id	sid	tid	token	lemma	pos
cana-2867	1	1	communications	communication	NOUN
cana-2867	1	2	on	on	ADP
cana-2867	1	3	applied	apply	VERB
cana-2867	1	4	nonlinear	nonlinear	ADJ
cana-2867	1	5	analysis	analysis	NOUN
cana-2867	1	6	issn	issn	NOUN
cana-2867	1	7	:	:	PUNCT
cana-2867	1	8	1074	1074	NUM
cana-2867	1	9	-	-	PUNCT
cana-2867	1	10	133x	133x	NUM
cana-2867	1	11	vol	vol	NOUN
cana-2867	1	12	32	32	NUM
cana-2867	1	13	no	no	NOUN
cana-2867	1	14	.	.	PUNCT
cana-2867	2	1	4s	4s	NUM
cana-2867	2	2	(	(	PUNCT
cana-2867	2	3	2025	2025	NUM
cana-2867	2	4	)	)	PUNCT
cana-2867	3	1	500	500	NUM
cana-2867	3	2	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-2867	3	3	multi	multi	X
cana-2867	3	4	fuzzy	fuzzy	ADJ
cana-2867	3	5	iterated	iterate	VERB
cana-2867	3	6	function	function	NOUN
cana-2867	3	7	system	system	NOUN
cana-2867	3	8	consisting	consist	VERB
cana-2867	3	9	of	of	ADP
cana-2867	3	10	generalized	generalized	ADJ
cana-2867	3	11	fuzzy	fuzzy	ADJ
cana-2867	3	12	contraction	contraction	NOUN
cana-2867	3	13	mapping	mapping	NOUN
cana-2867	3	14	in	in	ADP
cana-2867	3	15	fuzzy	fuzzy	ADJ
cana-2867	3	16	metric	metric	ADJ
cana-2867	3	17	spaces	space	NOUN
cana-2867	3	18	s.	s.	PROPN
cana-2867	3	19	c.	c.	PROPN
cana-2867	3	20	shrivastava1	shrivastava1	PROPN
cana-2867	3	21	,	,	PUNCT
cana-2867	3	22	r.	r.	PROPN
cana-2867	3	23	shrivastava2	shrivastava2	PROPN
cana-2867	3	24	,	,	PUNCT
cana-2867	3	25	jitendra	jitendra	PROPN
cana-2867	3	26	kumar1	kumar1	PROPN
cana-2867	4	1	1department	1department	NUM
cana-2867	4	2	of	of	ADP
cana-2867	4	3	applied	apply	VERB
cana-2867	4	4	mathematics	mathematic	NOUN
cana-2867	4	5	,	,	PUNCT
cana-2867	4	6	rungta	rungta	PROPN
cana-2867	4	7	college	college	NOUN
cana-2867	4	8	of	of	ADP
cana-2867	4	9	engineering	engineering	PROPN
cana-2867	4	10	&	&	CCONJ
cana-2867	4	11	technology	technology	PROPN
cana-2867	4	12	,	,	PUNCT
cana-2867	4	13	bhilai	bhilai	PROPN
cana-2867	4	14	,	,	PUNCT
cana-2867	4	15	(	(	PUNCT
cana-2867	4	16	c.g	c.g	PROPN
cana-2867	4	17	.	.	PROPN
cana-2867	4	18	)	)	PUNCT
cana-2867	4	19	,	,	PUNCT
cana-2867	4	20	india	india	PROPN
cana-2867	4	21	.	.	PUNCT
cana-2867	5	1	2department	2department	NUM
cana-2867	5	2	of	of	ADP
cana-2867	5	3	physics	physics	NOUN
cana-2867	5	4	,	,	PUNCT
cana-2867	5	5	govt	govt	PROPN
cana-2867	5	6	.	.	PUNCT
cana-2867	6	1	naveen	naveen	PROPN
cana-2867	6	2	college	college	PROPN
cana-2867	6	3	,	,	PUNCT
cana-2867	6	4	risali	risali	VERB
cana-2867	6	5	,	,	PUNCT
cana-2867	6	6	(	(	PUNCT
cana-2867	6	7	c.g.),india	c.g.),india	PROPN
cana-2867	6	8	.	.	PUNCT
cana-2867	7	1	article	article	PROPN
cana-2867	7	2	history	history	NOUN
cana-2867	7	3	:	:	PUNCT
cana-2867	7	4	received	receive	VERB
cana-2867	7	5	:	:	PUNCT
cana-2867	7	6	30	30	NUM
cana-2867	7	7	-	-	SYM
cana-2867	7	8	09	09	NUM
cana-2867	7	9	-	-	PUNCT
cana-2867	7	10	2024	2024	NUM
cana-2867	7	11	revised	revise	VERB
cana-2867	7	12	:	:	PUNCT
cana-2867	7	13	28	28	NUM
cana-2867	7	14	-	-	SYM
cana-2867	7	15	11	11	NUM
cana-2867	7	16	-	-	PUNCT
cana-2867	7	17	2024	2024	NUM
cana-2867	7	18	accepted	accept	VERB
cana-2867	7	19	:	:	PUNCT
cana-2867	7	20	09	09	NUM
cana-2867	7	21	-	-	SYM
cana-2867	7	22	12	12	NUM
cana-2867	7	23	-	-	PUNCT
cana-2867	7	24	2024	2024	NUM
cana-2867	7	25	abstract	abstract	NOUN
cana-2867	7	26	:	:	PUNCT
cana-2867	7	27	the	the	DET
cana-2867	7	28	research	research	NOUN
cana-2867	7	29	work	work	NOUN
cana-2867	7	30	in	in	ADP
cana-2867	7	31	this	this	DET
cana-2867	7	32	paper	paper	NOUN
cana-2867	7	33	combines	combine	VERB
cana-2867	7	34	two	two	NUM
cana-2867	7	35	concepts	concept	NOUN
cana-2867	7	36	iterated	iterate	VERB
cana-2867	7	37	function	function	NOUN
cana-2867	7	38	system	system	NOUN
cana-2867	7	39	and	and	CCONJ
cana-2867	7	40	multi	multi	ADJ
cana-2867	7	41	fuzzy	fuzzy	ADJ
cana-2867	7	42	fractal	fractal	ADJ
cana-2867	7	43	theory	theory	NOUN
cana-2867	7	44	in	in	ADP
cana-2867	7	45	the	the	DET
cana-2867	7	46	formulation	formulation	NOUN
cana-2867	7	47	of	of	ADP
cana-2867	7	48	a	a	DET
cana-2867	7	49	new	new	ADJ
cana-2867	7	50	iterated	iterated	ADJ
cana-2867	7	51	function	function	NOUN
cana-2867	7	52	system	system	NOUN
cana-2867	7	53	,	,	PUNCT
cana-2867	7	54	multi	multi	X
cana-2867	7	55	fuzzy	fuzzy	ADJ
cana-2867	7	56	fractal	fractal	ADJ
cana-2867	7	57	iterated	iterate	VERB
cana-2867	7	58	function	function	NOUN
cana-2867	7	59	system	system	NOUN
cana-2867	7	60	.	.	PUNCT
cana-2867	8	1	more	more	ADV
cana-2867	8	2	precisely	precisely	ADV
cana-2867	8	3	,	,	PUNCT
cana-2867	8	4	some	some	DET
cana-2867	8	5	uniqueness	uniqueness	NOUN
cana-2867	8	6	and	and	CCONJ
cana-2867	8	7	existence	existence	NOUN
cana-2867	8	8	of	of	ADP
cana-2867	8	9	the	the	DET
cana-2867	8	10	attractor	attractor	NOUN
cana-2867	8	11	were	be	AUX
cana-2867	8	12	obtained	obtain	VERB
cana-2867	8	13	in	in	ADP
cana-2867	8	14	the	the	DET
cana-2867	8	15	new	new	ADJ
cana-2867	8	16	multi	multi	ADJ
cana-2867	8	17	fuzzy	fuzzy	ADJ
cana-2867	8	18	fractal	fractal	ADJ
cana-2867	8	19	spaces	space	NOUN
cana-2867	8	20	.	.	PUNCT
cana-2867	9	1	some	some	DET
cana-2867	9	2	illustrative	illustrative	ADJ
cana-2867	9	3	examples	example	NOUN
cana-2867	9	4	are	be	AUX
cana-2867	9	5	given	give	VERB
cana-2867	9	6	that	that	PRON
cana-2867	9	7	will	will	AUX
cana-2867	9	8	be	be	AUX
cana-2867	9	9	useful	useful	ADJ
cana-2867	9	10	for	for	ADP
cana-2867	9	11	this	this	DET
cana-2867	9	12	research	research	NOUN
cana-2867	9	13	purpose	purpose	NOUN
cana-2867	9	14	.	.	PUNCT
cana-2867	10	1	from	from	ADP
cana-2867	10	2	then	then	ADV
cana-2867	10	3	on	on	ADV
cana-2867	10	4	,	,	PUNCT
cana-2867	10	5	our	our	PRON
cana-2867	10	6	results	result	NOUN
cana-2867	10	7	are	be	AUX
cana-2867	10	8	generalizations	generalization	NOUN
cana-2867	10	9	of	of	ADP
cana-2867	10	10	some	some	DET
cana-2867	10	11	recent	recent	ADJ
cana-2867	10	12	results	result	NOUN
cana-2867	10	13	provided	provide	VERB
cana-2867	10	14	in	in	ADP
cana-2867	10	15	the	the	DET
cana-2867	10	16	literature	literature	NOUN
cana-2867	10	17	.	.	PUNCT
cana-2867	11	1	we	we	PRON
cana-2867	11	2	define	define	VERB
cana-2867	11	3	a	a	DET
cana-2867	11	4	methodology	methodology	NOUN
cana-2867	11	5	for	for	ADP
cana-2867	11	6	the	the	DET
cana-2867	11	7	construction	construction	NOUN
cana-2867	11	8	of	of	ADP
cana-2867	11	9	multi	multi	X
cana-2867	11	10	fuzzy	fuzzy	ADJ
cana-2867	11	11	ifs	ifs	PROPN
cana-2867	11	12	in	in	ADP
cana-2867	11	13	multi	multi	X
cana-2867	11	14	fuzzy	fuzzy	ADJ
cana-2867	11	15	fractal	fractal	ADJ
cana-2867	11	16	space	space	NOUN
cana-2867	11	17	consisting	consist	VERB
cana-2867	11	18	of	of	ADP
cana-2867	11	19	a	a	DET
cana-2867	11	20	generalized	generalized	ADJ
cana-2867	11	21	contraction	contraction	NOUN
cana-2867	11	22	mapping	mapping	NOUN
cana-2867	11	23	.	.	PUNCT
cana-2867	12	1	keywords	keyword	NOUN
cana-2867	12	2	:	:	PUNCT
cana-2867	12	3	𝓗–contraction	𝓗–contraction	NOUN
cana-2867	12	4	;	;	PUNCT
cana-2867	12	5	attractor	attractor	NOUN
cana-2867	12	6	;	;	PUNCT
cana-2867	12	7	iterated	iterate	VERB
cana-2867	12	8	function	function	NOUN
cana-2867	12	9	system	system	NOUN
cana-2867	12	10	;	;	PUNCT
cana-2867	12	11	multi	multi	X
cana-2867	12	12	iterated	iterated	ADJ
cana-2867	12	13	function	function	NOUN
cana-2867	12	14	system	system	NOUN
cana-2867	12	15	;	;	PUNCT
cana-2867	12	16	multi	multi	X
cana-2867	12	17	fuzzy	fuzzy	ADJ
cana-2867	12	18	fractal	fractal	ADJ
cana-2867	12	19	space	space	NOUN
cana-2867	12	20	.	.	PUNCT
cana-2867	13	1	1	1	X
cana-2867	13	2	.	.	X
cana-2867	13	3	introduction	introduction	NOUN
cana-2867	13	4	a	a	DET
cana-2867	13	5	fuzzy	fuzzy	ADJ
cana-2867	13	6	set	set	NOUN
cana-2867	13	7	was	be	AUX
cana-2867	13	8	invented	invent	VERB
cana-2867	13	9	with	with	ADP
cana-2867	13	10	a	a	DET
cana-2867	13	11	new	new	ADJ
cana-2867	13	12	approach	approach	NOUN
cana-2867	13	13	in	in	ADP
cana-2867	13	14	1965	1965	NUM
cana-2867	13	15	by	by	ADP
cana-2867	13	16	zadeh	zadeh	PROPN
cana-2867	13	17	[	[	X
cana-2867	13	18	1	1	NUM
cana-2867	13	19	]	]	PUNCT
cana-2867	13	20	as	as	ADP
cana-2867	13	21	a	a	DET
cana-2867	13	22	development	development	NOUN
cana-2867	13	23	to	to	ADP
cana-2867	13	24	the	the	DET
cana-2867	13	25	standard	standard	ADJ
cana-2867	13	26	concept	concept	NOUN
cana-2867	13	27	of	of	ADP
cana-2867	13	28	sets	set	NOUN
cana-2867	13	29	and	and	CCONJ
cana-2867	13	30	to	to	PART
cana-2867	13	31	understand	understand	VERB
cana-2867	13	32	the	the	DET
cana-2867	13	33	degree	degree	NOUN
cana-2867	13	34	of	of	ADP
cana-2867	13	35	uncertainty	uncertainty	NOUN
cana-2867	13	36	in	in	ADP
cana-2867	13	37	daily	daily	ADJ
cana-2867	13	38	life	life	NOUN
cana-2867	13	39	in	in	ADP
cana-2867	13	40	purely	purely	ADV
cana-2867	13	41	mathematical	mathematical	ADJ
cana-2867	13	42	style	style	NOUN
cana-2867	13	43	.	.	PUNCT
cana-2867	14	1	kramosil	kramosil	NOUN
cana-2867	14	2	and	and	CCONJ
cana-2867	14	3	machalek	machalek	NOUN
cana-2867	15	1	[	[	X
cana-2867	15	2	2	2	X
cana-2867	15	3	]	]	PUNCT
cana-2867	15	4	estimated	estimate	VERB
cana-2867	15	5	the	the	DET
cana-2867	15	6	uncertainty	uncertainty	NOUN
cana-2867	15	7	in	in	ADP
cana-2867	15	8	measuring	measure	VERB
cana-2867	15	9	the	the	DET
cana-2867	15	10	distance	distance	NOUN
cana-2867	15	11	between	between	ADP
cana-2867	15	12	two	two	NUM
cana-2867	15	13	sets	set	NOUN
cana-2867	15	14	.	.	PUNCT
cana-2867	16	1	george	george	PROPN
cana-2867	16	2	and	and	CCONJ
cana-2867	16	3	veeramani	veeramani	NOUN
cana-2867	17	1	[	[	X
cana-2867	17	2	3	3	NUM
cana-2867	17	3	,	,	PUNCT
cana-2867	17	4	4	4	NUM
cana-2867	17	5	]	]	PUNCT
cana-2867	17	6	improved	improve	VERB
cana-2867	17	7	their	their	PRON
cana-2867	17	8	hypothesis	hypothesis	NOUN
cana-2867	17	9	by	by	ADP
cana-2867	17	10	defining	define	VERB
cana-2867	17	11	the	the	DET
cana-2867	17	12	best	good	ADJ
cana-2867	17	13	circumstances	circumstance	NOUN
cana-2867	17	14	in	in	ADP
cana-2867	17	15	the	the	DET
cana-2867	17	16	same	same	ADJ
cana-2867	17	17	setting	setting	NOUN
cana-2867	17	18	of	of	ADP
cana-2867	17	19	the	the	DET
cana-2867	17	20	space	space	NOUN
cana-2867	17	21	as	as	ADP
cana-2867	17	22	one	one	NUM
cana-2867	17	23	way	way	NOUN
cana-2867	17	24	to	to	PART
cana-2867	17	25	get	get	VERB
cana-2867	17	26	a	a	DET
cana-2867	17	27	countable	countable	ADJ
cana-2867	17	28	and	and	CCONJ
cana-2867	17	29	hausdorff	hausdorff	NOUN
cana-2867	17	30	topology	topology	NOUN
cana-2867	17	31	.	.	PUNCT
cana-2867	18	1	grabiec	grabiec	PROPN
cana-2867	18	2	[	[	X
cana-2867	18	3	5	5	NUM
cana-2867	18	4	]	]	X
cana-2867	18	5	generalized	generalize	VERB
cana-2867	18	6	and	and	CCONJ
cana-2867	18	7	celebrated	celebrate	VERB
cana-2867	18	8	the	the	DET
cana-2867	18	9	theory	theory	NOUN
cana-2867	18	10	of	of	ADP
cana-2867	18	11	banach	banach	NOUN
cana-2867	18	12	contraction	contraction	NOUN
cana-2867	18	13	for	for	ADP
cana-2867	18	14	fuzzy	fuzzy	ADJ
cana-2867	18	15	metric	metric	ADJ
cana-2867	18	16	space	space	NOUN
cana-2867	18	17	.	.	PUNCT
cana-2867	19	1	lopes	lope	NOUN
cana-2867	19	2	and	and	CCONJ
cana-2867	19	3	romaguera	romaguera	NOUN
cana-2867	20	1	[	[	X
cana-2867	20	2	6	6	X
cana-2867	20	3	]	]	PUNCT
cana-2867	20	4	established	establish	VERB
cana-2867	20	5	different	different	ADJ
cana-2867	20	6	axioms	axiom	NOUN
cana-2867	20	7	on	on	ADP
cana-2867	20	8	compact	compact	ADJ
cana-2867	20	9	sets	set	NOUN
cana-2867	20	10	for	for	ADP
cana-2867	20	11	hausdorff	hausdorff	X
cana-2867	20	12	fuzzy	fuzzy	ADJ
cana-2867	20	13	metric	metric	ADJ
cana-2867	20	14	space	space	NOUN
cana-2867	20	15	.	.	PUNCT
cana-2867	21	1	qui	qui	PROPN
cana-2867	21	2	and	and	CCONJ
cana-2867	21	3	zang	zang	PROPN
cana-2867	22	1	[	[	X
cana-2867	22	2	7	7	NUM
cana-2867	22	3	]	]	PUNCT
cana-2867	22	4	established	establish	VERB
cana-2867	22	5	that	that	SCONJ
cana-2867	22	6	for	for	ADP
cana-2867	22	7	a	a	DET
cana-2867	22	8	given	give	VERB
cana-2867	22	9	positive	positive	ADJ
cana-2867	22	10	continuous	continuous	ADJ
cana-2867	22	11	norm	norm	NOUN
cana-2867	22	12	,	,	PUNCT
cana-2867	22	13	there	there	PRON
cana-2867	22	14	exists	exist	VERB
cana-2867	22	15	a	a	DET
cana-2867	22	16	setting	setting	NOUN
cana-2867	22	17	of	of	ADP
cana-2867	22	18	space	space	NOUN
cana-2867	22	19	for	for	ADP
cana-2867	22	20	which	which	PRON
cana-2867	22	21	the	the	DET
cana-2867	22	22	given	give	VERB
cana-2867	22	23	norm	norm	NOUN
cana-2867	22	24	is	be	AUX
cana-2867	22	25	the	the	DET
cana-2867	22	26	strongest	strong	ADJ
cana-2867	22	27	.	.	PUNCT
cana-2867	23	1	qui	qui	PROPN
cana-2867	23	2	et	et	PROPN
cana-2867	23	3	al	al	PROPN
cana-2867	23	4	.	.	PUNCT
cana-2867	24	1	[	[	X
cana-2867	24	2	8	8	NUM
cana-2867	24	3	,	,	PUNCT
cana-2867	24	4	9	9	NUM
cana-2867	24	5	]	]	PUNCT
cana-2867	24	6	obtained	obtain	VERB
cana-2867	24	7	interesting	interesting	ADJ
cana-2867	24	8	results	result	NOUN
cana-2867	24	9	on	on	ADP
cana-2867	24	10	fixed	fix	VERB
cana-2867	24	11	points	point	NOUN
cana-2867	24	12	for	for	ADP
cana-2867	24	13	the	the	DET
cana-2867	24	14	same	same	ADJ
cana-2867	24	15	setting	setting	NOUN
cana-2867	24	16	of	of	ADP
cana-2867	24	17	space	space	NOUN
cana-2867	24	18	,	,	PUNCT
cana-2867	24	19	and	and	CCONJ
cana-2867	24	20	qui	qui	X
cana-2867	24	21	et	et	PROPN
cana-2867	24	22	al	al	PROPN
cana-2867	24	23	.	.	PUNCT
cana-2867	25	1	[	[	X
cana-2867	25	2	10	10	NUM
cana-2867	25	3	]	]	PUNCT
cana-2867	25	4	generalized	generalize	VERB
cana-2867	25	5	the	the	DET
cana-2867	25	6	standard	standard	ADJ
cana-2867	25	7	hausdorff	hausdorff	NOUN
cana-2867	25	8	metric	metric	ADJ
cana-2867	25	9	with	with	ADP
cana-2867	25	10	a	a	DET
cana-2867	25	11	norm	norm	NOUN
cana-2867	25	12	and	and	CCONJ
cana-2867	25	13	studied	study	VERB
cana-2867	25	14	its	its	PRON
cana-2867	25	15	standard	standard	ADJ
cana-2867	25	16	axioms	axiom	NOUN
cana-2867	25	17	.	.	PUNCT
cana-2867	26	1	consequently	consequently	ADV
cana-2867	26	2	,	,	PUNCT
cana-2867	26	3	various	various	ADJ
cana-2867	26	4	mathematicians	mathematician	NOUN
cana-2867	26	5	have	have	AUX
cana-2867	26	6	presented	present	VERB
cana-2867	26	7	and	and	CCONJ
cana-2867	26	8	analyzed	analyze	VERB
cana-2867	26	9	certain	certain	ADJ
cana-2867	26	10	concepts	concept	NOUN
cana-2867	26	11	of	of	ADP
cana-2867	26	12	the	the	DET
cana-2867	26	13	same	same	ADJ
cana-2867	26	14	framework	framework	NOUN
cana-2867	26	15	of	of	ADP
cana-2867	26	16	space	space	NOUN
cana-2867	26	17	in	in	ADP
cana-2867	26	18	distinct	distinct	ADJ
cana-2867	26	19	forms	form	NOUN
cana-2867	26	20	and	and	CCONJ
cana-2867	26	21	also	also	ADV
cana-2867	26	22	established	establish	VERB
cana-2867	26	23	fixed	fix	VERB
cana-2867	26	24	point	point	NOUN
cana-2867	26	25	results	result	NOUN
cana-2867	26	26	that	that	PRON
cana-2867	26	27	satisfy	satisfy	VERB
cana-2867	26	28	a	a	DET
cana-2867	26	29	few	few	ADJ
cana-2867	26	30	general	general	ADJ
cana-2867	26	31	contractive	contractive	ADJ
cana-2867	26	32	conditions	condition	NOUN
cana-2867	26	33	,	,	PUNCT
cana-2867	26	34	including	include	VERB
cana-2867	26	35	fascinating	fascinating	ADJ
cana-2867	26	36	consequent	consequent	ADJ
cana-2867	26	37	theories	theory	NOUN
cana-2867	26	38	in	in	ADP
cana-2867	26	39	the	the	DET
cana-2867	26	40	framework	framework	NOUN
cana-2867	26	41	of	of	ADP
cana-2867	26	42	the	the	DET
cana-2867	26	43	new	new	ADJ
cana-2867	26	44	notion	notion	NOUN
cana-2867	26	45	see	see	VERB
cana-2867	26	46	,	,	PUNCT
cana-2867	26	47	for	for	ADP
cana-2867	26	48	example	example	NOUN
cana-2867	26	49	,	,	PUNCT
cana-2867	26	50	[	[	X
cana-2867	26	51	11	11	NUM
cana-2867	26	52	,	,	PUNCT
cana-2867	26	53	12	12	NUM
cana-2867	26	54	,	,	PUNCT
cana-2867	26	55	13	13	NUM
cana-2867	26	56	,	,	PUNCT
cana-2867	26	57	14	14	NUM
cana-2867	26	58	,	,	PUNCT
cana-2867	26	59	15	15	NUM
cana-2867	26	60	,	,	PUNCT
cana-2867	26	61	16	16	NUM
cana-2867	26	62	,	,	PUNCT
cana-2867	26	63	17	17	NUM
cana-2867	26	64	]	]	PUNCT
cana-2867	26	65	.	.	PUNCT
cana-2867	27	1	fractals	fractal	NOUN
cana-2867	27	2	and	and	CCONJ
cana-2867	27	3	multivalued	multivalue	VERB
cana-2867	27	4	fractals	fractal	NOUN
cana-2867	27	5	have	have	VERB
cana-2867	27	6	broad	broad	ADJ
cana-2867	27	7	applications	application	NOUN
cana-2867	27	8	such	such	ADJ
cana-2867	27	9	as	as	ADP
cana-2867	27	10	image	image	NOUN
cana-2867	27	11	compression	compression	NOUN
cana-2867	27	12	,	,	PUNCT
cana-2867	27	13	computer	computer	NOUN
cana-2867	27	14	graphics	graphic	NOUN
cana-2867	27	15	,	,	PUNCT
cana-2867	27	16	soil	soil	NOUN
cana-2867	27	17	mechanics	mechanic	NOUN
cana-2867	27	18	,	,	PUNCT
cana-2867	27	19	the	the	DET
cana-2867	27	20	digital	digital	ADJ
cana-2867	27	21	photography	photography	NOUN
cana-2867	27	22	,	,	PUNCT
cana-2867	27	23	quantum	quantum	NOUN
cana-2867	27	24	physics	physics	NOUN
cana-2867	27	25	,	,	PUNCT
cana-2867	27	26	biomedicine	biomedicine	NOUN
cana-2867	27	27	,	,	PUNCT
cana-2867	27	28	fluid	fluid	ADJ
cana-2867	27	29	mechanics	mechanic	NOUN
cana-2867	27	30	etc	etc	X
cana-2867	27	31	.	.	PUNCT
cana-2867	28	1	the	the	DET
cana-2867	28	2	notion	notion	NOUN
cana-2867	28	3	of	of	ADP
cana-2867	28	4	fractal	fractal	ADJ
cana-2867	28	5	theory	theory	NOUN
cana-2867	28	6	was	be	AUX
cana-2867	28	7	introduced	introduce	VERB
cana-2867	28	8	in	in	ADP
cana-2867	28	9	1975	1975	NUM
cana-2867	28	10	by	by	ADP
cana-2867	28	11	mandelbrot	mandelbrot	PROPN
cana-2867	28	12	[	[	X
cana-2867	28	13	18	18	NUM
cana-2867	28	14	]	]	PUNCT
cana-2867	28	15	.	.	PUNCT
cana-2867	29	1	the	the	DET
cana-2867	29	2	terminologies	terminology	NOUN
cana-2867	29	3	fractals	fractal	NOUN
cana-2867	29	4	are	be	AUX
cana-2867	29	5	defined	define	VERB
cana-2867	29	6	as	as	ADP
cana-2867	29	7	a	a	DET
cana-2867	29	8	set	set	NOUN
cana-2867	29	9	of	of	ADP
cana-2867	29	10	a	a	DET
cana-2867	29	11	few	few	ADJ
cana-2867	29	12	self	self	NOUN
cana-2867	29	13	-	-	PUNCT
cana-2867	29	14	similar	similar	ADJ
cana-2867	29	15	pieces	piece	NOUN
cana-2867	29	16	that	that	PRON
cana-2867	29	17	acquire	acquire	VERB
cana-2867	29	18	similar	similar	ADJ
cana-2867	29	19	features	feature	NOUN
cana-2867	29	20	in	in	ADP
cana-2867	29	21	distinct	distinct	ADJ
cana-2867	29	22	measures	measure	NOUN
cana-2867	29	23	.	.	PUNCT
cana-2867	30	1	more	more	ADV
cana-2867	30	2	precisely	precisely	ADV
cana-2867	30	3	,	,	PUNCT
cana-2867	30	4	image	image	NOUN
cana-2867	30	5	compression	compression	NOUN
cana-2867	30	6	has	have	AUX
cana-2867	30	7	included	include	VERB
cana-2867	30	8	some	some	DET
cana-2867	30	9	usage	usage	NOUN
cana-2867	30	10	of	of	ADP
cana-2867	30	11	the	the	DET
cana-2867	30	12	scheme	scheme	NOUN
cana-2867	30	13	of	of	ADP
cana-2867	30	14	ifs	ifs	PROPN
cana-2867	30	15	on	on	ADP
cana-2867	30	16	complete	complete	ADJ
cana-2867	30	17	metric	metric	ADJ
cana-2867	30	18	space	space	NOUN
cana-2867	30	19	.	.	PUNCT
cana-2867	31	1	hutchinson	hutchinson	PROPN
cana-2867	31	2	in	in	ADP
cana-2867	31	3	1981	1981	NUM
cana-2867	31	4	gives	give	VERB
cana-2867	31	5	the	the	DET
cana-2867	31	6	definition	definition	NOUN
cana-2867	31	7	of	of	ADP
cana-2867	31	8	ifs	ifs	PROPN
cana-2867	31	9	[	[	X
cana-2867	31	10	19	19	NUM
cana-2867	31	11	]	]	PUNCT
cana-2867	31	12	which	which	PRON
cana-2867	31	13	is	be	AUX
cana-2867	31	14	further	far	ADV
cana-2867	31	15	investigated	investigate	VERB
cana-2867	31	16	by	by	ADP
cana-2867	31	17	barnsley	barnsley	NOUN
cana-2867	31	18	,	,	PUNCT
cana-2867	31	19	demko	demko	NOUN
cana-2867	31	20	and	and	CCONJ
cana-2867	31	21	others	other	NOUN
cana-2867	32	1	[	[	X
cana-2867	32	2	20	20	NUM
cana-2867	32	3	,	,	PUNCT
cana-2867	32	4	21	21	NUM
cana-2867	32	5	,	,	PUNCT
cana-2867	32	6	22	22	NUM
cana-2867	32	7	,	,	PUNCT
cana-2867	32	8	23	23	NUM
cana-2867	32	9	,	,	PUNCT
cana-2867	32	10	24	24	NUM
cana-2867	32	11	]	]	PUNCT
cana-2867	32	12	.	.	PUNCT
cana-2867	33	1	hutchinson	hutchinson	PROPN
cana-2867	33	2	-	-	PUNCT
cana-2867	33	3	barnsley	barnsley	PROPN
cana-2867	33	4	establish	establish	VERB
cana-2867	33	5	communications	communication	NOUN
cana-2867	33	6	on	on	ADP
cana-2867	33	7	applied	apply	VERB
cana-2867	33	8	nonlinear	nonlinear	ADJ
cana-2867	33	9	analysis	analysis	NOUN
cana-2867	33	10	issn	issn	NOUN
cana-2867	33	11	:	:	PUNCT
cana-2867	33	12	1074	1074	NUM
cana-2867	33	13	-	-	PUNCT
cana-2867	33	14	133x	133x	NUM
cana-2867	33	15	vol	vol	NOUN
cana-2867	33	16	32	32	NUM
cana-2867	33	17	no	no	NOUN
cana-2867	33	18	.	.	PUNCT
cana-2867	34	1	4s	4s	NUM
cana-2867	34	2	(	(	PUNCT
cana-2867	34	3	2025	2025	NUM
cana-2867	34	4	)	)	PUNCT
cana-2867	34	5	501	501	NUM
cana-2867	34	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	34	7	remarkable	remarkable	ADJ
cana-2867	34	8	concept	concept	NOUN
cana-2867	34	9	of	of	ADP
cana-2867	34	10	hutchinson	hutchinson	PROPN
cana-2867	34	11	-	-	PUNCT
cana-2867	34	12	barnsley	barnsley	PROPN
cana-2867	34	13	theory	theory	NOUN
cana-2867	34	14	and	and	CCONJ
cana-2867	34	15	presented	present	VERB
cana-2867	34	16	the	the	DET
cana-2867	34	17	hb	hb	PROPN
cana-2867	34	18	operator	operator	NOUN
cana-2867	34	19	[	[	X
cana-2867	34	20	19	19	NUM
cana-2867	34	21	,	,	PUNCT
cana-2867	34	22	20	20	NUM
cana-2867	34	23	]	]	PUNCT
cana-2867	34	24	.	.	PUNCT
cana-2867	35	1	by	by	ADP
cana-2867	35	2	using	use	VERB
cana-2867	35	3	banach	banach	NOUN
cana-2867	35	4	contraction	contraction	NOUN
cana-2867	35	5	principle	principle	NOUN
cana-2867	35	6	an	an	DET
cana-2867	35	7	attractor	attractor	NOUN
cana-2867	35	8	was	be	AUX
cana-2867	35	9	established	establish	VERB
cana-2867	35	10	through	through	ADP
cana-2867	35	11	ifs	ifs	PROPN
cana-2867	35	12	.	.	PUNCT
cana-2867	36	1	in	in	ADP
cana-2867	36	2	2009	2009	NUM
cana-2867	36	3	,	,	PUNCT
cana-2867	36	4	singh	singh	PROPN
cana-2867	36	5	et	et	PROPN
cana-2867	36	6	al	al	PROPN
cana-2867	36	7	.	.	PUNCT
cana-2867	37	1	[	[	X
cana-2867	37	2	25	25	NUM
cana-2867	37	3	]	]	PUNCT
cana-2867	37	4	presented	present	VERB
cana-2867	37	5	fractals	fractal	NOUN
cana-2867	37	6	on	on	ADP
cana-2867	37	7	ifs	ifs	PROPN
cana-2867	37	8	and	and	CCONJ
cana-2867	37	9	discussed	discuss	VERB
cana-2867	37	10	the	the	DET
cana-2867	37	11	novel	novel	ADJ
cana-2867	37	12	concept	concept	NOUN
cana-2867	37	13	of	of	ADP
cana-2867	37	14	multi	multi	NOUN
cana-2867	37	15	-	-	NOUN
cana-2867	37	16	functions	function	NOUN
cana-2867	37	17	.	.	PUNCT
cana-2867	38	1	sahu	sahu	PROPN
cana-2867	38	2	et.al	et.al	PROPN
cana-2867	38	3	.	.	PUNCT
cana-2867	38	4	establish	establish	VERB
cana-2867	38	5	kifs	kif	NOUN
cana-2867	38	6	and	and	CCONJ
cana-2867	38	7	collage	collage	NOUN
cana-2867	38	8	theorem	theorem	NOUN
cana-2867	38	9	for	for	ADP
cana-2867	38	10	kanan	kanan	PROPN
cana-2867	38	11	mapping	mapping	PROPN
cana-2867	38	12	in	in	ADP
cana-2867	38	13	complete	complete	ADJ
cana-2867	38	14	metric	metric	ADJ
cana-2867	38	15	space	space	NOUN
cana-2867	38	16	[	[	X
cana-2867	38	17	26	26	NUM
cana-2867	38	18	]	]	PUNCT
cana-2867	38	19	.	.	PUNCT
cana-2867	39	1	s	s	PART
cana-2867	39	2	c	c	PROPN
cana-2867	39	3	shrivastava	shrivastava	PROPN
cana-2867	39	4	et.al	et.al	PROPN
cana-2867	39	5	.	.	PROPN
cana-2867	40	1	established	establish	VERB
cana-2867	40	2	ifs	ifs	PROPN
cana-2867	40	3	for	for	ADP
cana-2867	40	4	two	two	NUM
cana-2867	40	5	mappings	mapping	NOUN
cana-2867	40	6	and	and	CCONJ
cana-2867	40	7	commuting	commute	VERB
cana-2867	40	8	mapping	mapping	NOUN
cana-2867	40	9	and	and	CCONJ
cana-2867	40	10	by	by	ADP
cana-2867	40	11	introducing	introduce	VERB
cana-2867	40	12	d	d	ADJ
cana-2867	40	13	-	-	ADJ
cana-2867	40	14	metric	metric	ADJ
cana-2867	40	15	space	space	NOUN
cana-2867	40	16	[	[	X
cana-2867	40	17	27	27	NUM
cana-2867	40	18	,	,	PUNCT
cana-2867	40	19	28	28	NUM
cana-2867	40	20	]	]	PUNCT
cana-2867	40	21	.	.	PUNCT
cana-2867	41	1	iterated	iterate	VERB
cana-2867	41	2	function	function	NOUN
cana-2867	41	3	system	system	NOUN
cana-2867	41	4	established	establish	VERB
cana-2867	41	5	for	for	ADP
cana-2867	41	6	riech	riech	NOUN
cana-2867	41	7	mapping	mapping	NOUN
cana-2867	41	8	by	by	ADP
cana-2867	41	9	many	many	ADJ
cana-2867	41	10	authors	author	NOUN
cana-2867	41	11	[	[	X
cana-2867	41	12	29	29	NUM
cana-2867	41	13	,	,	PUNCT
cana-2867	41	14	30	30	NUM
cana-2867	41	15	,	,	PUNCT
cana-2867	41	16	31	31	NUM
cana-2867	41	17	,	,	PUNCT
cana-2867	41	18	32	32	NUM
cana-2867	41	19	]	]	PUNCT
cana-2867	41	20	.	.	PUNCT
cana-2867	42	1	by	by	ADP
cana-2867	42	2	changing	change	VERB
cana-2867	42	3	different	different	ADJ
cana-2867	42	4	space	space	NOUN
cana-2867	42	5	and	and	CCONJ
cana-2867	42	6	mappings	mapping	VERB
cana-2867	42	7	different	different	ADJ
cana-2867	42	8	attractors	attractor	NOUN
cana-2867	42	9	and	and	CCONJ
cana-2867	42	10	ifs	ifs	PROPN
cana-2867	42	11	was	be	AUX
cana-2867	42	12	established	establish	VERB
cana-2867	42	13	by	by	ADP
cana-2867	42	14	many	many	ADJ
cana-2867	42	15	researches	research	NOUN
cana-2867	42	16	.	.	PUNCT
cana-2867	43	1	the	the	DET
cana-2867	43	2	notion	notion	NOUN
cana-2867	43	3	of	of	ADP
cana-2867	43	4	fuzzy	fuzzy	ADJ
cana-2867	43	5	theory	theory	NOUN
cana-2867	43	6	is	be	AUX
cana-2867	43	7	also	also	ADV
cana-2867	43	8	extended	extend	VERB
cana-2867	43	9	to	to	ADP
cana-2867	43	10	fractals	fractal	NOUN
cana-2867	43	11	.	.	PUNCT
cana-2867	44	1	recently	recently	ADV
cana-2867	44	2	,	,	PUNCT
cana-2867	44	3	easwara	easwara	PROPN
cana-2867	44	4	moorty	moorty	PROPN
cana-2867	44	5	and	and	CCONJ
cana-2867	44	6	uthaya	uthaya	NOUN
cana-2867	44	7	kumar	kumar	PROPN
cana-2867	45	1	[	[	X
cana-2867	45	2	33	33	NUM
cana-2867	45	3	,	,	PUNCT
cana-2867	45	4	34	34	NUM
cana-2867	45	5	]	]	PUNCT
cana-2867	45	6	studied	study	VERB
cana-2867	45	7	the	the	DET
cana-2867	45	8	hb	hb	PROPN
cana-2867	45	9	operator	operator	NOUN
cana-2867	45	10	in	in	ADP
cana-2867	45	11	fuzzy	fuzzy	ADJ
cana-2867	45	12	metric	metric	ADJ
cana-2867	45	13	space	space	NOUN
cana-2867	45	14	and	and	CCONJ
cana-2867	45	15	introduced	introduce	VERB
cana-2867	45	16	an	an	DET
cana-2867	45	17	investigation	investigation	NOUN
cana-2867	45	18	on	on	ADP
cana-2867	45	19	fractals	fractal	NOUN
cana-2867	45	20	in	in	ADP
cana-2867	45	21	such	such	ADJ
cana-2867	45	22	spaces	space	NOUN
cana-2867	45	23	.	.	PUNCT
cana-2867	46	1	a	a	DET
cana-2867	46	2	union	union	NOUN
cana-2867	46	3	of	of	ADP
cana-2867	46	4	a	a	DET
cana-2867	46	5	set	set	NOUN
cana-2867	46	6	of	of	ADP
cana-2867	46	7	many	many	ADJ
cana-2867	46	8	images	image	NOUN
cana-2867	46	9	with	with	ADP
cana-2867	46	10	distinct	distinct	ADJ
cana-2867	46	11	scales	scale	NOUN
cana-2867	46	12	is	be	AUX
cana-2867	46	13	known	know	VERB
cana-2867	46	14	as	as	ADP
cana-2867	46	15	a	a	DET
cana-2867	46	16	multi	multi	ADJ
cana-2867	46	17	-	-	ADJ
cana-2867	46	18	fractal	fractal	ADJ
cana-2867	46	19	.	.	PUNCT
cana-2867	47	1	initially	initially	ADV
cana-2867	47	2	,	,	PUNCT
cana-2867	47	3	multifractals	multifractal	NOUN
cana-2867	47	4	were	be	AUX
cana-2867	47	5	analyzed	analyze	VERB
cana-2867	47	6	by	by	ADP
cana-2867	47	7	physicists	physicist	NOUN
cana-2867	47	8	in	in	ADP
cana-2867	47	9	the	the	DET
cana-2867	47	10	1980s	1980s	NUM
cana-2867	47	11	.	.	PUNCT
cana-2867	48	1	many	many	ADJ
cana-2867	48	2	authors	author	NOUN
cana-2867	48	3	were	be	AUX
cana-2867	48	4	attracted	attract	VERB
cana-2867	48	5	to	to	ADP
cana-2867	48	6	this	this	DET
cana-2867	48	7	research	research	NOUN
cana-2867	48	8	area	area	NOUN
cana-2867	48	9	,	,	PUNCT
cana-2867	48	10	and	and	CCONJ
cana-2867	48	11	they	they	PRON
cana-2867	48	12	studied	study	VERB
cana-2867	48	13	and	and	CCONJ
cana-2867	48	14	published	publish	VERB
cana-2867	48	15	their	their	PRON
cana-2867	48	16	research	research	NOUN
cana-2867	48	17	work	work	NOUN
cana-2867	48	18	in	in	ADP
cana-2867	48	19	this	this	DET
cana-2867	48	20	field	field	NOUN
cana-2867	48	21	.	.	PUNCT
cana-2867	49	1	in	in	ADP
cana-2867	49	2	2001	2001	NUM
cana-2867	49	3	,	,	PUNCT
cana-2867	49	4	al	al	PROPN
cana-2867	49	5	-	-	PUNCT
cana-2867	49	6	shameri	shameri	PROPN
cana-2867	49	7	[	[	X
cana-2867	49	8	35	35	NUM
cana-2867	49	9	]	]	PUNCT
cana-2867	49	10	presented	present	VERB
cana-2867	49	11	a	a	DET
cana-2867	49	12	generalization	generalization	NOUN
cana-2867	49	13	of	of	ADP
cana-2867	49	14	ifs	ifs	PROPN
cana-2867	49	15	to	to	ADP
cana-2867	49	16	the	the	DET
cana-2867	49	17	new	new	ADJ
cana-2867	49	18	setup	setup	NOUN
cana-2867	49	19	of	of	ADP
cana-2867	49	20	the	the	DET
cana-2867	49	21	space	space	NOUN
cana-2867	49	22	.	.	PUNCT
cana-2867	50	1	in	in	ADP
cana-2867	50	2	2009	2009	NUM
cana-2867	50	3	,	,	PUNCT
cana-2867	50	4	al	al	PROPN
cana-2867	50	5	-	-	PUNCT
cana-2867	50	6	saidi	saidi	PROPN
cana-2867	50	7	et	et	PROPN
cana-2867	50	8	al	al	PROPN
cana-2867	50	9	.	.	PUNCT
cana-2867	51	1	[	[	X
cana-2867	51	2	36	36	NUM
cana-2867	51	3	]	]	PUNCT
cana-2867	51	4	presented	present	VERB
cana-2867	51	5	a	a	DET
cana-2867	51	6	novel	novel	ADJ
cana-2867	51	7	generalized	generalize	VERB
cana-2867	51	8	space	space	NOUN
cana-2867	51	9	known	know	VERB
cana-2867	51	10	as	as	ADP
cana-2867	51	11	the	the	DET
cana-2867	51	12	multi	multi	ADJ
cana-2867	51	13	-	-	ADJ
cana-2867	51	14	fuzzy	fuzzy	ADJ
cana-2867	51	15	fractal	fractal	ADJ
cana-2867	51	16	space	space	NOUN
cana-2867	51	17	.	.	PUNCT
cana-2867	52	1	in	in	ADP
cana-2867	52	2	2011	2011	NUM
cana-2867	52	3	,	,	PUNCT
cana-2867	52	4	uthayakumar	uthayakumar	PROPN
cana-2867	52	5	and	and	CCONJ
cana-2867	52	6	easwaramoorthy	easwaramoorthy	NOUN
cana-2867	52	7	[	[	X
cana-2867	52	8	37	37	NUM
cana-2867	52	9	]	]	PUNCT
cana-2867	52	10	proposed	propose	VERB
cana-2867	52	11	an	an	DET
cana-2867	52	12	improvement	improvement	NOUN
cana-2867	52	13	of	of	ADP
cana-2867	52	14	hb	hb	PROPN
cana-2867	52	15	theory	theory	NOUN
cana-2867	52	16	for	for	ADP
cana-2867	52	17	an	an	DET
cana-2867	52	18	ims	ims	NOUN
cana-2867	52	19	of	of	ADP
cana-2867	52	20	fuzzy	fuzzy	ADJ
cana-2867	52	21	contractions	contraction	NOUN
cana-2867	52	22	in	in	ADP
cana-2867	52	23	such	such	DET
cana-2867	52	24	a	a	DET
cana-2867	52	25	system	system	NOUN
cana-2867	52	26	.	.	PUNCT
cana-2867	53	1	recently	recently	ADV
cana-2867	53	2	,	,	PUNCT
cana-2867	53	3	prasad	prasad	PROPN
cana-2867	53	4	and	and	CCONJ
cana-2867	53	5	katiyar	katiyar	NOUN
cana-2867	54	1	[	[	X
cana-2867	54	2	38	38	NUM
cana-2867	54	3	]	]	PUNCT
cana-2867	54	4	studied	study	VERB
cana-2867	54	5	hb	hb	PROPN
cana-2867	54	6	operators	operator	NOUN
cana-2867	54	7	for	for	ADP
cana-2867	54	8	mifs	mif	NOUN
cana-2867	54	9	in	in	ADP
cana-2867	54	10	a	a	DET
cana-2867	54	11	general	general	ADJ
cana-2867	54	12	setting	setting	NOUN
cana-2867	54	13	,	,	PUNCT
cana-2867	54	14	and	and	CCONJ
cana-2867	54	15	they	they	PRON
cana-2867	54	16	obtained	obtain	VERB
cana-2867	54	17	existing	exist	VERB
cana-2867	54	18	results	result	NOUN
cana-2867	54	19	in	in	ADP
cana-2867	54	20	fuzzy	fuzzy	ADJ
cana-2867	54	21	fractal	fractal	ADJ
cana-2867	54	22	spaces	space	NOUN
cana-2867	54	23	and	and	CCONJ
cana-2867	54	24	multifuzzy	multifuzzy	VERB
cana-2867	54	25	fractal	fractal	ADJ
cana-2867	54	26	spaces	space	NOUN
cana-2867	54	27	.	.	PUNCT
cana-2867	55	1	the	the	DET
cana-2867	55	2	intention	intention	NOUN
cana-2867	55	3	of	of	ADP
cana-2867	55	4	this	this	DET
cana-2867	55	5	research	research	NOUN
cana-2867	55	6	is	be	AUX
cana-2867	55	7	to	to	PART
