id	sid	tid	token	lemma	pos
cana-5423	1	1	study	study	NOUN
cana-5423	1	2	of	of	ADP
cana-5423	1	3	generalized	generalized	ADJ
cana-5423	1	4	αadmissible	αadmissible	ADJ
cana-5423	1	5	modified	modify	VERB
cana-5423	1	6	almost	almost	ADV
cana-5423	1	7	zcontractions	zcontraction	NOUN
cana-5423	1	8	via	via	ADP
cana-5423	1	9	simulation	simulation	NOUN
cana-5423	1	10	functions	function	NOUN
cana-5423	1	11	surendra	surendra	PROPN
cana-5423	1	12	kumar	kumar	PROPN
cana-5423	1	13	tiwati	tiwati	PROPN
cana-5423	1	14	,	,	PUNCT
cana-5423	1	15	anand	anand	PROPN
cana-5423	1	16	mohan	mohan	PROPN
cana-5423	1	17	dubey	dubey	PROPN
cana-5423	1	18	,	,	PUNCT
cana-5423	1	19	a	a	DET
cana-5423	1	20	v	v	NOUN
cana-5423	1	21	senthil	senthil	NOUN
cana-5423	1	22	kumar	kumar	PROPN
cana-5423	1	23	1	1	NUM
cana-5423	1	24	.	.	PUNCT
cana-5423	1	25	introduction	introduction	NOUN
cana-5423	1	26	and	and	CCONJ
cana-5423	1	27	preliminaries	preliminary	NOUN
cana-5423	1	28	consider	consider	VERB
cana-5423	1	29	n⊬	n⊬	NOUN
cana-5423	1	30	=	=	NOUN
cana-5423	1	31	n	n	NOUN
cana-5423	1	32	∪	∪	X
cana-5423	1	33	{	{	PUNCT
cana-5423	1	34	0	0	NUM
cana-5423	1	35	}	}	PUNCT
cana-5423	1	36	,	,	PUNCT
cana-5423	1	37	where	where	SCONJ
cana-5423	1	38	n	n	PRON
cana-5423	1	39	denotes	denote	VERB
cana-5423	1	40	the	the	DET
cana-5423	1	41	set	set	NOUN
cana-5423	1	42	of	of	ADP
cana-5423	1	43	positive	positive	ADJ
cana-5423	1	44	integers	integer	NOUN
cana-5423	1	45	.	.	PUNCT
cana-5423	2	1	as	as	ADP
cana-5423	2	2	usual	usual	ADJ
cana-5423	2	3	r	r	NOUN
cana-5423	2	4	indicates	indicate	VERB
cana-5423	2	5	the	the	DET
cana-5423	2	6	set	set	NOUN
cana-5423	2	7	of	of	ADP
cana-5423	2	8	real	real	ADJ
cana-5423	2	9	numbers.furthermore	numbers.furthermore	ADV
cana-5423	2	10	,	,	PUNCT
cana-5423	2	11	we	we	PRON
cana-5423	2	12	set	set	VERB
cana-5423	2	13	r0	r0	NOUN
cana-5423	2	14	+	+	NOUN
cana-5423	3	1	=	=	PUNCT
cana-5423	4	1	[	[	X
cana-5423	4	2	0	0	NUM
cana-5423	4	3	,	,	PUNCT
cana-5423	4	4	∞	∞	PROPN
cana-5423	4	5	]	]	PUNCT
cana-5423	4	6	.	.	PUNCT
cana-5423	5	1	many	many	ADJ
cana-5423	5	2	problems	problem	NOUN
cana-5423	5	3	in	in	ADP
cana-5423	5	4	several	several	ADJ
cana-5423	5	5	branches	branch	NOUN
cana-5423	5	6	of	of	ADP
cana-5423	5	7	mathematics	mathematic	NOUN
cana-5423	5	8	are	be	AUX
cana-5423	5	9	well	well	ADV
cana-5423	5	10	known	known	ADJ
cana-5423	5	11	to	to	PART
cana-5423	5	12	be	be	AUX
cana-5423	5	13	transformed	transform	VERB
cana-5423	5	14	into	into	ADP
cana-5423	5	15	invariant	invariant	ADJ
cana-5423	5	16	point	point	NOUN
cana-5423	5	17	problems	problem	NOUN
cana-5423	5	18	in	in	ADP
cana-5423	5	19	the	the	DET
cana-5423	5	20	form	form	NOUN
cana-5423	5	21	t	t	NOUN
cana-5423	5	22	x	x	PUNCT
cana-5423	6	1	=	=	PUNCT
cana-5423	6	2	x	x	PROPN
cana-5423	6	3	for	for	ADP
cana-5423	6	4	self	self	NOUN
cana-5423	6	5	mapping	mapping	NOUN
cana-5423	6	6	t	t	NOUN
cana-5423	6	7	.	.	PUNCT
cana-5423	7	1	it	it	PRON
cana-5423	7	2	is	be	AUX
cana-5423	7	3	worth	worth	ADJ
cana-5423	7	4	noting	note	VERB
cana-5423	7	5	that	that	SCONJ
cana-5423	7	6	based	base	VERB
cana-5423	7	7	on	on	ADP
cana-5423	7	8	the	the	DET
cana-5423	7	9	work	work	NOUN
cana-5423	7	10	of	of	ADP
cana-5423	7	11	banach	banach	NOUN
cana-5423	7	12	s.	s.	PROPN
cana-5423	8	1	[	[	X
cana-5423	8	2	7	7	NUM
cana-5423	8	3	]	]	X
cana-5423	8	4	in1922	in1922	PROPN
cana-5423	8	5	,	,	PUNCT
cana-5423	8	6	known	know	VERB
cana-5423	8	7	as	as	ADP
cana-5423	8	8	the	the	DET
cana-5423	8	9	banach	banach	NOUN
cana-5423	8	10	contraction	contraction	NOUN
cana-5423	8	11	principle	principle	NOUN
cana-5423	8	12	(	(	PUNCT
cana-5423	8	13	bcp	bcp	PROPN
cana-5423	8	14	)	)	PUNCT
cana-5423	8	15	,	,	PUNCT
cana-5423	8	16	the	the	DET
cana-5423	8	17	metric	metric	ADJ
cana-5423	8	18	fixed	fix	VERB
cana-5423	8	19	point	point	NOUN
cana-5423	8	20	theory	theory	NOUN
cana-5423	8	21	took	take	VERB
cana-5423	8	22	off	off	ADP
cana-5423	8	23	.	.	PUNCT
cana-5423	9	1	alot	alot	NOUN
cana-5423	9	2	of	of	ADP
cana-5423	9	3	authrs	authrs	ADJ
cana-5423	9	4	studied	study	VERB
cana-5423	9	5	generalizations	generalization	NOUN
cana-5423	9	6	of	of	ADP
cana-5423	9	7	this	this	DET
cana-5423	9	8	principle	principle	NOUN
cana-5423	9	9	.	.	PUNCT
cana-5423	10	1	in	in	ADP
cana-5423	10	2	addition	addition	NOUN
cana-5423	10	3	,	,	PUNCT
cana-5423	10	4	berinde	berinde	NOUN
cana-5423	10	5	[	[	X
cana-5423	10	6	9	9	NUM
cana-5423	10	7	,	,	PUNCT
cana-5423	10	8	10	10	NUM
cana-5423	10	9	]	]	PUNCT
cana-5423	10	10	introduced	introduce	VERB
cana-5423	10	11	almost	almost	ADV
cana-5423	10	12	contractions	contraction	NOUN
cana-5423	10	13	which	which	PRON
cana-5423	10	14	exhibits	exhibit	VERB
cana-5423	10	15	new	new	ADJ
cana-5423	10	16	features	feature	NOUN
cana-5423	10	17	with	with	ADP
cana-5423	10	18	respect	respect	NOUN
cana-5423	10	19	to	to	ADP
cana-5423	10	20	the	the	DET
cana-5423	10	21	ones	one	NOUN
cana-5423	10	22	of	of	ADP
cana-5423	10	23	the	the	DET
cana-5423	10	24	particular	particular	ADJ
cana-5423	10	25	results	result	NOUN
cana-5423	10	26	in	in	SCONJ
cana-5423	10	27	coprated	coprate	VERB
cana-5423	10	28	as	as	SCONJ
cana-5423	10	29	follows	follow	VERB
cana-5423	10	30	:	:	PUNCT
cana-5423	10	31	definition	definition	NOUN
cana-5423	10	32	1.1	1.1	NUM
cana-5423	10	33	.	.	PUNCT
cana-5423	11	1	let	let	VERB
cana-5423	11	2	(	(	PUNCT
cana-5423	11	3	x	x	NOUN
cana-5423	11	4	,	,	PUNCT
cana-5423	11	5	d	d	NOUN
cana-5423	11	6	)	)	PUNCT
cana-5423	11	7	be	be	AUX
cana-5423	11	8	a	a	DET
cana-5423	11	9	metric	metric	ADJ
cana-5423	11	10	space	space	NOUN
cana-5423	11	11	.	.	PUNCT
cana-5423	12	1	a	a	DET
cana-5423	12	2	self	self	NOUN
cana-5423	12	3	mapping	map	VERB
cana-5423	12	4	γ	γ	NOUN
cana-5423	12	5	on	on	ADP
cana-5423	12	6	x	x	SYM
cana-5423	12	7	is	be	AUX
cana-5423	12	8	called	call	VERB
cana-5423	12	9	an	an	DET
cana-5423	12	10	almost	almost	ADV
cana-5423	12	11	contraction	contraction	NOUN
cana-5423	12	12	if	if	SCONJ
cana-5423	12	13	there	there	PRON
cana-5423	12	14	are	be	VERB
cana-5423	12	15	constants	constant	NOUN
cana-5423	12	16	δ	δ	PROPN
cana-5423	12	17	∈	∈	PROPN
cana-5423	13	1	[	[	X
cana-5423	13	2	0	0	NUM
cana-5423	13	3	,	,	PUNCT
cana-5423	13	4	1	1	NUM
cana-5423	13	5	)	)	PUNCT
cana-5423	13	6	and	and	CCONJ
cana-5423	13	7	∃l	∃l	PROPN
cana-5423	13	8	≥	≥	NUM
cana-5423	13	9	0	0	NUM
cana-5423	13	10	such	such	ADJ
cana-5423	13	11	that	that	SCONJ
cana-5423	13	12	d(γx	d(γx	PROPN
cana-5423	13	13	,	,	PUNCT
cana-5423	13	14	γy	γy	NOUN
cana-5423	13	15	)	)	PUNCT
cana-5423	13	16	≤	≤	NOUN
cana-5423	13	17	δd(x	δd(x	PUNCT
cana-5423	13	18	,	,	PUNCT
cana-5423	13	19	y	y	PROPN
cana-5423	13	20	)	)	PUNCT
cana-5423	13	21	+	+	CCONJ
cana-5423	13	22	ld(y	ld(y	X
cana-5423	13	23	,	,	PUNCT
cana-5423	13	24	γx),∀x	γx),∀x	PROPN
cana-5423	13	25	,	,	PUNCT
cana-5423	13	26	y	y	PROPN
cana-5423	13	27	∈	∈	PROPN
cana-5423	13	28	x.	x.	NOUN
cana-5423	13	29	(	(	PUNCT
cana-5423	13	30	1	1	X
cana-5423	13	31	)	)	PUNCT
cana-5423	13	32	berinde	berinde	NOUN
cana-5423	14	1	[	[	PUNCT
cana-5423	14	2	9	9	NUM
cana-5423	14	3	,	,	PUNCT
cana-5423	14	4	10	10	NUM
cana-5423	14	5	]	]	PUNCT
cana-5423	14	6	investigated	investigate	VERB
cana-5423	14	7	that	that	SCONJ
cana-5423	14	8	every	every	DET
cana-5423	14	9	almost	almost	ADV
cana-5423	14	10	contraction	contraction	NOUN
cana-5423	14	11	mapping	mapping	NOUN
cana-5423	14	12	defined	define	VERB
cana-5423	14	13	in	in	ADP
cana-5423	14	14	a	a	DET
cana-5423	14	15	complete	complete	ADJ
cana-5423	14	16	metric	metric	ADJ
cana-5423	14	17	space	space	NOUN
cana-5423	14	18	has	have	VERB
cana-5423	14	19	at	at	ADV
cana-5423	14	20	least	least	ADV
cana-5423	14	21	one	one	NUM
cana-5423	14	22	fixed	fix	VERB
cana-5423	14	23	point	point	NOUN
cana-5423	14	24	.	.	PUNCT
cana-5423	15	1	subsequently	subsequently	ADV
cana-5423	15	2	,	,	PUNCT
cana-5423	15	3	babu	babu	PROPN
cana-5423	15	4	et	et	PROPN
cana-5423	15	5	al.[6	al.[6	PROPN
cana-5423	15	6	]	]	PUNCT
cana-5423	15	7	defined	define	VERB
cana-5423	15	8	the	the	DET
cana-5423	15	9	class	class	NOUN
cana-5423	15	10	of	of	ADP
cana-5423	15	11	mapping	map	VERB
cana-5423	15	12	satisfying	satisfy	VERB
cana-5423	15	13	condition	condition	NOUN
cana-5423	15	14	(	(	PUNCT
cana-5423	15	15	b	b	NOUN
cana-5423	15	16	)	)	PUNCT
cana-5423	15	17	as	as	SCONJ
cana-5423	15	18	follows	follow	VERB
cana-5423	15	19	:	:	PUNCT
cana-5423	15	20	definition	definition	NOUN
cana-5423	15	21	1.2	1.2	NUM
cana-5423	15	22	.	.	PUNCT
cana-5423	16	1	let	let	VERB
cana-5423	16	2	(	(	PUNCT
cana-5423	16	3	x	x	NOUN
cana-5423	16	4	,	,	PUNCT
cana-5423	16	5	d	d	NOUN
cana-5423	16	6	)	)	PUNCT
cana-5423	16	7	be	be	AUX
cana-5423	16	8	a	a	DET
cana-5423	16	9	metric	metric	ADJ
cana-5423	16	10	space	space	NOUN
cana-5423	16	11	.	.	PUNCT
cana-5423	17	1	a	a	DET
cana-5423	17	2	self	self	NOUN
cana-5423	17	3	mapping	map	VERB
cana-5423	17	4	γ	γ	NOUN
cana-5423	17	5	on	on	ADP
cana-5423	17	6	x	x	PUNCT
cana-5423	17	7	issaid	issaid	VERB
cana-5423	17	8	to	to	PART
cana-5423	17	9	be	be	AUX
cana-5423	17	10	satisfy	satisfy	NOUN
cana-5423	17	11	condition	condition	NOUN
cana-5423	17	12	(	(	PUNCT
cana-5423	17	13	b	b	X
cana-5423	17	14	)	)	PUNCT
cana-5423	17	15	if	if	SCONJ
cana-5423	17	16	there	there	PRON
cana-5423	17	17	are	be	VERB
cana-5423	17	18	constants	constant	NOUN
cana-5423	17	19	δ	δ	PROPN
cana-5423	17	20	∈	∈	PROPN
cana-5423	18	1	[	[	X
cana-5423	18	2	0	0	NUM
cana-5423	18	3	,	,	PUNCT
cana-5423	18	4	1	1	NUM
cana-5423	18	5	)	)	PUNCT
cana-5423	18	6	and	and	CCONJ
cana-5423	18	7	∃l	∃l	PROPN
cana-5423	18	8	≥	≥	NUM
cana-5423	18	9	0	0	NUM
cana-5423	18	10	such	such	ADJ
cana-5423	18	11	that	that	SCONJ
cana-5423	18	12	d(γx	d(γx	PROPN
cana-5423	18	13	,	,	PUNCT
cana-5423	18	14	γy	γy	NOUN
cana-5423	18	15	)	)	PUNCT
cana-5423	18	16	≤	≤	NOUN
cana-5423	18	17	δd(x	δd(x	PUNCT
cana-5423	18	18	,	,	PUNCT
cana-5423	18	19	y	y	PROPN
cana-5423	18	20	)	)	PUNCT
cana-5423	18	21	+	+	CCONJ
cana-5423	18	22	lqd(y	lqd(y	PROPN
cana-5423	18	23	,	,	PUNCT
cana-5423	18	24	γx),∀x	γx),∀x	NOUN
cana-5423	18	25	,	,	PUNCT
cana-5423	18	26	y	y	PROPN
cana-5423	18	27	∈	∈	PROPN
cana-5423	18	28	x.	x.	NOUN
cana-5423	18	29	(	(	PUNCT
cana-5423	18	30	2	2	NUM
cana-5423	18	31	)	)	PUNCT
cana-5423	19	1	where	where	SCONJ
cana-5423	19	2	q(x	q(x	PROPN
cana-5423	19	3	,	,	PUNCT
cana-5423	19	4	y	y	NOUN
cana-5423	19	5	)	)	PUNCT
cana-5423	19	6	=	=	VERB
cana-5423	19	7	min{d(x	min{d(x	NOUN
cana-5423	19	8	,	,	PUNCT
cana-5423	19	9	γx	γx	NOUN
cana-5423	19	10	)	)	PUNCT
cana-5423	19	11	,	,	PUNCT
cana-5423	19	12	d(y	d(y	PROPN
cana-5423	19	13	,	,	PUNCT
cana-5423	19	14	γy	γy	NOUN
cana-5423	19	15	)	)	PUNCT
cana-5423	19	16	,	,	PUNCT
cana-5423	19	17	d(x	d(x	PROPN
cana-5423	19	18	,	,	PUNCT
cana-5423	19	19	γy	γy	PROPN
cana-5423	19	20	)	)	PUNCT
cana-5423	19	21	,	,	PUNCT
cana-5423	19	22	d(y	d(y	NOUN
cana-5423	19	23	,	,	PUNCT
cana-5423	19	24	γx	γx	NOUN
cana-5423	19	25	)	)	PUNCT
cana-5423	19	26	}	}	PUNCT
cana-5423	19	27	they	they	PRON
cana-5423	19	28	proved	prove	VERB
cana-5423	19	29	a	a	DET
cana-5423	19	30	fixed	fix	VERB
cana-5423	19	31	point	point	NOUN
cana-5423	19	32	theorem	theorem	NOUN
cana-5423	19	33	for	for	ADP
cana-5423	19	34	such	such	ADJ
cana-5423	19	35	mappings	mapping	NOUN
cana-5423	19	36	in	in	ADP
cana-5423	19	37	complete	complete	ADJ
cana-5423	19	38	metric	metric	ADJ
cana-5423	19	39	spaces	space	NOUN
cana-5423	19	40	.	.	PUNCT
cana-5423	20	1	they	they	PRON
cana-5423	20	2	also	also	ADV
cana-5423	20	3	discussed	discuss	VERB
cana-5423	20	4	quasi	quasi	NOUN
cana-5423	20	5	-	-	NOUN
cana-5423	20	6	contraction	contraction	NOUN
cana-5423	20	7	,	,	PUNCT
cana-5423	20	8	almost	almost	ADV
cana-5423	20	9	contraction	contraction	NOUN
cana-5423	20	10	and	and	CCONJ
cana-5423	20	11	the	the	DET
cana-5423	20	12	class	class	NOUN
cana-5423	20	13	of	of	ADP
cana-5423	20	14	mappingd	mappingd	NOUN
cana-5423	20	15	that	that	PRON
cana-5423	20	16	satisfy	satisfy	VERB
cana-5423	20	17	condition	condition	NOUN
cana-5423	20	18	(	(	PUNCT
cana-5423	20	19	b	b	NOUN
cana-5423	20	20	)	)	PUNCT
cana-5423	20	21	in	in	ADP
cana-5423	20	22	detail	detail	NOUN
cana-5423	20	23	.	.	PUNCT
cana-5423	21	1	iseki	iseki	PROPN
cana-5423	21	2	et	et	PROPN
cana-5423	21	3	al	al	PROPN
cana-5423	21	4	.	.	PUNCT
cana-5423	22	1	[	[	X
cana-5423	22	2	20	20	NUM
cana-5423	22	3	]	]	PUNCT
cana-5423	22	4	presented	present	VERB
cana-5423	22	5	definition	definition	NOUN
cana-5423	22	6	of	of	ADP
cana-5423	22	7	almost	almost	ADV
cana-5423	22	8	zcontraction	zcontraction	NOUN
cana-5423	22	9	as	as	SCONJ
cana-5423	22	10	follows	follow	VERB
cana-5423	22	11	:	:	PUNCT
cana-5423	22	12	date	date	NOUN
cana-5423	22	13	:	:	PUNCT
cana-5423	22	14	∗corresponding	∗corresponde	VERB
cana-5423	22	15	author	author	NOUN
cana-5423	22	16	:	:	PUNCT
cana-5423	22	17	surendra	surendra	PROPN
cana-5423	22	18	kumar	kumar	PROPN
cana-5423	22	19	tiwari	tiwari	PROPN
cana-5423	22	20	.	.	PROPN
cana-5423	23	1	2020	2020	NUM
cana-5423	23	2	mathematics	mathematic	NOUN
cana-5423	23	3	subject	subject	ADJ
cana-5423	23	4	classification	classification	NOUN
cana-5423	23	5	.	.	PUNCT
cana-5423	24	1	54h25,47h10,55m20	54h25,47h10,55m20	NUM
cana-5423	24	2	,	,	PUNCT
cana-5423	24	3	,	,	PUNCT
cana-5423	24	4	....	....	PUNCT
cana-5423	25	1	key	key	ADJ
cana-5423	25	2	words	word	NOUN
cana-5423	25	3	and	and	CCONJ
cana-5423	25	4	phrases	phrase	NOUN
cana-5423	25	5	.	.	PUNCT
cana-5423	26	1	almost	almost	ADV
cana-5423	26	2	z−contraction	z−contraction	NUM
cana-5423	26	3	,	,	PUNCT
cana-5423	26	4	modified	modify	VERB
cana-5423	26	5	almost	almost	ADV
cana-5423	26	6	z	z	NOUN
cana-5423	26	7	-	-	PUNCT
cana-5423	26	8	contraction	contraction	NOUN
cana-5423	26	9	,	,	PUNCT
cana-5423	26	10	simulation	simulation	NOUN
cana-5423	26	11	function	function	NOUN
cana-5423	26	12	,	,	PUNCT
cana-5423	26	13	αadmissible	αadmissible	ADJ
cana-5423	26	14	mapping	mapping	NOUN
cana-5423	26	15	.	.	PUNCT
cana-5423	27	1	communications	communication	NOUN
cana-5423	27	2	on	on	ADP
cana-5423	27	3	applied	apply	VERB
cana-5423	27	4	nonlinear	nonlinear	ADJ
cana-5423	27	5	analysis	analysis	NOUN
cana-5423	27	6	issn	issn	NOUN
cana-5423	27	7	:	:	PUNCT
cana-5423	27	8	1074	1074	NUM
cana-5423	27	9	-	-	PUNCT
cana-5423	27	10	133x	133x	NUM
cana-5423	27	11	vol	vol	NOUN
cana-5423	27	12	32	32	NUM
cana-5423	27	13	no	no	NOUN
cana-5423	27	14	.	.	PUNCT
cana-5423	28	1	10s(2025	10s(2025	NUM
cana-5423	28	2	)	)	PUNCT
cana-5423	29	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	29	2	2217	2217	NUM
cana-5423	29	3	abstract	abstract	NOUN
cana-5423	29	4	.	.	PUNCT
cana-5423	30	1	in	in	ADP
cana-5423	30	2	this	this	DET
cana-5423	30	3	paper	paper	NOUN
cana-5423	30	4	,	,	PUNCT
cana-5423	30	5	we	we	PRON
cana-5423	30	6	introduce	introduce	VERB
cana-5423	30	7	generalized	generalized	ADJ
cana-5423	30	8	α	α	NOUN
cana-5423	30	9	-admissible	-admissible	ADJ
cana-5423	30	10	modified	modify	VERB
cana-5423	30	11	almost	almost	ADV
cana-5423	30	12	z	z	NOUN
cana-5423	30	13	-contraction	-contraction	NOUN
cana-5423	30	14	with	with	ADP
cana-5423	30	15	the	the	DET
cana-5423	30	16	help	help	NOUN
cana-5423	30	17	of	of	ADP
cana-5423	30	18	simulation	simulation	NOUN
cana-5423	30	19	function	function	NOUN
cana-5423	30	20	and	and	CCONJ
cana-5423	30	21	obtain	obtain	VERB
cana-5423	30	22	fixed	fix	VERB
cana-5423	30	23	point	point	NOUN
cana-5423	30	24	results	result	NOUN
cana-5423	30	25	in	in	ADP
cana-5423	30	26	the	the	DET
cana-5423	30	27	setting	setting	NOUN
cana-5423	30	28	og	og	INTJ
cana-5423	30	29	metric	metric	ADJ
cana-5423	30	30	space	space	NOUN
cana-5423	30	31	and	and	CCONJ
cana-5423	30	32	verified	verify	VERB
cana-5423	30	33	with	with	ADP
cana-5423	30	34	an	an	DET
cana-5423	30	35	example	example	NOUN
cana-5423	30	36	.	.	PUNCT
cana-5423	31	1	the	the	DET
cana-5423	31	2	presented	present	VERB
cana-5423	31	3	results	result	NOUN
cana-5423	31	4	extend	extend	VERB
cana-5423	31	5	,	,	PUNCT
cana-5423	31	6	generalize	generalize	VERB
cana-5423	31	7	and	and	CCONJ
cana-5423	31	8	unify	unify	VERB
cana-5423	31	9	several	several	ADJ
cana-5423	31	10	related	related	ADJ
cana-5423	31	11	fixed	fix	VERB
cana-5423	31	12	point	point	NOUN
cana-5423	31	13	finding	finding	NOUN
cana-5423	31	14	in	in	ADP
cana-5423	31	15	the	the	DET
cana-5423	31	16	existing	exist	VERB
cana-5423	31	17	literature	literature	NOUN
cana-5423	31	18	.	.	PUNCT
cana-5423	32	1	article	article	NOUN
cana-5423	32	2	history	history	NOUN
cana-5423	32	3	:	:	PUNCT
cana-5423	32	4	received	receive	VERB
cana-5423	32	5	:	:	PUNCT
cana-5423	32	6	10	10	NUM
cana-5423	32	7	-	-	SYM
cana-5423	32	8	01	01	NUM
cana-5423	32	9	-	-	PUNCT
cana-5423	32	10	2025	2025	NUM
cana-5423	32	11	revised	revise	VERB
cana-5423	32	12	:	:	PUNCT
cana-5423	32	13	15	15	NUM
cana-5423	32	14	-	-	NUM
cana-5423	32	15	02	02	NUM
cana-5423	32	16	-	-	PUNCT
cana-5423	32	17	2025	2025	NUM
cana-5423	32	18	accepted	accept	VERB
cana-5423	32	19	:	:	PUNCT
cana-5423	32	20	08	08	NUM
cana-5423	32	21	-	-	SYM
cana-5423	32	22	03	03	NUM
cana-5423	32	23	-	-	PUNCT
cana-5423	32	24	2025	2025	NUM
cana-5423	32	25	definition	definition	NOUN
cana-5423	32	26	1.3	1.3	NUM
cana-5423	32	27	.	.	PUNCT
cana-5423	33	1	let	let	VERB
cana-5423	33	2	(	(	PUNCT
cana-5423	33	3	x	x	NOUN
cana-5423	33	4	,	,	PUNCT
cana-5423	33	5	d	d	NOUN
cana-5423	33	6	)	)	PUNCT
cana-5423	33	7	be	be	AUX
cana-5423	33	8	a	a	DET
cana-5423	33	9	metric	metric	ADJ
cana-5423	33	10	space	space	NOUN
cana-5423	33	11	and	and	CCONJ
cana-5423	33	12	ζ	ζ	NOUN
cana-5423	33	13	.	.	PUNCT
cana-5423	34	1	a	a	DET
cana-5423	34	2	self	self	NOUN
cana-5423	34	3	mapping	mapping	NOUN
cana-5423	34	4	γ	γ	NOUN
cana-5423	34	5	:	:	PUNCT
cana-5423	34	6	x	x	SYM
cana-5423	34	7	→	→	PUNCT
cana-5423	34	8	x	x	X
cana-5423	34	9	is	be	AUX
cana-5423	34	10	called	call	VERB
cana-5423	34	11	an	an	DET
cana-5423	34	12	almost	almost	ADV
cana-5423	34	13	z	z	NOUN
cana-5423	34	14	-	-	PUNCT
cana-5423	34	15	contraction	contraction	NOUN
cana-5423	34	16	if	if	SCONJ
cana-5423	34	17	there	there	PRON
cana-5423	34	18	are	be	VERB
cana-5423	34	19	constants	constant	NOUN
cana-5423	34	20	θ	θ	PROPN
cana-5423	34	21	≥	≥	NUM
cana-5423	34	22	o	o	NOUN
cana-5423	34	23	such	such	ADJ
cana-5423	34	24	that	that	SCONJ
cana-5423	34	25	ζ(d(γx	ζ(d(γx	NOUN
cana-5423	34	26	,	,	PUNCT
cana-5423	34	27	γy	γy	NOUN
cana-5423	34	28	)	)	PUNCT
cana-5423	34	29	,	,	PUNCT
cana-5423	34	30	d(x	d(x	PROPN
cana-5423	34	31	,	,	PUNCT
cana-5423	34	32	y	y	PROPN
cana-5423	34	33	)	)	PUNCT
cana-5423	35	1	+	+	NUM
cana-5423	35	2	lq(x	lq(x	NOUN
cana-5423	35	3	,	,	PUNCT
cana-5423	35	4	y),∀x	y),∀x	PROPN
cana-5423	35	5	,	,	PUNCT
cana-5423	35	6	y	y	PROPN
cana-5423	35	7	∈	∈	PROPN
cana-5423	35	8	x	x	X
cana-5423	35	9	,	,	PUNCT
cana-5423	35	10	(	(	PUNCT
cana-5423	35	11	3	3	X
cana-5423	35	12	)	)	PUNCT
cana-5423	35	13	where	where	SCONJ
cana-5423	35	14	q(x	q(x	PROPN
cana-5423	35	15	,	,	PUNCT
cana-5423	35	16	y	y	NOUN
cana-5423	35	17	)	)	PUNCT
cana-5423	35	18	is	be	AUX
cana-5423	35	19	defined	define	VERB
cana-5423	35	20	as	as	ADP
cana-5423	35	21	in	in	ADP
cana-5423	35	22	definition	definition	NOUN
cana-5423	35	23	1.2	1.2	NUM
cana-5423	35	24	.	.	PUNCT
cana-5423	36	1	also	also	ADV
cana-5423	36	2	they	they	PRON
cana-5423	36	3	investigated	investigate	VERB
cana-5423	36	4	the	the	DET
cana-5423	36	5	existence	existence	NOUN
cana-5423	36	6	and	and	CCONJ
cana-5423	36	7	uniqueness	uniqueness	NOUN
cana-5423	36	8	of	of	ADP
cana-5423	36	9	a	a	DET
cana-5423	36	10	fixed	fix	VERB
cana-5423	36	11	point	point	NOUN
cana-5423	36	12	of	of	ADP
cana-5423	36	13	an	an	DET
cana-5423	36	14	almost	almost	ADV
cana-5423	36	15	zcontraction	zcontraction	NOUN
cana-5423	36	16	in	in	ADP
cana-5423	36	17	metric	metric	ADJ
cana-5423	36	18	space	space	NOUN
cana-5423	36	19	with	with	ADP
cana-5423	36	20	simulation	simulation	NOUN
cana-5423	36	21	functions	function	NOUN
cana-5423	36	22	.	.	PUNCT
cana-5423	37	1	authos	authos	NOUN
cana-5423	38	1	[	[	X
cana-5423	38	2	25	25	NUM
cana-5423	38	3	,	,	PUNCT
cana-5423	38	4	26	26	NUM
cana-5423	38	5	,	,	PUNCT
cana-5423	38	6	27	27	NUM
cana-5423	38	7	,	,	PUNCT
cana-5423	38	8	28	28	NUM
cana-5423	38	9	]	]	PUNCT
cana-5423	38	10	demostrated	demostrate	VERB
cana-5423	38	11	that	that	SCONJ
cana-5423	38	12	almost	almost	ADV
cana-5423	38	13	contractions	contraction	NOUN
cana-5423	38	14	type	type	NOUN
cana-5423	38	15	mappings	mapping	NOUN
cana-5423	38	16	have	have	VERB
cana-5423	38	17	a	a	DET
cana-5423	38	18	unique	unique	ADJ
cana-5423	38	19	fixede	fixede	NOUN
cana-5423	38	20	point	point	NOUN
cana-5423	38	21	in	in	ADP
cana-5423	38	22	deferent	deferent	ADJ
cana-5423	38	23	metric	metric	ADJ
cana-5423	38	24	spaces	space	NOUN
cana-5423	38	25	.	.	PUNCT
cana-5423	39	1	in	in	ADP
cana-5423	39	2	sequel	sequel	NOUN
cana-5423	39	3	,	,	PUNCT
cana-5423	39	4	p.	p.	PROPN
cana-5423	39	5	bunpatcharacharoen	bunpatcharacharoen	PROPN
cana-5423	39	6	et	et	PROPN
cana-5423	39	7	al	al	PROPN
cana-5423	39	8	.	.	PUNCT
cana-5423	40	1	[	[	X
cana-5423	40	2	11	11	NUM
cana-5423	40	3	]	]	PUNCT
cana-5423	40	4	modified	modify	VERB
cana-5423	40	5	almost	almost	ADV
cana-5423	40	6	type	type	NOUN
cana-5423	40	7	zcontraction	zcontraction	NOUN
cana-5423	40	8	mapping	mapping	NOUN
cana-5423	40	9	in	in	ADP
cana-5423	40	10	metric	metric	ADJ
cana-5423	40	11	space	space	NOUN
cana-5423	40	12	and	and	CCONJ
cana-5423	40	13	obtained	obtain	VERB
cana-5423	40	14	fixed	fix	VERB
cana-5423	40	15	point	point	NOUN
cana-5423	40	16	.	.	PUNCT
cana-5423	41	1	khojsteh	khojsteh	PROPN
cana-5423	41	2	et	et	PROPN
cana-5423	41	3	al	al	PROPN
cana-5423	41	4	.	.	PUNCT
cana-5423	42	1	[	[	X
cana-5423	42	2	29	29	NUM
cana-5423	42	3	]	]	PUNCT
cana-5423	42	4	originated	originate	VERB
cana-5423	42	5	the	the	DET
cana-5423	42	6	notion	notion	NOUN
cana-5423	42	7	of	of	ADP
cana-5423	42	8	zcontractions	zcontraction	NOUN
cana-5423	42	9	by	by	ADP
cana-5423	42	10	usung	usung	PROPN
cana-5423	42	11	a	a	DET
cana-5423	42	12	specific	specific	ADJ
cana-5423	42	13	family	family	NOUN
cana-5423	42	14	of	of	ADP
cana-5423	42	15	functions	function	NOUN
cana-5423	42	16	called	call	VERB
cana-5423	42	17	simulation	simulation	NOUN
cana-5423	42	18	functions	function	NOUN
cana-5423	42	19	and	and	CCONJ
cana-5423	42	20	proved	prove	VERB
cana-5423	42	21	a	a	DET
cana-5423	42	22	version	version	NOUN
cana-5423	42	23	of	of	ADP
cana-5423	42	24	bcp	bcp	PROPN
cana-5423	42	25	.	.	PROPN
cana-5423	42	26	subsequently	subsequently	ADV
cana-5423	42	27	,	,	PUNCT
cana-5423	42	28	many	many	ADJ
cana-5423	42	29	researchers	researcher	NOUN
cana-5423	42	30	generalized	generalize	VERB
cana-5423	42	31	this	this	DET
cana-5423	42	32	idea	idea	NOUN
cana-5423	42	33	in	in	ADP
cana-5423	42	34	many	many	ADJ
cana-5423	42	35	ways	way	NOUN
cana-5423	42	36	(	(	PUNCT
cana-5423	42	37	see	see	VERB
cana-5423	42	38	[	[	X
cana-5423	42	39	30	30	NUM
cana-5423	42	40	,	,	PUNCT
cana-5423	42	41	31	31	NUM
cana-5423	42	42	,	,	PUNCT
cana-5423	42	43	32	32	NUM
cana-5423	42	44	,	,	PUNCT
cana-5423	42	45	33	33	NUM
cana-5423	42	46	,	,	PUNCT
cana-5423	42	47	34	34	NUM
cana-5423	42	48	,	,	PUNCT
cana-5423	42	49	35	35	NUM
cana-5423	42	50	,	,	PUNCT
cana-5423	42	51	36	36	NUM
cana-5423	42	52	,	,	PUNCT
cana-5423	42	53	37	37	NUM
cana-5423	42	54	,	,	PUNCT
cana-5423	42	55	38	38	NUM
cana-5423	42	56	]	]	PUNCT
cana-5423	42	57	)	)	PUNCT
cana-5423	42	58	and	and	CCONJ
cana-5423	42	59	proved	prove	VERB
cana-5423	42	60	various	various	ADJ
cana-5423	42	61	interesting	interesting	ADJ
cana-5423	42	62	results	result	NOUN
cana-5423	42	63	in	in	ADP
cana-5423	42	64	the	the	DET
cana-5423	42	65	arena	arena	NOUN
cana-5423	42	66	of	of	ADP
cana-5423	42	67	fixed	fix	VERB
cana-5423	42	68	point	point	NOUN
cana-5423	42	69	theory	theory	NOUN
cana-5423	42	70	by	by	ADP
cana-5423	42	71	using	use	VERB
cana-5423	42	72	simulation	simulation	NOUN
cana-5423	42	73	functions	function	NOUN
cana-5423	42	74	.	.	PUNCT
cana-5423	43	1	definition	definition	NOUN
cana-5423	43	2	1.4	1.4	NUM
cana-5423	43	3	.	.	PUNCT
cana-5423	44	1	[	[	X
cana-5423	44	2	29	29	NUM
cana-5423	44	3	]	]	PUNCT
cana-5423	44	4	a	a	DET
cana-5423	44	5	function	function	NOUN
cana-5423	44	6	ζ	ζ	NOUN
cana-5423	44	7	;	;	PUNCT
cana-5423	44	8	[	[	X
cana-5423	44	9	o,∞)2	o,∞)2	NOUN
cana-5423	44	10	→	→	SYM
cana-5423	44	11	r	r	NOUN
cana-5423	44	12	is	be	AUX
cana-5423	44	13	called	call	VERB
cana-5423	44	14	a	a	DET
cana-5423	44	15	simulation	simulation	NOUN
cana-5423	44	16	function	function	NOUN
cana-5423	44	17	if	if	SCONJ
cana-5423	44	18	ζ	ζ	NOUN
cana-5423	44	19	satisfies	satisfy	VERB
cana-5423	44	20	the	the	DET
cana-5423	44	21	following	follow	VERB
cana-5423	44	22	conditions	condition	NOUN
cana-5423	44	23	:	:	PUNCT
cana-5423	44	24	(	(	PUNCT
cana-5423	44	25	ζ1	ζ1	NOUN
cana-5423	44	26	)	)	PUNCT
cana-5423	44	27	ζ(o	ζ(o	PROPN
cana-5423	44	28	,	,	PUNCT
cana-5423	44	29	o	o	NOUN
cana-5423	44	30	)	)	PUNCT
cana-5423	45	1	=	=	SYM
cana-5423	45	2	o	o	NOUN
cana-5423	45	3	;	;	PUNCT
cana-5423	45	4	(	(	PUNCT
cana-5423	45	5	ζ2	ζ2	NOUN
cana-5423	45	6	)	)	PUNCT
cana-5423	45	7	(	(	PUNCT
cana-5423	45	8	t	t	PROPN
cana-5423	45	9	,	,	PUNCT
cana-5423	45	10	s	s	PART
cana-5423	45	11	)	)	PUNCT
cana-5423	45	12	<	<	X
cana-5423	45	13	s−	s−	PROPN
cana-5423	45	14	t	t	PROPN
cana-5423	45	15	for	for	ADP
cana-5423	45	16	all	all	DET
cana-5423	45	17	t	t	PROPN
cana-5423	45	18	,	,	PUNCT
cana-5423	45	19	s	s	PART
cana-5423	45	20	>	>	X
cana-5423	45	21	o	o	NOUN
cana-5423	45	22	;	;	PUNCT
cana-5423	45	23	(	(	PUNCT
cana-5423	45	24	ζ3	ζ3	NOUN
cana-5423	45	25	)	)	PUNCT
cana-5423	45	26	if	if	SCONJ
cana-5423	45	27	{	{	PUNCT
cana-5423	45	28	tn	tn	NOUN
cana-5423	45	29	}	}	PUNCT
cana-5423	45	30	,	,	PUNCT
cana-5423	45	31	{	{	PUNCT
cana-5423	45	32	sn	sn	X
cana-5423	45	33	}	}	PUNCT
cana-5423	45	34	are	be	AUX
cana-5423	45	35	sequence	sequence	NOUN
cana-5423	45	36	in	in	ADP
cana-5423	45	37	(	(	PUNCT
cana-5423	45	38	o,∞	o,∞	PROPN
cana-5423	45	39	)	)	PUNCT
cana-5423	45	40	such	such	ADJ
cana-5423	45	41	that	that	SCONJ
cana-5423	45	42	lim	lim	PROPN
cana-5423	45	43	n→∞	n→∞	NUM
cana-5423	45	44	tn	tn	PROPN
cana-5423	46	1	=	=	SYM
cana-5423	46	2	lim	lim	PROPN
cana-5423	46	3	n→∞	n→∞	X
cana-5423	46	4	sn	sn	PROPN
cana-5423	46	5	>	>	X
cana-5423	46	6	0	0	PROPN
cana-5423	46	7	,	,	PUNCT
cana-5423	46	8	then	then	ADV
cana-5423	46	9	lim	lim	PROPN
cana-5423	46	10	supn→∞ζ(tn	supn→∞ζ(tn	PROPN
cana-5423	46	11	,	,	PUNCT
cana-5423	46	12	sn	sn	PROPN
cana-5423	46	13	)	)	PUNCT
cana-5423	46	14	<	<	X
cana-5423	46	15	0	0	X
cana-5423	46	16	.	.	PUNCT
cana-5423	47	1	the	the	DET
cana-5423	47	2	following	follow	VERB
cana-5423	47	3	function	function	NOUN
cana-5423	47	4	ζ	ζ	NOUN
cana-5423	47	5	:	:	PUNCT
cana-5423	47	6	[	[	X
cana-5423	47	7	o,∞	o,∞	PROPN
cana-5423	47	8	)	)	PUNCT
cana-5423	47	9	×	×	NOUN
cana-5423	48	1	[	[	X
cana-5423	48	2	o,∞	o,∞	PROPN
cana-5423	48	3	)	)	PUNCT
cana-5423	48	4	→	→	SYM
cana-5423	48	5	r	r	NOUN
cana-5423	48	6	belongs	belong	VERB
cana-5423	48	7	to	to	ADP
cana-5423	48	8	z.	z.	PROPN
cana-5423	48	9	definition	definition	NOUN
cana-5423	48	10	1.5	1.5	NUM
cana-5423	48	11	.	.	PUNCT
cana-5423	49	1	[	[	X
cana-5423	49	2	29	29	NUM
cana-5423	49	3	]	]	PUNCT
cana-5423	49	4	a	a	DET
cana-5423	49	5	funtion	funtion	NOUN
cana-5423	49	6	γ	γ	X
cana-5423	49	7	:	:	PUNCT
cana-5423	49	8	x	x	SYM
cana-5423	49	9	→	→	PUNCT
cana-5423	49	10	x	x	X
cana-5423	49	11	is	be	AUX
cana-5423	49	12	called	call	VERB
cana-5423	49	13	a	a	DET
cana-5423	49	14	zcontraction	zcontraction	NOUN
cana-5423	49	15	with	with	ADP
cana-5423	49	16	respect	respect	NOUN
cana-5423	49	17	to	to	ADP
cana-5423	49	18	a	a	DET
cana-5423	49	19	simulation	simulation	NOUN
cana-5423	49	20	function	function	VERB
cana-5423	49	21	ζ	ζ	PROPN
cana-5423	49	22	∈	∈	PROPN
cana-5423	49	23	z	z	NOUN
cana-5423	49	24	on	on	ADP
cana-5423	49	25	metric	metric	PROPN
cana-5423	49	26	space(x	space(x	PROPN
cana-5423	49	27	,	,	PUNCT
cana-5423	49	28	d	d	PROPN
cana-5423	49	29	)	)	PUNCT
cana-5423	49	30	,	,	PUNCT
cana-5423	49	31	if	if	SCONJ
cana-5423	49	32	the	the	DET
cana-5423	49	33	following	follow	VERB
cana-5423	49	34	condition	condition	NOUN
cana-5423	49	35	is	be	AUX
cana-5423	49	36	satisfied	satisfied	ADJ
cana-5423	49	37	ζ(d(γx.γy	ζ(d(γx.γy	PROPN
cana-5423	49	38	)	)	PUNCT
cana-5423	49	39	,	,	PUNCT
cana-5423	49	40	d(x	d(x	PROPN
cana-5423	49	41	,	,	PUNCT
cana-5423	49	42	y	y	NOUN
cana-5423	49	43	)	)	PUNCT
cana-5423	49	44	)	)	PUNCT
cana-5423	49	45	≤	≤	NUM
cana-5423	49	46	∀x	∀x	NUM
cana-5423	49	47	,	,	PUNCT
cana-5423	49	48	y	y	PROPN
cana-5423	49	49	∈	∈	PROPN
cana-5423	49	50	x.	x.	NOUN
cana-5423	49	51	(	(	PUNCT
cana-5423	49	52	4	4	X
cana-5423	49	53	)	)	PUNCT
cana-5423	49	54	remark	remark	NOUN
cana-5423	49	55	1.6	1.6	NUM
cana-5423	49	56	.	.	PUNCT
cana-5423	50	1	[	[	X
cana-5423	50	2	29	29	NUM
cana-5423	50	3	]	]	X
cana-5423	50	4	it	it	PRON
cana-5423	50	5	is	be	AUX
cana-5423	50	6	clear	clear	ADJ
cana-5423	50	7	from	from	ADP
cana-5423	50	8	the	the	DET
cana-5423	50	9	defintion	defintion	NOUN
cana-5423	50	10	of	of	ADP
cana-5423	50	11	simulation	simulation	NOUN
cana-5423	50	12	function	function	VERB
cana-5423	50	13	thar	thar	PROPN
cana-5423	50	14	ζ(t	ζ(t	PROPN
cana-5423	50	15	,	,	PUNCT
cana-5423	50	16	s	s	PART
cana-5423	50	17	)	)	PUNCT
cana-5423	50	18	<	<	X
cana-5423	50	19	o	o	NOUN
cana-5423	50	20	for	for	ADP
cana-5423	50	21	all	all	DET
cana-5423	50	22	t	t	PROPN
cana-5423	50	23	≥	≥	NOUN
cana-5423	50	24	s	s	PART
cana-5423	50	25	>	>	X
cana-5423	50	26	o.	o.	PROPN
cana-5423	50	27	therefore	therefore	ADV
cana-5423	50	28	,	,	PUNCT
cana-5423	50	29	if	if	SCONJ
cana-5423	50	30	γ	γ	NOUN
cana-5423	50	31	is	be	AUX
cana-5423	50	32	a	a	DET
cana-5423	50	33	z	z	NOUN
cana-5423	50	34	-	-	PUNCT
cana-5423	50	35	contraction	contraction	NOUN
cana-5423	50	36	with	with	ADP
cana-5423	50	37	respect	respect	NOUN
cana-5423	50	38	to	to	ADP
cana-5423	50	39	simulation	simulation	NOUN
cana-5423	50	40	functiomn	functiomn	ADJ
cana-5423	50	41	ζ	ζ	PROPN
cana-5423	50	42	,	,	PUNCT
cana-5423	50	43	then	then	ADV
cana-5423	50	44	d(γx	d(γx	PROPN
cana-5423	50	45	,	,	PUNCT
cana-5423	50	46	γy	γy	NOUN
cana-5423	50	47	)	)	PUNCT
cana-5423	50	48	<	<	X
cana-5423	50	49	d(x	d(x	PROPN
cana-5423	50	50	,	,	PUNCT
cana-5423	50	51	y	y	NOUN
cana-5423	50	52	)	)	PUNCT
cana-5423	50	53	for	for	ADP
cana-5423	50	54	all	all	DET
cana-5423	50	55	x	x	NOUN
cana-5423	50	56	,	,	PUNCT
cana-5423	50	57	y	y	PROPN
cana-5423	50	58	∈	∈	PROPN
cana-5423	50	59	x.	x.	NOUN
cana-5423	50	60	(	(	PUNCT
cana-5423	50	61	5	5	NUM
cana-5423	50	62	)	)	PUNCT
cana-5423	50	63	theorem	theorem	VERB
cana-5423	50	64	1.7	1.7	NUM
cana-5423	50	65	.	.	PUNCT
cana-5423	51	1	[	[	X
cana-5423	51	2	29	29	NUM
cana-5423	51	3	]	]	X
cana-5423	51	4	let	let	AUX
cana-5423	51	5	(	(	PUNCT
cana-5423	51	6	x	x	NOUN
cana-5423	51	7	,	,	PUNCT
cana-5423	51	8	d	d	NOUN
cana-5423	51	9	)	)	PUNCT
cana-5423	51	10	be	be	AUX
cana-5423	51	11	a	a	DET
cana-5423	51	12	complete	complete	ADJ
cana-5423	51	13	metric	metric	ADJ
cana-5423	51	14	space	space	NOUN
cana-5423	51	15	and	and	CCONJ
cana-5423	51	16	γ	γ	X
cana-5423	51	17	:	:	PUNCT
cana-5423	51	18	x	x	SYM
cana-5423	51	19	→	→	PUNCT
cana-5423	51	20	x	x	PUNCT
cana-5423	51	21	be	be	AUX
cana-5423	51	22	a	a	DET
cana-5423	51	23	z	z	NOUN
cana-5423	51	24	-	-	PUNCT
cana-5423	51	25	contraction	contraction	NOUN
cana-5423	51	26	with	with	ADP
cana-5423	51	27	respect	respect	NOUN
cana-5423	51	28	to	to	ADP
cana-5423	51	29	ζ	ζ	NOUN
cana-5423	51	30	.	.	PUNCT
cana-5423	52	1	then	then	ADV
cana-5423	52	2	γ	γ	PROPN
cana-5423	52	3	has	have	VERB
cana-5423	52	4	a	a	DET
cana-5423	52	5	unique	unique	ADJ
cana-5423	52	6	fixed	fix	VERB
cana-5423	52	7	point	point	NOUN
cana-5423	52	8	u	u	NOUN
cana-5423	52	9	∈	∈	PROPN
cana-5423	52	10	x	x	X
cana-5423	52	11	and	and	CCONJ
cana-5423	52	12	for	for	ADP
cana-5423	52	13	every	every	DET
cana-5423	52	14	x0	x0	PROPN
cana-5423	52	15	∈	∈	PROPN
cana-5423	52	16	x	x	PRON
cana-5423	52	17	,	,	PUNCT
cana-5423	52	18	the	the	DET
cana-5423	52	19	picard	picard	NOUN
cana-5423	52	20	sequence	sequence	NOUN
cana-5423	52	21	{	{	PUNCT
cana-5423	52	22	xn	xn	NOUN
cana-5423	52	23	}	}	PUNCT
cana-5423	52	24	where	where	SCONJ
cana-5423	52	25	xn	xn	PUNCT
cana-5423	53	1	=	=	PUNCT
cana-5423	53	2	γxn−1	γxn−1	PROPN
cana-5423	53	3	for	for	ADP
cana-5423	53	4	all	all	DET
cana-5423	53	5	n	n	PRON
cana-5423	53	6	∈	∈	NOUN
cana-5423	53	7	n	n	PRON
cana-5423	53	8	converges	converge	VERB
cana-5423	53	9	to	to	ADP
cana-5423	53	10	this	this	DET
cana-5423	53	11	fixed	fix	VERB
cana-5423	53	12	point	point	NOUN
cana-5423	53	13	of	of	ADP
cana-5423	53	14	γ	γ	X
cana-5423	53	15	.	.	PUNCT
cana-5423	54	1	it	it	PRON
cana-5423	54	2	is	be	AUX
cana-5423	54	3	worth	worth	ADJ
cana-5423	54	4	mentioning	mention	VERB
cana-5423	54	5	that	that	SCONJ
cana-5423	54	6	the	the	DET
cana-5423	54	7	banch	banch	PROPN
cana-5423	54	8	contraction	contraction	NOUN
cana-5423	54	9	is	be	AUX
cana-5423	54	10	a	a	DET
cana-5423	54	11	perfect	perfect	ADJ
cana-5423	54	12	example	example	NOUN
cana-5423	54	13	of	of	ADP
cana-5423	54	14	zcontractions	zcontraction	NOUN
