id	sid	tid	token	lemma	pos
cana-5538	1	1	communications	communication	NOUN
cana-5538	1	2	on	on	ADP
cana-5538	1	3	applied	apply	VERB
cana-5538	1	4	nonlinear	nonlinear	ADJ
cana-5538	1	5	analysis	analysis	NOUN
cana-5538	1	6	issn	issn	NOUN
cana-5538	1	7	:	:	PUNCT
cana-5538	1	8	1074	1074	NUM
cana-5538	1	9	-	-	PUNCT
cana-5538	1	10	133x	133x	NUM
cana-5538	1	11	vol	vol	VERB
cana-5538	1	12	32	32	NUM
cana-5538	1	13	no	no	NOUN
cana-5538	1	14	.	.	PUNCT
cana-5538	2	1	10s	10	NOUN
cana-5538	2	2	(	(	PUNCT
cana-5538	2	3	2025	2025	NUM
cana-5538	2	4	)	)	PUNCT
cana-5538	2	5	2612	2612	NUM
cana-5538	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	3	1	some	some	PRON
cana-5538	3	2	results	result	VERB
cana-5538	3	3	on	on	ADP
cana-5538	3	4	partial	partial	ADJ
cana-5538	3	5	cone	cone	NOUN
cana-5538	3	6	metric	metric	ADJ
cana-5538	3	7	spaces	space	NOUN
cana-5538	3	8	with	with	ADP
cana-5538	3	9	an	an	DET
cana-5538	3	10	application	application	NOUN
cana-5538	3	11	heeramani	heeramani	NOUN
cana-5538	3	12	tiwari1	tiwari1	NOUN
cana-5538	3	13	,	,	PUNCT
cana-5538	3	14	*	*	PROPN
cana-5538	3	15	,	,	PUNCT
cana-5538	3	16	anil	anil	PROPN
cana-5538	3	17	mishra2	mishra2	PROPN
cana-5538	3	18	,	,	PUNCT
cana-5538	3	19	padmavati3	padmavati3	NOUN
cana-5538	3	20	1,2,3government	1,2,3government	NUM
cana-5538	3	21	v.y.t	v.y.t	NOUN
cana-5538	3	22	.	.	PUNCT
cana-5538	4	1	autonomous	autonomous	PROPN
cana-5538	4	2	p.g	p.g	PROPN
cana-5538	4	3	.	.	PROPN
cana-5538	4	4	college	college	PROPN
cana-5538	4	5	,	,	PUNCT
cana-5538	4	6	durg	durg	NOUN
cana-5538	4	7	,	,	PUNCT
cana-5538	4	8	chhattisgarh	chhattisgarh	NOUN
cana-5538	4	9	,	,	PUNCT
cana-5538	4	10	india	india	PROPN
cana-5538	4	11	toravi.tiwari@gmail.com	toravi.tiwari@gmail.com	PROPN
cana-5538	4	12	,	,	PUNCT
cana-5538	4	13	mshranil@gmail.com	mshranil@gmail.com	PROPN
cana-5538	4	14	,	,	PUNCT
cana-5538	4	15	padmavati.sudha62@gmail.com	padmavati.sudha62@gmail.com	X
cana-5538	4	16	article	article	NOUN
cana-5538	4	17	history	history	NOUN
cana-5538	4	18	:	:	PUNCT
cana-5538	4	19	received	receive	VERB
cana-5538	4	20	:	:	PUNCT
cana-5538	4	21	12	12	NUM
cana-5538	4	22	-	-	SYM
cana-5538	4	23	01	01	NUM
cana-5538	4	24	-	-	PUNCT
cana-5538	4	25	2025	2025	NUM
cana-5538	4	26	revised	revise	VERB
cana-5538	4	27	:	:	PUNCT
cana-5538	4	28	15	15	NUM
cana-5538	4	29	-	-	NUM
cana-5538	4	30	02	02	NUM
cana-5538	4	31	-	-	PUNCT
cana-5538	4	32	2025	2025	NUM
cana-5538	4	33	accepted	accept	VERB
cana-5538	4	34	:	:	PUNCT
cana-5538	4	35	01	01	NUM
cana-5538	4	36	-	-	SYM
cana-5538	4	37	03	03	NUM
cana-5538	4	38	-	-	PUNCT
cana-5538	4	39	2025	2025	NUM
cana-5538	4	40	abstract	abstract	NOUN
cana-5538	4	41	:	:	PUNCT
cana-5538	4	42	the	the	DET
cana-5538	4	43	objective	objective	NOUN
cana-5538	4	44	of	of	ADP
cana-5538	4	45	this	this	DET
cana-5538	4	46	paper	paper	NOUN
cana-5538	4	47	is	be	AUX
cana-5538	4	48	to	to	PART
cana-5538	4	49	determine	determine	VERB
cana-5538	4	50	some	some	DET
cana-5538	4	51	fixed	fix	VERB
cana-5538	4	52	point	point	NOUN
cana-5538	4	53	theorems	theorem	NOUN
cana-5538	4	54	for	for	ADP
cana-5538	4	55	generalized	generalized	ADJ
cana-5538	4	56	𝛼	𝛼	PRON
cana-5538	4	57	−	−	NOUN
cana-5538	4	58	𝜓	𝜓	ADP
cana-5538	4	59	contractive	contractive	ADJ
cana-5538	4	60	mappings	mapping	NOUN
cana-5538	4	61	in	in	ADP
cana-5538	4	62	the	the	DET
cana-5538	4	63	framework	framework	NOUN
cana-5538	4	64	of	of	ADP
cana-5538	4	65	partial	partial	ADJ
cana-5538	4	66	cone	cone	NOUN
cana-5538	4	67	metric	metric	ADJ
cana-5538	4	68	spaces	space	NOUN
cana-5538	4	69	.	.	PUNCT
cana-5538	5	1	in	in	ADP
cana-5538	5	2	addition	addition	NOUN
cana-5538	5	3	,	,	PUNCT
cana-5538	5	4	we	we	PRON
cana-5538	5	5	prove	prove	VERB
cana-5538	5	6	a	a	DET
cana-5538	5	7	unique	unique	ADJ
cana-5538	5	8	fixed	fix	VERB
cana-5538	5	9	point	point	NOUN
cana-5538	5	10	theorem	theorem	ADJ
cana-5538	5	11	using	use	VERB
cana-5538	5	12	a	a	DET
cana-5538	5	13	rational	rational	ADJ
cana-5538	5	14	contractive	contractive	ADJ
cana-5538	5	15	condition	condition	NOUN
cana-5538	5	16	.	.	PUNCT
cana-5538	6	1	our	our	PRON
cana-5538	6	2	findings	finding	NOUN
cana-5538	6	3	align	align	VERB
cana-5538	6	4	with	with	ADP
cana-5538	6	5	previous	previous	ADJ
cana-5538	6	6	research	research	NOUN
cana-5538	6	7	in	in	ADP
cana-5538	6	8	this	this	DET
cana-5538	6	9	area	area	NOUN
cana-5538	6	10	.	.	PUNCT
cana-5538	7	1	we	we	PRON
cana-5538	7	2	also	also	ADV
cana-5538	7	3	show	show	VERB
cana-5538	7	4	that	that	SCONJ
cana-5538	7	5	our	our	PRON
cana-5538	7	6	result	result	NOUN
cana-5538	7	7	can	can	AUX
cana-5538	7	8	be	be	AUX
cana-5538	7	9	applied	apply	VERB
cana-5538	7	10	to	to	ADP
cana-5538	7	11	the	the	DET
cana-5538	7	12	problem	problem	NOUN
cana-5538	7	13	of	of	ADP
cana-5538	7	14	determining	determine	VERB
cana-5538	7	15	the	the	DET
cana-5538	7	16	existence	existence	NOUN
cana-5538	7	17	of	of	ADP
cana-5538	7	18	solutions	solution	NOUN
cana-5538	7	19	to	to	ADP
cana-5538	7	20	second	second	ADJ
cana-5538	7	21	-	-	PUNCT
cana-5538	7	22	order	order	NOUN
cana-5538	7	23	differential	differential	ADJ
cana-5538	7	24	equations	equation	NOUN
cana-5538	7	25	.	.	PUNCT
cana-5538	8	1	keywords	keyword	NOUN
cana-5538	8	2	:	:	PUNCT
cana-5538	8	3	𝛼	𝛼	X
cana-5538	8	4	−	−	PROPN
cana-5538	8	5	𝜓	𝜓	ADP
cana-5538	8	6	contractive	contractive	ADJ
cana-5538	8	7	mappings	mapping	NOUN
cana-5538	8	8	,	,	PUNCT
cana-5538	8	9	partial	partial	ADJ
cana-5538	8	10	cone	cone	NOUN
cana-5538	8	11	metric	metric	ADJ
cana-5538	8	12	spaces	space	NOUN
cana-5538	8	13	,	,	PUNCT
cana-5538	8	14	α	α	DET
cana-5538	8	15	admissible	admissible	ADJ
cana-5538	8	16	mappings	mapping	NOUN
cana-5538	8	17	.	.	PUNCT
cana-5538	9	1	1	1	X
cana-5538	9	2	.	.	X
cana-5538	9	3	introduction	introduction	NOUN
cana-5538	9	4	the	the	DET
cana-5538	9	5	banach	banach	NOUN
cana-5538	9	6	contraction	contraction	NOUN
cana-5538	9	7	principle	principle	NOUN
cana-5538	10	1	[	[	X
cana-5538	10	2	1	1	X
cana-5538	10	3	]	]	PUNCT
cana-5538	10	4	was	be	AUX
cana-5538	10	5	a	a	DET
cana-5538	10	6	foundation	foundation	NOUN
cana-5538	10	7	for	for	ADP
cana-5538	10	8	a	a	DET
cana-5538	10	9	development	development	NOUN
cana-5538	10	10	of	of	ADP
cana-5538	10	11	metric	metric	ADJ
cana-5538	10	12	fixed	fix	VERB
cana-5538	10	13	point	point	NOUN
cana-5538	10	14	theory	theory	NOUN
cana-5538	10	15	which	which	PRON
cana-5538	10	16	has	have	AUX
cana-5538	10	17	been	be	AUX
cana-5538	10	18	generalized	generalize	VERB
cana-5538	10	19	by	by	ADP
cana-5538	10	20	utilizing	utilize	VERB
cana-5538	10	21	various	various	ADJ
cana-5538	10	22	contractive	contractive	ADJ
cana-5538	10	23	conditions	condition	NOUN
cana-5538	10	24	in	in	ADP
cana-5538	10	25	various	various	ADJ
cana-5538	10	26	contexts	contexts	NOUN
cana-5538	10	27	.	.	PUNCT
cana-5538	11	1	in	in	ADP
cana-5538	11	2	1906	1906	NUM
cana-5538	11	3	,	,	PUNCT
cana-5538	11	4	frechet	frechet	NOUN
cana-5538	12	1	[	[	X
cana-5538	12	2	2	2	X
cana-5538	12	3	]	]	PUNCT
cana-5538	12	4	introduced	introduce	VERB
cana-5538	12	5	the	the	DET
cana-5538	12	6	notion	notion	NOUN
cana-5538	12	7	of	of	ADP
cana-5538	12	8	metric	metric	ADJ
cana-5538	12	9	spaces	space	NOUN
cana-5538	12	10	.	.	PUNCT
cana-5538	13	1	in	in	ADP
cana-5538	13	2	2007	2007	NUM
cana-5538	13	3	,	,	PUNCT
cana-5538	13	4	huang	huang	PROPN
cana-5538	13	5	and	and	CCONJ
cana-5538	13	6	zhang	zhang	PROPN
cana-5538	13	7	[	[	X
cana-5538	13	8	7	7	X
cana-5538	13	9	]	]	PUNCT
cana-5538	13	10	introduced	introduce	VERB
cana-5538	13	11	the	the	DET
cana-5538	13	12	concept	concept	NOUN
cana-5538	13	13	of	of	ADP
cana-5538	13	14	cone	cone	NOUN
cana-5538	13	15	metric	metric	ADJ
cana-5538	13	16	space	space	NOUN
cana-5538	13	17	which	which	PRON
cana-5538	13	18	is	be	AUX
cana-5538	13	19	a	a	DET
cana-5538	13	20	generalization	generalization	NOUN
cana-5538	13	21	of	of	ADP
cana-5538	13	22	metric	metric	ADJ
cana-5538	13	23	space	space	NOUN
cana-5538	13	24	.	.	PUNCT
cana-5538	14	1	another	another	DET
cana-5538	14	2	generalization	generalization	NOUN
cana-5538	14	3	of	of	ADP
cana-5538	14	4	metric	metric	ADJ
cana-5538	14	5	spaces	space	NOUN
cana-5538	14	6	is	be	AUX
cana-5538	14	7	partial	partial	ADJ
cana-5538	14	8	metric	metric	ADJ
cana-5538	14	9	spaces	space	NOUN
cana-5538	14	10	which	which	PRON
cana-5538	14	11	was	be	AUX
cana-5538	14	12	introduced	introduce	VERB
cana-5538	14	13	by	by	ADP
cana-5538	14	14	matthews	matthews	PROPN
cana-5538	14	15	[	[	X
cana-5538	14	16	3	3	NUM
cana-5538	14	17	,	,	PUNCT
cana-5538	14	18	4	4	NUM
cana-5538	14	19	]	]	PUNCT
cana-5538	14	20	in	in	ADP
cana-5538	14	21	which	which	PRON
cana-5538	14	22	the	the	DET
cana-5538	14	23	self	self	NOUN
cana-5538	14	24	distance	distance	NOUN
cana-5538	14	25	need	need	AUX
cana-5538	14	26	not	not	PART
cana-5538	14	27	be	be	AUX
cana-5538	14	28	equal	equal	ADJ
cana-5538	14	29	to	to	ADP
cana-5538	14	30	zero	zero	NUM
cana-5538	14	31	and	and	CCONJ
cana-5538	14	32	proved	prove	VERB
cana-5538	14	33	the	the	DET
cana-5538	14	34	partial	partial	ADJ
cana-5538	14	35	metric	metric	ADJ
cana-5538	14	36	version	version	NOUN
cana-5538	14	37	of	of	ADP
cana-5538	14	38	banach	banach	ADV
cana-5538	14	39	fixed	fix	VERB
cana-5538	14	40	point	point	NOUN
cana-5538	14	41	theorem	theorem	VERB
cana-5538	14	42	.	.	PUNCT
cana-5538	15	1	partial	partial	ADJ
cana-5538	15	2	cone	cone	NOUN
cana-5538	15	3	metric	metric	ADJ
cana-5538	15	4	spaces	space	NOUN
cana-5538	15	5	have	have	AUX
cana-5538	15	6	been	be	AUX
cana-5538	15	7	investigated	investigate	VERB
cana-5538	15	8	by	by	ADP
cana-5538	15	9	mahlotra	mahlotra	PROPN
cana-5538	15	10	et	et	PROPN
cana-5538	15	11	al	al	PROPN
cana-5538	15	12	.	.	PUNCT
cana-5538	16	1	[	[	X
cana-5538	16	2	10	10	NUM
cana-5538	16	3	]	]	PUNCT
cana-5538	16	4	and	and	CCONJ
cana-5538	16	5	sonmez	sonmez	NOUN
cana-5538	16	6	.	.	PUNCT
cana-5538	17	1	they	they	PRON
cana-5538	17	2	proved	prove	VERB
cana-5538	17	3	some	some	DET
cana-5538	17	4	fixed	fix	VERB
cana-5538	17	5	point	point	NOUN
cana-5538	17	6	theorems	theorem	NOUN
cana-5538	17	7	in	in	ADP
cana-5538	17	8	this	this	DET
cana-5538	17	9	space	space	NOUN
cana-5538	17	10	.	.	PUNCT
cana-5538	18	1	recently	recently	ADV
cana-5538	18	2	many	many	ADJ
cana-5538	18	3	papers	paper	NOUN
cana-5538	18	4	on	on	ADP
cana-5538	18	5	cone	cone	NOUN
cana-5538	18	6	metric	metric	ADJ
cana-5538	18	7	spaces	space	NOUN
cana-5538	18	8	and	and	CCONJ
cana-5538	18	9	partial	partial	ADJ
cana-5538	18	10	cone	cone	NOUN
cana-5538	18	11	metric	metric	ADJ
cana-5538	18	12	spaces	space	NOUN
cana-5538	18	13	have	have	AUX
cana-5538	18	14	been	be	AUX
cana-5538	18	15	appeared	appear	VERB
cana-5538	18	16	e.g.	e.g.	ADV
cana-5538	18	17	see	see	VERB
cana-5538	18	18	.	.	PUNCT
cana-5538	19	1	[	[	X
cana-5538	19	2	8	8	NUM
cana-5538	19	3	,	,	PUNCT
cana-5538	19	4	9	9	NUM
cana-5538	19	5	,	,	PUNCT
cana-5538	19	6	13	13	NUM
cana-5538	19	7	,	,	PUNCT
cana-5538	19	8	14	14	NUM
cana-5538	19	9	,	,	PUNCT
cana-5538	19	10	15	15	NUM
cana-5538	19	11	,	,	PUNCT
cana-5538	19	12	16	16	NUM
cana-5538	19	13	,	,	PUNCT
cana-5538	19	14	17	17	NUM
cana-5538	19	15	,	,	PUNCT
cana-5538	19	16	18	18	NUM
cana-5538	19	17	]	]	PUNCT
cana-5538	19	18	.	.	PUNCT
cana-5538	20	1	on	on	ADP
cana-5538	20	2	the	the	DET
cana-5538	20	3	other	other	ADJ
cana-5538	20	4	hand	hand	NOUN
cana-5538	20	5	,	,	PUNCT
cana-5538	20	6	samet	samet	PROPN
cana-5538	20	7	et	et	PROPN
cana-5538	20	8	al	al	PROPN
cana-5538	20	9	.	.	PUNCT
cana-5538	21	1	[	[	X
cana-5538	21	2	5	5	NUM
cana-5538	21	3	]	]	PUNCT
cana-5538	21	4	extended	extended	ADJ
cana-5538	21	5	and	and	CCONJ
cana-5538	21	6	generalized	generalize	VERB
cana-5538	21	7	the	the	DET
cana-5538	21	8	banach	banach	NOUN
cana-5538	21	9	contraction	contraction	NOUN
cana-5538	21	10	principle	principle	NOUN
cana-5538	21	11	by	by	ADP
cana-5538	21	12	introducing	introduce	VERB
cana-5538	21	13	a	a	DET
cana-5538	21	14	new	new	ADJ
cana-5538	21	15	class	class	NOUN
cana-5538	21	16	of	of	ADP
cana-5538	21	17	contractive	contractive	ADJ
cana-5538	21	18	type	type	NOUN
cana-5538	21	19	mappings	mapping	NOUN
cana-5538	21	20	known	know	VERB
cana-5538	21	21	as	as	ADP
cana-5538	21	22	𝛼	𝛼	PRON
cana-5538	21	23	−	−	PROPN
cana-5538	21	24	𝜓	𝜓	PROPN
cana-5538	21	25	contractive	contractive	ADJ
cana-5538	21	26	type	type	NOUN
cana-5538	21	27	mappings	mapping	NOUN
cana-5538	21	28	.	.	PUNCT
cana-5538	22	1	karapinar	karapinar	NOUN
cana-5538	22	2	and	and	CCONJ
cana-5538	22	3	samet	samet	VERB
cana-5538	23	1	[	[	X
cana-5538	23	2	6	6	NUM
cana-5538	23	3	]	]	PUNCT
cana-5538	23	4	generalized	generalize	VERB
cana-5538	23	5	the	the	DET
cana-5538	23	6	𝛼	𝛼	PROPN
cana-5538	23	7	−	−	NOUN
cana-5538	23	8	𝜓	𝜓	PROPN
cana-5538	23	9	contractive	contractive	ADJ
cana-5538	23	10	type	type	NOUN
cana-5538	23	11	mappings	mapping	NOUN
cana-5538	23	12	and	and	CCONJ
cana-5538	23	13	established	establish	VERB
cana-5538	23	14	various	various	ADJ
cana-5538	23	15	fixed	fix	VERB
cana-5538	23	16	point	point	NOUN
cana-5538	23	17	theorems	theorem	NOUN
cana-5538	23	18	.	.	PUNCT
cana-5538	24	1	to	to	PART
cana-5538	24	2	begin	begin	VERB
cana-5538	24	3	,	,	PUNCT
cana-5538	24	4	we	we	PRON
cana-5538	24	5	will	will	AUX
cana-5538	24	6	define	define	VERB
cana-5538	24	7	partial	partial	ADJ
cana-5538	24	8	metric	metric	ADJ
cana-5538	24	9	spaces	space	NOUN
cana-5538	24	10	,	,	PUNCT
cana-5538	24	11	cone	cone	NOUN
cana-5538	24	12	metric	metric	ADJ
cana-5538	24	13	spaces	space	NOUN
cana-5538	24	14	,	,	PUNCT
cana-5538	24	15	and	and	CCONJ
cana-5538	24	16	partial	partial	ADJ
cana-5538	24	17	cone	cone	NOUN
cana-5538	24	18	metric	metric	ADJ
cana-5538	24	19	spaces	space	NOUN
cana-5538	24	20	as	as	ADV
cana-5538	24	21	well	well	ADV
cana-5538	24	22	as	as	ADP
cana-5538	24	23	their	their	PRON
cana-5538	24	24	properties	property	NOUN
cana-5538	24	25	:	:	PUNCT
cana-5538	24	26	definition	definition	NOUN
cana-5538	24	27	1.1	1.1	NUM
cana-5538	24	28	.	.	PUNCT
cana-5538	25	1	(	(	PUNCT
cana-5538	25	2	partial	partial	ADJ
cana-5538	25	3	metric	metric	ADJ
cana-5538	25	4	space	space	NOUN
cana-5538	25	5	)	)	PUNCT
cana-5538	25	6	a	a	DET
cana-5538	25	7	partial	partial	ADJ
cana-5538	25	8	metric	metric	NOUN
cana-5538	25	9	on	on	ADP
cana-5538	25	10	a	a	DET
cana-5538	25	11	non	non	ADJ
cana-5538	25	12	-	-	ADJ
cana-5538	25	13	empty	empty	ADJ
cana-5538	25	14	set	set	NOUN
cana-5538	25	15	x	x	PUNCT
cana-5538	25	16	is	be	AUX
cana-5538	25	17	a	a	DET
cana-5538	25	18	function	function	NOUN
cana-5538	25	19	𝜌	𝜌	ADP
cana-5538	25	20	:	:	PUNCT
cana-5538	25	21	𝑋	𝑋	NOUN
cana-5538	25	22	×	×	NOUN
cana-5538	25	23	𝑋	𝑋	PROPN
cana-5538	25	24	→	→	SYM
cana-5538	25	25	ℝ+	ℝ+	PUNCT
cana-5538	25	26	such	such	ADJ
cana-5538	25	27	that	that	DET
cana-5538	25	28	for	for	ADP
cana-5538	25	29	all	all	DET
cana-5538	25	30	𝑥	𝑥	PROPN
cana-5538	25	31	,	,	PUNCT
cana-5538	25	32	𝑦	𝑦	NOUN
cana-5538	25	33	,	,	PUNCT
cana-5538	25	34	𝑧	𝑧	DET
cana-5538	25	35	∈	∈	NOUN
cana-5538	25	36	𝑋	𝑋	NOUN
cana-5538	25	37	the	the	DET
cana-5538	25	38	following	follow	VERB
cana-5538	25	39	hold	hold	NOUN
cana-5538	25	40	1	1	NUM
cana-5538	25	41	.	.	PUNCT
cana-5538	26	1	𝑥	𝑥	NOUN
cana-5538	27	1	=	=	SYM
cana-5538	27	2	𝑦	𝑦	PROPN
cana-5538	27	3	⇔	⇔	PROPN
cana-5538	27	4	𝜌(𝑥	𝜌(𝑥	PROPN
cana-5538	27	5	,	,	PUNCT
cana-5538	27	6	𝑥	𝑥	NOUN
cana-5538	27	7	)	)	PUNCT
cana-5538	27	8	=	=	SYM
cana-5538	27	9	𝜌(𝑦	𝜌(𝑦	PROPN
cana-5538	27	10	,	,	PUNCT
cana-5538	27	11	𝑦	𝑦	NOUN
cana-5538	27	12	)	)	PUNCT
cana-5538	27	13	=	=	SYM
cana-5538	27	14	𝜌(𝑥	𝜌(𝑥	PROPN
cana-5538	27	15	,	,	PUNCT
cana-5538	27	16	𝑦	𝑦	NOUN
cana-5538	27	17	)	)	PUNCT
cana-5538	27	18	;	;	PUNCT
cana-5538	27	19	mailto:toravi.tiwari@gmail.com	mailto:toravi.tiwari@gmail.com	X
cana-5538	27	20	mailto:mshranil@gmail.com	mailto:mshranil@gmail.com	X
cana-5538	27	21	mailto:padmavati.sudha62@gmail.com	mailto:padmavati.sudha62@gmail.com	NOUN
cana-5538	27	22	communications	communication	NOUN
cana-5538	27	23	on	on	ADP
cana-5538	27	24	applied	apply	VERB
cana-5538	27	25	nonlinear	nonlinear	ADJ
cana-5538	27	26	analysis	analysis	NOUN
cana-5538	27	27	issn	issn	NOUN
cana-5538	27	28	:	:	PUNCT
cana-5538	27	29	1074	1074	NUM
cana-5538	27	30	-	-	PUNCT
cana-5538	27	31	133x	133x	NUM
cana-5538	27	32	vol	vol	VERB
cana-5538	27	33	32	32	NUM
cana-5538	27	34	no	no	NOUN
cana-5538	27	35	.	.	PUNCT
cana-5538	28	1	10s	10	NOUN
cana-5538	28	2	(	(	PUNCT
cana-5538	28	3	2025	2025	NUM
cana-5538	28	4	)	)	PUNCT
cana-5538	28	5	2613	2613	NUM
cana-5538	28	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	28	7	2	2	X
cana-5538	28	8	.	.	PUNCT
cana-5538	28	9	𝜌(𝑥	𝜌(𝑥	PROPN
cana-5538	28	10	,	,	PUNCT
cana-5538	28	11	𝑥	𝑥	NOUN
cana-5538	28	12	)	)	PUNCT
cana-5538	28	13	≤	≤	NOUN
cana-5538	28	14	𝜌(𝑥	𝜌(𝑥	PROPN
cana-5538	28	15	,	,	PUNCT
cana-5538	28	16	𝑦	𝑦	NOUN
cana-5538	28	17	)	)	PUNCT
cana-5538	28	18	;	;	PUNCT
cana-5538	28	19	3	3	X
cana-5538	28	20	.	.	X
cana-5538	28	21	𝜌(𝑥	𝜌(𝑥	PROPN
cana-5538	28	22	,	,	PUNCT
cana-5538	28	23	𝑦	𝑦	NOUN
cana-5538	28	24	)	)	PUNCT
cana-5538	28	25	=	=	SYM
cana-5538	28	26	𝜌(𝑦	𝜌(𝑦	PROPN
cana-5538	28	27	,	,	PUNCT
cana-5538	28	28	𝑥	𝑥	NOUN
cana-5538	28	29	)	)	PUNCT
cana-5538	28	30	;	;	PUNCT
cana-5538	28	31	4	4	X
cana-5538	28	32	.	.	X
cana-5538	28	33	𝜌(𝑥	𝜌(𝑥	PROPN
cana-5538	28	34	,	,	PUNCT
cana-5538	28	35	𝑦	𝑦	NOUN
cana-5538	28	36	)	)	PUNCT
cana-5538	28	37	≤	≤	NOUN
cana-5538	28	38	𝜌(𝑥	𝜌(𝑥	PROPN
cana-5538	28	39	,	,	PUNCT
cana-5538	28	40	𝑧	𝑧	NOUN
cana-5538	28	41	)	)	PUNCT
cana-5538	28	42	+	+	CCONJ
cana-5538	28	43	𝜌(𝑧	𝜌(𝑧	PROPN
cana-5538	28	44	,	,	PUNCT
cana-5538	28	45	𝑦	𝑦	NOUN
cana-5538	28	46	)	)	PUNCT
cana-5538	28	47	−	−	PROPN
cana-5538	29	1	𝜌(𝑧	𝜌(𝑧	PROPN
cana-5538	29	2	,	,	PUNCT
cana-5538	29	3	𝑧	𝑧	NOUN
cana-5538	29	4	)	)	PUNCT
cana-5538	29	5	.	.	PUNCT
cana-5538	30	1	for	for	ADP
cana-5538	30	2	all	all	DET
cana-5538	30	3	𝑥	𝑥	PROPN
cana-5538	30	4	,	,	PUNCT
cana-5538	30	5	𝑦	𝑦	NOUN
cana-5538	30	6	,	,	PUNCT
cana-5538	30	7	𝑧	𝑧	DET
cana-5538	30	8	∈	∈	PROPN
cana-5538	30	9	𝑋.	𝑋.	PROPN
cana-5538	30	10	then	then	ADV
cana-5538	30	11	the	the	DET
cana-5538	30	12	pair	pair	NOUN
cana-5538	30	13	(	(	PUNCT
cana-5538	30	14	x	x	NOUN
cana-5538	30	15	,	,	PUNCT
cana-5538	30	16	ρ	ρ	PROPN
cana-5538	30	17	)	)	PUNCT
cana-5538	30	18	is	be	AUX
cana-5538	30	19	called	call	VERB
cana-5538	30	20	a	a	DET
cana-5538	30	21	partial	partial	ADJ
cana-5538	30	22	metric	metric	ADJ
cana-5538	30	23	space	space	NOUN
cana-5538	30	24	.	.	PUNCT
cana-5538	31	1	it	it	PRON
cana-5538	31	2	is	be	AUX
cana-5538	31	3	clear	clear	ADJ
cana-5538	31	4	that	that	SCONJ
cana-5538	31	5	if	if	SCONJ
cana-5538	31	6	ρ(x	ρ(x	PROPN
cana-5538	31	7	,	,	PUNCT
cana-5538	31	8	y	y	NOUN
cana-5538	31	9	)	)	PUNCT
cana-5538	31	10	=	=	SYM
cana-5538	31	11	0	0	NUM
cana-5538	31	12	,	,	PUNCT
cana-5538	31	13	then	then	ADV
cana-5538	31	14	(	(	PUNCT
cana-5538	31	15	1	1	X
cana-5538	31	16	)	)	PUNCT
cana-5538	31	17	and	and	CCONJ
cana-5538	31	18	(	(	PUNCT
cana-5538	31	19	2	2	X
cana-5538	31	20	)	)	PUNCT
cana-5538	31	21	imply	imply	VERB
cana-5538	31	22	that	that	SCONJ
cana-5538	31	23	x	x	X
cana-5538	32	1	=	=	PUNCT
cana-5538	32	2	y.	y.	NOUN
cana-5538	32	3	but	but	CCONJ
cana-5538	32	4	if	if	SCONJ
cana-5538	32	5	x	x	PROPN
cana-5538	32	6	=	=	SYM
cana-5538	32	7	y	y	PROPN
cana-5538	32	8	,	,	PUNCT
cana-5538	32	9	ρ(x	ρ(x	PROPN
cana-5538	32	10	,	,	PUNCT
cana-5538	32	11	y	y	NOUN
cana-5538	32	12	)	)	PUNCT
cana-5538	32	13	may	may	AUX
cana-5538	32	14	not	not	PART
cana-5538	32	15	be	be	AUX
cana-5538	32	16	0	0	NUM
cana-5538	32	17	.	.	PUNCT
cana-5538	33	1	a	a	DET
cana-5538	33	2	basic	basic	ADJ
cana-5538	33	3	example	example	NOUN
cana-5538	33	4	of	of	ADP
cana-5538	33	5	partial	partial	ADJ
cana-5538	33	6	metric	metric	ADJ
cana-5538	33	7	space	space	NOUN
cana-5538	33	8	is	be	AUX
cana-5538	33	9	the	the	DET
cana-5538	33	10	pair	pair	NOUN
cana-5538	33	11	(	(	PUNCT
cana-5538	33	12	ℝ+	ℝ+	NOUN
cana-5538	33	13	,	,	PUNCT
cana-5538	33	14	𝜌	𝜌	NOUN
cana-5538	33	15	)	)	PUNCT
cana-5538	33	16	where	where	SCONJ
cana-5538	33	17	𝜌(𝑥	𝜌(𝑥	PROPN
cana-5538	33	18	,	,	PUNCT
cana-5538	33	19	𝑦	𝑦	NOUN
cana-5538	33	20	)	)	PUNCT
cana-5538	33	21	=	=	SYM
cana-5538	33	22	𝑚𝑎𝑥{𝑥	𝑚𝑎𝑥{𝑥	X
cana-5538	33	23	,	,	PUNCT
cana-5538	33	24	𝑦	𝑦	NOUN
cana-5538	33	25	}	}	PUNCT
cana-5538	33	26	for	for	ADP
cana-5538	33	27	all	all	PRON
cana-5538	33	28	𝑥	𝑥	PROPN
cana-5538	33	29	,	,	PUNCT
cana-5538	33	30	𝑦	𝑦	NOUN
cana-5538	33	31	∈	∈	NOUN
cana-5538	33	32	ℝ+	ℝ+	PUNCT
cana-5538	33	33	.	.	PUNCT
cana-5538	34	1	let	let	VERB
cana-5538	34	2	e	e	PRON
cana-5538	34	3	be	be	AUX
cana-5538	34	4	a	a	DET
cana-5538	34	5	real	real	ADJ
cana-5538	34	6	banach	banach	NOUN
cana-5538	34	7	space	space	NOUN
cana-5538	34	8	and	and	CCONJ
cana-5538	34	9	p	p	X
cana-5538	34	10	a	a	DET
cana-5538	34	11	subset	subset	NOUN
cana-5538	34	12	of	of	ADP
cana-5538	34	13	e.	e.	PROPN
cana-5538	34	14	p	p	PROPN
cana-5538	34	15	is	be	AUX
cana-5538	34	16	called	call	VERB
cana-5538	34	17	a	a	DET
cana-5538	34	18	cone	cone	NOUN
cana-5538	34	19	if	if	SCONJ
cana-5538	34	20	it	it	PRON
cana-5538	34	21	satisfies	satisfy	VERB
cana-5538	34	22	the	the	DET
cana-5538	34	23	following	following	NOUN
cana-5538	34	24	.	.	PUNCT
cana-5538	35	1	(	(	PUNCT
cana-5538	35	2	1	1	X
cana-5538	35	3	)	)	PUNCT
cana-5538	35	4	p	p	NOUN
cana-5538	35	5	is	be	AUX
cana-5538	35	6	closed	closed	ADJ
cana-5538	35	7	,	,	PUNCT
cana-5538	35	8	non	non	ADJ
cana-5538	35	9	-	-	ADJ
cana-5538	35	10	empty	empty	ADJ
cana-5538	35	11	,	,	PUNCT
cana-5538	35	12	and	and	CCONJ
cana-5538	35	13	𝑃	𝑃	VERB
cana-5538	35	14	≠	≠	PROPN
cana-5538	35	15	0	0	NUM
cana-5538	35	16	,	,	PUNCT
cana-5538	35	17	(	(	PUNCT
cana-5538	35	18	2	2	X
cana-5538	36	1	)	)	PUNCT
cana-5538	36	2	𝑎𝑥	𝑎𝑥	NOUN
cana-5538	37	1	+	+	CCONJ
cana-5538	37	2	𝑏𝑦	𝑏𝑦	NOUN
cana-5538	37	3	∈	∈	NOUN
cana-5538	37	4	𝑃	𝑃	NOUN
cana-5538	37	5	for	for	ADP
cana-5538	37	6	all	all	DET
cana-5538	37	7	𝑥	𝑥	PROPN
cana-5538	37	8	,	,	PUNCT
cana-5538	37	9	𝑦	𝑦	PRON
cana-5538	37	10	∈	∈	NOUN
cana-5538	37	11	𝑃	𝑃	NOUN
cana-5538	37	12	and	and	CCONJ
cana-5538	37	13	non	non	ADJ
cana-5538	37	14	-	-	ADJ
cana-5538	37	15	negative	negative	ADJ
cana-5538	37	16	real	real	ADJ
cana-5538	37	17	numbers	number	NOUN
cana-5538	37	18	𝑎	𝑎	ADP
cana-5538	37	19	,	,	PUNCT
cana-5538	37	20	𝑏	𝑏	PROPN
cana-5538	37	21	∈	∈	PROPN
cana-5538	37	22	ℝ	ℝ	PROPN
cana-5538	37	23	,	,	PUNCT
cana-5538	37	24	(	(	PUNCT
cana-5538	37	25	3	3	X
cana-5538	37	26	)	)	PUNCT
cana-5538	37	27	𝑃	𝑃	NOUN
cana-5538	37	28	∩	∩	NOUN
cana-5538	37	29	(	(	PUNCT
cana-5538	37	30	−𝑃	−𝑃	NOUN
cana-5538	37	31	)	)	PUNCT
cana-5538	37	32	=	=	PUNCT
cana-5538	37	33	{	{	PUNCT
cana-5538	37	34	0	0	NUM
cana-5538	37	35	}	}	PUNCT
cana-5538	37	36	.	.	PUNCT
cana-5538	38	1	for	for	ADP
cana-5538	38	2	a	a	DET
cana-5538	38	3	specified	specify	VERB
cana-5538	38	4	cone	cone	NOUN
cana-5538	38	5	𝑃	𝑃	PROPN
cana-5538	38	6	⊂	⊂	PROPN
cana-5538	38	7	𝐸	𝐸	PROPN
cana-5538	38	8	,	,	PUNCT
