id	sid	tid	token	lemma	pos
cana-787	1	1	communications	communication	NOUN
cana-787	1	2	on	on	ADP
cana-787	1	3	applied	apply	VERB
cana-787	1	4	nonlinear	nonlinear	ADJ
cana-787	1	5	analysis	analysis	NOUN
cana-787	1	6	issn	issn	NOUN
cana-787	1	7	:	:	PUNCT
cana-787	1	8	1074	1074	NUM
cana-787	1	9	-	-	PUNCT
cana-787	1	10	133x	133x	NUM
cana-787	1	11	vol	vol	NOUN
cana-787	1	12	31	31	NUM
cana-787	1	13	no	no	NOUN
cana-787	1	14	.	.	PUNCT
cana-787	2	1	3s	3s	NUM
cana-787	2	2	(	(	PUNCT
cana-787	2	3	2024	2024	NUM
cana-787	2	4	)	)	PUNCT
cana-787	2	5	367	367	NUM
cana-787	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	2	7	some	some	DET
cana-787	2	8	new	new	ADJ
cana-787	2	9	types	type	NOUN
cana-787	2	10	of	of	ADP
cana-787	2	11	𝜶𝜷−𝟏-admissible	𝜶𝜷−𝟏-admissible	ADJ
cana-787	2	12	mappings	mapping	NOUN
cana-787	2	13	and	and	CCONJ
cana-787	2	14	fixed	fix	VERB
cana-787	2	15	point	point	NOUN
cana-787	2	16	results	result	VERB
cana-787	2	17	dev	dev	PROPN
cana-787	2	18	raj	raj	PROPN
cana-787	2	19	joshi	joshi	PROPN
cana-787	2	20	,	,	PUNCT
cana-787	2	21	piyush	piyush	PROPN
cana-787	2	22	kumar	kumar	PROPN
cana-787	2	23	tripathi	tripathi	PROPN
cana-787	2	24	,	,	PUNCT
cana-787	2	25	alok	alok	PROPN
cana-787	2	26	kumar	kumar	PROPN
cana-787	2	27	agrawal	agrawal	PROPN
cana-787	2	28	and	and	CCONJ
cana-787	2	29	ashutosh	ashutosh	PROPN
cana-787	2	30	mishra	mishra	PROPN
cana-787	2	31	department	department	PROPN
cana-787	2	32	of	of	ADP
cana-787	2	33	mathematics	mathematics	PROPN
cana-787	2	34	,	,	PUNCT
cana-787	2	35	asas	asa	NOUN
cana-787	2	36	,	,	PUNCT
cana-787	2	37	amity	amity	NOUN
cana-787	2	38	university	university	NOUN
cana-787	2	39	uttar	uttar	PROPN
cana-787	2	40	pradesh	pradesh	PROPN
cana-787	2	41	lucknow	lucknow	PROPN
cana-787	2	42	campus	campus	PROPN
cana-787	2	43	,	,	PUNCT
cana-787	2	44	lucknow-226028	lucknow-226028	NOUN
cana-787	2	45	,	,	PUNCT
cana-787	2	46	india	india	PROPN
cana-787	2	47	email:dev.joshi@s.amity.edu	email:dev.joshi@s.amity.edu	PROPN
cana-787	2	48	,	,	PUNCT
cana-787	2	49	pktripathi@lko.amity.edu	pktripathi@lko.amity.edu	NOUN
cana-787	2	50	,	,	PUNCT
cana-787	2	51	akagraw@lko.amity.edu	akagraw@lko.amity.edu	PROPN
cana-787	2	52	and	and	CCONJ
cana-787	2	53	ashutosh.mishra8@s.amity.edu	ashutosh.mishra8@s.amity.edu	PROPN
cana-787	2	54	article	article	NOUN
cana-787	2	55	history	history	NOUN
cana-787	2	56	:	:	PUNCT
cana-787	2	57	received	receive	VERB
cana-787	2	58	:	:	PUNCT
cana-787	2	59	05	05	NUM
cana-787	2	60	-	-	PUNCT
cana-787	2	61	04	04	NUM
cana-787	2	62	-	-	PUNCT
cana-787	2	63	2024	2024	NUM
cana-787	2	64	revised	revise	VERB
cana-787	2	65	:	:	PUNCT
cana-787	2	66	30	30	NUM
cana-787	2	67	-	-	SYM
cana-787	2	68	05	05	NUM
cana-787	2	69	-	-	PUNCT
cana-787	2	70	2024	2024	NUM
cana-787	2	71	accepted	accept	VERB
cana-787	2	72	:	:	PUNCT
cana-787	2	73	14	14	NUM
cana-787	2	74	-	-	SYM
cana-787	2	75	06	06	NUM
cana-787	2	76	-	-	PUNCT
cana-787	2	77	2024	2024	NUM
cana-787	2	78	abstract	abstract	NOUN
cana-787	2	79	:	:	PUNCT
cana-787	2	80	in	in	ADP
cana-787	2	81	this	this	DET
cana-787	2	82	paper	paper	NOUN
cana-787	2	83	fixed	fix	VERB
cana-787	2	84	point	point	NOUN
cana-787	2	85	and	and	CCONJ
cana-787	2	86	common	common	ADJ
cana-787	2	87	fixed	fix	VERB
cana-787	2	88	point	point	NOUN
cana-787	2	89	theorems	theorem	NOUN
cana-787	2	90	are	be	AUX
cana-787	2	91	proved	prove	VERB
cana-787	2	92	for	for	ADP
cana-787	2	93	𝛼	𝛼	PROPN
cana-787	2	94	−	−	NOUN
cana-787	2	95	𝜑	𝜑	ADP
cana-787	2	96	contractions	contraction	NOUN
cana-787	2	97	under	under	ADP
cana-787	2	98	the	the	DET
cana-787	2	99	𝛼𝛽−1	𝛼𝛽−1	NOUN
cana-787	2	100	-	-	ADJ
cana-787	2	101	admissible	admissible	ADJ
cana-787	2	102	and	and	CCONJ
cana-787	2	103	𝜇	𝜇	ADP
cana-787	2	104	−	−	X
cana-787	2	105	𝛼𝛽−1	𝛼𝛽−1	ADP
cana-787	2	106	admissible	admissible	ADJ
cana-787	2	107	conditions	condition	NOUN
cana-787	2	108	.	.	PUNCT
cana-787	3	1	these	these	DET
cana-787	3	2	results	result	NOUN
cana-787	3	3	generalize	generalize	VERB
cana-787	3	4	the	the	DET
cana-787	3	5	𝛼	𝛼	ADJ
cana-787	3	6	–	–	PUNCT
cana-787	3	7	admissible	admissible	ADJ
cana-787	3	8	condition	condition	NOUN
cana-787	3	9	of	of	ADP
cana-787	3	10	self	self	NOUN
cana-787	3	11	-	-	PUNCT
cana-787	3	12	mappings	mapping	NOUN
cana-787	3	13	as	as	SCONJ
cana-787	3	14	explained	explain	VERB
cana-787	3	15	by	by	ADP
cana-787	3	16	many	many	ADJ
cana-787	3	17	researchers	researcher	NOUN
cana-787	3	18	in	in	ADP
cana-787	3	19	the	the	DET
cana-787	3	20	literature	literature	NOUN
cana-787	3	21	herein	herein	NOUN
cana-787	3	22	.	.	PUNCT
cana-787	4	1	in	in	ADP
cana-787	4	2	general	general	ADJ
cana-787	4	3	,	,	PUNCT
cana-787	4	4	many	many	ADJ
cana-787	4	5	research	research	NOUN
cana-787	4	6	reports	report	NOUN
cana-787	4	7	introduced	introduce	VERB
cana-787	4	8	the	the	DET
cana-787	4	9	𝛼	𝛼	ADJ
cana-787	4	10	–	–	PUNCT
cana-787	4	11	admissible	admissible	ADJ
cana-787	4	12	condition	condition	NOUN
cana-787	4	13	on	on	ADP
cana-787	4	14	the	the	DET
cana-787	4	15	basis	basis	NOUN
cana-787	4	16	of	of	ADP
cana-787	4	17	𝛼(𝑝	𝛼(𝑝	PROPN
cana-787	4	18	,	,	PUNCT
cana-787	4	19	𝑞	𝑞	PROPN
cana-787	4	20	)	)	PUNCT
cana-787	4	21	>	>	X
cana-787	5	1	1	1	X
cana-787	5	2	.	.	PUNCT
cana-787	6	1	however	however	ADV
cana-787	6	2	,	,	PUNCT
cana-787	6	3	this	this	DET
cana-787	6	4	condition	condition	NOUN
cana-787	6	5	is	be	AUX
cana-787	6	6	not	not	PART
cana-787	6	7	applicable	applicable	ADJ
cana-787	6	8	for	for	ADP
cana-787	6	9	finding	find	VERB
cana-787	6	10	the	the	DET
cana-787	6	11	fixed	fix	VERB
cana-787	6	12	points	point	NOUN
cana-787	6	13	of	of	ADP
cana-787	6	14	self	self	NOUN
cana-787	6	15	-	-	PUNCT
cana-787	6	16	mappings	mapping	NOUN
cana-787	6	17	in	in	ADP
cana-787	6	18	many	many	ADJ
cana-787	6	19	cases	case	NOUN
cana-787	6	20	.	.	PUNCT
cana-787	7	1	we	we	PRON
cana-787	7	2	propose	propose	VERB
cana-787	7	3	𝛼	𝛼	NOUN
cana-787	7	4	–	–	PUNCT
cana-787	7	5	admissible	admissible	ADJ
cana-787	7	6	on	on	ADP
cana-787	7	7	the	the	DET
cana-787	7	8	basis	basis	NOUN
cana-787	7	9	of	of	ADP
cana-787	7	10	𝛼(𝑝	𝛼(𝑝	PROPN
cana-787	7	11	,	,	PUNCT
cana-787	7	12	𝑞	𝑞	NOUN
cana-787	7	13	)	)	PUNCT
cana-787	7	14	>	>	X
cana-787	8	1	𝛽−1	𝛽−1	PROPN
cana-787	8	2	which	which	PRON
cana-787	8	3	generalize	generalize	VERB
cana-787	8	4	the	the	DET
cana-787	8	5	case	case	NOUN
cana-787	8	6	we	we	PRON
cana-787	8	7	mentioned	mention	VERB
cana-787	8	8	in	in	ADP
cana-787	8	9	the	the	DET
cana-787	8	10	literature	literature	NOUN
cana-787	8	11	.	.	PUNCT
cana-787	9	1	to	to	PART
cana-787	9	2	support	support	VERB
cana-787	9	3	our	our	PRON
cana-787	9	4	new	new	ADJ
cana-787	9	5	concepts	concept	NOUN
cana-787	9	6	,	,	PUNCT
cana-787	9	7	we	we	PRON
cana-787	9	8	present	present	VERB
cana-787	9	9	two	two	NUM
cana-787	9	10	examples	example	NOUN
cana-787	9	11	at	at	ADP
cana-787	9	12	the	the	DET
cana-787	9	13	end	end	NOUN
cana-787	9	14	of	of	ADP
cana-787	9	15	this	this	DET
cana-787	9	16	paper	paper	NOUN
cana-787	9	17	.	.	PUNCT
cana-787	10	1	keywords	keyword	NOUN
cana-787	10	2	:	:	PUNCT
cana-787	10	3	𝛼𝛽−1	𝛼𝛽−1	ADJ
cana-787	10	4	-	-	ADJ
cana-787	10	5	admissible	admissible	ADJ
cana-787	10	6	mappings	mapping	NOUN
cana-787	10	7	,	,	PUNCT
cana-787	10	8	𝜇	𝜇	ADP
cana-787	10	9	−	−	PROPN
cana-787	10	10	𝛼𝛽−1admissible	𝛼𝛽−1admissible	ADJ
cana-787	10	11	mapping	mapping	NOUN
cana-787	10	12	,	,	PUNCT
cana-787	10	13	𝜇	𝜇	ADP
cana-787	10	14	−	−	PROPN
cana-787	10	15	𝛼	𝛼	PROPN
cana-787	10	16	,	,	PUNCT
cana-787	10	17	𝜑	𝜑	PRON
cana-787	10	18	contraction	contraction	NOUN
cana-787	10	19	.	.	PUNCT
cana-787	11	1	mathematics	mathematic	NOUN
cana-787	11	2	subject	subject	ADJ
cana-787	11	3	classification	classification	NOUN
cana-787	11	4	:	:	PUNCT
cana-787	11	5	47h10	47h10	NUM
cana-787	11	6	,	,	PUNCT
cana-787	11	7	54h25	54h25	NUM
cana-787	11	8	.	.	PUNCT
cana-787	12	1	1	1	X
cana-787	12	2	.	.	X
cana-787	12	3	introduction	introduction	NOUN
cana-787	12	4	in	in	ADP
cana-787	12	5	the	the	DET
cana-787	12	6	field	field	NOUN
cana-787	12	7	of	of	ADP
cana-787	12	8	fixed	fix	VERB
cana-787	12	9	point	point	NOUN
cana-787	12	10	theory	theory	NOUN
cana-787	12	11	,	,	PUNCT
cana-787	12	12	banach	banach	NOUN
cana-787	12	13	contraction	contraction	NOUN
cana-787	12	14	principle	principle	NOUN
cana-787	12	15	[	[	X
cana-787	12	16	1	1	X
cana-787	12	17	]	]	PUNCT
cana-787	12	18	is	be	AUX
cana-787	12	19	one	one	NUM
cana-787	12	20	of	of	ADP
cana-787	12	21	the	the	DET
cana-787	12	22	fundamental	fundamental	ADJ
cana-787	12	23	tools	tool	NOUN
cana-787	12	24	to	to	PART
cana-787	12	25	explain	explain	VERB
cana-787	12	26	existence	existence	NOUN
cana-787	12	27	of	of	ADP
cana-787	12	28	fixed	fix	VERB
cana-787	12	29	point	point	NOUN
cana-787	12	30	in	in	ADP
cana-787	12	31	complete	complete	ADJ
cana-787	12	32	metric	metric	ADJ
cana-787	12	33	spaces	space	NOUN
cana-787	12	34	.	.	PUNCT
cana-787	13	1	after	after	ADP
cana-787	13	2	this	this	DET
cana-787	13	3	principle	principle	NOUN
cana-787	13	4	,	,	PUNCT
cana-787	13	5	many	many	ADJ
cana-787	13	6	researchers	researcher	NOUN
cana-787	13	7	have	have	AUX
cana-787	13	8	suggested	suggest	VERB
cana-787	13	9	different	different	ADJ
cana-787	13	10	generalizations	generalization	NOUN
cana-787	13	11	of	of	ADP
cana-787	13	12	contraction	contraction	NOUN
cana-787	13	13	principle	principle	NOUN
cana-787	13	14	and	and	CCONJ
cana-787	13	15	their	their	PRON
cana-787	13	16	applications	application	NOUN
cana-787	13	17	to	to	PART
cana-787	13	18	solve	solve	VERB
cana-787	13	19	many	many	ADJ
cana-787	13	20	mathematical	mathematical	ADJ
cana-787	13	21	problems	problem	NOUN
cana-787	13	22	such	such	ADJ
cana-787	13	23	as	as	ADP
cana-787	13	24	solution	solution	NOUN
cana-787	13	25	of	of	ADP
cana-787	13	26	differential	differential	ADJ
cana-787	13	27	equations	equation	NOUN
cana-787	13	28	,	,	PUNCT
cana-787	13	29	integral	integral	ADJ
cana-787	13	30	equations	equation	NOUN
cana-787	13	31	and	and	CCONJ
cana-787	13	32	non	non	ADJ
cana-787	13	33	-	-	ADJ
cana-787	13	34	linear	linear	ADJ
cana-787	13	35	analysis	analysis	NOUN
cana-787	13	36	.	.	PUNCT
cana-787	14	1	among	among	ADP
cana-787	14	2	the	the	DET
cana-787	14	3	generalizations	generalization	NOUN
cana-787	14	4	of	of	ADP
cana-787	14	5	banach	banach	NOUN
cana-787	14	6	contraction	contraction	NOUN
cana-787	14	7	principle	principle	NOUN
cana-787	14	8	,	,	PUNCT
cana-787	14	9	we	we	PRON
cana-787	14	10	discuss	discuss	VERB
cana-787	14	11	𝛼	𝛼	PRON
cana-787	14	12	−	−	PROPN
cana-787	14	13	𝜑	𝜑	PRON
cana-787	14	14	contraction	contraction	NOUN
cana-787	14	15	here	here	ADV
cana-787	14	16	.	.	PUNCT
cana-787	15	1	it	it	PRON
cana-787	15	2	was	be	AUX
cana-787	15	3	initiated	initiate	VERB
cana-787	15	4	by	by	ADP
cana-787	15	5	samet	samet	PROPN
cana-787	15	6	et	et	PROPN
cana-787	15	7	al	al	PROPN
cana-787	15	8	.	.	PUNCT
cana-787	16	1	[	[	X
cana-787	16	2	2	2	NUM
cana-787	16	3	]	]	PUNCT
cana-787	16	4	in	in	ADP
cana-787	16	5	which	which	PRON
cana-787	16	6	they	they	PRON
cana-787	16	7	introduced	introduce	VERB
cana-787	16	8	𝛼	𝛼	PRON
cana-787	16	9	−	−	PROPN
cana-787	16	10	𝜑	𝜑	PRON
cana-787	16	11	contraction	contraction	NOUN
cana-787	16	12	type	type	NOUN
cana-787	16	13	mappings	mapping	NOUN
cana-787	16	14	and	and	CCONJ
cana-787	16	15	proved	prove	VERB
cana-787	16	16	fixed	fix	VERB
cana-787	16	17	point	point	NOUN
cana-787	16	18	theorems	theorem	NOUN
cana-787	16	19	in	in	ADP
cana-787	16	20	complete	complete	ADJ
cana-787	16	21	metric	metric	ADJ
cana-787	16	22	spaces	space	NOUN
cana-787	16	23	.	.	PUNCT
cana-787	17	1	this	this	DET
cana-787	17	2	concept	concept	NOUN
cana-787	17	3	was	be	AUX
cana-787	17	4	further	far	ADV
cana-787	17	5	improved	improve	VERB
cana-787	17	6	and	and	CCONJ
cana-787	17	7	modified	modify	VERB
cana-787	17	8	by	by	ADP
cana-787	17	9	b.	b.	PROPN
cana-787	17	10	samet	samet	PROPN
cana-787	18	1	[	[	X
cana-787	18	2	3	3	NUM
cana-787	18	3	]	]	PUNCT
cana-787	18	4	alone	alone	ADV
cana-787	18	5	.	.	PUNCT
cana-787	19	1	after	after	ADP
cana-787	19	2	the	the	DET
cana-787	19	3	paper	paper	NOUN
cana-787	19	4	of	of	ADP
cana-787	19	5	samet	samet	PROPN
cana-787	19	6	et	et	PROPN
cana-787	19	7	al	al	PROPN
cana-787	19	8	.	.	PUNCT
cana-787	20	1	the	the	DET
cana-787	20	2	𝛼	𝛼	PROPN
cana-787	20	3	−	−	NOUN
cana-787	20	4	𝜑	𝜑	PRON
cana-787	20	5	contraction	contraction	NOUN
cana-787	20	6	is	be	AUX
cana-787	20	7	used	use	VERB
cana-787	20	8	by	by	ADP
cana-787	20	9	many	many	ADJ
cana-787	20	10	researchers	researcher	NOUN
cana-787	20	11	in	in	ADP
cana-787	20	12	different	different	ADJ
cana-787	20	13	metric	metric	ADJ
cana-787	20	14	spaces	space	NOUN
cana-787	20	15	.	.	PUNCT
cana-787	21	1	karapinar	karapinar	VERB
cana-787	21	2	et	et	PROPN
cana-787	21	3	al	al	PROPN
cana-787	21	4	.	.	PUNCT
cana-787	22	1	[	[	X
cana-787	22	2	4	4	X
cana-787	22	3	]	]	PUNCT
cana-787	22	4	used	use	VERB
cana-787	22	5	this	this	DET
cana-787	22	6	contraction	contraction	NOUN
cana-787	22	7	in	in	ADP
cana-787	22	8	b	b	ADJ
cana-787	22	9	-	-	PUNCT
cana-787	22	10	metric	metric	ADJ
cana-787	22	11	space	space	NOUN
cana-787	22	12	with	with	ADP
cana-787	22	13	𝛼-orbital	𝛼-orbital	PROPN
cana-787	22	14	admissible	admissible	ADJ
cana-787	22	15	condition	condition	NOUN
cana-787	22	16	.	.	PUNCT
cana-787	23	1	li	li	PROPN
cana-787	23	2	.	.	PROPN
cana-787	23	3	and	and	CCONJ
cana-787	23	4	guan	guan	PROPN
cana-787	24	1	[	[	X
cana-787	24	2	5	5	NUM
cana-787	24	3	]	]	PUNCT
cana-787	24	4	introduced	introduce	VERB
cana-787	24	5	𝛼𝑠𝑝	𝛼𝑠𝑝	PROPN
cana-787	24	6	and	and	CCONJ
cana-787	24	7	𝜌	𝜌	X
cana-787	24	8	−	−	PROPN
cana-787	24	9	𝛼𝑠𝑝	𝛼𝑠𝑝	PROPN
cana-787	24	10	admissible	admissible	ADJ
cana-787	24	11	condition	condition	NOUN
cana-787	24	12	in	in	ADP
cana-787	24	13	b	b	NOUN
cana-787	24	14	-	-	PUNCT
cana-787	24	15	metric	metric	ADJ
cana-787	24	16	space	space	NOUN
cana-787	24	17	.	.	PUNCT
cana-787	25	1	likewise	likewise	ADV
cana-787	25	2	,	,	PUNCT
cana-787	25	3	hussain	hussain	PROPN
cana-787	25	4	et	et	PROPN
cana-787	25	5	al	al	PROPN
cana-787	25	6	.	.	PUNCT
cana-787	26	1	[	[	X
cana-787	26	2	6	6	NUM
cana-787	26	3	]	]	PUNCT
cana-787	26	4	introduced	introduce	VERB
cana-787	26	5	𝛼	𝛼	NOUN
cana-787	26	6	–	–	PUNCT
cana-787	26	7	admissible	admissible	ADJ
cana-787	26	8	mapping	mapping	NOUN
cana-787	26	9	with	with	ADP
cana-787	26	10	respect	respect	NOUN
cana-787	26	11	to	to	ADP
cana-787	26	12	𝜂	𝜂	PROPN
cana-787	26	13	.	.	PUNCT
cana-787	27	1	zoto	zoto	PROPN
cana-787	27	2	et	et	PROPN
cana-787	27	3	al	al	PROPN
cana-787	27	4	.	.	PUNCT
cana-787	28	1	[	[	X
cana-787	28	2	7	7	X
cana-787	28	3	]	]	PUNCT
cana-787	28	4	introduced	introduce	VERB
cana-787	28	5	𝛼𝑞𝑠𝑝	𝛼𝑞𝑠𝑝	NOUN
cana-787	28	6	admissible	admissible	ADJ
cana-787	28	7	mapping	mapping	NOUN
cana-787	28	8	and	and	CCONJ
cana-787	28	9	prove	prove	VERB
cana-787	28	10	the	the	DET
cana-787	28	11	fixed	fix	VERB
cana-787	28	12	point	point	NOUN
cana-787	28	13	theorem	theorem	VERB
cana-787	28	14	in	in	ADP
cana-787	28	15	b	b	NOUN
cana-787	28	16	-	-	PUNCT
cana-787	28	17	metric	metric	ADJ
cana-787	28	18	space	space	NOUN
cana-787	28	19	.	.	PUNCT
cana-787	29	1	almost	almost	ADV
cana-787	29	2	researchers	researcher	NOUN
cana-787	29	3	in	in	ADP
cana-787	29	4	this	this	DET
cana-787	29	5	direction	direction	NOUN
cana-787	29	6	have	have	AUX
cana-787	29	7	used	use	VERB
cana-787	29	8	𝛼-admissible	𝛼-admissible	ADJ
cana-787	29	9	condition	condition	NOUN
cana-787	29	10	.	.	PUNCT
cana-787	30	1	for	for	ADP
cana-787	30	2	example	example	NOUN
cana-787	30	3	see	see	VERB
cana-787	30	4	[	[	X
cana-787	30	5	8],[9],[10	8],[9],[10	X
cana-787	30	6	]	]	PUNCT
cana-787	30	7	,	,	PUNCT
cana-787	30	8	[	[	X
cana-787	30	9	11],[12],[13],[14],[15	11],[12],[13],[14],[15	NUM
cana-787	30	10	]	]	PUNCT
cana-787	30	11	.	.	PUNCT
cana-787	31	1	as	as	SCONJ
cana-787	31	2	we	we	PRON
cana-787	31	3	motivated	motivate	VERB
cana-787	31	4	and	and	CCONJ
cana-787	31	5	inspired	inspire	VERB
cana-787	31	6	by	by	ADP
cana-787	31	7	the	the	DET
cana-787	31	8	papers	paper	NOUN
cana-787	31	9	[	[	X
cana-787	31	10	2],[3	2],[3	NUM
cana-787	31	11	]	]	PUNCT
cana-787	31	12	,	,	PUNCT
cana-787	31	13	[	[	X
cana-787	31	14	4	4	NUM
cana-787	31	15	]	]	PUNCT
cana-787	31	16	,	,	PUNCT
cana-787	31	17	[	[	X
cana-787	31	18	5	5	NUM
cana-787	31	19	]	]	PUNCT
cana-787	31	20	and	and	CCONJ
cana-787	31	21	[	[	X
cana-787	31	22	7	7	NUM
cana-787	31	23	]	]	PUNCT
cana-787	31	24	,	,	PUNCT
cana-787	31	25	our	our	PRON
cana-787	31	26	aim	aim	NOUN
cana-787	31	27	is	be	AUX
cana-787	31	28	to	to	PART
cana-787	31	29	introduce	introduce	VERB
cana-787	31	30	𝛼𝛽−1	𝛼𝛽−1	PROPN
cana-787	31	31	–	–	PUNCT
cana-787	31	32	admissible	admissible	ADJ
cana-787	31	33	mapping	mapping	NOUN
cana-787	31	34	to	to	PART
cana-787	31	35	prove	prove	VERB
cana-787	31	36	some	some	DET
cana-787	31	37	fixed	fix	VERB
cana-787	31	38	point	point	NOUN
cana-787	31	39	theorems	theorem	NOUN
cana-787	31	40	by	by	ADP
cana-787	31	41	using	use	VERB
cana-787	31	42	𝛼	𝛼	PRON
cana-787	31	43	−	−	PROPN
cana-787	31	44	𝜑	𝜑	PRON
cana-787	31	45	contraction	contraction	NOUN
cana-787	31	46	in	in	ADP
cana-787	31	47	complete	complete	ADJ
cana-787	31	48	communications	communication	NOUN
cana-787	31	49	on	on	ADP
cana-787	31	50	applied	apply	VERB
cana-787	31	51	nonlinear	nonlinear	ADJ
cana-787	31	52	analysis	analysis	NOUN
cana-787	31	53	issn	issn	NOUN
cana-787	31	54	:	:	PUNCT
cana-787	31	55	1074	1074	NUM
cana-787	31	56	-	-	PUNCT
cana-787	31	57	133x	133x	NUM
cana-787	31	58	vol	vol	NOUN
cana-787	31	59	31	31	NUM
cana-787	31	60	no	no	NOUN
cana-787	31	61	.	.	PUNCT
cana-787	32	1	3s	3s	NUM
cana-787	32	2	(	(	PUNCT
cana-787	32	3	2024	2024	NUM
cana-787	32	4	)	)	PUNCT
cana-787	32	5	368	368	NUM
cana-787	32	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	32	7	metric	metric	ADJ
cana-787	32	8	spaces	space	NOUN
cana-787	32	9	in	in	ADP
cana-787	32	10	this	this	DET
cana-787	32	11	paper	paper	NOUN
cana-787	32	12	.	.	PUNCT
cana-787	33	1	additionally	additionally	ADV
cana-787	33	2	,	,	PUNCT
cana-787	33	3	we	we	PRON
cana-787	33	4	introduce	introduce	VERB
cana-787	33	5	𝜇	𝜇	ADP
cana-787	33	6	−	−	PROPN
cana-787	33	7	𝛼𝛽−1admissible	𝛼𝛽−1admissible	ADJ
cana-787	33	8	mappings	mapping	NOUN
cana-787	33	9	to	to	PART
cana-787	33	10	prove	prove	VERB
cana-787	33	11	the	the	DET
cana-787	33	12	common	common	ADJ
cana-787	33	13	fixed	fix	VERB
cana-787	33	14	point	point	NOUN
cana-787	33	15	theorem	theorem	VERB
cana-787	33	16	by	by	ADP
cana-787	33	17	using	use	VERB
cana-787	33	18	(	(	PUNCT
cana-787	33	19	𝜇	𝜇	ADP
cana-787	33	20	−	−	NOUN
cana-787	33	21	𝛼	𝛼	NOUN
cana-787	33	22	,	,	PUNCT
cana-787	33	23	𝜑)-contraction	𝜑)-contraction	PUNCT
cana-787	33	24	in	in	ADP
cana-787	33	25	complete	complete	ADJ
cana-787	33	26	metric	metric	ADJ
cana-787	33	27	space	space	NOUN
cana-787	33	28	.	.	PUNCT
cana-787	34	1	2	2	X
cana-787	34	2	.	.	X
cana-787	34	3	preliminaries	preliminary	NOUN
cana-787	34	4	in	in	ADP
cana-787	34	5	this	this	DET
cana-787	34	6	section	section	NOUN
cana-787	34	7	,	,	PUNCT
cana-787	34	8	we	we	PRON
cana-787	34	9	present	present	VERB
cana-787	34	10	some	some	DET
cana-787	34	11	basic	basic	ADJ
cana-787	34	12	concepts	concept	NOUN
cana-787	34	13	and	and	CCONJ
cana-787	34	14	notations	notation	NOUN
cana-787	34	15	related	relate	VERB
cana-787	34	16	to	to	ADP
cana-787	34	17	𝛼	𝛼	PROPN
cana-787	34	18	−	−	PROPN
cana-787	34	19	𝜑	𝜑	ADP
cana-787	34	20	contractions	contraction	NOUN
cana-787	34	21	and	and	CCONJ
cana-787	34	22	𝛼	𝛼	PRON
cana-787	34	23	admissible	admissible	ADJ
cana-787	34	24	conditions	condition	NOUN
cana-787	34	25	.	.	PUNCT
cana-787	35	1	definition	definition	NOUN
cana-787	35	2	2.1	2.1	NUM
cana-787	35	3	:	:	PUNCT
cana-787	36	1	[	[	X
cana-787	36	2	16	16	NUM
cana-787	36	3	]	]	PUNCT
cana-787	36	4	a	a	DET
cana-787	36	5	mapping	mapping	NOUN
cana-787	36	6	𝜑	𝜑	NOUN
cana-787	36	7	:	:	PUNCT
cana-787	37	1	[	[	X
cana-787	37	2	0	0	NUM
cana-787	37	3	,	,	PUNCT
cana-787	37	4	∞	∞	PROPN
cana-787	37	5	)	)	PUNCT
cana-787	37	6	→	→	PUNCT
cana-787	37	7	[	[	X
cana-787	37	8	0	0	NUM
cana-787	37	9	,	,	PUNCT
cana-787	37	10	∞	∞	PROPN
cana-787	37	11	)	)	PUNCT
cana-787	37	12	is	be	AUX
cana-787	37	13	called	call	VERB
cana-787	37	14	comparison	comparison	NOUN
cana-787	37	15	function	function	NOUN
cana-787	37	16	if	if	SCONJ
cana-787	37	17	satisfy	satisfy	ADJ
cana-787	37	18	following	follow	VERB
cana-787	37	19	assumptions	assumption	NOUN
cana-787	37	20	:	:	PUNCT
cana-787	37	21	a	a	X
cana-787	37	22	)	)	PUNCT
cana-787	37	23	𝜑	𝜑	NOUN
cana-787	37	24	is	be	AUX
cana-787	37	25	non	non	ADJ
cana-787	37	26	-	-	ADJ
cana-787	37	27	decreasing	decrease	VERB
cana-787	37	28	.	.	PUNCT
cana-787	38	1	b	b	X
cana-787	38	2	)	)	PUNCT
cana-787	38	3	𝜑(𝑝	𝜑(𝑝	PROPN
cana-787	38	4	)	)	PUNCT
cana-787	39	1	<	<	X
cana-787	39	2	𝑝	𝑝	X
cana-787	39	3	∀	∀	X
cana-787	39	4	𝑝	𝑝	NOUN
cana-787	39	5	>	>	NOUN
cana-787	39	6	0	0	NUM
cana-787	39	7	.	.	PUNCT
cana-787	40	1	c	c	X
cana-787	40	2	)	)	PUNCT
cana-787	40	3	𝜑𝑚(𝑝	𝜑𝑚(𝑝	NOUN
cana-787	40	4	)	)	PUNCT
cana-787	40	5	→	→	SYM
cana-787	40	6	0	0	NUM
cana-787	40	7	as	as	ADP
cana-787	40	8	𝑚	𝑚	PROPN
cana-787	40	9	→	→	SYM
cana-787	40	10	∞	∞	NUM
cana-787	40	11	∀	∀	NOUN
cana-787	40	12	𝑝	𝑝	NOUN
cana-787	40	13	>	>	X
cana-787	40	14	0	0	NUM
cana-787	41	1	d	d	NOUN
cana-787	41	2	)	)	PUNCT
cana-787	41	3	𝜑(0	𝜑(0	NOUN
cana-787	41	4	)	)	PUNCT
cana-787	41	5	=	=	SYM
