id	sid	tid	token	lemma	pos
cana-1730	1	1	communications	communication	NOUN
cana-1730	1	2	on	on	ADP
cana-1730	1	3	applied	apply	VERB
cana-1730	1	4	nonlinear	nonlinear	ADJ
cana-1730	1	5	analysis	analysis	NOUN
cana-1730	1	6	issn	issn	NOUN
cana-1730	1	7	:	:	PUNCT
cana-1730	1	8	1074	1074	NUM
cana-1730	1	9	-	-	PUNCT
cana-1730	1	10	133x	133x	NUM
cana-1730	1	11	vol	vol	NOUN
cana-1730	1	12	32	32	NUM
cana-1730	1	13	no	no	NOUN
cana-1730	1	14	.	.	NOUN
cana-1730	1	15	2	2	NUM
cana-1730	1	16	(	(	PUNCT
cana-1730	1	17	2025	2025	NUM
cana-1730	1	18	)	)	PUNCT
cana-1730	1	19	159	159	NUM
cana-1730	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1730	1	21	contractive	contractive	ADJ
cana-1730	1	22	fixed	fix	VERB
cana-1730	1	23	point	point	NOUN
cana-1730	1	24	theorems	theorem	NOUN
cana-1730	1	25	in	in	ADP
cana-1730	1	26	cone	cone	NOUN
cana-1730	1	27	gmetric	gmetric	PROPN
cana-1730	1	28	spaces	space	VERB
cana-1730	1	29	m.	m.	NOUN
cana-1730	1	30	uma1	uma1	PROPN
cana-1730	1	31	,	,	PUNCT
cana-1730	1	32	p.thirunavukarasu2	p.thirunavukarasu2	NOUN
cana-1730	1	33	1	1	NUM
cana-1730	1	34	research	research	NOUN
cana-1730	1	35	scholar	scholar	NOUN
cana-1730	1	36	,	,	PUNCT
cana-1730	1	37	pg	pg	PROPN
cana-1730	1	38	&	&	CCONJ
cana-1730	1	39	research	research	PROPN
cana-1730	1	40	department	department	PROPN
cana-1730	1	41	of	of	ADP
cana-1730	1	42	mathematics	mathematic	NOUN
cana-1730	1	43	,	,	PUNCT
cana-1730	1	44	thanthai	thanthai	VERB
cana-1730	1	45	periyar	periyar	NOUN
cana-1730	1	46	government	government	NOUN
cana-1730	1	47	arts	art	NOUN
cana-1730	1	48	and	and	CCONJ
cana-1730	1	49	science	science	PROPN
cana-1730	1	50	college	college	PROPN
cana-1730	1	51	,	,	PUNCT
cana-1730	1	52	thiruchirapalli	thiruchirapalli	PROPN
cana-1730	1	53	,	,	PUNCT
cana-1730	1	54	tamil	tamil	PROPN
cana-1730	1	55	nadu	nadu	PROPN
cana-1730	1	56	,	,	PUNCT
cana-1730	1	57	india	india	PROPN
cana-1730	1	58	(	(	PUNCT
cana-1730	1	59	affiliated	affiliate	VERB
cana-1730	1	60	to	to	PART
cana-1730	1	61	bharathidasan	bharathidasan	VERB
cana-1730	1	62	university	university	NOUN
cana-1730	1	63	)	)	PUNCT
cana-1730	2	1	e	e	NOUN
cana-1730	2	2	-	-	NOUN
cana-1730	2	3	mail	mail	NOUN
cana-1730	2	4	:	:	PUNCT
cana-1730	3	1	umaleelu@gmail.com	umaleelu@gmail.com	X
cana-1730	3	2	2assistant	2assistant	NUM
cana-1730	3	3	professor	professor	NOUN
cana-1730	3	4	,	,	PUNCT
cana-1730	3	5	pg	pg	PROPN
cana-1730	3	6	&	&	CCONJ
cana-1730	3	7	research	research	PROPN
cana-1730	3	8	department	department	PROPN
cana-1730	3	9	of	of	ADP
cana-1730	3	10	mathematics	mathematic	NOUN
cana-1730	3	11	,	,	PUNCT
cana-1730	3	12	thanthai	thanthai	VERB
cana-1730	3	13	periyar	periyar	NOUN
cana-1730	3	14	government	government	NOUN
cana-1730	3	15	arts	art	NOUN
cana-1730	3	16	and	and	CCONJ
cana-1730	3	17	science	science	PROPN
cana-1730	3	18	college	college	PROPN
cana-1730	3	19	,	,	PUNCT
cana-1730	3	20	thiruchirapalli	thiruchirapalli	PROPN
cana-1730	3	21	,	,	PUNCT
cana-1730	3	22	tamil	tamil	PROPN
cana-1730	3	23	nadu	nadu	PROPN
cana-1730	3	24	,	,	PUNCT
cana-1730	3	25	india	india	PROPN
cana-1730	3	26	(	(	PUNCT
cana-1730	3	27	affiliated	affiliate	VERB
cana-1730	3	28	to	to	PART
cana-1730	3	29	bharathidasan	bharathidasan	VERB
cana-1730	3	30	university	university	NOUN
cana-1730	3	31	)	)	PUNCT
cana-1730	3	32	e	e	NOUN
cana-1730	3	33	-	-	NOUN
cana-1730	3	34	mail	mail	NOUN
cana-1730	3	35	:	:	PUNCT
cana-1730	3	36	ptavinash1967@gmail.com	ptavinash1967@gmail.com	NOUN
cana-1730	3	37	article	article	NOUN
cana-1730	3	38	history	history	NOUN
cana-1730	3	39	:	:	PUNCT
cana-1730	3	40	received	receive	VERB
cana-1730	3	41	:	:	PUNCT
cana-1730	3	42	28	28	NUM
cana-1730	3	43	-	-	SYM
cana-1730	3	44	07	07	NUM
cana-1730	3	45	-	-	PUNCT
cana-1730	3	46	2024	2024	NUM
cana-1730	3	47	revised	revise	VERB
cana-1730	3	48	:	:	PUNCT
cana-1730	3	49	09	09	NUM
cana-1730	3	50	-	-	SYM
cana-1730	3	51	09	09	NUM
cana-1730	3	52	-	-	PUNCT
cana-1730	3	53	2024	2024	NUM
cana-1730	3	54	accepted	accept	VERB
cana-1730	3	55	:	:	PUNCT
cana-1730	3	56	17	17	NUM
cana-1730	3	57	-	-	SYM
cana-1730	3	58	09	09	NUM
cana-1730	3	59	-	-	PUNCT
cana-1730	3	60	2024	2024	NUM
cana-1730	3	61	abstract	abstract	NOUN
cana-1730	3	62	:	:	PUNCT
cana-1730	3	63	in	in	ADP
cana-1730	3	64	partially	partially	ADV
cana-1730	3	65	ordered	order	VERB
cana-1730	3	66	cone	cone	NOUN
cana-1730	3	67	gmetre	gmetre	NOUN
cana-1730	3	68	spaces	space	NOUN
cana-1730	3	69	,	,	PUNCT
cana-1730	3	70	we	we	PRON
cana-1730	3	71	provide	provide	VERB
cana-1730	3	72	certain	certain	ADJ
cana-1730	3	73	fixed	fix	VERB
cana-1730	3	74	point	point	NOUN
cana-1730	3	75	and	and	CCONJ
cana-1730	3	76	coincidence	coincidence	NOUN
cana-1730	3	77	theorems	theorem	NOUN
cana-1730	3	78	for	for	ADP
cana-1730	3	79	mappings	mapping	NOUN
cana-1730	3	80	that	that	PRON
cana-1730	3	81	meet	meet	VERB
cana-1730	3	82	contractive	contractive	ADJ
cana-1730	3	83	criteria	criterion	NOUN
cana-1730	3	84	under	under	ADP
cana-1730	3	85	θ	θ	PROPN
cana-1730	3	86	-maps	-map	NOUN
cana-1730	3	87	.	.	PUNCT
cana-1730	4	1	keywords	keyword	NOUN
cana-1730	4	2	:	:	PUNCT
cana-1730	4	3	cone	cone	PROPN
cana-1730	4	4	gmetric	gmetric	ADJ
cana-1730	4	5	space	space	NOUN
cana-1730	4	6	,	,	PUNCT
cana-1730	4	7	complete	complete	ADJ
cana-1730	4	8	cone	cone	NOUN
cana-1730	4	9	gmetric	gmetric	ADJ
cana-1730	4	10	space	space	NOUN
cana-1730	4	11	,	,	PUNCT
cana-1730	4	12	partial	partial	ADJ
cana-1730	4	13	order	order	NOUN
cana-1730	4	14	comparable	comparable	ADJ
cana-1730	4	15	elements	element	NOUN
cana-1730	4	16	1	1	NUM
cana-1730	4	17	.	.	PUNCT
cana-1730	4	18	introduction	introduction	NOUN
cana-1730	4	19	a	a	DET
cana-1730	4	20	novel	novel	ADJ
cana-1730	4	21	concept	concept	NOUN
cana-1730	4	22	known	know	VERB
cana-1730	4	23	as	as	ADP
cana-1730	4	24	g	g	NOUN
cana-1730	4	25	-	-	PUNCT
cana-1730	4	26	metric	metric	ADJ
cana-1730	4	27	space	space	NOUN
cana-1730	4	28	was	be	AUX
cana-1730	4	29	introduced	introduce	VERB
cana-1730	4	30	in	in	ADP
cana-1730	4	31	2006	2006	NUM
cana-1730	4	32	by	by	ADP
cana-1730	4	33	z.	z.	PROPN
cana-1730	4	34	mustafa	mustafa	PROPN
cana-1730	4	35	and	and	CCONJ
cana-1730	4	36	b.	b.	PROPN
cana-1730	4	37	sims	sim	NOUN
cana-1730	4	38	as	as	ADP
cana-1730	4	39	a	a	DET
cana-1730	4	40	generalised	generalise	VERB
cana-1730	4	41	metric	metric	ADJ
cana-1730	4	42	space	space	NOUN
cana-1730	5	1	[	[	X
cana-1730	5	2	1	1	NUM
cana-1730	5	3	]	]	PUNCT
cana-1730	5	4	.	.	PUNCT
cana-1730	5	5	initiated	initiate	VERB
cana-1730	5	6	in	in	ADP
cana-1730	5	7	[	[	X
cana-1730	5	8	2	2	NUM
cana-1730	5	9	]	]	PUNCT
cana-1730	5	10	,	,	PUNCT
cana-1730	5	11	fixed	fix	VERB
cana-1730	5	12	point	point	NOUN
cana-1730	5	13	theory	theory	NOUN
cana-1730	5	14	in	in	ADP
cana-1730	5	15	such	such	ADJ
cana-1730	5	16	spaces	space	NOUN
cana-1730	5	17	was	be	AUX
cana-1730	5	18	investigated	investigate	VERB
cana-1730	5	19	in	in	ADP
cana-1730	5	20	[	[	X
cana-1730	5	21	3	3	NUM
cana-1730	5	22	]	]	PUNCT
cana-1730	5	23	.	.	PUNCT
cana-1730	6	1	specifically	specifically	ADV
cana-1730	6	2	,	,	PUNCT
cana-1730	6	3	these	these	DET
cana-1730	6	4	studies	study	NOUN
cana-1730	6	5	developed	develop	VERB
cana-1730	6	6	the	the	DET
cana-1730	6	7	principle	principle	NOUN
cana-1730	6	8	of	of	ADP
cana-1730	6	9	banach	banach	NOUN
cana-1730	6	10	contraction	contraction	NOUN
cana-1730	6	11	mapping	mapping	NOUN
cana-1730	6	12	.	.	PUNCT
cana-1730	7	1	cone	cone	PROPN
cana-1730	7	2	metric	metric	ADJ
cana-1730	7	3	spaces	space	NOUN
cana-1730	7	4	are	be	AUX
cana-1730	7	5	not	not	PART
cana-1730	7	6	a	a	DET
cana-1730	7	7	particularly	particularly	ADV
cana-1730	7	8	new	new	ADJ
cana-1730	7	9	idea	idea	NOUN
cana-1730	7	10	.	.	PUNCT
cana-1730	8	1	kurepa	kurepa	PROPN
cana-1730	8	2	proposed	propose	VERB
cana-1730	8	3	the	the	DET
cana-1730	8	4	concept	concept	NOUN
cana-1730	8	5	of	of	ADP
cana-1730	8	6	metric	metric	ADJ
cana-1730	8	7	spaces	space	NOUN
cana-1730	8	8	in	in	ADP
cana-1730	8	9	1934	1934	NUM
cana-1730	8	10	[	[	X
cana-1730	8	11	5	5	NUM
cana-1730	8	12	]	]	PUNCT
cana-1730	8	13	,	,	PUNCT
cana-1730	8	14	where	where	SCONJ
cana-1730	8	15	the	the	DET
cana-1730	8	16	metric	metric	NOUN
cana-1730	8	17	takes	take	VERB
cana-1730	8	18	values	value	NOUN
cana-1730	8	19	in	in	ADP
cana-1730	8	20	an	an	DET
cana-1730	8	21	ordered	order	VERB
cana-1730	8	22	space	space	NOUN
cana-1730	8	23	.	.	PUNCT
cana-1730	9	1	one	one	PRON
cana-1730	9	2	can	can	AUX
cana-1730	9	3	find	find	VERB
cana-1730	9	4	examples	example	NOUN
cana-1730	9	5	of	of	ADP
cana-1730	9	6	huangzhang	huangzhang	PROPN
cana-1730	9	7	's	's	PART
cana-1730	9	8	definition	definition	NOUN
cana-1730	9	9	[	[	X
cana-1730	9	10	6	6	NUM
cana-1730	9	11	]	]	PUNCT
cana-1730	9	12	of	of	ADP
cana-1730	9	13	a	a	DET
cana-1730	9	14	cone	cone	NOUN
cana-1730	9	15	metric	metric	ADJ
cana-1730	9	16	space	space	NOUN
cana-1730	9	17	in	in	ADP
cana-1730	9	18	chung	chung	PROPN
cana-1730	9	19	's	's	PART
cana-1730	9	20	works	work	NOUN
cana-1730	9	21	[	[	X
cana-1730	9	22	7	7	NUM
cana-1730	9	23	]	]	PUNCT
cana-1730	9	24	.	.	PUNCT
cana-1730	10	1	these	these	DET
cana-1730	10	2	spaces	space	NOUN
cana-1730	10	3	were	be	AUX
cana-1730	10	4	dubbed	dub	VERB
cana-1730	10	5	"	"	PUNCT
cana-1730	10	6	cone	cone	NOUN
cana-1730	10	7	-	-	PUNCT
cana-1730	10	8	valued	value	VERB
cana-1730	10	9	metric	metric	ADJ
cana-1730	10	10	spaces	space	NOUN
cana-1730	10	11	"	"	PUNCT
cana-1730	10	12	by	by	ADP
cana-1730	10	13	chung	chung	PROPN
cana-1730	10	14	in	in	ADP
cana-1730	10	15	such	such	ADJ
cana-1730	10	16	spaces	space	NOUN
cana-1730	10	17	,	,	PUNCT
cana-1730	10	18	additional	additional	ADJ
cana-1730	10	19	fixed	fix	VERB
cana-1730	10	20	point	point	NOUN
cana-1730	10	21	solutions	solution	NOUN
cana-1730	10	22	were	be	AUX
cana-1730	10	23	achieved	achieve	VERB
cana-1730	10	24	by	by	ADP
cana-1730	10	25	a	a	DET
cana-1730	10	26	number	number	NOUN
cana-1730	10	27	of	of	ADP
cana-1730	10	28	writers	writer	NOUN
cana-1730	10	29	[	[	X
cana-1730	10	30	9	9	NUM
cana-1730	10	31	,	,	PUNCT
cana-1730	10	32	10	10	NUM
cana-1730	10	33	]	]	PUNCT
cana-1730	10	34	.	.	PUNCT
cana-1730	11	1	cone	cone	NOUN
cana-1730	11	2	g	g	NOUN
cana-1730	11	3	-	-	PUNCT
cana-1730	11	4	metric	metric	ADJ
cana-1730	11	5	spaces	space	NOUN
cana-1730	11	6	,	,	PUNCT
cana-1730	11	7	a	a	DET
cana-1730	11	8	generalisation	generalisation	NOUN
cana-1730	11	9	of	of	ADP
cana-1730	11	10	g	g	NOUN
cana-1730	11	11	-	-	PUNCT
cana-1730	11	12	metric	metric	ADJ
cana-1730	11	13	spaces	space	NOUN
cana-1730	11	14	and	and	CCONJ
cana-1730	11	15	cone	cone	NOUN
cana-1730	11	16	metric	metric	ADJ
cana-1730	11	17	spaces	space	NOUN
cana-1730	11	18	,	,	PUNCT
cana-1730	11	19	were	be	AUX
cana-1730	11	20	recently	recently	ADV
cana-1730	11	21	introduced	introduce	VERB
cana-1730	11	22	by	by	ADP
cana-1730	11	23	beg	beg	NOUN
cana-1730	11	24	et	et	PROPN
cana-1730	11	25	al	al	PROPN
cana-1730	11	26	.	.	PUNCT
cana-1730	12	1	[	[	X
cana-1730	12	2	11	11	NUM
cana-1730	12	3	]	]	PUNCT
cana-1730	12	4	.	.	PUNCT
cana-1730	13	1	they	they	PRON
cana-1730	13	2	demonstrated	demonstrate	VERB
cana-1730	13	3	a	a	DET
cana-1730	13	4	few	few	ADJ
cana-1730	13	5	fixed	fix	VERB
cana-1730	13	6	point	point	NOUN
cana-1730	13	7	theorems	theorem	NOUN
cana-1730	13	8	in	in	ADP
cana-1730	13	9	terms	term	NOUN
cana-1730	13	10	of	of	ADP
cana-1730	13	11	specific	specific	ADJ
cana-1730	13	12	contractive	contractive	ADJ
cana-1730	13	13	requirements	requirement	NOUN
cana-1730	13	14	.	.	PUNCT
cana-1730	14	1	fixed	fix	VERB
cana-1730	14	2	points	point	NOUN
cana-1730	14	3	for	for	SCONJ
cana-1730	14	4	ϕ-maps	ϕ-map	NOUN
cana-1730	14	5	in	in	ADP
cana-1730	14	6	g	g	NOUN
cana-1730	14	7	-	-	PUNCT
cana-1730	14	8	metric	metric	ADJ
cana-1730	14	9	spaces	space	NOUN
cana-1730	14	10	were	be	AUX
cana-1730	14	11	studied	study	VERB
cana-1730	14	12	by	by	ADP
cana-1730	14	13	shatanawi	shatanawi	ADJ
cana-1730	14	14	[	[	X
cana-1730	14	15	4	4	NUM
cana-1730	14	16	]	]	PUNCT
cana-1730	14	17	,	,	PUNCT
cana-1730	14	18	and	and	CCONJ
cana-1730	14	19	these	these	DET
cana-1730	14	20	fixed	fix	VERB
cana-1730	14	21	points	point	NOUN
cana-1730	14	22	are	be	AUX
cana-1730	14	23	extended	extend	VERB
cana-1730	14	24	to	to	ADP
cana-1730	14	25	cone	cone	VERB
cana-1730	14	26	g	g	NOUN
cana-1730	14	27	-	-	PUNCT
cana-1730	14	28	metric	metric	ADJ
cana-1730	14	29	spaces	space	NOUN
cana-1730	14	30	for	for	ADP
cana-1730	14	31	two	two	NUM
cana-1730	14	32	maps	map	NOUN
cana-1730	14	33	by	by	ADP
cana-1730	14	34	ozturk	ozturk	NOUN
cana-1730	14	35	and	and	CCONJ
cana-1730	14	36	basarir	basarir	NOUN
cana-1730	15	1	[	[	X
cana-1730	15	2	12	12	NUM
cana-1730	15	3	]	]	PUNCT
cana-1730	15	4	.	.	PUNCT
cana-1730	16	1	additionally	additionally	ADV
cana-1730	16	2	,	,	PUNCT
cana-1730	16	3	partially	partially	ADV
cana-1730	16	4	ordered	order	VERB
cana-1730	16	5	g	g	NOUN
cana-1730	16	6	-	-	PUNCT
cana-1730	16	7	metric	metric	ADJ
cana-1730	16	8	spaces	space	NOUN
cana-1730	16	9	[	[	X
cana-1730	16	10	15	15	NUM
cana-1730	16	11	]	]	PUNCT
cana-1730	16	12	and	and	CCONJ
cana-1730	16	13	partially	partially	ADV
cana-1730	16	14	ordered	order	VERB
cana-1730	16	15	cone	cone	NOUN
cana-1730	16	16	metric	metric	ADJ
cana-1730	16	17	spaces	space	NOUN
cana-1730	16	18	[	[	X
cana-1730	16	19	14	14	NUM
cana-1730	16	20	]	]	PUNCT
cana-1730	16	21	have	have	AUX
cana-1730	16	22	been	be	AUX
cana-1730	16	23	studied	study	VERB
cana-1730	16	24	with	with	ADP
cana-1730	16	25	fixed	fix	VERB
cana-1730	16	26	point	point	NOUN
cana-1730	16	27	issues	issue	NOUN
cana-1730	16	28	.	.	PUNCT
cana-1730	17	1	in	in	ADP
cana-1730	17	2	this	this	DET
cana-1730	17	3	research	research	NOUN
cana-1730	17	4	,	,	PUNCT
cana-1730	17	5	we	we	PRON
cana-1730	17	6	investigate	investigate	VERB
cana-1730	17	7	common	common	ADJ
cana-1730	17	8	fixed	fix	VERB
cana-1730	17	9	point	point	NOUN
cana-1730	17	10	theorems	theorem	NOUN
cana-1730	17	11	in	in	ADP
cana-1730	17	12	partially	partially	ADV
cana-1730	17	13	ordered	order	VERB
cana-1730	17	14	cone	cone	NOUN
cana-1730	17	15	g	g	NOUN
cana-1730	17	16	-	-	PUNCT
cana-1730	17	17	metric	metric	ADJ
cana-1730	17	18	spaces	space	NOUN
cana-1730	17	19	for	for	ADP
cana-1730	17	20	mappings	mapping	NOUN
cana-1730	17	21	that	that	PRON
cana-1730	17	22	meet	meet	VERB
cana-1730	17	23	contractive	contractive	ADJ
cana-1730	17	24	criteria	criterion	NOUN
cana-1730	17	25	associated	associate	VERB
cana-1730	17	26	with	with	ADP
cana-1730	17	27	a	a	DET
cana-1730	17	28	nondecreasing	nondecreasing	ADJ
cana-1730	17	29	θ	θ	NOUN
cana-1730	17	30	-	-	NOUN
cana-1730	17	31	map	map	NOUN
cana-1730	17	32	[	[	X
cana-1730	17	33	8,9	8,9	NUM
cana-1730	17	34	]	]	PUNCT
cana-1730	17	35	.	.	PUNCT
cana-1730	18	1	our	our	PRON
cana-1730	18	2	findings	finding	NOUN
cana-1730	18	3	are	be	AUX
cana-1730	18	4	an	an	DET
cana-1730	18	5	ordered	order	VERB
cana-1730	18	6	cone	cone	NOUN
cana-1730	18	7	g	g	NOUN
cana-1730	18	8	-	-	PUNCT
cana-1730	18	9	version	version	NOUN
cana-1730	18	10	extension	extension	NOUN
cana-1730	18	11	of	of	ADP
cana-1730	18	12	research	research	NOUN
cana-1730	18	13	by	by	ADP
cana-1730	18	14	ozturk	ozturk	NOUN
cana-1730	18	15	and	and	CCONJ
cana-1730	18	16	basarir	basarir	NOUN
cana-1730	18	17	[	[	X
cana-1730	18	18	12	12	NUM
cana-1730	18	19	]	]	PUNCT
cana-1730	18	20	and	and	CCONJ
cana-1730	18	21	shatanawi	shatanawi	VERB
cana-1730	18	22	[	[	X
cana-1730	18	23	4	4	NUM
cana-1730	18	24	]	]	PUNCT
cana-1730	18	25	.	.	PUNCT
cana-1730	19	1	preliminaries	preliminary	NOUN
cana-1730	19	2	let	let	VERB
cana-1730	19	3	p	p	PRON
cana-1730	19	4	be	be	AUX
cana-1730	19	5	a	a	DET
cana-1730	19	6	real	real	ADJ
cana-1730	19	7	banach	banach	NOUN
cana-1730	19	8	space	space	NOUN
cana-1730	19	9	and	and	CCONJ
cana-1730	19	10	a	a	DET
cana-1730	19	11	be	be	AUX
