id	sid	tid	token	lemma	pos
cana-2862	1	1	communications	communication	NOUN
cana-2862	1	2	on	on	ADP
cana-2862	1	3	applied	apply	VERB
cana-2862	1	4	nonlinear	nonlinear	ADJ
cana-2862	1	5	analysis	analysis	NOUN
cana-2862	1	6	issn	issn	NOUN
cana-2862	1	7	:	:	PUNCT
cana-2862	1	8	1074	1074	NUM
cana-2862	1	9	-	-	PUNCT
cana-2862	1	10	133x	133x	NUM
cana-2862	1	11	vol	vol	NOUN
cana-2862	1	12	32	32	NUM
cana-2862	1	13	no	no	NOUN
cana-2862	1	14	.	.	PUNCT
cana-2862	2	1	4s	4s	NUM
cana-2862	2	2	(	(	PUNCT
cana-2862	2	3	2025	2025	NUM
cana-2862	2	4	)	)	PUNCT
cana-2862	2	5	432	432	NUM
cana-2862	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2862	2	7	common	common	ADJ
cana-2862	2	8	fixed	fix	VERB
cana-2862	2	9	point	point	NOUN
cana-2862	2	10	theorem	theorem	VERB
cana-2862	2	11	in	in	ADP
cana-2862	2	12	complex	complex	ADJ
cana-2862	2	13	valued	value	VERB
cana-2862	2	14	extended	extend	VERB
cana-2862	2	15	𝑩-metric	𝑩-metric	PROPN
cana-2862	2	16	space	space	NOUN
cana-2862	2	17	jitender	jitender	NOUN
cana-2862	2	18	kumar1	kumar1	PROPN
cana-2862	2	19	and	and	CCONJ
cana-2862	2	20	rajesh	rajesh	PROPN
cana-2862	2	21	kumar2	kumar2	PROPN
cana-2862	3	1	1department	1department	NUM
cana-2862	3	2	of	of	ADP
cana-2862	3	3	mathematics	mathematic	NOUN
cana-2862	3	4	,	,	PUNCT
cana-2862	3	5	govt	govt	NOUN
cana-2862	3	6	.	.	PUNCT
cana-2862	4	1	college	college	NOUN
cana-2862	4	2	for	for	ADP
cana-2862	4	3	girls	girl	NOUN
cana-2862	4	4	,	,	PUNCT
cana-2862	4	5	palwal	palwal	NOUN
cana-2862	4	6	,	,	PUNCT
cana-2862	4	7	kurukshetra	kurukshetra	PROPN
cana-2862	4	8	136118	136118	NUM
cana-2862	4	9	,	,	PUNCT
cana-2862	4	10	india	india	PROPN
cana-2862	4	11	.	.	PUNCT
cana-2862	4	12	email	email	NOUN
cana-2862	4	13	:	:	PUNCT
cana-2862	4	14	kumar.jmaths@gmail.com	kumar.jmaths@gmail.com	PROPN
cana-2862	4	15	2department	2department	NUM
cana-2862	4	16	of	of	ADP
cana-2862	4	17	mathematics	mathematic	NOUN
cana-2862	4	18	,	,	PUNCT
cana-2862	4	19	hindu	hindu	NOUN
cana-2862	4	20	college	college	NOUN
cana-2862	4	21	,	,	PUNCT
cana-2862	4	22	university	university	NOUN
cana-2862	4	23	of	of	ADP
cana-2862	4	24	delhi	delhi	PROPN
cana-2862	4	25	,	,	PUNCT
cana-2862	4	26	delhi	delhi	PROPN
cana-2862	4	27	110007	110007	NUM
cana-2862	4	28	,	,	PUNCT
cana-2862	4	29	india	india	PROPN
cana-2862	4	30	.	.	PUNCT
cana-2862	4	31	email	email	NOUN
cana-2862	4	32	:	:	PUNCT
cana-2862	5	1	rajeshhinducollege@gmail.com	rajeshhinducollege@gmail.com	X
cana-2862	5	2	article	article	NOUN
cana-2862	5	3	history	history	NOUN
cana-2862	5	4	:	:	PUNCT
cana-2862	5	5	received	receive	VERB
cana-2862	5	6	:	:	PUNCT
cana-2862	5	7	27	27	NUM
cana-2862	5	8	-	-	SYM
cana-2862	5	9	09	09	NUM
cana-2862	5	10	-	-	PUNCT
cana-2862	5	11	2024	2024	NUM
cana-2862	5	12	revised	revise	VERB
cana-2862	5	13	:	:	PUNCT
cana-2862	5	14	29	29	NUM
cana-2862	5	15	-	-	SYM
cana-2862	5	16	11	11	NUM
cana-2862	5	17	-	-	PUNCT
cana-2862	5	18	2024	2024	NUM
cana-2862	5	19	accepted	accept	VERB
cana-2862	5	20	:	:	PUNCT
cana-2862	5	21	09	09	NUM
cana-2862	5	22	-	-	SYM
cana-2862	5	23	12	12	NUM
cana-2862	5	24	-	-	PUNCT
cana-2862	5	25	2024	2024	NUM
cana-2862	5	26	abstract	abstract	NOUN
cana-2862	5	27	:	:	PUNCT
cana-2862	5	28	in	in	ADP
cana-2862	5	29	this	this	DET
cana-2862	5	30	paper	paper	NOUN
cana-2862	5	31	,	,	PUNCT
cana-2862	5	32	we	we	PRON
cana-2862	5	33	proved	prove	VERB
cana-2862	5	34	a	a	DET
cana-2862	5	35	common	common	ADJ
cana-2862	5	36	fixed	fix	VERB
cana-2862	5	37	point	point	NOUN
cana-2862	5	38	theorem	theorem	NOUN
cana-2862	5	39	for	for	ADP
cana-2862	5	40	generalized	generalized	ADJ
cana-2862	5	41	contractive	contractive	ADJ
cana-2862	5	42	type	type	NOUN
cana-2862	5	43	maps	map	NOUN
cana-2862	5	44	in	in	ADP
cana-2862	5	45	complex	complex	ADJ
cana-2862	5	46	valued	value	VERB
cana-2862	5	47	extended	extend	VERB
cana-2862	5	48	𝑏-metric	𝑏-metric	NOUN
cana-2862	5	49	space	space	NOUN
cana-2862	5	50	,	,	PUNCT
cana-2862	5	51	which	which	PRON
cana-2862	5	52	generalized	generalize	VERB
cana-2862	5	53	many	many	ADJ
cana-2862	5	54	results	result	NOUN
cana-2862	5	55	in	in	ADP
cana-2862	5	56	the	the	DET
cana-2862	5	57	literature	literature	NOUN
cana-2862	5	58	.	.	PUNCT
cana-2862	6	1	keywords	keyword	NOUN
cana-2862	6	2	:	:	PUNCT
cana-2862	6	3	fixed	fixed	ADJ
cana-2862	6	4	point	point	NOUN
cana-2862	6	5	theorem	theorem	VERB
cana-2862	6	6	,	,	PUNCT
cana-2862	6	7	contractive	contractive	ADJ
cana-2862	6	8	type	type	NOUN
cana-2862	6	9	mapping	mapping	NOUN
cana-2862	6	10	,	,	PUNCT
cana-2862	6	11	complex	complex	ADJ
cana-2862	6	12	valued	value	VERB
cana-2862	6	13	extended	extend	VERB
cana-2862	6	14	𝑏-metric	𝑏-metric	NOUN
cana-2862	6	15	space	space	NOUN
cana-2862	6	16	.	.	PUNCT
cana-2862	7	1	1	1	X
cana-2862	7	2	.	.	X
cana-2862	7	3	introduction	introduction	NOUN
cana-2862	7	4	in	in	ADP
cana-2862	7	5	2011	2011	NUM
cana-2862	7	6	,	,	PUNCT
cana-2862	7	7	azam	azam	PROPN
cana-2862	7	8	et	et	PROPN
cana-2862	7	9	al	al	PROPN
cana-2862	7	10	.	.	PUNCT
cana-2862	8	1	[	[	X
cana-2862	8	2	1	1	X
cana-2862	8	3	]	]	PUNCT
cana-2862	8	4	introduced	introduce	VERB
cana-2862	8	5	the	the	DET
cana-2862	8	6	notion	notion	NOUN
cana-2862	8	7	of	of	ADP
cana-2862	8	8	complex	complex	ADJ
cana-2862	8	9	valued	value	VERB
cana-2862	8	10	metric	metric	ADJ
cana-2862	8	11	spaces	space	NOUN
cana-2862	8	12	and	and	CCONJ
cana-2862	8	13	proved	prove	VERB
cana-2862	8	14	a	a	DET
cana-2862	8	15	common	common	ADJ
cana-2862	8	16	fixed	fix	VERB
cana-2862	8	17	point	point	NOUN
cana-2862	8	18	theorem	theorem	NOUN
cana-2862	8	19	for	for	ADP
cana-2862	8	20	a	a	DET
cana-2862	8	21	pair	pair	NOUN
cana-2862	8	22	of	of	ADP
cana-2862	8	23	contractive	contractive	ADJ
cana-2862	8	24	type	type	NOUN
cana-2862	8	25	maps	map	NOUN
cana-2862	8	26	involving	involve	VERB
cana-2862	8	27	rational	rational	ADJ
cana-2862	8	28	expressions	expression	NOUN
cana-2862	8	29	which	which	PRON
cana-2862	8	30	is	be	AUX
cana-2862	8	31	a	a	DET
cana-2862	8	32	generalization	generalization	NOUN
cana-2862	8	33	of	of	ADP
cana-2862	8	34	the	the	DET
cana-2862	8	35	classification	classification	NOUN
cana-2862	8	36	banach	banach	NOUN
cana-2862	8	37	fixed	fix	VERB
cana-2862	8	38	point	point	NOUN
cana-2862	8	39	theorem	theorem	VERB
cana-2862	8	40	.	.	PUNCT
cana-2862	9	1	in	in	ADP
cana-2862	9	2	2013	2013	NUM
cana-2862	9	3	,	,	PUNCT
cana-2862	9	4	rao	rao	PROPN
cana-2862	9	5	et	et	PROPN
cana-2862	9	6	al	al	PROPN
cana-2862	9	7	.	.	PUNCT
cana-2862	10	1	[	[	X
cana-2862	10	2	7	7	X
cana-2862	10	3	]	]	PUNCT
cana-2862	10	4	introduced	introduce	VERB
cana-2862	10	5	the	the	DET
cana-2862	10	6	concept	concept	NOUN
cana-2862	10	7	of	of	ADP
cana-2862	10	8	complex	complex	ADJ
cana-2862	10	9	valued	value	VERB
cana-2862	10	10	𝑏-metric	𝑏-metric	PROPN
cana-2862	10	11	space	space	NOUN
cana-2862	10	12	.	.	PUNCT
cana-2862	11	1	subsequently	subsequently	ADV
cana-2862	11	2	,	,	PUNCT
cana-2862	11	3	many	many	ADJ
cana-2862	11	4	authors	author	NOUN
cana-2862	11	5	have	have	AUX
cana-2862	11	6	studied	study	VERB
cana-2862	11	7	the	the	DET
cana-2862	11	8	existence	existence	NOUN
cana-2862	11	9	and	and	CCONJ
cana-2862	11	10	uniqueness	uniqueness	NOUN
cana-2862	11	11	of	of	ADP
cana-2862	11	12	common	common	ADJ
cana-2862	11	13	fixed	fix	VERB
cana-2862	11	14	point	point	NOUN
cana-2862	11	15	of	of	ADP
cana-2862	11	16	selfmappings	selfmapping	NOUN
cana-2862	11	17	in	in	ADP
cana-2862	11	18	view	view	NOUN
cana-2862	11	19	of	of	ADP
cana-2862	11	20	contractive	contractive	ADJ
cana-2862	11	21	conditions	condition	NOUN
cana-2862	11	22	.	.	PUNCT
cana-2862	12	1	some	some	PRON
cana-2862	12	2	of	of	ADP
cana-2862	12	3	these	these	DET
cana-2862	12	4	observations	observation	NOUN
cana-2862	12	5	are	be	AUX
cana-2862	12	6	described	describe	VERB
cana-2862	12	7	in	in	ADP
cana-2862	12	8	[	[	X
cana-2862	12	9	1,4	1,4	NUM
cana-2862	12	10	-	-	NOUN
cana-2862	12	11	6,8,9	6,8,9	NUM
cana-2862	12	12	]	]	PUNCT
cana-2862	12	13	.	.	PUNCT
cana-2862	13	1	in	in	ADP
cana-2862	13	2	2019	2019	NUM
cana-2862	13	3	,	,	PUNCT
cana-2862	13	4	n.	n.	PROPN
cana-2862	13	5	ullah	ullah	PROPN
cana-2862	13	6	et	et	PROPN
cana-2862	13	7	al	al	PROPN
cana-2862	13	8	.	.	PUNCT
cana-2862	14	1	[	[	X
cana-2862	14	2	11	11	NUM
cana-2862	14	3	]	]	PUNCT
cana-2862	14	4	extended	extend	VERB
cana-2862	14	5	the	the	DET
cana-2862	14	6	concept	concept	NOUN
cana-2862	14	7	of	of	ADP
cana-2862	14	8	complex	complex	ADJ
cana-2862	14	9	valued	value	VERB
cana-2862	14	10	𝑏-metric	𝑏-metric	PROPN
cana-2862	14	11	space	space	NOUN
cana-2862	14	12	to	to	PART
cana-2862	14	13	complex	complex	VERB
cana-2862	14	14	valued	value	VERB
cana-2862	14	15	extended	extend	VERB
cana-2862	14	16	𝑏-metric	𝑏-metric	NOUN
cana-2862	14	17	space	space	NOUN
cana-2862	14	18	.	.	PUNCT
cana-2862	15	1	the	the	DET
cana-2862	15	2	main	main	ADJ
cana-2862	15	3	purpose	purpose	NOUN
cana-2862	15	4	of	of	ADP
cana-2862	15	5	this	this	DET
cana-2862	15	6	paper	paper	NOUN
cana-2862	15	7	is	be	AUX
cana-2862	15	8	to	to	PART
cana-2862	15	9	present	present	VERB
cana-2862	15	10	a	a	DET
cana-2862	15	11	common	common	ADJ
cana-2862	15	12	fixed	fix	VERB
cana-2862	15	13	point	point	NOUN
cana-2862	15	14	result	result	NOUN
cana-2862	15	15	for	for	ADP
cana-2862	15	16	two	two	NUM
cana-2862	15	17	self	self	NOUN
cana-2862	15	18	maps	map	NOUN
cana-2862	15	19	satisfying	satisfy	VERB
cana-2862	15	20	a	a	DET
cana-2862	15	21	rational	rational	ADJ
cana-2862	15	22	inequality	inequality	NOUN
cana-2862	15	23	in	in	ADP
cana-2862	15	24	complex	complex	ADJ
cana-2862	15	25	valued	value	VERB
cana-2862	15	26	extended	extend	VERB
cana-2862	15	27	𝑏-metric	𝑏-metric	NOUN
cana-2862	15	28	space	space	NOUN
cana-2862	15	29	.	.	PUNCT
cana-2862	16	1	2	2	X
cana-2862	16	2	.	.	X
cana-2862	16	3	preliminaries	preliminary	NOUN
cana-2862	16	4	let	let	VERB
cana-2862	16	5	ℂ	ℂ	PROPN
cana-2862	16	6	be	be	AUX
cana-2862	16	7	the	the	DET
cana-2862	16	8	set	set	NOUN
cana-2862	16	9	of	of	ADP
cana-2862	16	10	complex	complex	ADJ
cana-2862	16	11	number	number	NOUN
cana-2862	16	12	and	and	CCONJ
cana-2862	16	13	𝑧1	𝑧1	NOUN
cana-2862	16	14	,	,	PUNCT
cana-2862	16	15	𝑧2	𝑧2	PROPN
cana-2862	16	16	∈	∈	PROPN
cana-2862	16	17	ℂ.	ℂ.	NOUN
cana-2862	16	18	define	define	VERB
cana-2862	16	19	a	a	DET
cana-2862	16	20	partial	partial	ADJ
cana-2862	16	21	order	order	NOUN
cana-2862	16	22	≾	≾	NOUN
cana-2862	16	23	on	on	ADP
cana-2862	16	24	ℂ	ℂ	PROPN
cana-2862	16	25	as	as	SCONJ
cana-2862	16	26	follows	follow	VERB
cana-2862	16	27	:	:	PUNCT
cana-2862	16	28	𝑧1	𝑧1	PROPN
cana-2862	16	29	≾	≾	PROPN
cana-2862	16	30	𝑧2	𝑧2	PROPN
cana-2862	16	31	iff	iff	PROPN
cana-2862	16	32	re(𝑧1	re(𝑧1	NOUN
cana-2862	16	33	)	)	PUNCT
cana-2862	16	34	≤	≤	NOUN
cana-2862	16	35	re(𝑧2	re(𝑧2	VERB
cana-2862	16	36	)	)	PUNCT
cana-2862	16	37	,	,	PUNCT
cana-2862	16	38	im(𝑧1	im(𝑧1	NOUN
cana-2862	16	39	)	)	PUNCT
cana-2862	16	40	≤	≤	NOUN
cana-2862	16	41	im(𝑧2	im(𝑧2	VERB
cana-2862	16	42	)	)	PUNCT
cana-2862	16	43	(	(	PUNCT
cana-2862	16	44	2.1	2.1	NUM
cana-2862	16	45	)	)	PUNCT
cana-2862	16	46	thus	thus	ADV
cana-2862	16	47	𝑧1	𝑧1	VERB
cana-2862	16	48	≾	≾	PROPN
cana-2862	16	49	𝑧2	𝑧2	PROPN
cana-2862	16	50	if	if	SCONJ
cana-2862	16	51	one	one	NUM
cana-2862	16	52	of	of	ADP
cana-2862	16	53	the	the	DET
cana-2862	16	54	following	follow	VERB
cana-2862	16	55	holds	hold	VERB
cana-2862	16	56	:	:	PUNCT
cana-2862	16	57	(	(	PUNCT
cana-2862	16	58	i)re(𝑧1	i)re(𝑧1	ADJ
cana-2862	16	59	)	)	PUNCT
cana-2862	16	60	=	=	PUNCT
cana-2862	16	61	re(𝑧2	re(𝑧2	VERB
cana-2862	16	62	)	)	PUNCT
cana-2862	16	63	and	and	CCONJ
cana-2862	16	64	im(𝑧1	im(𝑧1	NOUN
cana-2862	16	65	)	)	PUNCT
cana-2862	17	1	=	=	SYM
cana-2862	17	2	im(𝑧2	im(𝑧2	VERB
cana-2862	17	3	)	)	PUNCT
cana-2862	17	4	,	,	PUNCT
cana-2862	17	5	(	(	PUNCT
cana-2862	17	6	ii)re(𝑧1	ii)re(𝑧1	PROPN
cana-2862	17	7	)	)	PUNCT
cana-2862	17	8	<	<	X
cana-2862	17	9	re(𝑧2	re(𝑧2	X
cana-2862	17	10	)	)	PUNCT
cana-2862	17	11	and	and	CCONJ
cana-2862	17	12	im(𝑧1	im(𝑧1	NOUN
cana-2862	17	13	)	)	PUNCT
cana-2862	17	14	=	=	SYM
cana-2862	17	15	im(𝑧2	im(𝑧2	VERB
cana-2862	17	16	)	)	PUNCT
cana-2862	17	17	,	,	PUNCT
cana-2862	17	18	(	(	PUNCT
cana-2862	17	19	iii)re(𝑧1	iii)re(𝑧1	NOUN
cana-2862	17	20	)	)	PUNCT
cana-2862	17	21	=	=	PUNCT
cana-2862	17	22	re(𝑧2	re(𝑧2	VERB
cana-2862	17	23	)	)	PUNCT
cana-2862	17	24	and	and	CCONJ
cana-2862	17	25	im(𝑧1	im(𝑧1	NOUN
cana-2862	17	26	)	)	PUNCT
cana-2862	17	27	<	<	X
cana-2862	17	28	im(𝑧2	im(𝑧2	NOUN
cana-2862	17	29	)	)	PUNCT
cana-2862	17	30	,	,	PUNCT
cana-2862	17	31	(	(	PUNCT
cana-2862	17	32	iv)re(𝑧1	iv)re(𝑧1	ADV
cana-2862	17	33	)	)	PUNCT
cana-2862	17	34	<	<	X
cana-2862	17	35	re(𝑧2	re(𝑧2	X
cana-2862	17	36	)	)	PUNCT
cana-2862	17	37	and	and	CCONJ
cana-2862	17	38	im(𝑧1	im(𝑧1	NOUN
cana-2862	17	39	)	)	PUNCT
cana-2862	17	40	<	<	X
cana-2862	17	41	im(𝑧2	im(𝑧2	NOUN
cana-2862	17	42	)	)	PUNCT
cana-2862	17	43	.	.	PUNCT
cana-2862	18	1	communications	communication	NOUN
cana-2862	18	2	on	on	ADP
cana-2862	18	3	applied	apply	VERB
cana-2862	18	4	nonlinear	nonlinear	ADJ
cana-2862	18	5	analysis	analysis	NOUN
cana-2862	18	6	issn	issn	NOUN
cana-2862	18	7	:	:	PUNCT
cana-2862	18	8	1074	1074	NUM
cana-2862	18	9	-	-	PUNCT
cana-2862	18	10	133x	133x	NUM
cana-2862	18	11	vol	vol	NOUN
cana-2862	18	12	32	32	NUM
cana-2862	18	13	no	no	NOUN
cana-2862	18	14	.	.	PUNCT
cana-2862	19	1	4s	4s	NUM
cana-2862	19	2	(	(	PUNCT
cana-2862	19	3	2025	2025	NUM
cana-2862	19	4	)	)	PUNCT
cana-2862	19	5	433	433	NUM
cana-2862	19	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2862	19	7	we	we	PRON
cana-2862	19	8	will	will	AUX
cana-2862	19	9	write	write	VERB
cana-2862	19	10	if	if	SCONJ
cana-2862	19	11	𝑧1	𝑧1	NOUN
cana-2862	19	12	≺	≺	VERB
cana-2862	19	13	𝑧2	𝑧2	PROPN
cana-2862	19	14	if	if	SCONJ
cana-2862	19	15	𝑧1	𝑧1	NOUN
cana-2862	19	16	≠	≠	PROPN
cana-2862	19	17	𝑧2	𝑧2	NOUN
cana-2862	19	18	and	and	CCONJ
cana-2862	19	19	one	one	NUM
cana-2862	19	20	of	of	ADP
cana-2862	19	21	(	(	PUNCT
cana-2862	19	22	ii	ii	NOUN
cana-2862	19	23	)	)	PUNCT
cana-2862	19	24	,	,	PUNCT
cana-2862	19	25	(	(	PUNCT
cana-2862	19	26	iii	iii	NOUN
cana-2862	19	27	)	)	PUNCT
cana-2862	19	28	and	and	CCONJ
cana-2862	19	29	(	(	PUNCT
cana-2862	19	30	iv	iv	X
cana-2862	19	31	)	)	PUNCT
cana-2862	19	32	is	be	AUX
cana-2862	19	33	satisfied	satisfied	ADJ
cana-2862	19	34	:	:	PUNCT
cana-2862	19	35	also	also	ADV
cana-2862	19	36	we	we	PRON
cana-2862	19	37	will	will	AUX
cana-2862	19	38	write	write	VERB
cana-2862	19	39	𝑧1	𝑧1	NOUN
cana-2862	19	40	≺	≺	NOUN
cana-2862	19	41	𝑧2	𝑧2	PROPN
cana-2862	19	42	if	if	SCONJ
cana-2862	19	43	only	only	ADV
cana-2862	19	44	(	(	PUNCT
cana-2862	19	45	iv	iv	X
cana-2862	19	46	)	)	PUNCT
cana-2862	19	47	is	be	AUX
cana-2862	19	48	satisfied	satisfied	ADJ
cana-2862	19	49	.	.	PUNCT
cana-2862	20	1	we	we	PRON
cana-2862	20	2	can	can	AUX
cana-2862	20	3	easily	easily	ADV
cana-2862	20	4	check	check	VERB
cana-2862	20	5	that	that	SCONJ
cana-2862	20	6	the	the	DET
cana-2862	20	7	following	follow	VERB
cana-2862	20	8	statements	statement	NOUN
cana-2862	20	9	are	be	AUX
cana-2862	20	10	held	hold	VERB
cana-2862	20	11	:	:	PUNCT
cana-2862	20	12	(	(	PUNCT
cana-2862	20	13	i)if	i)if	NOUN
cana-2862	20	14	𝑎	𝑎	ADP
cana-2862	20	15	,	,	PUNCT
cana-2862	20	16	𝑏	𝑏	PROPN
cana-2862	20	17	∈	∈	PROPN
cana-2862	20	18	𝑅	𝑅	PROPN
cana-2862	20	19	and	and	CCONJ
cana-2862	20	20	𝑎	𝑎	NOUN
cana-2862	20	21	≤	≤	NOUN
cana-2862	20	22	𝑏	𝑏	NOUN
cana-2862	20	23	then	then	ADV
cana-2862	20	24	𝑎𝑧	𝑎𝑧	PROPN
cana-2862	20	25	≾	≾	PROPN
cana-2862	20	26	𝑏𝑧	𝑏𝑧	X
cana-2862	20	27	for	for	ADP
cana-2862	20	28	all	all	DET
cana-2862	20	29	𝑧	𝑧	DET
cana-2862	20	30	∈	∈	PROPN
cana-2862	20	31	ℂ	ℂ	PROPN
cana-2862	20	32	;	;	PUNCT
cana-2862	20	33	(	(	PUNCT
cana-2862	20	34	ii)if	ii)if	PROPN
cana-2862	20	35	0	0	X
cana-2862	20	36	≾	≾	NOUN
cana-2862	20	37	𝑧1	𝑧1	VERB
cana-2862	20	38	≺	≺	NOUN
cana-2862	20	39	𝑧2	𝑧2	NOUN
cana-2862	20	40	,	,	PUNCT
cana-2862	20	41	then	then	ADV
cana-2862	20	42	|𝑧1|	|𝑧1|	ADV
cana-2862	20	43	<	<	X
cana-2862	20	44	|𝑧2|	|𝑧2|	X
cana-2862	20	45	;	;	PUNCT
cana-2862	20	46	(	(	PUNCT
cana-2862	20	47	iii)if	iii)if	NOUN
cana-2862	20	48	𝑧1	𝑧1	VERB
cana-2862	20	49	≾	≾	PROPN
cana-2862	20	50	𝑧2	𝑧2	PROPN
cana-2862	20	51	and	and	CCONJ
cana-2862	20	52	𝑧2	𝑧2	PROPN
cana-2862	20	53	≺	≺	NOUN
cana-2862	20	54	𝑧3	𝑧3	PROPN
cana-2862	20	55	,	,	PUNCT
cana-2862	20	56	then	then	ADV
cana-2862	20	57	𝑧1	𝑧1	VERB
cana-2862	20	58	≺	≺	NOUN
cana-2862	20	59	𝑧3	𝑧3	PROPN
cana-2862	20	60	.	.	PUNCT
cana-2862	21	1	definition	definition	NOUN
cana-2862	21	2	2.1	2.1	NUM
cana-2862	21	3	(	(	PUNCT
cana-2862	21	4	[	[	X
cana-2862	21	5	1	1	NUM
cana-2862	21	6	]	]	PUNCT
cana-2862	21	7	)	)	PUNCT
cana-2862	21	8	.	.	PUNCT
cana-2862	22	1	let	let	VERB
cana-2862	22	2	𝑋	𝑋	NOUN
cana-2862	22	3	be	be	AUX
cana-2862	22	4	a	a	DET
cana-2862	22	5	nonempty	nonempty	ADV
cana-2862	22	6	set	set	VERB
cana-2862	22	7	.	.	PUNCT
cana-2862	23	1	a	a	DET
cana-2862	23	2	function	function	NOUN
cana-2862	23	3	𝑑	𝑑	NOUN
cana-2862	23	4	:	:	PUNCT
cana-2862	23	5	𝑋	𝑋	NOUN
cana-2862	23	6	×	×	NOUN
cana-2862	23	7	𝑋	𝑋	PROPN
cana-2862	23	8	→	→	SYM
cana-2862	23	9	ℂ	ℂ	PROPN
cana-2862	23	10	is	be	AUX
cana-2862	23	11	called	call	VERB
cana-2862	23	12	a	a	DET
cana-2862	23	13	complex	complex	NOUN
cana-2862	23	14	valued	value	VERB
cana-2862	23	15	metric	metric	NOUN
cana-2862	23	16	on	on	ADP
cana-2862	23	17	𝑋	𝑋	PROPN
cana-2862	23	18	if	if	SCONJ
cana-2862	23	19	for	for	SCONJ
cana-2862	23	20	all	all	PRON
cana-2862	23	21	𝑥	𝑥	PROPN
cana-2862	23	22	,	,	PUNCT
cana-2862	23	23	𝑦	𝑦	NOUN
cana-2862	23	24	,	,	PUNCT
cana-2862	23	25	𝑧	𝑧	DET
cana-2862	23	26	∈	∈	NOUN
cana-2862	23	27	𝑋	𝑋	NOUN
cana-2862	23	28	the	the	DET
cana-2862	23	29	following	follow	VERB
cana-2862	23	30	conditions	condition	NOUN
cana-2862	23	31	are	be	AUX
cana-2862	23	32	satisfied	satisfied	ADJ
cana-2862	23	33	:	:	PUNCT
cana-2862	23	34	(	(	PUNCT
cana-2862	23	35	i)0	i)0	PROPN
cana-2862	23	36	≾	≾	PROPN
cana-2862	23	37	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	23	38	,	,	PUNCT
cana-2862	23	39	𝑦	𝑦	NOUN
