id	sid	tid	token	lemma	pos
cana-760	1	1	communications	communication	NOUN
cana-760	1	2	on	on	ADP
cana-760	1	3	applied	apply	VERB
cana-760	1	4	nonlinear	nonlinear	ADJ
cana-760	1	5	analysis	analysis	NOUN
cana-760	1	6	issn	issn	NOUN
cana-760	1	7	:	:	PUNCT
cana-760	1	8	1074	1074	NUM
cana-760	1	9	-	-	PUNCT
cana-760	1	10	133x	133x	NUM
cana-760	1	11	vol	vol	NOUN
cana-760	1	12	31	31	NUM
cana-760	1	13	no	no	NOUN
cana-760	1	14	.	.	PUNCT
cana-760	2	1	3s	3s	NUM
cana-760	2	2	(	(	PUNCT
cana-760	2	3	2024	2024	NUM
cana-760	2	4	)	)	PUNCT
cana-760	2	5	212	212	NUM
cana-760	2	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-760	2	7	fixed	fix	VERB
cana-760	2	8	point	point	NOUN
cana-760	2	9	theorems	theorem	NOUN
cana-760	2	10	on	on	ADP
cana-760	2	11	modular	modular	ADJ
cana-760	2	12	revised	revise	VERB
cana-760	2	13	fuzzy	fuzzy	ADJ
cana-760	2	14	metric	metric	ADJ
cana-760	2	15	spaces	space	NOUN
cana-760	2	16	a.	a.	NOUN
cana-760	2	17	muraliraj1	muraliraj1	PROPN
cana-760	2	18	,	,	PUNCT
cana-760	2	19	p.	p.	PROPN
cana-760	2	20	shanmugavel2	shanmugavel2	PROPN
cana-760	2	21	,	,	PUNCT
cana-760	2	22	r.	r.	PROPN
cana-760	2	23	thangathamizh3	thangathamizh3	PROPN
cana-760	2	24	*	*	PROPN
cana-760	2	25	1pg	1pg	PROPN
cana-760	2	26	&	&	CCONJ
cana-760	2	27	research	research	PROPN
cana-760	2	28	department	department	PROPN
cana-760	2	29	of	of	ADP
cana-760	2	30	mathematics	mathematics	PROPN
cana-760	2	31	,	,	PUNCT
cana-760	2	32	urumu	urumu	PROPN
cana-760	2	33	dhanalakshmi	dhanalakshmi	PROPN
cana-760	2	34	college	college	NOUN
cana-760	2	35	,	,	PUNCT
cana-760	2	36	bharathidasan	bharathidasan	ADJ
cana-760	2	37	university	university	NOUN
cana-760	2	38	,	,	PUNCT
cana-760	2	39	trichy	trichy	PROPN
cana-760	2	40	,	,	PUNCT
cana-760	2	41	india	india	PROPN
cana-760	2	42	.	.	PUNCT
cana-760	3	1	email	email	NOUN
cana-760	4	1	i	i	PROPN
cana-760	4	2	d	d	PROPN
cana-760	4	3	:	:	PUNCT
cana-760	5	1	karguzali@gmail.com	karguzali@gmail.com	X
cana-760	5	2	.	.	PUNCT
cana-760	6	1	2department	2department	NUM
cana-760	6	2	of	of	ADP
cana-760	6	3	mathematics	mathematic	NOUN
cana-760	6	4	,	,	PUNCT
cana-760	6	5	selvamm	selvamm	PROPN
cana-760	6	6	arts	arts	PROPN
cana-760	6	7	and	and	CCONJ
cana-760	6	8	science	science	PROPN
cana-760	6	9	college	college	PROPN
cana-760	6	10	,	,	PUNCT
cana-760	6	11	periyar	periyar	PROPN
cana-760	6	12	university	university	NOUN
cana-760	6	13	,	,	PUNCT
cana-760	6	14	namakkal	namakkal	NOUN
cana-760	6	15	,	,	PUNCT
cana-760	6	16	india	india	PROPN
cana-760	6	17	.	.	PUNCT
cana-760	7	1	email	email	NOUN
cana-760	8	1	i	i	PROPN
cana-760	8	2	d	d	PROPN
cana-760	8	3	:	:	PUNCT
cana-760	8	4	p.sham1988@gmail.com	p.sham1988@gmail.com	X
cana-760	8	5	.	.	PUNCT
cana-760	9	1	3department	3department	NUM
cana-760	9	2	of	of	ADP
cana-760	9	3	mathematics	mathematic	NOUN
cana-760	9	4	,	,	PUNCT
cana-760	9	5	k.	k.	PROPN
cana-760	9	6	ramakrishnan	ramakrishnan	PROPN
cana-760	9	7	college	college	PROPN
cana-760	9	8	of	of	ADP
cana-760	9	9	engineering	engineering	PROPN
cana-760	9	10	,	,	PUNCT
cana-760	9	11	trichy	trichy	PROPN
cana-760	9	12	,	,	PUNCT
cana-760	9	13	india	india	PROPN
cana-760	9	14	.	.	PUNCT
cana-760	10	1	email	email	NOUN
cana-760	11	1	i	i	PROPN
cana-760	11	2	d	d	PROPN
cana-760	11	3	:	:	PUNCT
cana-760	11	4	thamizh1418@gmail.com	thamizh1418@gmail.com	X
cana-760	11	5	.	.	PUNCT
cana-760	12	1	*	*	PUNCT
cana-760	12	2	corresponding	correspond	VERB
cana-760	12	3	author	author	NOUN
cana-760	12	4	article	article	NOUN
cana-760	12	5	history	history	NOUN
cana-760	12	6	:	:	PUNCT
cana-760	12	7	received	receive	VERB
cana-760	12	8	:	:	PUNCT
cana-760	12	9	12	12	NUM
cana-760	12	10	-	-	PUNCT
cana-760	12	11	04	04	NUM
cana-760	12	12	-	-	PUNCT
cana-760	12	13	2024	2024	NUM
cana-760	12	14	revised	revise	VERB
cana-760	12	15	:	:	PUNCT
cana-760	12	16	27	27	NUM
cana-760	12	17	-	-	SYM
cana-760	12	18	05	05	NUM
cana-760	12	19	-	-	PUNCT
cana-760	12	20	2024	2024	NUM
cana-760	12	21	accepted	accept	VERB
cana-760	12	22	:	:	PUNCT
cana-760	12	23	14	14	NUM
cana-760	12	24	-	-	SYM
cana-760	12	25	06	06	NUM
cana-760	12	26	-	-	PUNCT
cana-760	12	27	2024	2024	NUM
cana-760	12	28	abstract	abstract	NOUN
cana-760	12	29	:	:	PUNCT
cana-760	12	30	in	in	ADP
cana-760	12	31	this	this	DET
cana-760	12	32	paper	paper	NOUN
cana-760	12	33	,	,	PUNCT
cana-760	12	34	we	we	PRON
cana-760	12	35	present	present	VERB
cana-760	12	36	a	a	DET
cana-760	12	37	new	new	ADJ
cana-760	12	38	space	space	NOUN
cana-760	12	39	which	which	PRON
cana-760	12	40	is	be	AUX
cana-760	12	41	a	a	DET
cana-760	12	42	melange	melange	NOUN
cana-760	12	43	between	between	ADP
cana-760	12	44	a	a	DET
cana-760	12	45	revised	revise	VERB
cana-760	12	46	fuzzy	fuzzy	ADJ
cana-760	12	47	metric	metric	ADJ
cana-760	12	48	space	space	NOUN
cana-760	12	49	and	and	CCONJ
cana-760	12	50	a	a	DET
cana-760	12	51	modular	modular	ADJ
cana-760	12	52	metric	metric	ADJ
cana-760	12	53	space	space	NOUN
cana-760	12	54	.	.	PUNCT
cana-760	13	1	we	we	PRON
cana-760	13	2	state	state	VERB
cana-760	13	3	some	some	DET
cana-760	13	4	properties	property	NOUN
cana-760	13	5	and	and	CCONJ
cana-760	13	6	examples	example	NOUN
cana-760	13	7	of	of	ADP
cana-760	13	8	our	our	PRON
cana-760	13	9	new	new	ADJ
cana-760	13	10	space	space	NOUN
cana-760	13	11	.	.	PUNCT
cana-760	14	1	then	then	ADV
cana-760	14	2	,	,	PUNCT
cana-760	14	3	we	we	PRON
cana-760	14	4	formulate	formulate	VERB
cana-760	14	5	and	and	CCONJ
cana-760	14	6	prove	prove	VERB
cana-760	14	7	the	the	DET
cana-760	14	8	existence	existence	NOUN
cana-760	14	9	and	and	CCONJ
cana-760	14	10	uniqueness	uniqueness	ADJ
cana-760	14	11	results	result	NOUN
cana-760	14	12	of	of	ADP
cana-760	14	13	a	a	DET
cana-760	14	14	fixed	fix	VERB
cana-760	14	15	point	point	NOUN
cana-760	14	16	for	for	ADP
cana-760	14	17	continuous	continuous	ADJ
cana-760	14	18	mappings	mapping	NOUN
cana-760	14	19	under	under	ADP
cana-760	14	20	this	this	DET
cana-760	14	21	new	new	ADJ
cana-760	14	22	space	space	NOUN
cana-760	14	23	.	.	PUNCT
cana-760	15	1	to	to	PART
cana-760	15	2	support	support	VERB
cana-760	15	3	our	our	PRON
cana-760	15	4	results	result	NOUN
cana-760	15	5	,	,	PUNCT
cana-760	15	6	we	we	PRON
cana-760	15	7	introduce	introduce	VERB
cana-760	15	8	some	some	DET
cana-760	15	9	examples	example	NOUN
cana-760	15	10	and	and	CCONJ
cana-760	15	11	an	an	DET
cana-760	15	12	application	application	NOUN
cana-760	15	13	.	.	PUNCT
cana-760	16	1	keywords	keyword	NOUN
cana-760	16	2	:	:	PUNCT
cana-760	16	3	modular	modular	ADJ
cana-760	16	4	metric	metric	ADJ
cana-760	16	5	space	space	NOUN
cana-760	16	6	,	,	PUNCT
cana-760	16	7	revised	revise	VERB
cana-760	16	8	fuzzy	fuzzy	ADJ
cana-760	16	9	metric	metric	ADJ
cana-760	16	10	space	space	NOUN
cana-760	16	11	,	,	PUNCT
cana-760	16	12	modular	modular	NOUN
cana-760	16	13	revised	revise	VERB
cana-760	16	14	fuzzy	fuzzy	ADJ
cana-760	16	15	metric	metric	ADJ
cana-760	16	16	space	space	NOUN
cana-760	16	17	,	,	PUNCT
cana-760	16	18	fixed	fix	VERB
cana-760	16	19	point	point	NOUN
cana-760	16	20	.	.	PUNCT
cana-760	17	1	msc2020	msc2020	PROPN
cana-760	17	2	:	:	PUNCT
cana-760	17	3	54h25	54h25	NUM
cana-760	17	4	,	,	PUNCT
cana-760	17	5	47h10	47h10	NUM
cana-760	17	6	.	.	NOUN
cana-760	18	1	1	1	NUM
cana-760	18	2	.	.	X
cana-760	18	3	introduction	introduction	NOUN
cana-760	18	4	in	in	ADP
cana-760	18	5	1965	1965	NUM
cana-760	18	6	,	,	PUNCT
cana-760	18	7	zadeh	zadeh	PROPN
cana-760	19	1	[	[	X
cana-760	19	2	24	24	NUM
cana-760	19	3	]	]	PUNCT
cana-760	19	4	presented	present	VERB
cana-760	19	5	the	the	DET
cana-760	19	6	concept	concept	NOUN
cana-760	19	7	of	of	ADP
cana-760	19	8	a	a	DET
cana-760	19	9	fuzzy	fuzzy	ADJ
cana-760	19	10	set	set	NOUN
cana-760	19	11	.	.	PUNCT
cana-760	20	1	ten	ten	NUM
cana-760	20	2	years	year	NOUN
cana-760	20	3	later	later	ADV
cana-760	20	4	,	,	PUNCT
cana-760	20	5	kramosil	kramosil	NOUN
cana-760	20	6	and	and	CCONJ
cana-760	20	7	michalek	michalek	VERB
cana-760	21	1	[	[	X
cana-760	21	2	12	12	NUM
cana-760	21	3	]	]	PUNCT
cana-760	21	4	stated	state	VERB
cana-760	21	5	the	the	DET
cana-760	21	6	definition	definition	NOUN
cana-760	21	7	of	of	ADP
cana-760	21	8	fuzzy	fuzzy	ADJ
cana-760	21	9	metric	metric	ADJ
cana-760	21	10	spaces	space	NOUN
cana-760	21	11	.	.	PUNCT
cana-760	22	1	in	in	ADP
cana-760	22	2	1988	1988	NUM
cana-760	22	3	,	,	PUNCT
cana-760	22	4	grabiec	grabiec	PROPN
cana-760	22	5	[	[	X
cana-760	22	6	7	7	X
cana-760	22	7	]	]	PUNCT
cana-760	22	8	implemented	implement	VERB
cana-760	22	9	the	the	DET
cana-760	22	10	notion	notion	NOUN
cana-760	22	11	of	of	ADP
cana-760	22	12	fuzzy	fuzzy	ADJ
cana-760	22	13	metric	metric	ADJ
cana-760	22	14	space	space	NOUN
cana-760	22	15	to	to	PART
cana-760	22	16	extend	extend	VERB
cana-760	22	17	the	the	DET
cana-760	22	18	banach	banach	NOUN
cana-760	22	19	contraction	contraction	NOUN
cana-760	22	20	theorem	theorem	VERB
cana-760	22	21	over	over	ADP
cana-760	22	22	this	this	DET
cana-760	22	23	space	space	NOUN
cana-760	22	24	.	.	PUNCT
cana-760	23	1	posteriorly	posteriorly	NOUN
cana-760	23	2	,	,	PUNCT
cana-760	23	3	george	george	NOUN
cana-760	23	4	and	and	CCONJ
cana-760	23	5	veeramani	veeramani	NOUN
cana-760	24	1	[	[	X
cana-760	24	2	6	6	NUM
cana-760	24	3	]	]	PUNCT
cana-760	24	4	employed	employ	VERB
cana-760	24	5	the	the	DET
cana-760	24	6	definition	definition	NOUN
cana-760	24	7	of	of	ADP
cana-760	24	8	t	t	PROPN
cana-760	24	9	-	-	PUNCT
cana-760	24	10	norm	norm	NOUN
cana-760	24	11	to	to	PART
cana-760	24	12	formulate	formulate	VERB
cana-760	24	13	and	and	CCONJ
cana-760	24	14	introduce	introduce	VERB
cana-760	24	15	some	some	DET
cana-760	24	16	results	result	NOUN
cana-760	24	17	on	on	ADP
cana-760	24	18	the	the	DET
cana-760	24	19	notion	notion	NOUN
cana-760	24	20	of	of	ADP
cana-760	24	21	a	a	DET
cana-760	24	22	fuzzy	fuzzy	ADJ
cana-760	24	23	metric	metric	ADJ
cana-760	24	24	space	space	NOUN
cana-760	24	25	.	.	PUNCT
cana-760	25	1	then	then	ADV
cana-760	25	2	,	,	PUNCT
cana-760	25	3	several	several	ADJ
cana-760	25	4	researchers	researcher	NOUN
cana-760	25	5	presented	present	VERB
cana-760	25	6	different	different	ADJ
cana-760	25	7	contraction	contraction	NOUN
cana-760	25	8	conditions	condition	NOUN
cana-760	25	9	over	over	ADP
cana-760	25	10	fuzzy	fuzzy	ADJ
cana-760	25	11	metric	metric	ADJ
cana-760	25	12	spaces	space	NOUN
cana-760	25	13	.	.	PUNCT
cana-760	26	1	in	in	ADP
cana-760	26	2	2010	2010	NUM
cana-760	26	3	,	,	PUNCT
cana-760	26	4	chistyakov	chistyakov	NOUN
cana-760	26	5	[	[	X
cana-760	26	6	3	3	NUM
cana-760	26	7	-	-	SYM
cana-760	26	8	5	5	NUM
cana-760	26	9	]	]	PUNCT
cana-760	26	10	introduced	introduce	VERB
cana-760	26	11	the	the	DET
cana-760	26	12	notion	notion	NOUN
cana-760	26	13	of	of	ADP
cana-760	26	14	modular	modular	ADJ
cana-760	26	15	metric	metric	ADJ
cana-760	26	16	spaces	space	NOUN
cana-760	26	17	.	.	PUNCT
cana-760	27	1	then	then	ADV
cana-760	27	2	,	,	PUNCT
cana-760	27	3	numerous	numerous	ADJ
cana-760	27	4	mathematicians	mathematician	NOUN
cana-760	27	5	discussed	discuss	VERB
cana-760	27	6	different	different	ADJ
cana-760	27	7	results	result	NOUN
cana-760	27	8	in	in	ADP
cana-760	27	9	their	their	PRON
cana-760	27	10	works	work	NOUN
cana-760	27	11	over	over	ADP
cana-760	27	12	modular	modular	ADJ
cana-760	27	13	metric	metric	ADJ
cana-760	27	14	spaces	space	NOUN
cana-760	27	15	,	,	PUNCT
cana-760	27	16	for	for	ADP
cana-760	27	17	example	example	NOUN
cana-760	27	18	,	,	PUNCT
cana-760	27	19	look	look	VERB
cana-760	27	20	at	at	ADP
cana-760	27	21	the	the	DET
cana-760	27	22	references	reference	NOUN
cana-760	27	23	[	[	X
cana-760	27	24	16	16	NUM
cana-760	27	25	,	,	PUNCT
cana-760	27	26	18	18	NUM
cana-760	27	27	-	-	SYM
cana-760	27	28	19	19	NUM
cana-760	27	29	,	,	PUNCT
cana-760	27	30	21,2	21,2	NUM
cana-760	27	31	-	-	PUNCT
cana-760	27	32	24].muraliraj	24].muraliraj	PROPN
cana-760	27	33	and	and	CCONJ
cana-760	27	34	thangathamizh	thangathamizh	ADJ
cana-760	27	35	[	[	X
cana-760	27	36	17	17	NUM
cana-760	27	37	]	]	PUNCT
cana-760	27	38	used	use	VERB
cana-760	27	39	the	the	DET
cana-760	27	40	revised	revise	VERB
cana-760	27	41	fuzzy	fuzzy	ADJ
cana-760	27	42	set	set	NOUN
cana-760	27	43	technique	technique	NOUN
cana-760	27	44	to	to	PART
cana-760	27	45	start	start	VERB
cana-760	27	46	a	a	DET
cana-760	27	47	family	family	NOUN
cana-760	27	48	of	of	ADP
cana-760	27	49	revised	revise	VERB
cana-760	27	50	fuzzy	fuzzy	ADJ
cana-760	27	51	mappings	mapping	NOUN
cana-760	27	52	that	that	PRON
cana-760	27	53	are	be	AUX
cana-760	27	54	extensions	extension	NOUN
cana-760	27	55	of	of	ADP
cana-760	27	56	multivalued	multivalue	VERB
cana-760	27	57	mappings	mapping	NOUN
cana-760	27	58	and	and	CCONJ
cana-760	27	59	produced	produce	VERB
cana-760	27	60	a	a	DET
cana-760	27	61	result	result	NOUN
cana-760	27	62	in	in	ADP
cana-760	27	63	revised	revise	VERB
cana-760	27	64	fuzzy	fuzzy	ADJ
cana-760	27	65	metric	metric	ADJ
cana-760	27	66	space	space	NOUN
cana-760	27	67	for	for	ADP
cana-760	27	68	these	these	DET
cana-760	27	69	mappings	mapping	NOUN
cana-760	27	70	in	in	ADP
cana-760	27	71	2021	2021	NUM
cana-760	27	72	.	.	PUNCT
cana-760	28	1	muraliraj	muraliraj	PROPN
cana-760	28	2	and	and	CCONJ
cana-760	28	3	thangathamizh	thangathamizh	PROPN
cana-760	29	1	[	[	X
cana-760	29	2	16	16	NUM
cana-760	29	3	]	]	PUNCT
cana-760	29	4	were	be	AUX
cana-760	29	5	the	the	DET
cana-760	29	6	first	first	ADJ
cana-760	29	7	to	to	PART
cana-760	29	8	establish	establish	VERB
cana-760	29	9	the	the	DET
cana-760	29	10	idea	idea	NOUN
cana-760	29	11	of	of	ADP
cana-760	29	12	revised	revise	VERB
cana-760	29	13	fuzzy	fuzzy	ADJ
cana-760	29	14	contractive	contractive	ADJ
cana-760	29	15	mappings	mapping	NOUN
cana-760	29	16	and	and	CCONJ
cana-760	29	17	show	show	VERB
cana-760	29	18	a	a	DET
cana-760	29	19	fixed	fix	VERB
cana-760	29	20	point	point	NOUN
cana-760	29	21	theorem	theorem	VERB
cana-760	29	22	in	in	ADP
cana-760	29	23	revised	revise	VERB
cana-760	29	24	fuzzy	fuzzy	ADJ
cana-760	29	25	metric	metric	ADJ
cana-760	29	26	spaces	space	NOUN
cana-760	29	27	for	for	ADP
cana-760	29	28	these	these	DET
cana-760	29	29	mappings	mapping	NOUN
cana-760	29	30	.	.	PUNCT
cana-760	30	1	the	the	DET
cana-760	30	2	rational	rational	ADJ
cana-760	30	3	type	type	NOUN
cana-760	30	4	revised	revise	VERB
cana-760	30	5	fuzzy	fuzzy	ADJ
cana-760	30	6	-	-	PUNCT
cana-760	30	7	contraction	contraction	NOUN
cana-760	30	8	condition	condition	NOUN
cana-760	30	9	in	in	ADP
cana-760	30	10	rfm	rfm	PROPN
cana-760	30	11	spaces	space	NOUN
cana-760	30	12	was	be	AUX
cana-760	30	13	recently	recently	ADV
cana-760	30	14	established	establish	VERB
cana-760	30	15	by	by	ADP
cana-760	30	16	muraliraj	muraliraj	PROPN
cana-760	30	17	et	et	PROPN
cana-760	30	18	al	al	PROPN
cana-760	30	19	.	.	PUNCT
cana-760	31	1	[	[	X
cana-760	31	2	14	14	NUM
cana-760	31	3	]	]	PUNCT
cana-760	31	4	,	,	PUNCT
cana-760	31	5	who	who	PRON
cana-760	31	6	also	also	ADV
cana-760	31	7	proved	prove	VERB
cana-760	31	8	other	other	ADJ
cana-760	31	9	fp	fp	NOUN
cana-760	31	10	theorems	theorem	NOUN
cana-760	31	11	with	with	ADP
cana-760	31	12	an	an	DET
cana-760	31	13	application	application	NOUN
cana-760	31	14	.	.	PUNCT
cana-760	32	1	the	the	DET
cana-760	32	2	concept	concept	NOUN
cana-760	32	3	of	of	ADP
cana-760	32	4	revised	revise	VERB
cana-760	32	5	fuzzy	fuzzy	ADJ
cana-760	32	6	cone	cone	NOUN
cana-760	32	7	metric	metric	NOUN
cana-760	32	8	(	(	PUNCT
cana-760	32	9	rfcm	rfcm	NOUN
cana-760	32	10	)	)	PUNCT
cana-760	32	11	communications	communication	NOUN
cana-760	32	12	on	on	ADP
cana-760	32	13	applied	apply	VERB
cana-760	32	14	nonlinear	nonlinear	ADJ
cana-760	32	15	analysis	analysis	NOUN
cana-760	32	16	issn	issn	NOUN
cana-760	32	17	:	:	PUNCT
cana-760	32	18	1074	1074	NUM
cana-760	32	19	-	-	PUNCT
cana-760	32	20	133x	133x	NUM
cana-760	32	21	vol	vol	NOUN
cana-760	32	22	31	31	NUM
cana-760	32	23	no	no	NOUN
cana-760	32	24	.	.	PUNCT
cana-760	33	1	3s	3s	NUM
cana-760	33	2	(	(	PUNCT
cana-760	33	3	2024	2024	NUM
cana-760	33	4	)	)	PUNCT
cana-760	33	5	213	213	NUM
cana-760	33	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	33	7	space	space	NOUN
cana-760	33	8	was	be	AUX
cana-760	33	9	first	first	ADV
cana-760	33	10	presented	present	VERB
cana-760	33	11	in	in	ADP
cana-760	33	12	2023	2023	NUM
cana-760	33	13	by	by	ADP
cana-760	33	14	thangathamizh	thangathamizh	PROPN
cana-760	33	15	et	et	PROPN
cana-760	33	16	al	al	PROPN
cana-760	33	17	.	.	PUNCT
cana-760	34	1	[	[	X
cana-760	34	2	15	15	NUM
cana-760	34	3	]	]	PUNCT
cana-760	34	4	.	.	PUNCT
cana-760	35	1	under	under	ADP
cana-760	35	2	the	the	DET
cana-760	35	3	presumption	presumption	NOUN
cana-760	35	4	that	that	SCONJ
cana-760	35	5	"	"	PUNCT
cana-760	35	6	the	the	DET
cana-760	35	7	revised	revise	VERB
cana-760	35	8	fuzzy	fuzzy	ADJ
cana-760	35	9	cone	cone	NOUN
cana-760	35	10	contractive	contractive	ADJ
cana-760	35	11	sequences	sequence	NOUN
cana-760	35	12	are	be	AUX
cana-760	35	13	cauchy	cauchy	NOUN
cana-760	35	14	,	,	PUNCT
cana-760	35	15	"	"	PUNCT
cana-760	35	16	they	they	PRON
cana-760	35	17	demonstrated	demonstrate	VERB
cana-760	35	18	a	a	DET
cana-760	35	19	few	few	ADJ
cana-760	35	20	fundamental	fundamental	ADJ
cana-760	35	21	features	feature	NOUN
cana-760	35	22	of	of	ADP
cana-760	35	23	fp	fp	INTJ
cana-760	35	24	as	as	ADV
cana-760	35	25	well	well	ADV
cana-760	35	26	as	as	ADP
cana-760	35	27	a	a	DET
cana-760	35	28	"	"	PUNCT
cana-760	35	29	revised	revise	VERB
cana-760	35	30	fuzzy	fuzzy	ADJ
cana-760	35	31	cone	cone	NOUN
cana-760	35	32	banach	banach	NOUN
cana-760	35	33	contraction	contraction	NOUN
cana-760	35	34	theorem	theorem	VERB
cana-760	35	35	.	.	PUNCT
cana-760	35	36	"	"	PUNCT
cana-760	36	1	afterwards	afterwards	ADV
cana-760	36	2	,	,	PUNCT
cana-760	36	3	several	several	ADJ
cana-760	36	4	fp	fp	ADJ
cana-760	36	5	theorems	theorem	NOUN
cana-760	36	6	in	in	ADP
cana-760	36	7	rfcms	rfcms	NOUN
cana-760	36	8	were	be	AUX
cana-760	36	9	proven	prove	VERB
cana-760	36	10	by	by	ADP
cana-760	36	11	muraliraj	muraliraj	PROPN
cana-760	36	12	and	and	CCONJ
cana-760	36	13	thangathamizh	thangathamizh	ADJ
cana-760	36	14	[	[	X
cana-760	36	15	20	20	NUM
cana-760	36	16	,	,	PUNCT
cana-760	36	17	25	25	NUM
cana-760	36	18	-	-	SYM
cana-760	36	19	27	27	NUM
cana-760	36	20	]	]	PUNCT
cana-760	36	21	without	without	ADP
cana-760	36	22	requiring	require	VERB
cana-760	36	23	that	that	SCONJ
cana-760	36	24	the	the	DET
cana-760	36	25	"	"	PUNCT
cana-760	36	26	revised	revise	VERB
cana-760	36	27	fuzzy	fuzzy	ADJ
cana-760	36	28	cone	cone	NOUN
cana-760	36	29	contractive	contractive	ADJ
cana-760	36	30	sequences	sequence	NOUN
cana-760	36	31	are	be	AUX
cana-760	36	32	cauchy	cauchy	NOUN
cana-760	36	33	.	.	PUNCT
cana-760	37	1	in	in	ADP
cana-760	37	2	this	this	DET
cana-760	37	3	paper	paper	NOUN
cana-760	37	4	,	,	PUNCT
cana-760	37	5	we	we	PRON
cana-760	37	6	introduce	introduce	VERB
cana-760	37	7	a	a	DET
cana-760	37	8	new	new	ADJ
cana-760	37	9	space	space	NOUN
cana-760	37	10	named	name	VERB
cana-760	37	11	modular	modular	ADJ
cana-760	37	12	revised	revise	VERB
cana-760	37	13	fuzzy	fuzzy	ADJ
cana-760	37	14	metric	metric	ADJ
cana-760	37	15	space	space	NOUN
cana-760	37	16	.	.	PUNCT
cana-760	38	1	we	we	PRON
cana-760	38	2	launch	launch	VERB
cana-760	38	3	some	some	DET
cana-760	38	4	fixed	fix	VERB
cana-760	38	5	point	point	NOUN
cana-760	38	6	results	result	NOUN
cana-760	38	7	over	over	ADP
cana-760	38	8	a	a	DET
cana-760	38	9	modular	modular	ADJ
cana-760	38	10	fuzzy	fuzzy	ADJ
cana-760	38	11	metric	metric	ADJ
cana-760	38	12	spaces	space	NOUN
cana-760	38	13	.	.	PUNCT
cana-760	39	1	to	to	PART
cana-760	39	2	analyse	analyse	VERB
cana-760	39	3	our	our	PRON
cana-760	39	4	work	work	NOUN
cana-760	39	5	,	,	PUNCT
cana-760	39	6	we	we	PRON
cana-760	39	7	state	state	VERB
cana-760	39	8	some	some	DET
cana-760	39	9	examples	example	NOUN
cana-760	39	10	,	,	PUNCT
cana-760	39	11	corollaries	corollary	NOUN
cana-760	39	12	,	,	PUNCT
cana-760	39	13	and	and	CCONJ
cana-760	39	14	an	an	DET
cana-760	39	15	application	application	NOUN
cana-760	39	16	.	.	PUNCT
cana-760	40	1	2	2	X
cana-760	40	2	.	.	X
cana-760	40	3	preliminaries	preliminary	NOUN
cana-760	40	4	in	in	ADP
cana-760	40	5	this	this	DET
cana-760	40	6	section	section	NOUN
cana-760	40	7	,	,	PUNCT
cana-760	40	8	we	we	PRON
cana-760	40	9	will	will	AUX
cana-760	40	10	recall	recall	VERB
cana-760	40	11	some	some	DET
cana-760	40	12	definitions	definition	NOUN
cana-760	40	13	which	which	PRON
cana-760	40	14	are	be	AUX
cana-760	40	15	crucial	crucial	ADJ
cana-760	40	16	in	in	ADP
cana-760	40	17	this	this	DET
cana-760	40	18	paper	paper	NOUN
cana-760	40	19	.	.	PUNCT
cana-760	41	1	definition	definition	NOUN
cana-760	41	2	2.1	2.1	NUM
cana-760	41	3	.	.	PUNCT
cana-760	42	1	[	[	X
cana-760	42	2	24	24	NUM
cana-760	42	3	]	]	PUNCT
cana-760	42	4	let	let	VERB
cana-760	42	5	y	y	PRON
cana-760	42	6	be	be	AUX
cana-760	42	7	any	any	DET
cana-760	42	8	set	set	NOUN
cana-760	42	9	.	.	PUNCT
cana-760	43	1	a	a	DET
cana-760	43	2	fuzzy	fuzzy	ADJ
cana-760	43	3	set	set	NOUN
cana-760	43	4	e	e	NOUN
cana-760	43	5	in	in	ADP
cana-760	43	6	y	y	PROPN
cana-760	43	7	is	be	AUX
cana-760	43	8	a	a	DET
cana-760	43	9	function	function	NOUN
cana-760	43	10	with	with	ADP
cana-760	43	11	domain	domain	NOUN
cana-760	43	12	y	y	PROPN
cana-760	43	13	and	and	CCONJ
cana-760	43	14	values	value	NOUN
cana-760	43	15	in	in	ADP
cana-760	43	16	[	[	X
cana-760	43	17	0,1	0,1	NUM
cana-760	43	18	]	]	PUNCT
cana-760	43	19	.	.	PUNCT
cana-760	44	1	definition	definition	NOUN
cana-760	44	2	2.2	2.2	NUM
cana-760	44	3	.	.	PUNCT
cana-760	45	1	[	[	X
cana-760	45	2	23	23	NUM
cana-760	45	3	]	]	PUNCT
cana-760	45	4	given	give	VERB
cana-760	45	5	a	a	DET
cana-760	45	6	binary	binary	ADJ
cana-760	45	7	operation	operation	NOUN
cana-760	45	8	⊛	⊛	NUM
cana-760	45	9	:	:	PUNCT
cana-760	46	1	[	[	X
cana-760	46	2	0,1]2	0,1]2	NUM
cana-760	46	3	→	→	SYM
cana-760	46	4	[	[	X
cana-760	46	5	0,1	0,1	NUM
cana-760	46	6	]	]	PUNCT
cana-760	46	7	.	.	PUNCT
cana-760	47	1	an	an	DET
cana-760	47	2	operator	operator	NOUN
cana-760	47	3	⊛	⊛	NUM
cana-760	47	4	is	be	AUX
cana-760	47	5	a	a	DET
cana-760	47	6	continuous	continuous	ADJ
cana-760	47	7	t	t	NOUN
cana-760	47	8	-	-	PUNCT
cana-760	47	9	conorm	conorm	NOUN
cana-760	47	10	if	if	SCONJ
cana-760	47	11	∀𝛼	∀𝛼	NOUN
cana-760	47	12	,	,	PUNCT
cana-760	47	13	𝛽	𝛽	NOUN
cana-760	47	14	,	,	PUNCT
cana-760	47	15	𝛾	𝛾	PROPN
cana-760	47	16	,	,	PUNCT
cana-760	47	17	𝛿	𝛿	PRON
cana-760	47	18	∈	∈	PROPN
cana-760	47	19	[	[	X
cana-760	47	20	0,1	0,1	NUM
cana-760	47	21	]	]	X
cana-760	47	22	satisfy	satisfy	NOUN
cana-760	47	23	:	:	PUNCT
cana-760	47	24	(	(	PUNCT
cana-760	47	25	1	1	X
cana-760	47	26	)	)	PUNCT
cana-760	47	27	𝛼	𝛼	SYM
cana-760	47	28	⊛	⊛	NUM
cana-760	47	29	𝛽	𝛽	NOUN
cana-760	47	30	=	=	SYM
cana-760	47	31	𝛽	𝛽	PROPN
cana-760	47	32	⊛	⊛	NUM
cana-760	47	33	𝛼.	𝛼.	NOUN
cana-760	47	34	(	(	PUNCT
cana-760	47	35	2	2	NUM
cana-760	47	36	)	)	PUNCT
cana-760	47	37	(	(	PUNCT
cana-760	47	38	𝛼	𝛼	PROPN
cana-760	47	39	⊛	⊛	NUM
cana-760	47	40	𝛽	𝛽	NOUN
cana-760	47	41	)	)	PUNCT
cana-760	47	42	⊛	⊛	NUM
cana-760	47	43	𝛾	𝛾	NOUN
cana-760	47	44	=	=	SYM
cana-760	47	45	𝛼	𝛼	X
cana-760	47	46	⊛	⊛	NUM
cana-760	47	47	(	(	PUNCT
cana-760	47	48	𝛽	𝛽	NOUN
cana-760	47	49	⊛	⊛	NUM
cana-760	47	50	𝛾	𝛾	NOUN
cana-760	47	51	)	)	PUNCT
cana-760	47	52	.	.	PUNCT
cana-760	48	1	(	(	PUNCT
cana-760	48	2	3	3	X
cana-760	48	3	)	)	PUNCT
cana-760	48	4	𝛼	𝛼	PROPN
cana-760	48	5	⊛	⊛	ADJ
cana-760	48	6	0	0	NUM
cana-760	48	7	=	=	SYM
cana-760	48	8	𝛼.	𝛼.	NOUN
cana-760	48	9	(	(	PUNCT
cana-760	48	10	4	4	X
cana-760	48	11	)	)	PUNCT
cana-760	48	12	if	if	SCONJ
cana-760	48	13	𝛼	𝛼	X
cana-760	48	14	≤	≤	NOUN
cana-760	48	15	𝛾	𝛾	ADP
cana-760	48	16	and	and	CCONJ
cana-760	48	17	𝛽	𝛽	NOUN
cana-760	48	18	≤	≤	NOUN
cana-760	48	19	𝛿	𝛿	X
cana-760	48	20	,	,	PUNCT
cana-760	48	21	then	then	ADV
cana-760	48	22	𝛼	𝛼	X
cana-760	48	23	⊛	⊛	ADJ
cana-760	48	24	𝛽	𝛽	NOUN
cana-760	48	25	≤	≤	NOUN
cana-760	48	26	𝛾	𝛾	ADP
cana-760	48	27	⊛	⊛	NUM
cana-760	48	28	𝛿.	𝛿.	ADJ
cana-760	48	29	a.	a.	NOUN
cana-760	48	30	sostak	sostak	NOUN
cana-760	49	1	[	[	X
cana-760	49	2	2	2	X
cana-760	49	3	]	]	PUNCT
cana-760	49	4	in	in	ADP
cana-760	49	5	2018	2018	NUM
cana-760	49	6	introduced	introduce	VERB
cana-760	49	7	the	the	DET
cana-760	49	8	concept	concept	NOUN
cana-760	49	9	of	of	ADP
cana-760	49	10	a	a	DET
cana-760	49	11	revised	revise	VERB
cana-760	49	12	fuzzy	fuzzy	ADJ
cana-760	49	13	metric	metric	ADJ
cana-760	49	14	space	space	NOUN
cana-760	49	15	using	use	VERB
cana-760	49	16	the	the	DET
cana-760	49	17	defintion	defintion	NOUN
cana-760	49	18	of	of	ADP
cana-760	49	19	t	t	PROPN
cana-760	49	20	-	-	PUNCT
cana-760	49	21	conorm	conorm	NOUN
cana-760	49	22	as	as	SCONJ
cana-760	49	23	follows	follow	VERB
cana-760	49	24	:	:	PUNCT
cana-760	49	25	definition	definition	NOUN
cana-760	49	26	2.3	2.3	NUM
cana-760	49	27	.	.	PUNCT
cana-760	50	1	[	[	X
cana-760	50	2	2	2	X
cana-760	50	3	]	]	PUNCT
cana-760	50	4	the	the	DET
cana-760	50	5	triplet	triplet	NOUN
cana-760	50	6	(	(	PUNCT
cana-760	50	7	𝑌	𝑌	PROPN
cana-760	50	8	,	,	PUNCT
cana-760	50	9	𝛬,⊛	𝛬,⊛	PROPN
cana-760	50	10	)	)	PUNCT
cana-760	50	11	is	be	AUX
cana-760	50	12	called	call	VERB
cana-760	50	13	a	a	DET
cana-760	50	14	revised	revise	VERB
cana-760	50	15	fuzzy	fuzzy	ADJ
cana-760	50	16	metric	metric	ADJ
cana-760	50	17	space	space	NOUN
cana-760	50	18	if	if	SCONJ
cana-760	50	19	y	y	PROPN
cana-760	50	20	is	be	AUX
cana-760	50	21	an	an	DET
cana-760	50	22	arbitrary	arbitrary	ADJ
cana-760	50	23	set	set	NOUN
cana-760	50	24	,	,	PUNCT
cana-760	50	25	⊛	⊛	NUM
cana-760	50	26	is	be	AUX
cana-760	50	27	a	a	DET
cana-760	50	28	continuous	continuous	ADJ
cana-760	50	29	t	t	NOUN
cana-760	50	30	-	-	PUNCT
cana-760	50	31	conorm	conorm	NOUN
cana-760	50	32	and	and	CCONJ
cana-760	50	33	λ	λ	PROPN
cana-760	50	34	is	be	AUX
cana-760	50	35	a	a	DET
cana-760	50	36	revised	revise	VERB
cana-760	50	37	fuzzy	fuzzy	ADJ
cana-760	50	38	metric	metric	ADJ
cana-760	50	39	on	on	ADP
cana-760	50	40	𝑌2	𝑌2	PROPN
cana-760	50	41	×	×	NOUN
cana-760	50	42	(	(	PUNCT
cana-760	50	43	0	0	NUM
cana-760	50	44	,	,	PUNCT
cana-760	50	45	∞	∞	NUM
cana-760	50	46	)	)	PUNCT
cana-760	50	47	→	→	PUNCT
cana-760	51	1	[	[	X
cana-760	51	2	0,1	0,1	NUM
cana-760	51	3	]	]	PUNCT
cana-760	51	4	for	for	ADP
cana-760	51	5	all	all	DET
cana-760	51	6	𝜄	𝜄	PROPN
cana-760	51	7	,	,	PUNCT
cana-760	51	8	휂	휂	PROPN
cana-760	51	9	,	,	PUNCT
cana-760	51	10	𝜗	𝜗	NOUN
cana-760	51	11	in	in	ADP
cana-760	51	12	𝑌	𝑌	PROPN
cana-760	51	13	,	,	PUNCT
cana-760	51	14	and	and	CCONJ
cana-760	51	15	for	for	ADP
cana-760	51	16	all	all	DET
cana-760	51	17	𝑠	𝑠	PROPN
cana-760	51	18	,	,	PUNCT
cana-760	51	19	𝑡	𝑡	X
cana-760	51	20	>	>	X
cana-760	51	21	0	0	PUNCT
cana-760	51	22	satisfying	satisfy	VERB
cana-760	51	23	the	the	DET
cana-760	51	24	following	follow	VERB
cana-760	51	25	conditions	condition	NOUN
cana-760	51	26	:	:	PUNCT
cana-760	51	27	(	(	PUNCT
cana-760	51	28	1	1	X
cana-760	51	29	)	)	PUNCT
cana-760	51	30	𝛬(𝜄	𝛬(𝜄	NUM
cana-760	51	31	,	,	PUNCT
cana-760	51	32	휂	휂	ADP
cana-760	51	33	,	,	PUNCT
cana-760	51	34	0	0	NUM
cana-760	51	35	)	)	PUNCT
cana-760	51	36	=	=	SYM
cana-760	51	37	0	0	NUM
cana-760	51	38	,	,	PUNCT
