id	sid	tid	token	lemma	pos
cana-533	1	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	1	2	communications	communication	NOUN
cana-533	1	3	on	on	ADP
cana-533	1	4	applied	apply	VERB
cana-533	1	5	nonlinear	nonlinear	ADJ
cana-533	1	6	analysis	analysis	NOUN
cana-533	1	7	issn	issn	NOUN
cana-533	1	8	:	:	PUNCT
cana-533	1	9	1074	1074	NUM
cana-533	1	10	-	-	PUNCT
cana-533	1	11	133x	133x	NUM
cana-533	1	12	vol	vol	NOUN
cana-533	1	13	31	31	NUM
cana-533	1	14	no	no	NOUN
cana-533	1	15	.	.	NOUN
cana-533	1	16	2	2	NUM
cana-533	1	17	(	(	PUNCT
cana-533	1	18	2024	2024	NUM
cana-533	1	19	)	)	PUNCT
cana-533	1	20	181	181	NUM
cana-533	1	21	altering	altering	NOUN
cana-533	1	22	point	point	NOUN
cana-533	1	23	results	result	NOUN
cana-533	1	24	involving	involve	VERB
cana-533	1	25	-class	-class	NOUN
cana-533	1	26	functions	function	NOUN
cana-533	1	27	in	in	ADP
cana-533	1	28	partial	partial	ADJ
cana-533	1	29	metric	metric	ADJ
cana-533	1	30	spaces	space	NOUN
cana-533	1	31	abdessalem	abdessalem	PROPN
cana-533	1	32	benterki	benterki	PROPN
cana-533	1	33	lmp2	lmp2	PROPN
cana-533	1	34	m	m	PROPN
cana-533	1	35	laboratory	laboratory	NOUN
cana-533	1	36	,	,	PUNCT
cana-533	1	37	department	department	NOUN
cana-533	1	38	of	of	ADP
cana-533	1	39	mathematics	mathematics	PROPN
cana-533	1	40	and	and	CCONJ
cana-533	1	41	computer	computer	NOUN
cana-533	1	42	science	science	NOUN
cana-533	1	43	,	,	PUNCT
cana-533	1	44	university	university	NOUN
cana-533	1	45	of	of	ADP
cana-533	1	46	medea	medea	PROPN
cana-533	1	47	,	,	PUNCT
cana-533	1	48	medea	medea	PROPN
cana-533	1	49	,	,	PUNCT
cana-533	1	50	algeria	algeria	PROPN
cana-533	1	51	email	email	NOUN
cana-533	1	52	:	:	PUNCT
cana-533	1	53	benterki.abdessalem@gmail.com	benterki.abdessalem@gmail.com	X
cana-533	1	54	article	article	NOUN
cana-533	1	55	history	history	NOUN
cana-533	1	56	:	:	PUNCT
cana-533	1	57	received	receive	VERB
cana-533	1	58	:	:	PUNCT
cana-533	1	59	28	28	NUM
cana-533	1	60	-	-	SYM
cana-533	1	61	01	01	NUM
cana-533	1	62	-	-	PUNCT
cana-533	1	63	2024	2024	NUM
cana-533	1	64	revised	revise	VERB
cana-533	1	65	:	:	PUNCT
cana-533	1	66	15	15	NUM
cana-533	1	67	-	-	PUNCT
cana-533	1	68	04	04	NUM
cana-533	1	69	-	-	PUNCT
cana-533	1	70	2024	2024	NUM
cana-533	1	71	accepted	accept	VERB
cana-533	1	72	:	:	PUNCT
cana-533	1	73	26	26	NUM
cana-533	1	74	-	-	PUNCT
cana-533	1	75	04	04	NUM
cana-533	1	76	-	-	PUNCT
cana-533	1	77	2024	2024	NUM
cana-533	1	78	abstract	abstract	NOUN
cana-533	1	79	:	:	PUNCT
cana-533	1	80	the	the	DET
cana-533	1	81	purpose	purpose	NOUN
cana-533	1	82	of	of	ADP
cana-533	1	83	this	this	DET
cana-533	1	84	paper	paper	NOUN
cana-533	1	85	is	be	AUX
cana-533	1	86	to	to	PART
cana-533	1	87	explore	explore	VERB
cana-533	1	88	the	the	DET
cana-533	1	89	existence	existence	NOUN
cana-533	1	90	of	of	ADP
cana-533	1	91	altering	alter	VERB
cana-533	1	92	points	point	NOUN
cana-533	1	93	satisfying	satisfy	VERB
cana-533	1	94	the	the	DET
cana-533	1	95	system	system	NOUN
cana-533	1	96	of	of	ADP
cana-533	1	97	equations	equation	NOUN
cana-533	1	98	using	use	VERB
cana-533	1	99	the	the	DET
cana-533	1	100	notion	notion	NOUN
cana-533	1	101	of	of	ADP
cana-533	1	102	-class	-class	PROPN
cana-533	1	103	functions	function	NOUN
cana-533	1	104	.	.	PUNCT
cana-533	2	1	here	here	ADV
cana-533	2	2	,	,	PUNCT
cana-533	2	3	and	and	CCONJ
cana-533	2	4	are	be	AUX
cana-533	2	5	two	two	NUM
cana-533	2	6	0	0	NUM
cana-533	2	7	-	-	PUNCT
cana-533	2	8	complete	complete	ADJ
cana-533	2	9	partial	partial	ADJ
cana-533	2	10	metric	metric	ADJ
cana-533	2	11	spaces	space	NOUN
cana-533	2	12	,	,	PUNCT
cana-533	2	13	and	and	CCONJ
cana-533	2	14	and	and	CCONJ
cana-533	2	15	are	be	AUX
cana-533	2	16	set	set	VERB
cana-533	2	17	-	-	PUNCT
cana-533	2	18	valued	value	VERB
cana-533	2	19	mappings	mapping	NOUN
cana-533	2	20	.	.	PUNCT
cana-533	3	1	the	the	DET
cana-533	3	2	key	key	ADJ
cana-533	3	3	idea	idea	NOUN
cana-533	3	4	is	be	AUX
cana-533	3	5	to	to	PART
cana-533	3	6	use	use	VERB
cana-533	3	7	the	the	DET
cana-533	3	8	properties	property	NOUN
cana-533	3	9	of	of	ADP
cana-533	3	10	-class	-class	NOUN
cana-533	3	11	functions	function	NOUN
cana-533	3	12	to	to	PART
cana-533	3	13	establish	establish	VERB
cana-533	3	14	the	the	DET
cana-533	3	15	existence	existence	NOUN
cana-533	3	16	of	of	ADP
cana-533	3	17	a	a	DET
cana-533	3	18	sequence	sequence	NOUN
cana-533	3	19	of	of	ADP
cana-533	3	20	points	point	NOUN
cana-533	3	21	that	that	PRON
cana-533	3	22	converge	converge	VERB
cana-533	3	23	to	to	ADP
cana-533	3	24	the	the	DET
cana-533	3	25	altering	altering	NOUN
cana-533	3	26	point	point	NOUN
cana-533	3	27	,	,	PUNCT
cana-533	3	28	thereby	thereby	ADV
cana-533	3	29	demonstrating	demonstrate	VERB
cana-533	3	30	the	the	DET
cana-533	3	31	existence	existence	NOUN
cana-533	3	32	of	of	ADP
cana-533	3	33	such	such	ADJ
cana-533	3	34	points	point	NOUN
cana-533	3	35	in	in	ADP
cana-533	3	36	the	the	DET
cana-533	3	37	first	first	ADJ
cana-533	3	38	place	place	NOUN
cana-533	3	39	.	.	PUNCT
cana-533	4	1	by	by	ADP
cana-533	4	2	analyzing	analyze	VERB
cana-533	4	3	the	the	DET
cana-533	4	4	properties	property	NOUN
cana-533	4	5	of	of	ADP
cana-533	4	6	and	and	CCONJ
cana-533	4	7	and	and	CCONJ
cana-533	4	8	their	their	PRON
cana-533	4	9	interaction	interaction	NOUN
cana-533	4	10	with	with	ADP
cana-533	4	11	-class	-class	PROPN
cana-533	4	12	functions	function	NOUN
cana-533	4	13	,	,	PUNCT
cana-533	4	14	we	we	PRON
cana-533	4	15	can	can	AUX
cana-533	4	16	gain	gain	VERB
cana-533	4	17	insight	insight	NOUN
cana-533	4	18	into	into	ADP
cana-533	4	19	the	the	DET
cana-533	4	20	nature	nature	NOUN
cana-533	4	21	of	of	ADP
cana-533	4	22	the	the	DET
cana-533	4	23	altering	altering	NOUN
cana-533	4	24	points	point	NOUN
cana-533	4	25	and	and	CCONJ
cana-533	4	26	the	the	DET
cana-533	4	27	underlying	underlie	VERB
cana-533	4	28	spaces	space	NOUN
cana-533	4	29	.	.	PUNCT
cana-533	5	1	the	the	DET
cana-533	5	2	findings	finding	NOUN
cana-533	5	3	of	of	ADP
cana-533	5	4	this	this	DET
cana-533	5	5	study	study	NOUN
cana-533	5	6	provide	provide	VERB
cana-533	5	7	a	a	DET
cana-533	5	8	generalization	generalization	NOUN
cana-533	5	9	and	and	CCONJ
cana-533	5	10	extension	extension	NOUN
cana-533	5	11	of	of	ADP
cana-533	5	12	various	various	ADJ
cana-533	5	13	results	result	NOUN
cana-533	5	14	in	in	ADP
cana-533	5	15	the	the	DET
cana-533	5	16	existing	exist	VERB
cana-533	5	17	literature	literature	NOUN
cana-533	5	18	,	,	PUNCT
cana-533	5	19	highlighting	highlight	VERB
cana-533	5	20	the	the	DET
cana-533	5	21	significance	significance	NOUN
cana-533	5	22	of	of	ADP
cana-533	5	23	the	the	DET
cana-533	5	24	proposed	propose	VERB
cana-533	5	25	approach	approach	NOUN
cana-533	5	26	.	.	PUNCT
cana-533	6	1	keywords	keyword	NOUN
cana-533	6	2	:	:	PUNCT
cana-533	6	3	altering	alter	VERB
cana-533	6	4	point	point	NOUN
cana-533	6	5	,	,	PUNCT
cana-533	6	6	partial	partial	ADJ
cana-533	6	7	metric	metric	ADJ
cana-533	6	8	space	space	NOUN
cana-533	6	9	,	,	PUNCT
cana-533	6	10	set	set	NOUN
cana-533	6	11	-	-	PUNCT
cana-533	6	12	valued	value	VERB
cana-533	6	13	mapping	mapping	NOUN
cana-533	6	14	,	,	PUNCT
cana-533	6	15	-class	-class	PROPN
cana-533	6	16	function	function	NOUN
cana-533	6	17	2010	2010	NUM
cana-533	6	18	mathematics	mathematic	NOUN
cana-533	6	19	subject	subject	ADJ
cana-533	6	20	classification	classification	NOUN
cana-533	6	21	.	.	PUNCT
cana-533	7	1	54h25	54h25	NUM
cana-533	7	2	,	,	PUNCT
cana-533	7	3	47h04	47h04	NUM
cana-533	7	4	.	.	NOUN
cana-533	8	1	1	1	X
cana-533	8	2	.	.	X
cana-533	8	3	introduction	introduction	NOUN
cana-533	8	4	since	since	SCONJ
cana-533	8	5	1922	1922	NUM
cana-533	8	6	,	,	PUNCT
cana-533	8	7	when	when	SCONJ
cana-533	8	8	the	the	DET
cana-533	8	9	celebrated	celebrated	ADJ
cana-533	8	10	banach	banach	NOUN
cana-533	8	11	contraction	contraction	NOUN
cana-533	8	12	principle	principle	NOUN
cana-533	8	13	was	be	AUX
cana-533	8	14	introduced	introduce	VERB
cana-533	8	15	,	,	PUNCT
cana-533	8	16	fixed	fix	VERB
cana-533	8	17	point	point	NOUN
cana-533	8	18	theory	theory	NOUN
cana-533	8	19	has	have	AUX
cana-533	8	20	fascinated	fascinate	VERB
cana-533	8	21	many	many	ADJ
cana-533	8	22	researchers	researcher	NOUN
cana-533	8	23	as	as	ADP
cana-533	8	24	one	one	NUM
cana-533	8	25	of	of	ADP
cana-533	8	26	the	the	DET
cana-533	8	27	most	most	ADV
cana-533	8	28	dynamic	dynamic	ADJ
cana-533	8	29	areas	area	NOUN
cana-533	8	30	.	.	PUNCT
cana-533	9	1	till	till	SCONJ
cana-533	9	2	now	now	ADV
cana-533	9	3	,	,	PUNCT
cana-533	9	4	it	it	PRON
cana-533	9	5	has	have	AUX
cana-533	9	6	been	be	AUX
cana-533	9	7	the	the	DET
cana-533	9	8	fastest	fast	ADJ
cana-533	9	9	growing	grow	VERB
cana-533	9	10	branch	branch	NOUN
cana-533	9	11	of	of	ADP
cana-533	9	12	mathematics	mathematic	NOUN
cana-533	9	13	,	,	PUNCT
cana-533	9	14	with	with	ADP
cana-533	9	15	several	several	ADJ
cana-533	9	16	applications	application	NOUN
cana-533	9	17	to	to	ADP
cana-533	9	18	real	real	ADJ
cana-533	9	19	-	-	PUNCT
cana-533	9	20	world	world	NOUN
cana-533	9	21	problems	problem	NOUN
cana-533	9	22	.	.	PUNCT
cana-533	10	1	there	there	PRON
cana-533	10	2	is	be	VERB
cana-533	10	3	a	a	DET
cana-533	10	4	vast	vast	ADJ
cana-533	10	5	literature	literature	NOUN
cana-533	10	6	on	on	ADP
cana-533	10	7	fixed	fix	VERB
cana-533	10	8	point	point	NOUN
cana-533	10	9	theory	theory	NOUN
cana-533	10	10	,	,	PUNCT
cana-533	10	11	and	and	CCONJ
cana-533	10	12	it	it	PRON
cana-533	10	13	is	be	AUX
cana-533	10	14	currently	currently	ADV
cana-533	10	15	a	a	DET
cana-533	10	16	very	very	ADV
cana-533	10	17	active	active	ADJ
cana-533	10	18	field	field	NOUN
cana-533	10	19	of	of	ADP
cana-533	10	20	research	research	NOUN
cana-533	10	21	.	.	PUNCT
cana-533	11	1	as	as	ADP
cana-533	11	2	a	a	DET
cana-533	11	3	tool	tool	NOUN
cana-533	11	4	for	for	ADP
cana-533	11	5	proving	prove	VERB
cana-533	11	6	the	the	DET
cana-533	11	7	existence	existence	NOUN
cana-533	11	8	and	and	CCONJ
cana-533	11	9	uniqueness	uniqueness	NOUN
cana-533	11	10	of	of	ADP
cana-533	11	11	solutions	solution	NOUN
cana-533	11	12	to	to	ADP
cana-533	11	13	different	different	ADJ
cana-533	11	14	mathematical	mathematical	ADJ
cana-533	11	15	problems	problem	NOUN
cana-533	11	16	,	,	PUNCT
cana-533	11	17	fixed	fix	VERB
cana-533	11	18	point	point	NOUN
cana-533	11	19	theorems	theorem	NOUN
cana-533	11	20	are	be	AUX
cana-533	11	21	essential	essential	ADJ
cana-533	11	22	in	in	ADP
cana-533	11	23	many	many	ADJ
cana-533	11	24	theoretical	theoretical	ADJ
cana-533	11	25	and	and	CCONJ
cana-533	11	26	applied	applied	ADJ
cana-533	11	27	branches	branch	NOUN
cana-533	11	28	of	of	ADP
cana-533	11	29	mathematics	mathematic	NOUN
cana-533	11	30	,	,	PUNCT
cana-533	11	31	including	include	VERB
cana-533	11	32	optimisation	optimisation	NOUN
cana-533	11	33	problems	problem	NOUN
cana-533	11	34	,	,	PUNCT
cana-533	11	35	dynamical	dynamical	ADJ
cana-533	11	36	systems	system	NOUN
cana-533	11	37	,	,	PUNCT
cana-533	11	38	nonlinear	nonlinear	ADJ
cana-533	11	39	analysis	analysis	NOUN
cana-533	11	40	,	,	PUNCT
cana-533	11	41	integral	integral	ADJ
cana-533	11	42	equations	equation	NOUN
cana-533	11	43	,	,	PUNCT
cana-533	11	44	partial	partial	ADJ
cana-533	11	45	differential	differential	NOUN
cana-533	11	46	equations	equation	NOUN
cana-533	11	47	,	,	PUNCT
cana-533	11	48	variational	variational	ADJ
cana-533	11	49	inequalities	inequality	NOUN
cana-533	11	50	,	,	PUNCT
cana-533	11	51	fractals	fractal	NOUN
cana-533	11	52	,	,	PUNCT
cana-533	11	53	economics	economic	NOUN
cana-533	11	54	,	,	PUNCT
cana-533	11	55	game	game	NOUN
cana-533	11	56	theory	theory	NOUN
cana-533	11	57	,	,	PUNCT
cana-533	11	58	and	and	CCONJ
cana-533	11	59	nonlinear	nonlinear	ADJ
cana-533	11	60	analysis	analysis	NOUN
cana-533	11	61	.	.	PUNCT
cana-533	12	1	in	in	ADP
cana-533	12	2	many	many	ADJ
cana-533	12	3	distinct	distinct	ADJ
cana-533	12	4	abstract	abstract	ADJ
cana-533	12	5	spaces	space	NOUN
cana-533	12	6	,	,	PUNCT
cana-533	12	7	diverse	diverse	ADJ
cana-533	12	8	results	result	NOUN
cana-533	12	9	about	about	ADP
cana-533	12	10	fixed	fix	VERB
cana-533	12	11	points	point	NOUN
cana-533	12	12	,	,	PUNCT
cana-533	12	13	common	common	ADJ
cana-533	12	14	fixed	fix	VERB
cana-533	12	15	points	point	NOUN
cana-533	12	16	,	,	PUNCT
cana-533	12	17	coincidence	coincidence	NOUN
cana-533	12	18	points	point	NOUN
cana-533	12	19	,	,	PUNCT
cana-533	12	20	altering	alter	VERB
cana-533	12	21	points	point	NOUN
cana-533	12	22	,	,	PUNCT
cana-533	12	23	and	and	CCONJ
cana-533	12	24	so	so	ADV
cana-533	12	25	on	on	ADV
cana-533	12	26	for	for	SCONJ
cana-533	12	27	single	single	ADV
cana-533	12	28	-	-	PUNCT
cana-533	12	29	valued	value	VERB
cana-533	12	30	and	and	CCONJ
cana-533	12	31	set	set	NOUN
cana-533	12	32	-	-	PUNCT
cana-533	12	33	valued	value	VERB
cana-533	12	34	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	12	35	communications	communication	NOUN
cana-533	12	36	on	on	ADP
cana-533	12	37	applied	apply	VERB
cana-533	12	38	nonlinear	nonlinear	ADJ
cana-533	12	39	analysis	analysis	NOUN
cana-533	12	40	issn	issn	NOUN
cana-533	12	41	:	:	PUNCT
cana-533	12	42	1074	1074	NUM
cana-533	12	43	-	-	PUNCT
cana-533	12	44	133x	133x	NUM
cana-533	12	45	vol	vol	NOUN
cana-533	12	46	31	31	NUM
cana-533	12	47	no	no	NOUN
cana-533	12	48	.	.	NOUN
cana-533	12	49	2	2	NUM
cana-533	12	50	(	(	PUNCT
cana-533	12	51	2024	2024	NUM
cana-533	12	52	)	)	PUNCT
cana-533	12	53	182	182	NUM
cana-533	12	54	mappings	mapping	NOUN
cana-533	12	55	have	have	AUX
cana-533	12	56	been	be	AUX
cana-533	12	57	investigated	investigate	VERB
cana-533	12	58	for	for	ADP
cana-533	12	59	various	various	ADJ
cana-533	12	60	contractive	contractive	ADJ
cana-533	12	61	conditions	condition	NOUN
cana-533	12	62	,	,	PUNCT
cana-533	12	63	and	and	CCONJ
cana-533	12	64	this	this	DET
cana-533	12	65	tradition	tradition	NOUN
cana-533	12	66	continues	continue	VERB
cana-533	12	67	.	.	PUNCT
cana-533	13	1	partial	partial	ADJ
cana-533	13	2	metric	metric	ADJ
cana-533	13	3	spaces	space	NOUN
cana-533	13	4	are	be	AUX
cana-533	13	5	an	an	DET
cana-533	13	6	extension	extension	NOUN
cana-533	13	7	of	of	ADP
cana-533	13	8	conventional	conventional	ADJ
cana-533	13	9	metric	metric	ADJ
cana-533	13	10	spaces	space	NOUN
cana-533	13	11	where	where	SCONJ
cana-533	13	12	the	the	DET
cana-533	13	13	value	value	NOUN
cana-533	13	14	of	of	ADP
cana-533	13	15	self	self	NOUN
cana-533	13	16	-	-	PUNCT
cana-533	13	17	distance	distance	NOUN
cana-533	13	18	for	for	ADP
cana-533	13	19	each	each	DET
cana-533	13	20	point	point	NOUN
cana-533	13	21	does	do	AUX
cana-533	13	22	not	not	PART
cana-533	13	23	have	have	VERB
cana-533	13	24	to	to	PART
cana-533	13	25	be	be	AUX
cana-533	13	26	zero	zero	NUM
cana-533	13	27	.	.	PUNCT
cana-533	14	1	matthews	matthews	PROPN
cana-533	14	2	first	first	ADV
cana-533	14	3	discussed	discuss	VERB
cana-533	14	4	partial	partial	ADJ
cana-533	14	5	metric	metric	ADJ
cana-533	14	6	spaces	space	NOUN
cana-533	14	7	in	in	ADP
cana-533	14	8	[	[	X
cana-533	14	9	5	5	NUM
