id	sid	tid	token	lemma	pos
cana-327	1	1	communications	communication	NOUN
cana-327	1	2	on	on	ADP
cana-327	1	3	applied	apply	VERB
cana-327	1	4	nonlinear	nonlinear	ADJ
cana-327	1	5	analysis	analysis	NOUN
cana-327	1	6	issn	issn	NOUN
cana-327	1	7	:	:	PUNCT
cana-327	1	8	1074	1074	NUM
cana-327	1	9	-	-	PUNCT
cana-327	1	10	133x	133x	NUM
cana-327	1	11	vol	vol	NOUN
cana-327	1	12	31	31	NUM
cana-327	1	13	no	no	NOUN
cana-327	1	14	.	.	NOUN
cana-327	1	15	1	1	NUM
cana-327	1	16	(	(	PUNCT
cana-327	1	17	2024	2024	NUM
cana-327	1	18	)	)	PUNCT
cana-327	2	1	82	82	NUM
cana-327	2	2	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-327	2	3	qualitative	qualitative	ADJ
cana-327	2	4	behavior	behavior	NOUN
cana-327	2	5	of	of	ADP
cana-327	2	6	second	second	ADJ
cana-327	2	7	order	order	NOUN
cana-327	2	8	difference	difference	NOUN
cana-327	2	9	equation	equation	NOUN
cana-327	2	10	with	with	ADP
cana-327	2	11	non	non	X
cana-327	2	12	positive	positive	ADJ
cana-327	2	13	neutral	neutral	ADJ
cana-327	2	14	term	term	NOUN
cana-327	2	15	s.	s.	PROPN
cana-327	2	16	kaleeswari	kaleeswari	PROPN
cana-327	2	17	1	1	NUM
cana-327	2	18	,	,	PUNCT
cana-327	2	19	j.	j.	PROPN
cana-327	2	20	gowri	gowri	PROPN
cana-327	2	21	2	2	NUM
cana-327	2	22	1,2	1,2	NUM
cana-327	2	23	department	department	NOUN
cana-327	2	24	of	of	ADP
cana-327	2	25	mathematics	mathematic	NOUN
cana-327	2	26	,	,	PUNCT
cana-327	2	27	nallamuthu	nallamuthu	ADJ
cana-327	2	28	gounder	gounder	PROPN
cana-327	2	29	mahalingam	mahalingam	PROPN
cana-327	2	30	college	college	PROPN
cana-327	2	31	,	,	PUNCT
cana-327	2	32	pollachi	pollachi	NOUN
cana-327	2	33	,	,	PUNCT
cana-327	2	34	tamilnadu	tamilnadu	PROPN
cana-327	2	35	,	,	PUNCT
cana-327	2	36	india	india	PROPN
cana-327	2	37	.	.	PUNCT
cana-327	3	1	e	e	X
cana-327	3	2	-	-	NOUN
cana-327	3	3	mail	mail	NOUN
cana-327	3	4	:	:	PUNCT
cana-327	3	5	kaleesdesika@gmail.com	kaleesdesika@gmail.com	X
cana-327	3	6	,	,	PUNCT
cana-327	3	7	e	e	NOUN
cana-327	3	8	-	-	NOUN
cana-327	3	9	mail	mail	NOUN
cana-327	3	10	:	:	PUNCT
cana-327	3	11	pranithagowri86@gmail.com	pranithagowri86@gmail.com	X
cana-327	3	12	article	article	NOUN
cana-327	3	13	history	history	NOUN
cana-327	3	14	:	:	PUNCT
cana-327	3	15	received	receive	VERB
cana-327	3	16	:	:	PUNCT
cana-327	3	17	15	15	NUM
cana-327	3	18	-	-	SYM
cana-327	3	19	09	09	NUM
cana-327	3	20	-	-	PUNCT
cana-327	3	21	2023	2023	NUM
cana-327	3	22	revised	revise	VERB
cana-327	3	23	:	:	PUNCT
cana-327	3	24	24	24	NUM
cana-327	3	25	-	-	SYM
cana-327	3	26	10	10	NUM
cana-327	3	27	-	-	PUNCT
cana-327	3	28	2023	2023	NUM
cana-327	3	29	accepted	accept	VERB
cana-327	3	30	:	:	PUNCT
cana-327	3	31	18	18	NUM
cana-327	3	32	-	-	SYM
cana-327	3	33	11	11	NUM
cana-327	3	34	-	-	SYM
cana-327	3	35	2023	2023	NUM
cana-327	3	36	abstract	abstract	NOUN
cana-327	3	37	:	:	PUNCT
cana-327	3	38	the	the	DET
cana-327	3	39	oscillation	oscillation	NOUN
cana-327	3	40	of	of	ADP
cana-327	3	41	second	second	ADJ
cana-327	3	42	order	order	NOUN
cana-327	3	43	difference	difference	NOUN
cana-327	3	44	equations	equation	NOUN
cana-327	3	45	with	with	ADP
cana-327	3	46	a	a	DET
cana-327	3	47	nonlinear	nonlinear	ADJ
cana-327	3	48	nonpositive	nonpositive	ADJ
cana-327	3	49	neutral	neutral	ADJ
cana-327	3	50	component	component	NOUN
cana-327	3	51	is	be	AUX
cana-327	3	52	the	the	DET
cana-327	3	53	subject	subject	NOUN
cana-327	3	54	of	of	ADP
cana-327	3	55	this	this	DET
cana-327	3	56	study	study	NOUN
cana-327	3	57	.	.	PUNCT
cana-327	4	1	we	we	PRON
cana-327	4	2	come	come	VERB
cana-327	4	3	up	up	ADP
cana-327	4	4	with	with	ADP
cana-327	4	5	a	a	DET
cana-327	4	6	sufficient	sufficient	ADJ
cana-327	4	7	condition	condition	NOUN
cana-327	4	8	that	that	PRON
cana-327	4	9	guarantees	guarantee	VERB
cana-327	4	10	that	that	SCONJ
cana-327	4	11	all	all	DET
cana-327	4	12	solutions	solution	NOUN
cana-327	4	13	to	to	ADP
cana-327	4	14	the	the	DET
cana-327	4	15	examined	examine	VERB
cana-327	4	16	equation	equation	NOUN
cana-327	4	17	are	be	AUX
cana-327	4	18	either	either	CCONJ
cana-327	4	19	oscillatory	oscillatory	ADJ
cana-327	4	20	or	or	CCONJ
cana-327	4	21	going	go	VERB
cana-327	4	22	towards	towards	ADP
cana-327	4	23	zero	zero	NUM
cana-327	4	24	.	.	PUNCT
cana-327	5	1	through	through	ADP
cana-327	5	2	examples	example	NOUN
cana-327	5	3	,	,	PUNCT
cana-327	5	4	the	the	DET
cana-327	5	5	improvement	improvement	NOUN
cana-327	5	6	of	of	ADP
cana-327	5	7	our	our	PRON
cana-327	5	8	primary	primary	ADJ
cana-327	5	9	findings	finding	NOUN
cana-327	5	10	is	be	AUX
cana-327	5	11	demonstrated	demonstrate	VERB
cana-327	5	12	.	.	PUNCT
cana-327	6	1	keywords	keyword	NOUN
cana-327	6	2	:	:	PUNCT
cana-327	6	3	oscillatory	oscillatory	ADJ
cana-327	6	4	,	,	PUNCT
cana-327	6	5	neutral	neutral	ADJ
cana-327	6	6	term	term	NOUN
cana-327	6	7	,	,	PUNCT
cana-327	6	8	non	non	X
cana-327	6	9	positive	positive	ADJ
cana-327	6	10	,	,	PUNCT
cana-327	6	11	second	second	ADJ
cana-327	6	12	order	order	NOUN
cana-327	6	13	.	.	PUNCT
cana-327	7	1	msc	msc	NOUN
cana-327	7	2	:	:	PUNCT
cana-327	7	3	39a10	39a10	NUM
cana-327	7	4	.	.	X
cana-327	8	1	1	1	X
cana-327	8	2	.	.	X
cana-327	8	3	introduction	introduction	NOUN
cana-327	8	4	in	in	ADP
cana-327	8	5	this	this	DET
cana-327	8	6	article	article	NOUN
cana-327	8	7	,	,	PUNCT
cana-327	8	8	we	we	PRON
cana-327	8	9	study	study	VERB
cana-327	8	10	some	some	DET
cana-327	8	11	oscillatory	oscillatory	ADJ
cana-327	8	12	manners	manner	NOUN
cana-327	8	13	of	of	ADP
cana-327	8	14	a	a	DET
cana-327	8	15	second	second	ADJ
cana-327	8	16	order	order	NOUN
cana-327	8	17	non	non	X
cana-327	8	18	linear	linear	VERB
cana-327	8	19	non	non	X
cana-327	8	20	positive	positive	ADJ
cana-327	8	21	neutral	neutral	ADJ
cana-327	8	22	delay	delay	NOUN
cana-327	8	23	difference	difference	NOUN
cana-327	8	24	equation	equation	NOUN
cana-327	8	25	of	of	ADP
cana-327	8	26	the	the	DET
cana-327	8	27	form	form	NOUN
cana-327	8	28	∆(𝑟(℘)∆(z(℘	∆(𝑟(℘)∆(z(℘	PROPN
cana-327	8	29	)	)	PUNCT
cana-327	9	1	−	−	ADP
cana-327	9	2	𝑝(℘)𝑧𝛾1(𝜏1(℘	𝑝(℘)𝑧𝛾1(𝜏1(℘	NOUN
cana-327	9	3	)	)	PUNCT
cana-327	9	4	)	)	PUNCT
cana-327	9	5	)	)	PUNCT
cana-327	9	6	)	)	PUNCT
cana-327	10	1	+	+	CCONJ
cana-327	10	2	𝑞(℘)𝑧𝛾2(𝜎1(℘	𝑞(℘)𝑧𝛾2(𝜎1(℘	NOUN
cana-327	10	3	)	)	PUNCT
cana-327	10	4	)	)	PUNCT
cana-327	10	5	,	,	PUNCT
cana-327	10	6	℘	℘	X
cana-327	10	7	≥	≥	NOUN
cana-327	10	8	℘0	℘0	NOUN
cana-327	10	9	>	>	X
cana-327	10	10	0	0	NUM
cana-327	10	11	.	.	PUNCT
cana-327	11	1	(	(	PUNCT
cana-327	11	2	1	1	X
cana-327	11	3	)	)	PUNCT
cana-327	11	4	subject	subject	NOUN
cana-327	11	5	to	to	ADP
cana-327	11	6	the	the	DET
cana-327	11	7	restrictions	restriction	NOUN
cana-327	11	8	outlined	outline	VERB