cana-787	41	6	0	0	NUM
cana-787	42	1	the	the	DET
cana-787	42	2	set	set	NOUN
cana-787	42	3	of	of	ADP
cana-787	42	4	all	all	DET
cana-787	42	5	comparison	comparison	NOUN
cana-787	42	6	functions	function	NOUN
cana-787	42	7	is	be	AUX
cana-787	42	8	denoted	denote	VERB
cana-787	42	9	by	by	ADP
cana-787	42	10	∅.	∅.	PRON
cana-787	42	11	definition	definition	NOUN
cana-787	42	12	2.2	2.2	NUM
cana-787	42	13	:	:	PUNCT
cana-787	43	1	[	[	X
cana-787	43	2	2	2	X
cana-787	43	3	]	]	PUNCT
cana-787	43	4	let	let	VERB
cana-787	43	5	𝑋	𝑋	NOUN
cana-787	43	6	be	be	AUX
cana-787	43	7	a	a	DET
cana-787	43	8	non	non	ADJ
cana-787	43	9	-	-	ADJ
cana-787	43	10	empty	empty	ADJ
cana-787	43	11	set	set	NOUN
cana-787	43	12	.	.	PUNCT
cana-787	44	1	the	the	DET
cana-787	44	2	mapping	mapping	NOUN
cana-787	44	3	𝜇	𝜇	X
cana-787	44	4	:	:	PUNCT
cana-787	44	5	𝑋	𝑋	PROPN
cana-787	44	6	→	→	SYM
cana-787	44	7	𝑋	𝑋	PROPN
cana-787	44	8	is	be	AUX
cana-787	44	9	called	call	VERB
cana-787	44	10	𝛼-admissible	𝛼-admissible	ADJ
cana-787	44	11	,	,	PUNCT
cana-787	44	12	if	if	SCONJ
cana-787	44	13	there	there	PRON
cana-787	44	14	exists	exist	VERB
cana-787	44	15	a	a	DET
cana-787	44	16	function	function	NOUN
cana-787	44	17	𝛼	𝛼	NOUN
cana-787	44	18	:	:	PUNCT
cana-787	44	19	𝑋	𝑋	NOUN
cana-787	44	20	×	×	NOUN
cana-787	44	21	𝑋	𝑋	PROPN
cana-787	44	22	→	→	SYM
cana-787	45	1	[	[	X
cana-787	45	2	0	0	NUM
cana-787	45	3	,	,	PUNCT
cana-787	45	4	∞	∞	NOUN
cana-787	45	5	)	)	PUNCT
cana-787	46	1	such	such	ADJ
cana-787	46	2	that	that	SCONJ
cana-787	46	3	𝛼(𝑎	𝛼(𝑎	NOUN
cana-787	46	4	,	,	PUNCT
cana-787	46	5	𝑏	𝑏	NOUN
cana-787	46	6	)	)	PUNCT
cana-787	46	7	≥	≥	NOUN
cana-787	46	8	1	1	NUM
cana-787	46	9	⇒	⇒	NOUN
cana-787	46	10	𝛼(𝜇(𝑎	𝛼(𝜇(𝑎	PROPN
cana-787	46	11	)	)	PUNCT
cana-787	46	12	,	,	PUNCT
cana-787	46	13	𝜇(𝑏	𝜇(𝑏	NOUN
cana-787	46	14	)	)	PUNCT
cana-787	46	15	)	)	PUNCT
cana-787	46	16	≥	≥	NOUN
cana-787	46	17	1	1	NUM
cana-787	46	18	for	for	ADP
cana-787	46	19	𝑎	𝑎	NOUN
cana-787	46	20	,	,	PUNCT
cana-787	46	21	𝑏	𝑏	PROPN
cana-787	46	22	∈	∈	PROPN
cana-787	46	23	𝑋.	𝑋.	PROPN
cana-787	46	24	definition	definition	NOUN
cana-787	46	25	2.3	2.3	NUM
cana-787	46	26	:	:	PUNCT
cana-787	46	27	[	[	X
cana-787	46	28	2	2	X
cana-787	46	29	]	]	PUNCT
cana-787	46	30	let	let	VERB
cana-787	46	31	𝜇	𝜇	PART
cana-787	46	32	be	be	AUX
cana-787	46	33	a	a	DET
cana-787	46	34	self	self	NOUN
cana-787	46	35	-	-	PUNCT
cana-787	46	36	map	map	NOUN
cana-787	46	37	on	on	ADP
cana-787	46	38	a	a	DET
cana-787	46	39	metric	metric	ADJ
cana-787	46	40	space	space	NOUN
cana-787	46	41	(	(	PUNCT
cana-787	46	42	𝑋	𝑋	PROPN
cana-787	46	43	,	,	PUNCT
cana-787	46	44	𝑑	𝑑	NOUN
cana-787	46	45	)	)	PUNCT
cana-787	46	46	.	.	PUNCT
cana-787	47	1	we	we	PRON
cana-787	47	2	say	say	VERB
cana-787	47	3	𝜇	𝜇	ADP
cana-787	47	4	is	be	AUX
cana-787	47	5	𝛼	𝛼	PRON
cana-787	47	6	−	−	NOUN
cana-787	47	7	𝜑	𝜑	PRON
cana-787	47	8	contraction	contraction	NOUN
cana-787	47	9	if	if	SCONJ
cana-787	47	10	there	there	PRON
cana-787	47	11	exist	exist	VERB
cana-787	47	12	𝛼	𝛼	NOUN
cana-787	47	13	:	:	PUNCT
cana-787	47	14	𝑋	𝑋	NOUN
cana-787	47	15	×	×	NOUN
cana-787	47	16	𝑋	𝑋	PROPN
cana-787	47	17	→	→	SYM
cana-787	47	18	[	[	X
cana-787	47	19	0	0	NUM
cana-787	47	20	,	,	PUNCT
cana-787	47	21	∞	∞	PROPN
cana-787	47	22	)	)	PUNCT
cana-787	47	23	and	and	CCONJ
cana-787	47	24	a	a	DET
cana-787	47	25	comparison	comparison	NOUN
cana-787	47	26	function	function	VERB
cana-787	47	27	𝜑	𝜑	PRON
cana-787	47	28	:	:	PUNCT
cana-787	48	1	[	[	X
cana-787	48	2	0	0	NUM
cana-787	48	3	,	,	PUNCT
cana-787	48	4	∞	∞	PROPN
cana-787	48	5	)	)	PUNCT
cana-787	48	6	→	→	PUNCT
cana-787	49	1	[	[	X
cana-787	49	2	0	0	NUM
cana-787	49	3	,	,	PUNCT
cana-787	49	4	∞	∞	NOUN
cana-787	49	5	)	)	PUNCT
cana-787	49	6	such	such	ADJ
cana-787	49	7	that	that	SCONJ
cana-787	49	8	,	,	PUNCT
cana-787	49	9	𝛼(𝑎	𝛼(𝑎	NOUN
cana-787	49	10	,	,	PUNCT
cana-787	49	11	𝑏)𝑑(𝜇(𝑎	𝑏)𝑑(𝜇(𝑎	PROPN
cana-787	49	12	)	)	PUNCT
cana-787	49	13	,	,	PUNCT
cana-787	49	14	𝜇(𝑏	𝜇(𝑏	NOUN
cana-787	49	15	)	)	PUNCT
cana-787	49	16	)	)	PUNCT
cana-787	49	17	≤	≤	PROPN
cana-787	49	18	𝜑(𝑑(𝑎	𝜑(𝑑(𝑎	PROPN
cana-787	49	19	,	,	PUNCT
cana-787	49	20	𝑏	𝑏	NOUN
cana-787	49	21	)	)	PUNCT
cana-787	49	22	)	)	PUNCT
cana-787	49	23	∀	∀	PUNCT
cana-787	50	1	𝑎	𝑎	NOUN
cana-787	50	2	,	,	PUNCT
cana-787	50	3	𝑏	𝑏	PROPN
cana-787	50	4	∈	∈	NOUN
cana-787	50	5	𝑋	𝑋	NOUN
cana-787	50	6	definition	definition	NOUN
cana-787	50	7	2.4	2.4	NUM
cana-787	50	8	:	:	PUNCT
cana-787	51	1	[	[	X
cana-787	51	2	3	3	X
cana-787	51	3	]	]	PUNCT
cana-787	51	4	for	for	ADP
cana-787	51	5	a	a	DET
cana-787	51	6	given	give	VERB
cana-787	51	7	𝜑	𝜑	PROPN
cana-787	51	8	∈	∈	NOUN
cana-787	51	9	∅	∅	NOUN
cana-787	51	10	,	,	PUNCT
cana-787	51	11	let	let	VERB
cana-787	51	12	us	we	PRON
cana-787	51	13	define	define	VERB
cana-787	51	14	a	a	DET
cana-787	51	15	set	set	NOUN
cana-787	51	16	𝑅𝜑	𝑅𝜑	PROPN
cana-787	51	17	=	=	PUNCT
cana-787	51	18	{	{	PUNCT
cana-787	51	19	𝛾	𝛾	PART
cana-787	51	20	∈	∈	PROPN
cana-787	52	1	[	[	X
cana-787	52	2	0	0	NUM
cana-787	52	3	,	,	PUNCT
cana-787	52	4	∞	∞	NUM
cana-787	52	5	):	):	PUNCT
cana-787	52	6	𝛾𝜑	𝛾𝜑	ADP
cana-787	52	7	∈	∈	NOUN
cana-787	52	8	∅	∅	NOUN
cana-787	52	9	}	}	PUNCT
cana-787	52	10	the	the	DET
cana-787	52	11	main	main	ADJ
cana-787	52	12	theorem	theorem	NOUN
cana-787	52	13	in	in	ADP
cana-787	52	14	[	[	X
cana-787	52	15	2	2	NUM
cana-787	52	16	]	]	PUNCT
cana-787	52	17	is	be	AUX
cana-787	52	18	re	re	AUX
cana-787	52	19	stated	state	VERB
cana-787	52	20	here	here	ADV
cana-787	52	21	.	.	PUNCT
cana-787	53	1	theorem	theorem	VERB
cana-787	53	2	2.1	2.1	NUM
cana-787	53	3	:	:	PUNCT
cana-787	53	4	let	let	VERB
cana-787	53	5	(	(	PUNCT
cana-787	53	6	𝑋	𝑋	NOUN
cana-787	53	7	,	,	PUNCT
cana-787	53	8	𝑑	𝑑	NOUN
cana-787	53	9	)	)	PUNCT
cana-787	53	10	be	be	AUX
cana-787	53	11	complete	complete	ADJ
cana-787	53	12	metric	metric	ADJ
cana-787	53	13	space	space	NOUN
cana-787	53	14	and	and	CCONJ
cana-787	53	15	𝜇	𝜇	SCONJ
cana-787	53	16	be	be	AUX
cana-787	53	17	a	a	DET
cana-787	53	18	given	give	VERB
cana-787	53	19	continuous	continuous	ADJ
cana-787	53	20	selfmap	selfmap	NOUN
cana-787	53	21	.	.	PUNCT
cana-787	54	1	suppose	suppose	VERB
cana-787	54	2	that	that	SCONJ
cana-787	54	3	there	there	PRON
cana-787	54	4	exist	exist	VERB
cana-787	54	5	two	two	NUM
cana-787	54	6	functions	function	NOUN
cana-787	54	7	𝛼	𝛼	NOUN
cana-787	54	8	:	:	PUNCT
cana-787	54	9	𝑋	𝑋	NOUN
cana-787	54	10	×	×	NOUN
cana-787	54	11	𝑋	𝑋	PROPN
cana-787	54	12	→	→	SYM
cana-787	54	13	[	[	X
cana-787	54	14	0	0	NUM
cana-787	54	15	,	,	PUNCT
cana-787	54	16	∞	∞	PROPN
cana-787	54	17	)	)	PUNCT
cana-787	54	18	and	and	CCONJ
cana-787	54	19	𝜑	𝜑	PROPN
cana-787	54	20	∈	∈	NOUN
cana-787	54	21	∅	∅	NOUN
cana-787	54	22	such	such	ADJ
cana-787	54	23	that	that	PRON
cana-787	54	24	,	,	PUNCT
cana-787	54	25	a	a	PRON
cana-787	54	26	)	)	PUNCT
cana-787	54	27	𝜇	𝜇	ADP
cana-787	54	28	is	be	AUX
cana-787	54	29	𝛼	𝛼	PRON
cana-787	54	30	−	−	NOUN
cana-787	54	31	𝜑	𝜑	PRON
cana-787	54	32	contraction	contraction	NOUN
cana-787	54	33	.	.	PUNCT
cana-787	55	1	b	b	X
cana-787	55	2	)	)	PUNCT
cana-787	55	3	𝜇	𝜇	ADP
cana-787	55	4	is	be	AUX
cana-787	55	5	𝛼-admissible	𝛼-admissible	ADJ
cana-787	55	6	.	.	PUNCT
cana-787	56	1	c	c	X
cana-787	56	2	)	)	PUNCT
cana-787	56	3	there	there	PRON
cana-787	56	4	exists	exist	VERB
cana-787	56	5	𝑎0	𝑎0	PROPN
cana-787	56	6	∈	∈	PROPN
cana-787	56	7	𝑋	𝑋	NOUN
cana-787	56	8	such	such	ADJ
cana-787	56	9	that	that	SCONJ
cana-787	56	10	𝛼(𝑎0	𝛼(𝑎0	ADJ
cana-787	56	11	,	,	PUNCT
cana-787	56	12	𝜇(𝑎0	𝜇(𝑎0	NOUN
cana-787	56	13	)	)	PUNCT
cana-787	56	14	≥	≥	NOUN
cana-787	56	15	1	1	NUM
cana-787	56	16	d	d	NOUN
cana-787	56	17	)	)	PUNCT
cana-787	56	18	𝜇	𝜇	ADP
cana-787	56	19	is	be	AUX
cana-787	56	20	continuous	continuous	ADJ
cana-787	56	21	or	or	CCONJ
cana-787	56	22	e	e	NOUN
cana-787	56	23	)	)	PUNCT
cana-787	56	24	for	for	ADP
cana-787	56	25	every	every	DET
cana-787	56	26	{	{	PUNCT
cana-787	56	27	𝑎𝑛	𝑎𝑛	NOUN
cana-787	56	28	}	}	PUNCT
cana-787	56	29	⊂	⊂	NOUN
cana-787	56	30	𝑋	𝑋	PROPN
cana-787	56	31	such	such	ADJ
cana-787	56	32	that	that	SCONJ
cana-787	56	33	𝑎𝑛	𝑎𝑛	PRON
cana-787	56	34	→	→	SYM
cana-787	56	35	𝑎	𝑎	X
cana-787	56	36	∈	∈	ADJ
cana-787	56	37	𝑋	𝑋	NOUN
cana-787	56	38	and	and	CCONJ
cana-787	56	39	𝛼(𝑎𝑛	𝛼(𝑎𝑛	NOUN
cana-787	56	40	,	,	PUNCT
cana-787	56	41	𝑎𝑛+1	𝑎𝑛+1	NOUN
cana-787	56	42	)	)	PUNCT
cana-787	56	43	≥	≥	NOUN
cana-787	56	44	1	1	NUM
cana-787	56	45	for	for	ADP
cana-787	56	46	𝑛	𝑛	DET
cana-787	56	47	∈	∈	PROPN
cana-787	56	48	𝑁	𝑁	PROPN
cana-787	56	49	,	,	PUNCT
cana-787	56	50	we	we	PRON
cana-787	56	51	have	have	VERB
cana-787	56	52	𝛼(𝑎𝑛	𝛼(𝑎𝑛	NUM
cana-787	56	53	,	,	PUNCT
cana-787	56	54	𝑎	𝑎	X
cana-787	56	55	)	)	PUNCT
cana-787	56	56	≥	≥	NOUN
cana-787	56	57	1	1	NUM
cana-787	56	58	for	for	ADP
cana-787	56	59	𝑛	𝑛	DET
cana-787	56	60	∈	∈	PROPN
cana-787	56	61	𝑁	𝑁	PROPN
cana-787	56	62	.	.	PUNCT
cana-787	57	1	then	then	ADV
cana-787	57	2	𝜇	𝜇	X
cana-787	57	3	has	have	VERB
cana-787	57	4	a	a	DET
cana-787	57	5	fixed	fix	VERB
cana-787	57	6	point	point	NOUN
cana-787	57	7	.	.	PUNCT
cana-787	58	1	moreover	moreover	ADV
cana-787	58	2	,	,	PUNCT
cana-787	58	3	if	if	SCONJ
cana-787	58	4	in	in	ADP
cana-787	58	5	addition	addition	NOUN
cana-787	58	6	we	we	PRON
cana-787	58	7	suppose	suppose	VERB
cana-787	58	8	that	that	SCONJ
cana-787	58	9	for	for	ADP
cana-787	58	10	every	every	DET
cana-787	58	11	pair	pair	NOUN
cana-787	58	12	𝑢	𝑢	NOUN
cana-787	58	13	,	,	PUNCT
cana-787	58	14	𝑣	𝑣	DET
cana-787	58	15	∈	∈	PROPN
cana-787	58	16	𝑋	𝑋	NOUN
cana-787	58	17	there	there	PRON
cana-787	58	18	exists	exist	VERB
cana-787	58	19	𝑤	𝑤	ADP
cana-787	58	20	∈	∈	NOUN
cana-787	58	21	𝑋	𝑋	NOUN
cana-787	58	22	such	such	ADJ
cana-787	58	23	that	that	SCONJ
cana-787	58	24	𝛼(𝑢	𝛼(𝑢	NOUN
cana-787	58	25	,	,	PUNCT
cana-787	58	26	𝑤	𝑤	X
cana-787	58	27	)	)	PUNCT
cana-787	58	28	≥	≥	NOUN
cana-787	58	29	1	1	NUM
cana-787	58	30	and	and	CCONJ
cana-787	58	31	𝛼(𝑣	𝛼(𝑣	PROPN
cana-787	58	32	,	,	PUNCT
cana-787	58	33	𝑤	𝑤	X
cana-787	58	34	)	)	PUNCT
cana-787	58	35	≥	≥	NOUN
cana-787	58	36	1	1	NUM
cana-787	58	37	,	,	PUNCT
cana-787	58	38	we	we	PRON
cana-787	58	39	have	have	VERB
cana-787	58	40	a	a	DET
cana-787	58	41	unique	unique	ADJ
cana-787	58	42	fixed	fix	VERB
cana-787	58	43	point	point	NOUN
cana-787	58	44	.	.	PUNCT
cana-787	59	1	following	follow	VERB
cana-787	59	2	are	be	AUX
cana-787	59	3	the	the	DET
cana-787	59	4	main	main	ADJ
cana-787	59	5	results	result	NOUN
cana-787	59	6	;	;	PUNCT
cana-787	59	7	we	we	PRON
cana-787	59	8	obtained	obtain	VERB
cana-787	59	9	in	in	ADP
cana-787	59	10	samt	samt	NOUN
cana-787	59	11	[	[	X
cana-787	59	12	3	3	NUM
cana-787	59	13	]	]	X
cana-787	59	14	theorem	theorem	VERB
cana-787	59	15	2.2	2.2	NUM
cana-787	59	16	:	:	PUNCT
cana-787	59	17	let	let	VERB
cana-787	59	18	(	(	PUNCT
cana-787	59	19	𝑋	𝑋	NOUN
cana-787	59	20	,	,	PUNCT
cana-787	59	21	𝑑	𝑑	NOUN
cana-787	59	22	)	)	PUNCT
cana-787	59	23	be	be	AUX
cana-787	59	24	complete	complete	ADJ
cana-787	59	25	metric	metric	ADJ
cana-787	59	26	space	space	NOUN
cana-787	59	27	and	and	CCONJ
cana-787	59	28	𝜇	𝜇	SCONJ
cana-787	59	29	be	be	AUX
cana-787	59	30	a	a	DET
cana-787	59	31	given	give	VERB
cana-787	59	32	continuous	continuous	ADJ
cana-787	59	33	selfmap	selfmap	NOUN
cana-787	59	34	.	.	PUNCT
cana-787	60	1	suppose	suppose	VERB
cana-787	60	2	that	that	SCONJ
cana-787	60	3	there	there	PRON
cana-787	60	4	exist	exist	VERB
cana-787	60	5	two	two	NUM
cana-787	60	6	functions	function	NOUN
cana-787	60	7	𝛼	𝛼	NOUN
cana-787	60	8	:	:	PUNCT
cana-787	60	9	𝑋	𝑋	NOUN
cana-787	60	10	×	×	NOUN
cana-787	60	11	𝑋	𝑋	PROPN
cana-787	60	12	→	→	SYM
cana-787	60	13	[	[	X
cana-787	60	14	0	0	NUM
cana-787	60	15	,	,	PUNCT
cana-787	60	16	∞	∞	PROPN
cana-787	60	17	)	)	PUNCT
cana-787	60	18	and	and	CCONJ
cana-787	60	19	𝜑	𝜑	PROPN
cana-787	60	20	∈	∈	NOUN
cana-787	60	21	∅	∅	NOUN
cana-787	60	22	such	such	ADJ
cana-787	60	23	that	that	DET
cana-787	60	24	𝜇	𝜇	ADV
cana-787	60	25	is	be	AUX
cana-787	60	26	𝛼	𝛼	PRON
cana-787	60	27	−	−	NOUN
cana-787	60	28	𝜑	𝜑	PRON
cana-787	60	29	contraction	contraction	NOUN
cana-787	60	30	.	.	PUNCT
cana-787	61	1	also	also	ADV
cana-787	61	2	let	let	VERB
cana-787	61	3	there	there	PRON
cana-787	61	4	exist	exist	VERB
cana-787	61	5	𝛾	𝛾	ADP
cana-787	61	6	∈	∈	NOUN
cana-787	61	7	𝑅𝜑	𝑅𝜑	PROPN
cana-787	61	8	and	and	CCONJ
cana-787	61	9	a	a	DET
cana-787	61	10	finite	finite	ADJ
cana-787	61	11	sequence	sequence	NOUN
cana-787	61	12	{	{	PUNCT
cana-787	61	13	𝜏𝑖	𝜏𝑖	NOUN
cana-787	61	14	}	}	PUNCT
cana-787	61	15	⊂	⊂	PROPN
cana-787	61	16	𝑋	𝑋	PROPN
cana-787	61	17	,	,	PUNCT
cana-787	61	18	𝑖	𝑖	PUNCT
cana-787	61	19	=	=	SYM
cana-787	61	20	1,2	1,2	NUM
cana-787	61	21	…	…	PUNCT
cana-787	61	22	.	.	PUNCT
cana-787	62	1	𝑝	𝑝	X
cana-787	62	2	such	such	ADJ
cana-787	62	3	that	that	DET
cana-787	62	4	𝜏0	𝜏0	PROPN
cana-787	62	5	=	=	SYM
cana-787	62	6	𝑥0	𝑥0	PROPN
cana-787	62	7	,	,	PUNCT
cana-787	62	8	𝜏𝑝	𝜏𝑝	NOUN
cana-787	62	9	=	=	NOUN
cana-787	62	10	𝜇(𝑥0	𝜇(𝑥0	NUM
cana-787	62	11	)	)	PUNCT
cana-787	62	12	,	,	PUNCT
cana-787	62	13	𝛼(𝜇𝑛(𝜏𝑖	𝛼(𝜇𝑛(𝜏𝑖	NUM
cana-787	62	14	)	)	PUNCT
cana-787	62	15	,	,	PUNCT
cana-787	62	16	𝜇𝑛(𝜏𝑖+1	𝜇𝑛(𝜏𝑖+1	NOUN
cana-787	62	17	)	)	PUNCT
cana-787	62	18	)	)	PUNCT
cana-787	62	19	≥	≥	PROPN
cana-787	62	20	𝛾−1	𝛾−1	PROPN
cana-787	62	21	∀𝑛	∀𝑛	NOUN
cana-787	62	22	∈	∈	PROPN
cana-787	62	23	𝑁	𝑁	PROPN
cana-787	62	24	,	,	PUNCT
cana-787	62	25	𝑖	𝑖	NOUN
cana-787	62	26	=	=	SYM
cana-787	62	27	1,2	1,2	NUM
cana-787	62	28	.	.	PUNCT
cana-787	62	29	.	.	PUNCT
cana-787	63	1	𝑝	𝑝	X
cana-787	63	2	−	−	NOUN
cana-787	63	3	1	1	NUM
cana-787	63	4	…	…	SYM
cana-787	63	5	……	……	NOUN
cana-787	63	6	.	.	PUNCT
cana-787	64	1	(	(	PUNCT
cana-787	64	2	2	2	NUM
cana-787	64	3	.	.	NOUN
cana-787	64	4	1	1	NUM
cana-787	64	5	)	)	PUNCT
cana-787	64	6	then	then	ADV
cana-787	64	7	there	there	PRON
cana-787	64	8	exist	exist	VERB
cana-787	64	9	a	a	DET
cana-787	64	10	fixed	fix	VERB
cana-787	64	11	point	point	NOUN
cana-787	64	12	of	of	ADP
cana-787	64	13	𝜇.	𝜇.	NOUN
cana-787	64	14	communications	communication	NOUN
cana-787	64	15	on	on	ADP
cana-787	64	16	applied	apply	VERB
cana-787	64	17	nonlinear	nonlinear	ADJ
cana-787	64	18	analysis	analysis	NOUN
cana-787	64	19	issn	issn	NOUN
cana-787	64	20	:	:	PUNCT
cana-787	64	21	1074	1074	NUM
cana-787	64	22	-	-	PUNCT
cana-787	64	23	133x	133x	NUM
cana-787	64	24	vol	vol	NOUN
cana-787	64	25	31	31	NUM
cana-787	64	26	no	no	NOUN
cana-787	64	27	.	.	PUNCT
cana-787	65	1	3s	3s	NUM
cana-787	65	2	(	(	PUNCT
cana-787	65	3	2024	2024	NUM
cana-787	65	4	)	)	PUNCT
cana-787	65	5	369	369	NUM
cana-787	65	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	65	7	the	the	DET
cana-787	65	8	uniqueness	uniqueness	NOUN
cana-787	65	9	of	of	ADP
cana-787	65	10	fixed	fix	VERB
cana-787	65	11	point	point	NOUN
cana-787	65	12	is	be	AUX
cana-787	65	13	guaranteed	guarantee	VERB
cana-787	65	14	by	by	ADP
cana-787	65	15	following	follow	VERB
cana-787	65	16	conditions	condition	NOUN
cana-787	65	17	theorem	theorem	VERB
cana-787	65	18	2.3	2.3	NUM
cana-787	65	19	:	:	PUNCT
cana-787	65	20	let	let	VERB
cana-787	65	21	(	(	PUNCT
cana-787	65	22	𝑋	𝑋	NOUN
cana-787	65	23	,	,	PUNCT
cana-787	65	24	𝑑	𝑑	NOUN
cana-787	65	25	)	)	PUNCT
cana-787	65	26	be	be	AUX
cana-787	65	27	complete	complete	ADJ
cana-787	65	28	metric	metric	ADJ
cana-787	65	29	space	space	NOUN
cana-787	65	30	and	and	CCONJ
cana-787	65	31	𝜇	𝜇	SCONJ
cana-787	65	32	be	be	AUX
cana-787	65	33	a	a	DET
cana-787	65	34	given	give	VERB
cana-787	65	35	selfmap	selfmap	NOUN
cana-787	65	36	.	.	PUNCT
cana-787	66	1	suppose	suppose	VERB
cana-787	66	2	that	that	SCONJ
cana-787	66	3	there	there	PRON
cana-787	66	4	exist	exist	VERB
cana-787	66	5	two	two	NUM
cana-787	66	6	functions	function	NOUN
cana-787	66	7	𝛼	𝛼	NOUN
cana-787	66	8	:	:	PUNCT
cana-787	66	9	𝑋	𝑋	NOUN
cana-787	66	10	×	×	NOUN
cana-787	66	11	𝑋	𝑋	PROPN
cana-787	66	12	→	→	SYM
cana-787	66	13	[	[	X
cana-787	66	14	0	0	NUM
cana-787	66	15	,	,	PUNCT
cana-787	66	16	∞	∞	PROPN
cana-787	66	17	)	)	PUNCT
cana-787	66	18	and	and	CCONJ
cana-787	66	19	𝜑	𝜑	PROPN
cana-787	66	20	∈	∈	NOUN
cana-787	66	21	∅	∅	NOUN
cana-787	66	22	such	such	ADJ
cana-787	66	23	that	that	DET
cana-787	66	24	𝜇	𝜇	ADV
cana-787	66	25	is	be	AUX
cana-787	66	26	𝛼	𝛼	PRON
cana-787	66	27	−	−	NOUN
cana-787	66	28	𝜑	𝜑	PRON
cana-787	66	29	contraction	contraction	NOUN
cana-787	66	30	.	.	PUNCT
cana-787	67	1	suppose	suppose	VERB
cana-787	67	2	also	also	ADV
cana-787	67	3	that	that	SCONJ
cana-787	67	4	a	a	X
cana-787	67	5	)	)	PUNCT
cana-787	67	6	the	the	DET
cana-787	67	7	fixed	fix	VERB
cana-787	67	8	point	point	NOUN
cana-787	67	9	set	set	VERB
cana-787	67	10	𝐹𝑖𝑥(𝜇	𝐹𝑖𝑥(𝜇	NOUN
cana-787	67	11	)	)	PUNCT
cana-787	67	12	is	be	AUX
cana-787	67	13	not	not	PART
cana-787	67	14	empty	empty	ADJ
cana-787	67	15	.	.	PUNCT
cana-787	68	1	b	b	X
cana-787	68	2	)	)	PUNCT
cana-787	68	3	for	for	ADP
cana-787	68	4	all	all	DET
cana-787	68	5	𝑢	𝑢	NOUN
cana-787	68	6	,	,	PUNCT
cana-787	68	7	𝑣	𝑣	PRON
cana-787	68	8	∈	∈	PROPN
cana-787	68	9	𝐹𝑖𝑥(𝜇	𝐹𝑖𝑥(𝜇	NOUN
cana-787	68	10	)	)	PUNCT
cana-787	68	11	with	with	ADP
cana-787	68	12	𝑢	𝑢	DET
cana-787	68	13	≠	≠	PROPN
cana-787	68	14	𝑣,if	𝑣,if	NUM
cana-787	68	15	𝛼(𝑢	𝛼(𝑢	NOUN
cana-787	68	16	,	,	PUNCT
cana-787	68	17	𝑣	𝑣	NOUN
cana-787	68	18	)	)	PUNCT
cana-787	68	19	<	<	X
cana-787	68	20	1	1	NUM
cana-787	68	21	then	then	ADV
cana-787	68	22	there	there	PRON
cana-787	68	23	exist	exist	VERB
cana-787	68	24	𝛾	𝛾	ADP
cana-787	68	25	∈	∈	NOUN
cana-787	68	26	𝑅𝜑	𝑅𝜑	PROPN
cana-787	68	27	and	and	CCONJ
cana-787	68	28	for	for	ADP
cana-787	68	29	some	some	DET
cana-787	68	30	𝑞	𝑞	PART
cana-787	68	31	∈	∈	PROPN
cana-787	68	32	𝑁	𝑁	PROPN
cana-787	68	33	there	there	PRON
cana-787	68	34	is	be	VERB
cana-787	68	35	a	a	DET
cana-787	68	36	finite	finite	ADJ
cana-787	68	37	sequence	sequence	NOUN
cana-787	68	38	{	{	PUNCT
cana-787	68	39	𝛿𝑖	𝛿𝑖	PROPN
cana-787	68	40	}	}	PUNCT
cana-787	68	41	⊂	⊂	PROPN
cana-787	68	42	𝑋	𝑋	PROPN
cana-787	68	43	,	,	PUNCT
cana-787	68	44	𝑖	𝑖	PUNCT
cana-787	68	45	=	=	SYM
cana-787	68	46	1,2	1,2	NUM
cana-787	68	47	…	…	PUNCT
cana-787	68	48	.	.	PUNCT
cana-787	69	1	𝑞	𝑞	X
cana-787	69	2	such	such	ADJ
cana-787	69	3	that	that	SCONJ
cana-787	69	4	𝛿0	𝛿0	NOUN
cana-787	69	5	=	=	SYM
cana-787	69	6	𝑢	𝑢	NOUN
cana-787	69	7	,	,	PUNCT
cana-787	69	8	𝛿𝑞	𝛿𝑞	PROPN
cana-787	69	9	=	=	SYM
cana-787	69	10	𝑣	𝑣	NOUN
cana-787	69	11	,	,	PUNCT
cana-787	69	12	𝛼(𝜇𝑛(𝛿𝑖	𝛼(𝜇𝑛(𝛿𝑖	NUM
cana-787	69	13	)	)	PUNCT
cana-787	69	14	,	,	PUNCT
cana-787	69	15	𝜇𝑛(𝛿𝑖+1	𝜇𝑛(𝛿𝑖+1	PROPN
cana-787	69	16	)	)	PUNCT
cana-787	69	17	)	)	PUNCT
cana-787	69	18	≥	≥	NOUN
cana-787	69	19	𝛾−1	𝛾−1	NOUN
cana-787	69	20	∀	∀	NOUN
cana-787	69	21	𝑛	𝑛	PRON
cana-787	69	22	∈	∈	NOUN
cana-787	69	23	𝑁	𝑁	PROPN
cana-787	69	24	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-787	69	25	𝑖	𝑖	NOUN
cana-787	70	1	=	=	NOUN
cana-787	70	2	0,1,2	0,1,2	NUM
cana-787	70	3	…	…	PUNCT
cana-787	70	4	𝑞	𝑞	NOUN
cana-787	70	5	−	−	PROPN
cana-787	70	6	1	1	NUM
cana-787	70	7	.	.	PUNCT
cana-787	70	8	then	then	ADV
cana-787	70	9	𝜇	𝜇	X
cana-787	70	10	has	have	VERB
cana-787	70	11	a	a	DET
cana-787	70	12	unique	unique	ADJ
cana-787	70	13	fixed	fix	VERB
cana-787	70	14	point	point	NOUN
cana-787	70	15	.	.	PUNCT
cana-787	71	1	definition	definition	NOUN
cana-787	71	2	2.5	2.5	NUM
cana-787	71	3	:	:	PUNCT
cana-787	72	1	[	[	X
cana-787	72	2	17	17	NUM
cana-787	72	3	]	]	PUNCT
cana-787	72	4	let	let	VERB
cana-787	72	5	𝜇	𝜇	ADP
cana-787	72	6	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-787	72	7	𝜌	𝜌	PART
cana-787	72	8	be	be	AUX
cana-787	72	9	two	two	NUM
cana-787	72	10	self	self	NOUN
cana-787	72	11	-	-	PUNCT
cana-787	72	12	mappings	mapping	NOUN
cana-787	72	13	of	of	ADP
cana-787	72	14	a	a	DET
cana-787	72	15	complete	complete	ADJ
cana-787	72	16	metric	metric	ADJ
cana-787	72	17	space	space	NOUN
cana-787	72	18	(	(	PUNCT
cana-787	72	19	𝑋	𝑋	NOUN
cana-787	72	20	,	,	PUNCT
cana-787	72	21	𝑑).we	𝑑).we	PROPN
cana-787	72	22	say	say	VERB
cana-787	72	23	they	they	PRON
cana-787	72	24	are	be	AUX
cana-787	72	25	weakly	weakly	ADV
cana-787	72	26	compatible	compatible	ADJ
cana-787	72	27	if	if	SCONJ
cana-787	72	28	they	they	PRON
cana-787	72	29	commute	commute	VERB
cana-787	72	30	at	at	ADP
cana-787	72	31	their	their	PRON
cana-787	72	32	coincident	coincident	ADJ
cana-787	72	33	point	point	NOUN
cana-787	72	34	.	.	PUNCT
cana-787	73	1	that	that	PRON
cana-787	73	2	is	be	AUX
cana-787	73	3	if	if	SCONJ
cana-787	73	4	𝜇(𝑠	𝜇(𝑠	X
cana-787	73	5	)	)	PUNCT
cana-787	73	6	=	=	SYM
cana-787	74	1	𝜌(𝑠	𝜌(𝑠	NOUN
cana-787	74	2	)	)	PUNCT
cana-787	74	3	for	for	ADP
cana-787	74	4	some	some	PRON
cana-787	74	5	𝑠	𝑠	PROPN
cana-787	74	6	∈	∈	PROPN
cana-787	74	7	𝑋	𝑋	PROPN
cana-787	74	8	implies	imply	VERB
cana-787	74	9	𝜌𝜇(𝑠	𝜌𝜇(𝑠	NUM
cana-787	74	10	)	)	PUNCT
cana-787	74	11	=	=	NOUN
cana-787	75	1	𝜇𝜌(𝑠	𝜇𝜌(𝑠	X
cana-787	75	2	)	)	PUNCT
cana-787	75	3	here	here	ADV
cana-787	75	4	,	,	PUNCT
cana-787	75	5	we	we	PRON
cana-787	75	6	restate	restate	VERB
cana-787	75	7	the	the	DET
cana-787	75	8	proposition	proposition	NOUN
cana-787	75	9	by	by	ADP
cana-787	75	10	m.abbas	m.abbas	PROPN
cana-787	75	11	and	and	CCONJ
cana-787	75	12	g.jungck	g.jungck	ADV
cana-787	75	13	[	[	X
cana-787	75	14	17	17	NUM
cana-787	75	15	]	]	PUNCT
cana-787	75	16	proposition	proposition	NOUN
cana-787	75	17	2.1	2.1	NUM
cana-787	75	18	:	:	PUNCT
cana-787	75	19	let	let	VERB
cana-787	75	20	(	(	PUNCT
cana-787	75	21	𝑋	𝑋	NOUN
cana-787	75	22	,	,	PUNCT
cana-787	75	23	𝑑	𝑑	NOUN
cana-787	75	24	)	)	PUNCT
cana-787	75	25	be	be	VERB
cana-787	75	26	a	a	DET
cana-787	75	27	complete	complete	ADJ
cana-787	75	28	metric	metric	ADJ