cana-5538	38	9	we	we	PRON
cana-5538	38	10	can	can	AUX
cana-5538	38	11	establish	establish	VERB
cana-5538	38	12	a	a	DET
cana-5538	38	13	partial	partial	ADJ
cana-5538	38	14	ordering	ordering	NOUN
cana-5538	38	15	≤	≤	NOUN
cana-5538	38	16	on	on	ADP
cana-5538	38	17	e	e	PROPN
cana-5538	38	18	in	in	ADP
cana-5538	38	19	relation	relation	NOUN
cana-5538	38	20	to	to	ADP
cana-5538	38	21	p	p	NOUN
cana-5538	38	22	by	by	ADP
cana-5538	38	23	defining	define	VERB
cana-5538	38	24	𝑥	𝑥	DET
cana-5538	38	25	≤	≤	NUM
cana-5538	38	26	𝑦	𝑦	NOUN
cana-5538	38	27	if	if	SCONJ
cana-5538	39	1	and	and	CCONJ
cana-5538	39	2	only	only	ADV
cana-5538	39	3	if	if	SCONJ
cana-5538	39	4	𝑦	𝑦	NUM
cana-5538	39	5	−	−	X
cana-5538	39	6	𝑥	𝑥	PRON
cana-5538	39	7	∈	∈	NOUN
cana-5538	39	8	𝑃	𝑃	NOUN
cana-5538	39	9	.	.	PUNCT
cana-5538	40	1	the	the	DET
cana-5538	40	2	notation	notation	NOUN
cana-5538	40	3	𝑥	𝑥	ADP
cana-5538	40	4	<	<	X
cana-5538	40	5	𝑦	𝑦	NOUN
cana-5538	40	6	is	be	AUX
cana-5538	40	7	used	use	VERB
cana-5538	40	8	to	to	PART
cana-5538	40	9	signify	signify	VERB
cana-5538	40	10	that	that	SCONJ
cana-5538	40	11	𝑥	𝑥	PROPN
cana-5538	40	12	≤	≤	NUM
cana-5538	40	13	𝑦	𝑦	NOUN
cana-5538	40	14	and	and	CCONJ
cana-5538	40	15	𝑥	𝑥	NOUN
cana-5538	40	16	≠	≠	PROPN
cana-5538	40	17	𝑦	𝑦	NOUN
cana-5538	40	18	,	,	PUNCT
cana-5538	40	19	while	while	SCONJ
cana-5538	40	20	𝑥	𝑥	DET
cana-5538	40	21	≪	≪	X
cana-5538	40	22	𝑦	𝑦	PRON
cana-5538	40	23	indicates	indicate	VERB
cana-5538	40	24	that	that	SCONJ
cana-5538	40	25	𝑦	𝑦	NOUN
cana-5538	40	26	−	−	NOUN
cana-5538	40	27	𝑥	𝑥	PRON
cana-5538	40	28	∈	∈	NOUN
cana-5538	40	29	𝑖𝑛𝑡𝑃	𝑖𝑛𝑡𝑃	NOUN
cana-5538	40	30	,	,	PUNCT
cana-5538	40	31	with	with	ADP
cana-5538	40	32	𝑖𝑛𝑡𝑃	𝑖𝑛𝑡𝑃	NOUN
cana-5538	40	33	representing	represent	VERB
cana-5538	40	34	the	the	DET
cana-5538	40	35	interior	interior	NOUN
cana-5538	40	36	of	of	ADP
cana-5538	40	37	p	p	PROPN
cana-5538	40	38	.	.	PUNCT
cana-5538	41	1	the	the	DET
cana-5538	41	2	cone	cone	NOUN
cana-5538	41	3	p	p	NOUN
cana-5538	41	4	is	be	AUX
cana-5538	41	5	termed	term	VERB
cana-5538	41	6	normal	normal	ADJ
cana-5538	41	7	if	if	SCONJ
cana-5538	41	8	there	there	PRON
cana-5538	41	9	exists	exist	VERB
cana-5538	41	10	a	a	DET
cana-5538	41	11	constant	constant	ADJ
cana-5538	41	12	k	k	X
cana-5538	41	13	>	>	X
cana-5538	41	14	0	0	NUM
cana-5538	41	15	such	such	ADJ
cana-5538	41	16	that	that	PRON
cana-5538	41	17	for	for	ADP
cana-5538	41	18	all	all	DET
cana-5538	41	19	x	x	NOUN
cana-5538	41	20	,	,	PUNCT
cana-5538	41	21	y	y	PROPN
cana-5538	41	22	∈	∈	PROPN
cana-5538	41	23	e	e	X
cana-5538	41	24	where	where	SCONJ
cana-5538	41	25	0	0	NUM
cana-5538	41	26	≤	≤	NUM
cana-5538	41	27	x	x	PUNCT
cana-5538	41	28	≤	≤	NUM
cana-5538	41	29	y	y	NOUN
cana-5538	41	30	,	,	PUNCT
cana-5538	41	31	it	it	PRON
cana-5538	41	32	follows	follow	VERB
cana-5538	41	33	that	that	SCONJ
cana-5538	41	34	||x||	||x||	ADV
cana-5538	41	35	≤	≤	NUM
cana-5538	41	36	k||y||	k||y||	NOUN
cana-5538	41	37	.	.	PUNCT
cana-5538	42	1	the	the	DET
cana-5538	42	2	smallest	small	ADJ
cana-5538	42	3	positive	positive	ADJ
cana-5538	42	4	value	value	NOUN
cana-5538	42	5	that	that	PRON
cana-5538	42	6	satisfies	satisfy	VERB
cana-5538	42	7	this	this	DET
cana-5538	42	8	condition	condition	NOUN
cana-5538	42	9	is	be	AUX
cana-5538	42	10	referred	refer	VERB
cana-5538	42	11	to	to	ADP
cana-5538	42	12	as	as	SCONJ
cana-5538	42	13	the	the	DET
cana-5538	42	14	normal	normal	ADJ
cana-5538	42	15	constant	constant	NOUN
cana-5538	42	16	of	of	ADP
cana-5538	42	17	p.	p.	NOUN
cana-5538	42	18	let	let	VERB
cana-5538	42	19	e	e	PRON
cana-5538	42	20	be	be	AUX
cana-5538	42	21	a	a	DET
cana-5538	42	22	banach	banach	NOUN
cana-5538	42	23	space	space	NOUN
cana-5538	42	24	,	,	PUNCT
cana-5538	42	25	p	p	X
cana-5538	42	26	a	a	DET
cana-5538	42	27	cone	cone	NOUN
cana-5538	42	28	in	in	ADP
cana-5538	42	29	e	e	NOUN
cana-5538	42	30	with	with	ADP
cana-5538	42	31	𝑖𝑛𝑡𝑃	𝑖𝑛𝑡𝑃	ADJ
cana-5538	42	32	≠	≠	ADJ
cana-5538	42	33	𝜙	𝜙	NOUN
cana-5538	42	34	and	and	CCONJ
cana-5538	42	35	≤	≤	NUM
cana-5538	42	36	is	be	AUX
cana-5538	42	37	partial	partial	ADJ
cana-5538	42	38	ordering	ordering	NOUN
cana-5538	42	39	with	with	ADP
cana-5538	42	40	respect	respect	NOUN
cana-5538	42	41	to	to	ADP
cana-5538	42	42	p.	p.	NOUN
cana-5538	42	43	definition	definition	NOUN
cana-5538	42	44	1.2	1.2	NUM
cana-5538	42	45	(	(	PUNCT
cana-5538	42	46	cone	cone	NOUN
cana-5538	42	47	metric	metric	ADJ
cana-5538	42	48	space	space	NOUN
cana-5538	42	49	)	)	PUNCT
cana-5538	42	50	let	let	VERB
cana-5538	42	51	x	x	PRON
cana-5538	42	52	be	be	AUX
cana-5538	42	53	a	a	DET
cana-5538	42	54	non	non	X
cana-5538	42	55	empty	empty	ADJ
cana-5538	42	56	set	set	NOUN
cana-5538	42	57	.	.	PUNCT
cana-5538	43	1	the	the	DET
cana-5538	43	2	mapping	mapping	NOUN
cana-5538	43	3	𝑑𝑐	𝑑𝑐	ADP
cana-5538	43	4	∶	∶	NOUN
cana-5538	43	5	𝑋	𝑋	NOUN
cana-5538	43	6	×	×	NOUN
cana-5538	43	7	𝑋	𝑋	PROPN
cana-5538	43	8	→	→	SYM
cana-5538	43	9	𝐸	𝐸	PROPN
cana-5538	43	10	is	be	AUX
cana-5538	43	11	said	say	VERB
cana-5538	43	12	to	to	PART
cana-5538	43	13	be	be	AUX
cana-5538	43	14	a	a	DET
cana-5538	43	15	cone	cone	NOUN
cana-5538	43	16	metric	metric	NOUN
cana-5538	43	17	on	on	ADP
cana-5538	43	18	x	x	SYM
cana-5538	43	19	if	if	SCONJ
cana-5538	43	20	for	for	ADP
cana-5538	43	21	all	all	PRON
cana-5538	43	22	𝑥	𝑥	PROPN
cana-5538	43	23	,	,	PUNCT
cana-5538	43	24	𝑦	𝑦	NOUN
cana-5538	43	25	,	,	PUNCT
cana-5538	43	26	𝑧	𝑧	DET
cana-5538	43	27	∈	∈	PROPN
cana-5538	43	28	𝑋.	𝑋.	PROPN
cana-5538	43	29	the	the	DET
cana-5538	43	30	followings	following	NOUN
cana-5538	43	31	hold	hold	VERB
cana-5538	43	32	:	:	PUNCT
cana-5538	43	33	(	(	PUNCT
cana-5538	43	34	1	1	X
cana-5538	43	35	)	)	PUNCT
cana-5538	43	36	0	0	NUM
cana-5538	43	37	≤	≤	NOUN
cana-5538	43	38	𝑑𝑐(𝑥	𝑑𝑐(𝑥	ADP
cana-5538	43	39	,	,	PUNCT
cana-5538	43	40	𝑦	𝑦	NOUN
cana-5538	43	41	)	)	PUNCT
cana-5538	43	42	and	and	CCONJ
cana-5538	43	43	𝑑𝑐(𝑥	𝑑𝑐(𝑥	ADJ
cana-5538	43	44	,	,	PUNCT
cana-5538	43	45	𝑦	𝑦	NOUN
cana-5538	43	46	)	)	PUNCT
cana-5538	43	47	=	=	SYM
cana-5538	43	48	0	0	PUNCT
cana-5538	44	1	if	if	SCONJ
cana-5538	44	2	and	and	CCONJ
cana-5538	44	3	only	only	ADV
cana-5538	44	4	if	if	SCONJ
cana-5538	44	5	𝑥	𝑥	PRON
cana-5538	44	6	=	=	SYM
cana-5538	44	7	𝑦	𝑦	NOUN
cana-5538	44	8	,	,	PUNCT
cana-5538	44	9	(	(	PUNCT
cana-5538	44	10	2	2	NUM
cana-5538	44	11	)	)	PUNCT
cana-5538	44	12	𝑑𝑐(𝑥	𝑑𝑐(𝑥	ADJ
cana-5538	44	13	,	,	PUNCT
cana-5538	44	14	𝑦	𝑦	NOUN
cana-5538	44	15	)	)	PUNCT
cana-5538	44	16	=	=	SYM
cana-5538	44	17	𝑑𝑐(𝑦	𝑑𝑐(𝑦	NUM
cana-5538	44	18	,	,	PUNCT
cana-5538	44	19	𝑥	𝑥	NOUN
cana-5538	44	20	)	)	PUNCT
cana-5538	44	21	,	,	PUNCT
cana-5538	44	22	(	(	PUNCT
cana-5538	44	23	3	3	X
cana-5538	44	24	)	)	PUNCT
cana-5538	44	25	𝑑𝑐(𝑥	𝑑𝑐(𝑥	ADJ
cana-5538	44	26	,	,	PUNCT
cana-5538	44	27	𝑦	𝑦	NOUN
cana-5538	44	28	)	)	PUNCT
cana-5538	44	29	≤	≤	NOUN
cana-5538	44	30	𝑑𝑐(𝑥	𝑑𝑐(𝑥	NOUN
cana-5538	44	31	,	,	PUNCT
cana-5538	44	32	𝑧	𝑧	NOUN
cana-5538	44	33	)	)	PUNCT
cana-5538	44	34	+	+	NUM
cana-5538	44	35	𝑑𝑐(𝑦	𝑑𝑐(𝑦	NOUN
cana-5538	44	36	,	,	PUNCT
cana-5538	44	37	𝑧	𝑧	NOUN
cana-5538	44	38	)	)	PUNCT
cana-5538	44	39	.	.	PUNCT
cana-5538	45	1	and	and	CCONJ
cana-5538	45	2	(	(	PUNCT
cana-5538	45	3	x	x	X
cana-5538	45	4	,	,	PUNCT
cana-5538	45	5	dc	dc	PROPN
cana-5538	45	6	)	)	PUNCT
cana-5538	45	7	is	be	AUX
cana-5538	45	8	called	call	VERB
cana-5538	45	9	a	a	DET
cana-5538	45	10	cone	cone	NOUN
cana-5538	45	11	metric	metric	ADJ
cana-5538	45	12	space	space	NOUN
cana-5538	45	13	.	.	PUNCT
cana-5538	46	1	mahlotra	mahlotra	PROPN
cana-5538	46	2	et	et	PROPN
cana-5538	46	3	al	al	PROPN
cana-5538	46	4	.	.	PUNCT
cana-5538	47	1	[	[	X
cana-5538	47	2	10	10	NUM
cana-5538	47	3	]	]	PUNCT
cana-5538	47	4	and	and	CCONJ
cana-5538	47	5	sonmez	sonmez	NOUN
cana-5538	48	1	[	[	X
cana-5538	48	2	11	11	NUM
cana-5538	48	3	]	]	PUNCT
cana-5538	48	4	introduced	introduce	VERB
cana-5538	48	5	the	the	DET
cana-5538	48	6	notion	notion	NOUN
cana-5538	48	7	of	of	ADP
cana-5538	48	8	partial	partial	ADJ
cana-5538	48	9	cone	cone	NOUN
cana-5538	48	10	metric	metric	ADJ
cana-5538	48	11	space	space	NOUN
cana-5538	48	12	and	and	CCONJ
cana-5538	48	13	its	its	PRON
cana-5538	48	14	topological	topological	ADJ
cana-5538	48	15	characterization	characterization	NOUN
cana-5538	48	16	.	.	PUNCT
cana-5538	49	1	we	we	PRON
cana-5538	49	2	now	now	ADV
cana-5538	49	3	state	state	VERB
cana-5538	49	4	the	the	DET
cana-5538	49	5	definition	definition	NOUN
cana-5538	49	6	of	of	ADP
cana-5538	49	7	partial	partial	ADJ
cana-5538	49	8	cone	cone	NOUN
cana-5538	49	9	metric	metric	ADJ
cana-5538	49	10	space	space	NOUN
cana-5538	49	11	.	.	PUNCT
cana-5538	50	1	definition	definition	NOUN
cana-5538	50	2	1.3	1.3	NUM
cana-5538	50	3	(	(	PUNCT
cana-5538	50	4	partial	partial	ADJ
cana-5538	50	5	cone	cone	NOUN
cana-5538	50	6	metric	metric	ADJ
cana-5538	50	7	space	space	NOUN
cana-5538	50	8	)	)	PUNCT
cana-5538	50	9	a	a	DET
cana-5538	50	10	partial	partial	ADJ
cana-5538	50	11	cone	cone	NOUN
cana-5538	50	12	metric	metric	NOUN
cana-5538	50	13	on	on	ADP
cana-5538	50	14	a	a	DET
cana-5538	50	15	non	non	ADJ
cana-5538	50	16	-	-	ADJ
cana-5538	50	17	empty	empty	ADJ
cana-5538	50	18	set	set	NOUN
cana-5538	50	19	x	x	PUNCT
cana-5538	50	20	is	be	AUX
cana-5538	50	21	a	a	DET
cana-5538	50	22	function	function	NOUN
cana-5538	50	23	𝜌𝑐	𝜌𝑐	NOUN
cana-5538	50	24	:	:	PUNCT
cana-5538	50	25	𝑋	𝑋	PROPN
cana-5538	50	26	×	×	NOUN
cana-5538	50	27	𝑋	𝑋	PROPN
cana-5538	50	28	→	→	SYM
cana-5538	50	29	𝐸	𝐸	PROPN
cana-5538	50	30	such	such	ADJ
cana-5538	50	31	that	that	PRON
cana-5538	50	32	for	for	ADP
cana-5538	50	33	all	all	DET
cana-5538	50	34	𝑥	𝑥	PROPN
cana-5538	50	35	,	,	PUNCT
cana-5538	50	36	𝑦	𝑦	NOUN
cana-5538	50	37	,	,	PUNCT
cana-5538	50	38	𝑧	𝑧	DET
cana-5538	50	39	∈	∈	PROPN
cana-5538	50	40	𝑋	𝑋	NOUN
cana-5538	50	41	(	(	PUNCT
cana-5538	50	42	1	1	NUM
cana-5538	50	43	)	)	PUNCT
cana-5538	50	44	0	0	NUM
cana-5538	50	45	≤	≤	NOUN
cana-5538	50	46	𝜌𝑐(𝑥	𝜌𝑐(𝑥	NUM
cana-5538	50	47	,	,	PUNCT
cana-5538	50	48	𝑥	𝑥	NOUN
cana-5538	50	49	)	)	PUNCT
cana-5538	50	50	≤	≤	NOUN
cana-5538	50	51	𝜌𝑐(𝑥	𝜌𝑐(𝑥	NUM
cana-5538	50	52	,	,	PUNCT
cana-5538	50	53	𝑦	𝑦	NOUN
cana-5538	50	54	)	)	PUNCT
cana-5538	50	55	,	,	PUNCT
cana-5538	50	56	(	(	PUNCT
cana-5538	50	57	2	2	X
cana-5538	50	58	)	)	PUNCT
cana-5538	50	59	𝑥	𝑥	NOUN
cana-5538	50	60	=	=	PUNCT
cana-5538	50	61	𝑦	𝑦	NOUN
cana-5538	51	1	if	if	SCONJ
cana-5538	51	2	and	and	CCONJ
cana-5538	51	3	only	only	ADV
cana-5538	51	4	if	if	SCONJ
cana-5538	51	5	𝜌𝑐(𝑥	𝜌𝑐(𝑥	ADJ
cana-5538	51	6	,	,	PUNCT
cana-5538	51	7	𝑥	𝑥	NOUN
cana-5538	51	8	)	)	PUNCT
cana-5538	51	9	=	=	SYM
cana-5538	52	1	𝜌𝑐(𝑥	𝜌𝑐(𝑥	ADJ
cana-5538	52	2	,	,	PUNCT
cana-5538	52	3	𝑦	𝑦	NOUN
cana-5538	52	4	)	)	PUNCT
cana-5538	52	5	=	=	SYM
cana-5538	52	6	𝜌𝑐(𝑦	𝜌𝑐(𝑦	X
cana-5538	52	7	,	,	PUNCT
cana-5538	52	8	𝑦	𝑦	NOUN
cana-5538	52	9	)	)	PUNCT
cana-5538	52	10	,	,	PUNCT
cana-5538	52	11	communications	communication	NOUN
cana-5538	52	12	on	on	ADP
cana-5538	52	13	applied	apply	VERB
cana-5538	52	14	nonlinear	nonlinear	ADJ
cana-5538	52	15	analysis	analysis	NOUN
cana-5538	52	16	issn	issn	NOUN
cana-5538	52	17	:	:	PUNCT
cana-5538	52	18	1074	1074	NUM
cana-5538	52	19	-	-	PUNCT
cana-5538	52	20	133x	133x	NUM
cana-5538	52	21	vol	vol	VERB
cana-5538	52	22	32	32	NUM
cana-5538	52	23	no	no	NOUN
cana-5538	52	24	.	.	PUNCT
cana-5538	53	1	10s	10	NOUN
cana-5538	53	2	(	(	PUNCT
cana-5538	53	3	2025	2025	NUM
cana-5538	53	4	)	)	PUNCT
cana-5538	53	5	2614	2614	NUM
cana-5538	53	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	53	7	(	(	PUNCT
cana-5538	53	8	3	3	NUM
cana-5538	53	9	)	)	PUNCT
cana-5538	53	10	𝜌𝑐(𝑥	𝜌𝑐(𝑥	ADJ
cana-5538	53	11	,	,	PUNCT
cana-5538	53	12	𝑦	𝑦	X
cana-5538	53	13	)	)	PUNCT
cana-5538	53	14	=	=	SYM
cana-5538	53	15	𝜌𝑐(𝑦	𝜌𝑐(𝑦	NOUN
cana-5538	53	16	,	,	PUNCT
cana-5538	53	17	𝑥	𝑥	NOUN
cana-5538	53	18	)	)	PUNCT
cana-5538	53	19	,	,	PUNCT
cana-5538	53	20	(	(	PUNCT
cana-5538	53	21	4	4	X
cana-5538	53	22	)	)	PUNCT
cana-5538	53	23	𝜌𝑐(𝑥	𝜌𝑐(𝑥	ADJ
cana-5538	53	24	,	,	PUNCT
cana-5538	53	25	𝑦	𝑦	NOUN
cana-5538	53	26	)	)	PUNCT
cana-5538	53	27	≤	≤	NOUN
cana-5538	53	28	𝜌𝑐(𝑥	𝜌𝑐(𝑥	NUM
cana-5538	53	29	,	,	PUNCT
cana-5538	53	30	𝑧	𝑧	NOUN
cana-5538	53	31	)	)	PUNCT
cana-5538	53	32	+	+	NUM
cana-5538	53	33	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	53	34	,	,	PUNCT
cana-5538	53	35	𝑦	𝑦	NOUN
cana-5538	53	36	)	)	PUNCT
cana-5538	53	37	−	−	NOUN
cana-5538	53	38	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	53	39	,	,	PUNCT
cana-5538	53	40	𝑧	𝑧	NOUN
cana-5538	53	41	)	)	PUNCT
cana-5538	53	42	.	.	PUNCT
cana-5538	54	1	a	a	DET
cana-5538	54	2	partial	partial	ADJ
cana-5538	54	3	cone	cone	NOUN
cana-5538	54	4	metric	metric	ADJ
cana-5538	54	5	space	space	NOUN
cana-5538	54	6	is	be	AUX
cana-5538	54	7	a	a	DET
cana-5538	54	8	pair	pair	NOUN
cana-5538	54	9	(	(	PUNCT
cana-5538	54	10	𝑋	𝑋	PROPN
cana-5538	54	11	,	,	PUNCT
cana-5538	54	12	𝜌𝑐	𝜌𝑐	NOUN
cana-5538	54	13	)	)	PUNCT
cana-5538	54	14	such	such	ADJ
cana-5538	54	15	that	that	SCONJ
cana-5538	54	16	x	x	PRON
cana-5538	54	17	is	be	AUX
cana-5538	54	18	a	a	DET
cana-5538	54	19	non	non	ADJ
cana-5538	54	20	-	-	ADJ
cana-5538	54	21	empty	empty	ADJ
cana-5538	54	22	set	set	NOUN
cana-5538	54	23	and	and	CCONJ
cana-5538	54	24	𝜌𝑐	𝜌𝑐	PRON
cana-5538	54	25	is	be	AUX
cana-5538	54	26	a	a	DET
cana-5538	54	27	partial	partial	ADJ
cana-5538	54	28	cone	cone	NOUN
cana-5538	54	29	metric	metric	NOUN
cana-5538	54	30	on	on	ADP
cana-5538	54	31	x	x	X
cana-5538	54	32	.	.	PUNCT
cana-5538	55	1	it	it	PRON
cana-5538	55	2	is	be	AUX
cana-5538	55	3	clear	clear	ADJ
cana-5538	55	4	that	that	SCONJ
cana-5538	55	5	,	,	PUNCT
cana-5538	55	6	if	if	SCONJ
cana-5538	55	7	𝜌𝑐(𝑥	𝜌𝑐(𝑥	ADP
cana-5538	55	8	,	,	PUNCT
cana-5538	55	9	𝑦	𝑦	NOUN
cana-5538	55	10	)	)	PUNCT
cana-5538	55	11	=	=	SYM
cana-5538	55	12	0	0	NUM
cana-5538	55	13	,	,	PUNCT
cana-5538	55	14	then	then	ADV
cana-5538	55	15	(	(	PUNCT
cana-5538	55	16	1	1	X
cana-5538	55	17	)	)	PUNCT
cana-5538	55	18	and	and	CCONJ
cana-5538	55	19	(	(	PUNCT
cana-5538	55	20	2	2	X
cana-5538	55	21	)	)	PUNCT
cana-5538	55	22	imply	imply	VERB
cana-5538	55	23	that	that	SCONJ
cana-5538	55	24	x	x	X
cana-5538	55	25	=	=	PUNCT
cana-5538	55	26	y.	y.	NOUN
cana-5538	55	27	but	but	CCONJ
cana-5538	55	28	the	the	DET
cana-5538	55	29	converse	converse	NOUN
cana-5538	55	30	is	be	AUX
cana-5538	55	31	not	not	PART
cana-5538	55	32	true	true	ADJ
cana-5538	55	33	in	in	ADP
cana-5538	55	34	general	general	ADJ
cana-5538	55	35	.	.	PUNCT
cana-5538	56	1	a	a	DET
cana-5538	56	2	cone	cone	NOUN
cana-5538	56	3	metric	metric	ADJ
cana-5538	56	4	space	space	NOUN
cana-5538	56	5	is	be	AUX
cana-5538	56	6	a	a	DET
cana-5538	56	7	partial	partial	ADJ
cana-5538	56	8	cone	cone	NOUN
cana-5538	56	9	metric	metric	ADJ
cana-5538	56	10	space	space	NOUN
cana-5538	56	11	,	,	PUNCT
cana-5538	56	12	but	but	CCONJ
cana-5538	56	13	there	there	PRON
cana-5538	56	14	exist	exist	VERB
cana-5538	56	15	partial	partial	ADJ
cana-5538	56	16	cone	cone	NOUN
cana-5538	56	17	metric	metric	ADJ
cana-5538	56	18	spaces	space	NOUN
cana-5538	56	19	which	which	PRON
cana-5538	56	20	are	be	AUX
cana-5538	56	21	not	not	PART
cana-5538	56	22	cone	cone	NOUN
cana-5538	56	23	metric	metric	ADJ
cana-5538	56	24	spaces	space	NOUN
cana-5538	56	25	.	.	PUNCT
cana-5538	57	1	we	we	PRON
cana-5538	57	2	give	give	VERB
cana-5538	57	3	the	the	DET
cana-5538	57	4	following	follow	VERB
cana-5538	57	5	example	example	NOUN
cana-5538	57	6	from	from	ADP
cana-5538	57	7	[	[	X
cana-5538	57	8	11	11	NUM
cana-5538	57	9	]	]	PUNCT
cana-5538	57	10	example	example	NOUN
cana-5538	57	11	1.4	1.4	NUM
cana-5538	57	12	consider	consider	VERB
cana-5538	57	13	a	a	DET
cana-5538	57	14	banach	banach	NOUN
cana-5538	57	15	space	space	NOUN
cana-5538	57	16	𝐸	𝐸	NOUN
cana-5538	57	17	=	=	SYM
cana-5538	57	18	ℝ2	ℝ2	PROPN
cana-5538	57	19	,	,	PUNCT
cana-5538	57	20	𝑃	𝑃	NOUN
cana-5538	57	21	=	=	SYM
cana-5538	57	22	{	{	PUNCT
cana-5538	57	23	(	(	PUNCT
cana-5538	57	24	𝑥	𝑥	NOUN
cana-5538	57	25	,	,	PUNCT
cana-5538	57	26	𝑦	𝑦	NOUN
cana-5538	57	27	)	)	PUNCT
cana-5538	57	28	∈	∈	PROPN
cana-5538	57	29	𝐸	𝐸	PROPN
cana-5538	57	30	∶	∶	NOUN
cana-5538	57	31	𝑥	𝑥	PROPN
cana-5538	57	32	,	,	PUNCT
cana-5538	57	33	𝑦	𝑦	PRON
cana-5538	57	34	≥	≥	NOUN
cana-5538	57	35	0	0	NUM
cana-5538	57	36	}	}	PUNCT
cana-5538	57	37	and	and	CCONJ
cana-5538	57	38	𝑋	𝑋	PROPN
cana-5538	57	39	=	=	PUNCT
cana-5538	57	40	ℝ+	ℝ+	PUNCT
cana-5538	57	41	and	and	CCONJ
cana-5538	57	42	𝜌𝑐	𝜌𝑐	PRON
cana-5538	57	43	∶	∶	NOUN
cana-5538	57	44	𝑋	𝑋	PROPN
cana-5538	57	45	×	×	NOUN
cana-5538	57	46	𝑋	𝑋	PROPN
cana-5538	57	47	→	→	SYM
cana-5538	57	48	𝐸	𝐸	PROPN
cana-5538	57	49	defined	define	VERB
cana-5538	57	50	𝑏𝑦	𝑏𝑦	NOUN
cana-5538	57	51	𝜌𝑐(𝑥	𝜌𝑐(𝑥	PROPN
cana-5538	57	52	,	,	PUNCT
cana-5538	57	53	𝑦	𝑦	NOUN
cana-5538	57	54	)	)	PUNCT
cana-5538	57	55	=	=	SYM
cana-5538	57	56	(	(	PUNCT
cana-5538	57	57	𝑚𝑎𝑥{𝑥	𝑚𝑎𝑥{𝑥	X
cana-5538	57	58	,	,	PUNCT
cana-5538	57	59	𝑦	𝑦	NOUN
cana-5538	57	60	}	}	PUNCT
cana-5538	57	61	,	,	PUNCT
cana-5538	57	62	𝑘𝑚𝑎𝑥{𝑥	𝑘𝑚𝑎𝑥{𝑥	PROPN
cana-5538	57	63	,	,	PUNCT
cana-5538	57	64	𝑦	𝑦	NOUN
cana-5538	57	65	}	}	PUNCT
cana-5538	57	66	)	)	PUNCT
cana-5538	57	67	where	where	SCONJ
cana-5538	57	68	𝑘	𝑘	PRON
cana-5538	57	69	≥	≥	NOUN
cana-5538	57	70	0	0	NUM
cana-5538	57	71	is	be	AUX
cana-5538	57	72	a	a	DET
cana-5538	57	73	constant	constant	ADJ
cana-5538	57	74	.	.	PUNCT
cana-5538	58	1	then	then	ADV
cana-5538	58	2	(	(	PUNCT
cana-5538	58	3	𝑋	𝑋	PROPN
cana-5538	58	4	,	,	PUNCT
cana-5538	58	5	𝜌𝑐	𝜌𝑐	PROPN
cana-5538	58	6	)	)	PUNCT
cana-5538	58	7	is	be	AUX
cana-5538	58	8	a	a	DET
cana-5538	58	9	partial	partial	ADJ
cana-5538	58	10	cone	cone	NOUN
cana-5538	58	11	metric	metric	ADJ
cana-5538	58	12	space	space	NOUN
cana-5538	58	13	which	which	PRON
cana-5538	58	14	is	be	AUX
cana-5538	58	15	not	not	PART
cana-5538	58	16	a	a	DET
cana-5538	58	17	cone	cone	NOUN
cana-5538	58	18	metric	metric	ADJ
cana-5538	58	19	space	space	NOUN
cana-5538	58	20	.	.	PUNCT
cana-5538	59	1	remark	remark	VERB
cana-5538	59	2	1.5	1.5	NUM
cana-5538	59	3	suppose	suppose	VERB
cana-5538	59	4	(	(	PUNCT
cana-5538	59	5	𝑋	𝑋	PROPN
cana-5538	59	6	,	,	PUNCT
cana-5538	59	7	𝜌𝑐	𝜌𝑐	PROPN
cana-5538	59	8	)	)	PUNCT
cana-5538	59	9	is	be	AUX
cana-5538	59	10	a	a	DET
cana-5538	59	11	partial	partial	ADJ
cana-5538	59	12	cone	cone	NOUN
cana-5538	59	13	metric	metric	ADJ
cana-5538	59	14	space	space	NOUN
cana-5538	59	15	,	,	PUNCT
cana-5538	59	16	then	then	ADV
cana-5538	59	17	𝑑𝑐(𝑥	𝑑𝑐(𝑥	PUNCT
cana-5538	59	18	,	,	PUNCT
cana-5538	59	19	𝑦	𝑦	NOUN
cana-5538	59	20	)	)	PUNCT
cana-5538	59	21	=	=	SYM
cana-5538	59	22	2𝜌𝑐(𝑥	2𝜌𝑐(𝑥	NUM
cana-5538	59	23	,	,	PUNCT
cana-5538	59	24	𝑦	𝑦	NOUN
cana-5538	59	25	)	)	PUNCT
cana-5538	59	26	–	–	PUNCT
cana-5538	59	27	𝜌𝑐(𝑥	𝜌𝑐(𝑥	NUM
cana-5538	59	28	,	,	PUNCT
cana-5538	59	29	𝑥	𝑥	NOUN
cana-5538	59	30	)	)	PUNCT
cana-5538	59	31	–	–	PUNCT
cana-5538	59	32	𝜌𝑐(𝑦	𝜌𝑐(𝑦	NOUN
cana-5538	59	33	,	,	PUNCT
cana-5538	59	34	𝑦	𝑦	NOUN
cana-5538	59	35	)	)	PUNCT
cana-5538	59	36	for	for	ADP
cana-5538	59	37	all	all	DET
cana-5538	59	38	𝑥	𝑥	PROPN
cana-5538	59	39	,	,	PUNCT
cana-5538	59	40	𝑦	𝑦	NOUN
cana-5538	59	41	,	,	PUNCT
cana-5538	59	42	𝑧	𝑧	DET
cana-5538	59	43	∈	∈	NOUN
cana-5538	59	44	𝑋	𝑋	NOUN
cana-5538	59	45	defines	define	VERB
cana-5538	59	46	a	a	DET
cana-5538	59	47	cone	cone	NOUN
cana-5538	59	48	metric	metric	NOUN
cana-5538	59	49	on	on	ADP
cana-5538	59	50	x.	x.	NOUN
cana-5538	59	51	theorem1.6	theorem1.6	X
cana-5538	59	52	every	every	DET
cana-5538	59	53	partial	partial	ADJ
cana-5538	59	54	cone	cone	NOUN
cana-5538	59	55	metric	metric	ADJ
cana-5538	59	56	space	space	NOUN
cana-5538	59	57	(	(	PUNCT
cana-5538	59	58	𝑋	𝑋	PROPN
cana-5538	59	59	,	,	PUNCT
cana-5538	59	60	𝜌𝑐	𝜌𝑐	PROPN
cana-5538	59	61	)	)	PUNCT
cana-5538	59	62	is	be	AUX
cana-5538	59	63	a	a	DET
cana-5538	59	64	topological	topological	ADJ
cana-5538	59	65	space	space	NOUN
cana-5538	59	66	.	.	PUNCT
cana-5538	60	1	following	follow	VERB
cana-5538	60	2	,	,	PUNCT
cana-5538	60	3	we	we	PRON
cana-5538	60	4	give	give	VERB
cana-5538	60	5	some	some	DET
cana-5538	60	6	properties	property	NOUN
cana-5538	60	7	of	of	ADP
cana-5538	60	8	partial	partial	ADJ
cana-5538	60	9	cone	cone	NOUN
cana-5538	60	10	metric	metric	ADJ
cana-5538	60	11	spaces	space	NOUN
cana-5538	60	12	,	,	PUNCT
cana-5538	60	13	for	for	SCONJ
cana-5538	60	14	more	more	ADJ
cana-5538	60	15	details	detail	NOUN
cana-5538	60	16	see	see	VERB
cana-5538	60	17	[	[	X
cana-5538	60	18	11	11	NUM
cana-5538	60	19	]	]	PUNCT
cana-5538	60	20	.	.	PUNCT
cana-5538	61	1	definition	definition	NOUN
cana-5538	61	2	1.7	1.7	NUM
cana-5538	61	3	let	let	VERB
cana-5538	61	4	(	(	PUNCT
cana-5538	61	5	𝑋	𝑋	PROPN
cana-5538	61	6	,	,	PUNCT
cana-5538	61	7	𝜌𝑐	𝜌𝑐	PRON
cana-5538	61	8	)	)	PUNCT
cana-5538	61	9	be	be	AUX
cana-5538	61	10	a	a	DET
cana-5538	61	11	partial	partial	ADJ
cana-5538	61	12	cone	cone	NOUN
cana-5538	61	13	metric	metric	ADJ
cana-5538	61	14	space	space	NOUN
cana-5538	61	15	.	.	PUNCT
cana-5538	62	1	let	let	VERB
cana-5538	62	2	{	{	PUNCT
cana-5538	62	3	𝑥𝑛	𝑥𝑛	AUX
cana-5538	62	4	}	}	PUNCT
cana-5538	62	5	be	be	AUX
cana-5538	62	6	a	a	DET
cana-5538	62	7	sequence	sequence	NOUN
cana-5538	62	8	in	in	ADP
cana-5538	62	9	x	x	PUNCT
cana-5538	62	10	and	and	CCONJ
cana-5538	62	11	𝑥	𝑥	DET
cana-5538	62	12	∈	∈	PROPN
cana-5538	62	13	𝑋	𝑋	NOUN
cana-5538	62	14	(	(	PUNCT
cana-5538	62	15	1	1	NUM
cana-5538	62	16	)	)	PUNCT
cana-5538	62	17	{	{	PUNCT
cana-5538	62	18	𝑥𝑛	𝑥𝑛	PRON
cana-5538	62	19	}	}	PUNCT
cana-5538	62	20	is	be	AUX
cana-5538	62	21	said	say	VERB
cana-5538	62	22	to	to	PART
cana-5538	62	23	be	be	AUX
cana-5538	62	24	convergent	convergent	ADJ
cana-5538	62	25	to	to	ADP
cana-5538	62	26	x	x	PUNCT
cana-5538	62	27	and	and	CCONJ
cana-5538	62	28	x	x	X
cana-5538	62	29	is	be	AUX
cana-5538	62	30	called	call	VERB
cana-5538	62	31	a	a	DET
cana-5538	62	32	limit	limit	NOUN
cana-5538	62	33	of	of	ADP
cana-5538	62	34	{	{	PUNCT
cana-5538	62	35	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	62	36	}	}	PUNCT
cana-5538	62	37	if	if	SCONJ
cana-5538	62	38	lim	lim	PROPN
cana-5538	62	39	𝑛→∞	𝑛→∞	NUM
cana-5538	62	40	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	62	41	,	,	PUNCT
cana-5538	62	42	𝑥	𝑥	NOUN
cana-5538	62	43	)	)	PUNCT
cana-5538	62	44	=	=	SYM
cana-5538	62	45	lim	lim	PROPN
cana-5538	62	46	𝑛→∞	𝑛→∞	NUM
cana-5538	62	47	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	62	48	,	,	PUNCT
cana-5538	62	49	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	62	50	)	)	PUNCT
cana-5538	62	51	=	=	SYM
cana-5538	63	1	𝜌𝑐(𝑥	𝜌𝑐(𝑥	ADJ
cana-5538	63	2	,	,	PUNCT
cana-5538	63	3	𝑥	𝑥	NOUN
cana-5538	63	4	)	)	PUNCT
cana-5538	63	5	(	(	PUNCT
cana-5538	63	6	2	2	X
cana-5538	63	7	)	)	PUNCT
cana-5538	63	8	{	{	PUNCT
cana-5538	63	9	𝑥𝑛	𝑥𝑛	PRON
cana-5538	63	10	}	}	PUNCT
cana-5538	63	11	is	be	AUX
cana-5538	63	12	cauchy	cauchy	ADJ
cana-5538	63	13	sequence	sequence	NOUN
cana-5538	63	14	if	if	SCONJ
cana-5538	63	15	there	there	PRON
cana-5538	63	16	is	be	VERB
cana-5538	63	17	𝑥	𝑥	PRON
cana-5538	63	18	∈	∈	NOUN
cana-5538	63	19	𝑃	𝑃	VERB