cana-533	14	10	]	]	PUNCT
cana-533	14	11	.	.	PUNCT
cana-533	15	1	romaguera	romaguera	PROPN
cana-533	15	2	presented	present	VERB
cana-533	15	3	the	the	DET
cana-533	15	4	concept	concept	NOUN
cana-533	15	5	of	of	ADP
cana-533	15	6	a	a	DET
cana-533	15	7	0	0	NUM
cana-533	15	8	-	-	PUNCT
cana-533	15	9	cauchy	cauchy	ADJ
cana-533	15	10	sequence	sequence	NOUN
cana-533	15	11	in	in	ADP
cana-533	15	12	a	a	DET
cana-533	15	13	partial	partial	ADJ
cana-533	15	14	metric	metric	ADJ
cana-533	15	15	space	space	NOUN
cana-533	15	16	,	,	PUNCT
cana-533	15	17	followed	follow	VERB
cana-533	15	18	by	by	ADP
cana-533	15	19	the	the	DET
cana-533	15	20	concept	concept	NOUN
cana-533	15	21	of	of	ADP
cana-533	15	22	a	a	DET
cana-533	15	23	0	0	NUM
cana-533	15	24	-	-	PUNCT
cana-533	15	25	complete	complete	ADJ
cana-533	15	26	partial	partial	ADJ
cana-533	15	27	metric	metric	ADJ
cana-533	15	28	space	space	NOUN
cana-533	15	29	in	in	ADP
cana-533	15	30	[	[	X
cana-533	15	31	8	8	NUM
cana-533	15	32	]	]	PUNCT
cana-533	15	33	.	.	PUNCT
cana-533	16	1	sahu	sahu	PROPN
cana-533	16	2	first	first	ADV
cana-533	16	3	proposes	propose	VERB
cana-533	16	4	the	the	DET
cana-533	16	5	idea	idea	NOUN
cana-533	16	6	of	of	ADP
cana-533	16	7	altering	alter	VERB
cana-533	16	8	points	point	NOUN
cana-533	16	9	in	in	ADP
cana-533	16	10	[	[	X
cana-533	16	11	9	9	NUM
cana-533	16	12	]	]	PUNCT
cana-533	16	13	for	for	ADP
cana-533	16	14	the	the	DET
cana-533	16	15	single	single	ADV
cana-533	16	16	-	-	PUNCT
cana-533	16	17	valued	value	VERB
cana-533	16	18	case	case	NOUN
cana-533	16	19	in	in	ADP
cana-533	16	20	order	order	NOUN
cana-533	16	21	to	to	PART
cana-533	16	22	introduce	introduce	VERB
cana-533	16	23	a	a	DET
cana-533	16	24	parallel	parallel	ADJ
cana-533	16	25	s	s	NOUN
cana-533	16	26	-	-	PUNCT
cana-533	16	27	iteration	iteration	NOUN
cana-533	16	28	process	process	NOUN
cana-533	16	29	to	to	PART
cana-533	16	30	solve	solve	VERB
cana-533	16	31	a	a	DET
cana-533	16	32	system	system	NOUN
cana-533	16	33	of	of	ADP
cana-533	16	34	operator	operator	NOUN
cana-533	16	35	equations	equation	NOUN
cana-533	16	36	.	.	PUNCT
cana-533	17	1	these	these	DET
cana-533	17	2	points	point	NOUN
cana-533	17	3	are	be	AUX
cana-533	17	4	more	more	ADV
cana-533	17	5	general	general	ADJ
cana-533	17	6	than	than	ADP
cana-533	17	7	the	the	DET
cana-533	17	8	fixed	fix	VERB
cana-533	17	9	points	point	NOUN
cana-533	17	10	,	,	PUNCT
cana-533	17	11	and	and	CCONJ
cana-533	17	12	they	they	PRON
cana-533	17	13	are	be	AUX
cana-533	17	14	extended	extend	VERB
cana-533	17	15	by	by	ADP
cana-533	17	16	petrusel	petrusel	NOUN
cana-533	17	17	,	,	PUNCT
cana-533	17	18	yao	yao	NOUN
cana-533	17	19	in	in	ADP
cana-533	17	20	[	[	X
cana-533	17	21	7	7	NUM
cana-533	17	22	]	]	PUNCT
cana-533	17	23	for	for	ADP
cana-533	17	24	the	the	DET
cana-533	17	25	set	set	NOUN
cana-533	17	26	-	-	PUNCT
cana-533	17	27	valued	value	VERB
cana-533	17	28	case	case	NOUN
cana-533	17	29	.	.	PUNCT
cana-533	18	1	for	for	ADP
cana-533	18	2	more	more	ADJ
cana-533	18	3	information	information	NOUN
cana-533	18	4	on	on	ADP
cana-533	18	5	altering	alter	VERB
cana-533	18	6	points	point	NOUN
cana-533	18	7	and	and	CCONJ
cana-533	18	8	applications	application	NOUN
cana-533	18	9	to	to	ADP
cana-533	18	10	variational	variational	ADJ
cana-533	18	11	problems	problem	NOUN
cana-533	18	12	one	one	PRON
cana-533	18	13	can	can	AUX
cana-533	18	14	refer	refer	VERB
cana-533	18	15	to	to	ADP
cana-533	18	16	[	[	X
cana-533	18	17	9,10	9,10	NUM
cana-533	18	18	]	]	X
cana-533	18	19	.	.	PUNCT
cana-533	19	1	nedelcheva	nedelcheva	PROPN
cana-533	20	1	[	[	X
cana-533	20	2	6	6	NUM
cana-533	20	3	]	]	PUNCT
cana-533	20	4	presented	present	VERB
cana-533	20	5	one	one	NUM
cana-533	20	6	of	of	ADP
cana-533	20	7	the	the	DET
cana-533	20	8	altering	alter	VERB
cana-533	20	9	point	point	NOUN
cana-533	20	10	results	result	NOUN
cana-533	20	11	and	and	CCONJ
cana-533	20	12	extended	extend	VERB
cana-533	20	13	the	the	DET
cana-533	20	14	fixed	fix	VERB
cana-533	20	15	point	point	NOUN
cana-533	20	16	result	result	NOUN
cana-533	20	17	mentioned	mention	VERB
cana-533	20	18	in	in	ADP
cana-533	20	19	[	[	X
cana-533	20	20	3	3	NUM
cana-533	20	21	]	]	PUNCT
cana-533	20	22	for	for	ADP
cana-533	20	23	the	the	DET
cana-533	20	24	objective	objective	NOUN
cana-533	20	25	of	of	ADP
cana-533	20	26	composing	compose	VERB
cana-533	20	27	two	two	NUM
cana-533	20	28	set	set	NOUN
cana-533	20	29	-	-	PUNCT
cana-533	20	30	valued	value	VERB
cana-533	20	31	mappings	mapping	NOUN
cana-533	20	32	on	on	ADP
cana-533	20	33	complete	complete	ADJ
cana-533	20	34	metric	metric	ADJ
cana-533	20	35	spaces	space	NOUN
cana-533	20	36	.	.	PUNCT
cana-533	21	1	the	the	DET
cana-533	21	2	following	follow	VERB
cana-533	21	3	is	be	AUX
cana-533	21	4	how	how	SCONJ
cana-533	21	5	this	this	DET
cana-533	21	6	work	work	NOUN
cana-533	21	7	is	be	AUX
cana-533	21	8	organized	organize	VERB
cana-533	21	9	:	:	PUNCT
cana-533	21	10	section	section	NOUN
cana-533	21	11	2	2	NUM
cana-533	21	12	presents	present	VERB
cana-533	21	13	some	some	DET
cana-533	21	14	preliminary	preliminary	ADJ
cana-533	21	15	observations	observation	NOUN
cana-533	21	16	and	and	CCONJ
cana-533	21	17	definitions	definition	NOUN
cana-533	21	18	.	.	PUNCT
cana-533	22	1	section	section	NOUN
cana-533	22	2	3	3	NUM
cana-533	22	3	establishes	establish	VERB
cana-533	22	4	the	the	DET
cana-533	22	5	primary	primary	ADJ
cana-533	22	6	conclusion	conclusion	NOUN
cana-533	22	7	by	by	ADP
cana-533	22	8	extending	extend	VERB
cana-533	22	9	both	both	CCONJ
cana-533	22	10	theorem	theorem	VERB
cana-533	22	11	11	11	NUM
cana-533	22	12	in	in	ADP
cana-533	22	13	[	[	X
cana-533	22	14	6	6	NUM
cana-533	22	15	]	]	PUNCT
cana-533	22	16	and	and	CCONJ
cana-533	22	17	theorem	theorem	VERB
cana-533	22	18	3.4	3.4	NUM
cana-533	22	19	in	in	ADP
cana-533	22	20	[	[	X
cana-533	22	21	2	2	NUM
cana-533	22	22	]	]	PUNCT
cana-533	22	23	for	for	ADP
cana-533	22	24	a	a	DET
cana-533	22	25	composition	composition	NOUN
cana-533	22	26	of	of	ADP
cana-533	22	27	two	two	NUM
cana-533	22	28	set	set	NOUN
cana-533	22	29	-	-	PUNCT
cana-533	22	30	valued	value	VERB
cana-533	22	31	mappings	mapping	NOUN
cana-533	22	32	using	use	VERB
cana-533	22	33	the	the	DET
cana-533	22	34	idea	idea	NOUN
cana-533	22	35	of	of	ADP
cana-533	22	36	-class	-class	PROPN
cana-533	22	37	functions	function	NOUN
cana-533	22	38	.	.	PUNCT
cana-533	23	1	finally	finally	ADV
cana-533	23	2	,	,	PUNCT
cana-533	23	3	we	we	PRON
cana-533	23	4	will	will	AUX
cana-533	23	5	discuss	discuss	VERB
cana-533	23	6	some	some	DET
cana-533	23	7	relevant	relevant	ADJ
cana-533	23	8	corollaries	corollary	NOUN
cana-533	23	9	.	.	PUNCT
cana-533	24	1	2	2	X
cana-533	24	2	.	.	X
cana-533	24	3	preliminary	preliminary	ADJ
cana-533	24	4	results	result	NOUN
cana-533	24	5	let	let	VERB
cana-533	24	6	us	we	PRON
cana-533	24	7	begin	begin	VERB
cana-533	24	8	by	by	ADP
cana-533	24	9	providing	provide	VERB
cana-533	24	10	an	an	DET
cana-533	24	11	overview	overview	NOUN
cana-533	24	12	of	of	ADP
cana-533	24	13	some	some	PRON
cana-533	24	14	of	of	ADP
cana-533	24	15	the	the	DET
cana-533	24	16	key	key	ADJ
cana-533	24	17	definitions	definition	NOUN
cana-533	24	18	and	and	CCONJ
cana-533	24	19	characteristics	characteristic	NOUN
cana-533	24	20	of	of	ADP
cana-533	24	21	partial	partial	ADJ
cana-533	24	22	metric	metric	ADJ
cana-533	24	23	spaces	space	NOUN
cana-533	24	24	.	.	PUNCT
cana-533	25	1	a	a	DET
cana-533	25	2	partial	partial	ADJ
cana-533	25	3	metric	metric	NOUN
cana-533	25	4	on	on	ADP
cana-533	25	5	a	a	DET
cana-533	25	6	non	non	ADJ
cana-533	25	7	-	-	ADJ
cana-533	25	8	empty	empty	ADJ
cana-533	25	9	set	set	NOUN
cana-533	25	10	is	be	AUX
cana-533	25	11	defined	define	VERB
cana-533	25	12	as	as	ADP
cana-533	25	13	a	a	DET
cana-533	25	14	function	function	NOUN
cana-533	25	15	that	that	PRON
cana-533	25	16	satisfies	satisfy	VERB
cana-533	25	17	the	the	DET
cana-533	25	18	following	follow	VERB
cana-533	25	19	conditions	condition	NOUN
cana-533	25	20	for	for	ADP
cana-533	25	21	all	all	PRON
cana-533	25	22	:	:	PUNCT
cana-533	25	23	if	if	SCONJ
cana-533	25	24	satisfies	satisfy	VERB
cana-533	25	25	the	the	DET
cana-533	25	26	above	above	ADJ
cana-533	25	27	conditions	condition	NOUN
cana-533	25	28	,	,	PUNCT
cana-533	25	29	then	then	ADV
cana-533	25	30	the	the	DET
cana-533	25	31	pair	pair	NOUN
cana-533	25	32	is	be	AUX
cana-533	25	33	referred	refer	VERB
cana-533	25	34	to	to	ADP
cana-533	25	35	as	as	ADP
cana-533	25	36	a	a	DET
cana-533	25	37	partial	partial	ADJ
cana-533	25	38	metric	metric	ADJ
cana-533	25	39	space	space	NOUN
cana-533	25	40	.	.	PUNCT
cana-533	26	1	this	this	DET
cana-533	26	2	definition	definition	NOUN
cana-533	26	3	was	be	AUX
cana-533	26	4	introduced	introduce	VERB
cana-533	26	5	by	by	ADP
cana-533	26	6	matthews	matthews	PROPN
cana-533	26	7	[	[	X
cana-533	26	8	5	5	NUM
cana-533	26	9	]	]	PUNCT
cana-533	26	10	.	.	PUNCT
cana-533	27	1	the	the	DET
cana-533	27	2	closed	closed	ADJ
cana-533	27	3	-ball	-ball	NOUN
cana-533	27	4	with	with	ADP
cana-533	27	5	radius	radius	NOUN
cana-533	27	6	and	and	CCONJ
cana-533	27	7	as	as	SCONJ
cana-533	27	8	its	its	PRON
cana-533	27	9	center	center	NOUN
cana-533	27	10	is	be	AUX
cana-533	27	11	denoted	denote	VERB
cana-533	27	12	by	by	ADP
cana-533	27	13	and	and	CCONJ
cana-533	27	14	opened	open	VERB
cana-533	27	15	-ball	-ball	NOUN
cana-533	27	16	by	by	ADP
cana-533	27	17	,	,	PUNCT
cana-533	27	18	where	where	SCONJ
cana-533	27	19	represents	represent	VERB
cana-533	27	20	for	for	ADP
cana-533	27	21	convenience	convenience	NOUN
cana-533	27	22	.	.	PUNCT
cana-533	28	1	the	the	DET
cana-533	28	2	function	function	NOUN
cana-533	28	3	provided	provide	VERB
cana-533	28	4	by	by	ADP
cana-533	28	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	28	6	communications	communication	NOUN
cana-533	28	7	on	on	ADP
cana-533	28	8	applied	apply	VERB
cana-533	28	9	nonlinear	nonlinear	ADJ
cana-533	28	10	analysis	analysis	NOUN
cana-533	28	11	issn	issn	NOUN
cana-533	28	12	:	:	PUNCT
cana-533	28	13	1074	1074	NUM
cana-533	28	14	-	-	PUNCT
cana-533	28	15	133x	133x	NUM
cana-533	28	16	vol	vol	NOUN
cana-533	28	17	31	31	NUM
cana-533	28	18	no	no	NOUN
cana-533	28	19	.	.	NOUN
cana-533	28	20	2	2	NUM
cana-533	28	21	(	(	PUNCT
cana-533	28	22	2024	2024	NUM
cana-533	28	23	)	)	PUNCT
cana-533	28	24	183	183	NUM
cana-533	28	25	is	be	AUX
cana-533	28	26	a	a	DET
cana-533	28	27	metric	metric	NOUN
cana-533	28	28	on	on	ADP
cana-533	28	29	if	if	SCONJ
cana-533	28	30	is	be	AUX
cana-533	28	31	a	a	DET
cana-533	28	32	partial	partial	ADJ
cana-533	28	33	metric	metric	NOUN
cana-533	28	34	on	on	ADP
cana-533	28	35	.	.	PUNCT
cana-533	29	1	consider	consider	VERB
cana-533	29	2	to	to	PART
cana-533	29	3	be	be	AUX
cana-533	29	4	a	a	DET
cana-533	29	5	partial	partial	ADJ
cana-533	29	6	metric	metric	ADJ
cana-533	29	7	space	space	NOUN
cana-533	29	8	.	.	PUNCT
cana-533	30	1	then	then	ADV
cana-533	30	2	:	:	PUNCT
cana-533	30	3	•	•	ADP
cana-533	30	4	if	if	SCONJ
cana-533	30	5	,	,	PUNCT
cana-533	30	6	then	then	ADV
cana-533	30	7	is	be	AUX
cana-533	30	8	said	say	VERB
cana-533	30	9	to	to	PART
cana-533	30	10	converge	converge	VERB
cana-533	30	11	to	to	ADP
cana-533	30	12	a	a	DET
cana-533	30	13	point	point	NOUN
cana-533	30	14	.	.	PUNCT
cana-533	31	1	•	•	INTJ
cana-533	31	2	if	if	SCONJ
cana-533	31	3	there	there	PRON
cana-533	31	4	exists	exist	VERB
cana-533	31	5	(	(	PUNCT
cana-533	31	6	and	and	CCONJ
cana-533	31	7	is	be	AUX
cana-533	31	8	finite	finite	ADJ
cana-533	31	9	)	)	PUNCT
cana-533	31	10	,	,	PUNCT
cana-533	31	11	a	a	DET
cana-533	31	12	sequence	sequence	NOUN
cana-533	31	13	is	be	AUX
cana-533	31	14	called	call	VERB
cana-533	31	15	a	a	DET
cana-533	31	16	cauchy	cauchy	ADJ
cana-533	31	17	sequence	sequence	NOUN
cana-533	31	18	.	.	PUNCT
cana-533	32	1	is	be	AUX
cana-533	32	2	referred	refer	VERB
cana-533	32	3	to	to	ADP
cana-533	32	4	as	as	ADP
cana-533	32	5	a	a	DET
cana-533	32	6	0	0	NUM
cana-533	32	7	-	-	PUNCT
cana-533	32	8	cauchy	cauchy	ADJ
cana-533	32	9	sequence	sequence	NOUN
cana-533	32	10	if	if	SCONJ
cana-533	32	11	.	.	PUNCT
cana-533	33	1	•	•	INTJ
cana-533	33	2	in	in	ADV
cana-533	33	3	,	,	PUNCT
cana-533	33	4	every	every	DET
cana-533	33	5	cauchy	cauchy	ADJ
cana-533	33	6	sequence	sequence	NOUN
cana-533	33	7	that	that	PRON
cana-533	33	8	converges	converge	VERB
cana-533	33	9	to	to	ADP
cana-533	33	10	a	a	DET
cana-533	33	11	point	point	NOUN
cana-533	33	12	in	in	ADP
cana-533	33	13	such	such	DET
cana-533	33	14	a	a	DET
cana-533	33	15	way	way	NOUN
cana-533	33	16	that	that	PRON
cana-533	33	17	indicates	indicate	VERB
cana-533	33	18	that	that	PRON
cana-533	33	19	is	be	AUX
cana-533	33	20	complete	complete	ADJ
cana-533	33	21	.	.	PUNCT
cana-533	34	1	•	•	NUM
cana-533	34	2	regarding	regard	VERB
cana-533	34	3	,	,	PUNCT
cana-533	34	4	every	every	DET
cana-533	34	5	0	0	NUM
cana-533	34	6	-	-	PUNCT
cana-533	34	7	cauchy	cauchy	ADJ
cana-533	34	8	sequence	sequence	NOUN
cana-533	34	9	in	in	ADP
cana-533	34	10	converges	converge	NOUN
cana-533	34	11	to	to	ADP
cana-533	34	12	a	a	DET
cana-533	34	13	point	point	NOUN
cana-533	34	14	where	where	SCONJ
cana-533	34	15	.	.	PUNCT
cana-533	35	1	a	a	DET
cana-533	35	2	0	0	NUM
cana-533	35	3	-	-	PUNCT
cana-533	35	4	complete	complete	ADJ
cana-533	35	5	is	be	AUX
cana-533	35	6	what	what	PRON
cana-533	35	7	this	this	PRON
cana-533	35	8	is	be	AUX
cana-533	35	9	called	call	VERB
cana-533	35	10	.	.	PUNCT
cana-533	36	1	•	•	INTJ
cana-533	36	2	in	in	ADV
cana-533	36	3	,	,	PUNCT
cana-533	36	4	any	any	DET
cana-533	36	5	0	0	NUM
cana-533	36	6	-	-	PUNCT
cana-533	36	7	cauchy	cauchy	ADJ
cana-533	36	8	sequence	sequence	NOUN
cana-533	36	9	is	be	AUX
cana-533	36	10	a	a	DET
cana-533	36	11	cauchy	cauchy	NOUN
cana-533	36	12	in	in	ADV
cana-533	36	13	.	.	PUNCT
cana-533	37	1	•	•	NUM
cana-533	37	2	complete	complete	ADJ
cana-533	37	3	partial	partial	ADJ
cana-533	37	4	metric	metric	ADJ
cana-533	37	5	spaces	space	NOUN
cana-533	37	6	are	be	AUX
cana-533	37	7	all	all	PRON
cana-533	37	8	0	0	NOUN
cana-533	37	9	-	-	PUNCT
cana-533	37	10	complete	complete	ADJ
cana-533	37	11	.	.	PUNCT
cana-533	38	1	however	however	ADV
cana-533	38	2	,	,	PUNCT
cana-533	38	3	the	the	DET
cana-533	38	4	reverse	reverse	NOUN
cana-533	38	5	is	be	AUX
cana-533	38	6	not	not	PART
cana-533	38	7	true	true	ADJ
cana-533	38	8	.	.	PUNCT
cana-533	39	1	in	in	ADP
cana-533	39	2	the	the	DET
cana-533	39	3	partial	partial	ADJ
cana-533	39	4	metric	metric	ADJ
cana-533	39	5	space	space	NOUN
cana-533	39	6	,	,	PUNCT
cana-533	39	7	the	the	PRON
cana-533	39	8	represents	represent	VERB
cana-533	39	9	the	the	DET
cana-533	39	10	family	family	NOUN
cana-533	39	11	of	of	ADP
cana-533	39	12	all	all	DET
cana-533	39	13	closed	closed	ADJ
cana-533	39	14	and	and	CCONJ
cana-533	39	15	nonempty	nonempty	ADJ
cana-533	39	16	subsets	subset	NOUN
cana-533	39	17	.	.	PUNCT
cana-533	40	1	with	with	ADP
cana-533	40	2	and	and	CCONJ
cana-533	40	3	,	,	PUNCT
cana-533	40	4	we	we	PRON
cana-533	40	5	establish	establish	VERB
cana-533	40	6	such	such	ADJ