cana-787	75	29	space	space	NOUN
cana-787	75	30	and	and	CCONJ
cana-787	75	31	𝜇	𝜇	X
cana-787	75	32	,	,	PUNCT
cana-787	75	33	𝜌	𝜌	X
cana-787	75	34	are	be	AUX
cana-787	75	35	weakly	weakly	ADV
cana-787	75	36	compatible	compatible	ADJ
cana-787	75	37	self	self	NOUN
cana-787	75	38	mappings	mapping	NOUN
cana-787	75	39	on	on	ADP
cana-787	75	40	𝑋.if	𝑋.if	NOUN
cana-787	75	41	there	there	PRON
cana-787	75	42	is	be	VERB
cana-787	75	43	unique	unique	ADJ
cana-787	75	44	coincident	coincident	ADJ
cana-787	75	45	point	point	NOUN
cana-787	75	46	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	75	47	)	)	PUNCT
cana-787	76	1	=	=	SYM
cana-787	76	2	𝜌(𝑧	𝜌(𝑧	X
cana-787	76	3	)	)	PUNCT
cana-787	76	4	=	=	SYM
cana-787	76	5	𝑢	𝑢	NOUN
cana-787	76	6	then	then	ADV
cana-787	76	7	𝑢	𝑢	PRON
cana-787	76	8	is	be	AUX
cana-787	76	9	the	the	DET
cana-787	76	10	unique	unique	ADJ
cana-787	76	11	common	common	ADJ
cana-787	76	12	fixed	fix	VERB
cana-787	76	13	point	point	NOUN
cana-787	76	14	of	of	ADP
cana-787	76	15	𝜇	𝜇	ADP
cana-787	76	16	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-787	76	17	𝜌.	𝜌.	NOUN
cana-787	76	18	the	the	DET
cana-787	76	19	definition	definition	NOUN
cana-787	76	20	2.2	2.2	NUM
cana-787	76	21	was	be	AUX
cana-787	76	22	generalized	generalize	VERB
cana-787	76	23	by	by	ADP
cana-787	76	24	h.aydi[18	h.aydi[18	PROPN
cana-787	76	25	]	]	PUNCT
cana-787	76	26	as	as	SCONJ
cana-787	76	27	follows	follow	VERB
cana-787	76	28	.	.	PUNCT
cana-787	77	1	definition	definition	NOUN
cana-787	77	2	2.6	2.6	NUM
cana-787	77	3	:	:	PUNCT
cana-787	77	4	let	let	VERB
cana-787	77	5	𝜇	𝜇	ADP
cana-787	77	6	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-787	77	7	𝜌	𝜌	PART
cana-787	77	8	be	be	AUX
cana-787	77	9	two	two	NUM
cana-787	77	10	self	self	NOUN
cana-787	77	11	-	-	PUNCT
cana-787	77	12	mappings	mapping	NOUN
cana-787	77	13	on	on	ADP
cana-787	77	14	a	a	DET
cana-787	77	15	non	non	ADJ
cana-787	77	16	-	-	ADJ
cana-787	77	17	empty	empty	ADJ
cana-787	77	18	set	set	ADJ
cana-787	77	19	𝑋.we	𝑋.we	NOUN
cana-787	77	20	say	say	VERB
cana-787	77	21	the	the	DET
cana-787	77	22	pair	pair	NOUN
cana-787	77	23	of	of	ADP
cana-787	77	24	mappings	mapping	NOUN
cana-787	77	25	(	(	PUNCT
cana-787	77	26	𝜇	𝜇	ADP
cana-787	77	27	,	,	PUNCT
cana-787	77	28	𝜌)is	𝜌)is	PROPN
cana-787	77	29	𝛼-admissible	𝛼-admissible	ADJ
cana-787	77	30	if	if	SCONJ
cana-787	77	31	there	there	PRON
cana-787	77	32	exist	exist	VERB
cana-787	77	33	a	a	DET
cana-787	77	34	mapping	mapping	NOUN
cana-787	77	35	𝛼	𝛼	NOUN
cana-787	77	36	:	:	PUNCT
cana-787	77	37	𝑋	𝑋	NOUN
cana-787	77	38	×	×	NOUN
cana-787	77	39	𝑋	𝑋	PROPN
cana-787	77	40	→	→	SYM
cana-787	77	41	[	[	X
cana-787	77	42	0	0	NUM
cana-787	77	43	,	,	PUNCT
cana-787	77	44	∞	∞	NOUN
cana-787	77	45	)	)	PUNCT
cana-787	77	46	such	such	ADJ
cana-787	77	47	that	that	SCONJ
cana-787	77	48	𝛼(𝑎	𝛼(𝑎	NOUN
cana-787	77	49	,	,	PUNCT
cana-787	77	50	𝑏	𝑏	NOUN
cana-787	77	51	)	)	PUNCT
cana-787	77	52	≥	≥	NOUN
cana-787	77	53	1	1	NUM
cana-787	77	54	⇒	⇒	NOUN
cana-787	77	55	𝛼(𝜇(𝑎	𝛼(𝜇(𝑎	PROPN
cana-787	77	56	)	)	PUNCT
cana-787	77	57	,	,	PUNCT
cana-787	77	58	𝜌(𝑏	𝜌(𝑏	NUM
cana-787	77	59	)	)	PUNCT
cana-787	77	60	)	)	PUNCT
cana-787	77	61	≥	≥	NOUN
cana-787	77	62	1	1	NUM
cana-787	77	63	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-787	77	64	𝛼(𝜌(𝑏	𝛼(𝜌(𝑏	NUM
cana-787	77	65	)	)	PUNCT
cana-787	77	66	,	,	PUNCT
cana-787	77	67	𝜇(𝑎	𝜇(𝑎	PROPN
cana-787	77	68	)	)	PUNCT
cana-787	77	69	)	)	PUNCT
cana-787	77	70	≥	≥	NOUN
cana-787	78	1	1	1	NUM
cana-787	78	2	.	.	PUNCT
cana-787	79	1	here	here	ADV
cana-787	79	2	,	,	PUNCT
cana-787	79	3	it	it	PRON
cana-787	79	4	is	be	AUX
cana-787	79	5	noted	note	VERB
cana-787	79	6	that	that	SCONJ
cana-787	79	7	there	there	PRON
cana-787	79	8	is	be	VERB
cana-787	79	9	no	no	DET
cana-787	79	10	any	any	DET
cana-787	79	11	mathematical	mathematical	ADJ
cana-787	79	12	support	support	NOUN
cana-787	79	13	to	to	PART
cana-787	79	14	find	find	VERB
cana-787	79	15	the	the	DET
cana-787	79	16	finite	finite	ADJ
cana-787	79	17	sequence	sequence	NOUN
cana-787	79	18	satisfying	satisfy	VERB
cana-787	79	19	(	(	PUNCT
cana-787	79	20	2.1).actually	2.1).actually	ADV
cana-787	79	21	we	we	PRON
cana-787	79	22	feel	feel	VERB
cana-787	79	23	,	,	PUNCT
cana-787	79	24	it	it	PRON
cana-787	79	25	is	be	AUX
cana-787	79	26	most	most	ADV
cana-787	79	27	serious	serious	ADJ
cana-787	79	28	task	task	NOUN
cana-787	79	29	to	to	PART
cana-787	79	30	get	get	VERB
cana-787	79	31	such	such	ADJ
cana-787	79	32	sequence	sequence	NOUN
cana-787	79	33	.	.	PUNCT
cana-787	80	1	similarly	similarly	ADV
cana-787	80	2	the	the	DET
cana-787	80	3	hypothesis	hypothesis	NOUN
cana-787	80	4	(	(	PUNCT
cana-787	80	5	b	b	NOUN
cana-787	80	6	)	)	PUNCT
cana-787	80	7	in	in	ADP
cana-787	80	8	theorem	theorem	ADJ
cana-787	80	9	2.3	2.3	NUM
cana-787	80	10	is	be	AUX
cana-787	80	11	not	not	PART
cana-787	80	12	suitable	suitable	ADJ
cana-787	80	13	for	for	ADP
cana-787	80	14	the	the	DET
cana-787	80	15	aim	aim	NOUN
cana-787	80	16	of	of	ADP
cana-787	80	17	paper	paper	NOUN
cana-787	80	18	[	[	X
cana-787	80	19	3	3	NUM
cana-787	80	20	]	]	PUNCT
cana-787	80	21	.	.	PUNCT
cana-787	81	1	because	because	SCONJ
cana-787	81	2	of	of	ADP
cana-787	81	3	these	these	DET
cana-787	81	4	reasons	reason	NOUN
cana-787	81	5	we	we	PRON
cana-787	81	6	introduce	introduce	VERB
cana-787	81	7	general	general	ADJ
cana-787	81	8	cases	case	NOUN
cana-787	81	9	to	to	PART
cana-787	81	10	extend	extend	VERB
cana-787	81	11	and	and	CCONJ
cana-787	81	12	improve	improve	VERB
cana-787	81	13	theorems	theorem	NOUN
cana-787	81	14	2.1	2.1	NUM
cana-787	81	15	,	,	PUNCT
cana-787	81	16	2.2	2.2	NUM
cana-787	81	17	and	and	CCONJ
cana-787	81	18	2.3	2.3	NUM
cana-787	81	19	without	without	ADP
cana-787	81	20	using	use	VERB
cana-787	81	21	condition	condition	NOUN
cana-787	81	22	defined	define	VERB
cana-787	81	23	in	in	ADP
cana-787	81	24	definition	definition	NOUN
cana-787	81	25	2.2	2.2	NUM
cana-787	81	26	.	.	PUNCT
cana-787	81	27	3	3	X
cana-787	81	28	.	.	X
cana-787	81	29	main	main	ADJ
cana-787	81	30	results	result	NOUN
cana-787	81	31	in	in	ADP
cana-787	81	32	this	this	DET
cana-787	81	33	section	section	NOUN
cana-787	81	34	we	we	PRON
cana-787	81	35	present	present	VERB
cana-787	81	36	some	some	DET
cana-787	81	37	new	new	ADJ
cana-787	81	38	concepts	concept	NOUN
cana-787	81	39	,	,	PUNCT
cana-787	81	40	fixed	fix	VERB
cana-787	81	41	point	point	NOUN
cana-787	81	42	theorems	theorem	NOUN
cana-787	81	43	and	and	CCONJ
cana-787	81	44	common	common	ADJ
cana-787	81	45	fixed	fix	VERB
cana-787	81	46	point	point	NOUN
cana-787	81	47	theorems	theorem	NOUN
cana-787	81	48	.	.	PUNCT
cana-787	82	1	from	from	ADP
cana-787	82	2	definition2.4	definition2.4	NUM
cana-787	82	3	,	,	PUNCT
cana-787	82	4	let	let	VERB
cana-787	82	5	us	we	PRON
cana-787	82	6	take	take	VERB
cana-787	82	7	the	the	DET
cana-787	82	8	set	set	NOUN
cana-787	82	9	𝑅𝜑+	𝑅𝜑+	NOUN
cana-787	82	10	=	=	PUNCT
cana-787	82	11	{	{	PUNCT
cana-787	82	12	𝜆	𝜆	PROPN
cana-787	82	13	∈	∈	PROPN
cana-787	82	14	𝑅𝜑	𝑅𝜑	PROPN
cana-787	82	15	∶	∶	PROPN
cana-787	82	16	𝜑(𝑝	𝜑(𝑝	NOUN
cana-787	82	17	)	)	PUNCT
cana-787	82	18	<	<	X
cana-787	82	19	𝜆𝜑(𝑝	𝜆𝜑(𝑝	X
cana-787	82	20	)	)	PUNCT
cana-787	82	21	<	<	X
cana-787	82	22	𝑝	𝑝	X
cana-787	82	23	∀	∀	X
cana-787	82	24	𝑝	𝑝	NOUN
cana-787	82	25	>	>	NOUN
cana-787	82	26	0	0	NUM
cana-787	82	27	}	}	PUNCT
cana-787	82	28	…	…	PUNCT
cana-787	82	29	……	……	NOUN
cana-787	82	30	(	(	PUNCT
cana-787	82	31	3.1	3.1	NUM
cana-787	82	32	)	)	PUNCT
cana-787	82	33	it	it	PRON
cana-787	82	34	is	be	AUX
cana-787	82	35	clear	clear	ADJ
cana-787	82	36	that	that	SCONJ
cana-787	82	37	this	this	DET
cana-787	82	38	set	set	NOUN
cana-787	82	39	(	(	PUNCT
cana-787	82	40	3.1	3.1	NUM
cana-787	82	41	)	)	PUNCT
cana-787	82	42	non	non	ADJ
cana-787	82	43	-	-	ADJ
cana-787	82	44	empty	empty	ADJ
cana-787	82	45	because	because	SCONJ
cana-787	82	46	𝜑(𝑝	𝜑(𝑝	NOUN
cana-787	82	47	)	)	PUNCT
cana-787	82	48	<	<	X
cana-787	82	49	𝑝	𝑝	X
cana-787	82	50	∀	∀	X
cana-787	82	51	𝑝	𝑝	NOUN
cana-787	82	52	>	>	X
cana-787	82	53	0	0	PUNCT
cana-787	82	54	and	and	CCONJ
cana-787	82	55	by	by	ADP
cana-787	82	56	properties	property	NOUN
cana-787	82	57	of	of	ADP
cana-787	82	58	real	real	ADJ
cana-787	82	59	numbers	number	NOUN
cana-787	82	60	there	there	PRON
cana-787	82	61	are	be	VERB
cana-787	82	62	infinite	infinite	ADJ
cana-787	82	63	real	real	ADJ
cana-787	82	64	numbers	number	NOUN
cana-787	82	65	between	between	ADP
cana-787	82	66	𝜑(𝑝)and	𝜑(𝑝)and	PROPN
cana-787	83	1	p.	p.	NOUN
cana-787	84	1	now	now	ADV
cana-787	84	2	we	we	PRON
cana-787	84	3	can	can	AUX
cana-787	84	4	take	take	VERB
cana-787	84	5	maximum	maximum	ADJ
cana-787	84	6	value	value	NOUN
cana-787	84	7	from	from	ADP
cana-787	84	8	(	(	PUNCT
cana-787	84	9	3.1	3.1	NUM
cana-787	84	10	)	)	PUNCT
cana-787	84	11	,	,	PUNCT
cana-787	84	12	say	say	VERB
cana-787	84	13	𝛽	𝛽	NOUN
cana-787	84	14	=	=	NOUN
cana-787	84	15	𝑚𝑎𝑥.	𝑚𝑎𝑥.	X
cana-787	84	16	{	{	PUNCT
cana-787	84	17	𝜆	𝜆	NOUN
cana-787	84	18	∈	∈	PROPN
cana-787	84	19	𝑅𝜑+}.obviously	𝑅𝜑+}.obviously	ADV
cana-787	84	20	,	,	PUNCT
cana-787	84	21	𝛽	𝛽	NOUN
cana-787	84	22	>	>	X
cana-787	84	23	1	1	NUM
cana-787	85	1	and	and	CCONJ
cana-787	85	2	it	it	PRON
cana-787	85	3	is	be	AUX
cana-787	85	4	maximum	maximum	ADJ
cana-787	85	5	value	value	NOUN
cana-787	85	6	for	for	ADP
cana-787	85	7	which	which	PRON
cana-787	85	8	𝛽𝜑	𝛽𝜑	X
cana-787	85	9	∈	∈	PROPN
cana-787	85	10	∅	∅	NOUN
cana-787	85	11	.thus	.thus	ADV
cana-787	85	12	𝛽−1	𝛽−1	PROPN
cana-787	85	13	<	<	X
cana-787	85	14	1	1	NUM
cana-787	85	15	is	be	AUX
cana-787	85	16	minimum	minimum	ADJ
cana-787	85	17	value	value	NOUN
cana-787	85	18	for	for	ADP
cana-787	85	19	which	which	PRON
cana-787	85	20	𝛼(𝑎	𝛼(𝑎	NOUN
cana-787	85	21	,	,	PUNCT
cana-787	85	22	𝑏	𝑏	NOUN
cana-787	85	23	)	)	PUNCT
cana-787	85	24	>	>	PUNCT
cana-787	85	25	𝛽−1	𝛽−1	PROPN
cana-787	85	26	for	for	ADP
cana-787	85	27	𝑎	𝑎	NOUN
cana-787	85	28	,	,	PUNCT
cana-787	85	29	𝑏	𝑏	PROPN
cana-787	85	30	∈	∈	PROPN
cana-787	85	31	𝑋.	𝑋.	PROPN
cana-787	85	32	now	now	ADV
cana-787	85	33	we	we	PRON
cana-787	85	34	can	can	AUX
cana-787	85	35	define	define	VERB
cana-787	85	36	𝛼-admissible	𝛼-admissible	ADJ
cana-787	85	37	mapping	mapping	NOUN
cana-787	85	38	on	on	ADP
cana-787	85	39	the	the	DET
cana-787	85	40	basis	basis	NOUN
cana-787	85	41	of	of	ADP
cana-787	85	42	𝛽	𝛽	NOUN
cana-787	85	43	as	as	SCONJ
cana-787	85	44	follows	follow	VERB
cana-787	85	45	:	:	PUNCT
cana-787	85	46	communications	communication	NOUN
cana-787	85	47	on	on	ADP
cana-787	85	48	applied	apply	VERB
cana-787	85	49	nonlinear	nonlinear	ADJ
cana-787	85	50	analysis	analysis	NOUN
cana-787	85	51	issn	issn	NOUN
cana-787	85	52	:	:	PUNCT
cana-787	85	53	1074	1074	NUM
cana-787	85	54	-	-	PUNCT
cana-787	85	55	133x	133x	NUM
cana-787	85	56	vol	vol	NOUN
cana-787	85	57	31	31	NUM
cana-787	85	58	no	no	NOUN
cana-787	85	59	.	.	PUNCT
cana-787	86	1	3s	3s	NUM
cana-787	86	2	(	(	PUNCT
cana-787	86	3	2024	2024	NUM
cana-787	86	4	)	)	PUNCT
cana-787	86	5	370	370	NUM
cana-787	86	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	86	7	definition	definition	NOUN
cana-787	86	8	3.1	3.1	NUM
cana-787	86	9	:	:	PUNCT
cana-787	86	10	let	let	VERB
cana-787	86	11	(	(	PUNCT
cana-787	86	12	𝑋	𝑋	NOUN
cana-787	86	13	,	,	PUNCT
cana-787	86	14	𝑑	𝑑	NOUN
cana-787	86	15	)	)	PUNCT
cana-787	86	16	be	be	VERB
cana-787	86	17	a	a	DET
cana-787	86	18	metric	metric	ADJ
cana-787	86	19	space	space	NOUN
cana-787	86	20	and	and	CCONJ
cana-787	86	21	𝜇	𝜇	ADP
cana-787	86	22	be	be	AUX
cana-787	86	23	a	a	DET
cana-787	86	24	given	give	VERB
cana-787	86	25	selfmap	selfmap	NOUN
cana-787	86	26	.	.	PUNCT
cana-787	87	1	we	we	PRON
cana-787	87	2	say	say	VERB
cana-787	87	3	𝜇	𝜇	ADP
cana-787	87	4	is	be	AUX
cana-787	87	5	𝛼𝛽−1	𝛼𝛽−1	ADJ
cana-787	87	6	–	–	PUNCT
cana-787	87	7	admissible	admissible	ADJ
cana-787	87	8	,	,	PUNCT
cana-787	87	9	if	if	SCONJ
cana-787	87	10	there	there	PRON
cana-787	87	11	exist	exist	VERB
cana-787	87	12	𝛼	𝛼	NOUN
cana-787	87	13	:	:	PUNCT
cana-787	87	14	𝑋	𝑋	NOUN
cana-787	87	15	×	×	NOUN
cana-787	87	16	𝑋	𝑋	PROPN
cana-787	87	17	→	→	SYM
cana-787	87	18	[	[	X
cana-787	87	19	0	0	NUM
cana-787	87	20	,	,	PUNCT
cana-787	87	21	∞	∞	NOUN
cana-787	87	22	)	)	PUNCT
cana-787	87	23	such	such	ADJ
cana-787	87	24	that	that	SCONJ
cana-787	87	25	,	,	PUNCT
cana-787	87	26	𝛼(𝑎	𝛼(𝑎	PROPN
cana-787	87	27	,	,	PUNCT
cana-787	87	28	𝑏	𝑏	NOUN
cana-787	87	29	)	)	PUNCT
cana-787	87	30	>	>	PUNCT
cana-787	87	31	𝛽−1	𝛽−1	PROPN
cana-787	87	32	⇒	⇒	NOUN
cana-787	87	33	𝛼(𝜇(𝑎	𝛼(𝜇(𝑎	PROPN
cana-787	87	34	)	)	PUNCT
cana-787	87	35	,	,	PUNCT
cana-787	87	36	𝜇(𝑏	𝜇(𝑏	NOUN
cana-787	87	37	)	)	PUNCT
cana-787	87	38	)	)	PUNCT
cana-787	87	39	>	>	PUNCT
cana-787	88	1	𝛽−1	𝛽−1	PROPN
cana-787	88	2	for	for	ADP
cana-787	88	3	𝑎	𝑎	NOUN
cana-787	88	4	,	,	PUNCT
cana-787	88	5	𝑏	𝑏	PROPN
cana-787	88	6	∈	∈	PROPN
cana-787	88	7	𝑋.	𝑋.	PROPN
cana-787	88	8	now	now	ADV
cana-787	88	9	we	we	PRON
cana-787	88	10	prove	prove	VERB
cana-787	88	11	following	follow	VERB
cana-787	88	12	fixed	fix	VERB
cana-787	88	13	point	point	NOUN
cana-787	88	14	theorems	theorem	NOUN
cana-787	88	15	as	as	ADP
cana-787	88	16	our	our	PRON
cana-787	88	17	main	main	ADJ
cana-787	88	18	findings	finding	NOUN
cana-787	88	19	.	.	PUNCT
cana-787	89	1	theorem	theorem	VERB
cana-787	89	2	3.1	3.1	NUM
cana-787	89	3	:	:	PUNCT
cana-787	89	4	let	let	VERB
cana-787	89	5	(	(	PUNCT
cana-787	89	6	𝑋	𝑋	NOUN
cana-787	89	7	,	,	PUNCT
cana-787	89	8	𝑑	𝑑	NOUN
cana-787	89	9	)	)	PUNCT
cana-787	89	10	be	be	VERB
cana-787	89	11	a	a	DET
cana-787	89	12	complete	complete	ADJ
cana-787	89	13	metric	metric	ADJ
cana-787	89	14	space	space	NOUN
cana-787	89	15	.	.	PUNCT
cana-787	90	1	let	let	VERB
cana-787	90	2	𝛽	𝛽	NOUN
cana-787	90	3	is	be	AUX
cana-787	90	4	maximum	maximum	ADJ
cana-787	90	5	of	of	ADP
cana-787	90	6	all	all	DET
cana-787	90	7	member	member	NOUN
cana-787	90	8	of	of	ADP
cana-787	90	9	𝑅𝜑+	𝑅𝜑+	PROPN
cana-787	90	10	.	.	PUNCT
cana-787	91	1	if	if	SCONJ
cana-787	91	2	a	a	DET
cana-787	91	3	self	self	NOUN
cana-787	91	4	-mapping	-mappe	VERB
cana-787	91	5	𝜇	𝜇	ADP
cana-787	91	6	satisfies	satisfie	NOUN
cana-787	91	7	following	follow	VERB
cana-787	91	8	conditions	condition	NOUN
cana-787	91	9	,	,	PUNCT
cana-787	91	10	a	a	PRON
cana-787	91	11	)	)	PUNCT
cana-787	91	12	𝜇	𝜇	ADP
cana-787	91	13	is	be	AUX
cana-787	91	14	𝛼	𝛼	PRON
cana-787	91	15	−	−	NOUN
cana-787	91	16	𝜑	𝜑	PRON
cana-787	91	17	contraction	contraction	NOUN
cana-787	91	18	on	on	ADP
cana-787	91	19	x.	x.	PROPN
cana-787	91	20	b	b	NOUN
cana-787	91	21	)	)	PUNCT
cana-787	91	22	𝜇	𝜇	ADP
cana-787	91	23	is	be	AUX
cana-787	91	24	𝛼𝛽−1admissible	𝛼𝛽−1admissible	ADJ
cana-787	91	25	c	c	NOUN
cana-787	91	26	)	)	PUNCT
cana-787	91	27	there	there	PRON
cana-787	91	28	exist	exist	VERB
cana-787	91	29	𝑥0	𝑥0	NOUN
cana-787	91	30	∈	∈	NOUN
cana-787	91	31	𝑋	𝑋	NOUN
cana-787	91	32	such	such	ADJ
cana-787	91	33	that	that	DET
cana-787	91	34	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-787	91	35	,	,	PUNCT
cana-787	91	36	𝜇(𝑥0	𝜇(𝑥0	ADJ
cana-787	91	37	)	)	PUNCT
cana-787	91	38	)	)	PUNCT
cana-787	92	1	>	>	PUNCT
cana-787	93	1	𝛽−1	𝛽−1	PROPN
cana-787	93	2	d	d	PROPN
cana-787	93	3	)	)	PUNCT
cana-787	93	4	𝜇	𝜇	ADP
cana-787	93	5	is	be	AUX
cana-787	93	6	continuous	continuous	ADJ
cana-787	93	7	in	in	ADP
cana-787	93	8	𝑋	𝑋	PROPN
cana-787	93	9	then	then	ADV
cana-787	93	10	there	there	PRON
cana-787	93	11	exist	exist	VERB
cana-787	93	12	a	a	DET
cana-787	93	13	fixed	fix	VERB
cana-787	93	14	point	point	NOUN
cana-787	93	15	of	of	ADP
cana-787	93	16	𝜇	𝜇	X
cana-787	93	17	in	in	ADP
cana-787	93	18	𝑋.	𝑋.	PROPN
cana-787	93	19	proof	proof	NOUN
cana-787	93	20	:	:	PUNCT
cana-787	93	21	-.let	-.let	NOUN
cana-787	93	22	there	there	PRON
cana-787	93	23	exist	exist	VERB
cana-787	93	24	𝑥0	𝑥0	NOUN
cana-787	93	25	∈	∈	NOUN
cana-787	93	26	𝑋	𝑋	NOUN
cana-787	93	27	such	such	ADJ
cana-787	93	28	that	that	DET
cana-787	93	29	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-787	93	30	,	,	PUNCT
cana-787	93	31	𝜇(𝑥0	𝜇(𝑥0	ADJ
cana-787	93	32	)	)	PUNCT
cana-787	93	33	)	)	PUNCT
cana-787	93	34	>	>	PUNCT
cana-787	94	1	𝛽−1	𝛽−1	PROPN
cana-787	94	2	.	.	PUNCT
cana-787	95	1	being	be	AUX
cana-787	95	2	𝜇	𝜇	ADV
cana-787	95	3	is	be	AUX
cana-787	95	4	𝛼	𝛼	PRON
cana-787	95	5	𝛽−1	𝛽−1	PROPN
cana-787	95	6	admissible	admissible	ADJ
cana-787	95	7	,	,	PUNCT
cana-787	95	8	so	so	CCONJ
cana-787	95	9	by	by	ADP
cana-787	95	10	induction	induction	NOUN
cana-787	95	11	we	we	PRON
cana-787	95	12	can	can	AUX
cana-787	95	13	get	get	VERB
cana-787	95	14	,	,	PUNCT
cana-787	95	15	𝛼(𝜇𝑛(𝑥0	𝛼(𝜇𝑛(𝑥0	ADJ
cana-787	95	16	)	)	PUNCT
cana-787	95	17	,	,	PUNCT
cana-787	95	18	𝜇𝑛+1(𝑥0	𝜇𝑛+1(𝑥0	NOUN
cana-787	95	19	)	)	PUNCT
cana-787	95	20	)	)	PUNCT
cana-787	95	21	>	>	PUNCT
cana-787	96	1	𝛽−1	𝛽−1	PROPN
cana-787	96	2	,	,	PUNCT
cana-787	96	3	∀	∀	X
cana-787	96	4	𝑛	𝑛	PRON
cana-787	96	5	∈	∈	PROPN
cana-787	96	6	𝑁	𝑁	PROPN
cana-787	96	7	…	…	X
cana-787	96	8	……	……	NOUN
cana-787	96	9	……	……	NOUN
cana-787	96	10	..	..	PUNCT
cana-787	96	11	(	(	PUNCT
cana-787	96	12	3.2	3.2	NUM
cana-787	96	13	)	)	PUNCT
cana-787	96	14	now	now	ADV
cana-787	96	15	we	we	PRON
cana-787	96	16	have	have	VERB
cana-787	96	17	from	from	ADP
cana-787	96	18	condition	condition	NOUN
cana-787	96	19	(	(	PUNCT
cana-787	96	20	a	a	NOUN
cana-787	96	21	)	)	PUNCT
cana-787	96	22	,	,	PUNCT
cana-787	96	23	𝛽−1𝑑(𝜇(𝑥0	𝛽−1𝑑(𝜇(𝑥0	PROPN
cana-787	96	24	)	)	PUNCT
cana-787	96	25	,	,	PUNCT
cana-787	96	26	𝜇2(𝑥0	𝜇2(𝑥0	ADJ
cana-787	96	27	)	)	PUNCT
cana-787	96	28	≤	≤	NUM
cana-787	96	29	𝛼	𝛼	X
cana-787	96	30	(	(	PUNCT
cana-787	96	31	𝑥0,𝜇(𝑥0	𝑥0,𝜇(𝑥0	PROPN
cana-787	96	32	)	)	PUNCT
cana-787	96	33	)	)	PUNCT
cana-787	97	1	𝑑(𝜇(𝑥0	𝑑(𝜇(𝑥0	PROPN
cana-787	97	2	)	)	PUNCT
cana-787	97	3	,	,	PUNCT
cana-787	97	4	𝜇2(𝑥0	𝜇2(𝑥0	ADJ
cana-787	97	5	)	)	PUNCT
cana-787	97	6	)	)	PUNCT
cana-787	98	1	≤	≤	NUM
cana-787	98	2	𝜑(𝑑(𝑥0,𝜇(𝑥0	𝜑(𝑑(𝑥0,𝜇(𝑥0	PROPN
cana-787	98	3	)	)	PUNCT
cana-787	98	4	)	)	PUNCT
cana-787	98	5	)	)	PUNCT
cana-787	98	6	⇒	⇒	PROPN
cana-787	98	7	𝑑(𝜇(𝑥0	𝑑(𝜇(𝑥0	PROPN
cana-787	98	8	)	)	PUNCT
cana-787	98	9	,	,	PUNCT
cana-787	98	10	𝜇2(𝑥0	𝜇2(𝑥0	ADJ
cana-787	98	11	)	)	PUNCT
cana-787	98	12	≤	≤	NOUN
cana-787	98	13	𝛽𝜑(𝑑(𝑥0,𝜇(𝑥0	𝛽𝜑(𝑑(𝑥0,𝜇(𝑥0	NUM
cana-787	98	14	)	)	PUNCT
cana-787	98	15	)	)	PUNCT
cana-787	98	16	)	)	PUNCT
cana-787	98	17	let	let	VERB
cana-787	98	18	𝛽𝜑	𝛽𝜑	NOUN
cana-787	98	19	=	=	SYM
cana-787	98	20	𝜎	𝜎	PROPN
cana-787	98	21	∈	∈	PROPN
cana-787	98	22	∅	∅	NOUN
cana-787	98	23	,	,	PUNCT
cana-787	98	24	then	then	ADV
cana-787	98	25	𝑑(𝜇(𝑥0	𝑑(𝜇(𝑥0	PROPN
cana-787	98	26	)	)	PUNCT
cana-787	98	27	,	,	PUNCT
cana-787	98	28	𝜇2(𝑥0	𝜇2(𝑥0	ADJ
cana-787	98	29	)	)	PUNCT
cana-787	98	30	≤	≤	NOUN
cana-787	98	31	𝜎(𝑑(𝑥0,𝜇(𝑥0	𝜎(𝑑(𝑥0,𝜇(𝑥0	NUM
cana-787	98	32	)	)	PUNCT
cana-787	98	33	)	)	PUNCT
cana-787	98	34	)	)	PUNCT
cana-787	99	1	…	…	PUNCT
cana-787	99	2	…	…	SYM
cana-787	99	3	……	……	NOUN
cana-787	99	4	.	.	PUNCT
cana-787	100	1	(	(	PUNCT
cana-787	100	2	3.3	3.3	NUM
cana-787	100	3	)	)	PUNCT
cana-787	100	4	again	again	ADV
cana-787	100	5	𝛽−1𝑑(𝜇2(𝑥0	𝛽−1𝑑(𝜇2(𝑥0	PROPN
cana-787	100	6	)	)	PUNCT
cana-787	100	7	,	,	PUNCT
cana-787	100	8	𝜇3(𝑥0	𝜇3(𝑥0	PROPN
cana-787	100	9	)	)	PUNCT
cana-787	100	10	≤	≤	NOUN
cana-787	101	1	𝛼(𝑓(𝑥0	𝛼(𝑓(𝑥0	PROPN
cana-787	101	2	)	)	PUNCT
cana-787	101	3	,	,	PUNCT
cana-787	101	4	𝜇2(𝑥0))𝑑(𝜇2(𝑥0	𝜇2(𝑥0))𝑑(𝜇2(𝑥0	NOUN
cana-787	101	5	)	)	PUNCT
cana-787	101	6	,	,	PUNCT
cana-787	101	7	𝜇3(𝑥0	𝜇3(𝑥0	PROPN
cana-787	101	8	)	)	PUNCT
cana-787	101	9	)	)	PUNCT
cana-787	101	10	≤	≤	NUM
cana-787	101	11	𝜑(𝑑(𝜇(𝑥0	𝜑(𝑑(𝜇(𝑥0	NOUN
cana-787	101	12	)	)	PUNCT
cana-787	101	13	,	,	PUNCT
cana-787	101	14	𝜇2(𝑥0	𝜇2(𝑥0	ADJ
cana-787	101	15	)	)	PUNCT
cana-787	101	16	)	)	PUNCT
cana-787	101	17	)	)	PUNCT
cana-787	101	18	⇒	⇒	NOUN
cana-787	101	19	𝑑(𝜇2(𝑥0	𝑑(𝜇2(𝑥0	NUM
cana-787	101	20	)	)	PUNCT
cana-787	101	21	,	,	PUNCT
cana-787	101	22	𝜇3(𝑥0	𝜇3(𝑥0	PROPN
cana-787	101	23	)	)	PUNCT
cana-787	101	24	≤	≤	NOUN
cana-787	101	25	𝜎(𝑑(𝜇(𝑥0	𝜎(𝑑(𝜇(𝑥0	NUM
cana-787	101	26	)	)	PUNCT
cana-787	101	27	,	,	PUNCT
cana-787	101	28	𝜇2(𝑥0	𝜇2(𝑥0	ADJ
cana-787	101	29	)	)	PUNCT
cana-787	101	30	)	)	PUNCT
cana-787	101	31	)	)	PUNCT
cana-787	101	32	,	,	PUNCT
cana-787	101	33	from	from	ADP
cana-787	101	34	(	(	PUNCT
cana-787	101	35	3.3	3.3	NUM
cana-787	101	36	)	)	PUNCT
cana-787	101	37	𝑑(𝜇2(𝑥0	𝑑(𝜇2(𝑥0	NUM
cana-787	101	38	)	)	PUNCT
cana-787	101	39	,	,	PUNCT
cana-787	101	40	𝜇3(𝑥0	𝜇3(𝑥0	PROPN
cana-787	101	41	)	)	PUNCT
cana-787	101	42	≤	≤	NOUN
cana-787	101	43	𝜎2(𝑑(𝑥0,𝜇(𝑥0	𝜎2(𝑑(𝑥0,𝜇(𝑥0	NOUN
cana-787	101	44	)	)	PUNCT
cana-787	101	45	)	)	PUNCT
cana-787	101	46	)	)	PUNCT
cana-787	102	1	continuing	continue	VERB
cana-787	102	2	this	this	DET
cana-787	102	3	process	process	NOUN
cana-787	102	4	,	,	PUNCT
cana-787	102	5	we	we	PRON
cana-787	102	6	have	have	VERB
cana-787	102	7	𝑑(𝜇𝑛(𝑥0	𝑑(𝜇𝑛(𝑥0	ADJ
cana-787	102	8	)	)	PUNCT
cana-787	102	9	,	,	PUNCT
cana-787	102	10	𝜇𝑛+1(𝑥0	𝜇𝑛+1(𝑥0	NOUN
cana-787	102	11	)	)	PUNCT
cana-787	102	12	≤	≤	NOUN
cana-787	102	13	𝜎𝑛(𝑑(𝑥0,𝜇(𝑥0	𝜎𝑛(𝑑(𝑥0,𝜇(𝑥0	NUM
cana-787	102	14	)	)	PUNCT
cana-787	102	15	)	)	PUNCT
cana-787	102	16	)	)	PUNCT
cana-787	102	17	……	……	NOUN
cana-787	102	18	..	..	PUNCT
cana-787	102	19	(	(	PUNCT
cana-787	102	20	3.4	3.4	NUM
cana-787	102	21	)	)	PUNCT
cana-787	102	22	now	now	ADV
cana-787	102	23	for	for	ADP
cana-787	102	24	𝑚	𝑚	NOUN
cana-787	102	25	,	,	PUNCT
cana-787	102	26	𝑛	𝑛	DET
cana-787	102	27	∈	∈	NOUN
cana-787	102	28	𝑁	𝑁	NOUN
cana-787	102	29	with	with	ADP
cana-787	102	30	𝑚	𝑚	ADP
cana-787	102	31	<	<	X
cana-787	102	32	𝑛	𝑛	PROPN
