4	let	let	VERB
cana-3308	67	5	𝑋	𝑋	NOUN
cana-3308	67	6	and	and	CCONJ
cana-3308	67	7	𝑌	𝑌	PROPN
cana-3308	67	8	be	be	VERB
cana-3308	67	9	two	two	NUM
cana-3308	67	10	non	non	ADJ
cana-3308	67	11	-	-	ADJ
cana-3308	67	12	empty	empty	ADJ
cana-3308	67	13	sets	set	NOUN
cana-3308	67	14	.	.	PUNCT
cana-3308	68	1	let	let	VERB
cana-3308	68	2	𝑇	𝑇	PROPN
cana-3308	68	3	∶	∶	NOUN
cana-3308	68	4	(	(	PUNCT
cana-3308	68	5	𝑋	𝑋	PROPN
cana-3308	68	6	,	,	PUNCT
cana-3308	68	7	𝑌	𝑌	PROPN
cana-3308	68	8	)	)	PUNCT
cana-3308	68	9	⤨	⤨	NUM
cana-3308	68	10	(	(	PUNCT
cana-3308	68	11	𝑋	𝑋	PROPN
cana-3308	68	12	,	,	PUNCT
cana-3308	68	13	𝑌	𝑌	PROPN
cana-3308	68	14	)	)	PUNCT
cana-3308	68	15	and	and	CCONJ
cana-3308	68	16	𝛼	𝛼	AUX
cana-3308	68	17	∶	∶	NOUN
cana-3308	68	18	𝑋	𝑋	NOUN
cana-3308	68	19	×	×	NOUN
cana-3308	68	20	𝑌	𝑌	PROPN
cana-3308	68	21	→	→	SYM
cana-3308	69	1	[	[	X
cana-3308	69	2	0	0	NUM
cana-3308	69	3	,	,	PUNCT
cana-3308	69	4	+	+	NOUN
cana-3308	69	5	∞	∞	NOUN
cana-3308	69	6	)	)	PUNCT
cana-3308	69	7	.	.	PUNCT
cana-3308	70	1	then	then	ADV
cana-3308	70	2	𝑇	𝑇	PROPN
cana-3308	70	3	is	be	AUX
cana-3308	70	4	called	call	VERB
cana-3308	70	5	𝛼-admissible	𝛼-admissible	ADJ
cana-3308	70	6	(	(	PUNCT
cana-3308	70	7	contravariant	contravariant	PROPN
cana-3308	70	8	)	)	PUNCT
cana-3308	70	9	if	if	SCONJ
cana-3308	70	10	𝛼(𝑥	𝛼(𝑥	PROPN
cana-3308	70	11	,	,	PUNCT
cana-3308	70	12	𝑦	𝑦	NOUN
cana-3308	70	13	)	)	PUNCT
cana-3308	70	14	≥	≥	NOUN
cana-3308	70	15	1	1	NUM
cana-3308	70	16	⇒	⇒	NOUN
cana-3308	70	17	𝛼(𝑇𝑦	𝛼(𝑇𝑦	NUM
cana-3308	70	18	,	,	PUNCT
cana-3308	70	19	𝑇𝑥	𝑇𝑥	PROPN
cana-3308	70	20	)	)	PUNCT
cana-3308	70	21	≥	≥	NOUN
cana-3308	70	22	1	1	NUM
cana-3308	70	23	for	for	ADP
cana-3308	70	24	all	all	PRON
cana-3308	70	25	𝑥	𝑥	DET
cana-3308	70	26	∈	∈	PROPN
cana-3308	70	27	𝑋	𝑋	NOUN
cana-3308	70	28	and	and	CCONJ
cana-3308	70	29	𝑦	𝑦	NOUN
cana-3308	70	30	∈	∈	NOUN
cana-3308	70	31	𝑌.	𝑌.	PROPN
cana-3308	70	32	let	let	VERB
cana-3308	70	33	ψ	ψ	PART
cana-3308	70	34	be	be	AUX
cana-3308	70	35	the	the	DET
cana-3308	70	36	family	family	NOUN
cana-3308	70	37	of	of	ADP
cana-3308	70	38	functions	function	NOUN
cana-3308	70	39	𝜓	𝜓	X
cana-3308	70	40	∶	∶	NOUN
cana-3308	70	41	[	[	X
cana-3308	70	42	0	0	NUM
cana-3308	70	43	,	,	PUNCT
cana-3308	70	44	∞	∞	PROPN
cana-3308	70	45	)	)	PUNCT
cana-3308	70	46	→	→	PUNCT
cana-3308	71	1	[	[	X
cana-3308	71	2	0	0	NUM
cana-3308	71	3	,	,	PUNCT
cana-3308	71	4	∞	∞	PROPN
cana-3308	71	5	)	)	PUNCT
cana-3308	71	6	which	which	PRON
cana-3308	71	7	satisfying	satisfy	VERB
cana-3308	71	8	the	the	DET
cana-3308	71	9	following	following	ADJ
cana-3308	71	10	conditions	condition	NOUN
cana-3308	71	11	:	:	PUNCT
cana-3308	71	12	(	(	PUNCT
cana-3308	71	13	i	i	NOUN
cana-3308	71	14	)	)	PUNCT
cana-3308	71	15	𝜓	𝜓	PROPN
cana-3308	71	16	is	be	AUX
cana-3308	71	17	continuous	continuous	ADJ
cana-3308	71	18	,	,	PUNCT
cana-3308	71	19	(	(	PUNCT
cana-3308	71	20	ii	ii	NOUN
cana-3308	71	21	)	)	PUNCT
cana-3308	71	22	𝜓	𝜓	PROPN
cana-3308	71	23	is	be	AUX
cana-3308	71	24	strictly	strictly	ADV
cana-3308	71	25	increasing	increase	VERB
cana-3308	71	26	,	,	PUNCT
cana-3308	71	27	(	(	PUNCT
cana-3308	71	28	iii	iii	NOUN
cana-3308	71	29	)	)	PUNCT
cana-3308	71	30	𝜓(0	𝜓(0	PROPN
cana-3308	71	31	)	)	PUNCT
cana-3308	71	32	=	=	SYM
cana-3308	72	1	0	0	X
cana-3308	72	2	.	.	PUNCT
cana-3308	72	3	consider	consider	VERB
cana-3308	72	4	θ	θ	NOUN
cana-3308	72	5	be	be	AUX
cana-3308	72	6	the	the	DET
cana-3308	72	7	family	family	NOUN
cana-3308	72	8	of	of	ADP
cana-3308	72	9	functions	function	NOUN
cana-3308	72	10	𝜃	𝜃	X
cana-3308	72	11	∶	∶	NOUN
cana-3308	72	12	[	[	X
cana-3308	72	13	0	0	NUM
cana-3308	72	14	,	,	PUNCT
cana-3308	72	15	∞	∞	PROPN
cana-3308	72	16	)	)	PUNCT
cana-3308	72	17	→	→	PUNCT
cana-3308	73	1	[	[	X
cana-3308	73	2	0,1	0,1	NUM
cana-3308	73	3	)	)	PUNCT
cana-3308	73	4	such	such	ADJ
cana-3308	73	5	that	that	PRON
cana-3308	73	6	for	for	ADP
cana-3308	73	7	any	any	DET
cana-3308	73	8	bounded	bounded	ADJ
cana-3308	73	9	sequence	sequence	NOUN
cana-3308	73	10	{	{	PUNCT
cana-3308	73	11	𝑡𝑛	𝑡𝑛	X
cana-3308	73	12	}	}	PUNCT
cana-3308	73	13	of	of	ADP
cana-3308	73	14	positive	positive	ADJ
cana-3308	73	15	reals	real	NOUN
cana-3308	73	16	,	,	PUNCT
cana-3308	73	17	𝜃(𝑡𝑛	𝜃(𝑡𝑛	NUM
cana-3308	73	18	)	)	PUNCT
cana-3308	73	19	→	→	SYM
cana-3308	73	20	1	1	NUM
cana-3308	73	21	implies	imply	VERB
cana-3308	73	22	that	that	SCONJ
cana-3308	73	23	𝑡𝑛	𝑡𝑛	VERB
cana-3308	73	24	→	→	SYM
cana-3308	73	25	0	0	NUM
cana-3308	73	26	,	,	PUNCT
cana-3308	73	27	as	as	SCONJ
cana-3308	73	28	𝑛	𝑛	PROPN
cana-3308	73	29	→	→	PUNCT
cana-3308	73	30	∞.	∞.	PROPN
cana-3308	73	31	let	let	VERB
cana-3308	73	32	θ𝑡	θ𝑡	PRON
cana-3308	73	33	be	be	AUX
cana-3308	73	34	the	the	DET
cana-3308	73	35	family	family	NOUN
cana-3308	73	36	of	of	ADP
cana-3308	73	37	functions	function	NOUN
cana-3308	73	38	𝜃	𝜃	X
cana-3308	73	39	∶	∶	NOUN
cana-3308	73	40	[	[	X
cana-3308	73	41	0	0	NUM
cana-3308	73	42	,	,	PUNCT
cana-3308	73	43	∞	∞	PROPN
cana-3308	73	44	)	)	PUNCT
cana-3308	73	45	→	→	PUNCT
cana-3308	74	1	[	[	X
cana-3308	74	2	0,1	0,1	NUM
cana-3308	74	3	)	)	PUNCT
cana-3308	74	4	such	such	ADJ
cana-3308	74	5	that	that	PRON
cana-3308	74	6	for	for	ADP
cana-3308	74	7	any	any	DET
cana-3308	74	8	bounded	bounded	ADJ
cana-3308	74	9	sequence	sequence	NOUN
cana-3308	74	10	{	{	PUNCT
cana-3308	74	11	𝑡𝑛	𝑡𝑛	X
cana-3308	74	12	}	}	PUNCT
cana-3308	74	13	of	of	ADP
cana-3308	74	14	positive	positive	ADJ
cana-3308	74	15	reals	real	NOUN
cana-3308	74	16	,	,	PUNCT
cana-3308	74	17	lim	lim	PROPN
cana-3308	74	18	𝑠𝑢𝑝	𝑠𝑢𝑝	PROPN
cana-3308	74	19	𝜃(𝑡𝑛	𝜃(𝑡𝑛	NUM
cana-3308	74	20	)	)	PUNCT
cana-3308	74	21	→	→	SYM
cana-3308	74	22	1	1	NUM
cana-3308	74	23	implies	imply	VERB
cana-3308	74	24	that	that	SCONJ
cana-3308	74	25	𝑡𝑛	𝑡𝑛	VERB
cana-3308	74	26	→	→	SYM
cana-3308	74	27	0	0	NUM
cana-3308	74	28	,	,	PUNCT
cana-3308	74	29	as	as	ADP
cana-3308	74	30	𝑛	𝑛	PROPN
cana-3308	74	31	→	→	SYM
cana-3308	74	32	∞.	∞.	PROPN
cana-3308	74	33	3	3	NUM
cana-3308	74	34	.	.	PUNCT
cana-3308	74	35	main	main	ADJ
cana-3308	74	36	results	result	NOUN
cana-3308	74	37	in	in	ADP
cana-3308	74	38	this	this	DET
cana-3308	74	39	section	section	NOUN
cana-3308	74	40	,	,	PUNCT
cana-3308	74	41	we	we	PRON
cana-3308	74	42	will	will	AUX
cana-3308	74	43	introduce	introduce	VERB
cana-3308	74	44	new	new	ADJ
cana-3308	74	45	notations	notation	NOUN
cana-3308	74	46	for	for	ADP
cana-3308	74	47	𝜑-geraghty	𝜑-geraghty	NUM
cana-3308	74	48	contraction	contraction	NOUN
cana-3308	74	49	mappings	mapping	NOUN
cana-3308	74	50	and	and	CCONJ
cana-3308	74	51	prove	prove	VERB
cana-3308	74	52	various	various	ADJ
cana-3308	74	53	fixed	fix	VERB
cana-3308	74	54	point	point	NOUN
cana-3308	74	55	theorems	theorem	NOUN
cana-3308	74	56	for	for	ADP
cana-3308	74	57	such	such	ADJ
cana-3308	74	58	type	type	NOUN
cana-3308	74	59	of	of	ADP
cana-3308	74	60	mappings	mapping	NOUN
cana-3308	74	61	in	in	ADP
cana-3308	74	62	complete	complete	ADJ
cana-3308	74	63	bipolar	bipolar	ADJ
cana-3308	74	64	metric	metric	ADJ
cana-3308	74	65	spaces	space	NOUN
cana-3308	74	66	.	.	PUNCT
cana-3308	75	1	definition	definition	NOUN
cana-3308	75	2	3.1	3.1	NUM
cana-3308	75	3	.	.	PUNCT
cana-3308	76	1	let	let	VERB
cana-3308	76	2	𝑋	𝑋	NOUN
cana-3308	76	3	and	and	CCONJ
cana-3308	76	4	𝑌	𝑌	PROPN
cana-3308	76	5	be	be	VERB
cana-3308	76	6	two	two	NUM
cana-3308	76	7	non	non	ADJ
cana-3308	76	8	-	-	ADJ
cana-3308	76	9	empty	empty	ADJ
cana-3308	76	10	sets	set	NOUN
cana-3308	76	11	.	.	PUNCT
cana-3308	77	1	consider	consider	VERB
cana-3308	77	2	(	(	PUNCT
cana-3308	77	3	𝑋	𝑋	PROPN
cana-3308	77	4	,	,	PUNCT
cana-3308	77	5	𝑌	𝑌	PROPN
cana-3308	77	6	,	,	PUNCT
cana-3308	77	7	𝑑	𝑑	NOUN
cana-3308	77	8	)	)	PUNCT
cana-3308	77	9	be	be	VERB
cana-3308	77	10	a	a	DET
cana-3308	77	11	bipolar	bipolar	ADJ
cana-3308	77	12	metric	metric	ADJ
cana-3308	77	13	space	space	NOUN
cana-3308	77	14	,	,	PUNCT
cana-3308	77	15	𝑇	𝑇	PROPN
cana-3308	77	16	∶	∶	PROPN
cana-3308	77	17	(	(	PUNCT
cana-3308	77	18	𝑋	𝑋	PROPN
cana-3308	77	19	,	,	PUNCT
cana-3308	77	20	𝑌	𝑌	PROPN
cana-3308	77	21	)	)	PUNCT
cana-3308	77	22	⇉	⇉	PUNCT
cana-3308	78	1	(	(	PUNCT
cana-3308	78	2	𝑋	𝑋	NOUN
cana-3308	78	3	,	,	PUNCT
cana-3308	78	4	𝑌	𝑌	PROPN
cana-3308	78	5	)	)	PUNCT
cana-3308	78	6	is	be	AUX
cana-3308	78	7	called	call	VERB
cana-3308	78	8	geraghty	geraghty	PROPN
cana-3308	78	9	contraction	contraction	NOUN
cana-3308	78	10	if	if	SCONJ
cana-3308	78	11	there	there	PRON
cana-3308	78	12	exist	exist	VERB
cana-3308	78	13	a	a	DET
cana-3308	78	14	function	function	NOUN
cana-3308	78	15	𝜃	𝜃	NOUN
cana-3308	78	16	∈	∈	NOUN
cana-3308	78	17	θ	θ	NOUN
cana-3308	78	18	which	which	PRON
cana-3308	78	19	satisfies	satisfy	VERB
cana-3308	78	20	the	the	DET
cana-3308	78	21	following	follow	VERB
cana-3308	78	22	condition	condition	NOUN
cana-3308	78	23	:	:	PUNCT
cana-3308	78	24	𝑑(𝑇𝑥	𝑑(𝑇𝑥	X
cana-3308	78	25	,	,	PUNCT
cana-3308	78	26	𝑇𝑦	𝑇𝑦	PROPN
cana-3308	78	27	)	)	PUNCT
cana-3308	78	28	≤	≤	NOUN
cana-3308	78	29	𝜃	𝜃	X
cana-3308	78	30	(	(	PUNCT
cana-3308	78	31	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	78	32	,	,	PUNCT
cana-3308	78	33	𝑦	𝑦	NOUN
cana-3308	78	34	)	)	PUNCT
cana-3308	78	35	)	)	PUNCT
cana-3308	78	36	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	78	37	,	,	PUNCT
cana-3308	78	38	𝑦	𝑦	NOUN
cana-3308	78	39	)	)	PUNCT
cana-3308	78	40	for	for	ADP
cana-3308	78	41	all	all	DET
cana-3308	78	42	𝑥	𝑥	DET
cana-3308	78	43	∈	∈	PROPN
cana-3308	78	44	𝑋	𝑋	NOUN
cana-3308	78	45	and	and	CCONJ
cana-3308	78	46	𝑦	𝑦	NOUN
cana-3308	78	47	∈	∈	NOUN
cana-3308	78	48	𝑌.	𝑌.	ADJ
cana-3308	78	49	communications	communication	NOUN
cana-3308	78	50	on	on	ADP
cana-3308	78	51	applied	apply	VERB
cana-3308	78	52	nonlinear	nonlinear	ADJ
cana-3308	78	53	analysis	analysis	NOUN
cana-3308	78	54	issn	issn	NOUN
cana-3308	78	55	:	:	PUNCT
cana-3308	78	56	1074	1074	NUM
cana-3308	78	57	-	-	PUNCT
cana-3308	78	58	133x	133x	NUM
cana-3308	78	59	vol	vol	NOUN
cana-3308	78	60	32	32	NUM
cana-3308	78	61	no	no	NOUN
cana-3308	78	62	.	.	PUNCT
cana-3308	79	1	6s	6s	NUM
cana-3308	79	2	(	(	PUNCT
cana-3308	79	3	2025	2025	NUM
cana-3308	79	4	)	)	PUNCT
cana-3308	79	5	442	442	NUM
cana-3308	79	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	79	7	definition	definition	NOUN
cana-3308	79	8	3.2	3.2	NUM
cana-3308	79	9	.	.	PUNCT
cana-3308	80	1	let	let	VERB
cana-3308	80	2	𝑋	𝑋	NOUN
cana-3308	80	3	and	and	CCONJ
cana-3308	80	4	𝑌	𝑌	PROPN
cana-3308	80	5	be	be	VERB
cana-3308	80	6	two	two	NUM
cana-3308	80	7	non	non	ADJ
cana-3308	80	8	-	-	ADJ
cana-3308	80	9	empty	empty	ADJ
cana-3308	80	10	sets	set	NOUN
cana-3308	80	11	and	and	CCONJ
cana-3308	80	12	(	(	PUNCT
cana-3308	80	13	𝑋	𝑋	PROPN
cana-3308	80	14	,	,	PUNCT
cana-3308	80	15	𝑌	𝑌	PROPN
cana-3308	80	16	,	,	PUNCT
cana-3308	80	17	𝑑	𝑑	NOUN
cana-3308	80	18	)	)	PUNCT
cana-3308	80	19	be	be	VERB
cana-3308	80	20	a	a	DET
cana-3308	80	21	bipolar	bipolar	ADJ
cana-3308	80	22	metric	metric	ADJ
cana-3308	80	23	space	space	NOUN
cana-3308	80	24	,	,	PUNCT
cana-3308	80	25	𝑇	𝑇	PROPN
cana-3308	80	26	∶	∶	PROPN
cana-3308	80	27	(	(	PUNCT
cana-3308	80	28	𝑋	𝑋	PROPN
cana-3308	80	29	,	,	PUNCT
cana-3308	80	30	𝑌	𝑌	PROPN
cana-3308	80	31	)	)	PUNCT
cana-3308	80	32	⇉	⇉	PUNCT
cana-3308	81	1	(	(	PUNCT
cana-3308	81	2	𝑋	𝑋	PROPN
cana-3308	81	3	,	,	PUNCT
cana-3308	81	4	𝑌	𝑌	PROPN
cana-3308	81	5	)	)	PUNCT
cana-3308	81	6	.	.	PUNCT
cana-3308	82	1	suppose	suppose	VERB
cana-3308	82	2	that	that	SCONJ
cana-3308	82	3	𝜑	𝜑	PROPN
cana-3308	82	4	∶	∶	NOUN
cana-3308	82	5	ℝ+	ℝ+	PUNCT
cana-3308	82	6	→	→	X
cana-3308	82	7	ℝ+	ℝ+	PUNCT
cana-3308	82	8	is	be	AUX
cana-3308	82	9	function	function	NOUN
cana-3308	82	10	and	and	CCONJ
cana-3308	82	11	𝜃	𝜃	NOUN
cana-3308	82	12	∈	∈	NOUN
cana-3308	82	13	θ	θ	NOUN
cana-3308	82	14	and	and	CCONJ
cana-3308	82	15	𝑇	𝑇	PROPN
cana-3308	82	16	is	be	AUX
cana-3308	82	17	called	call	VERB
cana-3308	82	18	𝜑	𝜑	DET
cana-3308	82	19	geraghty	geraghty	PROPN
cana-3308	82	20	contraction	contraction	NOUN
cana-3308	82	21	if	if	SCONJ
cana-3308	82	22	it	it	PRON
cana-3308	82	23	satisfies	satisfy	VERB
cana-3308	82	24	the	the	DET
cana-3308	82	25	following	follow	VERB
cana-3308	82	26	condition	condition	NOUN
cana-3308	82	27	:	:	PUNCT
cana-3308	82	28	(	(	PUNCT
cana-3308	82	29	i	i	NOUN
cana-3308	82	30	)	)	PUNCT
cana-3308	82	31	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	82	32	)	)	PUNCT
cana-3308	82	33	<	<	X
cana-3308	82	34	𝑡	𝑡	PROPN
cana-3308	82	35	for	for	ADP
cana-3308	82	36	any	any	DET
cana-3308	82	37	𝑡	𝑡	PROPN
cana-3308	82	38	∈	∈	PROPN
cana-3308	82	39	(	(	PUNCT
cana-3308	82	40	0	0	NUM
cana-3308	82	41	,	,	PUNCT
cana-3308	82	42	∞	∞	PROPN
cana-3308	82	43	)	)	PUNCT
cana-3308	82	44	,	,	PUNCT
cana-3308	82	45	(	(	PUNCT
cana-3308	82	46	ii	ii	NOUN
cana-3308	82	47	)	)	PUNCT
cana-3308	82	48	for	for	ADP
cana-3308	82	49	any	any	PRON
cana-3308	82	50	휀	휀	NOUN
cana-3308	82	51	>	>	X
cana-3308	82	52	0	0	NUM
cana-3308	82	53	,	,	PUNCT
cana-3308	82	54	there	there	PRON
cana-3308	82	55	exist	exist	VERB
cana-3308	82	56	𝛿	𝛿	PROPN
cana-3308	82	57	>	>	X
cana-3308	82	58	0	0	NUM
cana-3308	82	59	such	such	ADJ
cana-3308	82	60	that	that	SCONJ
cana-3308	82	61	휀	휀	X
cana-3308	82	62	<	<	X
cana-3308	82	63	𝑡	𝑡	X
cana-3308	82	64	<	<	X
cana-3308	82	65	휀	휀	NOUN
cana-3308	82	66	+	+	X
cana-3308	82	67	𝛿	𝛿	ADJ
cana-3308	82	68	⇒	⇒	NOUN
cana-3308	82	69	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	82	70	)	)	PUNCT
cana-3308	82	71	≤	≤	NUM
cana-3308	82	72	휀	휀	X
cana-3308	82	73	,	,	PUNCT
cana-3308	82	74	(	(	PUNCT
cana-3308	82	75	iii)𝑑(𝑇𝑥	iii)𝑑(𝑇𝑥	ADV
cana-3308	82	76	,	,	PUNCT
cana-3308	82	77	𝑇𝑦	𝑇𝑦	PROPN
cana-3308	82	78	)	)	PUNCT
cana-3308	82	79	≤	≤	NOUN
cana-3308	82	80	𝜃	𝜃	X
cana-3308	82	81	(	(	PUNCT
cana-3308	82	82	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	82	83	,	,	PUNCT
cana-3308	82	84	𝑦	𝑦	NOUN
cana-3308	82	85	)	)	PUNCT
cana-3308	82	86	)	)	PUNCT
cana-3308	82	87	𝜑(𝑑(𝑥	𝜑(𝑑(𝑥	PROPN
cana-3308	82	88	,	,	PUNCT
cana-3308	82	89	𝑦	𝑦	NOUN
cana-3308	82	90	)	)	PUNCT
cana-3308	82	91	)	)	PUNCT
cana-3308	82	92	for	for	ADP
cana-3308	82	93	all	all	PRON
cana-3308	82	94	𝑥	𝑥	DET
cana-3308	82	95	∈	∈	PROPN
cana-3308	82	96	𝑋	𝑋	NOUN
cana-3308	82	97	and	and	CCONJ
cana-3308	82	98	𝑦	𝑦	NOUN
cana-3308	82	99	∈	∈	NOUN
cana-3308	82	100	𝑌.	𝑌.	ADJ
cana-3308	82	101	definition	definition	NOUN
cana-3308	82	102	3.3	3.3	NUM
cana-3308	82	103	.	.	PUNCT
cana-3308	83	1	let	let	VERB
cana-3308	83	2	𝑋	𝑋	NOUN
cana-3308	83	3	and	and	CCONJ
cana-3308	83	4	𝑌	𝑌	PROPN
cana-3308	83	5	be	be	VERB
cana-3308	83	6	two	two	NUM
cana-3308	83	7	non	non	ADJ
cana-3308	83	8	-	-	ADJ
cana-3308	83	9	empty	empty	ADJ
cana-3308	83	10	sets	set	NOUN
cana-3308	83	11	and	and	CCONJ
cana-3308	83	12	(	(	PUNCT
cana-3308	83	13	𝑋	𝑋	PROPN
cana-3308	83	14	,	,	PUNCT
cana-3308	83	15	𝑌	𝑌	PROPN
cana-3308	83	16	,	,	PUNCT
cana-3308	83	17	𝑑	𝑑	NOUN
cana-3308	83	18	)	)	PUNCT
cana-3308	83	19	be	be	VERB
cana-3308	83	20	a	a	DET
cana-3308	83	21	bipolar	bipolar	ADJ
cana-3308	83	22	metric	metric	ADJ
cana-3308	83	23	space	space	NOUN
cana-3308	83	24	,	,	PUNCT
cana-3308	83	25	𝑇	𝑇	PROPN
cana-3308	83	26	∶	∶	PROPN
cana-3308	83	27	(	(	PUNCT
cana-3308	83	28	𝑋	𝑋	PROPN
cana-3308	83	29	,	,	PUNCT
cana-3308	83	30	𝑌	𝑌	PROPN
cana-3308	83	31	)	)	PUNCT
cana-3308	83	32	⇉	⇉	PUNCT
cana-3308	84	1	(	(	PUNCT
cana-3308	84	2	𝑋	𝑋	NOUN
cana-3308	84	3	,	,	PUNCT
cana-3308	84	4	𝑌	𝑌	PROPN
cana-3308	84	5	)	)	PUNCT
cana-3308	84	6	is	be	AUX
cana-3308	84	7	a	a	DET
cana-3308	84	8	self	self	NOUN
cana-3308	84	9	map	map	NOUN
cana-3308	84	10	and	and	CCONJ
cana-3308	84	11	𝛼	𝛼	ADP
cana-3308	84	12	∶	∶	NOUN
cana-3308	84	13	𝑋	𝑋	NOUN
cana-3308	84	14	×	×	NOUN
cana-3308	84	15	𝑌	𝑌	PROPN
cana-3308	84	16	→	→	SYM
cana-3308	84	17	[	[	X
cana-3308	84	18	0	0	NUM
cana-3308	84	19	,	,	PUNCT
cana-3308	84	20	∞	∞	PROPN
cana-3308	84	21	)	)	PUNCT
cana-3308	84	22	.	.	PUNCT
cana-3308	85	1	a	a	DET
cana-3308	85	2	mapping	mapping	NOUN
cana-3308	85	3	𝑇	𝑇	PROPN
cana-3308	85	4	is	be	AUX
cana-3308	85	5	said	say	VERB
cana-3308	85	6	to	to	PART
cana-3308	85	7	be	be	AUX
cana-3308	85	8	(	(	PUNCT
cana-3308	85	9	𝛼	𝛼	PROPN
cana-3308	85	10	,	,	PUNCT
cana-3308	85	11	𝜓	𝜓	NOUN
cana-3308	85	12	,	,	PUNCT
cana-3308	85	13	𝜑	𝜑	NOUN
cana-3308	85	14	)	)	PUNCT
cana-3308	85	15	geraghty	geraghty	PROPN
cana-3308	85	16	contraction	contraction	NOUN
cana-3308	85	17	mapping	mapping	NOUN
cana-3308	85	18	if	if	SCONJ
cana-3308	85	19	there	there	PRON
cana-3308	85	20	exist	exist	VERB
cana-3308	85	21	𝜑	𝜑	PRON
cana-3308	85	22	∶	∶	NOUN
cana-3308	85	23	ℝ+	ℝ+	PUNCT
cana-3308	85	24	→	→	SYM
cana-3308	85	25	ℝ+	ℝ+	PUNCT
cana-3308	85	26	,	,	PUNCT
cana-3308	85	27	𝜓	𝜓	PROPN
cana-3308	85	28	∈	∈	PROPN
cana-3308	85	29	ψ	ψ	NOUN
cana-3308	85	30	and	and	CCONJ
cana-3308	85	31	𝜃	𝜃	PRON
cana-3308	85	32	∈	∈	NOUN
cana-3308	85	33	θ	θ	NOUN
cana-3308	85	34	satisfies	satisfy	VERB
cana-3308	85	35	the	the	DET
cana-3308	85	36	following	follow	VERB
cana-3308	85	37	condition	condition	NOUN
cana-3308	85	38	:	:	PUNCT
cana-3308	85	39	(	(	PUNCT
cana-3308	85	40	i	i	NOUN
cana-3308	85	41	)	)	PUNCT
cana-3308	85	42	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	85	43	)	)	PUNCT
cana-3308	85	44	<	<	X
cana-3308	85	45	𝑡	𝑡	PROPN
cana-3308	85	46	for	for	ADP
cana-3308	85	47	any	any	DET
cana-3308	85	48	𝑡	𝑡	PROPN
cana-3308	85	49	∈	∈	PROPN
cana-3308	85	50	(	(	PUNCT
cana-3308	85	51	0	0	NUM
cana-3308	85	52	,	,	PUNCT
cana-3308	85	53	∞	∞	PROPN
cana-3308	85	54	)	)	PUNCT
cana-3308	85	55	,	,	PUNCT
cana-3308	85	56	(	(	PUNCT
cana-3308	85	57	3.1	3.1	NUM
cana-3308	85	58	)	)	PUNCT
cana-3308	85	59	(	(	PUNCT
cana-3308	85	60	ii	ii	NOUN
cana-3308	85	61	)	)	PUNCT
cana-3308	85	62	for	for	ADP
cana-3308	85	63	any	any	PRON
cana-3308	85	64	휀	휀	NOUN
cana-3308	85	65	>	>	X
cana-3308	85	66	0	0	NUM
cana-3308	85	67	,	,	PUNCT
cana-3308	85	68	there	there	PRON
cana-3308	85	69	exist	exist	VERB
cana-3308	85	70	𝛿	𝛿	PROPN
cana-3308	85	71	>	>	X
cana-3308	85	72	0	0	NUM
cana-3308	85	73	such	such	ADJ
cana-3308	85	74	that	that	SCONJ
cana-3308	85	75	휀	휀	X
cana-3308	85	76	<	<	X
cana-3308	85	77	𝑡	𝑡	X
cana-3308	85	78	<	<	X
cana-3308	85	79	휀	휀	NOUN
cana-3308	85	80	+	+	X
cana-3308	85	81	𝛿	𝛿	ADJ
cana-3308	85	82	⇒	⇒	NOUN
cana-3308	85	83	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	85	84	)	)	PUNCT
cana-3308	85	85	≤	≤	NUM
cana-3308	86	1	휀	휀	X
cana-3308	86	2	,	,	PUNCT
cana-3308	86	3	(	(	PUNCT
cana-3308	86	4	3.2	3.2	NUM
cana-3308	86	5	)	)	PUNCT
cana-3308	86	6	(	(	PUNCT
cana-3308	86	7	iii)𝛼(𝑥	iii)𝛼(𝑥	PROPN
cana-3308	86	8	,	,	PUNCT
cana-3308	86	9	𝑦)𝜓(𝑑(𝑇𝑥	𝑦)𝜓(𝑑(𝑇𝑥	PROPN
cana-3308	86	10	,	,	PUNCT
cana-3308	86	11	𝑇𝑦	𝑇𝑦	PROPN
cana-3308	86	12	)	)	PUNCT
cana-3308	86	13	)	)	PUNCT
cana-3308	86	14	≤	≤	PUNCT
cana-3308	87	1	𝜃(𝜓	𝜃(𝜓	PROPN
cana-3308	87	2	(	(	PUNCT
cana-3308	87	3	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	87	4	,	,	PUNCT
cana-3308	87	5	𝑦	𝑦	NOUN
cana-3308	87	6	)	)	PUNCT
cana-3308	87	7	)	)	PUNCT
cana-3308	87	8	)	)	PUNCT
cana-3308	88	1	𝜑(𝜓(𝑑(𝑥	𝜑(𝜓(𝑑(𝑥	PROPN
cana-3308	88	2	,	,	PUNCT
cana-3308	88	3	𝑦	𝑦	NOUN
cana-3308	88	4	)	)	PUNCT
cana-3308	88	5	)	)	PUNCT
cana-3308	88	6	)	)	PUNCT
cana-3308	88	7	,	,	PUNCT
cana-3308	88	8	for	for	ADP
cana-3308	88	9	all	all	PRON
cana-3308	88	10	𝑥	𝑥	DET
cana-3308	88	11	∈	∈	PROPN
cana-3308	88	12	𝑋	𝑋	NOUN
cana-3308	88	13	and	and	CCONJ
cana-3308	88	14	𝑦	𝑦	NOUN
cana-3308	88	15	∈	∈	PROPN
cana-3308	88	16	𝑌.	𝑌.	PROPN
cana-3308	88	17	(	(	PUNCT
cana-3308	88	18	3.3	3.3	NUM
cana-3308	88	19	)	)	PUNCT
cana-3308	88	20	theorem	theorem	VERB
cana-3308	88	21	3.4	3.4	NUM
cana-3308	88	22	.	.	PUNCT
cana-3308	89	1	let	let	VERB
cana-3308	89	2	(	(	PUNCT
cana-3308	89	3	𝑋	𝑋	PROPN
cana-3308	89	4	,	,	PUNCT
cana-3308	89	5	𝑌	𝑌	PROPN
cana-3308	89	6	,	,	PUNCT
cana-3308	89	7	𝑑	𝑑	NOUN
cana-3308	89	8	)	)	PUNCT
cana-3308	89	9	be	be	VERB
cana-3308	89	10	a	a	DET
cana-3308	89	11	complete	complete	ADJ
cana-3308	89	12	bipolar	bipolar	ADJ
cana-3308	89	13	metric	metric	ADJ
cana-3308	89	14	space	space	NOUN
cana-3308	89	15	,	,	PUNCT
cana-3308	89	16	𝑇	𝑇	PROPN
cana-3308	89	17	∶	∶	PROPN
cana-3308	89	18	(	(	PUNCT
cana-3308	89	19	𝑋	𝑋	PROPN
cana-3308	89	20	,	,	PUNCT
cana-3308	89	21	𝑌	𝑌	PROPN
cana-3308	89	22	)	)	PUNCT
cana-3308	89	23	⇉	⇉	PUNCT
cana-3308	90	1	(	(	PUNCT
cana-3308	90	2	𝑋	𝑋	NOUN
cana-3308	90	3	,	,	PUNCT
cana-3308	90	4	𝑌	𝑌	PROPN
cana-3308	90	5	)	)	PUNCT
cana-3308	90	6	is	be	AUX
cana-3308	90	7	a	a	DET
cana-3308	90	8	covariant	covariant	ADJ
cana-3308	90	9	mapping	mapping	NOUN
cana-3308	90	10	and	and	CCONJ
cana-3308	90	11	𝛼	𝛼	NOUN
cana-3308	90	12	∶	∶	NOUN
cana-3308	90	13	𝑋	𝑋	NOUN
cana-3308	90	14	×	×	NOUN
cana-3308	90	15	𝑌	𝑌	PROPN
cana-3308	90	16	→	→	SYM
cana-3308	90	17	[	[	X
cana-3308	90	18	0	0	NUM
cana-3308	90	19	,	,	PUNCT
cana-3308	90	20	∞	∞	PROPN
cana-3308	90	21	)	)	PUNCT
cana-3308	90	22	.	.	PUNCT
cana-3308	91	1	suppose	suppose	VERB
cana-3308	91	2	that	that	SCONJ
cana-3308	91	3	the	the	DET
cana-3308	91	4	following	follow	VERB
cana-3308	91	5	conditions	condition	NOUN
cana-3308	91	6	hold	hold	VERB
cana-3308	91	7	:	:	PUNCT
cana-3308	91	8	(	(	PUNCT
cana-3308	91	9	i	i	NOUN
cana-3308	91	10	)	)	PUNCT
cana-3308	91	11	𝑇	𝑇	PROPN
cana-3308	91	12	is	be	AUX
cana-3308	91	13	𝛼admissible	𝛼admissible	ADJ
cana-3308	91	14	mapping	mapping	NOUN
cana-3308	91	15	,	,	PUNCT
cana-3308	91	16	(	(	PUNCT
cana-3308	91	17	ii	ii	NOUN
cana-3308	91	18	)	)	PUNCT
cana-3308	91	19	𝑇	𝑇	PROPN
cana-3308	91	20	is	be	AUX
cana-3308	91	21	an	an	DET
cana-3308	91	22	(	(	PUNCT
cana-3308	91	23	𝛼	𝛼	PROPN
cana-3308	91	24	,	,	PUNCT
cana-3308	91	25	𝜓	𝜓	NOUN
cana-3308	91	26	,	,	PUNCT
cana-3308	91	27	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	91	28	contraction	contraction	NOUN
cana-3308	91	29	mapping	mapping	NOUN
cana-3308	91	30	,	,	PUNCT
cana-3308	91	31	(	(	PUNCT
cana-3308	91	32	iii	iii	X
cana-3308	91	33	)	)	PUNCT
cana-3308	91	34	there	there	PRON
cana-3308	91	35	exist	exist	VERB
cana-3308	91	36	𝑥0	𝑥0	NOUN
cana-3308	91	37	∈	∈	PROPN
cana-3308	91	38	𝑋and	𝑋and	PROPN
cana-3308	91	39	𝑦0	𝑦0	NOUN
cana-3308	91	40	∈	∈	NOUN
cana-3308	91	41	𝑌	𝑌	PROPN
cana-3308	91	42	such	such	ADJ
cana-3308	91	43	that	that	PRON
cana-3308	91	44	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	91	45	,	,	PUNCT
cana-3308	91	46	𝑦0	𝑦0	NOUN
cana-3308	91	47	)	)	PUNCT
cana-3308	91	48	≥	≥	NOUN
cana-3308	91	49	1	1	NUM
cana-3308	91	50	and	and	CCONJ
cana-3308	91	51	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	91	52	,	,	PUNCT
cana-3308	91	53	𝑇𝑦0	𝑇𝑦0	NOUN
cana-3308	91	54	)	)	PUNCT
cana-3308	91	55	≥	≥	NOUN
cana-3308	92	1	1	1	NUM
cana-3308	92	2	.	.	PUNCT
cana-3308	93	1	then	then	ADV
cana-3308	93	2	𝑇	𝑇	PROPN
cana-3308	93	3	has	have	AUX
cana-3308	93	4	fixed	fix	VERB
cana-3308	93	5	point	point	NOUN
cana-3308	93	6	.	.	PUNCT
cana-3308	94	1	proof	proof	NOUN
cana-3308	94	2	:	:	PUNCT
cana-3308	94	3	let	let	VERB
cana-3308	94	4	𝑥0	𝑥0	VERB
cana-3308	94	5	∈	∈	PROPN
cana-3308	94	6	𝑋and	𝑋and	PROPN
cana-3308	94	7	𝑦0	𝑦0	NOUN
cana-3308	94	8	∈	∈	NOUN
cana-3308	94	9	𝑌	𝑌	PROPN
cana-3308	94	10	such	such	ADJ
cana-3308	94	11	that	that	PRON
cana-3308	94	12	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	94	13	,	,	PUNCT
cana-3308	94	14	𝑦0	𝑦0	NOUN
cana-3308	94	15	)	)	PUNCT
cana-3308	94	16	≥	≥	NOUN
cana-3308	94	17	1	1	NUM
cana-3308	94	18	and	and	CCONJ
cana-3308	94	19	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	94	20	,	,	PUNCT
cana-3308	94	21	𝑇𝑦0	𝑇𝑦0	NOUN
cana-3308	94	22	)	)	PUNCT
cana-3308	94	23	≥	≥	NOUN
cana-3308	95	1	1	1	NUM
cana-3308	95	2	.	.	PUNCT
cana-3308	96	1	now	now	ADV
cana-3308	96	2	we	we	PRON
cana-3308	96	3	define	define	VERB
cana-3308	96	4	a	a	DET
cana-3308	96	5	bisequence	bisequence	NOUN
cana-3308	96	6	{	{	PUNCT
cana-3308	96	7	(	(	PUNCT
cana-3308	96	8	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	96	9	,	,	PUNCT
cana-3308	96	10	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	96	11	)	)	PUNCT
cana-3308	96	12	}	}	PUNCT
cana-3308	96	13	in	in	ADP
cana-3308	96	14	(	(	PUNCT
cana-3308	96	15	𝑋	𝑋	PROPN
cana-3308	96	16	,	,	PUNCT
cana-3308	96	17	𝑌	𝑌	PROPN
cana-3308	96	18	)	)	PUNCT
cana-3308	96	19	by	by	ADP
cana-3308	96	20	𝑇𝑥𝑛	𝑇𝑥𝑛	PROPN
cana-3308	96	21	=	=	PUNCT
cana-3308	96	22	𝑥𝑛+1	𝑥𝑛+1	PROPN
cana-3308	96	23	and	and	CCONJ
cana-3308	96	24	𝑇𝑦𝑛	𝑇𝑦𝑛	PROPN
cana-3308	96	25	=	=	SYM
cana-3308	96	26	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	96	27	for	for	ADP
cana-3308	96	28	all	all	DET
cana-3308	96	29	𝑛	𝑛	DET
cana-3308	96	30	∈	∈	PROPN
cana-3308	96	31	ℕ	ℕ	PROPN
cana-3308	96	32	∪	∪	X
cana-3308	96	33	{	{	PUNCT
cana-3308	96	34	0	0	NUM
cana-3308	96	35	}	}	PUNCT
cana-3308	96	36	.	.	PUNCT
cana-3308	97	1	since	since	SCONJ
cana-3308	97	2	𝑇	𝑇	PROPN
cana-3308	97	3	is	be	AUX
cana-3308	97	4	𝛼admissible	𝛼admissible	ADJ
cana-3308	97	5	mapping	mapping	NOUN
cana-3308	97	6	.	.	PUNCT
cana-3308	98	1	so	so	ADV
cana-3308	98	2	,	,	PUNCT