cana-5538	63	20	such	such	ADJ
cana-5538	63	21	that	that	PRON
cana-5538	63	22	for	for	ADP
cana-5538	63	23	every	every	DET
cana-5538	63	24	ϵ	ϵ	NOUN
cana-5538	63	25	>	>	X
cana-5538	63	26	0	0	PUNCT
cana-5538	64	1	there	there	PRON
cana-5538	64	2	is	be	VERB
cana-5538	64	3	ℕ	ℕ	PROPN
cana-5538	64	4	such	such	ADJ
cana-5538	64	5	that	that	PRON
cana-5538	64	6	for	for	ADP
cana-5538	64	7	all	all	DET
cana-5538	64	8	𝑛	𝑛	PROPN
cana-5538	64	9	,	,	PUNCT
cana-5538	64	10	𝑚	𝑚	PROPN
cana-5538	64	11	>	>	X
cana-5538	64	12	ℕ	ℕ	PROPN
cana-5538	64	13	,	,	PUNCT
cana-5538	64	14	||𝜌𝑐(𝑥𝑛	||𝜌𝑐(𝑥𝑛	ADJ
cana-5538	64	15	,	,	PUNCT
cana-5538	64	16	𝑥𝑚	𝑥𝑚	NOUN
cana-5538	64	17	)	)	PUNCT
cana-5538	64	18	−	−	PROPN
cana-5538	65	1	𝑥||	𝑥||	ADV
cana-5538	65	2	<	<	X
cana-5538	65	3	𝜖.	𝜖.	X
cana-5538	65	4	(	(	PUNCT
cana-5538	65	5	3	3	NUM
cana-5538	65	6	)	)	PUNCT
cana-5538	65	7	(	(	PUNCT
cana-5538	65	8	𝑋	𝑋	PROPN
cana-5538	65	9	,	,	PUNCT
cana-5538	65	10	𝜌𝑐	𝜌𝑐	PROPN
cana-5538	65	11	)	)	PUNCT
cana-5538	65	12	is	be	AUX
cana-5538	65	13	said	say	VERB
cana-5538	65	14	to	to	PART
cana-5538	65	15	be	be	AUX
cana-5538	65	16	complete	complete	ADJ
cana-5538	65	17	if	if	SCONJ
cana-5538	65	18	every	every	DET
cana-5538	65	19	cauchy	cauchy	ADJ
cana-5538	65	20	sequence	sequence	NOUN
cana-5538	65	21	in	in	ADP
cana-5538	65	22	(	(	PUNCT
cana-5538	65	23	𝑋	𝑋	PROPN
cana-5538	65	24	,	,	PUNCT
cana-5538	65	25	𝜌𝑐	𝜌𝑐	PRON
cana-5538	65	26	)	)	PUNCT
cana-5538	65	27	is	be	AUX
cana-5538	65	28	convergent	convergent	ADJ
cana-5538	65	29	in	in	ADP
cana-5538	65	30	(	(	PUNCT
cana-5538	65	31	𝑋	𝑋	PROPN
cana-5538	65	32	,	,	PUNCT
cana-5538	65	33	𝜌𝑐	𝜌𝑐	NOUN
cana-5538	65	34	)	)	PUNCT
cana-5538	65	35	.	.	PUNCT
cana-5538	66	1	in	in	ADP
cana-5538	66	2	2012	2012	NUM
cana-5538	66	3	,	,	PUNCT
cana-5538	66	4	samet	samet	PROPN
cana-5538	66	5	et	et	PROPN
cana-5538	66	6	al	al	PROPN
cana-5538	66	7	.	.	PUNCT
cana-5538	67	1	[	[	X
cana-5538	67	2	5	5	NUM
cana-5538	67	3	]	]	PUNCT
cana-5538	67	4	introduced	introduce	VERB
cana-5538	67	5	α	α	NUM
cana-5538	67	6	-	-	ADJ
cana-5538	67	7	admissible	admissible	ADJ
cana-5538	67	8	mapping	mapping	NOUN
cana-5538	67	9	as	as	SCONJ
cana-5538	67	10	follows	follow	VERB
cana-5538	67	11	:	:	PUNCT
cana-5538	67	12	definition	definition	NOUN
cana-5538	67	13	1.8	1.8	NUM
cana-5538	67	14	[	[	X
cana-5538	67	15	5	5	NUM
cana-5538	67	16	]	]	PUNCT
cana-5538	67	17	let	let	VERB
cana-5538	67	18	𝑇	𝑇	PROPN
cana-5538	67	19	∶	∶	VERB
cana-5538	67	20	𝑋	𝑋	PROPN
cana-5538	67	21	→	→	PUNCT
cana-5538	67	22	𝑋	𝑋	PROPN
cana-5538	67	23	and	and	CCONJ
cana-5538	67	24	𝛼	𝛼	ADP
cana-5538	67	25	∶	∶	NOUN
cana-5538	67	26	𝑋	𝑋	NOUN
cana-5538	67	27	×	×	NOUN
cana-5538	67	28	𝑋	𝑋	NOUN
cana-5538	67	29	→	→	SYM
cana-5538	67	30	[	[	X
cana-5538	67	31	0	0	NUM
cana-5538	67	32	,	,	PUNCT
cana-5538	67	33	∞	∞	PROPN
cana-5538	67	34	)	)	PUNCT
cana-5538	67	35	.	.	PUNCT
cana-5538	68	1	t	t	PROPN
cana-5538	68	2	is	be	AUX
cana-5538	68	3	said	say	VERB
cana-5538	68	4	to	to	ADP
cana-5538	68	5	α	α	NOUN
cana-5538	68	6	-	-	ADJ
cana-5538	68	7	admissible	admissible	ADJ
cana-5538	68	8	if	if	SCONJ
cana-5538	68	9	𝛼(𝑥	𝛼(𝑥	PROPN
cana-5538	68	10	,	,	PUNCT
cana-5538	68	11	𝑦	𝑦	NOUN
cana-5538	68	12	)	)	PUNCT
cana-5538	68	13	≥	≥	NOUN
cana-5538	68	14	1	1	NUM
cana-5538	68	15	⇒	⇒	NOUN
cana-5538	68	16	𝛼(𝑇	𝛼(𝑇	PROPN
cana-5538	68	17	𝑥	𝑥	PROPN
cana-5538	68	18	,	,	PUNCT
cana-5538	68	19	𝑇	𝑇	PROPN
cana-5538	68	20	𝑦	𝑦	NOUN
cana-5538	68	21	)	)	PUNCT
cana-5538	68	22	≥	≥	NOUN
cana-5538	68	23	1	1	NUM
cana-5538	68	24	for	for	ADP
cana-5538	68	25	all	all	PRON
cana-5538	68	26	𝑥	𝑥	PROPN
cana-5538	68	27	,	,	PUNCT
cana-5538	68	28	𝑦	𝑦	NOUN
cana-5538	68	29	∈	∈	PROPN
cana-5538	68	30	𝑋.	𝑋.	PROPN
cana-5538	68	31	communications	communication	NOUN
cana-5538	68	32	on	on	ADP
cana-5538	68	33	applied	apply	VERB
cana-5538	68	34	nonlinear	nonlinear	ADJ
cana-5538	68	35	analysis	analysis	NOUN
cana-5538	68	36	issn	issn	NOUN
cana-5538	68	37	:	:	PUNCT
cana-5538	68	38	1074	1074	NUM
cana-5538	68	39	-	-	PUNCT
cana-5538	68	40	133x	133x	NUM
cana-5538	68	41	vol	vol	VERB
cana-5538	68	42	32	32	NUM
cana-5538	68	43	no	no	NOUN
cana-5538	68	44	.	.	PUNCT
cana-5538	69	1	10s	10	NOUN
cana-5538	69	2	(	(	PUNCT
cana-5538	69	3	2025	2025	NUM
cana-5538	69	4	)	)	PUNCT
cana-5538	69	5	2615	2615	NUM
cana-5538	69	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	69	7	2	2	X
cana-5538	69	8	.	.	PUNCT
cana-5538	69	9	main	main	ADJ
cana-5538	69	10	results	result	NOUN
cana-5538	69	11	[	[	X
cana-5538	69	12	12	12	NUM
cana-5538	69	13	]	]	PUNCT
cana-5538	69	14	let	let	VERB
cana-5538	69	15	𝛹	𝛹	NOUN
cana-5538	69	16	be	be	AUX
cana-5538	69	17	the	the	DET
cana-5538	69	18	family	family	NOUN
cana-5538	69	19	of	of	ADP
cana-5538	69	20	non	non	ADJ
cana-5538	69	21	-	-	ADJ
cana-5538	69	22	decreasing	decrease	VERB
cana-5538	69	23	function	function	NOUN
cana-5538	70	1	𝜓	𝜓	PROPN
cana-5538	70	2	∶	∶	NOUN
cana-5538	70	3	[	[	X
cana-5538	70	4	0	0	NUM
cana-5538	70	5	,	,	PUNCT
cana-5538	70	6	∞	∞	PROPN
cana-5538	70	7	)	)	PUNCT
cana-5538	71	1	→	→	PUNCT
cana-5538	72	1	[	[	X
cana-5538	72	2	0	0	NUM
cana-5538	72	3	,	,	PUNCT
cana-5538	72	4	∞	∞	NOUN
cana-5538	72	5	)	)	PUNCT
cana-5538	72	6	such	such	ADJ
cana-5538	72	7	that	that	SCONJ
cana-5538	72	8	∑	∑	PROPN
cana-5538	72	9	𝜓𝑛(𝑡	𝜓𝑛(𝑡	NOUN
cana-5538	72	10	)	)	PUNCT
cana-5538	72	11	<	<	X
cana-5538	72	12	∞	∞	NUM
cana-5538	72	13	𝑛=1	𝑛=1	NOUN
cana-5538	72	14	∞	∞	PROPN
cana-5538	72	15	for	for	ADP
cana-5538	72	16	each	each	DET
cana-5538	72	17	𝑡	𝑡	X
cana-5538	72	18	>	>	X
cana-5538	72	19	0	0	NUM
cana-5538	72	20	where	where	SCONJ
cana-5538	72	21	𝜓𝑛	𝜓𝑛	ADV
cana-5538	72	22	is	be	AUX
cana-5538	72	23	nth	nth	NOUN
cana-5538	72	24	iterate	iterate	NOUN
cana-5538	72	25	of	of	ADP
cana-5538	72	26	ψ	ψ	SYM
cana-5538	72	27	.	.	PUNCT
cana-5538	73	1	lemma	lemma	PROPN
cana-5538	73	2	2.1	2.1	NUM
cana-5538	74	1	[	[	X
cana-5538	74	2	12	12	NUM
cana-5538	74	3	]	]	PUNCT
cana-5538	74	4	for	for	ADP
cana-5538	74	5	every	every	DET
cana-5538	74	6	function	function	NOUN
cana-5538	74	7	𝜓	𝜓	PROPN
cana-5538	74	8	∶	∶	NOUN
cana-5538	74	9	[	[	X
cana-5538	74	10	0	0	NUM
cana-5538	74	11	,	,	PUNCT
cana-5538	74	12	∞	∞	PROPN
cana-5538	74	13	)	)	PUNCT
cana-5538	74	14	→	→	PUNCT
cana-5538	75	1	[	[	X
cana-5538	75	2	0	0	NUM
cana-5538	75	3	,	,	PUNCT
cana-5538	75	4	∞	∞	PROPN
cana-5538	75	5	)	)	PUNCT
cana-5538	75	6	the	the	DET
cana-5538	75	7	following	follow	VERB
cana-5538	75	8	holds	hold	VERB
cana-5538	75	9	:	:	PUNCT
cana-5538	75	10	if	if	SCONJ
cana-5538	75	11	ψ	ψ	NOUN
cana-5538	75	12	is	be	AUX
cana-5538	75	13	non	non	NOUN
cana-5538	75	14	decreasing	decrease	VERB
cana-5538	75	15	,	,	PUNCT
cana-5538	75	16	then	then	ADV
cana-5538	75	17	for	for	ADP
cana-5538	75	18	each	each	DET
cana-5538	75	19	𝑡	𝑡	X
cana-5538	75	20	>	>	X
cana-5538	75	21	0	0	PROPN
cana-5538	75	22	,	,	PUNCT
cana-5538	75	23	lim	lim	NOUN
cana-5538	75	24	𝑛→∞	𝑛→∞	NUM
cana-5538	75	25	𝜓𝑛(𝑡	𝜓𝑛(𝑡	NUM
cana-5538	75	26	)	)	PUNCT
cana-5538	76	1	=	=	SYM
cana-5538	76	2	0	0	NUM
cana-5538	76	3	implies	imply	VERB
cana-5538	76	4	𝜓(𝑡	𝜓(𝑡	NOUN
cana-5538	76	5	)	)	PUNCT
cana-5538	76	6	<	<	X
cana-5538	76	7	𝑡	𝑡	PROPN
cana-5538	76	8	and	and	CCONJ
cana-5538	76	9	𝜓(0	𝜓(0	PROPN
cana-5538	76	10	)	)	PUNCT
cana-5538	76	11	=	=	NOUN
cana-5538	77	1	0	0	X
cana-5538	77	2	.	.	PUNCT
cana-5538	78	1	definition	definition	NOUN
cana-5538	78	2	2.2	2.2	NUM
cana-5538	78	3	let	let	VERB
cana-5538	78	4	(	(	PUNCT
cana-5538	78	5	𝑋	𝑋	PROPN
cana-5538	78	6	,	,	PUNCT
cana-5538	78	7	𝜌𝑐	𝜌𝑐	PRON
cana-5538	78	8	)	)	PUNCT
cana-5538	78	9	be	be	AUX
cana-5538	78	10	a	a	DET
cana-5538	78	11	partial	partial	ADJ
cana-5538	78	12	cone	cone	NOUN
cana-5538	78	13	metric	metric	ADJ
cana-5538	78	14	space	space	NOUN
cana-5538	78	15	p	p	NOUN
cana-5538	78	16	is	be	AUX
cana-5538	78	17	a	a	DET
cana-5538	78	18	normal	normal	ADJ
cana-5538	78	19	cone	cone	NOUN
cana-5538	78	20	with	with	ADP
cana-5538	78	21	constant	constant	ADJ
cana-5538	78	22	k.	k.	NOUN
cana-5538	78	23	let	let	VERB
cana-5538	78	24	𝑇	𝑇	PROPN
cana-5538	78	25	∶	∶	VERB
cana-5538	78	26	𝑋	𝑋	NOUN
cana-5538	78	27	→	→	PUNCT
cana-5538	78	28	𝑋	𝑋	PROPN
cana-5538	78	29	be	be	VERB
cana-5538	78	30	a	a	DET
cana-5538	78	31	self	self	NOUN
cana-5538	78	32	mapping	mapping	NOUN
cana-5538	78	33	.	.	PUNCT
cana-5538	79	1	then	then	ADV
cana-5538	79	2	t	t	PROPN
cana-5538	79	3	is	be	AUX
cana-5538	79	4	said	say	VERB
cana-5538	79	5	to	to	PART
cana-5538	79	6	be	be	AUX
cana-5538	79	7	generalized	generalize	VERB
cana-5538	79	8	𝛼	𝛼	PRON
cana-5538	79	9	−	−	NOUN
cana-5538	79	10	𝜓	𝜓	ADP
cana-5538	79	11	contractive	contractive	ADJ
cana-5538	79	12	mapping	mapping	NOUN
cana-5538	79	13	if	if	SCONJ
cana-5538	79	14	there	there	PRON
cana-5538	79	15	exists	exist	VERB
cana-5538	79	16	two	two	NUM
cana-5538	79	17	functions	function	NOUN
cana-5538	79	18	𝛼	𝛼	ADP
cana-5538	79	19	∶	∶	NOUN
cana-5538	79	20	𝑋	𝑋	NOUN
cana-5538	79	21	×	×	NOUN
cana-5538	79	22	𝑋	𝑋	NOUN
cana-5538	79	23	→	→	SYM
cana-5538	79	24	[	[	X
cana-5538	79	25	0	0	NUM
cana-5538	79	26	,	,	PUNCT
cana-5538	79	27	∞	∞	PROPN
cana-5538	79	28	)	)	PUNCT
cana-5538	79	29	and	and	CCONJ
cana-5538	79	30	𝜓	𝜓	ADP
cana-5538	79	31	∈	∈	PROPN
cana-5538	79	32	𝛹	𝛹	PROPN
cana-5538	79	33	for	for	ADP
cana-5538	79	34	all	all	DET
cana-5538	79	35	𝑥	𝑥	PROPN
cana-5538	79	36	,	,	PUNCT
cana-5538	79	37	𝑦	𝑦	NOUN
cana-5538	79	38	∈	∈	NOUN
cana-5538	79	39	𝑋	𝑋	NOUN
cana-5538	79	40	we	we	PRON
cana-5538	79	41	have	have	VERB
cana-5538	79	42	𝛼(𝑥	𝛼(𝑥	PROPN
cana-5538	79	43	,	,	PUNCT
cana-5538	79	44	𝑦)𝜌𝑐(𝑇	𝑦)𝜌𝑐(𝑇	PROPN
cana-5538	79	45	𝑥	𝑥	NOUN
cana-5538	79	46	,	,	PUNCT
cana-5538	79	47	𝑇	𝑇	PROPN
cana-5538	79	48	𝑦	𝑦	NOUN
cana-5538	79	49	)	)	PUNCT
cana-5538	79	50	≤	≤	NUM
cana-5538	79	51	𝜓(𝑀	𝜓(𝑀	NOUN
cana-5538	79	52	(	(	PUNCT
cana-5538	79	53	𝑥	𝑥	NOUN
cana-5538	79	54	,	,	PUNCT
cana-5538	79	55	𝑦	𝑦	NOUN
cana-5538	79	56	)	)	PUNCT
cana-5538	79	57	)	)	PUNCT
cana-5538	79	58	(	(	PUNCT
cana-5538	79	59	2.1	2.1	NUM
cana-5538	79	60	)	)	PUNCT
cana-5538	79	61	where	where	SCONJ
cana-5538	79	62	𝑀	𝑀	PROPN
cana-5538	79	63	(	(	PUNCT
cana-5538	79	64	𝑥	𝑥	PROPN
cana-5538	79	65	,	,	PUNCT
cana-5538	79	66	𝑦	𝑦	NOUN
cana-5538	79	67	)	)	PUNCT
cana-5538	79	68	=	=	SYM
cana-5538	79	69	𝑚𝑎𝑥{𝜌𝑐(𝑥	𝑚𝑎𝑥{𝜌𝑐(𝑥	NUM
cana-5538	79	70	,	,	PUNCT
cana-5538	79	71	𝑦	𝑦	NOUN
cana-5538	79	72	)	)	PUNCT
cana-5538	79	73	,	,	PUNCT
cana-5538	79	74	𝜌𝑐(𝑥	𝜌𝑐(𝑥	PROPN
cana-5538	79	75	,	,	PUNCT
cana-5538	79	76	𝑇	𝑇	PROPN
cana-5538	79	77	𝑥	𝑥	PROPN
cana-5538	79	78	)	)	PUNCT
cana-5538	79	79	,	,	PUNCT
cana-5538	79	80	𝜌𝑐(𝑦	𝜌𝑐(𝑦	PROPN
cana-5538	79	81	,	,	PUNCT
cana-5538	79	82	𝑇	𝑇	PROPN
cana-5538	79	83	𝑦	𝑦	NOUN
cana-5538	79	84	)	)	PUNCT
cana-5538	79	85	}	}	PUNCT
cana-5538	79	86	(	(	PUNCT
cana-5538	79	87	2.2	2.2	NUM
cana-5538	79	88	)	)	PUNCT
cana-5538	79	89	theorem	theorem	VERB
cana-5538	79	90	2.3	2.3	NUM
cana-5538	79	91	let	let	NOUN
cana-5538	79	92	(	(	PUNCT
cana-5538	79	93	𝑋	𝑋	PROPN
cana-5538	79	94	,	,	PUNCT
cana-5538	79	95	𝜌𝑐	𝜌𝑐	PROPN
cana-5538	79	96	)	)	PUNCT
cana-5538	79	97	be	be	AUX
cana-5538	79	98	a	a	DET
cana-5538	79	99	complete	complete	ADJ
cana-5538	79	100	partial	partial	ADJ
cana-5538	79	101	cone	cone	NOUN
cana-5538	79	102	metric	metric	ADJ
cana-5538	79	103	space	space	NOUN
cana-5538	79	104	and	and	CCONJ
cana-5538	79	105	𝑇	𝑇	PROPN
cana-5538	79	106	∶	∶	NOUN
cana-5538	79	107	𝑋	𝑋	NOUN
cana-5538	79	108	→	→	PUNCT
cana-5538	79	109	𝑋	𝑋	PROPN
cana-5538	79	110	be	be	VERB
cana-5538	79	111	self	self	NOUN
cana-5538	79	112	mapping	mapping	NOUN
cana-5538	79	113	.	.	PUNCT
cana-5538	80	1	suppose	suppose	VERB
cana-5538	80	2	𝛼	𝛼	PRON
cana-5538	80	3	∶	∶	NOUN
cana-5538	80	4	𝑋	𝑋	NOUN
cana-5538	80	5	×	×	NOUN
cana-5538	80	6	𝑋	𝑋	NOUN
cana-5538	80	7	→	→	SYM
cana-5538	80	8	[	[	X
cana-5538	80	9	0	0	NUM
cana-5538	80	10	,	,	PUNCT
cana-5538	80	11	∞	∞	PROPN
cana-5538	80	12	)	)	PUNCT
cana-5538	80	13	be	be	VERB
cana-5538	80	14	the	the	DET
cana-5538	80	15	mappings	mapping	NOUN
cana-5538	80	16	satisfying	satisfy	VERB
cana-5538	80	17	the	the	DET
cana-5538	80	18	conditions	condition	NOUN
cana-5538	80	19	:	:	PUNCT
cana-5538	80	20	(	(	PUNCT
cana-5538	80	21	i	i	NOUN
cana-5538	80	22	)	)	PUNCT
cana-5538	80	23	t	t	PROPN
cana-5538	80	24	is	be	AUX
cana-5538	80	25	α	α	PRON
cana-5538	80	26	admissible	admissible	ADJ
cana-5538	80	27	;	;	PUNCT
cana-5538	80	28	(	(	PUNCT
cana-5538	80	29	ii	ii	NOUN
cana-5538	80	30	)	)	PUNCT
cana-5538	80	31	t	t	PROPN
cana-5538	80	32	is	be	AUX
cana-5538	80	33	generalized	generalize	VERB
cana-5538	80	34	𝛼	𝛼	PRON
cana-5538	80	35	−	−	NOUN
cana-5538	80	36	𝜓	𝜓	ADP
cana-5538	80	37	contractive	contractive	ADJ
cana-5538	80	38	mapping	mapping	NOUN
cana-5538	80	39	;	;	PUNCT
cana-5538	80	40	(	(	PUNCT
cana-5538	80	41	iii	iii	X
cana-5538	80	42	)	)	PUNCT
cana-5538	80	43	there	there	PRON
cana-5538	80	44	exists	exist	VERB
cana-5538	80	45	𝑥0	𝑥0	NOUN
cana-5538	80	46	∈	∈	PROPN
cana-5538	80	47	𝑋	𝑋	PROPN
cana-5538	80	48	such	such	ADJ
cana-5538	80	49	that	that	DET
cana-5538	80	50	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-5538	80	51	,	,	PUNCT
cana-5538	80	52	𝑇	𝑇	PROPN
cana-5538	80	53	𝑥0	𝑥0	NOUN
cana-5538	80	54	)	)	PUNCT
cana-5538	80	55	≥	≥	NOUN
cana-5538	80	56	1	1	NUM
cana-5538	80	57	;	;	PUNCT
cana-5538	80	58	(	(	PUNCT
cana-5538	80	59	iv	iv	X
cana-5538	80	60	)	)	PUNCT
cana-5538	80	61	t	t	PROPN
cana-5538	80	62	is	be	AUX
cana-5538	80	63	continuous	continuous	ADJ
cana-5538	80	64	or	or	CCONJ
cana-5538	80	65	if	if	SCONJ
cana-5538	80	66	{	{	PUNCT
cana-5538	80	67	𝑥𝑛	𝑥𝑛	AUX
cana-5538	80	68	}	}	PUNCT
cana-5538	80	69	be	be	AUX
cana-5538	80	70	a	a	DET
cana-5538	80	71	sequence	sequence	NOUN
cana-5538	80	72	in	in	ADP
cana-5538	80	73	x	x	INTJ
cana-5538	80	74	such	such	ADJ
cana-5538	80	75	that	that	SCONJ
cana-5538	80	76	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-5538	80	77	,	,	PUNCT
cana-5538	80	78	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	80	79	)	)	PUNCT
cana-5538	80	80	≥	≥	NOUN
cana-5538	80	81	1	1	NUM
cana-5538	80	82	for	for	ADP
cana-5538	80	83	all	all	DET
cana-5538	80	84	n	n	NOUN
cana-5538	80	85	and	and	CCONJ
cana-5538	80	86	𝑥𝑛	𝑥𝑛	VERB
cana-5538	80	87	→	→	SYM
cana-5538	80	88	𝑥	𝑥	PROPN
cana-5538	80	89	as	as	ADP
cana-5538	80	90	𝑛	𝑛	PROPN
cana-5538	80	91	→	→	SYM
cana-5538	80	92	∞	∞	PROPN
cana-5538	80	93	then	then	ADV
cana-5538	80	94	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-5538	80	95	,	,	PUNCT
cana-5538	80	96	𝑥	𝑥	NOUN
cana-5538	80	97	)	)	PUNCT
cana-5538	80	98	≥	≥	NOUN
cana-5538	80	99	1	1	NUM
cana-5538	80	100	for	for	ADP
cana-5538	80	101	all	all	DET
cana-5538	80	102	n.	n.	NOUN
cana-5538	80	103	then	then	ADV
cana-5538	80	104	t	t	PROPN
cana-5538	80	105	has	have	VERB
cana-5538	80	106	a	a	DET
cana-5538	80	107	fixed	fix	VERB
cana-5538	80	108	point	point	NOUN
cana-5538	80	109	in	in	ADP
cana-5538	80	110	x.	x.	NOUN
cana-5538	80	111	proof	proof	NOUN
cana-5538	80	112	:	:	PUNCT
cana-5538	80	113	let	let	VERB
cana-5538	80	114	𝑥0	𝑥0	NOUN
cana-5538	80	115	be	be	AUX
cana-5538	80	116	an	an	DET
cana-5538	80	117	arbitrary	arbitrary	ADJ
cana-5538	80	118	point	point	NOUN
cana-5538	80	119	such	such	ADJ
cana-5538	80	120	that	that	DET
cana-5538	80	121	𝛼(𝑥0	𝛼(𝑥0	ADJ
cana-5538	80	122	,	,	PUNCT
cana-5538	80	123	𝑇	𝑇	PROPN
cana-5538	80	124	𝑥0	𝑥0	NOUN
cana-5538	80	125	)	)	PUNCT
cana-5538	80	126	≥	≥	NOUN
cana-5538	80	127	1	1	NUM
cana-5538	80	128	.	.	PUNCT
cana-5538	80	129	suppose	suppose	VERB
cana-5538	80	130	we	we	PRON
cana-5538	80	131	have	have	VERB
cana-5538	80	132	a	a	DET
cana-5538	80	133	sequence	sequence	NOUN
cana-5538	80	134	{	{	PUNCT
cana-5538	80	135	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	80	136	}	}	PUNCT
cana-5538	80	137	in	in	ADP
cana-5538	80	138	x	x	INTJ
cana-5538	80	139	such	such	ADJ
cana-5538	80	140	that	that	SCONJ
cana-5538	80	141	𝑥𝑛+1	𝑥𝑛+1	PROPN
cana-5538	80	142	=	=	SYM
cana-5538	80	143	𝑇	𝑇	PROPN
cana-5538	80	144	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	80	145	for	for	ADP
cana-5538	80	146	all	all	DET
cana-5538	80	147	𝑛	𝑛	DET
cana-5538	80	148	∈	∈	NOUN
cana-5538	80	149	ℕ.	ℕ.	PROPN
cana-5538	80	150	if	if	SCONJ
cana-5538	80	151	𝑥𝑛	𝑥𝑛	VERB
cana-5538	80	152	=	=	SYM
cana-5538	80	153	𝑥𝑛+1	𝑥𝑛+1	PROPN
cana-5538	80	154	for	for	ADP
cana-5538	80	155	some	some	DET
cana-5538	80	156	𝑛	𝑛	PRON
cana-5538	80	157	∈	∈	PROPN
cana-5538	80	158	ℕ	ℕ	PROPN
cana-5538	80	159	,	,	PUNCT
cana-5538	80	160	then	then	ADV
cana-5538	80	161	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	80	162	is	be	AUX
cana-5538	80	163	a	a	DET
cana-5538	80	164	fixed	fix	VERB
cana-5538	80	165	point	point	NOUN
cana-5538	80	166	of	of	ADP
cana-5538	80	167	t	t	PROPN
cana-5538	80	168	and	and	CCONJ
cana-5538	80	169	the	the	DET
cana-5538	80	170	existence	existence	NOUN
cana-5538	80	171	part	part	NOUN
cana-5538	80	172	of	of	ADP
cana-5538	80	173	the	the	DET
cana-5538	80	174	proof	proof	NOUN
cana-5538	80	175	is	be	AUX
cana-5538	80	176	finished	finish	VERB
cana-5538	80	177	.	.	PUNCT
cana-5538	81	1	suppose	suppose	VERB
cana-5538	81	2	𝑥𝑛	𝑥𝑛	VERB
cana-5538	81	3	≠	≠	PROPN
cana-5538	81	4	𝑥𝑛+1for	𝑥𝑛+1for	ADP
cana-5538	81	5	every	every	DET
cana-5538	81	6	𝑛	𝑛	PROPN
cana-5538	81	7	∈	∈	PROPN
cana-5538	81	8	ℕ	ℕ	PROPN
cana-5538	81	9	now	now	ADV
cana-5538	81	10	,	,	PUNCT
cana-5538	81	11	since	since	SCONJ
cana-5538	81	12	t	t	PROPN
cana-5538	81	13	is	be	AUX
cana-5538	81	14	α	α	PRON
cana-5538	81	15	-	-	ADJ
cana-5538	81	16	admissible	admissible	ADJ
cana-5538	81	17	,	,	PUNCT
cana-5538	81	18	so	so	SCONJ
cana-5538	81	19	𝛼(𝑇	𝛼(𝑇	NOUN
cana-5538	81	20	𝑥0	𝑥0	NOUN
cana-5538	81	21	,	,	PUNCT
cana-5538	81	22	𝑇	𝑇	PROPN
cana-5538	81	23	𝑥1	𝑥1	PROPN
cana-5538	81	24	)	)	PUNCT
cana-5538	81	25	=	=	SYM
cana-5538	81	26	𝛼(𝑥1	𝛼(𝑥1	ADJ
cana-5538	81	27	,	,	PUNCT
cana-5538	81	28	𝑥2	𝑥2	NOUN
cana-5538	81	29	)	)	PUNCT
cana-5538	81	30	≥	≥	NOUN
cana-5538	81	31	1	1	NUM
cana-5538	81	32	𝛼(𝑇	𝛼(𝑇	PROPN
cana-5538	81	33	𝑥1	𝑥1	PROPN
cana-5538	81	34	,	,	PUNCT
cana-5538	81	35	𝑇	𝑇	NOUN
cana-5538	81	36	𝑥2	𝑥2	NOUN
cana-5538	81	37	)	)	PUNCT
cana-5538	81	38	=	=	SYM
cana-5538	81	39	𝛼(𝑥2	𝛼(𝑥2	NOUN
cana-5538	81	40	,	,	PUNCT
cana-5538	81	41	𝑥3	𝑥3	NOUN
cana-5538	81	42	)	)	PUNCT
cana-5538	81	43	≥	≥	NOUN
cana-5538	81	44	1	1	NUM
cana-5538	81	45	and	and	CCONJ
cana-5538	81	46	using	use	VERB
cana-5538	81	47	induction	induction	NOUN
cana-5538	81	48	we	we	PRON
cana-5538	81	49	have	have	VERB
cana-5538	81	50	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-5538	81	51	,	,	PUNCT
cana-5538	81	52	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	81	53	)	)	PUNCT
cana-5538	81	54	≥	≥	NOUN
cana-5538	81	55	1	1	NUM
cana-5538	81	56	for	for	ADP
cana-5538	81	57	all	all	DET
cana-5538	81	58	𝑛	𝑛	DET
cana-5538	81	59	∈	∈	PROPN
cana-5538	81	60	ℕ.	ℕ.	PROPN
cana-5538	81	61	now	now	ADV
cana-5538	81	62	,	,	PUNCT
cana-5538	81	63	from	from	ADP
cana-5538	81	64	(	(	PUNCT
cana-5538	81	65	2.1	2.1	NUM
cana-5538	81	66	)	)	PUNCT
cana-5538	81	67	we	we	PRON
cana-5538	81	68	have	have	VERB
cana-5538	81	69	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	81	70	,	,	PUNCT
cana-5538	81	71	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	81	72	)	)	PUNCT
cana-5538	81	73	=	=	SYM
cana-5538	82	1	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	82	2	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-5538	82	3	,	,	PUNCT
cana-5538	82	4	𝑇	𝑇	PROPN
cana-5538	82	5	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	82	6	)	)	PUNCT
cana-5538	82	7	(	(	PUNCT
cana-5538	82	8	2.3	2.3	NUM
cana-5538	82	9	)	)	PUNCT
cana-5538	82	10	≤	≤	NOUN
cana-5538	82	11	𝛼(𝑥𝑛−1	𝛼(𝑥𝑛−1	NOUN
cana-5538	82	12	,	,	PUNCT
cana-5538	82	13	𝑥𝑛)𝜌𝑐(𝑇	𝑥𝑛)𝜌𝑐(𝑇	NOUN
cana-5538	82	14	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-5538	82	15	,	,	PUNCT
cana-5538	82	16	𝑇	𝑇	PROPN
cana-5538	82	17	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	82	18	)	)	PUNCT
cana-5538	82	19	≤	≤	NUM
cana-5538	82	20	𝜓(𝑀	𝜓(𝑀	PROPN
cana-5538	82	21	(	(	PUNCT
cana-5538	82	22	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-5538	82	23	,	,	PUNCT
cana-5538	82	24	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	82	25	)	)	PUNCT
cana-5538	82	26	)	)	PUNCT
cana-5538	82	27	(	(	PUNCT
cana-5538	82	28	2.4	2.4	NUM
cana-5538	82	29	)	)	PUNCT
cana-5538	83	1	where	where	SCONJ
cana-5538	83	2	𝑀	𝑀	PROPN
cana-5538	83	3	(	(	PUNCT
cana-5538	83	4	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-5538	83	5	,	,	PUNCT
cana-5538	83	6	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	83	7	)	)	PUNCT
cana-5538	83	8	=	=	SYM
cana-5538	83	9	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛−1	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	83	10	,	,	PUNCT
cana-5538	83	11	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	83	12	)	)	PUNCT
cana-5538	83	13	,	,	PUNCT
cana-5538	83	14	𝜌𝑐(𝑥𝑛−1	𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	83	15	,	,	PUNCT
cana-5538	83	16	𝑇	𝑇	PROPN
cana-5538	83	17	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-5538	83	18	)	)	PUNCT
cana-5538	83	19	,	,	PUNCT
cana-5538	83	20	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	83	21	,	,	PUNCT
cana-5538	83	22	𝑇	𝑇	PROPN
cana-5538	83	23	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	83	24	)	)	PUNCT
cana-5538	83	25	}	}	PUNCT
cana-5538	83	26	communications	communication	NOUN
cana-5538	83	27	on	on	ADP
cana-5538	83	28	applied	apply	VERB
cana-5538	83	29	nonlinear	nonlinear	ADJ
cana-5538	83	30	analysis	analysis	NOUN
cana-5538	83	31	issn	issn	NOUN
cana-5538	83	32	:	:	PUNCT
cana-5538	83	33	1074	1074	NUM
cana-5538	83	34	-	-	PUNCT
cana-5538	83	35	133x	133x	NUM
cana-5538	83	36	vol	vol	VERB
cana-5538	83	37	32	32	NUM
cana-5538	83	38	no	no	NOUN
cana-5538	83	39	.	.	PUNCT
cana-5538	84	1	10s	10	NOUN
cana-5538	84	2	(	(	PUNCT
cana-5538	84	3	2025	2025	NUM
cana-5538	84	4	)	)	PUNCT
cana-5538	84	5	2616	2616	NUM
cana-5538	84	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	84	7	=	=	SYM
cana-5538	84	8	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛−1	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	84	9	,	,	PUNCT
cana-5538	84	10	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	84	11	)	)	PUNCT
cana-5538	84	12	,	,	PUNCT
cana-5538	84	13	𝜌𝑐(𝑥𝑛−1	𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	84	14	,	,	PUNCT
cana-5538	84	15	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	84	16	)	)	PUNCT
cana-5538	84	17	,	,	PUNCT
cana-5538	84	18	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	84	19	,	,	PUNCT
cana-5538	84	20	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	84	21	)	)	PUNCT
cana-5538	84	22	}	}	PUNCT
cana-5538	84	23	(	(	PUNCT
cana-5538	84	24	2.5	2.5	NUM
cana-5538	84	25	)	)	PUNCT
cana-5538	84	26	now	now	ADV
cana-5538	84	27	,	,	PUNCT
cana-5538	84	28	if	if	SCONJ
cana-5538	84	29	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	84	30	,	,	PUNCT
cana-5538	84	31	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	84	32	)	)	PUNCT
cana-5538	84	33	>	>	X