cana-1730	19	12	a	a	DET
cana-1730	19	13	subset	subset	NOUN
cana-1730	19	14	of	of	ADP
cana-1730	19	15	p.	p.	NOUN
cana-1730	19	16	by	by	ADP
cana-1730	19	17	we	we	PRON
cana-1730	19	18	denote	denote	VERB
cana-1730	19	19	the	the	DET
cana-1730	19	20	zero	zero	NUM
cana-1730	19	21	element	element	NOUN
cana-1730	19	22	of	of	ADP
cana-1730	19	23	p	p	NOUN
cana-1730	19	24	and	and	CCONJ
cana-1730	19	25	by	by	ADV
cana-1730	19	26	in	in	ADP
cana-1730	19	27	a	a	DET
cana-1730	19	28	the	the	DET
cana-1730	19	29	interior	interior	NOUN
cana-1730	19	30	of	of	ADP
cana-1730	19	31	a	a	PRON
cana-1730	19	32	.	.	PUNCT
cana-1730	20	1	the	the	DET
cana-1730	20	2	subset	subset	NOUN
cana-1730	20	3	a	a	PRON
cana-1730	20	4	is	be	AUX
cana-1730	20	5	called	call	VERB
cana-1730	20	6	an	an	DET
cana-1730	20	7	order	order	NOUN
cana-1730	20	8	cone	cone	NOUN
cana-1730	20	9	if	if	SCONJ
cana-1730	20	10	:	:	PUNCT
cana-1730	21	1	1	1	X
cana-1730	21	2	.	.	X
cana-1730	21	3	a	a	PRON
cana-1730	21	4	is	be	AUX
cana-1730	21	5	closed	closed	ADJ
cana-1730	21	6	,	,	PUNCT
cana-1730	21	7	nonempty	nonempty	ADJ
cana-1730	21	8	and	and	CCONJ
cana-1730	21	9	a	a	DET
cana-1730	21	10	{	{	PUNCT
cana-1730	21	11	}	}	PUNCT
cana-1730	21	12	;	;	PUNCT
cana-1730	21	13	2	2	X
cana-1730	21	14	.	.	NUM
cana-1730	21	15	x	x	X
cana-1730	21	16	,	,	PUNCT
cana-1730	21	17	y	y	PROPN
cana-1730	21	18	∈	∈	PROPN
cana-1730	21	19	q	q	X
cana-1730	21	20	,	,	PUNCT
cana-1730	21	21	x	x	INTJ
cana-1730	21	22	,	,	PUNCT
cana-1730	21	23	y	y	PROPN
cana-1730	21	24	≥	≥	NUM
cana-1730	21	25	0	0	NUM
cana-1730	21	26	,	,	PUNCT
cana-1730	21	27	a	a	DET
cana-1730	21	28	,	,	PUNCT
cana-1730	21	29	b	b	X
cana-1730	21	30	∈	∈	PROPN
cana-1730	21	31	a	a	DET
cana-1730	21	32	⇒	⇒	NOUN
cana-1730	21	33	xa	xa	PROPN
cana-1730	22	1	+	+	CCONJ
cana-1730	22	2	yb	yb	PROPN
cana-1730	22	3	∈	∈	PROPN
cana-1730	22	4	a	a	PRON
cana-1730	22	5	;	;	PUNCT
cana-1730	22	6	3	3	X
cana-1730	22	7	.	.	X
cana-1730	22	8	a	a	DET
cana-1730	22	9	∈	∈	PROPN
cana-1730	22	10	a	a	DET
cana-1730	22	11	and	and	CCONJ
cana-1730	22	12	−a	−a	NOUN
cana-1730	22	13	∈	∈	PROPN
cana-1730	22	14	a	a	DET
cana-1730	22	15	⇒	⇒	NOUN
cana-1730	23	1	a	a	PRON
cana-1730	23	2	=	=	X
cana-1730	23	3	.	.	PUNCT
cana-1730	24	1	mailto:umaleelu@gmail.com	mailto:umaleelu@gmail.com	PROPN
cana-1730	24	2	communications	communication	NOUN
cana-1730	24	3	on	on	ADP
cana-1730	24	4	applied	apply	VERB
cana-1730	24	5	nonlinear	nonlinear	ADJ
cana-1730	24	6	analysis	analysis	NOUN
cana-1730	24	7	issn	issn	NOUN
cana-1730	24	8	:	:	PUNCT
cana-1730	24	9	1074	1074	NUM
cana-1730	24	10	-	-	PUNCT
cana-1730	24	11	133x	133x	NUM
cana-1730	24	12	vol	vol	NOUN
cana-1730	24	13	32	32	NUM
cana-1730	24	14	no	no	NOUN
cana-1730	24	15	.	.	NOUN
cana-1730	24	16	2	2	NUM
cana-1730	24	17	(	(	PUNCT
cana-1730	24	18	2025	2025	NUM
cana-1730	24	19	)	)	PUNCT
cana-1730	24	20	160	160	NUM
cana-1730	24	21	https://internationalpubls.com	https://internationalpubls.com	X
cana-1730	24	22	definition	definition	NOUN
cana-1730	24	23	1	1	NUM
cana-1730	24	24	let	let	VERB
cana-1730	24	25	s	s	PRON
cana-1730	24	26	be	be	AUX
cana-1730	24	27	a	a	DET
cana-1730	24	28	nonempty	nonempty	ADJ
cana-1730	24	29	set	set	VERB
cana-1730	24	30	,	,	PUNCT
cana-1730	24	31	p	p	PROPN
cana-1730	24	32	be	be	VERB
cana-1730	24	33	a	a	DET
cana-1730	24	34	real	real	ADJ
cana-1730	24	35	banach	banach	NOUN
cana-1730	24	36	space	space	NOUN
cana-1730	24	37	and	and	CCONJ
cana-1730	24	38	a	a	DET
cana-1730	24	39	⊂	⊂	PROPN
cana-1730	24	40	p	p	X
cana-1730	24	41	be	be	AUX
cana-1730	24	42	an	an	DET
cana-1730	24	43	order	order	NOUN
cana-1730	24	44	cone	cone	NOUN
cana-1730	24	45	.	.	PUNCT
cana-1730	25	1	suppose	suppose	VERB
cana-1730	25	2	a	a	DET
cana-1730	25	3	mapping	mapping	NOUN
cana-1730	25	4	t	t	NOUN
cana-1730	25	5	:	:	PUNCT
cana-1730	25	6	s	s	VERB
cana-1730	25	7	×	×	PROPN
cana-1730	25	8	s	s	PART
cana-1730	25	9	×	×	NOUN
cana-1730	25	10	s	s	X
cana-1730	25	11	→	→	SYM
cana-1730	25	12	p	p	ADJ
cana-1730	25	13	satisfies	satisfie	NOUN
cana-1730	25	14	(	(	PUNCT
cana-1730	25	15	t1	t1	NOUN
cana-1730	25	16	)	)	PUNCT
cana-1730	25	17	t(a	t(a	NOUN
cana-1730	25	18	,	,	PUNCT
cana-1730	25	19	b	b	NOUN
cana-1730	25	20	,	,	PUNCT
cana-1730	25	21	c	c	NOUN
cana-1730	25	22	)	)	PUNCT
cana-1730	26	1	=	=	NOUN
cana-1730	26	2	if	if	SCONJ
cana-1730	26	3	a	a	DET
cana-1730	26	4	=	=	SYM
cana-1730	26	5	b	b	NOUN
cana-1730	26	6	=	=	SYM
cana-1730	26	7	c	c	NOUN
cana-1730	26	8	;	;	PUNCT
cana-1730	26	9	(	(	PUNCT
cana-1730	26	10	t2	t2	NOUN
cana-1730	26	11	)	)	PUNCT
cana-1730	26	12	<	<	X
cana-1730	26	13	t(a	t(a	NOUN
cana-1730	26	14	,	,	PUNCT
cana-1730	26	15	a	a	DET
cana-1730	26	16	,	,	PUNCT
cana-1730	26	17	b	b	NOUN
cana-1730	26	18	)	)	PUNCT
cana-1730	26	19	for	for	ADP
cana-1730	26	20	all	all	DET
cana-1730	26	21	a	a	DET
cana-1730	26	22	,	,	PUNCT
cana-1730	26	23	b	b	X
cana-1730	26	24	∈	∈	NOUN
cana-1730	26	25	s	s	VERB
cana-1730	26	26	with	with	ADP
cana-1730	26	27	a	a	DET
cana-1730	26	28	b	b	NOUN
cana-1730	26	29	;	;	PUNCT
cana-1730	26	30	(	(	PUNCT
cana-1730	26	31	t3	t3	NOUN
cana-1730	26	32	)	)	PUNCT
cana-1730	26	33	t(a	t(a	NOUN
cana-1730	26	34	,	,	PUNCT
cana-1730	26	35	a	a	PRON
cana-1730	26	36	,	,	PUNCT
cana-1730	26	37	b	b	NOUN
cana-1730	26	38	)	)	PUNCT
cana-1730	26	39	≤	≤	NOUN
cana-1730	26	40	t(a	t(a	NOUN
cana-1730	26	41	,	,	PUNCT
cana-1730	26	42	b	b	NOUN
cana-1730	26	43	,	,	PUNCT
cana-1730	26	44	c	c	NOUN
cana-1730	26	45	)	)	PUNCT
cana-1730	26	46	for	for	ADP
cana-1730	26	47	all	all	DET
cana-1730	26	48	a	a	DET
cana-1730	26	49	,	,	PUNCT
cana-1730	26	50	b	b	NOUN
cana-1730	26	51	,	,	PUNCT
cana-1730	26	52	c	c	PROPN
cana-1730	26	53	∈	∈	PROPN
cana-1730	26	54	s	s	VERB
cana-1730	26	55	with	with	ADP
cana-1730	26	56	c	c	PROPN
cana-1730	26	57	b	b	PROPN
cana-1730	26	58	;	;	PUNCT
cana-1730	26	59	(	(	PUNCT
cana-1730	26	60	t4	t4	PROPN
cana-1730	26	61	)	)	PUNCT
cana-1730	26	62	t(a	t(a	PROPN
cana-1730	26	63	,	,	PUNCT
cana-1730	26	64	b	b	NOUN
cana-1730	26	65	,	,	PUNCT
cana-1730	26	66	c	c	NOUN
cana-1730	26	67	)	)	PUNCT
cana-1730	26	68	=	=	SYM
cana-1730	27	1	t(a	t(a	NOUN
cana-1730	27	2	,	,	PUNCT
cana-1730	27	3	c	c	NOUN
cana-1730	27	4	,	,	PUNCT
cana-1730	27	5	b	b	NOUN
cana-1730	27	6	)	)	PUNCT
cana-1730	27	7	=	=	SYM
cana-1730	27	8	t(b.c.a	t(b.c.a	NOUN
cana-1730	27	9	)	)	PUNCT
cana-1730	27	10	=	=	SYM
cana-1730	27	11	·	·	PUNCT
cana-1730	27	12	·	·	PUNCT
cana-1730	27	13	·	·	PUNCT
cana-1730	27	14	(	(	PUNCT
cana-1730	27	15	symmetry	symmetry	NOUN
cana-1730	27	16	in	in	ADP
cana-1730	27	17	all	all	DET
cana-1730	27	18	three	three	NUM
cana-1730	27	19	variables	variable	NOUN
cana-1730	27	20	)	)	PUNCT
cana-1730	27	21	;	;	PUNCT
cana-1730	27	22	(	(	PUNCT
cana-1730	27	23	t5	t5	PROPN
cana-1730	27	24	)	)	PUNCT
cana-1730	27	25	t(a	t(a	PROPN
cana-1730	27	26	,	,	PUNCT
cana-1730	27	27	b	b	NOUN
cana-1730	27	28	,	,	PUNCT
cana-1730	27	29	c	c	NOUN
cana-1730	27	30	)	)	PUNCT
cana-1730	27	31	≤	≤	NOUN
cana-1730	27	32	t(a	t(a	NOUN
cana-1730	27	33	,	,	PUNCT
cana-1730	27	34	x	x	X
cana-1730	27	35	,	,	PUNCT
cana-1730	27	36	x	x	X
cana-1730	27	37	)	)	PUNCT
cana-1730	27	38	+	+	CCONJ
cana-1730	28	1	t(x	t(x	PROPN
cana-1730	28	2	,	,	PUNCT
cana-1730	28	3	b	b	NOUN
cana-1730	28	4	,	,	PUNCT
cana-1730	28	5	c	c	NOUN
cana-1730	28	6	)	)	PUNCT
cana-1730	28	7	for	for	ADP
cana-1730	28	8	all	all	DET
cana-1730	28	9	a	a	DET
cana-1730	28	10	,	,	PUNCT
cana-1730	28	11	b	b	NOUN
cana-1730	28	12	,	,	PUNCT
cana-1730	28	13	c	c	NOUN
cana-1730	28	14	,	,	PUNCT
cana-1730	28	15	x	x	SYM
cana-1730	28	16	∈	∈	NOUN
cana-1730	28	17	s	s	PART
cana-1730	28	18	(	(	PUNCT
cana-1730	28	19	rectangle	rectangle	NOUN
cana-1730	28	20	inequality	inequality	NOUN
cana-1730	28	21	)	)	PUNCT
cana-1730	28	22	.	.	PUNCT
cana-1730	29	1	then	then	ADV
cana-1730	29	2	the	the	DET
cana-1730	29	3	function	function	NOUN
cana-1730	29	4	t	t	PROPN
cana-1730	29	5	is	be	AUX
cana-1730	29	6	called	call	VERB
cana-1730	29	7	a	a	DET
cana-1730	29	8	generalized	generalize	VERB
cana-1730	29	9	cone	cone	NOUN
cana-1730	29	10	metric	metric	NOUN
cana-1730	29	11	on	on	ADP
cana-1730	29	12	s	s	PRON
cana-1730	29	13	and	and	CCONJ
cana-1730	29	14	s	s	VERB
cana-1730	29	15	is	be	AUX
cana-1730	29	16	called	call	VERB
cana-1730	29	17	a	a	DET
cana-1730	29	18	generalized	generalized	ADJ
cana-1730	29	19	cone	cone	NOUN
cana-1730	29	20	metric	metric	ADJ
cana-1730	29	21	space	space	NOUN
cana-1730	29	22	or	or	CCONJ
cana-1730	29	23	,	,	PUNCT
cana-1730	29	24	shortly	shortly	ADV
cana-1730	29	25	,	,	PUNCT
cana-1730	29	26	a	a	DET
cana-1730	29	27	cone	cone	NOUN
cana-1730	29	28	gmetric	gmetric	ADJ
cana-1730	29	29	space	space	NOUN
cana-1730	29	30	.	.	PUNCT
cana-1730	30	1	it	it	PRON
cana-1730	30	2	is	be	AUX
cana-1730	30	3	obvious	obvious	ADJ
cana-1730	30	4	that	that	SCONJ
cana-1730	30	5	the	the	DET
cana-1730	30	6	concept	concept	NOUN
cana-1730	30	7	of	of	ADP
cana-1730	30	8	a	a	DET
cana-1730	30	9	cone	cone	NOUN
cana-1730	30	10	gmetric	gmetric	ADJ
cana-1730	30	11	space	space	NOUN
cana-1730	30	12	is	be	AUX
cana-1730	30	13	more	more	ADV
cana-1730	30	14	general	general	ADJ
cana-1730	30	15	than	than	ADP
cana-1730	30	16	that	that	PRON
cana-1730	30	17	of	of	ADP
cana-1730	30	18	a	a	DET
cana-1730	30	19	g	g	NOUN
cana-1730	30	20	-	-	PUNCT
cana-1730	30	21	metric	metric	ADJ
cana-1730	30	22	space	space	NOUN
cana-1730	30	23	or	or	CCONJ
cana-1730	30	24	a	a	DET
cana-1730	30	25	cone	cone	NOUN
cana-1730	30	26	metric	metric	ADJ
cana-1730	30	27	space	space	NOUN
cana-1730	30	28	.	.	PUNCT
cana-1730	31	1	if	if	SCONJ
cana-1730	31	2	p	p	NOUN
cana-1730	31	3	=	=	X
cana-1730	31	4	q	q	X
cana-1730	31	5	and	and	CCONJ
cana-1730	31	6	a	a	PRON
cana-1730	31	7	=	=	X
cana-1730	32	1	[	[	X
cana-1730	32	2	0	0	NUM
cana-1730	32	3	,	,	PUNCT
cana-1730	32	4	+	+	NOUN
cana-1730	32	5	∞	∞	NOUN
cana-1730	32	6	)	)	PUNCT
cana-1730	32	7	then	then	ADV
cana-1730	32	8	a	a	DET
cana-1730	32	9	cone	cone	NOUN
cana-1730	32	10	gmetric	gmetric	ADJ
cana-1730	32	11	space	space	NOUN
cana-1730	32	12	becomes	become	VERB
cana-1730	32	13	a	a	DET
cana-1730	32	14	gmetric	gmetric	ADJ
cana-1730	32	15	space	space	NOUN
cana-1730	32	16	.	.	PUNCT
cana-1730	33	1	definition	definition	NOUN
cana-1730	33	2	2	2	NUM
cana-1730	33	3	let	let	VERB
cana-1730	33	4	(	(	PUNCT
cana-1730	33	5	s	s	NOUN
cana-1730	33	6	,	,	PUNCT
cana-1730	33	7	t	t	PROPN
cana-1730	33	8	)	)	PUNCT
cana-1730	33	9	be	be	AUX
cana-1730	33	10	a	a	DET
cana-1730	33	11	cone	cone	NOUN
cana-1730	33	12	gmetric	gmetric	ADJ
cana-1730	33	13	space	space	NOUN
cana-1730	33	14	.	.	PUNCT
cana-1730	34	1	(	(	PUNCT
cana-1730	34	2	1	1	X
cana-1730	34	3	)	)	PUNCT
cana-1730	34	4	a	a	DET
cana-1730	34	5	sequence	sequence	NOUN
cana-1730	34	6	{	{	PUNCT
cana-1730	34	7	am	be	AUX
cana-1730	34	8	}	}	PUNCT
cana-1730	34	9	in	in	ADP
cana-1730	34	10	s	s	PROPN
cana-1730	34	11	is	be	AUX
cana-1730	34	12	said	say	VERB
cana-1730	34	13	to	to	PART
cana-1730	34	14	converge	converge	VERB
cana-1730	34	15	to	to	ADP
cana-1730	34	16	a	a	DET
cana-1730	34	17	∈	∈	NOUN
cana-1730	34	18	s	s	X
cana-1730	34	19	if	if	SCONJ
cana-1730	34	20	for	for	ADP
cana-1730	34	21	every	every	DET
cana-1730	34	22	z	z	NOUN
cana-1730	34	23	∈	∈	PROPN
cana-1730	34	24	p	p	NOUN
cana-1730	34	25	with	with	ADP
cana-1730	34	26	≪	≪	PUNCT
cana-1730	34	27	z	z	NOUN
cana-1730	34	28	there	there	PRON
cana-1730	34	29	is	be	VERB
cana-1730	34	30	n	n	DET
cana-1730	34	31	∈	∈	PROPN
cana-1730	34	32	n	n	PRON
cana-1730	34	33	such	such	ADJ
cana-1730	34	34	that	that	PRON
cana-1730	34	35	for	for	SCONJ
cana-1730	34	36	all	all	DET
cana-1730	34	37	n	n	CCONJ
cana-1730	34	38	,	,	PUNCT
cana-1730	34	39	m	m	VERB
cana-1730	34	40	≥	≥	NOUN
cana-1730	34	41	n	n	PROPN
cana-1730	34	42	,	,	PUNCT
cana-1730	34	43	t(an	t(an	PROPN
cana-1730	34	44	,	,	PUNCT
cana-1730	34	45	am	be	AUX
cana-1730	34	46	,	,	PUNCT
cana-1730	34	47	a	a	PRON
cana-1730	34	48	)	)	PUNCT
cana-1730	34	49	≪	≪	PROPN
cana-1730	34	50	z.	z.	PROPN
cana-1730	34	51	(	(	PUNCT
cana-1730	34	52	2	2	NUM
cana-1730	34	53	)	)	PUNCT
cana-1730	34	54	a	a	DET
cana-1730	34	55	sequence	sequence	NOUN
cana-1730	34	56	{	{	PUNCT
cana-1730	34	57	an	an	NOUN
cana-1730	34	58	}	}	PUNCT
cana-1730	34	59	in	in	ADP
cana-1730	34	60	s	s	PROPN
cana-1730	34	61	is	be	AUX
cana-1730	34	62	called	call	VERB
cana-1730	34	63	a	a	DET
cana-1730	34	64	cauchy	cauchy	ADJ
cana-1730	34	65	sequence	sequence	NOUN
cana-1730	34	66	if	if	SCONJ
cana-1730	34	67	for	for	ADP
cana-1730	34	68	every	every	DET
cana-1730	34	69	z	z	NOUN
cana-1730	34	70	∈	∈	PROPN
cana-1730	34	71	p	p	NOUN
cana-1730	34	72	with	with	ADP
cana-1730	34	73	≪	≪	PUNCT
cana-1730	34	74	z	z	NOUN
cana-1730	34	75	there	there	PRON
cana-1730	34	76	is	be	VERB
cana-1730	34	77	a	a	DET
cana-1730	34	78	positive	positive	ADJ
cana-1730	34	79	integer	integer	NOUN
cana-1730	34	80	n	n	CCONJ
cana-1730	34	81	such	such	ADJ
cana-1730	34	82	that	that	SCONJ
cana-1730	34	83	t(an	t(an	PROPN
cana-1730	34	84	,	,	PUNCT
cana-1730	34	85	am	be	AUX
cana-1730	34	86	,	,	PUNCT
cana-1730	34	87	aℓ	aℓ	PROPN
cana-1730	34	88	)	)	PUNCT
cana-1730	34	89	≪	≪	PROPN
cana-1730	35	1	z	z	NOUN
cana-1730	35	2	,	,	PUNCT
cana-1730	35	3	for	for	ADP
cana-1730	35	4	all	all	DET
cana-1730	35	5	n	n	CCONJ
cana-1730	35	6	,	,	PUNCT
cana-1730	35	7	m	m	PROPN
cana-1730	35	8	,	,	PUNCT
cana-1730	35	9	ℓ	ℓ	PROPN
cana-1730	35	10	≥	≥	NOUN
cana-1730	35	11	n	n	NOUN
cana-1730	35	12	.	.	PUNCT
cana-1730	36	1	(	(	PUNCT
cana-1730	36	2	3	3	X
cana-1730	36	3	)	)	PUNCT
cana-1730	36	4	(	(	PUNCT
cana-1730	36	5	s	s	X
cana-1730	36	6	,	,	PUNCT
cana-1730	36	7	t	t	PROPN
cana-1730	36	8	)	)	PUNCT
cana-1730	36	9	is	be	AUX
cana-1730	36	10	said	say	VERB
cana-1730	36	11	to	to	PART
cana-1730	36	12	be	be	AUX
cana-1730	36	13	complete	complete	ADJ
cana-1730	36	14	if	if	SCONJ
cana-1730	36	15	every	every	DET
cana-1730	36	16	cauchy	cauchy	ADJ
cana-1730	36	17	sequence	sequence	NOUN
cana-1730	36	18	in	in	ADP
cana-1730	36	19	s	s	PROPN
cana-1730	36	20	is	be	AUX
cana-1730	36	21	convergent	convergent	ADJ
cana-1730	36	22	in	in	ADP
cana-1730	36	23	s	s	PROPN
cana-1730	36	24	.	.	PUNCT
cana-1730	37	1	lemma	lemma	PROPN
cana-1730	37	2	1	1	NUM
cana-1730	38	1	[	[	X
cana-1730	38	2	11]let	11]let	NUM
cana-1730	38	3	s	s	AUX
cana-1730	38	4	be	be	AUX
cana-1730	38	5	a	a	DET
cana-1730	38	6	cone	cone	NOUN
cana-1730	38	7	gmetric	gmetric	ADJ
cana-1730	38	8	space	space	NOUN
cana-1730	38	9	over	over	ADP
cana-1730	38	10	a	a	DET
cana-1730	38	11	normal	normal	ADJ
cana-1730	38	12	cone	cone	NOUN
cana-1730	38	13	,	,	PUNCT
cana-1730	38	14	a	a	DET
cana-1730	38	15	∈	∈	NOUN
cana-1730	38	16	s	s	PART
cana-1730	38	17	and	and	CCONJ
cana-1730	38	18	let	let	VERB
cana-1730	38	19	{	{	PUNCT
cana-1730	38	20	an	an	PRON
cana-1730	38	21	}	}	PUNCT
cana-1730	38	22	be	be	AUX
cana-1730	38	23	a	a	DET
cana-1730	38	24	sequence	sequence	NOUN
cana-1730	38	25	in	in	ADP
cana-1730	38	26	s.	s.	PROPN
cana-1730	38	27	then	then	ADV
cana-1730	38	28	the	the	DET
cana-1730	38	29	following	follow	VERB
cana-1730	38	30	are	be	AUX
cana-1730	38	31	equivalent	equivalent	ADJ
cana-1730	38	32	:	:	PUNCT
cana-1730	38	33	(	(	PUNCT
cana-1730	38	34	1	1	X
cana-1730	38	35	)	)	PUNCT
cana-1730	38	36	{	{	PUNCT
cana-1730	38	37	an	an	PRON
cana-1730	38	38	}	}	PUNCT
cana-1730	38	39	is	be	AUX
cana-1730	38	40	convergent	convergent	ADJ
cana-1730	38	41	to	to	ADP
cana-1730	38	42	a	a	PRON
cana-1730	38	43	;	;	PUNCT
cana-1730	38	44	(	(	PUNCT
cana-1730	38	45	2	2	X
cana-1730	38	46	)	)	PUNCT
cana-1730	38	47	t(an	t(an	PROPN
cana-1730	38	48	,	,	PUNCT
cana-1730	38	49	an	an	DET
cana-1730	38	50	,	,	PUNCT
cana-1730	38	51	a	a	NOUN
cana-1730	38	52	)	)	PUNCT
cana-1730	38	53	→	→	SYM
cana-1730	38	54	as	as	ADP
cana-1730	38	55	n	n	PROPN
cana-1730	38	56	→	→	SYM
cana-1730	38	57	∞	∞	PROPN
cana-1730	38	58	;	;	PUNCT
cana-1730	38	59	(	(	PUNCT
cana-1730	38	60	3	3	X
cana-1730	38	61	)	)	PUNCT