cana-2862	23	40	)	)	PUNCT
cana-2862	23	41	and	and	CCONJ
cana-2862	23	42	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	23	43	,	,	PUNCT
cana-2862	23	44	𝑦	𝑦	X
cana-2862	23	45	)	)	PUNCT
cana-2862	23	46	=	=	SYM
cana-2862	23	47	0	0	PUNCT
cana-2862	24	1	if	if	SCONJ
cana-2862	24	2	and	and	CCONJ
cana-2862	24	3	only	only	ADV
cana-2862	24	4	for	for	ADP
cana-2862	24	5	𝑥	𝑥	NOUN
cana-2862	24	6	=	=	SYM
cana-2862	24	7	𝑦	𝑦	NUM
cana-2862	24	8	;	;	PUNCT
cana-2862	24	9	(	(	PUNCT
cana-2862	24	10	ii)𝑑(𝑥	ii)𝑑(𝑥	NOUN
cana-2862	24	11	,	,	PUNCT
cana-2862	24	12	𝑦	𝑦	NOUN
cana-2862	24	13	)	)	PUNCT
cana-2862	24	14	=	=	SYM
cana-2862	24	15	𝑑(𝑦	𝑑(𝑦	NOUN
cana-2862	24	16	,	,	PUNCT
cana-2862	24	17	𝑥	𝑥	NOUN
cana-2862	24	18	)	)	PUNCT
cana-2862	24	19	;	;	PUNCT
cana-2862	24	20	(	(	PUNCT
cana-2862	24	21	iii)𝑑(𝑥	iii)𝑑(𝑥	PROPN
cana-2862	24	22	,	,	PUNCT
cana-2862	24	23	𝑦	𝑦	NOUN
cana-2862	24	24	)	)	PUNCT
cana-2862	24	25	≾	≾	PROPN
cana-2862	24	26	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	24	27	,	,	PUNCT
cana-2862	24	28	𝑧	𝑧	NOUN
cana-2862	24	29	)	)	PUNCT
cana-2862	24	30	+	+	X
cana-2862	24	31	𝑑(𝑧	𝑑(𝑧	PROPN
cana-2862	24	32	,	,	PUNCT
cana-2862	24	33	𝑦	𝑦	NOUN
cana-2862	24	34	)	)	PUNCT
cana-2862	24	35	.	.	PUNCT
cana-2862	25	1	the	the	DET
cana-2862	25	2	pair	pair	NOUN
cana-2862	25	3	(	(	PUNCT
cana-2862	25	4	𝑋	𝑋	PROPN
cana-2862	25	5	,	,	PUNCT
cana-2862	25	6	𝑑	𝑑	NOUN
cana-2862	25	7	)	)	PUNCT
cana-2862	25	8	is	be	AUX
cana-2862	25	9	called	call	VERB
cana-2862	25	10	a	a	DET
cana-2862	25	11	complex	complex	ADJ
cana-2862	25	12	valued	value	VERB
cana-2862	25	13	metric	metric	ADJ
cana-2862	25	14	space	space	NOUN
cana-2862	25	15	.	.	PUNCT
cana-2862	26	1	example	example	NOUN
cana-2862	26	2	2.1	2.1	NUM
cana-2862	26	3	(	(	PUNCT
cana-2862	26	4	[	[	X
cana-2862	26	5	5	5	NUM
cana-2862	26	6	]	]	PUNCT
cana-2862	26	7	)	)	PUNCT
cana-2862	26	8	.	.	PUNCT
cana-2862	27	1	let	let	VERB
cana-2862	27	2	𝑋	𝑋	PROPN
cana-2862	27	3	=	=	PUNCT
cana-2862	27	4	ℂ	ℂ	PROPN
cana-2862	27	5	define	define	VERB
cana-2862	27	6	the	the	DET
cana-2862	27	7	mapping	mapping	NOUN
cana-2862	27	8	𝑑	𝑑	NOUN
cana-2862	27	9	:	:	PUNCT
cana-2862	27	10	𝑋	𝑋	NOUN
cana-2862	27	11	×	×	NOUN
cana-2862	27	12	𝑋	𝑋	PROPN
cana-2862	27	13	→	→	SYM
cana-2862	27	14	ℂ	ℂ	PROPN
cana-2862	27	15	by	by	ADP
cana-2862	27	16	𝑑(𝑧1	𝑑(𝑧1	NOUN
cana-2862	27	17	,	,	PUNCT
cana-2862	27	18	𝑧2	𝑧2	NOUN
cana-2862	27	19	)	)	PUNCT
cana-2862	27	20	=	=	SYM
cana-2862	28	1	𝑖|𝑧1	𝑖|𝑧1	PUNCT
cana-2862	29	1	−	−	PROPN
cana-2862	29	2	𝑧2|	𝑧2|	PROPN
cana-2862	29	3	with	with	ADP
cana-2862	29	4	𝑧1	𝑧1	NOUN
cana-2862	29	5	=	=	SYM
cana-2862	29	6	𝑥1	𝑥1	PROPN
cana-2862	29	7	+	+	CCONJ
cana-2862	29	8	𝑖𝑦1	𝑖𝑦1	ADJ
cana-2862	29	9	,	,	PUNCT
cana-2862	29	10	𝑧2	𝑧2	NOUN
cana-2862	29	11	=	=	SYM
cana-2862	29	12	𝑥2	𝑥2	NOUN
cana-2862	29	13	+	+	CCONJ
cana-2862	29	14	𝑖𝑦2	𝑖𝑦2	PROPN
cana-2862	29	15	(	(	PUNCT
cana-2862	29	16	2.2	2.2	NUM
cana-2862	29	17	)	)	PUNCT
cana-2862	29	18	(	(	PUNCT
cana-2862	29	19	𝑋	𝑋	PROPN
cana-2862	29	20	,	,	PUNCT
cana-2862	29	21	𝑑	𝑑	NOUN
cana-2862	29	22	)	)	PUNCT
cana-2862	29	23	is	be	AUX
cana-2862	29	24	complex	complex	ADJ
cana-2862	29	25	valued	value	VERB
cana-2862	29	26	metric	metric	ADJ
cana-2862	29	27	space	space	NOUN
cana-2862	29	28	.	.	PUNCT
cana-2862	30	1	example	example	NOUN
cana-2862	30	2	2.2	2.2	NUM
cana-2862	30	3	(	(	PUNCT
cana-2862	30	4	[	[	X
cana-2862	30	5	8	8	NUM
cana-2862	30	6	]	]	PUNCT
cana-2862	30	7	)	)	PUNCT
cana-2862	30	8	.	.	PUNCT
cana-2862	31	1	let	let	VERB
cana-2862	31	2	𝑋	𝑋	NOUN
cana-2862	31	3	=	=	SYM
cana-2862	31	4	ℂ.	ℂ.	NOUN
cana-2862	31	5	define	define	VERB
cana-2862	31	6	the	the	DET
cana-2862	31	7	mapping	mapping	NOUN
cana-2862	31	8	𝑑	𝑑	NOUN
cana-2862	31	9	:	:	PUNCT
cana-2862	31	10	𝑋	𝑋	NOUN
cana-2862	31	11	×	×	NOUN
cana-2862	31	12	𝑋	𝑋	PROPN
cana-2862	31	13	→	→	SYM
cana-2862	31	14	ℂ	ℂ	PROPN
cana-2862	31	15	by	by	ADP
cana-2862	31	16	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	31	17	,	,	PUNCT
cana-2862	31	18	𝑦	𝑦	X
cana-2862	31	19	)	)	PUNCT
cana-2862	31	20	=	=	VERB
cana-2862	31	21	𝑐𝑖𝑘|𝑥	𝑐𝑖𝑘|𝑥	NOUN
cana-2862	31	22	−	−	PROPN
cana-2862	31	23	𝑦|	𝑦|	PROPN
cana-2862	31	24	,	,	PUNCT
cana-2862	31	25	where	where	SCONJ
cana-2862	31	26	𝑘	𝑘	PROPN
cana-2862	31	27	∈	∈	PROPN
cana-2862	31	28	𝑅	𝑅	PROPN
cana-2862	31	29	,	,	PUNCT
cana-2862	31	30	∀𝑥	∀𝑥	PROPN
cana-2862	31	31	,	,	PUNCT
cana-2862	31	32	𝑦	𝑦	NOUN
cana-2862	31	33	∈	∈	PROPN
cana-2862	31	34	𝑋	𝑋	NOUN
cana-2862	31	35	(	(	PUNCT
cana-2862	31	36	2.3	2.3	NUM
cana-2862	31	37	)	)	PUNCT
cana-2862	31	38	then	then	ADV
cana-2862	31	39	(	(	PUNCT
cana-2862	31	40	𝑋	𝑋	PROPN
cana-2862	31	41	,	,	PUNCT
cana-2862	31	42	𝑑	𝑑	NOUN
cana-2862	31	43	)	)	PUNCT
cana-2862	31	44	is	be	AUX
cana-2862	31	45	complex	complex	ADJ
cana-2862	31	46	valued	value	VERB
cana-2862	31	47	metric	metric	ADJ
cana-2862	31	48	space	space	NOUN
cana-2862	31	49	.	.	PUNCT
cana-2862	32	1	definition	definition	NOUN
cana-2862	32	2	2.2	2.2	NUM
cana-2862	32	3	(	(	PUNCT
cana-2862	32	4	[	[	X
cana-2862	32	5	7	7	NUM
cana-2862	32	6	]	]	NUM
cana-2862	32	7	)	)	PUNCT
cana-2862	32	8	.	.	PUNCT
cana-2862	33	1	let	let	VERB
cana-2862	33	2	𝑋	𝑋	NOUN
cana-2862	33	3	be	be	AUX
cana-2862	33	4	a	a	DET
cana-2862	33	5	non	non	ADJ
cana-2862	33	6	-	-	ADJ
cana-2862	33	7	empty	empty	ADJ
cana-2862	33	8	set	set	NOUN
cana-2862	33	9	and	and	CCONJ
cana-2862	33	10	let	let	VERB
cana-2862	33	11	𝑠	𝑠	PRON
cana-2862	33	12	≥	≥	PRON
cana-2862	33	13	1	1	NUM
cana-2862	33	14	be	be	AUX
cana-2862	33	15	a	a	DET
cana-2862	33	16	given	give	VERB
cana-2862	33	17	real	real	ADJ
cana-2862	33	18	number	number	NOUN
cana-2862	33	19	.	.	PUNCT
cana-2862	34	1	a	a	DET
cana-2862	34	2	function	function	NOUN
cana-2862	34	3	𝑑	𝑑	NOUN
cana-2862	34	4	:	:	PUNCT
cana-2862	34	5	𝑋	𝑋	NOUN
cana-2862	34	6	×	×	NOUN
cana-2862	34	7	𝑋	𝑋	PROPN
cana-2862	34	8	→	→	SYM
cana-2862	34	9	ℂ	ℂ	PROPN
cana-2862	34	10	is	be	AUX
cana-2862	34	11	called	call	VERB
cana-2862	34	12	a	a	DET
cana-2862	34	13	complex	complex	ADJ
cana-2862	34	14	valued	value	VERB
cana-2862	34	15	𝑏-metric	𝑏-metric	NOUN
cana-2862	34	16	on	on	ADP
cana-2862	34	17	𝑋	𝑋	PROPN
cana-2862	34	18	it	it	PRON
cana-2862	34	19	for	for	ADP
cana-2862	34	20	all	all	DET
cana-2862	34	21	𝑥	𝑥	PROPN
cana-2862	34	22	,	,	PUNCT
cana-2862	34	23	𝑦	𝑦	NOUN
cana-2862	34	24	,	,	PUNCT
cana-2862	34	25	𝑧	𝑧	DET
cana-2862	34	26	∈	∈	NOUN
cana-2862	34	27	𝑋	𝑋	NOUN
cana-2862	34	28	the	the	DET
cana-2862	34	29	following	follow	VERB
cana-2862	34	30	conditions	condition	NOUN
cana-2862	34	31	are	be	AUX
cana-2862	34	32	satisfied	satisfied	ADJ
cana-2862	34	33	:	:	PUNCT
cana-2862	34	34	(	(	PUNCT
cana-2862	34	35	i)0	i)0	PROPN
cana-2862	34	36	≾	≾	PROPN
cana-2862	34	37	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	34	38	,	,	PUNCT
cana-2862	34	39	𝑦	𝑦	NOUN
cana-2862	34	40	)	)	PUNCT
cana-2862	34	41	and	and	CCONJ
cana-2862	34	42	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	34	43	,	,	PUNCT
cana-2862	34	44	𝑦	𝑦	X
cana-2862	34	45	)	)	PUNCT
cana-2862	34	46	=	=	SYM
cana-2862	34	47	0	0	PUNCT
cana-2862	35	1	if	if	SCONJ
cana-2862	35	2	and	and	CCONJ
cana-2862	35	3	only	only	ADV
cana-2862	35	4	if	if	SCONJ
cana-2862	35	5	𝑥	𝑥	PROPN
cana-2862	35	6	=	=	SYM
cana-2862	35	7	𝑦	𝑦	NUM
cana-2862	35	8	;	;	PUNCT
cana-2862	35	9	(	(	PUNCT
cana-2862	35	10	ii)𝑑(𝑥	ii)𝑑(𝑥	NOUN
cana-2862	35	11	,	,	PUNCT
cana-2862	35	12	𝑦	𝑦	NOUN
cana-2862	35	13	)	)	PUNCT
cana-2862	35	14	=	=	SYM
cana-2862	35	15	𝑑(𝑦	𝑑(𝑦	NOUN
cana-2862	35	16	,	,	PUNCT
cana-2862	35	17	𝑥	𝑥	NOUN
cana-2862	35	18	)	)	PUNCT
cana-2862	35	19	;	;	PUNCT
cana-2862	35	20	(	(	PUNCT
cana-2862	35	21	iii)𝑑(𝑥	iii)𝑑(𝑥	PROPN
cana-2862	35	22	,	,	PUNCT
cana-2862	35	23	𝑦	𝑦	X
cana-2862	35	24	)	)	PUNCT
cana-2862	35	25	≾	≾	NOUN
cana-2862	35	26	𝑠[𝑑(𝑥	𝑠[𝑑(𝑥	NUM
cana-2862	35	27	,	,	PUNCT
cana-2862	35	28	𝑧	𝑧	NOUN
cana-2862	35	29	)	)	PUNCT
cana-2862	35	30	+	+	X
cana-2862	35	31	𝑑(𝑧	𝑑(𝑧	PROPN
cana-2862	35	32	,	,	PUNCT
cana-2862	35	33	𝑦	𝑦	NOUN
cana-2862	35	34	)	)	PUNCT
cana-2862	35	35	]	]	PUNCT
cana-2862	35	36	.	.	PUNCT
cana-2862	36	1	the	the	DET
cana-2862	36	2	pair	pair	NOUN
cana-2862	36	3	(	(	PUNCT
cana-2862	36	4	𝑋	𝑋	PROPN
cana-2862	36	5	,	,	PUNCT
cana-2862	36	6	𝑑	𝑑	NOUN
cana-2862	36	7	)	)	PUNCT
cana-2862	36	8	is	be	AUX
cana-2862	36	9	called	call	VERB
cana-2862	36	10	a	a	DET
cana-2862	36	11	complex	complex	ADJ
cana-2862	36	12	valued	value	VERB
cana-2862	36	13	𝑏-metric	𝑏-metric	PROPN
cana-2862	36	14	space	space	NOUN
cana-2862	36	15	.	.	PUNCT
cana-2862	37	1	example	example	NOUN
cana-2862	37	2	2.3	2.3	NUM
cana-2862	37	3	(	(	PUNCT
cana-2862	37	4	[	[	X
cana-2862	37	5	7	7	NUM
cana-2862	37	6	]	]	NUM
cana-2862	37	7	)	)	PUNCT
cana-2862	37	8	.	.	PUNCT
cana-2862	38	1	let	let	VERB
cana-2862	38	2	𝑋	𝑋	NOUN
cana-2862	38	3	=	=	PUNCT
cana-2862	39	1	[	[	X
cana-2862	39	2	0,1	0,1	NUM
cana-2862	39	3	]	]	PUNCT
cana-2862	39	4	.	.	PUNCT
cana-2862	40	1	define	define	VERB
cana-2862	40	2	mapping	mapping	NOUN
cana-2862	40	3	𝑑	𝑑	NOUN
cana-2862	40	4	:	:	PUNCT
cana-2862	40	5	𝑋	𝑋	NOUN
cana-2862	40	6	×	×	NOUN
cana-2862	40	7	𝑋	𝑋	PROPN
cana-2862	40	8	→	→	SYM
cana-2862	40	9	ℂ	ℂ	PROPN
cana-2862	40	10	by	by	ADP
cana-2862	40	11	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	40	12	,	,	PUNCT
cana-2862	40	13	𝑦	𝑦	X
cana-2862	40	14	)	)	PUNCT
cana-2862	40	15	=	=	SYM
cana-2862	40	16	|𝑥	|𝑥	ADP
cana-2862	40	17	−	−	PROPN
cana-2862	40	18	𝑦|2	𝑦|2	PROPN
cana-2862	40	19	+	+	CCONJ
cana-2862	40	20	𝑖|𝑥	𝑖|𝑥	PUNCT
cana-2862	40	21	−	−	PROPN
cana-2862	40	22	𝑦|2	𝑦|2	PROPN
cana-2862	40	23	,	,	PUNCT
cana-2862	40	24	∀𝑥	∀𝑥	PROPN
cana-2862	40	25	,	,	PUNCT
cana-2862	40	26	𝑦	𝑦	NOUN
cana-2862	40	27	∈	∈	PROPN
cana-2862	40	28	𝑋	𝑋	NOUN
cana-2862	40	29	(	(	PUNCT
cana-2862	40	30	2.4	2.4	NUM
cana-2862	40	31	)	)	PUNCT
cana-2862	40	32	then	then	ADV
cana-2862	40	33	(	(	PUNCT
cana-2862	40	34	𝑋	𝑋	PROPN
cana-2862	40	35	,	,	PUNCT
cana-2862	40	36	𝑑	𝑑	NOUN
cana-2862	40	37	)	)	PUNCT
cana-2862	40	38	is	be	AUX
cana-2862	40	39	complex	complex	ADJ
cana-2862	40	40	valued	value	VERB
cana-2862	40	41	𝑏-metric	𝑏-metric	PROPN
cana-2862	40	42	space	space	NOUN
cana-2862	40	43	with	with	ADP
cana-2862	40	44	𝑠	𝑠	PROPN
cana-2862	40	45	=	=	SYM
cana-2862	40	46	2	2	NUM
cana-2862	40	47	.	.	PUNCT
cana-2862	40	48	definition	definition	NOUN
cana-2862	40	49	2.3	2.3	NUM
cana-2862	40	50	(	(	PUNCT
cana-2862	40	51	[	[	X
cana-2862	40	52	7	7	NUM
cana-2862	40	53	]	]	NUM
cana-2862	40	54	)	)	PUNCT
cana-2862	40	55	.	.	PUNCT
cana-2862	41	1	let	let	VERB
cana-2862	41	2	(	(	PUNCT
cana-2862	41	3	𝑋	𝑋	NOUN
cana-2862	41	4	,	,	PUNCT
cana-2862	41	5	𝑑	𝑑	NOUN
cana-2862	41	6	)	)	PUNCT
cana-2862	41	7	be	be	VERB
cana-2862	41	8	a	a	DET
cana-2862	41	9	complex	complex	ADJ
cana-2862	41	10	valued	value	VERB
cana-2862	41	11	𝑏-metric	𝑏-metric	PROPN
cana-2862	41	12	space	space	NOUN
cana-2862	41	13	.	.	PUNCT
cana-2862	42	1	consider	consider	VERB
cana-2862	42	2	the	the	DET
cana-2862	42	3	following	follow	VERB
cana-2862	42	4	(	(	PUNCT
cana-2862	42	5	i	i	NOUN
cana-2862	42	6	)	)	PUNCT
cana-2862	42	7	a	a	DET
cana-2862	42	8	point	point	NOUN
cana-2862	42	9	𝑥	𝑥	PRON
cana-2862	42	10	∈	∈	NOUN
cana-2862	42	11	𝑋	𝑋	NOUN
cana-2862	42	12	is	be	AUX
cana-2862	42	13	called	call	VERB
cana-2862	42	14	interior	interior	ADJ
cana-2862	42	15	point	point	NOUN
cana-2862	42	16	of	of	ADP
cana-2862	42	17	a	a	DET
cana-2862	42	18	set	set	ADJ
cana-2862	42	19	𝐴	𝐴	NOUN
cana-2862	42	20	⊆	⊆	NUM
cana-2862	42	21	𝑋	𝑋	NOUN
cana-2862	42	22	whenever	whenever	SCONJ
cana-2862	42	23	there	there	PRON
cana-2862	42	24	exists	exist	VERB
cana-2862	42	25	0	0	NUM
cana-2862	42	26	≺	≺	NOUN
cana-2862	42	27	𝑠	𝑠	SYM
cana-2862	42	28	∈	∈	PROPN
cana-2862	42	29	c	c	NOUN
cana-2862	42	30	such	such	ADJ
cana-2862	42	31	that	that	SCONJ
cana-2862	42	32	𝐵(𝑥	𝐵(𝑥	PROPN
cana-2862	42	33	,	,	PUNCT
cana-2862	42	34	𝑟	𝑟	NOUN
cana-2862	42	35	)	)	PUNCT
cana-2862	42	36	≔	≔	VERB
cana-2862	42	37	{	{	PUNCT
cana-2862	42	38	𝑦	𝑦	NOUN
cana-2862	42	39	∈	∈	PROPN
cana-2862	42	40	𝑋	𝑋	NOUN
cana-2862	42	41	:	:	PUNCT
cana-2862	42	42	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	42	43	,	,	PUNCT
cana-2862	42	44	𝑦	𝑦	NOUN
cana-2862	42	45	)	)	PUNCT
cana-2862	42	46	≺	≺	NOUN
cana-2862	42	47	𝑠	𝑠	NOUN
cana-2862	42	48	}	}	PUNCT
cana-2862	42	49	⊆	⊆	NUM
cana-2862	42	50	𝐴.	𝐴.	NOUN
cana-2862	42	51	communications	communication	NOUN
cana-2862	42	52	on	on	ADP
cana-2862	42	53	applied	apply	VERB
cana-2862	42	54	nonlinear	nonlinear	ADJ
cana-2862	42	55	analysis	analysis	NOUN
cana-2862	42	56	issn	issn	NOUN
cana-2862	42	57	:	:	PUNCT
cana-2862	42	58	1074	1074	NUM
cana-2862	42	59	-	-	PUNCT
cana-2862	42	60	133x	133x	NUM
cana-2862	42	61	vol	vol	NOUN
cana-2862	42	62	32	32	NUM
cana-2862	42	63	no	no	NOUN
cana-2862	42	64	.	.	PUNCT
cana-2862	43	1	4s	4s	NUM
cana-2862	43	2	(	(	PUNCT
cana-2862	43	3	2025	2025	NUM
cana-2862	43	4	)	)	PUNCT
cana-2862	43	5	434	434	NUM
cana-2862	43	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2862	43	7	(	(	PUNCT
cana-2862	43	8	ii	ii	NOUN
cana-2862	43	9	)	)	PUNCT
cana-2862	43	10	a	a	DET
cana-2862	43	11	point	point	NOUN
cana-2862	43	12	𝑥	𝑥	PRON
cana-2862	43	13	∈	∈	NOUN
cana-2862	43	14	𝑋	𝑋	NOUN
cana-2862	43	15	is	be	AUX
cana-2862	43	16	called	call	VERB
cana-2862	43	17	a	a	DET
cana-2862	43	18	limit	limit	NOUN
cana-2862	43	19	point	point	NOUN
cana-2862	43	20	of	of	ADP
cana-2862	43	21	a	a	DET
cana-2862	43	22	set	set	NOUN
cana-2862	43	23	𝐴	𝐴	NOUN
cana-2862	43	24	whenever	whenever	ADV
cana-2862	43	25	,	,	PUNCT
cana-2862	43	26	for	for	ADP
cana-2862	43	27	every	every	DET
cana-2862	43	28	0	0	NUM
cana-2862	43	29	≺	≺	NOUN
cana-2862	43	30	𝑟	𝑟	X
cana-2862	43	31	∈	∈	PROPN
cana-2862	43	32	ℂ	ℂ	PROPN
cana-2862	43	33	,	,	PUNCT
cana-2862	43	34	𝐵(𝑥	𝐵(𝑥	PRON
cana-2862	43	35	,	,	PUNCT
cana-2862	43	36	𝑟	𝑟	NOUN
cana-2862	43	37	)	)	PUNCT
cana-2862	43	38	∩	∩	ADJ
cana-2862	43	39	𝐴	𝐴	NOUN
cana-2862	43	40	−	−	PROPN
cana-2862	43	41	{	{	PUNCT
cana-2862	43	42	𝑥	𝑥	PROPN
cana-2862	43	43	}	}	PUNCT
cana-2862	43	44	≠	≠	PROPN
cana-2862	43	45	∅.	∅.	PRON
cana-2862	43	46	(	(	PUNCT
cana-2862	43	47	iii	iii	NOUN
cana-2862	43	48	)	)	PUNCT
cana-2862	43	49	a	a	DET
cana-2862	43	50	subset	subset	NOUN
cana-2862	43	51	𝐴	𝐴	NOUN
cana-2862	43	52	⊆	⊆	NUM
cana-2862	43	53	𝑋	𝑋	PROPN
cana-2862	43	54	is	be	AUX
cana-2862	43	55	called	call	VERB
cana-2862	43	56	open	open	ADJ
cana-2862	43	57	whenever	whenever	SCONJ
cana-2862	43	58	each	each	DET
cana-2862	43	59	element	element	NOUN
cana-2862	43	60	of	of	ADP
cana-2862	43	61	𝐴	𝐴	PROPN
cana-2862	43	62	is	be	AUX
cana-2862	43	63	an	an	DET
cana-2862	43	64	interior	interior	ADJ
cana-2862	43	65	point	point	NOUN
cana-2862	43	66	of	of	ADP
cana-2862	43	67	𝐴.	𝐴.	PROPN
cana-2862	43	68	(	(	PUNCT
cana-2862	43	69	iv	iv	X
cana-2862	43	70	)	)	PUNCT
cana-2862	43	71	a	a	DET
cana-2862	43	72	subbasis	subbasis	NOUN
cana-2862	43	73	for	for	ADP
cana-2862	43	74	a	a	DET
cana-2862	43	75	hausdorff	hausdorff	NOUN
cana-2862	43	76	topology	topology	NOUN
cana-2862	43	77	𝜏	𝜏	NOUN
cana-2862	43	78	on	on	ADP
cana-2862	43	79	𝑋	𝑋	PROPN
cana-2862	43	80	is	be	AUX
cana-2862	43	81	a	a	DET
cana-2862	43	82	family	family	NOUN
cana-2862	43	83	𝐹	𝐹	PROPN
cana-2862	43	84	=	=	PUNCT
cana-2862	43	85	{	{	PUNCT
cana-2862	43	86	𝐵(𝑥	𝐵(𝑥	PROPN
cana-2862	43	87	,	,	PUNCT
cana-2862	43	88	𝑟	𝑟	NOUN
cana-2862	43	89	):	):	PUNCT
cana-2862	44	1	𝑥	𝑥	PROPN
cana-2862	44	2	∈	∈	PROPN
cana-2862	44	3	𝑋	𝑋	NOUN
cana-2862	44	4	and	and	CCONJ
cana-2862	44	5	0	0	NUM
cana-2862	44	6	≺	≺	NOUN
cana-2862	44	7	𝑟	𝑟	NOUN
cana-2862	44	8	}	}	PUNCT
cana-2862	44	9	.	.	PUNCT
cana-2862	45	1	definition	definition	NOUN
cana-2862	45	2	2.4	2.4	NUM
cana-2862	45	3	(	(	PUNCT
cana-2862	45	4	[	[	X
cana-2862	45	5	11	11	NUM
cana-2862	45	6	]	]	NUM
cana-2862	45	7	)	)	PUNCT
cana-2862	45	8	.	.	PUNCT
cana-2862	46	1	let	let	VERB
cana-2862	46	2	𝑋	𝑋	NOUN
cana-2862	46	3	be	be	AUX
cana-2862	46	4	a	a	DET
cana-2862	46	5	non	non	ADJ
cana-2862	46	6	-	-	ADJ
cana-2862	46	7	empty	empty	ADJ
cana-2862	46	8	set	set	NOUN
cana-2862	46	9	and	and	CCONJ
cana-2862	46	10	𝜙	𝜙	NOUN
cana-2862	46	11	:	:	PUNCT
cana-2862	46	12	𝑋	𝑋	NOUN
cana-2862	46	13	×	×	NOUN
cana-2862	46	14	𝑋	𝑋	PROPN
cana-2862	46	15	→	→	SYM
cana-2862	47	1	[	[	X
cana-2862	47	2	1	1	NUM
cana-2862	47	3	,	,	PUNCT
cana-2862	47	4	∞	∞	PROPN
cana-2862	47	5	]	]	PUNCT
cana-2862	47	6	.	.	PUNCT
cana-2862	48	1	if	if	SCONJ
cana-2862	48	2	a	a	DET
cana-2862	48	3	mapping	mapping	NOUN
cana-2862	48	4	𝑑	𝑑	NOUN
cana-2862	48	5	:	:	PUNCT
cana-2862	48	6	𝑋	𝑋	NOUN
cana-2862	48	7	×	×	NOUN
cana-2862	48	8	𝑋	𝑋	PROPN
cana-2862	48	9	→	→	SYM
cana-2862	48	10	c	c	NOUN
cana-2862	48	11	satisfy	satisfy	NOUN
cana-2862	48	12	:	:	PUNCT
cana-2862	48	13	(	(	PUNCT
cana-2862	48	14	i)0	i)0	PROPN
cana-2862	48	15	≾	≾	PROPN
cana-2862	48	16	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	48	17	,	,	PUNCT
cana-2862	48	18	𝑦	𝑦	NOUN
cana-2862	48	19	)	)	PUNCT
cana-2862	48	20	and	and	CCONJ
cana-2862	48	21	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	48	22	,	,	PUNCT
cana-2862	48	23	𝑦	𝑦	X
cana-2862	48	24	)	)	PUNCT
cana-2862	48	25	=	=	SYM
cana-2862	48	26	0	0	PUNCT
cana-2862	49	1	if	if	SCONJ
cana-2862	49	2	and	and	CCONJ
cana-2862	49	3	only	only	ADV
cana-2862	49	4	if	if	SCONJ
cana-2862	49	5	𝑥	𝑥	PROPN
cana-2862	49	6	=	=	SYM
cana-2862	49	7	𝑦	𝑦	NUM
cana-2862	49	8	;	;	PUNCT
cana-2862	49	9	(	(	PUNCT
cana-2862	49	10	ii)𝑑(𝑥	ii)𝑑(𝑥	NOUN
cana-2862	49	11	,	,	PUNCT
cana-2862	49	12	𝑦	𝑦	NOUN
cana-2862	49	13	)	)	PUNCT
cana-2862	49	14	=	=	SYM
cana-2862	49	15	𝑑(𝑦	𝑑(𝑦	NOUN
cana-2862	49	16	,	,	PUNCT
cana-2862	49	17	𝑥	𝑥	NOUN
cana-2862	49	18	)	)	PUNCT
cana-2862	49	19	;	;	PUNCT
cana-2862	49	20	(	(	PUNCT
cana-2862	49	21	iii)𝑑(𝑥	iii)𝑑(𝑥	PROPN
cana-2862	49	22	,	,	PUNCT
cana-2862	49	23	𝑦	𝑦	X
cana-2862	49	24	)	)	PUNCT
cana-2862	49	25	≾	≾	PROPN
cana-2862	49	26	𝜙(𝑥	𝜙(𝑥	PROPN
cana-2862	49	27	,	,	PUNCT
cana-2862	49	28	𝑦)[𝑑(𝑥	𝑦)[𝑑(𝑥	PROPN
cana-2862	49	29	,	,	PUNCT
cana-2862	49	30	𝑧	𝑧	NOUN
cana-2862	49	31	)	)	PUNCT
cana-2862	49	32	+	+	X
cana-2862	49	33	𝑑(𝑧	𝑑(𝑧	PROPN
cana-2862	49	34	,	,	PUNCT
cana-2862	49	35	𝑦	𝑦	NOUN
cana-2862	49	36	)	)	PUNCT
cana-2862	49	37	]	]	PUNCT
cana-2862	49	38	;	;	PUNCT