cana-5538	85	1	𝜌𝑐(𝑥𝑛−1	𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	85	2	,	,	PUNCT
cana-5538	85	3	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	85	4	)	)	PUNCT
cana-5538	85	5	.	.	PUNCT
cana-5538	86	1	then	then	ADV
cana-5538	86	2	||𝜌𝑐(𝑥𝑛	||𝜌𝑐(𝑥𝑛	X
cana-5538	86	3	,	,	PUNCT
cana-5538	86	4	𝑥𝑛+1))||	𝑥𝑛+1))||	NOUN
cana-5538	86	5	≤	≤	PUNCT
cana-5538	86	6	𝜓(||𝜌𝑐(𝑥𝑛	𝜓(||𝜌𝑐(𝑥𝑛	PROPN
cana-5538	86	7	,	,	PUNCT
cana-5538	86	8	𝑥𝑛+1))||	𝑥𝑛+1))||	NOUN
cana-5538	86	9	)	)	PUNCT
cana-5538	86	10	<	<	X
cana-5538	86	11	||𝜌𝑐(𝑥𝑛	||𝜌𝑐(𝑥𝑛	ADJ
cana-5538	86	12	,	,	PUNCT
cana-5538	86	13	𝑥𝑛+1))||	𝑥𝑛+1))||	NUM
cana-5538	86	14	(	(	PUNCT
cana-5538	86	15	2.6	2.6	NUM
cana-5538	86	16	)	)	PUNCT
cana-5538	86	17	this	this	PRON
cana-5538	86	18	is	be	AUX
cana-5538	86	19	a	a	DET
cana-5538	86	20	contradiction	contradiction	NOUN
cana-5538	86	21	.	.	PUNCT
cana-5538	87	1	thus	thus	ADV
cana-5538	87	2	for	for	ADP
cana-5538	87	3	all	all	DET
cana-5538	87	4	𝑛	𝑛	DET
cana-5538	87	5	≥	≥	NUM
cana-5538	87	6	1	1	NUM
cana-5538	87	7	we	we	PRON
cana-5538	87	8	have	have	VERB
cana-5538	87	9	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛−1	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	87	10	,	,	PUNCT
cana-5538	87	11	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	87	12	)	)	PUNCT
cana-5538	87	13	,	,	PUNCT
cana-5538	87	14	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	87	15	,	,	PUNCT
cana-5538	87	16	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	87	17	)	)	PUNCT
cana-5538	87	18	}	}	PUNCT
cana-5538	88	1	=	=	SYM
cana-5538	88	2	𝜌𝑐(𝑥𝑛−1	𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	88	3	,	,	PUNCT
cana-5538	88	4	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	88	5	)	)	PUNCT
cana-5538	88	6	(	(	PUNCT
cana-5538	88	7	2.7	2.7	NUM
cana-5538	88	8	)	)	PUNCT
cana-5538	88	9	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	88	10	,	,	PUNCT
cana-5538	88	11	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	88	12	)	)	PUNCT
cana-5538	88	13	)	)	PUNCT
cana-5538	88	14	≤	≤	PROPN
cana-5538	89	1	𝜓(𝜌𝑐(𝑥𝑛−1	𝜓(𝜌𝑐(𝑥𝑛−1	PROPN
cana-5538	89	2	,	,	PUNCT
cana-5538	89	3	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	89	4	)	)	PUNCT
cana-5538	89	5	)	)	PUNCT
cana-5538	89	6	(	(	PUNCT
cana-5538	89	7	2.8	2.8	NUM
cana-5538	89	8	)	)	PUNCT
cana-5538	89	9	continuing	continue	VERB
cana-5538	89	10	this	this	DET
cana-5538	89	11	process	process	NOUN
cana-5538	89	12	inductively	inductively	ADV
cana-5538	89	13	,	,	PUNCT
cana-5538	89	14	we	we	PRON
cana-5538	89	15	obtain	obtain	VERB
cana-5538	89	16	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	89	17	,	,	PUNCT
cana-5538	89	18	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	89	19	)	)	PUNCT
cana-5538	89	20	)	)	PUNCT
cana-5538	90	1	≤	≤	NUM
cana-5538	90	2	𝜓𝑛	𝜓𝑛	NOUN
cana-5538	90	3	(	(	PUNCT
cana-5538	90	4	𝜌𝑐(𝑥0	𝜌𝑐(𝑥0	NUM
cana-5538	90	5	,	,	PUNCT
cana-5538	90	6	𝑥1	𝑥1	NOUN
cana-5538	90	7	)	)	PUNCT
cana-5538	90	8	)	)	PUNCT
cana-5538	90	9	(	(	PUNCT
cana-5538	90	10	2.9	2.9	NUM
cana-5538	90	11	)	)	PUNCT
cana-5538	90	12	now	now	ADV
cana-5538	90	13	for	for	ADP
cana-5538	90	14	𝑚	𝑚	PROPN
cana-5538	90	15	>	>	SYM
cana-5538	90	16	𝑛	𝑛	PROPN
cana-5538	90	17	,	,	PUNCT
cana-5538	90	18	using	use	VERB
cana-5538	90	19	(	(	PUNCT
cana-5538	90	20	2.9	2.9	NUM
cana-5538	90	21	)	)	PUNCT
cana-5538	90	22	and	and	CCONJ
cana-5538	90	23	triangular	triangular	NOUN
cana-5538	90	24	inequality	inequality	NOUN
cana-5538	90	25	,	,	PUNCT
cana-5538	90	26	we	we	PRON
cana-5538	90	27	obtain	obtain	VERB
cana-5538	90	28	𝜌𝑐(𝑥𝑚	𝜌𝑐(𝑥𝑚	PROPN
cana-5538	90	29	,	,	PUNCT
cana-5538	90	30	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	90	31	)	)	PUNCT
cana-5538	90	32	≤	≤	PUNCT
cana-5538	91	1	𝜌𝑐(𝑥𝑚	𝜌𝑐(𝑥𝑚	ADJ
cana-5538	91	2	,	,	PUNCT
cana-5538	91	3	𝑥𝑚−1	𝑥𝑚−1	PROPN
cana-5538	91	4	)	)	PUNCT
cana-5538	91	5	+	+	CCONJ
cana-5538	91	6	𝜌𝑐(𝑥𝑚−1	𝜌𝑐(𝑥𝑚−1	NOUN
cana-5538	91	7	,	,	PUNCT
cana-5538	91	8	𝑥𝑚−2	𝑥𝑚−2	NOUN
cana-5538	91	9	)	)	PUNCT
cana-5538	91	10	.	.	PUNCT
cana-5538	91	11	.	.	PUNCT
cana-5538	91	12	.	.	PUNCT
cana-5538	91	13	.	.	PUNCT
cana-5538	91	14	.	.	PUNCT
cana-5538	91	15	.	.	PUNCT
cana-5538	91	16	.	.	PUNCT
cana-5538	91	17	.	.	PUNCT
cana-5538	92	1	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	NOUN
cana-5538	92	2	,	,	PUNCT
cana-5538	92	3	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	92	4	)	)	PUNCT
cana-5538	92	5	–	–	PUNCT
cana-5538	92	6	∑	∑	PUNCT
cana-5538	92	7	𝜌𝑐(𝑥𝑚−𝑘	𝜌𝑐(𝑥𝑚−𝑘	NOUN
cana-5538	92	8	,	,	PUNCT
cana-5538	92	9	𝑥𝑚−𝑘	𝑥𝑚−𝑘	NOUN
cana-5538	92	10	)	)	PUNCT
cana-5538	92	11	𝑚−𝑛−1	𝑚−𝑛−1	PROPN
cana-5538	92	12	𝑘=1	𝑘=1	NOUN
cana-5538	92	13	≤	≤	NUM
cana-5538	92	14	𝜌𝑐(𝑥𝑚	𝜌𝑐(𝑥𝑚	NOUN
cana-5538	92	15	,	,	PUNCT
cana-5538	92	16	𝑥𝑚−1	𝑥𝑚−1	PROPN
cana-5538	92	17	)	)	PUNCT
cana-5538	92	18	+	+	CCONJ
cana-5538	92	19	𝜌𝑐(𝑥𝑚−1	𝜌𝑐(𝑥𝑚−1	NOUN
cana-5538	92	20	,	,	PUNCT
cana-5538	92	21	𝑥𝑚−2	𝑥𝑚−2	NOUN
cana-5538	92	22	)	)	PUNCT
cana-5538	92	23	.	.	PUNCT
cana-5538	92	24	.	.	PUNCT
cana-5538	92	25	.	.	PUNCT
cana-5538	92	26	.	.	PUNCT
cana-5538	92	27	.	.	PUNCT
cana-5538	92	28	.	.	PUNCT
cana-5538	92	29	.	.	PUNCT
cana-5538	92	30	.	.	PUNCT
cana-5538	93	1	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	NOUN
cana-5538	93	2	,	,	PUNCT
cana-5538	93	3	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	93	4	)	)	PUNCT
cana-5538	93	5	≤	≤	NOUN
cana-5538	93	6	(	(	PUNCT
cana-5538	93	7	𝜓𝑚−1	𝜓𝑚−1	NOUN
cana-5538	93	8	+	+	CCONJ
cana-5538	93	9	𝜓𝑚−2	𝜓𝑚−2	NOUN
cana-5538	93	10	+	+	CCONJ
cana-5538	93	11	.	.	PUNCT
cana-5538	93	12	.	.	PUNCT
cana-5538	93	13	.	.	PUNCT
cana-5538	93	14	.	.	PUNCT
cana-5538	93	15	.	.	PUNCT
cana-5538	93	16	.	.	PUNCT
cana-5538	93	17	.	.	PUNCT
cana-5538	94	1	𝜓𝑛)𝜌𝑐(𝑥0	𝜓𝑛)𝜌𝑐(𝑥0	NUM
cana-5538	94	2	,	,	PUNCT
cana-5538	94	3	𝑥1	𝑥1	NOUN
cana-5538	94	4	)	)	PUNCT
cana-5538	94	5	=	=	PUNCT
cana-5538	94	6	𝜓𝑛	𝜓𝑛	ADP
cana-5538	94	7	1−𝜓	1−𝜓	NUM
cana-5538	94	8	𝜌𝑐(𝑥0	𝜌𝑐(𝑥0	NUM
cana-5538	94	9	,	,	PUNCT
cana-5538	94	10	𝑥1	𝑥1	NOUN
cana-5538	94	11	)	)	PUNCT
cana-5538	94	12	(	(	PUNCT
cana-5538	94	13	2.10	2.10	NUM
cana-5538	94	14	)	)	PUNCT
cana-5538	94	15	since	since	SCONJ
cana-5538	94	16	p	p	NOUN
cana-5538	94	17	is	be	AUX
cana-5538	94	18	normal	normal	ADJ
cana-5538	94	19	cone	cone	NOUN
cana-5538	94	20	with	with	ADP
cana-5538	94	21	normal	normal	ADJ
cana-5538	94	22	constant	constant	ADJ
cana-5538	94	23	k	k	NOUN
cana-5538	94	24	,	,	PUNCT
cana-5538	94	25	we	we	PRON
cana-5538	94	26	find	find	VERB
cana-5538	94	27	that	that	SCONJ
cana-5538	94	28	||𝜌𝑐(𝑥𝑚	||𝜌𝑐(𝑥𝑚	NUM
cana-5538	94	29	,	,	PUNCT
cana-5538	94	30	𝑥𝑛))||	𝑥𝑛))||	PROPN
cana-5538	94	31	≤	≤	NUM
cana-5538	94	32	𝐾||	𝐾||	PUNCT
cana-5538	94	33	𝜓𝑛	𝜓𝑛	PROPN
cana-5538	94	34	1−𝜓	1−𝜓	NUM
cana-5538	94	35	𝜌𝑐(𝑥0	𝜌𝑐(𝑥0	NUM
cana-5538	94	36	,	,	PUNCT
cana-5538	94	37	𝑥1)||	𝑥1)||	PROPN
cana-5538	94	38	(	(	PUNCT
cana-5538	94	39	2.11	2.11	NUM
cana-5538	94	40	)	)	PUNCT
cana-5538	94	41	which	which	PRON
cana-5538	94	42	implies	imply	VERB
cana-5538	94	43	that	that	SCONJ
cana-5538	94	44	𝜌𝑐(𝑥𝑚	𝜌𝑐(𝑥𝑚	PROPN
cana-5538	94	45	,	,	PUNCT
cana-5538	94	46	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	94	47	)	)	PUNCT
cana-5538	94	48	→	→	SYM
cana-5538	94	49	0	0	NUM
cana-5538	94	50	as	as	ADP
cana-5538	94	51	𝑛	𝑛	PROPN
cana-5538	94	52	,	,	PUNCT
cana-5538	94	53	𝑚	𝑚	X
cana-5538	94	54	→	→	PUNCT
cana-5538	94	55	∞.	∞.	PROPN
cana-5538	94	56	hence	hence	ADV
cana-5538	94	57	{	{	PUNCT
cana-5538	94	58	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	94	59	}	}	PUNCT
cana-5538	94	60	a	a	DET
cana-5538	94	61	cauchy	cauchy	ADJ
cana-5538	94	62	sequence	sequence	NOUN
cana-5538	94	63	in	in	ADP
cana-5538	94	64	partial	partial	ADJ
cana-5538	94	65	cone	cone	NOUN
cana-5538	94	66	metric	metric	ADJ
cana-5538	94	67	space	space	NOUN
cana-5538	94	68	which	which	PRON
cana-5538	94	69	is	be	AUX
cana-5538	94	70	complete	complete	ADJ
cana-5538	94	71	hence	hence	ADV
cana-5538	94	72	it	it	PRON
cana-5538	94	73	must	must	AUX
cana-5538	94	74	be	be	AUX
cana-5538	94	75	convergent	convergent	ADJ
cana-5538	94	76	in	in	ADP
cana-5538	94	77	x	x	NOUN
cana-5538	94	78	,	,	PUNCT
cana-5538	94	79	let	let	VERB
cana-5538	94	80	lim	lim	PROPN
cana-5538	94	81	𝑛→∞	𝑛→∞	VERB
cana-5538	94	82	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	94	83	=	=	SYM
cana-5538	94	84	𝑧	𝑧	X
cana-5538	94	85	therefore	therefore	ADV
cana-5538	94	86	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	94	87	,	,	PUNCT
cana-5538	94	88	𝑧	𝑧	X
cana-5538	94	89	)	)	PUNCT
cana-5538	95	1	=	=	SYM
cana-5538	95	2	lim	lim	PROPN
cana-5538	95	3	𝑛→∞	𝑛→∞	NUM
cana-5538	95	4	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	95	5	,	,	PUNCT
cana-5538	95	6	𝑧	𝑧	NOUN
cana-5538	95	7	)	)	PUNCT
cana-5538	95	8	=	=	SYM
cana-5538	95	9	lim	lim	PROPN
cana-5538	95	10	𝑛→∞	𝑛→∞	NUM
cana-5538	95	11	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	95	12	,	,	PUNCT
cana-5538	95	13	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	95	14	)	)	PUNCT
cana-5538	95	15	=	=	SYM
cana-5538	95	16	0	0	NUM
cana-5538	95	17	case	case	NOUN
cana-5538	95	18	1	1	NUM
cana-5538	95	19	.	.	X
cana-5538	95	20	t	t	PROPN
cana-5538	95	21	is	be	AUX
cana-5538	95	22	continuous	continuous	ADJ
cana-5538	95	23	,	,	PUNCT
cana-5538	95	24	then	then	ADV
cana-5538	95	25	we	we	PRON
cana-5538	95	26	have	have	VERB
cana-5538	95	27	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	95	28	=	=	SYM
cana-5538	95	29	𝑇	𝑇	PROPN
cana-5538	95	30	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	95	31	→	→	SYM
cana-5538	95	32	𝑇	𝑇	PROPN
cana-5538	95	33	𝑧	𝑧	PRON
cana-5538	95	34	as	as	ADP
cana-5538	95	35	𝑛	𝑛	PROPN
cana-5538	95	36	→	→	PUNCT
cana-5538	95	37	∞.	∞.	PROPN
cana-5538	95	38	by	by	ADP
cana-5538	95	39	uniqueness	uniqueness	NOUN
cana-5538	95	40	of	of	ADP
cana-5538	95	41	limit	limit	NOUN
cana-5538	95	42	𝑇	𝑇	PROPN
cana-5538	95	43	𝑧	𝑧	NOUN
cana-5538	95	44	=	=	NOUN
cana-5538	95	45	𝑧.	𝑧.	NOUN
cana-5538	95	46	hence	hence	ADV
cana-5538	95	47	z	z	PROPN
cana-5538	95	48	is	be	AUX
cana-5538	95	49	a	a	DET
cana-5538	95	50	fixed	fix	VERB
cana-5538	95	51	point	point	NOUN
cana-5538	95	52	of	of	ADP
cana-5538	95	53	t	t	PROPN
cana-5538	95	54	.	.	PUNCT
cana-5538	96	1	case	case	NOUN
cana-5538	96	2	2	2	NUM
cana-5538	96	3	if	if	SCONJ
cana-5538	96	4	{	{	PUNCT
cana-5538	96	5	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	96	6	}	}	PUNCT
cana-5538	96	7	is	be	AUX
cana-5538	96	8	a	a	DET
cana-5538	96	9	sequence	sequence	NOUN
cana-5538	96	10	in	in	ADP
cana-5538	96	11	x	x	INTJ
cana-5538	96	12	such	such	ADJ
cana-5538	96	13	that	that	SCONJ
cana-5538	96	14	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-5538	96	15	,	,	PUNCT
cana-5538	96	16	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	96	17	)	)	PUNCT
cana-5538	96	18	≥	≥	NOUN
cana-5538	96	19	1	1	NUM
cana-5538	96	20	for	for	ADP
cana-5538	96	21	all	all	DET
cana-5538	96	22	n	n	NOUN
cana-5538	96	23	and	and	CCONJ
cana-5538	96	24	𝑥𝑛	𝑥𝑛	VERB
cana-5538	96	25	→	→	SYM
cana-5538	96	26	𝑧	𝑧	X
cana-5538	96	27	as	as	ADP
cana-5538	96	28	𝑛	𝑛	PROPN
cana-5538	96	29	→	→	SYM
cana-5538	96	30	∞.	∞.	PROPN
cana-5538	96	31	then	then	ADV
cana-5538	96	32	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-5538	96	33	,	,	PUNCT
cana-5538	96	34	𝑧	𝑧	NOUN
cana-5538	96	35	)	)	PUNCT
cana-5538	96	36	≥	≥	NOUN
cana-5538	96	37	1	1	NUM
cana-5538	96	38	for	for	ADP
cana-5538	96	39	all	all	DET
cana-5538	96	40	n.	n.	NOUN
cana-5538	96	41	now	now	ADV
cana-5538	96	42	we	we	PRON
cana-5538	96	43	show	show	VERB
cana-5538	96	44	that	that	SCONJ
cana-5538	96	45	||𝜌𝑐(𝑇	||𝜌𝑐(𝑇	ADV
cana-5538	96	46	𝑧	𝑧	ADJ
cana-5538	96	47	,	,	PUNCT
cana-5538	96	48	𝑧)||	𝑧)||	ADJ
cana-5538	96	49	≥	≥	NOUN
cana-5538	96	50	0	0	NUM
cana-5538	96	51	,	,	PUNCT
cana-5538	96	52	on	on	ADP
cana-5538	96	53	contrary	contrary	ADV
cana-5538	96	54	,	,	PUNCT
cana-5538	96	55	assume	assume	VERB
cana-5538	96	56	||𝜌𝑐(𝑇	||𝜌𝑐(𝑇	ADV
cana-5538	96	57	𝑧	𝑧	ADJ
cana-5538	96	58	,	,	PUNCT
cana-5538	96	59	𝑧)||	𝑧)||	ADJ
cana-5538	96	60	>	>	X
cana-5538	96	61	0	0	NUM
cana-5538	97	1	we	we	PRON
cana-5538	97	2	have	have	VERB
cana-5538	97	3	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	97	4	𝑧	𝑧	PROPN
cana-5538	97	5	,	,	PUNCT
cana-5538	97	6	𝑧	𝑧	NOUN
cana-5538	97	7	)	)	PUNCT
cana-5538	97	8	≤	≤	NOUN
cana-5538	97	9	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	97	10	𝑧	𝑧	PROPN
cana-5538	97	11	,	,	PUNCT
cana-5538	97	12	𝑇	𝑇	PROPN
cana-5538	97	13	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	97	14	)	)	PUNCT
cana-5538	97	15	+	+	CCONJ
cana-5538	97	16	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	97	17	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	97	18	,	,	PUNCT
cana-5538	97	19	𝑧	𝑧	NOUN
cana-5538	97	20	)	)	PUNCT
cana-5538	97	21	–	–	PUNCT
cana-5538	97	22	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	97	23	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	97	24	,	,	PUNCT
cana-5538	97	25	𝑇	𝑇	PROPN
cana-5538	97	26	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	97	27	)	)	PUNCT
cana-5538	97	28	≤	≤	NOUN
cana-5538	98	1	𝛼(𝑥𝑛	𝛼(𝑥𝑛	NUM
cana-5538	98	2	,	,	PUNCT
cana-5538	98	3	𝑧)𝜌𝑐(𝑇	𝑧)𝜌𝑐(𝑇	ADV
cana-5538	98	4	𝑧	𝑧	PRON
cana-5538	98	5	,	,	PUNCT
cana-5538	98	6	𝑇	𝑇	PROPN
cana-5538	98	7	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	98	8	)	)	PUNCT
cana-5538	98	9	+	+	CCONJ
cana-5538	98	10	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	98	11	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	98	12	,	,	PUNCT
cana-5538	98	13	𝑧	𝑧	NOUN
cana-5538	98	14	)	)	PUNCT
cana-5538	98	15	–	–	PUNCT
cana-5538	98	16	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	98	17	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	98	18	,	,	PUNCT
cana-5538	98	19	𝑇	𝑇	PROPN
cana-5538	98	20	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	98	21	)	)	PUNCT
cana-5538	98	22	≤	≤	NOUN
cana-5538	98	23	𝜓(𝑀	𝜓(𝑀	NOUN
cana-5538	98	24	(	(	PUNCT
cana-5538	98	25	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	98	26	,	,	PUNCT
cana-5538	98	27	𝑧	𝑧	NOUN
cana-5538	98	28	)	)	PUNCT
cana-5538	98	29	)	)	PUNCT
cana-5538	99	1	+	+	CCONJ
cana-5538	99	2	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	ADJ
cana-5538	99	3	,	,	PUNCT
cana-5538	99	4	𝑧	𝑧	NOUN
cana-5538	99	5	)	)	PUNCT
cana-5538	99	6	(	(	PUNCT
cana-5538	99	7	2.12	2.12	NUM
cana-5538	99	8	)	)	PUNCT
cana-5538	99	9	since	since	SCONJ
cana-5538	99	10	p	p	NOUN
cana-5538	99	11	is	be	AUX
cana-5538	99	12	normal	normal	ADJ
cana-5538	99	13	cone	cone	NOUN
cana-5538	99	14	with	with	ADP
cana-5538	99	15	normal	normal	ADJ
cana-5538	99	16	constant	constant	ADJ
cana-5538	99	17	k	k	NOUN
cana-5538	99	18	,	,	PUNCT
cana-5538	99	19	we	we	PRON
cana-5538	99	20	have	have	VERB
cana-5538	99	21	||𝜌𝑐(𝑇	||𝜌𝑐(𝑇	ADV
cana-5538	99	22	𝑧	𝑧	SYM
cana-5538	99	23	,	,	PUNCT
cana-5538	99	24	𝑧)||	𝑧)||	ADJ
cana-5538	99	25	≤	≤	NUM
cana-5538	99	26	𝐾||𝜓(𝑀	𝐾||𝜓(𝑀	NOUN
cana-5538	99	27	(	(	PUNCT
cana-5538	99	28	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	99	29	,	,	PUNCT
cana-5538	99	30	𝑧	𝑧	NOUN
cana-5538	99	31	)	)	PUNCT
cana-5538	99	32	)	)	PUNCT
cana-5538	100	1	+	+	CCONJ
cana-5538	100	2	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	ADJ
cana-5538	100	3	,	,	PUNCT
cana-5538	100	4	𝑧)||	𝑧)||	NUM
cana-5538	100	5	(	(	PUNCT
cana-5538	100	6	2.13	2.13	NUM
cana-5538	100	7	)	)	PUNCT
cana-5538	100	8	communications	communication	NOUN
cana-5538	100	9	on	on	ADP
cana-5538	100	10	applied	apply	VERB
cana-5538	100	11	nonlinear	nonlinear	ADJ
cana-5538	100	12	analysis	analysis	NOUN
cana-5538	100	13	issn	issn	NOUN
cana-5538	100	14	:	:	PUNCT
cana-5538	100	15	1074	1074	NUM
cana-5538	100	16	-	-	PUNCT
cana-5538	100	17	133x	133x	NUM
cana-5538	100	18	vol	vol	VERB
cana-5538	100	19	32	32	NUM
cana-5538	100	20	no	no	NOUN
cana-5538	100	21	.	.	PUNCT
cana-5538	101	1	10s	10	NOUN
cana-5538	101	2	(	(	PUNCT
cana-5538	101	3	2025	2025	NUM
cana-5538	101	4	)	)	PUNCT
cana-5538	101	5	2617	2617	NUM
cana-5538	101	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	101	7	where	where	SCONJ
cana-5538	101	8	𝑀	𝑀	PROPN
cana-5538	101	9	(	(	PUNCT
cana-5538	101	10	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	101	11	,	,	PUNCT
cana-5538	101	12	𝑧	𝑧	NOUN
cana-5538	101	13	)	)	PUNCT
cana-5538	101	14	=	=	SYM
cana-5538	102	1	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛	PROPN
cana-5538	102	2	,	,	PUNCT
cana-5538	102	3	𝑧	𝑧	NOUN
cana-5538	102	4	)	)	PUNCT
cana-5538	102	5	,	,	PUNCT
cana-5538	102	6	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	102	7	,	,	PUNCT
cana-5538	102	8	𝑇	𝑇	PROPN
cana-5538	102	9	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	102	10	)	)	PUNCT
cana-5538	102	11	,	,	PUNCT
cana-5538	102	12	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	102	13	,	,	PUNCT
cana-5538	102	14	𝑇	𝑇	PROPN
cana-5538	102	15	𝑧	𝑧	PART
cana-5538	102	16	)	)	PUNCT
cana-5538	102	17	}	}	PUNCT
cana-5538	102	18	=	=	PUNCT
cana-5538	103	1	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛	𝑚𝑎𝑥{𝜌𝑐(𝑥𝑛	X
cana-5538	103	2	,	,	PUNCT
cana-5538	103	3	𝑧	𝑧	NOUN
cana-5538	103	4	)	)	PUNCT
cana-5538	103	5	,	,	PUNCT
cana-5538	103	6	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	103	7	,	,	PUNCT
cana-5538	103	8	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	103	9	)	)	PUNCT
cana-5538	103	10	,	,	PUNCT
cana-5538	103	11	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	103	12	,	,	PUNCT
cana-5538	103	13	𝑇	𝑇	PROPN
cana-5538	103	14	𝑧	𝑧	PART
cana-5538	103	15	)	)	PUNCT
cana-5538	103	16	}	}	PUNCT
cana-5538	103	17	(	(	PUNCT
cana-5538	103	18	2.14	2.14	NUM
cana-5538	103	19	)	)	PUNCT
cana-5538	103	20	taking	take	VERB
cana-5538	103	21	𝑛	𝑛	PRON
cana-5538	103	22	→	→	SYM
cana-5538	103	23	∞	∞	NUM
cana-5538	103	24	we	we	PRON
cana-5538	103	25	get	get	VERB
cana-5538	103	26	𝑀	𝑀	PROPN
cana-5538	103	27	(	(	PUNCT
cana-5538	103	28	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	103	29	,	,	PUNCT
cana-5538	103	30	𝑧	𝑧	NOUN
cana-5538	103	31	)	)	PUNCT
cana-5538	103	32	=	=	SYM
cana-5538	103	33	𝜌𝑐(𝑧	𝜌𝑐(𝑧	PROPN
cana-5538	103	34	,	,	PUNCT
cana-5538	103	35	𝑇	𝑇	PROPN
cana-5538	103	36	𝑧	𝑧	VERB
cana-5538	103	37	)	)	PUNCT
cana-5538	103	38	(	(	PUNCT
cana-5538	103	39	2.15	2.15	NUM
cana-5538	103	40	)	)	PUNCT
cana-5538	103	41	now	now	ADV
cana-5538	103	42	,	,	PUNCT
cana-5538	103	43	taking	take	VERB
cana-5538	103	44	𝑛	𝑛	PRON
cana-5538	103	45	→	→	SYM
cana-5538	103	46	∞	∞	NUM
cana-5538	103	47	in	in	ADP
cana-5538	103	48	(	(	PUNCT
cana-5538	103	49	2.12	2.12	NUM
cana-5538	103	50	)	)	PUNCT
cana-5538	103	51	we	we	PRON
cana-5538	103	52	get	get	VERB
cana-5538	103	53	that	that	SCONJ
cana-5538	103	54	||𝜌𝑐(𝑇	||𝜌𝑐(𝑇	ADV
cana-5538	103	55	𝑧	𝑧	ADJ
cana-5538	103	56	,	,	PUNCT
cana-5538	103	57	𝑧)||	𝑧)||	ADJ
cana-5538	103	58	≤	≤	NOUN
cana-5538	103	59	𝐾||𝜓(𝜌𝑐(𝑧	𝐾||𝜓(𝜌𝑐(𝑧	PROPN
cana-5538	103	60	,	,	PUNCT
cana-5538	103	61	𝑇	𝑇	PROPN
cana-5538	103	62	𝑧)||≤	𝑧)||≤	PROPN
cana-5538	103	63	𝐾||𝜌𝑐(𝑧	𝐾||𝜌𝑐(𝑧	PROPN
cana-5538	103	64	,	,	PUNCT
cana-5538	103	65	𝑇	𝑇	PROPN
cana-5538	103	66	𝑧)||	𝑧)||	ADJ
cana-5538	103	67	(	(	PUNCT
cana-5538	103	68	2.16	2.16	NUM
cana-5538	103	69	)	)	PUNCT
cana-5538	103	70	which	which	PRON
cana-5538	103	71	is	be	AUX
cana-5538	103	72	not	not	PART
cana-5538	103	73	true	true	ADJ
cana-5538	103	74	for	for	ADP
cana-5538	103	75	all	all	DET
cana-5538	103	76	𝐾	𝐾	PROPN
cana-5538	103	77	>	>	X
cana-5538	103	78	0	0	PROPN
cana-5538	103	79	.	.	PUNCT
cana-5538	104	1	so	so	ADV
cana-5538	104	2	we	we	PRON
cana-5538	104	3	get	get	VERB
cana-5538	104	4	a	a	DET
cana-5538	104	5	contradiction	contradiction	NOUN
cana-5538	104	6	.	.	PUNCT
cana-5538	105	1	𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒	𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒	NOUN
cana-5538	105	2	||𝜌𝑐(𝑇	||𝜌𝑐(𝑇	ADV
cana-5538	105	3	𝑧	𝑧	PART
cana-5538	105	4	,	,	PUNCT
cana-5538	105	5	𝑧)||	𝑧)||	NUM
cana-5538	105	6	→	→	SYM
cana-5538	105	7	0	0	NUM
cana-5538	105	8	as	as	ADP
cana-5538	105	9	𝑛	𝑛	PROPN
cana-5538	105	10	→	→	SYM
cana-5538	105	11	∞.	∞.	PROPN
cana-5538	105	12	it	it	PRON
cana-5538	105	13	implies	imply	VERB
cana-5538	105	14	that	that	SCONJ
cana-5538	105	15	𝑇	𝑇	PROPN
cana-5538	105	16	𝑧	𝑧	NOUN
cana-5538	105	17	=	=	X
cana-5538	105	18	𝑧	𝑧	PROPN
cana-5538	105	19	and	and	CCONJ
cana-5538	105	20	hence	hence	ADV
cana-5538	105	21	z	z	PROPN
cana-5538	105	22	is	be	AUX
cana-5538	105	23	a	a	DET
cana-5538	105	24	fixed	fix	VERB
cana-5538	105	25	point	point	NOUN
cana-5538	105	26	of	of	ADP
cana-5538	105	27	t	t	PROPN
cana-5538	105	28	.this	.this	PRON
cana-5538	105	29	completes	complete	VERB
cana-5538	105	30	the	the	DET
cana-5538	105	31	proof	proof	NOUN
cana-5538	105	32	.	.	PUNCT
cana-5538	106	1	example	example	NOUN
cana-5538	106	2	2.4	2.4	NUM
cana-5538	106	3	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-5538	106	4	𝑋	𝑋	NOUN
cana-5538	106	5	=	=	PUNCT
cana-5538	107	1	[	[	X
cana-5538	107	2	0	0	NUM
cana-5538	107	3	,	,	PUNCT
cana-5538	107	4	∞	∞	PROPN
cana-5538	107	5	)	)	PUNCT
cana-5538	107	6	and	and	CCONJ
cana-5538	107	7	𝜌𝑐(𝑥	𝜌𝑐(𝑥	ADJ
cana-5538	107	8	,	,	PUNCT
cana-5538	107	9	𝑦	𝑦	NOUN
cana-5538	107	10	)	)	PUNCT
cana-5538	107	11	=	=	SYM
cana-5538	107	12	(	(	PUNCT
cana-5538	107	13	𝑚𝑎𝑥{𝑥	𝑚𝑎𝑥{𝑥	X
cana-5538	107	14	,	,	PUNCT
cana-5538	107	15	𝑦	𝑦	NOUN
cana-5538	107	16	}	}	PUNCT
cana-5538	107	17	,	,	PUNCT
cana-5538	107	18	𝑘	𝑘	PRON
cana-5538	107	19	𝑚𝑎𝑥{𝑥	𝑚𝑎𝑥{𝑥	PROPN
cana-5538	107	20	,	,	PUNCT
cana-5538	107	21	𝑦	𝑦	NOUN
cana-5538	107	22	}	}	PUNCT
cana-5538	107	23	)	)	PUNCT
cana-5538	107	24	.	.	PUNCT
cana-5538	108	1	then	then	ADV
cana-5538	108	2	(	(	PUNCT
cana-5538	108	3	𝑋	𝑋	PROPN
cana-5538	108	4	,	,	PUNCT
cana-5538	108	5	𝜌𝑐	𝜌𝑐	PROPN
cana-5538	108	6	)	)	PUNCT
cana-5538	108	7	is	be	AUX
cana-5538	108	8	a	a	DET
cana-5538	108	9	complete	complete	ADJ
cana-5538	108	10	partial	partial	ADJ
cana-5538	108	11	cone	cone	NOUN
cana-5538	108	12	metric	metric	ADJ
cana-5538	108	13	space	space	NOUN
cana-5538	108	14	.	.	PUNCT
cana-5538	109	1	consider	consider	VERB
cana-5538	109	2	the	the	DET
cana-5538	109	3	mapping	mapping	NOUN
cana-5538	109	4	𝑇	𝑇	PROPN
cana-5538	109	5	∶	∶	NOUN
cana-5538	109	6	𝑋	𝑋	PROPN
cana-5538	109	7	→	→	SYM
cana-5538	109	8	𝑋defined	𝑋define	VERB
cana-5538	109	9	by	by	ADP
cana-5538	109	10	𝑇(𝑥	𝑇(𝑥	NOUN
cana-5538	109	11	)	)	PUNCT
cana-5538	109	12	=	=	PRON
cana-5538	109	13	{	{	PUNCT
cana-5538	110	1	𝑥	𝑥	X
cana-5538	110	2	–	–	PUNCT
cana-5538	110	3	2	2	NUM
cana-5538	110	4	3	3	NUM
cana-5538	110	5	𝑥	𝑥	NOUN
cana-5538	110	6	>	>	SYM
cana-5538	110	7	1	1	NUM
cana-5538	110	8	𝑥	𝑥	DET
cana-5538	110	9	3	3	NUM
cana-5538	110	10	0	0	NUM
cana-5538	110	11	≤	≤	NUM
cana-5538	110	12	𝑥	𝑥	DET
cana-5538	110	13	≤	≤	NUM
cana-5538	110	14	1	1	NUM
cana-5538	110	15	(	(	PUNCT
cana-5538	110	16	2.17	2.17	NUM
cana-5538	110	17	)	)	PUNCT
cana-5538	111	1	and	and	CCONJ
cana-5538	111	2	let	let	VERB
cana-5538	111	3	𝜓	𝜓	PRON
cana-5538	111	4	∶	∶	VERB
cana-5538	111	5	[	[	X
cana-5538	111	6	0	0	NUM
cana-5538	111	7	,	,	PUNCT
cana-5538	111	8	∞	∞	PROPN
cana-5538	111	9	)	)	PUNCT
