id	sid	tid	token	lemma	pos
cana-3825	1	1	communications	communication	NOUN
cana-3825	1	2	on	on	ADP
cana-3825	1	3	applied	apply	VERB
cana-3825	1	4	nonlinear	nonlinear	ADJ
cana-3825	1	5	analysis	analysis	NOUN
cana-3825	1	6	issn	issn	NOUN
cana-3825	1	7	:	:	PUNCT
cana-3825	1	8	1074	1074	NUM
cana-3825	1	9	-	-	PUNCT
cana-3825	1	10	133x	133x	NUM
cana-3825	1	11	vol	vol	NOUN
cana-3825	1	12	32	32	NUM
cana-3825	1	13	no	no	NOUN
cana-3825	1	14	.	.	PUNCT
cana-3825	2	1	8s	8s	PROPN
cana-3825	2	2	(	(	PUNCT
cana-3825	2	3	2025	2025	NUM
cana-3825	2	4	)	)	PUNCT
cana-3825	2	5	829	829	NUM
cana-3825	2	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-3825	2	7	fractional	fractional	ADJ
cana-3825	2	8	difference	difference	NOUN
cana-3825	2	9	equations	equation	NOUN
cana-3825	2	10	with	with	ADP
cana-3825	2	11	initial	initial	ADJ
cana-3825	2	12	time	time	NOUN
cana-3825	2	13	difference	difference	NOUN
cana-3825	2	14	d.	d.	PROPN
cana-3825	2	15	naga	naga	PROPN
cana-3825	2	16	purnima	purnima	PROPN
cana-3825	2	17	1	1	NUM
cana-3825	2	18	*	*	PROPN
cana-3825	2	19	,	,	PUNCT
cana-3825	2	20	g.v.s.r	g.v.s.r	PROPN
cana-3825	2	21	.	.	PROPN
cana-3825	2	22	deekshitulu	deekshitulu	VERB
cana-3825	2	23	2	2	NUM
cana-3825	2	24	,	,	PUNCT
cana-3825	2	25	g.	g.	PROPN
cana-3825	2	26	v.	v.	ADP
cana-3825	2	27	ramana3	ramana3	PROPN
cana-3825	2	28	and	and	CCONJ
cana-3825	2	29	m.	m.	PROPN
cana-3825	2	30	bala	bala	PROPN
cana-3825	2	31	prabhakar4	prabhakar4	PROPN
cana-3825	3	1	1assistant	1assistant	NUM
cana-3825	3	2	professor	professor	NOUN
cana-3825	3	3	,	,	PUNCT
cana-3825	3	4	department	department	NOUN
cana-3825	3	5	of	of	ADP
cana-3825	3	6	mathematics	mathematics	PROPN
cana-3825	3	7	,	,	PUNCT
cana-3825	3	8	aditya	aditya	PROPN
cana-3825	3	9	university	university	PROPN
cana-3825	3	10	,	,	PUNCT
cana-3825	3	11	surampalem-533	surampalem-533	NOUN
cana-3825	3	12	437	437	NUM
cana-3825	3	13	,	,	PUNCT
cana-3825	3	14	a.	a.	NOUN
cana-3825	3	15	p.	p.	PROPN
cana-3825	3	16	,	,	PUNCT
cana-3825	3	17	india	india	PROPN
cana-3825	3	18	.	.	PUNCT
cana-3825	4	1	mail	mail	NOUN
cana-3825	4	2	:	:	PUNCT
cana-3825	4	3	nagapurnimad@adityauniversity.in	nagapurnimad@adityauniversity.in	PROPN
cana-3825	4	4	professor	professor	PROPN
cana-3825	4	5	,	,	PUNCT
cana-3825	4	6	department	department	NOUN
cana-3825	4	7	of	of	ADP
cana-3825	4	8	mathematics	mathematics	PROPN
cana-3825	4	9	,	,	PUNCT
cana-3825	4	10	jntuk	jntuk	PROPN
cana-3825	4	11	kakinada	kakinada	PROPN
cana-3825	4	12	,	,	PUNCT
cana-3825	4	13	surampalem-533	surampalem-533	VERB
cana-3825	4	14	437	437	NUM
cana-3825	4	15	,	,	PUNCT
cana-3825	4	16	a.	a.	NOUN
cana-3825	4	17	p.	p.	PROPN
cana-3825	4	18	,	,	PUNCT
cana-3825	4	19	india	india	PROPN
cana-3825	4	20	.	.	PUNCT
cana-3825	5	1	mail	mail	NOUN
cana-3825	5	2	:	:	PUNCT
cana-3825	6	1	dixitgvsr@gmail.com	dixitgvsr@gmail.com	PROPN
cana-3825	6	2	3associate	3associate	NUM
cana-3825	6	3	professor	professor	NOUN
cana-3825	6	4	,	,	PUNCT
cana-3825	6	5	department	department	NOUN
cana-3825	6	6	of	of	ADP
cana-3825	6	7	mathematics	mathematics	PROPN
cana-3825	6	8	,	,	PUNCT
cana-3825	6	9	aditya	aditya	PROPN
cana-3825	6	10	university	university	PROPN
cana-3825	6	11	,	,	PUNCT
cana-3825	6	12	surampalem-533	surampalem-533	NOUN
cana-3825	6	13	437	437	NUM
cana-3825	6	14	,	,	PUNCT
cana-3825	6	15	a.	a.	NOUN
cana-3825	6	16	p.	p.	PROPN
cana-3825	6	17	,	,	PUNCT
cana-3825	6	18	india	india	PROPN
cana-3825	6	19	.	.	PUNCT
cana-3825	7	1	mail	mail	NOUN
cana-3825	7	2	:	:	PUNCT
cana-3825	7	3	ramanaginjala9@gmail.com	ramanaginjala9@gmail.com	X
cana-3825	7	4	4associate	4associate	NUM
cana-3825	7	5	professor	professor	NOUN
cana-3825	7	6	,	,	PUNCT
cana-3825	7	7	department	department	NOUN
cana-3825	7	8	of	of	ADP
cana-3825	7	9	mathematics	mathematics	PROPN
cana-3825	7	10	,	,	PUNCT
cana-3825	7	11	aditya	aditya	PROPN
cana-3825	7	12	university	university	PROPN
cana-3825	7	13	,	,	PUNCT
cana-3825	7	14	surampalem-533	surampalem-533	NOUN
cana-3825	7	15	437	437	NUM
cana-3825	7	16	,	,	PUNCT
cana-3825	7	17	a.	a.	NOUN
cana-3825	7	18	p.	p.	PROPN
cana-3825	7	19	,	,	PUNCT
cana-3825	7	20	india	india	PROPN
cana-3825	7	21	.	.	PUNCT
cana-3825	8	1	mail	mail	NOUN
cana-3825	8	2	:	:	PUNCT
cana-3825	8	3	prabhakar_mb@yahoo.co.in	prabhakar_mb@yahoo.co.in	NOUN
cana-3825	8	4	*	*	PUNCT
cana-3825	8	5	corresponding	correspond	VERB
cana-3825	8	6	author	author	NOUN
cana-3825	8	7	:	:	PUNCT
cana-3825	8	8	d.	d.	PROPN
cana-3825	8	9	naga	naga	PROPN
cana-3825	8	10	purnima	purnima	PROPN
cana-3825	8	11	article	article	PROPN
cana-3825	8	12	history	history	NOUN
cana-3825	8	13	:	:	PUNCT
cana-3825	8	14	received	receive	VERB
cana-3825	8	15	:	:	PUNCT
cana-3825	8	16	09	09	NUM
cana-3825	8	17	-	-	SYM
cana-3825	8	18	11	11	NUM
cana-3825	8	19	-	-	PUNCT
cana-3825	8	20	2024	2024	NUM
cana-3825	8	21	revised:23	revised:23	ADJ
cana-3825	8	22	-	-	PUNCT
cana-3825	8	23	12	12	NUM
cana-3825	8	24	-	-	PUNCT
cana-3825	8	25	2024	2024	NUM
cana-3825	8	26	accepted:08	accepted:08	NOUN
cana-3825	8	27	-	-	PUNCT
cana-3825	8	28	01	01	NUM
cana-3825	8	29	-	-	PUNCT
cana-3825	8	30	2025	2025	NUM
cana-3825	8	31	abstract	abstract	NOUN
cana-3825	8	32	:	:	PUNCT
cana-3825	8	33	in	in	ADP
cana-3825	8	34	this	this	DET
cana-3825	8	35	paper	paper	NOUN
cana-3825	8	36	,	,	PUNCT
cana-3825	8	37	we	we	PRON
cana-3825	8	38	consider	consider	VERB
cana-3825	8	39	non	non	ADJ
cana-3825	8	40	-	-	ADJ
cana-3825	8	41	linear	linear	ADJ
cana-3825	8	42	fractional	fractional	ADJ
cana-3825	8	43	difference	difference	NOUN
cana-3825	8	44	equations	equation	NOUN
cana-3825	8	45	at	at	ADP
cana-3825	8	46	different	different	ADJ
cana-3825	8	47	initial	initial	ADJ
cana-3825	8	48	times	time	NOUN
cana-3825	8	49	and	and	CCONJ
cana-3825	8	50	establish	establish	VERB
cana-3825	8	51	the	the	DET
cana-3825	8	52	existence	existence	NOUN
cana-3825	8	53	of	of	ADP
cana-3825	8	54	solutions	solution	NOUN
cana-3825	8	55	using	use	VERB
cana-3825	8	56	monotone	monotone	ADJ
cana-3825	8	57	iterative	iterative	NOUN
cana-3825	8	58	technique	technique	NOUN
cana-3825	8	59	.	.	PUNCT
cana-3825	9	1	keywords	keyword	NOUN
cana-3825	9	2	:	:	PUNCT
cana-3825	9	3	fractional	fractional	ADJ
cana-3825	9	4	order	order	NOUN
cana-3825	9	5	,	,	PUNCT
cana-3825	9	6	monotone	monotone	ADJ
cana-3825	9	7	iterative	iterative	NOUN
cana-3825	9	8	technique	technique	NOUN
cana-3825	9	9	1	1	NUM
cana-3825	9	10	.	.	PUNCT
cana-3825	9	11	introduction	introduction	NOUN
cana-3825	9	12	.	.	PUNCT
cana-3825	10	1	fractional	fractional	ADJ
cana-3825	10	2	calculus	calculus	NOUN
cana-3825	10	3	gained	gain	VERB
cana-3825	10	4	importance	importance	NOUN
cana-3825	10	5	during	during	ADP
cana-3825	10	6	the	the	DET
cana-3825	10	7	past	past	ADJ
cana-3825	10	8	three	three	NUM
cana-3825	10	9	decades	decade	NOUN
cana-3825	10	10	due	due	ADP
cana-3825	10	11	to	to	ADP
cana-3825	10	12	its	its	PRON
cana-3825	10	13	applicability	applicability	NOUN
cana-3825	10	14	in	in	ADP
cana-3825	10	15	diverse	diverse	ADJ
cana-3825	10	16	fields	field	NOUN
cana-3825	10	17	of	of	ADP
cana-3825	10	18	science	science	NOUN
cana-3825	10	19	and	and	CCONJ
cana-3825	10	20	engineering	engineering	NOUN
cana-3825	10	21	.	.	PUNCT
cana-3825	11	1	the	the	DET
cana-3825	11	2	notions	notion	NOUN
cana-3825	11	3	of	of	ADP
cana-3825	11	4	fractional	fractional	ADJ
cana-3825	11	5	calculus	calculus	NOUN
cana-3825	11	6	may	may	AUX
cana-3825	11	7	be	be	AUX
cana-3825	11	8	traced	trace	VERB
cana-3825	11	9	back	back	ADV
cana-3825	11	10	to	to	ADP
cana-3825	11	11	the	the	DET
cana-3825	11	12	works	work	NOUN
cana-3825	11	13	of	of	ADP
cana-3825	11	14	euler	euler	NOUN
cana-3825	11	15	,	,	PUNCT
cana-3825	11	16	but	but	CCONJ
cana-3825	11	17	the	the	DET
cana-3825	11	18	idea	idea	NOUN
cana-3825	11	19	of	of	ADP
cana-3825	11	20	fractional	fractional	ADJ
cana-3825	11	21	difference	difference	NOUN
cana-3825	11	22	is	be	AUX
cana-3825	11	23	very	very	ADV
cana-3825	11	24	recent	recent	ADJ
cana-3825	11	25	.	.	PUNCT
cana-3825	12	1	g.v.s.r	g.v.s.r	PROPN
cana-3825	12	2	.	.	PROPN
cana-3825	12	3	deekshitulu	deekshitulu	PROPN
cana-3825	12	4	and	and	CCONJ
cana-3825	12	5	j.	j.	PROPN
cana-3825	12	6	jagan	jagan	PROPN
cana-3825	12	7	mohan	mohan	PROPN
cana-3825	13	1	[	[	X
cana-3825	13	2	7	7	NUM
cana-3825	13	3	]	]	X
cana-3825	13	4	modified	modify	VERB
cana-3825	13	5	the	the	DET
cana-3825	13	6	definition	definition	NOUN
cana-3825	13	7	of	of	ADP
cana-3825	13	8	fractional	fractional	ADJ
cana-3825	13	9	difference	difference	NOUN
cana-3825	13	10	given	give	VERB
cana-3825	13	11	by	by	ADP
cana-3825	13	12	nagai	nagai	NOUN
cana-3825	13	13	[	[	X
cana-3825	13	14	9	9	NUM
cana-3825	13	15	]	]	PUNCT
cana-3825	13	16	and	and	CCONJ
cana-3825	13	17	discussed	discuss	VERB
cana-3825	13	18	some	some	DET
cana-3825	13	19	basic	basic	ADJ
cana-3825	13	20	inequalities	inequality	NOUN
cana-3825	13	21	,	,	PUNCT
cana-3825	13	22	comparison	comparison	NOUN
cana-3825	13	23	theorems	theorem	NOUN
cana-3825	13	24	and	and	CCONJ
cana-3825	13	25	qualitative	qualitative	ADJ
cana-3825	13	26	properties	property	NOUN
cana-3825	13	27	of	of	ADP
cana-3825	13	28	the	the	DET
cana-3825	13	29	solutions	solution	NOUN
cana-3825	13	30	of	of	ADP
cana-3825	13	31	fractional	fractional	ADJ
cana-3825	13	32	difference	difference	NOUN
cana-3825	13	33	equations	equation	NOUN
cana-3825	13	34	[	[	X
cana-3825	13	35	2,3,4,5,7	2,3,4,5,7	NOUN
cana-3825	13	36	]	]	PUNCT
cana-3825	13	37	.	.	PUNCT
cana-3825	14	1	in	in	ADP
cana-3825	14	2	the	the	DET
cana-3825	14	3	study	study	NOUN
cana-3825	14	4	of	of	ADP
cana-3825	14	5	initial	initial	ADJ
cana-3825	14	6	and	and	CCONJ
cana-3825	14	7	boundary	boundary	ADJ
cana-3825	14	8	value	value	NOUN
cana-3825	14	9	problems	problem	NOUN
cana-3825	14	10	most	most	ADJ
cana-3825	14	11	of	of	ADP
cana-3825	14	12	the	the	DET
cana-3825	14	13	times	time	NOUN
cana-3825	14	14	,	,	PUNCT
cana-3825	14	15	it	it	PRON
cana-3825	14	16	is	be	AUX
cana-3825	14	17	assumed	assume	VERB
cana-3825	14	18	that	that	SCONJ
cana-3825	14	19	the	the	DET
cana-3825	14	20	independent	independent	ADJ
cana-3825	14	21	variable	variable	NOUN
cana-3825	14	22	is	be	AUX
cana-3825	14	23	unchanged	unchanged	ADJ
cana-3825	14	24	and	and	CCONJ
cana-3825	14	25	the	the	DET
cana-3825	14	26	dependent	dependent	ADJ
cana-3825	14	27	variable	variable	NOUN
cana-3825	14	28	or	or	CCONJ
cana-3825	14	29	space	space	NOUN
cana-3825	14	30	variable	variable	NOUN
cana-3825	14	31	is	be	AUX
cana-3825	14	32	perturbed	perturb	VERB
cana-3825	14	33	[	[	PUNCT
cana-3825	14	34	10,11,12	10,11,12	NUM
cana-3825	14	35	]	]	PUNCT
cana-3825	14	36	.	.	PUNCT
cana-3825	15	1	but	but	CCONJ
cana-3825	15	2	in	in	ADP
cana-3825	15	3	real	real	ADJ
cana-3825	15	4	world	world	NOUN
cana-3825	15	5	problems	problem	NOUN
cana-3825	15	6	,	,	PUNCT
cana-3825	15	7	it	it	PRON
cana-3825	15	8	is	be	AUX
cana-3825	15	9	almost	almost	ADV
cana-3825	15	10	impossible	impossible	ADJ
cana-3825	15	11	to	to	PART
cana-3825	15	12	measure	measure	VERB
cana-3825	15	13	the	the	DET
cana-3825	15	14	initial	initial	ADJ
cana-3825	15	15	time	time	NOUN
cana-3825	15	16	or	or	CCONJ
cana-3825	15	17	initial	initial	ADJ
cana-3825	15	18	value	value	NOUN
cana-3825	15	19	of	of	ADP
cana-3825	15	20	space	space	NOUN
cana-3825	15	21	variable	variable	NOUN
cana-3825	15	22	with	with	ADP
cana-3825	15	23	out	out	ADP
cana-3825	15	24	at	at	ADP
cana-3825	15	25	any	any	DET
cana-3825	15	26	error	error	NOUN
cana-3825	15	27	.	.	PUNCT
cana-3825	16	1	it	it	PRON
cana-3825	16	2	is	be	AUX
cana-3825	16	3	difficult	difficult	ADJ
cana-3825	16	4	to	to	PART
cana-3825	16	5	compare	compare	VERB
cana-3825	16	6	any	any	DET
cana-3825	16	7	two	two	NUM
cana-3825	16	8	solutions	solution	NOUN
cana-3825	16	9	if	if	SCONJ
cana-3825	16	10	the	the	DET
cana-3825	16	11	initial	initial	ADJ
cana-3825	16	12	times	time	NOUN
cana-3825	16	13	are	be	AUX
cana-3825	16	14	different	different	ADJ
cana-3825	16	15	.	.	PUNCT
cana-3825	17	1	this	this	PRON
cana-3825	17	2	has	have	AUX
cana-3825	17	3	attracted	attract	VERB
cana-3825	17	4	many	many	ADJ
cana-3825	17	5	mathematicians	mathematician	NOUN