cana-327	11	9	below	below	ADV
cana-327	11	10	:	:	PUNCT
cana-327	11	11	(	(	PUNCT
cana-327	11	12	r1	r1	NOUN
cana-327	11	13	)	)	PUNCT
cana-327	11	14	𝛾2	𝛾2	NOUN
cana-327	11	15	and	and	CCONJ
cana-327	11	16	0	0	NUM
cana-327	11	17	<	<	X
cana-327	11	18	𝛾1	𝛾1	PROPN
cana-327	11	19	≤	≤	ADV
cana-327	11	20	1	1	NUM
cana-327	11	21	are	be	AUX
cana-327	11	22	ratio	ratio	NOUN
cana-327	11	23	of	of	ADP
cana-327	11	24	odd	odd	ADJ
cana-327	11	25	positive	positive	ADJ
cana-327	11	26	integers	integer	NOUN
cana-327	11	27	;	;	PUNCT
cana-327	11	28	(	(	PUNCT
cana-327	11	29	r2	r2	PROPN
cana-327	11	30	)	)	PUNCT
cana-327	11	31	{	{	PUNCT
cana-327	11	32	𝑟(℘	𝑟(℘	NUM
cana-327	11	33	)	)	PUNCT
cana-327	11	34	}	}	PUNCT
cana-327	11	35	,	,	PUNCT
cana-327	11	36	{	{	PUNCT
cana-327	11	37	𝑞(℘	𝑞(℘	NOUN
cana-327	11	38	)	)	PUNCT
cana-327	11	39	}	}	PUNCT
cana-327	11	40	and	and	CCONJ
cana-327	11	41	{	{	PUNCT
cana-327	11	42	𝑝(℘	𝑝(℘	NOUN
cana-327	11	43	)	)	PUNCT
cana-327	11	44	}	}	PUNCT
cana-327	11	45	are	be	AUX
cana-327	11	46	positive	positive	ADJ
cana-327	11	47	real	real	ADJ
cana-327	11	48	sequences	sequence	NOUN
cana-327	11	49	such	such	ADJ
cana-327	11	50	that	that	SCONJ
cana-327	11	51	0	0	NUM
cana-327	11	52	<	<	X
cana-327	11	53	𝑝(℘	𝑝(℘	NOUN
cana-327	11	54	)	)	PUNCT
cana-327	11	55	≤	≤	NOUN
cana-327	11	56	𝑝	𝑝	ADP
cana-327	11	57	<	<	X
cana-327	11	58	1	1	NUM
cana-327	11	59	,	,	PUNCT
cana-327	11	60	∀	∀	NOUN
cana-327	11	61	℘	℘	NOUN
cana-327	11	62	≥	≥	NOUN
cana-327	11	63	℘0	℘0	NOUN
cana-327	11	64	and	and	CCONJ
cana-327	11	65	(	(	PUNCT
cana-327	11	66	r3	r3	PROPN
cana-327	11	67	)	)	PUNCT
cana-327	11	68	𝜎1	𝜎1	ADJ
cana-327	11	69	and	and	CCONJ
cana-327	11	70	𝜏1	𝜏1	NOUN
cana-327	11	71	are	be	AUX
cana-327	11	72	positive	positive	ADJ
cana-327	11	73	integers	integer	NOUN
cana-327	11	74	with	with	ADP
cana-327	11	75	𝜏1℘	𝜏1℘	NOUN
cana-327	11	76	)	)	PUNCT
cana-327	11	77	≤	≤	NUM
cana-327	11	78	℘	℘	PROPN
cana-327	11	79	,	,	PUNCT
cana-327	11	80	δ𝜏1(℘	δ𝜏1(℘	PROPN
cana-327	11	81	)	)	PUNCT
cana-327	11	82	>	>	X
cana-327	11	83	0	0	NUM
cana-327	11	84	,	,	PUNCT
cana-327	11	85	𝜎1(℘	𝜎1(℘	NOUN
cana-327	11	86	)	)	PUNCT
cana-327	11	87	≤	≤	NUM
cana-327	11	88	℘	℘	PROPN
cana-327	11	89	,	,	PUNCT
cana-327	11	90	δ𝜎1(℘	δ𝜎1(℘	NOUN
cana-327	11	91	)	)	PUNCT
cana-327	11	92	>	>	X
cana-327	11	93	0	0	NUM
cana-327	11	94	,	,	PUNCT
cana-327	11	95	lim℘→∞	lim℘→∞	ADJ
cana-327	11	96	 	 	SPACE
cana-327	11	97	𝜏1(℘	𝜏1(℘	NOUN
cana-327	11	98	)	)	PUNCT
cana-327	11	99	=	=	SYM
cana-327	11	100	lim℘→∞	lim℘→∞	ADJ
cana-327	11	101	 	 	SPACE
cana-327	11	102	𝜎1(℘	𝜎1(℘	NOUN
cana-327	11	103	)	)	PUNCT
cana-327	12	1	=	=	SYM
cana-327	12	2	∞.	∞.	PROPN
cana-327	12	3	a	a	DET
cana-327	12	4	real	real	ADJ
cana-327	12	5	sequence	sequence	NOUN
cana-327	12	6	{	{	PUNCT
cana-327	12	7	z(℘	z(℘	NUM
cana-327	12	8	)	)	PUNCT
cana-327	12	9	}	}	PUNCT
cana-327	12	10	is	be	AUX
cana-327	12	11	said	say	VERB
cana-327	12	12	to	to	PART
cana-327	12	13	be	be	AUX
cana-327	12	14	a	a	DET
cana-327	12	15	solution	solution	NOUN
cana-327	12	16	of	of	ADP
cana-327	12	17	(	(	PUNCT
cana-327	12	18	1	1	X
cana-327	12	19	)	)	PUNCT
cana-327	12	20	if	if	SCONJ
cana-327	12	21	it	it	PRON
cana-327	12	22	is	be	AUX
cana-327	12	23	defined	define	VERB
cana-327	12	24	for	for	ADP
cana-327	12	25	all	all	DET
cana-327	12	26	℘	℘	PROPN
cana-327	12	27	≥	≥	PRON
cana-327	12	28	℘_0	℘_0	PROPN
cana-327	12	29	.	.	PUNCT
cana-327	13	1	a	a	DET
cana-327	13	2	nontrivial	nontrivial	ADJ
cana-327	13	3	solution	solution	NOUN
cana-327	13	4	of	of	ADP
cana-327	13	5	(	(	PUNCT
cana-327	13	6	1	1	NUM
cana-327	13	7	)	)	PUNCT
cana-327	13	8	is	be	AUX
cana-327	13	9	called	call	VERB
cana-327	13	10	oscillatory	oscillatory	ADJ
cana-327	13	11	if	if	SCONJ
cana-327	13	12	it	it	PRON
cana-327	13	13	is	be	AUX
cana-327	13	14	neither	neither	CCONJ
cana-327	13	15	eventually	eventually	ADV
cana-327	13	16	positive	positive	ADJ
cana-327	13	17	nor	nor	CCONJ
cana-327	13	18	eventually	eventually	ADV
cana-327	13	19	negative	negative	ADJ
cana-327	13	20	.	.	PUNCT
cana-327	14	1	otherwise	otherwise	ADV
cana-327	14	2	,	,	PUNCT
cana-327	14	3	the	the	DET
cana-327	14	4	solution	solution	NOUN
cana-327	14	5	is	be	AUX
cana-327	14	6	said	say	VERB
cana-327	14	7	to	to	PART
cana-327	14	8	be	be	AUX
cana-327	14	9	non	non	X
cana-327	14	10	oscillatory	oscillatory	ADJ
cana-327	14	11	.	.	PUNCT
cana-327	15	1	an	an	DET
cana-327	15	2	equation	equation	NOUN
cana-327	15	3	is	be	AUX
cana-327	15	4	oscilltory	oscilltory	NOUN
cana-327	15	5	if	if	SCONJ
cana-327	15	6	all	all	DET
cana-327	15	7	its	its	PRON
cana-327	15	8	solutions	solution	NOUN
cana-327	15	9	oscillate	oscillate	VERB
cana-327	15	10	.	.	PUNCT
cana-327	16	1	since	since	SCONJ
cana-327	16	2	neutral	neutral	ADJ
cana-327	16	3	type	type	NOUN
cana-327	16	4	equations	equation	NOUN
cana-327	16	5	are	be	AUX
cana-327	16	6	prevalent	prevalent	ADJ
cana-327	16	7	in	in	ADP
cana-327	16	8	the	the	DET
cana-327	16	9	study	study	NOUN
cana-327	16	10	of	of	ADP
cana-327	16	11	economics	economic	NOUN
cana-327	16	12	,	,	PUNCT
cana-327	16	13	mathematical	mathematical	ADJ
cana-327	16	14	biology	biology	NOUN
cana-327	16	15	,	,	PUNCT
cana-327	16	16	and	and	CCONJ
cana-327	16	17	many	many	ADJ
cana-327	16	18	other	other	ADJ
cana-327	16	19	fields	field	NOUN
cana-327	16	20	of	of	ADP
cana-327	16	21	mathematics	mathematic	NOUN
cana-327	16	22	,	,	PUNCT
cana-327	16	23	determining	determine	VERB
cana-327	16	24	oscillation	oscillation	NOUN
cana-327	16	25	conditions	condition	NOUN
cana-327	16	26	for	for	ADP
cana-327	16	27	these	these	DET
cana-327	16	28	equations	equation	NOUN
cana-327	16	29	has	have	AUX
cana-327	16	30	garnered	garner	VERB
cana-327	16	31	a	a	DET
cana-327	16	32	lot	lot	NOUN
cana-327	16	33	of	of	ADP
cana-327	16	34	attention	attention	NOUN
cana-327	16	35	in	in	ADP
cana-327	16	36	recent	recent	ADJ
cana-327	16	37	years	year	NOUN
cana-327	16	38	.	.	PUNCT
cana-327	17	1	(	(	PUNCT
cana-327	17	2	see	see	VERB
cana-327	17	3	for	for	ADP
cana-327	17	4	example	example	NOUN
cana-327	17	5	[	[	X
cana-327	17	6	1	1	NUM
cana-327	17	7	]	]	PUNCT
cana-327	17	8	−	−	PROPN
cana-327	18	1	[	[	X
cana-327	18	2	9	9	NUM
cana-327	18	3	]	]	PUNCT
cana-327	18	4	)	)	PUNCT
cana-327	18	5	.	.	PUNCT
cana-327	19	1	to	to	ADP
cana-327	19	2	the	the	DET
cana-327	19	3	best	good	ADJ
cana-327	19	4	of	of	ADP
cana-327	19	5	our	our	PRON
cana-327	19	6	knowledge	knowledge	NOUN
cana-327	19	7	,	,	PUNCT
cana-327	19	8	there	there	PRON
cana-327	19	9	are	be	VERB
cana-327	19	10	no	no	DET
cana-327	19	11	results	result	NOUN
cana-327	19	12	in	in	ADP
cana-327	19	13	the	the	DET
cana-327	19	14	literature	literature	NOUN