cana-5423	54	15	by	by	ADP
cana-5423	54	16	taking	take	VERB
cana-5423	54	17	(	(	PUNCT
cana-5423	54	18	ζ	ζ	NOUN
cana-5423	54	19	,	,	PUNCT
cana-5423	54	20	s	s	NOUN
cana-5423	54	21	)	)	PUNCT
cana-5423	54	22	=	=	SYM
cana-5423	55	1	λs	λs	ADP
cana-5423	55	2	−	−	NUM
cana-5423	55	3	1	1	NUM
cana-5423	55	4	,	,	PUNCT
cana-5423	55	5	where	where	SCONJ
cana-5423	55	6	λ	λ	PROPN
cana-5423	55	7	∈	∈	PROPN
cana-5423	56	1	[	[	X
cana-5423	56	2	0	0	NUM
cana-5423	56	3	,	,	PUNCT
cana-5423	56	4	1	1	NUM
cana-5423	56	5	)	)	PUNCT
cana-5423	56	6	,	,	PUNCT
cana-5423	56	7	as	as	ADP
cana-5423	56	8	the	the	DET
cana-5423	56	9	corresponding	corresponding	ADJ
cana-5423	56	10	simulation	simulation	NOUN
cana-5423	56	11	function	function	NOUN
cana-5423	56	12	.	.	PUNCT
cana-5423	57	1	argoubi	argoubi	ADV
cana-5423	57	2	et	et	PROPN
cana-5423	57	3	al	al	PROPN
cana-5423	57	4	.	.	PUNCT
cana-5423	58	1	[	[	X
cana-5423	58	2	4	4	X
cana-5423	58	3	]	]	PUNCT
cana-5423	58	4	shown	show	VERB
cana-5423	58	5	that	that	SCONJ
cana-5423	58	6	the	the	DET
cana-5423	58	7	condition	condition	NOUN
cana-5423	58	8	(	(	PUNCT
cana-5423	58	9	ζ1	ζ1	NOUN
cana-5423	58	10	)	)	PUNCT
cana-5423	58	11	to	to	PART
cana-5423	58	12	be	be	AUX
cana-5423	58	13	reduntant	reduntant	ADJ
cana-5423	58	14	one	one	NUM
cana-5423	58	15	ine	ine	X
cana-5423	58	16	above	above	ADP
cana-5423	58	17	definition	definition	NOUN
cana-5423	58	18	1.4	1.4	NUM
cana-5423	58	19	of	of	ADP
cana-5423	58	20	simulation	simulation	NOUN
cana-5423	58	21	functionand	functionand	PROPN
cana-5423	58	22	so	so	ADV
cana-5423	58	23	redefined	redefine	VERB
cana-5423	58	24	it	it	PRON
cana-5423	58	25	as	as	ADP
cana-5423	58	26	:	:	PUNCT
cana-5423	58	27	definition	definition	NOUN
cana-5423	58	28	1.8	1.8	NUM
cana-5423	58	29	.	.	PUNCT
cana-5423	59	1	[	[	X
cana-5423	59	2	4	4	NUM
cana-5423	59	3	]	]	PUNCT
cana-5423	59	4	asimulation	asimulation	NOUN
cana-5423	59	5	function	function	NOUN
cana-5423	59	6	is	be	AUX
cana-5423	59	7	a	a	DET
cana-5423	59	8	mapping	mapping	NOUN
cana-5423	59	9	ζ	ζ	NOUN
cana-5423	59	10	:	:	PUNCT
cana-5423	60	1	[	[	X
cana-5423	60	2	o,∞)2	o,∞)2	NOUN
cana-5423	60	3	→	→	SYM
cana-5423	60	4	rsatisfies	rsatisfie	NOUN
cana-5423	60	5	the	the	DET
cana-5423	60	6	foolowing	foolowing	NOUN
cana-5423	60	7	conditions	condition	NOUN
cana-5423	60	8	:	:	PUNCT
cana-5423	60	9	(	(	PUNCT
cana-5423	60	10	i	i	NOUN
cana-5423	60	11	)	)	PUNCT
cana-5423	60	12	ζ(t	ζ(t	PROPN
cana-5423	60	13	,	,	PUNCT
cana-5423	60	14	s	s	PART
cana-5423	60	15	)	)	PUNCT
cana-5423	60	16	<	<	X
cana-5423	60	17	s−	s−	PROPN
cana-5423	60	18	t	t	PROPN
cana-5423	60	19	for	for	ADP
cana-5423	60	20	all	all	DET
cana-5423	60	21	t	t	PROPN
cana-5423	60	22	,	,	PUNCT
cana-5423	60	23	s	s	PART
cana-5423	60	24	>	>	X
cana-5423	60	25	0	0	NUM
cana-5423	60	26	;	;	PUNCT
cana-5423	60	27	(	(	PUNCT
cana-5423	60	28	ii	ii	NOUN
cana-5423	60	29	)	)	PUNCT
cana-5423	60	30	if	if	SCONJ
cana-5423	60	31	{	{	PUNCT
cana-5423	60	32	tn	tn	NOUN
cana-5423	60	33	}	}	PUNCT
cana-5423	60	34	and	and	CCONJ
cana-5423	60	35	{	{	PUNCT
cana-5423	60	36	sn	sn	NOUN
cana-5423	60	37	}	}	PUNCT
cana-5423	60	38	are	be	AUX
cana-5423	60	39	sequences	sequence	NOUN
cana-5423	60	40	in	in	ADP
cana-5423	60	41	(	(	PUNCT
cana-5423	60	42	0,∞	0,∞	NOUN
cana-5423	60	43	)	)	PUNCT
cana-5423	61	1	such	such	ADJ
cana-5423	61	2	that	that	SCONJ
cana-5423	61	3	lim	lim	PROPN
cana-5423	61	4	n→∞	n→∞	NUM
cana-5423	61	5	tn	tn	PROPN
cana-5423	62	1	=	=	SYM
cana-5423	62	2	lim	lim	PROPN
cana-5423	62	3	n→∞	n→∞	X
cana-5423	62	4	sn	sn	PROPN
cana-5423	62	5	>	>	X
cana-5423	62	6	0	0	NUM
cana-5423	62	7	,	,	PUNCT
cana-5423	62	8	and	and	CCONJ
cana-5423	62	9	tn	tn	NOUN
cana-5423	62	10	<	<	X
cana-5423	62	11	sn	sn	PROPN
cana-5423	62	12	,	,	PUNCT
cana-5423	62	13	then	then	ADV
cana-5423	62	14	lim	lim	PROPN
cana-5423	62	15	supn→∞	supn→∞	PROPN
cana-5423	62	16	ζ(tn	ζ(tn	PROPN
cana-5423	62	17	,	,	PUNCT
cana-5423	62	18	sn	sn	PROPN
cana-5423	62	19	)	)	PUNCT
cana-5423	62	20	<	<	X
cana-5423	62	21	0	0	X
cana-5423	62	22	.	.	PUNCT
cana-5423	62	23	communications	communication	NOUN
cana-5423	62	24	on	on	ADP
cana-5423	62	25	applied	apply	VERB
cana-5423	62	26	nonlinear	nonlinear	ADJ
cana-5423	62	27	analysis	analysis	NOUN
cana-5423	62	28	issn	issn	NOUN
cana-5423	62	29	:	:	PUNCT
cana-5423	62	30	1074	1074	NUM
cana-5423	62	31	-	-	PUNCT
cana-5423	62	32	133x	133x	NUM
cana-5423	62	33	vol	vol	NOUN
cana-5423	62	34	32	32	NUM
cana-5423	62	35	no	no	NOUN
cana-5423	62	36	.	.	PUNCT
cana-5423	63	1	10s(2025	10s(2025	NUM
cana-5423	63	2	)	)	PUNCT
cana-5423	63	3	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	63	4	2218	2218	NUM
cana-5423	63	5	n→∞	n→∞	NUM
cana-5423	63	6	tn	tn	NOUN
cana-5423	63	7	=	=	SYM
cana-5423	63	8	lim	lim	PROPN
cana-5423	63	9	n→∞	n→∞	NUM
cana-5423	64	1	roldan	roldan	PROPN
cana-5423	64	2	-	-	PUNCT
cana-5423	64	3	lopez	lopez	PROPN
cana-5423	64	4	-	-	PUNCT
cana-5423	64	5	de	de	X
cana-5423	64	6	hierrow	hierrow	PROPN
cana-5423	64	7	et	et	PROPN
cana-5423	64	8	al.[36	al.[36	PROPN
cana-5423	64	9	]	]	PUNCT
cana-5423	64	10	modified	modify	VERB
cana-5423	64	11	the	the	DET
cana-5423	64	12	notion	notion	NOUN
cana-5423	64	13	of	of	ADP
cana-5423	64	14	simulation	simulation	NOUN
cana-5423	64	15	function	function	NOUN
cana-5423	64	16	replacing	replace	VERB
cana-5423	64	17	(	(	PUNCT
cana-5423	64	18	ζ3	ζ3	NOUN
cana-5423	64	19	)	)	PUNCT
cana-5423	64	20	by(ζ	by(ζ	PUNCT
cana-5423	64	21	′	′	NUM
cana-5423	64	22	3	3	NUM
cana-5423	64	23	)	)	PUNCT
cana-5423	64	24	of	of	ADP
cana-5423	64	25	definition	definition	NOUN
cana-5423	64	26	1.4	1.4	NUM
cana-5423	64	27	as	as	ADP
cana-5423	64	28	:	:	PUNCT
cana-5423	64	29	(	(	PUNCT
cana-5423	64	30	ζ	ζ	NOUN
cana-5423	64	31	′	′	NOUN
cana-5423	64	32	3	3	NUM
cana-5423	64	33	)	)	PUNCT
cana-5423	64	34	if	if	SCONJ
cana-5423	64	35	{	{	PUNCT
cana-5423	64	36	tn	tn	NOUN
cana-5423	64	37	}	}	PUNCT
cana-5423	64	38	and	and	CCONJ
cana-5423	64	39	{	{	PUNCT
cana-5423	64	40	sn	sn	NOUN
cana-5423	64	41	}	}	PUNCT
cana-5423	64	42	are	be	AUX
cana-5423	64	43	sequences	sequence	NOUN
cana-5423	64	44	in	in	ADP
cana-5423	64	45	(	(	PUNCT
cana-5423	64	46	0	0	NUM
cana-5423	64	47	,	,	PUNCT
cana-5423	64	48	∞	∞	NOUN
cana-5423	64	49	)	)	PUNCT
cana-5423	64	50	such	such	ADJ
cana-5423	64	51	that	that	SCONJ
cana-5423	64	52	lim	lim	PROPN
cana-5423	64	53	sn	sn	PROPN
cana-5423	64	54	>	>	X
cana-5423	64	55	0	0	NUM
cana-5423	64	56	,	,	PUNCT
cana-5423	64	57	and	and	CCONJ
cana-5423	64	58	tn	tn	NOUN
cana-5423	64	59	<	<	X
cana-5423	64	60	sn	sn	PROPN
cana-5423	64	61	,	,	PUNCT
cana-5423	64	62	then	then	ADV
cana-5423	64	63	lim	lim	PROPN
cana-5423	64	64	supn→∞	supn→∞	PROPN
cana-5423	64	65	ζ(tn	ζ(tn	PROPN
cana-5423	64	66	,	,	PUNCT
cana-5423	64	67	sn	sn	PROPN
cana-5423	64	68	)	)	PUNCT
cana-5423	64	69	<	<	X
cana-5423	64	70	0	0	X
cana-5423	64	71	.	.	PUNCT
cana-5423	65	1	it	it	PRON
cana-5423	65	2	is	be	AUX
cana-5423	65	3	clear	clear	ADJ
cana-5423	65	4	that	that	SCONJ
cana-5423	65	5	,	,	PUNCT
cana-5423	65	6	if	if	SCONJ
cana-5423	65	7	the	the	DET
cana-5423	65	8	function	function	NOUN
cana-5423	65	9	ζ	ζ	NOUN
cana-5423	65	10	satisfies	satisfy	VERB
cana-5423	65	11	the	the	DET
cana-5423	65	12	conditions	condition	NOUN
cana-5423	65	13	(	(	PUNCT
cana-5423	65	14	ζ1	ζ1	NOUN
cana-5423	65	15	)	)	PUNCT
cana-5423	65	16	−	−	PROPN
cana-5423	65	17	(	(	PUNCT
cana-5423	65	18	ζ3	ζ3	NOUN
cana-5423	65	19	)	)	PUNCT
cana-5423	65	20	,	,	PUNCT
cana-5423	65	21	we	we	PRON
cana-5423	65	22	say	say	VERB
cana-5423	65	23	that	that	SCONJ
cana-5423	65	24	ζ	ζ	NOUN
cana-5423	65	25	is	be	AUX
cana-5423	65	26	a	a	DET
cana-5423	65	27	simulation	simulation	NOUN
cana-5423	65	28	function	function	NOUN
cana-5423	65	29	according	accord	VERB
cana-5423	65	30	to	to	ADP
cana-5423	65	31	the	the	DET
cana-5423	65	32	sense	sense	NOUN
cana-5423	65	33	of	of	ADP
cana-5423	65	34	khojasteh	khojasteh	X
cana-5423	65	35	et	et	PROPN
cana-5423	65	36	al	al	PROPN
cana-5423	66	1	[	[	X
cana-5423	66	2	29	29	NUM
cana-5423	66	3	]	]	PUNCT
cana-5423	66	4	.	.	PUNCT
cana-5423	67	1	if	if	SCONJ
cana-5423	67	2	it	it	PRON
cana-5423	67	3	satisfies	satisfy	VERB
cana-5423	67	4	(	(	PUNCT
cana-5423	67	5	ζ2	ζ2	NOUN
cana-5423	67	6	)	)	PUNCT
cana-5423	67	7	−	−	PROPN
cana-5423	68	1	(	(	PUNCT
cana-5423	68	2	ζ3),it	ζ3),it	PROPN
cana-5423	68	3	is	be	AUX
cana-5423	68	4	a	a	DET
cana-5423	68	5	simulation	simulation	NOUN
cana-5423	68	6	function	function	NOUN
cana-5423	68	7	according	accord	VERB
cana-5423	68	8	to	to	ADP
cana-5423	68	9	the	the	DET
cana-5423	68	10	sense	sense	NOUN
cana-5423	68	11	of	of	ADP
cana-5423	68	12	argoubi	argoubi	NOUN
cana-5423	68	13	et	et	NOUN
cana-5423	68	14	al.[4	al.[4	PROPN
cana-5423	68	15	]	]	PUNCT
cana-5423	68	16	.	.	PUNCT
cana-5423	69	1	and	and	CCONJ
cana-5423	69	2	if	if	SCONJ
cana-5423	69	3	it	it	PRON
cana-5423	69	4	satis	satis	NOUN
cana-5423	69	5	fies	fie	NOUN
cana-5423	69	6	(	(	PUNCT
cana-5423	69	7	ζ),(ζ2	ζ),(ζ2	PROPN
cana-5423	69	8	)	)	PUNCT
cana-5423	69	9	,	,	PUNCT
cana-5423	69	10	and	and	CCONJ
cana-5423	69	11	(	(	PUNCT
cana-5423	69	12	ζ	ζ	NOUN
cana-5423	69	13	′	′	NOUN
cana-5423	69	14	3	3	NUM
cana-5423	69	15	)	)	PUNCT
cana-5423	69	16	,	,	PUNCT
cana-5423	69	17	then	then	ADV
cana-5423	69	18	it	it	PRON
cana-5423	69	19	is	be	AUX
cana-5423	69	20	a	a	DET
cana-5423	69	21	simulation	simulation	NOUN
cana-5423	69	22	function	function	NOUN
cana-5423	69	23	according	accord	VERB
cana-5423	69	24	to	to	ADP
cana-5423	69	25	the	the	DET
cana-5423	69	26	sense	sense	NOUN
cana-5423	69	27	of	of	ADP
cana-5423	69	28	roldanlopezdehierro	roldanlopezdehierro	NOUN
cana-5423	69	29	et	et	PROPN
cana-5423	69	30	al	al	PROPN
cana-5423	69	31	.	.	PUNCT
cana-5423	70	1	[	[	X
cana-5423	70	2	36	36	NUM
cana-5423	70	3	]	]	PUNCT
cana-5423	70	4	.	.	PUNCT
cana-5423	71	1	samet	samet	PROPN
cana-5423	71	2	et	et	PROPN
cana-5423	71	3	al.[39	al.[39	PROPN
cana-5423	71	4	]	]	PUNCT
cana-5423	71	5	introduced	introduce	VERB
cana-5423	71	6	a	a	DET
cana-5423	71	7	new	new	ADJ
cana-5423	71	8	category	category	NOUN
cana-5423	71	9	of	of	ADP
cana-5423	71	10	contractive	contractive	ADJ
cana-5423	71	11	type	type	NOUN
cana-5423	71	12	mappings	mapping	NOUN
cana-5423	71	13	known	know	VERB
cana-5423	71	14	as	as	ADP
cana-5423	71	15	α−ψ	α−ψ	NOUN
cana-5423	71	16	contractive	contractive	ADJ
cana-5423	71	17	type	type	NOUN
cana-5423	71	18	mapping	mapping	NOUN
cana-5423	71	19	.	.	PUNCT
cana-5423	72	1	the	the	DET
cana-5423	72	2	results	result	NOUN
cana-5423	72	3	obtained	obtain	VERB
cana-5423	72	4	by	by	ADP
cana-5423	72	5	samet	samet	PROPN
cana-5423	72	6	et	et	PROPN
cana-5423	72	7	al.[39	al.[39	PROPN
cana-5423	72	8	]	]	PUNCT
cana-5423	72	9	extended	extended	ADJ
cana-5423	72	10	and	and	CCONJ
cana-5423	72	11	generalized	generalize	VERB
cana-5423	72	12	the	the	DET
cana-5423	72	13	existing	exist	VERB
cana-5423	72	14	fixed	fix	VERB
cana-5423	72	15	point	point	NOUN
cana-5423	72	16	results	result	NOUN
cana-5423	72	17	in	in	ADP
cana-5423	72	18	the	the	DET
cana-5423	72	19	literature	literature	NOUN
cana-5423	72	20	,	,	PUNCT
cana-5423	72	21	in	in	ADP
cana-5423	72	22	particular	particular	ADJ
cana-5423	72	23	the	the	DET
cana-5423	72	24	banach	banach	NOUN
cana-5423	72	25	contraction	contraction	NOUN
cana-5423	72	26	principle	principle	NOUN
cana-5423	72	27	.	.	PUNCT
cana-5423	73	1	further	far	ADV
cana-5423	73	2	,	,	PUNCT
cana-5423	73	3	karapinar	karapinar	PROPN
cana-5423	73	4	e.	e.	PROPN
cana-5423	73	5	and	and	CCONJ
cana-5423	73	6	samet[16	samet[16	PROPN
cana-5423	73	7	]	]	X
cana-5423	73	8	generalized	generalize	VERB
cana-5423	73	9	the	the	DET
cana-5423	73	10	α	α	NOUN
cana-5423	73	11	−	−	NOUN
cana-5423	73	12	ψ	ψ	ADP
cana-5423	73	13	comntractive	comntractive	ADJ
cana-5423	73	14	type	type	NOUN
cana-5423	73	15	mappings	mapping	NOUN
cana-5423	73	16	and	and	CCONJ
cana-5423	73	17	obtained	obtain	VERB
cana-5423	73	18	various	various	ADJ
cana-5423	73	19	fixed	fix	VERB
cana-5423	73	20	point	point	NOUN
cana-5423	73	21	theorems	theorem	NOUN
cana-5423	73	22	for	for	ADP
cana-5423	73	23	this	this	DET
cana-5423	73	24	generalized	generalized	ADJ
cana-5423	73	25	class	class	NOUN
cana-5423	73	26	of	of	ADP
cana-5423	73	27	contractive	contractive	ADJ
cana-5423	73	28	mappings	mapping	NOUN
cana-5423	73	29	.	.	PUNCT
cana-5423	74	1	in	in	ADP
cana-5423	74	2	2013	2013	NUM
cana-5423	74	3	,	,	PUNCT
cana-5423	74	4	hussain	hussain	PROPN
cana-5423	74	5	et	et	PROPN
cana-5423	74	6	al	al	PROPN
cana-5423	74	7	.	.	PUNCT
cana-5423	75	1	[	[	X
cana-5423	75	2	17	17	NUM
cana-5423	75	3	]	]	PUNCT
cana-5423	75	4	introduced	introduce	VERB
cana-5423	75	5	α	α	NUM
cana-5423	75	6	-	-	ADJ
cana-5423	75	7	admissible	admissible	ADJ
cana-5423	75	8	mappings	mapping	NOUN
cana-5423	75	9	and	and	CCONJ
cana-5423	75	10	proved	prove	VERB
cana-5423	75	11	fixed	fix	VERB
cana-5423	75	12	point	point	NOUN
cana-5423	75	13	theorems	theorem	NOUN
cana-5423	75	14	in	in	ADP
cana-5423	75	15	metric	metric	ADJ
cana-5423	75	16	space	space	NOUN
cana-5423	75	17	.	.	PUNCT
cana-5423	76	1	subsequently	subsequently	ADV
cana-5423	76	2	,	,	PUNCT
cana-5423	76	3	abdeljawad[1	abdeljawad[1	PROPN
cana-5423	76	4	]	]	PUNCT
cana-5423	76	5	introduced	introduce	VERB
cana-5423	76	6	a	a	DET
cana-5423	76	7	pair	pair	NOUN
cana-5423	76	8	of	of	ADP
cana-5423	76	9	α	α	NOUN
cana-5423	76	10	-	-	PUNCT
cana-5423	76	11	admissible	admissible	ADJ
cana-5423	76	12	mappings	mapping	NOUN
cana-5423	76	13	satisfying	satisfy	VERB
cana-5423	76	14	new	new	ADJ
cana-5423	76	15	sufficient	sufficient	ADJ
cana-5423	76	16	contractive	contractive	ADJ
cana-5423	76	17	conditions	condition	NOUN
cana-5423	76	18	,	,	PUNCT
cana-5423	76	19	whixh	whixh	PROPN
cana-5423	76	20	are	be	AUX
cana-5423	76	21	different	different	ADJ
cana-5423	76	22	from	from	ADP
cana-5423	76	23	those	those	DET
cana-5423	76	24	in[14	in[14	PROPN
cana-5423	76	25	,	,	PUNCT
cana-5423	76	26	16	16	NUM
cana-5423	76	27	]	]	PUNCT
cana-5423	76	28	and	and	CCONJ
cana-5423	76	29	obtained	obtain	VERB
cana-5423	76	30	fixed	fix	VERB
cana-5423	76	31	point	point	NOUN
cana-5423	76	32	and	and	CCONJ
cana-5423	76	33	common	common	ADJ
cana-5423	76	34	fixed	fix	VERB
cana-5423	76	35	point	point	NOUN
cana-5423	76	36	theorems	theorem	NOUN
cana-5423	76	37	.	.	PUNCT
cana-5423	77	1	afterward	afterward	ADV
cana-5423	77	2	,	,	PUNCT
cana-5423	77	3	some	some	DET
cana-5423	77	4	authors	author	NOUN
cana-5423	77	5	have	have	AUX
cana-5423	77	6	obtained	obtain	VERB
cana-5423	77	7	fixed	fix	VERB
cana-5423	77	8	point	point	NOUN
cana-5423	77	9	theorems	theorem	NOUN
cana-5423	77	10	for	for	ADP
cana-5423	77	11	some	some	DET
cana-5423	77	12	kinds	kind	NOUN
cana-5423	77	13	of	of	ADP
cana-5423	77	14	αadmissible	αadmissible	ADJ
cana-5423	77	15	mappins	mappin	NOUN
cana-5423	77	16	(	(	PUNCT
cana-5423	77	17	see	see	VERB
cana-5423	77	18	[	[	X
cana-5423	77	19	1	1	NUM
cana-5423	77	20	]	]	PUNCT
cana-5423	77	21	,	,	PUNCT
cana-5423	77	22	2	2	NUM
cana-5423	77	23	,	,	PUNCT
cana-5423	77	24	3	3	NUM
cana-5423	77	25	,	,	PUNCT
cana-5423	77	26	5	5	NUM
cana-5423	77	27	,	,	PUNCT
cana-5423	77	28	6,7	6,7	NUM
cana-5423	77	29	,	,	PUNCT
cana-5423	77	30	8,12,13,14	8,12,13,14	NUM
cana-5423	77	31	,	,	PUNCT
cana-5423	77	32	15,18	15,18	NUM
cana-5423	77	33	,	,	PUNCT
cana-5423	77	34	20	20	NUM
cana-5423	77	35	,	,	PUNCT
cana-5423	77	36	21,22	21,22	NOUN
cana-5423	77	37	,	,	PUNCT
cana-5423	77	38	24	24	NUM
cana-5423	77	39	,	,	PUNCT
cana-5423	77	40	38,40,41	38,40,41	NUM
cana-5423	77	41	and	and	CCONJ
cana-5423	77	42	42	42	NUM
cana-5423	77	43	)	)	PUNCT
cana-5423	77	44	.	.	PUNCT
cana-5423	78	1	definition	definition	NOUN
cana-5423	78	2	1.9	1.9	NUM
cana-5423	78	3	.	.	PUNCT
cana-5423	79	1	[	[	X
cana-5423	79	2	39	39	NUM
cana-5423	79	3	]	]	PUNCT
cana-5423	79	4	let	let	VERB
cana-5423	79	5	γ	γ	X
cana-5423	79	6	:	:	PUNCT
cana-5423	79	7	x	x	SYM
cana-5423	79	8	→	→	SYM
cana-5423	79	9	x	x	X
cana-5423	79	10	and	and	CCONJ
cana-5423	79	11	α	α	NOUN
cana-5423	79	12	:	:	PUNCT
cana-5423	79	13	x	x	SYM
cana-5423	79	14	×	×	NOUN
cana-5423	79	15	x	x	PUNCT
cana-5423	79	16	→	→	X
cana-5423	79	17	r+	r+	NOUN
cana-5423	79	18	be	be	AUX
cana-5423	79	19	the	the	DET
cana-5423	79	20	functions	function	NOUN
cana-5423	79	21	.	.	PUNCT
cana-5423	80	1	then	then	ADV
cana-5423	80	2	γ	γ	PROPN
cana-5423	80	3	is	be	AUX
cana-5423	80	4	called	call	VERB
cana-5423	80	5	α	α	ADV
cana-5423	80	6	-admissible	-admissible	ADJ
cana-5423	80	7	if	if	SCONJ
cana-5423	80	8	α(x	α(x	NOUN
cana-5423	80	9	,	,	PUNCT
cana-5423	80	10	y	y	PROPN
cana-5423	80	11	)	)	PUNCT
cana-5423	80	12	≥	≥	NOUN
cana-5423	80	13	1	1	NUM
cana-5423	80	14	⇒	⇒	PROPN
cana-5423	80	15	α(γx	α(γx	PROPN
cana-5423	80	16	,	,	PUNCT
cana-5423	80	17	γy	γy	PROPN
cana-5423	80	18	)	)	PUNCT
cana-5423	80	19	≥	≥	NOUN
cana-5423	80	20	1	1	NUM
cana-5423	80	21	,	,	PUNCT
cana-5423	80	22	karapinar	karapinar	PROPN
cana-5423	80	23	e.	e.	PROPN
cana-5423	81	1	[	[	X
cana-5423	81	2	23	23	NUM
cana-5423	81	3	]	]	PUNCT
cana-5423	81	4	introduced	introduce	VERB
cana-5423	81	5	the	the	DET
cana-5423	81	6	notion	notion	NOUN
cana-5423	81	7	of	of	ADP
cana-5423	81	8	αadmissible	αadmissible	ADJ
cana-5423	81	9	zcontraction	zcontraction	NOUN
cana-5423	81	10	and	and	CCONJ
cana-5423	81	11	generalized	generalize	VERB
cana-5423	81	12	the	the	DET
cana-5423	81	13	results	result	NOUN
cana-5423	81	14	of	of	ADP
cana-5423	81	15	samet	samet	PROPN
cana-5423	81	16	et	et	PROPN
cana-5423	81	17	al.[39]and	al.[39]and	PROPN
cana-5423	81	18	khojasteh	khojasteh	PROPN
cana-5423	81	19	et	et	PROPN
cana-5423	81	20	al	al	PROPN
cana-5423	81	21	.	.	PUNCT
cana-5423	82	1	[	[	X
cana-5423	82	2	29	29	NUM
cana-5423	82	3	]	]	PUNCT
cana-5423	82	4	.	.	PUNCT
cana-5423	83	1	very	very	ADV
cana-5423	83	2	recently	recently	ADV
cana-5423	83	3	,	,	PUNCT
cana-5423	83	4	dipti	dipti	PROPN
cana-5423	83	5	et	et	PROPN
cana-5423	83	6	al	al	PROPN
cana-5423	83	7	.	.	PUNCT
cana-5423	84	1	[	[	X
cana-5423	84	2	43	43	NUM
cana-5423	84	3	]	]	PUNCT
cana-5423	84	4	presented	present	VERB
cana-5423	84	5	some	some	DET
cana-5423	84	6	fixed	fix	VERB
cana-5423	84	7	point	point	NOUN
cana-5423	84	8	results	result	NOUN
cana-5423	84	9	in	in	ADP
cana-5423	84	10	complete	complete	ADJ
cana-5423	84	11	metric	metric	ADJ
cana-5423	84	12	spaces	space	NOUN
cana-5423	84	13	using	use	VERB
cana-5423	84	14	generalized	generalized	ADJ
cana-5423	84	15	α	α	PRON
cana-5423	84	16	admissible	admissible	ADJ
cana-5423	84	17	mappings	mapping	NOUN
cana-5423	84	18	embedded	embed	VERB
cana-5423	84	19	in	in	ADP
cana-5423	84	20	the	the	DET
cana-5423	84	21	simulation	simulation	NOUN
cana-5423	84	22	functions	function	NOUN
cana-5423	84	23	.	.	PUNCT
cana-5423	85	1	definition	definition	NOUN
cana-5423	85	2	1.10	1.10	NUM
cana-5423	85	3	.	.	PUNCT
cana-5423	86	1	[	[	X
cana-5423	86	2	15	15	NUM
cana-5423	86	3	]	]	X
cana-5423	86	4	letγ	letγ	NOUN
cana-5423	86	5	:	:	PUNCT
cana-5423	86	6	x	x	X
cana-5423	86	7	→	→	SYM
cana-5423	86	8	x	x	X
cana-5423	86	9	and	and	CCONJ
cana-5423	86	10	α	α	NOUN
cana-5423	86	11	:	:	PUNCT
cana-5423	86	12	x×x	x×x	PROPN
cana-5423	86	13	→	→	PUNCT
cana-5423	86	14	r+	r+	NOUN
cana-5423	86	15	be	be	AUX
cana-5423	86	16	a	a	DET
cana-5423	86	17	functions	function	NOUN
cana-5423	86	18	.	.	PUNCT
cana-5423	87	1	then	then	ADV
cana-5423	87	2	we	we	PRON
cana-5423	87	3	say	say	VERB
cana-5423	87	4	that	that	SCONJ
cana-5423	87	5	γ	γ	PROPN
cana-5423	87	6	is	be	AUX
cana-5423	87	7	an	an	DET
cana-5423	87	8	α	α	NOUN
cana-5423	87	9	-	-	ADJ
cana-5423	87	10	orbital	orbital	ADJ
cana-5423	87	11	admissible	admissible	NOUN
cana-5423	87	12	if	if	SCONJ
cana-5423	87	13	α(x	α(x	NOUN
cana-5423	87	14	,	,	PUNCT
cana-5423	87	15	γx	γx	NOUN
cana-5423	87	16	)	)	PUNCT
cana-5423	87	17	≥	≥	NOUN
cana-5423	87	18	1	1	NUM
cana-5423	87	19	implies	imply	VERB
cana-5423	87	20	α(γx	α(γx	NOUN
cana-5423	87	21	,	,	PUNCT
cana-5423	87	22	γ2x	γ2x	NUM
cana-5423	87	23	)	)	PUNCT
cana-5423	87	24	≥	≥	NOUN
cana-5423	88	1	1	1	NUM
cana-5423	88	2	.	.	PUNCT
cana-5423	89	1	moreover	moreover	ADV
cana-5423	89	2	,	,	PUNCT
cana-5423	89	3	γ	γ	PROPN
cana-5423	89	4	is	be	AUX
cana-5423	89	5	called	call	VERB
cana-5423	89	6	a	a	DET
cana-5423	89	7	triangular	triangular	NOUN
cana-5423	89	8	αorbital	αorbital	ADJ
cana-5423	89	9	admissible	admissible	ADJ
cana-5423	89	10	if	if	SCONJ
cana-5423	89	11	γ	γ	ADJ
cana-5423	89	12	-	-	ADJ
cana-5423	89	13	orbital	orbital	ADJ
cana-5423	89	14	admissible	admissible	ADJ
cana-5423	89	15	and	and	CCONJ
cana-5423	89	16	α(x	α(x	NOUN
cana-5423	89	17	,	,	PUNCT
cana-5423	89	18	y	y	PROPN
cana-5423	89	19	)	)	PUNCT
cana-5423	89	20	≥	≥	NOUN
cana-5423	89	21	1	1	NUM
cana-5423	89	22	and	and	CCONJ
cana-5423	89	23	α(y	α(y	NOUN
cana-5423	89	24	,	,	PUNCT
cana-5423	89	25	γy	γy	PROPN
cana-5423	89	26	)	)	PUNCT
cana-5423	89	27	≥	≥	NOUN
cana-5423	89	28	1	1	NUM
cana-5423	89	29	implies	imply	VERB
cana-5423	89	30	α(x	α(x	PROPN
cana-5423	89	31	,	,	PUNCT
cana-5423	89	32	γy	γy	PROPN
cana-5423	89	33	)	)	PUNCT
cana-5423	89	34	≥	≥	NOUN
cana-5423	89	35	1	1	NUM
cana-5423	89	36	,	,	PUNCT
cana-5423	89	37	forall	forall	NOUN
cana-5423	89	38	x	x	SYM
cana-5423	89	39	,	,	PUNCT
cana-5423	89	40	y	y	PROPN
cana-5423	89	41	∈	∈	PROPN
cana-5423	89	42	x.	x.	NOUN
cana-5423	89	43	definition	definition	NOUN
cana-5423	89	44	1.11	1.11	NUM
cana-5423	89	45	.	.	PUNCT
cana-5423	90	1	[	[	X
cana-5423	90	2	23	23	NUM
cana-5423	90	3	]	]	PUNCT
cana-5423	90	4	let	let	VERB
cana-5423	90	5	γ	γ	X
cana-5423	90	6	:	:	PUNCT
cana-5423	90	7	x	x	SYM
cana-5423	90	8	→	→	PUNCT
cana-5423	90	9	x	x	PUNCT
cana-5423	90	10	be	be	AUX
cana-5423	90	11	a	a	DET
cana-5423	90	12	self	self	NOUN
cana-5423	90	13	map	map	NOUN
cana-5423	90	14	defined	define	VERB
cana-5423	90	15	on	on	ADP
cana-5423	90	16	a	a	DET
cana-5423	90	17	metric	metric	ADJ
cana-5423	90	18	space(x	space(x	PROPN
cana-5423	90	19	,	,	PUNCT
cana-5423	90	20	d	d	NOUN
cana-5423	90	21	)	)	PUNCT
cana-5423	90	22	.	.	PUNCT
cana-5423	91	1	if	if	SCONJ
cana-5423	91	2	there	there	PRON
cana-5423	91	3	exist	exist	VERB
cana-5423	91	4	ζ	ζ	PROPN
cana-5423	91	5	∈	∈	PROPN
cana-5423	91	6	z	z	NOUN
cana-5423	91	7	and	and	CCONJ
cana-5423	91	8	α	α	NOUN
cana-5423	91	9	:	:	PUNCT
cana-5423	91	10	x	x	X
cana-5423	91	11	×x	×x	ADP
cana-5423	91	12	→	→	PUNCT
cana-5423	91	13	r+	r+	NOUN
cana-5423	91	14	such	such	ADJ
cana-5423	91	15	that	that	SCONJ
cana-5423	91	16	ζ(α(x	ζ(α(x	NOUN
cana-5423	91	17	,	,	PUNCT
cana-5423	91	18	y)d(γx	y)d(γx	NOUN
cana-5423	91	19	,	,	PUNCT
cana-5423	91	20	γy)d(x	γy)d(x	NOUN
cana-5423	91	21	,	,	PUNCT
cana-5423	91	22	y	y	NOUN
cana-5423	91	23	)	)	PUNCT
cana-5423	91	24	)	)	PUNCT
cana-5423	91	25	≥	≥	NOUN
cana-5423	91	26	0,∀x	0,∀x	NUM
cana-5423	91	27	,	,	PUNCT
cana-5423	91	28	y	y	PROPN
cana-5423	91	29	∈	∈	PROPN
cana-5423	91	30	x	x	X
cana-5423	91	31	(	(	PUNCT
cana-5423	91	32	6	6	NUM
cana-5423	91	33	)	)	PUNCT
cana-5423	91	34	then	then	ADV
cana-5423	91	35	γ	γ	PROPN
cana-5423	91	36	is	be	AUX
cana-5423	91	37	called	call	VERB
cana-5423	91	38	an	an	DET
cana-5423	91	39	α	α	NUM
cana-5423	91	40	-	-	ADJ
cana-5423	91	41	admiossible	admiossible	ADJ
cana-5423	91	42	zcontraction	zcontraction	NOUN
cana-5423	91	43	with	with	ADP
cana-5423	91	44	respect	respect	NOUN
cana-5423	91	45	to	to	ADP
cana-5423	91	46	ζ	ζ	PROPN
cana-5423	91	47	.	.	PUNCT
cana-5423	91	48	theorem	theorem	ADJ
cana-5423	91	49	1.12	1.12	NUM
cana-5423	91	50	.	.	PUNCT
cana-5423	92	1	[	[	X
cana-5423	92	2	23	23	NUM
cana-5423	92	3	]	]	X
cana-5423	92	4	let	let	VERB
cana-5423	92	5	(	(	PUNCT
cana-5423	92	6	x	x	NOUN
cana-5423	92	7	,	,	PUNCT
cana-5423	92	8	d	d	NOUN
cana-5423	92	9	)	)	PUNCT
cana-5423	92	10	be	be	AUX
cana-5423	92	11	a	a	DET
cana-5423	92	12	complete	complete	ADJ
cana-5423	92	13	metric	metric	ADJ
cana-5423	92	14	space	space	NOUN
cana-5423	92	15	and	and	CCONJ
cana-5423	92	16	let	let	VERB
cana-5423	92	17	γ	γ	X
cana-5423	92	18	:	:	PUNCT
cana-5423	92	19	x	x	SYM
cana-5423	92	20	→	→	PUNCT
cana-5423	92	21	x	x	PUNCT
cana-5423	92	22	be	be	AUX
cana-5423	92	23	an	an	DET
cana-5423	92	24	α	α	NOUN
cana-5423	92	25	-	-	ADJ
cana-5423	92	26	admissible	admissible	ADJ
cana-5423	92	27	zcontraction	zcontraction	NOUN
cana-5423	92	28	with	with	ADP
cana-5423	92	29	respect	respect	NOUN
cana-5423	92	30	to	to	ADP
cana-5423	92	31	ζ	ζ	PRON
cana-5423	92	32	.	.	PUNCT
cana-5423	93	1	suppose	suppose	VERB
cana-5423	93	2	that	that	SCONJ
cana-5423	93	3	(	(	PUNCT
cana-5423	93	4	i	i	NOUN
cana-5423	93	5	)	)	PUNCT
cana-5423	93	6	γ	γ	PROPN
cana-5423	93	7	is	be	AUX
cana-5423	93	8	triangular	triangular	NOUN
cana-5423	93	9	αorbital	αorbital	ADJ
cana-5423	93	10	admissble	admissble	ADJ
cana-5423	93	11	;	;	PUNCT
cana-5423	93	12	(	(	PUNCT
cana-5423	93	13	ii	ii	NOUN
cana-5423	93	14	)	)	PUNCT
cana-5423	93	15	there	there	PRON
cana-5423	93	16	exist	exist	VERB
cana-5423	93	17	x0	x0	PROPN
cana-5423	93	18	∈	∈	PROPN
cana-5423	93	19	x	x	PUNCT
cana-5423	93	20	such	such	ADJ
cana-5423	93	21	that	that	DET
cana-5423	93	22	α(x0,γx0	α(x0,γx0	PROPN
cana-5423	93	23	)	)	PUNCT
cana-5423	93	24	≥	≥	NOUN
cana-5423	93	25	1	1	NUM
cana-5423	93	26	;	;	PUNCT
cana-5423	93	27	(	(	PUNCT
cana-5423	93	28	iii	iii	X
cana-5423	93	29	)	)	PUNCT
cana-5423	93	30	γ	γ	NOUN
cana-5423	93	31	is	be	AUX
cana-5423	93	32	continuous	continuous	ADJ
cana-5423	93	33	.	.	PUNCT
cana-5423	94	1	then	then	ADV
cana-5423	94	2	there	there	PRON
cana-5423	94	3	exists	exist	VERB
cana-5423	94	4	u	u	NOUN
cana-5423	94	5	∈	∈	PROPN
cana-5423	94	6	x	x	PUNCT
cana-5423	94	7	such	such	ADJ
cana-5423	94	8	that	that	DET
cana-5423	94	9	γu	γu	NOUN
cana-5423	94	10	=	=	NOUN
cana-5423	94	11	u.	u.	NOUN
cana-5423	94	12	communications	communication	NOUN
cana-5423	94	13	on	on	ADP
cana-5423	94	14	applied	apply	VERB
cana-5423	94	15	nonlinear	nonlinear	ADJ
cana-5423	94	16	analysis	analysis	NOUN
cana-5423	94	17	issn	issn	NOUN
cana-5423	94	18	:	:	PUNCT
cana-5423	94	19	1074	1074	NUM
cana-5423	94	20	-	-	PUNCT
cana-5423	94	21	133x	133x	NUM
cana-5423	94	22	vol	vol	NOUN
cana-5423	94	23	32	32	NUM
cana-5423	94	24	no	no	NOUN
cana-5423	94	25	.	.	PUNCT
cana-5423	95	1	10s(2025	10s(2025	NUM
cana-5423	95	2	)	)	PUNCT
cana-5423	96	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	96	2	2219	2219	NUM
cana-5423	96	3	very	very	ADV
cana-5423	96	4	recently	recently	ADV
cana-5423	96	5	,	,	PUNCT
cana-5423	96	6	dipti	dipti	PROPN
cana-5423	96	7	et	et	PROPN
cana-5423	96	8	al	al	PROPN
cana-5423	96	9	.	.	PUNCT
cana-5423	97	1	[	[	X
cana-5423	97	2	43	43	NUM
cana-5423	97	3	]	]	PUNCT
cana-5423	97	4	presented	present	VERB
cana-5423	97	5	some	some	DET
cana-5423	97	6	fixed	fix	VERB
cana-5423	97	7	point	point	NOUN
cana-5423	97	8	results	result	NOUN
cana-5423	97	9	in	in	ADP
cana-5423	97	10	complete	complete	ADJ
cana-5423	97	11	metric	metric	ADJ
cana-5423	97	12	spaces	space	NOUN
cana-5423	97	13	using	use	VERB
cana-5423	97	14	generalized	generalized	ADJ
cana-5423	97	15	α	α	PRON
cana-5423	97	16	admissible	admissible	ADJ
cana-5423	97	17	mappings	mapping	NOUN
cana-5423	97	18	embedded	embed	VERB
cana-5423	97	19	in	in	ADP
cana-5423	97	20	the	the	DET
cana-5423	97	21	simulation	simulation	NOUN
cana-5423	97	22	functions	function	NOUN
cana-5423	97	23	.	.	PUNCT
cana-5423	98	1	definition	definition	NOUN
cana-5423	98	2	1.13	1.13	NUM
cana-5423	98	3	.	.	PUNCT
cana-5423	99	1	[	[	X
cana-5423	99	2	43	43	NUM
cana-5423	99	3	]	]	X
cana-5423	99	4	let(x	let(x	PROPN
cana-5423	99	5	,	,	PUNCT
cana-5423	99	6	d	d	X
cana-5423	99	7	)	)	PUNCT
cana-5423	99	8	be	be	AUX
cana-5423	99	9	a	a	DET
cana-5423	99	10	metric	metric	ADJ
cana-5423	99	11	space	space	NOUN
cana-5423	99	12	,	,	PUNCT
cana-5423	99	13	γ	γ	X
cana-5423	99	14	:	:	PUNCT
cana-5423	99	15	x	x	SYM
cana-5423	99	16	→	→	PUNCT
cana-5423	99	17	x	x	PUNCT
cana-5423	99	18	be	be	AUX
cana-5423	99	19	a	a	DET
cana-5423	99	20	self	self	NOUN
cana-5423	99	21	mapping	mapping	NOUN
cana-5423	99	22	,	,	PUNCT
cana-5423	99	23	there	there	PRON
cana-5423	99	24	exists	exist	VERB
cana-5423	99	25	ζ	ζ	PROPN
cana-5423	99	26	∈	∈	PROPN
cana-5423	99	27	z	z	NOUN
cana-5423	99	28	and	and	CCONJ
cana-5423	99	29	α	α	NOUN
cana-5423	99	30	:	:	PUNCT
cana-5423	100	1	x	x	SYM
cana-5423	100	2	×	×	NOUN
cana-5423	100	3	x	x	INTJ
cana-5423	100	4	→	→	SYM
cana-5423	100	5	x[0,∞	x[0,∞	NUM
cana-5423	100	6	)	)	PUNCT
cana-5423	100	7	.	.	PUNCT
cana-5423	101	1	then	then	ADV
cana-5423	101	2	continuous	continuous	ADJ
cana-5423	101	3	mapping	mapping	NOUN
cana-5423	101	4	γ	γ	NOUN
cana-5423	101	5	is	be	AUX
cana-5423	101	6	called	call	VERB
cana-5423	101	7	generalized	generalized	ADJ
cana-5423	101	8	α	α	NOUN
cana-5423	101	9	-admissible	-admissible	ADJ
cana-5423	101	10	almost	almost	ADV
cana-5423	101	11	zcontraction	zcontraction	NOUN
cana-5423	101	12	with	with	ADP
cana-5423	101	13	respect	respect	NOUN
cana-5423	101	14	to	to	ADP
cana-5423	101	15	ζ	ζ	NOUN
cana-5423	101	16	and	and	CCONJ
cana-5423	101	17	β	β	X
cana-5423	101	18	∈	∈	NOUN
cana-5423	101	19	g	g	NOUN
cana-5423	101	20	and	and	CCONJ
cana-5423	101	21	l	l	PROPN
cana-5423	101	22	≥	≥	NUM
cana-5423	101	23	0	0	NUM
cana-5423	101	24	such	such	ADJ
cana-5423	101	25	that	that	DET
cana-5423	101	26	for	for	ADP
cana-5423	101	27	all	all	DET
cana-5423	101	28	x	x	NOUN
cana-5423	101	29	,	,	PUNCT
cana-5423	101	30	y	y	PROPN
cana-5423	101	31	∈	∈	PROPN
cana-5423	101	32	x	x	PROPN
cana-5423	101	33	,	,	PUNCT
cana-5423	101	34	ζ(α(x	ζ(α(x	PROPN
cana-5423	101	35	,	,	PUNCT
cana-5423	101	36	γx)α(y	γx)α(y	NOUN
cana-5423	101	37	,	,	PUNCT
cana-5423	101	38	γy)d(γx	γy)d(γx	PROPN
cana-5423	101	39	,	,	PUNCT
cana-5423	101	40	γy),k(x	γy),k(x	NOUN
cana-5423	101	41	,	,	PUNCT
cana-5423	101	42	y	y	PROPN
cana-5423	101	43	)	)	PUNCT
cana-5423	101	44	+	+	NUM
cana-5423	101	45	lq(x	lq(x	X
cana-5423	101	46	,	,	PUNCT
cana-5423	101	47	y	y	NOUN
cana-5423	101	48	)	)	PUNCT
cana-5423	101	49	≥	≥	NOUN
cana-5423	101	50	0	0	NUM
cana-5423	101	51	(	(	PUNCT
cana-5423	101	52	7	7	NUM
cana-5423	101	53	)	)	PUNCT
cana-5423	101	54	for	for	ADP
cana-5423	101	55	all	all	DET
cana-5423	101	56	distinct	distinct	ADJ
cana-5423	101	57	x	x	NOUN
cana-5423	101	58	,	,	PUNCT
cana-5423	101	59	y	y	PROPN
cana-5423	101	60	∈	∈	PROPN
cana-5423	101	61	x	x	NOUN
cana-5423	101	62	,	,	PUNCT
cana-5423	101	63	where	where	SCONJ
cana-5423	101	64	zeta	zeta	NOUN
cana-5423	101	65	is	be	AUX
cana-5423	101	66	a	a	DET
cana-5423	101	67	simulation	simulation	NOUN
cana-5423	101	68	function	function	NOUN
cana-5423	101	69	in	in	ADP
cana-5423	101	70	the	the	DET
cana-5423	101	71	sense	sense	NOUN
cana-5423	101	72	of	of	ADP
cana-5423	101	73	definition	definition	NOUN
cana-5423	101	74	1	1	NUM
cana-5423	101	75	.	.	PUNCT
cana-5423	101	76	also	also	ADV
cana-5423	101	77	k(x	k(x	PROPN
cana-5423	101	78	,	,	PUNCT
cana-5423	101	79	y	y	NOUN
cana-5423	101	80	)	)	PUNCT
cana-5423	101	81	=	=	SYM
cana-5423	101	82	β(e(x	β(e(x	PROPN
cana-5423	101	83	,	,	PUNCT
cana-5423	101	84	y))e(x	y))e(x	PROPN
cana-5423	101	85	,	,	PUNCT
cana-5423	101	86	y	y	PROPN
cana-5423	101	87	)	)	PUNCT
cana-5423	102	1	+	+	CCONJ
cana-5423	102	2	ln(x	ln(x	X
cana-5423	102	3	,	,	PUNCT
cana-5423	102	4	y	y	NOUN
cana-5423	102	5	)	)	PUNCT
cana-5423	102	6	(	(	PUNCT
cana-5423	102	7	8)	8)	NUM
cana-5423	102	8	,	,	PUNCT
cana-5423	102	9	where	where	SCONJ
cana-5423	102	10	e(x	e(x	NUM
cana-5423	102	11	,	,	PUNCT
cana-5423	102	12	y	y	NOUN
cana-5423	102	13	)	)	PUNCT
cana-5423	102	14	=	=	SYM
cana-5423	102	15	d(x	d(x	PROPN
cana-5423	102	16	,	,	PUNCT
cana-5423	102	17	y	y	NOUN
cana-5423	102	18	)	)	PUNCT
cana-5423	102	19	+	+	X
cana-5423	102	20	|d(x	|d(x	ADJ
cana-5423	102	21	,	,	PUNCT
cana-5423	102	22	γx	γx	NOUN
cana-5423	102	23	)	)	PUNCT
cana-5423	102	24	−	−	ADP
cana-5423	102	25	d(y	d(y	PROPN
cana-5423	102	26	,	,	PUNCT
cana-5423	102	27	γy)|	γy)|	PROPN
cana-5423	102	28	(	(	PUNCT
cana-5423	102	29	9	9	NUM
cana-5423	102	30	)	)	PUNCT
cana-5423	102	31	and	and	CCONJ
cana-5423	102	32	n(x	n(x	PROPN
cana-5423	102	33	,	,	PUNCT
cana-5423	102	34	y	y	NOUN
cana-5423	102	35	)	)	PUNCT
cana-5423	103	1	=	=	VERB
cana-5423	103	2	min{d(x	min{d(x	NOUN
cana-5423	103	3	,	,	PUNCT
cana-5423	103	4	γx	γx	NOUN
cana-5423	103	5	)	)	PUNCT
cana-5423	103	6	,	,	PUNCT
cana-5423	103	7	d(y	d(y	PROPN
cana-5423	103	8	,	,	PUNCT
cana-5423	103	9	γy	γy	NOUN
cana-5423	103	10	)	)	PUNCT
cana-5423	103	11	,	,	PUNCT
cana-5423	103	12	d(x	d(x	PROPN
cana-5423	103	13	,	,	PUNCT
cana-5423	103	14	γy	γy	PROPN
cana-5423	103	15	)	)	PUNCT
cana-5423	103	16	,	,	PUNCT
cana-5423	103	17	d(y	d(y	NOUN
cana-5423	103	18	,	,	PUNCT
cana-5423	103	19	γx	γx	NOUN
cana-5423	103	20	)	)	PUNCT
cana-5423	103	21	}	}	PUNCT
cana-5423	103	22	.	.	PUNCT
cana-5423	104	1	(	(	PUNCT
cana-5423	104	2	10	10	NUM
cana-5423	104	3	)	)	PUNCT
cana-5423	104	4	definition	definition	NOUN
cana-5423	104	5	1.14	1.14	NUM
cana-5423	104	6	(	(	PUNCT
cana-5423	104	7	[	[	X
cana-5423	104	8	11	11	NUM
cana-5423	104	9	]	]	NUM
cana-5423	104	10	)	)	PUNCT
cana-5423	104	11	.	.	PUNCT
cana-5423	105	1	let	let	VERB
cana-5423	105	2	(	(	PUNCT
cana-5423	105	3	x	x	NOUN
cana-5423	105	4	,	,	PUNCT
cana-5423	105	5	d	d	NOUN
cana-5423	105	6	)	)	PUNCT
cana-5423	105	7	be	be	AUX
cana-5423	105	8	a	a	DET
cana-5423	105	9	metric	metric	ADJ