cana-3825	17	6	to	to	PART
cana-3825	17	7	study	study	VERB
cana-3825	17	8	the	the	DET
cana-3825	17	9	corresponding	corresponding	ADJ
cana-3825	17	10	problems	problem	NOUN
cana-3825	17	11	.	.	PUNCT
cana-3825	18	1	though	though	SCONJ
cana-3825	18	2	some	some	DET
cana-3825	18	3	literature	literature	NOUN
cana-3825	18	4	is	be	AUX
cana-3825	18	5	available	available	ADJ
cana-3825	18	6	on	on	ADP
cana-3825	18	7	fractional	fractional	ADJ
cana-3825	18	8	differential	differential	ADJ
cana-3825	18	9	equations	equation	NOUN
cana-3825	18	10	with	with	ADP
cana-3825	18	11	initial	initial	ADJ
cana-3825	18	12	time	time	NOUN
cana-3825	18	13	difference	difference	NOUN
cana-3825	18	14	,	,	PUNCT
cana-3825	18	15	not	not	PART
cana-3825	18	16	much	much	ADJ
cana-3825	18	17	of	of	ADP
cana-3825	18	18	work	work	NOUN
cana-3825	18	19	has	have	AUX
cana-3825	18	20	been	be	AUX
cana-3825	18	21	yet	yet	ADV
cana-3825	18	22	done	do	VERB
cana-3825	18	23	in	in	ADP
cana-3825	18	24	discrete	discrete	ADJ
cana-3825	18	25	case	case	NOUN
cana-3825	18	26	.	.	PUNCT
cana-3825	19	1	in	in	ADP
cana-3825	19	2	this	this	DET
cana-3825	19	3	paper	paper	NOUN
cana-3825	19	4	,	,	PUNCT
cana-3825	19	5	using	use	VERB
cana-3825	19	6	lower	low	ADJ
cana-3825	19	7	solutions	solution	NOUN
cana-3825	19	8	and	and	CCONJ
cana-3825	19	9	upper	upper	ADJ
cana-3825	19	10	solutions	solution	NOUN
cana-3825	19	11	starting	start	VERB
cana-3825	19	12	at	at	ADP
cana-3825	19	13	different	different	ADJ
cana-3825	19	14	initial	initial	ADJ
cana-3825	19	15	times	time	NOUN
cana-3825	19	16	,	,	PUNCT
cana-3825	19	17	comparison	comparison	NOUN
cana-3825	19	18	and	and	CCONJ
cana-3825	19	19	existence	existence	NOUN
cana-3825	19	20	results	result	VERB
cana-3825	19	21	for	for	ADP
cana-3825	19	22	fractional	fractional	ADJ
cana-3825	19	23	difference	difference	NOUN
cana-3825	19	24	equations	equation	NOUN
cana-3825	19	25	are	be	AUX
cana-3825	19	26	established	establish	VERB
cana-3825	19	27	.	.	PUNCT
cana-3825	20	1	2	2	X
cana-3825	20	2	.	.	X
cana-3825	20	3	fractional	fractional	ADJ
cana-3825	20	4	difference	difference	NOUN
cana-3825	20	5	equations	equation	NOUN
cana-3825	20	6	with	with	ADP
cana-3825	20	7	initial	initial	ADJ
cana-3825	20	8	time	time	NOUN
cana-3825	20	9	difference	difference	NOUN
cana-3825	20	10	in	in	ADP
cana-3825	20	11	this	this	DET
cana-3825	20	12	section	section	NOUN
cana-3825	20	13	,	,	PUNCT
cana-3825	20	14	we	we	PRON
cana-3825	20	15	consider	consider	VERB
cana-3825	20	16	the	the	DET
cana-3825	20	17	following	follow	VERB
cana-3825	20	18	initial	initial	ADJ
cana-3825	20	19	value	value	NOUN
cana-3825	20	20	problem	problem	NOUN
cana-3825	20	21	(	(	PUNCT
cana-3825	20	22	ivp	ivp	NOUN
cana-3825	20	23	)	)	PUNCT
cana-3825	20	24	∇μρ(n	∇μρ(n	PROPN
cana-3825	21	1	+	+	CCONJ
cana-3825	21	2	1	1	X
cana-3825	21	3	)	)	PUNCT
cana-3825	21	4	=	=	SYM
cana-3825	21	5	f(n	f(n	PROPN
cana-3825	21	6	,	,	PUNCT
cana-3825	21	7	ρ(n	ρ(n	PROPN
cana-3825	21	8	)	)	PUNCT
cana-3825	21	9	)	)	PUNCT
cana-3825	21	10	,	,	PUNCT
cana-3825	21	11	ρ(n0	ρ(n0	NUM
cana-3825	21	12	)	)	PUNCT
cana-3825	21	13	=	=	SYM
cana-3825	21	14	ρ0	ρ0	PROPN
cana-3825	21	15	,	,	PUNCT
cana-3825	21	16	(	(	PUNCT
cana-3825	21	17	2.1	2.1	NUM
cana-3825	21	18	)	)	PUNCT
cana-3825	21	19	mailto:dixitgvsr@gmail.com	mailto:dixitgvsr@gmail.com	X
cana-3825	21	20	mailto:ramanaginjala9@gmail.com	mailto:ramanaginjala9@gmail.com	X
cana-3825	22	1	mailto:prabhakar_mb@yahoo.co.in	mailto:prabhakar_mb@yahoo.co.in	PROPN
cana-3825	22	2	communications	communication	NOUN
cana-3825	22	3	on	on	ADP
cana-3825	22	4	applied	apply	VERB
cana-3825	22	5	nonlinear	nonlinear	ADJ
cana-3825	22	6	analysis	analysis	NOUN
cana-3825	22	7	issn	issn	NOUN
cana-3825	22	8	:	:	PUNCT
cana-3825	22	9	1074	1074	NUM
cana-3825	22	10	-	-	PUNCT
cana-3825	22	11	133x	133x	NUM
cana-3825	22	12	vol	vol	NOUN
cana-3825	22	13	32	32	NUM
cana-3825	22	14	no	no	NOUN
cana-3825	22	15	.	.	PUNCT
cana-3825	23	1	8s	8s	PROPN
cana-3825	23	2	(	(	PUNCT
cana-3825	23	3	2025	2025	NUM
cana-3825	23	4	)	)	PUNCT
cana-3825	23	5	830	830	NUM
cana-3825	23	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3825	23	7	for	for	ADP
cana-3825	23	8	n	n	PRON
cana-3825	23	9	∈nn0	∈nn0	NOUN
cana-3825	23	10	+	+	X
cana-3825	23	11	,	,	PUNCT
cana-3825	23	12	n0	n0	NUM
cana-3825	23	13	≥	≥	NOUN
cana-3825	23	14	0	0	NUM
cana-3825	23	15	.	.	PUNCT
cana-3825	23	16	theorem	theorem	VERB
cana-3825	23	17	2.1	2.1	NUM
cana-3825	23	18	.	.	PUNCT
cana-3825	24	1	[	[	X
cana-3825	24	2	6	6	NUM
cana-3825	24	3	]	]	PUNCT
cana-3825	24	4	let	let	VERB
cana-3825	24	5	ξ	ξ	PROPN
cana-3825	24	6	,	,	PUNCT
cana-3825	24	7	ω	ω	PROPN
cana-3825	24	8	:	:	PUNCT
cana-3825	24	9	n0	n0	PROPN
cana-3825	25	1	+	+	CCONJ
cana-3825	25	2	→	→	SYM
cana-3825	25	3	r	r	NOUN
cana-3825	25	4	be	be	VERB
cana-3825	25	5	l.s	l.s	PROPN
cana-3825	25	6	and	and	CCONJ
cana-3825	25	7	u.s	u.s	PROPN
cana-3825	25	8	of	of	ADP
cana-3825	25	9	(	(	PUNCT
cana-3825	25	10	2.1	2.1	NUM
cana-3825	25	11	)	)	PUNCT
cana-3825	25	12	.	.	PUNCT
cana-3825	26	1	further	far	ADV
cana-3825	26	2	assume	assume	VERB
cana-3825	26	3	that	that	SCONJ
cana-3825	26	4	f	f	PROPN
cana-3825	26	5	(	(	PUNCT
cana-3825	26	6	n	n	X
cana-3825	26	7	,	,	PUNCT
cana-3825	26	8	r	r	NOUN
cana-3825	26	9	)	)	PUNCT
cana-3825	26	10	,	,	PUNCT
cana-3825	26	11	for	for	ADP
cana-3825	26	12	r∈	r∈	PROPN
cana-3825	26	13	r	r	NOUN
cana-3825	26	14	is	be	AUX
cana-3825	26	15	strictly	strictly	ADV
cana-3825	26	16	non	non	ADJ
cana-3825	26	17	decreasing	decrease	VERB
cana-3825	26	18	in	in	ADP
cana-3825	26	19	r	r	NOUN
cana-3825	26	20	for	for	ADP
cana-3825	26	21	each	each	DET
cana-3825	26	22	n.	n.	NOUN
cana-3825	26	23	then	then	ADV
cana-3825	26	24	ξ(n	ξ(n	PROPN
cana-3825	26	25	)	)	PUNCT
cana-3825	26	26	≤	≤	NOUN
cana-3825	26	27	ω(n	ω(n	NUM
cana-3825	26	28	)	)	PUNCT
cana-3825	26	29	provided	provide	VERB
cana-3825	26	30	ξ	ξ	PROPN
cana-3825	26	31	(	(	PUNCT
cana-3825	26	32	0	0	NUM
cana-3825	26	33	)	)	PUNCT
cana-3825	26	34	≤	≤	NOUN
cana-3825	26	35	ω	ω	PROPN
cana-3825	26	36	(	(	PUNCT
cana-3825	26	37	0	0	NUM
cana-3825	26	38	)	)	PUNCT
cana-3825	26	39	.	.	PUNCT
cana-3825	27	1	theorem	theorem	VERB
cana-3825	27	2	2.2.[6	2.2.[6	NUM
cana-3825	27	3	]	]	PUNCT
cana-3825	27	4	suppose	suppose	VERB
cana-3825	27	5	that	that	SCONJ
cana-3825	27	6	∇µm	∇µm	PROPN
cana-3825	27	7	(	(	PUNCT
cana-3825	27	8	n	n	NOUN
cana-3825	27	9	+	+	CCONJ
cana-3825	27	10	1	1	NUM
cana-3825	27	11	)	)	PUNCT
cana-3825	27	12	≤	≤	NUM
cana-3825	27	13	−mm(n	−mm(n	NOUN
cana-3825	27	14	)	)	PUNCT
cana-3825	27	15	for	for	ADP
cana-3825	27	16	m>0	m>0	PROPN
cana-3825	27	17	and	and	CCONJ
cana-3825	27	18	m	m	PROPN
cana-3825	27	19	(	(	PUNCT
cana-3825	27	20	0	0	NUM
cana-3825	27	21	)	)	PUNCT
cana-3825	27	22	≤	≤	NOUN
cana-3825	27	23	m(n+1	m(n+1	NUM
cana-3825	27	24	)	)	PUNCT
cana-3825	27	25	,	,	PUNCT
cana-3825	27	26	then	then	ADV
cana-3825	27	27	m(n	m(n	PROPN
cana-3825	27	28	)	)	PUNCT
cana-3825	27	29	≤	≤	NOUN
cana-3825	27	30	0	0	NUM
cana-3825	27	31	.	.	PUNCT
cana-3825	27	32	theorem	theorem	VERB
cana-3825	27	33	2.3	2.3	NUM
cana-3825	27	34	.	.	PUNCT
cana-3825	27	35	assume	assume	VERB
cana-3825	27	36	that	that	SCONJ
cana-3825	27	37	1.ξ(n	1.ξ(n	NUM
cana-3825	27	38	)	)	PUNCT
cana-3825	27	39	and	and	CCONJ
cana-3825	27	40	ω(n	ω(n	NUM
cana-3825	27	41	)	)	PUNCT
cana-3825	27	42	be	be	AUX
cana-3825	27	43	such	such	ADJ
cana-3825	27	44	that	that	PRON
cana-3825	27	45	∇μξ(n	∇μξ(n	NOUN
cana-3825	27	46	+	+	CCONJ
cana-3825	27	47	1	1	NUM
cana-3825	27	48	)	)	PUNCT
cana-3825	27	49	≤	≤	NUM
cana-3825	27	50	f(n	f(n	PROPN
cana-3825	27	51	,	,	PUNCT
cana-3825	27	52	ξ(n	ξ(n	PROPN
cana-3825	27	53	)	)	PUNCT
cana-3825	27	54	)	)	PUNCT
cana-3825	27	55	,	,	PUNCT
cana-3825	27	56	ξ(n0	ξ(n0	NOUN
cana-3825	27	57	)	)	PUNCT
cana-3825	27	58	≤	≤	NOUN
cana-3825	27	59	ρ0	ρ0	PROPN
cana-3825	27	60	,	,	PUNCT
cana-3825	27	61	n	n	PROPN
cana-3825	27	62	ϵ	ϵ	X
cana-3825	27	63	nn0	nn0	PROPN
cana-3825	28	1	+	+	CCONJ
cana-3825	28	2	∇μω(n	∇μω(n	ADJ
cana-3825	29	1	+	+	NOUN
cana-3825	29	2	1	1	NUM
cana-3825	29	3	)	)	PUNCT
cana-3825	29	4	≥	≥	NOUN
cana-3825	29	5	f(n	f(n	PROPN
cana-3825	29	6	,	,	PUNCT
cana-3825	29	7	ω(n	ω(n	NUM
cana-3825	29	8	)	)	PUNCT
cana-3825	29	9	)	)	PUNCT
cana-3825	29	10	,	,	PUNCT
cana-3825	29	11	ω(s0	ω(s0	NOUN
cana-3825	29	12	)	)	PUNCT
cana-3825	29	13	≥	≥	NOUN
cana-3825	29	14	ρ0	ρ0	PROPN
cana-3825	29	15	,	,	PUNCT
cana-3825	29	16	n	n	PROPN
cana-3825	29	17	ϵ	ϵ	X
cana-3825	29	18	ns0	ns0	NOUN
cana-3825	30	1	+	+	X
cana-3825	30	2	with	with	ADP
cana-3825	30	3	ξ(n0	ξ(n0	NOUN
cana-3825	30	4	)	)	PUNCT
cana-3825	30	5	≤	≤	NUM
cana-3825	30	6	ω(s0	ω(s0	NOUN
cana-3825	30	7	)	)	PUNCT
cana-3825	30	8	2	2	NUM
cana-3825	30	9	.	.	X
cana-3825	31	1	for	for	ADP
cana-3825	31	2	x≥y	x≥y	PROPN
cana-3825	31	3	,	,	PUNCT
cana-3825	31	4	m>0	m>0	PROPN
cana-3825	31	5	,	,	PUNCT
cana-3825	31	6	f	f	PROPN
cana-3825	31	7	(	(	PUNCT
cana-3825	31	8	n	n	CCONJ
cana-3825	31	9	,	,	PUNCT
cana-3825	31	10	x)-f	x)-f	PROPN
cana-3825	31	11	(	(	PUNCT
cana-3825	31	12	n	n	CCONJ
cana-3825	31	13	,	,	PUNCT
cana-3825	31	14	y	y	NOUN
cana-3825	31	15	)	)	PUNCT
cana-3825	31	16	≤	≤	NOUN
cana-3825	31	17	m(x	m(x	PROPN
cana-3825	31	18	-	-	PUNCT
cana-3825	31	19	y	y	NOUN
cana-3825	31	20	)	)	PUNCT
cana-3825	31	21	.	.	PUNCT
cana-3825	32	1	3	3	X
cana-3825	32	2	.	.	X
cana-3825	32	3	s0	s0	PROPN
cana-3825	32	4	>	>	PROPN
cana-3825	32	5	n0	n0	PROPN
cana-3825	32	6	and	and	CCONJ
cana-3825	32	7	f	f	PROPN
cana-3825	32	8	(	(	PUNCT
cana-3825	32	9	n	n	CCONJ
cana-3825	32	10	,	,	PUNCT
cana-3825	32	11	ρ	ρ	PROPN
cana-3825	32	12	)	)	PUNCT
cana-3825	32	13	is	be	AUX
cana-3825	32	14	nondecreasing	nondecrease	VERB
cana-3825	32	15	in	in	ADP
cana-3825	32	16	n	n	PRON
cana-3825	32	17	for	for	ADP
cana-3825	32	18	each	each	DET
cana-3825	32	19	ρ	ρ	NOUN
cana-3825	32	20	.	.	PUNCT
cana-3825	33	1	then	then	ADV
cana-3825	33	2	a.	a.	NOUN
cana-3825	33	3	ξ(n	ξ(n	PROPN
cana-3825	33	4	)	)	PUNCT
cana-3825	33	5	≤	≤	NUM
cana-3825	33	6	ω	ω	PROPN
cana-3825	33	7	(	(	PUNCT
cana-3825	33	8	n	n	PROPN
cana-3825	33	9	+	+	CCONJ
cana-3825	33	10	η	η	PROPN
cana-3825	33	11	)	)	PUNCT
cana-3825	33	12	,	,	PUNCT
cana-3825	33	13	n	n	CCONJ
cana-3825	33	14	≥	≥	NOUN
cana-3825	33	15	n0	n0	PROPN
cana-3825	33	16	,	,	PUNCT
cana-3825	33	17	b.	b.	PROPN
cana-3825	34	1	ξ	ξ	PROPN
cana-3825	34	2	(	(	PUNCT
cana-3825	34	3	n	n	CCONJ
cana-3825	34	4	−	−	PROPN
cana-3825	34	5	η	η	PROPN
cana-3825	34	6	)	)	PUNCT
cana-3825	34	7	≤	≤	NOUN
cana-3825	34	8	ω(n	ω(n	NUM
cana-3825	34	9	)	)	PUNCT
cana-3825	34	10	,	,	PUNCT
cana-3825	34	11	n	n	PRON
cana-3825	34	12	≥	≥	NOUN
cana-3825	34	13	s0	s0	NOUN
cana-3825	34	14	,	,	PUNCT
cana-3825	34	15	and	and	CCONJ
cana-3825	34	16	η	η	PROPN
cana-3825	34	17	=	=	PROPN
cana-3825	34	18	s0	s0	PROPN
cana-3825	34	19	−	−	PROPN
cana-3825	34	20	n0	n0	PROPN
cana-3825	34	21	.	.	PUNCT
cana-3825	34	22	proof	proof	NOUN
cana-3825	34	23	.	.	PUNCT
cana-3825	35	1	(	(	PUNCT
cana-3825	35	2	a	a	X
cana-3825	35	3	)	)	PUNCT
cana-3825	35	4	let	let	VERB
cana-3825	35	5	ω̃	ω̃	NUM
cana-3825	35	6	(	(	PUNCT
cana-3825	35	7	n	n	CCONJ
cana-3825	35	8	)	)	PUNCT
cana-3825	36	1	=	=	SYM
cana-3825	36	2	ω	ω	PROPN
cana-3825	36	3	(	(	PUNCT
cana-3825	36	4	n	n	PROPN
cana-3825	36	5	+	+	CCONJ
cana-3825	36	6	η	η	PROPN
cana-3825	36	7	)	)	PUNCT
cana-3825	36	8	,	,	PUNCT
cana-3825	36	9	n	n	PRON
cana-3825	36	10	≥	≥	NOUN
cana-3825	36	11	nn0	nn0	PROPN
cana-3825	37	1	+	+	X
cana-3825	37	2	,	,	PUNCT
cana-3825	37	3	ω̃(n0	ω̃(n0	NUM
cana-3825	37	4	)	)	PUNCT
cana-3825	38	1	=	=	SYM
cana-3825	38	2	ω	ω	PROPN
cana-3825	38	3	(	(	PUNCT
cana-3825	38	4	n0	n0	X
cana-3825	38	5	+	+	CCONJ
cana-3825	38	6	η	η	PROPN
cana-3825	38	7	)	)	PUNCT
cana-3825	38	8	=	=	SYM
cana-3825	38	9	ω(s0	ω(s0	NOUN
cana-3825	38	10	)	)	PUNCT
cana-3825	38	11	≥	≥	NOUN
cana-3825	38	12	ρ0	ρ0	PROPN
cana-3825	38	13	.	.	PUNCT
cana-3825	39	1	since	since	SCONJ
cana-3825	39	2	f	f	PROPN
cana-3825	39	3	is	be	AUX
cana-3825	39	4	nondecreasing	nondecrease	VERB
cana-3825	39	5	in	in	ADP
cana-3825	39	6	n	n	PRON
cana-3825	39	7	for	for	ADP
cana-3825	39	8	each	each	DET
cana-3825	39	9	ρ	ρ	NOUN
cana-3825	39	10	.	.	PUNCT
cana-3825	40	1	also	also	ADV