cana-3308	98	3	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	98	4	,	,	PUNCT
cana-3308	98	5	𝑦0	𝑦0	NOUN
cana-3308	98	6	)	)	PUNCT
cana-3308	98	7	≥	≥	NOUN
cana-3308	98	8	1	1	NUM
cana-3308	98	9	⇒	⇒	PROPN
cana-3308	98	10	𝛼(𝑇𝑥0	𝛼(𝑇𝑥0	PROPN
cana-3308	98	11	,	,	PUNCT
cana-3308	98	12	𝑇𝑦0	𝑇𝑦0	NOUN
cana-3308	98	13	)	)	PUNCT
cana-3308	98	14	≥	≥	NOUN
cana-3308	98	15	1	1	NUM
cana-3308	98	16	,	,	PUNCT
cana-3308	98	17	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	98	18	,	,	PUNCT
cana-3308	98	19	𝑦1	𝑦1	NOUN
cana-3308	98	20	)	)	PUNCT
cana-3308	98	21	=	=	SYM
cana-3308	98	22	𝛼(𝑥0	𝛼(𝑥0	VERB
cana-3308	98	23	,	,	PUNCT
cana-3308	98	24	𝑇𝑦0	𝑇𝑦0	NOUN
cana-3308	98	25	)	)	PUNCT
cana-3308	98	26	≥	≥	NOUN
cana-3308	98	27	1	1	NUM
cana-3308	98	28	,	,	PUNCT
cana-3308	98	29	𝛼(𝑥1	𝛼(𝑥1	NOUN
cana-3308	98	30	,	,	PUNCT
cana-3308	98	31	𝑦1	𝑦1	NOUN
cana-3308	98	32	)	)	PUNCT
cana-3308	99	1	=	=	SYM
cana-3308	99	2	𝛼(𝑇𝑥0	𝛼(𝑇𝑥0	PROPN
cana-3308	99	3	,	,	PUNCT
cana-3308	99	4	𝑇𝑦0	𝑇𝑦0	NOUN
cana-3308	99	5	)	)	PUNCT
cana-3308	99	6	≥	≥	NOUN
cana-3308	99	7	1	1	NUM
cana-3308	99	8	,	,	PUNCT
cana-3308	99	9	using	use	VERB
cana-3308	99	10	mathematical	mathematical	ADJ
cana-3308	99	11	induction	induction	NOUN
cana-3308	99	12	,	,	PUNCT
cana-3308	99	13	we	we	PRON
cana-3308	99	14	get	get	VERB
cana-3308	99	15	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-3308	99	16	,	,	PUNCT
cana-3308	99	17	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	99	18	)	)	PUNCT
cana-3308	99	19	≥	≥	NOUN
cana-3308	99	20	1	1	NUM
cana-3308	99	21	and	and	CCONJ
cana-3308	99	22	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-3308	99	23	,	,	PUNCT
cana-3308	99	24	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	99	25	)	)	PUNCT
cana-3308	99	26	≥	≥	NOUN
cana-3308	99	27	1	1	NUM
cana-3308	99	28	for	for	ADP
cana-3308	99	29	all	all	DET
cana-3308	99	30	𝑛	𝑛	DET
cana-3308	99	31	∈	∈	PROPN
cana-3308	99	32	ℕ	ℕ	PROPN
cana-3308	99	33	∪	∪	X
cana-3308	99	34	{	{	PUNCT
cana-3308	99	35	0	0	NUM
cana-3308	99	36	}	}	PUNCT
cana-3308	99	37	.	.	PUNCT
cana-3308	100	1	(	(	PUNCT
cana-3308	100	2	3.4	3.4	NUM
cana-3308	100	3	)	)	PUNCT
cana-3308	100	4	putting	put	VERB
cana-3308	100	5	𝑥	𝑥	X
cana-3308	100	6	=	=	PUNCT
cana-3308	100	7	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-3308	100	8	and	and	CCONJ
cana-3308	100	9	𝑦	𝑦	NOUN
cana-3308	100	10	=	=	SYM
cana-3308	100	11	𝑦𝑛+2	𝑦𝑛+2	NUM
cana-3308	100	12	in	in	ADP
cana-3308	100	13	equation	equation	NOUN
cana-3308	100	14	(	(	PUNCT
cana-3308	100	15	3.3	3.3	NUM
cana-3308	100	16	)	)	PUNCT
cana-3308	100	17	,	,	PUNCT
cana-3308	100	18	using	use	VERB
cana-3308	100	19	equations	equation	NOUN
cana-3308	100	20	(	(	PUNCT
cana-3308	100	21	3.1	3.1	NUM
cana-3308	100	22	)	)	PUNCT
cana-3308	100	23	,	,	PUNCT
cana-3308	100	24	(	(	PUNCT
cana-3308	100	25	3.2	3.2	NUM
cana-3308	100	26	)	)	PUNCT
cana-3308	100	27	and	and	CCONJ
cana-3308	100	28	by	by	ADP
cana-3308	100	29	the	the	DET
cana-3308	100	30	properties	property	NOUN
cana-3308	100	31	of	of	ADP
cana-3308	100	32	𝜓	𝜓	PROPN
cana-3308	100	33	and	and	CCONJ
cana-3308	100	34	𝜃	𝜃	X
cana-3308	100	35	,	,	PUNCT
cana-3308	100	36	we	we	PRON
cana-3308	100	37	have	have	VERB
cana-3308	100	38	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	100	39	,	,	PUNCT
cana-3308	100	40	𝑦𝑛+2	𝑦𝑛+2	NUM
cana-3308	100	41	)	)	PUNCT
cana-3308	100	42	)	)	PUNCT
cana-3308	101	1	=	=	PUNCT
cana-3308	101	2	𝜓(𝑑(𝑇𝑥𝑛	𝜓(𝑑(𝑇𝑥𝑛	NOUN
cana-3308	101	3	,	,	PUNCT
cana-3308	101	4	𝑇𝑦𝑛+1	𝑇𝑦𝑛+1	PROPN
cana-3308	101	5	)	)	PUNCT
cana-3308	101	6	)	)	PUNCT
cana-3308	102	1	≤	≤	NOUN
cana-3308	102	2	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-3308	102	3	,	,	PUNCT
cana-3308	102	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	102	5	)	)	PUNCT
cana-3308	102	6	𝜓(𝑑(𝑇𝑥𝑛	𝜓(𝑑(𝑇𝑥𝑛	NOUN
cana-3308	102	7	,	,	PUNCT
cana-3308	102	8	𝑇𝑦𝑛+1	𝑇𝑦𝑛+1	PROPN
cana-3308	102	9	)	)	PUNCT
cana-3308	102	10	)	)	PUNCT
cana-3308	102	11	≤	≤	NUM
cana-3308	102	12	𝜃	𝜃	X
cana-3308	102	13	(	(	PUNCT
cana-3308	102	14	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	102	15	,	,	PUNCT
cana-3308	102	16	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	102	17	)	)	PUNCT
cana-3308	102	18	)	)	PUNCT
cana-3308	102	19	)	)	PUNCT
cana-3308	102	20	𝜑(𝜓(𝑑(𝑥𝑛	𝜑(𝜓(𝑑(𝑥𝑛	PROPN
cana-3308	102	21	,	,	PUNCT
cana-3308	102	22	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	102	23	)	)	PUNCT
cana-3308	102	24	)	)	PUNCT
cana-3308	102	25	)	)	PUNCT
cana-3308	102	26	communications	communication	NOUN
cana-3308	102	27	on	on	ADP
cana-3308	102	28	applied	apply	VERB
cana-3308	102	29	nonlinear	nonlinear	ADJ
cana-3308	102	30	analysis	analysis	NOUN
cana-3308	102	31	issn	issn	NOUN
cana-3308	102	32	:	:	PUNCT
cana-3308	102	33	1074	1074	NUM
cana-3308	102	34	-	-	PUNCT
cana-3308	102	35	133x	133x	NUM
cana-3308	102	36	vol	vol	NOUN
cana-3308	102	37	32	32	NUM
cana-3308	102	38	no	no	NOUN
cana-3308	102	39	.	.	PUNCT
cana-3308	103	1	6s	6s	NUM
cana-3308	103	2	(	(	PUNCT
cana-3308	103	3	2025	2025	NUM
cana-3308	103	4	)	)	PUNCT
cana-3308	103	5	443	443	NUM
cana-3308	103	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	103	7	≤	≤	NUM
cana-3308	103	8	𝜑(𝜓(𝑑(𝑥𝑛	𝜑(𝜓(𝑑(𝑥𝑛	PROPN
cana-3308	103	9	,	,	PUNCT
cana-3308	103	10	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	103	11	)	)	PUNCT
cana-3308	103	12	)	)	PUNCT
cana-3308	103	13	)	)	PUNCT
cana-3308	104	1	<	<	X
cana-3308	104	2	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	104	3	,	,	PUNCT
cana-3308	104	4	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	104	5	)	)	PUNCT
cana-3308	104	6	)	)	PUNCT
cana-3308	104	7	.	.	PUNCT
cana-3308	105	1	(	(	PUNCT
cana-3308	105	2	3.5	3.5	NUM
cana-3308	105	3	)	)	PUNCT
cana-3308	105	4	hence	hence	ADV
cana-3308	105	5	,	,	PUNCT
cana-3308	105	6	𝜓	𝜓	PROPN
cana-3308	105	7	is	be	AUX
cana-3308	105	8	strictly	strictly	ADV
cana-3308	105	9	increasing	increase	VERB
cana-3308	105	10	function	function	NOUN
cana-3308	105	11	so	so	ADV
cana-3308	105	12	,	,	PUNCT
cana-3308	105	13	we	we	PRON
cana-3308	105	14	get	get	VERB
cana-3308	105	15	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	105	16	,	,	PUNCT
cana-3308	105	17	𝑦𝑛+2	𝑦𝑛+2	NUM
cana-3308	105	18	)	)	PUNCT
cana-3308	105	19	<	<	X
cana-3308	105	20	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	105	21	,	,	PUNCT
cana-3308	105	22	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	105	23	)	)	PUNCT
cana-3308	105	24	for	for	ADP
cana-3308	105	25	all	all	DET
cana-3308	105	26	𝑛	𝑛	DET
cana-3308	105	27	≥	≥	NOUN
cana-3308	105	28	0	0	NUM
cana-3308	105	29	.	.	PUNCT
cana-3308	106	1	(	(	PUNCT
cana-3308	106	2	3.6	3.6	NUM
cana-3308	106	3	)	)	PUNCT
cana-3308	106	4	similarly	similarly	ADV
cana-3308	106	5	,	,	PUNCT
cana-3308	106	6	putting	put	VERB
cana-3308	106	7	𝑥	𝑥	X
cana-3308	106	8	=	=	PUNCT
cana-3308	106	9	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-3308	106	10	and	and	CCONJ
cana-3308	106	11	𝑦	𝑦	NOUN
cana-3308	106	12	=	=	SYM
cana-3308	106	13	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	106	14	in	in	ADP
cana-3308	106	15	equation	equation	NOUN
cana-3308	106	16	(	(	PUNCT
cana-3308	106	17	3.3	3.3	NUM
cana-3308	106	18	)	)	PUNCT
cana-3308	106	19	,	,	PUNCT
cana-3308	106	20	using	use	VERB
cana-3308	106	21	equations	equation	NOUN
cana-3308	106	22	(	(	PUNCT
cana-3308	106	23	3.1	3.1	NUM
cana-3308	106	24	)	)	PUNCT
cana-3308	106	25	,	,	PUNCT
cana-3308	106	26	(	(	PUNCT
cana-3308	106	27	3.2	3.2	NUM
cana-3308	106	28	)	)	PUNCT
cana-3308	106	29	and	and	CCONJ
cana-3308	106	30	by	by	ADP
cana-3308	106	31	the	the	DET
cana-3308	106	32	properties	property	NOUN
cana-3308	106	33	of	of	ADP
cana-3308	106	34	𝜓	𝜓	PROPN
cana-3308	106	35	and	and	CCONJ
cana-3308	106	36	𝜃	𝜃	X
cana-3308	106	37	,	,	PUNCT
cana-3308	106	38	we	we	PRON
cana-3308	106	39	have	have	VERB
cana-3308	106	40	the	the	DET
cana-3308	106	41	following	follow	VERB
cana-3308	106	42	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	106	43	,	,	PUNCT
cana-3308	106	44	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	106	45	)	)	PUNCT
cana-3308	106	46	)	)	PUNCT
cana-3308	107	1	=	=	PUNCT
cana-3308	107	2	𝜓(𝑑(𝑇𝑥𝑛	𝜓(𝑑(𝑇𝑥𝑛	NOUN
cana-3308	107	3	,	,	PUNCT
cana-3308	107	4	𝑇𝑦𝑛	𝑇𝑦𝑛	PROPN
cana-3308	107	5	)	)	PUNCT
cana-3308	107	6	)	)	PUNCT
cana-3308	108	1	≤	≤	NOUN
cana-3308	108	2	𝛼(𝑥𝑛	𝛼(𝑥𝑛	ADV
cana-3308	108	3	,	,	PUNCT
cana-3308	108	4	𝑦𝑛)𝜓(𝑑(𝑇𝑥𝑛	𝑦𝑛)𝜓(𝑑(𝑇𝑥𝑛	PROPN
cana-3308	108	5	,	,	PUNCT
cana-3308	108	6	𝑇𝑦𝑛	𝑇𝑦𝑛	PROPN
cana-3308	108	7	)	)	PUNCT
cana-3308	108	8	)	)	PUNCT
cana-3308	109	1	≤	≤	NUM
cana-3308	109	2	𝜃	𝜃	X
cana-3308	109	3	(	(	PUNCT
cana-3308	109	4	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	109	5	,	,	PUNCT
cana-3308	109	6	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	109	7	)	)	PUNCT
cana-3308	109	8	)	)	PUNCT
cana-3308	109	9	)	)	PUNCT
cana-3308	109	10	𝜑(𝜓(𝑑(𝑥𝑛	𝜑(𝜓(𝑑(𝑥𝑛	NOUN
cana-3308	109	11	,	,	PUNCT
cana-3308	109	12	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	109	13	)	)	PUNCT
cana-3308	109	14	)	)	PUNCT
cana-3308	109	15	)	)	PUNCT
cana-3308	109	16	≤	≤	NOUN
cana-3308	109	17	𝜑(𝜓(𝑑(𝑥𝑛	𝜑(𝜓(𝑑(𝑥𝑛	PUNCT
cana-3308	109	18	,	,	PUNCT
cana-3308	109	19	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	109	20	)	)	PUNCT
cana-3308	109	21	)	)	PUNCT
cana-3308	109	22	)	)	PUNCT
cana-3308	110	1	<	<	X
cana-3308	110	2	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	110	3	,	,	PUNCT
cana-3308	110	4	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	110	5	)	)	PUNCT
cana-3308	110	6	)	)	PUNCT
cana-3308	110	7	.	.	PUNCT
cana-3308	111	1	(	(	PUNCT
cana-3308	111	2	3.7	3.7	NUM
cana-3308	111	3	)	)	PUNCT
cana-3308	111	4	hence	hence	ADV
cana-3308	111	5	,	,	PUNCT
cana-3308	111	6	𝜓	𝜓	PROPN
cana-3308	111	7	is	be	AUX
cana-3308	111	8	strictly	strictly	ADV
cana-3308	111	9	increasing	increase	VERB
cana-3308	111	10	function	function	NOUN
cana-3308	111	11	so	so	ADV
cana-3308	111	12	,	,	PUNCT
cana-3308	111	13	we	we	PRON
cana-3308	111	14	obtain	obtain	VERB
cana-3308	111	15	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	111	16	,	,	PUNCT
cana-3308	111	17	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	111	18	)	)	PUNCT
cana-3308	111	19	<	<	X
cana-3308	111	20	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	111	21	,	,	PUNCT
cana-3308	111	22	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	111	23	)	)	PUNCT
cana-3308	111	24	for	for	ADP
cana-3308	111	25	all	all	DET
cana-3308	111	26	𝑛	𝑛	PRON
cana-3308	111	27	≥	≥	NOUN
cana-3308	111	28	0	0	NUM
cana-3308	111	29	.	.	PUNCT
cana-3308	112	1	(	(	PUNCT
cana-3308	112	2	3.8	3.8	NUM
cana-3308	112	3	)	)	PUNCT
cana-3308	112	4	from	from	ADP
cana-3308	112	5	the	the	DET
cana-3308	112	6	above	above	NOUN
cana-3308	112	7	,	,	PUNCT
cana-3308	112	8	we	we	PRON
cana-3308	112	9	conclude	conclude	VERB
cana-3308	112	10	that	that	SCONJ
cana-3308	112	11	the	the	DET
cana-3308	112	12	sequences	sequence	NOUN
cana-3308	112	13	{	{	PUNCT
cana-3308	112	14	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	112	15	,	,	PUNCT
cana-3308	112	16	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	112	17	)	)	PUNCT
cana-3308	112	18	}	}	PUNCT
cana-3308	112	19	and	and	CCONJ
cana-3308	112	20	{	{	PUNCT
cana-3308	112	21	𝑑(𝑥𝑛	𝑑(𝑥𝑛	NOUN
cana-3308	112	22	,	,	PUNCT
cana-3308	112	23	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	112	24	)	)	PUNCT
cana-3308	112	25	}	}	PUNCT
cana-3308	112	26	are	be	AUX
cana-3308	112	27	monotonically	monotonically	ADV
cana-3308	112	28	decreasing	decrease	VERB
cana-3308	112	29	and	and	CCONJ
cana-3308	112	30	for	for	ADP
cana-3308	112	31	the	the	DET
cana-3308	112	32	non	non	ADJ
cana-3308	112	33	-	-	ADJ
cana-3308	112	34	negative	negative	ADJ
cana-3308	112	35	monotonically	monotonically	ADV
cana-3308	112	36	decreasing	decrease	VERB
cana-3308	112	37	sequences	sequence	NOUN
cana-3308	112	38	{	{	PUNCT
cana-3308	112	39	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	112	40	,	,	PUNCT
cana-3308	112	41	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	112	42	)	)	PUNCT
cana-3308	112	43	}	}	PUNCT
cana-3308	112	44	and	and	CCONJ
cana-3308	112	45	{	{	PUNCT
cana-3308	112	46	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	112	47	,	,	PUNCT
cana-3308	112	48	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	112	49	)	)	PUNCT
cana-3308	112	50	}	}	PUNCT
cana-3308	112	51	,	,	PUNCT
cana-3308	112	52	there	there	PRON
cana-3308	112	53	exist	exist	VERB
cana-3308	112	54	some	some	DET
cana-3308	112	55	𝑟1	𝑟1	NOUN
cana-3308	112	56	≥	≥	NOUN
cana-3308	112	57	0	0	NUM
cana-3308	112	58	and	and	CCONJ
cana-3308	112	59	𝑟2	𝑟2	NOUN
cana-3308	112	60	≥	≥	NUM
cana-3308	112	61	0	0	NUM
cana-3308	112	62	,	,	PUNCT
cana-3308	112	63	such	such	ADJ
cana-3308	112	64	that	that	SCONJ
cana-3308	112	65	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	112	66	,	,	PUNCT
cana-3308	112	67	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	112	68	)	)	PUNCT
cana-3308	112	69	→	→	SYM
cana-3308	112	70	𝑟1	𝑟1	NOUN
cana-3308	112	71	,	,	PUNCT
cana-3308	112	72	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	112	73	,	,	PUNCT
cana-3308	112	74	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	112	75	)	)	PUNCT
cana-3308	112	76	→	→	SYM
cana-3308	112	77	𝑟2	𝑟2	NOUN
cana-3308	112	78	as	as	ADP
cana-3308	112	79	𝑛	𝑛	PROPN
cana-3308	112	80	→	→	SYM
cana-3308	112	81	∞	∞	NUM
cana-3308	112	82	.	.	PUNCT
cana-3308	113	1	(	(	PUNCT
cana-3308	113	2	3.9	3.9	NUM
cana-3308	113	3	)	)	PUNCT
cana-3308	113	4	we	we	PRON
cana-3308	113	5	suppose	suppose	VERB
cana-3308	113	6	on	on	ADP
cana-3308	113	7	the	the	DET
cana-3308	113	8	contrary	contrary	NOUN
cana-3308	113	9	that	that	SCONJ
cana-3308	113	10	𝑟1	𝑟1	PROPN
cana-3308	113	11	>	>	X
cana-3308	113	12	0	0	X
cana-3308	113	13	.	.	PUNCT
cana-3308	114	1	hence	hence	ADV
cana-3308	114	2	,	,	PUNCT
cana-3308	114	3	we	we	PRON
cana-3308	114	4	have	have	VERB
cana-3308	114	5	0	0	NUM
cana-3308	114	6	<	<	X
cana-3308	114	7	𝑟1	𝑟1	PROPN
cana-3308	114	8	<	<	X
cana-3308	114	9	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	114	10	,	,	PUNCT
cana-3308	114	11	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	114	12	)	)	PUNCT
cana-3308	114	13	for	for	ADP
cana-3308	114	14	all	all	DET
cana-3308	114	15	𝑛	𝑛	DET
cana-3308	114	16	≥	≥	NOUN
cana-3308	114	17	0	0	NUM
cana-3308	114	18	.	.	PUNCT
cana-3308	115	1	set	set	VERB
cana-3308	115	2	휀	휀	PRON
cana-3308	115	3	=	=	NOUN
cana-3308	115	4	𝑟1	𝑟1	PROPN
cana-3308	115	5	.	.	PUNCT
cana-3308	116	1	from	from	ADP
cana-3308	116	2	equation	equation	NOUN
cana-3308	116	3	(	(	PUNCT
cana-3308	116	4	3.2	3.2	NUM
cana-3308	116	5	)	)	PUNCT
cana-3308	116	6	,	,	PUNCT
cana-3308	116	7	there	there	PRON
cana-3308	116	8	exist	exist	VERB
cana-3308	116	9	𝛿	𝛿	PROPN
cana-3308	116	10	>	>	X
cana-3308	116	11	0	0	NUM
cana-3308	116	12	such	such	ADJ
cana-3308	116	13	that	that	SCONJ
cana-3308	116	14	휀	휀	X
cana-3308	116	15	<	<	X
cana-3308	116	16	𝑡	𝑡	X
cana-3308	116	17	<	<	X
cana-3308	116	18	휀	휀	NOUN
cana-3308	116	19	+	+	X
cana-3308	116	20	𝛿	𝛿	ADJ
cana-3308	116	21	⇒	⇒	NOUN
cana-3308	116	22	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	116	23	)	)	PUNCT
cana-3308	116	24	≤	≤	NOUN
cana-3308	117	1	휀	휀	NOUN
cana-3308	117	2	.	.	PUNCT
cana-3308	118	1	on	on	ADP
cana-3308	118	2	the	the	DET
cana-3308	118	3	other	other	ADJ
cana-3308	118	4	hand	hand	NOUN
cana-3308	118	5	,	,	PUNCT
cana-3308	118	6	by	by	ADP
cana-3308	118	7	the	the	DET
cana-3308	118	8	definition	definition	NOUN
cana-3308	118	9	of	of	ADP
cana-3308	118	10	휀	휀	NOUN
cana-3308	118	11	,	,	PUNCT
cana-3308	118	12	we	we	PRON
cana-3308	118	13	can	can	AUX
cana-3308	118	14	choose	choose	VERB
cana-3308	118	15	𝑛0	𝑛0	VERB
cana-3308	118	16	∈	∈	PROPN
cana-3308	118	17	ℕ	ℕ	PROPN
cana-3308	118	18	such	such	ADJ
cana-3308	118	19	that	that	SCONJ
cana-3308	118	20	휀	휀	PRON
cana-3308	118	21	<	<	X
cana-3308	118	22	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	PROPN
cana-3308	118	23	,	,	PUNCT
cana-3308	118	24	𝑦𝑛0	𝑦𝑛0	X
cana-3308	118	25	+	+	NOUN
cana-3308	118	26	1	1	NUM
cana-3308	118	27	)	)	PUNCT
cana-3308	118	28	<	<	X
cana-3308	119	1	휀	휀	X
cana-3308	119	2	+	+	X
cana-3308	119	3	𝛿	𝛿	ADJ
cana-3308	119	4	by	by	ADP
cana-3308	119	5	the	the	DET
cana-3308	119	6	properties	property	NOUN
cana-3308	119	7	of	of	ADP
cana-3308	119	8	𝜓	𝜓	NOUN
cana-3308	119	9	,	,	PUNCT
cana-3308	119	10	𝜃,using	𝜃,use	VERB
cana-3308	119	11	equations	equation	NOUN
cana-3308	119	12	(	(	PUNCT
cana-3308	119	13	3.1	3.1	NUM
cana-3308	119	14	)	)	PUNCT
cana-3308	119	15	,	,	PUNCT
cana-3308	119	16	(	(	PUNCT
cana-3308	119	17	3.2	3.2	NUM
cana-3308	119	18	)	)	PUNCT
cana-3308	119	19	and	and	CCONJ
cana-3308	119	20	(	(	PUNCT
cana-3308	119	21	3.3	3.3	NUM
cana-3308	119	22	)	)	PUNCT
cana-3308	119	23	,	,	PUNCT
cana-3308	119	24	we	we	PRON
cana-3308	119	25	have	have	VERB
cana-3308	119	26	𝜓(휀	𝜓(휀	NUM
cana-3308	119	27	)	)	PUNCT
cana-3308	120	1	<	<	X
cana-3308	120	2	𝜓	𝜓	X
cana-3308	120	3	(	(	PUNCT
cana-3308	120	4	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	120	5	,	,	PUNCT
cana-3308	120	6	𝑦𝑛0	𝑦𝑛0	X
cana-3308	120	7	+	+	NOUN
cana-3308	120	8	1	1	NUM
cana-3308	120	9	)	)	PUNCT
cana-3308	120	10	)	)	PUNCT
cana-3308	120	11	<	<	X
cana-3308	120	12	𝜓(휀	𝜓(휀	PROPN
cana-3308	120	13	+	+	PUNCT
cana-3308	120	14	𝛿	𝛿	X
cana-3308	120	15	)	)	PUNCT
cana-3308	120	16	=	=	SYM
cana-3308	120	17	𝜓(휀	𝜓(휀	NUM
cana-3308	120	18	)	)	PUNCT
cana-3308	121	1	+	+	CCONJ
cana-3308	121	2	𝜓(𝛿	𝜓(𝛿	PROPN
cana-3308	121	3	)	)	PUNCT
cana-3308	121	4	this	this	PRON
cana-3308	121	5	implies	imply	VERB
cana-3308	121	6	that	that	SCONJ
cana-3308	121	7	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	121	8	(	(	PUNCT
cana-3308	121	9	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	121	10	,	,	PUNCT
cana-3308	121	11	𝑦𝑛0	𝑦𝑛0	X
cana-3308	121	12	+	+	NOUN
cana-3308	121	13	1	1	NUM
cana-3308	121	14	)	)	PUNCT
cana-3308	121	15	)	)	PUNCT
cana-3308	121	16	)	)	PUNCT
cana-3308	121	17	≤	≤	NUM
cana-3308	121	18	𝜓(휀	𝜓(휀	NUM
cana-3308	121	19	)	)	PUNCT
cana-3308	121	20	.	.	PUNCT
cana-3308	122	1	(	(	PUNCT
cana-3308	122	2	3.10	3.10	NUM
cana-3308	122	3	)	)	PUNCT
cana-3308	122	4	we	we	PRON
cana-3308	122	5	have	have	VERB
cana-3308	122	6	also	also	ADV
cana-3308	122	7	휀	휀	X
cana-3308	122	8	<	<	X
cana-3308	122	9	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	122	10	+	+	PROPN
cana-3308	122	11	2	2	NUM
cana-3308	122	12	,	,	PUNCT
cana-3308	122	13	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	122	14	+	+	NOUN
cana-3308	122	15	3	3	NUM
cana-3308	122	16	)	)	PUNCT
cana-3308	122	17	<	<	X
cana-3308	123	1	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	PROPN
cana-3308	123	2	+	+	PROPN
cana-3308	123	3	1	1	NUM
cana-3308	123	4	,	,	PUNCT
cana-3308	123	5	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	123	6	+	+	NOUN
cana-3308	123	7	2	2	NUM
cana-3308	123	8	)	)	PUNCT
cana-3308	123	9	=	=	SYM
cana-3308	123	10	𝑑(𝑇𝑥𝑛0	𝑑(𝑇𝑥𝑛0	PROPN
cana-3308	123	11	,	,	PUNCT
cana-3308	123	12	𝑇𝑦𝑛0	𝑇𝑦𝑛0	X
cana-3308	123	13	+	+	NOUN
cana-3308	123	14	1	1	NUM
cana-3308	123	15	)	)	PUNCT
cana-3308	123	16	,	,	PUNCT
cana-3308	123	17	which	which	PRON
cana-3308	123	18	implies	imply	VERB
cana-3308	123	19	that	that	SCONJ
cana-3308	123	20	𝜓(휀	𝜓(휀	NUM
cana-3308	123	21	)	)	PUNCT
cana-3308	123	22	<	<	X
cana-3308	123	23	𝜓(𝑑(𝑥𝑛0	𝜓(𝑑(𝑥𝑛0	PROPN
cana-3308	123	24	+	+	PROPN
cana-3308	123	25	2	2	NUM
cana-3308	123	26	,	,	PUNCT
cana-3308	123	27	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	123	28	+	+	NOUN
cana-3308	123	29	3	3	NUM
cana-3308	123	30	)	)	PUNCT
cana-3308	123	31	)	)	PUNCT
cana-3308	123	32	communications	communication	NOUN
cana-3308	123	33	on	on	ADP
cana-3308	123	34	applied	apply	VERB
cana-3308	123	35	nonlinear	nonlinear	ADJ
cana-3308	123	36	analysis	analysis	NOUN
cana-3308	123	37	issn	issn	NOUN
cana-3308	123	38	:	:	PUNCT
cana-3308	123	39	1074	1074	NUM
cana-3308	123	40	-	-	PUNCT
cana-3308	123	41	133x	133x	NUM
cana-3308	123	42	vol	vol	NOUN
cana-3308	123	43	32	32	NUM
cana-3308	123	44	no	no	NOUN
cana-3308	123	45	.	.	PUNCT
cana-3308	124	1	6s	6s	NUM
cana-3308	124	2	(	(	PUNCT
cana-3308	124	3	2025	2025	NUM
cana-3308	124	4	)	)	PUNCT
cana-3308	124	5	444	444	NUM
cana-3308	124	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	124	7	<	<	X
cana-3308	124	8	𝜓(𝑑(𝑥𝑛0	𝜓(𝑑(𝑥𝑛0	PROPN
cana-3308	124	9	+	+	PROPN
cana-3308	124	10	1	1	NUM
cana-3308	124	11	,	,	PUNCT
cana-3308	124	12	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	124	13	+	+	NOUN
cana-3308	124	14	2	2	NUM
cana-3308	124	15	)	)	PUNCT
cana-3308	124	16	)	)	PUNCT
cana-3308	125	1	=	=	SYM
cana-3308	125	2	𝜓(𝑑(𝑇𝑥𝑛0	𝜓(𝑑(𝑇𝑥𝑛0	PROPN
cana-3308	125	3	,	,	PUNCT
cana-3308	125	4	𝑇𝑦𝑛0	𝑇𝑦𝑛0	X
cana-3308	125	5	+	+	NOUN
cana-3308	125	6	1	1	NUM
cana-3308	125	7	)	)	PUNCT
cana-3308	125	8	)	)	PUNCT
cana-3308	125	9	≤	≤	ADV
cana-3308	126	1	𝛼(𝑥𝑛0	𝛼(𝑥𝑛0	NOUN
cana-3308	126	2	,	,	PUNCT
cana-3308	126	3	𝑦𝑛0	𝑦𝑛0	X
cana-3308	126	4	+	+	NOUN
cana-3308	126	5	1	1	NUM
cana-3308	126	6	)	)	PUNCT
cana-3308	126	7	𝜓(𝑑(𝑇𝑥𝑛0	𝜓(𝑑(𝑇𝑥𝑛0	PROPN
cana-3308	126	8	,	,	PUNCT
cana-3308	126	9	𝑇𝑦𝑛0	𝑇𝑦𝑛0	X
cana-3308	126	10	+	+	NOUN
cana-3308	126	11	1	1	NUM
cana-3308	126	12	)	)	PUNCT
cana-3308	126	13	)	)	PUNCT
cana-3308	127	1	≤	≤	NUM
cana-3308	127	2	𝜃	𝜃	X
cana-3308	127	3	(	(	PUNCT
cana-3308	127	4	𝜓	𝜓	NOUN
cana-3308	127	5	(	(	PUNCT
cana-3308	127	6	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	127	7	,	,	PUNCT
cana-3308	127	8	𝑦𝑛0	𝑦𝑛0	X
cana-3308	127	9	+	+	NOUN
cana-3308	127	10	1	1	NUM
cana-3308	127	11	)	)	PUNCT
cana-3308	127	12	)	)	PUNCT
cana-3308	127	13	)	)	PUNCT
cana-3308	127	14	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	127	15	(	(	PUNCT
cana-3308	127	16	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	127	17	,	,	PUNCT
cana-3308	127	18	𝑦𝑛0	𝑦𝑛0	X
cana-3308	127	19	+	+	NOUN
cana-3308	127	20	1	1	NUM
cana-3308	127	21	)	)	PUNCT
cana-3308	127	22	)	)	PUNCT
cana-3308	127	23	)	)	PUNCT
cana-3308	128	1	<	<	X
cana-3308	128	2	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	128	3	(	(	PUNCT
cana-3308	128	4	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	128	5	,	,	PUNCT
cana-3308	128	6	𝑦𝑛0	𝑦𝑛0	X
cana-3308	128	7	+	+	NOUN
cana-3308	128	8	1	1	NUM
cana-3308	128	9	)	)	PUNCT
cana-3308	128	10	)	)	PUNCT
cana-3308	128	11	)	)	PUNCT
cana-3308	128	12	≤	≤	NUM
cana-3308	128	13	𝜓	𝜓	NOUN
cana-3308	128	14	(	(	PUNCT
cana-3308	128	15	휀	휀	NOUN
cana-3308	128	16	)	)	PUNCT
cana-3308	128	17	,	,	PUNCT
cana-3308	128	18	which	which	PRON
cana-3308	128	19	is	be	AUX
cana-3308	128	20	a	a	DET
cana-3308	128	21	contradiction	contradiction	NOUN
cana-3308	128	22	.	.	PUNCT
cana-3308	129	1	hence	hence	ADV
cana-3308	129	2	lim	lim	PROPN
cana-3308	129	3	𝑛→∞	𝑛→∞	NUM
cana-3308	129	4	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	129	5	,	,	PUNCT
cana-3308	129	6	𝑦𝑛+1	𝑦𝑛+1	ADJ
cana-3308	129	7	)	)	PUNCT
cana-3308	129	8	=	=	SYM
cana-3308	129	9	𝑟1	𝑟1	NOUN
cana-3308	129	10	=	=	SYM
cana-3308	129	11	0	0	X
cana-3308	129	12	.	.	PUNCT
cana-3308	130	1	(	(	PUNCT
cana-3308	130	2	3.11	3.11	NUM
cana-3308	130	3	)	)	PUNCT
cana-3308	130	4	similarly	similarly	ADV
cana-3308	130	5	,	,	PUNCT
cana-3308	130	6	lim	lim	PROPN
cana-3308	130	7	𝑛→∞	𝑛→∞	NUM
cana-3308	130	8	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	130	9	,	,	PUNCT
cana-3308	130	10	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	130	11	)	)	PUNCT
cana-3308	131	1	=	=	NOUN
cana-3308	131	2	𝑟2	𝑟2	NOUN
cana-3308	131	3	=	=	SYM
cana-3308	131	4	0	0	NUM
cana-3308	131	5	.	.	PUNCT
cana-3308	131	6	(	(	PUNCT
cana-3308	131	7	3.12	3.12	NUM
cana-3308	131	8	)	)	PUNCT
cana-3308	131	9	now	now	ADV
cana-3308	131	10	,	,	PUNCT
cana-3308	131	11	we	we	PRON
cana-3308	131	12	shall	shall	AUX
cana-3308	131	13	prove	prove	VERB
cana-3308	131	14	that	that	SCONJ
cana-3308	131	15	{	{	PUNCT
cana-3308	131	16	(	(	PUNCT
cana-3308	131	17	𝑥𝑛	𝑥𝑛	INTJ
cana-3308	131	18	,	,	PUNCT
cana-3308	131	19	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	131	20	)	)	PUNCT
cana-3308	131	21	}	}	PUNCT
cana-3308	131	22	is	be	AUX
cana-3308	131	23	a	a	DET
cana-3308	131	24	cauchy	cauchy	ADJ
cana-3308	131	25	bisequence	bisequence	NOUN
cana-3308	131	26	.	.	PUNCT
cana-3308	132	1	we	we	PRON
cana-3308	132	2	fix	fix	VERB
cana-3308	132	3	휀1	휀1	NOUN
cana-3308	132	4	>	>	X
cana-3308	132	5	0	0	NUM
cana-3308	132	6	,	,	PUNCT
cana-3308	132	7	then	then	ADV
cana-3308	132	8	by	by	ADP
cana-3308	132	9	(	(	PUNCT
cana-3308	132	10	3.2	3.2	NUM
cana-3308	132	11	)	)	PUNCT
cana-3308	132	12	there	there	PRON
cana-3308	132	13	exists	exist	VERB
cana-3308	132	14	𝛿1	𝛿1	NOUN
cana-3308	132	15	>	>	X
cana-3308	132	16	0	0	NUM
cana-3308	133	1	such	such	ADJ
cana-3308	133	2	that	that	SCONJ
cana-3308	133	3	𝑡	𝑡	ADP
cana-3308	133	4	<	<	X
cana-3308	133	5	휀1	휀1	NOUN
cana-3308	133	6	+	+	CCONJ
cana-3308	133	7	𝛿1	𝛿1	ADJ
cana-3308	133	8	⇒	⇒	NOUN
cana-3308	133	9	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	133	10	)	)	PUNCT
