id	sid	tid	token	lemma	pos
cana-2192	1	1	generalized	generalize	VERB
cana-2192	1	2	α	α	NOUN
cana-2192	1	3	-	-	ADJ
cana-2192	1	4	admissible	admissible	ADJ
cana-2192	1	5	almost	almost	ADV
cana-2192	1	6	z	z	NOUN
cana-2192	1	7	-	-	PUNCT
cana-2192	1	8	contractions	contraction	NOUN
cana-2192	1	9	involving	involve	VERB
cana-2192	1	10	simulation	simulation	NOUN
cana-2192	1	11	functions	function	NOUN
cana-2192	1	12	in	in	ADP
cana-2192	1	13	a	a	DET
cana-2192	1	14	metric	metric	ADJ
cana-2192	1	15	space	space	NOUN
cana-2192	1	16	dipti	dipti	NOUN
cana-2192	1	17	,	,	PUNCT
cana-2192	1	18	anil	anil	PROPN
cana-2192	1	19	kumar	kumar	PROPN
cana-2192	1	20	dubey	dubey	PROPN
cana-2192	1	21	,	,	PUNCT
cana-2192	1	22	urmila	urmila	PROPN
cana-2192	1	23	mishra	mishra	PROPN
cana-2192	1	24	abstract	abstract	PROPN
cana-2192	1	25	.	.	PUNCT
cana-2192	2	1	in	in	ADP
cana-2192	2	2	this	this	DET
cana-2192	2	3	paper	paper	NOUN
cana-2192	2	4	,	,	PUNCT
cana-2192	2	5	we	we	PRON
cana-2192	2	6	present	present	VERB
cana-2192	2	7	some	some	DET
cana-2192	2	8	fixed	fix	VERB
cana-2192	2	9	point	point	NOUN
cana-2192	2	10	results	result	NOUN
cana-2192	2	11	in	in	ADP
cana-2192	2	12	complete	complete	ADJ
cana-2192	2	13	metric	metric	ADJ
cana-2192	2	14	spaces	space	NOUN
cana-2192	2	15	using	use	VERB
cana-2192	2	16	generalized	generalized	ADJ
cana-2192	2	17	α	α	PRON
cana-2192	2	18	-	-	ADJ
cana-2192	2	19	admissible	admissible	ADJ
cana-2192	2	20	mappings	mapping	NOUN
cana-2192	2	21	embedded	embed	VERB
cana-2192	2	22	in	in	ADP
cana-2192	2	23	the	the	DET
cana-2192	2	24	simulation	simulation	NOUN
cana-2192	2	25	function	function	NOUN
cana-2192	2	26	.	.	PUNCT
cana-2192	3	1	these	these	DET
cana-2192	3	2	results	result	NOUN
cana-2192	3	3	serve	serve	VERB
cana-2192	3	4	to	to	PART
cana-2192	3	5	generalize	generalize	VERB
cana-2192	3	6	and	and	CCONJ
cana-2192	3	7	unify	unify	VERB
cana-2192	3	8	several	several	ADJ
cana-2192	3	9	related	related	ADJ
cana-2192	3	10	fixed	fix	VERB
cana-2192	3	11	point	point	NOUN
cana-2192	3	12	results	result	NOUN
cana-2192	3	13	found	find	VERB
cana-2192	3	14	in	in	ADP
cana-2192	3	15	the	the	DET
cana-2192	3	16	existing	exist	VERB
cana-2192	3	17	literature	literature	NOUN
cana-2192	3	18	.	.	PUNCT
cana-2192	4	1	to	to	PART
cana-2192	4	2	validate	validate	VERB
cana-2192	4	3	our	our	PRON
cana-2192	4	4	findings	finding	NOUN
cana-2192	4	5	,	,	PUNCT
cana-2192	4	6	we	we	PRON
cana-2192	4	7	provide	provide	VERB
cana-2192	4	8	a	a	DET
cana-2192	4	9	specific	specific	ADJ
cana-2192	4	10	example	example	NOUN
cana-2192	4	11	illustrating	illustrate	VERB
cana-2192	4	12	the	the	DET
cana-2192	4	13	application	application	NOUN
cana-2192	4	14	of	of	ADP
cana-2192	4	15	these	these	DET
cana-2192	4	16	results	result	NOUN
cana-2192	4	17	.	.	PUNCT
cana-2192	5	1	1	1	X
cana-2192	5	2	.	.	X
cana-2192	5	3	introduction	introduction	NOUN
cana-2192	5	4	let	let	VERB
cana-2192	5	5	n0	n0	X
cana-2192	5	6	=	=	PUNCT
cana-2192	5	7	n	n	PRON
cana-2192	5	8	∪	∪	X
cana-2192	5	9	{	{	PUNCT
cana-2192	5	10	0	0	NUM
cana-2192	5	11	}	}	PUNCT
cana-2192	5	12	,	,	PUNCT
cana-2192	5	13	where	where	SCONJ
cana-2192	5	14	n	n	PRON
cana-2192	5	15	represents	represent	VERB
cana-2192	5	16	the	the	DET
cana-2192	5	17	set	set	NOUN
cana-2192	5	18	of	of	ADP
cana-2192	5	19	positive	positive	ADJ
cana-2192	5	20	integers	integer	NOUN
cana-2192	5	21	.	.	PUNCT
cana-2192	6	1	as	as	ADP
cana-2192	6	2	usual	usual	ADJ
cana-2192	6	3	r	r	NOUN
cana-2192	6	4	indicates	indicate	VERB
cana-2192	6	5	the	the	DET
cana-2192	6	6	set	set	NOUN
cana-2192	6	7	of	of	ADP
cana-2192	6	8	all	all	DET
cana-2192	6	9	real	real	ADJ
cana-2192	6	10	numbers	number	NOUN
cana-2192	6	11	.	.	PUNCT
cana-2192	7	1	furthermore	furthermore	ADV
cana-2192	7	2	,	,	PUNCT
cana-2192	7	3	we	we	PRON
cana-2192	7	4	set	set	VERB
cana-2192	7	5	r+	r+	X
cana-2192	7	6	0	0	NUM
cana-2192	7	7	:	:	PUNCT
cana-2192	7	8	=	=	PUNCT
cana-2192	8	1	[	[	X
cana-2192	8	2	0,∞	0,∞	NUM
cana-2192	8	3	)	)	PUNCT
cana-2192	8	4	.	.	PUNCT
cana-2192	9	1	khojasteh	khojasteh	NOUN
cana-2192	9	2	et	et	PROPN
cana-2192	9	3	al.[14	al.[14	PROPN
cana-2192	9	4	]	]	PUNCT
cana-2192	9	5	introduced	introduce	VERB
cana-2192	9	6	the	the	DET
cana-2192	9	7	notion	notion	NOUN
cana-2192	9	8	of	of	ADP
cana-2192	9	9	z	z	NOUN
cana-2192	9	10	-	-	PUNCT
cana-2192	9	11	contraction	contraction	NOUN
cana-2192	9	12	by	by	ADP
cana-2192	9	13	using	use	VERB
cana-2192	9	14	a	a	DET
cana-2192	9	15	new	new	ADJ
cana-2192	9	16	class	class	NOUN
cana-2192	9	17	of	of	ADP
cana-2192	9	18	auxiliary	auxiliary	ADJ
cana-2192	9	19	function	function	NOUN
cana-2192	9	20	called	call	VERB
cana-2192	9	21	simulation	simulation	NOUN
cana-2192	9	22	function	function	NOUN
cana-2192	9	23	.	.	PUNCT
cana-2192	10	1	they	they	PRON
cana-2192	10	2	[	[	X
cana-2192	10	3	14	14	NUM
cana-2192	10	4	]	]	PUNCT
cana-2192	10	5	proved	prove	VERB
cana-2192	10	6	several	several	ADJ
cana-2192	10	7	fixed	fix	VERB
cana-2192	10	8	point	point	NOUN
cana-2192	10	9	theorems	theorem	NOUN
cana-2192	10	10	and	and	CCONJ
cana-2192	10	11	showed	show	VERB
cana-2192	10	12	that	that	SCONJ
cana-2192	10	13	many	many	ADJ
cana-2192	10	14	results	result	NOUN
cana-2192	10	15	in	in	ADP
cana-2192	10	16	the	the	DET
cana-2192	10	17	literature	literature	NOUN
cana-2192	10	18	are	be	AUX
cana-2192	10	19	simple	simple	ADJ
cana-2192	10	20	consequences	consequence	NOUN
cana-2192	10	21	of	of	ADP
cana-2192	10	22	their	their	PRON
cana-2192	10	23	obtained	obtain	VERB
cana-2192	10	24	results	result	NOUN
cana-2192	10	25	.	.	PUNCT
cana-2192	11	1	definition	definition	NOUN
cana-2192	11	2	1	1	NUM
cana-2192	11	3	.	.	PUNCT
cana-2192	12	1	[	[	X
cana-2192	12	2	14	14	NUM
cana-2192	12	3	]	]	PUNCT
cana-2192	12	4	a	a	DET
cana-2192	12	5	function	function	NOUN
cana-2192	12	6	ζ	ζ	NOUN
cana-2192	12	7	:	:	PUNCT
cana-2192	13	1	[	[	X
cana-2192	13	2	0,∞)×	0,∞)×	NUM
cana-2192	13	3	[	[	X
cana-2192	13	4	0,∞	0,∞	NUM
cana-2192	13	5	)	)	PUNCT
cana-2192	13	6	→	→	PUNCT
cana-2192	13	7	r	r	NOUN
cana-2192	13	8	is	be	AUX
cana-2192	13	9	called	call	VERB
cana-2192	13	10	a	a	DET
cana-2192	13	11	simulation	simulation	NOUN
cana-2192	13	12	function	function	NOUN
cana-2192	13	13	if	if	SCONJ
cana-2192	13	14	ζ	ζ	NOUN
cana-2192	13	15	satisfies	satisfy	VERB
cana-2192	13	16	the	the	DET
cana-2192	13	17	following	follow	VERB
cana-2192	13	18	conditions	condition	NOUN
cana-2192	13	19	:	:	PUNCT
cana-2192	13	20	(	(	PUNCT
cana-2192	13	21	ζ1	ζ1	NOUN
cana-2192	13	22	)	)	PUNCT
cana-2192	13	23	ζ(0	ζ(0	NOUN
cana-2192	13	24	,	,	PUNCT
cana-2192	13	25	0	0	NUM
cana-2192	13	26	)	)	PUNCT
cana-2192	13	27	=	=	SYM
cana-2192	14	1	0	0	X
cana-2192	14	2	.	.	PUNCT
cana-2192	15	1	(	(	PUNCT
cana-2192	15	2	ζ2	ζ2	NOUN
cana-2192	15	3	)	)	PUNCT
cana-2192	15	4	ζ(t	ζ(t	PROPN
cana-2192	15	5	,	,	PUNCT
cana-2192	15	6	s	s	PART
cana-2192	15	7	)	)	PUNCT
cana-2192	15	8	<	<	X
cana-2192	15	9	s−	s−	PROPN
cana-2192	15	10	t	t	PROPN
cana-2192	15	11	,	,	PUNCT
cana-2192	15	12	for	for	ADP
cana-2192	15	13	all	all	DET
cana-2192	15	14	t	t	PROPN
cana-2192	15	15	,	,	PUNCT
cana-2192	15	16	s	s	PART
cana-2192	15	17	>	>	X
cana-2192	15	18	0	0	NUM
cana-2192	15	19	.	.	PUNCT
cana-2192	16	1	(	(	PUNCT
cana-2192	16	2	ζ3	ζ3	NOUN
cana-2192	16	3	)	)	PUNCT
cana-2192	16	4	if	if	SCONJ
cana-2192	16	5	{	{	PUNCT
cana-2192	16	6	tn	tn	NOUN
cana-2192	16	7	}	}	PUNCT
cana-2192	16	8	,	,	PUNCT
cana-2192	16	9	{	{	PUNCT
cana-2192	16	10	sn	sn	X
cana-2192	16	11	}	}	PUNCT
cana-2192	16	12	are	be	AUX
cana-2192	16	13	sequences	sequence	NOUN
cana-2192	16	14	in	in	ADP
cana-2192	16	15	(	(	PUNCT
cana-2192	16	16	0,∞	0,∞	NOUN
cana-2192	16	17	)	)	PUNCT
cana-2192	17	1	such	such	ADJ
cana-2192	17	2	that	that	SCONJ
cana-2192	17	3	limn→∞	limn→∞	PROPN
cana-2192	17	4	tn	tn	NOUN
cana-2192	17	5	=	=	PUNCT
cana-2192	17	6	limn→∞sn	limn→∞sn	NOUN
cana-2192	17	7	=	=	SYM
cana-2192	17	8	l	l	NOUN
cana-2192	17	9	∈	∈	PROPN
cana-2192	17	10	(	(	PUNCT
cana-2192	17	11	0,∞	0,∞	NOUN
cana-2192	17	12	)	)	PUNCT
cana-2192	17	13	,	,	PUNCT
cana-2192	17	14	then	then	ADV
cana-2192	17	15	limn→∞	limn→∞	PROPN
cana-2192	17	16	sup	sup	NOUN
cana-2192	17	17	ζ(tn	ζ(tn	PROPN
cana-2192	17	18	,	,	PUNCT
cana-2192	17	19	sn	sn	PROPN
cana-2192	17	20	)	)	PUNCT
cana-2192	17	21	<	<	X
cana-2192	17	22	0	0	X
cana-2192	17	23	.	.	PUNCT
cana-2192	18	1	in	in	ADP
cana-2192	18	2	[	[	X
cana-2192	18	3	14	14	NUM
cana-2192	18	4	]	]	PUNCT
cana-2192	18	5	,	,	PUNCT
cana-2192	18	6	the	the	DET
cana-2192	18	7	following	follow	VERB
cana-2192	18	8	unique	unique	ADJ
cana-2192	18	9	fixed	fix	VERB
cana-2192	18	10	point	point	NOUN
cana-2192	18	11	theorem	theorem	NOUN
cana-2192	18	12	is	be	AUX
cana-2192	18	13	established	establish	VERB
cana-2192	18	14	.	.	PUNCT
cana-2192	19	1	theorem	theorem	NOUN
cana-2192	19	2	2	2	NUM
cana-2192	19	3	.	.	PUNCT
cana-2192	20	1	[	[	X
cana-2192	20	2	14	14	NUM
cana-2192	20	3	]	]	X
cana-2192	20	4	let	let	VERB
cana-2192	20	5	(	(	PUNCT
cana-2192	20	6	x	x	NOUN
cana-2192	20	7	,	,	PUNCT
cana-2192	20	8	d	d	NOUN
cana-2192	20	9	)	)	PUNCT
cana-2192	20	10	be	be	AUX
cana-2192	20	11	a	a	DET
cana-2192	20	12	metric	metric	ADJ
cana-2192	20	13	space	space	NOUN
cana-2192	20	14	and	and	CCONJ
cana-2192	20	15	t	t	NOUN
cana-2192	20	16	:	:	PUNCT
cana-2192	20	17	x	x	X
cana-2192	20	18	→	→	PUNCT
cana-2192	20	19	x	x	PUNCT
cana-2192	20	20	be	be	AUX
cana-2192	20	21	a	a	DET
cana-2192	20	22	zcontraction	zcontraction	NOUN
cana-2192	20	23	with	with	ADP
cana-2192	20	24	respect	respect	NOUN
cana-2192	20	25	to	to	ADP
cana-2192	20	26	a	a	DET
cana-2192	20	27	simulation	simulation	NOUN
cana-2192	20	28	function	function	NOUN
cana-2192	20	29	ζ	ζ	NOUN
cana-2192	20	30	,	,	PUNCT
cana-2192	20	31	that	that	PRON
cana-2192	20	32	is	be	AUX
cana-2192	20	33	ζ(d(tx	ζ(d(tx	NOUN
cana-2192	20	34	,	,	PUNCT
cana-2192	20	35	ty	ty	NOUN
cana-2192	20	36	)	)	PUNCT
cana-2192	20	37	,	,	PUNCT
cana-2192	20	38	d(x	d(x	PROPN
cana-2192	20	39	,	,	PUNCT
cana-2192	20	40	y	y	NOUN
cana-2192	20	41	)	)	PUNCT
cana-2192	20	42	)	)	PUNCT
cana-2192	20	43	≥	≥	NOUN
cana-2192	20	44	0	0	NUM
cana-2192	20	45	for	for	ADP
cana-2192	20	46	all	all	DET
cana-2192	20	47	x	x	NOUN
cana-2192	20	48	,	,	PUNCT
cana-2192	20	49	y	y	PROPN
cana-2192	20	50	∈	∈	PROPN
cana-2192	20	51	x.	x.	NOUN
cana-2192	20	52	then	then	ADV
cana-2192	20	53	t	t	PROPN
cana-2192	20	54	has	have	VERB
cana-2192	20	55	a	a	DET
cana-2192	20	56	unique	unique	ADJ
cana-2192	20	57	fixed	fix	VERB
cana-2192	20	58	point	point	NOUN
cana-2192	20	59	.	.	PUNCT
cana-2192	21	1	it	it	PRON
cana-2192	21	2	is	be	AUX
cana-2192	21	3	worth	worth	ADJ
cana-2192	21	4	mentioning	mention	VERB
cana-2192	21	5	that	that	SCONJ
cana-2192	21	6	the	the	DET
cana-2192	21	7	banach	banach	NOUN
cana-2192	21	8	contraction	contraction	NOUN
cana-2192	21	9	is	be	AUX
cana-2192	21	10	an	an	DET
cana-2192	21	11	example	example	NOUN
cana-2192	21	12	of	of	ADP
cana-2192	21	13	zcontractions	zcontraction	NOUN
cana-2192	21	14	by	by	ADP
cana-2192	21	15	defining	define	VERB
cana-2192	21	16	ζ	ζ	NOUN
cana-2192	21	17	:	:	PUNCT
cana-2192	21	18	[	[	X
cana-2192	21	19	0	0	NUM
cana-2192	21	20	,	,	PUNCT
cana-2192	21	21	∞	∞	PROPN
cana-2192	21	22	)	)	PUNCT
cana-2192	21	23	×	×	NOUN
cana-2192	22	1	[	[	X
cana-2192	22	2	0	0	NUM
cana-2192	22	3	,	,	PUNCT
cana-2192	22	4	∞	∞	PROPN
cana-2192	22	5	)	)	PUNCT
cana-2192	22	6	→	→	SYM
cana-2192	22	7	r	r	NOUN
cana-2192	22	8	via	via	ADP
cana-2192	22	9	ζ(t	ζ(t	PROPN
cana-2192	22	10	,	,	PUNCT
cana-2192	22	11	s	s	NOUN
cana-2192	22	12	)	)	PUNCT
cana-2192	22	13	=	=	PUNCT
cana-2192	22	14	λs	λs	ADP
cana-2192	22	15	−	−	PROPN
cana-2192	22	16	t	t	PROPN
cana-2192	22	17	,	,	PUNCT
cana-2192	22	18	for	for	ADP
cana-2192	22	19	all	all	DET
cana-2192	22	20	s	s	PROPN
cana-2192	22	21	,	,	PUNCT
cana-2192	22	22	t	t	PROPN
cana-2192	22	23	∈	∈	PROPN
cana-2192	23	1	[	[	X
cana-2192	23	2	0	0	NUM
cana-2192	23	3	,	,	PUNCT
cana-2192	23	4	∞	∞	PROPN
cana-2192	23	5	)	)	PUNCT
cana-2192	23	6	,	,	PUNCT
cana-2192	23	7	where	where	SCONJ
cana-2192	23	8	λ	λ	PROPN
cana-2192	23	9	∈	∈	PROPN
cana-2192	24	1	[	[	X
cana-2192	24	2	0	0	NUM
cana-2192	24	3	,	,	PUNCT
cana-2192	24	4	1	1	NUM
cana-2192	24	5	)	)	PUNCT
cana-2192	24	6	.	.	PUNCT
cana-2192	25	1	argoubi	argoubi	INTJ
cana-2192	25	2	et	et	PROPN
cana-2192	25	3	al	al	PROPN
cana-2192	25	4	.	.	PUNCT
cana-2192	26	1	[	[	X
cana-2192	26	2	2	2	NUM
cana-2192	26	3	]	]	PUNCT
cana-2192	26	4	modified	modified	ADJ
cana-2192	26	5	definition	definition	NOUN
cana-2192	26	6	(	(	PUNCT
cana-2192	26	7	1	1	NUM
cana-2192	26	8	)	)	PUNCT
cana-2192	26	9	as	as	SCONJ
cana-2192	26	10	follows	follow	VERB
cana-2192	26	11	.	.	PUNCT
cana-2192	27	1	definition	definition	NOUN
cana-2192	27	2	3	3	NUM
cana-2192	27	3	.	.	PUNCT
cana-2192	28	1	[	[	X
cana-2192	28	2	2	2	X
cana-2192	28	3	]	]	PUNCT
cana-2192	28	4	a	a	DET
cana-2192	28	5	simulation	simulation	NOUN
cana-2192	28	6	function	function	NOUN
cana-2192	28	7	is	be	AUX
cana-2192	28	8	a	a	DET
cana-2192	28	9	function	function	NOUN
cana-2192	28	10	ζ	ζ	NOUN
cana-2192	28	11	:	:	PUNCT
cana-2192	29	1	[	[	X
cana-2192	29	2	0	0	NUM
cana-2192	29	3	,	,	PUNCT
cana-2192	29	4	∞)×[0	∞)×[0	NUM
cana-2192	29	5	,	,	PUNCT
cana-2192	29	6	∞	∞	PROPN
cana-2192	29	7	)	)	PUNCT
cana-2192	29	8	→	→	SYM
cana-2192	29	9	r	r	NOUN
cana-2192	29	10	that	that	PRON
cana-2192	29	11	satisfies	satisfy	VERB
cana-2192	29	12	t	t	X
cana-2192	29	13	he	he	PRON
cana-2192	29	14	f	f	PROPN
cana-2192	29	15	ollowing	ollowe	VERB
cana-2192	29	16	conditions	condition	NOUN
cana-2192	29	17	:	:	PUNCT
cana-2192	29	18	2000	2000	NUM
cana-2192	29	19	mathematics	mathematic	NOUN
cana-2192	29	20	subject	subject	NOUN
cana-2192	29	21	classification	classification	NOUN
cana-2192	29	22	.	.	PUNCT
cana-2192	30	1	54h25	54h25	NUM
cana-2192	30	2	,	,	PUNCT
cana-2192	30	3	47h10	47h10	NUM
cana-2192	30	4	,	,	PUNCT
cana-2192	30	5	55m20	55m20	NUM
cana-2192	30	6	.	.	PUNCT
cana-2192	30	7	key	key	ADJ
cana-2192	30	8	words	word	NOUN
cana-2192	30	9	and	and	CCONJ
cana-2192	30	10	phrases	phrase	NOUN
cana-2192	30	11	.	.	PUNCT
cana-2192	31	1	almost	almost	ADV
cana-2192	31	2	z	z	NOUN
cana-2192	31	3	-	-	PUNCT
cana-2192	31	4	contraction	contraction	NOUN
cana-2192	31	5	,	,	PUNCT
cana-2192	31	6	simulation	simulation	NOUN
cana-2192	31	7	function	function	NOUN
cana-2192	31	8	,	,	PUNCT
cana-2192	31	9	α	α	NOUN
cana-2192	31	10	-	-	ADJ
cana-2192	31	11	admissible	admissible	ADJ
cana-2192	31	12	mapping	mapping	NOUN
cana-2192	31	13	.	.	PUNCT
cana-2192	32	1	corresponding	correspond	VERB
cana-2192	32	2	author	author	PROPN
cana-2192	32	3	-	-	PUNCT
cana-2192	32	4	dr	dr	PROPN
cana-2192	32	5	.	.	PROPN
cana-2192	32	6	urmila	urmila	PROPN
cana-2192	32	7	mishra	mishra	PROPN
cana-2192	32	8	.	.	PROPN
cana-2192	32	9	communications	communication	NOUN
cana-2192	32	10	on	on	ADP
cana-2192	32	11	applied	apply	VERB
cana-2192	32	12	nonlinear	nonlinear	ADJ
cana-2192	32	13	analysis	analysis	NOUN
cana-2192	32	14	issn	issn	NOUN
cana-2192	32	15	:	:	PUNCT
cana-2192	32	16	1074	1074	NUM
cana-2192	32	17	-	-	PUNCT
cana-2192	32	18	133x	133x	NUM
cana-2192	32	19	vol	vol	NOUN
cana-2192	32	20	32	32	NUM
cana-2192	32	21	no	no	NOUN
cana-2192	32	22	.	.	PUNCT
cana-2192	33	1	1s	1s	NUM
cana-2192	33	2	(	(	PUNCT
cana-2192	33	3	2025	2025	NUM
cana-2192	33	4	)	)	PUNCT
cana-2192	33	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-2192	33	6	354	354	NUM
cana-2192	33	7	received	receive	VERB
cana-2192	33	8	:	:	PUNCT
cana-2192	33	9	15	15	NUM
cana-2192	33	10	-	-	SYM
cana-2192	33	11	08	08	NUM
cana-2192	33	12	-	-	PUNCT
cana-2192	33	13	2024	2024	NUM
cana-2192	33	14	,	,	PUNCT
cana-2192	33	15	revised	revise	VERB
cana-2192	33	16	:	:	PUNCT
cana-2192	33	17	01	01	NUM
cana-2192	33	18	-	-	SYM
cana-2192	33	19	10	10	NUM
cana-2192	33	20	-	-	PUNCT
cana-2192	33	21	2024	2024	NUM
cana-2192	33	22	,	,	PUNCT
cana-2192	33	23	accepted	accept	VERB
cana-2192	33	24	:	:	PUNCT
cana-2192	33	25	20	20	NUM
cana-2192	33	26	-	-	SYM
cana-2192	33	27	10	10	NUM
cana-2192	33	28	-	-	PUNCT
cana-2192	33	29	2024	2024	NUM
cana-2192	33	30	dipti	dipti	NOUN
cana-2192	33	31	,	,	PUNCT
cana-2192	33	32	anil	anil	PROPN
cana-2192	33	33	kumar	kumar	PROPN
cana-2192	33	34	dubey	dubey	PROPN
cana-2192	33	35	,	,	PUNCT
cana-2192	33	36	urmila	urmila	PROPN
cana-2192	33	37	mishra	mishra	PROPN
cana-2192	33	38	,	,	PUNCT
cana-2192	33	39	(	(	PUNCT
cana-2192	33	40	i	i	NOUN
cana-2192	33	41	)	)	PUNCT
cana-2192	33	42	ζ(t	ζ(t	PROPN
cana-2192	33	43	,	,	PUNCT
cana-2192	33	44	s	s	PART
cana-2192	33	45	)	)	PUNCT
cana-2192	33	46	<	<	X
cana-2192	33	47	s−	s−	PROPN
cana-2192	33	48	t	t	PROPN
cana-2192	33	49	,	,	PUNCT
cana-2192	33	50	for	for	ADP
cana-2192	33	51	all	all	DET
cana-2192	33	52	s	s	PROPN
cana-2192	33	53	,	,	PUNCT
cana-2192	33	54	t	t	X
cana-2192	33	55	>	>	X
cana-2192	33	56	0	0	NUM
cana-2192	33	57	;	;	PUNCT
cana-2192	33	58	(	(	PUNCT
cana-2192	33	59	ii	ii	NOUN
cana-2192	33	60	)	)	PUNCT
cana-2192	33	61	if	if	SCONJ
cana-2192	33	62	{	{	PUNCT
cana-2192	33	63	tn	tn	NOUN
cana-2192	33	64	}	}	PUNCT
cana-2192	33	65	and	and	CCONJ
cana-2192	33	66	{	{	PUNCT
cana-2192	33	67	sn	sn	NOUN
cana-2192	33	68	}	}	PUNCT
cana-2192	33	69	are	be	AUX
cana-2192	33	70	sequences	sequence	NOUN
cana-2192	33	71	in	in	ADP
cana-2192	33	72	(	(	PUNCT
cana-2192	33	73	0,∞	0,∞	NOUN
cana-2192	33	74	)	)	PUNCT
cana-2192	33	75	such	such	ADJ
cana-2192	33	76	that	that	SCONJ
cana-2192	33	77	limn→∞	limn→∞	PROPN
cana-2192	34	1	tn	tn	NOUN
cana-2192	34	2	=	=	PUNCT
cana-2192	34	3	limn→∞sn	limn→∞sn	NOUN
cana-2192	34	4	=	=	SYM
cana-2192	34	5	l	l	NOUN
cana-2192	34	6	∈	∈	PROPN
cana-2192	34	7	(	(	PUNCT
cana-2192	34	8	0,∞	0,∞	NOUN
cana-2192	34	9	)	)	PUNCT
cana-2192	34	10	,	,	PUNCT
cana-2192	34	11	then	then	ADV
cana-2192	34	12	limn→∞	limn→∞	PROPN
cana-2192	34	13	sup	sup	NOUN
cana-2192	34	14	ζ(tn	ζ(tn	PROPN
cana-2192	34	15	,	,	PUNCT
cana-2192	34	16	sn	sn	PROPN
cana-2192	34	17	)	)	PUNCT
cana-2192	34	18	<	<	X
cana-2192	34	19	0	0	X
cana-2192	34	20	.	.	PUNCT
cana-2192	35	1	it	it	PRON
cana-2192	35	2	is	be	AUX
cana-2192	35	3	clear	clear	ADJ
cana-2192	35	4	that	that	SCONJ
cana-2192	35	5	any	any	DET
cana-2192	35	6	simulation	simulation	NOUN
cana-2192	35	7	function	function	NOUN
cana-2192	35	8	in	in	ADP
cana-2192	35	9	the	the	DET
cana-2192	35	10	sense	sense	NOUN
cana-2192	35	11	of	of	ADP
cana-2192	35	12	khojasteh	khojasteh	NOUN
cana-2192	35	13	et	et	PROPN
cana-2192	35	14	al	al	PROPN
cana-2192	35	15	.	.	PUNCT
cana-2192	36	1	[	[	X
cana-2192	36	2	14	14	NUM
cana-2192	36	3	]	]	PUNCT
cana-2192	36	4	(	(	PUNCT
cana-2192	36	5	definition	definition	NOUN
cana-2192	36	6	1	1	NUM
cana-2192	36	7	)	)	PUNCT
cana-2192	36	8	is	be	AUX
cana-2192	36	9	also	also	ADV
cana-2192	36	10	a	a	DET
cana-2192	36	11	simulation	simulation	NOUN
cana-2192	36	12	function	function	NOUN
cana-2192	36	13	in	in	ADP
cana-2192	36	14	the	the	DET
cana-2192	36	15	sense	sense	NOUN
cana-2192	36	16	of	of	ADP
cana-2192	36	17	argoubi	argoubi	NOUN
cana-2192	36	18	et	et	PROPN
cana-2192	36	19	al	al	PROPN
cana-2192	36	20	.	.	PUNCT
cana-2192	37	1	[	[	X
cana-2192	37	2	2	2	NUM
cana-2192	37	3	]	]	PUNCT
cana-2192	37	4	(	(	PUNCT
cana-2192	37	5	definition	definition	NOUN
cana-2192	37	6	3	3	NUM
cana-2192	37	7	)	)	PUNCT
cana-2192	37	8	.	.	PUNCT
cana-2192	38	1	the	the	DET
cana-2192	38	2	converse	converse	NOUN
cana-2192	38	3	is	be	AUX
cana-2192	38	4	not	not	PART
cana-2192	38	5	true	true	ADJ
cana-2192	38	6	.	.	PUNCT
cana-2192	39	1	very	very	ADV
cana-2192	39	2	recently	recently	ADV
cana-2192	39	3	many	many	ADJ
cana-2192	39	4	fixed	fix	VERB
cana-2192	39	5	point	point	NOUN
cana-2192	39	6	results	result	NOUN
cana-2192	39	7	by	by	ADP
cana-2192	39	8	using	use	VERB
cana-2192	39	9	simulation	simulation	NOUN
cana-2192	39	10	functions	function	NOUN
cana-2192	39	11	have	have	AUX
cana-2192	39	12	been	be	AUX
cana-2192	39	13	provided	provide	VERB
cana-2192	39	14	.	.	PUNCT
cana-2192	40	1	we	we	PRON
cana-2192	40	2	have	have	AUX
cana-2192	40	3	used	use	VERB
cana-2192	40	4	some	some	DET
cana-2192	40	5	important	important	ADJ
cana-2192	40	6	article	article	NOUN
cana-2192	40	7	related	relate	VERB
cana-2192	40	8	to	to	ADP
cana-2192	40	9	this	this	DET
cana-2192	40	10	paper	paper	NOUN
cana-2192	41	1	[	[	X
cana-2192	41	2	1	1	NUM
cana-2192	41	3	,	,	PUNCT
cana-2192	41	4	4	4	NUM
cana-2192	41	5	,	,	PUNCT
cana-2192	41	6	9	9	NUM
cana-2192	41	7	,	,	PUNCT
cana-2192	41	8	10	10	NUM
cana-2192	41	9	,	,	PUNCT
cana-2192	41	10	12	12	NUM
cana-2192	41	11	,	,	PUNCT
cana-2192	41	12	15	15	NUM