cana-327	19	15	that	that	PRON
cana-327	19	16	guarantee	guarantee	VERB
cana-327	19	17	that	that	SCONJ
cana-327	19	18	all	all	DET
cana-327	19	19	solutions	solution	NOUN
cana-327	19	20	for	for	ADP
cana-327	19	21	the	the	DET
cana-327	19	22	second	second	ADJ
cana-327	19	23	order	order	NOUN
cana-327	19	24	difference	difference	NOUN
cana-327	19	25	equation	equation	NOUN
cana-327	19	26	are	be	AUX
cana-327	19	27	just	just	ADV
cana-327	19	28	oscillatory	oscillatory	ADJ
cana-327	19	29	.	.	PUNCT
cana-327	20	1	this	this	DET
cana-327	20	2	conclusion	conclusion	NOUN
cana-327	20	3	is	be	AUX
cana-327	20	4	drawn	draw	VERB
cana-327	20	5	from	from	ADP
cana-327	20	6	a	a	DET
cana-327	20	7	review	review	NOUN
cana-327	20	8	of	of	ADP
cana-327	20	9	the	the	DET
cana-327	20	10	literature	literature	NOUN
cana-327	20	11	.	.	PUNCT
cana-327	21	1	all	all	DET
cana-327	21	2	results	result	NOUN
cana-327	21	3	(	(	PUNCT
cana-327	21	4	[	[	X
cana-327	21	5	10	10	NUM
cana-327	21	6	]	]	PUNCT
cana-327	21	7	−	−	PROPN
cana-327	22	1	[	[	X
cana-327	22	2	14	14	NUM
cana-327	22	3	]	]	PUNCT
cana-327	22	4	)	)	PUNCT
cana-327	22	5	established	establish	VERB
cana-327	22	6	for	for	ADP
cana-327	22	7	neutral	neutral	ADJ
cana-327	22	8	type	type	NOUN
cana-327	22	9	difference	difference	NOUN
cana-327	22	10	equations	equation	NOUN
cana-327	22	11	are	be	AUX
cana-327	22	12	guaranteed	guarantee	VERB
cana-327	22	13	that	that	SCONJ
cana-327	22	14	every	every	DET
cana-327	22	15	solution	solution	NOUN
cana-327	22	16	is	be	AUX
cana-327	22	17	either	either	CCONJ
cana-327	22	18	oscillatory	oscillatory	ADJ
cana-327	22	19	or	or	CCONJ
cana-327	22	20	tends	tend	VERB
cana-327	22	21	to	to	ADP
cana-327	22	22	zero	zero	NUM
cana-327	22	23	monotonically	monotonically	ADV
cana-327	22	24	.	.	PUNCT
cana-327	23	1	in	in	ADP
cana-327	23	2	order	order	NOUN
cana-327	23	3	to	to	PART
cana-327	23	4	define	define	VERB
cana-327	23	5	conditions	condition	NOUN
cana-327	23	6	for	for	ADP
cana-327	23	7	the	the	DET
cana-327	23	8	oscillation	oscillation	NOUN
cana-327	23	9	of	of	ADP
cana-327	23	10	all	all	DET
cana-327	23	11	solutions	solution	NOUN
cana-327	23	12	under	under	ADP
cana-327	23	13	the	the	DET
cana-327	23	14	following	follow	VERB
cana-327	23	15	condition	condition	NOUN
cana-327	23	16	,	,	PUNCT
cana-327	23	17	the	the	DET
cana-327	23	18	authors	author	NOUN
cana-327	23	19	considered	consider	VERB
cana-327	23	20	(	(	PUNCT
cana-327	23	21	1	1	NUM
cana-327	23	22	)	)	PUNCT
cana-327	23	23	with	with	ADP
cana-327	23	24	𝑝(℘	𝑝(℘	NOUN
cana-327	23	25	)	)	PUNCT
cana-327	23	26	<	<	X
cana-327	23	27	0	0	NUM
cana-327	23	28	,	,	PUNCT
cana-327	23	29	communications	communication	NOUN
cana-327	23	30	on	on	ADP
cana-327	23	31	applied	apply	VERB
cana-327	23	32	nonlinear	nonlinear	ADJ
cana-327	23	33	analysis	analysis	NOUN
cana-327	23	34	issn	issn	NOUN
cana-327	23	35	:	:	PUNCT
cana-327	23	36	1074	1074	NUM
cana-327	23	37	-	-	PUNCT
cana-327	23	38	133x	133x	NUM
cana-327	23	39	vol	vol	NOUN
cana-327	23	40	31	31	NUM
cana-327	23	41	no	no	NOUN
cana-327	23	42	.	.	NOUN
cana-327	23	43	1	1	NUM
cana-327	23	44	(	(	PUNCT
cana-327	23	45	2024	2024	NUM
cana-327	23	46	)	)	PUNCT
cana-327	23	47	83	83	NUM
cana-327	23	48	https://internationalpubls.com	https://internationalpubls.com	X
cana-327	23	49	∑	∑	PUNCT
cana-327	23	50	 	 	SPACE
cana-327	23	51	∞	∞	PROPN
cana-327	23	52	𝑖=℘0	𝑖=℘0	ADJ
cana-327	23	53	1	1	NUM
cana-327	23	54	𝑟(𝑖	𝑟(𝑖	PROPN
cana-327	23	55	)	)	PUNCT
cana-327	24	1	=	=	SYM
cana-327	24	2	∞	∞	PROPN
cana-327	24	3	(	(	PUNCT
cana-327	24	4	2	2	NUM
cana-327	24	5	)	)	PUNCT
cana-327	24	6	in	in	ADP
cana-327	24	7	this	this	DET
cana-327	24	8	article	article	NOUN
cana-327	24	9	we	we	PRON
cana-327	24	10	arrive	arrive	VERB
cana-327	24	11	at	at	ADP
cana-327	24	12	some	some	DET
cana-327	24	13	new	new	ADJ
cana-327	24	14	oscillation	oscillation	NOUN
cana-327	24	15	results	result	NOUN
cana-327	24	16	.	.	PUNCT
cana-327	25	1	2	2	X
cana-327	25	2	.	.	X
cana-327	25	3	oscillatory	oscillatory	ADJ
cana-327	25	4	results	result	NOUN
cana-327	25	5	we	we	PRON
cana-327	25	6	begin	begin	VERB
cana-327	25	7	with	with	ADP
cana-327	25	8	the	the	DET
cana-327	25	9	following	follow	VERB
cana-327	25	10	lemmas	lemmas	PROPN
cana-327	25	11	,	,	PUNCT
cana-327	25	12	which	which	PRON
cana-327	25	13	are	be	AUX
cana-327	25	14	critical	critical	ADJ
cana-327	25	15	in	in	ADP
cana-327	25	16	establishing	establish	VERB
cana-327	25	17	our	our	PRON
cana-327	25	18	key	key	ADJ
cana-327	25	19	results	result	NOUN
cana-327	25	20	.	.	PUNCT
cana-327	26	1	we	we	PRON
cana-327	26	2	represent	represent	VERB
cana-327	26	3	𝑠(℘	𝑠(℘	NOUN
cana-327	26	4	)	)	PUNCT
cana-327	26	5	=	=	SYM
cana-327	26	6	𝑧(℘	𝑧(℘	NUM
cana-327	26	7	)	)	PUNCT
cana-327	26	8	−	−	NOUN
cana-327	26	9	𝑝(℘)𝑧𝛾1(𝜏1(℘	𝑝(℘)𝑧𝛾1(𝜏1(℘	NOUN
cana-327	26	10	)	)	PUNCT
cana-327	26	11	)	)	PUNCT
cana-327	26	12	,	,	PUNCT
cana-327	26	13	𝑌(℘	𝑌(℘	NOUN
cana-327	26	14	)	)	PUNCT
cana-327	26	15	=	=	PUNCT
cana-327	27	1	∑	∑	PUNCT
cana-327	27	2	  	  	SPACE
cana-327	27	3	℘−1	℘−1	PROPN
cana-327	27	4	𝑖=℘1	𝑖=℘1	PROPN
cana-327	27	5	  	  	SPACE
cana-327	27	6	1	1	NUM
cana-327	27	7	𝑟(𝑖	𝑟(𝑖	PROPN
cana-327	27	8	)	)	PUNCT
cana-327	27	9	,	,	PUNCT
cana-327	27	10	for	for	SCONJ
cana-327	27	11	every	every	DET
cana-327	27	12	℘	℘	PROPN
cana-327	27	13	≥	≥	NOUN
cana-327	27	14	℘1	℘1	VERB
cana-327	27	15	≥	≥	NOUN
cana-327	27	16	℘0	℘0	NOUN
cana-327	27	17	.	.	PUNCT
cana-327	28	1	lemma	lemma	PROPN
cana-327	28	2	2.1	2.1	NUM
cana-327	28	3	.	.	PUNCT
cana-327	29	1	let	let	VERB
cana-327	29	2	(	(	PUNCT
cana-327	29	3	2	2	X
cana-327	29	4	)	)	PUNCT
cana-327	29	5	hold	hold	VERB
cana-327	29	6	and	and	CCONJ
cana-327	29	7	if	if	SCONJ
cana-327	29	8	z	z	NOUN
cana-327	29	9	is	be	AUX
cana-327	29	10	a	a	DET
cana-327	29	11	positive	positive	ADJ
cana-327	29	12	solution	solution	NOUN
cana-327	29	13	of	of	ADP
cana-327	29	14	(	(	PUNCT
cana-327	29	15	1	1	NUM
cana-327	29	16	)	)	PUNCT
cana-327	29	17	,	,	PUNCT
cana-327	29	18	then	then	ADV
cana-327	29	19	the	the	DET
cana-327	29	20	corresponding	correspond	VERB
cana-327	29	21	function	function	NOUN
cana-327	29	22	s	s	PART
cana-327	29	23	meets	meet	VERB
cana-327	29	24	one	one	NUM
cana-327	29	25	of	of	ADP
cana-327	29	26	the	the	DET
cana-327	29	27	following	follow	VERB
cana-327	29	28	two	two	NUM
cana-327	29	29	requirements	requirement	NOUN
cana-327	29	30	:	:	PUNCT
cana-327	29	31	(	(	PUNCT
cana-327	29	32	i	i	NOUN
cana-327	29	33	)	)	PUNCT
cana-327	29	34	𝑠(℘	𝑠(℘	NOUN
cana-327	29	35	)	)	PUNCT
cana-327	29	36	>	>	X
cana-327	29	37	0	0	NUM
cana-327	29	38	,	,	PUNCT
cana-327	29	39	δ𝑠(℘	δ𝑠(℘	PROPN