cana-533	40	7	that	that	SCONJ
cana-533	40	8	in	in	ADP
cana-533	40	9	conformity	conformity	NOUN
cana-533	40	10	with	with	ADP
cana-533	40	11	the	the	DET
cana-533	40	12	convention	convention	NOUN
cana-533	40	13	the	the	DET
cana-533	40	14	following	follow	VERB
cana-533	40	15	concepts	concept	NOUN
cana-533	40	16	will	will	AUX
cana-533	40	17	be	be	AUX
cana-533	40	18	used	use	VERB
cana-533	40	19	throughout	throughout	ADP
cana-533	40	20	the	the	DET
cana-533	40	21	article	article	NOUN
cana-533	40	22	.	.	PUNCT
cana-533	41	1	let	let	VERB
cana-533	41	2	denotes	denote	NOUN
cana-533	41	3	a	a	DET
cana-533	41	4	set	set	NOUN
cana-533	41	5	-	-	PUNCT
cana-533	41	6	valued	value	VERB
cana-533	41	7	mapping	mapping	NOUN
cana-533	41	8	defined	define	VERB
cana-533	41	9	in	in	ADP
cana-533	41	10	the	the	DET
cana-533	41	11	partial	partial	ADJ
cana-533	41	12	metric	metric	ADJ
cana-533	41	13	space	space	NOUN
cana-533	41	14	with	with	ADP
cana-533	41	15	closed	closed	ADJ
cana-533	41	16	values	value	NOUN
cana-533	41	17	in	in	ADP
cana-533	41	18	the	the	DET
cana-533	41	19	partial	partial	ADJ
cana-533	41	20	metric	metric	ADJ
cana-533	41	21	space	space	NOUN
cana-533	41	22	.	.	PUNCT
cana-533	42	1	we	we	PRON
cana-533	42	2	define	define	VERB
cana-533	42	3	the	the	DET
cana-533	42	4	composition	composition	NOUN
cana-533	42	5	of	of	ADP
cana-533	42	6	mappings	mapping	NOUN
cana-533	42	7	and	and	CCONJ
cana-533	42	8	as	as	ADP
cana-533	42	9	the	the	DET
cana-533	42	10	mapping	mapping	NOUN
cana-533	42	11	such	such	ADJ
cana-533	42	12	that	that	PRON
cana-533	42	13	for	for	ADP
cana-533	42	14	all	all	PRON
cana-533	42	15	,	,	PUNCT
cana-533	42	16	.	.	PUNCT
cana-533	43	1	the	the	DET
cana-533	43	2	sets	set	NOUN
cana-533	43	3	and	and	CCONJ
cana-533	43	4	in	in	ADP
cana-533	43	5	the	the	DET
cana-533	43	6	following	follow	VERB
cana-533	43	7	text	text	NOUN
cana-533	43	8	represent	represent	VERB
cana-533	43	9	intervals	interval	NOUN
cana-533	43	10	on	on	ADP
cana-533	43	11	that	that	DET
cana-533	43	12	contain	contain	VERB
cana-533	43	13	,	,	PUNCT
cana-533	43	14	like	like	ADP
cana-533	43	15	,	,	PUNCT
cana-533	43	16	,	,	PUNCT
cana-533	43	17	or	or	CCONJ
cana-533	43	18	.	.	PUNCT
cana-533	44	1	we	we	PRON
cana-533	44	2	also	also	ADV
cana-533	44	3	introduce	introduce	VERB
cana-533	44	4	,	,	PUNCT
cana-533	44	5	a	a	DET
cana-533	44	6	set	set	NOUN
cana-533	44	7	-	-	PUNCT
cana-533	44	8	valued	value	VERB
cana-533	44	9	mapping	mapping	NOUN
cana-533	44	10	defined	define	VERB
cana-533	44	11	on	on	ADP
cana-533	44	12	for	for	ADP
cana-533	44	13	.	.	PUNCT
cana-533	45	1	we	we	PRON
cana-533	45	2	denote	denote	VERB
cana-533	45	3	as	as	ADP
cana-533	45	4	lemma	lemma	PROPN
cana-533	45	5	2.1	2.1	NUM
cana-533	45	6	.	.	PUNCT
cana-533	46	1	[	[	X
cana-533	46	2	2	2	X
cana-533	46	3	]	]	PUNCT
cana-533	46	4	let	let	AUX
cana-533	46	5	be	be	AUX
cana-533	46	6	a	a	DET
cana-533	46	7	partial	partial	ADJ
cana-533	46	8	metric	metric	ADJ
cana-533	46	9	space	space	NOUN
cana-533	46	10	and	and	CCONJ
cana-533	46	11	let	let	VERB
cana-533	46	12	.	.	PUNCT
cana-533	47	1	then	then	ADV
cana-533	47	2	,	,	PUNCT
cana-533	47	3	the	the	DET
cana-533	47	4	following	follow	VERB
cana-533	47	5	statements	statement	NOUN
cana-533	47	6	hold	hold	VERB
cana-533	47	7	:	:	PUNCT
cana-533	47	8	1	1	NUM
cana-533	47	9	.	.	PUNCT
cana-533	47	10	.	.	PUNCT
cana-533	48	1	2	2	X
cana-533	48	2	.	.	X
cana-533	48	3	.	.	PUNCT
cana-533	49	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	49	2	communications	communication	NOUN
cana-533	49	3	on	on	ADP
cana-533	49	4	applied	apply	VERB
cana-533	49	5	nonlinear	nonlinear	ADJ
cana-533	49	6	analysis	analysis	NOUN
cana-533	49	7	issn	issn	NOUN
cana-533	49	8	:	:	PUNCT
cana-533	49	9	1074	1074	NUM
cana-533	49	10	-	-	PUNCT
cana-533	49	11	133x	133x	NUM
cana-533	49	12	vol	vol	NOUN
cana-533	49	13	31	31	NUM
cana-533	49	14	no	no	NOUN
cana-533	49	15	.	.	NOUN
cana-533	49	16	2	2	NUM
cana-533	49	17	(	(	PUNCT
cana-533	49	18	2024	2024	NUM
cana-533	49	19	)	)	PUNCT
cana-533	49	20	184	184	NUM
cana-533	49	21	given	give	VERB
cana-533	49	22	the	the	DET
cana-533	49	23	partial	partial	ADJ
cana-533	49	24	metric	metric	NOUN
cana-533	49	25	,	,	PUNCT
cana-533	49	26	the	the	DET
cana-533	49	27	closure	closure	NOUN
cana-533	49	28	of	of	ADP
cana-533	49	29	is	be	AUX
cana-533	49	30	represented	represent	VERB
cana-533	49	31	here	here	ADV
cana-533	49	32	as	as	ADP
cana-533	49	33	.	.	PUNCT
cana-533	50	1	it	it	PRON
cana-533	50	2	is	be	AUX
cana-533	50	3	worth	worth	ADJ
cana-533	50	4	mentioning	mention	VERB
cana-533	50	5	that	that	PRON
cana-533	50	6	is	be	AUX
cana-533	50	7	closed	close	VERB
cana-533	50	8	in	in	ADP
cana-533	50	9	if	if	SCONJ
cana-533	50	10	and	and	CCONJ
cana-533	50	11	only	only	ADV
cana-533	50	12	if	if	SCONJ
cana-533	50	13	.	.	PUNCT
cana-533	51	1	lemma	lemma	PROPN
cana-533	51	2	2.2	2.2	NUM
cana-533	51	3	.	.	PUNCT
cana-533	52	1	[	[	X
cana-533	52	2	3	3	NUM
cana-533	52	3	]	]	PUNCT
cana-533	52	4	consider	consider	VERB
cana-533	52	5	and	and	CCONJ
cana-533	52	6	in	in	ADP
cana-533	52	7	a	a	DET
cana-533	52	8	partial	partial	ADJ
cana-533	52	9	metric	metric	ADJ
cana-533	52	10	space	space	NOUN
cana-533	52	11	.	.	PUNCT
cana-533	53	1	if	if	SCONJ
cana-533	53	2	and	and	CCONJ
cana-533	53	3	we	we	PRON
cana-533	53	4	can	can	AUX
cana-533	53	5	find	find	VERB
cana-533	53	6	that	that	SCONJ
cana-533	53	7	there	there	PRON
cana-533	53	8	is	be	VERB
cana-533	53	9	an	an	DET
cana-533	53	10	element	element	NOUN
cana-533	53	11	such	such	ADJ
cana-533	53	12	that	that	PRON
cana-533	53	13	.	.	PUNCT
cana-533	54	1	definition	definition	NOUN
cana-533	54	2	1	1	NUM
cana-533	54	3	.	.	PUNCT
cana-533	55	1	[	[	X
cana-533	55	2	4	4	X
cana-533	55	3	]	]	PUNCT
cana-533	55	4	a	a	DET
cana-533	55	5	bianchini	bianchini	NOUN
cana-533	55	6	-	-	PUNCT
cana-533	55	7	grandolfi	grandolfi	ADJ
cana-533	55	8	gauge	gauge	NOUN
cana-533	55	9	function	function	NOUN
cana-533	55	10	on	on	ADP
cana-533	55	11	is	be	AUX
cana-533	55	12	defined	define	VERB
cana-533	55	13	as	as	ADP
cana-533	55	14	a	a	DET
cana-533	55	15	nondecreasing	nondecreasing	ADJ
cana-533	55	16	function	function	NOUN
cana-533	55	17	that	that	PRON
cana-533	55	18	satisfies	satisfy	VERB
cana-533	55	19	the	the	DET
cana-533	55	20	following	follow	VERB
cana-533	55	21	condition	condition	NOUN
cana-533	55	22	:	:	PUNCT
cana-533	55	23	where	where	SCONJ
cana-533	55	24	denotes	denote	VERB
cana-533	55	25	the	the	DET
cana-533	55	26	-th	-th	ADJ
cana-533	55	27	iteration	iteration	NOUN
cana-533	55	28	of	of	ADP
cana-533	55	29	the	the	DET
cana-533	55	30	function	function	NOUN
cana-533	55	31	and	and	CCONJ
cana-533	55	32	.	.	PUNCT
cana-533	56	1	in	in	ADP
cana-533	56	2	other	other	ADJ
cana-533	56	3	words	word	NOUN
cana-533	56	4	,	,	PUNCT
cana-533	56	5	is	be	AUX
cana-533	56	6	defined	define	VERB
cana-533	56	7	recursively	recursively	ADV
cana-533	56	8	as	as	ADP
cana-533	56	9	,	,	PUNCT
cana-533	56	10	,	,	PUNCT
cana-533	56	11	,	,	PUNCT
cana-533	56	12	and	and	CCONJ
cana-533	56	13	so	so	ADV
cana-533	56	14	on	on	ADP
cana-533	56	15	the	the	DET
cana-533	56	16	associated	associate	VERB
cana-533	56	17	estimate	estimate	NOUN
cana-533	56	18	function	function	NOUN
cana-533	56	19	is	be	AUX
cana-533	56	20	known	know	VERB
cana-533	56	21	as	as	ADP
cana-533	56	22	the	the	DET
cana-533	56	23	sum	sum	NOUN
cana-533	56	24	(	(	PUNCT
cana-533	56	25	2.3	2.3	NUM
cana-533	56	26	)	)	PUNCT
cana-533	56	27	,	,	PUNCT
cana-533	56	28	and	and	CCONJ
cana-533	56	29	it	it	PRON
cana-533	56	30	was	be	AUX
cana-533	56	31	observed	observe	VERB
cana-533	56	32	that	that	SCONJ
cana-533	56	33	fulfils	fulfil	VERB
cana-533	56	34	the	the	DET
cana-533	56	35	functional	functional	ADJ
cana-533	56	36	equation	equation	NOUN
cana-533	56	37	shown	show	VERB
cana-533	56	38	below	below	ADP
cana-533	56	39	in	in	ADP
cana-533	56	40	[	[	X
cana-533	56	41	6	6	NUM
cana-533	56	42	]	]	PUNCT
cana-533	56	43	,	,	PUNCT
cana-533	56	44	the	the	DET
cana-533	56	45	author	author	NOUN
cana-533	56	46	has	have	AUX
cana-533	56	47	defined	define	VERB
cana-533	56	48	as	as	ADP
cana-533	56	49	the	the	DET
cana-533	56	50	set	set	NOUN
cana-533	56	51	of	of	ADP
cana-533	56	52	pairs	pair	NOUN
cana-533	56	53	of	of	ADP
cana-533	56	54	increasing	increase	VERB
cana-533	56	55	functions	function	NOUN
cana-533	56	56	and	and	CCONJ
cana-533	56	57	that	that	DET
cana-533	56	58	map	map	NOUN
cana-533	56	59	from	from	ADP
cana-533	56	60	to	to	ADP
cana-533	56	61	,	,	PUNCT
cana-533	56	62	satisfying	satisfy	VERB
cana-533	56	63	the	the	DET
cana-533	56	64	following	follow	VERB
cana-533	56	65	three	three	NUM
cana-533	56	66	conditions	condition	NOUN
cana-533	56	67	:	:	PUNCT
cana-533	56	68	then	then	ADV
cana-533	56	69	the	the	DET
cana-533	56	70	theorem	theorem	NOUN
cana-533	56	71	mentioned	mention	VERB
cana-533	56	72	in	in	ADP
cana-533	56	73	[	[	X
cana-533	56	74	6	6	NUM
cana-533	56	75	]	]	PUNCT
cana-533	56	76	reads	read	NOUN
cana-533	56	77	as	as	SCONJ
cana-533	56	78	follows	follow	VERB
cana-533	56	79	:	:	PUNCT
cana-533	56	80	theorem	theorem	NOUN
cana-533	56	81	2.1	2.1	NUM
cana-533	56	82	.	.	PUNCT
cana-533	57	1	[	[	X
cana-533	57	2	6	6	NUM
cana-533	57	3	]	]	PUNCT
cana-533	57	4	let	let	VERB
cana-533	57	5	and	and	CCONJ
cana-533	57	6	be	be	AUX
cana-533	57	7	complete	complete	ADJ
cana-533	57	8	partial	partial	ADJ
cana-533	57	9	metric	metric	ADJ
cana-533	57	10	spaces	space	NOUN
cana-533	57	11	,	,	PUNCT
cana-533	57	12	and	and	CCONJ
cana-533	57	13	let	let	VERB
cana-533	57	14	and	and	CCONJ
cana-533	57	15	.	.	PUNCT
cana-533	58	1	let	let	AUX
cana-533	58	2	be	be	AUX
cana-533	58	3	a	a	DET
cana-533	58	4	constant	constant	ADJ
cana-533	58	5	,	,	PUNCT
cana-533	58	6	and	and	CCONJ
cana-533	58	7	let	let	VERB
cana-533	58	8	f	f	PRON
cana-533	58	9	and	and	CCONJ
cana-533	58	10	be	be	AUX
cana-533	58	11	two	two	NUM
cana-533	58	12	set	set	NOUN
cana-533	58	13	-	-	PUNCT
cana-533	58	14	valued	value	VERB
cana-533	58	15	mappings	mapping	NOUN
cana-533	58	16	such	such	ADJ
cana-533	58	17	that	that	PRON
cana-533	58	18	.	.	PUNCT
cana-533	59	1	let	let	VERB
cana-533	59	2	and	and	CCONJ
cana-533	59	3	be	be	AUX
cana-533	59	4	increasing	increase	VERB
cana-533	59	5	functions	function	NOUN
cana-533	59	6	such	such	ADJ
cana-533	59	7	that	that	SCONJ
cana-533	59	8	,	,	PUNCT
cana-533	59	9	where	where	SCONJ
cana-533	59	10	is	be	AUX
cana-533	59	11	a	a	DET
cana-533	59	12	collection	collection	NOUN
cana-533	59	13	of	of	ADP
cana-533	59	14	pairs	pair	NOUN
cana-533	59	15	of	of	ADP
cana-533	59	16	increasing	increase	VERB
cana-533	59	17	functions	function	NOUN
cana-533	59	18	from	from	ADP
cana-533	59	19	to	to	ADP
cana-533	59	20	,	,	PUNCT
cana-533	59	21	subject	subject	ADJ
cana-533	59	22	to	to	ADP
cana-533	59	23	the	the	DET
cana-533	59	24	following	follow	VERB
cana-533	59	25	three	three	NUM
cana-533	59	26	conditions	condition	NOUN
cana-533	59	27	.	.	PUNCT
cana-533	60	1	assume	assume	VERB
cana-533	60	2	there	there	PRON
cana-533	60	3	exists	exist	VERB
cana-533	60	4	such	such	ADJ
cana-533	60	5	that	that	SCONJ
cana-533	60	6	the	the	DET
cana-533	60	7	following	follow	VERB
cana-533	60	8	assumptions	assumption	NOUN
cana-533	60	9	hold	hold	VERB
cana-533	60	10	:	:	PUNCT
cana-533	60	11	then	then	ADV
cana-533	60	12	there	there	PRON
cana-533	60	13	exist	exist	VERB
cana-533	60	14	and	and	CCONJ
cana-533	60	15	such	such	ADJ
cana-533	60	16	that	that	PRON
cana-533	60	17	and	and	CCONJ
cana-533	60	18	i.e.	i.e.	X
cana-533	60	19	,	,	PUNCT
cana-533	60	20	is	be	AUX
cana-533	60	21	an	an	DET
cana-533	60	22	altering	alter	VERB
cana-533	60	23	point	point	NOUN
cana-533	60	24	of	of	ADP
cana-533	60	25	and	and	CCONJ
cana-533	60	26	.	.	PUNCT
cana-533	61	1	if	if	SCONJ
cana-533	61	2	and	and	CCONJ
cana-533	61	3	are	be	AUX
cana-533	61	4	single	single	ADJ
cana-533	61	5	valued	value	VERB
cana-533	61	6	mappings	mapping	NOUN
cana-533	61	7	,	,	PUNCT
cana-533	61	8	and	and	CCONJ
cana-533	61	9	,	,	PUNCT
cana-533	61	10	then	then	ADV
cana-533	61	11	is	be	AUX
cana-533	61	12	the	the	DET
cana-533	61	13	unique	unique	ADJ
cana-533	61	14	fixed	fix	VERB
cana-533	61	15	point	point	NOUN
cana-533	61	16	of	of	ADP
cana-533	61	17	in	in	ADV
cana-533	61	18	,	,	PUNCT
cana-533	61	19	and	and	CCONJ
cana-533	61	20	is	be	AUX
cana-533	61	21	the	the	DET
cana-533	61	22	unique	unique	ADJ
cana-533	61	23	fixed	fix	VERB
cana-533	61	24	point	point	NOUN
cana-533	61	25	of	of	ADP
cana-533	61	26	in	in	ADP
cana-533	61	27	.	.	PUNCT
cana-533	62	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	62	2	communications	communication	NOUN
cana-533	62	3	on	on	ADP
cana-533	62	4	applied	apply	VERB
cana-533	62	5	nonlinear	nonlinear	ADJ
cana-533	62	6	analysis	analysis	NOUN
cana-533	62	7	issn	issn	NOUN
cana-533	62	8	:	:	PUNCT
cana-533	62	9	1074	1074	NUM
cana-533	62	10	-	-	PUNCT
cana-533	62	11	133x	133x	NUM
cana-533	62	12	vol	vol	NOUN
cana-533	62	13	31	31	NUM
cana-533	62	14	no	no	NOUN
cana-533	62	15	.	.	NOUN
cana-533	62	16	2	2	NUM
cana-533	62	17	(	(	PUNCT
cana-533	62	18	2024	2024	NUM
cana-533	62	19	)	)	PUNCT
cana-533	62	20	185	185	NUM
cana-533	62	21	remark	remark	NOUN
cana-533	62	22	1	1	NUM
cana-533	62	23	.	.	PUNCT
cana-533	63	1	the	the	DET
cana-533	63	2	theorem	theorem	NOUN
cana-533	63	3	mentioned	mention	VERB
cana-533	63	4	above	above	ADV
cana-533	63	5	is	be	AUX
cana-533	63	6	satisfied	satisfied	ADJ
cana-533	63	7	without	without	ADP
cana-533	63	8	using	use	VERB
cana-533	63	9	the	the	DET
cana-533	63	10	assumption	assumption	NOUN
cana-533	63	11	.	.	PUNCT
cana-533	64	1	see	see	VERB
cana-533	64	2	the	the	DET
cana-533	64	3	proof	proof	NOUN
cana-533	64	4	of	of	ADP
cana-533	64	5	[	[	X
cana-533	64	6	6	6	NUM
cana-533	64	7	,	,	PUNCT
cana-533	64	8	theorem	theorem	VERB
cana-533	64	9	11	11	NUM
cana-533	64	10	]	]	PUNCT
cana-533	64	11	for	for	ADP
cana-533	64	12	further	further	ADJ
cana-533	64	13	details	detail	NOUN
cana-533	64	14	.	.	PUNCT
cana-533	65	1	for	for	ADP
cana-533	65	2	,	,	PUNCT
cana-533	65	3	,	,	PUNCT
cana-533	65	4	and	and	CCONJ
cana-533	65	5	where	where	SCONJ
cana-533	65	6	the	the	DET
cana-533	65	7	identity	identity	NOUN
cana-533	65	8	function	function	NOUN
cana-533	65	9	defined	define	VERB
cana-533	65	10	on	on	ADP
cana-533	65	11	,	,	PUNCT
cana-533	65	12	we	we	PRON
cana-533	65	13	have	have	AUX
cana-533	65	14	corollary	corollary	ADJ
cana-533	65	15	2.2	2.2	NUM
cana-533	65	16	.	.	PUNCT
cana-533	66	1	theorem	theorem	VERB
cana-533	66	2	3.2	3.2	NUM
cana-533	66	3	in	in	ADP
cana-533	66	4	[	[	X
cana-533	66	5	3	3	NUM
cana-533	66	6	]	]	PUNCT
cana-533	66	7	.	.	PUNCT
cana-533	67	1	a	a	DET
cana-533	67	2	notion	notion	NOUN
cana-533	67	3	of	of	ADP
cana-533	67	4	-class	-class	NOUN
cana-533	67	5	functions	function	NOUN
cana-533	67	6	was	be	AUX
cana-533	67	7	presented	present	VERB
cana-533	67	8	by	by	ADP
cana-533	67	9	a.h	a.h	PROPN
cana-533	67	10	.	.	PROPN
cana-533	67	11	ansari	ansari	PROPN
cana-533	67	12	in	in	ADP
cana-533	67	13	[	[	X
cana-533	67	14	1	1	NUM
cana-533	67	15	]	]	PUNCT