cana-2867	55	8	establish	establish	VERB
cana-2867	55	9	a	a	DET
cana-2867	55	10	novel	novel	NOUN
cana-2867	55	11	mfifs	mfif	NOUN
cana-2867	55	12	consisting	consist	VERB
cana-2867	55	13	of	of	ADP
cana-2867	55	14	generalized	generalized	ADJ
cana-2867	55	15	fuzzy	fuzzy	ADJ
cana-2867	55	16	contraction	contraction	NOUN
cana-2867	55	17	mapping	mapping	NOUN
cana-2867	55	18	in	in	ADP
cana-2867	55	19	a	a	DET
cana-2867	55	20	general	general	ADJ
cana-2867	55	21	setting	setting	NOUN
cana-2867	55	22	.	.	PUNCT
cana-2867	56	1	we	we	PRON
cana-2867	56	2	extend	extend	VERB
cana-2867	56	3	the	the	DET
cana-2867	56	4	result	result	NOUN
cana-2867	56	5	obtained	obtain	VERB
cana-2867	56	6	by	by	ADP
cana-2867	56	7	uthaya	uthaya	NOUN
cana-2867	56	8	kumar	kumar	PROPN
cana-2867	56	9	and	and	CCONJ
cana-2867	56	10	gowri	gowri	PROPN
cana-2867	56	11	shankar	shankar	PROPN
cana-2867	57	1	[	[	X
cana-2867	57	2	39	39	NUM
cana-2867	57	3	]	]	PUNCT
cana-2867	57	4	in	in	ADP
cana-2867	57	5	2015	2015	NUM
cana-2867	57	6	to	to	ADP
cana-2867	57	7	the	the	DET
cana-2867	57	8	condition	condition	NOUN
cana-2867	57	9	of	of	ADP
cana-2867	57	10	a	a	DET
cana-2867	57	11	new	new	ADJ
cana-2867	57	12	type	type	NOUN
cana-2867	57	13	multi	multi	ADJ
cana-2867	57	14	fuzzy	fuzzy	ADJ
cana-2867	57	15	ifs	if	NOUN
cana-2867	57	16	consisting	consist	VERB
cana-2867	57	17	of	of	ADP
cana-2867	57	18	-contractive	-contractive	ADJ
cana-2867	57	19	mapping	mapping	NOUN
cana-2867	57	20	defined	define	VERB
cana-2867	57	21	on	on	ADP
cana-2867	57	22	multifractal	multifractal	ADJ
cana-2867	57	23	space	space	NOUN
cana-2867	57	24	.	.	PUNCT
cana-2867	58	1	a	a	DET
cana-2867	58	2	generalized	generalized	ADJ
cana-2867	58	3	contraction	contraction	NOUN
cana-2867	58	4	was	be	AUX
cana-2867	58	5	initiated	initiate	VERB
cana-2867	58	6	and	and	CCONJ
cana-2867	58	7	developed	develop	VERB
cana-2867	58	8	by	by	ADP
cana-2867	58	9	wardowski	wardowski	NOUN
cana-2867	58	10	[	[	X
cana-2867	58	11	40	40	NUM
cana-2867	58	12	]	]	PUNCT
cana-2867	58	13	in	in	ADP
cana-2867	58	14	2013	2013	NUM
cana-2867	58	15	.	.	PUNCT
cana-2867	59	1	this	this	DET
cana-2867	59	2	paper	paper	NOUN
cana-2867	59	3	is	be	AUX
cana-2867	59	4	divided	divide	VERB
cana-2867	59	5	into	into	ADP
cana-2867	59	6	five	five	NUM
cana-2867	59	7	sections	section	NOUN
cana-2867	59	8	.	.	PUNCT
cana-2867	60	1	in	in	ADP
cana-2867	60	2	the	the	DET
cana-2867	60	3	second	second	ADJ
cana-2867	60	4	section	section	NOUN
cana-2867	60	5	includes	include	VERB
cana-2867	60	6	basic	basic	ADJ
cana-2867	60	7	concepts	concept	NOUN
cana-2867	60	8	,	,	PUNCT
cana-2867	60	9	result	result	NOUN
cana-2867	60	10	and	and	CCONJ
cana-2867	60	11	hypothesis	hypothesis	NOUN
cana-2867	60	12	for	for	ADP
cana-2867	60	13	fractals	fractal	NOUN
cana-2867	60	14	and	and	CCONJ
cana-2867	60	15	multi	multi	NOUN
cana-2867	60	16	fractals	fractal	NOUN
cana-2867	60	17	.	.	PUNCT
cana-2867	61	1	third	third	ADJ
cana-2867	61	2	section	section	NOUN
cana-2867	61	3	contains	contain	VERB
cana-2867	61	4	properties	property	NOUN
cana-2867	61	5	of	of	ADP
cana-2867	61	6	generalized	generalized	ADJ
cana-2867	61	7	contraction	contraction	NOUN
cana-2867	61	8	mapping	mapping	NOUN
cana-2867	61	9	and	and	CCONJ
cana-2867	61	10	their	their	PRON
cana-2867	61	11	attractor	attractor	NOUN
cana-2867	61	12	for	for	ADP
cana-2867	61	13	fuzzy	fuzzy	ADJ
cana-2867	61	14	metric	metric	ADJ
cana-2867	61	15	space	space	NOUN
cana-2867	61	16	.	.	PUNCT
cana-2867	62	1	the	the	DET
cana-2867	62	2	fourth	fourth	ADJ
cana-2867	62	3	section	section	NOUN
cana-2867	62	4	,	,	PUNCT
cana-2867	62	5	contains	contain	VERB
cana-2867	62	6	a	a	DET
cana-2867	62	7	methodology	methodology	NOUN
cana-2867	62	8	to	to	PART
cana-2867	62	9	construct	construct	VERB
cana-2867	62	10	the	the	DET
cana-2867	62	11	multi	multi	ADJ
cana-2867	62	12	fuzzy	fuzzy	ADJ
cana-2867	62	13	iterated	iterated	ADJ
cana-2867	62	14	function	function	NOUN
cana-2867	62	15	system	system	NOUN
cana-2867	62	16	(	(	PUNCT
cana-2867	62	17	mfifs	mfifs	PROPN
cana-2867	62	18	)	)	PUNCT
cana-2867	62	19	consisting	consist	VERB
cana-2867	62	20	of	of	ADP
cana-2867	62	21	generalized	generalized	ADJ
cana-2867	62	22	contraction	contraction	NOUN
cana-2867	62	23	mapping	mapping	NOUN
cana-2867	62	24	in	in	ADP
cana-2867	62	25	the	the	DET
cana-2867	62	26	sense	sense	NOUN
cana-2867	62	27	of	of	ADP
cana-2867	62	28	wardowski	wardowski	NOUN
cana-2867	62	29	[	[	X
cana-2867	62	30	40	40	NUM
cana-2867	62	31	]	]	PUNCT
cana-2867	62	32	,	,	PUNCT
cana-2867	62	33	al	al	PROPN
cana-2867	62	34	-	-	PUNCT
cana-2867	62	35	saidi	saidi	PROPN
cana-2867	62	36	et	et	PROPN
cana-2867	62	37	al	al	PROPN
cana-2867	62	38	.	.	PUNCT
cana-2867	63	1	[	[	X
cana-2867	63	2	36	36	NUM
cana-2867	63	3	]	]	PUNCT
cana-2867	63	4	,	,	PUNCT
cana-2867	63	5	and	and	CCONJ
cana-2867	63	6	prasad	prasad	PROPN
cana-2867	63	7	and	and	CCONJ
cana-2867	63	8	katiyar	katiyar	ADJ
cana-2867	64	1	[	[	X
cana-2867	64	2	38	38	NUM
cana-2867	64	3	]	]	PUNCT
cana-2867	64	4	and	and	CCONJ
cana-2867	64	5	prove	prove	VERB
cana-2867	64	6	their	their	PRON
cana-2867	64	7	some	some	DET
cana-2867	64	8	consequent	consequent	ADJ
cana-2867	64	9	results	result	NOUN
cana-2867	64	10	.	.	PUNCT
cana-2867	65	1	in	in	ADP
cana-2867	65	2	the	the	DET
cana-2867	65	3	fifth	fifth	ADJ
cana-2867	65	4	section	section	NOUN
cana-2867	65	5	,	,	PUNCT
cana-2867	65	6	we	we	PRON
cana-2867	65	7	conclude	conclude	VERB
cana-2867	65	8	our	our	PRON
cana-2867	65	9	outcome	outcome	NOUN
cana-2867	65	10	.	.	PUNCT
cana-2867	66	1	2	2	X
cana-2867	66	2	.	.	X
cana-2867	66	3	preliminaries	preliminary	NOUN
cana-2867	66	4	a.	a.	NOUN
cana-2867	66	5	attractor	attractor	NOUN
cana-2867	66	6	for	for	ADP
cana-2867	66	7	fuzzy	fuzzy	ADJ
cana-2867	66	8	metric	metric	ADJ
cana-2867	66	9	space	space	NOUN
cana-2867	66	10	in	in	ADP
cana-2867	66	11	this	this	DET
cana-2867	66	12	section	section	NOUN
cana-2867	66	13	we	we	PRON
cana-2867	66	14	discuss	discuss	VERB
cana-2867	66	15	required	require	VERB
cana-2867	66	16	concepts	concept	NOUN
cana-2867	66	17	and	and	CCONJ
cana-2867	66	18	properties	property	NOUN
cana-2867	66	19	of	of	ADP
cana-2867	66	20	fuzzy	fuzzy	ADJ
cana-2867	66	21	space	space	NOUN
cana-2867	66	22	and	and	CCONJ
cana-2867	66	23	fuzzy	fuzzy	ADJ
cana-2867	66	24	fractal	fractal	ADJ
cana-2867	66	25	space	space	NOUN
cana-2867	66	26	used	use	VERB
cana-2867	66	27	for	for	ADP
cana-2867	66	28	our	our	PRON
cana-2867	66	29	findings	finding	NOUN
cana-2867	66	30	.	.	PUNCT
cana-2867	67	1	definition	definition	NOUN
cana-2867	67	2	2.1	2.1	NUM
cana-2867	67	3	[	[	X
cana-2867	67	4	3	3	NUM
cana-2867	67	5	,	,	PUNCT
cana-2867	67	6	4	4	NUM
cana-2867	67	7	]	]	NOUN
cana-2867	67	8	:	:	PUNCT
cana-2867	67	9	fuzzy	fuzzy	ADJ
cana-2867	67	10	metric	metric	ADJ
cana-2867	67	11	space	space	NOUN
cana-2867	67	12	is	be	AUX
cana-2867	67	13	denoted	denote	VERB
cana-2867	67	14	by	by	ADP
cana-2867	67	15	the	the	DET
cana-2867	67	16	triplet	triplet	NOUN
cana-2867	67	17	(	(	PUNCT
cana-2867	67	18	x	x	NOUN
cana-2867	67	19	,	,	PUNCT
cana-2867	67	20	m,∗	m,∗	NOUN
cana-2867	67	21	)	)	PUNCT
cana-2867	67	22	if	if	SCONJ
cana-2867	67	23	x	x	PRON
cana-2867	67	24	is	be	AUX
cana-2867	67	25	an	an	DET
cana-2867	67	26	arbitrary	arbitrary	ADJ
cana-2867	67	27	set	set	NOUN
cana-2867	67	28	,	,	PUNCT
cana-2867	67	29	m	m	VERB
cana-2867	67	30	is	be	AUX
cana-2867	67	31	a	a	DET
cana-2867	67	32	fuzzy	fuzzy	ADJ
cana-2867	67	33	set	set	NOUN
cana-2867	67	34	and	and	CCONJ
cana-2867	67	35	∗	∗	NOUN
cana-2867	67	36	is	be	AUX
cana-2867	67	37	a	a	DET
cana-2867	67	38	continuous	continuous	ADJ
cana-2867	67	39	t	t	NOUN
cana-2867	67	40	-	-	PUNCT
cana-2867	67	41	norm	norm	NOUN
cana-2867	67	42	on	on	ADP
cana-2867	67	43	x	x	SYM
cana-2867	67	44	×	×	PROPN
cana-2867	67	45	x	x	SYM
cana-2867	67	46	×	×	NOUN
cana-2867	67	47	(	(	PUNCT
cana-2867	67	48	0	0	NUM
cana-2867	67	49	,	,	PUNCT
cana-2867	67	50	∞	∞	NUM
cana-2867	67	51	)	)	PUNCT
cana-2867	67	52	satisfies	satisfy	VERB
cana-2867	67	53	the	the	DET
cana-2867	67	54	following	follow	VERB
cana-2867	67	55	properties	property	NOUN
cana-2867	67	56	:	:	PUNCT
cana-2867	67	57	1	1	X
cana-2867	67	58	.	.	X
cana-2867	67	59	m(x	m(x	PROPN
cana-2867	67	60	,	,	PUNCT
cana-2867	67	61	y	y	PROPN
cana-2867	67	62	,	,	PUNCT
cana-2867	67	63	t	t	PROPN
cana-2867	67	64	)	)	PUNCT
cana-2867	67	65	>	>	X
cana-2867	67	66	0	0	X
cana-2867	67	67	.	.	NOUN
cana-2867	67	68	2	2	NUM
cana-2867	67	69	.	.	X
cana-2867	68	1	m(x	m(x	PROPN
cana-2867	68	2	,	,	PUNCT
cana-2867	68	3	m	m	PROPN
cana-2867	68	4	,	,	PUNCT
cana-2867	68	5	t	t	PROPN
cana-2867	68	6	)	)	PUNCT
cana-2867	68	7	=	=	SYM
cana-2867	68	8	1	1	NUM
cana-2867	68	9	iff	iff	NOUN
cana-2867	68	10	x	x	X
cana-2867	68	11	=	=	PUNCT
cana-2867	68	12	y.	y.	NOUN
cana-2867	68	13	3	3	NUM
cana-2867	68	14	.	.	PUNCT
cana-2867	69	1	m(x	m(x	PROPN
cana-2867	69	2	,	,	PUNCT
cana-2867	69	3	y	y	PROPN
cana-2867	69	4	,	,	PUNCT
cana-2867	69	5	t	t	PROPN
cana-2867	69	6	)	)	PUNCT
cana-2867	69	7	=	=	PUNCT
cana-2867	69	8	m(y	m(y	NOUN
cana-2867	69	9	,	,	PUNCT
cana-2867	69	10	x	x	PROPN
cana-2867	69	11	,	,	PUNCT
cana-2867	69	12	t	t	PROPN
cana-2867	69	13	)	)	PUNCT
cana-2867	69	14	.	.	PUNCT
cana-2867	70	1	4	4	X
cana-2867	70	2	.	.	X
cana-2867	71	1	m(x	m(x	PROPN
cana-2867	71	2	,	,	PUNCT
cana-2867	71	3	y	y	PROPN
cana-2867	71	4	,	,	PUNCT
cana-2867	71	5	t	t	PROPN
cana-2867	71	6	)	)	PUNCT
cana-2867	71	7	∗	∗	PROPN
cana-2867	71	8	m(y	m(y	PROPN
cana-2867	71	9	,	,	PUNCT
cana-2867	71	10	z	z	PROPN
cana-2867	71	11	,	,	PUNCT
cana-2867	71	12	s	s	NOUN
cana-2867	71	13	)	)	PUNCT
cana-2867	71	14	≤	≤	NOUN
cana-2867	71	15	m(x	m(x	PROPN
cana-2867	71	16	,	,	PUNCT
cana-2867	71	17	z	z	PROPN
cana-2867	71	18	,	,	PUNCT
cana-2867	71	19	t	t	PROPN
cana-2867	71	20	+	+	NUM
cana-2867	71	21	s	s	NOUN
cana-2867	71	22	)	)	PUNCT
cana-2867	71	23	.	.	PUNCT
cana-2867	72	1	5	5	X
cana-2867	72	2	.	.	X
cana-2867	72	3	m(x	m(x	PROPN
cana-2867	72	4	,	,	PUNCT
cana-2867	72	5	y	y	PROPN
cana-2867	72	6	,	,	PUNCT
cana-2867	72	7	t.	t.	NOUN
cana-2867	72	8	):	):	PUNCT
cana-2867	72	9	(	(	PUNCT
cana-2867	72	10	0	0	NUM
cana-2867	72	11	,	,	PUNCT
cana-2867	72	12	∞	∞	NUM
cana-2867	72	13	)	)	PUNCT
cana-2867	72	14	→	→	PUNCT
cana-2867	73	1	[	[	X
cana-2867	73	2	0,1	0,1	NUM
cana-2867	73	3	]	]	PUNCT
cana-2867	73	4	is	be	AUX
cana-2867	73	5	continuous	continuous	ADJ
cana-2867	73	6	x	x	X
cana-2867	73	7	,	,	PUNCT
cana-2867	73	8	y	y	PROPN
cana-2867	73	9	,	,	PUNCT
cana-2867	73	10	z	z	NOUN
cana-2867	73	11	∈	∈	PROPN
cana-2867	73	12	x	x	X
cana-2867	73	13	and	and	CCONJ
cana-2867	73	14	s	s	PROPN
cana-2867	73	15	,	,	PUNCT
cana-2867	73	16	t	t	X
cana-2867	73	17	>	>	X
cana-2867	73	18	0	0	X
cana-2867	73	19	.	.	PUNCT
cana-2867	74	1	communications	communication	NOUN
cana-2867	74	2	on	on	ADP
cana-2867	74	3	applied	apply	VERB
cana-2867	74	4	nonlinear	nonlinear	ADJ
cana-2867	74	5	analysis	analysis	NOUN
cana-2867	74	6	issn	issn	NOUN
cana-2867	74	7	:	:	PUNCT
cana-2867	74	8	1074	1074	NUM
cana-2867	74	9	-	-	PUNCT
cana-2867	74	10	133x	133x	NUM
cana-2867	74	11	vol	vol	NOUN
cana-2867	74	12	32	32	NUM
cana-2867	74	13	no	no	NOUN
cana-2867	74	14	.	.	PUNCT
cana-2867	75	1	4s	4s	NUM
cana-2867	75	2	(	(	PUNCT
cana-2867	75	3	2025	2025	NUM
cana-2867	75	4	)	)	PUNCT
cana-2867	76	1	502	502	NUM
cana-2867	76	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	76	3	definition	definition	NOUN
cana-2867	76	4	2.2	2.2	NUM
cana-2867	76	5	[	[	X
cana-2867	76	6	3	3	NUM
cana-2867	76	7	,	,	PUNCT
cana-2867	76	8	41	41	NUM
cana-2867	76	9	]	]	PUNCT
cana-2867	76	10	in	in	ADP
cana-2867	76	11	fuzzy	fuzzy	ADJ
cana-2867	76	12	metric	metric	ADJ
cana-2867	76	13	space	space	NOUN
cana-2867	76	14	(	(	PUNCT
cana-2867	76	15	x	x	NOUN
cana-2867	76	16	,	,	PUNCT
cana-2867	76	17	m,∗	m,∗	NOUN
cana-2867	76	18	)	)	PUNCT
cana-2867	76	19	,	,	PUNCT
cana-2867	76	20	(	(	PUNCT
cana-2867	76	21	i	i	NOUN
cana-2867	76	22	)	)	PUNCT
cana-2867	76	23	a	a	DET
cana-2867	76	24	sequence	sequence	NOUN
cana-2867	76	25	{	{	PUNCT
cana-2867	76	26	xn	xn	NOUN
cana-2867	76	27	}	}	PUNCT
cana-2867	76	28	is	be	AUX
cana-2867	76	29	a	a	DET
cana-2867	76	30	convergent	convergent	NOUN
cana-2867	76	31	sequence	sequence	NOUN
cana-2867	76	32	at	at	ADP
cana-2867	76	33	a	a	DET
cana-2867	76	34	point	point	NOUN
cana-2867	76	35	x	x	SYM
cana-2867	76	36	∈	∈	NOUN
cana-2867	76	37	x	x	PUNCT
cana-2867	76	38	with	with	ADP
cana-2867	76	39	respect	respect	NOUN
cana-2867	76	40	to	to	ADP
cana-2867	76	41	τm	τm	PROPN
cana-2867	76	42	ifflim	ifflim	PROPN
cana-2867	76	43	n→∞	n→∞	X
cana-2867	76	44	m(x	m(x	PROPN
cana-2867	76	45	,	,	PUNCT
cana-2867	76	46	xn	xn	PROPN
cana-2867	76	47	,	,	PUNCT
cana-2867	76	48	t	t	PROPN
cana-2867	76	49	)	)	PUNCT
cana-2867	76	50	=	=	SYM
cana-2867	76	51	1	1	NUM
cana-2867	76	52	,	,	PUNCT
cana-2867	76	53	for	for	ADP
cana-2867	76	54	every	every	DET
cana-2867	76	55	t	t	NOUN
cana-2867	76	56	>	>	X
cana-2867	76	57	0	0	PROPN
cana-2867	76	58	.	.	PUNCT
cana-2867	76	59	where	where	SCONJ
cana-2867	76	60	τm	τm	PROPN
cana-2867	76	61	is	be	AUX
cana-2867	76	62	the	the	DET
cana-2867	76	63	topology	topology	NOUN
cana-2867	76	64	produced	produce	VERB
cana-2867	76	65	by	by	ADP
cana-2867	76	66	the	the	DET
cana-2867	76	67	fuzzy	fuzzy	ADJ
cana-2867	76	68	metric	metric	ADJ
cana-2867	76	69	space	space	NOUN
cana-2867	76	70	(	(	PUNCT
cana-2867	76	71	x	x	NOUN
cana-2867	76	72	,	,	PUNCT
cana-2867	76	73	m,∗	m,∗	NOUN
cana-2867	76	74	)	)	PUNCT
cana-2867	76	75	)	)	PUNCT
cana-2867	76	76	(	(	PUNCT
cana-2867	76	77	ii	ii	X
cana-2867	76	78	)	)	PUNCT
cana-2867	76	79	a	a	DET
cana-2867	76	80	sequence{xn}is	sequence{xn}is	NOUN
cana-2867	76	81	known	know	VERB
cana-2867	76	82	as	as	ADP
cana-2867	76	83	cauchy	cauchy	NOUN
cana-2867	76	84	sequence	sequence	NOUN
cana-2867	76	85	if	if	SCONJ
cana-2867	76	86	for	for	ADP
cana-2867	76	87	anyε	anyε	PROPN
cana-2867	76	88	∈	∈	PROPN
cana-2867	76	89	(	(	PUNCT
cana-2867	76	90	0,1	0,1	NUM
cana-2867	76	91	)	)	PUNCT
cana-2867	76	92	,	,	PUNCT
cana-2867	76	93	t	t	X
cana-2867	76	94	>	>	PUNCT
cana-2867	76	95	0,∃	0,∃	NUM
cana-2867	76	96	n0	n0	NUM
cana-2867	76	97	∈	∈	PROPN
cana-2867	76	98	n	n	CCONJ
cana-2867	76	99	such	such	ADJ
cana-2867	76	100	that	that	SCONJ
cana-2867	76	101	m(xn	m(xn	PROPN
cana-2867	76	102	,	,	PUNCT
cana-2867	76	103	xm	xm	PROPN
cana-2867	76	104	,	,	PUNCT
cana-2867	76	105	t	t	PROPN
cana-2867	76	106	)	)	PUNCT
cana-2867	76	107	>	>	X
cana-2867	76	108	1	1	NUM
cana-2867	76	109	−	−	PROPN
cana-2867	76	110	ε	ε	PROPN
cana-2867	76	111	for	for	ADP
cana-2867	76	112	all	all	DET
cana-2867	76	113	n	n	CCONJ
cana-2867	76	114	,	,	PUNCT
cana-2867	76	115	m	m	VERB
cana-2867	76	116	>	>	X
cana-2867	76	117	n0	n0	PROPN
cana-2867	76	118	.	.	PUNCT
cana-2867	77	1	(	(	PUNCT
cana-2867	77	2	iii	iii	X
cana-2867	77	3	)	)	PUNCT
cana-2867	77	4	a	a	DET
cana-2867	77	5	fuzzy	fuzzy	ADJ
cana-2867	77	6	contraction	contraction	NOUN
cana-2867	77	7	serves	serve	VERB
cana-2867	77	8	as	as	ADP
cana-2867	77	9	a	a	DET
cana-2867	77	10	map	map	NOUN
cana-2867	77	11	w	w	NOUN
cana-2867	77	12	on	on	ADP
cana-2867	77	13	x	x	NOUN
cana-2867	77	14	,	,	PUNCT
cana-2867	77	15	such	such	ADJ
cana-2867	77	16	that	that	SCONJ
cana-2867	77	17	1	1	NUM
cana-2867	77	18	m(w(x	m(w(x	NOUN
cana-2867	77	19	)	)	PUNCT
cana-2867	77	20	,	,	PUNCT
cana-2867	77	21	w(y	w(y	PROPN
cana-2867	77	22	)	)	PUNCT
cana-2867	77	23	,	,	PUNCT
cana-2867	77	24	t	t	PROPN
cana-2867	77	25	)	)	PUNCT
cana-2867	77	26	−	−	PROPN
cana-2867	77	27	1	1	NUM
cana-2867	77	28	≤	≤	NUM
cana-2867	78	1	k	k	X
cana-2867	78	2	(	(	PUNCT
cana-2867	78	3	1	1	NUM
cana-2867	78	4	m(x	m(x	PROPN
cana-2867	78	5	,	,	PUNCT
cana-2867	78	6	y	y	PROPN
cana-2867	78	7	,	,	PUNCT
cana-2867	78	8	t	t	PROPN
cana-2867	78	9	)	)	PUNCT
cana-2867	78	10	−	−	PROPN
cana-2867	78	11	1	1	NUM
cana-2867	78	12	)	)	PUNCT
cana-2867	78	13	for	for	ADP
cana-2867	78	14	every	every	DET
cana-2867	78	15	x	x	PROPN
cana-2867	78	16	,	,	PUNCT
cana-2867	78	17	y	y	PROPN
cana-2867	78	18	∈	∈	PROPN
cana-2867	78	19	x	x	PROPN
cana-2867	78	20	,	,	PUNCT
cana-2867	78	21	t	t	PROPN
cana-2867	78	22	>	>	X
cana-2867	78	23	0	0	PROPN
cana-2867	78	24	,	,	PUNCT
cana-2867	78	25	where	where	SCONJ
cana-2867	78	26	0	0	PUNCT
cana-2867	78	27	<	<	X
cana-2867	78	28	k	k	X
cana-2867	78	29	<	<	X
cana-2867	78	30	1	1	X
cana-2867	78	31	.	.	PUNCT
cana-2867	79	1	the	the	DET
cana-2867	79	2	number	number	NOUN
cana-2867	79	3	k	k	PROPN
cana-2867	79	4	is	be	AUX
cana-2867	79	5	then	then	ADV
cana-2867	79	6	said	say	VERB
cana-2867	79	7	to	to	PART
cana-2867	79	8	be	be	AUX
cana-2867	79	9	a	a	DET
cana-2867	79	10	fuzzy	fuzzy	ADJ
cana-2867	79	11	contractivity	contractivity	NOUN
cana-2867	79	12	ratio	ratio	NOUN
cana-2867	79	13	of	of	ADP
cana-2867	79	14	w.	w.	PROPN
cana-2867	79	15	fuzzy	fuzzy	PROPN
cana-2867	79	16	banach	banach	NOUN
cana-2867	79	17	fixed	fix	VERB
cana-2867	79	18	point	point	NOUN
cana-2867	79	19	theorem	theorem	NOUN
cana-2867	79	20	is	be	AUX
cana-2867	79	21	presented	present	VERB
cana-2867	79	22	in	in	ADP
cana-2867	79	23	formal	formal	ADJ
cana-2867	79	24	way	way	NOUN
cana-2867	79	25	in	in	ADP
cana-2867	79	26	2002	2002	NUM
cana-2867	79	27	by	by	ADP
cana-2867	79	28	gregori	gregori	PROPN
cana-2867	79	29	and	and	CCONJ
cana-2867	79	30	sapena	sapena	ADJ
cana-2867	80	1	[	[	X
cana-2867	80	2	41	41	NUM
cana-2867	80	3	]	]	PUNCT
cana-2867	80	4	.	.	PUNCT
cana-2867	81	1	theorem	theorem	VERB
cana-2867	81	2	2.1[41	2.1[41	NUM
cana-2867	81	3	]	]	X
cana-2867	81	4	let(x	let(x	PROPN
cana-2867	81	5	,	,	PUNCT
cana-2867	81	6	m,∗	m,∗	NOUN
cana-2867	81	7	)	)	PUNCT
cana-2867	81	8	be	be	AUX
cana-2867	81	9	a	a	DET
cana-2867	81	10	complete	complete	ADJ
cana-2867	81	11	fuzzy	fuzzy	ADJ
cana-2867	81	12	metric	metric	ADJ
cana-2867	81	13	space	space	NOUN
cana-2867	81	14	for	for	ADP
cana-2867	81	15	fuzzy	fuzzy	ADJ
cana-2867	81	16	contractive	contractive	ADJ
cana-2867	81	17	mapping	mapping	NOUN
cana-2867	81	18	wn	wn	PROPN
cana-2867	81	19	:	:	PUNCT
cana-2867	81	20	x	x	SYM
cana-2867	81	21	→	→	SYM
cana-2867	81	22	x	x	X
cana-2867	81	23	with	with	ADP
cana-2867	81	24	contractivity	contractivity	NOUN
cana-2867	81	25	factor	factor	NOUN
cana-2867	82	1	k	k	PROPN
cana-2867	82	2	then	then	ADV
cana-2867	82	3	wn	wn	PROPN
cana-2867	82	4	possesses	possess	VERB
cana-2867	82	5	a	a	DET
cana-2867	82	6	unique	unique	ADJ
cana-2867	82	7	fixed	fix	VERB
cana-2867	82	8	point	point	NOUN
cana-2867	82	9	.	.	PUNCT
cana-2867	83	1	definition	definition	NOUN
cana-2867	83	2	2.3[6	2.3[6	NUM
cana-2867	83	3	]	]	PUNCT
cana-2867	83	4	let	let	AUX
cana-2867	83	5	𝒦(x	𝒦(x	PRON
cana-2867	83	6	)	)	PUNCT
cana-2867	83	7	be	be	AUX
cana-2867	83	8	a	a	DET
cana-2867	83	9	set	set	NOUN
cana-2867	83	10	of	of	ADP
cana-2867	83	11	all	all	DET
cana-2867	83	12	non	non	ADJ
cana-2867	83	13	-	-	ADJ
cana-2867	83	14	empty	empty	ADJ
cana-2867	83	15	compact	compact	ADJ
cana-2867	83	16	subsets	subset	NOUN
cana-2867	83	17	of	of	ADP
cana-2867	83	18	x	x	PUNCT
cana-2867	83	19	and	and	CCONJ
cana-2867	83	20	(	(	PUNCT
cana-2867	83	21	x	x	NOUN
cana-2867	83	22	,	,	PUNCT
cana-2867	83	23	m,∗	m,∗	NOUN
cana-2867	83	24	)	)	PUNCT
cana-2867	83	25	be	be	AUX
cana-2867	83	26	a	a	DET
cana-2867	83	27	fuzzy	fuzzy	ADJ
cana-2867	83	28	metric	metric	ADJ
cana-2867	83	29	space	space	NOUN
cana-2867	83	30	then	then	ADV
cana-2867	83	31	hausdorff	hausdorff	VERB
cana-2867	83	32	fuzzy	fuzzy	ADJ
cana-2867	83	33	metric	metric	ADJ
cana-2867	83	34	hm	hm	INTJ
cana-2867	83	35	:	:	PUNCT
cana-2867	83	36	𝒦(x	𝒦(x	NUM
cana-2867	83	37	)	)	PUNCT
cana-2867	83	38	×	×	NOUN
cana-2867	83	39	𝒦(x	𝒦(x	NOUN
cana-2867	83	40	)	)	PUNCT
cana-2867	83	41	×	×	NOUN
cana-2867	83	42	(	(	PUNCT
cana-2867	83	43	0	0	NUM
cana-2867	83	44	,	,	PUNCT
cana-2867	83	45	∞	∞	NUM
cana-2867	83	46	)	)	PUNCT
cana-2867	83	47	→	→	PUNCT
cana-2867	84	1	[	[	X
cana-2867	84	2	0,1	0,1	NUM
cana-2867	84	3	]	]	PUNCT
cana-2867	84	4	defined	define	VERB
cana-2867	84	5	by	by	ADP
cana-2867	84	6	hm(a	hm(a	NOUN
cana-2867	84	7	,	,	PUNCT
cana-2867	84	8	b	b	PROPN
cana-2867	84	9	,	,	PUNCT
cana-2867	84	10	t	t	PROPN
cana-2867	84	11	)	)	PUNCT
cana-2867	84	12	=	=	PUNCT
cana-2867	84	13	min{m(a	min{m(a	PROPN
cana-2867	84	14	,	,	PUNCT
cana-2867	84	15	b	b	PROPN
cana-2867	84	16	,	,	PUNCT
cana-2867	84	17	t	t	PROPN
cana-2867	84	18	)	)	PUNCT
cana-2867	84	19	,	,	PUNCT
cana-2867	84	20	m(b	m(b	PROPN
cana-2867	84	21	,	,	PUNCT
cana-2867	84	22	a	a	PRON
cana-2867	84	23	,	,	PUNCT
cana-2867	84	24	t	t	NOUN
cana-2867	84	25	)	)	PUNCT
cana-2867	84	26	}	}	PUNCT
cana-2867	84	27	where	where	SCONJ
cana-2867	84	28	m(x	m(x	PROPN
cana-2867	84	29	,	,	PUNCT
cana-2867	84	30	b	b	PROPN
cana-2867	84	31	,	,	PUNCT
cana-2867	84	32	t	t	PROPN
cana-2867	84	33	)	)	PUNCT
cana-2867	84	34	=	=	SYM
cana-2867	84	35	sup	sup	NUM
cana-2867	84	36	y∈b	y∈b	PROPN
cana-2867	84	37	m(x	m(x	PROPN
cana-2867	84	38	,	,	PUNCT
cana-2867	84	39	y	y	PROPN
cana-2867	84	40	,	,	PUNCT
cana-2867	84	41	t	t	PROPN
cana-2867	84	42	)	)	PUNCT
cana-2867	84	43	and	and	CCONJ
cana-2867	84	44	m(a	m(a	PROPN
cana-2867	84	45	,	,	PUNCT
cana-2867	84	46	b	b	NOUN
cana-2867	84	47	,	,	PUNCT
cana-2867	84	48	t	t	PROPN
cana-2867	84	49	)	)	PUNCT
cana-2867	84	50	=	=	SYM
cana-2867	84	51	infx∈am(x	infx∈am(x	X
cana-2867	84	52	,	,	PUNCT
cana-2867	84	53	b	b	PROPN
cana-2867	84	54	,	,	PUNCT
cana-2867	84	55	t	t	PROPN
cana-2867	84	56	)	)	PUNCT
cana-2867	84	57	for	for	ADP
cana-2867	84	58	every	every	DET
cana-2867	84	59	x	x	SYM
cana-2867	84	60	∈	∈	PROPN
cana-2867	84	61	x	x	X
cana-2867	84	62	and	and	CCONJ
cana-2867	84	63	a	a	DET
cana-2867	84	64	,	,	PUNCT
cana-2867	84	65	b	b	PROPN
cana-2867	84	66	∈	∈	PROPN
cana-2867	84	67	𝒦(x	𝒦(x	NOUN
cana-2867	84	68	)	)	PUNCT
cana-2867	84	69	.	.	PUNCT
cana-2867	85	1	then(𝒦(x	then(𝒦(x	NOUN
cana-2867	85	2	)	)	PUNCT
cana-2867	85	3	,	,	PUNCT
cana-2867	85	4	hm,∗	hm,∗	ADJ
cana-2867	85	5	)	)	PUNCT
cana-2867	85	6	is	be	AUX
cana-2867	85	7	known	know	VERB
cana-2867	85	8	as	as	ADP
cana-2867	85	9	hausdorff	hausdorff	X
cana-2867	85	10	fuzzy	fuzzy	ADJ
cana-2867	85	11	metric	metric	ADJ
cana-2867	85	12	space	space	NOUN
cana-2867	85	13	where	where	SCONJ
cana-2867	85	14	hm	hm	PRON
cana-2867	85	15	is	be	AUX
cana-2867	85	16	a	a	DET
cana-2867	85	17	fuzzy	fuzzy	ADJ
cana-2867	85	18	metric	metric	NOUN
cana-2867	85	19	on	on	ADP
cana-2867	85	20	𝒦(x	𝒦(x	NOUN
cana-2867	85	21	)	)	PUNCT
cana-2867	85	22	.	.	PUNCT
cana-2867	86	1	in	in	ADP
cana-2867	86	2	2011	2011	NUM
cana-2867	86	3	,	,	PUNCT
cana-2867	86	4	uthaya	uthaya	NOUN
cana-2867	86	5	kumar	kumar	PROPN
cana-2867	86	6	and	and	CCONJ
cana-2867	86	7	easwarmoorthy	easwarmoorthy	ADJ
cana-2867	86	8	[	[	X
cana-2867	86	9	33	33	NUM
cana-2867	86	10	]	]	PUNCT
cana-2867	86	11	discussed	discuss	VERB
cana-2867	86	12	basic	basic	ADJ
cana-2867	86	13	properties	property	NOUN
cana-2867	86	14	of	of	ADP
cana-2867	86	15	hb	hb	NOUN
cana-2867	86	16	operator	operator	NOUN
cana-2867	86	17	and	and	CCONJ
cana-2867	86	18	defined	define	VERB
cana-2867	86	19	the	the	DET
cana-2867	86	20	fifs	fif	NOUN
cana-2867	86	21	and	and	CCONJ
cana-2867	86	22	fhb	fhb	NOUN
cana-2867	86	23	operator	operator	NOUN
cana-2867	86	24	for	for	ADP
cana-2867	86	25	fuzzy	fuzzy	ADJ
cana-2867	86	26	metric	metric	ADJ
cana-2867	86	27	space	space	NOUN
cana-2867	86	28	.	.	PUNCT
cana-2867	87	1	lemma	lemma	PROPN
cana-2867	87	2	2.1[33	2.1[33	X
cana-2867	87	3	]	]	PUNCT
cana-2867	87	4	let	let	VERB
cana-2867	87	5	(	(	PUNCT
cana-2867	87	6	xj	xj	PROPN
cana-2867	87	7	,	,	PUNCT
cana-2867	87	8	mj,∗	mj,∗	PROPN
cana-2867	87	9	)	)	PUNCT
cana-2867	87	10	and	and	CCONJ
cana-2867	87	11	(	(	PUNCT
cana-2867	87	12	𝒦(xj	𝒦(xj	NOUN
cana-2867	87	13	)	)	PUNCT
cana-2867	87	14	,	,	PUNCT
cana-2867	87	15	hmj	hmj	PROPN
cana-2867	87	16	,	,	PUNCT
cana-2867	87	17	∗	∗	PROPN
cana-2867	87	18	)	)	PUNCT
cana-2867	87	19	for	for	ADP
cana-2867	87	20	j	j	PROPN
cana-2867	87	21	=	=	SYM
cana-2867	87	22	1	1	NUM
cana-2867	87	23	,	,	PUNCT
cana-2867	87	24	−	−	PROPN
cana-2867	87	25	−	−	PROPN
cana-2867	87	26	−−	−−	NOUN
cana-2867	87	27	,	,	PUNCT
cana-2867	87	28	n	n	PRON
cana-2867	87	29	be	be	AUX
cana-2867	87	30	fuzzy	fuzzy	ADJ
cana-2867	87	31	metric	metric	ADJ
cana-2867	87	32	space	space	NOUN
cana-2867	87	33	and	and	CCONJ
cana-2867	87	34	corresponding	correspond	VERB
cana-2867	87	35	hausdorff	hausdorff	NOUN
cana-2867	87	36	fuzzy	fuzzy	ADJ
cana-2867	87	37	metric	metric	ADJ
cana-2867	87	38	space	space	NOUN
cana-2867	87	39	.	.	PUNCT
cana-2867	88	1	if	if	SCONJ
cana-2867	88	2	aj	aj	PROPN
cana-2867	88	3	,	,	PUNCT
cana-2867	88	4	bj	bj	VERB
cana-2867	88	5	,	,	PUNCT
cana-2867	88	6	cj	cj	PROPN
cana-2867	88	7	,	,	PUNCT
cana-2867	88	8	dj	dj	PROPN
cana-2867	88	9	⊂	⊂	PROPN
cana-2867	88	10	xj	xj	PROPN
cana-2867	88	11	,	,	PUNCT
cana-2867	88	12	for	for	ADP
cana-2867	88	13	j	j	PROPN
cana-2867	88	14	=	=	SYM
cana-2867	88	15	1	1	NUM
cana-2867	88	16	,	,	PUNCT
cana-2867	88	17	−	−	PROPN
cana-2867	88	18	−	−	PROPN
cana-2867	88	19	−−	−−	NOUN
cana-2867	88	20	,	,	PUNCT
cana-2867	88	21	n	n	CCONJ
cana-2867	88	22	then	then	ADV
cana-2867	88	23	for	for	ADP
cana-2867	88	24	t	t	PROPN
cana-2867	88	25	>	>	X
cana-2867	88	26	0	0	PROPN
cana-2867	88	27	,	,	PUNCT
cana-2867	88	28	hmj	hmj	PROPN
cana-2867	88	29	(	(	PUNCT
cana-2867	88	30	aj	aj	PROPN
cana-2867	88	31	∪	∪	ADP
cana-2867	88	32	bj	bj	PROPN
cana-2867	88	33	,	,	PUNCT
cana-2867	88	34	cj	cj	NOUN
cana-2867	88	35	∪	∪	PROPN
cana-2867	88	36	dj	dj	PROPN
cana-2867	88	37	,	,	PUNCT
cana-2867	88	38	t	t	PROPN
cana-2867	88	39	)	)	PUNCT
cana-2867	88	40	≥	≥	PROPN
cana-2867	88	41	min	min	PROPN
cana-2867	88	42	{	{	PUNCT
cana-2867	88	43	hmj	hmj	PROPN
cana-2867	88	44	(	(	PUNCT
cana-2867	88	45	aj	aj	PROPN
cana-2867	88	46	,	,	PUNCT
cana-2867	88	47	cj	cj	PROPN
cana-2867	88	48	,	,	PUNCT
cana-2867	88	49	t	t	PROPN
cana-2867	88	50	)	)	PUNCT
cana-2867	88	51	,	,	PUNCT
cana-2867	88	52	hmj	hmj	PROPN
cana-2867	88	53	(	(	PUNCT
cana-2867	88	54	bj	bj	NOUN
cana-2867	88	55	,	,	PUNCT
cana-2867	88	56	dj	dj	PROPN
cana-2867	88	57	,	,	PUNCT
cana-2867	88	58	t	t	PROPN
cana-2867	88	59	)	)	PUNCT
cana-2867	88	60	}	}	PUNCT
cana-2867	88	61	,	,	PUNCT
cana-2867	88	62	j	j	PROPN
cana-2867	88	63	=	=	SYM
cana-2867	88	64	1	1	NUM
cana-2867	88	65	,	,	PUNCT
cana-2867	88	66	−	−	PROPN
cana-2867	88	67	−	−	PROPN
cana-2867	88	68	−−	−−	NOUN
cana-2867	88	69	,	,	PUNCT
cana-2867	88	70	n.	n.	NOUN
cana-2867	88	71	definition	definition	NOUN
cana-2867	88	72	2.4[33	2.4[33	X
cana-2867	88	73	]	]	PUNCT
cana-2867	88	74	the	the	DET
cana-2867	88	75	system	system	NOUN
cana-2867	88	76	{	{	PUNCT
cana-2867	88	77	x	x	NOUN
cana-2867	88	78	;	;	PUNCT
cana-2867	88	79	wn	wn	PROPN
cana-2867	88	80	,	,	PUNCT
cana-2867	88	81	n	n	NOUN
cana-2867	88	82	=	=	SYM
cana-2867	88	83	1	1	NUM
cana-2867	88	84	,	,	PUNCT
cana-2867	88	85	−	−	PROPN
cana-2867	88	86	−	−	PROPN
cana-2867	88	87	−−	−−	NOUN
cana-2867	88	88	,	,	PUNCT
cana-2867	88	89	n	n	CCONJ
cana-2867	88	90	}	}	PUNCT
cana-2867	88	91	is	be	AUX
cana-2867	88	92	known	know	VERB
cana-2867	88	93	as	as	ADP
cana-2867	88	94	fuzzy	fuzzy	ADJ
cana-2867	88	95	ifs	if	NOUN
cana-2867	88	96	for	for	ADP
cana-2867	88	97	fuzzy	fuzzy	ADJ
cana-2867	88	98	metric	metric	ADJ
cana-2867	88	99	space	space	NOUN
cana-2867	88	100	(	(	PUNCT
cana-2867	88	101	x	x	NOUN
cana-2867	88	102	,	,	PUNCT
cana-2867	88	103	m,∗	m,∗	NOUN
cana-2867	88	104	)	)	PUNCT
cana-2867	88	105	fuzzy	fuzzy	ADJ
cana-2867	88	106	contractive	contractive	ADJ
cana-2867	88	107	mapping	map	VERB
cana-2867	88	108	wn	wn	PROPN
cana-2867	88	109	:	:	PUNCT
cana-2867	88	110	x	x	SYM
cana-2867	88	111	→	→	SYM
cana-2867	88	112	x	x	SYM
cana-2867	88	113	,	,	PUNCT
cana-2867	88	114	with	with	ADP
cana-2867	88	115	the	the	DET
cana-2867	88	116	corresponding	corresponding	ADJ
cana-2867	88	117	contractivity	contractivity	NOUN
cana-2867	88	118	factors	factor	NOUN
cana-2867	88	119	0	0	PUNCT
cana-2867	88	120	<	<	X
cana-2867	88	121	kn	kn	X
cana-2867	88	122	<	<	X
cana-2867	88	123	1	1	NUM
cana-2867	88	124	,	,	PUNCT
cana-2867	88	125	where	where	SCONJ
cana-2867	88	126	n	n	NOUN
cana-2867	88	127	=	=	SYM
cana-2867	88	128	1	1	NUM
cana-2867	88	129	,	,	PUNCT
cana-2867	88	130	−	−	PROPN
cana-2867	88	131	−	−	PROPN
cana-2867	88	132	−−	−−	NOUN
cana-2867	88	133	,	,	PUNCT
cana-2867	88	134	n.	n.	NOUN
cana-2867	88	135	definition	definition	NOUN
cana-2867	88	136	2.5	2.5	NUM
cana-2867	88	137	[	[	SYM
cana-2867	88	138	33	33	NUM
cana-2867	88	139	]	]	PUNCT
cana-2867	88	140	suppose	suppose	VERB
cana-2867	88	141	that	that	SCONJ
cana-2867	88	142	(	(	PUNCT
cana-2867	88	143	x	x	NOUN
cana-2867	88	144	,	,	PUNCT
cana-2867	88	145	m,∗	m,∗	NOUN
cana-2867	88	146	)	)	PUNCT
cana-2867	88	147	be	be	AUX
cana-2867	88	148	fuzzy	fuzzy	ADJ
cana-2867	88	149	metric	metric	ADJ
cana-2867	88	150	spaces	space	NOUN
cana-2867	88	151	.	.	PUNCT
cana-2867	89	1	let	let	VERB
cana-2867	89	2	{	{	PUNCT
cana-2867	89	3	x	x	NOUN
cana-2867	89	4	;	;	PUNCT
cana-2867	89	5	wn	wn	PROPN
cana-2867	89	6	,	,	PUNCT
cana-2867	89	7	n	n	NOUN
cana-2867	89	8	=	=	SYM
cana-2867	89	9	1	1	NUM
cana-2867	89	10	,	,	PUNCT
cana-2867	89	11	−	−	PROPN
cana-2867	89	12	−	−	PROPN
cana-2867	89	13	−−	−−	NOUN
cana-2867	89	14	,	,	PUNCT
cana-2867	89	15	n	n	CCONJ
cana-2867	89	16	}	}	PUNCT
cana-2867	89	17	be	be	AUX
cana-2867	89	18	a	a	DET
cana-2867	89	19	fifs	fif	NOUN
cana-2867	89	20	.	.	PUNCT
cana-2867	90	1	then	then	ADV
cana-2867	90	2	the	the	DET
cana-2867	90	3	fuzzy	fuzzy	ADJ
cana-2867	90	4	hb	hb	NOUN
cana-2867	90	5	operator	operator	NOUN
cana-2867	90	6	is	be	AUX
cana-2867	90	7	a	a	DET
cana-2867	90	8	function	function	NOUN
cana-2867	90	9	w	w	NOUN
cana-2867	90	10	:	:	PUNCT
cana-2867	90	11	𝒦(x	𝒦(x	NUM
cana-2867	90	12	)	)	PUNCT
cana-2867	90	13	→	→	SYM
cana-2867	90	14	𝒦(x	𝒦(x	X
cana-2867	90	15	)	)	PUNCT
cana-2867	90	16	such	such	ADJ
cana-2867	90	17	that	that	SCONJ
cana-2867	90	18	w(b	w(b	NOUN
cana-2867	90	19	)	)	PUNCT
cana-2867	90	20	=	=	PUNCT
cana-2867	91	1	⋃	⋃	PROPN
cana-2867	91	2	wn(b)n	wn(b)n	PROPN
cana-2867	91	3	n=1	n=1	PROPN
cana-2867	91	4	for	for	ADP
cana-2867	91	5	all	all	DET
cana-2867	91	6	b	b	NOUN
cana-2867	91	7	∈	∈	PROPN
cana-2867	91	8	𝒦(x	𝒦(x	NOUN
cana-2867	91	9	)	)	PUNCT
cana-2867	91	10	.	.	PUNCT
cana-2867	92	1	definition	definition	NOUN
cana-2867	92	2	2.6	2.6	NUM
cana-2867	92	3	[	[	SYM
cana-2867	92	4	33	33	NUM
cana-2867	92	5	]	]	PUNCT
cana-2867	92	6	let	let	VERB
cana-2867	92	7	{	{	PUNCT
cana-2867	92	8	x	x	NOUN
cana-2867	92	9	,	,	PUNCT
cana-2867	92	10	wn	wn	PROPN
cana-2867	92	11	,	,	PUNCT
cana-2867	92	12	n	n	NOUN
cana-2867	92	13	=	=	SYM
cana-2867	92	14	1	1	NUM
cana-2867	92	15	,	,	PUNCT
cana-2867	92	16	−	−	PROPN
cana-2867	93	1	−	−	PROPN
cana-2867	93	2	−−	−−	NOUN
cana-2867	93	3	,	,	PUNCT
cana-2867	93	4	n	n	CCONJ
cana-2867	93	5	}	}	PUNCT
cana-2867	93	6	be	be	AUX
cana-2867	93	7	a	a	DET
cana-2867	93	8	fuzzy	fuzzy	ADJ
cana-2867	93	9	iterated	iterate	VERB
cana-2867	93	10	function	function	NOUN
cana-2867	93	11	system	system	NOUN
cana-2867	93	12	(	(	PUNCT
cana-2867	93	13	fifs	fif	NOUN
cana-2867	93	14	)	)	PUNCT
cana-2867	93	15	for	for	ADP
cana-2867	93	16	fuzzy	fuzzy	ADJ
cana-2867	93	17	metric	metric	ADJ
cana-2867	93	18	space	space	NOUN
cana-2867	93	19	(	(	PUNCT
cana-2867	93	20	x	x	NOUN
cana-2867	93	21	,	,	PUNCT
cana-2867	93	22	m,∗	m,∗	NOUN
cana-2867	93	23	)	)	PUNCT
cana-2867	93	24	and	and	CCONJ