cana-787	102	33	,	,	PUNCT
cana-787	102	34	𝑑(𝜇𝑚(𝑥0	𝑑(𝜇𝑚(𝑥0	ADJ
cana-787	102	35	)	)	PUNCT
cana-787	102	36	,	,	PUNCT
cana-787	102	37	𝜇𝑛(𝑥0	𝜇𝑛(𝑥0	NOUN
cana-787	102	38	)	)	PUNCT
cana-787	102	39	)	)	PUNCT
cana-787	102	40	≤	≤	NUM
cana-787	102	41	𝑑(𝜇𝑚(𝑥0	𝑑(𝜇𝑚(𝑥0	NOUN
cana-787	102	42	)	)	PUNCT
cana-787	102	43	,	,	PUNCT
cana-787	102	44	𝜇𝑚+1(𝑥0	𝜇𝑚+1(𝑥0	ADJ
cana-787	102	45	)	)	PUNCT
cana-787	102	46	)	)	PUNCT
cana-787	103	1	+	+	CCONJ
cana-787	103	2	𝑑(𝜇𝑚+1(𝑥0	𝑑(𝜇𝑚+1(𝑥0	PROPN
cana-787	103	3	)	)	PUNCT
cana-787	103	4	,	,	PUNCT
cana-787	103	5	𝜇𝑚+2(𝑥0	𝜇𝑚+2(𝑥0	NUM
cana-787	103	6	)	)	PUNCT
cana-787	103	7	)	)	PUNCT
cana-787	104	1	+	+	CCONJ
cana-787	104	2	⋯	⋯	VERB
cana-787	104	3	+	+	CCONJ
cana-787	104	4	𝑑(𝜇𝑛−1(𝑥0	𝑑(𝜇𝑛−1(𝑥0	NUM
cana-787	104	5	)	)	PUNCT
cana-787	104	6	,	,	PUNCT
cana-787	104	7	𝜇𝑛(𝑥0	𝜇𝑛(𝑥0	NOUN
cana-787	104	8	)	)	PUNCT
cana-787	104	9	)	)	PUNCT
cana-787	104	10	from	from	ADP
cana-787	104	11	(	(	PUNCT
cana-787	104	12	3.4	3.4	NUM
cana-787	104	13	)	)	PUNCT
cana-787	104	14	𝑑(𝜇𝑚(𝑥0	𝑑(𝜇𝑚(𝑥0	NOUN
cana-787	104	15	)	)	PUNCT
cana-787	104	16	,	,	PUNCT
cana-787	104	17	𝜇𝑛(𝑥0	𝜇𝑛(𝑥0	NOUN
cana-787	104	18	)	)	PUNCT
cana-787	104	19	)	)	PUNCT
cana-787	104	20	≤	≤	NUM
cana-787	105	1	𝜎𝑚	𝜎𝑚	X
cana-787	105	2	(	(	PUNCT
cana-787	105	3	𝑑	𝑑	PROPN
cana-787	105	4	(	(	PUNCT
cana-787	105	5	𝑥0,𝜇(𝑥0	𝑥0,𝜇(𝑥0	PROPN
cana-787	105	6	)	)	PUNCT
cana-787	105	7	)	)	PUNCT
cana-787	105	8	)	)	PUNCT
cana-787	106	1	+	+	CCONJ
cana-787	106	2	𝜎𝑚+1	𝜎𝑚+1	NUM
cana-787	106	3	(	(	PUNCT
cana-787	106	4	𝑑	𝑑	NOUN
cana-787	106	5	(	(	PUNCT
cana-787	106	6	𝑥0,𝜇(𝑥0	𝑥0,𝜇(𝑥0	PROPN
cana-787	106	7	)	)	PUNCT
cana-787	106	8	)	)	PUNCT
cana-787	106	9	)	)	PUNCT
cana-787	107	1	+	+	CCONJ
cana-787	107	2	⋯	⋯	VERB
cana-787	107	3	…	…	PUNCT
cana-787	107	4	…	…	PUNCT
cana-787	107	5	+	+	CCONJ
cana-787	107	6	𝜎𝑛−1(𝑑(𝑥0,𝜇(𝑥0	𝜎𝑛−1(𝑑(𝑥0,𝜇(𝑥0	NOUN
cana-787	107	7	)	)	PUNCT
cana-787	107	8	)	)	PUNCT
cana-787	107	9	)	)	PUNCT
cana-787	107	10	communications	communication	NOUN
cana-787	107	11	on	on	ADP
cana-787	107	12	applied	apply	VERB
cana-787	107	13	nonlinear	nonlinear	ADJ
cana-787	107	14	analysis	analysis	NOUN
cana-787	107	15	issn	issn	NOUN
cana-787	107	16	:	:	PUNCT
cana-787	107	17	1074	1074	NUM
cana-787	107	18	-	-	PUNCT
cana-787	107	19	133x	133x	NUM
cana-787	107	20	vol	vol	NOUN
cana-787	107	21	31	31	NUM
cana-787	107	22	no	no	NOUN
cana-787	107	23	.	.	PUNCT
cana-787	108	1	3s	3s	NUM
cana-787	108	2	(	(	PUNCT
cana-787	108	3	2024	2024	NUM
cana-787	108	4	)	)	PUNCT
cana-787	108	5	371	371	NUM
cana-787	108	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	108	7	taking	take	VERB
cana-787	108	8	𝑚	𝑚	PRON
cana-787	108	9	→	→	SYM
cana-787	108	10	∞	∞	NUM
cana-787	108	11	and	and	CCONJ
cana-787	108	12	by	by	ADP
cana-787	108	13	property	property	NOUN
cana-787	108	14	of	of	ADP
cana-787	108	15	comparison	comparison	NOUN
cana-787	108	16	mapping	mapping	NOUN
cana-787	108	17	,	,	PUNCT
cana-787	108	18	𝜎𝑚	𝜎𝑚	X
cana-787	108	19	(	(	PUNCT
cana-787	108	20	𝑑	𝑑	PROPN
cana-787	108	21	(	(	PUNCT
cana-787	108	22	𝑥0,𝜇(𝑥0	𝑥0,𝜇(𝑥0	PROPN
cana-787	108	23	)	)	PUNCT
cana-787	108	24	)	)	PUNCT
cana-787	108	25	)	)	PUNCT
cana-787	109	1	=	=	SYM
cana-787	109	2	0	0	NUM
cana-787	109	3	.	.	PUNCT
cana-787	110	1	thus	thus	ADV
cana-787	110	2	we	we	PRON
cana-787	110	3	have	have	VERB
cana-787	110	4	,	,	PUNCT
cana-787	110	5	𝑑(𝜇𝑚(𝑥0	𝑑(𝜇𝑚(𝑥0	ADJ
cana-787	110	6	)	)	PUNCT
cana-787	110	7	,	,	PUNCT
cana-787	110	8	𝜇𝑛(𝑥0	𝜇𝑛(𝑥0	NOUN
cana-787	110	9	)	)	PUNCT
cana-787	110	10	)	)	PUNCT
cana-787	111	1	=	=	SYM
cana-787	111	2	0	0	NUM
cana-787	112	1	this	this	PRON
cana-787	112	2	implies	imply	VERB
cana-787	112	3	{	{	PUNCT
cana-787	112	4	𝜇𝑛(𝑥0	𝜇𝑛(𝑥0	ADJ
cana-787	112	5	)	)	PUNCT
cana-787	112	6	}	}	PUNCT
cana-787	112	7	is	be	AUX
cana-787	112	8	cauchy	cauchy	ADJ
cana-787	112	9	sequence	sequence	NOUN
cana-787	112	10	in	in	ADP
cana-787	112	11	(	(	PUNCT
cana-787	112	12	𝑋	𝑋	PROPN
cana-787	112	13	,	,	PUNCT
cana-787	112	14	𝑑	𝑑	NOUN
cana-787	112	15	)	)	PUNCT
cana-787	112	16	.since	.since	NOUN
cana-787	113	1	(	(	PUNCT
cana-787	113	2	𝑋	𝑋	PROPN
cana-787	113	3	,	,	PUNCT
cana-787	113	4	𝑑	𝑑	NOUN
cana-787	113	5	)	)	PUNCT
cana-787	113	6	is	be	AUX
cana-787	113	7	complete	complete	ADJ
cana-787	113	8	,	,	PUNCT
cana-787	113	9	there	there	PRON
cana-787	113	10	exist	exist	VERB
cana-787	113	11	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	113	12	∈	∈	PROPN
cana-787	113	13	𝑋	𝑋	NOUN
cana-787	113	14	such	such	ADJ
cana-787	113	15	that	that	SCONJ
cana-787	113	16	lim	lim	PROPN
cana-787	113	17	𝑛→∞	𝑛→∞	NUM
cana-787	113	18	𝜇𝑛(𝑥0	𝜇𝑛(𝑥0	NOUN
cana-787	113	19	)	)	PUNCT
cana-787	113	20	=	=	VERB
cana-787	114	1	𝑥𝜇.	𝑥𝜇.	VERB
cana-787	114	2	the	the	DET
cana-787	114	3	continuity	continuity	NOUN
cana-787	114	4	of	of	ADP
cana-787	114	5	𝜇	𝜇	PRON
cana-787	114	6	implies	implie	NOUN
cana-787	114	7	that	that	SCONJ
cana-787	114	8	,	,	PUNCT
cana-787	114	9	lim	lim	PROPN
cana-787	114	10	𝑛→∞	𝑛→∞	NUM
cana-787	114	11	𝜇𝑛+1(𝑥0	𝜇𝑛+1(𝑥0	NOUN
cana-787	114	12	)	)	PUNCT
cana-787	114	13	=	=	SYM
cana-787	114	14	𝜇(𝑥𝜇	𝜇(𝑥𝜇	PROPN
cana-787	114	15	)	)	PUNCT
cana-787	114	16	.	.	PUNCT
cana-787	115	1	the	the	DET
cana-787	115	2	uniqueness	uniqueness	NOUN
cana-787	115	3	of	of	ADP
cana-787	115	4	limiting	limit	VERB
cana-787	115	5	value	value	NOUN
cana-787	115	6	,	,	PUNCT
cana-787	115	7	gives	give	VERB
cana-787	115	8	𝜇(𝑥𝜇	𝜇(𝑥𝜇	PROPN
cana-787	115	9	)	)	PUNCT
cana-787	115	10	=	=	SYM
cana-787	116	1	𝑥𝜇.	𝑥𝜇.	NOUN
cana-787	116	2	□	□	PUNCT
cana-787	116	3	it	it	PRON
cana-787	116	4	is	be	AUX
cana-787	116	5	clear	clear	ADJ
cana-787	116	6	that	that	SCONJ
cana-787	116	7	our	our	PRON
cana-787	116	8	suggested	suggested	ADJ
cana-787	116	9	hypothesis	hypothesis	NOUN
cana-787	116	10	,	,	PUNCT
cana-787	116	11	“	"	PUNCT
cana-787	116	12	there	there	PRON
cana-787	116	13	exist	exist	VERB
cana-787	116	14	𝑥0	𝑥0	NOUN
cana-787	116	15	∈	∈	NOUN
cana-787	116	16	𝑋	𝑋	NOUN
cana-787	116	17	such	such	ADJ
cana-787	116	18	that	that	DET
cana-787	116	19	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-787	116	20	,	,	PUNCT
cana-787	116	21	𝜇(𝑥0	𝜇(𝑥0	ADJ
cana-787	116	22	)	)	PUNCT
cana-787	116	23	)	)	PUNCT
cana-787	116	24	>	>	PUNCT
cana-787	117	1	𝛽−1	𝛽−1	PROPN
cana-787	117	2	”	"	PUNCT
cana-787	117	3	is	be	AUX
cana-787	117	4	more	more	ADV
cana-787	117	5	general	general	ADJ
cana-787	117	6	than	than	ADP
cana-787	117	7	the	the	DET
cana-787	117	8	hypothesis	hypothesis	NOUN
cana-787	117	9	given	give	VERB
cana-787	117	10	in	in	ADP
cana-787	117	11	theorem	theorem	ADJ
cana-787	117	12	2.1	2.1	NUM
cana-787	117	13	in	in	ADP
cana-787	117	14	preliminaries	preliminary	NOUN
cana-787	117	15	section	section	NOUN
cana-787	117	16	.	.	PUNCT
cana-787	118	1	next	next	ADV
cana-787	118	2	we	we	PRON
cana-787	118	3	show	show	VERB
cana-787	118	4	the	the	DET
cana-787	118	5	continuity	continuity	NOUN
cana-787	118	6	assumption	assumption	NOUN
cana-787	118	7	of	of	ADP
cana-787	118	8	𝜇	𝜇	PRON
cana-787	118	9	does	do	AUX
cana-787	118	10	not	not	PART
cana-787	118	11	require	require	VERB
cana-787	118	12	to	to	PART
cana-787	118	13	exist	exist	VERB
cana-787	118	14	a	a	DET
cana-787	118	15	fixed	fix	VERB
cana-787	118	16	point	point	NOUN
cana-787	118	17	of	of	ADP
cana-787	118	18	𝜇.	𝜇.	NOUN
cana-787	118	19	theorem	theorem	ADJ
cana-787	118	20	3.2	3.2	NUM
cana-787	118	21	:	:	PUNCT
cana-787	118	22	let	let	VERB
cana-787	118	23	(	(	PUNCT
cana-787	118	24	𝑋	𝑋	NOUN
cana-787	118	25	,	,	PUNCT
cana-787	118	26	𝑑	𝑑	NOUN
cana-787	118	27	)	)	PUNCT
cana-787	118	28	be	be	VERB
cana-787	118	29	a	a	DET
cana-787	118	30	complete	complete	ADJ
cana-787	118	31	metric	metric	ADJ
cana-787	118	32	space	space	NOUN
cana-787	118	33	.	.	PUNCT
cana-787	119	1	let	let	VERB
cana-787	119	2	𝛽	𝛽	NOUN
cana-787	119	3	is	be	AUX
cana-787	119	4	maximum	maximum	ADJ
cana-787	119	5	of	of	ADP
cana-787	119	6	all	all	DET
cana-787	119	7	member	member	NOUN
cana-787	119	8	of	of	ADP
cana-787	119	9	𝑅𝜑+	𝑅𝜑+	PROPN
cana-787	119	10	.	.	PUNCT
cana-787	120	1	if	if	SCONJ
cana-787	120	2	the	the	DET
cana-787	120	3	self	self	NOUN
cana-787	120	4	-mapping	-mappe	VERB
cana-787	120	5	𝜇	𝜇	ADP
cana-787	120	6	satisfies	satisfie	NOUN
cana-787	120	7	following	follow	VERB
cana-787	120	8	conditions	condition	NOUN
cana-787	120	9	,	,	PUNCT
cana-787	120	10	a	a	PRON
cana-787	120	11	)	)	PUNCT
cana-787	120	12	𝜇	𝜇	ADP
cana-787	120	13	is	be	AUX
cana-787	120	14	𝛼	𝛼	PRON
cana-787	120	15	−	−	NOUN
cana-787	120	16	𝜑	𝜑	PRON
cana-787	120	17	contraction	contraction	NOUN
cana-787	120	18	on	on	ADP
cana-787	120	19	x.	x.	PROPN
cana-787	120	20	b	b	NOUN
cana-787	120	21	)	)	PUNCT
cana-787	120	22	𝜇	𝜇	ADP
cana-787	120	23	is	be	AUX
cana-787	120	24	𝛼𝛽−1	𝛼𝛽−1	ADP
cana-787	120	25	-admissible	-admissible	ADJ
cana-787	120	26	.	.	PUNCT
cana-787	121	1	c	c	X
cana-787	121	2	)	)	PUNCT
cana-787	121	3	there	there	PRON
cana-787	121	4	exist	exist	VERB
cana-787	121	5	𝑥0	𝑥0	NOUN
cana-787	121	6	∈	∈	NOUN
cana-787	121	7	𝑋	𝑋	NOUN
cana-787	121	8	such	such	ADJ
cana-787	121	9	that	that	DET
cana-787	121	10	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-787	121	11	,	,	PUNCT
cana-787	121	12	𝜇(𝑥0	𝜇(𝑥0	ADJ
cana-787	121	13	)	)	PUNCT
cana-787	121	14	)	)	PUNCT
cana-787	122	1	>	>	PUNCT
cana-787	123	1	𝛽−1	𝛽−1	PROPN
cana-787	123	2	d	d	PROPN
cana-787	123	3	)	)	PUNCT
cana-787	123	4	if	if	SCONJ
cana-787	123	5	𝛼(𝜇𝑛(𝑥0	𝛼(𝜇𝑛(𝑥0	ADJ
cana-787	123	6	)	)	PUNCT
cana-787	123	7	,	,	PUNCT
cana-787	123	8	𝜇𝑛+1(𝑥0	𝜇𝑛+1(𝑥0	NOUN
cana-787	123	9	)	)	PUNCT
cana-787	123	10	)	)	PUNCT
cana-787	123	11	>	>	PUNCT
cana-787	124	1	𝛽−1	𝛽−1	PROPN
cana-787	124	2	,	,	PUNCT
cana-787	124	3	∀	∀	X
cana-787	124	4	𝑛	𝑛	PRON
cana-787	124	5	∈	∈	NOUN
cana-787	124	6	𝑁	𝑁	PROPN
cana-787	124	7	and	and	CCONJ
cana-787	124	8	lim	lim	PROPN
cana-787	124	9	𝑛→∞	𝑛→∞	NUM
cana-787	124	10	𝑑(𝜇𝑛(𝑥0	𝑑(𝜇𝑛(𝑥0	ADJ
cana-787	124	11	)	)	PUNCT
cana-787	124	12	,	,	PUNCT
cana-787	124	13	𝑧	𝑧	X
cana-787	124	14	)	)	PUNCT
cana-787	124	15	=	=	SYM
cana-787	124	16	0	0	NUM
cana-787	124	17	implies	imply	VERB
cana-787	124	18	𝛼(𝜇𝑛(𝑥0	𝛼(𝜇𝑛(𝑥0	ADJ
cana-787	124	19	)	)	PUNCT
cana-787	124	20	,	,	PUNCT
cana-787	124	21	𝑧	𝑧	X
cana-787	124	22	)	)	PUNCT
cana-787	124	23	>	>	PUNCT
cana-787	124	24	𝛽−1	𝛽−1	PROPN
cana-787	124	25	,	,	PUNCT
cana-787	124	26	∀	∀	X
cana-787	124	27	𝑛	𝑛	PRON
cana-787	124	28	∈	∈	NOUN
cana-787	124	29	𝑁	𝑁	NOUN
cana-787	124	30	then	then	ADV
cana-787	124	31	there	there	PRON
cana-787	124	32	exist	exist	VERB
cana-787	124	33	a	a	DET
cana-787	124	34	fixed	fix	VERB
cana-787	124	35	point	point	NOUN
cana-787	124	36	of	of	ADP
cana-787	124	37	𝜇	𝜇	X
cana-787	124	38	in	in	ADP
cana-787	124	39	𝑋.	𝑋.	PROPN
cana-787	124	40	proof	proof	NOUN
cana-787	124	41	:	:	PUNCT
cana-787	124	42	from	from	ADP
cana-787	124	43	theorem	theorem	VERB
cana-787	124	44	3.1	3.1	NUM
cana-787	124	45	the	the	DET
cana-787	124	46	sequence	sequence	NOUN
cana-787	124	47	{	{	PUNCT
cana-787	124	48	𝜇𝑛(𝑥0	𝜇𝑛(𝑥0	NOUN
cana-787	124	49	)	)	PUNCT
cana-787	124	50	}	}	PUNCT
cana-787	124	51	is	be	AUX
cana-787	124	52	cauchy	cauchy	ADJ
cana-787	124	53	sequence	sequence	NOUN
cana-787	124	54	in	in	ADP
cana-787	124	55	(	(	PUNCT
cana-787	124	56	𝑋	𝑋	PROPN
cana-787	124	57	,	,	PUNCT
cana-787	124	58	𝑑	𝑑	NOUN
cana-787	124	59	)	)	PUNCT
cana-787	124	60	.since	.since	NOUN
cana-787	125	1	(	(	PUNCT
cana-787	125	2	𝑋	𝑋	PROPN
cana-787	125	3	,	,	PUNCT
cana-787	125	4	𝑑	𝑑	NOUN
cana-787	125	5	)	)	PUNCT
cana-787	125	6	is	be	AUX
cana-787	125	7	complete	complete	ADJ
cana-787	125	8	,	,	PUNCT
cana-787	125	9	there	there	PRON
cana-787	125	10	exist	exist	VERB
cana-787	125	11	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	125	12	∈	∈	PROPN
cana-787	125	13	𝑋	𝑋	NOUN
cana-787	125	14	such	such	ADJ
cana-787	125	15	that	that	SCONJ
cana-787	125	16	lim	lim	PROPN
cana-787	125	17	𝑛→∞	𝑛→∞	NUM
cana-787	125	18	𝜇𝑛(𝑥0	𝜇𝑛(𝑥0	NOUN
cana-787	125	19	)	)	PUNCT
cana-787	125	20	=	=	SYM
cana-787	125	21	𝑥𝜇.thus	𝑥𝜇.thus	PART
cana-787	125	22	lim	lim	NOUN
cana-787	125	23	𝑛→∞	𝑛→∞	NUM
cana-787	125	24	𝑑(𝜇𝑛(𝑥0	𝑑(𝜇𝑛(𝑥0	ADJ
cana-787	125	25	)	)	PUNCT
cana-787	125	26	,	,	PUNCT
cana-787	125	27	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	125	28	)	)	PUNCT
cana-787	125	29	=	=	SYM
cana-787	126	1	0	0	X
cana-787	126	2	.	.	PUNCT
cana-787	127	1	now	now	ADV
cana-787	127	2	from	from	ADP
cana-787	127	3	given	give	VERB
cana-787	127	4	hypothesis	hypothesis	NOUN
cana-787	127	5	(	(	PUNCT
cana-787	127	6	d	d	NOUN
cana-787	127	7	)	)	PUNCT
cana-787	127	8	and	and	CCONJ
cana-787	127	9	from	from	ADP
cana-787	127	10	(	(	PUNCT
cana-787	127	11	3.2	3.2	NUM
cana-787	127	12	)	)	PUNCT
cana-787	127	13	,	,	PUNCT
cana-787	127	14	we	we	PRON
cana-787	127	15	can	can	AUX
cana-787	127	16	get	get	VERB
cana-787	127	17	,	,	PUNCT
cana-787	127	18	𝛼(𝜇𝑛(𝑥0	𝛼(𝜇𝑛(𝑥0	ADJ
cana-787	127	19	)	)	PUNCT
cana-787	127	20	,	,	PUNCT
cana-787	127	21	𝑥𝑓	𝑥𝑓	ADP
cana-787	127	22	)	)	PUNCT
cana-787	127	23	>	>	PUNCT
cana-787	128	1	𝛽−1	𝛽−1	PROPN
cana-787	128	2	,	,	PUNCT
cana-787	128	3	∀	∀	X
cana-787	128	4	𝑛	𝑛	PRON
cana-787	128	5	∈	∈	PROPN
cana-787	128	6	𝑁	𝑁	PROPN
cana-787	128	7	…	…	X
cana-787	128	8	……	……	NOUN
cana-787	128	9	……	……	NOUN
cana-787	128	10	(	(	PUNCT
cana-787	128	11	3.5	3.5	NUM
cana-787	128	12	)	)	PUNCT
cana-787	128	13	now	now	ADV
cana-787	128	14	,	,	PUNCT
cana-787	128	15	𝛽−1𝑑	𝛽−1𝑑	PROPN
cana-787	128	16	(	(	PUNCT
cana-787	128	17	𝜇𝑛+1(𝑥0	𝜇𝑛+1(𝑥0	ADJ
cana-787	128	18	)	)	PUNCT
cana-787	128	19	,	,	PUNCT
cana-787	128	20	𝜇(𝑥𝜇	𝜇(𝑥𝜇	PROPN
cana-787	128	21	)	)	PUNCT
cana-787	128	22	≤	≤	NUM
cana-787	128	23	𝛼	𝛼	X
cana-787	128	24	(	(	PUNCT
cana-787	128	25	𝜇𝑛(𝑥0	𝜇𝑛(𝑥0	ADJ
cana-787	128	26	)	)	PUNCT
cana-787	128	27	,	,	PUNCT
cana-787	128	28	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	128	29	)	)	PUNCT
cana-787	128	30	𝑑	𝑑	NOUN
cana-787	128	31	(	(	PUNCT
cana-787	128	32	𝜇𝑛+1(𝑥0	𝜇𝑛+1(𝑥0	ADJ
cana-787	128	33	)	)	PUNCT
cana-787	128	34	,	,	PUNCT
cana-787	128	35	𝜇(𝑥𝜇	𝜇(𝑥𝜇	PROPN
cana-787	128	36	)	)	PUNCT
cana-787	128	37	≤	≤	NUM
cana-787	128	38	𝜑(𝑑(𝜇𝑛(𝑥0	𝜑(𝑑(𝜇𝑛(𝑥0	NUM
cana-787	128	39	)	)	PUNCT
cana-787	128	40	,	,	PUNCT
cana-787	128	41	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	128	42	)	)	PUNCT
cana-787	128	43	)	)	PUNCT
cana-787	128	44	⇒	⇒	NOUN
cana-787	128	45	𝑑	𝑑	PROPN
cana-787	128	46	(	(	PUNCT
cana-787	128	47	𝜇𝑛+1(𝑥0	𝜇𝑛+1(𝑥0	ADJ
cana-787	128	48	)	)	PUNCT
cana-787	128	49	,	,	PUNCT
cana-787	128	50	𝜇(𝑥𝜇	𝜇(𝑥𝜇	PROPN
cana-787	128	51	)	)	PUNCT
cana-787	128	52	≤	≤	NOUN
cana-787	128	53	𝛽𝜑(𝑑(𝜇𝑛(𝑥0	𝛽𝜑(𝑑(𝜇𝑛(𝑥0	VERB
cana-787	128	54	)	)	PUNCT
cana-787	128	55	,	,	PUNCT
cana-787	128	56	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	128	57	)	)	PUNCT
cana-787	128	58	)	)	PUNCT
cana-787	129	1	⇒	⇒	NOUN
cana-787	129	2	𝑑	𝑑	PROPN
cana-787	129	3	(	(	PUNCT
cana-787	129	4	𝜇𝑛+1(𝑥0	𝜇𝑛+1(𝑥0	ADJ
cana-787	129	5	)	)	PUNCT
cana-787	129	6	,	,	PUNCT
cana-787	129	7	𝜇(𝑥𝜇	𝜇(𝑥𝜇	PROPN
cana-787	129	8	)	)	PUNCT
cana-787	129	9	≤	≤	NOUN
cana-787	129	10	𝜎(𝑑(𝜇𝑛(𝑥0	𝜎(𝑑(𝜇𝑛(𝑥0	NUM
cana-787	129	11	)	)	PUNCT
cana-787	129	12	,	,	PUNCT
cana-787	129	13	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	129	14	)	)	PUNCT
cana-787	129	15	)	)	PUNCT
cana-787	130	1	<	<	X
cana-787	130	2	𝑑(𝜇𝑛(𝑥0	𝑑(𝜇𝑛(𝑥0	ADJ
cana-787	130	3	)	)	PUNCT
cana-787	130	4	,	,	PUNCT
cana-787	130	5	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	130	6	)	)	PUNCT
cana-787	130	7	taking	take	VERB
cana-787	130	8	𝑛	𝑛	PRON
cana-787	130	9	→	→	SYM
cana-787	130	10	∞	∞	PROPN
cana-787	130	11	,	,	PUNCT
cana-787	130	12	we	we	PRON
cana-787	130	13	have	have	VERB
cana-787	130	14	𝑑	𝑑	X
cana-787	130	15	(	(	PUNCT
cana-787	130	16	𝑥𝜇	𝑥𝜇	ADV
cana-787	130	17	,	,	PUNCT
cana-787	130	18	𝜇(𝑥𝜇	𝜇(𝑥𝜇	PROPN
cana-787	130	19	)	)	PUNCT
cana-787	130	20	≤	≤	NOUN
cana-787	131	1	𝑑	𝑑	X
cana-787	131	2	(	(	PUNCT
cana-787	131	3	𝑥𝜇	𝑥𝜇	ADV
cana-787	131	4	,	,	PUNCT
cana-787	131	5	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	131	6	)	)	PUNCT
cana-787	131	7	⇒	⇒	NOUN
cana-787	131	8	𝑑	𝑑	PROPN
cana-787	131	9	(	(	PUNCT
cana-787	131	10	𝑥𝜇	𝑥𝜇	ADV
cana-787	131	11	,	,	PUNCT
cana-787	131	12	𝑓(𝑥𝜇	𝑓(𝑥𝜇	PROPN
cana-787	131	13	)	)	PUNCT
cana-787	131	14	=	=	SYM
cana-787	131	15	0	0	NUM
cana-787	131	16	communications	communication	NOUN
cana-787	131	17	on	on	ADP
cana-787	131	18	applied	apply	VERB
cana-787	131	19	nonlinear	nonlinear	ADJ
cana-787	131	20	analysis	analysis	NOUN
cana-787	131	21	issn	issn	NOUN
cana-787	131	22	:	:	PUNCT
cana-787	131	23	1074	1074	NUM
cana-787	131	24	-	-	PUNCT
cana-787	131	25	133x	133x	NUM
cana-787	131	26	vol	vol	NOUN
cana-787	131	27	31	31	NUM
cana-787	131	28	no	no	NOUN
cana-787	131	29	.	.	PUNCT
cana-787	132	1	3s	3s	NUM
cana-787	132	2	(	(	PUNCT
cana-787	132	3	2024	2024	NUM
cana-787	132	4	)	)	PUNCT
cana-787	132	5	372	372	NUM
cana-787	132	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	132	7	∴	∴	PROPN
cana-787	132	8	𝜇(𝑥𝜇	𝜇(𝑥𝜇	PROPN
cana-787	132	9	)	)	PUNCT
cana-787	132	10	=	=	PUNCT
cana-787	132	11	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	132	12	□	□	PUNCT
cana-787	132	13	the	the	DET
cana-787	132	14	uniqueness	uniqueness	NOUN
cana-787	132	15	of	of	ADP
cana-787	132	16	fixed	fix	VERB
cana-787	132	17	point	point	NOUN
cana-787	132	18	is	be	AUX
cana-787	132	19	given	give	VERB
cana-787	132	20	by	by	ADP
cana-787	132	21	following	follow	VERB
cana-787	132	22	theorem	theorem	VERB
cana-787	132	23	.	.	PUNCT
cana-787	132	24	theorem	theorem	VERB
cana-787	132	25	3.3	3.3	NUM
cana-787	132	26	:	:	PUNCT
cana-787	132	27	by	by	ADP
cana-787	132	28	adding	add	VERB
cana-787	132	29	the	the	DET
cana-787	132	30	condition	condition	NOUN
cana-787	132	31	:	:	PUNCT
cana-787	132	32	“	"	PUNCT
cana-787	132	33	if	if	SCONJ
cana-787	132	34	for	for	ADP
cana-787	132	35	every	every	DET
cana-787	132	36	pair	pair	NOUN
cana-787	132	37	𝑢	𝑢	NOUN
cana-787	132	38	,	,	PUNCT
cana-787	132	39	𝑣	𝑣	DET
cana-787	132	40	∈	∈	PROPN
cana-787	132	41	𝑋	𝑋	NOUN
cana-787	132	42	there	there	ADV
cana-787	132	43	exist	exist	VERB
cana-787	132	44	𝑤	𝑤	ADP
cana-787	132	45	∈	∈	NOUN
cana-787	132	46	𝑋	𝑋	NOUN
cana-787	132	47	such	such	ADJ
cana-787	132	48	that	that	SCONJ
cana-787	132	49	𝛼(𝑢	𝛼(𝑢	NOUN
cana-787	132	50	,	,	PUNCT
cana-787	132	51	𝑤	𝑤	ADP
cana-787	132	52	)	)	PUNCT
cana-787	132	53	>	>	PUNCT
cana-787	133	1	𝛽−1	𝛽−1	PROPN
cana-787	133	2	and	and	CCONJ
cana-787	133	3	𝛼(𝑣	𝛼(𝑣	PROPN
cana-787	133	4	,	,	PUNCT
cana-787	133	5	𝑤	𝑤	ADP
cana-787	133	6	)	)	PUNCT
cana-787	133	7	>	>	PUNCT
cana-787	133	8	𝛽−1	𝛽−1	PROPN
cana-787	133	9	"	"	PUNCT
cana-787	133	10	in	in	ADP
cana-787	133	11	the	the	DET
cana-787	133	12	theorem	theorem	ADJ
cana-787	133	13	3.1	3.1	NUM
cana-787	133	14	and	and	CCONJ
cana-787	133	15	3.2	3.2	NUM
cana-787	133	16	,	,	PUNCT
cana-787	133	17	then	then	ADV
cana-787	133	18	there	there	PRON
cana-787	133	19	exist	exist	VERB
cana-787	133	20	unique	unique	ADJ
cana-787	133	21	fixed	fix	VERB
cana-787	133	22	point	point	NOUN
cana-787	133	23	of	of	ADP
cana-787	133	24	𝜇.	𝜇.	NOUN
cana-787	133	25	proof	proof	NOUN
cana-787	133	26	:	:	PUNCT
cana-787	133	27	if	if	SCONJ
cana-787	133	28	possible	possible	ADJ
cana-787	133	29	suppose	suppose	VERB
cana-787	133	30	there	there	PRON
cana-787	133	31	are	be	VERB
cana-787	133	32	two	two	NUM
cana-787	133	33	fixed	fix	VERB
cana-787	133	34	points	point	NOUN
cana-787	133	35	𝑥𝑓	𝑥𝑓	VERB
cana-787	133	36	and	and	CCONJ
cana-787	133	37	𝑦𝑓	𝑦𝑓	INTJ
cana-787	133	38	.	.	PUNCT
cana-787	134	1	by	by	ADP
cana-787	134	2	our	our	PRON
cana-787	134	3	hypothesis	hypothesis	NOUN
cana-787	134	4	there	there	ADV
cana-787	134	5	exist	exist	VERB
cana-787	134	6	𝑤	𝑤	ADP
cana-787	134	7	∈	∈	NOUN
cana-787	134	8	𝑋	𝑋	NOUN
cana-787	134	9	such	such	ADJ
cana-787	134	10	that	that	DET
cana-787	134	11	𝛼(𝑥𝜇	𝛼(𝑥𝜇	NUM
cana-787	134	12	,	,	PUNCT
cana-787	134	13	𝑤	𝑤	ADP
cana-787	134	14	)	)	PUNCT
cana-787	134	15	>	>	PUNCT
cana-787	134	16	𝛽−1	𝛽−1	PROPN
cana-787	134	17	and	and	CCONJ
cana-787	134	18	𝛼(𝑦𝜇	𝛼(𝑦𝜇	NOUN
cana-787	134	19	,	,	PUNCT
cana-787	134	20	𝑤	𝑤	ADP
cana-787	134	21	)	)	PUNCT
cana-787	134	22	>	>	PUNCT
cana-787	135	1	𝛽−1	𝛽−1	PROPN
cana-787	135	2	.	.	PUNCT
cana-787	136	1	being	be	AUX
cana-787	136	2	𝜇	𝜇	PRON
cana-787	136	3	is	be	AUX
cana-787	136	4	𝛼-admissible	𝛼-admissible	NUM
cana-787	136	5	,	,	PUNCT
cana-787	136	6	𝛼(𝑥𝜇	𝛼(𝑥𝜇	NUM
cana-787	136	7	,	,	PUNCT
cana-787	136	8	𝑤	𝑤	ADP
cana-787	136	9	)	)	PUNCT
cana-787	136	10	>	>	PUNCT
cana-787	137	1	𝛽−1	𝛽−1	PROPN
cana-787	137	2	implies	imply	VERB
cana-787	137	3	𝛼(𝑥𝜇	𝛼(𝑥𝜇	PROPN
cana-787	137	4	,	,	PUNCT
cana-787	137	5	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	137	6	)	)	PUNCT
cana-787	137	7	)	)	PUNCT
cana-787	137	8	>	>	PUNCT
cana-787	138	1	𝛽−1	𝛽−1	PROPN
cana-787	138	2	.	.	PUNCT
cana-787	139	1	now	now	ADV
cana-787	139	2	we	we	PRON
cana-787	139	3	have	have	VERB
cana-787	139	4	,	,	PUNCT
cana-787	139	5	𝛽−1𝑑	𝛽−1𝑑	X
cana-787	139	6	(	(	PUNCT
cana-787	139	7	𝑥𝜇	𝑥𝜇	ADV
cana-787	139	8	,	,	PUNCT
cana-787	139	9	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	139	10	)	)	PUNCT
cana-787	139	11	)	)	PUNCT
cana-787	140	1	=	=	PUNCT
cana-787	140	2	𝛽−1𝑑(𝜇	𝛽−1𝑑(𝜇	ADJ
cana-787	140	3	(	(	PUNCT
cana-787	140	4	𝑥𝜇	𝑥𝜇	CCONJ
cana-787	140	5	)	)	PUNCT
cana-787	140	6	,	,	PUNCT
cana-787	140	7	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	140	8	)	)	PUNCT
cana-787	140	9	≤	≤	NUM
cana-787	140	10	𝛼(𝑥𝜇,𝜇	𝛼(𝑥𝜇,𝜇	PROPN
cana-787	140	11	𝑛−1(𝑤))𝑑(𝜇(𝑥𝜇	𝑛−1(𝑤))𝑑(𝜇(𝑥𝜇	NOUN
cana-787	140	12	)	)	PUNCT
cana-787	140	13	,	,	PUNCT
cana-787	140	14	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	140	15	)	)	PUNCT
cana-787	140	16	)	)	PUNCT
cana-787	141	1	≤	≤	NUM
cana-787	141	2	𝜑(𝑑(𝑥𝜇,𝜇	𝜑(𝑑(𝑥𝜇,𝜇	PROPN