cana-533	67	16	.	.	PUNCT
cana-533	68	1	definition	definition	NOUN
cana-533	68	2	2	2	NUM
cana-533	68	3	.	.	PUNCT
cana-533	69	1	(	(	PUNCT
cana-533	69	2	-class	-class	NOUN
cana-533	69	3	functions	function	NOUN
cana-533	69	4	)	)	PUNCT
cana-533	70	1	[	[	X
cana-533	70	2	1,2	1,2	NUM
cana-533	70	3	]	]	PUNCT
cana-533	70	4	assume	assume	VERB
cana-533	70	5	that	that	SCONJ
cana-533	70	6	there	there	PRON
cana-533	70	7	is	be	VERB
cana-533	70	8	a	a	DET
cana-533	70	9	continuous	continuous	ADJ
cana-533	70	10	mapping	mapping	NOUN
cana-533	70	11	.	.	PUNCT
cana-533	71	1	if	if	SCONJ
cana-533	71	2	satisfies	satisfy	VERB
cana-533	71	3	these	these	DET
cana-533	71	4	requirements	requirement	NOUN
cana-533	71	5	,	,	PUNCT
cana-533	71	6	we	we	PRON
cana-533	71	7	will	will	AUX
cana-533	71	8	classify	classify	VERB
cana-533	71	9	it	it	PRON
cana-533	71	10	as	as	ADP
cana-533	71	11	a	a	DET
cana-533	71	12	-class	-class	NOUN
cana-533	71	13	function	function	NOUN
cana-533	71	14	.	.	PUNCT
cana-533	72	1	and	and	CCONJ
cana-533	72	2	denotes	denote	VERB
cana-533	72	3	the	the	DET
cana-533	72	4	set	set	NOUN
cana-533	72	5	of	of	ADP
cana-533	72	6	all	all	DET
cana-533	72	7	-class	-class	NOUN
cana-533	72	8	functions	function	NOUN
cana-533	72	9	on	on	ADP
cana-533	72	10	.	.	PUNCT
cana-533	73	1	example	example	NOUN
cana-533	74	1	1	1	NUM
cana-533	74	2	.	.	PUNCT
cana-533	75	1	here	here	ADV
cana-533	75	2	are	be	AUX
cana-533	75	3	some	some	DET
cana-533	75	4	examples	example	NOUN
cana-533	75	5	of	of	ADP
cana-533	75	6	-class	-class	NOUN
cana-533	75	7	functions	function	NOUN
cana-533	75	8	with	with	ADP
cana-533	75	9	and	and	CCONJ
cana-533	75	10	,	,	PUNCT
cana-533	75	11	where	where	SCONJ
cana-533	75	12	:	:	PUNCT
cana-533	75	13	1	1	X
cana-533	75	14	.	.	X
cana-533	76	1	this	this	DET
cana-533	76	2	function	function	NOUN
cana-533	76	3	satisfies	satisfie	NOUN
cana-533	76	4	and	and	CCONJ
cana-533	76	5	,	,	PUNCT
cana-533	76	6	and	and	CCONJ
cana-533	76	7	it	it	PRON
cana-533	76	8	is	be	AUX
cana-533	76	9	known	know	VERB
cana-533	76	10	as	as	ADP
cana-533	76	11	the	the	DET
cana-533	76	12	harmonic	harmonic	ADJ
cana-533	76	13	mean	mean	NOUN
cana-533	76	14	.	.	PUNCT
cana-533	77	1	it	it	PRON
cana-533	77	2	is	be	AUX
cana-533	77	3	used	use	VERB
cana-533	77	4	,	,	PUNCT
cana-533	77	5	for	for	ADP
cana-533	77	6	example	example	NOUN
cana-533	77	7	,	,	PUNCT
cana-533	77	8	to	to	PART
cana-533	77	9	calculate	calculate	VERB
cana-533	77	10	the	the	DET
cana-533	77	11	average	average	ADJ
cana-533	77	12	rate	rate	NOUN
cana-533	77	13	of	of	ADP
cana-533	77	14	speed	speed	NOUN
cana-533	77	15	of	of	ADP
cana-533	77	16	two	two	NUM
cana-533	77	17	objects	object	NOUN
cana-533	77	18	moving	move	VERB
cana-533	77	19	at	at	ADP
cana-533	77	20	different	different	ADJ
cana-533	77	21	constant	constant	ADJ
cana-533	77	22	speeds	speed	NOUN
cana-533	77	23	.	.	PUNCT
cana-533	78	1	2	2	NUM
cana-533	78	2	.	.	PUNCT
cana-533	78	3	.	.	PUNCT
cana-533	79	1	this	this	DET
cana-533	79	2	function	function	NOUN
cana-533	79	3	satisfies	satisfie	NOUN
cana-533	79	4	and	and	CCONJ
cana-533	79	5	,	,	PUNCT
cana-533	79	6	and	and	CCONJ
cana-533	79	7	it	it	PRON
cana-533	79	8	is	be	AUX
cana-533	79	9	known	know	VERB
cana-533	79	10	as	as	ADP
cana-533	79	11	the	the	DET
cana-533	79	12	normalized	normalized	ADJ
cana-533	79	13	power	power	NOUN
cana-533	79	14	mean	mean	NOUN
cana-533	79	15	of	of	ADP
cana-533	79	16	order	order	NOUN
cana-533	79	17	1	1	X
cana-533	79	18	.	.	PUNCT
cana-533	80	1	it	it	PRON
cana-533	80	2	is	be	AUX
cana-533	80	3	used	use	VERB
cana-533	80	4	,	,	PUNCT
cana-533	80	5	for	for	ADP
cana-533	80	6	example	example	NOUN
cana-533	80	7	,	,	PUNCT
cana-533	80	8	in	in	ADP
cana-533	80	9	statistics	statistic	NOUN
cana-533	80	10	to	to	PART
cana-533	80	11	calculate	calculate	VERB
cana-533	80	12	the	the	DET
cana-533	80	13	average	average	NOUN
cana-533	80	14	of	of	ADP
cana-533	80	15	a	a	DET
cana-533	80	16	set	set	NOUN
cana-533	80	17	of	of	ADP
cana-533	80	18	non	non	ADJ
cana-533	80	19	-	-	ADJ
cana-533	80	20	negative	negative	ADJ
cana-533	80	21	numbers	number	NOUN
cana-533	80	22	.	.	PUNCT
cana-533	81	1	in	in	ADP
cana-533	81	2	[	[	X
cana-533	81	3	2	2	NUM
cana-533	81	4	]	]	PUNCT
cana-533	81	5	,	,	PUNCT
cana-533	81	6	the	the	DET
cana-533	81	7	authors	author	NOUN
cana-533	81	8	presented	present	VERB
cana-533	81	9	the	the	DET
cana-533	81	10	following	following	ADJ
cana-533	81	11	collections	collection	NOUN
cana-533	81	12	of	of	ADP
cana-533	81	13	-class	-class	NOUN
cana-533	81	14	functions	function	NOUN
cana-533	81	15	:	:	PUNCT
cana-533	81	16	definition	definition	NOUN
cana-533	81	17	3	3	NUM
cana-533	81	18	.	.	PUNCT
cana-533	82	1	[	[	X
cana-533	82	2	2	2	X
cana-533	82	3	]	]	PUNCT
cana-533	82	4	the	the	DET
cana-533	82	5	set	set	NOUN
cana-533	82	6	of	of	ADP
cana-533	82	7	functions	function	NOUN
cana-533	82	8	of	of	ADP
cana-533	82	9	the	the	DET
cana-533	82	10	-class	-class	NOUN
cana-533	82	11	that	that	PRON
cana-533	82	12	satisfy	satisfy	VERB
cana-533	82	13	these	these	DET
cana-533	82	14	criteria	criterion	NOUN
cana-533	82	15	is	be	AUX
cana-533	82	16	called	call	VERB
cana-533	82	17	:	:	PUNCT
cana-533	82	18	•	•	NUM
cana-533	82	19	is	be	AUX
cana-533	82	20	non	non	ADJ
cana-533	82	21	-	-	ADJ
cana-533	82	22	decreasing	decrease	VERB
cana-533	82	23	for	for	ADP
cana-533	82	24	both	both	PRON
cana-533	82	25	and	and	CCONJ
cana-533	82	26	when	when	SCONJ
cana-533	82	27	'	'	PUNCT
cana-533	82	28	.	.	PUNCT
cana-533	83	1	•	•	NOUN
cana-533	83	2	for	for	ADP
cana-533	83	3	every	every	PRON
cana-533	83	4	that	that	PRON
cana-533	83	5	are	be	AUX
cana-533	83	6	fixed	fix	VERB
cana-533	83	7	,	,	PUNCT
cana-533	83	8	the	the	DET
cana-533	83	9	series	series	NOUN
cana-533	83	10	converges	converge	VERB
cana-533	83	11	for	for	ADP
cana-533	83	12	all	all	PRON
cana-533	83	13	.	.	PUNCT
cana-533	84	1	the	the	DET
cana-533	84	2	function	function	NOUN
cana-533	84	3	is	be	AUX
cana-533	84	4	defined	define	VERB
cana-533	84	5	as	as	ADP
cana-533	84	6	follows	follow	VERB
cana-533	84	7	,	,	PUNCT
cana-533	84	8	and	and	CCONJ
cana-533	84	9	represents	represent	VERB
cana-533	84	10	the	the	DET
cana-533	84	11	-th	-th	ADJ
cana-533	84	12	iteration	iteration	NOUN
cana-533	84	13	of	of	ADP
cana-533	84	14	this	this	DET
cana-533	84	15	function	function	NOUN
cana-533	84	16	:	:	PUNCT
cana-533	84	17	.	.	PUNCT
cana-533	85	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	85	2	communications	communication	NOUN
cana-533	85	3	on	on	ADP
cana-533	85	4	applied	apply	VERB
cana-533	85	5	nonlinear	nonlinear	ADJ
cana-533	85	6	analysis	analysis	NOUN
cana-533	85	7	issn	issn	NOUN
cana-533	85	8	:	:	PUNCT
cana-533	85	9	1074	1074	NUM
cana-533	85	10	-	-	PUNCT
cana-533	85	11	133x	133x	NUM
cana-533	85	12	vol	vol	NOUN
cana-533	85	13	31	31	NUM
cana-533	85	14	no	no	NOUN
cana-533	85	15	.	.	NOUN
cana-533	85	16	2	2	NUM
cana-533	85	17	(	(	PUNCT
cana-533	85	18	2024	2024	NUM
cana-533	85	19	)	)	PUNCT
cana-533	85	20	186	186	NUM
cana-533	85	21	definition	definition	NOUN
cana-533	85	22	4	4	NUM
cana-533	85	23	.	.	PUNCT
cana-533	86	1	[	[	X
cana-533	86	2	2	2	X
cana-533	86	3	]	]	PUNCT
cana-533	86	4	comprises	comprise	VERB
cana-533	86	5	a	a	DET
cana-533	86	6	set	set	NOUN
cana-533	86	7	of	of	ADP
cana-533	86	8	-class	-class	NOUN
cana-533	86	9	functions	function	NOUN
cana-533	86	10	that	that	PRON
cana-533	86	11	adhere	adhere	VERB
cana-533	86	12	to	to	ADP
cana-533	86	13	the	the	DET
cana-533	86	14	following	following	ADJ
cana-533	86	15	specifications	specification	NOUN
cana-533	86	16	:	:	PUNCT
cana-533	86	17	exhibits	exhibit	VERB
cana-533	86	18	non	non	ADJ
cana-533	86	19	-	-	ADJ
cana-533	86	20	decreasing	decrease	VERB
cana-533	86	21	behavior	behavior	NOUN
cana-533	86	22	in	in	ADP
cana-533	86	23	and	and	CCONJ
cana-533	86	24	non	non	ADJ
cana-533	86	25	-	-	ADJ
cana-533	86	26	increasing	increase	VERB
cana-533	86	27	behavior	behavior	NOUN
cana-533	86	28	in	in	ADP
cana-533	86	29	.	.	PUNCT
cana-533	87	1	•	•	NUM
cana-533	87	2	for	for	ADP
cana-533	87	3	any	any	DET
cana-533	87	4	given	give	VERB
cana-533	87	5	,	,	PUNCT
cana-533	87	6	the	the	DET
cana-533	87	7	series	series	NOUN
cana-533	87	8	converges	converge	VERB
cana-533	87	9	for	for	ADP
cana-533	87	10	every	every	PRON
cana-533	87	11	,	,	PUNCT
cana-533	87	12	where	where	SCONJ
cana-533	87	13	the	the	DET
cana-533	87	14	-th	-th	ADJ
cana-533	87	15	iteration	iteration	NOUN
cana-533	87	16	of	of	ADP
cana-533	87	17	the	the	DET
cana-533	87	18	function	function	NOUN
cana-533	87	19	with	with	ADP
cana-533	87	20	the	the	DET
cana-533	87	21	following	follow	VERB
cana-533	87	22	recurrence	recurrence	NOUN
cana-533	87	23	relation	relation	NOUN
cana-533	87	24	is	be	AUX
cana-533	87	25	represented	represent	VERB
cana-533	87	26	as	as	ADP
cana-533	87	27	:	:	PUNCT
cana-533	87	28	for	for	ADP
cana-533	87	29	more	more	ADJ
cana-533	87	30	examples	example	NOUN
cana-533	87	31	and	and	CCONJ
cana-533	87	32	properties	property	NOUN
cana-533	87	33	on	on	ADP
cana-533	87	34	and	and	CCONJ
cana-533	87	35	one	one	PRON
cana-533	87	36	can	can	AUX
cana-533	87	37	refer	refer	VERB
cana-533	87	38	to	to	ADP
cana-533	87	39	[	[	X
cana-533	87	40	2	2	NUM
cana-533	87	41	]	]	PUNCT
cana-533	87	42	.	.	PUNCT
cana-533	88	1	we	we	PRON
cana-533	88	2	recall	recall	VERB
cana-533	88	3	a	a	DET
cana-533	88	4	class	class	NOUN
cana-533	88	5	of	of	ADP
cana-533	88	6	functions	function	NOUN
cana-533	88	7	that	that	PRON
cana-533	88	8	were	be	AUX
cana-533	88	9	mentioned	mention	VERB
cana-533	88	10	in	in	ADP
cana-533	88	11	[	[	X
cana-533	88	12	2	2	NUM
cana-533	88	13	]	]	PUNCT
cana-533	88	14	.	.	PUNCT
cana-533	89	1	one	one	NUM
cana-533	89	2	important	important	ADJ
cana-533	89	3	need	need	NOUN
cana-533	89	4	is	be	AUX
cana-533	89	5	satisfied	satisfied	ADJ
cana-533	89	6	by	by	ADP
cana-533	89	7	these	these	DET
cana-533	89	8	functions	function	NOUN
cana-533	89	9	,	,	PUNCT
cana-533	89	10	denoted	denote	VERB
cana-533	89	11	as	as	ADP
cana-533	89	12	.	.	PUNCT
cana-533	90	1	being	be	AUX
cana-533	90	2	more	more	ADV
cana-533	90	3	explicit	explicit	ADJ
cana-533	90	4	,	,	PUNCT
cana-533	90	5	means	mean	VERB
cana-533	90	6	that	that	SCONJ
cana-533	90	7	or	or	CCONJ
cana-533	90	8	is	be	AUX
cana-533	90	9	true	true	ADJ
cana-533	90	10	for	for	ADP
cana-533	90	11	every	every	PRON
cana-533	90	12	and	and	CCONJ
cana-533	90	13	.	.	PUNCT
cana-533	91	1	furthermore	furthermore	ADV
cana-533	91	2	,	,	PUNCT
cana-533	91	3	we	we	PRON
cana-533	91	4	prove	prove	VERB
cana-533	91	5	that	that	PRON
cana-533	91	6	is	be	AUX
cana-533	91	7	nondecreasing	nondecrease	VERB
cana-533	91	8	in	in	ADP
cana-533	91	9	the	the	DET
cana-533	91	10	space	space	NOUN
cana-533	91	11	,	,	PUNCT
cana-533	91	12	as	as	SCONJ
cana-533	91	13	proved	prove	VERB
cana-533	91	14	by	by	ADP
cana-533	91	15	the	the	DET
cana-533	91	16	following	follow	VERB
cana-533	91	17	inequality	inequality	NOUN
cana-533	91	18	:	:	PUNCT
cana-533	91	19	so	so	ADV
cana-533	91	20	,	,	PUNCT
cana-533	91	21	the	the	DET
cana-533	91	22	theorem	theorem	NOUN
cana-533	91	23	[	[	X
cana-533	91	24	2	2	NUM
cana-533	91	25	,	,	PUNCT
cana-533	91	26	theorem	theorem	VERB
cana-533	91	27	3.4	3.4	NUM
cana-533	91	28	]	]	PUNCT
cana-533	91	29	,	,	PUNCT
cana-533	91	30	which	which	PRON
cana-533	91	31	extends	extend	VERB
cana-533	91	32	theorem	theorem	VERB
cana-533	91	33	[	[	PUNCT
cana-533	91	34	3,theorem	3,theorem	NUM
cana-533	91	35	3.2	3.2	NUM
cana-533	91	36	]	]	PUNCT
cana-533	91	37	,	,	PUNCT
cana-533	91	38	is	be	AUX
cana-533	91	39	as	as	SCONJ
cana-533	91	40	follows	follow	VERB
cana-533	91	41	:	:	PUNCT
cana-533	91	42	theorem	theorem	VERB
cana-533	91	43	2.3	2.3	NUM
cana-533	91	44	.	.	PUNCT
cana-533	92	1	[	[	X
cana-533	92	2	2	2	X
cana-533	92	3	]	]	PUNCT
cana-533	92	4	assuming	assume	VERB
cana-533	92	5	is	be	AUX
cana-533	92	6	a	a	DET
cana-533	92	7	partial	partial	ADJ
cana-533	92	8	metric	metric	ADJ
cana-533	92	9	space	space	NOUN
cana-533	92	10	with	with	ADP
cana-533	92	11	and	and	CCONJ
cana-533	92	12	r>0	r>0	VERB
cana-533	92	13	such	such	ADJ
cana-533	92	14	that	that	PRON
cana-533	92	15	is	be	AUX
cana-533	92	16	a	a	DET
cana-533	92	17	0	0	NUM
cana-533	92	18	-	-	PUNCT
cana-533	92	19	complete	complete	ADJ
cana-533	92	20	subspace	subspace	NOUN
cana-533	92	21	of	of	ADP
cana-533	92	22	.	.	PUNCT
cana-533	93	1	let	let	AUX
cana-533	93	2	be	be	AUX
cana-533	93	3	a	a	DET
cana-533	93	4	setvalued	setvalue	VERB
cana-533	93	5	mapping	mapping	NOUN
cana-533	93	6	.	.	PUNCT
cana-533	94	1	consider	consider	VERB
cana-533	94	2	,	,	PUNCT
cana-533	94	3	and	and	CCONJ
cana-533	94	4	satisfying	satisfy	VERB
cana-533	94	5	one	one	NUM
cana-533	94	6	of	of	ADP
cana-533	94	7	the	the	DET
cana-533	94	8	following	following	ADJ
cana-533	94	9	conditions	condition	NOUN
cana-533	94	10	:	:	PUNCT
cana-533	94	11	•	•	NUM
cana-533	94	12	and	and	CCONJ
cana-533	94	13	is	be	AUX
cana-533	94	14	nondecreasing	nondecrease	VERB
cana-533	94	15	.	.	PUNCT
cana-533	95	1	•	•	NUM
cana-533	95	2	and	and	CCONJ
cana-533	95	3	for	for	ADP
cana-533	95	4	.	.	PUNCT
cana-533	96	1	we	we	PRON
cana-533	96	2	suppose	suppose	VERB
cana-533	96	3	the	the	DET
cana-533	96	4	following	follow	VERB
cana-533	96	5	two	two	NUM
cana-533	96	6	conditions	condition	NOUN
cana-533	96	7	are	be	AUX
cana-533	96	8	met	meet	VERB
cana-533	96	9	:	:	PUNCT
cana-533	96	10	hence	hence	ADV
cana-533	96	11	,	,	PUNCT
cana-533	96	12	in	in	ADV
cana-533	96	13	,	,	PUNCT
cana-533	96	14	there	there	PRON
cana-533	96	15	is	be	VERB
cana-533	96	16	a	a	DET
cana-533	96	17	fixed	fix	VERB
cana-533	96	18	point	point	NOUN
cana-533	96	19	in	in	ADP
cana-533	96	20	.	.	PUNCT
cana-533	97	1	in	in	ADP
cana-533	97	2	the	the	DET
cana-533	97	3	set	set	NOUN
cana-533	97	4	,	,	PUNCT
cana-533	97	5	the	the	DET
cana-533	97	6	unique	unique	ADJ
cana-533	97	7	fixed	fix	VERB
cana-533	97	8	point	point	NOUN
cana-533	97	9	of	of	ADP
cana-533	97	10	is	be	AUX
cana-533	97	11	if	if	SCONJ
cana-533	97	12	is	be	AUX
cana-533	97	13	a	a	DET
cana-533	97	14	single	single	ADV
cana-533	97	15	-	-	PUNCT
cana-533	97	16	valued	value	VERB
cana-533	97	17	mapping	mapping	NOUN
cana-533	97	18	and	and	CCONJ
cana-533	97	19	.	.	PUNCT
cana-533	98	1	3	3	X
cana-533	98	2	.	.	X
cana-533	98	3	main	main	ADJ
cana-533	98	4	results	result	NOUN
cana-533	98	5	we	we	PRON
cana-533	98	6	need	need	VERB
cana-533	98	7	to	to	PART
cana-533	98	8	review	review	VERB
cana-533	98	9	certain	certain	ADJ
cana-533	98	10	concepts	concept	NOUN
cana-533	98	11	and	and	CCONJ
cana-533	98	12	establish	establish	VERB
cana-533	98	13	some	some	DET
cana-533	98	14	definitions	definition	NOUN
cana-533	98	15	before	before	SCONJ
cana-533	98	16	we	we	PRON
cana-533	98	17	can	can	AUX
cana-533	98	18	present	present	VERB
cana-533	98	19	our	our	PRON
cana-533	98	20	major	major	ADJ
cana-533	98	21	finding	finding	NOUN
cana-533	98	22	.	.	PUNCT
cana-533	99	1	definition	definition	NOUN
cana-533	99	2	5	5	NUM
cana-533	99	3	.	.	PUNCT
cana-533	100	1	(	(	PUNCT
cana-533	100	2	altering	alter	VERB