cana-5538	111	10	→	→	PUNCT
cana-5538	112	1	[	[	X
cana-5538	112	2	0	0	NUM
cana-5538	112	3	,	,	PUNCT
cana-5538	112	4	∞	∞	PROPN
cana-5538	112	5	)	)	PUNCT
cana-5538	112	6	be	be	VERB
cana-5538	112	7	such	such	ADJ
cana-5538	112	8	that	that	SCONJ
cana-5538	112	9	𝜓(𝑡	𝜓(𝑡	PROPN
cana-5538	112	10	)	)	PUNCT
cana-5538	112	11	=	=	SYM
cana-5538	112	12	𝑡	𝑡	ADP
cana-5538	112	13	2	2	NUM
cana-5538	112	14	for	for	ADP
cana-5538	112	15	all	all	DET
cana-5538	112	16	𝑡	𝑡	PROPN
cana-5538	112	17	≥	≥	NOUN
cana-5538	112	18	0	0	NUM
cana-5538	112	19	.	.	PUNCT
cana-5538	113	1	if	if	SCONJ
cana-5538	113	2	we	we	PRON
cana-5538	113	3	define	define	VERB
cana-5538	113	4	the	the	DET
cana-5538	113	5	functions	function	NOUN
cana-5538	113	6	𝛼	𝛼	VERB
cana-5538	113	7	,	,	PUNCT
cana-5538	113	8	𝛽	𝛽	NOUN
cana-5538	113	9	∶	∶	NOUN
cana-5538	113	10	𝑋	𝑋	NOUN
cana-5538	113	11	×	×	NOUN
cana-5538	113	12	𝑋	𝑋	NOUN
cana-5538	113	13	→	→	SYM
cana-5538	113	14	[	[	X
cana-5538	113	15	0	0	NUM
cana-5538	113	16	,	,	PUNCT
cana-5538	113	17	∞	∞	PROPN
cana-5538	113	18	)	)	PUNCT
cana-5538	113	19	as	as	ADP
cana-5538	113	20	𝛼(𝑥	𝛼(𝑥	PROPN
cana-5538	113	21	,	,	PUNCT
cana-5538	113	22	𝑦	𝑦	NOUN
cana-5538	113	23	)	)	PUNCT
cana-5538	113	24	=	=	SYM
cana-5538	113	25	{	{	PUNCT
cana-5538	113	26	3	3	NUM
cana-5538	113	27	2	2	NUM
cana-5538	113	28	𝑥	𝑥	NOUN
cana-5538	113	29	,	,	PUNCT
cana-5538	113	30	𝑦	𝑦	NOUN
cana-5538	113	31	∈	∈	NOUN
cana-5538	113	32	[	[	X
cana-5538	113	33	0,1	0,1	NUM
cana-5538	113	34	]	]	SYM
cana-5538	113	35	0	0	NUM
cana-5538	113	36	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
cana-5538	113	37	(	(	PUNCT
cana-5538	113	38	2.18	2.18	NUM
cana-5538	113	39	)	)	PUNCT
cana-5538	113	40	we	we	PRON
cana-5538	113	41	show	show	VERB
cana-5538	113	42	that	that	SCONJ
cana-5538	113	43	contractive	contractive	ADJ
cana-5538	113	44	condition	condition	NOUN
cana-5538	113	45	of	of	ADP
cana-5538	113	46	theorem	theorem	ADJ
cana-5538	113	47	2.3	2.3	NUM
cana-5538	113	48	is	be	AUX
cana-5538	113	49	satisfied	satisfied	ADJ
cana-5538	113	50	.	.	PUNCT
cana-5538	114	1	without	without	ADP
cana-5538	114	2	loss	loss	NOUN
cana-5538	114	3	of	of	ADP
cana-5538	114	4	generality	generality	NOUN
cana-5538	114	5	we	we	PRON
cana-5538	114	6	assume	assume	VERB
cana-5538	114	7	that	that	SCONJ
cana-5538	114	8	𝑥	𝑥	PROPN
cana-5538	114	9	≥	≥	NOUN
cana-5538	114	10	𝑦.	𝑦.	VERB
cana-5538	114	11	then	then	ADV
cana-5538	114	12	for	for	ADP
cana-5538	114	13	𝑥	𝑥	PROPN
cana-5538	114	14	,	,	PUNCT
cana-5538	114	15	𝑦	𝑦	PRON
cana-5538	114	16	∈	∈	NOUN
cana-5538	115	1	[	[	X
cana-5538	115	2	0	0	NUM
cana-5538	115	3	,	,	PUNCT
cana-5538	115	4	1	1	NUM
cana-5538	115	5	]	]	PUNCT
cana-5538	115	6	we	we	PRON
cana-5538	115	7	get	get	VERB
cana-5538	115	8	𝛼(𝑥	𝛼(𝑥	PROPN
cana-5538	115	9	,	,	PUNCT
cana-5538	115	10	𝑦)𝜌𝑐(𝑇	𝑦)𝜌𝑐(𝑇	ADV
cana-5538	115	11	𝑥	𝑥	NOUN
cana-5538	115	12	,	,	PUNCT
cana-5538	115	13	𝑇	𝑇	PROPN
cana-5538	115	14	𝑦	𝑦	NOUN
cana-5538	115	15	)	)	PUNCT
cana-5538	115	16	=	=	SYM
cana-5538	116	1	𝛼(𝑥	𝛼(𝑥	PROPN
cana-5538	116	2	,	,	PUNCT
cana-5538	116	3	𝑦)𝜌𝑐	𝑦)𝜌𝑐	PROPN
cana-5538	116	4	(	(	PUNCT
cana-5538	116	5	𝑥	𝑥	PROPN
cana-5538	116	6	3	3	NUM
cana-5538	116	7	,	,	PUNCT
cana-5538	116	8	𝑦	𝑦	NOUN
cana-5538	116	9	3	3	NUM
cana-5538	116	10	)	)	PUNCT
cana-5538	116	11	=	=	SYM
cana-5538	116	12	3	3	NUM
cana-5538	116	13	2	2	NUM
cana-5538	116	14	(	(	PUNCT
cana-5538	116	15	𝑥	𝑥	PROPN
cana-5538	116	16	3	3	NUM
cana-5538	116	17	,	,	PUNCT
cana-5538	116	18	𝑘𝑥	𝑘𝑥	NOUN
cana-5538	116	19	3	3	NUM
cana-5538	116	20	)	)	PUNCT
cana-5538	116	21	=	=	SYM
cana-5538	116	22	1	1	NUM
cana-5538	116	23	4	4	NUM
cana-5538	116	24	(	(	PUNCT
cana-5538	116	25	𝑥	𝑥	NOUN
cana-5538	116	26	,	,	PUNCT
cana-5538	116	27	𝑘𝑥	𝑘𝑥	NOUN
cana-5538	116	28	)	)	PUNCT
cana-5538	116	29	≤	≤	NOUN
cana-5538	116	30	1	1	NUM
cana-5538	116	31	2	2	NUM
cana-5538	116	32	(	(	PUNCT
cana-5538	116	33	𝑥	𝑥	NOUN
cana-5538	116	34	,	,	PUNCT
cana-5538	116	35	𝑘𝑥	𝑘𝑥	NOUN
cana-5538	116	36	)	)	PUNCT
cana-5538	116	37	=	=	SYM
cana-5538	116	38	1	1	NUM
cana-5538	116	39	2	2	NUM
cana-5538	116	40	𝑚𝑎𝑥{(𝑥	𝑚𝑎𝑥{(𝑥	NOUN
cana-5538	116	41	,	,	PUNCT
cana-5538	116	42	𝑘𝑥	𝑘𝑥	NOUN
cana-5538	116	43	)	)	PUNCT
cana-5538	116	44	,	,	PUNCT
cana-5538	116	45	(	(	PUNCT
cana-5538	116	46	𝑥	𝑥	NOUN
cana-5538	116	47	,	,	PUNCT
cana-5538	116	48	𝑘𝑥	𝑘𝑥	NOUN
cana-5538	116	49	)	)	PUNCT
cana-5538	116	50	,	,	PUNCT
cana-5538	116	51	(	(	PUNCT
cana-5538	116	52	𝑦	𝑦	NOUN
cana-5538	116	53	,	,	PUNCT
cana-5538	116	54	𝑘𝑦	𝑘𝑦	NOUN
cana-5538	116	55	)	)	PUNCT
cana-5538	116	56	}	}	PUNCT
cana-5538	116	57	=	=	SYM
cana-5538	116	58	1	1	NUM
cana-5538	116	59	2	2	NUM
cana-5538	116	60	𝑚𝑎𝑥{𝜌𝑐(𝑥	𝑚𝑎𝑥{𝜌𝑐(𝑥	NUM
cana-5538	116	61	,	,	PUNCT
cana-5538	116	62	𝑦	𝑦	NOUN
cana-5538	116	63	)	)	PUNCT
cana-5538	116	64	,	,	PUNCT
cana-5538	116	65	𝜌𝑐(𝑥	𝜌𝑐(𝑥	PROPN
cana-5538	116	66	,	,	PUNCT
cana-5538	116	67	𝑇	𝑇	PROPN
cana-5538	116	68	𝑥	𝑥	PROPN
cana-5538	116	69	)	)	PUNCT
cana-5538	116	70	,	,	PUNCT
cana-5538	116	71	𝜌𝑐(𝑦	𝜌𝑐(𝑦	PROPN
cana-5538	116	72	,	,	PUNCT
cana-5538	116	73	𝑇	𝑇	PROPN
cana-5538	116	74	𝑦	𝑦	NOUN
cana-5538	116	75	)	)	PUNCT
cana-5538	116	76	}	}	PUNCT
cana-5538	116	77	=	=	SYM
cana-5538	116	78	𝜓(𝑚𝑎𝑥{𝜌𝑐(𝑥	𝜓(𝑚𝑎𝑥{𝜌𝑐(𝑥	ADJ
cana-5538	116	79	,	,	PUNCT
cana-5538	116	80	𝑦	𝑦	NOUN
cana-5538	116	81	)	)	PUNCT
cana-5538	116	82	,	,	PUNCT
cana-5538	116	83	𝜌𝑐(𝑥	𝜌𝑐(𝑥	PROPN
cana-5538	116	84	,	,	PUNCT
cana-5538	116	85	𝑇	𝑇	PROPN
cana-5538	116	86	𝑥	𝑥	PROPN
cana-5538	116	87	)	)	PUNCT
cana-5538	116	88	,	,	PUNCT
cana-5538	116	89	𝜌𝑐(𝑦	𝜌𝑐(𝑦	PROPN
cana-5538	116	90	,	,	PUNCT
cana-5538	116	91	𝑇	𝑇	PROPN
cana-5538	116	92	𝑦	𝑦	NOUN
cana-5538	116	93	)	)	PUNCT
cana-5538	116	94	}	}	PUNCT
cana-5538	116	95	)	)	PUNCT
cana-5538	116	96	(	(	PUNCT
cana-5538	116	97	2.19	2.19	NUM
cana-5538	116	98	)	)	PUNCT
cana-5538	116	99	theorem	theorem	VERB
cana-5538	116	100	2.5	2.5	NUM
cana-5538	116	101	let	let	VERB
cana-5538	116	102	(	(	PUNCT
cana-5538	116	103	𝑋	𝑋	PROPN
cana-5538	116	104	,	,	PUNCT
cana-5538	116	105	𝜌𝑐	𝜌𝑐	PRON
cana-5538	116	106	)	)	PUNCT
cana-5538	116	107	be	be	AUX
cana-5538	116	108	a	a	DET
cana-5538	116	109	complete	complete	ADJ
cana-5538	116	110	partial	partial	ADJ
cana-5538	116	111	cone	cone	NOUN
cana-5538	116	112	metric	metric	ADJ
cana-5538	116	113	space	space	NOUN
cana-5538	116	114	p	p	NOUN
cana-5538	116	115	is	be	AUX
cana-5538	116	116	a	a	DET
cana-5538	116	117	normal	normal	ADJ
cana-5538	116	118	cone	cone	NOUN
cana-5538	116	119	with	with	ADP
cana-5538	116	120	constant	constant	ADJ
cana-5538	116	121	k.	k.	PROPN
cana-5538	116	122	suppose	suppose	VERB
cana-5538	116	123	the	the	DET
cana-5538	116	124	mapping	mapping	NOUN
cana-5538	116	125	𝑇	𝑇	PROPN
cana-5538	116	126	∶	∶	NOUN
cana-5538	116	127	𝑋	𝑋	NOUN
cana-5538	116	128	→	→	PUNCT
cana-5538	116	129	𝑋	𝑋	PROPN
cana-5538	116	130	satisfies	satisfy	VERB
cana-5538	116	131	the	the	DET
cana-5538	116	132	contractive	contractive	ADJ
cana-5538	116	133	condition	condition	NOUN
cana-5538	116	134	communications	communication	NOUN
cana-5538	116	135	on	on	ADP
cana-5538	116	136	applied	apply	VERB
cana-5538	116	137	nonlinear	nonlinear	ADJ
cana-5538	116	138	analysis	analysis	NOUN
cana-5538	116	139	issn	issn	NOUN
cana-5538	116	140	:	:	PUNCT
cana-5538	116	141	1074	1074	NUM
cana-5538	116	142	-	-	PUNCT
cana-5538	116	143	133x	133x	NUM
cana-5538	116	144	vol	vol	VERB
cana-5538	116	145	32	32	NUM
cana-5538	116	146	no	no	NOUN
cana-5538	116	147	.	.	PUNCT
cana-5538	117	1	10s	10	NOUN
cana-5538	117	2	(	(	PUNCT
cana-5538	117	3	2025	2025	NUM
cana-5538	117	4	)	)	PUNCT
cana-5538	117	5	2618	2618	NUM
cana-5538	117	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	117	7	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	117	8	𝑥	𝑥	PROPN
cana-5538	117	9	,	,	PUNCT
cana-5538	117	10	𝑇	𝑇	PROPN
cana-5538	117	11	𝑦	𝑦	NOUN
cana-5538	117	12	)	)	PUNCT
cana-5538	117	13	)	)	PUNCT
cana-5538	118	1	≤	≤	NOUN
cana-5538	118	2	𝑎1	𝑎1	ADP
cana-5538	118	3	𝜌𝑐(𝑥	𝜌𝑐(𝑥	ADP
cana-5538	118	4	,	,	PUNCT
cana-5538	118	5	𝑦	𝑦	NOUN
cana-5538	118	6	)	)	PUNCT
cana-5538	118	7	+	+	CCONJ
cana-5538	118	8	𝑎2	𝑎2	NOUN
cana-5538	118	9	𝜌𝑐(𝑥,𝑇	𝜌𝑐(𝑥,𝑇	PUNCT
cana-5538	118	10	𝑥)𝜌𝑐(𝑦,𝑇	𝑥)𝜌𝑐(𝑦,𝑇	NUM
cana-5538	118	11	𝑦	𝑦	NUM
cana-5538	118	12	)	)	PUNCT
cana-5538	118	13	𝜌𝑐(𝑥,𝑦)+	𝜌𝑐(𝑥,𝑦)+	NUM
cana-5538	118	14	𝜌𝑐(𝑦,𝑇	𝜌𝑐(𝑦,𝑇	PROPN
cana-5538	118	15	𝑥)+	𝑥)+	VERB
cana-5538	118	16	𝜌𝑐(𝑥,𝑇	𝜌𝑐(𝑥,𝑇	PUNCT
cana-5538	118	17	𝑦	𝑦	X
cana-5538	118	18	)	)	PUNCT
cana-5538	118	19	+	+	CCONJ
cana-5538	118	20	𝑎3	𝑎3	PROPN
cana-5538	118	21	𝜌𝑐(𝑦,𝑇	𝜌𝑐(𝑦,𝑇	PROPN
cana-5538	118	22	𝑦	𝑦	NUM
cana-5538	118	23	)	)	PUNCT
cana-5538	118	24	𝜌𝑐(𝑥,𝑇	𝜌𝑐(𝑥,𝑇	PUNCT
cana-5538	118	25	𝑥	𝑥	X
cana-5538	118	26	)	)	PUNCT
cana-5538	118	27	𝜌𝑐(𝑥,𝑦	𝜌𝑐(𝑥,𝑦	NOUN
cana-5538	118	28	)	)	PUNCT
cana-5538	118	29	+	+	CCONJ
cana-5538	118	30	𝑎4	𝑎4	PROPN
cana-5538	118	31	𝜌𝑐(𝑥	𝜌𝑐(𝑥	VERB
cana-5538	118	32	,	,	PUNCT
cana-5538	118	33	𝑇	𝑇	PROPN
cana-5538	118	34	𝑦	𝑦	NOUN
cana-5538	118	35	)	)	PUNCT
cana-5538	118	36	+	+	CCONJ
cana-5538	118	37	𝑎5	𝑎5	PROPN
cana-5538	118	38	𝜌𝑐(𝑦	𝜌𝑐(𝑦	NOUN
cana-5538	118	39	,	,	PUNCT
cana-5538	118	40	𝑇	𝑇	PROPN
cana-5538	118	41	𝑥	𝑥	PROPN
cana-5538	118	42	)	)	PUNCT
cana-5538	118	43	(	(	PUNCT
cana-5538	118	44	2.20	2.20	NUM
cana-5538	118	45	)	)	PUNCT
cana-5538	118	46	where	where	SCONJ
cana-5538	118	47	𝑎1	𝑎1	NOUN
cana-5538	118	48	,	,	PUNCT
cana-5538	118	49	𝑎2	𝑎2	PROPN
cana-5538	118	50	,	,	PUNCT
cana-5538	118	51	𝑎3	𝑎3	PROPN
cana-5538	118	52	,	,	PUNCT
cana-5538	118	53	𝑎4	𝑎4	PROPN
cana-5538	118	54	,	,	PUNCT
cana-5538	118	55	𝑎5	𝑎5	PROPN
cana-5538	118	56	≥	≥	NUM
cana-5538	118	57	0	0	NUM
cana-5538	118	58	are	be	AUX
cana-5538	118	59	constants	constant	NOUN
cana-5538	118	60	such	such	ADJ
cana-5538	118	61	that	that	SCONJ
cana-5538	118	62	𝑎1	𝑎1	X
cana-5538	118	63	+	+	CCONJ
cana-5538	118	64	𝑎2	𝑎2	NOUN
cana-5538	118	65	+	+	CCONJ
cana-5538	118	66	𝑎3	𝑎3	PROPN
cana-5538	118	67	+	+	CCONJ
cana-5538	118	68	2𝑎4	2𝑎4	NUM
cana-5538	118	69	+	+	CCONJ
cana-5538	118	70	𝑎5	𝑎5	PROPN
cana-5538	118	71	<	<	X
cana-5538	118	72	1	1	NUM
cana-5538	118	73	.	.	PUNCT
cana-5538	119	1	then	then	ADV
cana-5538	119	2	t	t	PROPN
cana-5538	119	3	has	have	VERB
cana-5538	119	4	a	a	DET
cana-5538	119	5	unique	unique	ADJ
cana-5538	119	6	fixed	fix	VERB
cana-5538	119	7	point	point	NOUN
cana-5538	119	8	in	in	ADP
cana-5538	119	9	x.	x.	NOUN
cana-5538	119	10	proof	proof	NOUN
cana-5538	119	11	choose	choose	VERB
cana-5538	119	12	𝑥0	𝑥0	PROPN
cana-5538	119	13	∈	∈	NOUN
cana-5538	119	14	𝑋	𝑋	NOUN
cana-5538	119	15	such	such	ADJ
cana-5538	119	16	that	that	SCONJ
cana-5538	119	17	𝑇	𝑇	PROPN
cana-5538	119	18	𝑥0	𝑥0	NOUN
cana-5538	119	19	=	=	SYM
cana-5538	119	20	𝑥1	𝑥1	PROPN
cana-5538	119	21	,	,	PUNCT
cana-5538	119	22	𝑇	𝑇	PROPN
cana-5538	119	23	𝑥1	𝑥1	NOUN
cana-5538	119	24	=	=	SYM
cana-5538	119	25	𝑇2𝑥0	𝑇2𝑥0	PROPN
cana-5538	119	26	=	=	PUNCT
cana-5538	119	27	𝑥2	𝑥2	PROPN
cana-5538	119	28	.	.	PUNCT
cana-5538	119	29	.	.	PUNCT
cana-5538	119	30	.	.	PUNCT
cana-5538	120	1	𝑥𝑛	𝑥𝑛	VERB
cana-5538	120	2	=	=	SYM
cana-5538	120	3	𝑇	𝑇	PROPN
cana-5538	120	4	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-5538	120	5	=	=	SYM
cana-5538	120	6	𝑇𝑛𝑥0	𝑇𝑛𝑥0	PROPN
cana-5538	120	7	.	.	PUNCT
cana-5538	121	1	then	then	ADV
cana-5538	121	2	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	121	3	,	,	PUNCT
cana-5538	121	4	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	121	5	)	)	PUNCT
cana-5538	121	6	=	=	SYM
cana-5538	121	7	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	121	8	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-5538	121	9	,	,	PUNCT
cana-5538	121	10	𝑇	𝑇	PROPN
cana-5538	121	11	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	121	12	)	)	PUNCT
cana-5538	121	13	≤	≤	PUNCT
cana-5538	122	1	𝑎1	𝑎1	PROPN
cana-5538	122	2	𝜌𝑐(𝑥𝑛−1	𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	122	3	,	,	PUNCT
cana-5538	122	4	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	122	5	)	)	PUNCT
cana-5538	123	1	+	+	CCONJ
cana-5538	123	2	𝑎2	𝑎2	NOUN
cana-5538	123	3	𝜌𝑐(𝑥𝑛−1,𝑇	𝜌𝑐(𝑥𝑛−1,𝑇	ADJ
cana-5538	123	4	𝑥𝑛−1)𝜌𝑐(𝑥𝑛,𝑇	𝑥𝑛−1)𝜌𝑐(𝑥𝑛,𝑇	NOUN
cana-5538	123	5	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	123	6	)	)	PUNCT
cana-5538	123	7	𝜌𝑐(𝑥𝑛−1,𝑥𝑛)+	𝜌𝑐(𝑥𝑛−1,𝑥𝑛)+	VERB
cana-5538	123	8	𝜌𝑐(𝑥𝑛,𝑇	𝜌𝑐(𝑥𝑛,𝑇	PUNCT
cana-5538	123	9	𝑥𝑛−1	𝑥𝑛−1	PROPN
cana-5538	123	10	)	)	PUNCT
cana-5538	124	1	+	+	CCONJ
cana-5538	124	2	𝜌𝑐(𝑥𝑛−1,𝑇	𝜌𝑐(𝑥𝑛−1,𝑇	ADJ
cana-5538	124	3	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	124	4	)	)	PUNCT
cana-5538	125	1	+	+	CCONJ
cana-5538	125	2	𝑎3	𝑎3	ADJ
cana-5538	125	3	𝜌𝑐(𝑥𝑛,𝑇	𝜌𝑐(𝑥𝑛,𝑇	PUNCT
cana-5538	125	4	𝑥𝑛)𝜌𝑐(𝑥𝑛−1,𝑇	𝑥𝑛)𝜌𝑐(𝑥𝑛−1,𝑇	ADJ
cana-5538	125	5	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-5538	125	6	)	)	PUNCT
cana-5538	125	7	𝜌𝑐(𝑥𝑛−1,𝑥𝑛	𝜌𝑐(𝑥𝑛−1,𝑥𝑛	NOUN
cana-5538	125	8	)	)	PUNCT
cana-5538	126	1	+	+	CCONJ
cana-5538	126	2	𝑎4𝜌𝑐(𝑥𝑛−1	𝑎4𝜌𝑐(𝑥𝑛−1	PROPN
cana-5538	126	3	,	,	PUNCT
cana-5538	126	4	𝑇	𝑇	PROPN
cana-5538	126	5	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	126	6	)	)	PUNCT
cana-5538	127	1	+	+	CCONJ
cana-5538	127	2	𝑎5𝜌𝑐(𝑥𝑛	𝑎5𝜌𝑐(𝑥𝑛	PROPN
cana-5538	127	3	,	,	PUNCT
cana-5538	127	4	𝑇	𝑇	PROPN
cana-5538	127	5	𝑥𝑛−1	𝑥𝑛−1	NOUN
cana-5538	127	6	)	)	PUNCT
cana-5538	127	7	≤	≤	PUNCT
cana-5538	128	1	𝑎1	𝑎1	PROPN
cana-5538	128	2	𝜌𝑐(𝑥𝑛−1	𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	128	3	,	,	PUNCT
cana-5538	128	4	𝑥𝑛	𝑥𝑛	PRON
cana-5538	128	5	)	)	PUNCT
cana-5538	129	1	+	+	CCONJ
cana-5538	129	2	𝑎2	𝑎2	PROPN
cana-5538	129	3	𝜌𝑐(𝑥𝑛−1,𝑥𝑛)𝜌𝑐(𝑥𝑛,𝑥𝑛+1	𝜌𝑐(𝑥𝑛−1,𝑥𝑛)𝜌𝑐(𝑥𝑛,𝑥𝑛+1	NOUN
cana-5538	129	4	)	)	PUNCT
cana-5538	129	5	𝜌𝑐(𝑥𝑛−1,𝑥𝑛)+	𝜌𝑐(𝑥𝑛−1,𝑥𝑛)+	X
cana-5538	130	1	𝜌𝑐(𝑥𝑛,𝑥𝑛	𝜌𝑐(𝑥𝑛,𝑥𝑛	NUM
cana-5538	130	2	)	)	PUNCT
cana-5538	130	3	+	+	CCONJ
cana-5538	130	4	𝜌𝑐(𝑥𝑛−1,𝑥𝑛+1	𝜌𝑐(𝑥𝑛−1,𝑥𝑛+1	NOUN
cana-5538	130	5	)	)	PUNCT
cana-5538	130	6	+	+	CCONJ
cana-5538	130	7	𝑎3	𝑎3	PROPN
cana-5538	130	8	𝜌𝑐(𝑥𝑛,𝑥𝑛+1)𝜌𝑐(𝑥𝑛−1,𝑥𝑛	𝜌𝑐(𝑥𝑛,𝑥𝑛+1)𝜌𝑐(𝑥𝑛−1,𝑥𝑛	NOUN
cana-5538	130	9	)	)	PUNCT
cana-5538	130	10	𝜌𝑐(𝑥𝑛−1,𝑥𝑛	𝜌𝑐(𝑥𝑛−1,𝑥𝑛	NOUN
cana-5538	130	11	)	)	PUNCT
cana-5538	130	12	+	+	CCONJ
cana-5538	131	1	𝑎4𝜌𝑐(𝑥𝑛−1	𝑎4𝜌𝑐(𝑥𝑛−1	ADJ
cana-5538	131	2	,	,	PUNCT
cana-5538	131	3	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	131	4	)	)	PUNCT
cana-5538	132	1	+	+	CCONJ
cana-5538	132	2	𝑎5𝜌𝑐(𝑥𝑛	𝑎5𝜌𝑐(𝑥𝑛	PROPN
cana-5538	132	3	,	,	PUNCT
cana-5538	132	4	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	132	5	)	)	PUNCT
cana-5538	132	6	≤	≤	NOUN
cana-5538	132	7	𝑎1𝜌𝑐(𝑥𝑛−1	𝑎1𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	132	8	,	,	PUNCT
cana-5538	132	9	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	132	10	)	)	PUNCT
cana-5538	133	1	+	+	CCONJ
cana-5538	133	2	𝑎2𝜌𝑐(𝑥𝑛−1	𝑎2𝜌𝑐(𝑥𝑛−1	PROPN
cana-5538	133	3	,	,	PUNCT
cana-5538	133	4	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	133	5	)	)	PUNCT
cana-5538	133	6	+	+	CCONJ
cana-5538	133	7	𝑎3𝜌𝑐(𝑥𝑛	𝑎3𝜌𝑐(𝑥𝑛	PROPN
cana-5538	133	8	,	,	PUNCT
cana-5538	133	9	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	133	10	)	)	PUNCT
cana-5538	133	11	+	+	CCONJ
cana-5538	134	1	𝑎4𝜌𝑐(𝑥𝑛−1	𝑎4𝜌𝑐(𝑥𝑛−1	ADJ
cana-5538	134	2	,	,	PUNCT
cana-5538	134	3	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	134	4	)	)	PUNCT
cana-5538	135	1	+	+	CCONJ
cana-5538	135	2	𝑎5𝜌𝑐(𝑥𝑛	𝑎5𝜌𝑐(𝑥𝑛	PROPN
cana-5538	135	3	,	,	PUNCT
cana-5538	135	4	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	135	5	)	)	PUNCT
cana-5538	135	6	≤	≤	NOUN
cana-5538	135	7	𝑎1𝜌𝑐(𝑥𝑛−1	𝑎1𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	135	8	,	,	PUNCT
cana-5538	135	9	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	135	10	)	)	PUNCT
cana-5538	136	1	+	+	CCONJ
cana-5538	136	2	𝑎2𝜌𝑐(𝑥𝑛−1	𝑎2𝜌𝑐(𝑥𝑛−1	PROPN
cana-5538	136	3	,	,	PUNCT
cana-5538	136	4	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	136	5	)	)	PUNCT
cana-5538	136	6	+	+	CCONJ
cana-5538	136	7	𝑎3𝜌𝑐(𝑥𝑛	𝑎3𝜌𝑐(𝑥𝑛	PROPN
cana-5538	136	8	,	,	PUNCT
cana-5538	136	9	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	136	10	)	)	PUNCT
cana-5538	137	1	+	+	CCONJ
cana-5538	138	1	𝑎4𝜌𝑐(𝑥𝑛−1	𝑎4𝜌𝑐(𝑥𝑛−1	ADJ
cana-5538	138	2	,	,	PUNCT
cana-5538	138	3	𝑥𝑛	𝑥𝑛	PRON
cana-5538	138	4	)	)	PUNCT
cana-5538	139	1	+	+	CCONJ
cana-5538	139	2	𝑎4𝜌𝑐(𝑥𝑛	𝑎4𝜌𝑐(𝑥𝑛	PROPN
cana-5538	139	3	,	,	PUNCT
cana-5538	139	4	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	139	5	)	)	PUNCT
cana-5538	139	6	−	−	PRON
cana-5538	139	7	𝑎4𝜌𝑐(𝑥𝑛	𝑎4𝜌𝑐(𝑥𝑛	PROPN
cana-5538	139	8	,	,	PUNCT
cana-5538	139	9	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	139	10	)	)	PUNCT
cana-5538	140	1	+	+	CCONJ
cana-5538	140	2	𝑎5𝜌𝑐(𝑥𝑛	𝑎5𝜌𝑐(𝑥𝑛	PROPN
cana-5538	140	3	,	,	PUNCT
cana-5538	140	4	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	140	5	)	)	PUNCT
cana-5538	140	6	=	=	PUNCT
cana-5538	141	1	(	(	PUNCT
cana-5538	141	2	𝑎1	𝑎1	X
cana-5538	141	3	+	+	CCONJ
cana-5538	141	4	𝑎2	𝑎2	NOUN
cana-5538	141	5	+	+	CCONJ
cana-5538	141	6	𝑎4)𝜌𝑐(𝑥𝑛−1	𝑎4)𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	141	7	,	,	PUNCT
cana-5538	141	8	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	141	9	)	)	PUNCT
cana-5538	142	1	+	+	CCONJ
cana-5538	142	2	(	(	PUNCT
cana-5538	142	3	𝑎3	𝑎3	PROPN
cana-5538	142	4	+	+	PROPN
cana-5538	142	5	𝑎4)𝜌𝑐(𝑥𝑛	𝑎4)𝜌𝑐(𝑥𝑛	PROPN
cana-5538	142	6	,	,	PUNCT
cana-5538	142	7	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	142	8	)	)	PUNCT
cana-5538	142	9	+	+	CCONJ
cana-5538	142	10	(	(	PUNCT
cana-5538	142	11	𝑎5	𝑎5	PROPN
cana-5538	142	12	–	–	PUNCT
cana-5538	142	13	𝑎4)𝜌𝑐(𝑥𝑛	𝑎4)𝜌𝑐(𝑥𝑛	PROPN
cana-5538	142	14	,	,	PUNCT
cana-5538	142	15	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	142	16	)	)	PUNCT
cana-5538	142	17	=	=	PUNCT
cana-5538	143	1	(	(	PUNCT
cana-5538	143	2	𝑎1	𝑎1	X
cana-5538	143	3	+	+	CCONJ
cana-5538	143	4	𝑎2	𝑎2	NOUN
cana-5538	143	5	+	+	CCONJ
cana-5538	143	6	𝑎4)𝜌𝑐(𝑥𝑛−1	𝑎4)𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	143	7	,	,	PUNCT
cana-5538	143	8	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	143	9	)	)	PUNCT
cana-5538	144	1	+	+	CCONJ
cana-5538	144	2	(	(	PUNCT
cana-5538	144	3	𝑎3	𝑎3	PROPN
cana-5538	144	4	+	+	CCONJ
cana-5538	144	5	𝑎4	𝑎4	PROPN
cana-5538	144	6	+	+	CCONJ
cana-5538	144	7	𝑎5)𝜌𝑐(𝑥𝑛	𝑎5)𝜌𝑐(𝑥𝑛	PROPN
cana-5538	144	8	,	,	PUNCT
cana-5538	144	9	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	144	10	)	)	PUNCT
cana-5538	144	11	≤	≤	NOUN
cana-5538	144	12	(	(	PUNCT
cana-5538	144	13	𝑎1	𝑎1	NOUN
cana-5538	144	14	+	+	CCONJ
cana-5538	144	15	𝑎2	𝑎2	PROPN
cana-5538	144	16	+	+	CCONJ
cana-5538	144	17	𝑎4	𝑎4	PROPN
cana-5538	144	18	)	)	PUNCT
cana-5538	144	19	1	1	NUM
cana-5538	144	20	−	−	PROPN
cana-5538	145	1	(	(	PUNCT
cana-5538	145	2	𝑎3	𝑎3	PROPN
cana-5538	145	3	+	+	CCONJ
cana-5538	145	4	𝑎4	𝑎4	PROPN
cana-5538	145	5	+	+	CCONJ
cana-5538	145	6	𝑎5	𝑎5	PROPN
cana-5538	145	7	)	)	PUNCT
cana-5538	145	8	𝜌𝑐(𝑥𝑛−1	𝜌𝑐(𝑥𝑛−1	NOUN
cana-5538	145	9	,	,	PUNCT
cana-5538	145	10	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	145	11	)	)	PUNCT
cana-5538	145	12	(	(	PUNCT
cana-5538	145	13	2.21	2.21	NUM
cana-5538	145	14	)	)	PUNCT
cana-5538	146	1	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-5538	146	2	𝜆	𝜆	NOUN
cana-5538	146	3	=	=	PUNCT
cana-5538	146	4	(	(	PUNCT
cana-5538	146	5	𝑎1	𝑎1	X
cana-5538	146	6	+	+	CCONJ
cana-5538	146	7	𝑎2	𝑎2	PROPN
cana-5538	146	8	+	+	CCONJ
cana-5538	146	9	𝑎4	𝑎4	PROPN
cana-5538	146	10	)	)	PUNCT
cana-5538	146	11	1	1	NUM
cana-5538	146	12	−	−	PROPN
cana-5538	147	1	(	(	PUNCT
cana-5538	147	2	𝑎3	𝑎3	PROPN
cana-5538	147	3	+	+	CCONJ
cana-5538	147	4	𝑎4	𝑎4	PROPN
cana-5538	147	5	+	+	CCONJ
cana-5538	147	6	𝑎5	𝑎5	PROPN
cana-5538	147	7	)	)	PUNCT
cana-5538	147	8	since	since	SCONJ
cana-5538	147	9	𝑎1	𝑎1	PRON
cana-5538	147	10	+	+	CCONJ
cana-5538	147	11	𝑎2	𝑎2	NOUN
cana-5538	147	12	+	+	CCONJ
cana-5538	147	13	𝑎3	𝑎3	PROPN
cana-5538	147	14	+	+	CCONJ
cana-5538	147	15	2𝑎4	2𝑎4	NUM
cana-5538	147	16	+	+	CCONJ
cana-5538	147	17	𝑎5	𝑎5	PROPN
cana-5538	147	18	<	<	X
cana-5538	147	19	1	1	NUM
cana-5538	147	20	and	and	CCONJ
cana-5538	147	21	𝑎3	𝑎3	PROPN
cana-5538	147	22	+	+	CCONJ
cana-5538	147	23	𝑎4	𝑎4	PROPN
cana-5538	147	24	+	+	CCONJ
cana-5538	147	25	𝑎5	𝑎5	PROPN
cana-5538	147	26	<	<	X
cana-5538	147	27	1	1	NUM
cana-5538	147	28	implies	imply	VERB
cana-5538	147	29	that	that	SCONJ
cana-5538	147	30	𝜆	𝜆	X
cana-5538	147	31	<	<	X
cana-5538	147	32	1	1	NUM
cana-5538	147	33	.	.	PUNCT
cana-5538	147	34	hence	hence	ADV
cana-5538	147	35	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	147	36	,	,	PUNCT
cana-5538	147	37	𝑥𝑛+1	𝑥𝑛+1	NOUN
cana-5538	147	38	)	)	PUNCT
cana-5538	147	39	≤	≤	NOUN
cana-5538	147	40	𝜆𝜌𝑐(𝑥𝑛−1	𝜆𝜌𝑐(𝑥𝑛−1	PROPN
cana-5538	147	41	,	,	PUNCT
cana-5538	147	42	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	147	43	)	)	PUNCT
cana-5538	147	44	(	(	PUNCT
cana-5538	147	45	2.22	2.22	NUM
cana-5538	147	46	)	)	PUNCT
cana-5538	147	47	for	for	ADP
cana-5538	147	48	all	all	DET
cana-5538	147	49	𝑛	𝑛	DET
cana-5538	147	50	∈	∈	PROPN
cana-5538	147	51	ℕ.	ℕ.	PROPN
cana-5538	147	52	for	for	ADP
cana-5538	147	53	any	any	DET
cana-5538	147	54	𝑚	𝑚	PROPN
cana-5538	147	55	>	>	X
cana-5538	147	56	𝑛	𝑛	PROPN
cana-5538	147	57	where	where	SCONJ
cana-5538	147	58	𝑚	𝑚	NOUN
cana-5538	147	59	,	,	PUNCT
cana-5538	147	60	𝑛	𝑛	DET
cana-5538	147	61	∈	∈	PROPN
cana-5538	147	62	ℕ	ℕ	PROPN
cana-5538	147	63	we	we	PRON
cana-5538	147	64	have	have	VERB
cana-5538	147	65	𝜌𝑐(𝑥𝑚	𝜌𝑐(𝑥𝑚	PROPN
cana-5538	147	66	,	,	PUNCT
cana-5538	147	67	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	147	68	)	)	PUNCT
cana-5538	147	69	≤	≤	PUNCT
cana-5538	148	1	𝜌𝑐(𝑥𝑚	𝜌𝑐(𝑥𝑚	ADJ
cana-5538	148	2	,	,	PUNCT
cana-5538	148	3	𝑥𝑚−1	𝑥𝑚−1	PROPN
cana-5538	148	4	)	)	PUNCT
cana-5538	148	5	+	+	CCONJ
cana-5538	148	6	𝜌𝑐(𝑥𝑚−1	𝜌𝑐(𝑥𝑚−1	NOUN
cana-5538	148	7	,	,	PUNCT
cana-5538	148	8	𝑥𝑚−2	𝑥𝑚−2	NOUN
cana-5538	148	9	)	)	PUNCT
cana-5538	148	10	.	.	PUNCT
cana-5538	148	11	.	.	PUNCT
cana-5538	148	12	.	.	PUNCT
cana-5538	148	13	.	.	PUNCT
cana-5538	148	14	.	.	PUNCT
cana-5538	148	15	.	.	PUNCT
cana-5538	148	16	.	.	PUNCT
cana-5538	148	17	.	.	PUNCT
cana-5538	149	1	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	NOUN
cana-5538	149	2	,	,	PUNCT
cana-5538	149	3	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	149	4	)	)	PUNCT
cana-5538	149	5	−	−	PROPN
cana-5538	149	6	∑	∑	SYM
cana-5538	149	7	𝜌𝑐(𝑥𝑚−𝑘	𝜌𝑐(𝑥𝑚−𝑘	NOUN
cana-5538	149	8	,	,	PUNCT
cana-5538	149	9	𝑥𝑚−𝑘	𝑥𝑚−𝑘	NOUN
cana-5538	149	10	)	)	PUNCT
cana-5538	149	11	𝑚−𝑛−1	𝑚−𝑛−1	ADJ
cana-5538	149	12	𝑘	𝑘	X
cana-5538	149	13	≤	≤	ADJ
cana-5538	149	14	𝜌𝑐(𝑥𝑚	𝜌𝑐(𝑥𝑚	NOUN
cana-5538	149	15	,	,	PUNCT
cana-5538	149	16	𝑥𝑚−1	𝑥𝑚−1	PROPN
cana-5538	149	17	)	)	PUNCT
cana-5538	149	18	+	+	CCONJ
cana-5538	149	19	𝜌𝑐(𝑥𝑚−1	𝜌𝑐(𝑥𝑚−1	NOUN
cana-5538	149	20	,	,	PUNCT
cana-5538	149	21	𝑥𝑚−2	𝑥𝑚−2	NOUN
cana-5538	149	22	)	)	PUNCT
cana-5538	149	23	.	.	PUNCT
cana-5538	149	24	.	.	PUNCT
cana-5538	149	25	.	.	PUNCT
cana-5538	149	26	.	.	PUNCT
cana-5538	149	27	.	.	PUNCT
cana-5538	149	28	.	.	PUNCT
cana-5538	149	29	.	.	PUNCT
cana-5538	149	30	.	.	PUNCT
cana-5538	150	1	𝜌𝑐	𝜌𝑐	PRON
cana-5538	150	2	(	(	PUNCT
cana-5538	150	3	𝑥𝑛+1	𝑥𝑛+1	PROPN
cana-5538	150	4	,	,	PUNCT
cana-5538	150	5	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	150	6	)	)	PUNCT
cana-5538	150	7	≤	≤	NOUN
cana-5538	150	8	(	(	PUNCT
cana-5538	150	9	𝜆𝑚−1	𝜆𝑚−1	PROPN