cana-2192	41	13	,	,	PUNCT
cana-2192	41	14	16	16	NUM
cana-2192	41	15	,	,	PUNCT
cana-2192	41	16	17	17	NUM
cana-2192	41	17	,	,	PUNCT
cana-2192	41	18	19	19	NUM
cana-2192	41	19	]	]	PUNCT
cana-2192	41	20	.	.	PUNCT
cana-2192	42	1	in	in	ADP
cana-2192	42	2	2012	2012	NUM
cana-2192	42	3	,	,	PUNCT
cana-2192	42	4	samet	samet	NOUN
cana-2192	42	5	et	et	NOUN
cana-2192	42	6	al.[20	al.[20	NOUN
cana-2192	42	7	]	]	PUNCT
cana-2192	42	8	introduced	introduce	VERB
cana-2192	42	9	the	the	DET
cana-2192	42	10	concept	concept	NOUN
cana-2192	42	11	of	of	ADP
cana-2192	42	12	α	α	NOUN
cana-2192	42	13	-	-	NOUN
cana-2192	42	14	contraction	contraction	NOUN
cana-2192	42	15	and	and	CCONJ
cana-2192	42	16	α	α	NOUN
cana-2192	42	17	-	-	ADJ
cana-2192	42	18	admissible	admissible	ADJ
cana-2192	42	19	and	and	CCONJ
cana-2192	42	20	established	establish	VERB
cana-2192	42	21	various	various	ADJ
cana-2192	42	22	fixed	fix	VERB
cana-2192	42	23	point	point	NOUN
cana-2192	42	24	results	result	NOUN
cana-2192	42	25	for	for	ADP
cana-2192	42	26	such	such	ADJ
cana-2192	42	27	class	class	NOUN
cana-2192	42	28	of	of	ADP
cana-2192	42	29	mapping	mapping	NOUN
cana-2192	42	30	defined	define	VERB
cana-2192	42	31	on	on	ADP
cana-2192	42	32	complete	complete	ADJ
cana-2192	42	33	metric	metric	ADJ
cana-2192	42	34	space	space	NOUN
cana-2192	42	35	.	.	PUNCT
cana-2192	43	1	there	there	ADV
cana-2192	43	2	after	after	ADP
cana-2192	43	3	the	the	DET
cana-2192	43	4	existence	existence	NOUN
cana-2192	43	5	of	of	ADP
cana-2192	43	6	fixed	fix	VERB
cana-2192	43	7	point	point	NOUN
cana-2192	43	8	of	of	ADP
cana-2192	43	9	α	α	NOUN
cana-2192	43	10	-	-	PUNCT
cana-2192	43	11	admissible	admissible	ADJ
cana-2192	43	12	contraction	contraction	NOUN
cana-2192	43	13	type	type	NOUN
cana-2192	43	14	mappings	mapping	NOUN
cana-2192	43	15	in	in	ADP
cana-2192	43	16	different	different	ADJ
cana-2192	43	17	metric	metric	ADJ
cana-2192	43	18	spaces	space	NOUN
cana-2192	43	19	have	have	AUX
cana-2192	43	20	been	be	AUX
cana-2192	43	21	studied	study	VERB
cana-2192	43	22	by	by	ADP
cana-2192	43	23	several	several	ADJ
cana-2192	43	24	authors	author	NOUN
cana-2192	43	25	(	(	PUNCT
cana-2192	43	26	see	see	VERB
cana-2192	43	27	[	[	X
cana-2192	43	28	8	8	NUM
cana-2192	43	29	,	,	PUNCT
cana-2192	43	30	9	9	NUM
cana-2192	43	31	,	,	PUNCT
cana-2192	43	32	10	10	NUM
cana-2192	43	33	,	,	PUNCT
cana-2192	43	34	16	16	NUM
cana-2192	43	35	,	,	PUNCT
cana-2192	43	36	18	18	NUM
cana-2192	43	37	]	]	PUNCT
cana-2192	43	38	)	)	PUNCT
cana-2192	43	39	and	and	CCONJ
cana-2192	43	40	references	reference	NOUN
cana-2192	43	41	cited	cite	VERB
cana-2192	43	42	there	there	ADV
cana-2192	43	43	in	in	ADP
cana-2192	43	44	.	.	PUNCT
cana-2192	44	1	definition	definition	NOUN
cana-2192	44	2	4	4	NUM
cana-2192	44	3	.	.	PUNCT
cana-2192	45	1	[	[	X
cana-2192	45	2	20	20	NUM
cana-2192	45	3	]	]	PUNCT
cana-2192	45	4	consider	consider	VERB
cana-2192	45	5	two	two	NUM
cana-2192	45	6	mappings	mapping	NOUN
cana-2192	45	7	f	f	NOUN
cana-2192	45	8	:	:	PUNCT
cana-2192	45	9	x	x	X
cana-2192	45	10	→	→	SYM
cana-2192	45	11	x	x	X
cana-2192	45	12	and	and	CCONJ
cana-2192	45	13	α	α	NOUN
cana-2192	45	14	:	:	PUNCT
cana-2192	45	15	x	x	PROPN
cana-2192	45	16	×x	×x	X
cana-2192	45	17	→	→	X
cana-2192	45	18	[	[	X
cana-2192	45	19	0,∞	0,∞	NOUN
cana-2192	45	20	)	)	PUNCT
cana-2192	45	21	.	.	PUNCT
cana-2192	46	1	then	then	ADV
cana-2192	46	2	f	f	PROPN
cana-2192	46	3	is	be	AUX
cana-2192	46	4	called	call	VERB
cana-2192	46	5	α	α	DET
cana-2192	46	6	-	-	ADJ
cana-2192	46	7	admissible	admissible	ADJ
cana-2192	46	8	mapping	mapping	NOUN
cana-2192	46	9	if	if	SCONJ
cana-2192	46	10	for	for	ADP
cana-2192	46	11	all	all	DET
cana-2192	46	12	x	x	NOUN
cana-2192	46	13	,	,	PUNCT
cana-2192	46	14	y	y	PROPN
cana-2192	46	15	∈	∈	PROPN
cana-2192	46	16	x	x	PUNCT
cana-2192	46	17	with	with	ADP
cana-2192	46	18	α(x	α(x	PROPN
cana-2192	46	19	,	,	PUNCT
cana-2192	46	20	y	y	PROPN
cana-2192	46	21	)	)	PUNCT
cana-2192	46	22	≥	≥	NOUN
cana-2192	46	23	1	1	NUM
cana-2192	46	24	implies	imply	VERB
cana-2192	46	25	α(fx	α(fx	PROPN
cana-2192	46	26	,	,	PUNCT
cana-2192	46	27	fy	fy	PROPN
cana-2192	46	28	)	)	PUNCT
cana-2192	46	29	≥	≥	NOUN
cana-2192	47	1	1	1	NUM
cana-2192	47	2	.	.	PUNCT
cana-2192	48	1	in	in	ADP
cana-2192	48	2	this	this	DET
cana-2192	48	3	paper	paper	NOUN
cana-2192	48	4	,	,	PUNCT
cana-2192	48	5	we	we	PRON
cana-2192	48	6	introduce	introduce	VERB
cana-2192	48	7	the	the	DET
cana-2192	48	8	concept	concept	NOUN
cana-2192	48	9	of	of	ADP
cana-2192	48	10	generalized	generalized	ADJ
cana-2192	48	11	α	α	NOUN
cana-2192	48	12	-	-	ADJ
cana-2192	48	13	admissible	admissible	ADJ
cana-2192	48	14	almost	almost	ADV
cana-2192	48	15	z	z	NOUN
cana-2192	48	16	-	-	PUNCT
cana-2192	48	17	contraction	contraction	NOUN
cana-2192	48	18	with	with	ADP
cana-2192	48	19	respect	respect	NOUN
cana-2192	48	20	toζ	toζ	NOUN
cana-2192	48	21	.	.	PUNCT
cana-2192	49	1	we	we	PRON
cana-2192	49	2	also	also	ADV
cana-2192	49	3	establish	establish	VERB
cana-2192	49	4	the	the	DET
cana-2192	49	5	existence	existence	NOUN
cana-2192	49	6	of	of	ADP
cana-2192	49	7	fixed	fix	VERB
cana-2192	49	8	point	point	NOUN
cana-2192	49	9	for	for	ADP
cana-2192	49	10	this	this	DET
cana-2192	49	11	class	class	NOUN
cana-2192	49	12	of	of	ADP
cana-2192	49	13	mappings	mapping	NOUN
cana-2192	49	14	in	in	ADP
cana-2192	49	15	complete	complete	ADJ
cana-2192	49	16	metric	metric	ADJ
cana-2192	49	17	spaces	space	NOUN
cana-2192	49	18	.	.	PUNCT
cana-2192	50	1	the	the	DET
cana-2192	50	2	presented	present	VERB
cana-2192	50	3	theorems	theorems	PROPN
cana-2192	50	4	extends	extend	VERB
cana-2192	50	5	,	,	PUNCT
cana-2192	50	6	generalizes	generalize	VERB
cana-2192	50	7	and	and	CCONJ
cana-2192	50	8	improve	improve	VERB
cana-2192	50	9	many	many	ADJ
cana-2192	50	10	existing	exist	VERB
cana-2192	50	11	results	result	NOUN
cana-2192	50	12	in	in	ADP
cana-2192	50	13	the	the	DET
cana-2192	50	14	literature	literature	NOUN
cana-2192	50	15	,	,	PUNCT
cana-2192	50	16	in	in	ADP
cana-2192	50	17	particular	particular	ADJ
cana-2192	50	18	the	the	DET
cana-2192	50	19	results	result	NOUN
cana-2192	50	20	[	[	X
cana-2192	50	21	3	3	NUM
cana-2192	50	22	,	,	PUNCT
cana-2192	50	23	7	7	NUM
cana-2192	50	24	,	,	PUNCT
cana-2192	50	25	11	11	NUM
cana-2192	50	26	,	,	PUNCT
cana-2192	50	27	13	13	NUM
cana-2192	50	28	,	,	PUNCT
cana-2192	50	29	17	17	NUM
cana-2192	50	30	]	]	PUNCT
cana-2192	50	31	.	.	PUNCT
cana-2192	51	1	2	2	X
cana-2192	51	2	.	.	X
cana-2192	51	3	main	main	ADJ
cana-2192	51	4	results	result	NOUN
cana-2192	51	5	here	here	ADV
cana-2192	51	6	we	we	PRON
cana-2192	51	7	put	put	VERB
cana-2192	51	8	forward	forward	ADV
cana-2192	51	9	the	the	DET
cana-2192	51	10	notion	notion	NOUN
cana-2192	51	11	of	of	ADP
cana-2192	51	12	geraghty	geraghty	PROPN
cana-2192	51	13	functions	function	NOUN
cana-2192	51	14	and	and	CCONJ
cana-2192	51	15	geraghty	geraghty	PROPN
cana-2192	51	16	contractions	contraction	NOUN
cana-2192	51	17	were	be	AUX
cana-2192	51	18	discussed	discuss	VERB
cana-2192	51	19	by	by	ADP
cana-2192	51	20	geraghty	geraghty	PROPN
cana-2192	52	1	[	[	X
cana-2192	52	2	11	11	NUM
cana-2192	52	3	]	]	PUNCT
cana-2192	52	4	.	.	PUNCT
cana-2192	53	1	definition	definition	NOUN
cana-2192	53	2	5	5	NUM
cana-2192	53	3	.	.	PUNCT
cana-2192	54	1	[	[	X
cana-2192	54	2	11	11	NUM
cana-2192	54	3	]	]	PUNCT
cana-2192	54	4	a	a	DET
cana-2192	54	5	function	function	NOUN
cana-2192	54	6	β	β	NOUN
cana-2192	54	7	:	:	PUNCT
cana-2192	55	1	[	[	X
cana-2192	55	2	0,∞	0,∞	NOUN
cana-2192	55	3	)	)	PUNCT
cana-2192	55	4	→	→	SYM
cana-2192	55	5	(	(	PUNCT
cana-2192	55	6	0	0	NUM
cana-2192	55	7	,	,	PUNCT
cana-2192	55	8	1	1	NUM
cana-2192	55	9	)	)	PUNCT
cana-2192	55	10	is	be	AUX
cana-2192	55	11	called	call	VERB
cana-2192	55	12	geraghty	geraghty	PROPN
cana-2192	55	13	function	function	NOUN
cana-2192	55	14	if	if	SCONJ
cana-2192	55	15	{	{	PUNCT
cana-2192	55	16	rn	rn	NOUN
cana-2192	55	17	}	}	PUNCT
cana-2192	55	18	⊂	⊂	PROPN
cana-2192	56	1	[	[	X
cana-2192	56	2	0,∞	0,∞	NUM
cana-2192	56	3	)	)	PUNCT
cana-2192	56	4	and	and	CCONJ
cana-2192	56	5	limn→∞	limn→∞	PROPN
cana-2192	56	6	β(rn	β(rn	PART
cana-2192	56	7	)	)	PUNCT
cana-2192	56	8	=	=	SYM
cana-2192	56	9	1−	1−	NUM
cana-2192	56	10	implies	imply	VERB
cana-2192	56	11	rn	rn	PROPN
cana-2192	56	12	→	→	SYM
cana-2192	56	13	0	0	PROPN
cana-2192	57	1	+	+	PUNCT
cana-2192	57	2	as	as	ADP
cana-2192	57	3	n	n	NOUN
cana-2192	57	4	→	→	SYM
cana-2192	57	5	∞.	∞.	PROPN
cana-2192	57	6	definition	definition	NOUN
cana-2192	57	7	6	6	NUM
cana-2192	57	8	.	.	PUNCT
cana-2192	58	1	[	[	X
cana-2192	58	2	11	11	NUM
cana-2192	58	3	]	]	PUNCT
cana-2192	58	4	a	a	DET
cana-2192	58	5	mapping	mapping	NOUN
cana-2192	58	6	t	t	NOUN
cana-2192	58	7	:	:	PUNCT
cana-2192	58	8	x	x	X
cana-2192	58	9	→	→	PUNCT
cana-2192	58	10	x	x	X
cana-2192	58	11	is	be	AUX
cana-2192	58	12	called	call	VERB
cana-2192	58	13	geraghty	geraghty	PROPN
cana-2192	58	14	contraction	contraction	NOUN
cana-2192	58	15	if	if	SCONJ
cana-2192	58	16	there	there	PRON
cana-2192	58	17	exists	exist	VERB
cana-2192	58	18	a	a	DET
cana-2192	58	19	geraghty	geraghty	PROPN
cana-2192	58	20	function	function	NOUN
cana-2192	58	21	β	β	NOUN
cana-2192	58	22	such	such	ADJ
cana-2192	58	23	that	that	SCONJ
cana-2192	58	24	d(tx	d(tx	PROPN
cana-2192	58	25	,	,	PUNCT
cana-2192	58	26	ty	ty	NOUN
cana-2192	58	27	)	)	PUNCT
cana-2192	58	28	≤	≤	NOUN
cana-2192	58	29	β(d(x	β(d(x	ADV
cana-2192	58	30	,	,	PUNCT
cana-2192	58	31	y))d(x	y))d(x	PROPN
cana-2192	58	32	,	,	PUNCT
cana-2192	58	33	y	y	PROPN
cana-2192	58	34	)	)	PUNCT
cana-2192	58	35	,	,	PUNCT
cana-2192	58	36	for	for	ADP
cana-2192	58	37	all	all	DET
cana-2192	58	38	x	x	NOUN
cana-2192	58	39	,	,	PUNCT
cana-2192	58	40	y	y	PROPN
cana-2192	58	41	∈	∈	PROPN
cana-2192	58	42	x.	x.	NOUN
cana-2192	59	1	the	the	DET
cana-2192	59	2	concept	concept	NOUN
cana-2192	59	3	of	of	ADP
cana-2192	59	4	geraghty	geraghty	PROPN
cana-2192	59	5	contraction	contraction	PROPN
cana-2192	59	6	mapping	mapping	NOUN
cana-2192	59	7	has	have	AUX
cana-2192	59	8	been	be	AUX
cana-2192	59	9	used	use	VERB
cana-2192	59	10	in	in	ADP
cana-2192	59	11	many	many	ADJ
cana-2192	59	12	works	work	NOUN
cana-2192	59	13	for	for	ADP
cana-2192	59	14	example	example	NOUN
cana-2192	59	15	(	(	PUNCT
cana-2192	59	16	see[3	see[3	ADJ
cana-2192	59	17	,	,	PUNCT
cana-2192	59	18	8	8	NUM
cana-2192	59	19	,	,	PUNCT
cana-2192	59	20	18	18	NUM
cana-2192	59	21	]	]	PUNCT
cana-2192	59	22	)	)	PUNCT
cana-2192	59	23	.	.	PUNCT
cana-2192	60	1	berinde	berinde	NOUN
cana-2192	60	2	[	[	X
cana-2192	60	3	5	5	NUM
cana-2192	60	4	,	,	PUNCT
cana-2192	60	5	6	6	NUM
cana-2192	60	6	]	]	PUNCT
cana-2192	60	7	extended	extend	VERB
cana-2192	60	8	the	the	DET
cana-2192	60	9	class	class	NOUN
cana-2192	60	10	of	of	ADP
cana-2192	60	11	contractive	contractive	ADJ
cana-2192	60	12	mappings	mapping	NOUN
cana-2192	60	13	,	,	PUNCT
cana-2192	60	14	introducing	introduce	VERB
cana-2192	60	15	the	the	DET
cana-2192	60	16	notion	notion	NOUN
cana-2192	60	17	of	of	ADP
cana-2192	60	18	almost	almost	ADV
cana-2192	60	19	contractions	contraction	NOUN
cana-2192	60	20	as	as	SCONJ
cana-2192	60	21	follows	follow	VERB
cana-2192	60	22	.	.	PUNCT
cana-2192	61	1	definition	definition	NOUN
cana-2192	61	2	7	7	NUM
cana-2192	61	3	.	.	PUNCT
cana-2192	62	1	let	let	VERB
cana-2192	62	2	(	(	PUNCT
cana-2192	62	3	x	x	NOUN
cana-2192	62	4	,	,	PUNCT
cana-2192	62	5	d	d	NOUN
cana-2192	62	6	)	)	PUNCT
cana-2192	62	7	be	be	AUX
cana-2192	62	8	a	a	DET
cana-2192	62	9	metric	metric	ADJ
cana-2192	62	10	space	space	NOUN
cana-2192	62	11	.	.	PUNCT
cana-2192	63	1	a	a	DET
cana-2192	63	2	self	self	NOUN
cana-2192	63	3	mapping	mapping	NOUN
cana-2192	63	4	t	t	NOUN
cana-2192	63	5	on	on	ADP
cana-2192	63	6	x	x	VERB
cana-2192	63	7	is	be	AUX
cana-2192	63	8	called	call	VERB
cana-2192	63	9	an	an	DET
cana-2192	63	10	almost	almost	ADV
cana-2192	63	11	contraction	contraction	NOUN
cana-2192	63	12	if	if	SCONJ
cana-2192	63	13	there	there	PRON
cana-2192	63	14	are	be	VERB
cana-2192	63	15	constants	constant	NOUN
cana-2192	63	16	λ	λ	X
cana-2192	63	17	∈	∈	PROPN
cana-2192	63	18	(	(	PUNCT
cana-2192	63	19	0	0	NUM
cana-2192	63	20	,	,	PUNCT
cana-2192	63	21	1	1	NUM
cana-2192	63	22	)	)	PUNCT
cana-2192	63	23	and	and	CCONJ
cana-2192	63	24	θ	θ	PROPN
cana-2192	63	25	≥	≥	NUM
cana-2192	63	26	0	0	NUM
cana-2192	64	1	such	such	ADJ
cana-2192	64	2	that	that	SCONJ
cana-2192	64	3	d(tx	d(tx	PROPN
cana-2192	64	4	,	,	PUNCT
cana-2192	64	5	ty	ty	NOUN
cana-2192	64	6	)	)	PUNCT
cana-2192	64	7	≤	≤	NOUN
cana-2192	64	8	λd(x	λd(x	PUNCT
cana-2192	64	9	,	,	PUNCT
cana-2192	64	10	y	y	NOUN
cana-2192	64	11	)	)	PUNCT
cana-2192	64	12	+	+	NUM
cana-2192	64	13	θd(y	θd(y	NOUN
cana-2192	64	14	,	,	PUNCT
cana-2192	64	15	tx	tx	PROPN
cana-2192	64	16	)	)	PUNCT
cana-2192	64	17	for	for	ADP
cana-2192	64	18	all	all	DET
cana-2192	64	19	x	x	NOUN
cana-2192	64	20	,	,	PUNCT
cana-2192	64	21	y	y	PROPN
cana-2192	64	22	∈	∈	PROPN
cana-2192	64	23	x.	x.	NOUN
cana-2192	64	24	berinde	berinde	VERB
cana-2192	64	25	[	[	X
cana-2192	64	26	5	5	NUM
cana-2192	64	27	,	,	PUNCT
cana-2192	64	28	6	6	NUM
cana-2192	64	29	]	]	PUNCT
cana-2192	64	30	proved	prove	VERB
cana-2192	64	31	that	that	SCONJ
cana-2192	64	32	every	every	DET
cana-2192	64	33	almost	almost	ADV
cana-2192	64	34	contraction	contraction	NOUN
cana-2192	64	35	mapping	mapping	NOUN
cana-2192	64	36	defined	define	VERB
cana-2192	64	37	in	in	ADP
cana-2192	64	38	a	a	DET
cana-2192	64	39	complete	complete	ADJ
cana-2192	64	40	metric	metric	ADJ
cana-2192	64	41	space	space	NOUN
cana-2192	64	42	has	have	VERB
cana-2192	64	43	at	at	ADV
cana-2192	64	44	least	least	ADV
cana-2192	64	45	one	one	NUM
cana-2192	64	46	fixed	fix	VERB
cana-2192	64	47	point	point	NOUN
cana-2192	64	48	.	.	PUNCT
cana-2192	65	1	subsequently	subsequently	ADV
cana-2192	65	2	,	,	PUNCT
cana-2192	65	3	many	many	ADJ
cana-2192	65	4	authors	author	NOUN
cana-2192	65	5	[	[	X
cana-2192	65	6	7	7	NUM
cana-2192	65	7	,	,	PUNCT
cana-2192	65	8	9	9	NUM
cana-2192	65	9	,	,	PUNCT
cana-2192	65	10	13	13	NUM
cana-2192	65	11	]	]	PUNCT
cana-2192	65	12	demonstrated	demonstrate	VERB
cana-2192	65	13	that	that	SCONJ
cana-2192	65	14	almost	almost	ADV
cana-2192	65	15	contractions	contraction	NOUN
cana-2192	65	16	type	type	NOUN
cana-2192	65	17	mappings	mapping	NOUN
cana-2192	65	18	have	have	VERB
cana-2192	65	19	a	a	DET
cana-2192	65	20	unique	unique	ADJ
cana-2192	65	21	fixed	fix	VERB
cana-2192	65	22	point	point	NOUN
cana-2192	65	23	in	in	ADP
cana-2192	65	24	different	different	ADJ
cana-2192	65	25	metric	metric	ADJ
cana-2192	65	26	spaces	space	NOUN
cana-2192	65	27	.	.	PUNCT
cana-2192	66	1	by	by	ADP
cana-2192	66	2	using	use	VERB
cana-2192	66	3	the	the	DET
cana-2192	66	4	concept	concept	NOUN
cana-2192	66	5	of	of	ADP
cana-2192	66	6	geraghty	geraghty	PROPN
cana-2192	66	7	function	function	NOUN
cana-2192	66	8	(	(	PUNCT
cana-2192	66	9	β	β	NOUN
cana-2192	66	10	)	)	PUNCT
cana-2192	66	11	and	and	CCONJ
cana-2192	66	12	almost	almost	ADV
cana-2192	66	13	contractions	contraction	NOUN
cana-2192	66	14	,	,	PUNCT
cana-2192	66	15	we	we	PRON
cana-2192	66	16	introduce	introduce	VERB
cana-2192	66	17	the	the	DET
cana-2192	66	18	following	following	NOUN
cana-2192	66	19	:	:	PUNCT
cana-2192	66	20	communications	communication	NOUN
cana-2192	66	21	on	on	ADP
cana-2192	66	22	applied	apply	VERB
cana-2192	66	23	nonlinear	nonlinear	ADJ
cana-2192	66	24	analysis	analysis	NOUN
cana-2192	66	25	issn	issn	NOUN
cana-2192	66	26	:	:	PUNCT
cana-2192	66	27	1074	1074	NUM
cana-2192	66	28	-	-	PUNCT
cana-2192	66	29	133x	133x	NUM
cana-2192	66	30	vol	vol	NOUN
cana-2192	66	31	32	32	NUM
cana-2192	67	1	no	no	NOUN
cana-2192	67	2	.	.	PUNCT
cana-2192	68	1	1s	1s	NUM
cana-2192	68	2	(	(	PUNCT
cana-2192	68	3	2025	2025	NUM
cana-2192	68	4	)	)	PUNCT
cana-2192	68	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-2192	68	6	355	355	NUM
cana-2192	68	7	generalized	generalize	VERB
cana-2192	68	8	α	α	NOUN
cana-2192	68	9	-	-	ADJ
cana-2192	68	10	admissible	admissible	ADJ
cana-2192	68	11	almost	almost	ADV
cana-2192	68	12	......	......	PUNCT
cana-2192	68	13	functions	function	NOUN
cana-2192	68	14	in	in	ADP
cana-2192	68	15	a	a	DET
cana-2192	68	16	metric	metric	ADJ
cana-2192	68	17	space	space	NOUN
cana-2192	68	18	definition	definition	NOUN
cana-2192	68	19	8	8	NUM
cana-2192	68	20	.	.	PUNCT
cana-2192	69	1	let	let	VERB
cana-2192	69	2	(	(	PUNCT
cana-2192	69	3	x	x	NOUN
cana-2192	69	4	,	,	PUNCT
cana-2192	69	5	d	d	NOUN
cana-2192	69	6	)	)	PUNCT
cana-2192	69	7	be	be	AUX
cana-2192	69	8	a	a	DET
cana-2192	69	9	metric	metric	ADJ
cana-2192	69	10	space	space	NOUN
cana-2192	69	11	,	,	PUNCT
cana-2192	69	12	f	f	X
cana-2192	69	13	:	:	PUNCT
cana-2192	69	14	x	x	X
cana-2192	69	15	→	→	PUNCT
cana-2192	69	16	x	x	PUNCT
cana-2192	69	17	be	be	AUX
cana-2192	69	18	a	a	DET
cana-2192	69	19	self	self	NOUN
cana-2192	69	20	mapping	mapping	NOUN
cana-2192	69	21	,	,	PUNCT
cana-2192	69	22	there	there	PRON
cana-2192	69	23	exists	exist	VERB
cana-2192	69	24	ζ	ζ	PROPN
cana-2192	69	25	∈	∈	PROPN
cana-2192	69	26	z	z	NOUN
cana-2192	69	27	and	and	CCONJ
cana-2192	69	28	α	α	NOUN
cana-2192	69	29	:	:	PUNCT
cana-2192	69	30	x	x	SYM
cana-2192	69	31	×	×	NOUN
cana-2192	69	32	x	x	INTJ
cana-2192	69	33	→	→	X
cana-2192	69	34	[	[	X
cana-2192	69	35	0,∞	0,∞	NOUN
cana-2192	69	36	)	)	PUNCT
cana-2192	69	37	.	.	PUNCT
cana-2192	70	1	then	then	ADV
cana-2192	70	2	continuous	continuous	ADJ
cana-2192	70	3	mapping	mapping	NOUN
cana-2192	70	4	f	f	X
cana-2192	70	5	is	be	AUX
cana-2192	70	6	called	call	VERB
cana-2192	70	7	generalized	generalized	ADJ
cana-2192	70	8	α	α	NOUN
cana-2192	70	9	-	-	ADJ
cana-2192	70	10	admissible	admissible	ADJ
cana-2192	70	11	almost	almost	ADV
cana-2192	70	12	z	z	NOUN
cana-2192	70	13	-	-	PUNCT
cana-2192	70	14	contraction	contraction	NOUN
cana-2192	70	15	with	with	ADP
cana-2192	70	16	respect	respect	NOUN
cana-2192	70	17	to	to	ADP
cana-2192	70	18	ζ	ζ	NOUN
cana-2192	70	19	and	and	CCONJ
cana-2192	70	20	β	β	X
cana-2192	70	21	∈	∈	NOUN
cana-2192	70	22	g	g	NOUN
cana-2192	70	23	and	and	CCONJ
cana-2192	70	24	l	l	PROPN
cana-2192	70	25	≥	≥	NUM
cana-2192	70	26	0	0	NUM
cana-2192	70	27	such	such	ADJ
cana-2192	70	28	that	that	DET
cana-2192	70	29	for	for	ADP
cana-2192	70	30	all	all	DET
cana-2192	70	31	x	x	NOUN
cana-2192	70	32	,	,	PUNCT
cana-2192	70	33	y	y	PROPN
cana-2192	70	34	∈	∈	PROPN
cana-2192	70	35	x	x	PROPN
cana-2192	70	36	,	,	PUNCT
cana-2192	70	37	ζ(α(x	ζ(α(x	PROPN
cana-2192	70	38	,	,	PUNCT
cana-2192	70	39	fx)α(y	fx)α(y	NOUN
cana-2192	70	40	,	,	PUNCT
cana-2192	70	41	fy)d(fx	fy)d(fx	PROPN
cana-2192	70	42	,	,	PUNCT
cana-2192	70	43	fy),k(x	fy),k(x	NOUN
cana-2192	70	44	,	,	PUNCT
cana-2192	70	45	y	y	NOUN
cana-2192	70	46	)	)	PUNCT
cana-2192	70	47	)	)	PUNCT
cana-2192	70	48	≥	≥	NOUN
cana-2192	70	49	0	0	NUM
cana-2192	70	50	(	(	PUNCT
cana-2192	70	51	2.1	2.1	NUM
cana-2192	70	52	)	)	PUNCT
cana-2192	70	53	for	for	ADP
cana-2192	70	54	all	all	DET
cana-2192	70	55	distinct	distinct	ADJ
cana-2192	70	56	x	x	NOUN
cana-2192	70	57	,	,	PUNCT
cana-2192	70	58	y	y	PROPN
cana-2192	70	59	∈	∈	PROPN
cana-2192	70	60	x	x	NOUN
cana-2192	70	61	,	,	PUNCT
cana-2192	70	62	where	where	SCONJ
cana-2192	70	63	ζ	ζ	NOUN
cana-2192	70	64	is	be	AUX
cana-2192	70	65	a	a	DET
cana-2192	70	66	simulation	simulation	NOUN
cana-2192	70	67	function	function	NOUN
cana-2192	70	68	in	in	ADP
cana-2192	70	69	the	the	DET
cana-2192	70	70	sense	sense	NOUN
cana-2192	70	71	of	of	ADP
cana-2192	70	72	definition	definition	NOUN
cana-2192	70	73	1	1	NUM
cana-2192	70	74	.	.	PUNCT
cana-2192	70	75	also	also	ADV
cana-2192	70	76	k(x	k(x	PROPN
cana-2192	70	77	,	,	PUNCT
cana-2192	70	78	y	y	NOUN
cana-2192	70	79	)	)	PUNCT
cana-2192	71	1	=	=	SYM
cana-2192	71	2	β(e(x	β(e(x	PROPN
cana-2192	71	3	,	,	PUNCT
cana-2192	71	4	y))e(x	y))e(x	PROPN
cana-2192	71	5	,	,	PUNCT
cana-2192	71	6	y	y	PROPN
cana-2192	71	7	)	)	PUNCT
cana-2192	71	8	+	+	CCONJ
cana-2192	71	9	ln(x	ln(x	X
cana-2192	71	10	,	,	PUNCT
cana-2192	71	11	y	y	NOUN
cana-2192	71	12	)	)	PUNCT
cana-2192	71	13	,	,	PUNCT
cana-2192	71	14	(	(	PUNCT
cana-2192	71	15	2.2	2.2	NUM
cana-2192	71	16	)	)	PUNCT
cana-2192	71	17	where	where	SCONJ
cana-2192	71	18	e(x	e(x	NUM
cana-2192	71	19	,	,	PUNCT
cana-2192	71	20	y	y	NOUN
cana-2192	71	21	)	)	PUNCT
cana-2192	71	22	=	=	SYM
cana-2192	72	1	d(x	d(x	PROPN
cana-2192	72	2	,	,	PUNCT
cana-2192	72	3	y	y	NOUN
cana-2192	72	4	)	)	PUNCT
cana-2192	72	5	+	+	CCONJ
cana-2192	73	1	|d(x	|d(x	PROPN
cana-2192	73	2	,	,	PUNCT