cana-533	100	3	points	point	NOUN
cana-533	100	4	)	)	PUNCT
cana-533	100	5	.	.	PUNCT
cana-533	101	1	[	[	X
cana-533	101	2	7	7	NUM
cana-533	101	3	,	,	PUNCT
cana-533	101	4	9	9	NUM
cana-533	101	5	]	]	PUNCT
cana-533	101	6	let	let	VERB
cana-533	101	7	and	and	CCONJ
cana-533	101	8	be	be	AUX
cana-533	101	9	two	two	NUM
cana-533	101	10	partial	partial	ADJ
cana-533	101	11	metric	metric	ADJ
cana-533	101	12	spaces	space	NOUN
cana-533	101	13	and	and	CCONJ
cana-533	101	14	let	let	VERB
cana-533	101	15	,	,	PUNCT
cana-533	101	16	be	be	AUX
cana-533	101	17	two	two	NUM
cana-533	101	18	set	set	NOUN
cana-533	101	19	-	-	PUNCT
cana-533	101	20	valued	value	VERB
cana-533	101	21	mappings	mapping	NOUN
cana-533	101	22	.	.	PUNCT
cana-533	102	1	the	the	DET
cana-533	102	2	couple	couple	NOUN
cana-533	102	3	is	be	AUX
cana-533	102	4	called	call	VERB
cana-533	102	5	altering	alter	VERB
cana-533	102	6	point	point	NOUN
cana-533	102	7	of	of	ADP
cana-533	102	8	and	and	CCONJ
cana-533	102	9	if	if	SCONJ
cana-533	102	10	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	102	11	communications	communication	NOUN
cana-533	102	12	on	on	ADP
cana-533	102	13	applied	apply	VERB
cana-533	102	14	nonlinear	nonlinear	ADJ
cana-533	102	15	analysis	analysis	NOUN
cana-533	102	16	issn	issn	NOUN
cana-533	102	17	:	:	PUNCT
cana-533	102	18	1074	1074	NUM
cana-533	102	19	-	-	PUNCT
cana-533	102	20	133x	133x	NUM
cana-533	102	21	vol	vol	NOUN
cana-533	102	22	31	31	NUM
cana-533	102	23	no	no	NOUN
cana-533	102	24	.	.	NOUN
cana-533	102	25	2	2	NUM
cana-533	102	26	(	(	PUNCT
cana-533	102	27	2024	2024	NUM
cana-533	102	28	)	)	PUNCT
cana-533	102	29	187	187	NUM
cana-533	102	30	there	there	PRON
cana-533	102	31	are	be	VERB
cana-533	102	32	some	some	DET
cana-533	102	33	special	special	ADJ
cana-533	102	34	cases	case	NOUN
cana-533	102	35	that	that	PRON
cana-533	102	36	can	can	AUX
cana-533	102	37	be	be	AUX
cana-533	102	38	deduced	deduce	VERB
cana-533	102	39	from	from	ADP
cana-533	102	40	this	this	DET
cana-533	102	41	definition	definition	NOUN
cana-533	102	42	,	,	PUNCT
cana-533	102	43	which	which	PRON
cana-533	102	44	can	can	AUX
cana-533	102	45	be	be	AUX
cana-533	102	46	summarized	summarize	VERB
cana-533	102	47	in	in	ADP
cana-533	102	48	the	the	DET
cana-533	102	49	following	follow	VERB
cana-533	102	50	table	table	NOUN
cana-533	102	51	according	accord	VERB
cana-533	102	52	to	to	ADP
cana-533	102	53	theorem	theorem	ADJ
cana-533	102	54	2.3	2.3	NUM
cana-533	102	55	and	and	CCONJ
cana-533	102	56	definitions	definition	NOUN
cana-533	102	57	3	3	NUM
cana-533	102	58	,	,	PUNCT
cana-533	102	59	4	4	NUM
cana-533	102	60	,	,	PUNCT
cana-533	102	61	and	and	CCONJ
cana-533	102	62	5	5	NUM
cana-533	102	63	,	,	PUNCT
cana-533	102	64	we	we	PRON
cana-533	102	65	will	will	AUX
cana-533	102	66	establish	establish	VERB
cana-533	102	67	new	new	ADJ
cana-533	102	68	theorems	theorem	NOUN
cana-533	102	69	with	with	ADP
cana-533	102	70	respect	respect	NOUN
cana-533	102	71	to	to	ADP
cana-533	102	72	set	set	VERB
cana-533	102	73	-	-	PUNCT
cana-533	102	74	valued	value	VERB
cana-533	102	75	mappings	mapping	NOUN
cana-533	102	76	on	on	ADP
cana-533	102	77	0	0	NUM
cana-533	102	78	-	-	PUNCT
cana-533	102	79	complete	complete	ADJ
cana-533	102	80	partial	partial	ADJ
cana-533	102	81	metric	metric	ADJ
cana-533	102	82	spaces	space	NOUN
cana-533	102	83	that	that	PRON
cana-533	102	84	extend	extend	VERB
cana-533	102	85	and	and	CCONJ
cana-533	102	86	generalize	generalize	VERB
cana-533	102	87	theorem	theorem	VERB
cana-533	102	88	2.1	2.1	NUM
cana-533	102	89	for	for	ADP
cana-533	102	90	altering	alter	VERB
cana-533	102	91	points	point	NOUN
cana-533	102	92	of	of	ADP
cana-533	102	93	two	two	NUM
cana-533	102	94	set	set	NOUN
cana-533	102	95	-	-	PUNCT
cana-533	102	96	valued	value	VERB
cana-533	102	97	mappings	mapping	NOUN
cana-533	102	98	and	and	CCONJ
cana-533	102	99	generalized	generalize	VERB
cana-533	102	100	theorem	theorem	NOUN
cana-533	102	101	2.3	2.3	NUM
cana-533	103	1	[	[	X
cana-533	103	2	2,theorem	2,theorem	NUM
cana-533	103	3	3.4	3.4	NUM
cana-533	103	4	]	]	PUNCT
cana-533	103	5	and	and	CCONJ
cana-533	103	6	[	[	X
cana-533	103	7	3,theorem	3,theorem	NUM
cana-533	103	8	3.2]{benterki2016	3.2]{benterki2016	NUM
cana-533	103	9	}	}	PUNCT
cana-533	103	10	for	for	ADP
cana-533	103	11	the	the	DET
cana-533	103	12	fixed	fix	VERB
cana-533	103	13	points	point	NOUN
cana-533	103	14	of	of	ADP
cana-533	103	15	a	a	DET
cana-533	103	16	two	two	NUM
cana-533	103	17	-	-	PUNCT
cana-533	103	18	set	set	NOUN
cana-533	103	19	-	-	PUNCT
cana-533	103	20	valued	value	VERB
cana-533	103	21	mapping	mapping	NOUN
cana-533	103	22	composition	composition	NOUN
cana-533	103	23	.	.	PUNCT
cana-533	104	1	at	at	ADP
cana-533	104	2	beginning	begin	VERB
cana-533	104	3	,	,	PUNCT
cana-533	104	4	we	we	PRON
cana-533	104	5	set	set	VERB
cana-533	104	6	the	the	DET
cana-533	104	7	family	family	NOUN
cana-533	104	8	of	of	ADP
cana-533	104	9	pairs	pair	NOUN
cana-533	104	10	of	of	ADP
cana-533	104	11	an	an	DET
cana-533	104	12	increasing	increase	VERB
cana-533	104	13	function	function	NOUN
cana-533	104	14	and	and	CCONJ
cana-533	104	15	an	an	DET
cana-533	104	16	increasing	increase	VERB
cana-533	104	17	function	function	NOUN
cana-533	104	18	in	in	ADP
cana-533	104	19	first	first	ADJ
cana-533	104	20	variable	variable	NOUN
cana-533	104	21	satisfying	satisfy	VERB
cana-533	104	22	the	the	DET
cana-533	104	23	following	follow	VERB
cana-533	104	24	three	three	NUM
cana-533	104	25	assumptions	assumption	NOUN
cana-533	104	26	:	:	PUNCT
cana-533	104	27	remark	remark	NOUN
cana-533	104	28	2	2	NUM
cana-533	104	29	.	.	PUNCT
cana-533	105	1	if	if	SCONJ
cana-533	105	2	such	such	ADJ
cana-533	105	3	that	that	SCONJ
cana-533	105	4	then	then	ADV
cana-533	105	5	the	the	DET
cana-533	105	6	subset	subset	NOUN
cana-533	105	7	will	will	AUX
cana-533	105	8	be	be	AUX
cana-533	105	9	the	the	DET
cana-533	105	10	subset	subset	NOUN
cana-533	105	11	.	.	PUNCT
cana-533	106	1	as	as	ADP
cana-533	106	2	a	a	DET
cana-533	106	3	consequence	consequence	NOUN
cana-533	106	4	,	,	PUNCT
cana-533	106	5	we	we	PRON
cana-533	106	6	express	express	VERB
cana-533	106	7	and	and	CCONJ
cana-533	106	8	demonstrate	demonstrate	VERB
cana-533	106	9	our	our	PRON
cana-533	106	10	main	main	ADJ
cana-533	106	11	finding	finding	NOUN
cana-533	106	12	as	as	SCONJ
cana-533	106	13	follows	follow	NOUN
cana-533	106	14	theorem	theorem	VERB
cana-533	106	15	3.1	3.1	NUM
cana-533	106	16	.	.	PUNCT
cana-533	107	1	let	let	VERB
cana-533	107	2	and	and	CCONJ
cana-533	107	3	be	be	AUX
cana-533	107	4	partial	partial	ADJ
cana-533	107	5	metric	metric	ADJ
cana-533	107	6	spaces	space	NOUN
cana-533	107	7	with	with	ADP
cana-533	107	8	and	and	CCONJ
cana-533	107	9	as	as	ADP
cana-533	107	10	0complete	0complete	NUM
cana-533	107	11	subspaces	subspace	NOUN
cana-533	107	12	of	of	ADP
cana-533	107	13	and	and	CCONJ
cana-533	107	14	,	,	PUNCT
cana-533	107	15	respectively	respectively	ADV
cana-533	107	16	,	,	PUNCT
cana-533	107	17	where	where	SCONJ
cana-533	107	18	,	,	PUNCT
cana-533	107	19	,	,	PUNCT
cana-533	107	20	and	and	CCONJ
cana-533	107	21	.	.	PUNCT
cana-533	108	1	consider	consider	VERB
cana-533	108	2	setvalued	setvalue	VERB
cana-533	108	3	mappings	mapping	NOUN
cana-533	108	4	and	and	CCONJ
cana-533	108	5	such	such	ADJ
cana-533	108	6	that	that	PRON
cana-533	108	7	.	.	PUNCT
cana-533	109	1	let	let	VERB
cana-533	109	2	,	,	PUNCT
cana-533	109	3	on	on	ADV
cana-533	109	4	,	,	PUNCT
cana-533	109	5	and	and	CCONJ
cana-533	109	6	which	which	PRON
cana-533	109	7	satisfies	satisfy	VERB
cana-533	109	8	at	at	ADP
cana-533	109	9	least	least	ADJ
cana-533	109	10	one	one	NUM
cana-533	109	11	of	of	ADP
cana-533	109	12	the	the	DET
cana-533	109	13	conditions	condition	NOUN
cana-533	109	14	:	:	PUNCT
cana-533	109	15	given	give	VERB
cana-533	109	16	these	these	DET
cana-533	109	17	conditions	condition	NOUN
cana-533	109	18	,	,	PUNCT
cana-533	109	19	we	we	PRON
cana-533	109	20	are	be	AUX
cana-533	109	21	going	go	VERB
cana-533	109	22	to	to	PART
cana-533	109	23	suppose	suppose	VERB
cana-533	109	24	that	that	SCONJ
cana-533	109	25	the	the	DET
cana-533	109	26	following	follow	VERB
cana-533	109	27	is	be	AUX
cana-533	109	28	true	true	ADJ
cana-533	109	29	:	:	PUNCT
cana-533	109	30	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	109	31	communications	communication	NOUN
cana-533	109	32	on	on	ADP
cana-533	109	33	applied	apply	VERB
cana-533	109	34	nonlinear	nonlinear	ADJ
cana-533	109	35	analysis	analysis	NOUN
cana-533	109	36	issn	issn	NOUN
cana-533	109	37	:	:	PUNCT
cana-533	109	38	1074	1074	NUM
cana-533	109	39	-	-	PUNCT
cana-533	109	40	133x	133x	NUM
cana-533	109	41	vol	vol	NOUN
cana-533	109	42	31	31	NUM
cana-533	109	43	no	no	NOUN
cana-533	109	44	.	.	NOUN
cana-533	109	45	2	2	NUM
cana-533	109	46	(	(	PUNCT
cana-533	109	47	2024	2024	NUM
cana-533	109	48	)	)	PUNCT
cana-533	109	49	188	188	NUM
cana-533	109	50	then	then	ADV
cana-533	109	51	there	there	PRON
cana-533	109	52	exist	exist	VERB
cana-533	109	53	an	an	DET
cana-533	109	54	altering	altering	NOUN
cana-533	109	55	point	point	NOUN
cana-533	109	56	of	of	ADP
cana-533	109	57	and	and	CCONJ
cana-533	109	58	in	in	ADV
cana-533	109	59	.	.	PUNCT
cana-533	110	1	if	if	SCONJ
cana-533	110	2	and	and	CCONJ
cana-533	110	3	are	be	AUX
cana-533	110	4	both	both	PRON
cana-533	110	5	single	single	ADV
cana-533	110	6	-	-	PUNCT
cana-533	110	7	valued	value	VERB
cana-533	110	8	mappings	mapping	NOUN
cana-533	110	9	and	and	CCONJ
cana-533	110	10	,	,	PUNCT
cana-533	110	11	then	then	ADV
cana-533	110	12	is	be	AUX
cana-533	110	13	the	the	DET
cana-533	110	14	unique	unique	ADJ
cana-533	110	15	fixed	fix	VERB
cana-533	110	16	point	point	NOUN
cana-533	110	17	of	of	ADP
cana-533	110	18	in	in	ADP
cana-533	110	19	and	and	CCONJ
cana-533	110	20	is	be	AUX
cana-533	110	21	the	the	DET
cana-533	110	22	unique	unique	ADJ
cana-533	110	23	fixed	fix	VERB
cana-533	110	24	point	point	NOUN
cana-533	110	25	of	of	ADP
cana-533	110	26	in	in	ADP
cana-533	110	27	.	.	PUNCT
cana-533	111	1	proof	proof	NOUN
cana-533	111	2	.	.	PUNCT
cana-533	112	1	the	the	DET
cana-533	112	2	proof	proof	NOUN
cana-533	112	3	is	be	AUX
cana-533	112	4	complete	complete	ADJ
cana-533	112	5	if	if	SCONJ
cana-533	112	6	.	.	PUNCT
cana-533	113	1	so	so	ADV
cana-533	113	2	we	we	PRON
cana-533	113	3	assume	assume	VERB
cana-533	113	4	that	that	PRON
cana-533	113	5	.	.	PUNCT
cana-533	114	1	by	by	ADP
cana-533	114	2	assumption	assumption	NOUN
cana-533	114	3	(	(	PUNCT
cana-533	114	4	a	a	X
cana-533	114	5	)	)	PUNCT
cana-533	114	6	,	,	PUNCT
cana-533	114	7	we	we	PRON
cana-533	114	8	have	have	VERB
cana-533	114	9	now	now	ADV
cana-533	114	10	,	,	PUNCT
cana-533	114	11	using	use	VERB
cana-533	114	12	lemma	lemma	PROPN
cana-533	114	13	2.2	2.2	NUM
cana-533	114	14	,	,	PUNCT
cana-533	114	15	there	there	PRON
cana-533	114	16	exists	exist	VERB
cana-533	114	17	such	such	ADJ
cana-533	114	18	that	that	SCONJ
cana-533	114	19	then	then	ADV
cana-533	114	20	.	.	PUNCT
cana-533	115	1	now	now	ADV
cana-533	115	2	,	,	PUNCT
cana-533	115	3	denoting	denote	VERB
cana-533	115	4	and	and	CCONJ
cana-533	115	5	.	.	PUNCT
cana-533	116	1	the	the	DET
cana-533	116	2	proof	proof	NOUN
cana-533	116	3	is	be	AUX
cana-533	116	4	complete	complete	ADJ
cana-533	116	5	if	if	SCONJ
cana-533	116	6	.	.	PUNCT
cana-533	117	1	so	so	ADV
cana-533	117	2	we	we	PRON
cana-533	117	3	assume	assume	VERB
cana-533	117	4	that	that	SCONJ
cana-533	117	5	and	and	CCONJ
cana-533	117	6	by	by	ADP
cana-533	117	7	using	use	VERB
cana-533	117	8	(	(	PUNCT
cana-533	117	9	b	b	X
cana-533	117	10	)	)	PUNCT
cana-533	117	11	we	we	PRON
cana-533	117	12	have	have	VERB
cana-533	117	13	with	with	ADP
cana-533	117	14	however	however	ADV
cana-533	117	15	,	,	PUNCT
cana-533	117	16	we	we	PRON
cana-533	117	17	also	also	ADV
cana-533	117	18	have	have	VERB
cana-533	117	19	if	if	SCONJ
cana-533	117	20	and	and	CCONJ
cana-533	117	21	,	,	PUNCT
cana-533	117	22	then	then	ADV
cana-533	117	23	wich	wich	PRON
cana-533	117	24	implies	imply	VERB
cana-533	117	25	that	that	PRON
cana-533	117	26	or	or	CCONJ
cana-533	117	27	.	.	PUNCT
cana-533	118	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	118	2	communications	communication	NOUN
cana-533	118	3	on	on	ADP
cana-533	118	4	applied	apply	VERB
cana-533	118	5	nonlinear	nonlinear	ADJ
cana-533	118	6	analysis	analysis	NOUN
cana-533	118	7	issn	issn	NOUN
cana-533	118	8	:	:	PUNCT
cana-533	118	9	1074	1074	NUM
cana-533	118	10	-	-	PUNCT
cana-533	118	11	133x	133x	NUM
cana-533	118	12	vol	vol	NOUN
cana-533	118	13	31	31	NUM
cana-533	118	14	no	no	NOUN
cana-533	118	15	.	.	NOUN
cana-533	118	16	2	2	NUM
cana-533	118	17	(	(	PUNCT
cana-533	118	18	2024	2024	NUM
cana-533	118	19	)	)	PUNCT
cana-533	118	20	189	189	NUM
cana-533	118	21	thus	thus	ADV
cana-533	118	22	,	,	PUNCT
cana-533	118	23	or	or	CCONJ
cana-533	118	24	which	which	PRON
cana-533	118	25	is	be	AUX
cana-533	118	26	a	a	DET
cana-533	118	27	contradiction	contradiction	NOUN
cana-533	118	28	.	.	PUNCT
cana-533	119	1	so	so	ADV
cana-533	119	2	,	,	PUNCT
cana-533	119	3	we	we	PRON
cana-533	119	4	assume	assume	VERB
cana-533	119	5	that	that	PRON
cana-533	119	6	and	and	CCONJ
cana-533	119	7	.	.	PUNCT
cana-533	120	1	then	then	ADV
cana-533	120	2	,	,	PUNCT
cana-533	120	3	by	by	ADP
cana-533	120	4	using	use	VERB
cana-533	120	5	lemma	lemma	PROPN
cana-533	120	6	2.2	2.2	NUM
cana-533	120	7	,	,	PUNCT
cana-533	120	8	relation	relation	NOUN
cana-533	120	9	(	(	PUNCT
cana-533	120	10	3.1	3.1	NUM
cana-533	120	11	)	)	PUNCT
cana-533	120	12	and	and	CCONJ
cana-533	120	13	non	non	X
cana-533	120	14	empty	empty	ADJ
cana-533	120	15	of	of	ADP
cana-533	120	16	and	and	CCONJ
cana-533	120	17	,	,	PUNCT
cana-533	120	18	there	there	PRON
cana-533	120	19	exists	exist	VERB
cana-533	120	20	and	and	CCONJ
cana-533	120	21	such	such	ADJ
cana-533	120	22	that	that	PRON
cana-533	120	23	and	and	CCONJ
cana-533	120	24	consequently	consequently	ADV
cana-533	120	25	,	,	PUNCT
cana-533	120	26	this	this	PRON
cana-533	120	27	means	mean	VERB
cana-533	120	28	that	that	PRON
cana-533	120	29	.	.	PUNCT
cana-533	121	1	by	by	ADP
cana-533	121	2	induction	induction	NOUN
cana-533	121	3	we	we	PRON
cana-533	121	4	construct	construct	VERB
cana-533	121	5	two	two	NUM
cana-533	121	6	sequences	sequence	NOUN
cana-533	121	7	and	and	CCONJ
cana-533	121	8	satisfying	satisfying	NOUN
cana-533	121	9	:	:	PUNCT
cana-533	121	10	where	where	SCONJ
cana-533	121	11	assuming	assume	VERB
cana-533	121	12	that	that	PRON
cana-533	121	13	,	,	PUNCT
cana-533	121	14	and	and	CCONJ
cana-533	121	15	for	for	ADP
cana-533	121	16	all	all	PRON
cana-533	121	17	,	,	PUNCT
cana-533	121	18	we	we	PRON
cana-533	121	19	can	can	AUX
cana-533	121	20	conclude	conclude	VERB
cana-533	121	21	that	that	SCONJ
cana-533	121	22	if	if	SCONJ
cana-533	121	23	or	or	CCONJ
cana-533	121	24	for	for	ADP
cana-533	121	25	some	some	PRON
cana-533	121	26	,	,	PUNCT
cana-533	121	27	then	then	ADV
cana-533	121	28	we	we	PRON
cana-533	121	29	are	be	AUX