cana-3308	133	11	≤	≤	NOUN
cana-3308	133	12	휀1	휀1	NOUN
cana-3308	133	13	.	.	PUNCT
cana-3308	134	1	(	(	PUNCT
cana-3308	134	2	3.13	3.13	NUM
cana-3308	134	3	)	)	PUNCT
cana-3308	134	4	without	without	ADP
cana-3308	134	5	loss	loss	NOUN
cana-3308	134	6	of	of	ADP
cana-3308	134	7	generality	generality	NOUN
cana-3308	134	8	,	,	PUNCT
cana-3308	134	9	we	we	PRON
cana-3308	134	10	assume	assume	VERB
cana-3308	134	11	𝛿1	𝛿1	PROPN
cana-3308	134	12	<	<	X
cana-3308	134	13	휀1	휀1	NOUN
cana-3308	134	14	.	.	PUNCT
cana-3308	135	1	due	due	ADP
cana-3308	135	2	to	to	ADP
cana-3308	135	3	(	(	PUNCT
cana-3308	135	4	3.11	3.11	NUM
cana-3308	135	5	)	)	PUNCT
cana-3308	135	6	,	,	PUNCT
cana-3308	135	7	there	there	PRON
cana-3308	135	8	exist	exist	VERB
cana-3308	135	9	𝑛0	𝑛0	VERB
cana-3308	135	10	∈	∈	PROPN
cana-3308	135	11	ℕ	ℕ	PROPN
cana-3308	135	12	such	such	ADJ
cana-3308	135	13	that	that	SCONJ
cana-3308	135	14	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	135	15	,	,	PUNCT
cana-3308	135	16	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	135	17	)	)	PUNCT
cana-3308	135	18	<	<	X
cana-3308	135	19	𝛿1	𝛿1	PROPN
cana-3308	135	20	,	,	PUNCT
cana-3308	135	21	for	for	ADP
cana-3308	135	22	all	all	DET
cana-3308	135	23	𝑛	𝑛	DET
cana-3308	135	24	≥	≥	NOUN
cana-3308	135	25	𝑛0	𝑛0	VERB
cana-3308	135	26	,	,	PUNCT
cana-3308	135	27	(	(	PUNCT
cana-3308	135	28	3.14	3.14	NUM
cana-3308	135	29	)	)	PUNCT
cana-3308	135	30	which	which	PRON
cana-3308	135	31	implies	imply	VERB
cana-3308	135	32	that	that	SCONJ
cana-3308	135	33	𝜓	𝜓	PROPN
cana-3308	135	34	(	(	PUNCT
cana-3308	135	35	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	135	36	,	,	PUNCT
cana-3308	135	37	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	135	38	)	)	PUNCT
cana-3308	135	39	)	)	PUNCT
cana-3308	135	40	<	<	X
cana-3308	135	41	𝜓(𝛿1	𝜓(𝛿1	NOUN
cana-3308	135	42	)	)	PUNCT
cana-3308	135	43	.	.	PUNCT
cana-3308	136	1	by	by	ADP
cana-3308	136	2	mathematical	mathematical	ADJ
cana-3308	136	3	induction	induction	NOUN
cana-3308	136	4	,	,	PUNCT
cana-3308	136	5	we	we	PRON
cana-3308	136	6	show	show	VERB
cana-3308	136	7	that	that	SCONJ
cana-3308	136	8	for	for	ADP
cana-3308	136	9	any	any	DET
cana-3308	136	10	fixed	fix	VERB
cana-3308	136	11	𝑘	𝑘	DET
cana-3308	136	12	≥	≥	NOUN
cana-3308	136	13	𝑛0	𝑛0	VERB
cana-3308	136	14	𝑑(𝑥𝑘	𝑑(𝑥𝑘	NOUN
cana-3308	136	15	,	,	PUNCT
cana-3308	136	16	𝑦𝑘+𝑙	𝑦𝑘+𝑙	PROPN
cana-3308	136	17	)	)	PUNCT
cana-3308	136	18	<	<	X
cana-3308	136	19	휀1	휀1	PROPN
cana-3308	136	20	+	+	CCONJ
cana-3308	136	21	𝛿1	𝛿1	NOUN
cana-3308	136	22	,	,	PUNCT
cana-3308	136	23	for	for	ADP
cana-3308	136	24	all	all	DET
cana-3308	136	25	𝑙	𝑙	DET
cana-3308	136	26	∈	∈	PROPN
cana-3308	136	27	ℕ.	ℕ.	PROPN
cana-3308	136	28	(	(	PUNCT
cana-3308	136	29	3.15	3.15	NUM
cana-3308	136	30	)	)	PUNCT
cana-3308	136	31	for	for	ADP
cana-3308	136	32	𝑙	𝑙	NOUN
cana-3308	136	33	=	=	SYM
cana-3308	136	34	1	1	NUM
cana-3308	136	35	,	,	PUNCT
cana-3308	136	36	this	this	DET
cana-3308	136	37	inequality	inequality	NOUN
cana-3308	136	38	trivially	trivially	ADV
cana-3308	136	39	holds	hold	VERB
cana-3308	136	40	by	by	ADP
cana-3308	136	41	(	(	PUNCT
cana-3308	136	42	3.14	3.14	NUM
cana-3308	136	43	)	)	PUNCT
cana-3308	136	44	.	.	PUNCT
cana-3308	137	1	now	now	ADV
cana-3308	137	2	,	,	PUNCT
cana-3308	137	3	assume	assume	VERB
cana-3308	137	4	that	that	SCONJ
cana-3308	137	5	(	(	PUNCT
cana-3308	137	6	3.15	3.15	NUM
cana-3308	137	7	)	)	PUNCT
cana-3308	137	8	is	be	AUX
cana-3308	137	9	satisfied	satisfied	ADJ
cana-3308	137	10	for	for	ADP
cana-3308	137	11	some	some	DET
cana-3308	137	12	𝑗	𝑗	PRON
cana-3308	137	13	∈	∈	PROPN
cana-3308	137	14	ℕ	ℕ	PROPN
cana-3308	137	15	and	and	CCONJ
cana-3308	137	16	we	we	PRON
cana-3308	137	17	have	have	VERB
cana-3308	137	18	to	to	PART
cana-3308	137	19	show	show	VERB
cana-3308	137	20	that	that	SCONJ
cana-3308	137	21	it	it	PRON
cana-3308	137	22	holds	hold	VERB
cana-3308	137	23	for	for	ADP
cana-3308	137	24	𝑙	𝑙	PRON
cana-3308	137	25	=	=	SYM
cana-3308	137	26	𝑗	𝑗	PROPN
cana-3308	138	1	+	+	NOUN
cana-3308	138	2	1	1	NUM
cana-3308	138	3	.	.	PUNCT
cana-3308	138	4	from	from	ADP
cana-3308	138	5	the	the	DET
cana-3308	138	6	triangle	triangle	NOUN
cana-3308	138	7	inequality	inequality	NOUN
cana-3308	138	8	(	(	PUNCT
cana-3308	138	9	bp3	bp3	NOUN
cana-3308	138	10	)	)	PUNCT
cana-3308	138	11	,	,	PUNCT
cana-3308	138	12	properties	property	NOUN
cana-3308	138	13	of	of	ADP
cana-3308	138	14	𝜓	𝜓	PROPN
cana-3308	138	15	,	,	PUNCT
cana-3308	138	16	𝜃	𝜃	PROPN
cana-3308	138	17	,	,	PUNCT
cana-3308	138	18	equations	equation	NOUN
cana-3308	138	19	(	(	PUNCT
cana-3308	138	20	3.1	3.1	NUM
cana-3308	138	21	)	)	PUNCT
cana-3308	138	22	,	,	PUNCT
cana-3308	138	23	(	(	PUNCT
cana-3308	138	24	3.2	3.2	NUM
cana-3308	138	25	)	)	PUNCT
cana-3308	138	26	and	and	CCONJ
cana-3308	138	27	(	(	PUNCT
cana-3308	138	28	3.3	3.3	NUM
cana-3308	138	29	)	)	PUNCT
cana-3308	138	30	𝜓(𝑑(𝑥𝑘	𝜓(𝑑(𝑥𝑘	NOUN
cana-3308	138	31	,	,	PUNCT
cana-3308	138	32	𝑦𝑘+𝑗+1	𝑦𝑘+𝑗+1	NUM
cana-3308	138	33	)	)	PUNCT
cana-3308	138	34	)	)	PUNCT
cana-3308	138	35	≤	≤	NUM
cana-3308	138	36	𝜓	𝜓	NOUN
cana-3308	138	37	(	(	PUNCT
cana-3308	138	38	𝑑(𝑥𝑘	𝑑(𝑥𝑘	NUM
cana-3308	138	39	,	,	PUNCT
cana-3308	138	40	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	138	41	)	)	PUNCT
cana-3308	139	1	+	+	CCONJ
cana-3308	139	2	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	NUM
cana-3308	139	3	,	,	PUNCT
cana-3308	139	4	𝑦𝑘+1	𝑦𝑘+1	X
cana-3308	139	5	)	)	PUNCT
cana-3308	140	1	+	+	CCONJ
cana-3308	140	2	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	NUM
cana-3308	140	3	,	,	PUNCT
cana-3308	140	4	𝑦𝑘+𝑗+1	𝑦𝑘+𝑗+1	NUM
cana-3308	140	5	)	)	PUNCT
cana-3308	140	6	)	)	PUNCT
cana-3308	140	7	≤	≤	NUM
cana-3308	140	8	𝜓	𝜓	NOUN
cana-3308	140	9	(	(	PUNCT
cana-3308	140	10	𝑑(𝑥𝑘	𝑑(𝑥𝑘	NUM
cana-3308	140	11	,	,	PUNCT
cana-3308	140	12	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	140	13	)	)	PUNCT
cana-3308	140	14	)	)	PUNCT
cana-3308	141	1	+	+	CCONJ
cana-3308	141	2	𝜓	𝜓	X
cana-3308	141	3	(	(	PUNCT
cana-3308	141	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	141	5	,	,	PUNCT
cana-3308	141	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	141	7	)	)	PUNCT
cana-3308	141	8	)	)	PUNCT
cana-3308	142	1	+	+	CCONJ
cana-3308	142	2	𝜓	𝜓	X
cana-3308	142	3	(	(	PUNCT
cana-3308	142	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	NUM
cana-3308	142	5	,	,	PUNCT
cana-3308	142	6	𝑦𝑘+𝑗+1	𝑦𝑘+𝑗+1	NUM
cana-3308	142	7	)	)	PUNCT
cana-3308	142	8	)	)	PUNCT
cana-3308	142	9	≤	≤	NUM
cana-3308	143	1	𝜓	𝜓	NOUN
cana-3308	143	2	(	(	PUNCT
cana-3308	143	3	𝑑(𝑥𝑘	𝑑(𝑥𝑘	NUM
cana-3308	143	4	,	,	PUNCT
cana-3308	143	5	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	143	6	)	)	PUNCT
cana-3308	143	7	)	)	PUNCT
cana-3308	144	1	+	+	CCONJ
cana-3308	144	2	𝜓	𝜓	X
cana-3308	144	3	(	(	PUNCT
cana-3308	144	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	144	5	,	,	PUNCT
cana-3308	144	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	144	7	)	)	PUNCT
cana-3308	144	8	)	)	PUNCT
cana-3308	145	1	+	+	CCONJ
cana-3308	145	2	𝜓	𝜓	X
cana-3308	145	3	(	(	PUNCT
cana-3308	145	4	𝑑(𝑇𝑥𝑘	𝑑(𝑇𝑥𝑘	PROPN
cana-3308	145	5	,	,	PUNCT
cana-3308	145	6	𝑇𝑦𝑘+𝑗	𝑇𝑦𝑘+𝑗	PROPN
cana-3308	145	7	)	)	PUNCT
cana-3308	145	8	)	)	PUNCT
cana-3308	145	9	≤	≤	NUM
cana-3308	146	1	𝜓	𝜓	NOUN
cana-3308	146	2	(	(	PUNCT
cana-3308	146	3	𝑑(𝑥𝑘	𝑑(𝑥𝑘	NUM
cana-3308	146	4	,	,	PUNCT
cana-3308	146	5	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	146	6	)	)	PUNCT
cana-3308	146	7	)	)	PUNCT
cana-3308	147	1	+	+	CCONJ
cana-3308	147	2	𝜓	𝜓	X
cana-3308	147	3	(	(	PUNCT
cana-3308	147	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	147	5	,	,	PUNCT
cana-3308	147	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	147	7	)	)	PUNCT
cana-3308	147	8	)	)	PUNCT
cana-3308	148	1	+	+	ADV
cana-3308	148	2	𝛼(𝑥𝑘	𝛼(𝑥𝑘	ADJ
cana-3308	148	3	,	,	PUNCT
cana-3308	148	4	𝑦𝑘+𝑗	𝑦𝑘+𝑗	NUM
cana-3308	148	5	)	)	PUNCT
cana-3308	148	6	𝜓	𝜓	PROPN
cana-3308	148	7	(	(	PUNCT
cana-3308	148	8	𝑑(𝑇𝑥𝑘	𝑑(𝑇𝑥𝑘	NUM
cana-3308	148	9	,	,	PUNCT
cana-3308	148	10	𝑇𝑦𝑘+𝑗	𝑇𝑦𝑘+𝑗	PROPN
cana-3308	148	11	)	)	PUNCT
cana-3308	148	12	)	)	PUNCT
cana-3308	149	1	≤	≤	NUM
cana-3308	149	2	𝜓	𝜓	NOUN
cana-3308	149	3	(	(	PUNCT
cana-3308	149	4	𝑑(𝑥𝑘	𝑑(𝑥𝑘	NUM
cana-3308	149	5	,	,	PUNCT
cana-3308	149	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	149	7	)	)	PUNCT
cana-3308	149	8	)	)	PUNCT
cana-3308	150	1	+	+	CCONJ
cana-3308	150	2	𝜓	𝜓	X
cana-3308	150	3	(	(	PUNCT
cana-3308	150	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	150	5	,	,	PUNCT
cana-3308	150	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	150	7	)	)	PUNCT
cana-3308	150	8	)	)	PUNCT
cana-3308	151	1	+	+	ADV
cana-3308	151	2	𝜃(𝜓(𝑑(𝑥𝑘	𝜃(𝜓(𝑑(𝑥𝑘	ADJ
cana-3308	151	3	,	,	PUNCT
cana-3308	151	4	𝑦𝑘+𝑗	𝑦𝑘+𝑗	NUM
cana-3308	151	5	)	)	PUNCT
cana-3308	151	6	)	)	PUNCT
cana-3308	151	7	)	)	PUNCT
cana-3308	151	8	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	151	9	(	(	PUNCT
cana-3308	151	10	𝑑(𝑥𝑘	𝑑(𝑥𝑘	ADJ
cana-3308	151	11	,	,	PUNCT
cana-3308	151	12	𝑦𝑘+𝑗	𝑦𝑘+𝑗	NUM
cana-3308	151	13	)	)	PUNCT
cana-3308	151	14	)	)	PUNCT
cana-3308	151	15	)	)	PUNCT
cana-3308	151	16	<	<	X
cana-3308	151	17	𝜓	𝜓	X
cana-3308	151	18	(	(	PUNCT
cana-3308	151	19	𝑑(𝑥𝑘	𝑑(𝑥𝑘	NUM
cana-3308	151	20	,	,	PUNCT
cana-3308	151	21	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	151	22	)	)	PUNCT
cana-3308	151	23	)	)	PUNCT
cana-3308	152	1	+	+	CCONJ
cana-3308	152	2	𝜓	𝜓	X
cana-3308	152	3	(	(	PUNCT
cana-3308	152	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	152	5	,	,	PUNCT
cana-3308	152	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	152	7	)	)	PUNCT
cana-3308	152	8	)	)	PUNCT
cana-3308	153	1	+	+	CCONJ
cana-3308	153	2	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	153	3	(	(	PUNCT
cana-3308	153	4	𝑑(𝑥𝑘	𝑑(𝑥𝑘	ADJ
cana-3308	153	5	,	,	PUNCT
cana-3308	153	6	𝑦𝑘+𝑗	𝑦𝑘+𝑗	NUM
cana-3308	153	7	)	)	PUNCT
cana-3308	153	8	)	)	PUNCT
cana-3308	153	9	)	)	PUNCT
cana-3308	153	10	.	.	PUNCT
cana-3308	154	1	using	use	VERB
cana-3308	154	2	equations	equation	NOUN
cana-3308	154	3	(	(	PUNCT
cana-3308	154	4	3.14	3.14	NUM
cana-3308	154	5	)	)	PUNCT
cana-3308	154	6	and	and	CCONJ
cana-3308	154	7	(	(	PUNCT
cana-3308	154	8	3.15	3.15	NUM
cana-3308	154	9	)	)	PUNCT
cana-3308	154	10	,	,	PUNCT
cana-3308	154	11	we	we	PRON
cana-3308	154	12	get	get	VERB
cana-3308	154	13	communications	communication	NOUN
cana-3308	154	14	on	on	ADP
cana-3308	154	15	applied	apply	VERB
cana-3308	154	16	nonlinear	nonlinear	ADJ
cana-3308	154	17	analysis	analysis	NOUN
cana-3308	154	18	issn	issn	NOUN
cana-3308	154	19	:	:	PUNCT
cana-3308	154	20	1074	1074	NUM
cana-3308	154	21	-	-	PUNCT
cana-3308	154	22	133x	133x	NUM
cana-3308	154	23	vol	vol	NOUN
cana-3308	154	24	32	32	NUM
cana-3308	154	25	no	no	NOUN
cana-3308	154	26	.	.	PUNCT
cana-3308	155	1	6s	6s	NUM
cana-3308	155	2	(	(	PUNCT
cana-3308	155	3	2025	2025	NUM
cana-3308	155	4	)	)	PUNCT
cana-3308	155	5	445	445	NUM
cana-3308	156	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	156	2	𝜓	𝜓	PROPN
cana-3308	156	3	(	(	PUNCT
cana-3308	156	4	𝑑(𝑥𝑘	𝑑(𝑥𝑘	NUM
cana-3308	156	5	,	,	PUNCT
cana-3308	156	6	𝑦𝑘+𝑗+1	𝑦𝑘+𝑗+1	NUM
cana-3308	156	7	)	)	PUNCT
cana-3308	156	8	)	)	PUNCT
cana-3308	156	9	≤	≤	NUM
cana-3308	157	1	𝜓	𝜓	PROPN
cana-3308	157	2	(	(	PUNCT
cana-3308	157	3	𝛿1	𝛿1	NOUN
cana-3308	157	4	)	)	PUNCT
cana-3308	157	5	+	+	X
cana-3308	157	6	𝜓	𝜓	X
cana-3308	157	7	(	(	PUNCT
cana-3308	157	8	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	157	9	,	,	PUNCT
cana-3308	157	10	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	157	11	)	)	PUNCT
cana-3308	157	12	)	)	PUNCT
cana-3308	158	1	+	+	CCONJ
cana-3308	158	2	𝜓	𝜓	PROPN
cana-3308	158	3	(	(	PUNCT
cana-3308	158	4	휀1	휀1	NOUN
cana-3308	158	5	)	)	PUNCT
cana-3308	158	6	.	.	PUNCT
cana-3308	159	1	by	by	ADP
cana-3308	159	2	letting	let	VERB
cana-3308	159	3	𝑘	𝑘	X
cana-3308	159	4	→	→	SYM
cana-3308	159	5	∞	∞	PROPN
cana-3308	159	6	,	,	PUNCT
cana-3308	159	7	using	use	VERB
cana-3308	159	8	equation	equation	NOUN
cana-3308	159	9	(	(	PUNCT
cana-3308	159	10	3.12	3.12	NUM
cana-3308	159	11	)	)	PUNCT
cana-3308	159	12	,	,	PUNCT
cana-3308	159	13	we	we	PRON
cana-3308	159	14	obtain	obtain	VERB
cana-3308	159	15	𝜓	𝜓	ADP
cana-3308	159	16	(	(	PUNCT
cana-3308	159	17	𝑑(𝑥𝑘	𝑑(𝑥𝑘	ADJ
cana-3308	159	18	,	,	PUNCT
cana-3308	159	19	𝑦𝑘+𝑗+1	𝑦𝑘+𝑗+1	NUM
cana-3308	159	20	)	)	PUNCT
cana-3308	159	21	)	)	PUNCT
cana-3308	159	22	≤	≤	NUM
cana-3308	160	1	𝜓	𝜓	PROPN
cana-3308	160	2	(	(	PUNCT
cana-3308	160	3	𝛿1	𝛿1	NOUN
cana-3308	160	4	)	)	PUNCT
cana-3308	160	5	+	+	PUNCT
cana-3308	160	6	𝜓(휀1	𝜓(휀1	NOUN
cana-3308	160	7	)	)	PUNCT
cana-3308	160	8	=	=	SYM
cana-3308	160	9	𝜓(휀1	𝜓(휀1	X
cana-3308	160	10	+	+	SYM
cana-3308	160	11	𝛿1	𝛿1	NOUN
cana-3308	160	12	)	)	PUNCT
cana-3308	160	13	.	.	PUNCT
cana-3308	161	1	by	by	ADP
cana-3308	161	2	the	the	DET
cana-3308	161	3	property	property	NOUN
cana-3308	161	4	of	of	ADP
cana-3308	161	5	𝜓	𝜓	NOUN
cana-3308	161	6	,	,	PUNCT
cana-3308	161	7	we	we	PRON
cana-3308	161	8	get	get	VERB
cana-3308	161	9	𝑑(𝑥𝑘	𝑑(𝑥𝑘	ADJ
cana-3308	161	10	,	,	PUNCT
cana-3308	161	11	𝑦𝑘+𝑗+1	𝑦𝑘+𝑗+1	NUM
cana-3308	161	12	)	)	PUNCT
cana-3308	161	13	<	<	X
cana-3308	161	14	휀1	휀1	PROPN
cana-3308	161	15	+	+	CCONJ
cana-3308	161	16	𝛿1	𝛿1	NOUN
cana-3308	161	17	.	.	PUNCT
cana-3308	162	1	so	so	ADV
cana-3308	162	2	,	,	PUNCT
cana-3308	162	3	equation	equation	NOUN
cana-3308	162	4	(	(	PUNCT
cana-3308	162	5	3.15	3.15	NUM
cana-3308	162	6	)	)	PUNCT
cana-3308	162	7	is	be	AUX
cana-3308	162	8	holds	hold	NOUN
cana-3308	162	9	for	for	ADP
cana-3308	162	10	𝑙	𝑙	PRON
cana-3308	162	11	=	=	SYM
cana-3308	162	12	𝑗	𝑗	PROPN
cana-3308	163	1	+	+	NOUN
cana-3308	163	2	1	1	NUM
cana-3308	163	3	.	.	PUNCT
cana-3308	163	4	hence	hence	ADV
cana-3308	163	5	,	,	PUNCT
cana-3308	163	6	by	by	ADP
cana-3308	163	7	induction	induction	NOUN
cana-3308	163	8	we	we	PRON
cana-3308	163	9	prove	prove	VERB
cana-3308	163	10	that	that	SCONJ
cana-3308	163	11	𝑑(𝑥𝑘	𝑑(𝑥𝑘	ADJ
cana-3308	163	12	,	,	PUNCT
cana-3308	163	13	𝑦𝑘+𝑗+1	𝑦𝑘+𝑗+1	NUM
cana-3308	163	14	)	)	PUNCT
cana-3308	163	15	<	<	X
cana-3308	163	16	휀1	휀1	NOUN
cana-3308	163	17	+	+	CCONJ
cana-3308	163	18	𝛿1	𝛿1	NOUN
cana-3308	163	19	for	for	ADP
cana-3308	163	20	all	all	DET
cana-3308	163	21	𝑘	𝑘	DET
cana-3308	163	22	≥	≥	NOUN
cana-3308	163	23	𝑛0	𝑛0	VERB
cana-3308	163	24	and	and	CCONJ
cana-3308	163	25	𝑙	𝑙	DET
cana-3308	163	26	≥	≥	NUM
cana-3308	163	27	1	1	NUM
cana-3308	163	28	.	.	PUNCT
cana-3308	164	1	since	since	SCONJ
cana-3308	164	2	휀1	휀1	NOUN
cana-3308	164	3	is	be	AUX
cana-3308	164	4	arbitrary	arbitrary	ADJ
cana-3308	164	5	,	,	PUNCT
cana-3308	164	6	we	we	PRON
cana-3308	164	7	conclude	conclude	VERB
cana-3308	164	8	that	that	SCONJ
cana-3308	164	9	lim	lim	PROPN
cana-3308	164	10	𝑚,𝑛→∞	𝑚,𝑛→∞	VERB
cana-3308	164	11	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	164	12	,	,	PUNCT
cana-3308	164	13	𝑦𝑚	𝑦𝑚	NOUN
cana-3308	164	14	)	)	PUNCT
cana-3308	164	15	=	=	SYM
cana-3308	165	1	0	0	X
cana-3308	165	2	.	.	PUNCT
cana-3308	166	1	hence	hence	ADV
cana-3308	166	2	,	,	PUNCT
cana-3308	166	3	{	{	PUNCT
cana-3308	166	4	(	(	PUNCT
cana-3308	166	5	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	166	6	,	,	PUNCT
cana-3308	166	7	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	166	8	)	)	PUNCT
cana-3308	166	9	}	}	PUNCT
cana-3308	166	10	is	be	AUX
cana-3308	166	11	a	a	DET
cana-3308	166	12	cauchy	cauchy	ADJ
cana-3308	166	13	bisequence	bisequence	NOUN
cana-3308	166	14	and	and	CCONJ
cana-3308	166	15	(	(	PUNCT
cana-3308	166	16	𝑋	𝑋	PROPN
cana-3308	166	17	,	,	PUNCT
cana-3308	166	18	𝑌	𝑌	PROPN
cana-3308	166	19	,	,	PUNCT
cana-3308	166	20	𝑑	𝑑	NOUN
cana-3308	166	21	)	)	PUNCT
cana-3308	166	22	is	be	AUX
cana-3308	166	23	a	a	DET
cana-3308	166	24	complete	complete	ADJ
cana-3308	166	25	bipolar	bipolar	ADJ
cana-3308	166	26	metric	metric	ADJ
cana-3308	166	27	space	space	NOUN
cana-3308	166	28	.	.	PUNCT
cana-3308	167	1	so	so	ADV
cana-3308	167	2	,	,	PUNCT
cana-3308	167	3	{	{	PUNCT
cana-3308	167	4	(	(	PUNCT
cana-3308	167	5	𝑥𝑛	𝑥𝑛	INTJ
cana-3308	167	6	,	,	PUNCT
cana-3308	167	7	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	167	8	)	)	PUNCT
cana-3308	167	9	}	}	PUNCT
cana-3308	167	10	is	be	AUX
cana-3308	167	11	convergent	convergent	ADJ
cana-3308	167	12	and	and	CCONJ
cana-3308	167	13	in	in	ADP
cana-3308	167	14	fact	fact	NOUN
cana-3308	167	15	biconvergent	biconvergent	NOUN
cana-3308	167	16	.	.	PUNCT
cana-3308	168	1	so	so	ADV
cana-3308	168	2	,	,	PUNCT
cana-3308	168	3	there	there	PRON
cana-3308	168	4	exists	exist	VERB
cana-3308	168	5	𝑢	𝑢	PRON
cana-3308	168	6	∈	∈	PROPN
cana-3308	168	7	𝑋	𝑋	NOUN
cana-3308	168	8	∩	∩	NOUN
cana-3308	168	9	𝑌	𝑌	PROPN
cana-3308	168	10	such	such	ADJ
cana-3308	168	11	that	that	SCONJ
cana-3308	168	12	(	(	PUNCT
cana-3308	168	13	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	168	14	)	)	PUNCT
cana-3308	168	15	→	→	SYM
cana-3308	168	16	𝑢	𝑢	X
cana-3308	168	17	,	,	PUNCT
cana-3308	168	18	(	(	PUNCT
cana-3308	168	19	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	168	20	)	)	PUNCT
cana-3308	168	21	→	→	SYM
cana-3308	168	22	𝑢	𝑢	NOUN
cana-3308	168	23	as	as	ADP
cana-3308	168	24	𝑛	𝑛	PROPN
cana-3308	168	25	→	→	SYM
cana-3308	168	26	∞.	∞.	PROPN
cana-3308	168	27	we	we	PRON
cana-3308	168	28	claim	claim	VERB
cana-3308	168	29	that	that	SCONJ
cana-3308	168	30	𝑇𝑢	𝑇𝑢	PROPN
cana-3308	168	31	=	=	PUNCT
cana-3308	168	32	𝑢.	𝑢.	NOUN
cana-3308	168	33	let	let	VERB
cana-3308	168	34	,	,	PUNCT
cana-3308	168	35	if	if	SCONJ
cana-3308	168	36	possible	possible	ADJ
cana-3308	168	37	,	,	PUNCT
cana-3308	168	38	𝑇𝑢	𝑇𝑢	PROPN
cana-3308	168	39	≠	≠	PROPN
cana-3308	168	40	𝑢.	𝑢.	NOUN
cana-3308	168	41	then	then	ADV
cana-3308	168	42	there	there	PRON
cana-3308	168	43	exist	exist	VERB
cana-3308	168	44	𝑟′	𝑟′	PROPN
cana-3308	168	45	>	>	X
cana-3308	168	46	0	0	NUM
cana-3308	169	1	such	such	ADJ
cana-3308	169	2	that	that	SCONJ
cana-3308	169	3	𝑑(𝑢	𝑑(𝑢	NOUN
cana-3308	169	4	,	,	PUNCT
cana-3308	169	5	𝑇𝑢	𝑇𝑢	NOUN
cana-3308	169	6	)	)	PUNCT
cana-3308	169	7	=	=	SYM
cana-3308	169	8	𝑟′	𝑟′	PROPN
cana-3308	169	9	>	>	X
cana-3308	169	10	0	0	X
cana-3308	169	11	.	.	PUNCT
cana-3308	170	1	since	since	SCONJ
cana-3308	170	2	(	(	PUNCT
cana-3308	170	3	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	170	4	)	)	PUNCT
cana-3308	170	5	→	→	SYM
cana-3308	170	6	𝑢	𝑢	X
cana-3308	170	7	,	,	PUNCT
cana-3308	170	8	(	(	PUNCT
cana-3308	170	9	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	170	10	)	)	PUNCT
cana-3308	170	11	→	→	SYM
cana-3308	170	12	𝑢	𝑢	NOUN
cana-3308	170	13	as	as	ADP
cana-3308	170	14	𝑛	𝑛	PROPN
cana-3308	170	15	→	→	SYM
cana-3308	170	16	∞	∞	PROPN
cana-3308	170	17	,	,	PUNCT
cana-3308	170	18	we	we	PRON
cana-3308	170	19	can	can	AUX
cana-3308	170	20	choose	choose	VERB
cana-3308	170	21	𝑛0	𝑛0	VERB
cana-3308	170	22	∈	∈	PROPN
cana-3308	170	23	ℕ	ℕ	PROPN
cana-3308	170	24	such	such	ADJ
cana-3308	170	25	that	that	SCONJ
cana-3308	170	26	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	170	27	,	,	PUNCT
cana-3308	170	28	𝑢	𝑢	PROPN
cana-3308	170	29	)	)	PUNCT
cana-3308	170	30	<	<	X
cana-3308	170	31	𝑟′	𝑟′	PROPN
cana-3308	170	32	2	2	NUM
cana-3308	170	33	,	,	PUNCT
cana-3308	170	34	for	for	SCONJ
cana-3308	170	35	all	all	DET
cana-3308	170	36	𝑛	𝑛	DET
cana-3308	170	37	≥	≥	NOUN
cana-3308	170	38	𝑛0	𝑛0	VERB
cana-3308	170	39	and	and	CCONJ
cana-3308	170	40	𝑑(𝑦𝑛	𝑑(𝑦𝑛	NUM
cana-3308	170	41	,	,	PUNCT
cana-3308	170	42	𝑢	𝑢	X
cana-3308	170	43	)	)	PUNCT
cana-3308	170	44	<	<	X
cana-3308	170	45	𝑟′	𝑟′	PROPN
cana-3308	170	46	2	2	NUM
cana-3308	170	47	,	,	PUNCT
cana-3308	170	48	for	for	SCONJ
cana-3308	170	49	all	all	DET
cana-3308	170	50	𝑛	𝑛	DET
cana-3308	170	51	≥	≥	NOUN
cana-3308	170	52	𝑛0	𝑛0	VERB
cana-3308	170	53	.	.	PUNCT
cana-3308	171	1	(	(	PUNCT
cana-3308	171	2	3.16	3.16	NUM
cana-3308	171	3	)	)	PUNCT
cana-3308	171	4	from	from	ADP
cana-3308	171	5	the	the	DET
cana-3308	171	6	triangle	triangle	NOUN
cana-3308	171	7	inequality	inequality	NOUN
cana-3308	171	8	(	(	PUNCT
cana-3308	171	9	bp3	bp3	NOUN
cana-3308	171	10	)	)	PUNCT
cana-3308	171	11	,	,	PUNCT
cana-3308	171	12	properties	property	NOUN
cana-3308	171	13	of	of	ADP
cana-3308	171	14	𝜓	𝜓	PROPN
cana-3308	171	15	,	,	PUNCT
cana-3308	171	16	𝜃	𝜃	PROPN
cana-3308	171	17	,	,	PUNCT
cana-3308	171	18	equations	equation	NOUN
cana-3308	171	19	(	(	PUNCT
cana-3308	171	20	3.1	3.1	NUM
cana-3308	171	21	)	)	PUNCT
cana-3308	171	22	,	,	PUNCT
cana-3308	171	23	(	(	PUNCT
cana-3308	171	24	3.2	3.2	NUM
cana-3308	171	25	)	)	PUNCT
cana-3308	171	26	and	and	CCONJ
cana-3308	171	27	(	(	PUNCT
cana-3308	171	28	3.3	3.3	NUM
cana-3308	171	29	)	)	PUNCT
cana-3308	171	30	,	,	PUNCT
cana-3308	171	31	𝜓(𝑟	𝜓(𝑟	NOUN
cana-3308	171	32	′	′	NUM
cana-3308	171	33	)	)	PUNCT
cana-3308	171	34	=	=	PUNCT
cana-3308	172	1	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	NOUN
cana-3308	172	2	,	,	PUNCT
cana-3308	172	3	𝑇𝑢	𝑇𝑢	NOUN
cana-3308	172	4	)	)	PUNCT
cana-3308	172	5	)	)	PUNCT
cana-3308	172	6	≤	≤	NOUN
cana-3308	173	1	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	NUM
cana-3308	173	2	,	,	PUNCT
cana-3308	173	3	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	173	4	)	)	PUNCT
cana-3308	174	1	+	+	NUM
cana-3308	174	2	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	174	3	,	,	PUNCT
cana-3308	174	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	174	5	)	)	PUNCT
cana-3308	174	6	+	+	NUM
cana-3308	174	7	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	174	8	,	,	PUNCT
cana-3308	174	9	𝑇𝑢	𝑇𝑢	NOUN
cana-3308	174	10	)	)	PUNCT
cana-3308	174	11	)	)	PUNCT
cana-3308	175	1	≤	≤	NOUN
cana-3308	175	2	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	NUM
cana-3308	175	3	,	,	PUNCT
cana-3308	175	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	175	5	)	)	PUNCT
cana-3308	175	6	)	)	PUNCT
cana-3308	176	1	+	+	CCONJ
cana-3308	176	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	176	3	,	,	PUNCT
cana-3308	176	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	176	5	)	)	PUNCT
cana-3308	176	6	)	)	PUNCT
cana-3308	177	1	+	+	CCONJ
cana-3308	178	1	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	178	2	,	,	PUNCT
cana-3308	178	3	𝑇𝑢	𝑇𝑢	PROPN
cana-3308	178	4	)	)	PUNCT
cana-3308	178	5	)	)	PUNCT
cana-3308	178	6	≤	≤	NOUN
cana-3308	179	1	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	NUM
cana-3308	179	2	,	,	PUNCT
cana-3308	179	3	𝑦𝑛+1	𝑦𝑛+1	NOUN
cana-3308	179	4	)	)	PUNCT
cana-3308	179	5	)	)	PUNCT
cana-3308	180	1	+	+	CCONJ
cana-3308	180	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	180	3	,	,	PUNCT
cana-3308	180	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	180	5	)	)	PUNCT
cana-3308	180	6	)	)	PUNCT
cana-3308	181	1	+	+	CCONJ
cana-3308	181	2	𝜓(𝑑(𝑇𝑥𝑛	𝜓(𝑑(𝑇𝑥𝑛	NOUN
cana-3308	181	3	,	,	PUNCT
cana-3308	181	4	𝑇𝑢	𝑇𝑢	NOUN
cana-3308	181	5	)	)	PUNCT
cana-3308	181	6	)	)	PUNCT
cana-3308	182	1	≤	≤	NOUN
cana-3308	182	2	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	NUM
cana-3308	182	3	,	,	PUNCT
cana-3308	182	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	182	5	)	)	PUNCT
cana-3308	182	6	)	)	PUNCT
cana-3308	183	1	+	+	CCONJ
cana-3308	183	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	183	3	,	,	PUNCT
cana-3308	183	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	183	5	)	)	PUNCT
cana-3308	183	6	)	)	PUNCT
cana-3308	184	1	+	+	CCONJ
cana-3308	184	2	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-3308	184	3	,	,	PUNCT
cana-3308	184	4	𝑢	𝑢	NOUN
cana-3308	184	5	)	)	PUNCT
cana-3308	184	6	𝜓(𝑑(𝑇𝑥𝑛	𝜓(𝑑(𝑇𝑥𝑛	NOUN
cana-3308	184	7	,	,	PUNCT
cana-3308	184	8	𝑇𝑢	𝑇𝑢	NOUN
cana-3308	184	9	)	)	PUNCT
cana-3308	184	10	)	)	PUNCT
cana-3308	184	11	≤	≤	NOUN
cana-3308	185	1	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	NUM
cana-3308	185	2	,	,	PUNCT
cana-3308	185	3	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	185	4	)	)	PUNCT
cana-3308	185	5	)	)	PUNCT
cana-3308	186	1	+	+	CCONJ
cana-3308	186	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	186	3	,	,	PUNCT
cana-3308	186	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	186	5	)	)	PUNCT
cana-3308	186	6	)	)	PUNCT
cana-3308	187	1	+	+	CCONJ
cana-3308	188	1	𝜃(𝜓((𝑥𝑛	𝜃(𝜓((𝑥𝑛	X
cana-3308	188	2	,	,	PUNCT
cana-3308	188	3	𝑢	𝑢	NOUN
cana-3308	188	4	)	)	PUNCT
cana-3308	188	5	)	)	PUNCT
cana-3308	189	1	𝜑	𝜑	PROPN
cana-3308	189	2	(	(	PUNCT
cana-3308	189	3	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	189	4	,	,	PUNCT
cana-3308	189	5	𝑢	𝑢	PROPN
cana-3308	189	6	)	)	PUNCT
cana-3308	189	7	)	)	PUNCT
cana-3308	189	8	)	)	PUNCT
cana-3308	190	1	<	<	X
cana-3308	190	2	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	X
cana-3308	190	3	,	,	PUNCT
cana-3308	190	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	190	5	)	)	PUNCT
cana-3308	190	6	)	)	PUNCT
cana-3308	191	1	+	+	CCONJ
cana-3308	191	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	191	3	,	,	PUNCT