cana-760	51	39	𝛬(𝜄	𝛬(𝜄	PRON
cana-760	51	40	,	,	PUNCT
cana-760	51	41	휂	휂	ADP
cana-760	51	42	,	,	PUNCT
cana-760	51	43	𝑡	𝑡	NOUN
cana-760	51	44	)	)	PUNCT
cana-760	51	45	<	<	X
cana-760	51	46	1	1	NUM
cana-760	51	47	,	,	PUNCT
cana-760	51	48	∀𝑡	∀𝑡	NOUN
cana-760	51	49	>	>	X
cana-760	51	50	0	0	X
cana-760	51	51	.	.	PUNCT
cana-760	52	1	(	(	PUNCT
cana-760	52	2	2	2	NUM
cana-760	52	3	)	)	PUNCT
cana-760	52	4	𝛬(𝜄	𝛬(𝜄	NUM
cana-760	52	5	,	,	PUNCT
cana-760	52	6	휂	휂	ADP
cana-760	52	7	,	,	PUNCT
cana-760	52	8	𝑡	𝑡	NOUN
cana-760	52	9	)	)	PUNCT
cana-760	52	10	=	=	SYM
cana-760	52	11	0	0	PUNCT
cana-760	53	1	if	if	SCONJ
cana-760	53	2	and	and	CCONJ
cana-760	53	3	only	only	ADV
cana-760	53	4	if	if	SCONJ
cana-760	53	5	𝜄	𝜄	PROPN
cana-760	53	6	=	=	SYM
cana-760	53	7	휂	휂	PROPN
cana-760	53	8	,	,	PUNCT
cana-760	53	9	for	for	ADP
cana-760	53	10	all	all	DET
cana-760	53	11	𝑡	𝑡	NOUN
cana-760	53	12	>	>	X
cana-760	53	13	0	0	NUM
cana-760	53	14	.	.	PUNCT
cana-760	54	1	(	(	PUNCT
cana-760	54	2	3	3	NUM
cana-760	54	3	)	)	PUNCT
cana-760	54	4	𝛬(𝜄	𝛬(𝜄	NUM
cana-760	54	5	,	,	PUNCT
cana-760	54	6	휂	휂	ADP
cana-760	54	7	,	,	PUNCT
cana-760	54	8	𝑡	𝑡	NOUN
cana-760	54	9	)	)	PUNCT
cana-760	54	10	=	=	SYM
cana-760	55	1	𝛬(휂	𝛬(휂	NOUN
cana-760	55	2	,	,	PUNCT
cana-760	55	3	𝜄	𝜄	PROPN
cana-760	55	4	,	,	PUNCT
cana-760	55	5	𝑡	𝑡	PROPN
cana-760	55	6	)	)	PUNCT
cana-760	55	7	.	.	PUNCT
cana-760	56	1	(	(	PUNCT
cana-760	56	2	4	4	X
cana-760	56	3	)	)	PUNCT
cana-760	56	4	𝛬(𝜄	𝛬(𝜄	NUM
cana-760	56	5	,	,	PUNCT
cana-760	56	6	휂	휂	ADP
cana-760	56	7	,	,	PUNCT
cana-760	56	8	𝑡	𝑡	NOUN
cana-760	56	9	)	)	PUNCT
cana-760	56	10	⊛	⊛	PUNCT
cana-760	56	11	𝛬(휂	𝛬(휂	SYM
cana-760	56	12	,	,	PUNCT
cana-760	56	13	𝜗	𝜗	PROPN
cana-760	56	14	,	,	PUNCT
cana-760	56	15	𝑠	𝑠	NOUN
cana-760	56	16	)	)	PUNCT
cana-760	56	17	≥	≥	NOUN
cana-760	56	18	𝛬(𝜄	𝛬(𝜄	SYM
cana-760	56	19	,	,	PUNCT
cana-760	56	20	𝜗	𝜗	PROPN
cana-760	56	21	,	,	PUNCT
cana-760	56	22	𝑡	𝑡	PROPN
cana-760	56	23	+	+	PROPN
cana-760	56	24	𝑠	𝑠	NOUN
cana-760	56	25	)	)	PUNCT
cana-760	56	26	.	.	PUNCT
cana-760	57	1	(	(	PUNCT
cana-760	57	2	5	5	NUM
cana-760	57	3	)	)	PUNCT
cana-760	57	4	𝛬(𝜄	𝛬(𝜄	NUM
cana-760	57	5	,	,	PUNCT
cana-760	57	6	휂	휂	ADP
cana-760	57	7	,	,	PUNCT
cana-760	57	8	.	.	PUNCT
cana-760	57	9	):	):	PUNCT
cana-760	57	10	(	(	PUNCT
cana-760	57	11	0	0	NUM
cana-760	57	12	,	,	PUNCT
cana-760	57	13	∞	∞	NUM
cana-760	57	14	)	)	PUNCT
cana-760	57	15	→	→	PUNCT
cana-760	58	1	[	[	X
cana-760	58	2	0,1	0,1	NUM
cana-760	58	3	]	]	PUNCT
cana-760	58	4	is	be	AUX
cana-760	58	5	right	right	ADV
cana-760	58	6	continuous	continuous	ADJ
cana-760	58	7	.	.	PUNCT
cana-760	59	1	here	here	ADV
cana-760	59	2	,	,	PUNCT
cana-760	59	3	λ	λ	PROPN
cana-760	59	4	called	call	VERB
cana-760	59	5	a	a	DET
cana-760	59	6	revised	revise	VERB
cana-760	59	7	fuzzy	fuzzy	ADJ
cana-760	59	8	metric	metric	ADJ
cana-760	59	9	on	on	ADP
cana-760	59	10	y.	y.	PROPN
cana-760	59	11	example	example	NOUN
cana-760	59	12	2.1	2.1	NUM
cana-760	59	13	.	.	PUNCT
cana-760	60	1	[	[	X
cana-760	60	2	2	2	NUM
cana-760	60	3	]	]	X
cana-760	60	4	let	let	NOUN
cana-760	60	5	(	(	PUNCT
cana-760	60	6	𝑌	𝑌	PROPN
cana-760	60	7	,	,	PUNCT
cana-760	60	8	𝑑	𝑑	NOUN
cana-760	60	9	)	)	PUNCT
cana-760	60	10	be	be	VERB
cana-760	60	11	a	a	DET
cana-760	60	12	metric	metric	ADJ
cana-760	60	13	space	space	NOUN
cana-760	60	14	.	.	PUNCT
cana-760	61	1	define	define	VERB
cana-760	61	2	𝛼	𝛼	PRON
cana-760	61	3	⊛	⊛	NUM
cana-760	61	4	𝛽	𝛽	NOUN
cana-760	61	5	=	=	SYM
cana-760	61	6	𝛼	𝛼	PROPN
cana-760	61	7	+	+	NOUN
cana-760	62	1	𝛽	𝛽	NOUN
cana-760	62	2	−	−	NOUN
cana-760	62	3	𝛼𝛽	𝛼𝛽	NOUN
cana-760	62	4	for	for	ADP
cana-760	62	5	all	all	DET
cana-760	62	6	𝛼	𝛼	PROPN
cana-760	62	7	,	,	PUNCT
cana-760	62	8	𝛽	𝛽	PROPN
cana-760	62	9	∈	∈	NOUN
cana-760	63	1	[	[	X
cana-760	63	2	0,1	0,1	NUM
cana-760	63	3	]	]	PUNCT
cana-760	63	4	,	,	PUNCT
cana-760	63	5	and	and	CCONJ
cana-760	63	6	𝛬	𝛬	NOUN
cana-760	63	7	:	:	PUNCT
cana-760	63	8	𝑌2	𝑌2	PROPN
cana-760	63	9	×	×	NOUN
cana-760	63	10	(	(	PUNCT
cana-760	63	11	0	0	NUM
cana-760	63	12	,	,	PUNCT
cana-760	63	13	∞	∞	NUM
cana-760	63	14	)	)	PUNCT
cana-760	63	15	→	→	PUNCT
cana-760	64	1	[	[	X
cana-760	64	2	0,1	0,1	NUM
cana-760	64	3	]	]	PUNCT
cana-760	64	4	as	as	SCONJ
cana-760	64	5	communications	communication	NOUN
cana-760	64	6	on	on	ADP
cana-760	64	7	applied	apply	VERB
cana-760	64	8	nonlinear	nonlinear	ADJ
cana-760	64	9	analysis	analysis	NOUN
cana-760	64	10	issn	issn	NOUN
cana-760	64	11	:	:	PUNCT
cana-760	64	12	1074	1074	NUM
cana-760	64	13	-	-	PUNCT
cana-760	64	14	133x	133x	NUM
cana-760	64	15	vol	vol	NOUN
cana-760	64	16	31	31	NUM
cana-760	64	17	no	no	NOUN
cana-760	64	18	.	.	PUNCT
cana-760	65	1	3s	3s	NUM
cana-760	65	2	(	(	PUNCT
cana-760	65	3	2024	2024	NUM
cana-760	65	4	)	)	PUNCT
cana-760	65	5	214	214	NUM
cana-760	65	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	65	7	𝛬(𝜄	𝛬(𝜄	SYM
cana-760	65	8	,	,	PUNCT
cana-760	65	9	휂	휂	ADP
cana-760	65	10	,	,	PUNCT
cana-760	65	11	𝑡	𝑡	NOUN
cana-760	65	12	)	)	PUNCT
cana-760	65	13	=	=	SYM
cana-760	65	14	𝑑(𝜄,𝜂	𝑑(𝜄,𝜂	NOUN
cana-760	65	15	)	)	PUNCT
cana-760	65	16	𝑡+	𝑡+	X
cana-760	65	17	𝑑(𝜄,𝜂	𝑑(𝜄,𝜂	NUM
cana-760	65	18	)	)	PUNCT
cana-760	65	19	∀𝜄	∀𝜄	PROPN
cana-760	65	20	,	,	PUNCT
cana-760	65	21	휂	휂	ADP
cana-760	65	22	∈	∈	PROPN
cana-760	65	23	𝛯	𝛯	PROPN
cana-760	65	24	and	and	CCONJ
cana-760	65	25	𝑡	𝑡	X
cana-760	65	26	>	>	X
cana-760	65	27	0	0	X
cana-760	65	28	.	.	PUNCT
cana-760	66	1	then	then	ADV
cana-760	66	2	(	(	PUNCT
cana-760	66	3	𝛯	𝛯	NOUN
cana-760	66	4	,	,	PUNCT
cana-760	66	5	𝛬,⊛	𝛬,⊛	PROPN
cana-760	66	6	)	)	PUNCT
cana-760	66	7	is	be	AUX
cana-760	66	8	a	a	DET
cana-760	66	9	revised	revise	VERB
cana-760	66	10	fuzzy	fuzzy	ADJ
cana-760	66	11	metric	metric	ADJ
cana-760	66	12	space	space	NOUN
cana-760	66	13	;	;	PUNCT
cana-760	66	14	called	call	VERB
cana-760	66	15	revised	revise	VERB
cana-760	66	16	fuzzy	fuzzy	ADJ
cana-760	66	17	metric	metric	NOUN
cana-760	66	18	induced	induce	VERB
cana-760	66	19	by	by	ADP
cana-760	66	20	the	the	DET
cana-760	66	21	metric	metric	ADJ
cana-760	66	22	𝑑.	𝑑.	NOUN
cana-760	66	23	the	the	DET
cana-760	66	24	notions	notion	NOUN
cana-760	66	25	of	of	ADP
cana-760	66	26	convergence	convergence	NOUN
cana-760	66	27	,	,	PUNCT
cana-760	66	28	completeness	completeness	NOUN
cana-760	66	29	and	and	CCONJ
cana-760	66	30	compactness	compactness	NOUN
cana-760	66	31	on	on	ADP
cana-760	66	32	revised	revise	VERB
cana-760	66	33	fuzzy	fuzzy	ADJ
cana-760	66	34	metric	metric	ADJ
cana-760	66	35	spaces	space	NOUN
cana-760	66	36	were	be	AUX
cana-760	66	37	presented	present	VERB
cana-760	66	38	in	in	ADP
cana-760	66	39	[	[	X
cana-760	66	40	17	17	NUM
cana-760	66	41	]	]	PUNCT
cana-760	66	42	as	as	SCONJ
cana-760	66	43	follows	follow	VERB
cana-760	66	44	:	:	PUNCT
cana-760	66	45	definition	definition	NOUN
cana-760	66	46	2.4	2.4	NUM
cana-760	66	47	.	.	PUNCT
cana-760	67	1	[	[	X
cana-760	67	2	17	17	NUM
cana-760	67	3	]	]	X
cana-760	67	4	let	let	NOUN
cana-760	67	5	(	(	PUNCT
cana-760	67	6	𝛯	𝛯	NOUN
cana-760	67	7	,	,	PUNCT
cana-760	67	8	𝛬,⊛	𝛬,⊛	PROPN
cana-760	67	9	)	)	PUNCT
cana-760	67	10	be	be	AUX
cana-760	67	11	a	a	DET
cana-760	67	12	revised	revise	VERB
cana-760	67	13	fuzzy	fuzzy	ADJ
cana-760	67	14	metric	metric	ADJ
cana-760	67	15	space	space	NOUN
cana-760	67	16	.	.	PUNCT
cana-760	68	1	(	(	PUNCT
cana-760	68	2	1	1	X
cana-760	68	3	)	)	PUNCT
cana-760	68	4	a	a	DET
cana-760	68	5	sequence	sequence	NOUN
cana-760	68	6	{	{	PUNCT
cana-760	68	7	𝜄𝜅}𝜅∈𝑁	𝜄𝜅}𝜅∈𝑁	X
cana-760	68	8	in	in	ADP
cana-760	68	9	𝑌	𝑌	PROPN
cana-760	68	10	is	be	AUX
cana-760	68	11	convergent	convergent	ADJ
cana-760	68	12	to	to	ADP
cana-760	68	13	an	an	DET
cana-760	68	14	element	element	NOUN
cana-760	68	15	𝜄	𝜄	X
cana-760	68	16	∈	∈	PROPN
cana-760	68	17	𝑌	𝑌	PROPN
cana-760	68	18	if	if	SCONJ
cana-760	68	19	lim	lim	PROPN
cana-760	68	20	𝜅→∞	𝜅→∞	NUM
cana-760	68	21	𝛬(𝜄𝜅	𝛬(𝜄𝜅	NOUN
cana-760	68	22	,	,	PUNCT
cana-760	68	23	𝜄	𝜄	PROPN
cana-760	68	24	,	,	PUNCT
cana-760	68	25	𝑡	𝑡	PROPN
cana-760	68	26	)	)	PUNCT
cana-760	68	27	=	=	SYM
cana-760	68	28	0	0	NUM
cana-760	68	29	,	,	PUNCT
cana-760	68	30	for	for	ADP
cana-760	68	31	all	all	DET
cana-760	68	32	t	t	PROPN
cana-760	68	33	>	>	X
cana-760	68	34	0	0	X
cana-760	68	35	.	.	PUNCT
cana-760	69	1	(	(	PUNCT
cana-760	69	2	2	2	X
cana-760	69	3	)	)	PUNCT
cana-760	69	4	a	a	DET
cana-760	69	5	sequence	sequence	NOUN
cana-760	69	6	{	{	PUNCT
cana-760	69	7	𝜄𝜅}𝜅∈𝑁	𝜄𝜅}𝜅∈𝑁	X
cana-760	69	8	in	in	ADP
cana-760	69	9	𝑌	𝑌	PROPN
cana-760	69	10	is	be	AUX
cana-760	69	11	cauchy	cauchy	ADJ
cana-760	69	12	if	if	SCONJ
cana-760	69	13	for	for	ADP
cana-760	69	14	all	all	PRON
cana-760	69	15	0	0	NUM
cana-760	69	16	<	<	X
cana-760	69	17	휀	휀	X
cana-760	69	18	<	<	X
cana-760	69	19	1	1	NUM
cana-760	69	20	and	and	CCONJ
cana-760	69	21	for	for	ADP
cana-760	69	22	t	t	PROPN
cana-760	69	23	>	>	PROPN
cana-760	69	24	0	0	PROPN
cana-760	69	25	,	,	PUNCT
cana-760	69	26	there	there	PRON
cana-760	69	27	exists	exist	VERB
cana-760	69	28	a	a	DET
cana-760	69	29	number	number	NOUN
cana-760	69	30	𝜅0	𝜅0	ADP
cana-760	69	31	∈	∈	NOUN
cana-760	70	1	𝑁	𝑁	ADP
cana-760	70	2	such	such	ADJ
cana-760	70	3	that	that	SCONJ
cana-760	70	4	λ(lk	λ(lk	ADJ
cana-760	70	5	,	,	PUNCT
cana-760	70	6	ly	ly	PROPN
cana-760	70	7	,	,	PUNCT
cana-760	70	8	t	t	PROPN
cana-760	70	9	)	)	PUNCT
cana-760	70	10	<	<	X
cana-760	70	11	ε	ε	PROPN
cana-760	70	12	for	for	ADP
cana-760	70	13	each	each	DET
cana-760	70	14	𝜅	𝜅	NUM
cana-760	70	15	,	,	PUNCT
cana-760	70	16	𝑌	𝑌	PROPN
cana-760	70	17	≥	≥	NOUN
cana-760	70	18	𝜅0	𝜅0	PROPN
cana-760	70	19	.	.	PUNCT
cana-760	71	1	(	(	PUNCT
cana-760	71	2	3	3	X
cana-760	71	3	)	)	PUNCT
cana-760	71	4	a	a	DET
cana-760	71	5	revised	revise	VERB
cana-760	71	6	fuzzy	fuzzy	ADJ
cana-760	71	7	metric	metric	ADJ
cana-760	71	8	space	space	NOUN
cana-760	71	9	in	in	ADP
cana-760	71	10	which	which	PRON
cana-760	71	11	every	every	DET
cana-760	71	12	cauchy	cauchy	ADJ
cana-760	71	13	sequence	sequence	NOUN
cana-760	71	14	is	be	AUX
cana-760	71	15	convergent	convergent	NOUN
cana-760	71	16	is	be	AUX
cana-760	71	17	said	say	VERB
cana-760	71	18	to	to	PART
cana-760	71	19	be	be	AUX
cana-760	71	20	complete	complete	ADJ
cana-760	71	21	.	.	PUNCT
cana-760	72	1	(	(	PUNCT
cana-760	72	2	4	4	X
cana-760	72	3	)	)	PUNCT
cana-760	72	4	a	a	DET
cana-760	72	5	revised	revise	VERB
cana-760	72	6	fuzzy	fuzzy	ADJ
cana-760	72	7	metric	metric	ADJ
cana-760	72	8	space	space	NOUN
cana-760	72	9	in	in	ADP
cana-760	72	10	which	which	PRON
cana-760	72	11	every	every	DET
cana-760	72	12	sequence	sequence	NOUN
cana-760	72	13	has	have	VERB
cana-760	72	14	a	a	DET
cana-760	72	15	convergent	convergent	ADJ
cana-760	72	16	subsequence	subsequence	NOUN
cana-760	72	17	is	be	AUX
cana-760	72	18	said	say	VERB
cana-760	72	19	to	to	PART
cana-760	72	20	be	be	AUX
cana-760	72	21	compact	compact	ADJ
cana-760	72	22	.	.	PUNCT
cana-760	73	1	in	in	ADP
cana-760	73	2	2010	2010	NUM
cana-760	73	3	,	,	PUNCT
cana-760	73	4	chistyakov	chistyakov	NOUN
cana-760	74	1	[	[	X
cana-760	74	2	3	3	NUM
cana-760	74	3	-	-	SYM
cana-760	74	4	8	8	NUM
cana-760	74	5	]	]	PUNCT
cana-760	74	6	defined	define	VERB
cana-760	74	7	the	the	DET
cana-760	74	8	notion	notion	NOUN
cana-760	74	9	of	of	ADP
cana-760	74	10	modular	modular	ADJ
cana-760	74	11	metric	metric	ADJ
cana-760	74	12	spaces	space	NOUN
cana-760	74	13	as	as	SCONJ
cana-760	74	14	follows	follow	VERB
cana-760	74	15	:	:	PUNCT
cana-760	74	16	definition	definition	NOUN
cana-760	74	17	2.5	2.5	NUM
cana-760	74	18	.	.	PUNCT
cana-760	75	1	[	[	X
cana-760	75	2	3	3	X
cana-760	75	3	]	]	PUNCT
cana-760	75	4	a	a	DET
cana-760	75	5	modular	modular	ADJ
cana-760	75	6	metric	metric	NOUN
cana-760	75	7	on	on	ADP
cana-760	75	8	a	a	DET
cana-760	75	9	nonempty	nonempty	ADV
cana-760	75	10	set	set	VERB
cana-760	75	11	y	y	PROPN
cana-760	75	12	is	be	AUX
cana-760	75	13	a	a	DET
cana-760	75	14	function	function	NOUN
cana-760	75	15	𝛩	𝛩	NOUN
cana-760	75	16	:	:	PUNCT
cana-760	75	17	(	(	PUNCT
cana-760	75	18	0	0	NUM
cana-760	75	19	,	,	PUNCT
cana-760	75	20	∞	∞	PROPN
cana-760	75	21	)	)	PUNCT
cana-760	75	22	×	×	NOUN
cana-760	75	23	𝑌2	𝑌2	NOUN
cana-760	75	24	→	→	PUNCT
cana-760	75	25	[	[	X
cana-760	75	26	0	0	NUM
cana-760	75	27	,	,	PUNCT
cana-760	75	28	∞	∞	PROPN
cana-760	75	29	)	)	PUNCT
cana-760	75	30	that	that	PRON
cana-760	75	31	will	will	AUX
cana-760	75	32	be	be	AUX
cana-760	75	33	written	write	VERB
cana-760	75	34	as	as	ADP
cana-760	75	35	𝛩%(𝜄	𝛩%(𝜄	NOUN
cana-760	75	36	,	,	PUNCT
cana-760	75	37	휂	휂	NOUN
cana-760	75	38	)	)	PUNCT
cana-760	75	39	=	=	SYM
cana-760	75	40	𝛩(%	𝛩(%	NOUN
cana-760	75	41	,	,	PUNCT
cana-760	75	42	𝜄	𝜄	PROPN
cana-760	75	43	,	,	PUNCT
cana-760	75	44	휂	휂	NOUN
cana-760	75	45	)	)	PUNCT
cana-760	75	46	;	;	PUNCT
cana-760	75	47	for	for	ADP
cana-760	75	48	all	all	DET
cana-760	75	49	𝜄	𝜄	PROPN
cana-760	75	50	,	,	PUNCT
cana-760	75	51	휂	휂	ADP
cana-760	75	52	,	,	PUNCT
cana-760	75	53	𝜗	𝜗	PROPN
cana-760	75	54	∈	∈	PROPN
cana-760	75	55	𝑌	𝑌	PROPN
cana-760	75	56	and	and	CCONJ
cana-760	75	57	for	for	ADP
cana-760	75	58	all	all	DET
cana-760	75	59	%	%	NOUN
cana-760	75	60	,	,	PUNCT
cana-760	75	61	𝜎	𝜎	PROPN
cana-760	75	62	>	>	X
cana-760	75	63	0	0	PROPN
cana-760	75	64	,	,	PUNCT
cana-760	75	65	satisfy	satisfy	VERB
cana-760	75	66	the	the	DET
cana-760	75	67	following	follow	VERB
cana-760	75	68	three	three	NUM
cana-760	75	69	conditions	condition	NOUN
cana-760	75	70	:	:	PUNCT
cana-760	75	71	(	(	PUNCT
cana-760	75	72	1	1	X
cana-760	75	73	)	)	PUNCT
cana-760	75	74	𝛩%(𝜄	𝛩%(𝜄	NOUN
cana-760	75	75	,	,	PUNCT
cana-760	75	76	휂	휂	NOUN
cana-760	75	77	)	)	PUNCT
cana-760	75	78	=	=	SYM
cana-760	75	79	0	0	PUNCT
cana-760	76	1	if	if	SCONJ
cana-760	76	2	and	and	CCONJ
cana-760	76	3	only	only	ADV
cana-760	76	4	if	if	SCONJ
cana-760	76	5	𝜄	𝜄	PROPN
cana-760	76	6	=	=	SYM
cana-760	76	7	휂	휂	PROPN
cana-760	76	8	,	,	PUNCT
cana-760	76	9	∀	∀	X
cana-760	76	10	%	%	NOUN
cana-760	76	11	>	>	X
cana-760	76	12	0	0	PUNCT
cana-760	76	13	and	and	CCONJ
cana-760	76	14	𝜄	𝜄	X
cana-760	76	15	,	,	PUNCT
cana-760	76	16	휂	휂	ADP
cana-760	76	17	∈	∈	X
cana-760	76	18	𝑌.	𝑌.	PROPN
cana-760	76	19	(	(	PUNCT
cana-760	76	20	2	2	NUM
cana-760	76	21	)	)	PUNCT
cana-760	76	22	𝛩%(𝜄	𝛩%(𝜄	NOUN
cana-760	76	23	,	,	PUNCT
cana-760	76	24	휂	휂	NOUN
cana-760	76	25	)	)	PUNCT
cana-760	76	26	=	=	SYM
cana-760	76	27	𝛩%(휂	𝛩%(휂	PROPN
cana-760	76	28	,	,	PUNCT
cana-760	76	29	𝜄	𝜄	PROPN
cana-760	76	30	)	)	PUNCT
cana-760	76	31	,	,	PUNCT
cana-760	76	32	∀%	∀%	NOUN
cana-760	76	33	>	>	X
cana-760	76	34	0	0	PUNCT
cana-760	77	1	and	and	CCONJ
cana-760	77	2	𝜄	𝜄	X
cana-760	77	3	,	,	PUNCT
cana-760	77	4	휂	휂	ADP
cana-760	77	5	∈	∈	X
cana-760	77	6	𝑌.	𝑌.	PROPN
cana-760	77	7	(	(	PUNCT
cana-760	77	8	3	3	NUM
cana-760	77	9	)	)	PUNCT
cana-760	77	10	𝛩%+𝜎(𝜄	𝛩%+𝜎(𝜄	NOUN
cana-760	77	11	,	,	PUNCT
cana-760	77	12	휂	휂	NOUN
cana-760	77	13	)	)	PUNCT
cana-760	77	14	≤	≤	NUM
cana-760	77	15	𝛩%(𝜄	𝛩%(𝜄	NOUN
cana-760	77	16	,	,	PUNCT
cana-760	77	17	𝜗	𝜗	NOUN
cana-760	77	18	)	)	PUNCT
cana-760	78	1	+	+	CCONJ
cana-760	78	2	𝛩𝜎(𝜗	𝛩𝜎(𝜗	NOUN
cana-760	78	3	,	,	PUNCT
cana-760	78	4	휂	휂	NOUN
cana-760	78	5	)	)	PUNCT
cana-760	78	6	;	;	PUNCT
cana-760	78	7	for	for	ADP
cana-760	78	8	all	all	DET
cana-760	78	9	%	%	NOUN
cana-760	78	10	,	,	PUNCT
cana-760	78	11	𝜎	𝜎	PROPN
cana-760	78	12	>	>	X
cana-760	78	13	0	0	PUNCT
cana-760	78	14	and	and	CCONJ
cana-760	78	15	𝜄	𝜄	PROPN
cana-760	78	16	,	,	PUNCT
cana-760	78	17	휂	휂	PROPN
cana-760	78	18	,	,	PUNCT
cana-760	78	19	𝜗	𝜗	PROPN
cana-760	78	20	∈	∈	NOUN
cana-760	78	21	𝑌.	𝑌.	PROPN
cana-760	78	22	remark	remark	NOUN
cana-760	78	23	2.1	2.1	NUM
cana-760	78	24	.	.	PUNCT
cana-760	79	1	let	let	VERB
cana-760	79	2	𝛩	𝛩	PRON
cana-760	79	3	be	be	AUX
cana-760	79	4	modular	modular	ADJ
cana-760	79	5	on	on	ADP
cana-760	79	6	a	a	DET
cana-760	79	7	set	set	NOUN
cana-760	79	8	𝑌.	𝑌.	PROPN
cana-760	79	9	then	then	ADV
cana-760	79	10	for	for	ADP
cana-760	79	11	given	give	VERB
cana-760	79	12	𝜄	𝜄	PROPN
cana-760	79	13	,	,	PUNCT
cana-760	79	14	휂	휂	ADP
cana-760	79	15	∈	∈	PROPN
cana-760	79	16	𝑌	𝑌	PROPN
cana-760	79	17	,	,	PUNCT
cana-760	79	18	the	the	DET
cana-760	79	19	function	function	NOUN
cana-760	79	20	0	0	PUNCT
cana-760	80	1	<	<	X
cana-760	81	1	%	%	NOUN
cana-760	81	2	→	→	SYM
cana-760	81	3	𝛩%(𝜄	𝛩%(𝜄	PROPN
cana-760	81	4	,	,	PUNCT
cana-760	81	5	휂	휂	NOUN
cana-760	81	6	)	)	PUNCT
cana-760	81	7	∈	∈	NOUN
cana-760	81	8	(	(	PUNCT
cana-760	81	9	0	0	NUM
cana-760	81	10	,	,	PUNCT
cana-760	81	11	∞	∞	NUM
cana-760	81	12	)	)	PUNCT
cana-760	81	13	is	be	AUX
cana-760	81	14	non	non	ADJ
cana-760	81	15	increasing	increase	VERB
cana-760	81	16	on	on	ADP
cana-760	81	17	(	(	PUNCT
cana-760	81	18	0	0	NUM
cana-760	81	19	,	,	PUNCT
cana-760	81	20	∞	∞	PROPN
cana-760	81	21	)	)	PUNCT
cana-760	81	22	.	.	PUNCT
cana-760	82	1	in	in	ADP
cana-760	82	2	fact	fact	NOUN
cana-760	82	3	if	if	SCONJ
cana-760	82	4	0	0	NUM
cana-760	82	5	<	<	X
cana-760	82	6	%	%	X
cana-760	82	7	<	<	X
cana-760	82	8	𝜎	𝜎	PROPN
cana-760	82	9	,	,	PUNCT
cana-760	82	10	then	then	ADV
cana-760	82	11	by	by	ADP
cana-760	82	12	above	above	ADP
cana-760	82	13	definition	definition	NOUN
cana-760	82	14	𝛩𝜎(𝜄	𝛩𝜎(𝜄	PROPN
cana-760	82	15	,	,	PUNCT
cana-760	82	16	휂	휂	ADP
cana-760	82	17	)	)	PUNCT
cana-760	82	18	≤	≤	NUM
cana-760	82	19	𝛩𝜎−%(𝜄	𝛩𝜎−%(𝜄	PROPN
cana-760	82	20	,	,	PUNCT
cana-760	82	21	𝜄	𝜄	PROPN
cana-760	82	22	)	)	PUNCT
cana-760	82	23	+	+	CCONJ
cana-760	82	24	𝛩%(𝜄	𝛩%(𝜄	NOUN
cana-760	82	25	,	,	PUNCT
cana-760	82	26	휂	휂	NOUN
cana-760	82	27	)	)	PUNCT
cana-760	82	28	=	=	SYM
cana-760	82	29	𝛩%(𝜄	𝛩%(𝜄	NOUN
cana-760	82	30	,	,	PUNCT
cana-760	82	31	휂	휂	NOUN
cana-760	82	32	)	)	PUNCT
cana-760	82	33	for	for	ADP
cana-760	82	34	all	all	DET
cana-760	82	35	𝜄	𝜄	PROPN
cana-760	82	36	,	,	PUNCT
cana-760	82	37	휂	휂	ADP
cana-760	82	38	∈	∈	ADJ
cana-760	82	39	𝑌.	𝑌.	PROPN
cana-760	82	40	definition	definition	NOUN
cana-760	82	41	2.6	2.6	NUM
cana-760	82	42	.	.	PUNCT
cana-760	83	1	[	[	X
cana-760	83	2	5	5	NUM
cana-760	83	3	]	]	PUNCT
cana-760	83	4	given	give	VERB
cana-760	83	5	a	a	DET
cana-760	83	6	modular	modular	ADJ
cana-760	83	7	𝛩	𝛩	NOUN
cana-760	83	8	on	on	ADP
cana-760	83	9	𝑌	𝑌	PROPN
cana-760	83	10	,	,	PUNCT
cana-760	83	11	a	a	DET
cana-760	83	12	sequence	sequence	NOUN
cana-760	83	13	{	{	PUNCT
cana-760	83	14	𝜄𝜅}𝜅∈𝑁	𝜄𝜅}𝜅∈𝑁	ADV
cana-760	83	15	in	in	ADP
cana-760	83	16	𝑌𝛩	𝑌𝛩	PROPN
cana-760	83	17	is	be	AUX
cana-760	83	18	said	say	VERB
cana-760	83	19	to	to	PART
cana-760	83	20	be	be	AUX
cana-760	83	21	modular	modular	ADJ
cana-760	83	22	convergent	convergent	NOUN
cana-760	83	23	to	to	ADP
cana-760	83	24	an	an	DET
cana-760	83	25	element	element	NOUN
cana-760	83	26	𝜄	𝜄	PROPN
cana-760	83	27	∈	∈	PROPN
cana-760	83	28	𝑌𝛩	𝑌𝛩	PROPN
cana-760	83	29	if	if	SCONJ
cana-760	83	30	there	there	PRON
cana-760	83	31	exists	exist	VERB
cana-760	83	32	a	a	DET
cana-760	83	33	number	number	NOUN
cana-760	83	34	%	%	NOUN
cana-760	83	35	>	>	X
cana-760	83	36	0	0	NUM
cana-760	83	37	,	,	PUNCT
cana-760	83	38	possibly	possibly	ADV
cana-760	83	39	depending	depend	VERB
cana-760	83	40	on	on	ADP
cana-760	83	41	{	{	PUNCT
cana-760	83	42	𝜄𝜅	𝜄𝜅	X
cana-760	83	43	}	}	PUNCT
cana-760	83	44	and	and	CCONJ
cana-760	83	45	𝜄	𝜄	X
cana-760	83	46	,	,	PUNCT
cana-760	83	47	such	such	ADJ
cana-760	83	48	that	that	SCONJ
cana-760	83	49	lim	lim	PROPN
cana-760	83	50	𝜅→∞	𝜅→∞	NUM
cana-760	83	51	𝛩%(𝜄𝜅	𝛩%(𝜄𝜅	PROPN
cana-760	83	52	,	,	PUNCT
cana-760	83	53	𝜄	𝜄	PROPN
cana-760	83	54	)	)	PUNCT
cana-760	83	55	=	=	SYM
cana-760	84	1	0	0	X
cana-760	84	2	.	.	PUNCT
cana-760	85	1	i.e	i.e	PRON
cana-760	85	2	𝜄𝜅	𝜄𝜅	X
cana-760	85	3	→	→	PUNCT
cana-760	85	4	𝜄	𝜄	PROPN
cana-760	85	5	as	as	ADP
cana-760	85	6	κ	κ	PROPN
cana-760	85	7	→	→	SYM
cana-760	85	8	∞.	∞.	PROPN
cana-760	85	9	definition	definition	NOUN
cana-760	85	10	2.7	2.7	NUM
cana-760	85	11	.	.	PUNCT
cana-760	86	1	[	[	X
cana-760	86	2	5	5	NUM
cana-760	86	3	]	]	PUNCT
cana-760	86	4	given	give	VERB
cana-760	86	5	a	a	DET
cana-760	86	6	modular	modular	ADJ
cana-760	86	7	𝛩	𝛩	NOUN
cana-760	86	8	on	on	ADP
cana-760	86	9	𝑌	𝑌	PROPN
cana-760	86	10	,	,	PUNCT
cana-760	86	11	a	a	DET
cana-760	86	12	sequence	sequence	NOUN
cana-760	86	13	{	{	PUNCT
cana-760	86	14	𝜄𝜅}𝜅∈𝑁	𝜄𝜅}𝜅∈𝑁	ADV
cana-760	86	15	in	in	ADP
cana-760	86	16	𝑌𝛩	𝑌𝛩	PROPN
cana-760	86	17	is	be	AUX
cana-760	86	18	said	say	VERB
cana-760	86	19	to	to	PART
cana-760	86	20	be	be	AUX
cana-760	86	21	modular	modular	ADJ
cana-760	86	22	cauchy	cauchy	NOUN
cana-760	86	23	if	if	SCONJ
cana-760	86	24	there	there	PRON
cana-760	86	25	exists	exist	VERB
cana-760	87	1	a	a	DET
cana-760	87	2	number	number	NOUN
cana-760	87	3	%	%	NOUN
cana-760	87	4	=	=	SYM
cana-760	87	5	%	%	NOUN
cana-760	87	6	(	(	PUNCT
cana-760	87	7	{	{	PUNCT
cana-760	87	8	𝜄𝜅	𝜄𝜅	NOUN
cana-760	87	9	}	}	PUNCT
cana-760	87	10	)	)	PUNCT
cana-760	87	11	>	>	X
cana-760	87	12	0	0	PUNCT
cana-760	88	1	such	such	ADJ
cana-760	88	2	that	that	SCONJ
cana-760	88	3	lim	lim	PROPN
cana-760	88	4	𝜅,𝜉→∞	𝜅,𝜉→∞	VERB
cana-760	88	5	𝛩%(𝜄𝜅	𝛩%(𝜄𝜅	NOUN
cana-760	88	6	,	,	PUNCT
cana-760	88	7	𝜄𝜉	𝜄𝜉	INTJ
cana-760	88	8	)	)	PUNCT
cana-760	88	9	=	=	SYM
cana-760	88	10	0	0	X
cana-760	88	11	.	.	PUNCT
cana-760	89	1	communications	communication	NOUN
cana-760	89	2	on	on	ADP
cana-760	89	3	applied	apply	VERB
cana-760	89	4	nonlinear	nonlinear	ADJ
cana-760	89	5	analysis	analysis	NOUN
cana-760	89	6	issn	issn	NOUN
cana-760	89	7	:	:	PUNCT
cana-760	89	8	1074	1074	NUM
cana-760	89	9	-	-	PUNCT
cana-760	89	10	133x	133x	NUM
cana-760	89	11	vol	vol	NOUN
cana-760	89	12	31	31	NUM
cana-760	89	13	no	no	NOUN
cana-760	89	14	.	.	PUNCT
cana-760	90	1	3s	3s	NUM
cana-760	90	2	(	(	PUNCT
cana-760	90	3	2024	2024	NUM
cana-760	90	4	)	)	PUNCT
cana-760	90	5	215	215	NUM
cana-760	90	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	90	7	definition	definition	NOUN
cana-760	90	8	2.8	2.8	NUM
cana-760	90	9	.	.	PUNCT
cana-760	91	1	[	[	X
cana-760	91	2	5	5	NUM
cana-760	91	3	]	]	PUNCT
cana-760	91	4	a	a	DET
cana-760	91	5	modular	modular	ADJ
cana-760	91	6	space	space	NOUN
cana-760	91	7	𝑌𝛩	𝑌𝛩	PROPN
cana-760	91	8	is	be	AUX
cana-760	91	9	said	say	VERB
cana-760	91	10	to	to	PART
cana-760	91	11	be	be	AUX
cana-760	91	12	modular	modular	ADJ
cana-760	91	13	complete	complete	ADJ
cana-760	91	14	if	if	SCONJ
cana-760	91	15	each	each	DET
cana-760	91	16	cauchy	cauchy	ADJ
cana-760	91	17	sequence	sequence	NOUN
cana-760	91	18	in	in	ADP
cana-760	91	19	𝑌𝛩	𝑌𝛩	PROPN
cana-760	91	20	is	be	AUX
cana-760	91	21	modular	modular	ADJ
cana-760	91	22	convergent	convergent	NOUN
cana-760	91	23	.	.	PUNCT
cana-760	92	1	in	in	ADP
cana-760	92	2	fact	fact	NOUN
cana-760	92	3	,	,	PUNCT
cana-760	92	4	if	if	SCONJ
cana-760	92	5	{	{	PUNCT
cana-760	92	6	𝜄𝜅	𝜄𝜅	NOUN
cana-760	92	7	}	}	PUNCT
cana-760	92	8	⊂	⊂	PROPN
cana-760	92	9	𝑌𝛩	𝑌𝛩	PROPN
cana-760	92	10	and	and	CCONJ
cana-760	92	11	there	there	PRON
cana-760	92	12	exists	exist	VERB
cana-760	92	13	%	%	NOUN
cana-760	92	14	=	=	SYM
cana-760	92	15	%	%	NOUN
cana-760	92	16	(	(	PUNCT
cana-760	92	17	{	{	PUNCT
cana-760	92	18	𝜄𝜅	𝜄𝜅	NOUN
cana-760	92	19	}	}	PUNCT
cana-760	92	20	)	)	PUNCT
cana-760	92	21	>	>	X
cana-760	92	22	0	0	PUNCT
cana-760	93	1	such	such	ADJ
cana-760	93	2	that	that	SCONJ
cana-760	93	3	lim	lim	PROPN
cana-760	93	4	𝜅,𝑌→∞	𝜅,𝑌→∞	ADP
cana-760	93	5	𝛩%(𝜄𝜅	𝛩%(𝜄𝜅	NOUN
cana-760	93	6	,	,	PUNCT
cana-760	93	7	𝜄𝑌	𝜄𝑌	NOUN
cana-760	93	8	)	)	PUNCT
cana-760	93	9	=	=	SYM
cana-760	93	10	0	0	NUM
cana-760	93	11	,	,	PUNCT
cana-760	93	12	then	then	ADV
cana-760	93	13	there	there	PRON
cana-760	93	14	exists	exist	VERB
cana-760	93	15	𝜄	𝜄	PROPN
cana-760	93	16	∈	∈	PROPN
cana-760	93	17	𝑌𝛩	𝑌𝛩	PROPN
cana-760	93	18	,	,	PUNCT
cana-760	93	19	such	such	ADJ
cana-760	93	20	that	that	SCONJ
cana-760	93	21	lim	lim	PROPN
cana-760	93	22	𝜅→∞	𝜅→∞	NUM
cana-760	93	23	𝛩%(𝜄𝜅	𝛩%(𝜄𝜅	PROPN
cana-760	93	24	,	,	PUNCT
cana-760	93	25	𝜄	𝜄	PROPN
cana-760	93	26	)	)	PUNCT
cana-760	93	27	=	=	SYM
cana-760	94	1	0	0	X
cana-760	94	2	.	.	PUNCT
cana-760	94	3	definition	definition	NOUN
cana-760	94	4	2.9	2.9	NUM
cana-760	94	5	.	.	PUNCT
cana-760	95	1	a	a	DET
cana-760	95	2	modular	modular	ADJ
cana-760	95	3	𝛩	𝛩	NOUN
cana-760	95	4	on	on	ADP
cana-760	95	5	𝑌	𝑌	PROPN
cana-760	95	6	is	be	AUX
cana-760	95	7	said	say	VERB
cana-760	95	8	to	to	PART
cana-760	95	9	be	be	AUX
cana-760	95	10	satisfied	satisfy	VERB
cana-760	95	11	the	the	DET
cana-760	95	12	∆2	∆2	NOUN
cana-760	95	13	-	-	PUNCT
cana-760	95	14	condition	condition	NOUN
cana-760	95	15	if	if	SCONJ
cana-760	95	16	lim	lim	PROPN
cana-760	95	17	𝑛→∞	𝑛→∞	AUX
cana-760	95	18	𝛩%(𝜄𝜅	𝛩%(𝜄𝜅	NOUN
cana-760	95	19	,	,	PUNCT
cana-760	95	20	𝜄	𝜄	PROPN
cana-760	95	21	)	)	PUNCT
cana-760	95	22	=	=	SYM
cana-760	95	23	0	0	NUM
cana-760	95	24	,	,	PUNCT
cana-760	95	25	for	for	ADP
cana-760	95	26	some	some	DET
cana-760	95	27	%	%	NOUN
cana-760	95	28	>	>	X
cana-760	95	29	0	0	NUM
cana-760	95	30	implies	imply	VERB
cana-760	95	31	that	that	SCONJ
cana-760	95	32	lim	lim	PROPN
cana-760	95	33	𝑛→∞	𝑛→∞	NUM
cana-760	95	34	𝛩%(𝜄𝜅	𝛩%(𝜄𝜅	NOUN
cana-760	95	35	,	,	PUNCT
cana-760	95	36	𝜄	𝜄	PROPN
cana-760	95	37	)	)	PUNCT
cana-760	95	38	=	=	SYM
cana-760	95	39	0	0	NUM
cana-760	95	40	,	,	PUNCT
cana-760	95	41	for	for	ADP
cana-760	95	42	all	all	DET
cana-760	95	43	%	%	NOUN
cana-760	95	44	>	>	X
cana-760	95	45	0	0	NUM
cana-760	95	46	.	.	NOUN
cana-760	96	1	3	3	NUM
cana-760	96	2	.	.	X
cana-760	96	3	main	main	ADJ
cana-760	96	4	results	result	NOUN
cana-760	96	5	in	in	ADP
cana-760	96	6	this	this	DET
cana-760	96	7	section	section	NOUN
cana-760	96	8	,	,	PUNCT
cana-760	96	9	we	we	PRON
cana-760	96	10	construct	construct	VERB
cana-760	96	11	a	a	DET
cana-760	96	12	new	new	ADJ
cana-760	96	13	space	space	NOUN
cana-760	96	14	called	call	VERB
cana-760	96	15	a	a	DET
cana-760	96	16	modular	modular	ADJ
cana-760	96	17	revised	revise	VERB
cana-760	96	18	fuzzy	fuzzy	ADJ
cana-760	96	19	metric	metric	ADJ
cana-760	96	20	space	space	NOUN
cana-760	96	21	.	.	PUNCT
cana-760	97	1	we	we	PRON
cana-760	97	2	present	present	VERB
cana-760	97	3	some	some	DET
cana-760	97	4	examples	example	NOUN
cana-760	97	5	of	of	ADP
cana-760	97	6	this	this	DET
cana-760	97	7	space	space	NOUN
cana-760	97	8	.	.	PUNCT
cana-760	98	1	also	also	ADV
cana-760	98	2	,	,	PUNCT
cana-760	98	3	we	we	PRON
cana-760	98	4	formulate	formulate	VERB
cana-760	98	5	and	and	CCONJ
cana-760	98	6	prove	prove	VERB
cana-760	98	7	some	some	DET
cana-760	98	8	new	new	ADJ
cana-760	98	9	fixed	fix	VERB
cana-760	98	10	point	point	NOUN
cana-760	98	11	results	result	NOUN
cana-760	98	12	under	under	ADP
cana-760	98	13	this	this	DET
cana-760	98	14	space	space	NOUN
cana-760	98	15	.	.	PUNCT
cana-760	99	1	we	we	PRON
cana-760	99	2	start	start	VERB
cana-760	99	3	by	by	ADP
cana-760	99	4	presenting	present	VERB
cana-760	99	5	the	the	DET
cana-760	99	6	following	follow	VERB
cana-760	99	7	definitions	definition	NOUN
cana-760	99	8	.	.	PUNCT
cana-760	100	1	definition	definition	NOUN
cana-760	100	2	3.1	3.1	NUM
cana-760	100	3	.	.	PUNCT
cana-760	101	1	a	a	DET
cana-760	101	2	modular	modular	ADJ
cana-760	101	3	revised	revise	VERB
cana-760	101	4	fuzzy	fuzzy	ADJ
cana-760	101	5	metric	metric	ADJ
cana-760	101	6	space	space	NOUN
cana-760	101	7	is	be	AUX
cana-760	101	8	the	the	DET
cana-760	101	9	triplet	triplet	NOUN
cana-760	101	10	(	(	PUNCT
cana-760	101	11	𝑌	𝑌	PROPN
cana-760	101	12	,	,	PUNCT
cana-760	101	13	휁%,⊛	휁%,⊛	NOUN
cana-760	101	14	)	)	PUNCT
cana-760	101	15	such	such	ADJ
cana-760	101	16	that	that	SCONJ