cana-2867	93	25	w	w	PROPN
cana-2867	93	26	be	be	AUX
cana-2867	93	27	the	the	DET
cana-2867	93	28	fuzzy	fuzzy	ADJ
cana-2867	93	29	hb	hb	NOUN
cana-2867	93	30	operator	operator	NOUN
cana-2867	93	31	of	of	ADP
cana-2867	93	32	the	the	DET
cana-2867	93	33	fifs	fif	NOUN
cana-2867	93	34	.	.	PUNCT
cana-2867	94	1	then	then	ADV
cana-2867	94	2	there	there	PRON
cana-2867	94	3	exists	exist	VERB
cana-2867	94	4	unique	unique	ADJ
cana-2867	94	5	attractor	attractor	NOUN
cana-2867	94	6	(	(	PUNCT
cana-2867	94	7	fractal	fractal	NOUN
cana-2867	94	8	)	)	PUNCT
cana-2867	94	9	of	of	ADP
cana-2867	94	10	fifs	fif	NOUN
cana-2867	94	11	,	,	PUNCT
cana-2867	94	12	𝒜∞	𝒜∞	VERB
cana-2867	94	13	∈	∈	PROPN
cana-2867	94	14	𝒦(x	𝒦(x	NOUN
cana-2867	94	15	)	)	PUNCT
cana-2867	94	16	of	of	ADP
cana-2867	94	17	the	the	DET
cana-2867	94	18	fhb	fhb	NOUN
cana-2867	94	19	operator	operator	NOUN
cana-2867	94	20	w.	w.	NOUN
cana-2867	94	21	communications	communication	NOUN
cana-2867	94	22	on	on	ADP
cana-2867	94	23	applied	apply	VERB
cana-2867	94	24	nonlinear	nonlinear	ADJ
cana-2867	94	25	analysis	analysis	NOUN
cana-2867	94	26	issn	issn	NOUN
cana-2867	94	27	:	:	PUNCT
cana-2867	94	28	1074	1074	NUM
cana-2867	94	29	-	-	PUNCT
cana-2867	94	30	133x	133x	NUM
cana-2867	94	31	vol	vol	NOUN
cana-2867	94	32	32	32	NUM
cana-2867	94	33	no	no	NOUN
cana-2867	94	34	.	.	PUNCT
cana-2867	95	1	4s	4s	NUM
cana-2867	95	2	(	(	PUNCT
cana-2867	95	3	2025	2025	NUM
cana-2867	95	4	)	)	PUNCT
cana-2867	95	5	503	503	NUM
cana-2867	95	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	95	7	b.	b.	PROPN
cana-2867	95	8	attractor	attractor	NOUN
cana-2867	95	9	in	in	ADP
cana-2867	95	10	multi	multi	ADJ
cana-2867	95	11	fuzzy	fuzzy	ADJ
cana-2867	95	12	fractal	fractal	ADJ
cana-2867	95	13	space	space	NOUN
cana-2867	95	14	this	this	DET
cana-2867	95	15	segment	segment	NOUN
cana-2867	95	16	defines	define	VERB
cana-2867	95	17	concepts	concept	NOUN
cana-2867	95	18	,	,	PUNCT
cana-2867	95	19	properties	property	NOUN
cana-2867	95	20	and	and	CCONJ
cana-2867	95	21	consequences	consequence	NOUN
cana-2867	95	22	of	of	ADP
cana-2867	95	23	multi	multi	ADJ
cana-2867	95	24	-	-	ADJ
cana-2867	95	25	fractal	fractal	ADJ
cana-2867	95	26	spaces	space	NOUN
cana-2867	95	27	and	and	CCONJ
cana-2867	95	28	multi	multi	ADJ
cana-2867	95	29	-	-	ADJ
cana-2867	95	30	fuzzy	fuzzy	ADJ
cana-2867	95	31	fractal	fractal	ADJ
cana-2867	95	32	spaces	space	NOUN
cana-2867	95	33	which	which	PRON
cana-2867	95	34	is	be	AUX
cana-2867	95	35	required	require	VERB
cana-2867	95	36	for	for	ADP
cana-2867	95	37	our	our	PRON
cana-2867	95	38	outcome	outcome	NOUN
cana-2867	95	39	.	.	PUNCT
cana-2867	96	1	definition	definition	NOUN
cana-2867	96	2	2.7[36	2.7[36	PROPN
cana-2867	96	3	]	]	PUNCT
cana-2867	96	4	let	let	NOUN
cana-2867	96	5	(	(	PUNCT
cana-2867	96	6	xj	xj	PROPN
cana-2867	96	7	,	,	PUNCT
cana-2867	96	8	dj	dj	NOUN
cana-2867	96	9	)	)	PUNCT
cana-2867	96	10	be	be	AUX
cana-2867	96	11	a	a	DET
cana-2867	96	12	metric	metric	ADJ
cana-2867	96	13	space	space	NOUN
cana-2867	96	14	for	for	ADP
cana-2867	96	15	each	each	DET
cana-2867	96	16	j	j	PROPN
cana-2867	96	17	∈	∈	PROPN
cana-2867	96	18	j	j	PROPN
cana-2867	96	19	where	where	SCONJ
cana-2867	96	20	j	j	PROPN
cana-2867	96	21	is	be	AUX
cana-2867	96	22	an	an	DET
cana-2867	96	23	indexed	index	VERB
cana-2867	96	24	set	set	NOUN
cana-2867	96	25	.	.	PUNCT
cana-2867	97	1	the	the	DET
cana-2867	97	2	product	product	NOUN
cana-2867	97	3	space	space	NOUN
cana-2867	97	4	𝒳	𝒳	PROPN
cana-2867	97	5	=	=	SYM
cana-2867	97	6	∏	∏	PROPN
cana-2867	97	7	xjj∈j	xjj∈j	X
cana-2867	97	8	is	be	AUX
cana-2867	97	9	the	the	DET
cana-2867	97	10	space	space	NOUN
cana-2867	97	11	that	that	PRON
cana-2867	97	12	consists	consist	VERB
cana-2867	97	13	of	of	ADP
cana-2867	97	14	all	all	DET
cana-2867	97	15	j	j	NOUN
cana-2867	97	16	-	-	PUNCT
cana-2867	97	17	tuples	tuples	PROPN
cana-2867	97	18	{	{	PUNCT
cana-2867	97	19	xj}j∈j	xj}j∈j	NOUN
cana-2867	97	20	with	with	ADP
cana-2867	97	21	xj	xj	PROPN
cana-2867	97	22	∈	∈	PROPN
cana-2867	97	23	xj	xj	PROPN
cana-2867	97	24	.	.	PROPN
cana-2867	97	25	define	define	VERB
cana-2867	97	26	a	a	DET
cana-2867	97	27	metric	metric	ADJ
cana-2867	97	28	d̅	d̅	NOUN
cana-2867	97	29	:	:	PUNCT
cana-2867	97	30	𝒳	𝒳	PROPN
cana-2867	97	31	→	→	SYM
cana-2867	97	32	r	r	NOUN
cana-2867	97	33	as	as	ADP
cana-2867	97	34	d̅(x	d̅(x	NOUN
cana-2867	97	35	,	,	PUNCT
cana-2867	97	36	y	y	NOUN
cana-2867	97	37	)	)	PUNCT
cana-2867	98	1	=	=	SYM
cana-2867	98	2	max	max	PROPN
cana-2867	98	3	j=1,−−−−,n	j=1,−−−−,n	PROPN
cana-2867	98	4	dj	dj	X
cana-2867	98	5	(	(	PUNCT
cana-2867	98	6	xj	xj	PROPN
cana-2867	98	7	,	,	PUNCT
cana-2867	98	8	y	y	PROPN
cana-2867	98	9	j	j	PROPN
cana-2867	98	10	)	)	PUNCT
cana-2867	98	11	for	for	ADP
cana-2867	98	12	all	all	DET
cana-2867	98	13	x	x	NOUN
cana-2867	98	14	,	,	PUNCT
cana-2867	98	15	y	y	PROPN
cana-2867	98	16	∈	∈	PROPN
cana-2867	98	17	𝒳	𝒳	PROPN
cana-2867	98	18	where	where	SCONJ
cana-2867	98	19	x	x	X
cana-2867	98	20	=	=	PRON
cana-2867	98	21	(	(	PUNCT
cana-2867	98	22	x1	x1	PROPN
cana-2867	98	23	,	,	PUNCT
cana-2867	98	24	x2	x2	PROPN
cana-2867	98	25	,	,	PUNCT
cana-2867	98	26	−	−	PROPN
cana-2867	98	27	−	−	PROPN
cana-2867	98	28	−−	−−	NOUN
cana-2867	98	29	,	,	PUNCT
cana-2867	98	30	xn	xn	PROPN
cana-2867	98	31	)	)	PUNCT
cana-2867	98	32	,	,	PUNCT
cana-2867	98	33	y	y	PROPN
cana-2867	98	34	=	=	PRON
cana-2867	98	35	(	(	PUNCT
cana-2867	98	36	y	y	PROPN
cana-2867	98	37	1	1	NUM
cana-2867	98	38	,	,	PUNCT
cana-2867	98	39	y	y	PROPN
cana-2867	98	40	2	2	NUM
cana-2867	98	41	,	,	PUNCT
cana-2867	98	42	−	−	PROPN
cana-2867	98	43	−	−	PROPN
cana-2867	98	44	−−	−−	PROPN
cana-2867	98	45	,	,	PUNCT
cana-2867	98	46	y	y	PROPN
cana-2867	98	47	n	n	PROPN
cana-2867	98	48	)	)	PUNCT
cana-2867	98	49	and	and	CCONJ
cana-2867	98	50	xj	xj	PROPN
cana-2867	98	51	,	,	PUNCT
cana-2867	98	52	y	y	PROPN
cana-2867	98	53	j	j	PROPN
cana-2867	98	54	∈	∈	PROPN
cana-2867	98	55	xj	xj	PROPN
cana-2867	98	56	for	for	ADP
cana-2867	98	57	j	j	PROPN
cana-2867	98	58	=	=	SYM
cana-2867	98	59	1	1	NUM
cana-2867	98	60	,	,	PUNCT
cana-2867	98	61	−	−	PROPN
cana-2867	98	62	−	−	PROPN
cana-2867	98	63	−−	−−	NOUN
cana-2867	98	64	,	,	PUNCT
cana-2867	98	65	n.	n.	PROPN
cana-2867	98	66	then	then	ADV
cana-2867	98	67	(	(	PUNCT
cana-2867	98	68	𝒳	𝒳	PROPN
cana-2867	98	69	,	,	PUNCT
cana-2867	98	70	d̅	d̅	PROPN
cana-2867	98	71	)	)	PUNCT
cana-2867	98	72	is	be	AUX
cana-2867	98	73	a	a	DET
cana-2867	98	74	metric	metric	ADJ
cana-2867	98	75	space	space	NOUN
cana-2867	98	76	with	with	ADP
cana-2867	98	77	product	product	NOUN
cana-2867	98	78	metric	metric	ADJ
cana-2867	98	79	d̅.	d̅.	NUM
cana-2867	98	80	al	al	PROPN
cana-2867	98	81	-	-	PUNCT
cana-2867	98	82	saidi	saidi	PROPN
cana-2867	98	83	et	et	PROPN
cana-2867	98	84	al	al	PROPN
cana-2867	99	1	[	[	X
cana-2867	99	2	36	36	NUM
cana-2867	99	3	]	]	PUNCT
cana-2867	99	4	presented	present	VERB
cana-2867	99	5	a	a	DET
cana-2867	99	6	basic	basic	ADJ
cana-2867	99	7	definition	definition	NOUN
cana-2867	99	8	of	of	ADP
cana-2867	99	9	a	a	DET
cana-2867	99	10	multi	multi	ADJ
cana-2867	99	11	fuzzy	fuzzy	ADJ
cana-2867	99	12	fractal	fractal	ADJ
cana-2867	99	13	space	space	NOUN
cana-2867	99	14	in	in	ADP
cana-2867	99	15	the	the	DET
cana-2867	99	16	following	following	ADJ
cana-2867	99	17	way	way	NOUN
cana-2867	99	18	.	.	PUNCT
cana-2867	100	1	definition	definition	NOUN
cana-2867	100	2	2.8[36]suppose	2.8[36]suppose	NUM
cana-2867	100	3	𝒦(𝒳	𝒦(𝒳	PROPN
cana-2867	100	4	)	)	PUNCT
cana-2867	100	5	=	=	SYM
cana-2867	100	6	∏	∏	PROPN
cana-2867	100	7	𝒦(xj	𝒦(xj	NOUN
cana-2867	100	8	)	)	PUNCT
cana-2867	100	9	n	n	CCONJ
cana-2867	100	10	j=1	j=1	NOUN
cana-2867	100	11	=	=	SYM
cana-2867	100	12	𝒦(x1	𝒦(x1	PROPN
cana-2867	100	13	)	)	PUNCT
cana-2867	100	14	×	×	NOUN
cana-2867	100	15	𝒦(x2	𝒦(x2	NOUN
cana-2867	100	16	)	)	PUNCT
cana-2867	100	17	×	×	NOUN
cana-2867	100	18	−	−	PROPN
cana-2867	101	1	−	−	PROPN
cana-2867	101	2	−	−	PROPN
cana-2867	101	3	−×	−×	PROPN
cana-2867	101	4	𝒦(xn	𝒦(xn	NOUN
cana-2867	101	5	)	)	PUNCT
cana-2867	101	6	.	.	PUNCT
cana-2867	102	1	then	then	ADV
cana-2867	102	2	(	(	PUNCT
cana-2867	102	3	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	102	4	)	)	PUNCT
cana-2867	102	5	,	,	PUNCT
cana-2867	102	6	hd̅	hd̅	PROPN
cana-2867	102	7	)	)	PUNCT
cana-2867	102	8	is	be	AUX
cana-2867	102	9	said	say	VERB
cana-2867	102	10	to	to	PART
cana-2867	102	11	be	be	AUX
cana-2867	102	12	a	a	DET
cana-2867	102	13	multi	multi	ADJ
cana-2867	102	14	fuzzy	fuzzy	ADJ
cana-2867	102	15	fractal	fractal	ADJ
cana-2867	102	16	space	space	NOUN
cana-2867	102	17	with	with	ADP
cana-2867	102	18	metric	metric	ADJ
cana-2867	102	19	hd̅	hd̅	NOUN
cana-2867	102	20	represented	represent	VERB
cana-2867	102	21	as	as	ADP
cana-2867	102	22	hd̅(𝒜	hd̅(𝒜	PROPN
cana-2867	102	23	,	,	PUNCT
cana-2867	102	24	𝔅	𝔅	NOUN
cana-2867	102	25	)	)	PUNCT
cana-2867	102	26	=	=	SYM
cana-2867	102	27	max	max	PROPN
cana-2867	102	28	j=1,2,−−−,n	j=1,2,−−−,n	PROPN
cana-2867	102	29	{	{	PUNCT
cana-2867	102	30	hdj	hdj	PROPN
cana-2867	102	31	(	(	PUNCT
cana-2867	102	32	aj	aj	PROPN
cana-2867	102	33	,	,	PUNCT
cana-2867	102	34	bj	bj	NOUN
cana-2867	102	35	)	)	PUNCT
cana-2867	102	36	}	}	PUNCT
cana-2867	102	37	for	for	ADP
cana-2867	102	38	𝒜	𝒜	NOUN
cana-2867	102	39	,	,	PUNCT
cana-2867	102	40	𝔅	𝔅	PROPN
cana-2867	102	41	∈	∈	NOUN
cana-2867	102	42	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	102	43	)	)	PUNCT
cana-2867	102	44	,	,	PUNCT
cana-2867	102	45	where	where	SCONJ
cana-2867	102	46	𝒜	𝒜	NOUN
cana-2867	102	47	=	=	SYM
cana-2867	102	48	(	(	PUNCT
cana-2867	102	49	a1	a1	PROPN
cana-2867	102	50	,	,	PUNCT
cana-2867	102	51	−	−	PROPN
cana-2867	102	52	−	−	PROPN
cana-2867	102	53	−−	−−	NOUN
cana-2867	102	54	,	,	PUNCT
cana-2867	102	55	an	an	PRON
cana-2867	102	56	)	)	PUNCT
cana-2867	102	57	and	and	CCONJ
cana-2867	102	58	𝔅	𝔅	PROPN
cana-2867	102	59	=	=	SYM
cana-2867	102	60	(	(	PUNCT
cana-2867	102	61	b1	b1	NOUN
cana-2867	102	62	,	,	PUNCT
cana-2867	102	63	−	−	PROPN
cana-2867	102	64	−	−	PROPN
cana-2867	102	65	−−	−−	NOUN
cana-2867	102	66	,	,	PUNCT
cana-2867	102	67	bn	bn	ADJ
cana-2867	102	68	)	)	PUNCT
cana-2867	102	69	,	,	PUNCT
cana-2867	102	70	hdj	hdj	PROPN
cana-2867	102	71	is	be	AUX
cana-2867	102	72	hausdorff	hausdorff	NOUN
cana-2867	102	73	distance	distance	NOUN
cana-2867	102	74	between	between	ADP
cana-2867	102	75	two	two	NUM
cana-2867	102	76	sets	set	NOUN
cana-2867	102	77	aj	aj	PROPN
cana-2867	102	78	and	and	CCONJ
cana-2867	102	79	bj	bj	VERB
cana-2867	102	80	.	.	PUNCT
cana-2867	103	1	theorem2.2[42	theorem2.2[42	NOUN
cana-2867	103	2	]	]	SYM
cana-2867	103	3	:	:	PUNCT
cana-2867	103	4	(	(	PUNCT
cana-2867	103	5	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	103	6	)	)	PUNCT
cana-2867	103	7	,	,	PUNCT
cana-2867	103	8	hd̅	hd̅	PROPN
cana-2867	103	9	)	)	PUNCT
cana-2867	103	10	be	be	VERB
cana-2867	103	11	a	a	DET
cana-2867	103	12	complete	complete	ADJ
cana-2867	103	13	metric	metric	ADJ
cana-2867	103	14	space	space	NOUN
cana-2867	103	15	iff	iff	PROPN
cana-2867	103	16	(	(	PUNCT
cana-2867	103	17	𝒦(xj	𝒦(xj	NOUN
cana-2867	103	18	)	)	PUNCT
cana-2867	103	19	,	,	PUNCT
cana-2867	103	20	hdj	hdj	PROPN
cana-2867	103	21	)	)	PUNCT
cana-2867	103	22	,	,	PUNCT
cana-2867	103	23	where	where	SCONJ
cana-2867	103	24	j	j	PROPN
cana-2867	103	25	=	=	SYM
cana-2867	103	26	1	1	NUM
cana-2867	103	27	,	,	PUNCT
cana-2867	103	28	−	−	PROPN
cana-2867	103	29	−	−	PROPN
cana-2867	103	30	−	−	PROPN
cana-2867	103	31	,	,	PUNCT
cana-2867	103	32	n	n	PRON
cana-2867	103	33	is	be	AUX
cana-2867	103	34	a	a	DET
cana-2867	103	35	complete	complete	ADJ
cana-2867	103	36	metric	metric	ADJ
cana-2867	103	37	space	space	NOUN
cana-2867	103	38	.	.	PUNCT
cana-2867	104	1	then	then	ADV
cana-2867	104	2	theory	theory	NOUN
cana-2867	104	3	of	of	ADP
cana-2867	104	4	multi	multi	PROPN
cana-2867	104	5	ifs	ifs	PROPN
cana-2867	104	6	and	and	CCONJ
cana-2867	104	7	the	the	DET
cana-2867	104	8	hb	hb	PROPN
cana-2867	104	9	operator	operator	NOUN
cana-2867	104	10	of	of	ADP
cana-2867	104	11	mifs	mif	NOUN
cana-2867	104	12	may	may	AUX
cana-2867	104	13	be	be	AUX
cana-2867	104	14	expressed	express	VERB
cana-2867	104	15	in	in	ADP
cana-2867	104	16	the	the	DET
cana-2867	104	17	following	follow	VERB
cana-2867	104	18	sense	sense	NOUN
cana-2867	104	19	.	.	PUNCT
cana-2867	105	1	definition	definition	NOUN
cana-2867	105	2	2.9[36	2.9[36	NUM
cana-2867	105	3	]	]	X
cana-2867	105	4	:	:	PUNCT
cana-2867	105	5	the	the	DET
cana-2867	105	6	system	system	NOUN
cana-2867	105	7	{	{	PUNCT
cana-2867	105	8	xj	xj	PROPN
cana-2867	105	9	,	,	PUNCT
cana-2867	105	10	j	j	PROPN
cana-2867	105	11	=	=	SYM
cana-2867	105	12	1	1	NUM
cana-2867	105	13	,	,	PUNCT
cana-2867	105	14	−	−	PROPN
cana-2867	105	15	−	−	PROPN
cana-2867	105	16	−−	−−	NOUN
cana-2867	105	17	,	,	PUNCT
cana-2867	105	18	n	n	CCONJ
cana-2867	105	19	;	;	PUNCT
cana-2867	105	20	wij	wij	PROPN
cana-2867	105	21	l	l	NOUN
cana-2867	105	22	:	:	PUNCT
cana-2867	105	23	xj	xj	PROPN
cana-2867	105	24	→	→	SYM
cana-2867	105	25	xi	xi	PROPN
cana-2867	105	26	,	,	PUNCT
cana-2867	105	27	l	l	NOUN
cana-2867	105	28	=	=	SYM
cana-2867	105	29	1	1	NUM
cana-2867	105	30	,	,	PUNCT
cana-2867	105	31	−	−	PROPN
cana-2867	105	32	−	−	PROPN
cana-2867	105	33	−	−	NOUN
cana-2867	105	34	,	,	PUNCT
cana-2867	105	35	mijandi	mijandi	NOUN
cana-2867	105	36	,	,	PUNCT
cana-2867	105	37	j	j	PROPN
cana-2867	105	38	=	=	SYM
cana-2867	105	39	1	1	NUM
cana-2867	105	40	,	,	PUNCT
cana-2867	105	41	−	−	PROPN
cana-2867	105	42	−	−	PROPN
cana-2867	105	43	−	−	NOUN
cana-2867	105	44	,	,	PUNCT
cana-2867	105	45	n	n	CCONJ
cana-2867	105	46	}	}	PUNCT
cana-2867	105	47	is	be	AUX
cana-2867	105	48	multi	multi	X
cana-2867	105	49	iterated	iterated	ADJ
cana-2867	105	50	function	function	NOUN
cana-2867	105	51	system	system	NOUN
cana-2867	105	52	with	with	ADP
cana-2867	105	53	contractivity	contractivity	NOUN
cana-2867	105	54	factor	factor	NOUN
cana-2867	105	55	ratios	ratio	NOUN
cana-2867	105	56	=	=	SYM
cana-2867	105	57	ma	ma	PROPN
cana-2867	105	58	x{sij	x{sij	PROPN
cana-2867	105	59	l	l	NOUN
cana-2867	105	60	,	,	PUNCT
cana-2867	106	1	l	l	NOUN
cana-2867	106	2	=	=	SYM
cana-2867	106	3	1	1	NUM
cana-2867	106	4	,	,	PUNCT
cana-2867	106	5	−	−	PROPN
cana-2867	106	6	−	−	PROPN
cana-2867	106	7	−−	−−	NOUN
cana-2867	106	8	,	,	PUNCT
cana-2867	106	9	mij	mij	NOUN
cana-2867	106	10	and	and	CCONJ
cana-2867	106	11	i	i	PROPN
cana-2867	106	12	,	,	PUNCT
cana-2867	106	13	j	j	PROPN
cana-2867	106	14	=	=	SYM
cana-2867	106	15	1	1	NUM
cana-2867	106	16	,	,	PUNCT
cana-2867	106	17	−	−	PROPN
cana-2867	106	18	−	−	PROPN
cana-2867	106	19	−−	−−	NOUN
cana-2867	106	20	,	,	PUNCT
cana-2867	106	21	n	n	CCONJ
cana-2867	106	22	}	}	PUNCT
cana-2867	106	23	for	for	ADP
cana-2867	106	24	complete	complete	ADJ
cana-2867	106	25	metric	metric	ADJ
cana-2867	106	26	space	space	NOUN
cana-2867	106	27	(	(	PUNCT
cana-2867	106	28	xj	xj	PROPN
cana-2867	106	29	,	,	PUNCT
cana-2867	106	30	dj	dj	NOUN
cana-2867	106	31	)	)	PUNCT
cana-2867	106	32	,	,	PUNCT
cana-2867	106	33	j	j	X
cana-2867	106	34	=	=	SYM
cana-2867	106	35	1,2	1,2	NUM
cana-2867	106	36	,	,	PUNCT
cana-2867	106	37	−	−	PROPN
cana-2867	106	38	−	−	PROPN
cana-2867	106	39	−−	−−	NOUN
cana-2867	106	40	,	,	PUNCT
cana-2867	106	41	n	n	NOUN
cana-2867	106	42	and	and	CCONJ
cana-2867	106	43	contraction	contraction	VERB
cana-2867	106	44	mapping	map	VERB
cana-2867	106	45	wij	wij	PROPN
cana-2867	106	46	l	l	NOUN
cana-2867	106	47	:	:	PUNCT
cana-2867	106	48	xj	xj	PROPN
cana-2867	106	49	→	→	SYM
cana-2867	106	50	xi	xi	PROPN
cana-2867	106	51	,	,	PUNCT
cana-2867	106	52	l	l	NOUN
cana-2867	106	53	=	=	SYM
cana-2867	106	54	1	1	NUM
cana-2867	106	55	,	,	PUNCT
cana-2867	106	56	−	−	PROPN
cana-2867	107	1	−	−	PROPN
cana-2867	107	2	−	−	NOUN
cana-2867	107	3	,	,	PUNCT
cana-2867	107	4	mij	mij	NOUN
cana-2867	107	5	and	and	CCONJ
cana-2867	107	6	i	i	PROPN
cana-2867	107	7	,	,	PUNCT
cana-2867	107	8	j	j	PROPN
cana-2867	107	9	=	=	SYM
cana-2867	107	10	1	1	NUM
cana-2867	107	11	,	,	PUNCT
cana-2867	107	12	−	−	PROPN
cana-2867	108	1	−	−	PROPN
cana-2867	108	2	−	−	NOUN
cana-2867	108	3	,	,	PUNCT
cana-2867	108	4	n.	n.	ADJ
cana-2867	108	5	definition	definition	NOUN
cana-2867	108	6	2.10[36	2.10[36	PROPN
cana-2867	108	7	]	]	X
cana-2867	108	8	:	:	PUNCT
cana-2867	108	9	let	let	AUX
cana-2867	108	10	in	in	ADP
cana-2867	108	11	a	a	DET
cana-2867	108	12	complete	complete	ADJ
cana-2867	108	13	metric	metric	ADJ
cana-2867	108	14	space	space	NOUN
cana-2867	108	15	(	(	PUNCT
cana-2867	108	16	xj	xj	PROPN
cana-2867	108	17	,	,	PUNCT
cana-2867	108	18	dj	dj	NOUN
cana-2867	108	19	)	)	PUNCT
cana-2867	108	20	,	,	PUNCT
cana-2867	108	21	j	j	X
cana-2867	108	22	=	=	SYM
cana-2867	108	23	1,2	1,2	NUM
cana-2867	108	24	,	,	PUNCT
cana-2867	108	25	−	−	PROPN
cana-2867	108	26	−	−	PROPN
cana-2867	108	27	−−	−−	PROPN
cana-2867	108	28	,	,	PUNCT
cana-2867	108	29	n	n	CCONJ
cana-2867	108	30	,	,	PUNCT
cana-2867	108	31	xj	xj	PROPN
cana-2867	108	32	,	,	PUNCT
cana-2867	108	33	j	j	PROPN
cana-2867	108	34	=	=	SYM
cana-2867	108	35	1,2	1,2	NUM
cana-2867	108	36	,	,	PUNCT
cana-2867	108	37	−	−	PROPN
cana-2867	108	38	−	−	PROPN
cana-2867	108	39	−−	−−	NOUN
cana-2867	108	40	,	,	PUNCT
cana-2867	108	41	n	n	CCONJ
cana-2867	108	42	;	;	PUNCT
cana-2867	108	43	wij	wij	PROPN
cana-2867	108	44	l	l	NOUN
cana-2867	108	45	:	:	PUNCT
cana-2867	108	46	xj	xj	PROPN
cana-2867	108	47	→	→	SYM
cana-2867	108	48	xi	xi	PROPN
cana-2867	108	49	,	,	PUNCT
cana-2867	108	50	l	l	NOUN
cana-2867	108	51	=	=	SYM
cana-2867	108	52	1	1	NUM
cana-2867	108	53	,	,	PUNCT
cana-2867	108	54	−	−	PROPN
cana-2867	108	55	−	−	PROPN
cana-2867	108	56	−−	−−	NOUN
cana-2867	108	57	,	,	PUNCT
cana-2867	108	58	mij	mij	NOUN
cana-2867	108	59	and	and	CCONJ
cana-2867	108	60	i	i	PROPN
cana-2867	108	61	,	,	PUNCT
cana-2867	108	62	j	j	PROPN
cana-2867	108	63	=	=	SYM
cana-2867	108	64	1	1	NUM
cana-2867	108	65	,	,	PUNCT
cana-2867	108	66	−	−	PROPN
cana-2867	108	67	−	−	PROPN
cana-2867	108	68	−−	−−	NOUN
cana-2867	108	69	,	,	PUNCT
cana-2867	108	70	n	n	CCONJ
cana-2867	108	71	}	}	PUNCT
cana-2867	108	72	be	be	AUX
cana-2867	108	73	an	an	DET
cana-2867	108	74	multi	multi	ADJ
cana-2867	108	75	iterated	iterated	ADJ
cana-2867	108	76	function	function	NOUN
cana-2867	108	77	system	system	NOUN
cana-2867	108	78	.	.	PUNCT
cana-2867	109	1	then	then	ADV
cana-2867	109	2	,	,	PUNCT
cana-2867	109	3	the	the	DET
cana-2867	109	4	hb	hb	PROPN
cana-2867	109	5	operator	operator	NOUN
cana-2867	109	6	of	of	ADP
cana-2867	109	7	mifs	mif	NOUN
cana-2867	109	8	is	be	AUX
cana-2867	109	9	defined	define	VERB
cana-2867	109	10	by	by	ADP
cana-2867	109	11	w(𝔅	w(𝔅	PUNCT
cana-2867	109	12	)	)	PUNCT
cana-2867	110	1	=	=	SYM
cana-2867	110	2	∏	∏	PROPN
cana-2867	110	3	⋃	⋃	NOUN
cana-2867	110	4	⋃	⋃	NOUN
cana-2867	110	5	wij	wij	X
cana-2867	110	6	l	l	NOUN
cana-2867	110	7	(	(	PUNCT
cana-2867	110	8	bj	bj	NOUN
cana-2867	110	9	)	)	PUNCT
cana-2867	110	10	n	n	CCONJ
cana-2867	110	11	l=1	l=1	PROPN
cana-2867	110	12	n	n	CCONJ
cana-2867	110	13	j=1	j=1	PROPN
cana-2867	110	14	n	n	CCONJ
cana-2867	110	15	i=1	i=1	PROPN
cana-2867	110	16	,	,	PUNCT
cana-2867	110	17	for	for	ADP
cana-2867	110	18	all	all	DET
cana-2867	110	19	b	b	PROPN
cana-2867	110	20	∈	∈	NOUN
cana-2867	110	21	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	110	22	)	)	PUNCT
cana-2867	110	23	.	.	PUNCT
cana-2867	111	1	where	where	SCONJ
cana-2867	111	2	function	function	VERB
cana-2867	111	3	w	w	NOUN
cana-2867	111	4	:	:	PUNCT
cana-2867	111	5	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	111	6	)	)	PUNCT
cana-2867	111	7	→	→	SYM
cana-2867	111	8	𝒦(𝒳)al	𝒦(𝒳)al	NOUN
cana-2867	111	9	-	-	PUNCT
cana-2867	111	10	saidi	saidi	PROPN
cana-2867	111	11	et	et	PROPN
cana-2867	111	12	al	al	PROPN
cana-2867	112	1	[	[	X
cana-2867	112	2	36	36	NUM
cana-2867	112	3	]	]	PUNCT
cana-2867	112	4	consider	consider	VERB
cana-2867	112	5	the	the	DET
cana-2867	112	6	above	above	ADJ
cana-2867	112	7	notations	notation	NOUN
cana-2867	112	8	and	and	CCONJ
cana-2867	112	9	they	they	PRON
cana-2867	112	10	presented	present	VERB
cana-2867	112	11	the	the	DET
cana-2867	112	12	following	following	ADJ
cana-2867	112	13	result	result	NOUN
cana-2867	112	14	as	as	SCONJ
cana-2867	112	15	follows	follow	VERB
cana-2867	112	16	,	,	PUNCT
cana-2867	112	17	communications	communication	NOUN
cana-2867	112	18	on	on	ADP
cana-2867	112	19	applied	apply	VERB
cana-2867	112	20	nonlinear	nonlinear	ADJ
cana-2867	112	21	analysis	analysis	NOUN
cana-2867	112	22	issn	issn	NOUN
cana-2867	112	23	:	:	PUNCT
cana-2867	112	24	1074	1074	NUM
cana-2867	112	25	-	-	PUNCT
cana-2867	112	26	133x	133x	NUM
cana-2867	112	27	vol	vol	NOUN
cana-2867	112	28	32	32	NUM
cana-2867	112	29	no	no	NOUN
cana-2867	112	30	.	.	PUNCT
cana-2867	113	1	4s	4s	NUM
cana-2867	113	2	(	(	PUNCT
cana-2867	113	3	2025	2025	NUM
cana-2867	113	4	)	)	PUNCT
cana-2867	113	5	504	504	NUM
cana-2867	113	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	113	7	suppose	suppose	VERB
cana-2867	113	8	(	(	PUNCT
cana-2867	113	9	xj	xj	PROPN
cana-2867	113	10	,	,	PUNCT
cana-2867	113	11	dj	dj	NOUN
cana-2867	113	12	)	)	PUNCT
cana-2867	113	13	,	,	PUNCT
cana-2867	113	14	j	j	X
cana-2867	114	1	=	=	SYM
cana-2867	114	2	1,2	1,2	NUM
cana-2867	114	3	,	,	PUNCT
cana-2867	114	4	−	−	PROPN
cana-2867	114	5	−	−	PROPN
cana-2867	114	6	−−	−−	NOUN
cana-2867	114	7	,	,	PUNCT
cana-2867	114	8	n	n	X
cana-2867	114	9	be	be	AUX
cana-2867	114	10	given	give	VERB
cana-2867	114	11	complete	complete	ADJ
cana-2867	114	12	,	,	PUNCT
cana-2867	114	13	metric	metric	ADJ
cana-2867	114	14	spaces	space	NOUN
cana-2867	114	15	.	.	PUNCT
cana-2867	115	1	let	let	VERB
cana-2867	115	2	{	{	PUNCT
cana-2867	115	3	xj	xj	PROPN
cana-2867	115	4	,	,	PUNCT
cana-2867	115	5	j	j	PROPN
cana-2867	115	6	=	=	SYM
cana-2867	115	7	1,2	1,2	NUM
cana-2867	115	8	,	,	PUNCT
cana-2867	115	9	−	−	PROPN
cana-2867	115	10	−	−	PROPN
cana-2867	115	11	−−	−−	NOUN
cana-2867	115	12	,	,	PUNCT
cana-2867	115	13	n	n	CCONJ
cana-2867	115	14	;	;	PUNCT
cana-2867	115	15	wij	wij	PROPN
cana-2867	115	16	l	l	NOUN
cana-2867	115	17	:	:	PUNCT
cana-2867	115	18	xj	xj	PROPN
cana-2867	115	19	→	→	SYM
cana-2867	115	20	xi	xi	PROPN
cana-2867	115	21	,	,	PUNCT
cana-2867	115	22	l	l	NOUN
cana-2867	115	23	=	=	SYM
cana-2867	115	24	1	1	NUM
cana-2867	115	25	,	,	PUNCT
cana-2867	115	26	−	−	PROPN
cana-2867	115	27	−	−	PROPN
cana-2867	115	28	−−	−−	NOUN
cana-2867	115	29	,	,	PUNCT
cana-2867	115	30	mij	mij	NOUN
cana-2867	115	31	and	and	CCONJ
cana-2867	115	32	i	i	PROPN
cana-2867	115	33	,	,	PUNCT
cana-2867	115	34	j	j	PROPN
cana-2867	115	35	=	=	SYM
cana-2867	115	36	1	1	NUM
cana-2867	115	37	,	,	PUNCT
cana-2867	115	38	−	−	PROPN
cana-2867	115	39	−	−	PROPN
cana-2867	115	40	−−	−−	NOUN
cana-2867	115	41	,	,	PUNCT
cana-2867	115	42	n	n	CCONJ
cana-2867	115	43	}	}	PUNCT
cana-2867	115	44	be	be	AUX
cana-2867	115	45	an	an	DET
cana-2867	115	46	mifs	mif	NOUN
cana-2867	115	47	.	.	PUNCT
cana-2867	116	1	theorem	theorem	VERB
cana-2867	116	2	2.3[36	2.3[36	PROPN
cana-2867	116	3	]	]	PUNCT
cana-2867	116	4	:	:	PUNCT
cana-2867	116	5	consider	consider	VERB
cana-2867	116	6	n	n	PRON
cana-2867	116	7	complete	complete	VERB
cana-2867	116	8	metric	metric	ADJ
cana-2867	116	9	spaces	space	NOUN
cana-2867	116	10	(	(	PUNCT
cana-2867	116	11	xj	xj	PROPN
cana-2867	116	12	,	,	PUNCT
cana-2867	116	13	dj	dj	NOUN
cana-2867	116	14	)	)	PUNCT
cana-2867	116	15	,	,	PUNCT
cana-2867	116	16	j	j	PROPN
cana-2867	116	17	=	=	SYM
cana-2867	116	18	1	1	NUM
cana-2867	116	19	,	,	PUNCT
cana-2867	116	20	−	−	PROPN
cana-2867	116	21	−	−	PROPN
cana-2867	116	22	−−	−−	NOUN
cana-2867	116	23	,	,	PUNCT
cana-2867	116	24	n.	n.	PROPN
cana-2867	116	25	let	let	VERB
cana-2867	116	26	{	{	PUNCT
cana-2867	116	27	xj	xj	PROPN
cana-2867	116	28	,	,	PUNCT
cana-2867	116	29	j	j	PROPN
cana-2867	116	30	=	=	SYM
cana-2867	116	31	1	1	NUM
cana-2867	116	32	,	,	PUNCT
cana-2867	116	33	−	−	PROPN
cana-2867	116	34	−	−	PROPN
cana-2867	117	1	−	−	NOUN
cana-2867	117	2	,	,	PUNCT
cana-2867	117	3	n	n	CCONJ
cana-2867	117	4	;	;	PUNCT
cana-2867	117	5	wij	wij	PROPN
cana-2867	117	6	l	l	NOUN
cana-2867	117	7	:	:	PUNCT
cana-2867	117	8	xj	xj	PROPN
cana-2867	117	9	→	→	SYM
cana-2867	117	10	xi	xi	PROPN
cana-2867	117	11	,	,	PUNCT
cana-2867	117	12	l	l	NOUN
cana-2867	117	13	=	=	SYM
cana-2867	117	14	1	1	NUM
cana-2867	117	15	,	,	PUNCT
cana-2867	117	16	−	−	PROPN
cana-2867	118	1	−	−	PROPN
cana-2867	118	2	−−	−−	NOUN
cana-2867	118	3	,	,	PUNCT
cana-2867	118	4	mij	mij	NOUN
cana-2867	118	5	and	and	CCONJ
cana-2867	118	6	i	i	PROPN
cana-2867	118	7	,	,	PUNCT
cana-2867	118	8	j	j	PROPN
cana-2867	118	9	=	=	SYM
cana-2867	118	10	1	1	NUM
cana-2867	118	11	,	,	PUNCT
cana-2867	118	12	−	−	PROPN
cana-2867	118	13	−	−	PROPN
cana-2867	118	14	−−	−−	NOUN
cana-2867	118	15	,	,	PUNCT
cana-2867	118	16	n	n	CCONJ
cana-2867	118	17	}	}	PUNCT
cana-2867	118	18	be	be	AUX
cana-2867	118	19	an	an	DET
cana-2867	118	20	mifs	mif	NOUN
cana-2867	118	21	.	.	PUNCT
cana-2867	119	1	then	then	ADV
cana-2867	119	2	,	,	PUNCT
cana-2867	119	3	the	the	DET
cana-2867	119	4	hb	hb	PROPN
cana-2867	119	5	operator	operator	NOUN
cana-2867	119	6	w	w	NOUN
cana-2867	119	7	is	be	AUX
cana-2867	119	8	a	a	DET
cana-2867	119	9	contraction	contraction	NOUN
cana-2867	119	10	mapping	mapping	NOUN
cana-2867	119	11	on	on	ADP
cana-2867	119	12	(	(	PUNCT
cana-2867	119	13	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	119	14	)	)	PUNCT
cana-2867	119	15	,	,	PUNCT
cana-2867	119	16	hd̅	hd̅	PROPN
cana-2867	119	17	)	)	PUNCT
cana-2867	119	18	.	.	PUNCT
cana-2867	120	1	in	in	ADP
cana-2867	120	2	2017	2017	NUM
cana-2867	120	3	,	,	PUNCT
cana-2867	120	4	b.	b.	PROPN
cana-2867	120	5	prasad	prasad	PROPN
cana-2867	120	6	and	and	CCONJ
cana-2867	120	7	k.	k.	PROPN
cana-2867	120	8	katiyar	katiyar	PROPN
cana-2867	121	1	[	[	X
cana-2867	121	2	38	38	NUM
cana-2867	121	3	]	]	PUNCT
cana-2867	121	4	studied	study	VERB
cana-2867	121	5	the	the	DET
cana-2867	121	6	results	result	NOUN
cana-2867	121	7	of	of	ADP
cana-2867	121	8	al	al	PROPN
cana-2867	121	9	-	-	PUNCT
cana-2867	121	10	saidi	saidi	PROPN
cana-2867	121	11	et	et	PROPN
cana-2867	121	12	al	al	PROPN
cana-2867	121	13	.	.	PUNCT
cana-2867	122	1	[	[	X
cana-2867	122	2	36	36	NUM
cana-2867	122	3	]	]	PUNCT
cana-2867	122	4	and	and	CCONJ
cana-2867	122	5	established	establish	VERB
cana-2867	122	6	the	the	DET
cana-2867	122	7	following	follow	VERB
cana-2867	122	8	hb	hb	X
cana-2867	122	9	operators	operator	NOUN
cana-2867	122	10	in	in	ADP
cana-2867	122	11	multi	multi	ADJ
cana-2867	122	12	-	-	ADJ
cana-2867	122	13	fractal	fractal	ADJ
cana-2867	122	14	spaces	space	NOUN
cana-2867	122	15	.	.	PUNCT
cana-2867	123	1	theorem	theorem	ADJ
cana-2867	123	2	2.4[38	2.4[38	PROPN
cana-2867	123	3	]	]	PUNCT
cana-2867	123	4	suppose	suppose	VERB
cana-2867	123	5	(	(	PUNCT
cana-2867	123	6	xj	xj	PROPN
cana-2867	123	7	,	,	PUNCT
cana-2867	123	8	dj	dj	NOUN
cana-2867	123	9	)	)	PUNCT
cana-2867	123	10	,	,	PUNCT
cana-2867	123	11	j	j	PROPN
cana-2867	124	1	=	=	SYM
cana-2867	124	2	1	1	NUM
cana-2867	124	3	,	,	PUNCT
cana-2867	124	4	−	−	PROPN
cana-2867	125	1	−	−	PROPN
cana-2867	125	2	−−	−−	NOUN
cana-2867	125	3	,	,	PUNCT
cana-2867	125	4	n	n	PRON
cana-2867	125	5	are	be	AUX
cana-2867	125	6	complete	complete	ADJ
cana-2867	125	7	metric	metric	ADJ
cana-2867	125	8	spaces	space	NOUN
cana-2867	125	9	and	and	CCONJ
cana-2867	125	10	{	{	PUNCT
cana-2867	125	11	xj	xj	PROPN
cana-2867	125	12	,	,	PUNCT
cana-2867	125	13	j	j	PROPN
cana-2867	125	14	=	=	SYM
cana-2867	125	15	1	1	NUM
cana-2867	125	16	,	,	PUNCT
cana-2867	125	17	−	−	PROPN
cana-2867	125	18	−	−	PROPN
cana-2867	125	19	−	−	NOUN
cana-2867	125	20	−	−	PROPN
cana-2867	125	21	−	−	NOUN
cana-2867	125	22	,	,	PUNCT
cana-2867	125	23	mij	mij	NOUN
cana-2867	125	24	and	and	CCONJ
cana-2867	125	25	i	i	PROPN
cana-2867	125	26	,	,	PUNCT
cana-2867	125	27	j	j	PROPN
cana-2867	125	28	=	=	SYM
cana-2867	125	29	1	1	NUM
cana-2867	125	30	,	,	PUNCT
cana-2867	125	31	−	−	PROPN
cana-2867	125	32	−	−	PROPN
cana-2867	125	33	−−	−−	NOUN
cana-2867	125	34	,	,	PUNCT
cana-2867	125	35	n	n	CCONJ
cana-2867	125	36	;	;	PUNCT
cana-2867	125	37	wij	wij	PROPN
cana-2867	125	38	l	l	NOUN
cana-2867	125	39	:	:	PUNCT
cana-2867	125	40	xj	xj	PROPN
cana-2867	125	41	→	→	SYM
cana-2867	125	42	xi	xi	PROPN
cana-2867	125	43	,	,	PUNCT
cana-2867	125	44	l	l	NOUN
cana-2867	125	45	=	=	SYM
cana-2867	125	46	1	1	NUM
cana-2867	125	47	,	,	PUNCT
cana-2867	125	48	−	−	PROPN
cana-2867	125	49	−	−	PROPN
cana-2867	125	50	−−	−−	NOUN
cana-2867	125	51	,	,	PUNCT
cana-2867	125	52	mij	mij	NOUN
cana-2867	125	53	and	and	CCONJ
cana-2867	125	54	i	i	PROPN
cana-2867	125	55	,	,	PUNCT
cana-2867	125	56	j	j	PROPN
cana-2867	125	57	=	=	SYM
cana-2867	125	58	1	1	NUM
cana-2867	125	59	,	,	PUNCT
cana-2867	125	60	−	−	PROPN
cana-2867	125	61	−	−	PROPN
cana-2867	125	62	−−	−−	NOUN
cana-2867	125	63	,	,	PUNCT
cana-2867	125	64	n	n	CCONJ
cana-2867	125	65	}	}	PUNCT
cana-2867	125	66	be	be	AUX
cana-2867	125	67	an	an	DET
cana-2867	125	68	mfifs	mfif	NOUN
cana-2867	125	69	.	.	PUNCT
cana-2867	126	1	then	then	ADV
cana-2867	126	2	,	,	PUNCT
cana-2867	126	3	there	there	PRON
cana-2867	126	4	exists	exist	VERB
cana-2867	126	5	𝒜∞	𝒜∞	PROPN
cana-2867	126	6	∈	∈	PROPN
cana-2867	126	7	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	126	8	)	)	PUNCT
cana-2867	126	9	known	know	VERB
cana-2867	126	10	as	as	ADP
cana-2867	126	11	the	the	DET
cana-2867	126	12	attractor	attractor	NOUN
cana-2867	126	13	of	of	ADP
cana-2867	126	14	mfifs	mfifs	PROPN
cana-2867	126	15	,	,	PUNCT
cana-2867	126	16	of	of	ADP
cana-2867	126	17	hb	hb	PROPN
cana-2867	126	18	operator	operator	NOUN
cana-2867	126	19	w.	w.	PROPN
cana-2867	126	20	george	george	PROPN
cana-2867	126	21	and	and	CCONJ
cana-2867	126	22	veeramani	veeramani	NOUN
cana-2867	127	1	[	[	X
cana-2867	127	2	4	4	NUM
cana-2867	127	3	]	]	PUNCT
cana-2867	127	4	are	be	AUX
cana-2867	127	5	presented	present	VERB
cana-2867	127	6	a	a	DET
cana-2867	127	7	multi	multi	ADJ
cana-2867	127	8	fuzzy	fuzzy	ADJ
cana-2867	127	9	fractal	fractal	ADJ
cana-2867	127	10	space	space	NOUN
cana-2867	127	11	in	in	ADP
cana-2867	127	12	the	the	DET
cana-2867	127	13	following	following	ADJ
cana-2867	127	14	manner	manner	NOUN
cana-2867	127	15	.	.	PUNCT
cana-2867	128	1	definition	definition	NOUN
cana-2867	128	2	2.11[4	2.11[4	NUM
cana-2867	128	3	]	]	PUNCT
cana-2867	128	4	let	let	VERB
cana-2867	128	5	a	a	DET
cana-2867	128	6	complete	complete	ADJ
cana-2867	128	7	metric	metric	ADJ
cana-2867	128	8	space	space	NOUN
cana-2867	128	9	(	(	PUNCT
cana-2867	128	10	xj	xj	PROPN
cana-2867	128	11	,	,	PUNCT
cana-2867	128	12	mj,∗	mj,∗	PROPN
cana-2867	128	13	)	)	PUNCT
cana-2867	128	14	,	,	PUNCT
cana-2867	128	15	j	j	PROPN
cana-2867	128	16	=	=	SYM
cana-2867	128	17	1	1	NUM
cana-2867	128	18	,	,	PUNCT
cana-2867	128	19	−	−	PROPN
cana-2867	128	20	−	−	PROPN
cana-2867	128	21	−−	−−	NOUN
cana-2867	128	22	,	,	PUNCT
cana-2867	128	23	n	n	PROPN
cana-2867	128	24	and	and	CCONJ
cana-2867	128	25	(	(	PUNCT
cana-2867	128	26	𝒦(xj	𝒦(xj	NOUN
cana-2867	128	27	)	)	PUNCT
cana-2867	128	28	,	,	PUNCT
cana-2867	128	29	hmj	hmj	PROPN
cana-2867	128	30	,	,	PUNCT
cana-2867	128	31	∗	∗	PROPN
cana-2867	128	32	)	)	PUNCT
cana-2867	128	33	,	,	PUNCT
cana-2867	128	34	j	j	PROPN
cana-2867	128	35	=	=	SYM
cana-2867	128	36	1	1	NUM
cana-2867	128	37	,	,	PUNCT