cana-533	121	30	finished	finish	VERB
cana-533	121	31	.	.	PUNCT
cana-533	122	1	otherwise	otherwise	ADV
cana-533	122	2	,	,	PUNCT
cana-533	122	3	we	we	PRON
cana-533	122	4	have	have	VERB
cana-533	122	5	and	and	CCONJ
cana-533	122	6	.	.	PUNCT
cana-533	123	1	first	first	ADV
cana-533	123	2	,	,	PUNCT
cana-533	123	3	we	we	PRON
cana-533	123	4	demonstrate	demonstrate	VERB
cana-533	123	5	that	that	SCONJ
cana-533	123	6	the	the	DET
cana-533	123	7	sequence	sequence	NOUN
cana-533	123	8	and	and	CCONJ
cana-533	123	9	fulfilling	fulfil	VERB
cana-533	123	10	(	(	PUNCT
cana-533	123	11	3.2	3.2	NUM
cana-533	123	12	)	)	PUNCT
cana-533	123	13	provides	provide	VERB
cana-533	123	14	indeed	indeed	ADV
cana-533	123	15	,	,	PUNCT
cana-533	123	16	let	let	VERB
cana-533	123	17	then	then	ADV
cana-533	123	18	we	we	PRON
cana-533	123	19	have	have	VERB
cana-533	123	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	123	21	communications	communication	NOUN
cana-533	123	22	on	on	ADP
cana-533	123	23	applied	apply	VERB
cana-533	123	24	nonlinear	nonlinear	ADJ
cana-533	123	25	analysis	analysis	NOUN
cana-533	123	26	issn	issn	NOUN
cana-533	123	27	:	:	PUNCT
cana-533	123	28	1074	1074	NUM
cana-533	123	29	-	-	PUNCT
cana-533	123	30	133x	133x	NUM
cana-533	123	31	vol	vol	NOUN
cana-533	123	32	31	31	NUM
cana-533	123	33	no	no	NOUN
cana-533	123	34	.	.	NOUN
cana-533	123	35	2	2	NUM
cana-533	123	36	(	(	PUNCT
cana-533	123	37	2024	2024	NUM
cana-533	123	38	)	)	PUNCT
cana-533	123	39	190	190	NUM
cana-533	123	40	similarly	similarly	ADV
cana-533	123	41	,	,	PUNCT
cana-533	123	42	firstly	firstly	ADV
cana-533	123	43	,	,	PUNCT
cana-533	123	44	for	for	ADP
cana-533	123	45	the	the	DET
cana-533	123	46	other	other	ADJ
cana-533	123	47	inequalities	inequality	NOUN
cana-533	123	48	,	,	PUNCT
cana-533	123	49	we	we	PRON
cana-533	123	50	will	will	AUX
cana-533	123	51	handle	handle	VERB
cana-533	123	52	the	the	DET
cana-533	123	53	following	following	ADJ
cana-533	123	54	cases	case	NOUN
cana-533	123	55	:	:	PUNCT
cana-533	123	56	case	case	NOUN
cana-533	123	57	1	1	NUM
cana-533	123	58	:	:	PUNCT
cana-533	123	59	if	if	SCONJ
cana-533	123	60	and	and	CCONJ
cana-533	123	61	.	.	PUNCT
cana-533	124	1	we	we	PRON
cana-533	124	2	have	have	VERB
cana-533	124	3	this	this	PRON
cana-533	124	4	implies	imply	VERB
cana-533	124	5	that	that	SCONJ
cana-533	124	6	which	which	PRON
cana-533	124	7	give	give	VERB
cana-533	124	8	or	or	CCONJ
cana-533	124	9	.	.	PUNCT
cana-533	125	1	so	so	ADV
cana-533	125	2	,	,	PUNCT
cana-533	125	3	we	we	PRON
cana-533	125	4	get	get	VERB
cana-533	125	5	a	a	DET
cana-533	125	6	contradiction	contradiction	NOUN
cana-533	125	7	.	.	PUNCT
cana-533	126	1	case	case	NOUN
cana-533	126	2	2	2	NUM
cana-533	126	3	:	:	PUNCT
cana-533	126	4	if	if	SCONJ
cana-533	126	5	and	and	CCONJ
cana-533	126	6	.	.	PUNCT
cana-533	127	1	we	we	PRON
cana-533	127	2	have	have	VERB
cana-533	127	3	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	127	4	communications	communication	NOUN
cana-533	127	5	on	on	ADP
cana-533	127	6	applied	apply	VERB
cana-533	127	7	nonlinear	nonlinear	ADJ
cana-533	127	8	analysis	analysis	NOUN
cana-533	127	9	issn	issn	NOUN
cana-533	127	10	:	:	PUNCT
cana-533	127	11	1074	1074	NUM
cana-533	127	12	-	-	PUNCT
cana-533	127	13	133x	133x	NUM
cana-533	127	14	vol	vol	NOUN
cana-533	127	15	31	31	NUM
cana-533	127	16	no	no	NOUN
cana-533	127	17	.	.	NOUN
cana-533	127	18	2	2	NUM
cana-533	127	19	(	(	PUNCT
cana-533	127	20	2024	2024	NUM
cana-533	127	21	)	)	PUNCT
cana-533	127	22	191	191	NUM
cana-533	127	23	this	this	PRON
cana-533	127	24	implies	imply	VERB
cana-533	127	25	that	that	SCONJ
cana-533	127	26	which	which	PRON
cana-533	127	27	give	give	VERB
cana-533	127	28	or	or	CCONJ
cana-533	127	29	.	.	PUNCT
cana-533	128	1	so	so	ADV
cana-533	128	2	,	,	PUNCT
cana-533	128	3	we	we	PRON
cana-533	128	4	get	get	VERB
cana-533	128	5	a	a	DET
cana-533	128	6	contradiction	contradiction	NOUN
cana-533	128	7	.	.	PUNCT
cana-533	129	1	case	case	NOUN
cana-533	129	2	3	3	NUM
cana-533	129	3	:	:	PUNCT
cana-533	129	4	if	if	SCONJ
cana-533	129	5	and	and	CCONJ
cana-533	129	6	.	.	PUNCT
cana-533	130	1	then	then	ADV
cana-533	130	2	we	we	PRON
cana-533	130	3	get	get	VERB
cana-533	130	4	a	a	DET
cana-533	130	5	contradiction	contradiction	NOUN
cana-533	130	6	with	with	ADP
cana-533	130	7	the	the	DET
cana-533	130	8	decreasing	decreasing	NOUN
cana-533	130	9	of	of	ADP
cana-533	130	10	the	the	DET
cana-533	130	11	sequences	sequence	NOUN
cana-533	130	12	and	and	CCONJ
cana-533	130	13	.	.	PUNCT
cana-533	131	1	in	in	ADP
cana-533	131	2	the	the	DET
cana-533	131	3	second	second	ADJ
cana-533	131	4	place	place	NOUN
cana-533	131	5	,	,	PUNCT
cana-533	131	6	we	we	PRON
cana-533	131	7	realised	realise	VERB
cana-533	131	8	that	that	PRON
cana-533	131	9	and	and	CCONJ
cana-533	131	10	and	and	CCONJ
cana-533	131	11	then	then	ADV
cana-533	131	12	the	the	DET
cana-533	131	13	inequality	inequality	NOUN
cana-533	131	14	(	(	PUNCT
cana-533	131	15	3.3	3.3	NUM
cana-533	131	16	)	)	PUNCT
cana-533	131	17	.	.	PUNCT
cana-533	132	1	now	now	ADV
cana-533	132	2	we	we	PRON
cana-533	132	3	prove	prove	VERB
cana-533	132	4	the	the	DET
cana-533	132	5	second	second	ADJ
cana-533	132	6	step	step	NOUN
cana-533	132	7	of	of	ADP
cana-533	132	8	induction	induction	NOUN
cana-533	132	9	(	(	PUNCT
cana-533	132	10	3.2	3.2	NUM
cana-533	132	11	)	)	PUNCT
cana-533	132	12	.	.	PUNCT
cana-533	133	1	by	by	ADP
cana-533	133	2	using	use	VERB
cana-533	133	3	assumption	assumption	NOUN
cana-533	133	4	(	(	PUNCT
cana-533	133	5	c	c	NOUN
cana-533	133	6	)	)	PUNCT
cana-533	133	7	,	,	PUNCT
cana-533	133	8	inequality	inequality	NOUN
cana-533	133	9	(	(	PUNCT
cana-533	133	10	3.3	3.3	NUM
cana-533	133	11	)	)	PUNCT
cana-533	133	12	and	and	CCONJ
cana-533	133	13	the	the	DET
cana-533	133	14	nondecreasing	nondecreasing	NOUN
cana-533	133	15	of	of	ADP
cana-533	133	16	in	in	ADP
cana-533	133	17	first	first	ADJ
cana-533	133	18	variable	variable	NOUN
cana-533	133	19	,	,	PUNCT
cana-533	133	20	we	we	PRON
cana-533	133	21	have	have	VERB
cana-533	133	22	if	if	SCONJ
cana-533	133	23	we	we	PRON
cana-533	133	24	assume	assume	VERB
cana-533	133	25	that	that	SCONJ
cana-533	133	26	for	for	ADP
cana-533	133	27	some	some	PRON
cana-533	133	28	or	or	CCONJ
cana-533	133	29	,	,	PUNCT
cana-533	133	30	then	then	ADV
cana-533	133	31	we	we	PRON
cana-533	133	32	have	have	VERB
cana-533	133	33	which	which	PRON
cana-533	133	34	implies	imply	VERB
cana-533	133	35	that	that	PRON
cana-533	133	36	or	or	CCONJ
cana-533	133	37	and	and	CCONJ
cana-533	133	38	then	then	ADV
cana-533	133	39	or	or	CCONJ
cana-533	133	40	which	which	PRON
cana-533	133	41	is	be	AUX
cana-533	133	42	a	a	DET
cana-533	133	43	contradiction	contradiction	NOUN
cana-533	133	44	.	.	PUNCT
cana-533	134	1	so	so	ADV
cana-533	134	2	we	we	PRON
cana-533	134	3	assume	assume	VERB
cana-533	134	4	that	that	SCONJ
cana-533	134	5	for	for	ADP
cana-533	134	6	all	all	PRON
cana-533	134	7	and	and	CCONJ
cana-533	134	8	,	,	PUNCT
cana-533	134	9	then	then	ADV
cana-533	134	10	there	there	PRON
cana-533	134	11	exists	exist	VERB
cana-533	134	12	such	such	ADJ
cana-533	134	13	that	that	SCONJ
cana-533	134	14	moreover	moreover	ADV
cana-533	134	15	,	,	PUNCT
cana-533	134	16	if	if	SCONJ
cana-533	134	17	and	and	CCONJ
cana-533	134	18	is	be	AUX
cana-533	134	19	nondecreasing	nondecrease	VERB
cana-533	134	20	then	then	ADV
cana-533	134	21	we	we	PRON
cana-533	134	22	have	have	VERB
cana-533	134	23	else	else	ADV
cana-533	134	24	if	if	SCONJ
cana-533	135	1	and	and	CCONJ
cana-533	135	2	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-533	135	3	communications	communication	NOUN
cana-533	135	4	on	on	ADP
cana-533	135	5	applied	apply	VERB
cana-533	135	6	nonlinear	nonlinear	ADJ
cana-533	135	7	analysis	analysis	NOUN
cana-533	135	8	issn	issn	NOUN
cana-533	135	9	:	:	PUNCT
cana-533	135	10	1074	1074	NUM
cana-533	135	11	-	-	PUNCT
cana-533	135	12	133x	133x	NUM
cana-533	135	13	vol	vol	NOUN
cana-533	135	14	31	31	NUM
cana-533	135	15	no	no	NOUN
cana-533	135	16	.	.	NOUN
cana-533	135	17	2	2	NUM
cana-533	135	18	(	(	PUNCT
cana-533	135	19	2024	2024	NUM
cana-533	135	20	)	)	PUNCT
cana-533	135	21	192	192	NUM
cana-533	135	22	on	on	ADP
cana-533	135	23	the	the	DET
cana-533	135	24	other	other	ADJ
cana-533	135	25	hand	hand	NOUN
cana-533	135	26	,	,	PUNCT
cana-533	135	27	we	we	PRON
cana-533	135	28	have	have	VERB
cana-533	135	29	if	if	SCONJ
cana-533	135	30	we	we	PRON
cana-533	135	31	assume	assume	VERB
cana-533	135	32	that	that	SCONJ
cana-533	135	33	for	for	ADP
cana-533	135	34	some	some	PRON
cana-533	135	35	then	then	ADV
cana-533	135	36	we	we	PRON
cana-533	135	37	get	get	VERB
cana-533	135	38	a	a	DET
cana-533	135	39	contradiction	contradiction	NOUN
cana-533	135	40	with	with	ADP
cana-533	135	41	the	the	DET
cana-533	135	42	definition	definition	NOUN
cana-533	135	43	of	of	ADP
cana-533	135	44	and	and	CCONJ
cana-533	135	45	.	.	PUNCT
cana-533	136	1	then	then	ADV
cana-533	136	2	we	we	PRON
cana-533	136	3	suppose	suppose	VERB
cana-533	136	4	that	that	SCONJ
cana-533	136	5	and	and	CCONJ
cana-533	136	6	then	then	ADV
cana-533	136	7	there	there	PRON
cana-533	136	8	exists	exist	VERB
cana-533	136	9	such	such	ADJ
cana-533	136	10	that	that	PRON
cana-533	136	11	and	and	CCONJ
cana-533	136	12	with	with	ADP
cana-533	136	13	the	the	DET
cana-533	136	14	inequality	inequality	NOUN
cana-533	136	15	(	(	PUNCT
cana-533	136	16	3.3	3.3	NUM
cana-533	136	17	)	)	PUNCT
cana-533	136	18	,	,	PUNCT
cana-533	136	19	we	we	PRON
cana-533	136	20	have	have	VERB
cana-533	136	21	.	.	PUNCT
cana-533	137	1	on	on	ADP
cana-533	137	2	the	the	DET
cana-533	137	3	other	other	ADJ
cana-533	137	4	hand	hand	NOUN
cana-533	137	5	,	,	PUNCT
cana-533	137	6	is	be	AUX
cana-533	137	7	an	an	DET
cana-533	137	8	element	element	NOUN
cana-533	137	9	of	of	ADP
cana-533	137	10	the	the	DET
cana-533	137	11	opened	open	VERB
cana-533	137	12	-ball	-ball	NOUN
cana-533	137	13	and	and	CCONJ
cana-533	137	14	be	be	AUX
cana-533	137	15	an	an	DET
cana-533	137	16	element	element	NOUN
cana-533	137	17	of	of	ADP
cana-533	137	18	the	the	DET
cana-533	137	19	opened	open	VERB
cana-533	137	20	-ball	-ball	NOUN
cana-533	137	21	.	.	PUNCT
cana-533	138	1	indeed	indeed	ADV
cana-533	138	2	,	,	PUNCT
cana-533	138	3	and	and	CCONJ
cana-533	138	4	we	we	PRON
cana-533	138	5	have	have	VERB
cana-533	138	6	for	for	ADP
cana-533	138	7	all	all	DET
cana-533	138	8	integers	integer	NOUN
cana-533	138	9	and	and	CCONJ
cana-533	138	10	where	where	SCONJ
cana-533	138	11	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	138	12	communications	communication	NOUN
cana-533	138	13	on	on	ADP
cana-533	138	14	applied	apply	VERB
cana-533	138	15	nonlinear	nonlinear	ADJ
cana-533	138	16	analysis	analysis	NOUN
cana-533	138	17	issn	issn	NOUN
cana-533	138	18	:	:	PUNCT
cana-533	138	19	1074	1074	NUM
cana-533	138	20	-	-	PUNCT
cana-533	138	21	133x	133x	NUM
cana-533	138	22	vol	vol	NOUN
cana-533	138	23	31	31	NUM
cana-533	138	24	no	no	NOUN
cana-533	138	25	.	.	NOUN
cana-533	138	26	2	2	NUM
cana-533	138	27	(	(	PUNCT
cana-533	138	28	2024	2024	NUM
cana-533	138	29	)	)	PUNCT
cana-533	138	30	193	193	NUM
cana-533	138	31	we	we	PRON
cana-533	138	32	know	know	VERB
cana-533	138	33	that	that	PRON
cana-533	138	34	is	be	AUX
cana-533	138	35	a	a	DET
cana-533	138	36	0	0	NUM
cana-533	138	37	-	-	PUNCT
cana-533	138	38	cauchy	cauchy	ADJ
cana-533	138	39	sequence	sequence	NOUN
cana-533	138	40	in	in	ADV
cana-533	138	41	since	since	SCONJ
cana-533	138	42	converges	converge	NOUN
cana-533	138	43	for	for	ADP
cana-533	138	44	each	each	PRON
cana-533	138	45	.	.	PUNCT
cana-533	139	1	as	as	SCONJ
cana-533	139	2	is	be	AUX
cana-533	139	3	a	a	DET
cana-533	139	4	0	0	NUM
cana-533	139	5	-	-	PUNCT
cana-533	139	6	complete	complete	ADJ
cana-533	139	7	subspace	subspace	NOUN
cana-533	139	8	,	,	PUNCT
cana-533	139	9	converges	converge	VERB
cana-533	139	10	to	to	ADP
cana-533	139	11	with	with	ADP
cana-533	139	12	respect	respect	NOUN
cana-533	139	13	to	to	ADP
cana-533	139	14	.	.	PUNCT
cana-533	140	1	this	this	PRON
cana-533	140	2	implies	imply	VERB
cana-533	140	3	that	that	SCONJ
cana-533	140	4	analogously	analogously	ADV
cana-533	140	5	,	,	PUNCT
cana-533	140	6	for	for	ADP
cana-533	140	7	a	a	DET
cana-533	140	8	sequence	sequence	NOUN
cana-533	140	9	we	we	PRON
cana-533	140	10	have	have	AUX
cana-533	140	11	given	give	VERB
cana-533	140	12	that	that	PRON
cana-533	140	13	is	be	AUX
cana-533	140	14	convergent	convergent	NOUN
cana-533	140	15	for	for	ADP
cana-533	140	16	each	each	PRON
cana-533	140	17	and	and	CCONJ
cana-533	140	18	is	be	AUX
cana-533	140	19	a	a	DET
cana-533	140	20	0	0	NUM
cana-533	140	21	-	-	PUNCT
cana-533	140	22	complete	complete	ADJ
cana-533	140	23	subspace	subspace	NOUN
cana-533	140	24	,	,	PUNCT
cana-533	140	25	we	we	PRON
cana-533	140	26	can	can	AUX
cana-533	140	27	conclude	conclude	VERB
cana-533	140	28	that	that	PRON
cana-533	140	29	is	be	AUX
cana-533	140	30	a	a	DET
cana-533	140	31	0	0	NUM
cana-533	140	32	-	-	PUNCT
cana-533	140	33	cauchy	cauchy	ADJ
cana-533	140	34	sequence	sequence	NOUN
cana-533	140	35	in	in	ADV
cana-533	140	36	,	,	PUNCT
cana-533	140	37	and	and	CCONJ
cana-533	140	38	thus	thus	ADV
cana-533	140	39	converges	converge	VERB
cana-533	140	40	with	with	ADP
cana-533	140	41	respect	respect	NOUN
cana-533	140	42	to	to	ADP
cana-533	140	43	to	to	ADP
cana-533	140	44	a	a	DET
cana-533	140	45	point	point	NOUN
cana-533	140	46	.	.	PUNCT
cana-533	141	1	we	we	PRON
cana-533	141	2	assert	assert	VERB
cana-533	141	3	now	now	ADV
cana-533	141	4	that	that	PRON
cana-533	141	5	.	.	PUNCT
cana-533	142	1	the	the	DET
cana-533	142	2	modified	modify	VERB
cana-533	142	3	triangle	triangle	NOUN
cana-533	142	4	inequality	inequality	NOUN
cana-533	142	5	of	of	ADP
cana-533	142	6	,	,	PUNCT
cana-533	142	7	is	be	AUX
cana-533	142	8	a	a	DET
cana-533	142	9	-class	-class	ADJ
cana-533	142	10	function	function	NOUN
cana-533	142	11	and	and	CCONJ
cana-533	142	12	assumption	assumption	NOUN
cana-533	142	13	(	(	PUNCT
cana-533	142	14	c	c	X
cana-533	142	15	)	)	PUNCT
cana-533	142	16	give	give	VERB
cana-533	142	17	by	by	ADP
cana-533	142	18	taking	take	VERB
cana-533	142	19	the	the	DET
cana-533	142	20	limit	limit	NOUN
cana-533	142	21	as	as	ADP
cana-533	142	22	approaches	approach	NOUN
cana-533	142	23	infinity	infinity	NOUN
cana-533	142	24	,	,	PUNCT
cana-533	142	25	we	we	PRON
cana-533	142	26	get	get	VERB
cana-533	142	27	.	.	PUNCT
cana-533	143	1	applying	apply	VERB
cana-533	143	2	lemma	lemma	PROPN
cana-533	143	3	2.1	2.1	NUM
cana-533	143	4	,	,	PUNCT
cana-533	143	5	we	we	PRON
cana-533	143	6	conclude	conclude	VERB
cana-533	143	7	that	that	SCONJ
cana-533	143	8	analogously	analogously	ADV
cana-533	143	9	,	,	PUNCT
cana-533	143	10	we	we	PRON
cana-533	143	11	now	now	ADV
cana-533	143	12	claim	claim	VERB
cana-533	143	13	that	that	PRON
cana-533	143	14	.	.	PUNCT
cana-533	144	1	this	this	PRON
cana-533	144	2	follows	follow	VERB
cana-533	144	3	from	from	ADP
cana-533	144	4	the	the	DET