cana-3825	40	2	∇μω̃	∇μω̃	NUM
cana-3825	41	1	(	(	PUNCT
cana-3825	41	2	n	n	X
cana-3825	41	3	+	+	CCONJ
cana-3825	41	4	1	1	NUM
cana-3825	41	5	)	)	PUNCT
cana-3825	41	6	=	=	SYM
cana-3825	41	7	∇μω̃	∇μω̃	NOUN
cana-3825	41	8	(	(	PUNCT
cana-3825	41	9	n	n	X
cana-3825	41	10	+	+	CCONJ
cana-3825	41	11	η	η	PROPN
cana-3825	41	12	+	+	PROPN
cana-3825	41	13	1	1	NUM
cana-3825	41	14	)	)	PUNCT
cana-3825	41	15	=	=	SYM
cana-3825	41	16	f	f	X
cana-3825	41	17	(	(	PUNCT
cana-3825	41	18	n	n	PROPN
cana-3825	41	19	+	+	CCONJ
cana-3825	41	20	η	η	PROPN
cana-3825	41	21	,	,	PUNCT
cana-3825	41	22	ω	ω	PROPN
cana-3825	41	23	(	(	PUNCT
cana-3825	41	24	n	n	PROPN
cana-3825	41	25	+	+	CCONJ
cana-3825	41	26	η	η	NOUN
cana-3825	41	27	)	)	PUNCT
cana-3825	41	28	)	)	PUNCT
cana-3825	41	29	≥	≥	PROPN
cana-3825	42	1	f	f	X
cana-3825	42	2	(	(	PUNCT
cana-3825	42	3	n	n	CCONJ
cana-3825	42	4	,	,	PUNCT
cana-3825	42	5	ω̃	ω̃	PROPN
cana-3825	42	6	(	(	PUNCT
cana-3825	42	7	n	n	CCONJ
cana-3825	42	8	)	)	PUNCT
cana-3825	42	9	)	)	PUNCT
cana-3825	42	10	.	.	PUNCT
cana-3825	43	1	consolidating	consolidate	VERB
cana-3825	43	2	the	the	DET
cana-3825	43	3	above	above	NOUN
cana-3825	43	4	,	,	PUNCT
cana-3825	43	5	∇μω̃	∇μω̃	NUM
cana-3825	43	6	(	(	PUNCT
cana-3825	43	7	n	n	X
cana-3825	43	8	+	+	CCONJ
cana-3825	43	9	1	1	NUM
cana-3825	43	10	)	)	PUNCT
cana-3825	43	11	≥	≥	NOUN
cana-3825	43	12	f	f	X
cana-3825	43	13	(	(	PUNCT
cana-3825	43	14	n	n	CCONJ
cana-3825	43	15	,	,	PUNCT
cana-3825	43	16	ω̃	ω̃	PROPN
cana-3825	43	17	(	(	PUNCT
cana-3825	43	18	n	n	CCONJ
cana-3825	43	19	)	)	PUNCT
cana-3825	43	20	)	)	PUNCT
cana-3825	43	21	,	,	PUNCT
cana-3825	43	22	ω̃	ω̃	NUM
cana-3825	43	23	(	(	PUNCT
cana-3825	43	24	n0	n0	ADJ
cana-3825	43	25	)	)	PUNCT
cana-3825	43	26	≥ρ0	≥ρ0	NOUN
cana-3825	43	27	.	.	PUNCT
cana-3825	44	1	∇μξ	∇μξ	NOUN
cana-3825	44	2	(	(	PUNCT
cana-3825	44	3	n	n	NOUN
cana-3825	44	4	+	+	CCONJ
cana-3825	44	5	1	1	NUM
cana-3825	44	6	)	)	PUNCT
cana-3825	44	7	≤	≤	NUM
cana-3825	44	8	f	f	X
cana-3825	44	9	(	(	PUNCT
cana-3825	44	10	n	n	X
cana-3825	44	11	,	,	PUNCT
cana-3825	44	12	ξ(n	ξ(n	PROPN
cana-3825	44	13	)	)	PUNCT
cana-3825	44	14	)	)	PUNCT
cana-3825	44	15	,	,	PUNCT
cana-3825	44	16	ξ	ξ	PROPN
cana-3825	44	17	(	(	PUNCT
cana-3825	44	18	n0	n0	NUM
cana-3825	44	19	)	)	PUNCT
cana-3825	44	20	≤	≤	NOUN
cana-3825	44	21	ρ0	ρ0	PROPN
cana-3825	44	22	.	.	PUNCT
cana-3825	45	1	using	use	VERB
cana-3825	45	2	theorem	theorem	NOUN
cana-3825	45	3	2.1	2.1	NUM
cana-3825	45	4	,	,	PUNCT
cana-3825	45	5	clearly	clearly	ADV
cana-3825	45	6	ξ(n	ξ(n	NOUN
cana-3825	45	7	)	)	PUNCT
cana-3825	45	8	≤	≤	NUM
cana-3825	45	9	ω̃	ω̃	NUM
cana-3825	45	10	(	(	PUNCT
cana-3825	45	11	n	n	CCONJ
cana-3825	45	12	)	)	PUNCT
cana-3825	45	13	=	=	SYM
cana-3825	45	14	ω	ω	PROPN
cana-3825	45	15	(	(	PUNCT
cana-3825	45	16	n	n	PROPN
cana-3825	45	17	+	+	CCONJ
cana-3825	45	18	η	η	NOUN
cana-3825	45	19	)	)	PUNCT
cana-3825	45	20	.	.	PUNCT
cana-3825	46	1	(	(	PUNCT
cana-3825	46	2	b	b	X
cana-3825	46	3	)	)	PUNCT
cana-3825	46	4	let	let	VERB
cana-3825	46	5	ξ̃(n	ξ̃(n	PROPN
cana-3825	46	6	)	)	PUNCT
cana-3825	47	1	=	=	SYM
cana-3825	47	2	ξ(n	ξ(n	PROPN
cana-3825	47	3	−	−	PROPN
cana-3825	47	4	η	η	PROPN
cana-3825	47	5	)	)	PUNCT
cana-3825	47	6	,	,	PUNCT
cana-3825	47	7	n	n	PRON
cana-3825	47	8	≥	≥	NOUN
cana-3825	47	9	nn0	nn0	PROPN
cana-3825	48	1	+	+	X
cana-3825	48	2	for	for	ADP
cana-3825	48	3	n	n	NOUN
cana-3825	48	4	=	=	SYM
cana-3825	48	5	s0	s0	PROPN
cana-3825	48	6	,	,	PUNCT
cana-3825	48	7	ξ̃(s0	ξ̃(s0	PART
cana-3825	48	8	)	)	PUNCT
cana-3825	49	1	=	=	SYM
cana-3825	49	2	ξ	ξ	PROPN
cana-3825	49	3	(	(	PUNCT
cana-3825	49	4	s0	s0	PROPN
cana-3825	49	5	−	−	PROPN
cana-3825	49	6	η	η	PROPN
cana-3825	49	7	)	)	PUNCT
cana-3825	49	8	=	=	SYM
cana-3825	49	9	ξ	ξ	PROPN
cana-3825	49	10	(	(	PUNCT
cana-3825	49	11	n0	n0	NUM
cana-3825	49	12	)	)	PUNCT
cana-3825	49	13	≤	≤	NOUN
cana-3825	49	14	ρ0	ρ0	PROPN
cana-3825	49	15	.	.	PUNCT
cana-3825	50	1	now	now	ADV
cana-3825	50	2	,	,	PUNCT
cana-3825	50	3	since	since	SCONJ
cana-3825	50	4	f	f	PROPN
cana-3825	50	5	is	be	AUX
cana-3825	50	6	non	non	PRON
cana-3825	50	7	decreasing	decrease	VERB
cana-3825	50	8	in	in	ADP
cana-3825	50	9	n	n	CCONJ
cana-3825	50	10	,	,	PUNCT
cana-3825	50	11	∇μξ̃	∇μξ̃	NUM
cana-3825	50	12	(	(	PUNCT
cana-3825	50	13	n	n	X
cana-3825	50	14	+	+	CCONJ
cana-3825	50	15	1	1	NUM
cana-3825	51	1	)	)	PUNCT
cana-3825	51	2	=	=	SYM
cana-3825	51	3	f	f	X
cana-3825	51	4	(	(	PUNCT
cana-3825	51	5	n	n	CCONJ
cana-3825	51	6	−	−	PROPN
cana-3825	51	7	η	η	PROPN
cana-3825	51	8	,	,	PUNCT
cana-3825	51	9	ξ	ξ	PROPN
cana-3825	51	10	(	(	PUNCT
cana-3825	51	11	n	n	CCONJ
cana-3825	51	12	−	−	PROPN
cana-3825	51	13	η	η	PROPN
cana-3825	51	14	)	)	PUNCT
cana-3825	51	15	)	)	PUNCT
cana-3825	51	16	≤	≤	NUM
cana-3825	51	17	f(n	f(n	PROPN
cana-3825	51	18	,	,	PUNCT
cana-3825	51	19	ξ̃(n	ξ̃(n	PROPN
cana-3825	51	20	)	)	PUNCT
cana-3825	51	21	)	)	PUNCT
cana-3825	51	22	.	.	PUNCT
cana-3825	52	1	consolidating	consolidate	VERB
cana-3825	52	2	the	the	DET
cana-3825	52	3	above	above	NOUN
cana-3825	52	4	,	,	PUNCT
cana-3825	52	5	∇μξ̃	∇μξ̃	NUM
cana-3825	52	6	(	(	PUNCT
cana-3825	52	7	n	n	X
cana-3825	52	8	+	+	CCONJ
cana-3825	52	9	1	1	NUM
cana-3825	52	10	)	)	PUNCT
cana-3825	52	11	≤	≤	NUM
cana-3825	52	12	f	f	X
cana-3825	52	13	(	(	PUNCT
cana-3825	52	14	n	n	CCONJ
cana-3825	52	15	,	,	PUNCT
cana-3825	52	16	ξ̃	ξ̃	PROPN
cana-3825	52	17	(	(	PUNCT
cana-3825	52	18	n	n	CCONJ
cana-3825	52	19	)	)	PUNCT
cana-3825	52	20	)	)	PUNCT
cana-3825	52	21	,	,	PUNCT
cana-3825	52	22	ξ̃	ξ̃	PROPN
cana-3825	52	23	(	(	PUNCT
cana-3825	52	24	s0	s0	PROPN
cana-3825	52	25	)	)	PUNCT
cana-3825	52	26	≤ρ0	≤ρ0	NOUN
cana-3825	52	27	,	,	PUNCT
cana-3825	52	28	communications	communication	NOUN
cana-3825	52	29	on	on	ADP
cana-3825	52	30	applied	apply	VERB
cana-3825	52	31	nonlinear	nonlinear	ADJ
cana-3825	52	32	analysis	analysis	NOUN
cana-3825	52	33	issn	issn	NOUN
cana-3825	52	34	:	:	PUNCT
cana-3825	52	35	1074	1074	NUM
cana-3825	52	36	-	-	PUNCT
cana-3825	52	37	133x	133x	NUM
cana-3825	52	38	vol	vol	NOUN
cana-3825	52	39	32	32	NUM
cana-3825	52	40	no	no	NOUN
cana-3825	52	41	.	.	PUNCT
cana-3825	53	1	8s	8s	PROPN
cana-3825	53	2	(	(	PUNCT
cana-3825	53	3	2025	2025	NUM
cana-3825	53	4	)	)	PUNCT
cana-3825	53	5	831	831	NUM
cana-3825	53	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3825	53	7	∇μω(n	∇μω(n	NOUN
cana-3825	53	8	+	+	CCONJ
cana-3825	53	9	1	1	X
cana-3825	53	10	)	)	PUNCT
cana-3825	53	11	≥	≥	NOUN
cana-3825	53	12	f	f	X
cana-3825	53	13	(	(	PUNCT
cana-3825	53	14	n	n	CCONJ
cana-3825	53	15	,	,	PUNCT
cana-3825	53	16	ω(n	ω(n	NUM
cana-3825	53	17	)	)	PUNCT
cana-3825	53	18	)	)	PUNCT
cana-3825	53	19	,	,	PUNCT
cana-3825	53	20	ω	ω	PROPN
cana-3825	53	21	(	(	PUNCT
cana-3825	53	22	n0	n0	PROPN
cana-3825	53	23	)	)	PUNCT
cana-3825	53	24	≥	≥	NOUN
cana-3825	53	25	ρ0	ρ0	PROPN
cana-3825	53	26	.	.	PUNCT
cana-3825	54	1	by	by	ADP
cana-3825	54	2	using	use	VERB
cana-3825	54	3	theorem	theorem	NOUN
cana-3825	54	4	2.1	2.1	NUM
cana-3825	54	5	,	,	PUNCT
cana-3825	54	6	ξ	ξ	PROPN
cana-3825	54	7	(	(	PUNCT
cana-3825	54	8	n	n	CCONJ
cana-3825	54	9	−	−	PROPN
cana-3825	54	10	η	η	PROPN
cana-3825	54	11	)	)	PUNCT
cana-3825	54	12	=	=	SYM
cana-3825	54	13	ξ̃(n	ξ̃(n	PROPN
cana-3825	54	14	)	)	PUNCT
cana-3825	54	15	≤	≤	NOUN
cana-3825	54	16	ω(n	ω(n	NUM
cana-3825	54	17	)	)	PUNCT
cana-3825	54	18	,	,	PUNCT
cana-3825	54	19	for	for	ADP
cana-3825	54	20	n	n	PRON
cana-3825	54	21	≥s0	≥s0	ADJ
cana-3825	54	22	.	.	PUNCT
cana-3825	55	1	here	here	ADV
cana-3825	55	2	we	we	PRON
cana-3825	55	3	shall	shall	AUX
cana-3825	55	4	discuss	discuss	VERB
cana-3825	55	5	the	the	DET
cana-3825	55	6	monotone	monotone	ADJ
cana-3825	55	7	iterative	iterative	NOUN
cana-3825	55	8	method	method	NOUN
cana-3825	55	9	to	to	PART
cana-3825	55	10	obtain	obtain	VERB
cana-3825	55	11	extremal	extremal	ADJ
cana-3825	55	12	solutions	solution	NOUN
cana-3825	55	13	for	for	ADP
cana-3825	55	14	fractional	fractional	ADJ
cana-3825	55	15	difference	difference	NOUN
cana-3825	55	16	equations	equation	NOUN
cana-3825	55	17	with	with	ADP
cana-3825	55	18	initial	initial	ADJ
cana-3825	55	19	time	time	NOUN
cana-3825	55	20	difference	difference	NOUN
cana-3825	55	21	.	.	PUNCT
cana-3825	56	1	theorem	theorem	VERB
cana-3825	56	2	2.4	2.4	NUM
cana-3825	56	3	.	.	PUNCT
cana-3825	57	1	assume	assume	VERB
cana-3825	57	2	that	that	SCONJ
cana-3825	57	3	1	1	X
cana-3825	57	4	.	.	PUNCT
cana-3825	58	1	∇μξ	∇μξ	PROPN
cana-3825	58	2	(	(	PUNCT
cana-3825	58	3	n	n	NOUN
cana-3825	58	4	+	+	CCONJ
cana-3825	58	5	1	1	NUM
cana-3825	58	6	)	)	PUNCT
cana-3825	58	7	≤	≤	NUM
cana-3825	58	8	f	f	X
cana-3825	58	9	(	(	PUNCT
cana-3825	58	10	n	n	X
cana-3825	58	11	,	,	PUNCT
cana-3825	58	12	ξ(n	ξ(n	PROPN
cana-3825	58	13	)	)	PUNCT
cana-3825	58	14	)	)	PUNCT
cana-3825	58	15	,	,	PUNCT
cana-3825	58	16	ξ	ξ	PROPN
cana-3825	58	17	(	(	PUNCT
cana-3825	58	18	n0	n0	NUM
cana-3825	58	19	)	)	PUNCT
cana-3825	58	20	=	=	SYM
cana-3825	58	21	ρ0	ρ0	PROPN
cana-3825	58	22	,	,	PUNCT
cana-3825	58	23	n	n	NOUN
cana-3825	58	24	∈	∈	PROPN
cana-3825	58	25	nn0	nn0	NOUN
cana-3825	59	1	+	+	X
cana-3825	59	2	,	,	PUNCT
cana-3825	59	3	n0	n0	NUM
cana-3825	59	4	≥	≥	NOUN
cana-3825	59	5	0	0	NUM
cana-3825	59	6	,	,	PUNCT
cana-3825	59	7	∇μω	∇μω	NOUN
cana-3825	59	8	(	(	PUNCT
cana-3825	59	9	n	n	PROPN
cana-3825	59	10	+	+	CCONJ
cana-3825	59	11	1	1	NUM
cana-3825	59	12	)	)	PUNCT
cana-3825	59	13	≥	≥	NOUN
cana-3825	59	14	f	f	X
cana-3825	59	15	(	(	PUNCT
cana-3825	59	16	n	n	CCONJ
cana-3825	59	17	,	,	PUNCT
cana-3825	59	18	ω(n	ω(n	NUM
cana-3825	59	19	)	)	PUNCT
cana-3825	59	20	)	)	PUNCT
cana-3825	59	21	,	,	PUNCT
cana-3825	59	22	ω	ω	PROPN
cana-3825	59	23	(	(	PUNCT
cana-3825	59	24	s0	s0	PROPN
cana-3825	59	25	)	)	PUNCT
cana-3825	59	26	=	=	SYM
cana-3825	59	27	ρ0	ρ0	PROPN
cana-3825	59	28	,	,	PUNCT
cana-3825	59	29	n	n	NOUN
cana-3825	59	30	∈	∈	NOUN
cana-3825	59	31	ns0	ns0	NOUN
cana-3825	60	1	+	+	X
cana-3825	60	2	,	,	PUNCT
cana-3825	60	3	such	such	ADJ
cana-3825	60	4	that	that	SCONJ
cana-3825	60	5	ξ0	ξ0	PROPN
cana-3825	60	6	≤	≤	PROPN
cana-3825	60	7	ω0	ω0	NOUN
cana-3825	60	8	,	,	PUNCT
cana-3825	60	9	2	2	NUM
cana-3825	60	10	.	.	X
cana-3825	60	11	f	f	PROPN
cana-3825	60	12	(	(	PUNCT
cana-3825	60	13	n	n	CCONJ
cana-3825	60	14	,	,	PUNCT
cana-3825	60	15	x	x	NOUN
cana-3825	60	16	)	)	PUNCT
cana-3825	60	17	–	–	PUNCT
cana-3825	60	18	f	f	PROPN
cana-3825	60	19	(	(	PUNCT
cana-3825	60	20	n	n	CCONJ
cana-3825	60	21	,	,	PUNCT
cana-3825	60	22	y	y	PROPN
cana-3825	60	23	)	)	PUNCT
cana-3825	60	24	≥	≥	PROPN
cana-3825	60	25	−m	−m	NOUN
cana-3825	60	26	(	(	PUNCT
cana-3825	60	27	x	x	PROPN
cana-3825	60	28	−	−	PROPN
cana-3825	60	29	y	y	PROPN
cana-3825	60	30	)	)	PUNCT
cana-3825	60	31	,	,	PUNCT
cana-3825	60	32	where	where	SCONJ
cana-3825	60	33	m	m	VERB
cana-3825	60	34	>	>	X
cana-3825	60	35	0	0	PUNCT
cana-3825	60	36	and	and	CCONJ
cana-3825	60	37	y	y	PROPN
cana-3825	60	38	≤	≤	PROPN
cana-3825	60	39	x	x	X
cana-3825	60	40	,	,	PUNCT
cana-3825	60	41	3	3	NUM
cana-3825	60	42	.	.	PUNCT
cana-3825	60	43	s0	s0	PROPN
cana-3825	60	44	>	>	PROPN
cana-3825	60	45	n0	n0	PROPN
cana-3825	60	46	>	>	X
cana-3825	60	47	0	0	PUNCT
cana-3825	60	48	and	and	CCONJ
cana-3825	60	49	f	f	PROPN
cana-3825	60	50	(	(	PUNCT
cana-3825	60	51	n	n	CCONJ
cana-3825	60	52	,	,	PUNCT
cana-3825	60	53	ρ(n	ρ(n	PROPN
cana-3825	60	54	)	)	PUNCT
cana-3825	60	55	)	)	PUNCT
cana-3825	60	56	is	be	AUX
cana-3825	60	57	nondecreasing	nondecrease	VERB
cana-3825	60	58	n	n	INTJ
cana-3825	60	59	for	for	ADP
cana-3825	60	60	each	each	DET
cana-3825	60	61	ρ(n	ρ(n	PROPN
cana-3825	60	62	)	)	PUNCT
cana-3825	60	63	and	and	CCONJ
cana-3825	60	64	ξ(n	ξ(n	PROPN
cana-3825	60	65	)	)	PUNCT
cana-3825	60	66	≤	≤	NUM
cana-3825	60	67	ω	ω	PROPN
cana-3825	60	68	(	(	PUNCT
cana-3825	60	69	n	n	PROPN
cana-3825	60	70	+	+	CCONJ
cana-3825	60	71	η	η	PROPN
cana-3825	60	72	)	)	PUNCT
cana-3825	60	73	,	,	PUNCT
cana-3825	60	74	η	η	PROPN
cana-3825	60	75	=	=	PROPN
cana-3825	60	76	s0	s0	PROPN
cana-3825	60	77	−	−	PROPN
cana-3825	60	78	n0	n0	PROPN
cana-3825	60	79	.	.	PUNCT
cana-3825	61	1	then	then	ADV
cana-3825	61	2	there	there	PRON
cana-3825	61	3	exit	exit	NOUN
cana-3825	61	4	monotone	monotone	ADJ