cana-2867	128	38	−	−	PROPN
cana-2867	128	39	−	−	PROPN
cana-2867	128	40	−−	−−	NOUN
cana-2867	128	41	,	,	PUNCT
cana-2867	128	42	n	n	PRON
cana-2867	128	43	is	be	AUX
cana-2867	128	44	associated	associate	VERB
cana-2867	128	45	hausdorff	hausdorff	NOUN
cana-2867	128	46	fuzzy	fuzzy	ADJ
cana-2867	128	47	metric	metric	ADJ
cana-2867	128	48	spaces	space	NOUN
cana-2867	128	49	with	with	ADP
cana-2867	128	50	same	same	ADJ
cana-2867	128	51	t	t	NOUN
cana-2867	128	52	-	-	PUNCT
cana-2867	128	53	norm	norm	NOUN
cana-2867	128	54	then	then	ADV
cana-2867	128	55	the	the	DET
cana-2867	128	56	space	space	NOUN
cana-2867	128	57	(	(	PUNCT
cana-2867	128	58	𝒦(𝒳	𝒦(𝒳	NOUN
cana-2867	128	59	)	)	PUNCT
cana-2867	128	60	,	,	PUNCT
cana-2867	128	61	hℳ,∗	hℳ,∗	PROPN
cana-2867	128	62	)	)	PUNCT
cana-2867	128	63	with	with	ADP
cana-2867	128	64	metric	metric	ADJ
cana-2867	128	65	(	(	PUNCT
cana-2867	128	66	hm,∗	hm,∗	ADJ
cana-2867	128	67	)	)	PUNCT
cana-2867	128	68	is	be	AUX
cana-2867	128	69	called	call	VERB
cana-2867	128	70	a	a	DET
cana-2867	128	71	multi	multi	ADJ
cana-2867	128	72	fuzzy	fuzzy	ADJ
cana-2867	128	73	fractal	fractal	ADJ
cana-2867	128	74	space	space	PROPN
cana-2867	128	75	iff	iff	PROPN
cana-2867	128	76	𝒦(𝒳	𝒦(𝒳	PROPN
cana-2867	128	77	)	)	PUNCT
cana-2867	128	78	=	=	SYM
cana-2867	128	79	𝒦(x1	𝒦(x1	PROPN
cana-2867	128	80	)	)	PUNCT
cana-2867	128	81	×	×	NOUN
cana-2867	128	82	𝒦(x2	𝒦(x2	NOUN
cana-2867	128	83	)	)	PUNCT
cana-2867	128	84	×	×	NOUN
cana-2867	128	85	−	−	PROPN
cana-2867	128	86	−	−	PROPN
cana-2867	128	87	−	−	PROPN
cana-2867	128	88	−×	−×	PROPN
cana-2867	128	89	𝒦(xn	𝒦(xn	NOUN
cana-2867	128	90	)	)	PUNCT
cana-2867	128	91	represented	represent	VERB
cana-2867	128	92	by	by	ADP
cana-2867	128	93	hℳ(𝒜	hℳ(𝒜	PROPN
cana-2867	128	94	,	,	PUNCT
cana-2867	128	95	𝔅	𝔅	PROPN
cana-2867	128	96	,	,	PUNCT
cana-2867	128	97	t	t	PROPN
cana-2867	128	98	)	)	PUNCT
cana-2867	128	99	=	=	SYM
cana-2867	128	100	min	min	PROPN
cana-2867	128	101	{	{	PUNCT
cana-2867	128	102	hmj	hmj	PROPN
cana-2867	128	103	(	(	PUNCT
cana-2867	128	104	aj	aj	PROPN
cana-2867	128	105	,	,	PUNCT
cana-2867	128	106	bj	bj	NOUN
cana-2867	128	107	,	,	PUNCT
cana-2867	128	108	t	t	PROPN
cana-2867	128	109	)	)	PUNCT
cana-2867	128	110	,	,	PUNCT
cana-2867	128	111	j	j	PROPN
cana-2867	129	1	=	=	SYM
cana-2867	129	2	1	1	NUM
cana-2867	129	3	,	,	PUNCT
cana-2867	129	4	−	−	PROPN
cana-2867	129	5	−	−	PROPN
cana-2867	129	6	−−	−−	NOUN
cana-2867	129	7	,	,	PUNCT
cana-2867	129	8	n	n	CCONJ
cana-2867	129	9	}	}	PUNCT
cana-2867	129	10	for	for	ADP
cana-2867	129	11	all	all	DET
cana-2867	129	12	aj	aj	PROPN
cana-2867	129	13	,	,	PUNCT
cana-2867	129	14	bj	bj	ADP
cana-2867	129	15	∈	∈	PROPN
cana-2867	129	16	𝒦(xj	𝒦(xj	NOUN
cana-2867	129	17	)	)	PUNCT
cana-2867	129	18	,	,	PUNCT
cana-2867	129	19	𝒜	𝒜	NOUN
cana-2867	129	20	,	,	PUNCT
cana-2867	129	21	𝔅	𝔅	PROPN
cana-2867	129	22	∈	∈	NOUN
cana-2867	129	23	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	129	24	)	)	PUNCT
cana-2867	129	25	,	,	PUNCT
cana-2867	129	26	where	where	SCONJ
cana-2867	129	27	𝒜	𝒜	NOUN
cana-2867	129	28	=	=	SYM
cana-2867	129	29	(	(	PUNCT
cana-2867	129	30	a1	a1	PROPN
cana-2867	129	31	,	,	PUNCT
cana-2867	129	32	−	−	PROPN
cana-2867	129	33	−	−	PROPN
cana-2867	129	34	−−	−−	NOUN
cana-2867	129	35	,	,	PUNCT
cana-2867	129	36	an	an	PRON
cana-2867	129	37	)	)	PUNCT
cana-2867	129	38	,	,	PUNCT
cana-2867	129	39	𝔅	𝔅	NOUN
cana-2867	129	40	=	=	SYM
cana-2867	129	41	(	(	PUNCT
cana-2867	129	42	b1	b1	PROPN
cana-2867	129	43	,	,	PUNCT
cana-2867	129	44	b2	b2	NOUN
cana-2867	129	45	,	,	PUNCT
cana-2867	129	46	−	−	PROPN
cana-2867	129	47	−	−	PROPN
cana-2867	129	48	−−	−−	NOUN
cana-2867	129	49	,	,	PUNCT
cana-2867	129	50	bn	bn	ADJ
cana-2867	129	51	)	)	PUNCT
cana-2867	129	52	.	.	PUNCT
cana-2867	130	1	theorem	theorem	VERB
cana-2867	130	2	2.5[38	2.5[38	NUM
cana-2867	130	3	]	]	X
cana-2867	130	4	the	the	DET
cana-2867	130	5	metric	metric	ADJ
cana-2867	130	6	space	space	NOUN
cana-2867	130	7	(	(	PUNCT
cana-2867	130	8	𝒦(𝒳	𝒦(𝒳	NOUN
cana-2867	130	9	)	)	PUNCT
cana-2867	130	10	,	,	PUNCT
cana-2867	130	11	hℳ	hℳ	PROPN
cana-2867	130	12	,	,	PUNCT
cana-2867	130	13	∗	∗	NOUN
cana-2867	130	14	)	)	PUNCT
cana-2867	130	15	is	be	AUX
cana-2867	130	16	a	a	DET
cana-2867	130	17	complete	complete	ADJ
cana-2867	130	18	fuzzy	fuzzy	ADJ
cana-2867	130	19	metric	metric	ADJ
cana-2867	130	20	spaces	space	NOUN
cana-2867	130	21	iff	iff	PROPN
cana-2867	130	22	(	(	PUNCT
cana-2867	130	23	𝒦(xj	𝒦(xj	PROPN
cana-2867	130	24	)	)	PUNCT
cana-2867	130	25	,	,	PUNCT
cana-2867	130	26	hmj	hmj	PROPN
cana-2867	130	27	,	,	PUNCT
cana-2867	130	28	∗	∗	PROPN
cana-2867	130	29	)	)	PUNCT
cana-2867	130	30	is	be	AUX
cana-2867	130	31	complete	complete	ADJ
cana-2867	130	32	for	for	ADP
cana-2867	130	33	each	each	PRON
cana-2867	130	34	j	j	PROPN
cana-2867	130	35	=	=	PUNCT
cana-2867	130	36	1,2,3	1,2,3	NUM
cana-2867	130	37	,	,	PUNCT
cana-2867	130	38	−	−	PROPN
cana-2867	130	39	−	−	PROPN
cana-2867	130	40	−−	−−	NOUN
cana-2867	130	41	,	,	PUNCT
cana-2867	130	42	n.	n.	NOUN
cana-2867	130	43	prasad	prasad	PROPN
cana-2867	130	44	and	and	CCONJ
cana-2867	130	45	katiyar	katiyar	ADJ
cana-2867	131	1	[	[	X
cana-2867	131	2	38	38	NUM
cana-2867	131	3	]	]	PUNCT
cana-2867	131	4	extended	extend	VERB
cana-2867	131	5	the	the	DET
cana-2867	131	6	results	result	NOUN
cana-2867	131	7	of	of	ADP
cana-2867	131	8	al	al	PROPN
cana-2867	131	9	-	-	PUNCT
cana-2867	131	10	saidi	saidi	PROPN
cana-2867	131	11	et	et	PROPN
cana-2867	131	12	al	al	PROPN
cana-2867	132	1	[	[	X
cana-2867	132	2	36	36	NUM
cana-2867	132	3	]	]	PUNCT
cana-2867	132	4	and	and	CCONJ
cana-2867	132	5	they	they	PRON
cana-2867	132	6	define	define	VERB
cana-2867	132	7	the	the	DET
cana-2867	132	8	mfifs	mfif	NOUN
cana-2867	132	9	and	and	CCONJ
cana-2867	132	10	hb	hb	NOUN
cana-2867	132	11	operator	operator	NOUN
cana-2867	132	12	of	of	ADP
cana-2867	132	13	mfifs	mfif	NOUN
cana-2867	132	14	in	in	ADP
cana-2867	132	15	the	the	DET
cana-2867	132	16	following	follow	VERB
cana-2867	132	17	manner	manner	NOUN
cana-2867	132	18	.	.	PUNCT
cana-2867	133	1	definition	definition	NOUN
cana-2867	133	2	2.12[38]:suppose	2.12[38]:suppose	NUM
cana-2867	134	1	that	that	SCONJ
cana-2867	134	2	(	(	PUNCT
cana-2867	134	3	xj	xj	PROPN
cana-2867	134	4	,	,	PUNCT
cana-2867	134	5	mj,∗	mj,∗	PROPN
cana-2867	134	6	)	)	PUNCT
cana-2867	134	7	,	,	PUNCT
cana-2867	134	8	j	j	PROPN
cana-2867	134	9	=	=	SYM
cana-2867	134	10	1	1	NUM
cana-2867	134	11	,	,	PUNCT
cana-2867	134	12	−	−	PROPN
cana-2867	134	13	−	−	PROPN
cana-2867	134	14	−−	−−	NOUN
cana-2867	134	15	,	,	PUNCT
cana-2867	134	16	n	n	X
cana-2867	134	17	be	be	AUX
cana-2867	134	18	complete	complete	ADJ
cana-2867	134	19	fuzzy	fuzzy	ADJ
cana-2867	134	20	metric	metric	ADJ
cana-2867	134	21	spaces	space	NOUN
cana-2867	134	22	.	.	PUNCT
cana-2867	135	1	let	let	VERB
cana-2867	135	2	wij	wij	PROPN
cana-2867	135	3	l	l	NOUN
cana-2867	135	4	:	:	PUNCT
cana-2867	135	5	xj	xj	PROPN
cana-2867	135	6	→	→	SYM
cana-2867	135	7	xi	xi	PROPN
cana-2867	135	8	,	,	PUNCT
cana-2867	135	9	l	l	NOUN
cana-2867	135	10	=	=	SYM
cana-2867	135	11	1	1	NUM
cana-2867	135	12	,	,	PUNCT
cana-2867	135	13	−	−	PROPN
cana-2867	135	14	−	−	PROPN
cana-2867	135	15	−−	−−	NOUN
cana-2867	135	16	,	,	PUNCT
cana-2867	135	17	mij	mij	NOUN
cana-2867	135	18	and	and	CCONJ
cana-2867	135	19	i	i	PROPN
cana-2867	135	20	,	,	PUNCT
cana-2867	135	21	j	j	PROPN
cana-2867	135	22	=	=	SYM
cana-2867	135	23	1	1	NUM
cana-2867	135	24	,	,	PUNCT
cana-2867	135	25	−	−	PROPN
cana-2867	135	26	−	−	PROPN
cana-2867	135	27	−−	−−	NOUN
cana-2867	135	28	,	,	PUNCT
cana-2867	135	29	n	n	PRON
cana-2867	135	30	be	be	VERB
cana-2867	135	31	fuzzy	fuzzy	ADJ
cana-2867	135	32	b	b	NOUN
cana-2867	135	33	-	-	PUNCT
cana-2867	135	34	contractions	contraction	NOUN
cana-2867	135	35	with	with	ADP
cana-2867	135	36	the	the	DET
cana-2867	135	37	corresponding	correspond	VERB
cana-2867	135	38	contraction	contraction	NOUN
cana-2867	135	39	factors	factor	VERB
cana-2867	135	40	0	0	NUM
cana-2867	135	41	<	<	X
cana-2867	135	42	sij	sij	PROPN
cana-2867	135	43	l	l	X
cana-2867	135	44	<	<	X
cana-2867	135	45	1	1	NUM
cana-2867	135	46	,	,	PUNCT
cana-2867	135	47	where	where	SCONJ
cana-2867	135	48	l	l	NOUN
cana-2867	135	49	=	=	SYM
cana-2867	135	50	1	1	NUM
cana-2867	135	51	,	,	PUNCT
cana-2867	135	52	−	−	PROPN
cana-2867	135	53	−	−	PROPN
cana-2867	135	54	−−	−−	NOUN
cana-2867	135	55	,	,	PUNCT
cana-2867	135	56	mij	mij	NOUN
cana-2867	135	57	and	and	CCONJ
cana-2867	135	58	i	i	PROPN
cana-2867	135	59	,	,	PUNCT
cana-2867	135	60	j	j	PROPN
cana-2867	135	61	=	=	SYM
cana-2867	135	62	1	1	NUM
cana-2867	135	63	,	,	PUNCT
cana-2867	135	64	−	−	PROPN
cana-2867	135	65	−	−	PROPN
cana-2867	135	66	−−	−−	NOUN
cana-2867	135	67	,	,	PUNCT
cana-2867	135	68	n.	n.	PROPN
cana-2867	135	69	then	then	ADV
cana-2867	135	70	the	the	DET
cana-2867	135	71	system	system	NOUN
cana-2867	135	72	{	{	PUNCT
cana-2867	135	73	xj	xj	PROPN
cana-2867	135	74	,	,	PUNCT
cana-2867	135	75	j	j	PROPN
cana-2867	135	76	=	=	SYM
cana-2867	135	77	1,2	1,2	NUM
cana-2867	135	78	,	,	PUNCT
cana-2867	135	79	−	−	PROPN
cana-2867	135	80	−	−	PROPN
cana-2867	135	81	−−	−−	NOUN
cana-2867	135	82	,	,	PUNCT
cana-2867	135	83	n	n	CCONJ
cana-2867	135	84	;	;	PUNCT
cana-2867	135	85	wij	wij	PROPN
cana-2867	135	86	l	l	NOUN
cana-2867	135	87	:	:	PUNCT
cana-2867	135	88	xj	xj	PROPN
cana-2867	135	89	→	→	SYM
cana-2867	135	90	xi	xi	PROPN
cana-2867	135	91	,	,	PUNCT
cana-2867	135	92	l	l	NOUN
cana-2867	135	93	=	=	SYM
cana-2867	135	94	1	1	NUM
cana-2867	135	95	,	,	PUNCT
cana-2867	135	96	−	−	PROPN
cana-2867	135	97	−	−	PROPN
cana-2867	135	98	−−	−−	NOUN
cana-2867	135	99	,	,	PUNCT
cana-2867	135	100	mij	mij	NOUN
cana-2867	135	101	and	and	CCONJ
cana-2867	135	102	i	i	PROPN
cana-2867	135	103	,	,	PUNCT
cana-2867	135	104	j	j	PROPN
cana-2867	135	105	=	=	SYM
cana-2867	135	106	1,2	1,2	NUM
cana-2867	135	107	,	,	PUNCT
cana-2867	135	108	−	−	PROPN
cana-2867	135	109	−	−	PROPN
cana-2867	135	110	−−	−−	NOUN
cana-2867	135	111	,	,	PUNCT
cana-2867	135	112	n	n	CCONJ
cana-2867	135	113	}	}	PUNCT
cana-2867	135	114	is	be	AUX
cana-2867	135	115	said	say	VERB
cana-2867	135	116	to	to	PART
cana-2867	135	117	be	be	AUX
cana-2867	135	118	a	a	DET
cana-2867	135	119	mfifs	mfif	NOUN
cana-2867	135	120	along	along	ADP
cana-2867	135	121	contraction	contraction	NOUN
cana-2867	135	122	ratio	ratio	NOUN
cana-2867	135	123	s	s	PART
cana-2867	135	124	=	=	X
cana-2867	135	125	max{sij	max{sij	PROPN
cana-2867	135	126	l	l	NOUN
cana-2867	135	127	,	,	PUNCT
cana-2867	135	128	l	l	NOUN
cana-2867	136	1	=	=	SYM
cana-2867	136	2	1	1	NUM
cana-2867	136	3	,	,	PUNCT
cana-2867	136	4	−	−	PROPN
cana-2867	136	5	−	−	PROPN
cana-2867	136	6	−−	−−	NOUN
cana-2867	136	7	,	,	PUNCT
cana-2867	136	8	mij	mij	NOUN
cana-2867	136	9	,	,	PUNCT
cana-2867	136	10	i	i	PRON
cana-2867	136	11	,	,	PUNCT
cana-2867	136	12	j	j	PROPN
cana-2867	136	13	=	=	SYM
cana-2867	136	14	1	1	NUM
cana-2867	136	15	,	,	PUNCT
cana-2867	136	16	−	−	PROPN
cana-2867	136	17	−	−	PROPN
cana-2867	136	18	−−	−−	NOUN
cana-2867	136	19	,	,	PUNCT
cana-2867	136	20	n	n	CCONJ
cana-2867	136	21	}	}	PUNCT
cana-2867	136	22	.	.	PUNCT
cana-2867	137	1	definition	definition	NOUN
cana-2867	137	2	2.13[38	2.13[38	NUM
cana-2867	137	3	]	]	PUNCT
cana-2867	137	4	in	in	ADP
cana-2867	137	5	a	a	DET
cana-2867	137	6	complete	complete	ADJ
cana-2867	137	7	fuzzy	fuzzy	ADJ
cana-2867	137	8	metric	metric	ADJ
cana-2867	137	9	space	space	NOUN
cana-2867	137	10	(	(	PUNCT
cana-2867	137	11	xj	xj	PROPN
cana-2867	137	12	,	,	PUNCT
cana-2867	137	13	mj,∗	mj,∗	PROPN
cana-2867	137	14	)	)	PUNCT
cana-2867	137	15	,	,	PUNCT
cana-2867	137	16	j	j	PROPN
cana-2867	137	17	=	=	SYM
cana-2867	137	18	1	1	NUM
cana-2867	137	19	,	,	PUNCT
cana-2867	137	20	−	−	PROPN
cana-2867	137	21	−	−	PROPN
cana-2867	137	22	−−	−−	NOUN
cana-2867	137	23	,	,	PUNCT
cana-2867	137	24	n	n	PRON
cana-2867	137	25	assume	assume	VERB
cana-2867	137	26	that	that	SCONJ
cana-2867	137	27	{	{	PUNCT
cana-2867	137	28	xj	xj	PROPN
cana-2867	137	29	,	,	PUNCT
cana-2867	137	30	j	j	PROPN
cana-2867	137	31	=	=	SYM
cana-2867	137	32	1,2	1,2	NUM
cana-2867	137	33	,	,	PUNCT
cana-2867	137	34	−	−	PROPN
cana-2867	137	35	−	−	PROPN
cana-2867	137	36	−	−	NOUN
cana-2867	137	37	,	,	PUNCT
cana-2867	137	38	n	n	CCONJ
cana-2867	137	39	;	;	PUNCT
cana-2867	137	40	wij	wij	PROPN
cana-2867	137	41	l	l	NOUN
cana-2867	137	42	:	:	PUNCT
cana-2867	137	43	xj	xj	PROPN
cana-2867	137	44	→	→	SYM
cana-2867	137	45	xi	xi	PROPN
cana-2867	137	46	,	,	PUNCT
cana-2867	137	47	l	l	NOUN
cana-2867	137	48	=	=	SYM
cana-2867	137	49	1	1	NUM
cana-2867	137	50	,	,	PUNCT
cana-2867	137	51	−	−	PROPN
cana-2867	137	52	−	−	PROPN
cana-2867	137	53	−−	−−	NOUN
cana-2867	137	54	,	,	PUNCT
cana-2867	137	55	mij	mij	NOUN
cana-2867	137	56	and	and	CCONJ
cana-2867	137	57	i	i	PROPN
cana-2867	137	58	,	,	PUNCT
cana-2867	137	59	j	j	PROPN
cana-2867	137	60	=	=	SYM
cana-2867	137	61	1	1	NUM
cana-2867	137	62	,	,	PUNCT
cana-2867	137	63	−	−	PROPN
cana-2867	137	64	−	−	PROPN
cana-2867	137	65	−−	−−	NOUN
cana-2867	137	66	,	,	PUNCT
cana-2867	137	67	n	n	CCONJ
cana-2867	137	68	}	}	PUNCT
cana-2867	137	69	be	be	AUX
cana-2867	137	70	an	an	DET
cana-2867	137	71	mfifs	mfif	NOUN
cana-2867	137	72	.	.	PUNCT
cana-2867	138	1	then	then	ADV
cana-2867	138	2	w(𝔅	w(𝔅	PUNCT
cana-2867	138	3	)	)	PUNCT
cana-2867	138	4	=	=	SYM
cana-2867	138	5	∏	∏	PROPN
cana-2867	138	6	⋃	⋃	NOUN
cana-2867	138	7	⋃	⋃	NOUN
cana-2867	138	8	wij	wij	X
cana-2867	138	9	l	l	NOUN
cana-2867	138	10	(	(	PUNCT
cana-2867	138	11	bj	bj	NOUN
cana-2867	138	12	)	)	PUNCT
cana-2867	138	13	mij	mij	NOUN
cana-2867	138	14	l=1	l=1	NOUN
cana-2867	138	15	n	n	CCONJ
cana-2867	138	16	j=1	j=1	PROPN
cana-2867	138	17	n	n	CCONJ
cana-2867	138	18	i=1	i=1	PROPN
cana-2867	138	19	for	for	ADP
cana-2867	138	20	all	all	DET
cana-2867	138	21	𝔅	𝔅	PROPN
cana-2867	138	22	∈	∈	NOUN
cana-2867	138	23	𝒦(𝒦	𝒦(𝒦	NOUN
cana-2867	138	24	)	)	PUNCT
cana-2867	138	25	.	.	PUNCT
cana-2867	139	1	is	be	AUX
cana-2867	139	2	called	call	VERB
cana-2867	139	3	multi	multi	ADJ
cana-2867	139	4	fuzzy	fuzzy	ADJ
cana-2867	139	5	hutchinson	hutchinson	PROPN
cana-2867	139	6	-	-	PUNCT
cana-2867	139	7	barnsley	barnsley	PROPN
cana-2867	139	8	(	(	PUNCT
cana-2867	139	9	mfhb	mfhb	NOUN
cana-2867	139	10	)	)	PUNCT
cana-2867	139	11	operator	operator	NOUN
cana-2867	139	12	for	for	ADP
cana-2867	139	13	a	a	DET
cana-2867	139	14	function	function	NOUN
cana-2867	139	15	w	w	NOUN
cana-2867	139	16	:	:	PUNCT
cana-2867	139	17	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	139	18	)	)	PUNCT
cana-2867	139	19	→	→	SYM
cana-2867	139	20	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	139	21	)	)	PUNCT
cana-2867	139	22	.	.	PUNCT
cana-2867	140	1	communications	communication	NOUN
cana-2867	140	2	on	on	ADP
cana-2867	140	3	applied	apply	VERB
cana-2867	140	4	nonlinear	nonlinear	ADJ
cana-2867	140	5	analysis	analysis	NOUN
cana-2867	140	6	issn	issn	NOUN
cana-2867	140	7	:	:	PUNCT
cana-2867	140	8	1074	1074	NUM
cana-2867	140	9	-	-	PUNCT
cana-2867	140	10	133x	133x	NUM
cana-2867	140	11	vol	vol	NOUN
cana-2867	140	12	32	32	NUM
cana-2867	140	13	no	no	NOUN
cana-2867	140	14	.	.	PUNCT
cana-2867	141	1	4s	4s	NUM
cana-2867	141	2	(	(	PUNCT
cana-2867	141	3	2025	2025	NUM
cana-2867	141	4	)	)	PUNCT
cana-2867	141	5	505	505	NUM
cana-2867	141	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	141	7	b.	b.	PROPN
cana-2867	141	8	prasad	prasad	PROPN
cana-2867	141	9	and	and	CCONJ
cana-2867	141	10	k.	k.	PROPN
cana-2867	141	11	katyar	katyar	PROPN
cana-2867	142	1	[	[	X
cana-2867	142	2	38	38	NUM
cana-2867	142	3	]	]	PUNCT
cana-2867	142	4	presented	present	VERB
cana-2867	142	5	the	the	DET
cana-2867	142	6	fixed	fix	VERB
cana-2867	142	7	point	point	NOUN
cana-2867	142	8	theorem	theorem	VERB
cana-2867	142	9	in	in	ADP
cana-2867	142	10	these	these	DET
cana-2867	142	11	way	way	NOUN
cana-2867	142	12	.	.	PUNCT
cana-2867	143	1	theorem	theorem	VERB
cana-2867	143	2	2.6[38	2.6[38	NUM
cana-2867	143	3	]	]	PUNCT
cana-2867	143	4	:	:	PUNCT
cana-2867	143	5	let	let	VERB
cana-2867	143	6	(	(	PUNCT
cana-2867	143	7	xj	xj	PROPN
cana-2867	143	8	,	,	PUNCT
cana-2867	143	9	mj,∗	mj,∗	PROPN
cana-2867	143	10	)	)	PUNCT
cana-2867	143	11	,	,	PUNCT
cana-2867	143	12	j	j	PROPN
cana-2867	143	13	=	=	SYM
cana-2867	143	14	1	1	NUM
cana-2867	143	15	,	,	PUNCT
cana-2867	143	16	−	−	PROPN
cana-2867	143	17	−	−	PROPN
cana-2867	143	18	−−	−−	NOUN
cana-2867	143	19	,	,	PUNCT
cana-2867	143	20	n	n	X
cana-2867	143	21	be	be	AUX
cana-2867	143	22	complete	complete	ADJ
cana-2867	143	23	fuzzy	fuzzy	ADJ
cana-2867	143	24	metric	metric	ADJ
cana-2867	143	25	spaces	space	NOUN
cana-2867	143	26	.	.	PUNCT
cana-2867	144	1	let	let	VERB
cana-2867	144	2	{	{	PUNCT
cana-2867	144	3	xj	xj	PROPN
cana-2867	144	4	,	,	PUNCT
cana-2867	144	5	j	j	PROPN
cana-2867	144	6	=	=	SYM
cana-2867	144	7	1,2	1,2	NUM
cana-2867	144	8	,	,	PUNCT
cana-2867	144	9	−	−	PROPN
cana-2867	144	10	−	−	PROPN
cana-2867	144	11	−	−	NOUN
cana-2867	144	12	,	,	PUNCT
cana-2867	144	13	n	n	CCONJ
cana-2867	144	14	;	;	PUNCT
cana-2867	144	15	wij	wij	PROPN
cana-2867	144	16	l	l	NOUN
cana-2867	144	17	:	:	PUNCT
cana-2867	144	18	xj	xj	PROPN
cana-2867	144	19	→	→	SYM
cana-2867	144	20	xi	xi	PROPN
cana-2867	144	21	,	,	PUNCT
cana-2867	144	22	l	l	NOUN
cana-2867	144	23	=	=	SYM
cana-2867	144	24	1	1	NUM
cana-2867	144	25	,	,	PUNCT
cana-2867	144	26	−	−	PROPN
cana-2867	144	27	−	−	PROPN
cana-2867	144	28	−−	−−	NOUN
cana-2867	144	29	,	,	PUNCT
cana-2867	144	30	mij	mij	NOUN
cana-2867	144	31	and	and	CCONJ
cana-2867	144	32	i	i	PROPN
cana-2867	144	33	,	,	PUNCT
cana-2867	144	34	j	j	PROPN
cana-2867	144	35	=	=	SYM
cana-2867	144	36	1	1	NUM
cana-2867	144	37	,	,	PUNCT
cana-2867	144	38	−	−	PROPN
cana-2867	144	39	−	−	PROPN
cana-2867	144	40	−−	−−	NOUN
cana-2867	144	41	,	,	PUNCT
cana-2867	144	42	n	n	CCONJ
cana-2867	144	43	}	}	PUNCT
cana-2867	144	44	be	be	AUX
cana-2867	144	45	an	an	DET
cana-2867	144	46	mfifs	mfif	NOUN
cana-2867	144	47	with	with	ADP
cana-2867	144	48	contractivity	contractivity	NOUN
cana-2867	144	49	ratios	ratio	NOUN
cana-2867	144	50	=	=	SYM
cana-2867	144	51	max{sij	max{sij	NOUN
cana-2867	144	52	l	l	NOUN
cana-2867	144	53	,	,	PUNCT
cana-2867	145	1	l	l	NOUN
cana-2867	145	2	=	=	SYM
cana-2867	145	3	1	1	NUM
cana-2867	145	4	,	,	PUNCT
cana-2867	145	5	−	−	PROPN
cana-2867	145	6	−	−	PROPN
cana-2867	145	7	−−	−−	NOUN
cana-2867	145	8	,	,	PUNCT
cana-2867	145	9	mij	mij	NOUN
cana-2867	145	10	and	and	CCONJ
cana-2867	145	11	i	i	PROPN
cana-2867	145	12	,	,	PUNCT
cana-2867	145	13	j	j	PROPN
cana-2867	145	14	=	=	SYM
cana-2867	145	15	1	1	NUM
cana-2867	145	16	,	,	PUNCT
cana-2867	145	17	−	−	PROPN
cana-2867	145	18	−	−	PROPN
cana-2867	145	19	−−	−−	NOUN
cana-2867	145	20	,	,	PUNCT
cana-2867	145	21	n}and	n}and	ADV
cana-2867	146	1	w	w	AUX
cana-2867	146	2	be	be	AUX
cana-2867	146	3	the	the	DET
cana-2867	146	4	mfhb	mfhb	NOUN
cana-2867	146	5	operator	operator	NOUN
cana-2867	146	6	of	of	ADP
cana-2867	146	7	the	the	DET
cana-2867	146	8	mfifs	mfif	NOUN
cana-2867	146	9	.	.	PUNCT
cana-2867	147	1	then	then	ADV
cana-2867	147	2	,	,	PUNCT
cana-2867	147	3	there	there	PRON
cana-2867	147	4	exists	exist	VERB
cana-2867	147	5	only	only	ADV
cana-2867	147	6	one	one	NUM
cana-2867	147	7	compact	compact	ADJ
cana-2867	147	8	invariant	invariant	ADJ
cana-2867	147	9	set	set	NOUN
cana-2867	147	10	,	,	PUNCT
cana-2867	147	11	also	also	ADV
cana-2867	147	12	called	call	VERB
cana-2867	147	13	the	the	DET
cana-2867	147	14	attractor	attractor	NOUN
cana-2867	147	15	(	(	PUNCT
cana-2867	147	16	fractal	fractal	PROPN
cana-2867	147	17	)	)	PUNCT
cana-2867	147	18	of	of	ADP
cana-2867	147	19	mfifs	mfifs	PROPN
cana-2867	147	20	,	,	PUNCT
cana-2867	147	21	𝒜∞	𝒜∞	VERB
cana-2867	147	22	∈	∈	PROPN
cana-2867	147	23	𝒦(x	𝒦(x	NOUN
cana-2867	147	24	)	)	PUNCT
cana-2867	147	25	of	of	ADP
cana-2867	147	26	the	the	DET
cana-2867	147	27	mfhb	mfhb	NOUN
cana-2867	147	28	operator	operator	NOUN
cana-2867	147	29	w.	w.	PROPN
cana-2867	147	30	1	1	NUM
cana-2867	147	31	.	.	PUNCT
cana-2867	147	32	generalized	generalize	VERB
cana-2867	147	33	fuzzy	fuzzy	ADJ
cana-2867	147	34	contraction	contraction	NOUN
cana-2867	147	35	we	we	PRON
cana-2867	147	36	mainly	mainly	ADV
cana-2867	147	37	discuss	discuss	VERB
cana-2867	147	38	in	in	ADP
cana-2867	147	39	this	this	DET
cana-2867	147	40	section	section	NOUN
cana-2867	147	41	the	the	DET
cana-2867	147	42	notion	notion	NOUN
cana-2867	147	43	of	of	ADP
cana-2867	147	44	generalized	generalized	ADJ
cana-2867	147	45	fuzzy	fuzzy	ADJ
cana-2867	147	46	contraction	contraction	NOUN
cana-2867	147	47	presented	present	VERB
cana-2867	147	48	by	by	ADP
cana-2867	147	49	wardowski	wardowski	PROPN
cana-2867	147	50	[	[	X
cana-2867	147	51	40	40	NUM
cana-2867	147	52	]	]	PUNCT
cana-2867	147	53	.	.	PUNCT
cana-2867	148	1	we	we	PRON
cana-2867	148	2	also	also	ADV
cana-2867	148	3	discussed	discuss	VERB
cana-2867	148	4	the	the	DET
cana-2867	148	5	ℋ-ifs	ℋ-ifs	PROPN
cana-2867	148	6	theory	theory	NOUN
cana-2867	148	7	introduced	introduce	VERB
cana-2867	148	8	by	by	ADP
cana-2867	148	9	r.	r.	PROPN
cana-2867	148	10	uthayakumar	uthayakumar	PROPN
cana-2867	148	11	and	and	CCONJ
cana-2867	148	12	a.	a.	NOUN
cana-2867	148	13	gorishankar	gorishankar	NOUN
cana-2867	149	1	[	[	X
cana-2867	149	2	39	39	NUM
cana-2867	149	3	]	]	PUNCT
cana-2867	149	4	.	.	PUNCT
cana-2867	150	1	they	they	PRON
cana-2867	150	2	obtained	obtain	VERB
cana-2867	150	3	the	the	DET
cana-2867	150	4	attractor	attractor	NOUN
cana-2867	150	5	as	as	ADP
cana-2867	150	6	a	a	DET
cana-2867	150	7	compact	compact	ADJ
cana-2867	150	8	invariant	invariant	ADJ
cana-2867	150	9	subset	subset	NOUN
cana-2867	150	10	generated	generate	VERB
cana-2867	150	11	by	by	ADP
cana-2867	150	12	the	the	DET
cana-2867	150	13	banach	banach	ADV
cana-2867	150	14	fixed	fix	VERB
cana-2867	150	15	point	point	NOUN
cana-2867	150	16	theorem	theorem	VERB
cana-2867	150	17	[	[	X
cana-2867	150	18	43	43	NUM
cana-2867	150	19	]	]	PUNCT
cana-2867	150	20	in	in	ADP
cana-2867	150	21	m	m	ADJ
cana-2867	150	22	-	-	ADJ
cana-2867	150	23	complete	complete	ADJ
cana-2867	150	24	metric	metric	ADJ
cana-2867	150	25	space	space	NOUN
cana-2867	150	26	.	.	PUNCT
cana-2867	151	1	properties	property	NOUN
cana-2867	151	2	of	of	ADP
cana-2867	151	3	h	h	NOUN
cana-2867	151	4	-	-	PUNCT
cana-2867	151	5	contraction	contraction	NOUN
cana-2867	151	6	let	let	VERB
cana-2867	151	7	ℋ	ℋ	PROPN
cana-2867	151	8	represents	represent	VERB
cana-2867	151	9	a	a	DET
cana-2867	151	10	set	set	NOUN
cana-2867	151	11	of	of	ADP
cana-2867	151	12	mappings	mapping	NOUN
cana-2867	151	13	ƞ	ƞ	NOUN
cana-2867	151	14	:	:	PUNCT
cana-2867	151	15	(	(	PUNCT
cana-2867	151	16	0,1	0,1	NUM
cana-2867	151	17	]	]	PUNCT
cana-2867	151	18	→	→	PUNCT
cana-2867	152	1	[	[	X
cana-2867	152	2	0	0	NUM
cana-2867	152	3	,	,	PUNCT
cana-2867	152	4	∞	∞	NUM
cana-2867	152	5	)	)	PUNCT
cana-2867	152	6	and	and	CCONJ
cana-2867	152	7	s	s	VERB
cana-2867	152	8	>	>	X
cana-2867	152	9	t	t	PROPN
cana-2867	152	10	⇒	⇒	PROPN
cana-2867	152	11	ƞ(s	ƞ(s	PROPN
cana-2867	152	12	)	)	PUNCT
cana-2867	152	13	<	<	X
cana-2867	152	14	ƞ(t	ƞ(t	NOUN
cana-2867	152	15	)	)	PUNCT
cana-2867	152	16	for	for	ADP
cana-2867	152	17	everys	every	NOUN
cana-2867	152	18	,	,	PUNCT
cana-2867	152	19	t	t	PROPN
cana-2867	152	20	∈	∈	PROPN
cana-2867	152	21	(	(	PUNCT
cana-2867	152	22	0,1	0,1	NOUN
cana-2867	152	23	]	]	PUNCT
cana-2867	152	24	.	.	PUNCT
cana-2867	153	1	definition	definition	NOUN
cana-2867	153	2	3.1[40	3.1[40	NUM
cana-2867	153	3	]	]	PUNCT
cana-2867	153	4	in	in	ADP
cana-2867	153	5	a	a	DET
cana-2867	153	6	fuzzy	fuzzy	ADJ
cana-2867	153	7	metric	metric	ADJ
cana-2867	153	8	space	space	NOUN
cana-2867	153	9	(	(	PUNCT
cana-2867	153	10	x	x	NOUN
cana-2867	153	11	,	,	PUNCT
cana-2867	153	12	m,∗	m,∗	NOUN
cana-2867	153	13	)	)	PUNCT
cana-2867	153	14	contraction	contraction	NOUN
cana-2867	153	15	mapping	map	VERB
cana-2867	154	1	f	f	NOUN
cana-2867	154	2	:	:	PUNCT
cana-2867	154	3	x	x	SYM
cana-2867	154	4	→	→	PUNCT
cana-2867	154	5	x	x	X
cana-2867	154	6	is	be	AUX
cana-2867	154	7	known	know	VERB
cana-2867	154	8	as	as	ADP
cana-2867	154	9	ℋ	ℋ	PROPN
cana-2867	154	10	–	–	PUNCT
cana-2867	154	11	contractive	contractive	ADJ
cana-2867	154	12	such	such	ADJ
cana-2867	154	13	that	that	PRON
cana-2867	154	14	ƞ(m(f(x	ƞ(m(f(x	PROPN
cana-2867	154	15	)	)	PUNCT
cana-2867	154	16	,	,	PUNCT
cana-2867	154	17	f(y	f(y	NOUN
cana-2867	154	18	)	)	PUNCT
cana-2867	154	19	,	,	PUNCT
cana-2867	154	20	t	t	PROPN
cana-2867	154	21	)	)	PUNCT
cana-2867	154	22	)	)	PUNCT
cana-2867	155	1	≤	≤	NOUN
cana-2867	155	2	kƞ(m(x	kƞ(m(x	NUM
cana-2867	155	3	,	,	PUNCT
cana-2867	155	4	y	y	PROPN
cana-2867	155	5	,	,	PUNCT
cana-2867	155	6	t	t	PROPN
cana-2867	155	7	)	)	PUNCT
cana-2867	155	8	)	)	PUNCT
cana-2867	156	1	(	(	PUNCT
cana-2867	156	2	1	1	X
cana-2867	156	3	)	)	PUNCT
cana-2867	156	4	where	where	SCONJ
cana-2867	156	5	ƞ	ƞ	PROPN
cana-2867	156	6	∈	∈	PROPN
cana-2867	156	7	ℋ	ℋ	PROPN
cana-2867	156	8	and	and	CCONJ
cana-2867	156	9	k	k	PROPN
cana-2867	156	10	∈	∈	PROPN
cana-2867	156	11	(	(	PUNCT
cana-2867	156	12	0,1	0,1	NOUN
cana-2867	156	13	)	)	PUNCT
cana-2867	156	14	for	for	ADP
cana-2867	156	15	everyx	everyx	NOUN
cana-2867	156	16	,	,	PUNCT
cana-2867	156	17	y	y	PROPN
cana-2867	156	18	∈	∈	PROPN
cana-2867	156	19	x	x	X
cana-2867	156	20	and	and	CCONJ
cana-2867	156	21	t	t	X
cana-2867	156	22	>	>	X
cana-2867	156	23	0	0	PROPN
cana-2867	156	24	.	.	PUNCT
cana-2867	156	25	example	example	NOUN
cana-2867	157	1	3.1[40	3.1[40	NUM
cana-2867	157	2	]	]	X
cana-2867	157	3	let	let	VERB
cana-2867	157	4	ƞ	ƞ	NOUN
cana-2867	157	5	∈	∈	PROPN
cana-2867	157	6	ℋbe	ℋbe	PROPN
cana-2867	157	7	a	a	DET
cana-2867	157	8	mapping	mapping	NOUN
cana-2867	157	9	and	and	CCONJ
cana-2867	157	10	defined	define	VERB
cana-2867	157	11	byƞ(t	byƞ(t	PROPN
cana-2867	157	12	)	)	PUNCT
cana-2867	157	13	=	=	NOUN
cana-2867	157	14	1	1	NUM
cana-2867	157	15	t	t	NOUN
cana-2867	157	16	−	−	NOUN
cana-2867	157	17	1	1	NUM
cana-2867	157	18	,	,	PUNCT
cana-2867	157	19	t	t	PROPN
cana-2867	157	20	∈	∈	PROPN
cana-2867	157	21	(	(	PUNCT
cana-2867	157	22	0,1	0,1	NOUN
cana-2867	157	23	]	]	PUNCT
cana-2867	157	24	.	.	PUNCT
cana-2867	158	1	when	when	SCONJ
cana-2867	158	2	the	the	DET
cana-2867	158	3	equation	equation	NOUN
cana-2867	158	4	(	(	PUNCT
cana-2867	158	5	1	1	X
cana-2867	158	6	)	)	PUNCT
cana-2867	158	7	changes	change	NOUN
cana-2867	158	8	to	to	ADP
cana-2867	158	9	1	1	NUM
cana-2867	158	10	m(f(x	m(f(x	NOUN
cana-2867	158	11	)	)	PUNCT
cana-2867	158	12	,	,	PUNCT
cana-2867	158	13	f(y	f(y	NOUN
cana-2867	158	14	)	)	PUNCT
cana-2867	158	15	,	,	PUNCT
cana-2867	158	16	t	t	PROPN
cana-2867	158	17	)	)	PUNCT
cana-2867	158	18	−	−	PROPN
cana-2867	158	19	1	1	NUM
cana-2867	158	20	≤	≤	NUM
cana-2867	158	21	k	k	X
cana-2867	158	22	(	(	PUNCT
cana-2867	158	23	1	1	NUM
cana-2867	158	24	m(x	m(x	PROPN
cana-2867	158	25	,	,	PUNCT
cana-2867	158	26	y	y	PROPN
cana-2867	158	27	,	,	PUNCT
cana-2867	158	28	t	t	PROPN
cana-2867	158	29	)	)	PUNCT
cana-2867	158	30	−	−	PROPN
cana-2867	158	31	1	1	NUM
cana-2867	158	32	)	)	PUNCT
cana-2867	158	33	for	for	ADP
cana-2867	158	34	all	all	DET
cana-2867	158	35	x	x	NOUN
cana-2867	158	36	,	,	PUNCT
cana-2867	158	37	y	y	PROPN
cana-2867	158	38	∈	∈	PROPN
cana-2867	158	39	x	x	X
cana-2867	158	40	and	and	CCONJ
cana-2867	158	41	t	t	X
cana-2867	158	42	>	>	X
cana-2867	158	43	0	0	X
cana-2867	158	44	.	.	PUNCT
cana-2867	158	45	proposition	proposition	NOUN
cana-2867	158	46	3.1[40	3.1[40	NOUN
cana-2867	158	47	]	]	X
cana-2867	158	48	suppose	suppose	VERB
cana-2867	158	49	that(x	that(x	NUM
cana-2867	158	50	,	,	PUNCT
cana-2867	158	51	m,∗	m,∗	NOUN
cana-2867	158	52	)	)	PUNCT
cana-2867	158	53	be	be	AUX
cana-2867	158	54	a	a	DET
cana-2867	158	55	fuzzy	fuzzy	ADJ
cana-2867	158	56	metric	metric	ADJ
cana-2867	158	57	space	space	NOUN
cana-2867	158	58	and	and	CCONJ
cana-2867	158	59	ƞ	ƞ	PRON
cana-2867	158	60	∈	∈	PROPN
cana-2867	158	61	ℋ.	ℋ.	PROPN
cana-2867	158	62	a	a	DET
cana-2867	158	63	sequence	sequence	NOUN
cana-2867	158	64	{	{	PUNCT
cana-2867	158	65	xn}n∈n	xn}n∈n	PUNCT
cana-2867	158	66	⊂	⊂	PROPN
cana-2867	158	67	x	x	X
cana-2867	158	68	is	be	AUX
cana-2867	158	69	m	m	NOUN
cana-2867	158	70	-	-	ADJ
cana-2867	158	71	cauchy	cauchy	ADJ
cana-2867	158	72	iff	iff	NOUN
cana-2867	158	73	for	for	ADP
cana-2867	158	74	given	give	VERB
cana-2867	158	75	ε	ε	PROPN
cana-2867	158	76	>	>	X
cana-2867	158	77	0	0	PROPN
cana-2867	158	78	,	,	PUNCT
cana-2867	158	79	t	t	X
cana-2867	158	80	>	>	X
cana-2867	158	81	0	0	PUNCT
cana-2867	159	1	there	there	PRON
cana-2867	159	2	exists	exist	VERB
cana-2867	159	3	n0	n0	PROPN
cana-2867	159	4	∈	∈	PROPN
cana-2867	159	5	n	n	PRON
cana-2867	159	6	such	such	ADJ
cana-2867	159	7	that	that	DET
cana-2867	159	8	ƞ(m(xm	ƞ(m(xm	NOUN
cana-2867	159	9	,	,	PUNCT
cana-2867	159	10	xn	xn	PROPN
cana-2867	159	11	,	,	PUNCT
cana-2867	159	12	t	t	PROPN
cana-2867	159	13	)	)	PUNCT
cana-2867	159	14	)	)	PUNCT
cana-2867	160	1	<	<	X
cana-2867	160	2	ε	ε	PROPN
cana-2867	160	3	,	,	PUNCT
cana-2867	160	4	for	for	ADP
cana-2867	160	5	all	all	DET
cana-2867	160	6	m	m	PROPN
cana-2867	160	7	,	,	PUNCT
cana-2867	160	8	n	n	PRON
cana-2867	160	9	≥	≥	NOUN
cana-2867	160	10	n0	n0	NUM
cana-2867	160	11	.	.	PUNCT
cana-2867	161	1	proposition	proposition	NOUN
cana-2867	161	2	3.1[40	3.1[40	PROPN
cana-2867	161	3	]	]	PUNCT
cana-2867	161	4	suppose	suppose	VERB
cana-2867	161	5	that	that	SCONJ
cana-2867	161	6	(	(	PUNCT
cana-2867	161	7	x	x	NOUN
cana-2867	161	8	,	,	PUNCT
cana-2867	161	9	m,∗	m,∗	NOUN
cana-2867	161	10	)	)	PUNCT
cana-2867	161	11	be	be	AUX
cana-2867	161	12	a	a	DET
cana-2867	161	13	fuzzy	fuzzy	ADJ
cana-2867	161	14	metric	metric	ADJ
cana-2867	161	15	space	space	NOUN
cana-2867	161	16	and	and	CCONJ
cana-2867	161	17	ƞ	ƞ	PRON
cana-2867	161	18	∈	∈	PROPN
cana-2867	161	19	ℋ.	ℋ.	PROPN
cana-2867	161	20	a	a	DET
cana-2867	161	21	sequence	sequence	NOUN
cana-2867	161	22	{	{	PUNCT
cana-2867	161	23	xn}n∈n	xn}n∈n	PUNCT
cana-2867	161	24	⊂	⊂	PROPN
cana-2867	161	25	x	x	X
cana-2867	161	26	is	be	AUX
cana-2867	161	27	convergent	convergent	ADJ
cana-2867	161	28	to	to	ADP
cana-2867	161	29	x	x	PART
cana-2867	161	30	∈	∈	PROPN
cana-2867	161	31	x	x	PROPN
cana-2867	161	32	iff	iff	PROPN
cana-2867	161	33	lim	lim	PROPN
cana-2867	161	34	n→∞	n→∞	NUM
cana-2867	161	35	ƞ(m(xn	ƞ(m(xn	NUM
cana-2867	161	36	,	,	PUNCT
cana-2867	161	37	x	x	NOUN
cana-2867	161	38	,	,	PUNCT
cana-2867	161	39	t	t	PROPN
cana-2867	161	40	)	)	PUNCT
cana-2867	161	41	)	)	PUNCT
cana-2867	162	1	=	=	SYM
cana-2867	162	2	0	0	NUM
cana-2867	163	1	for	for	ADP
cana-2867	163	2	all	all	DET
cana-2867	163	3	t	t	PROPN
cana-2867	163	4	>	>	X
cana-2867	163	5	0	0	X
cana-2867	163	6	.	.	PUNCT
cana-2867	163	7	theorem	theorem	VERB
cana-2867	163	8	3.1[39	3.1[39	NUM
cana-2867	163	9	]	]	X
cana-2867	163	10	let	let	VERB
cana-2867	163	11	(	(	PUNCT
cana-2867	163	12	𝒦(x	𝒦(x	NUM
cana-2867	163	13	)	)	PUNCT
cana-2867	163	14	,	,	PUNCT
cana-2867	163	15	hm,∗	hm,∗	ADJ
cana-2867	163	16	)	)	PUNCT
cana-2867	163	17	is	be	AUX
cana-2867	163	18	a	a	DET
cana-2867	163	19	hausdorff	hausdorff	NOUN
cana-2867	163	20	fuzzy	fuzzy	ADJ
cana-2867	163	21	metric	metric	ADJ
cana-2867	163	22	space	space	NOUN
cana-2867	163	23	in	in	ADP
cana-2867	163	24	fuzzy	fuzzy	ADJ
cana-2867	163	25	metric	metric	ADJ
cana-2867	163	26	space	space	NOUN