cana-533	144	5	modified	modify	VERB
cana-533	144	6	triangle	triangle	NOUN
cana-533	144	7	inequality	inequality	NOUN
cana-533	144	8	and	and	CCONJ
cana-533	144	9	assumption	assumption	NOUN
cana-533	144	10	(	(	PUNCT
cana-533	144	11	b	b	NOUN
cana-533	144	12	)	)	PUNCT
cana-533	144	13	,	,	PUNCT
cana-533	144	14	which	which	PRON
cana-533	144	15	imply	imply	VERB
cana-533	144	16	that	that	SCONJ
cana-533	144	17	since	since	SCONJ
cana-533	144	18	,	,	PUNCT
cana-533	144	19	taking	take	VERB
cana-533	144	20	limit	limit	NOUN
cana-533	144	21	as	as	SCONJ
cana-533	144	22	,	,	PUNCT
cana-533	144	23	we	we	PRON
cana-533	144	24	obtain	obtain	VERB
cana-533	144	25	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	144	26	communications	communication	NOUN
cana-533	144	27	on	on	ADP
cana-533	144	28	applied	apply	VERB
cana-533	144	29	nonlinear	nonlinear	ADJ
cana-533	144	30	analysis	analysis	NOUN
cana-533	144	31	issn	issn	NOUN
cana-533	144	32	:	:	PUNCT
cana-533	144	33	1074	1074	NUM
cana-533	144	34	-	-	PUNCT
cana-533	144	35	133x	133x	NUM
cana-533	144	36	vol	vol	NOUN
cana-533	144	37	31	31	NUM
cana-533	144	38	no	no	NOUN
cana-533	144	39	.	.	NOUN
cana-533	144	40	2	2	NUM
cana-533	144	41	(	(	PUNCT
cana-533	144	42	2024	2024	NUM
cana-533	144	43	)	)	PUNCT
cana-533	144	44	194	194	NUM
cana-533	144	45	this	this	PRON
cana-533	144	46	,	,	PUNCT
cana-533	144	47	according	accord	VERB
cana-533	144	48	to	to	ADP
cana-533	144	49	lemma	lemma	PROPN
cana-533	144	50	2.1	2.1	NUM
cana-533	144	51	,	,	PUNCT
cana-533	144	52	implies	imply	VERB
cana-533	144	53	assuming	assume	VERB
cana-533	144	54	single	single	ADJ
cana-533	144	55	-	-	PUNCT
cana-533	144	56	valued	value	VERB
cana-533	144	57	mappings	mapping	NOUN
cana-533	144	58	and	and	CCONJ
cana-533	144	59	and	and	CCONJ
cana-533	144	60	,	,	PUNCT
cana-533	144	61	let	let	VERB
cana-533	144	62	and	and	CCONJ
cana-533	144	63	be	be	AUX
cana-533	144	64	two	two	NUM
cana-533	144	65	different	different	ADJ
cana-533	144	66	altering	altering	NOUN
cana-533	144	67	points	point	NOUN
cana-533	144	68	in	in	ADP
cana-533	144	69	for	for	ADP
cana-533	144	70	and	and	CCONJ
cana-533	144	71	,	,	PUNCT
cana-533	144	72	where	where	SCONJ
cana-533	144	73	,	,	PUNCT
cana-533	144	74	,	,	PUNCT
cana-533	144	75	,	,	PUNCT
cana-533	144	76	and	and	CCONJ
cana-533	144	77	.	.	PUNCT
cana-533	145	1	then	then	ADV
cana-533	145	2	,	,	PUNCT
cana-533	145	3	we	we	PRON
cana-533	145	4	have	have	VERB
cana-533	145	5	:	:	PUNCT
cana-533	145	6	and	and	CCONJ
cana-533	145	7	so	so	ADV
cana-533	145	8	,	,	PUNCT
cana-533	145	9	by	by	ADP
cana-533	145	10	using	use	VERB
cana-533	145	11	,	,	PUNCT
cana-533	145	12	we	we	PRON
cana-533	145	13	get	get	VERB
cana-533	145	14	then	then	ADV
cana-533	145	15	we	we	PRON
cana-533	145	16	have	have	AUX
cana-533	145	17	which	which	PRON
cana-533	145	18	implies	imply	VERB
cana-533	145	19	that	that	PRON
cana-533	145	20	or	or	CCONJ
cana-533	145	21	,	,	PUNCT
cana-533	145	22	thus	thus	ADV
cana-533	145	23	or	or	CCONJ
cana-533	145	24	which	which	PRON
cana-533	145	25	is	be	AUX
cana-533	145	26	a	a	DET
cana-533	145	27	contradiction	contradiction	NOUN
cana-533	145	28	and	and	CCONJ
cana-533	145	29	then	then	ADV
cana-533	145	30	there	there	PRON
cana-533	145	31	exist	exist	VERB
cana-533	145	32	a	a	DET
cana-533	145	33	unique	unique	ADJ
cana-533	145	34	altering	altering	NOUN
cana-533	145	35	points	point	NOUN
cana-533	145	36	such	such	ADJ
cana-533	145	37	that	that	PRON
cana-533	145	38	and	and	CCONJ
cana-533	145	39	.	.	PUNCT
cana-533	146	1	by	by	ADP
cana-533	146	2	consequence	consequence	NOUN
cana-533	146	3	,	,	PUNCT
cana-533	146	4	we	we	PRON
cana-533	146	5	get	get	VERB
cana-533	146	6	,	,	PUNCT
cana-533	146	7	4	4	X
cana-533	146	8	.	.	PUNCT
cana-533	147	1	corollaries	corollary	NOUN
cana-533	147	2	related	relate	VERB
cana-533	147	3	as	as	ADP
cana-533	147	4	corollaries	corollary	NOUN
cana-533	147	5	,	,	PUNCT
cana-533	147	6	we	we	PRON
cana-533	147	7	have	have	VERB
cana-533	147	8	an	an	DET
cana-533	147	9	extended	extended	ADJ
cana-533	147	10	version	version	NOUN
cana-533	147	11	of	of	ADP
cana-533	147	12	[	[	X
cana-533	147	13	6,theorem	6,theorem	NUM
cana-533	147	14	11	11	NUM
cana-533	147	15	]	]	PUNCT
cana-533	147	16	(	(	PUNCT
cana-533	147	17	i.e.	i.e.	X
cana-533	147	18	theorem	theorem	VERB
cana-533	147	19	2.1	2.1	NUM
cana-533	147	20	)	)	PUNCT
cana-533	147	21	,	,	PUNCT
cana-533	147	22	[	[	X
cana-533	147	23	7	7	NUM
cana-533	147	24	,	,	PUNCT
cana-533	147	25	theorem	theorem	VERB
cana-533	147	26	6.3	6.3	NUM
cana-533	147	27	]	]	PUNCT
cana-533	147	28	and	and	CCONJ
cana-533	147	29	[	[	X
cana-533	147	30	9	9	NUM
cana-533	147	31	,	,	PUNCT
cana-533	147	32	theorem	theorem	VERB
cana-533	147	33	3.1	3.1	NUM
cana-533	147	34	]	]	PUNCT
cana-533	147	35	within	within	ADP
cana-533	147	36	the	the	DET
cana-533	147	37	context	context	NOUN
cana-533	147	38	of	of	ADP
cana-533	147	39	partial	partial	ADJ
cana-533	147	40	metric	metric	ADJ
cana-533	147	41	spaces	space	NOUN
cana-533	147	42	that	that	PRON
cana-533	147	43	are	be	AUX
cana-533	147	44	0complete	0complete	PROPN
cana-533	147	45	.	.	PUNCT
cana-533	148	1	corollary	corollary	ADJ
cana-533	148	2	4.1	4.1	NUM
cana-533	148	3	.	.	PUNCT
cana-533	149	1	let	let	VERB
cana-533	149	2	and	and	CCONJ
cana-533	149	3	be	be	AUX
cana-533	149	4	partial	partial	ADJ
cana-533	149	5	metric	metric	ADJ
cana-533	149	6	spaces	space	NOUN
cana-533	149	7	where	where	SCONJ
cana-533	149	8	and	and	CCONJ
cana-533	149	9	are	be	AUX
cana-533	149	10	0	0	NUM
cana-533	149	11	-	-	PUNCT
cana-533	149	12	complete	complete	ADJ
cana-533	149	13	subspaces	subspace	NOUN
cana-533	149	14	of	of	ADP
cana-533	149	15	and	and	CCONJ
cana-533	149	16	respectively	respectively	ADV
cana-533	149	17	,	,	PUNCT
cana-533	149	18	for	for	ADP
cana-533	149	19	,	,	PUNCT
cana-533	149	20	,	,	PUNCT
cana-533	149	21	and	and	CCONJ
cana-533	149	22	.	.	PUNCT
cana-533	150	1	consider	consider	VERB
cana-533	150	2	setvalued	setvalue	VERB
cana-533	150	3	mappings	mapping	NOUN
cana-533	150	4	and	and	CCONJ
cana-533	150	5	such	such	ADJ
cana-533	150	6	that	that	PRON
cana-533	150	7	.	.	PUNCT
cana-533	151	1	let	let	AUX
cana-533	151	2	be	be	AUX
cana-533	151	3	increasing	increase	VERB
cana-533	151	4	functions	function	NOUN
cana-533	151	5	and	and	CCONJ
cana-533	151	6	.	.	PUNCT
cana-533	152	1	assume	assume	VERB
cana-533	152	2	the	the	DET
cana-533	152	3	following	follow	VERB
cana-533	152	4	:	:	PUNCT
cana-533	152	5	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	152	6	communications	communication	NOUN
cana-533	152	7	on	on	ADP
cana-533	152	8	applied	apply	VERB
cana-533	152	9	nonlinear	nonlinear	ADJ
cana-533	152	10	analysis	analysis	NOUN
cana-533	152	11	issn	issn	NOUN
cana-533	152	12	:	:	PUNCT
cana-533	152	13	1074	1074	NUM
cana-533	152	14	-	-	PUNCT
cana-533	152	15	133x	133x	NUM
cana-533	152	16	vol	vol	NOUN
cana-533	152	17	31	31	NUM
cana-533	152	18	no	no	NOUN
cana-533	152	19	.	.	NOUN
cana-533	152	20	2	2	NUM
cana-533	152	21	(	(	PUNCT
cana-533	152	22	2024	2024	NUM
cana-533	152	23	)	)	PUNCT
cana-533	152	24	195	195	NUM
cana-533	152	25	then	then	ADV
cana-533	152	26	there	there	PRON
cana-533	152	27	exist	exist	VERB
cana-533	152	28	an	an	DET
cana-533	152	29	altering	altering	NOUN
cana-533	152	30	point	point	NOUN
cana-533	152	31	of	of	ADP
cana-533	152	32	and	and	CCONJ
cana-533	152	33	in	in	ADV
cana-533	152	34	.	.	PUNCT
cana-533	153	1	if	if	SCONJ
cana-533	153	2	and	and	CCONJ
cana-533	153	3	are	be	AUX
cana-533	153	4	both	both	PRON
cana-533	153	5	single	single	ADV
cana-533	153	6	-	-	PUNCT
cana-533	153	7	valued	value	VERB
cana-533	153	8	mappings	mapping	NOUN
cana-533	153	9	and	and	CCONJ
cana-533	153	10	,	,	PUNCT
cana-533	153	11	then	then	ADV
cana-533	153	12	is	be	AUX
cana-533	153	13	the	the	DET
cana-533	153	14	unique	unique	ADJ
cana-533	153	15	fixed	fix	VERB
cana-533	153	16	point	point	NOUN
cana-533	153	17	of	of	ADP
cana-533	153	18	in	in	ADP
cana-533	153	19	and	and	CCONJ
cana-533	153	20	is	be	AUX
cana-533	153	21	the	the	DET
cana-533	153	22	unique	unique	ADJ
cana-533	153	23	fixed	fix	VERB
cana-533	153	24	point	point	NOUN
cana-533	153	25	of	of	ADP
cana-533	153	26	in	in	ADP
cana-533	153	27	.	.	PUNCT
cana-533	154	1	proof	proof	NOUN
cana-533	154	2	.	.	PUNCT
cana-533	155	1	take	take	VERB
cana-533	155	2	such	such	ADJ
cana-533	155	3	that	that	PRON
cana-533	155	4	and	and	CCONJ
cana-533	155	5	then	then	ADV
cana-533	155	6	the	the	DET
cana-533	155	7	subset	subset	NOUN
cana-533	155	8	will	will	AUX
cana-533	155	9	be	be	AUX
cana-533	155	10	the	the	DET
cana-533	155	11	subset	subset	NOUN
cana-533	155	12	.	.	PUNCT
cana-533	156	1	since	since	SCONJ
cana-533	156	2	the	the	DET
cana-533	156	3	function	function	NOUN
cana-533	156	4	does	do	AUX
cana-533	156	5	not	not	PART
cana-533	156	6	depend	depend	VERB
cana-533	156	7	on	on	ADP
cana-533	156	8	the	the	DET
cana-533	156	9	second	second	ADJ
cana-533	156	10	variable	variable	NOUN
cana-533	156	11	,	,	PUNCT
cana-533	156	12	we	we	PRON
cana-533	156	13	can	can	AUX
cana-533	156	14	choose	choose	VERB
cana-533	156	15	to	to	PART
cana-533	156	16	be	be	AUX
cana-533	156	17	non	non	ADJ
cana-533	156	18	-	-	ADJ
cana-533	156	19	decreasing	decrease	VERB
cana-533	156	20	or	or	CCONJ
cana-533	156	21	greater	great	ADJ
cana-533	156	22	than	than	ADP
cana-533	156	23	,	,	PUNCT
cana-533	156	24	and	and	CCONJ
cana-533	156	25	then	then	ADV
cana-533	156	26	apply	apply	VERB
cana-533	156	27	the	the	DET
cana-533	156	28	theorem	theorem	NOUN
cana-533	156	29	3.1	3.1	NUM
cana-533	156	30	.	.	PUNCT
cana-533	156	31	using	use	VERB
cana-533	156	32	theorem	theorem	NOUN
cana-533	156	33	3.1	3.1	NUM
cana-533	156	34	with	with	ADP
cana-533	156	35	,	,	PUNCT
cana-533	156	36	,	,	PUNCT
cana-533	156	37	,	,	PUNCT
cana-533	156	38	and	and	CCONJ
cana-533	156	39	,	,	PUNCT
cana-533	156	40	we	we	PRON
cana-533	156	41	can	can	AUX
cana-533	156	42	establish	establish	VERB
cana-533	156	43	the	the	DET
cana-533	156	44	following	following	ADJ
cana-533	156	45	fixed	fix	VERB
cana-533	156	46	point	point	NOUN
cana-533	156	47	theorem	theorem	ADJ
cana-533	156	48	:	:	PUNCT
cana-533	156	49	corollary	corollary	ADJ
cana-533	156	50	4.2	4.2	NUM
cana-533	156	51	.	.	PUNCT
cana-533	157	1	theorem	theorem	VERB
cana-533	157	2	3.4	3.4	NUM
cana-533	157	3	in	in	ADP
cana-533	157	4	[	[	X
cana-533	157	5	2	2	NUM
cana-533	157	6	]	]	PUNCT
cana-533	157	7	.	.	PUNCT
cana-533	158	1	for	for	ADP
cana-533	158	2	and	and	CCONJ
cana-533	158	3	where	where	SCONJ
cana-533	158	4	and	and	CCONJ
cana-533	158	5	applying	apply	VERB
cana-533	158	6	corollary	corollary	NOUN
cana-533	158	7	4.1	4.1	NUM
cana-533	158	8	then	then	ADV
cana-533	158	9	we	we	PRON
cana-533	158	10	get	get	VERB
cana-533	158	11	corollary	corollary	ADJ
cana-533	158	12	4.3	4.3	NUM
cana-533	158	13	.	.	PUNCT
cana-533	159	1	let	let	VERB
cana-533	159	2	and	and	CCONJ
cana-533	159	3	)	)	PUNCT
cana-533	159	4	be	be	AUX
cana-533	159	5	partial	partial	ADJ
cana-533	159	6	metric	metric	ADJ
cana-533	159	7	spaces	space	NOUN
cana-533	159	8	such	such	ADJ
cana-533	159	9	that	that	PRON
cana-533	159	10	and	and	CCONJ
cana-533	159	11	are	be	AUX
cana-533	159	12	0	0	NUM
cana-533	159	13	-	-	PUNCT
cana-533	159	14	complete	complete	ADJ
cana-533	159	15	subspaces	subspace	NOUN
cana-533	159	16	of	of	ADP
cana-533	159	17	and	and	CCONJ
cana-533	159	18	respectively	respectively	ADV
cana-533	159	19	,	,	PUNCT
cana-533	159	20	for	for	ADP
cana-533	159	21	,	,	PUNCT
cana-533	159	22	and	and	CCONJ
cana-533	159	23	.	.	PUNCT
cana-533	160	1	let	let	VERB
cana-533	160	2	and	and	CCONJ
cana-533	160	3	are	be	AUX
cana-533	160	4	set	set	VERB
cana-533	160	5	-	-	PUNCT
cana-533	160	6	valued	value	VERB
cana-533	160	7	mappings	mapping	NOUN
cana-533	160	8	such	such	ADJ
cana-533	160	9	that	that	PRON
cana-533	160	10	.	.	PUNCT
cana-533	161	1	suppose	suppose	VERB
cana-533	161	2	that	that	SCONJ
cana-533	161	3	the	the	DET
cana-533	161	4	following	follow	VERB
cana-533	161	5	assumptions	assumption	NOUN
cana-533	161	6	hold	hold	VERB
cana-533	161	7	:	:	PUNCT
cana-533	161	8	then	then	ADV
cana-533	161	9	there	there	PRON
cana-533	161	10	exist	exist	VERB
cana-533	161	11	an	an	DET
cana-533	161	12	altering	altering	NOUN
cana-533	161	13	point	point	NOUN
cana-533	161	14	of	of	ADP
cana-533	161	15	and	and	CCONJ
cana-533	161	16	in	in	ADV
cana-533	161	17	.	.	PUNCT
cana-533	162	1	if	if	SCONJ
cana-533	162	2	and	and	CCONJ
cana-533	162	3	are	be	AUX
cana-533	162	4	both	both	PRON
cana-533	162	5	single	single	ADV
cana-533	162	6	-	-	PUNCT
cana-533	162	7	valued	value	VERB
cana-533	162	8	mappings	mapping	NOUN
cana-533	162	9	and	and	CCONJ
cana-533	162	10	,	,	PUNCT
cana-533	162	11	then	then	ADV
cana-533	162	12	is	be	AUX
cana-533	162	13	the	the	DET
cana-533	162	14	unique	unique	ADJ
cana-533	162	15	fixed	fix	VERB
cana-533	162	16	point	point	NOUN
cana-533	162	17	of	of	ADP
cana-533	162	18	in	in	ADP
cana-533	162	19	and	and	CCONJ
cana-533	162	20	is	be	AUX
cana-533	162	21	the	the	DET
cana-533	162	22	unique	unique	ADJ
cana-533	162	23	fixed	fix	VERB
cana-533	162	24	point	point	NOUN
cana-533	162	25	of	of	ADP
cana-533	162	26	in	in	ADP
cana-533	162	27	.	.	PUNCT
cana-533	163	1	proof	proof	NOUN
cana-533	163	2	.	.	PUNCT
cana-533	164	1	for	for	ADP
cana-533	164	2	and	and	CCONJ
cana-533	164	3	the	the	DET
cana-533	164	4	corresponding	corresponding	ADJ
cana-533	164	5	estimate	estimate	NOUN
cana-533	164	6	function	function	NOUN
cana-533	164	7	,	,	PUNCT
cana-533	164	8	we	we	PRON
cana-533	164	9	select	select	VERB
cana-533	164	10	the	the	DET
cana-533	164	11	bianchini	bianchini	NOUN
cana-533	164	12	-	-	PUNCT
cana-533	164	13	grandolfi	grandolfi	ADJ
cana-533	164	14	gauge	gauge	NOUN
cana-533	164	15	function	function	NOUN
cana-533	164	16	in	in	ADP
cana-533	164	17	accordance	accordance	NOUN
cana-533	164	18	with	with	ADP
cana-533	164	19	corollary	corollary	ADJ
cana-533	164	20	4.1	4.1	NUM
cana-533	164	21	.	.	PUNCT
cana-533	165	1	then	then	ADV
cana-533	165	2	we	we	PRON
cana-533	165	3	set	set	VERB
cana-533	165	4	and	and	CCONJ
cana-533	165	5	apply	apply	VERB
cana-533	165	6	corollary	corollary	NOUN
cana-533	165	7	4.1	4.1	NUM
cana-533	165	8	.	.	PUNCT
cana-533	166	1	if	if	SCONJ
cana-533	166	2	,	,	PUNCT
cana-533	166	3	then	then	ADV
cana-533	166	4	,	,	PUNCT
cana-533	166	5	and	and	CCONJ
cana-533	166	6	we	we	PRON
cana-533	166	7	get	get	VERB
cana-533	166	8	the	the	DET
cana-533	166	9	extended	extended	ADJ
cana-533	166	10	version	version	NOUN
cana-533	166	11	of	of	ADP
cana-533	166	12	[	[	X
cana-533	166	13	7	7	NUM
cana-533	166	14	,	,	PUNCT
cana-533	166	15	theorem	theorem	VERB
cana-533	166	16	6.3	6.3	NUM
cana-533	166	17	]	]	PUNCT
cana-533	166	18	as	as	SCONJ
cana-533	166	19	follows	follow	VERB
cana-533	166	20	corollary	corollary	NOUN
cana-533	166	21	4.4	4.4	NUM
cana-533	166	22	.	.	PUNCT
cana-533	167	1	let	let	VERB
cana-533	167	2	and	and	CCONJ
cana-533	167	3	be	be	AUX
cana-533	167	4	0	0	NUM
cana-533	167	5	-	-	PUNCT
cana-533	167	6	complete	complete	ADJ
cana-533	167	7	partial	partial	ADJ