cana-327	29	40	)	)	PUNCT
cana-327	29	41	>	>	X
cana-327	29	42	0	0	PUNCT
cana-327	29	43	and	and	CCONJ
cana-327	29	44	δ(𝑟(℘)δ𝑠(℘	δ(𝑟(℘)δ𝑠(℘	PROPN
cana-327	29	45	)	)	PUNCT
cana-327	29	46	)	)	PUNCT
cana-327	30	1	<	<	X
cana-327	30	2	0	0	NUM
cana-327	30	3	;	;	PUNCT
cana-327	30	4	(	(	PUNCT
cana-327	30	5	ii	ii	NOUN
cana-327	30	6	)	)	PUNCT
cana-327	30	7	𝑠(℘	𝑠(℘	NOUN
cana-327	30	8	)	)	PUNCT
cana-327	30	9	<	<	X
cana-327	30	10	0	0	NUM
cana-327	30	11	,	,	PUNCT
cana-327	30	12	δ𝑠(℘	δ𝑠(℘	PROPN
cana-327	30	13	)	)	PUNCT
cana-327	30	14	>	>	X
cana-327	30	15	0	0	PUNCT
cana-327	30	16	and	and	CCONJ
cana-327	30	17	δ(𝑟(℘)δ𝑠(℘	δ(𝑟(℘)δ𝑠(℘	PROPN
cana-327	30	18	)	)	PUNCT
cana-327	30	19	)	)	PUNCT
cana-327	31	1	<	<	X
cana-327	31	2	0	0	NUM
cana-327	31	3	,	,	PUNCT
cana-327	31	4	for	for	ADP
cana-327	31	5	all	all	DET
cana-327	31	6	℘	℘	PROPN
cana-327	31	7	≥	≥	NUM
cana-327	31	8	℘1	℘1	NOUN
cana-327	31	9	,	,	PUNCT
cana-327	31	10	where	where	SCONJ
cana-327	31	11	℘1	℘1	NOUN
cana-327	31	12	≥	≥	NOUN
cana-327	31	13	℘0	℘0	NOUN
cana-327	31	14	is	be	AUX
cana-327	31	15	sufficiently	sufficiently	ADV
cana-327	31	16	large	large	ADJ
cana-327	31	17	.	.	PUNCT
cana-327	32	1	proof	proof	NOUN
cana-327	32	2	.	.	PUNCT
cana-327	33	1	it	it	PRON
cana-327	33	2	is	be	AUX
cana-327	33	3	sufficient	sufficient	ADJ
cana-327	33	4	to	to	PART
cana-327	33	5	state	state	VERB
cana-327	33	6	and	and	CCONJ
cana-327	33	7	prove	prove	VERB
cana-327	33	8	the	the	DET
cana-327	33	9	results	result	NOUN
cana-327	33	10	for	for	ADP
cana-327	33	11	positive	positive	ADJ
cana-327	33	12	solutions	solution	NOUN
cana-327	33	13	.	.	PUNCT
cana-327	34	1	because	because	SCONJ
cana-327	34	2	the	the	DET
cana-327	34	3	proof	proof	NOUN
cana-327	34	4	of	of	ADP
cana-327	34	5	the	the	DET
cana-327	34	6	other	other	ADJ
cana-327	34	7	case	case	NOUN
cana-327	34	8	is	be	AUX
cana-327	34	9	same	same	ADJ
cana-327	34	10	.	.	PUNCT
cana-327	35	1	suppose	suppose	VERB
cana-327	35	2	that	that	SCONJ
cana-327	35	3	𝑧(℘	𝑧(℘	NUM
cana-327	35	4	)	)	PUNCT
cana-327	35	5	>	>	X
cana-327	35	6	0	0	NUM
cana-327	35	7	,	,	PUNCT
cana-327	35	8	𝑧(𝜏1(℘	𝑧(𝜏1(℘	NOUN
cana-327	35	9	)	)	PUNCT
cana-327	35	10	)	)	PUNCT
cana-327	35	11	>	>	PUNCT
cana-327	35	12	0	0	NUM
cana-327	35	13	and	and	CCONJ
cana-327	35	14	𝑧(𝜎1(℘	𝑧(𝜎1(℘	NOUN
cana-327	35	15	)	)	PUNCT
cana-327	35	16	)	)	PUNCT
cana-327	36	1	>	>	X
cana-327	36	2	0	0	PUNCT
cana-327	37	1	for	for	ADP
cana-327	37	2	every	every	DET
cana-327	37	3	℘	℘	PROPN
cana-327	37	4	≥	≥	NOUN
cana-327	37	5	℘1	℘1	NOUN
cana-327	37	6	for	for	ADP
cana-327	37	7	some	some	DET
cana-327	37	8	℘1	℘1	NOUN
cana-327	37	9	≥	≥	NOUN
cana-327	37	10	℘0	℘0	NOUN
cana-327	37	11	.	.	PUNCT
cana-327	38	1	by	by	ADP
cana-327	38	2	the	the	DET
cana-327	38	3	representation	representation	NOUN
cana-327	38	4	of	of	ADP
cana-327	38	5	𝑠(℘	𝑠(℘	NOUN
cana-327	38	6	)	)	PUNCT
cana-327	38	7	and	and	CCONJ
cana-327	38	8	(	(	PUNCT
cana-327	38	9	1	1	NUM
cana-327	38	10	)	)	PUNCT
cana-327	38	11	,	,	PUNCT
cana-327	38	12	we	we	PRON
cana-327	38	13	get	get	VERB
cana-327	38	14	δ(𝑟(℘)δ𝑠(℘	δ(𝑟(℘)δ𝑠(℘	NOUN
cana-327	38	15	)	)	PUNCT
cana-327	38	16	)	)	PUNCT
cana-327	39	1	=	=	PUNCT
cana-327	39	2	−𝑞(℘)𝑧𝛾2(𝜎1(℘	−𝑞(℘)𝑧𝛾2(𝜎1(℘	NOUN
cana-327	39	3	)	)	PUNCT
cana-327	39	4	)	)	PUNCT
cana-327	40	1	<	<	X
cana-327	40	2	0	0	X
cana-327	40	3	.	.	PUNCT
cana-327	41	1	(	(	PUNCT
cana-327	41	2	3	3	NUM
cana-327	41	3	)	)	PUNCT
cana-327	41	4	hence	hence	ADV
cana-327	41	5	𝑟(℘)(δ𝑠(℘	𝑟(℘)(δ𝑠(℘	NUM
cana-327	41	6	)	)	PUNCT
cana-327	41	7	)	)	PUNCT
cana-327	41	8	is	be	AUX
cana-327	41	9	decreasing	decrease	VERB
cana-327	41	10	and	and	CCONJ
cana-327	41	11	of	of	ADP
cana-327	41	12	one	one	NUM
cana-327	41	13	sign	sign	NOUN
cana-327	41	14	for	for	ADP
cana-327	41	15	large	large	ADJ
cana-327	41	16	℘	℘	NOUN
cana-327	41	17	,	,	PUNCT
cana-327	41	18	that	that	PRON
cana-327	41	19	means	mean	VERB
cana-327	41	20	,	,	PUNCT
cana-327	41	21	∃	∃	PROPN
cana-327	41	22	℘2	℘2	PROPN
cana-327	41	23	≥	≥	NOUN
cana-327	41	24	℘1	℘1	PROPN
cana-327	41	25	and	and	CCONJ
cana-327	41	26	δ𝑠(℘	δ𝑠(℘	PROPN
cana-327	41	27	)	)	PUNCT
cana-327	41	28	>	>	X
cana-327	41	29	0	0	PUNCT
cana-327	42	1	(	(	PUNCT
cana-327	42	2	or	or	CCONJ
cana-327	42	3	)	)	PUNCT
cana-327	42	4	δ𝑠(℘	δ𝑠(℘	PROPN
cana-327	42	5	)	)	PUNCT
cana-327	42	6	<	<	X
cana-327	42	7	0	0	NUM
cana-327	42	8	for	for	ADP
cana-327	42	9	all	all	DET
cana-327	42	10	℘	℘	PROPN
cana-327	42	11	≥	≥	NOUN
cana-327	42	12	℘2	℘2	NOUN
cana-327	42	13	.	.	PUNCT
cana-327	43	1	if	if	SCONJ
cana-327	43	2	δ𝑠(℘	δ𝑠(℘	PROPN
cana-327	43	3	)	)	PUNCT
cana-327	43	4	<	<	X
cana-327	43	5	0	0	NUM
cana-327	43	6	for	for	ADP
cana-327	43	7	℘	℘	PROPN
cana-327	43	8	≥	≥	NUM
cana-327	43	9	℘2	℘2	NOUN
cana-327	43	10	,	,	PUNCT
cana-327	43	11	then	then	ADV
cana-327	43	12	𝑟(℘)(δ𝑠(℘	𝑟(℘)(δ𝑠(℘	NOUN
cana-327	43	13	)	)	PUNCT
cana-327	43	14	≤	≤	NOUN
cana-327	44	1	−𝑑1	−𝑑1	PROPN
cana-327	44	2	for	for	ADP
cana-327	44	3	℘	℘	PROPN
cana-327	44	4	≥	≥	NOUN
cana-327	44	5	℘2	℘2	NOUN
cana-327	44	6	where	where	SCONJ
cana-327	44	7	𝑑1	𝑑1	NOUN
cana-327	44	8	=	=	SYM
cana-327	44	9	−𝑟(℘2)δ𝑠(℘2	−𝑟(℘2)δ𝑠(℘2	PROPN
cana-327	44	10	)	)	PUNCT
cana-327	44	11	>	>	X
cana-327	44	12	0	0	X
cana-327	44	13	.	.	PUNCT
cana-327	45	1	then	then	ADV
cana-327	45	2	,	,	PUNCT
cana-327	45	3	we	we	PRON
cana-327	45	4	obtain	obtain	VERB
cana-327	45	5	𝑠(℘	𝑠(℘	NOUN
cana-327	45	6	)	)	PUNCT
cana-327	45	7	≤	≤	NUM
cana-327	45	8	𝑠(℘2	𝑠(℘2	PROPN
cana-327	45	9	)	)	PUNCT
cana-327	45	10	−	−	PROPN
cana-327	45	11	𝑑1	𝑑1	NOUN
cana-327	45	12	∑	∑	PUNCT
cana-327	45	13	  	  	SPACE
cana-327	45	14	℘−1	℘−1	PROPN
cana-327	45	15	𝑖=℘2	𝑖=℘2	VERB
cana-327	45	16	1	1	NUM
cana-327	45	17	𝑟(𝑖	𝑟(𝑖	PROPN
cana-327	45	18	)	)	PUNCT
cana-327	45	19	.	.	PUNCT
cana-327	46	1	by	by	ADP
cana-327	46	2	the	the	DET
cana-327	46	3	condition	condition	NOUN
cana-327	46	4	(	(	PUNCT
cana-327	46	5	2	2	NUM
cana-327	46	6	)	)	PUNCT
cana-327	46	7	,	,	PUNCT
cana-327	46	8	the	the	DET
cana-327	46	9	above	above	ADJ
cana-327	46	10	inequality	inequality	NOUN
cana-327	46	11	implies	imply	VERB
cana-327	46	12	lim℘→∞	lim℘→∞	PROPN
cana-327	46	13	 	 	SPACE
cana-327	46	14	𝑠(℘	𝑠(℘	NOUN
cana-327	46	15	)	)	PUNCT
cana-327	46	16	=	=	SYM
cana-327	46	17	−∞.	−∞.	NOUN
cana-327	46	18	we	we	PRON
cana-327	46	19	will	will	AUX
cana-327	46	20	now	now	ADV
cana-327	46	21	examine	examine	VERB
cana-327	46	22	each	each	PRON
cana-327	46	23	of	of	ADP
cana-327	46	24	the	the	DET
cana-327	46	25	next	next	ADJ
cana-327	46	26	two	two	NUM
cana-327	46	27	situations	situation	NOUN
cana-327	46	28	separately	separately	ADV
cana-327	46	29	.	.	PUNCT
cana-327	47	1	case	case	NOUN
cana-327	47	2	(	(	PUNCT
cana-327	47	3	i	i	NOUN