cana-2867	163	27	(	(	PUNCT
cana-2867	163	28	x	x	NOUN
cana-2867	163	29	,	,	PUNCT
cana-2867	163	30	m,∗	m,∗	NOUN
cana-2867	163	31	)	)	PUNCT
cana-2867	163	32	then	then	ADV
cana-2867	163	33	mapping	map	VERB
cana-2867	163	34	w	w	X
cana-2867	163	35	:	:	PUNCT
cana-2867	163	36	x	x	SYM
cana-2867	163	37	→	→	PUNCT
cana-2867	163	38	x	x	X
cana-2867	163	39	is	be	AUX
cana-2867	163	40	a	a	DET
cana-2867	163	41	fuzzy	fuzzy	ADJ
cana-2867	163	42	ℋ-contraction	ℋ-contraction	PROPN
cana-2867	163	43	mapping	mapping	NOUN
cana-2867	163	44	with	with	ADP
cana-2867	163	45	respect	respect	NOUN
cana-2867	163	46	to	to	ADP
cana-2867	163	47	ƞ	ƞ	PROPN
cana-2867	163	48	∈	∈	PROPN
cana-2867	163	49	ℋ	ℋ	PROPN
cana-2867	163	50	on(x	on(x	NUM
cana-2867	163	51	,	,	PUNCT
cana-2867	163	52	m,∗	m,∗	NOUN
cana-2867	163	53	)	)	PUNCT
cana-2867	163	54	.	.	PUNCT
cana-2867	164	1	communications	communication	NOUN
cana-2867	164	2	on	on	ADP
cana-2867	164	3	applied	apply	VERB
cana-2867	164	4	nonlinear	nonlinear	ADJ
cana-2867	164	5	analysis	analysis	NOUN
cana-2867	164	6	issn	issn	NOUN
cana-2867	164	7	:	:	PUNCT
cana-2867	164	8	1074	1074	NUM
cana-2867	164	9	-	-	PUNCT
cana-2867	164	10	133x	133x	NUM
cana-2867	164	11	vol	vol	NOUN
cana-2867	164	12	32	32	NUM
cana-2867	164	13	no	no	NOUN
cana-2867	164	14	.	.	PUNCT
cana-2867	165	1	4s	4s	NUM
cana-2867	165	2	(	(	PUNCT
cana-2867	165	3	2025	2025	NUM
cana-2867	165	4	)	)	PUNCT
cana-2867	165	5	506	506	NUM
cana-2867	165	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	165	7	a.	a.	NOUN
cana-2867	165	8	iterated	iterate	VERB
cana-2867	165	9	function	function	NOUN
cana-2867	165	10	system	system	NOUN
cana-2867	165	11	consisting	consist	VERB
cana-2867	165	12	of	of	ADP
cana-2867	165	13	𝐇-contraction	𝐇-contraction	PROPN
cana-2867	165	14	definition	definition	NOUN
cana-2867	165	15	3.2[39	3.2[39	NUM
cana-2867	165	16	]	]	PUNCT
cana-2867	165	17	suppose	suppose	VERB
cana-2867	165	18	that(x	that(x	NUM
cana-2867	165	19	,	,	PUNCT
cana-2867	165	20	m,∗	m,∗	NOUN
cana-2867	165	21	)	)	PUNCT
cana-2867	165	22	be	be	AUX
cana-2867	165	23	a	a	DET
cana-2867	165	24	fuzzy	fuzzy	ADJ
cana-2867	165	25	metric	metric	ADJ
cana-2867	165	26	space	space	NOUN
cana-2867	165	27	and	and	CCONJ
cana-2867	165	28	wn	wn	NOUN
cana-2867	165	29	:	:	PUNCT
cana-2867	165	30	x	x	SYM
cana-2867	165	31	→	→	SYM
cana-2867	165	32	x	x	SYM
cana-2867	165	33	,	,	PUNCT
cana-2867	165	34	n	n	NOUN
cana-2867	165	35	=	=	SYM
cana-2867	165	36	1	1	NUM
cana-2867	165	37	,	,	PUNCT
cana-2867	165	38	−	−	PROPN
cana-2867	165	39	−	−	PROPN
cana-2867	165	40	−	−	NOUN
cana-2867	165	41	,	,	PUNCT
cana-2867	165	42	n	n	PRON
cana-2867	165	43	be	be	VERB
cana-2867	165	44	n	n	PRON
cana-2867	165	45	−	−	PROPN
cana-2867	165	46	ℋ	ℋ	PROPN
cana-2867	165	47	contractive	contractive	ADJ
cana-2867	165	48	mappings	mapping	NOUN
cana-2867	165	49	.	.	PUNCT
cana-2867	166	1	then	then	ADV
cana-2867	166	2	the	the	DET
cana-2867	166	3	system	system	NOUN
cana-2867	166	4	{	{	PUNCT
cana-2867	166	5	x	x	NOUN
cana-2867	166	6	;	;	PUNCT
cana-2867	166	7	wn	wn	PROPN
cana-2867	166	8	,	,	PUNCT
cana-2867	166	9	n	n	NOUN
cana-2867	166	10	=	=	SYM
cana-2867	166	11	1	1	NUM
cana-2867	166	12	,	,	PUNCT
cana-2867	166	13	−	−	PROPN
cana-2867	166	14	−	−	PROPN
cana-2867	166	15	−	−	NOUN
cana-2867	166	16	,	,	PUNCT
cana-2867	166	17	n	n	CCONJ
cana-2867	166	18	}	}	PUNCT
cana-2867	166	19	is	be	AUX
cana-2867	166	20	called	call	VERB
cana-2867	166	21	a	a	DET
cana-2867	166	22	ℋ-ifs	ℋ-ifs	PROPN
cana-2867	166	23	of	of	ADP
cana-2867	166	24	ℋ	ℋ	PROPN
cana-2867	166	25	–	–	PUNCT
cana-2867	166	26	contractions	contraction	NOUN
cana-2867	166	27	in	in	ADP
cana-2867	166	28	(	(	PUNCT
cana-2867	166	29	x	x	NOUN
cana-2867	166	30	,	,	PUNCT
cana-2867	166	31	m,∗	m,∗	NOUN
cana-2867	166	32	)	)	PUNCT
cana-2867	166	33	.	.	PUNCT
cana-2867	167	1	the	the	DET
cana-2867	167	2	hb	hb	PROPN
cana-2867	167	3	operator	operator	NOUN
cana-2867	167	4	of	of	ADP
cana-2867	167	5	the	the	DET
cana-2867	167	6	ℋ-ifs	ℋ-ifs	PROPN
cana-2867	167	7	is	be	AUX
cana-2867	167	8	a	a	DET
cana-2867	167	9	function	function	NOUN
cana-2867	167	10	w	w	NOUN
cana-2867	167	11	:	:	PUNCT
cana-2867	167	12	𝒦(x	𝒦(x	NUM
cana-2867	167	13	)	)	PUNCT
cana-2867	167	14	→	→	SYM
cana-2867	167	15	𝒦(x	𝒦(x	X
cana-2867	167	16	)	)	PUNCT
cana-2867	167	17	defined	define	VERB
cana-2867	167	18	by	by	ADP
cana-2867	167	19	w(b	w(b	NOUN
cana-2867	167	20	)	)	PUNCT
cana-2867	167	21	=	=	SYM
cana-2867	167	22	⋃	⋃	NOUN
cana-2867	167	23	wn(b	wn(b	NOUN
cana-2867	167	24	)	)	PUNCT
cana-2867	167	25	n	n	CCONJ
cana-2867	167	26	n=1	n=1	PROPN
cana-2867	167	27	,	,	PUNCT
cana-2867	167	28	for	for	ADP
cana-2867	167	29	all	all	DET
cana-2867	167	30	b	b	NOUN
cana-2867	167	31	∈	∈	PROPN
cana-2867	167	32	𝒦(x	𝒦(x	NOUN
cana-2867	167	33	)	)	PUNCT
cana-2867	167	34	definition	definition	NOUN
cana-2867	167	35	3.3	3.3	NUM
cana-2867	167	36	[	[	SYM
cana-2867	167	37	39	39	NUM
cana-2867	167	38	]	]	PUNCT
cana-2867	167	39	suppose	suppose	VERB
cana-2867	167	40	that(x	that(x	NUM
cana-2867	167	41	,	,	PUNCT
cana-2867	167	42	m,∗	m,∗	NOUN
cana-2867	167	43	)	)	PUNCT
cana-2867	167	44	be	be	AUX
cana-2867	167	45	a	a	DET
cana-2867	167	46	fuzzy	fuzzy	ADJ
cana-2867	167	47	metric	metric	ADJ
cana-2867	167	48	space	space	NOUN
cana-2867	167	49	.	.	PUNCT
cana-2867	168	1	let	let	VERB
cana-2867	168	2	{	{	PUNCT
cana-2867	168	3	x	x	NOUN
cana-2867	168	4	,	,	PUNCT
cana-2867	168	5	wn	wn	PROPN
cana-2867	168	6	,	,	PUNCT
cana-2867	168	7	n	n	NOUN
cana-2867	168	8	=	=	SYM
cana-2867	168	9	1,2,3	1,2,3	NUM
cana-2867	168	10	,	,	PUNCT
cana-2867	168	11	−	−	PROPN
cana-2867	168	12	−	−	PROPN
cana-2867	169	1	−	−	NOUN
cana-2867	170	1	−	−	PROPN
cana-2867	170	2	−	−	PROPN
cana-2867	170	3	,	,	PUNCT
cana-2867	170	4	n	n	CCONJ
cana-2867	170	5	}	}	PUNCT
cana-2867	170	6	be	be	AUX
cana-2867	170	7	a	a	DET
cana-2867	170	8	ℋ-ifs	ℋ-ifs	PROPN
cana-2867	170	9	and	and	CCONJ
cana-2867	170	10	w	w	PROPN
cana-2867	170	11	be	be	AUX
cana-2867	170	12	the	the	DET
cana-2867	170	13	hb	hb	PROPN
cana-2867	170	14	operator	operator	NOUN
cana-2867	170	15	of	of	ADP
cana-2867	170	16	the	the	DET
cana-2867	170	17	ℋ-ifs	ℋ-ifs	PROPN
cana-2867	170	18	.	.	PUNCT
cana-2867	171	1	if	if	SCONJ
cana-2867	171	2	w	w	NOUN
cana-2867	171	3	has	have	VERB
cana-2867	171	4	a	a	DET
cana-2867	171	5	unique	unique	ADJ
cana-2867	171	6	fixed	fix	VERB
cana-2867	171	7	point	point	NOUN
cana-2867	171	8	a∞	a∞	PROPN
cana-2867	171	9	in	in	ADP
cana-2867	171	10	(	(	PUNCT
cana-2867	171	11	x	x	NOUN
cana-2867	171	12	,	,	PUNCT
cana-2867	171	13	m,∗	m,∗	NOUN
cana-2867	171	14	)	)	PUNCT
cana-2867	171	15	,	,	PUNCT
cana-2867	171	16	then	then	ADV
cana-2867	171	17	the	the	DET
cana-2867	171	18	set	set	NOUN
cana-2867	171	19	𝒜∞	𝒜∞	VERB
cana-2867	171	20	∈	∈	PROPN
cana-2867	171	21	k0(x	k0(x	NOUN
cana-2867	171	22	)	)	PUNCT
cana-2867	171	23	is	be	AUX
cana-2867	171	24	known	know	VERB
cana-2867	171	25	as	as	ADP
cana-2867	171	26	the	the	DET
cana-2867	171	27	attractor	attractor	NOUN
cana-2867	171	28	(	(	PUNCT
cana-2867	171	29	or	or	CCONJ
cana-2867	171	30	fractal	fractal	NOUN
cana-2867	171	31	)	)	PUNCT
cana-2867	171	32	produced	produce	VERB
cana-2867	171	33	by	by	ADP
cana-2867	171	34	the	the	DET
cana-2867	171	35	ℋ-ifs	ℋ-ifs	PROPN
cana-2867	171	36	of	of	ADP
cana-2867	171	37	ℋ-contraction	ℋ-contraction	PROPN
cana-2867	171	38	.	.	PUNCT
cana-2867	172	1	3	3	NUM
cana-2867	172	2	.	.	X
cana-2867	172	3	main	main	ADJ
cana-2867	172	4	results	result	NOUN
cana-2867	172	5	now	now	ADV
cana-2867	172	6	,	,	PUNCT
cana-2867	172	7	we	we	PRON
cana-2867	172	8	define	define	VERB
cana-2867	172	9	and	and	CCONJ
cana-2867	172	10	establish	establish	VERB
cana-2867	172	11	multifuzzy	multifuzzy	NOUN
cana-2867	172	12	fractals	fractal	NOUN
cana-2867	172	13	(	(	PUNCT
cana-2867	172	14	attractors	attractor	NOUN
cana-2867	172	15	)	)	PUNCT
cana-2867	172	16	consisting	consist	VERB
cana-2867	172	17	of	of	ADP
cana-2867	172	18	generalized	generalized	ADJ
cana-2867	172	19	fuzzy	fuzzy	ADJ
cana-2867	172	20	contractions	contraction	NOUN
cana-2867	172	21	defined	define	VERB
cana-2867	172	22	on	on	ADP
cana-2867	172	23	multifuzzy	multifuzzy	NOUN
cana-2867	172	24	fractal	fractal	ADJ
cana-2867	172	25	space	space	NOUN
cana-2867	172	26	.	.	PUNCT
cana-2867	173	1	our	our	PRON
cana-2867	173	2	research	research	NOUN
cana-2867	173	3	work	work	NOUN
cana-2867	173	4	starts	start	VERB
cana-2867	173	5	with	with	ADP
cana-2867	173	6	the	the	DET
cana-2867	173	7	corresponding	corresponding	ADJ
cana-2867	173	8	result	result	NOUN
cana-2867	173	9	.	.	PUNCT
cana-2867	174	1	theorem4.1	theorem4.1	NOUN
cana-2867	174	2	assume	assume	VERB
cana-2867	174	3	that	that	SCONJ
cana-2867	174	4	(	(	PUNCT
cana-2867	174	5	xj	xj	PROPN
cana-2867	174	6	,	,	PUNCT
cana-2867	174	7	mj,∗	mj,∗	PROPN
cana-2867	174	8	)	)	PUNCT
cana-2867	174	9	be	be	AUX
cana-2867	174	10	a	a	DET
cana-2867	174	11	complete	complete	ADJ
cana-2867	174	12	fuzzy	fuzzy	ADJ
cana-2867	174	13	metric	metric	ADJ
cana-2867	174	14	spaces	space	NOUN
cana-2867	174	15	for	for	ADP
cana-2867	174	16	j	j	PROPN
cana-2867	174	17	=	=	SYM
cana-2867	174	18	1	1	NUM
cana-2867	174	19	,	,	PUNCT
cana-2867	174	20	−	−	PROPN
cana-2867	174	21	−	−	PROPN
cana-2867	174	22	−−	−−	NOUN
cana-2867	174	23	,	,	PUNCT
cana-2867	174	24	n	n	CCONJ
cana-2867	174	25	with	with	ADP
cana-2867	174	26	same	same	ADJ
cana-2867	174	27	t	t	NOUN
cana-2867	174	28	-	-	PUNCT
cana-2867	174	29	norm	norm	NOUN
cana-2867	174	30	.	.	PUNCT
cana-2867	175	1	let	let	VERB
cana-2867	175	2	(	(	PUNCT
cana-2867	175	3	𝒦(xj	𝒦(xj	NOUN
cana-2867	175	4	)	)	PUNCT
cana-2867	175	5	,	,	PUNCT
cana-2867	175	6	hmj	hmj	PROPN
cana-2867	175	7	,	,	PUNCT
cana-2867	175	8	∗	∗	PROPN
cana-2867	175	9	)	)	PUNCT
cana-2867	175	10	be	be	VERB
cana-2867	175	11	the	the	DET
cana-2867	175	12	corresponding	corresponding	ADJ
cana-2867	175	13	hausdorff	hausdorff	NOUN
cana-2867	175	14	fuzzy	fuzzy	ADJ
cana-2867	175	15	fractal	fractal	ADJ
cana-2867	175	16	spaces	space	NOUN
cana-2867	175	17	.	.	PUNCT
cana-2867	176	1	if	if	SCONJ
cana-2867	176	2	wij	wij	PROPN
cana-2867	176	3	:	:	PUNCT
cana-2867	176	4	xj	xj	PROPN
cana-2867	176	5	→	→	SYM
cana-2867	176	6	xi	xi	PROPN
cana-2867	176	7	,	,	PUNCT
cana-2867	176	8	i	i	PRON
cana-2867	176	9	,	,	PUNCT
cana-2867	176	10	j	j	PROPN
cana-2867	176	11	=	=	SYM
cana-2867	176	12	1	1	NUM
cana-2867	176	13	,	,	PUNCT
cana-2867	176	14	−	−	PROPN
cana-2867	176	15	−	−	PROPN
cana-2867	176	16	−−	−−	NOUN
cana-2867	176	17	,	,	PUNCT
cana-2867	176	18	n	n	PRON
cana-2867	176	19	is	be	AUX
cana-2867	176	20	a	a	DET
cana-2867	176	21	fuzzy	fuzzy	ADJ
cana-2867	176	22	ℋ-contraction	ℋ-contraction	PROPN
cana-2867	176	23	withƞ	withƞ	NOUN
cana-2867	176	24	∈	∈	PROPN
cana-2867	176	25	ℋ	ℋ	PROPN
cana-2867	176	26	and	and	CCONJ
cana-2867	176	27	kij	kij	PROPN
cana-2867	176	28	∈	∈	PROPN
cana-2867	176	29	(	(	PUNCT
cana-2867	176	30	0,1	0,1	NUM
cana-2867	176	31	)	)	PUNCT
cana-2867	176	32	for	for	ADP
cana-2867	176	33	i	i	PRON
cana-2867	176	34	,	,	PUNCT
cana-2867	176	35	j	j	PROPN
cana-2867	176	36	=	=	SYM
cana-2867	176	37	1,2	1,2	NUM
cana-2867	176	38	,	,	PUNCT
cana-2867	176	39	−	−	PROPN
cana-2867	176	40	−	−	PROPN
cana-2867	176	41	−	−	NOUN
cana-2867	176	42	,	,	PUNCT
cana-2867	176	43	n	n	CCONJ
cana-2867	176	44	on	on	ADP
cana-2867	176	45	(	(	PUNCT
cana-2867	176	46	𝒦(xj	𝒦(xj	NOUN
cana-2867	176	47	)	)	PUNCT
cana-2867	176	48	,	,	PUNCT
cana-2867	176	49	hmj	hmj	PROPN
cana-2867	176	50	,	,	PUNCT
cana-2867	176	51	∗	∗	PROPN
cana-2867	176	52	)	)	PUNCT
cana-2867	176	53	.	.	PUNCT
cana-2867	177	1	then	then	ADV
cana-2867	177	2	wij	wij	PROPN
cana-2867	177	3	is	be	AUX
cana-2867	177	4	a	a	DET
cana-2867	177	5	fuzzy	fuzzy	ADJ
cana-2867	177	6	contraction	contraction	NOUN
cana-2867	177	7	with	with	ADP
cana-2867	177	8	ƞ	ƞ	PROPN
cana-2867	177	9	∈	∈	PROPN
cana-2867	177	10	ℋ	ℋ	PROPN
cana-2867	177	11	and	and	CCONJ
cana-2867	177	12	kij	kij	PROPN
cana-2867	177	13	∈	∈	PROPN
cana-2867	177	14	(	(	PUNCT
cana-2867	177	15	0,1	0,1	NOUN
cana-2867	177	16	)	)	PUNCT
cana-2867	177	17	on	on	ADP
cana-2867	177	18	(	(	PUNCT
cana-2867	177	19	𝒦(xj	𝒦(xj	NOUN
cana-2867	177	20	)	)	PUNCT
cana-2867	177	21	,	,	PUNCT
cana-2867	177	22	hmj	hmj	PROPN
cana-2867	177	23	,	,	PUNCT
cana-2867	177	24	∗	∗	PROPN
cana-2867	177	25	)	)	PUNCT
cana-2867	177	26	.	.	PUNCT
cana-2867	178	1	that	that	PRON
cana-2867	178	2	is	be	AUX
cana-2867	178	3	;	;	PUNCT
cana-2867	178	4	ƞ	ƞ	X
cana-2867	178	5	(	(	PUNCT
cana-2867	178	6	hmi	hmi	PROPN
cana-2867	178	7	(	(	PUNCT
cana-2867	178	8	wij(bj	wij(bj	NOUN
cana-2867	178	9	)	)	PUNCT
cana-2867	178	10	,	,	PUNCT
cana-2867	178	11	wij(aj	wij(aj	X
cana-2867	178	12	)	)	PUNCT
cana-2867	178	13	,	,	PUNCT
cana-2867	178	14	t	t	PROPN
cana-2867	178	15	)	)	PUNCT
cana-2867	178	16	)	)	PUNCT
cana-2867	178	17	≤	≤	NUM
cana-2867	178	18	kijƞ	kijƞ	X
cana-2867	178	19	(	(	PUNCT
cana-2867	178	20	hmi	hmi	PROPN
cana-2867	178	21	(	(	PUNCT
cana-2867	178	22	bi	bi	PROPN
cana-2867	178	23	,	,	PUNCT
cana-2867	178	24	ai	ai	VERB
cana-2867	178	25	,	,	PUNCT
cana-2867	178	26	t	t	PROPN
cana-2867	178	27	)	)	PUNCT
cana-2867	178	28	)	)	PUNCT
cana-2867	178	29	,	,	PUNCT
cana-2867	178	30	∀	∀	X
cana-2867	178	31	aj	aj	VERB
cana-2867	178	32	,	,	PUNCT
cana-2867	178	33	bj	bj	ADP
cana-2867	178	34	∈	∈	PROPN
cana-2867	178	35	𝒦(xj	𝒦(xj	NOUN
cana-2867	178	36	)	)	PUNCT
cana-2867	178	37	proof	proof	NOUN
cana-2867	178	38	:	:	PUNCT
cana-2867	178	39	let	let	VERB
cana-2867	178	40	t	t	PROPN
cana-2867	178	41	>	>	X
cana-2867	178	42	0	0	PUNCT
cana-2867	178	43	be	be	AUX
cana-2867	178	44	a	a	DET
cana-2867	178	45	fixed	fix	VERB
cana-2867	178	46	.	.	PUNCT
cana-2867	179	1	let	let	AUX
cana-2867	179	2	ai	ai	VERB
cana-2867	179	3	,	,	PUNCT
cana-2867	179	4	bi	bi	PROPN
cana-2867	179	5	∈	∈	PROPN
cana-2867	179	6	𝒦(xi	𝒦(xi	PROPN
cana-2867	179	7	)	)	PUNCT
cana-2867	180	1	and	and	CCONJ
cana-2867	180	2	aj	aj	PROPN
cana-2867	180	3	,	,	PUNCT
cana-2867	180	4	bj	bj	ADP
cana-2867	180	5	∈	∈	PROPN
cana-2867	180	6	𝒦(xj	𝒦(xj	NOUN
cana-2867	180	7	)	)	PUNCT
cana-2867	180	8	.	.	PUNCT
cana-2867	181	1	we	we	PRON
cana-2867	181	2	know	know	VERB
cana-2867	181	3	that	that	PRON
cana-2867	181	4	,	,	PUNCT
cana-2867	181	5	wij	wij	PROPN
cana-2867	181	6	is	be	AUX
cana-2867	181	7	a	a	DET
cana-2867	181	8	fuzzy	fuzzy	ADJ
cana-2867	181	9	ℋ-contraction	ℋ-contraction	PROPN
cana-2867	181	10	with	with	ADP
cana-2867	181	11	ƞ	ƞ	PROPN
cana-2867	181	12	∈	∈	PROPN
cana-2867	181	13	ℋ	ℋ	PROPN
cana-2867	181	14	on	on	ADP
cana-2867	181	15	(	(	PUNCT
cana-2867	181	16	x	x	NOUN
cana-2867	181	17	,	,	PUNCT
cana-2867	181	18	m,∗	m,∗	NOUN
cana-2867	181	19	)	)	PUNCT
cana-2867	181	20	.	.	PUNCT
cana-2867	182	1	thus	thus	ADV
cana-2867	182	2	,	,	PUNCT
cana-2867	182	3	∃kij	∃kij	PROPN
cana-2867	182	4	∈	∈	PROPN
cana-2867	182	5	(	(	PUNCT
cana-2867	182	6	0,1	0,1	NUM
cana-2867	182	7	)	)	PUNCT
cana-2867	183	1	so	so	SCONJ
cana-2867	183	2	that	that	SCONJ
cana-2867	183	3	ƞ	ƞ	X
cana-2867	183	4	(	(	PUNCT
cana-2867	183	5	mi	mi	PROPN
cana-2867	183	6	(	(	PUNCT
cana-2867	183	7	wij(xj	wij(xj	PROPN
cana-2867	183	8	)	)	PUNCT
cana-2867	183	9	,	,	PUNCT
cana-2867	183	10	wij	wij	PROPN
cana-2867	183	11	(	(	PUNCT
cana-2867	183	12	y	y	PROPN
cana-2867	183	13	j	j	PROPN
cana-2867	183	14	)	)	PUNCT
cana-2867	183	15	,	,	PUNCT
cana-2867	183	16	t	t	PROPN
cana-2867	183	17	)	)	PUNCT
cana-2867	183	18	)	)	PUNCT
cana-2867	183	19	≤	≤	NUM
cana-2867	183	20	kijƞ	kijƞ	X
cana-2867	183	21	(	(	PUNCT
cana-2867	183	22	mi(xi	mi(xi	PROPN
cana-2867	183	23	,	,	PUNCT
cana-2867	183	24	y	y	PROPN
cana-2867	183	25	i	i	PROPN
cana-2867	183	26	,	,	PUNCT
cana-2867	183	27	t	t	PROPN
cana-2867	183	28	)	)	PUNCT
cana-2867	183	29	)	)	PUNCT
cana-2867	183	30	,	,	PUNCT
cana-2867	183	31	∀	∀	X
cana-2867	183	32	xi	xi	X
cana-2867	183	33	,	,	PUNCT
cana-2867	183	34	y	y	PROPN
cana-2867	183	35	i	i	PRON
cana-2867	183	36	∈	∈	PROPN
cana-2867	183	37	xi	xi	PROPN
cana-2867	183	38	,	,	PUNCT
cana-2867	183	39	xj	xj	PROPN
cana-2867	183	40	,	,	PUNCT
cana-2867	183	41	y	y	PROPN
cana-2867	183	42	j	j	PROPN
cana-2867	183	43	∈	∈	PROPN
cana-2867	183	44	xj	xj	PROPN
cana-2867	183	45	ƞ	ƞ	PROPN
cana-2867	183	46	(	(	PUNCT
cana-2867	183	47	mi	mi	PROPN
cana-2867	183	48	(	(	PUNCT
cana-2867	183	49	wij(xj	wij(xj	PROPN
cana-2867	183	50	)	)	PUNCT
cana-2867	183	51	,	,	PUNCT
cana-2867	183	52	wij	wij	PROPN
cana-2867	183	53	(	(	PUNCT
cana-2867	183	54	y	y	PROPN
cana-2867	183	55	j	j	PROPN
cana-2867	183	56	)	)	PUNCT
cana-2867	183	57	,	,	PUNCT
cana-2867	183	58	t	t	PROPN
cana-2867	183	59	)	)	PUNCT
cana-2867	183	60	)	)	PUNCT
cana-2867	183	61	≤	≤	NUM
cana-2867	183	62	kijƞ	kijƞ	X
cana-2867	183	63	(	(	PUNCT
cana-2867	183	64	mi(xi	mi(xi	PROPN
cana-2867	183	65	,	,	PUNCT
cana-2867	183	66	y	y	PROPN
cana-2867	183	67	i	i	PROPN
cana-2867	183	68	,	,	PUNCT
cana-2867	183	69	t	t	PROPN
cana-2867	183	70	)	)	PUNCT
cana-2867	183	71	)	)	PUNCT
cana-2867	183	72	,	,	PUNCT
cana-2867	183	73	∀	∀	X
cana-2867	183	74	xi	xi	ADP
cana-2867	183	75	∈	∈	PROPN
cana-2867	183	76	ai	ai	VERB
cana-2867	183	77	,	,	PUNCT
cana-2867	183	78	y	y	PROPN
cana-2867	183	79	i	i	PRON
cana-2867	183	80	∈	∈	PROPN
cana-2867	183	81	bi	bi	NOUN
cana-2867	183	82	and	and	CCONJ
cana-2867	183	83	xi	xi	PROPN
cana-2867	183	84	∈	∈	PROPN
cana-2867	183	85	aj	aj	PROPN
cana-2867	183	86	,	,	PUNCT
cana-2867	183	87	y	y	PROPN
cana-2867	183	88	j	j	PROPN
cana-2867	183	89	∈	∈	PROPN
cana-2867	183	90	bj	bj	VERB
cana-2867	183	91	ƞ	ƞ	NOUN
cana-2867	183	92	(	(	PUNCT
cana-2867	183	93	sup	sup	PROPN
cana-2867	183	94	yj∈bj	yj∈bj	PROPN
cana-2867	183	95	mi	mi	PROPN
cana-2867	183	96	(	(	PUNCT
cana-2867	183	97	wij(xj	wij(xj	PROPN
cana-2867	183	98	)	)	PUNCT
cana-2867	183	99	,	,	PUNCT
cana-2867	183	100	wij	wij	PROPN
cana-2867	183	101	(	(	PUNCT
cana-2867	183	102	y	y	PROPN
cana-2867	183	103	j	j	PROPN
cana-2867	183	104	)	)	PUNCT
cana-2867	183	105	,	,	PUNCT
cana-2867	183	106	t	t	PROPN
cana-2867	183	107	)	)	PUNCT
cana-2867	183	108	)	)	PUNCT
cana-2867	184	1	≤	≤	NUM
cana-2867	184	2	kijƞ	kijƞ	X
cana-2867	184	3	(	(	PUNCT
cana-2867	184	4	sup	sup	NOUN
cana-2867	184	5	yi∈bi	yi∈bi	PUNCT
cana-2867	184	6	mi(xi	mi(xi	PROPN
cana-2867	184	7	,	,	PUNCT
cana-2867	184	8	y	y	PROPN
cana-2867	184	9	i	i	PROPN
cana-2867	184	10	,	,	PUNCT
cana-2867	184	11	t	t	PROPN
cana-2867	184	12	)	)	PUNCT
cana-2867	184	13	)	)	PUNCT
cana-2867	184	14	∀	∀	X
cana-2867	184	15	xi	xi	ADP
cana-2867	184	16	∈	∈	PROPN
cana-2867	184	17	ai	ai	VERB
cana-2867	184	18	,	,	PUNCT
cana-2867	184	19	y	y	PROPN
cana-2867	184	20	i	i	PROPN
cana-2867	184	21	∈	∈	PROPN
cana-2867	184	22	bi	bi	NOUN
cana-2867	184	23	,	,	PUNCT
cana-2867	184	24	and	and	CCONJ
cana-2867	184	25	xj	xj	PROPN
cana-2867	184	26	∈	∈	PROPN
cana-2867	184	27	aj	aj	PROPN
cana-2867	184	28	,	,	PUNCT
cana-2867	184	29	y	y	PROPN
cana-2867	184	30	j	j	PROPN
cana-2867	184	31	∈	∈	PROPN
cana-2867	184	32	bj	bj	VERB
cana-2867	184	33	ƞ	ƞ	PROPN
cana-2867	184	34	(	(	PUNCT
cana-2867	184	35	mi(wij(xj	mi(wij(xj	PROPN
cana-2867	184	36	)	)	PUNCT
cana-2867	184	37	,	,	PUNCT
cana-2867	184	38	wij(bj	wij(bj	NOUN
cana-2867	184	39	)	)	PUNCT
cana-2867	184	40	,	,	PUNCT
cana-2867	184	41	t	t	PROPN
cana-2867	184	42	)	)	PUNCT
cana-2867	184	43	)	)	PUNCT
cana-2867	185	1	≤	≤	NUM
cana-2867	185	2	kijƞ(mi(xi	kijƞ(mi(xi	NOUN
cana-2867	185	3	,	,	PUNCT
cana-2867	185	4	bi	bi	PROPN
cana-2867	185	5	,	,	PUNCT
cana-2867	185	6	t	t	PROPN
cana-2867	185	7	)	)	PUNCT
cana-2867	185	8	)	)	PUNCT
cana-2867	185	9	,	,	PUNCT
cana-2867	185	10	∀	∀	X
cana-2867	185	11	xi	xi	ADP
cana-2867	185	12	∈	∈	PROPN
cana-2867	185	13	ai	ai	VERB
cana-2867	185	14	,	,	PUNCT
cana-2867	185	15	xj	xj	PROPN
cana-2867	185	16	∈	∈	PROPN
cana-2867	185	17	aj	aj	PROPN
cana-2867	185	18	communications	communication	NOUN
cana-2867	185	19	on	on	ADP
cana-2867	185	20	applied	apply	VERB
cana-2867	185	21	nonlinear	nonlinear	ADJ
cana-2867	185	22	analysis	analysis	NOUN
cana-2867	185	23	issn	issn	NOUN
cana-2867	185	24	:	:	PUNCT
cana-2867	185	25	1074	1074	NUM
cana-2867	185	26	-	-	PUNCT
cana-2867	185	27	133x	133x	NUM
cana-2867	185	28	vol	vol	NOUN
cana-2867	185	29	32	32	NUM
cana-2867	185	30	no	no	NOUN
cana-2867	185	31	.	.	PUNCT
cana-2867	186	1	4s	4s	NUM
cana-2867	186	2	(	(	PUNCT
cana-2867	186	3	2025	2025	NUM
cana-2867	186	4	)	)	PUNCT
cana-2867	186	5	507	507	NUM
cana-2867	186	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	186	7	ƞ	ƞ	X
cana-2867	186	8	(	(	PUNCT
cana-2867	186	9	infxj∈aj	infxj∈aj	X
cana-2867	186	10	mi(wij(xj	mi(wij(xj	ADJ
cana-2867	186	11	)	)	PUNCT
cana-2867	186	12	,	,	PUNCT
cana-2867	186	13	wij(bj	wij(bj	NOUN
cana-2867	186	14	)	)	PUNCT
cana-2867	186	15	,	,	PUNCT
cana-2867	186	16	t	t	PROPN
cana-2867	186	17	)	)	PUNCT
cana-2867	186	18	)	)	PUNCT
cana-2867	186	19	≤	≤	NUM
cana-2867	186	20	kijƞ	kijƞ	X
cana-2867	186	21	(	(	PUNCT
cana-2867	186	22	infxi∈ai	infxi∈ai	X
cana-2867	186	23	mi(xi	mi(xi	PROPN
cana-2867	186	24	,	,	PUNCT
cana-2867	186	25	bi	bi	PROPN
cana-2867	186	26	,	,	PUNCT
cana-2867	186	27	t	t	PROPN
cana-2867	186	28	)	)	PUNCT
cana-2867	186	29	)	)	PUNCT
cana-2867	187	1	ƞ	ƞ	X
cana-2867	187	2	(	(	PUNCT
cana-2867	187	3	mi(wij(aj	mi(wij(aj	PROPN
cana-2867	187	4	)	)	PUNCT
cana-2867	187	5	,	,	PUNCT
cana-2867	187	6	wij(bj	wij(bj	NOUN
cana-2867	187	7	)	)	PUNCT
cana-2867	187	8	,	,	PUNCT
cana-2867	187	9	t	t	PROPN
cana-2867	187	10	)	)	PUNCT
cana-2867	187	11	)	)	PUNCT
cana-2867	187	12	≤	≤	NUM
cana-2867	187	13	kijƞ(mi(ai	kijƞ(mi(ai	PROPN
cana-2867	187	14	,	,	PUNCT
cana-2867	187	15	bi	bi	PROPN
cana-2867	187	16	,	,	PUNCT
cana-2867	187	17	t	t	PROPN
cana-2867	187	18	)	)	PUNCT
cana-2867	187	19	)	)	PUNCT
cana-2867	187	20	(	(	PUNCT
cana-2867	187	21	1	1	X
cana-2867	187	22	)	)	PUNCT
cana-2867	187	23	similarly	similarly	ADV
cana-2867	187	24	,	,	PUNCT
cana-2867	187	25	we	we	PRON
cana-2867	187	26	obtain	obtain	VERB
cana-2867	187	27	ƞ	ƞ	PRON
cana-2867	187	28	(	(	PUNCT
cana-2867	187	29	mi(wij(bj	mi(wij(bj	PROPN
cana-2867	187	30	)	)	PUNCT
cana-2867	187	31	,	,	PUNCT
cana-2867	187	32	wij(aj	wij(aj	X
cana-2867	187	33	)	)	PUNCT
cana-2867	187	34	,	,	PUNCT
cana-2867	187	35	t	t	PROPN
cana-2867	187	36	)	)	PUNCT
cana-2867	187	37	)	)	PUNCT
cana-2867	187	38	≤	≤	NUM
cana-2867	187	39	kijƞ(mi(ai	kijƞ(mi(ai	PROPN
cana-2867	187	40	,	,	PUNCT
cana-2867	187	41	bi	bi	PROPN
cana-2867	187	42	,	,	PUNCT
cana-2867	187	43	t	t	PROPN
cana-2867	187	44	)	)	PUNCT
cana-2867	187	45	)	)	PUNCT
cana-2867	187	46	(	(	PUNCT
cana-2867	187	47	2	2	X
cana-2867	187	48	)	)	PUNCT
cana-2867	187	49	therefore	therefore	ADV
cana-2867	187	50	,	,	PUNCT
cana-2867	187	51	from	from	ADP
cana-2867	187	52	(	(	PUNCT
cana-2867	187	53	1	1	NUM
cana-2867	187	54	)	)	PUNCT
cana-2867	187	55	and	and	CCONJ
cana-2867	187	56	(	(	PUNCT
cana-2867	187	57	2	2	NUM
cana-2867	187	58	)	)	PUNCT
cana-2867	187	59	,	,	PUNCT
cana-2867	187	60	we	we	PRON
cana-2867	187	61	can	can	AUX
cana-2867	187	62	say	say	VERB
cana-2867	187	63	ƞ	ƞ	X
cana-2867	187	64	(	(	PUNCT
cana-2867	187	65	hmi	hmi	PROPN
cana-2867	187	66	(	(	PUNCT
cana-2867	187	67	wij(bj	wij(bj	NOUN
cana-2867	187	68	)	)	PUNCT
cana-2867	187	69	,	,	PUNCT
cana-2867	187	70	wij(aj	wij(aj	X
cana-2867	187	71	)	)	PUNCT
cana-2867	187	72	,	,	PUNCT
cana-2867	187	73	t	t	PROPN
cana-2867	187	74	)	)	PUNCT
cana-2867	187	75	)	)	PUNCT
cana-2867	187	76	≤	≤	NUM
cana-2867	187	77	kijƞ	kijƞ	X
cana-2867	187	78	(	(	PUNCT
cana-2867	187	79	hmi	hmi	PROPN
cana-2867	187	80	(	(	PUNCT
cana-2867	187	81	bi	bi	PROPN
cana-2867	187	82	,	,	PUNCT
cana-2867	187	83	ai	ai	VERB
cana-2867	187	84	,	,	PUNCT
cana-2867	187	85	t	t	PROPN
cana-2867	187	86	)	)	PUNCT
cana-2867	187	87	)	)	PUNCT
cana-2867	187	88	.	.	PUNCT
cana-2867	188	1	thus	thus	ADV
cana-2867	188	2	,	,	PUNCT
cana-2867	188	3	the	the	DET
cana-2867	188	4	desired	desire	VERB
cana-2867	188	5	verification	verification	NOUN
cana-2867	188	6	completes	complete	VERB
cana-2867	188	7	.	.	PUNCT
cana-2867	189	1	corollary	corollary	ADJ
cana-2867	189	2	4.1consider(𝑋	4.1consider(𝑋	ADJ
cana-2867	189	3	,	,	PUNCT
cana-2867	189	4	𝑀,∗	𝑀,∗	PROPN
cana-2867	189	5	)	)	PUNCT
cana-2867	189	6	be	be	AUX
cana-2867	189	7	a	a	DET
cana-2867	189	8	fuzzy	fuzzy	ADJ
cana-2867	189	9	metric	metric	ADJ
cana-2867	189	10	space	space	NOUN
cana-2867	189	11	for	for	ADP
cana-2867	189	12	𝑗	𝑗	NOUN
cana-2867	189	13	=	=	SYM
cana-2867	189	14	1,2	1,2	NUM
cana-2867	189	15	,	,	PUNCT
cana-2867	189	16	−	−	PROPN
cana-2867	189	17	−	−	NOUN
cana-2867	190	1	−	−	NOUN
cana-2867	190	2	,	,	PUNCT
cana-2867	190	3	𝑁	𝑁	NOUN
cana-2867	190	4	along	along	ADP
cana-2867	190	5	same	same	ADJ
cana-2867	190	6	𝑡-norm	𝑡-norm	NOUN
cana-2867	190	7	.	.	PUNCT
cana-2867	191	1	suppose(𝒦(𝑋	suppose(𝒦(𝑋	PROPN
cana-2867	191	2	)	)	PUNCT
cana-2867	191	3	,	,	PUNCT
cana-2867	191	4	𝐻𝑀	𝐻𝑀	PROPN
cana-2867	191	5	,	,	PUNCT
cana-2867	191	6	∗	∗	NOUN
cana-2867	191	7	)	)	PUNCT
cana-2867	191	8	be	be	VERB
cana-2867	191	9	the	the	DET
cana-2867	191	10	corresponding	corresponding	ADJ
cana-2867	191	11	hausdorff	hausdorff	NOUN
cana-2867	191	12	fuzzy	fuzzy	ADJ
cana-2867	191	13	metric	metric	ADJ
cana-2867	191	14	space	space	NOUN
cana-2867	191	15	.	.	PUNCT
cana-2867	192	1	assume	assume	VERB
cana-2867	192	2	that𝑤	that𝑤	NOUN
cana-2867	192	3	:	:	PUNCT
cana-2867	192	4	𝑋	𝑋	PROPN
cana-2867	192	5	→	→	SYM
cana-2867	192	6	𝑋	𝑋	PROPN
cana-2867	192	7	be	be	VERB
cana-2867	192	8	a	a	DET
cana-2867	192	9	fuzzy	fuzzy	ADJ
cana-2867	192	10	ℋ-contraction	ℋ-contraction	PROPN
cana-2867	192	11	function	function	NOUN
cana-2867	192	12	with	with	ADP
cana-2867	192	13	ƞ	ƞ	PROPN
cana-2867	192	14	∈	∈	PROPN
cana-2867	193	1	ℋand	ℋand	PROPN
cana-2867	193	2	𝑘	𝑘	PRON
cana-2867	193	3	∈	∈	PROPN
cana-2867	193	4	(	(	PUNCT
cana-2867	193	5	0,1)on	0,1)on	NOUN
cana-2867	193	6	(	(	PUNCT
cana-2867	193	7	𝑋	𝑋	PROPN
cana-2867	193	8	,	,	PUNCT
cana-2867	193	9	𝑀,∗).if	𝑀,∗).if	ADJ
cana-2867	193	10	𝑤	𝑤	VERB
cana-2867	193	11	is	be	AUX
cana-2867	193	12	a	a	DET
cana-2867	193	13	fuzzy	fuzzy	ADJ
cana-2867	193	14	contraction	contraction	NOUN
cana-2867	193	15	withƞ	withƞ	NOUN
cana-2867	193	16	∈	∈	PROPN
cana-2867	194	1	ℋand	ℋand	PROPN
cana-2867	194	2	𝑘	𝑘	PROPN
cana-2867	194	3	∈	∈	PROPN
cana-2867	194	4	(	(	PUNCT
cana-2867	194	5	0,1)on	0,1)on	NOUN
cana-2867	194	6	(	(	PUNCT
cana-2867	194	7	𝑋	𝑋	NOUN
cana-2867	194	8	,	,	PUNCT
cana-2867	194	9	𝑀,∗	𝑀,∗	PROPN
cana-2867	194	10	)	)	PUNCT
cana-2867	194	11	.	.	PUNCT
cana-2867	195	1	then	then	ADV
cana-2867	195	2	𝑤	𝑤	INTJ
cana-2867	195	3	is	be	AUX
cana-2867	195	4	a	a	DET
cana-2867	195	5	fuzzy	fuzzy	ADJ
cana-2867	195	6	contraction	contraction	NOUN
cana-2867	195	7	with	with	ADP
cana-2867	195	8	ƞ	ƞ	PROPN
cana-2867	195	9	∈	∈	PROPN
cana-2867	195	10	ℋ	ℋ	PROPN
cana-2867	195	11	and	and	CCONJ
cana-2867	195	12	𝑘	𝑘	PRON
cana-2867	195	13	∈	∈	PROPN
cana-2867	195	14	(	(	PUNCT
cana-2867	195	15	0,1	0,1	NOUN
cana-2867	195	16	)	)	PUNCT
cana-2867	195	17	on	on	ADP
cana-2867	195	18	(	(	PUNCT
cana-2867	195	19	𝒦(𝑋𝑗	𝒦(𝑋𝑗	PROPN
cana-2867	195	20	)	)	PUNCT
cana-2867	195	21	,	,	PUNCT
cana-2867	195	22	𝐻𝑀𝑗	𝐻𝑀𝑗	NOUN
cana-2867	195	23	,	,	PUNCT
cana-2867	195	24	∗	∗	NOUN
cana-2867	195	25	)	)	PUNCT
cana-2867	195	26	.	.	PUNCT
cana-2867	196	1	that	that	PRON
cana-2867	196	2	is	be	AUX
cana-2867	196	3	ƞ(hm(w(b	ƞ(hm(w(b	NOUN
cana-2867	196	4	)	)	PUNCT
cana-2867	196	5	,	,	PUNCT
cana-2867	196	6	w(a	w(a	PROPN
cana-2867	196	7	)	)	PUNCT
cana-2867	196	8	,	,	PUNCT
cana-2867	196	9	t	t	PROPN
cana-2867	196	10	)	)	PUNCT
cana-2867	196	11	)	)	PUNCT
cana-2867	197	1	≤	≤	NUM
cana-2867	197	2	kƞ(hm(b	kƞ(hm(b	NOUN
cana-2867	197	3	,	,	PUNCT
cana-2867	197	4	a	a	PRON
cana-2867	197	5	,	,	PUNCT
cana-2867	197	6	t	t	PROPN
cana-2867	197	7	)	)	PUNCT
cana-2867	197	8	)	)	PUNCT
cana-2867	197	9	∀	∀	X
cana-2867	197	10	a	a	PRON
cana-2867	197	11	,	,	PUNCT
cana-2867	197	12	b	b	NOUN
cana-2867	197	13	∈	∈	PROPN
cana-2867	197	14	𝒦(x	𝒦(x	NOUN
cana-2867	197	15	)	)	PUNCT
cana-2867	197	16	and	and	CCONJ
cana-2867	197	17	t	t	X
cana-2867	197	18	>	>	X
cana-2867	197	19	0	0	X
cana-2867	197	20	.	.	PUNCT
cana-2867	198	1	now	now	ADV
cana-2867	198	2	,	,	PUNCT
cana-2867	198	3	we	we	PRON
cana-2867	198	4	define	define	VERB
cana-2867	198	5	the	the	DET
cana-2867	198	6	contractive	contractive	ADJ
cana-2867	198	7	operator	operator	NOUN
cana-2867	198	8	for	for	ADP
cana-2867	198	9	mfifs	mfif	NOUN
cana-2867	198	10	provided	provide	VERB
cana-2867	198	11	by	by	ADP
cana-2867	198	12	prasad	prasad	PROPN
cana-2867	198	13	and	and	CCONJ
cana-2867	198	14	katiyar	katiyar	NOUN
cana-2867	199	1	[	[	X
cana-2867	199	2	38	38	NUM
cana-2867	199	3	]	]	PUNCT
cana-2867	199	4	.	.	PUNCT
cana-2867	200	1	we	we	PRON
cana-2867	200	2	also	also	ADV
cana-2867	200	3	explain	explain	VERB
cana-2867	200	4	the	the	DET
cana-2867	200	5	uniqueness	uniqueness	NOUN
cana-2867	200	6	and	and	CCONJ
cana-2867	200	7	existence	existence	NOUN
cana-2867	200	8	results	result	NOUN
cana-2867	200	9	of	of	ADP
cana-2867	200	10	the	the	DET
cana-2867	200	11	mfifs	mfif	NOUN
cana-2867	200	12	in	in	ADP
cana-2867	200	13	the	the	DET
cana-2867	200	14	setup	setup	NOUN
cana-2867	200	15	of	of	ADP
cana-2867	200	16	fuzzy	fuzzy	ADJ
cana-2867	200	17	metric	metric	ADJ
cana-2867	200	18	spaces	space	NOUN
cana-2867	200	19	.	.	PUNCT
cana-2867	201	1	our	our	PRON
cana-2867	201	2	obtained	obtain	VERB
cana-2867	201	3	results	result	NOUN
cana-2867	201	4	are	be	AUX
cana-2867	201	5	inspired	inspire	VERB
cana-2867	201	6	by	by	ADP
cana-2867	201	7	the	the	DET
cana-2867	201	8	results	result	NOUN
cana-2867	201	9	of	of	ADP
cana-2867	201	10	al	al	PROPN
cana-2867	201	11	-	-	PUNCT
cana-2867	201	12	saidi	saidi	PROPN
cana-2867	201	13	[	[	X
cana-2867	201	14	36	36	NUM
cana-2867	201	15	]	]	PUNCT
cana-2867	201	16	and	and	CCONJ
cana-2867	201	17	prasad	prasad	PROPN
cana-2867	201	18	and	and	CCONJ
cana-2867	201	19	katiyar	katiyar	ADJ
cana-2867	202	1	[	[	X
cana-2867	202	2	38	38	NUM
cana-2867	202	3	]	]	PUNCT
cana-2867	202	4	.	.	PUNCT
cana-2867	203	1	definition	definition	NOUN
cana-2867	203	2	4.1assume	4.1assume	NUM
cana-2867	203	3	that	that	SCONJ
cana-2867	203	4	(	(	PUNCT
cana-2867	203	5	xj	xj	PROPN
cana-2867	203	6	,	,	PUNCT
cana-2867	203	7	mj,∗	mj,∗	PROPN
cana-2867	203	8	)	)	PUNCT
cana-2867	203	9	be	be	AUX
cana-2867	203	10	a	a	DET
cana-2867	203	11	complete	complete	ADJ
cana-2867	203	12	fuzzy	fuzzy	ADJ
cana-2867	203	13	metric	metric	ADJ
cana-2867	203	14	space	space	NOUN
cana-2867	203	15	for	for	ADP
cana-2867	203	16	j	j	PROPN
cana-2867	203	17	=	=	SYM
cana-2867	203	18	1	1	NUM
cana-2867	203	19	,	,	PUNCT
cana-2867	203	20	−	−	PROPN
cana-2867	203	21	−	−	PROPN
cana-2867	204	1	−	−	PROPN
cana-2867	204	2	,	,	PUNCT
cana-2867	204	3	n	n	CCONJ
cana-2867	204	4	with	with	ADP
cana-2867	204	5	same	same	ADJ
cana-2867	204	6	t	t	NOUN
cana-2867	204	7	-	-	PUNCT
cana-2867	204	8	norm	norm	NOUN
cana-2867	204	9	.	.	PUNCT
cana-2867	205	1	let	let	VERB
cana-2867	205	2	wij	wij	PROPN
cana-2867	205	3	l	l	NOUN
cana-2867	205	4	:	:	PUNCT
cana-2867	205	5	xj	xj	PROPN
cana-2867	205	6	→	→	SYM
cana-2867	205	7	xi	xi	PROPN
cana-2867	205	8	,	,	PUNCT
cana-2867	205	9	l	l	NOUN
cana-2867	205	10	=	=	SYM
cana-2867	205	11	1	1	NUM
cana-2867	205	12	,	,	PUNCT
cana-2867	205	13	−	−	PROPN
cana-2867	205	14	−	−	PROPN
cana-2867	205	15	−−	−−	NOUN
cana-2867	205	16	,	,	PUNCT
cana-2867	205	17	mij	mij	NOUN
cana-2867	205	18	and	and	CCONJ
cana-2867	205	19	i	i	PROPN
cana-2867	205	20	,	,	PUNCT
cana-2867	205	21	j	j	PROPN