cana-1730	38	62	t(an	t(an	PROPN
cana-1730	38	63	,	,	PUNCT
cana-1730	38	64	a	a	DET
cana-1730	38	65	,	,	PUNCT
cana-1730	38	66	a	a	NOUN
cana-1730	38	67	)	)	PUNCT
cana-1730	38	68	→	→	SYM
cana-1730	38	69	as	as	ADP
cana-1730	38	70	n	n	PROPN
cana-1730	38	71	→	→	SYM
cana-1730	38	72	∞	∞	PROPN
cana-1730	38	73	;	;	PUNCT
cana-1730	38	74	(	(	PUNCT
cana-1730	38	75	4	4	X
cana-1730	38	76	)	)	PUNCT
cana-1730	38	77	t(am	t(am	NUM
cana-1730	38	78	,	,	PUNCT
cana-1730	38	79	an	an	PRON
cana-1730	38	80	,	,	PUNCT
cana-1730	38	81	a	a	NOUN
cana-1730	38	82	)	)	PUNCT
cana-1730	38	83	→	→	PUNCT
cana-1730	38	84	as	as	SCONJ
cana-1730	38	85	m	m	PROPN
cana-1730	38	86	,	,	PUNCT
cana-1730	38	87	n	n	CCONJ
cana-1730	38	88	→	→	SYM
cana-1730	38	89	∞.	∞.	PROPN
cana-1730	38	90	definition	definition	NOUN
cana-1730	38	91	3	3	NUM
cana-1730	38	92	let	let	VERB
cana-1730	38	93	s	s	PRON
cana-1730	38	94	be	be	AUX
cana-1730	38	95	a	a	DET
cana-1730	38	96	nonempty	nonempty	ADJ
cana-1730	38	97	set	set	VERB
cana-1730	38	98	.	.	PUNCT
cana-1730	39	1	then	then	ADV
cana-1730	39	2	(	(	PUNCT
cana-1730	39	3	s	s	X
cana-1730	39	4	,	,	PUNCT
cana-1730	39	5	t	t	PROPN
cana-1730	39	6	,	,	PUNCT
cana-1730	39	7	)	)	PUNCT
cana-1730	39	8	is	be	AUX
cana-1730	39	9	called	call	VERB
cana-1730	39	10	an	an	DET
cana-1730	39	11	ordered	order	VERB
cana-1730	39	12	cone	cone	NOUN
cana-1730	39	13	g	g	NOUN
cana-1730	39	14	-	-	PUNCT
cana-1730	39	15	metric	metric	ADJ
cana-1730	39	16	space	space	NOUN
cana-1730	39	17	if	if	SCONJ
cana-1730	39	18	:	:	PUNCT
cana-1730	39	19	(	(	PUNCT
cana-1730	39	20	i	i	NOUN
cana-1730	39	21	)	)	PUNCT
cana-1730	39	22	(	(	PUNCT
cana-1730	39	23	s	s	X
cana-1730	39	24	,	,	PUNCT
cana-1730	39	25	t	t	PROPN
cana-1730	39	26	)	)	PUNCT
cana-1730	39	27	is	be	AUX
cana-1730	39	28	a	a	DET
cana-1730	39	29	cone	cone	NOUN
cana-1730	39	30	gmetric	gmetric	ADJ
cana-1730	39	31	space	space	NOUN
cana-1730	39	32	,	,	PUNCT
cana-1730	39	33	(	(	PUNCT
cana-1730	39	34	ii	ii	NOUN
cana-1730	39	35	)	)	PUNCT
cana-1730	39	36	(	(	PUNCT
cana-1730	39	37	s	s	NOUN
cana-1730	39	38	,	,	PUNCT
cana-1730	39	39	)	)	PUNCT
cana-1730	39	40	is	be	AUX
cana-1730	39	41	a	a	DET
cana-1730	39	42	partially	partially	ADV
cana-1730	39	43	ordered	order	VERB
cana-1730	39	44	set	set	NOUN
cana-1730	39	45	.	.	PUNCT
cana-1730	40	1	let	let	VERB
cana-1730	40	2	(	(	PUNCT
cana-1730	40	3	s	s	X
cana-1730	40	4	,	,	PUNCT
cana-1730	40	5	)	)	PUNCT
cana-1730	40	6	be	be	AUX
cana-1730	40	7	a	a	DET
cana-1730	40	8	partially	partially	ADV
cana-1730	40	9	ordered	order	VERB
cana-1730	40	10	set	set	NOUN
cana-1730	40	11	.	.	PUNCT
cana-1730	41	1	then	then	ADV
cana-1730	41	2	a	a	DET
cana-1730	41	3	,	,	PUNCT
cana-1730	41	4	b	b	X
cana-1730	41	5	∈	∈	NOUN
cana-1730	41	6	s	s	VERB
cana-1730	41	7	are	be	AUX
cana-1730	41	8	called	call	VERB
cana-1730	41	9	comparable	comparable	ADJ
cana-1730	41	10	if	if	SCONJ
cana-1730	41	11	a	a	DET
cana-1730	41	12	b	b	NOUN
cana-1730	41	13	or	or	CCONJ
cana-1730	41	14	b	b	NOUN
cana-1730	41	15	a	a	DET
cana-1730	41	16	holds	hold	NOUN
cana-1730	41	17	.	.	PUNCT
cana-1730	42	1	in	in	ADP
cana-1730	42	2	[	[	X
cana-1730	42	3	13	13	NUM
cana-1730	42	4	]	]	PUNCT
cana-1730	42	5	,	,	PUNCT
cana-1730	42	6	nashine	nashine	NOUN
cana-1730	42	7	and	and	CCONJ
cana-1730	42	8	samet	samet	PROPN
cana-1730	42	9	introduced	introduce	VERB
cana-1730	42	10	the	the	DET
cana-1730	42	11	following	follow	VERB
cana-1730	42	12	concept	concept	NOUN
cana-1730	42	13	.	.	PUNCT
cana-1730	43	1	communications	communication	NOUN
cana-1730	43	2	on	on	ADP
cana-1730	43	3	applied	apply	VERB
cana-1730	43	4	nonlinear	nonlinear	ADJ
cana-1730	43	5	analysis	analysis	NOUN
cana-1730	43	6	issn	issn	NOUN
cana-1730	43	7	:	:	PUNCT
cana-1730	43	8	1074	1074	NUM
cana-1730	43	9	-	-	PUNCT
cana-1730	43	10	133x	133x	NUM
cana-1730	43	11	vol	vol	NOUN
cana-1730	43	12	32	32	NUM
cana-1730	43	13	no	no	NOUN
cana-1730	43	14	.	.	NOUN
cana-1730	43	15	2	2	NUM
cana-1730	43	16	(	(	PUNCT
cana-1730	43	17	2025	2025	NUM
cana-1730	43	18	)	)	PUNCT
cana-1730	43	19	161	161	NUM
cana-1730	43	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1730	44	1	let	let	VERB
cana-1730	44	2	s	s	PRON
cana-1730	44	3	be	be	AUX
cana-1730	44	4	a	a	DET
cana-1730	44	5	non	non	ADJ
cana-1730	44	6	-	-	ADJ
cana-1730	44	7	empty	empty	ADJ
cana-1730	44	8	set	set	NOUN
cana-1730	44	9	and	and	CCONJ
cana-1730	44	10	let	let	VERB
cana-1730	44	11	q	q	NOUN
cana-1730	44	12	:	:	PUNCT
cana-1730	44	13	s	s	AUX
cana-1730	44	14	→	→	SYM
cana-1730	44	15	s	s	VERB
cana-1730	44	16	be	be	AUX
cana-1730	44	17	a	a	DET
cana-1730	44	18	given	give	VERB
cana-1730	44	19	mapping	mapping	NOUN
cana-1730	44	20	.	.	PUNCT
cana-1730	45	1	for	for	ADP
cana-1730	45	2	every	every	DET
cana-1730	45	3	a	a	DET
cana-1730	45	4	∈	∈	PROPN
cana-1730	45	5	s	s	PART
cana-1730	45	6	,	,	PUNCT
cana-1730	45	7	we	we	PRON
cana-1730	45	8	denote	denote	VERB
cana-1730	45	9	by	by	ADP
cana-1730	45	10	q−1(a	q−1(a	PROPN
cana-1730	45	11	)	)	PUNCT
cana-1730	45	12	the	the	DET
cana-1730	45	13	subset	subset	NOUN
cana-1730	45	14	of	of	ADP
cana-1730	45	15	s	s	PRON
cana-1730	45	16	defined	define	VERB
cana-1730	45	17	by	by	ADP
cana-1730	45	18	q−1(a	q−1(a	PROPN
cana-1730	45	19	)	)	PUNCT
cana-1730	45	20	=	=	PRON
cana-1730	46	1	{	{	PUNCT
cana-1730	46	2	p	p	X
cana-1730	46	3	∈	∈	PROPN
cana-1730	46	4	s	s	PART
cana-1730	46	5	:	:	PUNCT
cana-1730	46	6	qp	qp	ADP
cana-1730	46	7	=	=	PUNCT
cana-1730	46	8	a	a	PRON
cana-1730	46	9	}	}	PUNCT
cana-1730	46	10	.	.	PUNCT
cana-1730	47	1	definition	definition	NOUN
cana-1730	47	2	4	4	NUM
cana-1730	47	3	let	let	VERB
cana-1730	47	4	(	(	PUNCT
cana-1730	47	5	s	s	NOUN
cana-1730	47	6	,	,	PUNCT
cana-1730	47	7	)	)	PUNCT
cana-1730	47	8	be	be	AUX
cana-1730	47	9	a	a	DET
cana-1730	47	10	partially	partially	ADV
cana-1730	47	11	ordered	order	VERB
cana-1730	47	12	set	set	NOUN
cana-1730	47	13	and	and	CCONJ
cana-1730	47	14	let	let	VERB
cana-1730	47	15	g	g	PROPN
cana-1730	47	16	,	,	PUNCT
cana-1730	47	17	t	t	PROPN
cana-1730	47	18	,	,	PUNCT
cana-1730	47	19	q	q	X
cana-1730	47	20	:	:	PUNCT
cana-1730	47	21	s	s	X
cana-1730	47	22	→	→	SYM
cana-1730	47	23	s	s	AUX
cana-1730	47	24	be	be	AUX
cana-1730	47	25	given	give	VERB
cana-1730	47	26	mappings	mapping	NOUN
cana-1730	47	27	such	such	ADJ
cana-1730	47	28	that	that	DET
cana-1730	47	29	gs	gs	PROPN
cana-1730	47	30	⊆	⊆	NUM
cana-1730	47	31	qs	qs	NOUN
cana-1730	47	32	and	and	CCONJ
cana-1730	47	33	xs	xs	PROPN
cana-1730	47	34	⊆	⊆	NUM
cana-1730	47	35	qs	qs	NOUN
cana-1730	47	36	.	.	PUNCT
cana-1730	48	1	we	we	PRON
cana-1730	48	2	say	say	VERB
cana-1730	48	3	that	that	SCONJ
cana-1730	48	4	x	x	PROPN
cana-1730	48	5	and	and	CCONJ
cana-1730	48	6	g	g	PROPN
cana-1730	48	7	are	be	AUX
cana-1730	48	8	weakly	weakly	ADV
cana-1730	48	9	increasing	increase	VERB
cana-1730	48	10	with	with	ADP
cana-1730	48	11	respect	respect	NOUN
cana-1730	48	12	to	to	ADP
cana-1730	48	13	q	q	NOUN
cana-1730	48	14	if	if	SCONJ
cana-1730	48	15	for	for	ADP
cana-1730	48	16	all	all	DET
cana-1730	48	17	a	a	DET
cana-1730	48	18	∈	∈	NOUN
cana-1730	48	19	s	s	PART
cana-1730	48	20	,	,	PUNCT
cana-1730	48	21	we	we	PRON
cana-1730	48	22	have	have	AUX
cana-1730	48	23	:	:	PUNCT
cana-1730	48	24	ga	ga	PROPN
cana-1730	48	25	xb	xb	PROPN
cana-1730	48	26	,	,	PUNCT
cana-1730	48	27	∀	∀	PUNCT
cana-1730	48	28	b	b	X
cana-1730	48	29	∈	∈	PROPN
cana-1730	48	30	q−1(ga	q−1(ga	PROPN
cana-1730	48	31	)	)	PUNCT
cana-1730	48	32	and	and	CCONJ
cana-1730	48	33	xa	xa	PROPN
cana-1730	48	34	gb	gb	PROPN
cana-1730	48	35	,	,	PUNCT
cana-1730	48	36	∀	∀	NOUN
cana-1730	48	37	b	b	NOUN
cana-1730	48	38	∈	∈	PROPN
cana-1730	48	39	q−1(xa	q−1(xa	PROPN
cana-1730	48	40	)	)	PUNCT
cana-1730	48	41	.	.	PUNCT
cana-1730	49	1	if	if	SCONJ
cana-1730	49	2	g	g	NOUN
cana-1730	49	3	=	=	SYM
cana-1730	49	4	x	x	NOUN
cana-1730	49	5	,	,	PUNCT
cana-1730	49	6	we	we	PRON
cana-1730	49	7	say	say	VERB
cana-1730	49	8	that	that	SCONJ
cana-1730	49	9	g	g	PROPN
cana-1730	49	10	is	be	AUX
cana-1730	49	11	weakly	weakly	ADV
cana-1730	49	12	increasing	increase	VERB
cana-1730	49	13	with	with	ADP
cana-1730	49	14	respect	respect	NOUN
cana-1730	49	15	to	to	ADP
cana-1730	49	16	q.	q.	NOUN
cana-1730	49	17	definition	definition	NOUN
cana-1730	49	18	5	5	NUM
cana-1730	49	19	(	(	PUNCT
cana-1730	49	20	[	[	X
cana-1730	49	21	9,10	9,10	NUM
cana-1730	49	22	]	]	PUNCT
cana-1730	49	23	)	)	PUNCT
cana-1730	49	24	.	.	PUNCT
cana-1730	50	1	let	let	VERB
cana-1730	50	2	a	a	PRON
cana-1730	50	3	be	be	AUX
cana-1730	50	4	an	an	DET
cana-1730	50	5	order	order	NOUN
cana-1730	50	6	cone	cone	NOUN
cana-1730	50	7	.	.	PUNCT
cana-1730	51	1	a	a	DET
cana-1730	51	2	non	non	ADJ
cana-1730	51	3	decreasing	decrease	VERB
cana-1730	51	4	function	function	NOUN
cana-1730	51	5	:	:	PUNCT
cana-1730	51	6	p	p	X
cana-1730	51	7	→	→	PUNCT
cana-1730	51	8	p	p	X
cana-1730	51	9	is	be	AUX
cana-1730	51	10	called	call	VERB
cana-1730	51	11	a	a	DET
cana-1730	51	12	map	map	NOUN
cana-1730	51	13	if	if	SCONJ
cana-1730	51	14	:	:	PUNCT
cana-1730	51	15	(	(	PUNCT
cana-1730	51	16	i	i	NOUN
cana-1730	51	17	)	)	PUNCT
cana-1730	51	18	(	(	PUNCT
cana-1730	51	19	)	)	PUNCT
cana-1730	51	20	=	=	SYM
cana-1730	51	21	θ	θ	PROPN
cana-1730	51	22	and	and	CCONJ
cana-1730	51	23	<	<	X
cana-1730	51	24	(	(	PUNCT
cana-1730	51	25	ω	ω	NOUN
cana-1730	51	26	)	)	PUNCT
cana-1730	51	27	<	<	X
cana-1730	51	28	ω	ω	PROPN
cana-1730	51	29	for	for	ADP
cana-1730	51	30	ω	ω	PROPN
cana-1730	51	31	∈	∈	PROPN
cana-1730	51	32	a	a	DET
cana-1730	51	33	\	\	NOUN
cana-1730	51	34	{	{	PUNCT
cana-1730	51	35	}	}	PUNCT
cana-1730	51	36	,	,	PUNCT
cana-1730	51	37	(	(	PUNCT
cana-1730	51	38	ii	ii	NOUN
cana-1730	51	39	)	)	PUNCT
cana-1730	51	40	ω	ω	PROPN
cana-1730	51	41	∈	∈	PROPN
cana-1730	51	42	in	in	ADP
cana-1730	51	43	a	a	DET
cana-1730	51	44	implies	implie	NOUN
cana-1730	51	45	ω	ω	NUM
cana-1730	51	46	−	−	PROPN
cana-1730	51	47	(	(	PUNCT
cana-1730	51	48	ω	ω	NOUN
cana-1730	51	49	)	)	PUNCT
cana-1730	51	50	∈	∈	PROPN
cana-1730	51	51	in	in	ADP
cana-1730	51	52	a	a	DET
cana-1730	51	53	,	,	PUNCT
cana-1730	51	54	(	(	PUNCT
cana-1730	51	55	iii	iii	X
cana-1730	51	56	)	)	PUNCT
cana-1730	51	57	if	if	SCONJ
cana-1730	51	58	ω	ω	PROPN
cana-1730	51	59	∈	∈	PROPN
cana-1730	51	60	a	a	DET
cana-1730	51	61	\	\	NOUN
cana-1730	51	62	{	{	PUNCT
cana-1730	51	63	}	}	PUNCT
cana-1730	51	64	and	and	CCONJ
cana-1730	51	65	z	z	NOUN
cana-1730	51	66	∈	∈	PROPN
cana-1730	51	67	in	in	ADP
cana-1730	51	68	a	a	PRON
cana-1730	51	69	,	,	PUNCT
cana-1730	51	70	then	then	ADV
cana-1730	51	71	there	there	PRON
cana-1730	51	72	exists	exist	VERB
cana-1730	51	73	n0	n0	PROPN
cana-1730	51	74	∈	∈	PROPN
cana-1730	51	75	n	n	PRON
cana-1730	51	76	such	such	ADJ
cana-1730	51	77	that	that	DET
cana-1730	51	78	n(ω	n(ω	NOUN
cana-1730	51	79	)	)	PUNCT
cana-1730	51	80	≪	≪	VERB
cana-1730	51	81	z	z	NOUN
cana-1730	51	82	for	for	ADP
cana-1730	51	83	each	each	PRON
cana-1730	51	84	n	n	PROPN
cana-1730	51	85	n0	n0	PROPN
cana-1730	51	86	.	.	PUNCT
cana-1730	51	87	theorem	theorem	VERB
cana-1730	51	88	1	1	NUM
cana-1730	51	89	let	let	VERB
cana-1730	51	90	(	(	PUNCT
cana-1730	51	91	s	s	NOUN
cana-1730	51	92	,	,	PUNCT
cana-1730	51	93	)	)	PUNCT
cana-1730	51	94	be	be	AUX
cana-1730	51	95	a	a	DET
cana-1730	51	96	partially	partially	ADV
cana-1730	51	97	ordered	order	VERB
cana-1730	51	98	set	set	NOUN
cana-1730	51	99	,	,	PUNCT
cana-1730	51	100	a	a	PRON
cana-1730	51	101	be	be	AUX
cana-1730	51	102	an	an	DET
cana-1730	51	103	order	order	NOUN
cana-1730	51	104	cone	cone	NOUN
cana-1730	51	105	and	and	CCONJ
cana-1730	51	106	let	let	VERB
cana-1730	51	107	t	t	PROPN
cana-1730	51	108	be	be	AUX
cana-1730	51	109	a	a	DET
cana-1730	51	110	cone	cone	NOUN
cana-1730	51	111	gmetric	gmetric	NOUN
cana-1730	51	112	on	on	ADP
cana-1730	51	113	s.	s.	PROPN
cana-1730	51	114	let	let	VERB
cana-1730	51	115	g	g	NOUN
cana-1730	51	116	,	,	PUNCT
cana-1730	51	117	q	q	X
cana-1730	51	118	:	:	PUNCT
cana-1730	51	119	s	s	X
cana-1730	51	120	→	→	SYM
cana-1730	51	121	s	s	PART
cana-1730	51	122	be	be	AUX
cana-1730	51	123	two	two	NUM
cana-1730	51	124	mappings	mapping	NOUN
cana-1730	51	125	such	such	ADJ
cana-1730	51	126	that	that	SCONJ
cana-1730	51	127	t(ga	t(ga	NUM
cana-1730	51	128	,	,	PUNCT
cana-1730	51	129	gb	gb	NOUN
cana-1730	51	130	,	,	PUNCT
cana-1730	51	131	gc	gc	PROPN
cana-1730	51	132	)	)	PUNCT
cana-1730	51	133	(	(	PUNCT
cana-1730	51	134	t(qa	t(qa	PROPN
cana-1730	51	135	,	,	PUNCT
cana-1730	51	136	qb	qb	PROPN
cana-1730	51	137	,	,	PUNCT
cana-1730	51	138	qc	qc	PROPN
cana-1730	51	139	)	)	PUNCT
cana-1730	51	140	)	)	PUNCT
cana-1730	52	1	(	(	PUNCT
cana-1730	52	2	1	1	X
cana-1730	52	3	)	)	PUNCT
cana-1730	52	4	for	for	ADP
cana-1730	52	5	all	all	DET
cana-1730	52	6	a	a	DET
cana-1730	52	7	,	,	PUNCT
cana-1730	52	8	b	b	NOUN
cana-1730	52	9	,	,	PUNCT
cana-1730	52	10	c	c	PROPN
cana-1730	52	11	∈	∈	PROPN
cana-1730	52	12	s	s	VERB
cana-1730	52	13	with	with	ADP
cana-1730	52	14	qa	qa	PROPN
cana-1730	52	15	qb	qb	PROPN
cana-1730	52	16	qc	qc	PROPN
cana-1730	52	17	,	,	PUNCT
cana-1730	52	18	where	where	SCONJ
cana-1730	52	19	θ	θ	PROPN
cana-1730	52	20	is	be	AUX
cana-1730	52	21	a	a	DET
cana-1730	52	22	θ	θ	PROPN
cana-1730	52	23	-map	-map	NUM
cana-1730	52	24	.	.	PUNCT
cana-1730	53	1	we	we	PRON
cana-1730	53	2	suppose	suppose	VERB
cana-1730	53	3	the	the	DET
cana-1730	53	4	following	follow	VERB
cana-1730	53	5	:	:	PUNCT
cana-1730	53	6	(	(	PUNCT
cana-1730	53	7	i	i	NOUN
cana-1730	53	8	)	)	PUNCT
cana-1730	53	9	g	g	NOUN
cana-1730	53	10	is	be	AUX
cana-1730	53	11	weakly	weakly	ADV
cana-1730	53	12	increasing	increase	VERB
cana-1730	53	13	with	with	ADP
cana-1730	53	14	respect	respect	NOUN
cana-1730	53	15	to	to	ADP
cana-1730	53	16	q	q	NOUN
cana-1730	53	17	;	;	PUNCT
cana-1730	53	18	(	(	PUNCT
cana-1730	53	19	ii	ii	NOUN
cana-1730	53	20	)	)	PUNCT
cana-1730	53	21	qs	qs	PROPN
cana-1730	53	22	is	be	AUX
cana-1730	53	23	a	a	DET
cana-1730	53	24	complete	complete	ADJ
cana-1730	53	25	subspace	subspace	NOUN
cana-1730	53	26	of	of	ADP
cana-1730	53	27	s	s	PROPN
cana-1730	53	28	;	;	PUNCT
cana-1730	53	29	(	(	PUNCT
cana-1730	53	30	iii	iii	X
cana-1730	53	31	)	)	PUNCT
cana-1730	53	32	s	s	VERB
cana-1730	53	33	is	be	AUX
cana-1730	53	34	regular	regular	ADJ
cana-1730	53	35	.	.	PUNCT
cana-1730	54	1	then	then	ADV
cana-1730	54	2	g	g	PROPN
cana-1730	54	3	and	and	CCONJ
cana-1730	54	4	q	q	PROPN
cana-1730	54	5	have	have	VERB
cana-1730	54	6	a	a	DET
cana-1730	54	7	coincidence	coincidence	NOUN
cana-1730	54	8	point	point	NOUN
cana-1730	54	9	.	.	PUNCT
cana-1730	55	1	proof	proof	NOUN
cana-1730	55	2	.	.	PUNCT
cana-1730	56	1	let	let	VERB
cana-1730	56	2	a0	a0	PROPN
cana-1730	56	3	be	be	AUX
cana-1730	56	4	an	an	DET
cana-1730	56	5	arbitrary	arbitrary	ADJ
cana-1730	56	6	point	point	NOUN
cana-1730	56	7	in	in	ADP
cana-1730	56	8	s	s	PRON
cana-1730	56	9	.	.	PUNCT
cana-1730	57	1	since	since	SCONJ
cana-1730	57	2	gs	gs	PROPN
cana-1730	57	3	⊆	⊆	NUM
cana-1730	57	4	qs	qs	NOUN
cana-1730	57	5	(	(	PUNCT
cana-1730	57	6	by	by	ADP
cana-1730	57	7	definition	definition	NOUN
cana-1730	57	8	4	4	NUM
cana-1730	57	9	)	)	PUNCT
cana-1730	57	10	,	,	PUNCT
cana-1730	57	11	we	we	PRON
cana-1730	57	12	can	can	AUX
cana-1730	57	13	construct	construct	VERB
cana-1730	57	14	a	a	DET
cana-1730	57	15	sequence	sequence	NOUN
cana-1730	57	16	{	{	PUNCT
cana-1730	57	17	an	an	NOUN
cana-1730	57	18	}	}	PUNCT
cana-1730	57	19	in	in	ADP
cana-1730	57	20	s	s	PRON
cana-1730	57	21	defined	define	VERB
cana-1730	57	22	by	by	ADP
cana-1730	57	23	qan+1	qan+1	NOUN
cana-1730	57	24	=	=	SYM
cana-1730	57	25	gan	gan	PROPN
cana-1730	57	26	,	,	PUNCT
cana-1730	57	27	∀	∀	NOUN
cana-1730	57	28	n	n	PRON
cana-1730	57	29	∈	∈	PROPN
cana-1730	57	30	n0	n0	PROPN
cana-1730	57	31	.	.	PUNCT
cana-1730	58	1	now	now	ADV
cana-1730	58	2	,	,	PUNCT
cana-1730	58	3	since	since	SCONJ
cana-1730	58	4	a1	a1	NOUN
cana-1730	58	5	∈	∈	NOUN
cana-1730	58	6	q−1(ga0	q−1(ga0	NOUN