cana-5423	105	10	space	space	NOUN
cana-5423	105	11	and	and	CCONJ
cana-5423	105	12	ζ	ζ	NOUN
cana-5423	105	13	.	.	PUNCT
cana-5423	106	1	we	we	PRON
cana-5423	106	2	say	say	VERB
cana-5423	106	3	that	that	SCONJ
cana-5423	106	4	γ	γ	X
cana-5423	106	5	:	:	PUNCT
cana-5423	106	6	x	x	SYM
cana-5423	106	7	→	→	PUNCT
cana-5423	106	8	x	x	X
cana-5423	106	9	is	be	AUX
cana-5423	106	10	a	a	DET
cana-5423	106	11	modified	modify	VERB
cana-5423	106	12	almost	almost	ADV
cana-5423	106	13	type	type	NOUN
cana-5423	106	14	z	z	NOUN
cana-5423	106	15	-	-	NOUN
cana-5423	106	16	contraction	contraction	NOUN
cana-5423	106	17	if	if	SCONJ
cana-5423	106	18	there	there	PRON
cana-5423	106	19	are	be	VERB
cana-5423	106	20	constants	constant	NOUN
cana-5423	106	21	l	l	PROPN
cana-5423	106	22	≥	≥	NUM
cana-5423	106	23	o	o	NOUN
cana-5423	106	24	such	such	ADJ
cana-5423	106	25	that	that	SCONJ
cana-5423	106	26	ζ(d(γx	ζ(d(γx	PROPN
cana-5423	106	27	,	,	PUNCT
cana-5423	106	28	γy),k(x	γy),k(x	PROPN
cana-5423	106	29	,	,	PUNCT
cana-5423	106	30	y	y	PROPN
cana-5423	106	31	)	)	PUNCT
cana-5423	107	1	+	+	NUM
cana-5423	107	2	lq(x	lq(x	NOUN
cana-5423	107	3	,	,	PUNCT
cana-5423	107	4	y),∀x	y),∀x	PROPN
cana-5423	107	5	,	,	PUNCT
cana-5423	107	6	y	y	PROPN
cana-5423	107	7	∈	∈	PROPN
cana-5423	107	8	x	x	X
cana-5423	107	9	,	,	PUNCT
cana-5423	107	10	(	(	PUNCT
cana-5423	107	11	11	11	NUM
cana-5423	107	12	)	)	PUNCT
cana-5423	107	13	where	where	SCONJ
cana-5423	107	14	,	,	PUNCT
cana-5423	107	15	k(x	k(x	PROPN
cana-5423	107	16	,	,	PUNCT
cana-5423	107	17	y	y	NOUN
cana-5423	107	18	)	)	PUNCT
cana-5423	107	19	=	=	SYM
cana-5423	107	20	max	max	PROPN
cana-5423	107	21	{	{	PUNCT
cana-5423	107	22	d(x	d(x	PROPN
cana-5423	107	23	,	,	PUNCT
cana-5423	107	24	y	y	PROPN
cana-5423	107	25	)	)	PUNCT
cana-5423	107	26	,	,	PUNCT
cana-5423	108	1	[	[	X
cana-5423	108	2	1	1	NUM
cana-5423	108	3	+	+	NUM
cana-5423	108	4	d(x	d(x	NOUN
cana-5423	108	5	,	,	PUNCT
cana-5423	108	6	γx)d(y	γx)d(y	NOUN
cana-5423	108	7	,	,	PUNCT
cana-5423	108	8	γy	γy	PROPN
cana-5423	108	9	)	)	PUNCT
cana-5423	108	10	]	]	PUNCT
cana-5423	109	1	1	1	NUM
cana-5423	109	2	+	+	CCONJ
cana-5423	109	3	d(x	d(x	PROPN
cana-5423	109	4	,	,	PUNCT
cana-5423	109	5	y	y	NOUN
cana-5423	109	6	)	)	PUNCT
cana-5423	109	7	}	}	PUNCT
cana-5423	109	8	and	and	CCONJ
cana-5423	109	9	q(x	q(x	PROPN
cana-5423	109	10	,	,	PUNCT
cana-5423	109	11	y	y	NOUN
cana-5423	109	12	)	)	PUNCT
cana-5423	109	13	=	=	VERB
cana-5423	110	1	min{d(x	min{d(x	NOUN
cana-5423	110	2	,	,	PUNCT
cana-5423	110	3	γx	γx	NOUN
cana-5423	110	4	)	)	PUNCT
cana-5423	110	5	,	,	PUNCT
cana-5423	110	6	d(y	d(y	PROPN
cana-5423	110	7	,	,	PUNCT
cana-5423	110	8	γy	γy	NOUN
cana-5423	110	9	)	)	PUNCT
cana-5423	110	10	,	,	PUNCT
cana-5423	110	11	d(x	d(x	PROPN
cana-5423	110	12	,	,	PUNCT
cana-5423	110	13	γx	γx	NOUN
cana-5423	110	14	)	)	PUNCT
cana-5423	110	15	,	,	PUNCT
cana-5423	110	16	d(y	d(y	PROPN
cana-5423	110	17	,	,	PUNCT
cana-5423	110	18	γy	γy	NOUN
cana-5423	110	19	)	)	PUNCT
cana-5423	110	20	}	}	PUNCT
cana-5423	110	21	remark	remark	VERB
cana-5423	110	22	1.15	1.15	NUM
cana-5423	110	23	.	.	PUNCT
cana-5423	111	1	if	if	SCONJ
cana-5423	111	2	γ	γ	X
cana-5423	111	3	is	be	AUX
cana-5423	111	4	a	a	DET
cana-5423	111	5	modified	modify	VERB
cana-5423	111	6	almost	almost	ADV
cana-5423	111	7	type	type	NOUN
cana-5423	111	8	z	z	NOUN
cana-5423	111	9	-	-	NOUN
cana-5423	111	10	contraction	contraction	NOUN
cana-5423	111	11	with	with	ADP
cana-5423	111	12	respect	respect	NOUN
cana-5423	111	13	to	to	ADP
cana-5423	111	14	ζ	ζ	SYM
cana-5423	111	15	∈	∈	PROPN
cana-5423	111	16	z	z	NOUN
cana-5423	111	17	,	,	PUNCT
cana-5423	111	18	then	then	ADV
cana-5423	111	19	d(γx	d(γx	PROPN
cana-5423	111	20	,	,	PUNCT
cana-5423	111	21	γy	γy	PROPN
cana-5423	111	22	)	)	PUNCT
cana-5423	111	23	<	<	X
cana-5423	111	24	k(x	k(x	PROPN
cana-5423	111	25	,	,	PUNCT
cana-5423	111	26	y	y	NOUN
cana-5423	111	27	)	)	PUNCT
cana-5423	112	1	+	+	NUM
cana-5423	112	2	lq(x	lq(x	NOUN
cana-5423	112	3	,	,	PUNCT
cana-5423	112	4	y	y	NOUN
cana-5423	112	5	)	)	PUNCT
cana-5423	112	6	∀x	∀x	NUM
cana-5423	112	7	,	,	PUNCT
cana-5423	112	8	y	y	PROPN
cana-5423	112	9	∈	∈	PROPN
cana-5423	112	10	x	x	X
cana-5423	112	11	,	,	PUNCT
cana-5423	112	12	(	(	PUNCT
cana-5423	112	13	9	9	X
cana-5423	112	14	)	)	PUNCT
cana-5423	112	15	inspired	inspire	VERB
cana-5423	112	16	and	and	CCONJ
cana-5423	112	17	motivated	motivate	VERB
cana-5423	112	18	by	by	ADP
cana-5423	112	19	the	the	DET
cana-5423	112	20	combining	combine	VERB
cana-5423	112	21	the	the	DET
cana-5423	112	22	ideas	idea	NOUN
cana-5423	112	23	in[11],[20	in[11],[20	X
cana-5423	112	24	]	]	PUNCT
cana-5423	112	25	,	,	PUNCT
cana-5423	112	26	[	[	X
cana-5423	112	27	25	25	NUM
cana-5423	112	28	]	]	PUNCT
cana-5423	112	29	,	,	PUNCT
cana-5423	112	30	[	[	X
cana-5423	112	31	27	27	NUM
cana-5423	112	32	]	]	PUNCT
cana-5423	112	33	and	and	CCONJ
cana-5423	112	34	[	[	X
cana-5423	112	35	43	43	NUM
cana-5423	112	36	]	]	PUNCT
cana-5423	112	37	,	,	PUNCT
cana-5423	112	38	we	we	PRON
cana-5423	112	39	introduce	introduce	VERB
cana-5423	112	40	a	a	DET
cana-5423	112	41	new	new	ADJ
cana-5423	112	42	clas	cla	NOUN
cana-5423	112	43	of	of	ADP
cana-5423	112	44	mappings	mapping	NOUN
cana-5423	112	45	,	,	PUNCT
cana-5423	112	46	and	and	CCONJ
cana-5423	112	47	define	define	VERB
cana-5423	112	48	generalized	generalized	ADJ
cana-5423	112	49	α	α	NOUN
cana-5423	112	50	-admissible	-admissible	ADJ
cana-5423	112	51	modified	modify	VERB
cana-5423	112	52	almost	almost	ADV
cana-5423	112	53	z	z	NOUN
cana-5423	112	54	contraction	contraction	NOUN
cana-5423	112	55	with	with	ADP
cana-5423	112	56	respect	respect	NOUN
cana-5423	112	57	to	to	ADP
cana-5423	112	58	ζ	ζ	NOUN
cana-5423	112	59	in	in	ADP
cana-5423	112	60	the	the	DET
cana-5423	112	61	setting	setting	NOUN
cana-5423	112	62	of	of	ADP
cana-5423	112	63	metric	metric	ADJ
cana-5423	112	64	space	space	NOUN
cana-5423	112	65	and	and	CCONJ
cana-5423	112	66	obtain	obtain	VERB
cana-5423	112	67	the	the	DET
cana-5423	112	68	existence	existence	NOUN
cana-5423	112	69	and	and	CCONJ
cana-5423	112	70	uniqueness	uniqueness	NOUN
cana-5423	112	71	of	of	ADP
cana-5423	112	72	fixed	fix	VERB
cana-5423	112	73	point	point	NOUN
cana-5423	112	74	of	of	ADP
cana-5423	112	75	such	such	ADJ
cana-5423	112	76	map	map	NOUN
cana-5423	112	77	.	.	PUNCT
cana-5423	113	1	communications	communication	NOUN
cana-5423	113	2	on	on	ADP
cana-5423	113	3	applied	apply	VERB
cana-5423	113	4	nonlinear	nonlinear	ADJ
cana-5423	113	5	analysis	analysis	NOUN
cana-5423	113	6	issn	issn	NOUN
cana-5423	113	7	:	:	PUNCT
cana-5423	113	8	1074	1074	NUM
cana-5423	113	9	-	-	PUNCT
cana-5423	113	10	133x	133x	NUM
cana-5423	113	11	vol	vol	NOUN
cana-5423	113	12	32	32	NUM
cana-5423	113	13	no	no	NOUN
cana-5423	113	14	.	.	PUNCT
cana-5423	114	1	10s(2025	10s(2025	NUM
cana-5423	114	2	)	)	PUNCT
cana-5423	115	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	115	2	2220	2220	NUM
cana-5423	115	3	2	2	NUM
cana-5423	115	4	.	.	PUNCT
cana-5423	115	5	main	main	ADJ
cana-5423	115	6	results	result	NOUN
cana-5423	115	7	finaly	finaly	VERB
cana-5423	115	8	,	,	PUNCT
cana-5423	115	9	we	we	PRON
cana-5423	115	10	give	give	VERB
cana-5423	115	11	the	the	DET
cana-5423	115	12	following	follow	VERB
cana-5423	115	13	definition	definition	NOUN
cana-5423	115	14	which	which	PRON
cana-5423	115	15	will	will	AUX
cana-5423	115	16	be	be	AUX
cana-5423	115	17	used	use	VERB
cana-5423	115	18	in	in	ADP
cana-5423	115	19	our	our	PRON
cana-5423	115	20	main	main	ADJ
cana-5423	115	21	results	result	NOUN
cana-5423	115	22	.	.	PUNCT
cana-5423	116	1	definition	definition	NOUN
cana-5423	116	2	2.1	2.1	NUM
cana-5423	116	3	.	.	PUNCT
cana-5423	117	1	let(x	let(x	PROPN
cana-5423	117	2	,	,	PUNCT
cana-5423	117	3	d	d	X
cana-5423	117	4	)	)	PUNCT
cana-5423	117	5	be	be	AUX
cana-5423	117	6	a	a	DET
cana-5423	117	7	metric	metric	ADJ
cana-5423	117	8	space	space	NOUN
cana-5423	117	9	,	,	PUNCT
cana-5423	117	10	γ	γ	X
cana-5423	117	11	:	:	PUNCT
cana-5423	117	12	x	x	SYM
cana-5423	117	13	→	→	PUNCT
cana-5423	117	14	x	x	PUNCT
cana-5423	117	15	be	be	AUX
cana-5423	117	16	a	a	DET
cana-5423	117	17	self	self	NOUN
cana-5423	117	18	mapping	mapping	NOUN
cana-5423	117	19	,	,	PUNCT
cana-5423	117	20	there	there	PRON
cana-5423	117	21	exists	exist	VERB
cana-5423	117	22	ζ	ζ	PROPN
cana-5423	117	23	∈	∈	PROPN
cana-5423	117	24	z	z	NOUN
cana-5423	117	25	and	and	CCONJ
cana-5423	117	26	α	α	NOUN
cana-5423	117	27	:	:	PUNCT
cana-5423	118	1	x	x	SYM
cana-5423	118	2	×	×	NOUN
cana-5423	118	3	x	x	INTJ
cana-5423	118	4	→	→	SYM
cana-5423	118	5	x[0,∞	x[0,∞	NUM
cana-5423	118	6	)	)	PUNCT
cana-5423	118	7	.	.	PUNCT
cana-5423	119	1	then	then	ADV
cana-5423	119	2	continuous	continuous	ADJ
cana-5423	119	3	mapping	mapping	NOUN
cana-5423	119	4	γ	γ	NOUN
cana-5423	119	5	is	be	AUX
cana-5423	119	6	called	call	VERB
cana-5423	119	7	generalized	generalized	ADJ
cana-5423	119	8	α	α	NOUN
cana-5423	119	9	-	-	ADJ
cana-5423	119	10	admissible	admissible	ADJ
cana-5423	119	11	modified	modify	VERB
cana-5423	119	12	almost	almost	ADV
cana-5423	119	13	zcontraction	zcontraction	NOUN
cana-5423	119	14	with	with	ADP
cana-5423	119	15	respect	respect	NOUN
cana-5423	119	16	to	to	ADP
cana-5423	119	17	ζ	ζ	NOUN
cana-5423	119	18	and	and	CCONJ
cana-5423	119	19	l	l	NOUN
cana-5423	119	20	≥	≥	NOUN
cana-5423	119	21	0	0	NUM
cana-5423	119	22	such	such	ADJ
cana-5423	119	23	that	that	DET
cana-5423	119	24	for	for	ADP
cana-5423	119	25	all	all	DET
cana-5423	119	26	x	x	NOUN
cana-5423	119	27	,	,	PUNCT
cana-5423	119	28	y	y	PROPN
cana-5423	119	29	∈	∈	PROPN
cana-5423	119	30	x	x	PROPN
cana-5423	119	31	,	,	PUNCT
cana-5423	119	32	ζ(α(x	ζ(α(x	PROPN
cana-5423	119	33	,	,	PUNCT
cana-5423	119	34	γx)α(y	γx)α(y	NOUN
cana-5423	119	35	,	,	PUNCT
cana-5423	119	36	γy)d(γx	γy)d(γx	PROPN
cana-5423	119	37	,	,	PUNCT
cana-5423	119	38	γy),k(x	γy),k(x	NOUN
cana-5423	119	39	,	,	PUNCT
cana-5423	119	40	y	y	PROPN
cana-5423	119	41	)	)	PUNCT
cana-5423	119	42	+	+	NUM
cana-5423	119	43	lq(x	lq(x	NOUN
cana-5423	119	44	,	,	PUNCT
cana-5423	119	45	y	y	NOUN
cana-5423	119	46	)	)	PUNCT
cana-5423	119	47	)	)	PUNCT
cana-5423	119	48	≥	≥	NOUN
cana-5423	119	49	0	0	NUM
cana-5423	119	50	(	(	PUNCT
cana-5423	119	51	10	10	NUM
cana-5423	119	52	)	)	PUNCT
cana-5423	119	53	where	where	SCONJ
cana-5423	119	54	ζ	ζ	NOUN
cana-5423	119	55	is	be	AUX
cana-5423	119	56	a	a	DET
cana-5423	119	57	simulation	simulation	NOUN
cana-5423	119	58	function	function	NOUN
cana-5423	119	59	in	in	ADP
cana-5423	119	60	the	the	DET
cana-5423	119	61	sense	sense	NOUN
cana-5423	119	62	of	of	ADP
cana-5423	119	63	definition	definition	NOUN
cana-5423	119	64	1.4	1.4	NUM
cana-5423	119	65	.	.	PUNCT
cana-5423	120	1	also	also	ADV
cana-5423	120	2	k(x	k(x	PROPN
cana-5423	120	3	,	,	PUNCT
cana-5423	120	4	y	y	NOUN
cana-5423	120	5	)	)	PUNCT
cana-5423	121	1	=	=	SYM
cana-5423	121	2	max	max	PROPN
cana-5423	121	3	{	{	PUNCT
cana-5423	121	4	d(x	d(x	PROPN
cana-5423	121	5	,	,	PUNCT
cana-5423	121	6	y	y	PROPN
cana-5423	121	7	)	)	PUNCT
cana-5423	121	8	,	,	PUNCT
cana-5423	122	1	[	[	X
cana-5423	122	2	1	1	NUM
cana-5423	122	3	+	+	NUM
cana-5423	122	4	d(x	d(x	PROPN
cana-5423	122	5	,	,	PUNCT
cana-5423	122	6	γx)]d(y	γx)]d(y	NOUN
cana-5423	122	7	,	,	PUNCT
cana-5423	122	8	γy	γy	NOUN
cana-5423	122	9	)	)	PUNCT
cana-5423	122	10	1	1	NUM
cana-5423	123	1	+	+	CCONJ
cana-5423	123	2	d(x	d(x	PROPN
cana-5423	123	3	,	,	PUNCT
cana-5423	123	4	y	y	NOUN
cana-5423	123	5	)	)	PUNCT
cana-5423	123	6	,	,	PUNCT
cana-5423	124	1	[	[	X
cana-5423	124	2	1	1	NUM
cana-5423	124	3	+	+	NUM
cana-5423	124	4	d(x	d(x	PROPN
cana-5423	124	5	,	,	PUNCT
cana-5423	124	6	γy)]d(y	γy)]d(y	ADJ
cana-5423	124	7	,	,	PUNCT
cana-5423	124	8	γx	γx	NOUN
cana-5423	124	9	)	)	PUNCT
cana-5423	124	10	1	1	NUM
cana-5423	124	11	+	+	CCONJ
cana-5423	124	12	d(x	d(x	PROPN
cana-5423	124	13	,	,	PUNCT
cana-5423	124	14	y	y	NOUN
cana-5423	124	15	)	)	PUNCT
cana-5423	124	16	}	}	PUNCT
cana-5423	124	17	(	(	PUNCT
cana-5423	124	18	11	11	NUM
cana-5423	124	19	)	)	PUNCT
cana-5423	124	20	and	and	CCONJ
cana-5423	124	21	q(x	q(x	PROPN
cana-5423	124	22	,	,	PUNCT
cana-5423	124	23	y	y	NOUN
cana-5423	124	24	)	)	PUNCT
cana-5423	124	25	=	=	SYM
cana-5423	124	26	min	min	NOUN
cana-5423	124	27	{	{	PUNCT
cana-5423	124	28	d(x	d(x	PROPN
cana-5423	124	29	,	,	PUNCT
cana-5423	124	30	γx	γx	NOUN
cana-5423	124	31	)	)	PUNCT
cana-5423	124	32	,	,	PUNCT
cana-5423	124	33	d(y	d(y	PROPN
cana-5423	124	34	,	,	PUNCT
cana-5423	124	35	γy	γy	NOUN
cana-5423	124	36	)	)	PUNCT
cana-5423	124	37	,	,	PUNCT
cana-5423	124	38	d(x	d(x	PROPN
cana-5423	124	39	,	,	PUNCT
cana-5423	124	40	γy	γy	PROPN
cana-5423	124	41	)	)	PUNCT
cana-5423	124	42	,	,	PUNCT
cana-5423	124	43	d(y	d(y	NOUN
cana-5423	124	44	,	,	PUNCT
cana-5423	124	45	γx	γx	NOUN
cana-5423	124	46	)	)	PUNCT
cana-5423	124	47	,	,	PUNCT
cana-5423	124	48	d(x	d(x	PROPN
cana-5423	124	49	,	,	PUNCT
cana-5423	124	50	γy)d(y	γy)d(y	ADV
cana-5423	124	51	,	,	PUNCT
cana-5423	124	52	γx	γx	NOUN
cana-5423	124	53	)	)	PUNCT
cana-5423	124	54	1	1	NUM
cana-5423	124	55	+	+	CCONJ
cana-5423	124	56	d(x	d(x	PROPN
cana-5423	124	57	,	,	PUNCT
cana-5423	124	58	y	y	NOUN
cana-5423	124	59	)	)	PUNCT
cana-5423	124	60	,	,	PUNCT
cana-5423	124	61	d(x	d(x	PROPN
cana-5423	124	62	,	,	PUNCT
cana-5423	124	63	γx)d(y	γx)d(y	NOUN
cana-5423	124	64	,	,	PUNCT
cana-5423	124	65	γy	γy	NOUN
cana-5423	124	66	)	)	PUNCT
cana-5423	124	67	1	1	NUM
cana-5423	125	1	+	+	CCONJ
cana-5423	125	2	d(x	d(x	PROPN
cana-5423	125	3	,	,	PUNCT
cana-5423	125	4	y	y	NOUN
cana-5423	125	5	)	)	PUNCT
cana-5423	125	6	}	}	PUNCT
cana-5423	125	7	.	.	PUNCT
cana-5423	126	1	(	(	PUNCT
cana-5423	126	2	12	12	NUM
cana-5423	126	3	)	)	PUNCT
cana-5423	126	4	we	we	PRON
cana-5423	126	5	can	can	AUX
cana-5423	126	6	now	now	ADV
cana-5423	126	7	state	state	VERB
cana-5423	126	8	the	the	DET
cana-5423	126	9	main	main	ADJ
cana-5423	126	10	finding	finding	NOUN
cana-5423	126	11	of	of	ADP
cana-5423	126	12	this	this	DET
cana-5423	126	13	paper	paper	NOUN
cana-5423	126	14	.	.	PUNCT
cana-5423	127	1	theorem	theorem	VERB
cana-5423	127	2	2.2	2.2	NUM
cana-5423	127	3	.	.	PUNCT
cana-5423	128	1	let	let	AUX
cana-5423	128	2	(	(	PUNCT
cana-5423	128	3	x	x	NOUN
cana-5423	128	4	,	,	PUNCT
cana-5423	128	5	d	d	NOUN
cana-5423	128	6	)	)	PUNCT
cana-5423	128	7	be	be	AUX
cana-5423	128	8	a	a	DET
cana-5423	128	9	complete	complete	ADJ
cana-5423	128	10	metric	metric	ADJ
cana-5423	128	11	space	space	NOUN
cana-5423	128	12	and	and	CCONJ
cana-5423	128	13	γ	γ	X
cana-5423	128	14	:	:	PUNCT
cana-5423	128	15	x	x	SYM
cana-5423	128	16	→	→	PUNCT
cana-5423	128	17	x	x	X
cana-5423	128	18	is	be	AUX
cana-5423	128	19	a	a	DET
cana-5423	128	20	generalized	generalized	ADJ
cana-5423	128	21	α	α	NOUN
cana-5423	128	22	-	-	ADJ
cana-5423	128	23	admissible	admissible	ADJ
cana-5423	128	24	modified	modify	VERB
cana-5423	128	25	almost	almost	ADV
cana-5423	128	26	zcontraction	zcontraction	NOUN
cana-5423	128	27	with	with	ADP
cana-5423	128	28	respect	respect	NOUN
cana-5423	128	29	to	to	ADP
cana-5423	128	30	ζ	ζ	NOUN
cana-5423	128	31	.	.	PUNCT
cana-5423	129	1	furthermore	furthermore	ADV
cana-5423	129	2	,	,	PUNCT
cana-5423	129	3	we	we	PRON
cana-5423	129	4	suppose	suppose	VERB
cana-5423	129	5	for	for	ADP
cana-5423	129	6	all	all	DET
cana-5423	129	7	x	x	NOUN
cana-5423	129	8	,	,	PUNCT
cana-5423	129	9	y	y	PROPN
cana-5423	129	10	∈	∈	PROPN
cana-5423	129	11	x	x	PUNCT
cana-5423	129	12	such	such	ADJ
cana-5423	129	13	that	that	SCONJ
cana-5423	129	14	:	:	PUNCT
cana-5423	129	15	(	(	PUNCT
cana-5423	129	16	i	i	NOUN
cana-5423	129	17	)	)	PUNCT
cana-5423	129	18	γ	γ	PROPN
cana-5423	129	19	is	be	AUX
cana-5423	129	20	triangular	triangular	NOUN
cana-5423	129	21	αorbital	αorbital	ADJ
cana-5423	129	22	admissible	admissible	ADJ
cana-5423	129	23	;	;	PUNCT
cana-5423	129	24	(	(	PUNCT
cana-5423	129	25	ii	ii	NOUN
cana-5423	129	26	)	)	PUNCT
cana-5423	129	27	there	there	PRON
cana-5423	129	28	exists	exist	VERB
cana-5423	129	29	x0	x0	PROPN
cana-5423	129	30	∈	∈	PROPN
cana-5423	129	31	xsuch	xsuch	PROPN
cana-5423	129	32	that	that	SCONJ
cana-5423	129	33	α(x0	α(x0	ADJ
cana-5423	129	34	,	,	PUNCT
cana-5423	129	35	γx0	γx0	PROPN
cana-5423	129	36	)	)	PUNCT
cana-5423	129	37	≥	≥	NOUN
cana-5423	129	38	1	1	NUM
cana-5423	129	39	;	;	PUNCT
cana-5423	129	40	(	(	PUNCT
cana-5423	129	41	iii	iii	X
cana-5423	129	42	)	)	PUNCT
cana-5423	129	43	γ	γ	NOUN
cana-5423	129	44	is	be	AUX
cana-5423	129	45	continuous	continuous	ADJ
cana-5423	129	46	.	.	PUNCT
cana-5423	130	1	(	(	PUNCT
cana-5423	130	2	iv	iv	X
cana-5423	130	3	)	)	PUNCT
cana-5423	130	4	α(x	α(x	NOUN
cana-5423	130	5	,	,	PUNCT
cana-5423	130	6	γx	γx	NOUN
cana-5423	130	7	)	)	PUNCT
cana-5423	130	8	≥	≥	NOUN
cana-5423	130	9	1	1	NUM
cana-5423	130	10	then	then	ADV
cana-5423	130	11	γ	γ	PROPN
cana-5423	130	12	has	have	VERB
cana-5423	130	13	a	a	DET
cana-5423	130	14	unique	unique	ADJ
cana-5423	130	15	fixed	fix	VERB
cana-5423	130	16	point	point	NOUN
cana-5423	130	17	x∗	x∗	PROPN
cana-5423	130	18	∈	∈	PROPN
cana-5423	130	19	x.	x.	NOUN
cana-5423	130	20	proof	proof	NOUN
cana-5423	130	21	.	.	PUNCT
cana-5423	131	1	by(ii	by(ii	PROPN
cana-5423	131	2	)	)	PUNCT
cana-5423	131	3	,	,	PUNCT
cana-5423	131	4	there	there	PRON
cana-5423	131	5	exists	exist	VERB
cana-5423	131	6	x0	x0	PROPN
cana-5423	131	7	∈	∈	PROPN
cana-5423	131	8	x	x	PUNCT
cana-5423	131	9	such	such	ADJ
cana-5423	131	10	that	that	DET
cana-5423	131	11	α(x0.γx0	α(x0.γx0	NOUN
cana-5423	131	12	)	)	PUNCT
cana-5423	131	13	≥	≥	NOUN
cana-5423	132	1	1,and	1,and	NUM
cana-5423	132	2	let	let	VERB
cana-5423	132	3	{	{	PUNCT
cana-5423	132	4	xn	xn	VERB
cana-5423	132	5	}	}	PUNCT
cana-5423	132	6	be	be	VERB
cana-5423	132	7	the	the	DET
cana-5423	132	8	iterative	iterative	ADJ
cana-5423	132	9	sequence	sequence	NOUN
cana-5423	132	10	xdefined	xdefine	VERB
cana-5423	132	11	by	by	ADP
cana-5423	132	12	xn+1	xn+1	PROPN
cana-5423	132	13	=	=	SYM
cana-5423	132	14	γxn	γxn	PROPN
cana-5423	132	15	,	,	PUNCT
cana-5423	132	16	forall	forall	NOUN
cana-5423	132	17	n	n	CCONJ
cana-5423	132	18	∈	∈	PROPN
cana-5423	132	19	n	n	CCONJ
cana-5423	132	20	(	(	PUNCT
cana-5423	132	21	13	13	NUM
cana-5423	132	22	)	)	PUNCT
cana-5423	132	23	if	if	SCONJ
cana-5423	132	24	there	there	PRON
cana-5423	132	25	exists	exist	VERB
cana-5423	132	26	some	some	DET
cana-5423	132	27	nonnegative	nonnegative	ADJ
cana-5423	132	28	integer	integer	NOUN
cana-5423	132	29	n	n	PRON
cana-5423	132	30	such	such	ADJ
cana-5423	132	31	that	that	PRON
cana-5423	132	32	xn	xn	PUNCT
cana-5423	133	1	=	=	SYM
cana-5423	133	2	xn+1	xn+1	PROPN
cana-5423	133	3	=	=	SYM
cana-5423	133	4	γxn	γxn	PROPN
cana-5423	133	5	,	,	PUNCT
cana-5423	133	6	then	then	ADV
cana-5423	133	7	xn	xn	PROPN
cana-5423	133	8	is	be	AUX
cana-5423	133	9	a	a	DET
cana-5423	133	10	fixed	fix	VERB
cana-5423	133	11	point	point	NOUN
cana-5423	133	12	of	of	ADP
cana-5423	133	13	γ	γ	PROPN
cana-5423	133	14	.	.	PUNCT
cana-5423	133	15	therfore	therfore	ADV
cana-5423	133	16	,	,	PUNCT
cana-5423	133	17	to	to	PART
cana-5423	133	18	continue	continue	VERB
cana-5423	133	19	our	our	PRON
cana-5423	133	20	proof	proof	NOUN
cana-5423	133	21	,	,	PUNCT
cana-5423	133	22	we	we	PRON
cana-5423	133	23	assume	assume	VERB
cana-5423	133	24	that	that	SCONJ
cana-5423	133	25	xn	xn	PROPN
cana-5423	133	26	̸=	̸=	PROPN
cana-5423	133	27	xn+1	xn+1	PROPN
cana-5423	133	28	for	for	ADP
cana-5423	133	29	all	all	DET
cana-5423	133	30	n	n	PRON
cana-5423	133	31	∈	∈	PROPN
cana-5423	133	32	n.	n.	NOUN
cana-5423	133	33	since	since	SCONJ
cana-5423	133	34	γ	γ	PROPN
cana-5423	133	35	is	be	AUX
cana-5423	133	36	an	an	DET
cana-5423	133	37	αadmissible	αadmissible	ADJ
cana-5423	133	38	mapping	mapping	NOUN
cana-5423	133	39	,	,	PUNCT
cana-5423	133	40	we	we	PRON
cana-5423	133	41	have	have	VERB
cana-5423	133	42	α(x0	α(x0	ADJ
cana-5423	133	43	,	,	PUNCT
cana-5423	133	44	x1	x1	PROPN
cana-5423	133	45	)	)	PUNCT
cana-5423	133	46	=	=	SYM
cana-5423	133	47	α(x0,γx0	α(x0,γx0	PROPN
cana-5423	133	48	)	)	PUNCT
cana-5423	133	49	⇒	⇒	PROPN
cana-5423	133	50	α(γx0,γx1	α(γx0,γx1	PROPN
cana-5423	133	51	)	)	PUNCT
cana-5423	133	52	=	=	SYM
cana-5423	133	53	α(x1	α(x1	ADJ
cana-5423	133	54	,	,	PUNCT
cana-5423	133	55	x2	x2	PROPN
cana-5423	133	56	)	)	PUNCT
cana-5423	133	57	≥	≥	NOUN
cana-5423	133	58	1	1	NUM
cana-5423	133	59	.	.	PUNCT
cana-5423	134	1	(	(	PUNCT
cana-5423	134	2	14	14	NUM
cana-5423	134	3	)	)	PUNCT
cana-5423	134	4	by	by	ADP
cana-5423	134	5	induction	induction	NOUN
cana-5423	134	6	,	,	PUNCT
cana-5423	134	7	we	we	PRON
cana-5423	134	8	get	get	VERB
cana-5423	134	9	α(xn	α(xn	NOUN
cana-5423	134	10	,	,	PUNCT
cana-5423	134	11	xn+1	xn+1	NUM
cana-5423	134	12	)	)	PUNCT
cana-5423	134	13	≥	≥	NOUN
cana-5423	134	14	1	1	NUM
cana-5423	134	15	,	,	PUNCT
cana-5423	134	16	forall	forall	NOUN
cana-5423	134	17	n	n	CCONJ
cana-5423	134	18	∈	∈	PROPN
cana-5423	134	19	n	n	NOUN
cana-5423	134	20	∪	∪	X
cana-5423	134	21	{	{	PUNCT
cana-5423	134	22	0	0	NUM
cana-5423	134	23	}	}	PUNCT
cana-5423	134	24	.	.	PUNCT
cana-5423	135	1	(	(	PUNCT
cana-5423	135	2	15	15	X
cana-5423	135	3	)	)	PUNCT
cana-5423	135	4	applying	apply	VERB
cana-5423	135	5	the	the	DET
cana-5423	135	6	codition	codition	NOUN
cana-5423	135	7	(	(	PUNCT
cana-5423	135	8	10	10	NUM
cana-5423	135	9	)	)	PUNCT
cana-5423	135	10	,	,	PUNCT
cana-5423	135	11	putting	put	VERB
cana-5423	135	12	x	x	X
cana-5423	135	13	=	=	PUNCT
cana-5423	135	14	xn−1	xn−1	PROPN
cana-5423	135	15	and	and	CCONJ
cana-5423	135	16	y	y	PROPN
cana-5423	135	17	=	=	SYM
cana-5423	135	18	xn	xn	PROPN
cana-5423	135	19	and	and	CCONJ
cana-5423	135	20	by	by	ADP
cana-5423	135	21	using	use	VERB
cana-5423	135	22	(	(	PUNCT
cana-5423	135	23	15	15	NUM
cana-5423	135	24	)	)	PUNCT
cana-5423	135	25	,	,	PUNCT
cana-5423	135	26	we	we	PRON
cana-5423	135	27	have	have	VERB
cana-5423	135	28	0	0	NUM
cana-5423	135	29	≤	≤	NUM
cana-5423	135	30	ζ(α(xn−1,γxn−1	ζ(α(xn−1,γxn−1	PROPN
cana-5423	135	31	)	)	PUNCT
cana-5423	135	32	,	,	PUNCT
cana-5423	135	33	α(xn	α(xn	X
cana-5423	135	34	,	,	PUNCT
cana-5423	135	35	γxn	γxn	NOUN
cana-5423	135	36	)	)	PUNCT
cana-5423	135	37	,	,	PUNCT
cana-5423	135	38	d(γxn−1,γxn),k(xn−1	d(γxn−1,γxn),k(xn−1	PROPN
cana-5423	135	39	,	,	PUNCT
cana-5423	135	40	xn	xn	PUNCT
cana-5423	135	41	)	)	PUNCT
cana-5423	136	1	+	+	CCONJ
cana-5423	136	2	lq(xn−1	lq(xn−1	PROPN
cana-5423	136	3	,	,	PUNCT
cana-5423	136	4	xn	xn	NUM
cana-5423	136	5	)	)	PUNCT
cana-5423	136	6	)	)	PUNCT
cana-5423	137	1	=	=	SYM
cana-5423	137	2	ζ(α(xn−1	ζ(α(xn−1	PROPN
cana-5423	137	3	,	,	PUNCT
cana-5423	137	4	xn	xn	PROPN
cana-5423	137	5	)	)	PUNCT
cana-5423	137	6	,	,	PUNCT
cana-5423	137	7	α(xn	α(xn	X
cana-5423	137	8	,	,	PUNCT
cana-5423	137	9	xn+1	xn+1	NUM
cana-5423	137	10	)	)	PUNCT
cana-5423	137	11	,	,	PUNCT
cana-5423	137	12	d(xn	d(xn	PROPN
cana-5423	137	13	,	,	PUNCT
cana-5423	137	14	xn+1)k(xn−1	xn+1)k(xn−1	PROPN
cana-5423	137	15	,	,	PUNCT
cana-5423	137	16	xn	xn	PUNCT
cana-5423	137	17	)	)	PUNCT
cana-5423	138	1	+	+	CCONJ
cana-5423	138	2	lq(xn−1	lq(xn−1	PROPN
cana-5423	138	3	,	,	PUNCT
cana-5423	138	4	xn	xn	PUNCT
cana-5423	138	5	)	)	PUNCT
cana-5423	138	6	<	<	X
cana-5423	138	7	k(xn−1	k(xn−1	PROPN
cana-5423	138	8	,	,	PUNCT
cana-5423	138	9	xn	xn	PUNCT
cana-5423	138	10	)	)	PUNCT
cana-5423	139	1	+	+	CCONJ
cana-5423	139	2	lq(xn−1	lq(xn−1	PROPN
cana-5423	139	3	,	,	PUNCT
cana-5423	139	4	xn	xn	NUM
cana-5423	139	5	)	)	PUNCT
cana-5423	139	6	−	−	ADP
cana-5423	139	7	α(xn−1	α(xn−1	NUM
cana-5423	139	8	,	,	PUNCT
cana-5423	139	9	xn	xn	PROPN
cana-5423	139	10	)	)	PUNCT
cana-5423	139	11	,	,	PUNCT
cana-5423	139	12	α(xn	α(xn	X
cana-5423	139	13	,	,	PUNCT
cana-5423	139	14	xn+1	xn+1	NUM
cana-5423	139	15	)	)	PUNCT
cana-5423	139	16	,	,	PUNCT
cana-5423	139	17	d(xn	d(xn	PROPN
cana-5423	139	18	,	,	PUNCT
cana-5423	139	19	xn+1	xn+1	NUM
cana-5423	139	20	)	)	PUNCT
cana-5423	139	21	(	(	PUNCT
cana-5423	139	22	16	16	X
cana-5423	139	23	)	)	PUNCT
cana-5423	139	24	communications	communication	NOUN
cana-5423	139	25	on	on	ADP
cana-5423	139	26	applied	apply	VERB
cana-5423	139	27	nonlinear	nonlinear	ADJ
cana-5423	139	28	analysis	analysis	NOUN
cana-5423	139	29	issn	issn	NOUN
cana-5423	139	30	:	:	PUNCT
cana-5423	139	31	1074	1074	NUM
cana-5423	139	32	-	-	PUNCT
cana-5423	139	33	133x	133x	NUM
cana-5423	139	34	vol	vol	NOUN
cana-5423	139	35	32	32	NUM
cana-5423	139	36	no	no	NOUN
cana-5423	139	37	.	.	PUNCT
cana-5423	140	1	10s(2025	10s(2025	NUM
cana-5423	140	2	)	)	PUNCT
cana-5423	141	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	141	2	2221	2221	NUM
cana-5423	141	3	also	also	ADV
cana-5423	141	4	,	,	PUNCT
cana-5423	141	5	where	where	SCONJ
cana-5423	141	6	k(xn−1	k(xn−1	PROPN
cana-5423	141	7	,	,	PUNCT
cana-5423	141	8	xn	xn	PUNCT
cana-5423	141	9	)	)	PUNCT
cana-5423	142	1	=	=	PRON
cana-5423	142	2	max	max	X
cana-5423	142	3	{	{	PUNCT
cana-5423	142	4	d(xn−1	d(xn−1	PROPN
cana-5423	142	5	,	,	PUNCT
cana-5423	142	6	xn	xn	PROPN
cana-5423	142	7	)	)	PUNCT
cana-5423	142	8	,	,	PUNCT
cana-5423	142	9	[	[	X
cana-5423	142	10	1	1	NUM
cana-5423	142	11	+	+	CCONJ
cana-5423	142	12	d(xn−1,γxn−1)]d(xn	d(xn−1,γxn−1)]d(xn	NOUN
cana-5423	142	13	,	,	PUNCT
cana-5423	142	14	γxn	γxn	ADJ
cana-5423	142	15	)	)	PUNCT
cana-5423	142	16	1	1	NUM
cana-5423	143	1	+	+	CCONJ
cana-5423	143	2	d(x	d(x	PROPN
cana-5423	143	3	,	,	PUNCT
cana-5423	143	4	y	y	NOUN
cana-5423	143	5	)	)	PUNCT
cana-5423	143	6	,	,	PUNCT
cana-5423	143	7	[	[	X
cana-5423	143	8	1	1	NUM
cana-5423	143	9	+	+	CCONJ
cana-5423	143	10	d(xn−1,γxn)]d(xn	d(xn−1,γxn)]d(xn	NOUN
cana-5423	143	11	,	,	PUNCT
cana-5423	143	12	γxn−1	γxn−1	PROPN
cana-5423	143	13	)	)	PUNCT
cana-5423	143	14	1	1	NUM
cana-5423	144	1	+	+	CCONJ
cana-5423	144	2	d(x	d(x	PROPN
cana-5423	144	3	,	,	PUNCT
cana-5423	144	4	y	y	NOUN
cana-5423	144	5	)	)	PUNCT
cana-5423	144	6	}	}	PUNCT
cana-5423	144	7	≤	≤	NUM
cana-5423	144	8	max	max	PROPN
cana-5423	144	9	{	{	PUNCT
cana-5423	144	10	d(xn−1	d(xn−1	PROPN
cana-5423	144	11	,	,	PUNCT
cana-5423	144	12	xn	xn	PROPN
cana-5423	144	13	)	)	PUNCT
cana-5423	144	14	,	,	PUNCT
cana-5423	145	1	[	[	X
cana-5423	145	2	1	1	NUM
cana-5423	145	3	+	+	NUM
cana-5423	145	4	d(xn−1	d(xn−1	PROPN
cana-5423	145	5	,	,	PUNCT
cana-5423	145	6	xn)]d(xn	xn)]d(xn	PROPN
cana-5423	145	7	,	,	PUNCT
cana-5423	145	8	xn+1	xn+1	NUM
cana-5423	145	9	)	)	PUNCT
cana-5423	145	10	1	1	NUM
cana-5423	145	11	+	+	CCONJ
cana-5423	145	12	d(xn−1	d(xn−1	NOUN
cana-5423	145	13	,	,	PUNCT
cana-5423	145	14	xn	xn	PROPN
cana-5423	145	15	)	)	PUNCT
cana-5423	145	16	,	,	PUNCT
cana-5423	146	1	[	[	X
cana-5423	146	2	1	1	NUM
cana-5423	146	3	+	+	NUM
cana-5423	146	4	d(xn−1	d(xn−1	NOUN
cana-5423	146	5	,	,	PUNCT
cana-5423	146	6	xn+1)]d(xn	xn+1)]d(xn	PROPN
cana-5423	146	7	,	,	PUNCT
cana-5423	146	8	xn	xn	PROPN
cana-5423	146	9	)	)	PUNCT
cana-5423	146	10	1	1	NUM
cana-5423	146	11	+	+	CCONJ
cana-5423	146	12	d(xn−1	d(xn−1	NOUN
cana-5423	146	13	,	,	PUNCT
cana-5423	146	14	xn	xn	PROPN
cana-5423	146	15	)	)	PUNCT
cana-5423	146	16	}	}	PUNCT
cana-5423	146	17	=	=	SYM
cana-5423	146	18	max{d(xn−1	max{d(xn−1	ADJ
cana-5423	146	19	,	,	PUNCT
cana-5423	146	20	xn	xn	PROPN
cana-5423	146	21	)	)	PUNCT
cana-5423	146	22	,	,	PUNCT
cana-5423	146	23	d(xn	d(xn	PROPN
cana-5423	146	24	,	,	PUNCT
cana-5423	146	25	xn+1	xn+1	NUM
cana-5423	146	26	)	)	PUNCT
cana-5423	146	27	}	}	PUNCT
cana-5423	146	28	(	(	PUNCT
cana-5423	146	29	17	17	NUM
cana-5423	146	30	)	)	PUNCT
cana-5423	146	31	and	and	CCONJ
cana-5423	146	32	q(xn−1	q(xn−1	PROPN
cana-5423	146	33	,	,	PUNCT
cana-5423	146	34	xn	xn	PRON
cana-5423	146	35	)	)	PUNCT
cana-5423	146	36	=	=	SYM
cana-5423	146	37	min{d(xn−1,γxn−1	min{d(xn−1,γxn−1	ADJ
cana-5423	146	38	)	)	PUNCT
cana-5423	146	39	,	,	PUNCT
cana-5423	146	40	d(xn	d(xn	PROPN
cana-5423	146	41	,	,	PUNCT
cana-5423	146	42	γxn	γxn	ADJ
cana-5423	146	43	)	)	PUNCT
cana-5423	146	44	,	,	PUNCT
cana-5423	146	45	d(xn−1,γxn	d(xn−1,γxn	PROPN
cana-5423	146	46	)	)	PUNCT
cana-5423	146	47	,	,	PUNCT
cana-5423	146	48	d(xn	d(xn	PROPN
cana-5423	146	49	,	,	PUNCT
cana-5423	146	50	γxn−1	γxn−1	PROPN
cana-5423	146	51	)	)	PUNCT
cana-5423	146	52	,	,	PUNCT
cana-5423	146	53	d(xn−1,γxn)d(xn	d(xn−1,γxn)d(xn	PROPN
cana-5423	146	54	,	,	PUNCT
cana-5423	146	55	γxn−1	γxn−1	PROPN
cana-5423	146	56	)	)	PUNCT
cana-5423	146	57	1	1	NUM
cana-5423	147	1	+	+	CCONJ
cana-5423	147	2	d(xn−1	d(xn−1	NOUN
cana-5423	147	3	,	,	PUNCT
cana-5423	147	4	xn	xn	PROPN
cana-5423	147	5	)	)	PUNCT
cana-5423	147	6	,	,	PUNCT
cana-5423	147	7	d(xn−1,γxn−1)d(xn	d(xn−1,γxn−1)d(xn	X
cana-5423	147	8	,	,	PUNCT
cana-5423	147	9	γxn	γxn	ADJ
cana-5423	147	10	)	)	PUNCT
cana-5423	147	11	1	1	NUM
cana-5423	148	1	+	+	CCONJ
cana-5423	148	2	d(xn−1	d(xn−1	NOUN
cana-5423	148	3	,	,	PUNCT
cana-5423	148	4	xn	xn	PROPN
cana-5423	148	5	)	)	PUNCT
cana-5423	148	6	}	}	PUNCT
cana-5423	148	7	≤	≤	PROPN
cana-5423	148	8	min{d(xn−1	min{d(xn−1	PROPN
cana-5423	148	9	,	,	PUNCT
cana-5423	148	10	xn	xn	PROPN
cana-5423	148	11	)	)	PUNCT
cana-5423	148	12	,	,	PUNCT
cana-5423	148	13	d(xn	d(xn	PROPN
cana-5423	148	14	,	,	PUNCT
cana-5423	148	15	xn+1	xn+1	NUM
cana-5423	148	16	)	)	PUNCT
cana-5423	148	17	,	,	PUNCT
cana-5423	148	18	d(xn−1	d(xn−1	PROPN
cana-5423	148	19	,	,	PUNCT
cana-5423	148	20	xn+1	xn+1	NUM
cana-5423	148	21	)	)	PUNCT
cana-5423	148	22	,	,	PUNCT
cana-5423	148	23	d(xn	d(xn	PROPN
cana-5423	148	24	,	,	PUNCT
cana-5423	148	25	xn	xn	PROPN
cana-5423	148	26	)	)	PUNCT
cana-5423	148	27	,	,	PUNCT
cana-5423	148	28	d(xn−1	d(xn−1	PROPN
cana-5423	148	29	,	,	PUNCT
cana-5423	148	30	xn+1)d(xn	xn+1)d(xn	PROPN
cana-5423	148	31	,	,	PUNCT
cana-5423	148	32	xn	xn	PROPN
cana-5423	148	33	)	)	PUNCT
cana-5423	148	34	1	1	NUM
cana-5423	149	1	+	+	CCONJ
cana-5423	149	2	d(xn−1	d(xn−1	NOUN
cana-5423	149	3	,	,	PUNCT
cana-5423	149	4	xn	xn	PROPN
cana-5423	149	5	)	)	PUNCT
cana-5423	149	6	,	,	PUNCT
cana-5423	149	7	d(xn−1	d(xn−1	PROPN
cana-5423	149	8	,	,	PUNCT
cana-5423	149	9	xn)d(xn	xn)d(xn	PROPN
cana-5423	149	10	,	,	PUNCT
cana-5423	149	11	xn+1	xn+1	NUM
cana-5423	149	12	)	)	PUNCT
cana-5423	149	13	1	1	NUM
cana-5423	150	1	+	+	CCONJ
cana-5423	150	2	d(xn−1	d(xn−1	NOUN
cana-5423	150	3	,	,	PUNCT
cana-5423	150	4	xn	xn	PROPN
cana-5423	150	5	)	)	PUNCT
cana-5423	150	6	}	}	PUNCT
cana-5423	150	7	≤	≤	NUM
cana-5423	150	8	min	min	NOUN
cana-5423	150	9	{	{	PUNCT
cana-5423	150	10	d(xn−1	d(xn−1	PROPN
cana-5423	150	11	,	,	PUNCT
cana-5423	150	12	xn	xn	PROPN
cana-5423	150	13	)	)	PUNCT
cana-5423	150	14	,	,	PUNCT
cana-5423	150	15	d(xn	d(xn	PROPN
cana-5423	150	16	,	,	PUNCT
cana-5423	150	17	xn+1	xn+1	NUM
cana-5423	150	18	)	)	PUNCT
cana-5423	150	19	,	,	PUNCT
cana-5423	150	20	d(xn−1	d(xn−1	PROPN
cana-5423	150	21	,	,	PUNCT
cana-5423	150	22	xn+1	xn+1	NUM
cana-5423	150	23	)	)	PUNCT
cana-5423	150	24	,	,	PUNCT
cana-5423	150	25	0	0	NUM
cana-5423	150	26	,	,	PUNCT
cana-5423	150	27	0	0	NUM
cana-5423	150	28	,	,	PUNCT
cana-5423	150	29	d(xn−1	d(xn−1	PROPN
cana-5423	150	30	,	,	PUNCT
cana-5423	150	31	xn)d(xn	xn)d(xn	PROPN
cana-5423	150	32	,	,	PUNCT
cana-5423	150	33	xn+1	xn+1	NUM
cana-5423	150	34	)	)	PUNCT
cana-5423	150	35	1	1	NUM
cana-5423	151	1	+	+	CCONJ
cana-5423	151	2	d(xn−1	d(xn−1	NOUN
cana-5423	151	3	,	,	PUNCT
cana-5423	151	4	xn	xn	PROPN
cana-5423	151	5	)	)	PUNCT
cana-5423	151	6	}	}	PUNCT
cana-5423	152	1	=	=	NOUN
cana-5423	152	2	0	0	NUM
cana-5423	152	3	.	.	PUNCT
cana-5423	153	1	(	(	PUNCT
cana-5423	153	2	18	18	NUM
cana-5423	153	3	)	)	PUNCT
cana-5423	153	4	by	by	ADP
cana-5423	153	5	(	(	PUNCT
cana-5423	153	6	16	16	NUM
cana-5423	153	7	)	)	PUNCT
cana-5423	153	8	,	,	PUNCT
cana-5423	153	9	and	and	CCONJ
cana-5423	153	10	taking	take	VERB
cana-5423	153	11	in	in	ADP
cana-5423	153	12	account	account	NOUN
cana-5423	153	13	(	(	PUNCT
cana-5423	153	14	15),(17	15),(17	NUM
cana-5423	153	15	)	)	PUNCT
cana-5423	153	16	,	,	PUNCT
cana-5423	153	17	and(18	and(18	PROPN
cana-5423	153	18	)	)	PUNCT
cana-5423	153	19	,	,	PUNCT
cana-5423	153	20	we	we	PRON
cana-5423	153	21	derive	derive	VERB
cana-5423	153	22	that	that	SCONJ
cana-5423	153	23	0	0	NUM
cana-5423	153	24	<	<	X
cana-5423	153	25	max{d(xn−1	max{d(xn−1	PROPN
cana-5423	153	26	,	,	PUNCT
cana-5423	153	27	xn	xn	PROPN
cana-5423	153	28	)	)	PUNCT
cana-5423	153	29	,	,	PUNCT
cana-5423	153	30	d(xn	d(xn	PROPN
cana-5423	153	31	,	,	PUNCT
cana-5423	153	32	xn+1	xn+1	NUM
cana-5423	153	33	)	)	PUNCT
cana-5423	153	34	}	}	PUNCT
cana-5423	153	35	−	−	ADP
cana-5423	153	36	α(xn+1	α(xn+1	NUM
cana-5423	153	37	,	,	PUNCT
cana-5423	153	38	xn	xn	PROPN
cana-5423	153	39	)	)	PUNCT
cana-5423	153	40	,	,	PUNCT
cana-5423	153	41	α(xn	α(xn	PROPN
cana-5423	153	42	,	,	PUNCT
cana-5423	153	43	xn+1)d(xn	xn+1)d(xn	PROPN
cana-5423	153	44	,	,	PUNCT
cana-5423	153	45	xn+1	xn+1	NUM
cana-5423	153	46	)	)	PUNCT
cana-5423	153	47	(	(	PUNCT
cana-5423	153	48	19	19	NUM
cana-5423	153	49	)	)	PUNCT
cana-5423	153	50	which	which	PRON
cana-5423	153	51	implies	imply	VERB
cana-5423	153	52	that	that	SCONJ
cana-5423	153	53	d(xn	d(xn	PROPN
cana-5423	153	54	,	,	PUNCT
cana-5423	153	55	xn+1	xn+1	ADV
cana-5423	153	56	≤	≤	NUM
cana-5423	153	57	α(xn−1	α(xn−1	NUM
cana-5423	153	58	,	,	PUNCT
cana-5423	153	59	xn	xn	PROPN
cana-5423	153	60	)	)	PUNCT
cana-5423	153	61	,	,	PUNCT
cana-5423	153	62	α(xn	α(xn	PROPN
cana-5423	153	63	,	,	PUNCT