cana-2862	49	39	for	for	ADP
cana-2862	49	40	all	all	DET
cana-2862	49	41	𝑥	𝑥	PROPN
cana-2862	49	42	,	,	PUNCT
cana-2862	49	43	𝑦	𝑦	NOUN
cana-2862	49	44	,	,	PUNCT
cana-2862	49	45	𝑧	𝑧	DET
cana-2862	49	46	∈	∈	NOUN
cana-2862	49	47	𝑋	𝑋	NOUN
cana-2862	49	48	then	then	ADV
cana-2862	49	49	(	(	PUNCT
cana-2862	49	50	𝑋	𝑋	PROPN
cana-2862	49	51	,	,	PUNCT
cana-2862	49	52	𝑑	𝑑	NOUN
cana-2862	49	53	)	)	PUNCT
cana-2862	49	54	is	be	AUX
cana-2862	49	55	called	call	VERB
cana-2862	49	56	a	a	DET
cana-2862	49	57	complex	complex	NOUN
cana-2862	49	58	valued	value	VERB
cana-2862	49	59	extended	extend	VERB
cana-2862	49	60	𝑏-metric	𝑏-metric	PROPN
cana-2862	49	61	space	space	NOUN
cana-2862	49	62	.	.	PUNCT
cana-2862	50	1	example	example	NOUN
cana-2862	50	2	2.4	2.4	NUM
cana-2862	50	3	(	(	PUNCT
cana-2862	50	4	[	[	X
cana-2862	50	5	11	11	NUM
cana-2862	50	6	]	]	NUM
cana-2862	50	7	)	)	PUNCT
cana-2862	50	8	.	.	PUNCT
cana-2862	51	1	let	let	VERB
cana-2862	51	2	𝑋	𝑋	NOUN
cana-2862	51	3	=	=	PUNCT
cana-2862	52	1	[	[	X
cana-2862	52	2	0	0	NUM
cana-2862	52	3	,	,	PUNCT
cana-2862	52	4	∞	∞	PROPN
cana-2862	52	5	)	)	PUNCT
cana-2862	52	6	and	and	CCONJ
cana-2862	52	7	𝜙	𝜙	NOUN
cana-2862	52	8	:	:	PUNCT
cana-2862	52	9	𝑋	𝑋	NOUN
cana-2862	52	10	×	×	NOUN
cana-2862	52	11	𝑋	𝑋	PROPN
cana-2862	52	12	→	→	SYM
cana-2862	52	13	[	[	X
cana-2862	52	14	1	1	NUM
cana-2862	52	15	,	,	PUNCT
cana-2862	52	16	∞	∞	PROPN
cana-2862	52	17	)	)	PUNCT
cana-2862	52	18	be	be	VERB
cana-2862	52	19	a	a	DET
cana-2862	52	20	function	function	NOUN
cana-2862	52	21	defined	define	VERB
cana-2862	52	22	by	by	ADP
cana-2862	52	23	𝜙(𝑥	𝜙(𝑥	PROPN
cana-2862	52	24	,	,	PUNCT
cana-2862	52	25	𝑦	𝑦	NOUN
cana-2862	52	26	)	)	PUNCT
cana-2862	52	27	=	=	SYM
cana-2862	53	1	1	1	NUM
cana-2862	53	2	+	+	CCONJ
cana-2862	53	3	𝑥	𝑥	PROPN
cana-2862	54	1	+	+	NUM
cana-2862	54	2	𝑦	𝑦	NOUN
cana-2862	54	3	and	and	CCONJ
cana-2862	54	4	𝑑	𝑑	PROPN
cana-2862	54	5	:	:	PUNCT
cana-2862	54	6	𝑋	𝑋	NOUN
cana-2862	54	7	×	×	NOUN
cana-2862	54	8	𝑋	𝑋	PROPN
cana-2862	54	9	→	→	SYM
cana-2862	54	10	ℂ	ℂ	PROPN
cana-2862	54	11	by	by	ADP
cana-2862	54	12	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	54	13	,	,	PUNCT
cana-2862	54	14	𝑦	𝑦	X
cana-2862	54	15	)	)	PUNCT
cana-2862	54	16	=	=	SYM
cana-2862	54	17	{	{	PUNCT
cana-2862	54	18	0	0	NUM
cana-2862	55	1	if	if	SCONJ
cana-2862	55	2	𝑥	𝑥	PROPN
cana-2862	55	3	=	=	PUNCT
cana-2862	55	4	𝑦	𝑦	SYM
cana-2862	55	5	𝑖	𝑖	NOUN
cana-2862	55	6	if	if	SCONJ
cana-2862	55	7	𝑥	𝑥	PRON
cana-2862	55	8	≠	≠	PROPN
cana-2862	55	9	𝑦	𝑦	SYM
cana-2862	55	10	(	(	PUNCT
cana-2862	55	11	2.5	2.5	NUM
cana-2862	55	12	)	)	PUNCT
cana-2862	55	13	then	then	ADV
cana-2862	55	14	(	(	PUNCT
cana-2862	55	15	𝑋	𝑋	PROPN
cana-2862	55	16	,	,	PUNCT
cana-2862	55	17	𝑑	𝑑	NOUN
cana-2862	55	18	)	)	PUNCT
cana-2862	55	19	is	be	AUX
cana-2862	55	20	a	a	DET
cana-2862	55	21	complex	complex	ADJ
cana-2862	55	22	valued	value	VERB
cana-2862	55	23	extended	extend	VERB
cana-2862	55	24	𝑏-metric	𝑏-metric	PROPN
cana-2862	55	25	space	space	NOUN
cana-2862	55	26	.	.	PUNCT
cana-2862	56	1	theorem	theorem	VERB
cana-2862	56	2	2.1	2.1	NUM
cana-2862	56	3	(	(	PUNCT
cana-2862	56	4	[	[	X
cana-2862	56	5	1	1	NUM
cana-2862	56	6	]	]	PUNCT
cana-2862	56	7	)	)	PUNCT
cana-2862	56	8	.	.	PUNCT
cana-2862	57	1	let	let	VERB
cana-2862	57	2	(	(	PUNCT
cana-2862	57	3	𝑋	𝑋	NOUN
cana-2862	57	4	,	,	PUNCT
cana-2862	57	5	𝑑	𝑑	NOUN
cana-2862	57	6	)	)	PUNCT
cana-2862	57	7	be	be	VERB
cana-2862	57	8	a	a	DET
cana-2862	57	9	complete	complete	ADJ
cana-2862	57	10	complex	complex	NOUN
cana-2862	57	11	valued	value	VERB
cana-2862	57	12	metric	metric	ADJ
cana-2862	57	13	space	space	NOUN
cana-2862	57	14	and	and	CCONJ
cana-2862	57	15	𝜆	𝜆	NOUN
cana-2862	57	16	,	,	PUNCT
cana-2862	57	17	𝜇	𝜇	AUX
cana-2862	57	18	be	be	AUX
cana-2862	57	19	nonnegative	nonnegative	ADJ
cana-2862	57	20	real	real	ADJ
cana-2862	57	21	numbers	number	NOUN
cana-2862	57	22	such	such	ADJ
cana-2862	57	23	that	that	SCONJ
cana-2862	57	24	𝜆	𝜆	X
cana-2862	57	25	+	+	X
cana-2862	57	26	𝜇	𝜇	X
cana-2862	57	27	<	<	X
cana-2862	57	28	1	1	NUM
cana-2862	57	29	.	.	PUNCT
cana-2862	57	30	suppose	suppose	VERB
cana-2862	57	31	that	that	SCONJ
cana-2862	57	32	𝑆	𝑆	PROPN
cana-2862	57	33	,	,	PUNCT
cana-2862	57	34	𝑇	𝑇	PROPN
cana-2862	57	35	:	:	PUNCT
cana-2862	57	36	𝑋	𝑋	PROPN
cana-2862	57	37	→	→	SYM
cana-2862	57	38	𝑋	𝑋	PROPN
cana-2862	57	39	are	be	AUX
cana-2862	57	40	mapping	map	VERB
cana-2862	57	41	satisfying	satisfying	ADJ
cana-2862	57	42	:	:	PUNCT
cana-2862	57	43	𝑑(𝑆𝑥	𝑑(𝑆𝑥	ADJ
cana-2862	57	44	,	,	PUNCT
cana-2862	57	45	𝑇𝑦	𝑇𝑦	PROPN
cana-2862	57	46	)	)	PUNCT
cana-2862	57	47	≾	≾	NOUN
cana-2862	57	48	𝜆𝑑(𝑥	𝜆𝑑(𝑥	NOUN
cana-2862	57	49	,	,	PUNCT
cana-2862	57	50	𝑦	𝑦	NOUN
cana-2862	57	51	)	)	PUNCT
cana-2862	57	52	+	+	NUM
cana-2862	57	53	𝜇⋅𝑑(𝑥,𝑆𝑥)⋅𝑑(𝑦,𝑇𝑦	𝜇⋅𝑑(𝑥,𝑆𝑥)⋅𝑑(𝑦,𝑇𝑦	NOUN
cana-2862	57	54	)	)	PUNCT
cana-2862	57	55	1+𝑑(𝑥,𝑦	1+𝑑(𝑥,𝑦	NUM
cana-2862	57	56	)	)	PUNCT
cana-2862	57	57	(	(	PUNCT
cana-2862	57	58	2.6	2.6	NUM
cana-2862	57	59	)	)	PUNCT
cana-2862	57	60	for	for	ADP
cana-2862	57	61	all	all	PRON
cana-2862	57	62	𝑥	𝑥	PROPN
cana-2862	57	63	,	,	PUNCT
cana-2862	57	64	𝑦	𝑦	PROPN
cana-2862	57	65	∈	∈	PROPN
cana-2862	57	66	𝑋.	𝑋.	PROPN
cana-2862	57	67	then	then	ADV
cana-2862	57	68	𝑆	𝑆	PROPN
cana-2862	57	69	,	,	PUNCT
cana-2862	57	70	𝑇	𝑇	PROPN
cana-2862	57	71	have	have	VERB
cana-2862	57	72	a	a	DET
cana-2862	57	73	unique	unique	ADJ
cana-2862	57	74	common	common	ADJ
cana-2862	57	75	fired	fire	VERB
cana-2862	57	76	point	point	NOUN
cana-2862	57	77	in	in	ADP
cana-2862	57	78	𝑋.	𝑋.	PROPN
cana-2862	57	79	theorem	theorem	VERB
cana-2862	57	80	2.2	2.2	NUM
cana-2862	57	81	(	(	PUNCT
cana-2862	57	82	[	[	X
cana-2862	57	83	9	9	NUM
cana-2862	57	84	]	]	PUNCT
cana-2862	57	85	)	)	PUNCT
cana-2862	57	86	.	.	PUNCT
cana-2862	58	1	let	let	VERB
cana-2862	58	2	(	(	PUNCT
cana-2862	58	3	𝑋	𝑋	NOUN
cana-2862	58	4	,	,	PUNCT
cana-2862	58	5	𝑑	𝑑	NOUN
cana-2862	58	6	)	)	PUNCT
cana-2862	58	7	be	be	VERB
cana-2862	58	8	a	a	DET
cana-2862	58	9	complete	complete	ADJ
cana-2862	58	10	complex	complex	NOUN
cana-2862	58	11	valued	value	VERB
cana-2862	58	12	b	b	X
cana-2862	58	13	-	-	PUNCT
cana-2862	58	14	metric	metric	ADJ
cana-2862	58	15	space	space	NOUN
cana-2862	58	16	with	with	ADP
cana-2862	58	17	the	the	DET
cana-2862	58	18	coefficient	coefficient	NOUN
cana-2862	58	19	𝑠	𝑠	PROPN
cana-2862	58	20	≥	≥	NOUN
cana-2862	58	21	1	1	NUM
cana-2862	58	22	and	and	CCONJ
cana-2862	58	23	𝑓	𝑓	ADV
cana-2862	58	24	,	,	PUNCT
cana-2862	58	25	𝑔	𝑔	NOUN
cana-2862	58	26	:	:	PUNCT
cana-2862	58	27	𝑋	𝑋	PROPN
cana-2862	58	28	→	→	SYM
cana-2862	58	29	𝑋	𝑋	PROPN
cana-2862	58	30	be	be	AUX
cana-2862	58	31	mapping	map	VERB
cana-2862	58	32	satisfying	satisfying	ADJ
cana-2862	58	33	:	:	PUNCT
cana-2862	58	34	𝑑(𝑓𝑥	𝑑(𝑓𝑥	PROPN
cana-2862	58	35	,	,	PUNCT
cana-2862	58	36	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	58	37	)	)	PUNCT
cana-2862	58	38	≾	≾	NOUN
cana-2862	58	39	𝜆𝑑(𝑥	𝜆𝑑(𝑥	NOUN
cana-2862	58	40	,	,	PUNCT
cana-2862	58	41	𝑦	𝑦	NOUN
cana-2862	58	42	)	)	PUNCT
cana-2862	58	43	+	+	CCONJ
cana-2862	58	44	𝜇⋅𝑑(𝑥,𝑓𝑥)⋅𝑑(𝑦,𝑔𝑦	𝜇⋅𝑑(𝑥,𝑓𝑥)⋅𝑑(𝑦,𝑔𝑦	ADJ
cana-2862	58	45	)	)	PUNCT
cana-2862	58	46	1+𝑑(𝑥,𝑦	1+𝑑(𝑥,𝑦	NOUN
cana-2862	58	47	)	)	PUNCT
cana-2862	59	1	+	+	CCONJ
cana-2862	59	2	𝛿⋅𝑑(𝑦,𝑓𝑥)⋅𝑑(𝑥,𝑔𝑦	𝛿⋅𝑑(𝑦,𝑓𝑥)⋅𝑑(𝑥,𝑔𝑦	NOUN
cana-2862	59	3	)	)	PUNCT
cana-2862	59	4	1+𝑑(𝑥,𝑦	1+𝑑(𝑥,𝑦	NOUN
cana-2862	59	5	)	)	PUNCT
cana-2862	59	6	(	(	PUNCT
cana-2862	59	7	2.7	2.7	NUM
cana-2862	59	8	)	)	PUNCT
cana-2862	59	9	where	where	SCONJ
cana-2862	59	10	𝜆	𝜆	X
cana-2862	59	11	,	,	PUNCT
cana-2862	59	12	𝜇	𝜇	ADP
cana-2862	59	13	,	,	PUNCT
cana-2862	59	14	𝛿	𝛿	DET
cana-2862	59	15	nonnegative	nonnegative	ADJ
cana-2862	59	16	real	real	ADJ
cana-2862	59	17	numbers	number	NOUN
cana-2862	59	18	with	with	ADP
cana-2862	59	19	𝑠𝜆	𝑠𝜆	NOUN
cana-2862	59	20	+	+	X
cana-2862	59	21	𝜇	𝜇	X
cana-2862	59	22	+	+	NOUN
cana-2862	59	23	𝛿	𝛿	ADJ
cana-2862	59	24	<	<	X
cana-2862	59	25	1	1	NUM
cana-2862	59	26	.	.	PUNCT
cana-2862	60	1	then	then	ADV
cana-2862	60	2	𝑓	𝑓	X
cana-2862	60	3	,	,	PUNCT
cana-2862	60	4	𝑔	𝑔	PROPN
cana-2862	60	5	have	have	VERB
cana-2862	60	6	a	a	DET
cana-2862	60	7	unique	unique	ADJ
cana-2862	60	8	common	common	ADJ
cana-2862	60	9	fixed	fix	VERB
cana-2862	60	10	point	point	NOUN
cana-2862	60	11	in	in	ADP
cana-2862	60	12	𝑋.	𝑋.	PROPN
cana-2862	60	13	3	3	NUM
cana-2862	60	14	.	.	PUNCT
cana-2862	60	15	main	main	ADJ
cana-2862	60	16	result	result	NOUN
cana-2862	60	17	theorem	theorem	VERB
cana-2862	60	18	3.1	3.1	NUM
cana-2862	60	19	.	.	PUNCT
cana-2862	61	1	let	let	VERB
cana-2862	61	2	(	(	PUNCT
cana-2862	61	3	𝑋	𝑋	NOUN
cana-2862	61	4	,	,	PUNCT
cana-2862	61	5	𝑑	𝑑	NOUN
cana-2862	61	6	)	)	PUNCT
cana-2862	61	7	be	be	VERB
cana-2862	61	8	a	a	DET
cana-2862	61	9	complete	complete	ADJ
cana-2862	61	10	cvebms	cvebms	NOUN
cana-2862	61	11	with	with	ADP
cana-2862	61	12	𝜙	𝜙	NOUN
cana-2862	61	13	:	:	PUNCT
cana-2862	61	14	𝑋	𝑋	PROPN
cana-2862	61	15	×	×	NOUN
cana-2862	61	16	𝑋	𝑋	PROPN
cana-2862	61	17	→	→	SYM
cana-2862	61	18	[	[	X
cana-2862	61	19	1	1	NUM
cana-2862	61	20	,	,	PUNCT
cana-2862	61	21	∞	∞	PROPN
cana-2862	61	22	)	)	PUNCT
cana-2862	61	23	and	and	CCONJ
cana-2862	61	24	𝑓	𝑓	X
cana-2862	61	25	,	,	PUNCT
cana-2862	61	26	𝑔	𝑔	NOUN
cana-2862	61	27	:	:	PUNCT
cana-2862	61	28	𝑋	𝑋	PROPN
cana-2862	61	29	→	→	SYM
cana-2862	61	30	𝑋	𝑋	PROPN
cana-2862	61	31	be	be	VERB
cana-2862	61	32	two	two	NUM
cana-2862	61	33	selfmaps	selfmap	NOUN
cana-2862	61	34	satisfying	satisfy	VERB
cana-2862	61	35	𝑑(𝑓𝑥	𝑑(𝑓𝑥	NOUN
cana-2862	61	36	,	,	PUNCT
cana-2862	61	37	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	61	38	)	)	PUNCT
cana-2862	61	39	≾	≾	PROPN
cana-2862	61	40	𝐴	𝐴	PROPN
cana-2862	61	41	⋅	⋅	PROPN
cana-2862	61	42	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	61	43	,	,	PUNCT
cana-2862	61	44	𝑦	𝑦	NOUN
cana-2862	61	45	)	)	PUNCT
cana-2862	62	1	+	+	CCONJ
cana-2862	62	2	𝐵	𝐵	PROPN
cana-2862	62	3	⋅	⋅	PROPN
cana-2862	62	4	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	62	5	,	,	PUNCT
cana-2862	62	6	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	62	7	)	)	PUNCT
cana-2862	62	8	⋅	⋅	PROPN
cana-2862	62	9	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	62	10	,	,	PUNCT
cana-2862	62	11	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	62	12	)	)	PUNCT
cana-2862	62	13	1	1	NUM
cana-2862	63	1	+	+	CCONJ
cana-2862	63	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	63	3	,	,	PUNCT
cana-2862	63	4	𝑦	𝑦	NOUN
cana-2862	63	5	)	)	PUNCT
cana-2862	63	6	+	+	CCONJ
cana-2862	63	7	𝐶	𝐶	PROPN
cana-2862	63	8	⋅	⋅	PROPN
cana-2862	63	9	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	63	10	,	,	PUNCT
cana-2862	63	11	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	63	12	)	)	PUNCT
cana-2862	63	13	⋅	⋅	PROPN
cana-2862	63	14	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	63	15	,	,	PUNCT
cana-2862	63	16	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	63	17	)	)	PUNCT
cana-2862	63	18	1	1	NUM
cana-2862	64	1	+	+	CCONJ
cana-2862	64	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	64	3	,	,	PUNCT
cana-2862	64	4	𝑦	𝑦	NOUN
cana-2862	64	5	)	)	PUNCT
cana-2862	64	6	+	+	ADJ
cana-2862	64	7	𝐷	𝐷	PROPN
cana-2862	64	8	⋅	⋅	PROPN
cana-2862	64	9	𝑑(𝑥,𝑓𝑥)⋅𝑑(𝑥,𝑔𝑦	𝑑(𝑥,𝑓𝑥)⋅𝑑(𝑥,𝑔𝑦	PROPN
cana-2862	64	10	)	)	PUNCT
cana-2862	64	11	1+𝑑(𝑥,𝑦	1+𝑑(𝑥,𝑦	NOUN
cana-2862	64	12	)	)	PUNCT
cana-2862	64	13	+	+	CCONJ
cana-2862	64	14	𝐸	𝐸	PROPN
cana-2862	64	15	⋅	⋅	PROPN
cana-2862	64	16	𝑑(𝑦,𝑓𝑥)⋅𝑑(𝑦,𝑔𝑦	𝑑(𝑦,𝑓𝑥)⋅𝑑(𝑦,𝑔𝑦	NOUN
cana-2862	64	17	)	)	PUNCT
cana-2862	64	18	1+𝑑(𝑥,𝑦	1+𝑑(𝑥,𝑦	NOUN
cana-2862	64	19	)	)	PUNCT
cana-2862	64	20	(	(	PUNCT
cana-2862	64	21	3.1	3.1	NUM
cana-2862	64	22	)	)	PUNCT
cana-2862	64	23	communications	communication	NOUN
cana-2862	64	24	on	on	ADP
cana-2862	64	25	applied	apply	VERB
cana-2862	64	26	nonlinear	nonlinear	ADJ
cana-2862	64	27	analysis	analysis	NOUN
cana-2862	64	28	issn	issn	NOUN
cana-2862	64	29	:	:	PUNCT
cana-2862	64	30	1074	1074	NUM
cana-2862	64	31	-	-	PUNCT
cana-2862	64	32	133x	133x	NUM
cana-2862	64	33	vol	vol	NOUN
cana-2862	64	34	32	32	NUM
cana-2862	64	35	no	no	NOUN
cana-2862	64	36	.	.	PUNCT
cana-2862	65	1	4s	4s	NUM
cana-2862	65	2	(	(	PUNCT
cana-2862	65	3	2025	2025	NUM
cana-2862	65	4	)	)	PUNCT
cana-2862	65	5	435	435	NUM
cana-2862	65	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2862	65	7	where	where	SCONJ
cana-2862	65	8	𝐴	𝐴	PROPN
cana-2862	65	9	,	,	PUNCT
cana-2862	65	10	𝐵	𝐵	PROPN
cana-2862	65	11	,	,	PUNCT
cana-2862	65	12	𝐶	𝐶	PROPN
cana-2862	65	13	,	,	PUNCT
cana-2862	65	14	𝐷	𝐷	PROPN
cana-2862	65	15	,	,	PUNCT
cana-2862	65	16	𝐸	𝐸	PROPN
cana-2862	65	17	nonnegative	nonnegative	ADJ
cana-2862	65	18	real	real	ADJ
cana-2862	65	19	numbers	number	NOUN
cana-2862	65	20	,	,	PUNCT
cana-2862	65	21	with	with	ADP
cana-2862	65	22	𝐴	𝐴	PROPN
cana-2862	65	23	+	+	CCONJ
cana-2862	65	24	𝐵	𝐵	PROPN
cana-2862	65	25	+	+	CCONJ
cana-2862	65	26	𝐶	𝐶	PROPN
cana-2862	65	27	+	+	CCONJ
cana-2862	65	28	2𝐷	2𝐷	NOUN
cana-2862	65	29	+	+	CCONJ
cana-2862	65	30	2𝐸	2𝐸	ADJ
cana-2862	65	31	<	<	X
cana-2862	65	32	1	1	NUM
cana-2862	65	33	.	.	PUNCT
cana-2862	66	1	then	then	ADV
cana-2862	66	2	𝑓	𝑓	X
cana-2862	66	3	and	and	CCONJ
cana-2862	66	4	𝑔	𝑔	AUX
cana-2862	66	5	have	have	VERB
cana-2862	66	6	a	a	DET
cana-2862	66	7	unique	unique	ADJ
cana-2862	66	8	common	common	ADJ
cana-2862	66	9	fixed	fix	VERB
cana-2862	66	10	point	point	NOUN
cana-2862	66	11	in	in	ADP
cana-2862	66	12	𝑋.	𝑋.	PROPN
cana-2862	66	13	proof	proof	NOUN
cana-2862	66	14	.	.	PUNCT
cana-2862	67	1	for	for	ADP
cana-2862	67	2	any	any	DET
cana-2862	67	3	arbitrary	arbitrary	ADJ
cana-2862	67	4	point	point	NOUN
cana-2862	67	5	,	,	PUNCT
cana-2862	67	6	𝑥0	𝑥0	PROPN
cana-2862	67	7	∈	∈	PROPN
cana-2862	67	8	𝑋.	𝑋.	PROPN
cana-2862	67	9	define	define	VERB
cana-2862	67	10	a	a	DET
cana-2862	67	11	sequence	sequence	NOUN
cana-2862	67	12	{	{	PUNCT
cana-2862	67	13	𝑥𝑛	𝑥𝑛	NOUN
cana-2862	67	14	}	}	PUNCT
cana-2862	67	15	in	in	ADP
cana-2862	67	16	𝑋	𝑋	PROPN
cana-2862	67	17	such	such	DET
cana-2862	67	18	that	that	PRON
cana-2862	67	19	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	67	20	=	=	SYM
cana-2862	67	21	𝑓𝑥2𝑛	𝑓𝑥2𝑛	PROPN
cana-2862	67	22	,	,	PUNCT
cana-2862	67	23	𝑥2𝑛+2	𝑥2𝑛+2	PROPN
cana-2862	67	24	=	=	PROPN
cana-2862	67	25	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	67	26	,	,	PUNCT
cana-2862	67	27	for	for	ADP
cana-2862	67	28	𝑛	𝑛	NOUN
cana-2862	67	29	=	=	SYM
cana-2862	67	30	0,1,2,3	0,1,2,3	NUM
cana-2862	67	31	,	,	PUNCT
cana-2862	67	32	…	…	PUNCT
cana-2862	67	33	(	(	PUNCT
cana-2862	67	34	3.2	3.2	NUM
cana-2862	67	35	)	)	PUNCT
cana-2862	67	36	now	now	ADV
cana-2862	67	37	,	,	PUNCT
cana-2862	67	38	we	we	PRON
cana-2862	67	39	show	show	VERB
cana-2862	67	40	that	that	SCONJ
cana-2862	67	41	the	the	DET
cana-2862	67	42	sequence	sequence	NOUN
cana-2862	67	43	{	{	PUNCT
cana-2862	67	44	𝑥𝑛	𝑥𝑛	NOUN
cana-2862	67	45	}	}	PUNCT
cana-2862	67	46	is	be	AUX
cana-2862	67	47	a	a	DET
cana-2862	67	48	cauchy	cauchy	ADJ
cana-2862	67	49	sequence	sequence	NOUN
cana-2862	67	50	.	.	PUNCT
cana-2862	68	1	let	let	VERB
cana-2862	68	2	𝑥	𝑥	X
cana-2862	68	3	=	=	PUNCT
cana-2862	68	4	𝑥2𝑛	𝑥2𝑛	PROPN
cana-2862	68	5	and	and	CCONJ
cana-2862	68	6	𝑦	𝑦	NOUN
cana-2862	68	7	=	=	SYM
cana-2862	68	8	𝑥2𝑛+1	𝑥2𝑛+1	PROPN
cana-2862	68	9	in	in	ADP
cana-2862	68	10	(	(	PUNCT
cana-2862	68	11	3.1	3.1	NUM
cana-2862	68	12	)	)	PUNCT
cana-2862	68	13	,	,	PUNCT
cana-2862	68	14	we	we	PRON
cana-2862	68	15	have	have	VERB
cana-2862	68	16	𝑑(𝑓𝑥2𝑛	𝑑(𝑓𝑥2𝑛	NOUN
cana-2862	68	17	,	,	PUNCT
cana-2862	68	18	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	68	19	)	)	PUNCT
cana-2862	68	20	=	=	SYM
cana-2862	69	1	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	2	,	,	PUNCT
cana-2862	69	3	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	69	4	)	)	PUNCT
cana-2862	69	5	≾	≾	PROPN
cana-2862	69	6	𝐴	𝐴	PROPN
cana-2862	69	7	⋅	⋅	PROPN
cana-2862	69	8	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	PROPN
cana-2862	69	9	,	,	PUNCT
cana-2862	69	10	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	69	11	)	)	PUNCT
cana-2862	69	12	+	+	CCONJ
cana-2862	69	13	𝐵	𝐵	PROPN
cana-2862	69	14	⋅	⋅	PROPN
cana-2862	69	15	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	16	,	,	PUNCT
cana-2862	69	17	𝑓𝑥2𝑛	𝑓𝑥2𝑛	ADJ
cana-2862	69	18	)	)	PUNCT
cana-2862	69	19	⋅	⋅	PROPN
cana-2862	69	20	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	21	,	,	PUNCT
cana-2862	69	22	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	69	23	)	)	PUNCT
cana-2862	69	24	1	1	NUM
cana-2862	69	25	+	+	PUNCT
cana-2862	69	26	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	27	,	,	PUNCT
cana-2862	69	28	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	29	)	)	PUNCT
cana-2862	69	30	+	+	PROPN
cana-2862	69	31	𝐶	𝐶	PROPN
cana-2862	69	32	⋅	⋅	PROPN
cana-2862	69	33	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	34	,	,	PUNCT
cana-2862	69	35	𝑓𝑥2𝑛	𝑓𝑥2𝑛	ADJ
cana-2862	69	36	)	)	PUNCT
cana-2862	69	37	⋅	⋅	PROPN
cana-2862	69	38	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	39	,	,	PUNCT
cana-2862	69	40	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	69	41	)	)	PUNCT
cana-2862	69	42	1	1	NUM
cana-2862	69	43	+	+	PUNCT
cana-2862	69	44	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	45	,	,	PUNCT
cana-2862	69	46	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	47	)	)	PUNCT
cana-2862	69	48	+	+	ADJ
cana-2862	69	49	𝐷	𝐷	PROPN
cana-2862	69	50	⋅	⋅	PROPN
cana-2862	69	51	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	52	,	,	PUNCT
cana-2862	69	53	𝑓𝑥2𝑛	𝑓𝑥2𝑛	ADJ
cana-2862	69	54	)	)	PUNCT
cana-2862	69	55	⋅	⋅	PROPN
cana-2862	69	56	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	57	,	,	PUNCT
cana-2862	69	58	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	69	59	)	)	PUNCT
cana-2862	69	60	1	1	NUM
cana-2862	69	61	+	+	PUNCT
cana-2862	69	62	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	63	,	,	PUNCT
cana-2862	69	64	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	69	65	)	)	PUNCT
cana-2862	69	66	+	+	NOUN
cana-2862	69	67	𝐸	𝐸	PROPN
cana-2862	69	68	⋅	⋅	PROPN
cana-2862	69	69	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	70	,	,	PUNCT