cana-1730	58	7	)	)	PUNCT
cana-1730	58	8	and	and	CCONJ
cana-1730	58	9	a2	a2	PROPN
cana-1730	58	10	∈	∈	PROPN
cana-1730	58	11	q−1(ga1	q−1(ga1	NOUN
cana-1730	58	12	)	)	PUNCT
cana-1730	58	13	,	,	PUNCT
cana-1730	58	14	using	use	VERB
cana-1730	58	15	that	that	SCONJ
cana-1730	58	16	g	g	PROPN
cana-1730	58	17	is	be	AUX
cana-1730	58	18	weakly	weakly	ADV
cana-1730	58	19	increasing	increase	VERB
cana-1730	58	20	with	with	ADP
cana-1730	58	21	respect	respect	NOUN
cana-1730	58	22	to	to	ADP
cana-1730	58	23	q	q	X
cana-1730	58	24	,	,	PUNCT
cana-1730	58	25	we	we	PRON
cana-1730	58	26	obtain	obtain	VERB
cana-1730	58	27	that	that	DET
cana-1730	58	28	qa1	qa1	PROPN
cana-1730	58	29	=	=	NOUN
cana-1730	58	30	ga0	ga0	PROPN
cana-1730	58	31	ga1	ga1	NOUN
cana-1730	58	32	=	=	NOUN
cana-1730	59	1	qa2	qa2	PROPN
cana-1730	59	2	ga2	ga2	NOUN
cana-1730	59	3	=	=	SYM
cana-1730	59	4	qa3	qa3	PROPN
cana-1730	59	5	.	.	PUNCT
cana-1730	60	1	continuing	continue	VERB
cana-1730	60	2	this	this	DET
cana-1730	60	3	process	process	NOUN
cana-1730	60	4	,	,	PUNCT
cana-1730	60	5	we	we	PRON
cana-1730	60	6	get	get	VERB
cana-1730	60	7	that	that	PRON
cana-1730	61	1	qa1	qa1	PROPN
cana-1730	61	2	qa2	qa2	INTJ
cana-1730	61	3	qa3	qa3	X
cana-1730	61	4	·	·	PUNCT
cana-1730	61	5	·	·	PUNCT
cana-1730	61	6	·	·	PUNCT
cana-1730	61	7	qan	qan	X
cana-1730	61	8	qan+1	qan+1	PROPN
cana-1730	61	9	·	·	PUNCT
cana-1730	61	10	·	·	PUNCT
cana-1730	61	11	·	·	PUNCT
cana-1730	61	12	.	.	PUNCT
cana-1730	62	1	we	we	PRON
cana-1730	62	2	will	will	AUX
cana-1730	62	3	prove	prove	VERB
cana-1730	62	4	that	that	SCONJ
cana-1730	62	5	{	{	PUNCT
cana-1730	62	6	qan	qan	X
cana-1730	62	7	}	}	PUNCT
cana-1730	62	8	is	be	AUX
cana-1730	62	9	a	a	DET
cana-1730	62	10	cauchy	cauchy	ADJ
cana-1730	62	11	sequence	sequence	NOUN
cana-1730	62	12	in	in	ADP
cana-1730	62	13	(	(	PUNCT
cana-1730	62	14	q(s	q(s	PROPN
cana-1730	62	15	)	)	PUNCT
cana-1730	62	16	,	,	PUNCT
cana-1730	62	17	t	t	PROPN
cana-1730	62	18	)	)	PUNCT
cana-1730	62	19	.	.	PUNCT
cana-1730	63	1	we	we	PRON
cana-1730	63	2	distinguish	distinguish	VERB
cana-1730	63	3	two	two	NUM
cana-1730	63	4	cases	case	NOUN
cana-1730	63	5	.	.	PUNCT
cana-1730	64	1	first	first	ADJ
cana-1730	64	2	case	case	NOUN
cana-1730	64	3	.	.	PUNCT
cana-1730	65	1	there	there	PRON
cana-1730	65	2	exists	exist	VERB
cana-1730	65	3	n	n	PRON
cana-1730	65	4	∈	∈	PROPN
cana-1730	65	5	n	n	PRON
cana-1730	65	6	such	such	ADJ
cana-1730	65	7	that	that	DET
cana-1730	65	8	qan	qan	PROPN
cana-1730	65	9	=	=	SYM
cana-1730	65	10	qan+1	qan+1	NOUN
cana-1730	65	11	.	.	PUNCT
cana-1730	66	1	using	use	VERB
cana-1730	66	2	the	the	DET
cana-1730	66	3	considered	consider	VERB
cana-1730	66	4	contractive	contractive	ADJ
cana-1730	66	5	condition	condition	NOUN
cana-1730	66	6	,	,	PUNCT
cana-1730	66	7	we	we	PRON
cana-1730	66	8	get	get	VERB
cana-1730	66	9	gan	gan	ADJ
cana-1730	66	10	=	=	SYM
cana-1730	66	11	gan+1	gan+1	PROPN
cana-1730	66	12	,	,	PUNCT
cana-1730	66	13	that	that	PRON
cana-1730	66	14	s	s	VERB
cana-1730	66	15	is	be	AUX
cana-1730	66	16	,	,	PUNCT
cana-1730	66	17	qan+1	qan+1	NOUN
cana-1730	66	18	=	=	SYM
cana-1730	66	19	qan+2	qan+2	NOUN
cana-1730	66	20	.	.	PUNCT
cana-1730	67	1	so	so	ADV
cana-1730	67	2	,	,	PUNCT
cana-1730	67	3	for	for	ADP
cana-1730	67	4	every	every	DET
cana-1730	67	5	m	m	PROPN
cana-1730	67	6	≥	≥	NOUN
cana-1730	67	7	n	n	CCONJ
cana-1730	67	8	,	,	PUNCT
cana-1730	67	9	we	we	PRON
cana-1730	67	10	have	have	VERB
cana-1730	67	11	qam	qam	PROPN
cana-1730	67	12	=	=	SYM
cana-1730	67	13	qan	qan	PROPN
cana-1730	67	14	.	.	PUNCT
cana-1730	68	1	this	this	PRON
cana-1730	68	2	implies	imply	VERB
cana-1730	68	3	that	that	SCONJ
cana-1730	68	4	{	{	PUNCT
cana-1730	68	5	qan	qan	X
cana-1730	68	6	}	}	PUNCT
cana-1730	68	7	is	be	AUX
cana-1730	68	8	a	a	DET
cana-1730	68	9	cauchy	cauchy	ADJ
cana-1730	68	10	sequence	sequence	NOUN
cana-1730	68	11	.	.	PUNCT
cana-1730	69	1	second	second	ADJ
cana-1730	69	2	case	case	NOUN
cana-1730	69	3	.	.	PUNCT
cana-1730	70	1	the	the	DET
cana-1730	70	2	successive	successive	ADJ
cana-1730	70	3	terms	term	NOUN
cana-1730	70	4	of	of	ADP
cana-1730	70	5	{	{	PUNCT
cana-1730	70	6	qan	qan	NOUN
cana-1730	70	7	}	}	PUNCT
cana-1730	70	8	are	be	AUX
cana-1730	70	9	different	different	ADJ
cana-1730	70	10	.	.	PUNCT
cana-1730	71	1	from	from	ADP
cana-1730	71	2	(	(	PUNCT
cana-1730	71	3	1	1	NUM
cana-1730	71	4	)	)	PUNCT
cana-1730	71	5	,	,	PUNCT
cana-1730	71	6	we	we	PRON
cana-1730	71	7	have	have	VERB
cana-1730	71	8	t(qan	t(qan	ADV
cana-1730	71	9	,	,	PUNCT
cana-1730	71	10	qan+1	qan+1	NOUN
cana-1730	71	11	,	,	PUNCT
cana-1730	71	12	qan+1	qan+1	NOUN
cana-1730	71	13	)	)	PUNCT
cana-1730	71	14	=	=	SYM
cana-1730	71	15	t(gan−1	t(gan−1	PROPN
cana-1730	71	16	,	,	PUNCT
cana-1730	71	17	gan	gan	PROPN
cana-1730	71	18	,	,	PUNCT
cana-1730	71	19	qan	qan	PROPN
cana-1730	71	20	)	)	PUNCT
cana-1730	71	21	≤	≤	NUM
cana-1730	71	22	θ	θ	PROPN
cana-1730	71	23	(	(	PUNCT
cana-1730	71	24	t(qan−1	t(qan−1	PROPN
cana-1730	71	25	,	,	PUNCT
cana-1730	71	26	qan	qan	PROPN
cana-1730	71	27	,	,	PUNCT
cana-1730	71	28	qan	qan	PROPN
cana-1730	71	29	)	)	PUNCT
cana-1730	71	30	)	)	PUNCT
cana-1730	71	31	≤	≤	NUM
cana-1730	72	1	θ	θ	PROPN
cana-1730	72	2	2(t(qan−2	2(t(qan−2	NUM
cana-1730	72	3	,	,	PUNCT
cana-1730	72	4	qan−1	qan−1	PROPN
cana-1730	72	5	,	,	PUNCT
cana-1730	72	6	qan−1	qan−1	PROPN
cana-1730	72	7	)	)	PUNCT
cana-1730	72	8	)	)	PUNCT
cana-1730	72	9	…	…	PUNCT
cana-1730	73	1	communications	communication	NOUN
cana-1730	73	2	on	on	ADP
cana-1730	73	3	applied	apply	VERB
cana-1730	73	4	nonlinear	nonlinear	ADJ
cana-1730	73	5	analysis	analysis	NOUN
cana-1730	73	6	issn	issn	NOUN
cana-1730	73	7	:	:	PUNCT
cana-1730	73	8	1074	1074	NUM
cana-1730	73	9	-	-	PUNCT
cana-1730	73	10	133x	133x	NUM
cana-1730	73	11	vol	vol	NOUN
cana-1730	73	12	32	32	NUM
cana-1730	73	13	no	no	NOUN
cana-1730	73	14	.	.	NOUN
cana-1730	73	15	2	2	NUM
cana-1730	73	16	(	(	PUNCT
cana-1730	73	17	2025	2025	NUM
cana-1730	73	18	)	)	PUNCT
cana-1730	73	19	162	162	NUM
cana-1730	73	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1730	73	21	≤	≤	NUM
cana-1730	73	22	θ	θ	PROPN
cana-1730	73	23	n(t(qa0	n(t(qa0	PROPN
cana-1730	73	24	,	,	PUNCT
cana-1730	73	25	qa1	qa1	PROPN
cana-1730	73	26	,	,	PUNCT
cana-1730	73	27	qa1	qa1	NOUN
cana-1730	73	28	)	)	PUNCT
cana-1730	73	29	)	)	PUNCT
cana-1730	73	30	.	.	PUNCT
cana-1730	74	1	fix	fix	VERB
cana-1730	74	2	z	z	PROPN
cana-1730	74	3	,	,	PUNCT
cana-1730	74	4	≪	≪	SCONJ
cana-1730	74	5	z.	z.	PROPN
cana-1730	74	6	according	accord	VERB
cana-1730	74	7	to	to	ADP
cana-1730	74	8	property	property	NOUN
cana-1730	74	9	(	(	PUNCT
cana-1730	74	10	iii	iii	NOUN
cana-1730	74	11	)	)	PUNCT
cana-1730	74	12	of	of	ADP
cana-1730	74	13	function	function	NOUN
cana-1730	74	14	θ	θ	PROPN
cana-1730	74	15	,	,	PUNCT
cana-1730	74	16	there	there	PRON
cana-1730	74	17	is	be	VERB
cana-1730	74	18	n0	n0	NUM
cana-1730	74	19	∈	∈	PROPN
cana-1730	74	20	n	n	PRON
cana-1730	74	21	such	such	ADJ
cana-1730	74	22	that	that	SCONJ
cana-1730	74	23	θ	θ	PROPN
cana-1730	74	24	n(t(qa0	n(t(qa0	PROPN
cana-1730	74	25	,	,	PUNCT
cana-1730	74	26	qa1,gqa1	qa1,gqa1	PROPN
cana-1730	74	27	)	)	PUNCT
cana-1730	74	28	)	)	PUNCT
cana-1730	75	1	≪	≪	PUNCT
cana-1730	75	2	z	z	NOUN
cana-1730	75	3	for	for	ADP
cana-1730	75	4	n	n	PRON
cana-1730	75	5	≥	≥	NOUN
cana-1730	75	6	n0	n0	NUM
cana-1730	75	7	.	.	PUNCT
cana-1730	76	1	we	we	PRON
cana-1730	76	2	get	get	VERB
cana-1730	76	3	that	that	DET
cana-1730	76	4	t(qan	t(qan	ADV
cana-1730	76	5	,	,	PUNCT
cana-1730	76	6	qan+1	qan+1	NOUN
cana-1730	76	7	,	,	PUNCT
cana-1730	76	8	qan+1	qan+1	NOUN
cana-1730	76	9	)	)	PUNCT
cana-1730	76	10	≪	≪	PUNCT
cana-1730	76	11	z	z	NOUN
cana-1730	76	12	for	for	ADP
cana-1730	76	13	n	n	PRON
cana-1730	76	14	≥	≥	NOUN
cana-1730	76	15	n0	n0	NUM
cana-1730	76	16	.	.	PUNCT
cana-1730	77	1	in	in	ADP
cana-1730	77	2	a	a	DET
cana-1730	77	3	similar	similar	ADJ
cana-1730	77	4	way	way	NOUN
cana-1730	77	5	,	,	PUNCT
cana-1730	77	6	there	there	PRON
cana-1730	77	7	is	be	VERB
cana-1730	77	8	n1	n1	PROPN
cana-1730	77	9	∈	∈	NOUN
cana-1730	77	10	n	n	PRON
cana-1730	77	11	such	such	ADJ
cana-1730	77	12	that	that	SCONJ
cana-1730	77	13	t(qam	t(qam	PROPN
cana-1730	77	14	,	,	PUNCT
cana-1730	77	15	qam+1	qam+1	X
cana-1730	77	16	,	,	PUNCT
cana-1730	77	17	qam+1	qam+1	X
cana-1730	77	18	)	)	PUNCT
cana-1730	77	19	<	<	X
cana-1730	77	20	z	z	NOUN
cana-1730	77	21	−	−	PROPN
cana-1730	77	22	θ	θ	NOUN
cana-1730	77	23	(	(	PUNCT
cana-1730	77	24	c	c	NOUN
cana-1730	77	25	)	)	PUNCT
cana-1730	77	26	for	for	ADP
cana-1730	77	27	all	all	DET
cana-1730	77	28	m	m	PROPN
cana-1730	77	29	≥	≥	NOUN
cana-1730	77	30	n1	n1	NOUN
cana-1730	77	31	.	.	PUNCT
cana-1730	78	1	(	(	PUNCT
cana-1730	78	2	2	2	X
cana-1730	78	3	)	)	PUNCT
cana-1730	78	4	we	we	PRON
cana-1730	78	5	claim	claim	VERB
cana-1730	78	6	that	that	SCONJ
cana-1730	78	7	t(qan	t(qan	ADV
cana-1730	78	8	,	,	PUNCT
cana-1730	78	9	qam	qam	PROPN
cana-1730	78	10	,	,	PUNCT
cana-1730	78	11	qam	qam	PROPN
cana-1730	78	12	)	)	PUNCT
cana-1730	78	13	≪	≪	PUNCT
cana-1730	78	14	z	z	X
cana-1730	78	15	∀	∀	X
cana-1730	78	16	m	m	VERB
cana-1730	78	17	>	>	X
cana-1730	78	18	n	n	CCONJ
cana-1730	78	19	≥	≥	PROPN
cana-1730	78	20	n1	n1	PROPN
cana-1730	78	21	(	(	PUNCT
cana-1730	78	22	3	3	NUM
cana-1730	78	23	)	)	PUNCT
cana-1730	78	24	and	and	CCONJ
cana-1730	78	25	prove	prove	VERB
cana-1730	78	26	it	it	PRON
cana-1730	78	27	by	by	ADP
cana-1730	78	28	induction	induction	NOUN
cana-1730	78	29	on	on	ADP
cana-1730	78	30	m.	m.	NOUN
cana-1730	78	31	the	the	DET
cana-1730	78	32	inequality	inequality	NOUN
cana-1730	78	33	(	(	PUNCT
cana-1730	78	34	3	3	X
cana-1730	78	35	)	)	PUNCT
cana-1730	78	36	holds	hold	VERB
cana-1730	78	37	for	for	ADP
cana-1730	78	38	m	m	PROPN
cana-1730	78	39	=	=	SYM
cana-1730	78	40	n	n	PROPN
cana-1730	78	41	+	+	NOUN
cana-1730	78	42	1	1	NUM
cana-1730	78	43	by	by	ADP
cana-1730	78	44	using	use	VERB
cana-1730	78	45	(	(	PUNCT
cana-1730	78	46	2	2	NUM
cana-1730	78	47	)	)	PUNCT
cana-1730	78	48	and	and	CCONJ
cana-1730	78	49	the	the	DET
cana-1730	78	50	fact	fact	NOUN
cana-1730	78	51	that	that	SCONJ
cana-1730	78	52	z	z	NOUN
cana-1730	79	1	−	−	NUM
cana-1730	79	2	θ	θ	PROPN
cana-1730	79	3	(	(	PUNCT
cana-1730	79	4	z	z	NOUN
cana-1730	79	5	)	)	PUNCT
cana-1730	79	6	<	<	X
cana-1730	79	7	z.	z.	PROPN
cana-1730	79	8	assume	assume	VERB
cana-1730	79	9	that	that	SCONJ
cana-1730	79	10	(	(	PUNCT
cana-1730	79	11	3	3	X
cana-1730	79	12	)	)	PUNCT
cana-1730	79	13	holds	hold	VERB
cana-1730	79	14	for	for	ADP
cana-1730	79	15	m	m	PROPN
cana-1730	79	16	=	=	SYM
cana-1730	79	17	d.	d.	PROPN
cana-1730	79	18	for	for	ADP
cana-1730	79	19	m	m	PROPN
cana-1730	80	1	=	=	SYM
cana-1730	80	2	d	d	PROPN
cana-1730	80	3	+	+	NOUN
cana-1730	80	4	1	1	NUM
cana-1730	80	5	,	,	PUNCT
cana-1730	80	6	t(qan	t(qan	ADV
cana-1730	80	7	,	,	PUNCT
cana-1730	80	8	qad+1	qad+1	X
cana-1730	80	9	,	,	PUNCT
cana-1730	80	10	qad+1	qad+1	NOUN
cana-1730	80	11	)	)	PUNCT
cana-1730	80	12	≤	≤	NOUN
cana-1730	80	13	q(qan	q(qan	NOUN
cana-1730	80	14	,	,	PUNCT
cana-1730	80	15	qan+1	qan+1	NOUN
cana-1730	80	16	,	,	PUNCT
cana-1730	80	17	qan+1	qan+1	NOUN
cana-1730	80	18	)	)	PUNCT
cana-1730	80	19	+	+	SYM
cana-1730	80	20	t(qan+1	t(qan+1	NOUN
cana-1730	80	21	,	,	PUNCT
cana-1730	80	22	qad+1	qad+1	X
cana-1730	80	23	,	,	PUNCT
cana-1730	80	24	qad+1	qad+1	NOUN
cana-1730	80	25	)	)	PUNCT
cana-1730	80	26	≪	≪	PUNCT
cana-1730	81	1	z	z	NOUN
cana-1730	81	2	−	−	NOUN
cana-1730	81	3	θ	θ	PROPN
cana-1730	81	4	(	(	PUNCT
cana-1730	81	5	z	z	NOUN
cana-1730	81	6	)	)	PUNCT
cana-1730	81	7	+	+	NUM
cana-1730	81	8	θ	θ	PROPN
cana-1730	81	9	(	(	PUNCT
cana-1730	81	10	t(qan	t(qan	PROPN
cana-1730	81	11	,	,	PUNCT
cana-1730	81	12	qad	qad	PROPN
cana-1730	81	13	,	,	PUNCT
cana-1730	81	14	qad	qad	PROPN
cana-1730	81	15	)	)	PUNCT
cana-1730	81	16	)	)	PUNCT
cana-1730	82	1	≪	≪	PUNCT
cana-1730	82	2	z	z	NOUN
cana-1730	82	3	−	−	NOUN
cana-1730	82	4	θ	θ	PROPN
cana-1730	82	5	(	(	PUNCT
cana-1730	82	6	z	z	NOUN
cana-1730	82	7	)	)	PUNCT
cana-1730	82	8	+	+	NUM
cana-1730	82	9	θ	θ	PROPN
cana-1730	82	10	(	(	PUNCT
cana-1730	82	11	z	z	NOUN
cana-1730	82	12	)	)	PUNCT
cana-1730	82	13	=	=	SYM
cana-1730	83	1	z.	z.	X
cana-1730	83	2	by	by	ADP
cana-1730	83	3	induction	induction	NOUN
cana-1730	83	4	on	on	ADP
cana-1730	83	5	m	m	PROPN
cana-1730	83	6	,	,	PUNCT
cana-1730	83	7	we	we	PRON
cana-1730	83	8	conclude	conclude	VERB
cana-1730	83	9	that	that	SCONJ
cana-1730	83	10	(	(	PUNCT
cana-1730	83	11	3	3	X
cana-1730	83	12	)	)	PUNCT
cana-1730	83	13	holds	hold	VERB
cana-1730	83	14	for	for	ADP
cana-1730	83	15	all	all	DET
cana-1730	83	16	m	m	PRON
cana-1730	83	17	>	>	PUNCT
cana-1730	83	18	n	n	PRON
cana-1730	83	19	≥	≥	NOUN
cana-1730	83	20	n1	n1	NOUN
cana-1730	83	21	.	.	PUNCT
cana-1730	84	1	now	now	ADV
cana-1730	84	2	axiom	axiom	NOUN
cana-1730	84	3	(	(	PUNCT
cana-1730	84	4	t5	t5	PROPN
cana-1730	84	5	)	)	PUNCT
cana-1730	84	6	of	of	ADP
cana-1730	84	7	g	g	NOUN
cana-1730	84	8	-	-	PUNCT
cana-1730	84	9	metric	metric	ADJ
cana-1730	84	10	implies	imply	VERB
cana-1730	84	11	that	that	SCONJ
cana-1730	84	12	t(am	t(am	NOUN
cana-1730	84	13	,	,	PUNCT
cana-1730	84	14	an	an	PRON
cana-1730	84	15	,	,	PUNCT
cana-1730	84	16	aℓ	aℓ	PROPN
cana-1730	84	17	)	)	PUNCT
cana-1730	84	18	≤	≤	NUM
cana-1730	84	19	t(am	t(am	NUM
cana-1730	84	20	,	,	PUNCT
cana-1730	84	21	an	an	PRON
cana-1730	84	22	,	,	PUNCT
cana-1730	84	23	an	an	NOUN
cana-1730	84	24	)	)	PUNCT
cana-1730	84	25	+	+	NOUN
cana-1730	84	26	t(an	t(an	NUM
cana-1730	84	27	,	,	PUNCT
cana-1730	84	28	an	an	DET
cana-1730	84	29	,	,	PUNCT
cana-1730	84	30	aℓ	aℓ	PROPN
cana-1730	84	31	)	)	PUNCT
cana-1730	84	32	≪	≪	PUNCT
cana-1730	84	33	2z	2z	NOUN
cana-1730	84	34	holds	hold	VERB
cana-1730	84	35	for	for	ADP
cana-1730	84	36	m	m	PROPN
cana-1730	84	37	,	,	PUNCT
cana-1730	84	38	n	n	CCONJ
cana-1730	84	39	,	,	PUNCT
cana-1730	84	40	ℓ	ℓ	PROPN
cana-1730	84	41	≥	≥	NOUN
cana-1730	84	42	n1	n1	NOUN
cana-1730	84	43	.	.	PUNCT
cana-1730	85	1	hence	hence	ADV
cana-1730	85	2	{	{	PUNCT
cana-1730	85	3	qan	qan	PROPN
cana-1730	85	4	}	}	PUNCT
cana-1730	85	5	is	be	AUX
cana-1730	85	6	a	a	DET
cana-1730	85	7	g	g	NOUN
cana-1730	85	8	-	-	PUNCT
cana-1730	85	9	cauchy	cauchy	ADJ
cana-1730	85	10	sequence	sequence	NOUN
cana-1730	85	11	in	in	ADP
cana-1730	85	12	(	(	PUNCT
cana-1730	85	13	qs	qs	PROPN
cana-1730	85	14	,	,	PUNCT
cana-1730	85	15	t	t	PROPN
cana-1730	85	16	)	)	PUNCT
cana-1730	85	17	which	which	PRON
cana-1730	85	18	is	be	AUX
cana-1730	85	19	complete	complete	ADJ
cana-1730	85	20	by	by	ADP
cana-1730	85	21	assumption	assumption	NOUN
cana-1730	85	22	.	.	PUNCT
cana-1730	86	1	then	then	ADV
cana-1730	86	2	,	,	PUNCT
cana-1730	86	3	there	there	PRON
cana-1730	86	4	exist	exist	VERB
cana-1730	86	5	p	p	X
cana-1730	86	6	=	=	SYM
cana-1730	86	7	qq	qq	X
cana-1730	86	8	,	,	PUNCT
cana-1730	86	9	c	c	PROPN
cana-1730	86	10	∈	∈	PROPN
cana-1730	86	11	s	s	VERB
cana-1730	86	12	such	such	ADJ
cana-1730	86	13	that	that	SCONJ
cana-1730	86	14	lim	lim	PROPN
cana-1730	86	15	qan	qan	PROPN
cana-1730	87	1	=	=	PUNCT
cana-1730	88	1	p	p	X
cana-1730	88	2	=	=	PROPN
cana-1730	88	3	qc	qc	PROPN
cana-1730	88	4	.	.	PROPN
cana-1730	88	5	(	(	PUNCT
cana-1730	88	6	4	4	X
cana-1730	88	7	)	)	PUNCT
cana-1730	88	8	n→∞	n→∞	NOUN
cana-1730	88	9	since	since	SCONJ
cana-1730	88	10	{	{	PUNCT
cana-1730	88	11	qan	qan	NOUN
cana-1730	88	12	}	}	PUNCT
cana-1730	88	13	is	be	AUX
cana-1730	88	14	a	a	DET
cana-1730	88	15	non	non	ADJ
cana-1730	88	16	-	-	ADJ
cana-1730	88	17	decreasing	decrease	VERB
cana-1730	88	18	sequence	sequence	NOUN
cana-1730	88	19	and	and	CCONJ
cana-1730	88	20	s	s	NOUN
cana-1730	88	21	is	be	AUX
cana-1730	88	22	regular	regular	ADJ
cana-1730	88	23	,	,	PUNCT
cana-1730	88	24	it	it	PRON
cana-1730	88	25	follows	follow	VERB
cana-1730	88	26	from	from	ADP
cana-1730	88	27	(	(	PUNCT
cana-1730	88	28	4	4	NUM