cana-5423	153	64	xn+1)d(xn	xn+1)d(xn	PROPN
cana-5423	153	65	,	,	PUNCT
cana-5423	153	66	xn+1	xn+1	NUM
cana-5423	153	67	)	)	PUNCT
cana-5423	153	68	<	<	X
cana-5423	153	69	max{d(xn−1	max{d(xn−1	PROPN
cana-5423	153	70	,	,	PUNCT
cana-5423	153	71	xn	xn	PROPN
cana-5423	153	72	)	)	PUNCT
cana-5423	153	73	,	,	PUNCT
cana-5423	153	74	d(xn	d(xn	PROPN
cana-5423	153	75	,	,	PUNCT
cana-5423	153	76	xn+1	xn+1	NUM
cana-5423	153	77	)	)	PUNCT
cana-5423	153	78	}	}	PUNCT
cana-5423	153	79	∀n	∀n	NUM
cana-5423	153	80	≥	≥	NOUN
cana-5423	153	81	1	1	NUM
cana-5423	153	82	.	.	PUNCT
cana-5423	154	1	(	(	PUNCT
cana-5423	154	2	20	20	NUM
cana-5423	154	3	)	)	PUNCT
cana-5423	154	4	if	if	SCONJ
cana-5423	154	5	max	max	PROPN
cana-5423	154	6	{	{	PUNCT
cana-5423	154	7	d(xn−1	d(xn−1	PROPN
cana-5423	154	8	,	,	PUNCT
cana-5423	154	9	xn	xn	PROPN
cana-5423	154	10	)	)	PUNCT
cana-5423	154	11	,	,	PUNCT
cana-5423	154	12	d(xn	d(xn	PROPN
cana-5423	154	13	,	,	PUNCT
cana-5423	154	14	xn+1	xn+1	NUM
cana-5423	154	15	)	)	PUNCT
cana-5423	154	16	}	}	PUNCT
cana-5423	154	17	=	=	PUNCT
cana-5423	155	1	d(xn	d(xn	X
cana-5423	155	2	,	,	PUNCT
cana-5423	155	3	xn+1	xn+1	NUM
cana-5423	155	4	)	)	PUNCT
cana-5423	155	5	for	for	ADP
cana-5423	155	6	some	some	DET
cana-5423	155	7	n	n	PRON
cana-5423	155	8	≥	≥	NOUN
cana-5423	155	9	1	1	NUM
cana-5423	155	10	,	,	PUNCT
cana-5423	155	11	then	then	ADV
cana-5423	155	12	from	from	ADP
cana-5423	155	13	(	(	PUNCT
cana-5423	155	14	20	20	NUM
cana-5423	155	15	)	)	PUNCT
cana-5423	155	16	,	,	PUNCT
cana-5423	155	17	we	we	PRON
cana-5423	155	18	get	get	VERB
cana-5423	155	19	d(xn	d(xn	NOUN
cana-5423	155	20	,	,	PUNCT
cana-5423	155	21	xn+1	xn+1	NUM
cana-5423	155	22	)	)	PUNCT
cana-5423	155	23	≤	≤	NOUN
cana-5423	155	24	α(xn−1	α(xn−1	NUM
cana-5423	155	25	,	,	PUNCT
cana-5423	155	26	xn	xn	PROPN
cana-5423	155	27	)	)	PUNCT
cana-5423	155	28	,	,	PUNCT
cana-5423	155	29	α(xn	α(xn	PROPN
cana-5423	155	30	,	,	PUNCT
cana-5423	155	31	xn+1)d(xn	xn+1)d(xn	PROPN
cana-5423	155	32	,	,	PUNCT
cana-5423	155	33	xn+1	xn+1	NUM
cana-5423	155	34	)	)	PUNCT
cana-5423	155	35	<	<	X
cana-5423	156	1	d(xn	d(xn	PROPN
cana-5423	156	2	,	,	PUNCT
cana-5423	156	3	xn+1	xn+1	NUM
cana-5423	156	4	)	)	PUNCT
cana-5423	156	5	.	.	PUNCT
cana-5423	157	1	(	(	PUNCT
cana-5423	157	2	21	21	NUM
cana-5423	157	3	)	)	PUNCT
cana-5423	157	4	which	which	PRON
cana-5423	157	5	is	be	AUX
cana-5423	157	6	contradiction	contradiction	NOUN
cana-5423	157	7	,	,	PUNCT
cana-5423	157	8	therefore	therefore	ADV
cana-5423	157	9	,	,	PUNCT
cana-5423	157	10	max{d(xn−1	max{d(xn−1	PROPN
cana-5423	157	11	,	,	PUNCT
cana-5423	157	12	xn	xn	PROPN
cana-5423	157	13	)	)	PUNCT
cana-5423	157	14	,	,	PUNCT
cana-5423	157	15	d(xn	d(xn	PROPN
cana-5423	157	16	,	,	PUNCT
cana-5423	157	17	xn+1	xn+1	NUM
cana-5423	157	18	)	)	PUNCT
cana-5423	157	19	}	}	PUNCT
cana-5423	157	20	=	=	SYM
cana-5423	157	21	d(xn−1	d(xn−1	NOUN
cana-5423	157	22	,	,	PUNCT
cana-5423	157	23	xn	xn	PROPN
cana-5423	157	24	)	)	PUNCT
cana-5423	157	25	(	(	PUNCT
cana-5423	157	26	22	22	NUM
cana-5423	157	27	)	)	PUNCT
cana-5423	157	28	hence	hence	ADV
cana-5423	157	29	d(xn	d(xn	PROPN
cana-5423	157	30	,	,	PUNCT
cana-5423	157	31	xn+1	xn+1	NUM
cana-5423	157	32	)	)	PUNCT
cana-5423	157	33	≤	≤	NOUN
cana-5423	157	34	α(xn−1	α(xn−1	NUM
cana-5423	157	35	,	,	PUNCT
cana-5423	157	36	xn	xn	PROPN
cana-5423	157	37	)	)	PUNCT
cana-5423	157	38	,	,	PUNCT
cana-5423	157	39	α(xn	α(xn	X
cana-5423	157	40	,	,	PUNCT
cana-5423	157	41	xn+1	xn+1	NUM
cana-5423	157	42	)	)	PUNCT
cana-5423	157	43	,	,	PUNCT
cana-5423	157	44	d(xn	d(xn	PROPN
cana-5423	157	45	,	,	PUNCT
cana-5423	157	46	xn+1	xn+1	NUM
cana-5423	157	47	)	)	PUNCT
cana-5423	157	48	<	<	X
cana-5423	157	49	d(xn−1	d(xn−1	PROPN
cana-5423	157	50	,	,	PUNCT
cana-5423	157	51	xn	xn	PROPN
cana-5423	157	52	)	)	PUNCT
cana-5423	157	53	.	.	PUNCT
cana-5423	158	1	(	(	PUNCT
cana-5423	158	2	23	23	NUM
cana-5423	158	3	)	)	PUNCT
cana-5423	158	4	consequently	consequently	ADV
cana-5423	158	5	,	,	PUNCT
cana-5423	158	6	we	we	PRON
cana-5423	158	7	deduce	deduce	VERB
cana-5423	158	8	that	that	SCONJ
cana-5423	158	9	{	{	PUNCT
cana-5423	158	10	d(xn−1	d(xn−1	PROPN
cana-5423	158	11	,	,	PUNCT
cana-5423	158	12	xn	xn	PROPN
cana-5423	158	13	)	)	PUNCT
cana-5423	158	14	}	}	PUNCT
cana-5423	158	15	is	be	AUX
cana-5423	158	16	a	a	DET
cana-5423	158	17	monotonically	monotonically	ADV
cana-5423	158	18	decreasing	decrease	VERB
cana-5423	158	19	sequence	sequence	NOUN
cana-5423	158	20	for	for	ADP
cana-5423	158	21	nonnegative	nonnegative	ADJ
cana-5423	158	22	reals	real	NOUN
cana-5423	158	23	and	and	CCONJ
cana-5423	158	24	bounded	bound	VERB
cana-5423	158	25	below	below	ADV
cana-5423	158	26	by	by	ADP
cana-5423	158	27	zero	zero	NUM
cana-5423	158	28	.	.	PUNCT
cana-5423	159	1	so	so	ADV
cana-5423	159	2	,	,	PUNCT
cana-5423	159	3	there	there	PRON
cana-5423	159	4	esists	esist	VERB
cana-5423	159	5	r	r	NOUN
cana-5423	159	6	≥	≥	NOUN
cana-5423	159	7	0	0	NUM
cana-5423	159	8	such	such	ADJ
cana-5423	159	9	that	that	SCONJ
cana-5423	159	10	lim	lim	PROPN
cana-5423	159	11	n→∞	n→∞	PRON
cana-5423	159	12	d(xn−1	d(xn−1	NOUN
cana-5423	159	13	,	,	PUNCT
cana-5423	159	14	xn	xn	PUNCT
cana-5423	159	15	)	)	PUNCT
cana-5423	160	1	=	=	SYM
cana-5423	160	2	r	r	NOUN
cana-5423	160	3	(	(	PUNCT
cana-5423	160	4	24	24	NUM
cana-5423	160	5	)	)	PUNCT
cana-5423	160	6	communications	communication	NOUN
cana-5423	160	7	on	on	ADP
cana-5423	160	8	applied	apply	VERB
cana-5423	160	9	nonlinear	nonlinear	ADJ
cana-5423	160	10	analysis	analysis	NOUN
cana-5423	160	11	issn	issn	NOUN
cana-5423	160	12	:	:	PUNCT
cana-5423	160	13	1074	1074	NUM
cana-5423	160	14	-	-	PUNCT
cana-5423	160	15	133x	133x	NUM
cana-5423	160	16	vol	vol	NOUN
cana-5423	160	17	32	32	NUM
cana-5423	160	18	no	no	NOUN
cana-5423	160	19	.	.	PUNCT
cana-5423	161	1	10s(2025	10s(2025	NUM
cana-5423	161	2	)	)	PUNCT
cana-5423	162	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	162	2	2222	2222	NUM
cana-5423	162	3	we	we	PRON
cana-5423	162	4	claim	claim	VERB
cana-5423	162	5	that	that	SCONJ
cana-5423	162	6	lim	lim	PROPN
cana-5423	162	7	n→∞	n→∞	PRON
cana-5423	162	8	d(xn−1	d(xn−1	NOUN
cana-5423	162	9	,	,	PUNCT
cana-5423	162	10	xn	xn	PRON
cana-5423	162	11	)	)	PUNCT
cana-5423	162	12	=	=	SYM
cana-5423	163	1	0	0	X
cana-5423	163	2	.	.	PUNCT
cana-5423	164	1	(	(	PUNCT
cana-5423	164	2	25	25	NUM
cana-5423	164	3	)	)	PUNCT
cana-5423	164	4	(	(	PUNCT
cana-5423	164	5	26	26	NUM
cana-5423	164	6	)	)	PUNCT
cana-5423	164	7	on	on	ADP
cana-5423	164	8	the	the	DET
cana-5423	164	9	contrary	contrary	NOUN
cana-5423	164	10	,	,	PUNCT
cana-5423	164	11	assume	assume	VERB
cana-5423	164	12	that	that	SCONJ
cana-5423	164	13	r	r	NOUN
cana-5423	164	14	>	>	X
cana-5423	164	15	0	0	PUNCT
cana-5423	164	16	and	and	CCONJ
cana-5423	164	17	using	use	VERB
cana-5423	164	18	equation(23	equation(23	NOUN
cana-5423	164	19	)	)	PUNCT
cana-5423	164	20	we	we	PRON
cana-5423	164	21	have	have	VERB
cana-5423	164	22	the	the	DET
cana-5423	164	23	following	follow	VERB
cana-5423	164	24	lim	lim	PROPN
cana-5423	164	25	n→∞	n→∞	NUM
cana-5423	164	26	α(xn−1	α(xn−1	NUM
cana-5423	164	27	,	,	PUNCT
cana-5423	164	28	xn	xn	NUM
cana-5423	164	29	)	)	PUNCT
cana-5423	164	30	,	,	PUNCT
cana-5423	164	31	α(xn	α(xn	X
cana-5423	164	32	,	,	PUNCT
cana-5423	164	33	xn+1	xn+1	NUM
cana-5423	164	34	)	)	PUNCT
cana-5423	164	35	,	,	PUNCT
cana-5423	164	36	d(xn	d(xn	PROPN
cana-5423	164	37	,	,	PUNCT
cana-5423	164	38	xn+1	xn+1	NUM
cana-5423	164	39	)	)	PUNCT
cana-5423	165	1	=	=	SYM
cana-5423	165	2	r	r	NOUN
cana-5423	165	3	(	(	PUNCT
cana-5423	165	4	27	27	NUM
cana-5423	165	5	)	)	PUNCT
cana-5423	165	6	now	now	ADV
cana-5423	165	7	,	,	PUNCT
cana-5423	165	8	we	we	PRON
cana-5423	165	9	take	take	VERB
cana-5423	165	10	tn	tn	NOUN
cana-5423	165	11	=	=	SYM
cana-5423	165	12	{	{	PUNCT
cana-5423	165	13	α(xn−1	α(xn−1	PROPN
cana-5423	165	14	,	,	PUNCT
cana-5423	165	15	xn	xn	PROPN
cana-5423	165	16	)	)	PUNCT
cana-5423	165	17	,	,	PUNCT
cana-5423	165	18	α(xn	α(xn	X
cana-5423	165	19	,	,	PUNCT
cana-5423	165	20	xn+1	xn+1	NUM
cana-5423	165	21	)	)	PUNCT
cana-5423	165	22	,	,	PUNCT
cana-5423	165	23	d(xn	d(xn	PROPN
cana-5423	165	24	,	,	PUNCT
cana-5423	165	25	xn+1	xn+1	NUM
cana-5423	165	26	)	)	PUNCT
cana-5423	165	27	and	and	CCONJ
cana-5423	165	28	sn	sn	NOUN
cana-5423	165	29	=	=	SYM
cana-5423	165	30	{	{	PUNCT
cana-5423	165	31	d(xn−1	d(xn−1	PROPN
cana-5423	165	32	,	,	PUNCT
cana-5423	165	33	xn	xn	PROPN
cana-5423	165	34	)	)	PUNCT
cana-5423	165	35	}	}	PUNCT
cana-5423	165	36	.	.	PUNCT
cana-5423	166	1	then	then	ADV
cana-5423	166	2	lim	lim	PROPN
cana-5423	166	3	n→∞	n→∞	NUM
cana-5423	166	4	tn	tn	PROPN
cana-5423	166	5	=	=	SYM
cana-5423	166	6	lim	lim	PROPN
cana-5423	166	7	n→∞	n→∞	X
cana-5423	166	8	sn	sn	PROPN
cana-5423	166	9	=	=	SYM
cana-5423	166	10	r	r	NOUN
cana-5423	166	11	(	(	PUNCT
cana-5423	166	12	28	28	NUM
cana-5423	166	13	)	)	PUNCT
cana-5423	166	14	since	since	SCONJ
cana-5423	166	15	t	t	PROPN
cana-5423	166	16	is	be	AUX
cana-5423	166	17	a	a	DET
cana-5423	166	18	generalized	generalized	ADJ
cana-5423	166	19	αadmissible	αadmissible	ADJ
cana-5423	166	20	modified	modify	VERB
cana-5423	166	21	almost	almost	ADV
cana-5423	166	22	zcontraction	zcontraction	NOUN
cana-5423	166	23	with	with	ADP
cana-5423	166	24	respect	respect	NOUN
cana-5423	166	25	to	to	ADP
cana-5423	166	26	ζ	ζ	SYM
cana-5423	166	27	∈	∈	NOUN
cana-5423	166	28	z.	z.	X
cana-5423	166	29	therfore	therfore	ADJ
cana-5423	166	30	,	,	PUNCT
cana-5423	166	31	by(ζ3	by(ζ3	NOUN
cana-5423	166	32	)	)	PUNCT
cana-5423	166	33	and	and	CCONJ
cana-5423	166	34	equation	equation	NOUN
cana-5423	166	35	(	(	PUNCT
cana-5423	166	36	21	21	NUM
cana-5423	166	37	)	)	PUNCT
cana-5423	166	38	and	and	CCONJ
cana-5423	166	39	taking	take	VERB
cana-5423	166	40	limit	limit	NOUN
cana-5423	166	41	as	as	ADP
cana-5423	166	42	n	n	PROPN
cana-5423	166	43	→	→	SYM
cana-5423	166	44	∞	∞	PROPN
cana-5423	166	45	,	,	PUNCT
cana-5423	166	46	we	we	PRON
cana-5423	166	47	have	have	VERB
cana-5423	166	48	lim	lim	PROPN
cana-5423	166	49	supn→∞	supn→∞	PROPN
cana-5423	166	50	ζ(tn	ζ(tn	PROPN
cana-5423	166	51	,	,	PUNCT
cana-5423	166	52	sn	sn	PROPN
cana-5423	166	53	)	)	PUNCT
cana-5423	166	54	<	<	X
cana-5423	166	55	0	0	PUNCT
cana-5423	166	56	i.e.	i.e.	X
cana-5423	166	57	0	0	NUM
cana-5423	166	58	≤	≤	NOUN
cana-5423	166	59	lim	lim	PROPN
cana-5423	166	60	supn→∞	supn→∞	PROPN
cana-5423	166	61	ζ(α(xn−1	ζ(α(xn−1	PROPN
cana-5423	166	62	,	,	PUNCT
cana-5423	166	63	xn	xn	PROPN
cana-5423	166	64	)	)	PUNCT
cana-5423	166	65	,	,	PUNCT
cana-5423	166	66	α(xn	α(xn	X
cana-5423	166	67	,	,	PUNCT
cana-5423	166	68	xn+1	xn+1	NUM
cana-5423	166	69	)	)	PUNCT
cana-5423	166	70	,	,	PUNCT
cana-5423	166	71	d(xn	d(xn	PROPN
cana-5423	166	72	,	,	PUNCT
cana-5423	166	73	xn+1	xn+1	NUM
cana-5423	166	74	)	)	PUNCT
cana-5423	166	75	,	,	PUNCT
cana-5423	166	76	d(xn−1	d(xn−1	PROPN
cana-5423	166	77	,	,	PUNCT
cana-5423	166	78	xn	xn	PUNCT
cana-5423	166	79	)	)	PUNCT
cana-5423	166	80	<	<	X
cana-5423	166	81	0	0	PUNCT
cana-5423	166	82	(	(	PUNCT
cana-5423	166	83	29	29	NUM
cana-5423	166	84	)	)	PUNCT
cana-5423	166	85	this	this	PRON
cana-5423	166	86	is	be	AUX
cana-5423	166	87	a	a	DET
cana-5423	166	88	contradiction	contradiction	NOUN
cana-5423	166	89	.	.	PUNCT
cana-5423	167	1	then	then	ADV
cana-5423	167	2	we	we	PRON
cana-5423	167	3	deduce	deduce	VERB
cana-5423	167	4	that	that	SCONJ
cana-5423	167	5	r	r	NOUN
cana-5423	167	6	=	=	SYM
cana-5423	167	7	0	0	NUM
cana-5423	167	8	,	,	PUNCT
cana-5423	167	9	that	that	ADV
cana-5423	167	10	is	is	ADV
cana-5423	167	11	,	,	PUNCT
cana-5423	167	12	we	we	PRON
cana-5423	167	13	have	have	AUX
cana-5423	167	14	following	follow	VERB
cana-5423	167	15	lim	lim	PROPN
cana-5423	167	16	n→∞	n→∞	X
cana-5423	167	17	d(xn−1	d(xn−1	NOUN
cana-5423	167	18	,	,	PUNCT
cana-5423	167	19	xn	xn	PROPN
cana-5423	167	20	)	)	PUNCT
cana-5423	167	21	=	=	SYM
cana-5423	167	22	0	0	NUM
cana-5423	167	23	(	(	PUNCT
cana-5423	167	24	30	30	NUM
cana-5423	167	25	)	)	PUNCT
cana-5423	167	26	now	now	ADV
cana-5423	167	27	,	,	PUNCT
cana-5423	167	28	we	we	PRON
cana-5423	167	29	will	will	AUX
cana-5423	167	30	show	show	VERB
cana-5423	167	31	that	that	DET
cana-5423	167	32	sequence	sequence	NOUN
cana-5423	167	33	{	{	PUNCT
cana-5423	167	34	xn	xn	PUNCT
cana-5423	167	35	}	}	PUNCT
cana-5423	167	36	is	be	AUX
cana-5423	167	37	acauchy	acauchy	ADJ
cana-5423	167	38	sequence	sequence	NOUN
cana-5423	167	39	in	in	ADP
cana-5423	167	40	x.assume	x.assume	NOUN
cana-5423	167	41	that	that	SCONJ
cana-5423	167	42	{	{	PUNCT
cana-5423	167	43	xn	xn	X
cana-5423	167	44	}	}	PUNCT
cana-5423	167	45	is	be	AUX
cana-5423	167	46	not	not	PART
cana-5423	167	47	a	a	DET
cana-5423	167	48	cauchy	cauchy	ADJ
cana-5423	167	49	sequence	sequence	NOUN
cana-5423	167	50	,	,	PUNCT
cana-5423	167	51	then	then	ADV
cana-5423	167	52	there	there	PRON
cana-5423	167	53	exists	exist	VERB
cana-5423	167	54	ϵ	ϵ	X
cana-5423	167	55	>	>	X
cana-5423	167	56	0	0	NUM
cana-5423	168	1	and	and	CCONJ
cana-5423	168	2	two	two	NUM
cana-5423	168	3	sequences	sequence	NOUN
cana-5423	168	4	{	{	PUNCT
cana-5423	168	5	xnk	xnk	NOUN
cana-5423	168	6	}	}	PUNCT
cana-5423	168	7	,	,	PUNCT
cana-5423	168	8	{	{	PUNCT
cana-5423	168	9	xmk	xmk	PROPN
cana-5423	168	10	}	}	PUNCT
cana-5423	168	11	:	:	PUNCT
cana-5423	168	12	mk	mk	PROPN
cana-5423	168	13	>	>	X
cana-5423	168	14	nk	nk	PROPN
cana-5423	168	15	>	>	X
cana-5423	168	16	k	k	X
cana-5423	168	17	such	such	ADJ
cana-5423	168	18	that	that	DET
cana-5423	168	19	d(xmk	d(xmk	NOUN
cana-5423	168	20	,	,	PUNCT
cana-5423	168	21	xnk	xnk	PROPN
cana-5423	168	22	)	)	PUNCT
cana-5423	168	23	≤	≤	NUM
cana-5423	169	1	ϵ.	ϵ.	NOUN
cana-5423	169	2	(	(	PUNCT
cana-5423	169	3	31	31	NUM
cana-5423	169	4	)	)	PUNCT
cana-5423	169	5	and	and	CCONJ
cana-5423	169	6	d(xmk	d(xmk	PROPN
cana-5423	169	7	,	,	PUNCT
cana-5423	169	8	xnk−1	xnk−1	PROPN
cana-5423	169	9	)	)	PUNCT
cana-5423	169	10	≤	≤	NOUN
cana-5423	170	1	ϵ	ϵ	ADP
cana-5423	170	2	,	,	PUNCT
cana-5423	170	3	for	for	ADP
cana-5423	170	4	all	all	DET
cana-5423	170	5	m	m	PROPN
cana-5423	170	6	,	,	PUNCT
cana-5423	170	7	n	n	CCONJ
cana-5423	170	8	,	,	PUNCT
cana-5423	170	9	k	k	PROPN
cana-5423	170	10	∈	∈	PROPN
cana-5423	170	11	n	n	CCONJ
cana-5423	170	12	(	(	PUNCT
cana-5423	170	13	32	32	NUM
cana-5423	170	14	)	)	PUNCT
cana-5423	170	15	by	by	ADP
cana-5423	170	16	applying	apply	VERB
cana-5423	170	17	the	the	DET
cana-5423	170	18	triangal	triangal	ADJ
cana-5423	170	19	inequality	inequality	NOUN
cana-5423	170	20	and	and	CCONJ
cana-5423	170	21	using	use	VERB
cana-5423	170	22	equations	equation	NOUN
cana-5423	170	23	(	(	PUNCT
cana-5423	170	24	30	30	NUM
cana-5423	170	25	)	)	PUNCT
cana-5423	170	26	and	and	CCONJ
cana-5423	170	27	(	(	PUNCT
cana-5423	170	28	31	31	NUM
cana-5423	170	29	)	)	PUNCT
cana-5423	170	30	,	,	PUNCT
cana-5423	170	31	we	we	PRON
cana-5423	170	32	get	get	VERB
cana-5423	170	33	the	the	DET
cana-5423	170	34	following	following	NOUN
cana-5423	170	35	ϵ	ϵ	X
cana-5423	170	36	<	<	X
cana-5423	170	37	d(xmk	d(xmk	PROPN
cana-5423	170	38	,	,	PUNCT
cana-5423	170	39	xnk	xnk	PROPN
cana-5423	170	40	)	)	PUNCT
cana-5423	170	41	≤	≤	NUM
cana-5423	170	42	d(xmk	d(xmk	NOUN
cana-5423	170	43	,	,	PUNCT
cana-5423	170	44	xnk−1	xnk−1	PROPN
cana-5423	170	45	)	)	PUNCT
cana-5423	171	1	+	+	SYM
cana-5423	171	2	d(xnk−1	d(xnk−1	NOUN
cana-5423	171	3	,	,	PUNCT
cana-5423	171	4	xnk	xnk	NOUN
cana-5423	171	5	)	)	PUNCT
cana-5423	171	6	≤	≤	PROPN
cana-5423	171	7	d(xnk−1	d(xnk−1	PROPN
cana-5423	171	8	,	,	PUNCT
cana-5423	171	9	xnk	xnk	PROPN
cana-5423	171	10	)	)	PUNCT
cana-5423	172	1	+	+	CCONJ
cana-5423	172	2	ϵ	ϵ	X
cana-5423	172	3	(	(	PUNCT
cana-5423	172	4	33	33	NUM
cana-5423	172	5	)	)	PUNCT
cana-5423	172	6	taking	take	VERB
cana-5423	172	7	k	k	X
cana-5423	172	8	→	→	SYM
cana-5423	172	9	∞	∞	NUM
cana-5423	172	10	in	in	ADP
cana-5423	172	11	equation	equation	NOUN
cana-5423	172	12	(	(	PUNCT
cana-5423	172	13	33	33	NUM
cana-5423	172	14	)	)	PUNCT
cana-5423	172	15	and	and	CCONJ
cana-5423	172	16	using	use	VERB
cana-5423	172	17	equation(30	equation(30	NOUN
cana-5423	172	18	)	)	PUNCT
cana-5423	172	19	,	,	PUNCT
cana-5423	172	20	we	we	PRON
cana-5423	172	21	get	get	VERB
cana-5423	172	22	lim	lim	PROPN
cana-5423	172	23	n→∞	n→∞	NUM
cana-5423	173	1	d(xmk	d(xmk	PROPN
cana-5423	173	2	,	,	PUNCT
cana-5423	173	3	xnk	xnk	PROPN
cana-5423	173	4	)	)	PUNCT
cana-5423	174	1	=	=	PUNCT
cana-5423	174	2	ϵ.	ϵ.	NOUN
cana-5423	174	3	(	(	PUNCT
cana-5423	174	4	34	34	NUM
cana-5423	174	5	)	)	PUNCT
cana-5423	174	6	again	again	ADV
cana-5423	174	7	,	,	PUNCT
cana-5423	174	8	using	use	VERB
cana-5423	174	9	the	the	DET
cana-5423	174	10	triangal	triangal	ADJ
cana-5423	174	11	inequality	inequality	NOUN
cana-5423	174	12	,	,	PUNCT
cana-5423	174	13	we	we	PRON
cana-5423	174	14	have	have	VERB
cana-5423	174	15	d(xmk	d(xmk	NOUN
cana-5423	174	16	,	,	PUNCT
cana-5423	174	17	xnk	xnk	PROPN
cana-5423	174	18	)	)	PUNCT
cana-5423	174	19	≤	≤	NUM
cana-5423	174	20	d(xmk	d(xmk	NOUN
cana-5423	174	21	,	,	PUNCT
cana-5423	174	22	xnk−1	xnk−1	PROPN
cana-5423	174	23	)	)	PUNCT
cana-5423	175	1	+	+	SYM
cana-5423	175	2	d(xnk−1	d(xnk−1	NOUN
cana-5423	175	3	,	,	PUNCT
cana-5423	175	4	xnk	xnk	PROPN
cana-5423	175	5	)	)	PUNCT
cana-5423	175	6	≤	≤	NUM
cana-5423	175	7	d(xmk	d(xmk	NOUN
cana-5423	175	8	,	,	PUNCT
cana-5423	175	9	xnk−1	xnk−1	PROPN
cana-5423	175	10	)	)	PUNCT
cana-5423	176	1	+	+	X
cana-5423	176	2	d(xmk−1	d(xmk−1	ADJ
cana-5423	176	3	,	,	PUNCT
cana-5423	176	4	xnk−1	xnk−1	PROPN
cana-5423	176	5	)	)	PUNCT
cana-5423	177	1	+	+	SYM
cana-5423	177	2	d(xnk−1	d(xnk−1	NOUN
cana-5423	177	3	,	,	PUNCT
cana-5423	177	4	xnk	xnk	PROPN
cana-5423	177	5	)	)	PUNCT
cana-5423	177	6	(	(	PUNCT
cana-5423	177	7	35	35	NUM
cana-5423	177	8	)	)	PUNCT
cana-5423	177	9	again	again	ADV
cana-5423	177	10	,	,	PUNCT
cana-5423	177	11	we	we	PRON
cana-5423	177	12	have	have	VERB
cana-5423	177	13	d(xmk−1	d(xmk−1	PROPN
cana-5423	177	14	,	,	PUNCT
cana-5423	177	15	xnk−1	xnk−1	PROPN
cana-5423	177	16	)	)	PUNCT
cana-5423	177	17	≤	≤	PUNCT
cana-5423	177	18	d(xmk−1	d(xmk−1	PROPN
cana-5423	177	19	,	,	PUNCT
cana-5423	177	20	xmk	xmk	PROPN
cana-5423	177	21	)	)	PUNCT
cana-5423	178	1	+	+	CCONJ
cana-5423	178	2	d(xmk	d(xmk	NOUN
cana-5423	178	3	,	,	PUNCT
cana-5423	178	4	xnk−1	xnk−1	PROPN
cana-5423	178	5	)	)	PUNCT
cana-5423	179	1	+	+	X
cana-5423	179	2	d(xmk−1	d(xmk−1	PROPN
cana-5423	179	3	,	,	PUNCT
cana-5423	179	4	xmk	xmk	PROPN
cana-5423	179	5	)	)	PUNCT
cana-5423	179	6	(	(	PUNCT
cana-5423	179	7	36	36	NUM
cana-5423	179	8	)	)	PUNCT
cana-5423	179	9	communications	communication	NOUN
cana-5423	179	10	on	on	ADP
cana-5423	179	11	applied	apply	VERB
cana-5423	179	12	nonlinear	nonlinear	ADJ
cana-5423	179	13	analysis	analysis	NOUN
cana-5423	179	14	issn	issn	NOUN
cana-5423	179	15	:	:	PUNCT
cana-5423	179	16	1074	1074	NUM
cana-5423	179	17	-	-	PUNCT
cana-5423	179	18	133x	133x	NUM
cana-5423	179	19	vol	vol	NOUN
cana-5423	179	20	32	32	NUM
cana-5423	179	21	no	no	NOUN
cana-5423	179	22	.	.	PUNCT
cana-5423	180	1	10s(2025	10s(2025	NUM
cana-5423	180	2	)	)	PUNCT
cana-5423	181	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	181	2	2223	2223	NUM
cana-5423	181	3	by	by	ADP
cana-5423	181	4	taking	take	VERB
cana-5423	181	5	the	the	DET
cana-5423	181	6	limit	limit	NOUN
cana-5423	181	7	as	as	ADP
cana-5423	181	8	k	k	PROPN
cana-5423	181	9	→	→	SYM
cana-5423	181	10	∞	∞	PROPN
cana-5423	181	11	in	in	ADP
cana-5423	181	12	equation	equation	NOUN
cana-5423	181	13	(	(	PUNCT
cana-5423	181	14	35	35	NUM
cana-5423	181	15	)	)	PUNCT
cana-5423	181	16	,	,	PUNCT
cana-5423	181	17	(	(	PUNCT
cana-5423	181	18	36	36	NUM
cana-5423	181	19	)	)	PUNCT
cana-5423	181	20	,	,	PUNCT
cana-5423	181	21	and	and	CCONJ
cana-5423	181	22	using	use	VERB
cana-5423	181	23	(	(	PUNCT
cana-5423	181	24	30	30	NUM
cana-5423	181	25	)	)	PUNCT
cana-5423	181	26	we	we	PRON
cana-5423	181	27	deduce	deduce	VERB
cana-5423	181	28	that	that	SCONJ
cana-5423	181	29	lim	lim	PROPN
cana-5423	181	30	n→∞	n→∞	X
cana-5423	181	31	d(xmk−1	d(xmk−1	PROPN
cana-5423	181	32	,	,	PUNCT
cana-5423	181	33	xnk−1	xnk−1	PROPN
cana-5423	181	34	)	)	PUNCT
cana-5423	181	35	<	<	X
cana-5423	181	36	ϵ.	ϵ.	NOUN
cana-5423	181	37	(	(	PUNCT
cana-5423	181	38	37	37	NUM
cana-5423	181	39	)	)	PUNCT
cana-5423	181	40	by	by	ADP
cana-5423	181	41	the	the	DET
cana-5423	181	42	same	same	ADJ
cana-5423	181	43	reasoning	reasoning	NOUN
cana-5423	181	44	as	as	ADP
cana-5423	181	45	above	above	ADV
cana-5423	181	46	,	,	PUNCT
cana-5423	181	47	we	we	PRON
cana-5423	181	48	get	get	VERB
cana-5423	181	49	that	that	DET
cana-5423	181	50	lim	lim	PROPN
cana-5423	181	51	n→∞	n→∞	NUM
cana-5423	181	52	d(xmk	d(xmk	PROPN
cana-5423	181	53	,	,	PUNCT
cana-5423	181	54	xnk−1	xnk−1	PROPN
cana-5423	181	55	)	)	PUNCT
cana-5423	182	1	=	=	VERB
cana-5423	182	2	lim	lim	PROPN
cana-5423	182	3	n→∞	n→∞	X
cana-5423	182	4	d(xmk−1	d(xmk−1	PROPN
cana-5423	182	5	,	,	PUNCT
cana-5423	182	6	xnk	xnk	PROPN
cana-5423	182	7	)	)	PUNCT
cana-5423	183	1	=	=	PUNCT
cana-5423	183	2	ϵ.	ϵ.	NOUN
cana-5423	183	3	(	(	PUNCT
cana-5423	183	4	38	38	NUM
cana-5423	183	5	)	)	PUNCT
cana-5423	183	6	since	since	SCONJ
cana-5423	183	7	t	t	PROPN
cana-5423	183	8	is	be	AUX
cana-5423	183	9	triangular	triangular	NOUN
cana-5423	183	10	α	α	PRON
cana-5423	183	11	-	-	ADJ
cana-5423	183	12	orbital	orbital	ADJ
cana-5423	183	13	admisssible	admisssible	NOUN
cana-5423	183	14	,	,	PUNCT
cana-5423	183	15	we	we	PRON
cana-5423	183	16	have	have	VERB
cana-5423	183	17	αd(xmk−1	αd(xmk−1	NOUN
cana-5423	183	18	,	,	PUNCT
cana-5423	183	19	xnk−1	xnk−1	PROPN
cana-5423	183	20	)	)	PUNCT
cana-5423	183	21	≤	≤	NUM
cana-5423	183	22	1	1	NUM
cana-5423	183	23	.	.	PUNCT
cana-5423	184	1	(	(	PUNCT
cana-5423	184	2	39	39	NUM
cana-5423	184	3	)	)	PUNCT
cana-5423	184	4	moreover	moreover	ADV
cana-5423	184	5	,	,	PUNCT
cana-5423	184	6	since	since	SCONJ
cana-5423	184	7	t	t	PROPN
cana-5423	184	8	is	be	AUX
cana-5423	184	9	a	a	DET
cana-5423	184	10	generalized	generalized	ADJ
cana-5423	184	11	α	α	NOUN
cana-5423	184	12	-admissible	-admissible	ADJ
cana-5423	184	13	modified	modify	VERB
cana-5423	184	14	almost	almost	ADV
cana-5423	184	15	z	z	NOUN
cana-5423	184	16	-	-	PUNCT
cana-5423	184	17	contraction	contraction	NOUN
cana-5423	184	18	with	with	ADP
cana-5423	184	19	respect	respect	NOUN
cana-5423	184	20	to	to	ADP
cana-5423	184	21	ζ	ζ	NOUN
cana-5423	184	22	,	,	PUNCT
cana-5423	184	23	0	0	NUM
cana-5423	184	24	≤	≤	NUM
cana-5423	184	25	ζ(α(xmk−1,γxmk−1	ζ(α(xmk−1,γxmk−1	PROPN
cana-5423	184	26	)	)	PUNCT
cana-5423	184	27	,	,	PUNCT
cana-5423	184	28	α(xnk−1,γxnk−1	α(xnk−1,γxnk−1	NUM
cana-5423	184	29	)	)	PUNCT
cana-5423	184	30	,	,	PUNCT
cana-5423	184	31	d(γxmk−1,γxnk−1)k(xmk−1	d(γxmk−1,γxnk−1)k(xmk−1	PROPN
cana-5423	184	32	,	,	PUNCT
cana-5423	184	33	xnk−1	xnk−1	PROPN
cana-5423	184	34	)	)	PUNCT
cana-5423	185	1	+	+	CCONJ
cana-5423	185	2	lq(xmk−1	lq(xmk−1	NOUN
cana-5423	185	3	,	,	PUNCT
cana-5423	185	4	xnk−1	xnk−1	PROPN
cana-5423	185	5	)	)	PUNCT
cana-5423	186	1	=	=	SYM
cana-5423	186	2	ζ(α(xmk−1	ζ(α(xmk−1	NOUN
cana-5423	186	3	,	,	PUNCT
cana-5423	186	4	xmk	xmk	PROPN
cana-5423	186	5	)	)	PUNCT
cana-5423	186	6	,	,	PUNCT
cana-5423	186	7	α(xnk−1	α(xnk−1	NOUN
cana-5423	186	8	,	,	PUNCT
cana-5423	186	9	xnk	xnk	PROPN
cana-5423	186	10	)	)	PUNCT
cana-5423	186	11	,	,	PUNCT
cana-5423	186	12	d(xmk	d(xmk	PROPN
cana-5423	186	13	,	,	PUNCT
cana-5423	186	14	xnk	xnk	PROPN
cana-5423	186	15	)	)	PUNCT
cana-5423	186	16	,	,	PUNCT
cana-5423	186	17	k(xmk−1	k(xmk−1	PROPN
cana-5423	186	18	,	,	PUNCT
cana-5423	186	19	xnk−1	xnk−1	PROPN
cana-5423	186	20	)	)	PUNCT
cana-5423	187	1	+	+	CCONJ
cana-5423	187	2	lq(xmk−1	lq(xmk−1	NOUN
cana-5423	187	3	,	,	PUNCT
cana-5423	187	4	xnk−1	xnk−1	PROPN
cana-5423	187	5	)	)	PUNCT
cana-5423	187	6	it	it	PRON
cana-5423	187	7	follows	follow	VERB
cana-5423	187	8	from	from	ADP
cana-5423	187	9	condition(ζ2	condition(ζ2	PROPN
cana-5423	187	10	)	)	PUNCT
cana-5423	187	11	,	,	PUNCT
cana-5423	187	12	we	we	PRON
cana-5423	187	13	get	get	VERB
cana-5423	187	14	0	0	NUM
cana-5423	187	15	<	<	X
cana-5423	187	16	k(xmk−1	k(xmk−1	PROPN
cana-5423	187	17	,	,	PUNCT
cana-5423	187	18	xnk−1	xnk−1	PROPN
cana-5423	187	19	)	)	PUNCT
cana-5423	188	1	+	+	CCONJ
cana-5423	188	2	lq(xmk−1	lq(xmk−1	NOUN
cana-5423	188	3	,	,	PUNCT
cana-5423	188	4	xnk−1	xnk−1	PROPN
cana-5423	188	5	)	)	PUNCT
cana-5423	189	1	−	−	PROPN
cana-5423	189	2	ζ(α(xmk−1	ζ(α(xmk−1	NOUN
cana-5423	189	3	,	,	PUNCT
cana-5423	189	4	xmk	xmk	PROPN
cana-5423	189	5	)	)	PUNCT
cana-5423	189	6	,	,	PUNCT
cana-5423	189	7	α(xnk−1	α(xnk−1	NOUN
cana-5423	189	8	,	,	PUNCT
cana-5423	189	9	xnk	xnk	PROPN
cana-5423	189	10	)	)	PUNCT
cana-5423	189	11	,	,	PUNCT
cana-5423	189	12	d(xmk	d(xmk	PROPN
cana-5423	189	13	,	,	PUNCT
cana-5423	189	14	xnk	xnk	PROPN
cana-5423	189	15	)	)	PUNCT
cana-5423	189	16	(	(	PUNCT
cana-5423	189	17	40	40	NUM
cana-5423	189	18	)	)	PUNCT
cana-5423	189	19	hence	hence	ADV
cana-5423	189	20	,	,	PUNCT
cana-5423	189	21	0	0	NUM
cana-5423	189	22	<	<	X
cana-5423	189	23	d(xmk	d(xmk	PROPN
cana-5423	189	24	,	,	PUNCT
cana-5423	189	25	xnk	xnk	PROPN
cana-5423	189	26	)	)	PUNCT
cana-5423	189	27	≤	≤	NOUN
cana-5423	189	28	α(xmk−1	α(xmk−1	NOUN
cana-5423	189	29	,	,	PUNCT
cana-5423	189	30	xmk	xmk	PROPN
cana-5423	189	31	)	)	PUNCT
cana-5423	189	32	,	,	PUNCT
cana-5423	189	33	α(xnk−1	α(xnk−1	NOUN
cana-5423	189	34	,	,	PUNCT
cana-5423	189	35	xnk	xnk	PROPN
cana-5423	189	36	)	)	PUNCT
cana-5423	189	37	d(xmk	d(xmk	PROPN
cana-5423	189	38	,	,	PUNCT
cana-5423	189	39	xnk	xnk	PROPN
cana-5423	189	40	)	)	PUNCT
cana-5423	189	41	<	<	X
cana-5423	189	42	k(xmk−1	k(xmk−1	PROPN
cana-5423	189	43	,	,	PUNCT
cana-5423	189	44	xnk−1),+lq(xmk−1	xnk−1),+lq(xmk−1	PROPN
cana-5423	189	45	,	,	PUNCT
cana-5423	189	46	xnk−1	xnk−1	PROPN
cana-5423	189	47	)	)	PUNCT
cana-5423	189	48	(	(	PUNCT
cana-5423	189	49	41	41	NUM
cana-5423	189	50	)	)	PUNCT
cana-5423	189	51	also	also	ADV
cana-5423	189	52	,	,	PUNCT
cana-5423	189	53	where	where	SCONJ
cana-5423	189	54	k(xmk−1	k(xmk−1	PROPN
cana-5423	189	55	,	,	PUNCT
cana-5423	189	56	xnk−1	xnk−1	PROPN
cana-5423	189	57	)	)	PUNCT
cana-5423	189	58	=	=	SYM
cana-5423	190	1	max{d(xmk−1	max{d(xmk−1	PROPN
cana-5423	190	2	,	,	PUNCT
cana-5423	190	3	xnk−1	xnk−1	PROPN
cana-5423	190	4	)	)	PUNCT
cana-5423	190	5	,	,	PUNCT
cana-5423	191	1	[	[	X
cana-5423	191	2	1	1	NUM
cana-5423	191	3	+	+	NUM
cana-5423	191	4	d(xmk−1,γxmk−1)]d(xnk−1,γxnk−1	d(xmk−1,γxmk−1)]d(xnk−1,γxnk−1	NOUN
cana-5423	191	5	)	)	PUNCT
cana-5423	191	6	1	1	NUM
cana-5423	192	1	+	+	CCONJ
cana-5423	192	2	d(xmk−1	d(xmk−1	ADJ
cana-5423	192	3	,	,	PUNCT
cana-5423	192	4	xnk−1	xnk−1	PROPN
cana-5423	192	5	)	)	PUNCT
cana-5423	193	1	[	[	X
cana-5423	193	2	1	1	NUM
cana-5423	193	3	+	+	NUM
cana-5423	193	4	d(xmk−1,γxnk−1)]d(xnk−1,γxmk−1	d(xmk−1,γxnk−1)]d(xnk−1,γxmk−1	NOUN
cana-5423	193	5	)	)	PUNCT
cana-5423	193	6	1	1	NUM
cana-5423	193	7	+	+	CCONJ
cana-5423	193	8	d(xmk−1	d(xmk−1	ADJ
cana-5423	193	9	,	,	PUNCT
cana-5423	193	10	xnk−1	xnk−1	PROPN
cana-5423	193	11	)	)	PUNCT
cana-5423	193	12	}	}	PUNCT
cana-5423	193	13	=	=	PUNCT
cana-5423	193	14	max{d(xmk−1	max{d(xmk−1	PROPN
cana-5423	193	15	,	,	PUNCT
cana-5423	193	16	xnk−1	xnk−1	PROPN
cana-5423	193	17	)	)	PUNCT
cana-5423	193	18	,	,	PUNCT
cana-5423	193	19	[	[	X
cana-5423	193	20	1	1	NUM
cana-5423	193	21	+	+	CCONJ
cana-5423	193	22	d(xmk−1	d(xmk−1	PROPN
cana-5423	193	23	,	,	PUNCT
cana-5423	193	24	xmk	xmk	PROPN
cana-5423	193	25	)	)	PUNCT
cana-5423	193	26	]	]	X
cana-5423	193	27	d(xnk−1	d(xnk−1	NOUN
cana-5423	193	28	,	,	PUNCT
cana-5423	193	29	xnk	xnk	PROPN
cana-5423	193	30	)	)	PUNCT
cana-5423	193	31	1	1	NUM
cana-5423	193	32	+	+	CCONJ
cana-5423	193	33	d(xmk−1	d(xmk−1	ADJ
cana-5423	193	34	,	,	PUNCT
cana-5423	193	35	xnk−1	xnk−1	PROPN
cana-5423	193	36	)	)	PUNCT
cana-5423	194	1	[	[	X
cana-5423	194	2	1	1	NUM
cana-5423	194	3	+	+	CCONJ
cana-5423	194	4	d(xmk−1	d(xmk−1	PROPN
cana-5423	194	5	,	,	PUNCT
cana-5423	194	6	xnk	xnk	PROPN
cana-5423	194	7	)	)	PUNCT
cana-5423	195	1	]	]	X
cana-5423	195	2	d(xnk−1	d(xnk−1	PROPN
cana-5423	195	3	,	,	PUNCT
cana-5423	195	4	xmk	xmk	PROPN
cana-5423	195	5	)	)	PUNCT
cana-5423	195	6	1	1	NUM
cana-5423	196	1	+	+	CCONJ
cana-5423	196	2	d(xmk−1	d(xmk−1	ADJ
cana-5423	196	3	,	,	PUNCT
cana-5423	196	4	xnk−1	xnk−1	PROPN
cana-5423	196	5	)	)	PUNCT
cana-5423	196	6	}	}	PUNCT
cana-5423	196	7	.	.	PUNCT
cana-5423	197	1	(	(	PUNCT
cana-5423	197	2	42	42	NUM
cana-5423	197	3	)	)	PUNCT
cana-5423	197	4	and	and	CCONJ
cana-5423	197	5	,	,	PUNCT
cana-5423	197	6	q(xmk−1	q(xmk−1	NOUN
cana-5423	197	7	,	,	PUNCT
cana-5423	197	8	xnk−1	xnk−1	PROPN
cana-5423	197	9	)	)	PUNCT
cana-5423	197	10	=	=	SYM
cana-5423	197	11	min{d(xmk−1,γxmk−1	min{d(xmk−1,γxmk−1	NOUN
cana-5423	197	12	)	)	PUNCT
cana-5423	197	13	,	,	PUNCT
cana-5423	197	14	d(xnk−1,γxnk−1	d(xnk−1,γxnk−1	PROPN
cana-5423	197	15	)	)	PUNCT
cana-5423	197	16	,	,	PUNCT
cana-5423	197	17	d(xmk−1,γxnk−1	d(xmk−1,γxnk−1	PROPN
cana-5423	197	18	)	)	PUNCT
cana-5423	197	19	,	,	PUNCT
cana-5423	197	20	d(xnk−1,γxmk−1	d(xnk−1,γxmk−1	PROPN
cana-5423	197	21	)	)	PUNCT
cana-5423	197	22	,	,	PUNCT
cana-5423	197	23	d(xmk−1,γxnk−1)d(xmk−1,γxmk−1	d(xmk−1,γxnk−1)d(xmk−1,γxmk−1	NUM
cana-5423	197	24	)	)	PUNCT
cana-5423	197	25	1	1	NUM
cana-5423	198	1	+	+	CCONJ
cana-5423	198	2	d(xmk−1	d(xmk−1	ADJ
cana-5423	198	3	,	,	PUNCT
cana-5423	198	4	xnk−1	xnk−1	PROPN
cana-5423	198	5	)	)	PUNCT
cana-5423	198	6	,	,	PUNCT
cana-5423	198	7	d(xmk−1,γxmk−1)d(xnk−1,γxnk−1	d(xmk−1,γxmk−1)d(xnk−1,γxnk−1	X
cana-5423	198	8	)	)	PUNCT
cana-5423	198	9	1	1	NUM
cana-5423	199	1	+	+	CCONJ
cana-5423	199	2	d(xmk−1	d(xmk−1	ADJ
cana-5423	199	3	,	,	PUNCT
cana-5423	199	4	xnk−1	xnk−1	PROPN
cana-5423	199	5	)	)	PUNCT
cana-5423	199	6	}	}	PUNCT
cana-5423	200	1	=	=	PUNCT
cana-5423	200	2	min{d(xmk−1	min{d(xmk−1	PROPN
cana-5423	200	3	,	,	PUNCT
cana-5423	200	4	xmk	xmk	PROPN
cana-5423	200	5	)	)	PUNCT
cana-5423	200	6	,	,	PUNCT
cana-5423	200	7	d(xnk−1	d(xnk−1	PROPN
cana-5423	200	8	,	,	PUNCT
cana-5423	200	9	xnk	xnk	PROPN
cana-5423	200	10	)	)	PUNCT
cana-5423	200	11	,	,	PUNCT
cana-5423	200	12	d(xmk−1	d(xmk−1	PROPN
cana-5423	200	13	,	,	PUNCT
cana-5423	200	14	xnk	xnk	PROPN
cana-5423	200	15	)	)	PUNCT
cana-5423	200	16	,	,	PUNCT
cana-5423	200	17	d(xnk−1	d(xnk−1	PROPN
cana-5423	200	18	,	,	PUNCT
cana-5423	200	19	xmk	xmk	PROPN
cana-5423	200	20	)	)	PUNCT
cana-5423	200	21	,	,	PUNCT
cana-5423	200	22	d(xmk−1	d(xmk−1	PROPN
cana-5423	200	23	,	,	PUNCT
cana-5423	200	24	xnk	xnk	NOUN
cana-5423	200	25	)	)	PUNCT
cana-5423	200	26	d(xmk−1	d(xmk−1	PROPN
cana-5423	200	27	,	,	PUNCT
cana-5423	200	28	xmk	xmk	PROPN
cana-5423	200	29	)	)	PUNCT
cana-5423	200	30	1	1	NUM
cana-5423	201	1	+	+	CCONJ
cana-5423	201	2	d(xmk−1	d(xmk−1	ADJ
cana-5423	201	3	,	,	PUNCT
cana-5423	201	4	xnk−1	xnk−1	PROPN
cana-5423	201	5	)	)	PUNCT
cana-5423	201	6	,	,	PUNCT
cana-5423	201	7	d(xmk−1	d(xmk−1	PROPN
cana-5423	201	8	,	,	PUNCT
cana-5423	201	9	xmk	xmk	PROPN
cana-5423	201	10	)	)	PUNCT
cana-5423	201	11	d(xnk−1	d(xnk−1	PROPN
cana-5423	201	12	,	,	PUNCT
cana-5423	201	13	xnk	xnk	PROPN
cana-5423	201	14	)	)	PUNCT
cana-5423	201	15	1	1	NUM
cana-5423	202	1	+	+	CCONJ
cana-5423	202	2	d(xmk−1	d(xmk−1	ADJ
cana-5423	202	3	,	,	PUNCT
cana-5423	202	4	xnk−1	xnk−1	PROPN
cana-5423	202	5	)	)	PUNCT
cana-5423	202	6	}	}	PUNCT
cana-5423	202	7	(	(	PUNCT
cana-5423	202	8	43	43	X
cana-5423	202	9	)	)	PUNCT
cana-5423	202	10	taking	take	VERB
cana-5423	202	11	limit	limit	NOUN
cana-5423	202	12	ask	ask	NOUN
cana-5423	202	13	→	→	SYM
cana-5423	202	14	∞	∞	NUM
cana-5423	202	15	in	in	ADP
cana-5423	202	16	(	(	PUNCT
cana-5423	202	17	42),(43	42),(43	NOUN
cana-5423	202	18	)	)	PUNCT
cana-5423	202	19	using	use	VERB
cana-5423	202	20	(	(	PUNCT
cana-5423	202	21	30	30	NUM
cana-5423	202	22	)	)	PUNCT
cana-5423	202	23	,	,	PUNCT
cana-5423	202	24	(	(	PUNCT
cana-5423	202	25	34	34	NUM
cana-5423	202	26	)	)	PUNCT
cana-5423	202	27	,	,	PUNCT
cana-5423	202	28	(	(	PUNCT
cana-5423	202	29	37	37	NUM
cana-5423	202	30	)	)	PUNCT
cana-5423	202	31	and	and	CCONJ
cana-5423	202	32	(	(	PUNCT
cana-5423	202	33	38	38	NUM
cana-5423	202	34	)	)	PUNCT
cana-5423	202	35	,	,	PUNCT
cana-5423	202	36	we	we	PRON
cana-5423	202	37	get	get	VERB
cana-5423	202	38	lim	lim	PROPN
cana-5423	202	39	k→∞	k→∞	PROPN
cana-5423	202	40	k(xmk−1	k(xmk−1	PROPN
cana-5423	202	41	,	,	PUNCT
cana-5423	202	42	xnk−1	xnk−1	PROPN
cana-5423	202	43	)	)	PUNCT
cana-5423	203	1	=	=	SYM
cana-5423	203	2	ϵ.	ϵ.	NOUN
cana-5423	203	3	(	(	PUNCT
cana-5423	203	4	44	44	NUM
cana-5423	203	5	)	)	PUNCT
cana-5423	203	6	and	and	CCONJ
cana-5423	203	7	,	,	PUNCT
cana-5423	203	8	lim	lim	PROPN
cana-5423	203	9	k→∞	k→∞	NOUN
cana-5423	203	10	q(xmk−1	q(xmk−1	PROPN
cana-5423	203	11	,	,	PUNCT
cana-5423	203	12	xnk−1	xnk−1	PROPN
cana-5423	203	13	)	)	PUNCT
cana-5423	203	14	=	=	SYM
cana-5423	204	1	0	0	X
cana-5423	204	2	.	.	PUNCT
cana-5423	205	1	(	(	PUNCT
cana-5423	205	2	45	45	NUM
cana-5423	205	3	)	)	PUNCT
cana-5423	205	4	communications	communication	NOUN
cana-5423	205	5	on	on	ADP
cana-5423	205	6	applied	apply	VERB
cana-5423	205	7	nonlinear	nonlinear	ADJ
cana-5423	205	8	analysis	analysis	NOUN
cana-5423	205	9	issn	issn	NOUN
cana-5423	205	10	:	:	PUNCT
cana-5423	205	11	1074	1074	NUM