cana-2192	73	3	fx)−	fx)−	NOUN
cana-2192	73	4	d(y	d(y	NOUN
cana-2192	73	5	,	,	PUNCT
cana-2192	73	6	fy)|	fy)|	PROPN
cana-2192	73	7	and	and	CCONJ
cana-2192	73	8	n(x	n(x	PROPN
cana-2192	73	9	,	,	PUNCT
cana-2192	73	10	y	y	NOUN
cana-2192	73	11	)	)	PUNCT
cana-2192	73	12	=	=	VERB
cana-2192	74	1	min{d(x	min{d(x	PROPN
cana-2192	74	2	,	,	PUNCT
cana-2192	74	3	fx	fx	PROPN
cana-2192	74	4	)	)	PUNCT
cana-2192	74	5	,	,	PUNCT
cana-2192	74	6	d(y	d(y	PROPN
cana-2192	74	7	,	,	PUNCT
cana-2192	74	8	fy	fy	PROPN
cana-2192	74	9	)	)	PUNCT
cana-2192	74	10	,	,	PUNCT
cana-2192	74	11	d(x	d(x	PROPN
cana-2192	74	12	,	,	PUNCT
cana-2192	74	13	fy	fy	PROPN
cana-2192	74	14	)	)	PUNCT
cana-2192	74	15	,	,	PUNCT
cana-2192	74	16	d(y	d(y	PROPN
cana-2192	74	17	,	,	PUNCT
cana-2192	74	18	fx	fx	PROPN
cana-2192	74	19	)	)	PUNCT
cana-2192	74	20	}	}	PUNCT
cana-2192	74	21	.	.	PUNCT
cana-2192	75	1	now	now	ADV
cana-2192	75	2	we	we	PRON
cana-2192	75	3	prove	prove	VERB
cana-2192	75	4	our	our	PRON
cana-2192	75	5	main	main	ADJ
cana-2192	75	6	result	result	NOUN
cana-2192	75	7	.	.	PUNCT
cana-2192	76	1	theorem	theorem	NOUN
cana-2192	76	2	9	9	NUM
cana-2192	76	3	.	.	PUNCT
cana-2192	77	1	let	let	AUX
cana-2192	77	2	(	(	PUNCT
cana-2192	77	3	x	x	NOUN
cana-2192	77	4	,	,	PUNCT
cana-2192	77	5	d	d	NOUN
cana-2192	77	6	)	)	PUNCT
cana-2192	77	7	be	be	AUX
cana-2192	77	8	a	a	DET
cana-2192	77	9	complete	complete	ADJ
cana-2192	77	10	metric	metric	ADJ
cana-2192	77	11	space	space	NOUN
cana-2192	77	12	,	,	PUNCT
cana-2192	77	13	f	f	PROPN
cana-2192	77	14	is	be	AUX
cana-2192	77	15	a	a	DET
cana-2192	77	16	generalized	generalize	VERB
cana-2192	77	17	αadmissible	αadmissible	ADJ
cana-2192	77	18	almost	almost	ADV
cana-2192	77	19	z	z	NOUN
cana-2192	77	20	-	-	PUNCT
cana-2192	77	21	contraction	contraction	NOUN
cana-2192	77	22	with	with	ADP
cana-2192	77	23	respect	respect	NOUN
cana-2192	77	24	to	to	ADP
cana-2192	77	25	ζ	ζ	NOUN
cana-2192	77	26	furthermore	furthermore	ADV
cana-2192	77	27	,	,	PUNCT
cana-2192	77	28	we	we	PRON
cana-2192	77	29	suppose	suppose	VERB
cana-2192	77	30	for	for	ADP
cana-2192	77	31	all	all	DET
cana-2192	77	32	x	x	NOUN
cana-2192	77	33	,	,	PUNCT
cana-2192	77	34	y	y	PROPN
cana-2192	77	35	∈	∈	PROPN
cana-2192	77	36	x	x	PUNCT
cana-2192	77	37	such	such	ADJ
cana-2192	77	38	that	that	SCONJ
cana-2192	77	39	:	:	PUNCT
cana-2192	77	40	(	(	PUNCT
cana-2192	77	41	i	i	NOUN
cana-2192	77	42	)	)	PUNCT
cana-2192	77	43	f	f	PROPN
cana-2192	77	44	is	be	AUX
cana-2192	77	45	α	α	PRON
cana-2192	77	46	-	-	ADJ
cana-2192	77	47	admissible	admissible	ADJ
cana-2192	77	48	;	;	PUNCT
cana-2192	77	49	(	(	PUNCT
cana-2192	77	50	ii	ii	NOUN
cana-2192	77	51	)	)	PUNCT
cana-2192	77	52	there	there	PRON
cana-2192	77	53	exists	exist	VERB
cana-2192	77	54	x0	x0	PROPN
cana-2192	77	55	∈	∈	PROPN
cana-2192	77	56	x	x	PUNCT
cana-2192	77	57	such	such	ADJ
cana-2192	77	58	that	that	DET
cana-2192	77	59	α(x0	α(x0	ADJ
cana-2192	77	60	,	,	PUNCT
cana-2192	77	61	fx0	fx0	PROPN
cana-2192	77	62	)	)	PUNCT
cana-2192	77	63	≥	≥	NOUN
cana-2192	77	64	1	1	NUM
cana-2192	77	65	;	;	PUNCT
cana-2192	77	66	(	(	PUNCT
cana-2192	77	67	iii	iii	NOUN
cana-2192	77	68	)	)	PUNCT
cana-2192	77	69	for	for	ADP
cana-2192	77	70	every	every	DET
cana-2192	77	71	sequence	sequence	NOUN
cana-2192	77	72	{	{	PUNCT
cana-2192	77	73	xn	xn	NOUN
cana-2192	77	74	}	}	PUNCT
cana-2192	77	75	∈	∈	PROPN
cana-2192	77	76	x	x	NOUN
cana-2192	77	77	such	such	ADJ
cana-2192	77	78	that	that	SCONJ
cana-2192	77	79	α(xn	α(xn	PROPN
cana-2192	77	80	,	,	PUNCT
cana-2192	77	81	fxn	fxn	NOUN
cana-2192	77	82	)	)	PUNCT
cana-2192	77	83	≥	≥	NOUN
cana-2192	77	84	1	1	NUM
cana-2192	77	85	for	for	ADP
cana-2192	77	86	all	all	PRON
cana-2192	77	87	n	n	PRON
cana-2192	77	88	∈	∈	NOUN
cana-2192	77	89	n	n	NOUN
cana-2192	77	90	∪	∪	X
cana-2192	77	91	{	{	PUNCT
cana-2192	77	92	0	0	NUM
cana-2192	77	93	}	}	PUNCT
cana-2192	77	94	and	and	CCONJ
cana-2192	77	95	{	{	PUNCT
cana-2192	77	96	xn	xn	NOUN
cana-2192	77	97	}	}	PUNCT
cana-2192	77	98	converges	converge	NOUN
cana-2192	77	99	to	to	ADP
cana-2192	77	100	x	x	PRON
cana-2192	77	101	,	,	PUNCT
cana-2192	77	102	then	then	ADV
cana-2192	77	103	α(x	α(x	NOUN
cana-2192	77	104	,	,	PUNCT
cana-2192	77	105	fx	fx	PROPN
cana-2192	77	106	)	)	PUNCT
cana-2192	77	107	≥	≥	NOUN
cana-2192	77	108	1	1	NUM
cana-2192	77	109	;	;	PUNCT
cana-2192	77	110	(	(	PUNCT
cana-2192	77	111	iv	iv	X
cana-2192	77	112	)	)	PUNCT
cana-2192	77	113	α(x	α(x	NOUN
cana-2192	77	114	,	,	PUNCT
cana-2192	77	115	fx	fx	PROPN
cana-2192	77	116	)	)	PUNCT
cana-2192	77	117	≥	≥	NOUN
cana-2192	77	118	1	1	NUM
cana-2192	77	119	for	for	ADP
cana-2192	77	120	all	all	DET
cana-2192	77	121	x	x	SYM
cana-2192	77	122	∈	∈	PROPN
cana-2192	77	123	fix(f	fix(f	PROPN
cana-2192	77	124	)	)	PUNCT
cana-2192	77	125	.	.	PUNCT
cana-2192	78	1	then	then	ADV
cana-2192	78	2	f	f	PROPN
cana-2192	78	3	has	have	VERB
cana-2192	78	4	a	a	DET
cana-2192	78	5	unique	unique	ADJ
cana-2192	78	6	fixed	fix	VERB
cana-2192	78	7	point	point	NOUN
cana-2192	78	8	x∗	x∗	PROPN
cana-2192	78	9	in	in	ADP
cana-2192	78	10	x.	x.	NOUN
cana-2192	78	11	proof	proof	NOUN
cana-2192	78	12	.	.	PUNCT
cana-2192	79	1	on	on	ADP
cana-2192	79	2	account	account	NOUN
cana-2192	79	3	of	of	ADP
cana-2192	79	4	(	(	PUNCT
cana-2192	79	5	ii	ii	NOUN
cana-2192	79	6	)	)	PUNCT
cana-2192	79	7	,	,	PUNCT
cana-2192	79	8	there	there	PRON
cana-2192	79	9	is	be	VERB
cana-2192	79	10	a	a	DET
cana-2192	79	11	point	point	NOUN
cana-2192	79	12	x0	x0	PROPN
cana-2192	79	13	∈	∈	PROPN
cana-2192	79	14	x	x	PUNCT
cana-2192	79	15	such	such	ADJ
cana-2192	79	16	that	that	DET
cana-2192	79	17	α(x0	α(x0	ADJ
cana-2192	79	18	,	,	PUNCT
cana-2192	79	19	fx0	fx0	PROPN
cana-2192	79	20	)	)	PUNCT
cana-2192	79	21	≥	≥	NOUN
cana-2192	80	1	1	1	NUM
cana-2192	80	2	.	.	PUNCT
cana-2192	80	3	there	there	PRON
cana-2192	80	4	exists	exist	VERB
cana-2192	80	5	xn	xn	PROPN
cana-2192	80	6	∈	∈	PROPN
cana-2192	80	7	x	x	PUNCT
cana-2192	80	8	such	such	ADJ
cana-2192	80	9	that	that	PRON
cana-2192	80	10	xn	xn	PROPN
cana-2192	81	1	=	=	PUNCT
cana-2192	81	2	fxn−1	fxn−1	PROPN
cana-2192	81	3	for	for	ADP
cana-2192	81	4	all	all	PRON
cana-2192	81	5	n	n	DET
cana-2192	81	6	∈	∈	PROPN
cana-2192	81	7	n.	n.	NOUN
cana-2192	81	8	since	since	SCONJ
cana-2192	81	9	f	f	PROPN
cana-2192	81	10	is	be	AUX
cana-2192	81	11	αadmissible	αadmissible	ADJ
cana-2192	81	12	,	,	PUNCT
cana-2192	81	13	we	we	PRON
cana-2192	81	14	obtain	obtain	VERB
cana-2192	81	15	α(fx0	α(fx0	ADV
cana-2192	81	16	,	,	PUNCT
cana-2192	81	17	fx1	fx1	NOUN
cana-2192	81	18	)	)	PUNCT
cana-2192	81	19	=	=	SYM
cana-2192	81	20	α(x1	α(x1	ADJ
cana-2192	81	21	,	,	PUNCT
cana-2192	81	22	x2	x2	PROPN
cana-2192	81	23	)	)	PUNCT
cana-2192	81	24	≥	≥	NOUN
cana-2192	81	25	1	1	NUM
cana-2192	81	26	implies	imply	VERB
cana-2192	81	27	α(fx1	α(fx1	PROPN
cana-2192	81	28	,	,	PUNCT
cana-2192	81	29	fx2	fx2	PROPN
cana-2192	81	30	)	)	PUNCT
cana-2192	81	31	=	=	SYM
cana-2192	81	32	α(x2	α(x2	NOUN
cana-2192	81	33	,	,	PUNCT
cana-2192	81	34	x3	x3	ADJ
cana-2192	81	35	)	)	PUNCT
cana-2192	81	36	≥	≥	NOUN
cana-2192	81	37	1	1	NUM
cana-2192	81	38	.	.	PUNCT
cana-2192	82	1	by	by	ADP
cana-2192	82	2	induction	induction	NOUN
cana-2192	82	3	,	,	PUNCT
cana-2192	82	4	we	we	PRON
cana-2192	82	5	get	get	VERB
cana-2192	82	6	α(xn	α(xn	NOUN
cana-2192	82	7	,	,	PUNCT
cana-2192	82	8	xn+1	xn+1	NUM
cana-2192	82	9	)	)	PUNCT
cana-2192	82	10	≥	≥	NOUN
cana-2192	82	11	1	1	NUM
cana-2192	82	12	,	,	PUNCT
cana-2192	82	13	for	for	ADP
cana-2192	82	14	all	all	DET
cana-2192	82	15	n	n	PRON
cana-2192	82	16	∈	∈	NOUN
cana-2192	82	17	n	n	NOUN
cana-2192	82	18	∪	∪	X
cana-2192	82	19	{	{	PUNCT
cana-2192	82	20	0	0	NUM
cana-2192	82	21	}	}	PUNCT
cana-2192	82	22	.	.	PUNCT
cana-2192	83	1	(	(	PUNCT
cana-2192	83	2	2.3	2.3	NUM
cana-2192	83	3	)	)	PUNCT
cana-2192	83	4	if	if	SCONJ
cana-2192	83	5	xn	xn	NOUN
cana-2192	83	6	=	=	SYM
cana-2192	83	7	xn+1	xn+1	PROPN
cana-2192	83	8	for	for	ADP
cana-2192	83	9	some	some	DET
cana-2192	83	10	n	n	PRON
cana-2192	83	11	∈	∈	PROPN
cana-2192	83	12	n	n	NOUN
cana-2192	83	13	∪	∪	X
cana-2192	83	14	{	{	PUNCT
cana-2192	83	15	0	0	NUM
cana-2192	83	16	}	}	PUNCT
cana-2192	83	17	,	,	PUNCT
cana-2192	83	18	then	then	ADV
cana-2192	83	19	xn	xn	PUNCT
cana-2192	84	1	=	=	SYM
cana-2192	84	2	xn+1	xn+1	PROPN
cana-2192	84	3	=	=	SYM
cana-2192	84	4	fxn	fxn	NOUN
cana-2192	84	5	and	and	CCONJ
cana-2192	84	6	hence	hence	ADV
cana-2192	84	7	xn	xn	PROPN
cana-2192	84	8	is	be	AUX
cana-2192	84	9	a	a	DET
cana-2192	84	10	fixed	fix	VERB
cana-2192	84	11	point	point	NOUN
cana-2192	84	12	of	of	ADP
cana-2192	84	13	f	f	PROPN
cana-2192	84	14	.	.	PUNCT
cana-2192	85	1	therefore	therefore	ADV
cana-2192	85	2	,	,	PUNCT
cana-2192	85	3	we	we	PRON
cana-2192	85	4	can	can	AUX
cana-2192	85	5	assume	assume	VERB
cana-2192	85	6	that	that	SCONJ
cana-2192	85	7	xn	xn	PUNCT
cana-2192	86	1	=	=	PUNCT
cana-2192	86	2	̸	̸	X
cana-2192	86	3	xn+1	xn+1	ADJ
cana-2192	86	4	for	for	ADP
cana-2192	86	5	all	all	DET
cana-2192	86	6	n	n	DET
cana-2192	86	7	∈	∈	PROPN
cana-2192	86	8	n.	n.	NOUN
cana-2192	86	9	then	then	ADV
cana-2192	86	10	we	we	PRON
cana-2192	86	11	get	get	VERB
cana-2192	86	12	d(xn	d(xn	NOUN
cana-2192	86	13	,	,	PUNCT
cana-2192	86	14	xn+1	xn+1	NUM
cana-2192	86	15	)	)	PUNCT
cana-2192	86	16	>	>	X
cana-2192	86	17	0	0	NUM
cana-2192	86	18	,	,	PUNCT
cana-2192	86	19	so	so	ADV
cana-2192	86	20	by	by	ADP
cana-2192	86	21	(	(	PUNCT
cana-2192	86	22	2.1	2.1	NUM
cana-2192	86	23	)	)	PUNCT
cana-2192	86	24	,	,	PUNCT
cana-2192	86	25	we	we	PRON
cana-2192	86	26	have	have	VERB
cana-2192	86	27	0	0	NUM
cana-2192	86	28	≤	≤	NOUN
cana-2192	86	29	ζ(α(xn	ζ(α(xn	NOUN
cana-2192	86	30	,	,	PUNCT
cana-2192	86	31	fxn)α(xn−1	fxn)α(xn−1	PROPN
cana-2192	86	32	,	,	PUNCT
cana-2192	86	33	fxn−1)d(fxn	fxn−1)d(fxn	PROPN
cana-2192	86	34	,	,	PUNCT
cana-2192	86	35	fxn−1),k(xn	fxn−1),k(xn	NOUN
cana-2192	86	36	,	,	PUNCT
cana-2192	86	37	xn−1	xn−1	PROPN
cana-2192	86	38	)	)	PUNCT
cana-2192	86	39	)	)	PUNCT
cana-2192	87	1	=	=	PUNCT
cana-2192	88	1	ζ(α(xn	ζ(α(xn	X
cana-2192	88	2	,	,	PUNCT
cana-2192	88	3	xn+1)α(xn−1	xn+1)α(xn−1	NUM
cana-2192	88	4	,	,	PUNCT
cana-2192	88	5	xn)d(xn+1	xn)d(xn+1	NOUN
cana-2192	88	6	,	,	PUNCT
cana-2192	88	7	xn),k(xn	xn),k(xn	PROPN
cana-2192	88	8	,	,	PUNCT
cana-2192	88	9	xn−1	xn−1	PROPN
cana-2192	88	10	)	)	PUNCT
cana-2192	88	11	)	)	PUNCT
cana-2192	89	1	<	<	X
cana-2192	89	2	k(xn	k(xn	PROPN
cana-2192	89	3	,	,	PUNCT
cana-2192	89	4	xn−1)−	xn−1)−	PROPN
cana-2192	89	5	α(xn	α(xn	PROPN
cana-2192	89	6	,	,	PUNCT
cana-2192	89	7	xn+1)α(xn−1	xn+1)α(xn−1	NUM
cana-2192	89	8	,	,	PUNCT
cana-2192	89	9	xn)d(xn+1	xn)d(xn+1	NOUN
cana-2192	89	10	,	,	PUNCT
cana-2192	89	11	xn	xn	PROPN
cana-2192	89	12	)	)	PUNCT
cana-2192	89	13	,	,	PUNCT
cana-2192	89	14	(	(	PUNCT
cana-2192	89	15	2.4	2.4	NUM
cana-2192	89	16	)	)	PUNCT
cana-2192	89	17	where	where	SCONJ
cana-2192	89	18	k(xn	k(xn	PROPN
cana-2192	89	19	,	,	PUNCT
cana-2192	89	20	xn−1	xn−1	PROPN
cana-2192	89	21	)	)	PUNCT
cana-2192	89	22	=	=	PUNCT
cana-2192	90	1	β(e(xn	β(e(xn	NOUN
cana-2192	90	2	,	,	PUNCT
cana-2192	90	3	xn−1))e(xn	xn−1))e(xn	NUM
cana-2192	90	4	,	,	PUNCT
cana-2192	90	5	xn−1	xn−1	PROPN
cana-2192	90	6	)	)	PUNCT
cana-2192	91	1	+	+	CCONJ
cana-2192	91	2	ln(xn	ln(xn	NOUN
cana-2192	91	3	,	,	PUNCT
cana-2192	91	4	xn−1	xn−1	PROPN
cana-2192	91	5	)	)	PUNCT
cana-2192	91	6	.	.	PUNCT
cana-2192	92	1	also	also	ADV
cana-2192	92	2	,	,	PUNCT
cana-2192	92	3	n(xn	n(xn	NUM
cana-2192	92	4	,	,	PUNCT
cana-2192	92	5	xn−1	xn−1	PROPN
cana-2192	92	6	)	)	PUNCT
cana-2192	92	7	=	=	SYM
cana-2192	92	8	min{d(xn	min{d(xn	ADJ
cana-2192	92	9	,	,	PUNCT
cana-2192	92	10	fxn	fxn	NOUN
cana-2192	92	11	)	)	PUNCT
cana-2192	92	12	,	,	PUNCT
cana-2192	92	13	d(xn−1	d(xn−1	PROPN
cana-2192	92	14	,	,	PUNCT
cana-2192	92	15	fxn−1	fxn−1	PROPN
cana-2192	92	16	)	)	PUNCT
cana-2192	92	17	,	,	PUNCT
cana-2192	92	18	d(xn	d(xn	PROPN
cana-2192	92	19	,	,	PUNCT
cana-2192	92	20	fxn−1	fxn−1	PROPN
cana-2192	92	21	)	)	PUNCT
cana-2192	92	22	,	,	PUNCT
cana-2192	92	23	d(xn−1	d(xn−1	PROPN
cana-2192	92	24	,	,	PUNCT
cana-2192	92	25	fxn	fxn	NOUN
cana-2192	92	26	)	)	PUNCT
cana-2192	92	27	}	}	PUNCT
cana-2192	93	1	=	=	SYM
cana-2192	93	2	min{d(xn	min{d(xn	X
cana-2192	93	3	,	,	PUNCT
cana-2192	93	4	xn+1	xn+1	NUM
cana-2192	93	5	)	)	PUNCT
cana-2192	93	6	,	,	PUNCT
cana-2192	93	7	d(xn−1	d(xn−1	PROPN
cana-2192	93	8	,	,	PUNCT
cana-2192	93	9	xn	xn	PROPN
cana-2192	93	10	)	)	PUNCT
cana-2192	93	11	,	,	PUNCT
cana-2192	93	12	d(xn	d(xn	PROPN
cana-2192	93	13	,	,	PUNCT
cana-2192	93	14	xn	xn	PROPN
cana-2192	93	15	)	)	PUNCT
cana-2192	93	16	,	,	PUNCT
cana-2192	93	17	d(xn−1	d(xn−1	PROPN
cana-2192	93	18	,	,	PUNCT
cana-2192	93	19	xn+1	xn+1	NUM
cana-2192	93	20	)	)	PUNCT
cana-2192	93	21	}	}	PUNCT
cana-2192	93	22	=	=	SYM
cana-2192	93	23	0	0	NUM
cana-2192	93	24	,	,	PUNCT
cana-2192	93	25	communications	communication	NOUN
cana-2192	93	26	on	on	ADP
cana-2192	93	27	applied	apply	VERB
cana-2192	93	28	nonlinear	nonlinear	ADJ
cana-2192	93	29	analysis	analysis	NOUN
cana-2192	93	30	issn	issn	NOUN
cana-2192	93	31	:	:	PUNCT
cana-2192	93	32	1074	1074	NUM
cana-2192	93	33	-	-	PUNCT
cana-2192	93	34	133x	133x	NUM
cana-2192	93	35	vol	vol	NOUN
cana-2192	93	36	32	32	NUM
cana-2192	93	37	no	no	NOUN
cana-2192	93	38	.	.	PUNCT
cana-2192	94	1	1s	1s	NUM
cana-2192	94	2	(	(	PUNCT
cana-2192	94	3	2025	2025	NUM
cana-2192	94	4	)	)	PUNCT
cana-2192	94	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-2192	94	6	356	356	NUM
cana-2192	94	7	dipti	dipti	NOUN
cana-2192	94	8	,	,	PUNCT
cana-2192	94	9	anil	anil	PROPN
cana-2192	94	10	kumar	kumar	PROPN
cana-2192	94	11	dubey	dubey	PROPN
cana-2192	94	12	,	,	PUNCT
cana-2192	94	13	urmila	urmila	PROPN
cana-2192	94	14	mishra	mishra	PROPN
cana-2192	94	15	,	,	PUNCT
cana-2192	94	16	and	and	CCONJ
cana-2192	94	17	e(xn	e(xn	PRON
cana-2192	94	18	,	,	PUNCT
cana-2192	94	19	xn−1	xn−1	PROPN
cana-2192	94	20	)	)	PUNCT
cana-2192	94	21	=	=	PUNCT
cana-2192	95	1	d(xn	d(xn	X
cana-2192	95	2	,	,	PUNCT
cana-2192	95	3	xn−1	xn−1	PROPN
cana-2192	95	4	)	)	PUNCT
cana-2192	95	5	+	+	CCONJ
cana-2192	96	1	|d(xn	|d(xn	PROPN
cana-2192	96	2	,	,	PUNCT
cana-2192	96	3	fxn)−	fxn)−	PROPN
cana-2192	96	4	d(xn−1	d(xn−1	PROPN
cana-2192	96	5	,	,	PUNCT
cana-2192	96	6	fxn−1)|	fxn−1)|	X
cana-2192	96	7	=	=	SYM
cana-2192	96	8	d(xn	d(xn	PROPN
cana-2192	96	9	,	,	PUNCT
cana-2192	96	10	xn−1	xn−1	PROPN
cana-2192	96	11	)	)	PUNCT
cana-2192	97	1	+	+	CCONJ
cana-2192	97	2	|d(xn	|d(xn	PROPN
cana-2192	97	3	,	,	PUNCT
cana-2192	97	4	xn+1)−	xn+1)−	PROPN
cana-2192	97	5	d(xn−1	d(xn−1	PROPN
cana-2192	97	6	,	,	PUNCT
cana-2192	97	7	xn)|	xn)|	PROPN
cana-2192	97	8	=	=	PUNCT
cana-2192	97	9	d(xn	d(xn	X
cana-2192	97	10	,	,	PUNCT
cana-2192	97	11	xn+1	xn+1	NUM
cana-2192	97	12	)	)	PUNCT
cana-2192	97	13	.	.	PUNCT
cana-2192	98	1	therefore	therefore	ADV
cana-2192	98	2	,	,	PUNCT
cana-2192	98	3	k(xn	k(xn	PROPN
cana-2192	98	4	,	,	PUNCT
cana-2192	98	5	xn−1	xn−1	PROPN
cana-2192	98	6	)	)	PUNCT
cana-2192	98	7	=	=	PUNCT
cana-2192	99	1	β(d(xn	β(d(xn	X
cana-2192	99	2	,	,	PUNCT
cana-2192	99	3	xn+1))d(xn	xn+1))d(xn	PROPN
cana-2192	99	4	,	,	PUNCT
cana-2192	99	5	xn+1	xn+1	NUM
cana-2192	99	6	)	)	PUNCT
cana-2192	99	7	,	,	PUNCT
cana-2192	99	8	from	from	ADP
cana-2192	99	9	(	(	PUNCT
cana-2192	99	10	2.4	2.4	NUM
cana-2192	99	11	)	)	PUNCT
cana-2192	99	12	,	,	PUNCT
cana-2192	99	13	we	we	PRON
cana-2192	99	14	get	get	VERB
cana-2192	99	15	0	0	NUM
cana-2192	99	16	≤	≤	NUM
cana-2192	100	1	β(d(xn	β(d(xn	NOUN
cana-2192	100	2	,	,	PUNCT
cana-2192	100	3	xn+1))d(xn	xn+1))d(xn	PROPN
cana-2192	100	4	,	,	PUNCT
cana-2192	100	5	xn+1)−	xn+1)−	PROPN
cana-2192	100	6	α(xn	α(xn	PROPN
cana-2192	100	7	,	,	PUNCT
cana-2192	100	8	xn+1)α(xn−1	xn+1)α(xn−1	NUM
cana-2192	100	9	,	,	PUNCT
cana-2192	100	10	xn)d(xn+1	xn)d(xn+1	NOUN
cana-2192	100	11	,	,	PUNCT
cana-2192	100	12	xn	xn	PROPN
cana-2192	100	13	)	)	PUNCT
cana-2192	100	14	which	which	PRON
cana-2192	100	15	implies	imply	VERB
cana-2192	100	16	that	that	SCONJ
cana-2192	100	17	d(xn	d(xn	PROPN
cana-2192	100	18	,	,	PUNCT
cana-2192	100	19	xn+1	xn+1	NUM
cana-2192	100	20	)	)	PUNCT
cana-2192	100	21	≤	≤	NOUN
cana-2192	101	1	β(d(xn	β(d(xn	NOUN
cana-2192	101	2	,	,	PUNCT
cana-2192	101	3	xn+1))d(xn	xn+1))d(xn	PROPN
cana-2192	101	4	,	,	PUNCT
cana-2192	101	5	xn+1	xn+1	NUM
cana-2192	101	6	)	)	PUNCT
cana-2192	101	7	<	<	X
cana-2192	102	1	d(xn	d(xn	PROPN
cana-2192	102	2	,	,	PUNCT
cana-2192	102	3	xn+1	xn+1	NUM
cana-2192	102	4	)	)	PUNCT
cana-2192	102	5	,	,	PUNCT
cana-2192	102	6	(	(	PUNCT
cana-2192	102	7	2.5	2.5	NUM
cana-2192	102	8	)	)	PUNCT
cana-2192	102	9	a	a	DET
cana-2192	102	10	contradiction	contradiction	NOUN
cana-2192	102	11	.	.	PUNCT
cana-2192	103	1	consequently	consequently	ADV
cana-2192	103	2	,	,	PUNCT
cana-2192	103	3	we	we	PRON
cana-2192	103	4	deduce	deduce	VERB
cana-2192	103	5	that	that	SCONJ
cana-2192	103	6	d(xn	d(xn	PROPN
cana-2192	103	7	,	,	PUNCT
cana-2192	103	8	xn+1	xn+1	NUM
cana-2192	103	9	)	)	PUNCT
cana-2192	103	10	<	<	X
cana-2192	103	11	d(xn−1	d(xn−1	PROPN
cana-2192	103	12	,	,	PUNCT
cana-2192	103	13	xn	xn	PROPN
cana-2192	103	14	)	)	PUNCT
cana-2192	103	15	for	for	ADP
cana-2192	103	16	each	each	DET
cana-2192	103	17	n	n	PRON
cana-2192	103	18	∈	∈	PROPN
cana-2192	103	19	n.	n.	NOUN
cana-2192	103	20	thus	thus	ADV
cana-2192	103	21	,	,	PUNCT
cana-2192	103	22	we	we	PRON
cana-2192	103	23	conclude	conclude	VERB
cana-2192	103	24	that	that	SCONJ
cana-2192	103	25	the	the	DET
cana-2192	103	26	sequence	sequence	NOUN
cana-2192	103	27	{	{	PUNCT
cana-2192	103	28	d(xn−1	d(xn−1	PROPN
cana-2192	103	29	,	,	PUNCT
cana-2192	103	30	xn	xn	PROPN
cana-2192	103	31	)	)	PUNCT
cana-2192	103	32	}	}	PUNCT
cana-2192	103	33	is	be	AUX
cana-2192	103	34	a	a	DET
cana-2192	103	35	monotonically	monotonically	ADV
cana-2192	103	36	decreasing	decrease	VERB
cana-2192	103	37	sequence	sequence	NOUN
cana-2192	103	38	of	of	ADP
cana-2192	103	39	non	non	ADJ
cana-2192	103	40	-	-	ADJ
cana-2192	103	41	negative	negative	ADJ
cana-2192	103	42	reals	real	NOUN
cana-2192	103	43	and	and	CCONJ
cana-2192	103	44	bounded	bound	VERB
cana-2192	103	45	from	from	ADP
cana-2192	103	46	below	below	ADV
cana-2192	103	47	by	by	ADP
cana-2192	103	48	zero	zero	NUM
cana-2192	103	49	.	.	PUNCT
cana-2192	104	1	so	so	ADV
cana-2192	104	2	,	,	PUNCT
cana-2192	104	3	there	there	PRON
cana-2192	104	4	is	be	VERB
cana-2192	104	5	some	some	DET
cana-2192	104	6	r	r	NOUN
cana-2192	104	7	≥	≥	NOUN
cana-2192	104	8	0	0	NUM
cana-2192	104	9	such	such	ADJ
cana-2192	104	10	that	that	SCONJ
cana-2192	104	11	limn→∞	limn→∞	PROPN
cana-2192	104	12	d(xn−1	d(xn−1	X
cana-2192	104	13	,	,	PUNCT
cana-2192	104	14	xn	xn	PRON
cana-2192	104	15	)	)	PUNCT
cana-2192	105	1	=	=	VERB
cana-2192	106	1	r.	r.	NOUN
cana-2192	106	2	it	it	PRON
cana-2192	106	3	is	be	AUX
cana-2192	106	4	evident	evident	ADJ
cana-2192	106	5	that	that	SCONJ
cana-2192	106	6	limn→∞e(xn−1	limn→∞e(xn−1	NOUN
cana-2192	106	7	,	,	PUNCT
cana-2192	106	8	xn	xn	PROPN
cana-2192	106	9	)	)	PUNCT
cana-2192	106	10	=	=	VERB
cana-2192	106	11	r.	r.	NOUN
cana-2192	106	12	as	as	ADP
cana-2192	106	13	a	a	DET
cana-2192	106	14	next	next	ADJ
cana-2192	106	15	step	step	NOUN
cana-2192	106	16	,	,	PUNCT
cana-2192	106	17	we	we	PRON
cana-2192	106	18	will	will	AUX
cana-2192	106	19	show	show	VERB
cana-2192	106	20	that	that	SCONJ
cana-2192	106	21	limn→∞	limn→∞	PROPN
cana-2192	106	22	d(xn	d(xn	X
cana-2192	106	23	,	,	PUNCT
cana-2192	106	24	xn−1	xn−1	PROPN
cana-2192	106	25	)	)	PUNCT
cana-2192	106	26	=	=	SYM
cana-2192	107	1	0	0	X
cana-2192	107	2	.	.	PUNCT
cana-2192	108	1	we	we	PRON
cana-2192	108	2	assert	assert	VERB
cana-2192	108	3	that	that	SCONJ
cana-2192	108	4	r	r	NOUN
cana-2192	108	5	=	=	SYM
cana-2192	108	6	0	0	X
cana-2192	108	7	.	.	PUNCT
cana-2192	108	8	suppose	suppose	VERB
cana-2192	108	9	,	,	PUNCT
cana-2192	108	10	in	in	ADP
cana-2192	108	11	contrast	contrast	NOUN
cana-2192	108	12	that	that	SCONJ
cana-2192	108	13	r	r	NOUN
cana-2192	108	14	̸=	̸=	PROPN
cana-2192	108	15	0	0	NUM
cana-2192	108	16	,	,	PUNCT
cana-2192	108	17	then	then	ADV
cana-2192	108	18	since	since	SCONJ
cana-2192	108	19	f	f	PROPN
cana-2192	108	20	is	be	AUX
cana-2192	108	21	generalized	generalize	VERB
cana-2192	108	22	α	α	PRON
cana-2192	108	23	-	-	ADJ
cana-2192	108	24	admissible	admissible	ADJ
cana-2192	108	25	almost	almost	ADV
cana-2192	108	26	z	z	NOUN
cana-2192	108	27	-	-	PUNCT
cana-2192	108	28	contraction	contraction	NOUN
cana-2192	108	29	with	with	ADP
cana-2192	108	30	respect	respect	NOUN
cana-2192	108	31	to	to	ADP