cana-3308	191	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	191	5	)	)	PUNCT
cana-3308	191	6	)	)	PUNCT
cana-3308	192	1	+	+	CCONJ
cana-3308	192	2	𝜑	𝜑	X
cana-3308	192	3	(	(	PUNCT
cana-3308	192	4	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	192	5	,	,	PUNCT
cana-3308	192	6	𝑢	𝑢	PROPN
cana-3308	192	7	)	)	PUNCT
cana-3308	192	8	)	)	PUNCT
cana-3308	192	9	)	)	PUNCT
cana-3308	193	1	<	<	X
cana-3308	193	2	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	X
cana-3308	193	3	,	,	PUNCT
cana-3308	193	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	193	5	)	)	PUNCT
cana-3308	193	6	)	)	PUNCT
cana-3308	194	1	+	+	CCONJ
cana-3308	194	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	194	3	,	,	PUNCT
cana-3308	194	4	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	194	5	)	)	PUNCT
cana-3308	194	6	)	)	PUNCT
cana-3308	195	1	+	+	CCONJ
cana-3308	195	2	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	195	3	,	,	PUNCT
cana-3308	195	4	𝑢	𝑢	PROPN
cana-3308	195	5	)	)	PUNCT
cana-3308	195	6	)	)	PUNCT
cana-3308	195	7	.	.	PUNCT
cana-3308	196	1	using	use	VERB
cana-3308	196	2	equations	equation	NOUN
cana-3308	196	3	(	(	PUNCT
cana-3308	196	4	3.12	3.12	NUM
cana-3308	196	5	)	)	PUNCT
cana-3308	196	6	and	and	CCONJ
cana-3308	196	7	(	(	PUNCT
cana-3308	196	8	3.16	3.16	NUM
cana-3308	196	9	)	)	PUNCT
cana-3308	196	10	,	,	PUNCT
cana-3308	196	11	we	we	PRON
cana-3308	196	12	get	get	VERB
cana-3308	196	13	𝜓(𝑟	𝜓(𝑟	NOUN
cana-3308	196	14	′	′	NUM
cana-3308	196	15	)	)	PUNCT
cana-3308	196	16	<	<	X
cana-3308	196	17	𝜓	𝜓	X
cana-3308	196	18	(	(	PUNCT
cana-3308	196	19	𝑟	𝑟	NOUN
cana-3308	196	20	′	′	NOUN
cana-3308	196	21	2	2	NUM
cana-3308	196	22	)	)	PUNCT
cana-3308	197	1	+	+	CCONJ
cana-3308	197	2	𝜓	𝜓	X
cana-3308	197	3	(	(	PUNCT
cana-3308	197	4	𝑟	𝑟	NOUN
cana-3308	197	5	′	′	NOUN
cana-3308	197	6	2	2	X
cana-3308	197	7	)	)	PUNCT
cana-3308	197	8	communications	communication	NOUN
cana-3308	197	9	on	on	ADP
cana-3308	197	10	applied	apply	VERB
cana-3308	197	11	nonlinear	nonlinear	ADJ
cana-3308	197	12	analysis	analysis	NOUN
cana-3308	197	13	issn	issn	NOUN
cana-3308	197	14	:	:	PUNCT
cana-3308	197	15	1074	1074	NUM
cana-3308	197	16	-	-	PUNCT
cana-3308	197	17	133x	133x	NUM
cana-3308	197	18	vol	vol	NOUN
cana-3308	197	19	32	32	NUM
cana-3308	197	20	no	no	NOUN
cana-3308	197	21	.	.	PUNCT
cana-3308	198	1	6s	6s	NUM
cana-3308	198	2	(	(	PUNCT
cana-3308	198	3	2025	2025	NUM
cana-3308	198	4	)	)	PUNCT
cana-3308	198	5	446	446	NUM
cana-3308	198	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	198	7	=	=	SYM
cana-3308	198	8	𝜓	𝜓	PROPN
cana-3308	198	9	(	(	PUNCT
cana-3308	198	10	𝑟′	𝑟′	PROPN
cana-3308	198	11	2	2	NUM
cana-3308	198	12	+	+	CCONJ
cana-3308	198	13	𝑟′	𝑟′	NUM
cana-3308	198	14	2	2	NUM
cana-3308	198	15	)	)	PUNCT
cana-3308	198	16	=	=	PUNCT
cana-3308	198	17	𝜓(𝑟	𝜓(𝑟	PROPN
cana-3308	198	18	′	′	NUM
cana-3308	198	19	)	)	PUNCT
cana-3308	198	20	,	,	PUNCT
cana-3308	198	21	which	which	PRON
cana-3308	198	22	is	be	AUX
cana-3308	198	23	a	a	DET
cana-3308	198	24	contradiction	contradiction	NOUN
cana-3308	198	25	.	.	PUNCT
cana-3308	199	1	thus	thus	ADV
cana-3308	199	2	𝑇𝑢	𝑇𝑢	ADP
cana-3308	199	3	=	=	PUNCT
cana-3308	199	4	𝑢.	𝑢.	NOUN
cana-3308	199	5	i.e.	i.e.	X
cana-3308	199	6	,	,	PUNCT
cana-3308	199	7	𝑢	𝑢	PRON
cana-3308	199	8	is	be	AUX
cana-3308	199	9	the	the	DET
cana-3308	199	10	fixed	fixed	ADJ
cana-3308	199	11	point	point	NOUN
cana-3308	199	12	of	of	ADP
cana-3308	199	13	𝑇.	𝑇.	PROPN
cana-3308	199	14	example	example	NOUN
cana-3308	199	15	3.5	3.5	NUM
cana-3308	199	16	.	.	PUNCT
cana-3308	200	1	let	let	VERB
cana-3308	200	2	𝑋	𝑋	NOUN
cana-3308	200	3	=	=	PUNCT
cana-3308	201	1	[	[	X
cana-3308	201	2	0	0	NUM
cana-3308	201	3	,	,	PUNCT
cana-3308	201	4	+	+	CCONJ
cana-3308	201	5	∞	∞	NUM
cana-3308	201	6	)	)	PUNCT
cana-3308	201	7	and	and	CCONJ
cana-3308	201	8	𝑌	𝑌	PROPN
cana-3308	201	9	=	=	PUNCT
cana-3308	202	1	[	[	X
cana-3308	202	2	−1,1	−1,1	X
cana-3308	202	3	]	]	PUNCT
cana-3308	202	4	and	and	CCONJ
cana-3308	202	5	let	let	VERB
cana-3308	202	6	𝑑	𝑑	PRON
cana-3308	202	7	∶	∶	VERB
cana-3308	202	8	𝑋	𝑋	NOUN
cana-3308	202	9	×	×	NOUN
cana-3308	202	10	𝑌	𝑌	PROPN
cana-3308	202	11	→	→	SYM
cana-3308	202	12	[	[	X
cana-3308	202	13	0	0	NUM
cana-3308	202	14	,	,	PUNCT
cana-3308	202	15	+	+	NOUN
cana-3308	202	16	∞	∞	NOUN
cana-3308	202	17	)	)	PUNCT
cana-3308	202	18	be	be	VERB
cana-3308	202	19	a	a	DET
cana-3308	202	20	function	function	NOUN
cana-3308	202	21	such	such	ADJ
cana-3308	202	22	that	that	SCONJ
cana-3308	202	23	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	202	24	,	,	PUNCT
cana-3308	202	25	𝑦	𝑦	X
cana-3308	202	26	)	)	PUNCT
cana-3308	202	27	=	=	NOUN
cana-3308	203	1	|𝑥2	|𝑥2	X
cana-3308	203	2	−	−	PROPN
cana-3308	203	3	𝑦2|	𝑦2|	NOUN
cana-3308	203	4	for	for	ADP
cana-3308	203	5	all	all	DET
cana-3308	203	6	(	(	PUNCT
cana-3308	203	7	𝑥	𝑥	PROPN
cana-3308	203	8	,	,	PUNCT
cana-3308	203	9	𝑦	𝑦	X
cana-3308	203	10	)	)	PUNCT
cana-3308	203	11	∈	∈	NOUN
cana-3308	203	12	𝑋	𝑋	NOUN
cana-3308	203	13	×	×	NOUN
cana-3308	203	14	𝑌.	𝑌.	PROPN
cana-3308	203	15	then	then	ADV
cana-3308	203	16	,	,	PUNCT
cana-3308	203	17	clearly	clearly	ADV
cana-3308	203	18	(	(	PUNCT
cana-3308	203	19	𝑋	𝑋	PROPN
cana-3308	203	20	,	,	PUNCT
cana-3308	203	21	𝑌	𝑌	PROPN
cana-3308	203	22	,	,	PUNCT
cana-3308	203	23	𝑑	𝑑	NOUN
cana-3308	203	24	)	)	PUNCT
cana-3308	203	25	be	be	VERB
cana-3308	203	26	a	a	DET
cana-3308	203	27	complete	complete	ADJ
cana-3308	203	28	bipolar	bipolar	ADJ
cana-3308	203	29	metric	metric	ADJ
cana-3308	203	30	space	space	NOUN
cana-3308	203	31	.	.	PUNCT
cana-3308	204	1	define	define	VERB
cana-3308	204	2	𝑇	𝑇	PROPN
cana-3308	204	3	∶	∶	PROPN
cana-3308	204	4	(	(	PUNCT
cana-3308	204	5	𝑋	𝑋	PROPN
cana-3308	204	6	,	,	PUNCT
cana-3308	204	7	𝑌	𝑌	PROPN
cana-3308	204	8	)	)	PUNCT
cana-3308	204	9	⇉	⇉	PUNCT
cana-3308	205	1	(	(	PUNCT
cana-3308	205	2	𝑋	𝑋	PROPN
cana-3308	205	3	,	,	PUNCT
cana-3308	205	4	𝑌	𝑌	PROPN
cana-3308	205	5	)	)	PUNCT
cana-3308	205	6	such	such	ADJ
cana-3308	205	7	that	that	PRON
cana-3308	205	8	𝑇𝑥	𝑇𝑥	ADP
cana-3308	205	9	=	=	SYM
cana-3308	205	10	𝑥	𝑥	PRON
cana-3308	205	11	2	2	NUM
cana-3308	205	12	is	be	AUX
cana-3308	205	13	a	a	DET
cana-3308	205	14	mapping	mapping	NOUN
cana-3308	205	15	and	and	CCONJ
cana-3308	205	16	𝛼	𝛼	NOUN
cana-3308	205	17	∶	∶	NOUN
cana-3308	205	18	𝑋	𝑋	NOUN
cana-3308	205	19	×	×	NOUN
cana-3308	205	20	𝑌	𝑌	PROPN
cana-3308	205	21	→	→	SYM
cana-3308	206	1	[	[	X
cana-3308	206	2	0	0	NUM
cana-3308	206	3	,	,	PUNCT
cana-3308	206	4	∞	∞	NOUN
cana-3308	206	5	)	)	PUNCT
cana-3308	206	6	such	such	ADJ
cana-3308	206	7	that	that	SCONJ
cana-3308	206	8	𝛼(𝑥	𝛼(𝑥	PROPN
cana-3308	206	9	,	,	PUNCT
cana-3308	206	10	𝑦	𝑦	NOUN
cana-3308	206	11	)	)	PUNCT
cana-3308	206	12	=	=	SYM
cana-3308	206	13	3	3	NUM
cana-3308	206	14	2	2	NUM
cana-3308	206	15	for	for	ADP
cana-3308	206	16	(	(	PUNCT
cana-3308	206	17	𝑥	𝑥	PROPN
cana-3308	206	18	,	,	PUNCT
cana-3308	206	19	𝑦	𝑦	X
cana-3308	206	20	)	)	PUNCT
cana-3308	206	21	∈	∈	NOUN
cana-3308	206	22	𝑋	𝑋	PROPN
cana-3308	206	23	×	×	PROPN
cana-3308	206	24	𝑌.	𝑌.	PROPN
cana-3308	206	25	clearly	clearly	ADV
cana-3308	206	26	,	,	PUNCT
cana-3308	206	27	𝑇	𝑇	PROPN
cana-3308	206	28	is	be	AUX
cana-3308	206	29	𝛼-admissible	𝛼-admissible	ADJ
cana-3308	206	30	mapping	mapping	NOUN
cana-3308	206	31	and	and	CCONJ
cana-3308	206	32	there	there	PRON
cana-3308	206	33	exist	exist	VERB
cana-3308	206	34	(	(	PUNCT
cana-3308	206	35	𝑥0	𝑥0	NOUN
cana-3308	206	36	,	,	PUNCT
cana-3308	206	37	𝑦0	𝑦0	NOUN
cana-3308	206	38	)	)	PUNCT
cana-3308	206	39	∈	∈	PROPN
cana-3308	206	40	𝑋	𝑋	PROPN
cana-3308	206	41	×	×	PROPN
cana-3308	206	42	𝑌	𝑌	PROPN
cana-3308	206	43	such	such	ADJ
cana-3308	206	44	that	that	PRON
cana-3308	206	45	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	206	46	,	,	PUNCT
cana-3308	206	47	𝑇𝑦0	𝑇𝑦0	NOUN
cana-3308	206	48	)	)	PUNCT
cana-3308	206	49	≥	≥	NOUN
cana-3308	206	50	1	1	NUM
cana-3308	206	51	,	,	PUNCT
cana-3308	206	52	𝑋	𝑋	NOUN
cana-3308	206	53	∩	∩	NOUN
cana-3308	206	54	𝑌	𝑌	PROPN
cana-3308	206	55	=	=	SYM
cana-3308	206	56	{	{	PUNCT
cana-3308	206	57	0	0	NUM
cana-3308	206	58	}	}	PUNCT
cana-3308	206	59	and	and	CCONJ
cana-3308	206	60	𝑇0	𝑇0	X
cana-3308	206	61	=	=	SYM
cana-3308	206	62	0	0	X
cana-3308	206	63	.	.	PUNCT
cana-3308	206	64	taking	take	VERB
cana-3308	206	65	𝜓(𝑡	𝜓(𝑡	NOUN
cana-3308	206	66	)	)	PUNCT
cana-3308	206	67	=	=	SYM
cana-3308	206	68	𝑡	𝑡	ADP
cana-3308	206	69	4	4	NUM
cana-3308	206	70	,	,	PUNCT
cana-3308	206	71	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	206	72	)	)	PUNCT
cana-3308	206	73	=	=	SYM
cana-3308	206	74	𝑡	𝑡	ADP
cana-3308	206	75	2	2	NUM
cana-3308	206	76	and	and	CCONJ
cana-3308	206	77	𝜃(𝑡	𝜃(𝑡	PROPN
cana-3308	206	78	)	)	PUNCT
cana-3308	207	1	=	=	PUNCT
cana-3308	207	2	3	3	NUM
cana-3308	207	3	4	4	NUM
cana-3308	207	4	.	.	PUNCT
cana-3308	208	1	left	leave	VERB
cana-3308	208	2	hand	hand	NOUN
cana-3308	208	3	side	side	NOUN
cana-3308	208	4	of	of	ADP
cana-3308	208	5	equation	equation	NOUN
cana-3308	208	6	(	(	PUNCT
cana-3308	208	7	3.3	3.3	NUM
cana-3308	208	8	)	)	PUNCT
cana-3308	208	9	becomes	become	VERB
cana-3308	208	10	𝛼(𝑥	𝛼(𝑥	PROPN
cana-3308	208	11	,	,	PUNCT
cana-3308	208	12	𝑦	𝑦	NOUN
cana-3308	208	13	)	)	PUNCT
cana-3308	208	14	𝜓(𝑑(𝑇𝑥	𝜓(𝑑(𝑇𝑥	PROPN
cana-3308	208	15	,	,	PUNCT
cana-3308	208	16	𝑇𝑦	𝑇𝑦	NOUN
cana-3308	208	17	)	)	PUNCT
cana-3308	208	18	)	)	PUNCT
cana-3308	209	1	=	=	SYM
cana-3308	209	2	3	3	NUM
cana-3308	209	3	2	2	NUM
cana-3308	209	4	|𝑥2−𝑦2|	|𝑥2−𝑦2|	NUM
cana-3308	209	5	16	16	NUM
cana-3308	209	6	.	.	PUNCT
cana-3308	210	1	right	right	ADJ
cana-3308	210	2	hand	hand	NOUN
cana-3308	210	3	side	side	NOUN
cana-3308	210	4	of	of	ADP
cana-3308	210	5	equation	equation	NOUN
cana-3308	210	6	(	(	PUNCT
cana-3308	210	7	3.3	3.3	NUM
cana-3308	210	8	)	)	PUNCT
cana-3308	210	9	becomes	become	VERB
cana-3308	210	10	𝜃	𝜃	PROPN
cana-3308	210	11	(	(	PUNCT
cana-3308	210	12	𝜓(𝑑(𝑥	𝜓(𝑑(𝑥	PROPN
cana-3308	210	13	,	,	PUNCT
cana-3308	210	14	𝑦	𝑦	NOUN
cana-3308	210	15	)	)	PUNCT
cana-3308	210	16	)	)	PUNCT
cana-3308	210	17	)	)	PUNCT
cana-3308	211	1	𝜑(𝜓(𝑑(𝑥	𝜑(𝜓(𝑑(𝑥	PROPN
cana-3308	211	2	,	,	PUNCT
cana-3308	211	3	𝑦	𝑦	NOUN
cana-3308	211	4	)	)	PUNCT
cana-3308	211	5	)	)	PUNCT
cana-3308	211	6	)	)	PUNCT
cana-3308	212	1	=	=	SYM
cana-3308	212	2	3	3	NUM
cana-3308	212	3	2	2	NUM
cana-3308	212	4	|𝑥2−𝑦2|	|𝑥2−𝑦2|	NOUN
cana-3308	212	5	16	16	NUM
cana-3308	212	6	,	,	PUNCT
cana-3308	212	7	for	for	ADP
cana-3308	212	8	all	all	DET
cana-3308	212	9	(	(	PUNCT
cana-3308	212	10	𝑥	𝑥	PROPN
cana-3308	212	11	,	,	PUNCT
cana-3308	212	12	𝑦	𝑦	X
cana-3308	212	13	)	)	PUNCT
cana-3308	212	14	∈	∈	PROPN
cana-3308	212	15	𝑋	𝑋	PROPN
cana-3308	212	16	×	×	PROPN
cana-3308	212	17	𝑌	𝑌	PROPN
cana-3308	212	18	,	,	PUNCT
cana-3308	212	19	which	which	PRON
cana-3308	212	20	implies	imply	VERB
cana-3308	212	21	equation	equation	NOUN
cana-3308	212	22	(	(	PUNCT
cana-3308	212	23	3.3	3.3	NUM
cana-3308	212	24	)	)	PUNCT
cana-3308	212	25	holds	hold	VERB
cana-3308	212	26	.	.	PUNCT
cana-3308	213	1	hence	hence	ADV
cana-3308	213	2	,	,	PUNCT
cana-3308	213	3	𝑇	𝑇	PROPN
cana-3308	213	4	is	be	AUX
cana-3308	213	5	an	an	DET
cana-3308	213	6	(	(	PUNCT
cana-3308	213	7	𝛼	𝛼	PROPN
cana-3308	213	8	,	,	PUNCT
cana-3308	213	9	𝜓	𝜓	NOUN
cana-3308	213	10	,	,	PUNCT
cana-3308	213	11	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	213	12	contraction	contraction	NOUN
cana-3308	213	13	mapping	mapping	NOUN
cana-3308	213	14	.	.	PUNCT
cana-3308	214	1	all	all	DET
cana-3308	214	2	the	the	DET
cana-3308	214	3	conditions	condition	NOUN
cana-3308	214	4	of	of	ADP
cana-3308	214	5	theorem	theorem	NOUN
cana-3308	214	6	3.4	3.4	NUM
cana-3308	214	7	.	.	PUNCT
cana-3308	214	8	are	be	AUX
cana-3308	214	9	satisfied	satisfied	ADJ
cana-3308	214	10	.	.	PUNCT
cana-3308	215	1	so	so	ADV
cana-3308	215	2	,	,	PUNCT
cana-3308	215	3	𝑇	𝑇	PROPN
cana-3308	215	4	has	have	VERB
cana-3308	215	5	a	a	DET
cana-3308	215	6	fixed	fix	VERB
cana-3308	215	7	point	point	NOUN
cana-3308	215	8	and	and	CCONJ
cana-3308	215	9	𝑥	𝑥	NOUN
cana-3308	215	10	=	=	SYM
cana-3308	215	11	0	0	NUM
cana-3308	215	12	is	be	AUX
cana-3308	215	13	the	the	DET
cana-3308	215	14	fixed	fixed	ADJ
cana-3308	215	15	point	point	NOUN
cana-3308	215	16	of	of	ADP
cana-3308	215	17	𝑇.	𝑇.	PROPN
cana-3308	215	18	theorem	theorem	VERB
cana-3308	215	19	3.6	3.6	NUM
cana-3308	215	20	.	.	PUNCT
cana-3308	216	1	let	let	AUX
cana-3308	216	2	(	(	PUNCT
cana-3308	216	3	𝑋	𝑋	PROPN
cana-3308	216	4	,	,	PUNCT
cana-3308	216	5	𝑌	𝑌	PROPN
cana-3308	216	6	,	,	PUNCT
cana-3308	216	7	𝑑	𝑑	NOUN
cana-3308	216	8	)	)	PUNCT
cana-3308	216	9	be	be	VERB
cana-3308	216	10	a	a	DET
cana-3308	216	11	complete	complete	ADJ
cana-3308	216	12	bipolar	bipolar	ADJ
cana-3308	216	13	metric	metric	ADJ
cana-3308	216	14	space	space	NOUN
cana-3308	216	15	,	,	PUNCT
cana-3308	216	16	𝑇	𝑇	PROPN
cana-3308	216	17	∶	∶	PROPN
cana-3308	216	18	(	(	PUNCT
cana-3308	216	19	𝑋	𝑋	PROPN
cana-3308	216	20	,	,	PUNCT
cana-3308	216	21	𝑌	𝑌	PROPN
cana-3308	216	22	)	)	PUNCT
cana-3308	216	23	⇉	⇉	PUNCT
cana-3308	217	1	(	(	PUNCT
cana-3308	217	2	𝑋	𝑋	NOUN
cana-3308	217	3	,	,	PUNCT
cana-3308	217	4	𝑌	𝑌	PROPN
cana-3308	217	5	)	)	PUNCT
cana-3308	217	6	is	be	AUX
cana-3308	217	7	a	a	DET
cana-3308	217	8	covariant	covariant	ADJ
cana-3308	217	9	mapping	mapping	NOUN
cana-3308	217	10	and	and	CCONJ
cana-3308	217	11	𝛼	𝛼	NOUN
cana-3308	217	12	∶	∶	NOUN
cana-3308	217	13	𝑋	𝑋	NOUN
cana-3308	217	14	×	×	NOUN
cana-3308	217	15	𝑌	𝑌	PROPN
cana-3308	217	16	→	→	SYM
cana-3308	217	17	[	[	X
cana-3308	217	18	0	0	NUM
cana-3308	217	19	,	,	PUNCT
cana-3308	217	20	∞	∞	PROPN
cana-3308	217	21	)	)	PUNCT
cana-3308	217	22	.	.	PUNCT
cana-3308	218	1	suppose	suppose	VERB
cana-3308	218	2	that	that	SCONJ
cana-3308	218	3	the	the	DET
cana-3308	218	4	following	follow	VERB
cana-3308	218	5	conditions	condition	NOUN
cana-3308	218	6	hold	hold	VERB
cana-3308	218	7	:	:	PUNCT
cana-3308	218	8	(	(	PUNCT
cana-3308	218	9	i	i	NOUN
cana-3308	218	10	)	)	PUNCT
cana-3308	218	11	𝑇	𝑇	PROPN
cana-3308	218	12	is	be	AUX
cana-3308	218	13	𝛼-admissible	𝛼-admissible	ADJ
cana-3308	218	14	mapping	mapping	NOUN
cana-3308	218	15	,	,	PUNCT
cana-3308	218	16	(	(	PUNCT
cana-3308	218	17	ii	ii	NOUN
cana-3308	218	18	)	)	PUNCT
cana-3308	218	19	𝑇	𝑇	PROPN
cana-3308	218	20	is	be	AUX
cana-3308	218	21	an	an	DET
cana-3308	218	22	(	(	PUNCT
cana-3308	218	23	𝛼	𝛼	PROPN
cana-3308	218	24	,	,	PUNCT
cana-3308	218	25	𝜓	𝜓	NOUN
cana-3308	218	26	,	,	PUNCT
cana-3308	218	27	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	218	28	contraction	contraction	NOUN
cana-3308	218	29	mapping	mapping	NOUN
cana-3308	218	30	,	,	PUNCT
cana-3308	218	31	(	(	PUNCT
cana-3308	218	32	iii)there	iii)there	NOUN
cana-3308	218	33	exist	exist	VERB
cana-3308	218	34	𝑥0	𝑥0	NOUN
cana-3308	218	35	∈	∈	PROPN
cana-3308	218	36	𝑋and	𝑋and	PROPN
cana-3308	218	37	𝑦0	𝑦0	NOUN
cana-3308	218	38	∈	∈	NOUN
cana-3308	218	39	𝑌	𝑌	PROPN
cana-3308	218	40	such	such	ADJ
cana-3308	218	41	that	that	PRON
cana-3308	218	42	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	218	43	,	,	PUNCT
cana-3308	218	44	𝑦0	𝑦0	NOUN
cana-3308	218	45	)	)	PUNCT
cana-3308	218	46	≥	≥	NOUN
cana-3308	218	47	1	1	NUM
cana-3308	218	48	and	and	CCONJ
cana-3308	218	49	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	218	50	,	,	PUNCT
cana-3308	218	51	𝑇𝑦0	𝑇𝑦0	NOUN
cana-3308	218	52	)	)	PUNCT
cana-3308	218	53	≥	≥	NOUN
cana-3308	219	1	1	1	NUM
cana-3308	219	2	.	.	PUNCT
cana-3308	220	1	then	then	ADV
cana-3308	220	2	𝑇	𝑇	PROPN
cana-3308	220	3	has	have	VERB
cana-3308	220	4	a	a	DET
cana-3308	220	5	unique	unique	ADJ
cana-3308	220	6	fixed	fix	VERB
cana-3308	220	7	point	point	NOUN
cana-3308	220	8	.	.	PUNCT
cana-3308	221	1	proof	proof	NOUN
cana-3308	221	2	:	:	PUNCT
cana-3308	221	3	following	follow	VERB
cana-3308	221	4	the	the	DET
cana-3308	221	5	proof	proof	NOUN
cana-3308	221	6	of	of	ADP
cana-3308	221	7	theorem	theorem	ADJ
cana-3308	221	8	3.4	3.4	NUM
cana-3308	221	9	.	.	PUNCT
cana-3308	222	1	𝑇	𝑇	PROPN
cana-3308	222	2	has	have	AUX
cana-3308	222	3	fixed	fix	VERB
cana-3308	222	4	point	point	NOUN
cana-3308	222	5	.	.	PUNCT
cana-3308	223	1	to	to	PART
cana-3308	223	2	prove	prove	VERB
cana-3308	223	3	the	the	DET
cana-3308	223	4	uniqueness	uniqueness	NOUN
cana-3308	223	5	of	of	ADP
cana-3308	223	6	fixed	fix	VERB
cana-3308	223	7	point	point	NOUN
cana-3308	223	8	of	of	ADP
cana-3308	223	9	covariant	covariant	ADJ
cana-3308	223	10	mapping	mapping	NOUN
cana-3308	223	11	𝑇	𝑇	PROPN
cana-3308	223	12	in	in	ADP
cana-3308	223	13	complete	complete	ADJ
cana-3308	223	14	bipolar	bipolar	ADJ
cana-3308	223	15	metric	metric	ADJ
cana-3308	223	16	space	space	NOUN
cana-3308	223	17	,	,	PUNCT
cana-3308	223	18	let	let	VERB
cana-3308	223	19	us	we	PRON
cana-3308	223	20	assume	assume	VERB
cana-3308	223	21	,	,	PUNCT
cana-3308	223	22	if	if	SCONJ
cana-3308	223	23	possible	possible	ADJ
cana-3308	223	24	,	,	PUNCT
cana-3308	223	25	𝑢	𝑢	PROPN
cana-3308	223	26	and	and	CCONJ
cana-3308	223	27	𝑣	𝑣	PROPN
cana-3308	223	28	are	be	AUX
cana-3308	223	29	two	two	NUM
cana-3308	223	30	distinct	distinct	ADJ
cana-3308	223	31	fixed	fix	VERB
cana-3308	223	32	point	point	NOUN
cana-3308	223	33	of	of	ADP
cana-3308	223	34	𝑇.	𝑇.	PROPN
cana-3308	223	35	i.e.	i.e.	X
cana-3308	223	36	𝑇𝑢	𝑇𝑢	PROPN
cana-3308	223	37	=	=	PUNCT
cana-3308	223	38	𝑢	𝑢	X
cana-3308	223	39	and	and	CCONJ
cana-3308	223	40	𝑇𝑣	𝑇𝑣	PROPN
cana-3308	223	41	=	=	PUNCT
cana-3308	223	42	𝑣.	𝑣.	NOUN
cana-3308	223	43	by	by	ADP
cana-3308	223	44	using	use	VERB
cana-3308	223	45	the	the	DET
cana-3308	223	46	properties	property	NOUN
cana-3308	223	47	of	of	ADP
cana-3308	223	48	𝜓	𝜓	PROPN
cana-3308	223	49	,	,	PUNCT
cana-3308	223	50	𝜃	𝜃	PROPN
cana-3308	223	51	,	,	PUNCT
cana-3308	223	52	equations	equation	NOUN
cana-3308	223	53	(	(	PUNCT
cana-3308	223	54	3.1	3.1	NUM
cana-3308	223	55	)	)	PUNCT
cana-3308	223	56	,	,	PUNCT
cana-3308	223	57	(	(	PUNCT
cana-3308	223	58	3.2	3.2	NUM
cana-3308	223	59	)	)	PUNCT
cana-3308	223	60	and	and	CCONJ
cana-3308	223	61	(	(	PUNCT
cana-3308	223	62	3.3	3.3	NUM
cana-3308	223	63	)	)	PUNCT
cana-3308	223	64	,	,	PUNCT
cana-3308	223	65	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	NUM
cana-3308	223	66	,	,	PUNCT
cana-3308	223	67	𝑣	𝑣	NOUN
cana-3308	223	68	)	)	PUNCT
cana-3308	223	69	)	)	PUNCT
cana-3308	224	1	=	=	SYM
cana-3308	224	2	𝜓(𝑑(𝑇𝑢	𝜓(𝑑(𝑇𝑢	NOUN
cana-3308	224	3	,	,	PUNCT
cana-3308	224	4	𝑇𝑣	𝑇𝑣	PROPN
cana-3308	224	5	)	)	PUNCT
cana-3308	224	6	)	)	PUNCT
cana-3308	224	7	communications	communication	NOUN
cana-3308	224	8	on	on	ADP
cana-3308	224	9	applied	apply	VERB
cana-3308	224	10	nonlinear	nonlinear	ADJ
cana-3308	224	11	analysis	analysis	NOUN
cana-3308	224	12	issn	issn	NOUN
cana-3308	224	13	:	:	PUNCT
cana-3308	224	14	1074	1074	NUM
cana-3308	224	15	-	-	PUNCT
cana-3308	224	16	133x	133x	NUM
cana-3308	224	17	vol	vol	NOUN
cana-3308	224	18	32	32	NUM
cana-3308	224	19	no	no	NOUN
cana-3308	224	20	.	.	PUNCT
cana-3308	225	1	6s	6s	NUM
cana-3308	225	2	(	(	PUNCT
cana-3308	225	3	2025	2025	NUM
cana-3308	225	4	)	)	PUNCT
cana-3308	225	5	447	447	NUM
cana-3308	225	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	225	7	≤	≤	PROPN
cana-3308	225	8	𝛼(𝑢	𝛼(𝑢	PROPN
cana-3308	225	9	,	,	PUNCT
cana-3308	225	10	𝑣)𝜓(𝑑(𝑇𝑢	𝑣)𝜓(𝑑(𝑇𝑢	PROPN
cana-3308	225	11	,	,	PUNCT
cana-3308	225	12	𝑇𝑣	𝑇𝑣	PROPN
cana-3308	225	13	)	)	PUNCT
cana-3308	225	14	)	)	PUNCT
cana-3308	225	15	≤	≤	NOUN
cana-3308	226	1	𝜃(𝜓(𝑑(𝑢	𝜃(𝜓(𝑑(𝑢	NUM
cana-3308	226	2	,	,	PUNCT
cana-3308	226	3	𝑣)))𝜑(𝜓(𝑑(𝑢	𝑣)))𝜑(𝜓(𝑑(𝑢	PROPN
cana-3308	226	4	,	,	PUNCT
cana-3308	226	5	𝑣	𝑣	NOUN
cana-3308	226	6	)	)	PUNCT
cana-3308	226	7	)	)	PUNCT
cana-3308	226	8	)	)	PUNCT
cana-3308	227	1	≤	≤	NOUN
cana-3308	227	2	𝜑(𝜓(𝑑(𝑢	𝜑(𝜓(𝑑(𝑢	PROPN
cana-3308	227	3	,	,	PUNCT
cana-3308	227	4	𝑣	𝑣	NOUN
cana-3308	227	5	)	)	PUNCT
cana-3308	227	6	)	)	PUNCT
cana-3308	227	7	)	)	PUNCT
cana-3308	228	1	<	<	X
cana-3308	228	2	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	X
cana-3308	228	3	,	,	PUNCT
cana-3308	228	4	𝑣	𝑣	NOUN
cana-3308	228	5	)	)	PUNCT
cana-3308	228	6	)	)	PUNCT
cana-3308	228	7	.	.	PUNCT
cana-3308	229	1	this	this	PRON
cana-3308	229	2	implies	imply	VERB
cana-3308	229	3	that	that	SCONJ
cana-3308	229	4	𝑑(𝑢	𝑑(𝑢	ADJ
cana-3308	229	5	,	,	PUNCT
cana-3308	229	6	𝑣	𝑣	NOUN
cana-3308	229	7	)	)	PUNCT
cana-3308	229	8	<	<	X
cana-3308	229	9	𝑑(𝑢	𝑑(𝑢	PROPN
cana-3308	229	10	,	,	PUNCT
cana-3308	229	11	𝑣	𝑣	NOUN
cana-3308	229	12	)	)	PUNCT
cana-3308	229	13	,	,	PUNCT
cana-3308	229	14	which	which	PRON
cana-3308	229	15	is	be	AUX
cana-3308	229	16	a	a	DET
cana-3308	229	17	contradiction	contradiction	NOUN
cana-3308	229	18	.	.	PUNCT
cana-3308	230	1	thus	thus	ADV
cana-3308	230	2	,	,	PUNCT
cana-3308	230	3	𝑢	𝑢	PRON
cana-3308	230	4	is	be	AUX
cana-3308	230	5	the	the	DET
cana-3308	230	6	unique	unique	ADJ
cana-3308	230	7	fixed	fix	VERB
cana-3308	230	8	point	point	NOUN
cana-3308	230	9	of	of	ADP
cana-3308	230	10	𝑇.	𝑇.	PROPN
cana-3308	230	11	example	example	NOUN
cana-3308	230	12	3.7	3.7	NUM
cana-3308	230	13	.	.	PUNCT
cana-3308	231	1	in	in	ADP
cana-3308	231	2	the	the	DET
cana-3308	231	3	example	example	NOUN
cana-3308	231	4	3.5	3.5	NUM
cana-3308	231	5	,	,	PUNCT
cana-3308	231	6	we	we	PRON
cana-3308	231	7	can	can	AUX
cana-3308	231	8	easily	easily	ADV
cana-3308	231	9	say	say	VERB
cana-3308	231	10	that	that	SCONJ
cana-3308	231	11	𝑇	𝑇	PROPN
cana-3308	231	12	satisfies	satisfy	VERB
cana-3308	231	13	all	all	DET
cana-3308	231	14	the	the	DET
cana-3308	231	15	conditions	condition	NOUN
cana-3308	231	16	of	of	ADP
cana-3308	231	17	theorem	theorem	NOUN
cana-3308	231	18	3.6	3.6	NUM
cana-3308	231	19	.	.	PUNCT
cana-3308	232	1	so	so	ADV
cana-3308	232	2	,	,	PUNCT
cana-3308	232	3	𝑇	𝑇	PROPN
cana-3308	232	4	has	have	VERB
cana-3308	232	5	a	a	DET
cana-3308	232	6	unique	unique	ADJ
cana-3308	232	7	fixed	fix	VERB
cana-3308	232	8	point	point	NOUN
cana-3308	232	9	.	.	PUNCT
cana-3308	233	1	clearly	clearly	ADV
cana-3308	233	2	,	,	PUNCT
cana-3308	233	3	‘	'	PUNCT
cana-3308	233	4	0	0	NUM
cana-3308	233	5	’	'	PUNCT
cana-3308	233	6	is	be	AUX
cana-3308	233	7	unique	unique	ADJ
cana-3308	233	8	fixed	fix	VERB
cana-3308	233	9	point	point	NOUN
cana-3308	233	10	of	of	ADP
cana-3308	233	11	𝑇.	𝑇.	PROPN
cana-3308	233	12	definition	definition	NOUN
cana-3308	233	13	3.8	3.8	NUM
cana-3308	233	14	.	.	PUNCT
cana-3308	234	1	let	let	VERB
cana-3308	234	2	𝑋	𝑋	NOUN
cana-3308	234	3	and	and	CCONJ
cana-3308	234	4	𝑌	𝑌	PROPN
cana-3308	234	5	be	be	VERB
cana-3308	234	6	two	two	NUM
cana-3308	234	7	non	non	ADJ
cana-3308	234	8	-	-	ADJ
cana-3308	234	9	empty	empty	ADJ
cana-3308	234	10	sets	set	NOUN
cana-3308	234	11	.	.	PUNCT
cana-3308	235	1	consider	consider	VERB
cana-3308	235	2	(	(	PUNCT
cana-3308	235	3	𝑋	𝑋	PROPN
cana-3308	235	4	,	,	PUNCT
cana-3308	235	5	𝑌	𝑌	PROPN
cana-3308	235	6	,	,	PUNCT
cana-3308	235	7	𝑑	𝑑	NOUN
cana-3308	235	8	)	)	PUNCT
cana-3308	235	9	be	be	VERB
cana-3308	235	10	a	a	DET
cana-3308	235	11	bipolar	bipolar	ADJ
cana-3308	235	12	metric	metric	ADJ
cana-3308	235	13	space	space	NOUN
cana-3308	235	14	,	,	PUNCT
cana-3308	235	15	a	a	DET
cana-3308	235	16	contravariant	contravariant	ADJ
cana-3308	235	17	mapping	mapping	NOUN
cana-3308	235	18	𝑇	𝑇	PROPN
cana-3308	235	19	∶	∶	NOUN
cana-3308	235	20	(	(	PUNCT
cana-3308	235	21	𝑋	𝑋	PROPN
cana-3308	235	22	,	,	PUNCT
cana-3308	235	23	𝑌	𝑌	PROPN
cana-3308	235	24	)	)	PUNCT
cana-3308	235	25	⤨	⤨	NUM
cana-3308	235	26	(	(	PUNCT
cana-3308	235	27	𝑋	𝑋	PROPN
cana-3308	235	28	,	,	PUNCT
cana-3308	235	29	𝑌	𝑌	PROPN
cana-3308	235	30	)	)	PUNCT
cana-3308	235	31	is	be	AUX
cana-3308	235	32	called	call	VERB
cana-3308	235	33	geraghty	geraghty	PROPN
cana-3308	235	34	contraction	contraction	NOUN
cana-3308	235	35	if	if	SCONJ
cana-3308	235	36	there	there	PRON
cana-3308	235	37	exist	exist	VERB
cana-3308	235	38	a	a	DET
cana-3308	235	39	function	function	NOUN
cana-3308	235	40	𝜃	𝜃	NOUN
cana-3308	235	41	∈	∈	NOUN
cana-3308	235	42	θ	θ	NOUN
cana-3308	235	43	which	which	PRON
cana-3308	235	44	satisfies	satisfy	VERB
cana-3308	235	45	the	the	DET
cana-3308	235	46	following	follow	VERB
cana-3308	235	47	condition	condition	NOUN
cana-3308	235	48	:	:	PUNCT
cana-3308	235	49	𝑑(𝑇𝑦	𝑑(𝑇𝑦	NUM
cana-3308	235	50	,	,	PUNCT
cana-3308	235	51	𝑇𝑥	𝑇𝑥	NOUN
cana-3308	235	52	)	)	PUNCT
cana-3308	235	53	≤	≤	NOUN
cana-3308	235	54	𝜃	𝜃	PROPN
cana-3308	235	55	(	(	PUNCT
cana-3308	235	56	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	235	57	,	,	PUNCT
cana-3308	235	58	𝑦	𝑦	NOUN
cana-3308	235	59	)	)	PUNCT
cana-3308	235	60	)	)	PUNCT
cana-3308	235	61	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	235	62	,	,	PUNCT
cana-3308	235	63	𝑦	𝑦	NOUN
cana-3308	235	64	)	)	PUNCT
cana-3308	235	65	for	for	ADP
cana-3308	235	66	all	all	DET
cana-3308	235	67	𝑥	𝑥	DET
cana-3308	235	68	∈	∈	PROPN
cana-3308	235	69	𝑋	𝑋	NOUN
cana-3308	235	70	and	and	CCONJ
cana-3308	235	71	𝑦	𝑦	NOUN
cana-3308	235	72	∈	∈	NOUN
cana-3308	235	73	𝑌.	𝑌.	ADJ