cana-1730	88	29	)	)	PUNCT
cana-1730	88	30	that	that	PRON
cana-1730	88	31	qan	qan	PROPN
cana-1730	88	32	≤	≤	X
cana-1730	88	33	qc	qc	PROPN
cana-1730	88	34	for	for	ADP
cana-1730	88	35	all	all	DET
cana-1730	88	36	n	n	PRON
cana-1730	88	37	∈	∈	PROPN
cana-1730	88	38	n.	n.	NOUN
cana-1730	88	39	assume	assume	VERB
cana-1730	88	40	qan	qan	PROPN
cana-1730	88	41	qc	qc	PROPN
cana-1730	88	42	.	.	PUNCT
cana-1730	89	1	fix	fix	PROPN
cana-1730	89	2	z	z	PROPN
cana-1730	89	3	,	,	PUNCT
cana-1730	89	4	≪	≪	ADJ
cana-1730	89	5	z	z	NOUN
cana-1730	89	6	,	,	PUNCT
cana-1730	89	7	and	and	CCONJ
cana-1730	89	8	choose	choose	VERB
cana-1730	89	9	a	a	DET
cana-1730	89	10	natural	natural	ADJ
cana-1730	89	11	number	number	NOUN
cana-1730	89	12	n	n	ADP
cana-1730	89	13	such	such	ADJ
cana-1730	89	14	that	that	SCONJ
cana-1730	89	15	t(qan	t(qan	PROPN
cana-1730	89	16	,	,	PUNCT
cana-1730	89	17	qan	qan	PROPN
cana-1730	89	18	,	,	PUNCT
cana-1730	89	19	qc	qc	PROPN
cana-1730	89	20	)	)	PUNCT
cana-1730	89	21	≪	≪	NOUN
cana-1730	89	22	and	and	CCONJ
cana-1730	89	23	t(qan+1	t(qan+1	PROPN
cana-1730	89	24	,	,	PUNCT
cana-1730	89	25	qc	qc	PROPN
cana-1730	89	26	,	,	PUNCT
cana-1730	89	27	qc	qc	PROPN
cana-1730	89	28	)	)	PUNCT
cana-1730	89	29	≪	≪	NOUN
cana-1730	89	30	.	.	PUNCT
cana-1730	90	1	hence	hence	ADV
cana-1730	90	2	,	,	PUNCT
cana-1730	90	3	we	we	PRON
cana-1730	90	4	can	can	AUX
cana-1730	90	5	apply	apply	VERB
cana-1730	90	6	the	the	DET
cana-1730	90	7	considered	consider	VERB
cana-1730	90	8	contractive	contractive	ADJ
cana-1730	90	9	condition	condition	NOUN
cana-1730	90	10	to	to	PART
cana-1730	90	11	obtain	obtain	VERB
cana-1730	90	12	t(gc	t(gc	NUM
cana-1730	90	13	,	,	PUNCT
cana-1730	90	14	qc	qc	PROPN
cana-1730	90	15	,	,	PUNCT
cana-1730	90	16	qc	qc	PROPN
cana-1730	90	17	)	)	PUNCT
cana-1730	90	18	≤	≤	NOUN
cana-1730	90	19	t(gc	t(gc	ADV
cana-1730	90	20	,	,	PUNCT
cana-1730	90	21	gan	gan	PROPN
cana-1730	90	22	,	,	PUNCT
cana-1730	90	23	gan	gan	NOUN
cana-1730	90	24	)	)	PUNCT
cana-1730	90	25	+	+	CCONJ
cana-1730	90	26	t(gan	t(gan	PROPN
cana-1730	90	27	,	,	PUNCT
cana-1730	90	28	qc	qc	PROPN
cana-1730	90	29	,	,	PUNCT
cana-1730	90	30	qc	qc	PROPN
cana-1730	90	31	)	)	PUNCT
cana-1730	90	32	≤	≤	NUM
cana-1730	90	33	θ	θ	PROPN
cana-1730	90	34	(	(	PUNCT
cana-1730	90	35	t(qan	t(qan	PROPN
cana-1730	90	36	,	,	PUNCT
cana-1730	90	37	qan	qan	PROPN
cana-1730	90	38	,	,	PUNCT
cana-1730	90	39	qc	qc	PROPN
cana-1730	90	40	)	)	PUNCT
cana-1730	90	41	)	)	PUNCT
cana-1730	91	1	+	+	CCONJ
cana-1730	91	2	t(qan+1	t(qan+1	PROPN
cana-1730	91	3	,	,	PUNCT
cana-1730	91	4	qc	qc	PROPN
cana-1730	91	5	,	,	PUNCT
cana-1730	91	6	qc	qc	PROPN
cana-1730	91	7	)	)	PUNCT
cana-1730	91	8	(	(	PUNCT
cana-1730	91	9	by	by	ADP
cana-1730	91	10	(	(	PUNCT
cana-1730	91	11	1	1	NUM
cana-1730	91	12	)	)	PUNCT
cana-1730	91	13	)	)	PUNCT
cana-1730	92	1	<	<	X
cana-1730	93	1	t(qan	t(qan	PROPN
cana-1730	93	2	,	,	PUNCT
cana-1730	93	3	qan	qan	PROPN
cana-1730	93	4	,	,	PUNCT
cana-1730	93	5	qc	qc	PROPN
cana-1730	93	6	)	)	PUNCT
cana-1730	93	7	+	+	SYM
cana-1730	94	1	t(qan+1	t(qan+1	PROPN
cana-1730	94	2	,	,	PUNCT
cana-1730	94	3	qc	qc	PROPN
cana-1730	94	4	,	,	PUNCT
cana-1730	94	5	qc	qc	PROPN
cana-1730	94	6	)	)	PUNCT
cana-1730	94	7	≪	≪	PUNCT
cana-1730	94	8	+	+	PUNCT
cana-1730	95	1	=	=	SYM
cana-1730	95	2	z	z	NOUN
cana-1730	95	3	.	.	PUNCT
cana-1730	96	1	since	since	SCONJ
cana-1730	96	2	z	z	PROPN
cana-1730	96	3	∈	∈	PROPN
cana-1730	96	4	in	in	ADP
cana-1730	96	5	a	a	DET
cana-1730	96	6	is	be	AUX
cana-1730	96	7	arbitrary	arbitrary	ADJ
cana-1730	96	8	,	,	PUNCT
cana-1730	96	9	it	it	PRON
cana-1730	96	10	follows	follow	VERB
cana-1730	96	11	that	that	SCONJ
cana-1730	96	12	t(gc	t(gc	ADV
cana-1730	96	13	,	,	PUNCT
cana-1730	96	14	qc	qc	PROPN
cana-1730	96	15	,	,	PUNCT
cana-1730	96	16	qc	qc	PROPN
cana-1730	96	17	)	)	PUNCT
cana-1730	96	18	=	=	PUNCT
cana-1730	96	19	which	which	PRON
cana-1730	96	20	by	by	ADP
cana-1730	96	21	axiom	axiom	NOUN
cana-1730	96	22	(	(	PUNCT
cana-1730	96	23	t2	t2	NOUN
cana-1730	96	24	)	)	PUNCT
cana-1730	96	25	implies	imply	VERB
cana-1730	96	26	that	that	SCONJ
cana-1730	96	27	gc	gc	PROPN
cana-1730	96	28	=	=	PROPN
cana-1730	96	29	qc	qc	PROPN
cana-1730	96	30	.	.	PUNCT
cana-1730	97	1	then	then	ADV
cana-1730	97	2	c	c	PROPN
cana-1730	97	3	is	be	AUX
cana-1730	97	4	a	a	DET
cana-1730	97	5	coincidence	coincidence	NOUN
cana-1730	97	6	point	point	NOUN
cana-1730	97	7	for	for	SCONJ
cana-1730	97	8	the	the	DET
cana-1730	97	9	mappings	mapping	NOUN
cana-1730	97	10	g	g	NOUN
cana-1730	97	11	and	and	CCONJ
cana-1730	97	12	q.	q.	PROPN
cana-1730	97	13	communications	communication	NOUN
cana-1730	97	14	on	on	ADP
cana-1730	97	15	applied	apply	VERB
cana-1730	97	16	nonlinear	nonlinear	ADJ
cana-1730	97	17	analysis	analysis	NOUN
cana-1730	97	18	issn	issn	NOUN
cana-1730	97	19	:	:	PUNCT
cana-1730	97	20	1074	1074	NUM
cana-1730	97	21	-	-	PUNCT
cana-1730	97	22	133x	133x	NUM
cana-1730	97	23	vol	vol	NOUN
cana-1730	97	24	32	32	NUM
cana-1730	97	25	no	no	NOUN
cana-1730	97	26	.	.	NOUN
cana-1730	97	27	2	2	NUM
cana-1730	97	28	(	(	PUNCT
cana-1730	97	29	2025	2025	NUM
cana-1730	97	30	)	)	PUNCT
cana-1730	97	31	163	163	NUM
cana-1730	98	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1730	98	2	corollary	corollary	NOUN
cana-1730	98	3	1	1	NUM
cana-1730	98	4	let	let	VERB
cana-1730	98	5	(	(	PUNCT
cana-1730	98	6	s	s	NOUN
cana-1730	98	7	,	,	PUNCT
cana-1730	98	8	≤	≤	NUM
cana-1730	98	9	)	)	PUNCT
cana-1730	98	10	be	be	VERB
cana-1730	98	11	a	a	DET
cana-1730	98	12	partially	partially	ADV
cana-1730	98	13	ordered	order	VERB
cana-1730	98	14	set	set	NOUN
cana-1730	98	15	,	,	PUNCT
cana-1730	98	16	a	a	PRON
cana-1730	98	17	be	be	AUX
cana-1730	98	18	an	an	DET
cana-1730	98	19	order	order	NOUN
cana-1730	98	20	cone	cone	NOUN
cana-1730	98	21	and	and	CCONJ
cana-1730	98	22	suppose	suppose	VERB
cana-1730	98	23	there	there	PRON
cana-1730	98	24	is	be	VERB
cana-1730	98	25	a	a	DET
cana-1730	98	26	metric	metric	ADJ
cana-1730	98	27	t	t	NOUN
cana-1730	98	28	on	on	ADP
cana-1730	98	29	s	s	PRON
cana-1730	98	30	such	such	ADJ
cana-1730	98	31	that	that	SCONJ
cana-1730	98	32	(	(	PUNCT
cana-1730	98	33	s	s	X
cana-1730	98	34	,	,	PUNCT
cana-1730	98	35	t	t	PROPN
cana-1730	98	36	)	)	PUNCT
cana-1730	98	37	is	be	AUX
cana-1730	98	38	a	a	DET
cana-1730	98	39	complete	complete	ADJ
cana-1730	98	40	cone	cone	NOUN
cana-1730	98	41	gmetric	gmetric	ADJ
cana-1730	98	42	space	space	NOUN
cana-1730	98	43	.	.	PUNCT
cana-1730	99	1	let	let	VERB
cana-1730	99	2	g	g	NOUN
cana-1730	99	3	:	:	PUNCT
cana-1730	99	4	s	s	AUX
cana-1730	99	5	→	→	SYM
cana-1730	99	6	s	s	PART
cana-1730	99	7	be	be	AUX
cana-1730	99	8	a	a	DET
cana-1730	99	9	mapping	mapping	NOUN
cana-1730	99	10	such	such	ADJ
cana-1730	99	11	that	that	SCONJ
cana-1730	99	12	t(ga	t(ga	NUM
cana-1730	99	13	,	,	PUNCT
cana-1730	99	14	gb	gb	NOUN
cana-1730	99	15	,	,	PUNCT
cana-1730	99	16	gc	gc	PROPN
cana-1730	99	17	)	)	PUNCT
cana-1730	99	18	≤	≤	NUM
cana-1730	99	19	θ	θ	PROPN
cana-1730	99	20	(	(	PUNCT
cana-1730	99	21	t(a	t(a	NOUN
cana-1730	99	22	,	,	PUNCT
cana-1730	99	23	b	b	NOUN
cana-1730	99	24	,	,	PUNCT
cana-1730	99	25	c	c	NOUN
cana-1730	99	26	)	)	PUNCT
cana-1730	99	27	)	)	PUNCT
cana-1730	99	28	holds	hold	VERB
cana-1730	99	29	for	for	ADP
cana-1730	99	30	all	all	DET
cana-1730	99	31	a	a	DET
cana-1730	99	32	,	,	PUNCT
cana-1730	99	33	b	b	NOUN
cana-1730	99	34	,	,	PUNCT
cana-1730	99	35	c	c	PROPN
cana-1730	99	36	∈	∈	PROPN
cana-1730	99	37	s	s	VERB
cana-1730	99	38	with	with	ADP
cana-1730	99	39	a	a	DET
cana-1730	99	40	≥	≥	NOUN
cana-1730	99	41	b	b	NOUN
cana-1730	99	42	≥	≥	X
cana-1730	99	43	c	c	NOUN
cana-1730	99	44	where	where	SCONJ
cana-1730	99	45	θ	θ	PROPN
cana-1730	99	46	is	be	AUX
cana-1730	99	47	a	a	DET
cana-1730	99	48	θ	θ	PROPN
cana-1730	99	49	-map	-map	NUM
cana-1730	99	50	.	.	PUNCT
cana-1730	100	1	we	we	PRON
cana-1730	100	2	suppose	suppose	VERB
cana-1730	100	3	the	the	DET
cana-1730	100	4	following	follow	VERB
cana-1730	100	5	:	:	PUNCT
cana-1730	100	6	(	(	PUNCT
cana-1730	100	7	i	i	NOUN
cana-1730	100	8	)	)	PUNCT
cana-1730	100	9	ga	ga	PROPN
cana-1730	100	10	≤	≤	PROPN
cana-1730	100	11	g	g	PROPN
cana-1730	100	12	(	(	PUNCT
cana-1730	100	13	ga	ga	PROPN
cana-1730	100	14	)	)	PUNCT
cana-1730	100	15	for	for	ADP
cana-1730	100	16	all	all	DET
cana-1730	100	17	a	a	DET
cana-1730	100	18	∈	∈	ADJ
cana-1730	100	19	s	s	NOUN
cana-1730	100	20	;	;	PUNCT
cana-1730	100	21	(	(	PUNCT
cana-1730	100	22	ii	ii	NOUN
cana-1730	100	23	)	)	PUNCT
cana-1730	100	24	s	s	VERB
cana-1730	100	25	is	be	AUX
cana-1730	100	26	regular	regular	ADJ
cana-1730	100	27	.	.	PUNCT
cana-1730	101	1	then	then	ADV
cana-1730	101	2	g	g	PROPN
cana-1730	101	3	has	have	VERB
cana-1730	101	4	a	a	DET
cana-1730	101	5	fixed	fix	VERB
cana-1730	101	6	point	point	NOUN
cana-1730	101	7	.	.	PUNCT
cana-1730	102	1	theorem	theorem	ADJ
cana-1730	102	2	2	2	NUM
cana-1730	102	3	let	let	VERB
cana-1730	102	4	(	(	PUNCT
cana-1730	102	5	s	s	NOUN
cana-1730	102	6	,	,	PUNCT
cana-1730	102	7	≤	≤	NUM
cana-1730	102	8	)	)	PUNCT
cana-1730	102	9	be	be	AUX
cana-1730	102	10	a	a	DET
cana-1730	102	11	partially	partially	ADV
cana-1730	102	12	ordered	order	VERB
cana-1730	102	13	set	set	NOUN
cana-1730	102	14	,	,	PUNCT
cana-1730	102	15	a	a	PRON
cana-1730	102	16	be	be	AUX
cana-1730	102	17	an	an	DET
cana-1730	102	18	order	order	NOUN
cana-1730	102	19	cone	cone	NOUN
cana-1730	102	20	and	and	CCONJ
cana-1730	102	21	suppose	suppose	VERB
cana-1730	102	22	there	there	PRON
cana-1730	102	23	is	be	VERB
cana-1730	102	24	a	a	DET
cana-1730	102	25	cone	cone	NOUN
cana-1730	102	26	gmetric	gmetric	PROPN
cana-1730	102	27	t	t	PROPN
cana-1730	102	28	on	on	ADP
cana-1730	102	29	s	s	PRON
cana-1730	102	30	such	such	ADJ
cana-1730	102	31	that	that	SCONJ
cana-1730	102	32	(	(	PUNCT
cana-1730	102	33	s	s	X
cana-1730	102	34	,	,	PUNCT
cana-1730	102	35	t	t	PROPN
cana-1730	102	36	)	)	PUNCT
cana-1730	102	37	is	be	AUX
cana-1730	102	38	a	a	DET
cana-1730	102	39	complete	complete	ADJ
cana-1730	102	40	cone	cone	NOUN
cana-1730	102	41	gmetric	gmetric	ADJ
cana-1730	102	42	space	space	NOUN
cana-1730	102	43	.	.	PUNCT
cana-1730	103	1	let	let	VERB
cana-1730	103	2	g	g	NOUN
cana-1730	103	3	,	,	PUNCT
cana-1730	103	4	q	q	X
cana-1730	103	5	:	:	PUNCT
cana-1730	103	6	s	s	X
cana-1730	103	7	→	→	SYM
cana-1730	103	8	s	s	AUX
cana-1730	103	9	be	be	AUX
cana-1730	103	10	nondecreasing	nondecrease	VERB
cana-1730	103	11	mappings	mapping	NOUN
cana-1730	103	12	such	such	ADJ
cana-1730	103	13	that	that	PRON
cana-1730	103	14	for	for	ADP
cana-1730	103	15	all	all	DET
cana-1730	103	16	a	a	DET
cana-1730	103	17	,	,	PUNCT
cana-1730	103	18	b	b	NOUN
cana-1730	103	19	,	,	PUNCT
cana-1730	103	20	c	c	PROPN
cana-1730	103	21	∈	∈	PROPN
cana-1730	103	22	s	s	VERB
cana-1730	103	23	with	with	ADP
cana-1730	103	24	qa	qa	PROPN
cana-1730	103	25	≥	≥	PROPN
cana-1730	104	1	qq	qq	X
cana-1730	104	2	≥	≥	PROPN
cana-1730	104	3	qc	qc	PROPN
cana-1730	104	4	there	there	PRON
cana-1730	104	5	exists	exist	VERB
cana-1730	104	6	θ(a	θ(a	PROPN
cana-1730	104	7	,	,	PUNCT
cana-1730	104	8	b	b	NOUN
cana-1730	104	9	,	,	PUNCT
cana-1730	104	10	c	c	NOUN
cana-1730	104	11	)	)	PUNCT
cana-1730	104	12	∈	∈	PROPN
cana-1730	104	13	{	{	PUNCT
cana-1730	104	14	t(qa	t(qa	PROPN
cana-1730	104	15	,	,	PUNCT
cana-1730	104	16	qb	qb	PROPN
cana-1730	104	17	,	,	PUNCT
cana-1730	104	18	qc	qc	PROPN
cana-1730	104	19	)	)	PUNCT
cana-1730	104	20	,	,	PUNCT
cana-1730	104	21	t(qa	t(qa	PROPN
cana-1730	104	22	,	,	PUNCT
cana-1730	104	23	ga	ga	PROPN
cana-1730	104	24	,	,	PUNCT
cana-1730	104	25	ga	ga	PROPN
cana-1730	104	26	)	)	PUNCT
cana-1730	104	27	,	,	PUNCT
cana-1730	104	28	t(qb	t(qb	ADV
cana-1730	104	29	,	,	PUNCT
cana-1730	104	30	gb	gb	PRON
cana-1730	104	31	,	,	PUNCT
cana-1730	104	32	gb	gb	PROPN
cana-1730	104	33	)	)	PUNCT
cana-1730	104	34	,	,	PUNCT
cana-1730	104	35	t(ga	t(ga	NOUN
cana-1730	104	36	,	,	PUNCT
cana-1730	104	37	gb	gb	PROPN
cana-1730	104	38	,	,	PUNCT
cana-1730	104	39	gc	gc	PROPN
cana-1730	104	40	)	)	PUNCT
cana-1730	104	41	}	}	PUNCT
cana-1730	104	42	such	such	ADJ
cana-1730	104	43	that	that	SCONJ
cana-1730	104	44	t(ga	t(ga	NUM
cana-1730	104	45	,	,	PUNCT
cana-1730	104	46	gb	gb	NOUN
cana-1730	104	47	,	,	PUNCT
cana-1730	104	48	gc	gc	PROPN
cana-1730	104	49	)	)	PUNCT
cana-1730	104	50	≤	≤	NUM
cana-1730	104	51	θ	θ	PROPN
cana-1730	104	52	(	(	PUNCT
cana-1730	104	53	θ(a	θ(a	PROPN
cana-1730	104	54	,	,	PUNCT
cana-1730	104	55	b	b	NOUN
cana-1730	104	56	,	,	PUNCT
cana-1730	104	57	c	c	NOUN
cana-1730	104	58	)	)	PUNCT
cana-1730	104	59	)	)	PUNCT
cana-1730	104	60	,	,	PUNCT
cana-1730	104	61	where	where	SCONJ
cana-1730	104	62	θ	θ	PROPN
cana-1730	104	63	is	be	AUX
cana-1730	104	64	a	a	DET
cana-1730	104	65	θ	θ	PROPN
cana-1730	104	66	-map	-map	NUM
cana-1730	104	67	.	.	PUNCT
cana-1730	105	1	we	we	PRON
cana-1730	105	2	suppose	suppose	VERB
cana-1730	105	3	the	the	DET
cana-1730	105	4	following	follow	VERB
cana-1730	105	5	:	:	PUNCT
cana-1730	105	6	(	(	PUNCT
cana-1730	105	7	i	i	NOUN
cana-1730	105	8	)	)	PUNCT
cana-1730	105	9	g	g	NOUN
cana-1730	105	10	is	be	AUX
cana-1730	105	11	weakly	weakly	ADV
cana-1730	105	12	increasing	increase	VERB
cana-1730	105	13	with	with	ADP
cana-1730	105	14	respect	respect	NOUN
cana-1730	105	15	to	to	ADP
cana-1730	105	16	q	q	NOUN
cana-1730	105	17	,	,	PUNCT
cana-1730	105	18	(	(	PUNCT
cana-1730	105	19	ii	ii	NOUN
cana-1730	105	20	)	)	PUNCT
cana-1730	105	21	s	s	VERB
cana-1730	105	22	is	be	AUX
cana-1730	105	23	regular	regular	ADJ
cana-1730	105	24	.	.	PUNCT
cana-1730	106	1	then	then	ADV
cana-1730	106	2	g	g	PROPN
cana-1730	106	3	and	and	CCONJ
cana-1730	106	4	q	q	PROPN
cana-1730	106	5	have	have	VERB
cana-1730	106	6	a	a	DET
cana-1730	106	7	coincidence	coincidence	NOUN
cana-1730	106	8	point	point	NOUN
cana-1730	106	9	.	.	PUNCT
cana-1730	107	1	proof	proof	NOUN
cana-1730	107	2	.	.	PUNCT
cana-1730	108	1	let	let	VERB
cana-1730	108	2	a0	a0	PROPN
cana-1730	108	3	be	be	AUX
cana-1730	108	4	an	an	DET
cana-1730	108	5	arbitrary	arbitrary	ADJ
cana-1730	108	6	point	point	NOUN
cana-1730	108	7	in	in	ADP
cana-1730	108	8	s	s	PRON
cana-1730	108	9	.	.	PUNCT
cana-1730	109	1	since	since	SCONJ
cana-1730	109	2	gs	gs	PROPN
cana-1730	109	3	⊆	⊆	NUM
cana-1730	109	4	qs	qs	NOUN
cana-1730	109	5	(	(	PUNCT
cana-1730	109	6	by	by	ADP
cana-1730	109	7	definition	definition	NOUN
cana-1730	109	8	4	4	NUM
cana-1730	109	9	)	)	PUNCT
cana-1730	109	10	,	,	PUNCT
cana-1730	109	11	we	we	PRON
cana-1730	109	12	can	can	AUX
cana-1730	109	13	construct	construct	VERB
cana-1730	109	14	a	a	DET
cana-1730	109	15	sequence	sequence	NOUN
cana-1730	109	16	{	{	PUNCT
cana-1730	109	17	an	an	NOUN
cana-1730	109	18	}	}	PUNCT
cana-1730	109	19	in	in	ADP
cana-1730	109	20	s	s	PRON
cana-1730	109	21	defined	define	VERB
cana-1730	109	22	by	by	ADP
cana-1730	109	23	:	:	PUNCT
cana-1730	109	24	san+1	san+1	PROPN
cana-1730	109	25	=	=	SYM
cana-1730	109	26	gan	gan	PROPN
cana-1730	109	27	,	,	PUNCT
cana-1730	109	28	∀	∀	NOUN
cana-1730	109	29	n	n	PRON
cana-1730	109	30	∈	∈	PROPN
cana-1730	109	31	n.	n.	NOUN
cana-1730	109	32	now	now	ADV
cana-1730	109	33	,	,	PUNCT
cana-1730	109	34	since	since	SCONJ
cana-1730	109	35	a1	a1	NOUN
cana-1730	109	36	∈	∈	NOUN
cana-1730	109	37	q−1(ga0	q−1(ga0	NOUN
cana-1730	109	38	)	)	PUNCT
cana-1730	109	39	and	and	CCONJ
cana-1730	109	40	a2	a2	PROPN
cana-1730	109	41	∈	∈	PROPN
cana-1730	109	42	q−1(ga1	q−1(ga1	NOUN
cana-1730	109	43	)	)	PUNCT
cana-1730	109	44	,	,	PUNCT
cana-1730	109	45	using	use	VERB
cana-1730	109	46	that	that	SCONJ
cana-1730	109	47	g	g	PROPN
cana-1730	109	48	is	be	AUX
cana-1730	109	49	weakly	weakly	ADV
cana-1730	109	50	increasing	increase	VERB
cana-1730	109	51	with	with	ADP
cana-1730	109	52	respect	respect	NOUN
cana-1730	109	53	to	to	ADP
cana-1730	109	54	q	q	X
cana-1730	109	55	,	,	PUNCT
cana-1730	109	56	we	we	PRON
cana-1730	109	57	obtain	obtain	VERB