cana-2192	108	32	ζ	ζ	SYM
cana-2192	108	33	∈	∈	NOUN
cana-2192	108	34	z	z	AUX
cana-2192	108	35	therefore	therefore	ADV
cana-2192	108	36	by	by	ADP
cana-2192	108	37	(	(	PUNCT
cana-2192	108	38	ζ3	ζ3	NOUN
cana-2192	108	39	)	)	PUNCT
cana-2192	108	40	and	and	CCONJ
cana-2192	108	41	equation	equation	NOUN
cana-2192	108	42	(	(	PUNCT
cana-2192	108	43	2.5	2.5	NUM
cana-2192	108	44	)	)	PUNCT
cana-2192	108	45	,	,	PUNCT
cana-2192	108	46	and	and	CCONJ
cana-2192	108	47	taking	take	VERB
cana-2192	108	48	limit	limit	NOUN
cana-2192	108	49	as	as	ADP
cana-2192	108	50	n	n	PROPN
cana-2192	108	51	→	→	SYM
cana-2192	108	52	∞	∞	PROPN
cana-2192	108	53	,	,	PUNCT
cana-2192	108	54	we	we	PRON
cana-2192	108	55	have	have	VERB
cana-2192	108	56	0	0	NUM
cana-2192	108	57	≤	≤	NOUN
cana-2192	108	58	lim	lim	PROPN
cana-2192	108	59	n→∞	n→∞	NUM
cana-2192	108	60	sup	sup	PROPN
cana-2192	108	61	ζ(α(xn	ζ(α(xn	NOUN
cana-2192	108	62	,	,	PUNCT
cana-2192	108	63	xn+1)α(xn−1	xn+1)α(xn−1	NUM
cana-2192	108	64	,	,	PUNCT
cana-2192	108	65	xn)d(xn+1	xn)d(xn+1	NOUN
cana-2192	108	66	,	,	PUNCT
cana-2192	108	67	xn),k(xn	xn),k(xn	PROPN
cana-2192	108	68	,	,	PUNCT
cana-2192	108	69	xn−1	xn−1	PROPN
cana-2192	108	70	)	)	PUNCT
cana-2192	108	71	)	)	PUNCT
cana-2192	109	1	<	<	X
cana-2192	109	2	0	0	X
cana-2192	109	3	.	.	PUNCT
cana-2192	110	1	therefore	therefore	ADV
cana-2192	110	2	lim	lim	PROPN
cana-2192	110	3	n→∞	n→∞	X
cana-2192	110	4	β(e(xn−1	β(e(xn−1	PROPN
cana-2192	110	5	,	,	PUNCT
cana-2192	110	6	xn	xn	NUM
cana-2192	110	7	)	)	PUNCT
cana-2192	110	8	)	)	PUNCT
cana-2192	110	9	=	=	SYM
cana-2192	110	10	1	1	NUM
cana-2192	110	11	⇒	⇒	NOUN
cana-2192	110	12	lim	lim	PROPN
cana-2192	110	13	n→∞	n→∞	X
cana-2192	110	14	e(xn−1	e(xn−1	PROPN
cana-2192	110	15	,	,	PUNCT
cana-2192	110	16	xn	xn	PROPN
cana-2192	110	17	)	)	PUNCT
cana-2192	110	18	=	=	SYM
cana-2192	110	19	0	0	X
cana-2192	110	20	.	.	PUNCT
cana-2192	111	1	attendantly	attendantly	ADV
cana-2192	111	2	,	,	PUNCT
cana-2192	111	3	r	r	NOUN
cana-2192	111	4	=	=	SYM
cana-2192	111	5	0	0	NUM
cana-2192	111	6	and	and	CCONJ
cana-2192	111	7	also	also	ADV
cana-2192	111	8	r	r	NOUN
cana-2192	111	9	=	=	SYM
cana-2192	111	10	lim	lim	NOUN
cana-2192	111	11	n→∞	n→∞	X
cana-2192	111	12	d(xn	d(xn	PROPN
cana-2192	111	13	,	,	PUNCT
cana-2192	111	14	xn−1	xn−1	PROPN
cana-2192	111	15	)	)	PUNCT
cana-2192	111	16	=	=	SYM
cana-2192	112	1	0	0	X
cana-2192	112	2	.	.	PUNCT
cana-2192	112	3	(	(	PUNCT
cana-2192	112	4	2.6	2.6	NUM
cana-2192	112	5	)	)	PUNCT
cana-2192	112	6	now	now	ADV
cana-2192	112	7	,	,	PUNCT
cana-2192	112	8	we	we	PRON
cana-2192	112	9	will	will	AUX
cana-2192	112	10	show	show	VERB
cana-2192	112	11	that	that	DET
cana-2192	112	12	sequence	sequence	NOUN
cana-2192	112	13	{	{	PUNCT
cana-2192	112	14	xn	xn	PUNCT
cana-2192	112	15	}	}	PUNCT
cana-2192	112	16	is	be	AUX
cana-2192	112	17	a	a	DET
cana-2192	112	18	cauchy	cauchy	ADJ
cana-2192	112	19	sequence	sequence	NOUN
cana-2192	112	20	.	.	PUNCT
cana-2192	113	1	assume	assume	VERB
cana-2192	113	2	that	that	SCONJ
cana-2192	113	3	{	{	PUNCT
cana-2192	113	4	xn	xn	X
cana-2192	113	5	}	}	PUNCT
cana-2192	113	6	is	be	AUX
cana-2192	113	7	not	not	PART
cana-2192	113	8	a	a	DET
cana-2192	113	9	cauchy	cauchy	ADJ
cana-2192	113	10	sequence	sequence	NOUN
cana-2192	113	11	,	,	PUNCT
cana-2192	113	12	then	then	ADV
cana-2192	113	13	there	there	PRON
cana-2192	113	14	exists	exist	VERB
cana-2192	113	15	ϵ	ϵ	X
cana-2192	113	16	>	>	X
cana-2192	113	17	0	0	PUNCT
cana-2192	114	1	and	and	CCONJ
cana-2192	114	2	sequences	sequence	NOUN
cana-2192	114	3	{	{	PUNCT
cana-2192	114	4	xnk	xnk	PROPN
cana-2192	114	5	}	}	PUNCT
cana-2192	114	6	,	,	PUNCT
cana-2192	114	7	{	{	PUNCT
cana-2192	114	8	xmk	xmk	PROPN
cana-2192	114	9	}	}	PUNCT
cana-2192	114	10	:	:	PUNCT
cana-2192	114	11	mk	mk	PROPN
cana-2192	114	12	>	>	X
cana-2192	114	13	nk	nk	PROPN
cana-2192	114	14	>	>	X
cana-2192	115	1	k	k	X
cana-2192	115	2	such	such	ADJ
cana-2192	115	3	that	that	DET
cana-2192	115	4	d(xmk	d(xmk	NOUN
cana-2192	115	5	,	,	PUNCT
cana-2192	115	6	xnk	xnk	PROPN
cana-2192	115	7	)	)	PUNCT
cana-2192	115	8	>	>	X
cana-2192	116	1	ϵ	ϵ	X
cana-2192	116	2	and	and	CCONJ
cana-2192	116	3	d(xmk−1	d(xmk−1	PROPN
cana-2192	116	4	,	,	PUNCT
cana-2192	116	5	xnk	xnk	PROPN
cana-2192	116	6	)	)	PUNCT
cana-2192	116	7	≤	≤	PUNCT
cana-2192	117	1	ϵ	ϵ	X
cana-2192	117	2	for	for	ADP
cana-2192	117	3	all	all	DET
cana-2192	117	4	m	m	PROPN
cana-2192	117	5	,	,	PUNCT
cana-2192	117	6	n	n	CCONJ
cana-2192	117	7	,	,	PUNCT
cana-2192	117	8	k	k	PROPN
cana-2192	117	9	∈	∈	PROPN
cana-2192	117	10	n.	n.	PROPN
cana-2192	117	11	therefore	therefore	ADV
cana-2192	117	12	,	,	PUNCT
cana-2192	117	13	by	by	ADP
cana-2192	117	14	the	the	DET
cana-2192	117	15	triangle	triangle	NOUN
cana-2192	117	16	inequality	inequality	NOUN
cana-2192	117	17	,	,	PUNCT
cana-2192	117	18	we	we	PRON
cana-2192	117	19	have	have	VERB
cana-2192	117	20	that	that	PRON
cana-2192	117	21	ϵ	ϵ	X
cana-2192	117	22	<	<	X
cana-2192	117	23	d(xmk	d(xmk	PROPN
cana-2192	117	24	,	,	PUNCT
cana-2192	117	25	xnk	xnk	PROPN
cana-2192	117	26	)	)	PUNCT
cana-2192	117	27	≤	≤	NUM
cana-2192	117	28	d(xmk	d(xmk	NOUN
cana-2192	117	29	,	,	PUNCT
cana-2192	117	30	xmk−1	xmk−1	PROPN
cana-2192	117	31	)	)	PUNCT
cana-2192	118	1	+	+	X
cana-2192	118	2	d(xmk−1	d(xmk−1	PROPN
cana-2192	118	3	,	,	PUNCT
cana-2192	118	4	xnk	xnk	PROPN
cana-2192	118	5	)	)	PUNCT
cana-2192	118	6	≤	≤	NUM
cana-2192	118	7	d(xmk	d(xmk	NOUN
cana-2192	118	8	,	,	PUNCT
cana-2192	118	9	xmk−1	xmk−1	PROPN
cana-2192	118	10	)	)	PUNCT
cana-2192	119	1	+	+	CCONJ
cana-2192	119	2	ϵ.	ϵ.	NOUN
cana-2192	119	3	(	(	PUNCT
cana-2192	119	4	2.7	2.7	NUM
cana-2192	119	5	)	)	PUNCT
cana-2192	119	6	letting	let	VERB
cana-2192	119	7	k	k	X
cana-2192	119	8	→	→	SYM
cana-2192	119	9	∞	∞	PROPN
cana-2192	119	10	,	,	PUNCT
cana-2192	119	11	using	use	VERB
cana-2192	119	12	(	(	PUNCT
cana-2192	119	13	2.6	2.6	NUM
cana-2192	119	14	)	)	PUNCT
cana-2192	119	15	and	and	CCONJ
cana-2192	119	16	(	(	PUNCT
cana-2192	119	17	2.7	2.7	NUM
cana-2192	119	18	)	)	PUNCT
cana-2192	119	19	,	,	PUNCT
cana-2192	119	20	we	we	PRON
cana-2192	119	21	get	get	VERB
cana-2192	119	22	lim	lim	PROPN
cana-2192	119	23	n→∞	n→∞	NUM
cana-2192	120	1	d(xmk	d(xmk	PROPN
cana-2192	120	2	,	,	PUNCT
cana-2192	120	3	xnk	xnk	PROPN
cana-2192	120	4	)	)	PUNCT
cana-2192	121	1	=	=	PUNCT
cana-2192	121	2	ϵ.	ϵ.	NOUN
cana-2192	121	3	(	(	PUNCT
cana-2192	121	4	2.8	2.8	NUM
cana-2192	121	5	)	)	PUNCT
cana-2192	121	6	since	since	SCONJ
cana-2192	121	7	f	f	PROPN
cana-2192	121	8	is	be	AUX
cana-2192	121	9	a	a	DET
cana-2192	121	10	generalized	generalized	ADJ
cana-2192	121	11	α	α	NOUN
cana-2192	121	12	-	-	ADJ
cana-2192	121	13	admissible	admissible	ADJ
cana-2192	121	14	almost	almost	ADV
cana-2192	121	15	z	z	NOUN
cana-2192	121	16	-	-	PUNCT
cana-2192	121	17	contraction	contraction	NOUN
cana-2192	121	18	with	with	ADP
cana-2192	121	19	respect	respect	NOUN
cana-2192	121	20	to	to	ADP
cana-2192	121	21	ζ	ζ	NOUN
cana-2192	121	22	,	,	PUNCT
cana-2192	121	23	0	0	NUM
cana-2192	121	24	≤	≤	NUM
cana-2192	121	25	ζ(α(xmk−1	ζ(α(xmk−1	NOUN
cana-2192	121	26	,	,	PUNCT
cana-2192	121	27	xmk	xmk	NOUN
cana-2192	121	28	)	)	PUNCT
cana-2192	121	29	α(xnk−1	α(xnk−1	NOUN
cana-2192	121	30	,	,	PUNCT
cana-2192	121	31	xnk	xnk	PROPN
cana-2192	121	32	)	)	PUNCT
cana-2192	121	33	d(xmk	d(xmk	PROPN
cana-2192	121	34	,	,	PUNCT
cana-2192	121	35	xnk	xnk	PROPN
cana-2192	121	36	)	)	PUNCT
cana-2192	121	37	,	,	PUNCT
cana-2192	121	38	k(xmk−1	k(xmk−1	PROPN
cana-2192	121	39	,	,	PUNCT
cana-2192	121	40	xnk−1	xnk−1	PROPN
cana-2192	121	41	)	)	PUNCT
cana-2192	121	42	)	)	PUNCT
cana-2192	122	1	it	it	PRON
cana-2192	122	2	follows	follow	VERB
cana-2192	122	3	from	from	ADP
cana-2192	122	4	condition	condition	NOUN
cana-2192	122	5	(	(	PUNCT
cana-2192	122	6	ζ2	ζ2	NOUN
cana-2192	122	7	)	)	PUNCT
cana-2192	122	8	,	,	PUNCT
cana-2192	122	9	we	we	PRON
cana-2192	122	10	get	get	VERB
cana-2192	122	11	0	0	NUM
cana-2192	122	12	<	<	X
cana-2192	122	13	k(xmk−1	k(xmk−1	PROPN
cana-2192	122	14	,	,	PUNCT
cana-2192	122	15	xnk−1)−	xnk−1)−	PROPN
cana-2192	122	16	α(xmk−1	α(xmk−1	PROPN
cana-2192	122	17	,	,	PUNCT
cana-2192	122	18	xmk	xmk	PROPN
cana-2192	122	19	)	)	PUNCT
cana-2192	122	20	α(xnk−1	α(xnk−1	NOUN
cana-2192	122	21	,	,	PUNCT
cana-2192	122	22	xnk	xnk	PROPN
cana-2192	122	23	)	)	PUNCT
cana-2192	122	24	d(xmk	d(xmk	PROPN
cana-2192	122	25	,	,	PUNCT
cana-2192	123	1	xnk	xnk	PROPN
cana-2192	123	2	)	)	PUNCT
cana-2192	123	3	d(xmk	d(xmk	PROPN
cana-2192	123	4	,	,	PUNCT
cana-2192	123	5	xnk	xnk	PROPN
cana-2192	123	6	)	)	PUNCT
cana-2192	124	1	=	=	PUNCT
cana-2192	124	2	d(fxmk−1	d(fxmk−1	NOUN
cana-2192	124	3	,	,	PUNCT
cana-2192	124	4	fxnk−1	fxnk−1	NOUN
cana-2192	124	5	)	)	PUNCT
cana-2192	124	6	<	<	X
cana-2192	124	7	k(xmk−1	k(xmk−1	PROPN
cana-2192	124	8	,	,	PUNCT
cana-2192	124	9	xnk−1	xnk−1	PROPN
cana-2192	124	10	)	)	PUNCT
cana-2192	124	11	.	.	PUNCT
cana-2192	125	1	(	(	PUNCT
cana-2192	125	2	2.9	2.9	NUM
cana-2192	125	3	)	)	PUNCT
cana-2192	125	4	communications	communication	NOUN
cana-2192	125	5	on	on	ADP
cana-2192	125	6	applied	apply	VERB
cana-2192	125	7	nonlinear	nonlinear	ADJ
cana-2192	125	8	analysis	analysis	NOUN
cana-2192	125	9	issn	issn	NOUN
cana-2192	125	10	:	:	PUNCT
cana-2192	125	11	1074	1074	NUM
cana-2192	125	12	-	-	PUNCT
cana-2192	125	13	133x	133x	NUM
cana-2192	125	14	vol	vol	NOUN
cana-2192	125	15	32	32	NUM
cana-2192	125	16	no	no	NOUN
cana-2192	125	17	.	.	PUNCT
cana-2192	126	1	1s	1s	NUM
cana-2192	126	2	(	(	PUNCT
cana-2192	126	3	2025	2025	NUM
cana-2192	126	4	)	)	PUNCT
cana-2192	126	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-2192	126	6	357	357	NUM
cana-2192	126	7	generalized	generalize	VERB
cana-2192	126	8	α	α	NOUN
cana-2192	126	9	-	-	ADJ
cana-2192	126	10	admissible	admissible	ADJ
cana-2192	126	11	almost	almost	ADV
cana-2192	126	12	......	......	PUNCT
cana-2192	126	13	functions	function	NOUN
cana-2192	126	14	in	in	ADP
cana-2192	126	15	a	a	DET
cana-2192	126	16	metric	metric	ADJ
cana-2192	126	17	space	space	NOUN
cana-2192	126	18	also	also	ADV
cana-2192	126	19	,	,	PUNCT
cana-2192	126	20	k(xmk−1	k(xmk−1	PROPN
cana-2192	126	21	,	,	PUNCT
cana-2192	126	22	xnk−1	xnk−1	PROPN
cana-2192	126	23	)	)	PUNCT
cana-2192	126	24	=	=	SYM
cana-2192	126	25	β(e(xmk−1	β(e(xmk−1	X
cana-2192	126	26	,	,	PUNCT
cana-2192	126	27	xnk−1))e(xmk−1	xnk−1))e(xmk−1	NUM
cana-2192	126	28	,	,	PUNCT
cana-2192	126	29	xnk−1	xnk−1	PROPN
cana-2192	126	30	)	)	PUNCT
cana-2192	127	1	+	+	VERB
cana-2192	127	2	ln(xmk−1	ln(xmk−1	ADJ
cana-2192	127	3	,	,	PUNCT
cana-2192	127	4	xnk−1	xnk−1	PROPN
cana-2192	127	5	)	)	PUNCT
cana-2192	127	6	,	,	PUNCT
cana-2192	127	7	(	(	PUNCT
cana-2192	127	8	2.10	2.10	NUM
cana-2192	127	9	)	)	PUNCT
cana-2192	127	10	where	where	SCONJ
cana-2192	127	11	n(xmk−1	n(xmk−1	NOUN
cana-2192	127	12	,	,	PUNCT
cana-2192	127	13	xnk−1	xnk−1	PROPN
cana-2192	127	14	)	)	PUNCT
cana-2192	128	1	=	=	SYM
cana-2192	128	2	min{d(xmk−1	min{d(xmk−1	PROPN
cana-2192	128	3	,	,	PUNCT
cana-2192	128	4	xmk	xmk	PROPN
cana-2192	128	5	)	)	PUNCT
cana-2192	128	6	,	,	PUNCT
cana-2192	128	7	d(xnk−1	d(xnk−1	PROPN
cana-2192	128	8	,	,	PUNCT
cana-2192	128	9	xnk	xnk	PROPN
cana-2192	128	10	)	)	PUNCT
cana-2192	128	11	,	,	PUNCT
cana-2192	128	12	d(xmk−1	d(xmk−1	PROPN
cana-2192	128	13	,	,	PUNCT
cana-2192	128	14	xnk	xnk	PROPN
cana-2192	128	15	)	)	PUNCT
cana-2192	128	16	,	,	PUNCT
cana-2192	128	17	d(xnk−1	d(xnk−1	PROPN
cana-2192	128	18	,	,	PUNCT
cana-2192	128	19	xmk	xmk	PROPN
cana-2192	128	20	)	)	PUNCT
cana-2192	128	21	}	}	PUNCT
cana-2192	128	22	and	and	CCONJ
cana-2192	128	23	e(xmk−1	e(xmk−1	ADP
cana-2192	128	24	,	,	PUNCT
cana-2192	128	25	xnk−1	xnk−1	PROPN
cana-2192	128	26	)	)	PUNCT
cana-2192	128	27	=	=	SYM
cana-2192	128	28	d(xmk−1	d(xmk−1	PROPN
cana-2192	128	29	,	,	PUNCT
cana-2192	128	30	xnk−1	xnk−1	PROPN
cana-2192	128	31	)	)	PUNCT
cana-2192	129	1	+	+	CCONJ
cana-2192	129	2	|d(xmk−1	|d(xmk−1	ADJ
cana-2192	129	3	,	,	PUNCT
cana-2192	129	4	xmk	xmk	PROPN
cana-2192	129	5	)	)	PUNCT
cana-2192	129	6	−	−	PROPN
cana-2192	129	7	d(xnk−1	d(xnk−1	PROPN
cana-2192	129	8	,	,	PUNCT
cana-2192	129	9	xnk	xnk	PROPN
cana-2192	129	10	)	)	PUNCT
cana-2192	129	11	|	|	ADV
cana-2192	129	12	.	.	PUNCT
cana-2192	130	1	letting	let	VERB
cana-2192	130	2	k	k	X
cana-2192	130	3	→	→	SYM
cana-2192	130	4	∞	∞	PROPN
cana-2192	130	5	,	,	PUNCT
cana-2192	130	6	using	use	VERB
cana-2192	130	7	(	(	PUNCT
cana-2192	130	8	2.6	2.6	NUM
cana-2192	130	9	)	)	PUNCT
cana-2192	130	10	and	and	CCONJ
cana-2192	130	11	(	(	PUNCT
cana-2192	130	12	2.8	2.8	NUM
cana-2192	130	13	)	)	PUNCT
cana-2192	130	14	,	,	PUNCT
cana-2192	130	15	we	we	PRON
cana-2192	130	16	get	get	VERB
cana-2192	130	17	lim	lim	PROPN
cana-2192	130	18	k→∞	k→∞	PROPN
cana-2192	130	19	k(xmk−1	k(xmk−1	PROPN
cana-2192	130	20	,	,	PUNCT
cana-2192	130	21	xnk−1	xnk−1	PROPN
cana-2192	130	22	)	)	PUNCT
cana-2192	131	1	=	=	SYM
cana-2192	131	2	ϵ.	ϵ.	NOUN
cana-2192	131	3	(	(	PUNCT
cana-2192	131	4	2.11	2.11	NUM
cana-2192	131	5	)	)	PUNCT
cana-2192	131	6	by	by	ADP
cana-2192	131	7	(	(	PUNCT
cana-2192	131	8	2.8	2.8	NUM
cana-2192	131	9	)	)	PUNCT
cana-2192	131	10	,	,	PUNCT
cana-2192	131	11	(	(	PUNCT
cana-2192	131	12	2.9	2.9	NUM
cana-2192	131	13	)	)	PUNCT
cana-2192	131	14	,	,	PUNCT
cana-2192	131	15	(	(	PUNCT
cana-2192	131	16	2.11	2.11	NUM
cana-2192	131	17	)	)	PUNCT
cana-2192	131	18	and	and	CCONJ
cana-2192	131	19	the	the	DET
cana-2192	131	20	condition	condition	NOUN
cana-2192	131	21	(	(	PUNCT
cana-2192	131	22	ζ3	ζ3	NOUN
cana-2192	131	23	)	)	PUNCT
cana-2192	131	24	,	,	PUNCT
cana-2192	131	25	we	we	PRON
cana-2192	131	26	get	get	VERB
cana-2192	131	27	0	0	NUM
cana-2192	131	28	≤	≤	NOUN
cana-2192	131	29	lim	lim	PROPN
cana-2192	131	30	n→∞	n→∞	NUM
cana-2192	131	31	sup	sup	PROPN
cana-2192	131	32	ζ((α(xmk−1	ζ((α(xmk−1	PROPN
cana-2192	131	33	,	,	PUNCT
cana-2192	131	34	xmk	xmk	NOUN
cana-2192	131	35	)	)	PUNCT
cana-2192	131	36	α(xnk−1	α(xnk−1	NOUN
cana-2192	131	37	,	,	PUNCT
cana-2192	131	38	xnk	xnk	PROPN
cana-2192	131	39	)	)	PUNCT
cana-2192	131	40	d(xmk	d(xmk	PROPN
cana-2192	131	41	,	,	PUNCT
cana-2192	131	42	xnk	xnk	PROPN
cana-2192	131	43	)	)	PUNCT
cana-2192	131	44	,	,	PUNCT
cana-2192	131	45	k(xmk−1	k(xmk−1	PROPN
cana-2192	131	46	,	,	PUNCT
cana-2192	131	47	xnk−1	xnk−1	PROPN
cana-2192	131	48	)	)	PUNCT
cana-2192	131	49	)	)	PUNCT
cana-2192	132	1	<	<	X
cana-2192	132	2	0	0	X
cana-2192	132	3	.	.	PUNCT
cana-2192	133	1	this	this	PRON
cana-2192	133	2	is	be	AUX
cana-2192	133	3	a	a	DET
cana-2192	133	4	contradiction	contradiction	NOUN
cana-2192	133	5	.	.	PUNCT
cana-2192	134	1	hence	hence	ADV
cana-2192	134	2	{	{	PUNCT
cana-2192	134	3	xn	xn	X
cana-2192	134	4	}	}	PUNCT
cana-2192	134	5	is	be	AUX
cana-2192	134	6	a	a	DET
cana-2192	134	7	cauchy	cauchy	ADJ
cana-2192	134	8	sequence	sequence	NOUN
cana-2192	134	9	.	.	PUNCT
cana-2192	135	1	thus	thus	ADV
cana-2192	135	2	limm	limm	NOUN
cana-2192	135	3	,	,	PUNCT
cana-2192	135	4	n→∞	n→∞	X
cana-2192	135	5	d(xn	d(xn	PROPN
cana-2192	135	6	,	,	PUNCT
cana-2192	135	7	xm	xm	NOUN
cana-2192	135	8	)	)	PUNCT
cana-2192	135	9	exists	exist	VERB
cana-2192	135	10	and	and	CCONJ
cana-2192	135	11	is	be	AUX
cana-2192	135	12	equal	equal	ADJ
cana-2192	135	13	to	to	ADP
cana-2192	135	14	zero	zero	NUM
cana-2192	135	15	.	.	PUNCT
cana-2192	136	1	since	since	SCONJ
cana-2192	136	2	(	(	PUNCT
cana-2192	136	3	x	x	X
cana-2192	136	4	,	,	PUNCT
cana-2192	136	5	d	d	NOUN
cana-2192	136	6	)	)	PUNCT
cana-2192	136	7	is	be	AUX
cana-2192	136	8	complete	complete	ADJ
cana-2192	136	9	,	,	PUNCT
cana-2192	136	10	there	there	PRON
cana-2192	136	11	exists	exist	VERB
cana-2192	136	12	x∗	x∗	PROPN
cana-2192	136	13	∈	∈	PROPN
cana-2192	136	14	x	x	PUNCT
cana-2192	136	15	such	such	ADJ
cana-2192	136	16	that	that	SCONJ
cana-2192	136	17	lim	lim	PROPN
cana-2192	136	18	n→∞	n→∞	X
cana-2192	137	1	d(xn	d(xn	PROPN
cana-2192	137	2	,	,	PUNCT
cana-2192	137	3	x	x	SYM
cana-2192	137	4	∗	∗	NOUN
cana-2192	137	5	)	)	PUNCT
cana-2192	137	6	=	=	SYM
cana-2192	137	7	0	0	X
cana-2192	137	8	.	.	PUNCT
cana-2192	138	1	(	(	PUNCT
cana-2192	138	2	2.12	2.12	NUM
cana-2192	138	3	)	)	PUNCT
cana-2192	138	4	now	now	ADV
cana-2192	138	5	we	we	PRON
cana-2192	138	6	shall	shall	AUX
cana-2192	138	7	show	show	VERB
cana-2192	138	8	that	that	PRON
cana-2192	138	9	fx∗	fx∗	PROPN
cana-2192	139	1	=	=	PUNCT
cana-2192	139	2	x∗.	x∗.	PROPN
cana-2192	139	3	since	since	SCONJ
cana-2192	139	4	f	f	PROPN
cana-2192	139	5	is	be	AUX
cana-2192	139	6	continuous	continuous	ADJ
cana-2192	139	7	,	,	PUNCT
cana-2192	139	8	we	we	PRON
cana-2192	139	9	drive	drive	VERB
cana-2192	139	10	the	the	DET
cana-2192	139	11	desired	desire	VERB
cana-2192	139	12	results	result	NOUN
cana-2192	139	13	obviously	obviously	ADV
cana-2192	139	14	,	,	PUNCT
cana-2192	139	15	that	that	PRON
cana-2192	139	16	is	be	AUX
cana-2192	139	17	fx∗	fx∗	ADJ
cana-2192	140	1	=	=	PUNCT
cana-2192	140	2	f	f	X
cana-2192	140	3	(	(	PUNCT
cana-2192	140	4	lim	lim	PROPN
cana-2192	140	5	n→∞	n→∞	NUM
cana-2192	140	6	xn	xn	PUNCT
cana-2192	140	7	)	)	PUNCT
cana-2192	141	1	=	=	VERB
cana-2192	141	2	lim	lim	PROPN
cana-2192	141	3	n→∞	n→∞	NUM
cana-2192	141	4	f(xn	f(xn	PROPN
cana-2192	141	5	)	)	PUNCT
cana-2192	142	1	=	=	VERB
cana-2192	142	2	lim	lim	PROPN
cana-2192	142	3	n→∞	n→∞	X
cana-2192	142	4	xn+1	xn+1	PROPN
cana-2192	142	5	=	=	SYM
cana-2192	142	6	x∗.	x∗.	PROPN
cana-2192	142	7	suppose	suppose	VERB
cana-2192	142	8	we	we	PRON
cana-2192	142	9	have	have	VERB
cana-2192	142	10	(	(	PUNCT
cana-2192	142	11	iii	iii	NOUN
cana-2192	142	12	)	)	PUNCT
cana-2192	142	13	,	,	PUNCT
cana-2192	142	14	0	0	X
cana-2192	143	1	=	=	SYM
cana-2192	143	2	lim	lim	PROPN
cana-2192	143	3	m	m	PROPN
cana-2192	143	4	,	,	PUNCT
cana-2192	143	5	n→∞	n→∞	X
cana-2192	143	6	d(xm	d(xm	PROPN
cana-2192	143	7	,	,	PUNCT
cana-2192	143	8	xn	xn	PUNCT
cana-2192	143	9	)	)	PUNCT
cana-2192	143	10	=	=	VERB
cana-2192	144	1	lim	lim	PROPN
cana-2192	144	2	n→∞	n→∞	X
cana-2192	145	1	d(xn	d(xn	PROPN
cana-2192	145	2	,	,	PUNCT
cana-2192	145	3	x	x	SYM
cana-2192	145	4	∗	∗	NOUN
cana-2192	145	5	)	)	PUNCT
cana-2192	145	6	=	=	SYM
cana-2192	145	7	d(x∗	d(x∗	NOUN
cana-2192	145	8	,	,	PUNCT
cana-2192	145	9	x∗	x∗	PROPN
cana-2192	145	10	)	)	PUNCT
cana-2192	145	11	and	and	CCONJ
cana-2192	145	12	α(x∗	α(x∗	NOUN
cana-2192	145	13	,	,	PUNCT
cana-2192	145	14	fx∗	fx∗	PROPN
cana-2192	145	15	)	)	PUNCT
cana-2192	145	16	≥	≥	NOUN
cana-2192	145	17	1	1	NUM
cana-2192	145	18	.	.	PUNCT
cana-2192	146	1	moreover	moreover	ADV
cana-2192	146	2	,	,	PUNCT
cana-2192	146	3	0	0	NUM
cana-2192	146	4	≤	≤	NUM
cana-2192	146	5	ζ(α(xn	ζ(α(xn	NOUN
cana-2192	146	6	,	,	PUNCT
cana-2192	146	7	fxn)α(x	fxn)α(x	PROPN
cana-2192	146	8	∗	∗	NOUN
cana-2192	146	9	,	,	PUNCT
cana-2192	146	10	fx∗)d(fxn	fx∗)d(fxn	PROPN
cana-2192	146	11	,	,	PUNCT
cana-2192	146	12	fx	fx	NOUN
cana-2192	146	13	∗),k(xn	∗),k(xn	NUM
cana-2192	146	14	,	,	PUNCT
cana-2192	146	15	x	x	NOUN
cana-2192	146	16	∗	∗	NOUN
cana-2192	146	17	)	)	PUNCT
cana-2192	146	18	)	)	PUNCT
cana-2192	147	1	=	=	PUNCT
cana-2192	147	2	ζ(α(xn	ζ(α(xn	X
cana-2192	147	3	,	,	PUNCT
cana-2192	147	4	xn+1)α(x	xn+1)α(x	PROPN
cana-2192	147	5	∗	∗	NOUN
cana-2192	147	6	,	,	PUNCT
cana-2192	147	7	fx∗)d(xn+1	fx∗)d(xn+1	PROPN
cana-2192	147	8	,	,	PUNCT
cana-2192	147	9	fx	fx	NOUN
cana-2192	147	10	∗),k(xn	∗),k(xn	NUM
cana-2192	147	11	,	,	PUNCT
cana-2192	147	12	x	x	NOUN
cana-2192	147	13	∗	∗	NOUN
cana-2192	147	14	)	)	PUNCT
cana-2192	147	15	)	)	PUNCT
cana-2192	147	16	<	<	X
cana-2192	147	17	k(xn	k(xn	X
cana-2192	147	18	,	,	PUNCT
cana-2192	147	19	x	x	X
cana-2192	147	20	∗)−	∗)−	ADP
cana-2192	147	21	α(xn	α(xn	PROPN
cana-2192	147	22	,	,	PUNCT
cana-2192	147	23	xn+1)α(x	xn+1)α(x	PROPN
cana-2192	147	24	∗	∗	NOUN
cana-2192	147	25	,	,	PUNCT
cana-2192	147	26	fx∗)d(xn+1	fx∗)d(xn+1	PROPN
cana-2192	147	27	,	,	PUNCT
cana-2192	147	28	fx	fx	NOUN
cana-2192	147	29	∗	∗	NOUN
cana-2192	147	30	)	)	PUNCT
cana-2192	147	31	,	,	PUNCT
cana-2192	147	32	(	(	PUNCT
cana-2192	147	33	2.13	2.13	NUM
cana-2192	147	34	)	)	PUNCT
cana-2192	147	35	where	where	SCONJ
cana-2192	147	36	k(xn	k(xn	X
cana-2192	147	37	,	,	PUNCT
cana-2192	147	38	x	x	NOUN
cana-2192	147	39	∗	∗	NOUN
cana-2192	147	40	)	)	PUNCT
cana-2192	147	41	=	=	SYM
cana-2192	148	1	β(e(xn	β(e(xn	NOUN
cana-2192	148	2	,	,	PUNCT
cana-2192	148	3	x	x	X
cana-2192	148	4	∗))e(xn	∗))e(xn	NOUN
cana-2192	148	5	,	,	PUNCT
cana-2192	148	6	x	x	NOUN
cana-2192	148	7	∗	∗	NOUN
cana-2192	148	8	)	)	PUNCT
cana-2192	149	1	+	+	CCONJ
cana-2192	149	2	ln(xn	ln(xn	NOUN
cana-2192	149	3	,	,	PUNCT
cana-2192	149	4	x	x	NOUN