cana-3308	235	74	definition	definition	NOUN
cana-3308	235	75	3.9	3.9	NUM
cana-3308	235	76	.	.	PUNCT
cana-3308	236	1	let	let	VERB
cana-3308	236	2	(	(	PUNCT
cana-3308	236	3	𝑋	𝑋	PROPN
cana-3308	236	4	,	,	PUNCT
cana-3308	236	5	𝑌	𝑌	PROPN
cana-3308	236	6	,	,	PUNCT
cana-3308	236	7	𝑑	𝑑	NOUN
cana-3308	236	8	)	)	PUNCT
cana-3308	236	9	be	be	VERB
cana-3308	236	10	a	a	DET
cana-3308	236	11	bipolar	bipolar	ADJ
cana-3308	236	12	metric	metric	ADJ
cana-3308	236	13	space	space	NOUN
cana-3308	236	14	and	and	CCONJ
cana-3308	236	15	𝑇	𝑇	PROPN
cana-3308	236	16	∶	∶	NOUN
cana-3308	236	17	(	(	PUNCT
cana-3308	236	18	𝑋	𝑋	PROPN
cana-3308	236	19	,	,	PUNCT
cana-3308	236	20	𝑌	𝑌	PROPN
cana-3308	236	21	)	)	PUNCT
cana-3308	236	22	⤨	⤨	NUM
cana-3308	236	23	(	(	PUNCT
cana-3308	236	24	𝑋	𝑋	PROPN
cana-3308	236	25	,	,	PUNCT
cana-3308	236	26	𝑌	𝑌	PROPN
cana-3308	236	27	)	)	PUNCT
cana-3308	236	28	is	be	AUX
cana-3308	236	29	a	a	DET
cana-3308	236	30	contravariant	contravariant	ADJ
cana-3308	236	31	mapping	mapping	NOUN
cana-3308	236	32	where	where	SCONJ
cana-3308	236	33	𝑋	𝑋	PROPN
cana-3308	236	34	and	and	CCONJ
cana-3308	236	35	𝑌	𝑌	PROPN
cana-3308	236	36	are	be	AUX
cana-3308	236	37	two	two	NUM
cana-3308	236	38	non	non	ADJ
cana-3308	236	39	-	-	ADJ
cana-3308	236	40	empty	empty	ADJ
cana-3308	236	41	sets	set	NOUN
cana-3308	236	42	.	.	PUNCT
cana-3308	237	1	suppose	suppose	VERB
cana-3308	237	2	that	that	SCONJ
cana-3308	237	3	𝜑	𝜑	PROPN
cana-3308	237	4	∶	∶	NOUN
cana-3308	237	5	ℝ+	ℝ+	PUNCT
cana-3308	237	6	→	→	X
cana-3308	237	7	ℝ+	ℝ+	PUNCT
cana-3308	237	8	is	be	AUX
cana-3308	237	9	function	function	NOUN
cana-3308	237	10	and	and	CCONJ
cana-3308	237	11	𝜃	𝜃	NOUN
cana-3308	237	12	∈	∈	NOUN
cana-3308	237	13	θ	θ	NOUN
cana-3308	237	14	and	and	CCONJ
cana-3308	237	15	𝑇	𝑇	PROPN
cana-3308	237	16	is	be	AUX
cana-3308	237	17	called	call	VERB
cana-3308	237	18	𝜑geraghty	𝜑geraghty	ADJ
cana-3308	237	19	contraction	contraction	NOUN
cana-3308	237	20	if	if	SCONJ
cana-3308	237	21	it	it	PRON
cana-3308	237	22	satisfies	satisfy	VERB
cana-3308	237	23	the	the	DET
cana-3308	237	24	following	follow	VERB
cana-3308	237	25	condition	condition	NOUN
cana-3308	237	26	:	:	PUNCT
cana-3308	237	27	(	(	PUNCT
cana-3308	237	28	i)𝜑(𝑡	i)𝜑(𝑡	NOUN
cana-3308	237	29	)	)	PUNCT
cana-3308	237	30	<	<	X
cana-3308	237	31	𝑡	𝑡	PROPN
cana-3308	237	32	for	for	ADP
cana-3308	237	33	any	any	DET
cana-3308	237	34	𝑡	𝑡	PROPN
cana-3308	237	35	∈	∈	PROPN
cana-3308	237	36	(	(	PUNCT
cana-3308	237	37	0	0	NUM
cana-3308	237	38	,	,	PUNCT
cana-3308	237	39	∞	∞	PROPN
cana-3308	237	40	)	)	PUNCT
cana-3308	237	41	,	,	PUNCT
cana-3308	237	42	(	(	PUNCT
cana-3308	237	43	ii)for	ii)for	ADP
cana-3308	237	44	any	any	PRON
cana-3308	237	45	휀	휀	NOUN
cana-3308	237	46	>	>	X
cana-3308	237	47	0	0	NUM
cana-3308	237	48	,	,	PUNCT
cana-3308	237	49	there	there	PRON
cana-3308	237	50	exist	exist	VERB
cana-3308	237	51	𝛿	𝛿	PROPN
cana-3308	237	52	>	>	X
cana-3308	237	53	0	0	NUM
cana-3308	237	54	such	such	ADJ
cana-3308	237	55	that	that	SCONJ
cana-3308	237	56	휀	휀	X
cana-3308	237	57	<	<	X
cana-3308	237	58	𝑡	𝑡	X
cana-3308	237	59	<	<	X
cana-3308	237	60	휀	휀	NOUN
cana-3308	237	61	+	+	X
cana-3308	237	62	𝛿	𝛿	ADJ
cana-3308	237	63	⇒	⇒	NOUN
cana-3308	237	64	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	237	65	)	)	PUNCT
cana-3308	237	66	≤	≤	NUM
cana-3308	237	67	휀	휀	X
cana-3308	237	68	,	,	PUNCT
cana-3308	237	69	(	(	PUNCT
cana-3308	237	70	iii)𝑑(𝑇𝑦	iii)𝑑(𝑇𝑦	PROPN
cana-3308	237	71	,	,	PUNCT
cana-3308	237	72	𝑇𝑥	𝑇𝑥	NOUN
cana-3308	237	73	)	)	PUNCT
cana-3308	237	74	≤	≤	NOUN
cana-3308	237	75	𝜃	𝜃	PROPN
cana-3308	237	76	(	(	PUNCT
cana-3308	237	77	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	237	78	,	,	PUNCT
cana-3308	237	79	𝑦	𝑦	NOUN
cana-3308	237	80	)	)	PUNCT
cana-3308	237	81	)	)	PUNCT
cana-3308	237	82	𝜑(𝑑(𝑥	𝜑(𝑑(𝑥	PROPN
cana-3308	237	83	,	,	PUNCT
cana-3308	237	84	𝑦	𝑦	NOUN
cana-3308	237	85	)	)	PUNCT
cana-3308	237	86	)	)	PUNCT
cana-3308	237	87	for	for	ADP
cana-3308	237	88	all	all	PRON
cana-3308	237	89	𝑥	𝑥	DET
cana-3308	237	90	∈	∈	PROPN
cana-3308	237	91	𝑋	𝑋	NOUN
cana-3308	237	92	and	and	CCONJ
cana-3308	237	93	𝑦	𝑦	NOUN
cana-3308	237	94	∈	∈	NOUN
cana-3308	237	95	𝑌.	𝑌.	ADJ
cana-3308	237	96	definition	definition	NOUN
cana-3308	237	97	3.10	3.10	NUM
cana-3308	237	98	.	.	PUNCT
cana-3308	238	1	let	let	VERB
cana-3308	238	2	𝑋	𝑋	NOUN
cana-3308	238	3	and	and	CCONJ
cana-3308	238	4	𝑌	𝑌	PROPN
cana-3308	238	5	be	be	VERB
cana-3308	238	6	two	two	NUM
cana-3308	238	7	non	non	ADJ
cana-3308	238	8	-	-	ADJ
cana-3308	238	9	empty	empty	ADJ
cana-3308	238	10	sets	set	NOUN
cana-3308	238	11	and	and	CCONJ
cana-3308	238	12	(	(	PUNCT
cana-3308	238	13	𝑋	𝑋	PROPN
cana-3308	238	14	,	,	PUNCT
cana-3308	238	15	𝑌	𝑌	PROPN
cana-3308	238	16	,	,	PUNCT
cana-3308	238	17	𝑑	𝑑	NOUN
cana-3308	238	18	)	)	PUNCT
cana-3308	238	19	be	be	VERB
cana-3308	238	20	a	a	DET
cana-3308	238	21	bipolar	bipolar	ADJ
cana-3308	238	22	metric	metric	ADJ
cana-3308	238	23	space	space	NOUN
cana-3308	238	24	,	,	PUNCT
cana-3308	238	25	𝑇	𝑇	PROPN
cana-3308	238	26	∶	∶	PROPN
cana-3308	238	27	(	(	PUNCT
cana-3308	238	28	𝑋	𝑋	PROPN
cana-3308	238	29	,	,	PUNCT
cana-3308	238	30	𝑌	𝑌	PROPN
cana-3308	238	31	)	)	PUNCT
cana-3308	238	32	⤨	⤨	NUM
cana-3308	238	33	(	(	PUNCT
cana-3308	238	34	𝑋	𝑋	PROPN
cana-3308	238	35	,	,	PUNCT
cana-3308	238	36	𝑌	𝑌	PROPN
cana-3308	238	37	)	)	PUNCT
cana-3308	238	38	is	be	AUX
cana-3308	238	39	a	a	DET
cana-3308	238	40	contravariant	contravariant	ADJ
cana-3308	238	41	self	self	NOUN
cana-3308	238	42	map	map	NOUN
cana-3308	238	43	and	and	CCONJ
cana-3308	238	44	𝛼	𝛼	ADP
cana-3308	238	45	∶	∶	NOUN
cana-3308	238	46	𝑋	𝑋	NOUN
cana-3308	238	47	×	×	NOUN
cana-3308	238	48	𝑌	𝑌	PROPN
cana-3308	238	49	→	→	SYM
cana-3308	239	1	[	[	X
cana-3308	239	2	0	0	NUM
cana-3308	239	3	,	,	PUNCT
cana-3308	239	4	∞	∞	PROPN
cana-3308	239	5	)	)	PUNCT
cana-3308	239	6	.	.	PUNCT
cana-3308	240	1	a	a	DET
cana-3308	240	2	mapping	mapping	NOUN
cana-3308	240	3	𝑇	𝑇	PROPN
cana-3308	240	4	is	be	AUX
cana-3308	240	5	said	say	VERB
cana-3308	240	6	to	to	PART
cana-3308	240	7	be	be	AUX
cana-3308	240	8	(	(	PUNCT
cana-3308	240	9	𝛼	𝛼	PROPN
cana-3308	240	10	,	,	PUNCT
cana-3308	240	11	𝜓	𝜓	NOUN
cana-3308	240	12	,	,	PUNCT
cana-3308	240	13	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	240	14	contraction	contraction	NOUN
cana-3308	240	15	mapping	mapping	NOUN
cana-3308	240	16	if	if	SCONJ
cana-3308	240	17	there	there	PRON
cana-3308	240	18	exist	exist	VERB
cana-3308	240	19	𝜑	𝜑	PRON
cana-3308	240	20	∶	∶	NOUN
cana-3308	240	21	ℝ+	ℝ+	PUNCT
cana-3308	240	22	→	→	SYM
cana-3308	240	23	ℝ+	ℝ+	PUNCT
cana-3308	240	24	,	,	PUNCT
cana-3308	240	25	𝜓	𝜓	PROPN
cana-3308	240	26	∈	∈	PROPN
cana-3308	240	27	ψ	ψ	NOUN
cana-3308	240	28	and	and	CCONJ
cana-3308	240	29	𝜃	𝜃	PRON
cana-3308	240	30	∈	∈	NOUN
cana-3308	240	31	θ	θ	NOUN
cana-3308	240	32	satisfies	satisfy	VERB
cana-3308	240	33	the	the	DET
cana-3308	240	34	following	follow	VERB
cana-3308	240	35	condition	condition	NOUN
cana-3308	240	36	:	:	PUNCT
cana-3308	240	37	(	(	PUNCT
cana-3308	240	38	i)𝜑(𝑡	i)𝜑(𝑡	NOUN
cana-3308	240	39	)	)	PUNCT
cana-3308	240	40	<	<	X
cana-3308	241	1	𝑡	𝑡	PROPN
cana-3308	241	2	for	for	ADP
cana-3308	241	3	any	any	DET
cana-3308	241	4	𝑡	𝑡	PROPN
cana-3308	241	5	∈	∈	PROPN
cana-3308	241	6	(	(	PUNCT
cana-3308	241	7	0	0	NUM
cana-3308	241	8	,	,	PUNCT
cana-3308	241	9	∞	∞	PROPN
cana-3308	241	10	)	)	PUNCT
cana-3308	241	11	,	,	PUNCT
cana-3308	241	12	(	(	PUNCT
cana-3308	241	13	ii)for	ii)for	ADP
cana-3308	241	14	any	any	PRON
cana-3308	241	15	휀	휀	NOUN
cana-3308	241	16	>	>	X
cana-3308	241	17	0	0	NUM
cana-3308	241	18	,	,	PUNCT
cana-3308	241	19	there	there	PRON
cana-3308	241	20	exist	exist	VERB
cana-3308	241	21	𝛿	𝛿	PROPN
cana-3308	241	22	>	>	X
cana-3308	241	23	0	0	NUM
cana-3308	241	24	such	such	ADJ
cana-3308	241	25	that	that	SCONJ
cana-3308	241	26	휀	휀	X
cana-3308	241	27	<	<	X
cana-3308	241	28	𝑡	𝑡	X
cana-3308	241	29	<	<	X
cana-3308	241	30	휀	휀	NOUN
cana-3308	241	31	+	+	X
cana-3308	241	32	𝛿	𝛿	ADJ
cana-3308	241	33	⇒	⇒	NOUN
cana-3308	241	34	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	241	35	)	)	PUNCT
cana-3308	241	36	≤	≤	NUM
cana-3308	242	1	휀	휀	X
cana-3308	242	2	,	,	PUNCT
cana-3308	242	3	(	(	PUNCT
cana-3308	242	4	iii)𝛼(𝑥	iii)𝛼(𝑥	PROPN
cana-3308	242	5	,	,	PUNCT
cana-3308	242	6	𝑦)𝜓(𝑑(𝑇𝑦	𝑦)𝜓(𝑑(𝑇𝑦	PROPN
cana-3308	242	7	,	,	PUNCT
cana-3308	242	8	𝑇𝑥	𝑇𝑥	NOUN
cana-3308	242	9	)	)	PUNCT
cana-3308	242	10	)	)	PUNCT
cana-3308	242	11	≤	≤	PUNCT
cana-3308	243	1	𝜃(𝜓	𝜃(𝜓	PROPN
cana-3308	243	2	(	(	PUNCT
cana-3308	243	3	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	243	4	,	,	PUNCT
cana-3308	243	5	𝑦	𝑦	NOUN
cana-3308	243	6	)	)	PUNCT
cana-3308	243	7	)	)	PUNCT
cana-3308	243	8	)	)	PUNCT
cana-3308	244	1	𝜑(𝜓(𝑑(𝑥	𝜑(𝜓(𝑑(𝑥	PROPN
cana-3308	244	2	,	,	PUNCT
cana-3308	244	3	𝑦))),∀	𝑦))),∀	PROPN
cana-3308	244	4	𝑥	𝑥	DET
cana-3308	244	5	∈	∈	PROPN
cana-3308	244	6	𝑋	𝑋	NOUN
cana-3308	244	7	and	and	CCONJ
cana-3308	244	8	𝑦	𝑦	NOUN
cana-3308	244	9	∈	∈	PROPN
cana-3308	244	10	𝑌	𝑌	PROPN
cana-3308	244	11	(	(	PUNCT
cana-3308	244	12	3.17	3.17	NUM
cana-3308	244	13	)	)	PUNCT
cana-3308	244	14	theorem	theorem	NOUN
cana-3308	244	15	3.11	3.11	NUM
cana-3308	244	16	.	.	PUNCT
cana-3308	245	1	let	let	AUX
cana-3308	245	2	(	(	PUNCT
cana-3308	245	3	𝑋	𝑋	PROPN
cana-3308	245	4	,	,	PUNCT
cana-3308	245	5	𝑌	𝑌	PROPN
cana-3308	245	6	,	,	PUNCT
cana-3308	245	7	𝑑	𝑑	NOUN
cana-3308	245	8	)	)	PUNCT
cana-3308	245	9	be	be	VERB
cana-3308	245	10	a	a	DET
cana-3308	245	11	complete	complete	ADJ
cana-3308	245	12	bipolar	bipolar	ADJ
cana-3308	245	13	metric	metric	ADJ
cana-3308	245	14	space	space	NOUN
cana-3308	245	15	,	,	PUNCT
cana-3308	245	16	𝑇	𝑇	PROPN
cana-3308	245	17	∶	∶	PROPN
cana-3308	245	18	(	(	PUNCT
cana-3308	245	19	𝑋	𝑋	PROPN
cana-3308	245	20	,	,	PUNCT
cana-3308	245	21	𝑌	𝑌	PROPN
cana-3308	245	22	)	)	PUNCT
cana-3308	245	23	⤨	⤨	NUM
cana-3308	245	24	(	(	PUNCT
cana-3308	245	25	𝑋	𝑋	PROPN
cana-3308	245	26	,	,	PUNCT
cana-3308	245	27	𝑌	𝑌	PROPN
cana-3308	245	28	)	)	PUNCT
cana-3308	245	29	is	be	AUX
cana-3308	245	30	a	a	DET
cana-3308	245	31	contravariant	contravariant	ADJ
cana-3308	245	32	mapping	mapping	NOUN
cana-3308	245	33	and	and	CCONJ
cana-3308	245	34	𝛼	𝛼	ADP
cana-3308	245	35	∶	∶	NOUN
cana-3308	245	36	𝑋	𝑋	NOUN
cana-3308	245	37	×	×	NOUN
cana-3308	245	38	𝑌	𝑌	PROPN
cana-3308	245	39	→	→	SYM
cana-3308	245	40	[	[	X
cana-3308	245	41	0	0	NUM
cana-3308	245	42	,	,	PUNCT
cana-3308	245	43	∞	∞	PROPN
cana-3308	245	44	)	)	PUNCT
cana-3308	245	45	.	.	PUNCT
cana-3308	246	1	suppose	suppose	VERB
cana-3308	246	2	that	that	SCONJ
cana-3308	246	3	the	the	DET
cana-3308	246	4	following	follow	VERB
cana-3308	246	5	conditions	condition	NOUN
cana-3308	246	6	hold	hold	VERB
cana-3308	246	7	:	:	PUNCT
cana-3308	246	8	(	(	PUNCT
cana-3308	246	9	i)𝑇	i)𝑇	NOUN
cana-3308	246	10	is	be	AUX
cana-3308	246	11	𝛼admissible	𝛼admissible	ADJ
cana-3308	246	12	mapping	mapping	NOUN
cana-3308	246	13	,	,	PUNCT
cana-3308	246	14	(	(	PUNCT
cana-3308	246	15	ii)𝑇	ii)𝑇	NOUN
cana-3308	246	16	is	be	AUX
cana-3308	246	17	an	an	DET
cana-3308	246	18	(	(	PUNCT
cana-3308	246	19	𝛼	𝛼	PROPN
cana-3308	246	20	,	,	PUNCT
cana-3308	246	21	𝜓	𝜓	NOUN
cana-3308	246	22	,	,	PUNCT
cana-3308	246	23	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	246	24	contraction	contraction	NOUN
cana-3308	246	25	mapping	mapping	NOUN
cana-3308	246	26	,	,	PUNCT
cana-3308	246	27	(	(	PUNCT
cana-3308	246	28	iii	iii	X
cana-3308	246	29	)	)	PUNCT
cana-3308	246	30	there	there	PRON
cana-3308	246	31	exist	exist	VERB
cana-3308	246	32	𝑥0	𝑥0	NOUN
cana-3308	246	33	∈	∈	NOUN
cana-3308	246	34	𝑋	𝑋	NOUN
cana-3308	246	35	such	such	ADJ
cana-3308	246	36	that	that	PRON
cana-3308	246	37	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	246	38	,	,	PUNCT
cana-3308	246	39	𝑇𝑥0	𝑇𝑥0	NOUN
cana-3308	246	40	)	)	PUNCT
cana-3308	246	41	≥	≥	NOUN
cana-3308	247	1	1	1	NUM
cana-3308	247	2	.	.	PUNCT
cana-3308	248	1	then	then	ADV
cana-3308	248	2	𝑇	𝑇	PROPN
cana-3308	248	3	has	have	AUX
cana-3308	248	4	fixed	fix	VERB
cana-3308	248	5	point	point	NOUN
cana-3308	248	6	.	.	PUNCT
cana-3308	249	1	proof	proof	NOUN
cana-3308	249	2	:	:	PUNCT
cana-3308	249	3	let	let	VERB
cana-3308	249	4	𝑥0	𝑥0	VERB
cana-3308	249	5	∈	∈	PROPN
cana-3308	249	6	𝑋and	𝑋and	PROPN
cana-3308	249	7	𝑦0	𝑦0	NOUN
cana-3308	249	8	∈	∈	NOUN
cana-3308	249	9	𝑌	𝑌	PROPN
cana-3308	249	10	such	such	ADJ
cana-3308	249	11	that	that	DET
cana-3308	249	12	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	249	13	,	,	PUNCT
cana-3308	249	14	𝑦0	𝑦0	NOUN
cana-3308	249	15	)	)	PUNCT
cana-3308	249	16	≥	≥	NOUN
cana-3308	249	17	1	1	NUM
cana-3308	249	18	and	and	CCONJ
cana-3308	249	19	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	249	20	,	,	PUNCT
cana-3308	249	21	𝑇𝑥0	𝑇𝑥0	NOUN
cana-3308	249	22	)	)	PUNCT
cana-3308	249	23	≥	≥	NOUN
cana-3308	250	1	1	1	NUM
cana-3308	250	2	.	.	PUNCT
cana-3308	251	1	now	now	ADV
cana-3308	251	2	we	we	PRON
cana-3308	251	3	define	define	VERB
cana-3308	251	4	a	a	DET
cana-3308	251	5	bisequence	bisequence	NOUN
cana-3308	251	6	{	{	PUNCT
cana-3308	251	7	(	(	PUNCT
cana-3308	251	8	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	251	9	,	,	PUNCT
cana-3308	251	10	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	251	11	)	)	PUNCT
cana-3308	251	12	}	}	PUNCT
cana-3308	251	13	in	in	ADP
cana-3308	251	14	(	(	PUNCT
cana-3308	251	15	𝑋	𝑋	PROPN
cana-3308	251	16	,	,	PUNCT
cana-3308	251	17	𝑌	𝑌	PROPN
cana-3308	251	18	)	)	PUNCT
cana-3308	251	19	by	by	ADP
cana-3308	251	20	𝑇𝑥𝑛	𝑇𝑥𝑛	PROPN
cana-3308	251	21	=	=	PUNCT
cana-3308	251	22	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	251	23	and	and	CCONJ
cana-3308	251	24	𝑇𝑦𝑛	𝑇𝑦𝑛	PROPN
cana-3308	251	25	=	=	SYM
cana-3308	251	26	𝑥𝑛+1	𝑥𝑛+1	PROPN
cana-3308	251	27	for	for	SCONJ
cana-3308	251	28	all	all	DET
cana-3308	251	29	𝑛	𝑛	DET
cana-3308	251	30	∈	∈	PROPN
cana-3308	251	31	ℕ	ℕ	PROPN
cana-3308	251	32	∪	∪	X
cana-3308	251	33	{	{	PUNCT
cana-3308	251	34	0	0	NUM
cana-3308	251	35	}	}	PUNCT
cana-3308	251	36	.	.	PUNCT
cana-3308	252	1	since	since	SCONJ
cana-3308	252	2	𝑇	𝑇	PROPN
cana-3308	252	3	is	be	AUX
cana-3308	252	4	𝛼admissible	𝛼admissible	ADJ
cana-3308	252	5	mapping	mapping	NOUN
cana-3308	252	6	.	.	PUNCT
cana-3308	253	1	communications	communication	NOUN
cana-3308	253	2	on	on	ADP
cana-3308	253	3	applied	apply	VERB
cana-3308	253	4	nonlinear	nonlinear	ADJ
cana-3308	253	5	analysis	analysis	NOUN
cana-3308	253	6	issn	issn	NOUN
cana-3308	253	7	:	:	PUNCT
cana-3308	253	8	1074	1074	NUM
cana-3308	253	9	-	-	PUNCT
cana-3308	253	10	133x	133x	NUM
cana-3308	253	11	vol	vol	NOUN
cana-3308	253	12	32	32	NUM
cana-3308	253	13	no	no	NOUN
cana-3308	253	14	.	.	PUNCT
cana-3308	254	1	6s	6s	NUM
cana-3308	254	2	(	(	PUNCT
cana-3308	254	3	2025	2025	NUM
cana-3308	254	4	)	)	PUNCT
cana-3308	254	5	448	448	NUM
cana-3308	254	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	255	1	so	so	ADV
cana-3308	255	2	,	,	PUNCT
cana-3308	255	3	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	255	4	,	,	PUNCT
cana-3308	255	5	𝑦0	𝑦0	NOUN
cana-3308	255	6	)	)	PUNCT
cana-3308	255	7	=	=	SYM
cana-3308	255	8	𝛼(𝑥0	𝛼(𝑥0	VERB
cana-3308	255	9	,	,	PUNCT
cana-3308	255	10	𝑇𝑥0	𝑇𝑥0	NOUN
cana-3308	255	11	)	)	PUNCT
cana-3308	255	12	≥	≥	PROPN
cana-3308	255	13	1	1	NUM
cana-3308	255	14	,	,	PUNCT
cana-3308	255	15	𝛼(𝑥1	𝛼(𝑥1	ADJ
cana-3308	255	16	,	,	PUNCT
cana-3308	255	17	𝑦0	𝑦0	NOUN
cana-3308	255	18	)	)	PUNCT
cana-3308	256	1	=	=	SYM
cana-3308	256	2	𝛼(𝑇𝑦0	𝛼(𝑇𝑦0	PROPN
cana-3308	256	3	,	,	PUNCT
cana-3308	256	4	𝑇𝑥0	𝑇𝑥0	NOUN
cana-3308	256	5	)	)	PUNCT
cana-3308	256	6	≥	≥	PROPN
cana-3308	256	7	1	1	NUM
cana-3308	256	8	,	,	PUNCT
cana-3308	256	9	𝛼(𝑥1	𝛼(𝑥1	ADJ
cana-3308	256	10	,	,	PUNCT
cana-3308	256	11	𝑦1	𝑦1	NOUN
cana-3308	256	12	)	)	PUNCT
cana-3308	256	13	=	=	SYM
cana-3308	256	14	𝛼(𝑥1	𝛼(𝑥1	X
cana-3308	256	15	,	,	PUNCT
cana-3308	256	16	𝑇𝑥1	𝑇𝑥1	ADJ
cana-3308	256	17	)	)	PUNCT
cana-3308	256	18	≥	≥	PROPN
cana-3308	256	19	1	1	NUM
cana-3308	256	20	.	.	PUNCT
cana-3308	257	1	using	use	VERB
cana-3308	257	2	mathematical	mathematical	ADJ
cana-3308	257	3	induction	induction	NOUN
cana-3308	257	4	,	,	PUNCT
cana-3308	257	5	we	we	PRON
cana-3308	257	6	get	get	VERB
cana-3308	257	7	𝛼(𝑥𝑛+1	𝛼(𝑥𝑛+1	ADJ
cana-3308	257	8	,	,	PUNCT
cana-3308	257	9	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	257	10	)	)	PUNCT
cana-3308	257	11	≥	≥	NOUN
cana-3308	257	12	1	1	NUM
cana-3308	257	13	and	and	CCONJ
cana-3308	257	14	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-3308	257	15	,	,	PUNCT
cana-3308	257	16	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	257	17	)	)	PUNCT
cana-3308	257	18	≥	≥	NOUN
cana-3308	257	19	1	1	NUM
cana-3308	257	20	for	for	ADP
cana-3308	257	21	all	all	DET
cana-3308	257	22	𝑛	𝑛	DET
cana-3308	257	23	∈	∈	PROPN
cana-3308	257	24	ℕ	ℕ	PROPN
cana-3308	257	25	∪	∪	X
cana-3308	257	26	{	{	PUNCT
cana-3308	257	27	0	0	NUM
cana-3308	257	28	}	}	PUNCT
cana-3308	257	29	.	.	PUNCT
cana-3308	258	1	(	(	PUNCT
cana-3308	258	2	3.18	3.18	NUM
cana-3308	258	3	)	)	PUNCT
cana-3308	258	4	putting	put	VERB
cana-3308	258	5	𝑥	𝑥	X
cana-3308	258	6	=	=	PUNCT
cana-3308	258	7	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-3308	258	8	and	and	CCONJ
cana-3308	258	9	𝑦	𝑦	NOUN
cana-3308	258	10	=	=	PUNCT
cana-3308	258	11	𝑦𝑛	𝑦𝑛	ADP
cana-3308	258	12	in	in	ADP
cana-3308	258	13	equation	equation	NOUN
cana-3308	258	14	(	(	PUNCT
cana-3308	258	15	3.17	3.17	NUM
cana-3308	258	16	)	)	PUNCT
cana-3308	258	17	,	,	PUNCT
cana-3308	258	18	using	use	VERB
cana-3308	258	19	equations	equation	NOUN
cana-3308	258	20	(	(	PUNCT
cana-3308	258	21	3.1	3.1	NUM
cana-3308	258	22	)	)	PUNCT
cana-3308	258	23	,	,	PUNCT
cana-3308	258	24	(	(	PUNCT
cana-3308	258	25	3.2	3.2	NUM
cana-3308	258	26	)	)	PUNCT
cana-3308	258	27	and	and	CCONJ
cana-3308	258	28	by	by	ADP
cana-3308	258	29	the	the	DET
cana-3308	258	30	properties	property	NOUN
cana-3308	258	31	of	of	ADP
cana-3308	258	32	𝜓	𝜓	PROPN
cana-3308	258	33	and	and	CCONJ
cana-3308	258	34	𝜃	𝜃	X
cana-3308	258	35	,	,	PUNCT
cana-3308	258	36	we	we	PRON
cana-3308	258	37	have	have	VERB
cana-3308	258	38	the	the	DET
cana-3308	258	39	following	follow	VERB
cana-3308	258	40	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	258	41	,	,	PUNCT
cana-3308	258	42	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	258	43	)	)	PUNCT
cana-3308	258	44	)	)	PUNCT
cana-3308	259	1	=	=	SYM
cana-3308	259	2	𝜓(𝑑(𝑇𝑦𝑛	𝜓(𝑑(𝑇𝑦𝑛	NOUN
cana-3308	259	3	,	,	PUNCT
cana-3308	259	4	𝑇𝑥𝑛	𝑇𝑥𝑛	PROPN
cana-3308	259	5	)	)	PUNCT
cana-3308	259	6	)	)	PUNCT
cana-3308	259	7	≤	≤	NOUN
cana-3308	260	1	𝛼(𝑥𝑛	𝛼(𝑥𝑛	ADV
cana-3308	260	2	,	,	PUNCT
cana-3308	260	3	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	260	4	)	)	PUNCT
cana-3308	260	5	𝜓(𝑑(𝑇𝑦𝑛	𝜓(𝑑(𝑇𝑦𝑛	NOUN
cana-3308	260	6	,	,	PUNCT
cana-3308	260	7	𝑇𝑥𝑛	𝑇𝑥𝑛	PROPN
cana-3308	260	8	)	)	PUNCT
cana-3308	260	9	)	)	PUNCT
cana-3308	261	1	≤	≤	PROPN
cana-3308	261	2	𝜃	𝜃	X
cana-3308	261	3	(	(	PUNCT
cana-3308	261	4	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	261	5	,	,	PUNCT
cana-3308	261	6	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	261	7	)	)	PUNCT
cana-3308	261	8	)	)	PUNCT
cana-3308	261	9	)	)	PUNCT
cana-3308	261	10	𝜑(𝜓(𝑑(𝑥𝑛	𝜑(𝜓(𝑑(𝑥𝑛	NOUN
cana-3308	261	11	,	,	PUNCT
cana-3308	261	12	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	261	13	)	)	PUNCT
cana-3308	261	14	)	)	PUNCT
cana-3308	261	15	)	)	PUNCT
cana-3308	261	16	≤	≤	NOUN
cana-3308	261	17	𝜑(𝜓(𝑑(𝑥𝑛	𝜑(𝜓(𝑑(𝑥𝑛	PUNCT
cana-3308	261	18	,	,	PUNCT
cana-3308	261	19	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	261	20	)	)	PUNCT
cana-3308	261	21	)	)	PUNCT
cana-3308	261	22	)	)	PUNCT
cana-3308	262	1	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	CCONJ
cana-3308	262	2	,	,	PUNCT
cana-3308	262	3	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	262	4	)	)	PUNCT
cana-3308	262	5	)	)	PUNCT
cana-3308	263	1	<	<	X
cana-3308	263	2	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	263	3	,	,	PUNCT
cana-3308	263	4	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	263	5	)	)	PUNCT
cana-3308	263	6	)	)	PUNCT
cana-3308	263	7	.	.	PUNCT
cana-3308	264	1	(	(	PUNCT
cana-3308	264	2	3.19	3.19	NUM
cana-3308	264	3	)	)	PUNCT
cana-3308	264	4	hence	hence	ADV
cana-3308	264	5	,	,	PUNCT
cana-3308	264	6	𝜓	𝜓	PROPN
cana-3308	264	7	is	be	AUX
cana-3308	264	8	strictly	strictly	ADV
cana-3308	264	9	increasing	increase	VERB
cana-3308	264	10	function	function	NOUN
cana-3308	264	11	so	so	ADV
cana-3308	264	12	,	,	PUNCT
cana-3308	264	13	we	we	PRON
cana-3308	264	14	get	get	VERB
cana-3308	264	15	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	264	16	,	,	PUNCT
cana-3308	264	17	𝑦𝑛	𝑦𝑛	ADP
cana-3308	264	18	)	)	PUNCT
cana-3308	264	19	<	<	X
cana-3308	264	20	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	264	21	,	,	PUNCT
cana-3308	264	22	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	264	23	)	)	PUNCT
cana-3308	264	24	for	for	ADP
cana-3308	264	25	all	all	DET
cana-3308	264	26	𝑛	𝑛	DET
cana-3308	264	27	≥	≥	NOUN
cana-3308	264	28	0	0	NUM
cana-3308	264	29	.	.	PUNCT
cana-3308	265	1	(	(	PUNCT
cana-3308	265	2	3.20	3.20	NUM
cana-3308	265	3	)	)	PUNCT
cana-3308	265	4	similarly	similarly	ADV
cana-3308	265	5	,	,	PUNCT
cana-3308	265	6	putting	put	VERB
cana-3308	265	7	𝑥	𝑥	X
cana-3308	265	8	=	=	PUNCT
cana-3308	265	9	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-3308	265	10	and	and	CCONJ
cana-3308	265	11	𝑦	𝑦	NOUN
cana-3308	265	12	=	=	SYM
cana-3308	265	13	𝑦𝑛+1	𝑦𝑛+1	PROPN
cana-3308	265	14	in	in	ADP
cana-3308	265	15	equation	equation	NOUN
cana-3308	265	16	(	(	PUNCT
cana-3308	265	17	3.17	3.17	NUM
cana-3308	265	18	)	)	PUNCT
cana-3308	265	19	,	,	PUNCT
cana-3308	265	20	using	use	VERB
cana-3308	265	21	equations	equation	NOUN
cana-3308	265	22	(	(	PUNCT
cana-3308	265	23	3.1	3.1	NUM
cana-3308	265	24	)	)	PUNCT
cana-3308	265	25	,	,	PUNCT
cana-3308	265	26	(	(	PUNCT
cana-3308	265	27	3.2	3.2	NUM
cana-3308	265	28	)	)	PUNCT
cana-3308	265	29	and	and	CCONJ
cana-3308	265	30	by	by	ADP
cana-3308	265	31	the	the	DET
cana-3308	265	32	properties	property	NOUN
cana-3308	265	33	of	of	ADP
cana-3308	265	34	𝜓	𝜓	PROPN
cana-3308	265	35	and	and	CCONJ
cana-3308	265	36	𝜃	𝜃	X
cana-3308	265	37	,	,	PUNCT
cana-3308	265	38	we	we	PRON
cana-3308	265	39	have	have	VERB
cana-3308	265	40	the	the	DET
cana-3308	265	41	following	follow	VERB
cana-3308	265	42	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	265	43	,	,	PUNCT
cana-3308	265	44	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	265	45	)	)	PUNCT
cana-3308	265	46	)	)	PUNCT
cana-3308	266	1	=	=	SYM
cana-3308	266	2	𝜓(𝑑(𝑇𝑦𝑛	𝜓(𝑑(𝑇𝑦𝑛	NOUN
cana-3308	266	3	,	,	PUNCT
cana-3308	266	4	𝑇𝑥𝑛+1	𝑇𝑥𝑛+1	PROPN
cana-3308	266	5	)	)	PUNCT
cana-3308	266	6	)	)	PUNCT
cana-3308	266	7	≤	≤	NUM
cana-3308	266	8	𝛼(𝑥𝑛+1	𝛼(𝑥𝑛+1	NUM
cana-3308	266	9	,	,	PUNCT
cana-3308	266	10	𝑦𝑛)𝜓(𝑑(𝑇𝑦𝑛	𝑦𝑛)𝜓(𝑑(𝑇𝑦𝑛	PROPN
cana-3308	266	11	,	,	PUNCT
cana-3308	266	12	𝑇𝑥𝑛+1	𝑇𝑥𝑛+1	PROPN
cana-3308	266	13	)	)	PUNCT
cana-3308	266	14	)	)	PUNCT
cana-3308	267	1	≤	≤	NUM
cana-3308	267	2	𝜃	𝜃	X
cana-3308	267	3	(	(	PUNCT
cana-3308	267	4	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NOUN
cana-3308	267	5	,	,	PUNCT
cana-3308	267	6	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	267	7	)	)	PUNCT
cana-3308	267	8	)	)	PUNCT
cana-3308	267	9	)	)	PUNCT
cana-3308	267	10	𝜑(𝜓(𝑑(𝑥𝑛+1	𝜑(𝜓(𝑑(𝑥𝑛+1	ADJ
cana-3308	267	11	,	,	PUNCT
cana-3308	267	12	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	267	13	)	)	PUNCT
cana-3308	267	14	)	)	PUNCT
cana-3308	267	15	)	)	PUNCT
cana-3308	267	16	≤	≤	ADV
cana-3308	267	17	𝜑(𝜓(𝑑(𝑥𝑛+1	𝜑(𝜓(𝑑(𝑥𝑛+1	ADV
cana-3308	267	18	,	,	PUNCT
cana-3308	267	19	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	267	20	)	)	PUNCT
cana-3308	267	21	)	)	PUNCT
cana-3308	267	22	)	)	PUNCT
cana-3308	268	1	<	<	X
cana-3308	268	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	268	3	,	,	PUNCT
cana-3308	268	4	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	268	5	)	)	PUNCT
cana-3308	268	6	)	)	PUNCT
cana-3308	268	7	.	.	PUNCT
cana-3308	269	1	(	(	PUNCT
cana-3308	269	2	3.21	3.21	NUM
cana-3308	269	3	)	)	PUNCT
cana-3308	269	4	hence	hence	ADV
cana-3308	269	5	,	,	PUNCT
cana-3308	269	6	𝜓	𝜓	PROPN
cana-3308	269	7	is	be	AUX
cana-3308	269	8	strictly	strictly	ADV
cana-3308	269	9	increasing	increase	VERB
cana-3308	269	10	function	function	NOUN
cana-3308	269	11	so	so	ADV
cana-3308	269	12	,	,	PUNCT
cana-3308	269	13	we	we	PRON
cana-3308	269	14	obtain	obtain	VERB
cana-3308	269	15	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	269	16	,	,	PUNCT
cana-3308	269	17	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	269	18	)	)	PUNCT
cana-3308	269	19	<	<	X
cana-3308	269	20	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	269	21	,	,	PUNCT
cana-3308	269	22	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	269	23	)	)	PUNCT
cana-3308	269	24	for	for	ADP