cana-327	47	4	):	):	PUNCT
cana-327	47	5	if	if	SCONJ
cana-327	47	6	z	z	NOUN
cana-327	47	7	is	be	AUX
cana-327	47	8	unbounded	unbounded	ADJ
cana-327	47	9	,	,	PUNCT
cana-327	47	10	then	then	ADV
cana-327	47	11	∃	∃	PROPN
cana-327	47	12	a	a	DET
cana-327	47	13	𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒	𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒	PROPN
cana-327	47	14	{	{	PUNCT
cana-327	47	15	℘𝑛	℘𝑛	NOUN
cana-327	47	16	}	}	PUNCT
cana-327	47	17	such	such	ADJ
cana-327	47	18	that	that	DET
cana-327	47	19	lim𝑛→∞	lim𝑛→∞	PROPN
cana-327	47	20	 	 	SPACE
cana-327	47	21	℘𝑛	℘𝑛	NOUN
cana-327	47	22	=	=	NOUN
cana-327	47	23	∞	∞	PROPN
cana-327	47	24	and	and	CCONJ
cana-327	47	25	lim𝑛→∞	lim𝑛→∞	NOUN
cana-327	47	26	 	 	SPACE
cana-327	47	27	𝑧(℘𝑛	𝑧(℘𝑛	NOUN
cana-327	47	28	)	)	PUNCT
cana-327	48	1	=	=	SYM
cana-327	48	2	∞	∞	PROPN
cana-327	48	3	,	,	PUNCT
cana-327	48	4	where	where	SCONJ
cana-327	48	5	𝑧(℘𝑛	𝑧(℘𝑛	NOUN
cana-327	48	6	)	)	PUNCT
cana-327	48	7	=	=	PUNCT
cana-327	48	8	max{𝑧(𝑖	max{𝑧(𝑖	NOUN
cana-327	48	9	)	)	PUNCT
cana-327	48	10	,	,	PUNCT
cana-327	48	11	℘0	℘0	VERB
cana-327	48	12	≤	≤	NOUN
cana-327	48	13	𝑖	𝑖	SYM
cana-327	48	14	≤	≤	NUM
cana-327	48	15	℘𝑛	℘𝑛	NOUN
cana-327	48	16	}	}	PUNCT
cana-327	48	17	.	.	PUNCT
cana-327	49	1	since	since	SCONJ
cana-327	49	2	lim℘→∞	lim℘→∞	PROPN
cana-327	49	3	 	 	SPACE
cana-327	49	4	𝜏1(℘	𝜏1(℘	NOUN
cana-327	49	5	)	)	PUNCT
cana-327	49	6	=	=	SYM
cana-327	49	7	∞	∞	PROPN
cana-327	49	8	,	,	PUNCT
cana-327	49	9	𝜏1(℘𝑛	𝜏1(℘𝑛	NOUN
cana-327	49	10	)	)	PUNCT
cana-327	49	11	>	>	PUNCT
cana-327	49	12	℘0	℘0	NOUN
cana-327	49	13	for	for	ADP
cana-327	49	14	large	large	ADJ
cana-327	49	15	℘	℘	NOUN
cana-327	49	16	and	and	CCONJ
cana-327	49	17	𝜏1(℘	𝜏1(℘	NOUN
cana-327	49	18	)	)	PUNCT
cana-327	49	19	≤	≤	NUM
cana-327	49	20	℘	℘	NOUN
cana-327	49	21	,	,	PUNCT
cana-327	49	22	then	then	ADV
cana-327	49	23	we	we	PRON
cana-327	49	24	obtain	obtain	VERB
cana-327	49	25	communications	communication	NOUN
cana-327	49	26	on	on	ADP
cana-327	49	27	applied	apply	VERB
cana-327	49	28	nonlinear	nonlinear	ADJ
cana-327	49	29	analysis	analysis	NOUN
cana-327	49	30	issn	issn	NOUN
cana-327	49	31	:	:	PUNCT
cana-327	49	32	1074	1074	NUM
cana-327	49	33	-	-	PUNCT
cana-327	49	34	133x	133x	NUM
cana-327	49	35	vol	vol	NOUN
cana-327	49	36	31	31	NUM
cana-327	49	37	no	no	NOUN
cana-327	49	38	.	.	NOUN
cana-327	49	39	1	1	NUM
cana-327	49	40	(	(	PUNCT
cana-327	49	41	2024	2024	NUM
cana-327	49	42	)	)	PUNCT
cana-327	49	43	84	84	NUM
cana-327	49	44	https://internationalpubls.com	https://internationalpubls.com	X
cana-327	49	45	𝑧(𝜏1(℘𝑛	𝑧(𝜏1(℘𝑛	NOUN
cana-327	49	46	)	)	PUNCT
cana-327	49	47	)	)	PUNCT
cana-327	50	1	=	=	PUNCT
cana-327	51	1	max{𝑧(𝑖	max{𝑧(𝑖	NOUN
cana-327	51	2	):	):	PUNCT
cana-327	51	3	℘0	℘0	NOUN
cana-327	51	4	≤	≤	NUM
cana-327	51	5	𝑖	𝑖	SYM
cana-327	51	6	≤	≤	NUM
cana-327	51	7	𝜏1(℘𝑛	𝜏1(℘𝑛	NOUN
cana-327	51	8	)	)	PUNCT
cana-327	51	9	}	}	PUNCT
cana-327	51	10	≤	≤	NOUN
cana-327	51	11	max{𝑧(𝑖	max{𝑧(𝑖	NOUN
cana-327	51	12	):	):	PUNCT
cana-327	51	13	℘0	℘0	NOUN
cana-327	51	14	≤	≤	NOUN
cana-327	51	15	𝑖	𝑖	SYM
cana-327	51	16	≤	≤	NUM
cana-327	51	17	℘𝑛	℘𝑛	NOUN
cana-327	51	18	}	}	PUNCT
cana-327	51	19	=	=	SYM
cana-327	51	20	𝑧(℘𝑛	𝑧(℘𝑛	ADJ
cana-327	51	21	)	)	PUNCT
cana-327	51	22	.	.	PUNCT
cana-327	52	1	that	that	PRON
cana-327	52	2	is	be	AUX
cana-327	52	3	,	,	PUNCT
cana-327	52	4	𝑧(𝜏1(℘𝑛	𝑧(𝜏1(℘𝑛	NOUN
cana-327	52	5	)	)	PUNCT
cana-327	52	6	)	)	PUNCT
cana-327	53	1	≤	≤	NOUN
cana-327	53	2	𝑧(℘𝑛	𝑧(℘𝑛	NOUN
cana-327	53	3	)	)	PUNCT
cana-327	53	4	.	.	PUNCT
cana-327	54	1	consequently	consequently	ADV
cana-327	54	2	,	,	PUNCT
cana-327	54	3	𝑠(℘𝑛	𝑠(℘𝑛	NOUN
cana-327	54	4	)	)	PUNCT
cana-327	54	5	=	=	SYM
cana-327	54	6	𝑧(℘𝑛	𝑧(℘𝑛	ADJ
cana-327	54	7	)	)	PUNCT
cana-327	54	8	−	−	PROPN
cana-327	54	9	𝑝(℘𝑛)𝑧𝛾1(𝜏1(℘𝑛	𝑝(℘𝑛)𝑧𝛾1(𝜏1(℘𝑛	NOUN
cana-327	54	10	)	)	PUNCT
cana-327	54	11	)	)	PUNCT
cana-327	54	12	≥	≥	NOUN
cana-327	55	1	𝑧(℘𝑛)[1	𝑧(℘𝑛)[1	PROPN
cana-327	55	2	−	−	PROPN
cana-327	55	3	𝑝(℘𝑛)𝑧𝛾1−1(℘𝑛	𝑝(℘𝑛)𝑧𝛾1−1(℘𝑛	NOUN
cana-327	55	4	)	)	PUNCT
cana-327	55	5	]	]	PUNCT
cana-327	56	1	→	→	PUNCT
cana-327	56	2	∞	∞	PROPN
cana-327	56	3	as	as	ADP
cana-327	56	4	𝑛	𝑛	PROPN
cana-327	56	5	→	→	SYM
cana-327	56	6	∞	∞	PROPN
cana-327	56	7	,	,	PUNCT
cana-327	56	8	since	since	SCONJ
cana-327	56	9	𝛾1	𝛾1	PROPN
cana-327	56	10	∈	∈	PROPN
cana-327	56	11	(	(	PUNCT
cana-327	56	12	0,1	0,1	NOUN
cana-327	56	13	]	]	PUNCT
cana-327	56	14	and	and	CCONJ
cana-327	56	15	𝑝(℘	𝑝(℘	NUM
cana-327	56	16	)	)	PUNCT
cana-327	56	17	is	be	AUX
cana-327	56	18	bounded	bound	VERB
cana-327	56	19	,	,	PUNCT
cana-327	56	20	which	which	PRON
cana-327	56	21	contradicts	contradict	VERB
cana-327	56	22	lim℘→∞	lim℘→∞	PROPN
cana-327	56	23	 	 	SPACE
cana-327	56	24	𝑠(℘	𝑠(℘	NOUN
cana-327	56	25	)	)	PUNCT
cana-327	56	26	=	=	SYM
cana-327	56	27	−∞.	−∞.	ADJ
cana-327	56	28	case	case	NOUN
cana-327	56	29	(	(	PUNCT
cana-327	56	30	ii	ii	NOUN
cana-327	56	31	):	):	PUNCT
cana-327	56	32	if	if	SCONJ
cana-327	56	33	z	z	NOUN
cana-327	56	34	is	be	AUX
cana-327	56	35	bounded	bound	VERB
cana-327	56	36	,	,	PUNCT
cana-327	56	37	then	then	ADV
cana-327	56	38	s	s	VERB
cana-327	56	39	is	be	AUX
cana-327	56	40	also	also	ADV
cana-327	56	41	bounded	bound	VERB
cana-327	56	42	,	,	PUNCT
cana-327	56	43	because	because	SCONJ
cana-327	56	44	𝑝(℘	𝑝(℘	NUM
cana-327	56	45	)	)	PUNCT
cana-327	56	46	is	be	AUX
cana-327	56	47	bounded	bound	VERB
cana-327	56	48	,	,	PUNCT
cana-327	56	49	which	which	PRON
cana-327	56	50	contradicts	contradict	VERB
cana-327	56	51	that	that	SCONJ
cana-327	56	52	lim℘→∞	lim℘→∞	PROPN
cana-327	56	53	 	 	SPACE
cana-327	56	54	𝑠(℘	𝑠(℘	NOUN
cana-327	56	55	)	)	PUNCT
cana-327	56	56	=	=	SYM
cana-327	56	57	−∞.	−∞.	PROPN
cana-327	56	58	so	so	SCONJ
cana-327	56	59	𝑠(℘	𝑠(℘	NOUN
cana-327	56	60	)	)	PUNCT
cana-327	56	61	fulfills	fulfill	VERB
cana-327	56	62	one	one	NUM
cana-327	56	63	of	of	ADP
cana-327	56	64	the	the	DET
cana-327	56	65	cases	case	NOUN
cana-327	56	66	(	(	PUNCT
cana-327	56	67	i	i	NOUN
cana-327	56	68	)	)	PUNCT
cana-327	56	69	and	and	CCONJ
cana-327	56	70	(	(	PUNCT
cana-327	56	71	ii	ii	NOUN
cana-327	56	72	)	)	PUNCT
cana-327	56	73	.	.	PUNCT
cana-327	57	1	lemma	lemma	PROPN
cana-327	57	2	2.2	2.2	NUM
cana-327	57	3	.	.	PUNCT
cana-327	58	1	let	let	VERB
cana-327	58	2	the	the	DET
cana-327	58	3	condition	condition	NOUN
cana-327	58	4	(	(	PUNCT
cana-327	58	5	2	2	X
cana-327	58	6	)	)	PUNCT
cana-327	58	7	be	be	AUX
cana-327	58	8	true	true	ADJ
cana-327	58	9	.	.	PUNCT
cana-327	59	1	assume	assume	VERB
cana-327	59	2	𝑧	𝑧	PRON
cana-327	59	3	be	be	AUX
cana-327	59	4	a	a	DET
cana-327	59	5	positive	positive	ADJ
cana-327	59	6	solution	solution	NOUN
cana-327	59	7	of	of	ADP