cana-787	141	3	𝑛−1(𝑤	𝑛−1(𝑤	NUM
cana-787	141	4	)	)	PUNCT
cana-787	141	5	)	)	PUNCT
cana-787	141	6	⇒	⇒	PROPN
cana-787	141	7	𝑑(𝑥𝜇	𝑑(𝑥𝜇	PROPN
cana-787	141	8	,	,	PUNCT
cana-787	141	9	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	141	10	)	)	PUNCT
cana-787	141	11	)	)	PUNCT
cana-787	141	12	≤	≤	NUM
cana-787	141	13	𝜎(𝑑(𝑥𝜇,𝜇	𝜎(𝑑(𝑥𝜇,𝜇	PROPN
cana-787	141	14	𝑛−1(𝑤	𝑛−1(𝑤	NUM
cana-787	141	15	)	)	PUNCT
cana-787	141	16	)	)	PUNCT
cana-787	141	17	.	.	PUNCT
cana-787	142	1	since	since	SCONJ
cana-787	142	2	𝜎	𝜎	PROPN
cana-787	142	3	is	be	AUX
cana-787	142	4	non	non	ADJ
cana-787	142	5	-	-	ADJ
cana-787	142	6	decreasing	decrease	VERB
cana-787	142	7	,	,	PUNCT
cana-787	142	8	then	then	ADV
cana-787	142	9	by	by	ADP
cana-787	142	10	induction	induction	NOUN
cana-787	142	11	we	we	PRON
cana-787	142	12	can	can	AUX
cana-787	142	13	have	have	VERB
cana-787	142	14	,	,	PUNCT
cana-787	142	15	𝑑(𝑥𝜇	𝑑(𝑥𝜇	PROPN
cana-787	142	16	,	,	PUNCT
cana-787	142	17	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	142	18	)	)	PUNCT
cana-787	142	19	)	)	PUNCT
cana-787	142	20	≤	≤	NUM
cana-787	143	1	𝜎𝑛(𝑑(𝑥𝜇,𝑤	𝜎𝑛(𝑑(𝑥𝜇,𝑤	PUNCT
cana-787	143	2	)	)	PUNCT
cana-787	143	3	)	)	PUNCT
cana-787	143	4	.	.	PUNCT
cana-787	144	1	but	but	CCONJ
cana-787	144	2	lim	lim	PROPN
cana-787	144	3	𝑛→∞	𝑛→∞	NUM
cana-787	144	4	𝜎𝑛(𝑑(𝑥𝜇,𝑤	𝜎𝑛(𝑑(𝑥𝜇,𝑤	PUNCT
cana-787	144	5	)	)	PUNCT
cana-787	144	6	)	)	PUNCT
cana-787	145	1	=	=	SYM
cana-787	145	2	0.hence	0.hence	PROPN
cana-787	145	3	,	,	PUNCT
cana-787	145	4	lim	lim	PROPN
cana-787	145	5	𝑛→∞	𝑛→∞	NUM
cana-787	145	6	𝑑(𝑥𝜇	𝑑(𝑥𝜇	PROPN
cana-787	145	7	,	,	PUNCT
cana-787	145	8	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	145	9	)	)	PUNCT
cana-787	145	10	)	)	PUNCT
cana-787	146	1	=	=	SYM
cana-787	146	2	0	0	NUM
cana-787	147	1	similarly	similarly	ADV
cana-787	147	2	we	we	PRON
cana-787	147	3	can	can	AUX
cana-787	147	4	show	show	VERB
cana-787	147	5	that	that	SCONJ
cana-787	147	6	lim	lim	PROPN
cana-787	147	7	𝑛→∞	𝑛→∞	NUM
cana-787	147	8	𝑑(𝑦𝜇	𝑑(𝑦𝜇	NOUN
cana-787	147	9	,	,	PUNCT
cana-787	147	10	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	147	11	)	)	PUNCT
cana-787	147	12	)	)	PUNCT
cana-787	148	1	=	=	PUNCT
cana-787	148	2	0	0	X
cana-787	148	3	.	.	PUNCT
cana-787	149	1	by	by	ADP
cana-787	149	2	triangular	triangular	NOUN
cana-787	149	3	property	property	NOUN
cana-787	149	4	of	of	ADP
cana-787	149	5	metric	metric	ADJ
cana-787	149	6	𝑑(𝑥𝜇	𝑑(𝑥𝜇	PROPN
cana-787	149	7	,	,	PUNCT
cana-787	149	8	𝑦𝜇	𝑦𝜇	PROPN
cana-787	149	9	)	)	PUNCT
cana-787	149	10	≤	≤	NOUN
cana-787	149	11	𝑑	𝑑	PROPN
cana-787	149	12	(	(	PUNCT
cana-787	149	13	𝑥𝜇	𝑥𝜇	ADV
cana-787	149	14	,	,	PUNCT
cana-787	149	15	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	149	16	)	)	PUNCT
cana-787	149	17	)	)	PUNCT
cana-787	150	1	+	+	CCONJ
cana-787	150	2	𝑑(𝑦𝜇	𝑑(𝑦𝜇	NOUN
cana-787	150	3	,	,	PUNCT
cana-787	150	4	𝜇𝑛(𝑤	𝜇𝑛(𝑤	NUM
cana-787	150	5	)	)	PUNCT
cana-787	150	6	)	)	PUNCT
cana-787	150	7	.	.	PUNCT
cana-787	151	1	now	now	ADV
cana-787	151	2	,	,	PUNCT
cana-787	151	3	taking	take	VERB
cana-787	151	4	𝑛	𝑛	PRON
cana-787	151	5	→	→	SYM
cana-787	151	6	∞	∞	PROPN
cana-787	151	7	𝑑(𝑥𝜇	𝑑(𝑥𝜇	PROPN
cana-787	151	8	,	,	PUNCT
cana-787	151	9	𝑦𝜇	𝑦𝜇	PROPN
cana-787	151	10	)	)	PUNCT
cana-787	151	11	=	=	SYM
cana-787	151	12	0	0	X
cana-787	151	13	.	.	PUNCT
cana-787	152	1	hence	hence	ADV
cana-787	152	2	fixed	fix	VERB
cana-787	152	3	point	point	NOUN
cana-787	152	4	is	be	AUX
cana-787	152	5	unique	unique	ADJ
cana-787	152	6	.	.	PUNCT
cana-787	153	1	□	□	PUNCT
cana-787	153	2	next	next	ADV
cana-787	153	3	,	,	PUNCT
cana-787	153	4	we	we	PRON
cana-787	153	5	introduce	introduce	VERB
cana-787	153	6	some	some	DET
cana-787	153	7	new	new	ADJ
cana-787	153	8	types	type	NOUN
cana-787	153	9	of	of	ADP
cana-787	153	10	𝛼-admissible	𝛼-admissible	ADJ
cana-787	153	11	mappings	mapping	NOUN
cana-787	153	12	and	and	CCONJ
cana-787	153	13	contraction	contraction	NOUN
cana-787	153	14	conditions	condition	NOUN
cana-787	153	15	to	to	PART
cana-787	153	16	prove	prove	VERB
cana-787	153	17	common	common	ADJ
cana-787	153	18	fixed	fix	VERB
cana-787	153	19	point	point	NOUN
cana-787	153	20	theorem	theorem	VERB
cana-787	153	21	.	.	PUNCT
cana-787	154	1	first	first	ADV
cana-787	154	2	of	of	ADP
cana-787	154	3	all	all	PRON
cana-787	154	4	let	let	VERB
cana-787	154	5	us	we	PRON
cana-787	154	6	denote	denote	VERB
cana-787	154	7	the	the	DET
cana-787	154	8	set	set	NOUN
cana-787	154	9	of	of	ADP
cana-787	154	10	all	all	DET
cana-787	154	11	coincident	coincident	ADJ
cana-787	154	12	points	point	NOUN
cana-787	154	13	of	of	ADP
cana-787	154	14	pair	pair	NOUN
cana-787	154	15	of	of	ADP
cana-787	154	16	mappings	mapping	NOUN
cana-787	154	17	(	(	PUNCT
cana-787	154	18	𝜇	𝜇	ADP
cana-787	154	19	,	,	PUNCT
cana-787	154	20	𝜌	𝜌	X
cana-787	154	21	)	)	PUNCT
cana-787	154	22	by	by	ADP
cana-787	154	23	𝐶(𝜇	𝐶(𝜇	PROPN
cana-787	154	24	,	,	PUNCT
cana-787	154	25	𝜌	𝜌	ADP
cana-787	154	26	)	)	PUNCT
cana-787	154	27	.	.	PUNCT
cana-787	155	1	definition	definition	NOUN
cana-787	155	2	3.2	3.2	NUM
cana-787	155	3	:	:	PUNCT
cana-787	155	4	let	let	VERB
cana-787	155	5	(	(	PUNCT
cana-787	155	6	𝑋	𝑋	NOUN
cana-787	155	7	,	,	PUNCT
cana-787	155	8	𝑑	𝑑	NOUN
cana-787	155	9	)	)	PUNCT
cana-787	155	10	be	be	VERB
cana-787	155	11	a	a	DET
cana-787	155	12	complete	complete	ADJ
cana-787	155	13	metric	metric	ADJ
cana-787	155	14	space	space	NOUN
cana-787	155	15	and	and	CCONJ
cana-787	155	16	𝜇	𝜇	X
cana-787	155	17	,	,	PUNCT
cana-787	155	18	𝜌	𝜌	X
cana-787	155	19	are	be	AUX
cana-787	155	20	selfmappings	selfmapping	NOUN
cana-787	155	21	on	on	ADP
cana-787	155	22	𝑋	𝑋	PROPN
cana-787	155	23	with	with	ADP
cana-787	155	24	𝜌(𝑋	𝜌(𝑋	NUM
cana-787	155	25	)	)	PUNCT
cana-787	156	1	⊂	⊂	PROPN
cana-787	156	2	𝜇(𝑋	𝜇(𝑋	NUM
cana-787	156	3	)	)	PUNCT
cana-787	156	4	.	.	PUNCT
cana-787	157	1	we	we	PRON
cana-787	157	2	say	say	VERB
cana-787	157	3	𝜌	𝜌	X
cana-787	157	4	is	be	AUX
cana-787	157	5	𝜇	𝜇	ADP
cana-787	157	6	−	−	NOUN
cana-787	157	7	𝛼	𝛼	NOUN
cana-787	157	8	,	,	PUNCT
cana-787	157	9	𝜑	𝜑	PROPN
cana-787	157	10	-contraction	-contraction	NOUN
cana-787	157	11	if	if	SCONJ
cana-787	157	12	there	there	PRON
cana-787	157	13	exist	exist	VERB
cana-787	157	14	𝛼	𝛼	NOUN
cana-787	157	15	:	:	PUNCT
cana-787	157	16	𝑋	𝑋	NOUN
cana-787	157	17	×	×	NOUN
cana-787	157	18	𝑋	𝑋	PROPN
cana-787	157	19	→	→	SYM
cana-787	157	20	[	[	X
cana-787	157	21	0	0	NUM
cana-787	157	22	,	,	PUNCT
cana-787	157	23	∞	∞	PROPN
cana-787	157	24	)	)	PUNCT
cana-787	157	25	and	and	CCONJ
cana-787	157	26	a	a	DET
cana-787	157	27	comparison	comparison	NOUN
cana-787	157	28	function	function	VERB
cana-787	157	29	𝜑	𝜑	PRON
cana-787	157	30	:	:	PUNCT
cana-787	158	1	[	[	X
cana-787	158	2	0	0	NUM
cana-787	158	3	,	,	PUNCT
cana-787	158	4	∞	∞	PROPN
cana-787	158	5	)	)	PUNCT
cana-787	158	6	→	→	PUNCT
cana-787	159	1	[	[	X
cana-787	159	2	0	0	NUM
cana-787	159	3	,	,	PUNCT
cana-787	159	4	∞	∞	NOUN
cana-787	159	5	)	)	PUNCT
cana-787	159	6	such	such	ADJ
cana-787	159	7	that	that	SCONJ
cana-787	159	8	,	,	PUNCT
cana-787	159	9	𝛼(𝜇(𝑎	𝛼(𝜇(𝑎	PROPN
cana-787	159	10	)	)	PUNCT
cana-787	159	11	,	,	PUNCT
cana-787	159	12	𝜇(𝑏))𝑑(𝜌(𝑎	𝜇(𝑏))𝑑(𝜌(𝑎	PROPN
cana-787	159	13	)	)	PUNCT
cana-787	159	14	,	,	PUNCT
cana-787	159	15	𝜌(𝑏	𝜌(𝑏	NUM
cana-787	159	16	)	)	PUNCT
cana-787	159	17	)	)	PUNCT
cana-787	159	18	≤	≤	NUM
cana-787	159	19	𝜑(𝑑(𝜇(𝑎	𝜑(𝑑(𝜇(𝑎	PROPN
cana-787	159	20	)	)	PUNCT
cana-787	159	21	,	,	PUNCT
cana-787	159	22	𝜇(𝑏	𝜇(𝑏	NOUN
cana-787	159	23	)	)	PUNCT
cana-787	159	24	)	)	PUNCT
cana-787	159	25	)	)	PUNCT
cana-787	159	26	∀	∀	PUNCT
cana-787	160	1	𝑎	𝑎	X
cana-787	160	2	,	,	PUNCT
cana-787	160	3	𝑏	𝑏	PROPN
cana-787	160	4	∈	∈	PROPN
cana-787	160	5	𝑋.	𝑋.	PROPN
cana-787	160	6	definition3.3	definition3.3	NOUN
cana-787	160	7	:	:	PUNCT
cana-787	160	8	let	let	VERB
cana-787	160	9	(	(	PUNCT
cana-787	160	10	𝑋	𝑋	NOUN
cana-787	160	11	,	,	PUNCT
cana-787	160	12	𝑑	𝑑	NOUN
cana-787	160	13	)	)	PUNCT
cana-787	160	14	be	be	VERB
cana-787	160	15	a	a	DET
cana-787	160	16	metric	metric	ADJ
cana-787	160	17	space	space	NOUN
cana-787	160	18	and	and	CCONJ
cana-787	160	19	𝜇	𝜇	X
cana-787	160	20	,	,	PUNCT
cana-787	160	21	𝜌	𝜌	PART
cana-787	160	22	selfmappings	selfmapping	NOUN
cana-787	160	23	on	on	ADP
cana-787	160	24	𝑋	𝑋	NOUN
cana-787	160	25	with	with	ADP
cana-787	160	26	𝜌(𝑋	𝜌(𝑋	NOUN
cana-787	160	27	)	)	PUNCT
cana-787	160	28	⊂	⊂	PROPN
cana-787	160	29	𝜇(𝑋).we	𝜇(𝑋).we	PROPN
cana-787	160	30	say	say	VERB
cana-787	160	31	𝜌	𝜌	PRON
cana-787	160	32	is	be	AUX
cana-787	160	33	𝜇	𝜇	ADP
cana-787	160	34	−	−	PROPN
cana-787	160	35	𝛼𝛽−1	𝛼𝛽−1	ADP
cana-787	160	36	-admissible	-admissible	ADJ
cana-787	160	37	if	if	SCONJ
cana-787	160	38	there	there	PRON
cana-787	160	39	exists	exist	VERB
cana-787	160	40	a	a	DET
cana-787	160	41	function	function	NOUN
cana-787	160	42	𝛼	𝛼	NOUN
cana-787	160	43	:	:	PUNCT
cana-787	160	44	𝑋	𝑋	NOUN
cana-787	160	45	×	×	NOUN
cana-787	160	46	𝑋	𝑋	PROPN
cana-787	160	47	→	→	SYM
cana-787	160	48	[	[	X
cana-787	160	49	0	0	NUM
cana-787	160	50	,	,	PUNCT
cana-787	160	51	∞	∞	NOUN
cana-787	160	52	)	)	PUNCT
cana-787	160	53	such	such	ADJ
cana-787	160	54	that	that	SCONJ
cana-787	160	55	,	,	PUNCT
cana-787	160	56	𝛼(𝜇(𝑎	𝛼(𝜇(𝑎	PROPN
cana-787	160	57	)	)	PUNCT
cana-787	160	58	,	,	PUNCT
cana-787	160	59	𝜇(𝑏	𝜇(𝑏	NOUN
cana-787	160	60	)	)	PUNCT
cana-787	160	61	)	)	PUNCT
cana-787	160	62	>	>	PUNCT
cana-787	161	1	𝛽−1	𝛽−1	PROPN
cana-787	161	2	⇒	⇒	VERB
cana-787	161	3	𝛼(𝜌(𝑎	𝛼(𝜌(𝑎	PROPN
cana-787	161	4	)	)	PUNCT
cana-787	161	5	,	,	PUNCT
cana-787	161	6	𝜌(𝑏	𝜌(𝑏	NUM
cana-787	161	7	)	)	PUNCT
cana-787	161	8	)	)	PUNCT
cana-787	162	1	>	>	PUNCT
cana-787	163	1	𝛽−1	𝛽−1	PROPN
cana-787	163	2	for	for	ADP
cana-787	163	3	𝑎	𝑎	NOUN
cana-787	163	4	,	,	PUNCT
cana-787	163	5	𝑏	𝑏	PROPN
cana-787	163	6	∈	∈	PROPN
cana-787	163	7	𝑋	𝑋	NOUN
cana-787	163	8	communications	communication	NOUN
cana-787	163	9	on	on	ADP
cana-787	163	10	applied	apply	VERB
cana-787	163	11	nonlinear	nonlinear	ADJ
cana-787	163	12	analysis	analysis	NOUN
cana-787	163	13	issn	issn	NOUN
cana-787	163	14	:	:	PUNCT
cana-787	163	15	1074	1074	NUM
cana-787	163	16	-	-	PUNCT
cana-787	163	17	133x	133x	NUM
cana-787	163	18	vol	vol	NOUN
cana-787	163	19	31	31	NUM
cana-787	163	20	no	no	NOUN
cana-787	163	21	.	.	PUNCT
cana-787	164	1	3s	3s	NUM
cana-787	164	2	(	(	PUNCT
cana-787	164	3	2024	2024	NUM
cana-787	164	4	)	)	PUNCT
cana-787	164	5	373	373	NUM
cana-787	164	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	164	7	now	now	ADV
cana-787	164	8	we	we	PRON
cana-787	164	9	apply	apply	VERB
cana-787	164	10	above	above	ADP
cana-787	164	11	results	result	NOUN
cana-787	164	12	to	to	PART
cana-787	164	13	prove	prove	VERB
cana-787	164	14	the	the	DET
cana-787	164	15	common	common	ADJ
cana-787	164	16	fixed	fix	VERB
cana-787	164	17	point	point	NOUN
cana-787	164	18	in	in	ADP
cana-787	164	19	complete	complete	ADJ
cana-787	164	20	metric	metric	ADJ
cana-787	164	21	spaces	space	NOUN
cana-787	164	22	.	.	PUNCT
cana-787	165	1	theorem	theorem	VERB
cana-787	165	2	3.4	3.4	NUM
cana-787	165	3	:	:	PUNCT
cana-787	165	4	let	let	VERB
cana-787	165	5	(	(	PUNCT
cana-787	165	6	𝑋	𝑋	PROPN
cana-787	165	7	,	,	PUNCT
cana-787	165	8	𝑑	𝑑	NOUN
cana-787	165	9	)	)	PUNCT
cana-787	165	10	be	be	VERB
cana-787	165	11	a	a	DET
cana-787	165	12	complete	complete	ADJ
cana-787	165	13	metric	metric	ADJ
cana-787	165	14	space	space	NOUN
cana-787	165	15	and	and	CCONJ
cana-787	165	16	𝜇	𝜇	X
cana-787	165	17	,	,	PUNCT
cana-787	165	18	𝜌	𝜌	X
cana-787	165	19	are	be	AUX
cana-787	165	20	weakly	weakly	ADV
cana-787	165	21	compatible	compatible	ADJ
cana-787	165	22	selfmappings	selfmapping	NOUN
cana-787	165	23	on	on	ADP
cana-787	165	24	𝑋	𝑋	PROPN
cana-787	165	25	with	with	ADP
cana-787	165	26	𝜌(𝑋	𝜌(𝑋	NOUN
cana-787	165	27	)	)	PUNCT
cana-787	165	28	⊂	⊂	PROPN
cana-787	166	1	𝜇(𝑋).also	𝜇(𝑋).also	ADV
cana-787	166	2	let	let	VERB
cana-787	166	3	𝜇(𝑋	𝜇(𝑋	NUM
cana-787	166	4	)	)	PUNCT
cana-787	166	5	is	be	AUX
cana-787	166	6	closed	close	VERB
cana-787	166	7	subset	subset	NOUN
cana-787	166	8	of	of	ADP
cana-787	166	9	𝑋.there	𝑋.there	X
cana-787	166	10	exist	exist	VERB
cana-787	166	11	𝛼	𝛼	NOUN
cana-787	166	12	:	:	PUNCT
cana-787	166	13	𝑋	𝑋	NOUN
cana-787	166	14	×	×	NOUN
cana-787	166	15	𝑋	𝑋	PROPN
cana-787	166	16	→	→	SYM
cana-787	166	17	[	[	X
cana-787	166	18	0	0	NUM
cana-787	166	19	,	,	PUNCT
cana-787	166	20	∞	∞	PROPN
cana-787	166	21	)	)	PUNCT
cana-787	166	22	,	,	PUNCT
cana-787	166	23	𝜑	𝜑	PROPN
cana-787	166	24	∈	∈	PROPN
cana-787	166	25	∅	∅	NOUN
cana-787	166	26	and	and	CCONJ
cana-787	166	27	𝛽	𝛽	NOUN
cana-787	166	28	is	be	AUX
cana-787	166	29	maximum	maximum	ADJ
cana-787	166	30	of	of	ADP
cana-787	166	31	all	all	DET
cana-787	166	32	member	member	NOUN
cana-787	166	33	of	of	ADP
cana-787	166	34	𝑅𝜑+	𝑅𝜑+	NOUN
cana-787	166	35	such	such	ADJ
cana-787	166	36	that	that	SCONJ
cana-787	166	37	𝜇	𝜇	ADP
cana-787	166	38	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-787	166	39	𝜌	𝜌	PART
cana-787	166	40	satisfy	satisfy	VERB
cana-787	166	41	following	follow	VERB
cana-787	166	42	conditions	condition	NOUN
cana-787	166	43	:	:	PUNCT
cana-787	166	44	a	a	X
cana-787	166	45	)	)	PUNCT
cana-787	166	46	𝜌	𝜌	PRON
cana-787	166	47	is	be	AUX
cana-787	166	48	𝜇	𝜇	ADP
cana-787	166	49	−	−	NOUN
cana-787	166	50	𝛼	𝛼	NOUN
cana-787	166	51	,	,	PUNCT
cana-787	166	52	𝜑	𝜑	PRON
cana-787	166	53	contraction	contraction	NOUN
cana-787	166	54	.	.	PUNCT
cana-787	167	1	b	b	X
cana-787	167	2	)	)	PUNCT
cana-787	167	3	𝜌	𝜌	PRON
cana-787	167	4	is	be	AUX
cana-787	167	5	𝜇	𝜇	ADP
cana-787	167	6	−	−	PROPN
cana-787	167	7	𝛼𝛽−1	𝛼𝛽−1	PROPN
cana-787	167	8	admissible	admissible	NOUN
cana-787	167	9	.	.	PUNCT
cana-787	168	1	c	c	X
cana-787	168	2	)	)	PUNCT
cana-787	168	3	there	there	PRON
cana-787	168	4	exist	exist	VERB
cana-787	168	5	𝑥0	𝑥0	NOUN
cana-787	168	6	∈	∈	NOUN
cana-787	168	7	𝑋	𝑋	NOUN
cana-787	168	8	such	such	ADJ
cana-787	168	9	that	that	DET
cana-787	168	10	𝛼(𝜇(𝑥0	𝛼(𝜇(𝑥0	NOUN
cana-787	168	11	)	)	PUNCT
cana-787	168	12	,	,	PUNCT
cana-787	168	13	𝜌(𝑥0	𝜌(𝑥0	NOUN
cana-787	168	14	)	)	PUNCT
cana-787	168	15	)	)	PUNCT
cana-787	169	1	>	>	PUNCT
cana-787	170	1	𝛽−1	𝛽−1	PROPN
cana-787	170	2	d	d	PROPN
cana-787	170	3	)	)	PUNCT
cana-787	170	4	if	if	SCONJ
cana-787	170	5	for	for	ADP
cana-787	170	6	a	a	DET
cana-787	170	7	sequence	sequence	NOUN
cana-787	170	8	{	{	PUNCT
cana-787	170	9	𝑥𝑛	𝑥𝑛	NOUN
cana-787	170	10	}	}	PUNCT
cana-787	170	11	in	in	ADP
cana-787	170	12	𝑋	𝑋	PROPN
cana-787	170	13	,	,	PUNCT
cana-787	170	14	the	the	DET
cana-787	170	15	sequence	sequence	NOUN
cana-787	170	16	𝜇(𝑥𝑛	𝜇(𝑥𝑛	NOUN
cana-787	170	17	)	)	PUNCT
cana-787	170	18	converges	converge	VERB
cana-787	170	19	to	to	ADP
cana-787	170	20	𝜇(𝑥	𝜇(𝑥	PROPN
cana-787	170	21	)	)	PUNCT
cana-787	170	22	and	and	CCONJ
cana-787	170	23	𝛼(𝜇(𝑥𝑛	𝛼(𝜇(𝑥𝑛	NOUN
cana-787	170	24	)	)	PUNCT
cana-787	170	25	,	,	PUNCT
cana-787	170	26	𝜇(𝑥𝑛+1	𝜇(𝑥𝑛+1	PROPN
cana-787	170	27	)	)	PUNCT
cana-787	170	28	>	>	PUNCT
cana-787	171	1	𝛽−1	𝛽−1	PROPN
cana-787	171	2	then	then	ADV
cana-787	171	3	there	there	PRON
cana-787	171	4	exist	exist	VERB
cana-787	171	5	sub	sub	ADJ
cana-787	171	6	-	-	NOUN
cana-787	171	7	sequence	sequence	ADJ
cana-787	171	8	𝜇(𝑥𝑛𝑘	𝜇(𝑥𝑛𝑘	ADJ
cana-787	171	9	)	)	PUNCT
cana-787	171	10	of	of	ADP
cana-787	171	11	𝜇(𝑥𝑛	𝜇(𝑥𝑛	PROPN
cana-787	171	12	)	)	PUNCT
cana-787	171	13	such	such	ADJ
cana-787	171	14	that	that	DET
cana-787	171	15	𝛼(𝜇(𝑥𝑛𝑘	𝛼(𝜇(𝑥𝑛𝑘	NUM
cana-787	171	16	)	)	PUNCT
cana-787	171	17	,	,	PUNCT
cana-787	171	18	𝜇(𝑥	𝜇(𝑥	PROPN
cana-787	171	19	)	)	PUNCT
cana-787	171	20	)	)	PUNCT
cana-787	171	21	>	>	PUNCT
cana-787	172	1	𝛽−1	𝛽−1	PROPN
cana-787	172	2	.	.	PUNCT
cana-787	173	1	e	e	X
cana-787	173	2	)	)	PUNCT
cana-787	173	3	for	for	ADP
cana-787	173	4	all	all	DET
cana-787	173	5	𝑝	𝑝	NOUN
cana-787	173	6	,	,	PUNCT
cana-787	173	7	𝑞	𝑞	X
cana-787	173	8	∈	∈	PROPN
cana-787	173	9	𝐶(𝜇	𝐶(𝜇	PROPN
cana-787	173	10	,	,	PUNCT
cana-787	173	11	𝜌)the	𝜌)the	DET
cana-787	173	12	condition	condition	NOUN
cana-787	173	13	𝛼(𝜇(𝑝	𝛼(𝜇(𝑝	PROPN
cana-787	173	14	)	)	PUNCT
cana-787	173	15	,	,	PUNCT
cana-787	173	16	𝜇(𝑞	𝜇(𝑞	PROPN
cana-787	173	17	)	)	PUNCT
cana-787	173	18	)	)	PUNCT
cana-787	174	1	>	>	PUNCT
cana-787	174	2	𝛽−1	𝛽−1	PROPN
cana-787	174	3	or	or	CCONJ
cana-787	174	4	𝛼(𝜇(𝑞	𝛼(𝜇(𝑞	NOUN
cana-787	174	5	)	)	PUNCT
cana-787	174	6	,	,	PUNCT
cana-787	174	7	𝜇(𝑝	𝜇(𝑝	PROPN
cana-787	174	8	)	)	PUNCT
cana-787	174	9	)	)	PUNCT
cana-787	174	10	>	>	PUNCT
cana-787	175	1	𝛽−1	𝛽−1	PROPN
cana-787	175	2	exist	exist	VERB
cana-787	175	3	.	.	PUNCT
cana-787	176	1	then	then	ADV
cana-787	176	2	𝜇	𝜇	ADV
cana-787	176	3	and	and	CCONJ
cana-787	176	4	𝜌	𝜌	PART
cana-787	176	5	have	have	VERB
cana-787	176	6	a	a	DET
cana-787	176	7	unique	unique	ADJ
cana-787	176	8	point	point	NOUN
cana-787	176	9	of	of	ADP
cana-787	176	10	coincident	coincident	NOUN
cana-787	176	11	.	.	PUNCT
cana-787	177	1	further	far	ADV
cana-787	177	2	this	this	DET
cana-787	177	3	point	point	NOUN
cana-787	177	4	is	be	AUX
cana-787	177	5	unique	unique	ADJ
cana-787	177	6	common	common	ADJ
cana-787	177	7	fixed	fix	VERB
cana-787	177	8	point	point	NOUN
cana-787	177	9	.	.	PUNCT
cana-787	178	1	proof	proof	NOUN
cana-787	178	2	:	:	PUNCT
cana-787	178	3	since	since	SCONJ
cana-787	178	4	𝜌(𝑋	𝜌(𝑋	NUM
cana-787	178	5	)	)	PUNCT
cana-787	178	6	⊂	⊂	PROPN
cana-787	178	7	𝜇(𝑋	𝜇(𝑋	NUM
cana-787	178	8	)	)	PUNCT
cana-787	178	9	,	,	PUNCT
cana-787	178	10	for	for	ADP
cana-787	178	11	𝑥0	𝑥0	NOUN
cana-787	178	12	∈	∈	PROPN
cana-787	178	13	𝑋	𝑋	PROPN
cana-787	178	14	there	there	ADV
cana-787	178	15	exist	exist	VERB
cana-787	178	16	𝑥1	𝑥1	NOUN
cana-787	178	17	∈	∈	PROPN
cana-787	178	18	𝑋	𝑋	PROPN
cana-787	178	19	such	such	ADJ
cana-787	178	20	that	that	SCONJ
cana-787	178	21	𝜌(𝑥0	𝜌(𝑥0	NOUN
cana-787	178	22	)	)	PUNCT
cana-787	178	23	=	=	SYM
cana-787	178	24	𝜇(𝑥1	𝜇(𝑥1	X
cana-787	178	25	)	)	PUNCT
cana-787	178	26	.	.	PUNCT
cana-787	179	1	in	in	ADP
cana-787	179	2	general	general	ADJ
cana-787	179	3	we	we	PRON
cana-787	179	4	can	can	AUX
cana-787	179	5	write	write	VERB
cana-787	179	6	𝑠𝑛	𝑠𝑛	NOUN
cana-787	179	7	=	=	PUNCT
cana-787	179	8	𝜌(𝑥𝑛	𝜌(𝑥𝑛	PROPN
cana-787	179	9	)	)	PUNCT
cana-787	180	1	=	=	SYM
cana-787	180	2	𝜇(𝑥𝑛+1).if	𝜇(𝑥𝑛+1).if	NOUN
cana-787	180	3	𝑠𝑛	𝑠𝑛	NOUN
cana-787	180	4	=	=	SYM
cana-787	180	5	𝑠𝑛+1	𝑠𝑛+1	NOUN
cana-787	180	6	for	for	ADP
cana-787	180	7	some	some	DET
cana-787	180	8	𝑛	𝑛	PRON
cana-787	180	9	∈	∈	PROPN
cana-787	180	10	𝑁	𝑁	PROPN
cana-787	180	11	,	,	PUNCT
cana-787	180	12	then	then	ADV
cana-787	180	13	𝑠𝑛	𝑠𝑛	NOUN
cana-787	180	14	=	=	SYM
cana-787	180	15	𝜌(𝑥𝑛+1	𝜌(𝑥𝑛+1	ADJ
cana-787	180	16	)	)	PUNCT
cana-787	180	17	=	=	PUNCT
cana-787	181	1	𝜇(𝑥𝑛+1).this	𝜇(𝑥𝑛+1).this	PROPN
cana-787	181	2	implies	imply	VERB
cana-787	181	3	there	there	PRON
cana-787	181	4	is	be	VERB
cana-787	181	5	a	a	DET
cana-787	181	6	point	point	NOUN
cana-787	181	7	of	of	ADP
cana-787	181	8	coincident	coincident	NOUN
cana-787	181	9	.	.	PUNCT
cana-787	182	1	thus	thus	ADV
cana-787	182	2	let	let	VERB
cana-787	182	3	𝑠𝑛	𝑠𝑛	NOUN
cana-787	182	4	≠	≠	PROPN
cana-787	182	5	𝑠𝑛+1	𝑠𝑛+1	ADP
cana-787	182	6	∀𝑛	∀𝑛	NOUN
cana-787	182	7	∈	∈	NOUN
cana-787	182	8	𝑁	𝑁	PROPN
cana-787	182	9	.	.	PUNCT
cana-787	183	1	by	by	ADP
cana-787	183	2	condition	condition	NOUN
cana-787	183	3	(	(	PUNCT
cana-787	183	4	c	c	NOUN
cana-787	183	5	)	)	PUNCT
cana-787	183	6	,	,	PUNCT
cana-787	183	7	there	there	PRON
cana-787	183	8	exist	exist	VERB
cana-787	183	9	𝑥0	𝑥0	NOUN
cana-787	183	10	∈	∈	NOUN
cana-787	183	11	𝑋	𝑋	NOUN
cana-787	183	12	such	such	ADJ
cana-787	183	13	that	that	DET
cana-787	183	14	𝛼(𝜇(𝑥0	𝛼(𝜇(𝑥0	NOUN
cana-787	183	15	)	)	PUNCT
cana-787	183	16	,	,	PUNCT
cana-787	183	17	𝜌(𝑥0	𝜌(𝑥0	NOUN
cana-787	183	18	)	)	PUNCT
cana-787	183	19	)	)	PUNCT
cana-787	183	20	>	>	PUNCT
cana-787	184	1	𝛽−1	𝛽−1	PROPN
cana-787	184	2	⇒	⇒	VERB
cana-787	184	3	𝛼(𝜇(𝑥0	𝛼(𝜇(𝑥0	NOUN
cana-787	184	4	)	)	PUNCT
cana-787	184	5	,	,	PUNCT
cana-787	184	6	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	184	7	)	)	PUNCT
cana-787	184	8	)	)	PUNCT
cana-787	185	1	>	>	PUNCT
cana-787	185	2	𝛽−1	𝛽−1	PROPN
cana-787	185	3	since	since	SCONJ
cana-787	185	4	𝑔	𝑔	PROPN
cana-787	185	5	is	be	AUX
cana-787	185	6	𝜇	𝜇	ADP
cana-787	185	7	−	−	NOUN
cana-787	185	8	𝛼	𝛼	NOUN
cana-787	185	9	admissible	admissible	ADJ
cana-787	185	10	,	,	PUNCT
cana-787	185	11	so	so	ADV
cana-787	185	12	𝛼(𝜇(𝑥0	𝛼(𝜇(𝑥0	PROPN
cana-787	185	13	)	)	PUNCT
cana-787	185	14	,	,	PUNCT
cana-787	185	15	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	185	16	)	)	PUNCT
cana-787	185	17	)	)	PUNCT
cana-787	185	18	>	>	PUNCT
cana-787	186	1	𝛽−1	𝛽−1	PROPN
cana-787	186	2	⇒	⇒	VERB
cana-787	186	3	𝛼(𝜌(𝑥0	𝛼(𝜌(𝑥0	VERB
cana-787	186	4	)	)	PUNCT
cana-787	186	5	,	,	PUNCT
cana-787	186	6	𝜌(𝑥1	𝜌(𝑥1	X
cana-787	186	7	)	)	PUNCT
cana-787	186	8	)	)	PUNCT
cana-787	186	9	>	>	PUNCT
cana-787	187	1	𝛽−1	𝛽−1	PROPN
cana-787	187	2	⇒	⇒	VERB
cana-787	187	3	𝛼(𝜇(𝑥1	𝛼(𝜇(𝑥1	PROPN
cana-787	187	4	)	)	PUNCT
cana-787	187	5	,	,	PUNCT
cana-787	187	6	𝜇(𝑥2	𝜇(𝑥2	NOUN
cana-787	187	7	)	)	PUNCT
cana-787	187	8	)	)	PUNCT
cana-787	187	9	>	>	PUNCT
cana-787	188	1	𝛽−1	𝛽−1	PROPN
cana-787	188	2	⇒	⇒	VERB
cana-787	188	3	𝛼(𝜌(𝑥1	𝛼(𝜌(𝑥1	NOUN
cana-787	188	4	)	)	PUNCT
cana-787	188	5	,	,	PUNCT
cana-787	188	6	𝜌(𝑥2	𝜌(𝑥2	NOUN
cana-787	188	7	)	)	PUNCT
cana-787	188	8	)	)	PUNCT
cana-787	189	1	>	>	PUNCT
cana-787	189	2	𝛽−1	𝛽−1	PROPN
cana-787	189	3	⇒	⇒	VERB
cana-787	189	4	𝛼(𝜇(𝑥2	𝛼(𝜇(𝑥2	NUM
cana-787	189	5	)	)	PUNCT
cana-787	189	6	,	,	PUNCT
cana-787	189	7	𝜇(𝑥3	𝜇(𝑥3	NOUN
cana-787	189	8	)	)	PUNCT
cana-787	189	9	)	)	PUNCT
cana-787	189	10	>	>	PUNCT
cana-787	190	1	𝛽−1	𝛽−1	PROPN
cana-787	190	2	continuing	continue	VERB