cana-5538	150	10	+	+	CCONJ
cana-5538	150	11	𝜆𝑚−1	𝜆𝑚−1	PROPN
cana-5538	150	12	+	+	X
cana-5538	150	13	.	.	PUNCT
cana-5538	150	14	.	.	PUNCT
cana-5538	150	15	.	.	PUNCT
cana-5538	150	16	.	.	PUNCT
cana-5538	150	17	.	.	PUNCT
cana-5538	150	18	.	.	PUNCT
cana-5538	150	19	.	.	PUNCT
cana-5538	151	1	𝜆𝑛)𝜌𝑐(𝑥0	𝜆𝑛)𝜌𝑐(𝑥0	PROPN
cana-5538	151	2	,	,	PUNCT
cana-5538	151	3	𝑥1	𝑥1	NOUN
cana-5538	151	4	)	)	PUNCT
cana-5538	151	5	=	=	PUNCT
cana-5538	152	1	𝜆𝑛	𝜆𝑛	ADP
cana-5538	152	2	1	1	NUM
cana-5538	152	3	−	−	NOUN
cana-5538	152	4	𝜆	𝜆	DET
cana-5538	152	5	𝜌𝑐(𝑥0	𝜌𝑐(𝑥0	NUM
cana-5538	152	6	,	,	PUNCT
cana-5538	152	7	𝑥1	𝑥1	NOUN
cana-5538	152	8	)	)	PUNCT
cana-5538	152	9	(	(	PUNCT
cana-5538	152	10	2.23	2.23	NUM
cana-5538	152	11	)	)	PUNCT
cana-5538	152	12	since	since	SCONJ
cana-5538	152	13	p	p	NOUN
cana-5538	152	14	is	be	AUX
cana-5538	152	15	normal	normal	ADJ
cana-5538	152	16	cone	cone	NOUN
cana-5538	152	17	with	with	ADP
cana-5538	152	18	normal	normal	ADJ
cana-5538	152	19	constant	constant	ADJ
cana-5538	152	20	k	k	NOUN
cana-5538	152	21	,	,	PUNCT
cana-5538	152	22	we	we	PRON
cana-5538	152	23	have	have	VERB
cana-5538	152	24	communications	communication	NOUN
cana-5538	152	25	on	on	ADP
cana-5538	152	26	applied	apply	VERB
cana-5538	152	27	nonlinear	nonlinear	ADJ
cana-5538	152	28	analysis	analysis	NOUN
cana-5538	152	29	issn	issn	NOUN
cana-5538	152	30	:	:	PUNCT
cana-5538	152	31	1074	1074	NUM
cana-5538	152	32	-	-	PUNCT
cana-5538	152	33	133x	133x	NUM
cana-5538	152	34	vol	vol	VERB
cana-5538	152	35	32	32	NUM
cana-5538	152	36	no	no	NOUN
cana-5538	152	37	.	.	PUNCT
cana-5538	153	1	10s	10	NOUN
cana-5538	153	2	(	(	PUNCT
cana-5538	153	3	2025	2025	NUM
cana-5538	153	4	)	)	PUNCT
cana-5538	153	5	2619	2619	NUM
cana-5538	154	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	154	2	||𝜌𝑐(𝑥𝑚	||𝜌𝑐(𝑥𝑚	NUM
cana-5538	154	3	,	,	PUNCT
cana-5538	154	4	𝑥𝑛))||	𝑥𝑛))||	PROPN
cana-5538	154	5	≤	≤	NUM
cana-5538	154	6	𝐾||	𝐾||	PUNCT
cana-5538	155	1	𝜆𝑛	𝜆𝑛	ADP
cana-5538	155	2	1	1	NUM
cana-5538	155	3	−	−	NOUN
cana-5538	155	4	𝜆	𝜆	PRON
cana-5538	155	5	𝜌𝑐(𝑥0	𝜌𝑐(𝑥0	NUM
cana-5538	155	6	,	,	PUNCT
cana-5538	155	7	𝑥1)||	𝑥1)||	PROPN
cana-5538	155	8	(	(	PUNCT
cana-5538	155	9	2.24	2.24	NUM
cana-5538	155	10	)	)	PUNCT
cana-5538	155	11	now	now	ADV
cana-5538	155	12	since	since	SCONJ
cana-5538	155	13	𝜆	𝜆	PRON
cana-5538	155	14	<	<	X
cana-5538	155	15	1	1	NUM
cana-5538	155	16	,	,	PUNCT
cana-5538	155	17	||𝜌𝑐(𝑥𝑚	||𝜌𝑐(𝑥𝑚	NUM
cana-5538	155	18	,	,	PUNCT
cana-5538	155	19	𝑥𝑛))||	𝑥𝑛))||	PROPN
cana-5538	155	20	≤	≤	NUM
cana-5538	155	21	𝐾||	𝐾||	PUNCT
cana-5538	156	1	𝜆𝑛	𝜆𝑛	PROPN
cana-5538	156	2	1−𝜆	1−𝜆	NUM
cana-5538	156	3	𝜌𝑐(𝑥0	𝜌𝑐(𝑥0	NOUN
cana-5538	156	4	,	,	PUNCT
cana-5538	156	5	𝑥1)||	𝑥1)||	PROPN
cana-5538	156	6	→	→	SYM
cana-5538	156	7	0	0	NUM
cana-5538	156	8	as	as	SCONJ
cana-5538	156	9	𝑛	𝑛	PROPN
cana-5538	156	10	→	→	SYM
cana-5538	156	11	∞	∞	NOUN
cana-5538	156	12	hence	hence	ADV
cana-5538	156	13	{	{	PUNCT
cana-5538	156	14	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	156	15	}	}	PUNCT
cana-5538	156	16	is	be	AUX
cana-5538	156	17	a	a	DET
cana-5538	156	18	cauchy	cauchy	ADJ
cana-5538	156	19	sequence	sequence	NOUN
cana-5538	156	20	in	in	ADP
cana-5538	156	21	a	a	DET
cana-5538	156	22	partial	partial	ADJ
cana-5538	156	23	cone	cone	NOUN
cana-5538	156	24	metric	metric	ADJ
cana-5538	156	25	space	space	NOUN
cana-5538	156	26	which	which	PRON
cana-5538	156	27	is	be	AUX
cana-5538	156	28	complete	complete	ADJ
cana-5538	156	29	hence	hence	ADV
cana-5538	156	30	it	it	PRON
cana-5538	156	31	must	must	AUX
cana-5538	156	32	be	be	AUX
cana-5538	156	33	convergent	convergent	ADJ
cana-5538	156	34	in	in	ADP
cana-5538	156	35	x	x	NOUN
cana-5538	156	36	,	,	PUNCT
cana-5538	156	37	let	let	VERB
cana-5538	156	38	lim	lim	PROPN
cana-5538	156	39	𝑛→∞	𝑛→∞	VERB
cana-5538	156	40	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	156	41	=	=	SYM
cana-5538	156	42	𝑧	𝑧	X
cana-5538	156	43	therefore	therefore	ADV
cana-5538	156	44	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	156	45	,	,	PUNCT
cana-5538	156	46	𝑧	𝑧	X
cana-5538	156	47	)	)	PUNCT
cana-5538	156	48	=	=	SYM
cana-5538	156	49	lim	lim	PROPN
cana-5538	156	50	𝑛→∞	𝑛→∞	NUM
cana-5538	156	51	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	156	52	,	,	PUNCT
cana-5538	156	53	𝑧	𝑧	NOUN
cana-5538	156	54	)	)	PUNCT
cana-5538	156	55	=	=	SYM
cana-5538	156	56	lim	lim	PROPN
cana-5538	156	57	𝑛→∞	𝑛→∞	NUM
cana-5538	156	58	𝜌𝑐(𝑥𝑛	𝜌𝑐(𝑥𝑛	PROPN
cana-5538	156	59	,	,	PUNCT
cana-5538	156	60	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	156	61	)	)	PUNCT
cana-5538	156	62	=	=	SYM
cana-5538	157	1	0	0	PUNCT
cana-5538	158	1	now	now	ADV
cana-5538	158	2	we	we	PRON
cana-5538	158	3	show	show	VERB
cana-5538	158	4	that	that	SCONJ
cana-5538	158	5	||𝜌𝑐(𝑇	||𝜌𝑐(𝑇	ADV
cana-5538	158	6	𝑧	𝑧	ADJ
cana-5538	158	7	,	,	PUNCT
cana-5538	158	8	𝑧)||	𝑧)||	ADJ
cana-5538	158	9	≥	≥	NOUN
cana-5538	158	10	0	0	NUM
cana-5538	158	11	,	,	PUNCT
cana-5538	158	12	on	on	ADP
cana-5538	158	13	contrary	contrary	ADV
cana-5538	158	14	,	,	PUNCT
cana-5538	158	15	assume	assume	VERB
cana-5538	158	16	||𝜌𝑐(𝑇	||𝜌𝑐(𝑇	ADV
cana-5538	158	17	𝑧	𝑧	ADJ
cana-5538	158	18	,	,	PUNCT
cana-5538	158	19	𝑧)||	𝑧)||	ADJ
cana-5538	158	20	>	>	X
cana-5538	158	21	0	0	NUM
cana-5538	159	1	we	we	PRON
cana-5538	159	2	have	have	VERB
cana-5538	159	3	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	159	4	𝑧	𝑧	PROPN
cana-5538	159	5	,	,	PUNCT
cana-5538	159	6	𝑧	𝑧	NOUN
cana-5538	159	7	)	)	PUNCT
cana-5538	159	8	≤	≤	NOUN
cana-5538	159	9	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	159	10	𝑧	𝑧	PROPN
cana-5538	159	11	,	,	PUNCT
cana-5538	159	12	𝑇	𝑇	PROPN
cana-5538	159	13	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	159	14	)	)	PUNCT
cana-5538	159	15	+	+	CCONJ
cana-5538	159	16	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	159	17	𝑥𝑛	𝑥𝑛	VERB
cana-5538	159	18	,	,	PUNCT
cana-5538	159	19	𝑧	𝑧	NOUN
cana-5538	159	20	)	)	PUNCT
cana-5538	159	21	–	–	PUNCT
cana-5538	159	22	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	159	23	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	159	24	,	,	PUNCT
cana-5538	159	25	𝑇	𝑇	PROPN
cana-5538	159	26	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	159	27	)	)	PUNCT
cana-5538	159	28	≤𝑎1𝜌𝑐(𝑧	≤𝑎1𝜌𝑐(𝑧	PROPN
cana-5538	159	29	,	,	PUNCT
cana-5538	159	30	𝑥𝑛	𝑥𝑛	PRON
cana-5538	159	31	)	)	PUNCT
cana-5538	160	1	+	+	CCONJ
cana-5538	160	2	𝑎2	𝑎2	NOUN
cana-5538	160	3	𝜌𝑐(𝑧,𝑇	𝜌𝑐(𝑧,𝑇	PROPN
cana-5538	160	4	𝑧)𝜌𝑐(𝑥𝑛,𝑇	𝑧)𝜌𝑐(𝑥𝑛,𝑇	PROPN
cana-5538	160	5	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	160	6	)	)	PUNCT
cana-5538	160	7	𝜌𝑐(𝑧,𝑥𝑛)+	𝜌𝑐(𝑧,𝑥𝑛)+	PROPN
cana-5538	160	8	𝜌𝑐(𝑥𝑛,𝑇	𝜌𝑐(𝑥𝑛,𝑇	NUM
cana-5538	160	9	𝑧)+	𝑧)+	NUM
cana-5538	160	10	𝜌𝑐(𝑧,𝑇	𝜌𝑐(𝑧,𝑇	PROPN
cana-5538	160	11	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	160	12	)	)	PUNCT
cana-5538	161	1	+	+	CCONJ
cana-5538	161	2	𝑎3	𝑎3	PROPN
cana-5538	161	3	𝜌𝑐(𝑥𝑛,𝑇	𝜌𝑐(𝑥𝑛,𝑇	PUNCT
cana-5538	161	4	𝑥𝑛)𝜌𝑐(𝑧,𝑇	𝑥𝑛)𝜌𝑐(𝑧,𝑇	NOUN
cana-5538	161	5	𝑧	𝑧	NOUN
cana-5538	161	6	)	)	PUNCT
cana-5538	161	7	𝜌𝑐(𝑧,𝑥𝑛	𝜌𝑐(𝑧,𝑥𝑛	NOUN
cana-5538	161	8	)	)	PUNCT
cana-5538	162	1	+	+	ADJ
cana-5538	162	2	𝑎4𝜌𝑐(𝑧	𝑎4𝜌𝑐(𝑧	PROPN
cana-5538	162	3	,	,	PUNCT
cana-5538	162	4	𝑇	𝑇	PROPN
cana-5538	162	5	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	162	6	)	)	PUNCT
cana-5538	163	1	+	+	CCONJ
cana-5538	163	2	𝑎5𝜌𝑐(𝑥𝑛	𝑎5𝜌𝑐(𝑥𝑛	PROPN
cana-5538	163	3	,	,	PUNCT
cana-5538	163	4	𝑇	𝑇	PROPN
cana-5538	163	5	𝑧	𝑧	PART
cana-5538	163	6	)	)	PUNCT
cana-5538	163	7	+	+	CCONJ
cana-5538	163	8	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	163	9	𝑥𝑛	𝑥𝑛	VERB
cana-5538	163	10	,	,	PUNCT
cana-5538	163	11	𝑧	𝑧	NOUN
cana-5538	163	12	)	)	PUNCT
cana-5538	163	13	–	–	PUNCT
cana-5538	163	14	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	163	15	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	163	16	,	,	PUNCT
cana-5538	163	17	𝑇	𝑇	PROPN
cana-5538	163	18	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	163	19	)	)	PUNCT
cana-5538	163	20	≤𝑎1𝜌𝑐(𝑧	≤𝑎1𝜌𝑐(𝑧	PROPN
cana-5538	163	21	,	,	PUNCT
cana-5538	163	22	𝑥𝑛	𝑥𝑛	PRON
cana-5538	163	23	)	)	PUNCT
cana-5538	164	1	+	+	CCONJ
cana-5538	164	2	𝑎2	𝑎2	NOUN
cana-5538	164	3	𝜌𝑐(𝑧,𝑇	𝜌𝑐(𝑧,𝑇	PROPN
cana-5538	164	4	𝑧)𝜌𝑐(𝑥𝑛,𝑥𝑛+1	𝑧)𝜌𝑐(𝑥𝑛,𝑥𝑛+1	NOUN
cana-5538	164	5	)	)	PUNCT
cana-5538	164	6	𝜌𝑐(𝑧,𝑥𝑛)+	𝜌𝑐(𝑧,𝑥𝑛)+	PROPN
cana-5538	164	7	𝜌𝑐(𝑥𝑛,𝑇	𝜌𝑐(𝑥𝑛,𝑇	NUM
cana-5538	164	8	𝑧)+	𝑧)+	NUM
cana-5538	164	9	𝜌𝑐(𝑧,𝑥𝑛+1	𝜌𝑐(𝑧,𝑥𝑛+1	PROPN
cana-5538	164	10	)	)	PUNCT
cana-5538	165	1	+	+	CCONJ
cana-5538	165	2	𝑎3	𝑎3	PROPN
cana-5538	165	3	𝜌𝑐(𝑥𝑛,𝑥𝑛+1)𝜌𝑐(𝑧,𝑇	𝜌𝑐(𝑥𝑛,𝑥𝑛+1)𝜌𝑐(𝑧,𝑇	PROPN
cana-5538	165	4	𝑧	𝑧	NOUN
cana-5538	165	5	)	)	PUNCT
cana-5538	165	6	𝜌𝑐(𝑧,𝑥𝑛	𝜌𝑐(𝑧,𝑥𝑛	NOUN
cana-5538	165	7	)	)	PUNCT
cana-5538	166	1	+	+	ADJ
cana-5538	166	2	𝑎4𝜌𝑐(𝑧	𝑎4𝜌𝑐(𝑧	PROPN
cana-5538	166	3	,	,	PUNCT
cana-5538	166	4	𝑇	𝑇	PROPN
cana-5538	166	5	𝑥𝑛	𝑥𝑛	PROPN
cana-5538	166	6	)	)	PUNCT
cana-5538	167	1	+	+	CCONJ
cana-5538	167	2	𝑎5𝜌𝑐(𝑥𝑛	𝑎5𝜌𝑐(𝑥𝑛	PROPN
cana-5538	167	3	,	,	PUNCT
cana-5538	167	4	𝑇	𝑇	PROPN
cana-5538	167	5	𝑧	𝑧	PART
cana-5538	167	6	)	)	PUNCT
cana-5538	168	1	+	+	NUM
cana-5538	168	2	𝜌𝑐	𝜌𝑐	X
cana-5538	168	3	(	(	PUNCT
cana-5538	168	4	𝑥𝑛+1	𝑥𝑛+1	PROPN
cana-5538	168	5	,	,	PUNCT
cana-5538	168	6	𝑧	𝑧	NOUN
cana-5538	168	7	)	)	PUNCT
cana-5538	168	8	≤	≤	NOUN
cana-5538	168	9	𝑎1𝜌𝑐(𝑧	𝑎1𝜌𝑐(𝑧	PROPN
cana-5538	168	10	,	,	PUNCT
cana-5538	168	11	𝑥𝑛	𝑥𝑛	PRON
cana-5538	168	12	)	)	PUNCT
cana-5538	168	13	+	+	CCONJ
cana-5538	169	1	+	+	ADJ
cana-5538	169	2	𝑎4𝜌𝑐(𝑧	𝑎4𝜌𝑐(𝑧	PROPN
cana-5538	169	3	,	,	PUNCT
cana-5538	169	4	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	169	5	)	)	PUNCT
cana-5538	169	6	+	+	CCONJ
cana-5538	169	7	𝑎5𝜌𝑐(𝑥𝑛	𝑎5𝜌𝑐(𝑥𝑛	PROPN
cana-5538	169	8	,	,	PUNCT
cana-5538	169	9	𝑇	𝑇	PROPN
cana-5538	169	10	𝑧	𝑧	PART
cana-5538	169	11	)	)	PUNCT
cana-5538	169	12	+	+	CCONJ
cana-5538	169	13	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	PROPN
cana-5538	169	14	,	,	PUNCT
cana-5538	169	15	𝑧	𝑧	NOUN
cana-5538	169	16	)	)	PUNCT
cana-5538	169	17	≤	≤	NOUN
cana-5538	170	1	𝑎1𝜌𝑐(𝑧	𝑎1𝜌𝑐(𝑧	PROPN
cana-5538	170	2	,	,	PUNCT
cana-5538	170	3	𝑥𝑛	𝑥𝑛	PRON
cana-5538	170	4	)	)	PUNCT
cana-5538	170	5	+	+	CCONJ
cana-5538	170	6	+	+	ADJ
cana-5538	170	7	𝑎4	𝑎4	NOUN
cana-5538	170	8	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	170	9	,	,	PUNCT
cana-5538	170	10	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-5538	170	11	)	)	PUNCT
cana-5538	170	12	+	+	CCONJ
cana-5538	170	13	𝑎5𝜌𝑐(𝑇	𝑎5𝜌𝑐(𝑇	ADJ
cana-5538	170	14	𝑧	𝑧	ADP
cana-5538	170	15	,	,	PUNCT
cana-5538	170	16	𝑧	𝑧	NOUN
cana-5538	170	17	)	)	PUNCT
cana-5538	170	18	+	+	CCONJ
cana-5538	170	19	𝑎5𝜌𝑐(𝑧	𝑎5𝜌𝑐(𝑧	PROPN
cana-5538	170	20	,	,	PUNCT
cana-5538	170	21	𝑥𝑛	𝑥𝑛	NOUN
cana-5538	170	22	)	)	PUNCT
cana-5538	170	23	–	–	PUNCT
cana-5538	170	24	𝑎5𝜌𝑐(𝑧	𝑎5𝜌𝑐(𝑧	PROPN
cana-5538	170	25	,	,	PUNCT
cana-5538	170	26	𝑧	𝑧	ADJ
cana-5538	170	27	)	)	PUNCT
cana-5538	170	28	+	+	CCONJ
cana-5538	170	29	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	ADJ
cana-5538	170	30	,	,	PUNCT
cana-5538	170	31	𝑧	𝑧	NOUN
cana-5538	170	32	)	)	PUNCT
cana-5538	170	33	≤	≤	NOUN
cana-5538	170	34	(	(	PUNCT
cana-5538	170	35	𝑎1	𝑎1	NOUN
cana-5538	170	36	+	+	CCONJ
cana-5538	170	37	𝑎5)𝜌𝑐(𝑧	𝑎5)𝜌𝑐(𝑧	PROPN
cana-5538	170	38	,	,	PUNCT
cana-5538	170	39	𝑥𝑛	𝑥𝑛	VERB
cana-5538	170	40	)	)	PUNCT
cana-5538	171	1	+	+	PUNCT
cana-5538	172	1	+	+	PROPN
cana-5538	172	2	[	[	X
cana-5538	172	3	𝑎4𝜌𝑐(𝑥𝑛+1	𝑎4𝜌𝑐(𝑥𝑛+1	X
cana-5538	172	4	,	,	PUNCT
cana-5538	172	5	𝑧	𝑧	NOUN
cana-5538	172	6	)	)	PUNCT
cana-5538	172	7	+	+	CCONJ
cana-5538	172	8	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	PROPN
cana-5538	172	9	,	,	PUNCT
cana-5538	172	10	𝑧	𝑧	NOUN
cana-5538	172	11	)	)	PUNCT
cana-5538	172	12	]	]	PUNCT
cana-5538	173	1	+	+	CCONJ
cana-5538	173	2	𝑎5𝜌𝑐(𝑇	𝑎5𝜌𝑐(𝑇	ADJ
cana-5538	173	3	𝑧	𝑧	ADP
cana-5538	173	4	,	,	PUNCT
cana-5538	173	5	𝑧	𝑧	NOUN
cana-5538	173	6	)	)	PUNCT
cana-5538	173	7	(	(	PUNCT
cana-5538	173	8	2.25	2.25	NUM
cana-5538	173	9	)	)	PUNCT
cana-5538	173	10	so	so	ADV
cana-5538	173	11	using	use	VERB
cana-5538	173	12	(	(	PUNCT
cana-5538	173	13	2.25	2.25	NUM
cana-5538	173	14	)	)	PUNCT
cana-5538	173	15	we	we	PRON
cana-5538	173	16	have	have	VERB
cana-5538	173	17	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	173	18	𝑧	𝑧	PROPN
cana-5538	173	19	,	,	PUNCT
cana-5538	173	20	𝑧	𝑧	NOUN
cana-5538	173	21	)	)	PUNCT
cana-5538	173	22	≤	≤	NOUN
cana-5538	173	23	(	(	PUNCT
cana-5538	173	24	𝑎1	𝑎1	NOUN
cana-5538	173	25	+	+	CCONJ
cana-5538	173	26	𝑎5	𝑎5	PROPN
cana-5538	173	27	)	)	PUNCT
cana-5538	173	28	1	1	NUM
cana-5538	173	29	–	–	PUNCT
cana-5538	173	30	𝑎5	𝑎5	PROPN
cana-5538	173	31	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	173	32	,	,	PUNCT
cana-5538	173	33	𝑥𝑛	𝑥𝑛	PRON
cana-5538	173	34	)	)	PUNCT
cana-5538	174	1	+	+	CCONJ
cana-5538	174	2	𝑎4	𝑎4	PROPN
cana-5538	174	3	1	1	NUM
cana-5538	174	4	–	–	PUNCT
cana-5538	174	5	𝑎5	𝑎5	PROPN
cana-5538	174	6	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	NOUN
cana-5538	174	7	,	,	PUNCT
cana-5538	174	8	𝑧	𝑧	PROPN
cana-5538	174	9	)	)	PUNCT
cana-5538	174	10	+	+	CCONJ
cana-5538	174	11	1	1	NUM
cana-5538	174	12	1	1	NUM
cana-5538	174	13	–	–	PUNCT
cana-5538	174	14	𝑎5	𝑎5	PROPN
cana-5538	174	15	𝜌𝑐(𝑥𝑛+1	𝜌𝑐(𝑥𝑛+1	NOUN
cana-5538	174	16	,	,	PUNCT
cana-5538	174	17	𝑧	𝑧	NOUN
cana-5538	174	18	)	)	PUNCT
cana-5538	174	19	(	(	PUNCT
cana-5538	174	20	2.26	2.26	NUM
cana-5538	174	21	)	)	PUNCT
cana-5538	174	22	hence	hence	ADV
cana-5538	174	23	||𝜌𝑐(𝑇	||𝜌𝑐(𝑇	ADV
cana-5538	174	24	𝑧	𝑧	PART
cana-5538	174	25	,	,	PUNCT
cana-5538	174	26	𝑧)||	𝑧)||	ADJ
cana-5538	174	27	≤	≤	NOUN
cana-5538	174	28	(	(	PUNCT
cana-5538	174	29	𝑎1	𝑎1	NOUN
cana-5538	174	30	+	+	CCONJ
cana-5538	174	31	𝑎5	𝑎5	PROPN
cana-5538	174	32	)	)	PUNCT
cana-5538	174	33	1	1	NUM
cana-5538	174	34	–	–	PUNCT
cana-5538	174	35	𝑎5	𝑎5	PROPN
cana-5538	174	36	𝐾||𝜌𝑐(𝑧	𝐾||𝜌𝑐(𝑧	NUM
cana-5538	174	37	,	,	PUNCT
cana-5538	174	38	𝑥𝑛)||	𝑥𝑛)||	NOUN
cana-5538	174	39	+	+	CCONJ
cana-5538	174	40	𝑎4	𝑎4	PROPN
cana-5538	174	41	1	1	NUM
cana-5538	174	42	–	–	PUNCT
cana-5538	174	43	𝑎5	𝑎5	PROPN
cana-5538	174	44	𝐾||𝜌𝑐(𝑥𝑛+1	𝐾||𝜌𝑐(𝑥𝑛+1	PROPN
cana-5538	174	45	,	,	PUNCT
cana-5538	174	46	𝑧)||	𝑧)||	ADJ
cana-5538	174	47	+	+	CCONJ
cana-5538	174	48	1	1	NUM
cana-5538	174	49	1	1	NUM
cana-5538	174	50	–	–	PUNCT
cana-5538	174	51	𝑎5	𝑎5	PROPN
cana-5538	174	52	𝐾||𝜌𝑐(𝑥𝑛+1	𝐾||𝜌𝑐(𝑥𝑛+1	PROPN
cana-5538	174	53	,	,	PUNCT
cana-5538	174	54	𝑧)||	𝑧)||	NUM
cana-5538	174	55	→	→	SYM
cana-5538	174	56	0	0	NUM
cana-5538	174	57	(	(	PUNCT
cana-5538	174	58	2.27	2.27	NUM
cana-5538	174	59	)	)	PUNCT
cana-5538	174	60	so	so	SCONJ
cana-5538	174	61	we	we	PRON
cana-5538	174	62	have	have	VERB
cana-5538	174	63	||𝜌𝑐(𝑇	||𝜌𝑐(𝑇	ADV
cana-5538	174	64	𝑧	𝑧	ADJ
cana-5538	174	65	,	,	PUNCT
cana-5538	174	66	𝑧)||	𝑧)||	ADJ
cana-5538	174	67	=	=	SYM
cana-5538	174	68	0	0	NUM
cana-5538	174	69	therefore	therefore	ADV
cana-5538	174	70	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	174	71	𝑧	𝑧	PROPN
cana-5538	174	72	,	,	PUNCT
cana-5538	174	73	𝑧	𝑧	NOUN
cana-5538	174	74	)	)	PUNCT
cana-5538	174	75	=	=	SYM
cana-5538	174	76	0	0	NUM
cana-5538	174	77	or	or	CCONJ
cana-5538	174	78	t	t	PROPN
cana-5538	174	79	z	z	NOUN
cana-5538	174	80	=	=	PUNCT
cana-5538	174	81	z.	z.	PROPN
cana-5538	174	82	uniqueness	uniqueness	PROPN
cana-5538	174	83	if	if	SCONJ
cana-5538	174	84	𝑧1	𝑧1	NOUN
cana-5538	174	85	is	be	AUX
cana-5538	174	86	s	s	PRON
cana-5538	174	87	another	another	DET
cana-5538	174	88	fixed	fix	VERB
cana-5538	174	89	point	point	NOUN
cana-5538	174	90	of	of	ADP
cana-5538	174	91	t	t	PROPN
cana-5538	174	92	,	,	PUNCT
cana-5538	174	93	then	then	ADV
cana-5538	174	94	𝑇	𝑇	PROPN
cana-5538	174	95	𝑧1	𝑧1	NOUN
cana-5538	174	96	=	=	NOUN
cana-5538	174	97	𝑧1	𝑧1	NOUN
cana-5538	174	98	replacing	replace	VERB
cana-5538	174	99	x	x	PUNCT
cana-5538	174	100	by	by	ADP
cana-5538	174	101	z	z	PROPN
cana-5538	174	102	and	and	CCONJ
cana-5538	174	103	y	y	PROPN
cana-5538	174	104	by	by	ADP
cana-5538	174	105	𝑧1	𝑧1	NOUN
cana-5538	174	106	in	in	ADP
cana-5538	174	107	(	(	PUNCT
cana-5538	174	108	2.1	2.1	NUM
cana-5538	174	109	)	)	PUNCT
cana-5538	174	110	we	we	PRON
cana-5538	174	111	get	get	VERB
cana-5538	174	112	𝜌𝑐(𝑧	𝜌𝑐(𝑧	PUNCT
cana-5538	174	113	,	,	PUNCT
cana-5538	174	114	𝑧1	𝑧1	NOUN
cana-5538	174	115	)	)	PUNCT
cana-5538	174	116	=	=	PUNCT
cana-5538	174	117	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	174	118	𝑧	𝑧	PROPN
cana-5538	174	119	,	,	PUNCT
cana-5538	174	120	𝑇	𝑇	PROPN
cana-5538	174	121	𝑧1	𝑧1	PROPN
cana-5538	174	122	)	)	PUNCT
cana-5538	174	123	≤	≤	NOUN
cana-5538	175	1	𝑎1𝜌𝑐(𝑧	𝑎1𝜌𝑐(𝑧	PROPN
cana-5538	175	2	,	,	PUNCT
cana-5538	175	3	𝑧1	𝑧1	NOUN
cana-5538	175	4	)	)	PUNCT
cana-5538	175	5	+	+	CCONJ
cana-5538	175	6	𝑎2	𝑎2	PROPN
cana-5538	175	7	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	175	8	,	,	PUNCT
cana-5538	175	9	𝑇	𝑇	PROPN
cana-5538	175	10	𝑧)𝜌𝑐(𝑧1	𝑧)𝜌𝑐(𝑧1	NOUN
cana-5538	175	11	,	,	PUNCT
cana-5538	175	12	𝑇	𝑇	PROPN
cana-5538	175	13	𝑧1	𝑧1	NOUN
cana-5538	175	14	)	)	PUNCT
cana-5538	175	15	𝜌𝑐(𝑧	𝜌𝑐(𝑧	PUNCT
cana-5538	175	16	,	,	PUNCT
cana-5538	175	17	𝑧1	𝑧1	NOUN
cana-5538	175	18	)	)	PUNCT
cana-5538	175	19	+	+	CCONJ
cana-5538	175	20	𝜌𝑐(𝑧1	𝜌𝑐(𝑧1	ADV
cana-5538	175	21	,	,	PUNCT
cana-5538	175	22	𝑇	𝑇	PROPN
cana-5538	175	23	𝑧	𝑧	PART
cana-5538	175	24	)	)	PUNCT
cana-5538	175	25	+	+	NOUN
cana-5538	175	26	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	175	27	,	,	PUNCT
cana-5538	175	28	𝑇	𝑇	PROPN
cana-5538	175	29	𝑧1	𝑧1	NOUN
cana-5538	175	30	)	)	PUNCT
cana-5538	176	1	+	+	CCONJ
cana-5538	176	2	𝑎3	𝑎3	PROPN
cana-5538	176	3	𝜌𝑐(𝑧1	𝜌𝑐(𝑧1	ADV
cana-5538	176	4	,	,	PUNCT
cana-5538	176	5	𝑇	𝑇	PROPN
cana-5538	176	6	𝑧1)𝜌𝑐(𝑧	𝑧1)𝜌𝑐(𝑧	PROPN
cana-5538	176	7	,	,	PUNCT
cana-5538	176	8	𝑇	𝑇	PROPN
cana-5538	176	9	𝑧	𝑧	PART
cana-5538	176	10	)	)	PUNCT
cana-5538	176	11	𝜌𝑐(𝑧	𝜌𝑐(𝑧	PUNCT
cana-5538	176	12	,	,	PUNCT
cana-5538	176	13	𝑧1	𝑧1	NOUN
cana-5538	176	14	)	)	PUNCT
cana-5538	176	15	+	+	CCONJ
cana-5538	176	16	𝑎4𝜌𝑐(𝑧	𝑎4𝜌𝑐(𝑧	PROPN
cana-5538	176	17	,	,	PUNCT
cana-5538	176	18	𝑇	𝑇	PROPN
cana-5538	176	19	𝑧1	𝑧1	NOUN
cana-5538	176	20	)	)	PUNCT
cana-5538	176	21	+	+	CCONJ
cana-5538	176	22	𝑎5𝜌𝑐(𝑧1	𝑎5𝜌𝑐(𝑧1	ADV
cana-5538	176	23	,	,	PUNCT
cana-5538	176	24	𝑇	𝑇	PROPN
cana-5538	176	25	𝑧	𝑧	PART
cana-5538	176	26	)	)	PUNCT
cana-5538	176	27	communications	communication	NOUN
cana-5538	176	28	on	on	ADP
cana-5538	176	29	applied	apply	VERB
cana-5538	176	30	nonlinear	nonlinear	ADJ
cana-5538	176	31	analysis	analysis	NOUN
cana-5538	176	32	issn	issn	NOUN
cana-5538	176	33	:	:	PUNCT
cana-5538	176	34	1074	1074	NUM
cana-5538	176	35	-	-	PUNCT
cana-5538	176	36	133x	133x	NUM
cana-5538	176	37	vol	vol	VERB
cana-5538	176	38	32	32	NUM
cana-5538	176	39	no	no	NOUN
cana-5538	176	40	.	.	PUNCT
cana-5538	177	1	10s	10	NOUN
cana-5538	177	2	(	(	PUNCT
cana-5538	177	3	2025	2025	NUM
cana-5538	177	4	)	)	PUNCT
cana-5538	177	5	2620	2620	NUM
cana-5538	177	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	178	1	≤	≤	X
cana-5538	178	2	𝑎1𝜌𝑐(𝑧	𝑎1𝜌𝑐(𝑧	PROPN
cana-5538	178	3	,	,	PUNCT
cana-5538	178	4	𝑧1	𝑧1	NOUN
cana-5538	178	5	)	)	PUNCT
cana-5538	179	1	+	+	CCONJ
cana-5538	179	2	𝑎2	𝑎2	NOUN
cana-5538	179	3	𝜌𝑐(𝑧,𝑧)𝜌𝑐(𝑧1,𝑧1	𝜌𝑐(𝑧,𝑧)𝜌𝑐(𝑧1,𝑧1	NUM
cana-5538	179	4	)	)	PUNCT
cana-5538	179	5	𝜌𝑐(𝑧,𝑧1	𝜌𝑐(𝑧,𝑧1	PROPN
cana-5538	179	6	)	)	PUNCT
cana-5538	180	1	+	+	NUM
cana-5538	180	2	𝜌𝑐(𝑧1,𝑧	𝜌𝑐(𝑧1,𝑧	NOUN
cana-5538	180	3	)	)	PUNCT
cana-5538	181	1	+	+	CCONJ
cana-5538	181	2	𝜌𝑐(𝑧,𝑧1	𝜌𝑐(𝑧,𝑧1	PROPN
cana-5538	181	3	)	)	PUNCT
cana-5538	182	1	+	+	CCONJ
cana-5538	182	2	𝑎3	𝑎3	PROPN
cana-5538	182	3	𝜌𝑐(𝑧1,𝑧1)𝜌𝑐(𝑧,𝑧	𝜌𝑐(𝑧1,𝑧1)𝜌𝑐(𝑧,𝑧	NOUN
cana-5538	182	4	)	)	PUNCT
cana-5538	182	5	𝜌𝑐(𝑧,𝑧1	𝜌𝑐(𝑧,𝑧1	PROPN
cana-5538	182	6	)	)	PUNCT
cana-5538	183	1	+	+	CCONJ
cana-5538	183	2	𝑎4𝜌𝑐(𝑧	𝑎4𝜌𝑐(𝑧	PROPN
cana-5538	183	3	,	,	PUNCT
cana-5538	183	4	𝑧1	𝑧1	NOUN
cana-5538	183	5	)	)	PUNCT
cana-5538	184	1	+	+	CCONJ
cana-5538	184	2	𝑎5𝜌𝑐(𝑧1	𝑎5𝜌𝑐(𝑧1	ADJ
cana-5538	184	3	,	,	PUNCT
cana-5538	184	4	𝑧	𝑧	NOUN
cana-5538	184	5	)	)	PUNCT
cana-5538	184	6	(	(	PUNCT
cana-5538	184	7	2.28	2.28	NUM
cana-5538	184	8	)	)	PUNCT
cana-5538	184	9	therefore	therefore	ADV
cana-5538	184	10	𝜌𝑐(𝑧	𝜌𝑐(𝑧	NUM
cana-5538	184	11	,	,	PUNCT
cana-5538	184	12	𝑧1	𝑧1	NOUN
cana-5538	184	13	)	)	PUNCT
cana-5538	184	14	=	=	SYM
cana-5538	184	15	0	0	NUM
cana-5538	184	16	or	or	CCONJ
cana-5538	184	17	𝑧	𝑧	PRON
cana-5538	184	18	=	=	NOUN
cana-5538	184	19	𝑧1	𝑧1	NOUN
cana-5538	184	20	.	.	PUNCT
cana-5538	185	1	3	3	X
cana-5538	185	2	.	.	X
cana-5538	185	3	application	application	NOUN
cana-5538	185	4	this	this	DET
cana-5538	185	5	section	section	NOUN
cana-5538	185	6	is	be	AUX
cana-5538	185	7	influenced	influence	VERB
cana-5538	185	8	by	by	ADP
cana-5538	185	9	the	the	DET
cana-5538	185	10	findings	finding	NOUN
cana-5538	185	11	presented	present	VERB
cana-5538	185	12	in	in	ADP
cana-5538	185	13	the	the	DET
cana-5538	185	14	papers	paper	NOUN
cana-5538	185	15	[	[	X
cana-5538	185	16	19	19	NUM
cana-5538	185	17	,	,	PUNCT
cana-5538	185	18	20	20	NUM
cana-5538	185	19	]	]	PUNCT
cana-5538	185	20	which	which	PRON
cana-5538	185	21	aims	aim	VERB
cana-5538	185	22	to	to	PART
cana-5538	185	23	offer	offer	VERB
cana-5538	185	24	an	an	DET
cana-5538	185	25	application	application	NOUN
cana-5538	185	26	of	of	ADP
cana-5538	185	27	theorem	theorem	ADJ
cana-5538	185	28	2.3	2.3	NUM
cana-5538	185	29	to	to	ADP
cana-5538	185	30	the	the	DET
cana-5538	185	31	solution	solution	NOUN
cana-5538	185	32	of	of	ADP
cana-5538	185	33	second	second	ADJ
cana-5538	185	34	order	order	NOUN
cana-5538	185	35	differential	differential	ADJ
cana-5538	185	36	equation	equation	NOUN
cana-5538	185	37	of	of	ADP
cana-5538	185	38	the	the	DET
cana-5538	185	39	form	form	NOUN
cana-5538	185	40	𝑥′′(𝑡	𝑥′′(𝑡	NOUN
cana-5538	185	41	)	)	PUNCT
cana-5538	185	42	=	=	SYM
cana-5538	185	43	−𝑓	−𝑓	ADJ
cana-5538	185	44	(	(	PUNCT
cana-5538	185	45	𝑡	𝑡	NOUN
cana-5538	185	46	,	,	PUNCT
cana-5538	185	47	𝑥(𝑡	𝑥(𝑡	NOUN
cana-5538	185	48	)	)	PUNCT
cana-5538	185	49	)	)	PUNCT
cana-5538	185	50	,	,	PUNCT
cana-5538	186	1	𝑡	𝑡	PROPN
cana-5538	186	2	∈	∈	PROPN
cana-5538	186	3	𝐼	𝐼	ADP
cana-5538	186	4	𝑥(0	𝑥(0	PROPN
cana-5538	186	5	)	)	PUNCT
cana-5538	187	1	=	=	PUNCT
cana-5538	187	2	𝑥(1	𝑥(1	NOUN
cana-5538	187	3	)	)	PUNCT
cana-5538	187	4	=	=	SYM
cana-5538	187	5	0	0	NUM
cana-5538	187	6	(	(	PUNCT
cana-5538	187	7	3.1	3.1	NUM
cana-5538	187	8	)	)	PUNCT
cana-5538	187	9	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-5538	187	10	𝐼	𝐼	NOUN
cana-5538	187	11	=	=	SYM
cana-5538	188	1	[	[	X
cana-5538	188	2	0	0	NUM
cana-5538	188	3	,	,	PUNCT
cana-5538	188	4	1	1	NUM
cana-5538	188	5	]	]	PUNCT
cana-5538	188	6	,	,	PUNCT
cana-5538	188	7	𝑓	𝑓	DET
cana-5538	188	8	∶	∶	NOUN
cana-5538	188	9	𝐼	𝐼	ADP
cana-5538	188	10	×	×	NOUN
cana-5538	188	11	ℝ	ℝ	PROPN
cana-5538	188	12	→	→	PUNCT
cana-5538	188	13	ℝ	ℝ	PROPN
cana-5538	188	14	is	be	AUX
cana-5538	188	15	a	a	DET
cana-5538	188	16	continuous	continuous	ADJ
cana-5538	188	17	function.consider	function.consider	NOUN
cana-5538	188	18	the	the	DET
cana-5538	188	19	space	space	NOUN
cana-5538	188	20	x	x	PUNCT
cana-5538	188	21	=	=	SYM
cana-5538	188	22	c(i	c(i	PROPN