cana-3825	61	5	sequences	sequence	NOUN
cana-3825	61	6	ξ̃n	ξ̃n	PROPN
cana-3825	61	7	,	,	PUNCT
cana-3825	61	8	ω̃n	ω̃n	PUNCT
cana-3825	61	9	which	which	PRON
cana-3825	61	10	converge	converge	VERB
cana-3825	61	11	uniformly	uniformly	ADV
cana-3825	61	12	and	and	CCONJ
cana-3825	61	13	monotonically	monotonically	ADV
cana-3825	61	14	for	for	ADP
cana-3825	61	15	n	n	DET
cana-3825	61	16	∈	∈	PROPN
cana-3825	61	17	nn0	nn0	NOUN
cana-3825	61	18	+	+	CCONJ
cana-3825	61	19	such	such	ADJ
cana-3825	61	20	that	that	SCONJ
cana-3825	61	21	ξ̃m	ξ̃m	NOUN
cana-3825	61	22	→	→	SYM
cana-3825	61	23	ξ̃	ξ̃	PROPN
cana-3825	61	24	and	and	CCONJ
cana-3825	61	25	ω̃m	ω̃m	PRON
cana-3825	61	26	→	→	SYM
cana-3825	61	27	ω̃	ω̃	NUM
cana-3825	61	28	for	for	ADP
cana-3825	61	29	m	m	PROPN
cana-3825	61	30	→	→	SYM
cana-3825	61	31	∞.	∞.	PROPN
cana-3825	61	32	moreover	moreover	ADV
cana-3825	61	33	ξ̃	ξ̃	PROPN
cana-3825	61	34	and	and	CCONJ
cana-3825	61	35	ω̃	ω̃	NUM
cana-3825	61	36	are	be	AUX
cana-3825	61	37	minimal	minimal	ADJ
cana-3825	61	38	and	and	CCONJ
cana-3825	61	39	maximal	maximal	ADJ
cana-3825	61	40	solutions	solution	NOUN
cana-3825	61	41	of	of	ADP
cana-3825	61	42	(	(	PUNCT
cana-3825	61	43	2.1	2.1	NUM
cana-3825	61	44	)	)	PUNCT
cana-3825	61	45	.	.	PUNCT
cana-3825	62	1	proof	proof	NOUN
cana-3825	62	2	:	:	PUNCT
cana-3825	62	3	let	let	VERB
cana-3825	62	4	ω̃0(n	ω̃0(n	X
cana-3825	62	5	)	)	PUNCT
cana-3825	63	1	=	=	SYM
cana-3825	63	2	ω	ω	PROPN
cana-3825	63	3	(	(	PUNCT
cana-3825	63	4	n	n	PROPN
cana-3825	63	5	+	+	CCONJ
cana-3825	63	6	η	η	NOUN
cana-3825	63	7	)	)	PUNCT
cana-3825	63	8	and	and	CCONJ
cana-3825	63	9	ξ̃0(n	ξ̃0(n	PROPN
cana-3825	63	10	)	)	PUNCT
cana-3825	63	11	=	=	SYM
cana-3825	63	12	ξ(n	ξ(n	PROPN
cana-3825	63	13	)	)	PUNCT
cana-3825	63	14	for	for	ADP
cana-3825	63	15	n	n	PRON
cana-3825	63	16	∈	∈	PROPN
cana-3825	63	17	nn0	nn0	NOUN
cana-3825	63	18	+	+	X
cana-3825	63	19	where	where	SCONJ
cana-3825	63	20	η	η	PROPN
cana-3825	63	21	=	=	PROPN
cana-3825	63	22	s0	s0	PROPN
cana-3825	63	23	−	−	PROPN
cana-3825	63	24	n0	n0	PROPN
cana-3825	63	25	.	.	PUNCT
cana-3825	64	1	let	let	VERB
cana-3825	64	2	θ	θ	X
cana-3825	64	3	:	:	PUNCT
cana-3825	64	4	nn0	nn0	PROPN
cana-3825	65	1	+	+	X
cana-3825	65	2	→	→	SYM
cana-3825	65	3	r	r	NOUN
cana-3825	65	4	be	be	VERB
cana-3825	65	5	such	such	ADJ
cana-3825	65	6	that	that	SCONJ
cana-3825	65	7	ξ̃0(n	ξ̃0(n	NOUN
cana-3825	65	8	)	)	PUNCT
cana-3825	65	9	≤	≤	NUM
cana-3825	66	1	θ	θ	PROPN
cana-3825	66	2	≤	≤	NUM
cana-3825	66	3	ξ̃0(n	ξ̃0(n	PROPN
cana-3825	66	4	)	)	PUNCT
cana-3825	66	5	,	,	PUNCT
cana-3825	66	6	and	and	CCONJ
cana-3825	66	7	the	the	DET
cana-3825	66	8	following	follow	VERB
cana-3825	66	9	linear	linear	ADJ
cana-3825	66	10	fractional	fractional	ADJ
cana-3825	66	11	difference	difference	NOUN
cana-3825	66	12	equation	equation	NOUN
cana-3825	66	13	for	for	ADP
cana-3825	66	14	0	0	NUM
cana-3825	66	15	<	<	X
cana-3825	66	16	µ	µ	X
cana-3825	66	17	<	<	X
cana-3825	66	18	1	1	NUM
cana-3825	66	19	,	,	PUNCT
cana-3825	66	20	∇μρ	∇μρ	PROPN
cana-3825	66	21	(	(	PUNCT
cana-3825	66	22	n	n	NOUN
cana-3825	66	23	+	+	NOUN
cana-3825	66	24	1	1	NUM
cana-3825	66	25	)	)	PUNCT
cana-3825	66	26	=	=	SYM
cana-3825	66	27	f	f	PROPN
cana-3825	66	28	(	(	PUNCT
cana-3825	66	29	n	n	CCONJ
cana-3825	66	30	,	,	PUNCT
cana-3825	66	31	θ(n	θ(n	NOUN
cana-3825	66	32	)	)	PUNCT
cana-3825	66	33	)	)	PUNCT
cana-3825	66	34	−	−	ADP
cana-3825	67	1	m[ρ(n	m[ρ(n	X
cana-3825	67	2	)	)	PUNCT
cana-3825	67	3	−	−	PRON
cana-3825	67	4	θ(n	θ(n	NOUN
cana-3825	67	5	)	)	PUNCT
cana-3825	67	6	]	]	PUNCT
cana-3825	67	7	,	,	PUNCT
cana-3825	67	8	ρ	ρ	PROPN
cana-3825	67	9	(	(	PUNCT
cana-3825	67	10	n0	n0	NUM
cana-3825	67	11	)	)	PUNCT
cana-3825	67	12	=	=	SYM
cana-3825	67	13	ρ0	ρ0	PROPN
cana-3825	67	14	.	.	PUNCT
cana-3825	68	1	(	(	PUNCT
cana-3825	68	2	2.2	2.2	NUM
cana-3825	68	3	)	)	PUNCT
cana-3825	68	4	it	it	PRON
cana-3825	68	5	is	be	AUX
cana-3825	68	6	clear	clear	ADJ
cana-3825	68	7	that	that	SCONJ
cana-3825	68	8	,	,	PUNCT
cana-3825	68	9	if	if	SCONJ
cana-3825	68	10	θ(n	θ(n	NOUN
cana-3825	68	11	)	)	PUNCT
cana-3825	68	12	=	=	SYM
cana-3825	68	13	ρ(n	ρ(n	PROPN
cana-3825	68	14	)	)	PUNCT
cana-3825	68	15	,	,	PUNCT
cana-3825	68	16	ρ(n	ρ(n	PROPN
cana-3825	68	17	)	)	PUNCT
cana-3825	68	18	is	be	AUX
cana-3825	68	19	the	the	DET
cana-3825	68	20	unique	unique	ADJ
cana-3825	68	21	solution	solution	NOUN
cana-3825	68	22	of	of	ADP
cana-3825	68	23	(	(	PUNCT
cana-3825	68	24	3.1	3.1	NUM
cana-3825	68	25	)	)	PUNCT
cana-3825	68	26	on	on	ADP
cana-3825	68	27	n	n	DET
cana-3825	68	28	∈	∈	PROPN
cana-3825	68	29	n0	n0	PROPN
cana-3825	68	30	+	+	X
cana-3825	68	31	.	.	PUNCT
cana-3825	69	1	if	if	SCONJ
cana-3825	69	2	θ(n	θ(n	NOUN
cana-3825	69	3	)	)	PUNCT
cana-3825	69	4	≠	≠	PROPN
cana-3825	69	5	ρ(n	ρ(n	PROPN
cana-3825	69	6	)	)	PUNCT
cana-3825	69	7	,	,	PUNCT
cana-3825	69	8	the	the	DET
cana-3825	69	9	non	non	ADJ
cana-3825	69	10	-	-	ADJ
cana-3825	69	11	homogeneous	homogeneous	ADJ
cana-3825	69	12	fractional	fractional	ADJ
cana-3825	69	13	difference	difference	NOUN
cana-3825	69	14	equation	equation	NOUN
cana-3825	69	15	(	(	PUNCT
cana-3825	69	16	2.2	2.2	NUM
cana-3825	69	17	)	)	PUNCT
cana-3825	69	18	has	have	VERB
cana-3825	69	19	unique	unique	ADJ
cana-3825	69	20	solution	solution	NOUN
cana-3825	69	21	ρn	ρn	ADP
cana-3825	70	1	[	[	X
cana-3825	70	2	1	1	NUM
cana-3825	70	3	]	]	PUNCT
cana-3825	70	4	.	.	PUNCT
cana-3825	71	1	in	in	ADP
cana-3825	71	2	order	order	NOUN
cana-3825	71	3	to	to	PART
cana-3825	71	4	construct	construct	VERB
cana-3825	71	5	monotone	monotone	ADJ
cana-3825	71	6	sequences	sequence	NOUN
cana-3825	71	7	ξ̃	ξ̃	PROPN
cana-3825	71	8	(	(	PUNCT
cana-3825	71	9	n	n	CCONJ
cana-3825	71	10	)	)	PUNCT
cana-3825	71	11	,	,	PUNCT
cana-3825	71	12	ω̃	ω̃	NUM
cana-3825	71	13	(	(	PUNCT
cana-3825	71	14	n	n	CCONJ
cana-3825	71	15	)	)	PUNCT
cana-3825	71	16	,	,	PUNCT
cana-3825	71	17	a	a	DET
cana-3825	71	18	mapping	mapping	NOUN
cana-3825	71	19	a	a	X
cana-3825	71	20	:	:	PUNCT
cana-3825	71	21	r	r	NOUN
cana-3825	71	22	→	→	SYM
cana-3825	71	23	r	r	NOUN
cana-3825	71	24	is	be	AUX
cana-3825	71	25	defined	define	VERB
cana-3825	71	26	such	such	ADJ
cana-3825	71	27	that	that	SCONJ
cana-3825	71	28	aθ(n	aθ(n	X
cana-3825	71	29	)	)	PUNCT
cana-3825	71	30	=	=	SYM
cana-3825	71	31	ρ(n	ρ(n	PROPN
cana-3825	71	32	)	)	PUNCT
cana-3825	71	33	.	.	PUNCT
cana-3825	72	1	now	now	ADV
cana-3825	72	2	the	the	DET
cana-3825	72	3	following	follow	VERB
cana-3825	72	4	properties	property	NOUN
cana-3825	72	5	of	of	ADP
cana-3825	72	6	a	a	PRON
cana-3825	72	7	are	be	AUX
cana-3825	72	8	proved	prove	VERB
cana-3825	72	9	below	below	ADV
cana-3825	72	10	.	.	PUNCT
cana-3825	73	1	(	(	PUNCT
cana-3825	73	2	a	a	NOUN
cana-3825	73	3	)	)	PUNCT
cana-3825	73	4	.	.	PUNCT
cana-3825	74	1	ξ̃0(n	ξ̃0(n	X
cana-3825	74	2	)	)	PUNCT
cana-3825	74	3	≤	≤	NUM
cana-3825	74	4	aξ̃0(n	aξ̃0(n	X
cana-3825	74	5	)	)	PUNCT
cana-3825	74	6	,	,	PUNCT
cana-3825	74	7	ω̃0(n	ω̃0(n	PROPN
cana-3825	74	8	)	)	PUNCT
cana-3825	74	9	≥	≥	NOUN
cana-3825	74	10	aω̃0(n	aω̃0(n	NOUN
cana-3825	74	11	)	)	PUNCT
cana-3825	74	12	.	.	PUNCT
cana-3825	75	1	(	(	PUNCT
cana-3825	75	2	b	b	NOUN
cana-3825	75	3	)	)	PUNCT
cana-3825	75	4	.	.	PUNCT
cana-3825	76	1	a	a	PRON
cana-3825	76	2	is	be	AUX
cana-3825	76	3	monotone	monotone	ADJ
cana-3825	76	4	operator	operator	NOUN
cana-3825	76	5	on	on	ADP
cana-3825	76	6	[	[	X
cana-3825	76	7	ξ̃0	ξ̃0	PROPN
cana-3825	76	8	,	,	PUNCT
cana-3825	76	9	ω̃0	ω̃0	PROPN
cana-3825	76	10	]	]	X
cana-3825	77	1	=	=	SYM
cana-3825	77	2	{	{	PUNCT
cana-3825	77	3	ρ(n)/	ρ(n)/	NOUN
cana-3825	77	4	ξ̃0	ξ̃0	PROPN
cana-3825	77	5	≤	≤	PUNCT
cana-3825	77	6	ρ(n	ρ(n	PROPN
cana-3825	77	7	)	)	PUNCT
cana-3825	77	8	≤	≤	NOUN
cana-3825	77	9	ω̃0	ω̃0	NOUN
cana-3825	77	10	}	}	PUNCT
cana-3825	77	11	.	.	PUNCT
cana-3825	78	1	to	to	PART
cana-3825	78	2	prove	prove	VERB
cana-3825	78	3	(	(	PUNCT
cana-3825	78	4	a	a	X
cana-3825	78	5	)	)	PUNCT
cana-3825	78	6	,	,	PUNCT
cana-3825	78	7	set	set	VERB
cana-3825	78	8	aξ̃0	aξ̃0	PRON
cana-3825	78	9	(	(	PUNCT
cana-3825	78	10	n	n	CCONJ
cana-3825	78	11	)	)	PUNCT
cana-3825	79	1	=	=	SYM
cana-3825	79	2	ξ̃1	ξ̃1	PROPN
cana-3825	79	3	(	(	PUNCT
cana-3825	79	4	n	n	CCONJ
cana-3825	79	5	)	)	PUNCT
cana-3825	79	6	,	,	PUNCT
cana-3825	79	7	where	where	SCONJ
cana-3825	79	8	ξ̃1(n	ξ̃1(n	NOUN
cana-3825	79	9	)	)	PUNCT
cana-3825	79	10	is	be	AUX
cana-3825	79	11	the	the	DET
cana-3825	79	12	unique	unique	ADJ
cana-3825	79	13	solution	solution	NOUN
cana-3825	79	14	of	of	ADP
cana-3825	79	15	(	(	PUNCT
cana-3825	79	16	2.2	2.2	NUM
cana-3825	79	17	)	)	PUNCT
cana-3825	79	18	.	.	PUNCT
cana-3825	80	1	with	with	ADP
cana-3825	80	2	θ(n	θ(n	NOUN
cana-3825	80	3	)	)	PUNCT
cana-3825	80	4	=	=	SYM
cana-3825	80	5	ξ̃0	ξ̃0	PROPN
cana-3825	80	6	(	(	PUNCT
cana-3825	80	7	n	n	CCONJ
cana-3825	80	8	)	)	PUNCT
cana-3825	80	9	.	.	PUNCT
cana-3825	81	1	set	set	VERB
cana-3825	81	2	p	p	NOUN
cana-3825	81	3	(	(	PUNCT
cana-3825	81	4	n	n	CCONJ
cana-3825	81	5	)	)	PUNCT
cana-3825	81	6	=	=	SYM
cana-3825	81	7	ξ̃0(n	ξ̃0(n	X
cana-3825	81	8	)	)	PUNCT
cana-3825	82	1	−	−	PROPN
cana-3825	82	2	ξ̃1(n	ξ̃1(n	NOUN
cana-3825	82	3	)	)	PUNCT
cana-3825	82	4	.	.	PUNCT
cana-3825	83	1	consider	consider	VERB
cana-3825	83	2	∇μp	∇μp	PROPN
cana-3825	83	3	(	(	PUNCT
cana-3825	83	4	n	n	NOUN
cana-3825	83	5	+	+	CCONJ
cana-3825	83	6	1	1	NUM
cana-3825	83	7	)	)	PUNCT
cana-3825	83	8	=	=	SYM
cana-3825	84	1	∇μ	∇μ	PROPN
cana-3825	85	1	[	[	X
cana-3825	85	2	ξ̃0(n	ξ̃0(n	NOUN
cana-3825	85	3	+	+	PUNCT
cana-3825	85	4	1	1	X
cana-3825	85	5	)	)	PUNCT
cana-3825	85	6	−	−	PROPN
cana-3825	86	1	ξ̃1(n	ξ̃1(n	NOUN
cana-3825	86	2	+	+	NOUN
cana-3825	86	3	1	1	NUM
cana-3825	86	4	)	)	PUNCT
cana-3825	86	5	]	]	PUNCT
cana-3825	86	6	≤	≤	NUM
cana-3825	86	7	f	f	X
cana-3825	86	8	(	(	PUNCT
cana-3825	86	9	n	n	CCONJ
cana-3825	86	10	,	,	PUNCT
cana-3825	86	11	ξ̃0(n	ξ̃0(n	PROPN
cana-3825	86	12	)	)	PUNCT
cana-3825	86	13	)	)	PUNCT
cana-3825	86	14	–	–	PUNCT
cana-3825	87	1	f	f	X
cana-3825	87	2	(	(	PUNCT
cana-3825	87	3	n	n	CCONJ
cana-3825	87	4	,	,	PUNCT
cana-3825	87	5	ξ̃0(n	ξ̃0(n	PROPN
cana-3825	87	6	)	)	PUNCT
cana-3825	87	7	)	)	PUNCT
cana-3825	88	1	+	+	CCONJ
cana-3825	88	2	m[ξ̃1(n	m[ξ̃1(n	NOUN
cana-3825	88	3	)	)	PUNCT
cana-3825	88	4	−	−	PROPN
cana-3825	88	5	ξ̃0(n	ξ̃0(n	PROPN
cana-3825	88	6	)	)	PUNCT
cana-3825	88	7	]	]	PUNCT
cana-3825	89	1	=	=	PUNCT
cana-3825	89	2	−m[ξ̃0(n	−m[ξ̃0(n	X
cana-3825	89	3	)	)	PUNCT
cana-3825	90	1	−	−	PROPN
cana-3825	90	2	ξ̃1(n	ξ̃1(n	NOUN
cana-3825	90	3	)	)	PUNCT
cana-3825	90	4	]	]	PUNCT
cana-3825	90	5	or	or	CCONJ
cana-3825	90	6	∇μ	∇μ	PROPN
cana-3825	90	7	p	p	PROPN
cana-3825	90	8	(	(	PUNCT
cana-3825	90	9	n	n	PROPN
cana-3825	90	10	+	+	CCONJ
cana-3825	90	11	1	1	NUM
cana-3825	90	12	)	)	PUNCT
cana-3825	90	13	≤	≤	NUM
cana-3825	90	14	−m	−m	NOUN
cana-3825	90	15	p(n	p(n	PROPN
cana-3825	90	16	)	)	PUNCT
cana-3825	90	17	.	.	PUNCT
cana-3825	91	1	communications	communication	NOUN
cana-3825	91	2	on	on	ADP
cana-3825	91	3	applied	apply	VERB
cana-3825	91	4	nonlinear	nonlinear	ADJ
cana-3825	91	5	analysis	analysis	NOUN
cana-3825	91	6	issn	issn	NOUN
cana-3825	91	7	:	:	PUNCT
cana-3825	91	8	1074	1074	NUM
cana-3825	91	9	-	-	PUNCT
cana-3825	91	10	133x	133x	NUM
cana-3825	91	11	vol	vol	NOUN
cana-3825	91	12	32	32	NUM
cana-3825	91	13	no	no	NOUN
cana-3825	91	14	.	.	PUNCT
cana-3825	92	1	8s	8s	PROPN
cana-3825	92	2	(	(	PUNCT
cana-3825	92	3	2025	2025	NUM
cana-3825	92	4	)	)	PUNCT
cana-3825	92	5	832	832	NUM
cana-3825	92	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3825	92	7	also	also	ADV
cana-3825	92	8	,	,	PUNCT
cana-3825	92	9	ρ	ρ	PROPN
cana-3825	92	10	(	(	PUNCT
cana-3825	92	11	n0	n0	PROPN
cana-3825	92	12	)	)	PUNCT
cana-3825	92	13	=	=	SYM
cana-3825	92	14	ξ̃0(n0	ξ̃0(n0	X
cana-3825	92	15	)	)	PUNCT
cana-3825	92	16	−ξ̃1(n0	−ξ̃1(n0	NOUN
cana-3825	92	17	)	)	PUNCT
cana-3825	92	18	≤	≤	PUNCT
cana-3825	93	1	ρ0	ρ0	PROPN
cana-3825	93	2	−	−	PROPN
cana-3825	93	3	ρ0	ρ0	PROPN
cana-3825	93	4	≤	≤	NUM
cana-3825	93	5	0	0	NUM
cana-3825	93	6	.	.	PUNCT
cana-3825	94	1	by	by	ADP
cana-3825	94	2	using	use	VERB
cana-3825	94	3	theorem	theorem	ADJ
cana-3825	94	4	2.2	2.2	NUM
cana-3825	94	5	,	,	PUNCT
cana-3825	94	6	p(n	p(n	NOUN
cana-3825	94	7	)	)	PUNCT