cana-2192	149	5	∗	∗	NOUN
cana-2192	149	6	)	)	PUNCT
cana-2192	149	7	.	.	PUNCT
cana-2192	150	1	also	also	ADV
cana-2192	150	2	,	,	PUNCT
cana-2192	150	3	n(xn	n(xn	NUM
cana-2192	150	4	,	,	PUNCT
cana-2192	150	5	x	x	NOUN
cana-2192	150	6	∗	∗	NOUN
cana-2192	150	7	)	)	PUNCT
cana-2192	150	8	=	=	SYM
cana-2192	150	9	min{d(xn	min{d(xn	ADJ
cana-2192	150	10	,	,	PUNCT
cana-2192	150	11	fxn	fxn	NOUN
cana-2192	150	12	)	)	PUNCT
cana-2192	150	13	,	,	PUNCT
cana-2192	150	14	d(x∗	d(x∗	NOUN
cana-2192	150	15	,	,	PUNCT
cana-2192	150	16	fx∗	fx∗	PROPN
cana-2192	150	17	)	)	PUNCT
cana-2192	150	18	,	,	PUNCT
cana-2192	150	19	d(xn	d(xn	PROPN
cana-2192	150	20	,	,	PUNCT
cana-2192	150	21	fx∗	fx∗	ADJ
cana-2192	150	22	)	)	PUNCT
cana-2192	150	23	,	,	PUNCT
cana-2192	150	24	d(x∗	d(x∗	NOUN
cana-2192	150	25	,	,	PUNCT
cana-2192	150	26	fxn	fxn	NOUN
cana-2192	150	27	)	)	PUNCT
cana-2192	150	28	}	}	PUNCT
cana-2192	150	29	=	=	SYM
cana-2192	150	30	min{d(xn	min{d(xn	X
cana-2192	150	31	,	,	PUNCT
cana-2192	150	32	xn+1	xn+1	NUM
cana-2192	150	33	)	)	PUNCT
cana-2192	150	34	,	,	PUNCT
cana-2192	150	35	d(x	d(x	PROPN
cana-2192	150	36	∗	∗	NOUN
cana-2192	150	37	,	,	PUNCT
cana-2192	150	38	fx∗	fx∗	NUM
cana-2192	150	39	)	)	PUNCT
cana-2192	150	40	,	,	PUNCT
cana-2192	150	41	d(xn	d(xn	PROPN
cana-2192	150	42	,	,	PUNCT
cana-2192	150	43	fx	fx	ADP
cana-2192	150	44	∗	∗	NOUN
cana-2192	150	45	)	)	PUNCT
cana-2192	150	46	,	,	PUNCT
cana-2192	150	47	d(x∗	d(x∗	NOUN
cana-2192	150	48	,	,	PUNCT
cana-2192	150	49	xn+1	xn+1	NUM
cana-2192	150	50	)	)	PUNCT
cana-2192	150	51	}	}	PUNCT
cana-2192	150	52	=	=	SYM
cana-2192	150	53	0	0	X
cana-2192	150	54	.	.	PUNCT
cana-2192	151	1	(	(	PUNCT
cana-2192	151	2	2.14	2.14	NUM
cana-2192	151	3	)	)	PUNCT
cana-2192	151	4	and	and	CCONJ
cana-2192	151	5	e(xn	e(xn	PRON
cana-2192	151	6	,	,	PUNCT
cana-2192	151	7	x	x	NOUN
cana-2192	151	8	∗	∗	NOUN
cana-2192	151	9	)	)	PUNCT
cana-2192	151	10	=	=	PUNCT
cana-2192	152	1	d(xn	d(xn	ADJ
cana-2192	152	2	,	,	PUNCT
cana-2192	152	3	x	x	SYM
cana-2192	152	4	∗	∗	NOUN
cana-2192	152	5	)	)	PUNCT
cana-2192	153	1	+	+	CCONJ
cana-2192	153	2	|d(xn	|d(xn	PROPN
cana-2192	153	3	,	,	PUNCT
cana-2192	153	4	fxn)−	fxn)−	PROPN
cana-2192	153	5	d(x∗	d(x∗	NOUN
cana-2192	153	6	,	,	PUNCT
cana-2192	153	7	fx∗)|	fx∗)|	AUX
cana-2192	153	8	=	=	PUNCT
cana-2192	153	9	0	0	PUNCT
cana-2192	154	1	+	+	NUM
cana-2192	154	2	|0−	|0−	NOUN
cana-2192	154	3	d(x∗	d(x∗	NOUN
cana-2192	154	4	,	,	PUNCT
cana-2192	154	5	fx∗)|	fx∗)|	PROPN
cana-2192	154	6	=	=	SYM
cana-2192	154	7	d(x∗	d(x∗	PROPN
cana-2192	154	8	,	,	PUNCT
cana-2192	154	9	fx∗	fx∗	PROPN
cana-2192	154	10	)	)	PUNCT
cana-2192	154	11	,	,	PUNCT
cana-2192	154	12	as	as	SCONJ
cana-2192	154	13	n	n	PROPN
cana-2192	154	14	→	→	SYM
cana-2192	154	15	∞.	∞.	PROPN
cana-2192	154	16	(	(	PUNCT
cana-2192	154	17	2.15	2.15	NUM
cana-2192	154	18	)	)	PUNCT
cana-2192	154	19	communications	communication	NOUN
cana-2192	154	20	on	on	ADP
cana-2192	154	21	applied	apply	VERB
cana-2192	154	22	nonlinear	nonlinear	ADJ
cana-2192	154	23	analysis	analysis	NOUN
cana-2192	154	24	issn	issn	NOUN
cana-2192	154	25	:	:	PUNCT
cana-2192	154	26	1074	1074	NUM
cana-2192	154	27	-	-	PUNCT
cana-2192	154	28	133x	133x	NUM
cana-2192	154	29	vol	vol	NOUN
cana-2192	154	30	32	32	NUM
cana-2192	154	31	no	no	NOUN
cana-2192	154	32	.	.	PUNCT
cana-2192	155	1	1s	1s	NUM
cana-2192	155	2	(	(	PUNCT
cana-2192	155	3	2025	2025	NUM
cana-2192	155	4	)	)	PUNCT
cana-2192	155	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-2192	155	6	358	358	NUM
cana-2192	155	7	dipti	dipti	NOUN
cana-2192	155	8	,	,	PUNCT
cana-2192	155	9	anil	anil	PROPN
cana-2192	155	10	kumar	kumar	PROPN
cana-2192	155	11	dubey	dubey	PROPN
cana-2192	155	12	,	,	PUNCT
cana-2192	155	13	urmila	urmila	PROPN
cana-2192	155	14	mishra	mishra	PROPN
cana-2192	155	15	,	,	PUNCT
cana-2192	155	16	by	by	ADP
cana-2192	155	17	(	(	PUNCT
cana-2192	155	18	2.13	2.13	NUM
cana-2192	155	19	)	)	PUNCT
cana-2192	155	20	,	,	PUNCT
cana-2192	155	21	(	(	PUNCT
cana-2192	155	22	2.14	2.14	NUM
cana-2192	155	23	)	)	PUNCT
cana-2192	155	24	and	and	CCONJ
cana-2192	155	25	(	(	PUNCT
cana-2192	155	26	2.15	2.15	NUM
cana-2192	155	27	)	)	PUNCT
cana-2192	155	28	,	,	PUNCT
cana-2192	155	29	we	we	PRON
cana-2192	155	30	get	get	VERB
cana-2192	155	31	d(xn+1	d(xn+1	PROPN
cana-2192	155	32	,	,	PUNCT
cana-2192	155	33	fx	fx	NOUN
cana-2192	155	34	∗	∗	NOUN
cana-2192	155	35	)	)	PUNCT
cana-2192	156	1	=	=	SYM
cana-2192	156	2	d(fxn	d(fxn	PROPN
cana-2192	156	3	,	,	PUNCT
cana-2192	156	4	fx	fx	NOUN
cana-2192	156	5	∗	∗	NOUN
cana-2192	156	6	)	)	PUNCT
cana-2192	156	7	≤	≤	NOUN
cana-2192	156	8	α(xn	α(xn	NUM
cana-2192	156	9	,	,	PUNCT
cana-2192	156	10	xn+1)α(x	xn+1)α(x	PROPN
cana-2192	156	11	∗	∗	NOUN
cana-2192	156	12	,	,	PUNCT
cana-2192	156	13	fx∗)d(fxn	fx∗)d(fxn	PROPN
cana-2192	156	14	,	,	PUNCT
cana-2192	156	15	fx	fx	NOUN
cana-2192	156	16	∗	∗	NOUN
cana-2192	156	17	)	)	PUNCT
cana-2192	156	18	<	<	X
cana-2192	156	19	d(x∗	d(x∗	PROPN
cana-2192	156	20	,	,	PUNCT
cana-2192	156	21	fx∗	fx∗	PROPN
cana-2192	156	22	)	)	PUNCT
cana-2192	156	23	.	.	PUNCT
cana-2192	157	1	(	(	PUNCT
cana-2192	157	2	2.16	2.16	NUM
cana-2192	157	3	)	)	PUNCT
cana-2192	157	4	by	by	ADP
cana-2192	157	5	letting	let	VERB
cana-2192	157	6	n	n	PRON
cana-2192	157	7	→	→	SYM
cana-2192	157	8	∞	∞	NUM
cana-2192	157	9	in	in	ADP
cana-2192	157	10	(	(	PUNCT
cana-2192	157	11	2.13	2.13	NUM
cana-2192	157	12	)	)	PUNCT
cana-2192	157	13	,	,	PUNCT
cana-2192	157	14	together	together	ADV
cana-2192	157	15	with	with	ADP
cana-2192	157	16	the	the	DET
cana-2192	157	17	observation	observation	NOUN
cana-2192	157	18	above	above	ADV
cana-2192	157	19	,	,	PUNCT
cana-2192	157	20	we	we	PRON
cana-2192	157	21	have	have	VERB
cana-2192	157	22	0	0	NUM
cana-2192	157	23	≤	≤	NUM
cana-2192	157	24	lim	lim	PROPN
cana-2192	157	25	n→∞	n→∞	NUM
cana-2192	157	26	supζ(α(xn	supζ(α(xn	PROPN
cana-2192	157	27	,	,	PUNCT
cana-2192	157	28	fxn)α(x	fxn)α(x	PROPN
cana-2192	157	29	∗	∗	NOUN
cana-2192	157	30	,	,	PUNCT
cana-2192	157	31	fx∗)d(fxn	fx∗)d(fxn	PROPN
cana-2192	157	32	,	,	PUNCT
cana-2192	157	33	fx	fx	NOUN
cana-2192	157	34	∗),k(xn	∗),k(xn	NUM
cana-2192	157	35	,	,	PUNCT
cana-2192	157	36	x	x	NOUN
cana-2192	157	37	∗	∗	NOUN
cana-2192	157	38	)	)	PUNCT
cana-2192	157	39	)	)	PUNCT
cana-2192	158	1	<	<	X
cana-2192	158	2	0	0	X
cana-2192	158	3	.	.	PUNCT
cana-2192	159	1	this	this	PRON
cana-2192	159	2	is	be	AUX
cana-2192	159	3	a	a	DET
cana-2192	159	4	contradiction	contradiction	NOUN
cana-2192	159	5	.	.	PUNCT
cana-2192	160	1	hence	hence	ADV
cana-2192	160	2	,	,	PUNCT
cana-2192	160	3	therefore	therefore	ADV
cana-2192	160	4	x∗	x∗	PROPN
cana-2192	160	5	is	be	AUX
cana-2192	160	6	a	a	DET
cana-2192	160	7	fixed	fix	VERB
cana-2192	160	8	point	point	NOUN
cana-2192	160	9	of	of	ADP
cana-2192	160	10	f	f	PROPN
cana-2192	160	11	i.e.	i.e.	X
cana-2192	160	12	fx∗	fx∗	X
cana-2192	161	1	=	=	PUNCT
cana-2192	161	2	x∗.	x∗.	PROPN
cana-2192	161	3	suppose	suppose	VERB
cana-2192	161	4	that	that	SCONJ
cana-2192	161	5	x∗	x∗	PROPN
cana-2192	161	6	and	and	CCONJ
cana-2192	161	7	u∗	u∗	ADV
cana-2192	161	8	be	be	AUX
cana-2192	161	9	two	two	NUM
cana-2192	161	10	fixed	fix	VERB
cana-2192	161	11	points	point	NOUN
cana-2192	161	12	of	of	ADP
cana-2192	161	13	f	f	PROPN
cana-2192	161	14	and	and	CCONJ
cana-2192	161	15	hence	hence	ADV
cana-2192	161	16	x∗	x∗	PROPN
cana-2192	161	17	,	,	PUNCT
cana-2192	161	18	u∗	u∗	PROPN
cana-2192	161	19	∈	∈	PROPN
cana-2192	161	20	fix(f	fix(f	PROPN
cana-2192	161	21	)	)	PUNCT
cana-2192	161	22	which	which	PRON
cana-2192	161	23	is	be	AUX
cana-2192	161	24	a	a	DET
cana-2192	161	25	generalized	generalized	ADJ
cana-2192	161	26	α	α	NOUN
cana-2192	161	27	-	-	ADJ
cana-2192	161	28	admissible	admissible	ADJ
cana-2192	161	29	almost	almost	ADV
cana-2192	161	30	z	z	NOUN
cana-2192	161	31	-	-	PUNCT
cana-2192	161	32	contraction	contraction	NOUN
cana-2192	161	33	self	self	NOUN
cana-2192	161	34	mappings	mapping	NOUN
cana-2192	161	35	of	of	ADP
cana-2192	161	36	a	a	DET
cana-2192	161	37	metric	metric	ADJ
cana-2192	161	38	space	space	NOUN
cana-2192	161	39	(	(	PUNCT
cana-2192	161	40	x	x	X
cana-2192	161	41	,	,	PUNCT
cana-2192	161	42	d	d	NOUN
cana-2192	161	43	)	)	PUNCT
cana-2192	161	44	.	.	PUNCT
cana-2192	162	1	by	by	ADP
cana-2192	162	2	(	(	PUNCT
cana-2192	162	3	2.1	2.1	NUM
cana-2192	162	4	)	)	PUNCT
cana-2192	162	5	,	,	PUNCT
cana-2192	162	6	we	we	PRON
cana-2192	162	7	have	have	VERB
cana-2192	162	8	that	that	PRON
cana-2192	162	9	0	0	NUM
cana-2192	162	10	≤	≤	NUM
cana-2192	162	11	ζ(α(x∗	ζ(α(x∗	PROPN
cana-2192	162	12	,	,	PUNCT
cana-2192	162	13	fx∗)α(u∗	fx∗)α(u∗	PROPN
cana-2192	162	14	,	,	PUNCT
cana-2192	162	15	fu∗)d(fx∗	fu∗)d(fx∗	ADJ
cana-2192	162	16	,	,	PUNCT
cana-2192	162	17	fu∗),k(x∗	fu∗),k(x∗	NOUN
cana-2192	162	18	,	,	PUNCT
cana-2192	162	19	u∗	u∗	PROPN
cana-2192	162	20	)	)	PUNCT
cana-2192	162	21	)	)	PUNCT
cana-2192	162	22	,	,	PUNCT
cana-2192	162	23	(	(	PUNCT
cana-2192	162	24	2.17	2.17	NUM
cana-2192	162	25	)	)	PUNCT
cana-2192	162	26	where	where	SCONJ
cana-2192	162	27	k(x∗	k(x∗	NOUN
cana-2192	162	28	,	,	PUNCT
cana-2192	162	29	u∗	u∗	PROPN
cana-2192	162	30	)	)	PUNCT
cana-2192	162	31	=	=	SYM
cana-2192	162	32	β(e(x∗	β(e(x∗	PROPN
cana-2192	162	33	,	,	PUNCT
cana-2192	162	34	u∗))e(x∗	u∗))e(x∗	NOUN
cana-2192	162	35	,	,	PUNCT
cana-2192	162	36	u∗	u∗	ADJ
cana-2192	162	37	)	)	PUNCT
cana-2192	163	1	+	+	NUM
cana-2192	163	2	ln(x∗	ln(x∗	NOUN
cana-2192	163	3	,	,	PUNCT
cana-2192	163	4	u∗	u∗	PROPN
cana-2192	163	5	)	)	PUNCT
cana-2192	163	6	.	.	PUNCT
cana-2192	164	1	(	(	PUNCT
cana-2192	164	2	2.18	2.18	NUM
cana-2192	164	3	)	)	PUNCT
cana-2192	164	4	also	also	ADV
cana-2192	164	5	e(x∗	e(x∗	ADV
cana-2192	164	6	,	,	PUNCT
cana-2192	164	7	u∗	u∗	ADJ
cana-2192	164	8	)	)	PUNCT
cana-2192	164	9	=	=	SYM
cana-2192	164	10	d(x∗	d(x∗	NOUN
cana-2192	164	11	,	,	PUNCT
cana-2192	164	12	u∗	u∗	ADJ
cana-2192	164	13	)	)	PUNCT
cana-2192	165	1	+	+	CCONJ
cana-2192	165	2	|d(x∗	|d(x∗	ADV
cana-2192	165	3	,	,	PUNCT
cana-2192	165	4	fx∗)−	fx∗)−	NOUN
cana-2192	165	5	d(u∗	d(u∗	NOUN
cana-2192	165	6	,	,	PUNCT
cana-2192	165	7	fu∗)|	fu∗)|	PROPN
cana-2192	165	8	=	=	PUNCT
cana-2192	165	9	d(x∗	d(x∗	NOUN
cana-2192	165	10	,	,	PUNCT
cana-2192	165	11	u∗	u∗	PROPN
cana-2192	165	12	)	)	PUNCT
cana-2192	165	13	(	(	PUNCT
cana-2192	165	14	2.19	2.19	NUM
cana-2192	165	15	)	)	PUNCT
cana-2192	165	16	and	and	CCONJ
cana-2192	165	17	n(x∗	n(x∗	NOUN
cana-2192	165	18	,	,	PUNCT
cana-2192	165	19	u∗	u∗	ADJ
cana-2192	165	20	)	)	PUNCT
cana-2192	165	21	=	=	SYM
cana-2192	165	22	min{d(x∗	min{d(x∗	PROPN
cana-2192	165	23	,	,	PUNCT
cana-2192	165	24	fx∗	fx∗	PROPN
cana-2192	165	25	)	)	PUNCT
cana-2192	165	26	,	,	PUNCT
cana-2192	165	27	d(u∗	d(u∗	NUM
cana-2192	165	28	,	,	PUNCT
cana-2192	165	29	fu∗	fu∗	ADJ
cana-2192	165	30	)	)	PUNCT
cana-2192	165	31	,	,	PUNCT
cana-2192	165	32	d(x∗	d(x∗	NOUN
cana-2192	165	33	,	,	PUNCT
cana-2192	165	34	fu∗	fu∗	ADJ
cana-2192	165	35	)	)	PUNCT
cana-2192	165	36	,	,	PUNCT
cana-2192	165	37	d(u∗	d(u∗	NUM
cana-2192	165	38	,	,	PUNCT
cana-2192	165	39	fx∗	fx∗	ADJ
cana-2192	165	40	)	)	PUNCT
cana-2192	165	41	}	}	PUNCT
cana-2192	165	42	=	=	SYM
cana-2192	165	43	0	0	X
cana-2192	165	44	.	.	PUNCT
cana-2192	165	45	(	(	PUNCT
cana-2192	165	46	2.20	2.20	NUM
cana-2192	165	47	)	)	PUNCT
cana-2192	165	48	therefore	therefore	ADV
cana-2192	165	49	,	,	PUNCT
cana-2192	165	50	from	from	ADP
cana-2192	165	51	(	(	PUNCT
cana-2192	165	52	2.17	2.17	NUM
cana-2192	165	53	)	)	PUNCT
cana-2192	165	54	,	,	PUNCT
cana-2192	165	55	(	(	PUNCT
cana-2192	165	56	2.18	2.18	NUM
cana-2192	165	57	)	)	PUNCT
cana-2192	165	58	,	,	PUNCT
cana-2192	165	59	(	(	PUNCT
cana-2192	165	60	2.19	2.19	NUM
cana-2192	165	61	)	)	PUNCT
cana-2192	165	62	and	and	CCONJ
cana-2192	165	63	(	(	PUNCT
cana-2192	165	64	2.20	2.20	NUM
cana-2192	165	65	)	)	PUNCT
cana-2192	165	66	we	we	PRON
cana-2192	165	67	get	get	VERB
cana-2192	165	68	that	that	PRON
cana-2192	165	69	0	0	NUM
cana-2192	165	70	≤	≤	NUM
cana-2192	165	71	ζ(α(x∗	ζ(α(x∗	PROPN
cana-2192	165	72	,	,	PUNCT
cana-2192	165	73	fx∗)α(u∗	fx∗)α(u∗	PROPN
cana-2192	165	74	,	,	PUNCT
cana-2192	165	75	fu∗)d(fx∗	fu∗)d(fx∗	ADJ
cana-2192	165	76	,	,	PUNCT
cana-2192	165	77	fu∗	fu∗	ADJ
cana-2192	165	78	)	)	PUNCT
cana-2192	165	79	,	,	PUNCT
cana-2192	165	80	d(x∗	d(x∗	NOUN
cana-2192	165	81	,	,	PUNCT
cana-2192	165	82	u∗	u∗	PROPN
cana-2192	165	83	)	)	PUNCT
cana-2192	165	84	)	)	PUNCT
cana-2192	166	1	=	=	SYM
cana-2192	166	2	ζ(α(x∗	ζ(α(x∗	PROPN
cana-2192	166	3	,	,	PUNCT
cana-2192	166	4	x∗)α(u∗	x∗)α(u∗	PROPN
cana-2192	166	5	,	,	PUNCT
cana-2192	166	6	u∗)d(x∗	u∗)d(x∗	PROPN
cana-2192	166	7	,	,	PUNCT
cana-2192	166	8	u∗	u∗	PROPN
cana-2192	166	9	)	)	PUNCT
cana-2192	166	10	,	,	PUNCT
cana-2192	166	11	d(x∗	d(x∗	NOUN
cana-2192	166	12	,	,	PUNCT
cana-2192	166	13	u∗	u∗	PROPN
cana-2192	166	14	)	)	PUNCT
cana-2192	166	15	)	)	PUNCT
cana-2192	166	16	.	.	PUNCT
cana-2192	167	1	this	this	PRON
cana-2192	167	2	is	be	AUX
cana-2192	167	3	a	a	DET
cana-2192	167	4	contradiction	contradiction	NOUN
cana-2192	167	5	.	.	PUNCT
cana-2192	168	1	thus	thus	ADV
cana-2192	168	2	,	,	PUNCT
cana-2192	168	3	we	we	PRON
cana-2192	168	4	have	have	VERB
cana-2192	168	5	x∗	x∗	PROPN
cana-2192	168	6	=	=	PRON
cana-2192	168	7	u∗.	u∗.	PROPN
cana-2192	168	8	hence	hence	ADV
cana-2192	168	9	f	f	PROPN
cana-2192	168	10	is	be	AUX
cana-2192	168	11	a	a	DET
cana-2192	168	12	unique	unique	ADJ
cana-2192	168	13	fixed	fix	VERB
cana-2192	168	14	point	point	NOUN
cana-2192	168	15	.	.	PUNCT
cana-2192	169	1	□	□	PUNCT
cana-2192	169	2	theorem	theorem	ADJ
cana-2192	169	3	10	10	NUM
cana-2192	169	4	.	.	PUNCT
cana-2192	170	1	let	let	AUX
cana-2192	170	2	(	(	PUNCT
cana-2192	170	3	x	x	NOUN
cana-2192	170	4	,	,	PUNCT
cana-2192	170	5	d	d	NOUN
cana-2192	170	6	)	)	PUNCT
cana-2192	170	7	be	be	AUX
cana-2192	170	8	a	a	DET
cana-2192	170	9	complete	complete	ADJ
cana-2192	170	10	metric	metric	ADJ
cana-2192	170	11	space	space	NOUN
cana-2192	170	12	,	,	PUNCT
cana-2192	170	13	f	f	PROPN
cana-2192	170	14	is	be	AUX
cana-2192	170	15	a	a	DET
cana-2192	170	16	generalized	generalized	ADJ
cana-2192	170	17	α	α	NOUN
cana-2192	170	18	-	-	ADJ
cana-2192	170	19	admissible	admissible	ADJ
cana-2192	170	20	almost	almost	ADV
cana-2192	170	21	z	z	NOUN
cana-2192	170	22	-	-	PUNCT
cana-2192	170	23	contraction	contraction	NOUN
cana-2192	170	24	with	with	ADP
cana-2192	170	25	respect	respect	NOUN
cana-2192	170	26	to	to	ADP
cana-2192	170	27	ζ	ζ	PROPN
cana-2192	170	28	.	.	PUNCT
cana-2192	170	29	assume	assume	VERB
cana-2192	170	30	that	that	SCONJ
cana-2192	170	31	(	(	PUNCT
cana-2192	170	32	i	i	NOUN
cana-2192	170	33	)	)	PUNCT
cana-2192	170	34	f	f	PROPN
cana-2192	170	35	is	be	AUX
cana-2192	170	36	a	a	DET
cana-2192	170	37	α	α	NOUN
cana-2192	170	38	-	-	ADJ
cana-2192	170	39	admissible	admissible	ADJ
cana-2192	170	40	,	,	PUNCT
cana-2192	170	41	(	(	PUNCT
cana-2192	170	42	ii	ii	NOUN
cana-2192	170	43	)	)	PUNCT
cana-2192	170	44	there	there	PRON
cana-2192	170	45	exists	exist	VERB
cana-2192	170	46	x0	x0	PROPN
cana-2192	170	47	∈	∈	PROPN
cana-2192	170	48	x	x	PUNCT
cana-2192	170	49	such	such	ADJ
cana-2192	170	50	that	that	DET
cana-2192	170	51	α(x0	α(x0	ADJ
cana-2192	170	52	,	,	PUNCT
cana-2192	170	53	fx0	fx0	PROPN
cana-2192	170	54	)	)	PUNCT
cana-2192	170	55	≥	≥	NOUN
cana-2192	170	56	1	1	NUM
cana-2192	170	57	,	,	PUNCT
cana-2192	170	58	(	(	PUNCT
cana-2192	170	59	iii	iii	NOUN
cana-2192	170	60	)	)	PUNCT
cana-2192	170	61	x	x	X
cana-2192	170	62	is	be	AUX
cana-2192	170	63	a	a	DET
cana-2192	170	64	regular	regular	ADJ
cana-2192	170	65	and	and	CCONJ
cana-2192	170	66	for	for	ADP
cana-2192	170	67	every	every	DET
cana-2192	170	68	sequence	sequence	NOUN
cana-2192	170	69	{	{	PUNCT
cana-2192	170	70	xn	xn	NOUN
cana-2192	170	71	}	}	PUNCT
cana-2192	170	72	in	in	ADP
cana-2192	170	73	x	x	SYM
cana-2192	170	74	such	such	ADJ
cana-2192	170	75	that	that	SCONJ
cana-2192	170	76	α(xn	α(xn	NOUN
cana-2192	170	77	,	,	PUNCT
cana-2192	170	78	xn+1	xn+1	NUM
cana-2192	170	79	)	)	PUNCT
cana-2192	170	80	≥	≥	NOUN
cana-2192	170	81	1	1	NUM
cana-2192	170	82	for	for	ADP
cana-2192	170	83	all	all	PRON
cana-2192	170	84	n	n	PRON
cana-2192	170	85	∈	∈	NOUN
cana-2192	170	86	n	n	NOUN
cana-2192	170	87	∪	∪	X
cana-2192	170	88	{	{	PUNCT
cana-2192	170	89	0	0	NUM
cana-2192	170	90	}	}	PUNCT
cana-2192	170	91	and	and	CCONJ
cana-2192	170	92	we	we	PRON
cana-2192	170	93	have	have	VERB
cana-2192	170	94	α(xm	α(xm	NUM
cana-2192	170	95	,	,	PUNCT
cana-2192	170	96	xn	xn	PROPN
cana-2192	170	97	)	)	PUNCT
cana-2192	170	98	≥	≥	NOUN
cana-2192	170	99	1	1	NUM
cana-2192	170	100	for	for	ADP
cana-2192	170	101	all	all	DET
cana-2192	170	102	m	m	PROPN
cana-2192	170	103	,	,	PUNCT
cana-2192	170	104	n	n	PROPN
cana-2192	170	105	∈	∈	PROPN
cana-2192	170	106	n	n	X
cana-2192	170	107	with	with	ADP
cana-2192	170	108	m	m	PROPN
cana-2192	170	109	<	<	X
cana-2192	170	110	n	n	CCONJ
cana-2192	170	111	,	,	PUNCT
cana-2192	170	112	(	(	PUNCT
cana-2192	170	113	iv	iv	X
cana-2192	170	114	)	)	PUNCT
cana-2192	170	115	α(x	α(x	PROPN
cana-2192	170	116	,	,	PUNCT
cana-2192	170	117	y	y	PROPN
cana-2192	170	118	)	)	PUNCT
cana-2192	170	119	≥	≥	NOUN
cana-2192	170	120	1	1	NUM
cana-2192	170	121	,	,	PUNCT
cana-2192	170	122	for	for	ADP
cana-2192	170	123	all	all	DET
cana-2192	170	124	x	x	NOUN
cana-2192	170	125	,	,	PUNCT
cana-2192	170	126	y	y	PROPN
cana-2192	170	127	∈	∈	PROPN
cana-2192	170	128	fix(f	fix(f	PROPN
cana-2192	170	129	)	)	PUNCT
cana-2192	170	130	.	.	PUNCT
cana-2192	171	1	then	then	ADV
cana-2192	171	2	f	f	PROPN
cana-2192	171	3	has	have	VERB
cana-2192	171	4	a	a	DET
cana-2192	171	5	unique	unique	ADJ
cana-2192	171	6	fixed	fix	VERB
cana-2192	171	7	point	point	NOUN
cana-2192	171	8	x∗	x∗	PROPN
cana-2192	171	9	in	in	ADP
cana-2192	171	10	x.	x.	NOUN
cana-2192	171	11	proof	proof	NOUN
cana-2192	171	12	.	.	PUNCT
cana-2192	172	1	by	by	ADP
cana-2192	172	2	(	(	PUNCT
cana-2192	172	3	ii	ii	NOUN
cana-2192	172	4	)	)	PUNCT
cana-2192	172	5	,	,	PUNCT
cana-2192	172	6	let	let	VERB
cana-2192	172	7	x0	x0	PROPN
cana-2192	172	8	∈	∈	PROPN
cana-2192	172	9	x	x	PUNCT
cana-2192	172	10	such	such	ADJ
cana-2192	172	11	that	that	DET
cana-2192	172	12	α(x0	α(x0	ADJ
cana-2192	172	13	,	,	PUNCT
cana-2192	172	14	fx0	fx0	PROPN
cana-2192	172	15	)	)	PUNCT
cana-2192	172	16	≥	≥	NOUN
cana-2192	172	17	1	1	NUM
cana-2192	172	18	.	.	PUNCT
cana-2192	173	1	there	there	PRON
cana-2192	173	2	exist	exist	VERB
cana-2192	173	3	xn	xn	X
cana-2192	173	4	∈	∈	PROPN
cana-2192	173	5	x	x	PUNCT
cana-2192	174	1	such	such	ADJ
cana-2192	174	2	that	that	PRON
cana-2192	174	3	xn	xn	PROPN
cana-2192	175	1	=	=	PUNCT
cana-2192	176	1	fxn−1	fxn−1	PROPN
cana-2192	176	2	for	for	ADP
cana-2192	176	3	all	all	PRON
cana-2192	176	4	n	n	DET
cana-2192	176	5	∈	∈	PROPN
cana-2192	176	6	n.	n.	NOUN
cana-2192	176	7	we	we	PRON
cana-2192	176	8	have	have	VERB
cana-2192	176	9	by	by	ADP
cana-2192	176	10	theorem	theorem	NOUN
cana-2192	176	11	9	9	NUM
cana-2192	176	12	,	,	PUNCT
cana-2192	176	13	{	{	PUNCT
cana-2192	176	14	xn	xn	X
cana-2192	176	15	}	}	PUNCT
cana-2192	176	16	is	be	AUX
cana-2192	176	17	a	a	DET
cana-2192	176	18	cauchy	cauchy	ADJ
cana-2192	176	19	sequence	sequence	NOUN
cana-2192	176	20	such	such	ADJ
cana-2192	176	21	that	that	SCONJ
cana-2192	176	22	limn→∞	limn→∞	PROPN
cana-2192	176	23	d(xnxn+1	d(xnxn+1	PROPN
cana-2192	176	24	)	)	PUNCT
cana-2192	176	25	=	=	SYM
cana-2192	177	1	0	0	X
cana-2192	177	2	.	.	PUNCT
cana-2192	178	1	thus	thus	ADV
cana-2192	178	2	limm	limm	NOUN
cana-2192	178	3	,	,	PUNCT
cana-2192	178	4	n→∞	n→∞	X
cana-2192	178	5	d(xn	d(xn	PROPN
cana-2192	178	6	,	,	PUNCT
cana-2192	178	7	xm	xm	NOUN
cana-2192	178	8	)	)	PUNCT
cana-2192	178	9	exists	exist	VERB
cana-2192	178	10	and	and	CCONJ
cana-2192	178	11	is	be	AUX
cana-2192	178	12	equal	equal	ADJ
cana-2192	178	13	to	to	ADP
cana-2192	178	14	0	0	NUM
cana-2192	178	15	.	.	PUNCT
cana-2192	179	1	since	since	SCONJ
cana-2192	179	2	(	(	PUNCT
cana-2192	179	3	x	x	X
cana-2192	179	4	,	,	PUNCT
cana-2192	179	5	d	d	NOUN
cana-2192	179	6	)	)	PUNCT
cana-2192	179	7	is	be	AUX
cana-2192	179	8	complete	complete	ADJ
cana-2192	179	9	,	,	PUNCT
cana-2192	179	10	there	there	PRON
cana-2192	179	11	exists	exist	VERB
cana-2192	179	12	x∗	x∗	PROPN
cana-2192	179	13	∈	∈	PROPN
cana-2192	179	14	x	x	PUNCT
cana-2192	179	15	such	such	ADJ
cana-2192	179	16	that	that	SCONJ
cana-2192	179	17	lim	lim	PROPN
cana-2192	179	18	n→∞	n→∞	X
cana-2192	180	1	d(xn	d(xn	PROPN
cana-2192	180	2	,	,	PUNCT
cana-2192	180	3	x	x	SYM
cana-2192	180	4	∗	∗	NOUN
cana-2192	180	5	)	)	PUNCT