cana-760	101	17	y	y	PROPN
cana-760	101	18	is	be	AUX
cana-760	101	19	an	an	DET
cana-760	101	20	arbitrary	arbitrary	ADJ
cana-760	101	21	set	set	NOUN
cana-760	101	22	,	,	PUNCT
cana-760	101	23	(	(	PUNCT
cana-760	101	24	⊛	⊛	NUM
cana-760	101	25	)	)	PUNCT
cana-760	101	26	is	be	AUX
cana-760	101	27	a	a	DET
cana-760	101	28	continuous	continuous	ADJ
cana-760	101	29	t	t	NOUN
cana-760	101	30	-	-	PUNCT
cana-760	101	31	conorm	conorm	NOUN
cana-760	101	32	and	and	CCONJ
cana-760	101	33	휁%	휁%	NOUN
cana-760	101	34	is	be	AUX
cana-760	101	35	a	a	DET
cana-760	101	36	revised	revise	VERB
cana-760	101	37	fuzzy	fuzzy	ADJ
cana-760	101	38	metric	metric	NOUN
cana-760	101	39	on	on	ADP
cana-760	101	40	(	(	PUNCT
cana-760	101	41	0	0	NUM
cana-760	101	42	,	,	PUNCT
cana-760	101	43	∞	∞	PROPN
cana-760	101	44	)	)	PUNCT
cana-760	101	45	×	×	NOUN
cana-760	101	46	𝑌2	𝑌2	NOUN
cana-760	101	47	×	×	NOUN
cana-760	101	48	(	(	PUNCT
cana-760	101	49	0	0	NUM
cana-760	101	50	,	,	PUNCT
cana-760	101	51	∞	∞	NUM
cana-760	101	52	)	)	PUNCT
cana-760	101	53	→	→	PUNCT
cana-760	102	1	[	[	X
cana-760	102	2	0,1	0,1	NUM
cana-760	102	3	]	]	PUNCT
cana-760	102	4	;	;	PUNCT
cana-760	102	5	for	for	ADP
cana-760	102	6	all	all	DET
cana-760	102	7	𝜄	𝜄	PROPN
cana-760	102	8	,	,	PUNCT
cana-760	102	9	휂	휂	PROPN
cana-760	102	10	,	,	PUNCT
cana-760	102	11	𝜗	𝜗	NOUN
cana-760	102	12	in	in	ADP
cana-760	102	13	𝑌	𝑌	PROPN
cana-760	102	14	,	,	PUNCT
cana-760	102	15	and	and	CCONJ
cana-760	102	16	𝑠	𝑠	INTJ
cana-760	102	17	,	,	PUNCT
cana-760	102	18	𝑡	𝑡	X
cana-760	102	19	>	>	X
cana-760	102	20	0	0	PUNCT
cana-760	102	21	satisfying	satisfy	VERB
cana-760	102	22	the	the	DET
cana-760	102	23	following	follow	VERB
cana-760	102	24	conditions	condition	NOUN
cana-760	102	25	:	:	PUNCT
cana-760	102	26	(	(	PUNCT
cana-760	102	27	1	1	X
cana-760	102	28	)	)	PUNCT
cana-760	102	29	휁%(𝜄	휁%(𝜄	PROPN
cana-760	102	30	,	,	PUNCT
cana-760	102	31	휂	휂	ADP
cana-760	102	32	,	,	PUNCT
cana-760	102	33	0	0	NUM
cana-760	102	34	)	)	PUNCT
cana-760	102	35	=	=	SYM
cana-760	102	36	0	0	NUM
cana-760	102	37	,	,	PUNCT
cana-760	102	38	휁%(𝜄	휁%(𝜄	PROPN
cana-760	102	39	,	,	PUNCT
cana-760	102	40	휂	휂	NOUN
cana-760	102	41	,	,	PUNCT
cana-760	102	42	𝑡	𝑡	NOUN
cana-760	102	43	)	)	PUNCT
cana-760	102	44	<	<	X
cana-760	102	45	1	1	NUM
cana-760	102	46	,	,	PUNCT
cana-760	102	47	for	for	ADP
cana-760	102	48	all	all	DET
cana-760	102	49	𝑡	𝑡	NOUN
cana-760	102	50	,	,	PUNCT
cana-760	102	51	%	%	INTJ
cana-760	102	52	>	>	X
cana-760	102	53	0	0	X
cana-760	102	54	.	.	PUNCT
cana-760	103	1	(	(	PUNCT
cana-760	103	2	2	2	X
cana-760	103	3	)	)	PUNCT
cana-760	103	4	휁%(𝜄	휁%(𝜄	PROPN
cana-760	103	5	,	,	PUNCT
cana-760	103	6	휂	휂	NOUN
cana-760	103	7	,	,	PUNCT
cana-760	103	8	𝑡	𝑡	NOUN
cana-760	103	9	)	)	PUNCT
cana-760	103	10	=	=	SYM
cana-760	103	11	0	0	PUNCT
cana-760	104	1	if	if	SCONJ
cana-760	104	2	and	and	CCONJ
cana-760	104	3	only	only	ADV
cana-760	104	4	if	if	SCONJ
cana-760	104	5	𝜄	𝜄	PROPN
cana-760	104	6	=	=	SYM
cana-760	104	7	휂	휂	PROPN
cana-760	104	8	,	,	PUNCT
cana-760	104	9	for	for	ADP
cana-760	104	10	all	all	DET
cana-760	104	11	𝑡	𝑡	NOUN
cana-760	104	12	,	,	PUNCT
cana-760	104	13	%	%	INTJ
cana-760	104	14	>	>	X
cana-760	104	15	0	0	X
cana-760	104	16	.	.	PUNCT
cana-760	105	1	(	(	PUNCT
cana-760	105	2	3	3	X
cana-760	105	3	)	)	PUNCT
cana-760	105	4	휁%(𝜄	휁%(𝜄	PROPN
cana-760	105	5	,	,	PUNCT
cana-760	105	6	휂	휂	NOUN
cana-760	105	7	,	,	PUNCT
cana-760	105	8	𝑡	𝑡	NOUN
cana-760	105	9	)	)	PUNCT
cana-760	106	1	=	=	SYM
cana-760	106	2	휁%(휂	휁%(휂	PROPN
cana-760	106	3	,	,	PUNCT
cana-760	106	4	𝜄	𝜄	PROPN
cana-760	106	5	,	,	PUNCT
cana-760	106	6	𝑡	𝑡	NOUN
cana-760	106	7	)	)	PUNCT
cana-760	106	8	,	,	PUNCT
cana-760	106	9	for	for	ADP
cana-760	106	10	all	all	DET
cana-760	106	11	𝑡	𝑡	NOUN
cana-760	106	12	,	,	PUNCT
cana-760	106	13	%	%	INTJ
cana-760	106	14	>	>	X
cana-760	106	15	0	0	X
cana-760	106	16	.	.	PUNCT
cana-760	106	17	(	(	PUNCT
cana-760	106	18	4	4	NUM
cana-760	106	19	)	)	PUNCT
cana-760	106	20	휁𝜎(𝜄	휁𝜎(𝜄	NOUN
cana-760	106	21	,	,	PUNCT
cana-760	106	22	휂	휂	ADP
cana-760	106	23	,	,	PUNCT
cana-760	106	24	𝑡	𝑡	NOUN
cana-760	106	25	)	)	PUNCT
cana-760	106	26	⊛	⊛	VERB
cana-760	107	1	휁%(휂	휁%(휂	PROPN
cana-760	107	2	,	,	PUNCT
cana-760	107	3	𝜗	𝜗	NOUN
cana-760	107	4	,	,	PUNCT
cana-760	107	5	𝑠	𝑠	NOUN
cana-760	107	6	)	)	PUNCT
cana-760	107	7	≤	≤	PROPN
cana-760	107	8	휁𝜎+%(𝜄	휁𝜎+%(𝜄	PROPN
cana-760	107	9	,	,	PUNCT
cana-760	107	10	𝜗	𝜗	PROPN
cana-760	107	11	,	,	PUNCT
cana-760	107	12	𝑡	𝑡	PROPN
cana-760	107	13	+	+	PROPN
cana-760	107	14	𝑠	𝑠	NOUN
cana-760	107	15	)	)	PUNCT
cana-760	107	16	,	,	PUNCT
cana-760	107	17	for	for	ADP
cana-760	107	18	all	all	DET
cana-760	107	19	𝑡	𝑡	PROPN
cana-760	107	20	,	,	PUNCT
cana-760	107	21	𝑠	𝑠	PROPN
cana-760	107	22	,	,	PUNCT
cana-760	107	23	𝜎	𝜎	PROPN
cana-760	107	24	,	,	PUNCT
cana-760	107	25	%	%	INTJ
cana-760	107	26	>	>	X
cana-760	107	27	0	0	X
cana-760	107	28	.	.	PUNCT
cana-760	108	1	(	(	PUNCT
cana-760	108	2	5	5	X
cana-760	108	3	)	)	PUNCT
cana-760	108	4	휁%(𝜄	휁%(𝜄	PROPN
cana-760	108	5	,	,	PUNCT
cana-760	108	6	휂	휂	ADP
cana-760	108	7	,	,	PUNCT
cana-760	108	8	.	.	PUNCT
cana-760	108	9	):	):	PUNCT
cana-760	108	10	(	(	PUNCT
cana-760	108	11	0	0	NUM
cana-760	108	12	,	,	PUNCT
cana-760	108	13	∞	∞	NUM
cana-760	108	14	)	)	PUNCT
cana-760	108	15	→	→	PUNCT
cana-760	109	1	[	[	X
cana-760	109	2	0,1	0,1	NUM
cana-760	109	3	]	]	PUNCT
cana-760	109	4	is	be	AUX
cana-760	109	5	right	right	ADV
cana-760	109	6	continuous	continuous	ADJ
cana-760	109	7	.	.	PUNCT
cana-760	110	1	here	here	ADV
cana-760	110	2	,	,	PUNCT
cana-760	110	3	휁%	휁%	PUNCT
cana-760	110	4	is	be	AUX
cana-760	110	5	called	call	VERB
cana-760	110	6	a	a	DET
cana-760	110	7	modular	modular	ADJ
cana-760	110	8	revised	revise	VERB
cana-760	110	9	fuzzy	fuzzy	ADJ
cana-760	110	10	metric	metric	NOUN
cana-760	110	11	.	.	PUNCT
cana-760	111	1	definition	definition	NOUN
cana-760	111	2	3.2	3.2	NUM
cana-760	111	3	.	.	PUNCT
cana-760	112	1	let	let	AUX
cana-760	112	2	(	(	PUNCT
cana-760	112	3	𝑌	𝑌	PROPN
cana-760	112	4	,	,	PUNCT
cana-760	112	5	휁%,⊛	휁%,⊛	NOUN
cana-760	112	6	)	)	PUNCT
cana-760	112	7	be	be	VERB
cana-760	112	8	a	a	DET
cana-760	112	9	modular	modular	ADJ
cana-760	112	10	revised	revise	VERB
cana-760	112	11	fuzzy	fuzzy	ADJ
cana-760	112	12	metric	metric	ADJ
cana-760	112	13	space	space	NOUN
cana-760	112	14	.	.	PUNCT
cana-760	113	1	(	(	PUNCT
cana-760	113	2	1	1	X
cana-760	113	3	)	)	PUNCT
cana-760	113	4	a	a	DET
cana-760	113	5	sequence	sequence	NOUN
cana-760	113	6	{	{	PUNCT
cana-760	113	7	𝜄𝜅}𝜅∈𝑁	𝜄𝜅}𝜅∈𝑁	X
cana-760	113	8	in	in	ADP
cana-760	113	9	𝑌	𝑌	PROPN
cana-760	113	10	is	be	AUX
cana-760	113	11	convergent	convergent	ADJ
cana-760	113	12	to	to	ADP
cana-760	113	13	an	an	DET
cana-760	113	14	element	element	NOUN
cana-760	113	15	𝜄	𝜄	PROPN
cana-760	113	16	∈	∈	PROPN
cana-760	113	17	𝑌	𝑌	PROPN
cana-760	113	18	𝑖𝑓	𝑖𝑓	PROPN
cana-760	113	19	lim	lim	PROPN
cana-760	113	20	𝜅→∞	𝜅→∞	NUM
cana-760	113	21	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	113	22	,	,	PUNCT
cana-760	113	23	𝜄	𝜄	PROPN
cana-760	113	24	,	,	PUNCT
cana-760	113	25	𝑡	𝑡	PROPN
cana-760	113	26	)	)	PUNCT
cana-760	113	27	=	=	SYM
cana-760	113	28	0	0	NUM
cana-760	113	29	for	for	ADP
cana-760	113	30	all	all	DET
cana-760	113	31	𝑡	𝑡	NOUN
cana-760	113	32	>	>	PUNCT
cana-760	113	33	0	0	PUNCT
cana-760	113	34	and	and	CCONJ
cana-760	113	35	some	some	DET
cana-760	113	36	%	%	NOUN
cana-760	113	37	>	>	X
cana-760	113	38	0	0	X
cana-760	113	39	.	.	PUNCT
cana-760	114	1	(	(	PUNCT
cana-760	114	2	2	2	X
cana-760	114	3	)	)	PUNCT
cana-760	114	4	a	a	DET
cana-760	114	5	sequence	sequence	NOUN
cana-760	114	6	{	{	PUNCT
cana-760	114	7	𝜄𝜅}𝜅∈𝑁	𝜄𝜅}𝜅∈𝑁	X
cana-760	114	8	in	in	ADP
cana-760	114	9	𝑌	𝑌	PROPN
cana-760	114	10	is	be	AUX
cana-760	114	11	cauchy	cauchy	ADJ
cana-760	114	12	if	if	SCONJ
cana-760	114	13	for	for	ADP
cana-760	114	14	all	all	PRON
cana-760	114	15	0	0	NUM
cana-760	114	16	<	<	X
cana-760	114	17	휀	휀	X
cana-760	114	18	<	<	X
cana-760	114	19	1	1	NUM
cana-760	114	20	,	,	PUNCT
cana-760	114	21	there	there	PRON
cana-760	114	22	exists	exist	VERB
cana-760	114	23	a	a	DET
cana-760	114	24	number	number	NOUN
cana-760	114	25	𝜅0	𝜅0	ADP
cana-760	114	26	∈	∈	NOUN
cana-760	115	1	𝑁	𝑁	PROPN
cana-760	115	2	such	such	ADJ
cana-760	115	3	that	that	SCONJ
cana-760	115	4	휁𝜚(𝜄𝜅	휁𝜚(𝜄𝜅	PROPN
cana-760	115	5	,	,	PUNCT
cana-760	115	6	𝜄𝜉	𝜄𝜉	INTJ
cana-760	115	7	,	,	PUNCT
cana-760	115	8	𝑡	𝑡	PROPN
cana-760	115	9	)	)	PUNCT
cana-760	115	10	<	<	X
cana-760	115	11	휀	휀	X
cana-760	115	12	,	,	PUNCT
cana-760	115	13	for	for	ADP
cana-760	115	14	each	each	DET
cana-760	115	15	𝜅	𝜅	NUM
cana-760	115	16	,	,	PUNCT
cana-760	115	17	𝑌	𝑌	PROPN
cana-760	115	18	≥	≥	NOUN
cana-760	115	19	𝜅0	𝜅0	NOUN
cana-760	115	20	and	and	CCONJ
cana-760	115	21	some	some	DET
cana-760	115	22	%	%	NOUN
cana-760	115	23	>	>	X
cana-760	115	24	0	0	X
cana-760	115	25	.	.	PUNCT
cana-760	116	1	(	(	PUNCT
cana-760	116	2	3	3	X
cana-760	116	3	)	)	PUNCT
cana-760	116	4	a	a	DET
cana-760	116	5	modular	modular	NOUN
cana-760	116	6	revised	revise	VERB
cana-760	116	7	fuzzy	fuzzy	ADJ
cana-760	116	8	metric	metric	ADJ
cana-760	116	9	space	space	NOUN
cana-760	116	10	in	in	ADP
cana-760	116	11	which	which	PRON
cana-760	116	12	every	every	DET
cana-760	116	13	cauchy	cauchy	ADJ
cana-760	116	14	sequence	sequence	NOUN
cana-760	116	15	is	be	AUX
cana-760	116	16	convergent	convergent	NOUN
cana-760	116	17	is	be	AUX
cana-760	116	18	said	say	VERB
cana-760	116	19	to	to	PART
cana-760	116	20	be	be	AUX
cana-760	116	21	complete	complete	ADJ
cana-760	116	22	.	.	PUNCT
cana-760	117	1	(	(	PUNCT
cana-760	117	2	4	4	X
cana-760	117	3	)	)	PUNCT
cana-760	117	4	a	a	DET
cana-760	117	5	modular	modular	NOUN
cana-760	117	6	revised	revise	VERB
cana-760	117	7	fuzzy	fuzzy	ADJ
cana-760	117	8	metric	metric	ADJ
cana-760	117	9	space	space	NOUN
cana-760	117	10	in	in	ADP
cana-760	117	11	which	which	PRON
cana-760	117	12	every	every	DET
cana-760	117	13	sequence	sequence	NOUN
cana-760	117	14	has	have	VERB
cana-760	117	15	a	a	DET
cana-760	117	16	convergent	convergent	ADJ
cana-760	117	17	subsequence	subsequence	NOUN
cana-760	117	18	is	be	AUX
cana-760	117	19	said	say	VERB
cana-760	117	20	to	to	PART
cana-760	117	21	be	be	AUX
cana-760	117	22	compact	compact	ADJ
cana-760	117	23	.	.	PUNCT
cana-760	118	1	communications	communication	NOUN
cana-760	118	2	on	on	ADP
cana-760	118	3	applied	apply	VERB
cana-760	118	4	nonlinear	nonlinear	ADJ
cana-760	118	5	analysis	analysis	NOUN
cana-760	118	6	issn	issn	NOUN
cana-760	118	7	:	:	PUNCT
cana-760	118	8	1074	1074	NUM
cana-760	118	9	-	-	PUNCT
cana-760	118	10	133x	133x	NUM
cana-760	118	11	vol	vol	NOUN
cana-760	118	12	31	31	NUM
cana-760	118	13	no	no	NOUN
cana-760	118	14	.	.	PUNCT
cana-760	119	1	3s	3s	NUM
cana-760	119	2	(	(	PUNCT
cana-760	119	3	2024	2024	NUM
cana-760	119	4	)	)	PUNCT
cana-760	119	5	216	216	NUM
cana-760	119	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	119	7	definition	definition	NOUN
cana-760	119	8	3.3	3.3	NUM
cana-760	119	9	.	.	PUNCT
cana-760	120	1	a	a	DET
cana-760	120	2	revised	revise	VERB
cana-760	120	3	fuzzy	fuzzy	ADJ
cana-760	120	4	modular	modular	ADJ
cana-760	120	5	metric	metric	ADJ
cana-760	120	6	휁%	휁%	NOUN
cana-760	120	7	on	on	ADP
cana-760	120	8	𝑌	𝑌	PROPN
cana-760	120	9	is	be	AUX
cana-760	120	10	said	say	VERB
cana-760	120	11	to	to	PART
cana-760	120	12	be	be	AUX
cana-760	120	13	satisfied	satisfy	VERB
cana-760	120	14	the	the	DET
cana-760	120	15	∆2t	∆2t	NOUN
cana-760	120	16	-	-	PUNCT
cana-760	120	17	condition	condition	NOUN
cana-760	120	18	if	if	SCONJ
cana-760	120	19	lim	lim	PROPN
cana-760	120	20	𝜅→∞	𝜅→∞	NUM
cana-760	120	21	𝛬(𝜄𝜅	𝛬(𝜄𝜅	NOUN
cana-760	120	22	,	,	PUNCT
cana-760	120	23	𝜄	𝜄	PROPN
cana-760	120	24	,	,	PUNCT
cana-760	120	25	𝑡	𝑡	PROPN
cana-760	120	26	)	)	PUNCT
cana-760	120	27	=	=	SYM
cana-760	120	28	0	0	NUM
cana-760	120	29	,	,	PUNCT
cana-760	120	30	for	for	ADP
cana-760	120	31	some	some	DET
cana-760	120	32	%	%	NOUN
cana-760	120	33	>	>	X
cana-760	120	34	0	0	PUNCT
cana-760	120	35	and	and	CCONJ
cana-760	120	36	for	for	ADP
cana-760	120	37	some	some	DET
cana-760	120	38	𝑡	𝑡	NOUN
cana-760	120	39	>	>	X
cana-760	120	40	0	0	PUNCT
cana-760	120	41	imply	imply	VERB
cana-760	120	42	that	that	SCONJ
cana-760	120	43	lim	lim	PROPN
cana-760	120	44	𝜅→∞	𝜅→∞	NUM
cana-760	120	45	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	120	46	,	,	PUNCT
cana-760	120	47	𝜄	𝜄	PROPN
cana-760	120	48	,	,	PUNCT
cana-760	120	49	𝑡	𝑡	PROPN
cana-760	120	50	)	)	PUNCT
cana-760	120	51	=	=	SYM
cana-760	120	52	0	0	NUM
cana-760	120	53	,	,	PUNCT
cana-760	120	54	for	for	ADP
cana-760	120	55	all	all	DET
cana-760	120	56	%	%	NOUN
cana-760	120	57	>	>	X
cana-760	120	58	0	0	NUM
cana-760	120	59	,	,	PUNCT
cana-760	120	60	and	and	CCONJ
cana-760	120	61	for	for	ADP
cana-760	120	62	all	all	DET
cana-760	120	63	𝑡	𝑡	X
cana-760	120	64	>	>	X
cana-760	120	65	0	0	NUM
cana-760	120	66	.	.	PUNCT
cana-760	120	67	example	example	NOUN
cana-760	121	1	3.1	3.1	NUM
cana-760	121	2	.	.	PUNCT
cana-760	122	1	let	let	AUX
cana-760	122	2	(	(	PUNCT
cana-760	122	3	𝑌	𝑌	PROPN
cana-760	122	4	,	,	PUNCT
cana-760	122	5	𝛩%	𝛩%	PRON
cana-760	122	6	)	)	PUNCT
cana-760	122	7	be	be	AUX
cana-760	122	8	a	a	DET
cana-760	122	9	modular	modular	ADJ
cana-760	122	10	metric	metric	ADJ
cana-760	122	11	space	space	NOUN
cana-760	122	12	.	.	PUNCT
cana-760	123	1	define	define	VERB
cana-760	123	2	𝛼	𝛼	NOUN
cana-760	123	3	∗	∗	NOUN
cana-760	123	4	𝛽	𝛽	NOUN
cana-760	123	5	=	=	SYM
cana-760	123	6	𝛼	𝛼	NOUN
cana-760	123	7	+	+	NOUN
cana-760	124	1	𝛽	𝛽	NOUN
cana-760	124	2	−	−	NOUN
cana-760	124	3	𝛼𝛽	𝛼𝛽	NOUN
cana-760	124	4	for	for	ADP
cana-760	124	5	all	all	DET
cana-760	124	6	𝛼	𝛼	PROPN
cana-760	124	7	,	,	PUNCT
cana-760	124	8	𝛽	𝛽	PROPN
cana-760	124	9	∈	∈	NOUN
cana-760	125	1	[	[	X
cana-760	125	2	0,1	0,1	NUM
cana-760	125	3	]	]	PUNCT
cana-760	125	4	,	,	PUNCT
cana-760	125	5	and	and	CCONJ
cana-760	125	6	휁%	휁%	NOUN
cana-760	125	7	:	:	PUNCT
cana-760	125	8	(	(	PUNCT
cana-760	125	9	0	0	NUM
cana-760	125	10	,	,	PUNCT
cana-760	125	11	∞	∞	PROPN
cana-760	125	12	)	)	PUNCT
cana-760	125	13	×	×	NOUN
cana-760	125	14	𝑌2	𝑌2	NOUN
cana-760	125	15	×	×	NOUN
cana-760	125	16	(	(	PUNCT
cana-760	125	17	0	0	NUM
cana-760	125	18	,	,	PUNCT
cana-760	125	19	∞	∞	NUM
cana-760	125	20	)	)	PUNCT
cana-760	125	21	→	→	PUNCT
cana-760	126	1	[	[	X
cana-760	126	2	0,1	0,1	NUM
cana-760	126	3	]	]	PUNCT
cana-760	126	4	by	by	ADP
cana-760	126	5	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	126	6	,	,	PUNCT
cana-760	126	7	휂	휂	ADP
cana-760	126	8	,	,	PUNCT
cana-760	126	9	𝑡	𝑡	NOUN
cana-760	126	10	)	)	PUNCT
cana-760	126	11	=	=	SYM
cana-760	127	1	𝛩𝜚(𝜄	𝛩𝜚(𝜄	PROPN
cana-760	127	2	,	,	PUNCT
cana-760	127	3	휂	휂	NOUN
cana-760	127	4	)	)	PUNCT
cana-760	127	5	𝑡	𝑡	PROPN
cana-760	128	1	+	+	PROPN
cana-760	128	2	𝛩𝜚(𝜄	𝛩𝜚(𝜄	PROPN
cana-760	128	3	,	,	PUNCT
cana-760	128	4	휂	휂	ADP
cana-760	128	5	)	)	PUNCT
cana-760	128	6	then	then	ADV
cana-760	128	7	(	(	PUNCT
cana-760	128	8	𝑌	𝑌	PROPN
cana-760	128	9	,	,	PUNCT
cana-760	128	10	휁%,⊛	휁%,⊛	NOUN
cana-760	128	11	)	)	PUNCT
cana-760	128	12	is	be	AUX
cana-760	128	13	a	a	DET
cana-760	128	14	modular	modular	ADJ
cana-760	128	15	revised	revise	VERB
cana-760	128	16	fuzzy	fuzzy	ADJ
cana-760	128	17	metric	metric	ADJ
cana-760	128	18	space	space	NOUN
cana-760	128	19	.	.	PUNCT
cana-760	129	1	remark	remark	PROPN
cana-760	129	2	3.1	3.1	NUM
cana-760	129	3	.	.	PUNCT
cana-760	130	1	(	(	PUNCT
cana-760	130	2	1	1	X
cana-760	130	3	)	)	PUNCT
cana-760	130	4	for	for	ADP
cana-760	130	5	𝛩𝜚(𝜄	𝛩𝜚(𝜄	PROPN
cana-760	130	6	,	,	PUNCT
cana-760	130	7	휂	휂	PROPN
cana-760	130	8	)	)	PUNCT
cana-760	130	9	=	=	SYM
cana-760	131	1	|𝜄−𝜂|	|𝜄−𝜂|	PUNCT
cana-760	131	2	𝜚	𝜚	INTJ
cana-760	131	3	,	,	PUNCT
cana-760	132	1	we	we	PRON
cana-760	132	2	have휁𝜚(𝜄	have휁𝜚(𝜄	PROPN
cana-760	132	3	,	,	PUNCT
cana-760	132	4	휂	휂	PROPN
cana-760	132	5	,	,	PUNCT
cana-760	132	6	𝑡	𝑡	NOUN
cana-760	132	7	)	)	PUNCT
cana-760	132	8	=	=	PUNCT
cana-760	133	1	|𝜄−𝜂|	|𝜄−𝜂|	PUNCT
cana-760	133	2	𝜚	𝜚	NOUN
cana-760	133	3	𝑡+	𝑡+	X
cana-760	133	4	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	133	5	𝜚	𝜚	NOUN
cana-760	133	6	is	be	AUX
cana-760	133	7	a	a	DET
cana-760	133	8	modular	modular	ADJ
cana-760	133	9	revised	revise	VERB
cana-760	133	10	fuzzy	fuzzy	ADJ
cana-760	133	11	metric	metric	NOUN
cana-760	133	12	.	.	PUNCT
cana-760	134	1	(	(	PUNCT
cana-760	134	2	2	2	NUM
cana-760	134	3	)	)	PUNCT
cana-760	134	4	for	for	ADP
cana-760	134	5	𝛩𝜚(𝜄	𝛩𝜚(𝜄	PROPN
cana-760	134	6	,	,	PUNCT
cana-760	134	7	휂	휂	PROPN
cana-760	134	8	)	)	PUNCT
cana-760	134	9	=	=	PUNCT
cana-760	135	1	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	135	2	𝜚+1	𝜚+1	PRON
cana-760	135	3	,	,	PUNCT
cana-760	135	4	we	we	PRON
cana-760	135	5	have	have	VERB
cana-760	135	6	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	135	7	,	,	PUNCT
cana-760	135	8	휂	휂	ADP
cana-760	135	9	,	,	PUNCT
cana-760	135	10	𝑡	𝑡	NOUN
cana-760	135	11	)	)	PUNCT
cana-760	135	12	=	=	PUNCT
cana-760	136	1	|𝜄−𝜂|	|𝜄−𝜂|	PUNCT
cana-760	136	2	𝜚+1	𝜚+1	PRON
cana-760	136	3	𝑡+	𝑡+	PUNCT
cana-760	136	4	|𝜄−𝜂|	|𝜄−𝜂|	PUNCT
cana-760	136	5	𝜚+1	𝜚+1	PRON
cana-760	136	6	is	be	AUX
cana-760	136	7	a	a	DET
cana-760	136	8	modular	modular	ADJ
cana-760	136	9	revised	revise	VERB
cana-760	136	10	fuzzy	fuzzy	ADJ
cana-760	136	11	metric	metric	NOUN
cana-760	136	12	.	.	PUNCT
cana-760	136	13	example	example	NOUN
cana-760	137	1	3.2	3.2	NUM
cana-760	137	2	.	.	PUNCT
cana-760	138	1	let	let	AUX
cana-760	138	2	(	(	PUNCT
cana-760	138	3	𝑌	𝑌	PROPN
cana-760	138	4	,	,	PUNCT
cana-760	138	5	𝛩%	𝛩%	PRON
cana-760	138	6	)	)	PUNCT
cana-760	138	7	be	be	AUX
cana-760	138	8	a	a	DET
cana-760	138	9	modular	modular	ADJ
cana-760	138	10	metric	metric	ADJ
cana-760	138	11	space	space	NOUN
cana-760	138	12	.	.	PUNCT
cana-760	139	1	define	define	VERB
cana-760	139	2	𝛼	𝛼	NOUN
cana-760	139	3	∗	∗	NOUN
cana-760	139	4	𝛽	𝛽	NOUN
cana-760	139	5	=	=	SYM
cana-760	139	6	𝛼	𝛼	NOUN
cana-760	139	7	+	+	NOUN
cana-760	140	1	𝛽	𝛽	NOUN
cana-760	140	2	−	−	NOUN
cana-760	140	3	𝛼𝛽	𝛼𝛽	NOUN
cana-760	140	4	for	for	ADP
cana-760	140	5	all	all	DET
cana-760	140	6	𝛼	𝛼	PROPN
cana-760	140	7	,	,	PUNCT
cana-760	140	8	𝛽	𝛽	PROPN
cana-760	140	9	∈	∈	NOUN
cana-760	141	1	[	[	X
cana-760	141	2	0,1	0,1	NUM
cana-760	141	3	]	]	PUNCT
cana-760	141	4	,	,	PUNCT
cana-760	141	5	and	and	CCONJ
cana-760	141	6	휁%	휁%	NOUN
cana-760	141	7	:	:	PUNCT
cana-760	141	8	(	(	PUNCT
cana-760	141	9	0	0	NUM
cana-760	141	10	,	,	PUNCT
cana-760	141	11	∞	∞	PROPN
cana-760	141	12	)	)	PUNCT
cana-760	141	13	×	×	NOUN
cana-760	141	14	𝑌2	𝑌2	NOUN
cana-760	141	15	×	×	NOUN
cana-760	141	16	(	(	PUNCT
cana-760	141	17	0	0	NUM
cana-760	141	18	,	,	PUNCT
cana-760	141	19	∞	∞	NUM
cana-760	141	20	)	)	PUNCT
cana-760	141	21	→	→	PUNCT
cana-760	141	22	[	[	X
cana-760	141	23	0,1	0,1	NUM
cana-760	141	24	]	]	PUNCT
cana-760	141	25	by	by	ADP
cana-760	141	26	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	141	27	,	,	PUNCT
cana-760	141	28	휂	휂	ADP
cana-760	141	29	,	,	PUNCT
cana-760	141	30	𝑡	𝑡	NOUN
cana-760	141	31	)	)	PUNCT
cana-760	141	32	=	=	SYM
cana-760	142	1	𝑒𝑥𝑝−{𝛩𝜚(𝜄,𝜂)}(1	𝑒𝑥𝑝−{𝛩𝜚(𝜄,𝜂)}(1	PROPN
cana-760	142	2	−	−	PROPN
cana-760	143	1	𝑒𝑥𝑝{𝛩𝜚(𝜄,𝜂	𝑒𝑥𝑝{𝛩𝜚(𝜄,𝜂	NOUN
cana-760	143	2	)	)	PUNCT
cana-760	143	3	}	}	PUNCT
cana-760	143	4	)	)	PUNCT
cana-760	143	5	then	then	ADV
cana-760	143	6	(	(	PUNCT
cana-760	143	7	𝑌	𝑌	PROPN
cana-760	143	8	,	,	PUNCT
cana-760	143	9	휁%,⊛	휁%,⊛	NOUN
cana-760	143	10	)	)	PUNCT
cana-760	143	11	is	be	AUX
cana-760	143	12	a	a	DET
cana-760	143	13	modular	modular	ADJ
cana-760	143	14	revised	revise	VERB
cana-760	143	15	fuzzy	fuzzy	ADJ
cana-760	143	16	metric	metric	ADJ
cana-760	143	17	space	space	NOUN
cana-760	143	18	.	.	PUNCT
cana-760	144	1	remark	remark	PROPN
cana-760	144	2	3.2	3.2	NUM
cana-760	144	3	.	.	PUNCT
cana-760	145	1	(	(	PUNCT
cana-760	145	2	1	1	X
cana-760	145	3	)	)	PUNCT
cana-760	145	4	for	for	ADP
cana-760	145	5	𝛩𝜚(𝜄	𝛩𝜚(𝜄	PROPN
cana-760	145	6	,	,	PUNCT
cana-760	145	7	휂	휂	PROPN
cana-760	145	8	)	)	PUNCT
cana-760	145	9	=	=	SYM
cana-760	145	10	|𝜄−𝜂|	|𝜄−𝜂|	PUNCT
cana-760	145	11	𝜚	𝜚	NOUN
cana-760	145	12	,	,	PUNCT
cana-760	145	13	we	we	PRON
cana-760	145	14	have	have	VERB
cana-760	145	15	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	145	16	,	,	PUNCT
cana-760	145	17	휂	휂	ADP
cana-760	145	18	,	,	PUNCT
cana-760	145	19	𝑡	𝑡	NOUN
cana-760	145	20	)	)	PUNCT
cana-760	145	21	=	=	NOUN
cana-760	146	1	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	146	2	−	−	PROPN
cana-760	146	3	{	{	PUNCT
cana-760	146	4	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	146	5	𝜚	𝜚	NOUN
cana-760	146	6	}	}	PUNCT
cana-760	146	7	(	(	PUNCT
cana-760	146	8	1	1	NUM
cana-760	146	9	−	−	NOUN
cana-760	146	10	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	146	11	{	{	PUNCT
cana-760	146	12	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	146	13	𝜚	𝜚	NOUN
cana-760	146	14	}	}	PUNCT
cana-760	146	15	)	)	PUNCT
cana-760	146	16	is	be	AUX
cana-760	146	17	a	a	DET
cana-760	146	18	modular	modular	ADJ
cana-760	146	19	revised	revise	VERB
cana-760	146	20	fuzzy	fuzzy	ADJ
cana-760	146	21	metric	metric	NOUN
cana-760	146	22	.	.	PUNCT
cana-760	147	1	(	(	PUNCT
cana-760	147	2	2	2	NUM
cana-760	147	3	)	)	PUNCT
cana-760	147	4	for	for	ADP
cana-760	147	5	𝛩𝜚(𝜄	𝛩𝜚(𝜄	PROPN
cana-760	147	6	,	,	PUNCT
cana-760	147	7	휂	휂	PROPN
cana-760	147	8	)	)	PUNCT
cana-760	147	9	=	=	PUNCT
cana-760	148	1	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	148	2	𝜚+1	𝜚+1	PRON
cana-760	148	3	,	,	PUNCT
cana-760	148	4	we	we	PRON
cana-760	148	5	have	have	VERB
cana-760	148	6	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	148	7	,	,	PUNCT
cana-760	148	8	휂	휂	ADP
cana-760	148	9	,	,	PUNCT
cana-760	148	10	𝑡	𝑡	NOUN
cana-760	148	11	)	)	PUNCT
cana-760	148	12	=	=	NOUN
cana-760	149	1	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	149	2	−	−	PROPN
cana-760	149	3	{	{	PUNCT
cana-760	149	4	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	149	5	𝜚+1	𝜚+1	PRON
cana-760	149	6	}	}	PUNCT
cana-760	149	7	(	(	PUNCT
cana-760	149	8	1	1	NUM
cana-760	149	9	−	−	NOUN
cana-760	149	10	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	149	11	{	{	PUNCT
cana-760	149	12	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	149	13	𝜚+1	𝜚+1	PROPN
cana-760	149	14	}	}	PUNCT
cana-760	149	15	)	)	PUNCT
cana-760	149	16	is	be	AUX
cana-760	149	17	a	a	DET
cana-760	149	18	modular	modular	ADJ
cana-760	149	19	revised	revise	VERB
cana-760	149	20	fuzzy	fuzzy	ADJ
cana-760	149	21	metric	metric	NOUN
cana-760	149	22	.	.	PUNCT
cana-760	150	1	theorem	theorem	VERB
cana-760	150	2	3.1	3.1	NUM
cana-760	150	3	.	.	PUNCT
cana-760	151	1	on	on	ADP
cana-760	151	2	a	a	DET
cana-760	151	3	complete	complete	ADJ
cana-760	151	4	modular	modular	NOUN
cana-760	151	5	revised	revise	VERB
cana-760	151	6	fuzzy	fuzzy	ADJ
cana-760	151	7	metric	metric	ADJ
cana-760	151	8	space	space	NOUN
cana-760	151	9	(	(	PUNCT
cana-760	151	10	𝑌	𝑌	PROPN
cana-760	151	11	,	,	PUNCT
cana-760	151	12	휁%,⊛	휁%,⊛	NOUN
cana-760	151	13	)	)	PUNCT
cana-760	151	14	,	,	PUNCT
cana-760	151	15	consider	consider	VERB
cana-760	151	16	a	a	DET
cana-760	151	17	continuous	continuous	ADJ
cana-760	151	18	mapping	mapping	NOUN
cana-760	151	19	𝛤	𝛤	NOUN
cana-760	151	20	:	:	PUNCT
cana-760	151	21	𝑌	𝑌	PROPN
cana-760	151	22	→	→	SYM
cana-760	151	23	𝑌.	𝑌.	PROPN
cana-760	151	24	suppose	suppose	VERB
cana-760	151	25	there	there	PRON
cana-760	151	26	exist	exist	VERB
cana-760	151	27	a	a	DET
cana-760	151	28	strictly	strictly	ADV
cana-760	151	29	non	non	ADJ
cana-760	151	30	-	-	ADJ
cana-760	151	31	decreasing	decrease	VERB
cana-760	151	32	,	,	PUNCT
cana-760	151	33	continuous	continuous	ADJ
cana-760	151	34	function	function	NOUN
cana-760	151	35	𝛶	𝛶	PROPN
cana-760	151	36	:	:	PUNCT
cana-760	151	37	(	(	PUNCT
cana-760	151	38	0,1	0,1	NUM
cana-760	151	39	]	]	PUNCT
cana-760	151	40	→	→	PUNCT
cana-760	152	1	[	[	X
cana-760	152	2	0	0	NUM
cana-760	152	3	,	,	PUNCT
cana-760	152	4	∞	∞	PROPN
cana-760	152	5	)	)	PUNCT
cana-760	152	6	with	with	ADP
cana-760	152	7	𝛶(1	𝛶(1	ADP
cana-760	152	8	)	)	PUNCT
cana-760	152	9	=	=	SYM
cana-760	152	10	0	0	NUM
cana-760	152	11	and	and	CCONJ
cana-760	152	12	a	a	DET
cana-760	152	13	real	real	ADJ
cana-760	152	14	number	number	NOUN
cana-760	152	15	𝐻	𝐻	NOUN
cana-760	152	16	with	with	ADP
cana-760	152	17	0	0	NUM
cana-760	152	18	<	<	X
cana-760	152	19	𝐻	𝐻	NOUN
cana-760	152	20	<	<	X
cana-760	152	21	1	1	NUM
cana-760	152	22	such	such	ADJ
cana-760	152	23	that	that	SCONJ
cana-760	152	24	𝛶	𝛶	PROPN
cana-760	152	25	(	(	PUNCT
cana-760	152	26	1	1	NUM
cana-760	152	27	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	NOUN
cana-760	152	28	)	)	PUNCT
cana-760	152	29	−	−	ADP
cana-760	152	30	1	1	X
cana-760	152	31	)	)	PUNCT
cana-760	152	32	≤	≤	NOUN
cana-760	152	33	𝐻𝛶	𝐻𝛶	PROPN
cana-760	152	34	(	(	PUNCT
cana-760	152	35	1	1	NUM
cana-760	152	36	𝜁%(𝜄,𝜂,𝑡	𝜁%(𝜄,𝜂,𝑡	NUM
cana-760	152	37	)	)	PUNCT
cana-760	152	38	−	−	PROPN
cana-760	152	39	1	1	NUM
cana-760	152	40	)	)	PUNCT
cana-760	152	41	,	,	PUNCT
cana-760	152	42	for	for	ADP
cana-760	152	43	all	all	DET
cana-760	152	44	𝜄	𝜄	PROPN
cana-760	152	45	,	,	PUNCT
cana-760	152	46	휂	휂	ADP
cana-760	152	47	∈	∈	PROPN
cana-760	152	48	𝑌	𝑌	PROPN
cana-760	152	49	,	,	PUNCT
cana-760	152	50	𝜄	𝜄	PROPN
cana-760	152	51	≠	≠	PROPN
cana-760	152	52	휂	휂	PROPN
cana-760	152	53	.	.	PUNCT
cana-760	152	54	(	(	PUNCT
cana-760	152	55	3.1	3.1	NUM
cana-760	152	56	)	)	PUNCT
cana-760	152	57	then	then	ADV
cana-760	152	58	𝛤	𝛤	PRON
cana-760	152	59	has	have	VERB
cana-760	152	60	a	a	DET
cana-760	152	61	unique	unique	ADJ
cana-760	152	62	fixed	fix	VERB
cana-760	152	63	point	point	NOUN
cana-760	152	64	in	in	ADP
cana-760	152	65	𝑌.	𝑌.	PROPN
cana-760	152	66	communications	communication	NOUN
cana-760	152	67	on	on	ADP
cana-760	152	68	applied	apply	VERB
cana-760	152	69	nonlinear	nonlinear	ADJ
cana-760	152	70	analysis	analysis	NOUN
cana-760	152	71	issn	issn	NOUN
cana-760	152	72	:	:	PUNCT
cana-760	152	73	1074	1074	NUM
cana-760	152	74	-	-	PUNCT
cana-760	152	75	133x	133x	NUM
cana-760	152	76	vol	vol	NOUN
cana-760	152	77	31	31	NUM
cana-760	152	78	no	no	NOUN
cana-760	152	79	.	.	PUNCT
cana-760	153	1	3s	3s	NUM
cana-760	153	2	(	(	PUNCT
cana-760	153	3	2024	2024	NUM
cana-760	153	4	)	)	PUNCT
cana-760	153	5	217	217	NUM
cana-760	153	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	153	7	proof	proof	NOUN
cana-760	153	8	.	.	PUNCT
cana-760	154	1	let	let	VERB
cana-760	154	2	𝜄0	𝜄0	PROPN
cana-760	154	3	be	be	AUX
cana-760	154	4	an	an	DET
cana-760	154	5	arbitrary	arbitrary	ADJ
cana-760	154	6	point	point	NOUN
cana-760	154	7	in	in	ADP
cana-760	154	8	𝑌.	𝑌.	PROPN
cana-760	154	9	choose	choose	VERB
cana-760	154	10	𝜄1	𝜄1	NOUN
cana-760	154	11	∈	∈	PROPN
cana-760	154	12	𝑌	𝑌	PROPN
cana-760	154	13	such	such	ADJ
cana-760	154	14	that	that	DET
cana-760	154	15	𝜄1	𝜄1	NOUN
cana-760	154	16	=	=	NOUN
cana-760	154	17	𝛤𝜄0	𝛤𝜄0	NOUN
cana-760	154	18	.	.	PUNCT
cana-760	155	1	continuing	continue	VERB
cana-760	155	2	this	this	DET
cana-760	155	3	process	process	NOUN
cana-760	155	4	,	,	PUNCT
cana-760	155	5	we	we	PRON
cana-760	155	6	construct	construct	VERB
cana-760	155	7	a	a	DET
cana-760	155	8	sequence	sequence	NOUN
cana-760	155	9	(	(	PUNCT
cana-760	155	10	𝜄𝜅	𝜄𝜅	X
cana-760	155	11	)	)	PUNCT
cana-760	156	1	such	such	ADJ
cana-760	156	2	that	that	SCONJ