cana-1730	109	58	that	that	DET
cana-1730	109	59	qa1	qa1	NOUN
cana-1730	110	1	=	=	NOUN
cana-1730	110	2	ga0	ga0	X
cana-1730	110	3	≤	≤	NUM
cana-1730	110	4	ga1	ga1	NOUN
cana-1730	110	5	=	=	SYM
cana-1730	110	6	qa2	qa2	PROPN
cana-1730	110	7	≤	≤	ADJ
cana-1730	110	8	ga2	ga2	NOUN
cana-1730	110	9	=	=	SYM
cana-1730	110	10	qa3	qa3	PROPN
cana-1730	110	11	.	.	PUNCT
cana-1730	111	1	continuing	continue	VERB
cana-1730	111	2	this	this	DET
cana-1730	111	3	process	process	NOUN
cana-1730	111	4	,	,	PUNCT
cana-1730	111	5	we	we	PRON
cana-1730	111	6	get	get	VERB
cana-1730	111	7	that	that	DET
cana-1730	111	8	qa1	qa1	PROPN
cana-1730	111	9	≤	≤	X
cana-1730	111	10	qa2	qa2	VERB
cana-1730	111	11	≤	≤	NUM
cana-1730	112	1	qa3	qa3	CCONJ
cana-1730	112	2	≤	≤	NOUN
cana-1730	112	3	·	·	PUNCT
cana-1730	112	4	·	·	PUNCT
cana-1730	113	1	·	·	PUNCT
cana-1730	113	2	≤	≤	NUM
cana-1730	113	3	qan	qan	PROPN
cana-1730	113	4	≤	≤	NUM
cana-1730	113	5	qan+1	qan+1	NOUN
cana-1730	113	6	≤	≤	NOUN
cana-1730	113	7	·	·	PUNCT
cana-1730	113	8	·	·	PUNCT
cana-1730	113	9	·	·	PUNCT
cana-1730	113	10	.	.	PUNCT
cana-1730	114	1	if	if	SCONJ
cana-1730	114	2	there	there	PRON
cana-1730	114	3	exists	exist	VERB
cana-1730	114	4	n0	n0	PROPN
cana-1730	114	5	∈	∈	PROPN
cana-1730	114	6	{	{	PUNCT
cana-1730	114	7	1	1	NUM
cana-1730	114	8	,	,	PUNCT
cana-1730	114	9	2	2	NUM
cana-1730	114	10	,	,	PUNCT
cana-1730	114	11	.	.	PUNCT
cana-1730	114	12	.	.	PUNCT
cana-1730	115	1	.	.	PUNCT
cana-1730	115	2	}	}	PUNCT
cana-1730	116	1	such	such	ADJ
cana-1730	116	2	that	that	SCONJ
cana-1730	116	3	θ(an0	θ(an0	PROPN
cana-1730	116	4	,	,	PUNCT
cana-1730	116	5	an0	an0	PROPN
cana-1730	116	6	−1	−1	PROPN
cana-1730	116	7	,	,	PUNCT
cana-1730	116	8	an0	an0	PROPN
cana-1730	116	9	−1	−1	NOUN
cana-1730	116	10	)	)	PUNCT
cana-1730	117	1	=	=	SYM
cana-1730	117	2	θ	θ	NOUN
cana-1730	117	3	then	then	ADV
cana-1730	117	4	it	it	PRON
cana-1730	117	5	is	be	AUX
cana-1730	117	6	clear	clear	ADJ
cana-1730	117	7	that	that	SCONJ
cana-1730	117	8	qan0	qan0	PROPN
cana-1730	117	9	−1	−1	NOUN
cana-1730	117	10	=	=	PUNCT
cana-1730	117	11	qan0	qan0	PROPN
cana-1730	117	12	=	=	PUNCT
cana-1730	117	13	gan0	gan0	VERB
cana-1730	117	14	−1	−1	NOUN
cana-1730	118	1	and	and	CCONJ
cana-1730	118	2	so	so	ADV
cana-1730	118	3	we	we	PRON
cana-1730	118	4	are	be	AUX
cana-1730	118	5	finished	finish	VERB
cana-1730	118	6	.	.	PUNCT
cana-1730	119	1	now	now	ADV
cana-1730	119	2	we	we	PRON
cana-1730	119	3	can	can	AUX
cana-1730	119	4	suppose	suppose	VERB
cana-1730	119	5	θ(an	θ(an	PROPN
cana-1730	119	6	,	,	PUNCT
cana-1730	119	7	an−1	an−1	ADJ
cana-1730	119	8	,	,	PUNCT
cana-1730	119	9	an−1	an−1	ADJ
cana-1730	119	10	)	)	PUNCT
cana-1730	119	11	>	>	PUNCT
cana-1730	119	12	for	for	ADP
cana-1730	119	13	all	all	DET
cana-1730	119	14	n	n	PRON
cana-1730	119	15	≥	≥	NUM
cana-1730	119	16	1	1	NUM
cana-1730	119	17	.	.	PUNCT
cana-1730	119	18	assume	assume	VERB
cana-1730	119	19	qan	qan	PROPN
cana-1730	119	20	qan−1	qan−1	PROPN
cana-1730	119	21	,	,	PUNCT
cana-1730	119	22	for	for	ADP
cana-1730	119	23	each	each	DET
cana-1730	119	24	n	n	PRON
cana-1730	119	25	∈	∈	PROPN
cana-1730	119	26	n.	n.	NOUN
cana-1730	119	27	thus	thus	ADV
cana-1730	119	28	for	for	ADP
cana-1730	119	29	n	n	PRON
cana-1730	119	30	∈	∈	PROPN
cana-1730	119	31	n	n	CCONJ
cana-1730	119	32	,	,	PUNCT
cana-1730	119	33	we	we	PRON
cana-1730	119	34	have	have	VERB
cana-1730	119	35	communications	communication	NOUN
cana-1730	119	36	on	on	ADP
cana-1730	119	37	applied	apply	VERB
cana-1730	119	38	nonlinear	nonlinear	ADJ
cana-1730	119	39	analysis	analysis	NOUN
cana-1730	119	40	issn	issn	NOUN
cana-1730	119	41	:	:	PUNCT
cana-1730	119	42	1074	1074	NUM
cana-1730	119	43	-	-	PUNCT
cana-1730	119	44	133x	133x	NUM
cana-1730	119	45	vol	vol	NOUN
cana-1730	119	46	32	32	NUM
cana-1730	120	1	no	no	NOUN
cana-1730	120	2	.	.	NOUN
cana-1730	120	3	2	2	NUM
cana-1730	120	4	(	(	PUNCT
cana-1730	120	5	2025	2025	NUM
cana-1730	120	6	)	)	PUNCT
cana-1730	120	7	164	164	NUM
cana-1730	120	8	https://internationalpubls.com	https://internationalpubls.com	X
cana-1730	120	9	t(qan	t(qan	ADV
cana-1730	120	10	,	,	PUNCT
cana-1730	120	11	qan+1	qan+1	NOUN
cana-1730	120	12	,	,	PUNCT
cana-1730	120	13	qan+1	qan+1	NOUN
cana-1730	120	14	)	)	PUNCT
cana-1730	120	15	=	=	SYM
cana-1730	120	16	t(gan−1	t(gan−1	PROPN
cana-1730	120	17	,	,	PUNCT
cana-1730	120	18	gan	gan	PROPN
cana-1730	120	19	,	,	PUNCT
cana-1730	120	20	gan	gan	NOUN
cana-1730	120	21	)	)	PUNCT
cana-1730	120	22	≤	≤	NUM
cana-1730	120	23	θ	θ	PROPN
cana-1730	120	24	(	(	PUNCT
cana-1730	120	25	θ(an−1	θ(an−1	PROPN
cana-1730	120	26	,	,	PUNCT
cana-1730	120	27	an	an	PRON
cana-1730	120	28	,	,	PUNCT
cana-1730	120	29	an	an	NOUN
cana-1730	120	30	)	)	PUNCT
cana-1730	120	31	)	)	PUNCT
cana-1730	120	32	where	where	SCONJ
cana-1730	120	33	θ(an−1	θ(an−1	ADP
cana-1730	120	34	,	,	PUNCT
cana-1730	120	35	an	an	PRON
cana-1730	120	36	,	,	PUNCT
cana-1730	120	37	an	an	PRON
cana-1730	120	38	)	)	PUNCT
cana-1730	120	39	∈	∈	PROPN
cana-1730	120	40	{	{	PUNCT
cana-1730	120	41	t(qan−1	t(qan−1	PROPN
cana-1730	120	42	,	,	PUNCT
cana-1730	120	43	qan	qan	PROPN
cana-1730	120	44	,	,	PUNCT
cana-1730	120	45	qan	qan	PROPN
cana-1730	120	46	)	)	PUNCT
cana-1730	120	47	,	,	PUNCT
cana-1730	120	48	t(qan−1	t(qan−1	PROPN
cana-1730	120	49	,	,	PUNCT
cana-1730	120	50	gan−1	gan−1	PROPN
cana-1730	120	51	,	,	PUNCT
cana-1730	120	52	gan−1	gan−1	PROPN
cana-1730	120	53	)	)	PUNCT
cana-1730	120	54	,	,	PUNCT
cana-1730	120	55	t(qan	t(qan	ADV
cana-1730	120	56	,	,	PUNCT
cana-1730	120	57	gan	gan	PROPN
cana-1730	120	58	,	,	PUNCT
cana-1730	120	59	gan	gan	PROPN
cana-1730	120	60	)	)	PUNCT
cana-1730	120	61	,	,	PUNCT
cana-1730	120	62	t(gan−1	t(gan−1	PROPN
cana-1730	120	63	,	,	PUNCT
cana-1730	120	64	qan	qan	PROPN
cana-1730	120	65	,	,	PUNCT
cana-1730	120	66	qan	qan	PROPN
cana-1730	120	67	)	)	PUNCT
cana-1730	120	68	}	}	PUNCT
cana-1730	120	69	=	=	SYM
cana-1730	120	70	{	{	PUNCT
cana-1730	120	71	t(qan−1	t(qan−1	PROPN
cana-1730	120	72	,	,	PUNCT
cana-1730	120	73	qan	qan	PROPN
cana-1730	120	74	,	,	PUNCT
cana-1730	120	75	qan	qan	PROPN
cana-1730	120	76	)	)	PUNCT
cana-1730	120	77	,	,	PUNCT
cana-1730	120	78	t(qan−1	t(qan−1	PROPN
cana-1730	120	79	,	,	PUNCT
cana-1730	120	80	qan	qan	PROPN
cana-1730	120	81	,	,	PUNCT
cana-1730	120	82	qan	qan	PROPN
cana-1730	120	83	)	)	PUNCT
cana-1730	120	84	,	,	PUNCT
cana-1730	120	85	t(qan	t(qan	ADV
cana-1730	120	86	,	,	PUNCT
cana-1730	120	87	qan+1	qan+1	NOUN
cana-1730	120	88	,	,	PUNCT
cana-1730	120	89	qan+1	qan+1	NOUN
cana-1730	120	90	)	)	PUNCT
cana-1730	120	91	,	,	PUNCT
cana-1730	120	92	t(qan	t(qan	PROPN
cana-1730	120	93	,	,	PUNCT
cana-1730	120	94	qan	qan	PROPN
cana-1730	120	95	,	,	PUNCT
cana-1730	120	96	qan	qan	PROPN
cana-1730	120	97	)	)	PUNCT
cana-1730	120	98	}	}	PUNCT
cana-1730	120	99	=	=	SYM
cana-1730	120	100	{	{	PUNCT
cana-1730	120	101	t(qan−1	t(qan−1	PROPN
cana-1730	120	102	,	,	PUNCT
cana-1730	120	103	qan	qan	PROPN
cana-1730	120	104	,	,	PUNCT
cana-1730	120	105	qan	qan	PROPN
cana-1730	120	106	)	)	PUNCT
cana-1730	120	107	,	,	PUNCT
cana-1730	120	108	t(qan	t(qan	ADV
cana-1730	120	109	,	,	PUNCT
cana-1730	120	110	qan+1	qan+1	NOUN
cana-1730	120	111	,	,	PUNCT
cana-1730	120	112	qan+1	qan+1	NOUN
cana-1730	120	113	)	)	PUNCT
cana-1730	120	114	,	,	PUNCT
cana-1730	120	115	}	}	PUNCT
cana-1730	120	116	.	.	PUNCT
cana-1730	121	1	•	•	NOUN
cana-1730	121	2	if	if	SCONJ
cana-1730	121	3	θ(an−1	θ(an−1	NOUN
cana-1730	121	4	,	,	PUNCT
cana-1730	121	5	an	an	PRON
cana-1730	121	6	,	,	PUNCT
cana-1730	121	7	an	an	NOUN
cana-1730	121	8	)	)	PUNCT
cana-1730	121	9	=	=	SYM
cana-1730	122	1	t(qan	t(qan	ADJ
cana-1730	122	2	,	,	PUNCT
cana-1730	122	3	qan+1	qan+1	NOUN
cana-1730	122	4	,	,	PUNCT
cana-1730	122	5	qan+1	qan+1	NOUN
cana-1730	122	6	)	)	PUNCT
cana-1730	122	7	,	,	PUNCT
cana-1730	122	8	then	then	ADV
cana-1730	122	9	t(qan	t(qan	ADV
cana-1730	122	10	,	,	PUNCT
cana-1730	122	11	qan+1	qan+1	NOUN
cana-1730	122	12	,	,	PUNCT
cana-1730	122	13	qan+1	qan+1	NOUN
cana-1730	122	14	)	)	PUNCT
cana-1730	123	1	≤	≤	NUM
cana-1730	123	2	θ	θ	PROPN
cana-1730	123	3	(	(	PUNCT
cana-1730	123	4	t(qan	t(qan	ADV
cana-1730	123	5	,	,	PUNCT
cana-1730	123	6	qan+1	qan+1	NOUN
cana-1730	123	7	,	,	PUNCT
cana-1730	123	8	qan+1	qan+1	NOUN
cana-1730	123	9	)	)	PUNCT
cana-1730	123	10	)	)	PUNCT
cana-1730	123	11	and	and	CCONJ
cana-1730	123	12	by	by	ADP
cana-1730	123	13	the	the	DET
cana-1730	123	14	property	property	NOUN
cana-1730	123	15	of	of	ADP
cana-1730	123	16	θ	θ	NOUN
cana-1730	123	17	we	we	PRON
cana-1730	123	18	have	have	VERB
cana-1730	123	19	t(qan	t(qan	ADV
cana-1730	123	20	,	,	PUNCT
cana-1730	123	21	qan+1	qan+1	NOUN
cana-1730	123	22	,	,	PUNCT
cana-1730	123	23	qan+1	qan+1	NOUN
cana-1730	123	24	)	)	PUNCT
cana-1730	123	25	<	<	X
cana-1730	123	26	t(qan	t(qan	ADJ
cana-1730	123	27	,	,	PUNCT
cana-1730	123	28	qan+1	qan+1	NOUN
cana-1730	123	29	,	,	PUNCT
cana-1730	123	30	qan+1	qan+1	NOUN
cana-1730	123	31	)	)	PUNCT
cana-1730	123	32	which	which	PRON
cana-1730	123	33	is	be	AUX
cana-1730	123	34	impossible	impossible	ADJ
cana-1730	123	35	.	.	PUNCT
cana-1730	124	1	•	•	INTJ
cana-1730	124	2	if	if	SCONJ
cana-1730	124	3	θ(an−1	θ(an−1	NOUN
cana-1730	124	4	,	,	PUNCT
cana-1730	124	5	an	an	PRON
cana-1730	124	6	,	,	PUNCT
cana-1730	124	7	an	an	NOUN
cana-1730	124	8	)	)	PUNCT
cana-1730	124	9	=	=	SYM
cana-1730	124	10	,	,	PUNCT
cana-1730	124	11	then	then	ADV
cana-1730	124	12	t(qan	t(qan	ADV
cana-1730	124	13	,	,	PUNCT
cana-1730	124	14	qan+1	qan+1	NOUN
cana-1730	124	15	,	,	PUNCT
cana-1730	124	16	qan+1	qan+1	NOUN
cana-1730	124	17	)	)	PUNCT
cana-1730	124	18	≤	≤	NOUN
cana-1730	124	19	(	(	PUNCT
cana-1730	124	20	θ	θ	NOUN
cana-1730	124	21	)	)	PUNCT
cana-1730	125	1	<	<	X
cana-1730	125	2	which	which	PRON
cana-1730	125	3	is	be	AUX
cana-1730	125	4	a	a	DET
cana-1730	125	5	contradiction	contradiction	NOUN
cana-1730	125	6	.	.	PUNCT
cana-1730	126	1	therefore	therefore	ADV
cana-1730	126	2	,	,	PUNCT
cana-1730	126	3	θ(an−1	θ(an−1	PROPN
cana-1730	126	4	,	,	PUNCT
cana-1730	126	5	an	an	PRON
cana-1730	126	6	,	,	PUNCT
cana-1730	126	7	an	an	NOUN
cana-1730	126	8	)	)	PUNCT
cana-1730	126	9	=	=	SYM
cana-1730	126	10	t(qan−1	t(qan−1	PROPN
cana-1730	126	11	,	,	PUNCT
cana-1730	126	12	qan	qan	PROPN
cana-1730	126	13	,	,	PUNCT
cana-1730	126	14	qan	qan	PROPN
cana-1730	126	15	)	)	PUNCT
cana-1730	126	16	,	,	PUNCT
cana-1730	126	17	and	and	CCONJ
cana-1730	126	18	then	then	ADV
cana-1730	126	19	t(qan	t(qan	ADV
cana-1730	126	20	,	,	PUNCT
cana-1730	126	21	qan+1	qan+1	NOUN
cana-1730	126	22	,	,	PUNCT
cana-1730	126	23	qan+1	qan+1	NOUN
cana-1730	126	24	)	)	PUNCT
cana-1730	126	25	≤	≤	NUM
cana-1730	126	26	θ	θ	PROPN
cana-1730	126	27	(	(	PUNCT
cana-1730	126	28	t(qan−1	t(qan−1	PROPN
cana-1730	126	29	,	,	PUNCT
cana-1730	126	30	qan	qan	PROPN
cana-1730	126	31	,	,	PUNCT
cana-1730	126	32	qan	qan	PROPN
cana-1730	126	33	)	)	PUNCT
cana-1730	126	34	)	)	PUNCT
cana-1730	126	35	.	.	PUNCT
cana-1730	127	1	thus	thus	ADV
cana-1730	127	2	for	for	ADP
cana-1730	127	3	n	n	PRON
cana-1730	127	4	∈	∈	PROPN
cana-1730	127	5	n	n	NOUN
cana-1730	127	6	,	,	PUNCT
cana-1730	127	7	we	we	PRON
cana-1730	127	8	have	have	VERB
cana-1730	127	9	t(qan	t(qan	ADV
cana-1730	127	10	,	,	PUNCT
cana-1730	127	11	qan+1	qan+1	NOUN
cana-1730	127	12	,	,	PUNCT
cana-1730	127	13	qan+1	qan+1	NOUN
cana-1730	127	14	)	)	PUNCT
cana-1730	127	15	=	=	SYM
cana-1730	127	16	t(gan−1	t(gan−1	PROPN
cana-1730	127	17	,	,	PUNCT
cana-1730	127	18	gan	gan	PROPN
cana-1730	127	19	,	,	PUNCT
cana-1730	127	20	gan	gan	NOUN
cana-1730	127	21	)	)	PUNCT
cana-1730	127	22	≤	≤	NUM
cana-1730	127	23	θ	θ	PROPN
cana-1730	127	24	(	(	PUNCT
cana-1730	127	25	t(qan−1	t(qan−1	PROPN
cana-1730	127	26	,	,	PUNCT
cana-1730	127	27	qan	qan	PROPN
cana-1730	127	28	,	,	PUNCT
cana-1730	127	29	qan	qan	PROPN
cana-1730	127	30	)	)	PUNCT
cana-1730	127	31	)	)	PUNCT
cana-1730	127	32	≤	≤	NUM
cana-1730	128	1	θ	θ	PROPN
cana-1730	128	2	2(t(qan−2	2(t(qan−2	NUM
cana-1730	128	3	,	,	PUNCT
cana-1730	128	4	qan−1	qan−1	PROPN
cana-1730	128	5	,	,	PUNCT
cana-1730	128	6	qan−1	qan−1	PROPN
cana-1730	128	7	)	)	PUNCT
cana-1730	128	8	)	)	PUNCT
cana-1730	128	9	≤	≤	NUM
cana-1730	128	10	θ	θ	X
cana-1730	128	11	n(t(qa0	n(t(qa0	PROPN
cana-1730	128	12	,	,	PUNCT
cana-1730	128	13	qa1	qa1	PROPN
cana-1730	128	14	,	,	PUNCT
cana-1730	128	15	qa1	qa1	NOUN
cana-1730	128	16	)	)	PUNCT
cana-1730	128	17	)	)	PUNCT
cana-1730	128	18	.	.	PUNCT
cana-1730	129	1	by	by	ADP
cana-1730	129	2	an	an	DET
cana-1730	129	3	argument	argument	NOUN
cana-1730	129	4	similar	similar	ADJ
cana-1730	129	5	to	to	ADP
cana-1730	129	6	that	that	PRON
cana-1730	129	7	in	in	ADP
cana-1730	129	8	the	the	DET
cana-1730	129	9	proof	proof	NOUN
cana-1730	129	10	of	of	ADP
cana-1730	129	11	theorem	theorem	ADJ
cana-1730	129	12	,	,	PUNCT
cana-1730	129	13	one	one	PRON
cana-1730	129	14	can	can	AUX
cana-1730	129	15	show	show	VERB
cana-1730	129	16	that	that	SCONJ
cana-1730	129	17	{	{	PUNCT
cana-1730	129	18	qan	qan	X
cana-1730	129	19	}	}	PUNCT
cana-1730	129	20	is	be	AUX
cana-1730	129	21	a	a	DET
cana-1730	129	22	cauchy	cauchy	ADJ
cana-1730	129	23	sequence	sequence	NOUN
cana-1730	129	24	.	.	PUNCT
cana-1730	130	1	since	since	SCONJ
cana-1730	130	2	s	s	PROPN
cana-1730	130	3	is	be	AUX
cana-1730	130	4	g	g	NOUN
cana-1730	130	5	-	-	PUNCT
cana-1730	130	6	complete	complete	ADJ
cana-1730	130	7	,	,	PUNCT
cana-1730	130	8	qan	qan	PROPN
cana-1730	130	9	is	be	AUX
cana-1730	130	10	convergent	convergent	ADJ
cana-1730	130	11	to	to	ADP
cana-1730	130	12	p	p	PROPN
cana-1730	130	13	∈	∈	PROPN
cana-1730	130	14	s	s	PART
cana-1730	130	15	.	.	PUNCT
cana-1730	131	1	now	now	ADV
cana-1730	131	2	we	we	PRON
cana-1730	131	3	show	show	VERB
cana-1730	131	4	that	that	SCONJ
cana-1730	131	5	qp	qp	ADV
cana-1730	131	6	=	=	SYM
cana-1730	131	7	gp	gp	NOUN
cana-1730	131	8	.	.	PUNCT
cana-1730	132	1	since	since	SCONJ
cana-1730	132	2	{	{	PUNCT
cana-1730	132	3	qan	qan	NOUN
cana-1730	132	4	}	}	PUNCT
cana-1730	132	5	is	be	AUX
cana-1730	132	6	a	a	DET
cana-1730	132	7	nondecreasing	nondecrease	VERB
cana-1730	132	8	sequence	sequence	NOUN
cana-1730	132	9	and	and	CCONJ
cana-1730	132	10	qan	qan	PROPN
cana-1730	132	11	→	→	SYM
cana-1730	132	12	p	p	X
cana-1730	132	13	,	,	PUNCT
cana-1730	132	14	by	by	ADP
cana-1730	132	15	regularity	regularity	NOUN
cana-1730	132	16	of	of	ADP
cana-1730	132	17	s	s	PRON
cana-1730	132	18	we	we	PRON
cana-1730	132	19	have	have	VERB
cana-1730	132	20	qan	qan	PROPN
cana-1730	132	21	≤	≤	PROPN
cana-1730	132	22	p	p	NOUN
cana-1730	132	23	for	for	ADP
cana-1730	132	24	all	all	DET
cana-1730	132	25	n.	n.	NOUN
cana-1730	132	26	if	if	SCONJ
cana-1730	132	27	qan	qan	PROPN
cana-1730	133	1	=	=	PUNCT
cana-1730	133	2	p	p	NOUN
cana-1730	133	3	for	for	ADP
cana-1730	133	4	some	some	DET
cana-1730	133	5	n	n	CCONJ
cana-1730	133	6	,	,	PUNCT
cana-1730	133	7	then	then	ADV
cana-1730	133	8	,	,	PUNCT
cana-1730	133	9	by	by	ADP
cana-1730	133	10	construction	construction	NOUN
cana-1730	133	11	,	,	PUNCT
cana-1730	133	12	qan+1	qan+1	NOUN
cana-1730	133	13	=	=	SYM
cana-1730	133	14	p	p	NOUN
cana-1730	133	15	and	and	CCONJ
cana-1730	133	16	p	p	NOUN
cana-1730	133	17	is	be	AUX
cana-1730	133	18	a	a	DET
cana-1730	133	19	fixed	fix	VERB
cana-1730	133	20	point	point	NOUN
cana-1730	133	21	.	.	PUNCT
cana-1730	134	1	so	so	ADV
cana-1730	134	2	we	we	PRON
cana-1730	134	3	assume	assume	VERB
cana-1730	134	4	that	that	SCONJ
cana-1730	134	5	qan	qan	PROPN
cana-1730	134	6	.	.	PUNCT
cana-1730	135	1	then	then	ADV
cana-1730	135	2	,	,	PUNCT
cana-1730	135	3	for	for	ADP
cana-1730	135	4	n	n	PRON
cana-1730	135	5	∈	∈	PROPN
cana-1730	135	6	n	n	CCONJ
cana-1730	135	7	,	,	PUNCT
cana-1730	135	8	we	we	PRON
cana-1730	135	9	have	have	VERB
cana-1730	135	10	t(qp	t(qp	NUM
cana-1730	135	11	,	,	PUNCT
cana-1730	135	12	qp	qp	NOUN
cana-1730	135	13	,	,	PUNCT
cana-1730	135	14	gp	gp	NOUN