cana-2192	180	6	=	=	SYM
cana-2192	180	7	0	0	NUM
cana-2192	180	8	,	,	PUNCT
cana-2192	180	9	(	(	PUNCT
cana-2192	180	10	2.21	2.21	NUM
cana-2192	180	11	)	)	PUNCT
cana-2192	180	12	communications	communication	NOUN
cana-2192	180	13	on	on	ADP
cana-2192	180	14	applied	apply	VERB
cana-2192	180	15	nonlinear	nonlinear	ADJ
cana-2192	180	16	analysis	analysis	NOUN
cana-2192	180	17	issn	issn	NOUN
cana-2192	180	18	:	:	PUNCT
cana-2192	180	19	1074	1074	NUM
cana-2192	180	20	-	-	PUNCT
cana-2192	180	21	133x	133x	NUM
cana-2192	180	22	vol	vol	NOUN
cana-2192	180	23	32	32	NUM
cana-2192	180	24	no	no	NOUN
cana-2192	180	25	.	.	PUNCT
cana-2192	181	1	1s	1s	NUM
cana-2192	181	2	(	(	PUNCT
cana-2192	181	3	2025	2025	NUM
cana-2192	181	4	)	)	PUNCT
cana-2192	182	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2192	182	2	359	359	NUM
cana-2192	182	3	generalized	generalize	VERB
cana-2192	182	4	α	α	NOUN
cana-2192	182	5	-	-	ADJ
cana-2192	182	6	admissible	admissible	ADJ
cana-2192	182	7	almost	almost	ADV
cana-2192	182	8	......	......	PUNCT
cana-2192	182	9	functions	function	NOUN
cana-2192	182	10	in	in	ADP
cana-2192	182	11	a	a	DET
cana-2192	182	12	metric	metric	ADJ
cana-2192	182	13	space	space	NOUN
cana-2192	182	14	then	then	ADV
cana-2192	182	15	lim	lim	PROPN
cana-2192	182	16	m	m	PROPN
cana-2192	182	17	,	,	PUNCT
cana-2192	182	18	n→∞	n→∞	X
cana-2192	182	19	d(xm	d(xm	PROPN
cana-2192	182	20	,	,	PUNCT
cana-2192	182	21	xn	xn	PUNCT
cana-2192	182	22	)	)	PUNCT
cana-2192	183	1	=	=	VERB
cana-2192	183	2	lim	lim	PROPN
cana-2192	183	3	n→∞	n→∞	X
cana-2192	184	1	d(xn	d(xn	PROPN
cana-2192	184	2	,	,	PUNCT
cana-2192	184	3	x	x	SYM
cana-2192	184	4	∗	∗	NOUN
cana-2192	184	5	)	)	PUNCT
cana-2192	184	6	=	=	SYM
cana-2192	184	7	d(x∗	d(x∗	NOUN
cana-2192	184	8	,	,	PUNCT
cana-2192	184	9	x∗	x∗	PROPN
cana-2192	184	10	)	)	PUNCT
cana-2192	184	11	=	=	SYM
cana-2192	185	1	0	0	X
cana-2192	185	2	.	.	PUNCT
cana-2192	186	1	since	since	SCONJ
cana-2192	186	2	x	x	PRON
cana-2192	186	3	is	be	AUX
cana-2192	186	4	regular	regular	ADJ
cana-2192	186	5	,	,	PUNCT
cana-2192	186	6	therefore	therefore	ADV
cana-2192	186	7	there	there	PRON
cana-2192	186	8	exists	exist	VERB
cana-2192	186	9	a	a	DET
cana-2192	186	10	subsequence	subsequence	NOUN
cana-2192	186	11	{	{	PUNCT
cana-2192	186	12	xnk	xnk	PROPN
cana-2192	186	13	}	}	PUNCT
cana-2192	186	14	of	of	ADP
cana-2192	186	15	{	{	PUNCT
cana-2192	186	16	xn	xn	NOUN
cana-2192	186	17	}	}	PUNCT
cana-2192	186	18	such	such	ADJ
cana-2192	186	19	that	that	SCONJ
cana-2192	186	20	α(xnk	α(xnk	NOUN
cana-2192	186	21	,	,	PUNCT
cana-2192	186	22	x∗	x∗	PROPN
cana-2192	186	23	)	)	PUNCT
cana-2192	186	24	≥	≥	NOUN
cana-2192	186	25	1	1	NUM
cana-2192	186	26	for	for	ADP
cana-2192	186	27	all	all	DET
cana-2192	186	28	k	k	PROPN
cana-2192	186	29	∈	∈	PROPN
cana-2192	186	30	n.	n.	NOUN
cana-2192	186	31	therefore	therefore	ADV
cana-2192	186	32	0	0	NUM
cana-2192	186	33	≤	≤	NUM
cana-2192	186	34	ζ(α(xnk	ζ(α(xnk	NOUN
cana-2192	186	35	,	,	PUNCT
cana-2192	186	36	fxnk	fxnk	NOUN
cana-2192	186	37	)	)	PUNCT
cana-2192	186	38	α(x∗	α(x∗	NOUN
cana-2192	186	39	,	,	PUNCT
cana-2192	186	40	fx∗)d(fxnk	fx∗)d(fxnk	INTJ
cana-2192	186	41	,	,	PUNCT
cana-2192	186	42	fx∗),k(xnk	fx∗),k(xnk	NOUN
cana-2192	186	43	,	,	PUNCT
cana-2192	186	44	x∗	x∗	PROPN
cana-2192	186	45	)	)	PUNCT
cana-2192	186	46	)	)	PUNCT
cana-2192	187	1	=	=	PUNCT
cana-2192	187	2	ζ(α(xnk	ζ(α(xnk	NOUN
cana-2192	187	3	,	,	PUNCT
cana-2192	187	4	xnk+1	xnk+1	NUM
cana-2192	187	5	)	)	PUNCT
cana-2192	187	6	α(x∗	α(x∗	NOUN
cana-2192	187	7	,	,	PUNCT
cana-2192	187	8	fx∗)d(xnk+1	fx∗)d(xnk+1	NOUN
cana-2192	187	9	,	,	PUNCT
cana-2192	187	10	fx∗),k(xnk	fx∗),k(xnk	NOUN
cana-2192	187	11	,	,	PUNCT
cana-2192	187	12	x∗	x∗	PROPN
cana-2192	187	13	)	)	PUNCT
cana-2192	187	14	)	)	PUNCT
cana-2192	188	1	<	<	X
cana-2192	188	2	k(xnk	k(xnk	NOUN
cana-2192	188	3	,	,	PUNCT
cana-2192	188	4	x∗)−	x∗)−	PRON
cana-2192	188	5	α(xnk	α(xnk	PROPN
cana-2192	188	6	,	,	PUNCT
cana-2192	188	7	xnk+1	xnk+1	PROPN
cana-2192	188	8	)	)	PUNCT
cana-2192	188	9	α(x∗	α(x∗	NOUN
cana-2192	188	10	,	,	PUNCT
cana-2192	188	11	fx∗)d(xnk+1	fx∗)d(xnk+1	NOUN
cana-2192	188	12	,	,	PUNCT
cana-2192	188	13	fx∗	fx∗	PROPN
cana-2192	188	14	)	)	PUNCT
cana-2192	188	15	,	,	PUNCT
cana-2192	188	16	(	(	PUNCT
cana-2192	188	17	2.22	2.22	NUM
cana-2192	188	18	)	)	PUNCT
cana-2192	189	1	where	where	SCONJ
cana-2192	189	2	k(xnk	k(xnk	NOUN
cana-2192	189	3	,	,	PUNCT
cana-2192	189	4	x∗	x∗	PROPN
cana-2192	189	5	)	)	PUNCT
cana-2192	189	6	=	=	PUNCT
cana-2192	190	1	β(e(xnk	β(e(xnk	INTJ
cana-2192	190	2	,	,	PUNCT
cana-2192	190	3	x∗))e(xnk	x∗))e(xnk	PRON
cana-2192	190	4	,	,	PUNCT
cana-2192	190	5	x∗	x∗	PROPN
cana-2192	190	6	)	)	PUNCT
cana-2192	191	1	+	+	CCONJ
cana-2192	191	2	ln(xnk	ln(xnk	INTJ
cana-2192	191	3	,	,	PUNCT
cana-2192	191	4	x∗	x∗	PROPN
cana-2192	191	5	)	)	PUNCT
cana-2192	191	6	.	.	PUNCT
cana-2192	192	1	(	(	PUNCT
cana-2192	192	2	2.23	2.23	NUM
cana-2192	192	3	)	)	PUNCT
cana-2192	192	4	also	also	ADV
cana-2192	192	5	e(xnk	e(xnk	PROPN
cana-2192	192	6	,	,	PUNCT
cana-2192	192	7	x∗	x∗	PROPN
cana-2192	192	8	)	)	PUNCT
cana-2192	192	9	=	=	SYM
cana-2192	192	10	d(xnk	d(xnk	PROPN
cana-2192	192	11	,	,	PUNCT
cana-2192	192	12	x∗	x∗	PROPN
cana-2192	192	13	)	)	PUNCT
cana-2192	193	1	+	+	NUM
cana-2192	193	2	|d(xnk	|d(xnk	NUM
cana-2192	193	3	,	,	PUNCT
cana-2192	193	4	fxnk	fxnk	NOUN
cana-2192	193	5	)	)	PUNCT
cana-2192	193	6	−	−	PROPN
cana-2192	193	7	d(x∗	d(x∗	NOUN
cana-2192	193	8	,	,	PUNCT
cana-2192	193	9	fx∗)|	fx∗)|	PROPN
cana-2192	193	10	=	=	PUNCT
cana-2192	193	11	d(xnk	d(xnk	PROPN
cana-2192	193	12	,	,	PUNCT
cana-2192	193	13	x∗	x∗	PROPN
cana-2192	193	14	)	)	PUNCT
cana-2192	194	1	+	+	NUM
cana-2192	194	2	|d(xnk	|d(xnk	NUM
cana-2192	194	3	,	,	PUNCT
cana-2192	194	4	xnk+1	xnk+1	NUM
cana-2192	194	5	)	)	PUNCT
cana-2192	194	6	−	−	PROPN
cana-2192	194	7	d(x∗	d(x∗	NOUN
cana-2192	194	8	,	,	PUNCT
cana-2192	194	9	fx∗)|	fx∗)|	PROPN
cana-2192	194	10	=	=	SYM
cana-2192	194	11	d(x∗	d(x∗	PROPN
cana-2192	194	12	,	,	PUNCT
cana-2192	194	13	fx∗	fx∗	PROPN
cana-2192	194	14	)	)	PUNCT
cana-2192	194	15	for	for	ADP
cana-2192	194	16	large	large	ADJ
cana-2192	194	17	k	k	NOUN
cana-2192	194	18	,	,	PUNCT
cana-2192	194	19	(	(	PUNCT
cana-2192	194	20	2.24	2.24	NUM
cana-2192	194	21	)	)	PUNCT
cana-2192	194	22	and	and	CCONJ
cana-2192	194	23	n(xnk	n(xnk	PROPN
cana-2192	194	24	,	,	PUNCT
cana-2192	194	25	x∗	x∗	PROPN
cana-2192	194	26	)	)	PUNCT
cana-2192	194	27	=	=	SYM
cana-2192	194	28	min{d(xnk	min{d(xnk	NOUN
cana-2192	194	29	,	,	PUNCT
cana-2192	194	30	fxnk	fxnk	NOUN
cana-2192	194	31	)	)	PUNCT
cana-2192	194	32	,	,	PUNCT
cana-2192	194	33	d(x∗	d(x∗	NOUN
cana-2192	194	34	,	,	PUNCT
cana-2192	194	35	fx∗	fx∗	PROPN
cana-2192	194	36	)	)	PUNCT
cana-2192	194	37	,	,	PUNCT
cana-2192	194	38	d(xnk	d(xnk	PROPN
cana-2192	194	39	,	,	PUNCT
cana-2192	194	40	fx∗	fx∗	PROPN
cana-2192	194	41	)	)	PUNCT
cana-2192	194	42	,	,	PUNCT
cana-2192	194	43	d(x∗	d(x∗	NOUN
cana-2192	194	44	,	,	PUNCT
cana-2192	194	45	fxnk	fxnk	NOUN
cana-2192	194	46	)	)	PUNCT
cana-2192	194	47	}	}	PUNCT
cana-2192	194	48	=	=	SYM
cana-2192	194	49	min{d(xnk	min{d(xnk	PROPN
cana-2192	194	50	,	,	PUNCT
cana-2192	194	51	xnk+1	xnk+1	PROPN
cana-2192	194	52	)	)	PUNCT
cana-2192	194	53	,	,	PUNCT
cana-2192	194	54	d(x∗	d(x∗	NOUN
cana-2192	194	55	,	,	PUNCT
cana-2192	194	56	fx∗	fx∗	PROPN
cana-2192	194	57	)	)	PUNCT
cana-2192	194	58	,	,	PUNCT
cana-2192	194	59	d(xnk	d(xnk	PROPN
cana-2192	194	60	,	,	PUNCT
cana-2192	194	61	fx∗	fx∗	PROPN
cana-2192	194	62	)	)	PUNCT
cana-2192	194	63	,	,	PUNCT
cana-2192	194	64	d(x∗	d(x∗	NOUN
cana-2192	194	65	,	,	PUNCT
cana-2192	194	66	xnk+1	xnk+1	X
cana-2192	194	67	)	)	PUNCT
cana-2192	194	68	}	}	PUNCT
cana-2192	195	1	=	=	SYM
cana-2192	195	2	0	0	X
cana-2192	195	3	.	.	PUNCT
cana-2192	196	1	(	(	PUNCT
cana-2192	196	2	2.25	2.25	NUM
cana-2192	196	3	)	)	PUNCT
cana-2192	196	4	therefore	therefore	ADV
cana-2192	196	5	k(xnk	k(xnk	NOUN
cana-2192	196	6	,	,	PUNCT
cana-2192	196	7	x∗	x∗	PROPN
cana-2192	196	8	)	)	PUNCT
cana-2192	196	9	=	=	SYM
cana-2192	196	10	d(x∗	d(x∗	NOUN
cana-2192	196	11	,	,	PUNCT
cana-2192	196	12	fx∗	fx∗	PROPN
cana-2192	196	13	)	)	PUNCT
cana-2192	196	14	.	.	PUNCT
cana-2192	197	1	consequently	consequently	ADV
cana-2192	197	2	,	,	PUNCT
cana-2192	197	3	we	we	PRON
cana-2192	197	4	have	have	VERB
cana-2192	197	5	d(xnk+1	d(xnk+1	NOUN
cana-2192	197	6	,	,	PUNCT
cana-2192	197	7	fx	fx	NOUN
cana-2192	197	8	∗	∗	NOUN
cana-2192	197	9	)	)	PUNCT
cana-2192	197	10	=	=	SYM
cana-2192	197	11	d(fxnk	d(fxnk	NOUN
cana-2192	197	12	,	,	PUNCT
cana-2192	197	13	fx∗	fx∗	PROPN
cana-2192	197	14	)	)	PUNCT
cana-2192	197	15	≤	≤	NOUN
cana-2192	198	1	α(xnk	α(xnk	NOUN
cana-2192	198	2	,	,	PUNCT
cana-2192	198	3	fxnk	fxnk	NOUN
cana-2192	198	4	)	)	PUNCT
cana-2192	198	5	α(x∗	α(x∗	NOUN
cana-2192	198	6	,	,	PUNCT
cana-2192	198	7	fx∗)d(fxnk	fx∗)d(fxnk	INTJ
cana-2192	198	8	,	,	PUNCT
cana-2192	198	9	fx∗	fx∗	PROPN
cana-2192	198	10	)	)	PUNCT
cana-2192	198	11	<	<	X
cana-2192	198	12	d(x∗	d(x∗	PROPN
cana-2192	198	13	,	,	PUNCT
cana-2192	198	14	fx∗	fx∗	ADJ
cana-2192	198	15	)	)	PUNCT
cana-2192	198	16	for	for	ADP
cana-2192	198	17	all	all	DET
cana-2192	198	18	k	k	PROPN
cana-2192	198	19	∈	∈	PROPN
cana-2192	198	20	n.	n.	NOUN
cana-2192	198	21	(	(	PUNCT
cana-2192	198	22	2.26	2.26	NUM
cana-2192	198	23	)	)	PUNCT
cana-2192	198	24	by	by	ADP
cana-2192	198	25	(	(	PUNCT
cana-2192	198	26	2.22	2.22	NUM
cana-2192	198	27	)	)	PUNCT
cana-2192	198	28	,	,	PUNCT
cana-2192	198	29	(	(	PUNCT
cana-2192	198	30	2.26	2.26	NUM
cana-2192	198	31	)	)	PUNCT
cana-2192	198	32	and	and	CCONJ
cana-2192	198	33	the	the	DET
cana-2192	198	34	condition	condition	NOUN
cana-2192	198	35	(	(	PUNCT
cana-2192	198	36	ζ3	ζ3	NOUN
cana-2192	198	37	)	)	PUNCT
cana-2192	198	38	,	,	PUNCT
cana-2192	198	39	we	we	PRON
cana-2192	198	40	get	get	VERB
cana-2192	198	41	0	0	NUM
cana-2192	198	42	≤	≤	NUM
cana-2192	198	43	limn→∞supζ(α(xn	limn→∞supζ(α(xn	PROPN
cana-2192	198	44	,	,	PUNCT
cana-2192	198	45	fxn)α(x	fxn)α(x	PROPN
cana-2192	198	46	∗	∗	NOUN
cana-2192	198	47	,	,	PUNCT
cana-2192	198	48	fx∗)d(fxn	fx∗)d(fxn	PROPN
cana-2192	198	49	,	,	PUNCT
cana-2192	198	50	fx	fx	NOUN
cana-2192	198	51	∗),k(xn	∗),k(xn	NUM
cana-2192	198	52	,	,	PUNCT
cana-2192	198	53	x	x	NOUN
cana-2192	198	54	∗	∗	NOUN
cana-2192	198	55	)	)	PUNCT
cana-2192	198	56	)	)	PUNCT
cana-2192	199	1	<	<	X
cana-2192	199	2	0	0	X
cana-2192	199	3	.	.	PUNCT
cana-2192	200	1	this	this	PRON
cana-2192	200	2	is	be	AUX
cana-2192	200	3	a	a	DET
cana-2192	200	4	contradiction	contradiction	NOUN
cana-2192	200	5	.	.	PUNCT
cana-2192	201	1	hence	hence	ADV
cana-2192	201	2	,	,	PUNCT
cana-2192	201	3	therefore	therefore	ADV
cana-2192	201	4	x∗	x∗	PROPN
cana-2192	201	5	is	be	AUX
cana-2192	201	6	a	a	DET
cana-2192	201	7	fixed	fix	VERB
cana-2192	201	8	point	point	NOUN
cana-2192	201	9	of	of	ADP
cana-2192	201	10	f	f	PROPN
cana-2192	201	11	.	.	PUNCT
cana-2192	202	1	suppose	suppose	VERB
cana-2192	202	2	that	that	SCONJ
cana-2192	202	3	x∗	x∗	PROPN
cana-2192	202	4	and	and	CCONJ
cana-2192	202	5	u∗	u∗	ADV
cana-2192	202	6	be	be	AUX
cana-2192	202	7	two	two	NUM
cana-2192	202	8	fixed	fix	VERB
cana-2192	202	9	points	point	NOUN
cana-2192	202	10	of	of	ADP
cana-2192	202	11	f	f	PROPN
cana-2192	202	12	and	and	CCONJ
cana-2192	202	13	hence	hence	ADV
cana-2192	202	14	x∗	x∗	PROPN
cana-2192	202	15	,	,	PUNCT
cana-2192	202	16	u∗	u∗	PROPN
cana-2192	202	17	∈	∈	PROPN
cana-2192	202	18	fix(f	fix(f	PROPN
cana-2192	202	19	)	)	PUNCT
cana-2192	202	20	which	which	PRON
cana-2192	202	21	is	be	AUX
cana-2192	202	22	a	a	DET
cana-2192	202	23	generalized	generalized	ADJ
cana-2192	202	24	α	α	NOUN
cana-2192	202	25	-	-	ADJ
cana-2192	202	26	admissible	admissible	ADJ
cana-2192	202	27	almost	almost	ADV
cana-2192	202	28	z	z	NOUN
cana-2192	202	29	-	-	PUNCT
cana-2192	202	30	contraction	contraction	NOUN
cana-2192	202	31	self	self	NOUN
cana-2192	202	32	-mappings	-mapping	NOUN
cana-2192	202	33	of	of	ADP
cana-2192	202	34	a	a	DET
cana-2192	202	35	metric	metric	ADJ
cana-2192	202	36	space	space	NOUN
cana-2192	202	37	(	(	PUNCT
cana-2192	202	38	x	x	X
cana-2192	202	39	,	,	PUNCT
cana-2192	202	40	d	d	NOUN
cana-2192	202	41	)	)	PUNCT
cana-2192	202	42	.	.	PUNCT
cana-2192	203	1	by	by	ADP
cana-2192	203	2	(	(	PUNCT
cana-2192	203	3	2.1	2.1	NUM
cana-2192	203	4	)	)	PUNCT
cana-2192	203	5	,	,	PUNCT
cana-2192	203	6	we	we	PRON
cana-2192	203	7	have	have	VERB
cana-2192	203	8	that	that	DET
cana-2192	203	9	0	0	NUM
cana-2192	203	10	≤	≤	NUM
cana-2192	203	11	ζ(α(x∗	ζ(α(x∗	PROPN
cana-2192	203	12	,	,	PUNCT
cana-2192	203	13	fx∗)α(u∗	fx∗)α(u∗	PROPN
cana-2192	203	14	,	,	PUNCT
cana-2192	203	15	fu∗)d(fx∗	fu∗)d(fx∗	ADJ
cana-2192	203	16	,	,	PUNCT
cana-2192	203	17	fu∗),k(x∗	fu∗),k(x∗	NOUN
cana-2192	203	18	,	,	PUNCT
cana-2192	203	19	u∗	u∗	PROPN
cana-2192	203	20	)	)	PUNCT
cana-2192	203	21	)	)	PUNCT
cana-2192	203	22	,	,	PUNCT
cana-2192	203	23	(	(	PUNCT
cana-2192	203	24	2.27	2.27	NUM
cana-2192	203	25	)	)	PUNCT
cana-2192	203	26	where	where	SCONJ
cana-2192	203	27	k(x∗	k(x∗	NOUN
cana-2192	203	28	,	,	PUNCT
cana-2192	203	29	u∗	u∗	ADJ
cana-2192	203	30	)	)	PUNCT
cana-2192	203	31	=	=	SYM
cana-2192	203	32	d(x∗	d(x∗	NOUN
cana-2192	203	33	,	,	PUNCT
cana-2192	203	34	u∗	u∗	PROPN
cana-2192	203	35	)	)	PUNCT
cana-2192	203	36	,	,	PUNCT
cana-2192	203	37	by	by	ADP
cana-2192	203	38	using	use	VERB
cana-2192	203	39	(	(	PUNCT
cana-2192	203	40	2.18	2.18	NUM
cana-2192	203	41	)	)	PUNCT
cana-2192	203	42	,	,	PUNCT
cana-2192	203	43	(	(	PUNCT
cana-2192	203	44	2.19	2.19	NUM
cana-2192	203	45	)	)	PUNCT
cana-2192	203	46	and	and	CCONJ
cana-2192	203	47	(	(	PUNCT
cana-2192	203	48	2.20	2.20	NUM
cana-2192	203	49	)	)	PUNCT
cana-2192	203	50	.	.	PUNCT
cana-2192	204	1	this	this	PRON
cana-2192	204	2	together	together	ADV
cana-2192	204	3	with	with	ADP
cana-2192	204	4	(	(	PUNCT
cana-2192	204	5	2.27	2.27	NUM
cana-2192	204	6	)	)	PUNCT
cana-2192	204	7	shows	show	VERB
cana-2192	204	8	that	that	SCONJ
cana-2192	204	9	0	0	NUM
cana-2192	204	10	≤	≤	NUM
cana-2192	204	11	ζ(α(x∗	ζ(α(x∗	PROPN
cana-2192	204	12	,	,	PUNCT
cana-2192	204	13	fx∗)α(u∗	fx∗)α(u∗	PROPN
cana-2192	204	14	,	,	PUNCT
cana-2192	204	15	fu∗)d(fx∗	fu∗)d(fx∗	ADJ
cana-2192	204	16	,	,	PUNCT
cana-2192	204	17	fu∗),k(x∗	fu∗),k(x∗	NOUN
cana-2192	204	18	,	,	PUNCT
cana-2192	204	19	u∗	u∗	PROPN
cana-2192	204	20	)	)	PUNCT
cana-2192	204	21	)	)	PUNCT
cana-2192	205	1	=	=	SYM
cana-2192	205	2	ζ(α(x∗	ζ(α(x∗	PROPN
cana-2192	205	3	,	,	PUNCT
cana-2192	205	4	x∗)α(u∗	x∗)α(u∗	PROPN
cana-2192	205	5	,	,	PUNCT
cana-2192	205	6	u∗)d(x∗	u∗)d(x∗	PROPN
cana-2192	205	7	,	,	PUNCT
cana-2192	205	8	u∗	u∗	PROPN
cana-2192	205	9	)	)	PUNCT
cana-2192	205	10	,	,	PUNCT
cana-2192	205	11	d(x∗	d(x∗	NOUN
cana-2192	205	12	,	,	PUNCT
cana-2192	205	13	u∗	u∗	PROPN
cana-2192	205	14	)	)	PUNCT
cana-2192	205	15	)	)	PUNCT
cana-2192	205	16	.	.	PUNCT
cana-2192	206	1	this	this	PRON
cana-2192	206	2	is	be	AUX
cana-2192	206	3	a	a	DET
cana-2192	206	4	contradiction	contradiction	NOUN
cana-2192	206	5	.	.	PUNCT
cana-2192	207	1	thus	thus	ADV
cana-2192	207	2	,	,	PUNCT
cana-2192	207	3	we	we	PRON
cana-2192	207	4	have	have	VERB
cana-2192	207	5	x∗	x∗	PROPN
cana-2192	207	6	=	=	PRON
cana-2192	207	7	u∗.	u∗.	PROPN
cana-2192	207	8	hence	hence	ADV
cana-2192	207	9	f	f	PROPN
cana-2192	207	10	has	have	VERB
cana-2192	207	11	a	a	DET
cana-2192	207	12	unique	unique	ADJ
cana-2192	207	13	fixed	fix	VERB
cana-2192	207	14	point	point	NOUN
cana-2192	207	15	.	.	PUNCT
cana-2192	208	1	□	□	PUNCT
cana-2192	208	2	communications	communication	NOUN
cana-2192	208	3	on	on	ADP
cana-2192	208	4	applied	apply	VERB
cana-2192	208	5	nonlinear	nonlinear	ADJ
cana-2192	208	6	analysis	analysis	NOUN
cana-2192	208	7	issn	issn	NOUN
cana-2192	208	8	:	:	PUNCT
cana-2192	208	9	1074	1074	NUM
cana-2192	208	10	-	-	PUNCT
cana-2192	208	11	133x	133x	NUM
cana-2192	208	12	vol	vol	NOUN
cana-2192	208	13	32	32	NUM
cana-2192	208	14	no	no	NOUN
cana-2192	208	15	.	.	PUNCT
cana-2192	209	1	1s	1s	NUM
cana-2192	209	2	(	(	PUNCT
cana-2192	209	3	2025	2025	NUM
cana-2192	209	4	)	)	PUNCT
cana-2192	210	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2192	210	2	360	360	NUM
cana-2192	210	3	dipti	dipti	NOUN
cana-2192	210	4	,	,	PUNCT
cana-2192	210	5	anil	anil	PROPN
cana-2192	210	6	kumar	kumar	PROPN
cana-2192	210	7	dubey	dubey	PROPN
cana-2192	210	8	,	,	PUNCT
cana-2192	210	9	urmila	urmila	PROPN
cana-2192	210	10	mishra	mishra	PROPN
cana-2192	210	11	,	,	PUNCT
cana-2192	210	12	corollary	corollary	ADJ
cana-2192	210	13	11	11	NUM
cana-2192	210	14	.	.	PUNCT
cana-2192	211	1	let	let	VERB
cana-2192	211	2	(	(	PUNCT
cana-2192	211	3	x	x	NOUN
cana-2192	211	4	,	,	PUNCT
cana-2192	211	5	d	d	NOUN
cana-2192	211	6	)	)	PUNCT
cana-2192	211	7	be	be	AUX
cana-2192	211	8	a	a	DET
cana-2192	211	9	complete	complete	ADJ
cana-2192	211	10	metric	metric	ADJ
cana-2192	211	11	space	space	NOUN
cana-2192	211	12	,	,	PUNCT
cana-2192	211	13	f	f	X
cana-2192	211	14	:	:	PUNCT
cana-2192	211	15	x	x	X
cana-2192	211	16	→	→	PUNCT
cana-2192	211	17	x	x	PUNCT
cana-2192	211	18	be	be	AUX
cana-2192	211	19	a	a	DET
cana-2192	211	20	self	self	NOUN
cana-2192	211	21	-	-	PUNCT
cana-2192	211	22	mapping	mapping	NOUN
cana-2192	211	23	,	,	PUNCT
cana-2192	211	24	there	there	PRON
cana-2192	211	25	exists	exist	VERB
cana-2192	211	26	ζ	ζ	PROPN
cana-2192	211	27	∈	∈	PROPN
cana-2192	211	28	z	z	NOUN
cana-2192	211	29	and	and	CCONJ
cana-2192	211	30	α	α	NOUN
cana-2192	211	31	:	:	PUNCT
cana-2192	212	1	x	x	PROPN
cana-2192	212	2	×x	×x	X
cana-2192	212	3	→	→	X
cana-2192	212	4	[	[	X
cana-2192	212	5	0,∞	0,∞	NOUN
cana-2192	212	6	)	)	PUNCT
cana-2192	212	7	be	be	VERB
cana-2192	212	8	a	a	DET
cana-2192	212	9	function	function	NOUN
cana-2192	212	10	with	with	ADP
cana-2192	212	11	α(x	α(x	PROPN
cana-2192	212	12	,	,	PUNCT
cana-2192	212	13	y	y	PROPN
cana-2192	212	14	)	)	PUNCT
cana-2192	212	15	=	=	SYM
cana-2192	212	16	1	1	NUM
cana-2192	212	17	for	for	ADP
cana-2192	212	18	all	all	DET
cana-2192	212	19	x	x	NOUN
cana-2192	212	20	,	,	PUNCT
cana-2192	212	21	y	y	PROPN
cana-2192	212	22	∈	∈	PROPN
cana-2192	212	23	x	x	PUNCT
cana-2192	212	24	such	such	ADJ
cana-2192	212	25	that	that	SCONJ
cana-2192	212	26	ζ(d(fx	ζ(d(fx	NOUN
cana-2192	212	27	,	,	PUNCT
cana-2192	212	28	fy),k(x	fy),k(x	PROPN
cana-2192	212	29	,	,	PUNCT
cana-2192	212	30	y	y	NOUN
cana-2192	212	31	)	)	PUNCT
cana-2192	212	32	)	)	PUNCT
cana-2192	212	33	≥	≥	NOUN
cana-2192	212	34	0	0	NUM
cana-2192	212	35	for	for	ADP
cana-2192	212	36	all	all	DET
cana-2192	212	37	distinct	distinct	ADJ
cana-2192	212	38	x	x	NOUN
cana-2192	212	39	,	,	PUNCT
cana-2192	212	40	y	y	PROPN
cana-2192	212	41	∈	∈	PROPN
cana-2192	213	1	x	x	NOUN
cana-2192	213	2	,	,	PUNCT
cana-2192	213	3	where	where	SCONJ
cana-2192	213	4	k(x	k(x	PROPN
cana-2192	213	5	,	,	PUNCT
cana-2192	213	6	y	y	NOUN
cana-2192	213	7	)	)	PUNCT
cana-2192	213	8	=	=	SYM
cana-2192	213	9	β(e(x	β(e(x	PROPN
cana-2192	213	10	,	,	PUNCT
cana-2192	213	11	y))e(x	y))e(x	PROPN
cana-2192	213	12	,	,	PUNCT
cana-2192	213	13	y	y	PROPN
cana-2192	213	14	)	)	PUNCT
cana-2192	214	1	+	+	CCONJ
cana-2192	214	2	ln(x	ln(x	X
cana-2192	214	3	,	,	PUNCT
cana-2192	214	4	y	y	NOUN
cana-2192	214	5	)	)	PUNCT
cana-2192	214	6	and	and	CCONJ
cana-2192	214	7	e(x	e(x	NUM
cana-2192	214	8	,	,	PUNCT
cana-2192	214	9	y	y	NOUN
cana-2192	214	10	)	)	PUNCT
cana-2192	214	11	=	=	SYM
cana-2192	214	12	d(x	d(x	PROPN
cana-2192	214	13	,	,	PUNCT
cana-2192	214	14	y	y	NOUN
cana-2192	214	15	)	)	PUNCT
cana-2192	214	16	+	+	CCONJ
cana-2192	215	1	|d(x	|d(x	PROPN
cana-2192	215	2	,	,	PUNCT
cana-2192	215	3	fx)−	fx)−	NOUN
cana-2192	215	4	d(y	d(y	NOUN
cana-2192	215	5	,	,	PUNCT
cana-2192	215	6	fy)|	fy)|	PROPN
cana-2192	215	7	and	and	CCONJ
cana-2192	215	8	n(x	n(x	PROPN
cana-2192	215	9	,	,	PUNCT
cana-2192	215	10	y	y	NOUN
cana-2192	215	11	)	)	PUNCT
cana-2192	215	12	=	=	VERB
cana-2192	216	1	min{d(x	min{d(x	PROPN
cana-2192	216	2	,	,	PUNCT
cana-2192	216	3	fx	fx	PROPN
cana-2192	216	4	)	)	PUNCT
cana-2192	216	5	,	,	PUNCT
cana-2192	216	6	d(y	d(y	PROPN
cana-2192	216	7	,	,	PUNCT
cana-2192	216	8	fy	fy	PROPN
cana-2192	216	9	)	)	PUNCT
cana-2192	216	10	,	,	PUNCT
cana-2192	216	11	d(x	d(x	PROPN
cana-2192	216	12	,	,	PUNCT
cana-2192	216	13	fy	fy	PROPN
cana-2192	216	14	)	)	PUNCT
cana-2192	216	15	,	,	PUNCT
cana-2192	216	16	d(y	d(y	PROPN
cana-2192	216	17	,	,	PUNCT
cana-2192	216	18	fx	fx	PROPN
cana-2192	216	19	)	)	PUNCT
cana-2192	216	20	}	}	PUNCT
cana-2192	216	21	.	.	PUNCT
cana-2192	217	1	then	then	ADV
cana-2192	217	2	f	f	PROPN