cana-760	156	3	𝜄𝜅+1	𝜄𝜅+1	ADJ
cana-760	156	4	=	=	SYM
cana-760	156	5	𝛤𝜄𝜅	𝛤𝜄𝜅	PROPN
cana-760	156	6	,	,	PUNCT
cana-760	156	7	for	for	ADP
cana-760	156	8	𝜅	𝜅	PRON
cana-760	156	9	=	=	SYM
cana-760	156	10	0,1,2	0,1,2	X
cana-760	156	11	..	..	PUNCT
cana-760	156	12	let	let	VERB
cana-760	156	13	𝜄	𝜄	PROPN
cana-760	156	14	=	=	PUNCT
cana-760	156	15	𝜄𝜅−1	𝜄𝜅−1	PROPN
cana-760	156	16	and	and	CCONJ
cana-760	156	17	휂	휂	ADP
cana-760	156	18	=	=	PUNCT
cana-760	156	19	𝜄𝜅.	𝜄𝜅.	VERB
cana-760	156	20	replacing	replace	VERB
cana-760	156	21	this	this	PRON
cana-760	156	22	in	in	ADP
cana-760	156	23	(	(	PUNCT
cana-760	156	24	3.1	3.1	NUM
cana-760	156	25	)	)	PUNCT
cana-760	156	26	,	,	PUNCT
cana-760	156	27	we	we	PRON
cana-760	156	28	get	get	VERB
cana-760	156	29	𝛶	𝛶	PROPN
cana-760	156	30	(	(	PUNCT
cana-760	156	31	1	1	NUM
cana-760	156	32	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	NOUN
cana-760	156	33	)	)	PUNCT
cana-760	156	34	−	−	ADP
cana-760	157	1	1	1	X
cana-760	157	2	)	)	PUNCT
cana-760	157	3	=	=	PUNCT
cana-760	157	4	𝛶	𝛶	PROPN
cana-760	157	5	(	(	PUNCT
cana-760	157	6	1	1	NUM
cana-760	157	7	𝜁%(𝛤𝜄𝜅−1	𝜁%(𝛤𝜄𝜅−1	PRON
cana-760	157	8	,	,	PUNCT
cana-760	157	9	𝛤𝜄𝜅	𝛤𝜄𝜅	PROPN
cana-760	157	10	,	,	PUNCT
cana-760	157	11	𝑡	𝑡	NOUN
cana-760	157	12	)	)	PUNCT
cana-760	157	13	−	−	NUM
cana-760	157	14	1	1	NUM
cana-760	157	15	)	)	PUNCT
cana-760	157	16	=	=	PUNCT
cana-760	157	17	𝛶	𝛶	PROPN
cana-760	157	18	(	(	PUNCT
cana-760	157	19	1	1	NUM
cana-760	157	20	𝜁%(𝜄𝜅,𝜄𝜅+1,𝑡	𝜁%(𝜄𝜅,𝜄𝜅+1,𝑡	NOUN
cana-760	157	21	)	)	PUNCT
cana-760	157	22	−	−	NOUN
cana-760	157	23	1	1	X
cana-760	157	24	)	)	PUNCT
cana-760	157	25	≤	≤	NOUN
cana-760	157	26	𝐻𝛶	𝐻𝛶	PROPN
cana-760	157	27	(	(	PUNCT
cana-760	157	28	1	1	NUM
cana-760	157	29	휁%(𝜄𝜅−1	휁%(𝜄𝜅−1	ADV
cana-760	157	30	,	,	PUNCT
cana-760	157	31	𝜄𝜅	𝜄𝜅	INTJ
cana-760	157	32	,	,	PUNCT
cana-760	157	33	𝑡	𝑡	PROPN
cana-760	157	34	)	)	PUNCT
cana-760	157	35	−	−	PROPN
cana-760	157	36	1	1	NUM
cana-760	157	37	)	)	PUNCT
cana-760	157	38	<	<	X
cana-760	157	39	𝛶	𝛶	PROPN
cana-760	157	40	(	(	PUNCT
cana-760	157	41	1	1	NUM
cana-760	157	42	휁%(𝜄𝜅−1	휁%(𝜄𝜅−1	ADV
cana-760	157	43	,	,	PUNCT
cana-760	157	44	𝜄𝜅	𝜄𝜅	INTJ
cana-760	157	45	,	,	PUNCT
cana-760	157	46	𝑡	𝑡	PROPN
cana-760	157	47	)	)	PUNCT
cana-760	157	48	−	−	PROPN
cana-760	157	49	1	1	NUM
cana-760	157	50	)	)	PUNCT
cana-760	157	51	.	.	PUNCT
cana-760	158	1	since	since	SCONJ
cana-760	158	2	υ	υ	PROPN
cana-760	158	3	is	be	AUX
cana-760	158	4	a	a	DET
cana-760	158	5	strictly	strictly	ADV
cana-760	158	6	non	non	ADJ
cana-760	158	7	-	-	ADJ
cana-760	158	8	decreasing	decrease	VERB
cana-760	158	9	function	function	NOUN
cana-760	158	10	,	,	PUNCT
cana-760	158	11	we	we	PRON
cana-760	158	12	obtain	obtain	VERB
cana-760	158	13	(	(	PUNCT
cana-760	158	14	3.2	3.2	NUM
cana-760	158	15	)	)	PUNCT
cana-760	158	16	1	1	NUM
cana-760	158	17	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	158	18	,	,	PUNCT
cana-760	158	19	𝜄𝜅+1	𝜄𝜅+1	NOUN
cana-760	158	20	,	,	PUNCT
cana-760	158	21	𝑡	𝑡	PROPN
cana-760	158	22	)	)	PUNCT
cana-760	158	23	−	−	PROPN
cana-760	158	24	1	1	NUM
cana-760	158	25	<	<	X
cana-760	158	26	1	1	NUM
cana-760	158	27	휁%(𝜄𝜅−1	휁%(𝜄𝜅−1	PRON
cana-760	158	28	,	,	PUNCT
cana-760	158	29	𝜄𝜅	𝜄𝜅	INTJ
cana-760	158	30	,	,	PUNCT
cana-760	158	31	𝑡	𝑡	PROPN
cana-760	158	32	)	)	PUNCT
cana-760	158	33	−	−	PROPN
cana-760	159	1	1	1	X
cana-760	159	2	.	.	PUNCT
cana-760	160	1	we	we	PRON
cana-760	160	2	use	use	VERB
cana-760	160	3	the	the	DET
cana-760	160	4	same	same	ADJ
cana-760	160	5	method	method	NOUN
cana-760	160	6	for	for	ADP
cana-760	160	7	𝜄	𝜄	PROPN
cana-760	160	8	=	=	PUNCT
cana-760	160	9	𝜄𝜅−2	𝜄𝜅−2	PROPN
cana-760	160	10	and	and	CCONJ
cana-760	160	11	휂	휂	ADP
cana-760	160	12	=	=	SYM
cana-760	160	13	𝜄𝜅−1	𝜄𝜅−1	PROPN
cana-760	160	14	,	,	PUNCT
cana-760	160	15	we	we	PRON
cana-760	160	16	get	get	VERB
cana-760	160	17	(	(	PUNCT
cana-760	160	18	3.3	3.3	NUM
cana-760	160	19	)	)	PUNCT
cana-760	160	20	1	1	NUM
cana-760	160	21	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	160	22	,	,	PUNCT
cana-760	160	23	𝜄𝜅−1	𝜄𝜅−1	PROPN
cana-760	160	24	,	,	PUNCT
cana-760	160	25	𝑡	𝑡	PROPN
cana-760	160	26	)	)	PUNCT
cana-760	160	27	−	−	PROPN
cana-760	160	28	1	1	NUM
cana-760	160	29	<	<	X
cana-760	160	30	1	1	NUM
cana-760	160	31	휁%(𝜄𝜅−1	휁%(𝜄𝜅−1	ADV
cana-760	160	32	,	,	PUNCT
cana-760	160	33	𝜄𝜅−2	𝜄𝜅−2	PROPN
cana-760	160	34	,	,	PUNCT
cana-760	160	35	𝑡	𝑡	PROPN
cana-760	160	36	)	)	PUNCT
cana-760	160	37	−	−	PROPN
cana-760	161	1	1	1	NUM
cana-760	161	2	.	.	PUNCT
cana-760	161	3	(	(	PUNCT
cana-760	161	4	3.4	3.4	NUM
cana-760	161	5	)	)	PUNCT
cana-760	161	6	therefore	therefore	ADV
cana-760	161	7	,	,	PUNCT
cana-760	161	8	(	(	PUNCT
cana-760	161	9	3.3	3.3	NUM
cana-760	161	10	)	)	PUNCT
cana-760	161	11	and	and	CCONJ
cana-760	161	12	(	(	PUNCT
cana-760	161	13	3.4	3.4	NUM
cana-760	161	14	)	)	PUNCT
cana-760	161	15	imply	imply	VERB
cana-760	161	16	that	that	SCONJ
cana-760	161	17	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	161	18	,	,	PUNCT
cana-760	161	19	𝜄𝜅−1	𝜄𝜅−1	PROPN
cana-760	161	20	,	,	PUNCT
cana-760	161	21	𝑡	𝑡	PROPN
cana-760	161	22	)	)	PUNCT
cana-760	161	23	is	be	AUX
cana-760	161	24	a	a	DET
cana-760	161	25	strictly	strictly	ADV
cana-760	161	26	non	non	ADJ
cana-760	161	27	-	-	ADJ
cana-760	161	28	increasing	increasing	ADJ
cana-760	161	29	sequence	sequence	NOUN
cana-760	161	30	of	of	ADP
cana-760	161	31	positive	positive	ADJ
cana-760	161	32	real	real	ADJ
cana-760	161	33	numbers	number	NOUN
cana-760	161	34	in	in	ADP
cana-760	161	35	[	[	X
cana-760	161	36	0,1	0,1	NUM
cana-760	161	37	]	]	PUNCT
cana-760	161	38	.	.	PUNCT
cana-760	162	1	put	put	VERB
cana-760	162	2	𝛴𝜅(%	𝛴𝜅(%	NUM
cana-760	162	3	,	,	PUNCT
cana-760	162	4	𝑡	𝑡	X
cana-760	162	5	)	)	PUNCT
cana-760	162	6	=	=	SYM
cana-760	163	1	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	163	2	,	,	PUNCT
cana-760	163	3	𝜄𝜅+1	𝜄𝜅+1	NOUN
cana-760	163	4	,	,	PUNCT
cana-760	163	5	𝑡	𝑡	PROPN
cana-760	163	6	)	)	PUNCT
cana-760	163	7	.	.	PUNCT
cana-760	164	1	then	then	ADV
cana-760	164	2	{	{	PUNCT
cana-760	164	3	𝛴𝜅(%	𝛴𝜅(%	PROPN
cana-760	164	4	,	,	PUNCT
cana-760	164	5	𝑡	𝑡	NOUN
cana-760	164	6	)	)	PUNCT
cana-760	164	7	}	}	PUNCT
cana-760	164	8	is	be	AUX
cana-760	164	9	a	a	DET
cana-760	164	10	strictly	strictly	ADV
cana-760	164	11	non	non	ADJ
cana-760	164	12	-	-	ADJ
cana-760	164	13	increasing	increase	VERB
cana-760	164	14	sequence	sequence	NOUN
cana-760	164	15	.	.	PUNCT
cana-760	165	1	so	so	ADV
cana-760	165	2	∃	∃	PROPN
cana-760	165	3	𝛴(%	𝛴(%	PROPN
cana-760	165	4	,	,	PUNCT
cana-760	165	5	𝑡	𝑡	PROPN
cana-760	165	6	)	)	PUNCT
cana-760	165	7	such	such	ADJ
cana-760	165	8	that	that	SCONJ
cana-760	165	9	lim	lim	PROPN
cana-760	165	10	𝜅→∞	𝜅→∞	NUM
cana-760	165	11	𝛴𝜅(%	𝛴𝜅(%	PROPN
cana-760	165	12	,	,	PUNCT
cana-760	165	13	𝑡	𝑡	NOUN
cana-760	165	14	)	)	PUNCT
cana-760	165	15	=	=	SYM
cana-760	165	16	𝛴(%	𝛴(%	NOUN
cana-760	165	17	,	,	PUNCT
cana-760	165	18	𝑡	𝑡	PROPN
cana-760	165	19	)	)	PUNCT
cana-760	165	20	.	.	PUNCT
cana-760	166	1	assume	assume	VERB
cana-760	166	2	that	that	SCONJ
cana-760	166	3	0	0	NUM
cana-760	166	4	<	<	X
cana-760	166	5	𝛴(%	𝛴(%	NOUN
cana-760	166	6	,	,	PUNCT
cana-760	166	7	𝑡	𝑡	PROPN
cana-760	166	8	)	)	PUNCT
cana-760	166	9	<	<	X
cana-760	166	10	1	1	X
cana-760	166	11	.	.	PUNCT
cana-760	166	12	by	by	ADP
cana-760	166	13	(	(	PUNCT
cana-760	166	14	3.2	3.2	NUM
cana-760	166	15	)	)	PUNCT
cana-760	166	16	,	,	PUNCT
cana-760	166	17	we	we	PRON
cana-760	166	18	have	have	VERB
cana-760	166	19	𝛶(𝛴𝜅(%	𝛶(𝛴𝜅(%	NOUN
cana-760	166	20	,	,	PUNCT
cana-760	166	21	𝑡	𝑡	NOUN
cana-760	166	22	)	)	PUNCT
cana-760	166	23	)	)	PUNCT
cana-760	166	24	≤	≤	NUM
cana-760	166	25	𝐻𝛶(𝛴𝜅−1(%	𝐻𝛶(𝛴𝜅−1(%	NOUN
cana-760	166	26	,	,	PUNCT
cana-760	166	27	𝑡	𝑡	NOUN
cana-760	166	28	)	)	PUNCT
cana-760	166	29	)	)	PUNCT
cana-760	166	30	.	.	PUNCT
cana-760	167	1	so	so	ADV
cana-760	167	2	,	,	PUNCT
cana-760	167	3	lim	lim	PROPN
cana-760	167	4	𝜅→∞	𝜅→∞	NUM
cana-760	167	5	𝛶(𝛴𝜅(%	𝛶(𝛴𝜅(%	NOUN
cana-760	167	6	,	,	PUNCT
cana-760	167	7	𝑡	𝑡	NOUN
cana-760	167	8	)	)	PUNCT
cana-760	167	9	)	)	PUNCT
cana-760	167	10	≤	≤	PROPN
cana-760	167	11	lim	lim	PROPN
cana-760	167	12	𝜅→∞	𝜅→∞	NUM
cana-760	167	13	𝐻𝛶(𝛴𝜅−1(%	𝐻𝛶(𝛴𝜅−1(%	PROPN
cana-760	167	14	,	,	PUNCT
cana-760	167	15	𝑡	𝑡	NOUN
cana-760	167	16	)	)	PUNCT
cana-760	167	17	)	)	PUNCT
cana-760	167	18	.	.	PUNCT
cana-760	168	1	the	the	DET
cana-760	168	2	continuity	continuity	NOUN
cana-760	168	3	of	of	ADP
cana-760	168	4	𝛶	𝛶	PROPN
cana-760	168	5	implies	imply	VERB
cana-760	168	6	that	that	SCONJ
cana-760	168	7	𝛶(𝛴(%	𝛶(𝛴(%	ADJ
cana-760	168	8	,	,	PUNCT
cana-760	168	9	𝑡	𝑡	NOUN
cana-760	168	10	)	)	PUNCT
cana-760	168	11	)	)	PUNCT
cana-760	168	12	≤	≤	NUM
cana-760	168	13	𝐻𝛶(𝛴(%	𝐻𝛶(𝛴(%	NOUN
cana-760	168	14	,	,	PUNCT
cana-760	168	15	𝑡	𝑡	NOUN
cana-760	168	16	)	)	PUNCT
cana-760	168	17	)	)	PUNCT
cana-760	168	18	,	,	PUNCT
cana-760	168	19	a	a	DET
cana-760	168	20	contradiction	contradiction	NOUN
cana-760	168	21	.	.	PUNCT
cana-760	169	1	then	then	ADV
cana-760	169	2	𝛴(%	𝛴(%	PROPN
cana-760	169	3	,	,	PUNCT
cana-760	169	4	𝑡	𝑡	PROPN
cana-760	169	5	)	)	PUNCT
cana-760	169	6	=	=	SYM
cana-760	169	7	0	0	X
cana-760	169	8	.	.	PUNCT
cana-760	170	1	now	now	ADV
cana-760	170	2	,	,	PUNCT
cana-760	170	3	we	we	PRON
cana-760	170	4	will	will	AUX
cana-760	170	5	prove	prove	VERB
cana-760	170	6	that	that	SCONJ
cana-760	170	7	{	{	PUNCT
cana-760	170	8	𝜄𝜅	𝜄𝜅	NOUN
cana-760	170	9	}	}	PUNCT
cana-760	170	10	is	be	AUX
cana-760	170	11	a	a	DET
cana-760	170	12	cauchy	cauchy	ADJ
cana-760	170	13	sequence	sequence	NOUN
cana-760	170	14	.	.	PUNCT
cana-760	171	1	assume	assume	VERB
cana-760	171	2	not	not	PART
cana-760	171	3	,	,	PUNCT
cana-760	171	4	then	then	ADV
cana-760	171	5	for	for	ADP
cana-760	171	6	0	0	NUM
cana-760	171	7	<	<	X
cana-760	171	8	휀	휀	X
cana-760	171	9	<	<	X
cana-760	171	10	1	1	NUM
cana-760	171	11	,	,	PUNCT
cana-760	171	12	there	there	PRON
cana-760	171	13	exist	exist	VERB
cana-760	171	14	two	two	NUM
cana-760	171	15	sub	sub	NOUN
cana-760	171	16	-	-	NOUN
cana-760	171	17	sequences	sequence	NOUN
cana-760	171	18	{	{	PUNCT
cana-760	171	19	𝜄𝑌(𝑖	𝜄𝑌(𝑖	ADV
cana-760	171	20	)	)	PUNCT
cana-760	171	21	}	}	PUNCT
cana-760	171	22	and	and	CCONJ
cana-760	171	23	{	{	PUNCT
cana-760	171	24	𝜄𝜅(𝑖	𝜄𝜅(𝑖	NOUN
cana-760	171	25	)	)	PUNCT
cana-760	171	26	}	}	PUNCT
cana-760	171	27	such	such	ADJ
cana-760	171	28	that	that	SCONJ
cana-760	171	29	for	for	SCONJ
cana-760	171	30	each	each	PRON
cana-760	171	31	𝑖	𝑖	ADP
cana-760	171	32	∈	∈	PROPN
cana-760	171	33	𝑁.	𝑁.	PROPN
cana-760	171	34	let	let	VERB
cana-760	171	35	𝜅(𝑖	𝜅(𝑖	NOUN
cana-760	171	36	)	)	PUNCT
cana-760	171	37	,	,	PUNCT
cana-760	171	38	𝑌(𝑖	𝑌(𝑖	X
cana-760	171	39	)	)	PUNCT
cana-760	171	40	∈	∈	NOUN
cana-760	172	1	𝑁	𝑁	NOUN
cana-760	172	2	satisfying	satisfy	VERB
cana-760	172	3	𝜅(𝑖	𝜅(𝑖	NOUN
cana-760	172	4	)	)	PUNCT
cana-760	172	5	,	,	PUNCT
cana-760	172	6	𝑌(𝑖	𝑌(𝑖	NUM
cana-760	172	7	)	)	PUNCT
cana-760	172	8	≥	≥	NOUN
cana-760	172	9	𝜅	𝜅	NOUN
cana-760	172	10	and	and	CCONJ
cana-760	172	11	𝜅(𝑖	𝜅(𝑖	X
cana-760	172	12	)	)	PUNCT
cana-760	172	13	<	<	X
cana-760	172	14	𝑌(𝑖	𝑌(𝑖	X
cana-760	172	15	)	)	PUNCT
cana-760	172	16	<	<	X
cana-760	172	17	𝑖	𝑖	SYM
cana-760	172	18	,	,	PUNCT
cana-760	172	19	such	such	ADJ
cana-760	172	20	that	that	SCONJ
cana-760	172	21	1	1	NUM
cana-760	172	22	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	NOUN
cana-760	172	23	)	)	PUNCT
cana-760	172	24	−	−	PROPN
cana-760	172	25	1	1	NUM
cana-760	172	26	≥	≥	NOUN
cana-760	172	27	휀	휀	NOUN
cana-760	172	28	,	,	PUNCT
cana-760	172	29	1	1	NUM
cana-760	172	30	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡	NUM
cana-760	172	31	)	)	PUNCT
cana-760	172	32	−	−	NOUN
cana-760	172	33	1	1	NUM
cana-760	172	34	<	<	X
cana-760	172	35	휀	휀	NOUN
cana-760	172	36	,	,	PUNCT
cana-760	172	37	1	1	NUM
cana-760	172	38	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖),𝑡	ADJ
cana-760	172	39	)	)	PUNCT
cana-760	172	40	−	−	NOUN
cana-760	172	41	1	1	NUM
cana-760	172	42	<	<	X
cana-760	172	43	휀	휀	X
cana-760	172	44	.	.	PUNCT
cana-760	172	45	(	(	PUNCT
cana-760	172	46	3.5	3.5	NUM
cana-760	172	47	)	)	PUNCT
cana-760	172	48	consider	consider	VERB
cana-760	172	49	휀	휀	NOUN
cana-760	172	50	≤	≤	NUM
cana-760	172	51	1	1	NUM
cana-760	172	52	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	NOUN
cana-760	172	53	)	)	PUNCT
cana-760	172	54	−	−	ADP
cana-760	172	55	1	1	NUM
cana-760	172	56	≤	≤	NUM
cana-760	172	57	1	1	NUM
cana-760	172	58	𝜁𝜚	𝜁𝜚	ADP
cana-760	172	59	2	2	NUM
cana-760	172	60	(	(	PUNCT
cana-760	172	61	𝜄𝜅(𝑖),𝜄𝜉(𝑖)−1	𝜄𝜅(𝑖),𝜄𝜉(𝑖)−1	NOUN
cana-760	172	62	,	,	PUNCT
cana-760	172	63	𝑡	𝑡	X
cana-760	172	64	2	2	NUM
cana-760	172	65	)	)	PUNCT
cana-760	172	66	−	−	PROPN
cana-760	172	67	1	1	NUM
cana-760	172	68	⊛	⊛	NUM
cana-760	172	69	1	1	NUM
cana-760	172	70	𝜁𝜚	𝜁𝜚	ADP
cana-760	172	71	2	2	NUM
cana-760	172	72	(	(	PUNCT
cana-760	172	73	𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖	𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖	NOUN
cana-760	172	74	)	)	PUNCT
cana-760	172	75	,	,	PUNCT
cana-760	172	76	𝑡	𝑡	PROPN
cana-760	172	77	2	2	NUM
cana-760	172	78	)	)	PUNCT
cana-760	172	79	−	−	PROPN
cana-760	172	80	1	1	X
cana-760	172	81	.	.	PUNCT
cana-760	173	1	by	by	ADP
cana-760	173	2	definition	definition	NOUN
cana-760	173	3	of	of	ADP
cana-760	173	4	∆2t	∆2t	NOUN
cana-760	173	5	-	-	PUNCT
cana-760	173	6	condition	condition	NOUN
cana-760	173	7	on	on	ADP
cana-760	173	8	y	y	PROPN
cana-760	173	9	and	and	CCONJ
cana-760	173	10	(	(	PUNCT
cana-760	173	11	3.5	3.5	NUM
cana-760	173	12	)	)	PUNCT
cana-760	173	13	,	,	PUNCT
cana-760	173	14	we	we	PRON
cana-760	173	15	have	have	VERB
cana-760	173	16	communications	communication	NOUN
cana-760	173	17	on	on	ADP
cana-760	173	18	applied	apply	VERB
cana-760	173	19	nonlinear	nonlinear	ADJ
cana-760	173	20	analysis	analysis	NOUN
cana-760	173	21	issn	issn	NOUN
cana-760	173	22	:	:	PUNCT
cana-760	173	23	1074	1074	NUM
cana-760	173	24	-	-	PUNCT
cana-760	173	25	133x	133x	NUM
cana-760	173	26	vol	vol	NOUN
cana-760	173	27	31	31	NUM
cana-760	173	28	no	no	NOUN
cana-760	173	29	.	.	PUNCT
cana-760	174	1	3s	3s	NUM
cana-760	174	2	(	(	PUNCT
cana-760	174	3	2024	2024	NUM
cana-760	174	4	)	)	PUNCT
cana-760	174	5	218	218	NUM
cana-760	174	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	174	7	1	1	NUM
cana-760	174	8	𝜁𝜚	𝜁𝜚	ADP
cana-760	174	9	2	2	NUM
cana-760	174	10	(	(	PUNCT
cana-760	174	11	𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖	𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖	NOUN
cana-760	174	12	)	)	PUNCT
cana-760	174	13	,	,	PUNCT
cana-760	174	14	𝑡	𝑡	PROPN
cana-760	174	15	2	2	NUM
cana-760	174	16	)	)	PUNCT
cana-760	174	17	−	−	NOUN
cana-760	175	1	1	1	NUM
cana-760	175	2	<	<	X
cana-760	175	3	휀	휀	NOUN
cana-760	175	4	.	.	PUNCT
cana-760	175	5	hence	hence	ADV
cana-760	175	6	,	,	PUNCT
cana-760	175	7	휀	휀	NOUN
cana-760	175	8	≤	≤	NUM
cana-760	175	9	1	1	NUM
cana-760	175	10	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	NOUN
cana-760	175	11	)	)	PUNCT
cana-760	175	12	−	−	ADP
cana-760	175	13	1	1	NUM
cana-760	175	14	≤	≤	NUM
cana-760	175	15	1	1	NUM
cana-760	175	16	𝜁𝜚	𝜁𝜚	ADP
cana-760	175	17	2	2	NUM
cana-760	175	18	(	(	PUNCT
cana-760	175	19	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	NOUN
cana-760	175	20	,	,	PUNCT
cana-760	175	21	𝑡	𝑡	X
cana-760	175	22	2	2	NUM
cana-760	175	23	)	)	PUNCT
cana-760	175	24	−	−	PROPN
cana-760	175	25	1	1	NUM
cana-760	175	26	⊛	⊛	NUM
cana-760	175	27	휀	휀	NOUN
cana-760	175	28	.	.	PUNCT
cana-760	176	1	if	if	SCONJ
cana-760	176	2	𝑖	𝑖	X
cana-760	176	3	→	→	SYM
cana-760	176	4	∞	∞	PROPN
cana-760	176	5	,	,	PUNCT
cana-760	176	6	we	we	PRON
cana-760	176	7	have	have	VERB
cana-760	176	8	𝛴𝜅(𝑖	𝛴𝜅(𝑖	VERB
cana-760	176	9	)	)	PUNCT
cana-760	176	10	(	(	PUNCT
cana-760	176	11	𝜚	𝜚	NOUN
cana-760	176	12	2	2	NUM
cana-760	176	13	,	,	PUNCT
cana-760	176	14	𝑡	𝑡	X
cana-760	176	15	2	2	NUM
cana-760	176	16	)	)	PUNCT
cana-760	176	17	=	=	SYM
cana-760	176	18	1	1	NUM
cana-760	176	19	𝜁𝜚	𝜁𝜚	NUM
cana-760	176	20	2	2	NUM
cana-760	176	21	(	(	PUNCT
cana-760	176	22	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	NOUN
cana-760	176	23	,	,	PUNCT
cana-760	176	24	𝑡	𝑡	X
cana-760	176	25	2	2	NUM
cana-760	176	26	)	)	PUNCT
cana-760	176	27	−	−	PROPN
cana-760	176	28	1	1	NUM
cana-760	176	29	→	→	SYM
cana-760	176	30	0	0	NUM
cana-760	176	31	.	.	PUNCT
cana-760	177	1	so	so	ADV
cana-760	177	2	,	,	PUNCT
cana-760	177	3	1	1	NUM
cana-760	177	4	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	NOUN
cana-760	177	5	)	)	PUNCT
cana-760	178	1	−	−	PROPN
cana-760	178	2	1	1	NUM
cana-760	178	3	→	→	SYM
cana-760	178	4	휀	휀	X
cana-760	178	5	.	.	PUNCT
cana-760	178	6	then	then	ADV
cana-760	178	7	by	by	ADP
cana-760	178	8	(	(	PUNCT
cana-760	178	9	3.1	3.1	NUM
cana-760	178	10	)	)	PUNCT
cana-760	178	11	,	,	PUNCT
cana-760	178	12	we	we	PRON
cana-760	178	13	have	have	VERB
cana-760	178	14	𝛶	𝛶	PROPN
cana-760	178	15	(	(	PUNCT
cana-760	178	16	1	1	NUM
cana-760	178	17	𝜁%(𝜄𝜅(𝑖),𝜄𝑌(𝑖),𝑡	𝜁%(𝜄𝜅(𝑖),𝜄𝑌(𝑖),𝑡	NUM
cana-760	178	18	)	)	PUNCT
cana-760	178	19	−	−	ADP
cana-760	178	20	1	1	X
cana-760	178	21	)	)	PUNCT
cana-760	178	22	≤	≤	NOUN
cana-760	178	23	𝐻𝛶	𝐻𝛶	PROPN
cana-760	178	24	(	(	PUNCT
cana-760	178	25	1	1	NUM
cana-760	178	26	𝜁%(𝜄𝜅(𝑖)−1,𝜄𝑌(𝑖)−1,𝑡	𝜁%(𝜄𝜅(𝑖)−1,𝜄𝑌(𝑖)−1,𝑡	PROPN
cana-760	178	27	)	)	PUNCT
cana-760	178	28	−	−	NOUN
cana-760	178	29	1	1	NUM
cana-760	178	30	)	)	PUNCT
cana-760	178	31	<	<	X
cana-760	179	1	𝛶	𝛶	PROPN
cana-760	179	2	(	(	PUNCT
cana-760	179	3	1	1	NUM
cana-760	179	4	𝜁%(𝜄𝜅(𝑖)−1,𝜄𝑌(𝑖)−1,𝑡	𝜁%(𝜄𝜅(𝑖)−1,𝜄𝑌(𝑖)−1,𝑡	PROPN
cana-760	179	5	)	)	PUNCT
cana-760	179	6	−	−	NOUN
cana-760	179	7	1	1	NUM
cana-760	179	8	)	)	PUNCT
cana-760	179	9	.	.	PUNCT
cana-760	180	1	thus	thus	ADV
cana-760	180	2	휀	휀	DET
cana-760	180	3	≤	≤	NUM
cana-760	180	4	1	1	NUM
cana-760	180	5	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	NOUN
cana-760	180	6	)	)	PUNCT
cana-760	180	7	−	−	ADP
cana-760	180	8	1	1	NUM
cana-760	180	9	<	<	SYM
cana-760	180	10	1	1	NUM
cana-760	180	11	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖)−1	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖)−1	NOUN
cana-760	180	12	,	,	PUNCT
cana-760	180	13	𝑡	𝑡	X
cana-760	180	14	2	2	NUM
cana-760	180	15	)	)	PUNCT
cana-760	180	16	−	−	NOUN
cana-760	180	17	1	1	NUM
cana-760	180	18	<	<	X
cana-760	180	19	휀	휀	NOUN
cana-760	180	20	,	,	PUNCT
cana-760	180	21	which	which	PRON
cana-760	180	22	is	be	AUX
cana-760	180	23	impossible	impossible	ADJ
cana-760	180	24	.	.	PUNCT
cana-760	181	1	hence	hence	ADV
cana-760	181	2	{	{	PUNCT
cana-760	181	3	𝜄𝜅	𝜄𝜅	X
cana-760	181	4	}	}	PUNCT
cana-760	181	5	is	be	AUX
cana-760	181	6	a	a	DET
cana-760	181	7	cauchy	cauchy	ADJ
cana-760	181	8	sequence	sequence	NOUN
cana-760	181	9	in	in	ADP
cana-760	181	10	a	a	DET
cana-760	181	11	complete	complete	ADJ
cana-760	181	12	modular	modular	NOUN
cana-760	181	13	revised	revise	VERB
cana-760	181	14	fuzzy	fuzzy	ADJ
cana-760	181	15	metric	metric	ADJ
cana-760	181	16	space	space	NOUN
cana-760	181	17	.	.	PUNCT
cana-760	182	1	so	so	ADV
cana-760	182	2	∃$	∃$	X
cana-760	182	3	∈	∈	PROPN
cana-760	182	4	𝑌	𝑌	PROPN
cana-760	182	5	such	such	ADJ
cana-760	182	6	that	that	SCONJ
cana-760	182	7	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-760	182	8	𝜄𝜅	𝜄𝜅	NOUN
cana-760	182	9	=	=	PUNCT
cana-760	182	10	$	$	SYM
cana-760	182	11	,	,	PUNCT
cana-760	182	12	that	that	PRON
cana-760	182	13	means	mean	VERB
cana-760	182	14	lim	lim	PROPN
cana-760	182	15	𝜅→∞	𝜅→∞	NUM
cana-760	182	16	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	182	17	,	,	PUNCT
cana-760	182	18	$	$	SYM
cana-760	182	19	,	,	PUNCT
cana-760	182	20	𝑡	𝑡	NOUN
cana-760	182	21	)	)	PUNCT
cana-760	182	22	=	=	SYM
cana-760	182	23	0	0	X
cana-760	182	24	.	.	PUNCT
cana-760	182	25	to	to	PART
cana-760	182	26	show	show	VERB
cana-760	182	27	$	$	SYM
cana-760	182	28	is	be	AUX
cana-760	182	29	a	a	DET
cana-760	182	30	fixed	fix	VERB
cana-760	182	31	point	point	NOUN
cana-760	182	32	of	of	ADP
cana-760	182	33	𝛤	𝛤	PROPN
cana-760	182	34	,	,	PUNCT
cana-760	182	35	we	we	PRON
cana-760	182	36	have	have	VERB
cana-760	182	37	:	:	PUNCT
cana-760	182	38	𝛤	𝛤	PROPN
cana-760	182	39	is	be	AUX
cana-760	182	40	continuous	continuous	ADJ
cana-760	182	41	:	:	PUNCT
cana-760	182	42	𝜄𝜅	𝜄𝜅	NOUN
cana-760	182	43	→	→	SYM
cana-760	182	44	$	$	SYM
cana-760	182	45	⇒	⇒	NOUN
cana-760	182	46	𝛤𝜄𝜅	𝛤𝜄𝜅	PROPN
cana-760	182	47	→	→	SYM
cana-760	182	48	𝛤$.	𝛤$.	X
cana-760	182	49	by	by	ADP
cana-760	182	50	(	(	PUNCT
cana-760	182	51	3.1	3.1	NUM
cana-760	182	52	)	)	PUNCT
cana-760	182	53	,	,	PUNCT
cana-760	182	54	we	we	PRON
cana-760	182	55	have	have	VERB
cana-760	182	56	𝛶	𝛶	PROPN
cana-760	182	57	(	(	PUNCT
cana-760	182	58	1	1	NUM
cana-760	182	59	𝜁%(𝜄𝜅,𝛤𝜄𝜅,𝑡	𝜁%(𝜄𝜅,𝛤𝜄𝜅,𝑡	NOUN
cana-760	182	60	)	)	PUNCT
cana-760	183	1	−	−	ADP
cana-760	183	2	1	1	X
cana-760	183	3	)	)	PUNCT
cana-760	183	4	≤	≤	NOUN
cana-760	183	5	𝐻𝛶	𝐻𝛶	PROPN
cana-760	183	6	(	(	PUNCT
cana-760	183	7	1	1	NUM
cana-760	183	8	𝜁%(𝜄𝜅−1,𝜄𝜅,𝑡	𝜁%(𝜄𝜅−1,𝜄𝜅,𝑡	NUM
cana-760	183	9	)	)	PUNCT
cana-760	184	1	−	−	ADP
cana-760	184	2	1	1	NUM
cana-760	184	3	)	)	PUNCT
cana-760	184	4	.	.	PUNCT
cana-760	185	1	since	since	SCONJ
cana-760	185	2	𝛶(1	𝛶(1	ADP
cana-760	185	3	)	)	PUNCT
cana-760	185	4	=	=	SYM
cana-760	185	5	0	0	NUM
cana-760	185	6	and	and	CCONJ
cana-760	185	7	for	for	ADP
cana-760	185	8	𝜅	𝜅	PRON
cana-760	185	9	→	→	SYM
cana-760	185	10	∞	∞	PROPN
cana-760	185	11	,	,	PUNCT
cana-760	185	12	we	we	PRON
cana-760	185	13	get	get	VERB
cana-760	185	14	𝛶	𝛶	PROPN
cana-760	185	15	(	(	PUNCT
cana-760	185	16	1	1	NUM
cana-760	185	17	𝜁%($,𝛤$,𝑡	𝜁%($,𝛤$,𝑡	NOUN
cana-760	185	18	)	)	PUNCT
cana-760	185	19	−	−	ADP
cana-760	185	20	1	1	X
cana-760	185	21	)	)	PUNCT
cana-760	185	22	≤	≤	NOUN
cana-760	185	23	𝐻𝛶	𝐻𝛶	PROPN
cana-760	185	24	(	(	PUNCT
cana-760	185	25	1	1	NUM
cana-760	185	26	𝜁%($,$,𝑡	𝜁%($,$,𝑡	NOUN
cana-760	185	27	)	)	PUNCT
cana-760	186	1	−	−	ADP
cana-760	187	1	1	1	NUM
cana-760	187	2	)	)	PUNCT
cana-760	187	3	=	=	SYM
cana-760	187	4	𝐻𝛶(1	𝐻𝛶(1	ADJ
cana-760	187	5	)	)	PUNCT
cana-760	187	6	=	=	SYM
cana-760	188	1	0	0	X
cana-760	188	2	.	.	PUNCT
cana-760	189	1	so	so	ADV
cana-760	189	2	휁%($	휁%($	PROPN
cana-760	189	3	,	,	PUNCT
cana-760	189	4	𝛤$	𝛤$	NOUN
cana-760	189	5	,	,	PUNCT
cana-760	189	6	𝑡	𝑡	NOUN
cana-760	189	7	)	)	PUNCT
cana-760	189	8	=	=	SYM
cana-760	189	9	0	0	X
cana-760	189	10	.	.	PUNCT
cana-760	190	1	hence	hence	ADV
cana-760	190	2	,	,	PUNCT
cana-760	190	3	휁%($	휁%($	PROPN
cana-760	190	4	,	,	PUNCT
cana-760	190	5	𝛤$	𝛤$	VERB
cana-760	190	6	,	,	PUNCT
cana-760	190	7	𝑡	𝑡	NOUN
cana-760	190	8	)	)	PUNCT
cana-760	190	9	=	=	SYM
cana-760	190	10	0	0	NUM
cana-760	190	11	⇒	⇒	PROPN
cana-760	190	12	𝛤$	𝛤$	PROPN
cana-760	190	13	=	=	SYM
cana-760	190	14	$	$	SYM
cana-760	190	15	.	.	PUNCT
cana-760	191	1	thus	thus	ADV
cana-760	191	2	$	$	PRON
cana-760	191	3	is	be	AUX
cana-760	191	4	a	a	DET
cana-760	191	5	fixed	fix	VERB
cana-760	191	6	point	point	NOUN
cana-760	191	7	of	of	ADP
cana-760	191	8	𝛤.	𝛤.	PROPN
cana-760	191	9	now	now	ADV
cana-760	191	10	,	,	PUNCT
cana-760	191	11	we	we	PRON
cana-760	191	12	will	will	AUX
cana-760	191	13	prove	prove	VERB
cana-760	191	14	that	that	SCONJ
cana-760	191	15	$	$	PRON
cana-760	191	16	is	be	AUX
cana-760	191	17	unique	unique	ADJ
cana-760	191	18	.	.	PUNCT
cana-760	192	1	assume	assume	VERB
cana-760	192	2	not	not	PART
cana-760	192	3	,	,	PUNCT
cana-760	192	4	∃𝜔	∃𝜔	PROPN
cana-760	192	5	∈	∈	PROPN
cana-760	192	6	𝑌	𝑌	PROPN
cana-760	192	7	,	,	PUNCT
cana-760	192	8	such	such	ADJ
cana-760	192	9	that	that	SCONJ
cana-760	192	10	𝛤𝜔	𝛤𝜔	PROPN
cana-760	192	11	=	=	PUNCT
cana-760	192	12	𝜔	𝜔	X
cana-760	192	13	where	where	SCONJ
cana-760	192	14	𝜔	𝜔	ADP
cana-760	192	15	≠	≠	PROPN
cana-760	192	16	$	$	NOUN
cana-760	192	17	and	and	CCONJ
cana-760	192	18	lim	lim	PROPN
cana-760	192	19	𝜅→∞	𝜅→∞	X
cana-760	192	20	𝜄𝜅	𝜄𝜅	X
cana-760	192	21	=	=	PUNCT
cana-760	192	22	𝜔.	𝜔.	PROPN
cana-760	192	23	then	then	ADV
cana-760	192	24	𝛶	𝛶	PROPN
cana-760	192	25	(	(	PUNCT
cana-760	192	26	1	1	NUM
cana-760	192	27	𝜁𝜚(𝜔,𝜛,𝑡	𝜁𝜚(𝜔,𝜛,𝑡	NUM
cana-760	192	28	)	)	PUNCT
cana-760	192	29	−	−	NOUN
cana-760	192	30	1	1	NUM
cana-760	192	31	)	)	PUNCT
cana-760	192	32	=	=	SYM
cana-760	193	1	𝛶	𝛶	PROPN
cana-760	193	2	(	(	PUNCT
cana-760	193	3	1	1	NUM
cana-760	193	4	𝜁𝜚(𝛤𝜔,𝛤𝜛,𝑡	𝜁𝜚(𝛤𝜔,𝛤𝜛,𝑡	NOUN
cana-760	193	5	)	)	PUNCT
cana-760	193	6	−	−	NOUN
cana-760	193	7	1	1	NUM
cana-760	193	8	)	)	PUNCT
cana-760	193	9	≤	≤	NOUN
cana-760	193	10	𝐻𝛶	𝐻𝛶	PROPN
cana-760	193	11	(	(	PUNCT
cana-760	193	12	1	1	NUM
cana-760	193	13	𝜁𝜚(𝜔,𝜛,𝑡	𝜁𝜚(𝜔,𝜛,𝑡	NUM
cana-760	193	14	)	)	PUNCT
cana-760	193	15	−	−	NOUN
cana-760	193	16	1	1	X
cana-760	193	17	)	)	PUNCT
cana-760	193	18	≤	≤	NOUN
cana-760	193	19	𝐻𝛶	𝐻𝛶	PROPN
cana-760	193	20	(	(	PUNCT
cana-760	193	21	휁𝜚	휁𝜚	ADP
cana-760	193	22	2	2	NUM
cana-760	193	23	(	(	PUNCT
cana-760	193	24	𝜔	𝜔	PROPN
cana-760	193	25	,	,	PUNCT
cana-760	193	26	𝜄𝜅	𝜄𝜅	INTJ
cana-760	193	27	,	,	PUNCT
cana-760	193	28	𝑡	𝑡	PROPN
cana-760	193	29	2	2	NUM
cana-760	193	30	)	)	PUNCT
cana-760	193	31	⊛	⊛	NUM
cana-760	193	32	휁𝜚	휁𝜚	ADP
cana-760	193	33	2	2	NUM
cana-760	193	34	(	(	PUNCT
cana-760	193	35	𝜄𝜅	𝜄𝜅	INTJ
cana-760	193	36	,	,	PUNCT
cana-760	193	37	𝜛	𝜛	PROPN
cana-760	193	38	,	,	PUNCT
cana-760	193	39	𝑡	𝑡	PROPN
cana-760	193	40	2	2	NUM
cana-760	193	41	)	)	PUNCT
cana-760	193	42	)	)	PUNCT
cana-760	193	43	.	.	PUNCT
cana-760	194	1	since	since	SCONJ
cana-760	194	2	𝛶(1	𝛶(1	ADP
cana-760	194	3	)	)	PUNCT
cana-760	194	4	=	=	SYM
cana-760	194	5	0	0	NUM
cana-760	194	6	and	and	CCONJ
cana-760	194	7	for	for	ADP
cana-760	194	8	𝜅	𝜅	PRON
cana-760	194	9	→	→	SYM
cana-760	194	10	∞	∞	NUM
cana-760	194	11	on	on	ADP
cana-760	194	12	both	both	DET
cana-760	194	13	sides	side	NOUN
cana-760	194	14	,	,	PUNCT
cana-760	194	15	we	we	PRON
cana-760	194	16	have	have	VERB
cana-760	194	17	𝛶	𝛶	PROPN
cana-760	194	18	(	(	PUNCT
cana-760	194	19	1	1	NUM
cana-760	194	20	𝜁𝜚(𝜔,𝜛,𝑡	𝜁𝜚(𝜔,𝜛,𝑡	NUM
cana-760	194	21	)	)	PUNCT
cana-760	194	22	−	−	NOUN
cana-760	194	23	1	1	X
cana-760	194	24	)	)	PUNCT
cana-760	194	25	≤	≤	NOUN
cana-760	194	26	𝐻𝛶	𝐻𝛶	PROPN
cana-760	194	27	(	(	PUNCT
cana-760	194	28	휁𝜚	휁𝜚	ADP
cana-760	194	29	2	2	NUM
cana-760	194	30	(	(	PUNCT
cana-760	194	31	𝜔	𝜔	PROPN
cana-760	194	32	,	,	PUNCT
cana-760	194	33	𝜄𝜅	𝜄𝜅	INTJ
cana-760	194	34	,	,	PUNCT
cana-760	194	35	𝑡	𝑡	PROPN
cana-760	194	36	2	2	NUM
cana-760	194	37	)	)	PUNCT
cana-760	194	38	⊛	⊛	NUM
cana-760	194	39	휁𝜚	휁𝜚	ADP
cana-760	194	40	2	2	NUM
cana-760	194	41	(	(	PUNCT
cana-760	194	42	𝜄𝜅	𝜄𝜅	INTJ
cana-760	194	43	,	,	PUNCT
cana-760	194	44	𝜛	𝜛	PROPN
cana-760	194	45	,	,	PUNCT
cana-760	194	46	𝑡	𝑡	PROPN
cana-760	194	47	2	2	NUM
cana-760	194	48	)	)	PUNCT
cana-760	194	49	)	)	PUNCT
cana-760	195	1	=	=	SYM
cana-760	195	2	𝐻𝛶(1	𝐻𝛶(1	ADJ
cana-760	195	3	)	)	PUNCT
cana-760	195	4	=	=	SYM
cana-760	195	5	0	0	X
cana-760	195	6	.	.	PUNCT
cana-760	195	7	communications	communication	NOUN
cana-760	195	8	on	on	ADP
cana-760	195	9	applied	apply	VERB
cana-760	195	10	nonlinear	nonlinear	ADJ
cana-760	195	11	analysis	analysis	NOUN
cana-760	195	12	issn	issn	NOUN
cana-760	195	13	:	:	PUNCT
cana-760	195	14	1074	1074	NUM
cana-760	195	15	-	-	PUNCT
cana-760	195	16	133x	133x	NUM
cana-760	195	17	vol	vol	NOUN
cana-760	195	18	31	31	NUM
cana-760	195	19	no	no	NOUN
cana-760	195	20	.	.	PUNCT
cana-760	196	1	3s	3s	NUM
cana-760	196	2	(	(	PUNCT
cana-760	196	3	2024	2024	NUM
cana-760	196	4	)	)	PUNCT
cana-760	196	5	219	219	NUM