cana-3308	269	25	all	all	DET
cana-3308	269	26	𝑛	𝑛	DET
cana-3308	269	27	≥	≥	NOUN
cana-3308	269	28	0	0	NUM
cana-3308	269	29	.	.	PUNCT
cana-3308	270	1	(	(	PUNCT
cana-3308	270	2	3.22	3.22	NUM
cana-3308	270	3	)	)	PUNCT
cana-3308	270	4	from	from	ADP
cana-3308	270	5	the	the	DET
cana-3308	270	6	above	above	NOUN
cana-3308	270	7	,	,	PUNCT
cana-3308	270	8	we	we	PRON
cana-3308	270	9	conclude	conclude	VERB
cana-3308	270	10	that	that	SCONJ
cana-3308	270	11	the	the	DET
cana-3308	270	12	sequences	sequence	NOUN
cana-3308	270	13	{	{	PUNCT
cana-3308	270	14	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	270	15	,	,	PUNCT
cana-3308	270	16	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	270	17	)	)	PUNCT
cana-3308	270	18	}	}	PUNCT
cana-3308	270	19	and	and	CCONJ
cana-3308	270	20	{	{	PUNCT
cana-3308	270	21	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	PROPN
cana-3308	270	22	,	,	PUNCT
cana-3308	270	23	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	270	24	)	)	PUNCT
cana-3308	270	25	}	}	PUNCT
cana-3308	270	26	are	be	AUX
cana-3308	270	27	monotonically	monotonically	ADV
cana-3308	270	28	decreasing	decrease	VERB
cana-3308	270	29	and	and	CCONJ
cana-3308	270	30	for	for	ADP
cana-3308	270	31	the	the	DET
cana-3308	270	32	non	non	ADJ
cana-3308	270	33	-	-	ADJ
cana-3308	270	34	negative	negative	ADJ
cana-3308	270	35	monotonically	monotonically	ADV
cana-3308	270	36	decreasing	decrease	VERB
cana-3308	270	37	sequences	sequence	NOUN
cana-3308	270	38	{	{	PUNCT
cana-3308	270	39	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	270	40	,	,	PUNCT
cana-3308	270	41	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	270	42	)	)	PUNCT
cana-3308	270	43	}	}	PUNCT
cana-3308	270	44	and	and	CCONJ
cana-3308	270	45	{	{	PUNCT
cana-3308	270	46	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	PROPN
cana-3308	270	47	,	,	PUNCT
cana-3308	270	48	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	270	49	)	)	PUNCT
cana-3308	270	50	}	}	PUNCT
cana-3308	270	51	,	,	PUNCT
cana-3308	270	52	there	there	PRON
cana-3308	270	53	exist	exist	VERB
cana-3308	270	54	some	some	DET
cana-3308	270	55	𝑟1	𝑟1	NOUN
cana-3308	270	56	≥	≥	NOUN
cana-3308	270	57	0	0	NUM
cana-3308	270	58	and	and	CCONJ
cana-3308	270	59	𝑟2	𝑟2	NOUN
cana-3308	270	60	≥	≥	NUM
cana-3308	270	61	0	0	NUM
cana-3308	270	62	,	,	PUNCT
cana-3308	270	63	such	such	ADJ
cana-3308	270	64	that	that	DET
cana-3308	270	65	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	270	66	,	,	PUNCT
cana-3308	270	67	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	270	68	)	)	PUNCT
cana-3308	270	69	→	→	SYM
cana-3308	270	70	𝑟1	𝑟1	NOUN
cana-3308	270	71	,	,	PUNCT
cana-3308	270	72	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	270	73	,	,	PUNCT
cana-3308	270	74	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	270	75	)	)	PUNCT
cana-3308	270	76	→	→	SYM
cana-3308	270	77	𝑟2	𝑟2	NOUN
cana-3308	270	78	as	as	ADP
cana-3308	270	79	𝑛	𝑛	PROPN
cana-3308	270	80	→	→	SYM
cana-3308	270	81	∞.	∞.	PROPN
cana-3308	270	82	(	(	PUNCT
cana-3308	270	83	3.23	3.23	NUM
cana-3308	270	84	)	)	PUNCT
cana-3308	270	85	we	we	PRON
cana-3308	270	86	suppose	suppose	VERB
cana-3308	270	87	on	on	ADP
cana-3308	270	88	the	the	DET
cana-3308	270	89	contrary	contrary	NOUN
cana-3308	270	90	that	that	SCONJ
cana-3308	270	91	𝑟1	𝑟1	PROPN
cana-3308	270	92	>	>	X
cana-3308	270	93	0	0	X
cana-3308	270	94	.	.	PUNCT
cana-3308	271	1	hence	hence	ADV
cana-3308	271	2	,	,	PUNCT
cana-3308	271	3	we	we	PRON
cana-3308	271	4	have	have	VERB
cana-3308	271	5	0	0	NUM
cana-3308	271	6	<	<	X
cana-3308	271	7	𝑟1	𝑟1	PROPN
cana-3308	271	8	<	<	X
cana-3308	271	9	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	PROPN
cana-3308	271	10	,	,	PUNCT
cana-3308	271	11	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	271	12	)	)	PUNCT
cana-3308	271	13	for	for	ADP
cana-3308	271	14	all	all	DET
cana-3308	271	15	𝑛	𝑛	DET
cana-3308	271	16	≥	≥	NOUN
cana-3308	271	17	0	0	NUM
cana-3308	271	18	.	.	PUNCT
cana-3308	272	1	set	set	VERB
cana-3308	272	2	휀	휀	PRON
cana-3308	272	3	=	=	NOUN
cana-3308	272	4	𝑟1	𝑟1	PROPN
cana-3308	272	5	.	.	PUNCT
cana-3308	273	1	from	from	ADP
cana-3308	273	2	equation	equation	NOUN
cana-3308	273	3	(	(	PUNCT
cana-3308	273	4	3.2	3.2	NUM
cana-3308	273	5	)	)	PUNCT
cana-3308	273	6	,	,	PUNCT
cana-3308	273	7	there	there	PRON
cana-3308	273	8	exist	exist	VERB
cana-3308	273	9	𝛿	𝛿	PROPN
cana-3308	273	10	>	>	X
cana-3308	273	11	0	0	NUM
cana-3308	273	12	such	such	ADJ
cana-3308	273	13	that	that	SCONJ
cana-3308	273	14	휀	휀	X
cana-3308	273	15	<	<	X
cana-3308	273	16	𝑡	𝑡	X
cana-3308	273	17	<	<	X
cana-3308	273	18	휀	휀	NOUN
cana-3308	273	19	+	+	X
cana-3308	273	20	𝛿	𝛿	ADJ
cana-3308	273	21	⇒	⇒	NOUN
cana-3308	273	22	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	273	23	)	)	PUNCT
cana-3308	273	24	≤	≤	NUM
cana-3308	274	1	휀	휀	NOUN
cana-3308	274	2	.	.	PUNCT
cana-3308	274	3	communications	communication	NOUN
cana-3308	274	4	on	on	ADP
cana-3308	274	5	applied	apply	VERB
cana-3308	274	6	nonlinear	nonlinear	ADJ
cana-3308	274	7	analysis	analysis	NOUN
cana-3308	274	8	issn	issn	NOUN
cana-3308	274	9	:	:	PUNCT
cana-3308	274	10	1074	1074	NUM
cana-3308	274	11	-	-	PUNCT
cana-3308	274	12	133x	133x	NUM
cana-3308	274	13	vol	vol	NOUN
cana-3308	274	14	32	32	NUM
cana-3308	274	15	no	no	NOUN
cana-3308	274	16	.	.	PUNCT
cana-3308	275	1	6s	6s	NUM
cana-3308	275	2	(	(	PUNCT
cana-3308	275	3	2025	2025	NUM
cana-3308	275	4	)	)	PUNCT
cana-3308	275	5	449	449	NUM
cana-3308	275	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	275	7	on	on	ADP
cana-3308	275	8	the	the	DET
cana-3308	275	9	other	other	ADJ
cana-3308	275	10	hand	hand	NOUN
cana-3308	275	11	,	,	PUNCT
cana-3308	275	12	by	by	ADP
cana-3308	275	13	the	the	DET
cana-3308	275	14	definition	definition	NOUN
cana-3308	275	15	of	of	ADP
cana-3308	275	16	휀	휀	NOUN
cana-3308	275	17	,	,	PUNCT
cana-3308	275	18	we	we	PRON
cana-3308	275	19	can	can	AUX
cana-3308	275	20	choose	choose	VERB
cana-3308	275	21	𝑛0	𝑛0	VERB
cana-3308	275	22	∈	∈	PROPN
cana-3308	275	23	ℕ	ℕ	PROPN
cana-3308	275	24	such	such	ADJ
cana-3308	275	25	that	that	SCONJ
cana-3308	275	26	휀	휀	DET
cana-3308	275	27	<	<	X
cana-3308	275	28	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	275	29	+	+	PROPN
cana-3308	275	30	1	1	NUM
cana-3308	275	31	,	,	PUNCT
cana-3308	275	32	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	275	33	)	)	PUNCT
cana-3308	275	34	<	<	X
cana-3308	276	1	휀	휀	X
cana-3308	276	2	+	+	X
cana-3308	276	3	𝛿.	𝛿.	NOUN
cana-3308	276	4	by	by	ADP
cana-3308	276	5	the	the	DET
cana-3308	276	6	properties	property	NOUN
cana-3308	276	7	of	of	ADP
cana-3308	276	8	𝜓	𝜓	NOUN
cana-3308	276	9	,	,	PUNCT
cana-3308	276	10	𝜃,using	𝜃,use	VERB
cana-3308	276	11	equations	equation	NOUN
cana-3308	276	12	(	(	PUNCT
cana-3308	276	13	3.1	3.1	NUM
cana-3308	276	14	)	)	PUNCT
cana-3308	276	15	,	,	PUNCT
cana-3308	276	16	(	(	PUNCT
cana-3308	276	17	3.2	3.2	NUM
cana-3308	276	18	)	)	PUNCT
cana-3308	276	19	and	and	CCONJ
cana-3308	276	20	(	(	PUNCT
cana-3308	276	21	3.17	3.17	NUM
cana-3308	276	22	)	)	PUNCT
cana-3308	276	23	,	,	PUNCT
cana-3308	276	24	we	we	PRON
cana-3308	276	25	have	have	VERB
cana-3308	276	26	𝜓(휀	𝜓(휀	NUM
cana-3308	276	27	)	)	PUNCT
cana-3308	276	28	<	<	X
cana-3308	276	29	𝜓	𝜓	PROPN
cana-3308	276	30	(	(	PUNCT
cana-3308	276	31	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	276	32	+	+	NOUN
cana-3308	276	33	1	1	NUM
cana-3308	276	34	,	,	PUNCT
cana-3308	276	35	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	276	36	)	)	PUNCT
cana-3308	276	37	)	)	PUNCT
cana-3308	277	1	<	<	X
cana-3308	277	2	𝜓(휀	𝜓(휀	PROPN
cana-3308	277	3	+	+	PUNCT
cana-3308	277	4	𝛿	𝛿	X
cana-3308	277	5	)	)	PUNCT
cana-3308	277	6	=	=	SYM
cana-3308	277	7	𝜓(휀	𝜓(휀	NUM
cana-3308	277	8	)	)	PUNCT
cana-3308	278	1	+	+	CCONJ
cana-3308	279	1	𝜓(𝛿	𝜓(𝛿	PROPN
cana-3308	279	2	)	)	PUNCT
cana-3308	279	3	.	.	PUNCT
cana-3308	280	1	this	this	PRON
cana-3308	280	2	implies	imply	VERB
cana-3308	280	3	that	that	SCONJ
cana-3308	280	4	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	280	5	(	(	PUNCT
cana-3308	280	6	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	280	7	+	+	NOUN
cana-3308	280	8	1	1	NUM
cana-3308	280	9	,	,	PUNCT
cana-3308	280	10	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	280	11	)	)	PUNCT
cana-3308	280	12	)	)	PUNCT
cana-3308	280	13	)	)	PUNCT
cana-3308	280	14	≤	≤	NUM
cana-3308	280	15	𝜓(휀	𝜓(휀	NUM
cana-3308	280	16	)	)	PUNCT
cana-3308	280	17	.	.	PUNCT
cana-3308	281	1	(	(	PUNCT
cana-3308	281	2	3.24	3.24	NUM
cana-3308	281	3	)	)	PUNCT
cana-3308	281	4	we	we	PRON
cana-3308	281	5	have	have	VERB
cana-3308	281	6	also	also	ADV
cana-3308	281	7	휀	휀	X
cana-3308	281	8	<	<	X
cana-3308	281	9	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	281	10	+	+	PROPN
cana-3308	281	11	2	2	NUM
cana-3308	281	12	,	,	PUNCT
cana-3308	281	13	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	281	14	+	+	NOUN
cana-3308	281	15	1	1	NUM
cana-3308	281	16	)	)	PUNCT
cana-3308	281	17	<	<	X
cana-3308	282	1	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	X
cana-3308	282	2	+	+	PROPN
cana-3308	282	3	1	1	NUM
cana-3308	282	4	,	,	PUNCT
cana-3308	282	5	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	282	6	+	+	NOUN
cana-3308	282	7	1	1	NUM
cana-3308	282	8	)	)	PUNCT
cana-3308	282	9	=	=	SYM
cana-3308	282	10	𝑑(𝑇𝑦𝑛0	𝑑(𝑇𝑦𝑛0	PROPN
cana-3308	282	11	,	,	PUNCT
cana-3308	282	12	𝑇𝑥𝑛0	𝑇𝑥𝑛0	NUM
cana-3308	282	13	+	+	NOUN
cana-3308	282	14	1	1	NUM
cana-3308	282	15	)	)	PUNCT
cana-3308	282	16	,	,	PUNCT
cana-3308	282	17	which	which	PRON
cana-3308	282	18	implies	imply	VERB
cana-3308	282	19	that	that	SCONJ
cana-3308	282	20	𝜓(휀	𝜓(휀	NUM
cana-3308	282	21	)	)	PUNCT
cana-3308	282	22	<	<	X
cana-3308	282	23	𝜓(𝑑(𝑥𝑛0	𝜓(𝑑(𝑥𝑛0	PROPN
cana-3308	282	24	+	+	PROPN
cana-3308	282	25	2	2	NUM
cana-3308	282	26	,	,	PUNCT
cana-3308	282	27	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	282	28	+	+	NOUN
cana-3308	282	29	1	1	NUM
cana-3308	282	30	)	)	PUNCT
cana-3308	282	31	)	)	PUNCT
cana-3308	282	32	<	<	X
cana-3308	282	33	𝜓(𝑑(𝑥𝑛0	𝜓(𝑑(𝑥𝑛0	PROPN
cana-3308	282	34	+	+	PROPN
cana-3308	282	35	1	1	NUM
cana-3308	282	36	,	,	PUNCT
cana-3308	282	37	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	282	38	+	+	NOUN
cana-3308	282	39	1	1	NUM
cana-3308	282	40	)	)	PUNCT
cana-3308	282	41	)	)	PUNCT
cana-3308	283	1	=	=	SYM
cana-3308	283	2	𝜓(𝑑(𝑇𝑦𝑛0	𝜓(𝑑(𝑇𝑦𝑛0	PROPN
cana-3308	283	3	,	,	PUNCT
cana-3308	283	4	𝑇𝑥𝑛0	𝑇𝑥𝑛0	PROPN
cana-3308	283	5	+	+	NOUN
cana-3308	283	6	1	1	NUM
cana-3308	283	7	)	)	PUNCT
cana-3308	283	8	)	)	PUNCT
cana-3308	283	9	≤	≤	PUNCT
cana-3308	284	1	𝛼(𝑥𝑛0	𝛼(𝑥𝑛0	PROPN
cana-3308	285	1	+	+	PROPN
cana-3308	285	2	1	1	NUM
cana-3308	285	3	,	,	PUNCT
cana-3308	285	4	𝑦𝑛0	𝑦𝑛0	X
cana-3308	285	5	)	)	PUNCT
cana-3308	286	1	𝜓(𝑑(𝑇𝑦𝑛0	𝜓(𝑑(𝑇𝑦𝑛0	PROPN
cana-3308	286	2	,	,	PUNCT
cana-3308	286	3	𝑇𝑥𝑛0	𝑇𝑥𝑛0	PROPN
cana-3308	286	4	+	+	NOUN
cana-3308	286	5	1	1	NUM
cana-3308	286	6	)	)	PUNCT
cana-3308	286	7	)	)	PUNCT
cana-3308	287	1	≤	≤	NUM
cana-3308	287	2	𝜃	𝜃	X
cana-3308	287	3	(	(	PUNCT
cana-3308	287	4	𝜓	𝜓	PROPN
cana-3308	287	5	(	(	PUNCT
cana-3308	287	6	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	287	7	+	+	NOUN
cana-3308	287	8	1	1	NUM
cana-3308	287	9	,	,	PUNCT
cana-3308	287	10	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	287	11	)	)	PUNCT
cana-3308	287	12	)	)	PUNCT
cana-3308	287	13	)	)	PUNCT
cana-3308	287	14	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	287	15	(	(	PUNCT
cana-3308	287	16	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	287	17	+	+	NOUN
cana-3308	287	18	1	1	NUM
cana-3308	287	19	,	,	PUNCT
cana-3308	287	20	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	287	21	)	)	PUNCT
cana-3308	287	22	)	)	PUNCT
cana-3308	287	23	)	)	PUNCT
cana-3308	288	1	<	<	X
cana-3308	288	2	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	288	3	(	(	PUNCT
cana-3308	288	4	𝑑(𝑥𝑛0	𝑑(𝑥𝑛0	NOUN
cana-3308	288	5	+	+	NOUN
cana-3308	288	6	1	1	NUM
cana-3308	288	7	,	,	PUNCT
cana-3308	288	8	𝑦𝑛0	𝑦𝑛0	NOUN
cana-3308	288	9	)	)	PUNCT
cana-3308	288	10	)	)	PUNCT
cana-3308	288	11	)	)	PUNCT
cana-3308	288	12	≤	≤	NUM
cana-3308	288	13	𝜓	𝜓	NOUN
cana-3308	288	14	(	(	PUNCT
cana-3308	288	15	휀	휀	NOUN
cana-3308	288	16	)	)	PUNCT
cana-3308	288	17	,	,	PUNCT
cana-3308	288	18	which	which	PRON
cana-3308	288	19	is	be	AUX
cana-3308	288	20	a	a	DET
cana-3308	288	21	contradiction	contradiction	NOUN
cana-3308	288	22	.	.	PUNCT
cana-3308	289	1	hence	hence	ADV
cana-3308	289	2	lim	lim	PROPN
cana-3308	289	3	𝑛→∞	𝑛→∞	NUM
cana-3308	289	4	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	289	5	,	,	PUNCT
cana-3308	289	6	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	289	7	)	)	PUNCT
cana-3308	289	8	=	=	SYM
cana-3308	289	9	𝑟1	𝑟1	NOUN
cana-3308	289	10	=	=	SYM
cana-3308	289	11	0	0	X
cana-3308	289	12	.	.	PUNCT
cana-3308	290	1	(	(	PUNCT
cana-3308	290	2	3.25	3.25	NUM
cana-3308	290	3	)	)	PUNCT
cana-3308	290	4	similarly	similarly	ADV
cana-3308	290	5	,	,	PUNCT
cana-3308	290	6	lim	lim	PROPN
cana-3308	290	7	𝑛→∞	𝑛→∞	NUM
cana-3308	290	8	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	290	9	,	,	PUNCT
cana-3308	290	10	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	290	11	)	)	PUNCT
cana-3308	291	1	=	=	NOUN
cana-3308	291	2	𝑟2	𝑟2	NOUN
cana-3308	291	3	=	=	SYM
cana-3308	291	4	0	0	NUM
cana-3308	291	5	.	.	PUNCT
cana-3308	292	1	(	(	PUNCT
cana-3308	292	2	3.26	3.26	NUM
cana-3308	292	3	)	)	PUNCT
cana-3308	292	4	now	now	ADV
cana-3308	292	5	,	,	PUNCT
cana-3308	292	6	we	we	PRON
cana-3308	292	7	shall	shall	AUX
cana-3308	292	8	prove	prove	VERB
cana-3308	292	9	that	that	SCONJ
cana-3308	292	10	{	{	PUNCT
cana-3308	292	11	(	(	PUNCT
cana-3308	292	12	𝑥𝑛	𝑥𝑛	INTJ
cana-3308	292	13	,	,	PUNCT
cana-3308	292	14	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	292	15	)	)	PUNCT
cana-3308	292	16	}	}	PUNCT
cana-3308	292	17	is	be	AUX
cana-3308	292	18	a	a	DET
cana-3308	292	19	cauchy	cauchy	ADJ
cana-3308	292	20	bisequence	bisequence	NOUN
cana-3308	292	21	.	.	PUNCT
cana-3308	293	1	we	we	PRON
cana-3308	293	2	fix	fix	VERB
cana-3308	293	3	휀1	휀1	NOUN
cana-3308	293	4	>	>	X
cana-3308	293	5	0	0	NUM
cana-3308	293	6	,	,	PUNCT
cana-3308	293	7	then	then	ADV
cana-3308	293	8	by	by	ADP
cana-3308	293	9	(	(	PUNCT
cana-3308	293	10	3.2	3.2	NUM
cana-3308	293	11	)	)	PUNCT
cana-3308	293	12	there	there	PRON
cana-3308	293	13	exists	exist	VERB
cana-3308	293	14	𝛿1	𝛿1	NOUN
cana-3308	293	15	>	>	X
cana-3308	293	16	0	0	NUM
cana-3308	294	1	such	such	ADJ
cana-3308	294	2	that	that	SCONJ
cana-3308	294	3	𝑡	𝑡	ADP
cana-3308	294	4	<	<	X
cana-3308	294	5	휀1	휀1	NOUN
cana-3308	294	6	+	+	CCONJ
cana-3308	294	7	𝛿1	𝛿1	ADJ
cana-3308	294	8	⇒	⇒	NOUN
cana-3308	294	9	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	294	10	)	)	PUNCT
cana-3308	294	11	≤	≤	NOUN
cana-3308	294	12	휀1	휀1	NOUN
cana-3308	294	13	.	.	PUNCT
cana-3308	295	1	(	(	PUNCT
cana-3308	295	2	3.27	3.27	NUM
cana-3308	295	3	)	)	PUNCT
cana-3308	295	4	without	without	ADP
cana-3308	295	5	loss	loss	NOUN
cana-3308	295	6	of	of	ADP
cana-3308	295	7	generality	generality	NOUN
cana-3308	295	8	,	,	PUNCT
cana-3308	295	9	we	we	PRON
cana-3308	295	10	assume	assume	VERB
cana-3308	295	11	𝛿1	𝛿1	PROPN
cana-3308	295	12	<	<	X
cana-3308	295	13	휀1	휀1	NOUN
cana-3308	295	14	.	.	PUNCT
cana-3308	296	1	due	due	ADP
cana-3308	296	2	to	to	ADP
cana-3308	296	3	(	(	PUNCT
cana-3308	296	4	3.25	3.25	NUM
cana-3308	296	5	)	)	PUNCT
cana-3308	296	6	,	,	PUNCT
cana-3308	296	7	there	there	PRON
cana-3308	296	8	exist	exist	VERB
cana-3308	296	9	𝑛0	𝑛0	VERB
cana-3308	296	10	∈	∈	PROPN
cana-3308	296	11	ℕ	ℕ	PROPN
cana-3308	296	12	such	such	ADJ
cana-3308	296	13	that	that	DET
cana-3308	296	14	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	296	15	,	,	PUNCT
cana-3308	296	16	𝑦𝑛	𝑦𝑛	ADP
cana-3308	296	17	)	)	PUNCT
cana-3308	296	18	<	<	X
cana-3308	296	19	𝛿1	𝛿1	PROPN
cana-3308	296	20	,	,	PUNCT
cana-3308	296	21	for	for	ADP
cana-3308	296	22	all	all	DET
cana-3308	296	23	𝑛	𝑛	DET
cana-3308	296	24	≥	≥	NOUN
cana-3308	296	25	𝑛0	𝑛0	VERB
cana-3308	296	26	,	,	PUNCT
cana-3308	296	27	(	(	PUNCT
cana-3308	296	28	3.28	3.28	NUM
cana-3308	296	29	)	)	PUNCT
cana-3308	296	30	which	which	PRON
cana-3308	296	31	implies	imply	VERB
cana-3308	296	32	that	that	SCONJ
cana-3308	296	33	𝜓	𝜓	PROPN
cana-3308	296	34	(	(	PUNCT
cana-3308	296	35	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	296	36	,	,	PUNCT
cana-3308	296	37	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	296	38	)	)	PUNCT
cana-3308	296	39	)	)	PUNCT
cana-3308	296	40	<	<	X
cana-3308	296	41	𝜓(𝛿1	𝜓(𝛿1	NOUN
cana-3308	296	42	)	)	PUNCT
cana-3308	296	43	.	.	PUNCT
cana-3308	297	1	by	by	ADP
cana-3308	297	2	mathematical	mathematical	ADJ
cana-3308	297	3	induction	induction	NOUN
cana-3308	297	4	,	,	PUNCT
cana-3308	297	5	we	we	PRON
cana-3308	297	6	show	show	VERB
cana-3308	297	7	that	that	SCONJ
cana-3308	297	8	for	for	ADP
cana-3308	297	9	any	any	DET
cana-3308	297	10	fixed	fix	VERB
cana-3308	297	11	𝑘	𝑘	PRON
cana-3308	297	12	≥	≥	NOUN
cana-3308	297	13	𝑛0	𝑛0	VERB
cana-3308	297	14	𝑑(𝑥𝑘+𝑙	𝑑(𝑥𝑘+𝑙	PROPN
cana-3308	297	15	,	,	PUNCT
cana-3308	297	16	𝑦𝑘	𝑦𝑘	PROPN
cana-3308	297	17	)	)	PUNCT
cana-3308	297	18	<	<	X
cana-3308	297	19	휀1	휀1	PROPN
cana-3308	297	20	+	+	CCONJ
cana-3308	297	21	𝛿1	𝛿1	NOUN
cana-3308	297	22	,	,	PUNCT
cana-3308	297	23	for	for	ADP
cana-3308	297	24	all	all	DET
cana-3308	297	25	𝑙	𝑙	DET
cana-3308	297	26	∈	∈	PROPN
cana-3308	297	27	ℕ.	ℕ.	PROPN
cana-3308	297	28	(	(	PUNCT
cana-3308	297	29	3.29	3.29	NUM
cana-3308	297	30	)	)	PUNCT
cana-3308	297	31	for	for	ADP
cana-3308	297	32	𝑙	𝑙	NOUN
cana-3308	297	33	=	=	SYM
cana-3308	297	34	1	1	NUM
cana-3308	297	35	,	,	PUNCT
cana-3308	297	36	this	this	DET
cana-3308	297	37	inequality	inequality	NOUN
cana-3308	297	38	trivially	trivially	ADV
cana-3308	297	39	holds	hold	VERB
cana-3308	297	40	by	by	ADP
cana-3308	297	41	(	(	PUNCT
cana-3308	297	42	3.28	3.28	NUM
cana-3308	297	43	)	)	PUNCT
cana-3308	297	44	.	.	PUNCT
cana-3308	298	1	communications	communication	NOUN
cana-3308	298	2	on	on	ADP
cana-3308	298	3	applied	apply	VERB
cana-3308	298	4	nonlinear	nonlinear	ADJ
cana-3308	298	5	analysis	analysis	NOUN
cana-3308	298	6	issn	issn	NOUN
cana-3308	298	7	:	:	PUNCT
cana-3308	298	8	1074	1074	NUM
cana-3308	298	9	-	-	PUNCT
cana-3308	298	10	133x	133x	NUM
cana-3308	298	11	vol	vol	NOUN
cana-3308	298	12	32	32	NUM
cana-3308	298	13	no	no	NOUN
cana-3308	298	14	.	.	PUNCT
cana-3308	299	1	6s	6s	NUM
cana-3308	299	2	(	(	PUNCT
cana-3308	299	3	2025	2025	NUM
cana-3308	299	4	)	)	PUNCT
cana-3308	299	5	450	450	NUM
cana-3308	299	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	299	7	let	let	VERB
cana-3308	299	8	us	we	PRON
cana-3308	299	9	assume	assume	VERB
cana-3308	299	10	that	that	SCONJ
cana-3308	299	11	(	(	PUNCT
cana-3308	299	12	3.29	3.29	NUM
cana-3308	299	13	)	)	PUNCT
cana-3308	299	14	is	be	AUX
cana-3308	299	15	satisfied	satisfied	ADJ
cana-3308	299	16	for	for	ADP
cana-3308	299	17	some	some	DET
cana-3308	299	18	𝑗	𝑗	PRON
cana-3308	299	19	∈	∈	PROPN
cana-3308	299	20	ℕ	ℕ	PROPN
cana-3308	299	21	and	and	CCONJ
cana-3308	299	22	we	we	PRON
cana-3308	299	23	will	will	AUX
cana-3308	299	24	show	show	VERB
cana-3308	299	25	that	that	SCONJ
cana-3308	299	26	it	it	PRON
cana-3308	299	27	holds	hold	VERB
cana-3308	299	28	for	for	ADP
cana-3308	299	29	𝑙	𝑙	PRON
cana-3308	299	30	=	=	SYM
cana-3308	299	31	𝑗	𝑗	PROPN
cana-3308	300	1	+	+	NOUN
cana-3308	300	2	1	1	NUM
cana-3308	300	3	.	.	PUNCT
cana-3308	300	4	from	from	ADP
cana-3308	300	5	the	the	DET
cana-3308	300	6	triangle	triangle	NOUN
cana-3308	300	7	inequality	inequality	NOUN
cana-3308	300	8	(	(	PUNCT
cana-3308	300	9	bp3	bp3	NOUN
cana-3308	300	10	)	)	PUNCT
cana-3308	300	11	,	,	PUNCT
cana-3308	300	12	properties	property	NOUN
cana-3308	300	13	of	of	ADP
cana-3308	300	14	𝜓	𝜓	PROPN
cana-3308	300	15	,	,	PUNCT
cana-3308	300	16	𝜃	𝜃	PROPN
cana-3308	300	17	,	,	PUNCT
cana-3308	300	18	equations	equation	NOUN
cana-3308	300	19	(	(	PUNCT
cana-3308	300	20	3.1	3.1	NUM
cana-3308	300	21	)	)	PUNCT
cana-3308	300	22	,	,	PUNCT
cana-3308	300	23	(	(	PUNCT
cana-3308	300	24	3.2	3.2	NUM
cana-3308	300	25	)	)	PUNCT
cana-3308	300	26	and	and	CCONJ
cana-3308	300	27	(	(	PUNCT
cana-3308	300	28	3.3	3.3	NUM
cana-3308	300	29	)	)	PUNCT
cana-3308	300	30	𝜓	𝜓	PROPN
cana-3308	300	31	(	(	PUNCT
cana-3308	300	32	𝑑(𝑥𝑘+𝑗+1	𝑑(𝑥𝑘+𝑗+1	PROPN
cana-3308	300	33	,	,	PUNCT
cana-3308	300	34	𝑦𝑘	𝑦𝑘	NOUN
cana-3308	300	35	)	)	PUNCT
cana-3308	300	36	)	)	PUNCT
cana-3308	300	37	≤	≤	NUM
cana-3308	300	38	𝜓	𝜓	PROPN
cana-3308	300	39	(	(	PUNCT
cana-3308	300	40	𝑑(𝑥𝑘+𝑗+1	𝑑(𝑥𝑘+𝑗+1	PROPN
cana-3308	300	41	,	,	PUNCT
cana-3308	300	42	𝑦𝑘+1	𝑦𝑘+1	NOUN
cana-3308	300	43	)	)	PUNCT
cana-3308	300	44	+	+	CCONJ
cana-3308	300	45	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	NUM
cana-3308	300	46	,	,	PUNCT
cana-3308	300	47	𝑦𝑘+1	𝑦𝑘+1	X
cana-3308	300	48	)	)	PUNCT
cana-3308	301	1	+	+	CCONJ
cana-3308	301	2	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	NUM
cana-3308	301	3	,	,	PUNCT
cana-3308	301	4	𝑦𝑘	𝑦𝑘	NOUN
cana-3308	301	5	)	)	PUNCT
cana-3308	301	6	)	)	PUNCT
cana-3308	301	7	≤	≤	NUM
cana-3308	301	8	𝜓	𝜓	PROPN
cana-3308	301	9	(	(	PUNCT
cana-3308	301	10	𝑑(𝑥𝑘+𝑗+1	𝑑(𝑥𝑘+𝑗+1	PROPN
cana-3308	301	11	,	,	PUNCT
cana-3308	301	12	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	301	13	)	)	PUNCT
cana-3308	301	14	)	)	PUNCT
cana-3308	302	1	+	+	CCONJ
cana-3308	302	2	𝜓	𝜓	X
cana-3308	302	3	(	(	PUNCT
cana-3308	302	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	302	5	,	,	PUNCT
cana-3308	302	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	302	7	)	)	PUNCT
cana-3308	302	8	)	)	PUNCT
cana-3308	303	1	+	+	CCONJ
cana-3308	303	2	𝜓	𝜓	X
cana-3308	303	3	(	(	PUNCT
cana-3308	303	4	𝑑(𝑥𝑘+	𝑑(𝑥𝑘+	PROPN
cana-3308	303	5	1	1	NUM
cana-3308	303	6	,	,	PUNCT
cana-3308	303	7	𝑦𝑘	𝑦𝑘	INTJ
cana-3308	303	8	)	)	PUNCT
cana-3308	303	9	)	)	PUNCT
cana-3308	303	10	≤	≤	NUM
cana-3308	303	11	𝜓	𝜓	NOUN
cana-3308	303	12	(	(	PUNCT
cana-3308	303	13	𝑑(𝑇𝑦𝑘+𝑗	𝑑(𝑇𝑦𝑘+𝑗	NOUN
cana-3308	303	14	,	,	PUNCT
cana-3308	303	15	𝑇𝑥𝑘+1	𝑇𝑥𝑘+1	NOUN
cana-3308	303	16	)	)	PUNCT
cana-3308	303	17	)	)	PUNCT
cana-3308	304	1	+	+	CCONJ
cana-3308	304	2	𝜓	𝜓	X
cana-3308	304	3	(	(	PUNCT
cana-3308	304	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	304	5	,	,	PUNCT
cana-3308	304	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	304	7	)	)	PUNCT
cana-3308	304	8	)	)	PUNCT
cana-3308	305	1	+	+	CCONJ
cana-3308	305	2	𝜓	𝜓	X
cana-3308	305	3	(	(	PUNCT
cana-3308	305	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	305	5	,	,	PUNCT
cana-3308	305	6	𝑦𝑘	𝑦𝑘	NOUN
cana-3308	305	7	)	)	PUNCT
cana-3308	305	8	)	)	PUNCT
cana-3308	305	9	≤	≤	NUM
cana-3308	306	1	𝜓	𝜓	PROPN
cana-3308	306	2	(	(	PUNCT
cana-3308	306	3	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	NUM
cana-3308	306	4	,	,	PUNCT
cana-3308	306	5	𝑦𝑘	𝑦𝑘	NOUN
cana-3308	306	6	)	)	PUNCT
cana-3308	306	7	)	)	PUNCT
cana-3308	307	1	+	+	CCONJ
cana-3308	307	2	𝜓	𝜓	X
cana-3308	307	3	(	(	PUNCT
cana-3308	307	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	307	5	,	,	PUNCT
cana-3308	307	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	307	7	)	)	PUNCT
cana-3308	307	8	)	)	PUNCT
cana-3308	308	1	+	+	VERB
cana-3308	308	2	𝛼(𝑥𝑘+1	𝛼(𝑥𝑘+1	NOUN
cana-3308	308	3	,	,	PUNCT
cana-3308	308	4	𝑦𝑘+𝑗	𝑦𝑘+𝑗	NOUN
cana-3308	308	5	)	)	PUNCT
cana-3308	308	6	𝜓	𝜓	PROPN
cana-3308	308	7	(	(	PUNCT
cana-3308	308	8	𝑑(𝑇𝑦𝑘+𝑗	𝑑(𝑇𝑦𝑘+𝑗	NOUN
cana-3308	308	9	,	,	PUNCT
cana-3308	308	10	𝑇𝑥𝑘	𝑇𝑥𝑘	PROPN
cana-3308	308	11	)	)	PUNCT
cana-3308	308	12	)	)	PUNCT
cana-3308	308	13	≤	≤	NUM
cana-3308	309	1	𝜓	𝜓	PROPN
cana-3308	309	2	(	(	PUNCT
cana-3308	309	3	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	NUM
cana-3308	309	4	,	,	PUNCT
cana-3308	309	5	𝑦𝑘	𝑦𝑘	NOUN
cana-3308	309	6	)	)	PUNCT
cana-3308	309	7	)	)	PUNCT
cana-3308	310	1	+	+	CCONJ
cana-3308	310	2	𝜓	𝜓	X
cana-3308	310	3	(	(	PUNCT
cana-3308	310	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	310	5	,	,	PUNCT
cana-3308	310	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	310	7	)	)	PUNCT
cana-3308	310	8	)	)	PUNCT
cana-3308	311	1	+	+	ADV
cana-3308	311	2	𝜃(𝜓(𝑑(𝑥𝑘	𝜃(𝜓(𝑑(𝑥𝑘	ADJ
cana-3308	311	3	,	,	PUNCT
cana-3308	311	4	𝑦𝑘+𝑗	𝑦𝑘+𝑗	NUM
cana-3308	311	5	)	)	PUNCT
cana-3308	311	6	)	)	PUNCT
cana-3308	311	7	)	)	PUNCT
cana-3308	311	8	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	311	9	(	(	PUNCT
cana-3308	311	10	𝑑(𝑥𝑘	𝑑(𝑥𝑘	ADJ
cana-3308	311	11	,	,	PUNCT
cana-3308	311	12	𝑦𝑘+𝑗	𝑦𝑘+𝑗	NUM
cana-3308	311	13	)	)	PUNCT
cana-3308	311	14	)	)	PUNCT
cana-3308	311	15	)	)	PUNCT
cana-3308	311	16	<	<	X
cana-3308	311	17	𝜓	𝜓	PROPN
cana-3308	311	18	(	(	PUNCT
cana-3308	311	19	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	311	20	,	,	PUNCT
cana-3308	311	21	𝑦𝑘	𝑦𝑘	NOUN
cana-3308	311	22	)	)	PUNCT
cana-3308	311	23	)	)	PUNCT
cana-3308	312	1	+	+	CCONJ