cana-5423	205	12	-	-	PUNCT
cana-5423	205	13	133x	133x	NUM
cana-5423	205	14	vol	vol	NOUN
cana-5423	205	15	32	32	NUM
cana-5423	205	16	no	no	NOUN
cana-5423	205	17	.	.	PUNCT
cana-5423	206	1	10s(2025	10s(2025	NUM
cana-5423	206	2	)	)	PUNCT
cana-5423	206	3	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	206	4	2224	2224	NUM
cana-5423	206	5	from	from	ADP
cana-5423	206	6	(	(	PUNCT
cana-5423	206	7	34	34	NUM
cana-5423	206	8	)	)	PUNCT
cana-5423	206	9	,	,	PUNCT
cana-5423	206	10	(	(	PUNCT
cana-5423	206	11	37	37	NUM
cana-5423	206	12	)	)	PUNCT
cana-5423	206	13	,	,	PUNCT
cana-5423	206	14	(	(	PUNCT
cana-5423	206	15	40	40	NUM
cana-5423	206	16	)	)	PUNCT
cana-5423	206	17	,	,	PUNCT
cana-5423	206	18	and	and	CCONJ
cana-5423	206	19	(	(	PUNCT
cana-5423	206	20	44	44	NUM
cana-5423	206	21	)	)	PUNCT
cana-5423	206	22	(	(	PUNCT
cana-5423	206	23	45	45	NUM
cana-5423	206	24	)	)	PUNCT
cana-5423	206	25	,	,	PUNCT
cana-5423	206	26	also	also	ADV
cana-5423	206	27	condition	condition	NOUN
cana-5423	206	28	(	(	PUNCT
cana-5423	206	29	ζ3	ζ3	NOUN
cana-5423	206	30	)	)	PUNCT
cana-5423	206	31	,	,	PUNCT
cana-5423	206	32	we	we	PRON
cana-5423	206	33	get	get	VERB
cana-5423	206	34	0	0	NUM
cana-5423	206	35	≤	≤	NOUN
cana-5423	206	36	lim	lim	PROPN
cana-5423	206	37	supk→∞ζ(α(xmk−1	supk→∞ζ(α(xmk−1	PROPN
cana-5423	206	38	,	,	PUNCT
cana-5423	206	39	xmk	xmk	PROPN
cana-5423	206	40	)	)	PUNCT
cana-5423	206	41	,	,	PUNCT
cana-5423	206	42	α(xnk−1	α(xnk−1	NOUN
cana-5423	206	43	,	,	PUNCT
cana-5423	206	44	xnk	xnk	PROPN
cana-5423	206	45	)	)	PUNCT
cana-5423	206	46	,	,	PUNCT
cana-5423	206	47	d(xmk	d(xmk	PROPN
cana-5423	206	48	,	,	PUNCT
cana-5423	206	49	xnk	xnk	PROPN
cana-5423	206	50	)	)	PUNCT
cana-5423	206	51	k(xmk−1	k(xmk−1	PROPN
cana-5423	206	52	,	,	PUNCT
cana-5423	206	53	xnk−1	xnk−1	PROPN
cana-5423	206	54	)	)	PUNCT
cana-5423	207	1	+	+	CCONJ
cana-5423	207	2	lq(xmk−1	lq(xmk−1	NOUN
cana-5423	207	3	,	,	PUNCT
cana-5423	207	4	xnk−1	xnk−1	PROPN
cana-5423	207	5	)	)	PUNCT
cana-5423	207	6	≤	≤	NOUN
cana-5423	207	7	lim	lim	PROPN
cana-5423	207	8	supk→∞ζ(α(xmk−1	supk→∞ζ(α(xmk−1	PROPN
cana-5423	207	9	,	,	PUNCT
cana-5423	207	10	xmk	xmk	PROPN
cana-5423	207	11	)	)	PUNCT
cana-5423	207	12	,	,	PUNCT
cana-5423	207	13	α(xnk−1	α(xnk−1	NOUN
cana-5423	207	14	,	,	PUNCT
cana-5423	207	15	xnk	xnk	PROPN
cana-5423	207	16	)	)	PUNCT
cana-5423	207	17	,	,	PUNCT
cana-5423	207	18	d(xmk	d(xmk	PROPN
cana-5423	207	19	,	,	PUNCT
cana-5423	207	20	xnk	xnk	PROPN
cana-5423	207	21	)	)	PUNCT
cana-5423	207	22	k(xmk−1	k(xmk−1	PROPN
cana-5423	207	23	,	,	PUNCT
cana-5423	207	24	xnk−1	xnk−1	PROPN
cana-5423	207	25	)	)	PUNCT
cana-5423	207	26	<	<	X
cana-5423	207	27	0	0	X
cana-5423	207	28	.	.	PUNCT
cana-5423	208	1	this	this	PRON
cana-5423	208	2	is	be	AUX
cana-5423	208	3	a	a	DET
cana-5423	208	4	contradiction	contradiction	NOUN
cana-5423	208	5	.	.	PUNCT
cana-5423	209	1	hence	hence	ADV
cana-5423	209	2	{	{	PUNCT
cana-5423	209	3	xn	xn	X
cana-5423	209	4	}	}	PUNCT
cana-5423	209	5	is	be	AUX
cana-5423	209	6	a	a	DET
cana-5423	209	7	cauchy	cauchy	ADJ
cana-5423	209	8	sequence	sequence	NOUN
cana-5423	209	9	.	.	PUNCT
cana-5423	210	1	since	since	SCONJ
cana-5423	210	2	(	(	PUNCT
cana-5423	210	3	x	x	X
cana-5423	210	4	,	,	PUNCT
cana-5423	210	5	d	d	NOUN
cana-5423	210	6	)	)	PUNCT
cana-5423	210	7	is	be	AUX
cana-5423	210	8	complete	complete	ADJ
cana-5423	210	9	metric	metric	ADJ
cana-5423	210	10	space	space	NOUN
cana-5423	210	11	,	,	PUNCT
cana-5423	210	12	there	there	PRON
cana-5423	210	13	exists	exist	VERB
cana-5423	210	14	x∗	x∗	PROPN
cana-5423	210	15	∈	∈	PROPN
cana-5423	210	16	x	x	PUNCT
cana-5423	210	17	such	such	ADJ
cana-5423	210	18	that	that	SCONJ
cana-5423	210	19	lim	lim	PROPN
cana-5423	210	20	n→∞	n→∞	X
cana-5423	210	21	d(xn	d(xn	PROPN
cana-5423	210	22	,	,	PUNCT
cana-5423	210	23	x	x	SYM
cana-5423	210	24	∗	∗	NOUN
cana-5423	210	25	)	)	PUNCT
cana-5423	210	26	=	=	SYM
cana-5423	211	1	0	0	X
cana-5423	211	2	.	.	PUNCT
cana-5423	212	1	(	(	PUNCT
cana-5423	212	2	46	46	NUM
cana-5423	212	3	)	)	PUNCT
cana-5423	212	4	now	now	ADV
cana-5423	212	5	,	,	PUNCT
cana-5423	212	6	we	we	PRON
cana-5423	212	7	shall	shall	AUX
cana-5423	212	8	show	show	VERB
cana-5423	212	9	that	that	SCONJ
cana-5423	212	10	γx∗	γx∗	NOUN
cana-5423	213	1	=	=	PUNCT
cana-5423	214	1	x∗.	x∗.	PROPN
cana-5423	215	1	since	since	SCONJ
cana-5423	215	2	γ	γ	PROPN
cana-5423	215	3	is	be	AUX
cana-5423	215	4	continuous	continuous	ADJ
cana-5423	215	5	,	,	PUNCT
cana-5423	215	6	we	we	PRON
cana-5423	215	7	obtain	obtain	VERB
cana-5423	215	8	that	that	PRON
cana-5423	215	9	γx∗	γx∗	NOUN
cana-5423	216	1	=	=	PUNCT
cana-5423	217	1	γ	γ	X
cana-5423	217	2	(	(	PUNCT
cana-5423	217	3	lim	lim	PROPN
cana-5423	217	4	n→∞	n→∞	NUM
cana-5423	217	5	xn	xn	PUNCT
cana-5423	217	6	)	)	PUNCT
cana-5423	218	1	=	=	SYM
cana-5423	218	2	lim	lim	PROPN
cana-5423	218	3	n→∞	n→∞	NUM
cana-5423	218	4	γ(xn	γ(xn	NUM
cana-5423	218	5	)	)	PUNCT
cana-5423	218	6	=	=	VERB
cana-5423	218	7	lim	lim	PROPN
cana-5423	218	8	n→∞	n→∞	X
cana-5423	218	9	xn+1	xn+1	PROPN
cana-5423	218	10	=	=	SYM
cana-5423	218	11	x∗.	x∗.	PROPN
cana-5423	219	1	thus	thus	ADV
cana-5423	219	2	,	,	PUNCT
cana-5423	219	3	x∗	x∗	PROPN
cana-5423	219	4	is	be	AUX
cana-5423	219	5	a	a	DET
cana-5423	219	6	fixed	fix	VERB
cana-5423	219	7	point	point	NOUN
cana-5423	219	8	of	of	ADP
cana-5423	219	9	γ	γ	PROPN
cana-5423	219	10	.	.	PROPN
cana-5423	219	11	to	to	PART
cana-5423	219	12	prove	prove	VERB
cana-5423	219	13	the	the	DET
cana-5423	219	14	uniqueness	uniqueness	NOUN
cana-5423	219	15	of	of	ADP
cana-5423	219	16	the	the	DET
cana-5423	219	17	fixed	fix	VERB
cana-5423	219	18	point.assume	point.assume	PROPN
cana-5423	219	19	that	that	SCONJ
cana-5423	219	20	,	,	PUNCT
cana-5423	219	21	x∗	x∗	PROPN
cana-5423	219	22	andy∗	andy∗	PROPN
cana-5423	219	23	be	be	AUX
cana-5423	219	24	two	two	NUM
cana-5423	219	25	fixed	fix	VERB
cana-5423	219	26	point	point	NOUN
cana-5423	219	27	of	of	ADP
cana-5423	219	28	γ	γ	NOUN
cana-5423	219	29	and	and	CCONJ
cana-5423	219	30	hence	hence	ADV
cana-5423	219	31	x∗	x∗	PROPN
cana-5423	219	32	,	,	PUNCT
cana-5423	219	33	y∗	y∗	PROPN
cana-5423	219	34	∈	∈	PROPN
cana-5423	219	35	fix(γ	fix(γ	PROPN
cana-5423	219	36	)	)	PUNCT
cana-5423	219	37	,	,	PUNCT
cana-5423	219	38	which	which	PRON
cana-5423	219	39	is	be	AUX
cana-5423	219	40	a	a	DET
cana-5423	219	41	generalized	generalized	ADJ
cana-5423	219	42	α	α	NOUN
cana-5423	219	43	-	-	ADJ
cana-5423	219	44	admissible	admissible	ADJ
cana-5423	219	45	modified	modified	ADJ
cana-5423	219	46	self	self	NOUN
cana-5423	219	47	mapping	mapping	NOUN
cana-5423	219	48	of	of	ADP
cana-5423	219	49	metric	metric	PROPN
cana-5423	219	50	spacex	spacex	PROPN
cana-5423	219	51	,	,	PUNCT
cana-5423	219	52	d	d	NOUN
cana-5423	219	53	)	)	PUNCT
cana-5423	219	54	.	.	PUNCT
cana-5423	220	1	then	then	ADV
cana-5423	220	2	d(x∗	d(x∗	ADV
cana-5423	220	3	,	,	PUNCT
cana-5423	220	4	y∗	y∗	PROPN
cana-5423	220	5	)	)	PUNCT
cana-5423	220	6	>	>	X
cana-5423	221	1	0	0	X
cana-5423	221	2	.	.	PUNCT
cana-5423	221	3	by	by	ADP
cana-5423	221	4	(	(	PUNCT
cana-5423	221	5	10	10	NUM
cana-5423	221	6	)	)	PUNCT
cana-5423	221	7	,	,	PUNCT
cana-5423	221	8	we	we	PRON
cana-5423	221	9	have	have	VERB
cana-5423	221	10	that	that	DET
cana-5423	221	11	0	0	NUM
cana-5423	221	12	≤	≤	NUM
cana-5423	221	13	ζ(α(x∗,γx∗	ζ(α(x∗,γx∗	NOUN
cana-5423	221	14	)	)	PUNCT
cana-5423	221	15	,	,	PUNCT
cana-5423	221	16	α(y∗,γy∗)d(γx∗,γy∗),k(x∗	α(y∗,γy∗)d(γx∗,γy∗),k(x∗	NOUN
cana-5423	221	17	,	,	PUNCT
cana-5423	221	18	y∗	y∗	PROPN
cana-5423	221	19	)	)	PUNCT
cana-5423	222	1	+	+	NUM
cana-5423	222	2	lq(x∗	lq(x∗	NOUN
cana-5423	222	3	,	,	PUNCT
cana-5423	222	4	y∗	y∗	PROPN
cana-5423	222	5	)	)	PUNCT
cana-5423	222	6	(	(	PUNCT
cana-5423	222	7	47	47	NUM
cana-5423	222	8	)	)	PUNCT
cana-5423	222	9	where	where	SCONJ
cana-5423	222	10	k(x∗	k(x∗	NOUN
cana-5423	222	11	,	,	PUNCT
cana-5423	222	12	y∗	y∗	PROPN
cana-5423	222	13	)	)	PUNCT
cana-5423	222	14	=	=	SYM
cana-5423	222	15	d(x∗	d(x∗	NOUN
cana-5423	222	16	,	,	PUNCT
cana-5423	222	17	y∗	y∗	PROPN
cana-5423	222	18	)	)	PUNCT
cana-5423	222	19	and	and	CCONJ
cana-5423	222	20	q(x∗	q(x∗	NOUN
cana-5423	222	21	,	,	PUNCT
cana-5423	222	22	y∗	y∗	PROPN
cana-5423	222	23	)	)	PUNCT
cana-5423	222	24	=	=	SYM
cana-5423	223	1	0	0	X
cana-5423	223	2	.	.	PUNCT
cana-5423	224	1	then	then	ADV
cana-5423	224	2	,	,	PUNCT
cana-5423	224	3	by	by	ADP
cana-5423	224	4	(	(	PUNCT
cana-5423	224	5	47	47	NUM
cana-5423	224	6	)	)	PUNCT
cana-5423	224	7	,	,	PUNCT
cana-5423	224	8	we	we	PRON
cana-5423	224	9	get	get	VERB
cana-5423	224	10	0	0	NUM
cana-5423	224	11	≤	≤	NUM
cana-5423	224	12	ζ(α(x∗	ζ(α(x∗	PROPN
cana-5423	224	13	,	,	PUNCT
cana-5423	224	14	x∗	x∗	PROPN
cana-5423	224	15	)	)	PUNCT
cana-5423	224	16	,	,	PUNCT
cana-5423	224	17	α(y∗	α(y∗	NUM
cana-5423	224	18	,	,	PUNCT
cana-5423	224	19	y∗	y∗	PROPN
cana-5423	224	20	)	)	PUNCT
cana-5423	224	21	,	,	PUNCT
cana-5423	224	22	d(x∗	d(x∗	NOUN
cana-5423	224	23	,	,	PUNCT
cana-5423	224	24	y∗)k(x∗	y∗)k(x∗	PROPN
cana-5423	224	25	,	,	PUNCT
cana-5423	224	26	y∗	y∗	PROPN
cana-5423	224	27	)	)	PUNCT
cana-5423	225	1	+	+	NUM
cana-5423	225	2	lq(x∗	lq(x∗	NOUN
cana-5423	225	3	,	,	PUNCT
cana-5423	225	4	y∗	y∗	PROPN
cana-5423	225	5	)	)	PUNCT
cana-5423	226	1	<	<	X
cana-5423	226	2	k(x∗	k(x∗	PROPN
cana-5423	226	3	,	,	PUNCT
cana-5423	226	4	y∗	y∗	PROPN
cana-5423	226	5	)	)	PUNCT
cana-5423	226	6	+	+	NUM
cana-5423	226	7	lq(x∗	lq(x∗	NOUN
cana-5423	226	8	,	,	PUNCT
cana-5423	226	9	y∗	y∗	PROPN
cana-5423	226	10	)	)	PUNCT
cana-5423	226	11	−	−	PROPN
cana-5423	226	12	α(x∗	α(x∗	NOUN
cana-5423	226	13	,	,	PUNCT
cana-5423	226	14	x∗	x∗	PROPN
cana-5423	226	15	)	)	PUNCT
cana-5423	226	16	,	,	PUNCT
cana-5423	226	17	α(y∗	α(y∗	NUM
cana-5423	226	18	,	,	PUNCT
cana-5423	226	19	y∗	y∗	PROPN
cana-5423	226	20	)	)	PUNCT
cana-5423	226	21	,	,	PUNCT
cana-5423	226	22	d(x∗	d(x∗	NOUN
cana-5423	226	23	,	,	PUNCT
cana-5423	226	24	y∗	y∗	PROPN
cana-5423	226	25	)	)	PUNCT
cana-5423	226	26	0	0	PUNCT
cana-5423	227	1	<	<	X
cana-5423	227	2	d(x∗	d(x∗	PROPN
cana-5423	227	3	,	,	PUNCT
cana-5423	227	4	y∗	y∗	PROPN
cana-5423	227	5	)	)	PUNCT
cana-5423	227	6	−	−	PROPN
cana-5423	227	7	d(x∗	d(x∗	NOUN
cana-5423	227	8	,	,	PUNCT
cana-5423	227	9	y∗	y∗	PROPN
cana-5423	227	10	)	)	PUNCT
cana-5423	227	11	=	=	SYM
cana-5423	228	1	0	0	X
cana-5423	228	2	.	.	PUNCT
cana-5423	229	1	(	(	PUNCT
cana-5423	229	2	48	48	NUM
cana-5423	229	3	)	)	PUNCT
cana-5423	229	4	which	which	PRON
cana-5423	229	5	is	be	AUX
cana-5423	229	6	a	a	DET
cana-5423	229	7	contradiction.thus	contradiction.thus	PROPN
cana-5423	229	8	we	we	PRON
cana-5423	229	9	have	have	VERB
cana-5423	229	10	x∗	x∗	NOUN
cana-5423	229	11	=	=	SYM
cana-5423	230	1	y∗.	y∗.	PROPN
cana-5423	230	2	hence	hence	ADV
cana-5423	230	3	γ	γ	PROPN
cana-5423	230	4	has	have	VERB
cana-5423	230	5	a	a	DET
cana-5423	230	6	unique	unique	ADJ
cana-5423	230	7	fixed	fix	VERB
cana-5423	230	8	point	point	NOUN
cana-5423	230	9	.	.	PUNCT
cana-5423	231	1	□	□	PUNCT
cana-5423	231	2	theorem	theorem	ADJ
cana-5423	231	3	2.3	2.3	NUM
cana-5423	231	4	.	.	PUNCT
cana-5423	232	1	let	let	AUX
cana-5423	232	2	(	(	PUNCT
cana-5423	232	3	x	x	NOUN
cana-5423	232	4	,	,	PUNCT
cana-5423	232	5	d	d	NOUN
cana-5423	232	6	)	)	PUNCT
cana-5423	232	7	be	be	AUX
cana-5423	232	8	a	a	DET
cana-5423	232	9	complete	complete	ADJ
cana-5423	232	10	metric	metric	ADJ
cana-5423	232	11	space	space	NOUN
cana-5423	232	12	and	and	CCONJ
cana-5423	232	13	γ	γ	X
cana-5423	232	14	:	:	PUNCT
cana-5423	232	15	x	x	SYM
cana-5423	232	16	→	→	PUNCT
cana-5423	232	17	x	x	X
cana-5423	232	18	is	be	AUX
cana-5423	232	19	a	a	DET
cana-5423	232	20	generalized	generalized	ADJ
cana-5423	232	21	α	α	NOUN
cana-5423	232	22	-	-	ADJ
cana-5423	232	23	admissible	admissible	ADJ
cana-5423	232	24	modified	modify	VERB
cana-5423	232	25	almost	almost	ADV
cana-5423	232	26	zcontraction	zcontraction	NOUN
cana-5423	232	27	with	with	ADP
cana-5423	232	28	respect	respect	NOUN
cana-5423	232	29	to	to	ADP
cana-5423	232	30	ζsatisfying	ζsatisfye	VERB
cana-5423	232	31	the	the	DET
cana-5423	232	32	following	follow	VERB
cana-5423	232	33	conditions	condition	NOUN
cana-5423	232	34	:	:	PUNCT
cana-5423	232	35	(	(	PUNCT
cana-5423	232	36	i	i	NOUN
cana-5423	232	37	)	)	PUNCT
cana-5423	232	38	γ	γ	PROPN
cana-5423	232	39	is	be	AUX
cana-5423	232	40	triangular	triangular	NOUN
cana-5423	232	41	αorbital	αorbital	ADJ
cana-5423	232	42	admissible	admissible	ADJ
cana-5423	232	43	;	;	PUNCT
cana-5423	232	44	(	(	PUNCT
cana-5423	232	45	ii	ii	NOUN
cana-5423	232	46	)	)	PUNCT
cana-5423	232	47	there	there	PRON
cana-5423	232	48	exists	exist	VERB
cana-5423	232	49	x0	x0	PROPN
cana-5423	232	50	∈	∈	PROPN
cana-5423	232	51	xsuch	xsuch	PROPN
cana-5423	233	1	that	that	PRON
cana-5423	233	2	α(x0,γx0	α(x0,γx0	VERB
cana-5423	233	3	)	)	PUNCT
cana-5423	233	4	≥	≥	NOUN
cana-5423	233	5	1	1	NUM
cana-5423	233	6	;	;	PUNCT
cana-5423	233	7	(	(	PUNCT
cana-5423	233	8	iii	iii	X
cana-5423	233	9	)	)	PUNCT
cana-5423	233	10	if	if	SCONJ
cana-5423	233	11	{	{	PUNCT
cana-5423	233	12	xn	xn	X
cana-5423	233	13	}	}	PUNCT
cana-5423	233	14	is	be	AUX
cana-5423	233	15	a	a	DET
cana-5423	233	16	sequence	sequence	NOUN
cana-5423	233	17	in	in	ADP
cana-5423	233	18	x	x	INTJ
cana-5423	233	19	such	such	ADJ
cana-5423	233	20	that	that	SCONJ
cana-5423	233	21	α(xn	α(xn	NOUN
cana-5423	233	22	,	,	PUNCT
cana-5423	233	23	xn+1	xn+1	NUM
cana-5423	233	24	)	)	PUNCT
cana-5423	233	25	≤	≤	NUM
cana-5423	233	26	1	1	NUM
cana-5423	233	27	for	for	ADP
cana-5423	233	28	all	all	PRON
cana-5423	233	29	n	n	PRON
cana-5423	233	30	∈	∈	NOUN
cana-5423	233	31	n	n	NOUN
cana-5423	233	32	∪	∪	X
cana-5423	233	33	{	{	PUNCT
cana-5423	233	34	0	0	NUM
cana-5423	233	35	}	}	PUNCT
cana-5423	233	36	and	and	CCONJ
cana-5423	233	37	xn	xn	PROPN
cana-5423	234	1	→	→	SYM
cana-5423	234	2	x	x	PUNCT
cana-5423	234	3	∈	∈	PROPN
cana-5423	234	4	x	x	PUNCT
cana-5423	234	5	as	as	ADP
cana-5423	234	6	n→	n→	PROPN
cana-5423	234	7	∞	∞	PROPN
cana-5423	234	8	,	,	PUNCT
cana-5423	234	9	then	then	ADV
cana-5423	234	10	there	there	PRON
cana-5423	234	11	exists	exist	VERB
cana-5423	234	12	a	a	DET
cana-5423	234	13	subsequence	subsequence	NOUN
cana-5423	234	14	{	{	PUNCT
cana-5423	234	15	xn(k	xn(k	NUM
cana-5423	234	16	)	)	PUNCT
cana-5423	234	17	}	}	PUNCT
cana-5423	234	18	of	of	ADP
cana-5423	234	19	{	{	PUNCT
cana-5423	234	20	xn	xn	NOUN
cana-5423	234	21	}	}	PUNCT
cana-5423	234	22	such	such	ADJ
cana-5423	234	23	that	that	SCONJ
cana-5423	234	24	α(xn(k	α(xn(k	NUM
cana-5423	234	25	)	)	PUNCT
cana-5423	234	26	,	,	PUNCT
cana-5423	234	27	x∗	x∗	PROPN
cana-5423	234	28	)	)	PUNCT
cana-5423	234	29	≤	≤	NUM
cana-5423	234	30	1	1	NUM
cana-5423	234	31	;	;	PUNCT
cana-5423	234	32	(	(	PUNCT
cana-5423	234	33	iv	iv	X
cana-5423	234	34	)	)	PUNCT
cana-5423	234	35	α(x	α(x	PROPN
cana-5423	234	36	,	,	PUNCT
cana-5423	234	37	y	y	PROPN
cana-5423	234	38	)	)	PUNCT
cana-5423	234	39	≥	≥	NOUN
cana-5423	234	40	1	1	NUM
cana-5423	234	41	,	,	PUNCT
cana-5423	234	42	for	for	ADP
cana-5423	234	43	all	all	DET
cana-5423	234	44	x	x	NOUN
cana-5423	234	45	,	,	PUNCT
cana-5423	234	46	y	y	PROPN
cana-5423	234	47	∈	∈	PROPN
cana-5423	234	48	fix(γ	fix(γ	PROPN
cana-5423	234	49	)	)	PUNCT
cana-5423	234	50	,	,	PUNCT
cana-5423	234	51	where	where	SCONJ
cana-5423	234	52	fix(γ	fix(γ	PROPN
cana-5423	234	53	)	)	PUNCT
cana-5423	234	54	denotes	denote	VERB
cana-5423	234	55	the	the	DET
cana-5423	234	56	set	set	NOUN
cana-5423	234	57	of	of	ADP
cana-5423	234	58	fixed	fix	VERB
cana-5423	234	59	point	point	NOUN
cana-5423	234	60	of	of	ADP
cana-5423	234	61	γ	γ	PROPN
cana-5423	234	62	.	.	PUNCT
cana-5423	235	1	then	then	ADV
cana-5423	235	2	,	,	PUNCT
cana-5423	235	3	γ	γ	PROPN
cana-5423	235	4	has	have	VERB
cana-5423	235	5	a	a	DET
cana-5423	235	6	unique	unique	ADJ
cana-5423	235	7	fixed	fix	VERB
cana-5423	235	8	point	point	NOUN
cana-5423	235	9	x∗	x∗	PROPN
cana-5423	235	10	∈	∈	PROPN
cana-5423	235	11	x.	x.	NOUN
cana-5423	235	12	proof	proof	NOUN
cana-5423	235	13	.	.	PUNCT
cana-5423	236	1	by(ii	by(ii	PROPN
cana-5423	236	2	)	)	PUNCT
cana-5423	236	3	,	,	PUNCT
cana-5423	236	4	suppose	suppose	VERB
cana-5423	236	5	x0	x0	PROPN
cana-5423	236	6	∈	∈	PROPN
cana-5423	236	7	x	x	PUNCT
cana-5423	236	8	such	such	ADJ
cana-5423	236	9	that	that	DET
cana-5423	236	10	α(x0,γx0	α(x0,γx0	PROPN
cana-5423	236	11	)	)	PUNCT
cana-5423	236	12	≥	≥	NOUN
cana-5423	236	13	1	1	NUM
cana-5423	236	14	.	.	PUNCT
cana-5423	237	1	there	there	PRON
cana-5423	237	2	exists	exist	VERB
cana-5423	237	3	xn	xn	PROPN
cana-5423	237	4	∈	∈	PROPN
cana-5423	237	5	x	x	PUNCT
cana-5423	237	6	such	such	ADJ
cana-5423	237	7	that	that	PRON
cana-5423	237	8	xn+1	xn+1	PROPN
cana-5423	237	9	=	=	SYM
cana-5423	237	10	γxn	γxn	PROPN
cana-5423	237	11	,	,	PUNCT
cana-5423	237	12	for	for	ADP
cana-5423	237	13	all	all	PRON
cana-5423	237	14	n	n	DET
cana-5423	237	15	∈	∈	PROPN
cana-5423	237	16	n.	n.	NOUN
cana-5423	237	17	we	we	PRON
cana-5423	237	18	have	have	VERB
cana-5423	237	19	by	by	ADP
cana-5423	237	20	theorem	theorem	NOUN
cana-5423	237	21	2.2	2.2	NUM
cana-5423	237	22	,	,	PUNCT
cana-5423	237	23	{	{	PUNCT
cana-5423	237	24	xn	xn	X
cana-5423	237	25	}	}	PUNCT
cana-5423	237	26	is	be	AUX
cana-5423	237	27	a	a	DET
cana-5423	237	28	cauchy	cauchy	ADJ
cana-5423	237	29	sequence	sequence	NOUN
cana-5423	237	30	such	such	ADJ
cana-5423	237	31	that	that	SCONJ
cana-5423	237	32	lim	lim	PROPN
cana-5423	237	33	n→∞	n→∞	X
cana-5423	237	34	d(xn	d(xn	PROPN
cana-5423	237	35	,	,	PUNCT
cana-5423	237	36	xn+1	xn+1	NUM
cana-5423	237	37	)	)	PUNCT
cana-5423	238	1	=	=	SYM
cana-5423	238	2	0	0	X
cana-5423	238	3	.	.	PUNCT
cana-5423	239	1	since	since	SCONJ
cana-5423	239	2	(	(	PUNCT
cana-5423	239	3	x	x	X
cana-5423	239	4	,	,	PUNCT
cana-5423	239	5	d	d	NOUN
cana-5423	239	6	)	)	PUNCT
cana-5423	239	7	is	be	AUX
cana-5423	239	8	complete	complete	ADJ
cana-5423	239	9	,	,	PUNCT
cana-5423	239	10	there	there	PRON
cana-5423	239	11	exists	exist	VERB
cana-5423	239	12	x∗	x∗	PROPN
cana-5423	239	13	∈	∈	PROPN
cana-5423	239	14	x	x	PUNCT
cana-5423	240	1	such	such	ADJ
cana-5423	240	2	that	that	PRON
cana-5423	240	3	xn	xn	PROPN
cana-5423	240	4	→	→	SYM
cana-5423	240	5	x∗.by	x∗.by	PROPN
cana-5423	240	6	(	(	PUNCT
cana-5423	240	7	15	15	NUM
cana-5423	240	8	)	)	PUNCT
cana-5423	240	9	and	and	CCONJ
cana-5423	240	10	the	the	DET
cana-5423	240	11	condition	condition	NOUN
cana-5423	240	12	(	(	PUNCT
cana-5423	240	13	iii	iii	NOUN
cana-5423	240	14	)	)	PUNCT
cana-5423	240	15	,	,	PUNCT
cana-5423	240	16	there	there	PRON
cana-5423	240	17	exists	exist	VERB
cana-5423	240	18	a	a	DET
cana-5423	240	19	subsequence	subsequence	NOUN
cana-5423	240	20	{	{	PUNCT
cana-5423	240	21	xn(k	xn(k	NUM
cana-5423	240	22	)	)	PUNCT
cana-5423	240	23	}	}	PUNCT
cana-5423	240	24	of	of	ADP
cana-5423	240	25	{	{	PUNCT
cana-5423	240	26	xn	xn	NOUN
cana-5423	240	27	}	}	PUNCT
cana-5423	240	28	such	such	ADJ
cana-5423	240	29	that	that	SCONJ
cana-5423	240	30	α(xn(k	α(xn(k	NUM
cana-5423	240	31	)	)	PUNCT
cana-5423	240	32	,	,	PUNCT
cana-5423	240	33	x∗	x∗	PROPN
cana-5423	240	34	)	)	PUNCT
cana-5423	240	35	≤	≤	NUM
cana-5423	240	36	1	1	NUM
cana-5423	240	37	for	for	ADP
cana-5423	240	38	all	all	DET
cana-5423	240	39	k	k	PROPN
cana-5423	240	40	∈	∈	PROPN
cana-5423	240	41	n.	n.	NOUN
cana-5423	240	42	using	use	VERB
cana-5423	240	43	(	(	PUNCT
cana-5423	240	44	10	10	NUM
cana-5423	240	45	)	)	PUNCT
cana-5423	240	46	,	,	PUNCT
cana-5423	240	47	communications	communication	NOUN
cana-5423	240	48	on	on	ADP
cana-5423	240	49	applied	apply	VERB
cana-5423	240	50	nonlinear	nonlinear	ADJ
cana-5423	240	51	analysis	analysis	NOUN
cana-5423	240	52	issn	issn	NOUN
cana-5423	240	53	:	:	PUNCT
cana-5423	240	54	1074	1074	NUM
cana-5423	240	55	-	-	PUNCT
cana-5423	240	56	133x	133x	NUM
cana-5423	240	57	vol	vol	NOUN
cana-5423	240	58	32	32	NUM
cana-5423	240	59	no	no	NOUN
cana-5423	240	60	.	.	PUNCT
cana-5423	241	1	10s(2025	10s(2025	NUM
cana-5423	241	2	)	)	PUNCT
cana-5423	241	3	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	241	4	2225	2225	NUM
cana-5423	241	5	we	we	PRON
cana-5423	241	6	have	have	VERB
cana-5423	241	7	0	0	NUM
cana-5423	241	8	≤	≤	NUM
cana-5423	241	9	ζ(α(xnk	ζ(α(xnk	NOUN
cana-5423	241	10	,	,	PUNCT
cana-5423	241	11	γxnk	γxnk	PROPN
cana-5423	241	12	)	)	PUNCT
cana-5423	241	13	,	,	PUNCT
cana-5423	241	14	α(x∗,γx∗	α(x∗,γx∗	PROPN
cana-5423	241	15	)	)	PUNCT
cana-5423	241	16	,	,	PUNCT
cana-5423	241	17	d(γxnk	d(γxnk	NOUN
cana-5423	241	18	,	,	PUNCT
cana-5423	241	19	γx∗)k(xnk	γx∗)k(xnk	PROPN
cana-5423	241	20	,	,	PUNCT
cana-5423	241	21	x∗	x∗	PROPN
cana-5423	241	22	)	)	PUNCT
cana-5423	242	1	+	+	CCONJ
cana-5423	242	2	lq(xnk	lq(xnk	NOUN
cana-5423	242	3	,	,	PUNCT
cana-5423	242	4	x∗	x∗	PROPN
cana-5423	242	5	)	)	PUNCT
cana-5423	242	6	=	=	SYM
cana-5423	242	7	ζ(α(xnk	ζ(α(xnk	NOUN
cana-5423	242	8	,	,	PUNCT
cana-5423	242	9	xnk+1	xnk+1	PROPN
cana-5423	242	10	)	)	PUNCT
cana-5423	242	11	,	,	PUNCT
cana-5423	242	12	α(x∗,γx∗)d(xnk+1,γx	α(x∗,γx∗)d(xnk+1,γx	PROPN
cana-5423	242	13	∗),k(xnk	∗),k(xnk	PROPN
cana-5423	242	14	,	,	PUNCT
cana-5423	242	15	x∗	x∗	PROPN
cana-5423	242	16	)	)	PUNCT
cana-5423	243	1	+	+	CCONJ
cana-5423	243	2	lq(xnk	lq(xnk	NOUN
cana-5423	243	3	,	,	PUNCT
cana-5423	243	4	x∗	x∗	PROPN
cana-5423	243	5	)	)	PUNCT
cana-5423	244	1	<	<	X
cana-5423	244	2	k(xnk	k(xnk	NOUN
cana-5423	244	3	,	,	PUNCT
cana-5423	244	4	x∗	x∗	PROPN
cana-5423	244	5	)	)	PUNCT
cana-5423	245	1	+	+	CCONJ
cana-5423	245	2	lq(xnk	lq(xnk	NOUN
cana-5423	245	3	,	,	PUNCT
cana-5423	245	4	x∗	x∗	PROPN
cana-5423	245	5	)	)	PUNCT
cana-5423	245	6	−	−	PROPN
cana-5423	245	7	α(xnk	α(xnk	INTJ
cana-5423	245	8	,	,	PUNCT
cana-5423	245	9	xnk+1	xnk+1	PROPN
cana-5423	245	10	)	)	PUNCT
cana-5423	245	11	,	,	PUNCT
cana-5423	245	12	α(x∗,γx∗	α(x∗,γx∗	PROPN
cana-5423	245	13	)	)	PUNCT
cana-5423	245	14	,	,	PUNCT
cana-5423	245	15	d(xnk+1,γx	d(xnk+1,γx	VERB
cana-5423	245	16	∗	∗	NOUN
cana-5423	245	17	)	)	PUNCT
cana-5423	245	18	.	.	PUNCT
cana-5423	246	1	(	(	PUNCT
cana-5423	246	2	49	49	NUM
cana-5423	246	3	)	)	PUNCT
cana-5423	246	4	hence	hence	ADV
cana-5423	246	5	,	,	PUNCT
cana-5423	246	6	d(xnk+1,γx	d(xnk+1,γx	NOUN
cana-5423	246	7	∗	∗	NOUN
cana-5423	246	8	)	)	PUNCT
cana-5423	246	9	≤	≤	NUM
cana-5423	246	10	α(xnk	α(xnk	NOUN
cana-5423	246	11	,	,	PUNCT
cana-5423	246	12	xnk+1	xnk+1	PROPN
cana-5423	246	13	)	)	PUNCT
cana-5423	246	14	,	,	PUNCT
cana-5423	246	15	α(x∗,γx∗)d(xnk+1,γx	α(x∗,γx∗)d(xnk+1,γx	PROPN
cana-5423	246	16	∗	∗	NOUN
cana-5423	246	17	)	)	PUNCT
cana-5423	246	18	<	<	X
cana-5423	246	19	k(xnk	k(xnk	NOUN
cana-5423	246	20	,	,	PUNCT
cana-5423	246	21	x∗	x∗	PROPN
cana-5423	246	22	)	)	PUNCT
cana-5423	247	1	+	+	CCONJ
cana-5423	247	2	lq(xnk	lq(xnk	NOUN
cana-5423	247	3	,	,	PUNCT
cana-5423	247	4	x∗	x∗	PROPN
cana-5423	247	5	)	)	PUNCT
cana-5423	247	6	.	.	PUNCT
cana-5423	248	1	(	(	PUNCT
cana-5423	248	2	50	50	NUM
cana-5423	248	3	)	)	PUNCT
cana-5423	248	4	also	also	ADV
cana-5423	248	5	,	,	PUNCT
cana-5423	248	6	where	where	SCONJ
cana-5423	248	7	k(xnk	k(xnk	NOUN
cana-5423	248	8	,	,	PUNCT
cana-5423	248	9	x∗	x∗	PROPN
cana-5423	248	10	)	)	PUNCT
cana-5423	248	11	=	=	SYM
cana-5423	248	12	max{d(xnk	max{d(xnk	ADJ
cana-5423	248	13	,	,	PUNCT
cana-5423	248	14	x∗	x∗	PROPN
cana-5423	248	15	)	)	PUNCT
cana-5423	248	16	,	,	PUNCT
cana-5423	249	1	[	[	X
cana-5423	249	2	1	1	NUM
cana-5423	249	3	+	+	NUM
cana-5423	249	4	d(xnk	d(xnk	NOUN
cana-5423	249	5	,	,	PUNCT
cana-5423	249	6	γxnk	γxnk	PROPN
cana-5423	249	7	)	)	PUNCT
cana-5423	249	8	]	]	PUNCT
cana-5423	249	9	d(x∗,γx∗	d(x∗,γx∗	X
cana-5423	249	10	)	)	PUNCT
cana-5423	249	11	1	1	NUM
cana-5423	250	1	+	+	CCONJ
cana-5423	250	2	d(xnk	d(xnk	PROPN
cana-5423	250	3	,	,	PUNCT
cana-5423	250	4	x∗	x∗	PROPN
cana-5423	250	5	)	)	PUNCT
cana-5423	251	1	[	[	X
cana-5423	251	2	1	1	NUM
cana-5423	251	3	+	+	NUM
cana-5423	251	4	d(xnk	d(xnk	NOUN
cana-5423	251	5	,	,	PUNCT
cana-5423	251	6	γx∗)]d(x∗,γxnk	γx∗)]d(x∗,γxnk	NOUN
cana-5423	251	7	)	)	PUNCT
cana-5423	251	8	1	1	NUM
cana-5423	252	1	+	+	CCONJ
cana-5423	252	2	d(xnk	d(xnk	PROPN
cana-5423	252	3	,	,	PUNCT
cana-5423	252	4	x∗	x∗	PROPN
cana-5423	252	5	)	)	PUNCT
cana-5423	252	6	}	}	PUNCT
cana-5423	253	1	=	=	SYM
cana-5423	253	2	max{d(xnk	max{d(xnk	ADJ
cana-5423	253	3	,	,	PUNCT
cana-5423	253	4	x∗	x∗	PROPN
cana-5423	253	5	)	)	PUNCT
cana-5423	253	6	,	,	PUNCT
cana-5423	254	1	[	[	X
cana-5423	254	2	1	1	NUM
cana-5423	254	3	+	+	NUM
cana-5423	254	4	d(xnk	d(xnk	PROPN
cana-5423	254	5	,	,	PUNCT
cana-5423	254	6	xnk+1)]d(x∗,γx∗	xnk+1)]d(x∗,γx∗	PROPN
cana-5423	254	7	)	)	PUNCT
cana-5423	254	8	1	1	NUM
cana-5423	255	1	+	+	CCONJ
cana-5423	255	2	d(xnk	d(xnk	PROPN
cana-5423	255	3	,	,	PUNCT
cana-5423	255	4	x∗	x∗	PROPN
cana-5423	255	5	)	)	PUNCT
cana-5423	255	6	,	,	PUNCT
cana-5423	256	1	[	[	X
cana-5423	256	2	1	1	NUM
cana-5423	256	3	+	+	NUM
cana-5423	256	4	d(xnk	d(xnk	PROPN
cana-5423	256	5	,	,	PUNCT
cana-5423	256	6	γx∗)]d(x∗	γx∗)]d(x∗	PROPN
cana-5423	256	7	,	,	PUNCT
cana-5423	256	8	xnk+1	xnk+1	PROPN
cana-5423	256	9	)	)	PUNCT
cana-5423	256	10	1	1	NUM
cana-5423	257	1	+	+	CCONJ
cana-5423	257	2	d(xnk	d(xnk	PROPN
cana-5423	257	3	,	,	PUNCT
cana-5423	257	4	x∗	x∗	PROPN
cana-5423	257	5	)	)	PUNCT
cana-5423	257	6	}	}	PUNCT
cana-5423	257	7	.	.	PUNCT
cana-5423	258	1	(	(	PUNCT
cana-5423	258	2	51	51	NUM
cana-5423	258	3	)	)	PUNCT
cana-5423	258	4	and	and	CCONJ
cana-5423	258	5	q(xnk	q(xnk	VERB
cana-5423	258	6	,	,	PUNCT
cana-5423	258	7	x∗	x∗	PROPN
cana-5423	258	8	)	)	PUNCT
cana-5423	258	9	=	=	SYM
cana-5423	258	10	min{d(xnk	min{d(xnk	PROPN
cana-5423	258	11	,	,	PUNCT
cana-5423	258	12	γxnk	γxnk	PROPN
cana-5423	258	13	)	)	PUNCT
cana-5423	258	14	,	,	PUNCT
cana-5423	258	15	d(x∗,γx∗	d(x∗,γx∗	PROPN
cana-5423	258	16	)	)	PUNCT
cana-5423	258	17	,	,	PUNCT
cana-5423	258	18	d(xnk	d(xnk	PROPN
cana-5423	258	19	,	,	PUNCT
cana-5423	258	20	γx∗	γx∗	PROPN
cana-5423	258	21	)	)	PUNCT
cana-5423	258	22	,	,	PUNCT
cana-5423	258	23	d(x∗,γxnk	d(x∗,γxnk	NOUN
cana-5423	258	24	)	)	PUNCT
cana-5423	258	25	,	,	PUNCT
cana-5423	258	26	d(xnk	d(xnk	PROPN
cana-5423	258	27	,	,	PUNCT
cana-5423	258	28	γx∗)d(x∗,γxnk	γx∗)d(x∗,γxnk	PROPN
cana-5423	258	29	)	)	PUNCT
cana-5423	258	30	1	1	NUM
cana-5423	259	1	+	+	CCONJ
cana-5423	259	2	d(xnk	d(xnk	PROPN
cana-5423	259	3	,	,	PUNCT
cana-5423	259	4	x∗	x∗	PROPN
cana-5423	259	5	)	)	PUNCT
cana-5423	259	6	,	,	PUNCT
cana-5423	259	7	d(xnk	d(xnk	PROPN
cana-5423	259	8	,	,	PUNCT
cana-5423	259	9	γx∗)d(x∗,γx∗	γx∗)d(x∗,γx∗	PROPN
cana-5423	259	10	)	)	PUNCT
cana-5423	259	11	1	1	NUM
cana-5423	260	1	+	+	CCONJ
cana-5423	260	2	d(xnk	d(xnk	PROPN
cana-5423	260	3	,	,	PUNCT
cana-5423	260	4	x∗	x∗	PROPN
cana-5423	260	5	)	)	PUNCT
cana-5423	260	6	}	}	PUNCT
cana-5423	260	7	=	=	PUNCT
cana-5423	260	8	min{d(xnk	min{d(xnk	PROPN
cana-5423	260	9	,	,	PUNCT
cana-5423	260	10	xnk+1	xnk+1	PROPN
cana-5423	260	11	)	)	PUNCT
cana-5423	260	12	,	,	PUNCT
cana-5423	260	13	d(x∗,γx∗	d(x∗,γx∗	PROPN
cana-5423	260	14	)	)	PUNCT
cana-5423	260	15	,	,	PUNCT
cana-5423	260	16	d(xnk	d(xnk	PROPN
cana-5423	260	17	,	,	PUNCT
cana-5423	260	18	γx∗	γx∗	NOUN
cana-5423	260	19	)	)	PUNCT
cana-5423	260	20	,	,	PUNCT
cana-5423	260	21	d(x∗	d(x∗	NOUN
cana-5423	260	22	,	,	PUNCT
cana-5423	260	23	xnk+1	xnk+1	PROPN
cana-5423	260	24	)	)	PUNCT
cana-5423	260	25	,	,	PUNCT
cana-5423	260	26	d(xnk	d(xnk	PROPN
cana-5423	260	27	,	,	PUNCT
cana-5423	260	28	γx∗)d(x∗	γx∗)d(x∗	PROPN
cana-5423	260	29	,	,	PUNCT
cana-5423	260	30	xnk+1	xnk+1	PROPN
cana-5423	260	31	)	)	PUNCT
cana-5423	260	32	1	1	NUM
cana-5423	261	1	+	+	CCONJ
cana-5423	261	2	d(xnk	d(xnk	PROPN
cana-5423	261	3	,	,	PUNCT
cana-5423	261	4	x∗	x∗	PROPN
cana-5423	261	5	)	)	PUNCT
cana-5423	261	6	,	,	PUNCT
cana-5423	261	7	d(xnk	d(xnk	PROPN
cana-5423	261	8	,	,	PUNCT
cana-5423	261	9	γx∗)d(x∗,γx∗	γx∗)d(x∗,γx∗	PROPN
cana-5423	261	10	)	)	PUNCT
cana-5423	261	11	1	1	NUM
cana-5423	262	1	+	+	CCONJ
cana-5423	262	2	d(xnk	d(xnk	PROPN
cana-5423	262	3	,	,	PUNCT
cana-5423	262	4	x∗	x∗	PROPN
cana-5423	262	5	)	)	PUNCT
cana-5423	262	6	}	}	PUNCT
cana-5423	262	7	.	.	PUNCT
cana-5423	263	1	(	(	PUNCT
cana-5423	263	2	52	52	X
cana-5423	263	3	)	)	PUNCT
cana-5423	263	4	taking	take	VERB
cana-5423	263	5	k	k	X
cana-5423	263	6	→	→	SYM
cana-5423	263	7	0	0	NUM
cana-5423	263	8	in	in	ADP
cana-5423	263	9	the	the	DET
cana-5423	263	10	equation(50	equation(50	ADJ
cana-5423	263	11	)	)	PUNCT
cana-5423	263	12	and(51	and(51	PROPN
cana-5423	263	13	)	)	PUNCT
cana-5423	263	14	we	we	PRON
cana-5423	263	15	derive	derive	VERB
cana-5423	263	16	that	that	SCONJ
cana-5423	263	17	k(xnk	k(xnk	NOUN
cana-5423	263	18	,	,	PUNCT
cana-5423	263	19	x	x	NOUN
cana-5423	263	20	∗	∗	NOUN
cana-5423	263	21	)	)	PUNCT
cana-5423	263	22	=	=	SYM
cana-5423	263	23	d(x∗,γx∗	d(x∗,γx∗	PROPN
cana-5423	263	24	)	)	PUNCT
cana-5423	263	25	and	and	CCONJ
cana-5423	263	26	q(xnk	q(xnk	PROPN
cana-5423	263	27	,	,	PUNCT
cana-5423	263	28	x∗	x∗	PROPN
cana-5423	263	29	)	)	PUNCT
cana-5423	263	30	=	=	SYM
cana-5423	264	1	0	0	X
cana-5423	264	2	.	.	PUNCT
cana-5423	265	1	(	(	PUNCT
cana-5423	265	2	53	53	NUM
cana-5423	265	3	)	)	PUNCT
cana-5423	265	4	from	from	ADP
cana-5423	265	5	(	(	PUNCT
cana-5423	265	6	50	50	NUM
cana-5423	265	7	)	)	PUNCT
cana-5423	265	8	,	,	PUNCT
cana-5423	265	9	by	by	ADP
cana-5423	265	10	using	use	VERB
cana-5423	265	11	(	(	PUNCT
cana-5423	265	12	53	53	NUM
cana-5423	265	13	)	)	PUNCT
cana-5423	265	14	,	,	PUNCT
cana-5423	265	15	we	we	PRON
cana-5423	265	16	get	get	VERB
cana-5423	265	17	d(xnk+1,γx	d(xnk+1,γx	NOUN
cana-5423	265	18	∗	∗	NOUN
cana-5423	265	19	)	)	PUNCT
cana-5423	265	20	<	<	X
cana-5423	265	21	d(x∗,γx∗)for	d(x∗,γx∗)for	VERB
cana-5423	265	22	all	all	DET
cana-5423	265	23	k	k	PROPN
cana-5423	265	24	∈	∈	PROPN
cana-5423	265	25	n	n	CCONJ
cana-5423	265	26	(	(	PUNCT
cana-5423	265	27	54	54	NUM
cana-5423	265	28	)	)	PUNCT
cana-5423	265	29	by	by	ADP
cana-5423	265	30	(	(	PUNCT
cana-5423	265	31	49	49	NUM
cana-5423	265	32	)	)	PUNCT
cana-5423	265	33	,	,	PUNCT
cana-5423	265	34	(	(	PUNCT
cana-5423	265	35	54	54	NUM
cana-5423	265	36	)	)	PUNCT
cana-5423	265	37	,	,	PUNCT
cana-5423	265	38	and	and	CCONJ
cana-5423	265	39	the	the	DET
cana-5423	265	40	condition	condition	NOUN
cana-5423	265	41	(	(	PUNCT
cana-5423	265	42	ζ3	ζ3	NOUN
cana-5423	265	43	)	)	PUNCT
cana-5423	265	44	,	,	PUNCT
cana-5423	265	45	we	we	PRON
cana-5423	265	46	have	have	VERB
cana-5423	265	47	0	0	NUM
cana-5423	265	48	≤	≤	NUM
cana-5423	266	1	lim	lim	PROPN
cana-5423	266	2	supn→∞ζ(α(xnk	supn→∞ζ(α(xnk	PROPN
cana-5423	266	3	,	,	PUNCT
cana-5423	266	4	γxnk	γxnk	PROPN
cana-5423	266	5	)	)	PUNCT
cana-5423	266	6	,	,	PUNCT
cana-5423	266	7	α(x∗,γx∗	α(x∗,γx∗	PROPN
cana-5423	266	8	)	)	PUNCT
cana-5423	266	9	,	,	PUNCT
cana-5423	266	10	d(γxnk	d(γxnk	NOUN
cana-5423	266	11	,	,	PUNCT
cana-5423	266	12	γx∗)k(xnk	γx∗)k(xnk	PROPN
cana-5423	266	13	,	,	PUNCT
cana-5423	266	14	x∗	x∗	PROPN
cana-5423	266	15	)	)	PUNCT
cana-5423	267	1	+	+	CCONJ
cana-5423	267	2	lq(xnk	lq(xnk	NOUN
cana-5423	267	3	,	,	PUNCT
cana-5423	267	4	x∗	x∗	PROPN
cana-5423	267	5	)	)	PUNCT
cana-5423	267	6	<	<	X
cana-5423	267	7	0	0	X
cana-5423	267	8	.	.	PUNCT
cana-5423	268	1	this	this	PRON
cana-5423	268	2	is	be	AUX
cana-5423	268	3	contradiction.hence	contradiction.hence	NOUN
cana-5423	268	4	therefore	therefore	ADV
cana-5423	268	5	,	,	PUNCT
cana-5423	268	6	x∗	x∗	PROPN
cana-5423	268	7	is	be	AUX
cana-5423	268	8	a	a	DET
cana-5423	268	9	fixed	fix	VERB
cana-5423	268	10	point	point	NOUN
cana-5423	268	11	og	og	PROPN
cana-5423	268	12	γ	γ	PROPN
cana-5423	268	13	.	.	PUNCT
cana-5423	269	1	now	now	ADV
cana-5423	269	2	,	,	PUNCT
cana-5423	269	3	assume	assume	VERB
cana-5423	269	4	that	that	SCONJ
cana-5423	269	5	,	,	PUNCT
cana-5423	269	6	there	there	PRON
cana-5423	269	7	exists	exist	VERB
cana-5423	269	8	x∗	x∗	PROPN
cana-5423	269	9	,	,	PUNCT
cana-5423	269	10	y∗	y∗	PROPN
cana-5423	269	11	∈	∈	PROPN
cana-5423	269	12	x	x	PUNCT
cana-5423	269	13	such	such	ADJ
cana-5423	269	14	that	that	DET
cana-5423	269	15	x∗	x∗	PROPN
cana-5423	269	16	=	=	PUNCT
cana-5423	269	17	γx∗	γx∗	NOUN
cana-5423	269	18	and	and	CCONJ
cana-5423	269	19	y∗	y∗	PROPN
cana-5423	269	20	=	=	PUNCT
cana-5423	269	21	γy∗	γy∗	VERB
cana-5423	269	22	with	with	ADP
cana-5423	269	23	x∗	x∗	PROPN
cana-5423	269	24	̸=	̸=	PROPN
cana-5423	269	25	y∗.	y∗.	NOUN
cana-5423	269	26	since	since	SCONJ
cana-5423	269	27	γ	γ	PROPN
cana-5423	269	28	is	be	AUX
cana-5423	269	29	generalized	generalize	VERB
cana-5423	269	30	α	α	PRON
cana-5423	269	31	-	-	ADJ
cana-5423	269	32	admissible	admissible	ADJ
cana-5423	269	33	modified	modify	VERB
cana-5423	269	34	almost	almost	ADV
cana-5423	269	35	z	z	NOUN
cana-5423	269	36	contraction	contraction	NOUN
cana-5423	269	37	selfmapping	selfmappe	VERB
cana-5423	269	38	of	of	ADP
cana-5423	269	39	a	a	DET
cana-5423	269	40	metric	metric	ADJ
cana-5423	269	41	space	space	NOUN
cana-5423	269	42	(	(	PUNCT
cana-5423	269	43	x	x	X
cana-5423	269	44	,	,	PUNCT
cana-5423	269	45	d	d	NOUN
cana-5423	269	46	)	)	PUNCT
cana-5423	269	47	.	.	PUNCT
cana-5423	270	1	so	so	ADV
cana-5423	270	2	by	by	ADP
cana-5423	270	3	assumption	assumption	NOUN
cana-5423	270	4	(	(	PUNCT
cana-5423	270	5	iv	iv	X
cana-5423	270	6	)	)	PUNCT
cana-5423	270	7	from	from	ADP
cana-5423	270	8	theorem	theorem	ADJ
cana-5423	270	9	2.3	2.3	NUM