cana-760	196	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	197	1	so	so	ADV
cana-760	197	2	,	,	PUNCT
cana-760	197	3	𝛶	𝛶	PROPN
cana-760	197	4	(	(	PUNCT
cana-760	197	5	1	1	NUM
cana-760	197	6	𝜁%(𝜔,$,𝑡	𝜁%(𝜔,$,𝑡	NOUN
cana-760	197	7	)	)	PUNCT
cana-760	197	8	−	−	PROPN
cana-760	197	9	1	1	X
cana-760	197	10	)	)	PUNCT
cana-760	197	11	=	=	SYM
cana-760	198	1	0	0	X
cana-760	198	2	.	.	PUNCT
cana-760	199	1	hence	hence	ADV
cana-760	199	2	,	,	PUNCT
cana-760	199	3	휁%(𝜔	휁%(𝜔	PROPN
cana-760	199	4	,	,	PUNCT
cana-760	199	5	$	$	SYM
cana-760	199	6	,	,	PUNCT
cana-760	199	7	𝑡	𝑡	NOUN
cana-760	199	8	)	)	PUNCT
cana-760	199	9	=	=	SYM
cana-760	199	10	0	0	NUM
cana-760	199	11	⇒	⇒	NOUN
cana-760	199	12	𝑤	𝑤	ADP
cana-760	199	13	=	=	PUNCT
cana-760	199	14	$	$	SYM
cana-760	199	15	.	.	PUNCT
cana-760	200	1	thus	thus	ADV
cana-760	200	2	γ	γ	X
cana-760	200	3	has	have	VERB
cana-760	200	4	a	a	DET
cana-760	200	5	unique	unique	ADJ
cana-760	200	6	fixed	fix	VERB
cana-760	200	7	point	point	NOUN
cana-760	200	8	$	$	SYM
cana-760	200	9	.	.	PUNCT
cana-760	201	1	theorem	theorem	VERB
cana-760	201	2	3.2	3.2	NUM
cana-760	201	3	.	.	PUNCT
cana-760	202	1	on	on	ADP
cana-760	202	2	a	a	DET
cana-760	202	3	complete	complete	ADJ
cana-760	202	4	modular	modular	NOUN
cana-760	202	5	revised	revise	VERB
cana-760	202	6	fuzzy	fuzzy	ADJ
cana-760	202	7	metric	metric	ADJ
cana-760	202	8	space	space	NOUN
cana-760	202	9	(	(	PUNCT
cana-760	202	10	𝑌	𝑌	PROPN
cana-760	202	11	,	,	PUNCT
cana-760	202	12	휁%,⊛	휁%,⊛	NOUN
cana-760	202	13	)	)	PUNCT
cana-760	202	14	,	,	PUNCT
cana-760	202	15	consider	consider	VERB
cana-760	202	16	a	a	DET
cana-760	202	17	continuous	continuous	ADJ
cana-760	202	18	mapping	mapping	NOUN
cana-760	202	19	𝛤	𝛤	NOUN
cana-760	202	20	:	:	PUNCT
cana-760	202	21	𝑌	𝑌	PROPN
cana-760	202	22	→	→	SYM
cana-760	202	23	𝑌.	𝑌.	PROPN
cana-760	202	24	suppose	suppose	VERB
cana-760	202	25	there	there	PRON
cana-760	202	26	exist	exist	VERB
cana-760	202	27	a	a	DET
cana-760	202	28	strictly	strictly	ADV
cana-760	202	29	non	non	ADJ
cana-760	202	30	-	-	ADJ
cana-760	202	31	decreasing	decrease	VERB
cana-760	202	32	,	,	PUNCT
cana-760	202	33	continuous	continuous	ADJ
cana-760	202	34	function	function	NOUN
cana-760	202	35	𝛶	𝛶	PROPN
cana-760	202	36	:	:	PUNCT
cana-760	202	37	(	(	PUNCT
cana-760	202	38	0,1	0,1	NUM
cana-760	202	39	]	]	PUNCT
cana-760	202	40	→	→	PUNCT
cana-760	203	1	[	[	X
cana-760	203	2	0	0	NUM
cana-760	203	3	,	,	PUNCT
cana-760	203	4	∞	∞	PROPN
cana-760	203	5	)	)	PUNCT
cana-760	203	6	with	with	ADP
cana-760	203	7	𝛶(1	𝛶(1	ADP
cana-760	203	8	)	)	PUNCT
cana-760	203	9	=	=	SYM
cana-760	203	10	0	0	NUM
cana-760	203	11	and	and	CCONJ
cana-760	203	12	a	a	DET
cana-760	203	13	real	real	ADJ
cana-760	203	14	number	number	NOUN
cana-760	203	15	𝐻	𝐻	NOUN
cana-760	203	16	with	with	ADP
cana-760	203	17	0	0	NUM
cana-760	203	18	<	<	X
cana-760	203	19	𝐻	𝐻	NOUN
cana-760	203	20	<	<	X
cana-760	203	21	1	1	NUM
cana-760	203	22	such	such	ADJ
cana-760	203	23	that	that	SCONJ
cana-760	203	24	𝛶	𝛶	PROPN
cana-760	203	25	(	(	PUNCT
cana-760	203	26	1	1	NUM
cana-760	203	27	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	NOUN
cana-760	203	28	)	)	PUNCT
cana-760	203	29	−	−	ADP
cana-760	204	1	1	1	X
cana-760	204	2	)	)	PUNCT
cana-760	204	3	≤	≤	NOUN
cana-760	204	4	𝐻	𝐻	PROPN
cana-760	204	5	(	(	PUNCT
cana-760	204	6	1	1	NUM
cana-760	204	7	𝜁𝜚(𝜄,𝜂,𝑡)+𝜁𝜚(𝛤𝜄,𝜄,𝑡	𝜁𝜚(𝜄,𝜂,𝑡)+𝜁𝜚(𝛤𝜄,𝜄,𝑡	ADV
cana-760	204	8	)	)	PUNCT
cana-760	204	9	4	4	NUM
cana-760	204	10	−	−	NOUN
cana-760	204	11	1	1	NUM
cana-760	204	12	+	+	SYM
cana-760	204	13	1	1	NUM
cana-760	204	14	𝜁𝜚(𝜂,𝛤𝜂,𝑡	𝜁𝜚(𝜂,𝛤𝜂,𝑡	NOUN
cana-760	204	15	)	)	PUNCT
cana-760	204	16	2	2	NUM
cana-760	204	17	−	−	NOUN
cana-760	204	18	1	1	NUM
cana-760	204	19	)	)	PUNCT
cana-760	204	20	(	(	PUNCT
cana-760	204	21	3.6	3.6	NUM
cana-760	204	22	)	)	PUNCT
cana-760	204	23	for	for	ADP
cana-760	204	24	all	all	DET
cana-760	204	25	𝜄	𝜄	PROPN
cana-760	204	26	,	,	PUNCT
cana-760	204	27	휂	휂	ADP
cana-760	204	28	∈	∈	PROPN
cana-760	204	29	𝑌	𝑌	PROPN
cana-760	204	30	,	,	PUNCT
cana-760	204	31	𝜄	𝜄	PROPN
cana-760	204	32	≠	≠	PROPN
cana-760	204	33	휂	휂	PROPN
cana-760	204	34	.	.	PUNCT
cana-760	204	35	then	then	ADV
cana-760	204	36	𝛤	𝛤	PROPN
cana-760	204	37	has	have	VERB
cana-760	204	38	a	a	DET
cana-760	204	39	unique	unique	ADJ
cana-760	204	40	fixed	fix	VERB
cana-760	204	41	point	point	NOUN
cana-760	204	42	in	in	ADP
cana-760	204	43	𝑌.	𝑌.	PROPN
cana-760	204	44	proof	proof	NOUN
cana-760	204	45	.	.	PUNCT
cana-760	205	1	let	let	VERB
cana-760	205	2	𝜄0	𝜄0	PROPN
cana-760	205	3	be	be	AUX
cana-760	205	4	an	an	DET
cana-760	205	5	arbitrary	arbitrary	ADJ
cana-760	205	6	point	point	NOUN
cana-760	205	7	in	in	ADP
cana-760	205	8	y.	y.	PROPN
cana-760	205	9	choose	choose	VERB
cana-760	205	10	𝜄1	𝜄1	PROPN
cana-760	205	11	∈	∈	PROPN
cana-760	205	12	𝑌	𝑌	PROPN
cana-760	205	13	such	such	ADJ
cana-760	205	14	that	that	DET
cana-760	205	15	𝜄1	𝜄1	NOUN
cana-760	205	16	=	=	NOUN
cana-760	205	17	𝛤𝜄0	𝛤𝜄0	NOUN
cana-760	205	18	.	.	PUNCT
cana-760	206	1	continuing	continue	VERB
cana-760	206	2	this	this	DET
cana-760	206	3	process	process	NOUN
cana-760	206	4	,	,	PUNCT
cana-760	206	5	we	we	PRON
cana-760	206	6	construct	construct	VERB
cana-760	206	7	a	a	DET
cana-760	206	8	sequence	sequence	NOUN
cana-760	206	9	(	(	PUNCT
cana-760	206	10	𝜄𝜅	𝜄𝜅	X
cana-760	206	11	)	)	PUNCT
cana-760	207	1	such	such	ADJ
cana-760	207	2	that	that	SCONJ
cana-760	207	3	𝜄𝜅+1	𝜄𝜅+1	ADJ
cana-760	207	4	=	=	SYM
cana-760	207	5	𝛤𝜄𝜅	𝛤𝜄𝜅	PROPN
cana-760	207	6	,	,	PUNCT
cana-760	207	7	for	for	ADP
cana-760	207	8	κ	κ	NOUN
cana-760	207	9	=	=	SYM
cana-760	207	10	0,1,2	0,1,2	X
cana-760	207	11	..	..	PUNCT
cana-760	207	12	let	let	VERB
cana-760	207	13	𝜄	𝜄	PROPN
cana-760	207	14	=	=	PUNCT
cana-760	207	15	𝜄𝜅−1	𝜄𝜅−1	PROPN
cana-760	207	16	and	and	CCONJ
cana-760	207	17	휂	휂	ADP
cana-760	207	18	=	=	PUNCT
cana-760	207	19	𝜄𝜅.	𝜄𝜅.	VERB
cana-760	207	20	replacing	replace	VERB
cana-760	207	21	this	this	PRON
cana-760	207	22	in	in	ADP
cana-760	207	23	(	(	PUNCT
cana-760	207	24	3.6	3.6	NUM
cana-760	207	25	)	)	PUNCT
cana-760	207	26	,	,	PUNCT
cana-760	207	27	we	we	PRON
cana-760	207	28	get	get	VERB
cana-760	207	29	𝛶	𝛶	PROPN
cana-760	207	30	(	(	PUNCT
cana-760	207	31	1	1	NUM
cana-760	207	32	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	NOUN
cana-760	207	33	)	)	PUNCT
cana-760	207	34	−	−	ADP
cana-760	208	1	1	1	X
cana-760	208	2	)	)	PUNCT
cana-760	208	3	=	=	SYM
cana-760	208	4	𝛶	𝛶	PROPN
cana-760	208	5	(	(	PUNCT
cana-760	208	6	1	1	NUM
cana-760	208	7	𝜁%(𝛤𝜄𝜅−1,𝛤𝜄𝜅,𝑡	𝜁%(𝛤𝜄𝜅−1,𝛤𝜄𝜅,𝑡	NOUN
cana-760	208	8	)	)	PUNCT
cana-760	208	9	−	−	ADP
cana-760	208	10	1	1	NUM
cana-760	208	11	)	)	PUNCT
cana-760	208	12	=	=	PUNCT
cana-760	208	13	𝛶	𝛶	PROPN
cana-760	208	14	(	(	PUNCT
cana-760	208	15	1	1	NUM
cana-760	208	16	𝜁%(𝜄𝜅,𝜄𝜅+1,𝑡	𝜁%(𝜄𝜅,𝜄𝜅+1,𝑡	NOUN
cana-760	208	17	)	)	PUNCT
cana-760	208	18	−	−	NOUN
cana-760	208	19	1	1	X
cana-760	208	20	)	)	PUNCT
cana-760	208	21	≤	≤	NOUN
cana-760	208	22	𝛶𝐻	𝛶𝐻	PROPN
cana-760	208	23	(	(	PUNCT
cana-760	208	24	1	1	NUM
cana-760	208	25	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝛤𝜄𝜅−1,𝜄𝜅−1,𝑡	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝛤𝜄𝜅−1,𝜄𝜅−1,𝑡	NOUN
cana-760	208	26	)	)	PUNCT
cana-760	208	27	4	4	NUM
cana-760	208	28	−	−	NOUN
cana-760	208	29	1	1	NUM
cana-760	208	30	+	+	CCONJ
cana-760	208	31	1	1	NUM
cana-760	208	32	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	ADJ
cana-760	208	33	)	)	PUNCT
cana-760	208	34	2	2	NUM
cana-760	208	35	−	−	NOUN
cana-760	208	36	1	1	NUM
cana-760	208	37	)	)	PUNCT
cana-760	208	38	<	<	X
cana-760	208	39	𝛶	𝛶	PROPN
cana-760	208	40	(	(	PUNCT
cana-760	208	41	1	1	NUM
cana-760	208	42	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝛤𝜄𝜅−1,𝜄𝜅−1,𝑡	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝛤𝜄𝜅−1,𝜄𝜅−1,𝑡	NOUN
cana-760	208	43	)	)	PUNCT
cana-760	208	44	4	4	NUM
cana-760	208	45	−	−	NOUN
cana-760	208	46	1	1	NUM
cana-760	208	47	+	+	CCONJ
cana-760	208	48	1	1	NUM
cana-760	208	49	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	ADJ
cana-760	208	50	)	)	PUNCT
cana-760	208	51	2	2	NUM
cana-760	208	52	−	−	NOUN
cana-760	208	53	1	1	NUM
cana-760	208	54	)	)	PUNCT
cana-760	208	55	=	=	PUNCT
cana-760	208	56	𝛶	𝛶	PROPN
cana-760	208	57	(	(	PUNCT
cana-760	208	58	1	1	NUM
cana-760	208	59	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝜄𝜅,𝜄𝜅−1,𝑡	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝜄𝜅,𝜄𝜅−1,𝑡	ADJ
cana-760	208	60	)	)	PUNCT
cana-760	208	61	4	4	NUM
cana-760	208	62	−	−	NOUN
cana-760	208	63	1	1	NUM
cana-760	208	64	+	+	NUM
cana-760	208	65	1	1	NUM
cana-760	208	66	𝜁𝜚(𝜄𝜅,𝜄𝜅+1,𝑡	𝜁𝜚(𝜄𝜅,𝜄𝜅+1,𝑡	NOUN
cana-760	208	67	)	)	PUNCT
cana-760	208	68	2	2	NUM
cana-760	208	69	−	−	NOUN
cana-760	208	70	1	1	NUM
cana-760	208	71	)	)	PUNCT
cana-760	208	72	(	(	PUNCT
cana-760	208	73	3.7	3.7	NUM
cana-760	208	74	)	)	PUNCT
cana-760	208	75	since	since	SCONJ
cana-760	208	76	𝛶	𝛶	PROPN
cana-760	208	77	is	be	AUX
cana-760	208	78	a	a	DET
cana-760	208	79	strictly	strictly	ADV
cana-760	208	80	non	non	ADJ
cana-760	208	81	-	-	ADJ
cana-760	208	82	decreasing	decrease	VERB
cana-760	208	83	function	function	NOUN
cana-760	208	84	,	,	PUNCT
cana-760	208	85	we	we	PRON
cana-760	208	86	obtain	obtain	VERB
cana-760	208	87	1	1	NUM
cana-760	208	88	𝜁𝜚(𝜄𝜅,𝜄𝜅+1,𝑡	𝜁𝜚(𝜄𝜅,𝜄𝜅+1,𝑡	NOUN
cana-760	208	89	)	)	PUNCT
cana-760	208	90	−	−	NOUN
cana-760	209	1	1	1	NUM
cana-760	209	2	<	<	X
cana-760	209	3	𝛶	𝛶	PROPN
cana-760	209	4	(	(	PUNCT
cana-760	209	5	1	1	NUM
cana-760	209	6	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝜄𝜅,𝜄𝜅−1,𝑡	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝜄𝜅,𝜄𝜅−1,𝑡	ADJ
cana-760	209	7	)	)	PUNCT
cana-760	209	8	4	4	NUM
cana-760	209	9	−	−	NOUN
cana-760	209	10	1	1	NUM
cana-760	209	11	+	+	NUM
cana-760	209	12	1	1	NUM
cana-760	209	13	𝜁𝜚(𝜄𝜅,𝜄𝜅+1,𝑡	𝜁𝜚(𝜄𝜅,𝜄𝜅+1,𝑡	NOUN
cana-760	209	14	)	)	PUNCT
cana-760	209	15	2	2	NUM
cana-760	209	16	−	−	NOUN
cana-760	209	17	1	1	NUM
cana-760	209	18	)	)	PUNCT
cana-760	209	19	.	.	PUNCT
cana-760	210	1	hence	hence	ADV
cana-760	210	2	1	1	NUM
cana-760	210	3	𝜁%(𝜄𝜅,𝜄𝜅+1,𝑡	𝜁%(𝜄𝜅,𝜄𝜅+1,𝑡	ADJ
cana-760	210	4	)	)	PUNCT
cana-760	210	5	−	−	NOUN
cana-760	210	6	1	1	NUM
cana-760	210	7	<	<	SYM
cana-760	210	8	1	1	NUM
cana-760	210	9	𝜁%(𝜄𝜅,𝜄𝜅−1,𝑡	𝜁%(𝜄𝜅,𝜄𝜅−1,𝑡	ADJ
cana-760	210	10	)	)	PUNCT
cana-760	211	1	−	−	NOUN
cana-760	212	1	1	1	X
cana-760	212	2	.	.	PUNCT
cana-760	213	1	we	we	PRON
cana-760	213	2	use	use	VERB
cana-760	213	3	the	the	DET
cana-760	213	4	same	same	ADJ
cana-760	213	5	method	method	NOUN
cana-760	213	6	for	for	ADP
cana-760	213	7	𝜄	𝜄	PROPN
cana-760	213	8	=	=	PUNCT
cana-760	213	9	𝜄𝜅−2	𝜄𝜅−2	PROPN
cana-760	213	10	and	and	CCONJ
cana-760	213	11	휂	휂	ADP
cana-760	213	12	=	=	SYM
cana-760	213	13	𝜄𝜅−1	𝜄𝜅−1	PROPN
cana-760	213	14	,	,	PUNCT
cana-760	213	15	we	we	PRON
cana-760	213	16	get	get	VERB
cana-760	213	17	(	(	PUNCT
cana-760	213	18	3.8	3.8	NUM
cana-760	213	19	)	)	PUNCT
cana-760	213	20	1	1	NUM
cana-760	213	21	𝜁%(𝜄𝜅,𝜄𝜅−1,𝑡	𝜁%(𝜄𝜅,𝜄𝜅−1,𝑡	ADJ
cana-760	213	22	)	)	PUNCT
cana-760	213	23	−	−	NOUN
cana-760	214	1	1	1	NUM
cana-760	214	2	<	<	SYM
cana-760	214	3	1	1	NUM
cana-760	214	4	𝜁%(𝜄𝜅−1,𝜄𝜅−2,𝑡	𝜁%(𝜄𝜅−1,𝜄𝜅−2,𝑡	NOUN
cana-760	214	5	)	)	PUNCT
cana-760	215	1	−	−	PROPN
cana-760	215	2	1	1	X
cana-760	215	3	.	.	PUNCT
cana-760	215	4	(	(	PUNCT
cana-760	215	5	3.9	3.9	NUM
cana-760	215	6	)	)	PUNCT
cana-760	215	7	therefore	therefore	ADV
cana-760	215	8	,	,	PUNCT
cana-760	215	9	(	(	PUNCT
cana-760	215	10	3.8	3.8	NUM
cana-760	215	11	)	)	PUNCT
cana-760	215	12	and	and	CCONJ
cana-760	215	13	(	(	PUNCT
cana-760	215	14	3.9	3.9	NUM
cana-760	215	15	)	)	PUNCT
cana-760	215	16	imply	imply	VERB
cana-760	215	17	that	that	SCONJ
cana-760	215	18	{	{	PUNCT
cana-760	215	19	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	215	20	,	,	PUNCT
cana-760	215	21	𝜄𝜅+1	𝜄𝜅+1	NUM
cana-760	215	22	,	,	PUNCT
cana-760	215	23	𝑡	𝑡	NOUN
cana-760	215	24	)	)	PUNCT
cana-760	215	25	}	}	PUNCT
cana-760	215	26	is	be	AUX
cana-760	215	27	a	a	DET
cana-760	215	28	strictly	strictly	ADV
cana-760	215	29	non	non	ADJ
cana-760	215	30	-	-	ADJ
cana-760	215	31	increasing	increasing	ADJ
cana-760	215	32	sequence	sequence	NOUN
cana-760	215	33	of	of	ADP
cana-760	215	34	positive	positive	ADJ
cana-760	215	35	real	real	ADJ
cana-760	215	36	numbers	number	NOUN
cana-760	215	37	in	in	ADP
cana-760	215	38	[	[	X
cana-760	215	39	0,1	0,1	NUM
cana-760	215	40	]	]	PUNCT
cana-760	215	41	.	.	PUNCT
cana-760	216	1	put	put	VERB
cana-760	216	2	𝛴𝜅(%	𝛴𝜅(%	NUM
cana-760	216	3	,	,	PUNCT
cana-760	216	4	𝑡	𝑡	X
cana-760	216	5	)	)	PUNCT
cana-760	216	6	=	=	SYM
cana-760	217	1	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	217	2	,	,	PUNCT
cana-760	217	3	𝜄𝜅+1	𝜄𝜅+1	NOUN
cana-760	217	4	,	,	PUNCT
cana-760	217	5	𝑡	𝑡	PROPN
cana-760	217	6	)	)	PUNCT
cana-760	217	7	.	.	PUNCT
cana-760	218	1	then	then	ADV
cana-760	218	2	{	{	PUNCT
cana-760	218	3	𝛴𝜅(%	𝛴𝜅(%	PROPN
cana-760	218	4	,	,	PUNCT
cana-760	218	5	𝑡	𝑡	NOUN
cana-760	218	6	)	)	PUNCT
cana-760	218	7	}	}	PUNCT
cana-760	218	8	is	be	AUX
cana-760	218	9	a	a	DET
cana-760	218	10	strictly	strictly	ADV
cana-760	218	11	non	non	ADJ
cana-760	218	12	-	-	ADJ
cana-760	218	13	increasing	increasing	ADJ
cana-760	218	14	sequence	sequence	NOUN
cana-760	218	15	.	.	PUNCT
cana-760	219	1	communications	communication	NOUN
cana-760	219	2	on	on	ADP
cana-760	219	3	applied	apply	VERB
cana-760	219	4	nonlinear	nonlinear	ADJ
cana-760	219	5	analysis	analysis	NOUN
cana-760	219	6	issn	issn	NOUN
cana-760	219	7	:	:	PUNCT
cana-760	219	8	1074	1074	NUM
cana-760	219	9	-	-	PUNCT
cana-760	219	10	133x	133x	NUM
cana-760	219	11	vol	vol	NOUN
cana-760	219	12	31	31	NUM
cana-760	219	13	no	no	NOUN
cana-760	219	14	.	.	PUNCT
cana-760	220	1	3s	3s	NUM
cana-760	220	2	(	(	PUNCT
cana-760	220	3	2024	2024	NUM
cana-760	220	4	)	)	PUNCT
cana-760	220	5	220	220	NUM
cana-760	220	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	220	7	so	so	ADV
cana-760	220	8	∃	∃	PROPN
cana-760	220	9	𝛴(𝑡	𝛴(𝑡	PROPN
cana-760	220	10	,	,	PUNCT
cana-760	220	11	%	%	INTJ
cana-760	220	12	)	)	PUNCT
cana-760	220	13	such	such	ADJ
cana-760	220	14	that	that	SCONJ
cana-760	220	15	lim	lim	PROPN
cana-760	220	16	𝛴𝜅(%	𝛴𝜅(%	PROPN
cana-760	220	17	,	,	PUNCT
cana-760	220	18	𝑡	𝑡	X
cana-760	220	19	)	)	PUNCT
cana-760	220	20	=	=	SYM
cana-760	220	21	𝛴(𝑡	𝛴(𝑡	NOUN
cana-760	220	22	,	,	PUNCT
cana-760	220	23	%	%	NOUN
cana-760	220	24	)	)	PUNCT
cana-760	220	25	.	.	PUNCT
cana-760	220	26	assume	assume	VERB
cana-760	220	27	that	that	SCONJ
cana-760	220	28	0	0	NUM
cana-760	220	29	<	<	X
cana-760	220	30	𝛴(𝑡	𝛴(𝑡	NOUN
cana-760	220	31	,	,	PUNCT
cana-760	220	32	%	%	INTJ
cana-760	220	33	)	)	PUNCT
cana-760	220	34	<	<	X
cana-760	221	1	1	1	X
cana-760	221	2	.	.	PUNCT
cana-760	221	3	by	by	ADP
cana-760	221	4	(	(	PUNCT
cana-760	221	5	3.7	3.7	NUM
cana-760	221	6	)	)	PUNCT
cana-760	221	7	,	,	PUNCT
cana-760	221	8	we	we	PRON
cana-760	221	9	have	have	AUX
cana-760	221	10	𝛶(𝛴𝜅(𝜚	𝛶(𝛴𝜅(𝜚	NOUN
cana-760	221	11	,	,	PUNCT
cana-760	221	12	𝑡	𝑡	NOUN
cana-760	221	13	)	)	PUNCT
cana-760	221	14	)	)	PUNCT
cana-760	222	1	≤	≤	NUM
cana-760	222	2	𝛶𝐻	𝛶𝐻	PROPN
cana-760	222	3	(	(	PUNCT
cana-760	222	4	𝛴𝜅−1(𝜚,𝑡	𝛴𝜅−1(𝜚,𝑡	NUM
cana-760	222	5	)	)	PUNCT
cana-760	222	6	2	2	NUM
cana-760	222	7	+	+	SYM
cana-760	222	8	𝛴𝜅(𝜚,𝑡	𝛴𝜅(𝜚,𝑡	NOUN
cana-760	222	9	)	)	PUNCT
cana-760	222	10	2	2	NUM
cana-760	222	11	)	)	PUNCT
cana-760	222	12	.	.	PUNCT
cana-760	223	1	so	so	ADV
cana-760	223	2	,	,	PUNCT
cana-760	223	3	lim	lim	PROPN
cana-760	223	4	𝜅→∞	𝜅→∞	NUM
cana-760	223	5	𝛶(𝛴𝜅(𝜚	𝛶(𝛴𝜅(𝜚	PROPN
cana-760	223	6	,	,	PUNCT
cana-760	223	7	𝑡	𝑡	NOUN
cana-760	223	8	)	)	PUNCT
cana-760	223	9	)	)	PUNCT
cana-760	223	10	≤	≤	PROPN
cana-760	223	11	lim	lim	PROPN
cana-760	223	12	𝜅→∞	𝜅→∞	NUM
cana-760	223	13	𝛶𝐻	𝛶𝐻	PROPN
cana-760	223	14	(	(	PUNCT
cana-760	223	15	𝛴𝜅−1(𝜚,𝑡	𝛴𝜅−1(𝜚,𝑡	NUM
cana-760	223	16	)	)	PUNCT
cana-760	223	17	2	2	NUM
cana-760	224	1	+	+	SYM
cana-760	224	2	𝛴𝜅(𝜚,𝑡	𝛴𝜅(𝜚,𝑡	NOUN
cana-760	224	3	)	)	PUNCT
cana-760	224	4	2	2	NUM
cana-760	224	5	)	)	PUNCT
cana-760	224	6	by	by	ADP
cana-760	224	7	the	the	DET
cana-760	224	8	continuity	continuity	NOUN
cana-760	224	9	of	of	ADP
cana-760	224	10	𝛶	𝛶	PROPN
cana-760	224	11	,	,	PUNCT
cana-760	224	12	we	we	PRON
cana-760	224	13	have	have	VERB
cana-760	224	14	𝛶(𝛴(𝜚	𝛶(𝛴(𝜚	NOUN
cana-760	224	15	,	,	PUNCT
cana-760	224	16	𝑡	𝑡	NOUN
cana-760	224	17	)	)	PUNCT
cana-760	224	18	)	)	PUNCT
cana-760	224	19	≤	≤	NUM
cana-760	225	1	𝛶𝐻	𝛶𝐻	PROPN
cana-760	225	2	(	(	PUNCT
cana-760	225	3	𝛴(𝜚,𝑡	𝛴(𝜚,𝑡	NOUN
cana-760	225	4	)	)	PUNCT
cana-760	225	5	2	2	NUM
cana-760	225	6	+	+	NUM
cana-760	225	7	𝛴(𝜚,𝑡	𝛴(𝜚,𝑡	NOUN
cana-760	225	8	)	)	PUNCT
cana-760	225	9	2	2	NUM
cana-760	225	10	)	)	PUNCT
cana-760	225	11	,	,	PUNCT
cana-760	225	12	a	a	DET
cana-760	225	13	contradiction	contradiction	NOUN
cana-760	225	14	.	.	PUNCT
cana-760	226	1	then	then	ADV
cana-760	226	2	𝛴(%	𝛴(%	PROPN
cana-760	226	3	,	,	PUNCT
cana-760	226	4	𝑡	𝑡	PROPN
cana-760	226	5	)	)	PUNCT
cana-760	226	6	=	=	SYM
cana-760	226	7	0	0	X
cana-760	226	8	.	.	PUNCT
cana-760	227	1	now	now	ADV
cana-760	227	2	,	,	PUNCT
cana-760	227	3	we	we	PRON
cana-760	227	4	will	will	AUX
cana-760	227	5	prove	prove	VERB
cana-760	227	6	that	that	SCONJ
cana-760	227	7	{	{	PUNCT
cana-760	227	8	𝜄𝜅	𝜄𝜅	NOUN
cana-760	227	9	}	}	PUNCT
cana-760	227	10	is	be	AUX
cana-760	227	11	a	a	DET
cana-760	227	12	cauchy	cauchy	ADJ
cana-760	227	13	sequence	sequence	NOUN
cana-760	227	14	.	.	PUNCT
cana-760	228	1	assume	assume	VERB
cana-760	228	2	not	not	PART
cana-760	228	3	,	,	PUNCT
cana-760	228	4	then	then	ADV
cana-760	228	5	for	for	ADP
cana-760	228	6	0	0	NUM
cana-760	228	7	<	<	X
cana-760	228	8	휀	휀	X
cana-760	228	9	<	<	X
cana-760	228	10	1	1	NUM
cana-760	228	11	,	,	PUNCT
cana-760	228	12	there	there	PRON
cana-760	228	13	exists	exist	VERB
cana-760	228	14	two	two	NUM
cana-760	228	15	sub	sub	NOUN
cana-760	228	16	-	-	NOUN
cana-760	228	17	sequences	sequence	NOUN
cana-760	228	18	{	{	PUNCT
cana-760	228	19	𝜄𝑌(𝑖	𝜄𝑌(𝑖	ADV
cana-760	228	20	)	)	PUNCT
cana-760	228	21	}	}	PUNCT
cana-760	228	22	and	and	CCONJ
cana-760	228	23	{	{	PUNCT
cana-760	228	24	𝜄𝜅(𝑖	𝜄𝜅(𝑖	NOUN
cana-760	228	25	)	)	PUNCT
cana-760	228	26	}	}	PUNCT
cana-760	228	27	such	such	ADJ
cana-760	228	28	that	that	SCONJ
cana-760	228	29	for	for	ADP
cana-760	228	30	each	each	DET
cana-760	228	31	𝑖	𝑖	SYM
cana-760	228	32	∈	∈	PROPN
cana-760	228	33	𝑁	𝑁	PROPN
cana-760	228	34	,	,	PUNCT
cana-760	228	35	let	let	VERB
cana-760	228	36	𝜅(𝑖	𝜅(𝑖	NOUN
cana-760	228	37	)	)	PUNCT
cana-760	228	38	,	,	PUNCT
cana-760	228	39	𝑌(𝑖	𝑌(𝑖	X
cana-760	228	40	)	)	PUNCT
cana-760	228	41	∈	∈	NOUN
cana-760	229	1	𝑁	𝑁	NOUN
cana-760	229	2	satisfying	satisfy	VERB
cana-760	229	3	𝜅(𝑖	𝜅(𝑖	NOUN
cana-760	229	4	)	)	PUNCT
cana-760	229	5	,	,	PUNCT
cana-760	229	6	𝑌(𝑖	𝑌(𝑖	NUM
cana-760	229	7	)	)	PUNCT
cana-760	229	8	≥	≥	NOUN
cana-760	229	9	𝜅	𝜅	NOUN
cana-760	229	10	and	and	CCONJ
cana-760	229	11	𝜅(𝑖	𝜅(𝑖	X
cana-760	229	12	)	)	PUNCT
cana-760	229	13	>	>	X
cana-760	229	14	𝑌(𝑖	𝑌(𝑖	PROPN
cana-760	229	15	)	)	PUNCT
cana-760	229	16	>	>	X
cana-760	230	1	𝑖	𝑖	ADP
cana-760	230	2	,	,	PUNCT
cana-760	230	3	such	such	ADJ
cana-760	230	4	that	that	SCONJ
cana-760	230	5	1	1	NUM
cana-760	230	6	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	NOUN
cana-760	230	7	)	)	PUNCT
cana-760	230	8	−	−	PROPN
cana-760	230	9	1	1	NUM
cana-760	230	10	≥	≥	NOUN
cana-760	230	11	휀	휀	NOUN
cana-760	230	12	,	,	PUNCT
cana-760	230	13	1	1	NUM
cana-760	230	14	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡	NUM
cana-760	230	15	)	)	PUNCT
cana-760	230	16	−	−	NOUN
cana-760	230	17	1	1	NUM
cana-760	230	18	<	<	X
cana-760	230	19	휀	휀	NOUN
cana-760	230	20	,	,	PUNCT
cana-760	230	21	1	1	NUM
cana-760	230	22	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖),𝑡	ADJ
cana-760	230	23	)	)	PUNCT
cana-760	230	24	−	−	NOUN
cana-760	230	25	1	1	NUM
cana-760	230	26	<	<	X
cana-760	230	27	휀	휀	X
cana-760	230	28	.	.	PUNCT
cana-760	230	29	(	(	PUNCT
cana-760	230	30	3.10	3.10	NUM
cana-760	230	31	)	)	PUNCT
cana-760	230	32	consider	consider	VERB
cana-760	230	33	휀	휀	NOUN
cana-760	230	34	≤	≤	NUM
cana-760	230	35	(	(	PUNCT
cana-760	230	36	1	1	NUM
cana-760	230	37	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	NOUN
cana-760	230	38	)	)	PUNCT
cana-760	230	39	−	−	ADP
cana-760	230	40	1	1	X
cana-760	230	41	)	)	PUNCT
cana-760	230	42	≤	≤	NOUN
cana-760	230	43	(	(	PUNCT
cana-760	230	44	1	1	NUM
cana-760	230	45	𝜁𝜚	𝜁𝜚	ADP
cana-760	230	46	2	2	NUM
cana-760	230	47	(	(	PUNCT
cana-760	230	48	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	NOUN
cana-760	230	49	,	,	PUNCT
cana-760	230	50	𝑡	𝑡	X
cana-760	230	51	2	2	NUM
cana-760	230	52	)	)	PUNCT
cana-760	230	53	−	−	PROPN
cana-760	230	54	1	1	NUM
cana-760	230	55	)	)	PUNCT
cana-760	230	56	⊛	⊛	NOUN
cana-760	230	57	(	(	PUNCT
cana-760	230	58	1	1	NUM
cana-760	230	59	𝜁𝜚	𝜁𝜚	ADP
cana-760	230	60	2	2	NUM
cana-760	230	61	(	(	PUNCT
cana-760	230	62	𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖	𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖	NOUN
cana-760	230	63	)	)	PUNCT
cana-760	230	64	,	,	PUNCT
cana-760	230	65	𝑡	𝑡	PROPN
cana-760	230	66	2	2	NUM
cana-760	230	67	)	)	PUNCT
cana-760	230	68	−	−	PROPN
cana-760	230	69	1	1	NUM
cana-760	230	70	)	)	PUNCT
cana-760	230	71	.	.	PUNCT
cana-760	231	1	by	by	ADP
cana-760	231	2	definition	definition	NOUN
cana-760	231	3	of	of	ADP
cana-760	231	4	∆2t	∆2t	NOUN
cana-760	231	5	-	-	PUNCT
cana-760	231	6	condition	condition	NOUN
cana-760	231	7	on	on	ADP
cana-760	231	8	y	y	PROPN
cana-760	231	9	and	and	CCONJ
cana-760	231	10	(	(	PUNCT
cana-760	231	11	3.10	3.10	NUM
cana-760	231	12	)	)	PUNCT
cana-760	231	13	,	,	PUNCT
cana-760	231	14	we	we	PRON
cana-760	231	15	have	have	VERB
cana-760	231	16	휁𝜚	휁𝜚	X
cana-760	231	17	2	2	NUM
cana-760	231	18	(	(	PUNCT
cana-760	231	19	𝜄𝜅(𝑖)−1	𝜄𝜅(𝑖)−1	NOUN
cana-760	231	20	,	,	PUNCT
cana-760	231	21	𝜄𝜉(𝑖	𝜄𝜉(𝑖	ADJ
cana-760	231	22	)	)	PUNCT
cana-760	231	23	,	,	PUNCT
cana-760	231	24	𝑡	𝑡	PROPN
cana-760	231	25	2	2	NUM
cana-760	231	26	)	)	PUNCT
cana-760	231	27	<	<	X
cana-760	232	1	휀	휀	X
cana-760	232	2	.	.	PUNCT
cana-760	232	3	thus	thus	ADV
cana-760	232	4	휀	휀	DET
cana-760	232	5	≤	≤	NOUN
cana-760	232	6	(	(	PUNCT
cana-760	232	7	1	1	NUM
cana-760	232	8	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	NOUN
cana-760	232	9	)	)	PUNCT
cana-760	232	10	−	−	ADP
cana-760	232	11	1	1	X
cana-760	232	12	)	)	PUNCT
cana-760	232	13	≤	≤	NOUN
cana-760	232	14	(	(	PUNCT
cana-760	232	15	1	1	NUM
cana-760	232	16	𝜁𝜚	𝜁𝜚	ADP
cana-760	232	17	2	2	NUM
cana-760	232	18	(	(	PUNCT
cana-760	232	19	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	NOUN
cana-760	232	20	,	,	PUNCT
cana-760	232	21	𝑡	𝑡	X
cana-760	232	22	2	2	NUM
cana-760	232	23	)	)	PUNCT
cana-760	232	24	−	−	PROPN
cana-760	232	25	1	1	NUM
cana-760	232	26	)	)	PUNCT
cana-760	232	27	⊛	⊛	NUM
cana-760	233	1	휀	휀	NOUN
cana-760	233	2	.	.	PUNCT
cana-760	234	1	if	if	SCONJ
cana-760	234	2	i	i	PRON
cana-760	234	3	→	→	SYM
cana-760	234	4	∞	∞	PROPN
cana-760	234	5	,	,	PUNCT
cana-760	234	6	we	we	PRON
cana-760	234	7	have	have	VERB
cana-760	234	8	𝛴𝜅(𝑖	𝛴𝜅(𝑖	VERB
cana-760	234	9	)	)	PUNCT
cana-760	234	10	(	(	PUNCT
cana-760	234	11	𝜚	𝜚	NOUN
cana-760	234	12	2	2	NUM
cana-760	234	13	,	,	PUNCT
cana-760	234	14	𝑡	𝑡	X
cana-760	234	15	2	2	NUM
cana-760	234	16	)	)	PUNCT
cana-760	234	17	≤	≤	NOUN
cana-760	234	18	1	1	NUM
cana-760	234	19	𝜁𝜚	𝜁𝜚	ADP
cana-760	234	20	2	2	NUM
cana-760	234	21	(	(	PUNCT
cana-760	234	22	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1	NOUN
cana-760	234	23	,	,	PUNCT
cana-760	234	24	𝑡	𝑡	X
cana-760	234	25	2	2	NUM
cana-760	234	26	)	)	PUNCT
cana-760	234	27	−	−	PROPN
cana-760	234	28	1	1	NUM
cana-760	234	29	→	→	SYM
cana-760	234	30	0	0	NUM
cana-760	234	31	.	.	PUNCT
cana-760	234	32	.	.	PUNCT
cana-760	235	1	so	so	ADV
cana-760	235	2	휁𝜚(𝜄𝜅(𝑖	휁𝜚(𝜄𝜅(𝑖	PROPN
cana-760	235	3	)	)	PUNCT
cana-760	235	4	,	,	PUNCT
cana-760	235	5	𝜄𝜉(𝑖	𝜄𝜉(𝑖	NOUN
cana-760	235	6	)	)	PUNCT
cana-760	235	7	,	,	PUNCT
cana-760	235	8	𝑡	𝑡	X
cana-760	235	9	)	)	PUNCT
cana-760	235	10	→	→	SYM
cana-760	235	11	휀	휀	X
cana-760	235	12	.	.	PUNCT
cana-760	235	13	then	then	ADV
cana-760	235	14	by	by	ADP
cana-760	235	15	(	(	PUNCT
cana-760	235	16	3.6	3.6	NUM
cana-760	235	17	)	)	PUNCT
cana-760	235	18	,	,	PUNCT
cana-760	235	19	we	we	PRON
cana-760	235	20	have	have	VERB
cana-760	235	21	𝛶	𝛶	PROPN
cana-760	235	22	(	(	PUNCT
cana-760	235	23	1	1	NUM
cana-760	235	24	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜉(𝑖),𝑡	NOUN
cana-760	235	25	)	)	PUNCT
cana-760	235	26	−	−	ADP
cana-760	235	27	1	1	X
cana-760	235	28	)	)	PUNCT
cana-760	235	29	≤	≤	NOUN
cana-760	235	30	𝛶𝐻	𝛶𝐻	PROPN
cana-760	235	31	(	(	PUNCT
cana-760	235	32	1	1	NUM
cana-760	235	33	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)+𝜁𝜚(𝛤𝜄𝜅(𝑖)−1,𝜄𝜅(𝑖)−1,𝑡	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)+𝜁𝜚(𝛤𝜄𝜅(𝑖)−1,𝜄𝜅(𝑖)−1,𝑡	ADJ
cana-760	235	34	)	)	PUNCT
cana-760	235	35	4	4	NUM
cana-760	235	36	−	−	NOUN
cana-760	235	37	1	1	NUM
cana-760	235	38	+	+	CCONJ
cana-760	235	39	1	1	NUM
cana-760	235	40	𝜁𝜚(𝜄𝜉(𝑖)−1,𝛤𝜄𝜉(𝑖)−1,𝑡	𝜁𝜚(𝜄𝜉(𝑖)−1,𝛤𝜄𝜉(𝑖)−1,𝑡	NOUN
cana-760	235	41	)	)	PUNCT
cana-760	235	42	2	2	NUM
cana-760	235	43	−	−	NUM
cana-760	235	44	1	1	NUM
cana-760	235	45	)	)	PUNCT
cana-760	235	46	communications	communication	NOUN
cana-760	235	47	on	on	ADP
cana-760	235	48	applied	apply	VERB
cana-760	235	49	nonlinear	nonlinear	ADJ
cana-760	235	50	analysis	analysis	NOUN
cana-760	235	51	issn	issn	NOUN
cana-760	235	52	:	:	PUNCT
cana-760	235	53	1074	1074	NUM
cana-760	235	54	-	-	PUNCT
cana-760	235	55	133x	133x	NUM
cana-760	235	56	vol	vol	NOUN
cana-760	235	57	31	31	NUM
cana-760	235	58	no	no	NOUN
cana-760	235	59	.	.	PUNCT
cana-760	236	1	3s	3s	NUM
cana-760	236	2	(	(	PUNCT
cana-760	236	3	2024	2024	NUM
cana-760	236	4	)	)	PUNCT
cana-760	236	5	221	221	NUM
cana-760	236	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	236	7	<	<	X
cana-760	236	8	𝛶	𝛶	PROPN
cana-760	236	9	(	(	PUNCT
cana-760	236	10	1	1	NUM
cana-760	236	11	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)+𝜁𝜚(𝛤𝜄𝜅(𝑖)−1,𝜄𝜅(𝑖)−1,𝑡	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)+𝜁𝜚(𝛤𝜄𝜅(𝑖)−1,𝜄𝜅(𝑖)−1,𝑡	ADJ
cana-760	236	12	)	)	PUNCT
cana-760	236	13	4	4	NUM
cana-760	236	14	−	−	NOUN
cana-760	236	15	1	1	NUM
cana-760	236	16	+	+	CCONJ
cana-760	236	17	1	1	NUM
cana-760	236	18	𝜁𝜚(𝜄𝜉(𝑖)−1,𝛤𝜄𝜉(𝑖)−1,𝑡	𝜁𝜚(𝜄𝜉(𝑖)−1,𝛤𝜄𝜉(𝑖)−1,𝑡	NOUN
cana-760	236	19	)	)	PUNCT
cana-760	236	20	2	2	NUM
cana-760	236	21	−	−	NOUN
cana-760	236	22	1	1	NUM
cana-760	236	23	)	)	PUNCT
cana-760	236	24	=	=	SYM
cana-760	237	1	𝛶	𝛶	PROPN
cana-760	237	2	(	(	PUNCT
cana-760	237	3	1	1	NUM
cana-760	237	4	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)+𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1,𝑡	𝜁𝜚(𝜄𝜅(𝑖)−1,𝜄𝜉(𝑖)−1,𝑡)+𝜁𝜚(𝜄𝜅(𝑖),𝜄𝜅(𝑖)−1,𝑡	NOUN
cana-760	237	5	)	)	PUNCT
cana-760	237	6	4	4	NUM
cana-760	237	7	−	−	NOUN
cana-760	237	8	1	1	NUM
cana-760	237	9	+	+	CCONJ
cana-760	237	10	1	1	NUM
cana-760	237	11	𝜁𝜚(𝜄𝜉(𝑖)−1,𝜄𝜉(𝑖),𝑡	𝜁𝜚(𝜄𝜉(𝑖)−1,𝜄𝜉(𝑖),𝑡	ADJ
cana-760	237	12	)	)	PUNCT
cana-760	237	13	2	2	NUM
cana-760	237	14	−	−	NOUN
cana-760	237	15	1	1	NUM
cana-760	237	16	)	)	PUNCT
cana-760	237	17	since	since	SCONJ
cana-760	237	18	𝛶	𝛶	PROPN
cana-760	237	19	is	be	AUX
cana-760	237	20	a	a	DET
cana-760	237	21	strictly	strictly	ADV
cana-760	237	22	decreasing	decrease	VERB
cana-760	237	23	function	function	NOUN
cana-760	237	24	,	,	PUNCT
cana-760	237	25	(	(	PUNCT