cana-2192	217	3	has	have	VERB
cana-2192	217	4	a	a	DET
cana-2192	217	5	unique	unique	ADJ
cana-2192	217	6	fixed	fix	VERB
cana-2192	217	7	point	point	NOUN
cana-2192	217	8	x∗	x∗	PROPN
cana-2192	217	9	in	in	ADP
cana-2192	217	10	x.	x.	PROPN
cana-2192	217	11	example	example	NOUN
cana-2192	217	12	12	12	NUM
cana-2192	217	13	.	.	PUNCT
cana-2192	218	1	let	let	VERB
cana-2192	218	2	x	x	PUNCT
cana-2192	218	3	=	=	PUNCT
cana-2192	219	1	[	[	X
cana-2192	219	2	0	0	NUM
cana-2192	219	3	,	,	PUNCT
cana-2192	219	4	1	1	NUM
cana-2192	219	5	]	]	PUNCT
cana-2192	219	6	endowed	endow	VERB
cana-2192	219	7	with	with	ADP
cana-2192	219	8	metric	metric	ADJ
cana-2192	219	9	d(x	d(x	PROPN
cana-2192	219	10	,	,	PUNCT
cana-2192	219	11	y	y	NOUN
cana-2192	219	12	)	)	PUNCT
cana-2192	219	13	=	=	NOUN
cana-2192	219	14	|x	|x	NOUN
cana-2192	220	1	−	−	VERB
cana-2192	220	2	y|	y|	NOUN
cana-2192	220	3	for	for	ADP
cana-2192	220	4	all	all	DET
cana-2192	220	5	x	x	NOUN
cana-2192	220	6	,	,	PUNCT
cana-2192	220	7	y	y	PROPN
cana-2192	220	8	∈	∈	PROPN
cana-2192	220	9	x.	x.	NOUN
cana-2192	220	10	let	let	VERB
cana-2192	220	11	ζ(t	ζ(t	PROPN
cana-2192	220	12	,	,	PUNCT
cana-2192	220	13	s	s	PART
cana-2192	220	14	)	)	PUNCT
cana-2192	220	15	=	=	SYM
cana-2192	220	16	s−t	s−t	NOUN
cana-2192	220	17	and	and	CCONJ
cana-2192	220	18	considering	consider	VERB
cana-2192	220	19	β	β	X
cana-2192	220	20	:	:	PUNCT
cana-2192	221	1	[	[	X
cana-2192	221	2	0,∞	0,∞	NUM
cana-2192	221	3	)	)	PUNCT
cana-2192	221	4	→	→	PUNCT
cana-2192	222	1	[	[	X
cana-2192	222	2	0	0	NUM
cana-2192	222	3	,	,	PUNCT
cana-2192	222	4	1	1	NUM
cana-2192	222	5	)	)	PUNCT
cana-2192	222	6	as	as	ADP
cana-2192	222	7	β(t	β(t	NOUN
cana-2192	222	8	)	)	PUNCT
cana-2192	222	9	=	=	PUNCT
cana-2192	222	10	1	1	NUM
cana-2192	222	11	1+t	1+t	NUM
cana-2192	222	12	for	for	ADP
cana-2192	222	13	all	all	DET
cana-2192	222	14	t	t	PROPN
cana-2192	222	15	≥	≥	NOUN
cana-2192	222	16	0	0	NUM
cana-2192	222	17	and	and	CCONJ
cana-2192	222	18	l	l	PROPN
cana-2192	222	19	≥	≥	NOUN
cana-2192	222	20	0	0	NUM
cana-2192	222	21	.	.	PUNCT
cana-2192	223	1	let	let	VERB
cana-2192	223	2	f	f	NOUN
cana-2192	223	3	:	:	PUNCT
cana-2192	223	4	x	x	X
cana-2192	223	5	→	→	PUNCT
cana-2192	223	6	x	x	AUX
cana-2192	223	7	be	be	AUX
cana-2192	223	8	defined	define	VERB
cana-2192	223	9	by	by	ADP
cana-2192	223	10	f(x	f(x	PROPN
cana-2192	223	11	)	)	PUNCT
cana-2192	224	1	=	=	PUNCT
cana-2192	224	2	x	x	SYM
cana-2192	224	3	3	3	NUM
cana-2192	224	4	for	for	ADP
cana-2192	224	5	all	all	DET
cana-2192	224	6	x	x	SYM
cana-2192	224	7	∈	∈	PROPN
cana-2192	225	1	[	[	X
cana-2192	225	2	0	0	NUM
cana-2192	225	3	,	,	PUNCT
cana-2192	225	4	1	1	NUM
cana-2192	225	5	]	]	PUNCT
cana-2192	225	6	and	and	CCONJ
cana-2192	225	7	α	α	X
cana-2192	225	8	:	:	PUNCT
cana-2192	225	9	x	x	PROPN
cana-2192	225	10	×x	×x	X
cana-2192	225	11	→	→	X
cana-2192	225	12	[	[	X
cana-2192	225	13	0,∞	0,∞	X
cana-2192	225	14	)	)	PUNCT
cana-2192	225	15	be	be	AUX
cana-2192	225	16	defined	define	VERB
cana-2192	225	17	by	by	ADP
cana-2192	225	18	α(x	α(x	PROPN
cana-2192	225	19	,	,	PUNCT
cana-2192	225	20	y	y	PROPN
cana-2192	225	21	)	)	PUNCT
cana-2192	225	22	=	=	NOUN
cana-2192	225	23	{	{	PUNCT
cana-2192	226	1	1	1	NUM
cana-2192	226	2	,	,	PUNCT
cana-2192	226	3	if	if	SCONJ
cana-2192	226	4	,	,	PUNCT
cana-2192	226	5	x	x	X
cana-2192	226	6	,	,	PUNCT
cana-2192	226	7	y	y	PROPN
cana-2192	226	8	∈	∈	PROPN
cana-2192	227	1	[	[	X
cana-2192	227	2	0	0	NUM
cana-2192	227	3	,	,	PUNCT
cana-2192	227	4	1	1	NUM
cana-2192	227	5	]	]	PUNCT
cana-2192	227	6	;	;	PUNCT
cana-2192	227	7	0	0	NUM
cana-2192	227	8	,	,	PUNCT
cana-2192	227	9	otherwise	otherwise	ADV
cana-2192	227	10	.	.	PUNCT
cana-2192	228	1	note	note	VERB
cana-2192	228	2	that	that	SCONJ
cana-2192	228	3	f	f	PROPN
cana-2192	228	4	is	be	AUX
cana-2192	228	5	an	an	DET
cana-2192	228	6	α	α	NOUN
cana-2192	228	7	-	-	ADJ
cana-2192	228	8	admissible	admissible	ADJ
cana-2192	228	9	if	if	SCONJ
cana-2192	228	10	α(x	α(x	NOUN
cana-2192	228	11	,	,	PUNCT
cana-2192	228	12	fx	fx	PROPN
cana-2192	228	13	)	)	PUNCT
cana-2192	228	14	≥	≥	NOUN
cana-2192	228	15	1	1	NUM
cana-2192	228	16	implies	imply	VERB
cana-2192	228	17	α(fx	α(fx	ADV
cana-2192	228	18	,	,	PUNCT
cana-2192	228	19	f2x	f2x	X
cana-2192	228	20	)	)	PUNCT
cana-2192	228	21	≥	≥	NOUN
cana-2192	229	1	1	1	NUM
cana-2192	229	2	.	.	PUNCT
cana-2192	229	3	now	now	ADV
cana-2192	229	4	by	by	ADP
cana-2192	229	5	definition	definition	NOUN
cana-2192	229	6	of	of	ADP
cana-2192	229	7	α	α	PROPN
cana-2192	229	8	and	and	CCONJ
cana-2192	229	9	x	x	NOUN
cana-2192	229	10	,	,	PUNCT
cana-2192	229	11	y	y	PROPN
cana-2192	229	12	∈	∈	PROPN
cana-2192	230	1	[	[	X
cana-2192	230	2	0	0	NUM
cana-2192	230	3	,	,	PUNCT
cana-2192	230	4	1	1	NUM
cana-2192	230	5	]	]	PUNCT
cana-2192	230	6	,	,	PUNCT
cana-2192	230	7	we	we	PRON
cana-2192	230	8	have	have	VERB
cana-2192	230	9	α(x	α(x	NOUN
cana-2192	230	10	,	,	PUNCT
cana-2192	230	11	fx	fx	ADJ
cana-2192	230	12	)	)	PUNCT
cana-2192	230	13	=	=	SYM
cana-2192	230	14	α(x	α(x	NOUN
cana-2192	230	15	,	,	PUNCT
cana-2192	230	16	x3	x3	ADJ
cana-2192	230	17	)	)	PUNCT
cana-2192	231	1	=	=	SYM
cana-2192	231	2	1	1	X
cana-2192	231	3	.	.	X
cana-2192	231	4	similarly	similarly	ADV
cana-2192	231	5	α(y	α(y	NOUN
cana-2192	231	6	,	,	PUNCT
cana-2192	231	7	fy	fy	PROPN
cana-2192	231	8	)	)	PUNCT
cana-2192	231	9	=	=	SYM
cana-2192	231	10	1	1	NUM
cana-2192	231	11	for	for	ADP
cana-2192	231	12	all	all	DET
cana-2192	231	13	x	x	NOUN
cana-2192	231	14	,	,	PUNCT
cana-2192	231	15	y	y	PROPN
cana-2192	231	16	∈	∈	PROPN
cana-2192	231	17	x.	x.	NOUN
cana-2192	231	18	from	from	ADP
cana-2192	231	19	above	above	ADV
cana-2192	231	20	,	,	PUNCT
cana-2192	231	21	it	it	PRON
cana-2192	231	22	is	be	AUX
cana-2192	231	23	clear	clear	ADJ
cana-2192	231	24	that	that	SCONJ
cana-2192	231	25	f	f	PROPN
cana-2192	231	26	is	be	AUX
cana-2192	231	27	a	a	DET
cana-2192	231	28	generalized	generalized	ADJ
cana-2192	231	29	α	α	NOUN
cana-2192	231	30	-	-	ADJ
cana-2192	231	31	admissible	admissible	ADJ
cana-2192	231	32	mapping	mapping	NOUN
cana-2192	231	33	.	.	PUNCT
cana-2192	232	1	now	now	ADV
cana-2192	232	2	ζ(d(fx	ζ(d(fx	VERB
cana-2192	232	3	,	,	PUNCT
cana-2192	232	4	fy),k(x	fy),k(x	PROPN
cana-2192	232	5	,	,	PUNCT
cana-2192	232	6	y	y	NOUN
cana-2192	232	7	)	)	PUNCT
cana-2192	232	8	)	)	PUNCT
cana-2192	233	1	=	=	SYM
cana-2192	233	2	k(x	k(x	PROPN
cana-2192	233	3	,	,	PUNCT
cana-2192	233	4	y)−	y)−	PROPN
cana-2192	233	5	d(fx	d(fx	PROPN
cana-2192	233	6	,	,	PUNCT
cana-2192	233	7	fy	fy	NOUN
cana-2192	233	8	)	)	PUNCT
cana-2192	233	9	=	=	SYM
cana-2192	233	10	β(e(x	β(e(x	PROPN
cana-2192	233	11	,	,	PUNCT
cana-2192	233	12	y))e(x	y))e(x	PROPN
cana-2192	233	13	,	,	PUNCT
cana-2192	233	14	y	y	PROPN
cana-2192	233	15	)	)	PUNCT
cana-2192	234	1	+	+	CCONJ
cana-2192	234	2	ln(x	ln(x	X
cana-2192	234	3	,	,	PUNCT
cana-2192	234	4	y)−	y)−	PROPN
cana-2192	234	5	1	1	NUM
cana-2192	234	6	3	3	NUM
cana-2192	234	7	|x−	|x−	NOUN
cana-2192	234	8	y|	y|	NOUN
cana-2192	234	9	=	=	SYM
cana-2192	234	10	e(x	e(x	NUM
cana-2192	234	11	,	,	PUNCT
cana-2192	234	12	y	y	NOUN
cana-2192	234	13	)	)	PUNCT
cana-2192	234	14	1	1	NUM
cana-2192	235	1	+	+	NUM
cana-2192	235	2	e(x	e(x	NUM
cana-2192	235	3	,	,	PUNCT
cana-2192	235	4	y	y	NOUN
cana-2192	235	5	)	)	PUNCT
cana-2192	235	6	+	+	CCONJ
cana-2192	235	7	ln(x	ln(x	X
cana-2192	235	8	,	,	PUNCT
cana-2192	235	9	y)−	y)−	PROPN
cana-2192	235	10	1	1	NUM
cana-2192	235	11	3	3	NUM
cana-2192	235	12	|x−	|x−	NOUN
cana-2192	235	13	y|	y|	NOUN
cana-2192	235	14	≤	≤	NUM
cana-2192	235	15	5	5	NUM
cana-2192	235	16	3d(x	3d(x	NUM
cana-2192	235	17	,	,	PUNCT
cana-2192	235	18	y	y	NOUN
cana-2192	235	19	)	)	PUNCT
cana-2192	235	20	1	1	NUM
cana-2192	236	1	+	+	CCONJ
cana-2192	236	2	5	5	NUM
cana-2192	236	3	3d(x	3d(x	NUM
cana-2192	236	4	,	,	PUNCT
cana-2192	236	5	y	y	NOUN
cana-2192	236	6	)	)	PUNCT
cana-2192	237	1	+	+	CCONJ
cana-2192	237	2	ln(x	ln(x	X
cana-2192	237	3	,	,	PUNCT
cana-2192	237	4	y)−	y)−	PROPN
cana-2192	237	5	1	1	NUM
cana-2192	237	6	3	3	NUM
cana-2192	237	7	|x−	|x−	NOUN
cana-2192	237	8	y|	y|	NOUN
cana-2192	237	9	=	=	NOUN
cana-2192	237	10	5	5	NUM
cana-2192	237	11	3	3	NUM
cana-2192	237	12	|x−	|x−	NOUN
cana-2192	237	13	y|	y|	NOUN
cana-2192	237	14	1	1	NUM
cana-2192	237	15	+	+	CCONJ
cana-2192	237	16	5	5	NUM
cana-2192	237	17	3	3	NUM
cana-2192	237	18	|x−	|x−	NOUN
cana-2192	237	19	y|	y|	NOUN
cana-2192	237	20	+	+	CCONJ
cana-2192	237	21	ln(x	ln(x	X
cana-2192	237	22	,	,	PUNCT
cana-2192	237	23	y)−	y)−	PROPN
cana-2192	237	24	1	1	NUM
cana-2192	237	25	3	3	NUM
cana-2192	237	26	|x−	|x−	NOUN
cana-2192	237	27	y|	y|	NOUN
cana-2192	237	28	≥	≥	NOUN
cana-2192	237	29	0	0	NUM
cana-2192	237	30	.	.	PUNCT
cana-2192	238	1	therefore	therefore	ADV
cana-2192	238	2	,	,	PUNCT
cana-2192	238	3	f	f	PROPN
cana-2192	238	4	is	be	AUX
cana-2192	238	5	generalized	generalize	VERB
cana-2192	238	6	α	α	PRON
cana-2192	238	7	-	-	ADJ
cana-2192	238	8	admissible	admissible	ADJ
cana-2192	238	9	almost	almost	ADV
cana-2192	238	10	z	z	NOUN
cana-2192	238	11	-	-	PUNCT
cana-2192	238	12	contraction	contraction	NOUN
cana-2192	238	13	with	with	ADP
cana-2192	238	14	respect	respect	NOUN
cana-2192	238	15	to	to	ADP
cana-2192	238	16	ζ	ζ	SYM
cana-2192	238	17	∈	∈	NOUN
cana-2192	238	18	z.	z.	NOUN
cana-2192	238	19	hence	hence	ADV
cana-2192	238	20	all	all	DET
cana-2192	238	21	the	the	DET
cana-2192	238	22	assumptions	assumption	NOUN
cana-2192	238	23	of	of	ADP
cana-2192	238	24	theorem	theorem	ADJ
cana-2192	238	25	9	9	NUM
cana-2192	238	26	and	and	CCONJ
cana-2192	238	27	corollary	corollary	ADJ
cana-2192	238	28	11	11	NUM
cana-2192	238	29	are	be	AUX
cana-2192	238	30	satisfied	satisfied	ADJ
cana-2192	238	31	and	and	CCONJ
cana-2192	238	32	hence	hence	ADV
cana-2192	238	33	f	f	PROPN
cana-2192	238	34	has	have	VERB
cana-2192	238	35	a	a	DET
cana-2192	238	36	unique	unique	ADJ
cana-2192	238	37	fixed	fix	VERB
cana-2192	238	38	point	point	NOUN
cana-2192	238	39	.	.	PUNCT
cana-2192	239	1	references	reference	NOUN
cana-2192	239	2	[	[	X
cana-2192	239	3	1	1	NUM
cana-2192	239	4	]	]	PUNCT
cana-2192	239	5	a.	a.	PROPN
cana-2192	239	6	s.	s.	PROPN
cana-2192	239	7	alharbi	alharbi	PROPN
cana-2192	239	8	,	,	PUNCT
cana-2192	239	9	h.	h.	PROPN
cana-2192	239	10	alsulami	alsulami	PROPN
cana-2192	239	11	and	and	CCONJ
cana-2192	239	12	e.	e.	PROPN
cana-2192	239	13	karapinar	karapinar	PROPN
cana-2192	239	14	:	:	PUNCT
cana-2192	239	15	on	on	ADP
cana-2192	239	16	the	the	DET
cana-2192	239	17	power	power	NOUN
cana-2192	239	18	of	of	ADP
cana-2192	239	19	simulation	simulation	NOUN
cana-2192	239	20	and	and	CCONJ
cana-2192	239	21	admissible	admissible	ADJ
cana-2192	239	22	functions	function	NOUN
cana-2192	239	23	in	in	ADP
cana-2192	239	24	metric	metric	ADJ
cana-2192	239	25	fixed	fix	VERB
cana-2192	239	26	point	point	NOUN
cana-2192	239	27	theory	theory	NOUN
cana-2192	239	28	,	,	PUNCT
cana-2192	239	29	jour	jour	X
cana-2192	239	30	.	.	PROPN
cana-2192	239	31	func	func	PROPN
cana-2192	239	32	.	.	PUNCT
cana-2192	240	1	spaces	space	NOUN
cana-2192	240	2	,	,	PUNCT
cana-2192	240	3	volume	volume	NOUN
cana-2192	240	4	2017	2017	NUM
cana-2192	240	5	,	,	PUNCT
cana-2192	240	6	article	article	NOUN
cana-2192	240	7	i	i	PROPN
cana-2192	240	8	d	d	PROPN
cana-2192	240	9	2068163	2068163	NUM
cana-2192	240	10	,	,	PUNCT
cana-2192	240	11	7	7	NUM
cana-2192	240	12	pages	page	NOUN
cana-2192	240	13	.	.	PUNCT
cana-2192	241	1	https://doi.org/10.1155/2017/2068163	https://doi.org/10.1155/2017/2068163	PROPN
cana-2192	241	2	.	.	PUNCT
cana-2192	242	1	[	[	X
cana-2192	242	2	2	2	X
cana-2192	242	3	]	]	PUNCT
cana-2192	242	4	h.	h.	NOUN
cana-2192	242	5	argoubi	argoubi	PROPN
cana-2192	242	6	,	,	PUNCT
cana-2192	242	7	b.	b.	PROPN
cana-2192	242	8	samet	samet	PROPN
cana-2192	242	9	,	,	PUNCT
cana-2192	242	10	c.	c.	PROPN
cana-2192	242	11	vetro	vetro	PROPN
cana-2192	242	12	:	:	PUNCT
cana-2192	242	13	nonlinear	nonlinear	ADJ
cana-2192	242	14	contractions	contraction	NOUN
cana-2192	242	15	involving	involve	VERB
cana-2192	242	16	simulation	simulation	NOUN
cana-2192	242	17	functions	function	NOUN
cana-2192	242	18	in	in	ADP
cana-2192	242	19	a	a	DET
cana-2192	242	20	metric	metric	ADJ
cana-2192	242	21	space	space	NOUN
cana-2192	242	22	with	with	ADP
cana-2192	242	23	a	a	DET
cana-2192	242	24	partial	partial	ADJ
cana-2192	242	25	order	order	NOUN
cana-2192	242	26	,	,	PUNCT
cana-2192	242	27	jour	jour	X
cana-2192	242	28	.	.	PUNCT
cana-2192	242	29	nonlinear	nonlinear	PROPN
cana-2192	242	30	sci	sci	PROPN
cana-2192	242	31	.	.	PUNCT
cana-2192	242	32	appl	appl	PROPN
cana-2192	242	33	.	.	PROPN
cana-2192	242	34	,	,	PUNCT
cana-2192	242	35	8	8	NUM
cana-2192	242	36	(	(	PUNCT
cana-2192	242	37	2015	2015	NUM
cana-2192	242	38	)	)	PUNCT
cana-2192	242	39	,	,	PUNCT
cana-2192	242	40	1082	1082	NUM
cana-2192	242	41	-	-	SYM
cana-2192	242	42	1094	1094	NUM
cana-2192	242	43	.	.	PUNCT
cana-2192	243	1	communications	communication	NOUN
cana-2192	243	2	on	on	ADP
cana-2192	243	3	applied	apply	VERB
cana-2192	243	4	nonlinear	nonlinear	ADJ
cana-2192	243	5	analysis	analysis	NOUN
cana-2192	243	6	issn	issn	NOUN
cana-2192	243	7	:	:	PUNCT
cana-2192	243	8	1074	1074	NUM
cana-2192	243	9	-	-	PUNCT
cana-2192	243	10	133x	133x	NUM
cana-2192	243	11	vol	vol	NOUN
cana-2192	243	12	32	32	NUM
cana-2192	243	13	no	no	NOUN
cana-2192	243	14	.	.	PUNCT
cana-2192	244	1	1s	1s	NUM
cana-2192	244	2	(	(	PUNCT
cana-2192	244	3	2025	2025	NUM
cana-2192	244	4	)	)	PUNCT
cana-2192	244	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-2192	244	6	361	361	NUM
cana-2192	244	7	generalized	generalize	VERB
cana-2192	244	8	α	α	NOUN
cana-2192	244	9	-	-	ADJ
cana-2192	244	10	admissible	admissible	ADJ
cana-2192	244	11	almost	almost	ADV
cana-2192	244	12	......	......	PUNCT
cana-2192	244	13	functions	function	NOUN
cana-2192	244	14	in	in	ADP
cana-2192	244	15	a	a	DET
cana-2192	244	16	metric	metric	ADJ
cana-2192	244	17	space	space	NOUN
cana-2192	244	18	[	[	X
cana-2192	244	19	3	3	X
cana-2192	244	20	]	]	X
cana-2192	244	21	h.	h.	PROPN
cana-2192	244	22	aydi	aydi	PROPN
cana-2192	244	23	,	,	PUNCT
cana-2192	244	24	a.	a.	PROPN
cana-2192	244	25	felhi	felhi	PROPN
cana-2192	244	26	,	,	PUNCT
cana-2192	244	27	h.	h.	PROPN
cana-2192	244	28	afshari	afshari	PROPN
cana-2192	244	29	:	:	PUNCT
cana-2192	244	30	new	new	ADJ
cana-2192	244	31	geraghty	geraghty	VERB
cana-2192	244	32	type	type	NOUN
cana-2192	244	33	contractions	contraction	NOUN
cana-2192	244	34	on	on	ADP
cana-2192	244	35	metric	metric	ADJ
cana-2192	244	36	-	-	PUNCT
cana-2192	244	37	like	like	ADJ
cana-2192	244	38	spaces	space	NOUN
cana-2192	244	39	,	,	PUNCT
cana-2192	244	40	jour	jour	X
cana-2192	244	41	.	.	PROPN
cana-2192	244	42	nonlinear	nonlinear	PROPN
cana-2192	244	43	sci	sci	PROPN
cana-2192	244	44	.	.	PUNCT
cana-2192	244	45	appl	appl	PROPN
cana-2192	244	46	.	.	PROPN
cana-2192	244	47	,	,	PUNCT
cana-2192	244	48	10	10	NUM
cana-2192	244	49	(	(	PUNCT
cana-2192	244	50	2017	2017	NUM
cana-2192	244	51	)	)	PUNCT
cana-2192	244	52	,	,	PUNCT
cana-2192	244	53	780	780	NUM
cana-2192	244	54	-	-	SYM
cana-2192	244	55	788	788	NUM
cana-2192	244	56	.	.	PUNCT
cana-2192	245	1	[	[	X
cana-2192	245	2	4	4	X
cana-2192	245	3	]	]	X
cana-2192	245	4	h.	h.	PROPN
cana-2192	245	5	aydi	aydi	PROPN
cana-2192	245	6	,	,	PUNCT
cana-2192	245	7	a.	a.	PROPN
cana-2192	245	8	felhi	felhi	PROPN
cana-2192	245	9	,	,	PUNCT
cana-2192	245	10	e.	e.	PROPN
cana-2192	245	11	karapinar	karapinar	PROPN
cana-2192	245	12	and	and	CCONJ
cana-2192	245	13	f.	f.	PROPN
cana-2192	245	14	a.	a.	PROPN
cana-2192	245	15	alojail	alojail	PROPN
cana-2192	245	16	:	:	PUNCT
cana-2192	245	17	fixed	fix	VERB
cana-2192	245	18	points	point	NOUN
cana-2192	245	19	on	on	ADP
cana-2192	245	20	quasi	quasi	ADJ
cana-2192	245	21	-	-	ADJ
cana-2192	245	22	metric	metric	ADJ
cana-2192	245	23	spaces	space	NOUN
cana-2192	245	24	via	via	ADP
cana-2192	245	25	simulation	simulation	NOUN
cana-2192	245	26	functions	function	NOUN
cana-2192	245	27	and	and	CCONJ
cana-2192	245	28	consequences	consequence	NOUN
cana-2192	245	29	,	,	PUNCT
cana-2192	245	30	jour	jour	X
cana-2192	245	31	.	.	PUNCT
cana-2192	245	32	math	math	PROPN
cana-2192	245	33	.	.	PUNCT
cana-2192	246	1	anal	anal	PROPN
cana-2192	246	2	.	.	PUNCT
cana-2192	246	3	,	,	PUNCT
cana-2192	246	4	9(2018	9(2018	NUM
cana-2192	246	5	)	)	PUNCT
cana-2192	246	6	(	(	PUNCT
cana-2192	246	7	2	2	NUM
cana-2192	246	8	)	)	PUNCT
cana-2192	246	9	,	,	PUNCT
cana-2192	246	10	10	10	NUM
cana-2192	246	11	-	-	SYM
cana-2192	246	12	24	24	NUM
cana-2192	246	13	.	.	PUNCT
cana-2192	247	1	[	[	X
cana-2192	247	2	5	5	X
cana-2192	247	3	]	]	PUNCT
cana-2192	247	4	v.	v.	ADP
cana-2192	247	5	berinde	berinde	NOUN
cana-2192	247	6	:	:	PUNCT
cana-2192	247	7	approximating	approximate	VERB
cana-2192	247	8	fixed	fix	VERB
cana-2192	247	9	point	point	NOUN
cana-2192	247	10	of	of	ADP
cana-2192	247	11	weak	weak	ADJ
cana-2192	247	12	contractions	contraction	NOUN
cana-2192	247	13	using	use	VERB
cana-2192	247	14	the	the	DET
cana-2192	247	15	picard	picard	NOUN
cana-2192	247	16	iteration	iteration	NOUN
cana-2192	247	17	,	,	PUNCT
cana-2192	247	18	nonlinear	nonlinear	ADJ
cana-2192	247	19	anal	anal	PROPN
cana-2192	247	20	.	.	PUNCT
cana-2192	248	1	forum	forum	PROPN
cana-2192	248	2	,	,	PUNCT
cana-2192	248	3	9	9	NUM
cana-2192	248	4	(	(	PUNCT
cana-2192	248	5	1)(2004	1)(2004	NUM
cana-2192	248	6	)	)	PUNCT
cana-2192	248	7	,	,	PUNCT
cana-2192	248	8	43	43	NUM
cana-2192	248	9	-	-	SYM
cana-2192	248	10	53	53	NUM
cana-2192	248	11	.	.	PUNCT
cana-2192	249	1	[	[	X
cana-2192	249	2	6	6	NUM
cana-2192	249	3	]	]	PUNCT
cana-2192	249	4	v.	v.	ADP
cana-2192	249	5	berinde	berinde	PROPN
cana-2192	249	6	:	:	PUNCT
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cana-2192	249	8	constructive	constructive	ADJ
cana-2192	249	9	fixed	fix	VERB
cana-2192	249	10	point	point	NOUN
cana-2192	249	11	theorems	theorem	NOUN
cana-2192	249	12	for	for	ADP
cana-2192	249	13	ciric	ciric	ADJ
cana-2192	249	14	-	-	PUNCT
cana-2192	249	15	type	type	NOUN
cana-2192	249	16	almost	almost	ADV
cana-2192	249	17	contractions	contraction	NOUN
cana-2192	249	18	in	in	ADP
cana-2192	249	19	metric	metric	ADJ
cana-2192	249	20	spaces	space	NOUN
cana-2192	249	21	,	,	PUNCT
cana-2192	249	22	carpathian	carpathian	ADJ
cana-2192	249	23	jour	jour	X
cana-2192	249	24	.	.	PUNCT
cana-2192	249	25	math	math	PROPN
cana-2192	249	26	,	,	PUNCT
cana-2192	249	27	24	24	NUM
cana-2192	249	28	(	(	PUNCT
cana-2192	249	29	2)(2018	2)(2018	NUM
cana-2192	249	30	)	)	PUNCT
cana-2192	249	31	,	,	PUNCT
cana-2192	249	32	10	10	NUM
cana-2192	249	33	-	-	SYM
cana-2192	249	34	19	19	NUM
cana-2192	249	35	.	.	PUNCT
cana-2192	250	1	[	[	X
cana-2192	250	2	7	7	X
cana-2192	250	3	]	]	PUNCT
cana-2192	250	4	p.	p.	NOUN
cana-2192	250	5	bunpatcharacharoen	bunpatcharacharoen	PROPN
cana-2192	250	6	,	,	PUNCT
cana-2192	250	7	s.	s.	PROPN
cana-2192	250	8	saelee	saelee	PROPN
cana-2192	250	9	,	,	PUNCT
cana-2192	250	10	and	and	CCONJ
cana-2192	250	11	p.	p.	NOUN
cana-2192	250	12	saipara	saipara	PROPN
cana-2192	250	13	:	:	PUNCT
cana-2192	250	14	modified	modify	VERB
cana-2192	250	15	almost	almost	ADV
cana-2192	250	16	type	type	NOUN
cana-2192	250	17	zcontraction	zcontraction	NOUN
cana-2192	250	18	,	,	PUNCT
cana-2192	250	19	tahi	tahi	X
cana-2192	250	20	jour	jour	PROPN
cana-2192	250	21	.	.	PUNCT
cana-2192	250	22	math	math	PROPN
cana-2192	250	23	.	.	PUNCT
cana-2192	250	24	,	,	PUNCT
cana-2192	250	25	18(1	18(1	X
cana-2192	250	26	)	)	PUNCT
cana-2192	250	27	(	(	PUNCT
cana-2192	250	28	2020	2020	NUM
cana-2192	250	29	)	)	PUNCT
cana-2192	250	30	,	,	PUNCT
cana-2192	250	31	252	252	NUM
cana-2192	250	32	-	-	SYM
cana-2192	250	33	260	260	NUM
cana-2192	250	34	.	.	PUNCT
cana-2192	251	1	[	[	X
cana-2192	251	2	8	8	X
cana-2192	251	3	]	]	PUNCT
cana-2192	251	4	s.	s.	PROPN
cana-2192	251	5	chandok	chandok	PROPN
cana-2192	251	6	:	:	PUNCT
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cana-2192	251	8	fixed	fix	VERB
cana-2192	251	9	point	point	NOUN
cana-2192	251	10	theorems	theorem	NOUN
cana-2192	251	11	for	for	ADP
cana-2192	251	12	(	(	PUNCT
cana-2192	251	13	α	α	X
cana-2192	251	14	,	,	PUNCT
cana-2192	251	15	β)-admissible	β)-admissible	PUNCT
cana-2192	251	16	geraghty	geraghty	PROPN
cana-2192	251	17	type	type	VERB
cana-2192	251	18	contractive	contractive	ADJ
cana-2192	251	19	mappings	mapping	NOUN
cana-2192	251	20	and	and	CCONJ
cana-2192	251	21	related	related	ADJ
cana-2192	251	22	results	result	NOUN
cana-2192	251	23	,	,	PUNCT
cana-2192	251	24	mathematical	mathematical	ADJ