cana-3825	94	8	≤	≤	NOUN
cana-3825	94	9	0	0	NUM
cana-3825	94	10	or	or	CCONJ
cana-3825	94	11	ξ̃0(n	ξ̃0(n	PROPN
cana-3825	94	12	)	)	PUNCT
cana-3825	94	13	≤	≤	NUM
cana-3825	94	14	ξ̃1(n	ξ̃1(n	NOUN
cana-3825	94	15	)	)	PUNCT
cana-3825	94	16	=	=	PUNCT
cana-3825	94	17	aξ̃0(n	aξ̃0(n	X
cana-3825	94	18	)	)	PUNCT
cana-3825	94	19	.	.	PUNCT
cana-3825	95	1	similarly	similarly	ADV
cana-3825	95	2	,	,	PUNCT
cana-3825	95	3	it	it	PRON
cana-3825	95	4	can	can	AUX
cana-3825	95	5	be	be	AUX
cana-3825	95	6	proved	prove	VERB
cana-3825	95	7	that	that	SCONJ
cana-3825	95	8	ω̃0(n	ω̃0(n	PROPN
cana-3825	95	9	)	)	PUNCT
cana-3825	95	10	≥	≥	NOUN
cana-3825	95	11	a{ω̃0(n	a{ω̃0(n	PROPN
cana-3825	95	12	)	)	PUNCT
cana-3825	95	13	}	}	PUNCT
cana-3825	95	14	.	.	PUNCT
cana-3825	96	1	to	to	PART
cana-3825	96	2	prove	prove	VERB
cana-3825	96	3	(	(	PUNCT
cana-3825	96	4	b	b	NOUN
cana-3825	96	5	)	)	PUNCT
cana-3825	96	6	,	,	PUNCT
cana-3825	96	7	let	let	VERB
cana-3825	96	8	k	k	PROPN
cana-3825	96	9	∈	∈	PROPN
cana-3825	96	10	n0	n0	PROPN
cana-3825	96	11	+	+	CCONJ
cana-3825	96	12	and	and	CCONJ
cana-3825	96	13	θk	θk	NOUN
cana-3825	96	14	and	and	CCONJ
cana-3825	96	15	θk+1∈	θk+1∈	NUM
cana-3825	97	1	[	[	PUNCT
cana-3825	97	2	ξ̃0	ξ̃0	PROPN
cana-3825	97	3	(	(	PUNCT
cana-3825	97	4	n	n	CCONJ
cana-3825	97	5	)	)	PUNCT
cana-3825	97	6	,	,	PUNCT
cana-3825	97	7	ω̃0	ω̃0	PROPN
cana-3825	97	8	(	(	PUNCT
cana-3825	97	9	n	n	CCONJ
cana-3825	97	10	)	)	PUNCT
cana-3825	97	11	]	]	PUNCT
cana-3825	97	12	such	such	ADJ
cana-3825	97	13	that	that	DET
cana-3825	97	14	θk	θk	PROPN
cana-3825	97	15	≤	≤	NUM
cana-3825	97	16	θk+1	θk+1	AUX
cana-3825	97	17	.	.	PUNCT
cana-3825	97	18	suppose	suppose	VERB
cana-3825	97	19	a{θk(n	a{θk(n	NOUN
cana-3825	97	20	)	)	PUNCT
cana-3825	97	21	}	}	PUNCT
cana-3825	97	22	=	=	SYM
cana-3825	97	23	ξ̃k	ξ̃k	X
cana-3825	97	24	(	(	PUNCT
cana-3825	97	25	n	n	CCONJ
cana-3825	97	26	)	)	PUNCT
cana-3825	97	27	and	and	CCONJ
cana-3825	97	28	a	a	DET
cana-3825	97	29	{	{	PUNCT
cana-3825	97	30	θk+1(n	θk+1(n	PROPN
cana-3825	97	31	)	)	PUNCT
cana-3825	97	32	}	}	PUNCT
cana-3825	98	1	=	=	SYM
cana-3825	98	2	ξ̃k+1	ξ̃k+1	NOUN
cana-3825	98	3	(	(	PUNCT
cana-3825	98	4	n	n	CCONJ
cana-3825	98	5	)	)	PUNCT
cana-3825	98	6	.	.	PUNCT
cana-3825	99	1	take	take	VERB
cana-3825	99	2	q(n	q(n	NOUN
cana-3825	99	3	)	)	PUNCT
cana-3825	99	4	=	=	PUNCT
cana-3825	99	5	ξ̃k	ξ̃k	VERB
cana-3825	99	6	−	−	PROPN
cana-3825	99	7	ξ̃k+1	ξ̃k+1	PROPN
cana-3825	99	8	.	.	PUNCT
cana-3825	100	1	consider	consider	VERB
cana-3825	100	2	∇μ	∇μ	PROPN
cana-3825	100	3	q	q	PROPN
cana-3825	100	4	(	(	PUNCT
cana-3825	100	5	n	n	NOUN
cana-3825	100	6	+	+	NOUN
cana-3825	100	7	1	1	NUM
cana-3825	100	8	)	)	PUNCT
cana-3825	100	9	=	=	SYM
cana-3825	101	1	∇μ	∇μ	PROPN
cana-3825	101	2	[	[	X
cana-3825	101	3	ξ̃k	ξ̃k	X
cana-3825	101	4	(	(	PUNCT
cana-3825	101	5	n	n	X
cana-3825	101	6	+	+	NUM
cana-3825	101	7	1	1	NUM
cana-3825	101	8	)	)	PUNCT
cana-3825	101	9	−	−	PROPN
cana-3825	101	10	ξ̃k+1	ξ̃k+1	NOUN
cana-3825	101	11	(	(	PUNCT
cana-3825	101	12	n	n	PROPN
cana-3825	101	13	+	+	NOUN
cana-3825	101	14	1	1	NUM
cana-3825	101	15	)	)	PUNCT
cana-3825	101	16	]	]	PUNCT
cana-3825	102	1	=	=	SYM
cana-3825	102	2	f	f	X
cana-3825	102	3	(	(	PUNCT
cana-3825	102	4	n	n	CCONJ
cana-3825	102	5	,	,	PUNCT
cana-3825	102	6	θk(n	θk(n	NOUN
cana-3825	102	7	)	)	PUNCT
cana-3825	102	8	)	)	PUNCT
cana-3825	102	9	–	–	PUNCT
cana-3825	102	10	m	m	VERB
cana-3825	102	11	[	[	X
cana-3825	102	12	ξ̃k	ξ̃k	X
cana-3825	102	13	(	(	PUNCT
cana-3825	102	14	n	n	CCONJ
cana-3825	102	15	)	)	PUNCT
cana-3825	102	16	−	−	NOUN
cana-3825	102	17	θk	θk	NOUN
cana-3825	102	18	(	(	PUNCT
cana-3825	102	19	n	n	CCONJ
cana-3825	102	20	)	)	PUNCT
cana-3825	102	21	]	]	PUNCT
cana-3825	102	22	–	–	PUNCT
cana-3825	102	23	f	f	X
cana-3825	102	24	(	(	PUNCT
cana-3825	102	25	n	n	CCONJ
cana-3825	102	26	,	,	PUNCT
cana-3825	102	27	θk+1	θk+1	X
cana-3825	102	28	(	(	PUNCT
cana-3825	102	29	n	n	CCONJ
cana-3825	102	30	)	)	PUNCT
cana-3825	102	31	)	)	PUNCT
cana-3825	103	1	+	+	CCONJ
cana-3825	103	2	m[ξ̃k+1	m[ξ̃k+1	ADJ
cana-3825	103	3	(	(	PUNCT
cana-3825	103	4	n	n	CCONJ
cana-3825	103	5	)	)	PUNCT
cana-3825	103	6	−	−	PROPN
cana-3825	103	7	θk+1	θk+1	X
cana-3825	103	8	(	(	PUNCT
cana-3825	103	9	n	n	X
cana-3825	103	10	)	)	PUNCT
cana-3825	103	11	]	]	PUNCT
cana-3825	104	1	=	=	PUNCT
cana-3825	104	2	m	m	VERB
cana-3825	105	1	[	[	X
cana-3825	105	2	θk	θk	X
cana-3825	105	3	(	(	PUNCT
cana-3825	105	4	n	n	CCONJ
cana-3825	105	5	)	)	PUNCT
cana-3825	105	6	−	−	PROPN
cana-3825	105	7	θk+1	θk+1	X
cana-3825	105	8	(	(	PUNCT
cana-3825	105	9	n	n	X
cana-3825	105	10	)	)	PUNCT
cana-3825	105	11	]	]	PUNCT
cana-3825	106	1	+	+	CCONJ
cana-3825	106	2	m[ξ̃k+1	m[ξ̃k+1	ADJ
cana-3825	106	3	(	(	PUNCT
cana-3825	106	4	n	n	CCONJ
cana-3825	106	5	)	)	PUNCT
cana-3825	106	6	−	−	PROPN
cana-3825	106	7	ξ̃k	ξ̃k	PROPN
cana-3825	106	8	(	(	PUNCT
cana-3825	106	9	n	n	X
cana-3825	106	10	)	)	PUNCT
cana-3825	106	11	]	]	PUNCT
cana-3825	107	1	+	+	CCONJ
cana-3825	107	2	f	f	X
cana-3825	107	3	(	(	PUNCT
cana-3825	107	4	n	n	CCONJ
cana-3825	107	5	,	,	PUNCT
cana-3825	107	6	θk	θk	PROPN
cana-3825	107	7	)	)	PUNCT
cana-3825	107	8	–	–	PUNCT
cana-3825	107	9	f	f	PROPN
cana-3825	107	10	(	(	PUNCT
cana-3825	107	11	n	n	CCONJ
cana-3825	107	12	,	,	PUNCT
cana-3825	107	13	θk+1	θk+1	X
cana-3825	107	14	)	)	PUNCT
cana-3825	107	15	≤	≤	NUM
cana-3825	107	16	−m	−m	ADJ
cana-3825	107	17	q(n	q(n	PROPN
cana-3825	107	18	)	)	PUNCT
cana-3825	107	19	also	also	ADV
cana-3825	107	20	,	,	PUNCT
cana-3825	107	21	q	q	X
cana-3825	107	22	(	(	PUNCT
cana-3825	107	23	n0	n0	NUM
cana-3825	107	24	)	)	PUNCT
cana-3825	107	25	=	=	SYM
cana-3825	107	26	ξ̃k	ξ̃k	PROPN
cana-3825	107	27	(	(	PUNCT
cana-3825	107	28	n0	n0	NUM
cana-3825	107	29	)	)	PUNCT
cana-3825	107	30	−	−	PROPN
cana-3825	107	31	ξ̃k+1	ξ̃k+1	PROPN
cana-3825	107	32	(	(	PUNCT
cana-3825	107	33	n0	n0	NUM
cana-3825	107	34	)	)	PUNCT
cana-3825	107	35	≤	≤	PUNCT
cana-3825	107	36	ρ0	ρ0	PROPN
cana-3825	107	37	−ρ0	−ρ0	NOUN
cana-3825	107	38	≤	≤	ADJ
cana-3825	107	39	0	0	NUM
cana-3825	107	40	.	.	PUNCT
cana-3825	108	1	by	by	ADP
cana-3825	108	2	using	use	VERB
cana-3825	108	3	theorem	theorem	ADJ
cana-3825	108	4	2.2	2.2	NUM
cana-3825	108	5	,	,	PUNCT
cana-3825	108	6	hence	hence	ADV
cana-3825	108	7	q(n	q(n	ADJ
cana-3825	108	8	)	)	PUNCT
cana-3825	108	9	≤	≤	NOUN
cana-3825	108	10	0	0	NUM
cana-3825	108	11	or	or	CCONJ
cana-3825	108	12	ξ̃k	ξ̃k	PROPN
cana-3825	108	13	(	(	PUNCT
cana-3825	108	14	n	n	CCONJ
cana-3825	108	15	)	)	PUNCT
cana-3825	108	16	≤	≤	NOUN
cana-3825	108	17	ξ̃k+1	ξ̃k+1	NOUN
cana-3825	108	18	(	(	PUNCT
cana-3825	108	19	n	n	CCONJ
cana-3825	108	20	)	)	PUNCT
cana-3825	108	21	=	=	SYM
cana-3825	108	22	aξ̃k	aξ̃k	PROPN
cana-3825	108	23	(	(	PUNCT
cana-3825	108	24	n	n	CCONJ
cana-3825	108	25	)	)	PUNCT
cana-3825	108	26	.	.	PUNCT
cana-3825	109	1	similarly	similarly	ADV
cana-3825	109	2	,	,	PUNCT
cana-3825	109	3	it	it	PRON
cana-3825	109	4	can	can	AUX
cana-3825	109	5	be	be	AUX
cana-3825	109	6	proved	prove	VERB
cana-3825	109	7	that	that	PRON
cana-3825	109	8	ωk(n	ωk(n	NUM
cana-3825	109	9	)	)	PUNCT
cana-3825	109	10	≥	≥	NOUN
cana-3825	109	11	ωk+1(n	ωk+1(n	NOUN
cana-3825	109	12	)	)	PUNCT
cana-3825	109	13	.	.	PUNCT
cana-3825	110	1	that	that	PRON
cana-3825	110	2	means	mean	VERB
cana-3825	110	3	,	,	PUNCT
cana-3825	110	4	a	a	DET
cana-3825	110	5	sequence	sequence	NOUN
cana-3825	110	6	can	can	AUX
cana-3825	110	7	be	be	AUX
cana-3825	110	8	constructed	construct	VERB
cana-3825	110	9	as	as	ADP
cana-3825	110	10	ξ̃0	ξ̃0	PROPN
cana-3825	110	11	(	(	PUNCT
cana-3825	110	12	n)≤	n)≤	NOUN
cana-3825	110	13	ξ̃1	ξ̃1	PROPN
cana-3825	110	14	(	(	PUNCT
cana-3825	110	15	n	n	CCONJ
cana-3825	110	16	)	)	PUNCT
cana-3825	110	17	≤	≤	NOUN
cana-3825	111	1	ξ̃2	ξ̃2	PROPN
cana-3825	111	2	(	(	PUNCT
cana-3825	111	3	n	n	CCONJ
cana-3825	111	4	)	)	PUNCT
cana-3825	111	5	≤	≤	NOUN
cana-3825	111	6	..........	..........	PUNCT
cana-3825	112	1	≤	≤	ADJ
cana-3825	112	2	ξ̃m	ξ̃m	NOUN
cana-3825	112	3	(	(	PUNCT
cana-3825	112	4	n	n	CCONJ
cana-3825	112	5	)	)	PUNCT
cana-3825	112	6	≤	≤	NOUN
cana-3825	112	7	ω̃m(n	ω̃m(n	NOUN
cana-3825	112	8	)	)	PUNCT
cana-3825	112	9	≤	≤	NOUN
cana-3825	112	10	.............	.............	PUNCT
cana-3825	112	11	≤	≤	NUM
cana-3825	112	12	ω̃2(n	ω̃2(n	PROPN
cana-3825	112	13	)	)	PUNCT
cana-3825	112	14	≤	≤	NUM
cana-3825	112	15	ω̃1(n	ω̃1(n	NOUN
cana-3825	112	16	)	)	PUNCT
cana-3825	112	17	≤	≤	NUM
cana-3825	112	18	ω̃0(n	ω̃0(n	PROPN
cana-3825	112	19	)	)	PUNCT
cana-3825	112	20	for	for	ADP
cana-3825	112	21	n	n	DET
cana-3825	112	22	∈	∈	PROPN
cana-3825	112	23	nn0	nn0	NOUN
cana-3825	112	24	+	+	X
cana-3825	112	25	.	.	PUNCT
cana-3825	113	1	by	by	ADP
cana-3825	113	2	dini	dini	NOUN
cana-3825	113	3	’s	’s	PART
cana-3825	113	4	theorem	theorem	NOUN
cana-3825	113	5	as	as	ADP
cana-3825	113	6	m	m	PROPN
cana-3825	113	7	→	→	SYM
cana-3825	113	8	∞	∞	PROPN
cana-3825	113	9	,	,	PUNCT
cana-3825	113	10	ξ̃m	ξ̃m	X
cana-3825	113	11	→	→	SYM
cana-3825	113	12	ξ̃	ξ̃	PROPN
cana-3825	113	13	and	and	CCONJ
cana-3825	113	14	ω̃m→	ω̃m→	PROPN
cana-3825	113	15	ω̃	ω̃	PROPN
cana-3825	113	16	where	where	SCONJ
cana-3825	113	17	ξ̃	ξ̃	PROPN
cana-3825	113	18	and	and	CCONJ
cana-3825	113	19	ω̃	ω̃	NUM
cana-3825	113	20	are	be	AUX
cana-3825	113	21	any	any	DET
cana-3825	113	22	two	two	NUM
cana-3825	113	23	functions	function	NOUN
cana-3825	113	24	defined	define	VERB
cana-3825	113	25	for	for	ADP
cana-3825	113	26	n	n	PRON
cana-3825	113	27	∈	∈	PROPN
cana-3825	113	28	nn0	nn0	NOUN
cana-3825	114	1	+	+	X
cana-3825	114	2	.	.	PUNCT
cana-3825	115	1	also	also	ADV
cana-3825	115	2	ξ̃m	ξ̃m	X
cana-3825	115	3	(	(	PUNCT
cana-3825	115	4	n	n	CCONJ
cana-3825	115	5	)	)	PUNCT
cana-3825	115	6	and	and	CCONJ
cana-3825	115	7	ω̃m(n	ω̃m(n	NOUN
cana-3825	115	8	)	)	PUNCT
cana-3825	115	9	satisfy	satisfy	NOUN
cana-3825	115	10	∇μξ̃m	∇μξ̃m	PUNCT
cana-3825	116	1	(	(	PUNCT
cana-3825	116	2	n	n	X
cana-3825	116	3	+	+	CCONJ
cana-3825	116	4	1	1	NUM
cana-3825	116	5	)	)	PUNCT
cana-3825	116	6	=	=	SYM
cana-3825	116	7	f	f	PROPN
cana-3825	116	8	(	(	PUNCT
cana-3825	116	9	n	n	CCONJ
cana-3825	116	10	,	,	PUNCT
cana-3825	116	11	ξ̃m−1	ξ̃m−1	PROPN
cana-3825	116	12	(	(	PUNCT
cana-3825	116	13	n	n	CCONJ
cana-3825	116	14	)	)	PUNCT
cana-3825	116	15	−	−	PROPN
cana-3825	117	1	m	m	PRON
cana-3825	117	2	[	[	PUNCT
cana-3825	117	3	ξ̃m	ξ̃m	X
cana-3825	117	4	(	(	PUNCT
cana-3825	117	5	n	n	CCONJ
cana-3825	117	6	)	)	PUNCT
cana-3825	117	7	−ξ̃m−1	−ξ̃m−1	PROPN
cana-3825	117	8	(	(	PUNCT
cana-3825	117	9	n	n	CCONJ
cana-3825	117	10	)	)	PUNCT
cana-3825	118	1	]	]	PUNCT
cana-3825	118	2	,	,	PUNCT
cana-3825	118	3	ξ̃m	ξ̃m	X
cana-3825	118	4	(	(	PUNCT
cana-3825	118	5	n0)=	n0)=	NUM
cana-3825	118	6	ρ0	ρ0	PROPN
cana-3825	118	7	,	,	PUNCT
cana-3825	118	8	(	(	PUNCT
cana-3825	118	9	2.3	2.3	NUM
cana-3825	118	10	)	)	PUNCT
cana-3825	118	11	∇μω̃m	∇μω̃m	NOUN
cana-3825	118	12	(	(	PUNCT
cana-3825	118	13	n	n	NOUN
cana-3825	118	14	+	+	CCONJ
cana-3825	118	15	1	1	NUM
cana-3825	118	16	)	)	PUNCT
cana-3825	118	17	=	=	SYM
cana-3825	118	18	f	f	PROPN
cana-3825	118	19	(	(	PUNCT
cana-3825	118	20	n	n	CCONJ
cana-3825	118	21	,	,	PUNCT
cana-3825	118	22	ω̃m−1	ω̃m−1	PROPN
cana-3825	118	23	(	(	PUNCT
cana-3825	118	24	n	n	CCONJ
cana-3825	118	25	)	)	PUNCT
cana-3825	118	26	)	)	PUNCT
cana-3825	119	1	−	−	PROPN
cana-3825	119	2	m[ω̃m(n)−	m[ω̃m(n)−	PROPN
cana-3825	119	3	ω̃m−1(n	ω̃m−1(n	NUM
cana-3825	119	4	)	)	PUNCT
cana-3825	119	5	]	]	PUNCT
cana-3825	119	6	,	,	PUNCT
cana-3825	119	7	ω̃m	ω̃m	NUM
cana-3825	119	8	(	(	PUNCT
cana-3825	119	9	n0)=	n0)=	NUM
cana-3825	119	10	ρ0	ρ0	PROPN
cana-3825	119	11	(	(	PUNCT
cana-3825	119	12	2.4	2.4	NUM
cana-3825	119	13	)	)	PUNCT
cana-3825	119	14	as	as	ADP
cana-3825	119	15	m	m	PROPN
cana-3825	119	16	→	→	SYM
cana-3825	119	17	∞	∞	PROPN
cana-3825	119	18	,	,	PUNCT
cana-3825	119	19	ξ̃(n	ξ̃(n	PROPN
cana-3825	119	20	)	)	PUNCT
cana-3825	119	21	→	→	SYM
cana-3825	119	22	ξ̃	ξ̃	PROPN
cana-3825	119	23	and	and	CCONJ
cana-3825	119	24	ω̃(n	ω̃(n	ADJ
cana-3825	119	25	)	)	PUNCT
cana-3825	119	26	→	→	SYM
cana-3825	119	27	ω̃	ω̃	PROPN
cana-3825	119	28	,	,	PUNCT
cana-3825	119	29	hence	hence	ADV
cana-3825	119	30	∇μξ̃	∇μξ̃	NUM
cana-3825	119	31	=	=	SYM