cana-2862	69	71	𝑓𝑥2𝑛	𝑓𝑥2𝑛	ADJ
cana-2862	69	72	)	)	PUNCT
cana-2862	69	73	⋅	⋅	PROPN
cana-2862	69	74	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	75	,	,	PUNCT
cana-2862	69	76	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	69	77	)	)	PUNCT
cana-2862	69	78	1	1	NUM
cana-2862	69	79	+	+	PUNCT
cana-2862	69	80	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	81	,	,	PUNCT
cana-2862	69	82	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	69	83	)	)	PUNCT
cana-2862	69	84	i.e.	i.e.	X
cana-2862	69	85	,	,	PUNCT
cana-2862	69	86	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	ADJ
cana-2862	69	87	,	,	PUNCT
cana-2862	69	88	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	69	89	)	)	PUNCT
cana-2862	69	90	≾	≾	PROPN
cana-2862	69	91	𝐴	𝐴	PROPN
cana-2862	69	92	⋅	⋅	PROPN
cana-2862	69	93	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	PROPN
cana-2862	69	94	,	,	PUNCT
cana-2862	69	95	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	69	96	)	)	PUNCT
cana-2862	69	97	+	+	CCONJ
cana-2862	69	98	𝐵	𝐵	PROPN
cana-2862	69	99	⋅	⋅	PROPN
cana-2862	69	100	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	101	,	,	PUNCT
cana-2862	69	102	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	103	)	)	PUNCT
cana-2862	69	104	⋅	⋅	PROPN
cana-2862	69	105	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	106	,	,	PUNCT
cana-2862	69	107	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	69	108	)	)	PUNCT
cana-2862	69	109	1	1	NUM
cana-2862	69	110	+	+	CCONJ
cana-2862	69	111	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	112	,	,	PUNCT
cana-2862	69	113	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	114	)	)	PUNCT
cana-2862	69	115	+	+	PROPN
cana-2862	69	116	𝐶	𝐶	PROPN
cana-2862	69	117	⋅	⋅	PROPN
cana-2862	69	118	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	119	,	,	PUNCT
cana-2862	69	120	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	121	)	)	PUNCT
cana-2862	69	122	⋅	⋅	PROPN
cana-2862	69	123	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	124	,	,	PUNCT
cana-2862	69	125	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	69	126	)	)	PUNCT
cana-2862	69	127	1	1	NUM
cana-2862	69	128	+	+	CCONJ
cana-2862	69	129	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	130	,	,	PUNCT
cana-2862	69	131	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	132	)	)	PUNCT
cana-2862	69	133	+	+	ADJ
cana-2862	69	134	𝐷	𝐷	PROPN
cana-2862	69	135	⋅	⋅	PROPN
cana-2862	69	136	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	137	,	,	PUNCT
cana-2862	69	138	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	139	)	)	PUNCT
cana-2862	69	140	⋅	⋅	PROPN
cana-2862	69	141	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	142	,	,	PUNCT
cana-2862	69	143	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	69	144	)	)	PUNCT
cana-2862	69	145	1	1	NUM
cana-2862	69	146	+	+	CCONJ
cana-2862	69	147	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	148	,	,	PUNCT
cana-2862	69	149	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	69	150	)	)	PUNCT
cana-2862	69	151	+	+	NOUN
cana-2862	69	152	𝐸	𝐸	PROPN
cana-2862	69	153	⋅	⋅	PROPN
cana-2862	69	154	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	155	,	,	PUNCT
cana-2862	69	156	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	157	)	)	PUNCT
cana-2862	69	158	⋅	⋅	PROPN
cana-2862	69	159	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	160	,	,	PUNCT
cana-2862	69	161	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	69	162	)	)	PUNCT
cana-2862	69	163	1	1	NUM
cana-2862	69	164	+	+	CCONJ
cana-2862	69	165	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	166	,	,	PUNCT
cana-2862	69	167	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	69	168	)	)	PUNCT
cana-2862	69	169	⇒	⇒	NOUN
cana-2862	69	170	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	ADP
cana-2862	69	171	,	,	PUNCT
cana-2862	69	172	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	69	173	)	)	PUNCT
cana-2862	69	174	≾	≾	PROPN
cana-2862	69	175	𝐴	𝐴	PROPN
cana-2862	69	176	⋅	⋅	PROPN
cana-2862	69	177	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	PROPN
cana-2862	69	178	,	,	PUNCT
cana-2862	69	179	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	69	180	)	)	PUNCT
cana-2862	69	181	+	+	CCONJ
cana-2862	69	182	𝐵	𝐵	PROPN
cana-2862	69	183	⋅	⋅	PROPN
cana-2862	69	184	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	185	,	,	PUNCT
cana-2862	69	186	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	187	)	)	PUNCT
cana-2862	69	188	⋅	⋅	PROPN
cana-2862	69	189	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	69	190	,	,	PUNCT
cana-2862	69	191	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	69	192	)	)	PUNCT
cana-2862	69	193	1	1	NUM
cana-2862	69	194	+	+	CCONJ
cana-2862	69	195	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	196	,	,	PUNCT
cana-2862	69	197	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	198	)	)	PUNCT
cana-2862	69	199	+	+	ADJ
cana-2862	69	200	𝐷	𝐷	PROPN
cana-2862	69	201	⋅	⋅	PROPN
cana-2862	69	202	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	203	,	,	PUNCT
cana-2862	69	204	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	69	205	)	)	PUNCT
cana-2862	69	206	⋅	⋅	PROPN
cana-2862	69	207	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	NOUN
cana-2862	69	208	,	,	PUNCT
cana-2862	69	209	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	69	210	)	)	PUNCT
cana-2862	69	211	1	1	NUM
cana-2862	69	212	+	+	CCONJ
cana-2862	69	213	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	214	,	,	PUNCT
cana-2862	69	215	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	69	216	)	)	PUNCT
cana-2862	69	217	(	(	PUNCT
cana-2862	69	218	3.3	3.3	NUM
cana-2862	69	219	)	)	PUNCT
cana-2862	69	220	which	which	PRON
cana-2862	69	221	implies	imply	VERB
cana-2862	69	222	that	that	SCONJ
cana-2862	69	223	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	69	224	,	,	PUNCT
cana-2862	69	225	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	69	226	≤	≤	PROPN
cana-2862	69	227	𝐴	𝐴	PROPN
cana-2862	69	228	⋅	⋅	PROPN
cana-2862	69	229	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	PROPN
cana-2862	69	230	,	,	PUNCT
cana-2862	69	231	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	69	232	+	+	CCONJ
cana-2862	69	233	𝐵	𝐵	PROPN
cana-2862	69	234	⋅	⋅	PROPN
cana-2862	69	235	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	NOUN
cana-2862	69	236	,	,	PUNCT
cana-2862	69	237	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	69	238	⋅	⋅	PROPN
cana-2862	69	239	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	69	240	,	,	PUNCT
cana-2862	69	241	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	69	242	|1	|1	PRON
cana-2862	69	243	+	+	PUNCT
cana-2862	69	244	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	245	,	,	PUNCT
cana-2862	69	246	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	69	247	+	+	NOUN
cana-2862	69	248	𝐷	𝐷	PROPN
cana-2862	69	249	⋅	⋅	X
cana-2862	69	250	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	NOUN
cana-2862	69	251	,	,	PUNCT
cana-2862	69	252	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	69	253	⋅	⋅	X
cana-2862	69	254	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	NOUN
cana-2862	69	255	,	,	PUNCT
cana-2862	69	256	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	69	257	|1	|1	PRON
cana-2862	69	258	+	+	PUNCT
cana-2862	69	259	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	260	,	,	PUNCT
cana-2862	69	261	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	69	262	(	(	PUNCT
cana-2862	69	263	3.4	3.4	NUM
cana-2862	69	264	)	)	PUNCT
cana-2862	69	265	since	since	SCONJ
cana-2862	69	266	|1	|1	PRON
cana-2862	69	267	+	+	PUNCT
cana-2862	69	268	𝑑(𝑥2𝑛	𝑑(𝑥2𝑛	X
cana-2862	69	269	,	,	PUNCT
cana-2862	69	270	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	69	271	>	>	SYM
cana-2862	69	272	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	NOUN
cana-2862	69	273	,	,	PUNCT
cana-2862	69	274	𝑥2𝑛+1)|	𝑥2𝑛+1)|	VERB
cana-2862	69	275	we	we	PRON
cana-2862	69	276	get	get	VERB
cana-2862	69	277	communications	communication	NOUN
cana-2862	69	278	on	on	ADP
cana-2862	69	279	applied	apply	VERB
cana-2862	69	280	nonlinear	nonlinear	ADJ
cana-2862	69	281	analysis	analysis	NOUN
cana-2862	69	282	issn	issn	NOUN
cana-2862	69	283	:	:	PUNCT
cana-2862	69	284	1074	1074	NUM
cana-2862	69	285	-	-	PUNCT
cana-2862	69	286	133x	133x	NUM
cana-2862	69	287	vol	vol	NOUN
cana-2862	69	288	32	32	NUM
cana-2862	69	289	no	no	NOUN
cana-2862	69	290	.	.	PUNCT
cana-2862	70	1	4s	4s	NUM
cana-2862	70	2	(	(	PUNCT
cana-2862	70	3	2025	2025	NUM
cana-2862	70	4	)	)	PUNCT
cana-2862	70	5	436	436	NUM
cana-2862	71	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2862	71	2	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	ADJ
cana-2862	71	3	,	,	PUNCT
cana-2862	71	4	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	71	5	≤	≤	PROPN
cana-2862	71	6	𝐴	𝐴	PROPN
cana-2862	71	7	⋅	⋅	PROPN
cana-2862	71	8	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	PROPN
cana-2862	71	9	,	,	PUNCT
cana-2862	71	10	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	71	11	+	+	CCONJ
cana-2862	71	12	𝐵	𝐵	PROPN
cana-2862	71	13	⋅	⋅	PROPN
cana-2862	71	14	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	71	15	,	,	PUNCT
cana-2862	71	16	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	71	17	+	+	CCONJ
cana-2862	71	18	𝐷	𝐷	PROPN
cana-2862	71	19	⋅	⋅	PROPN
cana-2862	71	20	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	NOUN
cana-2862	71	21	,	,	PUNCT
cana-2862	71	22	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	71	23	+	+	ADJ
cana-2862	71	24	𝐷.	𝐷.	PROPN
cana-2862	71	25	|𝑑2𝑛+1	|𝑑2𝑛+1	PROPN
cana-2862	71	26	,	,	PUNCT
cana-2862	71	27	𝑥2𝑛+2|	𝑥2𝑛+2|	NOUN
cana-2862	71	28	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	NOUN
cana-2862	71	29	,	,	PUNCT
cana-2862	71	30	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	71	31	≤	≤	PROPN
cana-2862	71	32	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	NOUN
cana-2862	71	33	,	,	PUNCT
cana-2862	71	34	𝑥2𝑛+1)|+∣	𝑥2𝑛+1)|+∣	PROPN
cana-2862	71	35	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	71	36	,	,	PUNCT
cana-2862	71	37	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	71	38	)	)	PUNCT
cana-2862	71	39	⇒	⇒	NOUN
cana-2862	71	40	(	(	PUNCT
cana-2862	71	41	1	1	NUM
cana-2862	71	42	−	−	PROPN
cana-2862	71	43	𝐵	𝐵	NOUN
cana-2862	71	44	−	−	PROPN
cana-2862	71	45	𝐷	𝐷	PROPN
cana-2862	71	46	)	)	PUNCT
cana-2862	71	47	⋅	⋅	PROPN
cana-2862	71	48	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	71	49	,	,	PUNCT
cana-2862	71	50	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	71	51	≤	≤	NOUN
cana-2862	71	52	(	(	PUNCT
cana-2862	71	53	𝐴	𝐴	PROPN
cana-2862	71	54	+	+	CCONJ
cana-2862	71	55	𝐷	𝐷	PROPN
cana-2862	71	56	)	)	PUNCT
cana-2862	71	57	⋅	⋅	PROPN
cana-2862	71	58	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	NOUN
cana-2862	71	59	,	,	PUNCT
cana-2862	71	60	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	71	61	⇒	⇒	VERB
cana-2862	71	62	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	71	63	,	,	PUNCT
cana-2862	71	64	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	71	65	≤	≤	PROPN
cana-2862	71	66	𝐴	𝐴	PROPN
cana-2862	71	67	+	+	CCONJ
cana-2862	71	68	𝐷	𝐷	PROPN
cana-2862	71	69	1	1	NUM
cana-2862	71	70	−	−	PROPN
cana-2862	71	71	𝐵	𝐵	NOUN
cana-2862	71	72	−	−	PROPN
cana-2862	71	73	𝐷	𝐷	PROPN
cana-2862	71	74	|𝑑(𝑥2𝑛	|𝑑(𝑥2𝑛	NOUN
cana-2862	71	75	,	,	PUNCT
cana-2862	71	76	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	71	77	.	.	PUNCT
cana-2862	72	1	(	(	PUNCT
cana-2862	72	2	3.5	3.5	NUM
cana-2862	72	3	)	)	PUNCT
cana-2862	72	4	similarly	similarly	ADV
cana-2862	72	5	,	,	PUNCT
cana-2862	72	6	we	we	PRON
cana-2862	72	7	get	get	VERB
cana-2862	72	8	|𝑑(𝑥2𝑛+2	|𝑑(𝑥2𝑛+2	ADJ
cana-2862	72	9	,	,	PUNCT
cana-2862	72	10	𝑥2𝑛+3)|	𝑥2𝑛+3)|	PROPN
cana-2862	72	11	≤	≤	NUM
cana-2862	72	12	𝐴+𝐷	𝐴+𝐷	ADJ
cana-2862	72	13	1−𝐵−𝐷	1−𝐵−𝐷	NUM
cana-2862	72	14	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	ADJ
cana-2862	72	15	,	,	PUNCT
cana-2862	72	16	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	72	17	.	.	PUNCT
cana-2862	73	1	(	(	PUNCT
cana-2862	73	2	3.6	3.6	NUM
cana-2862	73	3	)	)	PUNCT
cana-2862	73	4	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	73	5	,	,	PUNCT
cana-2862	73	6	𝑥2𝑛+3	𝑥2𝑛+3	NOUN
cana-2862	73	7	)	)	PUNCT
cana-2862	73	8	=	=	SYM
cana-2862	73	9	𝑑(𝑥2𝑛+3	𝑑(𝑥2𝑛+3	PROPN
cana-2862	73	10	,	,	PUNCT
cana-2862	73	11	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	73	12	)	)	PUNCT
cana-2862	73	13	=	=	SYM
cana-2862	73	14	𝑑(𝑓𝑥2𝑛+2	𝑑(𝑓𝑥2𝑛+2	NOUN
cana-2862	73	15	,	,	PUNCT
cana-2862	73	16	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	NOUN
cana-2862	73	17	)	)	PUNCT
cana-2862	73	18	≾	≾	PROPN
cana-2862	73	19	𝐴	𝐴	PROPN
cana-2862	73	20	⋅	⋅	PROPN
cana-2862	73	21	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	73	22	,	,	PUNCT
cana-2862	73	23	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	73	24	)	)	PUNCT
cana-2862	73	25	+	+	CCONJ
cana-2862	73	26	𝐵	𝐵	PROPN
cana-2862	73	27	⋅	⋅	PROPN
cana-2862	73	28	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	73	29	,	,	PUNCT
cana-2862	73	30	𝑓𝑥2𝑛+2	𝑓𝑥2𝑛+2	PROPN
cana-2862	73	31	)	)	PUNCT
cana-2862	73	32	⋅	⋅	PROPN
cana-2862	73	33	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	73	34	,	,	PUNCT
cana-2862	73	35	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	73	36	)	)	PUNCT
cana-2862	73	37	1	1	NUM
cana-2862	74	1	+	+	CCONJ
cana-2862	74	2	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	74	3	,	,	PUNCT
cana-2862	74	4	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	74	5	)	)	PUNCT
cana-2862	75	1	+	+	PROPN
cana-2862	75	2	𝐶	𝐶	PROPN
cana-2862	75	3	⋅	⋅	PROPN
cana-2862	75	4	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	75	5	,	,	PUNCT
cana-2862	75	6	𝑓𝑥2𝑛+2	𝑓𝑥2𝑛+2	PROPN
cana-2862	75	7	)	)	PUNCT
cana-2862	75	8	⋅	⋅	PROPN
cana-2862	75	9	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	75	10	,	,	PUNCT
cana-2862	75	11	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	75	12	)	)	PUNCT
cana-2862	75	13	1	1	NUM
cana-2862	76	1	+	+	CCONJ
cana-2862	76	2	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	76	3	,	,	PUNCT
cana-2862	76	4	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	76	5	)	)	PUNCT
cana-2862	76	6	+	+	NUM
cana-2862	76	7	𝐷	𝐷	PROPN
cana-2862	76	8	⋅	⋅	PROPN
cana-2862	76	9	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	76	10	,	,	PUNCT
cana-2862	76	11	𝑓𝑥2𝑛+2	𝑓𝑥2𝑛+2	PROPN
cana-2862	76	12	)	)	PUNCT
cana-2862	76	13	⋅	⋅	PROPN
cana-2862	76	14	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	76	15	,	,	PUNCT
cana-2862	76	16	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	76	17	)	)	PUNCT
cana-2862	76	18	1	1	NUM
cana-2862	77	1	+	+	CCONJ
cana-2862	77	2	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	77	3	,	,	PUNCT
cana-2862	77	4	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	77	5	)	)	PUNCT
cana-2862	77	6	+	+	NOUN
cana-2862	77	7	𝐸	𝐸	PROPN
cana-2862	77	8	⋅	⋅	PROPN
cana-2862	77	9	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	77	10	,	,	PUNCT
cana-2862	77	11	𝑓𝑥2𝑛+2	𝑓𝑥2𝑛+2	PROPN
cana-2862	77	12	)	)	PUNCT
cana-2862	77	13	⋅	⋅	PROPN
cana-2862	77	14	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	77	15	,	,	PUNCT
cana-2862	77	16	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	77	17	)	)	PUNCT
cana-2862	77	18	1	1	NUM
cana-2862	78	1	+	+	CCONJ
cana-2862	78	2	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	78	3	,	,	PUNCT
cana-2862	78	4	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	78	5	)	)	PUNCT
cana-2862	78	6	⇒	⇒	NOUN
cana-2862	78	7	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	78	8	,	,	PUNCT
cana-2862	78	9	𝑥2𝑛+3	𝑥2𝑛+3	NOUN
cana-2862	78	10	)	)	PUNCT
cana-2862	78	11	=	=	SYM
cana-2862	78	12	𝑑(𝑓𝑥2𝑛+2	𝑑(𝑓𝑥2𝑛+2	NOUN
cana-2862	78	13	,	,	PUNCT
cana-2862	78	14	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	NOUN
cana-2862	78	15	)	)	PUNCT
cana-2862	78	16	≾	≾	PROPN
cana-2862	78	17	𝐴	𝐴	PROPN
cana-2862	78	18	⋅	⋅	PROPN
cana-2862	78	19	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	78	20	,	,	PUNCT
cana-2862	78	21	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	78	22	)	)	PUNCT
cana-2862	78	23	+	+	CCONJ
cana-2862	78	24	𝐵	𝐵	PROPN
cana-2862	78	25	⋅	⋅	PROPN
cana-2862	78	26	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	78	27	,	,	PUNCT
cana-2862	78	28	𝑥2𝑛+3	𝑥2𝑛+3	NOUN
cana-2862	78	29	)	)	PUNCT
cana-2862	78	30	⋅	⋅	PROPN
cana-2862	78	31	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	78	32	,	,	PUNCT
cana-2862	78	33	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	78	34	)	)	PUNCT
cana-2862	78	35	1	1	NUM
cana-2862	79	1	+	+	CCONJ
cana-2862	79	2	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	79	3	,	,	PUNCT
cana-2862	79	4	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	79	5	)	)	PUNCT
cana-2862	79	6	+	+	PROPN
cana-2862	79	7	𝐶	𝐶	PROPN
cana-2862	79	8	⋅	⋅	PROPN
cana-2862	79	9	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	79	10	,	,	PUNCT
cana-2862	79	11	𝑥2𝑛+3	𝑥2𝑛+3	NOUN
cana-2862	79	12	)	)	PUNCT
cana-2862	79	13	⋅	⋅	PROPN
cana-2862	79	14	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	79	15	,	,	PUNCT
cana-2862	79	16	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	79	17	)	)	PUNCT
cana-2862	79	18	1	1	NUM
cana-2862	80	1	+	+	CCONJ
cana-2862	80	2	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	80	3	,	,	PUNCT
cana-2862	80	4	𝑥2𝑛+1	𝑥2𝑛+1	ADJ
cana-2862	80	5	)	)	PUNCT
cana-2862	80	6	+	+	PROPN
cana-2862	80	7	𝐷	𝐷	PROPN
cana-2862	80	8	⋅	⋅	X
cana-2862	80	9	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	80	10	,	,	PUNCT
cana-2862	80	11	𝑥2𝑛+3	𝑥2𝑛+3	NOUN
cana-2862	80	12	)	)	PUNCT
cana-2862	80	13	⋅	⋅	PROPN
cana-2862	80	14	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	80	15	,	,	PUNCT
cana-2862	80	16	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	80	17	)	)	PUNCT
cana-2862	80	18	1	1	NUM
cana-2862	81	1	+	+	CCONJ