cana-1730	135	15	)	)	PUNCT
cana-1730	135	16	≤	≤	NOUN
cana-1730	136	1	t(qp	t(qp	PROPN
cana-1730	136	2	,	,	PUNCT
cana-1730	136	3	qp	qp	PROPN
cana-1730	136	4	,	,	PUNCT
cana-1730	136	5	qan	qan	PROPN
cana-1730	136	6	)	)	PUNCT
cana-1730	137	1	+	+	CCONJ
cana-1730	137	2	t(qan	t(qan	ADV
cana-1730	137	3	,	,	PUNCT
cana-1730	137	4	qan	qan	PROPN
cana-1730	137	5	,	,	PUNCT
cana-1730	137	6	gp	gp	NOUN
cana-1730	137	7	)	)	PUNCT
cana-1730	137	8	=	=	SYM
cana-1730	137	9	t(qp	t(qp	PROPN
cana-1730	137	10	,	,	PUNCT
cana-1730	137	11	qp	qp	PROPN
cana-1730	137	12	,	,	PUNCT
cana-1730	137	13	qan	qan	PROPN
cana-1730	137	14	)	)	PUNCT
cana-1730	138	1	+	+	CCONJ
cana-1730	138	2	t(gan−1	t(gan−1	PROPN
cana-1730	138	3	,	,	PUNCT
cana-1730	138	4	gan−1	gan−1	PROPN
cana-1730	138	5	,	,	PUNCT
cana-1730	138	6	gp	gp	NOUN
cana-1730	138	7	)	)	PUNCT
cana-1730	138	8	≤	≤	NOUN
cana-1730	138	9	t(qp	t(qp	PROPN
cana-1730	138	10	,	,	PUNCT
cana-1730	138	11	qp	qp	PROPN
cana-1730	138	12	,	,	PUNCT
cana-1730	138	13	qan	qan	PROPN
cana-1730	138	14	)	)	PUNCT
cana-1730	139	1	+	+	NUM
cana-1730	139	2	θ	θ	PROPN
cana-1730	139	3	(	(	PUNCT
cana-1730	139	4	θ(an−1	θ(an−1	PROPN
cana-1730	139	5	,	,	PUNCT
cana-1730	139	6	an−1	an−1	ADJ
cana-1730	139	7	,	,	PUNCT
cana-1730	139	8	p	p	NOUN
cana-1730	139	9	)	)	PUNCT
cana-1730	139	10	)	)	PUNCT
cana-1730	139	11	communications	communication	NOUN
cana-1730	139	12	on	on	ADP
cana-1730	139	13	applied	apply	VERB
cana-1730	139	14	nonlinear	nonlinear	ADJ
cana-1730	139	15	analysis	analysis	NOUN
cana-1730	139	16	issn	issn	NOUN
cana-1730	139	17	:	:	PUNCT
cana-1730	139	18	1074	1074	NUM
cana-1730	139	19	-	-	PUNCT
cana-1730	139	20	133x	133x	NUM
cana-1730	139	21	vol	vol	NOUN
cana-1730	139	22	32	32	NUM
cana-1730	139	23	no	no	NOUN
cana-1730	139	24	.	.	NOUN
cana-1730	139	25	2	2	NUM
cana-1730	139	26	(	(	PUNCT
cana-1730	139	27	2025	2025	NUM
cana-1730	139	28	)	)	PUNCT
cana-1730	139	29	165	165	NUM
cana-1730	139	30	https://internationalpubls.com	https://internationalpubls.com	X
cana-1730	139	31	where	where	SCONJ
cana-1730	139	32	θ(an−1	θ(an−1	ADP
cana-1730	139	33	,	,	PUNCT
cana-1730	139	34	an−1	an−1	ADJ
cana-1730	139	35	,	,	PUNCT
cana-1730	139	36	p	p	ADJ
cana-1730	139	37	)	)	PUNCT
cana-1730	139	38	∈	∈	PROPN
cana-1730	139	39	{	{	PUNCT
cana-1730	139	40	t(qan−1	t(qan−1	PROPN
cana-1730	139	41	,	,	PUNCT
cana-1730	139	42	qan−1	qan−1	PROPN
cana-1730	139	43	,	,	PUNCT
cana-1730	139	44	qp	qp	NOUN
cana-1730	139	45	)	)	PUNCT
cana-1730	139	46	,	,	PUNCT
cana-1730	139	47	t(qan−1	t(qan−1	PROPN
cana-1730	139	48	,	,	PUNCT
cana-1730	139	49	gan−1	gan−1	PROPN
cana-1730	139	50	,	,	PUNCT
cana-1730	139	51	gan−1	gan−1	PROPN
cana-1730	139	52	)	)	PUNCT
cana-1730	139	53	,	,	PUNCT
cana-1730	139	54	t(qan−1	t(qan−1	PROPN
cana-1730	139	55	,	,	PUNCT
cana-1730	139	56	gan−1	gan−1	PROPN
cana-1730	139	57	,	,	PUNCT
cana-1730	139	58	gan−1	gan−1	PROPN
cana-1730	139	59	)	)	PUNCT
cana-1730	139	60	,	,	PUNCT
cana-1730	139	61	t(gan−1	t(gan−1	PROPN
cana-1730	139	62	,	,	PUNCT
cana-1730	139	63	qan−1	qan−1	PROPN
cana-1730	139	64	,	,	PUNCT
cana-1730	139	65	qp	qp	NOUN
cana-1730	139	66	)	)	PUNCT
cana-1730	139	67	}	}	PUNCT
cana-1730	139	68	=	=	SYM
cana-1730	139	69	{	{	PUNCT
cana-1730	139	70	t(qan−1	t(qan−1	PROPN
cana-1730	139	71	,	,	PUNCT
cana-1730	139	72	qan−1	qan−1	PROPN
cana-1730	139	73	,	,	PUNCT
cana-1730	139	74	qp	qp	NOUN
cana-1730	139	75	)	)	PUNCT
cana-1730	139	76	,	,	PUNCT
cana-1730	139	77	t(qan−1	t(qan−1	PROPN
cana-1730	139	78	,	,	PUNCT
cana-1730	139	79	qan	qan	PROPN
cana-1730	139	80	,	,	PUNCT
cana-1730	139	81	qan	qan	PROPN
cana-1730	139	82	)	)	PUNCT
cana-1730	139	83	,	,	PUNCT
cana-1730	139	84	t(qan	t(qan	ADV
cana-1730	139	85	,	,	PUNCT
cana-1730	139	86	qan−1	qan−1	PROPN
cana-1730	139	87	,	,	PUNCT
cana-1730	139	88	qp	qp	NOUN
cana-1730	139	89	)	)	PUNCT
cana-1730	139	90	}	}	PUNCT
cana-1730	139	91	.	.	PUNCT
cana-1730	140	1	fix	fix	VERB
cana-1730	140	2	z	z	NOUN
cana-1730	140	3	,	,	PUNCT
cana-1730	140	4	≪	≪	SCONJ
cana-1730	140	5	z.	z.	PROPN
cana-1730	140	6	choose	choose	VERB
cana-1730	140	7	a	a	DET
cana-1730	140	8	natural	natural	ADJ
cana-1730	140	9	number	number	NOUN
cana-1730	140	10	n1	n1	NOUN
cana-1730	140	11	such	such	ADJ
cana-1730	141	1	that	that	SCONJ
cana-1730	141	2	t(qp	t(qp	PROPN
cana-1730	141	3	,	,	PUNCT
cana-1730	141	4	qp	qp	PROPN
cana-1730	141	5	,	,	PUNCT
cana-1730	141	6	qan	qan	PROPN
cana-1730	141	7	)	)	PUNCT
cana-1730	141	8	≪	≪	ADJ
cana-1730	141	9	and	and	CCONJ
cana-1730	141	10	t(qan−1	t(qan−1	PRON
cana-1730	141	11	,	,	PUNCT
cana-1730	141	12	qan−1	qan−1	PROPN
cana-1730	141	13	,	,	PUNCT
cana-1730	141	14	qp	qp	NOUN
cana-1730	141	15	)	)	PUNCT
cana-1730	141	16	≪	≪	VERB
cana-1730	141	17	,	,	PUNCT
cana-1730	141	18	for	for	ADP
cana-1730	141	19	all	all	DET
cana-1730	141	20	n	n	PRON
cana-1730	141	21	≥	≥	NOUN
cana-1730	141	22	n1	n1	NOUN
cana-1730	141	23	.	.	PUNCT
cana-1730	142	1	we	we	PRON
cana-1730	142	2	investigate	investigate	VERB
cana-1730	142	3	these	these	DET
cana-1730	142	4	situations	situation	NOUN
cana-1730	142	5	as	as	SCONJ
cana-1730	142	6	follows	follow	VERB
cana-1730	142	7	:	:	PUNCT
cana-1730	142	8	case	case	NOUN
cana-1730	142	9	1	1	NUM
cana-1730	142	10	.	.	PUNCT
cana-1730	143	1	if	if	SCONJ
cana-1730	143	2	θ(an−1	θ(an−1	NOUN
cana-1730	143	3	,	,	PUNCT
cana-1730	143	4	an−1	an−1	ADJ
cana-1730	143	5	,	,	PUNCT
cana-1730	143	6	p	p	NOUN
cana-1730	143	7	)	)	PUNCT
cana-1730	143	8	=	=	SYM
cana-1730	143	9	t(qan−1	t(qan−1	PROPN
cana-1730	143	10	,	,	PUNCT
cana-1730	143	11	qan−1	qan−1	PROPN
cana-1730	143	12	,	,	PUNCT
cana-1730	143	13	qp	qp	NOUN
cana-1730	143	14	)	)	PUNCT
cana-1730	143	15	,	,	PUNCT
cana-1730	143	16	then	then	ADV
cana-1730	143	17	we	we	PRON
cana-1730	143	18	have	have	VERB
cana-1730	143	19	t(qp	t(qp	NUM
cana-1730	143	20	,	,	PUNCT
cana-1730	143	21	qp	qp	NOUN
cana-1730	143	22	,	,	PUNCT
cana-1730	143	23	gp	gp	NOUN
cana-1730	143	24	)	)	PUNCT
cana-1730	143	25	≤	≤	NOUN
cana-1730	143	26	t	t	PROPN
cana-1730	143	27	(	(	PUNCT
cana-1730	143	28	qp	qp	PROPN
cana-1730	143	29	,	,	PUNCT
cana-1730	143	30	qp	qp	PROPN
cana-1730	143	31	,	,	PUNCT
cana-1730	143	32	qan	qan	PROPN
cana-1730	143	33	)	)	PUNCT
cana-1730	144	1	+	+	NUM
cana-1730	144	2	θ	θ	PROPN
cana-1730	144	3	(	(	PUNCT
cana-1730	144	4	t(qan−1	t(qan−1	PROPN
cana-1730	144	5	,	,	PUNCT
cana-1730	144	6	qan−1	qan−1	PROPN
cana-1730	144	7	,	,	PUNCT
cana-1730	144	8	qp	qp	NOUN
cana-1730	144	9	)	)	PUNCT
cana-1730	144	10	)	)	PUNCT
cana-1730	145	1	<	<	X
cana-1730	145	2	t(qp	t(qp	PROPN
cana-1730	145	3	,	,	PUNCT
cana-1730	145	4	qp	qp	PROPN
cana-1730	145	5	,	,	PUNCT
cana-1730	145	6	qan	qan	PROPN
cana-1730	145	7	)	)	PUNCT
cana-1730	146	1	+	+	CCONJ
cana-1730	146	2	t(qan−1	t(qan−1	PROPN
cana-1730	146	3	,	,	PUNCT
cana-1730	146	4	qan−1	qan−1	PROPN
cana-1730	146	5	,	,	PUNCT
cana-1730	146	6	qp	qp	NOUN
cana-1730	146	7	)	)	PUNCT
cana-1730	146	8	≪	≪	VERB
cana-1730	146	9	+	+	PUNCT
cana-1730	147	1	=	=	SYM
cana-1730	147	2	z	z	NOUN
cana-1730	147	3	.	.	PUNCT
cana-1730	148	1	case	case	NOUN
cana-1730	148	2	2	2	NUM
cana-1730	148	3	.	.	X
cana-1730	149	1	if	if	SCONJ
cana-1730	149	2	θ(an−1	θ(an−1	NOUN
cana-1730	149	3	,	,	PUNCT
cana-1730	149	4	an−1	an−1	ADJ
cana-1730	149	5	,	,	PUNCT
cana-1730	149	6	p	p	NOUN
cana-1730	149	7	)	)	PUNCT
cana-1730	149	8	=	=	SYM
cana-1730	149	9	t(qan−1	t(qan−1	PROPN
cana-1730	149	10	,	,	PUNCT
cana-1730	149	11	qan	qan	PROPN
cana-1730	149	12	,	,	PUNCT
cana-1730	149	13	qan	qan	PROPN
cana-1730	149	14	)	)	PUNCT
cana-1730	149	15	,	,	PUNCT
cana-1730	149	16	then	then	ADV
cana-1730	149	17	we	we	PRON
cana-1730	149	18	have	have	VERB
cana-1730	149	19	t(qp	t(qp	NUM
cana-1730	149	20	,	,	PUNCT
cana-1730	149	21	qp	qp	PROPN
cana-1730	149	22	,	,	PUNCT
cana-1730	149	23	qp	qp	NOUN
cana-1730	149	24	)	)	PUNCT
cana-1730	149	25	≤	≤	NOUN
cana-1730	150	1	t(qp	t(qp	PROPN
cana-1730	150	2	,	,	PUNCT
cana-1730	150	3	qp	qp	PROPN
cana-1730	150	4	,	,	PUNCT
cana-1730	150	5	qan	qan	PROPN
cana-1730	150	6	)	)	PUNCT
cana-1730	151	1	+	+	NUM
cana-1730	151	2	θ	θ	PROPN
cana-1730	151	3	(	(	PUNCT
cana-1730	151	4	t(qan−1	t(qan−1	PROPN
cana-1730	151	5	,	,	PUNCT
cana-1730	151	6	qan	qan	PROPN
cana-1730	151	7	,	,	PUNCT
cana-1730	151	8	qan	qan	PROPN
cana-1730	151	9	)	)	PUNCT
cana-1730	151	10	)	)	PUNCT
cana-1730	152	1	<	<	X
cana-1730	152	2	t(qp	t(qp	PROPN
cana-1730	152	3	,	,	PUNCT
cana-1730	152	4	qp	qp	PROPN
cana-1730	152	5	,	,	PUNCT
cana-1730	152	6	qan	qan	PROPN
cana-1730	152	7	)	)	PUNCT
cana-1730	152	8	+	+	CCONJ
cana-1730	152	9	t(qan−1	t(qan−1	PROPN
cana-1730	152	10	,	,	PUNCT
cana-1730	152	11	qan	qan	PROPN
cana-1730	152	12	,	,	PUNCT
cana-1730	152	13	qan	qan	PROPN
cana-1730	152	14	)	)	PUNCT
cana-1730	152	15	≪	≪	PUNCT
cana-1730	152	16	z.	z.	PROPN
cana-1730	152	17	case	case	NOUN
cana-1730	152	18	3	3	X
cana-1730	152	19	.	.	PUNCT
cana-1730	153	1	if	if	SCONJ
cana-1730	153	2	θ(an−1	θ(an−1	NOUN
cana-1730	153	3	,	,	PUNCT
cana-1730	153	4	an−1	an−1	ADJ
cana-1730	153	5	,	,	PUNCT
cana-1730	153	6	p	p	NOUN
cana-1730	153	7	)	)	PUNCT
cana-1730	153	8	=	=	SYM
cana-1730	154	1	t(qan	t(qan	ADV
cana-1730	154	2	,	,	PUNCT
cana-1730	154	3	qan−1	qan−1	PROPN
cana-1730	154	4	,	,	PUNCT
cana-1730	154	5	qp	qp	NOUN
cana-1730	154	6	)	)	PUNCT
cana-1730	154	7	,	,	PUNCT
cana-1730	154	8	then	then	ADV
cana-1730	154	9	we	we	PRON
cana-1730	154	10	have	have	VERB
cana-1730	154	11	t(qp	t(qp	NUM
cana-1730	154	12	,	,	PUNCT
cana-1730	154	13	qp	qp	NOUN
cana-1730	154	14	,	,	PUNCT
cana-1730	154	15	gp	gp	NOUN
cana-1730	154	16	)	)	PUNCT
cana-1730	154	17	≤	≤	NOUN
cana-1730	155	1	t(qp	t(qp	PROPN
cana-1730	155	2	,	,	PUNCT
cana-1730	155	3	qp	qp	PROPN
cana-1730	155	4	,	,	PUNCT
cana-1730	155	5	qan	qan	PROPN
cana-1730	155	6	)	)	PUNCT
cana-1730	156	1	+	+	NUM
cana-1730	156	2	θ	θ	PROPN
cana-1730	156	3	(	(	PUNCT
cana-1730	156	4	t(qan	t(qan	ADV
cana-1730	156	5	,	,	PUNCT
cana-1730	156	6	qan−1	qan−1	PROPN
cana-1730	156	7	,	,	PUNCT
cana-1730	156	8	qp	qp	NOUN
cana-1730	156	9	)	)	PUNCT
cana-1730	156	10	)	)	PUNCT
cana-1730	157	1	<	<	X
cana-1730	157	2	t(qp	t(qp	PROPN
cana-1730	157	3	,	,	PUNCT
cana-1730	157	4	qp	qp	PROPN
cana-1730	157	5	,	,	PUNCT
cana-1730	157	6	qan	qan	PROPN
cana-1730	157	7	)	)	PUNCT
cana-1730	158	1	+	+	CCONJ
cana-1730	158	2	t(qan	t(qan	ADV
cana-1730	158	3	,	,	PUNCT
cana-1730	158	4	qan−1	qan−1	PROPN
cana-1730	158	5	,	,	PUNCT
cana-1730	158	6	qp	qp	NOUN
cana-1730	158	7	)	)	PUNCT
cana-1730	158	8	≤	≤	NOUN
cana-1730	158	9	t(qp	t(qp	PROPN
cana-1730	158	10	,	,	PUNCT
cana-1730	158	11	qp	qp	PROPN
cana-1730	158	12	,	,	PUNCT
cana-1730	158	13	qan	qan	PROPN
cana-1730	158	14	)	)	PUNCT
cana-1730	159	1	+	+	CCONJ
cana-1730	159	2	t	t	PROPN
cana-1730	159	3	(	(	PUNCT
cana-1730	159	4	qan	qan	PROPN
cana-1730	159	5	,	,	PUNCT
cana-1730	159	6	qan−1	qan−1	PROPN
cana-1730	159	7	,	,	PUNCT
cana-1730	159	8	qan−1	qan−1	PROPN
cana-1730	159	9	)	)	PUNCT
cana-1730	159	10	+	+	CCONJ
cana-1730	159	11	t(qan−1	t(qan−1	PROPN
cana-1730	159	12	,	,	PUNCT
cana-1730	159	13	qan−1	qan−1	PROPN
cana-1730	159	14	,	,	PUNCT
cana-1730	159	15	qp	qp	NOUN
cana-1730	159	16	)	)	PUNCT
cana-1730	159	17	≪	≪	PUNCT
cana-1730	159	18	z	z	NOUN
cana-1730	159	19	whenever	whenever	SCONJ
cana-1730	159	20	n	n	DET
cana-1730	159	21	∈	∈	PROPN
cana-1730	159	22	n.	n.	NOUN
cana-1730	159	23	thus	thus	ADV
cana-1730	159	24	in	in	ADP
cana-1730	159	25	all	all	DET
cana-1730	159	26	cases	case	NOUN
cana-1730	159	27	t(qp	t(qp	PROPN
cana-1730	159	28	,	,	PUNCT
cana-1730	159	29	qp	qp	NOUN
cana-1730	159	30	,	,	PUNCT
cana-1730	159	31	gp	gp	NOUN
cana-1730	159	32	)	)	PUNCT
cana-1730	159	33	≪	≪	PUNCT
cana-1730	159	34	z	z	NOUN
cana-1730	159	35	for	for	ADP
cana-1730	159	36	arbitrary	arbitrary	ADJ
cana-1730	159	37	z	z	NOUN
cana-1730	159	38	∈	∈	PROPN
cana-1730	159	39	in	in	ADP
cana-1730	159	40	a	a	PRON
cana-1730	159	41	.	.	PUNCT
cana-1730	160	1	it	it	PRON
cana-1730	160	2	follows	follow	VERB
cana-1730	160	3	that	that	SCONJ
cana-1730	160	4	t(qp	t(qp	PROPN
cana-1730	160	5	,	,	PUNCT
cana-1730	160	6	qp	qp	NOUN
cana-1730	160	7	,	,	PUNCT
cana-1730	160	8	gp)=	gp)=	NOUN
cana-1730	160	9	θ	θ	PROPN
cana-1730	160	10	which	which	PRON
cana-1730	160	11	implies	imply	VERB
cana-1730	160	12	that	that	DET
cana-1730	160	13	gp	gp	NOUN
cana-1730	160	14	=	=	SYM
cana-1730	160	15	qp	qp	PROPN
cana-1730	160	16	.	.	PUNCT
cana-1730	161	1	then	then	ADV
cana-1730	161	2	p	p	PROPN
cana-1730	161	3	is	be	AUX
cana-1730	161	4	a	a	DET
cana-1730	161	5	coincidence	coincidence	NOUN
cana-1730	161	6	point	point	NOUN
cana-1730	161	7	for	for	ADP
cana-1730	161	8	the	the	DET
cana-1730	161	9	mappings	mapping	NOUN
cana-1730	161	10	g	g	NOUN
cana-1730	161	11	and	and	CCONJ
cana-1730	161	12	q.	q.	PROPN
cana-1730	161	13	conclusion	conclusion	NOUN
cana-1730	161	14	we	we	PRON
cana-1730	161	15	examine	examine	VERB
cana-1730	161	16	common	common	ADJ
cana-1730	161	17	fixed	fix	VERB
cana-1730	161	18	point	point	NOUN
cana-1730	161	19	theorems	theorem	NOUN
cana-1730	161	20	in	in	ADP
cana-1730	161	21	partially	partially	ADV
cana-1730	161	22	ordered	order	VERB
cana-1730	161	23	cone	cone	NOUN
cana-1730	161	24	g	g	NOUN
cana-1730	161	25	-	-	PUNCT
cana-1730	161	26	metric	metric	ADJ
cana-1730	161	27	spaces	space	NOUN
cana-1730	161	28	for	for	ADP
cana-1730	161	29	mappings	mapping	NOUN
cana-1730	161	30	that	that	PRON
cana-1730	161	31	meet	meet	VERB
cana-1730	161	32	contractive	contractive	ADJ
cana-1730	161	33	criteria	criterion	NOUN
cana-1730	161	34	associated	associate	VERB
cana-1730	161	35	with	with	ADP
cana-1730	161	36	a	a	DET
cana-1730	161	37	nondecreasing	nondecreasing	ADJ
cana-1730	161	38	θ	θ	NOUN
cana-1730	161	39	-map	-map	PUNCT
cana-1730	162	1	[	[	X
cana-1730	162	2	8,9	8,9	NUM
cana-1730	162	3	]	]	PUNCT
cana-1730	162	4	.	.	PUNCT
cana-1730	163	1	the	the	DET
cana-1730	163	2	work	work	NOUN
cana-1730	163	3	by	by	ADP
cana-1730	163	4	shatanawi	shatanawi	NOUN
cana-1730	163	5	[	[	X
cana-1730	163	6	4	4	NUM
cana-1730	163	7	]	]	PUNCT
cana-1730	163	8	and	and	CCONJ
cana-1730	163	9	ozturk	ozturk	NOUN
cana-1730	163	10	and	and	CCONJ
cana-1730	163	11	basarir	basarir	NOUN
cana-1730	163	12	[	[	X
cana-1730	163	13	12	12	NUM
cana-1730	163	14	]	]	PUNCT
cana-1730	163	15	is	be	AUX
cana-1730	163	16	extended	extend	VERB
cana-1730	163	17	to	to	ADP
cana-1730	163	18	our	our	PRON
cana-1730	163	19	results	result	NOUN
cana-1730	163	20	in	in	ADP
cana-1730	163	21	an	an	DET
cana-1730	163	22	ordered	order	VERB
cana-1730	163	23	cone	cone	NOUN
cana-1730	163	24	g	g	NOUN
cana-1730	163	25	-	-	PUNCT
cana-1730	163	26	version	version	NOUN
cana-1730	163	27	.	.	PUNCT
cana-1730	164	1	references	reference	NOUN
cana-1730	164	2	:	:	PUNCT
cana-1730	165	1	[	[	X
cana-1730	165	2	1	1	X
cana-1730	165	3	]	]	PUNCT
cana-1730	165	4	z.	z.	PROPN
cana-1730	165	5	mustafa	mustafa	PROPN
cana-1730	165	6	,	,	PUNCT
cana-1730	165	7	b.	b.	PROPN
cana-1730	165	8	sims	sims	PROPN
cana-1730	165	9	,	,	PUNCT
cana-1730	165	10	a	a	DET
cana-1730	165	11	new	new	ADJ
cana-1730	165	12	approach	approach	NOUN
cana-1730	165	13	to	to	ADP
cana-1730	165	14	generalized	generalize	VERB
cana-1730	165	15	metric	metric	ADJ
cana-1730	165	16	spaces	space	NOUN
cana-1730	165	17	,	,	PUNCT
cana-1730	165	18	j.	j.	PROPN
cana-1730	165	19	nonlinear	nonlinear	PROPN
cana-1730	165	20	convex	convex	PROPN
cana-1730	165	21	anal	anal	NOUN
cana-1730	165	22	.	.	PUNCT
cana-1730	165	23	7	7	NUM
cana-1730	165	24	(	(	PUNCT
cana-1730	165	25	2006	2006	NUM
cana-1730	165	26	)	)	PUNCT
cana-1730	166	1	289–297	289–297	NUM
cana-1730	166	2	.	.	PUNCT
cana-1730	167	1	[	[	X
cana-1730	167	2	2	2	NUM
cana-1730	167	3	]	]	PUNCT
cana-1730	167	4	z.	z.	PROPN
cana-1730	167	5	mustafa	mustafa	PROPN
cana-1730	167	6	,	,	PUNCT
cana-1730	167	7	h.	h.	PROPN
cana-1730	167	8	obiedat	obiedat	PROPN
cana-1730	167	9	,	,	PUNCT
cana-1730	167	10	f.	f.	PROPN
cana-1730	167	11	awawdeh	awawdeh	PROPN