cana-3308	312	2	𝜓	𝜓	X
cana-3308	312	3	(	(	PUNCT
cana-3308	312	4	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	312	5	,	,	PUNCT
cana-3308	312	6	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	312	7	)	)	PUNCT
cana-3308	312	8	)	)	PUNCT
cana-3308	313	1	+	+	CCONJ
cana-3308	313	2	𝜑(𝜓	𝜑(𝜓	PROPN
cana-3308	313	3	(	(	PUNCT
cana-3308	313	4	𝑑(𝑥𝑘	𝑑(𝑥𝑘	ADJ
cana-3308	313	5	,	,	PUNCT
cana-3308	313	6	𝑦𝑘+𝑗	𝑦𝑘+𝑗	NUM
cana-3308	313	7	)	)	PUNCT
cana-3308	313	8	)	)	PUNCT
cana-3308	313	9	)	)	PUNCT
cana-3308	313	10	.	.	PUNCT
cana-3308	314	1	using	use	VERB
cana-3308	314	2	equations	equation	NOUN
cana-3308	314	3	(	(	PUNCT
cana-3308	314	4	3.28	3.28	NUM
cana-3308	314	5	)	)	PUNCT
cana-3308	314	6	and	and	CCONJ
cana-3308	314	7	(	(	PUNCT
cana-3308	314	8	3.29	3.29	NUM
cana-3308	314	9	)	)	PUNCT
cana-3308	314	10	,	,	PUNCT
cana-3308	314	11	we	we	PRON
cana-3308	314	12	get	get	VERB
cana-3308	314	13	𝜓	𝜓	NOUN
cana-3308	314	14	(	(	PUNCT
cana-3308	314	15	𝑑(𝑥𝑘+𝑗+1	𝑑(𝑥𝑘+𝑗+1	NOUN
cana-3308	314	16	,	,	PUNCT
cana-3308	314	17	𝑦𝑘	𝑦𝑘	NOUN
cana-3308	314	18	)	)	PUNCT
cana-3308	314	19	)	)	PUNCT
cana-3308	314	20	≤	≤	NUM
cana-3308	315	1	𝜓	𝜓	PROPN
cana-3308	315	2	(	(	PUNCT
cana-3308	315	3	𝛿1	𝛿1	NOUN
cana-3308	315	4	)	)	PUNCT
cana-3308	315	5	+	+	X
cana-3308	315	6	𝜓	𝜓	X
cana-3308	315	7	(	(	PUNCT
cana-3308	315	8	𝑑(𝑥𝑘+1	𝑑(𝑥𝑘+1	PRON
cana-3308	315	9	,	,	PUNCT
cana-3308	315	10	𝑦𝑘+1	𝑦𝑘+1	NUM
cana-3308	315	11	)	)	PUNCT
cana-3308	315	12	)	)	PUNCT
cana-3308	316	1	+	+	CCONJ
cana-3308	316	2	𝜓	𝜓	PROPN
cana-3308	316	3	(	(	PUNCT
cana-3308	316	4	휀1	휀1	NOUN
cana-3308	316	5	)	)	PUNCT
cana-3308	316	6	.	.	PUNCT
cana-3308	317	1	by	by	ADP
cana-3308	317	2	letting	let	VERB
cana-3308	317	3	𝑘	𝑘	X
cana-3308	317	4	→	→	SYM
cana-3308	317	5	∞	∞	PROPN
cana-3308	317	6	,	,	PUNCT
cana-3308	317	7	using	use	VERB
cana-3308	317	8	equation	equation	NOUN
cana-3308	317	9	(	(	PUNCT
cana-3308	317	10	3.26	3.26	NUM
cana-3308	317	11	)	)	PUNCT
cana-3308	317	12	,	,	PUNCT
cana-3308	317	13	we	we	PRON
cana-3308	317	14	obtain	obtain	VERB
cana-3308	317	15	𝜓	𝜓	PROPN
cana-3308	317	16	(	(	PUNCT
cana-3308	317	17	𝑑(𝑥𝑘+𝑗+1	𝑑(𝑥𝑘+𝑗+1	NOUN
cana-3308	317	18	,	,	PUNCT
cana-3308	317	19	𝑦𝑘	𝑦𝑘	NOUN
cana-3308	317	20	)	)	PUNCT
cana-3308	317	21	)	)	PUNCT
cana-3308	317	22	≤	≤	NUM
cana-3308	318	1	𝜓	𝜓	PROPN
cana-3308	318	2	(	(	PUNCT
cana-3308	318	3	𝛿1	𝛿1	NOUN
cana-3308	318	4	)	)	PUNCT
cana-3308	318	5	+	+	PUNCT
cana-3308	318	6	𝜓(휀1	𝜓(휀1	NOUN
cana-3308	318	7	)	)	PUNCT
cana-3308	318	8	,	,	PUNCT
cana-3308	318	9	=	=	SYM
cana-3308	318	10	𝜓(휀1	𝜓(휀1	X
cana-3308	318	11	+	+	CCONJ
cana-3308	318	12	𝛿1	𝛿1	NOUN
cana-3308	318	13	)	)	PUNCT
cana-3308	318	14	.	.	PUNCT
cana-3308	319	1	by	by	ADP
cana-3308	319	2	the	the	DET
cana-3308	319	3	property	property	NOUN
cana-3308	319	4	of	of	ADP
cana-3308	319	5	𝜓	𝜓	NOUN
cana-3308	319	6	,	,	PUNCT
cana-3308	319	7	we	we	PRON
cana-3308	319	8	get	get	VERB
cana-3308	319	9	𝑑(𝑥𝑘+𝑗+1	𝑑(𝑥𝑘+𝑗+1	NOUN
cana-3308	319	10	,	,	PUNCT
cana-3308	319	11	𝑦𝑘	𝑦𝑘	INTJ
cana-3308	319	12	)	)	PUNCT
cana-3308	319	13	<	<	X
cana-3308	319	14	휀1	휀1	PROPN
cana-3308	319	15	+	+	CCONJ
cana-3308	319	16	𝛿1	𝛿1	NOUN
cana-3308	319	17	.	.	PUNCT
cana-3308	320	1	so	so	ADV
cana-3308	320	2	,	,	PUNCT
cana-3308	320	3	equation	equation	NOUN
cana-3308	320	4	(	(	PUNCT
cana-3308	320	5	3.29	3.29	NUM
cana-3308	320	6	)	)	PUNCT
cana-3308	320	7	is	be	AUX
cana-3308	320	8	holds	hold	NOUN
cana-3308	320	9	for	for	ADP
cana-3308	320	10	𝑙	𝑙	PRON
cana-3308	320	11	=	=	SYM
cana-3308	320	12	𝑗	𝑗	PROPN
cana-3308	321	1	+	+	NOUN
cana-3308	321	2	1	1	NUM
cana-3308	321	3	.	.	PUNCT
cana-3308	321	4	hence	hence	ADV
cana-3308	321	5	,	,	PUNCT
cana-3308	321	6	by	by	ADP
cana-3308	321	7	induction	induction	NOUN
cana-3308	321	8	we	we	PRON
cana-3308	321	9	prove	prove	VERB
cana-3308	321	10	that	that	SCONJ
cana-3308	321	11	𝑑(𝑥𝑘+𝑗+1	𝑑(𝑥𝑘+𝑗+1	VERB
cana-3308	321	12	,	,	PUNCT
cana-3308	321	13	𝑦𝑘	𝑦𝑘	INTJ
cana-3308	321	14	)	)	PUNCT
cana-3308	321	15	<	<	X
cana-3308	321	16	휀1	휀1	NOUN
cana-3308	321	17	+	+	CCONJ
cana-3308	321	18	𝛿1	𝛿1	NOUN
cana-3308	321	19	for	for	ADP
cana-3308	321	20	all	all	DET
cana-3308	321	21	𝑘	𝑘	DET
cana-3308	321	22	≥	≥	NOUN
cana-3308	321	23	𝑛0	𝑛0	VERB
cana-3308	321	24	and	and	CCONJ
cana-3308	321	25	𝑙	𝑙	DET
cana-3308	321	26	≥	≥	NUM
cana-3308	321	27	1	1	NUM
cana-3308	321	28	.	.	PUNCT
cana-3308	322	1	since	since	SCONJ
cana-3308	322	2	휀1	휀1	NOUN
cana-3308	322	3	is	be	AUX
cana-3308	322	4	arbitrary	arbitrary	ADJ
cana-3308	322	5	,	,	PUNCT
cana-3308	322	6	we	we	PRON
cana-3308	322	7	conclude	conclude	VERB
cana-3308	322	8	that	that	SCONJ
cana-3308	322	9	lim	lim	PROPN
cana-3308	322	10	𝑚,𝑛→∞	𝑚,𝑛→∞	VERB
cana-3308	322	11	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	322	12	,	,	PUNCT
cana-3308	322	13	𝑦𝑚	𝑦𝑚	NOUN
cana-3308	322	14	)	)	PUNCT
cana-3308	322	15	=	=	SYM
cana-3308	323	1	0	0	X
cana-3308	323	2	.	.	PUNCT
cana-3308	324	1	hence	hence	ADV
cana-3308	324	2	,	,	PUNCT
cana-3308	324	3	{	{	PUNCT
cana-3308	324	4	(	(	PUNCT
cana-3308	324	5	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	324	6	,	,	PUNCT
cana-3308	324	7	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	324	8	)	)	PUNCT
cana-3308	324	9	}	}	PUNCT
cana-3308	324	10	is	be	AUX
cana-3308	324	11	a	a	DET
cana-3308	324	12	cauchy	cauchy	ADJ
cana-3308	324	13	bisequence	bisequence	NOUN
cana-3308	324	14	and	and	CCONJ
cana-3308	324	15	(	(	PUNCT
cana-3308	324	16	𝑋	𝑋	PROPN
cana-3308	324	17	,	,	PUNCT
cana-3308	324	18	𝑌	𝑌	PROPN
cana-3308	324	19	,	,	PUNCT
cana-3308	324	20	𝑑	𝑑	NOUN
cana-3308	324	21	)	)	PUNCT
cana-3308	324	22	is	be	AUX
cana-3308	324	23	a	a	DET
cana-3308	324	24	complete	complete	ADJ
cana-3308	324	25	bipolar	bipolar	ADJ
cana-3308	324	26	metric	metric	ADJ
cana-3308	324	27	space	space	NOUN
cana-3308	324	28	.	.	PUNCT
cana-3308	325	1	so	so	ADV
cana-3308	325	2	,	,	PUNCT
cana-3308	325	3	{	{	PUNCT
cana-3308	325	4	(	(	PUNCT
cana-3308	325	5	𝑥𝑛	𝑥𝑛	INTJ
cana-3308	325	6	,	,	PUNCT
cana-3308	325	7	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	325	8	)	)	PUNCT
cana-3308	325	9	}	}	PUNCT
cana-3308	325	10	is	be	AUX
cana-3308	325	11	convergent	convergent	ADJ
cana-3308	325	12	and	and	CCONJ
cana-3308	325	13	in	in	ADP
cana-3308	325	14	fact	fact	NOUN
cana-3308	325	15	biconvergent	biconvergent	NOUN
cana-3308	325	16	.	.	PUNCT
cana-3308	326	1	so	so	ADV
cana-3308	326	2	,	,	PUNCT
cana-3308	326	3	there	there	PRON
cana-3308	326	4	exists	exist	VERB
cana-3308	326	5	𝑢	𝑢	PRON
cana-3308	326	6	∈	∈	PROPN
cana-3308	326	7	𝑋	𝑋	NOUN
cana-3308	326	8	∩	∩	NOUN
cana-3308	326	9	𝑌such	𝑌such	PROPN
cana-3308	326	10	that	that	SCONJ
cana-3308	326	11	(	(	PUNCT
cana-3308	326	12	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	326	13	)	)	PUNCT
cana-3308	326	14	→	→	SYM
cana-3308	326	15	𝑢	𝑢	X
cana-3308	326	16	,	,	PUNCT
cana-3308	326	17	(	(	PUNCT
cana-3308	326	18	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	326	19	)	)	PUNCT
cana-3308	326	20	→	→	SYM
cana-3308	326	21	𝑢	𝑢	NOUN
cana-3308	326	22	as	as	ADP
cana-3308	326	23	𝑛	𝑛	PROPN
cana-3308	326	24	→	→	SYM
cana-3308	326	25	∞.	∞.	PROPN
cana-3308	326	26	we	we	PRON
cana-3308	326	27	claim	claim	VERB
cana-3308	326	28	that	that	SCONJ
cana-3308	326	29	𝑇𝑢	𝑇𝑢	PROPN
cana-3308	326	30	=	=	PUNCT
cana-3308	326	31	𝑢.	𝑢.	NOUN
cana-3308	326	32	let	let	VERB
cana-3308	326	33	,	,	PUNCT
cana-3308	326	34	if	if	SCONJ
cana-3308	326	35	possible	possible	ADJ
cana-3308	326	36	,	,	PUNCT
cana-3308	326	37	𝑇𝑢	𝑇𝑢	PROPN
cana-3308	326	38	≠	≠	PROPN
cana-3308	326	39	𝑢.	𝑢.	NOUN
cana-3308	326	40	then	then	ADV
cana-3308	326	41	there	there	PRON
cana-3308	326	42	exist	exist	VERB
cana-3308	326	43	𝑟′	𝑟′	PROPN
cana-3308	326	44	>	>	X
cana-3308	326	45	0	0	NUM
cana-3308	326	46	such	such	ADJ
cana-3308	326	47	that	that	DET
cana-3308	326	48	𝑑(𝑇𝑢	𝑑(𝑇𝑢	NOUN
cana-3308	326	49	,	,	PUNCT
cana-3308	326	50	𝑢	𝑢	X
cana-3308	326	51	)	)	PUNCT
cana-3308	326	52	=	=	SYM
cana-3308	326	53	𝑟′	𝑟′	PROPN
cana-3308	326	54	>	>	X
cana-3308	327	1	0	0	X
cana-3308	327	2	.	.	PUNCT
cana-3308	328	1	since	since	SCONJ
cana-3308	328	2	(	(	PUNCT
cana-3308	328	3	𝑥𝑛	𝑥𝑛	PROPN
cana-3308	328	4	)	)	PUNCT
cana-3308	328	5	→	→	SYM
cana-3308	328	6	𝑢	𝑢	X
cana-3308	328	7	,	,	PUNCT
cana-3308	328	8	(	(	PUNCT
cana-3308	328	9	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	328	10	)	)	PUNCT
cana-3308	328	11	→	→	SYM
cana-3308	328	12	𝑢	𝑢	NOUN
cana-3308	328	13	as	as	ADP
cana-3308	328	14	𝑛	𝑛	PROPN
cana-3308	328	15	→	→	SYM
cana-3308	328	16	∞	∞	PROPN
cana-3308	328	17	,	,	PUNCT
cana-3308	328	18	we	we	PRON
cana-3308	328	19	can	can	AUX
cana-3308	328	20	choose	choose	VERB
cana-3308	328	21	𝑛0	𝑛0	VERB
cana-3308	328	22	∈	∈	PROPN
cana-3308	328	23	ℕ	ℕ	PROPN
cana-3308	328	24	such	such	ADJ
cana-3308	328	25	that	that	SCONJ
cana-3308	328	26	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-3308	328	27	,	,	PUNCT
cana-3308	328	28	𝑢	𝑢	PROPN
cana-3308	328	29	)	)	PUNCT
cana-3308	328	30	<	<	X
cana-3308	328	31	𝑟′	𝑟′	PROPN
cana-3308	328	32	2	2	NUM
cana-3308	328	33	,	,	PUNCT
cana-3308	328	34	for	for	SCONJ
cana-3308	328	35	all	all	DET
cana-3308	328	36	𝑛	𝑛	DET
cana-3308	328	37	≥	≥	NOUN
cana-3308	328	38	𝑛0	𝑛0	VERB
cana-3308	328	39	and	and	CCONJ
cana-3308	328	40	𝑑(𝑦𝑛	𝑑(𝑦𝑛	NUM
cana-3308	328	41	,	,	PUNCT
cana-3308	328	42	𝑢	𝑢	X
cana-3308	328	43	)	)	PUNCT
cana-3308	328	44	<	<	X
cana-3308	328	45	𝑟′	𝑟′	PROPN
cana-3308	328	46	2	2	NUM
cana-3308	328	47	,	,	PUNCT
cana-3308	328	48	for	for	SCONJ
cana-3308	328	49	all	all	DET
cana-3308	328	50	𝑛	𝑛	DET
cana-3308	328	51	≥	≥	NOUN
cana-3308	328	52	𝑛0	𝑛0	VERB
cana-3308	328	53	.	.	PUNCT
cana-3308	329	1	(	(	PUNCT
cana-3308	329	2	3.30	3.30	NUM
cana-3308	329	3	)	)	PUNCT
cana-3308	329	4	from	from	ADP
cana-3308	329	5	the	the	DET
cana-3308	329	6	triangle	triangle	NOUN
cana-3308	329	7	inequality	inequality	NOUN
cana-3308	329	8	(	(	PUNCT
cana-3308	329	9	bp3	bp3	NOUN
cana-3308	329	10	)	)	PUNCT
cana-3308	329	11	,	,	PUNCT
cana-3308	329	12	properties	property	NOUN
cana-3308	329	13	of	of	ADP
cana-3308	329	14	𝜓	𝜓	PROPN
cana-3308	329	15	,	,	PUNCT
cana-3308	329	16	𝜃	𝜃	PROPN
cana-3308	329	17	,	,	PUNCT
cana-3308	329	18	equations	equation	NOUN
cana-3308	329	19	(	(	PUNCT
cana-3308	329	20	3.1	3.1	NUM
cana-3308	329	21	)	)	PUNCT
cana-3308	329	22	,	,	PUNCT
cana-3308	329	23	(	(	PUNCT
cana-3308	329	24	3.2	3.2	NUM
cana-3308	329	25	)	)	PUNCT
cana-3308	329	26	and	and	CCONJ
cana-3308	329	27	(	(	PUNCT
cana-3308	329	28	3.17	3.17	NUM
cana-3308	329	29	)	)	PUNCT
cana-3308	329	30	communications	communication	NOUN
cana-3308	329	31	on	on	ADP
cana-3308	329	32	applied	apply	VERB
cana-3308	329	33	nonlinear	nonlinear	ADJ
cana-3308	329	34	analysis	analysis	NOUN
cana-3308	329	35	issn	issn	NOUN
cana-3308	329	36	:	:	PUNCT
cana-3308	329	37	1074	1074	NUM
cana-3308	329	38	-	-	PUNCT
cana-3308	329	39	133x	133x	NUM
cana-3308	329	40	vol	vol	NOUN
cana-3308	329	41	32	32	NUM
cana-3308	329	42	no	no	NOUN
cana-3308	329	43	.	.	PUNCT
cana-3308	330	1	6s	6s	NUM
cana-3308	330	2	(	(	PUNCT
cana-3308	330	3	2025	2025	NUM
cana-3308	330	4	)	)	PUNCT
cana-3308	330	5	451	451	NUM
cana-3308	331	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	331	2	𝜓(𝑟	𝜓(𝑟	NOUN
cana-3308	331	3	′	′	NUM
cana-3308	331	4	)	)	PUNCT
cana-3308	331	5	=	=	SYM
cana-3308	331	6	𝜓(𝑑(𝑇𝑢	𝜓(𝑑(𝑇𝑢	NOUN
cana-3308	331	7	,	,	PUNCT
cana-3308	331	8	𝑢	𝑢	NOUN
cana-3308	331	9	)	)	PUNCT
cana-3308	331	10	)	)	PUNCT
cana-3308	331	11	,	,	PUNCT
cana-3308	331	12	≤	≤	ADJ
cana-3308	331	13	𝜓(𝑑(𝑇𝑢	𝜓(𝑑(𝑇𝑢	NOUN
cana-3308	331	14	,	,	PUNCT
cana-3308	331	15	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	331	16	)	)	PUNCT
cana-3308	331	17	+	+	NUM
cana-3308	331	18	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	331	19	,	,	PUNCT
cana-3308	331	20	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	331	21	)	)	PUNCT
cana-3308	332	1	+	+	CCONJ
cana-3308	332	2	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NOUN
cana-3308	332	3	,	,	PUNCT
cana-3308	332	4	𝑢	𝑢	NOUN
cana-3308	332	5	)	)	PUNCT
cana-3308	332	6	)	)	PUNCT
cana-3308	332	7	≤	≤	ADJ
cana-3308	332	8	𝜓(𝑑(𝑇𝑢	𝜓(𝑑(𝑇𝑢	NOUN
cana-3308	332	9	,	,	PUNCT
cana-3308	332	10	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	332	11	)	)	PUNCT
cana-3308	332	12	)	)	PUNCT
cana-3308	333	1	+	+	CCONJ
cana-3308	333	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	333	3	,	,	PUNCT
cana-3308	333	4	𝑦𝑛	𝑦𝑛	NOUN
cana-3308	333	5	)	)	PUNCT
cana-3308	333	6	)	)	PUNCT
cana-3308	334	1	+	+	CCONJ
cana-3308	335	1	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	335	2	,	,	PUNCT
cana-3308	335	3	𝑢	𝑢	NOUN
cana-3308	335	4	)	)	PUNCT
cana-3308	335	5	)	)	PUNCT
cana-3308	335	6	≤	≤	ADJ
cana-3308	335	7	𝜓(𝑑(𝑇𝑢	𝜓(𝑑(𝑇𝑢	NOUN
cana-3308	335	8	,	,	PUNCT
cana-3308	335	9	𝑇𝑥𝑛	𝑇𝑥𝑛	PROPN
cana-3308	335	10	)	)	PUNCT
cana-3308	335	11	)	)	PUNCT
cana-3308	336	1	+	+	CCONJ
cana-3308	336	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	336	3	,	,	PUNCT
cana-3308	336	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	336	5	)	)	PUNCT
cana-3308	336	6	)	)	PUNCT
cana-3308	337	1	+	+	CCONJ
cana-3308	338	1	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	338	2	,	,	PUNCT
cana-3308	338	3	𝑢	𝑢	NOUN
cana-3308	338	4	)	)	PUNCT
cana-3308	338	5	)	)	PUNCT
cana-3308	338	6	≤	≤	NOUN
cana-3308	339	1	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-3308	339	2	,	,	PUNCT
cana-3308	339	3	𝑢	𝑢	NOUN
cana-3308	339	4	)	)	PUNCT
cana-3308	339	5	𝜓(𝑑(𝑇𝑢	𝜓(𝑑(𝑇𝑢	NOUN
cana-3308	339	6	,	,	PUNCT
cana-3308	339	7	𝑇𝑥𝑛	𝑇𝑥𝑛	PROPN
cana-3308	339	8	)	)	PUNCT
cana-3308	339	9	)	)	PUNCT
cana-3308	340	1	+	+	CCONJ
cana-3308	340	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	340	3	,	,	PUNCT
cana-3308	340	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	340	5	)	)	PUNCT
cana-3308	340	6	)	)	PUNCT
cana-3308	341	1	+	+	CCONJ
cana-3308	342	1	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	342	2	,	,	PUNCT
cana-3308	342	3	𝑢	𝑢	NOUN
cana-3308	342	4	)	)	PUNCT
cana-3308	342	5	)	)	PUNCT
cana-3308	342	6	≤	≤	PUNCT
cana-3308	343	1	𝜃(𝜓((𝑥𝑛	𝜃(𝜓((𝑥𝑛	PROPN
cana-3308	343	2	,	,	PUNCT
cana-3308	343	3	𝑢	𝑢	NOUN
cana-3308	343	4	)	)	PUNCT
cana-3308	343	5	)	)	PUNCT
cana-3308	344	1	𝜑	𝜑	PROPN
cana-3308	344	2	(	(	PUNCT
cana-3308	344	3	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	344	4	,	,	PUNCT
cana-3308	344	5	𝑢	𝑢	PROPN
cana-3308	344	6	)	)	PUNCT
cana-3308	344	7	)	)	PUNCT
cana-3308	344	8	)	)	PUNCT
cana-3308	345	1	+	+	CCONJ
cana-3308	345	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	345	3	,	,	PUNCT
cana-3308	345	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	345	5	)	)	PUNCT
cana-3308	345	6	)	)	PUNCT
cana-3308	346	1	+	+	CCONJ
cana-3308	347	1	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	347	2	,	,	PUNCT
cana-3308	347	3	𝑢	𝑢	PROPN
cana-3308	347	4	)	)	PUNCT
cana-3308	347	5	)	)	PUNCT
cana-3308	348	1	<	<	X
cana-3308	348	2	𝜑	𝜑	X
cana-3308	348	3	(	(	PUNCT
cana-3308	348	4	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	348	5	,	,	PUNCT
cana-3308	348	6	𝑢	𝑢	PROPN
cana-3308	348	7	)	)	PUNCT
cana-3308	348	8	)	)	PUNCT
cana-3308	348	9	)	)	PUNCT
cana-3308	349	1	+	+	CCONJ
cana-3308	349	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	349	3	,	,	PUNCT
cana-3308	349	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	349	5	)	)	PUNCT
cana-3308	349	6	)	)	PUNCT
cana-3308	350	1	+	+	CCONJ
cana-3308	350	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	350	3	,	,	PUNCT
cana-3308	350	4	𝑢	𝑢	PROPN
cana-3308	350	5	)	)	PUNCT
cana-3308	350	6	)	)	PUNCT
cana-3308	350	7	<	<	X
cana-3308	350	8	𝜓(𝑑(𝑥𝑛	𝜓(𝑑(𝑥𝑛	X
cana-3308	350	9	,	,	PUNCT
cana-3308	350	10	𝑢	𝑢	PROPN
cana-3308	350	11	)	)	PUNCT
cana-3308	350	12	)	)	PUNCT
cana-3308	351	1	+	+	CCONJ
cana-3308	351	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	PROPN
cana-3308	351	3	,	,	PUNCT
cana-3308	351	4	𝑦𝑛+1	𝑦𝑛+1	NUM
cana-3308	351	5	)	)	PUNCT
cana-3308	351	6	)	)	PUNCT
cana-3308	352	1	+	+	CCONJ
cana-3308	352	2	𝜓(𝑑(𝑥𝑛+1	𝜓(𝑑(𝑥𝑛+1	NUM
cana-3308	352	3	,	,	PUNCT
cana-3308	352	4	𝑢	𝑢	PROPN
cana-3308	352	5	)	)	PUNCT
cana-3308	352	6	)	)	PUNCT
cana-3308	352	7	.	.	PUNCT
cana-3308	353	1	using	use	VERB
cana-3308	353	2	equations	equation	NOUN
cana-3308	353	3	(	(	PUNCT
cana-3308	353	4	3.26	3.26	NUM
cana-3308	353	5	)	)	PUNCT
cana-3308	353	6	and	and	CCONJ
cana-3308	353	7	(	(	PUNCT
cana-3308	353	8	3.30	3.30	NUM
cana-3308	353	9	)	)	PUNCT
cana-3308	353	10	,	,	PUNCT
cana-3308	353	11	we	we	PRON
cana-3308	353	12	get	get	VERB
cana-3308	353	13	𝜓(𝑟	𝜓(𝑟	NOUN
cana-3308	353	14	′	′	NUM
cana-3308	353	15	)	)	PUNCT
cana-3308	353	16	<	<	X
cana-3308	353	17	𝜓	𝜓	X
cana-3308	353	18	(	(	PUNCT
cana-3308	353	19	𝑟	𝑟	NOUN
cana-3308	353	20	′	′	NOUN
cana-3308	353	21	2	2	NUM
cana-3308	353	22	)	)	PUNCT
cana-3308	354	1	+	+	CCONJ
cana-3308	354	2	𝜓	𝜓	X
cana-3308	354	3	(	(	PUNCT
cana-3308	354	4	𝑟	𝑟	NOUN
cana-3308	354	5	′	′	NOUN
cana-3308	354	6	2	2	X
cana-3308	354	7	)	)	PUNCT
cana-3308	354	8	=	=	SYM
cana-3308	354	9	𝜓	𝜓	PROPN
cana-3308	354	10	(	(	PUNCT
cana-3308	354	11	𝑟′	𝑟′	PROPN
cana-3308	354	12	2	2	NUM
cana-3308	354	13	+	+	CCONJ
cana-3308	354	14	𝑟′	𝑟′	NUM
cana-3308	354	15	2	2	NUM
cana-3308	354	16	)	)	PUNCT
cana-3308	354	17	=	=	PUNCT
cana-3308	354	18	𝜓(𝑟	𝜓(𝑟	PROPN
cana-3308	354	19	′	′	NUM
cana-3308	354	20	)	)	PUNCT
cana-3308	354	21	,	,	PUNCT
cana-3308	354	22	which	which	PRON
cana-3308	354	23	is	be	AUX
cana-3308	354	24	a	a	DET
cana-3308	354	25	contradiction	contradiction	NOUN
cana-3308	354	26	.	.	PUNCT
cana-3308	355	1	thus	thus	ADV
cana-3308	355	2	𝑇𝑢	𝑇𝑢	ADP
cana-3308	355	3	=	=	PUNCT
cana-3308	355	4	𝑢.	𝑢.	NOUN
cana-3308	355	5	i.e.	i.e.	X
cana-3308	355	6	,	,	PUNCT
cana-3308	355	7	𝑢	𝑢	PRON
cana-3308	355	8	is	be	AUX
cana-3308	355	9	the	the	DET
cana-3308	355	10	fixed	fixed	ADJ
cana-3308	355	11	point	point	NOUN
cana-3308	355	12	of	of	ADP
cana-3308	355	13	𝑇.	𝑇.	PROPN
cana-3308	355	14	example	example	NOUN
cana-3308	355	15	3.12	3.12	NUM
cana-3308	355	16	.	.	PUNCT
cana-3308	356	1	let	let	VERB
cana-3308	356	2	𝑋	𝑋	NOUN
cana-3308	356	3	=	=	PUNCT
cana-3308	357	1	[	[	X
cana-3308	357	2	0	0	NUM
cana-3308	357	3	,	,	PUNCT
cana-3308	357	4	+	+	CCONJ
cana-3308	357	5	∞	∞	NUM
cana-3308	357	6	)	)	PUNCT
cana-3308	357	7	and	and	CCONJ
cana-3308	357	8	𝑌	𝑌	PROPN
cana-3308	357	9	=	=	PUNCT
cana-3308	358	1	[	[	X
cana-3308	358	2	−1,1	−1,1	X
cana-3308	358	3	]	]	PUNCT
cana-3308	358	4	and	and	CCONJ
cana-3308	358	5	let	let	VERB
cana-3308	358	6	𝑑	𝑑	PRON
cana-3308	358	7	∶	∶	VERB
cana-3308	358	8	𝑋	𝑋	NOUN
cana-3308	358	9	×	×	NOUN
cana-3308	358	10	𝑌	𝑌	PROPN
cana-3308	358	11	→	→	SYM
cana-3308	358	12	[	[	X
cana-3308	358	13	0	0	NUM
cana-3308	358	14	,	,	PUNCT
cana-3308	358	15	+	+	NOUN
cana-3308	358	16	∞	∞	NOUN
cana-3308	358	17	)	)	PUNCT
cana-3308	358	18	be	be	VERB
cana-3308	358	19	a	a	DET
cana-3308	358	20	function	function	NOUN
cana-3308	358	21	such	such	ADJ
cana-3308	358	22	that	that	SCONJ
cana-3308	358	23	𝑑(𝑥	𝑑(𝑥	PROPN
cana-3308	358	24	,	,	PUNCT
cana-3308	358	25	𝑦	𝑦	X
cana-3308	358	26	)	)	PUNCT
cana-3308	358	27	=	=	NOUN
cana-3308	359	1	|𝑥2	|𝑥2	X
cana-3308	359	2	−	−	PROPN
cana-3308	359	3	𝑦2|	𝑦2|	NOUN
cana-3308	359	4	for	for	ADP
cana-3308	359	5	all	all	DET
cana-3308	359	6	(	(	PUNCT
cana-3308	359	7	𝑥	𝑥	PROPN
cana-3308	359	8	,	,	PUNCT
cana-3308	359	9	𝑦	𝑦	X
cana-3308	359	10	)	)	PUNCT
cana-3308	359	11	∈	∈	NOUN
cana-3308	359	12	𝑋	𝑋	NOUN
cana-3308	359	13	×	×	NOUN
cana-3308	359	14	𝑌.	𝑌.	PROPN
cana-3308	359	15	then	then	ADV
cana-3308	359	16	,	,	PUNCT
cana-3308	359	17	clearly	clearly	ADV
cana-3308	359	18	(	(	PUNCT
cana-3308	359	19	𝑋	𝑋	PROPN
cana-3308	359	20	,	,	PUNCT
cana-3308	359	21	𝑌	𝑌	PROPN
cana-3308	359	22	,	,	PUNCT
cana-3308	359	23	𝑑	𝑑	NOUN
cana-3308	359	24	)	)	PUNCT
cana-3308	359	25	be	be	VERB
cana-3308	359	26	a	a	DET
cana-3308	359	27	complete	complete	ADJ
cana-3308	359	28	bipolar	bipolar	ADJ
cana-3308	359	29	metric	metric	ADJ
cana-3308	359	30	space	space	NOUN
cana-3308	359	31	.	.	PUNCT
cana-3308	360	1	define	define	VERB
cana-3308	360	2	𝑇	𝑇	PROPN
cana-3308	360	3	∶	∶	PROPN
cana-3308	360	4	(	(	PUNCT
cana-3308	360	5	𝑋	𝑋	PROPN
cana-3308	360	6	,	,	PUNCT
cana-3308	360	7	𝑌	𝑌	PROPN
cana-3308	360	8	)	)	PUNCT
cana-3308	360	9	⤨	⤨	NUM
cana-3308	360	10	(	(	PUNCT
cana-3308	360	11	𝑋	𝑋	PROPN
cana-3308	360	12	,	,	PUNCT
cana-3308	360	13	𝑌	𝑌	PROPN
cana-3308	360	14	)	)	PUNCT
cana-3308	361	1	such	such	ADJ
cana-3308	361	2	that	that	SCONJ
cana-3308	361	3	𝑇𝑥	𝑇𝑥	ADV
cana-3308	361	4	=	=	PUNCT
cana-3308	361	5	−𝑥	−𝑥	PUNCT
cana-3308	361	6	2	2	NUM
cana-3308	361	7	is	be	AUX
cana-3308	361	8	a	a	DET
cana-3308	361	9	mapping	mapping	NOUN
cana-3308	361	10	and	and	CCONJ
cana-3308	361	11	𝛼	𝛼	NOUN
cana-3308	361	12	∶	∶	NOUN
cana-3308	361	13	𝑋	𝑋	NOUN
cana-3308	361	14	×	×	NOUN
cana-3308	361	15	𝑌	𝑌	PROPN
cana-3308	361	16	→	→	SYM
cana-3308	361	17	[	[	X
cana-3308	361	18	0	0	NUM
cana-3308	361	19	,	,	PUNCT
cana-3308	361	20	∞	∞	NOUN
cana-3308	361	21	)	)	PUNCT
cana-3308	361	22	such	such	ADJ
cana-3308	361	23	that	that	SCONJ
cana-3308	361	24	𝛼(𝑥	𝛼(𝑥	PROPN
cana-3308	361	25	,	,	PUNCT
cana-3308	361	26	𝑦	𝑦	NOUN
cana-3308	361	27	)	)	PUNCT
cana-3308	361	28	=	=	SYM
cana-3308	361	29	3	3	NUM
cana-3308	361	30	2	2	NUM
cana-3308	361	31	for	for	ADP
cana-3308	361	32	(	(	PUNCT
cana-3308	361	33	𝑥	𝑥	PROPN
cana-3308	361	34	,	,	PUNCT
cana-3308	361	35	𝑦	𝑦	X
cana-3308	361	36	)	)	PUNCT
cana-3308	361	37	∈	∈	NOUN
cana-3308	361	38	𝑋	𝑋	PROPN
cana-3308	361	39	×	×	PROPN
cana-3308	361	40	𝑌.	𝑌.	PROPN
cana-3308	361	41	clearly	clearly	ADV
cana-3308	361	42	,	,	PUNCT
cana-3308	361	43	𝑇	𝑇	PROPN
cana-3308	361	44	is	be	AUX
cana-3308	361	45	𝛼-admissible	𝛼-admissible	ADJ
cana-3308	361	46	mapping	mapping	NOUN
cana-3308	361	47	and	and	CCONJ
cana-3308	361	48	there	there	PRON
cana-3308	361	49	exist	exist	VERB
cana-3308	361	50	𝑥0	𝑥0	NOUN
cana-3308	361	51	∈	∈	NOUN
cana-3308	361	52	𝑋	𝑋	NOUN
cana-3308	361	53	such	such	ADJ
cana-3308	361	54	that	that	PRON
cana-3308	361	55	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	361	56	,	,	PUNCT
cana-3308	361	57	𝑇𝑥0	𝑇𝑥0	NOUN
cana-3308	361	58	)	)	PUNCT
cana-3308	361	59	≥	≥	NOUN
cana-3308	361	60	1	1	NUM
cana-3308	361	61	and	and	CCONJ
cana-3308	361	62	𝑋	𝑋	PROPN
cana-3308	361	63	∩	∩	ADJ
cana-3308	361	64	𝑌	𝑌	PROPN
cana-3308	361	65	=	=	SYM
cana-3308	361	66	{	{	PUNCT
cana-3308	361	67	0	0	NUM
cana-3308	361	68	}	}	PUNCT
cana-3308	361	69	and	and	CCONJ
cana-3308	361	70	𝑇0	𝑇0	X
cana-3308	361	71	=	=	SYM
cana-3308	361	72	0	0	X
cana-3308	361	73	.	.	PUNCT
cana-3308	362	1	taking	take	VERB
cana-3308	362	2	𝜓(𝑡	𝜓(𝑡	NOUN
cana-3308	362	3	)	)	PUNCT
cana-3308	362	4	=	=	SYM
cana-3308	362	5	𝑡	𝑡	ADP
cana-3308	362	6	4	4	NUM
cana-3308	362	7	,	,	PUNCT
cana-3308	362	8	𝜑(𝑡	𝜑(𝑡	PROPN
cana-3308	362	9	)	)	PUNCT
cana-3308	362	10	=	=	SYM
cana-3308	362	11	𝑡	𝑡	ADP
cana-3308	362	12	2	2	NUM
cana-3308	362	13	and	and	CCONJ
cana-3308	362	14	𝜃(𝑡	𝜃(𝑡	PROPN
cana-3308	362	15	)	)	PUNCT
cana-3308	363	1	=	=	PUNCT
cana-3308	363	2	3	3	NUM
cana-3308	363	3	4	4	NUM
cana-3308	363	4	.	.	PUNCT
cana-3308	364	1	left	leave	VERB
cana-3308	364	2	hand	hand	NOUN
cana-3308	364	3	side	side	NOUN
cana-3308	364	4	of	of	ADP
cana-3308	364	5	equation	equation	NOUN
cana-3308	364	6	(	(	PUNCT
cana-3308	364	7	3.17	3.17	NUM
cana-3308	364	8	)	)	PUNCT
cana-3308	364	9	becomes	become	VERB
cana-3308	364	10	𝛼(𝑥	𝛼(𝑥	PROPN
cana-3308	364	11	,	,	PUNCT
cana-3308	364	12	𝑦	𝑦	NOUN
cana-3308	364	13	)	)	PUNCT
cana-3308	364	14	𝜓(𝑑(𝑇𝑦	𝜓(𝑑(𝑇𝑦	PROPN
cana-3308	364	15	,	,	PUNCT
cana-3308	364	16	𝑇𝑥	𝑇𝑥	NOUN
cana-3308	364	17	)	)	PUNCT
cana-3308	364	18	)	)	PUNCT
cana-3308	365	1	=	=	SYM
cana-3308	365	2	3	3	NUM
cana-3308	365	3	2	2	NUM
cana-3308	365	4	|𝑥2−𝑦2|	|𝑥2−𝑦2|	NUM
cana-3308	365	5	16	16	NUM
cana-3308	365	6	.	.	PUNCT
cana-3308	366	1	right	right	ADJ
cana-3308	366	2	hand	hand	NOUN
cana-3308	366	3	side	side	NOUN
cana-3308	366	4	becomes	become	VERB
cana-3308	366	5	𝜃	𝜃	PROPN
cana-3308	366	6	(	(	PUNCT
cana-3308	366	7	𝜓(𝑑(𝑥	𝜓(𝑑(𝑥	PROPN
cana-3308	366	8	,	,	PUNCT
cana-3308	366	9	𝑦	𝑦	NOUN
cana-3308	366	10	)	)	PUNCT
cana-3308	366	11	)	)	PUNCT
cana-3308	366	12	)	)	PUNCT
cana-3308	367	1	𝜑(𝜓(𝑑(𝑥	𝜑(𝜓(𝑑(𝑥	PROPN
cana-3308	367	2	,	,	PUNCT
cana-3308	367	3	𝑦	𝑦	NOUN
cana-3308	367	4	)	)	PUNCT
cana-3308	367	5	)	)	PUNCT
cana-3308	367	6	)	)	PUNCT
cana-3308	368	1	=	=	SYM
cana-3308	368	2	3	3	NUM
cana-3308	368	3	2	2	NUM
cana-3308	368	4	|𝑥2−𝑦2|	|𝑥2−𝑦2|	NOUN
cana-3308	368	5	16	16	NUM
cana-3308	368	6	,	,	PUNCT
cana-3308	368	7	for	for	ADP
cana-3308	368	8	all	all	DET
cana-3308	368	9	(	(	PUNCT
cana-3308	368	10	𝑥	𝑥	PROPN
cana-3308	368	11	,	,	PUNCT
cana-3308	368	12	𝑦	𝑦	X
cana-3308	368	13	)	)	PUNCT
cana-3308	368	14	∈	∈	PROPN
cana-3308	368	15	𝑋	𝑋	PROPN
cana-3308	368	16	×	×	PROPN
cana-3308	368	17	𝑌	𝑌	PROPN
cana-3308	368	18	,	,	PUNCT
cana-3308	368	19	which	which	PRON
cana-3308	368	20	implies	imply	VERB
cana-3308	368	21	equation	equation	NOUN
cana-3308	368	22	(	(	PUNCT
cana-3308	368	23	3.17	3.17	NUM
cana-3308	368	24	)	)	PUNCT
cana-3308	368	25	holds	hold	VERB