cana-3825	119	32	f(n	f(n	PROPN
cana-3825	119	33	,	,	PUNCT
cana-3825	119	34	ξ̃	ξ̃	PROPN
cana-3825	119	35	)	)	PUNCT
cana-3825	119	36	,	,	PUNCT
cana-3825	119	37	ξ̃(n0)=	ξ̃(n0)=	PROPN
cana-3825	119	38	ρ0	ρ0	PROPN
cana-3825	119	39	,	,	PUNCT
cana-3825	119	40	(	(	PUNCT
cana-3825	119	41	2.5	2.5	NUM
cana-3825	119	42	)	)	PUNCT
cana-3825	119	43	∇μω̃	∇μω̃	NOUN
cana-3825	120	1	=	=	SYM
cana-3825	120	2	f(n	f(n	PROPN
cana-3825	120	3	,	,	PUNCT
cana-3825	120	4	ω̃	ω̃	PROPN
cana-3825	120	5	)	)	PUNCT
cana-3825	120	6	,	,	PUNCT
cana-3825	120	7	ω̃(n0)=	ω̃(n0)=	NUM
cana-3825	120	8	ρ0	ρ0	PROPN
cana-3825	120	9	.	.	PUNCT
cana-3825	121	1	(	(	PUNCT
cana-3825	121	2	2.6	2.6	NUM
cana-3825	121	3	)	)	PUNCT
cana-3825	121	4	it	it	PRON
cana-3825	121	5	implies	imply	VERB
cana-3825	121	6	that	that	SCONJ
cana-3825	121	7	the	the	DET
cana-3825	121	8	functions	function	NOUN
cana-3825	121	9	ξ̃	ξ̃	PROPN
cana-3825	121	10	and	and	CCONJ
cana-3825	121	11	ω̃	ω̃	NUM
cana-3825	121	12	defined	define	VERB
cana-3825	121	13	on	on	ADP
cana-3825	121	14	n	n	DET
cana-3825	121	15	∈	∈	PROPN
cana-3825	121	16	nn0	nn0	NOUN
cana-3825	121	17	+	+	CCONJ
cana-3825	121	18	are	be	AUX
cana-3825	121	19	solutions	solution	NOUN
cana-3825	121	20	of	of	ADP
cana-3825	121	21	(	(	PUNCT
cana-3825	121	22	2.1	2.1	NUM
cana-3825	121	23	)	)	PUNCT
cana-3825	121	24	.	.	PUNCT
cana-3825	122	1	now	now	ADV
cana-3825	122	2	to	to	PART
cana-3825	122	3	prove	prove	VERB
cana-3825	122	4	that	that	SCONJ
cana-3825	122	5	ξ̃	ξ̃	PROPN
cana-3825	122	6	and	and	CCONJ
cana-3825	122	7	ω̃	ω̃	NUM
cana-3825	122	8	are	be	AUX
cana-3825	122	9	minimal	minimal	ADJ
cana-3825	122	10	and	and	CCONJ
cana-3825	122	11	maximal	maximal	ADJ
cana-3825	122	12	solutions	solution	NOUN
cana-3825	122	13	of	of	ADP
cana-3825	122	14	(	(	PUNCT
cana-3825	122	15	2.1	2.1	NUM
cana-3825	122	16	)	)	PUNCT
cana-3825	122	17	respectively	respectively	ADV
cana-3825	122	18	,	,	PUNCT
cana-3825	122	19	it	it	PRON
cana-3825	122	20	is	be	AUX
cana-3825	122	21	sufficient	sufficient	ADJ
cana-3825	122	22	to	to	PART
cana-3825	122	23	prove	prove	VERB
cana-3825	122	24	that	that	SCONJ
cana-3825	122	25	ξ̃	ξ̃	PROPN
cana-3825	122	26	≤	≤	PUNCT
cana-3825	122	27	ρ(n	ρ(n	PROPN
cana-3825	122	28	)	)	PUNCT
cana-3825	123	1	≤	≤	NOUN
cana-3825	123	2	ω̃	ω̃	NUM
cana-3825	123	3	(	(	PUNCT
cana-3825	123	4	2.7	2.7	NUM
cana-3825	123	5	)	)	PUNCT
cana-3825	123	6	if	if	SCONJ
cana-3825	123	7	ρ(n	ρ(n	PROPN
cana-3825	123	8	)	)	PUNCT
cana-3825	123	9	is	be	AUX
cana-3825	123	10	any	any	DET
cana-3825	123	11	solution	solution	NOUN
cana-3825	123	12	of	of	ADP
cana-3825	123	13	(	(	PUNCT
cana-3825	123	14	2.1	2.1	NUM
cana-3825	123	15	)	)	PUNCT
cana-3825	123	16	such	such	ADJ
cana-3825	123	17	that	that	SCONJ
cana-3825	123	18	ξ̃0	ξ̃0	PROPN
cana-3825	123	19	≤	≤	PROPN
cana-3825	123	20	ρ	ρ	NUM
cana-3825	123	21	≤	≤	NUM
cana-3825	123	22	ω̃0	ω̃0	PROPN
cana-3825	123	23	.	.	PUNCT
cana-3825	124	1	(	(	PUNCT
cana-3825	124	2	2.8	2.8	NUM
cana-3825	124	3	)	)	PUNCT
cana-3825	124	4	communications	communication	NOUN
cana-3825	124	5	on	on	ADP
cana-3825	124	6	applied	apply	VERB
cana-3825	124	7	nonlinear	nonlinear	ADJ
cana-3825	124	8	analysis	analysis	NOUN
cana-3825	124	9	issn	issn	NOUN
cana-3825	124	10	:	:	PUNCT
cana-3825	124	11	1074	1074	NUM
cana-3825	124	12	-	-	PUNCT
cana-3825	124	13	133x	133x	NUM
cana-3825	124	14	vol	vol	NOUN
cana-3825	124	15	32	32	NUM
cana-3825	124	16	no	no	NOUN
cana-3825	124	17	.	.	PUNCT
cana-3825	125	1	8s	8s	PROPN
cana-3825	125	2	(	(	PUNCT
cana-3825	125	3	2025	2025	NUM
cana-3825	125	4	)	)	PUNCT
cana-3825	125	5	833	833	NUM
cana-3825	125	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3825	125	7	let	let	VERB
cana-3825	125	8	the	the	DET
cana-3825	125	9	statement	statement	NOUN
cana-3825	125	10	ξ̃k	ξ̃k	VERB
cana-3825	125	11	≤	≤	ADJ
cana-3825	125	12	ρ(n	ρ(n	PROPN
cana-3825	125	13	)	)	PUNCT
cana-3825	125	14	≤	≤	NUM
cana-3825	125	15	ω̃k	ω̃k	NOUN
cana-3825	125	16	.	.	PUNCT
cana-3825	126	1	(	(	PUNCT
cana-3825	126	2	2.9	2.9	NUM
cana-3825	126	3	)	)	PUNCT
cana-3825	126	4	be	be	AUX
cana-3825	126	5	true	true	ADJ
cana-3825	126	6	.	.	PUNCT
cana-3825	127	1	let	let	VERB
cana-3825	127	2	ξ̃k	ξ̃k	VERB
cana-3825	127	3	satisfy	satisfy	VERB
cana-3825	127	4	(	(	PUNCT
cana-3825	127	5	2.2	2.2	NUM
cana-3825	127	6	)	)	PUNCT
cana-3825	127	7	and	and	CCONJ
cana-3825	127	8	consider	consider	VERB
cana-3825	127	9	r(n	r(n	NOUN
cana-3825	127	10	)	)	PUNCT
cana-3825	128	1	=	=	SYM
cana-3825	128	2	ξ̃k+1	ξ̃k+1	NOUN
cana-3825	128	3	(	(	PUNCT
cana-3825	128	4	n	n	CCONJ
cana-3825	128	5	)	)	PUNCT
cana-3825	128	6	−	−	PROPN
cana-3825	128	7	ρ(n	ρ(n	PROPN
cana-3825	128	8	)	)	PUNCT
cana-3825	128	9	.	.	PUNCT
cana-3825	129	1	consider	consider	VERB
cana-3825	129	2	∇μr	∇μr	PROPN
cana-3825	129	3	(	(	PUNCT
cana-3825	129	4	n	n	NOUN
cana-3825	129	5	+	+	NOUN
cana-3825	129	6	1	1	NUM
cana-3825	129	7	)	)	PUNCT
cana-3825	129	8	=	=	PUNCT
cana-3825	130	1	∇μ	∇μ	PROPN
cana-3825	131	1	[	[	X
cana-3825	131	2	ξ̃k+1	ξ̃k+1	PROPN
cana-3825	131	3	(	(	PUNCT
cana-3825	131	4	n	n	NOUN
cana-3825	131	5	+	+	CCONJ
cana-3825	131	6	1	1	NUM
cana-3825	131	7	)	)	PUNCT
cana-3825	131	8	–	–	PUNCT
cana-3825	131	9	ρ	ρ	PROPN
cana-3825	131	10	(	(	PUNCT
cana-3825	131	11	n	n	NOUN
cana-3825	131	12	+	+	NOUN
cana-3825	131	13	1	1	NUM
cana-3825	131	14	)	)	PUNCT
cana-3825	131	15	]	]	PUNCT
cana-3825	132	1	=	=	SYM
cana-3825	132	2	f	f	X
cana-3825	132	3	(	(	PUNCT
cana-3825	132	4	n	n	CCONJ
cana-3825	132	5	,	,	PUNCT
cana-3825	132	6	ξ̃k	ξ̃k	PROPN
cana-3825	132	7	(	(	PUNCT
cana-3825	132	8	n	n	CCONJ
cana-3825	132	9	)	)	PUNCT
cana-3825	132	10	)	)	PUNCT
cana-3825	132	11	–	–	PUNCT
cana-3825	132	12	f	f	X
cana-3825	132	13	(	(	PUNCT
cana-3825	132	14	n	n	CCONJ
cana-3825	132	15	,	,	PUNCT
cana-3825	132	16	ρ(n	ρ(n	PROPN
cana-3825	132	17	)	)	PUNCT
cana-3825	133	1	−	−	NOUN
cana-3825	133	2	m[ξ̃k+1	m[ξ̃k+1	ADJ
cana-3825	133	3	(	(	PUNCT
cana-3825	133	4	n	n	CCONJ
cana-3825	133	5	)	)	PUNCT
cana-3825	134	1	−	−	PROPN
cana-3825	134	2	ξ̃k	ξ̃k	PROPN
cana-3825	134	3	(	(	PUNCT
cana-3825	134	4	n	n	CCONJ
cana-3825	134	5	)	)	PUNCT
cana-3825	134	6	]	]	PUNCT
cana-3825	134	7	≤	≤	NUM
cana-3825	134	8	−m[ξ̃k	−m[ξ̃k	SYM
cana-3825	134	9	(	(	PUNCT
cana-3825	134	10	n	n	CCONJ
cana-3825	134	11	)	)	PUNCT
cana-3825	134	12	−	−	PROPN
cana-3825	134	13	ρ(n	ρ(n	PROPN
cana-3825	134	14	)	)	PUNCT
cana-3825	134	15	]	]	PUNCT
cana-3825	135	1	−	−	PROPN
cana-3825	135	2	m[ξ̃k+1	m[ξ̃k+1	ADJ
cana-3825	135	3	(	(	PUNCT
cana-3825	135	4	n	n	CCONJ
cana-3825	135	5	)	)	PUNCT
cana-3825	135	6	−ξ̃k	−ξ̃k	NOUN
cana-3825	135	7	(	(	PUNCT
cana-3825	135	8	n	n	CCONJ
cana-3825	135	9	)	)	PUNCT
cana-3825	135	10	]	]	PUNCT
cana-3825	136	1	=	=	PUNCT
cana-3825	136	2	−mr(n	−mr(n	X
cana-3825	136	3	)	)	PUNCT
cana-3825	136	4	.	.	PUNCT
cana-3825	137	1	also	also	ADV
cana-3825	137	2	,	,	PUNCT
cana-3825	137	3	r	r	NOUN
cana-3825	137	4	(	(	PUNCT
cana-3825	137	5	n0	n0	NUM
cana-3825	137	6	)	)	PUNCT
cana-3825	137	7	=	=	SYM
cana-3825	137	8	ξ̃k+1(n0)−	ξ̃k+1(n0)−	ADJ
cana-3825	137	9	ρ	ρ	PROPN
cana-3825	137	10	(	(	PUNCT
cana-3825	137	11	n0	n0	NUM
cana-3825	137	12	)	)	PUNCT
cana-3825	137	13	≤	≤	PUNCT
cana-3825	138	1	ρ0	ρ0	PROPN
cana-3825	138	2	−	−	PROPN
cana-3825	138	3	ρ0	ρ0	PROPN
cana-3825	138	4	≤	≤	NUM
cana-3825	138	5	0	0	NUM
cana-3825	138	6	.	.	PUNCT
cana-3825	139	1	by	by	ADP
cana-3825	139	2	using	use	VERB
cana-3825	139	3	theorem	theorem	NOUN
cana-3825	139	4	2.2	2.2	NUM
cana-3825	139	5	,	,	PUNCT
cana-3825	139	6	we	we	PRON
cana-3825	139	7	have	have	VERB
cana-3825	139	8	ξ̃k+1	ξ̃k+1	NOUN
cana-3825	139	9	(	(	PUNCT
cana-3825	139	10	n	n	CCONJ
cana-3825	139	11	)	)	PUNCT
cana-3825	139	12	≤	≤	NUM
cana-3825	139	13	ρ(n	ρ(n	PROPN
cana-3825	139	14	)	)	PUNCT
cana-3825	139	15	.	.	PUNCT
cana-3825	140	1	similarly	similarly	ADV
cana-3825	140	2	,	,	PUNCT
cana-3825	140	3	it	it	PRON
cana-3825	140	4	can	can	AUX
cana-3825	140	5	be	be	AUX
cana-3825	140	6	proved	prove	VERB
cana-3825	140	7	that	that	SCONJ
cana-3825	140	8	ρ(n	ρ(n	PROPN
cana-3825	140	9	)	)	PUNCT
cana-3825	140	10	≤	≤	NOUN
cana-3825	140	11	ωk+1(n	ωk+1(n	NOUN
cana-3825	140	12	)	)	PUNCT
cana-3825	140	13	.	.	PUNCT
cana-3825	141	1	therefore	therefore	ADV
cana-3825	141	2	ξ̃k+1	ξ̃k+1	PROPN
cana-3825	141	3	(	(	PUNCT
cana-3825	141	4	n	n	CCONJ
cana-3825	141	5	)	)	PUNCT
cana-3825	141	6	≤	≤	NUM
cana-3825	141	7	ρ(n	ρ(n	PROPN
cana-3825	141	8	)	)	PUNCT
cana-3825	141	9	≤	≤	NOUN
cana-3825	141	10	ωk+1(n	ωk+1(n	NOUN
cana-3825	141	11	)	)	PUNCT
cana-3825	141	12	.	.	PUNCT
cana-3825	142	1	hence	hence	ADV
cana-3825	142	2	the	the	DET
cana-3825	142	3	statement	statement	NOUN
cana-3825	142	4	(	(	PUNCT
cana-3825	142	5	2.9	2.9	NUM
cana-3825	142	6	)	)	PUNCT
cana-3825	142	7	is	be	AUX
cana-3825	142	8	true	true	ADJ
cana-3825	142	9	for	for	ADP
cana-3825	142	10	m	m	PROPN
cana-3825	142	11	=	=	SYM
cana-3825	142	12	k	k	PROPN
cana-3825	143	1	+	+	NOUN
cana-3825	143	2	1	1	X
cana-3825	143	3	.	.	PUNCT
cana-3825	143	4	by	by	ADP
cana-3825	143	5	the	the	DET
cana-3825	143	6	principle	principle	NOUN
cana-3825	143	7	of	of	ADP
cana-3825	143	8	mathematical	mathematical	ADJ
cana-3825	143	9	induction	induction	NOUN
cana-3825	143	10	,	,	PUNCT
cana-3825	143	11	the	the	DET
cana-3825	143	12	statement	statement	NOUN
cana-3825	143	13	(	(	PUNCT
cana-3825	143	14	2.9	2.9	NUM
cana-3825	143	15	)	)	PUNCT
cana-3825	143	16	is	be	AUX
cana-3825	143	17	true	true	ADJ
cana-3825	143	18	for	for	ADP
cana-3825	143	19	every	every	DET
cana-3825	143	20	m	m	PROPN
cana-3825	143	21	∈	∈	PROPN
cana-3825	143	22	n0	n0	NOUN
cana-3825	143	23	+	+	CCONJ
cana-3825	143	24	thus	thus	ADV
cana-3825	143	25	ξ̃m	ξ̃m	X
cana-3825	143	26	(	(	PUNCT
cana-3825	143	27	n	n	CCONJ
cana-3825	143	28	)	)	PUNCT
cana-3825	143	29	≤	≤	NUM
cana-3825	143	30	ρ(n	ρ(n	PROPN
cana-3825	143	31	)	)	PUNCT
cana-3825	143	32	≤	≤	NUM
cana-3825	143	33	ω̃m	ω̃m	NUM
cana-3825	143	34	(	(	PUNCT
cana-3825	143	35	n	n	CCONJ
cana-3825	143	36	)	)	PUNCT
cana-3825	143	37	.	.	PUNCT
cana-3825	144	1	taking	take	VERB
cana-3825	144	2	the	the	DET
cana-3825	144	3	limit	limit	NOUN
cana-3825	144	4	as	as	ADP
cana-3825	144	5	m	m	PROPN
cana-3825	144	6	→	→	SYM
cana-3825	144	7	∞	∞	PROPN
cana-3825	144	8	,	,	PUNCT
cana-3825	144	9	we	we	PRON
cana-3825	144	10	get	get	VERB
cana-3825	144	11	ξ̃	ξ̃	PROPN
cana-3825	144	12	≤	≤	ADV
cana-3825	144	13	ρ(n	ρ(n	PROPN
cana-3825	144	14	)	)	PUNCT
cana-3825	144	15	≤	≤	NOUN
cana-3825	145	1	ω̃	ω̃	NUM
cana-3825	145	2	such	such	ADJ
cana-3825	145	3	that	that	SCONJ
cana-3825	145	4	ξ̃0	ξ̃0	PROPN
cana-3825	145	5	(	(	PUNCT
cana-3825	145	6	n	n	CCONJ
cana-3825	145	7	)	)	PUNCT
cana-3825	145	8	≤	≤	NUM
cana-3825	145	9	ρ(n	ρ(n	PROPN
cana-3825	145	10	)	)	PUNCT
cana-3825	145	11	≤	≤	NUM
cana-3825	145	12	ω̃0	ω̃0	PROPN
cana-3825	145	13	.	.	PUNCT
cana-3825	146	1	hence	hence	ADV
cana-3825	146	2	the	the	DET
cana-3825	146	3	functions	function	NOUN
cana-3825	146	4	ξ̃	ξ̃	PROPN
cana-3825	146	5	and	and	CCONJ
cana-3825	146	6	ω̃	ω̃	NUM
cana-3825	146	7	defined	define	VERB
cana-3825	146	8	for	for	ADP
cana-3825	146	9	n	n	DET
cana-3825	146	10	∈	∈	PROPN
cana-3825	146	11	nn0	nn0	NOUN
cana-3825	146	12	+	+	CCONJ
cana-3825	146	13	are	be	AUX
cana-3825	146	14	minimal	minimal	ADJ
cana-3825	146	15	and	and	CCONJ
cana-3825	146	16	maximal	maximal	ADJ
cana-3825	146	17	solutions	solution	NOUN
cana-3825	146	18	of	of	ADP
cana-3825	146	19	(	(	PUNCT
cana-3825	146	20	2.1	2.1	NUM
cana-3825	146	21	)	)	PUNCT
cana-3825	146	22	respectively	respectively	ADV
cana-3825	146	23	.	.	PUNCT
cana-3825	147	1	theorem	theorem	VERB
cana-3825	147	2	2.5	2.5	NUM
cana-3825	147	3	.	.	PUNCT
cana-3825	148	1	in	in	ADP
cana-3825	148	2	addition	addition	NOUN
cana-3825	148	3	to	to	ADP
cana-3825	148	4	the	the	DET
cana-3825	148	5	hypothesis	hypothesis	NOUN
cana-3825	148	6	of	of	ADP
cana-3825	148	7	theorem	theorem	NOUN
cana-3825	148	8	(	(	PUNCT
cana-3825	148	9	2.4	2.4	NUM
cana-3825	148	10	)	)	PUNCT
cana-3825	148	11	,	,	PUNCT
cana-3825	148	12	if	if	SCONJ
cana-3825	148	13	we	we	PRON
cana-3825	148	14	assume	assume	VERB
cana-3825	148	15	that	that	SCONJ
cana-3825	148	16	f(n	f(n	PROPN
cana-3825	148	17	,	,	PUNCT
cana-3825	148	18	x	x	NOUN
cana-3825	148	19	)	)	PUNCT