cana-2867	205	22	=	=	SYM
cana-2867	205	23	1	1	NUM
cana-2867	205	24	,	,	PUNCT
cana-2867	205	25	−	−	PROPN
cana-2867	205	26	−	−	PROPN
cana-2867	205	27	−−	−−	NOUN
cana-2867	205	28	,	,	PUNCT
cana-2867	205	29	nbe	nbe	VERB
cana-2867	205	30	fuzzy	fuzzy	PROPN
cana-2867	205	31	ℋ-contractions	ℋ-contractions	PROPN
cana-2867	205	32	withƞ	withƞ	VERB
cana-2867	205	33	∈	∈	PROPN
cana-2867	205	34	ℋ	ℋ	PROPN
cana-2867	205	35	and	and	CCONJ
cana-2867	205	36	kij	kij	PROPN
cana-2867	205	37	l	l	PROPN
cana-2867	205	38	∈	∈	PROPN
cana-2867	205	39	(	(	PUNCT
cana-2867	205	40	0,1	0,1	NOUN
cana-2867	205	41	)	)	PUNCT
cana-2867	206	1	where	where	SCONJ
cana-2867	206	2	l	l	NOUN
cana-2867	206	3	=	=	SYM
cana-2867	206	4	1,2	1,2	NUM
cana-2867	206	5	,	,	PUNCT
cana-2867	206	6	−	−	PROPN
cana-2867	206	7	−	−	PROPN
cana-2867	206	8	−	−	NOUN
cana-2867	206	9	,	,	PUNCT
cana-2867	206	10	mij	mij	NOUN
cana-2867	206	11	;	;	PUNCT
cana-2867	206	12	i	i	PRON
cana-2867	206	13	,	,	PUNCT
cana-2867	206	14	j	j	PROPN
cana-2867	206	15	=	=	SYM
cana-2867	206	16	1	1	NUM
cana-2867	206	17	,	,	PUNCT
cana-2867	206	18	−	−	PROPN
cana-2867	206	19	−	−	PROPN
cana-2867	206	20	−−	−−	NOUN
cana-2867	206	21	,	,	PUNCT
cana-2867	206	22	n.	n.	PROPN
cana-2867	206	23	thenthesystem	thenthesystem	PROPN
cana-2867	206	24	{	{	PUNCT
cana-2867	206	25	xj	xj	PROPN
cana-2867	206	26	,	,	PUNCT
cana-2867	206	27	j	j	PROPN
cana-2867	206	28	=	=	SYM
cana-2867	206	29	1	1	NUM
cana-2867	206	30	,	,	PUNCT
cana-2867	206	31	−	−	PROPN
cana-2867	206	32	−	−	PROPN
cana-2867	206	33	−−	−−	NOUN
cana-2867	206	34	,	,	PUNCT
cana-2867	206	35	n	n	CCONJ
cana-2867	206	36	;	;	PUNCT
cana-2867	206	37	wij	wij	PROPN
cana-2867	206	38	l	l	NOUN
cana-2867	206	39	:	:	PUNCT
cana-2867	206	40	xj	xj	PROPN
cana-2867	206	41	→	→	SYM
cana-2867	206	42	xi	xi	PROPN
cana-2867	206	43	,	,	PUNCT
cana-2867	206	44	l	l	NOUN
cana-2867	206	45	=	=	SYM
cana-2867	206	46	1	1	NUM
cana-2867	206	47	,	,	PUNCT
cana-2867	206	48	−	−	PROPN
cana-2867	206	49	−	−	PROPN
cana-2867	206	50	−−	−−	NOUN
cana-2867	206	51	,	,	PUNCT
cana-2867	206	52	mij	mij	NOUN
cana-2867	206	53	and	and	CCONJ
cana-2867	206	54	i	i	PROPN
cana-2867	206	55	,	,	PUNCT
cana-2867	206	56	j	j	PROPN
cana-2867	206	57	=	=	SYM
cana-2867	206	58	1	1	NUM
cana-2867	206	59	,	,	PUNCT
cana-2867	206	60	−	−	PROPN
cana-2867	206	61	−	−	PROPN
cana-2867	206	62	−−	−−	NOUN
cana-2867	206	63	,	,	PUNCT
cana-2867	206	64	n	n	CCONJ
cana-2867	206	65	}	}	PUNCT
cana-2867	206	66	is	be	AUX
cana-2867	206	67	said	say	VERB
cana-2867	206	68	to	to	ADP
cana-2867	206	69	bea	bea	PROPN
cana-2867	206	70	ℋ-mfifs	ℋ-mfifs	PROPN
cana-2867	206	71	on(x	on(x	ADV
cana-2867	206	72	,	,	PUNCT
cana-2867	206	73	m,∗)with	m,∗)with	X
cana-2867	206	74	ƞ	ƞ	PROPN
cana-2867	206	75	∈	∈	PROPN
cana-2867	206	76	ℋ	ℋ	PROPN
cana-2867	206	77	and	and	CCONJ
cana-2867	206	78	k	k	PROPN
cana-2867	206	79	=	=	PROPN
cana-2867	206	80	max{kij	max{kij	PROPN
cana-2867	206	81	l	l	NOUN
cana-2867	206	82	,	,	PUNCT
cana-2867	206	83	l	l	NOUN
cana-2867	206	84	=	=	SYM
cana-2867	206	85	1,2	1,2	NUM
cana-2867	206	86	,	,	PUNCT
cana-2867	206	87	−	−	PROPN
cana-2867	206	88	−	−	PROPN
cana-2867	206	89	−−	−−	NOUN
cana-2867	206	90	,	,	PUNCT
cana-2867	206	91	mij	mij	NOUN
cana-2867	206	92	,	,	PUNCT
cana-2867	206	93	i	i	PRON
cana-2867	206	94	,	,	PUNCT
cana-2867	206	95	j	j	PROPN
cana-2867	206	96	=	=	SYM
cana-2867	206	97	1,2	1,2	NUM
cana-2867	206	98	,	,	PUNCT
cana-2867	206	99	−	−	PROPN
cana-2867	206	100	−	−	PROPN
cana-2867	206	101	−−	−−	NOUN
cana-2867	206	102	,	,	PUNCT
cana-2867	206	103	n	n	CCONJ
cana-2867	206	104	}	}	PUNCT
cana-2867	206	105	.	.	PUNCT
cana-2867	207	1	definition4.2	definition4.2	NOUN
cana-2867	207	2	suppose	suppose	VERB
cana-2867	207	3	that(xj	that(xj	PROPN
cana-2867	207	4	,	,	PUNCT
cana-2867	207	5	mj,∗	mj,∗	PROPN
cana-2867	207	6	)	)	PUNCT
cana-2867	207	7	be	be	AUX
cana-2867	207	8	a	a	DET
cana-2867	207	9	complete	complete	ADJ
cana-2867	207	10	fuzzy	fuzzy	ADJ
cana-2867	207	11	metric	metric	ADJ
cana-2867	207	12	space	space	NOUN
cana-2867	207	13	for	for	ADP
cana-2867	207	14	j	j	PROPN
cana-2867	207	15	=	=	SYM
cana-2867	207	16	1	1	NUM
cana-2867	207	17	,	,	PUNCT
cana-2867	207	18	−	−	PROPN
cana-2867	207	19	−	−	PROPN
cana-2867	207	20	−−	−−	NOUN
cana-2867	207	21	,	,	PUNCT
cana-2867	207	22	n	n	X
cana-2867	207	23	be	be	AUX
cana-2867	207	24	complete	complete	ADJ
cana-2867	207	25	metric	metric	ADJ
cana-2867	207	26	spaces	space	NOUN
cana-2867	207	27	.	.	PUNCT
cana-2867	208	1	let	let	VERB
cana-2867	208	2	{	{	PUNCT
cana-2867	208	3	xj	xj	PROPN
cana-2867	208	4	,	,	PUNCT
cana-2867	208	5	j	j	PROPN
cana-2867	208	6	=	=	SYM
cana-2867	208	7	1	1	NUM
cana-2867	208	8	,	,	PUNCT
cana-2867	208	9	−	−	PROPN
cana-2867	208	10	−	−	PROPN
cana-2867	208	11	−−	−−	NOUN
cana-2867	208	12	,	,	PUNCT
cana-2867	208	13	n	n	CCONJ
cana-2867	208	14	;	;	PUNCT
cana-2867	208	15	wij	wij	PROPN
cana-2867	208	16	l	l	NOUN
cana-2867	208	17	:	:	PUNCT
cana-2867	208	18	xj	xj	PROPN
cana-2867	208	19	→	→	SYM
cana-2867	208	20	xi	xi	PROPN
cana-2867	208	21	,	,	PUNCT
cana-2867	208	22	l	l	NOUN
cana-2867	208	23	=	=	SYM
cana-2867	208	24	1	1	NUM
cana-2867	208	25	,	,	PUNCT
cana-2867	208	26	−	−	PROPN
cana-2867	208	27	−	−	PROPN
cana-2867	208	28	−−	−−	NOUN
cana-2867	208	29	,	,	PUNCT
cana-2867	208	30	mij	mij	NOUN
cana-2867	208	31	and	and	CCONJ
cana-2867	208	32	i	i	PROPN
cana-2867	208	33	,	,	PUNCT
cana-2867	208	34	j	j	PROPN
cana-2867	208	35	=	=	SYM
cana-2867	208	36	1	1	NUM
cana-2867	208	37	,	,	PUNCT
cana-2867	208	38	−	−	PROPN
cana-2867	208	39	−	−	PROPN
cana-2867	208	40	−−	−−	NOUN
cana-2867	208	41	,	,	PUNCT
cana-2867	208	42	n	n	CCONJ
cana-2867	208	43	}	}	PUNCT
cana-2867	208	44	be	be	AUX
cana-2867	208	45	an	an	DET
cana-2867	208	46	ℋ	ℋ	PROPN
cana-2867	208	47	–	–	PUNCT
cana-2867	208	48	mfifs	mfifs	PROPN
cana-2867	208	49	.	.	PUNCT
cana-2867	209	1	then	then	ADV
cana-2867	209	2	the	the	DET
cana-2867	209	3	mfhb	mfhb	NOUN
cana-2867	209	4	operator	operator	NOUN
cana-2867	209	5	of	of	ADP
cana-2867	209	6	the	the	DET
cana-2867	209	7	ℋ-ifs	ℋ-ifs	PROPN
cana-2867	209	8	is	be	AUX
cana-2867	209	9	a	a	DET
cana-2867	209	10	function	function	NOUN
cana-2867	209	11	w	w	NOUN
cana-2867	209	12	:	:	PUNCT
cana-2867	209	13	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	209	14	)	)	PUNCT
cana-2867	209	15	→	→	SYM
cana-2867	209	16	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	209	17	)	)	PUNCT
cana-2867	209	18	such	such	ADJ
cana-2867	209	19	that	that	PRON
cana-2867	209	20	w(𝔅	w(𝔅	NOUN
cana-2867	209	21	)	)	PUNCT
cana-2867	210	1	=	=	SYM
cana-2867	210	2	∏	∏	PROPN
cana-2867	210	3	⋃	⋃	NOUN
cana-2867	210	4	⋃	⋃	NOUN
cana-2867	210	5	wij	wij	PROPN
cana-2867	210	6	l	l	X
cana-2867	210	7	mij	mij	NOUN
cana-2867	210	8	l=1	l=1	PROPN
cana-2867	210	9	n	n	CCONJ
cana-2867	210	10	j=1	j=1	PROPN
cana-2867	210	11	n	n	CCONJ
cana-2867	210	12	i=1	i=1	PROPN
cana-2867	210	13	(	(	PUNCT
cana-2867	210	14	bj	bj	NOUN
cana-2867	210	15	)	)	PUNCT
cana-2867	210	16	∀	∀	PUNCT
cana-2867	210	17	𝔅	𝔅	NOUN
cana-2867	210	18	∈	∈	NOUN
cana-2867	210	19	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	210	20	)	)	PUNCT
cana-2867	210	21	.	.	PUNCT
cana-2867	211	1	theorem4.2	theorem4.2	NOUN
cana-2867	211	2	suppose	suppose	VERB
cana-2867	211	3	that	that	SCONJ
cana-2867	211	4	(	(	PUNCT
cana-2867	211	5	xj	xj	PROPN
cana-2867	211	6	,	,	PUNCT
cana-2867	211	7	mj,∗	mj,∗	PROPN
cana-2867	211	8	)	)	PUNCT
cana-2867	211	9	,	,	PUNCT
cana-2867	211	10	j	j	X
cana-2867	211	11	=	=	SYM
cana-2867	211	12	1,2	1,2	NUM
cana-2867	211	13	,	,	PUNCT
cana-2867	211	14	−	−	PROPN
cana-2867	211	15	−	−	PROPN
cana-2867	211	16	−	−	NOUN
cana-2867	211	17	−	−	PROPN
cana-2867	211	18	−	−	NOUN
cana-2867	211	19	,	,	PUNCT
cana-2867	211	20	n	n	CCONJ
cana-2867	211	21	;	;	PUNCT
cana-2867	211	22	be	be	AUX
cana-2867	211	23	complete	complete	ADJ
cana-2867	211	24	fuzzy	fuzzy	ADJ
cana-2867	211	25	metric	metric	ADJ
cana-2867	211	26	spaces	space	NOUN
cana-2867	211	27	.	.	PUNCT
cana-2867	212	1	let	let	VERB
cana-2867	212	2	{	{	PUNCT
cana-2867	212	3	xj	xj	PROPN
cana-2867	212	4	,	,	PUNCT
cana-2867	212	5	j	j	PROPN
cana-2867	212	6	=	=	SYM
cana-2867	212	7	1,2	1,2	NUM
cana-2867	212	8	,	,	PUNCT
cana-2867	212	9	−	−	PROPN
cana-2867	212	10	−	−	PROPN
cana-2867	212	11	−−	−−	NOUN
cana-2867	212	12	,	,	PUNCT
cana-2867	212	13	n	n	CCONJ
cana-2867	212	14	;	;	PUNCT
cana-2867	212	15	wij	wij	PROPN
cana-2867	212	16	l	l	NOUN
cana-2867	212	17	:	:	PUNCT
cana-2867	212	18	xj	xj	PROPN
cana-2867	212	19	→	→	SYM
cana-2867	212	20	xi	xi	PROPN
cana-2867	212	21	,	,	PUNCT
cana-2867	212	22	l	l	NOUN
cana-2867	212	23	=	=	SYM
cana-2867	212	24	1	1	NUM
cana-2867	212	25	,	,	PUNCT
cana-2867	212	26	−	−	PROPN
cana-2867	212	27	−	−	PROPN
cana-2867	212	28	−−	−−	NOUN
cana-2867	212	29	,	,	PUNCT
cana-2867	212	30	mij	mij	NOUN
cana-2867	212	31	and	and	CCONJ
cana-2867	212	32	i	i	PROPN
cana-2867	212	33	,	,	PUNCT
cana-2867	212	34	j	j	PROPN
cana-2867	212	35	=	=	SYM
cana-2867	212	36	1	1	NUM
cana-2867	212	37	,	,	PUNCT
cana-2867	212	38	−	−	PROPN
cana-2867	212	39	−	−	PROPN
cana-2867	212	40	−−	−−	NOUN
cana-2867	212	41	,	,	PUNCT
cana-2867	212	42	n	n	CCONJ
cana-2867	212	43	}	}	PUNCT
cana-2867	212	44	be	be	AUX
cana-2867	212	45	an	an	DET
cana-2867	212	46	ℋ-mfifs	ℋ-mfifs	PROPN
cana-2867	212	47	with	with	ADP
cana-2867	212	48	ƞ	ƞ	PROPN
cana-2867	212	49	∈	∈	PROPN
cana-2867	212	50	ℋ	ℋ	PROPN
cana-2867	212	51	and	and	CCONJ
cana-2867	212	52	k	k	PROPN
cana-2867	212	53	=	=	PROPN
cana-2867	212	54	max{kij	max{kij	PROPN
cana-2867	212	55	l	l	NOUN
cana-2867	212	56	,	,	PUNCT
cana-2867	212	57	l	l	NOUN
cana-2867	212	58	=	=	SYM
cana-2867	212	59	1	1	NUM
cana-2867	212	60	,	,	PUNCT
cana-2867	212	61	−	−	PROPN
cana-2867	212	62	−	−	PROPN
cana-2867	212	63	−−	−−	NOUN
cana-2867	212	64	,	,	PUNCT
cana-2867	212	65	mij	mij	NOUN
cana-2867	212	66	and	and	CCONJ
cana-2867	212	67	i	i	PROPN
cana-2867	212	68	,	,	PUNCT
cana-2867	212	69	j	j	PROPN
cana-2867	212	70	=	=	SYM
cana-2867	212	71	1	1	NUM
cana-2867	212	72	,	,	PUNCT
cana-2867	212	73	−	−	PROPN
cana-2867	212	74	−	−	PROPN
cana-2867	212	75	−−	−−	NOUN
cana-2867	212	76	,	,	PUNCT
cana-2867	212	77	n}.then	n}.then	ADV
cana-2867	212	78	,	,	PUNCT
cana-2867	212	79	w	w	PROPN
cana-2867	212	80	is	be	AUX
cana-2867	212	81	a	a	DET
cana-2867	212	82	fuzzy	fuzzy	ADJ
cana-2867	212	83	ℋcontraction	ℋcontraction	PROPN
cana-2867	212	84	on	on	ADP
cana-2867	212	85	(	(	PUNCT
cana-2867	212	86	𝒦(x	𝒦(x	NOUN
cana-2867	212	87	)	)	PUNCT
cana-2867	212	88	,	,	PUNCT
cana-2867	212	89	hℳ,∗	hℳ,∗	PROPN
cana-2867	212	90	)	)	PUNCT
cana-2867	212	91	.	.	PUNCT
cana-2867	213	1	communications	communication	NOUN
cana-2867	213	2	on	on	ADP
cana-2867	213	3	applied	apply	VERB
cana-2867	213	4	nonlinear	nonlinear	ADJ
cana-2867	213	5	analysis	analysis	NOUN
cana-2867	213	6	issn	issn	NOUN
cana-2867	213	7	:	:	PUNCT
cana-2867	213	8	1074	1074	NUM
cana-2867	213	9	-	-	PUNCT
cana-2867	213	10	133x	133x	NUM
cana-2867	213	11	vol	vol	NOUN
cana-2867	213	12	32	32	NUM
cana-2867	213	13	no	no	NOUN
cana-2867	213	14	.	.	PUNCT
cana-2867	214	1	4s	4s	NUM
cana-2867	214	2	(	(	PUNCT
cana-2867	214	3	2025	2025	NUM
cana-2867	214	4	)	)	PUNCT
cana-2867	214	5	508	508	NUM
cana-2867	215	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	215	2	proof	proof	NOUN
cana-2867	215	3	:	:	PUNCT
cana-2867	215	4	fix	fix	VERB
cana-2867	215	5	t	t	X
cana-2867	215	6	>	>	X
cana-2867	215	7	0	0	X
cana-2867	215	8	.	.	PUNCT
cana-2867	215	9	assume	assume	VERB
cana-2867	215	10	that	that	SCONJ
cana-2867	215	11	𝔅	𝔅	NOUN
cana-2867	215	12	,	,	PUNCT
cana-2867	215	13	𝒞	𝒞	PROPN
cana-2867	215	14	∈	∈	PROPN
cana-2867	215	15	𝒦(x	𝒦(x	NOUN
cana-2867	215	16	)	)	PUNCT
cana-2867	215	17	.	.	PUNCT
cana-2867	216	1	then	then	ADV
cana-2867	216	2	we	we	PRON
cana-2867	216	3	know	know	VERB
cana-2867	216	4	that	that	SCONJ
cana-2867	216	5	ƞ(hℳ(w(𝔅	ƞ(hℳ(w(𝔅	PROPN
cana-2867	216	6	)	)	PUNCT
cana-2867	216	7	,	,	PUNCT
cana-2867	216	8	w(𝒞	w(𝒞	PROPN
cana-2867	216	9	)	)	PUNCT
cana-2867	216	10	,	,	PUNCT
cana-2867	216	11	t	t	PROPN
cana-2867	216	12	)	)	PUNCT
cana-2867	216	13	)	)	PUNCT
cana-2867	217	1	=	=	SYM
cana-2867	217	2	ƞ(hℳ(w(b1	ƞ(hℳ(w(b1	NOUN
cana-2867	217	3	,	,	PUNCT
cana-2867	217	4	−	−	PROPN
cana-2867	217	5	−	−	PROPN
cana-2867	217	6	−−	−−	NOUN
cana-2867	217	7	,	,	PUNCT
cana-2867	217	8	bn	bn	ADJ
cana-2867	217	9	)	)	PUNCT
cana-2867	217	10	,	,	PUNCT
cana-2867	217	11	w(c1	w(c1	NOUN
cana-2867	217	12	,	,	PUNCT
cana-2867	217	13	−	−	PROPN
cana-2867	217	14	−	−	PROPN
cana-2867	217	15	−−	−−	NOUN
cana-2867	217	16	,	,	PUNCT
cana-2867	217	17	cn	cn	PROPN
cana-2867	217	18	)	)	PUNCT
cana-2867	217	19	,	,	PUNCT
cana-2867	217	20	t	t	PROPN
cana-2867	217	21	)	)	PUNCT
cana-2867	217	22	)	)	PUNCT
cana-2867	218	1	=	=	PUNCT
cana-2867	218	2	ƞ	ƞ	X
cana-2867	218	3	(	(	PUNCT
cana-2867	218	4	hℳ	hℳ	PROPN
cana-2867	218	5	(	(	PUNCT
cana-2867	218	6	∏	∏	PROPN
cana-2867	218	7	⋃	⋃	NOUN
cana-2867	218	8	⋃	⋃	NOUN
cana-2867	218	9	wij	wij	PROPN
cana-2867	218	10	l	l	X
cana-2867	218	11	mij	mij	NOUN
cana-2867	218	12	l=1	l=1	PROPN
cana-2867	218	13	n	n	CCONJ
cana-2867	218	14	j=1	j=1	PROPN
cana-2867	218	15	n	n	CCONJ
cana-2867	218	16	i=1	i=1	PROPN
cana-2867	218	17	(	(	PUNCT
cana-2867	218	18	bj	bj	NOUN
cana-2867	218	19	)	)	PUNCT
cana-2867	218	20	,	,	PUNCT
cana-2867	218	21	∏	∏	PROPN
cana-2867	218	22	⋃	⋃	PROPN
cana-2867	218	23	⋃	⋃	PROPN
cana-2867	218	24	wij	wij	X
cana-2867	218	25	l	l	X
cana-2867	218	26	(	(	PUNCT
cana-2867	218	27	cj	cj	NOUN
cana-2867	218	28	)	)	PUNCT
cana-2867	218	29	,	,	PUNCT
cana-2867	218	30	t	t	PROPN
cana-2867	218	31	mij	mij	NOUN
cana-2867	218	32	l=1	l=1	PROPN
cana-2867	218	33	n	n	CCONJ
cana-2867	218	34	j=1	j=1	PROPN
cana-2867	218	35	n	n	CCONJ
cana-2867	218	36	i=1	i=1	PROPN
cana-2867	218	37	)	)	PUNCT
cana-2867	218	38	)	)	PUNCT
cana-2867	218	39	∀	∀	PUNCT
cana-2867	218	40	bj	bj	VERB
cana-2867	218	41	,	,	PUNCT
cana-2867	218	42	cj	cj	PROPN
cana-2867	218	43	∈	∈	PROPN
cana-2867	218	44	xj	xj	PROPN
cana-2867	218	45	=	=	SYM
cana-2867	218	46	min	min	PROPN
cana-2867	218	47	i∈{1−,−−−,n	i∈{1−,−−−,n	PROPN
cana-2867	218	48	}	}	PUNCT
cana-2867	218	49	ƞ	ƞ	PROPN
cana-2867	218	50	(	(	PUNCT
cana-2867	218	51	hmi	hmi	PROPN
cana-2867	218	52	(	(	PUNCT
cana-2867	218	53	⋃	⋃	PROPN
cana-2867	218	54	⋃	⋃	NOUN
cana-2867	218	55	wij	wij	X
cana-2867	218	56	l	l	NOUN
cana-2867	218	57	(	(	PUNCT
cana-2867	218	58	bj	bj	NOUN
cana-2867	218	59	)	)	PUNCT
cana-2867	218	60	mij	mij	NOUN
cana-2867	218	61	l=1	l=1	NOUN
cana-2867	218	62	n	n	CCONJ
cana-2867	218	63	j=1	j=1	NOUN
cana-2867	218	64	,	,	PUNCT
cana-2867	218	65	⋃	⋃	PROPN
cana-2867	218	66	⋃	⋃	NOUN
cana-2867	218	67	wij	wij	PROPN
cana-2867	218	68	l	l	X
cana-2867	218	69	mij	mij	X
cana-2867	218	70	l=1	l=1	PROPN
cana-2867	218	71	n	n	CCONJ
cana-2867	218	72	j=1	j=1	NOUN
cana-2867	218	73	(	(	PUNCT
cana-2867	218	74	cj	cj	NOUN
cana-2867	218	75	)	)	PUNCT
cana-2867	218	76	,	,	PUNCT
cana-2867	218	77	t	t	PROPN
cana-2867	218	78	)	)	PUNCT
cana-2867	218	79	)	)	PUNCT
cana-2867	218	80	∀	∀	PUNCT
cana-2867	218	81	bj	bj	VERB
cana-2867	218	82	,	,	PUNCT
cana-2867	218	83	cj	cj	PROPN
cana-2867	218	84	∈	∈	PROPN
cana-2867	218	85	xj	xj	PROPN
cana-2867	218	86	(	(	PUNCT
cana-2867	218	87	by	by	ADP
cana-2867	218	88	definition	definition	NOUN
cana-2867	218	89	of	of	ADP
cana-2867	218	90	hℳ	hℳ	NOUN
cana-2867	218	91	)	)	PUNCT
cana-2867	218	92	=	=	SYM
cana-2867	218	93	min	min	NOUN
cana-2867	218	94	i∈{1,−−−−,n	i∈{1,−−−−,n	ADV
cana-2867	218	95	}	}	PUNCT
cana-2867	218	96	(	(	PUNCT
cana-2867	218	97	min	min	NOUN
cana-2867	218	98	j∈{1,−−−−,n	j∈{1,−−−−,n	ADJ
cana-2867	218	99	}	}	PUNCT
cana-2867	218	100	ƞ	ƞ	NOUN
cana-2867	218	101	(	(	PUNCT
cana-2867	218	102	hmij	hmij	NOUN
cana-2867	218	103	(	(	PUNCT
cana-2867	218	104	⋃	⋃	NOUN
cana-2867	218	105	wij	wij	X
cana-2867	218	106	l	l	NOUN
cana-2867	218	107	mij	mij	NOUN
cana-2867	218	108	l=1	l=1	PROPN
cana-2867	218	109	(	(	PUNCT
cana-2867	218	110	bj	bj	NOUN
cana-2867	218	111	)	)	PUNCT
cana-2867	218	112	,	,	PUNCT
cana-2867	218	113	⋃	⋃	PROPN
cana-2867	218	114	wij	wij	PROPN
cana-2867	218	115	l	l	X
cana-2867	218	116	mij	mij	NOUN
cana-2867	218	117	l=1	l=1	PROPN
cana-2867	218	118	(	(	PUNCT
cana-2867	218	119	cj	cj	NOUN
cana-2867	218	120	)	)	PUNCT
cana-2867	218	121	,	,	PUNCT
cana-2867	218	122	t	t	PROPN
cana-2867	218	123	)	)	PUNCT
cana-2867	218	124	)	)	PUNCT
cana-2867	218	125	)	)	PUNCT
cana-2867	218	126	∀	∀	PUNCT
cana-2867	218	127	bj	bj	VERB
cana-2867	218	128	,	,	PUNCT
cana-2867	218	129	cj	cj	PROPN
cana-2867	218	130	∈	∈	PROPN
cana-2867	218	131	xj	xj	PROPN
cana-2867	218	132	(	(	PUNCT
cana-2867	218	133	from	from	ADP
cana-2867	218	134	lemma	lemma	PROPN
cana-2867	218	135	(	(	PUNCT
cana-2867	218	136	2.1	2.1	NUM
cana-2867	218	137	)	)	PUNCT
cana-2867	218	138	)	)	PUNCT
cana-2867	219	1	=	=	SYM
cana-2867	219	2	min	min	NOUN
cana-2867	219	3	i∈{1,−−−−,n	i∈{1,−−−−,n	ADV
cana-2867	219	4	}	}	PUNCT
cana-2867	219	5	(	(	PUNCT
cana-2867	219	6	min	min	NOUN
cana-2867	219	7	j∈{1,−−−−,n	j∈{1,−−−−,n	ADV
cana-2867	219	8	}	}	PUNCT
cana-2867	219	9	(	(	PUNCT
cana-2867	219	10	min	min	NOUN
cana-2867	219	11	l∈{1,−−−−,mij	l∈{1,−−−−,mij	PROPN
cana-2867	219	12	}	}	PUNCT
cana-2867	219	13	ƞ	ƞ	X
cana-2867	219	14	(	(	PUNCT
cana-2867	219	15	h	h	NOUN
cana-2867	219	16	mij	mij	NOUN
cana-2867	219	17	l	l	NOUN
cana-2867	219	18	(	(	PUNCT
cana-2867	219	19	wij	wij	PROPN
cana-2867	219	20	l	l	NOUN
cana-2867	219	21	(	(	PUNCT
cana-2867	219	22	bj	bj	NOUN
cana-2867	219	23	)	)	PUNCT
cana-2867	219	24	,	,	PUNCT
cana-2867	219	25	wij	wij	PROPN
cana-2867	219	26	l	l	X
cana-2867	219	27	(	(	PUNCT
cana-2867	219	28	cj	cj	NOUN
cana-2867	219	29	)	)	PUNCT
cana-2867	219	30	,	,	PUNCT
cana-2867	219	31	t	t	PROPN
cana-2867	219	32	)	)	PUNCT
cana-2867	219	33	)	)	PUNCT
cana-2867	219	34	)	)	PUNCT
cana-2867	219	35	)	)	PUNCT
cana-2867	219	36	∀	∀	PUNCT
cana-2867	219	37	bj	bj	VERB
cana-2867	219	38	,	,	PUNCT
cana-2867	219	39	cj	cj	PROPN
cana-2867	219	40	∈	∈	PROPN
cana-2867	219	41	xj	xj	PROPN
cana-2867	219	42	(	(	PUNCT
cana-2867	219	43	from	from	ADP
cana-2867	219	44	lemma	lemma	PROPN
cana-2867	219	45	(	(	PUNCT
cana-2867	219	46	2.1	2.1	NUM
cana-2867	219	47	)	)	PUNCT
cana-2867	219	48	)	)	PUNCT
cana-2867	219	49	≤	≤	NUM
cana-2867	219	50	min	min	NOUN
cana-2867	219	51	i∈{1,−−−−,n	i∈{1,−−−−,n	ADV
cana-2867	219	52	}	}	PUNCT
cana-2867	219	53	(	(	PUNCT
cana-2867	219	54	min	min	NOUN
cana-2867	219	55	j∈{1,−−−−,n	j∈{1,−−−−,n	ADV
cana-2867	219	56	}	}	PUNCT
cana-2867	219	57	(	(	PUNCT
cana-2867	219	58	min	min	NOUN
cana-2867	219	59	l∈{1,−−−−,mij	l∈{1,−−−−,mij	PROPN
cana-2867	219	60	}	}	PUNCT
cana-2867	219	61	kij	kij	PROPN
cana-2867	219	62	l	l	PROPN
cana-2867	219	63	ƞ	ƞ	PROPN
cana-2867	219	64	(	(	PUNCT
cana-2867	219	65	h	h	NOUN
cana-2867	219	66	mij	mij	PROPN
cana-2867	219	67	l	l	NOUN
cana-2867	219	68	(	(	PUNCT
cana-2867	219	69	bi	bi	PROPN
cana-2867	219	70	,	,	PUNCT
cana-2867	219	71	ci	ci	PROPN
cana-2867	219	72	,	,	PUNCT
cana-2867	219	73	t	t	PROPN
cana-2867	219	74	)	)	PUNCT
cana-2867	219	75	)	)	PUNCT
cana-2867	219	76	)	)	PUNCT
cana-2867	219	77	)	)	PUNCT
cana-2867	219	78	∀	∀	PUNCT
cana-2867	219	79	bi	bi	NOUN
cana-2867	219	80	,	,	PUNCT
cana-2867	219	81	ci	ci	PROPN
cana-2867	219	82	∈	∈	PROPN
cana-2867	219	83	xi	xi	X
cana-2867	219	84	(	(	PUNCT
cana-2867	219	85	by	by	ADP
cana-2867	219	86	theorem	theorem	NOUN
cana-2867	219	87	(	(	PUNCT
cana-2867	219	88	4.1	4.1	NUM
cana-2867	219	89	)	)	PUNCT
cana-2867	219	90	)	)	PUNCT
cana-2867	219	91	≤	≤	NUM
cana-2867	219	92	min	min	NOUN
cana-2867	219	93	i∈{1,−−−−,n	i∈{1,−−−−,n	ADV
cana-2867	219	94	}	}	PUNCT
cana-2867	219	95	(	(	PUNCT
cana-2867	219	96	min	min	NOUN
cana-2867	219	97	j∈{1,−−−−,n	j∈{1,−−−−,n	ADJ
cana-2867	219	98	}	}	PUNCT
cana-2867	219	99	kijƞ	kijƞ	NOUN
cana-2867	219	100	(	(	PUNCT
cana-2867	219	101	hmij	hmij	NOUN
cana-2867	219	102	(	(	PUNCT
cana-2867	219	103	bi	bi	NOUN
cana-2867	219	104	,	,	PUNCT
cana-2867	219	105	ci	ci	PROPN
cana-2867	219	106	,	,	PUNCT
cana-2867	219	107	t	t	PROPN
cana-2867	219	108	)	)	PUNCT
cana-2867	219	109	)	)	PUNCT
cana-2867	219	110	)	)	PUNCT
cana-2867	219	111	∀	∀	PUNCT
cana-2867	220	1	bi	bi	NOUN
cana-2867	220	2	,	,	PUNCT
cana-2867	220	3	ci	ci	PROPN
cana-2867	220	4	∈	∈	PROPN
cana-2867	220	5	xi	xi	PUNCT
cana-2867	220	6	≤	≤	NUM
cana-2867	220	7	min	min	NOUN
cana-2867	220	8	i∈{1,−−−−,n	i∈{1,−−−−,n	ADV
cana-2867	220	9	}	}	PUNCT
cana-2867	220	10	kiƞ	kiƞ	PROPN
cana-2867	220	11	(	(	PUNCT
cana-2867	220	12	hmi	hmi	PROPN
cana-2867	220	13	(	(	PUNCT
cana-2867	220	14	bi	bi	PROPN
cana-2867	220	15	,	,	PUNCT
cana-2867	220	16	ci	ci	PROPN
cana-2867	220	17	,	,	PUNCT
cana-2867	220	18	t	t	PROPN
cana-2867	220	19	)	)	PUNCT
cana-2867	220	20	)	)	PUNCT
cana-2867	220	21	∀	∀	PUNCT
cana-2867	221	1	bi	bi	NOUN
cana-2867	221	2	,	,	PUNCT
cana-2867	221	3	ci	ci	PROPN
cana-2867	221	4	∈	∈	PROPN
cana-2867	221	5	xi	xi	NOUN
cana-2867	221	6	≤	≤	PROPN
cana-2867	221	7	kƞ(hℳ(𝔅	kƞ(hℳ(𝔅	NOUN
cana-2867	221	8	,	,	PUNCT
cana-2867	221	9	𝒞	𝒞	PROPN
cana-2867	221	10	,	,	PUNCT
cana-2867	221	11	t	t	PROPN
cana-2867	221	12	)	)	PUNCT
cana-2867	221	13	)	)	PUNCT
cana-2867	222	1	thus	thus	ADV
cana-2867	222	2	,	,	PUNCT
cana-2867	222	3	the	the	DET
cana-2867	222	4	desired	desire	VERB
cana-2867	222	5	verification	verification	NOUN
cana-2867	222	6	completes	complete	VERB
cana-2867	222	7	.	.	PUNCT
cana-2867	223	1	corollary	corollary	ADJ
cana-2867	223	2	4.2	4.2	NUM
cana-2867	223	3	suppose	suppose	VERB
cana-2867	223	4	that	that	SCONJ
cana-2867	223	5	(	(	PUNCT
cana-2867	223	6	𝑋	𝑋	PROPN
cana-2867	223	7	,	,	PUNCT
cana-2867	223	8	𝑀.∗	𝑀.∗	PROPN
cana-2867	223	9	)	)	PUNCT
cana-2867	223	10	be	be	VERB
cana-2867	223	11	a	a	DET
cana-2867	223	12	complete	complete	ADJ
cana-2867	223	13	fuzzy	fuzzy	ADJ
cana-2867	223	14	metric	metric	ADJ
cana-2867	223	15	space	space	NOUN
cana-2867	223	16	and	and	CCONJ
cana-2867	223	17	(	(	PUNCT
cana-2867	223	18	𝒦(𝑋	𝒦(𝑋	PROPN
cana-2867	223	19	)	)	PUNCT
cana-2867	223	20	,	,	PUNCT
cana-2867	223	21	𝐻𝑀,∗	𝐻𝑀,∗	PROPN
cana-2867	223	22	)	)	PUNCT
cana-2867	223	23	be	be	VERB
cana-2867	223	24	the	the	DET
cana-2867	223	25	corresponding	corresponding	ADJ
cana-2867	223	26	hausdorff	hausdorff	NOUN
cana-2867	223	27	fuzzy	fuzzy	ADJ
cana-2867	223	28	metric	metric	ADJ
cana-2867	223	29	space	space	NOUN
cana-2867	223	30	.	.	PUNCT
cana-2867	224	1	let	let	VERB
cana-2867	224	2	𝑤𝑖	𝑤𝑖	NOUN
cana-2867	224	3	:	:	PUNCT
cana-2867	224	4	𝑋	𝑋	PROPN
cana-2867	224	5	→	→	SYM
cana-2867	224	6	𝑋	𝑋	PROPN
cana-2867	224	7	,	,	PUNCT
cana-2867	224	8	𝑖	𝑖	SYM
cana-2867	224	9	=	=	SYM
cana-2867	224	10	1	1	NUM
cana-2867	224	11	,	,	PUNCT
cana-2867	224	12	−	−	PROPN
cana-2867	224	13	−	−	PROPN
cana-2867	224	14	−−	−−	NOUN
cana-2867	224	15	,	,	PUNCT
cana-2867	224	16	𝑁	𝑁	PROPN
cana-2867	224	17	be	be	VERB
cana-2867	224	18	a	a	DET
cana-2867	224	19	fuzzy	fuzzy	ADJ
cana-2867	224	20	ℋcontraction	ℋcontraction	PROPN
cana-2867	224	21	on	on	ADP
cana-2867	224	22	(	(	PUNCT
cana-2867	224	23	𝑋	𝑋	NOUN
cana-2867	224	24	,	,	PUNCT
cana-2867	224	25	𝑀,∗	𝑀,∗	PROPN
cana-2867	224	26	)	)	PUNCT
cana-2867	224	27	.	.	PUNCT
cana-2867	225	1	then	then	ADV
cana-2867	225	2	the	the	DET
cana-2867	225	3	fuzzy	fuzzy	ADJ
cana-2867	225	4	hutchinson	hutchinson	PROPN
cana-2867	225	5	-	-	PUNCT
cana-2867	225	6	barnsley	barnsley	NOUN
cana-2867	225	7	operator	operator	NOUN
cana-2867	225	8	is	be	AUX
cana-2867	225	9	also	also	ADV
cana-2867	225	10	a	a	DET
cana-2867	225	11	ℋ-contraction	ℋ-contraction	PROPN
cana-2867	225	12	on	on	ADP
cana-2867	225	13	(	(	PUNCT
cana-2867	225	14	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	225	15	)	)	PUNCT
cana-2867	225	16	,	,	PUNCT
cana-2867	225	17	𝐻𝑀,∗	𝐻𝑀,∗	PROPN
cana-2867	225	18	)	)	PUNCT
cana-2867	225	19	.	.	PUNCT
cana-2867	226	1	theorem	theorem	VERB
cana-2867	226	2	4.3	4.3	NUM
cana-2867	226	3	let	let	VERB
cana-2867	226	4	(	(	PUNCT
cana-2867	226	5	𝑋𝑗	𝑋𝑗	NOUN
cana-2867	226	6	,	,	PUNCT
cana-2867	226	7	𝑀𝑗	𝑀𝑗	PROPN
cana-2867	226	8	,	,	PUNCT
cana-2867	226	9	∗	∗	NOUN
cana-2867	226	10	)	)	PUNCT
cana-2867	226	11	,	,	PUNCT
cana-2867	226	12	𝑗	𝑗	NOUN
cana-2867	226	13	=	=	SYM
cana-2867	226	14	1,2	1,2	NUM
cana-2867	226	15	,	,	PUNCT
cana-2867	226	16	−	−	PROPN
cana-2867	226	17	−	−	NOUN
cana-2867	226	18	−	−	NOUN
cana-2867	226	19	,	,	PUNCT
cana-2867	226	20	𝑁	𝑁	PROPN
cana-2867	226	21	be	be	AUX
cana-2867	226	22	complete	complete	ADJ
cana-2867	226	23	fuzzy	fuzzy	ADJ
cana-2867	226	24	metric	metric	ADJ
cana-2867	226	25	spaces	space	NOUN
cana-2867	226	26	.	.	PUNCT
cana-2867	227	1	let	let	VERB
cana-2867	227	2	{	{	PUNCT
cana-2867	227	3	𝑋𝑗	𝑋𝑗	VERB
cana-2867	227	4	,	,	PUNCT
cana-2867	227	5	𝑗	𝑗	NOUN
cana-2867	227	6	=	=	SYM
cana-2867	227	7	1,2	1,2	NUM
cana-2867	227	8	,	,	PUNCT
cana-2867	227	9	−	−	PROPN
cana-2867	227	10	−	−	PROPN
cana-2867	227	11	−	−	NOUN
cana-2867	227	12	,	,	PUNCT
cana-2867	227	13	𝑁	𝑁	PROPN
cana-2867	227	14	;	;	PUNCT
cana-2867	227	15	𝑤𝑖𝑗	𝑤𝑖𝑗	PROPN
cana-2867	227	16	𝑙	𝑙	X
cana-2867	227	17	:	:	PUNCT
cana-2867	228	1	𝑋𝑗	𝑋𝑗	PROPN
cana-2867	228	2	→	→	SYM
cana-2867	228	3	𝑋𝑖	𝑋𝑖	PROPN
cana-2867	228	4	,	,	PUNCT
cana-2867	228	5	𝑙	𝑙	NOUN
cana-2867	228	6	=	=	SYM
cana-2867	228	7	1	1	NUM
cana-2867	228	8	,	,	PUNCT
cana-2867	228	9	−	−	PROPN
cana-2867	228	10	−	−	PROPN
cana-2867	228	11	−−	−−	NOUN
cana-2867	228	12	,	,	PUNCT
cana-2867	228	13	𝑚𝑖𝑗	𝑚𝑖𝑗	VERB
cana-2867	228	14	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-2867	228	15	𝑖	𝑖	PROPN
cana-2867	228	16	,	,	PUNCT
cana-2867	228	17	𝑗	𝑗	NOUN
cana-2867	228	18	=	=	ADJ
cana-2867	228	19	1	1	NUM
cana-2867	228	20	,	,	PUNCT
cana-2867	228	21	−	−	PROPN
cana-2867	228	22	−	−	PROPN
cana-2867	228	23	−−	−−	NOUN
cana-2867	228	24	,	,	PUNCT
cana-2867	228	25	𝑁	𝑁	PROPN
cana-2867	228	26	}	}	PUNCT
cana-2867	228	27	be	be	AUX
cana-2867	228	28	a	a	DET
cana-2867	228	29	ℋ-mfifs	ℋ-mfifs	PROPN
cana-2867	228	30	with	with	ADP
cana-2867	228	31	ƞ	ƞ	PROPN
cana-2867	228	32	∈	∈	PROPN
cana-2867	228	33	ℋ	ℋ	PROPN
cana-2867	228	34	and𝑘	and𝑘	NOUN
cana-2867	228	35	=	=	PUNCT
cana-2867	228	36	𝑚𝑎𝑥{𝑘𝑖𝑗	𝑚𝑎𝑥{𝑘𝑖𝑗	NOUN
cana-2867	228	37	𝑙	𝑙	NOUN
cana-2867	228	38	,	,	PUNCT
cana-2867	228	39	𝑙	𝑙	X
cana-2867	228	40	=	=	SYM
cana-2867	228	41	1	1	NUM
cana-2867	228	42	,	,	PUNCT
cana-2867	228	43	−	−	PROPN
cana-2867	228	44	−	−	PROPN
cana-2867	228	45	−−	−−	NOUN
cana-2867	228	46	,	,	PUNCT
cana-2867	228	47	𝑚𝑖𝑗	𝑚𝑖𝑗	VERB
cana-2867	228	48	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-2867	228	49	𝑖	𝑖	PROPN
cana-2867	228	50	,	,	PUNCT
cana-2867	228	51	𝑗	𝑗	NOUN
cana-2867	228	52	=	=	ADJ
cana-2867	228	53	1	1	NUM
cana-2867	228	54	,	,	PUNCT
cana-2867	228	55	−	−	PROPN
cana-2867	228	56	−	−	PROPN
cana-2867	228	57	−	−	NOUN
cana-2867	228	58	−	−	PROPN
cana-2867	228	59	𝑁	𝑁	NOUN
cana-2867	228	60	}	}	PUNCT
cana-2867	228	61	.	.	PUNCT
cana-2867	229	1	let	let	VERB
cana-2867	229	2	𝑊	𝑊	PRON
cana-2867	229	3	be	be	AUX
cana-2867	229	4	the	the	DET
cana-2867	229	5	ℋ-mfhb	ℋ-mfhb	PROPN
cana-2867	229	6	operator	operator	NOUN
cana-2867	229	7	of	of	ADP
cana-2867	229	8	the	the	DET
cana-2867	229	9	ℋ-mfifs	ℋ-mfifs	PROPN
cana-2867	229	10	.	.	PUNCT
cana-2867	230	1	then	then	ADV
cana-2867	230	2	there	there	PRON
cana-2867	230	3	exists	exist	VERB
cana-2867	230	4	only	only	ADV
cana-2867	230	5	one	one	NUM
cana-2867	230	6	compact	compact	ADJ
cana-2867	230	7	invariant	invariant	ADJ
cana-2867	230	8	set𝒜∞	set𝒜∞	NOUN
cana-2867	230	9	∈	∈	PROPN
cana-2867	230	10	𝒦(𝑋	𝒦(𝑋	PROPN
cana-2867	230	11	)	)	PUNCT
cana-2867	230	12	,	,	PUNCT
cana-2867	230	13	also	also	ADV
cana-2867	230	14	called	call	VERB
cana-2867	230	15	the	the	DET
cana-2867	230	16	attractor	attractor	NOUN
cana-2867	230	17	(	(	PUNCT
cana-2867	230	18	fractal	fractal	PROPN
cana-2867	230	19	)	)	PUNCT
cana-2867	230	20	of	of	ADP
cana-2867	230	21	the	the	DET
cana-2867	230	22	ℋ-mfhb	ℋ-mfhb	PROPN
cana-2867	230	23	operator	operator	NOUN
cana-2867	230	24	𝑊	𝑊	PROPN
cana-2867	230	25	or	or	CCONJ
cana-2867	230	26	,	,	PUNCT
cana-2867	230	27	equivalently	equivalently	ADV
cana-2867	230	28	,	,	PUNCT
cana-2867	230	29	𝑊	𝑊	PROPN
cana-2867	230	30	has	have	VERB
cana-2867	230	31	a	a	DET
cana-2867	230	32	unique	unique	ADJ
cana-2867	230	33	fixed	fix	VERB
cana-2867	230	34	point	point	NOUN
cana-2867	230	35	namely	namely	ADV
cana-2867	230	36	𝒜∞	𝒜∞	VERB
cana-2867	230	37	∈	∈	PROPN
cana-2867	230	38	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	230	39	)	)	PUNCT
cana-2867	230	40	.	.	PUNCT
cana-2867	231	1	communications	communication	NOUN
cana-2867	231	2	on	on	ADP
cana-2867	231	3	applied	apply	VERB
cana-2867	231	4	nonlinear	nonlinear	ADJ
cana-2867	231	5	analysis	analysis	NOUN
cana-2867	231	6	issn	issn	NOUN
cana-2867	231	7	:	:	PUNCT
cana-2867	231	8	1074	1074	NUM
cana-2867	231	9	-	-	PUNCT
cana-2867	231	10	133x	133x	NUM
cana-2867	231	11	vol	vol	NOUN
cana-2867	231	12	32	32	NUM
cana-2867	231	13	no	no	NOUN
cana-2867	231	14	.	.	PUNCT
cana-2867	232	1	4s	4s	NUM
cana-2867	232	2	(	(	PUNCT
cana-2867	232	3	2025	2025	NUM
cana-2867	232	4	)	)	PUNCT
cana-2867	232	5	509	509	NUM
cana-2867	232	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	232	7	proof	proof	NOUN
cana-2867	232	8	:	:	PUNCT
cana-2867	232	9	let(𝑋𝑗	let(𝑋𝑗	ADJ
cana-2867	232	10	,	,	PUNCT
cana-2867	232	11	𝑀𝑗	𝑀𝑗	PROPN
cana-2867	232	12	,	,	PUNCT
cana-2867	232	13	∗	∗	NOUN
cana-2867	232	14	)	)	PUNCT
cana-2867	232	15	,	,	PUNCT
cana-2867	232	16	𝑗	𝑗	NOUN
cana-2867	232	17	=	=	SYM
cana-2867	232	18	1,2	1,2	NUM
cana-2867	232	19	,	,	PUNCT
cana-2867	232	20	−	−	PROPN
cana-2867	232	21	−	−	PROPN
cana-2867	232	22	−−	−−	NOUN
cana-2867	232	23	,	,	PUNCT
cana-2867	232	24	𝑁	𝑁	PROPN
cana-2867	232	25	are	be	AUX
cana-2867	232	26	complete	complete	ADJ
cana-2867	232	27	fuzzy	fuzzy	ADJ
cana-2867	232	28	metric	metric	ADJ
cana-2867	232	29	spaces	space	NOUN
cana-2867	232	30	.	.	PUNCT
cana-2867	233	1	then(𝒦(𝒳	then(𝒦(𝒳	VERB
cana-2867	233	2	)	)	PUNCT
cana-2867	233	3	,	,	PUNCT
cana-2867	233	4	𝐻ℳ,∗	𝐻ℳ,∗	PROPN
cana-2867	233	5	)	)	PUNCT
cana-2867	233	6	is	be	AUX
cana-2867	233	7	also	also	ADV
cana-2867	233	8	a	a	DET
cana-2867	233	9	complete	complete	ADJ
cana-2867	233	10	hausdorff	hausdorff	NOUN
cana-2867	233	11	multi	multi	X
cana-2867	233	12	fuzzy	fuzzy	ADJ
cana-2867	233	13	metric	metric	ADJ
cana-2867	233	14	space	space	NOUN
cana-2867	233	15	by	by	ADP
cana-2867	233	16	theorem	theorem	NOUN
cana-2867	233	17	(	(	PUNCT
cana-2867	233	18	2.5	2.5	NUM
cana-2867	233	19	)	)	PUNCT
cana-2867	233	20	.	.	PUNCT
cana-2867	234	1	also	also	ADV
cana-2867	234	2	the	the	DET
cana-2867	234	3	mfhb	mfhb	NOUN
cana-2867	234	4	operator	operator	NOUN
cana-2867	234	5	𝑊	𝑊	PROPN
cana-2867	234	6	is	be	AUX
cana-2867	234	7	a	a	DET
cana-2867	234	8	fuzzy	fuzzy	ADJ
cana-2867	234	9	ℋ-contraction	ℋ-contraction	PROPN
cana-2867	234	10	by	by	ADP
cana-2867	234	11	theorem	theorem	NOUN
cana-2867	234	12	(	(	PUNCT
cana-2867	234	13	4.2	4.2	NUM
cana-2867	234	14	)	)	PUNCT
cana-2867	234	15	.	.	PUNCT
cana-2867	235	1	hence	hence	ADV
cana-2867	235	2	,	,	PUNCT
cana-2867	235	3	we	we	PRON
cana-2867	235	4	conclude	conclude	VERB
cana-2867	235	5	that	that	SCONJ
cana-2867	235	6	𝑊	𝑊	PROPN
cana-2867	235	7	has	have	VERB
cana-2867	235	8	a	a	DET
cana-2867	235	9	unique	unique	ADJ
cana-2867	235	10	fixed	fix	VERB
cana-2867	235	11	point	point	NOUN
cana-2867	235	12	,	,	PUNCT
cana-2867	235	13	namely	namely	ADV
cana-2867	235	14	𝒜∞	𝒜∞	VERB
cana-2867	235	15	∈	∈	PROPN
cana-2867	235	16	𝐾0(𝑋	𝐾0(𝑋	PROPN
cana-2867	235	17	)	)	PUNCT