cana-2862	81	2	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	81	3	,	,	PUNCT
cana-2862	81	4	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	81	5	)	)	PUNCT
cana-2862	82	1	+	+	NOUN
cana-2862	82	2	𝐸	𝐸	PROPN
cana-2862	82	3	⋅	⋅	PROPN
cana-2862	82	4	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	82	5	,	,	PUNCT
cana-2862	82	6	𝑥2𝑛+3	𝑥2𝑛+3	NOUN
cana-2862	82	7	)	)	PUNCT
cana-2862	82	8	⋅	⋅	PROPN
cana-2862	82	9	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	82	10	,	,	PUNCT
cana-2862	82	11	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	82	12	)	)	PUNCT
cana-2862	82	13	1	1	NUM
cana-2862	82	14	+	+	CCONJ
cana-2862	82	15	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	NOUN
cana-2862	82	16	,	,	PUNCT
cana-2862	82	17	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	82	18	)	)	PUNCT
cana-2862	82	19	.	.	PUNCT
cana-2862	83	1	(	(	PUNCT
cana-2862	83	2	3.7	3.7	NUM
cana-2862	83	3	)	)	PUNCT
cana-2862	83	4	which	which	PRON
cana-2862	83	5	implies	imply	VERB
cana-2862	83	6	that	that	SCONJ
cana-2862	83	7	|𝑑	|𝑑	PROPN
cana-2862	83	8	⋅	⋅	X
cana-2862	83	9	(	(	PUNCT
cana-2862	83	10	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	83	11	,	,	PUNCT
cana-2862	83	12	𝑥2𝑛+3)|	𝑥2𝑛+3)|	PROPN
cana-2862	83	13	≤	≤	PROPN
cana-2862	83	14	𝐴	𝐴	PROPN
cana-2862	83	15	⋅	⋅	PROPN
cana-2862	83	16	|𝑑(𝑥2𝑛+2	|𝑑(𝑥2𝑛+2	PROPN
cana-2862	83	17	,	,	PUNCT
cana-2862	83	18	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	83	19	+	+	CCONJ
cana-2862	83	20	𝐵	𝐵	PROPN
cana-2862	83	21	⋅	⋅	PROPN
cana-2862	83	22	|𝑑(𝑥2𝑛+2	|𝑑(𝑥2𝑛+2	NOUN
cana-2862	83	23	,	,	PUNCT
cana-2862	83	24	𝑥2𝑛+3)|	𝑥2𝑛+3)|	PROPN
cana-2862	83	25	⋅	⋅	PROPN
cana-2862	83	26	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	83	27	,	,	PUNCT
cana-2862	83	28	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	83	29	|1	|1	NUM
cana-2862	83	30	+	+	SYM
cana-2862	83	31	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	83	32	,	,	PUNCT
cana-2862	83	33	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	83	34	+	+	NOUN
cana-2862	83	35	𝐸	𝐸	ADJ
cana-2862	83	36	⋅	⋅	ADJ
cana-2862	83	37	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	83	38	,	,	PUNCT
cana-2862	83	39	𝑥2𝑛+3)|	𝑥2𝑛+3)|	PROPN
cana-2862	83	40	⋅	⋅	PROPN
cana-2862	83	41	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	83	42	,	,	PUNCT
cana-2862	83	43	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	83	44	|1	|1	NUM
cana-2862	83	45	+	+	SYM
cana-2862	83	46	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	83	47	,	,	PUNCT
cana-2862	83	48	𝑥2𝑛+1)|	𝑥2𝑛+1)|	PROPN
cana-2862	83	49	(	(	PUNCT
cana-2862	83	50	3.8	3.8	NUM
cana-2862	83	51	)	)	PUNCT
cana-2862	83	52	since	since	SCONJ
cana-2862	83	53	|1	|1	PRON
cana-2862	83	54	+	+	SYM
cana-2862	83	55	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	ADJ
cana-2862	83	56	,	,	PUNCT
cana-2862	83	57	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	83	58	>	>	X
cana-2862	83	59	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	PROPN
cana-2862	83	60	,	,	PUNCT
cana-2862	83	61	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	83	62	.	.	PUNCT
cana-2862	84	1	so	so	ADV
cana-2862	84	2	,	,	PUNCT
cana-2862	84	3	we	we	PRON
cana-2862	84	4	get	get	VERB
cana-2862	84	5	|𝑑(𝑥2𝑛+2	|𝑑(𝑥2𝑛+2	ADJ
cana-2862	84	6	,	,	PUNCT
cana-2862	84	7	𝑥2𝑛+3)|	𝑥2𝑛+3)|	PROPN
cana-2862	84	8	<	<	X
cana-2862	84	9	𝐴	𝐴	PROPN
cana-2862	84	10	⋅	⋅	PROPN
cana-2862	84	11	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	PROPN
cana-2862	84	12	,	,	PUNCT
cana-2862	84	13	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	84	14	+	+	CCONJ
cana-2862	84	15	𝐵	𝐵	PROPN
cana-2862	84	16	⋅	⋅	PROPN
cana-2862	84	17	|𝑑(𝑥2𝑛+2	|𝑑(𝑥2𝑛+2	NOUN
cana-2862	84	18	,	,	PUNCT
cana-2862	84	19	𝑥2𝑛+3)|	𝑥2𝑛+3)|	PROPN
cana-2862	84	20	communications	communication	NOUN
cana-2862	84	21	on	on	ADP
cana-2862	84	22	applied	apply	VERB
cana-2862	84	23	nonlinear	nonlinear	ADJ
cana-2862	84	24	analysis	analysis	NOUN
cana-2862	84	25	issn	issn	NOUN
cana-2862	84	26	:	:	PUNCT
cana-2862	84	27	1074	1074	NUM
cana-2862	84	28	-	-	PUNCT
cana-2862	84	29	133x	133x	NUM
cana-2862	84	30	vol	vol	NOUN
cana-2862	84	31	32	32	NUM
cana-2862	84	32	no	no	NOUN
cana-2862	84	33	.	.	PUNCT
cana-2862	85	1	4s	4s	NUM
cana-2862	85	2	(	(	PUNCT
cana-2862	85	3	2025	2025	NUM
cana-2862	85	4	)	)	PUNCT
cana-2862	85	5	437	437	NUM
cana-2862	86	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2862	86	2	+	+	PUNCT
cana-2862	86	3	𝐸	𝐸	ADJ
cana-2862	86	4	⋅	⋅	ADJ
cana-2862	86	5	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	86	6	,	,	PUNCT
cana-2862	86	7	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	86	8	+	+	CCONJ
cana-2862	86	9	𝐸	𝐸	PROPN
cana-2862	86	10	⋅	⋅	PROPN
cana-2862	86	11	|𝑑(𝑥2𝑛+2	|𝑑(𝑥2𝑛+2	NOUN
cana-2862	86	12	,	,	PUNCT
cana-2862	86	13	𝑥2𝑛+3)|	𝑥2𝑛+3)|	PROPN
cana-2862	86	14	⇒	⇒	NOUN
cana-2862	86	15	(	(	PUNCT
cana-2862	86	16	1	1	NUM
cana-2862	86	17	−	−	PROPN
cana-2862	86	18	𝐵	𝐵	NOUN
cana-2862	86	19	−	−	PROPN
cana-2862	86	20	𝐸)|𝑑.	𝐸)|𝑑.	SYM
cana-2862	86	21	(	(	PUNCT
cana-2862	86	22	𝑥2𝑛+2	𝑥2𝑛+2	ADJ
cana-2862	86	23	,	,	PUNCT
cana-2862	86	24	𝑥2𝑛+3)|	𝑥2𝑛+3)|	PROPN
cana-2862	86	25	≤	≤	PROPN
cana-2862	86	26	𝐴	𝐴	PROPN
cana-2862	86	27	⋅	⋅	PROPN
cana-2862	86	28	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	PROPN
cana-2862	86	29	,	,	PUNCT
cana-2862	86	30	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	86	31	+	+	CCONJ
cana-2862	86	32	𝐸	𝐸	PROPN
cana-2862	86	33	⋅	⋅	PROPN
cana-2862	86	34	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	86	35	,	,	PUNCT
cana-2862	86	36	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	86	37	⇒	⇒	VERB
cana-2862	86	38	|𝑑.	|𝑑.	NOUN
cana-2862	86	39	(	(	PUNCT
cana-2862	86	40	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	86	41	,	,	PUNCT
cana-2862	86	42	𝑥2𝑛+3)|	𝑥2𝑛+3)|	PROPN
cana-2862	86	43	≤	≤	PROPN
cana-2862	86	44	𝐴	𝐴	PROPN
cana-2862	86	45	+	+	CCONJ
cana-2862	86	46	𝐸	𝐸	PROPN
cana-2862	86	47	1	1	NUM
cana-2862	86	48	−	−	PROPN
cana-2862	86	49	𝐵	𝐵	NOUN
cana-2862	86	50	−	−	PROPN
cana-2862	86	51	𝐸	𝐸	PROPN
cana-2862	86	52	⋅	⋅	PROPN
cana-2862	86	53	|𝑑(𝑥2𝑛+1	|𝑑(𝑥2𝑛+1	NOUN
cana-2862	86	54	,	,	PUNCT
cana-2862	86	55	𝑥2𝑛+2)|	𝑥2𝑛+2)|	PROPN
cana-2862	86	56	putting	put	VERB
cana-2862	86	57	𝜆	𝜆	PRON
cana-2862	86	58	=	=	X
cana-2862	86	59	max	max	PROPN
cana-2862	86	60	{	{	PUNCT
cana-2862	86	61	𝐴	𝐴	PROPN
cana-2862	86	62	+	+	CCONJ
cana-2862	86	63	𝐷	𝐷	PROPN
cana-2862	86	64	1	1	NUM
cana-2862	86	65	−	−	PROPN
cana-2862	86	66	𝐵	𝐵	NOUN
cana-2862	86	67	−	−	PROPN
cana-2862	86	68	𝐷	𝐷	PROPN
cana-2862	86	69	,	,	PUNCT
cana-2862	86	70	𝐴	𝐴	PROPN
cana-2862	86	71	+	+	CCONJ
cana-2862	86	72	𝐸	𝐸	PROPN
cana-2862	86	73	1	1	NUM
cana-2862	86	74	−	−	PROPN
cana-2862	86	75	𝐵	𝐵	NOUN
cana-2862	86	76	−	−	PROPN
cana-2862	86	77	𝐸	𝐸	PROPN
cana-2862	86	78	}	}	PUNCT
cana-2862	86	79	we	we	PRON
cana-2862	86	80	obtain	obtain	VERB
cana-2862	86	81	that	that	SCONJ
cana-2862	86	82	|𝑑(𝑥𝑘	|𝑑(𝑥𝑘	PROPN
cana-2862	86	83	,	,	PUNCT
cana-2862	86	84	,	,	PUNCT
cana-2862	86	85	𝑥𝑘+1)|	𝑥𝑘+1)|	PRON
cana-2862	86	86	≤	≤	ADV
cana-2862	86	87	𝜆𝑘|𝑑(𝑥0	𝜆𝑘|𝑑(𝑥0	ADJ
cana-2862	86	88	,	,	PUNCT
cana-2862	86	89	𝑥1)|	𝑥1)|	PROPN
cana-2862	86	90	,	,	PUNCT
cana-2862	86	91	for	for	ADP
cana-2862	86	92	some	some	DET
cana-2862	86	93	𝑘	𝑘	PRON
cana-2862	86	94	∈	∈	NOUN
cana-2862	86	95	𝑁	𝑁	PROPN
cana-2862	86	96	(	(	PUNCT
cana-2862	86	97	3.9	3.9	NUM
cana-2862	86	98	)	)	PUNCT
cana-2862	86	99	now	now	ADV
cana-2862	86	100	,	,	PUNCT
cana-2862	86	101	for	for	ADP
cana-2862	86	102	𝑚	𝑚	PROPN
cana-2862	86	103	>	>	X
cana-2862	86	104	𝑛	𝑛	PROPN
cana-2862	86	105	and	and	CCONJ
cana-2862	86	106	by	by	ADP
cana-2862	86	107	triangular	triangular	NOUN
cana-2862	86	108	inequality	inequality	NOUN
cana-2862	86	109	,	,	PUNCT
cana-2862	86	110	we	we	PRON
cana-2862	86	111	have	have	VERB
cana-2862	86	112	𝑑(𝑥𝑛	𝑑(𝑥𝑛	NOUN
cana-2862	86	113	,	,	PUNCT
cana-2862	86	114	𝑥𝑚	𝑥𝑚	ADJ
cana-2862	86	115	)	)	PUNCT
cana-2862	86	116	≾	≾	PROPN
cana-2862	86	117	𝜙(𝑥𝑛	𝜙(𝑥𝑛	PROPN
cana-2862	86	118	,	,	PUNCT
cana-2862	86	119	𝑥𝑚)[𝑑(𝑥𝑛	𝑥𝑚)[𝑑(𝑥𝑛	PROPN
cana-2862	86	120	,	,	PUNCT
cana-2862	86	121	𝑥𝑛+1	𝑥𝑛+1	ADJ
cana-2862	86	122	)	)	PUNCT
cana-2862	86	123	+	+	CCONJ
cana-2862	86	124	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NUM
cana-2862	86	125	,	,	PUNCT
cana-2862	86	126	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	86	127	)	)	PUNCT
cana-2862	86	128	]	]	PUNCT
cana-2862	87	1	≾	≾	PROPN
cana-2862	87	2	𝜙(𝑥𝑛	𝜙(𝑥𝑛	PROPN
cana-2862	87	3	,	,	PUNCT
cana-2862	87	4	𝑥𝑚)𝜆𝑛𝑑(𝑥0	𝑥𝑚)𝜆𝑛𝑑(𝑥0	PROPN
cana-2862	87	5	,	,	PUNCT
cana-2862	87	6	𝑥1	𝑥1	NOUN
cana-2862	87	7	)	)	PUNCT
cana-2862	87	8	+	+	X
cana-2862	87	9	𝜙(𝑥𝑛	𝜙(𝑥𝑛	NUM
cana-2862	87	10	,	,	PUNCT
cana-2862	87	11	𝑥𝑚)𝑑(𝑥𝑛+1	𝑥𝑚)𝑑(𝑥𝑛+1	PROPN
cana-2862	87	12	,	,	PUNCT
cana-2862	87	13	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	87	14	)	)	PUNCT
cana-2862	87	15	≾	≾	PROPN
cana-2862	87	16	𝜙(𝑥𝑛	𝜙(𝑥𝑛	PROPN
cana-2862	87	17	,	,	PUNCT
cana-2862	87	18	𝑥𝑚)𝜆𝑛𝑑(𝑥0	𝑥𝑚)𝜆𝑛𝑑(𝑥0	PROPN
cana-2862	87	19	,	,	PUNCT
cana-2862	87	20	𝑥1	𝑥1	NOUN
cana-2862	87	21	)	)	PUNCT
cana-2862	87	22	+	+	X
cana-2862	87	23	𝜙(𝑥𝑛	𝜙(𝑥𝑛	NUM
cana-2862	87	24	,	,	PUNCT
cana-2862	87	25	𝑥𝑚)𝜙(𝑥𝑛+1	𝑥𝑚)𝜙(𝑥𝑛+1	NOUN
cana-2862	87	26	,	,	PUNCT
cana-2862	87	27	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	87	28	)	)	PUNCT
cana-2862	87	29	⋅	⋅	PROPN
cana-2862	88	1	[	[	X
cana-2862	88	2	𝑑(𝑥𝑛+1	𝑑(𝑥𝑛+1	NUM
cana-2862	88	3	,	,	PUNCT
cana-2862	88	4	𝑥𝑛+2	𝑥𝑛+2	NUM
cana-2862	88	5	)	)	PUNCT
cana-2862	88	6	+	+	CCONJ
cana-2862	88	7	𝑑(𝑥𝑛+2	𝑑(𝑥𝑛+2	PROPN
cana-2862	88	8	,	,	PUNCT
cana-2862	88	9	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	88	10	)	)	PUNCT
cana-2862	88	11	]	]	PUNCT
cana-2862	89	1	≾	≾	PROPN
cana-2862	89	2	𝜙(𝑥𝑛	𝜙(𝑥𝑛	PROPN
cana-2862	89	3	,	,	PUNCT
cana-2862	89	4	𝑥𝑚)𝜆𝑛𝑑(𝑥0	𝑥𝑚)𝜆𝑛𝑑(𝑥0	PROPN
cana-2862	89	5	,	,	PUNCT
cana-2862	89	6	𝑥1	𝑥1	NOUN
cana-2862	89	7	)	)	PUNCT
cana-2862	89	8	+	+	X
cana-2862	89	9	𝜙(𝑥𝑛	𝜙(𝑥𝑛	NUM
cana-2862	89	10	,	,	PUNCT
cana-2862	89	11	𝑥𝑚)𝜙(𝑥𝑛+1	𝑥𝑚)𝜙(𝑥𝑛+1	NOUN
cana-2862	89	12	,	,	PUNCT
cana-2862	89	13	𝑥𝑚	𝑥𝑚	ADJ
cana-2862	89	14	)	)	PUNCT
cana-2862	89	15	⋅	⋅	PROPN
cana-2862	89	16	𝜆𝑛+1	𝜆𝑛+1	NUM
cana-2862	89	17	⋅	⋅	PROPN
cana-2862	89	18	𝑑[(𝑥0	𝑑[(𝑥0	NUM
cana-2862	89	19	,	,	PUNCT
cana-2862	89	20	𝑥1	𝑥1	NOUN
cana-2862	89	21	)	)	PUNCT
cana-2862	89	22	+	+	X
cana-2862	89	23	𝜙(𝑥𝑛	𝜙(𝑥𝑛	NUM
cana-2862	89	24	,	,	PUNCT
cana-2862	89	25	𝑥𝑚)𝜙(𝑥𝑛+1	𝑥𝑚)𝜙(𝑥𝑛+1	NOUN
cana-2862	89	26	,	,	PUNCT
cana-2862	89	27	𝑥𝑚	𝑥𝑚	ADJ
cana-2862	89	28	)	)	PUNCT
cana-2862	89	29	⋅	⋅	PROPN
cana-2862	89	30	𝜆𝑛+1	𝜆𝑛+1	NUM
cana-2862	89	31	⋅	⋅	PROPN
cana-2862	89	32	𝑑(𝑥𝑛+2	𝑑(𝑥𝑛+2	PROPN
cana-2862	89	33	,	,	PUNCT
cana-2862	89	34	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	89	35	)	)	PUNCT
cana-2862	89	36	]	]	PUNCT
cana-2862	89	37	.	.	PUNCT
cana-2862	90	1	this	this	PRON
cana-2862	90	2	implies	imply	VERB
cana-2862	90	3	that	that	SCONJ
cana-2862	90	4	𝑑(𝑥𝑛	𝑑(𝑥𝑛	PROPN
cana-2862	90	5	,	,	PUNCT
cana-2862	90	6	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	90	7	)	)	PUNCT
cana-2862	90	8	≾	≾	PROPN
cana-2862	90	9	𝜙(𝑥𝑛	𝜙(𝑥𝑛	PROPN
cana-2862	90	10	,	,	PUNCT
cana-2862	90	11	𝑥𝑚)𝜆𝑛𝑑(𝑥0	𝑥𝑚)𝜆𝑛𝑑(𝑥0	PROPN
cana-2862	90	12	,	,	PUNCT
cana-2862	90	13	𝑥1	𝑥1	NOUN
cana-2862	90	14	)	)	PUNCT
cana-2862	90	15	+	+	X
cana-2862	90	16	𝜙(𝑥𝑛	𝜙(𝑥𝑛	NUM
cana-2862	90	17	,	,	PUNCT
cana-2862	90	18	𝑥𝑚)𝜙(𝑥𝑛+1	𝑥𝑚)𝜙(𝑥𝑛+1	NOUN
cana-2862	90	19	,	,	PUNCT
cana-2862	90	20	𝑥𝑚)𝜆𝑛+1	𝑥𝑚)𝜆𝑛+1	PROPN
cana-2862	90	21	⋅	⋅	PROPN
cana-2862	90	22	𝑑(𝑥0	𝑑(𝑥0	NOUN
cana-2862	90	23	,	,	PUNCT
cana-2862	90	24	𝑥1	𝑥1	NOUN
cana-2862	90	25	)	)	PUNCT
cana-2862	90	26	+	+	CCONJ
cana-2862	90	27	⋯	⋯	ADP
cana-2862	90	28	+	+	ADJ
cana-2862	90	29	𝜙(𝑥𝑛	𝜙(𝑥𝑛	PROPN
cana-2862	90	30	,	,	PUNCT
cana-2862	90	31	𝑥𝑚)𝜙(𝑥𝑛+1	𝑥𝑚)𝜙(𝑥𝑛+1	NOUN
cana-2862	90	32	,	,	PUNCT
cana-2862	90	33	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	90	34	)	)	PUNCT
cana-2862	90	35	…	…	PUNCT
cana-2862	90	36	𝜙(𝑥𝑚−1	𝜙(𝑥𝑚−1	NUM
cana-2862	90	37	,	,	PUNCT
cana-2862	90	38	𝑥𝑚)𝜆𝑚−1	𝑥𝑚)𝜆𝑚−1	PROPN
cana-2862	90	39	⋅	⋅	PROPN
cana-2862	90	40	𝑑(𝑥0	𝑑(𝑥0	NOUN
cana-2862	90	41	,	,	PUNCT
cana-2862	90	42	𝑥1	𝑥1	PROPN
cana-2862	90	43	)	)	PUNCT
cana-2862	90	44	(	(	PUNCT
cana-2862	90	45	3.10	3.10	NUM
cana-2862	90	46	)	)	PUNCT
cana-2862	90	47	which	which	PRON
cana-2862	90	48	implies	imply	VERB
cana-2862	90	49	that	that	SCONJ
cana-2862	90	50	|𝑑(𝑥𝑛	|𝑑(𝑥𝑛	NOUN
cana-2862	90	51	,	,	PUNCT
cana-2862	90	52	𝑥𝑚)|	𝑥𝑚)|	PROPN
cana-2862	90	53	≤	≤	NUM
cana-2862	90	54	|𝑑(𝑥0	|𝑑(𝑥0	NUM
cana-2862	90	55	,	,	PUNCT
cana-2862	90	56	𝑥1)|[𝜙(𝑥𝑛	𝑥1)|[𝜙(𝑥𝑛	PROPN
cana-2862	90	57	,	,	PUNCT
cana-2862	90	58	𝑥𝑚)𝜆𝑛	𝑥𝑚)𝜆𝑛	PUNCT
cana-2862	90	59	+	+	NUM
cana-2862	90	60	𝜙(𝑥𝑛	𝜙(𝑥𝑛	NUM
cana-2862	90	61	,	,	PUNCT
cana-2862	90	62	𝑥𝑚)𝜙(𝑥𝑛+1	𝑥𝑚)𝜙(𝑥𝑛+1	NOUN
cana-2862	90	63	,	,	PUNCT
cana-2862	90	64	𝑥𝑚)𝜆𝑛+1	𝑥𝑚)𝜆𝑛+1	NOUN
cana-2862	90	65	+	+	CCONJ
cana-2862	90	66	⋯	⋯	ADP
cana-2862	90	67	+	+	NOUN
cana-2862	90	68	𝜙(𝑥𝑛	𝜙(𝑥𝑛	PROPN
cana-2862	90	69	,	,	PUNCT
cana-2862	90	70	𝑥𝑚)𝜙(𝑥𝑛+1	𝑥𝑚)𝜙(𝑥𝑛+1	NOUN
cana-2862	90	71	,	,	PUNCT
cana-2862	90	72	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	90	73	)	)	PUNCT
cana-2862	90	74	…	…	PUNCT
cana-2862	90	75	𝜙(𝑥𝑚−1	𝜙(𝑥𝑚−1	NOUN
cana-2862	90	76	,	,	PUNCT
cana-2862	90	77	𝑥𝑚)𝜆𝑚−1	𝑥𝑚)𝜆𝑚−1	PROPN
cana-2862	90	78	]	]	PUNCT
cana-2862	90	79	.	.	PUNCT
cana-2862	91	1	since	since	SCONJ
cana-2862	91	2	limit	limit	NOUN
cana-2862	91	3	𝜙(𝑥𝑛	𝜙(𝑥𝑛	NUM
cana-2862	91	4	,	,	PUNCT
cana-2862	91	5	𝑥𝑚)𝜆	𝑥𝑚)𝜆	NOUN
cana-2862	91	6	<	<	X
cana-2862	91	7	1	1	NUM
cana-2862	91	8	,	,	PUNCT
cana-2862	91	9	𝑛	𝑛	PROPN
cana-2862	91	10	,	,	PUNCT
cana-2862	91	11	𝑚	𝑚	X
cana-2862	91	12	→	→	SYM
cana-2862	91	13	∞	∞	PROPN
cana-2862	91	14	,	,	PUNCT
cana-2862	91	15	so	so	CCONJ
cana-2862	91	16	the	the	DET
cana-2862	91	17	series	series	NOUN
cana-2862	91	18	∑	∑	PUNCT
cana-2862	91	19	  	  	SPACE
cana-2862	91	20	∞	∞	NUM
cana-2862	91	21	𝑛=1	𝑛=1	NOUN
cana-2862	91	22	𝜆𝑛	𝜆𝑛	ADP
cana-2862	91	23	∏	∏	PROPN
cana-2862	91	24	  	  	SPACE
cana-2862	91	25	𝐾	𝐾	PROPN
cana-2862	91	26	𝑖=1	𝑖=1	PROPN
cana-2862	91	27	𝜙(𝑥𝑖	𝜙(𝑥𝑖	PROPN
cana-2862	91	28	,	,	PUNCT
cana-2862	91	29	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	91	30	)	)	PUNCT
cana-2862	91	31	converges	converge	NOUN
cana-2862	91	32	by	by	ADP
cana-2862	91	33	ratio	ratio	NOUN
cana-2862	91	34	test	test	NOUN
cana-2862	91	35	for	for	ADP
cana-2862	91	36	each	each	DET
cana-2862	91	37	𝑚	𝑚	ADP
cana-2862	91	38	∈	∈	NOUN
cana-2862	91	39	𝑁	𝑁	PROPN
cana-2862	91	40	let	let	VERB
cana-2862	91	41	𝑥	𝑥	X
cana-2862	91	42	=	=	PUNCT
cana-2862	91	43	∑	∑	PART
cana-2862	91	44	 	 	SPACE
cana-2862	91	45	∞	∞	NUM
cana-2862	91	46	𝑛=1	𝑛=1	NOUN
cana-2862	91	47	𝜆𝑛	𝜆𝑛	ADP
cana-2862	91	48	∏	∏	PROPN
cana-2862	91	49	 	 	SPACE
cana-2862	91	50	𝐾	𝐾	PROPN
cana-2862	91	51	𝑖=1	𝑖=1	PROPN
cana-2862	91	52	𝜙(𝑥𝑖	𝜙(𝑥𝑖	PROPN
cana-2862	91	53	,	,	PUNCT
cana-2862	91	54	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	91	55	)	)	PUNCT
cana-2862	91	56	,	,	PUNCT
cana-2862	91	57	𝑥𝑛	𝑥𝑛	PROPN
cana-2862	91	58	=	=	PUNCT
cana-2862	91	59	∑	∑	PART
cana-2862	91	60	 	 	SPACE
cana-2862	91	61	𝑛	𝑛	PROPN
cana-2862	91	62	𝑗=1	𝑗=1	PROPN
cana-2862	91	63	𝜆𝑗	𝜆𝑗	X
cana-2862	91	64	∏	∏	PROPN
cana-2862	91	65	 	 	SPACE
cana-2862	91	66	𝐾	𝐾	PROPN
cana-2862	91	67	𝑖=1	𝑖=1	PROPN
cana-2862	91	68	𝜙(𝑥𝑖	𝜙(𝑥𝑖	PROPN
cana-2862	91	69	,	,	PUNCT
cana-2862	91	70	𝑥𝑚	𝑥𝑚	NOUN
cana-2862	91	71	)	)	PUNCT
cana-2862	91	72	,	,	PUNCT
cana-2862	91	73	(	(	PUNCT
cana-2862	91	74	3.11	3.11	NUM
cana-2862	91	75	)	)	PUNCT
cana-2862	91	76	thus	thus	ADV
cana-2862	91	77	for	for	ADP
cana-2862	91	78	𝑚	𝑚	PROPN
cana-2862	91	79	>	>	SYM
cana-2862	91	80	𝑛	𝑛	PROPN
cana-2862	91	81	,	,	PUNCT
cana-2862	91	82	the	the	DET
cana-2862	91	83	above	above	ADJ
cana-2862	91	84	inequality	inequality	NOUN
cana-2862	91	85	can	can	AUX
cana-2862	91	86	be	be	AUX
cana-2862	91	87	written	write	VERB
cana-2862	91	88	as	as	ADP
cana-2862	91	89	|𝑑(𝑥𝑛	|𝑑(𝑥𝑛	NOUN
cana-2862	91	90	,	,	PUNCT
cana-2862	91	91	𝑥𝑚)|	𝑥𝑚)|	NOUN
cana-2862	91	92	≤	≤	NUM
cana-2862	91	93	|𝑑(𝑥0	|𝑑(𝑥0	NUM
cana-2862	91	94	,	,	PUNCT
cana-2862	91	95	𝑥1)|	𝑥1)|	PROPN
cana-2862	91	96	⋅	⋅	PROPN
cana-2862	91	97	|𝑥𝑚−1	|𝑥𝑚−1	X
cana-2862	92	1	−	−	PROPN
cana-2862	93	1	𝑥𝑛|	𝑥𝑛|	PROPN
cana-2862	93	2	.	.	PUNCT
cana-2862	94	1	now	now	ADV
cana-2862	94	2	,	,	PUNCT
cana-2862	94	3	by	by	ADP
cana-2862	94	4	taking	take	VERB
cana-2862	94	5	the	the	DET
cana-2862	94	6	limit	limit	NOUN
cana-2862	94	7	as	as	ADP
cana-2862	94	8	𝑛	𝑛	PROPN
cana-2862	94	9	,	,	PUNCT
cana-2862	94	10	𝑚	𝑚	X
cana-2862	94	11	→	→	SYM
cana-2862	94	12	∞	∞	NUM
cana-2862	94	13	we	we	PRON
cana-2862	94	14	get	get	VERB
cana-2862	94	15	|𝑑(𝑥𝑛	|𝑑(𝑥𝑛	NOUN
cana-2862	94	16	,	,	PUNCT
cana-2862	94	17	𝑥𝑚)|	𝑥𝑚)|	NOUN
cana-2862	94	18	→	→	SYM
cana-2862	94	19	0	0	NUM
cana-2862	94	20	as	as	ADP
cana-2862	94	21	𝑛	𝑛	PROPN
cana-2862	94	22	,	,	PUNCT
cana-2862	94	23	𝑚	𝑚	X
cana-2862	94	24	→	→	SYM
cana-2862	94	25	∞	∞	NUM
cana-2862	94	26	thus	thus	ADV
cana-2862	94	27	{	{	PUNCT
cana-2862	94	28	𝑥𝑛	𝑥𝑛	NOUN
cana-2862	94	29	}	}	PUNCT
cana-2862	94	30	is	be	AUX
cana-2862	94	31	a	a	DET
cana-2862	94	32	cauchy	cauchy	ADJ
cana-2862	94	33	sequence	sequence	NOUN
cana-2862	94	34	in	in	ADP
cana-2862	94	35	𝑋.	𝑋.	PROPN
cana-2862	94	36	but	but	CCONJ
cana-2862	94	37	𝑋	𝑋	PROPN
cana-2862	94	38	is	be	AUX
cana-2862	94	39	a	a	DET
cana-2862	94	40	complete	complete	ADJ
cana-2862	94	41	metric	metric	ADJ
cana-2862	94	42	space	space	NOUN
cana-2862	94	43	,	,	PUNCT
cana-2862	94	44	so	so	SCONJ
cana-2862	94	45	this	this	DET
cana-2862	94	46	cauchy	cauchy	ADJ
cana-2862	94	47	sequence	sequence	NOUN
cana-2862	94	48	convergent	convergent	NOUN
cana-2862	94	49	and	and	CCONJ
cana-2862	94	50	say	say	VERB
cana-2862	94	51	converges	converge	NOUN
cana-2862	94	52	to	to	ADP
cana-2862	94	53	𝑥.	𝑥.	ADJ
cana-2862	94	54	communications	communication	NOUN
cana-2862	94	55	on	on	ADP
cana-2862	94	56	applied	apply	VERB
cana-2862	94	57	nonlinear	nonlinear	ADJ
cana-2862	94	58	analysis	analysis	NOUN
cana-2862	94	59	issn	issn	NOUN
cana-2862	94	60	:	:	PUNCT
cana-2862	94	61	1074	1074	NUM
cana-2862	94	62	-	-	PUNCT
cana-2862	94	63	133x	133x	NUM
cana-2862	94	64	vol	vol	NOUN
cana-2862	94	65	32	32	NUM