cana-3825	148	20	−	−	PROPN
cana-3825	148	21	f(n	f(n	PROPN
cana-3825	148	22	,	,	PUNCT
cana-3825	148	23	y	y	PROPN
cana-3825	148	24	)	)	PUNCT
cana-3825	148	25	≥	≥	PROPN
cana-3825	148	26	−m(x	−m(x	PUNCT
cana-3825	149	1	−	−	PROPN
cana-3825	149	2	y	y	NOUN
cana-3825	149	3	)	)	PUNCT
cana-3825	149	4	,	,	PUNCT
cana-3825	149	5	(	(	PUNCT
cana-3825	149	6	2.10	2.10	NUM
cana-3825	149	7	)	)	PUNCT
cana-3825	149	8	for	for	ADP
cana-3825	149	9	ξ	ξ	PROPN
cana-3825	149	10	≤	≤	NUM
cana-3825	149	11	y	y	NOUN
cana-3825	149	12	≤	≤	NUM
cana-3825	149	13	x	x	PUNCT
cana-3825	149	14	≤	≤	NUM
cana-3825	149	15	ω	ω	NUM
cana-3825	149	16	and	and	CCONJ
cana-3825	149	17	m	m	PROPN
cana-3825	149	18	≥	≥	NOUN
cana-3825	149	19	0	0	NUM
cana-3825	149	20	.	.	PUNCT
cana-3825	150	1	then	then	ADV
cana-3825	150	2	ξ	ξ	X
cana-3825	150	3	=	=	SYM
cana-3825	150	4	ω	ω	PROPN
cana-3825	150	5	=	=	SYM
cana-3825	150	6	ρ	ρ	PROPN
cana-3825	150	7	is	be	AUX
cana-3825	150	8	the	the	DET
cana-3825	150	9	unique	unique	ADJ
cana-3825	150	10	solution	solution	NOUN
cana-3825	150	11	of	of	ADP
cana-3825	150	12	(	(	PUNCT
cana-3825	150	13	2.1	2.1	NUM
cana-3825	150	14	)	)	PUNCT
cana-3825	150	15	.	.	PUNCT
cana-3825	151	1	proof	proof	NOUN
cana-3825	151	2	:	:	PUNCT
cana-3825	151	3	since	since	SCONJ
cana-3825	151	4	ξ(n	ξ(n	PROPN
cana-3825	151	5	)	)	PUNCT
cana-3825	151	6	≤	≤	NOUN
cana-3825	151	7	ω(n	ω(n	NUM
cana-3825	151	8	)	)	PUNCT
cana-3825	151	9	,	,	PUNCT
cana-3825	151	10	it	it	PRON
cana-3825	151	11	is	be	AUX
cana-3825	151	12	enough	enough	ADJ
cana-3825	151	13	to	to	PART
cana-3825	151	14	prove	prove	VERB
cana-3825	151	15	that	that	SCONJ
cana-3825	151	16	ξ(n	ξ(n	PROPN
cana-3825	151	17	)	)	PUNCT
cana-3825	151	18	≥	≥	NOUN
cana-3825	151	19	ω(n	ω(n	NUM
cana-3825	151	20	)	)	PUNCT
cana-3825	151	21	.	.	PUNCT
cana-3825	152	1	take	take	VERB
cana-3825	152	2	s(n	s(n	NOUN
cana-3825	152	3	)	)	PUNCT
cana-3825	152	4	=	=	PUNCT
cana-3825	152	5	ξ(n	ξ(n	NOUN
cana-3825	152	6	)	)	PUNCT
cana-3825	152	7	−	−	PROPN
cana-3825	152	8	ω(n	ω(n	NUM
cana-3825	152	9	)	)	PUNCT
cana-3825	152	10	.	.	PUNCT
cana-3825	153	1	consider	consider	VERB
cana-3825	153	2	∇μ	∇μ	PROPN
cana-3825	153	3	s	s	PROPN
cana-3825	153	4	(	(	PUNCT
cana-3825	153	5	n	n	PROPN
cana-3825	153	6	+	+	NOUN
cana-3825	153	7	1	1	NUM
cana-3825	153	8	)	)	PUNCT
cana-3825	153	9	=	=	PUNCT
cana-3825	154	1	∇μ	∇μ	PROPN
cana-3825	154	2	[	[	X
cana-3825	154	3	ξ	ξ	X
cana-3825	154	4	(	(	PUNCT
cana-3825	154	5	n	n	X
cana-3825	154	6	+	+	NOUN
cana-3825	154	7	1	1	NUM
cana-3825	154	8	)	)	PUNCT
cana-3825	154	9	–	–	PUNCT
cana-3825	154	10	ω	ω	PROPN
cana-3825	154	11	(	(	PUNCT
cana-3825	154	12	n	n	PROPN
cana-3825	154	13	+	+	NOUN
cana-3825	154	14	1	1	NUM
cana-3825	154	15	)	)	PUNCT
cana-3825	154	16	]	]	PUNCT
cana-3825	155	1	=	=	PUNCT
cana-3825	155	2	−	−	PROPN
cana-3825	155	3	(	(	PUNCT
cana-3825	155	4	f	f	PROPN
cana-3825	155	5	(	(	PUNCT
cana-3825	155	6	n	n	CCONJ
cana-3825	155	7	,	,	PUNCT
cana-3825	155	8	ω(n	ω(n	NUM
cana-3825	155	9	)	)	PUNCT
cana-3825	155	10	)	)	PUNCT
cana-3825	155	11	–	–	PUNCT
cana-3825	155	12	f	f	X
cana-3825	155	13	(	(	PUNCT
cana-3825	155	14	n	n	X
cana-3825	155	15	,	,	PUNCT
cana-3825	155	16	ξ(n	ξ(n	PROPN
cana-3825	155	17	)	)	PUNCT
cana-3825	155	18	)	)	PUNCT
cana-3825	155	19	)	)	PUNCT
cana-3825	155	20	≤	≤	NOUN
cana-3825	155	21	m[ω(n	m[ω(n	PROPN
cana-3825	155	22	)	)	PUNCT
cana-3825	155	23	−	−	ADP
cana-3825	155	24	ξ(n	ξ(n	NOUN
cana-3825	155	25	)	)	PUNCT
cana-3825	155	26	]	]	PUNCT
cana-3825	155	27	=	=	PUNCT
cana-3825	155	28	−ms(n	−ms(n	NOUN
cana-3825	155	29	)	)	PUNCT
cana-3825	155	30	.	.	PUNCT
cana-3825	156	1	also	also	ADV
cana-3825	156	2	r	r	NOUN
cana-3825	156	3	(	(	PUNCT
cana-3825	156	4	n0	n0	NUM
cana-3825	156	5	)	)	PUNCT
cana-3825	156	6	=	=	SYM
cana-3825	156	7	ξ	ξ	PROPN
cana-3825	156	8	(	(	PUNCT
cana-3825	156	9	n0	n0	NUM
cana-3825	156	10	)	)	PUNCT
cana-3825	156	11	–	–	PUNCT
cana-3825	156	12	ρ	ρ	PROPN
cana-3825	156	13	(	(	PUNCT
cana-3825	156	14	n0	n0	NUM
cana-3825	156	15	)	)	PUNCT
cana-3825	156	16	≤	≤	PUNCT
cana-3825	157	1	ρ0	ρ0	PROPN
cana-3825	157	2	−	−	PROPN
cana-3825	157	3	ρ0	ρ0	PROPN
cana-3825	157	4	≤	≤	NUM
cana-3825	157	5	0	0	NUM
cana-3825	157	6	.	.	PUNCT
cana-3825	158	1	by	by	ADP
cana-3825	158	2	using	use	VERB
cana-3825	158	3	theorem2.2	theorem2.2	NOUN
cana-3825	158	4	,	,	PUNCT
cana-3825	158	5	hence	hence	ADV
cana-3825	158	6	s(n	s(n	NOUN
cana-3825	158	7	)	)	PUNCT
cana-3825	158	8	≤	≤	NOUN
cana-3825	158	9	0	0	NUM
cana-3825	158	10	for	for	ADP
cana-3825	158	11	n	n	PRON
cana-3825	158	12	∈	∈	PROPN
cana-3825	158	13	nn0	nn0	NOUN
cana-3825	158	14	+	+	X
cana-3825	158	15	.	.	PUNCT
cana-3825	158	16	thus	thus	ADV
cana-3825	158	17	ω(n	ω(n	NUM
cana-3825	158	18	)	)	PUNCT
cana-3825	158	19	≤	≤	NUM
cana-3825	158	20	ξ(n	ξ(n	NOUN
cana-3825	158	21	)	)	PUNCT
cana-3825	158	22	.	.	PUNCT
cana-3825	159	1	hence	hence	ADV
cana-3825	159	2	ξ(n	ξ(n	PROPN
cana-3825	159	3	)	)	PUNCT
cana-3825	159	4	=	=	PUNCT
cana-3825	159	5	ω(n	ω(n	NUM
cana-3825	159	6	)	)	PUNCT
cana-3825	159	7	=	=	SYM
cana-3825	159	8	ρ(n	ρ(n	PROPN
cana-3825	159	9	)	)	PUNCT
cana-3825	159	10	is	be	AUX
cana-3825	159	11	the	the	DET
cana-3825	159	12	unique	unique	ADJ
cana-3825	159	13	solution	solution	NOUN
cana-3825	159	14	of	of	ADP
cana-3825	159	15	(	(	PUNCT
cana-3825	159	16	2.1	2.1	NUM
cana-3825	159	17	)	)	PUNCT
cana-3825	159	18	.	.	PUNCT
cana-3825	160	1	communications	communication	NOUN
cana-3825	160	2	on	on	ADP
cana-3825	160	3	applied	apply	VERB
cana-3825	160	4	nonlinear	nonlinear	ADJ
cana-3825	160	5	analysis	analysis	NOUN
cana-3825	160	6	issn	issn	NOUN
cana-3825	160	7	:	:	PUNCT
cana-3825	160	8	1074	1074	NUM
cana-3825	160	9	-	-	PUNCT
cana-3825	160	10	133x	133x	NUM
cana-3825	160	11	vol	vol	NOUN
cana-3825	160	12	32	32	NUM
cana-3825	160	13	no	no	NOUN
cana-3825	160	14	.	.	PUNCT
cana-3825	161	1	8s	8s	PROPN
cana-3825	161	2	(	(	PUNCT
cana-3825	161	3	2025	2025	NUM
cana-3825	161	4	)	)	PUNCT
cana-3825	161	5	834	834	NUM
cana-3825	161	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3825	161	7	conclusion	conclusion	NOUN
cana-3825	161	8	in	in	ADP
cana-3825	161	9	this	this	DET
cana-3825	161	10	chapter	chapter	NOUN
cana-3825	161	11	,	,	PUNCT
cana-3825	161	12	using	use	VERB
cana-3825	161	13	monotone	monotone	ADJ
cana-3825	161	14	iterative	iterative	NOUN
cana-3825	161	15	technique	technique	NOUN
cana-3825	161	16	on	on	ADP
cana-3825	161	17	solutions	solution	NOUN
cana-3825	161	18	of	of	ADP
cana-3825	161	19	fractional	fractional	ADJ
cana-3825	161	20	difference	difference	NOUN
cana-3825	161	21	equations	equation	NOUN
cana-3825	161	22	at	at	ADP
cana-3825	161	23	different	different	ADJ
cana-3825	161	24	initial	initial	ADJ
cana-3825	161	25	times	time	NOUN
cana-3825	161	26	the	the	DET
cana-3825	161	27	convergence	convergence	NOUN
cana-3825	161	28	of	of	ADP
cana-3825	161	29	monotone	monotone	ADJ
cana-3825	161	30	sequences	sequence	NOUN
cana-3825	161	31	to	to	ADP
cana-3825	161	32	maximum	maximum	ADJ
cana-3825	161	33	and	and	CCONJ
cana-3825	161	34	minimum	minimum	ADJ
cana-3825	161	35	solutions	solution	NOUN
cana-3825	161	36	is	be	AUX
cana-3825	161	37	established	establish	VERB
cana-3825	161	38	.	.	PUNCT
cana-3825	162	1	using	use	VERB
cana-3825	162	2	lower	low	ADJ
cana-3825	162	3	and	and	CCONJ
cana-3825	162	4	upper	upper	ADJ
cana-3825	162	5	solutions	solution	NOUN
cana-3825	162	6	,	,	PUNCT
cana-3825	162	7	existence	existence	NOUN
cana-3825	162	8	and	and	CCONJ
cana-3825	162	9	uniqueness	uniqueness	NOUN
cana-3825	162	10	of	of	ADP
cana-3825	162	11	solution	solution	NOUN
cana-3825	162	12	to	to	ADP
cana-3825	162	13	fractional	fractional	ADJ
cana-3825	162	14	difference	difference	NOUN
cana-3825	162	15	equations	equation	NOUN
cana-3825	162	16	at	at	ADP
cana-3825	162	17	different	different	ADJ
cana-3825	162	18	initial	initial	ADJ
cana-3825	162	19	times	time	NOUN
cana-3825	162	20	is	be	AUX
cana-3825	162	21	also	also	ADV
cana-3825	162	22	obtained	obtain	VERB
cana-3825	162	23	.	.	PUNCT
cana-3825	163	1	future	future	ADJ
cana-3825	163	2	work	work	NOUN
cana-3825	163	3	:	:	PUNCT
cana-3825	163	4	this	this	DET
cana-3825	163	5	work	work	NOUN
cana-3825	163	6	can	can	AUX
cana-3825	163	7	be	be	AUX
cana-3825	163	8	extended	extend	VERB
cana-3825	163	9	onto	onto	ADP
cana-3825	163	10	the	the	DET
cana-3825	163	11	systems	system	NOUN
cana-3825	163	12	of	of	ADP
cana-3825	163	13	fractional	fractional	ADJ
cana-3825	163	14	difference	difference	NOUN
cana-3825	163	15	equations	equation	NOUN
cana-3825	163	16	this	this	DET
cana-3825	163	17	work	work	NOUN
cana-3825	163	18	may	may	AUX
cana-3825	163	19	make	make	VERB
cana-3825	163	20	better	well	ADJ
cana-3825	163	21	some	some	DET
cana-3825	163	22	processes	process	NOUN
cana-3825	163	23	concerned	concern	VERB
cana-3825	163	24	with	with	ADP
cana-3825	163	25	signal	signal	NOUN
cana-3825	163	26	processing	processing	NOUN
cana-3825	163	27	and	and	CCONJ
cana-3825	163	28	image	image	NOUN
cana-3825	163	29	processing	processing	NOUN
cana-3825	163	30	,	,	PUNCT
cana-3825	163	31	system	system	NOUN
cana-3825	163	32	trajectories	trajectory	NOUN
cana-3825	163	33	and	and	CCONJ
cana-3825	163	34	various	various	ADJ
cana-3825	163	35	other	other	ADJ
cana-3825	163	36	fields	field	NOUN
cana-3825	163	37	.	.	PUNCT
cana-3825	164	1	acknowledgement	acknowledgement	NOUN
cana-3825	164	2	:	:	PUNCT
cana-3825	164	3	the	the	DET
cana-3825	164	4	authors	author	NOUN
cana-3825	164	5	thank	thank	VERB
cana-3825	164	6	their	their	PRON
cana-3825	164	7	respective	respective	ADJ
cana-3825	164	8	college	college	NOUN
cana-3825	164	9	managements	management	NOUN
cana-3825	164	10	for	for	ADP
cana-3825	164	11	their	their	PRON
cana-3825	164	12	continuous	continuous	ADJ
cana-3825	164	13	support	support	NOUN
cana-3825	164	14	and	and	CCONJ
cana-3825	164	15	constant	constant	ADJ
cana-3825	164	16	encouragement	encouragement	NOUN
cana-3825	164	17	.	.	PUNCT
cana-3825	165	1	reference	reference	NOUN
cana-3825	165	2	[	[	X
cana-3825	165	3	1	1	NUM
cana-3825	165	4	]	]	X
cana-3825	165	5	deekshitulu	deekshitulu	ADJ
cana-3825	165	6	gvsr	gvsr	NOUN
cana-3825	165	7	and	and	CCONJ
cana-3825	165	8	jj	jj	PROPN
cana-3825	165	9	mohan	mohan	PROPN
cana-3825	165	10	:	:	PUNCT
cana-3825	165	11	solutions	solution	NOUN
cana-3825	165	12	of	of	ADP
cana-3825	165	13	perturbed	perturb	VERB
cana-3825	165	14	non	non	ADJ
cana-3825	165	15	-	-	ADJ
cana-3825	165	16	linear	linear	ADJ
cana-3825	165	17	nabla	nabla	PROPN
cana-3825	165	18	fractionaldifference	fractionaldifference	NOUN
cana-3825	165	19	equations	equation	NOUN
cana-3825	165	20	of	of	ADP
cana-3825	165	21	order	order	NOUN
cana-3825	165	22	0	0	PUNCT
cana-3825	165	23	<	<	X
cana-3825	165	24	α	α	X
cana-3825	165	25	<	<	X
cana-3825	165	26	1	1	NUM
cana-3825	165	27	,	,	PUNCT
cana-3825	165	28	mathematica	mathematica	PROPN
cana-3825	165	29	aeterna	aeterna	PROPN
cana-3825	165	30	,	,	PUNCT
cana-3825	165	31	vol	vol	NOUN
cana-3825	165	32	.	.	PROPN
cana-3825	166	1	3	3	NUM
cana-3825	166	2	,	,	PUNCT
cana-3825	166	3	2013	2013	NUM
cana-3825	166	4	,	,	PUNCT
cana-3825	166	5	no.2	no.2	PROPN
cana-3825	166	6	,	,	PUNCT
cana-3825	166	7	139	139	NUM
cana-3825	166	8	-	-	SYM
cana-3825	166	9	150	150	NUM
cana-3825	166	10	.	.	PUNCT
cana-3825	167	1	[	[	X
cana-3825	167	2	2	2	NUM
cana-3825	167	3	]	]	PUNCT
cana-3825	167	4	deekshitulu	deekshitulu	VERB
cana-3825	167	5	,	,	PUNCT
cana-3825	167	6	g.v.s.r	g.v.s.r	PROPN
cana-3825	167	7	.	.	PROPN
cana-3825	167	8	and	and	CCONJ
cana-3825	167	9	jagan	jagan	PROPN
cana-3825	167	10	mohan	mohan	PROPN
cana-3825	167	11	,	,	PUNCT
cana-3825	167	12	j.	j.	PROPN
cana-3825	167	13	,	,	PUNCT
cana-3825	167	14	fractional	fractional	ADJ
cana-3825	167	15	difference	difference	NOUN
cana-3825	167	16	inequalities	inequality	NOUN
cana-3825	167	17	of	of	ADP
cana-3825	167	18	bihari	bihari	PROPN
cana-3825	167	19	type	type	NOUN
cana-3825	167	20	,	,	PUNCT
cana-3825	167	21	communications	communication	NOUN
cana-3825	167	22	in	in	ADP
cana-3825	167	23	applied	apply	VERB
cana-3825	167	24	analysis	analysis	NOUN
cana-3825	167	25	,	,	PUNCT
cana-3825	167	26	14	14	NUM
cana-3825	167	27	,	,	PUNCT