cana-787	190	3	this	this	DET
cana-787	190	4	process	process	NOUN
cana-787	190	5	we	we	PRON
cana-787	190	6	have	have	VERB
cana-787	190	7	,	,	PUNCT
cana-787	190	8	𝛼(𝜇(𝑥𝑛	𝛼(𝜇(𝑥𝑛	NOUN
cana-787	190	9	)	)	PUNCT
cana-787	190	10	,	,	PUNCT
cana-787	190	11	𝜇(𝑥𝑛+1	𝜇(𝑥𝑛+1	PROPN
cana-787	190	12	)	)	PUNCT
cana-787	190	13	)	)	PUNCT
cana-787	191	1	>	>	PUNCT
cana-787	191	2	𝛽−1	𝛽−1	PROPN
cana-787	191	3	…	…	PUNCT
cana-787	191	4	…	…	PUNCT
cana-787	191	5	(	(	PUNCT
cana-787	191	6	3.6	3.6	NUM
cana-787	191	7	)	)	PUNCT
cana-787	191	8	from	from	ADP
cana-787	191	9	condition	condition	NOUN
cana-787	191	10	(	(	PUNCT
cana-787	191	11	a	a	NOUN
cana-787	191	12	)	)	PUNCT
cana-787	191	13	𝛽−1𝑑(𝜌(𝑥0	𝛽−1𝑑(𝜌(𝑥0	PROPN
cana-787	191	14	)	)	PUNCT
cana-787	191	15	,	,	PUNCT
cana-787	191	16	𝜌(𝑥1	𝜌(𝑥1	X
cana-787	191	17	)	)	PUNCT
cana-787	191	18	≤	≤	NOUN
cana-787	191	19	𝛼(𝜇(𝑥0	𝛼(𝜇(𝑥0	NOUN
cana-787	191	20	)	)	PUNCT
cana-787	191	21	,	,	PUNCT
cana-787	191	22	𝜇(𝑥1))𝑑(𝜌(𝑥0	𝜇(𝑥1))𝑑(𝜌(𝑥0	NOUN
cana-787	191	23	)	)	PUNCT
cana-787	191	24	,	,	PUNCT
cana-787	191	25	𝜌(𝑥1	𝜌(𝑥1	X
cana-787	191	26	)	)	PUNCT
cana-787	191	27	≤	≤	NUM
cana-787	191	28	𝜑(𝑑(𝜇(𝑥0	𝜑(𝑑(𝜇(𝑥0	NOUN
cana-787	191	29	)	)	PUNCT
cana-787	191	30	,	,	PUNCT
cana-787	191	31	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	191	32	)	)	PUNCT
cana-787	191	33	)	)	PUNCT
cana-787	191	34	⇒	⇒	PROPN
cana-787	191	35	𝑑(𝜌(𝑥0	𝑑(𝜌(𝑥0	PROPN
cana-787	191	36	)	)	PUNCT
cana-787	191	37	,	,	PUNCT
cana-787	191	38	𝜌(𝑥1	𝜌(𝑥1	X
cana-787	191	39	)	)	PUNCT
cana-787	191	40	≤	≤	NOUN
cana-787	191	41	𝛽𝜑(𝑑(𝜇(𝑥0	𝛽𝜑(𝑑(𝜇(𝑥0	VERB
cana-787	191	42	)	)	PUNCT
cana-787	191	43	,	,	PUNCT
cana-787	191	44	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	191	45	)	)	PUNCT
cana-787	191	46	)	)	PUNCT
cana-787	192	1	[	[	PUNCT
cana-787	192	2	∵	∵	INTJ
cana-787	192	3	𝛽𝜑	𝛽𝜑	PROPN
cana-787	192	4	=	=	SYM
cana-787	192	5	𝜎	𝜎	PROPN
cana-787	192	6	∈	∈	PROPN
cana-787	192	7	∅	∅	NOUN
cana-787	192	8	]	]	PUNCT
cana-787	192	9	⇒	⇒	NOUN
cana-787	192	10	𝑑(𝜌(𝑥0	𝑑(𝜌(𝑥0	PROPN
cana-787	192	11	)	)	PUNCT
cana-787	192	12	,	,	PUNCT
cana-787	192	13	𝜌(𝑥1	𝜌(𝑥1	X
cana-787	192	14	)	)	PUNCT
cana-787	192	15	≤	≤	NOUN
cana-787	192	16	𝜎(𝑑(𝜇(𝑥0	𝜎(𝑑(𝜇(𝑥0	NUM
cana-787	192	17	)	)	PUNCT
cana-787	192	18	,	,	PUNCT
cana-787	192	19	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	192	20	)	)	PUNCT
cana-787	192	21	)	)	PUNCT
cana-787	192	22	……	……	NOUN
cana-787	192	23	……	……	NOUN
cana-787	192	24	.	.	PUNCT
cana-787	193	1	(	(	PUNCT
cana-787	193	2	3.7	3.7	NUM
cana-787	193	3	)	)	PUNCT
cana-787	193	4	⇒	⇒	NOUN
cana-787	193	5	𝑑(𝜇(𝑥1	𝑑(𝜇(𝑥1	VERB
cana-787	193	6	)	)	PUNCT
cana-787	193	7	,	,	PUNCT
cana-787	193	8	𝜇(𝑥2	𝜇(𝑥2	NOUN
cana-787	193	9	)	)	PUNCT
cana-787	193	10	≤	≤	NOUN
cana-787	193	11	𝜎(𝑑(𝜇(𝑥0	𝜎(𝑑(𝜇(𝑥0	NUM
cana-787	193	12	)	)	PUNCT
cana-787	193	13	,	,	PUNCT
cana-787	193	14	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	193	15	)	)	PUNCT
cana-787	193	16	)	)	PUNCT
cana-787	194	1	……	……	NOUN
cana-787	194	2	……	……	NOUN
cana-787	194	3	……	……	NOUN
cana-787	194	4	.	.	PUNCT
cana-787	195	1	(	(	PUNCT
cana-787	195	2	3.8	3.8	NUM
cana-787	195	3	)	)	PUNCT
cana-787	195	4	similarly	similarly	ADV
cana-787	195	5	we	we	PRON
cana-787	195	6	can	can	AUX
cana-787	195	7	show	show	VERB
cana-787	195	8	𝑑(𝜇(𝑥2	𝑑(𝜇(𝑥2	PROPN
cana-787	195	9	)	)	PUNCT
cana-787	195	10	,	,	PUNCT
cana-787	195	11	𝜇(𝑥3	𝜇(𝑥3	NOUN
cana-787	195	12	)	)	PUNCT
cana-787	195	13	≤	≤	NUM
cana-787	195	14	𝜎(𝑑(𝜇(𝑥1	𝜎(𝑑(𝜇(𝑥1	NOUN
cana-787	195	15	)	)	PUNCT
cana-787	195	16	,	,	PUNCT
cana-787	195	17	𝜇(𝑥2	𝜇(𝑥2	NOUN
cana-787	195	18	)	)	PUNCT
cana-787	195	19	)	)	PUNCT
cana-787	195	20	communications	communication	NOUN
cana-787	195	21	on	on	ADP
cana-787	195	22	applied	apply	VERB
cana-787	195	23	nonlinear	nonlinear	ADJ
cana-787	195	24	analysis	analysis	NOUN
cana-787	195	25	issn	issn	NOUN
cana-787	195	26	:	:	PUNCT
cana-787	195	27	1074	1074	NUM
cana-787	195	28	-	-	PUNCT
cana-787	195	29	133x	133x	NUM
cana-787	195	30	vol	vol	NOUN
cana-787	195	31	31	31	NUM
cana-787	195	32	no	no	NOUN
cana-787	195	33	.	.	PUNCT
cana-787	196	1	3s	3s	NUM
cana-787	196	2	(	(	PUNCT
cana-787	196	3	2024	2024	NUM
cana-787	196	4	)	)	PUNCT
cana-787	196	5	374	374	NUM
cana-787	196	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	196	7	from	from	ADP
cana-787	196	8	(	(	PUNCT
cana-787	196	9	3.8	3.8	NUM
cana-787	196	10	)	)	PUNCT
cana-787	196	11	and	and	CCONJ
cana-787	196	12	non	non	ADJ
cana-787	196	13	-	-	ADJ
cana-787	196	14	decreasing	decrease	VERB
cana-787	196	15	property	property	NOUN
cana-787	196	16	of	of	ADP
cana-787	196	17	𝜎	𝜎	PROPN
cana-787	196	18	,	,	PUNCT
cana-787	196	19	we	we	PRON
cana-787	196	20	have	have	VERB
cana-787	196	21	𝑑(𝜇(𝑥2	𝑑(𝜇(𝑥2	NOUN
cana-787	196	22	)	)	PUNCT
cana-787	196	23	,	,	PUNCT
cana-787	196	24	𝜇(𝑥3	𝜇(𝑥3	NOUN
cana-787	196	25	)	)	PUNCT
cana-787	196	26	≤	≤	NUM
cana-787	196	27	𝜎2(𝑑(𝜇(𝑥0	𝜎2(𝑑(𝜇(𝑥0	NUM
cana-787	196	28	)	)	PUNCT
cana-787	196	29	,	,	PUNCT
cana-787	196	30	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	196	31	)	)	PUNCT
cana-787	196	32	)	)	PUNCT
cana-787	197	1	continuing	continue	VERB
cana-787	197	2	this	this	DET
cana-787	197	3	process	process	NOUN
cana-787	197	4	we	we	PRON
cana-787	197	5	have	have	VERB
cana-787	197	6	,	,	PUNCT
cana-787	197	7	𝑑(𝜇(𝑥𝑛	𝑑(𝜇(𝑥𝑛	NOUN
cana-787	197	8	)	)	PUNCT
cana-787	197	9	,	,	PUNCT
cana-787	197	10	𝜇(𝑥𝑛+1	𝜇(𝑥𝑛+1	PROPN
cana-787	197	11	)	)	PUNCT
cana-787	197	12	≤	≤	NOUN
cana-787	197	13	𝜎𝑛(𝑑(𝜇(𝑥0	𝜎𝑛(𝑑(𝜇(𝑥0	ADJ
cana-787	197	14	)	)	PUNCT
cana-787	197	15	,	,	PUNCT
cana-787	197	16	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	197	17	)	)	PUNCT
cana-787	197	18	)	)	PUNCT
cana-787	198	1	…	…	PUNCT
cana-787	198	2	……	……	NOUN
cana-787	198	3	(	(	PUNCT
cana-787	198	4	3.9	3.9	NUM
cana-787	198	5	)	)	PUNCT
cana-787	198	6	now	now	ADV
cana-787	198	7	for	for	ADP
cana-787	198	8	any	any	DET
cana-787	198	9	𝑚	𝑚	NOUN
cana-787	198	10	,	,	PUNCT
cana-787	198	11	𝑛	𝑛	DET
cana-787	198	12	∈	∈	NOUN
cana-787	198	13	𝑁	𝑁	NOUN
cana-787	198	14	with	with	ADP
cana-787	198	15	𝑚	𝑚	NOUN
cana-787	198	16	<	<	X
cana-787	198	17	𝑛	𝑛	PROPN
cana-787	198	18	,	,	PUNCT
cana-787	198	19	𝑑(𝜇(𝑥𝑚	𝑑(𝜇(𝑥𝑚	NOUN
cana-787	198	20	)	)	PUNCT
cana-787	198	21	,	,	PUNCT
cana-787	198	22	𝜇(𝑥𝑛	𝜇(𝑥𝑛	PROPN
cana-787	198	23	)	)	PUNCT
cana-787	198	24	≤	≤	NUM
cana-787	198	25	𝑑(𝜇(𝑥𝑚	𝑑(𝜇(𝑥𝑚	NOUN
cana-787	198	26	)	)	PUNCT
cana-787	198	27	,	,	PUNCT
cana-787	198	28	𝜇(𝑥𝑚+1	𝜇(𝑥𝑚+1	PUNCT
cana-787	198	29	)	)	PUNCT
cana-787	198	30	+	+	CCONJ
cana-787	198	31	𝑑(𝜇(𝑥𝑚+1	𝑑(𝜇(𝑥𝑚+1	ADJ
cana-787	198	32	)	)	PUNCT
cana-787	198	33	,	,	PUNCT
cana-787	198	34	𝜇(𝑥𝑛+2	𝜇(𝑥𝑛+2	PROPN
cana-787	198	35	)	)	PUNCT
cana-787	198	36	+	+	CCONJ
cana-787	198	37	⋯	⋯	VERB
cana-787	198	38	…	…	PUNCT
cana-787	198	39	+	+	NUM
cana-787	198	40	𝑑(𝜇(𝑥𝑛−1	𝑑(𝜇(𝑥𝑛−1	NOUN
cana-787	198	41	)	)	PUNCT
cana-787	198	42	,	,	PUNCT
cana-787	198	43	𝜇(𝑥𝑛	𝜇(𝑥𝑛	PROPN
cana-787	198	44	)	)	PUNCT
cana-787	198	45	from	from	ADP
cana-787	198	46	(	(	PUNCT
cana-787	198	47	3.9	3.9	NUM
cana-787	198	48	)	)	PUNCT
cana-787	198	49	𝑑(𝜇(𝑥𝑚	𝑑(𝜇(𝑥𝑚	NOUN
cana-787	198	50	)	)	PUNCT
cana-787	198	51	,	,	PUNCT
cana-787	198	52	𝜇(𝑥𝑛	𝜇(𝑥𝑛	PROPN
cana-787	198	53	)	)	PUNCT
cana-787	198	54	≤	≤	NOUN
cana-787	198	55	𝜎𝑚(𝑑(𝜇(𝑥0	𝜎𝑚(𝑑(𝜇(𝑥0	NOUN
cana-787	198	56	)	)	PUNCT
cana-787	198	57	,	,	PUNCT
cana-787	198	58	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	198	59	)	)	PUNCT
cana-787	198	60	)	)	PUNCT
cana-787	199	1	+	+	CCONJ
cana-787	199	2	𝜎𝑚+1(𝑑(𝜇(𝑥0	𝜎𝑚+1(𝑑(𝜇(𝑥0	ADJ
cana-787	199	3	)	)	PUNCT
cana-787	199	4	,	,	PUNCT
cana-787	199	5	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	199	6	)	)	PUNCT
cana-787	199	7	)	)	PUNCT
cana-787	200	1	+	+	CCONJ
cana-787	200	2	⋯	⋯	NOUN
cana-787	200	3	…	…	PUNCT
cana-787	200	4	.	.	PUNCT
cana-787	200	5	.	.	PUNCT
cana-787	201	1	+	+	NOUN
cana-787	201	2	𝜎𝑛−1(𝑑(𝜇(𝑥0	𝜎𝑛−1(𝑑(𝜇(𝑥0	ADJ
cana-787	201	3	)	)	PUNCT
cana-787	201	4	,	,	PUNCT
cana-787	201	5	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-787	201	6	)	)	PUNCT
cana-787	201	7	)	)	PUNCT
cana-787	201	8	using	use	VERB
cana-787	201	9	the	the	DET
cana-787	201	10	property	property	NOUN
cana-787	201	11	𝜎𝑚(𝑡	𝜎𝑚(𝑡	PUNCT
cana-787	201	12	)	)	PUNCT
cana-787	202	1	=	=	SYM
cana-787	202	2	0	0	NUM
cana-787	202	3	∀	∀	X
cana-787	202	4	𝑡	𝑡	X
cana-787	202	5	>	>	X
cana-787	202	6	0	0	PUNCT
cana-787	202	7	for	for	ADP
cana-787	202	8	𝑚	𝑚	X
cana-787	202	9	→	→	SYM
cana-787	202	10	∞	∞	NUM
cana-787	202	11	𝑑(𝜇(𝑥𝑚	𝑑(𝜇(𝑥𝑚	NOUN
cana-787	202	12	)	)	PUNCT
cana-787	202	13	,	,	PUNCT
cana-787	202	14	𝜇(𝑥𝑛	𝜇(𝑥𝑛	PROPN
cana-787	202	15	)	)	PUNCT
cana-787	202	16	=	=	SYM
cana-787	202	17	0	0	NUM
cana-787	202	18	…	…	PUNCT
cana-787	202	19	…	…	PUNCT
cana-787	202	20	…	…	SYM
cana-787	202	21	……	……	NOUN
cana-787	202	22	(	(	PUNCT
cana-787	202	23	3.10	3.10	NUM
cana-787	202	24	)	)	PUNCT
cana-787	202	25	this	this	DET
cana-787	202	26	yield	yield	NOUN
cana-787	202	27	{	{	PUNCT
cana-787	202	28	𝜇(𝑥𝑛	𝜇(𝑥𝑛	PROPN
cana-787	202	29	)	)	PUNCT
cana-787	202	30	}	}	PUNCT
cana-787	202	31	is	be	AUX
cana-787	202	32	chauchy	chauchy	ADJ
cana-787	202	33	sequence	sequence	NOUN
cana-787	202	34	in	in	ADP
cana-787	202	35	𝑋.	𝑋.	PROPN
cana-787	202	36	being	be	AUX
cana-787	202	37	(	(	PUNCT
cana-787	202	38	𝑋	𝑋	PROPN
cana-787	202	39	,	,	PUNCT
cana-787	202	40	𝑑	𝑑	NOUN
cana-787	202	41	)	)	PUNCT
cana-787	202	42	is	be	AUX
cana-787	202	43	complete	complete	ADJ
cana-787	202	44	metric	metric	ADJ
cana-787	202	45	space	space	NOUN
cana-787	202	46	,	,	PUNCT
cana-787	202	47	there	there	PRON
cana-787	202	48	exist	exist	VERB
cana-787	202	49	𝑢	𝑢	DET
cana-787	202	50	∈	∈	NOUN
cana-787	202	51	𝑋	𝑋	NOUN
cana-787	202	52	such	such	ADJ
cana-787	202	53	that	that	SCONJ
cana-787	202	54	lim	lim	PROPN
cana-787	202	55	𝑛→∞	𝑛→∞	NUM
cana-787	202	56	𝜇(𝑥𝑛	𝜇(𝑥𝑛	PROPN
cana-787	202	57	)	)	PUNCT
cana-787	202	58	=	=	SYM
cana-787	202	59	𝑢.	𝑢.	NOUN
cana-787	202	60	here	here	ADV
cana-787	202	61	𝜌(𝑥𝑛	𝜌(𝑥𝑛	NUM
cana-787	202	62	)	)	PUNCT
cana-787	202	63	=	=	SYM
cana-787	202	64	𝜇(𝑥𝑛+1	𝜇(𝑥𝑛+1	PROPN
cana-787	202	65	)	)	PUNCT
cana-787	202	66	,	,	PUNCT
cana-787	202	67	implies	imply	VERB
cana-787	202	68	lim	lim	PROPN
cana-787	202	69	𝑛→∞	𝑛→∞	NUM
cana-787	202	70	𝜌(𝑥𝑛	𝜌(𝑥𝑛	PROPN
cana-787	202	71	)	)	PUNCT
cana-787	202	72	=	=	SYM
cana-787	203	1	𝑢.	𝑢.	NOUN
cana-787	203	2	as	as	ADP
cana-787	203	3	our	our	PRON
cana-787	203	4	hypothesis	hypothesis	NOUN
cana-787	203	5	𝜇(𝑋	𝜇(𝑋	NOUN
cana-787	203	6	)	)	PUNCT
cana-787	203	7	is	be	AUX
cana-787	203	8	closed	close	VERB
cana-787	203	9	sub	sub	ADJ
cana-787	203	10	-	-	ADJ
cana-787	203	11	set	set	ADJ
cana-787	203	12	of	of	ADP
cana-787	203	13	𝑋	𝑋	PROPN
cana-787	203	14	,	,	PUNCT
cana-787	203	15	so	so	SCONJ
cana-787	203	16	𝑢	𝑢	X
cana-787	203	17	∈	∈	NOUN
cana-787	203	18	𝜇(𝑋	𝜇(𝑋	NUM
cana-787	203	19	)	)	PUNCT
cana-787	203	20	and	and	CCONJ
cana-787	203	21	there	there	PRON
cana-787	203	22	exist	exist	VERB
cana-787	203	23	𝑧	𝑧	DET
cana-787	203	24	∈	∈	NOUN
cana-787	203	25	𝑋	𝑋	NOUN
cana-787	203	26	such	such	ADJ
cana-787	203	27	that	that	SCONJ
cana-787	203	28	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	203	29	)	)	PUNCT
cana-787	203	30	=	=	PUNCT
cana-787	204	1	𝑢	𝑢	NOUN
cana-787	204	2	thus	thus	ADV
cana-787	204	3	lim	lim	PROPN
cana-787	204	4	𝑛→∞	𝑛→∞	NUM
cana-787	204	5	𝑑	𝑑	PROPN
cana-787	204	6	(	(	PUNCT
cana-787	204	7	𝜇(𝑥𝑛	𝜇(𝑥𝑛	ADJ
cana-787	204	8	)	)	PUNCT
cana-787	204	9	−	−	ADP
cana-787	204	10	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	204	11	)	)	PUNCT
cana-787	204	12	)	)	PUNCT
cana-787	205	1	=	=	PUNCT
cana-787	205	2	0	0	NUM
cana-787	205	3	…	…	SYM
cana-787	205	4	……	……	NOUN
cana-787	205	5	.	.	PUNCT
cana-787	206	1	(	(	PUNCT
cana-787	206	2	3.11	3.11	NUM
cana-787	206	3	)	)	PUNCT
cana-787	206	4	now	now	ADV
cana-787	206	5	by	by	ADP
cana-787	206	6	hypothesis	hypothesis	NOUN
cana-787	206	7	(	(	PUNCT
cana-787	206	8	d	d	NOUN
cana-787	206	9	)	)	PUNCT
cana-787	206	10	,	,	PUNCT
cana-787	206	11	from	from	ADP
cana-787	206	12	(	(	PUNCT
cana-787	206	13	3.6	3.6	NUM
cana-787	206	14	)	)	PUNCT
cana-787	206	15	and	and	CCONJ
cana-787	206	16	(	(	PUNCT
cana-787	206	17	3.11	3.11	NUM
cana-787	206	18	)	)	PUNCT
cana-787	206	19	;	;	PUNCT
cana-787	206	20	there	there	PRON
cana-787	206	21	exist	exist	VERB
cana-787	206	22	sub	sub	ADJ
cana-787	206	23	-	-	NOUN
cana-787	206	24	sequence	sequence	ADJ
cana-787	206	25	𝜇(𝑥𝑛𝑘	𝜇(𝑥𝑛𝑘	ADJ
cana-787	206	26	)	)	PUNCT
cana-787	206	27	of	of	ADP
cana-787	206	28	𝜇(𝑥𝑛	𝜇(𝑥𝑛	PROPN
cana-787	206	29	)	)	PUNCT
cana-787	206	30	such	such	ADJ
cana-787	206	31	that	that	SCONJ
cana-787	206	32	𝛼	𝛼	PROPN
cana-787	206	33	(	(	PUNCT
cana-787	206	34	𝜇(𝑥𝑛𝑘	𝜇(𝑥𝑛𝑘	ADJ
cana-787	206	35	)	)	PUNCT
cana-787	206	36	,	,	PUNCT
cana-787	206	37	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	206	38	)	)	PUNCT
cana-787	206	39	)	)	PUNCT
cana-787	206	40	>	>	PUNCT
cana-787	207	1	𝛽−1	𝛽−1	PROPN
cana-787	207	2	∀	∀	X
cana-787	207	3	𝑘	𝑘	DET
cana-787	207	4	∈	∈	NOUN
cana-787	207	5	𝑁	𝑁	NOUN
cana-787	207	6	now	now	ADV
cana-787	207	7	by	by	ADP
cana-787	207	8	(	(	PUNCT
cana-787	207	9	3.7	3.7	NUM
cana-787	207	10	)	)	PUNCT
cana-787	207	11	we	we	PRON
cana-787	207	12	have	have	VERB
cana-787	207	13	,	,	PUNCT
cana-787	207	14	𝛽−1𝑑(𝜌(𝑥𝑛𝑘	𝛽−1𝑑(𝜌(𝑥𝑛𝑘	PROPN
cana-787	207	15	)	)	PUNCT
cana-787	207	16	,	,	PUNCT
cana-787	207	17	𝜌(𝑧	𝜌(𝑧	PROPN
cana-787	207	18	)	)	PUNCT
cana-787	207	19	)	)	PUNCT
cana-787	207	20	≤	≤	NUM
cana-787	208	1	𝛼	𝛼	X
cana-787	208	2	(	(	PUNCT
cana-787	208	3	𝜇(𝑥𝑛𝑘	𝜇(𝑥𝑛𝑘	ADJ
cana-787	208	4	)	)	PUNCT
cana-787	208	5	,	,	PUNCT
cana-787	208	6	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	208	7	)	)	PUNCT
cana-787	208	8	)	)	PUNCT
cana-787	209	1	𝑑(𝜌(𝑥𝑛𝑘	𝑑(𝜌(𝑥𝑛𝑘	NUM
cana-787	209	2	)	)	PUNCT
cana-787	209	3	,	,	PUNCT
cana-787	209	4	𝜌(𝑧	𝜌(𝑧	PROPN
cana-787	209	5	)	)	PUNCT
cana-787	209	6	≤	≤	NUM
cana-787	209	7	𝜑(𝑑(𝜇(𝑥𝑛𝑘	𝜑(𝑑(𝜇(𝑥𝑛𝑘	NOUN
cana-787	209	8	)	)	PUNCT
cana-787	209	9	,	,	PUNCT
cana-787	209	10	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	209	11	)	)	PUNCT
cana-787	209	12	)	)	PUNCT
cana-787	209	13	)	)	PUNCT
cana-787	210	1	𝑑(𝜌(𝑥𝑛𝑘	𝑑(𝜌(𝑥𝑛𝑘	NUM
cana-787	210	2	)	)	PUNCT
cana-787	210	3	,	,	PUNCT
cana-787	210	4	𝜌(𝑧	𝜌(𝑧	PROPN
cana-787	210	5	)	)	PUNCT
cana-787	210	6	)	)	PUNCT
cana-787	211	1	≤	≤	NUM
cana-787	211	2	𝜎(𝑑(𝜇(𝑥𝑛𝑘	𝜎(𝑑(𝜇(𝑥𝑛𝑘	NOUN
cana-787	211	3	)	)	PUNCT
cana-787	211	4	,	,	PUNCT
cana-787	211	5	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	211	6	)	)	PUNCT
cana-787	211	7	)	)	PUNCT
cana-787	211	8	)	)	PUNCT
cana-787	211	9	taking	take	VERB
cana-787	211	10	𝑘	𝑘	PRON
cana-787	211	11	→	→	SYM
cana-787	211	12	∞	∞	PROPN
cana-787	211	13	𝑑(𝑢	𝑑(𝑢	NOUN
cana-787	211	14	,	,	PUNCT
cana-787	211	15	𝜌(𝑧	𝜌(𝑧	PROPN
cana-787	211	16	)	)	PUNCT
cana-787	211	17	)	)	PUNCT
cana-787	211	18	≤	≤	NUM
cana-787	211	19	𝜎(𝑑(𝜇(𝑧	𝜎(𝑑(𝜇(𝑧	NUM
cana-787	211	20	)	)	PUNCT
cana-787	211	21	,	,	PUNCT
cana-787	211	22	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	211	23	)	)	PUNCT
cana-787	211	24	)	)	PUNCT
cana-787	211	25	⇒	⇒	NOUN
cana-787	211	26	𝑑(𝑢	𝑑(𝑢	PROPN
cana-787	211	27	,	,	PUNCT
cana-787	211	28	𝜌(𝑧	𝜌(𝑧	PROPN
cana-787	211	29	)	)	PUNCT
cana-787	211	30	)	)	PUNCT
cana-787	212	1	=	=	SYM
cana-787	212	2	0	0	NUM
cana-787	212	3	∴	∴	PROPN
cana-787	212	4	𝜌(𝑧	𝜌(𝑧	PROPN
cana-787	212	5	)	)	PUNCT
cana-787	212	6	=	=	SYM
cana-787	212	7	𝑢	𝑢	NOUN
cana-787	212	8	=	=	SYM
cana-787	212	9	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	212	10	)	)	PUNCT
cana-787	212	11	therefore	therefore	ADV
cana-787	212	12	𝑧	𝑧	PROPN
cana-787	212	13	is	be	AUX
cana-787	212	14	a	a	DET
cana-787	212	15	coincident	coincident	NOUN
cana-787	212	16	of	of	ADP
cana-787	212	17	𝜇	𝜇	ADP
cana-787	212	18	and	and	CCONJ
cana-787	212	19	𝜌.	𝜌.	NOUN
cana-787	212	20	we	we	PRON
cana-787	212	21	show	show	VERB
cana-787	212	22	𝑧	𝑧	ADP
cana-787	212	23	is	be	AUX
cana-787	212	24	unique	unique	ADJ
cana-787	212	25	coincident	coincident	ADJ
cana-787	212	26	point	point	NOUN
cana-787	212	27	.	.	PUNCT
cana-787	213	1	if	if	SCONJ
cana-787	213	2	possible	possible	ADJ
cana-787	213	3	suppose	suppose	VERB
cana-787	213	4	𝑤	𝑤	ADP
cana-787	213	5	is	be	AUX
cana-787	213	6	another	another	DET
cana-787	213	7	coincident	coincident	ADJ
cana-787	213	8	point	point	NOUN
cana-787	213	9	then	then	ADV
cana-787	213	10	by	by	ADP
cana-787	213	11	hypothesis	hypothesis	NOUN
cana-787	213	12	(	(	PUNCT
cana-787	213	13	e	e	NOUN
cana-787	213	14	)	)	PUNCT
cana-787	213	15	𝛼(𝜇(𝑧	𝛼(𝜇(𝑧	NOUN
cana-787	213	16	)	)	PUNCT
cana-787	213	17	,	,	PUNCT
cana-787	213	18	𝜇(𝑤	𝜇(𝑤	NUM
cana-787	213	19	)	)	PUNCT
cana-787	213	20	)	)	PUNCT
cana-787	214	1	>	>	PUNCT
cana-787	214	2	𝛽−1	𝛽−1	PROPN
cana-787	214	3	…	…	PUNCT
cana-787	214	4	…	…	PUNCT
cana-787	214	5	..	..	PUNCT
cana-787	214	6	(	(	PUNCT
cana-787	214	7	3.12	3.12	NUM
cana-787	214	8	)	)	PUNCT
cana-787	214	9	applying	apply	VERB
cana-787	214	10	condition	condition	NOUN
cana-787	214	11	(	(	PUNCT
cana-787	214	12	a	a	NOUN
cana-787	214	13	)	)	PUNCT
cana-787	214	14	and	and	CCONJ
cana-787	214	15	from	from	ADP
cana-787	214	16	(	(	PUNCT
cana-787	214	17	3.7	3.7	NUM
cana-787	214	18	)	)	PUNCT
cana-787	214	19	and	and	CCONJ
cana-787	214	20	(	(	PUNCT
cana-787	214	21	3.12	3.12	NUM
cana-787	214	22	)	)	PUNCT
cana-787	214	23	we	we	PRON
cana-787	214	24	have	have	VERB
cana-787	214	25	,	,	PUNCT
cana-787	214	26	𝛽−1𝑑(𝜌(𝑧	𝛽−1𝑑(𝜌(𝑧	PROPN
cana-787	214	27	)	)	PUNCT
cana-787	214	28	,	,	PUNCT
cana-787	214	29	𝜌(𝑤	𝜌(𝑤	PROPN
cana-787	214	30	)	)	PUNCT
cana-787	214	31	)	)	PUNCT
cana-787	214	32	≤	≤	NUM
cana-787	214	33	𝛼(𝜇(𝑧	𝛼(𝜇(𝑧	NOUN
cana-787	214	34	)	)	PUNCT
cana-787	214	35	,	,	PUNCT
cana-787	214	36	𝜇(𝑤))𝑑(𝜌(𝑧	𝜇(𝑤))𝑑(𝜌(𝑧	PROPN
cana-787	214	37	)	)	PUNCT
cana-787	214	38	,	,	PUNCT
cana-787	214	39	𝜌(𝑤	𝜌(𝑤	PROPN
cana-787	214	40	)	)	PUNCT
cana-787	214	41	≤	≤	NUM
cana-787	214	42	𝜑(𝑑(𝜇(𝑧	𝜑(𝑑(𝜇(𝑧	PROPN
cana-787	214	43	)	)	PUNCT
cana-787	214	44	,	,	PUNCT
cana-787	214	45	𝜇(𝑤	𝜇(𝑤	NUM
cana-787	214	46	)	)	PUNCT
cana-787	214	47	)	)	PUNCT
cana-787	214	48	)	)	PUNCT
cana-787	214	49	⇒	⇒	PROPN
cana-787	214	50	𝑑(𝜌(𝑧	𝑑(𝜌(𝑧	PROPN
cana-787	214	51	)	)	PUNCT
cana-787	214	52	,	,	PUNCT
cana-787	214	53	𝜌(𝑤	𝜌(𝑤	PROPN
cana-787	214	54	)	)	PUNCT
cana-787	214	55	)	)	PUNCT
cana-787	214	56	≤	≤	NUM
cana-787	214	57	𝜎(𝑑(𝜇(𝑧	𝜎(𝑑(𝜇(𝑧	NUM
cana-787	214	58	)	)	PUNCT
cana-787	214	59	,	,	PUNCT
cana-787	214	60	𝜇(𝑤	𝜇(𝑤	NUM
cana-787	214	61	)	)	PUNCT
cana-787	214	62	)	)	PUNCT
cana-787	214	63	≤	≤	NUM
cana-787	214	64	𝑑(𝜇(𝑧	𝑑(𝜇(𝑧	PROPN
cana-787	214	65	)	)	PUNCT
cana-787	214	66	,	,	PUNCT
cana-787	214	67	𝜇(𝑤	𝜇(𝑤	NUM
cana-787	214	68	)	)	PUNCT
cana-787	214	69	)	)	PUNCT
cana-787	215	1	communications	communication	NOUN
cana-787	215	2	on	on	ADP
cana-787	215	3	applied	apply	VERB
cana-787	215	4	nonlinear	nonlinear	ADJ
cana-787	215	5	analysis	analysis	NOUN
cana-787	215	6	issn	issn	NOUN
cana-787	215	7	:	:	PUNCT
cana-787	215	8	1074	1074	NUM
cana-787	215	9	-	-	PUNCT
cana-787	215	10	133x	133x	NUM
cana-787	215	11	vol	vol	NOUN
cana-787	215	12	31	31	NUM
cana-787	215	13	no	no	NOUN
cana-787	215	14	.	.	PUNCT
cana-787	216	1	3s	3s	NUM
cana-787	216	2	(	(	PUNCT
cana-787	216	3	2024	2024	NUM
cana-787	216	4	)	)	PUNCT
cana-787	216	5	375	375	NUM
cana-787	216	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	216	7	⇒	⇒	PROPN
cana-787	216	8	𝑑(𝜌(𝑧	𝑑(𝜌(𝑧	PROPN
cana-787	216	9	)	)	PUNCT
cana-787	216	10	,	,	PUNCT
cana-787	216	11	𝜌(𝑤	𝜌(𝑤	PROPN
cana-787	216	12	)	)	PUNCT
cana-787	216	13	)	)	PUNCT
cana-787	217	1	≤	≤	NUM
cana-787	217	2	𝑑(𝜌(𝑧	𝑑(𝜌(𝑧	PROPN
cana-787	217	3	)	)	PUNCT
cana-787	217	4	,	,	PUNCT
cana-787	217	5	𝜌(𝑤	𝜌(𝑤	PROPN
cana-787	217	6	)	)	PUNCT
cana-787	217	7	)	)	PUNCT
cana-787	217	8	⇒	⇒	PROPN
cana-787	217	9	𝑑(𝜌(𝑧	𝑑(𝜌(𝑧	PROPN
cana-787	217	10	)	)	PUNCT
cana-787	217	11	,	,	PUNCT
cana-787	217	12	𝜌(𝑤	𝜌(𝑤	PROPN
cana-787	217	13	)	)	PUNCT
cana-787	217	14	)	)	PUNCT
cana-787	218	1	=	=	SYM
cana-787	218	2	0	0	NUM
cana-787	218	3	⇒	⇒	PROPN
cana-787	218	4	𝜌(𝑧	𝜌(𝑧	PROPN
cana-787	218	5	)	)	PUNCT
cana-787	218	6	=	=	SYM
cana-787	218	7	𝜌(𝑤	𝜌(𝑤	NOUN
cana-787	218	8	)	)	PUNCT
cana-787	218	9	∴	∴	PROPN
cana-787	218	10	𝜌(𝑧	𝜌(𝑧	PROPN
cana-787	218	11	)	)	PUNCT
cana-787	218	12	=	=	SYM
cana-787	218	13	𝜇(𝑧	𝜇(𝑧	PROPN
cana-787	218	14	)	)	PUNCT
cana-787	218	15	=	=	SYM
cana-787	218	16	𝜌(𝑤	𝜌(𝑤	NOUN
cana-787	218	17	)	)	PUNCT
cana-787	218	18	=	=	SYM
cana-787	218	19	𝜇(𝑤	𝜇(𝑤	NOUN
cana-787	218	20	)	)	PUNCT
cana-787	218	21	hence	hence	ADV
cana-787	218	22	𝑧	𝑧	NOUN
cana-787	218	23	is	be	AUX
cana-787	218	24	unique	unique	ADJ
cana-787	218	25	coincident	coincident	ADJ
cana-787	218	26	point	point	NOUN
cana-787	218	27	and	and	CCONJ
cana-787	218	28	𝑢	𝑢	PRON
cana-787	218	29	is	be	AUX
cana-787	218	30	unique	unique	ADJ
cana-787	218	31	point	point	NOUN
cana-787	218	32	of	of	ADP
cana-787	218	33	coincident	coincident	NOUN
cana-787	218	34	.	.	PUNCT
cana-787	219	1	from	from	ADP
cana-787	219	2	proposition	proposition	NOUN
cana-787	219	3	2.1	2.1	NUM
cana-787	219	4	,	,	PUNCT
cana-787	219	5	it	it	PRON
cana-787	219	6	is	be	AUX
cana-787	219	7	clear	clear	ADJ
cana-787	219	8	that	that	SCONJ
cana-787	219	9	𝑢	𝑢	NOUN
cana-787	219	10	is	be	AUX
cana-787	219	11	unique	unique	ADJ
cana-787	219	12	common	common	ADJ
cana-787	219	13	fixed	fix	VERB
cana-787	219	14	point	point	NOUN
cana-787	219	15	of	of	ADP
cana-787	219	16	𝜇	𝜇	ADP
cana-787	219	17	and	and	CCONJ
cana-787	219	18	𝜌	𝜌	X
cana-787	219	19	□	□	PUNCT
cana-787	219	20	example	example	NOUN
cana-787	219	21	1	1	NUM
cana-787	219	22	:	:	PUNCT
cana-787	219	23	let	let	VERB
cana-787	219	24	𝑋	𝑋	PROPN
cana-787	219	25	=	=	PUNCT
cana-787	220	1	[	[	X
cana-787	220	2	0	0	NUM
cana-787	220	3	,	,	PUNCT
cana-787	220	4	1	1	NUM