cana-760	237	26	3.10	3.10	NUM
cana-760	237	27	)	)	PUNCT
cana-760	237	28	implies	imply	VERB
cana-760	237	29	that	that	SCONJ
cana-760	237	30	휀	휀	PRON
cana-760	237	31	<	<	X
cana-760	237	32	𝜀+0	𝜀+0	NUM
cana-760	237	33	4	4	NUM
cana-760	237	34	+	+	CCONJ
cana-760	237	35	1	1	NUM
cana-760	237	36	2	2	NUM
cana-760	237	37	=	=	SYM
cana-760	237	38	𝜀	𝜀	DET
cana-760	237	39	4	4	NUM
cana-760	237	40	+	+	SYM
cana-760	237	41	1	1	NUM
cana-760	237	42	2	2	NUM
cana-760	237	43	<	<	X
cana-760	237	44	휀	휀	NOUN
cana-760	237	45	,	,	PUNCT
cana-760	237	46	which	which	PRON
cana-760	237	47	is	be	AUX
cana-760	237	48	impossible	impossible	ADJ
cana-760	237	49	.	.	PUNCT
cana-760	238	1	hence	hence	ADV
cana-760	238	2	{	{	PUNCT
cana-760	238	3	𝜄𝜅	𝜄𝜅	X
cana-760	238	4	}	}	PUNCT
cana-760	238	5	is	be	AUX
cana-760	238	6	a	a	DET
cana-760	238	7	cauchy	cauchy	ADJ
cana-760	238	8	sequence	sequence	NOUN
cana-760	238	9	in	in	ADP
cana-760	238	10	a	a	DET
cana-760	238	11	complete	complete	ADJ
cana-760	238	12	modular	modular	NOUN
cana-760	238	13	revised	revise	VERB
cana-760	238	14	fuzzy	fuzzy	ADJ
cana-760	238	15	metric	metric	ADJ
cana-760	238	16	space	space	NOUN
cana-760	238	17	.	.	PUNCT
cana-760	239	1	so	so	ADV
cana-760	239	2	∃$	∃$	X
cana-760	239	3	∈	∈	PROPN
cana-760	239	4	𝑌	𝑌	PROPN
cana-760	239	5	such	such	ADJ
cana-760	239	6	that	that	SCONJ
cana-760	239	7	lim	lim	PROPN
cana-760	239	8	𝜅→∞	𝜅→∞	X
cana-760	239	9	𝜄𝜅	𝜄𝜅	NOUN
cana-760	239	10	=	=	PUNCT
cana-760	239	11	$	$	SYM
cana-760	239	12	,	,	PUNCT
cana-760	239	13	that	that	PRON
cana-760	239	14	means	mean	VERB
cana-760	239	15	lim	lim	PROPN
cana-760	239	16	𝜅→∞	𝜅→∞	NUM
cana-760	239	17	휁%(𝜄𝜅	휁%(𝜄𝜅	PROPN
cana-760	239	18	,	,	PUNCT
cana-760	239	19	$	$	SYM
cana-760	239	20	,	,	PUNCT
cana-760	239	21	𝑡	𝑡	NOUN
cana-760	239	22	)	)	PUNCT
cana-760	239	23	=	=	SYM
cana-760	239	24	0	0	X
cana-760	239	25	.	.	PUNCT
cana-760	239	26	to	to	PART
cana-760	239	27	show	show	VERB
cana-760	239	28	$	$	SYM
cana-760	239	29	is	be	AUX
cana-760	239	30	a	a	DET
cana-760	239	31	fixed	fix	VERB
cana-760	239	32	point	point	NOUN
cana-760	239	33	of	of	ADP
cana-760	239	34	𝛤	𝛤	PROPN
cana-760	239	35	,	,	PUNCT
cana-760	239	36	we	we	PRON
cana-760	239	37	have	have	VERB
cana-760	239	38	:	:	PUNCT
cana-760	239	39	𝛤	𝛤	PROPN
cana-760	239	40	is	be	AUX
cana-760	239	41	continuous	continuous	ADJ
cana-760	239	42	:	:	PUNCT
cana-760	239	43	𝜄𝜅	𝜄𝜅	NOUN
cana-760	239	44	→	→	SYM
cana-760	239	45	$	$	SYM
cana-760	239	46	⇒	⇒	NOUN
cana-760	239	47	𝛤𝜄𝜅	𝛤𝜄𝜅	PROPN
cana-760	239	48	→	→	SYM
cana-760	239	49	𝛤$.	𝛤$.	X
cana-760	239	50	by	by	ADP
cana-760	239	51	(	(	PUNCT
cana-760	239	52	3.6	3.6	NUM
cana-760	239	53	)	)	PUNCT
cana-760	239	54	,	,	PUNCT
cana-760	239	55	we	we	PRON
cana-760	239	56	have	have	VERB
cana-760	239	57	𝛶	𝛶	PROPN
cana-760	239	58	(	(	PUNCT
cana-760	239	59	1	1	NUM
cana-760	239	60	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	ADJ
cana-760	239	61	)	)	PUNCT
cana-760	239	62	−	−	ADP
cana-760	239	63	1	1	X
cana-760	239	64	)	)	PUNCT
cana-760	239	65	≤	≤	NOUN
cana-760	239	66	𝛶𝐻	𝛶𝐻	PROPN
cana-760	239	67	(	(	PUNCT
cana-760	239	68	1	1	NUM
cana-760	239	69	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝛤𝜄𝜅−1,𝜄𝜅−1,𝑡	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝛤𝜄𝜅−1,𝜄𝜅−1,𝑡	NOUN
cana-760	239	70	)	)	PUNCT
cana-760	239	71	4	4	NUM
cana-760	239	72	−	−	NOUN
cana-760	239	73	1	1	NUM
cana-760	239	74	+	+	CCONJ
cana-760	239	75	1	1	NUM
cana-760	239	76	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	ADJ
cana-760	239	77	)	)	PUNCT
cana-760	240	1	2	2	NUM
cana-760	240	2	−	−	NOUN
cana-760	240	3	1	1	NUM
cana-760	240	4	)	)	PUNCT
cana-760	240	5	=	=	SYM
cana-760	240	6	𝐻𝛶	𝐻𝛶	PROPN
cana-760	240	7	(	(	PUNCT
cana-760	240	8	1	1	NUM
cana-760	240	9	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝜄𝜅,𝜄𝜅−1,𝑡	𝜁𝜚(𝜄𝜅−1,𝜄𝜅,𝑡)+𝜁𝜚(𝜄𝜅,𝜄𝜅−1,𝑡	ADJ
cana-760	240	10	)	)	PUNCT
cana-760	240	11	4	4	NUM
cana-760	240	12	−	−	NOUN
cana-760	240	13	1	1	NUM
cana-760	240	14	+	+	CCONJ
cana-760	240	15	1	1	NUM
cana-760	240	16	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	𝜁𝜚(𝜄𝜅,𝛤𝜄𝜅,𝑡	ADJ
cana-760	240	17	)	)	PUNCT
cana-760	240	18	2	2	NUM
cana-760	240	19	−	−	NOUN
cana-760	240	20	1	1	NUM
cana-760	240	21	)	)	PUNCT
cana-760	240	22	since	since	SCONJ
cana-760	240	23	υ(1	υ(1	PROPN
cana-760	240	24	)	)	PUNCT
cana-760	240	25	=	=	SYM
cana-760	240	26	0	0	NUM
cana-760	240	27	and	and	CCONJ
cana-760	240	28	for	for	ADP
cana-760	240	29	κ	κ	PROPN
cana-760	240	30	→	→	SYM
cana-760	240	31	∞	∞	PROPN
cana-760	240	32	,	,	PUNCT
cana-760	240	33	we	we	PRON
cana-760	240	34	get	get	VERB
cana-760	240	35	𝛶	𝛶	PROPN
cana-760	240	36	(	(	PUNCT
cana-760	240	37	1	1	NUM
cana-760	240	38	𝜁𝜚(𝜔,𝛤𝜛,𝑡	𝜁𝜚(𝜔,𝛤𝜛,𝑡	NOUN
cana-760	240	39	)	)	PUNCT
cana-760	240	40	−	−	NOUN
cana-760	240	41	1	1	NUM
cana-760	240	42	)	)	PUNCT
cana-760	240	43	=	=	SYM
cana-760	240	44	𝐻𝛶	𝐻𝛶	PROPN
cana-760	240	45	(	(	PUNCT
cana-760	240	46	(	(	PUNCT
cana-760	240	47	1	1	NUM
cana-760	240	48	𝜁𝜚(𝜔,𝜛,𝑡)+𝜁𝜚(𝜔,𝜛,𝑡	𝜁𝜚(𝜔,𝜛,𝑡)+𝜁𝜚(𝜔,𝜛,𝑡	NUM
cana-760	240	49	)	)	PUNCT
cana-760	240	50	4	4	NUM
cana-760	240	51	−	−	NOUN
cana-760	240	52	1	1	NUM
cana-760	240	53	)	)	PUNCT
cana-760	240	54	+	+	CCONJ
cana-760	240	55	(	(	PUNCT
cana-760	240	56	1	1	NUM
cana-760	240	57	𝜁𝜚(𝜔,𝜛,𝑡	𝜁𝜚(𝜔,𝜛,𝑡	NUM
cana-760	240	58	)	)	PUNCT
cana-760	240	59	2	2	NUM
cana-760	240	60	−	−	NOUN
cana-760	240	61	1	1	NUM
cana-760	240	62	)	)	PUNCT
cana-760	240	63	)	)	PUNCT
cana-760	241	1	=	=	SYM
cana-760	241	2	𝐻𝛶	𝐻𝛶	PROPN
cana-760	241	3	(	(	PUNCT
cana-760	241	4	1	1	NUM
cana-760	241	5	𝜁𝜚(𝜔,𝛤𝜛,𝑡	𝜁𝜚(𝜔,𝛤𝜛,𝑡	NOUN
cana-760	241	6	)	)	PUNCT
cana-760	241	7	−	−	NOUN
cana-760	241	8	1	1	NUM
cana-760	241	9	)	)	PUNCT
cana-760	241	10	=	=	SYM
cana-760	241	11	𝐻𝛶(1	𝐻𝛶(1	ADJ
cana-760	241	12	)	)	PUNCT
cana-760	241	13	=	=	SYM
cana-760	242	1	0	0	X
cana-760	242	2	.	.	PUNCT
cana-760	243	1	hence	hence	ADV
cana-760	243	2	,	,	PUNCT
cana-760	243	3	휁%($	휁%($	PROPN
cana-760	243	4	,	,	PUNCT
cana-760	243	5	𝛤$	𝛤$	VERB
cana-760	243	6	,	,	PUNCT
cana-760	243	7	𝑡	𝑡	NOUN
cana-760	243	8	)	)	PUNCT
cana-760	243	9	=	=	SYM
cana-760	243	10	0	0	NUM
cana-760	243	11	⇒	⇒	PROPN
cana-760	243	12	𝛤$	𝛤$	PROPN
cana-760	243	13	=	=	SYM
cana-760	243	14	$	$	SYM
cana-760	243	15	.	.	PUNCT
cana-760	244	1	thus	thus	ADV
cana-760	244	2	$	$	PRON
cana-760	244	3	is	be	AUX
cana-760	244	4	a	a	DET
cana-760	244	5	fixed	fix	VERB
cana-760	244	6	point	point	NOUN
cana-760	244	7	of	of	ADP
cana-760	244	8	𝛤.	𝛤.	PROPN
cana-760	244	9	now	now	ADV
cana-760	244	10	,	,	PUNCT
cana-760	244	11	we	we	PRON
cana-760	244	12	will	will	AUX
cana-760	244	13	prove	prove	VERB
cana-760	244	14	that	that	SCONJ
cana-760	244	15	$	$	PRON
cana-760	244	16	is	be	AUX
cana-760	244	17	unique	unique	ADJ
cana-760	244	18	.	.	PUNCT
cana-760	245	1	assume	assume	VERB
cana-760	245	2	not	not	PART
cana-760	245	3	,	,	PUNCT
cana-760	245	4	∃𝑤	∃𝑤	PROPN
cana-760	245	5	∈	∈	PROPN
cana-760	245	6	𝑌	𝑌	PROPN
cana-760	245	7	,	,	PUNCT
cana-760	245	8	such	such	ADJ
cana-760	245	9	that	that	SCONJ
cana-760	245	10	𝛤𝑤	𝛤𝑤	PROPN
cana-760	245	11	=	=	SYM
cana-760	245	12	𝑤	𝑤	X
cana-760	245	13	where	where	SCONJ
cana-760	245	14	𝑤	𝑤	ADP
cana-760	245	15	≠	≠	PROPN
cana-760	245	16	$	$	NOUN
cana-760	245	17	and	and	CCONJ
cana-760	245	18	lim	lim	PROPN
cana-760	245	19	𝜅→∞	𝜅→∞	X
cana-760	245	20	𝜄𝜅	𝜄𝜅	NOUN
cana-760	245	21	=	=	PUNCT
cana-760	245	22	𝑤.	𝑤.	NOUN
cana-760	245	23	by	by	ADP
cana-760	245	24	(	(	PUNCT
cana-760	245	25	3.6	3.6	NUM
cana-760	245	26	)	)	PUNCT
cana-760	245	27	,	,	PUNCT
cana-760	245	28	we	we	PRON
cana-760	245	29	have	have	VERB
cana-760	245	30	𝛶	𝛶	PROPN
cana-760	245	31	(	(	PUNCT
cana-760	245	32	1	1	NUM
cana-760	245	33	𝜁𝜚(𝜔,𝛤𝜛,𝑡	𝜁𝜚(𝜔,𝛤𝜛,𝑡	NOUN
cana-760	245	34	)	)	PUNCT
cana-760	245	35	−	−	NOUN
cana-760	245	36	1	1	NUM
cana-760	245	37	)	)	PUNCT
cana-760	245	38	=	=	SYM
cana-760	246	1	𝛶	𝛶	PROPN
cana-760	246	2	(	(	PUNCT
cana-760	246	3	1	1	NUM
cana-760	246	4	𝜁𝜚(𝛤𝜔,𝛤𝜛,𝑡	𝜁𝜚(𝛤𝜔,𝛤𝜛,𝑡	NOUN
cana-760	246	5	)	)	PUNCT
cana-760	246	6	−	−	NOUN
cana-760	246	7	1	1	X
cana-760	246	8	)	)	PUNCT
cana-760	246	9	≤	≤	NOUN
cana-760	246	10	𝐻𝛶	𝐻𝛶	PROPN
cana-760	246	11	(	(	PUNCT
cana-760	246	12	(	(	PUNCT
cana-760	246	13	1	1	NUM
cana-760	246	14	𝜁𝜚(𝜔,𝜛,𝑡)+𝜁𝜚(𝛤𝜔,𝜛,𝑡	𝜁𝜚(𝜔,𝜛,𝑡)+𝜁𝜚(𝛤𝜔,𝜛,𝑡	NOUN
cana-760	246	15	)	)	PUNCT
cana-760	246	16	4	4	NUM
cana-760	246	17	−	−	NOUN
cana-760	246	18	1	1	NUM
cana-760	246	19	)	)	PUNCT
cana-760	246	20	+	+	CCONJ
cana-760	246	21	(	(	PUNCT
cana-760	246	22	1	1	NUM
cana-760	246	23	𝜁𝜚(𝜔,𝛤𝜛,𝑡	𝜁𝜚(𝜔,𝛤𝜛,𝑡	NOUN
cana-760	246	24	)	)	PUNCT
cana-760	246	25	2	2	NUM
cana-760	246	26	−	−	NOUN
cana-760	246	27	1	1	NUM
cana-760	246	28	)	)	PUNCT
cana-760	246	29	)	)	PUNCT
cana-760	247	1	=	=	SYM
cana-760	247	2	𝐻𝛶	𝐻𝛶	PROPN
cana-760	247	3	(	(	PUNCT
cana-760	247	4	1	1	NUM
cana-760	247	5	𝜁𝜚(𝜔,𝛤𝜛,𝑡	𝜁𝜚(𝜔,𝛤𝜛,𝑡	NOUN
cana-760	247	6	)	)	PUNCT
cana-760	247	7	4	4	NUM
cana-760	247	8	−	−	NOUN
cana-760	247	9	1	1	NUM
cana-760	247	10	2	2	NUM
cana-760	247	11	)	)	PUNCT
cana-760	247	12	communications	communication	NOUN
cana-760	247	13	on	on	ADP
cana-760	247	14	applied	apply	VERB
cana-760	247	15	nonlinear	nonlinear	ADJ
cana-760	247	16	analysis	analysis	NOUN
cana-760	247	17	issn	issn	NOUN
cana-760	247	18	:	:	PUNCT
cana-760	247	19	1074	1074	NUM
cana-760	247	20	-	-	PUNCT
cana-760	247	21	133x	133x	NUM
cana-760	247	22	vol	vol	NOUN
cana-760	247	23	31	31	NUM
cana-760	247	24	no	no	NOUN
cana-760	247	25	.	.	PUNCT
cana-760	248	1	3s	3s	NUM
cana-760	248	2	(	(	PUNCT
cana-760	248	3	2024	2024	NUM
cana-760	248	4	)	)	PUNCT
cana-760	248	5	222	222	NUM
cana-760	248	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	248	7	hence	hence	ADV
cana-760	248	8	,	,	PUNCT
cana-760	248	9	휁%(𝑤	휁%(𝑤	PROPN
cana-760	248	10	,	,	PUNCT
cana-760	248	11	$	$	SYM
cana-760	248	12	,	,	PUNCT
cana-760	248	13	𝑡	𝑡	NOUN
cana-760	248	14	)	)	PUNCT
cana-760	248	15	≤	≤	NOUN
cana-760	248	16	0	0	NUM
cana-760	248	17	.	.	PUNCT
cana-760	249	1	thus	thus	ADV
cana-760	249	2	,	,	PUNCT
cana-760	249	3	휁%(𝑤	휁%(𝑤	PROPN
cana-760	249	4	,	,	PUNCT
cana-760	249	5	$	$	SYM
cana-760	249	6	,	,	PUNCT
cana-760	249	7	𝑡	𝑡	NOUN
cana-760	249	8	)	)	PUNCT
cana-760	249	9	=	=	SYM
cana-760	249	10	0	0	NUM
cana-760	249	11	⇒	⇒	NOUN
cana-760	249	12	𝑤	𝑤	ADP
cana-760	249	13	=	=	PUNCT
cana-760	249	14	$	$	SYM
cana-760	249	15	.	.	PUNCT
cana-760	250	1	so	so	ADV
cana-760	250	2	𝛤	𝛤	PROPN
cana-760	250	3	has	have	VERB
cana-760	250	4	a	a	DET
cana-760	250	5	unique	unique	ADJ
cana-760	250	6	fixed	fix	VERB
cana-760	250	7	point	point	NOUN
cana-760	250	8	$	$	SYM
cana-760	250	9	.	.	PUNCT
cana-760	251	1	the	the	DET
cana-760	251	2	two	two	NUM
cana-760	251	3	following	follow	VERB
cana-760	251	4	examples	example	NOUN
cana-760	251	5	satisfy	satisfy	NOUN
cana-760	251	6	theorem	theorem	ADJ
cana-760	251	7	(	(	PUNCT
cana-760	251	8	3.1	3.1	NUM
cana-760	251	9	)	)	PUNCT
cana-760	251	10	.	.	PUNCT
cana-760	252	1	example	example	NOUN
cana-760	252	2	3.3	3.3	NUM
cana-760	252	3	.	.	PUNCT
cana-760	253	1	let	let	VERB
cana-760	253	2	𝑌	𝑌	PROPN
cana-760	253	3	=	=	PUNCT
cana-760	254	1	[	[	X
cana-760	254	2	0,1	0,1	NUM
cana-760	254	3	]	]	PUNCT
cana-760	254	4	and	and	CCONJ
cana-760	254	5	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	254	6	,	,	PUNCT
cana-760	254	7	휂	휂	ADP
cana-760	254	8	,	,	PUNCT
cana-760	254	9	𝑡	𝑡	NOUN
cana-760	254	10	)	)	PUNCT
cana-760	254	11	=	=	PUNCT
cana-760	255	1	|𝜄−𝜂|	|𝜄−𝜂|	PUNCT
cana-760	255	2	𝜚	𝜚	NOUN
cana-760	255	3	𝑡+	𝑡+	X
cana-760	255	4	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	255	5	𝜚	𝜚	NOUN
cana-760	255	6	.	.	PUNCT
cana-760	256	1	define	define	VERB
cana-760	256	2	𝛤	𝛤	PROPN
cana-760	256	3	:	:	PUNCT
cana-760	256	4	[	[	X
cana-760	256	5	0,1	0,1	NUM
cana-760	256	6	]	]	PUNCT
cana-760	256	7	→	→	PUNCT
cana-760	257	1	[	[	X
cana-760	257	2	0,1	0,1	NUM
cana-760	257	3	]	]	PUNCT
cana-760	257	4	via	via	ADP
cana-760	257	5	𝛤(𝜄	𝛤(𝜄	NOUN
cana-760	257	6	)	)	PUNCT
cana-760	257	7	=	=	PUNCT
cana-760	257	8	𝜄	𝜄	X
cana-760	257	9	3	3	NUM
cana-760	257	10	.	.	PUNCT
cana-760	257	11	also	also	ADV
cana-760	257	12	,	,	PUNCT
cana-760	257	13	define	define	VERB
cana-760	257	14	𝛶	𝛶	NOUN
cana-760	257	15	:	:	PUNCT
cana-760	257	16	(	(	PUNCT
cana-760	257	17	0,1	0,1	NUM
cana-760	257	18	]	]	PUNCT
cana-760	257	19	→	→	PUNCT
cana-760	258	1	[	[	X
cana-760	258	2	0	0	NUM
cana-760	258	3	,	,	PUNCT
cana-760	258	4	∞	∞	NUM
cana-760	258	5	)	)	PUNCT
cana-760	258	6	via	via	ADP
cana-760	258	7	𝛶(𝜄	𝛶(𝜄	X
cana-760	258	8	)	)	PUNCT
cana-760	258	9	=	=	SYM
cana-760	259	1	1	1	NUM
cana-760	259	2	𝜄	𝜄	PRON
cana-760	259	3	−	−	PROPN
cana-760	259	4	1	1	X
cana-760	259	5	.	.	PUNCT
cana-760	259	6	note	note	VERB
cana-760	259	7	that	that	SCONJ
cana-760	259	8	𝛶	𝛶	PROPN
cana-760	259	9	is	be	AUX
cana-760	259	10	a	a	DET
cana-760	259	11	strictly	strictly	ADV
cana-760	259	12	non	non	ADJ
cana-760	259	13	-	-	ADJ
cana-760	259	14	decreasing	decrease	VERB
cana-760	259	15	,	,	PUNCT
cana-760	259	16	continuous	continuous	ADJ
cana-760	259	17	function	function	NOUN
cana-760	259	18	and	and	CCONJ
cana-760	259	19	𝛶(1	𝛶(1	ADP
cana-760	259	20	)	)	PUNCT
cana-760	259	21	=	=	SYM
cana-760	259	22	0	0	X
cana-760	259	23	.	.	PUNCT
cana-760	260	1	now	now	ADV
cana-760	260	2	,	,	PUNCT
cana-760	260	3	we	we	PRON
cana-760	260	4	have	have	VERB
cana-760	260	5	:	:	PUNCT
cana-760	260	6	휁𝜚(𝛤𝜄	휁𝜚(𝛤𝜄	ADJ
cana-760	260	7	,	,	PUNCT
cana-760	260	8	𝛤휂	𝛤휂	PROPN
cana-760	260	9	,	,	PUNCT
cana-760	260	10	𝑡	𝑡	NOUN
cana-760	260	11	)	)	PUNCT
cana-760	260	12	=	=	PUNCT
cana-760	261	1	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	261	2	3𝜚𝑡+|𝜄−𝜂|	3𝜚𝑡+|𝜄−𝜂|	NUM
cana-760	261	3	,	,	PUNCT
cana-760	261	4	휁𝜚(𝜄	휁𝜚(𝜄	NUM
cana-760	261	5	,	,	PUNCT
cana-760	261	6	휂	휂	ADP
cana-760	261	7	,	,	PUNCT
cana-760	261	8	𝑡	𝑡	NOUN
cana-760	261	9	)	)	PUNCT
cana-760	261	10	=	=	PUNCT
cana-760	262	1	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	262	2	3𝜚𝑡+|𝜄−𝜂|	3𝜚𝑡+|𝜄−𝜂|	PROPN
cana-760	262	3	𝛶	𝛶	PROPN
cana-760	262	4	(	(	PUNCT
cana-760	262	5	1	1	NUM
cana-760	262	6	𝜁𝜚(𝛤𝜄,𝛤𝜂,𝑡	𝜁𝜚(𝛤𝜄,𝛤𝜂,𝑡	NOUN
cana-760	262	7	)	)	PUNCT
cana-760	262	8	−	−	PROPN
cana-760	262	9	1	1	NUM
cana-760	262	10	)	)	PUNCT
cana-760	262	11	=	=	SYM
cana-760	263	1	휁𝜚(𝛤𝜄	휁𝜚(𝛤𝜄	ADJ
cana-760	263	2	,	,	PUNCT
cana-760	263	3	𝛤휂	𝛤휂	PROPN
cana-760	263	4	,	,	PUNCT
cana-760	263	5	𝑡	𝑡	NOUN
cana-760	263	6	)	)	PUNCT
cana-760	263	7	=	=	PUNCT
cana-760	263	8	|𝜄−𝜂|	|𝜄−𝜂|	NOUN
cana-760	263	9	3𝜚𝑡	3𝜚𝑡	NOUN
cana-760	263	10	;	;	PUNCT
cana-760	263	11	𝛶	𝛶	PROPN
cana-760	263	12	(	(	PUNCT
cana-760	263	13	1	1	NUM
cana-760	263	14	𝜁𝜚(𝜄,𝜂,𝑡	𝜁𝜚(𝜄,𝜂,𝑡	NOUN
cana-760	263	15	)	)	PUNCT
cana-760	263	16	−	−	ADP
cana-760	263	17	1	1	X
cana-760	263	18	)	)	PUNCT
cana-760	263	19	=	=	NOUN
cana-760	263	20	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	263	21	,	,	PUNCT
cana-760	263	22	휂	휂	ADP
cana-760	263	23	,	,	PUNCT
cana-760	263	24	𝑡	𝑡	NOUN
cana-760	263	25	)	)	PUNCT
cana-760	263	26	=	=	PUNCT
cana-760	263	27	|𝜄−𝜂|	|𝜄−𝜂|	AUX
cana-760	263	28	𝜚𝑡	𝜚𝑡	VERB
cana-760	263	29	so	so	ADV
cana-760	263	30	,	,	PUNCT
cana-760	263	31	hence	hence	ADV
cana-760	263	32	,	,	PUNCT
cana-760	263	33	for	for	ADP
cana-760	263	34	𝐻	𝐻	PROPN
cana-760	263	35	=	=	SYM
cana-760	263	36	1	1	NUM
cana-760	263	37	3	3	NUM
cana-760	263	38	,	,	PUNCT
cana-760	263	39	we	we	PRON
cana-760	263	40	get	get	VERB
cana-760	263	41	𝛶	𝛶	PROPN
cana-760	263	42	(	(	PUNCT
cana-760	263	43	1	1	NUM
cana-760	263	44	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	NOUN
cana-760	263	45	)	)	PUNCT
cana-760	263	46	−	−	ADP
cana-760	264	1	1	1	X
cana-760	264	2	)	)	PUNCT
cana-760	264	3	=	=	SYM
cana-760	264	4	𝐻𝛶	𝐻𝛶	PROPN
cana-760	264	5	(	(	PUNCT
cana-760	264	6	1	1	NUM
cana-760	264	7	𝜁%(𝜄,𝜂,𝑡	𝜁%(𝜄,𝜂,𝑡	NUM
cana-760	264	8	)	)	PUNCT
cana-760	264	9	−	−	PROPN
cana-760	264	10	1	1	NUM
cana-760	264	11	)	)	PUNCT
cana-760	264	12	.	.	PUNCT
cana-760	265	1	thus	thus	ADV
cana-760	265	2	,	,	PUNCT
cana-760	265	3	theorem	theorem	ADJ
cana-760	265	4	(	(	PUNCT
cana-760	265	5	3.1	3.1	NUM
cana-760	265	6	)	)	PUNCT
cana-760	265	7	implies	imply	VERB
cana-760	265	8	that	that	SCONJ
cana-760	265	9	𝛤	𝛤	PROPN
cana-760	265	10	has	have	VERB
cana-760	265	11	a	a	DET
cana-760	265	12	unique	unique	ADJ
cana-760	265	13	fixed	fix	VERB
cana-760	265	14	point	point	NOUN
cana-760	265	15	0	0	PUNCT
cana-760	265	16	∈	∈	PROPN
cana-760	265	17	𝑌.	𝑌.	PROPN
cana-760	265	18	example	example	NOUN
cana-760	265	19	3.4	3.4	NUM
cana-760	265	20	.	.	PUNCT
cana-760	266	1	let	let	VERB
cana-760	266	2	𝑌	𝑌	PROPN
cana-760	266	3	=	=	PUNCT
cana-760	267	1	[	[	X
cana-760	267	2	0,1	0,1	NUM
cana-760	267	3	]	]	PUNCT
cana-760	267	4	and	and	CCONJ
cana-760	267	5	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	267	6	,	,	PUNCT
cana-760	267	7	휂	휂	ADP
cana-760	267	8	,	,	PUNCT
cana-760	267	9	𝑡	𝑡	NOUN
cana-760	267	10	)	)	PUNCT
cana-760	267	11	=	=	SYM
cana-760	267	12	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	267	13	{	{	PUNCT
cana-760	267	14	−	−	NOUN
cana-760	267	15	|𝜄−𝜂|	|𝜄−𝜂|	INTJ
cana-760	267	16	𝑡𝜚	𝑡𝜚	NOUN
cana-760	267	17	}	}	PUNCT
cana-760	267	18	(	(	PUNCT
cana-760	267	19	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	267	20	{	{	PUNCT
cana-760	267	21	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	267	22	𝑡𝜚	𝑡𝜚	NOUN
cana-760	267	23	}	}	PUNCT
cana-760	267	24	−	−	PROPN
cana-760	267	25	1	1	NUM
cana-760	267	26	)	)	PUNCT
cana-760	267	27	.	.	PUNCT
cana-760	268	1	define	define	VERB
cana-760	268	2	𝛤	𝛤	PROPN
cana-760	268	3	:	:	PUNCT
cana-760	268	4	[	[	X
cana-760	268	5	0,1	0,1	NUM
cana-760	268	6	]	]	PUNCT
cana-760	268	7	→	→	PUNCT
cana-760	269	1	[	[	X
cana-760	269	2	0,1	0,1	NUM
cana-760	269	3	]	]	PUNCT
cana-760	269	4	via	via	ADP
cana-760	269	5	𝛤(𝜄	𝛤(𝜄	NOUN
cana-760	269	6	)	)	PUNCT
cana-760	269	7	=	=	PUNCT
cana-760	269	8	𝜄	𝜄	X
cana-760	269	9	5	5	NUM
cana-760	269	10	.	.	PUNCT
cana-760	269	11	also	also	ADV
cana-760	269	12	,	,	PUNCT
cana-760	269	13	define	define	VERB
cana-760	269	14	𝛶	𝛶	NOUN
cana-760	269	15	:	:	PUNCT
cana-760	269	16	(	(	PUNCT
cana-760	269	17	0,1	0,1	NUM
cana-760	269	18	]	]	PUNCT
cana-760	269	19	→	→	PUNCT
cana-760	270	1	[	[	X
cana-760	270	2	0	0	NUM
cana-760	270	3	,	,	PUNCT
cana-760	270	4	∞	∞	NUM
cana-760	270	5	)	)	PUNCT
cana-760	270	6	via	via	ADP
cana-760	270	7	𝛶(𝜄	𝛶(𝜄	X
cana-760	270	8	)	)	PUNCT
cana-760	270	9	=	=	SYM
cana-760	270	10	−𝑙𝑛𝜄.	−𝑙𝑛𝜄.	NOUN
cana-760	270	11	note	note	VERB
cana-760	270	12	that	that	SCONJ
cana-760	270	13	υ	υ	NOUN
cana-760	270	14	is	be	AUX
cana-760	270	15	a	a	DET
cana-760	270	16	strictly	strictly	ADV
cana-760	270	17	non	non	ADJ
cana-760	270	18	-	-	ADJ
cana-760	270	19	decreasing	decrease	VERB
cana-760	270	20	,	,	PUNCT
cana-760	270	21	continuous	continuous	ADJ
cana-760	270	22	function	function	NOUN
cana-760	270	23	and	and	CCONJ
cana-760	270	24	𝛶(1	𝛶(1	ADP
cana-760	270	25	)	)	PUNCT
cana-760	270	26	=	=	SYM
cana-760	271	1	0	0	X
cana-760	271	2	.	.	PUNCT
cana-760	272	1	now	now	ADV
cana-760	272	2	,	,	PUNCT
cana-760	272	3	we	we	PRON
cana-760	272	4	have	have	VERB
cana-760	272	5	:	:	PUNCT
cana-760	272	6	휁𝜚(𝛤𝜄	휁𝜚(𝛤𝜄	ADJ
cana-760	272	7	,	,	PUNCT
cana-760	272	8	𝛤휂	𝛤휂	PROPN
cana-760	272	9	,	,	PUNCT
cana-760	272	10	𝑡	𝑡	NOUN
cana-760	272	11	)	)	PUNCT
cana-760	272	12	=	=	SYM
cana-760	272	13	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	272	14	{	{	PUNCT
cana-760	272	15	−	−	NOUN
cana-760	272	16	|𝜄−𝜂|	|𝜄−𝜂|	NUM
cana-760	272	17	5𝑡𝜚	5𝑡𝜚	NOUN
cana-760	272	18	}	}	PUNCT
cana-760	272	19	(	(	PUNCT
cana-760	272	20	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	272	21	{	{	PUNCT
cana-760	272	22	|𝜄−𝜂|	|𝜄−𝜂|	ADJ
cana-760	272	23	5𝑡𝜚	5𝑡𝜚	ADJ
cana-760	272	24	}	}	PUNCT
cana-760	272	25	−	−	PROPN
cana-760	272	26	1	1	NUM
cana-760	272	27	)	)	PUNCT
cana-760	272	28	,	,	PUNCT
cana-760	272	29	휁𝜚(𝜄	휁𝜚(𝜄	NUM
cana-760	272	30	,	,	PUNCT
cana-760	272	31	휂	휂	ADP
cana-760	272	32	,	,	PUNCT
cana-760	272	33	𝑡	𝑡	NOUN
cana-760	272	34	)	)	PUNCT
cana-760	272	35	=	=	SYM
cana-760	272	36	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	272	37	{	{	PUNCT
cana-760	272	38	−	−	NOUN
cana-760	272	39	|𝜄−𝜂|	|𝜄−𝜂|	INTJ
cana-760	272	40	𝑡𝜚	𝑡𝜚	NOUN
cana-760	272	41	}	}	PUNCT
cana-760	272	42	(	(	PUNCT
cana-760	272	43	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-760	272	44	{	{	PUNCT
cana-760	272	45	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	272	46	𝑡𝜚	𝑡𝜚	NOUN
cana-760	272	47	}	}	PUNCT
cana-760	272	48	−	−	PROPN
cana-760	272	49	1	1	NUM
cana-760	272	50	)	)	PUNCT
cana-760	272	51	.	.	PUNCT
cana-760	273	1	so	so	ADV
cana-760	273	2	,	,	PUNCT
cana-760	273	3	υ	υ	NOUN
cana-760	273	4	(	(	PUNCT
cana-760	273	5	1	1	NUM
cana-760	273	6	𝜁𝜚(𝛤𝜄,𝛤𝜂,𝑡	𝜁𝜚(𝛤𝜄,𝛤𝜂,𝑡	NOUN
cana-760	273	7	)	)	PUNCT
cana-760	273	8	−	−	PROPN
cana-760	274	1	1	1	NUM
cana-760	274	2	)	)	PUNCT
cana-760	274	3	=	=	PUNCT
cana-760	275	1	|𝜄−𝜂|	|𝜄−𝜂|	NOUN
cana-760	275	2	5𝑡𝜚	5𝑡𝜚	NOUN
cana-760	275	3	;	;	PUNCT
cana-760	275	4	υ	υ	X
cana-760	275	5	(	(	PUNCT
cana-760	275	6	1	1	NUM
cana-760	275	7	𝜁𝜚(𝜄,𝜂,𝑡	𝜁𝜚(𝜄,𝜂,𝑡	NOUN
cana-760	275	8	)	)	PUNCT
cana-760	275	9	−	−	ADP
cana-760	276	1	1	1	NUM
cana-760	276	2	)	)	PUNCT
cana-760	276	3	=	=	PUNCT
cana-760	277	1	|𝜄−𝜂|	|𝜄−𝜂|	NOUN
cana-760	277	2	𝑡𝜚	𝑡𝜚	INTJ
cana-760	277	3	.	.	PUNCT
cana-760	278	1	hence	hence	ADV
cana-760	278	2	,	,	PUNCT
cana-760	278	3	for	for	ADP
cana-760	278	4	𝐻	𝐻	PROPN
cana-760	278	5	=	=	SYM
cana-760	278	6	1	1	NUM
cana-760	278	7	5	5	NUM
cana-760	278	8	,	,	PUNCT
cana-760	278	9	we	we	PRON
cana-760	278	10	get	get	VERB
cana-760	278	11	𝛶	𝛶	PROPN
cana-760	278	12	(	(	PUNCT
cana-760	278	13	1	1	NUM
cana-760	278	14	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	NOUN
cana-760	278	15	)	)	PUNCT
cana-760	278	16	−	−	ADP
cana-760	278	17	1	1	X
cana-760	278	18	)	)	PUNCT
cana-760	278	19	=	=	SYM
cana-760	278	20	𝐻𝛶	𝐻𝛶	PROPN
cana-760	278	21	(	(	PUNCT
cana-760	278	22	1	1	NUM
cana-760	278	23	𝜁%(𝜄,𝜂,𝑡	𝜁%(𝜄,𝜂,𝑡	NUM
cana-760	278	24	)	)	PUNCT
cana-760	278	25	−	−	PROPN
cana-760	278	26	1	1	NUM
cana-760	278	27	)	)	PUNCT
cana-760	278	28	.	.	PUNCT
cana-760	279	1	thus	thus	ADV
cana-760	279	2	theorem	theorem	VERB
cana-760	279	3	(	(	PUNCT
cana-760	279	4	3.1	3.1	NUM
cana-760	279	5	)	)	PUNCT
cana-760	279	6	implies	imply	VERB
cana-760	279	7	that	that	SCONJ
cana-760	279	8	𝛤	𝛤	PROPN
cana-760	279	9	has	have	VERB
cana-760	279	10	a	a	DET
cana-760	279	11	unique	unique	ADJ
cana-760	279	12	fixed	fix	VERB
cana-760	279	13	point	point	NOUN
cana-760	279	14	0	0	PUNCT
cana-760	280	1	∈	∈	PROPN
cana-760	280	2	𝑌.	𝑌.	PROPN
cana-760	280	3	the	the	DET
cana-760	280	4	following	follow	VERB
cana-760	280	5	example	example	NOUN
cana-760	280	6	satisfies	satisfie	NOUN
cana-760	280	7	theorem	theorem	VERB
cana-760	280	8	(	(	PUNCT
cana-760	280	9	3.2	3.2	NUM
cana-760	280	10	)	)	PUNCT
cana-760	280	11	.	.	PUNCT
cana-760	281	1	communications	communication	NOUN
cana-760	281	2	on	on	ADP
cana-760	281	3	applied	apply	VERB
cana-760	281	4	nonlinear	nonlinear	ADJ
cana-760	281	5	analysis	analysis	NOUN
cana-760	281	6	issn	issn	NOUN
cana-760	281	7	:	:	PUNCT
cana-760	281	8	1074	1074	NUM
cana-760	281	9	-	-	PUNCT
cana-760	281	10	133x	133x	NUM
cana-760	281	11	vol	vol	NOUN
cana-760	281	12	31	31	NUM
cana-760	281	13	no	no	NOUN
cana-760	281	14	.	.	PUNCT
cana-760	282	1	3s	3s	NUM
cana-760	282	2	(	(	PUNCT
cana-760	282	3	2024	2024	NUM
cana-760	282	4	)	)	PUNCT
cana-760	282	5	223	223	NUM
cana-760	283	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	283	2	example	example	NOUN
cana-760	283	3	3.5	3.5	NUM
cana-760	283	4	.	.	PUNCT
cana-760	284	1	let	let	VERB
cana-760	284	2	𝑌	𝑌	PROPN
cana-760	284	3	=	=	PUNCT
cana-760	285	1	[	[	X
cana-760	285	2	0,1	0,1	NUM
cana-760	285	3	]	]	PUNCT
cana-760	285	4	and	and	CCONJ
cana-760	285	5	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	285	6	,	,	PUNCT
cana-760	285	7	휂	휂	ADP
cana-760	285	8	,	,	PUNCT
cana-760	285	9	𝑡	𝑡	NOUN
cana-760	285	10	)	)	PUNCT
cana-760	285	11	=	=	PUNCT
cana-760	286	1	|𝜄−𝜂|	|𝜄−𝜂|	PUNCT
cana-760	286	2	𝜚	𝜚	NOUN
cana-760	286	3	𝑡+	𝑡+	X
cana-760	286	4	|𝜄−𝜂|	|𝜄−𝜂|	PROPN
cana-760	286	5	𝜚	𝜚	NOUN
cana-760	286	6	.	.	PUNCT
cana-760	287	1	define	define	VERB
cana-760	287	2	𝛤	𝛤	PROPN
cana-760	287	3	∶	∶	NOUN
cana-760	287	4	[	[	X
cana-760	287	5	0,1	0,1	NUM
cana-760	287	6	]	]	PUNCT
cana-760	287	7	→	→	PUNCT
cana-760	287	8	[	[	X
cana-760	287	9	0,1	0,1	NUM
cana-760	287	10	]	]	PUNCT
cana-760	287	11	via	via	ADP
cana-760	287	12	γ(𝜄	γ(𝜄	PROPN
cana-760	287	13	)	)	PUNCT
cana-760	288	1	=	=	SYM
cana-760	288	2	𝑐	𝑐	PROPN
cana-760	288	3	,	,	PUNCT
cana-760	288	4	𝑐	𝑐	PROPN
cana-760	288	5	∈	∈	PROPN
cana-760	289	1	[	[	X
cana-760	289	2	0,1	0,1	NUM
cana-760	289	3	]	]	PUNCT
cana-760	289	4	.	.	PUNCT
cana-760	290	1	also	also	ADV
cana-760	290	2	,	,	PUNCT
cana-760	290	3	define	define	VERB
cana-760	290	4	𝛶	𝛶	NOUN
cana-760	290	5	:	:	PUNCT
cana-760	290	6	(	(	PUNCT
cana-760	290	7	0,1	0,1	NUM
cana-760	290	8	]	]	PUNCT
cana-760	290	9	→	→	PUNCT
cana-760	291	1	[	[	X
cana-760	291	2	0	0	NUM
cana-760	291	3	,	,	PUNCT
cana-760	291	4	∞	∞	NUM
cana-760	291	5	)	)	PUNCT
cana-760	291	6	via	via	ADP
cana-760	291	7	𝛶(𝜄	𝛶(𝜄	X
cana-760	291	8	)	)	PUNCT
cana-760	291	9	=	=	SYM
cana-760	292	1	1	1	NUM
cana-760	292	2	𝜄	𝜄	PRON
cana-760	292	3	−	−	PROPN
cana-760	292	4	1	1	X
cana-760	292	5	.	.	PUNCT
cana-760	292	6	note	note	VERB
cana-760	292	7	that	that	SCONJ
cana-760	292	8	𝛶	𝛶	PROPN
cana-760	292	9	is	be	AUX
cana-760	292	10	a	a	DET
cana-760	292	11	strictly	strictly	ADV
cana-760	292	12	nondecreasing	nondecreasing	ADJ
cana-760	292	13	,	,	PUNCT
cana-760	292	14	continuous	continuous	ADJ
cana-760	292	15	function	function	NOUN
cana-760	292	16	and	and	CCONJ
cana-760	292	17	𝛶(1	𝛶(1	ADP
cana-760	292	18	)	)	PUNCT
cana-760	292	19	=	=	SYM
cana-760	292	20	0	0	X
cana-760	292	21	.	.	PUNCT
cana-760	293	1	now	now	ADV
cana-760	293	2	,	,	PUNCT
cana-760	293	3	we	we	PRON
cana-760	293	4	have	have	VERB
cana-760	293	5	:	:	PUNCT
cana-760	293	6	휁𝜚(γ𝜄	휁𝜚(γ𝜄	PROPN
cana-760	293	7	,	,	PUNCT
cana-760	293	8	γ휂	γ휂	ADV
cana-760	293	9	,	,	PUNCT
cana-760	293	10	𝑡	𝑡	NOUN
cana-760	293	11	)	)	PUNCT
cana-760	293	12	=	=	SYM
cana-760	293	13	|𝑐	|𝑐	PROPN
cana-760	294	1	−	−	PROPN
cana-760	294	2	𝑐|	𝑐|	PROPN
cana-760	294	3	𝑡	𝑡	PROPN
cana-760	294	4	+	+	CCONJ
cana-760	294	5	|𝑐	|𝑐	PUNCT
cana-760	294	6	−	−	PROPN
cana-760	294	7	𝑐|	𝑐|	NOUN
cana-760	294	8	=	=	SYM
cana-760	294	9	0	0	NUM
cana-760	294	10	;	;	PUNCT
cana-760	294	11	휁𝜚(𝜄	휁𝜚(𝜄	NUM
cana-760	294	12	,	,	PUNCT
cana-760	294	13	휂	휂	ADP
cana-760	294	14	,	,	PUNCT
cana-760	294	15	𝑡	𝑡	NOUN
cana-760	294	16	)	)	PUNCT
cana-760	294	17	4	4	NUM
cana-760	294	18	=	=	SYM
cana-760	294	19	1	1	NUM
cana-760	294	20	4	4	NUM
cana-760	294	21	×	×	NOUN
cana-760	294	22	|𝜄	|𝜄	NOUN
cana-760	294	23	−	−	PROPN
cana-760	294	24	휂|	휂|	PUNCT
cana-760	294	25	𝜚𝑡	𝜚𝑡	VERB
cana-760	294	26	+	+	CCONJ
cana-760	294	27	|𝜄	|𝜄	PROPN
cana-760	294	28	−	−	PROPN
cana-760	294	29	휂|	휂|	PROPN
cana-760	295	1	휁𝜚(γ𝜄	휁𝜚(γ𝜄	PROPN
cana-760	295	2	,	,	PUNCT
cana-760	295	3	𝜄	𝜄	PROPN
cana-760	295	4	,	,	PUNCT
cana-760	295	5	𝑡	𝑡	PROPN
cana-760	295	6	)	)	PUNCT
cana-760	295	7	=	=	SYM
cana-760	295	8	1	1	NUM
cana-760	295	9	4	4	NUM
cana-760	295	10	×	×	NOUN
cana-760	295	11	|𝑐−𝜄|	|𝑐−𝜄|	PUNCT
cana-760	295	12	𝜚𝑡+|𝑐−𝜄|	𝜚𝑡+|𝑐−𝜄|	NOUN
cana-760	295	13	and	and	CCONJ
cana-760	295	14	휁𝜚(휂	휁𝜚(휂	NOUN
cana-760	295	15	,	,	PUNCT
cana-760	295	16	γ휂	γ휂	ADV
cana-760	295	17	,	,	PUNCT
cana-760	295	18	𝑡	𝑡	NOUN
cana-760	295	19	)	)	PUNCT
cana-760	295	20	=	=	SYM
cana-760	295	21	1	1	NUM
cana-760	295	22	2	2	NUM