cana-3825	167	28	no	no	INTJ
cana-3825	167	29	.	.	NOUN
cana-3825	167	30	4	4	NUM
cana-3825	167	31	(	(	PUNCT
cana-3825	167	32	2010	2010	NUM
cana-3825	167	33	)	)	PUNCT
cana-3825	167	34	,	,	PUNCT
cana-3825	167	35	343	343	NUM
cana-3825	167	36	-	-	SYM
cana-3825	167	37	354	354	NUM
cana-3825	167	38	.	.	PUNCT
cana-3825	168	1	[	[	X
cana-3825	168	2	3	3	NUM
cana-3825	168	3	]	]	X
cana-3825	168	4	deekshitulu	deekshitulu	VERB
cana-3825	168	5	,	,	PUNCT
cana-3825	168	6	g.v.s.r	g.v.s.r	PROPN
cana-3825	168	7	.	.	PROPN
cana-3825	168	8	and	and	CCONJ
cana-3825	168	9	jagan	jagan	PROPN
cana-3825	168	10	mohan	mohan	PROPN
cana-3825	168	11	,	,	PUNCT
cana-3825	168	12	j.	j.	PROPN
cana-3825	168	13	,	,	PUNCT
cana-3825	168	14	fractional	fractional	ADJ
cana-3825	168	15	difference	difference	NOUN
cana-3825	168	16	inequalities	inequality	NOUN
cana-3825	168	17	of	of	ADP
cana-3825	168	18	volterra	volterra	PROPN
cana-3825	168	19	type	type	NOUN
cana-3825	168	20	,	,	PUNCT
cana-3825	168	21	international	international	ADJ
cana-3825	168	22	journal	journal	NOUN
cana-3825	168	23	of	of	ADP
cana-3825	168	24	pure	pure	ADJ
cana-3825	168	25	and	and	CCONJ
cana-3825	168	26	applied	applied	ADJ
cana-3825	168	27	mathematics	mathematic	NOUN
cana-3825	168	28	,	,	PUNCT
cana-3825	168	29	70	70	NUM
cana-3825	168	30	,	,	PUNCT
cana-3825	168	31	no	no	INTJ
cana-3825	168	32	.	.	NOUN
cana-3825	168	33	2	2	NUM
cana-3825	168	34	(	(	PUNCT
cana-3825	168	35	2011	2011	NUM
cana-3825	168	36	)	)	PUNCT
cana-3825	168	37	,	,	PUNCT
cana-3825	168	38	137	137	NUM
cana-3825	168	39	-	-	SYM
cana-3825	168	40	149	149	NUM
cana-3825	168	41	.	.	PUNCT
cana-3825	169	1	[	[	X
cana-3825	169	2	4	4	NUM
cana-3825	169	3	]	]	X
cana-3825	169	4	deekshitulu	deekshitulu	VERB
cana-3825	169	5	,	,	PUNCT
cana-3825	169	6	g.v.s.r	g.v.s.r	PROPN
cana-3825	169	7	.	.	PROPN
cana-3825	169	8	and	and	CCONJ
cana-3825	169	9	jagan	jagan	PROPN
cana-3825	169	10	mohan	mohan	PROPN
cana-3825	169	11	,	,	PUNCT
cana-3825	169	12	j.	j.	PROPN
cana-3825	169	13	,	,	PUNCT
cana-3825	169	14	fractional	fractional	ADJ
cana-3825	169	15	difference	difference	NOUN
cana-3825	169	16	inequalities	inequality	NOUN
cana-3825	169	17	of	of	ADP
cana-3825	169	18	opial	opial	ADJ
cana-3825	169	19	type	type	NOUN
cana-3825	169	20	and	and	CCONJ
cana-3825	169	21	initial	initial	ADJ
cana-3825	169	22	value	value	NOUN
cana-3825	169	23	problem	problem	NOUN
cana-3825	169	24	,	,	PUNCT
cana-3825	169	25	fractional	fractional	ADJ
cana-3825	169	26	differential	differential	NOUN
cana-3825	169	27	calculus	calculus	NOUN
cana-3825	169	28	,	,	PUNCT
cana-3825	169	29	2	2	NUM
cana-3825	169	30	,	,	PUNCT
cana-3825	169	31	no	no	INTJ
cana-3825	169	32	.	.	NOUN
cana-3825	169	33	1	1	NUM
cana-3825	169	34	(	(	PUNCT
cana-3825	169	35	2012	2012	NUM
cana-3825	169	36	)	)	PUNCT
cana-3825	169	37	,	,	PUNCT
cana-3825	169	38	73	73	NUM
cana-3825	169	39	-	-	SYM
cana-3825	169	40	86	86	NUM
cana-3825	169	41	.	.	PUNCT
cana-3825	170	1	[	[	X
cana-3825	170	2	5	5	NUM
cana-3825	170	3	]	]	PUNCT
cana-3825	170	4	deekshitulu	deekshitulu	VERB
cana-3825	170	5	,	,	PUNCT
cana-3825	170	6	g.v.s.r	g.v.s.r	PROPN
cana-3825	170	7	.	.	PROPN
cana-3825	170	8	and	and	CCONJ
cana-3825	170	9	jagan	jagan	PROPN
cana-3825	170	10	mohan	mohan	PROPN
cana-3825	170	11	,	,	PUNCT
cana-3825	170	12	j.	j.	PROPN
cana-3825	170	13	,	,	PUNCT
cana-3825	170	14	some	some	DET
cana-3825	170	15	new	new	ADJ
cana-3825	170	16	fractional	fractional	ADJ
cana-3825	170	17	difference	difference	NOUN
cana-3825	170	18	inequalities	inequality	NOUN
cana-3825	170	19	of	of	ADP
cana-3825	170	20	gronwall	gronwall	ADJ
cana-3825	170	21	bellman	bellman	NOUN
cana-3825	170	22	type	type	NOUN
cana-3825	170	23	,	,	PUNCT
cana-3825	170	24	mathematical	mathematical	ADJ
cana-3825	170	25	sciences	science	NOUN
cana-3825	170	26	,	,	PUNCT
cana-3825	170	27	springer	springer	NOUN
cana-3825	170	28	open	open	NOUN
cana-3825	170	29	,	,	PUNCT
cana-3825	170	30	volume	volume	NOUN
cana-3825	170	31	6	6	NUM
cana-3825	170	32	,	,	PUNCT
cana-3825	170	33	number	number	NOUN
cana-3825	170	34	69	69	NUM
cana-3825	170	35	,	,	PUNCT
cana-3825	170	36	doi	doi	NOUN
cana-3825	170	37	:	:	PUNCT
cana-3825	170	38	10.1186/2251	10.1186/2251	NUM
cana-3825	170	39	-	-	SYM
cana-3825	170	40	7456	7456	NUM
cana-3825	170	41	-	-	PUNCT
cana-3825	170	42	6	6	NUM
cana-3825	170	43	-	-	NUM
cana-3825	170	44	69	69	NUM
cana-3825	170	45	.	.	PUNCT
cana-3825	171	1	[	[	X
cana-3825	171	2	6	6	NUM
cana-3825	171	3	]	]	SYM
cana-3825	171	4	d.n	d.n	PROPN
cana-3825	171	5	.	.	PROPN
cana-3825	171	6	purnima	purnima	PROPN
cana-3825	171	7	,	,	PUNCT
cana-3825	171	8	g.v.s.r	g.v.s.r	PROPN
cana-3825	171	9	.	.	PROPN
cana-3825	171	10	deekshitulu	deekshitulu	VERB
cana-3825	171	11	,	,	PUNCT
cana-3825	171	12	monotone	monotone	ADJ
cana-3825	171	13	iterative	iterative	NOUN
cana-3825	171	14	technique	technique	NOUN
cana-3825	171	15	for	for	ADP
cana-3825	171	16	finite	finite	ADJ
cana-3825	171	17	system	system	NOUN
cana-3825	171	18	of	of	ADP
cana-3825	171	19	fractional	fractional	ADJ
cana-3825	171	20	difference	difference	NOUN
cana-3825	171	21	equations	equation	NOUN
cana-3825	171	22	,	,	PUNCT
cana-3825	171	23	international	international	ADJ
cana-3825	171	24	journal	journal	NOUN
cana-3825	171	25	of	of	ADP
cana-3825	171	26	pure	pure	ADJ
cana-3825	171	27	and	and	CCONJ
cana-3825	171	28	applied	applied	ADJ
cana-3825	171	29	mathematics	mathematic	NOUN
cana-3825	171	30	,	,	PUNCT
cana-3825	171	31	volume	volume	NOUN
cana-3825	171	32	112	112	NUM
cana-3825	172	1	no	no	NOUN
cana-3825	172	2	.	.	NOUN
cana-3825	172	3	4	4	NUM
cana-3825	172	4	2017	2017	NUM
cana-3825	172	5	,	,	PUNCT
cana-3825	172	6	673	673	NUM
cana-3825	172	7	-	-	SYM
cana-3825	172	8	682	682	NUM
cana-3825	172	9	,	,	PUNCT
cana-3825	172	10	doi	doi	NOUN
cana-3825	172	11	:	:	PUNCT
cana-3825	172	12	10.12732	10.12732	NUM
cana-3825	172	13	/	/	SYM
cana-3825	172	14	ijpam	ijpam	NOUN
cana-3825	172	15	.	.	PUNCT
cana-3825	173	1	v112i4.2	v112i4.2	NOUN
cana-3825	173	2	[	[	X
cana-3825	173	3	7	7	X
cana-3825	173	4	]	]	X
cana-3825	173	5	g.v.s.r	g.v.s.r	PROPN
cana-3825	173	6	.	.	PROPN
cana-3825	173	7	deekshitulu	deekshitulu	PROPN
cana-3825	173	8	,	,	PUNCT
cana-3825	173	9	j.	j.	PROPN
cana-3825	173	10	jagan	jagan	PROPN
cana-3825	173	11	mohan	mohan	PROPN
cana-3825	173	12	,	,	PUNCT
cana-3825	173	13	some	some	DET
cana-3825	173	14	new	new	ADJ
cana-3825	173	15	fractional	fractional	ADJ
cana-3825	173	16	difference	difference	NOUN
cana-3825	173	17	inequalities	inequality	NOUN
cana-3825	173	18	,	,	PUNCT
cana-3825	173	19	icmmsc	icmmsc	NOUN
cana-3825	173	20	2012	2012	NUM
cana-3825	173	21	,	,	PUNCT
cana-3825	173	22	ccis	ccis	PROPN
cana-3825	173	23	,	,	PUNCT
cana-3825	173	24	283	283	NUM
cana-3825	173	25	,	,	PUNCT
cana-3825	173	26	springer	springer	NOUN
cana-3825	173	27	-	-	PUNCT
cana-3825	173	28	verlag	verlag	PROPN
cana-3825	173	29	,	,	PUNCT
cana-3825	173	30	berlinheidelberg	berlinheidelberg	PROPN
cana-3825	173	31	,	,	PUNCT
cana-3825	173	32	403	403	NUM
cana-3825	173	33	-	-	SYM
cana-3825	173	34	12	12	NUM
cana-3825	173	35	(	(	PUNCT
cana-3825	173	36	2012	2012	NUM
cana-3825	173	37	)	)	PUNCT
cana-3825	173	38	.	.	PUNCT
cana-3825	174	1	[	[	X
cana-3825	174	2	8	8	NUM
cana-3825	174	3	]	]	X
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cana-3825	174	5	,	,	PUNCT
cana-3825	174	6	lectures	lecture	VERB
cana-3825	174	7	on	on	ADP
cana-3825	174	8	difference	difference	NOUN
cana-3825	174	9	equations	equation	NOUN
cana-3825	174	10	,	,	PUNCT
cana-3825	174	11	science	science	NOUN
cana-3825	174	12	-	-	PUNCT
cana-3825	174	13	sha	sha	PROPN
cana-3825	174	14	,	,	PUNCT
cana-3825	174	15	(	(	PUNCT
cana-3825	174	16	in	in	ADP
cana-3825	174	17	japanese	japanese	PROPN
cana-3825	174	18	)	)	PUNCT
cana-3825	174	19	,	,	PUNCT
cana-3825	174	20	(	(	PUNCT
cana-3825	174	21	2000	2000	NUM
cana-3825	174	22	)	)	PUNCT
cana-3825	174	23	.	.	PUNCT
cana-3825	175	1	[	[	X
cana-3825	175	2	9	9	NUM
cana-3825	175	3	]	]	SYM
cana-3825	175	4	nagai	nagai	PROPN
cana-3825	175	5	:	:	PUNCT
cana-3825	175	6	fractional	fractional	ADJ
cana-3825	175	7	logistic	logistic	ADJ
cana-3825	175	8	map	map	NOUN
cana-3825	175	9	,	,	PUNCT
cana-3825	175	10	arxiv	arxiv	PROPN
cana-3825	175	11	:	:	PUNCT
cana-3825	175	12	nlin/0206018v1	nlin/0206018v1	PROPN
cana-3825	176	1	[	[	X
cana-3825	176	2	nlin.si	nlin.si	X
cana-3825	176	3	]	]	X
cana-3825	176	4	,	,	PUNCT
cana-3825	176	5	(	(	PUNCT
cana-3825	176	6	2002	2002	NUM
cana-3825	176	7	)	)	PUNCT
cana-3825	176	8	.	.	PUNCT
cana-3825	177	1	[	[	X
cana-3825	177	2	10	10	NUM
cana-3825	177	3	]	]	X
cana-3825	177	4	ramana	ramana	PROPN
cana-3825	177	5	,	,	PUNCT
cana-3825	177	6	g.v	g.v	PROPN
cana-3825	177	7	and	and	CCONJ
cana-3825	177	8	et	et	PROPN
cana-3825	177	9	al	al	PROPN
cana-3825	177	10	.	.	PROPN
cana-3825	177	11	(	(	PUNCT
cana-3825	177	12	2025	2025	NUM
cana-3825	177	13	)	)	PUNCT
cana-3825	177	14	.	.	PUNCT
cana-3825	178	1	stability	stability	NOUN
cana-3825	178	2	analysis	analysis	NOUN
cana-3825	178	3	of	of	ADP
cana-3825	178	4	two	two	NUM
cana-3825	178	5	point	point	NOUN
cana-3825	178	6	boundary	boundary	ADJ
cana-3825	178	7	value	value	NOUN
cana-3825	178	8	problem	problem	NOUN
cana-3825	178	9	on	on	ADP
cana-3825	178	10	time	time	NOUN
cana-3825	178	11	scales	scale	NOUN
cana-3825	178	12	,	,	PUNCT
cana-3825	178	13	communications	communication	NOUN
cana-3825	178	14	on	on	ADP
cana-3825	178	15	applied	apply	VERB
cana-3825	178	16	nonlinear	nonlinear	ADJ
cana-3825	178	17	analysis	analysis	NOUN
cana-3825	178	18	.	.	PUNCT
cana-3825	179	1	[	[	X
cana-3825	179	2	11	11	NUM
cana-3825	179	3	]	]	X
cana-3825	179	4	ramana	ramana	PROPN
cana-3825	179	5	,	,	PUNCT
cana-3825	179	6	g.v	g.v	PROPN
cana-3825	179	7	;	;	PUNCT
cana-3825	179	8	deekshitulu	deekshitulu	VERB
cana-3825	179	9	,	,	PUNCT
cana-3825	179	10	g.v.s.r	g.v.s.r	PROPN
cana-3825	179	11	.	.	PROPN
cana-3825	179	12	(	(	PUNCT
cana-3825	179	13	2017	2017	NUM
cana-3825	179	14	)	)	PUNCT
cana-3825	179	15	.	.	PUNCT
cana-3825	179	16	controllability	controllability	NOUN
cana-3825	179	17	,	,	PUNCT
cana-3825	179	18	observability	observability	NOUN
cana-3825	179	19	and	and	CCONJ
cana-3825	179	20	stability	stability	NOUN
cana-3825	179	21	of	of	ADP
cana-3825	179	22	volterra	volterra	PROPN
cana-3825	179	23	type	type	PROPN
cana-3825	179	24	nonlinear	nonlinear	ADJ
cana-3825	179	25	matrix	matrix	NOUN
cana-3825	179	26	integro	integro	ADJ
cana-3825	179	27	-	-	PUNCT
cana-3825	179	28	dynamic	dynamic	ADJ
cana-3825	179	29	system	system	NOUN
cana-3825	179	30	on	on	ADP
cana-3825	179	31	time	time	NOUN
cana-3825	179	32	scales	scale	NOUN
cana-3825	179	33	.	.	PUNCT
cana-3825	180	1	ijet	ijet	PROPN
cana-3825	180	2	,	,	PUNCT
cana-3825	180	3	7	7	NUM
cana-3825	180	4	,	,	PUNCT
cana-3825	180	5	115	115	NUM
cana-3825	180	6	-	-	SYM
cana-3825	180	7	120	120	NUM
cana-3825	180	8	.	.	PUNCT
cana-3825	181	1	[	[	X
cana-3825	181	2	12	12	NUM
cana-3825	181	3	]	]	X
cana-3825	181	4	ramana	ramana	PROPN
cana-3825	181	5	,	,	PUNCT
cana-3825	181	6	g.v	g.v	PROPN
cana-3825	181	7	;	;	PUNCT
cana-3825	181	8	deekshitulu	deekshitulu	VERB
cana-3825	181	9	,	,	PUNCT
cana-3825	181	10	g.v.s.r	g.v.s.r	PROPN
cana-3825	181	11	.	.	PROPN
cana-3825	181	12	(	(	PUNCT
cana-3825	181	13	2017	2017	NUM
cana-3825	181	14	)	)	PUNCT
cana-3825	181	15	.	.	PUNCT
cana-3825	182	1	asymptotic	asymptotic	ADJ
cana-3825	182	2	stability	stability	NOUN
cana-3825	182	3	of	of	ADP
cana-3825	182	4	solution	solution	NOUN
cana-3825	182	5	of	of	ADP
cana-3825	182	6	lyapunov	lyapunov	ADJ
cana-3825	182	7	type	type	NOUN
cana-3825	182	8	matrix	matrix	NOUN
cana-3825	182	9	volterra	volterra	NOUN
cana-3825	182	10	integro	integro	PROPN
cana-3825	182	11	-	-	PUNCT
cana-3825	182	12	dynamic	dynamic	ADJ
cana-3825	182	13	system	system	NOUN
cana-3825	182	14	on	on	ADP
cana-3825	182	15	time	time	NOUN
cana-3825	182	16	scales	scale	NOUN
cana-3825	182	17	.	.	PUNCT
cana-3825	183	1	ijet	ijet	PROPN
cana-3825	183	2	,	,	PUNCT
cana-3825	183	3	7	7	NUM
cana-3825	183	4	,	,	PUNCT
cana-3825	183	5	179	179	NUM
cana-3825	183	6	-	-	SYM
cana-3825	183	7	185	185	NUM
cana-3825	183	8	.	.	PUNCT