cana-1730	167	12	,	,	PUNCT
cana-1730	167	13	some	some	PRON
cana-1730	167	14	of	of	ADP
cana-1730	167	15	fixed	fix	VERB
cana-1730	167	16	point	point	NOUN
cana-1730	167	17	theorem	theorem	NOUN
cana-1730	167	18	for	for	ADP
cana-1730	167	19	mapping	mapping	NOUN
cana-1730	167	20	on	on	ADP
cana-1730	167	21	complete	complete	ADJ
cana-1730	167	22	g	g	NOUN
cana-1730	167	23	-	-	PUNCT
cana-1730	167	24	metric	metric	ADJ
cana-1730	167	25	spaces	space	NOUN
cana-1730	167	26	,	,	PUNCT
cana-1730	167	27	fixed	fix	VERB
cana-1730	167	28	point	point	NOUN
cana-1730	167	29	theory	theory	NOUN
cana-1730	167	30	appl	appl	PROPN
cana-1730	167	31	.	.	PROPN
cana-1730	167	32	2008	2008	NUM
cana-1730	167	33	(	(	PUNCT
cana-1730	167	34	2008	2008	NUM
cana-1730	167	35	)	)	PUNCT
cana-1730	167	36	12	12	NUM
cana-1730	167	37	.	.	PUNCT
cana-1730	168	1	article	article	NOUN
cana-1730	168	2	i	i	PROPN
cana-1730	168	3	d	d	PROPN
cana-1730	168	4	189870	189870	NUM
cana-1730	168	5	.	.	PUNCT
cana-1730	169	1	[	[	X
cana-1730	169	2	3	3	X
cana-1730	169	3	]	]	PUNCT
cana-1730	169	4	z.	z.	PROPN
cana-1730	169	5	mustafa	mustafa	PROPN
cana-1730	169	6	,	,	PUNCT
cana-1730	169	7	b.	b.	PROPN
cana-1730	169	8	sims	sim	NOUN
cana-1730	169	9	,	,	PUNCT
cana-1730	169	10	fixed	fix	VERB
cana-1730	169	11	point	point	NOUN
cana-1730	169	12	theorems	theorem	NOUN
cana-1730	169	13	for	for	ADP
cana-1730	169	14	contractive	contractive	ADJ
cana-1730	169	15	mappings	mapping	NOUN
cana-1730	169	16	in	in	ADP
cana-1730	169	17	complete	complete	ADJ
cana-1730	169	18	g	g	NOUN
cana-1730	169	19	-	-	PUNCT
cana-1730	169	20	metric	metric	ADJ
cana-1730	169	21	space	space	NOUN
cana-1730	169	22	,	,	PUNCT
cana-1730	169	23	fixed	fix	VERB
cana-1730	169	24	point	point	NOUN
cana-1730	169	25	theory	theory	NOUN
cana-1730	169	26	appl	appl	NOUN
cana-1730	169	27	.	.	PUNCT
cana-1730	169	28	2009	2009	NUM
cana-1730	169	29	(	(	PUNCT
cana-1730	169	30	2009	2009	NUM
cana-1730	169	31	)	)	PUNCT
cana-1730	169	32	10	10	NUM
cana-1730	169	33	.	.	PUNCT
cana-1730	170	1	article	article	NOUN
cana-1730	170	2	i	i	PROPN
cana-1730	170	3	d	d	PROPN
cana-1730	170	4	917175	917175	NUM
cana-1730	170	5	.	.	PUNCT
cana-1730	171	1	communications	communication	NOUN
cana-1730	171	2	on	on	ADP
cana-1730	171	3	applied	apply	VERB
cana-1730	171	4	nonlinear	nonlinear	ADJ
cana-1730	171	5	analysis	analysis	NOUN
cana-1730	171	6	issn	issn	NOUN
cana-1730	171	7	:	:	PUNCT
cana-1730	171	8	1074	1074	NUM
cana-1730	171	9	-	-	PUNCT
cana-1730	171	10	133x	133x	NUM
cana-1730	171	11	vol	vol	NOUN
cana-1730	171	12	32	32	NUM
cana-1730	171	13	no	no	NOUN
cana-1730	171	14	.	.	NOUN
cana-1730	171	15	2	2	NUM
cana-1730	171	16	(	(	PUNCT
cana-1730	171	17	2025	2025	NUM
cana-1730	171	18	)	)	PUNCT
cana-1730	171	19	166	166	NUM
cana-1730	171	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1730	172	1	[	[	X
cana-1730	172	2	4	4	X
cana-1730	172	3	]	]	X
cana-1730	172	4	w.	w.	PROPN
cana-1730	172	5	shatanawi	shatanawi	PROPN
cana-1730	172	6	,	,	PUNCT
cana-1730	172	7	fixed	fix	VERB
cana-1730	172	8	point	point	NOUN
cana-1730	172	9	theory	theory	NOUN
cana-1730	172	10	for	for	ADP
cana-1730	172	11	contractive	contractive	ADJ
cana-1730	172	12	mappings	mapping	NOUN
cana-1730	172	13	satisfying	satisfy	VERB
cana-1730	172	14	φ	φ	NOUN
cana-1730	172	15	-	-	NOUN
cana-1730	172	16	maps	map	NOUN
cana-1730	172	17	in	in	ADP
cana-1730	172	18	g	g	NOUN
cana-1730	172	19	-	-	PUNCT
cana-1730	172	20	metric	metric	ADJ
cana-1730	172	21	spaces	space	NOUN
cana-1730	172	22	,	,	PUNCT
cana-1730	172	23	fixed	fix	VERB
cana-1730	172	24	point	point	NOUN
cana-1730	172	25	theory	theory	NOUN
cana-1730	172	26	appl	appl	NOUN
cana-1730	172	27	.	.	PUNCT
cana-1730	173	1	2010	2010	NUM
cana-1730	173	2	(	(	PUNCT
cana-1730	173	3	2010	2010	NUM
cana-1730	173	4	)	)	PUNCT
cana-1730	173	5	9	9	NUM
cana-1730	173	6	.	.	PUNCT
cana-1730	174	1	article	article	NOUN
cana-1730	174	2	i	i	PROPN
cana-1730	174	3	d	d	PROPN
cana-1730	174	4	181650	181650	NUM
cana-1730	174	5	.	.	PUNCT
cana-1730	175	1	[	[	X
cana-1730	175	2	5	5	NUM
cana-1730	175	3	]	]	X
cana-1730	175	4	ð.r	ð.r	PROPN
cana-1730	175	5	.	.	PROPN
cana-1730	175	6	kurepa	kurepa	PROPN
cana-1730	175	7	,	,	PUNCT
cana-1730	175	8	tableaux	tableaux	ADJ
cana-1730	175	9	ramifiés	ramifiés	NOUN
cana-1730	175	10	d’ensembles	d’ensemble	NOUN
cana-1730	175	11	.	.	PUNCT
cana-1730	176	1	espace	espace	PROPN
cana-1730	176	2	pseudo	pseudo	NOUN
cana-1730	176	3	-	-	NOUN
cana-1730	176	4	distanciés	distancié	NOUN
cana-1730	176	5	,	,	PUNCT
cana-1730	176	6	c.r	c.r	PROPN
cana-1730	176	7	.	.	PROPN
cana-1730	176	8	acad	acad	PROPN
cana-1730	176	9	.	.	PUNCT
cana-1730	177	1	sci	sci	PROPN
cana-1730	177	2	.	.	PROPN
cana-1730	178	1	paris	paris	PROPN
cana-1730	178	2	198	198	NUM
cana-1730	178	3	(	(	PUNCT
cana-1730	178	4	1934	1934	NUM
cana-1730	178	5	)	)	PUNCT
cana-1730	178	6	1563–1565	1563–1565	NUM
cana-1730	178	7	.	.	PUNCT
cana-1730	179	1	[	[	X
cana-1730	179	2	6	6	NUM
cana-1730	179	3	]	]	X
cana-1730	179	4	l.g	l.g	PROPN
cana-1730	179	5	.	.	PROPN
cana-1730	179	6	huang	huang	PROPN
cana-1730	179	7	,	,	PUNCT
cana-1730	179	8	x.	x.	PROPN
cana-1730	179	9	zhang	zhang	PROPN
cana-1730	179	10	,	,	PUNCT
cana-1730	179	11	cone	cone	NOUN
cana-1730	179	12	metric	metric	ADJ
cana-1730	179	13	spaces	space	NOUN
cana-1730	179	14	and	and	CCONJ
cana-1730	179	15	fixed	fix	VERB
cana-1730	179	16	point	point	NOUN
cana-1730	179	17	theorems	theorem	NOUN
cana-1730	179	18	of	of	ADP
cana-1730	179	19	contractive	contractive	ADJ
cana-1730	179	20	mappings	mapping	NOUN
cana-1730	179	21	,	,	PUNCT
cana-1730	179	22	j.	j.	PROPN
cana-1730	179	23	math	math	PROPN
cana-1730	179	24	.	.	PUNCT
cana-1730	180	1	anal	anal	PROPN
cana-1730	180	2	.	.	PUNCT
cana-1730	180	3	appl	appl	PROPN
cana-1730	180	4	.	.	PUNCT
cana-1730	181	1	332	332	NUM
cana-1730	181	2	(	(	PUNCT
cana-1730	181	3	2007	2007	NUM
cana-1730	181	4	)	)	PUNCT
cana-1730	181	5	1468–1476	1468–1476	NUM
cana-1730	181	6	.	.	PUNCT
cana-1730	182	1	[	[	X
cana-1730	182	2	7	7	X
cana-1730	182	3	]	]	X
cana-1730	182	4	k.j	k.j	PROPN
cana-1730	182	5	.	.	PROPN
cana-1730	182	6	chung	chung	PROPN
cana-1730	182	7	,	,	PUNCT
cana-1730	182	8	remarks	remark	NOUN
cana-1730	182	9	on	on	ADP
cana-1730	182	10	nonlinear	nonlinear	ADJ
cana-1730	182	11	contractions	contraction	NOUN
cana-1730	182	12	,	,	PUNCT
cana-1730	182	13	pacific	pacific	PROPN
cana-1730	182	14	j.	j.	PROPN
cana-1730	182	15	math	math	PROPN
cana-1730	182	16	.	.	PUNCT
cana-1730	183	1	101	101	NUM
cana-1730	183	2	(	(	PUNCT
cana-1730	183	3	1982	1982	NUM
cana-1730	183	4	)	)	PUNCT
cana-1730	183	5	41–48	41–48	NUM
cana-1730	183	6	.	.	PUNCT
cana-1730	184	1	[	[	X
cana-1730	184	2	8	8	NUM
cana-1730	184	3	]	]	X
cana-1730	184	4	i.	i.	PROPN
cana-1730	184	5	aranđelović	aranđelović	PROPN
cana-1730	184	6	,	,	PUNCT
cana-1730	184	7	z.	z.	PROPN
cana-1730	184	8	kadelburg	kadelburg	PROPN
cana-1730	184	9	,	,	PUNCT
cana-1730	184	10	s.	s.	PROPN
cana-1730	184	11	radenović	radenović	PROPN
cana-1730	184	12	,	,	PUNCT
cana-1730	184	13	boyd	boyd	PROPN
cana-1730	184	14	-	-	PUNCT
cana-1730	184	15	wong	wong	PROPN
cana-1730	184	16	-	-	PUNCT
cana-1730	184	17	type	type	NOUN
cana-1730	184	18	common	common	ADJ
cana-1730	184	19	fixed	fix	VERB
cana-1730	184	20	point	point	NOUN
cana-1730	184	21	results	result	NOUN
cana-1730	184	22	in	in	ADP
cana-1730	184	23	cone	cone	NOUN
cana-1730	184	24	metric	metric	ADJ
cana-1730	184	25	spaces	space	NOUN
cana-1730	184	26	,	,	PUNCT
cana-1730	184	27	appl	appl	PROPN
cana-1730	184	28	.	.	PROPN
cana-1730	184	29	math	math	PROPN
cana-1730	184	30	.	.	PUNCT
cana-1730	185	1	comput	comput	NOUN
cana-1730	185	2	.	.	PUNCT
cana-1730	186	1	217	217	NUM
cana-1730	186	2	(	(	PUNCT
cana-1730	186	3	2011	2011	NUM
cana-1730	186	4	)	)	PUNCT
cana-1730	186	5	7167–7171	7167–7171	NUM
cana-1730	186	6	.	.	PUNCT
cana-1730	187	1	[	[	X
cana-1730	187	2	9	9	NUM
cana-1730	187	3	]	]	X
cana-1730	187	4	c.	c.	PROPN
cana-1730	187	5	di	di	PROPN
cana-1730	187	6	bari	bari	PROPN
cana-1730	187	7	,	,	PUNCT
cana-1730	187	8	p.	p.	NOUN
cana-1730	187	9	vetro	vetro	NOUN
cana-1730	187	10	,	,	PUNCT
cana-1730	187	11	ϕ-pairs	ϕ-pair	NOUN
cana-1730	187	12	and	and	CCONJ
cana-1730	187	13	common	common	ADJ
cana-1730	187	14	fixed	fix	VERB
cana-1730	187	15	points	point	NOUN
cana-1730	187	16	in	in	ADP
cana-1730	187	17	cone	cone	NOUN
cana-1730	187	18	metric	metric	ADJ
cana-1730	187	19	spaces	space	NOUN
cana-1730	187	20	,	,	PUNCT
cana-1730	187	21	rend	rend	VERB
cana-1730	187	22	.	.	PUNCT
cana-1730	188	1	circolo	circolo	PROPN
cana-1730	188	2	mat	mat	PROPN
cana-1730	188	3	.	.	PUNCT
cana-1730	188	4	palermo	palermo	PROPN
cana-1730	188	5	57	57	NUM
cana-1730	188	6	(	(	PUNCT
cana-1730	188	7	2008	2008	NUM
cana-1730	188	8	)	)	PUNCT
cana-1730	188	9	279–285	279–285	NUM
cana-1730	188	10	.	.	PUNCT
cana-1730	189	1	[	[	X
cana-1730	189	2	10	10	NUM
cana-1730	189	3	]	]	X
cana-1730	189	4	z.	z.	PROPN
cana-1730	189	5	kadelburg	kadelburg	PROPN
cana-1730	189	6	,	,	PUNCT
cana-1730	189	7	s.	s.	PROPN
cana-1730	189	8	radenović	radenović	PROPN
cana-1730	189	9	,	,	PUNCT
cana-1730	189	10	v.	v.	PROPN
cana-1730	189	11	rakočević	rakočević	PROPN
cana-1730	189	12	,	,	PUNCT
cana-1730	189	13	a	a	DET
cana-1730	189	14	note	note	NOUN
cana-1730	189	15	on	on	ADP
cana-1730	189	16	the	the	DET
cana-1730	189	17	equivalence	equivalence	NOUN
cana-1730	189	18	of	of	ADP
cana-1730	189	19	some	some	DET
cana-1730	189	20	metric	metric	NOUN
cana-1730	189	21	and	and	CCONJ
cana-1730	189	22	cone	cone	NOUN
cana-1730	189	23	metric	metric	ADJ
cana-1730	189	24	fixed	fix	VERB
cana-1730	189	25	point	point	NOUN
cana-1730	189	26	results	result	NOUN
cana-1730	189	27	,	,	PUNCT
cana-1730	189	28	appl	appl	PROPN
cana-1730	189	29	.	.	PROPN
cana-1730	189	30	math	math	PROPN
cana-1730	189	31	.	.	PUNCT
cana-1730	190	1	lett	lett	PROPN
cana-1730	190	2	.	.	PROPN
cana-1730	191	1	24	24	NUM
cana-1730	191	2	(	(	PUNCT
cana-1730	191	3	2011	2011	NUM
cana-1730	191	4	)	)	PUNCT
cana-1730	192	1	370–374	370–374	NUM
cana-1730	192	2	.	.	PUNCT
cana-1730	193	1	[	[	X
cana-1730	193	2	11	11	NUM
cana-1730	193	3	]	]	PUNCT
cana-1730	193	4	i.	i.	NOUN
cana-1730	193	5	beg	beg	PROPN
cana-1730	193	6	,	,	PUNCT
cana-1730	193	7	m.	m.	NOUN
cana-1730	193	8	abbas	abbas	PROPN
cana-1730	193	9	,	,	PUNCT
cana-1730	193	10	t.	t.	PROPN
cana-1730	193	11	nazir	nazir	PROPN
cana-1730	193	12	,	,	PUNCT
cana-1730	193	13	generalized	generalize	VERB
cana-1730	193	14	cone	cone	NOUN
cana-1730	193	15	metric	metric	ADJ
cana-1730	193	16	spaces	space	NOUN
cana-1730	193	17	,	,	PUNCT
cana-1730	193	18	j.	j.	PROPN
cana-1730	193	19	nonlinear	nonlinear	PROPN
cana-1730	193	20	sci	sci	PROPN
cana-1730	193	21	.	.	PUNCT
cana-1730	193	22	appl	appl	PROPN
cana-1730	193	23	.	.	PROPN
cana-1730	194	1	3	3	NUM
cana-1730	194	2	(	(	PUNCT
cana-1730	194	3	2010	2010	NUM
cana-1730	194	4	)	)	PUNCT
cana-1730	194	5	21–31	21–31	NUM
cana-1730	194	6	.	.	PUNCT
cana-1730	195	1	[	[	X
cana-1730	195	2	12	12	NUM
cana-1730	195	3	]	]	PUNCT
cana-1730	195	4	m.	m.	NOUN
cana-1730	195	5	ozturk	ozturk	PROPN
cana-1730	195	6	,	,	PUNCT
cana-1730	195	7	m.	m.	NOUN
cana-1730	195	8	basarir	basarir	NOUN
cana-1730	195	9	,	,	PUNCT
cana-1730	195	10	on	on	ADP
cana-1730	195	11	some	some	DET
cana-1730	195	12	common	common	ADJ
cana-1730	195	13	fixed	fix	VERB
cana-1730	195	14	point	point	NOUN
cana-1730	195	15	theorems	theorem	NOUN
cana-1730	195	16	with	with	ADP
cana-1730	195	17	ϕ-maps	ϕ-map	NOUN
cana-1730	195	18	on	on	ADP
cana-1730	195	19	g	g	NOUN
cana-1730	195	20	-	-	PUNCT
cana-1730	195	21	cone	cone	NOUN
cana-1730	195	22	metric	metric	ADJ
cana-1730	195	23	spaces	space	NOUN
cana-1730	195	24	,	,	PUNCT
cana-1730	195	25	bull	bull	NOUN
cana-1730	195	26	.	.	PUNCT
cana-1730	196	1	math	math	NOUN
cana-1730	196	2	.	.	PUNCT
cana-1730	197	1	anal	anal	PROPN
cana-1730	197	2	.	.	PUNCT
cana-1730	197	3	appl	appl	PROPN
cana-1730	197	4	.	.	PROPN
cana-1730	198	1	3	3	NUM
cana-1730	198	2	(	(	PUNCT
cana-1730	198	3	2011	2011	NUM
cana-1730	198	4	)	)	PUNCT
cana-1730	199	1	121–133	121–133	NUM
cana-1730	199	2	.	.	PUNCT
cana-1730	200	1	[	[	X
cana-1730	200	2	13	13	NUM
cana-1730	200	3	]	]	SYM
cana-1730	200	4	h.k	h.k	PROPN
cana-1730	200	5	.	.	PROPN
cana-1730	200	6	nashine	nashine	PROPN
cana-1730	200	7	,	,	PUNCT
cana-1730	200	8	b.	b.	PROPN
cana-1730	200	9	samet	samet	PROPN
cana-1730	200	10	,	,	PUNCT
cana-1730	200	11	fixed	fix	VERB
cana-1730	200	12	point	point	NOUN
cana-1730	200	13	results	result	NOUN
cana-1730	200	14	for	for	ADP
cana-1730	200	15	mappings	mapping	NOUN
cana-1730	200	16	satisfying	satisfy	VERB
cana-1730	200	17	(	(	PUNCT
cana-1730	200	18	ψ	ψ	NOUN
cana-1730	200	19	,	,	PUNCT
cana-1730	200	20	ϕ)-weakly	ϕ)-weakly	PUNCT
cana-1730	200	21	contractive	contractive	ADJ
cana-1730	200	22	condition	condition	NOUN
cana-1730	200	23	in	in	ADP
cana-1730	200	24	partially	partially	ADV
cana-1730	200	25	ordered	order	VERB
cana-1730	200	26	metric	metric	ADJ
cana-1730	200	27	spaces	space	NOUN
cana-1730	200	28	,	,	PUNCT
cana-1730	200	29	nonlinear	nonlinear	ADJ
cana-1730	200	30	anal	anal	NOUN
cana-1730	200	31	.	.	PUNCT
cana-1730	201	1	74	74	NUM
cana-1730	201	2	(	(	PUNCT
cana-1730	201	3	2011	2011	NUM
cana-1730	201	4	)	)	PUNCT
cana-1730	201	5	2201–2209	2201–2209	NUM
cana-1730	201	6	.	.	PUNCT
cana-1730	202	1	[	[	X
cana-1730	202	2	14	14	NUM
cana-1730	202	3	]	]	X
cana-1730	202	4	z.	z.	PROPN
cana-1730	202	5	kadelburg	kadelburg	PROPN
cana-1730	202	6	,	,	PUNCT
cana-1730	202	7	m.	m.	NOUN
cana-1730	202	8	pavlović	pavlović	NOUN
cana-1730	202	9	,	,	PUNCT
cana-1730	202	10	s.	s.	PROPN
cana-1730	202	11	radenović	radenović	PROPN
cana-1730	202	12	,	,	PUNCT
cana-1730	202	13	common	common	ADJ
cana-1730	202	14	fixed	fix	VERB
cana-1730	202	15	point	point	NOUN
cana-1730	202	16	theorems	theorem	NOUN
cana-1730	202	17	for	for	ADP
cana-1730	202	18	ordered	order	VERB
cana-1730	202	19	contractions	contraction	NOUN
cana-1730	202	20	and	and	CCONJ
cana-1730	202	21	quasicontractions	quasicontraction	NOUN
cana-1730	202	22	in	in	ADP
cana-1730	202	23	ordered	order	VERB
cana-1730	202	24	cone	cone	NOUN
cana-1730	202	25	metric	metric	ADJ
cana-1730	202	26	spaces	space	NOUN
cana-1730	202	27	,	,	PUNCT
cana-1730	202	28	comput	comput	NOUN
cana-1730	202	29	.	.	PUNCT
cana-1730	203	1	math	math	NOUN
cana-1730	203	2	.	.	PUNCT
cana-1730	204	1	appl	appl	PROPN
cana-1730	204	2	.	.	PUNCT
cana-1730	205	1	59	59	NUM
cana-1730	205	2	(	(	PUNCT
cana-1730	205	3	2010	2010	NUM
cana-1730	205	4	)	)	PUNCT
cana-1730	205	5	3148–3159	3148–3159	NUM
cana-1730	205	6	.	.	PUNCT
cana-1730	206	1	[	[	X
cana-1730	206	2	15	15	NUM
cana-1730	206	3	]	]	X
cana-1730	206	4	saadati	saadati	PROPN
cana-1730	206	5	,	,	PUNCT
cana-1730	206	6	s.m	s.m	PROPN
cana-1730	206	7	.	.	PROPN
cana-1730	206	8	vaezpour	vaezpour	NOUN
cana-1730	206	9	,	,	PUNCT
cana-1730	206	10	p.	p.	NOUN
cana-1730	206	11	vetro	vetro	PROPN
cana-1730	206	12	,	,	PUNCT
cana-1730	206	13	b.e	b.e	PROPN
cana-1730	206	14	.	.	PROPN
cana-1730	206	15	rhoades	rhoade	NOUN
cana-1730	206	16	,	,	PUNCT
cana-1730	206	17	fixed	fix	VERB
cana-1730	206	18	point	point	NOUN
cana-1730	206	19	theorems	theorem	NOUN
cana-1730	206	20	in	in	ADP
cana-1730	206	21	generalized	generalize	VERB
cana-1730	206	22	partially	partially	ADV
cana-1730	206	23	ordered	order	VERB
cana-1730	206	24	g	g	NOUN
cana-1730	206	25	-	-	PUNCT
cana-1730	206	26	metric	metric	ADJ
cana-1730	206	27	spaces	space	NOUN
cana-1730	206	28	,	,	PUNCT
cana-1730	206	29	math	math	NOUN
cana-1730	206	30	.	.	PUNCT
cana-1730	207	1	comput	comput	NOUN
cana-1730	207	2	.	.	PUNCT
cana-1730	208	1	modelling	model	VERB
cana-1730	208	2	52	52	NUM
cana-1730	208	3	(	(	PUNCT
cana-1730	208	4	2010	2010	NUM
cana-1730	208	5	)	)	PUNCT
cana-1730	208	6	797–801	797–801	NUM
cana-1730	208	7	.	.	PUNCT