cana-327	59	8	(	(	PUNCT
cana-327	59	9	1	1	X
cana-327	59	10	)	)	PUNCT
cana-327	59	11	there	there	PRON
cana-327	59	12	exists	exist	VERB
cana-327	59	13	case	case	NOUN
cana-327	59	14	(	(	PUNCT
cana-327	59	15	i	i	NOUN
cana-327	59	16	)	)	PUNCT
cana-327	59	17	of	of	ADP
cana-327	59	18	lemma	lemma	PROPN
cana-327	59	19	2.1	2.1	NUM
cana-327	59	20	.	.	PUNCT
cana-327	60	1	then	then	ADV
cana-327	60	2	𝑧(℘	𝑧(℘	NUM
cana-327	60	3	)	)	PUNCT
cana-327	60	4	>	>	X
cana-327	60	5	𝑠(℘	𝑠(℘	PROPN
cana-327	60	6	)	)	PUNCT
cana-327	60	7	>	>	X
cana-327	60	8	𝑌(℘)𝑟(℘)δ𝑠(℘	𝑌(℘)𝑟(℘)δ𝑠(℘	PROPN
cana-327	60	9	)	)	PUNCT
cana-327	60	10	(	(	PUNCT
cana-327	60	11	4	4	NUM
cana-327	60	12	)	)	PUNCT
cana-327	60	13	for	for	ADP
cana-327	60	14	℘	℘	NUM
cana-327	60	15	≥	≥	NUM
cana-327	60	16	℘1	℘1	NOUN
cana-327	60	17	and	and	CCONJ
cana-327	60	18	𝑠(℘)/𝑌(℘	𝑠(℘)/𝑌(℘	NUM
cana-327	60	19	)	)	PUNCT
cana-327	60	20	is	be	AUX
cana-327	60	21	eventually	eventually	ADV
cana-327	60	22	decreasing	decrease	VERB
cana-327	60	23	.	.	PUNCT
cana-327	61	1	proof	proof	NOUN
cana-327	61	2	.	.	PUNCT
cana-327	62	1	by	by	ADP
cana-327	62	2	the	the	DET
cana-327	62	3	representation	representation	NOUN
cana-327	62	4	of	of	ADP
cana-327	62	5	𝑠(℘	𝑠(℘	NOUN
cana-327	62	6	)	)	PUNCT
cana-327	62	7	and	and	CCONJ
cana-327	62	8	above	above	ADV
cana-327	62	9	(	(	PUNCT
cana-327	62	10	𝑅2	𝑅2	NOUN
cana-327	62	11	)	)	PUNCT
cana-327	62	12	,	,	PUNCT
cana-327	62	13	we	we	PRON
cana-327	62	14	can	can	AUX
cana-327	62	15	write	write	VERB
cana-327	62	16	𝑧(℘	𝑧(℘	NUM
cana-327	62	17	)	)	PUNCT
cana-327	62	18	>	>	X
cana-327	62	19	𝑠(℘	𝑠(℘	NOUN
cana-327	62	20	)	)	PUNCT
cana-327	62	21	for	for	ADP
cana-327	62	22	℘	℘	NUM
cana-327	62	23	≥	≥	NUM
cana-327	62	24	℘1	℘1	VERB
cana-327	62	25	≥	≥	NOUN
cana-327	62	26	℘0	℘0	NOUN
cana-327	62	27	.	.	PUNCT
cana-327	63	1	from	from	ADP
cana-327	63	2	the	the	DET
cana-327	63	3	case	case	NOUN
cana-327	63	4	(	(	PUNCT
cana-327	63	5	i	i	NOUN
cana-327	63	6	)	)	PUNCT
cana-327	63	7	,	,	PUNCT
cana-327	63	8	we	we	PRON
cana-327	63	9	get	get	VERB
cana-327	63	10	𝑠(℘	𝑠(℘	NOUN
cana-327	63	11	)	)	PUNCT
cana-327	63	12	=	=	SYM
cana-327	63	13	𝑠(℘1	𝑠(℘1	NOUN
cana-327	63	14	)	)	PUNCT
cana-327	64	1	+	+	CCONJ
cana-327	64	2	∑	∑	PUNCT
cana-327	64	3	 	 	SPACE
cana-327	64	4	℘−1	℘−1	PROPN
cana-327	64	5	𝑖=℘1	𝑖=℘1	PROPN
cana-327	64	6	  	  	SPACE
cana-327	64	7	𝑟(𝑖)δ𝑠(𝑖	𝑟(𝑖)δ𝑠(𝑖	X
cana-327	64	8	)	)	PUNCT
cana-327	64	9	𝑟(𝑖	𝑟(𝑖	PROPN
cana-327	64	10	)	)	PUNCT
cana-327	64	11	,	,	PUNCT
cana-327	64	12	>	>	X
cana-327	64	13	𝑌(℘)𝑟(℘)δ𝑠(℘	𝑌(℘)𝑟(℘)δ𝑠(℘	NOUN
cana-327	64	14	)	)	PUNCT
cana-327	64	15	,	,	PUNCT
cana-327	64	16	℘	℘	PROPN
cana-327	64	17	≥	≥	NUM
cana-327	64	18	℘1	℘1	VERB
cana-327	64	19	.	.	PUNCT
cana-327	65	1	(	(	PUNCT
cana-327	65	2	5	5	NUM
cana-327	65	3	)	)	PUNCT
cana-327	65	4	also	also	ADV
cana-327	65	5	,	,	PUNCT
cana-327	65	6	δ	δ	PROPN
cana-327	65	7	(	(	PUNCT
cana-327	65	8	𝑠(℘	𝑠(℘	NOUN
cana-327	65	9	)	)	PUNCT
cana-327	65	10	𝑌(℘	𝑌(℘	NOUN
cana-327	65	11	)	)	PUNCT
cana-327	65	12	)	)	PUNCT
cana-327	66	1	=	=	PUNCT
cana-327	66	2	𝑌(℘)𝑟(℘)δ𝑠(℘	𝑌(℘)𝑟(℘)δ𝑠(℘	X
cana-327	66	3	)	)	PUNCT
cana-327	66	4	−	−	PROPN
cana-327	66	5	𝑠(℘	𝑠(℘	NOUN
cana-327	66	6	)	)	PUNCT
cana-327	66	7	𝑟(℘)𝑌(℘)𝑌(𝑙	𝑟(℘)𝑌(℘)𝑌(𝑙	NOUN
cana-327	66	8	+	+	CCONJ
cana-327	66	9	1	1	NUM
cana-327	66	10	)	)	PUNCT
cana-327	66	11	<	<	X
cana-327	66	12	0	0	NUM
cana-327	66	13	,	,	PUNCT
cana-327	66	14	℘	℘	X
cana-327	66	15	≥	≥	NUM
cana-327	66	16	℘1	℘1	NOUN
cana-327	66	17	.	.	PUNCT
cana-327	67	1	thus	thus	ADV
cana-327	67	2	{	{	PUNCT
cana-327	67	3	𝑠(℘	𝑠(℘	NOUN
cana-327	67	4	)	)	PUNCT
cana-327	67	5	𝑌(℘	𝑌(℘	NOUN
cana-327	67	6	)	)	PUNCT
cana-327	67	7	}	}	PUNCT
cana-327	67	8	is	be	AUX
cana-327	67	9	strictly	strictly	ADV
cana-327	67	10	decreasing	decrease	VERB
cana-327	67	11	for	for	ADP
cana-327	67	12	all	all	DET
cana-327	67	13	℘	℘	PROPN
cana-327	67	14	≥	≥	NUM
cana-327	67	15	℘1	℘1	NOUN
cana-327	67	16	.	.	PUNCT
cana-327	68	1	theorem	theorem	VERB
cana-327	68	2	2.3	2.3	NUM
cana-327	68	3	.	.	PUNCT
cana-327	69	1	let	let	AUX
cana-327	69	2	𝛾2	𝛾2	VERB
cana-327	69	3	<	<	X
cana-327	69	4	𝛾1	𝛾1	PROPN
cana-327	69	5	,	,	PUNCT
cana-327	69	6	𝜎1(℘	𝜎1(℘	NOUN
cana-327	69	7	)	)	PUNCT
cana-327	69	8	<	<	X
cana-327	69	9	𝜏1(℘	𝜏1(℘	NOUN
cana-327	69	10	)	)	PUNCT
cana-327	69	11	and	and	CCONJ
cana-327	69	12	condition	condition	NOUN
cana-327	69	13	(	(	PUNCT
cana-327	69	14	2	2	X
cana-327	69	15	)	)	PUNCT
cana-327	69	16	hold	hold	NOUN
cana-327	69	17	.	.	PUNCT
cana-327	70	1	if	if	SCONJ
cana-327	70	2	∑	∑	ADP
cana-327	70	3	 	 	SPACE
cana-327	70	4	∞	∞	PROPN
cana-327	70	5	℘1	℘1	PROPN
cana-327	70	6	𝑞(℘)𝑌𝛾2(𝜎1(℘	𝑞(℘)𝑌𝛾2(𝜎1(℘	NOUN
cana-327	70	7	)	)	PUNCT
cana-327	70	8	)	)	PUNCT
cana-327	71	1	=	=	SYM
cana-327	71	2	∞	∞	PROPN
cana-327	71	3	(	(	PUNCT
cana-327	71	4	6	6	NUM
cana-327	71	5	)	)	PUNCT
cana-327	71	6	and	and	CCONJ
cana-327	71	7	lim	lim	PROPN
cana-327	71	8	℘→∞	℘→∞	PROPN
cana-327	71	9	 	 	SPACE
cana-327	71	10	sup	sup	PROPN
cana-327	71	11	∑	∑	ADP
cana-327	71	12	  	  	SPACE
cana-327	71	13	℘−1	℘−1	PROPN
cana-327	71	14	𝑗=𝜏1	𝑗=𝜏1	PUNCT
cana-327	71	15	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	71	16	)	)	PUNCT
cana-327	71	17	)	)	PUNCT
cana-327	71	18	1	1	NUM
cana-327	71	19	𝑟(𝑗	𝑟(𝑗	PROPN
cana-327	71	20	)	)	PUNCT
cana-327	71	21	∑	∑	ADP
cana-327	71	22	  	  	SPACE
cana-327	71	23	𝑗−1	𝑗−1	PROPN
cana-327	71	24	𝑢=𝑗3	𝑢=𝑗3	VERB
cana-327	71	25	𝑞(𝑢	𝑞(𝑢	ADV
cana-327	71	26	)	)	PUNCT
cana-327	71	27	𝑝	𝑝	NOUN
cana-327	71	28	𝛾2	𝛾2	VERB
cana-327	71	29	𝛾1(𝜏1	𝛾1(𝜏1	PRON
cana-327	71	30	−1(𝜎1(𝑢	−1(𝜎1(𝑢	VERB
cana-327	71	31	)	)	PUNCT
cana-327	71	32	)	)	PUNCT
cana-327	71	33	)	)	PUNCT
cana-327	72	1	>	>	X
cana-327	72	2	0	0	PUNCT
cana-327	72	3	,	,	PUNCT
cana-327	72	4	(	(	PUNCT
cana-327	72	5	7	7	X
cana-327	72	6	)	)	PUNCT
cana-327	72	7	then	then	ADV
cana-327	72	8	every	every	DET
cana-327	72	9	solution	solution	NOUN
cana-327	72	10	of	of	ADP
cana-327	72	11	equation	equation	NOUN
cana-327	72	12	(	(	PUNCT
cana-327	72	13	1	1	X
cana-327	72	14	)	)	PUNCT
cana-327	72	15	is	be	AUX
cana-327	72	16	oscillatory	oscillatory	ADJ
cana-327	72	17	.	.	PUNCT
cana-327	73	1	proof	proof	NOUN
cana-327	73	2	.	.	PUNCT
cana-327	74	1	let	let	VERB
cana-327	74	2	z	z	PRON
cana-327	74	3	be	be	AUX
cana-327	74	4	a	a	DET
cana-327	74	5	non	non	X
cana-327	74	6	oscillatory	oscillatory	ADJ
cana-327	74	7	solution	solution	NOUN
cana-327	74	8	of	of	ADP
cana-327	74	9	(	(	PUNCT
cana-327	74	10	1	1	NUM
cana-327	74	11	)	)	PUNCT
cana-327	74	12	.	.	PUNCT
cana-327	75	1	then	then	ADV
cana-327	75	2	𝑧(℘	𝑧(℘	NUM
cana-327	75	3	)	)	PUNCT
cana-327	75	4	>	>	X
cana-327	75	5	0	0	NUM
cana-327	75	6	,	,	PUNCT
cana-327	75	7	𝑧(𝜎1(℘	𝑧(𝜎1(℘	NOUN