cana-5423	270	10	,	,	PUNCT
cana-5423	270	11	we	we	PRON
cana-5423	270	12	hav	hav	VERB
cana-5423	270	13	α(x⋆	α(x⋆	NOUN
cana-5423	270	14	,	,	PUNCT
cana-5423	270	15	y⋆	y⋆	SYM
cana-5423	270	16	)	)	PUNCT
cana-5423	270	17	≥	≥	NOUN
cana-5423	270	18	1	1	NUM
cana-5423	270	19	.	.	PUNCT
cana-5423	271	1	(	(	PUNCT
cana-5423	271	2	55	55	NUM
cana-5423	271	3	)	)	PUNCT
cana-5423	271	4	therefore	therefore	ADV
cana-5423	271	5	,	,	PUNCT
cana-5423	271	6	from	from	ADP
cana-5423	271	7	(	(	PUNCT
cana-5423	271	8	10	10	NUM
cana-5423	271	9	)	)	PUNCT
cana-5423	271	10	and	and	CCONJ
cana-5423	271	11	|zeta2	|zeta2	NOUN
cana-5423	271	12	that	that	DET
cana-5423	271	13	,	,	PUNCT
cana-5423	271	14	0	0	NUM
cana-5423	271	15	≤	≤	NOUN
cana-5423	271	16	ζ(α(x⋆,γx⋆	ζ(α(x⋆,γx⋆	NOUN
cana-5423	271	17	)	)	PUNCT
cana-5423	271	18	,	,	PUNCT
cana-5423	271	19	α(y⋆,γy⋆)d(γx⋆,γy⋆),k(x⋆	α(y⋆,γy⋆)d(γx⋆,γy⋆),k(x⋆	PROPN
cana-5423	271	20	,	,	PUNCT
cana-5423	271	21	y⋆	y⋆	PRON
cana-5423	271	22	)	)	PUNCT
cana-5423	271	23	+	+	X
cana-5423	271	24	lq(x⋆	lq(x⋆	PROPN
cana-5423	271	25	,	,	PUNCT
cana-5423	271	26	y⋆	y⋆	NUM
cana-5423	271	27	)	)	PUNCT
cana-5423	271	28	=	=	SYM
cana-5423	272	1	ζ(α(x⋆	ζ(α(x⋆	NOUN
cana-5423	272	2	,	,	PUNCT
cana-5423	272	3	x⋆	x⋆	NUM
cana-5423	272	4	)	)	PUNCT
cana-5423	272	5	,	,	PUNCT
cana-5423	272	6	α(y⋆	α(y⋆	NOUN
cana-5423	272	7	,	,	PUNCT
cana-5423	272	8	y⋆	y⋆	ADV
cana-5423	272	9	)	)	PUNCT
cana-5423	272	10	,	,	PUNCT
cana-5423	272	11	d(x⋆	d(x⋆	NOUN
cana-5423	272	12	,	,	PUNCT
cana-5423	272	13	y⋆)k(x⋆	y⋆)k(x⋆	PROPN
cana-5423	272	14	,	,	PUNCT
cana-5423	272	15	y⋆	y⋆	CCONJ
cana-5423	272	16	)	)	PUNCT
cana-5423	272	17	+	+	X
cana-5423	272	18	lq(x⋆	lq(x⋆	PROPN
cana-5423	272	19	,	,	PUNCT
cana-5423	272	20	y⋆	y⋆	NUM
cana-5423	272	21	)	)	PUNCT
cana-5423	272	22	<	<	X
cana-5423	272	23	k(x⋆	k(x⋆	NOUN
cana-5423	272	24	,	,	PUNCT
cana-5423	272	25	y⋆	y⋆	CCONJ
cana-5423	272	26	)	)	PUNCT
cana-5423	272	27	+	+	X
cana-5423	272	28	lq(x⋆	lq(x⋆	PROPN
cana-5423	272	29	,	,	PUNCT
cana-5423	272	30	y⋆	y⋆	NUM
cana-5423	272	31	)	)	PUNCT
cana-5423	272	32	−	−	NOUN
cana-5423	272	33	α(x⋆	α(x⋆	NOUN
cana-5423	272	34	,	,	PUNCT
cana-5423	272	35	x⋆	x⋆	NUM
cana-5423	272	36	)	)	PUNCT
cana-5423	272	37	,	,	PUNCT
cana-5423	272	38	α(y⋆	α(y⋆	NOUN
cana-5423	272	39	,	,	PUNCT
cana-5423	272	40	y⋆	y⋆	ADV
cana-5423	272	41	)	)	PUNCT
cana-5423	272	42	,	,	PUNCT
cana-5423	272	43	d(x⋆	d(x⋆	NOUN
cana-5423	272	44	,	,	PUNCT
cana-5423	272	45	y⋆	y⋆	CCONJ
cana-5423	272	46	)	)	PUNCT
cana-5423	272	47	(	(	PUNCT
cana-5423	272	48	56	56	X
cana-5423	272	49	)	)	PUNCT
cana-5423	272	50	communications	communication	NOUN
cana-5423	272	51	on	on	ADP
cana-5423	272	52	applied	apply	VERB
cana-5423	272	53	nonlinear	nonlinear	ADJ
cana-5423	272	54	analysis	analysis	NOUN
cana-5423	272	55	issn	issn	NOUN
cana-5423	272	56	:	:	PUNCT
cana-5423	272	57	1074	1074	NUM
cana-5423	272	58	-	-	PUNCT
cana-5423	272	59	133x	133x	NUM
cana-5423	272	60	vol	vol	NOUN
cana-5423	272	61	32	32	NUM
cana-5423	272	62	no	no	NOUN
cana-5423	272	63	.	.	PUNCT
cana-5423	273	1	10s(2025	10s(2025	NUM
cana-5423	273	2	)	)	PUNCT
cana-5423	274	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	274	2	2226	2226	NUM
cana-5423	274	3	also	also	ADV
cana-5423	274	4	,	,	PUNCT
cana-5423	274	5	where	where	SCONJ
cana-5423	274	6	k(x⋆	k(x⋆	NOUN
cana-5423	274	7	,	,	PUNCT
cana-5423	274	8	y⋆	y⋆	SYM
cana-5423	274	9	)	)	PUNCT
cana-5423	275	1	=	=	SYM
cana-5423	275	2	max	max	PROPN
cana-5423	275	3	{	{	PUNCT
cana-5423	275	4	d(x∗	d(x∗	NOUN
cana-5423	275	5	,	,	PUNCT
cana-5423	275	6	y∗	y∗	PROPN
cana-5423	275	7	)	)	PUNCT
cana-5423	276	1	[	[	X
cana-5423	276	2	1	1	NUM
cana-5423	276	3	+	+	NUM
cana-5423	276	4	d(x∗	d(x∗	PROPN
cana-5423	276	5	,	,	PUNCT
cana-5423	276	6	x∗)]d(y∗	x∗)]d(y∗	PROPN
cana-5423	276	7	,	,	PUNCT
cana-5423	276	8	y∗	y∗	PROPN
cana-5423	276	9	)	)	PUNCT
cana-5423	276	10	1	1	NUM
cana-5423	277	1	+	+	CCONJ
cana-5423	277	2	d(x∗	d(x∗	PROPN
cana-5423	277	3	,	,	PUNCT
cana-5423	277	4	y∗	y∗	PROPN
cana-5423	277	5	)	)	PUNCT
cana-5423	277	6	,	,	PUNCT
cana-5423	278	1	[	[	X
cana-5423	278	2	1	1	NUM
cana-5423	278	3	+	+	NUM
cana-5423	278	4	d(x∗	d(x∗	NOUN
cana-5423	278	5	,	,	PUNCT
cana-5423	278	6	y∗)]d(y∗	y∗)]d(y∗	PROPN
cana-5423	278	7	,	,	PUNCT
cana-5423	278	8	x∗	x∗	PROPN
cana-5423	278	9	)	)	PUNCT
cana-5423	278	10	1	1	NUM
cana-5423	279	1	+	+	CCONJ
cana-5423	279	2	d(x∗	d(x∗	PROPN
cana-5423	279	3	,	,	PUNCT
cana-5423	279	4	y∗	y∗	PROPN
cana-5423	279	5	)	)	PUNCT
cana-5423	279	6	}	}	PUNCT
cana-5423	279	7	=	=	SYM
cana-5423	279	8	max{d(x∗	max{d(x∗	NOUN
cana-5423	279	9	,	,	PUNCT
cana-5423	279	10	y∗	y∗	PROPN
cana-5423	279	11	)	)	PUNCT
cana-5423	279	12	,	,	PUNCT
cana-5423	279	13	d(y	d(y	PROPN
cana-5423	279	14	∗	∗	NOUN
cana-5423	279	15	,	,	PUNCT
cana-5423	279	16	x∗	x∗	PROPN
cana-5423	279	17	)	)	PUNCT
cana-5423	279	18	}	}	PUNCT
cana-5423	279	19	=	=	SYM
cana-5423	279	20	d(x∗	d(x∗	NOUN
cana-5423	279	21	,	,	PUNCT
cana-5423	279	22	y∗	y∗	PROPN
cana-5423	279	23	)	)	PUNCT
cana-5423	279	24	(	(	PUNCT
cana-5423	279	25	57	57	NUM
cana-5423	279	26	)	)	PUNCT
cana-5423	279	27	and	and	CCONJ
cana-5423	279	28	q(x⋆	q(x⋆	NOUN
cana-5423	279	29	,	,	PUNCT
cana-5423	279	30	y⋆	y⋆	SYM
cana-5423	279	31	)	)	PUNCT
cana-5423	279	32	=	=	SYM
cana-5423	279	33	min{d(x∗	min{d(x∗	NOUN
cana-5423	279	34	,	,	PUNCT
cana-5423	279	35	x∗	x∗	PROPN
cana-5423	279	36	)	)	PUNCT
cana-5423	279	37	,	,	PUNCT
cana-5423	279	38	d(y∗	d(y∗	NOUN
cana-5423	279	39	,	,	PUNCT
cana-5423	279	40	y∗	y∗	PROPN
cana-5423	279	41	)	)	PUNCT
cana-5423	279	42	,	,	PUNCT
cana-5423	279	43	d(x∗	d(x∗	NOUN
cana-5423	279	44	,	,	PUNCT
cana-5423	279	45	y∗	y∗	PROPN
cana-5423	279	46	)	)	PUNCT
cana-5423	279	47	,	,	PUNCT
cana-5423	279	48	d(y∗	d(y∗	NOUN
cana-5423	279	49	,	,	PUNCT
cana-5423	279	50	x∗	x∗	PROPN
cana-5423	279	51	)	)	PUNCT
cana-5423	279	52	,	,	PUNCT
cana-5423	279	53	d(x∗	d(x∗	PROPN
cana-5423	279	54	,	,	PUNCT
cana-5423	279	55	y∗)d(y∗	y∗)d(y∗	PROPN
cana-5423	279	56	,	,	PUNCT
cana-5423	279	57	x∗	x∗	PROPN
cana-5423	279	58	)	)	PUNCT
cana-5423	279	59	1	1	NUM
cana-5423	280	1	+	+	CCONJ
cana-5423	280	2	d(x∗	d(x∗	PROPN
cana-5423	280	3	,	,	PUNCT
cana-5423	280	4	y∗	y∗	PROPN
cana-5423	280	5	)	)	PUNCT
cana-5423	280	6	,	,	PUNCT
cana-5423	280	7	d(x∗	d(x∗	PROPN
cana-5423	280	8	,	,	PUNCT
cana-5423	280	9	x∗)d(y∗	x∗)d(y∗	PROPN
cana-5423	280	10	,	,	PUNCT
cana-5423	280	11	y∗	y∗	PROPN
cana-5423	280	12	)	)	PUNCT
cana-5423	280	13	1	1	NUM
cana-5423	281	1	+	+	CCONJ
cana-5423	281	2	d(x∗	d(x∗	PROPN
cana-5423	281	3	,	,	PUNCT
cana-5423	281	4	y∗	y∗	PROPN
cana-5423	281	5	)	)	PUNCT
cana-5423	281	6	}	}	PUNCT
cana-5423	281	7	=	=	SYM
cana-5423	281	8	min	min	NOUN
cana-5423	281	9	{	{	PUNCT
cana-5423	281	10	0	0	NUM
cana-5423	281	11	,	,	PUNCT
cana-5423	281	12	0	0	NUM
cana-5423	281	13	,	,	PUNCT
cana-5423	281	14	d(x∗	d(x∗	NOUN
cana-5423	281	15	,	,	PUNCT
cana-5423	281	16	y∗	y∗	PROPN
cana-5423	281	17	)	)	PUNCT
cana-5423	281	18	,	,	PUNCT
cana-5423	281	19	d(y∗	d(y∗	NOUN
cana-5423	281	20	,	,	PUNCT
cana-5423	281	21	x∗	x∗	PROPN
cana-5423	281	22	)	)	PUNCT
cana-5423	281	23	,	,	PUNCT
cana-5423	281	24	d(x∗	d(x∗	PROPN
cana-5423	281	25	,	,	PUNCT
cana-5423	281	26	y∗)d(y∗	y∗)d(y∗	PROPN
cana-5423	281	27	,	,	PUNCT
cana-5423	281	28	x∗	x∗	PROPN
cana-5423	281	29	)	)	PUNCT
cana-5423	281	30	1	1	NUM
cana-5423	282	1	+	+	CCONJ
cana-5423	282	2	d(x∗	d(x∗	PROPN
cana-5423	282	3	,	,	PUNCT
cana-5423	282	4	y∗	y∗	PROPN
cana-5423	282	5	)	)	PUNCT
cana-5423	282	6	,	,	PUNCT
cana-5423	282	7	0	0	NUM
cana-5423	282	8	,	,	PUNCT
cana-5423	282	9	}	}	PUNCT
cana-5423	282	10	=	=	SYM
cana-5423	282	11	0	0	X
cana-5423	282	12	.	.	PUNCT
cana-5423	283	1	(	(	PUNCT
cana-5423	283	2	58	58	NUM
cana-5423	283	3	)	)	PUNCT
cana-5423	283	4	from	from	ADP
cana-5423	283	5	(	(	PUNCT
cana-5423	283	6	56	56	NUM
cana-5423	283	7	)	)	PUNCT
cana-5423	283	8	together	together	ADV
cana-5423	283	9	with	with	ADP
cana-5423	283	10	(	(	PUNCT
cana-5423	283	11	57	57	NUM
cana-5423	283	12	)	)	PUNCT
cana-5423	283	13	and	and	CCONJ
cana-5423	283	14	(	(	PUNCT
cana-5423	283	15	58	58	NUM
cana-5423	283	16	)	)	PUNCT
cana-5423	283	17	,	,	PUNCT
cana-5423	283	18	we	we	PRON
cana-5423	283	19	deduce	deduce	VERB
cana-5423	283	20	that	that	SCONJ
cana-5423	283	21	0	0	NUM
cana-5423	283	22	<	<	X
cana-5423	283	23	d(x∗	d(x∗	NOUN
cana-5423	283	24	,	,	PUNCT
cana-5423	283	25	y∗	y∗	PROPN
cana-5423	283	26	)	)	PUNCT
cana-5423	283	27	≤	≤	NOUN
cana-5423	283	28	α(x∗	α(x∗	NOUN
cana-5423	283	29	,	,	PUNCT
cana-5423	283	30	x∗	x∗	PROPN
cana-5423	283	31	)	)	PUNCT
cana-5423	283	32	,	,	PUNCT
cana-5423	283	33	α(y∗	α(y∗	NUM
cana-5423	283	34	,	,	PUNCT
cana-5423	283	35	y∗)d(x∗	y∗)d(x∗	PROPN
cana-5423	283	36	,	,	PUNCT
cana-5423	283	37	y∗	y∗	PROPN
cana-5423	283	38	)	)	PUNCT
cana-5423	283	39	<	<	X
cana-5423	283	40	d(x∗	d(x∗	PROPN
cana-5423	283	41	,	,	PUNCT
cana-5423	283	42	y∗	y∗	PROPN
cana-5423	283	43	)	)	PUNCT
cana-5423	283	44	.	.	PUNCT
cana-5423	284	1	this	this	PRON
cana-5423	284	2	is	be	AUX
cana-5423	284	3	contradiction	contradiction	NOUN
cana-5423	284	4	.	.	PUNCT
cana-5423	285	1	thus	thus	ADV
cana-5423	285	2	,	,	PUNCT
cana-5423	285	3	we	we	PRON
cana-5423	285	4	have	have	VERB
cana-5423	285	5	x⋆	x⋆	ADJ
cana-5423	286	1	=	=	PUNCT
cana-5423	287	1	y⋆.	y⋆.	NOUN
cana-5423	288	1	hence	hence	ADV
cana-5423	288	2	,	,	PUNCT
cana-5423	288	3	γ	γ	PROPN
cana-5423	288	4	has	have	VERB
cana-5423	288	5	a	a	DET
cana-5423	288	6	unique	unique	ADJ
cana-5423	288	7	fixed	fix	VERB
cana-5423	288	8	point	point	NOUN
cana-5423	288	9	.	.	PUNCT
cana-5423	289	1	□	□	PUNCT
cana-5423	289	2	corollary	corollary	ADJ
cana-5423	289	3	2.4	2.4	NUM
cana-5423	289	4	.	.	PUNCT
cana-5423	290	1	let	let	VERB
cana-5423	290	2	(	(	PUNCT
cana-5423	290	3	x	x	NOUN
cana-5423	290	4	,	,	PUNCT
cana-5423	290	5	d	d	NOUN
cana-5423	290	6	)	)	PUNCT
cana-5423	290	7	be	be	AUX
cana-5423	290	8	a	a	DET
cana-5423	290	9	metric	metric	ADJ
cana-5423	290	10	space	space	NOUN
cana-5423	290	11	,	,	PUNCT
cana-5423	290	12	γ	γ	X
cana-5423	290	13	:	:	PUNCT
cana-5423	290	14	x	x	SYM
cana-5423	290	15	→	→	PUNCT
cana-5423	290	16	x	x	X
cana-5423	290	17	is	be	AUX
cana-5423	290	18	a	a	DET
cana-5423	290	19	generalized	generalized	ADJ
cana-5423	290	20	αadmissible	αadmissible	ADJ
cana-5423	290	21	modified	modify	VERB
cana-5423	290	22	almost	almost	ADV
cana-5423	290	23	zcontraction	zcontraction	NOUN
cana-5423	290	24	self	self	NOUN
cana-5423	290	25	mapping	mapping	NOUN
cana-5423	290	26	.	.	PUNCT
cana-5423	291	1	there	there	PRON
cana-5423	291	2	exists	exist	VERB
cana-5423	291	3	ζ	ζ	PROPN
cana-5423	291	4	∈	∈	PROPN
cana-5423	291	5	z	z	NOUN
cana-5423	291	6	and	and	CCONJ
cana-5423	291	7	α	α	NOUN
cana-5423	291	8	:	:	PUNCT
cana-5423	292	1	x	x	SYM
cana-5423	292	2	×	×	NOUN
cana-5423	292	3	x	x	INTJ
cana-5423	292	4	→	→	X
cana-5423	292	5	[	[	X
cana-5423	292	6	0,∞	0,∞	NOUN
cana-5423	292	7	)	)	PUNCT
cana-5423	292	8	be	be	VERB
cana-5423	292	9	a	a	DET
cana-5423	292	10	function	function	NOUN
cana-5423	292	11	with	with	ADP
cana-5423	292	12	α(x	α(x	NOUN
cana-5423	292	13	,	,	PUNCT
cana-5423	292	14	γx	γx	NOUN
cana-5423	292	15	)	)	PUNCT
cana-5423	292	16	=	=	SYM
cana-5423	292	17	1	1	NUM
cana-5423	292	18	,	,	PUNCT
cana-5423	292	19	and	and	CCONJ
cana-5423	292	20	α(y	α(y	NOUN
cana-5423	292	21	,	,	PUNCT
cana-5423	292	22	γy	γy	NOUN
cana-5423	292	23	)	)	PUNCT
cana-5423	292	24	=	=	SYM
cana-5423	292	25	1	1	NUM
cana-5423	292	26	,	,	PUNCT
cana-5423	292	27	for	for	ADP
cana-5423	292	28	all	all	DET
cana-5423	292	29	x	x	NOUN
cana-5423	292	30	,	,	PUNCT
cana-5423	292	31	y	y	PROPN
cana-5423	292	32	∈	∈	PROPN
cana-5423	292	33	x	x	PUNCT
cana-5423	292	34	such	such	ADJ
cana-5423	292	35	that	that	DET
cana-5423	292	36	ζ(d(γx	ζ(d(γx	NOUN
cana-5423	292	37	,	,	PUNCT
cana-5423	292	38	γy)k(x	γy)k(x	PROPN
cana-5423	292	39	,	,	PUNCT
cana-5423	292	40	y	y	NOUN
cana-5423	292	41	)	)	PUNCT
cana-5423	293	1	+	+	NUM
cana-5423	293	2	lq(x	lq(x	X
cana-5423	293	3	,	,	PUNCT
cana-5423	293	4	y	y	NOUN
cana-5423	293	5	)	)	PUNCT
cana-5423	293	6	≥	≥	NOUN
cana-5423	293	7	0	0	NUM
cana-5423	293	8	,	,	PUNCT
cana-5423	293	9	forall	forall	NOUN
cana-5423	293	10	x	x	X
cana-5423	293	11	,	,	PUNCT
cana-5423	293	12	y	y	PROPN
cana-5423	293	13	∈	∈	PROPN
cana-5423	293	14	x	x	NOUN
cana-5423	293	15	,	,	PUNCT
cana-5423	293	16	andl	andl	NOUN
cana-5423	293	17	≥	≥	NOUN
cana-5423	293	18	0	0	NUM
cana-5423	293	19	,	,	PUNCT
cana-5423	293	20	also	also	ADV
cana-5423	293	21	where	where	SCONJ
cana-5423	293	22	k(x	k(x	PROPN
cana-5423	293	23	,	,	PUNCT
cana-5423	293	24	y	y	NOUN
cana-5423	293	25	)	)	PUNCT
cana-5423	293	26	=	=	SYM
cana-5423	293	27	max	max	PROPN
cana-5423	293	28	{	{	PUNCT
cana-5423	293	29	d(x	d(x	PROPN
cana-5423	293	30	,	,	PUNCT
cana-5423	293	31	y	y	PROPN
cana-5423	293	32	)	)	PUNCT
cana-5423	293	33	,	,	PUNCT
cana-5423	294	1	[	[	X
cana-5423	294	2	1	1	NUM
cana-5423	294	3	+	+	NUM
cana-5423	294	4	d(x	d(x	PROPN
cana-5423	294	5	,	,	PUNCT
cana-5423	294	6	γx)]d(y	γx)]d(y	NOUN
cana-5423	294	7	,	,	PUNCT
cana-5423	294	8	γy	γy	NOUN
cana-5423	294	9	)	)	PUNCT
cana-5423	294	10	1	1	NUM
cana-5423	295	1	+	+	CCONJ
cana-5423	295	2	d(x	d(x	PROPN
cana-5423	295	3	,	,	PUNCT
cana-5423	295	4	y	y	NOUN
cana-5423	295	5	)	)	PUNCT
cana-5423	295	6	,	,	PUNCT
cana-5423	296	1	[	[	X
cana-5423	296	2	1	1	NUM
cana-5423	296	3	+	+	NUM
cana-5423	296	4	d(x	d(x	PROPN
cana-5423	296	5	,	,	PUNCT
cana-5423	296	6	γy)]d(y	γy)]d(y	ADJ
cana-5423	296	7	,	,	PUNCT
cana-5423	296	8	γx	γx	NOUN
cana-5423	296	9	)	)	PUNCT
cana-5423	296	10	1	1	NUM
cana-5423	296	11	+	+	CCONJ
cana-5423	296	12	d(x	d(x	PROPN
cana-5423	296	13	,	,	PUNCT
cana-5423	296	14	y	y	NOUN
cana-5423	296	15	)	)	PUNCT
cana-5423	296	16	}	}	PUNCT
cana-5423	296	17	and	and	CCONJ
cana-5423	296	18	q(x	q(x	PROPN
cana-5423	296	19	,	,	PUNCT
cana-5423	296	20	y	y	NOUN
cana-5423	296	21	)	)	PUNCT
cana-5423	296	22	=	=	SYM
cana-5423	296	23	min	min	NOUN
cana-5423	296	24	{	{	PUNCT
cana-5423	296	25	d(x	d(x	PROPN
cana-5423	296	26	,	,	PUNCT
cana-5423	296	27	γx	γx	NOUN
cana-5423	296	28	)	)	PUNCT
cana-5423	296	29	,	,	PUNCT
cana-5423	296	30	d(y	d(y	PROPN
cana-5423	296	31	,	,	PUNCT
cana-5423	296	32	γy	γy	NOUN
cana-5423	296	33	)	)	PUNCT
cana-5423	296	34	,	,	PUNCT
cana-5423	296	35	d(x	d(x	PROPN
cana-5423	296	36	,	,	PUNCT
cana-5423	296	37	γy	γy	PROPN
cana-5423	296	38	)	)	PUNCT
cana-5423	296	39	,	,	PUNCT
cana-5423	296	40	d(y	d(y	NOUN
cana-5423	296	41	,	,	PUNCT
cana-5423	296	42	γx	γx	NOUN
cana-5423	296	43	)	)	PUNCT
cana-5423	296	44	,	,	PUNCT
cana-5423	296	45	d(x	d(x	PROPN
cana-5423	296	46	,	,	PUNCT
cana-5423	296	47	γy)d(y	γy)d(y	ADV
cana-5423	296	48	,	,	PUNCT
cana-5423	296	49	γx	γx	NOUN
cana-5423	296	50	)	)	PUNCT
cana-5423	296	51	1	1	NUM
cana-5423	297	1	+	+	CCONJ
cana-5423	297	2	d(x	d(x	PROPN
cana-5423	297	3	,	,	PUNCT
cana-5423	297	4	y	y	NOUN
cana-5423	297	5	)	)	PUNCT
cana-5423	297	6	,	,	PUNCT
cana-5423	297	7	d(x	d(x	PROPN
cana-5423	297	8	,	,	PUNCT
cana-5423	297	9	γx)d(y	γx)d(y	NOUN
cana-5423	297	10	,	,	PUNCT
cana-5423	297	11	γy	γy	NOUN
cana-5423	297	12	)	)	PUNCT
cana-5423	297	13	1	1	NUM
cana-5423	298	1	+	+	CCONJ
cana-5423	298	2	d(x	d(x	PROPN
cana-5423	298	3	,	,	PUNCT
cana-5423	298	4	y	y	NOUN
cana-5423	298	5	)	)	PUNCT
cana-5423	298	6	}	}	PUNCT
cana-5423	298	7	.	.	PUNCT
cana-5423	299	1	then	then	ADV
cana-5423	299	2	γ	γ	PROPN
cana-5423	299	3	has	have	VERB
cana-5423	299	4	a	a	DET
cana-5423	299	5	unique	unique	ADJ
cana-5423	299	6	fixed	fix	VERB
cana-5423	299	7	point	point	NOUN
cana-5423	299	8	x⋆	x⋆	PUNCT
cana-5423	299	9	∈	∈	PROPN
cana-5423	299	10	x.	x.	NOUN
cana-5423	299	11	corollary	corollary	NOUN
cana-5423	299	12	2.5	2.5	NUM
cana-5423	299	13	.	.	PUNCT
cana-5423	300	1	let	let	VERB
cana-5423	300	2	(	(	PUNCT
cana-5423	300	3	x	x	NOUN
cana-5423	300	4	,	,	PUNCT
cana-5423	300	5	d	d	NOUN
cana-5423	300	6	)	)	PUNCT
cana-5423	300	7	be	be	AUX
cana-5423	300	8	a	a	DET
cana-5423	300	9	metric	metric	ADJ
cana-5423	300	10	space	space	NOUN
cana-5423	300	11	,	,	PUNCT
cana-5423	300	12	γ	γ	X
cana-5423	300	13	:	:	PUNCT
cana-5423	300	14	x	x	SYM
cana-5423	300	15	→	→	PUNCT
cana-5423	300	16	x	x	X
cana-5423	300	17	is	be	AUX
cana-5423	300	18	a	a	DET
cana-5423	300	19	generalized	generalized	ADJ
cana-5423	300	20	αadmissible	αadmissible	ADJ
cana-5423	300	21	modified	modify	VERB
cana-5423	300	22	almost	almost	ADV
cana-5423	300	23	zcontraction	zcontraction	NOUN
cana-5423	300	24	self	self	NOUN
cana-5423	300	25	mapping	mapping	NOUN
cana-5423	300	26	.	.	PUNCT
cana-5423	301	1	there	there	PRON
cana-5423	301	2	exists	exist	VERB
cana-5423	301	3	ζ	ζ	PROPN
cana-5423	301	4	∈	∈	PROPN
cana-5423	301	5	z	z	NOUN
cana-5423	301	6	and	and	CCONJ
cana-5423	301	7	α	α	NOUN
cana-5423	301	8	:	:	PUNCT
cana-5423	301	9	x	x	PROPN
cana-5423	301	10	×x	×x	X
cana-5423	301	11	→	→	X
cana-5423	301	12	[	[	X
cana-5423	301	13	0,∞	0,∞	NOUN
cana-5423	301	14	)	)	PUNCT
cana-5423	301	15	be	be	VERB
cana-5423	301	16	a	a	DET
cana-5423	301	17	function	function	NOUN
cana-5423	301	18	with	with	ADP
cana-5423	301	19	α(x	α(x	NOUN
cana-5423	301	20	,	,	PUNCT
cana-5423	301	21	γx	γx	NOUN
cana-5423	301	22	)	)	PUNCT
cana-5423	301	23	=	=	SYM
cana-5423	301	24	1	1	NUM
cana-5423	301	25	,	,	PUNCT
cana-5423	301	26	α(y	α(y	NOUN
cana-5423	301	27	,	,	PUNCT
cana-5423	301	28	γy	γy	NOUN
cana-5423	301	29	)	)	PUNCT
cana-5423	301	30	=	=	PUNCT
cana-5423	302	1	1,and	1,and	NUM
cana-5423	302	2	q	q	NOUN
cana-5423	302	3	=	=	NOUN
cana-5423	302	4	0	0	NUM
cana-5423	302	5	.	.	PUNCT
cana-5423	303	1	for	for	ADP
cana-5423	303	2	all	all	DET
cana-5423	303	3	x	x	NOUN
cana-5423	303	4	,	,	PUNCT
cana-5423	303	5	y	y	PROPN
cana-5423	303	6	∈	∈	PROPN
cana-5423	303	7	x	x	PUNCT
cana-5423	303	8	such	such	ADJ
cana-5423	303	9	that	that	DET
cana-5423	303	10	ζ(d(γx	ζ(d(γx	NOUN
cana-5423	303	11	,	,	PUNCT
cana-5423	303	12	γy)k(x	γy)k(x	PROPN
cana-5423	303	13	,	,	PUNCT
cana-5423	303	14	y	y	PROPN
cana-5423	303	15	)	)	PUNCT
cana-5423	303	16	≥	≥	NOUN
cana-5423	303	17	0	0	NUM
cana-5423	303	18	,	,	PUNCT
cana-5423	303	19	forall	forall	NOUN
cana-5423	303	20	x	x	X
cana-5423	303	21	,	,	PUNCT
cana-5423	303	22	y	y	PROPN
cana-5423	303	23	∈	∈	PROPN
cana-5423	303	24	x	x	NOUN
cana-5423	303	25	,	,	PUNCT
cana-5423	303	26	andl	andl	NOUN
cana-5423	303	27	≥	≥	NOUN
cana-5423	303	28	0	0	NUM
cana-5423	303	29	,	,	PUNCT
cana-5423	303	30	also	also	ADV
cana-5423	303	31	where	where	SCONJ
cana-5423	303	32	k(x	k(x	PROPN
cana-5423	303	33	,	,	PUNCT
cana-5423	303	34	y	y	NOUN
cana-5423	303	35	)	)	PUNCT
cana-5423	303	36	=	=	SYM
cana-5423	303	37	max	max	PROPN
cana-5423	303	38	{	{	PUNCT
cana-5423	303	39	d(x	d(x	PROPN
cana-5423	303	40	,	,	PUNCT
cana-5423	303	41	y	y	PROPN
cana-5423	303	42	)	)	PUNCT
cana-5423	303	43	,	,	PUNCT
cana-5423	304	1	[	[	X
cana-5423	304	2	1	1	NUM
cana-5423	304	3	+	+	NUM
cana-5423	304	4	d(x	d(x	PROPN
cana-5423	304	5	,	,	PUNCT
cana-5423	304	6	γx)]d(y	γx)]d(y	NOUN
cana-5423	304	7	,	,	PUNCT
cana-5423	304	8	γy	γy	NOUN
cana-5423	304	9	)	)	PUNCT
cana-5423	304	10	1	1	NUM
cana-5423	305	1	+	+	CCONJ
cana-5423	305	2	d(x	d(x	PROPN
cana-5423	305	3	,	,	PUNCT
cana-5423	305	4	y	y	NOUN
cana-5423	305	5	)	)	PUNCT
cana-5423	305	6	,	,	PUNCT
cana-5423	306	1	[	[	X
cana-5423	306	2	1	1	NUM
cana-5423	306	3	+	+	NUM
cana-5423	306	4	d(x	d(x	PROPN
cana-5423	306	5	,	,	PUNCT
cana-5423	306	6	γy)]d(y	γy)]d(y	ADJ
cana-5423	306	7	,	,	PUNCT
cana-5423	306	8	γx	γx	NOUN
cana-5423	306	9	)	)	PUNCT
cana-5423	306	10	1	1	NUM
cana-5423	307	1	+	+	CCONJ
cana-5423	307	2	d(x	d(x	PROPN
cana-5423	307	3	,	,	PUNCT
cana-5423	307	4	y	y	NOUN
cana-5423	307	5	)	)	PUNCT
cana-5423	307	6	}	}	PUNCT
cana-5423	307	7	then	then	ADV
cana-5423	307	8	γ	γ	PROPN
cana-5423	307	9	has	have	VERB
cana-5423	307	10	a	a	DET
cana-5423	307	11	unique	unique	ADJ
cana-5423	307	12	fixed	fix	VERB
cana-5423	307	13	point	point	NOUN
cana-5423	307	14	x⋆	x⋆	PUNCT
cana-5423	307	15	∈	∈	PROPN
cana-5423	307	16	x.	x.	NOUN
cana-5423	307	17	communications	communication	NOUN
cana-5423	307	18	on	on	ADP
cana-5423	307	19	applied	apply	VERB
cana-5423	307	20	nonlinear	nonlinear	ADJ
cana-5423	307	21	analysis	analysis	NOUN
cana-5423	307	22	issn	issn	NOUN
cana-5423	307	23	:	:	PUNCT
cana-5423	307	24	1074	1074	NUM
cana-5423	307	25	-	-	PUNCT
cana-5423	307	26	133x	133x	NUM
cana-5423	307	27	vol	vol	NOUN
cana-5423	307	28	32	32	NUM
cana-5423	307	29	no	no	NOUN
cana-5423	307	30	.	.	PUNCT
cana-5423	308	1	10s(2025	10s(2025	NUM
cana-5423	308	2	)	)	PUNCT
cana-5423	309	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	309	2	2227	2227	NUM
cana-5423	309	3	example	example	NOUN
cana-5423	309	4	2.6	2.6	NUM
cana-5423	309	5	.	.	PUNCT
cana-5423	310	1	let	let	VERB
cana-5423	310	2	[	[	X
cana-5423	310	3	0	0	NUM
cana-5423	310	4	,	,	PUNCT
cana-5423	310	5	4	4	NUM
cana-5423	310	6	]	]	PUNCT
cana-5423	310	7	be	be	AUX
cana-5423	310	8	endowed	endow	VERB
cana-5423	310	9	with	with	ADP
cana-5423	310	10	metric	metric	ADJ
cana-5423	310	11	space	space	NOUN
cana-5423	310	12	d(x	d(x	PROPN
cana-5423	310	13	,	,	PUNCT
cana-5423	310	14	y	y	NOUN
cana-5423	311	1	)	)	PUNCT
cana-5423	311	2	=	=	NOUN
cana-5423	311	3	|x	|x	NOUN
cana-5423	312	1	−	−	VERB
cana-5423	312	2	y|	y|	NOUN
cana-5423	312	3	for	for	ADP
cana-5423	312	4	all	all	DET
cana-5423	312	5	x	x	NOUN
cana-5423	312	6	,	,	PUNCT
cana-5423	312	7	y	y	PROPN
cana-5423	312	8	∈	∈	PROPN
cana-5423	312	9	x	x	X
cana-5423	312	10	and	and	CCONJ
cana-5423	312	11	γ	γ	X
cana-5423	312	12	:	:	PUNCT
cana-5423	312	13	x	x	SYM
cana-5423	312	14	→	→	PUNCT
cana-5423	312	15	x	x	AUX
cana-5423	312	16	be	be	AUX
cana-5423	312	17	defined	define	VERB
cana-5423	312	18	by	by	ADP
cana-5423	312	19	γx	γx	NOUN
cana-5423	312	20	=	=	SYM
cana-5423	312	21	4	4	NUM
cana-5423	312	22	−	−	NOUN
cana-5423	312	23	x.	x.	NOUN
cana-5423	312	24	consider	consider	VERB
cana-5423	312	25	ζ(t	ζ(t	NOUN
cana-5423	312	26	,	,	PUNCT
cana-5423	312	27	s	s	PART
cana-5423	312	28	)	)	PUNCT
cana-5423	312	29	=	=	SYM
cana-5423	313	1	αs	αs	ADP
cana-5423	313	2	−	−	PROPN
cana-5423	313	3	t	t	PROPN
cana-5423	313	4	,	,	PUNCT
cana-5423	313	5	where	where	SCONJ
cana-5423	313	6	α	α	PRON
cana-5423	313	7	∈	∈	PROPN
cana-5423	314	1	[	[	X
cana-5423	314	2	0	0	NUM
cana-5423	314	3	,	,	PUNCT
cana-5423	314	4	1	1	NUM
cana-5423	314	5	)	)	PUNCT
cana-5423	314	6	,	,	PUNCT
cana-5423	314	7	for	for	ADP
cana-5423	314	8	all	all	DET
cana-5423	314	9	t	t	PROPN
cana-5423	314	10	≥	≥	NOUN
cana-5423	314	11	0	0	NUM
cana-5423	314	12	and	and	CCONJ
cana-5423	314	13	l	l	PROPN
cana-5423	314	14	≥	≥	NOUN
cana-5423	314	15	0	0	NUM
cana-5423	314	16	and	and	CCONJ
cana-5423	314	17	α	α	NOUN
cana-5423	314	18	:	:	PUNCT
cana-5423	314	19	x	x	PROPN
cana-5423	314	20	×x	×x	X
cana-5423	314	21	→	→	X
cana-5423	314	22	[	[	X
cana-5423	314	23	0,∞	0,∞	X
cana-5423	314	24	)	)	PUNCT
cana-5423	314	25	be	be	AUX
cana-5423	314	26	defined	define	VERB
cana-5423	314	27	by	by	ADP
cana-5423	314	28	α(x	α(x	PROPN
cana-5423	314	29	,	,	PUNCT
cana-5423	314	30	y	y	PROPN
cana-5423	314	31	)	)	PUNCT
cana-5423	314	32	=	=	PRON
cana-5423	314	33	{	{	PUNCT
cana-5423	314	34	1	1	NUM
cana-5423	314	35	if	if	SCONJ
cana-5423	314	36	,	,	PUNCT
cana-5423	314	37	x	x	X
cana-5423	315	1	,	,	PUNCT
cana-5423	315	2	y	y	PROPN
cana-5423	315	3	∈	∈	PROPN
cana-5423	316	1	[	[	X
cana-5423	316	2	0	0	NUM
cana-5423	316	3	,	,	PUNCT
cana-5423	316	4	1	1	NUM
cana-5423	316	5	]	]	SYM
cana-5423	316	6	0	0	NUM
cana-5423	316	7	,	,	PUNCT
cana-5423	316	8	otherwise	otherwise	ADV
cana-5423	316	9	}	}	PUNCT
cana-5423	316	10	note	note	VERB
cana-5423	316	11	that	that	SCONJ
cana-5423	316	12	γ	γ	PROPN
cana-5423	316	13	is	be	AUX
cana-5423	316	14	triangular	triangular	ADJ
cana-5423	316	15	α	α	PRON
cana-5423	316	16	-	-	ADJ
cana-5423	316	17	orbital	orbital	ADJ
cana-5423	316	18	admissible	admissible	NOUN
cana-5423	316	19	if	if	SCONJ
cana-5423	316	20	α(x	α(x	NOUN
cana-5423	316	21	,	,	PUNCT
cana-5423	316	22	γx	γx	NOUN
cana-5423	316	23	)	)	PUNCT
cana-5423	316	24	≥	≥	NOUN
cana-5423	316	25	1	1	NUM
cana-5423	316	26	⇒	⇒	PROPN
cana-5423	316	27	α(γx	α(γx	NUM
cana-5423	316	28	,	,	PUNCT
cana-5423	316	29	γ2x	γ2x	NUM
cana-5423	316	30	)	)	PUNCT
cana-5423	316	31	≥	≥	NOUN
cana-5423	316	32	1	1	NUM
cana-5423	316	33	,	,	PUNCT
cana-5423	316	34	and	and	CCONJ
cana-5423	316	35	α(x	α(x	PROPN
cana-5423	316	36	,	,	PUNCT
cana-5423	316	37	y	y	PROPN
cana-5423	316	38	)	)	PUNCT
cana-5423	316	39	≥	≥	NOUN
cana-5423	316	40	1	1	NUM
cana-5423	316	41	and	and	CCONJ
cana-5423	316	42	α(y	α(y	NOUN
cana-5423	316	43	,	,	PUNCT
cana-5423	316	44	γy	γy	NOUN
cana-5423	316	45	)	)	PUNCT
cana-5423	316	46	⇒	⇒	PROPN
cana-5423	316	47	α(x	α(x	PROPN
cana-5423	316	48	,	,	PUNCT
cana-5423	316	49	γy	γy	PROPN
cana-5423	316	50	)	)	PUNCT
cana-5423	316	51	≥	≥	NOUN
cana-5423	316	52	1	1	NUM
cana-5423	316	53	.	.	PUNCT
cana-5423	317	1	since	since	SCONJ
cana-5423	317	2	α(x	α(x	PROPN
cana-5423	317	3	,	,	PUNCT
cana-5423	317	4	y	y	PROPN
cana-5423	317	5	)	)	PUNCT
cana-5423	317	6	>	>	X
cana-5423	317	7	1	1	NUM
cana-5423	317	8	and	and	CCONJ
cana-5423	317	9	x	x	NOUN
cana-5423	317	10	,	,	PUNCT
cana-5423	317	11	y	y	PROPN
cana-5423	317	12	∈	∈	PROPN
cana-5423	318	1	[	[	X
cana-5423	318	2	0	0	NUM
cana-5423	318	3	,	,	PUNCT
cana-5423	318	4	1	1	NUM
cana-5423	318	5	]	]	PUNCT
cana-5423	318	6	.	.	PUNCT
cana-5423	319	1	then	then	ADV
cana-5423	319	2	,	,	PUNCT
cana-5423	319	3	we	we	PRON
cana-5423	319	4	have	have	VERB
cana-5423	319	5	α(x	α(x	NOUN
cana-5423	319	6	,	,	PUNCT
cana-5423	319	7	γx	γx	NOUN
cana-5423	319	8	)	)	PUNCT
cana-5423	319	9	=	=	SYM
cana-5423	320	1	(	(	PUNCT
cana-5423	320	2	x	x	X
cana-5423	320	3	,	,	PUNCT
cana-5423	320	4	4	4	NUM
cana-5423	320	5	)	)	PUNCT
cana-5423	320	6	=	=	SYM
cana-5423	320	7	1	1	NUM
cana-5423	320	8	and	and	CCONJ
cana-5423	320	9	α(y	α(y	NOUN
cana-5423	320	10	,	,	PUNCT
cana-5423	320	11	γy	γy	NOUN
cana-5423	320	12	)	)	PUNCT
cana-5423	320	13	=	=	SYM
cana-5423	320	14	1	1	NUM
cana-5423	320	15	for	for	ADP
cana-5423	320	16	all	all	DET
cana-5423	320	17	x	x	NOUN
cana-5423	320	18	,	,	PUNCT
cana-5423	320	19	y	y	PROPN
cana-5423	320	20	∈	∈	PROPN
cana-5423	320	21	x.	x.	NOUN
cana-5423	320	22	in	in	ADP
cana-5423	320	23	fact	fact	NOUN
cana-5423	320	24	,	,	PUNCT
cana-5423	320	25	for	for	ADP
cana-5423	320	26	all	all	PRON
cana-5423	320	27	x	x	PUNCT
cana-5423	320	28	̸=	̸=	PROPN
cana-5423	320	29	y	y	PROPN
cana-5423	320	30	,	,	PUNCT
cana-5423	320	31	then	then	ADV
cana-5423	320	32	we	we	PRON
cana-5423	320	33	have	have	VERB
cana-5423	320	34	ζ(d(γx	ζ(d(γx	NOUN
cana-5423	320	35	,	,	PUNCT
cana-5423	320	36	γy)d(x	γy)d(x	NOUN
cana-5423	320	37	,	,	PUNCT
cana-5423	320	38	y	y	NOUN
cana-5423	320	39	)	)	PUNCT
cana-5423	320	40	)	)	PUNCT
cana-5423	321	1	=	=	PUNCT
cana-5423	322	1	α	α	PROPN
cana-5423	322	2	|x−	|x−	NOUN
cana-5423	322	3	y|	y|	NOUN
cana-5423	322	4	−	−	PROPN
cana-5423	322	5	|4	|4	SYM
cana-5423	322	6	−	−	PROPN
cana-5423	322	7	x−	x−	PROPN
cana-5423	322	8	(	(	PUNCT
cana-5423	322	9	4	4	NUM
cana-5423	322	10	−	−	NOUN
cana-5423	323	1	y)|	y)|	NOUN
cana-5423	323	2	=	=	PROPN
cana-5423	323	3	α	α	PROPN
cana-5423	323	4	|x−	|x−	PROPN
cana-5423	323	5	y|	y|	NOUN
cana-5423	323	6	−	−	PROPN
cana-5423	323	7	|x−	|x−	NOUN
cana-5423	323	8	y|	y|	NOUN
cana-5423	323	9	<	<	X
cana-5423	323	10	|x−	|x−	PROPN
cana-5423	323	11	y|	y|	NOUN
cana-5423	323	12	−	−	PROPN
cana-5423	323	13	|x−	|x−	NOUN
cana-5423	323	14	y|	y|	NOUN
cana-5423	323	15	=	=	NOUN
cana-5423	323	16	0	0	PUNCT
cana-5423	324	1	now	now	ADV
cana-5423	324	2	we	we	PRON
cana-5423	324	3	show	show	VERB
cana-5423	324	4	that	that	SCONJ
cana-5423	324	5	γ	γ	PROPN
cana-5423	324	6	is	be	AUX
cana-5423	324	7	a	a	DET
cana-5423	324	8	generalized	generalized	ADJ
cana-5423	324	9	α	α	NOUN
cana-5423	324	10	admissible	admissible	ADJ
cana-5423	324	11	modified	modify	VERB
cana-5423	324	12	almost	almost	ADV
cana-5423	324	13	z	z	NOUN
cana-5423	324	14	-contractionwith	-contractionwith	PROPN
cana-5423	324	15	respect	respect	NOUN
cana-5423	324	16	to	to	ADP
cana-5423	324	17	ζ	ζ	SYM
cana-5423	324	18	∈	∈	PROPN
cana-5423	324	19	z	z	NOUN
cana-5423	324	20	.now	.now	PUNCT
cana-5423	325	1	ζ(d(γx	ζ(d(γx	NOUN
cana-5423	325	2	,	,	PUNCT
cana-5423	325	3	γy)k(x	γy)k(x	PROPN
cana-5423	325	4	,	,	PUNCT
cana-5423	325	5	y	y	NOUN
cana-5423	325	6	)	)	PUNCT
cana-5423	325	7	)	)	PUNCT
cana-5423	326	1	+	+	CCONJ
cana-5423	326	2	lq(x	lq(x	X
cana-5423	326	3	,	,	PUNCT
cana-5423	326	4	y	y	NOUN
cana-5423	326	5	)	)	PUNCT
cana-5423	326	6	=	=	SYM
cana-5423	326	7	k(x	k(x	PROPN
cana-5423	326	8	,	,	PUNCT
cana-5423	326	9	y	y	NOUN
cana-5423	326	10	)	)	PUNCT
cana-5423	326	11	+	+	NUM
cana-5423	326	12	lq(x	lq(x	X
cana-5423	326	13	,	,	PUNCT
cana-5423	326	14	y	y	NOUN
cana-5423	326	15	)	)	PUNCT
cana-5423	326	16	−	−	PROPN
cana-5423	326	17	d(γx	d(γx	PROPN
cana-5423	326	18	,	,	PUNCT
cana-5423	326	19	γy	γy	NOUN
cana-5423	326	20	)	)	PUNCT
cana-5423	326	21	=	=	SYM
cana-5423	326	22	α	α	PRON
cana-5423	326	23	[	[	X
cana-5423	326	24	|k(x	|k(x	PROPN
cana-5423	326	25	,	,	PUNCT
cana-5423	326	26	y	y	NOUN
cana-5423	326	27	)	)	PUNCT
cana-5423	326	28	+	+	NUM
cana-5423	326	29	lq(x	lq(x	NOUN
cana-5423	326	30	,	,	PUNCT
cana-5423	326	31	y)|	y)|	PROPN
cana-5423	326	32	]	]	X
cana-5423	326	33	−	−	PROPN
cana-5423	326	34	|4	|4	SYM
cana-5423	326	35	−	−	PROPN
cana-5423	326	36	x−	x−	PROPN
cana-5423	326	37	(	(	PUNCT
cana-5423	326	38	4	4	NUM
cana-5423	326	39	−	−	NOUN
cana-5423	326	40	y)|	y)|	INTJ
cana-5423	326	41	,	,	PUNCT
cana-5423	326	42	=	=	PUNCT
cana-5423	326	43	α	α	PROPN
cana-5423	327	1	[	[	X
cana-5423	327	2	|k(x	|k(x	PROPN
cana-5423	327	3	,	,	PUNCT
cana-5423	327	4	y	y	NOUN
cana-5423	327	5	)	)	PUNCT
cana-5423	327	6	+	+	NUM
cana-5423	327	7	lq(x	lq(x	NOUN
cana-5423	327	8	,	,	PUNCT
cana-5423	327	9	y)|	y)|	PROPN
cana-5423	327	10	]	]	PUNCT
cana-5423	327	11	−	−	PROPN
cana-5423	327	12	|x−	|x−	PROPN
cana-5423	327	13	y|	y|	NOUN
cana-5423	327	14	,	,	PUNCT
cana-5423	327	15	wher	wher	PROPN
cana-5423	327	16	k(x	k(x	PROPN
cana-5423	327	17	,	,	PUNCT
cana-5423	327	18	y	y	NOUN
cana-5423	327	19	)	)	PUNCT
cana-5423	327	20	=	=	SYM
cana-5423	327	21	max	max	PROPN
cana-5423	327	22	{	{	PUNCT
cana-5423	327	23	|x−	|x−	NOUN
cana-5423	327	24	y|	y|	NOUN
cana-5423	327	25	,	,	PUNCT
cana-5423	328	1	[	[	X
cana-5423	328	2	1	1	NUM
cana-5423	328	3	+	+	NUM
cana-5423	328	4	|x−	|x−	NOUN
cana-5423	328	5	(	(	PUNCT
cana-5423	328	6	4	4	NUM
cana-5423	328	7	−	−	PROPN
cana-5423	328	8	x)|	x)|	PROPN
cana-5423	328	9	]	]	PUNCT
cana-5423	328	10	|y	|y	NOUN
cana-5423	328	11	−	−	PROPN
cana-5423	328	12	(	(	PUNCT
cana-5423	328	13	4	4	NUM
cana-5423	328	14	−	−	NOUN
cana-5423	328	15	y)|	y)|	NOUN
cana-5423	328	16	1	1	NUM
cana-5423	328	17	+	+	NUM
cana-5423	328	18	|x−	|x−	NOUN
cana-5423	328	19	y|	y|	NOUN
cana-5423	328	20	,	,	PUNCT
cana-5423	329	1	[	[	X
cana-5423	329	2	1	1	NUM
cana-5423	329	3	+	+	NUM
cana-5423	329	4	|x−	|x−	NOUN
cana-5423	329	5	(	(	PUNCT
cana-5423	329	6	4	4	NUM
cana-5423	329	7	−	−	NOUN
cana-5423	329	8	y)|	y)|	NOUN
cana-5423	329	9	]	]	PUNCT
cana-5423	329	10	|y	|y	NOUN
cana-5423	329	11	−	−	PROPN
cana-5423	329	12	(	(	PUNCT
cana-5423	329	13	4	4	NUM
cana-5423	329	14	−	−	NOUN
cana-5423	329	15	x)|	x)|	NOUN
cana-5423	329	16	1	1	NUM
cana-5423	329	17	+	+	NUM
cana-5423	329	18	|x−	|x−	NOUN
cana-5423	329	19	y|	y|	NOUN
cana-5423	329	20	}	}	PUNCT
cana-5423	329	21	=	=	SYM
cana-5423	329	22	max	max	X
cana-5423	329	23	{	{	PUNCT
cana-5423	329	24	|x−	|x−	NOUN
cana-5423	329	25	y|	y|	NOUN
cana-5423	329	26	,	,	PUNCT
cana-5423	330	1	[	[	X
cana-5423	330	2	1	1	NUM
cana-5423	330	3	+	+	NUM
cana-5423	330	4	|2x−	|2x−	NOUN
cana-5423	330	5	4|	4|	NUM
cana-5423	330	6	]	]	PUNCT
cana-5423	330	7	|2y	|2y	NOUN
cana-5423	330	8	−	−	NOUN
cana-5423	331	1	4|	4|	NUM
cana-5423	331	2	1	1	NUM
cana-5423	332	1	+	+	NUM
cana-5423	333	1	|x−	|x−	NOUN
cana-5423	334	1	y|	y|	NOUN
cana-5423	334	2	,	,	PUNCT
cana-5423	334	3	[	[	X
cana-5423	334	4	1	1	NUM
cana-5423	334	5	+	+	NUM
cana-5423	334	6	|x−	|x−	NOUN
cana-5423	334	7	4	4	NUM
cana-5423	334	8	+	+	CCONJ
cana-5423	334	9	y|	y|	NOUN
cana-5423	334	10	]	]	SYM
cana-5423	334	11	|y	|y	NOUN
cana-5423	334	12	−	−	PROPN
cana-5423	334	13	4	4	NUM
cana-5423	334	14	−	−	NOUN
cana-5423	334	15	x|	x|	NOUN
cana-5423	334	16	1	1	NUM
cana-5423	334	17	+	+	NUM
cana-5423	334	18	|x−	|x−	NOUN
cana-5423	334	19	y|	y|	NOUN
cana-5423	334	20	}	}	PUNCT
cana-5423	334	21	and	and	CCONJ
cana-5423	334	22	,	,	PUNCT
cana-5423	334	23	q(x	q(x	PROPN
cana-5423	334	24	,	,	PUNCT
cana-5423	334	25	y	y	NOUN
cana-5423	334	26	)	)	PUNCT
cana-5423	334	27	=	=	SYM
cana-5423	334	28	min{|x−	min{|x−	PROPN
cana-5423	334	29	(	(	PUNCT
cana-5423	334	30	4	4	NUM
cana-5423	334	31	−	−	PROPN
cana-5423	334	32	x)|	x)|	NOUN
cana-5423	334	33	,	,	PUNCT
cana-5423	334	34	|y	|y	NOUN
cana-5423	334	35	−	−	PROPN
cana-5423	334	36	(	(	PUNCT
cana-5423	334	37	4	4	NUM
cana-5423	334	38	−	−	NOUN
cana-5423	334	39	y)|	y)|	PROPN
cana-5423	334	40	,	,	PUNCT
cana-5423	334	41	|x−	|x−	PROPN
cana-5423	334	42	(	(	PUNCT
cana-5423	334	43	4	4	NUM
cana-5423	334	44	−	−	NOUN
cana-5423	334	45	y)|	y)|	NOUN
cana-5423	334	46	,	,	PUNCT
cana-5423	334	47	|y	|y	ADJ
cana-5423	334	48	−	−	PROPN
cana-5423	334	49	(	(	PUNCT
cana-5423	334	50	4	4	NUM
cana-5423	334	51	−	−	PROPN
cana-5423	334	52	x)|	x)|	PROPN
cana-5423	334	53	,	,	PUNCT
cana-5423	334	54	|x−	|x−	PROPN