cana-787	220	5	]	]	PUNCT
cana-787	220	6	and	and	CCONJ
cana-787	220	7	metric	metric	ADJ
cana-787	220	8	on	on	ADP
cana-787	220	9	it	it	PRON
cana-787	220	10	is	be	AUX
cana-787	220	11	𝑑(𝑥	𝑑(𝑥	PROPN
cana-787	220	12	,	,	PUNCT
cana-787	220	13	𝑦	𝑦	X
cana-787	220	14	)	)	PUNCT
cana-787	220	15	=	=	PUNCT
cana-787	220	16	⌊𝑥	⌊𝑥	PROPN
cana-787	221	1	−	−	PROPN
cana-787	222	1	𝑦⌋.take	𝑦⌋.take	ADP
cana-787	222	2	the	the	DET
cana-787	222	3	mapping	mapping	NOUN
cana-787	222	4	𝜇	𝜇	ADP
cana-787	222	5	:	:	PUNCT
cana-787	222	6	𝑋	𝑋	PROPN
cana-787	222	7	→	→	SYM
cana-787	222	8	𝑋	𝑋	PROPN
cana-787	222	9	defined	define	VERB
cana-787	222	10	by	by	ADP
cana-787	222	11	𝜇(𝑥	𝜇(𝑥	PROPN
cana-787	222	12	)	)	PUNCT
cana-787	223	1	=	=	SYM
cana-787	223	2	𝑥3	𝑥3	NOUN
cana-787	223	3	2	2	NUM
cana-787	224	1	+	+	CCONJ
cana-787	224	2	7	7	NUM
cana-787	224	3	16	16	NUM
cana-787	224	4	.it	.it	PUNCT
cana-787	224	5	is	be	AUX
cana-787	224	6	clear	clear	ADJ
cana-787	224	7	that	that	SCONJ
cana-787	224	8	𝜇	𝜇	ADP
cana-787	224	9	is	be	AUX
cana-787	224	10	continuous	continuous	ADJ
cana-787	224	11	but	but	CCONJ
cana-787	224	12	not	not	PART
cana-787	224	13	banach	banach	NOUN
cana-787	224	14	contraction	contraction	NOUN
cana-787	224	15	at	at	ADP
cana-787	224	16	𝑥	𝑥	NOUN
cana-787	224	17	=	=	SYM
cana-787	224	18	1	1	NUM
cana-787	224	19	and	and	CCONJ
cana-787	224	20	𝑦	𝑦	NOUN
cana-787	224	21	=	=	SYM
cana-787	224	22	0.9.actualy	0.9.actualy	X
cana-787	225	1	it	it	PRON
cana-787	225	2	is	be	AUX
cana-787	225	3	𝛼	𝛼	PRON
cana-787	225	4	−	−	NOUN
cana-787	225	5	𝜑	𝜑	PRON
cana-787	225	6	contraction	contraction	NOUN
cana-787	225	7	.	.	PUNCT
cana-787	226	1	where	where	SCONJ
cana-787	226	2	,	,	PUNCT
cana-787	226	3	𝛼	𝛼	X
cana-787	226	4	:	:	PUNCT
cana-787	226	5	𝑋	𝑋	NOUN
cana-787	226	6	×	×	NOUN
cana-787	226	7	𝑋	𝑋	PROPN
cana-787	226	8	→	→	SYM
cana-787	226	9	[	[	X
cana-787	226	10	0	0	NUM
cana-787	226	11	,	,	PUNCT
cana-787	226	12	∞	∞	PROPN
cana-787	226	13	)	)	PUNCT
cana-787	226	14	defined	define	VERB
cana-787	226	15	by	by	ADP
cana-787	226	16	𝛼(𝑥	𝛼(𝑥	PROPN
cana-787	226	17	,	,	PUNCT
cana-787	226	18	𝑦	𝑦	NOUN
cana-787	226	19	)	)	PUNCT
cana-787	226	20	=	=	SYM
cana-787	226	21	1	1	NUM
cana-787	226	22	𝑥2+𝑥𝑦+𝑦2	𝑥2+𝑥𝑦+𝑦2	NOUN
cana-787	226	23	and	and	CCONJ
cana-787	226	24	𝜑	𝜑	NOUN
cana-787	226	25	:	:	PUNCT
cana-787	226	26	𝑋	𝑋	NOUN
cana-787	226	27	×	×	NOUN
cana-787	226	28	𝑋	𝑋	PROPN
cana-787	226	29	→	→	SYM
cana-787	226	30	[	[	X
cana-787	226	31	0	0	NUM
cana-787	226	32	,	,	PUNCT
cana-787	226	33	∞	∞	PROPN
cana-787	226	34	)	)	PUNCT
cana-787	226	35	defined	define	VERB
cana-787	226	36	by	by	ADP
cana-787	226	37	𝜑(𝑡	𝜑(𝑡	PROPN
cana-787	226	38	)	)	PUNCT
cana-787	226	39	=	=	PUNCT
cana-787	226	40	𝑡	𝑡	NOUN
cana-787	226	41	2	2	NUM
cana-787	226	42	.	.	PUNCT
cana-787	227	1	take	take	VERB
cana-787	227	2	𝛽	𝛽	NOUN
cana-787	227	3	=	=	NOUN
cana-787	227	4	1.999	1.999	NUM
cana-787	227	5	…	…	PUNCT
cana-787	227	6	.	.	PUNCT
cana-787	228	1	so	so	ADV
cana-787	228	2	that	that	SCONJ
cana-787	228	3	𝛽𝜑	𝛽𝜑	PROPN
cana-787	228	4	is	be	AUX
cana-787	228	5	again	again	ADV
cana-787	228	6	a	a	DET
cana-787	228	7	comparison	comparison	NOUN
cana-787	228	8	function	function	NOUN
cana-787	228	9	.	.	PUNCT
cana-787	229	1	there	there	PRON
cana-787	229	2	exist	exist	VERB
cana-787	229	3	𝑥0	𝑥0	NOUN
cana-787	229	4	=	=	NOUN
cana-787	229	5	0.1	0.1	NUM
cana-787	229	6	such	such	ADJ
cana-787	229	7	that	that	DET
cana-787	229	8	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-787	229	9	,	,	PUNCT
cana-787	229	10	𝜇(𝑥0	𝜇(𝑥0	ADJ
cana-787	229	11	)	)	PUNCT
cana-787	229	12	)	)	PUNCT
cana-787	230	1	>	>	PUNCT
cana-787	230	2	𝛽−1.by	𝛽−1.by	PROPN
cana-787	230	3	calculation	calculation	NOUN
cana-787	230	4	we	we	PRON
cana-787	230	5	can	can	AUX
cana-787	230	6	easily	easily	ADV
cana-787	230	7	find	find	VERB
cana-787	230	8	that	that	SCONJ
cana-787	230	9	𝜇	𝜇	ADP
cana-787	230	10	is	be	AUX
cana-787	230	11	𝛼	𝛼	PRON
cana-787	230	12	𝛽−1	𝛽−1	PROPN
cana-787	230	13	-	-	PUNCT
cana-787	230	14	admissible	admissible	ADJ
cana-787	230	15	.	.	PUNCT
cana-787	231	1	more	more	ADV
cana-787	231	2	over	over	ADV
cana-787	231	3	for	for	ADP
cana-787	231	4	every	every	DET
cana-787	231	5	pair	pair	NOUN
cana-787	231	6	𝑢	𝑢	NOUN
cana-787	231	7	,	,	PUNCT
cana-787	231	8	𝑣	𝑣	DET
cana-787	231	9	∈	∈	PROPN
cana-787	231	10	𝑋	𝑋	NOUN
cana-787	231	11	there	there	ADV
cana-787	231	12	exist	exist	VERB
cana-787	231	13	𝑤	𝑤	ADP
cana-787	231	14	∈	∈	NOUN
cana-787	231	15	𝑋	𝑋	NOUN
cana-787	231	16	such	such	ADJ
cana-787	231	17	that	that	SCONJ
cana-787	231	18	𝛼(𝑢	𝛼(𝑢	NOUN
cana-787	231	19	,	,	PUNCT
cana-787	231	20	𝑤	𝑤	ADP
cana-787	231	21	)	)	PUNCT
cana-787	231	22	>	>	PUNCT
cana-787	232	1	𝛽−1	𝛽−1	PROPN
cana-787	232	2	and	and	CCONJ
cana-787	232	3	𝛼(𝑣	𝛼(𝑣	PROPN
cana-787	232	4	,	,	PUNCT
cana-787	232	5	𝑤	𝑤	ADP
cana-787	232	6	)	)	PUNCT
cana-787	232	7	>	>	X
cana-787	232	8	𝛽−1.here	𝛽−1.here	PROPN
cana-787	232	9	𝜇	𝜇	ADP
cana-787	232	10	satisfy	satisfy	NOUN
cana-787	232	11	all	all	DET
cana-787	232	12	condition	condition	NOUN
cana-787	232	13	of	of	ADP
cana-787	232	14	theorem	theorem	ADJ
cana-787	232	15	3.1	3.1	NUM
cana-787	232	16	and	and	CCONJ
cana-787	232	17	3.3.hence	3.3.hence	NUM
cana-787	232	18	there	there	PRON
cana-787	232	19	exist	exist	VERB
cana-787	232	20	unique	unique	ADJ
cana-787	232	21	fixed	fix	VERB
cana-787	232	22	point	point	NOUN
cana-787	232	23	𝑥	𝑥	NOUN
cana-787	232	24	=	=	SYM
cana-787	232	25	1	1	NUM
cana-787	232	26	2	2	NUM
cana-787	232	27	.	.	PUNCT
cana-787	232	28	example	example	NOUN
cana-787	232	29	2	2	NUM
cana-787	232	30	:	:	PUNCT
cana-787	232	31	let	let	VERB
cana-787	232	32	𝑋	𝑋	PROPN
cana-787	232	33	=	=	PUNCT
cana-787	233	1	[	[	X
cana-787	233	2	0	0	NUM
cana-787	233	3	,	,	PUNCT
cana-787	233	4	1	1	NUM
cana-787	233	5	]	]	PUNCT
cana-787	233	6	and	and	CCONJ
cana-787	233	7	metric	metric	ADJ
cana-787	233	8	on	on	ADP
cana-787	233	9	it	it	PRON
cana-787	233	10	is	be	AUX
cana-787	233	11	𝑑(𝑥	𝑑(𝑥	PROPN
cana-787	233	12	,	,	PUNCT
cana-787	233	13	𝑦	𝑦	X
cana-787	233	14	)	)	PUNCT
cana-787	233	15	=	=	PUNCT
cana-787	233	16	⌊𝑥	⌊𝑥	PROPN
cana-787	233	17	−	−	PROPN
cana-787	233	18	𝑦⌋.take	𝑦⌋.take	NUM
cana-787	233	19	two	two	NUM
cana-787	233	20	self	self	NOUN
cana-787	233	21	-	-	PUNCT
cana-787	233	22	mappings	mapping	NOUN
cana-787	233	23	defined	define	VERB
cana-787	233	24	by	by	ADP
cana-787	233	25	𝜇(𝑥	𝜇(𝑥	PROPN
cana-787	233	26	)	)	PUNCT
cana-787	234	1	=	=	SYM
cana-787	234	2	𝑥3	𝑥3	NOUN
cana-787	234	3	2	2	NUM
cana-787	235	1	+	+	CCONJ
cana-787	235	2	7	7	NUM
cana-787	235	3	16	16	NUM
cana-787	235	4	and	and	CCONJ
cana-787	235	5	𝜌(𝑥	𝜌(𝑥	NUM
cana-787	235	6	)	)	PUNCT
cana-787	236	1	=	=	NOUN
cana-787	236	2	𝑥4	𝑥4	NOUN
cana-787	236	3	3	3	NUM
cana-787	237	1	+	+	CCONJ
cana-787	237	2	23	23	NUM
cana-787	237	3	48	48	NUM
cana-787	237	4	.	.	PUNCT
cana-787	238	1	it	it	PRON
cana-787	238	2	is	be	AUX
cana-787	238	3	clear	clear	ADJ
cana-787	238	4	that	that	SCONJ
cana-787	238	5	𝜇	𝜇	ADP
cana-787	238	6	and	and	CCONJ
cana-787	238	7	𝜌	𝜌	X
cana-787	238	8	are	be	AUX
cana-787	238	9	continuous	continuous	ADJ
cana-787	238	10	but	but	CCONJ
cana-787	238	11	neither	neither	PRON
cana-787	238	12	of	of	ADP
cana-787	238	13	them	they	PRON
cana-787	238	14	are	be	AUX
cana-787	238	15	banach	banach	NOUN
cana-787	238	16	contraction	contraction	NOUN
cana-787	238	17	at	at	ADP
cana-787	238	18	𝑥	𝑥	NOUN
cana-787	238	19	=	=	SYM
cana-787	238	20	1	1	NUM
cana-787	238	21	and	and	CCONJ
cana-787	238	22	𝑦	𝑦	NOUN
cana-787	238	23	=	=	SYM
cana-787	238	24	0.9.actualy	0.9.actualy	X
cana-787	239	1	they	they	PRON
cana-787	239	2	satisfy	satisfy	VERB
cana-787	239	3	𝑓	𝑓	DET
cana-787	239	4	−	−	PROPN
cana-787	239	5	𝛼	𝛼	PROPN
cana-787	239	6	,	,	PUNCT
cana-787	239	7	𝜑	𝜑	PRON
cana-787	239	8	contraction	contraction	NOUN
cana-787	239	9	.	.	PUNCT
cana-787	240	1	where	where	SCONJ
cana-787	240	2	,	,	PUNCT
cana-787	240	3	𝛼	𝛼	X
cana-787	240	4	:	:	PUNCT
cana-787	240	5	𝑋	𝑋	NOUN
cana-787	240	6	×	×	NOUN
cana-787	240	7	𝑋	𝑋	PROPN
cana-787	240	8	→	→	SYM
cana-787	240	9	[	[	X
cana-787	240	10	0	0	NUM
cana-787	240	11	,	,	PUNCT
cana-787	240	12	∞	∞	PROPN
cana-787	240	13	)	)	PUNCT
cana-787	240	14	defined	define	VERB
cana-787	240	15	by	by	ADP
cana-787	240	16	𝛼(𝑥	𝛼(𝑥	PROPN
cana-787	240	17	,	,	PUNCT
cana-787	240	18	𝑦	𝑦	NOUN
cana-787	240	19	)	)	PUNCT
cana-787	240	20	=	=	NOUN
cana-787	240	21	𝑥2+𝑥𝑦+𝑦2	𝑥2+𝑥𝑦+𝑦2	VERB
cana-787	240	22	𝑥3+𝑥𝑦2+𝑥2𝑦+𝑦3	𝑥3+𝑥𝑦2+𝑥2𝑦+𝑦3	NUM
cana-787	240	23	and	and	CCONJ
cana-787	240	24	𝜑	𝜑	NOUN
cana-787	240	25	:	:	PUNCT
cana-787	240	26	𝑋	𝑋	NOUN
cana-787	240	27	×	×	NOUN
cana-787	240	28	𝑋	𝑋	PROPN
cana-787	240	29	→	→	SYM
cana-787	240	30	[	[	X
cana-787	240	31	0	0	NUM
cana-787	240	32	,	,	PUNCT
cana-787	240	33	∞	∞	PROPN
cana-787	240	34	)	)	PUNCT
cana-787	240	35	defined	define	VERB
cana-787	240	36	by	by	ADP
cana-787	240	37	𝜑(𝑡	𝜑(𝑡	PROPN
cana-787	240	38	)	)	PUNCT
cana-787	241	1	=	=	PUNCT
cana-787	241	2	2𝑡	2𝑡	NOUN
cana-787	241	3	3	3	X
cana-787	241	4	.	.	PUNCT
cana-787	242	1	take	take	VERB
cana-787	242	2	𝛽	𝛽	NOUN
cana-787	242	3	=	=	PROPN
cana-787	242	4	1.499	1.499	NUM
cana-787	242	5	…	…	PUNCT
cana-787	242	6	.	.	PUNCT
cana-787	243	1	so	so	ADV
cana-787	243	2	that	that	SCONJ
cana-787	243	3	𝛽𝜑	𝛽𝜑	PROPN
cana-787	243	4	is	be	AUX
cana-787	243	5	again	again	ADV
cana-787	243	6	a	a	DET
cana-787	243	7	comparison	comparison	NOUN
cana-787	243	8	function	function	NOUN
cana-787	243	9	.	.	PUNCT
cana-787	244	1	there	there	PRON
cana-787	244	2	exist	exist	VERB
cana-787	244	3	𝑥0	𝑥0	NOUN
cana-787	244	4	=	=	NOUN
cana-787	244	5	0.1	0.1	NUM
cana-787	244	6	such	such	ADJ
cana-787	244	7	that	that	DET
cana-787	244	8	𝛼(𝜇(𝑥0	𝛼(𝜇(𝑥0	NOUN
cana-787	244	9	)	)	PUNCT
cana-787	244	10	,	,	PUNCT
cana-787	244	11	𝜌(𝑥0	𝜌(𝑥0	NOUN
cana-787	244	12	)	)	PUNCT
cana-787	244	13	)	)	PUNCT
cana-787	245	1	>	>	PUNCT
cana-787	245	2	𝛽−1.by	𝛽−1.by	PROPN
cana-787	245	3	calculation	calculation	NOUN
cana-787	245	4	we	we	PRON
cana-787	245	5	can	can	AUX
cana-787	245	6	easily	easily	ADV
cana-787	245	7	find	find	VERB
cana-787	245	8	that	that	SCONJ
cana-787	245	9	𝜌	𝜌	PART
cana-787	245	10	is	be	AUX
cana-787	245	11	𝜇	𝜇	ADP
cana-787	245	12	−	−	NOUN
cana-787	245	13	𝛼	𝛼	PRON
cana-787	245	14	𝛽−1	𝛽−1	PROPN
cana-787	245	15	-	-	PUNCT
cana-787	245	16	admissible	admissible	ADJ
cana-787	245	17	.	.	PUNCT
cana-787	246	1	more	more	ADV
cana-787	246	2	over	over	ADV
cana-787	246	3	for	for	ADP
cana-787	246	4	every	every	DET
cana-787	246	5	pair	pair	NOUN
cana-787	246	6	𝑢	𝑢	NOUN
cana-787	246	7	,	,	PUNCT
cana-787	246	8	𝑣	𝑣	PRON
cana-787	246	9	∈	∈	PROPN
cana-787	246	10	𝐶(𝜇	𝐶(𝜇	NOUN
cana-787	246	11	,	,	PUNCT
cana-787	246	12	𝜌	𝜌	ADP
cana-787	246	13	)	)	PUNCT
cana-787	246	14	such	such	ADJ
cana-787	246	15	that	that	DET
cana-787	246	16	𝛼(𝜇(𝑢	𝛼(𝜇(𝑢	PROPN
cana-787	246	17	)	)	PUNCT
cana-787	246	18	,	,	PUNCT
cana-787	246	19	𝜇(𝑣	𝜇(𝑣	PROPN
cana-787	246	20	)	)	PUNCT
cana-787	246	21	)	)	PUNCT
cana-787	246	22	>	>	PUNCT
cana-787	247	1	𝛽−1	𝛽−1	PROPN
cana-787	247	2	or	or	CCONJ
cana-787	247	3	𝛼(𝜇(𝑣	𝛼(𝜇(𝑣	PROPN
cana-787	247	4	)	)	PUNCT
cana-787	247	5	,	,	PUNCT
cana-787	247	6	𝜇(𝑢	𝜇(𝑢	PROPN
cana-787	247	7	)	)	PUNCT
cana-787	247	8	)	)	PUNCT
cana-787	247	9	>	>	X
cana-787	248	1	𝛽−1.here	𝛽−1.here	PROPN
cana-787	248	2	𝜇	𝜇	X
cana-787	248	3	and	and	CCONJ
cana-787	248	4	𝜌	𝜌	AUX
cana-787	248	5	commute	commute	NOUN
cana-787	248	6	at	at	ADP
cana-787	248	7	only	only	ADV
cana-787	248	8	coincident	coincident	ADJ
cana-787	248	9	point	point	NOUN
cana-787	248	10	.hence	.hence	VERB
cana-787	248	11	there	there	PRON
cana-787	248	12	exist	exist	VERB
cana-787	248	13	unique	unique	ADJ
cana-787	248	14	common	common	ADJ
cana-787	248	15	fixed	fix	VERB
cana-787	248	16	point	point	NOUN
cana-787	248	17	𝑥	𝑥	NOUN
cana-787	248	18	=	=	SYM
cana-787	248	19	1	1	NUM
cana-787	248	20	2	2	NUM
cana-787	248	21	.	.	PUNCT
cana-787	249	1	4	4	X
cana-787	249	2	.	.	X
cana-787	249	3	remarks	remark	VERB
cana-787	249	4	a	a	PRON
cana-787	249	5	)	)	PUNCT
cana-787	249	6	if	if	SCONJ
cana-787	249	7	𝛽	𝛽	NOUN
cana-787	249	8	=	=	NOUN
cana-787	249	9	1	1	NUM
cana-787	249	10	then	then	ADV
cana-787	249	11	the	the	DET
cana-787	249	12	theorem	theorem	ADJ
cana-787	249	13	3.1	3.1	NUM
cana-787	249	14	becomes	become	NOUN
cana-787	249	15	theorem	theorem	ADJ
cana-787	249	16	2.1	2.1	NUM
cana-787	249	17	.	.	PUNCT
cana-787	250	1	b	b	X
cana-787	250	2	)	)	PUNCT
cana-787	250	3	for	for	ADP
cana-787	250	4	𝛽	𝛽	NOUN
cana-787	250	5	=	=	SYM
cana-787	250	6	1	1	NUM
cana-787	250	7	the	the	DET
cana-787	250	8	mapping	mapping	NOUN
cana-787	250	9	𝜇	𝜇	ADP
cana-787	250	10	becomes	become	VERB
cana-787	250	11	𝛼-admissible	𝛼-admissible	ADJ
cana-787	250	12	.	.	PUNCT
cana-787	251	1	c	c	X
cana-787	251	2	)	)	PUNCT
cana-787	251	3	if	if	SCONJ
cana-787	251	4	𝜇	𝜇	ADV
cana-787	251	5	is	be	AUX
cana-787	251	6	identity	identity	NOUN
cana-787	251	7	map	map	NOUN
cana-787	251	8	and	and	CCONJ
cana-787	251	9	𝛽	𝛽	NOUN
cana-787	251	10	=	=	NOUN
cana-787	251	11	1	1	NUM
cana-787	251	12	then	then	ADV
cana-787	251	13	the	the	DET
cana-787	251	14	theorem	theorem	ADJ
cana-787	251	15	3.4	3.4	NUM
cana-787	251	16	becomes	become	VERB
cana-787	251	17	theorem	theorem	ADJ
cana-787	251	18	2.1	2.1	NUM
cana-787	251	19	.	.	PUNCT
cana-787	252	1	refrences	refrence	NOUN
cana-787	253	1	[	[	X
cana-787	253	2	1	1	X
cana-787	253	3	]	]	PUNCT
cana-787	253	4	s.	s.	PROPN
cana-787	253	5	banach	banach	PROPN
cana-787	253	6	,	,	PUNCT
cana-787	253	7	“	"	PUNCT
cana-787	253	8	sur	sur	X
cana-787	253	9	les	les	X
cana-787	253	10	opérations	opération	NOUN
cana-787	253	11	dans	dan	NOUN
cana-787	253	12	les	les	X
cana-787	253	13	ensembles	ensemble	NOUN
cana-787	253	14	abstraits	abstrait	NOUN
cana-787	253	15	et	et	PROPN
cana-787	253	16	leur	leur	X
cana-787	253	17	application	application	PROPN
cana-787	253	18	aux	aux	PROPN
cana-787	253	19	équations	équations	PROPN
cana-787	253	20	intégrales	intégrale	NOUN
cana-787	253	21	,	,	PUNCT
cana-787	253	22	”	"	PUNCT
cana-787	253	23	fundamenta	fundamenta	PROPN
cana-787	253	24	mathematicae	mathematicae	PROPN
cana-787	253	25	,	,	PUNCT
cana-787	253	26	vol	vol	NOUN
cana-787	253	27	.	.	PROPN
cana-787	254	1	3	3	NUM
cana-787	254	2	,	,	PUNCT
cana-787	254	3	no	no	INTJ
cana-787	254	4	.	.	NOUN
cana-787	254	5	1	1	NUM
cana-787	254	6	,	,	PUNCT
cana-787	254	7	pp	pp	ADJ
cana-787	254	8	.	.	PUNCT
cana-787	255	1	133–181	133–181	NUM
cana-787	255	2	,	,	PUNCT
cana-787	255	3	1922	1922	NUM
cana-787	255	4	.	.	PUNCT
cana-787	256	1	[	[	X
cana-787	256	2	2	2	NUM
cana-787	256	3	]	]	PUNCT
cana-787	256	4	b.	b.	PROPN
cana-787	256	5	samet	samet	PROPN
cana-787	256	6	,	,	PUNCT
cana-787	256	7	c.	c.	PROPN
cana-787	256	8	vetro	vetro	PROPN
cana-787	256	9	,	,	PUNCT
cana-787	256	10	and	and	CCONJ
cana-787	256	11	p.	p.	PROPN
cana-787	256	12	vetro	vetro	PROPN
cana-787	256	13	,	,	PUNCT
cana-787	256	14	“	"	PUNCT
cana-787	256	15	fixed	fix	VERB
cana-787	256	16	point	point	NOUN
cana-787	256	17	theorems	theorem	NOUN
cana-787	256	18	for	for	ADP
cana-787	256	19	α	α	NOUN
cana-787	256	20	–	–	PUNCT
cana-787	256	21	ψ	ψ	ADJ
cana-787	256	22	-	-	ADJ
cana-787	256	23	contractive	contractive	ADJ
cana-787	256	24	type	type	NOUN
cana-787	256	25	mappings	mapping	NOUN
cana-787	256	26	,	,	PUNCT
cana-787	256	27	”	"	PUNCT
cana-787	256	28	nonlinear	nonlinear	ADJ
cana-787	256	29	analysis	analysis	NOUN
cana-787	256	30	:	:	PUNCT
cana-787	256	31	theory	theory	NOUN
cana-787	256	32	,	,	PUNCT
cana-787	256	33	methods	method	NOUN
cana-787	256	34	&	&	CCONJ
cana-787	256	35	applications	application	NOUN
cana-787	256	36	,	,	PUNCT
cana-787	256	37	vol	vol	NOUN
cana-787	256	38	.	.	PROPN
cana-787	256	39	75	75	NUM
cana-787	256	40	,	,	PUNCT
cana-787	256	41	no	no	INTJ
cana-787	256	42	.	.	NOUN
cana-787	256	43	4	4	NUM
cana-787	256	44	,	,	PUNCT
cana-787	256	45	pp	pp	ADJ
cana-787	256	46	.	.	PUNCT
cana-787	257	1	2154–2165	2154–2165	NUM
cana-787	257	2	,	,	PUNCT
cana-787	257	3	2012	2012	NUM
cana-787	257	4	.	.	PUNCT
cana-787	258	1	[	[	X
cana-787	258	2	3	3	X
cana-787	258	3	]	]	X
cana-787	258	4	b.	b.	PROPN
cana-787	258	5	samet	samet	PROPN
cana-787	258	6	,	,	PUNCT
cana-787	258	7	“	"	PUNCT
cana-787	258	8	fixed	fix	VERB
cana-787	258	9	points	point	NOUN
cana-787	258	10	for	for	ADP
cana-787	258	11	α	α	NOUN
cana-787	258	12	-	-	PUNCT
cana-787	258	13	ψ	ψ	NOUN
cana-787	258	14	contractive	contractive	ADJ
cana-787	258	15	mappings	mapping	NOUN
cana-787	258	16	with	with	ADP
cana-787	258	17	an	an	DET
cana-787	258	18	application	application	NOUN
cana-787	258	19	to	to	ADP
cana-787	258	20	quadratic	quadratic	ADJ
cana-787	258	21	integral	integral	ADJ
cana-787	258	22	equations	equation	NOUN
cana-787	258	23	,	,	PUNCT
cana-787	258	24	”	"	PUNCT
cana-787	258	25	vol	vol	NOUN
cana-787	258	26	.	.	PUNCT
cana-787	258	27	1914	1914	NUM
cana-787	258	28	,	,	PUNCT
cana-787	258	29	pp	pp	ADV
cana-787	258	30	.	.	PUNCT
cana-787	258	31	1–18	1–18	NUM
cana-787	258	32	,	,	PUNCT
cana-787	258	33	1914	1914	NUM
cana-787	258	34	.	.	PUNCT
cana-787	259	1	communications	communication	NOUN
cana-787	259	2	on	on	ADP
cana-787	259	3	applied	apply	VERB
cana-787	259	4	nonlinear	nonlinear	ADJ
cana-787	259	5	analysis	analysis	NOUN
cana-787	259	6	issn	issn	NOUN
cana-787	259	7	:	:	PUNCT
cana-787	259	8	1074	1074	NUM
cana-787	259	9	-	-	PUNCT
cana-787	259	10	133x	133x	NUM
cana-787	259	11	vol	vol	NOUN
cana-787	259	12	31	31	NUM
cana-787	259	13	no	no	NOUN
cana-787	259	14	.	.	PUNCT
cana-787	260	1	3s	3s	NUM
cana-787	260	2	(	(	PUNCT
cana-787	260	3	2024	2024	NUM
cana-787	260	4	)	)	PUNCT
cana-787	260	5	376	376	NUM
cana-787	260	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-787	261	1	[	[	X
cana-787	261	2	4	4	X
cana-787	261	3	]	]	PUNCT
cana-787	261	4	e.	e.	PROPN
cana-787	261	5	karapinar	karapinar	PROPN
cana-787	261	6	,	,	PUNCT
cana-787	261	7	s.	s.	PROPN
cana-787	261	8	czerwik	czerwik	PROPN
cana-787	261	9	,	,	PUNCT
cana-787	261	10	and	and	CCONJ
cana-787	261	11	h.	h.	PROPN
cana-787	261	12	aydi	aydi	VERB
cana-787	261	13	,	,	PUNCT
cana-787	261	14	“	"	PUNCT
cana-787	261	15	research	research	NOUN
cana-787	261	16	article	article	NOUN
cana-787	261	17	(	(	PUNCT
cana-787	261	18	𝛼	𝛼	NOUN
cana-787	261	19	,	,	PUNCT
cana-787	261	20	𝜓)-meir	𝜓)-meir	ADJ
cana-787	261	21	-	-	PUNCT
cana-787	261	22	keeler	keeler	NOUN
cana-787	261	23	contraction	contraction	NOUN
cana-787	261	24	mappings	mapping	NOUN
cana-787	261	25	in	in	ADP
cana-787	261	26	generalized	generalized	ADJ
cana-787	261	27	𝑏-metric	𝑏-metric	PROPN
cana-787	261	28	spaces	space	NOUN
cana-787	261	29	,	,	PUNCT
cana-787	261	30	”	"	PUNCT
cana-787	261	31	2018	2018	NUM
cana-787	261	32	,	,	PUNCT
cana-787	261	33	accessed	access	VERB
cana-787	261	34	:	:	PUNCT
cana-787	261	35	may	may	PROPN
cana-787	261	36	18	18	NUM
cana-787	261	37	,	,	PUNCT
cana-787	261	38	2024	2024	NUM
cana-787	261	39	.	.	PUNCT
cana-787	262	1	[	[	X
cana-787	262	2	online	online	X
cana-787	262	3	]	]	X
cana-787	262	4	.	.	PUNCT
cana-787	263	1	available	available	ADJ
cana-787	263	2	:	:	PUNCT
cana-787	264	1	https://www.academia.edu/download/72105343/3264620.pdf	https://www.academia.edu/download/72105343/3264620.pdf	PROPN
cana-787	264	2	[	[	X
cana-787	264	3	5	5	NUM
cana-787	264	4	]	]	PUNCT
cana-787	264	5	j.	j.	PROPN
cana-787	264	6	li	li	PROPN
cana-787	264	7	and	and	CCONJ
cana-787	264	8	h.	h.	PROPN
cana-787	264	9	guan	guan	PROPN
cana-787	264	10	,	,	PUNCT
cana-787	264	11	“	"	PUNCT
cana-787	264	12	common	common	ADJ
cana-787	264	13	fixed	fix	VERB
cana-787	264	14	point	point	NOUN
cana-787	264	15	results	result	NOUN
cana-787	264	16	for	for	ADP
cana-787	264	17	generalized	generalized	ADJ
cana-787	264	18	-contractive	-contractive	ADJ
cana-787	264	19	mappings	mapping	NOUN
cana-787	264	20	with	with	ADP
cana-787	264	21	applications	application	NOUN
cana-787	264	22	,	,	PUNCT
cana-787	264	23	”	"	PUNCT
cana-787	264	24	journal	journal	NOUN
cana-787	264	25	of	of	ADP
cana-787	264	26	function	function	NOUN
cana-787	264	27	spaces	space	NOUN
cana-787	264	28	,	,	PUNCT
cana-787	264	29	vol	vol	NOUN
cana-787	264	30	.	.	PROPN
cana-787	264	31	2021	2021	NUM
cana-787	264	32	,	,	PUNCT
cana-787	264	33	p.	p.	PROPN
cana-787	264	34	e5020027	e5020027	PROPN
cana-787	264	35	,	,	PUNCT
cana-787	264	36	jun	jun	PROPN
cana-787	264	37	.	.	PROPN
cana-787	264	38	2021	2021	NUM
cana-787	264	39	,	,	PUNCT
cana-787	264	40	doi	doi	NOUN
cana-787	264	41	:	:	PUNCT
cana-787	264	42	10.1155/2021/5020027	10.1155/2021/5020027	NUM
cana-787	264	43	.	.	PUNCT
cana-787	265	1	[	[	X
cana-787	265	2	6	6	NUM
cana-787	265	3	]	]	PUNCT
cana-787	265	4	n.	n.	NOUN
cana-787	265	5	hussain	hussain	PROPN
cana-787	265	6	,	,	PUNCT
cana-787	265	7	p.	p.	PROPN
cana-787	265	8	salimi	salimi	PROPN
cana-787	265	9	,	,	PUNCT
cana-787	265	10	and	and	CCONJ
cana-787	265	11	a.	a.	PROPN
cana-787	265	12	latif	latif	PROPN
cana-787	265	13	,	,	PUNCT
cana-787	265	14	“	"	PUNCT
cana-787	265	15	fixed	fix	VERB
cana-787	265	16	point	point	NOUN
cana-787	265	17	results	result	NOUN
cana-787	265	18	for	for	ADP
cana-787	265	19	single	single	ADJ
cana-787	265	20	and	and	CCONJ
cana-787	265	21	set	set	NOUN
cana-787	265	22	-	-	PUNCT
cana-787	265	23	valued	value	VERB
cana-787	265	24	α	α	PROPN
cana-787	265	25	-	-	PUNCT
cana-787	265	26	η	η	NOUN
cana-787	265	27	-	-	PUNCT
cana-787	265	28	ψ	ψ	ADP
cana-787	265	29	-	-	ADJ
cana-787	265	30	contractive	contractive	ADJ
cana-787	265	31	mappings	mapping	NOUN
cana-787	265	32	,	,	PUNCT
cana-787	265	33	”	"	PUNCT
cana-787	265	34	fixed	fix	VERB