cana-760	295	23	×	×	NOUN
cana-760	295	24	|𝑐−𝜂|	|𝑐−𝜂|	NUM
cana-760	295	25	𝜚𝑡+|𝑐−𝜂|	𝜚𝑡+|𝑐−𝜂|	PROPN
cana-760	295	26	.	.	PUNCT
cana-760	296	1	so	so	ADV
cana-760	296	2	,	,	PUNCT
cana-760	296	3	(	(	PUNCT
cana-760	296	4	1	1	NUM
cana-760	296	5	𝜁𝜚(𝜄,𝜂,𝑡)+𝜁𝜚(γ𝜄,𝜄,𝑡	𝜁𝜚(𝜄,𝜂,𝑡)+𝜁𝜚(γ𝜄,𝜄,𝑡	NOUN
cana-760	296	6	)	)	PUNCT
cana-760	296	7	4	4	NUM
cana-760	296	8	−	−	NOUN
cana-760	296	9	1	1	NUM
cana-760	296	10	)	)	PUNCT
cana-760	296	11	+	+	CCONJ
cana-760	296	12	(	(	PUNCT
cana-760	296	13	1	1	NUM
cana-760	296	14	𝜁𝜚(𝜂,γ𝜂,𝑡	𝜁𝜚(𝜂,γ𝜂,𝑡	NOUN
cana-760	296	15	)	)	PUNCT
cana-760	296	16	2	2	NUM
cana-760	296	17	−	−	NOUN
cana-760	296	18	1	1	NUM
cana-760	296	19	)	)	PUNCT
cana-760	296	20	=	=	SYM
cana-760	296	21	0	0	PUNCT
cana-760	297	1	=	=	SYM
cana-760	297	2	휁𝜚(γ𝜄	휁𝜚(γ𝜄	PROPN
cana-760	297	3	,	,	PUNCT
cana-760	297	4	γ휂	γ휂	ADV
cana-760	297	5	,	,	PUNCT
cana-760	297	6	𝑡	𝑡	NOUN
cana-760	297	7	)	)	PUNCT
cana-760	297	8	.	.	PUNCT
cana-760	298	1	so	so	ADV
cana-760	298	2	,	,	PUNCT
cana-760	298	3	for	for	ADP
cana-760	298	4	𝐻	𝐻	PRON
cana-760	298	5	such	such	ADJ
cana-760	298	6	that	that	SCONJ
cana-760	298	7	0	0	NUM
cana-760	298	8	<	<	X
cana-760	298	9	𝐻	𝐻	X
cana-760	298	10	<	<	X
cana-760	298	11	1	1	NUM
cana-760	298	12	,	,	PUNCT
cana-760	298	13	we	we	PRON
cana-760	298	14	have	have	VERB
cana-760	298	15	(	(	PUNCT
cana-760	298	16	1	1	NUM
cana-760	298	17	𝜁𝜚(𝜄,𝜂,𝑡)+𝜁𝜚(γ𝜄,𝜄,𝑡	𝜁𝜚(𝜄,𝜂,𝑡)+𝜁𝜚(γ𝜄,𝜄,𝑡	NOUN
cana-760	298	18	)	)	PUNCT
cana-760	298	19	4	4	NUM
cana-760	298	20	−	−	NOUN
cana-760	298	21	1	1	NUM
cana-760	298	22	)	)	PUNCT
cana-760	298	23	+	+	CCONJ
cana-760	298	24	(	(	PUNCT
cana-760	298	25	1	1	NUM
cana-760	298	26	𝜁𝜚(𝜂,γ𝜂,𝑡	𝜁𝜚(𝜂,γ𝜂,𝑡	NOUN
cana-760	298	27	)	)	PUNCT
cana-760	298	28	2	2	NUM
cana-760	298	29	−	−	NOUN
cana-760	298	30	1	1	NUM
cana-760	298	31	)	)	PUNCT
cana-760	298	32	≤	≤	NOUN
cana-760	298	33	0	0	NUM
cana-760	299	1	=	=	SYM
cana-760	299	2	휁𝜚(γ𝜄	휁𝜚(γ𝜄	PROPN
cana-760	299	3	,	,	PUNCT
cana-760	299	4	γ휂	γ휂	ADV
cana-760	299	5	,	,	PUNCT
cana-760	299	6	𝑡	𝑡	NOUN
cana-760	299	7	)	)	PUNCT
cana-760	299	8	.	.	PUNCT
cana-760	300	1	thus	thus	ADV
cana-760	300	2	theorem	theorem	VERB
cana-760	300	3	(	(	PUNCT
cana-760	300	4	3.2	3.2	NUM
cana-760	300	5	)	)	PUNCT
cana-760	300	6	implies	imply	VERB
cana-760	300	7	that	that	SCONJ
cana-760	300	8	𝛤	𝛤	PROPN
cana-760	300	9	has	have	VERB
cana-760	300	10	a	a	DET
cana-760	300	11	unique	unique	ADJ
cana-760	300	12	fixed	fix	VERB
cana-760	300	13	point	point	NOUN
cana-760	300	14	𝑐	𝑐	PROPN
cana-760	300	15	∈	∈	PROPN
cana-760	300	16	𝑌.	𝑌.	PROPN
cana-760	300	17	4	4	NUM
cana-760	300	18	.	.	PUNCT
cana-760	300	19	application	application	NOUN
cana-760	300	20	in	in	ADP
cana-760	300	21	this	this	DET
cana-760	300	22	section	section	NOUN
cana-760	300	23	,	,	PUNCT
cana-760	300	24	we	we	PRON
cana-760	300	25	use	use	VERB
cana-760	300	26	our	our	PRON
cana-760	300	27	obtained	obtain	VERB
cana-760	300	28	results	result	NOUN
cana-760	300	29	to	to	PART
cana-760	300	30	show	show	VERB
cana-760	300	31	that	that	SCONJ
cana-760	300	32	the	the	DET
cana-760	300	33	following	follow	VERB
cana-760	300	34	integral	integral	ADJ
cana-760	300	35	equation	equation	NOUN
cana-760	300	36	has	have	VERB
cana-760	300	37	a	a	DET
cana-760	300	38	solution	solution	NOUN
cana-760	300	39	:	:	PUNCT
cana-760	300	40	𝜄(𝜅	𝜄(𝜅	NOUN
cana-760	300	41	)	)	PUNCT
cana-760	301	1	=	=	SYM
cana-760	301	2	ℎ(𝜅	ℎ(𝜅	PROPN
cana-760	301	3	)	)	PUNCT
cana-760	301	4	+	+	NUM
cana-760	301	5	∫	∫	PROPN
cana-760	301	6	ω(𝑘	ω(𝑘	NUM
cana-760	301	7	,	,	PUNCT
cana-760	301	8	𝑠)𝜍	𝑠)𝜍	ADJ
cana-760	301	9	1	1	NUM
cana-760	301	10	0	0	NUM
cana-760	301	11	(	(	PUNCT
cana-760	301	12	𝑠	𝑠	PROPN
cana-760	301	13	,	,	PUNCT
cana-760	301	14	𝜄(𝑠))𝑑𝑠	𝜄(𝑠))𝑑𝑠	PROPN
cana-760	301	15	,	,	PUNCT
cana-760	301	16	𝜅	𝜅	NOUN
cana-760	301	17	∈	∈	PROPN
cana-760	302	1	[	[	X
cana-760	302	2	0,1	0,1	NUM
cana-760	302	3	]	]	PUNCT
cana-760	302	4	.	.	PUNCT
cana-760	303	1	(	(	PUNCT
cana-760	303	2	4.1	4.1	NUM
cana-760	303	3	)	)	PUNCT
cana-760	303	4	let	let	VERB
cana-760	303	5	𝑌	𝑌	PROPN
cana-760	303	6	=	=	SYM
cana-760	303	7	𝐶([0,1	𝐶([0,1	NOUN
cana-760	303	8	]	]	PUNCT
cana-760	303	9	)	)	PUNCT
cana-760	303	10	be	be	AUX
cana-760	303	11	the	the	DET
cana-760	303	12	space	space	NOUN
cana-760	303	13	of	of	ADP
cana-760	303	14	all	all	DET
cana-760	303	15	continuous	continuous	ADJ
cana-760	303	16	functions	function	NOUN
cana-760	303	17	defined	define	VERB
cana-760	303	18	on	on	ADP
cana-760	303	19	[	[	X
cana-760	303	20	0,1	0,1	NUM
cana-760	303	21	]	]	PUNCT
cana-760	303	22	.	.	PUNCT
cana-760	304	1	define	define	VERB
cana-760	304	2	a	a	DET
cana-760	304	3	modular	modular	ADJ
cana-760	304	4	revised	revise	VERB
cana-760	304	5	fuzzy	fuzzy	ADJ
cana-760	304	6	metric	metric	NOUN
cana-760	304	7	:	:	PUNCT
cana-760	304	8	휁%(𝜄	휁%(𝜄	PROPN
cana-760	304	9	,	,	PUNCT
cana-760	304	10	휂	휂	ADP
cana-760	304	11	,	,	PUNCT
cana-760	304	12	𝑡	𝑡	NOUN
cana-760	304	13	)	)	PUNCT
cana-760	304	14	∶	∶	NOUN
cana-760	304	15	(	(	PUNCT
cana-760	304	16	0	0	NUM
cana-760	304	17	,	,	PUNCT
cana-760	304	18	∞)2	∞)2	PROPN
cana-760	304	19	×	×	NOUN
cana-760	304	20	𝐶([0,1])2	𝐶([0,1])2	NOUN
cana-760	304	21	→	→	SYM
cana-760	304	22	[	[	X
cana-760	304	23	0,1	0,1	NUM
cana-760	304	24	]	]	PUNCT
cana-760	304	25	,	,	PUNCT
cana-760	304	26	by	by	ADP
cana-760	304	27	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	304	28	,	,	PUNCT
cana-760	304	29	휂	휂	ADP
cana-760	304	30	,	,	PUNCT
cana-760	304	31	𝑡	𝑡	NOUN
cana-760	304	32	)	)	PUNCT
cana-760	304	33	=	=	PUNCT
cana-760	304	34	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-760	304	35	𝜅	𝜅	X
cana-760	304	36	∈	∈	PROPN
cana-760	305	1	[	[	X
cana-760	305	2	0,1	0,1	NUM
cana-760	305	3	]	]	X
cana-760	305	4	|𝜄(𝜅	|𝜄(𝜅	PROPN
cana-760	305	5	)	)	PUNCT
cana-760	305	6	−	−	NOUN
cana-760	305	7	휂(𝜅)|	휂(𝜅)|	X
cana-760	306	1	𝜚	𝜚	NOUN
cana-760	306	2	𝑡	𝑡	X
cana-760	306	3	+	+	CCONJ
cana-760	306	4	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	306	5	𝜅	𝜅	PROPN
cana-760	306	6	∈	∈	PROPN
cana-760	307	1	[	[	X
cana-760	307	2	0,1	0,1	NUM
cana-760	307	3	]	]	X
cana-760	307	4	|𝜄(𝜅	|𝜄(𝜅	PROPN
cana-760	307	5	)	)	PUNCT
cana-760	307	6	−	−	NOUN
cana-760	307	7	휂(𝜅)|	휂(𝜅)|	X
cana-760	307	8	𝜚	𝜚	NOUN
cana-760	307	9	then	then	ADV
cana-760	307	10	(	(	PUNCT
cana-760	307	11	𝑌	𝑌	PROPN
cana-760	307	12	,	,	PUNCT
cana-760	307	13	휁%,⊛	휁%,⊛	NOUN
cana-760	307	14	)	)	PUNCT
cana-760	307	15	is	be	AUX
cana-760	307	16	a	a	DET
cana-760	307	17	complete	complete	ADJ
cana-760	307	18	modular	modular	NOUN
cana-760	307	19	revised	revise	VERB
cana-760	307	20	fuzzy	fuzzy	ADJ
cana-760	307	21	metric	metric	ADJ
cana-760	307	22	space	space	NOUN
cana-760	307	23	.	.	PUNCT
cana-760	308	1	theorem	theorem	VERB
cana-760	308	2	4.1	4.1	NUM
cana-760	308	3	.	.	PUNCT
cana-760	309	1	suppose	suppose	VERB
cana-760	309	2	we	we	PRON
cana-760	309	3	have	have	VERB
cana-760	309	4	the	the	DET
cana-760	309	5	following	follow	VERB
cana-760	309	6	hypotheses	hypothesis	NOUN
cana-760	309	7	:	:	PUNCT
cana-760	309	8	(	(	PUNCT
cana-760	310	1	1	1	X
cana-760	310	2	)	)	PUNCT
cana-760	310	3	∃𝑎	∃𝑎	PROPN
cana-760	310	4	continuous	continuous	ADJ
cana-760	310	5	function	function	NOUN
cana-760	310	6	𝑔	𝑔	NOUN
cana-760	310	7	:	:	PUNCT
cana-760	311	1	[	[	X
cana-760	311	2	0,1	0,1	NUM
cana-760	311	3	]	]	PUNCT
cana-760	311	4	→	→	PUNCT
cana-760	312	1	[	[	X
cana-760	312	2	0,1	0,1	NUM
cana-760	312	3	]	]	PUNCT
cana-760	312	4	such	such	ADJ
cana-760	312	5	that	that	SCONJ
cana-760	312	6	|(𝑠	|(𝑠	PROPN
cana-760	312	7	,	,	PUNCT
cana-760	312	8	𝜄	𝜄	PROPN
cana-760	312	9	)	)	PUNCT
cana-760	312	10	−	−	PROPN
cana-760	312	11	(	(	PUNCT
cana-760	312	12	𝑠	𝑠	PROPN
cana-760	312	13	,	,	PUNCT
cana-760	312	14	휂)|	휂)|	NOUN
cana-760	312	15	≤	≤	NUM
cana-760	312	16	𝑔(𝑠)|𝜄	𝑔(𝑠)|𝜄	PROPN
cana-760	312	17	−	−	PROPN
cana-760	312	18	휂|	휂|	PROPN
cana-760	312	19	.	.	PUNCT
cana-760	313	1	(	(	PUNCT
cana-760	313	2	4.2	4.2	NUM
cana-760	313	3	)	)	PUNCT
cana-760	313	4	communications	communication	NOUN
cana-760	313	5	on	on	ADP
cana-760	313	6	applied	apply	VERB
cana-760	313	7	nonlinear	nonlinear	ADJ
cana-760	313	8	analysis	analysis	NOUN
cana-760	313	9	issn	issn	NOUN
cana-760	313	10	:	:	PUNCT
cana-760	313	11	1074	1074	NUM
cana-760	313	12	-	-	PUNCT
cana-760	313	13	133x	133x	NUM
cana-760	313	14	vol	vol	NOUN
cana-760	313	15	31	31	NUM
cana-760	313	16	no	no	NOUN
cana-760	313	17	.	.	PUNCT
cana-760	314	1	3s	3s	NUM
cana-760	314	2	(	(	PUNCT
cana-760	314	3	2024	2024	NUM
cana-760	314	4	)	)	PUNCT
cana-760	314	5	224	224	NUM
cana-760	314	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	314	7	and	and	CCONJ
cana-760	314	8	∫	∫	PROPN
cana-760	314	9	𝑔(𝑘)𝑑𝑘	𝑔(𝑘)𝑑𝑘	PART
cana-760	314	10	≤	≤	NUM
cana-760	314	11	1	1	NUM
cana-760	314	12	3	3	NUM
cana-760	314	13	1	1	NUM
cana-760	314	14	0	0	NUM
cana-760	314	15	(	(	PUNCT
cana-760	314	16	4.3	4.3	NUM
cana-760	314	17	)	)	PUNCT
cana-760	314	18	(	(	PUNCT
cana-760	314	19	2	2	NUM
cana-760	314	20	)	)	PUNCT
cana-760	314	21	ω(𝑘	ω(𝑘	NUM
cana-760	314	22	,	,	PUNCT
cana-760	314	23	𝑠	𝑠	NUM
cana-760	314	24	)	)	PUNCT
cana-760	314	25	≥	≥	NOUN
cana-760	314	26	0	0	NUM
cana-760	314	27	,	,	PUNCT
cana-760	314	28	∀	∀	X
cana-760	314	29	𝑘	𝑘	NOUN
cana-760	314	30	,	,	PUNCT
cana-760	314	31	𝑠	𝑠	PROPN
cana-760	314	32	∈	∈	PROPN
cana-760	315	1	[	[	X
cana-760	315	2	0,1	0,1	NUM
cana-760	315	3	]	]	PUNCT
cana-760	315	4	(	(	PUNCT
cana-760	315	5	4.4	4.4	NUM
cana-760	315	6	)	)	PUNCT
cana-760	315	7	then	then	ADV
cana-760	315	8	the	the	DET
cana-760	315	9	integral	integral	ADJ
cana-760	315	10	equation	equation	NOUN
cana-760	315	11	(	(	PUNCT
cana-760	315	12	4.1	4.1	NUM
cana-760	315	13	)	)	PUNCT
cana-760	315	14	has	have	VERB
cana-760	315	15	a	a	DET
cana-760	315	16	solution	solution	NOUN
cana-760	315	17	𝜄⊛	𝜄⊛	PROPN
cana-760	315	18	∈	∈	PROPN
cana-760	315	19	𝐶2([0,1	𝐶2([0,1	NOUN
cana-760	315	20	]	]	PUNCT
cana-760	315	21	)	)	PUNCT
cana-760	315	22	.	.	PUNCT
cana-760	316	1	proof	proof	NOUN
cana-760	316	2	.	.	PUNCT
cana-760	317	1	take	take	VERB
cana-760	317	2	the	the	DET
cana-760	317	3	operator	operator	NOUN
cana-760	317	4	:	:	PUNCT
cana-760	317	5	γ𝜄(𝜅	γ𝜄(𝜅	NUM
cana-760	317	6	)	)	PUNCT
cana-760	317	7	=	=	SYM
cana-760	317	8	ℎ(𝜅	ℎ(𝜅	PROPN
cana-760	317	9	)	)	PUNCT
cana-760	318	1	+	+	NUM
cana-760	318	2	∫	∫	PROPN
cana-760	318	3	ω(𝑘	ω(𝑘	NUM
cana-760	318	4	,	,	PUNCT
cana-760	318	5	𝑠)𝜍	𝑠)𝜍	ADJ
cana-760	318	6	1	1	NUM
cana-760	318	7	0	0	NUM
cana-760	318	8	(	(	PUNCT
cana-760	318	9	𝑠	𝑠	PROPN
cana-760	318	10	,	,	PUNCT
cana-760	318	11	𝜄(𝑠))𝑑𝑠	𝜄(𝑠))𝑑𝑠	PROPN
cana-760	318	12	,	,	PUNCT
cana-760	318	13	𝜅	𝜅	NOUN
cana-760	318	14	∈	∈	PROPN
cana-760	318	15	[	[	X
cana-760	318	16	0,1	0,1	NUM
cana-760	318	17	]	]	PUNCT
cana-760	318	18	for	for	ADP
cana-760	318	19	all	all	DET
cana-760	318	20	𝜄	𝜄	PROPN
cana-760	318	21	,	,	PUNCT
cana-760	318	22	휂	휂	ADP
cana-760	318	23	∈	∈	PROPN
cana-760	318	24	𝐶([0,1	𝐶([0,1	NOUN
cana-760	318	25	]	]	PUNCT
cana-760	318	26	)	)	PUNCT
cana-760	318	27	,	,	PUNCT
cana-760	318	28	we	we	PRON
cana-760	318	29	have	have	VERB
cana-760	318	30	휁𝜚(𝜄	휁𝜚(𝜄	NOUN
cana-760	318	31	,	,	PUNCT
cana-760	318	32	휂	휂	ADP
cana-760	318	33	,	,	PUNCT
cana-760	318	34	𝑡	𝑡	NOUN
cana-760	318	35	)	)	PUNCT
cana-760	318	36	=	=	PUNCT
cana-760	318	37	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-760	318	38	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	318	39	]	]	PUNCT
cana-760	318	40	|𝜄(𝜅)−𝜂(𝜅)|	|𝜄(𝜅)−𝜂(𝜅)|	ADP
cana-760	318	41	𝜚	𝜚	NOUN
cana-760	318	42	𝑡+	𝑡+	PUNCT
cana-760	318	43	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	318	44	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	318	45	]	]	PUNCT
cana-760	319	1	|𝜄(𝜅)−𝜂(𝜅)|	|𝜄(𝜅)−𝜂(𝜅)|	NOUN
cana-760	319	2	𝜚	𝜚	NOUN
cana-760	319	3	=	=	PUNCT
cana-760	319	4	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-760	319	5	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	319	6	]	]	PUNCT
cana-760	319	7	1	1	NUM
cana-760	319	8	𝜚	𝜚	PROPN
cana-760	319	9	|ℎ(𝜅)+∫	|ℎ(𝜅)+∫	PROPN
cana-760	319	10	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	319	11	1	1	NUM
cana-760	319	12	0	0	NUM
cana-760	319	13	(	(	PUNCT
cana-760	319	14	𝑠,𝜄(𝑠))𝑑𝑠−ℎ(𝜅)−∫	𝑠,𝜄(𝑠))𝑑𝑠−ℎ(𝜅)−∫	NUM
cana-760	319	15	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	319	16	1	1	NUM
cana-760	319	17	0	0	NUM
cana-760	319	18	(	(	PUNCT
cana-760	319	19	𝑠,𝜂(𝑠))𝑑𝑠|	𝑠,𝜂(𝑠))𝑑𝑠|	NOUN
cana-760	319	20	𝑡+	𝑡+	X
cana-760	319	21	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	319	22	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	319	23	]	]	PUNCT
cana-760	319	24	1	1	NUM
cana-760	319	25	𝜚	𝜚	PROPN
cana-760	319	26	|ℎ(𝜅)+∫	|ℎ(𝜅)+∫	PROPN
cana-760	319	27	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	319	28	1	1	NUM
cana-760	319	29	0	0	NUM
cana-760	319	30	(	(	PUNCT
cana-760	319	31	𝑠,𝜄(𝑠))𝑑𝑠−ℎ(𝜅)−∫	𝑠,𝜄(𝑠))𝑑𝑠−ℎ(𝜅)−∫	NUM
cana-760	319	32	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	319	33	1	1	NUM
cana-760	319	34	0	0	NUM
cana-760	319	35	(	(	PUNCT
cana-760	319	36	𝑠,𝜂(𝑠))𝑑𝑠|	𝑠,𝜂(𝑠))𝑑𝑠|	NOUN
cana-760	319	37	=	=	PUNCT
cana-760	319	38	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	319	39	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	319	40	]	]	PUNCT
cana-760	319	41	1	1	NUM
cana-760	319	42	𝜚	𝜚	NOUN
cana-760	319	43	|∫	|∫	NOUN
cana-760	319	44	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	319	45	1	1	NUM
cana-760	319	46	0	0	NUM
cana-760	319	47	(	(	PUNCT
cana-760	319	48	𝑠,𝜄(𝑠))𝑑𝑠−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	𝑠,𝜄(𝑠))𝑑𝑠−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	NUM
cana-760	319	49	𝑡+	𝑡+	VERB
cana-760	319	50	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	319	51	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	319	52	]	]	PUNCT
cana-760	319	53	1	1	NUM
cana-760	319	54	𝜚	𝜚	NOUN
cana-760	319	55	|∫	|∫	NOUN
cana-760	319	56	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	319	57	1	1	NUM
cana-760	319	58	0	0	NUM
cana-760	319	59	(	(	PUNCT
cana-760	319	60	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	NOUN
cana-760	319	61	by	by	ADP
cana-760	319	62	(	(	PUNCT
cana-760	319	63	4.2	4.2	NUM
cana-760	319	64	)	)	PUNCT
cana-760	319	65	and	and	CCONJ
cana-760	319	66	(	(	PUNCT
cana-760	319	67	4.4	4.4	NUM
cana-760	319	68	)	)	PUNCT
cana-760	319	69	,	,	PUNCT
cana-760	319	70	we	we	PRON
cana-760	319	71	have	have	VERB
cana-760	319	72	|∫	|∫	ADJ
cana-760	319	73	ω(𝑘	ω(𝑘	NOUN
cana-760	319	74	,	,	PUNCT
cana-760	319	75	𝑠)𝜍	𝑠)𝜍	ADJ
cana-760	319	76	1	1	NUM
cana-760	319	77	0	0	NUM
cana-760	319	78	(	(	PUNCT
cana-760	319	79	𝑠	𝑠	PROPN
cana-760	319	80	,	,	PUNCT
cana-760	319	81	𝜄(𝑠	𝜄(𝑠	PROPN
cana-760	319	82	)	)	PUNCT
cana-760	319	83	)	)	PUNCT
cana-760	320	1	−	−	PROPN
cana-760	320	2	𝜍(𝑠	𝜍(𝑠	PROPN
cana-760	320	3	,	,	PUNCT
cana-760	320	4	휂(𝑠))𝑑𝑠|	휂(𝑠))𝑑𝑠|	ADJ
cana-760	320	5	≤	≤	NUM
cana-760	320	6	∫	∫	PROPN
cana-760	320	7	ω(𝑘	ω(𝑘	PROPN
cana-760	320	8	,	,	PUNCT
cana-760	320	9	𝑠	𝑠	NOUN
cana-760	320	10	)	)	PUNCT
cana-760	320	11	1	1	NUM
cana-760	320	12	0	0	NUM
cana-760	320	13	|𝜍(𝑠	|𝜍(𝑠	PROPN
cana-760	320	14	,	,	PUNCT
cana-760	320	15	𝜄(𝑠	𝜄(𝑠	PROPN
cana-760	320	16	)	)	PUNCT
cana-760	320	17	)	)	PUNCT
cana-760	321	1	−	−	PROPN
cana-760	322	1	𝜍(𝑠	𝜍(𝑠	PROPN
cana-760	322	2	,	,	PUNCT
cana-760	322	3	휂(𝑠))𝑑𝑠|	휂(𝑠))𝑑𝑠|	ADJ
cana-760	322	4	≤	≤	NUM
cana-760	322	5	∫	∫	PROPN
cana-760	322	6	𝑔(𝑠	𝑔(𝑠	NOUN
cana-760	322	7	)	)	PUNCT
cana-760	322	8	1	1	NUM
cana-760	322	9	0	0	NUM
cana-760	322	10	|𝜄(𝑠	|𝜄(𝑠	PROPN
cana-760	322	11	)	)	PUNCT
cana-760	322	12	−	−	NOUN
cana-760	322	13	휂(𝑠)|𝑑𝑠.	휂(𝑠)|𝑑𝑠.	NOUN
cana-760	322	14	hence	hence	ADV
cana-760	322	15	by	by	ADP
cana-760	322	16	(	(	PUNCT
cana-760	322	17	4.3	4.3	NUM
cana-760	322	18	)	)	PUNCT
cana-760	322	19	,	,	PUNCT
cana-760	322	20	we	we	PRON
cana-760	322	21	get	get	VERB
cana-760	322	22	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	322	23	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	322	24	]	]	PUNCT
cana-760	322	25	1	1	NUM
cana-760	322	26	𝜚	𝜚	NOUN
cana-760	322	27	|∫	|∫	NOUN
cana-760	322	28	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	322	29	1	1	NUM
cana-760	322	30	0	0	NUM
cana-760	322	31	(	(	PUNCT
cana-760	322	32	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	NOUN
cana-760	322	33	𝑡+	𝑡+	PUNCT
cana-760	322	34	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	322	35	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	322	36	]	]	PUNCT
cana-760	322	37	1	1	NUM
cana-760	322	38	𝜚	𝜚	NOUN
cana-760	322	39	|∫	|∫	NOUN
cana-760	322	40	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	322	41	1	1	NUM
cana-760	322	42	0	0	NUM
cana-760	322	43	(	(	PUNCT
cana-760	322	44	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	NOUN
cana-760	322	45	≤	≤	PROPN
cana-760	322	46	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	322	47	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	322	48	]	]	PUNCT
cana-760	322	49	1	1	NUM
cana-760	322	50	𝜚	𝜚	NOUN
cana-760	322	51	∫	∫	NOUN
cana-760	322	52	𝑔(𝑠	𝑔(𝑠	NOUN
cana-760	322	53	)	)	PUNCT
cana-760	322	54	1	1	NUM
cana-760	322	55	0	0	NUM
cana-760	322	56	|𝜄(𝑠)−𝜂(𝑠)|𝑑𝑠	|𝜄(𝑠)−𝜂(𝑠)|𝑑𝑠	X
cana-760	322	57	𝑡+	𝑡+	X
cana-760	322	58	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-760	322	59	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	322	60	]	]	PUNCT
cana-760	322	61	1	1	NUM
cana-760	322	62	𝜚	𝜚	NOUN
cana-760	322	63	∫	∫	NOUN
cana-760	322	64	𝑔(𝑠	𝑔(𝑠	NOUN
cana-760	322	65	)	)	PUNCT
cana-760	322	66	1	1	NUM
cana-760	322	67	0	0	NUM
cana-760	322	68	|𝜄(𝑠)−𝜂(𝑠)|𝑑𝑠	|𝜄(𝑠)−𝜂(𝑠)|𝑑𝑠	X
cana-760	322	69	=	=	PUNCT
cana-760	322	70	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-760	322	71	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	322	72	]	]	PUNCT
cana-760	322	73	1	1	NUM
cana-760	322	74	3𝜚	3𝜚	NUM
cana-760	322	75	|𝜄(𝜅)−𝜂(𝜅)|	|𝜄(𝜅)−𝜂(𝜅)|	NOUN
cana-760	322	76	𝑡+	𝑡+	PUNCT
cana-760	322	77	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-760	322	78	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	322	79	]	]	PUNCT
cana-760	322	80	1	1	NUM
cana-760	322	81	3𝜚	3𝜚	NUM
cana-760	322	82	|𝜄(𝜅)−𝜂(𝜅)|	|𝜄(𝜅)−𝜂(𝜅)|	ADV
cana-760	322	83	thus	thus	ADV
cana-760	322	84	𝑡	𝑡	X
cana-760	322	85	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-760	322	86	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	322	87	]	]	PUNCT
cana-760	322	88	1	1	NUM
cana-760	322	89	3𝜚	3𝜚	NUM
cana-760	322	90	|𝜄(𝜅)−𝜂(𝜅)|	|𝜄(𝜅)−𝜂(𝜅)|	NOUN
cana-760	322	91	≤	≤	NUM
cana-760	323	1	𝑡	𝑡	PROPN
cana-760	323	2	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	323	3	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	323	4	]	]	PUNCT
cana-760	323	5	1	1	NUM
cana-760	323	6	𝜚	𝜚	NOUN
cana-760	323	7	|∫	|∫	NOUN
cana-760	323	8	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	323	9	1	1	NUM
cana-760	323	10	0	0	NUM
cana-760	323	11	(	(	PUNCT
cana-760	323	12	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	NOUN
cana-760	323	13	so	so	ADV
cana-760	323	14	3𝑡	3𝑡	PROPN
cana-760	323	15	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	323	16	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	323	17	]	]	PUNCT
cana-760	323	18	1	1	NUM
cana-760	323	19	𝜚	𝜚	NOUN
cana-760	323	20	|𝜄(𝜅)−𝜂(𝜅)|	|𝜄(𝜅)−𝜂(𝜅)|	ADV
cana-760	323	21	≤	≤	NUM
cana-760	323	22	𝑡	𝑡	PROPN
cana-760	323	23	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-760	323	24	𝜅∈[0,1	𝜅∈[0,1	NUM
cana-760	323	25	]	]	PUNCT
cana-760	323	26	1	1	NUM
cana-760	323	27	𝜚	𝜚	NOUN
cana-760	323	28	|∫	|∫	NOUN
cana-760	323	29	ω(𝑘,𝑠)𝜍	ω(𝑘,𝑠)𝜍	NUM
cana-760	323	30	1	1	NUM
cana-760	323	31	0	0	NUM
cana-760	323	32	(	(	PUNCT
cana-760	323	33	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	𝑠,𝜄(𝑠))−𝜍(𝑠,𝜂(𝑠))𝑑𝑠|	NOUN
cana-760	323	34	define	define	VERB
cana-760	323	35	𝛶	𝛶	NOUN
cana-760	323	36	:	:	PUNCT
cana-760	323	37	(	(	PUNCT
cana-760	323	38	0,1	0,1	NUM
cana-760	323	39	]	]	PUNCT
cana-760	323	40	→	→	PUNCT
cana-760	324	1	[	[	X
cana-760	324	2	0	0	NUM
cana-760	324	3	,	,	PUNCT
cana-760	324	4	+	+	NOUN
cana-760	324	5	∞	∞	NOUN
cana-760	324	6	)	)	PUNCT
cana-760	324	7	by	by	ADP
cana-760	324	8	𝛶(𝑡	𝛶(𝑡	NOUN
cana-760	324	9	)	)	PUNCT
cana-760	324	10	=	=	SYM
cana-760	324	11	1	1	NUM
cana-760	324	12	𝜄	𝜄	PRON
cana-760	324	13	−	−	PROPN
cana-760	325	1	1	1	NUM
cana-760	325	2	.	.	PUNCT
cana-760	326	1	then	then	ADV
cana-760	326	2	𝛶	𝛶	PROPN
cana-760	326	3	is	be	AUX
cana-760	326	4	a	a	DET
cana-760	326	5	strictly	strictly	ADV
cana-760	326	6	non	non	ADJ
cana-760	326	7	-	-	ADJ
cana-760	326	8	decreasing	decrease	VERB
cana-760	326	9	,	,	PUNCT
cana-760	326	10	continuous	continuous	ADJ
cana-760	326	11	function	function	NOUN
cana-760	326	12	and	and	CCONJ
cana-760	326	13	𝛶(1	𝛶(1	ADP
cana-760	326	14	)	)	PUNCT
cana-760	326	15	=	=	SYM
cana-760	326	16	0	0	X
cana-760	326	17	.	.	X
cana-760	327	1	for	for	ADP
cana-760	327	2	𝐻	𝐻	PROPN
cana-760	327	3	=	=	SYM
cana-760	327	4	1	1	NUM
cana-760	327	5	3	3	NUM
cana-760	327	6	,	,	PUNCT
cana-760	327	7	we	we	PRON
cana-760	327	8	obtain	obtain	VERB
cana-760	327	9	𝛶	𝛶	PROPN
cana-760	327	10	(	(	PUNCT
cana-760	327	11	1	1	NUM
cana-760	327	12	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	𝜁%(𝛤𝜄,𝛤𝜂,𝑡	NOUN
cana-760	327	13	)	)	PUNCT
cana-760	327	14	−	−	ADP
cana-760	327	15	1	1	X
cana-760	327	16	)	)	PUNCT
cana-760	327	17	≤	≤	NOUN
cana-760	327	18	𝐻𝛶	𝐻𝛶	PROPN
cana-760	327	19	(	(	PUNCT
cana-760	327	20	1	1	NUM
cana-760	327	21	𝜁%(𝜄,𝜂,𝑡	𝜁%(𝜄,𝜂,𝑡	NUM
cana-760	327	22	)	)	PUNCT
cana-760	327	23	−	−	PROPN
cana-760	327	24	1	1	NUM
cana-760	327	25	)	)	PUNCT
cana-760	327	26	.	.	PUNCT
cana-760	328	1	therefore	therefore	ADV
cana-760	328	2	theorem	theorem	ADJ
cana-760	328	3	(	(	PUNCT
cana-760	328	4	3.1	3.1	NUM
cana-760	328	5	)	)	PUNCT
cana-760	328	6	implies	imply	VERB
cana-760	328	7	𝛤	𝛤	PROPN
cana-760	328	8	has	have	VERB
cana-760	328	9	a	a	DET
cana-760	328	10	unique	unique	ADJ
cana-760	328	11	fixed	fix	VERB
cana-760	328	12	point	point	NOUN
cana-760	328	13	and	and	CCONJ
cana-760	328	14	hence	hence	ADV
cana-760	328	15	the	the	DET
cana-760	328	16	integral	integral	ADJ
cana-760	328	17	equation	equation	NOUN
cana-760	328	18	(	(	PUNCT
cana-760	328	19	4.1	4.1	NUM
cana-760	328	20	)	)	PUNCT
cana-760	328	21	has	have	VERB
cana-760	328	22	a	a	DET
cana-760	328	23	solution	solution	NOUN
cana-760	328	24	𝜄⊛	𝜄⊛	PROPN
cana-760	328	25	∈	∈	PROPN
cana-760	328	26	𝐶2([0,1	𝐶2([0,1	NOUN
cana-760	328	27	]	]	PUNCT
cana-760	328	28	)	)	PUNCT
cana-760	328	29	.	.	PUNCT
cana-760	329	1	communications	communication	NOUN
cana-760	329	2	on	on	ADP
cana-760	329	3	applied	apply	VERB
cana-760	329	4	nonlinear	nonlinear	ADJ
cana-760	329	5	analysis	analysis	NOUN
cana-760	329	6	issn	issn	NOUN
cana-760	329	7	:	:	PUNCT
cana-760	329	8	1074	1074	NUM
cana-760	329	9	-	-	PUNCT
cana-760	329	10	133x	133x	NUM
cana-760	329	11	vol	vol	NOUN
cana-760	329	12	31	31	NUM
cana-760	329	13	no	no	NOUN
cana-760	329	14	.	.	PUNCT
cana-760	330	1	3s	3s	NUM
cana-760	330	2	(	(	PUNCT
cana-760	330	3	2024	2024	NUM
cana-760	330	4	)	)	PUNCT
cana-760	330	5	225	225	NUM
cana-760	330	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	330	7	conclusion	conclusion	NOUN
cana-760	330	8	:	:	PUNCT
cana-760	330	9	in	in	ADP
cana-760	330	10	this	this	DET
cana-760	330	11	paper	paper	NOUN
cana-760	330	12	,	,	PUNCT
cana-760	330	13	we	we	PRON
cana-760	330	14	defined	define	VERB
cana-760	330	15	a	a	DET
cana-760	330	16	new	new	ADJ
cana-760	330	17	space	space	NOUN
cana-760	330	18	called	call	VERB
cana-760	330	19	modular	modular	ADJ
cana-760	330	20	revised	revise	VERB
cana-760	330	21	fuzzy	fuzzy	ADJ
cana-760	330	22	metric	metric	ADJ
cana-760	330	23	space	space	NOUN
cana-760	330	24	and	and	CCONJ
cana-760	330	25	stated	state	VERB
cana-760	330	26	some	some	DET
cana-760	330	27	examples	example	NOUN
cana-760	330	28	of	of	ADP
cana-760	330	29	this	this	DET
cana-760	330	30	space	space	NOUN
cana-760	330	31	.	.	PUNCT
cana-760	331	1	also	also	ADV
cana-760	331	2	,	,	PUNCT
cana-760	331	3	we	we	PRON
cana-760	331	4	formulate	formulate	VERB
cana-760	331	5	and	and	CCONJ
cana-760	331	6	prove	prove	VERB
cana-760	331	7	some	some	DET
cana-760	331	8	new	new	ADJ
cana-760	331	9	fixed	fix	VERB
cana-760	331	10	point	point	NOUN
cana-760	331	11	results	result	NOUN
cana-760	331	12	under	under	ADP
cana-760	331	13	this	this	DET
cana-760	331	14	space	space	NOUN
cana-760	331	15	.	.	PUNCT
cana-760	332	1	in	in	ADP
cana-760	332	2	addition	addition	NOUN
cana-760	332	3	,	,	PUNCT
cana-760	332	4	we	we	PRON
cana-760	332	5	provided	provide	VERB
cana-760	332	6	some	some	DET
cana-760	332	7	examples	example	NOUN
cana-760	332	8	and	and	CCONJ
cana-760	332	9	an	an	DET
cana-760	332	10	application	application	NOUN
cana-760	332	11	for	for	ADP
cana-760	332	12	showing	show	VERB
cana-760	332	13	the	the	DET
cana-760	332	14	validity	validity	NOUN
cana-760	332	15	of	of	ADP
cana-760	332	16	our	our	PRON
cana-760	332	17	results	result	NOUN
cana-760	332	18	.	.	PUNCT
cana-760	333	1	references	reference	NOUN
cana-760	333	2	[	[	X
cana-760	333	3	1	1	NUM
cana-760	333	4	]	]	PUNCT
cana-760	333	5	a.	a.	PROPN
cana-760	333	6	abdou	abdou	PROPN
cana-760	333	7	and	and	CCONJ
cana-760	333	8	m.a	m.a	PROPN
cana-760	333	9	.	.	PROPN
cana-760	333	10	khamsi	khamsi	PROPN
cana-760	333	11	,	,	PUNCT
cana-760	333	12	“	"	PUNCT
cana-760	333	13	on	on	ADP
cana-760	333	14	the	the	DET
cana-760	333	15	fixed	fix	VERB
cana-760	333	16	points	point	NOUN
cana-760	333	17	of	of	ADP
cana-760	333	18	non	non	PRON
cana-760	333	19	expansive	expansive	ADJ
cana-760	333	20	mappings	mapping	NOUN
cana-760	333	21	in	in	ADP
cana-760	333	22	modular	modular	ADJ
cana-760	333	23	metric	metric	ADJ
cana-760	333	24	spaces	space	NOUN
cana-760	333	25	”	"	PUNCT
cana-760	333	26	,	,	PUNCT
cana-760	333	27	fixed	fix	VERB
cana-760	333	28	point	point	NOUN
cana-760	333	29	theory	theory	NOUN
cana-760	333	30	appl	appl	NOUN
cana-760	333	31	.	.	PUNCT
cana-760	334	1	2013	2013	NUM
cana-760	334	2	,	,	PUNCT
cana-760	334	3	no	no	INTJ
cana-760	334	4	.	.	NOUN
cana-760	334	5	1	1	NUM
cana-760	334	6	,	,	PUNCT
cana-760	334	7	1	1	NUM
cana-760	334	8	-	-	SYM
cana-760	334	9	13	13	NUM
cana-760	334	10	.	.	PUNCT
cana-760	335	1	[	[	X
cana-760	335	2	2	2	NUM
cana-760	335	3	]	]	X
cana-760	335	4	alexander	alexander	PROPN
cana-760	335	5	sostak	sostak	PROPN
cana-760	335	6	“	"	PUNCT
cana-760	335	7	george	george	NOUN
cana-760	335	8	-	-	PUNCT
cana-760	335	9	veeramani	veeramani	NOUN
cana-760	335	10	fuzzy	fuzzy	ADJ
cana-760	335	11	metrics	metric	NOUN
cana-760	335	12	revised	revise	VERB
cana-760	335	13	”	"	PUNCT
cana-760	335	14	axioms	axiom	NOUN
cana-760	335	15	2018,7,60	2018,7,60	NUM
cana-760	335	16	.	.	PUNCT
cana-760	336	1	[	[	X
cana-760	336	2	3	3	X
cana-760	336	3	]	]	X
cana-760	336	4	v.	v.	CCONJ
cana-760	336	5	chistyakov	chistyakov	NOUN
cana-760	336	6	,	,	PUNCT
cana-760	336	7	modular	modular	ADJ
cana-760	336	8	metric	metric	ADJ
cana-760	336	9	spaces	space	NOUN
cana-760	336	10	,	,	PUNCT
cana-760	336	11	i	i	PRON
cana-760	336	12	:	:	PUNCT
cana-760	336	13	basic	basic	ADJ
cana-760	336	14	concepts	concept	NOUN
cana-760	336	15	,	,	PUNCT
cana-760	336	16	nonlinear	nonlinear	ADJ
cana-760	336	17	anal	anal	NOUN
cana-760	336	18	.	.	PUNCT
cana-760	337	1	72(2010),1	72(2010),1	NUM
cana-760	337	2	-	-	PUNCT
cana-760	337	3	14	14	NUM
cana-760	337	4	.	.	PUNCT
cana-760	338	1	[	[	X
cana-760	338	2	4	4	X
cana-760	338	3	]	]	X