cana-2867	235	18	by	by	ADP
cana-2867	235	19	generalized	generalized	ADJ
cana-2867	235	20	fuzzy	fuzzy	ADJ
cana-2867	235	21	banach	banach	NOUN
cana-2867	235	22	contraction	contraction	NOUN
cana-2867	235	23	theorem	theorem	NOUN
cana-2867	235	24	(	(	PUNCT
cana-2867	235	25	3.3	3.3	NUM
cana-2867	235	26	)	)	PUNCT
cana-2867	235	27	.	.	PUNCT
cana-2867	236	1	example	example	NOUN
cana-2867	236	2	4.1suppose	4.1suppose	NUM
cana-2867	237	1	that	that	PRON
cana-2867	237	2	𝑋1	𝑋1	NOUN
cana-2867	238	1	=	=	SYM
cana-2867	239	1	{	{	PUNCT
cana-2867	239	2	33	33	NUM
cana-2867	239	3	𝑛	𝑛	NOUN
cana-2867	239	4	:	:	PUNCT
cana-2867	239	5	𝑛	𝑛	PRON
cana-2867	239	6	∈	∈	PROPN
cana-2867	239	7	𝑁	𝑁	PROPN
cana-2867	239	8	}	}	PUNCT
cana-2867	239	9	∪	∪	NOUN
cana-2867	239	10	{	{	PUNCT
cana-2867	239	11	3	3	NUM
cana-2867	239	12	}	}	PUNCT
cana-2867	239	13	,	,	PUNCT
cana-2867	239	14	𝑋2	𝑋2	VERB
cana-2867	239	15	=	=	PUNCT
cana-2867	239	16	{	{	PUNCT
cana-2867	239	17	44	44	NUM
cana-2867	239	18	𝑛	𝑛	NOUN
cana-2867	239	19	:	:	PUNCT
cana-2867	239	20	𝑛	𝑛	PRON
cana-2867	239	21	∈	∈	PROPN
cana-2867	239	22	𝑁	𝑁	PROPN
cana-2867	239	23	}	}	PUNCT
cana-2867	239	24	∪	∪	NOUN
cana-2867	239	25	{	{	PUNCT
cana-2867	239	26	4	4	NUM
cana-2867	239	27	}	}	PUNCT
cana-2867	239	28	and	and	CCONJ
cana-2867	239	29	let	let	VERB
cana-2867	239	30	𝑡-norm	𝑡-norm	NOUN
cana-2867	239	31	∗	∗	NOUN
cana-2867	239	32	be	be	AUX
cana-2867	239	33	the	the	DET
cana-2867	239	34	frequent	frequent	ADJ
cana-2867	239	35	product	product	NOUN
cana-2867	239	36	.	.	PUNCT
cana-2867	240	1	assume	assume	VERB
cana-2867	240	2	that	that	SCONJ
cana-2867	240	3	,	,	PUNCT
cana-2867	240	4	a	a	DET
cana-2867	240	5	fuzzy	fuzzy	ADJ
cana-2867	240	6	set	set	NOUN
cana-2867	240	7	𝑀1	𝑀1	NOUN
cana-2867	240	8	:	:	PUNCT
cana-2867	240	9	𝑋1	𝑋1	PROPN
cana-2867	240	10	2	2	NUM
cana-2867	240	11	×	×	NOUN
cana-2867	240	12	(	(	PUNCT
cana-2867	240	13	0,∞	0,∞	NOUN
cana-2867	240	14	]	]	PUNCT
cana-2867	240	15	→	→	PUNCT
cana-2867	240	16	[	[	X
cana-2867	240	17	0,1	0,1	NUM
cana-2867	240	18	]	]	PUNCT
cana-2867	240	19	and	and	CCONJ
cana-2867	240	20	𝑀2	𝑀2	PROPN
cana-2867	240	21	:	:	PUNCT
cana-2867	240	22	𝑋2	𝑋2	VERB
cana-2867	240	23	2	2	NUM
cana-2867	240	24	×	×	NOUN
cana-2867	240	25	(	(	PUNCT
cana-2867	240	26	0,∞	0,∞	NOUN
cana-2867	240	27	]	]	PUNCT
cana-2867	240	28	→	→	PUNCT
cana-2867	241	1	[	[	X
cana-2867	241	2	0,1	0,1	NUM
cana-2867	241	3	]	]	PUNCT
cana-2867	241	4	,	,	PUNCT
cana-2867	241	5	given	give	VERB
cana-2867	241	6	by	by	ADP
cana-2867	241	7	the	the	DET
cana-2867	241	8	formula	formula	NOUN
cana-2867	241	9	;	;	PUNCT
cana-2867	241	10	𝑀1(𝑥	𝑀1(𝑥	NOUN
cana-2867	241	11	,	,	PUNCT
cana-2867	241	12	𝑦	𝑦	NOUN
cana-2867	241	13	,	,	PUNCT
cana-2867	241	14	𝑡	𝑡	NOUN
cana-2867	241	15	)	)	PUNCT
cana-2867	241	16	=	=	PRON
cana-2867	241	17	{	{	PUNCT
cana-2867	242	1	𝑥	𝑥	PRON
cana-2867	242	2	𝑦	𝑦	NOUN
cana-2867	242	3	;	;	PUNCT
cana-2867	242	4	𝑖𝑓𝑥	𝑖𝑓𝑥	NOUN
cana-2867	242	5	≤	≤	NUM
cana-2867	242	6	𝑦𝑓𝑜𝑟𝑎𝑙𝑙𝑡	𝑦𝑓𝑜𝑟𝑎𝑙𝑙𝑡	VERB
cana-2867	242	7	≥	≥	PROPN
cana-2867	242	8	0	0	NUM
cana-2867	242	9	𝑦	𝑦	NOUN
cana-2867	242	10	𝑥	𝑥	NOUN
cana-2867	242	11	;	;	PUNCT
cana-2867	242	12	𝑖𝑓𝑦	𝑖𝑓𝑦	ADJ
cana-2867	242	13	≤	≤	NUM
cana-2867	242	14	𝑥𝑓𝑜𝑟𝑎𝑙𝑙𝑥	𝑥𝑓𝑜𝑟𝑎𝑙𝑙𝑥	NOUN
cana-2867	242	15	,	,	PUNCT
cana-2867	242	16	𝑦	𝑦	NOUN
cana-2867	242	17	∈	∈	NOUN
cana-2867	242	18	𝑋1	𝑋1	NOUN
cana-2867	242	19	and	and	CCONJ
cana-2867	242	20	𝑀2(𝑥	𝑀2(𝑥	PROPN
cana-2867	242	21	,	,	PUNCT
cana-2867	242	22	𝑦	𝑦	NOUN
cana-2867	242	23	,	,	PUNCT
cana-2867	242	24	𝑡	𝑡	NOUN
cana-2867	242	25	)	)	PUNCT
cana-2867	242	26	=	=	PRON
cana-2867	242	27	{	{	PUNCT
cana-2867	243	1	𝑥	𝑥	PRON
cana-2867	243	2	𝑦	𝑦	NOUN
cana-2867	243	3	;	;	PUNCT
cana-2867	243	4	𝑖𝑓𝑥	𝑖𝑓𝑥	NOUN
cana-2867	243	5	≤	≤	NUM
cana-2867	243	6	𝑦𝑓𝑜𝑟𝑎𝑙𝑙𝑡	𝑦𝑓𝑜𝑟𝑎𝑙𝑙𝑡	VERB
cana-2867	243	7	≥	≥	PROPN
cana-2867	243	8	0	0	NUM
cana-2867	243	9	𝑦	𝑦	NOUN
cana-2867	243	10	𝑥	𝑥	NOUN
cana-2867	243	11	;	;	PUNCT
cana-2867	243	12	𝑖𝑓𝑦	𝑖𝑓𝑦	ADJ
cana-2867	243	13	≤	≤	NUM
cana-2867	243	14	𝑥𝑓𝑜𝑟𝑎𝑙𝑙𝑥	𝑥𝑓𝑜𝑟𝑎𝑙𝑙𝑥	NOUN
cana-2867	243	15	,	,	PUNCT
cana-2867	243	16	𝑦	𝑦	PROPN
cana-2867	243	17	∈	∈	PROPN
cana-2867	243	18	𝑋2	𝑋2	VERB
cana-2867	243	19	then	then	ADV
cana-2867	243	20	,	,	PUNCT
cana-2867	243	21	(	(	PUNCT
cana-2867	243	22	𝑋𝑗	𝑋𝑗	NOUN
cana-2867	243	23	,	,	PUNCT
cana-2867	243	24	𝑀𝑗	𝑀𝑗	PROPN
cana-2867	243	25	,	,	PUNCT
cana-2867	243	26	∗	∗	NOUN
cana-2867	243	27	)	)	PUNCT
cana-2867	243	28	,	,	PUNCT
cana-2867	243	29	𝑗	𝑗	NOUN
cana-2867	243	30	=	=	SYM
cana-2867	243	31	1,2	1,2	NUM
cana-2867	243	32	are	be	AUX
cana-2867	243	33	complete	complete	ADJ
cana-2867	243	34	fuzzy	fuzzy	ADJ
cana-2867	243	35	metric	metric	ADJ
cana-2867	243	36	spaces	space	NOUN
cana-2867	243	37	.	.	PUNCT
cana-2867	244	1	now	now	ADV
cana-2867	244	2	,	,	PUNCT
cana-2867	244	3	define	define	VERB
cana-2867	244	4	𝑤11	𝑤11	NOUN
cana-2867	244	5	1	1	NUM
cana-2867	244	6	:	:	PUNCT
cana-2867	244	7	𝑋1	𝑋1	PROPN
cana-2867	244	8	→	→	SYM
cana-2867	244	9	𝑋1	𝑋1	PROPN
cana-2867	244	10	as	as	ADP
cana-2867	244	11	𝑤11	𝑤11	NOUN
cana-2867	244	12	1	1	NUM
cana-2867	244	13	(	(	PUNCT
cana-2867	244	14	33	33	NUM
cana-2867	244	15	𝑛	𝑛	NOUN
cana-2867	244	16	)	)	PUNCT
cana-2867	244	17	=	=	SYM
cana-2867	245	1	3	3	NUM
cana-2867	245	2	3𝑛−1	3𝑛−1	NUM
cana-2867	245	3	and	and	CCONJ
cana-2867	245	4	𝑤22	𝑤22	NOUN
cana-2867	245	5	1	1	NUM
cana-2867	245	6	:	:	PUNCT
cana-2867	245	7	𝑋2	𝑋2	VERB
cana-2867	245	8	→	→	SYM
cana-2867	245	9	𝑋2	𝑋2	VERB
cana-2867	245	10	as	as	ADP
cana-2867	245	11	𝑤22	𝑤22	NOUN
cana-2867	245	12	1	1	NUM
cana-2867	245	13	(	(	PUNCT
cana-2867	245	14	44	44	NUM
cana-2867	245	15	𝑛	𝑛	NOUN
cana-2867	245	16	)	)	PUNCT
cana-2867	246	1	=	=	SYM
cana-2867	246	2	44	44	NUM
cana-2867	246	3	𝑛−1	𝑛−1	NUM
cana-2867	246	4	.	.	PUNCT
cana-2867	247	1	moreover	moreover	ADV
cana-2867	247	2	,	,	PUNCT
cana-2867	247	3	for	for	ADP
cana-2867	247	4	every	every	DET
cana-2867	247	5	𝑚	𝑚	NOUN
cana-2867	247	6	,	,	PUNCT
cana-2867	247	7	𝑛	𝑛	DET
cana-2867	247	8	∈	∈	PROPN
cana-2867	247	9	𝑁	𝑁	PROPN
cana-2867	247	10	,	,	PUNCT
cana-2867	247	11	𝑚	𝑚	X
cana-2867	247	12	<	<	X
cana-2867	247	13	𝑛	𝑛	PROPN
cana-2867	247	14	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-2867	247	15	𝑡	𝑡	PROPN
cana-2867	247	16	>	>	X
cana-2867	247	17	0	0	NUM
cana-2867	247	18	we	we	PRON
cana-2867	247	19	have	have	VERB
cana-2867	247	20	ƞ	ƞ	NOUN
cana-2867	247	21	(	(	PUNCT
cana-2867	247	22	𝐻𝑀	𝐻𝑀	PROPN
cana-2867	247	23	(	(	PUNCT
cana-2867	247	24	𝑤11	𝑤11	PROPN
cana-2867	247	25	1	1	NUM
cana-2867	247	26	(	(	PUNCT
cana-2867	247	27	33	33	NUM
cana-2867	247	28	𝑚	𝑚	NOUN
cana-2867	247	29	)	)	PUNCT
cana-2867	247	30	,	,	PUNCT
cana-2867	247	31	𝑤11	𝑤11	VERB
cana-2867	247	32	1	1	NUM
cana-2867	247	33	(	(	PUNCT
cana-2867	247	34	33	33	NUM
cana-2867	247	35	𝑛	𝑛	NOUN
cana-2867	247	36	)	)	PUNCT
cana-2867	247	37	,	,	PUNCT
cana-2867	247	38	𝑡	𝑡	NOUN
cana-2867	247	39	)	)	PUNCT
cana-2867	247	40	)	)	PUNCT
cana-2867	248	1	=	=	SYM
cana-2867	248	2	𝑙𝑛	𝑙𝑛	NOUN
cana-2867	248	3	(	(	PUNCT
cana-2867	248	4	33	33	NUM
cana-2867	248	5	𝑛−1	𝑛−1	NUM
cana-2867	248	6	/33	/33	SYM
cana-2867	248	7	𝑚−1	𝑚−1	PROPN
cana-2867	248	8	)	)	PUNCT
cana-2867	248	9	=	=	SYM
cana-2867	248	10	1	1	NUM
cana-2867	248	11	3	3	NUM
cana-2867	248	12	𝑙𝑛	𝑙𝑛	NOUN
cana-2867	248	13	(	(	PUNCT
cana-2867	248	14	33	33	NUM
cana-2867	248	15	𝑛	𝑛	PRON
cana-2867	248	16	/33	/33	PUNCT
cana-2867	248	17	𝑚	𝑚	X
cana-2867	248	18	)	)	PUNCT
cana-2867	248	19	=	=	SYM
cana-2867	248	20	1	1	NUM
cana-2867	248	21	3	3	NUM
cana-2867	248	22	ƞ	ƞ	X
cana-2867	248	23	(	(	PUNCT
cana-2867	248	24	𝐻𝑀	𝐻𝑀	PROPN
cana-2867	248	25	(	(	PUNCT
cana-2867	248	26	33	33	NUM
cana-2867	248	27	𝑚	𝑚	NOUN
cana-2867	248	28	,	,	PUNCT
cana-2867	248	29	33	33	NUM
cana-2867	248	30	𝑛	𝑛	PROPN
cana-2867	248	31	,	,	PUNCT
cana-2867	248	32	𝑡	𝑡	NOUN
cana-2867	248	33	)	)	PUNCT
cana-2867	248	34	)	)	PUNCT
cana-2867	249	1	and	and	CCONJ
cana-2867	249	2	,	,	PUNCT
cana-2867	249	3	ƞ	ƞ	X
cana-2867	249	4	(	(	PUNCT
cana-2867	249	5	𝐻𝑀	𝐻𝑀	PROPN
cana-2867	249	6	(	(	PUNCT
cana-2867	249	7	𝑤11	𝑤11	PROPN
cana-2867	249	8	1	1	NUM
cana-2867	249	9	(	(	PUNCT
cana-2867	249	10	3	3	NUM
cana-2867	249	11	)	)	PUNCT
cana-2867	249	12	,	,	PUNCT
cana-2867	249	13	𝑤11	𝑤11	VERB
cana-2867	249	14	1	1	NUM
cana-2867	249	15	(	(	PUNCT
cana-2867	249	16	33	33	NUM
cana-2867	249	17	𝑛	𝑛	NOUN
cana-2867	249	18	)	)	PUNCT
cana-2867	249	19	,	,	PUNCT
cana-2867	249	20	𝑡	𝑡	NOUN
cana-2867	249	21	)	)	PUNCT
cana-2867	249	22	)	)	PUNCT
cana-2867	250	1	=	=	SYM
cana-2867	250	2	𝑙𝑛	𝑙𝑛	NOUN
cana-2867	250	3	(	(	PUNCT
cana-2867	250	4	33	33	NUM
cana-2867	250	5	𝑛−1	𝑛−1	NUM
cana-2867	250	6	/3	/3	PUNCT
cana-2867	250	7	)	)	PUNCT
cana-2867	250	8	<	<	X
cana-2867	250	9	1	1	NUM
cana-2867	250	10	3	3	NUM
cana-2867	250	11	ƞ	ƞ	X
cana-2867	250	12	(	(	PUNCT
cana-2867	250	13	𝐻𝑀	𝐻𝑀	PROPN
cana-2867	250	14	(	(	PUNCT
cana-2867	250	15	3	3	NUM
cana-2867	250	16	,	,	PUNCT
cana-2867	250	17	33	33	NUM
cana-2867	250	18	𝑛	𝑛	PROPN
cana-2867	250	19	,	,	PUNCT
cana-2867	250	20	𝑡	𝑡	NOUN
cana-2867	250	21	)	)	PUNCT
cana-2867	250	22	)	)	PUNCT
cana-2867	250	23	similarly	similarly	ADV
cana-2867	250	24	;	;	PUNCT
cana-2867	250	25	ƞ	ƞ	X
cana-2867	250	26	(	(	PUNCT
cana-2867	250	27	𝐻𝑀	𝐻𝑀	PROPN
cana-2867	250	28	(	(	PUNCT
cana-2867	250	29	𝑤22	𝑤22	NOUN
cana-2867	250	30	1	1	NUM
cana-2867	250	31	(	(	PUNCT
cana-2867	250	32	44	44	NUM
cana-2867	250	33	𝑚	𝑚	NOUN
cana-2867	250	34	)	)	PUNCT
cana-2867	250	35	,	,	PUNCT
cana-2867	250	36	𝑤22	𝑤22	NOUN
cana-2867	250	37	1	1	NUM
cana-2867	250	38	(	(	PUNCT
cana-2867	250	39	44	44	NUM
cana-2867	250	40	𝑛	𝑛	NOUN
cana-2867	250	41	)	)	PUNCT
cana-2867	250	42	,	,	PUNCT
cana-2867	250	43	𝑡	𝑡	NOUN
cana-2867	250	44	)	)	PUNCT
cana-2867	250	45	)	)	PUNCT
cana-2867	251	1	=	=	SYM
cana-2867	251	2	𝑙𝑛	𝑙𝑛	NOUN
cana-2867	251	3	(	(	PUNCT
cana-2867	251	4	44	44	NUM
cana-2867	251	5	𝑛−1	𝑛−1	NUM
cana-2867	251	6	/44	/44	SYM
cana-2867	251	7	𝑚−1	𝑚−1	NOUN
cana-2867	251	8	)	)	PUNCT
cana-2867	251	9	=	=	SYM
cana-2867	251	10	1	1	NUM
cana-2867	251	11	4	4	NUM
cana-2867	251	12	𝑙𝑛	𝑙𝑛	NOUN
cana-2867	251	13	(	(	PUNCT
cana-2867	251	14	44	44	NUM
cana-2867	251	15	𝑛	𝑛	PROPN
cana-2867	251	16	/44	/44	PUNCT
cana-2867	251	17	𝑚	𝑚	NOUN
cana-2867	251	18	)	)	PUNCT
cana-2867	251	19	=	=	SYM
cana-2867	251	20	1	1	NUM
cana-2867	251	21	4	4	NUM
cana-2867	251	22	ƞ	ƞ	X
cana-2867	251	23	(	(	PUNCT
cana-2867	251	24	𝐻𝑀	𝐻𝑀	PROPN
cana-2867	251	25	(	(	PUNCT
cana-2867	251	26	44	44	NUM
cana-2867	251	27	𝑚	𝑚	NOUN
cana-2867	251	28	,	,	PUNCT
cana-2867	251	29	44	44	NUM
cana-2867	251	30	𝑛	𝑛	PROPN
cana-2867	251	31	,	,	PUNCT
cana-2867	251	32	𝑡	𝑡	NOUN
cana-2867	251	33	)	)	PUNCT
cana-2867	251	34	)	)	PUNCT
cana-2867	251	35	communications	communication	NOUN
cana-2867	251	36	on	on	ADP
cana-2867	251	37	applied	apply	VERB
cana-2867	251	38	nonlinear	nonlinear	ADJ
cana-2867	251	39	analysis	analysis	NOUN
cana-2867	251	40	issn	issn	NOUN
cana-2867	251	41	:	:	PUNCT
cana-2867	251	42	1074	1074	NUM
cana-2867	251	43	-	-	PUNCT
cana-2867	251	44	133x	133x	NUM
cana-2867	251	45	vol	vol	NOUN
cana-2867	251	46	32	32	NUM
cana-2867	251	47	no	no	NOUN
cana-2867	251	48	.	.	PUNCT
cana-2867	252	1	4s	4s	NUM
cana-2867	252	2	(	(	PUNCT
cana-2867	252	3	2025	2025	NUM
cana-2867	252	4	)	)	PUNCT
cana-2867	252	5	510	510	NUM
cana-2867	252	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	252	7	and	and	CCONJ
cana-2867	252	8	,	,	PUNCT
cana-2867	252	9	ƞ	ƞ	PROPN
cana-2867	252	10	(	(	PUNCT
cana-2867	252	11	𝐻𝑀	𝐻𝑀	PROPN
cana-2867	252	12	(	(	PUNCT
cana-2867	252	13	𝑤22	𝑤22	NOUN
cana-2867	252	14	1	1	NUM
cana-2867	252	15	(	(	PUNCT
cana-2867	252	16	4	4	NUM
cana-2867	252	17	)	)	PUNCT
cana-2867	252	18	,	,	PUNCT
cana-2867	252	19	𝑤22	𝑤22	NOUN
cana-2867	252	20	1	1	NUM
cana-2867	252	21	(	(	PUNCT
cana-2867	252	22	44	44	NUM
cana-2867	252	23	𝑛	𝑛	NOUN
cana-2867	252	24	)	)	PUNCT
cana-2867	252	25	,	,	PUNCT
cana-2867	252	26	𝑡	𝑡	NOUN
cana-2867	252	27	)	)	PUNCT
cana-2867	252	28	)	)	PUNCT
cana-2867	253	1	=	=	SYM
cana-2867	253	2	𝑙𝑛	𝑙𝑛	NOUN
cana-2867	253	3	(	(	PUNCT
cana-2867	253	4	44	44	NUM
cana-2867	253	5	𝑛−1	𝑛−1	NUM
cana-2867	253	6	/4	/4	NOUN
cana-2867	253	7	)	)	PUNCT
cana-2867	253	8	<	<	X
cana-2867	253	9	1	1	NUM
cana-2867	253	10	4	4	NUM
cana-2867	253	11	ƞ	ƞ	X
cana-2867	253	12	(	(	PUNCT
cana-2867	253	13	𝐻𝑀	𝐻𝑀	PROPN
cana-2867	253	14	(	(	PUNCT
cana-2867	253	15	4	4	NUM
cana-2867	253	16	,	,	PUNCT
cana-2867	253	17	44	44	NUM
cana-2867	253	18	𝑛	𝑛	PROPN
cana-2867	253	19	,	,	PUNCT
cana-2867	253	20	𝑡	𝑡	PROPN
cana-2867	253	21	)	)	PUNCT
cana-2867	253	22	)	)	PUNCT
cana-2867	253	23	since	since	SCONJ
cana-2867	253	24	;	;	PUNCT
cana-2867	253	25	𝑤𝑗𝑗	𝑤𝑗𝑗	PROPN
cana-2867	253	26	𝑙	𝑙	NUM
cana-2867	253	27	,	,	PUNCT
cana-2867	253	28	𝑙	𝑙	X
cana-2867	253	29	=	=	SYM
cana-2867	253	30	1	1	NUM
cana-2867	253	31	,	,	PUNCT
cana-2867	253	32	𝑗	𝑗	NOUN
cana-2867	253	33	=	=	NOUN
cana-2867	253	34	1,2	1,2	NUM
cana-2867	253	35	are	be	AUX
cana-2867	253	36	ℋ-fuzzy	ℋ-fuzzy	PROPN
cana-2867	253	37	contractive	contractive	ADJ
cana-2867	253	38	with	with	ADP
cana-2867	253	39	𝑘	𝑘	PROPN
cana-2867	253	40	=	=	SYM
cana-2867	253	41	1	1	NUM
cana-2867	253	42	3	3	NUM
cana-2867	253	43	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-2867	253	44	1	1	NUM
cana-2867	253	45	4	4	NUM
cana-2867	253	46	.	.	PUNCT
cana-2867	254	1	therefore	therefore	ADV
cana-2867	254	2	,	,	PUNCT
cana-2867	254	3	the	the	DET
cana-2867	254	4	ℋ	ℋ	PROPN
cana-2867	254	5	–	–	PUNCT
cana-2867	254	6	fhb	fhb	NOUN
cana-2867	254	7	operator	operator	NOUN
cana-2867	254	8	𝑊	𝑊	NOUN
cana-2867	254	9	for	for	ADP
cana-2867	254	10	ℋ-mfifs	ℋ-mfifs	PROPN
cana-2867	254	11	{	{	PUNCT
cana-2867	254	12	𝑋𝑗	𝑋𝑗	PROPN
cana-2867	254	13	,	,	PUNCT
cana-2867	254	14	𝑗	𝑗	NOUN
cana-2867	254	15	=	=	SYM
cana-2867	254	16	1,2	1,2	NUM
cana-2867	254	17	;	;	PUNCT
cana-2867	254	18	𝑤𝑗𝑗	𝑤𝑗𝑗	VERB
cana-2867	254	19	𝑙	𝑙	DET
cana-2867	254	20	:	:	PUNCT
cana-2867	254	21	𝑋𝑗	𝑋𝑗	PROPN
cana-2867	254	22	→	→	SYM
cana-2867	254	23	𝑋𝑗	𝑋𝑗	PROPN
cana-2867	254	24	,	,	PUNCT
cana-2867	254	25	𝑙	𝑙	X
cana-2867	254	26	=	=	SYM
cana-2867	254	27	1	1	NUM
cana-2867	254	28	,	,	PUNCT
cana-2867	254	29	𝑗	𝑗	NOUN
cana-2867	254	30	=	=	SYM
cana-2867	254	31	1,2	1,2	NUM
cana-2867	254	32	}	}	PUNCT
cana-2867	254	33	has	have	VERB
cana-2867	254	34	a	a	DET
cana-2867	254	35	unique	unique	ADJ
cana-2867	254	36	attractor	attractor	NOUN
cana-2867	254	37	(	(	PUNCT
cana-2867	254	38	3,4	3,4	NUM
cana-2867	254	39	)	)	PUNCT
cana-2867	254	40	∈	∈	PROPN
cana-2867	254	41	𝑋1	𝑋1	PROPN
cana-2867	254	42	×	×	NOUN
cana-2867	254	43	𝑋2	𝑋2	ADJ
cana-2867	254	44	.	.	PUNCT
cana-2867	255	1	remark	remark	VERB
cana-2867	255	2	4.1	4.1	NUM
cana-2867	255	3	in	in	ADP
cana-2867	255	4	theorem	theorem	NOUN
cana-2867	255	5	(	(	PUNCT
cana-2867	255	6	4.1),if	4.1),if	NUM
cana-2867	255	7	we	we	PRON
cana-2867	255	8	substituting	substitute	VERB
cana-2867	255	9	i	i	PRON
cana-2867	255	10	,	,	PUNCT
cana-2867	255	11	j	j	PROPN
cana-2867	255	12	=	=	SYM
cana-2867	255	13	1.then	1.then	NUM
cana-2867	255	14	we	we	PRON
cana-2867	255	15	get	get	VERB
cana-2867	255	16	the	the	DET
cana-2867	255	17	consequent	consequent	ADJ
cana-2867	255	18	result	result	NOUN
cana-2867	255	19	of	of	ADP
cana-2867	255	20	uthayakumar	uthayakumar	PROPN
cana-2867	255	21	and	and	CCONJ
cana-2867	255	22	gowrishankar	gowrishankar	PROPN
cana-2867	255	23	(	(	PUNCT
cana-2867	255	24	theorem	theorem	NOUN
cana-2867	255	25	(	(	PUNCT
cana-2867	255	26	3.1	3.1	NUM
cana-2867	255	27	)	)	PUNCT
cana-2867	255	28	,	,	PUNCT
cana-2867	256	1	[	[	X
cana-2867	256	2	39	39	NUM
cana-2867	256	3	]	]	PUNCT
cana-2867	256	4	)	)	PUNCT
cana-2867	256	5	.	.	PUNCT
cana-2867	257	1	remark	remark	VERB
cana-2867	257	2	4.2	4.2	NUM
cana-2867	257	3	we	we	PRON
cana-2867	257	4	establishes	establish	VERB
cana-2867	257	5	the	the	DET
cana-2867	257	6	attractor	attractor	NOUN
cana-2867	257	7	for	for	ADP
cana-2867	257	8	multi	multi	X
cana-2867	257	9	fuzzy	fuzzy	ADJ
cana-2867	257	10	ifs	ifs	PROPN
cana-2867	257	11	in	in	ADP
cana-2867	257	12	theorem	theorem	PROPN
cana-2867	257	13	(	(	PUNCT
cana-2867	257	14	4.3	4.3	NUM
cana-2867	257	15	)	)	PUNCT
cana-2867	257	16	which	which	PRON
cana-2867	257	17	is	be	AUX
cana-2867	257	18	an	an	DET
cana-2867	257	19	improvement	improvement	NOUN
cana-2867	257	20	of	of	ADP
cana-2867	257	21	the	the	DET
cana-2867	257	22	leading	lead	VERB
cana-2867	257	23	results	result	NOUN
cana-2867	257	24	concerning	concern	VERB
cana-2867	257	25	uthayakumar	uthayakumar	PROPN
cana-2867	257	26	and	and	CCONJ
cana-2867	257	27	gowrishankar	gowrishankar	VERB
cana-2867	258	1	[	[	X
cana-2867	258	2	39	39	NUM
cana-2867	258	3	]	]	PUNCT
cana-2867	258	4	.	.	PUNCT
cana-2867	259	1	remark	remark	VERB
cana-2867	259	2	4.3	4.3	NUM
cana-2867	259	3	every	every	DET
cana-2867	259	4	mifs	mif	NOUN
cana-2867	259	5	consisting	consist	VERB
cana-2867	259	6	of	of	ADP
cana-2867	259	7	ℋ	ℋ	PROPN
cana-2867	259	8	−	−	NOUN
cana-2867	259	9	contractions	contraction	NOUN
cana-2867	259	10	on	on	ADP
cana-2867	259	11	a	a	DET
cana-2867	259	12	fuzzy	fuzzy	ADJ
cana-2867	259	13	metric	metric	ADJ
cana-2867	259	14	space	space	NOUN
cana-2867	259	15	has	have	VERB
cana-2867	259	16	a	a	DET
cana-2867	259	17	unique	unique	ADJ
cana-2867	259	18	attractor	attractor	NOUN
cana-2867	259	19	.	.	PUNCT
cana-2867	260	1	remark	remark	PROPN
cana-2867	260	2	4.4	4.4	NUM
cana-2867	260	3	theorem	theorem	NOUN
cana-2867	260	4	(	(	PUNCT
cana-2867	260	5	4.3	4.3	NUM
cana-2867	260	6	)	)	PUNCT
cana-2867	260	7	explains	explain	VERB
cana-2867	260	8	that	that	SCONJ
cana-2867	260	9	multi	multi	VERB
cana-2867	260	10	fuzzy	fuzzy	ADJ
cana-2867	260	11	hb	hb	NOUN
cana-2867	260	12	operator	operator	NOUN
cana-2867	260	13	for	for	ADP
cana-2867	260	14	multi	multi	X
cana-2867	260	15	fuzzy	fuzzy	ADJ
cana-2867	260	16	ifs	ifs	PROPN
cana-2867	260	17	and	and	CCONJ
cana-2867	260	18	its	its	PRON
cana-2867	260	19	unique	unique	ADJ
cana-2867	260	20	‘	'	PUNCT
cana-2867	260	21	attractor	attractor	NOUN
cana-2867	260	22	’	'	PUNCT
cana-2867	260	23	is	be	AUX
cana-2867	260	24	a	a	DET
cana-2867	260	25	multi	multi	ADJ
cana-2867	260	26	-	-	ADJ
cana-2867	260	27	fuzzy	fuzzy	ADJ
cana-2867	260	28	fractal	fractal	NOUN
cana-2867	260	29	.	.	PUNCT
cana-2867	261	1	that	that	PRON
cana-2867	261	2	is	be	AUX
cana-2867	261	3	known	know	VERB
cana-2867	261	4	as	as	ADP
cana-2867	261	5	multi	multi	X
cana-2867	261	6	fuzzy	fuzzy	ADJ
cana-2867	261	7	fractal	fractal	NOUN
cana-2867	261	8	for	for	ADP
cana-2867	261	9	generalized	generalized	ADJ
cana-2867	261	10	fuzzy	fuzzy	ADJ
cana-2867	261	11	contractions	contraction	NOUN
cana-2867	261	12	.	.	PUNCT
cana-2867	262	1	remark	remark	VERB
cana-2867	262	2	4.5	4.5	NUM
cana-2867	262	3	our	our	PRON
cana-2867	262	4	analysis	analysis	NOUN
cana-2867	262	5	explains	explain	VERB
cana-2867	262	6	why	why	SCONJ
cana-2867	262	7	𝒜∞	𝒜∞	PROPN
cana-2867	262	8	is	be	AUX
cana-2867	262	9	called	call	VERB
cana-2867	262	10	the	the	DET
cana-2867	262	11	attractor	attractor	NOUN
cana-2867	262	12	of	of	ADP
cana-2867	262	13	w	w	PROPN
cana-2867	262	14	,	,	PUNCT
cana-2867	262	15	namely	namely	ADV
cana-2867	262	16	because	because	SCONJ
cana-2867	262	17	it	it	PRON
cana-2867	262	18	“	"	PUNCT
cana-2867	262	19	attracts	attract	VERB
cana-2867	262	20	”	"	PUNCT
cana-2867	262	21	all	all	DET
cana-2867	262	22	the	the	DET
cana-2867	262	23	elements	element	NOUN
cana-2867	262	24	of	of	ADP
cana-2867	262	25	𝒦(𝒳	𝒦(𝒳	NUM
cana-2867	262	26	)	)	PUNCT
cana-2867	262	27	.	.	PUNCT
cana-2867	263	1	remark	remark	VERB
cana-2867	263	2	4.6	4.6	NUM
cana-2867	263	3	theorem	theorem	NOUN
cana-2867	263	4	(	(	PUNCT
cana-2867	263	5	4.3	4.3	NUM
cana-2867	263	6	)	)	PUNCT
cana-2867	263	7	is	be	AUX
cana-2867	263	8	a	a	DET
cana-2867	263	9	valuable	valuable	ADJ
cana-2867	263	10	addition	addition	NOUN
cana-2867	263	11	in	in	ADP
cana-2867	263	12	our	our	PRON
cana-2867	263	13	main	main	ADJ
cana-2867	263	14	result	result	NOUN
cana-2867	263	15	.	.	PUNCT
cana-2867	264	1	as	as	ADP
cana-2867	264	2	an	an	DET
cana-2867	264	3	application	application	NOUN
cana-2867	264	4	,	,	PUNCT
cana-2867	264	5	which	which	PRON
cana-2867	264	6	will	will	AUX
cana-2867	264	7	be	be	AUX
cana-2867	264	8	useful	useful	ADJ
cana-2867	264	9	in	in	ADP
cana-2867	264	10	area	area	NOUN
cana-2867	264	11	of	of	ADP
cana-2867	264	12	image	image	NOUN
cana-2867	264	13	representation	representation	NOUN
cana-2867	264	14	,	,	PUNCT
cana-2867	264	15	image	image	NOUN
cana-2867	264	16	processing	processing	NOUN
cana-2867	264	17	and	and	CCONJ
cana-2867	264	18	image	image	NOUN
cana-2867	264	19	compression	compression	NOUN
cana-2867	264	20	.	.	PUNCT
cana-2867	265	1	remark	remark	VERB
cana-2867	265	2	4.7	4.7	NUM
cana-2867	265	3	our	our	PRON
cana-2867	265	4	leading	lead	VERB
cana-2867	265	5	outcomes	outcome	NOUN
cana-2867	265	6	are	be	AUX
cana-2867	265	7	the	the	DET
cana-2867	265	8	generalization	generalization	NOUN
cana-2867	265	9	of	of	ADP
cana-2867	265	10	the	the	DET
cana-2867	265	11	leading	lead	VERB
cana-2867	265	12	consequence	consequence	NOUN
cana-2867	265	13	of	of	ADP
cana-2867	265	14	prasad	prasad	PROPN
cana-2867	265	15	and	and	CCONJ
cana-2867	265	16	katiyar	katiyar	NOUN
cana-2867	266	1	[	[	X
cana-2867	266	2	38	38	NUM
cana-2867	266	3	]	]	PUNCT
cana-2867	266	4	in	in	ADP
cana-2867	266	5	which	which	PRON
cana-2867	266	6	we	we	PRON
cana-2867	266	7	have	have	AUX
cana-2867	266	8	replaced	replace	VERB
cana-2867	266	9	b	b	PRON
cana-2867	266	10	-contraction	-contraction	NOUN
cana-2867	266	11	by	by	ADP
cana-2867	266	12	generalized	generalized	ADJ
cana-2867	266	13	contraction	contraction	NOUN
cana-2867	266	14	mapping	mapping	NOUN
cana-2867	266	15	.	.	PUNCT
cana-2867	267	1	remark	remark	VERB
cana-2867	267	2	4.8	4.8	NUM
cana-2867	267	3	it	it	PRON
cana-2867	267	4	should	should	AUX
cana-2867	267	5	be	be	AUX
cana-2867	267	6	noticed	notice	VERB
cana-2867	267	7	that	that	SCONJ
cana-2867	267	8	,	,	PUNCT
cana-2867	267	9	example	example	NOUN
cana-2867	267	10	(	(	PUNCT
cana-2867	267	11	4.1	4.1	NUM
cana-2867	267	12	)	)	PUNCT
cana-2867	267	13	will	will	AUX
cana-2867	267	14	be	be	AUX
cana-2867	267	15	helpful	helpful	ADJ
cana-2867	267	16	for	for	ADP
cana-2867	267	17	finding	find	VERB
cana-2867	267	18	a	a	DET
cana-2867	267	19	unique	unique	ADJ
cana-2867	267	20	attractor	attractor	NOUN
cana-2867	267	21	for	for	ADP
cana-2867	267	22	generalized	generalized	ADJ
cana-2867	267	23	multi	multi	ADJ
cana-2867	267	24	fuzzy	fuzzy	ADJ
cana-2867	267	25	iterated	iterate	VERB
cana-2867	267	26	function	function	NOUN
cana-2867	267	27	system	system	NOUN
cana-2867	267	28	.	.	PUNCT
cana-2867	268	1	4	4	X
cana-2867	268	2	.	.	X
cana-2867	268	3	conclusions	conclusion	NOUN
cana-2867	268	4	in	in	ADP
cana-2867	268	5	this	this	DET
cana-2867	268	6	research	research	NOUN
cana-2867	268	7	,	,	PUNCT
cana-2867	268	8	we	we	PRON
cana-2867	268	9	have	have	AUX
cana-2867	268	10	studied	study	VERB
cana-2867	268	11	multi	multi	ADJ
cana-2867	268	12	-	-	ADJ
cana-2867	268	13	fuzzy	fuzzy	ADJ
cana-2867	268	14	fractal	fractal	ADJ
cana-2867	268	15	spaces	space	NOUN
cana-2867	268	16	and	and	CCONJ
cana-2867	268	17	the	the	DET
cana-2867	268	18	properties	property	NOUN
cana-2867	268	19	of	of	ADP
cana-2867	268	20	generalized	generalized	ADJ
cana-2867	268	21	contraction	contraction	NOUN
cana-2867	268	22	mapping	mapping	NOUN
cana-2867	268	23	.	.	PUNCT
cana-2867	269	1	multi	multi	ADJ
cana-2867	269	2	fuzzy	fuzzy	ADJ
cana-2867	269	3	fractal	fractal	ADJ
cana-2867	269	4	spaces	space	NOUN
cana-2867	269	5	are	be	AUX
cana-2867	269	6	the	the	DET
cana-2867	269	7	set	set	NOUN
cana-2867	269	8	of	of	ADP
cana-2867	269	9	fuzzy	fuzzy	ADJ
cana-2867	269	10	bounded	bounded	ADJ
cana-2867	269	11	subsets	subset	NOUN
cana-2867	269	12	,	,	PUNCT
cana-2867	269	13	compact	compact	ADJ
cana-2867	269	14	,	,	PUNCT
cana-2867	269	15	nonempty	nonempty	ADV
cana-2867	269	16	closed	close	VERB
cana-2867	269	17	,	,	PUNCT
cana-2867	269	18	respectively	respectively	ADV
cana-2867	269	19	,	,	PUNCT
cana-2867	269	20	and	and	CCONJ
cana-2867	269	21	supplied	supply	VERB
cana-2867	269	22	along	along	ADP
cana-2867	269	23	the	the	DET
cana-2867	269	24	hausdorff	hausdorff	NOUN
cana-2867	269	25	multi	multi	X
cana-2867	269	26	fuzzy	fuzzy	ADJ
cana-2867	269	27	fractal	fractal	ADJ
cana-2867	269	28	spaces	space	NOUN
cana-2867	269	29	.	.	PUNCT
cana-2867	270	1	moreover	moreover	ADV
cana-2867	270	2	,	,	PUNCT
cana-2867	270	3	with	with	ADP
cana-2867	270	4	the	the	DET
cana-2867	270	5	completeness	completeness	NOUN
cana-2867	270	6	of	of	ADP
cana-2867	270	7	multi	multi	ADJ
cana-2867	270	8	-	-	ADJ
cana-2867	270	9	fuzzy	fuzzy	ADJ
cana-2867	270	10	fractal	fractal	ADJ
cana-2867	270	11	spaces	space	NOUN
cana-2867	270	12	,	,	PUNCT
cana-2867	270	13	we	we	PRON
cana-2867	270	14	have	have	VERB
cana-2867	270	15	numerous	numerous	ADJ
cana-2867	270	16	existing	exist	VERB
cana-2867	270	17	results	result	NOUN
cana-2867	270	18	of	of	ADP
cana-2867	270	19	attractors	attractor	NOUN
cana-2867	270	20	reported	report	VERB
cana-2867	270	21	in	in	ADP
cana-2867	270	22	the	the	DET
cana-2867	270	23	literature	literature	NOUN
cana-2867	270	24	.	.	PUNCT
cana-2867	271	1	this	this	PRON
cana-2867	271	2	will	will	AUX
cana-2867	271	3	help	help	VERB
cana-2867	271	4	us	we	PRON
cana-2867	271	5	build	build	VERB
cana-2867	271	6	our	our	PRON
cana-2867	271	7	direction	direction	NOUN
cana-2867	271	8	to	to	PART
cana-2867	271	9	construct	construct	VERB
cana-2867	271	10	the	the	DET
cana-2867	271	11	multi	multi	ADJ
cana-2867	271	12	-	-	ADJ
cana-2867	271	13	fuzzy	fuzzy	ADJ
cana-2867	271	14	ifs	if	NOUN
cana-2867	271	15	consisting	consist	VERB
cana-2867	271	16	of	of	ADP
cana-2867	271	17	generalized	generalized	ADJ
cana-2867	271	18	contraction	contraction	NOUN
cana-2867	271	19	mapping	mapping	NOUN
cana-2867	271	20	in	in	ADP
cana-2867	271	21	the	the	DET
cana-2867	271	22	multi	multi	ADJ
cana-2867	271	23	-	-	ADJ
cana-2867	271	24	fuzzy	fuzzy	ADJ
cana-2867	271	25	fractal	fractal	ADJ
cana-2867	271	26	spaces	space	NOUN
cana-2867	271	27	.	.	PUNCT
cana-2867	272	1	consequently	consequently	ADV
cana-2867	272	2	,	,	PUNCT
cana-2867	272	3	we	we	PRON
cana-2867	272	4	have	have	AUX
cana-2867	272	5	defined	define	VERB
cana-2867	272	6	multifuzzy	multifuzzy	NOUN
cana-2867	272	7	fractals	fractal	NOUN
cana-2867	272	8	in	in	ADP
cana-2867	272	9	multifuzzy	multifuzzy	NOUN
cana-2867	272	10	fractal	fractal	ADJ
cana-2867	272	11	spaces	space	NOUN
cana-2867	272	12	consisting	consist	VERB
cana-2867	272	13	of	of	ADP
cana-2867	272	14	generalized	generalized	ADJ
cana-2867	272	15	contraction	contraction	NOUN
cana-2867	272	16	mappings	mapping	NOUN
cana-2867	272	17	for	for	ADP
cana-2867	272	18	an	an	DET
cana-2867	272	19	mfifs	mfif	NOUN
cana-2867	272	20	.	.	PUNCT
cana-2867	273	1	roughly	roughly	ADV
cana-2867	273	2	,	,	PUNCT
cana-2867	273	3	we	we	PRON
cana-2867	273	4	say	say	VERB
cana-2867	273	5	that	that	SCONJ
cana-2867	273	6	an	an	DET
cana-2867	273	7	mfifs	mfif	NOUN
cana-2867	273	8	consisting	consist	VERB
cana-2867	273	9	of	of	ADP
cana-2867	273	10	generalized	generalized	ADJ
cana-2867	273	11	contraction	contraction	NOUN
cana-2867	273	12	mapping	mapping	NOUN
cana-2867	273	13	in	in	ADP
cana-2867	273	14	a	a	DET
cana-2867	273	15	complete	complete	ADJ
cana-2867	273	16	multifuzzy	multifuzzy	NOUN
cana-2867	273	17	fractal	fractal	ADJ
cana-2867	273	18	space	space	NOUN
cana-2867	273	19	possesses	possess	VERB
cana-2867	273	20	a	a	DET
cana-2867	273	21	unique	unique	ADJ
cana-2867	273	22	attractor	attractor	NOUN
cana-2867	273	23	,	,	PUNCT
cana-2867	273	24	and	and	CCONJ
cana-2867	273	25	also	also	ADV
cana-2867	273	26	that	that	SCONJ
cana-2867	273	27	the	the	DET
cana-2867	273	28	sequence	sequence	NOUN
cana-2867	273	29	of	of	ADP
cana-2867	273	30	iterative	iterative	ADJ
cana-2867	273	31	mappings	mapping	NOUN
cana-2867	273	32	produced	produce	VERB
cana-2867	273	33	by	by	ADP
cana-2867	273	34	any	any	DET
cana-2867	273	35	closed	closed	ADJ
cana-2867	273	36	bounded	bounded	ADJ
cana-2867	273	37	set	set	NOUN
cana-2867	273	38	or	or	CCONJ
cana-2867	273	39	compact	compact	ADJ
cana-2867	273	40	can	can	AUX
cana-2867	273	41	converge	converge	VERB
cana-2867	273	42	to	to	ADP
cana-2867	273	43	this	this	DET
cana-2867	273	44	communications	communication	NOUN
cana-2867	273	45	on	on	ADP
cana-2867	273	46	applied	apply	VERB
cana-2867	273	47	nonlinear	nonlinear	ADJ
cana-2867	273	48	analysis	analysis	NOUN
cana-2867	273	49	issn	issn	NOUN
cana-2867	273	50	:	:	PUNCT
cana-2867	273	51	1074	1074	NUM
cana-2867	273	52	-	-	PUNCT
cana-2867	273	53	133x	133x	NUM
cana-2867	273	54	vol	vol	NOUN
cana-2867	273	55	32	32	NUM
cana-2867	273	56	no	no	NOUN
cana-2867	273	57	.	.	PUNCT
cana-2867	274	1	4s	4s	NUM
cana-2867	274	2	(	(	PUNCT
cana-2867	274	3	2025	2025	NUM
cana-2867	274	4	)	)	PUNCT
cana-2867	274	5	511	511	NUM
cana-2867	274	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2867	274	7	attractor	attractor	NOUN
cana-2867	274	8	.	.	PUNCT
cana-2867	275	1	a	a	DET
cana-2867	275	2	methodology	methodology	NOUN
cana-2867	275	3	for	for	ADP
cana-2867	275	4	constructing	construct	VERB
cana-2867	275	5	multifuzzy	multifuzzy	NOUN
cana-2867	275	6	ifs	if	NOUN
cana-2867	275	7	in	in	ADP
cana-2867	275	8	multifuzzy	multifuzzy	NOUN
cana-2867	275	9	fractal	fractal	ADJ
cana-2867	275	10	spaces	space	NOUN
cana-2867	275	11	consisting	consist	VERB
cana-2867	275	12	of	of	ADP
cana-2867	275	13	a	a	DET
cana-2867	275	14	generalized	generalized	ADJ
cana-2867	275	15	contraction	contraction	NOUN
cana-2867	275	16	mapping	mapping	NOUN
cana-2867	275	17	is	be	AUX
cana-2867	275	18	presented	present	VERB
cana-2867	275	19	.	.	PUNCT
cana-2867	276	1	references	reference	NOUN
cana-2867	276	2	[	[	X
cana-2867	276	3	1	1	NUM
cana-2867	276	4	]	]	PUNCT
cana-2867	276	5	l.	l.	PROPN
cana-2867	276	6	a.	a.	PROPN
cana-2867	276	7	zadeh	zadeh	PROPN
cana-2867	276	8	,	,	PUNCT
cana-2867	276	9	fuzzy	fuzzy	ADJ
cana-2867	276	10	sets	set	NOUN
cana-2867	276	11	,	,	PUNCT
cana-2867	276	12	information	information	NOUN
cana-2867	276	13	and	and	CCONJ
cana-2867	276	14	control	control	NOUN
cana-2867	276	15	,	,	PUNCT
cana-2867	276	16	8(1965	8(1965	NUM
cana-2867	276	17	)	)	PUNCT
cana-2867	276	18	,	,	PUNCT
cana-2867	276	19	338	338	NUM
cana-2867	276	20	-	-	SYM
cana-2867	276	21	353	353	NUM
cana-2867	276	22	.	.	PUNCT
cana-2867	277	1	https://doi.org/10.1016/s00199958(65)90241-x	https://doi.org/10.1016/s00199958(65)90241-x	X
cana-2867	278	1	[	[	X
cana-2867	278	2	2	2	NUM
cana-2867	278	3	]	]	PUNCT
cana-2867	278	4	i.	i.	NOUN
cana-2867	278	5	kramosil	kramosil	PROPN
cana-2867	278	6	,	,	PUNCT
cana-2867	278	7	j.	j.	PROPN
cana-2867	278	8	michalek	michalek	PROPN
cana-2867	278	9	,	,	PUNCT
cana-2867	278	10	fuzzy	fuzzy	ADJ
cana-2867	278	11	metrics	metric	NOUN
cana-2867	278	12	and	and	CCONJ
cana-2867	278	13	statistical	statistical	ADJ
cana-2867	278	14	metric	metric	ADJ
cana-2867	278	15	spaces	space	NOUN
cana-2867	278	16	,	,	PUNCT
cana-2867	278	17	kybernetika	kybernetika	NOUN
cana-2867	278	18	,	,	PUNCT
cana-2867	278	19	11(5)(1975	11(5)(1975	NUM
cana-2867	278	20	)	)	PUNCT
cana-2867	278	21	,	,	PUNCT
cana-2867	278	22	336	336	NUM
cana-2867	278	23	-	-	SYM
cana-2867	278	24	344	344	NUM
cana-2867	278	25	.	.	PUNCT
cana-2867	279	1	[	[	X
cana-2867	279	2	3	3	NUM
cana-2867	279	3	]	]	PUNCT
cana-2867	279	4	a.	a.	NOUN
cana-2867	279	5	george	george	PROPN