cana-2192	251	25	sciences	science	NOUN
cana-2192	251	26	,	,	PUNCT
cana-2192	251	27	9	9	NUM
cana-2192	251	28	(	(	PUNCT
cana-2192	251	29	2015	2015	NUM
cana-2192	251	30	)	)	PUNCT
cana-2192	251	31	,	,	PUNCT
cana-2192	251	32	127	127	NUM
cana-2192	251	33	-	-	SYM
cana-2192	251	34	135	135	NUM
cana-2192	251	35	.	.	PUNCT
cana-2192	252	1	[	[	X
cana-2192	252	2	9	9	NUM
cana-2192	252	3	]	]	PUNCT
cana-2192	252	4	a.	a.	NOUN
cana-2192	252	5	dewangan	dewangan	PROPN
cana-2192	252	6	,	,	PUNCT
cana-2192	252	7	a.	a.	PROPN
cana-2192	252	8	k.	k.	PROPN
cana-2192	252	9	dubey	dubey	PROPN
cana-2192	252	10	,	,	PUNCT
cana-2192	252	11	u.	u.	PROPN
cana-2192	252	12	mishra	mishra	PROPN
cana-2192	252	13	and	and	CCONJ
cana-2192	252	14	r.	r.	PROPN
cana-2192	252	15	p.	p.	PROPN
cana-2192	252	16	dubey	dubey	PROPN
cana-2192	252	17	:	:	PUNCT
cana-2192	252	18	fixed	fix	VERB
cana-2192	252	19	point	point	NOUN
cana-2192	252	20	results	result	VERB
cana-2192	252	21	for	for	ADP
cana-2192	252	22	(	(	PUNCT
cana-2192	252	23	α	α	NOUN
cana-2192	252	24	,	,	PUNCT
cana-2192	252	25	β)-admissible	β)-admissible	PUNCT
cana-2192	252	26	almost	almost	ADV
cana-2192	252	27	z	z	NOUN
cana-2192	252	28	-	-	PUNCT
cana-2192	252	29	contractions	contraction	NOUN
cana-2192	252	30	in	in	ADP
cana-2192	252	31	metric	metric	ADJ
cana-2192	252	32	-	-	PUNCT
cana-2192	252	33	like	like	ADJ
cana-2192	252	34	space	space	NOUN
cana-2192	252	35	via	via	ADP
cana-2192	252	36	simulation	simulation	NOUN
cana-2192	252	37	function	function	NOUN
cana-2192	252	38	,	,	PUNCT
cana-2192	252	39	facta	facta	PROPN
cana-2192	252	40	universitatis	universitatis	PROPN
cana-2192	252	41	(	(	PUNCT
cana-2192	252	42	nis	nis	NOUN
cana-2192	252	43	)	)	PUNCT
cana-2192	252	44	,	,	PUNCT
cana-2192	252	45	ser	ser	PROPN
cana-2192	252	46	.	.	PROPN
cana-2192	252	47	math	math	PROPN
cana-2192	252	48	.	.	PUNCT
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cana-2192	253	2	,	,	PUNCT
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cana-2192	253	4	)	)	PUNCT
cana-2192	253	5	(	(	PUNCT
cana-2192	253	6	2022	2022	NUM
cana-2192	253	7	)	)	PUNCT
cana-2192	253	8	,	,	PUNCT
cana-2192	253	9	529	529	NUM
cana-2192	253	10	-	-	SYM
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cana-2192	253	12	.	.	PUNCT
cana-2192	254	1	[	[	X
cana-2192	254	2	10	10	NUM
cana-2192	254	3	]	]	X
cana-2192	254	4	a.	a.	NOUN
cana-2192	254	5	felehi	felehi	PROPN
cana-2192	254	6	,	,	PUNCT
cana-2192	254	7	h.	h.	PROPN
cana-2192	254	8	aydi	aydi	PROPN
cana-2192	254	9	,	,	PUNCT
cana-2192	254	10	d.	d.	PROPN
cana-2192	254	11	zhang	zhang	PROPN
cana-2192	254	12	:	:	PUNCT
cana-2192	254	13	fixed	fix	VERB
cana-2192	254	14	point	point	NOUN
cana-2192	254	15	for	for	ADP
cana-2192	254	16	α	α	NOUN
cana-2192	254	17	-	-	ADJ
cana-2192	254	18	admissible	admissible	ADJ
cana-2192	254	19	contractive	contractive	ADJ
cana-2192	254	20	mappings	mapping	NOUN
cana-2192	254	21	via	via	ADP
cana-2192	254	22	simulation	simulation	NOUN
cana-2192	254	23	functions	function	NOUN
cana-2192	254	24	,	,	PUNCT
cana-2192	254	25	jour	jour	X
cana-2192	254	26	.	.	PROPN
cana-2192	254	27	nonlinear	nonlinear	PROPN
cana-2192	254	28	sci	sci	PROPN
cana-2192	254	29	.	.	PUNCT
cana-2192	254	30	appl	appl	PROPN
cana-2192	254	31	.	.	PROPN
cana-2192	254	32	,	,	PUNCT
cana-2192	254	33	9(10	9(10	NUM
cana-2192	254	34	)	)	PUNCT
cana-2192	254	35	(	(	PUNCT
cana-2192	254	36	2016	2016	NUM
cana-2192	254	37	)	)	PUNCT
cana-2192	254	38	,	,	PUNCT
cana-2192	254	39	5544	5544	NUM
cana-2192	254	40	-	-	SYM
cana-2192	254	41	5560	5560	NUM
cana-2192	254	42	.	.	PUNCT
cana-2192	255	1	[	[	X
cana-2192	255	2	11	11	NUM
cana-2192	255	3	]	]	PUNCT
cana-2192	255	4	m.	m.	NOUN
cana-2192	255	5	geraghty	geraghty	PROPN
cana-2192	255	6	:	:	PUNCT
cana-2192	255	7	on	on	ADP
cana-2192	255	8	contractive	contractive	ADJ
cana-2192	255	9	mappings	mapping	NOUN
cana-2192	255	10	,	,	PUNCT
cana-2192	255	11	proc	proc	NOUN
cana-2192	255	12	.	.	PUNCT
cana-2192	256	1	amer	amer	PROPN
cana-2192	256	2	.	.	PUNCT
cana-2192	256	3	math	math	PROPN
cana-2192	256	4	.	.	PUNCT
cana-2192	257	1	soc	soc	PROPN
cana-2192	257	2	.	.	PROPN
cana-2192	257	3	,	,	PUNCT
cana-2192	257	4	40(2	40(2	NUM
cana-2192	257	5	)	)	PUNCT
cana-2192	257	6	(	(	PUNCT
cana-2192	257	7	1973	1973	NUM
cana-2192	257	8	)	)	PUNCT
cana-2192	257	9	,	,	PUNCT
cana-2192	257	10	604	604	NUM
cana-2192	257	11	-	-	SYM
cana-2192	257	12	608	608	NUM
cana-2192	257	13	.	.	PUNCT
cana-2192	258	1	[	[	X
cana-2192	258	2	12	12	NUM
cana-2192	258	3	]	]	X
cana-2192	258	4	e.	e.	PROPN
cana-2192	258	5	karapinar	karapinar	PROPN
cana-2192	258	6	:	:	PUNCT
cana-2192	258	7	fixed	fix	VERB
cana-2192	258	8	point	point	NOUN
cana-2192	258	9	results	result	NOUN
cana-2192	258	10	via	via	ADP
cana-2192	258	11	simulation	simulation	NOUN
cana-2192	258	12	functions	function	NOUN
cana-2192	258	13	,	,	PUNCT
cana-2192	258	14	filomat	filomat	NOUN
cana-2192	258	15	,	,	PUNCT
cana-2192	258	16	30(8	30(8	NUM
cana-2192	258	17	)	)	PUNCT
cana-2192	258	18	(	(	PUNCT
cana-2192	258	19	2016	2016	NUM
cana-2192	258	20	)	)	PUNCT
cana-2192	258	21	,	,	PUNCT
cana-2192	258	22	2343	2343	NUM
cana-2192	258	23	-	-	SYM
cana-2192	258	24	2350	2350	NUM
cana-2192	258	25	.	.	PUNCT
cana-2192	259	1	[	[	X
cana-2192	259	2	13	13	NUM
cana-2192	259	3	]	]	X
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cana-2192	259	6	and	and	CCONJ
cana-2192	259	7	v.	v.	ADP
cana-2192	259	8	m.	m.	NOUN
cana-2192	259	9	l.	l.	PROPN
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cana-2192	259	12	:	:	PUNCT
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cana-2192	259	14	on	on	ADP
cana-2192	259	15	the	the	DET
cana-2192	259	16	almost	almost	ADV
cana-2192	259	17	z	z	NOUN
cana-2192	259	18	-	-	PUNCT
cana-2192	259	19	contraction	contraction	NOUN
cana-2192	259	20	,	,	PUNCT
cana-2192	259	21	open	open	ADJ
cana-2192	259	22	mathematics	mathematic	NOUN
cana-2192	259	23	,	,	PUNCT
cana-2192	259	24	18	18	NUM
cana-2192	259	25	(	(	PUNCT
cana-2192	259	26	2020	2020	NUM
cana-2192	259	27	)	)	PUNCT
cana-2192	259	28	,	,	PUNCT
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cana-2192	259	30	-	-	SYM
cana-2192	259	31	457	457	NUM
cana-2192	259	32	.	.	PUNCT
cana-2192	260	1	[	[	X
cana-2192	260	2	14	14	NUM
cana-2192	260	3	]	]	X
cana-2192	260	4	f.	f.	PROPN
cana-2192	260	5	khojasteh	khojasteh	PROPN
cana-2192	260	6	,	,	PUNCT
cana-2192	260	7	s.	s.	PROPN
cana-2192	260	8	shukla	shukla	PROPN
cana-2192	260	9	and	and	CCONJ
cana-2192	260	10	s.	s.	PROPN
cana-2192	260	11	radenovic	radenovic	PROPN
cana-2192	260	12	:	:	PUNCT
cana-2192	260	13	a	a	DET
cana-2192	260	14	new	new	ADJ
cana-2192	260	15	approach	approach	NOUN
cana-2192	260	16	to	to	ADP
cana-2192	260	17	the	the	DET
cana-2192	260	18	study	study	NOUN
cana-2192	260	19	of	of	ADP
cana-2192	260	20	fixed	fix	VERB
cana-2192	260	21	point	point	NOUN
cana-2192	260	22	theorems	theorem	NOUN
cana-2192	260	23	via	via	ADP
cana-2192	260	24	simulation	simulation	NOUN
cana-2192	260	25	functions	function	NOUN
cana-2192	260	26	,	,	PUNCT
cana-2192	260	27	filomat	filomat	NOUN
cana-2192	260	28	,	,	PUNCT
cana-2192	260	29	29	29	NUM
cana-2192	260	30	(	(	PUNCT
cana-2192	260	31	6)(2015	6)(2015	NOUN
cana-2192	260	32	)	)	PUNCT
cana-2192	260	33	,	,	PUNCT
cana-2192	260	34	1189	1189	NUM
cana-2192	260	35	-	-	SYM
cana-2192	260	36	1194	1194	NUM
cana-2192	260	37	.	.	PUNCT
cana-2192	261	1	[	[	X
cana-2192	261	2	15	15	NUM
cana-2192	261	3	]	]	X
cana-2192	261	4	s.	s.	PROPN
cana-2192	261	5	mishra	mishra	PROPN
cana-2192	261	6	,	,	PUNCT
cana-2192	261	7	a.	a.	PROPN
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cana-2192	261	11	u.	u.	PROPN
cana-2192	261	12	mishra	mishra	PROPN
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cana-2192	261	14	h.	h.	PROPN
cana-2192	261	15	g.	g.	PROPN
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cana-2192	261	17	:	:	PUNCT
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cana-2192	261	21	theorems	theorem	NOUN
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cana-2192	261	24	(	(	PUNCT
cana-2192	261	25	α	α	NOUN
cana-2192	261	26	,	,	PUNCT
cana-2192	261	27	β	β	X
cana-2192	261	28	,	,	PUNCT
cana-2192	261	29	z)-contraction	z)-contraction	PRON
cana-2192	261	30	mappings	mapping	NOUN
cana-2192	261	31	under	under	ADP
cana-2192	261	32	simulation	simulation	NOUN
cana-2192	261	33	functions	function	NOUN
cana-2192	261	34	and	and	CCONJ
cana-2192	261	35	cyclic	cyclic	NOUN
cana-2192	261	36	(	(	PUNCT
cana-2192	261	37	α	α	NOUN
cana-2192	261	38	,	,	PUNCT
cana-2192	261	39	β)admissibility	β)admissibility	NOUN
cana-2192	261	40	,	,	PUNCT
cana-2192	261	41	nonlinear	nonlinear	ADJ
cana-2192	261	42	func	func	NOUN
cana-2192	261	43	.	.	PUNCT
cana-2192	262	1	anal	anal	PROPN
cana-2192	262	2	.	.	PUNCT
cana-2192	262	3	appl	appl	PROPN
cana-2192	262	4	.	.	PROPN
cana-2192	262	5	,	,	PUNCT
cana-2192	262	6	27	27	NUM
cana-2192	262	7	(	(	PUNCT
cana-2192	262	8	4)(2022	4)(2022	NOUN
cana-2192	262	9	)	)	PUNCT
cana-2192	262	10	,	,	PUNCT
cana-2192	262	11	751	751	NUM
cana-2192	262	12	-	-	SYM
cana-2192	262	13	771	771	NUM
cana-2192	262	14	.	.	PUNCT
cana-2192	263	1	[	[	X
cana-2192	263	2	16	16	NUM
cana-2192	263	3	]	]	PUNCT
cana-2192	263	4	s.	s.	PROPN
cana-2192	263	5	mishra	mishra	PROPN
cana-2192	263	6	,	,	PUNCT
cana-2192	263	7	a.	a.	PROPN
cana-2192	263	8	k.	k.	PROPN
cana-2192	263	9	dubey	dubey	PROPN
cana-2192	263	10	,	,	PUNCT
cana-2192	263	11	u.	u.	PROPN
cana-2192	263	12	mishra	mishra	PROPN
cana-2192	263	13	and	and	CCONJ
cana-2192	263	14	r.	r.	PROPN
cana-2192	263	15	p.	p.	PROPN
cana-2192	263	16	dubey	dubey	PROPN
cana-2192	263	17	:	:	PUNCT
cana-2192	263	18	on	on	ADP
cana-2192	263	19	some	some	DET
cana-2192	263	20	fixed	fix	VERB
cana-2192	263	21	point	point	NOUN
cana-2192	263	22	results	result	NOUN
cana-2192	263	23	for	for	ADP
cana-2192	263	24	cyclic	cyclic	ADJ
cana-2192	263	25	(	(	PUNCT
cana-2192	263	26	α	α	NOUN
cana-2192	263	27	,	,	PUNCT
cana-2192	263	28	β)-admissible	β)-admissible	PUNCT
cana-2192	263	29	almost	almost	ADV
cana-2192	263	30	z	z	NOUN
cana-2192	263	31	-	-	PUNCT
cana-2192	263	32	contraction	contraction	NOUN
cana-2192	263	33	in	in	ADP
cana-2192	263	34	metric	metric	ADJ
cana-2192	263	35	-	-	PUNCT
cana-2192	263	36	like	like	ADJ
cana-2192	263	37	space	space	NOUN
cana-2192	263	38	with	with	ADP
cana-2192	263	39	simulation	simulation	NOUN
cana-2192	263	40	function	function	NOUN
cana-2192	263	41	,	,	PUNCT
cana-2192	263	42	communication	communication	NOUN
cana-2192	263	43	in	in	ADP
cana-2192	263	44	math	math	NOUN
cana-2192	263	45	.	.	PUNCT
cana-2192	264	1	appl	appl	PROPN
cana-2192	264	2	.	.	PROPN
cana-2192	265	1	13	13	NUM
cana-2192	265	2	(	(	PUNCT
cana-2192	265	3	1)(2022	1)(2022	NUM
cana-2192	265	4	)	)	PUNCT
cana-2192	265	5	,	,	PUNCT
cana-2192	265	6	223	223	NUM
cana-2192	265	7	-	-	SYM
cana-2192	265	8	233	233	NUM
cana-2192	265	9	.	.	PUNCT
cana-2192	266	1	[	[	X
cana-2192	266	2	17	17	NUM
cana-2192	266	3	]	]	PUNCT
cana-2192	266	4	a.	a.	NOUN
cana-2192	266	5	padcharoen	padcharoen	NOUN
cana-2192	266	6	and	and	CCONJ
cana-2192	266	7	p.	p.	PROPN
cana-2192	266	8	sukprasert	sukprasert	PROPN
cana-2192	266	9	:	:	PUNCT
cana-2192	266	10	on	on	ADP
cana-2192	266	11	admissible	admissible	ADJ
cana-2192	266	12	mapping	mapping	NOUN
cana-2192	266	13	via	via	ADP
cana-2192	266	14	simulation	simulation	NOUN
cana-2192	266	15	functions	function	NOUN
cana-2192	266	16	,	,	PUNCT
cana-2192	266	17	aust	aust	PROPN
cana-2192	266	18	.	.	PUNCT
cana-2192	266	19	jour	jour	PROPN
cana-2192	266	20	.	.	PUNCT
cana-2192	266	21	math	math	PROPN
cana-2192	266	22	.	.	PUNCT
cana-2192	267	1	anal	anal	PROPN
cana-2192	267	2	.	.	PUNCT
cana-2192	267	3	appl	appl	PROPN
cana-2192	267	4	.	.	PROPN
cana-2192	268	1	,	,	PUNCT
cana-2192	268	2	13	13	NUM
cana-2192	268	3	(	(	PUNCT
cana-2192	268	4	1)(2021	1)(2021	NUM
cana-2192	268	5	)	)	PUNCT
cana-2192	268	6	,	,	PUNCT
cana-2192	268	7	1	1	NUM
cana-2192	268	8	-	-	SYM
cana-2192	268	9	10	10	NUM
cana-2192	268	10	.	.	PUNCT
cana-2192	269	1	[	[	X
cana-2192	269	2	18	18	NUM
cana-2192	269	3	]	]	X
cana-2192	269	4	o.	o.	PROPN
cana-2192	269	5	popescu	popescu	PROPN
cana-2192	269	6	:	:	PUNCT
cana-2192	269	7	some	some	DET
cana-2192	269	8	new	new	ADJ
cana-2192	269	9	fixed	fix	VERB
cana-2192	269	10	point	point	NOUN
cana-2192	269	11	theorems	theorem	NOUN
cana-2192	269	12	for	for	ADP
cana-2192	269	13	α	α	NOUN
cana-2192	269	14	-	-	PUNCT
cana-2192	269	15	geraghty	geraghty	VERB
cana-2192	269	16	contractive	contractive	ADJ
cana-2192	269	17	type	type	NOUN
cana-2192	269	18	maps	map	NOUN
cana-2192	269	19	in	in	ADP
cana-2192	269	20	metric	metric	ADJ
cana-2192	269	21	spaces	space	NOUN
cana-2192	269	22	,	,	PUNCT
cana-2192	269	23	fixed	fix	VERB
cana-2192	269	24	point	point	NOUN
cana-2192	269	25	theory	theory	NOUN
cana-2192	269	26	appl	appl	PROPN
cana-2192	269	27	.	.	PROPN
cana-2192	269	28	,	,	PUNCT
cana-2192	269	29	2014	2014	NUM
cana-2192	269	30	(	(	PUNCT
cana-2192	269	31	2014	2014	NUM
cana-2192	269	32	)	)	PUNCT
cana-2192	269	33	,	,	PUNCT
cana-2192	269	34	art	art	NOUN
cana-2192	269	35	.	.	PUNCT
cana-2192	270	1	i	i	PRON
cana-2192	270	2	d	d	PROPN
cana-2192	270	3	190	190	NUM
cana-2192	270	4	.	.	PUNCT
cana-2192	271	1	[	[	X
cana-2192	271	2	19	19	NUM
cana-2192	271	3	]	]	PUNCT
cana-2192	271	4	a.	a.	PROPN
cana-2192	271	5	f.	f.	PROPN
cana-2192	271	6	roldan	roldan	PROPN
cana-2192	271	7	-	-	PUNCT
cana-2192	271	8	lopez	lopez	PROPN
cana-2192	271	9	-	-	PUNCT
cana-2192	271	10	de	de	X
cana-2192	271	11	hierro	hierro	PROPN
cana-2192	271	12	,	,	PUNCT
cana-2192	271	13	e.	e.	PROPN
cana-2192	271	14	karapinar	karapinar	PROPN
cana-2192	271	15	,	,	PUNCT
cana-2192	271	16	c.	c.	PROPN
cana-2192	271	17	roldan	roldan	PROPN
cana-2192	271	18	-	-	PUNCT
cana-2192	271	19	lopez	lopez	PROPN
cana-2192	271	20	-	-	PUNCT
cana-2192	271	21	de	de	NOUN
cana-2192	271	22	-	-	NOUN
cana-2192	271	23	hierro	hierro	ADJ
cana-2192	271	24	,	,	PUNCT
cana-2192	271	25	j.	j.	PROPN
cana-2192	271	26	martinez	martinez	PROPN
cana-2192	271	27	-	-	PROPN
cana-2192	271	28	moreno	moreno	PROPN
cana-2192	271	29	:	:	PUNCT
cana-2192	271	30	coincidence	coincidence	NOUN
cana-2192	271	31	point	point	NOUN
cana-2192	271	32	theorems	theorem	NOUN
cana-2192	271	33	on	on	ADP
cana-2192	271	34	metric	metric	ADJ
cana-2192	271	35	spaces	space	NOUN
cana-2192	271	36	via	via	ADP
cana-2192	271	37	simulation	simulation	NOUN
cana-2192	271	38	functions	function	NOUN
cana-2192	271	39	,	,	PUNCT
cana-2192	271	40	jour	jour	X
cana-2192	271	41	.	.	PUNCT
cana-2192	272	1	comp	comp	PROPN
cana-2192	272	2	.	.	PUNCT
cana-2192	273	1	appl	appl	PROPN
cana-2192	273	2	.	.	PROPN
cana-2192	273	3	math	math	PROPN
cana-2192	273	4	.	.	PUNCT
cana-2192	274	1	,	,	PUNCT
cana-2192	274	2	275	275	NUM
cana-2192	274	3	(	(	PUNCT
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cana-2192	274	5	)	)	PUNCT
cana-2192	274	6	,	,	PUNCT
cana-2192	274	7	345	345	NUM
cana-2192	274	8	-	-	SYM
cana-2192	274	9	355	355	NUM
cana-2192	274	10	.	.	PUNCT
cana-2192	275	1	[	[	X
cana-2192	275	2	20	20	NUM
cana-2192	275	3	]	]	PUNCT
cana-2192	275	4	b.	b.	PROPN
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cana-2192	275	7	c.	c.	PROPN
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cana-2192	275	9	and	and	CCONJ
cana-2192	275	10	p.	p.	PROPN
cana-2192	275	11	vetro	vetro	PROPN
cana-2192	275	12	:	:	PUNCT
cana-2192	275	13	fixed	fix	VERB
cana-2192	275	14	point	point	NOUN
cana-2192	275	15	theorems	theorem	VERB
cana-2192	275	16	for	for	ADP
cana-2192	275	17	(	(	PUNCT
cana-2192	275	18	α−ψ	α−ψ	NOUN
cana-2192	275	19	)	)	PUNCT
cana-2192	275	20	contractive	contractive	ADJ
cana-2192	275	21	type	type	NOUN
cana-2192	275	22	mappings	mapping	NOUN
cana-2192	275	23	,	,	PUNCT
cana-2192	275	24	jour	jour	X
cana-2192	275	25	.	.	PUNCT
cana-2192	275	26	nonlinear	nonlinear	PROPN
cana-2192	275	27	anal	anal	PROPN
cana-2192	275	28	.	.	PUNCT
cana-2192	275	29	,	,	PUNCT
cana-2192	275	30	75	75	NUM
cana-2192	275	31	(	(	PUNCT
cana-2192	275	32	4)(2012	4)(2012	NUM
cana-2192	275	33	)	)	PUNCT
cana-2192	275	34	,	,	PUNCT
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cana-2192	275	36	-	-	SYM
cana-2192	275	37	2165	2165	NUM
cana-2192	275	38	.	.	PUNCT
cana-2192	276	1	(	(	PUNCT
cana-2192	276	2	dipti	dipti	PROPN
cana-2192	276	3	)	)	PUNCT
cana-2192	276	4	department	department	NOUN
cana-2192	276	5	of	of	ADP
cana-2192	276	6	mathematics	mathematics	PROPN
cana-2192	276	7	,	,	PUNCT
cana-2192	276	8	dr	dr	PROPN
cana-2192	276	9	.	.	PROPN
cana-2192	276	10	c.	c.	PROPN
cana-2192	276	11	v.	v.	PROPN
cana-2192	276	12	raman	raman	PROPN
cana-2192	276	13	university	university	PROPN
cana-2192	276	14	,	,	PUNCT
cana-2192	276	15	kargi	kargi	PROPN
cana-2192	276	16	road	road	PROPN
cana-2192	276	17	kota	kota	PROPN
cana-2192	276	18	,	,	PUNCT
cana-2192	276	19	bilaspur	bilaspur	PROPN
cana-2192	276	20	(	(	PUNCT
cana-2192	276	21	c.g	c.g	PROPN
cana-2192	276	22	.	.	PROPN
cana-2192	276	23	)	)	PUNCT
cana-2192	277	1	email	email	NOUN
cana-2192	277	2	address	address	NOUN
cana-2192	277	3	:	:	PUNCT
cana-2192	277	4	diptisharma2704@gmail.com	diptisharma2704@gmail.com	X
cana-2192	277	5	(	(	PUNCT
cana-2192	277	6	anil	anil	PROPN
cana-2192	277	7	kumar	kumar	PROPN
cana-2192	277	8	dubey	dubey	PROPN
cana-2192	277	9	)	)	PUNCT
cana-2192	277	10	department	department	NOUN
cana-2192	277	11	of	of	ADP
cana-2192	277	12	applied	apply	VERB
cana-2192	277	13	mathematics	mathematic	NOUN
cana-2192	277	14	,	,	PUNCT
cana-2192	277	15	bhilai	bhilai	PROPN
cana-2192	277	16	institute	institute	PROPN
cana-2192	277	17	of	of	ADP
cana-2192	277	18	technology	technology	PROPN
cana-2192	277	19	,	,	PUNCT
cana-2192	277	20	bhilai	bhilai	PROPN
cana-2192	277	21	house	house	PROPN
cana-2192	277	22	,	,	PUNCT
cana-2192	277	23	durg	durg	NOUN
cana-2192	277	24	(	(	PUNCT
cana-2192	277	25	chhattisgarh	chhattisgarh	NOUN
cana-2192	277	26	)	)	PUNCT
cana-2192	277	27	,	,	PUNCT
cana-2192	277	28	india	india	PROPN
cana-2192	277	29	email	email	NOUN
cana-2192	277	30	address	address	NOUN
cana-2192	277	31	:	:	PUNCT
cana-2192	277	32	anilkumardby70@gmail.com	anilkumardby70@gmail.com	X
cana-2192	277	33	(	(	PUNCT
cana-2192	277	34	urmila	urmila	PROPN
cana-2192	277	35	mishra	mishra	PROPN
cana-2192	277	36	)	)	PUNCT
cana-2192	277	37	department	department	PROPN
cana-2192	277	38	of	of	ADP
cana-2192	277	39	mathematics	mathematic	NOUN
cana-2192	277	40	,	,	PUNCT
cana-2192	277	41	vishwavidyalaya	vishwavidyalaya	PROPN
cana-2192	277	42	engg	engg	PROPN
cana-2192	277	43	.	.	PUNCT
cana-2192	278	1	college	college	NOUN
cana-2192	278	2	,	,	PUNCT
cana-2192	278	3	ambikapur	ambikapur	NOUN
cana-2192	278	4	(	(	PUNCT
cana-2192	278	5	chhattisgarh	chhattisgarh	NOUN
cana-2192	278	6	)	)	PUNCT
cana-2192	278	7	(	(	PUNCT
cana-2192	278	8	a	a	DET
cana-2192	278	9	constituent	constituent	ADJ
cana-2192	278	10	college	college	NOUN
cana-2192	278	11	of	of	ADP
cana-2192	278	12	csvtu	csvtu	PROPN
cana-2192	278	13	,	,	PUNCT
cana-2192	278	14	bhilai	bhilai	PROPN
cana-2192	278	15	)	)	PUNCT
cana-2192	278	16	,	,	PUNCT
cana-2192	278	17	india	india	PROPN
cana-2192	278	18	email	email	NOUN
cana-2192	278	19	address	address	NOUN
cana-2192	278	20	:	:	PUNCT
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cana-2192	278	22	communications	communication	NOUN
cana-2192	278	23	on	on	ADP
cana-2192	278	24	applied	apply	VERB
cana-2192	278	25	nonlinear	nonlinear	ADJ
cana-2192	278	26	analysis	analysis	NOUN
cana-2192	278	27	issn	issn	NOUN
cana-2192	278	28	:	:	PUNCT
cana-2192	278	29	1074	1074	NUM
cana-2192	278	30	-	-	PUNCT
cana-2192	278	31	133x	133x	NUM
cana-2192	278	32	vol	vol	NOUN
cana-2192	278	33	32	32	NUM
cana-2192	278	34	no	no	NOUN
cana-2192	278	35	.	.	PUNCT
cana-2192	279	1	1s	1s	NUM
cana-2192	279	2	(	(	PUNCT
cana-2192	279	3	2025	2025	NUM
cana-2192	279	4	)	)	PUNCT
cana-2192	280	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2192	280	2	362	362	NUM
cana-2192	280	3	diptisharma2704@gmail.com	diptisharma2704@gmail.com	X
cana-2192	280	4	mailto:anilkumardby70@gmail.com	mailto:anilkumardby70@gmail.com	PROPN
cana-2192	280	5	mailto:mishra.urmila22@csvtu.ac.in	mailto:mishra.urmila22@csvtu.ac.in	PROPN
cana-2192	280	6	1	1	NUM
cana-2192	280	7	.	.	PUNCT
cana-2192	281	1	introduction	introduction	NOUN
cana-2192	281	2	2	2	NUM
cana-2192	281	3	.	.	PUNCT
cana-2192	281	4	main	main	ADJ
cana-2192	281	5	results	result	NOUN
cana-2192	281	6	references	reference	NOUN