cana-760	338	4	v.	v.	ADP
cana-760	338	5	chistyakov	chistyakov	NOUN
cana-760	338	6	,	,	PUNCT
cana-760	338	7	modular	modular	ADJ
cana-760	338	8	metric	metric	ADJ
cana-760	338	9	spaces	space	NOUN
cana-760	338	10	,	,	PUNCT
cana-760	338	11	ii	ii	PROPN
cana-760	338	12	:	:	PUNCT
cana-760	338	13	application	application	NOUN
cana-760	338	14	to	to	ADP
cana-760	338	15	superposition	superposition	NOUN
cana-760	338	16	operators	operator	NOUN
cana-760	338	17	,	,	PUNCT
cana-760	338	18	nonlinear	nonlinear	ADJ
cana-760	338	19	anal	anal	NOUN
cana-760	338	20	.	.	PUNCT
cana-760	339	1	72(2010),15	72(2010),15	NUM
cana-760	339	2	-	-	SYM
cana-760	339	3	30	30	NUM
cana-760	339	4	.	.	PUNCT
cana-760	340	1	[	[	X
cana-760	340	2	5	5	NUM
cana-760	340	3	]	]	PUNCT
cana-760	340	4	v.	v.	CCONJ
cana-760	340	5	chistyakov	chistyakov	PROPN
cana-760	340	6	,	,	PUNCT
cana-760	340	7	a	a	DET
cana-760	340	8	fixed	fix	VERB
cana-760	340	9	point	point	NOUN
cana-760	340	10	theorem	theorem	NOUN
cana-760	340	11	for	for	ADP
cana-760	340	12	contractions	contraction	NOUN
cana-760	340	13	in	in	ADP
cana-760	340	14	modular	modular	ADJ
cana-760	340	15	metric	metric	ADJ
cana-760	340	16	spaces	space	NOUN
cana-760	340	17	,	,	PUNCT
cana-760	340	18	arxiv	arxiv	PROPN
cana-760	340	19	(	(	PUNCT
cana-760	340	20	2011	2011	NUM
cana-760	340	21	)	)	PUNCT
cana-760	340	22	.	.	PUNCT
cana-760	341	1	[	[	X
cana-760	341	2	6	6	NUM
cana-760	341	3	]	]	X
cana-760	341	4	george.a	george.a	PROPN
cana-760	341	5	,	,	PUNCT
cana-760	341	6	veeramani	veeramani	PROPN
cana-760	341	7	.	.	PUNCT
cana-760	342	1	p.	p.	NOUN
cana-760	342	2	“	"	PUNCT
cana-760	342	3	on	on	ADP
cana-760	342	4	some	some	DET
cana-760	342	5	results	result	NOUN
cana-760	342	6	in	in	ADP
cana-760	342	7	fuzzy	fuzzy	ADJ
cana-760	342	8	metric	metric	ADJ
cana-760	342	9	spaces	space	NOUN
cana-760	342	10	”	"	PUNCT
cana-760	342	11	,	,	PUNCT
cana-760	342	12	fuzzy	fuzzy	ADJ
cana-760	342	13	sets	set	NOUN
cana-760	342	14	and	and	CCONJ
cana-760	342	15	syst	syst	NOUN
cana-760	342	16	.	.	PUNCT
cana-760	343	1	64(1994	64(1994	NUM
cana-760	343	2	)	)	PUNCT
cana-760	343	3	,	,	PUNCT
cana-760	343	4	395	395	NUM
cana-760	343	5	-	-	SYM
cana-760	343	6	399	399	NUM
cana-760	343	7	.	.	PUNCT
cana-760	344	1	[	[	X
cana-760	344	2	7	7	NUM
cana-760	344	3	]	]	X
cana-760	344	4	grabiec	grabiec	PROPN
cana-760	344	5	.	.	PUNCT
cana-760	345	1	m	m	PROPN
cana-760	345	2	,	,	PUNCT
cana-760	345	3	fixed	fix	VERB
cana-760	345	4	points	point	NOUN
cana-760	345	5	in	in	ADP
cana-760	345	6	fuzzy	fuzzy	ADJ
cana-760	345	7	metric	metric	ADJ
cana-760	345	8	spaces	space	NOUN
cana-760	345	9	.	.	PUNCT
cana-760	346	1	fuzzy	fuzzy	ADJ
cana-760	346	2	sets	set	NOUN
cana-760	346	3	and	and	CCONJ
cana-760	346	4	syst	syst	NOUN
cana-760	346	5	.	.	PUNCT
cana-760	346	6	27(1988	27(1988	NUM
cana-760	346	7	)	)	PUNCT
cana-760	346	8	,	,	PUNCT
cana-760	346	9	385	385	NUM
cana-760	346	10	-	-	SYM
cana-760	346	11	389	389	NUM
cana-760	346	12	.	.	PUNCT
cana-760	347	1	[	[	X
cana-760	347	2	8	8	NUM
cana-760	347	3	]	]	X
cana-760	347	4	gregori	gregori	X
cana-760	347	5	,	,	PUNCT
cana-760	347	6	v.	v.	PROPN
cana-760	347	7	,	,	PUNCT
cana-760	347	8	miñana	miñana	PROPN
cana-760	347	9	,	,	PUNCT
cana-760	347	10	j.-j	j.-j	PROPN
cana-760	347	11	.	.	PROPN
cana-760	347	12	,	,	PUNCT
cana-760	347	13	miravet	miravet	PROPN
cana-760	347	14	,	,	PUNCT
cana-760	347	15	d.	d.	PROPN
cana-760	347	16	contractive	contractive	ADJ
cana-760	347	17	sequences	sequence	NOUN
cana-760	347	18	in	in	ADP
cana-760	347	19	fuzzy	fuzzy	ADJ
cana-760	347	20	metric	metric	ADJ
cana-760	347	21	spaces	space	NOUN
cana-760	347	22	.	.	PUNCT
cana-760	348	1	fuzzy	fuzzy	ADJ
cana-760	348	2	sets	set	NOUN
cana-760	348	3	syst	syst	PROPN
cana-760	348	4	.	.	PUNCT
cana-760	349	1	2020	2020	NUM
cana-760	349	2	,	,	PUNCT
cana-760	349	3	379	379	NUM
cana-760	349	4	,	,	PUNCT
cana-760	349	5	125–133	125–133	NUM
cana-760	349	6	.	.	PUNCT
cana-760	350	1	[	[	X
cana-760	350	2	9	9	NUM
cana-760	350	3	]	]	X
cana-760	350	4	hanaa	hanaa	PROPN
cana-760	350	5	kerim	kerim	PROPN
cana-760	350	6	,	,	PUNCT
cana-760	350	7	wasfi	wasfi	NOUN
cana-760	350	8	shatanawi	shatanawi	PROPN
cana-760	350	9	,	,	PUNCT
cana-760	350	10	abdalla	abdalla	PROPN
cana-760	350	11	tallafha	tallafha	NOUN
cana-760	350	12	and	and	CCONJ
cana-760	350	13	shatawi	shatawi	NOUN
cana-760	350	14	,	,	PUNCT
cana-760	350	15	“	"	PUNCT
cana-760	350	16	fixed	fix	VERB
cana-760	350	17	point	point	NOUN
cana-760	350	18	theorems	theorem	NOUN
cana-760	350	19	on	on	ADP
cana-760	350	20	modular	modular	ADJ
cana-760	350	21	fuzzy	fuzzy	ADJ
cana-760	350	22	metric	metric	ADJ
cana-760	350	23	spaces	space	NOUN
cana-760	350	24	”	"	PUNCT
cana-760	350	25	,	,	PUNCT
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cana-760	350	27	.	.	PROPN
cana-760	350	28	,	,	PUNCT
cana-760	350	29	series	series	PROPN
cana-760	350	30	a	a	NOUN
cana-760	350	31	,	,	PUNCT
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cana-760	350	33	,	,	PUNCT
cana-760	350	34	iss	iss	PROPN
cana-760	350	35	.	.	PROPN
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cana-760	350	37	,	,	PUNCT
cana-760	350	38	2022	2022	NUM
cana-760	350	39	.	.	PUNCT
cana-760	351	1	[	[	X
cana-760	351	2	10	10	NUM
cana-760	351	3	]	]	X
cana-760	351	4	jehad	jehad	PROPN
cana-760	351	5	r	r	PROPN
cana-760	351	6	,	,	PUNCT
cana-760	351	7	madhan	madhan	NOUN
cana-760	351	8	kider	kider	NOUN
cana-760	351	9	“	"	PUNCT
cana-760	351	10	some	some	DET
cana-760	351	11	properties	property	NOUN
cana-760	351	12	of	of	ADP
cana-760	351	13	algebra	algebra	NOUN
cana-760	351	14	fuzzy	fuzzy	ADJ
cana-760	351	15	metric	metric	ADJ
cana-760	351	16	space	space	NOUN
cana-760	351	17	”	"	PUNCT
cana-760	351	18	,	,	PUNCT
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cana-760	351	20	of	of	ADP
cana-760	351	21	al	al	PROPN
cana-760	351	22	-	-	PUNCT
cana-760	351	23	qadisiyah	qadisiyah	NOUN
cana-760	351	24	for	for	ADP
cana-760	351	25	computer	computer	NOUN
cana-760	351	26	science	science	NOUN
cana-760	351	27	and	and	CCONJ
cana-760	351	28	mathematics	mathematic	NOUN
cana-760	351	29	vol.12(2	vol.12(2	ADV
cana-760	351	30	)	)	PUNCT
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cana-760	351	32	,	,	PUNCT
cana-760	351	33	pp	pp	ADP
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cana-760	351	37	.	.	PUNCT
cana-760	352	1	[	[	X
cana-760	352	2	11	11	NUM
cana-760	352	3	]	]	X
cana-760	352	4	jehad	jehad	PROPN
cana-760	352	5	r	r	PROPN
cana-760	352	6	,	,	PUNCT
cana-760	352	7	madhan	madhan	NOUN
cana-760	352	8	kider	kider	NOUN
cana-760	352	9	“	"	PUNCT
cana-760	352	10	application	application	NOUN
cana-760	352	11	of	of	ADP
cana-760	352	12	fixed	fix	VERB
cana-760	352	13	points	point	NOUN
cana-760	352	14	in	in	ADP
cana-760	352	15	algebra	algebra	PROPN
cana-760	352	16	fuzzy	fuzzy	ADJ
cana-760	352	17	normed	norme	VERB
cana-760	352	18	spaces	space	NOUN
cana-760	352	19	”	"	PUNCT
cana-760	352	20	,	,	PUNCT
cana-760	352	21	journal	journal	NOUN
cana-760	352	22	of	of	ADP
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cana-760	352	24	:	:	PUNCT
cana-760	352	25	conference	conference	NOUN
cana-760	352	26	series	series	NOUN
cana-760	352	27	1879	1879	NUM
cana-760	352	28	(	(	PUNCT
cana-760	352	29	2021	2021	NUM
cana-760	352	30	)	)	PUNCT
cana-760	352	31	022099	022099	NUM
cana-760	352	32	,	,	PUNCT
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cana-760	352	34	publishing	publishing	NOUN
cana-760	352	35	.	.	PUNCT
cana-760	353	1	[	[	X
cana-760	353	2	12	12	NUM
cana-760	353	3	]	]	PUNCT
cana-760	353	4	j.	j.	PROPN
cana-760	353	5	kramosil	kramosil	PROPN
cana-760	353	6	,	,	PUNCT
cana-760	353	7	j.	j.	PROPN
cana-760	353	8	michalek	michalek	PROPN
cana-760	353	9	:	:	PUNCT
cana-760	353	10	fuzzy	fuzzy	ADJ
cana-760	353	11	metric	metric	ADJ
cana-760	353	12	and	and	CCONJ
cana-760	353	13	statistical	statistical	ADJ
cana-760	353	14	metric	metric	ADJ
cana-760	353	15	spaces	space	NOUN
cana-760	353	16	.	.	PUNCT
cana-760	354	1	kibernetika.11(1975	kibernetika.11(1975	PROPN
cana-760	354	2	)	)	PUNCT
cana-760	354	3	,	,	PUNCT
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cana-760	355	2	.	.	PUNCT
cana-760	356	1	[	[	X
cana-760	356	2	13	13	NUM
cana-760	356	3	]	]	X
cana-760	356	4	mihet	mihet	PROPN
cana-760	356	5	d.	d.	PROPN
cana-760	356	6	erratum	erratum	PROPN
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cana-760	356	8	“	"	PUNCT
cana-760	356	9	fuzzy	fuzzy	ADJ
cana-760	356	10	y	y	PROPN
cana-760	356	11	-	-	PUNCT
cana-760	356	12	contractive	contractive	ADJ
cana-760	356	13	mappings	mapping	NOUN
cana-760	356	14	in	in	ADP
cana-760	356	15	non	non	ADJ
cana-760	356	16	-	-	ADJ
cana-760	356	17	archimedean	archimedean	ADJ
cana-760	356	18	fuzzy	fuzzy	ADJ
cana-760	356	19	metric	metric	ADJ
cana-760	356	20	spaces	space	NOUN
cana-760	356	21	,	,	PUNCT
cana-760	356	22	fuzzy	fuzzy	ADJ
cana-760	356	23	sets	set	NOUN
cana-760	356	24	and	and	CCONJ
cana-760	356	25	systems	system	NOUN
cana-760	356	26	.	.	PUNCT
cana-760	357	1	159(2008	159(2008	NUM
cana-760	357	2	)	)	PUNCT
cana-760	357	3	,	,	PUNCT
cana-760	357	4	739	739	NUM
cana-760	357	5	-	-	SYM
cana-760	357	6	744	744	NUM
cana-760	357	7	]	]	PUNCT
cana-760	357	8	”	"	PUNCT
cana-760	357	9	.	.	PUNCT
cana-760	358	1	fuzzy	fuzzy	ADJ
cana-760	358	2	sets	set	VERB
cana-760	358	3	syst	syst	PROPN
cana-760	358	4	.	.	PUNCT
cana-760	359	1	2010	2010	NUM
cana-760	359	2	,	,	PUNCT
cana-760	359	3	161	161	NUM
cana-760	359	4	,	,	PUNCT
cana-760	359	5	1150–1151	1150–1151	NOUN
cana-760	359	6	.	.	PUNCT
cana-760	360	1	[	[	X
cana-760	360	2	14	14	NUM
cana-760	360	3	]	]	X
cana-760	360	4	muraliraj	muraliraj	VERB
cana-760	360	5	a	a	DET
cana-760	360	6	,	,	PUNCT
cana-760	360	7	thangathamizh	thangathamizh	ADJ
cana-760	360	8	r	r	NOUN
cana-760	360	9	,	,	PUNCT
cana-760	360	10	popovic	popovic	NOUN
cana-760	360	11	n	n	CCONJ
cana-760	360	12	,	,	PUNCT
cana-760	360	13	savic	savic	PROPN
cana-760	360	14	a	a	X
cana-760	360	15	,	,	PUNCT
cana-760	360	16	radenovic	radenovic	PROPN
cana-760	360	17	s.	s.	PROPN
cana-760	360	18	the	the	DET
cana-760	360	19	first	first	ADJ
cana-760	360	20	rational	rational	ADJ
cana-760	360	21	type	type	NOUN
cana-760	360	22	revisd	revisd	NOUN
cana-760	360	23	fuzzycontractions	fuzzycontraction	NOUN
cana-760	360	24	in	in	ADP
cana-760	360	25	revisd	revisd	NOUN
cana-760	360	26	fuzzy	fuzzy	ADJ
cana-760	360	27	metric	metric	ADJ
cana-760	360	28	spaces	space	NOUN
cana-760	360	29	with	with	ADP
cana-760	360	30	an	an	DET
cana-760	360	31	applications	application	NOUN
cana-760	360	32	.	.	PUNCT
cana-760	361	1	mathematics	mathematic	NOUN
cana-760	361	2	.	.	PUNCT
cana-760	362	1	2023;11(10):2244	2023;11(10):2244	X
cana-760	362	2	.	.	PUNCT
cana-760	363	1	[	[	X
cana-760	363	2	15	15	NUM
cana-760	363	3	]	]	X
cana-760	363	4	muraliraj	muraliraj	NOUN
cana-760	363	5	.	.	PUNCT
cana-760	364	1	a	a	PRON
cana-760	364	2	and	and	CCONJ
cana-760	364	3	thangathamizh	thangathamizh	ADJ
cana-760	364	4	.	.	PUNCT
cana-760	365	1	r	r	X
cana-760	365	2	,	,	PUNCT
cana-760	365	3	“	"	PUNCT
cana-760	365	4	some	some	DET
cana-760	365	5	topological	topological	ADJ
cana-760	365	6	properties	property	NOUN
cana-760	365	7	of	of	ADP
cana-760	365	8	revised	revise	VERB
cana-760	365	9	fuzzy	fuzzy	ADJ
cana-760	365	10	cone	cone	NOUN
cana-760	365	11	metric	metric	ADJ
cana-760	365	12	space	space	NOUN
cana-760	365	13	”	"	PUNCT
cana-760	365	14	,	,	PUNCT
cana-760	365	15	ratio	ratio	PROPN
cana-760	365	16	mathematica	mathematica	PROPN
cana-760	365	17	,	,	PUNCT
cana-760	365	18	volume	volume	NOUN
cana-760	365	19	26	26	NUM
cana-760	365	20	,	,	PUNCT
cana-760	365	21	number	number	NOUN
cana-760	365	22	2	2	NUM
cana-760	365	23	,	,	PUNCT
cana-760	365	24	(	(	PUNCT
cana-760	365	25	2023	2023	NUM
cana-760	365	26	)	)	PUNCT
cana-760	365	27	.	.	PUNCT
cana-760	366	1	[	[	X
cana-760	366	2	16	16	NUM
cana-760	366	3	]	]	X
cana-760	366	4	muraliraj	muraliraj	VERB
cana-760	366	5	a	a	PRON
cana-760	366	6	and	and	CCONJ
cana-760	366	7	thangathamizh	thangathamizh	ADJ
cana-760	366	8	,	,	PUNCT
cana-760	366	9	“	"	PUNCT
cana-760	366	10	new	new	ADJ
cana-760	366	11	relation	relation	NOUN
cana-760	366	12	-	-	PUNCT
cana-760	366	13	theoretic	theoretic	NOUN
cana-760	366	14	fixed	fix	VERB
cana-760	366	15	point	point	NOUN
cana-760	366	16	theorems	theorem	NOUN
cana-760	366	17	in	in	ADP
cana-760	366	18	revised	revise	VERB
cana-760	366	19	fuzzy	fuzzy	ADJ
cana-760	366	20	metric	metric	ADJ
cana-760	366	21	spaces	space	NOUN
cana-760	366	22	with	with	ADP
cana-760	366	23	an	an	DET
cana-760	366	24	application	application	NOUN
cana-760	366	25	to	to	ADP
cana-760	366	26	fractional	fractional	ADJ
cana-760	366	27	differential	differential	ADJ
cana-760	366	28	equations	equation	NOUN
cana-760	366	29	”	"	PUNCT
cana-760	366	30	,	,	PUNCT
cana-760	366	31	communications	communication	NOUN
cana-760	366	32	in	in	ADP
cana-760	366	33	mathematics	mathematic	NOUN
cana-760	366	34	and	and	CCONJ
cana-760	366	35	applications	application	NOUN
cana-760	366	36	,	,	PUNCT
cana-760	366	37	vol.12	vol.12	NOUN
cana-760	366	38	,	,	PUNCT
cana-760	366	39	no	no	DET
cana-760	366	40	22	22	NUM
cana-760	366	41	.	.	PUNCT
cana-760	366	42	(	(	PUNCT
cana-760	366	43	2023	2023	NUM
cana-760	366	44	)	)	PUNCT
cana-760	366	45	.	.	PUNCT
cana-760	367	1	[	[	X
cana-760	367	2	17	17	NUM
cana-760	367	3	]	]	X
cana-760	367	4	muraliraj	muraliraj	PROPN
cana-760	367	5	.	.	PUNCT
cana-760	368	1	a	a	PRON
cana-760	368	2	and	and	CCONJ
cana-760	368	3	thangathamizh	thangathamizh	ADJ
cana-760	368	4	.	.	PUNCT
cana-760	369	1	r	r	X
cana-760	369	2	,	,	PUNCT
cana-760	369	3	“	"	PUNCT
cana-760	369	4	fixed	fix	VERB
cana-760	369	5	point	point	NOUN
cana-760	369	6	theorems	theorem	NOUN
cana-760	369	7	in	in	ADP
cana-760	369	8	revised	revise	VERB
cana-760	369	9	fuzzy	fuzzy	ADJ
cana-760	369	10	metric	metric	ADJ
cana-760	369	11	space	space	NOUN
cana-760	369	12	”	"	PUNCT
cana-760	369	13	,	,	PUNCT
cana-760	369	14	advances	advance	NOUN
cana-760	369	15	in	in	ADP
cana-760	369	16	fuzzy	fuzzy	ADJ
cana-760	369	17	sets	set	NOUN
cana-760	369	18	and	and	CCONJ
cana-760	369	19	systems	system	NOUN
cana-760	369	20	,	,	PUNCT
cana-760	369	21	volume	volume	NOUN
cana-760	369	22	26	26	NUM
cana-760	369	23	,	,	PUNCT
cana-760	369	24	number	number	NOUN
cana-760	369	25	2	2	NUM
cana-760	369	26	,	,	PUNCT
cana-760	369	27	2021	2021	NUM
cana-760	369	28	.	.	PUNCT
cana-760	370	1	[	[	X
cana-760	370	2	18	18	NUM
cana-760	370	3	]	]	PUNCT
cana-760	370	4	a.	a.	NOUN
cana-760	370	5	muraliraj	muraliraj	PROPN
cana-760	370	6	.	.	PUNCT
cana-760	371	1	and	and	CCONJ
cana-760	371	2	r.	r.	PROPN
cana-760	371	3	thangathamizh	thangathamizh	PROPN
cana-760	371	4	,	,	PUNCT
cana-760	371	5	“	"	PUNCT
cana-760	371	6	introduction	introduction	NOUN
cana-760	371	7	on	on	ADP
cana-760	371	8	revised	revise	VERB
cana-760	371	9	fuzzy	fuzzy	ADJ
cana-760	371	10	modular	modular	ADJ
cana-760	371	11	spaces	space	NOUN
cana-760	371	12	”	"	PUNCT
cana-760	371	13	,	,	PUNCT
cana-760	371	14	issn	issn	PROPN
cana-760	371	15	0973	0973	NUM
cana-760	371	16	-	-	SYM
cana-760	371	17	1768	1768	NUM
cana-760	371	18	volume	volume	NOUN
cana-760	371	19	17	17	NUM
cana-760	371	20	,	,	PUNCT
cana-760	371	21	number	number	NOUN
cana-760	371	22	2	2	NUM
cana-760	371	23	(	(	PUNCT
cana-760	371	24	2021	2021	NUM
cana-760	371	25	)	)	PUNCT
cana-760	371	26	,	,	PUNCT
cana-760	371	27	pp	pp	ADP
cana-760	371	28	.	.	PUNCT
cana-760	372	1	303	303	NUM
cana-760	372	2	-	-	SYM
cana-760	372	3	317	317	NUM
cana-760	372	4	.	.	PUNCT
cana-760	373	1	[	[	X
cana-760	373	2	19	19	NUM
cana-760	373	3	]	]	X
cana-760	373	4	muraliraj	muraliraj	VERB
cana-760	373	5	a	a	PRON
cana-760	373	6	and	and	CCONJ
cana-760	373	7	shanmugavel	shanmugavel	NOUN
cana-760	373	8	p	p	X
cana-760	373	9	,	,	PUNCT
cana-760	373	10	thangathamizh	thangathamizh	ADJ
cana-760	373	11	r	r	NOUN
cana-760	373	12	“	"	PUNCT
cana-760	373	13	existence	existence	NOUN
cana-760	373	14	of	of	ADP
cana-760	373	15	fixed	fix	VERB
cana-760	373	16	point	point	NOUN
cana-760	373	17	theorems	theorem	NOUN
cana-760	373	18	in	in	ADP
cana-760	373	19	revised	revise	VERB
cana-760	373	20	fuzzy	fuzzy	ADJ
cana-760	373	21	modular	modular	ADJ
cana-760	373	22	spaces	space	NOUN
cana-760	373	23	”	"	PUNCT
cana-760	373	24	,	,	PUNCT
cana-760	373	25	advances	advance	NOUN
cana-760	373	26	in	in	ADP
cana-760	373	27	nonlinear	nonlinear	ADJ
cana-760	373	28	variational	variational	ADJ
cana-760	373	29	inequalities	inequality	NOUN
cana-760	373	30	,	,	PUNCT
cana-760	373	31	2024	2024	NUM
cana-760	373	32	.	.	PUNCT
cana-760	374	1	(	(	PUNCT
cana-760	374	2	accepted	accept	VERB
cana-760	374	3	for	for	ADP
cana-760	374	4	publication	publication	NOUN
cana-760	374	5	)	)	PUNCT
cana-760	375	1	[	[	X
cana-760	375	2	20	20	NUM
cana-760	375	3	]	]	PUNCT
cana-760	375	4	thangathamizh	thangathamizh	PROPN
cana-760	375	5	r	r	NOUN
cana-760	375	6	,	,	PUNCT
cana-760	375	7	muraliraj	muraliraj	VERB
cana-760	375	8	a	a	PRON
cana-760	375	9	and	and	CCONJ
cana-760	375	10	shanmugavel	shanmugavel	NOUN
cana-760	375	11	p	p	X
cana-760	375	12	“	"	PUNCT
cana-760	375	13	new	new	ADJ
cana-760	375	14	approach	approach	NOUN
cana-760	375	15	of	of	ADP
cana-760	375	16	lebesgue	lebesgue	PROPN
cana-760	375	17	integral	integral	ADJ
cana-760	375	18	in	in	ADP
cana-760	375	19	revised	revise	VERB
cana-760	375	20	fuzzy	fuzzy	ADJ
cana-760	375	21	cone	cone	NOUN
cana-760	375	22	metric	metric	ADJ
cana-760	375	23	spaces	space	NOUN
cana-760	375	24	vie	vie	X
cana-760	375	25	unique	unique	ADJ
cana-760	375	26	coupled	couple	VERB
cana-760	375	27	fixed	fix	VERB
cana-760	375	28	point	point	NOUN
cana-760	375	29	theorems	theorem	NOUN
cana-760	375	30	”	"	PUNCT
cana-760	375	31	,	,	PUNCT
cana-760	375	32	military	military	ADJ
cana-760	375	33	technical	technical	ADJ
cana-760	375	34	courier	courier	NOUN
cana-760	375	35	,	,	PUNCT
cana-760	375	36	3	3	NUM
cana-760	375	37	,	,	PUNCT
cana-760	375	38	2024	2024	NUM
cana-760	375	39	.	.	PUNCT
cana-760	376	1	doi	doi	NOUN
cana-760	376	2	:	:	PUNCT
cana-760	376	3	10.5937	10.5937	NUM
cana-760	376	4	/	/	SYM
cana-760	376	5	vojtechg72	vojtechg72	NOUN
cana-760	376	6	-	-	PUNCT
cana-760	376	7	48816	48816	NUM
cana-760	376	8	.	.	PUNCT
cana-760	377	1	[	[	X
cana-760	377	2	21	21	NUM
cana-760	377	3	]	]	X
cana-760	377	4	thangathamizh	thangathamizh	PROPN
cana-760	377	5	r	r	NOUN
cana-760	377	6	,	,	PUNCT
cana-760	377	7	balamurugan	balamurugan	VERB
cana-760	377	8	k	k	PROPN
cana-760	377	9	,	,	PUNCT
cana-760	377	10	karnan	karnan	PROPN
cana-760	377	11	c	c	NOUN
cana-760	377	12	,	,	PUNCT
cana-760	377	13	shanmugavel	shanmugavel	NOUN
cana-760	377	14	p	p	NOUN
cana-760	377	15	,	,	PUNCT
cana-760	377	16	and	and	CCONJ
cana-760	377	17	balraj	balraj	X
cana-760	377	18	d	d	NOUN
cana-760	377	19	,	,	PUNCT
cana-760	377	20	“	"	PUNCT
cana-760	377	21	revised	revise	VERB
cana-760	377	22	fuzzy	fuzzy	ADJ
cana-760	377	23	differential	differential	ADJ
cana-760	377	24	equations	equation	NOUN
cana-760	377	25	using	use	VERB
cana-760	377	26	weakly	weakly	ADJ
cana-760	377	27	compatible	compatible	ADJ
cana-760	377	28	self	self	NOUN
cana-760	377	29	-	-	PUNCT
cana-760	377	30	mappings	mapping	NOUN
cana-760	377	31	in	in	ADP
cana-760	377	32	revised	revise	VERB
cana-760	377	33	fuzzy	fuzzy	ADJ
cana-760	377	34	metric	metric	ADJ
cana-760	377	35	spaces	space	NOUN
cana-760	377	36	”	"	PUNCT
cana-760	377	37	,	,	PUNCT
cana-760	377	38	advances	advance	NOUN
cana-760	377	39	in	in	ADP
cana-760	377	40	nonlinear	nonlinear	ADJ
cana-760	377	41	variational	variational	ADJ
cana-760	377	42	inequalities	inequality	NOUN
cana-760	377	43	,	,	PUNCT
cana-760	377	44	2024	2024	NUM
cana-760	377	45	.	.	PUNCT
cana-760	378	1	(	(	PUNCT
cana-760	378	2	accepted	accept	VERB
cana-760	378	3	for	for	ADP
cana-760	378	4	publication	publication	NOUN
cana-760	378	5	)	)	PUNCT
cana-760	379	1	[	[	X
cana-760	379	2	22	22	NUM
cana-760	379	3	]	]	X
cana-760	379	4	rhoades	rhoade	NOUN
cana-760	379	5	.	.	PUNCT
cana-760	380	1	e.b	e.b	PROPN
cana-760	380	2	,	,	PUNCT
cana-760	380	3	a	a	DET
cana-760	380	4	comparison	comparison	NOUN
cana-760	380	5	of	of	ADP
cana-760	380	6	various	various	ADJ
cana-760	380	7	definitions	definition	NOUN
cana-760	380	8	of	of	ADP
cana-760	380	9	contractive	contractive	ADJ
cana-760	380	10	mappings	mapping	NOUN
cana-760	380	11	,	,	PUNCT
cana-760	380	12	trans	trans	PROPN
cana-760	380	13	.	.	PROPN
cana-760	381	1	amer	amer	PROPN
cana-760	381	2	.	.	PUNCT
cana-760	381	3	math	math	PROPN
cana-760	381	4	.	.	PUNCT
cana-760	382	1	soc	soc	PROPN
cana-760	382	2	.	.	PUNCT
cana-760	383	1	226	226	NUM
cana-760	383	2	(	(	PUNCT
cana-760	383	3	1977	1977	NUM
cana-760	383	4	)	)	PUNCT
cana-760	383	5	,	,	PUNCT
cana-760	383	6	257	257	NUM
cana-760	383	7	-	-	SYM
cana-760	383	8	290	290	NUM
cana-760	383	9	.	.	PUNCT
cana-760	384	1	[	[	X
cana-760	384	2	23	23	NUM
cana-760	384	3	]	]	PUNCT
cana-760	384	4	b.	b.	PROPN
cana-760	384	5	schweizer	schweizer	PROPN
cana-760	384	6	,	,	PUNCT
cana-760	384	7	a.	a.	NOUN
cana-760	384	8	sklar	sklar	NOUN
cana-760	384	9	:	:	PUNCT
cana-760	384	10	statistical	statistical	ADJ
cana-760	384	11	metric	metric	ADJ
cana-760	384	12	spaces	space	NOUN
cana-760	384	13	.	.	PUNCT
cana-760	385	1	pacific	pacific	PROPN
cana-760	385	2	j.	j.	PROPN
cana-760	385	3	math	math	PROPN
cana-760	385	4	.	.	PUNCT
cana-760	386	1	10(1960	10(1960	NUM
cana-760	386	2	)	)	PUNCT
cana-760	386	3	,	,	PUNCT
cana-760	387	1	314–334	314–334	NUM
cana-760	387	2	.	.	PUNCT
cana-760	388	1	[	[	X
cana-760	388	2	24	24	NUM
cana-760	388	3	]	]	PUNCT
cana-760	388	4	l.	l.	PROPN
cana-760	388	5	a.	a.	PROPN
cana-760	388	6	zadeh	zadeh	PROPN
cana-760	388	7	:	:	PUNCT
cana-760	388	8	fuzzy	fuzzy	ADJ
cana-760	388	9	sets	set	NOUN
cana-760	388	10	.	.	PUNCT
cana-760	389	1	inf	inf	PROPN
cana-760	389	2	.	.	PUNCT
cana-760	389	3	control	control	PROPN
cana-760	389	4	.	.	PUNCT
cana-760	389	5	,	,	PUNCT
cana-760	389	6	8(1965	8(1965	NUM
cana-760	389	7	)	)	PUNCT
cana-760	389	8	,	,	PUNCT
cana-760	389	9	338–353	338–353	NUM
cana-760	389	10	.	.	PUNCT
cana-760	390	1	communications	communication	NOUN
cana-760	390	2	on	on	ADP
cana-760	390	3	applied	apply	VERB
cana-760	390	4	nonlinear	nonlinear	ADJ
cana-760	390	5	analysis	analysis	NOUN
cana-760	390	6	issn	issn	NOUN
cana-760	390	7	:	:	PUNCT
cana-760	390	8	1074	1074	NUM
cana-760	390	9	-	-	PUNCT
cana-760	390	10	133x	133x	NUM
cana-760	390	11	vol	vol	NOUN
cana-760	390	12	31	31	NUM
cana-760	390	13	no	no	NOUN
cana-760	390	14	.	.	PUNCT
cana-760	391	1	3s	3s	NUM
cana-760	391	2	(	(	PUNCT
cana-760	391	3	2024	2024	NUM
cana-760	391	4	)	)	PUNCT
cana-760	391	5	226	226	NUM
cana-760	391	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-760	392	1	[	[	X
cana-760	392	2	25	25	NUM
cana-760	392	3	]	]	X
cana-760	392	4	muraliraj	muraliraj	VERB
cana-760	392	5	a	a	PRON
cana-760	392	6	and	and	CCONJ
cana-760	392	7	shanmugavel	shanmugavel	NOUN
cana-760	392	8	p	p	X
cana-760	392	9	,	,	PUNCT
cana-760	392	10	thangathamizh	thangathamizh	ADJ
cana-760	392	11	r	r	NOUN
cana-760	392	12	“	"	PUNCT
cana-760	392	13	existence	existence	NOUN
cana-760	392	14	of	of	ADP
cana-760	392	15	fixed	fix	VERB
cana-760	392	16	point	point	NOUN
cana-760	392	17	theorems	theorem	NOUN
cana-760	392	18	in	in	ADP
cana-760	392	19	revised	revise	VERB
cana-760	392	20	fuzzy	fuzzy	ADJ
cana-760	392	21	modular	modular	ADJ
cana-760	392	22	spaces	space	NOUN
cana-760	392	23	”	"	PUNCT
cana-760	392	24	,	,	PUNCT
cana-760	392	25	advances	advance	NOUN
cana-760	392	26	in	in	ADP
cana-760	392	27	nonlinear	nonlinear	ADJ
cana-760	392	28	variational	variational	ADJ
cana-760	392	29	inequalities	inequality	NOUN
cana-760	392	30	,	,	PUNCT
cana-760	392	31	2024	2024	NUM
cana-760	392	32	.	.	PUNCT
cana-760	393	1	(	(	PUNCT
cana-760	393	2	accepted	accept	VERB
cana-760	393	3	for	for	ADP
cana-760	393	4	publication	publication	NOUN
cana-760	393	5	)	)	PUNCT
cana-760	394	1	[	[	X
cana-760	394	2	26	26	NUM
cana-760	394	3	]	]	PUNCT
cana-760	394	4	thangathamizh	thangathamizh	PROPN
cana-760	394	5	r	r	NOUN
cana-760	394	6	,	,	PUNCT
cana-760	394	7	abdelhamid	abdelhamid	ADP
cana-760	394	8	moussaoui	moussaoui	NOUN
cana-760	394	9	,	,	PUNCT
cana-760	394	10	tatjana	tatjana	PROPN
cana-760	394	11	dosenovic	dosenovic	PROPN
cana-760	394	12	,	,	PUNCT
cana-760	394	13	stojan	stojan	ADP
cana-760	394	14	radenovic	radenovic	PROPN
cana-760	394	15	“	"	PUNCT
cana-760	394	16	fixed	fix	VERB
cana-760	394	17	point	point	NOUN
cana-760	394	18	results	result	NOUN
cana-760	394	19	in	in	ADP
cana-760	394	20	controlled	control	VERB
cana-760	394	21	revised	revise	VERB
cana-760	394	22	fuzzy	fuzzy	ADJ
cana-760	394	23	metric	metric	ADJ
cana-760	394	24	spaces	space	NOUN
cana-760	394	25	with	with	ADP
cana-760	394	26	an	an	DET
cana-760	394	27	application	application	NOUN
cana-760	394	28	to	to	ADP
cana-760	394	29	the	the	DET
cana-760	394	30	transformation	transformation	NOUN
cana-760	394	31	of	of	ADP
cana-760	394	32	solar	solar	ADJ
cana-760	394	33	energy	energy	NOUN
cana-760	394	34	to	to	ADP
cana-760	394	35	electric	electric	ADJ
cana-760	394	36	power	power	NOUN
cana-760	394	37	”	"	PUNCT
cana-760	394	38	,	,	PUNCT
cana-760	394	39	military	military	ADJ
cana-760	394	40	technical	technical	ADJ
cana-760	394	41	courier	courier	NOUN
cana-760	394	42	,	,	PUNCT
cana-760	394	43	2024	2024	NUM
cana-760	394	44	.	.	PUNCT
cana-760	395	1	doi.10.5937	doi.10.5937	NOUN
cana-760	395	2	/	/	SYM
cana-760	395	3	vojtehg72	vojtehg72	NOUN
cana-760	395	4	-	-	PUNCT
cana-760	395	5	49064	49064	NUM
cana-760	395	6	.	.	PUNCT
cana-760	396	1	[	[	X
cana-760	396	2	27	27	NUM
cana-760	396	3	]	]	PUNCT
cana-760	396	4	parakath	parakath	PROPN
cana-760	396	5	nisha	nisha	PROPN
cana-760	396	6	bagam	bagam	PROPN
cana-760	396	7	p	p	PROPN
cana-760	396	8	,	,	PUNCT
cana-760	396	9	sandhya	sandhya	PROPN
cana-760	396	10	p	p	PROPN
cana-760	396	11	,	,	PUNCT
cana-760	396	12	thangathamizh	thangathamizh	ADJ
cana-760	396	13	r	r	NOUN
cana-760	396	14	,	,	PUNCT
cana-760	396	15	shanmugavel	shanmugavel	NOUN
cana-760	396	16	p	p	NOUN
cana-760	396	17	,	,	PUNCT
cana-760	396	18	sarathbabu	sarathbabu	PROPN
cana-760	396	19	k	k	NOUN
cana-760	396	20	,	,	PUNCT
cana-760	396	21	anusuya	anusuya	PROPN
cana-760	396	22	r	r	NOUN
cana-760	396	23	,	,	PUNCT
cana-760	396	24	“	"	PUNCT
cana-760	396	25	fixed	fix	VERB
cana-760	396	26	point	point	NOUN
cana-760	396	27	theorems	theorem	NOUN
cana-760	396	28	in	in	ADP
cana-760	396	29	revised	revise	VERB
cana-760	396	30	fuzzy	fuzzy	ADJ
cana-760	396	31	metric	metric	ADJ
cana-760	396	32	space	space	NOUN
cana-760	396	33	via	via	ADP
cana-760	396	34	𝑅𝐹	𝑅𝐹	PROPN
cana-760	396	35	−contraction	−contraction	PROPN
cana-760	396	36	”	"	PUNCT
cana-760	396	37	,	,	PUNCT
cana-760	396	38	communications	communication	NOUN
cana-760	396	39	on	on	ADP
cana-760	396	40	applied	apply	VERB
cana-760	396	41	nonlinear	nonlinear	ADJ
cana-760	396	42	analysis	analysis	NOUN
cana-760	396	43	,	,	PUNCT
cana-760	396	44	2024	2024	NUM
cana-760	396	45	.	.	PUNCT
cana-760	397	1	(	(	PUNCT
cana-760	397	2	accepted	accept	VERB
cana-760	397	3	for	for	ADP
cana-760	397	4	publication	publication	NOUN
cana-760	397	5	)	)	PUNCT
cana-760	397	6	.	.	PUNCT