cana-5423	334	55	(	(	PUNCT
cana-5423	334	56	4	4	NUM
cana-5423	334	57	−	−	NOUN
cana-5423	334	58	y)|	y)|	INTJ
cana-5423	334	59	.	.	PUNCT
cana-5423	335	1	|y	|y	NOUN
cana-5423	335	2	−	−	PROPN
cana-5423	335	3	(	(	PUNCT
cana-5423	335	4	4	4	NUM
cana-5423	335	5	−	−	NOUN
cana-5423	335	6	x)|	x)|	NOUN
cana-5423	335	7	1	1	NUM
cana-5423	335	8	+	+	NUM
cana-5423	335	9	|x−	|x−	NOUN
cana-5423	335	10	y|	y|	NOUN
cana-5423	335	11	|x−	|x−	PROPN
cana-5423	335	12	(	(	PUNCT
cana-5423	335	13	4	4	NUM
cana-5423	335	14	−	−	NOUN
cana-5423	335	15	y)|	y)|	INTJ
cana-5423	335	16	.	.	PUNCT
cana-5423	336	1	|y	|y	NOUN
cana-5423	336	2	−	−	PROPN
cana-5423	336	3	(	(	PUNCT
cana-5423	336	4	4	4	NUM
cana-5423	336	5	−	−	NOUN
cana-5423	336	6	x)|	x)|	NOUN
cana-5423	336	7	1	1	NUM
cana-5423	336	8	+	+	NUM
cana-5423	336	9	|x−	|x−	NOUN
cana-5423	336	10	y|	y|	NOUN
cana-5423	336	11	}	}	PUNCT
cana-5423	336	12	=	=	SYM
cana-5423	336	13	min	min	NOUN
cana-5423	336	14	{	{	PUNCT
cana-5423	336	15	|2x−	|2x−	PROPN
cana-5423	336	16	4|	4|	NUM
cana-5423	336	17	,	,	PUNCT
cana-5423	336	18	|2y	|2y	NOUN
cana-5423	336	19	−	−	PROPN
cana-5423	336	20	4|	4|	NUM
cana-5423	336	21	,	,	PUNCT
cana-5423	336	22	|x+	|x+	PROPN
cana-5423	336	23	y	y	PROPN
cana-5423	336	24	−	−	PROPN
cana-5423	336	25	4|	4|	NUM
cana-5423	336	26	,	,	PUNCT
cana-5423	336	27	|x+	|x+	PROPN
cana-5423	336	28	y	y	NOUN
cana-5423	336	29	−	−	NOUN
cana-5423	336	30	4|	4|	NUM
cana-5423	336	31	1	1	NUM
cana-5423	337	1	+	+	NUM
cana-5423	337	2	|x−	|x−	NOUN
cana-5423	337	3	y|	y|	NOUN
cana-5423	337	4	,	,	PUNCT
cana-5423	337	5	|2x−	|2x−	VERB
cana-5423	337	6	4|	4|	NUM
cana-5423	337	7	.	.	PUNCT
cana-5423	338	1	|2y	|2y	NOUN
cana-5423	338	2	−	−	NOUN
cana-5423	338	3	4|	4|	NUM
cana-5423	338	4	1	1	NUM
cana-5423	338	5	+	+	NUM
cana-5423	338	6	|x−	|x−	NOUN
cana-5423	338	7	y|	y|	NOUN
cana-5423	338	8	}	}	PUNCT
cana-5423	338	9	.	.	PUNCT
cana-5423	339	1	we	we	PRON
cana-5423	339	2	deduce	deduce	VERB
cana-5423	339	3	that	that	PRON
cana-5423	339	4	ζ(d(γx	ζ(d(γx	NOUN
cana-5423	339	5	,	,	PUNCT
cana-5423	339	6	γy)k(x	γy)k(x	PROPN
cana-5423	339	7	,	,	PUNCT
cana-5423	339	8	y	y	NOUN
cana-5423	339	9	)	)	PUNCT
cana-5423	339	10	)	)	PUNCT
cana-5423	340	1	+	+	CCONJ
cana-5423	340	2	lq(x	lq(x	X
cana-5423	340	3	,	,	PUNCT
cana-5423	340	4	y	y	NOUN
cana-5423	340	5	)	)	PUNCT
cana-5423	340	6	=	=	PUNCT
cana-5423	340	7	α[max	α[max	PUNCT
cana-5423	340	8	{	{	PUNCT
cana-5423	340	9	|x−	|x−	NOUN
cana-5423	340	10	y|	y|	NOUN
cana-5423	340	11	,	,	PUNCT
cana-5423	340	12	[	[	X
cana-5423	340	13	1	1	NUM
cana-5423	340	14	+	+	NUM
cana-5423	340	15	|2x−	|2x−	NOUN
cana-5423	340	16	4|	4|	NUM
cana-5423	340	17	]	]	PUNCT
cana-5423	340	18	|2y	|2y	NOUN
cana-5423	340	19	−	−	NOUN
cana-5423	340	20	4|	4|	NUM
cana-5423	340	21	1	1	NUM
cana-5423	340	22	+	+	NUM
cana-5423	340	23	|x−	|x−	NOUN
cana-5423	340	24	y|	y|	NOUN
cana-5423	340	25	,	,	PUNCT
cana-5423	341	1	[	[	X
cana-5423	341	2	1	1	NUM
cana-5423	341	3	+	+	X
cana-5423	341	4	|x+	|x+	ADJ
cana-5423	341	5	y	y	NOUN
cana-5423	341	6	−	−	NOUN
cana-5423	341	7	4|	4|	NUM
cana-5423	341	8	]	]	X
cana-5423	341	9	|x+	|x+	PROPN
cana-5423	341	10	y	y	PROPN
cana-5423	341	11	−	−	NOUN
cana-5423	341	12	4|	4|	NUM
cana-5423	341	13	1	1	NUM
cana-5423	341	14	+	+	NUM
cana-5423	341	15	|x−	|x−	NOUN
cana-5423	341	16	y|	y|	NOUN
cana-5423	341	17	}	}	PUNCT
cana-5423	341	18	+	+	CCONJ
cana-5423	341	19	lmin	lmin	X
cana-5423	341	20	{	{	PUNCT
cana-5423	341	21	|2x−	|2x−	PROPN
cana-5423	341	22	4|	4|	NUM
cana-5423	341	23	,	,	PUNCT
cana-5423	341	24	|2y	|2y	NOUN
cana-5423	341	25	−	−	PROPN
cana-5423	341	26	4|	4|	NUM
cana-5423	341	27	,	,	PUNCT
cana-5423	341	28	|x+	|x+	PROPN
cana-5423	341	29	y	y	PROPN
cana-5423	341	30	−	−	PROPN
cana-5423	341	31	4|	4|	NUM
cana-5423	341	32	,	,	PUNCT
cana-5423	341	33	|x+	|x+	PROPN
cana-5423	341	34	y	y	NOUN
cana-5423	341	35	−	−	NOUN
cana-5423	341	36	4|	4|	NUM
cana-5423	341	37	1	1	NUM
cana-5423	342	1	+	+	NUM
cana-5423	342	2	|x−	|x−	NOUN
cana-5423	342	3	y|	y|	NOUN
cana-5423	342	4	,	,	PUNCT
cana-5423	342	5	|2x−	|2x−	VERB
cana-5423	342	6	4|	4|	NUM
cana-5423	342	7	.	.	PUNCT
cana-5423	343	1	|2y	|2y	NOUN
cana-5423	343	2	−	−	NOUN
cana-5423	343	3	4|	4|	NUM
cana-5423	343	4	1	1	NUM
cana-5423	343	5	+	+	NUM
cana-5423	343	6	|x−	|x−	NOUN
cana-5423	343	7	y|	y|	NOUN
cana-5423	343	8	}	}	PUNCT
cana-5423	343	9	]	]	PUNCT
cana-5423	343	10	−	−	PROPN
cana-5423	343	11	|x−	|x−	PROPN
cana-5423	343	12	y|	y|	NOUN
cana-5423	343	13	.	.	PUNCT
cana-5423	344	1	hence	hence	ADV
cana-5423	344	2	,	,	PUNCT
cana-5423	344	3	we	we	PRON
cana-5423	344	4	get	get	VERB
cana-5423	344	5	two	two	NUM
cana-5423	344	6	cases	case	NOUN
cana-5423	344	7	:	:	PUNCT
cana-5423	344	8	case(i	case(i	NOUN
cana-5423	344	9	):	):	PUNCT
cana-5423	344	10	if	if	SCONJ
cana-5423	344	11	x	x	X
cana-5423	344	12	=	=	SYM
cana-5423	344	13	y	y	PROPN
cana-5423	344	14	,	,	PUNCT
cana-5423	344	15	then	then	ADV
cana-5423	344	16	,	,	PUNCT
cana-5423	344	17	ζ(d(γx	ζ(d(γx	NOUN
cana-5423	344	18	,	,	PUNCT
cana-5423	344	19	γy)k(x	γy)k(x	PROPN
cana-5423	344	20	,	,	PUNCT
cana-5423	344	21	y	y	NOUN
cana-5423	344	22	)	)	PUNCT
cana-5423	344	23	)	)	PUNCT
cana-5423	345	1	+	+	CCONJ
cana-5423	345	2	lq(x	lq(x	X
cana-5423	345	3	,	,	PUNCT
cana-5423	345	4	y	y	NOUN
cana-5423	345	5	)	)	PUNCT
cana-5423	345	6	=	=	SYM
cana-5423	345	7	α[{[1	α[{[1	VERB
cana-5423	345	8	+	+	X
cana-5423	345	9	|2x−	|2x−	NOUN
cana-5423	345	10	4|	4|	NUM
cana-5423	345	11	]	]	PUNCT
cana-5423	345	12	|2x−	|2x−	PROPN
cana-5423	345	13	4|	4|	NUM
cana-5423	345	14	,	,	PUNCT
cana-5423	346	1	[	[	X
cana-5423	346	2	1	1	NUM
cana-5423	346	3	+	+	NUM
cana-5423	346	4	|2x−	|2x−	NOUN
cana-5423	346	5	4|	4|	NUM
cana-5423	346	6	]	]	PUNCT
cana-5423	346	7	|2x−	|2x−	NOUN
cana-5423	346	8	4|	4|	NUM
cana-5423	346	9	}	}	PUNCT
cana-5423	346	10	+	+	NUM
cana-5423	346	11	l	l	NOUN
cana-5423	346	12	{	{	PUNCT
cana-5423	346	13	|2x−	|2x−	PROPN
cana-5423	346	14	4|	4|	NUM
cana-5423	346	15	,	,	PUNCT
cana-5423	346	16	|2y	|2y	NOUN
cana-5423	346	17	−	−	PROPN
cana-5423	346	18	4|	4|	NUM
cana-5423	346	19	,	,	PUNCT
cana-5423	346	20	|2x−	|2x−	PROPN
cana-5423	346	21	4|	4|	NUM
cana-5423	346	22	,	,	PUNCT
cana-5423	346	23	|2x−	|2x−	PROPN
cana-5423	346	24	4|	4|	NUM
cana-5423	346	25	,	,	PUNCT
cana-5423	346	26	|2x−	|2x−	VERB
cana-5423	346	27	4|	4|	NUM
cana-5423	346	28	.	.	PUNCT
cana-5423	346	29	|2y	|2y	NOUN
cana-5423	346	30	−	−	NOUN
cana-5423	346	31	4|	4|	NUM
cana-5423	346	32	}	}	PUNCT
cana-5423	346	33	]	]	PUNCT
cana-5423	346	34	;	;	PUNCT
cana-5423	346	35	=	=	SYM
cana-5423	346	36	α	α	X
cana-5423	346	37	{	{	PUNCT
cana-5423	346	38	[	[	X
cana-5423	346	39	1	1	NUM
cana-5423	346	40	+	+	NUM
cana-5423	346	41	|2x−	|2x−	NOUN
cana-5423	346	42	4|	4|	NUM
cana-5423	346	43	]	]	PUNCT
cana-5423	346	44	|2x−	|2x−	VERB
cana-5423	346	45	4|	4|	NUM
cana-5423	346	46	+	+	NUM
cana-5423	346	47	l	l	X
cana-5423	346	48	|2x−	|2x−	PROPN
cana-5423	346	49	4|	4|	NUM
cana-5423	346	50	}	}	PUNCT
cana-5423	346	51	≥	≥	NOUN
cana-5423	346	52	0	0	NUM
cana-5423	346	53	.	.	PUNCT
cana-5423	347	1	communications	communication	NOUN
cana-5423	347	2	on	on	ADP
cana-5423	347	3	applied	apply	VERB
cana-5423	347	4	nonlinear	nonlinear	ADJ
cana-5423	347	5	analysis	analysis	NOUN
cana-5423	347	6	issn	issn	NOUN
cana-5423	347	7	:	:	PUNCT
cana-5423	347	8	1074	1074	NUM
cana-5423	347	9	-	-	PUNCT
cana-5423	347	10	133x	133x	NUM
cana-5423	347	11	vol	vol	NOUN
cana-5423	347	12	32	32	NUM
cana-5423	347	13	no	no	NOUN
cana-5423	347	14	.	.	PUNCT
cana-5423	348	1	10s(2025	10s(2025	NUM
cana-5423	348	2	)	)	PUNCT
cana-5423	349	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	349	2	2228	2228	NUM
cana-5423	349	3	case(ii	case(ii	PROPN
cana-5423	349	4	):	):	PUNCT
cana-5423	349	5	without	without	ADP
cana-5423	349	6	loss	loss	NOUN
cana-5423	349	7	of	of	ADP
cana-5423	349	8	generality	generality	NOUN
cana-5423	349	9	,	,	PUNCT
cana-5423	349	10	suppose	suppose	VERB
cana-5423	349	11	thatx	thatx	ADJ
cana-5423	349	12	>	>	X
cana-5423	349	13	y	y	PROPN
cana-5423	349	14	,	,	PUNCT
cana-5423	349	15	then	then	ADV
cana-5423	349	16	ζ(d(γx	ζ(d(γx	NOUN
cana-5423	349	17	,	,	PUNCT
cana-5423	349	18	γy)k(x	γy)k(x	PROPN
cana-5423	349	19	,	,	PUNCT
cana-5423	349	20	y	y	NOUN
cana-5423	349	21	)	)	PUNCT
cana-5423	349	22	)	)	PUNCT
cana-5423	350	1	+	+	CCONJ
cana-5423	350	2	lq(x	lq(x	X
cana-5423	350	3	,	,	PUNCT
cana-5423	350	4	y	y	NOUN
cana-5423	350	5	)	)	PUNCT
cana-5423	350	6	=	=	SYM
cana-5423	350	7	α	α	X
cana-5423	350	8	[	[	PUNCT
cana-5423	350	9	{	{	PUNCT
cana-5423	350	10	[	[	X
cana-5423	350	11	1	1	NUM
cana-5423	350	12	+	+	NUM
cana-5423	350	13	|2x−	|2x−	NOUN
cana-5423	350	14	4|	4|	NUM
cana-5423	350	15	]	]	PUNCT
cana-5423	350	16	|2y	|2y	NOUN
cana-5423	350	17	−	−	NOUN
cana-5423	351	1	4|	4|	NUM
cana-5423	351	2	1	1	NUM
cana-5423	351	3	+	+	NUM
cana-5423	351	4	|x−	|x−	NOUN
cana-5423	351	5	y|	y|	NOUN
cana-5423	351	6	,	,	PUNCT
cana-5423	351	7	[	[	X
cana-5423	351	8	1	1	NUM
cana-5423	351	9	+	+	NUM
cana-5423	351	10	|2x−	|2x−	NOUN
cana-5423	351	11	4|	4|	NUM
cana-5423	351	12	]	]	PUNCT
cana-5423	351	13	|2y	|2y	NOUN
cana-5423	351	14	−	−	NOUN
cana-5423	351	15	4|	4|	NUM
cana-5423	351	16	1	1	NUM
cana-5423	352	1	+	+	NUM
cana-5423	352	2	|x−	|x−	NOUN
cana-5423	352	3	y|	y|	NOUN
cana-5423	352	4	}	}	PUNCT
cana-5423	353	1	+	+	CCONJ
cana-5423	353	2	l	l	NOUN
cana-5423	353	3	{	{	PUNCT
cana-5423	353	4	|2y	|2y	NOUN
cana-5423	353	5	−	−	PROPN
cana-5423	353	6	4|	4|	NUM
cana-5423	353	7	,	,	PUNCT
cana-5423	353	8	|2y	|2y	NOUN
cana-5423	353	9	−	−	PROPN
cana-5423	353	10	4|	4|	NUM
cana-5423	353	11	,	,	PUNCT
cana-5423	353	12	|2y	|2y	NOUN
cana-5423	353	13	−	−	PROPN
cana-5423	353	14	4|	4|	NUM
cana-5423	353	15	,	,	PUNCT
cana-5423	353	16	|2y	|2y	NOUN
cana-5423	353	17	−	−	PROPN
cana-5423	353	18	4|	4|	NUM
cana-5423	353	19	1	1	NUM
cana-5423	353	20	+	+	NUM
cana-5423	353	21	|x−	|x−	NOUN
cana-5423	353	22	y|	y|	NOUN
cana-5423	353	23	,	,	PUNCT
cana-5423	353	24	|2y	|2y	NOUN
cana-5423	353	25	−	−	PROPN
cana-5423	353	26	4|	4|	NUM
cana-5423	353	27	.	.	PUNCT
cana-5423	354	1	|2y	|2y	NOUN
cana-5423	354	2	−	−	NOUN
cana-5423	354	3	4|	4|	NUM
cana-5423	354	4	1	1	NUM
cana-5423	355	1	+	+	NUM
cana-5423	355	2	|x−	|x−	NOUN
cana-5423	355	3	y|	y|	NOUN
cana-5423	355	4	}	}	PUNCT
cana-5423	355	5	]	]	PUNCT
cana-5423	356	1	−	−	PROPN
cana-5423	356	2	|x−	|x−	PROPN
cana-5423	356	3	y|	y|	NOUN
cana-5423	356	4	.	.	PUNCT
cana-5423	357	1	=	=	PUNCT
cana-5423	357	2	α	α	PRON
cana-5423	358	1	[	[	X
cana-5423	358	2	1	1	NUM
cana-5423	358	3	+	+	NUM
cana-5423	358	4	|2x−	|2x−	NOUN
cana-5423	358	5	4|	4|	NUM
cana-5423	358	6	]	]	PUNCT
cana-5423	358	7	|2y	|2y	NOUN
cana-5423	358	8	−	−	NOUN
cana-5423	358	9	4|	4|	NUM
cana-5423	358	10	1	1	NUM
cana-5423	359	1	+	+	NUM
cana-5423	359	2	|x−	|x−	NOUN
cana-5423	359	3	y|	y|	NOUN
cana-5423	359	4	+	+	CCONJ
cana-5423	359	5	αl	αl	ADP
cana-5423	359	6	|2y	|2y	NOUN
cana-5423	359	7	−	−	PROPN
cana-5423	359	8	4|	4|	NUM
cana-5423	359	9	−	−	ADP
cana-5423	359	10	|x−	|x−	PROPN
cana-5423	359	11	y|	y|	NOUN
cana-5423	359	12	.	.	PUNCT
cana-5423	360	1	if	if	SCONJ
cana-5423	360	2	α	α	PRON
cana-5423	360	3	=	=	NOUN
cana-5423	360	4	1	1	NUM
cana-5423	360	5	2	2	NUM
cana-5423	360	6	and	and	CCONJ
cana-5423	360	7	l	l	NOUN
cana-5423	360	8	=	=	SYM
cana-5423	360	9	10	10	NUM
cana-5423	360	10	,	,	PUNCT
cana-5423	360	11	then	then	ADV
cana-5423	360	12	we	we	PRON
cana-5423	360	13	get	get	VERB
cana-5423	360	14	ζ(d(γx	ζ(d(γx	NOUN
cana-5423	360	15	,	,	PUNCT
cana-5423	360	16	γy)k(x	γy)k(x	PROPN
cana-5423	360	17	,	,	PUNCT
cana-5423	360	18	y	y	NOUN
cana-5423	360	19	)	)	PUNCT
cana-5423	360	20	)	)	PUNCT
cana-5423	361	1	+	+	CCONJ
cana-5423	361	2	lq(x	lq(x	X
cana-5423	361	3	,	,	PUNCT
cana-5423	361	4	y	y	NOUN
cana-5423	361	5	)	)	PUNCT
cana-5423	361	6	=	=	SYM
cana-5423	361	7	1	1	NUM
cana-5423	361	8	2	2	NUM
cana-5423	361	9	[	[	SYM
cana-5423	361	10	1	1	NUM
cana-5423	361	11	+	+	NUM
cana-5423	361	12	|2x−	|2x−	NOUN
cana-5423	361	13	4|	4|	NUM
cana-5423	361	14	]	]	PUNCT
cana-5423	361	15	|2y	|2y	NOUN
cana-5423	361	16	−	−	NOUN
cana-5423	361	17	4|	4|	NUM
cana-5423	361	18	1	1	NUM
cana-5423	362	1	+	+	NUM
cana-5423	363	1	|x−	|x−	NOUN
cana-5423	364	1	y|	y|	NOUN
cana-5423	365	1	+	+	CCONJ
cana-5423	365	2	5	5	NUM
cana-5423	365	3	|2y	|2y	NOUN
cana-5423	365	4	−	−	NUM
cana-5423	365	5	4|	4|	NUM
cana-5423	365	6	−	−	ADP
cana-5423	365	7	|x−	|x−	PROPN
cana-5423	365	8	y|	y|	NOUN
cana-5423	365	9	.	.	PUNCT
cana-5423	366	1	(	(	PUNCT
cana-5423	366	2	59	59	NUM
cana-5423	366	3	)	)	PUNCT
cana-5423	366	4	therefore	therefore	ADV
cana-5423	366	5	,	,	PUNCT
cana-5423	366	6	γ	γ	PROPN
cana-5423	366	7	is	be	AUX
cana-5423	366	8	generalized	generalize	VERB
cana-5423	366	9	α	α	PRON
cana-5423	366	10	admissible	admissible	ADJ
cana-5423	366	11	modified	modify	VERB
cana-5423	366	12	almostz	almostz	NOUN
cana-5423	366	13	-	-	PUNCT
cana-5423	366	14	contraction	contraction	NOUN
cana-5423	366	15	with	with	ADP
cana-5423	366	16	respect	respect	NOUN
cana-5423	366	17	to	to	ADP
cana-5423	366	18	ζ	ζ	SYM
cana-5423	366	19	∈	∈	NOUN
cana-5423	366	20	z.	z.	NOUN
cana-5423	366	21	hence	hence	ADV
cana-5423	366	22	,	,	PUNCT
cana-5423	366	23	all	all	DET
cana-5423	366	24	the	the	DET
cana-5423	366	25	assumptions	assumption	NOUN
cana-5423	366	26	of	of	ADP
cana-5423	366	27	theorem	theorem	NOUN
cana-5423	366	28	2.2	2.2	NUM
cana-5423	366	29	with	with	ADP
cana-5423	366	30	corollary	corollary	ADJ
cana-5423	366	31	2.4	2.4	NUM
cana-5423	366	32	and	and	CCONJ
cana-5423	366	33	corollary	corollary	ADJ
cana-5423	366	34	2.5	2.5	NUM
cana-5423	366	35	are	be	AUX
cana-5423	366	36	satisfied	satisfied	ADJ
cana-5423	366	37	.	.	PUNCT
cana-5423	367	1	hence	hence	ADV
cana-5423	367	2	,	,	PUNCT
cana-5423	367	3	γ	γ	PROPN
cana-5423	367	4	has	have	VERB
cana-5423	367	5	a	a	DET
cana-5423	367	6	unique	unique	ADJ
cana-5423	367	7	fixed	fix	VERB
cana-5423	367	8	point	point	NOUN
cana-5423	367	9	.	.	PUNCT
cana-5423	368	1	references	reference	NOUN
cana-5423	368	2	1	1	NUM
cana-5423	368	3	.	.	PUNCT
cana-5423	369	1	t.	t.	PROPN
cana-5423	369	2	abdeljawad	abdeljawad	NOUN
cana-5423	369	3	:	:	PUNCT
cana-5423	369	4	meir	meir	NOUN
cana-5423	369	5	-keeler	-keeler	PROPN
cana-5423	369	6	α	α	PROPN
cana-5423	369	7	contractive	contractive	ADJ
cana-5423	369	8	fixed	fix	VERB
cana-5423	369	9	and	and	CCONJ
cana-5423	369	10	common	common	ADJ
cana-5423	369	11	fixed	fix	VERB
cana-5423	369	12	point	point	NOUN
cana-5423	369	13	theorems	theorem	NOUN
cana-5423	369	14	,	,	PUNCT
cana-5423	369	15	fixed	fix	VERB
cana-5423	369	16	point	point	NOUN
cana-5423	369	17	theory	theory	NOUN
cana-5423	369	18	and	and	CCONJ
cana-5423	369	19	appl	appl	NOUN
cana-5423	369	20	.	.	PROPN
cana-5423	370	1	18	18	NUM
cana-5423	370	2	(	(	PUNCT
cana-5423	370	3	2013	2013	NUM
cana-5423	370	4	)	)	PUNCT
cana-5423	370	5	.	.	PUNCT
cana-5423	371	1	2	2	X
cana-5423	371	2	.	.	X
cana-5423	371	3	t.	t.	NOUN
cana-5423	371	4	abdeljawad	abdeljawad	PROPN
cana-5423	371	5	and	and	CCONJ
cana-5423	371	6	d.	d.	PROPN
cana-5423	371	7	gopal	gopal	PROPN
cana-5423	371	8	:	:	PUNCT
cana-5423	371	9	erratum	erratum	PROPN
cana-5423	371	10	to	to	ADP
cana-5423	371	11	meir	meir	PROPN
cana-5423	371	12	-keeler	-keeler	PROPN
cana-5423	371	13	α	α	PROPN
cana-5423	371	14	contractive	contractive	ADJ
cana-5423	371	15	fixed	fix	VERB
cana-5423	371	16	and	and	CCONJ
cana-5423	371	17	common	common	ADJ
cana-5423	371	18	fixed	fix	VERB
cana-5423	371	19	point	point	NOUN
cana-5423	371	20	theorems	theorem	NOUN
cana-5423	371	21	,	,	PUNCT
cana-5423	371	22	fixed	fix	VERB
cana-5423	371	23	point	point	NOUN
cana-5423	371	24	theory	theory	NOUN
cana-5423	371	25	and	and	CCONJ
cana-5423	371	26	appl.110	appl.110	PROPN
cana-5423	371	27	(	(	PUNCT
cana-5423	371	28	2013	2013	NUM
cana-5423	371	29	)	)	PUNCT
cana-5423	371	30	.	.	PUNCT
cana-5423	372	1	3	3	X
cana-5423	372	2	.	.	X
cana-5423	372	3	a.s	a.s	PROPN
cana-5423	372	4	.	.	PROPN
cana-5423	372	5	alharbi	alharbi	PROPN
cana-5423	372	6	,	,	PUNCT
cana-5423	372	7	h.	h.	PROPN
cana-5423	372	8	alsulami	alsulami	PROPN
cana-5423	372	9	and	and	CCONJ
cana-5423	372	10	e.	e.	PROPN
cana-5423	372	11	karapinar	karapinar	PROPN
cana-5423	372	12	:	:	PUNCT
cana-5423	372	13	on	on	ADP
cana-5423	372	14	the	the	DET
cana-5423	372	15	power	power	NOUN
cana-5423	372	16	of	of	ADP
cana-5423	372	17	simulation	simulation	NOUN
cana-5423	372	18	and	and	CCONJ
cana-5423	372	19	admissible	admissible	ADJ
cana-5423	372	20	functions	function	NOUN
cana-5423	372	21	in	in	ADP
cana-5423	372	22	metric	metric	ADJ
cana-5423	372	23	fixed	fix	VERB
cana-5423	372	24	point	point	NOUN
cana-5423	372	25	theory	theory	NOUN
cana-5423	372	26	,	,	PUNCT
cana-5423	372	27	jour	jour	X
cana-5423	372	28	.	.	PROPN
cana-5423	372	29	func	func	PROPN
cana-5423	372	30	.	.	PUNCT
cana-5423	373	1	spaces	space	NOUN
cana-5423	373	2	,	,	PUNCT
cana-5423	373	3	vol	vol	NOUN
cana-5423	373	4	.	.	PROPN
cana-5423	373	5	2017	2017	NUM
cana-5423	373	6	,	,	PUNCT
cana-5423	373	7	article	article	NOUN
cana-5423	373	8	i	i	PROPN
cana-5423	373	9	d	d	PROPN
cana-5423	373	10	2068163	2068163	NUM
cana-5423	373	11	,	,	PUNCT
cana-5423	373	12	7	7	NUM
cana-5423	373	13	pages	page	NOUN
cana-5423	373	14	.	.	PUNCT
cana-5423	374	1	4	4	X
cana-5423	374	2	.	.	X
cana-5423	374	3	h.argoubi	h.argoubi	ADJ
cana-5423	374	4	,	,	PUNCT
cana-5423	374	5	b.samet	b.samet	NOUN
cana-5423	374	6	,	,	PUNCT
cana-5423	374	7	and	and	CCONJ
cana-5423	374	8	c.	c.	PROPN
cana-5423	374	9	vetro	vetro	PROPN
cana-5423	374	10	:	:	PUNCT
cana-5423	375	1	nonlinear	nonlinear	ADJ
cana-5423	375	2	contractions	contraction	NOUN
cana-5423	375	3	involving	involve	VERB
cana-5423	375	4	simulation	simulation	NOUN
cana-5423	375	5	functions	function	NOUN
cana-5423	375	6	in	in	ADP
cana-5423	375	7	a	a	DET
cana-5423	375	8	metric	metric	ADJ
cana-5423	375	9	space	space	NOUN
cana-5423	375	10	with	with	ADP
cana-5423	375	11	a	a	DET
cana-5423	375	12	partial	partial	ADJ
cana-5423	375	13	order	order	NOUN
cana-5423	375	14	,	,	PUNCT
cana-5423	375	15	jour	jour	X
cana-5423	375	16	.	.	PROPN
cana-5423	375	17	nonlinear	nonlinear	PROPN
cana-5423	375	18	sci	sci	PROPN
cana-5423	375	19	.	.	PUNCT
cana-5423	375	20	appl	appl	PROPN
cana-5423	375	21	.	.	PROPN
cana-5423	375	22	,	,	PUNCT
cana-5423	375	23	8(2015	8(2015	NUM
cana-5423	375	24	)	)	PUNCT
cana-5423	375	25	,	,	PUNCT
cana-5423	375	26	1082	1082	NUM
cana-5423	375	27	-	-	SYM
cana-5423	375	28	1094	1094	NUM
cana-5423	375	29	.	.	PUNCT
cana-5423	376	1	5	5	NUM
cana-5423	376	2	.	.	X
cana-5423	376	3	h.	h.	PROPN
cana-5423	376	4	aydi	aydi	PROPN
cana-5423	376	5	,	,	PUNCT
cana-5423	376	6	a.felhii	a.felhii	ADJ
cana-5423	376	7	,	,	PUNCT
cana-5423	376	8	e.	e.	PROPN
cana-5423	376	9	karapinar	karapinar	PROPN
cana-5423	376	10	and	and	CCONJ
cana-5423	376	11	f.a	f.a	PROPN
cana-5423	376	12	.	.	PROPN
cana-5423	376	13	alojail	alojail	PROPN
cana-5423	376	14	:	:	PUNCT
cana-5423	376	15	fixed	fix	VERB
cana-5423	376	16	points	point	NOUN
cana-5423	376	17	on	on	ADP
cana-5423	376	18	quasi	quasi	ADJ
cana-5423	376	19	-	-	ADJ
cana-5423	376	20	metric	metric	ADJ
cana-5423	376	21	spaces	space	NOUN
cana-5423	376	22	via	via	ADP
cana-5423	376	23	simulation	simulation	NOUN
cana-5423	376	24	functions	function	NOUN
cana-5423	376	25	and	and	CCONJ
cana-5423	376	26	conesequences	conesequence	NOUN
cana-5423	376	27	,	,	PUNCT
cana-5423	376	28	jour	jour	X
cana-5423	376	29	.	.	PUNCT
cana-5423	376	30	math	math	PROPN
cana-5423	376	31	.	.	PUNCT
cana-5423	377	1	anal	anal	PROPN
cana-5423	377	2	.	.	PROPN
cana-5423	377	3	,	,	PUNCT
cana-5423	377	4	9	9	NUM
cana-5423	377	5	(	(	PUNCT
cana-5423	377	6	2	2	NUM
cana-5423	377	7	)	)	PUNCT
cana-5423	377	8	(	(	PUNCT
cana-5423	377	9	2018	2018	NUM
cana-5423	377	10	)	)	PUNCT
cana-5423	377	11	.	.	PUNCT
cana-5423	378	1	10	10	NUM
cana-5423	378	2	-	-	SYM
cana-5423	378	3	24	24	NUM
cana-5423	378	4	.	.	PUNCT
cana-5423	379	1	6	6	NUM
cana-5423	379	2	.	.	X
cana-5423	379	3	g.v.r	g.v.r	NOUN
cana-5423	379	4	.	.	PUNCT
cana-5423	380	1	babu	babu	PROPN
cana-5423	380	2	,	,	PUNCT
cana-5423	380	3	m.l	m.l	PROPN
cana-5423	380	4	.	.	PROPN
cana-5423	380	5	sandhya	sandhya	PROPN
cana-5423	380	6	and	and	CCONJ
cana-5423	380	7	m.v.r	m.v.r	NOUN
cana-5423	380	8	.	.	PUNCT
cana-5423	380	9	kameshwari	kameshwari	PROPN
cana-5423	380	10	:	:	PUNCT
cana-5423	380	11	note	note	VERB
cana-5423	380	12	on	on	ADP
cana-5423	380	13	a	a	DET
cana-5423	380	14	fixed	fix	VERB
cana-5423	380	15	point	point	NOUN
cana-5423	380	16	theorems	theorem	NOUN
cana-5423	380	17	of	of	ADP
cana-5423	380	18	berinde	berinde	NOUN
cana-5423	380	19	on	on	ADP
cana-5423	380	20	weak	weak	ADJ
cana-5423	380	21	contractions	contraction	NOUN
cana-5423	380	22	,	,	PUNCT
cana-5423	380	23	carpathian	carpathian	PROPN
cana-5423	380	24	j.	j.	PROPN
cana-5423	380	25	math	math	PROPN
cana-5423	380	26	.	.	PUNCT
cana-5423	381	1	24	24	NUM
cana-5423	381	2	(	(	PUNCT
cana-5423	381	3	2008	2008	NUM
cana-5423	381	4	)	)	PUNCT
cana-5423	381	5	,	,	PUNCT
cana-5423	381	6	8	8	NUM
cana-5423	381	7	-	-	SYM
cana-5423	381	8	12	12	NUM
cana-5423	381	9	.	.	PUNCT
cana-5423	382	1	7	7	X
cana-5423	382	2	.	.	X
cana-5423	382	3	s.banach	s.banach	NOUN
cana-5423	382	4	:	:	PUNCT
cana-5423	382	5	sur	sur	PROPN
cana-5423	382	6	less	less	ADJ
cana-5423	382	7	operattions	operattion	NOUN
cana-5423	382	8	dans	dan	NOUN
cana-5423	382	9	les	les	X
cana-5423	382	10	ensembles	ensemble	NOUN
cana-5423	382	11	abstaits	abstait	NOUN
cana-5423	382	12	et	et	PROPN
cana-5423	382	13	leur	leur	X
cana-5423	382	14	application	application	PROPN
cana-5423	382	15	aux	aux	PROPN
cana-5423	382	16	equations	equation	NOUN
cana-5423	382	17	integrales	integrale	NOUN
cana-5423	382	18	,	,	PUNCT
cana-5423	382	19	fund	fund	NOUN
cana-5423	382	20	.	.	PUNCT
cana-5423	383	1	math	math	NOUN
cana-5423	383	2	.	.	PUNCT
cana-5423	383	3	,	,	PUNCT
cana-5423	383	4	2(1922	2(1922	NUM
cana-5423	383	5	)	)	PUNCT
cana-5423	383	6	,	,	PUNCT
cana-5423	383	7	133	133	NUM
cana-5423	383	8	-	-	SYM
cana-5423	383	9	181	181	NUM
cana-5423	383	10	.	.	NOUN
cana-5423	383	11	8	8	NUM
cana-5423	383	12	.	.	X
cana-5423	383	13	m.	m.	NOUN
cana-5423	383	14	berzing	berzing	NOUN
cana-5423	383	15	,	,	PUNCT
cana-5423	383	16	and	and	CCONJ
cana-5423	383	17	m.	m.	PROPN
cana-5423	383	18	d.	d.	PROPN
cana-5423	383	19	rus	rus	PROPN
cana-5423	383	20	:	:	PUNCT
cana-5423	383	21	fixed	fix	VERB
cana-5423	383	22	point	point	NOUN
cana-5423	383	23	theorems	theorem	NOUN
cana-5423	383	24	for	for	ADP
cana-5423	383	25	α	α	NOUN
cana-5423	383	26	-	-	ADJ
cana-5423	383	27	contractive	contractive	ADJ
cana-5423	383	28	mappings	mapping	NOUN
cana-5423	383	29	of	of	ADP
cana-5423	383	30	meir	meir	PROPN
cana-5423	383	31	-	-	PUNCT
cana-5423	383	32	keeler	keeler	PROPN
cana-5423	383	33	type	type	NOUN
cana-5423	383	34	and	and	CCONJ
cana-5423	383	35	application	application	NOUN
cana-5423	383	36	,	,	PUNCT
cana-5423	383	37	non	non	ADJ
cana-5423	383	38	linear	linear	PROPN
cana-5423	383	39	anal	anal	PROPN
cana-5423	383	40	.	.	PUNCT
cana-5423	384	1	model	model	PROPN
cana-5423	384	2	.	.	PUNCT
cana-5423	385	1	control	control	PROPN
cana-5423	385	2	.	.	PUNCT
cana-5423	386	1	,19(2),(2014),178	,19(2),(2014),178	PROPN
cana-5423	386	2	-	-	PUNCT
cana-5423	386	3	198	198	NUM
cana-5423	386	4	.	.	PUNCT
cana-5423	387	1	9	9	NUM
cana-5423	387	2	.	.	X
cana-5423	388	1	v.	v.	ADP
cana-5423	388	2	berinde	berinde	NOUN
cana-5423	388	3	:	:	PUNCT
cana-5423	388	4	approximating	approximate	VERB
cana-5423	388	5	fixed	fix	VERB
cana-5423	388	6	points	point	NOUN
cana-5423	388	7	of	of	ADP
cana-5423	388	8	weak	weak	ADJ
cana-5423	388	9	contractions	contraction	NOUN
cana-5423	388	10	using	use	VERB
cana-5423	388	11	the	the	DET
cana-5423	388	12	picard	picard	NOUN
cana-5423	388	13	iteration	iteration	NOUN
cana-5423	388	14	,	,	PUNCT
cana-5423	388	15	nonlinear	nonlinear	ADJ
cana-5423	388	16	anal	anal	PROPN
cana-5423	388	17	.	.	PUNCT
cana-5423	389	1	forum	forum	PROPN
cana-5423	389	2	,	,	PUNCT
cana-5423	389	3	9(1	9(1	NUM
cana-5423	389	4	)	)	PUNCT
cana-5423	389	5	(	(	PUNCT
cana-5423	389	6	2004	2004	NUM
cana-5423	389	7	)	)	PUNCT
cana-5423	389	8	,	,	PUNCT
cana-5423	389	9	43	43	NUM
cana-5423	389	10	-	-	SYM
cana-5423	389	11	53	53	NUM
cana-5423	389	12	.	.	PUNCT
cana-5423	389	13	10	10	NUM
cana-5423	389	14	.	.	PUNCT
cana-5423	390	1	v.	v.	ADP
cana-5423	390	2	berinde	berinde	PROPN
cana-5423	390	3	:	:	PUNCT
cana-5423	390	4	general	general	ADJ
cana-5423	390	5	constructive	constructive	ADJ
cana-5423	390	6	fixed	fix	VERB
cana-5423	390	7	point	point	NOUN
cana-5423	390	8	theorems	theorem	NOUN
cana-5423	390	9	for	for	ADP
cana-5423	390	10	ciric	ciric	ADJ
cana-5423	390	11	type	type	NOUN
cana-5423	390	12	almost	almost	ADV
cana-5423	390	13	contractions	contraction	NOUN
cana-5423	390	14	in	in	ADP
cana-5423	390	15	metric	metric	ADJ
cana-5423	390	16	spaces	space	NOUN
cana-5423	390	17	,	,	PUNCT
cana-5423	390	18	carpathian	carpathian	PROPN
cana-5423	390	19	j.	j.	PROPN
cana-5423	390	20	math	math	PROPN
cana-5423	390	21	.	.	PUNCT
cana-5423	391	1	24(2	24(2	NUM
cana-5423	391	2	)	)	PUNCT
cana-5423	391	3	(	(	PUNCT
cana-5423	391	4	2008	2008	NUM
cana-5423	391	5	)	)	PUNCT
cana-5423	391	6	,	,	PUNCT
cana-5423	391	7	10	10	NUM
cana-5423	391	8	-	-	SYM
cana-5423	391	9	19	19	NUM
cana-5423	391	10	.	.	NOUN
cana-5423	391	11	11	11	NUM
cana-5423	391	12	.	.	PUNCT
cana-5423	392	1	p.	p.	NOUN
cana-5423	392	2	bunpatcharacharoen	bunpatcharacharoen	PROPN
cana-5423	392	3	,	,	PUNCT
cana-5423	392	4	s.	s.	PROPN
cana-5423	392	5	saelee	saelee	PROPN
cana-5423	392	6	and	and	CCONJ
cana-5423	392	7	p.sipara	p.sipara	NOUN
cana-5423	392	8	:	:	PUNCT
cana-5423	392	9	modified	modify	VERB
cana-5423	392	10	almost	almost	ADV
cana-5423	392	11	type	type	NOUN
cana-5423	392	12	z−	z−	PROPN
cana-5423	392	13	contraction	contraction	NOUN
cana-5423	392	14	,	,	PUNCT
cana-5423	392	15	thai	thai	PROPN
cana-5423	392	16	journal	journal	PROPN
cana-5423	392	17	of	of	ADP
cana-5423	392	18	mathematics	mathematic	NOUN
cana-5423	392	19	,	,	PUNCT
cana-5423	392	20	18(1)(2020	18(1)(2020	NUM
cana-5423	392	21	)	)	PUNCT
cana-5423	392	22	,	,	PUNCT
cana-5423	392	23	252	252	NUM
cana-5423	392	24	-	-	SYM
cana-5423	392	25	260	260	NUM
cana-5423	392	26	.	.	PUNCT
cana-5423	392	27	12	12	NUM
cana-5423	392	28	.	.	PUNCT
cana-5423	392	29	a.	a.	PROPN
cana-5423	392	30	chanda	chanda	PROPN
cana-5423	392	31	,	,	PUNCT
cana-5423	392	32	b.	b.	PROPN
cana-5423	392	33	damjanovic	damjanovic	PROPN
cana-5423	392	34	,	,	PUNCT
cana-5423	392	35	and	and	CCONJ
cana-5423	392	36	l.k.dey	l.k.dey	PROPN
cana-5423	392	37	:	:	PUNCT
cana-5423	392	38	fixed	fix	VERB
cana-5423	392	39	point	point	NOUN
cana-5423	392	40	results	result	NOUN
cana-5423	392	41	on	on	ADP
cana-5423	392	42	θmetric	θmetric	ADJ
cana-5423	392	43	spaces	space	NOUN
cana-5423	392	44	via	via	ADP
cana-5423	392	45	simulation	simulation	NOUN
cana-5423	392	46	functions	function	NOUN
cana-5423	392	47	,	,	PUNCT
cana-5423	392	48	filomat	filomat	NOUN
cana-5423	392	49	,	,	PUNCT
cana-5423	392	50	31(11),(2017	31(11),(2017	NUM
cana-5423	392	51	)	)	PUNCT
cana-5423	392	52	,	,	PUNCT
cana-5423	392	53	3365	3365	NUM
cana-5423	392	54	-	-	SYM
cana-5423	392	55	3375	3375	NUM
cana-5423	392	56	.	.	PUNCT
cana-5423	393	1	13	13	NUM
cana-5423	393	2	.	.	X
cana-5423	394	1	s	s	X
cana-5423	394	2	-	-	PUNCT
cana-5423	394	3	h	h	NOUN
cana-5423	394	4	,	,	PUNCT
cana-5423	394	5	cho	cho	PROPN
cana-5423	394	6	:	:	PUNCT
cana-5423	394	7	fixed	fixed	ADJ
cana-5423	394	8	point	point	NOUN
cana-5423	394	9	theorem	theorem	NOUN
cana-5423	394	10	for	for	ADP
cana-5423	394	11	(	(	PUNCT
cana-5423	394	12	α	α	NOUN
cana-5423	394	13	,	,	PUNCT
cana-5423	394	14	β)−z	β)−z	NOUN
cana-5423	394	15	-	-	PUNCT
cana-5423	394	16	contractions	contraction	NOUN
cana-5423	394	17	in	in	ADP
cana-5423	394	18	metric	metric	ADJ
cana-5423	394	19	spaces	space	NOUN
cana-5423	394	20	,	,	PUNCT
cana-5423	394	21	int	int	NOUN
cana-5423	394	22	.	.	PUNCT
cana-5423	395	1	j.	j.	PROPN
cana-5423	395	2	of	of	ADP
cana-5423	395	3	math	math	PROPN
cana-5423	395	4	.	.	PUNCT
cana-5423	396	1	anal	anal	PROPN
cana-5423	396	2	.	.	PUNCT
cana-5423	397	1	13(4	13(4	NUM
cana-5423	397	2	)	)	PUNCT
cana-5423	397	3	,	,	PUNCT
cana-5423	397	4	(	(	PUNCT
cana-5423	397	5	2019	2019	NUM
cana-5423	397	6	)	)	PUNCT
cana-5423	397	7	,	,	PUNCT
cana-5423	397	8	161	161	NUM
cana-5423	397	9	-	-	SYM
cana-5423	397	10	174	174	NUM
cana-5423	397	11	.	.	PUNCT
cana-5423	398	1	14	14	NUM
cana-5423	398	2	.	.	PUNCT
cana-5423	398	3	a.	a.	PROPN
cana-5423	398	4	dewangan	dewangan	PROPN
cana-5423	398	5	,	,	PUNCT
cana-5423	398	6	a.k.dubey	a.k.dubey	INTJ
cana-5423	398	7	,	,	PUNCT
cana-5423	398	8	m.d	m.d	PROPN
cana-5423	398	9	.	.	PROPN
cana-5423	398	10	pandey	pandey	PROPN
cana-5423	398	11	and	and	CCONJ
cana-5423	398	12	r.p.pandey	r.p.pandey	PROPN
cana-5423	398	13	:	:	PUNCT
cana-5423	398	14	fixed	fix	VERB
cana-5423	398	15	point	point	NOUN
cana-5423	398	16	for	for	ADP
cana-5423	398	17	(	(	PUNCT
cana-5423	398	18	α	α	NOUN
cana-5423	398	19	,	,	PUNCT
cana-5423	398	20	β)admissible	β)admissible	ADJ
cana-5423	398	21	mapping	mapping	NOUN
cana-5423	398	22	via	via	ADP
cana-5423	398	23	simulation	simulation	NOUN
cana-5423	398	24	functions	function	NOUN
cana-5423	398	25	,	,	PUNCT
cana-5423	398	26	communications	communication	NOUN
cana-5423	398	27	in	in	ADP
cana-5423	398	28	mathematics	mathematic	NOUN
cana-5423	398	29	and	and	CCONJ
cana-5423	398	30	applications	application	NOUN
cana-5423	398	31	,	,	PUNCT
cana-5423	398	32	12(4),(2021	12(4),(2021	NUM
cana-5423	398	33	)	)	PUNCT
cana-5423	398	34	,	,	PUNCT
cana-5423	398	35	1101	1101	NUM
cana-5423	398	36	-	-	SYM
cana-5423	398	37	1111	1111	NUM
cana-5423	398	38	.	.	PUNCT
cana-5423	399	1	15	15	NUM
cana-5423	399	2	.	.	PUNCT
cana-5423	399	3	a.	a.	PROPN
cana-5423	399	4	felhi	felhi	PROPN
cana-5423	399	5	,	,	PUNCT
cana-5423	399	6	h.	h.	PROPN
cana-5423	399	7	adyi	adyi	PROPN
cana-5423	399	8	and	and	CCONJ
cana-5423	399	9	d.	d.	PROPN
cana-5423	399	10	zhang	zhang	PROPN
cana-5423	399	11	:	:	PUNCT
cana-5423	399	12	fixed	fix	VERB
cana-5423	399	13	points	point	NOUN
cana-5423	399	14	for	for	ADP
cana-5423	399	15	α	α	DET
cana-5423	399	16	admissible	admissible	ADJ
cana-5423	399	17	contraction	contraction	NOUN
cana-5423	399	18	mapping	mapping	NOUN
cana-5423	399	19	via	via	ADP
cana-5423	399	20	simulation	simulation	NOUN
cana-5423	399	21	functions	function	NOUN
cana-5423	399	22	,	,	PUNCT
cana-5423	399	23	journal	journal	NOUN
cana-5423	399	24	of	of	ADP
cana-5423	399	25	nonlinear	nonlinear	PROPN
cana-5423	399	26	sciences	sciences	PROPN
cana-5423	399	27	and	and	CCONJ
cana-5423	399	28	applications,9(10	applications,9(10	NOUN
cana-5423	399	29	)	)	PUNCT
cana-5423	399	30	,	,	PUNCT
cana-5423	399	31	(	(	PUNCT
cana-5423	399	32	2016	2016	NUM
cana-5423	399	33	)	)	PUNCT
cana-5423	399	34	,	,	PUNCT
cana-5423	399	35	55445560	55445560	NUM
cana-5423	399	36	.	.	PUNCT
cana-5423	400	1	communications	communication	NOUN
cana-5423	400	2	on	on	ADP
cana-5423	400	3	applied	apply	VERB
cana-5423	400	4	nonlinear	nonlinear	ADJ
cana-5423	400	5	analysis	analysis	NOUN
cana-5423	400	6	issn	issn	NOUN
cana-5423	400	7	:	:	PUNCT
cana-5423	400	8	1074	1074	NUM
cana-5423	400	9	-	-	PUNCT
cana-5423	400	10	133x	133x	NUM
cana-5423	400	11	vol	vol	NOUN
cana-5423	400	12	32	32	NUM
cana-5423	400	13	no	no	NOUN
cana-5423	400	14	.	.	PUNCT
cana-5423	401	1	10s(2025	10s(2025	NUM
cana-5423	401	2	)	)	PUNCT
cana-5423	402	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	402	2	2229	2229	NUM
cana-5423	402	3	16	16	NUM
cana-5423	402	4	.	.	PUNCT
cana-5423	403	1	d.gopal	d.gopal	DET
cana-5423	403	2	:	:	PUNCT
cana-5423	403	3	fixed	fix	VERB
cana-5423	403	4	points	point	NOUN
cana-5423	403	5	of	of	ADP
cana-5423	403	6	α	α	NOUN
cana-5423	403	7	-	-	PUNCT
cana-5423	403	8	type	type	NOUN
cana-5423	403	9	f	f	NOUN
cana-5423	403	10	-contractive	-contractive	ADJ
cana-5423	403	11	mappingss	mappingss	NOUN
cana-5423	403	12	with	with	ADP
cana-5423	403	13	an	an	DET
cana-5423	403	14	application	application	NOUN
cana-5423	403	15	to	to	ADP
cana-5423	403	16	nonlinear	nonlinear	ADJ
cana-5423	403	17	fractional	fractional	ADJ
cana-5423	403	18	differential	differential	NOUN
cana-5423	403	19	equation	equation	NOUN
cana-5423	403	20	,	,	PUNCT
cana-5423	403	21	acta	acta	PROPN
cana-5423	403	22	mathematica	mathematica	PROPN
cana-5423	403	23	scientica,36(2016),957	scientica,36(2016),957	PROPN
cana-5423	403	24	-	-	SYM
cana-5423	403	25	970	970	NUM
cana-5423	403	26	.	.	PUNCT
cana-5423	403	27	17	17	NUM
cana-5423	403	28	.	.	PUNCT
cana-5423	404	1	n.	n.	PROPN
cana-5423	404	2	hussain	hussain	PROPN
cana-5423	404	3	,	,	PUNCT
cana-5423	404	4	p.	p.	PROPN
cana-5423	404	5	salimi	salimi	PROPN
cana-5423	404	6	,	,	PUNCT
cana-5423	404	7	and	and	CCONJ
cana-5423	404	8	a.	a.	PROPN
cana-5423	404	9	latif	latif	PROPN
cana-5423	404	10	:	:	PUNCT
cana-5423	404	11	fixed	fix	VERB
cana-5423	404	12	point	point	NOUN
cana-5423	404	13	results	result	NOUN
cana-5423	404	14	for	for	ADP
cana-5423	404	15	single	single	ADJ
cana-5423	404	16	and	and	CCONJ
cana-5423	404	17	set	set	VERB
cana-5423	404	18	valued	value	VERB