cana-5538	188	23	)	)	PUNCT
cana-5538	188	24	of	of	ADP
cana-5538	188	25	continuous	continuous	ADJ
cana-5538	188	26	function	function	NOUN
cana-5538	188	27	defined	define	VERB
cana-5538	188	28	on	on	ADP
cana-5538	188	29	i.	i.	NOUN
cana-5538	188	30	it	it	PRON
cana-5538	188	31	is	be	AUX
cana-5538	188	32	well	well	ADV
cana-5538	188	33	-	-	PUNCT
cana-5538	188	34	known	know	VERB
cana-5538	188	35	that	that	SCONJ
cana-5538	188	36	the	the	DET
cana-5538	188	37	problem	problem	NOUN
cana-5538	188	38	(	(	PUNCT
cana-5538	188	39	3.1	3.1	NUM
cana-5538	188	40	)	)	PUNCT
cana-5538	188	41	is	be	AUX
cana-5538	188	42	equivalent	equivalent	ADJ
cana-5538	188	43	to	to	ADP
cana-5538	188	44	the	the	DET
cana-5538	188	45	integral	integral	ADJ
cana-5538	188	46	equation	equation	NOUN
cana-5538	188	47	𝑥(𝑡	𝑥(𝑡	NOUN
cana-5538	188	48	)	)	PUNCT
cana-5538	188	49	=	=	SYM
cana-5538	189	1	∫	∫	PROPN
cana-5538	189	2	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	189	3	,	,	PUNCT
cana-5538	189	4	𝑠)𝑓	𝑠)𝑓	NUM
cana-5538	189	5	(	(	PUNCT
cana-5538	189	6	𝑠	𝑠	INTJ
cana-5538	189	7	,	,	PUNCT
cana-5538	189	8	𝑥(𝑠	𝑥(𝑠	PROPN
cana-5538	189	9	)	)	PUNCT
cana-5538	189	10	)	)	PUNCT
cana-5538	190	1	𝑑𝑠	𝑑𝑠	ADP
cana-5538	190	2	1	1	NUM
cana-5538	190	3	0	0	NUM
cana-5538	190	4	(	(	PUNCT
cana-5538	190	5	3.2	3.2	NUM
cana-5538	190	6	)	)	PUNCT
cana-5538	190	7	for	for	ADP
cana-5538	190	8	all	all	DET
cana-5538	190	9	𝑡	𝑡	ADP
cana-5538	190	10	∈	∈	PROPN
cana-5538	191	1	[	[	X
cana-5538	191	2	0	0	NUM
cana-5538	191	3	,	,	PUNCT
cana-5538	191	4	1	1	NUM
cana-5538	191	5	]	]	PUNCT
cana-5538	191	6	where	where	SCONJ
cana-5538	191	7	g	g	PROPN
cana-5538	191	8	is	be	AUX
cana-5538	191	9	the	the	DET
cana-5538	191	10	green	green	ADJ
cana-5538	191	11	function	function	NOUN
cana-5538	191	12	defined	define	VERB
cana-5538	191	13	by	by	ADP
cana-5538	191	14	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	191	15	,	,	PUNCT
cana-5538	191	16	𝑠	𝑠	PROPN
cana-5538	191	17	)	)	PUNCT
cana-5538	191	18	=	=	PRON
cana-5538	191	19	{	{	PUNCT
cana-5538	191	20	(	(	PUNCT
cana-5538	191	21	1	1	NUM
cana-5538	191	22	−	−	NOUN
cana-5538	191	23	𝑠)𝑡	𝑠)𝑡	NOUN
cana-5538	191	24	0	0	NUM
cana-5538	191	25	≤	≤	NUM
cana-5538	191	26	t	t	PROPN
cana-5538	191	27	≤	≤	PROPN
cana-5538	191	28	s	s	PART
cana-5538	191	29	≤	≤	NUM
cana-5538	191	30	1	1	NUM
cana-5538	191	31	(	(	PUNCT
cana-5538	191	32	1	1	NUM
cana-5538	191	33	−	−	NOUN
cana-5538	191	34	𝑡)𝑠	𝑡)𝑠	NOUN
cana-5538	191	35	0	0	NUM
cana-5538	191	36	≤	≤	NUM
cana-5538	191	37	s	s	PART
cana-5538	191	38	≤	≤	NUM
cana-5538	191	39	t	t	NOUN
cana-5538	191	40	≤	≤	NUM
cana-5538	191	41	1	1	NUM
cana-5538	191	42	(	(	PUNCT
cana-5538	191	43	3.3	3.3	NUM
cana-5538	191	44	)	)	PUNCT
cana-5538	191	45	then	then	ADV
cana-5538	191	46	solving	solve	VERB
cana-5538	191	47	problem	problem	NOUN
cana-5538	191	48	(	(	PUNCT
cana-5538	191	49	3.1	3.1	NUM
cana-5538	191	50	)	)	PUNCT
cana-5538	191	51	is	be	AUX
cana-5538	191	52	equivalent	equivalent	ADJ
cana-5538	191	53	to	to	ADP
cana-5538	191	54	finding	find	VERB
cana-5538	191	55	fixed	fix	VERB
cana-5538	191	56	point	point	NOUN
cana-5538	191	57	of	of	ADP
cana-5538	191	58	t	t	PROPN
cana-5538	191	59	in	in	ADP
cana-5538	191	60	c(i	c(i	NOUN
cana-5538	191	61	)	)	PUNCT
cana-5538	191	62	.	.	PUNCT
cana-5538	192	1	theorem	theorem	VERB
cana-5538	192	2	3.1	3.1	NUM
cana-5538	192	3	let	let	VERB
cana-5538	192	4	𝑋	𝑋	PROPN
cana-5538	192	5	=	=	PUNCT
cana-5538	192	6	𝐶(𝐼	𝐶(𝐼	PROPN
cana-5538	192	7	)	)	PUNCT
cana-5538	192	8	and	and	CCONJ
cana-5538	192	9	𝑇	𝑇	PROPN
cana-5538	192	10	∶	∶	NOUN
cana-5538	192	11	𝑋	𝑋	NOUN
cana-5538	192	12	→	→	PUNCT
cana-5538	192	13	𝑋	𝑋	PROPN
cana-5538	192	14	be	be	VERB
cana-5538	192	15	an	an	DET
cana-5538	192	16	operator	operator	NOUN
cana-5538	192	17	given	give	VERB
cana-5538	192	18	by	by	ADP
cana-5538	192	19	𝑇	𝑇	PROPN
cana-5538	192	20	𝑥(𝑡	𝑥(𝑡	PROPN
cana-5538	192	21	)	)	PUNCT
cana-5538	192	22	=	=	SYM
cana-5538	192	23	∫	∫	PROPN
cana-5538	192	24	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	192	25	,	,	PUNCT
cana-5538	192	26	𝑠)𝑓	𝑠)𝑓	NUM
cana-5538	192	27	(	(	PUNCT
cana-5538	192	28	𝑠	𝑠	INTJ
cana-5538	192	29	,	,	PUNCT
cana-5538	192	30	𝑥(𝑠	𝑥(𝑠	PROPN
cana-5538	192	31	)	)	PUNCT
cana-5538	192	32	)	)	PUNCT
cana-5538	193	1	𝑑𝑠	𝑑𝑠	ADP
cana-5538	193	2	1	1	NUM
cana-5538	193	3	0	0	NUM
cana-5538	193	4	(	(	PUNCT
cana-5538	193	5	3.4	3.4	NUM
cana-5538	193	6	)	)	PUNCT
cana-5538	193	7	for	for	ADP
cana-5538	193	8	all	all	DET
cana-5538	193	9	𝑥	𝑥	PRON
cana-5538	193	10	∈	∈	PROPN
cana-5538	193	11	𝑋	𝑋	NOUN
cana-5538	193	12	and	and	CCONJ
cana-5538	193	13	𝑡	𝑡	NOUN
cana-5538	193	14	∈	∈	NOUN
cana-5538	193	15	𝐼	𝐼	ADP
cana-5538	193	16	=	=	SYM
cana-5538	194	1	[	[	X
cana-5538	194	2	0	0	NUM
cana-5538	194	3	,	,	PUNCT
cana-5538	194	4	1	1	NUM
cana-5538	194	5	]	]	PUNCT
cana-5538	194	6	.	.	PUNCT
cana-5538	195	1	suppose	suppose	VERB
cana-5538	195	2	the	the	DET
cana-5538	195	3	following	follow	VERB
cana-5538	195	4	conditions	condition	NOUN
cana-5538	195	5	hold	hold	VERB
cana-5538	195	6	:	:	PUNCT
cana-5538	195	7	(	(	PUNCT
cana-5538	195	8	i	i	NOUN
cana-5538	195	9	)	)	PUNCT
cana-5538	195	10	for	for	ADP
cana-5538	195	11	all	all	PRON
cana-5538	195	12	𝑡	𝑡	ADP
cana-5538	195	13	∈	∈	PROPN
cana-5538	195	14	𝐼	𝐼	PROPN
cana-5538	195	15	,	,	PUNCT
cana-5538	195	16	for	for	ADP
cana-5538	195	17	all	all	DET
cana-5538	195	18	𝑎	𝑎	NOUN
cana-5538	195	19	,	,	PUNCT
cana-5538	195	20	𝑏	𝑏	PROPN
cana-5538	195	21	∈	∈	PROPN
cana-5538	195	22	ℝ	ℝ	PROPN
cana-5538	195	23	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-5538	195	24	||𝑎||	||𝑎||	NOUN
cana-5538	195	25	,	,	PUNCT
cana-5538	195	26	||𝑏||	||𝑏||	NOUN
cana-5538	195	27	≤	≤	NOUN
cana-5538	195	28	1	1	NUM
cana-5538	195	29	,	,	PUNCT
cana-5538	195	30	we	we	PRON
cana-5538	195	31	have	have	VERB
cana-5538	195	32	|𝑓	|𝑓	NOUN
cana-5538	195	33	(	(	PUNCT
cana-5538	195	34	𝑡	𝑡	X
cana-5538	195	35	,	,	PUNCT
cana-5538	195	36	𝑎	𝑎	NOUN
cana-5538	195	37	)	)	PUNCT
cana-5538	195	38	−	−	PROPN
cana-5538	195	39	𝑓	𝑓	PROPN
cana-5538	195	40	(	(	PUNCT
cana-5538	195	41	𝑡	𝑡	NOUN
cana-5538	195	42	,	,	PUNCT
cana-5538	195	43	𝑏)|	𝑏)|	NOUN
cana-5538	195	44	≤	≤	NOUN
cana-5538	196	1	8𝜇(|𝑎	8𝜇(|𝑎	NUM
cana-5538	196	2	−	−	PROPN
cana-5538	196	3	𝑏|	𝑏|	PROPN
cana-5538	196	4	)	)	PUNCT
cana-5538	196	5	(	(	PUNCT
cana-5538	196	6	3.5	3.5	NUM
cana-5538	196	7	)	)	PUNCT
cana-5538	196	8	(	(	PUNCT
cana-5538	196	9	ii	ii	NOUN
cana-5538	196	10	)	)	PUNCT
cana-5538	196	11	there	there	PRON
cana-5538	196	12	exists	exist	VERB
cana-5538	196	13	𝑥0	𝑥0	PROPN
cana-5538	196	14	∈	∈	PROPN
cana-5538	196	15	𝐶(𝐼	𝐶(𝐼	PROPN
cana-5538	196	16	)	)	PUNCT
cana-5538	196	17	such	such	ADJ
cana-5538	196	18	that	that	DET
cana-5538	196	19	||𝑥0||	||𝑥0||	NOUN
cana-5538	196	20	∞	∞	NOUN
cana-5538	196	21	≤	≤	NUM
cana-5538	196	22	1	1	NUM
cana-5538	196	23	,	,	PUNCT
cana-5538	196	24	(	(	PUNCT
cana-5538	196	25	iii	iii	NOUN
cana-5538	196	26	)	)	PUNCT
cana-5538	196	27	for	for	ADP
cana-5538	196	28	all	all	DET
cana-5538	196	29	𝑥	𝑥	DET
cana-5538	196	30	∈	∈	PROPN
cana-5538	196	31	𝐶(𝐼	𝐶(𝐼	NOUN
cana-5538	196	32	)	)	PUNCT
cana-5538	196	33	|𝑥||∞	|𝑥||∞	VERB
cana-5538	196	34	≤	≤	NUM
cana-5538	196	35	1	1	NUM
cana-5538	196	36	→	→	SYM
cana-5538	196	37	||	||	NUM
cana-5538	196	38	∫	∫	PROPN
cana-5538	196	39	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	196	40	,	,	PUNCT
cana-5538	196	41	𝑠)𝑓	𝑠)𝑓	NUM
cana-5538	196	42	(	(	PUNCT
cana-5538	196	43	𝑠	𝑠	INTJ
cana-5538	196	44	,	,	PUNCT
cana-5538	196	45	𝑥(𝑠))𝑑𝑠||	𝑥(𝑠))𝑑𝑠||	PROPN
cana-5538	196	46	1	1	NUM
cana-5538	196	47	0	0	NUM
cana-5538	196	48	∞	∞	NUM
cana-5538	196	49	≤	≤	NUM
cana-5538	196	50	1	1	NUM
cana-5538	196	51	(	(	PUNCT
cana-5538	196	52	3.6	3.6	NUM
cana-5538	196	53	)	)	PUNCT
cana-5538	196	54	then	then	ADV
cana-5538	196	55	the	the	DET
cana-5538	196	56	second	second	ADJ
cana-5538	196	57	order	order	NOUN
cana-5538	196	58	differential	differential	NOUN
cana-5538	196	59	equation	equation	NOUN
cana-5538	196	60	(	(	PUNCT
cana-5538	196	61	3.1	3.1	NUM
cana-5538	196	62	)	)	PUNCT
cana-5538	196	63	has	have	VERB
cana-5538	196	64	a	a	DET
cana-5538	196	65	solution	solution	NOUN
cana-5538	196	66	.	.	PUNCT
cana-5538	197	1	proof	proof	NOUN
cana-5538	197	2	consider	consider	VERB
cana-5538	197	3	c(i	c(i	NOUN
cana-5538	197	4	)	)	PUNCT
cana-5538	197	5	endowed	endow	VERB
cana-5538	197	6	with	with	ADP
cana-5538	197	7	the	the	DET
cana-5538	197	8	partial	partial	ADJ
cana-5538	197	9	metric	metric	NOUN
cana-5538	197	10	given	give	VERB
cana-5538	197	11	by	by	ADP
cana-5538	197	12	𝜌𝑐(𝑥	𝜌𝑐(𝑥	ADJ
cana-5538	197	13	,	,	PUNCT
cana-5538	197	14	𝑦	𝑦	NOUN
cana-5538	197	15	)	)	PUNCT
cana-5538	197	16	=	=	PRON
cana-5538	197	17	{	{	PUNCT
cana-5538	197	18	||𝑥	||𝑥	PROPN
cana-5538	197	19	−	−	PROPN
cana-5538	197	20	𝑦||	𝑦||	PROPN
cana-5538	197	21	∞	∞	PROPN
cana-5538	197	22	||𝑥||	||𝑥||	PROPN
cana-5538	197	23	≤	≤	ADV
cana-5538	197	24	1	1	NUM
cana-5538	197	25	,	,	PUNCT
cana-5538	197	26	||𝑦||	||𝑦||	ADJ
cana-5538	197	27	≤	≤	ADJ
cana-5538	197	28	1	1	NUM
cana-5538	197	29	||𝑥	||𝑥	NOUN
cana-5538	197	30	−	−	PROPN
cana-5538	197	31	𝑦||	𝑦||	SYM
cana-5538	197	32	∞	∞	PROPN
cana-5538	197	33	+	+	CCONJ
cana-5538	197	34	𝜏	𝜏	PROPN
cana-5538	197	35	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
cana-5538	197	36	(	(	PUNCT
cana-5538	197	37	3.7	3.7	NUM
cana-5538	197	38	)	)	PUNCT
cana-5538	197	39	where	where	SCONJ
cana-5538	197	40	𝜏	𝜏	PRON
cana-5538	197	41	>	>	X
cana-5538	197	42	0	0	NUM
cana-5538	197	43	.	.	PUNCT
cana-5538	198	1	then	then	ADV
cana-5538	198	2	(	(	PUNCT
cana-5538	198	3	𝐶(𝐼	𝐶(𝐼	NOUN
cana-5538	198	4	)	)	PUNCT
cana-5538	198	5	,	,	PUNCT
cana-5538	198	6	𝜌𝑐	𝜌𝑐	PRON
cana-5538	198	7	)	)	PUNCT
cana-5538	198	8	is	be	AUX
cana-5538	198	9	a	a	DET
cana-5538	198	10	partial	partial	ADJ
cana-5538	198	11	metric	metric	ADJ
cana-5538	198	12	space	space	NOUN
cana-5538	198	13	.	.	PUNCT
cana-5538	199	1	now	now	ADV
cana-5538	199	2	we	we	PRON
cana-5538	199	3	define	define	VERB
cana-5538	199	4	partial	partial	ADJ
cana-5538	199	5	cone	cone	NOUN
cana-5538	199	6	metric	metric	NOUN
cana-5538	199	7	as	as	SCONJ
cana-5538	199	8	communications	communication	NOUN
cana-5538	199	9	on	on	ADP
cana-5538	199	10	applied	apply	VERB
cana-5538	199	11	nonlinear	nonlinear	ADJ
cana-5538	199	12	analysis	analysis	NOUN
cana-5538	199	13	issn	issn	NOUN
cana-5538	199	14	:	:	PUNCT
cana-5538	199	15	1074	1074	NUM
cana-5538	199	16	-	-	PUNCT
cana-5538	199	17	133x	133x	NUM
cana-5538	199	18	vol	vol	VERB
cana-5538	199	19	32	32	NUM
cana-5538	199	20	no	no	NOUN
cana-5538	199	21	.	.	PUNCT
cana-5538	200	1	10s	10	NOUN
cana-5538	200	2	(	(	PUNCT
cana-5538	200	3	2025	2025	NUM
cana-5538	200	4	)	)	PUNCT
cana-5538	200	5	2621	2621	NUM
cana-5538	200	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	200	7	𝜌𝑐(𝑥	𝜌𝑐(𝑥	PROPN
cana-5538	200	8	,	,	PUNCT
cana-5538	200	9	𝑦	𝑦	NOUN
cana-5538	200	10	)	)	PUNCT
cana-5538	200	11	=	=	SYM
cana-5538	200	12	(	(	PUNCT
cana-5538	200	13	𝜌𝑐(𝑥	𝜌𝑐(𝑥	PROPN
cana-5538	200	14	,	,	PUNCT
cana-5538	200	15	𝑦	𝑦	NOUN
cana-5538	200	16	)	)	PUNCT
cana-5538	200	17	,	,	PUNCT
cana-5538	200	18	𝛽𝜌𝑐(𝑥	𝛽𝜌𝑐(𝑥	PROPN
cana-5538	200	19	,	,	PUNCT
cana-5538	200	20	𝑦	𝑦	NOUN
cana-5538	200	21	)	)	PUNCT
cana-5538	200	22	)	)	PUNCT
cana-5538	200	23	(	(	PUNCT
cana-5538	200	24	3.8	3.8	NUM
cana-5538	200	25	)	)	PUNCT
cana-5538	200	26	where	where	SCONJ
cana-5538	200	27	𝛽	𝛽	NOUN
cana-5538	200	28	≥	≥	NOUN
cana-5538	200	29	0	0	NUM
cana-5538	200	30	.	.	PUNCT
cana-5538	201	1	now	now	ADV
cana-5538	201	2	,	,	PUNCT
cana-5538	201	3	let	let	VERB
cana-5538	201	4	𝑥	𝑥	PRON
cana-5538	201	5	,	,	PUNCT
cana-5538	201	6	𝑦	𝑦	PROPN
cana-5538	201	7	∈	∈	PROPN
cana-5538	201	8	𝐶(𝐼	𝐶(𝐼	NOUN
cana-5538	201	9	)	)	PUNCT
cana-5538	201	10	such	such	ADJ
cana-5538	201	11	that	that	SCONJ
cana-5538	201	12	||𝑥||	||𝑥||	NOUN
cana-5538	201	13	≤	≤	ADV
cana-5538	201	14	1	1	NUM
cana-5538	201	15	,	,	PUNCT
cana-5538	201	16	||𝑦||	||𝑦||	ADJ
cana-5538	201	17	≤	≤	ADV
cana-5538	201	18	1	1	NUM
cana-5538	201	19	,	,	PUNCT
cana-5538	201	20	then	then	ADV
cana-5538	201	21	we	we	PRON
cana-5538	201	22	have	have	VERB
cana-5538	201	23	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	201	24	𝑥	𝑥	PROPN
cana-5538	201	25	,	,	PUNCT
cana-5538	201	26	𝑇	𝑇	PROPN
cana-5538	201	27	𝑦	𝑦	NOUN
cana-5538	201	28	)	)	PUNCT
cana-5538	201	29	=	=	SYM
cana-5538	201	30	(	(	PUNCT
cana-5538	201	31	||𝑇	||𝑇	NOUN
cana-5538	202	1	𝑥	𝑥	NOUN
cana-5538	202	2	−	−	PROPN
cana-5538	202	3	𝑇	𝑇	PROPN
cana-5538	202	4	𝑦||	𝑦||	PRON
cana-5538	202	5	∞	∞	PROPN
cana-5538	202	6	,	,	PUNCT
cana-5538	202	7	𝛽||𝑇	𝛽||𝑇	VERB
cana-5538	202	8	𝑥	𝑥	PRON
cana-5538	202	9	−	−	PROPN
cana-5538	202	10	𝑇	𝑇	PROPN
cana-5538	202	11	𝑦||	𝑦||	PRON
cana-5538	202	12	∞	∞	NUM
cana-5538	202	13	)	)	PUNCT
cana-5538	203	1	=	=	PRON
cana-5538	203	2	(	(	PUNCT
cana-5538	203	3	sup	sup	NOUN
cana-5538	203	4	𝑡∈[0,1	𝑡∈[0,1	NOUN
cana-5538	203	5	]	]	PUNCT
cana-5538	203	6	∫	∫	PROPN
cana-5538	203	7	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	203	8	,	,	PUNCT
cana-5538	203	9	𝑠)|𝑓	𝑠)|𝑓	ADJ
cana-5538	203	10	(	(	PUNCT
cana-5538	203	11	𝑠	𝑠	NOUN
cana-5538	203	12	,	,	PUNCT
cana-5538	203	13	𝑥(𝑠	𝑥(𝑠	PROPN
cana-5538	203	14	)	)	PUNCT
cana-5538	203	15	)	)	PUNCT
cana-5538	204	1	−	−	PROPN
cana-5538	204	2	𝑓	𝑓	PRON
cana-5538	204	3	(	(	PUNCT
cana-5538	204	4	𝑠	𝑠	PROPN
cana-5538	204	5	,	,	PUNCT
cana-5538	204	6	𝑦(𝑠))|𝑑𝑠	𝑦(𝑠))|𝑑𝑠	VERB
cana-5538	204	7	1	1	NUM
cana-5538	204	8	0	0	NUM
cana-5538	204	9	,	,	PUNCT
cana-5538	204	10	𝛽	𝛽	PROPN
cana-5538	204	11	sup	sup	NOUN
cana-5538	204	12	𝑡∈[0,1	𝑡∈[0,1	NOUN
cana-5538	204	13	]	]	PUNCT
cana-5538	204	14	∫	∫	PROPN
cana-5538	204	15	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	204	16	,	,	PUNCT
cana-5538	204	17	𝑠)|𝑓	𝑠)|𝑓	ADJ
cana-5538	204	18	(	(	PUNCT
cana-5538	204	19	𝑠	𝑠	NOUN
cana-5538	204	20	,	,	PUNCT
cana-5538	204	21	𝑥(𝑠	𝑥(𝑠	PROPN
cana-5538	204	22	)	)	PUNCT
cana-5538	204	23	)	)	PUNCT
cana-5538	205	1	−	−	PROPN
cana-5538	205	2	𝑓	𝑓	PRON
cana-5538	205	3	(	(	PUNCT
cana-5538	205	4	𝑠	𝑠	PROPN
cana-5538	205	5	,	,	PUNCT
cana-5538	205	6	𝑦(𝑠))|𝑑𝑠	𝑦(𝑠))|𝑑𝑠	VERB
cana-5538	205	7	1	1	NUM
cana-5538	205	8	0	0	NUM
cana-5538	205	9	)	)	PUNCT
cana-5538	206	1	=	=	PRON
cana-5538	206	2	(	(	PUNCT
cana-5538	206	3	sup	sup	NOUN
cana-5538	206	4	𝑡∈[0,1	𝑡∈[0,1	NOUN
cana-5538	206	5	]	]	PUNCT
cana-5538	206	6	∫	∫	PROPN
cana-5538	206	7	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	206	8	,	,	PUNCT
cana-5538	206	9	𝑠)8	𝑠)8	ADJ
cana-5538	206	10	𝜇|𝑥(𝑠	𝜇|𝑥(𝑠	PROPN
cana-5538	206	11	)	)	PUNCT
cana-5538	206	12	−	−	NOUN
cana-5538	206	13	𝑦(𝑠)|𝑑𝑠	𝑦(𝑠)|𝑑𝑠	VERB
cana-5538	206	14	1	1	NUM
cana-5538	206	15	0	0	NUM
cana-5538	206	16	,	,	PUNCT
cana-5538	206	17	𝛽	𝛽	PROPN
cana-5538	206	18	sup	sup	NOUN
cana-5538	206	19	𝑡∈[0,1	𝑡∈[0,1	NOUN
cana-5538	206	20	]	]	PUNCT
cana-5538	206	21	∫	∫	PROPN
cana-5538	206	22	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	206	23	,	,	PUNCT
cana-5538	206	24	𝑠)8	𝑠)8	ADJ
cana-5538	206	25	𝜇|𝑥(𝑠	𝜇|𝑥(𝑠	PROPN
cana-5538	206	26	)	)	PUNCT
cana-5538	206	27	−	−	NOUN
cana-5538	207	1	𝑦(𝑠)|𝑑𝑠	𝑦(𝑠)|𝑑𝑠	VERB
cana-5538	207	2	1	1	NUM
cana-5538	207	3	0	0	NUM
cana-5538	207	4	)	)	PUNCT
cana-5538	208	1	=	=	PRON
cana-5538	208	2	(	(	PUNCT
cana-5538	208	3	sup	sup	NOUN
cana-5538	208	4	𝑡∈[0,1	𝑡∈[0,1	NOUN
cana-5538	208	5	]	]	PUNCT
cana-5538	208	6	∫	∫	PROPN
cana-5538	208	7	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	208	8	,	,	PUNCT
cana-5538	208	9	𝑠	𝑠	NOUN
cana-5538	208	10	)	)	PUNCT
cana-5538	208	11	×	×	NOUN
cana-5538	208	12	(	(	PUNCT
cana-5538	208	13	8	8	NUM
cana-5538	208	14	𝜇||𝑥	𝜇||𝑥	PROPN
cana-5538	208	15	−	−	PROPN
cana-5538	208	16	𝑦||	𝑦||	PROPN
cana-5538	208	17	∞	∞	NUM
cana-5538	208	18	)	)	PUNCT
cana-5538	208	19	𝑑𝑠	𝑑𝑠	ADP
cana-5538	208	20	1	1	NUM
cana-5538	208	21	0	0	NUM
cana-5538	208	22	,	,	PUNCT
cana-5538	208	23	𝛽	𝛽	PROPN
cana-5538	208	24	sup	sup	NOUN
cana-5538	208	25	𝑡∈[0,1	𝑡∈[0,1	NOUN
cana-5538	208	26	]	]	PUNCT
cana-5538	208	27	∫	∫	PROPN
cana-5538	208	28	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	208	29	,	,	PUNCT
cana-5538	208	30	𝑠	𝑠	NOUN
cana-5538	208	31	)	)	PUNCT
cana-5538	208	32	×	×	NOUN
cana-5538	208	33	(	(	PUNCT
cana-5538	208	34	8	8	NUM
cana-5538	208	35	𝜇||𝑥	𝜇||𝑥	PROPN
cana-5538	208	36	−	−	PROPN
cana-5538	208	37	𝑦||	𝑦||	PROPN
cana-5538	208	38	∞	∞	NUM
cana-5538	208	39	)	)	PUNCT
cana-5538	208	40	𝑑𝑠	𝑑𝑠	ADP
cana-5538	208	41	1	1	NUM
cana-5538	208	42	0	0	NUM
cana-5538	208	43	)	)	PUNCT
cana-5538	208	44	(	(	PUNCT
cana-5538	208	45	3.9	3.9	NUM
cana-5538	208	46	)	)	PUNCT
cana-5538	208	47	now	now	ADV
cana-5538	208	48	,	,	PUNCT
cana-5538	208	49	as	as	SCONJ
cana-5538	208	50	we	we	PRON
cana-5538	208	51	know	know	VERB
cana-5538	208	52	sup	sup	NOUN
cana-5538	208	53	𝑡∈[0,1	𝑡∈[0,1	NOUN
cana-5538	208	54	]	]	PUNCT
cana-5538	208	55	∫	∫	PROPN
cana-5538	208	56	𝑔(𝑡	𝑔(𝑡	PROPN
cana-5538	208	57	,	,	PUNCT
cana-5538	208	58	𝑠)𝑑𝑠	𝑠)𝑑𝑠	PROPN
cana-5538	208	59	1	1	NUM
cana-5538	208	60	0	0	NUM
cana-5538	208	61	=	=	SYM
cana-5538	208	62	1	1	NUM
cana-5538	208	63	8	8	NUM
cana-5538	208	64	and	and	CCONJ
cana-5538	208	65	taking	take	VERB
cana-5538	208	66	𝜓(𝑡	𝜓(𝑡	NOUN
cana-5538	208	67	)	)	PUNCT
cana-5538	208	68	=	=	PRON
cana-5538	208	69	𝜇𝑡	𝜇𝑡	VERB
cana-5538	208	70	𝜌𝑐(𝑇	𝜌𝑐(𝑇	ADV
cana-5538	208	71	𝑥	𝑥	PROPN
cana-5538	208	72	,	,	PUNCT
cana-5538	208	73	𝑇	𝑇	PROPN
cana-5538	208	74	𝑦	𝑦	NOUN
cana-5538	208	75	)	)	PUNCT
cana-5538	208	76	≤	≤	PUNCT
cana-5538	209	1	𝜇(𝜌𝑐(𝑥	𝜇(𝜌𝑐(𝑥	PROPN
cana-5538	209	2	,	,	PUNCT
cana-5538	209	3	𝑦	𝑦	NOUN
cana-5538	209	4	)	)	PUNCT
cana-5538	209	5	)	)	PUNCT
cana-5538	209	6	,	,	PUNCT
cana-5538	209	7	𝛽(𝜌𝑐(𝑥	𝛽(𝜌𝑐(𝑥	PROPN
cana-5538	209	8	,	,	PUNCT
cana-5538	209	9	𝑦	𝑦	NOUN
cana-5538	209	10	)	)	PUNCT
cana-5538	209	11	)	)	PUNCT
cana-5538	209	12	)	)	PUNCT
cana-5538	210	1	≤	≤	PUNCT
cana-5538	211	1	𝜇𝜌𝑐(𝑥	𝜇𝜌𝑐(𝑥	PROPN
cana-5538	211	2	,	,	PUNCT
cana-5538	211	3	𝑦	𝑦	NOUN
cana-5538	211	4	)	)	PUNCT
cana-5538	211	5	≤	≤	NOUN
cana-5538	212	1	𝜓(𝜌𝑐(𝑥	𝜓(𝜌𝑐(𝑥	PROPN
cana-5538	212	2	,	,	PUNCT
cana-5538	212	3	𝑦	𝑦	NOUN
cana-5538	212	4	)	)	PUNCT
cana-5538	212	5	)	)	PUNCT
cana-5538	212	6	≤	≤	NUM
cana-5538	212	7	𝜓(𝑚𝑎𝑥{𝜌𝑐(𝑥	𝜓(𝑚𝑎𝑥{𝜌𝑐(𝑥	NOUN
cana-5538	212	8	,	,	PUNCT
cana-5538	212	9	𝑦	𝑦	NOUN
cana-5538	212	10	)	)	PUNCT
cana-5538	212	11	,	,	PUNCT
cana-5538	212	12	𝜌𝑐(𝑥	𝜌𝑐(𝑥	PROPN
cana-5538	212	13	,	,	PUNCT
cana-5538	212	14	𝑇	𝑇	PROPN
cana-5538	212	15	𝑥	𝑥	PROPN
cana-5538	212	16	)	)	PUNCT
cana-5538	212	17	,	,	PUNCT
cana-5538	212	18	𝜌𝑐(𝑦	𝜌𝑐(𝑦	PROPN
cana-5538	212	19	,	,	PUNCT
cana-5538	212	20	𝑇	𝑇	PROPN
cana-5538	212	21	𝑦	𝑦	NOUN
cana-5538	212	22	)	)	PUNCT
cana-5538	212	23	}	}	PUNCT
cana-5538	212	24	)	)	PUNCT
cana-5538	212	25	(	(	PUNCT
cana-5538	212	26	3.10	3.10	NUM
cana-5538	212	27	)	)	PUNCT
cana-5538	212	28	define	define	VERB
cana-5538	212	29	the	the	DET
cana-5538	212	30	function	function	NOUN
cana-5538	212	31	𝛼	𝛼	PROPN
cana-5538	212	32	∶	∶	NOUN
cana-5538	212	33	𝐶(𝐼	𝐶(𝐼	NOUN
cana-5538	212	34	)	)	PUNCT
cana-5538	212	35	×	×	PROPN
cana-5538	212	36	𝐶(𝐼	𝐶(𝐼	NOUN
cana-5538	212	37	)	)	PUNCT
cana-5538	212	38	→	→	PUNCT
cana-5538	213	1	[	[	X
cana-5538	213	2	0	0	NUM
cana-5538	213	3	,	,	PUNCT
cana-5538	213	4	∞	∞	PROPN
cana-5538	213	5	)	)	PUNCT
cana-5538	213	6	as	as	ADP
cana-5538	213	7	𝛼(𝑥	𝛼(𝑥	PROPN
cana-5538	213	8	,	,	PUNCT
cana-5538	213	9	𝑦	𝑦	NOUN
cana-5538	213	10	)	)	PUNCT
cana-5538	213	11	=	=	SYM
cana-5538	213	12	{	{	PUNCT
cana-5538	213	13	1	1	NUM
cana-5538	213	14	||𝑥||	||𝑥||	PROPN
cana-5538	213	15	≤	≤	NUM
cana-5538	213	16	1	1	NUM
cana-5538	213	17	,	,	PUNCT
cana-5538	213	18	||𝑦||	||𝑦||	ADJ
cana-5538	213	19	≤	≤	ADV
cana-5538	213	20	1	1	NUM
cana-5538	213	21	0	0	NUM
cana-5538	213	22	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
cana-5538	213	23	(	(	PUNCT
cana-5538	213	24	3.11	3.11	NUM
cana-5538	213	25	)	)	PUNCT
cana-5538	213	26	for	for	ADP
cana-5538	213	27	all	all	DET
cana-5538	213	28	𝑥	𝑥	PROPN
cana-5538	213	29	,	,	PUNCT
cana-5538	213	30	𝑦	𝑦	PROPN
cana-5538	213	31	∈	∈	PROPN
cana-5538	213	32	𝐶(𝐼	𝐶(𝐼	NOUN
cana-5538	213	33	)	)	PUNCT
cana-5538	213	34	𝛼(𝑥	𝛼(𝑥	PROPN
cana-5538	213	35	,	,	PUNCT
cana-5538	213	36	𝑦)𝜌𝑐(𝑇	𝑦)𝜌𝑐(𝑇	PROPN
cana-5538	213	37	𝑥	𝑥	NOUN
cana-5538	213	38	,	,	PUNCT
cana-5538	213	39	𝑇	𝑇	PROPN
cana-5538	213	40	𝑦	𝑦	NOUN
cana-5538	213	41	)	)	PUNCT
cana-5538	213	42	≤	≤	NUM
cana-5538	213	43	𝜓(𝑚𝑎𝑥{𝜌𝑐(𝑥	𝜓(𝑚𝑎𝑥{𝜌𝑐(𝑥	NOUN
cana-5538	213	44	,	,	PUNCT
cana-5538	213	45	𝑦	𝑦	NOUN
cana-5538	213	46	)	)	PUNCT
cana-5538	213	47	,	,	PUNCT
cana-5538	213	48	𝜌𝑐(𝑥	𝜌𝑐(𝑥	PROPN
cana-5538	213	49	,	,	PUNCT
cana-5538	213	50	𝑇	𝑇	PROPN
cana-5538	213	51	𝑥	𝑥	PROPN
cana-5538	213	52	)	)	PUNCT
cana-5538	213	53	,	,	PUNCT
cana-5538	213	54	𝜌𝑐(𝑦	𝜌𝑐(𝑦	PROPN
cana-5538	213	55	,	,	PUNCT
cana-5538	213	56	𝑇	𝑇	PROPN
cana-5538	213	57	𝑦	𝑦	NOUN
cana-5538	213	58	)	)	PUNCT
cana-5538	213	59	}	}	PUNCT
cana-5538	213	60	)	)	PUNCT
cana-5538	213	61	(	(	PUNCT
cana-5538	213	62	3.12	3.12	NUM
cana-5538	213	63	)	)	PUNCT
cana-5538	213	64	clearly	clearly	ADV
cana-5538	213	65	,	,	PUNCT
cana-5538	213	66	all	all	DET
cana-5538	213	67	the	the	DET
cana-5538	213	68	conditions	condition	NOUN
cana-5538	213	69	of	of	ADP
cana-5538	213	70	theorem	theorem	ADJ
cana-5538	213	71	2.3	2.3	NUM
cana-5538	213	72	are	be	AUX
cana-5538	213	73	satisfied	satisfied	ADJ
cana-5538	213	74	and	and	CCONJ
cana-5538	213	75	so	so	ADV
cana-5538	213	76	γ	γ	PROPN
cana-5538	213	77	has	have	VERB
cana-5538	213	78	a	a	DET
cana-5538	213	79	fixed	fix	VERB
cana-5538	213	80	point	point	NOUN
cana-5538	213	81	.	.	PUNCT
cana-5538	214	1	thus	thus	ADV
cana-5538	214	2	the	the	DET
cana-5538	214	3	system	system	NOUN
cana-5538	214	4	of	of	ADP
cana-5538	214	5	integral	integral	ADJ
cana-5538	214	6	equations	equation	NOUN
cana-5538	214	7	(	(	PUNCT
cana-5538	214	8	3.2	3.2	NUM
cana-5538	214	9	)	)	PUNCT
cana-5538	214	10	has	have	VERB
cana-5538	214	11	a	a	DET
cana-5538	214	12	solution	solution	NOUN
cana-5538	214	13	.	.	PUNCT
cana-5538	215	1	references	reference	NOUN
cana-5538	215	2	[	[	X
cana-5538	215	3	1	1	X
cana-5538	215	4	]	]	PUNCT
cana-5538	215	5	s.	s.	PROPN
cana-5538	215	6	banach	banach	PROPN
cana-5538	215	7	,	,	PUNCT
cana-5538	215	8	sur	sur	PROPN
cana-5538	215	9	les	les	PROPN
cana-5538	215	10	operations	operation	NOUN
cana-5538	215	11	dans	dan	NOUN
cana-5538	215	12	les	le	NOUN
cana-5538	215	13	ensembles	ensemble	NOUN
cana-5538	215	14	abstraits	abstrait	NOUN
cana-5538	215	15	et	et	PROPN
cana-5538	215	16	leur	leur	X
cana-5538	215	17	application	application	PROPN
cana-5538	215	18	aux	aux	PROPN
cana-5538	215	19	equations	equations	PROPN
cana-5538	215	20	integrals	integral	NOUN
cana-5538	215	21	,	,	PUNCT
cana-5538	215	22	fund	fund	NOUN
cana-5538	215	23	.	.	PUNCT
cana-5538	215	24	math	math	NOUN
cana-5538	215	25	.	.	PUNCT
cana-5538	215	26	,	,	PUNCT
cana-5538	215	27	3	3	NUM
cana-5538	215	28	(	(	PUNCT
cana-5538	215	29	1922	1922	NUM
cana-5538	215	30	)	)	PUNCT
cana-5538	215	31	,	,	PUNCT
cana-5538	215	32	133	133	NUM
cana-5538	215	33	-	-	SYM
cana-5538	215	34	181	181	NUM
cana-5538	215	35	.	.	PUNCT
cana-5538	216	1	[	[	X