cana-327	75	8	)	)	PUNCT
cana-327	75	9	)	)	PUNCT
cana-327	75	10	>	>	X
cana-327	75	11	0	0	NUM
cana-327	75	12	,	,	PUNCT
cana-327	75	13	𝑧(𝜏1(℘	𝑧(𝜏1(℘	NOUN
cana-327	75	14	)	)	PUNCT
cana-327	75	15	)	)	PUNCT
cana-327	76	1	>	>	X
cana-327	76	2	0	0	NUM
cana-327	76	3	,	,	PUNCT
cana-327	76	4	℘	℘	X
cana-327	76	5	≥	≥	NUM
cana-327	76	6	℘1	℘1	VERB
cana-327	76	7	≥	≥	NOUN
cana-327	76	8	℘0	℘0	NOUN
cana-327	76	9	.	.	PUNCT
cana-327	77	1	by	by	ADP
cana-327	77	2	lemma	lemma	PROPN
cana-327	77	3	2.1	2.1	NUM
cana-327	77	4	,	,	PUNCT
cana-327	77	5	the	the	DET
cana-327	77	6	corresponding	corresponding	ADJ
cana-327	77	7	function	function	PROPN
cana-327	77	8	𝑠(℘	𝑠(℘	NOUN
cana-327	77	9	)	)	PUNCT
cana-327	77	10	fullfills	fullfill	VERB
cana-327	77	11	either	either	DET
cana-327	77	12	case	case	NOUN
cana-327	77	13	(	(	PUNCT
cana-327	77	14	i	i	NOUN
cana-327	77	15	)	)	PUNCT
cana-327	77	16	or	or	CCONJ
cana-327	77	17	case	case	NOUN
cana-327	77	18	(	(	PUNCT
cana-327	77	19	ii	ii	NOUN
cana-327	77	20	)	)	PUNCT
cana-327	77	21	.	.	PUNCT
cana-327	78	1	communications	communication	NOUN
cana-327	78	2	on	on	ADP
cana-327	78	3	applied	apply	VERB
cana-327	78	4	nonlinear	nonlinear	ADJ
cana-327	78	5	analysis	analysis	NOUN
cana-327	78	6	issn	issn	NOUN
cana-327	78	7	:	:	PUNCT
cana-327	78	8	1074	1074	NUM
cana-327	78	9	-	-	PUNCT
cana-327	78	10	133x	133x	NUM
cana-327	78	11	vol	vol	NOUN
cana-327	78	12	31	31	NUM
cana-327	78	13	no	no	NOUN
cana-327	78	14	.	.	NOUN
cana-327	78	15	1	1	NUM
cana-327	78	16	(	(	PUNCT
cana-327	78	17	2024	2024	NUM
cana-327	78	18	)	)	PUNCT
cana-327	78	19	85	85	NUM
cana-327	78	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-327	78	21	first	first	ADV
cana-327	78	22	,	,	PUNCT
cana-327	78	23	we	we	PRON
cana-327	78	24	assume	assume	VERB
cana-327	78	25	that	that	SCONJ
cana-327	78	26	𝑠(℘	𝑠(℘	NOUN
cana-327	78	27	)	)	PUNCT
cana-327	78	28	satisfie	satisfie	NOUN
cana-327	78	29	s	s	PART
cana-327	78	30	case	case	NOUN
cana-327	78	31	(	(	PUNCT
cana-327	78	32	i	i	NOUN
cana-327	78	33	)	)	PUNCT
cana-327	78	34	.	.	PUNCT
cana-327	79	1	from	from	ADP
cana-327	79	2	the	the	DET
cana-327	79	3	representation	representation	NOUN
cana-327	79	4	of	of	ADP
cana-327	79	5	𝑠(℘	𝑠(℘	NOUN
cana-327	79	6	)	)	PUNCT
cana-327	79	7	,	,	PUNCT
cana-327	79	8	we	we	PRON
cana-327	79	9	get	get	VERB
cana-327	79	10	𝑧(℘	𝑧(℘	NUM
cana-327	79	11	)	)	PUNCT
cana-327	79	12	≥	≥	PROPN
cana-327	79	13	𝑠(℘	𝑠(℘	NOUN
cana-327	79	14	)	)	PUNCT
cana-327	79	15	,	,	PUNCT
cana-327	79	16	𝑧𝛾2(𝜎𝑙(℘	𝑧𝛾2(𝜎𝑙(℘	ADJ
cana-327	79	17	)	)	PUNCT
cana-327	79	18	)	)	PUNCT
cana-327	79	19	≥	≥	NOUN
cana-327	79	20	𝑠𝛾2(𝜎𝑙(℘	𝑠𝛾2(𝜎𝑙(℘	VERB
cana-327	79	21	)	)	PUNCT
cana-327	79	22	)	)	PUNCT
cana-327	79	23	.	.	PUNCT
cana-327	80	1	applying	apply	VERB
cana-327	80	2	above	above	ADP
cana-327	80	3	inequality	inequality	NOUN
cana-327	80	4	in	in	ADP
cana-327	80	5	(	(	PUNCT
cana-327	80	6	1	1	NUM
cana-327	80	7	)	)	PUNCT
cana-327	80	8	,	,	PUNCT
cana-327	80	9	we	we	PRON
cana-327	80	10	get	get	VERB
cana-327	80	11	δ(𝑟(℘)δ𝑠(℘	δ(𝑟(℘)δ𝑠(℘	NOUN
cana-327	80	12	)	)	PUNCT
cana-327	80	13	)	)	PUNCT
cana-327	81	1	+	+	CCONJ
cana-327	81	2	𝑞(℘)𝑠𝛾2(𝜎1(℘	𝑞(℘)𝑠𝛾2(𝜎1(℘	NOUN
cana-327	81	3	)	)	PUNCT
cana-327	81	4	)	)	PUNCT
cana-327	81	5	≤	≤	ADV
cana-327	81	6	0	0	NUM
cana-327	81	7	.	.	PUNCT
cana-327	82	1	(	(	PUNCT
cana-327	82	2	8)	8)	NUM
cana-327	82	3	substituting	substituting	NOUN
cana-327	82	4	(	(	PUNCT
cana-327	82	5	4	4	NUM
cana-327	82	6	)	)	PUNCT
cana-327	82	7	in	in	ADP
cana-327	82	8	(	(	PUNCT
cana-327	82	9	8)	8)	NUM
cana-327	82	10	and	and	CCONJ
cana-327	82	11	taking	take	VERB
cana-327	82	12	𝑦(℘	𝑦(℘	NUM
cana-327	82	13	)	)	PUNCT
cana-327	82	14	=	=	SYM
cana-327	82	15	𝑟(℘)δ𝑠(℘	𝑟(℘)δ𝑠(℘	NOUN
cana-327	82	16	)	)	PUNCT
cana-327	82	17	,	,	PUNCT
cana-327	82	18	we	we	PRON
cana-327	82	19	clear	clear	VERB
cana-327	82	20	that	that	SCONJ
cana-327	82	21	𝑦(℘	𝑦(℘	NUM
cana-327	82	22	)	)	PUNCT
cana-327	82	23	is	be	AUX
cana-327	82	24	a	a	DET
cana-327	82	25	positive	positive	ADJ
cana-327	82	26	solution	solution	NOUN
cana-327	82	27	of	of	ADP
cana-327	82	28	the	the	DET
cana-327	82	29	inequality	inequality	NOUN
cana-327	82	30	δ𝑦(℘	δ𝑦(℘	NUM
cana-327	82	31	)	)	PUNCT
cana-327	83	1	+	+	NUM
cana-327	83	2	𝑞(℘)𝑌𝛾2(𝜎1(℘))(𝑟(℘)δ𝑠(℘))𝛾2	𝑞(℘)𝑌𝛾2(𝜎1(℘))(𝑟(℘)δ𝑠(℘))𝛾2	PUNCT
cana-327	83	3	≤	≤	X
cana-327	83	4	0	0	NUM
cana-327	83	5	,	,	PUNCT
cana-327	83	6	δ𝑦(℘	δ𝑦(℘	NUM
cana-327	83	7	)	)	PUNCT
cana-327	83	8	+	+	NUM
cana-327	83	9	𝑞(℘)𝑌𝛾2(𝜎1(℘))𝑦𝛾2(𝜎1(℘	𝑞(℘)𝑌𝛾2(𝜎1(℘))𝑦𝛾2(𝜎1(℘	NOUN
cana-327	83	10	)	)	PUNCT
cana-327	83	11	)	)	PUNCT
cana-327	83	12	≤	≤	NOUN
cana-327	83	13	0	0	NUM
cana-327	83	14	,	,	PUNCT
cana-327	83	15	℘	℘	X
cana-327	83	16	≥	≥	NUM
cana-327	83	17	℘1	℘1	NOUN
cana-327	83	18	(	(	PUNCT
cana-327	83	19	9	9	NUM
cana-327	83	20	)	)	PUNCT
cana-327	83	21	on	on	ADP
cana-327	83	22	the	the	DET
cana-327	83	23	other	other	ADJ
cana-327	83	24	hand	hand	NOUN
cana-327	83	25	,	,	PUNCT
cana-327	83	26	from	from	ADP
cana-327	83	27	[	[	X
cana-327	83	28	6	6	NUM
cana-327	83	29	]	]	PUNCT
cana-327	83	30	,	,	PUNCT
cana-327	83	31	we	we	PRON
cana-327	83	32	can	can	AUX
cana-327	83	33	see	see	VERB
cana-327	83	34	that	that	DET
cana-327	83	35	condition	condition	NOUN
cana-327	83	36	(	(	PUNCT
cana-327	83	37	6	6	NUM
cana-327	83	38	)	)	PUNCT
cana-327	83	39	assures	assure	VERB
cana-327	83	40	that	that	SCONJ
cana-327	83	41	(	(	PUNCT
cana-327	83	42	9	9	X
cana-327	83	43	)	)	PUNCT
cana-327	83	44	has	have	VERB
cana-327	83	45	no	no	DET
cana-327	83	46	eventually	eventually	ADV
cana-327	83	47	positive	positive	ADJ
cana-327	83	48	solution	solution	NOUN
cana-327	83	49	,	,	PUNCT
cana-327	83	50	which	which	PRON
cana-327	83	51	is	be	AUX
cana-327	83	52	contradiction	contradiction	NOUN
cana-327	83	53	.	.	PUNCT
cana-327	84	1	next	next	ADV
cana-327	84	2	,	,	PUNCT
cana-327	84	3	assume	assume	VERB
cana-327	84	4	that	that	SCONJ
cana-327	84	5	𝑠(℘	𝑠(℘	NOUN
cana-327	84	6	)	)	PUNCT
cana-327	84	7	satisfies	satisfie	NOUN
cana-327	84	8	case	case	NOUN
cana-327	84	9	(	(	PUNCT
cana-327	84	10	ii	ii	NOUN
cana-327	84	11	)	)	PUNCT
cana-327	84	12	of	of	ADP
cana-327	84	13	lemma	lemma	PROPN
cana-327	84	14	2.1	2.1	NUM
cana-327	84	15	.	.	PUNCT
cana-327	85	1	then	then	ADV
cana-327	85	2	,	,	PUNCT
cana-327	85	3	by	by	ADP
cana-327	85	4	the	the	DET
cana-327	85	5	representation	representation	NOUN
cana-327	85	6	of	of	ADP
cana-327	85	7	𝑠(℘	𝑠(℘	NOUN
cana-327	85	8	)	)	PUNCT
cana-327	85	9	,	,	PUNCT
cana-327	85	10	we	we	PRON
cana-327	85	11	get	get	VERB
cana-327	85	12	𝑧(𝜏1(℘	𝑧(𝜏1(℘	NOUN
cana-327	85	13	)	)	PUNCT
cana-327	85	14	)	)	PUNCT
cana-327	85	15	>	>	PUNCT
cana-327	86	1	(	(	PUNCT
cana-327	86	2	−𝑠(℘	−𝑠(℘	ADP