cana-2867	279	6	,	,	PUNCT
cana-2867	279	7	p.	p.	NOUN
cana-2867	279	8	veeramani	veeramani	PROPN
cana-2867	279	9	,	,	PUNCT
cana-2867	279	10	on	on	ADP
cana-2867	279	11	some	some	DET
cana-2867	279	12	results	result	NOUN
cana-2867	279	13	in	in	ADP
cana-2867	279	14	fuzzy	fuzzy	ADJ
cana-2867	279	15	metric	metric	ADJ
cana-2867	279	16	spaces	space	NOUN
cana-2867	279	17	,	,	PUNCT
cana-2867	279	18	fuzzy	fuzzy	ADJ
cana-2867	279	19	sets	set	NOUN
cana-2867	279	20	and	and	CCONJ
cana-2867	279	21	systems	system	NOUN
cana-2867	279	22	,	,	PUNCT
cana-2867	279	23	64(1994	64(1994	NUM
cana-2867	279	24	)	)	PUNCT
cana-2867	279	25	,	,	PUNCT
cana-2867	279	26	395	395	NUM
cana-2867	279	27	-	-	SYM
cana-2867	279	28	399	399	NUM
cana-2867	279	29	.	.	PUNCT
cana-2867	280	1	https://doi.org/10.1016/0165-0114(94)90162-7	https://doi.org/10.1016/0165-0114(94)90162-7	PRON
cana-2867	280	2	[	[	X
cana-2867	280	3	4	4	NUM
cana-2867	280	4	]	]	PUNCT
cana-2867	280	5	a.	a.	NOUN
cana-2867	280	6	george	george	PROPN
cana-2867	280	7	,	,	PUNCT
cana-2867	280	8	p.	p.	NOUN
cana-2867	280	9	veeramani	veeramani	PROPN
cana-2867	280	10	,	,	PUNCT
cana-2867	280	11	on	on	ADP
cana-2867	280	12	some	some	DET
cana-2867	280	13	results	result	NOUN
cana-2867	280	14	of	of	ADP
cana-2867	280	15	analysis	analysis	NOUN
cana-2867	280	16	for	for	ADP
cana-2867	280	17	fuzzy	fuzzy	ADJ
cana-2867	280	18	metric	metric	ADJ
cana-2867	280	19	spaces	space	NOUN
cana-2867	280	20	,	,	PUNCT
cana-2867	280	21	fuzzy	fuzzy	ADJ
cana-2867	280	22	sets	set	NOUN
cana-2867	280	23	and	and	CCONJ
cana-2867	280	24	systems	system	NOUN
cana-2867	280	25	,	,	PUNCT
cana-2867	280	26	90(1997	90(1997	NUM
cana-2867	280	27	)	)	PUNCT
cana-2867	280	28	,	,	PUNCT
cana-2867	280	29	365	365	NUM
cana-2867	280	30	-	-	SYM
cana-2867	280	31	368	368	NUM
cana-2867	280	32	.	.	PUNCT
cana-2867	281	1	https://doi.org/10.1016/s0165-0114(96)00207-2	https://doi.org/10.1016/s0165-0114(96)00207-2	NOUN
cana-2867	281	2	[	[	X
cana-2867	281	3	5	5	NUM
cana-2867	281	4	]	]	PUNCT
cana-2867	281	5	m.	m.	NOUN
cana-2867	281	6	grabiec	grabiec	PROPN
cana-2867	281	7	,	,	PUNCT
cana-2867	281	8	fixed	fix	VERB
cana-2867	281	9	points	point	NOUN
cana-2867	281	10	in	in	ADP
cana-2867	281	11	fuzzy	fuzzy	ADJ
cana-2867	281	12	metric	metric	ADJ
cana-2867	281	13	space	space	NOUN
cana-2867	281	14	,	,	PUNCT
cana-2867	281	15	fuzzy	fuzzy	ADJ
cana-2867	281	16	sets	set	NOUN
cana-2867	281	17	and	and	CCONJ
cana-2867	281	18	systems	system	NOUN
cana-2867	281	19	,	,	PUNCT
cana-2867	281	20	27(1988	27(1988	NUM
cana-2867	281	21	)	)	PUNCT
cana-2867	281	22	,	,	PUNCT
cana-2867	281	23	385	385	NUM
cana-2867	281	24	-	-	SYM
cana-2867	281	25	389	389	NUM
cana-2867	281	26	.	.	PUNCT
cana-2867	282	1	https://doi.org/10.1016/0165-0114(88)90064-4	https://doi.org/10.1016/0165-0114(88)90064-4	PROPN
cana-2867	283	1	[	[	X
cana-2867	283	2	6	6	NUM
cana-2867	283	3	]	]	PUNCT
cana-2867	283	4	j.	j.	PROPN
cana-2867	283	5	rodriguez	rodriguez	PROPN
cana-2867	283	6	-	-	PUNCT
cana-2867	283	7	lopez	lopez	PROPN
cana-2867	283	8	,	,	PUNCT
cana-2867	283	9	s.	s.	PROPN
cana-2867	283	10	romaguera	romaguera	PROPN
cana-2867	283	11	,	,	PUNCT
cana-2867	283	12	the	the	DET
cana-2867	283	13	hausdorff	hausdorff	NOUN
cana-2867	283	14	fuzzy	fuzzy	ADJ
cana-2867	283	15	metric	metric	ADJ
cana-2867	283	16	on	on	ADP
cana-2867	283	17	compact	compact	ADJ
cana-2867	283	18	sets	set	NOUN
cana-2867	283	19	,	,	PUNCT
cana-2867	283	20	fuzzy	fuzzy	ADJ
cana-2867	283	21	sets	set	NOUN
cana-2867	283	22	and	and	CCONJ
cana-2867	283	23	systems	system	NOUN
cana-2867	283	24	147(2004	147(2004	NUM
cana-2867	283	25	)	)	PUNCT
cana-2867	283	26	,	,	PUNCT
cana-2867	283	27	273	273	NUM
cana-2867	283	28	-	-	SYM
cana-2867	283	29	283	283	NUM
cana-2867	283	30	.	.	PUNCT
cana-2867	284	1	https://doi.org/10.1016/j.fss.2003.09.007	https://doi.org/10.1016/j.fss.2003.09.007	ADJ
cana-2867	284	2	[	[	X
cana-2867	284	3	7	7	NUM
cana-2867	284	4	]	]	X
cana-2867	284	5	d.	d.	PROPN
cana-2867	284	6	qui	qui	PROPN
cana-2867	284	7	,	,	PUNCT
cana-2867	284	8	w.	w.	PROPN
cana-2867	284	9	zhang	zhang	PROPN
cana-2867	284	10	,	,	PUNCT
cana-2867	284	11	the	the	DET
cana-2867	284	12	strongest	strong	ADJ
cana-2867	284	13	𝑡	𝑡	NOUN
cana-2867	284	14	–	–	PUNCT
cana-2867	284	15	norm	norm	NOUN
cana-2867	284	16	for	for	ADP
cana-2867	284	17	fuzzy	fuzzy	ADJ
cana-2867	284	18	metric	metric	ADJ
cana-2867	284	19	spaces	space	NOUN
cana-2867	284	20	,	,	PUNCT
cana-2867	284	21	kybernetika	kybernetika	NOUN
cana-2867	284	22	,	,	PUNCT
cana-2867	284	23	49(1)(2013),141	49(1)(2013),141	NOUN
cana-2867	284	24	-	-	SYM
cana-2867	284	25	148	148	NUM
cana-2867	284	26	.	.	PUNCT
cana-2867	285	1	[	[	X
cana-2867	285	2	8	8	NUM
cana-2867	285	3	]	]	X
cana-2867	285	4	d.	d.	PROPN
cana-2867	285	5	qui	qui	PROPN
cana-2867	285	6	,	,	PUNCT
cana-2867	285	7	c.	c.	PROPN
cana-2867	285	8	lu	lu	PROPN
cana-2867	285	9	,	,	PUNCT
cana-2867	285	10	w.	w.	PROPN
cana-2867	285	11	zhang	zhang	PROPN
cana-2867	285	12	,	,	PUNCT
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cana-2867	285	14	fixed	fix	VERB
cana-2867	285	15	point	point	NOUN
cana-2867	285	16	theorems	theorem	NOUN
cana-2867	285	17	for	for	ADP
cana-2867	285	18	contractive	contractive	ADJ
cana-2867	285	19	-	-	PUNCT
cana-2867	285	20	type	type	NOUN
cana-2867	285	21	mappings	mapping	NOUN
cana-2867	285	22	in	in	ADP
cana-2867	285	23	fuzzy	fuzzy	ADJ
cana-2867	285	24	metric	metric	ADJ
cana-2867	285	25	spaces	space	NOUN
cana-2867	285	26	,	,	PUNCT
cana-2867	285	27	iranian	iranian	ADJ
cana-2867	285	28	journal	journal	NOUN
cana-2867	285	29	of	of	ADP
cana-2867	285	30	fuzzy	fuzzy	ADJ
cana-2867	285	31	systems	system	NOUN
cana-2867	285	32	,	,	PUNCT
cana-2867	285	33	11(6)(2014	11(6)(2014	NUM
cana-2867	285	34	)	)	PUNCT
cana-2867	285	35	,	,	PUNCT
cana-2867	285	36	123	123	NUM
cana-2867	285	37	-	-	SYM
cana-2867	285	38	130	130	NUM
cana-2867	285	39	.	.	PUNCT
cana-2867	286	1	https://doi.org/10.22111/ijfs.2014.1752	https://doi.org/10.22111/ijfs.2014.1752	PROPN
cana-2867	286	2	[	[	X
cana-2867	286	3	9	9	NUM
cana-2867	286	4	]	]	X
cana-2867	286	5	d.	d.	PROPN
cana-2867	286	6	qui	qui	PROPN
cana-2867	286	7	,	,	PUNCT
cana-2867	286	8	h.	h.	PROPN
cana-2867	286	9	li	li	PROPN
cana-2867	286	10	,	,	PUNCT
cana-2867	286	11	c.	c.	PROPN
cana-2867	286	12	lu	lu	PROPN
cana-2867	286	13	,	,	PUNCT
cana-2867	286	14	complete	complete	ADJ
cana-2867	286	15	stationary	stationary	ADJ
cana-2867	286	16	fuzzy	fuzzy	ADJ
cana-2867	286	17	metric	metric	ADJ
cana-2867	286	18	space	space	NOUN
cana-2867	286	19	of	of	ADP
cana-2867	286	20	the	the	DET
cana-2867	286	21	bounded	bound	VERB
cana-2867	286	22	closed	close	VERB
cana-2867	286	23	fuzzy	fuzzy	ADJ
cana-2867	286	24	sets	set	NOUN
cana-2867	286	25	and	and	CCONJ
cana-2867	286	26	common	common	ADJ
cana-2867	286	27	fixed	fix	VERB
cana-2867	286	28	point	point	NOUN
cana-2867	286	29	theorems	theorem	NOUN
cana-2867	286	30	,	,	PUNCT
cana-2867	286	31	iaeng	iaeng	PROPN
cana-2867	286	32	international	international	PROPN
cana-2867	286	33	journal	journal	NOUN
cana-2867	286	34	of	of	ADP
cana-2867	286	35	applied	apply	VERB
cana-2867	286	36	mathematics	mathematic	NOUN
cana-2867	286	37	,	,	PUNCT
cana-2867	286	38	45(4)(2015	45(4)(2015	NUM
cana-2867	286	39	)	)	PUNCT
cana-2867	286	40	,	,	PUNCT
cana-2867	286	41	286	286	NUM
cana-2867	286	42	-	-	SYM
cana-2867	286	43	292	292	NUM
cana-2867	286	44	.	.	PUNCT
cana-2867	287	1	[	[	X
cana-2867	287	2	10	10	NUM
cana-2867	287	3	]	]	X
cana-2867	287	4	d.	d.	PROPN
cana-2867	287	5	qui	qui	PROPN
cana-2867	287	6	,	,	PUNCT
cana-2867	287	7	c.	c.	PROPN
cana-2867	287	8	lu	lu	PROPN
cana-2867	287	9	,	,	PUNCT
cana-2867	287	10	s.	s.	PROPN
cana-2867	287	11	deng	deng	PROPN
cana-2867	287	12	,	,	PUNCT
cana-2867	287	13	l.	l.	PROPN
cana-2867	287	14	wang	wang	PROPN
cana-2867	287	15	,	,	PUNCT
cana-2867	287	16	on	on	ADP
cana-2867	287	17	the	the	DET
cana-2867	287	18	hyperspace	hyperspace	NOUN
cana-2867	287	19	of	of	ADP
cana-2867	287	20	bounded	bound	VERB
cana-2867	287	21	closed	close	VERB
cana-2867	287	22	sets	set	NOUN
cana-2867	287	23	under	under	ADP
cana-2867	287	24	a	a	DET
cana-2867	287	25	generalized	generalized	ADJ
cana-2867	287	26	hausdorff	hausdorff	NOUN
cana-2867	287	27	stationary	stationary	ADJ
cana-2867	287	28	fuzzy	fuzzy	ADJ
cana-2867	287	29	metric	metric	ADJ
cana-2867	287	30	,	,	PUNCT
cana-2867	287	31	kybernetika	kybernetika	NOUN
cana-2867	287	32	,	,	PUNCT
cana-2867	287	33	50(5)(2014	50(5)(2014	NUM
cana-2867	287	34	)	)	PUNCT
cana-2867	287	35	,	,	PUNCT
cana-2867	287	36	758	758	NUM
cana-2867	287	37	-	-	SYM
cana-2867	287	38	773	773	NUM
cana-2867	287	39	.	.	PUNCT
cana-2867	287	40	http://eudml.org/doc/262158	http://eudml.org/doc/262158	X
cana-2867	288	1	[	[	X
cana-2867	288	2	11	11	NUM
cana-2867	288	3	]	]	PUNCT
cana-2867	288	4	r.	r.	PROPN
cana-2867	288	5	farnoosh	farnoosh	PROPN
cana-2867	288	6	,	,	PUNCT
cana-2867	288	7	a.	a.	NOUN
cana-2867	288	8	aghajani	aghajani	PROPN
cana-2867	288	9	,	,	PUNCT
cana-2867	288	10	p.	p.	NOUN
cana-2867	288	11	azhadari	azhadari	NOUN
cana-2867	288	12	,	,	PUNCT
cana-2867	288	13	contraction	contraction	NOUN
cana-2867	288	14	theorems	theorem	NOUN
cana-2867	288	15	in	in	ADP
cana-2867	288	16	fuzzy	fuzzy	ADJ
cana-2867	288	17	metric	metric	ADJ
cana-2867	288	18	space	space	NOUN
cana-2867	288	19	,	,	PUNCT
cana-2867	288	20	chaos	chaos	NOUN
cana-2867	288	21	,	,	PUNCT
cana-2867	288	22	solitons	soliton	NOUN
cana-2867	288	23	and	and	CCONJ
cana-2867	288	24	fractals	fractal	NOUN
cana-2867	288	25	,	,	PUNCT
cana-2867	288	26	41(2)(2009	41(2)(2009	NUM
cana-2867	288	27	)	)	PUNCT
cana-2867	288	28	,	,	PUNCT
cana-2867	288	29	854	854	NUM
cana-2867	288	30	-	-	SYM
cana-2867	288	31	858	858	NUM
cana-2867	288	32	.	.	PUNCT
cana-2867	289	1	http://eudml.org/doc/252499	http://eudml.org/doc/252499	VERB
cana-2867	290	1	[	[	X
cana-2867	290	2	12	12	NUM
cana-2867	290	3	]	]	X
cana-2867	290	4	s.	s.	PROPN
cana-2867	290	5	sedghi	sedghi	PROPN
cana-2867	290	6	,	,	PUNCT
cana-2867	290	7	i.	i.	NOUN
cana-2867	290	8	altun	altun	PROPN
cana-2867	290	9	,	,	PUNCT
cana-2867	290	10	n.	n.	PROPN
cana-2867	290	11	shobe	shobe	PROPN
cana-2867	290	12	,	,	PUNCT
cana-2867	290	13	coupled	couple	VERB
cana-2867	290	14	fixed	fix	VERB
cana-2867	290	15	point	point	NOUN
cana-2867	290	16	theorems	theorem	NOUN
cana-2867	290	17	for	for	ADP
cana-2867	290	18	contractions	contraction	NOUN
cana-2867	290	19	in	in	ADP
cana-2867	290	20	fuzzy	fuzzy	ADJ
cana-2867	290	21	metric	metric	ADJ
cana-2867	290	22	spaces	space	NOUN
cana-2867	290	23	,	,	PUNCT
cana-2867	290	24	nonlinear	nonlinear	ADJ
cana-2867	290	25	analysis	analysis	NOUN
cana-2867	290	26	:	:	PUNCT
cana-2867	290	27	theory	theory	NOUN
cana-2867	290	28	,	,	PUNCT
cana-2867	290	29	methods	method	NOUN
cana-2867	290	30	and	and	CCONJ
cana-2867	290	31	applications	application	NOUN
cana-2867	290	32	,	,	PUNCT
cana-2867	290	33	72(3)(4	72(3)(4	NUM
cana-2867	290	34	)	)	PUNCT
cana-2867	290	35	,	,	PUNCT
cana-2867	290	36	1298	1298	NUM
cana-2867	290	37	-	-	SYM
cana-2867	290	38	1304	1304	NUM
cana-2867	290	39	.	.	PUNCT
cana-2867	291	1	http://dx.doi.org/10.1016/j.na.2009.08.018	http://dx.doi.org/10.1016/j.na.2009.08.018	NOUN
cana-2867	292	1	[	[	X
cana-2867	292	2	13	13	NUM
cana-2867	292	3	]	]	PUNCT
cana-2867	292	4	v.	v.	CCONJ
cana-2867	292	5	gregori	gregori	PROPN
cana-2867	292	6	,	,	PUNCT
cana-2867	292	7	s.	s.	PROPN
cana-2867	292	8	romaguera	romaguera	PROPN
cana-2867	292	9	,	,	PUNCT
cana-2867	292	10	some	some	DET
cana-2867	292	11	properties	property	NOUN
cana-2867	292	12	of	of	ADP
cana-2867	292	13	fuzzy	fuzzy	ADJ
cana-2867	292	14	metric	metric	ADJ
cana-2867	292	15	spaces	space	NOUN
cana-2867	292	16	,	,	PUNCT
cana-2867	292	17	fuzzy	fuzzy	ADJ
cana-2867	292	18	sets	set	NOUN
cana-2867	292	19	and	and	CCONJ
cana-2867	292	20	systems	system	NOUN
cana-2867	292	21	,	,	PUNCT
cana-2867	292	22	115(3)(2000),485	115(3)(2000),485	NUM
cana-2867	292	23	-	-	SYM
cana-2867	292	24	489	489	NUM
cana-2867	292	25	.	.	PUNCT
cana-2867	293	1	https://doi.org/10.1016/s0165-0114(98)00281-4	https://doi.org/10.1016/s0165-0114(98)00281-4	NUM
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cana-2867	294	2	14	14	NUM
cana-2867	294	3	]	]	X
cana-2867	294	4	v.n	v.n	PROPN
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cana-2867	294	8	j.j	j.j	PROPN
cana-2867	294	9	.	.	PROPN
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cana-2867	294	11	,	,	PUNCT
cana-2867	294	12	s.	s.	PROPN
cana-2867	294	13	morillas	morillas	PROPN
cana-2867	294	14	,	,	PUNCT
cana-2867	294	15	some	some	DET
cana-2867	294	16	questions	question	NOUN
cana-2867	294	17	in	in	ADP
cana-2867	294	18	fuzzy	fuzzy	ADJ
cana-2867	294	19	metric	metric	ADJ
cana-2867	294	20	spaces	space	NOUN
cana-2867	294	21	,	,	PUNCT
cana-2867	294	22	fuzzy	fuzzy	ADJ
cana-2867	294	23	sets	set	NOUN
cana-2867	294	24	and	and	CCONJ
cana-2867	294	25	systems	system	NOUN
cana-2867	294	26	,	,	PUNCT
cana-2867	294	27	204(2012	204(2012	NUM
cana-2867	294	28	)	)	PUNCT
cana-2867	294	29	,	,	PUNCT
cana-2867	294	30	71	71	NUM
cana-2867	294	31	-	-	SYM
cana-2867	294	32	85	85	NUM
cana-2867	294	33	.	.	PUNCT
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cana-2867	296	1	[	[	X
cana-2867	296	2	15	15	NUM
cana-2867	296	3	]	]	X
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cana-2867	296	5	gupta	gupta	PROPN
cana-2867	296	6	,	,	PUNCT
cana-2867	296	7	a.	a.	PROPN
cana-2867	296	8	kanwar	kanwar	PROPN
cana-2867	296	9	,	,	PUNCT
cana-2867	296	10	fixed	fix	VERB
cana-2867	296	11	point	point	NOUN
cana-2867	296	12	theorem	theorem	VERB
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cana-2867	296	14	fuzzy	fuzzy	ADJ
cana-2867	296	15	metric	metric	ADJ
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cana-2867	296	17	satisfying	satisfy	VERB
cana-2867	296	18	e.a.property	e.a.property	NOUN
cana-2867	296	19	,	,	PUNCT
cana-2867	296	20	indian	indian	ADJ
cana-2867	296	21	journal	journal	NOUN
cana-2867	296	22	of	of	ADP
cana-2867	296	23	science	science	NOUN
cana-2867	296	24	and	and	CCONJ
cana-2867	296	25	technology	technology	NOUN
cana-2867	296	26	,	,	PUNCT
cana-2867	296	27	5(12)(2012	5(12)(2012	NUM
cana-2867	296	28	)	)	PUNCT
cana-2867	296	29	,	,	PUNCT
cana-2867	296	30	3767	3767	NUM
cana-2867	296	31	-	-	SYM
cana-2867	296	32	3769	3769	NUM
cana-2867	296	33	.	.	PUNCT
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cana-2867	298	1	[	[	X
cana-2867	298	2	16	16	NUM
cana-2867	298	3	]	]	X
cana-2867	298	4	v.	v.	CCONJ
cana-2867	298	5	gupta	gupta	PROPN
cana-2867	298	6	,	,	PUNCT
cana-2867	298	7	r.	r.	PROPN
cana-2867	298	8	deep	deep	ADJ
cana-2867	298	9	,	,	PUNCT
cana-2867	298	10	coupled	couple	VERB
cana-2867	298	11	fixed	fix	VERB
cana-2867	298	12	points	point	NOUN
cana-2867	298	13	results	result	NOUN
cana-2867	298	14	for	for	ADP
cana-2867	298	15	nonlinear	nonlinear	ADJ
cana-2867	298	16	contractions	contraction	NOUN
cana-2867	298	17	in	in	ADP
cana-2867	298	18	quasi	quasi	ADJ
cana-2867	298	19	-	-	ADJ
cana-2867	298	20	ordered	order	VERB
cana-2867	298	21	metric	metric	ADJ
cana-2867	298	22	spaces	space	NOUN
cana-2867	298	23	,	,	PUNCT
cana-2867	298	24	indian	indian	ADJ
cana-2867	298	25	journal	journal	NOUN
cana-2867	298	26	of	of	ADP
cana-2867	298	27	science	science	NOUN
cana-2867	298	28	and	and	CCONJ
cana-2867	298	29	technology	technology	NOUN
cana-2867	298	30	,	,	PUNCT
cana-2867	298	31	8(13)(2015	8(13)(2015	NUM
cana-2867	298	32	)	)	PUNCT
cana-2867	298	33	,	,	PUNCT
cana-2867	298	34	1	1	NUM
cana-2867	298	35	-	-	SYM
cana-2867	298	36	7	7	NUM
cana-2867	298	37	.	.	PUNCT
cana-2867	299	1	https://dx.doi.org/10.17485/ijst/2015/v8i13/51702	https://dx.doi.org/10.17485/ijst/2015/v8i13/51702	PROPN
cana-2867	299	2	[	[	X
cana-2867	299	3	17	17	NUM
cana-2867	299	4	]	]	SYM
cana-2867	299	5	b.prasad	b.prasad	ADJ
cana-2867	299	6	,	,	PUNCT
cana-2867	299	7	r.sahani	r.sahani	NOUN
cana-2867	299	8	,	,	PUNCT
cana-2867	299	9	common	common	ADJ
cana-2867	299	10	fixed	fix	VERB
cana-2867	299	11	point	point	NOUN
cana-2867	299	12	theorems	theorem	NOUN
cana-2867	299	13	for	for	ADP
cana-2867	299	14	𝛹-weakly	𝛹-weakly	ADJ
cana-2867	299	15	commuting	commuting	NOUN
cana-2867	299	16	maps	map	NOUN
cana-2867	299	17	in	in	ADP
cana-2867	299	18	fuzzy	fuzzy	ADJ
cana-2867	299	19	metric	metric	ADJ
cana-2867	299	20	spaces	space	NOUN
cana-2867	299	21	.	.	PUNCT
cana-2867	300	1	acta	acta	PROPN
cana-2867	300	2	et	et	PROPN
cana-2867	300	3	commentationes	commentatione	VERB
cana-2867	300	4	universities	university	NOUN
cana-2867	300	5	tartuensis	tartuensis	PROPN
cana-2867	300	6	de	de	X
cana-2867	300	7	mathematica	mathematica	PROPN
cana-2867	300	8	,	,	PUNCT
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cana-2867	300	10	-	-	SYM
cana-2867	300	11	125	125	NUM
cana-2867	300	12	.	.	PUNCT
cana-2867	300	13	https://doi.org/10.12697/acutm.2013.17.10	https://doi.org/10.12697/acutm.2013.17.10	PROPN
cana-2867	301	1	[	[	X
cana-2867	301	2	18	18	NUM
cana-2867	301	3	]	]	X
cana-2867	301	4	b.	b.	PROPN
cana-2867	301	5	b.	b.	PROPN
cana-2867	301	6	mandelbrot	mandelbrot	PROPN
cana-2867	301	7	,	,	PUNCT
cana-2867	301	8	the	the	DET
cana-2867	301	9	fractal	fractal	ADJ
cana-2867	301	10	geometry	geometry	NOUN
cana-2867	301	11	of	of	ADP
cana-2867	301	12	nature	nature	NOUN
cana-2867	301	13	,	,	PUNCT
cana-2867	301	14	w.	w.	PROPN
cana-2867	301	15	h.	h.	PROPN
cana-2867	301	16	freeman	freeman	PROPN
cana-2867	301	17	and	and	CCONJ
cana-2867	301	18	company	company	NOUN
cana-2867	301	19	,	,	PUNCT
cana-2867	301	20	new	new	PROPN
cana-2867	301	21	york	york	PROPN
cana-2867	301	22	,	,	PUNCT
cana-2867	301	23	(	(	PUNCT
cana-2867	301	24	1983	1983	NUM
cana-2867	301	25	)	)	PUNCT
cana-2867	301	26	.	.	PUNCT
cana-2867	302	1	[	[	X
cana-2867	302	2	19	19	NUM
cana-2867	302	3	]	]	X
cana-2867	302	4	j.	j.	PROPN
cana-2867	302	5	e.	e.	PROPN
cana-2867	302	6	hutchinson	hutchinson	PROPN
cana-2867	302	7	,	,	PUNCT
cana-2867	302	8	fractals	fractal	NOUN
cana-2867	302	9	and	and	CCONJ
cana-2867	302	10	self	self	NOUN
cana-2867	302	11	similarity	similarity	NOUN
cana-2867	302	12	,	,	PUNCT
cana-2867	302	13	indiana	indiana	PROPN
cana-2867	302	14	university	university	PROPN
cana-2867	302	15	mathematics	mathematics	PROPN
cana-2867	302	16	journal	journal	NOUN
cana-2867	302	17	,	,	PUNCT
cana-2867	302	18	30	30	NUM
cana-2867	302	19	(	(	PUNCT
cana-2867	302	20	5	5	NUM
cana-2867	302	21	)	)	PUNCT
cana-2867	302	22	(	(	PUNCT
cana-2867	302	23	1981	1981	NUM
cana-2867	302	24	)	)	PUNCT
cana-2867	302	25	,	,	PUNCT
cana-2867	302	26	713	713	NUM
cana-2867	302	27	-	-	SYM
cana-2867	302	28	747	747	NUM
cana-2867	302	29	.	.	PUNCT
cana-2867	303	1	[	[	X
cana-2867	303	2	20	20	NUM
cana-2867	303	3	]	]	X
cana-2867	303	4	m.f	m.f	PROPN
cana-2867	303	5	.	.	PROPN
cana-2867	303	6	baransley	baransley	NOUN
cana-2867	303	7	,	,	PUNCT
cana-2867	303	8	fractals	fractal	NOUN
cana-2867	303	9	everywhere	everywhere	ADV
cana-2867	303	10	,	,	PUNCT
cana-2867	303	11	new	new	PROPN
cana-2867	303	12	york	york	PROPN
cana-2867	303	13	,	,	PUNCT
cana-2867	303	14	academic	academic	ADJ
cana-2867	303	15	press	press	NOUN
cana-2867	303	16	,	,	PUNCT
cana-2867	303	17	(	(	PUNCT
cana-2867	303	18	1988	1988	NUM
cana-2867	303	19	)	)	PUNCT
cana-2867	303	20	.	.	PUNCT
cana-2867	304	1	[	[	X
cana-2867	304	2	21	21	NUM
cana-2867	304	3	]	]	PUNCT
cana-2867	304	4	m.	m.	PROPN
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cana-2867	304	6	baransley	baransley	PROPN
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cana-2867	304	8	s.	s.	PROPN
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cana-2867	304	10	,	,	PUNCT
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cana-2867	304	12	function	function	NOUN
cana-2867	304	13	systems	system	NOUN
cana-2867	304	14	and	and	CCONJ
cana-2867	304	15	global	global	ADJ
cana-2867	304	16	construction	construction	NOUN
cana-2867	304	17	of	of	ADP
cana-2867	304	18	fractals	fractal	NOUN
cana-2867	304	19	,	,	PUNCT
cana-2867	304	20	proc	proc	NOUN
cana-2867	304	21	.	.	PUNCT
cana-2867	305	1	roy	roy	PROPN
cana-2867	305	2	.	.	PROPN
cana-2867	305	3	soc	soc	PROPN
cana-2867	305	4	.	.	PUNCT
cana-2867	306	1	london	london	PROPN
cana-2867	306	2	.	.	PUNCT
cana-2867	307	1	ser	ser	PROPN
cana-2867	307	2	a	a	X
cana-2867	307	3	,	,	PUNCT
cana-2867	307	4	399	399	NUM
cana-2867	307	5	(	(	PUNCT
cana-2867	307	6	1985	1985	NUM
cana-2867	307	7	)	)	PUNCT
cana-2867	307	8	,	,	PUNCT
cana-2867	307	9	243	243	NUM
cana-2867	307	10	-	-	SYM
cana-2867	307	11	275	275	NUM
cana-2867	307	12	.	.	PUNCT
cana-2867	308	1	https://doi.org/10.1098/rspa.1985.0057	https://doi.org/10.1098/rspa.1985.0057	VERB
cana-2867	308	2	[	[	X
cana-2867	308	3	22	22	NUM
cana-2867	308	4	]	]	PUNCT
cana-2867	308	5	e.	e.	PROPN
cana-2867	308	6	e.	e.	PROPN
cana-2867	308	7	vrscay	vrscay	PROPN
cana-2867	308	8	,	,	PUNCT
cana-2867	308	9	iterated	iterate	VERB
cana-2867	308	10	function	function	NOUN
cana-2867	308	11	systems	system	NOUN
cana-2867	308	12	:	:	PUNCT
cana-2867	308	13	theory	theory	NOUN
cana-2867	308	14	,	,	PUNCT
cana-2867	308	15	applications	application	NOUN
cana-2867	308	16	and	and	CCONJ
cana-2867	308	17	the	the	DET
cana-2867	308	18	inverse	inverse	NOUN
cana-2867	308	19	problem	problem	NOUN
cana-2867	308	20	,	,	PUNCT
cana-2867	308	21	in	in	ADP
cana-2867	308	22	:	:	PUNCT
cana-2867	308	23	proceedings	proceeding	NOUN
cana-2867	308	24	of	of	ADP
cana-2867	308	25	the	the	DET
cana-2867	308	26	nato	nato	PROPN
cana-2867	308	27	advanced	advanced	ADJ
cana-2867	308	28	study	study	NOUN
cana-2867	308	29	institute	institute	NOUN
cana-2867	308	30	on	on	ADP
cana-2867	308	31	fractal	fractal	ADJ
cana-2867	308	32	geometry	geometry	NOUN
cana-2867	308	33	,	,	PUNCT
cana-2867	308	34	montreal	montreal	PROPN
cana-2867	308	35	,	,	PUNCT
cana-2867	308	36	july	july	PROPN
cana-2867	308	37	1989	1989	NUM
cana-2867	308	38	,	,	PUNCT
cana-2867	308	39	kluwer	kluwer	PROPN
cana-2867	308	40	academic	academic	PROPN
cana-2867	308	41	publishers(1990	publishers(1990	PROPN
cana-2867	308	42	)	)	PUNCT
cana-2867	308	43	.	.	PUNCT
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cana-2867	309	2	[	[	X
cana-2867	309	3	23	23	NUM
cana-2867	309	4	]	]	PUNCT
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cana-2867	309	7	,	,	PUNCT
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cana-2867	309	10	,	,	PUNCT
cana-2867	309	11	some	some	DET
cana-2867	309	12	nonlinear	nonlinear	ADJ
cana-2867	309	13	iterated	iterate	VERB
cana-2867	309	14	function	function	NOUN
cana-2867	309	15	system	system	NOUN
cana-2867	309	16	,	,	PUNCT
cana-2867	309	17	comput	comput	NOUN
cana-2867	309	18	.	.	PUNCT
cana-2867	309	19	graph	graph	NOUN
cana-2867	309	20	.	.	PUNCT
cana-2867	309	21	,18(5)(1994	,18(5)(1994	PUNCT
cana-2867	309	22	)	)	PUNCT
cana-2867	309	23	,	,	PUNCT
cana-2867	309	24	119	119	NUM
cana-2867	309	25	-	-	SYM
cana-2867	309	26	125	125	NUM
cana-2867	309	27	.	.	PUNCT
cana-2867	310	1	[	[	X
cana-2867	310	2	24	24	NUM
cana-2867	310	3	]	]	X
cana-2867	310	4	d.	d.	PROPN
cana-2867	310	5	la	la	PROPN
cana-2867	310	6	torre	torre	PROPN
cana-2867	310	7	,	,	PUNCT
cana-2867	310	8	f.	f.	PROPN
cana-2867	310	9	mendivil	mendivil	PROPN
cana-2867	310	10	,	,	PUNCT
cana-2867	310	11	e.r	e.r	PROPN
cana-2867	310	12	.	.	PROPN
cana-2867	310	13	vrscay	vrscay	PROPN
cana-2867	310	14	,	,	PUNCT
cana-2867	310	15	iterated	iterate	VERB
cana-2867	310	16	function	function	NOUN
cana-2867	310	17	system	system	NOUN
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cana-2867	310	19	multi	multi	ADJ
cana-2867	310	20	-	-	NOUN
cana-2867	310	21	functions	function	NOUN
cana-2867	310	22	,	,	PUNCT
cana-2867	310	23	in	in	ADP
cana-2867	310	24	:	:	PUNCT
cana-2867	310	25	math	math	NOUN
cana-2867	310	26	.	.	PUNCT
cana-2867	311	1	everywhere	everywhere	ADV
cana-2867	311	2	,	,	PUNCT
cana-2867	311	3	springer	springer	NOUN
cana-2867	311	4	,	,	PUNCT
cana-2867	311	5	berlin	berlin	PROPN
cana-2867	311	6	-	-	PUNCT
cana-2867	311	7	heidelberg(2006	heidelberg(2006	NOUN
cana-2867	311	8	)	)	PUNCT
cana-2867	311	9	.	.	PUNCT
cana-2867	312	1	[	[	X
cana-2867	312	2	25	25	NUM
cana-2867	312	3	]	]	PUNCT
cana-2867	312	4	s.	s.	PROPN
cana-2867	312	5	l.	l.	PROPN
cana-2867	312	6	singh	singh	PROPN
cana-2867	312	7	,	,	PUNCT
cana-2867	312	8	b.	b.	PROPN
cana-2867	312	9	prasad	prasad	PROPN
cana-2867	312	10	,	,	PUNCT
cana-2867	312	11	a.	a.	PROPN
cana-2867	312	12	kumar	kumar	PROPN
cana-2867	312	13	,	,	PUNCT
cana-2867	312	14	fractals	fractal	NOUN
cana-2867	312	15	via	via	ADP
cana-2867	312	16	iterated	iterated	ADJ
cana-2867	312	17	functions	function	NOUN
cana-2867	312	18	and	and	CCONJ
cana-2867	312	19	multifunctions	multifunction	NOUN
cana-2867	312	20	.	.	PUNCT
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cana-2867	313	2	solitions	solition	NOUN
cana-2867	313	3	fractals,39(2009	fractals,39(2009	NUM
cana-2867	313	4	)	)	PUNCT
cana-2867	313	5	,	,	PUNCT
cana-2867	313	6	1224	1224	NUM
cana-2867	313	7	-	-	SYM
cana-2867	313	8	1231	1231	NUM
cana-2867	313	9	.	.	PUNCT
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cana-2867	314	2	communications	communication	NOUN
cana-2867	314	3	on	on	ADP
cana-2867	314	4	applied	apply	VERB
cana-2867	314	5	nonlinear	nonlinear	ADJ
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cana-2867	314	7	issn	issn	NOUN
cana-2867	314	8	:	:	PUNCT
cana-2867	314	9	1074	1074	NUM
cana-2867	314	10	-	-	PUNCT
cana-2867	314	11	133x	133x	NUM
cana-2867	314	12	vol	vol	NOUN
cana-2867	314	13	32	32	NUM
cana-2867	314	14	no	no	NOUN
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cana-2867	315	4	)	)	PUNCT
cana-2867	315	5	512	512	NUM
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cana-2867	316	5	r.	r.	PROPN
cana-2867	316	6	sahu	sahu	PROPN
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cana-2867	316	9	chakraborty	chakraborty	PROPN
cana-2867	316	10	,	,	PUNCT
cana-2867	316	11	k	k	ADJ
cana-2867	316	12	-	-	ADJ
cana-2867	316	13	iterated	iterated	ADJ
cana-2867	316	14	function	function	NOUN
cana-2867	316	15	system	system	NOUN
cana-2867	316	16	,	,	PUNCT
cana-2867	316	17	fractals	fractal	VERB
cana-2867	316	18	18(1	18(1	NOUN
cana-2867	316	19	)	)	PUNCT
cana-2867	316	20	(	(	PUNCT
cana-2867	316	21	2010	2010	NUM
cana-2867	316	22	)	)	PUNCT
cana-2867	316	23	,	,	PUNCT
cana-2867	316	24	139	139	NUM
cana-2867	316	25	-	-	SYM
cana-2867	316	26	144	144	NUM
cana-2867	316	27	.	.	PUNCT
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cana-2867	318	3	]	]	PUNCT
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cana-2867	318	12	system	system	NOUN
cana-2867	318	13	for	for	ADP
cana-2867	318	14	commuting	commute	VERB
cana-2867	318	15	mapping	mapping	NOUN
cana-2867	318	16	,	,	PUNCT
cana-2867	318	17	international	international	ADJ
cana-2867	318	18	journal	journal	NOUN
cana-2867	318	19	of	of	ADP
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cana-2867	318	21	and	and	CCONJ
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cana-2867	318	26	)	)	PUNCT
cana-2867	318	27	(	(	PUNCT
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cana-2867	318	29	)	)	PUNCT
cana-2867	318	30	,	,	PUNCT
cana-2867	318	31	975	975	NUM
cana-2867	318	32	-	-	SYM
cana-2867	318	33	982	982	NUM
cana-2867	318	34	.	.	PUNCT
cana-2867	318	35	https://www.researchgate.net/publication/267206785_an_itereted_function_system_for_commuting_mapping	https://www.researchgate.net/publication/267206785_an_itereted_function_system_for_commuting_mapping	NOUN
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cana-2867	319	2	28	28	NUM
cana-2867	319	3	]	]	X
cana-2867	319	4	s.c	s.c	PROPN
cana-2867	319	5	.	.	PROPN
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cana-2867	319	7	,	,	PUNCT
cana-2867	319	8	padmavati	padmavati	NOUN
cana-2867	319	9	,	,	PUNCT
cana-2867	319	10	iterated	iterate	VERB
cana-2867	319	11	function	function	NOUN
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cana-2867	319	13	in	in	ADP
cana-2867	319	14	d	d	ADJ
cana-2867	319	15	-	-	ADJ
cana-2867	319	16	metric	metric	ADJ
cana-2867	319	17	space	space	NOUN
cana-2867	319	18	,	,	PUNCT
cana-2867	319	19	applied	apply	VERB
cana-2867	319	20	mathematical	mathematical	ADJ
cana-2867	319	21	sciences	science	NOUN
cana-2867	319	22	,	,	PUNCT
cana-2867	319	23	5(35	5(35	NUM
cana-2867	319	24	)	)	PUNCT
cana-2867	319	25	(	(	PUNCT
cana-2867	319	26	2011	2011	NUM
cana-2867	319	27	)	)	PUNCT
cana-2867	319	28	,	,	PUNCT
cana-2867	319	29	1705	1705	NUM
cana-2867	319	30	-	-	SYM
cana-2867	319	31	1711	1711	NUM
cana-2867	319	32	.	.	PUNCT
cana-2867	320	1	[	[	X
cana-2867	320	2	29	29	NUM
cana-2867	320	3	]	]	PUNCT
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cana-2867	320	7	,	,	PUNCT
cana-2867	320	8	padmavati	padmavati	PROPN
cana-2867	320	9	,	,	PUNCT
cana-2867	320	10	an	an	DET
cana-2867	320	11	iterated	iterated	ADJ
cana-2867	320	12	function	function	NOUN
cana-2867	320	13	system	system	NOUN
cana-2867	320	14	due	due	ADP
cana-2867	320	15	to	to	ADP
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cana-2867	320	17	,	,	PUNCT
cana-2867	320	18	bulletin	bulletin	NOUN
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cana-2867	320	22	,	,	PUNCT
cana-2867	320	23	104(3	104(3	NUM
cana-2867	320	24	)	)	PUNCT
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cana-2867	320	27	)	)	PUNCT
cana-2867	320	28	,	,	PUNCT
cana-2867	320	29	211	211	NUM
cana-2867	320	30	-	-	SYM
cana-2867	320	31	218	218	NUM
cana-2867	320	32	.	.	PUNCT
cana-2867	321	1	[	[	X
cana-2867	321	2	30	30	NUM
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cana-2867	321	5	miculescu	miculescu	PROPN
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cana-2867	321	29	-	-	PUNCT
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cana-2867	321	31	-	-	PUNCT
cana-2867	321	32	0264	0264	NUM
cana-2867	321	33	-	-	PUNCT
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cana-2867	321	36	(	(	PUNCT
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cana-2867	321	38	)	)	PUNCT
cana-2867	321	39	,	,	PUNCT
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cana-2867	321	41	-	-	SYM
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cana-2867	321	43	.	.	PUNCT
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cana-2867	322	2	31	31	NUM
cana-2867	322	3	]	]	PUNCT
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cana-2867	322	43	-	-	PUNCT
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cana-2867	322	51	.	.	PUNCT
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cana-2867	328	7	(	(	PUNCT
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cana-2867	328	9	)	)	PUNCT
cana-2867	328	10	.	.	PUNCT
cana-2867	329	1	[	[	X
cana-2867	329	2	36	36	NUM
cana-2867	329	3	]	]	X
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cana-2867	329	5	m.	m.	NOUN
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cana-2867	329	17	m.	m.	PROPN
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cana-2867	329	39	)	)	PUNCT
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cana-2867	330	3	]	]	X
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cana-2867	330	15	,	,	PUNCT
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