cana-533	167	8	metric	metric	ADJ
cana-533	167	9	spaces	space	NOUN
cana-533	167	10	.	.	PUNCT
cana-533	168	1	let	let	VERB
cana-533	168	2	.	.	PUNCT
cana-533	169	1	let	let	VERB
cana-533	169	2	and	and	CCONJ
cana-533	169	3	be	be	AUX
cana-533	169	4	set	set	VERB
cana-533	169	5	-	-	PUNCT
cana-533	169	6	valued	value	VERB
cana-533	169	7	mappings	mapping	NOUN
cana-533	169	8	.	.	PUNCT
cana-533	170	1	suppose	suppose	VERB
cana-533	170	2	the	the	DET
cana-533	170	3	following	follow	VERB
cana-533	170	4	assumptions	assumption	NOUN
cana-533	170	5	hold	hold	VERB
cana-533	170	6	:	:	PUNCT
cana-533	170	7	https://internationalpubls.com	https://internationalpubls.com	X
cana-533	170	8	communications	communication	NOUN
cana-533	170	9	on	on	ADP
cana-533	170	10	applied	apply	VERB
cana-533	170	11	nonlinear	nonlinear	ADJ
cana-533	170	12	analysis	analysis	NOUN
cana-533	170	13	issn	issn	NOUN
cana-533	170	14	:	:	PUNCT
cana-533	170	15	1074	1074	NUM
cana-533	170	16	-	-	PUNCT
cana-533	170	17	133x	133x	NUM
cana-533	170	18	vol	vol	NOUN
cana-533	170	19	31	31	NUM
cana-533	170	20	no	no	NOUN
cana-533	170	21	.	.	NOUN
cana-533	170	22	2	2	NUM
cana-533	170	23	(	(	PUNCT
cana-533	170	24	2024	2024	NUM
cana-533	170	25	)	)	PUNCT
cana-533	170	26	196	196	NUM
cana-533	170	27	then	then	ADV
cana-533	170	28	there	there	PRON
cana-533	170	29	exist	exist	VERB
cana-533	170	30	an	an	DET
cana-533	170	31	altering	altering	NOUN
cana-533	170	32	point	point	NOUN
cana-533	170	33	of	of	ADP
cana-533	170	34	and	and	CCONJ
cana-533	170	35	in	in	ADV
cana-533	170	36	.	.	PUNCT
cana-533	171	1	if	if	SCONJ
cana-533	171	2	and	and	CCONJ
cana-533	171	3	are	be	AUX
cana-533	171	4	both	both	PRON
cana-533	171	5	single	single	ADV
cana-533	171	6	-	-	PUNCT
cana-533	171	7	valued	value	VERB
cana-533	171	8	mappings	mapping	NOUN
cana-533	171	9	,	,	PUNCT
cana-533	171	10	then	then	ADV
cana-533	171	11	is	be	AUX
cana-533	171	12	the	the	DET
cana-533	171	13	unique	unique	ADJ
cana-533	171	14	fixed	fix	VERB
cana-533	171	15	point	point	NOUN
cana-533	171	16	of	of	ADP
cana-533	171	17	in	in	ADP
cana-533	171	18	and	and	CCONJ
cana-533	171	19	is	be	AUX
cana-533	171	20	the	the	DET
cana-533	171	21	unique	unique	ADJ
cana-533	171	22	fixed	fix	VERB
cana-533	171	23	point	point	NOUN
cana-533	171	24	of	of	ADP
cana-533	171	25	in	in	ADP
cana-533	171	26	.	.	PUNCT
cana-533	172	1	proof	proof	NOUN
cana-533	172	2	.	.	PUNCT
cana-533	173	1	given	give	VERB
cana-533	173	2	and	and	CCONJ
cana-533	173	3	such	such	ADJ
cana-533	173	4	that	that	PRON
cana-533	173	5	,	,	PUNCT
cana-533	173	6	let	let	AUX
cana-533	173	7	be	be	AUX
cana-533	173	8	chosen	choose	VERB
cana-533	173	9	such	such	ADJ
cana-533	173	10	that	that	SCONJ
cana-533	173	11	then	then	ADV
cana-533	173	12	,	,	PUNCT
cana-533	173	13	we	we	PRON
cana-533	173	14	can	can	AUX
cana-533	173	15	apply	apply	VERB
cana-533	173	16	corollary	corollary	ADJ
cana-533	173	17	4.3	4.3	NUM
cana-533	173	18	.	.	PUNCT
cana-533	174	1	references	reference	NOUN
cana-533	174	2	[	[	X
cana-533	174	3	1	1	NUM
cana-533	174	4	]	]	PUNCT
cana-533	174	5	a.	a.	NOUN
cana-533	174	6	h.	h.	PROPN
cana-533	174	7	ansari	ansari	PROPN
cana-533	174	8	,	,	PUNCT
cana-533	174	9	note	note	NOUN
cana-533	174	10	on	on	ADP
cana-533	174	11	φ	φ	PROPN
cana-533	174	12	−	−	PROPN
cana-533	174	13	ψ	ψ	X
cana-533	174	14	-	-	ADJ
cana-533	174	15	contractive	contractive	ADJ
cana-533	174	16	type	type	NOUN
cana-533	174	17	mappings	mapping	NOUN
cana-533	174	18	and	and	CCONJ
cana-533	174	19	related	relate	VERB
cana-533	174	20	fixed	fix	VERB
cana-533	174	21	point	point	NOUN
cana-533	174	22	,	,	PUNCT
cana-533	174	23	in	in	ADP
cana-533	174	24	:	:	PUNCT
cana-533	174	25	the	the	DET
cana-533	174	26	2nd	2nd	ADJ
cana-533	174	27	regional	regional	ADJ
cana-533	174	28	conference	conference	NOUN
cana-533	174	29	on	on	ADP
cana-533	174	30	mathematics	mathematic	NOUN
cana-533	174	31	and	and	CCONJ
cana-533	174	32	applications	application	NOUN
cana-533	174	33	,	,	PUNCT
cana-533	174	34	pnu	pnu	PROPN
cana-533	174	35	(	(	PUNCT
cana-533	174	36	vol.377380)(2014	vol.377380)(2014	NOUN
cana-533	174	37	)	)	PUNCT
cana-533	174	38	.	.	PUNCT
cana-533	175	1	[	[	X
cana-533	175	2	2	2	NUM
cana-533	175	3	]	]	PUNCT
cana-533	175	4	a.	a.	NOUN
cana-533	175	5	h.	h.	PROPN
cana-533	175	6	ansari	ansari	PROPN
cana-533	175	7	and	and	CCONJ
cana-533	175	8	a.	a.	NOUN
cana-533	175	9	benterki	benterki	NOUN
cana-533	175	10	and	and	CCONJ
cana-533	175	11	m.	m.	NOUN
cana-533	175	12	rouaki	rouaki	NOUN
cana-533	175	13	,	,	PUNCT
cana-533	175	14	some	some	DET
cana-533	175	15	local	local	ADJ
cana-533	175	16	fixed	fix	VERB
cana-533	175	17	point	point	NOUN
cana-533	175	18	results	result	NOUN
cana-533	175	19	under	under	ADP
cana-533	175	20	-class	-class	NOUN
cana-533	175	21	functions	function	NOUN
cana-533	175	22	with	with	ADP
cana-533	175	23	applications	application	NOUN
cana-533	175	24	to	to	ADP
cana-533	175	25	coupled	couple	VERB
cana-533	175	26	elliptic	elliptic	ADJ
cana-533	175	27	systems	system	NOUN
cana-533	175	28	,	,	PUNCT
cana-533	175	29	journal	journal	NOUN
cana-533	175	30	of	of	ADP
cana-533	175	31	linear	linear	PROPN
cana-533	175	32	and	and	CCONJ
cana-533	175	33	topological	topological	ADJ
cana-533	175	34	algebra	algebra	NOUN
cana-533	175	35	(	(	PUNCT
cana-533	175	36	jlta	jlta	PROPN
cana-533	175	37	)	)	PUNCT
cana-533	175	38	,	,	PUNCT
cana-533	175	39	7(3	7(3	NUM
cana-533	175	40	)	)	PUNCT
cana-533	175	41	,	,	PUNCT
cana-533	175	42	(	(	PUNCT
cana-533	175	43	2018	2018	NUM
cana-533	175	44	)	)	PUNCT
cana-533	175	45	,	,	PUNCT
cana-533	175	46	169	169	NUM
cana-533	175	47	-	-	SYM
cana-533	175	48	182	182	NUM
cana-533	175	49	.	.	PUNCT
cana-533	176	1	[	[	X
cana-533	176	2	3	3	NUM
cana-533	176	3	]	]	PUNCT
cana-533	176	4	a.	a.	NOUN
cana-533	176	5	benterki	benterki	NOUN
cana-533	176	6	,	,	PUNCT
cana-533	176	7	a	a	DET
cana-533	176	8	local	local	ADJ
cana-533	176	9	fixed	fix	VERB
cana-533	176	10	point	point	NOUN
cana-533	176	11	theorem	theorem	NOUN
cana-533	176	12	for	for	ADP
cana-533	176	13	set	set	NOUN
cana-533	176	14	-	-	PUNCT
cana-533	176	15	valued	value	VERB
cana-533	176	16	mappings	mapping	NOUN
cana-533	176	17	on	on	ADP
cana-533	176	18	partial	partial	ADJ
cana-533	176	19	metric	metric	ADJ
cana-533	176	20	spaces	space	NOUN
cana-533	176	21	,	,	PUNCT
cana-533	176	22	applied	apply	VERB
cana-533	176	23	general	general	ADJ
cana-533	176	24	topology	topology	NOUN
cana-533	176	25	,	,	PUNCT
cana-533	176	26	17(1	17(1	NUM
cana-533	176	27	)	)	PUNCT
cana-533	176	28	,	,	PUNCT
cana-533	176	29	(	(	PUNCT
cana-533	176	30	2016	2016	NUM
cana-533	176	31	)	)	PUNCT
cana-533	176	32	,	,	PUNCT
cana-533	176	33	37	37	NUM
cana-533	176	34	-	-	SYM
cana-533	176	35	49	49	NUM
cana-533	176	36	.	.	PUNCT
cana-533	177	1	[	[	X
cana-533	177	2	4	4	NUM
cana-533	177	3	]	]	X
cana-533	177	4	r.	r.	PROPN
cana-533	177	5	m.	m.	PROPN
cana-533	177	6	bianchini	bianchini	PROPN
cana-533	177	7	and	and	CCONJ
cana-533	177	8	m.	m.	NOUN
cana-533	177	9	grandolfi	grandolfi	NOUN
cana-533	177	10	,	,	PUNCT
cana-533	177	11	transformazioni	transformazioni	PROPN
cana-533	177	12	di	di	X
cana-533	177	13	tipo	tipo	PROPN
cana-533	177	14	contracttivo	contracttivo	PROPN
cana-533	177	15	generalizzato	generalizzato	ADJ
cana-533	177	16	in	in	ADP
cana-533	177	17	uno	uno	PROPN
cana-533	177	18	spazio	spazio	PROPN
cana-533	177	19	metrico	metrico	PROPN
cana-533	177	20	,	,	PUNCT
cana-533	177	21	atti	atti	PROPN
cana-533	177	22	accad	accad	PROPN
cana-533	177	23	.	.	PUNCT
cana-533	178	1	naz	naz	PROPN
cana-533	178	2	.	.	PUNCT
cana-533	179	1	lincei	lincei	NOUN
cana-533	179	2	,	,	PUNCT
cana-533	179	3	rend	rend	VERB
cana-533	179	4	.	.	PUNCT
cana-533	180	1	cl	cl	NOUN
cana-533	180	2	.	.	PUNCT
cana-533	181	1	sci	sci	PROPN
cana-533	181	2	.	.	PROPN
cana-533	181	3	fis	fis	PROPN
cana-533	181	4	.	.	PUNCT
cana-533	181	5	mat	mat	PROPN
cana-533	181	6	.	.	PUNCT
cana-533	181	7	nat	nat	PROPN
cana-533	181	8	,	,	PUNCT
cana-533	181	9	45	45	NUM
cana-533	181	10	,	,	PUNCT
cana-533	181	11	(	(	PUNCT
cana-533	181	12	1968	1968	NUM
cana-533	181	13	)	)	PUNCT
cana-533	181	14	,	,	PUNCT
cana-533	181	15	212	212	NUM
cana-533	181	16	-	-	SYM
cana-533	181	17	216	216	NUM
cana-533	181	18	.	.	PUNCT
cana-533	182	1	[	[	X
cana-533	182	2	5	5	X
cana-533	182	3	]	]	PUNCT
cana-533	182	4	s.	s.	PROPN
cana-533	182	5	g.	g.	PROPN
cana-533	182	6	matthews	matthews	PROPN
cana-533	182	7	,	,	PUNCT
cana-533	182	8	partial	partial	ADJ
cana-533	182	9	metric	metric	ADJ
cana-533	182	10	topology	topology	NOUN
cana-533	182	11	,	,	PUNCT
cana-533	182	12	annals	annal	NOUN
cana-533	182	13	of	of	ADP
cana-533	182	14	the	the	DET
cana-533	182	15	new	new	PROPN
cana-533	182	16	york	york	PROPN
cana-533	182	17	academy	academy	PROPN
cana-533	182	18	of	of	ADP
cana-533	182	19	sciences	sciences	PROPN
cana-533	182	20	-	-	PUNCT
cana-533	182	21	paper	paper	NOUN
cana-533	182	22	edition	edition	NOUN
cana-533	182	23	,	,	PUNCT
cana-533	182	24	728	728	NUM
cana-533	182	25	,	,	PUNCT
cana-533	182	26	(	(	PUNCT
cana-533	182	27	1994	1994	NUM
cana-533	182	28	)	)	PUNCT
cana-533	182	29	,	,	PUNCT
cana-533	182	30	183	183	NUM
cana-533	182	31	-	-	SYM
cana-533	182	32	197	197	NUM
cana-533	182	33	.	.	PUNCT
cana-533	183	1	[	[	X
cana-533	183	2	6	6	NUM
cana-533	183	3	]	]	PUNCT
cana-533	183	4	d.	d.	PROPN
cana-533	183	5	k.	k.	PROPN
cana-533	183	6	nedelcheva	nedelcheva	PROPN
cana-533	183	7	,	,	PUNCT
cana-533	183	8	altering	alter	VERB
cana-533	183	9	points	point	NOUN
cana-533	183	10	in	in	ADP
cana-533	183	11	partial	partial	ADJ
cana-533	183	12	metric	metric	ADJ
cana-533	183	13	space	space	NOUN
cana-533	183	14	,	,	PUNCT
cana-533	183	15	in	in	ADP
cana-533	183	16	:	:	PUNCT
cana-533	183	17	proceedings	proceeding	NOUN
cana-533	183	18	of	of	ADP
cana-533	183	19	the	the	DET
cana-533	183	20	twenty	twenty	NUM
cana-533	183	21	-	-	PUNCT
cana-533	183	22	first	first	ADJ
cana-533	183	23	international	international	ADJ
cana-533	183	24	conference	conference	NOUN
cana-533	183	25	on	on	ADP
cana-533	183	26	geometry	geometry	NOUN
cana-533	183	27	,	,	PUNCT
cana-533	183	28	integrability	integrability	NOUN
cana-533	183	29	and	and	CCONJ
cana-533	183	30	quantization	quantization	NOUN
cana-533	183	31	,	,	PUNCT
cana-533	183	32	(	(	PUNCT
cana-533	183	33	2020	2020	NUM
cana-533	183	34	)	)	PUNCT
cana-533	183	35	,	,	PUNCT
cana-533	183	36	221	221	NUM
cana-533	183	37	-	-	SYM
cana-533	183	38	231	231	NUM
cana-533	183	39	.	.	PUNCT
cana-533	184	1	[	[	X
cana-533	184	2	7	7	X
cana-533	184	3	]	]	PUNCT
cana-533	184	4	a.	a.	NOUN
cana-533	184	5	petrusel	petrusel	NOUN
cana-533	184	6	and	and	CCONJ
cana-533	184	7	g.	g.	PROPN
cana-533	184	8	petrusel	petrusel	PROPN
cana-533	184	9	and	and	CCONJ
cana-533	184	10	j.	j.	PROPN
cana-533	184	11	c.	c.	PROPN
cana-533	184	12	yao	yao	PROPN
cana-533	184	13	,	,	PUNCT
cana-533	184	14	multi	multi	ADJ
cana-533	184	15	-	-	ADJ
cana-533	184	16	valued	value	VERB
cana-533	184	17	graph	graph	NOUN
cana-533	184	18	contraction	contraction	NOUN
cana-533	184	19	principle	principle	NOUN
cana-533	184	20	with	with	ADP
cana-533	184	21	applications	application	NOUN
cana-533	184	22	,	,	PUNCT
cana-533	184	23	optimization	optimization	NOUN
cana-533	184	24	,	,	PUNCT
cana-533	184	25	69	69	NUM
cana-533	184	26	(	(	PUNCT
cana-533	184	27	7	7	NUM
cana-533	184	28	-	-	SYM
cana-533	184	29	8)	8)	NUM
cana-533	184	30	,	,	PUNCT
cana-533	184	31	(	(	PUNCT
cana-533	184	32	2020	2020	NUM
cana-533	184	33	)	)	PUNCT
cana-533	184	34	,	,	PUNCT
cana-533	184	35	1541	1541	NUM
cana-533	184	36	-	-	SYM
cana-533	184	37	1556	1556	NUM
cana-533	184	38	.	.	PUNCT
cana-533	185	1	[	[	X
cana-533	185	2	8	8	NUM
cana-533	185	3	]	]	X
cana-533	185	4	s.	s.	PROPN
cana-533	185	5	romaguera	romaguera	PROPN
cana-533	185	6	and	and	CCONJ
cana-533	185	7	a.	a.	PROPN
cana-533	185	8	kirk	kirk	PROPN
cana-533	185	9	,	,	PUNCT
cana-533	185	10	type	type	NOUN
cana-533	185	11	characterization	characterization	NOUN
cana-533	185	12	of	of	ADP
cana-533	185	13	completeness	completeness	NOUN
cana-533	185	14	for	for	ADP
cana-533	185	15	partial	partial	ADJ
cana-533	185	16	metric	metric	ADJ
cana-533	185	17	spaces	space	NOUN
cana-533	185	18	,	,	PUNCT
cana-533	185	19	fixed	fix	VERB
cana-533	185	20	point	point	NOUN
cana-533	185	21	theory	theory	NOUN
cana-533	185	22	appl	appl	PROPN
cana-533	185	23	.	.	PROPN
cana-533	185	24	,	,	PUNCT
cana-533	185	25	2010	2010	NUM
cana-533	185	26	,	,	PUNCT
cana-533	185	27	(	(	PUNCT
cana-533	185	28	2009	2009	NUM
cana-533	185	29	)	)	PUNCT
cana-533	185	30	,	,	PUNCT
cana-533	185	31	article	article	NOUN
cana-533	185	32	i	i	PROPN
cana-533	185	33	d	d	PROPN
cana-533	185	34	493298	493298	NUM
cana-533	185	35	,	,	PUNCT
cana-533	185	36	6	6	NUM
cana-533	185	37	pages	page	NOUN
cana-533	185	38	.	.	PUNCT
cana-533	186	1	[	[	X
cana-533	186	2	9	9	NUM
cana-533	186	3	]	]	X
cana-533	186	4	d.	d.	PROPN
cana-533	186	5	sahu	sahu	PROPN
cana-533	186	6	,	,	PUNCT
cana-533	186	7	altering	alter	VERB
cana-533	186	8	points	point	NOUN
cana-533	186	9	and	and	CCONJ
cana-533	186	10	applications	application	NOUN
cana-533	186	11	,	,	PUNCT
cana-533	186	12	nonlinear	nonlinear	ADJ
cana-533	186	13	stud	stud	NOUN
cana-533	186	14	.	.	PUNCT
cana-533	186	15	,	,	PUNCT
cana-533	186	16	21	21	NUM
cana-533	186	17	,	,	PUNCT
cana-533	186	18	(	(	PUNCT
cana-533	186	19	2014	2014	NUM
cana-533	186	20	)	)	PUNCT
cana-533	186	21	,	,	PUNCT
cana-533	186	22	349–365	349–365	NUM
cana-533	186	23	.	.	PUNCT
cana-533	187	1	[	[	X
cana-533	187	2	10	10	NUM
cana-533	187	3	]	]	X
cana-533	187	4	j.	j.	PROPN
cana-533	187	5	c.	c.	PROPN
cana-533	187	6	yong	yong	PROPN
cana-533	187	7	and	and	CCONJ
cana-533	187	8	s.	s.	PROPN
cana-533	187	9	kumar	kumar	PROPN
cana-533	187	10	and	and	CCONJ
cana-533	187	11	s.	s.	PROPN
cana-533	187	12	m.	m.	PROPN
cana-533	187	13	k.	k.	PROPN
cana-533	187	14	convergence	convergence	PROPN
cana-533	187	15	analysis	analysis	NOUN
cana-533	187	16	of	of	ADP
cana-533	187	17	parallel	parallel	ADJ
cana-533	187	18	s	s	NOUN
cana-533	187	19	-	-	PUNCT
cana-533	187	20	iteration	iteration	NOUN
cana-533	187	21	process	process	NOUN
cana-533	187	22	for	for	ADP
cana-533	187	23	a	a	DET
cana-533	187	24	system	system	NOUN
cana-533	187	25	of	of	ADP
cana-533	187	26	variational	variational	ADJ
cana-533	187	27	inequalities	inequality	NOUN
cana-533	187	28	using	use	VERB
cana-533	187	29	altering	alter	VERB
cana-533	187	30	points	point	NOUN
cana-533	187	31	,	,	PUNCT
cana-533	187	32	j.	j.	PROPN
cana-533	187	33	appl	appl	PROPN
cana-533	187	34	.	.	PROPN
cana-533	187	35	math	math	PROPN
cana-533	187	36	.	.	PUNCT
cana-533	188	1	&	&	CCONJ
cana-533	188	2	informatics	informatics	PROPN
cana-533	188	3	36(5	36(5	PROPN
cana-533	188	4	-	-	SYM
cana-533	188	5	6	6	NUM
cana-533	188	6	)	)	PUNCT
cana-533	188	7	,	,	PUNCT
cana-533	188	8	(	(	PUNCT
cana-533	188	9	2018	2018	NUM
cana-533	188	10	)	)	PUNCT
cana-533	188	11	,	,	PUNCT
cana-533	188	12	381	381	NUM
cana-533	188	13	-	-	SYM
cana-533	188	14	396	396	NUM
cana-533	188	15	.	.	PUNCT