cana-5423	404	19	α−	α−	ADP
cana-5423	404	20	η−ψ	η−ψ	PROPN
cana-5423	404	21	contraction	contraction	NOUN
cana-5423	404	22	mappings	mapping	NOUN
cana-5423	404	23	,	,	PUNCT
cana-5423	404	24	fixed	fix	VERB
cana-5423	404	25	pont	pont	NOUN
cana-5423	404	26	theory	theory	NOUN
cana-5423	404	27	application	application	NOUN
cana-5423	404	28	,	,	PUNCT
cana-5423	404	29	212(2013	212(2013	NUM
cana-5423	404	30	)	)	PUNCT
cana-5423	404	31	.	.	PUNCT
cana-5423	405	1	18	18	NUM
cana-5423	405	2	.	.	PUNCT
cana-5423	406	1	n.hussain	n.hussain	NOUN
cana-5423	406	2	,	,	PUNCT
cana-5423	406	3	e.karapinar	e.karapinar	NOUN
cana-5423	406	4	,	,	PUNCT
cana-5423	406	5	p.salimi	p.salimi	NOUN
cana-5423	406	6	,	,	PUNCT
cana-5423	406	7	and	and	CCONJ
cana-5423	406	8	f.akbar	f.akbar	ADJ
cana-5423	406	9	:	:	PUNCT
cana-5423	406	10	αadmissible	αadmissible	ADJ
cana-5423	406	11	mappings	mapping	NOUN
cana-5423	406	12	and	and	CCONJ
cana-5423	406	13	related	relate	VERB
cana-5423	406	14	fixed	fix	VERB
cana-5423	406	15	point	point	NOUN
cana-5423	406	16	theorem	theorem	VERB
cana-5423	406	17	,	,	PUNCT
cana-5423	406	18	fixed	fix	VERB
cana-5423	406	19	point	point	NOUN
cana-5423	406	20	theory	theory	NOUN
cana-5423	406	21	appl	appl	PROPN
cana-5423	406	22	.	.	PROPN
cana-5423	406	23	,	,	PUNCT
cana-5423	406	24	2012	2012	NUM
cana-5423	406	25	(	(	PUNCT
cana-5423	406	26	2013	2013	NUM
cana-5423	406	27	)	)	PUNCT
cana-5423	406	28	.	.	PUNCT
cana-5423	407	1	19	19	NUM
cana-5423	407	2	.	.	PUNCT
cana-5423	407	3	n.	n.	PROPN
cana-5423	407	4	hussain	hussain	PROPN
cana-5423	407	5	,	,	PUNCT
cana-5423	407	6	m.	m.	PROPN
cana-5423	407	7	arshad	arshad	PROPN
cana-5423	407	8	,	,	PUNCT
cana-5423	407	9	a.	a.	PROPN
cana-5423	407	10	shoaib	shoaib	PROPN
cana-5423	407	11	,	,	PUNCT
cana-5423	407	12	and	and	CCONJ
cana-5423	407	13	fabimuddin	fabimuddin	NOUN
cana-5423	407	14	:	:	PUNCT
cana-5423	407	15	common	common	ADJ
cana-5423	407	16	fixed	fix	VERB
cana-5423	407	17	point	point	NOUN
cana-5423	407	18	results	result	NOUN
cana-5423	407	19	for	for	ADP
cana-5423	407	20	α−ψcontractions	α−ψcontraction	NOUN
cana-5423	407	21	on	on	ADP
cana-5423	407	22	a	a	DET
cana-5423	407	23	metric	metric	ADJ
cana-5423	407	24	space	space	NOUN
cana-5423	407	25	endowed	endow	VERB
cana-5423	407	26	with	with	ADP
cana-5423	407	27	a	a	DET
cana-5423	407	28	graph	graph	NOUN
cana-5423	407	29	,	,	PUNCT
cana-5423	407	30	j.	j.	PROPN
cana-5423	407	31	inequality	inequality	PROPN
cana-5423	407	32	appl	appl	PROPN
cana-5423	407	33	.	.	PROPN
cana-5423	407	34	,	,	PUNCT
cana-5423	407	35	136(2014	136(2014	NUM
cana-5423	407	36	)	)	PUNCT
cana-5423	407	37	.	.	PUNCT
cana-5423	408	1	20	20	NUM
cana-5423	408	2	.	.	PUNCT
cana-5423	409	1	h.	h.	PROPN
cana-5423	409	2	isik	isik	PROPN
cana-5423	409	3	,	,	PUNCT
cana-5423	409	4	n.b.gungor	n.b.gungor	VERB
cana-5423	409	5	,	,	PUNCT
cana-5423	409	6	c.	c.	NOUN
cana-5423	409	7	park	park	NOUN
cana-5423	409	8	,	,	PUNCT
cana-5423	409	9	and	and	CCONJ
cana-5423	409	10	s.y.jang	s.y.jang	ADJ
cana-5423	409	11	:	:	PUNCT
cana-5423	409	12	fixe	fixe	ADJ
cana-5423	409	13	point	point	NOUN
cana-5423	409	14	theorems	theorem	VERB
cana-5423	409	15	for	for	ADP
cana-5423	409	16	almost	almost	ADV
cana-5423	409	17	z	z	NOUN
cana-5423	409	18	-	-	PUNCT
cana-5423	409	19	contractions	contraction	NOUN
cana-5423	409	20	with	with	ADP
cana-5423	409	21	an	an	DET
cana-5423	409	22	application	application	NOUN
cana-5423	409	23	,	,	PUNCT
cana-5423	409	24	∑	∑	PUNCT
cana-5423	409	25	mathematics,6(208	mathematics,6(208	X
cana-5423	409	26	)	)	PUNCT
cana-5423	409	27	,	,	PUNCT
cana-5423	409	28	37,1	37,1	NUM
cana-5423	409	29	-	-	PUNCT
cana-5423	409	30	8.doi:10.3390	8.doi:10.3390	NUM
cana-5423	409	31	/	/	SYM
cana-5423	409	32	math6030037	math6030037	PROPN
cana-5423	409	33	21	21	NUM
cana-5423	409	34	.	.	PUNCT
cana-5423	410	1	e.	e.	PROPN
cana-5423	410	2	karapinar	karapinar	PROPN
cana-5423	410	3	,	,	PUNCT
cana-5423	410	4	and	and	CCONJ
cana-5423	410	5	b.samet	b.samet	VERB
cana-5423	410	6	:	:	PUNCT
cana-5423	410	7	generalized	generalize	VERB
cana-5423	410	8	(	(	PUNCT
cana-5423	410	9	α	α	X
cana-5423	410	10	,	,	PUNCT
cana-5423	410	11	ψ	ψ	NOUN
cana-5423	410	12	)	)	PUNCT
cana-5423	410	13	contractive	contractive	ADJ
cana-5423	410	14	type	type	NOUN
cana-5423	410	15	mappings	mapping	NOUN
cana-5423	410	16	and	and	CCONJ
cana-5423	410	17	related	relate	VERB
cana-5423	410	18	fixed	fix	VERB
cana-5423	410	19	point	point	NOUN
cana-5423	410	20	theorems	theorem	NOUN
cana-5423	410	21	with	with	ADP
cana-5423	410	22	applications	application	NOUN
cana-5423	410	23	,	,	PUNCT
cana-5423	410	24	abstra.appl	abstra.appl	PROPN
cana-5423	410	25	.	.	PROPN
cana-5423	410	26	anal	anal	PROPN
cana-5423	410	27	.	.	PUNCT
cana-5423	411	1	2012,(2012	2012,(2012	NUM
cana-5423	411	2	)	)	PUNCT
cana-5423	411	3	article	article	NOUN
cana-5423	411	4	i	i	PROPN
cana-5423	411	5	d	d	PROPN
cana-5423	411	6	793486	793486	NUM
cana-5423	411	7	.	.	PUNCT
cana-5423	412	1	22	22	NUM
cana-5423	412	2	.	.	PUNCT
cana-5423	413	1	e.	e.	PROPN
cana-5423	413	2	karapinar	karapinar	PROPN
cana-5423	413	3	,	,	PUNCT
cana-5423	413	4	p.	p.	NOUN
cana-5423	413	5	kumam	kumam	PROPN
cana-5423	413	6	,	,	PUNCT
cana-5423	413	7	and	and	CCONJ
cana-5423	413	8	p.	p.	PROPN
cana-5423	413	9	salimi	salimi	PROPN
cana-5423	413	10	:	:	PUNCT
cana-5423	413	11	on	on	ADP
cana-5423	413	12	α	α	PROPN
cana-5423	413	13	−	−	PROPN
cana-5423	413	14	ψmeir	ψmeir	NOUN
cana-5423	413	15	-keeler	-keeler	PROPN
cana-5423	413	16	contractive	contractive	ADJ
cana-5423	413	17	mappings	mapping	NOUN
cana-5423	413	18	,	,	PUNCT
cana-5423	413	19	fixed	fix	VERB
cana-5423	413	20	point	point	NOUN
cana-5423	413	21	theory	theory	NOUN
cana-5423	413	22	and	and	CCONJ
cana-5423	413	23	application	application	NOUN
cana-5423	413	24	,	,	PUNCT
cana-5423	413	25	(	(	PUNCT
cana-5423	413	26	2013	2013	NUM
cana-5423	413	27	)	)	PUNCT
cana-5423	413	28	,	,	PUNCT
cana-5423	413	29	article	article	NOUN
cana-5423	413	30	i	i	PROPN
cana-5423	413	31	d	d	PROPN
cana-5423	413	32	94	94	NUM
cana-5423	413	33	(	(	PUNCT
cana-5423	413	34	2013	2013	NUM
cana-5423	413	35	)	)	PUNCT
cana-5423	413	36	.	.	PUNCT
cana-5423	414	1	23	23	NUM
cana-5423	414	2	.	.	PUNCT
cana-5423	415	1	e.karapinar	e.karapinar	NOUN
cana-5423	415	2	:	:	PUNCT
cana-5423	415	3	fixe	fixe	NOUN
cana-5423	415	4	point	point	NOUN
cana-5423	415	5	results	result	NOUN
cana-5423	415	6	via	via	ADP
cana-5423	415	7	simulation	simulation	NOUN
cana-5423	415	8	functions	function	NOUN
cana-5423	415	9	,	,	PUNCT
cana-5423	415	10	filomat	filomat	PROPN
cana-5423	415	11	30(8)(2016),2343	30(8)(2016),2343	PROPN
cana-5423	415	12	-	-	PUNCT
cana-5423	415	13	2350	2350	NUM
cana-5423	415	14	.	.	PUNCT
cana-5423	416	1	24	24	NUM
cana-5423	416	2	.	.	PUNCT
cana-5423	417	1	e.	e.	PROPN
cana-5423	417	2	karapinar	karapinar	PROPN
cana-5423	417	3	:	:	PUNCT
cana-5423	417	4	α	α	PRON
cana-5423	417	5	−	−	NOUN
cana-5423	417	6	ψ	ψ	X
cana-5423	417	7	-	-	ADJ
cana-5423	417	8	geraghty	geraghty	ADJ
cana-5423	417	9	contraction	contraction	NOUN
cana-5423	417	10	type	type	NOUN
cana-5423	417	11	mappings	mapping	NOUN
cana-5423	417	12	and	and	CCONJ
cana-5423	417	13	some	some	DET
cana-5423	417	14	related	related	ADJ
cana-5423	417	15	fixed	fix	VERB
cana-5423	417	16	point	point	NOUN
cana-5423	417	17	results	result	NOUN
cana-5423	417	18	,	,	PUNCT
cana-5423	417	19	filomat	filomat	NOUN
cana-5423	417	20	28(1),(2014),37	28(1),(2014),37	PROPN
cana-5423	417	21	-	-	SYM
cana-5423	417	22	48	48	NUM
cana-5423	417	23	.	.	PUNCT
cana-5423	418	1	25	25	NUM
cana-5423	418	2	.	.	PUNCT
cana-5423	419	1	e.	e.	PROPN
cana-5423	419	2	karapinar	karapinar	PROPN
cana-5423	419	3	v.m.l	v.m.l	PROPN
cana-5423	419	4	.	.	PUNCT
cana-5423	420	1	hima	hima	PROPN
cana-5423	420	2	bindu	bindu	PROPN
cana-5423	420	3	:	:	PUNCT
cana-5423	420	4	discussion	discussion	NOUN
cana-5423	420	5	on	on	ADP
cana-5423	420	6	the	the	DET
cana-5423	420	7	almost	almost	ADV
cana-5423	420	8	z	z	NOUN
cana-5423	420	9	contraction	contraction	NOUN
cana-5423	420	10	,	,	PUNCT
cana-5423	420	11	open	open	ADJ
cana-5423	420	12	mathematics	mathematic	NOUN
cana-5423	420	13	,	,	PUNCT
cana-5423	420	14	18(2020	18(2020	NUM
cana-5423	420	15	)	)	PUNCT
cana-5423	420	16	,	,	PUNCT
cana-5423	420	17	448	448	NUM
cana-5423	420	18	-	-	SYM
cana-5423	420	19	457	457	NUM
cana-5423	420	20	.	.	PUNCT
cana-5423	421	1	26	26	NUM
cana-5423	421	2	.	.	PUNCT
cana-5423	421	3	a.	a.	PROPN
cana-5423	421	4	dewangan	dewangan	PROPN
cana-5423	421	5	,	,	PUNCT
cana-5423	421	6	a.k.dubey	a.k.dubey	VERB
cana-5423	421	7	u.mishra	u.mishra	NOUN
cana-5423	421	8	and	and	CCONJ
cana-5423	421	9	r.p.dubey	r.p.dubey	PROPN
cana-5423	421	10	:	:	PUNCT
cana-5423	421	11	fixed	fix	VERB
cana-5423	421	12	point	point	NOUN
cana-5423	421	13	results	result	VERB
cana-5423	421	14	for	for	ADP
cana-5423	421	15	(	(	PUNCT
cana-5423	421	16	α−β	α−β	X
cana-5423	421	17	)	)	PUNCT
cana-5423	421	18	admissible	admissible	ADJ
cana-5423	421	19	almost	almost	ADV
cana-5423	421	20	zcontractions	zcontraction	NOUN
cana-5423	421	21	in	in	ADP
cana-5423	421	22	metyric	metyric	ADJ
cana-5423	421	23	like	like	ADP
cana-5423	421	24	space	space	NOUN
cana-5423	421	25	via	via	ADP
cana-5423	421	26	simulation	simulation	NOUN
cana-5423	421	27	function	function	NOUN
cana-5423	421	28	,	,	PUNCT
cana-5423	421	29	facta	facta	NOUN
cana-5423	421	30	universitats(nis	universitats(nis	PROPN
cana-5423	421	31	)	)	PUNCT
cana-5423	421	32	ser	ser	NOUN
cana-5423	421	33	.	.	PROPN
cana-5423	421	34	math	math	PROPN
cana-5423	421	35	.	.	PUNCT
cana-5423	422	1	inform	inform	NOUN
cana-5423	422	2	.	.	PUNCT
cana-5423	423	1	,	,	PUNCT
cana-5423	423	2	37(3	37(3	NUM
cana-5423	423	3	)	)	PUNCT
cana-5423	423	4	,	,	PUNCT
cana-5423	423	5	(	(	PUNCT
cana-5423	423	6	2022	2022	NUM
cana-5423	423	7	)	)	PUNCT
cana-5423	423	8	,	,	PUNCT
cana-5423	423	9	529	529	NUM
cana-5423	423	10	-	-	SYM
cana-5423	423	11	540	540	NUM
cana-5423	423	12	.	.	PUNCT
cana-5423	424	1	27	27	NUM
cana-5423	424	2	.	.	PUNCT
cana-5423	425	1	s.	s.	PROPN
cana-5423	425	2	melliani	melliani	PROPN
cana-5423	425	3	,	,	PUNCT
cana-5423	425	4	a.	a.	NOUN
cana-5423	425	5	moussaoui	moussaoui	NOUN
cana-5423	425	6	,	,	PUNCT
cana-5423	425	7	and	and	CCONJ
cana-5423	425	8	l	l	PROPN
cana-5423	425	9	..	..	PROPN
cana-5423	425	10	s.	s.	PROPN
cana-5423	425	11	chadli	chadli	PROPN
cana-5423	425	12	:	:	PUNCT
cana-5423	425	13	admissible	admissible	ADJ
cana-5423	425	14	almost	almost	ADV
cana-5423	425	15	type	type	NOUN
cana-5423	425	16	contraction	contraction	NOUN
cana-5423	425	17	an	an	DET
cana-5423	425	18	fixed	fix	VERB
cana-5423	425	19	point	point	NOUN
cana-5423	425	20	results	result	NOUN
cana-5423	425	21	,	,	PUNCT
cana-5423	425	22	int	int	NOUN
cana-5423	425	23	.	.	PUNCT
cana-5423	426	1	j.	j.	PROPN
cana-5423	426	2	ot	ot	PROPN
cana-5423	426	3	mathematics	mathematics	PROPN
cana-5423	426	4	and	and	CCONJ
cana-5423	426	5	mathematical	mathematical	ADJ
cana-5423	426	6	sciences	science	NOUN
cana-5423	426	7	,	,	PUNCT
cana-5423	426	8	vol.2020	vol.2020	ADV
cana-5423	426	9	,	,	PUNCT
cana-5423	426	10	article	article	NOUN
cana-5423	426	11	i	i	PROPN
cana-5423	426	12	d	d	PROPN
cana-5423	426	13	:	:	PUNCT
cana-5423	426	14	9104909	9104909	NUM
cana-5423	426	15	(	(	PUNCT
cana-5423	426	16	2020),1	2020),1	NUM
cana-5423	426	17	-	-	SYM
cana-5423	426	18	7	7	NUM
cana-5423	426	19	.	.	NOUN
cana-5423	426	20	28	28	NUM
cana-5423	426	21	.	.	PUNCT
cana-5423	427	1	s.g	s.g	PROPN
cana-5423	427	2	.	.	PROPN
cana-5423	427	3	teweldemedhin	teweldemedhin	PROPN
cana-5423	427	4	and	and	CCONJ
cana-5423	427	5	k.k.tola	k.k.tola	PROPN
cana-5423	427	6	:	:	PUNCT
cana-5423	427	7	fixe	fixe	NOUN
cana-5423	427	8	point	point	NOUN
cana-5423	427	9	results	result	VERB
cana-5423	427	10	for	for	ADP
cana-5423	427	11	an	an	DET
cana-5423	427	12	almost	almost	ADV
cana-5423	427	13	generalized	generalize	VERB
cana-5423	427	14	αadmissible	αadmissible	ADJ
cana-5423	427	15	z	z	NOUN
cana-5423	427	16	-contraction	-contraction	NOUN
cana-5423	427	17	in	in	ADP
cana-5423	427	18	the	the	DET
cana-5423	427	19	settinmg	settinmg	NOUN
cana-5423	427	20	of	of	ADP
cana-5423	427	21	partially	partially	ADV
cana-5423	427	22	ordered	order	VERB
cana-5423	427	23	bmetric	bmetric	ADJ
cana-5423	427	24	spaces	space	NOUN
cana-5423	427	25	,	,	PUNCT
cana-5423	427	26	abstract	abstract	ADJ
cana-5423	427	27	and	and	CCONJ
cana-5423	427	28	applied	apply	VERB
cana-5423	427	29	analysis	analysis	NOUN
cana-5423	427	30	,	,	PUNCT
cana-5423	427	31	vol	vol	NOUN
cana-5423	427	32	.	.	NOUN
cana-5423	427	33	2021	2021	NUM
cana-5423	427	34	,	,	PUNCT
cana-5423	427	35	article	article	NOUN
cana-5423	427	36	id:5988007	id:5988007	NOUN
cana-5423	427	37	,	,	PUNCT
cana-5423	427	38	(	(	PUNCT
cana-5423	427	39	2021),1	2021),1	NUM
cana-5423	427	40	-	-	SYM
cana-5423	427	41	11	11	NUM
cana-5423	427	42	.	.	PUNCT
cana-5423	428	1	29	29	NUM
cana-5423	428	2	.	.	PUNCT
cana-5423	429	1	f.	f.	PROPN
cana-5423	429	2	khojasteh	khojasteh	PROPN
cana-5423	429	3	,	,	PUNCT
cana-5423	429	4	s.	s.	PROPN
cana-5423	429	5	shukla	shukla	PROPN
cana-5423	429	6	,	,	PUNCT
cana-5423	429	7	s.	s.	PROPN
cana-5423	429	8	radenovic	radenovic	PROPN
cana-5423	429	9	,	,	PUNCT
cana-5423	429	10	a	a	DET
cana-5423	429	11	new	new	ADJ
cana-5423	429	12	approach	approach	NOUN
cana-5423	429	13	to	to	ADP
cana-5423	429	14	the	the	DET
cana-5423	429	15	study	study	NOUN
cana-5423	429	16	of	of	ADP
cana-5423	429	17	fixed	fix	VERB
cana-5423	429	18	point	point	NOUN
cana-5423	429	19	theory	theory	NOUN
cana-5423	429	20	for	for	ADP
cana-5423	429	21	simulation	simulation	NOUN
cana-5423	429	22	functions	function	NOUN
cana-5423	429	23	,	,	PUNCT
cana-5423	429	24	filomat	filomat	NOUN
cana-5423	429	25	29(6),(2015	29(6),(2015	NUM
cana-5423	429	26	)	)	PUNCT
cana-5423	429	27	,	,	PUNCT
cana-5423	429	28	1189	1189	NUM
cana-5423	429	29	-	-	SYM
cana-5423	429	30	1194	1194	NUM
cana-5423	429	31	.	.	PUNCT
cana-5423	430	1	30	30	NUM
cana-5423	430	2	.	.	PUNCT
cana-5423	431	1	s.	s.	PROPN
cana-5423	431	2	komal	komal	PROPN
cana-5423	431	3	,	,	PUNCT
cana-5423	431	4	p.	p.	PROPN
cana-5423	431	5	kumam	kumam	PROPN
cana-5423	431	6	,	,	PUNCT
cana-5423	431	7	d.	d.	PROPN
cana-5423	431	8	gopal	gopal	PROPN
cana-5423	431	9	,	,	PUNCT
cana-5423	431	10	best	good	ADJ
cana-5423	431	11	proximity	proximity	NOUN
cana-5423	431	12	point	point	NOUN
cana-5423	431	13	z	z	NOUN
cana-5423	431	14	-	-	PUNCT
cana-5423	431	15	contraction	contraction	NOUN
cana-5423	431	16	and	and	CCONJ
cana-5423	431	17	suzuki	suzuki	NOUN
cana-5423	431	18	type	type	NOUN
cana-5423	431	19	z	z	PROPN
cana-5423	431	20	contraction	contraction	NOUN
cana-5423	431	21	mappings	mapping	NOUN
cana-5423	431	22	with	with	ADP
cana-5423	431	23	an	an	DET
cana-5423	431	24	application	application	NOUN
cana-5423	431	25	to	to	ADP
cana-5423	431	26	fractional	fractional	ADJ
cana-5423	431	27	calculus	calculus	NOUN
cana-5423	431	28	,	,	PUNCT
cana-5423	431	29	appl	appl	PROPN
cana-5423	431	30	.	.	PUNCT
cana-5423	432	1	gen	gen	PROPN
cana-5423	432	2	.	.	PROPN
cana-5423	432	3	topol	topol	PROPN
cana-5423	432	4	.	.	PUNCT
cana-5423	433	1	17(2	17(2	NUM
cana-5423	433	2	)	)	PUNCT
cana-5423	433	3	(	(	PUNCT
cana-5423	433	4	2016	2016	NUM
cana-5423	433	5	)	)	PUNCT
cana-5423	433	6	,	,	PUNCT
cana-5423	433	7	185	185	NUM
cana-5423	433	8	-	-	SYM
cana-5423	433	9	198	198	NUM
cana-5423	433	10	.	.	PUNCT
cana-5423	434	1	31	31	NUM
cana-5423	434	2	.	.	PUNCT
cana-5423	435	1	p.kumam	p.kumam	NOUN
cana-5423	435	2	,	,	PUNCT
cana-5423	435	3	d.	d.	PROPN
cana-5423	435	4	gopal	gopal	PROPN
cana-5423	435	5	,	,	PUNCT
cana-5423	435	6	l.	l.	PROPN
cana-5423	435	7	budhiya	budhiya	PROPN
cana-5423	435	8	:	:	PUNCT
cana-5423	435	9	a	a	DET
cana-5423	435	10	new	new	ADJ
cana-5423	435	11	fixed	fix	VERB
cana-5423	435	12	point	point	NOUN
cana-5423	435	13	theorem	theorem	VERB
cana-5423	435	14	under	under	ADP
cana-5423	435	15	suzuki	suzuki	NOUN
cana-5423	435	16	type	type	NOUN
cana-5423	435	17	zcontraction	zcontraction	NOUN
cana-5423	435	18	mappings	mapping	NOUN
cana-5423	435	19	,	,	PUNCT
cana-5423	435	20	j.	j.	PROPN
cana-5423	435	21	math	math	PROPN
cana-5423	435	22	.	.	PUNCT
cana-5423	436	1	anal	anal	PROPN
cana-5423	436	2	.	.	PUNCT
cana-5423	437	1	8(1	8(1	NOUN
cana-5423	437	2	)	)	PUNCT
cana-5423	437	3	,	,	PUNCT
cana-5423	437	4	(	(	PUNCT
cana-5423	437	5	2017	2017	NUM
cana-5423	437	6	)	)	PUNCT
cana-5423	437	7	,	,	PUNCT
cana-5423	437	8	113	113	NUM
cana-5423	437	9	-	-	SYM
cana-5423	437	10	119	119	NUM
cana-5423	437	11	.	.	PUNCT
cana-5423	438	1	32	32	NUM
cana-5423	438	2	.	.	PUNCT
cana-5423	439	1	m.olgun	m.olgun	NOUN
cana-5423	439	2	,	,	PUNCT
cana-5423	439	3	o.	o.	NOUN
cana-5423	439	4	bicer	bicer	NOUN
cana-5423	439	5	,	,	PUNCT
cana-5423	439	6	and	and	CCONJ
cana-5423	439	7	t.	t.	PROPN
cana-5423	439	8	alyildiz	alyildiz	PROPN
cana-5423	439	9	:	:	PUNCT
cana-5423	439	10	anew	anew	ADJ
cana-5423	439	11	aspect	aspect	NOUN
cana-5423	439	12	to	to	ADP
cana-5423	439	13	picard	picard	NOUN
cana-5423	439	14	operators	operator	NOUN
cana-5423	439	15	with	with	ADP
cana-5423	439	16	simulation	simulation	NOUN
cana-5423	439	17	functions	function	NOUN
cana-5423	439	18	,	,	PUNCT
cana-5423	439	19	turk	turk	PROPN
cana-5423	439	20	j.	j.	PROPN
cana-5423	439	21	math	math	PROPN
cana-5423	439	22	.	.	PUNCT
cana-5423	440	1	40(2016),832	40(2016),832	NUM
cana-5423	440	2	-	-	SYM
cana-5423	440	3	837	837	NUM
cana-5423	440	4	.	.	PUNCT
cana-5423	441	1	33	33	NUM
cana-5423	441	2	.	.	PUNCT
cana-5423	441	3	a.	a.	NOUN
cana-5423	441	4	padcharoen	padcharoen	PROPN
cana-5423	441	5	,	,	PUNCT
cana-5423	441	6	p.	p.	PROPN
cana-5423	441	7	kumam	kumam	PROPN
cana-5423	441	8	,	,	PUNCT
cana-5423	441	9	p.	p.	NOUN
cana-5423	441	10	saipara	saipara	PROPN
cana-5423	441	11	,	,	PUNCT
cana-5423	441	12	p.	p.	PROPN
cana-5423	441	13	chaipunya	chaipunya	PROPN
cana-5423	441	14	:	:	PUNCT
cana-5423	441	15	generalized	generalized	ADJ
cana-5423	441	16	suzuki	suzuki	PROPN
cana-5423	441	17	type	type	PROPN
cana-5423	441	18	z	z	PROPN
cana-5423	441	19	contraction	contraction	NOUN
cana-5423	441	20	in	in	ADP
cana-5423	441	21	complete	complete	ADJ
cana-5423	441	22	metric	metric	ADJ
cana-5423	441	23	space	space	NOUN
cana-5423	441	24	,	,	PUNCT
cana-5423	441	25	kragujevac	kragujevac	PROPN
cana-5423	441	26	journal	journal	NOUN
cana-5423	441	27	of	of	ADP
cana-5423	441	28	mathematics	mathematics	PROPN
cana-5423	441	29	,	,	PUNCT
cana-5423	441	30	43(3)(2018),419430	43(3)(2018),419430	PROPN
cana-5423	441	31	.	.	PUNCT
cana-5423	441	32	34	34	NUM
cana-5423	441	33	.	.	PUNCT
cana-5423	441	34	s.radenovic	s.radenovic	NUM
cana-5423	441	35	,	,	PUNCT
cana-5423	441	36	and	and	CCONJ
cana-5423	441	37	s.	s.	PROPN
cana-5423	441	38	chandok	chandok	PROPN
cana-5423	441	39	:	:	PUNCT
cana-5423	441	40	simulation	simulation	NOUN
cana-5423	441	41	type	type	NOUN
cana-5423	441	42	functions	function	NOUN
cana-5423	441	43	and	and	CCONJ
cana-5423	441	44	coicidence	coicidence	NOUN
cana-5423	441	45	points	point	NOUN
cana-5423	441	46	,	,	PUNCT
cana-5423	441	47	filomat	filomat	NOUN
cana-5423	441	48	,	,	PUNCT
cana-5423	441	49	32(1)(2018),141	32(1)(2018),141	PROPN
cana-5423	441	50	-	-	SYM
cana-5423	441	51	147	147	NUM
cana-5423	441	52	.	.	PUNCT
cana-5423	441	53	35	35	NUM
cana-5423	441	54	.	.	PUNCT
cana-5423	442	1	s.	s.	PROPN
cana-5423	442	2	radenovic	radenovic	PROPN
cana-5423	442	3	,	,	PUNCT
cana-5423	442	4	f.	f.	PROPN
cana-5423	442	5	vetro	vetro	PROPN
cana-5423	442	6	,	,	PUNCT
cana-5423	442	7	j.	j.	PROPN
cana-5423	442	8	vujakovic	vujakovic	PROPN
cana-5423	442	9	:	:	PUNCT
cana-5423	442	10	an	an	DET
cana-5423	442	11	alternative	alternative	ADJ
cana-5423	442	12	and	and	CCONJ
cana-5423	442	13	easy	easy	ADJ
cana-5423	442	14	approach	approach	NOUN
cana-5423	442	15	to	to	ADP
cana-5423	442	16	fixed	fix	VERB
cana-5423	442	17	point	point	NOUN
cana-5423	442	18	results	result	NOUN
cana-5423	442	19	via	via	ADP
cana-5423	442	20	simulation	simulation	NOUN
cana-5423	442	21	functions	function	NOUN
cana-5423	442	22	,	,	PUNCT
cana-5423	442	23	demonstrate	demonstrate	NOUN
cana-5423	442	24	.	.	PUNCT
cana-5423	443	1	math	math	NOUN
cana-5423	443	2	.	.	PUNCT
cana-5423	444	1	50(1)(2017),223	50(1)(2017),223	NUM
cana-5423	444	2	-	-	SYM
cana-5423	444	3	230	230	NUM
cana-5423	444	4	.	.	PUNCT
cana-5423	445	1	36	36	NUM
cana-5423	445	2	.	.	PUNCT
cana-5423	446	1	a.f.roldan	a.f.roldan	NOUN
cana-5423	446	2	-	-	PUNCT
cana-5423	446	3	lopez	lopez	NOUN
cana-5423	446	4	-	-	PUNCT
cana-5423	446	5	de	de	NOUN
cana-5423	446	6	-	-	NOUN
cana-5423	446	7	hierro	hierro	ADJ
cana-5423	446	8	,	,	PUNCT
cana-5423	446	9	e.	e.	PROPN
cana-5423	446	10	karapinar	karapinar	PROPN
cana-5423	446	11	,	,	PUNCT
cana-5423	446	12	c.	c.	PROPN
cana-5423	446	13	roldan	roldan	PROPN
cana-5423	446	14	-	-	PUNCT
cana-5423	446	15	lopez	lopez	PROPN
cana-5423	446	16	-	-	PUNCT
cana-5423	446	17	de	de	NOUN
cana-5423	446	18	-	-	NOUN
cana-5423	446	19	hierro	hierro	ADJ
cana-5423	446	20	,	,	PUNCT
cana-5423	446	21	and	and	CCONJ
cana-5423	446	22	j.	j.	PROPN
cana-5423	446	23	martinez	martinez	PROPN
cana-5423	446	24	moreno	moreno	PROPN
cana-5423	446	25	:	:	PUNCT
cana-5423	447	1	coicidence	coicidence	NOUN
cana-5423	447	2	point	point	NOUN
cana-5423	447	3	theorem	theorem	VERB
cana-5423	447	4	on	on	ADP
cana-5423	447	5	metric	metric	ADJ
cana-5423	447	6	spaces	space	NOUN
cana-5423	447	7	via	via	ADP
cana-5423	447	8	simulation	simulation	NOUN
cana-5423	447	9	functions	function	NOUN
cana-5423	447	10	,	,	PUNCT
cana-5423	447	11	j.	j.	PROPN
cana-5423	447	12	comput.appl	comput.appl	PROPN
cana-5423	447	13	.	.	PUNCT
cana-5423	447	14	math	math	NOUN
cana-5423	447	15	.	.	PUNCT
cana-5423	448	1	275(2015),345	275(2015),345	ADJ
cana-5423	448	2	-	-	SYM
cana-5423	448	3	355	355	NUM
cana-5423	448	4	.	.	PUNCT
cana-5423	449	1	37	37	NUM
cana-5423	449	2	.	.	PUNCT
cana-5423	450	1	a.f.roldan	a.f.roldan	NOUN
cana-5423	450	2	-	-	PUNCT
cana-5423	450	3	lopez	lopez	NOUN
cana-5423	450	4	-	-	PUNCT
cana-5423	450	5	de	de	NOUN
cana-5423	450	6	-	-	NOUN
cana-5423	450	7	hierro	hierro	ADJ
cana-5423	450	8	,	,	PUNCT
cana-5423	450	9	and	and	CCONJ
cana-5423	450	10	b.samet	b.samet	NUM
cana-5423	451	1	:	:	PUNCT
cana-5423	451	2	ψadmissibility	ψadmissibility	NOUN
cana-5423	451	3	results	result	NOUN
cana-5423	451	4	via	via	ADP
cana-5423	451	5	extended	extended	ADJ
cana-5423	451	6	simulation	simulation	NOUN
cana-5423	451	7	functions	function	NOUN
cana-5423	451	8	,	,	PUNCT
cana-5423	451	9	j.	j.	PROPN
cana-5423	451	10	fixedvpoint	fixedvpoint	PROPN
cana-5423	451	11	theory	theory	NOUN
cana-5423	451	12	appl	appl	PROPN
cana-5423	451	13	.	.	PUNCT
cana-5423	451	14	,19(3),(2017	,19(3),(2017	PROPN
cana-5423	451	15	)	)	PUNCT
cana-5423	451	16	,	,	PUNCT
cana-5423	451	17	1997	1997	NUM
cana-5423	451	18	-	-	SYM
cana-5423	451	19	2015	2015	NUM
cana-5423	451	20	.	.	PUNCT
cana-5423	452	1	38	38	NUM
cana-5423	452	2	.	.	PUNCT
cana-5423	453	1	v.l	v.l	PROPN
cana-5423	453	2	.	.	PROPN
cana-5423	453	3	rosa	rosa	PROPN
cana-5423	453	4	and	and	CCONJ
cana-5423	453	5	p.vetro	p.vetro	PROPN
cana-5423	453	6	:	:	PUNCT
cana-5423	453	7	common	common	ADJ
cana-5423	453	8	fixed	fix	VERB
cana-5423	453	9	points	point	NOUN
cana-5423	453	10	for	for	ADP
cana-5423	453	11	α	α	NOUN
cana-5423	453	12	,	,	PUNCT
cana-5423	453	13	φ	φ	PROPN
cana-5423	453	14	,	,	PUNCT
cana-5423	453	15	ϕcontracttions	ϕcontracttion	NOUN
cana-5423	453	16	in	in	ADP
cana-5423	453	17	generalized	generalized	ADJ
cana-5423	453	18	metric	metric	ADJ
cana-5423	453	19	space	space	NOUN
cana-5423	453	20	,	,	PUNCT
cana-5423	453	21	nnlinear	nnlinear	ADJ
cana-5423	453	22	anal	anal	PROPN
cana-5423	453	23	.	.	PUNCT
cana-5423	454	1	model	model	PROPN
cana-5423	454	2	.	.	PUNCT
cana-5423	455	1	control	control	PROPN
cana-5423	455	2	,	,	PUNCT
cana-5423	455	3	19(1)(2014	19(1)(2014	NUM
cana-5423	455	4	)	)	PUNCT
cana-5423	455	5	,	,	PUNCT
cana-5423	455	6	43	43	NUM
cana-5423	455	7	-	-	SYM
cana-5423	455	8	54	54	NUM
cana-5423	455	9	.	.	NUM
cana-5423	455	10	39	39	NUM
cana-5423	455	11	.	.	PUNCT
cana-5423	456	1	b.samet	b.samet	PUNCT
cana-5423	456	2	,	,	PUNCT
cana-5423	456	3	c.	c.	PROPN
cana-5423	456	4	vetro	vetro	PROPN
cana-5423	456	5	,	,	PUNCT
cana-5423	456	6	and	and	CCONJ
cana-5423	456	7	p.	p.	NOUN
cana-5423	456	8	vetro	vetro	NOUN
cana-5423	456	9	:	:	PUNCT
cana-5423	456	10	fixed	fix	VERB
cana-5423	456	11	point	point	NOUN
cana-5423	456	12	theorems	theorem	NOUN
cana-5423	456	13	for	for	ADP
cana-5423	456	14	α−ψ	α−ψ	NOUN
cana-5423	456	15	contractive	contractive	ADJ
cana-5423	456	16	type	type	NOUN
cana-5423	456	17	mappings	mapping	NOUN
cana-5423	456	18	,	,	PUNCT
cana-5423	456	19	nonlinear	nonlinear	ADJ
cana-5423	456	20	anal	anal	NOUN
cana-5423	456	21	.	.	PUNCT
cana-5423	457	1	75	75	NUM
cana-5423	457	2	,	,	PUNCT
cana-5423	457	3	(	(	PUNCT
cana-5423	457	4	20120	20120	NUM
cana-5423	457	5	,	,	PUNCT
cana-5423	457	6	2154	2154	NUM
cana-5423	457	7	-	-	SYM
cana-5423	457	8	2165	2165	NUM
cana-5423	457	9	.	.	PUNCT
cana-5423	458	1	40	40	NUM
cana-5423	458	2	.	.	PUNCT
cana-5423	459	1	p.salimi	p.salimi	NOUN
cana-5423	459	2	,	,	PUNCT
cana-5423	459	3	a.latif	a.latif	NOUN
cana-5423	459	4	and	and	CCONJ
cana-5423	459	5	n.hussain	n.hussain	NOUN
cana-5423	459	6	:	:	PUNCT
cana-5423	459	7	modified	modify	VERB
cana-5423	459	8	α−	α−	ADP
cana-5423	459	9	ψcontractive	ψcontractive	ADJ
cana-5423	459	10	mappings	mapping	NOUN
cana-5423	459	11	with	with	ADP
cana-5423	459	12	applications	application	NOUN
cana-5423	459	13	,	,	PUNCT
cana-5423	459	14	fixed	fix	VERB
cana-5423	459	15	point	point	NOUN
cana-5423	459	16	theory	theory	NOUN
cana-5423	459	17	aplli	aplli	PROPN
cana-5423	459	18	.	.	PUNCT
cana-5423	459	19	,15	,15	PUNCT
cana-5423	460	1	(	(	PUNCT
cana-5423	460	2	2015	2015	NUM
cana-5423	460	3	)	)	PUNCT
cana-5423	460	4	41	41	NUM
cana-5423	460	5	.	.	PUNCT
cana-5423	460	6	p.shahi	p.shahi	PROPN
cana-5423	460	7	,	,	PUNCT
cana-5423	460	8	j.	j.	PROPN
cana-5423	460	9	kaur	kaur	PROPN
cana-5423	460	10	,	,	PUNCT
cana-5423	460	11	and	and	CCONJ
cana-5423	460	12	s.s	s.s	PROPN
cana-5423	460	13	.	.	PROPN
cana-5423	460	14	bhatia	bhatia	PROPN
cana-5423	460	15	:	:	PUNCT
cana-5423	461	1	coicedence	coicedence	NOUN
cana-5423	461	2	and	and	CCONJ
cana-5423	461	3	common	common	ADJ
cana-5423	461	4	fixed	fix	VERB
cana-5423	461	5	point	point	NOUN
cana-5423	461	6	results	result	NOUN
cana-5423	461	7	for	for	ADP
cana-5423	461	8	generalized	generalized	ADJ
cana-5423	461	9	α−	α−	ADP
cana-5423	461	10	ψ	ψ	ADP
cana-5423	461	11	contractive	contractive	ADJ
cana-5423	461	12	mappings	mapping	NOUN
cana-5423	461	13	with	with	ADP
cana-5423	461	14	applications	application	NOUN
cana-5423	461	15	,	,	PUNCT
cana-5423	461	16	bull	bull	NOUN
cana-5423	461	17	.	.	PUNCT
cana-5423	462	1	belg	belg	PROPN
cana-5423	462	2	.	.	PUNCT
cana-5423	463	1	math	math	NOUN
cana-5423	463	2	.	.	PUNCT
cana-5423	464	1	soc	soc	PROPN
cana-5423	464	2	.	.	PUNCT
cana-5423	465	1	simon	simon	PROPN
cana-5423	465	2	stevin	stevin	PROPN
cana-5423	465	3	,	,	PUNCT
cana-5423	465	4	22(2)(2015),299	22(2)(2015),299	NUM
cana-5423	465	5	-	-	SYM
cana-5423	465	6	318	318	NUM
cana-5423	465	7	.	.	PUNCT
cana-5423	466	1	communications	communication	NOUN
cana-5423	466	2	on	on	ADP
cana-5423	466	3	applied	apply	VERB
cana-5423	466	4	nonlinear	nonlinear	ADJ
cana-5423	466	5	analysis	analysis	NOUN
cana-5423	466	6	issn	issn	NOUN
cana-5423	466	7	:	:	PUNCT
cana-5423	466	8	1074	1074	NUM
cana-5423	466	9	-	-	PUNCT
cana-5423	466	10	133x	133x	NUM
cana-5423	466	11	vol	vol	NOUN
cana-5423	466	12	32	32	NUM
cana-5423	466	13	no	no	NOUN
cana-5423	466	14	.	.	PUNCT
cana-5423	467	1	10s(2025	10s(2025	NUM
cana-5423	467	2	)	)	PUNCT
cana-5423	468	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	468	2	2230	2230	NUM
cana-5423	468	3	42	42	NUM
cana-5423	468	4	.	.	PUNCT
cana-5423	469	1	h.	h.	PROPN
cana-5423	469	2	qawagneh	qawagneh	PROPN
cana-5423	469	3	,	,	PUNCT
cana-5423	469	4	m.s.m	m.s.m	PROPN
cana-5423	469	5	noorani	noorani	PROPN
cana-5423	469	6	,	,	PUNCT
cana-5423	469	7	w.	w.	PROPN
cana-5423	469	8	shatanawi	shatanawi	PROPN
cana-5423	469	9	,	,	PUNCT
cana-5423	469	10	and	and	CCONJ
cana-5423	469	11	h.	h.	PROPN
cana-5423	469	12	alsamir	alsamir	PROPN
cana-5423	469	13	:	:	PUNCT
cana-5423	469	14	common	common	ADJ
cana-5423	469	15	fixed	fix	VERB
cana-5423	469	16	points	point	NOUN
cana-5423	469	17	for	for	ADP
cana-5423	469	18	pair	pair	NOUN
cana-5423	469	19	of	of	ADP
cana-5423	469	20	triangular	triangular	NOUN
cana-5423	469	21	α	α	PRON
cana-5423	469	22	-admissible	-admissible	ADJ
cana-5423	469	23	mappings	mapping	NOUN
cana-5423	469	24	,	,	PUNCT
cana-5423	469	25	journal	journal	NOUN
cana-5423	469	26	of	of	ADP
cana-5423	469	27	nonlinear	nonlinear	PROPN
cana-5423	469	28	sciences	sciences	PROPN
cana-5423	469	29	and	and	CCONJ
cana-5423	469	30	applications	application	NOUN
cana-5423	469	31	,	,	PUNCT
cana-5423	469	32	10(12)(2017),6192	10(12)(2017),6192	NUM
cana-5423	469	33	-	-	SYM
cana-5423	469	34	6204	6204	NUM
cana-5423	469	35	.	.	PUNCT
cana-5423	470	1	43	43	NUM
cana-5423	470	2	.	.	PUNCT
cana-5423	470	3	dipti	dipti	PROPN
cana-5423	470	4	,	,	PUNCT
cana-5423	470	5	a.k.dubey	a.k.dubey	NOUN
cana-5423	470	6	and	and	CCONJ
cana-5423	470	7	u.mishra	u.mishra	ADJ
cana-5423	470	8	:	:	PUNCT
cana-5423	470	9	generalized	generalize	VERB
cana-5423	470	10	αadmissible	αadmissible	ADJ
cana-5423	470	11	almost	almost	ADV
cana-5423	470	12	zcontractions	zcontraction	NOUN
cana-5423	470	13	involving	involve	VERB
cana-5423	470	14	simulation	simulation	NOUN
cana-5423	470	15	function	function	NOUN
cana-5423	470	16	in	in	ADP
cana-5423	470	17	a	a	DET
cana-5423	470	18	metric	metric	ADJ
cana-5423	470	19	space	space	NOUN
cana-5423	470	20	,	,	PUNCT
cana-5423	470	21	communications	communication	NOUN
cana-5423	470	22	on	on	ADP
cana-5423	470	23	applied	apply	VERB
cana-5423	470	24	nonlinear	nonlinear	ADJ
cana-5423	470	25	analysis	analysis	NOUN
cana-5423	470	26	,	,	PUNCT
cana-5423	470	27	vol	vol	NOUN
cana-5423	470	28	.	.	PROPN
cana-5423	470	29	32	32	NUM
cana-5423	471	1	no	no	NOUN
cana-5423	471	2	.	.	NOUN
cana-5423	471	3	1	1	NUM
cana-5423	471	4	,	,	PUNCT
cana-5423	471	5	(	(	PUNCT
cana-5423	471	6	2025),354	2025),354	NOUN
cana-5423	471	7	-	-	NUM
cana-5423	471	8	362	362	NUM
cana-5423	471	9	.	.	PUNCT
cana-5423	472	1	1department	1department	NUM
cana-5423	472	2	of	of	ADP
cana-5423	472	3	mathematics	mathematic	NOUN
cana-5423	472	4	,	,	PUNCT
cana-5423	472	5	dr	dr	PROPN
cana-5423	472	6	.	.	PROPN
cana-5423	472	7	c.v	c.v	PROPN
cana-5423	472	8	.	.	PROPN
cana-5423	472	9	raman	raman	PROPN
cana-5423	472	10	university	university	PROPN
cana-5423	472	11	,	,	PUNCT
cana-5423	472	12	bilaspur	bilaspur	NOUN
cana-5423	472	13	,	,	PUNCT
cana-5423	472	14	chhattisgarhindia	chhattisgarhindia	PROPN
cana-5423	472	15	.	.	PUNCT
cana-5423	473	1	email	email	NOUN
cana-5423	473	2	address	address	NOUN
cana-5423	473	3	:	:	PUNCT
cana-5423	473	4	sk10tiwari@gmail.com	sk10tiwari@gmail.com	PROPN
cana-5423	473	5	2	2	NUM
cana-5423	473	6	teacher	teacher	NOUN
cana-5423	473	7	,	,	PUNCT
cana-5423	473	8	school	school	NOUN
cana-5423	473	9	education	education	NOUN
cana-5423	473	10	department	department	NOUN
cana-5423	473	11	,	,	PUNCT
cana-5423	473	12	takhatpur	takhatpur	NOUN
cana-5423	473	13	,	,	PUNCT
cana-5423	473	14	bilaspur	bilaspur	NOUN
cana-5423	473	15	,	,	PUNCT
cana-5423	473	16	chhattisgarh	chhattisgarh	NOUN
cana-5423	473	17	,	,	PUNCT
cana-5423	473	18	-india	-india	PROPN
cana-5423	473	19	.	.	PUNCT
cana-5423	474	1	email	email	NOUN
cana-5423	474	2	address	address	NOUN
cana-5423	474	3	:	:	PUNCT
cana-5423	474	4	dubeyanandmohan767@gmail.com	dubeyanandmohan767@gmail.com	X
cana-5423	474	5	2	2	NUM
cana-5423	474	6	tdepartment	tdepartment	NOUN
cana-5423	474	7	of	of	ADP
cana-5423	474	8	mca	mca	PROPN
cana-5423	474	9	,	,	PUNCT
cana-5423	474	10	hidustan	hidustan	PROPN
cana-5423	474	11	college	college	PROPN
cana-5423	474	12	of	of	ADP
cana-5423	474	13	arts	art	NOUN
cana-5423	474	14	and	and	CCONJ
cana-5423	474	15	science	science	NOUN
cana-5423	474	16	,	,	PUNCT
cana-5423	474	17	coimbatore	coimbatore	NOUN
cana-5423	474	18	tamilnadu	tamilnadu	NOUN
cana-5423	474	19	,	,	PUNCT
cana-5423	474	20	-india	-india	PROPN
cana-5423	474	21	.	.	PUNCT
cana-5423	475	1	email	email	NOUN
cana-5423	475	2	address	address	NOUN
cana-5423	475	3	:	:	PUNCT
cana-5423	475	4	avsenthilkumar@yahoo.com	avsenthilkumar@yahoo.com	X
cana-5423	475	5	communications	communication	NOUN
cana-5423	475	6	on	on	ADP
cana-5423	475	7	applied	apply	VERB
cana-5423	475	8	nonlinear	nonlinear	ADJ
cana-5423	475	9	analysis	analysis	NOUN
cana-5423	475	10	issn	issn	NOUN
cana-5423	475	11	:	:	PUNCT
cana-5423	475	12	1074	1074	NUM
cana-5423	475	13	-	-	PUNCT
cana-5423	475	14	133x	133x	NUM
cana-5423	475	15	vol	vol	NOUN
cana-5423	475	16	32	32	NUM
cana-5423	475	17	no	no	NOUN
cana-5423	475	18	.	.	PUNCT
cana-5423	476	1	10s(2025	10s(2025	NUM
cana-5423	476	2	)	)	PUNCT
cana-5423	477	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5423	477	2	2231	2231	NUM