cana-2862	94	66	no	no	NOUN
cana-2862	94	67	.	.	PUNCT
cana-2862	95	1	4s	4s	NUM
cana-2862	95	2	(	(	PUNCT
cana-2862	95	3	2025	2025	NUM
cana-2862	95	4	)	)	PUNCT
cana-2862	95	5	438	438	NUM
cana-2862	95	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2862	95	7	i.e.	i.e.	X
cana-2862	95	8	,	,	PUNCT
cana-2862	95	9	lim𝑛→∞	lim𝑛→∞	NOUN
cana-2862	95	10	 	 	SPACE
cana-2862	95	11	𝑥𝑛	𝑥𝑛	PROPN
cana-2862	96	1	=	=	PUNCT
cana-2862	96	2	𝑥.	𝑥.	ADV
cana-2862	96	3	now	now	ADV
cana-2862	96	4	,	,	PUNCT
cana-2862	96	5	we	we	PRON
cana-2862	96	6	prove	prove	VERB
cana-2862	96	7	that	that	SCONJ
cana-2862	96	8	𝑓𝑥	𝑓𝑥	ADP
cana-2862	96	9	=	=	PUNCT
cana-2862	96	10	𝑥	𝑥	X
cana-2862	96	11	i.e.	i.e.	X
cana-2862	96	12	,	,	PUNCT
cana-2862	96	13	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	96	14	,	,	PUNCT
cana-2862	96	15	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	96	16	)	)	PUNCT
cana-2862	96	17	=	=	SYM
cana-2862	96	18	0	0	X
cana-2862	96	19	.	.	PUNCT
cana-2862	97	1	on	on	ADP
cana-2862	97	2	the	the	DET
cana-2862	97	3	contrary	contrary	NOUN
cana-2862	97	4	,	,	PUNCT
cana-2862	97	5	suppose	suppose	VERB
cana-2862	97	6	that	that	SCONJ
cana-2862	97	7	0	0	NUM
cana-2862	97	8	≺	≺	NOUN
cana-2862	97	9	𝑣	𝑣	X
cana-2862	97	10	=	=	SYM
cana-2862	97	11	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	97	12	,	,	PUNCT
cana-2862	97	13	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	97	14	)	)	PUNCT
cana-2862	97	15	=	=	SYM
cana-2862	97	16	𝑑(𝑓𝑥	𝑑(𝑓𝑥	NOUN
cana-2862	97	17	,	,	PUNCT
cana-2862	97	18	𝑥	𝑥	NOUN
cana-2862	97	19	)	)	PUNCT
cana-2862	97	20	,	,	PUNCT
cana-2862	97	21	0	0	NUM
cana-2862	97	22	≺	≺	NOUN
cana-2862	97	23	𝑣	𝑣	ADP
cana-2862	97	24	≾	≾	PROPN
cana-2862	97	25	𝑑(𝑓𝑥	𝑑(𝑓𝑥	NOUN
cana-2862	97	26	,	,	PUNCT
cana-2862	97	27	𝑥	𝑥	NOUN
cana-2862	97	28	)	)	PUNCT
cana-2862	97	29	≾	≾	NOUN
cana-2862	97	30	𝜙(𝑓𝑥	𝜙(𝑓𝑥	NOUN
cana-2862	97	31	,	,	PUNCT
cana-2862	97	32	𝑥)[𝑑(𝑓𝑥	𝑥)[𝑑(𝑓𝑥	NOUN
cana-2862	97	33	,	,	PUNCT
cana-2862	97	34	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	97	35	)	)	PUNCT
cana-2862	97	36	+	+	CCONJ
cana-2862	97	37	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	97	38	,	,	PUNCT
cana-2862	97	39	𝑥	𝑥	NOUN
cana-2862	97	40	)	)	PUNCT
cana-2862	97	41	]	]	PUNCT
cana-2862	97	42	≾	≾	PROPN
cana-2862	97	43	𝜙(𝑓𝑥	𝜙(𝑓𝑥	PROPN
cana-2862	97	44	,	,	PUNCT
cana-2862	97	45	𝑥	𝑥	NOUN
cana-2862	97	46	)	)	PUNCT
cana-2862	97	47	⋅	⋅	PROPN
cana-2862	98	1	[	[	X
cana-2862	98	2	𝑑(𝑓𝑥	𝑑(𝑓𝑥	X
cana-2862	98	3	,	,	PUNCT
cana-2862	98	4	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	98	5	)	)	PUNCT
cana-2862	98	6	+	+	CCONJ
cana-2862	98	7	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	98	8	,	,	PUNCT
cana-2862	98	9	𝑥	𝑥	NOUN
cana-2862	98	10	)	)	PUNCT
cana-2862	98	11	]	]	PUNCT
cana-2862	98	12	i.e.	i.e.	X
cana-2862	98	13	,	,	PUNCT
cana-2862	98	14	0	0	NUM
cana-2862	98	15	≺	≺	NOUN
cana-2862	98	16	𝑣	𝑣	ADP
cana-2862	98	17	≾	≾	PROPN
cana-2862	98	18	𝜙(𝑓𝑥	𝜙(𝑓𝑥	NUM
cana-2862	98	19	,	,	PUNCT
cana-2862	98	20	𝑥	𝑥	NOUN
cana-2862	98	21	)	)	PUNCT
cana-2862	99	1	[	[	X
cana-2862	99	2	𝐴	𝐴	PROPN
cana-2862	99	3	⋅	⋅	PROPN
cana-2862	99	4	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	99	5	,	,	PUNCT
cana-2862	99	6	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	99	7	)	)	PUNCT
cana-2862	99	8	+	+	CCONJ
cana-2862	99	9	𝐵	𝐵	PROPN
cana-2862	99	10	⋅	⋅	PROPN
cana-2862	99	11	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	99	12	,	,	PUNCT
cana-2862	99	13	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	99	14	)	)	PUNCT
cana-2862	99	15	⋅	⋅	PROPN
cana-2862	99	16	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	99	17	,	,	PUNCT
cana-2862	99	18	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	99	19	)	)	PUNCT
cana-2862	99	20	1	1	NUM
cana-2862	99	21	+	+	CCONJ
cana-2862	99	22	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	99	23	,	,	PUNCT
cana-2862	99	24	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	99	25	)	)	PUNCT
cana-2862	99	26	+	+	PROPN
cana-2862	99	27	𝐶	𝐶	PROPN
cana-2862	99	28	⋅	⋅	PROPN
cana-2862	99	29	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	99	30	,	,	PUNCT
cana-2862	99	31	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	99	32	)	)	PUNCT
cana-2862	99	33	⋅	⋅	PROPN
cana-2862	99	34	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	99	35	,	,	PUNCT
cana-2862	99	36	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	99	37	)	)	PUNCT
cana-2862	99	38	1	1	NUM
cana-2862	99	39	+	+	CCONJ
cana-2862	99	40	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	99	41	,	,	PUNCT
cana-2862	99	42	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	99	43	)	)	PUNCT
cana-2862	99	44	+	+	NUM
cana-2862	99	45	𝐷	𝐷	PROPN
cana-2862	99	46	⋅	⋅	PROPN
cana-2862	99	47	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	99	48	,	,	PUNCT
cana-2862	99	49	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	99	50	)	)	PUNCT
cana-2862	99	51	⋅	⋅	PROPN
cana-2862	99	52	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	99	53	,	,	PUNCT
cana-2862	99	54	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	99	55	)	)	PUNCT
cana-2862	99	56	1	1	NUM
cana-2862	99	57	+	+	CCONJ
cana-2862	99	58	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	99	59	,	,	PUNCT
cana-2862	99	60	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	99	61	)	)	PUNCT
cana-2862	99	62	+	+	NOUN
cana-2862	99	63	𝐸	𝐸	PROPN
cana-2862	99	64	⋅	⋅	PROPN
cana-2862	99	65	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	99	66	,	,	PUNCT
cana-2862	99	67	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	99	68	)	)	PUNCT
cana-2862	99	69	⋅	⋅	PROPN
cana-2862	99	70	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	99	71	,	,	PUNCT
cana-2862	99	72	𝑔𝑥2𝑛+1	𝑔𝑥2𝑛+1	PROPN
cana-2862	99	73	)	)	PUNCT
cana-2862	99	74	1	1	NUM
cana-2862	99	75	+	+	CCONJ
cana-2862	99	76	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	99	77	,	,	PUNCT
cana-2862	99	78	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	99	79	)	)	PUNCT
cana-2862	99	80	+	+	CCONJ
cana-2862	99	81	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	99	82	,	,	PUNCT
cana-2862	99	83	𝑥	𝑥	NOUN
cana-2862	99	84	)	)	PUNCT
cana-2862	99	85	]	]	PUNCT
cana-2862	99	86	i.e.	i.e.	X
cana-2862	99	87	0	0	NUM
cana-2862	99	88	≺	≺	NOUN
cana-2862	99	89	𝑣	𝑣	ADP
cana-2862	99	90	≾	≾	PROPN
cana-2862	99	91	𝜙(𝑓𝑥	𝜙(𝑓𝑥	NUM
cana-2862	99	92	,	,	PUNCT
cana-2862	99	93	𝑥	𝑥	NOUN
cana-2862	99	94	)	)	PUNCT
cana-2862	100	1	[	[	X
cana-2862	100	2	𝐴	𝐴	PROPN
cana-2862	100	3	⋅	⋅	PROPN
cana-2862	100	4	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	100	5	,	,	PUNCT
cana-2862	100	6	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	100	7	)	)	PUNCT
cana-2862	100	8	+	+	CCONJ
cana-2862	100	9	𝐵	𝐵	PROPN
cana-2862	100	10	⋅	⋅	PROPN
cana-2862	100	11	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	100	12	,	,	PUNCT
cana-2862	100	13	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	100	14	)	)	PUNCT
cana-2862	100	15	⋅	⋅	PROPN
cana-2862	100	16	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	100	17	,	,	PUNCT
cana-2862	100	18	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	100	19	)	)	PUNCT
cana-2862	100	20	1	1	NUM
cana-2862	100	21	+	+	CCONJ
cana-2862	100	22	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	100	23	,	,	PUNCT
cana-2862	100	24	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	100	25	)	)	PUNCT
cana-2862	100	26	+	+	PROPN
cana-2862	100	27	𝐶	𝐶	PROPN
cana-2862	100	28	⋅	⋅	PROPN
cana-2862	100	29	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	100	30	,	,	PUNCT
cana-2862	100	31	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	100	32	)	)	PUNCT
cana-2862	100	33	⋅	⋅	PROPN
cana-2862	100	34	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	100	35	,	,	PUNCT
cana-2862	100	36	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	100	37	)	)	PUNCT
cana-2862	100	38	1	1	NUM
cana-2862	100	39	+	+	CCONJ
cana-2862	100	40	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	100	41	,	,	PUNCT
cana-2862	100	42	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	100	43	)	)	PUNCT
cana-2862	100	44	+	+	NUM
cana-2862	100	45	𝐷	𝐷	PROPN
cana-2862	100	46	⋅	⋅	PROPN
cana-2862	100	47	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	100	48	,	,	PUNCT
cana-2862	100	49	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	100	50	)	)	PUNCT
cana-2862	100	51	⋅	⋅	PROPN
cana-2862	100	52	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	100	53	,	,	PUNCT
cana-2862	100	54	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	100	55	)	)	PUNCT
cana-2862	100	56	1	1	NUM
cana-2862	100	57	+	+	CCONJ
cana-2862	100	58	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	100	59	,	,	PUNCT
cana-2862	100	60	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	100	61	)	)	PUNCT
cana-2862	100	62	+	+	NOUN
cana-2862	100	63	𝐸	𝐸	PROPN
cana-2862	100	64	⋅	⋅	PROPN
cana-2862	100	65	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	100	66	,	,	PUNCT
cana-2862	100	67	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	100	68	)	)	PUNCT
cana-2862	100	69	⋅	⋅	PROPN
cana-2862	100	70	𝑑(𝑥2𝑛+1	𝑑(𝑥2𝑛+1	PROPN
cana-2862	100	71	,	,	PUNCT
cana-2862	100	72	𝑥2𝑛+2	𝑥2𝑛+2	NOUN
cana-2862	100	73	)	)	PUNCT
cana-2862	100	74	1	1	NUM
cana-2862	100	75	+	+	CCONJ
cana-2862	100	76	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	100	77	,	,	PUNCT
cana-2862	100	78	𝑥2𝑛+1	𝑥2𝑛+1	NOUN
cana-2862	100	79	)	)	PUNCT
cana-2862	100	80	+	+	CCONJ
cana-2862	100	81	𝑑(𝑥2𝑛+2	𝑑(𝑥2𝑛+2	X
cana-2862	100	82	,	,	PUNCT
cana-2862	100	83	𝑥	𝑥	NOUN
cana-2862	100	84	)	)	PUNCT
cana-2862	100	85	]	]	PUNCT
cana-2862	100	86	.	.	PUNCT
cana-2862	101	1	now	now	ADV
cana-2862	101	2	,	,	PUNCT
cana-2862	101	3	taking	take	VERB
cana-2862	101	4	the	the	DET
cana-2862	101	5	limit	limit	NOUN
cana-2862	101	6	as	as	ADP
cana-2862	101	7	𝑛	𝑛	PROPN
cana-2862	101	8	→	→	SYM
cana-2862	101	9	∞	∞	PROPN
cana-2862	101	10	we	we	PRON
cana-2862	101	11	get	get	VERB
cana-2862	101	12	0	0	NUM
cana-2862	101	13	≺	≺	NOUN
cana-2862	101	14	𝑣	𝑣	ADP
cana-2862	101	15	≺	≺	NOUN
cana-2862	101	16	0	0	NUM
cana-2862	101	17	,	,	PUNCT
cana-2862	101	18	which	which	PRON
cana-2862	101	19	implies	imply	VERB
cana-2862	101	20	that	that	SCONJ
cana-2862	101	21	𝑣	𝑣	ADP
cana-2862	101	22	=	=	SYM
cana-2862	101	23	0	0	PROPN
cana-2862	101	24	.	.	PUNCT
cana-2862	102	1	i.e.	i.e.	X
cana-2862	102	2	𝑑(𝑓𝑥	𝑑(𝑓𝑥	PROPN
cana-2862	102	3	,	,	PUNCT
cana-2862	102	4	𝑥	𝑥	NOUN
cana-2862	102	5	)	)	PUNCT
cana-2862	102	6	=	=	SYM
cana-2862	102	7	0	0	NUM
cana-2862	102	8	⇒	⇒	NOUN
cana-2862	102	9	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	102	10	=	=	PUNCT
cana-2862	102	11	𝑥.	𝑥.	ADV
cana-2862	102	12	similarly	similarly	ADV
cana-2862	102	13	,	,	PUNCT
cana-2862	102	14	we	we	PRON
cana-2862	102	15	can	can	AUX
cana-2862	102	16	prove	prove	VERB
cana-2862	102	17	that	that	SCONJ
cana-2862	102	18	𝑔𝑥	𝑔𝑥	PROPN
cana-2862	102	19	=	=	PUNCT
cana-2862	102	20	𝑥.	𝑥.	ADV
cana-2862	102	21	now	now	ADV
cana-2862	102	22	,	,	PUNCT
cana-2862	102	23	we	we	PRON
cana-2862	102	24	can	can	AUX
cana-2862	102	25	show	show	VERB
cana-2862	102	26	that	that	SCONJ
cana-2862	102	27	𝑓	𝑓	PROPN
cana-2862	102	28	and	and	CCONJ
cana-2862	102	29	𝑔	𝑔	PROPN
cana-2862	102	30	have	have	VERB
cana-2862	102	31	unique	unique	ADJ
cana-2862	102	32	common	common	ADJ
cana-2862	102	33	fixed	fix	VERB
cana-2862	102	34	point	point	NOUN
cana-2862	102	35	.	.	PUNCT
cana-2862	103	1	on	on	ADP
cana-2862	103	2	the	the	DET
cana-2862	103	3	contrary	contrary	NOUN
cana-2862	103	4	,	,	PUNCT
cana-2862	103	5	suppose	suppose	VERB
cana-2862	103	6	that	that	SCONJ
cana-2862	103	7	𝑥	𝑥	PROPN
cana-2862	103	8	and	and	CCONJ
cana-2862	103	9	𝑦	𝑦	NOUN
cana-2862	103	10	be	be	AUX
cana-2862	103	11	two	two	NUM
cana-2862	103	12	common	common	ADJ
cana-2862	103	13	fixed	fix	VERB
cana-2862	103	14	point	point	NOUN
cana-2862	103	15	of	of	ADP
cana-2862	103	16	𝑓	𝑓	PRON
cana-2862	103	17	and	and	CCONJ
cana-2862	103	18	𝑔.	𝑔.	NOUN
cana-2862	103	19	now	now	ADV
cana-2862	103	20	,	,	PUNCT
cana-2862	103	21	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	103	22	,	,	PUNCT
cana-2862	103	23	𝑦	𝑦	X
cana-2862	103	24	)	)	PUNCT
cana-2862	103	25	=	=	SYM
cana-2862	103	26	𝑑(𝑓𝑥	𝑑(𝑓𝑥	PROPN
cana-2862	103	27	,	,	PUNCT
cana-2862	103	28	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	103	29	)	)	PUNCT
cana-2862	103	30	≾	≾	PROPN
cana-2862	103	31	𝐴	𝐴	PROPN
cana-2862	103	32	⋅	⋅	PROPN
cana-2862	103	33	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	103	34	,	,	PUNCT
cana-2862	103	35	𝑦	𝑦	NOUN
cana-2862	103	36	)	)	PUNCT
cana-2862	103	37	+	+	CCONJ
cana-2862	103	38	𝐵	𝐵	PROPN
cana-2862	103	39	⋅	⋅	PROPN
cana-2862	103	40	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	103	41	,	,	PUNCT
cana-2862	103	42	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	103	43	)	)	PUNCT
cana-2862	103	44	⋅	⋅	PROPN
cana-2862	103	45	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	103	46	,	,	PUNCT
cana-2862	103	47	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	103	48	)	)	PUNCT
cana-2862	103	49	1	1	NUM
cana-2862	104	1	+	+	CCONJ
cana-2862	104	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	104	3	,	,	PUNCT
cana-2862	104	4	𝑦	𝑦	NOUN
cana-2862	104	5	)	)	PUNCT
cana-2862	104	6	+	+	CCONJ
cana-2862	104	7	𝐶	𝐶	PROPN
cana-2862	104	8	⋅	⋅	PROPN
cana-2862	104	9	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	104	10	,	,	PUNCT
cana-2862	104	11	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	104	12	)	)	PUNCT
cana-2862	104	13	⋅	⋅	PROPN
cana-2862	104	14	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	104	15	,	,	PUNCT
cana-2862	104	16	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	104	17	)	)	PUNCT
cana-2862	104	18	1	1	NUM
cana-2862	105	1	+	+	CCONJ
cana-2862	105	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	105	3	,	,	PUNCT
cana-2862	105	4	𝑦	𝑦	NOUN
cana-2862	105	5	)	)	PUNCT
cana-2862	105	6	+	+	ADJ
cana-2862	105	7	𝐷	𝐷	PROPN
cana-2862	105	8	⋅	⋅	PROPN
cana-2862	105	9	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	105	10	,	,	PUNCT
cana-2862	105	11	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	105	12	)	)	PUNCT
cana-2862	105	13	⋅	⋅	PROPN
cana-2862	105	14	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	105	15	,	,	PUNCT
cana-2862	105	16	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	105	17	)	)	PUNCT
cana-2862	105	18	1	1	NUM
cana-2862	105	19	+	+	CCONJ
cana-2862	105	20	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	105	21	,	,	PUNCT
cana-2862	105	22	𝑦	𝑦	NOUN
cana-2862	105	23	)	)	PUNCT
cana-2862	105	24	+	+	CCONJ
cana-2862	105	25	𝐸	𝐸	PROPN
cana-2862	105	26	⋅	⋅	PROPN
cana-2862	105	27	𝑑(𝑦	𝑑(𝑦	NOUN
cana-2862	105	28	,	,	PUNCT
cana-2862	105	29	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	105	30	)	)	PUNCT
cana-2862	105	31	⋅	⋅	PROPN
cana-2862	105	32	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	105	33	,	,	PUNCT
cana-2862	105	34	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	105	35	)	)	PUNCT
cana-2862	105	36	1	1	NUM
cana-2862	106	1	+	+	CCONJ
cana-2862	106	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	106	3	,	,	PUNCT
cana-2862	106	4	𝑦	𝑦	X
cana-2862	106	5	)	)	PUNCT
cana-2862	106	6	communications	communication	NOUN
cana-2862	106	7	on	on	ADP
cana-2862	106	8	applied	apply	VERB
cana-2862	106	9	nonlinear	nonlinear	ADJ
cana-2862	106	10	analysis	analysis	NOUN
cana-2862	106	11	issn	issn	NOUN
cana-2862	106	12	:	:	PUNCT
cana-2862	106	13	1074	1074	NUM
cana-2862	106	14	-	-	PUNCT
cana-2862	106	15	133x	133x	NUM
cana-2862	106	16	vol	vol	NOUN
cana-2862	106	17	32	32	NUM
cana-2862	106	18	no	no	NOUN
cana-2862	106	19	.	.	PUNCT
cana-2862	107	1	4s	4s	NUM
cana-2862	107	2	(	(	PUNCT
cana-2862	107	3	2025	2025	NUM
cana-2862	107	4	)	)	PUNCT
cana-2862	107	5	439	439	NUM
cana-2862	107	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2862	107	7	i.e.	i.e.	X
cana-2862	107	8	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	107	9	,	,	PUNCT
cana-2862	107	10	𝑦	𝑦	X
cana-2862	107	11	)	)	PUNCT
cana-2862	107	12	=	=	SYM
cana-2862	107	13	𝑑(𝑓𝑥	𝑑(𝑓𝑥	PROPN
cana-2862	107	14	,	,	PUNCT
cana-2862	107	15	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	107	16	)	)	PUNCT
cana-2862	107	17	≾	≾	PROPN
cana-2862	107	18	𝐴	𝐴	PROPN
cana-2862	107	19	⋅	⋅	PROPN
cana-2862	107	20	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	107	21	,	,	PUNCT
cana-2862	107	22	𝑦	𝑦	NOUN
cana-2862	107	23	)	)	PUNCT
cana-2862	107	24	+	+	CCONJ
cana-2862	107	25	𝐵	𝐵	PROPN
cana-2862	107	26	⋅	⋅	PROPN
cana-2862	107	27	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	107	28	,	,	PUNCT
cana-2862	107	29	𝑥	𝑥	NOUN
cana-2862	107	30	)	)	PUNCT
cana-2862	107	31	⋅	⋅	PROPN
cana-2862	107	32	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	107	33	,	,	PUNCT
cana-2862	107	34	𝑦	𝑦	NOUN
cana-2862	107	35	)	)	PUNCT
cana-2862	107	36	1	1	NUM
cana-2862	107	37	+	+	CCONJ
cana-2862	107	38	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	107	39	,	,	PUNCT
cana-2862	107	40	𝑦	𝑦	NOUN
cana-2862	107	41	)	)	PUNCT
cana-2862	107	42	+	+	CCONJ
cana-2862	107	43	𝐶	𝐶	PROPN
cana-2862	107	44	⋅	⋅	PROPN
cana-2862	107	45	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	107	46	,	,	PUNCT
cana-2862	107	47	𝑥	𝑥	NOUN
cana-2862	107	48	)	)	PUNCT
cana-2862	107	49	⋅	⋅	PROPN
cana-2862	107	50	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	107	51	,	,	PUNCT
cana-2862	107	52	𝑦	𝑦	NOUN
cana-2862	107	53	)	)	PUNCT
cana-2862	107	54	1	1	NUM
cana-2862	108	1	+	+	CCONJ
cana-2862	108	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	108	3	,	,	PUNCT
cana-2862	108	4	𝑦	𝑦	NOUN
cana-2862	108	5	)	)	PUNCT
cana-2862	108	6	+	+	ADJ
cana-2862	108	7	𝐷	𝐷	PROPN
cana-2862	108	8	⋅	⋅	PROPN
cana-2862	108	9	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	108	10	,	,	PUNCT
cana-2862	108	11	𝑥	𝑥	NOUN
cana-2862	108	12	)	)	PUNCT
cana-2862	108	13	⋅	⋅	PROPN
cana-2862	108	14	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	108	15	,	,	PUNCT
cana-2862	108	16	𝑦	𝑦	NOUN
cana-2862	108	17	)	)	PUNCT
cana-2862	108	18	1	1	NUM