cana-3308	368	26	.	.	PUNCT
cana-3308	369	1	hence	hence	ADV
cana-3308	369	2	,	,	PUNCT
cana-3308	369	3	𝑇	𝑇	PROPN
cana-3308	369	4	is	be	AUX
cana-3308	369	5	an	an	DET
cana-3308	369	6	(	(	PUNCT
cana-3308	369	7	𝛼	𝛼	PROPN
cana-3308	369	8	,	,	PUNCT
cana-3308	369	9	𝜓	𝜓	NOUN
cana-3308	369	10	,	,	PUNCT
cana-3308	369	11	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	369	12	contraction	contraction	NOUN
cana-3308	369	13	mapping	mapping	NOUN
cana-3308	369	14	.	.	PUNCT
cana-3308	370	1	communications	communication	NOUN
cana-3308	370	2	on	on	ADP
cana-3308	370	3	applied	apply	VERB
cana-3308	370	4	nonlinear	nonlinear	ADJ
cana-3308	370	5	analysis	analysis	NOUN
cana-3308	370	6	issn	issn	NOUN
cana-3308	370	7	:	:	PUNCT
cana-3308	370	8	1074	1074	NUM
cana-3308	370	9	-	-	PUNCT
cana-3308	370	10	133x	133x	NUM
cana-3308	370	11	vol	vol	NOUN
cana-3308	370	12	32	32	NUM
cana-3308	370	13	no	no	NOUN
cana-3308	370	14	.	.	PUNCT
cana-3308	371	1	6s	6s	NUM
cana-3308	371	2	(	(	PUNCT
cana-3308	371	3	2025	2025	NUM
cana-3308	371	4	)	)	PUNCT
cana-3308	371	5	452	452	NUM
cana-3308	371	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3308	371	7	all	all	DET
cana-3308	371	8	the	the	DET
cana-3308	371	9	conditions	condition	NOUN
cana-3308	371	10	of	of	ADP
cana-3308	371	11	theorem	theorem	NOUN
cana-3308	371	12	3.11	3.11	NUM
cana-3308	371	13	.	.	PUNCT
cana-3308	371	14	are	be	AUX
cana-3308	371	15	satisfied	satisfied	ADJ
cana-3308	371	16	.	.	PUNCT
cana-3308	372	1	so	so	ADV
cana-3308	372	2	,	,	PUNCT
cana-3308	372	3	𝑇	𝑇	PROPN
cana-3308	372	4	has	have	VERB
cana-3308	372	5	a	a	DET
cana-3308	372	6	fixed	fix	VERB
cana-3308	372	7	point	point	NOUN
cana-3308	372	8	and	and	CCONJ
cana-3308	372	9	𝑥	𝑥	NOUN
cana-3308	372	10	=	=	SYM
cana-3308	372	11	0	0	NUM
cana-3308	372	12	is	be	AUX
cana-3308	372	13	the	the	DET
cana-3308	372	14	fixed	fixed	ADJ
cana-3308	372	15	point	point	NOUN
cana-3308	372	16	of	of	ADP
cana-3308	372	17	𝑇.	𝑇.	PROPN
cana-3308	372	18	theorem	theorem	VERB
cana-3308	372	19	3.13	3.13	NUM
cana-3308	372	20	.	.	PUNCT
cana-3308	373	1	let	let	VERB
cana-3308	373	2	(	(	PUNCT
cana-3308	373	3	𝑋	𝑋	PROPN
cana-3308	373	4	,	,	PUNCT
cana-3308	373	5	𝑌	𝑌	PROPN
cana-3308	373	6	,	,	PUNCT
cana-3308	373	7	𝑑	𝑑	NOUN
cana-3308	373	8	)	)	PUNCT
cana-3308	373	9	be	be	VERB
cana-3308	373	10	a	a	DET
cana-3308	373	11	complete	complete	ADJ
cana-3308	373	12	bipolar	bipolar	ADJ
cana-3308	373	13	metric	metric	ADJ
cana-3308	373	14	space	space	NOUN
cana-3308	373	15	,	,	PUNCT
cana-3308	373	16	𝑇	𝑇	PROPN
cana-3308	373	17	∶	∶	PROPN
cana-3308	373	18	(	(	PUNCT
cana-3308	373	19	𝑋	𝑋	PROPN
cana-3308	373	20	,	,	PUNCT
cana-3308	373	21	𝑌	𝑌	PROPN
cana-3308	373	22	)	)	PUNCT
cana-3308	373	23	⤨	⤨	NUM
cana-3308	373	24	(	(	PUNCT
cana-3308	373	25	𝑋	𝑋	PROPN
cana-3308	373	26	,	,	PUNCT
cana-3308	373	27	𝑌	𝑌	PROPN
cana-3308	373	28	)	)	PUNCT
cana-3308	373	29	is	be	AUX
cana-3308	373	30	a	a	DET
cana-3308	373	31	contravariant	contravariant	ADJ
cana-3308	373	32	mapping	mapping	NOUN
cana-3308	373	33	and	and	CCONJ
cana-3308	373	34	𝛼	𝛼	ADP
cana-3308	373	35	∶	∶	NOUN
cana-3308	373	36	𝑋	𝑋	NOUN
cana-3308	373	37	×	×	NOUN
cana-3308	373	38	𝑌	𝑌	PROPN
cana-3308	373	39	→	→	SYM
cana-3308	373	40	[	[	X
cana-3308	373	41	0	0	NUM
cana-3308	373	42	,	,	PUNCT
cana-3308	373	43	∞	∞	PROPN
cana-3308	373	44	)	)	PUNCT
cana-3308	373	45	.	.	PUNCT
cana-3308	374	1	suppose	suppose	VERB
cana-3308	374	2	that	that	SCONJ
cana-3308	374	3	the	the	DET
cana-3308	374	4	following	follow	VERB
cana-3308	374	5	conditions	condition	NOUN
cana-3308	374	6	hold	hold	VERB
cana-3308	374	7	:	:	PUNCT
cana-3308	374	8	(	(	PUNCT
cana-3308	374	9	i)𝑇	i)𝑇	NOUN
cana-3308	374	10	is	be	AUX
cana-3308	374	11	𝛼admissible	𝛼admissible	ADJ
cana-3308	374	12	mapping	mapping	NOUN
cana-3308	374	13	,	,	PUNCT
cana-3308	374	14	(	(	PUNCT
cana-3308	374	15	ii)𝑇	ii)𝑇	NOUN
cana-3308	374	16	is	be	AUX
cana-3308	374	17	an	an	DET
cana-3308	374	18	(	(	PUNCT
cana-3308	374	19	𝛼	𝛼	PROPN
cana-3308	374	20	,	,	PUNCT
cana-3308	374	21	𝜓	𝜓	NOUN
cana-3308	374	22	,	,	PUNCT
cana-3308	374	23	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	374	24	contraction	contraction	NOUN
cana-3308	374	25	mapping	mapping	NOUN
cana-3308	374	26	,	,	PUNCT
cana-3308	374	27	(	(	PUNCT
cana-3308	374	28	iii)there	iii)there	NOUN
cana-3308	374	29	exist	exist	VERB
cana-3308	374	30	𝑥0	𝑥0	NOUN
cana-3308	374	31	∈	∈	NOUN
cana-3308	374	32	𝑋	𝑋	NOUN
cana-3308	374	33	such	such	ADJ
cana-3308	374	34	that	that	PRON
cana-3308	374	35	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-3308	374	36	,	,	PUNCT
cana-3308	374	37	𝑇𝑥0	𝑇𝑥0	NOUN
cana-3308	374	38	)	)	PUNCT
cana-3308	374	39	≥	≥	NOUN
cana-3308	375	1	1	1	NUM
cana-3308	375	2	.	.	PUNCT
cana-3308	376	1	then	then	ADV
cana-3308	376	2	𝑇	𝑇	PROPN
cana-3308	376	3	has	have	VERB
cana-3308	376	4	a	a	DET
cana-3308	376	5	unique	unique	ADJ
cana-3308	376	6	fixed	fix	VERB
cana-3308	376	7	point	point	NOUN
cana-3308	376	8	.	.	PUNCT
cana-3308	377	1	proof	proof	NOUN
cana-3308	377	2	:	:	PUNCT
cana-3308	377	3	following	follow	VERB
cana-3308	377	4	the	the	DET
cana-3308	377	5	proof	proof	NOUN
cana-3308	377	6	of	of	ADP
cana-3308	377	7	theorem	theorem	NOUN
cana-3308	377	8	3.13	3.13	NUM
cana-3308	377	9	.	.	PUNCT
cana-3308	378	1	𝑇	𝑇	PROPN
cana-3308	378	2	has	have	AUX
cana-3308	378	3	fixed	fix	VERB
cana-3308	378	4	point	point	NOUN
cana-3308	378	5	.	.	PUNCT
cana-3308	379	1	to	to	PART
cana-3308	379	2	prove	prove	VERB
cana-3308	379	3	the	the	DET
cana-3308	379	4	uniqueness	uniqueness	NOUN
cana-3308	379	5	of	of	ADP
cana-3308	379	6	fixed	fix	VERB
cana-3308	379	7	point	point	NOUN
cana-3308	379	8	of	of	ADP
cana-3308	379	9	contravariant	contravariant	ADJ
cana-3308	379	10	mapping	mapping	NOUN
cana-3308	379	11	𝑇	𝑇	PROPN
cana-3308	379	12	in	in	ADP
cana-3308	379	13	complete	complete	ADJ
cana-3308	379	14	bipolar	bipolar	ADJ
cana-3308	379	15	metric	metric	ADJ
cana-3308	379	16	space	space	NOUN
cana-3308	379	17	,	,	PUNCT
cana-3308	379	18	let	let	VERB
cana-3308	379	19	if	if	SCONJ
cana-3308	379	20	possible	possible	ADJ
cana-3308	379	21	,	,	PUNCT
cana-3308	379	22	𝑢	𝑢	PROPN
cana-3308	379	23	and	and	CCONJ
cana-3308	379	24	𝑣	𝑣	PROPN
cana-3308	379	25	are	be	AUX
cana-3308	379	26	two	two	NUM
cana-3308	379	27	distinct	distinct	ADJ
cana-3308	379	28	fixed	fix	VERB
cana-3308	379	29	point	point	NOUN
cana-3308	379	30	of	of	ADP
cana-3308	379	31	𝑇.	𝑇.	PROPN
cana-3308	379	32	i.e.	i.e.	X
cana-3308	379	33	,	,	PUNCT
cana-3308	379	34	𝑇𝑢	𝑇𝑢	PROPN
cana-3308	379	35	=	=	PUNCT
cana-3308	379	36	𝑢	𝑢	NOUN
cana-3308	379	37	and	and	CCONJ
cana-3308	379	38	𝑇𝑣	𝑇𝑣	PROPN
cana-3308	379	39	=	=	PUNCT
cana-3308	379	40	𝑣.	𝑣.	NOUN
cana-3308	379	41	by	by	ADP
cana-3308	379	42	using	use	VERB
cana-3308	379	43	the	the	DET
cana-3308	379	44	properties	property	NOUN
cana-3308	379	45	of	of	ADP
cana-3308	379	46	𝜓	𝜓	PROPN
cana-3308	379	47	,	,	PUNCT
cana-3308	379	48	𝜃	𝜃	PROPN
cana-3308	379	49	,	,	PUNCT
cana-3308	379	50	equations	equation	NOUN
cana-3308	379	51	(	(	PUNCT
cana-3308	379	52	3.1	3.1	NUM
cana-3308	379	53	)	)	PUNCT
cana-3308	379	54	,	,	PUNCT
cana-3308	379	55	(	(	PUNCT
cana-3308	379	56	3.2	3.2	NUM
cana-3308	379	57	)	)	PUNCT
cana-3308	379	58	and	and	CCONJ
cana-3308	379	59	(	(	PUNCT
cana-3308	379	60	3.17	3.17	NUM
cana-3308	379	61	)	)	PUNCT
cana-3308	379	62	,	,	PUNCT
cana-3308	379	63	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	NUM
cana-3308	379	64	,	,	PUNCT
cana-3308	379	65	𝑣	𝑣	NOUN
cana-3308	379	66	)	)	PUNCT
cana-3308	379	67	)	)	PUNCT
cana-3308	380	1	=	=	SYM
cana-3308	380	2	𝜓(𝑑(𝑇𝑢	𝜓(𝑑(𝑇𝑢	NOUN
cana-3308	380	3	,	,	PUNCT
cana-3308	380	4	𝑇𝑣	𝑇𝑣	PROPN
cana-3308	380	5	)	)	PUNCT
cana-3308	380	6	)	)	PUNCT
cana-3308	380	7	≤	≤	NUM
cana-3308	380	8	𝛼(𝑢	𝛼(𝑢	NOUN
cana-3308	380	9	,	,	PUNCT
cana-3308	380	10	𝑣)𝜓(𝑑(𝑇𝑢	𝑣)𝜓(𝑑(𝑇𝑢	PROPN
cana-3308	380	11	,	,	PUNCT
cana-3308	380	12	𝑇𝑣	𝑇𝑣	PROPN
cana-3308	380	13	)	)	PUNCT
cana-3308	380	14	)	)	PUNCT
cana-3308	380	15	≤	≤	NOUN
cana-3308	381	1	𝜃(𝜓(𝑑(𝑢	𝜃(𝜓(𝑑(𝑢	NUM
cana-3308	381	2	,	,	PUNCT
cana-3308	381	3	𝑣)))𝜑(𝜓(𝑑(𝑢	𝑣)))𝜑(𝜓(𝑑(𝑢	PROPN
cana-3308	381	4	,	,	PUNCT
cana-3308	381	5	𝑣	𝑣	NOUN
cana-3308	381	6	)	)	PUNCT
cana-3308	381	7	)	)	PUNCT
cana-3308	381	8	)	)	PUNCT
cana-3308	382	1	≤	≤	NOUN
cana-3308	382	2	𝜑(𝜓(𝑑(𝑢	𝜑(𝜓(𝑑(𝑢	PROPN
cana-3308	382	3	,	,	PUNCT
cana-3308	382	4	𝑣	𝑣	NOUN
cana-3308	382	5	)	)	PUNCT
cana-3308	382	6	)	)	PUNCT
cana-3308	382	7	)	)	PUNCT
cana-3308	383	1	<	<	X
cana-3308	383	2	𝜓(𝑑(𝑢	𝜓(𝑑(𝑢	X
cana-3308	383	3	,	,	PUNCT
cana-3308	383	4	𝑣	𝑣	NOUN
cana-3308	383	5	)	)	PUNCT
cana-3308	383	6	)	)	PUNCT
cana-3308	383	7	.	.	PUNCT
cana-3308	384	1	this	this	PRON
cana-3308	384	2	implies	imply	VERB
cana-3308	384	3	that	that	SCONJ
cana-3308	384	4	𝑑(𝑢	𝑑(𝑢	ADJ
cana-3308	384	5	,	,	PUNCT
cana-3308	384	6	𝑣	𝑣	NOUN
cana-3308	384	7	)	)	PUNCT
cana-3308	384	8	<	<	X
cana-3308	384	9	𝑑(𝑢	𝑑(𝑢	PROPN
cana-3308	384	10	,	,	PUNCT
cana-3308	384	11	𝑣	𝑣	NOUN
cana-3308	384	12	)	)	PUNCT
cana-3308	384	13	,	,	PUNCT
cana-3308	384	14	which	which	PRON
cana-3308	384	15	is	be	AUX
cana-3308	384	16	a	a	DET
cana-3308	384	17	contradiction	contradiction	NOUN
cana-3308	384	18	.	.	PUNCT
cana-3308	385	1	hence	hence	ADV
cana-3308	385	2	,	,	PUNCT
cana-3308	385	3	𝑇	𝑇	PROPN
cana-3308	385	4	has	have	VERB
cana-3308	385	5	a	a	DET
cana-3308	385	6	unique	unique	ADJ
cana-3308	385	7	fixed	fix	VERB
cana-3308	385	8	point	point	NOUN
cana-3308	385	9	.	.	PUNCT
cana-3308	385	10	example	example	NOUN
cana-3308	386	1	3.14	3.14	NUM
cana-3308	386	2	.	.	PUNCT
cana-3308	387	1	in	in	ADP
cana-3308	387	2	the	the	DET
cana-3308	387	3	example	example	NOUN
cana-3308	387	4	3.12	3.12	NUM
cana-3308	387	5	,	,	PUNCT
cana-3308	387	6	we	we	PRON
cana-3308	387	7	can	can	AUX
cana-3308	387	8	easily	easily	ADV
cana-3308	387	9	say	say	VERB
cana-3308	387	10	that	that	SCONJ
cana-3308	387	11	𝑇	𝑇	PROPN
cana-3308	387	12	satisfies	satisfy	VERB
cana-3308	387	13	all	all	DET
cana-3308	387	14	the	the	DET
cana-3308	387	15	conditions	condition	NOUN
cana-3308	387	16	of	of	ADP
cana-3308	387	17	theorem	theorem	NOUN
cana-3308	387	18	3.13	3.13	NUM
cana-3308	387	19	.	.	PUNCT
cana-3308	388	1	so	so	ADV
cana-3308	388	2	,	,	PUNCT
cana-3308	388	3	𝑇	𝑇	PROPN
cana-3308	388	4	has	have	VERB
cana-3308	388	5	a	a	DET
cana-3308	388	6	unique	unique	ADJ
cana-3308	388	7	fixed	fix	VERB
cana-3308	388	8	point	point	NOUN
cana-3308	388	9	.	.	PUNCT
cana-3308	389	1	clearly	clearly	ADV
cana-3308	389	2	,	,	PUNCT
cana-3308	389	3	‘	'	PUNCT
cana-3308	389	4	0	0	NUM
cana-3308	389	5	’	'	PUNCT
cana-3308	389	6	is	be	AUX
cana-3308	389	7	unique	unique	ADJ
cana-3308	389	8	fixed	fix	VERB
cana-3308	389	9	point	point	NOUN
cana-3308	389	10	of	of	ADP
cana-3308	389	11	𝑇.	𝑇.	PROPN
cana-3308	389	12	references	reference	NOUN
cana-3308	389	13	[	[	X
cana-3308	389	14	1	1	NUM
cana-3308	389	15	]	]	PUNCT
cana-3308	389	16	abduletif	abduletif	NOUN
cana-3308	389	17	m.	m.	NOUN
cana-3308	389	18	,	,	PUNCT
cana-3308	389	19	koyas	koyas	PROPN
cana-3308	389	20	k.	k.	PROPN
cana-3308	389	21	and	and	CCONJ
cana-3308	389	22	gebregiorgis	gebregiorgis	PROPN
cana-3308	389	23	s.	s.	PROPN
cana-3308	389	24	,	,	PUNCT
cana-3308	389	25	“	"	PUNCT
cana-3308	389	26	fixed	fix	VERB
cana-3308	389	27	point	point	NOUN
cana-3308	389	28	results	result	NOUN
cana-3308	389	29	for	for	ADP
cana-3308	389	30	generalized	generalized	ADJ
cana-3308	389	31	(	(	PUNCT
cana-3308	389	32	𝛼	𝛼	PROPN
cana-3308	389	33	,	,	PUNCT
cana-3308	389	34	𝜓	𝜓	NOUN
cana-3308	389	35	,	,	PUNCT
cana-3308	389	36	𝜑)geraghty	𝜑)geraghty	ADJ
cana-3308	389	37	contraction	contraction	NOUN
cana-3308	389	38	in	in	ADP
cana-3308	389	39	𝑏-metric	𝑏-metric	PROPN
cana-3308	389	40	spaces	space	NOUN
cana-3308	389	41	”	"	PUNCT
cana-3308	389	42	,	,	PUNCT
cana-3308	389	43	int	int	NOUN
cana-3308	389	44	.	.	PUNCT
cana-3308	390	1	j.	j.	PROPN
cana-3308	390	2	nonlinear	nonlinear	PROPN
cana-3308	390	3	anal	anal	PROPN
cana-3308	390	4	.	.	PUNCT
cana-3308	391	1	appl	appl	PROPN
cana-3308	391	2	.	.	PROPN
cana-3308	391	3	,	,	PUNCT
cana-3308	391	4	14(1	14(1	NUM
cana-3308	391	5	)	)	PUNCT
cana-3308	391	6	(	(	PUNCT
cana-3308	391	7	2023	2023	NUM
cana-3308	391	8	)	)	PUNCT
cana-3308	391	9	,	,	PUNCT
cana-3308	391	10	965	965	NUM
cana-3308	391	11	-	-	SYM
cana-3308	391	12	977	977	NUM
cana-3308	391	13	.	.	PUNCT
cana-3308	392	1	[	[	X
cana-3308	392	2	2	2	X
cana-3308	392	3	]	]	X
cana-3308	392	4	banach	banach	NOUN
cana-3308	392	5	s.	s.	PROPN
cana-3308	392	6	,	,	PUNCT
cana-3308	392	7	“	"	PUNCT
cana-3308	392	8	sur	sur	X
cana-3308	392	9	les	les	X
cana-3308	392	10	opérations	opération	NOUN
cana-3308	392	11	dans	dan	NOUN
cana-3308	392	12	les	les	X
cana-3308	392	13	ensembles	ensemble	NOUN
cana-3308	392	14	abstraits	abstrait	NOUN
cana-3308	392	15	et	et	PROPN
cana-3308	392	16	leur	leur	X
cana-3308	392	17	application	application	PROPN
cana-3308	392	18	aux	aux	PROPN
cana-3308	392	19	équations	équations	PROPN
cana-3308	392	20	integrals	integral	NOUN
cana-3308	392	21	”	"	PUNCT
cana-3308	392	22	,	,	PUNCT
cana-3308	392	23	fundam	fundam	PROPN
cana-3308	392	24	.	.	PUNCT
cana-3308	392	25	math	math	NOUN
cana-3308	392	26	.	.	PUNCT
cana-3308	392	27	,	,	PUNCT
cana-3308	392	28	3(1	3(1	NUM
cana-3308	392	29	)	)	PUNCT
cana-3308	392	30	(	(	PUNCT
cana-3308	392	31	1922	1922	NUM
cana-3308	392	32	)	)	PUNCT
cana-3308	392	33	,	,	PUNCT
cana-3308	392	34	133	133	NUM
cana-3308	392	35	-	-	SYM
cana-3308	392	36	181	181	NUM
cana-3308	392	37	.	.	PUNCT
cana-3308	393	1	[	[	X
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cana-3308	393	3	]	]	X
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cana-3308	393	12	with	with	ADP
cana-3308	393	13	applications	application	NOUN
cana-3308	393	14	to	to	ADP
cana-3308	393	15	economics	economic	NOUN
cana-3308	393	16	and	and	CCONJ
cana-3308	393	17	game	game	NOUN
cana-3308	393	18	theory	theory	NOUN
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cana-3308	393	20	,	,	PUNCT
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cana-3308	394	13	(	(	PUNCT
cana-3308	394	14	𝛼	𝛼	PROPN
cana-3308	394	15	,	,	PUNCT
cana-3308	394	16	𝛽	𝛽	NOUN
cana-3308	394	17	)	)	PUNCT
cana-3308	394	18	−	−	PROPN
cana-3308	394	19	admissible	admissible	ADJ
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cana-3308	396	2	5	5	X
cana-3308	396	3	]	]	X
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cana-3308	396	17	bipolar	bipolar	ADJ
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cana-3308	396	21	,	,	PUNCT
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cana-3308	396	27	(	(	PUNCT
cana-3308	396	28	2021	2021	NUM
cana-3308	396	29	)	)	PUNCT
cana-3308	396	30	,	,	PUNCT
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cana-3308	396	32	.	.	PUNCT
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cana-3308	397	7	“	"	PUNCT
cana-3308	397	8	𝛼	𝛼	PROPN
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cana-3308	397	10	𝜓-geraghty	𝜓-geraghty	NOUN
cana-3308	397	11	contraction	contraction	NOUN
cana-3308	397	12	type	type	NOUN
cana-3308	397	13	mappings	mapping	NOUN
cana-3308	397	14	and	and	CCONJ
cana-3308	397	15	some	some	DET
cana-3308	397	16	related	related	ADJ
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cana-3308	397	18	point	point	NOUN
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cana-3308	397	25	)	)	PUNCT
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cana-3308	397	28	)	)	PUNCT
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cana-3308	398	3	]	]	X
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cana-3308	398	8	and	and	CCONJ
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cana-3308	398	14	ciric	ciric	ADJ
cana-3308	398	15	type𝜓-geraghty	type𝜓-geraghty	ADJ
cana-3308	398	16	contractions	contraction	NOUN
cana-3308	398	17	”	"	PUNCT
cana-3308	398	18	,	,	PUNCT
cana-3308	398	19	thai	thai	PROPN
cana-3308	398	20	j.	j.	PROPN
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cana-3308	398	23	,	,	PUNCT
cana-3308	398	24	17(1	17(1	NUM
cana-3308	398	25	)	)	PUNCT
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cana-3308	398	28	)	)	PUNCT
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cana-3308	399	2	8	8	NUM
cana-3308	399	3	]	]	X
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cana-3308	399	7	“	"	PUNCT
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cana-3308	399	11	”	"	PUNCT
cana-3308	399	12	,	,	PUNCT
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cana-3308	402	5	)	)	PUNCT
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cana-3308	402	8	-	-	SYM
cana-3308	402	9	608	608	NUM
cana-3308	402	10	.	.	PUNCT
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cana-3308	403	2	on	on	ADP
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cana-3308	403	7	:	:	PUNCT
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cana-3308	403	10	133x	133x	NUM
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cana-3308	403	12	32	32	NUM
cana-3308	403	13	no	no	NOUN
cana-3308	403	14	.	.	PUNCT
cana-3308	404	1	6s	6s	NUM
cana-3308	404	2	(	(	PUNCT
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cana-3308	404	4	)	)	PUNCT
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cana-3308	405	8	r.	r.	PROPN
cana-3308	405	9	,	,	PUNCT
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cana-3308	405	11	a.j	a.j	PROPN
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cana-3308	405	34	f	f	NOUN
cana-3308	405	35	-	-	PUNCT
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cana-3308	407	3	]	]	PUNCT
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cana-3308	408	9	)	)	PUNCT
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cana-3308	408	12	)	)	PUNCT
cana-3308	408	13	,	,	PUNCT
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cana-3308	409	2	11	11	NUM
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cana-3308	409	20	𝜓	𝜓	PROPN
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cana-3308	409	23	in	in	ADP
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cana-3308	410	3	,	,	PUNCT
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cana-3308	410	5	)	)	PUNCT
cana-3308	410	6	(	(	PUNCT
cana-3308	410	7	2020	2020	NUM
cana-3308	410	8	)	)	PUNCT
cana-3308	410	9	,	,	PUNCT
cana-3308	410	10	64	64	NUM
cana-3308	410	11	-	-	SYM
cana-3308	410	12	75	75	NUM
cana-3308	410	13	.	.	PUNCT
cana-3308	411	1	[	[	X
cana-3308	411	2	12	12	NUM
cana-3308	411	3	]	]	X
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cana-3308	411	5	r.	r.	PROPN
cana-3308	411	6	,	,	PUNCT
cana-3308	411	7	mani	mani	PROPN
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cana-3308	411	9	,	,	PUNCT
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cana-3308	411	11	a.j	a.j	PROPN
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cana-3308	411	13	,	,	PUNCT
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cana-3308	411	15	o.a.a	o.a.a	PROPN
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cana-3308	411	17	,	,	PUNCT
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cana-3308	411	32	and	and	CCONJ
cana-3308	411	33	contravariant	contravariant	PROPN
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cana-3308	411	35	with	with	ADP
cana-3308	411	36	an	an	DET
cana-3308	411	37	application	application	NOUN
cana-3308	411	38	”	"	PUNCT
cana-3308	411	39	,	,	PUNCT
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cana-3308	411	41	,	,	PUNCT
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cana-3308	411	43	(	(	PUNCT
cana-3308	411	44	2022	2022	NUM
cana-3308	411	45	)	)	PUNCT
cana-3308	411	46	,	,	PUNCT
cana-3308	411	47	4385	4385	NUM
cana-3308	411	48	.	.	PUNCT
cana-3308	412	1	[	[	X
cana-3308	412	2	13	13	NUM
cana-3308	412	3	]	]	X
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cana-3308	412	5	b.s	b.s	PROPN
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cana-3308	412	7	,	,	PUNCT
cana-3308	412	8	kishore	kishore	PROPN
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cana-3308	412	10	.	.	PUNCT
cana-3308	413	1	and	and	CCONJ
cana-3308	413	2	kumar	kumar	PROPN
cana-3308	413	3	g.k	g.k	PROPN
cana-3308	413	4	.	.	PROPN
cana-3308	413	5	,	,	PUNCT
cana-3308	413	6	“	"	PUNCT
cana-3308	413	7	geraghty	geraghty	PROPN
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cana-3308	413	9	contraction	contraction	NOUN
cana-3308	413	10	and	and	CCONJ
cana-3308	413	11	common	common	ADJ
cana-3308	413	12	coupled	couple	VERB
cana-3308	413	13	fixed	fix	VERB
cana-3308	413	14	point	point	NOUN
cana-3308	413	15	theorems	theorem	NOUN
cana-3308	413	16	in	in	ADP
cana-3308	413	17	bipolar	bipolar	ADJ
cana-3308	413	18	metric	metric	ADJ
cana-3308	413	19	spaces	space	NOUN
cana-3308	413	20	with	with	ADP
cana-3308	413	21	applications	application	NOUN
cana-3308	413	22	to	to	PART
cana-3308	413	23	homotopy	homotopy	VERB
cana-3308	413	24	”	"	PUNCT
cana-3308	413	25	,	,	PUNCT
cana-3308	413	26	int	int	NOUN
cana-3308	413	27	.	.	PUNCT
cana-3308	414	1	j.	j.	PROPN
cana-3308	414	2	math	math	PROPN
cana-3308	414	3	.	.	PUNCT
cana-3308	415	1	trends	trend	NOUN
cana-3308	415	2	technol	technol	ADJ
cana-3308	415	3	.	.	PROPN
cana-3308	415	4	,	,	PUNCT
cana-3308	415	5	63	63	NUM
cana-3308	415	6	(	(	PUNCT
cana-3308	415	7	2018	2018	NUM
cana-3308	415	8	)	)	PUNCT
cana-3308	415	9	,	,	PUNCT
cana-3308	415	10	25–34	25–34	NUM
cana-3308	415	11	.	.	PUNCT
cana-3308	416	1	[	[	X
cana-3308	416	2	14	14	NUM
cana-3308	416	3	]	]	X
cana-3308	416	4	samet	samet	PROPN
cana-3308	416	5	b.	b.	PROPN
cana-3308	416	6	,	,	PUNCT
cana-3308	416	7	vetro	vetro	PROPN
cana-3308	416	8	c.	c.	NOUN
cana-3308	416	9	and	and	CCONJ
cana-3308	416	10	vetro	vetro	PROPN
cana-3308	416	11	p.	p.	NOUN
cana-3308	416	12	,	,	PUNCT
cana-3308	416	13	“	"	PUNCT
cana-3308	416	14	fixed	fix	VERB
cana-3308	416	15	point	point	NOUN
cana-3308	416	16	theorems	theorem	NOUN
cana-3308	416	17	for	for	ADP
cana-3308	416	18	𝛼	𝛼	NOUN
cana-3308	416	19	−	−	PROPN
cana-3308	416	20	𝜓	𝜓	NOUN
cana-3308	416	21	−	−	ADP
cana-3308	416	22	contractive	contractive	ADJ
cana-3308	416	23	type	type	NOUN
cana-3308	416	24	mappings	mapping	NOUN
cana-3308	416	25	”	"	PUNCT
cana-3308	416	26	,	,	PUNCT
cana-3308	416	27	nonlinear	nonlinear	ADJ
cana-3308	416	28	anal	anal	NOUN
cana-3308	416	29	.	.	PUNCT
cana-3308	417	1	theory	theory	NOUN
cana-3308	417	2	methods	method	NOUN
cana-3308	417	3	appl	appl	PROPN
cana-3308	417	4	.	.	PUNCT
cana-3308	417	5	,	,	PUNCT
cana-3308	417	6	75(4	75(4	NOUN
cana-3308	417	7	)	)	PUNCT
cana-3308	417	8	(	(	PUNCT
cana-3308	417	9	2012	2012	NUM
cana-3308	417	10	)	)	PUNCT
cana-3308	417	11	,	,	PUNCT
cana-3308	417	12	2154	2154	NUM
cana-3308	417	13	-	-	SYM
cana-3308	417	14	2165	2165	NUM
cana-3308	417	15	.	.	PUNCT
cana-3308	418	1	[	[	X
cana-3308	418	2	15	15	NUM
cana-3308	418	3	]	]	X
cana-3308	418	4	shahi	shahi	PROPN
cana-3308	418	5	p.	p.	PROPN
cana-3308	418	6	,	,	PUNCT
cana-3308	418	7	kaur	kaur	PROPN
cana-3308	418	8	j.	j.	PROPN
cana-3308	418	9	and	and	CCONJ
cana-3308	418	10	bhatia	bhatia	PROPN
cana-3308	418	11	s	s	PROPN
cana-3308	418	12	s	s	PROPN
cana-3308	418	13	,	,	PUNCT
cana-3308	418	14	“	"	PUNCT
cana-3308	418	15	fixed	fix	VERB
cana-3308	418	16	point	point	NOUN
cana-3308	418	17	theorems	theorem	NOUN
cana-3308	418	18	for	for	ADP
cana-3308	418	19	(	(	PUNCT
cana-3308	418	20	𝜉	𝜉	X
cana-3308	418	21	,	,	PUNCT
cana-3308	418	22	𝛼	𝛼	NOUN
cana-3308	418	23	)	)	PUNCT
cana-3308	418	24	−	−	ADP
cana-3308	418	25	expansive	expansive	ADJ
cana-3308	418	26	mappings	mapping	NOUN
cana-3308	418	27	in	in	ADP
cana-3308	418	28	complete	complete	ADJ
cana-3308	418	29	metric	metric	ADJ
cana-3308	418	30	space	space	NOUN
cana-3308	418	31	”	"	PUNCT
cana-3308	418	32	,	,	PUNCT
cana-3308	418	33	fixed	fix	VERB
cana-3308	418	34	point	point	NOUN
cana-3308	418	35	theory	theory	NOUN
cana-3308	418	36	appl	appl	PROPN
cana-3308	418	37	.	.	PROPN
cana-3308	418	38	,	,	PUNCT
cana-3308	418	39	2012	2012	NUM
cana-3308	418	40	(	(	PUNCT
cana-3308	418	41	2012	2012	NUM
cana-3308	418	42	)	)	PUNCT
cana-3308	418	43	,	,	PUNCT
cana-3308	418	44	157	157	NUM
cana-3308	418	45	.	.	PUNCT
cana-3308	419	1	[	[	X
cana-3308	419	2	16	16	NUM
cana-3308	419	3	]	]	X
cana-3308	419	4	zeidler	zeidler	PROPN
cana-3308	419	5	e.	e.	PROPN
cana-3308	419	6	,	,	PUNCT
cana-3308	419	7	“	"	PUNCT
cana-3308	419	8	nonlinear	nonlinear	ADJ
cana-3308	419	9	functinal	functinal	ADJ
cana-3308	419	10	analysis	analysis	NOUN
cana-3308	419	11	and	and	CCONJ
cana-3308	419	12	its	its	PRON
cana-3308	419	13	applications	application	NOUN
cana-3308	419	14	”	"	PUNCT
cana-3308	419	15	,	,	PUNCT
cana-3308	419	16	springer	springer	NOUN
cana-3308	419	17	new	new	PROPN
cana-3308	419	18	york	york	PROPN
cana-3308	419	19	,	,	PUNCT
cana-3308	419	20	(	(	PUNCT
cana-3308	419	21	1989	1989	NUM
cana-3308	419	22	)	)	PUNCT
cana-3308	419	23	.	.	PUNCT