cana-5538	216	2	2	2	NUM
cana-5538	216	3	]	]	PUNCT
cana-5538	216	4	m.	m.	NOUN
cana-5538	216	5	frechet	frechet	PROPN
cana-5538	216	6	,	,	PUNCT
cana-5538	216	7	sur	sur	PROPN
cana-5538	216	8	quelques	quelques	PROPN
cana-5538	216	9	points	point	NOUN
cana-5538	216	10	du	du	PROPN
cana-5538	216	11	calcul	calcul	PROPN
cana-5538	216	12	fonctionnel	fonctionnel	PROPN
cana-5538	216	13	,	,	PUNCT
cana-5538	216	14	rendic	rendic	ADJ
cana-5538	216	15	.	.	PUNCT
cana-5538	217	1	circ	circ	PROPN
cana-5538	217	2	.	.	PUNCT
cana-5538	218	1	mat	mat	PROPN
cana-5538	218	2	.	.	PUNCT
cana-5538	218	3	palermo	palermo	PROPN
cana-5538	218	4	22	22	NUM
cana-5538	218	5	(	(	PUNCT
cana-5538	218	6	1906	1906	NUM
cana-5538	218	7	)	)	PUNCT
cana-5538	218	8	1	1	NUM
cana-5538	218	9	-	-	SYM
cana-5538	218	10	74	74	NUM
cana-5538	218	11	.	.	PUNCT
cana-5538	219	1	[	[	X
cana-5538	219	2	3	3	X
cana-5538	219	3	]	]	PUNCT
cana-5538	219	4	s.	s.	PROPN
cana-5538	219	5	g.	g.	PROPN
cana-5538	219	6	matthews	matthews	PROPN
cana-5538	219	7	,	,	PUNCT
cana-5538	219	8	partial	partial	ADJ
cana-5538	219	9	metric	metric	ADJ
cana-5538	219	10	topology	topology	NOUN
cana-5538	219	11	,	,	PUNCT
cana-5538	219	12	research	research	NOUN
cana-5538	219	13	report	report	NOUN
cana-5538	219	14	212	212	NUM
cana-5538	219	15	,	,	PUNCT
cana-5538	219	16	department	department	NOUN
cana-5538	219	17	of	of	ADP
cana-5538	219	18	computer	computer	NOUN
cana-5538	219	19	science	science	NOUN
cana-5538	219	20	,	,	PUNCT
cana-5538	219	21	university	university	PROPN
cana-5538	219	22	of	of	ADP
cana-5538	219	23	warwick	warwick	PROPN
cana-5538	219	24	,	,	PUNCT
cana-5538	219	25	(	(	PUNCT
cana-5538	219	26	1992	1992	NUM
cana-5538	219	27	)	)	PUNCT
cana-5538	219	28	.	.	PUNCT
cana-5538	220	1	[	[	X
cana-5538	220	2	4	4	X
cana-5538	220	3	]	]	PUNCT
cana-5538	220	4	s.	s.	PROPN
cana-5538	220	5	g.	g.	PROPN
cana-5538	220	6	matthews	matthews	PROPN
cana-5538	220	7	,	,	PUNCT
cana-5538	220	8	partial	partial	ADJ
cana-5538	220	9	metric	metric	ADJ
cana-5538	220	10	topology	topology	NOUN
cana-5538	220	11	,	,	PUNCT
cana-5538	220	12	proceedings	proceeding	NOUN
cana-5538	220	13	of	of	ADP
cana-5538	220	14	the	the	DET
cana-5538	220	15	8th	8th	ADJ
cana-5538	220	16	summer	summer	NOUN
cana-5538	220	17	conference	conference	NOUN
cana-5538	220	18	on	on	ADP
cana-5538	220	19	topology	topology	NOUN
cana-5538	220	20	and	and	CCONJ
cana-5538	220	21	its	its	PRON
cana-5538	220	22	applications	application	NOUN
cana-5538	220	23	,	,	PUNCT
cana-5538	220	24	annals	annal	NOUN
cana-5538	220	25	of	of	ADP
cana-5538	220	26	the	the	DET
cana-5538	220	27	new	new	PROPN
cana-5538	220	28	york	york	PROPN
cana-5538	220	29	academy	academy	PROPN
cana-5538	220	30	of	of	ADP
cana-5538	220	31	sciences	sciences	PROPN
cana-5538	220	32	,	,	PUNCT
cana-5538	220	33	728	728	NUM
cana-5538	220	34	(	(	PUNCT
cana-5538	220	35	1994	1994	NUM
cana-5538	220	36	)	)	PUNCT
cana-5538	220	37	,	,	PUNCT
cana-5538	220	38	183	183	NUM
cana-5538	220	39	-	-	SYM
cana-5538	220	40	197	197	NUM
cana-5538	220	41	.	.	PUNCT
cana-5538	221	1	[	[	X
cana-5538	221	2	5	5	X
cana-5538	221	3	]	]	PUNCT
cana-5538	221	4	b.	b.	PROPN
cana-5538	221	5	samet	samet	PROPN
cana-5538	221	6	,	,	PUNCT
cana-5538	221	7	c.	c.	PROPN
cana-5538	221	8	vetro	vetro	PROPN
cana-5538	221	9	and	and	CCONJ
cana-5538	221	10	p.	p.	PROPN
cana-5538	221	11	vetro	vetro	PROPN
cana-5538	221	12	,	,	PUNCT
cana-5538	221	13	fixed	fix	VERB
cana-5538	221	14	point	point	NOUN
cana-5538	221	15	theorem	theorem	NOUN
cana-5538	221	16	for	for	ADP
cana-5538	221	17	𝛼	𝛼	PROPN
cana-5538	221	18	−	−	PROPN
cana-5538	221	19	𝜓	𝜓	PROPN
cana-5538	221	20	contractive	contractive	ADJ
cana-5538	221	21	type	type	NOUN
cana-5538	221	22	mappings	mapping	NOUN
cana-5538	221	23	,	,	PUNCT
cana-5538	221	24	nonlinear	nonlinear	ADJ
cana-5538	221	25	anal	anal	NOUN
cana-5538	221	26	.	.	PUNCT
cana-5538	221	27	,	,	PUNCT
cana-5538	221	28	75	75	NUM
cana-5538	221	29	(	(	PUNCT
cana-5538	221	30	2012	2012	NUM
cana-5538	221	31	)	)	PUNCT
cana-5538	221	32	,	,	PUNCT
cana-5538	221	33	2154	2154	NUM
cana-5538	221	34	-	-	SYM
cana-5538	221	35	2165	2165	NUM
cana-5538	221	36	.	.	PUNCT
cana-5538	222	1	communications	communication	NOUN
cana-5538	222	2	on	on	ADP
cana-5538	222	3	applied	apply	VERB
cana-5538	222	4	nonlinear	nonlinear	ADJ
cana-5538	222	5	analysis	analysis	NOUN
cana-5538	222	6	issn	issn	NOUN
cana-5538	222	7	:	:	PUNCT
cana-5538	222	8	1074	1074	NUM
cana-5538	222	9	-	-	PUNCT
cana-5538	222	10	133x	133x	NUM
cana-5538	222	11	vol	vol	VERB
cana-5538	222	12	32	32	NUM
cana-5538	222	13	no	no	NOUN
cana-5538	222	14	.	.	PUNCT
cana-5538	223	1	10s	10	NOUN
cana-5538	223	2	(	(	PUNCT
cana-5538	223	3	2025	2025	NUM
cana-5538	223	4	)	)	PUNCT
cana-5538	223	5	2622	2622	NUM
cana-5538	223	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5538	224	1	[	[	X
cana-5538	224	2	6	6	NUM
cana-5538	224	3	]	]	PUNCT
cana-5538	224	4	e.	e.	PROPN
cana-5538	224	5	karapinar	karapinar	PROPN
cana-5538	224	6	and	and	CCONJ
cana-5538	224	7	b.	b.	PROPN
cana-5538	224	8	samet	samet	PROPN
cana-5538	224	9	,	,	PUNCT
cana-5538	224	10	generalized	generalize	VERB
cana-5538	224	11	α	α	NOUN
cana-5538	224	12	−	−	NOUN
cana-5538	224	13	ψ	ψ	SYM
cana-5538	224	14	-contractive	-contractive	ADJ
cana-5538	224	15	type	type	NOUN
cana-5538	224	16	mappings	mapping	NOUN
cana-5538	224	17	and	and	CCONJ
cana-5538	224	18	related	relate	VERB
cana-5538	224	19	fixed	fix	VERB
cana-5538	224	20	point	point	NOUN
cana-5538	224	21	theorems	theorem	NOUN
cana-5538	224	22	with	with	ADP
cana-5538	224	23	applications	application	NOUN
cana-5538	224	24	,	,	PUNCT
cana-5538	224	25	abstract	abstract	ADJ
cana-5538	224	26	and	and	CCONJ
cana-5538	224	27	applied	apply	VERB
cana-5538	224	28	analysis	analysis	NOUN
cana-5538	224	29	2012	2012	NUM
cana-5538	224	30	(	(	PUNCT
cana-5538	224	31	2012	2012	NUM
cana-5538	224	32	)	)	PUNCT
cana-5538	224	33	,	,	PUNCT
cana-5538	224	34	article	article	NOUN
cana-5538	224	35	i	i	PROPN
cana-5538	224	36	d	d	PROPN
cana-5538	224	37	793486	793486	NUM
cana-5538	224	38	.	.	PUNCT
cana-5538	225	1	[	[	X
cana-5538	225	2	7	7	X
cana-5538	225	3	]	]	X
cana-5538	225	4	l.	l.	PROPN
cana-5538	225	5	g.	g.	PROPN
cana-5538	225	6	huang	huang	PROPN
cana-5538	225	7	,	,	PUNCT
cana-5538	225	8	x.	x.	PROPN
cana-5538	225	9	zhang	zhang	PROPN
cana-5538	225	10	,	,	PUNCT
cana-5538	225	11	cone	cone	NOUN
cana-5538	225	12	metric	metric	ADJ
cana-5538	225	13	spaces	space	NOUN
cana-5538	225	14	and	and	CCONJ
cana-5538	225	15	fixed	fix	VERB
cana-5538	225	16	point	point	NOUN
cana-5538	225	17	theorems	theorem	NOUN
cana-5538	225	18	of	of	ADP
cana-5538	225	19	contractive	contractive	ADJ
cana-5538	225	20	mappings	mapping	NOUN
cana-5538	225	21	,	,	PUNCT
cana-5538	225	22	j.	j.	PROPN
cana-5538	225	23	math	math	PROPN
cana-5538	225	24	.	.	PUNCT
cana-5538	226	1	anal	anal	PROPN
cana-5538	226	2	.	.	PUNCT
cana-5538	227	1	app	app	PROPN
cana-5538	227	2	.	.	PUNCT
cana-5538	228	1	332	332	NUM
cana-5538	228	2	(	(	PUNCT
cana-5538	228	3	2007	2007	NUM
cana-5538	228	4	)	)	PUNCT
cana-5538	228	5	,	,	PUNCT
cana-5538	228	6	1468	1468	NUM
cana-5538	228	7	-	-	SYM
cana-5538	228	8	1476	1476	NUM
cana-5538	228	9	.	.	PUNCT
cana-5538	229	1	[	[	X
cana-5538	229	2	8	8	NUM
cana-5538	229	3	]	]	PUNCT
cana-5538	229	4	m.	m.	NOUN
cana-5538	229	5	abbas	abbas	PROPN
cana-5538	229	6	,	,	PUNCT
cana-5538	229	7	g.	g.	PROPN
cana-5538	229	8	jungck	jungck	PROPN
cana-5538	229	9	,	,	PUNCT
cana-5538	229	10	common	common	ADJ
cana-5538	229	11	fixed	fix	VERB
cana-5538	229	12	point	point	NOUN
cana-5538	229	13	results	result	NOUN
cana-5538	229	14	for	for	ADP
cana-5538	229	15	non	non	ADJ
cana-5538	229	16	commuting	commuting	NOUN
cana-5538	229	17	mappings	mapping	NOUN
cana-5538	229	18	without	without	ADP
cana-5538	229	19	continuity	continuity	NOUN
cana-5538	229	20	in	in	ADP
cana-5538	229	21	cone	cone	NOUN
cana-5538	229	22	metric	metric	ADJ
cana-5538	229	23	spaces	space	NOUN
cana-5538	229	24	,	,	PUNCT
cana-5538	229	25	j.	j.	PROPN
cana-5538	229	26	math	math	PROPN
cana-5538	229	27	.	.	PUNCT
cana-5538	230	1	anal	anal	PROPN
cana-5538	230	2	.	.	PUNCT
cana-5538	230	3	appl	appl	PROPN
cana-5538	230	4	.	.	PROPN
cana-5538	230	5	,	,	PUNCT
cana-5538	230	6	341	341	NUM
cana-5538	230	7	(	(	PUNCT
cana-5538	230	8	2008	2008	NUM
cana-5538	230	9	)	)	PUNCT
cana-5538	230	10	,	,	PUNCT
cana-5538	230	11	416–420	416–420	NUM
cana-5538	230	12	.	.	PUNCT
cana-5538	231	1	[	[	X
cana-5538	231	2	9	9	NUM
cana-5538	231	3	]	]	PUNCT
cana-5538	231	4	m.	m.	NOUN
cana-5538	231	5	abbas	abbas	PROPN
cana-5538	231	6	and	and	CCONJ
cana-5538	231	7	b.	b.	PROPN
cana-5538	231	8	e.	e.	PROPN
cana-5538	231	9	rhoades	rhoades	PROPN
cana-5538	231	10	,	,	PUNCT
cana-5538	231	11	fixed	fix	VERB
cana-5538	231	12	and	and	CCONJ
cana-5538	231	13	periodic	periodic	ADJ
cana-5538	231	14	point	point	NOUN
cana-5538	231	15	results	result	NOUN
cana-5538	231	16	in	in	ADP
cana-5538	231	17	cone	cone	NOUN
cana-5538	231	18	metric	metric	ADJ
cana-5538	231	19	space	space	NOUN
cana-5538	231	20	,	,	PUNCT
cana-5538	231	21	appl	appl	PROPN
cana-5538	231	22	.	.	PROPN
cana-5538	231	23	math	math	PROPN
cana-5538	231	24	.	.	PUNCT
cana-5538	232	1	lett	lett	PROPN
cana-5538	232	2	.	.	PROPN
cana-5538	232	3	,	,	PUNCT
cana-5538	232	4	22(4	22(4	NUM
cana-5538	232	5	)	)	PUNCT
cana-5538	232	6	(	(	PUNCT
cana-5538	232	7	2009	2009	NUM
cana-5538	232	8	)	)	PUNCT
cana-5538	232	9	,	,	PUNCT
cana-5538	232	10	511–515	511–515	NUM
cana-5538	232	11	.	.	PUNCT
cana-5538	233	1	[	[	X
cana-5538	233	2	10	10	NUM
cana-5538	233	3	]	]	PUNCT
cana-5538	233	4	s.	s.	PROPN
cana-5538	233	5	k.	k.	PROPN
cana-5538	233	6	mahlotra	mahlotra	PROPN
cana-5538	233	7	,	,	PUNCT
cana-5538	233	8	s.	s.	PROPN
cana-5538	233	9	shukla	shukla	PROPN
cana-5538	233	10	,	,	PUNCT
cana-5538	233	11	r.	r.	PROPN
cana-5538	233	12	sen	sen	PROPN
cana-5538	233	13	,	,	PUNCT
cana-5538	233	14	n.	n.	PROPN
cana-5538	233	15	verma	verma	PROPN
cana-5538	233	16	,	,	PUNCT
cana-5538	233	17	fixed	fix	VERB
cana-5538	233	18	point	point	NOUN
cana-5538	233	19	theorems	theorem	NOUN
cana-5538	233	20	in	in	ADP
cana-5538	233	21	partial	partial	ADJ
cana-5538	233	22	cone	cone	NOUN
cana-5538	233	23	metric	metric	ADJ
cana-5538	233	24	spaces	space	NOUN
cana-5538	233	25	,	,	PUNCT
cana-5538	233	26	inter	inter	PROPN
cana-5538	233	27	.	.	PUNCT
cana-5538	234	1	j.	j.	PROPN
cana-5538	234	2	math	math	PROPN
cana-5538	234	3	.	.	PUNCT
cana-5538	235	1	arch	arch	PROPN
cana-5538	235	2	.	.	PUNCT
cana-5538	235	3	,	,	PUNCT
cana-5538	235	4	2(4	2(4	NUM
cana-5538	235	5	)	)	PUNCT
cana-5538	235	6	(	(	PUNCT
cana-5538	235	7	2011	2011	NUM
cana-5538	235	8	)	)	PUNCT
cana-5538	235	9	,	,	PUNCT
cana-5538	235	10	610–616	610–616	NUM
cana-5538	235	11	.	.	PUNCT
cana-5538	236	1	[	[	X
cana-5538	236	2	11	11	NUM
cana-5538	236	3	]	]	PUNCT
cana-5538	236	4	a.	a.	NOUN
cana-5538	236	5	sonmez	sonmez	NOUN
cana-5538	236	6	,	,	PUNCT
cana-5538	236	7	fixed	fix	VERB
cana-5538	236	8	point	point	NOUN
cana-5538	236	9	theorems	theorem	NOUN
cana-5538	236	10	in	in	ADP
cana-5538	236	11	partial	partial	ADJ
cana-5538	236	12	cone	cone	NOUN
cana-5538	236	13	metric	metric	ADJ
cana-5538	236	14	spaces	space	NOUN
cana-5538	236	15	,	,	PUNCT
cana-5538	236	16	arxiv:1101.2741v1	arxiv:1101.2741v1	PROPN
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cana-5538	236	18	math.gn	math.gn	X
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cana-5538	237	1	[	[	X
cana-5538	237	2	12	12	NUM
cana-5538	237	3	]	]	PUNCT
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cana-5538	237	7	n.	n.	PROPN
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cana-5538	237	13	generalized	generalized	ADJ
cana-5538	237	14	α	α	NOUN
cana-5538	237	15	−	−	NOUN
cana-5538	237	16	ψ	ψ	ADP
cana-5538	237	17	contractive	contractive	ADJ
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cana-5538	237	19	in	in	ADP
cana-5538	237	20	cone	cone	NOUN
cana-5538	237	21	metric	metric	ADJ
cana-5538	237	22	spaces	space	NOUN
cana-5538	237	23	,	,	PUNCT
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cana-5538	237	27	point	point	NOUN
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cana-5538	237	29	,	,	PUNCT
cana-5538	237	30	6(3	6(3	NUM
cana-5538	237	31	)	)	PUNCT
cana-5538	237	32	(	(	PUNCT
cana-5538	237	33	2016	2016	NUM
cana-5538	237	34	)	)	PUNCT
cana-5538	237	35	,	,	PUNCT
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cana-5538	237	37	-	-	SYM
cana-5538	237	38	253	253	NUM
cana-5538	237	39	.	.	PUNCT
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cana-5538	238	2	13	13	NUM
cana-5538	238	3	]	]	PUNCT
cana-5538	238	4	s.	s.	PROPN
cana-5538	238	5	k.	k.	PROPN
cana-5538	238	6	malhotra	malhotra	PROPN
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cana-5538	238	19	fixed	fix	VERB
cana-5538	238	20	point	point	NOUN
cana-5538	238	21	theorems	theorem	NOUN
cana-5538	238	22	in	in	ADP
cana-5538	238	23	partial	partial	ADJ
cana-5538	238	24	cone	cone	NOUN
cana-5538	238	25	metric	metric	ADJ
cana-5538	238	26	spaces	space	NOUN
cana-5538	238	27	,	,	PUNCT
cana-5538	238	28	int	int	NOUN
cana-5538	238	29	.	.	PUNCT
cana-5538	239	1	j.	j.	PROPN
cana-5538	239	2	math	math	PROPN
cana-5538	239	3	.	.	PUNCT
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cana-5538	240	2	.	.	PUNCT
cana-5538	241	1	2(4	2(4	NUM
cana-5538	241	2	)	)	PUNCT
cana-5538	241	3	,	,	PUNCT
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cana-5538	241	5	.	.	PUNCT
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cana-5538	242	2	14	14	NUM
cana-5538	242	3	]	]	X
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cana-5538	242	8	partial	partial	ADJ
cana-5538	242	9	metric	metric	ADJ
cana-5538	242	10	spaces	space	NOUN
cana-5538	242	11	and	and	CCONJ
cana-5538	242	12	partial	partial	ADJ
cana-5538	242	13	cone	cone	NOUN
cana-5538	242	14	metric	metric	ADJ
cana-5538	242	15	spaces	space	NOUN
cana-5538	242	16	,	,	PUNCT
cana-5538	242	17	hacettepe	hacettepe	ADJ
cana-5538	242	18	journal	journal	NOUN
cana-5538	242	19	of	of	ADP
cana-5538	242	20	mathematics	mathematics	PROPN
cana-5538	242	21	and	and	CCONJ
cana-5538	242	22	statistics	statistic	NOUN
cana-5538	242	23	volume	volume	NOUN
cana-5538	242	24	46	46	NUM
cana-5538	242	25	(	(	PUNCT
cana-5538	242	26	6	6	NUM
cana-5538	242	27	)	)	PUNCT
cana-5538	242	28	(	(	PUNCT
cana-5538	242	29	2017	2017	NUM
cana-5538	242	30	)	)	PUNCT
cana-5538	242	31	,	,	PUNCT
cana-5538	242	32	1069	1069	NUM
cana-5538	242	33	–	–	PUNCT
cana-5538	242	34	1075	1075	NUM
cana-5538	242	35	.	.	PUNCT
cana-5538	243	1	[	[	X
cana-5538	243	2	15	15	NUM
cana-5538	243	3	]	]	X
cana-5538	243	4	j.	j.	PROPN
cana-5538	243	5	fernandez	fernandez	PROPN
cana-5538	243	6	,	,	PUNCT
cana-5538	243	7	some	some	DET
cana-5538	243	8	fixed	fix	VERB
cana-5538	243	9	-	-	PUNCT
cana-5538	243	10	point	point	NOUN
cana-5538	243	11	results	result	NOUN
cana-5538	243	12	of	of	ADP
cana-5538	243	13	α	α	NOUN
cana-5538	243	14	-	-	PUNCT
cana-5538	243	15	admissible	admissible	ADJ
cana-5538	243	16	mappings	mapping	NOUN
cana-5538	243	17	in	in	ADP
cana-5538	243	18	partial	partial	ADJ
cana-5538	243	19	cone	cone	NOUN
cana-5538	243	20	metric	metric	ADJ
cana-5538	243	21	spaces	space	NOUN
cana-5538	243	22	over	over	ADP
cana-5538	243	23	banach	banach	NOUN
cana-5538	243	24	algebra	algebra	NOUN
cana-5538	243	25	,	,	PUNCT
cana-5538	243	26	proceedings	proceeding	NOUN
cana-5538	243	27	of	of	ADP
cana-5538	243	28	the	the	DET
cana-5538	243	29	1st	1st	ADJ
cana-5538	243	30	international	international	ADJ
cana-5538	243	31	conference	conference	NOUN
cana-5538	243	32	on	on	ADP
cana-5538	243	33	artificial	artificial	ADJ
cana-5538	243	34	intelligence	intelligence	NOUN
cana-5538	243	35	for	for	ADP
cana-5538	243	36	internet	internet	NOUN
cana-5538	243	37	of	of	ADP
cana-5538	243	38	things	thing	NOUN
cana-5538	243	39	:	:	PUNCT
cana-5538	243	40	accelerating	accelerate	VERB
cana-5538	243	41	innovation	innovation	NOUN
cana-5538	243	42	in	in	ADP
cana-5538	243	43	industry	industry	NOUN
cana-5538	243	44	and	and	CCONJ
cana-5538	243	45	consumer	consumer	NOUN
cana-5538	243	46	electronics	electronic	NOUN
cana-5538	243	47	(	(	PUNCT
cana-5538	243	48	ai4iot	ai4iot	NOUN
cana-5538	243	49	2023	2023	NUM
cana-5538	243	50	)	)	PUNCT
cana-5538	243	51	,	,	PUNCT
cana-5538	243	52	pages	page	NOUN
cana-5538	243	53	192	192	NUM
cana-5538	243	54	-	-	SYM
cana-5538	243	55	194	194	NUM
cana-5538	243	56	.	.	PUNCT
cana-5538	244	1	[	[	X
cana-5538	244	2	16	16	NUM
cana-5538	244	3	]	]	PUNCT
cana-5538	244	4	j.	j.	PROPN
cana-5538	244	5	fernandez	fernandez	PROPN
cana-5538	244	6	,	,	PUNCT
cana-5538	244	7	k.	k.	PROPN
cana-5538	244	8	saxena	saxena	PROPN
cana-5538	244	9	and	and	CCONJ
cana-5538	244	10	n.	n.	PROPN
cana-5538	244	11	malviya	malviya	PROPN
cana-5538	244	12	,	,	PUNCT
cana-5538	244	13	fixed	fix	VERB
cana-5538	244	14	points	point	NOUN
cana-5538	244	15	of	of	ADP
cana-5538	244	16	expansive	expansive	ADJ
cana-5538	244	17	maps	map	NOUN
cana-5538	244	18	in	in	ADP
cana-5538	244	19	partial	partial	ADJ
cana-5538	244	20	cone	cone	NOUN
cana-5538	244	21	metric	metric	ADJ
cana-5538	244	22	spaces	space	NOUN
cana-5538	244	23	,	,	PUNCT
cana-5538	244	24	gu	gu	PROPN
cana-5538	244	25	j	j	PROPN
cana-5538	244	26	sci	sci	PROPN
cana-5538	244	27	27(4):1085	27(4):1085	PROPN
cana-5538	244	28	-	-	SYM
cana-5538	244	29	1091	1091	NUM
cana-5538	244	30	(	(	PUNCT
cana-5538	244	31	2014	2014	NUM
cana-5538	244	32	)	)	PUNCT
cana-5538	244	33	.	.	PUNCT
cana-5538	245	1	[	[	X
cana-5538	245	2	17	17	NUM
cana-5538	245	3	]	]	PUNCT
cana-5538	245	4	j.	j.	PROPN
cana-5538	245	5	fernandez	fernandez	PROPN
cana-5538	245	6	,	,	PUNCT
cana-5538	245	7	s.	s.	PROPN
cana-5538	245	8	saelee	saelee	PROPN
cana-5538	245	9	,	,	PUNCT
cana-5538	245	10	k.	k.	PROPN
cana-5538	245	11	saxena	saxena	PROPN
cana-5538	245	12	,	,	PUNCT
cana-5538	245	13	n.	n.	PROPN
cana-5538	245	14	malviya	malviya	PROPN
cana-5538	245	15	and	and	CCONJ
cana-5538	245	16	p.	p.	PROPN
cana-5538	245	17	kumam	kumam	PROPN
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cana-5538	245	20	a	a	DET
cana-5538	245	21	-	-	PUNCT
cana-5538	245	22	cone	cone	NOUN
cana-5538	245	23	metric	metric	ADJ
cana-5538	245	24	space	space	NOUN
cana-5538	245	25	over	over	ADP
cana-5538	245	26	banach	banach	NOUN
cana-5538	245	27	algebra	algebra	NOUN
cana-5538	245	28	with	with	ADP
cana-5538	245	29	applications	application	NOUN
cana-5538	245	30	,	,	PUNCT
cana-5538	245	31	,	,	PUNCT
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cana-5538	245	34	,	,	PUNCT
cana-5538	245	35	4:1	4:1	NUM
cana-5538	245	36	,	,	PUNCT
cana-5538	245	37	1282690	1282690	NUM
cana-5538	245	38	.	.	PUNCT
cana-5538	246	1	[	[	X
cana-5538	246	2	18	18	NUM
cana-5538	246	3	]	]	X
cana-5538	246	4	d.	d.	PROPN
cana-5538	246	5	turkoglu	turkoglu	PROPN
cana-5538	246	6	,	,	PUNCT
cana-5538	246	7	m.	m.	NOUN
cana-5538	246	8	abuloha	abuloha	PROPN
cana-5538	246	9	,	,	PUNCT
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cana-5538	246	11	metric	metric	ADJ
cana-5538	246	12	spaces	space	NOUN
cana-5538	246	13	and	and	CCONJ
cana-5538	246	14	fixed	fix	VERB
cana-5538	246	15	point	point	NOUN
cana-5538	246	16	theorems	theorem	NOUN
cana-5538	246	17	in	in	ADP
cana-5538	246	18	diametrically	diametrically	ADV
cana-5538	246	19	contractive	contractive	ADJ
cana-5538	246	20	mappings	mapping	NOUN
cana-5538	246	21	,	,	PUNCT
cana-5538	246	22	acta	acta	PROPN
cana-5538	246	23	math	math	PROPN
cana-5538	246	24	.	.	PUNCT
cana-5538	247	1	sin	sin	NOUN
cana-5538	247	2	.	.	PUNCT
cana-5538	248	1	engl	engl	PROPN
cana-5538	248	2	.	.	PUNCT
cana-5538	248	3	series	series	PROPN
cana-5538	248	4	,	,	PUNCT
cana-5538	248	5	26(3	26(3	NUM
cana-5538	248	6	)	)	PUNCT
cana-5538	248	7	(	(	PUNCT
cana-5538	248	8	2010	2010	NUM
cana-5538	248	9	)	)	PUNCT
cana-5538	248	10	,	,	PUNCT
cana-5538	248	11	489–496	489–496	NUM
cana-5538	248	12	.	.	PUNCT
cana-5538	249	1	[	[	X
cana-5538	249	2	19	19	NUM
cana-5538	249	3	]	]	X
cana-5538	249	4	r.a	r.a	PROPN
cana-5538	249	5	.	.	PROPN
cana-5538	249	6	rashwan	rashwan	PROPN
cana-5538	249	7	,	,	PUNCT
cana-5538	249	8	h.a	h.a	PROPN
cana-5538	249	9	.	.	PROPN
cana-5538	249	10	hammad	hammad	PROPN
cana-5538	249	11	,	,	PUNCT
cana-5538	249	12	m.	m.	PROPN
cana-5538	249	13	gamal	gamal	PROPN
cana-5538	249	14	,	,	PUNCT
cana-5538	249	15	s.	s.	PROPN
cana-5538	249	16	omran	omran	PROPN
cana-5538	249	17	,	,	PUNCT
cana-5538	249	18	,	,	PUNCT
cana-5538	249	19	m.	m.	NOUN
cana-5538	249	20	de	de	PROPN
cana-5538	249	21	la	la	PROPN
cana-5538	249	22	sen	sen	PROPN
cana-5538	249	23	,	,	PUNCT
cana-5538	249	24	fixed	fix	VERB
cana-5538	249	25	point	point	NOUN
cana-5538	249	26	method	method	NOUN
cana-5538	249	27	ologies	ology	NOUN
cana-5538	249	28	for	for	ADP
cana-5538	249	29	ψ	ψ	NOUN
cana-5538	249	30	-	-	NOUN
cana-5538	249	31	contraction	contraction	NOUN
cana-5538	249	32	mappings	mapping	NOUN
cana-5538	249	33	in	in	ADP
cana-5538	249	34	cone	cone	NOUN
cana-5538	249	35	metric	metric	ADJ
cana-5538	249	36	spaces	space	NOUN
cana-5538	249	37	over	over	ADP
cana-5538	249	38	banach	banach	NOUN
cana-5538	249	39	algebra	algebra	NOUN
cana-5538	249	40	with	with	ADP
cana-5538	249	41	sup	sup	PROPN
cana-5538	249	42	portive	portive	ADJ
cana-5538	249	43	application	application	NOUN
cana-5538	249	44	,	,	PUNCT
cana-5538	249	45	int	int	NOUN
cana-5538	249	46	.	.	PUNCT
cana-5538	250	1	j.	j.	PROPN
cana-5538	250	2	anal	anal	PROPN
cana-5538	250	3	.	.	PUNCT
cana-5538	251	1	appl	appl	PROPN
cana-5538	251	2	.	.	PUNCT
cana-5538	252	1	(	(	PUNCT
cana-5538	252	2	2024	2024	NUM
cana-5538	252	3	)	)	PUNCT
cana-5538	252	4	,	,	PUNCT
cana-5538	252	5	22:120	22:120	NUM
cana-5538	252	6	.	.	PUNCT
cana-5538	253	1	[	[	X
cana-5538	253	2	20	20	NUM
cana-5538	253	3	]	]	PUNCT
cana-5538	253	4	p.	p.	NOUN
cana-5538	253	5	kumam	kumam	PROPN
cana-5538	253	6	,	,	PUNCT
cana-5538	253	7	c.	c.	PROPN
cana-5538	253	8	vetro	vetro	PROPN
cana-5538	253	9	and	and	CCONJ
cana-5538	253	10	f.	f.	PROPN
cana-5538	253	11	vetro	vetro	PROPN
cana-5538	253	12	,	,	PUNCT
cana-5538	253	13	fixed	fix	VERB
cana-5538	253	14	points	point	NOUN
cana-5538	253	15	for	for	ADP
cana-5538	253	16	weak	weak	ADJ
cana-5538	253	17	ψ	ψ	NOUN
cana-5538	253	18	-	-	NOUN
cana-5538	253	19	contractions	contraction	NOUN
cana-5538	253	20	in	in	ADP
cana-5538	253	21	partial	partial	ADJ
cana-5538	253	22	metric	metric	ADJ
cana-5538	253	23	spaces	space	NOUN
cana-5538	253	24	,	,	PUNCT
cana-5538	253	25	abstract	abstract	ADJ
cana-5538	253	26	and	and	CCONJ
cana-5538	253	27	applied	apply	VERB
cana-5538	253	28	analysis	analysis	NOUN
cana-5538	253	29	volume	volume	NOUN
cana-5538	253	30	2013	2013	NUM
cana-5538	253	31	,	,	PUNCT
cana-5538	253	32	article	article	NOUN
cana-5538	253	33	i	i	PROPN
cana-5538	253	34	d	d	PROPN
cana-5538	253	35	986028	986028	NUM
cana-5538	253	36	.	.	PUNCT