cana-2862	108	19	+	+	CCONJ
cana-2862	108	20	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	108	21	,	,	PUNCT
cana-2862	108	22	𝑦	𝑦	NOUN
cana-2862	108	23	)	)	PUNCT
cana-2862	108	24	+	+	CCONJ
cana-2862	108	25	𝐸	𝐸	PROPN
cana-2862	108	26	⋅	⋅	PROPN
cana-2862	108	27	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	108	28	,	,	PUNCT
cana-2862	108	29	𝑥	𝑥	NOUN
cana-2862	108	30	)	)	PUNCT
cana-2862	108	31	⋅	⋅	PROPN
cana-2862	108	32	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	108	33	,	,	PUNCT
cana-2862	108	34	𝑦	𝑦	NOUN
cana-2862	108	35	)	)	PUNCT
cana-2862	108	36	1	1	NUM
cana-2862	108	37	+	+	CCONJ
cana-2862	108	38	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	108	39	,	,	PUNCT
cana-2862	108	40	𝑦	𝑦	NOUN
cana-2862	108	41	)	)	PUNCT
cana-2862	108	42	which	which	PRON
cana-2862	108	43	implies	imply	VERB
cana-2862	108	44	that	that	SCONJ
cana-2862	108	45	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	108	46	,	,	PUNCT
cana-2862	108	47	𝑦)|	𝑦)|	PROPN
cana-2862	108	48	=	=	PUNCT
cana-2862	108	49	|𝑑(𝑓𝑥	|𝑑(𝑓𝑥	PROPN
cana-2862	108	50	,	,	PUNCT
cana-2862	108	51	𝑔𝑦)|	𝑔𝑦)|	PROPN
cana-2862	108	52	≤	≤	PROPN
cana-2862	108	53	𝐴	𝐴	PROPN
cana-2862	108	54	⋅	⋅	PROPN
cana-2862	108	55	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	108	56	,	,	PUNCT
cana-2862	108	57	𝑦)|	𝑦)|	PROPN
cana-2862	108	58	+	+	CCONJ
cana-2862	108	59	𝐵	𝐵	PROPN
cana-2862	108	60	⋅	⋅	PROPN
cana-2862	108	61	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	108	62	,	,	PUNCT
cana-2862	108	63	𝑥	𝑥	NOUN
cana-2862	108	64	)	)	PUNCT
cana-2862	108	65	⋅	⋅	PROPN
cana-2862	108	66	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	108	67	,	,	PUNCT
cana-2862	108	68	𝑦)|	𝑦)|	PROPN
cana-2862	108	69	|1	|1	NUM
cana-2862	108	70	+	+	CCONJ
cana-2862	108	71	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	108	72	,	,	PUNCT
cana-2862	108	73	𝑦)|	𝑦)|	PROPN
cana-2862	108	74	+	+	CCONJ
cana-2862	108	75	𝐶	𝐶	PROPN
cana-2862	108	76	⋅	⋅	PROPN
cana-2862	108	77	|𝑑(𝑦	|𝑑(𝑦	PROPN
cana-2862	108	78	,	,	PUNCT
cana-2862	108	79	𝑥	𝑥	NOUN
cana-2862	108	80	)	)	PUNCT
cana-2862	108	81	⋅	⋅	PROPN
cana-2862	108	82	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	108	83	,	,	PUNCT
cana-2862	108	84	𝑦)|	𝑦)|	PROPN
cana-2862	108	85	|1	|1	NUM
cana-2862	109	1	+	+	CCONJ
cana-2862	109	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	109	3	,	,	PUNCT
cana-2862	109	4	𝑦)|	𝑦)|	PROPN
cana-2862	109	5	+	+	PROPN
cana-2862	109	6	𝐷	𝐷	PROPN
cana-2862	109	7	⋅	⋅	PROPN
cana-2862	109	8	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	109	9	,	,	PUNCT
cana-2862	109	10	𝑥	𝑥	NOUN
cana-2862	109	11	)	)	PUNCT
cana-2862	109	12	⋅	⋅	PROPN
cana-2862	109	13	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	109	14	,	,	PUNCT
cana-2862	109	15	𝑦)|	𝑦)|	PROPN
cana-2862	109	16	|1	|1	NUM
cana-2862	109	17	+	+	CCONJ
cana-2862	109	18	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	109	19	,	,	PUNCT
cana-2862	109	20	𝑦)|	𝑦)|	PROPN
cana-2862	109	21	+	+	CCONJ
cana-2862	109	22	𝐸	𝐸	PROPN
cana-2862	109	23	⋅	⋅	PROPN
cana-2862	109	24	|𝑑(𝑦	|𝑑(𝑦	PROPN
cana-2862	109	25	,	,	PUNCT
cana-2862	109	26	𝑥	𝑥	NOUN
cana-2862	109	27	)	)	PUNCT
cana-2862	109	28	⋅	⋅	PROPN
cana-2862	109	29	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	109	30	,	,	PUNCT
cana-2862	109	31	𝑦)|	𝑦)|	PROPN
cana-2862	109	32	|1	|1	NUM
cana-2862	109	33	+	+	CCONJ
cana-2862	109	34	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	109	35	,	,	PUNCT
cana-2862	109	36	𝑦)|	𝑦)|	PROPN
cana-2862	109	37	.	.	PUNCT
cana-2862	110	1	since	since	SCONJ
cana-2862	110	2	|1	|1	PRON
cana-2862	110	3	+	+	CCONJ
cana-2862	110	4	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	110	5	,	,	PUNCT
cana-2862	110	6	𝑦)|	𝑦)|	PROPN
cana-2862	110	7	>	>	X
cana-2862	110	8	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	110	9	,	,	PUNCT
cana-2862	110	10	𝑦)|	𝑦)|	PROPN
cana-2862	110	11	i.e.	i.e.	X
cana-2862	110	12	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	110	13	,	,	PUNCT
cana-2862	110	14	𝑦)|	𝑦)|	PROPN
cana-2862	110	15	|1	|1	NUM
cana-2862	111	1	+	+	CCONJ
cana-2862	111	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	111	3	,	,	PUNCT
cana-2862	111	4	𝑦)|	𝑦)|	PROPN
cana-2862	111	5	<	<	X
cana-2862	111	6	1	1	NUM
cana-2862	111	7	,	,	PUNCT
cana-2862	111	8	so	so	SCONJ
cana-2862	111	9	we	we	PRON
cana-2862	111	10	get	get	VERB
cana-2862	111	11	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	111	12	,	,	PUNCT
cana-2862	111	13	𝑦)|	𝑦)|	PROPN
cana-2862	111	14	=	=	PUNCT
cana-2862	112	1	|𝑑(𝑓𝑥	|𝑑(𝑓𝑥	PROPN
cana-2862	112	2	,	,	PUNCT
cana-2862	112	3	𝑔𝑦)|	𝑔𝑦)|	PROPN
cana-2862	112	4	<	<	X
cana-2862	112	5	𝐴.	𝐴.	PROPN
cana-2862	112	6	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	112	7	,	,	PUNCT
cana-2862	112	8	𝑦)|	𝑦)|	PROPN
cana-2862	112	9	+	+	PROPN
cana-2862	112	10	𝐵.	𝐵.	PROPN
cana-2862	112	11	0	0	PUNCT
cana-2862	113	1	+	+	CCONJ
cana-2862	113	2	𝐶.	𝐶.	PROPN
cana-2862	113	3	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	113	4	,	,	PUNCT
cana-2862	113	5	𝑦)|	𝑦)|	PROPN
cana-2862	113	6	+	+	CCONJ
cana-2862	113	7	𝐷.	𝐷.	PROPN
cana-2862	113	8	0	0	PUNCT
cana-2862	114	1	+	+	CCONJ
cana-2862	114	2	𝐸.	𝐸.	PROPN
cana-2862	114	3	0	0	PUNCT
cana-2862	114	4	i.e.	i.e.	X
cana-2862	114	5	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	114	6	,	,	PUNCT
cana-2862	114	7	𝑦)|	𝑦)|	PROPN
cana-2862	114	8	=	=	PUNCT
cana-2862	114	9	|𝑑(𝑓𝑥	|𝑑(𝑓𝑥	PROPN
cana-2862	114	10	,	,	PUNCT
cana-2862	114	11	𝑔𝑦)|	𝑔𝑦)|	PROPN
cana-2862	114	12	<	<	X
cana-2862	114	13	𝐴	𝐴	PROPN
cana-2862	114	14	⋅	⋅	PROPN
cana-2862	114	15	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	114	16	,	,	PUNCT
cana-2862	114	17	𝑦)|	𝑦)|	PROPN
cana-2862	114	18	+	+	CCONJ
cana-2862	114	19	𝐶.	𝐶.	PROPN
cana-2862	114	20	|𝑑(𝑥	|𝑑(𝑥	PROPN
cana-2862	114	21	,	,	PUNCT
cana-2862	114	22	𝑦)|	𝑦)|	PROPN
cana-2862	114	23	i.e.	i.e.	X
cana-2862	114	24	(	(	PUNCT
cana-2862	114	25	1	1	NUM
cana-2862	114	26	−	−	PROPN
cana-2862	114	27	𝐴	𝐴	PROPN
cana-2862	114	28	−	−	PROPN
cana-2862	114	29	𝐶)|𝑑(𝑥	𝐶)|𝑑(𝑥	PROPN
cana-2862	114	30	,	,	PUNCT
cana-2862	114	31	𝑦)|	𝑦)|	PROPN
cana-2862	114	32	<	<	X
cana-2862	114	33	0	0	PROPN
cana-2862	114	34	,	,	PUNCT
cana-2862	114	35	a	a	DET
cana-2862	114	36	contradiction	contradiction	NOUN
cana-2862	114	37	since	since	SCONJ
cana-2862	114	38	𝐴	𝐴	PROPN
cana-2862	114	39	+	+	CCONJ
cana-2862	114	40	𝐵	𝐵	PROPN
cana-2862	114	41	+	+	CCONJ
cana-2862	114	42	𝐶	𝐶	PROPN
cana-2862	114	43	+	+	CCONJ
cana-2862	114	44	2𝐷	2𝐷	NOUN
cana-2862	114	45	+	+	CCONJ
cana-2862	114	46	2𝐸	2𝐸	ADJ
cana-2862	114	47	<	<	X
cana-2862	114	48	1	1	NUM
cana-2862	114	49	⇒	⇒	NOUN
cana-2862	114	50	𝐴	𝐴	PROPN
cana-2862	114	51	+	+	CCONJ
cana-2862	114	52	𝐶	𝐶	PROPN
cana-2862	114	53	<	<	X
cana-2862	114	54	1	1	NUM
cana-2862	114	55	so	so	ADV
cana-2862	114	56	,	,	PUNCT
cana-2862	114	57	𝑥	𝑥	PROPN
cana-2862	114	58	=	=	SYM
cana-2862	114	59	𝑦	𝑦	NOUN
cana-2862	114	60	,	,	PUNCT
cana-2862	114	61	which	which	PRON
cana-2862	114	62	proves	prove	VERB
cana-2862	114	63	the	the	DET
cana-2862	114	64	uniqueness	uniqueness	NOUN
cana-2862	114	65	of	of	ADP
cana-2862	114	66	common	common	ADJ
cana-2862	114	67	fixed	fix	VERB
cana-2862	114	68	point	point	NOUN
cana-2862	114	69	of	of	ADP
cana-2862	114	70	𝑓	𝑓	PRON
cana-2862	114	71	and	and	CCONJ
cana-2862	114	72	𝑔	𝑔	PROPN
cana-2862	114	73	in	in	ADP
cana-2862	114	74	𝑋.	𝑋.	PROPN
cana-2862	114	75	corollary	corollary	NOUN
cana-2862	114	76	3.1	3.1	NUM
cana-2862	114	77	.	.	PUNCT
cana-2862	115	1	let	let	VERB
cana-2862	115	2	(	(	PUNCT
cana-2862	115	3	𝑋	𝑋	NOUN
cana-2862	115	4	,	,	PUNCT
cana-2862	115	5	𝑑	𝑑	NOUN
cana-2862	115	6	)	)	PUNCT
cana-2862	115	7	be	be	VERB
cana-2862	115	8	a	a	DET
cana-2862	115	9	complete	complete	ADJ
cana-2862	115	10	cvebms	cvebms	NOUN
cana-2862	115	11	with	with	ADP
cana-2862	115	12	𝜙	𝜙	NOUN
cana-2862	115	13	:	:	PUNCT
cana-2862	115	14	𝑋	𝑋	PROPN
cana-2862	115	15	×	×	NOUN
cana-2862	115	16	𝑋	𝑋	PROPN
cana-2862	115	17	→	→	SYM
cana-2862	115	18	[	[	X
cana-2862	115	19	1	1	NUM
cana-2862	115	20	,	,	PUNCT
cana-2862	115	21	∞	∞	PROPN
cana-2862	115	22	)	)	PUNCT
cana-2862	115	23	and	and	CCONJ
cana-2862	115	24	𝑓	𝑓	X
cana-2862	115	25	:	:	PUNCT
cana-2862	115	26	𝑋	𝑋	PROPN
cana-2862	115	27	→	→	SYM
cana-2862	115	28	𝑋	𝑋	PROPN
cana-2862	115	29	be	be	VERB
cana-2862	115	30	self	self	NOUN
cana-2862	115	31	-	-	PUNCT
cana-2862	115	32	map	map	NOUN
cana-2862	115	33	satisfying	satisfy	VERB
cana-2862	115	34	𝑑(𝑓𝑥	𝑑(𝑓𝑥	NOUN
cana-2862	115	35	,	,	PUNCT
cana-2862	115	36	𝑓𝑦	𝑓𝑦	NOUN
cana-2862	115	37	)	)	PUNCT
cana-2862	115	38	≾	≾	PROPN
cana-2862	115	39	𝐴	𝐴	PROPN
cana-2862	115	40	⋅	⋅	PROPN
cana-2862	115	41	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	115	42	,	,	PUNCT
cana-2862	115	43	𝑦	𝑦	NOUN
cana-2862	115	44	)	)	PUNCT
cana-2862	116	1	+	+	CCONJ
cana-2862	116	2	𝐵	𝐵	PROPN
cana-2862	116	3	⋅	⋅	PROPN
cana-2862	116	4	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	116	5	,	,	PUNCT
cana-2862	116	6	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	116	7	)	)	PUNCT
cana-2862	116	8	⋅	⋅	PROPN
cana-2862	116	9	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	116	10	,	,	PUNCT
cana-2862	116	11	𝑓𝑦	𝑓𝑦	NOUN
cana-2862	116	12	)	)	PUNCT
cana-2862	116	13	1	1	NUM
cana-2862	117	1	+	+	CCONJ
cana-2862	117	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	117	3	,	,	PUNCT
cana-2862	117	4	𝑦	𝑦	NOUN
cana-2862	117	5	)	)	PUNCT
cana-2862	117	6	+	+	CCONJ
cana-2862	117	7	𝐶	𝐶	PROPN
cana-2862	117	8	⋅	⋅	PROPN
cana-2862	117	9	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	117	10	,	,	PUNCT
cana-2862	117	11	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	117	12	)	)	PUNCT
cana-2862	117	13	⋅	⋅	PROPN
cana-2862	117	14	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	117	15	,	,	PUNCT
cana-2862	117	16	𝑓𝑦	𝑓𝑦	NOUN
cana-2862	117	17	)	)	PUNCT
cana-2862	117	18	1	1	NUM
cana-2862	118	1	+	+	CCONJ
cana-2862	118	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	118	3	,	,	PUNCT
cana-2862	118	4	𝑦	𝑦	NOUN
cana-2862	118	5	)	)	PUNCT
cana-2862	118	6	+	+	ADJ
cana-2862	118	7	𝐷	𝐷	PROPN
cana-2862	118	8	⋅	⋅	PROPN
cana-2862	118	9	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	118	10	,	,	PUNCT
cana-2862	118	11	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	118	12	)	)	PUNCT
cana-2862	118	13	⋅	⋅	PROPN
cana-2862	118	14	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	118	15	,	,	PUNCT
cana-2862	118	16	𝑓𝑦	𝑓𝑦	NOUN
cana-2862	118	17	)	)	PUNCT
cana-2862	118	18	1	1	NUM
cana-2862	118	19	+	+	CCONJ
cana-2862	118	20	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	118	21	,	,	PUNCT
cana-2862	118	22	𝑦	𝑦	NOUN
cana-2862	118	23	)	)	PUNCT
cana-2862	118	24	+	+	CCONJ
cana-2862	118	25	𝐸	𝐸	PROPN
cana-2862	118	26	⋅	⋅	PROPN
cana-2862	118	27	𝑑(𝑦	𝑑(𝑦	NOUN
cana-2862	118	28	,	,	PUNCT
cana-2862	118	29	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	118	30	)	)	PUNCT
cana-2862	118	31	⋅	⋅	PROPN
cana-2862	118	32	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	118	33	,	,	PUNCT
cana-2862	118	34	𝑓𝑦	𝑓𝑦	NOUN
cana-2862	118	35	)	)	PUNCT
cana-2862	118	36	1	1	NUM
cana-2862	119	1	+	+	CCONJ
cana-2862	119	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	119	3	,	,	PUNCT
cana-2862	119	4	𝑦	𝑦	NOUN
cana-2862	119	5	)	)	PUNCT
cana-2862	119	6	where	where	SCONJ
cana-2862	119	7	𝐴	𝐴	PROPN
cana-2862	119	8	,	,	PUNCT
cana-2862	119	9	𝐵	𝐵	PROPN
cana-2862	119	10	,	,	PUNCT
cana-2862	119	11	𝐶	𝐶	PROPN
cana-2862	119	12	,	,	PUNCT
cana-2862	119	13	𝐷	𝐷	PROPN
cana-2862	119	14	,	,	PUNCT
cana-2862	119	15	𝐸	𝐸	PROPN
cana-2862	119	16	nonnegative	nonnegative	ADJ
cana-2862	119	17	real	real	ADJ
cana-2862	119	18	numbers	number	NOUN
cana-2862	119	19	,	,	PUNCT
cana-2862	119	20	with	with	ADP
cana-2862	119	21	𝐴	𝐴	PROPN
cana-2862	119	22	+	+	CCONJ
cana-2862	119	23	𝐵	𝐵	PROPN
cana-2862	119	24	+	+	CCONJ
cana-2862	119	25	𝐶	𝐶	PROPN
cana-2862	119	26	+	+	CCONJ
cana-2862	119	27	2𝐷	2𝐷	NOUN
cana-2862	119	28	+	+	CCONJ
cana-2862	119	29	2𝐸	2𝐸	ADJ
cana-2862	119	30	<	<	X
cana-2862	119	31	1	1	NUM
cana-2862	119	32	.	.	PUNCT
cana-2862	120	1	then	then	ADV
cana-2862	120	2	𝑓	𝑓	PRON
cana-2862	120	3	has	have	VERB
cana-2862	120	4	a	a	DET
cana-2862	120	5	unique	unique	ADJ
cana-2862	120	6	fires	fire	NOUN
cana-2862	120	7	point	point	VERB
cana-2862	120	8	in	in	ADP
cana-2862	120	9	𝑋.	𝑋.	PROPN
cana-2862	120	10	proof	proof	NOUN
cana-2862	120	11	.	.	PUNCT
cana-2862	121	1	taking	take	VERB
cana-2862	121	2	𝑔	𝑔	PART
cana-2862	121	3	=	=	PUNCT
cana-2862	121	4	𝑓	𝑓	PROPN
cana-2862	121	5	in	in	ADP
cana-2862	121	6	theorem	theorem	NOUN
cana-2862	121	7	3.1	3.1	NUM
cana-2862	121	8	.	.	PUNCT
cana-2862	121	9	communications	communication	NOUN
cana-2862	121	10	on	on	ADP
cana-2862	121	11	applied	apply	VERB
cana-2862	121	12	nonlinear	nonlinear	ADJ
cana-2862	121	13	analysis	analysis	NOUN
cana-2862	121	14	issn	issn	NOUN
cana-2862	121	15	:	:	PUNCT
cana-2862	121	16	1074	1074	NUM
cana-2862	121	17	-	-	PUNCT
cana-2862	121	18	133x	133x	NUM
cana-2862	121	19	vol	vol	NOUN
cana-2862	121	20	32	32	NUM
cana-2862	121	21	no	no	NOUN
cana-2862	121	22	.	.	PUNCT
cana-2862	122	1	4s	4s	NUM
cana-2862	122	2	(	(	PUNCT
cana-2862	122	3	2025	2025	NUM
cana-2862	122	4	)	)	PUNCT
cana-2862	122	5	440	440	NUM
cana-2862	123	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2862	123	2	corollary	corollary	ADJ
cana-2862	123	3	3.2	3.2	NUM
cana-2862	123	4	.	.	PUNCT
cana-2862	124	1	let	let	VERB
cana-2862	124	2	(	(	PUNCT
cana-2862	124	3	𝑋	𝑋	NOUN
cana-2862	124	4	,	,	PUNCT
cana-2862	124	5	𝑑	𝑑	NOUN
cana-2862	124	6	)	)	PUNCT
cana-2862	124	7	be	be	VERB
cana-2862	124	8	a	a	DET
cana-2862	124	9	complete	complete	ADJ
cana-2862	124	10	cvebms	cvebms	NOUN
cana-2862	124	11	with	with	ADP
cana-2862	124	12	𝜙	𝜙	NOUN
cana-2862	124	13	:	:	PUNCT
cana-2862	124	14	𝑋	𝑋	PROPN
cana-2862	124	15	×	×	NOUN
cana-2862	124	16	𝑋	𝑋	PROPN
cana-2862	124	17	→	→	SYM
cana-2862	125	1	[	[	X
cana-2862	125	2	1	1	NUM
cana-2862	125	3	,	,	PUNCT
cana-2862	125	4	∞	∞	PROPN
cana-2862	125	5	)	)	PUNCT
cana-2862	125	6	and	and	CCONJ
cana-2862	125	7	𝑓	𝑓	X
cana-2862	125	8	,	,	PUNCT
cana-2862	125	9	𝑔	𝑔	NOUN
cana-2862	125	10	:	:	PUNCT
cana-2862	125	11	𝑋	𝑋	PROPN
cana-2862	125	12	→	→	SYM
cana-2862	125	13	𝑋	𝑋	PROPN
cana-2862	125	14	be	be	AUX
cana-2862	125	15	selfmaps	selfmap	NOUN
cana-2862	125	16	satisfying	satisfy	VERB
cana-2862	125	17	𝑑(𝑓𝑥	𝑑(𝑓𝑥	NOUN
cana-2862	125	18	,	,	PUNCT
cana-2862	125	19	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	125	20	)	)	PUNCT
cana-2862	125	21	≾	≾	PROPN
cana-2862	125	22	𝐴	𝐴	PROPN
cana-2862	125	23	⋅	⋅	PROPN
cana-2862	125	24	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	125	25	,	,	PUNCT
cana-2862	125	26	𝑦	𝑦	NOUN
cana-2862	125	27	)	)	PUNCT
cana-2862	125	28	+	+	CCONJ
cana-2862	125	29	𝐵	𝐵	PROPN
cana-2862	125	30	⋅	⋅	PROPN
cana-2862	125	31	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	125	32	,	,	PUNCT
cana-2862	125	33	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	125	34	)	)	PUNCT
cana-2862	125	35	⋅	⋅	PROPN
cana-2862	125	36	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	125	37	,	,	PUNCT
cana-2862	125	38	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	125	39	)	)	PUNCT
cana-2862	125	40	1	1	NUM
cana-2862	126	1	+	+	CCONJ
cana-2862	126	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	126	3	,	,	PUNCT
cana-2862	126	4	𝑦	𝑦	NOUN
cana-2862	126	5	)	)	PUNCT
cana-2862	126	6	+	+	CCONJ
cana-2862	126	7	𝐶	𝐶	PROPN
cana-2862	126	8	⋅	⋅	PROPN
cana-2862	126	9	𝑑(𝑦	𝑑(𝑦	PROPN
cana-2862	126	10	,	,	PUNCT
cana-2862	126	11	𝑓𝑥	𝑓𝑥	NOUN
cana-2862	126	12	)	)	PUNCT
cana-2862	126	13	⋅	⋅	PROPN
cana-2862	126	14	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	126	15	,	,	PUNCT
cana-2862	126	16	𝑔𝑦	𝑔𝑦	NOUN
cana-2862	126	17	)	)	PUNCT
cana-2862	126	18	1	1	NUM
cana-2862	127	1	+	+	CCONJ
cana-2862	127	2	𝑑(𝑥	𝑑(𝑥	PROPN
cana-2862	127	3	,	,	PUNCT
cana-2862	127	4	𝑦	𝑦	NOUN
cana-2862	127	5	)	)	PUNCT
cana-2862	127	6	where	where	SCONJ
cana-2862	127	7	𝐴	𝐴	PROPN
cana-2862	127	8	,	,	PUNCT
cana-2862	127	9	𝐵	𝐵	PROPN
cana-2862	127	10	,	,	PUNCT
cana-2862	127	11	𝐶	𝐶	PROPN
cana-2862	127	12	nonnegative	nonnegative	ADJ
cana-2862	127	13	real	real	ADJ
cana-2862	127	14	numbers	number	NOUN
cana-2862	127	15	,	,	PUNCT
cana-2862	127	16	with	with	ADP
cana-2862	127	17	𝐴	𝐴	PROPN
cana-2862	127	18	+	+	CCONJ
cana-2862	127	19	𝐵	𝐵	PROPN
cana-2862	127	20	+	+	CCONJ
cana-2862	127	21	𝐶	𝐶	PROPN
cana-2862	127	22	<	<	X
cana-2862	127	23	1	1	NUM
cana-2862	127	24	.	.	PUNCT
cana-2862	128	1	then	then	ADV
cana-2862	128	2	𝑓	𝑓	X
cana-2862	128	3	and	and	CCONJ
cana-2862	128	4	𝑔	𝑔	AUX
cana-2862	128	5	have	have	VERB
cana-2862	128	6	a	a	DET
cana-2862	128	7	unique	unique	ADJ
cana-2862	128	8	common	common	ADJ
cana-2862	128	9	fixed	fix	VERB
cana-2862	128	10	point	point	NOUN
cana-2862	128	11	in	in	ADP
cana-2862	128	12	𝑋.	𝑋.	PROPN
cana-2862	128	13	proof	proof	NOUN
cana-2862	128	14	.	.	PUNCT
cana-2862	129	1	taking	take	VERB
cana-2862	129	2	𝐷	𝐷	NOUN
cana-2862	129	3	=	=	SYM
cana-2862	129	4	𝐸	𝐸	PROPN
cana-2862	129	5	=	=	SYM
cana-2862	129	6	0	0	NUM
cana-2862	129	7	in	in	ADP
cana-2862	129	8	theorem	theorem	ADJ
cana-2862	129	9	3.1	3.1	NUM
cana-2862	129	10	.	.	PUNCT
cana-2862	129	11	remark	remark	PROPN
cana-2862	129	12	3.1	3.1	NUM
cana-2862	129	13	.	.	PUNCT
cana-2862	130	1	(	(	PUNCT
cana-2862	130	2	i	i	NOUN
cana-2862	130	3	)	)	PUNCT
cana-2862	130	4	theorem	theorem	VERB
cana-2862	130	5	3.1	3.1	NUM
cana-2862	130	6	generalized	generalize	VERB
cana-2862	130	7	theorem	theorem	NOUN
cana-2862	130	8	15	15	NUM
cana-2862	130	9	of	of	ADP
cana-2862	130	10	[	[	X
cana-2862	130	11	6	6	NUM
cana-2862	130	12	]	]	PUNCT
cana-2862	130	13	after	after	ADP
cana-2862	130	14	substituting	substitute	VERB
cana-2862	130	15	𝐶	𝐶	PROPN
cana-2862	130	16	=	=	PROPN
cana-2862	130	17	𝐷	𝐷	PROPN
cana-2862	130	18	=	=	NOUN
cana-2862	130	19	𝐸	𝐸	PROPN
cana-2862	130	20	=	=	SYM
cana-2862	130	21	0	0	NUM
cana-2862	130	22	and	and	CCONJ
cana-2862	130	23	𝜙(𝑥	𝜙(𝑥	PROPN
cana-2862	130	24	,	,	PUNCT
cana-2862	130	25	𝑦	𝑦	NOUN
cana-2862	130	26	)	)	PUNCT
cana-2862	130	27	=	=	SYM
cana-2862	130	28	𝑠	𝑠	PROPN
cana-2862	130	29	≥	≥	NUM