cana-787	265	35	point	point	NOUN
cana-787	265	36	theory	theory	NOUN
cana-787	265	37	appl	appl	NOUN
cana-787	265	38	,	,	PUNCT
cana-787	265	39	vol	vol	NOUN
cana-787	265	40	.	.	PROPN
cana-787	265	41	2013	2013	NUM
cana-787	265	42	,	,	PUNCT
cana-787	265	43	no	no	INTJ
cana-787	265	44	.	.	NOUN
cana-787	265	45	1	1	NUM
cana-787	265	46	,	,	PUNCT
cana-787	265	47	p.	p.	NOUN
cana-787	265	48	212	212	NUM
cana-787	265	49	,	,	PUNCT
cana-787	265	50	dec	dec	PROPN
cana-787	265	51	.	.	PROPN
cana-787	265	52	2013	2013	NUM
cana-787	265	53	,	,	PUNCT
cana-787	265	54	doi	doi	NOUN
cana-787	265	55	:	:	PUNCT
cana-787	265	56	10.1186/1687	10.1186/1687	NUM
cana-787	265	57	-	-	SYM
cana-787	265	58	1812	1812	NUM
cana-787	265	59	-	-	PUNCT
cana-787	265	60	2013	2013	NUM
cana-787	265	61	-	-	PUNCT
cana-787	265	62	212	212	NUM
cana-787	265	63	.	.	PUNCT
cana-787	266	1	[	[	X
cana-787	266	2	7	7	X
cana-787	266	3	]	]	PUNCT
cana-787	266	4	k.	k.	PROPN
cana-787	266	5	zoto	zoto	PROPN
cana-787	266	6	,	,	PUNCT
cana-787	266	7	b.	b.	PROPN
cana-787	266	8	rhoades	rhoades	PROPN
cana-787	266	9	,	,	PUNCT
cana-787	266	10	and	and	CCONJ
cana-787	266	11	s.	s.	PROPN
cana-787	266	12	radenović	radenović	PROPN
cana-787	266	13	,	,	PUNCT
cana-787	266	14	“	"	PUNCT
cana-787	266	15	some	some	DET
cana-787	266	16	generalizations	generalization	NOUN
cana-787	266	17	for	for	ADP
cana-787	266	18	(	(	PUNCT
cana-787	266	19	α	α	NOUN
cana-787	266	20	-	-	PUNCT
cana-787	266	21	ψ	ψ	NOUN
cana-787	266	22	,	,	PUNCT
cana-787	266	23	φ)-contractions	φ)-contraction	NOUN
cana-787	266	24	in	in	ADP
cana-787	266	25	b	b	NOUN
cana-787	266	26	-	-	PUNCT
cana-787	266	27	metric	metric	ADJ
cana-787	266	28	-	-	PUNCT
cana-787	266	29	like	like	ADJ
cana-787	266	30	spaces	space	NOUN
cana-787	266	31	and	and	CCONJ
cana-787	266	32	an	an	DET
cana-787	266	33	application	application	NOUN
cana-787	266	34	,	,	PUNCT
cana-787	266	35	”	"	PUNCT
cana-787	266	36	fixed	fix	VERB
cana-787	266	37	point	point	NOUN
cana-787	266	38	theory	theory	NOUN
cana-787	266	39	and	and	CCONJ
cana-787	266	40	applications	application	NOUN
cana-787	266	41	,	,	PUNCT
cana-787	266	42	vol	vol	NOUN
cana-787	266	43	.	.	PROPN
cana-787	266	44	2017	2017	NUM
cana-787	266	45	,	,	PUNCT
cana-787	266	46	pp	pp	ADJ
cana-787	266	47	.	.	PUNCT
cana-787	267	1	13663–2017	13663–2017	NUM
cana-787	267	2	,	,	PUNCT
cana-787	267	3	nov	nov	PROPN
cana-787	267	4	.	.	PROPN
cana-787	267	5	2017	2017	NUM
cana-787	267	6	,	,	PUNCT
cana-787	267	7	doi	doi	NOUN
cana-787	267	8	:	:	PUNCT
cana-787	267	9	10.1186	10.1186	NUM
cana-787	267	10	/	/	SYM
cana-787	267	11	s13663	s13663	NOUN
cana-787	267	12	-	-	PUNCT
cana-787	267	13	017	017	NUM
cana-787	267	14	-	-	PUNCT
cana-787	267	15	0620	0620	NUM
cana-787	267	16	-	-	PUNCT
cana-787	267	17	1	1	NUM
cana-787	267	18	.	.	PUNCT
cana-787	268	1	[	[	X
cana-787	268	2	8	8	NUM
cana-787	268	3	]	]	X
cana-787	268	4	b.	b.	PROPN
cana-787	268	5	n.	n.	PROPN
cana-787	268	6	abagaro	abagaro	PROPN
cana-787	268	7	,	,	PUNCT
cana-787	268	8	k.	k.	PROPN
cana-787	268	9	k.	k.	PROPN
cana-787	268	10	tola	tola	PROPN
cana-787	268	11	,	,	PUNCT
cana-787	268	12	and	and	CCONJ
cana-787	268	13	m.	m.	NOUN
cana-787	268	14	a.	a.	NOUN
cana-787	268	15	mamud	mamud	PROPN
cana-787	268	16	,	,	PUNCT
cana-787	268	17	“	"	PUNCT
cana-787	268	18	fixed	fix	VERB
cana-787	268	19	point	point	NOUN
cana-787	268	20	theorems	theorem	NOUN
cana-787	268	21	for	for	ADP
cana-787	268	22	generalized	generalized	ADJ
cana-787	268	23	$	$	SYM
cana-787	268	24	(	(	PUNCT
cana-787	268	25	\alpha	\alpha	ADV
cana-787	268	26	,	,	PUNCT
cana-787	268	27	\psi	\psi	PROPN
cana-787	268	28	)	)	PUNCT
cana-787	268	29	$	$	SYM
cana-787	268	30	contraction	contraction	NOUN
cana-787	268	31	mappings	mapping	NOUN
cana-787	268	32	in	in	ADP
cana-787	268	33	rectangular	rectangular	ADJ
cana-787	268	34	quasi	quasi	X
cana-787	268	35	b	b	NOUN
cana-787	268	36	-	-	ADJ
cana-787	268	37	metric	metric	ADJ
cana-787	268	38	spaces	space	NOUN
cana-787	268	39	,	,	PUNCT
cana-787	268	40	”	"	PUNCT
cana-787	268	41	fixed	fix	VERB
cana-787	268	42	point	point	NOUN
cana-787	268	43	theory	theory	NOUN
cana-787	268	44	algorithms	algorithm	VERB
cana-787	268	45	sci	sci	PROPN
cana-787	268	46	eng	eng	PROPN
cana-787	268	47	,	,	PUNCT
cana-787	268	48	vol	vol	NOUN
cana-787	268	49	.	.	PROPN
cana-787	268	50	2022	2022	NUM
cana-787	268	51	,	,	PUNCT
cana-787	268	52	no	no	INTJ
cana-787	268	53	.	.	NOUN
cana-787	268	54	1	1	NUM
cana-787	268	55	,	,	PUNCT
cana-787	268	56	p.	p.	NOUN
cana-787	268	57	13	13	NUM
cana-787	268	58	,	,	PUNCT
cana-787	268	59	dec	dec	PROPN
cana-787	268	60	.	.	PROPN
cana-787	268	61	2022	2022	NUM
cana-787	268	62	,	,	PUNCT
cana-787	268	63	doi	doi	NOUN
cana-787	268	64	:	:	PUNCT
cana-787	268	65	10.1186	10.1186	NUM
cana-787	268	66	/	/	SYM
cana-787	268	67	s13663	s13663	PROPN
cana-787	268	68	-	-	PUNCT
cana-787	268	69	022	022	NUM
cana-787	268	70	-	-	PUNCT
cana-787	268	71	00723	00723	NUM
cana-787	268	72	-	-	PUNCT
cana-787	268	73	w.	w.	NOUN
cana-787	269	1	[	[	X
cana-787	269	2	9	9	NUM
cana-787	269	3	]	]	PUNCT
cana-787	269	4	j.	j.	PROPN
cana-787	269	5	r.	r.	PROPN
cana-787	269	6	morales	morales	PROPN
cana-787	269	7	and	and	CCONJ
cana-787	269	8	e.	e.	PROPN
cana-787	269	9	m.	m.	PROPN
cana-787	269	10	rojas	rojas	PROPN
cana-787	269	11	,	,	PUNCT
cana-787	269	12	“	"	PUNCT
cana-787	269	13	common	common	ADJ
cana-787	269	14	fixed	fix	VERB
cana-787	269	15	points	point	NOUN
cana-787	269	16	for	for	ADP
cana-787	269	17	$	$	SYM
cana-787	269	18	$	$	SYM
cana-787	269	19	(	(	PUNCT
cana-787	269	20	\psi	\psi	PROPN
cana-787	269	21	-\varphi	-\varphi	VERB
cana-787	269	22	)	)	PUNCT
cana-787	269	23	$	$	SYM
cana-787	269	24	$	$	NUM
cana-787	269	25	-weak	-weak	NOUN
cana-787	269	26	contractions	contraction	NOUN
cana-787	269	27	type	type	NOUN
cana-787	269	28	in	in	ADP
cana-787	269	29	bmetric	bmetric	ADJ
cana-787	269	30	spaces	space	NOUN
cana-787	269	31	,	,	PUNCT
cana-787	269	32	”	"	PUNCT
cana-787	269	33	arab	arab	PROPN
cana-787	269	34	.	.	PUNCT
cana-787	270	1	j.	j.	PROPN
cana-787	270	2	math	math	PROPN
cana-787	270	3	.	.	PUNCT
cana-787	270	4	,	,	PUNCT
cana-787	270	5	vol	vol	NOUN
cana-787	270	6	.	.	PROPN
cana-787	271	1	10	10	NUM
cana-787	271	2	,	,	PUNCT
cana-787	271	3	no	no	INTJ
cana-787	271	4	.	.	NOUN
cana-787	271	5	3	3	NUM
cana-787	271	6	,	,	PUNCT
cana-787	271	7	pp	pp	ADJ
cana-787	271	8	.	.	PUNCT
cana-787	272	1	639–658	639–658	NUM
cana-787	272	2	,	,	PUNCT
cana-787	272	3	dec	dec	PROPN
cana-787	272	4	.	.	PROPN
cana-787	272	5	2021	2021	NUM
cana-787	272	6	,	,	PUNCT
cana-787	272	7	doi	doi	NOUN
cana-787	272	8	:	:	PUNCT
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cana-787	272	10	/	/	SYM
cana-787	272	11	s40065	s40065	NOUN
cana-787	272	12	-	-	PUNCT
cana-787	272	13	021	021	NUM
cana-787	272	14	-	-	PUNCT
cana-787	272	15	00347	00347	NUM
cana-787	272	16	-	-	SYM
cana-787	272	17	9	9	NUM
cana-787	272	18	.	.	PUNCT
cana-787	273	1	[	[	X
cana-787	273	2	10	10	NUM
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cana-787	273	5	manthena	manthena	PROPN
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cana-787	273	7	r.	r.	PROPN
cana-787	273	8	manchala	manchala	PROPN
cana-787	273	9	,	,	PUNCT
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cana-787	273	11	fixed	fix	VERB
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cana-787	273	13	results	result	VERB
cana-787	273	14	for	for	ADP
cana-787	273	15	$	$	SYM
cana-787	273	16	\backslashmathscr	\backslashmathscr	PROPN
cana-787	273	17	{	{	PUNCT
cana-787	273	18	h	h	NOUN
cana-787	273	19	}	}	PUNCT
cana-787	273	20	$	$	SYM
cana-787	273	21	-contractions	-contraction	NOUN
cana-787	273	22	in	in	ADP
cana-787	273	23	fuzzy	fuzzy	ADJ
cana-787	273	24	metric	metric	ADJ
cana-787	273	25	spaces	space	NOUN
cana-787	273	26	via	via	ADP
cana-787	273	27	admissible	admissible	ADJ
cana-787	273	28	,	,	PUNCT
cana-787	273	29	”	"	PUNCT
cana-787	273	30	malaya	malaya	PROPN
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cana-787	273	32	of	of	ADP
cana-787	273	33	matematik	matematik	PROPN
cana-787	273	34	,	,	PUNCT
cana-787	273	35	vol	vol	NOUN
cana-787	273	36	.	.	PROPN
cana-787	273	37	6	6	NUM
cana-787	273	38	,	,	PUNCT
cana-787	273	39	no	no	INTJ
cana-787	273	40	.	.	NOUN
cana-787	273	41	03	03	NUM
cana-787	273	42	,	,	PUNCT
cana-787	273	43	pp	pp	ADJ
cana-787	273	44	.	.	PUNCT
cana-787	274	1	588–594	588–594	NUM
cana-787	274	2	,	,	PUNCT
cana-787	274	3	2018	2018	NUM
cana-787	274	4	.	.	PUNCT
cana-787	275	1	[	[	X
cana-787	275	2	11	11	NUM
cana-787	275	3	]	]	X
cana-787	275	4	p.	p.	NOUN
cana-787	275	5	salimi	salimi	PROPN
cana-787	275	6	,	,	PUNCT
cana-787	275	7	n.	n.	PROPN
cana-787	275	8	hussain	hussain	PROPN
cana-787	275	9	,	,	PUNCT
cana-787	275	10	a.	a.	PROPN
cana-787	275	11	roldan	roldan	PROPN
cana-787	275	12	,	,	PUNCT
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cana-787	275	14	e.	e.	PROPN
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cana-787	275	16	,	,	PUNCT
cana-787	275	17	“	"	PUNCT
cana-787	275	18	on	on	ADP
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cana-787	275	20	𝛼-ϕ-asymmetric	𝛼-ϕ-asymmetric	PROPN
cana-787	275	21	meir	meir	PROPN
cana-787	275	22	-	-	PUNCT
cana-787	275	23	keeler	keeler	PROPN
cana-787	275	24	contractive	contractive	ADJ
cana-787	275	25	mappings	mapping	NOUN
cana-787	275	26	,	,	PUNCT
cana-787	275	27	”	"	PUNCT
cana-787	275	28	filomat	filomat	NOUN
cana-787	275	29	,	,	PUNCT
cana-787	275	30	vol	vol	NOUN
cana-787	275	31	.	.	PROPN
cana-787	275	32	28	28	NUM
cana-787	275	33	,	,	PUNCT
cana-787	275	34	no	no	INTJ
cana-787	275	35	.	.	NOUN
cana-787	275	36	9	9	NUM
cana-787	275	37	,	,	PUNCT
cana-787	275	38	pp	pp	ADJ
cana-787	275	39	.	.	PUNCT
cana-787	276	1	1855–1869	1855–1869	NUM
cana-787	276	2	,	,	PUNCT
cana-787	276	3	2014	2014	NUM
cana-787	276	4	.	.	PUNCT
cana-787	277	1	[	[	X
cana-787	277	2	12	12	NUM
cana-787	277	3	]	]	PUNCT
cana-787	277	4	a.	a.	PROPN
cana-787	277	5	latif	latif	PROPN
cana-787	277	6	,	,	PUNCT
cana-787	277	7	m.	m.	PROPN
cana-787	277	8	e.	e.	PROPN
cana-787	277	9	gordji	gordji	PROPN
cana-787	277	10	,	,	PUNCT
cana-787	277	11	e.	e.	PROPN
cana-787	277	12	karapınar	karapınar	PROPN
cana-787	277	13	,	,	PUNCT
cana-787	277	14	and	and	CCONJ
cana-787	277	15	w.	w.	PROPN
cana-787	277	16	sintunavarat	sintunavarat	PROPN
cana-787	277	17	,	,	PUNCT
cana-787	277	18	“	"	PUNCT
cana-787	277	19	fixed	fix	VERB
cana-787	277	20	point	point	NOUN
cana-787	277	21	results	result	NOUN
cana-787	277	22	for	for	ADP
cana-787	277	23	generalized	generalized	ADJ
cana-787	277	24	(	(	PUNCT
cana-787	277	25	α	α	NOUN
cana-787	277	26	,	,	PUNCT
cana-787	277	27	ψ	ψ	NOUN
cana-787	277	28	)	)	PUNCT
cana-787	277	29	-meirkeeler	-meirkeeler	NOUN
cana-787	277	30	contractive	contractive	ADJ
cana-787	277	31	mappings	mapping	NOUN
cana-787	277	32	and	and	CCONJ
cana-787	277	33	applications	application	NOUN
cana-787	277	34	,	,	PUNCT
cana-787	277	35	”	"	PUNCT
cana-787	277	36	j	j	PROPN
cana-787	277	37	inequal	inequal	PROPN
cana-787	277	38	appl	appl	PROPN
cana-787	277	39	,	,	PUNCT
cana-787	277	40	vol	vol	NOUN
cana-787	277	41	.	.	PROPN
cana-787	277	42	2014	2014	NUM
cana-787	277	43	,	,	PUNCT
cana-787	277	44	no	no	INTJ
cana-787	277	45	.	.	NOUN
cana-787	277	46	1	1	NUM
cana-787	277	47	,	,	PUNCT
cana-787	277	48	p.	p.	NOUN
cana-787	277	49	68	68	NUM
cana-787	277	50	,	,	PUNCT
cana-787	277	51	dec	dec	PROPN
cana-787	277	52	.	.	PROPN
cana-787	277	53	2014	2014	NUM
cana-787	277	54	,	,	PUNCT
cana-787	277	55	doi	doi	NOUN
cana-787	277	56	:	:	PUNCT
cana-787	277	57	10.1186/1029	10.1186/1029	NUM
cana-787	277	58	-	-	SYM
cana-787	277	59	242x-2014	242x-2014	NUM
cana-787	277	60	-	-	PUNCT
cana-787	277	61	68	68	NUM
cana-787	277	62	.	.	PUNCT
cana-787	278	1	[	[	X
cana-787	278	2	13	13	NUM
cana-787	278	3	]	]	X
cana-787	278	4	n.	n.	PROPN
cana-787	278	5	hussain	hussain	PROPN
cana-787	278	6	,	,	PUNCT
cana-787	278	7	e.	e.	PROPN
cana-787	278	8	karapınar	karapınar	PROPN
cana-787	278	9	,	,	PUNCT
cana-787	278	10	p.	p.	PROPN
cana-787	278	11	salimi	salimi	PROPN
cana-787	278	12	,	,	PUNCT
cana-787	278	13	and	and	CCONJ
cana-787	278	14	p.	p.	PROPN
cana-787	278	15	vetro	vetro	PROPN
cana-787	278	16	,	,	PUNCT
cana-787	278	17	“	"	PUNCT
cana-787	278	18	fixed	fix	VERB
cana-787	278	19	point	point	NOUN
cana-787	278	20	results	result	NOUN
cana-787	278	21	for	for	ADP
cana-787	278	22	g	g	PROPN
cana-787	278	23	m	m	PROPN
cana-787	278	24	-meir	-meir	NOUN
cana-787	278	25	-	-	PUNCT
cana-787	278	26	keeler	keeler	NOUN
cana-787	278	27	contractive	contractive	NOUN
cana-787	278	28	and	and	CCONJ
cana-787	278	29	g	g	PROPN
cana-787	278	30	(	(	PUNCT
cana-787	278	31	α	α	NOUN
cana-787	278	32	,	,	PUNCT
cana-787	278	33	ψ	ψ	NOUN
cana-787	278	34	)	)	PUNCT
cana-787	278	35	-meir	-meir	ADJ
cana-787	278	36	-	-	PUNCT
cana-787	278	37	keeler	keeler	NOUN
cana-787	278	38	contractive	contractive	ADJ
cana-787	278	39	mappings	mapping	NOUN
cana-787	278	40	,	,	PUNCT
cana-787	278	41	”	"	PUNCT
cana-787	278	42	fixed	fix	VERB
cana-787	278	43	point	point	NOUN
cana-787	278	44	theory	theory	NOUN
cana-787	278	45	appl	appl	NOUN
cana-787	278	46	,	,	PUNCT
cana-787	278	47	vol	vol	NOUN
cana-787	278	48	.	.	PROPN
cana-787	279	1	2013	2013	NUM
cana-787	279	2	,	,	PUNCT
cana-787	279	3	no	no	INTJ
cana-787	279	4	.	.	NOUN
cana-787	279	5	1	1	NUM
cana-787	279	6	,	,	PUNCT
cana-787	279	7	p.	p.	NOUN
cana-787	279	8	34	34	NUM
cana-787	279	9	,	,	PUNCT
cana-787	279	10	dec	dec	PROPN
cana-787	279	11	.	.	PROPN
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cana-787	279	13	,	,	PUNCT
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cana-787	279	17	-	-	SYM
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cana-787	279	19	-	-	PUNCT
cana-787	279	20	2013	2013	NUM
cana-787	279	21	-	-	SYM
cana-787	279	22	34	34	NUM
cana-787	279	23	.	.	PUNCT
cana-787	280	1	[	[	X
cana-787	280	2	14	14	NUM
cana-787	280	3	]	]	X
cana-787	280	4	n.	n.	PROPN
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cana-787	280	6	,	,	PUNCT
cana-787	280	7	m.	m.	NOUN
cana-787	280	8	a.	a.	NOUN
cana-787	280	9	alghamdi	alghamdi	PROPN
cana-787	280	10	,	,	PUNCT
cana-787	280	11	s.	s.	PROPN
cana-787	280	12	alshehri	alshehri	PROPN
cana-787	280	13	,	,	PUNCT
cana-787	280	14	and	and	CCONJ
cana-787	280	15	i.	i.	PROPN
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cana-787	280	17	,	,	PUNCT
cana-787	280	18	“	"	PUNCT
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cana-787	280	20	-	-	ADJ
cana-787	280	21	metric	metric	ADJ
cana-787	280	22	spaces	space	NOUN
cana-787	280	23	and	and	CCONJ
cana-787	280	24	fixed	fix	VERB
cana-787	280	25	points	point	NOUN
cana-787	280	26	of	of	ADP
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cana-787	280	30	,	,	PUNCT
cana-787	280	31	”	"	PUNCT
cana-787	280	32	j.	j.	PROPN
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cana-787	280	36	appl	appl	PROPN
cana-787	280	37	.	.	PROPN
cana-787	280	38	,	,	PUNCT
cana-787	280	39	vol	vol	NOUN
cana-787	280	40	.	.	PROPN
cana-787	280	41	09	09	NUM
cana-787	280	42	,	,	PUNCT
cana-787	280	43	no	no	INTJ
cana-787	280	44	.	.	NOUN
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cana-787	280	46	,	,	PUNCT
cana-787	280	47	pp	pp	ADJ
cana-787	280	48	.	.	PUNCT
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cana-787	281	2	,	,	PUNCT
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cana-787	281	5	,	,	PUNCT
cana-787	281	6	doi	doi	NOUN
cana-787	281	7	:	:	PUNCT
cana-787	281	8	10.22436	10.22436	NUM
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cana-787	281	11	.	.	PUNCT
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cana-787	282	2	15	15	NUM
cana-787	282	3	]	]	X
cana-787	282	4	a.	a.	NOUN
cana-787	282	5	felhi	felhi	PROPN
cana-787	282	6	,	,	PUNCT
cana-787	282	7	s.	s.	PROPN
cana-787	282	8	sahmim	sahmim	PROPN
cana-787	282	9	,	,	PUNCT
cana-787	282	10	and	and	CCONJ
cana-787	282	11	h.	h.	PROPN
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cana-787	282	13	,	,	PUNCT
cana-787	282	14	“	"	PUNCT
cana-787	282	15	on	on	ADP
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cana-787	282	17	fixed	fix	VERB
cana-787	282	18	points	point	NOUN
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cana-787	282	20	(	(	PUNCT
cana-787	282	21	α	α	NOUN
cana-787	282	22	,	,	PUNCT
cana-787	282	23	ψ)-contractions	ψ)-contraction	NOUN
cana-787	282	24	and	and	CCONJ
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cana-787	282	27	contractions	contraction	NOUN
cana-787	282	28	in	in	ADP
cana-787	282	29	b	b	NOUN
cana-787	282	30	-	-	PUNCT
cana-787	282	31	metric	metric	ADJ
cana-787	282	32	-	-	PUNCT
cana-787	282	33	like	like	ADJ
cana-787	282	34	spaces	space	NOUN
cana-787	282	35	and	and	CCONJ
cana-787	282	36	consequences	consequence	NOUN
cana-787	282	37	,	,	PUNCT
cana-787	282	38	”	"	PUNCT
cana-787	282	39	2016	2016	NUM
cana-787	282	40	,	,	PUNCT
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cana-787	282	42	:	:	PUNCT
cana-787	282	43	may	may	PROPN
cana-787	282	44	18	18	NUM
cana-787	282	45	,	,	PUNCT
cana-787	282	46	2024	2024	NUM
cana-787	282	47	.	.	PUNCT
cana-787	283	1	[	[	X
cana-787	283	2	online	online	X
cana-787	283	3	]	]	X
cana-787	283	4	.	.	PUNCT
cana-787	284	1	available	available	ADJ
cana-787	284	2	:	:	PUNCT
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cana-787	284	5	,	,	PUNCT
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cana-787	284	9	]	]	PUNCT
cana-787	284	10	i.	i.	PROPN
cana-787	284	11	a.	a.	PROPN
cana-787	284	12	rus	rus	PROPN
cana-787	284	13	,	,	PUNCT
cana-787	284	14	a.	a.	NOUN
cana-787	284	15	petruşel	petruşel	NOUN
cana-787	284	16	,	,	PUNCT
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cana-787	284	18	g.	g.	PROPN
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cana-787	284	20	,	,	PUNCT
cana-787	284	21	fixed	fix	VERB
cana-787	284	22	point	point	NOUN
cana-787	284	23	theory	theory	NOUN
cana-787	284	24	,	,	PUNCT
cana-787	284	25	vol	vol	NOUN
cana-787	284	26	.	.	PROPN
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cana-787	285	2	.	.	X
cana-787	285	3	cluj	cluj	PROPN
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cana-787	285	5	press	press	PROPN
cana-787	285	6	cluj	cluj	PROPN
cana-787	285	7	-	-	PUNCT
cana-787	285	8	napoca	napoca	NOUN
cana-787	285	9	,	,	PUNCT
cana-787	285	10	2008	2008	NUM
cana-787	285	11	.	.	PUNCT
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cana-787	286	2	17	17	NUM
cana-787	286	3	]	]	PUNCT
cana-787	286	4	m.	m.	NOUN
cana-787	286	5	abbas	abbas	PROPN
cana-787	286	6	and	and	CCONJ
cana-787	286	7	g.	g.	PROPN
cana-787	286	8	jungck	jungck	PROPN
cana-787	286	9	,	,	PUNCT
cana-787	286	10	“	"	PUNCT
cana-787	286	11	common	common	ADJ
cana-787	286	12	fixed	fix	VERB
cana-787	286	13	point	point	NOUN
cana-787	286	14	results	result	NOUN
cana-787	286	15	for	for	ADP
cana-787	286	16	noncommuting	noncommute	VERB
cana-787	286	17	mappings	mapping	NOUN
cana-787	286	18	without	without	ADP
cana-787	286	19	continuity	continuity	NOUN
cana-787	286	20	in	in	ADP
cana-787	286	21	cone	cone	NOUN
cana-787	286	22	metric	metric	ADJ
cana-787	286	23	spaces	space	NOUN
cana-787	286	24	,	,	PUNCT
cana-787	286	25	”	"	PUNCT
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cana-787	286	28	mathematical	mathematical	ADJ
cana-787	286	29	analysis	analysis	NOUN
cana-787	286	30	and	and	CCONJ
cana-787	286	31	applications	application	NOUN
cana-787	286	32	,	,	PUNCT
cana-787	286	33	vol	vol	NOUN
cana-787	286	34	.	.	PROPN
cana-787	287	1	341	341	NUM
cana-787	287	2	,	,	PUNCT
cana-787	287	3	no	no	INTJ
cana-787	287	4	.	.	NOUN
cana-787	287	5	1	1	NUM
cana-787	287	6	,	,	PUNCT
cana-787	287	7	pp	pp	ADJ
cana-787	287	8	.	.	PUNCT
cana-787	288	1	416–420	416–420	NUM
cana-787	288	2	,	,	PUNCT
cana-787	288	3	may	may	PROPN
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cana-787	288	5	,	,	PUNCT
cana-787	288	6	doi	doi	NOUN
cana-787	288	7	:	:	PUNCT
cana-787	288	8	10.1016	10.1016	NUM
cana-787	288	9	/	/	SYM
cana-787	288	10	j.jmaa.2007.09.070	j.jmaa.2007.09.070	NOUN
cana-787	288	11	.	.	PUNCT
cana-787	289	1	[	[	X
cana-787	289	2	18	18	NUM
cana-787	289	3	]	]	X
cana-787	289	4	h.	h.	PROPN
cana-787	289	5	aydi	aydi	PROPN
cana-787	289	6	,	,	PUNCT
cana-787	289	7	“	"	PUNCT
cana-787	289	8	α	α	NOUN
cana-787	289	9	-	-	ADJ
cana-787	289	10	implicit	implicit	ADJ
cana-787	289	11	contractive	contractive	ADJ
cana-787	289	12	pair	pair	NOUN
cana-787	289	13	of	of	ADP
cana-787	289	14	mappings	mapping	NOUN
cana-787	289	15	on	on	ADP
cana-787	289	16	quasi	quasi	X
cana-787	289	17	b	b	X
cana-787	289	18	-	-	ADJ
cana-787	289	19	metric	metric	ADJ
cana-787	289	20	spaces	space	NOUN
cana-787	289	21	and	and	CCONJ
cana-787	289	22	an	an	DET
cana-787	289	23	application	application	NOUN
cana-787	289	24	to	to	ADP
cana-787	289	25	integral	integral	ADJ
cana-787	289	26	equations	equation	NOUN
cana-787	289	27	,	,	PUNCT
cana-787	289	28	”	"	PUNCT
cana-787	289	29	j.	j.	PROPN
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cana-787	289	34	vol	vol	NOUN
cana-787	289	35	.	.	PROPN
cana-787	289	36	17	17	NUM
cana-787	289	37	,	,	PUNCT
cana-787	289	38	no	no	INTJ
cana-787	289	39	.	.	NOUN
cana-787	289	40	12	12	NUM
cana-787	289	41	,	,	PUNCT
cana-787	289	42	pp	pp	ADJ
cana-787	289	43	.	.	PUNCT
cana-787	290	1	2417–2433	2417–2433	NUM
cana-787	290	2	,	,	PUNCT
cana-787	290	3	2016	2016	NUM
cana-787	290	4	.	.	PUNCT