cana-2862	130	30	1	1	NUM
cana-2862	130	31	,	,	PUNCT
cana-2862	130	32	(	(	PUNCT
cana-2862	130	33	ii	ii	NOUN
cana-2862	130	34	)	)	PUNCT
cana-2862	130	35	theorem	theorem	VERB
cana-2862	130	36	3.1	3.1	NUM
cana-2862	130	37	generalized	generalize	VERB
cana-2862	130	38	theorem	theorem	NOUN
cana-2862	130	39	10	10	NUM
cana-2862	130	40	of	of	ADP
cana-2862	130	41	[	[	X
cana-2862	130	42	9	9	NUM
cana-2862	130	43	]	]	PUNCT
cana-2862	130	44	after	after	ADP
cana-2862	130	45	substituting	substitute	VERB
cana-2862	130	46	𝐷	𝐷	PROPN
cana-2862	130	47	=	=	SYM
cana-2862	130	48	𝐸	𝐸	PROPN
cana-2862	130	49	=	=	SYM
cana-2862	130	50	0	0	NUM
cana-2862	130	51	and	and	CCONJ
cana-2862	130	52	𝜙(𝑥	𝜙(𝑥	PROPN
cana-2862	130	53	,	,	PUNCT
cana-2862	130	54	𝑦	𝑦	NOUN
cana-2862	130	55	)	)	PUNCT
cana-2862	130	56	=	=	SYM
cana-2862	130	57	𝑠	𝑠	PROPN
cana-2862	130	58	≥	≥	NOUN
cana-2862	130	59	1	1	NUM
cana-2862	130	60	.	.	PUNCT
cana-2862	131	1	(	(	PUNCT
cana-2862	131	2	iii	iii	NOUN
cana-2862	131	3	)	)	PUNCT
cana-2862	131	4	theorem	theorem	VERB
cana-2862	131	5	3.1	3.1	NUM
cana-2862	131	6	generalized	generalize	VERB
cana-2862	131	7	theorem	theorem	NOUN
cana-2862	131	8	1	1	NUM
cana-2862	131	9	of	of	ADP
cana-2862	131	10	[	[	X
cana-2862	131	11	4	4	X
cana-2862	131	12	]	]	PUNCT
cana-2862	131	13	after	after	ADP
cana-2862	131	14	substituting	substitute	VERB
cana-2862	131	15	𝐷	𝐷	PROPN
cana-2862	131	16	=	=	SYM
cana-2862	131	17	𝐸	𝐸	PROPN
cana-2862	131	18	=	=	SYM
cana-2862	131	19	0	0	NUM
cana-2862	131	20	and	and	CCONJ
cana-2862	131	21	𝜙(𝑥	𝜙(𝑥	PROPN
cana-2862	131	22	,	,	PUNCT
cana-2862	131	23	𝑦	𝑦	NOUN
cana-2862	131	24	)	)	PUNCT
cana-2862	131	25	=	=	SYM
cana-2862	131	26	1	1	X
cana-2862	131	27	.	.	PUNCT
cana-2862	131	28	(	(	PUNCT
cana-2862	131	29	iv	iv	X
cana-2862	131	30	)	)	PUNCT
cana-2862	131	31	theorem	theorem	VERB
cana-2862	131	32	3.1	3.1	NUM
cana-2862	131	33	generalized	generalize	VERB
cana-2862	131	34	theorem	theorem	NOUN
cana-2862	131	35	4	4	NUM
cana-2862	131	36	of	of	ADP
cana-2862	131	37	[	[	X
cana-2862	131	38	1	1	NUM
cana-2862	131	39	]	]	PUNCT
cana-2862	131	40	after	after	ADP
cana-2862	131	41	substituting	substitute	VERB
cana-2862	131	42	𝐶	𝐶	PROPN
cana-2862	131	43	=	=	PROPN
cana-2862	131	44	𝐷	𝐷	PROPN
cana-2862	131	45	=	=	NOUN
cana-2862	131	46	𝐸	𝐸	PROPN
cana-2862	131	47	=	=	SYM
cana-2862	131	48	0	0	NUM
cana-2862	131	49	and	and	CCONJ
cana-2862	131	50	𝜙(𝑥	𝜙(𝑥	PROPN
cana-2862	131	51	,	,	PUNCT
cana-2862	131	52	𝑦	𝑦	NOUN
cana-2862	131	53	)	)	PUNCT
cana-2862	131	54	=	=	SYM
cana-2862	131	55	1	1	X
cana-2862	131	56	.	.	PUNCT
cana-2862	131	57	(	(	PUNCT
cana-2862	131	58	v	v	NOUN
cana-2862	131	59	)	)	PUNCT
cana-2862	131	60	theorem	theorem	VERB
cana-2862	131	61	3.1	3.1	NUM
cana-2862	131	62	generalized	generalize	VERB
cana-2862	131	63	theorem	theorem	NOUN
cana-2862	131	64	2	2	NUM
cana-2862	131	65	of	of	ADP
cana-2862	131	66	[	[	X
cana-2862	131	67	10	10	NUM
cana-2862	131	68	]	]	PUNCT
cana-2862	131	69	after	after	ADP
cana-2862	131	70	substituting	substitute	VERB
cana-2862	131	71	𝐵	𝐵	NOUN
cana-2862	131	72	=	=	SYM
cana-2862	131	73	𝐶	𝐶	PROPN
cana-2862	131	74	=	=	PROPN
cana-2862	131	75	𝐷	𝐷	PROPN
cana-2862	131	76	=	=	NOUN
cana-2862	131	77	𝐸	𝐸	PROPN
cana-2862	131	78	=	=	SYM
cana-2862	131	79	0	0	NUM
cana-2862	131	80	and	and	CCONJ
cana-2862	131	81	ℂ	ℂ	PROPN
cana-2862	131	82	=	=	PUNCT
cana-2862	132	1	𝑅.	𝑅.	NOUN
cana-2862	132	2	refrences	refrence	NOUN
cana-2862	132	3	[	[	X
cana-2862	132	4	1	1	NUM
cana-2862	132	5	]	]	PUNCT
cana-2862	132	6	a.	a.	PROPN
cana-2862	132	7	azam	azam	PROPN
cana-2862	132	8	,	,	PUNCT
cana-2862	132	9	b.	b.	PROPN
cana-2862	132	10	fisher	fisher	PROPN
cana-2862	132	11	and	and	CCONJ
cana-2862	132	12	m.	m.	PROPN
cana-2862	132	13	khan	khan	PROPN
cana-2862	132	14	,	,	PUNCT
cana-2862	132	15	common	common	ADJ
cana-2862	132	16	fixed	fix	VERB
cana-2862	132	17	point	point	NOUN
cana-2862	132	18	theorem	theorem	VERB
cana-2862	132	19	in	in	ADP
cana-2862	132	20	complex	complex	ADJ
cana-2862	132	21	valued	value	VERB
cana-2862	132	22	metric	metric	ADJ
cana-2862	132	23	spaces	space	NOUN
cana-2862	132	24	,	,	PUNCT
cana-2862	132	25	numerical	numerical	ADJ
cana-2862	132	26	functional	functional	ADJ
cana-2862	132	27	analysis	analysis	NOUN
cana-2862	132	28	and	and	CCONJ
cana-2862	132	29	optimization	optimization	NOUN
cana-2862	132	30	,	,	PUNCT
cana-2862	132	31	2011	2011	NUM
cana-2862	132	32	,	,	PUNCT
cana-2862	132	33	32(3	32(3	NUM
cana-2862	132	34	)	)	PUNCT
cana-2862	132	35	,	,	PUNCT
cana-2862	132	36	243	243	NUM
cana-2862	132	37	253	253	NUM
cana-2862	132	38	.	.	PUNCT
cana-2862	133	1	doi	doi	NOUN
cana-2862	133	2	:	:	PUNCT
cana-2862	133	3	10.1080/01630563.2011.533046	10.1080/01630563.2011.533046	NUM
cana-2862	133	4	[	[	X
cana-2862	133	5	2	2	X
cana-2862	133	6	]	]	PUNCT
cana-2862	133	7	s.	s.	PROPN
cana-2862	133	8	banach	banach	PROPN
cana-2862	133	9	,	,	PUNCT
cana-2862	133	10	sur	sur	PROPN
cana-2862	133	11	les	les	PROPN
cana-2862	133	12	operations	operation	NOUN
cana-2862	133	13	dons	don	NOUN
cana-2862	133	14	les	les	PART
cana-2862	133	15	ensembles	ensemble	NOUN
cana-2862	133	16	abstrats	abstrat	NOUN
cana-2862	133	17	et	et	PROPN
cana-2862	133	18	leur	leur	X
cana-2862	133	19	application	application	PROPN
cana-2862	133	20	aux	aux	PROPN
cana-2862	133	21	equations	equation	NOUN
cana-2862	133	22	integrals	integral	NOUN
cana-2862	133	23	,	,	PUNCT
cana-2862	133	24	fundamenta	fundamenta	PROPN
cana-2862	133	25	mathematica	mathematica	PROPN
cana-2862	133	26	,	,	PUNCT
cana-2862	133	27	1922	1922	NUM
cana-2862	133	28	,	,	PUNCT
cana-2862	133	29	3	3	NUM
cana-2862	133	30	,	,	PUNCT
cana-2862	133	31	133	133	NUM
cana-2862	133	32	181	181	NUM
cana-2862	133	33	.	.	PUNCT
cana-2862	134	1	[	[	X
cana-2862	134	2	3	3	X
cana-2862	134	3	]	]	X
cana-2862	134	4	s.	s.	PROPN
cana-2862	134	5	bhatt	bhatt	PROPN
cana-2862	134	6	,	,	PUNCT
cana-2862	134	7	s.	s.	PROPN
cana-2862	134	8	chaukiyal	chaukiyal	PROPN
cana-2862	134	9	and	and	CCONJ
cana-2862	134	10	r.c	r.c	PROPN
cana-2862	134	11	.	.	PROPN
cana-2862	134	12	dimri	dimri	PROPN
cana-2862	134	13	,	,	PUNCT
cana-2862	134	14	common	common	ADJ
cana-2862	134	15	fixed	fix	VERB
cana-2862	134	16	point	point	NOUN
cana-2862	134	17	of	of	ADP
cana-2862	134	18	mappings	mapping	NOUN
cana-2862	134	19	satisfying	satisfy	VERB
cana-2862	134	20	rational	rational	ADJ
cana-2862	134	21	inequality	inequality	NOUN
cana-2862	134	22	in	in	ADP
cana-2862	134	23	complex	complex	ADJ
cana-2862	134	24	valued	value	VERB
cana-2862	134	25	metric	metric	ADJ
cana-2862	134	26	space	space	NOUN
cana-2862	134	27	,	,	PUNCT
cana-2862	134	28	international	international	ADJ
cana-2862	134	29	journal	journal	NOUN
cana-2862	134	30	of	of	ADP
cana-2862	134	31	pure	pure	ADJ
cana-2862	134	32	and	and	CCONJ
cana-2862	134	33	applied	applied	ADJ
cana-2862	134	34	mathematics	mathematic	NOUN
cana-2862	134	35	,	,	PUNCT
cana-2862	134	36	2011	2011	NUM
cana-2862	134	37	,	,	PUNCT
cana-2862	134	38	73(2	73(2	NUM
cana-2862	134	39	)	)	PUNCT
cana-2862	134	40	,	,	PUNCT
cana-2862	134	41	159	159	NUM
cana-2862	134	42	164	164	NUM
cana-2862	134	43	.	.	PUNCT
cana-2862	135	1	[	[	X
cana-2862	135	2	4	4	NUM
cana-2862	135	3	]	]	X
cana-2862	135	4	f.	f.	PROPN
cana-2862	135	5	fouzkard	fouzkard	PROPN
cana-2862	135	6	and	and	CCONJ
cana-2862	135	7	m.	m.	PROPN
cana-2862	135	8	imdad	imdad	PROPN
cana-2862	135	9	,	,	PUNCT
cana-2862	135	10	some	some	DET
cana-2862	135	11	common	common	ADJ
cana-2862	135	12	fixed	fix	VERB
cana-2862	135	13	point	point	NOUN
cana-2862	135	14	theorems	theorem	NOUN
cana-2862	135	15	on	on	ADP
cana-2862	135	16	complex	complex	ADJ
cana-2862	135	17	valued	value	VERB
cana-2862	135	18	metric	metric	ADJ
cana-2862	135	19	spaces	space	NOUN
cana-2862	135	20	,	,	PUNCT
cana-2862	135	21	computers	computer	NOUN
cana-2862	135	22	of	of	ADP
cana-2862	135	23	mathematics	mathematic	NOUN
cana-2862	135	24	with	with	ADP
cana-2862	135	25	applications	application	NOUN
cana-2862	135	26	,	,	PUNCT
cana-2862	135	27	2012	2012	NUM
cana-2862	135	28	,	,	PUNCT
cana-2862	135	29	64(6	64(6	NUM
cana-2862	135	30	)	)	PUNCT
cana-2862	135	31	,	,	PUNCT
cana-2862	135	32	1866	1866	NUM
cana-2862	135	33	1874	1874	NUM
cana-2862	135	34	.	.	PUNCT
cana-2862	136	1	[	[	X
cana-2862	136	2	5	5	X
cana-2862	136	3	]	]	PUNCT
cana-2862	136	4	j.	j.	PROPN
cana-2862	136	5	kumar	kumar	PROPN
cana-2862	136	6	and	and	CCONJ
cana-2862	136	7	s.	s.	PROPN
cana-2862	136	8	vashistha	vashistha	PROPN
cana-2862	136	9	,	,	PUNCT
cana-2862	136	10	coupled	couple	VERB
cana-2862	136	11	fixed	fix	VERB
cana-2862	136	12	point	point	NOUN
cana-2862	136	13	theorem	theorem	NOUN
cana-2862	136	14	for	for	ADP
cana-2862	136	15	generalized	generalized	ADJ
cana-2862	136	16	contraction	contraction	NOUN
cana-2862	136	17	in	in	ADP
cana-2862	136	18	complex	complex	ADV
cana-2862	136	19	-	-	PUNCT
cana-2862	136	20	valued	value	VERB
cana-2862	136	21	metric	metric	ADJ
cana-2862	136	22	spaces	space	NOUN
cana-2862	136	23	,	,	PUNCT
cana-2862	136	24	int	int	PROPN
cana-2862	136	25	.	.	PUNCT
cana-2862	137	1	journal	journal	PROPN
cana-2862	137	2	of	of	ADP
cana-2862	137	3	comp	comp	PROPN
cana-2862	137	4	.	.	PUNCT
cana-2862	138	1	appl	appl	PROPN
cana-2862	138	2	.	.	PROPN
cana-2862	138	3	,	,	PUNCT
cana-2862	138	4	2013	2013	NUM
cana-2862	138	5	,	,	PUNCT
cana-2862	138	6	83(7	83(7	NUM
cana-2862	138	7	)	)	PUNCT
cana-2862	138	8	,	,	PUNCT
cana-2862	138	9	36	36	NUM
cana-2862	138	10	40	40	NUM
cana-2862	138	11	.	.	PUNCT
cana-2862	139	1	doi:10.5120/14463	doi:10.5120/14463	PROPN
cana-2862	139	2	-	-	SYM
cana-2862	139	3	2745	2745	NUM
cana-2862	139	4	[	[	X
cana-2862	139	5	6	6	NUM
cana-2862	139	6	]	]	SYM
cana-2862	139	7	a.a	a.a	PROPN
cana-2862	139	8	.	.	PROPN
cana-2862	139	9	mukheimer	mukheimer	PROPN
cana-2862	139	10	,	,	PUNCT
cana-2862	139	11	some	some	DET
cana-2862	139	12	common	common	ADJ
cana-2862	139	13	fixed	fix	VERB
cana-2862	139	14	point	point	NOUN
cana-2862	139	15	theorems	theorem	NOUN
cana-2862	139	16	in	in	ADP
cana-2862	139	17	complex	complex	ADJ
cana-2862	139	18	valued	value	VERB
cana-2862	139	19	b	b	PROPN
cana-2862	139	20	metric	metric	ADJ
cana-2862	139	21	spaces	space	NOUN
cana-2862	139	22	,	,	PUNCT
cana-2862	139	23	the	the	DET
cana-2862	139	24	scientific	scientific	ADJ
cana-2862	139	25	world	world	NOUN
cana-2862	139	26	journal	journal	NOUN
cana-2862	139	27	,	,	PUNCT
cana-2862	139	28	2014	2014	NUM
cana-2862	139	29	,	,	PUNCT
cana-2862	139	30	(	(	PUNCT
cana-2862	139	31	2014	2014	NUM
cana-2862	139	32	)	)	PUNCT
cana-2862	139	33	,	,	PUNCT
cana-2862	139	34	1	1	NUM
cana-2862	139	35	6	6	NUM
cana-2862	139	36	.	.	PUNCT
cana-2862	140	1	doi	doi	NOUN
cana-2862	140	2	:	:	PUNCT
cana-2862	140	3	10.1155/2014/587825	10.1155/2014/587825	NUM
cana-2862	141	1	[	[	X
cana-2862	141	2	7	7	NUM
cana-2862	141	3	]	]	PUNCT
cana-2862	141	4	k.	k.	PROPN
cana-2862	141	5	rao	rao	PROPN
cana-2862	141	6	,	,	PUNCT
cana-2862	141	7	p.	p.	PROPN
cana-2862	141	8	swamy	swamy	PROPN
cana-2862	141	9	and	and	CCONJ
cana-2862	141	10	j.	j.	PROPN
cana-2862	141	11	prasad	prasad	PROPN
cana-2862	141	12	,	,	PUNCT
cana-2862	141	13	a	a	DET
cana-2862	141	14	common	common	ADJ
cana-2862	141	15	fixed	fix	VERB
cana-2862	141	16	point	point	NOUN
cana-2862	141	17	theorem	theorem	VERB
cana-2862	141	18	in	in	ADP
cana-2862	141	19	complex	complex	ADJ
cana-2862	141	20	valued	value	VERB
cana-2862	141	21	𝑏-metric	𝑏-metric	PROPN
cana-2862	141	22	spaces	space	NOUN
cana-2862	141	23	,	,	PUNCT
cana-2862	141	24	bulletin	bulletin	NOUN
cana-2862	141	25	of	of	ADP
cana-2862	141	26	mathematics	mathematic	NOUN
cana-2862	141	27	and	and	CCONJ
cana-2862	141	28	statistics	statistic	NOUN
cana-2862	141	29	research	research	NOUN
cana-2862	141	30	,	,	PUNCT
cana-2862	141	31	1	1	NUM
cana-2862	141	32	(	(	PUNCT
cana-2862	141	33	2013	2013	NUM
cana-2862	141	34	)	)	PUNCT
cana-2862	141	35	,	,	PUNCT
cana-2862	141	36	1	1	NUM
cana-2862	141	37	8	8	NUM
cana-2862	141	38	.	.	PUNCT
cana-2862	142	1	[	[	X
cana-2862	142	2	8	8	X
cana-2862	142	3	]	]	X
cana-2862	142	4	w.	w.	PROPN
cana-2862	142	5	sintunavarat	sintunavarat	PROPN
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cana-2862	142	7	p.	p.	PROPN
cana-2862	142	8	kumam	kumam	PROPN
cana-2862	142	9	,	,	PUNCT
cana-2862	142	10	generalized	generalize	VERB
cana-2862	142	11	common	common	ADJ
cana-2862	142	12	fixed	fix	VERB
cana-2862	142	13	point	point	NOUN
cana-2862	142	14	theorems	theorem	NOUN
cana-2862	142	15	in	in	ADP
cana-2862	142	16	complex	complex	ADJ
cana-2862	142	17	valued	value	VERB
cana-2862	142	18	metric	metric	ADJ
cana-2862	142	19	spaces	space	NOUN
cana-2862	142	20	and	and	CCONJ
cana-2862	142	21	applications	application	NOUN
cana-2862	142	22	,	,	PUNCT
cana-2862	142	23	journal	journal	NOUN
cana-2862	142	24	of	of	ADP
cana-2862	142	25	inequalities	inequality	NOUN
cana-2862	142	26	and	and	CCONJ
cana-2862	142	27	applications	application	NOUN
cana-2862	142	28	,	,	PUNCT
cana-2862	142	29	2012	2012	NUM
cana-2862	142	30	,	,	PUNCT
cana-2862	142	31	(	(	PUNCT
cana-2862	142	32	2012	2012	NUM
cana-2862	142	33	)	)	PUNCT
cana-2862	142	34	,	,	PUNCT
cana-2862	142	35	84	84	NUM
cana-2862	142	36	.	.	PUNCT
cana-2862	143	1	[	[	X
cana-2862	143	2	9	9	NUM
cana-2862	143	3	]	]	PUNCT
cana-2862	143	4	j.	j.	PROPN
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cana-2862	143	7	s.	s.	PROPN
cana-2862	143	8	vashistha	vashistha	PROPN
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cana-2862	143	21	b	b	X
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cana-2862	143	23	metric	metric	ADJ
cana-2862	143	24	spaces	space	NOUN
cana-2862	143	25	,	,	PUNCT
cana-2862	143	26	int	int	PROPN
cana-2862	143	27	.	.	PUNCT
cana-2862	144	1	journal	journal	PROPN
cana-2862	144	2	math	math	PROPN
cana-2862	144	3	.	.	PUNCT
cana-2862	145	1	anal	anal	PROPN
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cana-2862	145	3	,	,	PUNCT
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cana-2862	145	5	,	,	PUNCT
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cana-2862	145	7	)	)	PUNCT
cana-2862	145	8	,	,	PUNCT
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cana-2862	145	10	2334	2334	NUM
cana-2862	145	11	.	.	PUNCT
cana-2862	146	1	doi	doi	NOUN
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cana-2862	146	7	10	10	NUM
cana-2862	146	8	]	]	PUNCT
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cana-2862	146	21	of	of	ADP
cana-2862	146	22	b	b	NOUN
cana-2862	146	23	-	-	PUNCT
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cana-2862	146	25	space	space	NOUN
cana-2862	146	26	and	and	CCONJ
cana-2862	146	27	some	some	DET
cana-2862	146	28	fixed	fix	VERB
cana-2862	146	29	point	point	NOUN
cana-2862	146	30	theorem	theorem	ADJ
cana-2862	146	31	,	,	PUNCT
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cana-2862	146	33	,	,	PUNCT
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cana-2862	146	35	,	,	PUNCT
cana-2862	146	36	5(19	5(19	NUM
cana-2862	146	37	)	)	PUNCT
cana-2862	146	38	,	,	PUNCT
cana-2862	146	39	doi	doi	NOUN
cana-2862	146	40	:	:	PUNCT
cana-2862	146	41	10.3390	10.3390	NUM
cana-2862	146	42	/	/	SYM
cana-2862	146	43	math5020019	math5020019	NOUN
cana-2862	147	1	[	[	X
cana-2862	147	2	11	11	NUM
cana-2862	147	3	]	]	X
cana-2862	147	4	n.	n.	PROPN
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cana-2862	147	6	,	,	PUNCT
cana-2862	147	7	s.s	s.s	PROPN
cana-2862	147	8	.	.	PROPN
cana-2862	147	9	mohammed	mohammed	PROPN
cana-2862	147	10	and	and	CCONJ
cana-2862	147	11	a	a	DET
cana-2862	147	12	azam	azam	PROPN
cana-2862	147	13	,	,	PUNCT
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cana-2862	147	15	point	point	NOUN
cana-2862	147	16	theorems	theorem	NOUN
cana-2862	147	17	in	in	ADP
cana-2862	147	18	complex	complex	ADJ
cana-2862	147	19	extended	extended	ADJ
cana-2862	147	20	b	b	NOUN
cana-2862	147	21	-	-	PUNCT
cana-2862	147	22	metric	metric	ADJ
cana-2862	147	23	space	space	NOUN
cana-2862	147	24	,	,	PUNCT
cana-2862	147	25	moroccan	moroccan	PROPN
cana-2862	147	26	j.	j.	PROPN
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cana-2862	147	28	pure	pure	ADJ
cana-2862	147	29	and	and	CCONJ
cana-2862	147	30	appl	appl	ADJ
cana-2862	147	31	.	.	PUNCT
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cana-2862	148	2	.	.	PROPN
cana-2862	148	3	,	,	PUNCT
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cana-2862	148	5	,	,	PUNCT
cana-2862	148	6	5(2	5(2	NUM
cana-2862	148	7	)	)	PUNCT
cana-2862	148	8	,	,	PUNCT
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cana-2862	148	10	163	163	NUM
cana-2862	148	11	.	.	PUNCT
cana-2862	149	1	doi	doi	NOUN
cana-2862	149	2	:	:	PUNCT
cana-2862	149	3	102478	102478	NUM
cana-2862	149	4	/	/	SYM
cana-2862	149	5	mjpaa-2019	mjpaa-2019	NOUN
cana-2862	149	6	-	-	SYM
cana-2862	149	7	0011	0011	NUM
cana-2862	149	8	[	[	X
cana-2862	149	9	12	12	NUM
cana-2862	149	10	]	]	X
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cana-2862	149	16	fixed	fix	VERB
cana-2862	149	17	point	point	NOUN
cana-2862	149	18	theorems	theorem	NOUN
cana-2862	149	19	on	on	ADP
cana-2862	149	20	complex	complex	ADJ
cana-2862	149	21	valued	value	VERB
cana-2862	149	22	extended	extend	VERB
cana-2862	149	23	b	b	NOUN
cana-2862	149	24	metric	metric	ADJ
cana-2862	149	25	spaces	space	NOUN
cana-2862	149	26	for	for	ADP
cana-2862	149	27	rational	rational	ADJ
cana-2862	149	28	contractions	contraction	NOUN
cana-2862	149	29	with	with	ADP
cana-2862	149	30	application	application	NOUN
cana-2862	149	31	,	,	PUNCT
cana-2862	149	32	aims	aim	VERB
cana-2862	149	33	mathematics	mathematic	NOUN
cana-2862	149	34	,	,	PUNCT
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cana-2862	149	36	,	,	PUNCT
cana-2862	149	37	8(1	8(1	NOUN
cana-2862	149	38	)	)	PUNCT
cana-2862	149	39	,	,	PUNCT
cana-2862	149	40	1360–1374	1360–1374	NUM
cana-2862	149	41	.	.	PUNCT
cana-2862	150	1	doi	doi	NOUN
cana-2862	150	2	:	:	PUNCT
cana-2862	150	3	10.3934	10.3934	NUM
cana-2862	150	4	/	/	SYM
cana-2862	151	1	math.2023068	math.2023068	VERB