cana-327	86	3	)	)	PUNCT
cana-327	86	4	𝑝(℘	𝑝(℘	NOUN
cana-327	86	5	)	)	PUNCT
cana-327	86	6	)	)	PUNCT
cana-327	87	1	1	1	NUM
cana-327	87	2	𝛾1	𝛾1	NOUN
cana-327	87	3	.	.	PUNCT
cana-327	88	1	(	(	PUNCT
cana-327	88	2	10	10	NUM
cana-327	88	3	)	)	PUNCT
cana-327	88	4	applying	apply	VERB
cana-327	88	5	(	(	PUNCT
cana-327	88	6	10	10	NUM
cana-327	88	7	)	)	PUNCT
cana-327	88	8	in	in	ADP
cana-327	88	9	(	(	PUNCT
cana-327	88	10	1	1	NUM
cana-327	88	11	)	)	PUNCT
cana-327	88	12	,	,	PUNCT
cana-327	88	13	we	we	PRON
cana-327	88	14	get	get	VERB
cana-327	88	15	δ(𝑟(℘)δ𝑠(℘	δ(𝑟(℘)δ𝑠(℘	NOUN
cana-327	88	16	)	)	PUNCT
cana-327	88	17	)	)	PUNCT
cana-327	89	1	−	−	PROPN
cana-327	89	2	1	1	NUM
cana-327	89	3	𝑝	𝑝	NOUN
cana-327	89	4	𝛾2	𝛾2	VERB
cana-327	89	5	𝛾1(𝜏1	𝛾1(𝜏1	VERB
cana-327	89	6	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	89	7	)	)	PUNCT
cana-327	89	8	)	)	PUNCT
cana-327	89	9	)	)	PUNCT
cana-327	90	1	𝑞(℘)𝑠	𝑞(℘)𝑠	VERB
cana-327	90	2	𝛾2	𝛾2	VERB
cana-327	90	3	𝛾1(𝜏1	𝛾1(𝜏1	VERB
cana-327	90	4	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	90	5	)	)	PUNCT
cana-327	90	6	)	)	PUNCT
cana-327	90	7	)	)	PUNCT
cana-327	91	1	≤	≤	ADV
cana-327	91	2	0	0	NUM
cana-327	91	3	.	.	PUNCT
cana-327	92	1	(	(	PUNCT
cana-327	92	2	11	11	NUM
cana-327	92	3	)	)	PUNCT
cana-327	92	4	since	since	SCONJ
cana-327	92	5	𝑠(℘	𝑠(℘	NOUN
cana-327	92	6	)	)	PUNCT
cana-327	92	7	is	be	AUX
cana-327	92	8	negative	negative	ADJ
cana-327	92	9	and	and	CCONJ
cana-327	92	10	increasing	increase	VERB
cana-327	92	11	,	,	PUNCT
cana-327	92	12	we	we	PRON
cana-327	92	13	obtain	obtain	VERB
cana-327	92	14	lim℘→∞	lim℘→∞	ADJ
cana-327	92	15	 	 	SPACE
cana-327	92	16	𝑠(℘	𝑠(℘	NOUN
cana-327	92	17	)	)	PUNCT
cana-327	92	18	=	=	PUNCT
cana-327	92	19	𝑐1	𝑐1	NOUN
cana-327	92	20	≤	≤	NUM
cana-327	92	21	0	0	X
cana-327	92	22	.	.	PUNCT
cana-327	93	1	we	we	PRON
cana-327	93	2	prove	prove	VERB
cana-327	93	3	that	that	SCONJ
cana-327	93	4	𝑐1	𝑐1	NOUN
cana-327	93	5	=	=	NOUN
cana-327	93	6	0	0	X
cana-327	93	7	.	.	PUNCT
cana-327	94	1	if	if	SCONJ
cana-327	94	2	not	not	PART
cana-327	94	3	,	,	PUNCT
cana-327	94	4	then	then	ADV
cana-327	94	5	𝑐1	𝑐1	VERB
cana-327	94	6	<	<	X
cana-327	94	7	0	0	PUNCT
cana-327	94	8	and	and	CCONJ
cana-327	94	9	𝑠(℘	𝑠(℘	NOUN
cana-327	94	10	)	)	PUNCT
cana-327	94	11	≤	≤	NOUN
cana-327	94	12	𝑐1	𝑐1	NOUN
cana-327	94	13	and	and	CCONJ
cana-327	94	14	𝑠(𝜏1	𝑠(𝜏1	ADJ
cana-327	94	15	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	94	16	)	)	PUNCT
cana-327	94	17	)	)	PUNCT
cana-327	94	18	)	)	PUNCT
cana-327	94	19	≤	≤	NOUN
cana-327	95	1	𝑐1	𝑐1	NOUN
cana-327	95	2	for	for	ADP
cana-327	95	3	large	large	ADJ
cana-327	95	4	℘.	℘.	PROPN
cana-327	95	5	therefore	therefore	ADV
cana-327	95	6	,	,	PUNCT
cana-327	95	7	𝑠	𝑠	PROPN
cana-327	95	8	𝛾2	𝛾2	VERB
cana-327	95	9	𝛾1(𝜏1	𝛾1(𝜏1	VERB
cana-327	95	10	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	95	11	)	)	PUNCT
cana-327	95	12	)	)	PUNCT
cana-327	95	13	)	)	PUNCT
cana-327	96	1	≤	≤	NUM
cana-327	96	2	𝑐1	𝑐1	NOUN
cana-327	96	3	𝛾2	𝛾2	VERB
cana-327	96	4	𝛾1	𝛾1	PROPN
cana-327	96	5	(	(	PUNCT
cana-327	96	6	12	12	NUM
cana-327	96	7	)	)	PUNCT
cana-327	96	8	summing	sum	VERB
cana-327	96	9	(	(	PUNCT
cana-327	96	10	11	11	NUM
cana-327	96	11	)	)	PUNCT
cana-327	96	12	from	from	ADP
cana-327	96	13	℘	℘	PROPN
cana-327	96	14	to	to	ADP
cana-327	96	15	∞	∞	NUM
cana-327	96	16	and	and	CCONJ
cana-327	96	17	using	use	VERB
cana-327	96	18	(	(	PUNCT
cana-327	96	19	12	12	NUM
cana-327	96	20	)	)	PUNCT
cana-327	96	21	,	,	PUNCT
cana-327	96	22	we	we	PRON
cana-327	96	23	get	get	VERB
cana-327	96	24	𝑟(℘)δ𝑠(℘	𝑟(℘)δ𝑠(℘	NOUN
cana-327	96	25	)	)	PUNCT
cana-327	96	26	−	−	PROPN
cana-327	96	27	𝑟(℘1)δ𝑠(℘1	𝑟(℘1)δ𝑠(℘1	NOUN
cana-327	96	28	)	)	PUNCT
cana-327	96	29	≤	≤	NOUN
cana-327	96	30	∑	∑	ADP
cana-327	96	31	  	  	SPACE
cana-327	96	32	∞	∞	NUM
cana-327	96	33	𝑖=𝑙	𝑖=𝑙	NOUN
cana-327	96	34	  	  	SPACE
cana-327	96	35	𝑞(𝑖	𝑞(𝑖	NUM
cana-327	96	36	)	)	PUNCT
cana-327	96	37	𝑝	𝑝	NOUN
cana-327	96	38	𝛾2	𝛾2	VERB
cana-327	96	39	𝛾1(𝜏1	𝛾1(𝜏1	PRON
cana-327	96	40	−1(𝜎1(𝑖	−1(𝜎1(𝑖	PROPN
cana-327	96	41	)	)	PUNCT
cana-327	96	42	)	)	PUNCT
cana-327	96	43	)	)	PUNCT
cana-327	97	1	𝑠	𝑠	X
cana-327	97	2	𝛾2	𝛾2	VERB
cana-327	97	3	𝛾1(𝜏1	𝛾1(𝜏1	NUM
cana-327	97	4	−1(𝜎1(𝑖	−1(𝜎1(𝑖	PROPN
cana-327	97	5	)	)	PUNCT
cana-327	97	6	)	)	PUNCT
cana-327	97	7	)	)	PUNCT
cana-327	97	8	,	,	PUNCT
cana-327	97	9	−𝑟(℘)δ𝑠(℘	−𝑟(℘)δ𝑠(℘	NOUN
cana-327	97	10	)	)	PUNCT
cana-327	97	11	≤	≤	NOUN
cana-327	97	12	𝑐1	𝑐1	NOUN
cana-327	97	13	𝛾2	𝛾2	VERB
cana-327	97	14	𝛾1	𝛾1	NOUN
cana-327	97	15	∑	∑	ADP
cana-327	97	16	  	  	SPACE
cana-327	97	17	∞	∞	NUM
cana-327	97	18	𝑖=𝑙	𝑖=𝑙	NOUN
cana-327	97	19	  	  	SPACE
cana-327	97	20	𝑞(𝑖	𝑞(𝑖	NUM
cana-327	97	21	)	)	PUNCT
cana-327	97	22	𝑝	𝑝	NOUN
cana-327	97	23	𝛾2	𝛾2	VERB
cana-327	97	24	𝛾1(𝜏1	𝛾1(𝜏1	PRON
cana-327	97	25	−1(𝜎1(𝑖	−1(𝜎1(𝑖	PROPN
cana-327	97	26	)	)	PUNCT
cana-327	97	27	)	)	PUNCT
cana-327	97	28	)	)	PUNCT
cana-327	97	29	.	.	PUNCT
cana-327	98	1	again	again	ADV
cana-327	98	2	summing	sum	VERB
cana-327	98	3	from	from	ADP
cana-327	98	4	℘1	℘1	NOUN
cana-327	98	5	to	to	ADP
cana-327	98	6	∞	∞	PROPN
cana-327	98	7	,	,	PUNCT
cana-327	98	8	we	we	PRON
cana-327	98	9	have	have	VERB
cana-327	98	10	𝑠(℘1	𝑠(℘1	NOUN
cana-327	98	11	)	)	PUNCT
cana-327	98	12	≤	≤	NOUN
cana-327	98	13	𝑐1	𝑐1	NOUN
cana-327	98	14	𝛾2	𝛾2	VERB
cana-327	98	15	𝛾1	𝛾1	NOUN
cana-327	98	16	∑	∑	ADP
cana-327	98	17	 	 	SPACE
cana-327	98	18	∞	∞	NUM
cana-327	98	19	𝑗=℘1	𝑗=℘1	PROPN
cana-327	98	20	1	1	NUM
cana-327	98	21	𝑟(𝑗	𝑟(𝑗	PROPN
cana-327	98	22	)	)	PUNCT
cana-327	98	23	∑	∑	ADP
cana-327	98	24	 	 	SPACE
cana-327	98	25	∞	∞	PROPN
cana-327	98	26	𝑚=𝑗1	𝑚=𝑗1	PROPN
cana-327	98	27	𝑞(𝑚	𝑞(𝑚	PROPN
cana-327	98	28	)	)	PUNCT
cana-327	98	29	𝑝	𝑝	NOUN
cana-327	98	30	𝛾2	𝛾2	VERB
cana-327	98	31	𝛾1(𝜏1	𝛾1(𝜏1	PRON
cana-327	98	32	−1(𝜎1(𝑚	−1(𝜎1(𝑚	NOUN
cana-327	98	33	)	)	PUNCT
cana-327	98	34	)	)	PUNCT
cana-327	98	35	)	)	PUNCT
cana-327	99	1	communications	communication	NOUN
cana-327	99	2	on	on	ADP
cana-327	99	3	applied	apply	VERB
cana-327	99	4	nonlinear	nonlinear	ADJ
cana-327	99	5	analysis	analysis	NOUN
cana-327	99	6	issn	issn	NOUN
cana-327	99	7	:	:	PUNCT
cana-327	99	8	1074	1074	NUM
cana-327	99	9	-	-	PUNCT
cana-327	99	10	133x	133x	NUM
cana-327	99	11	vol	vol	NOUN
cana-327	99	12	31	31	NUM
cana-327	99	13	no	no	NOUN
cana-327	99	14	.	.	NOUN
cana-327	99	15	1	1	NUM
cana-327	99	16	(	(	PUNCT
cana-327	99	17	2024	2024	NUM
cana-327	99	18	)	)	PUNCT
cana-327	99	19	86	86	NUM
cana-327	99	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-327	99	21	which	which	PRON
cana-327	99	22	is	be	AUX
cana-327	99	23	contradiction	contradiction	NOUN
cana-327	99	24	with	with	ADP
cana-327	99	25	(	(	PUNCT
cana-327	99	26	7	7	NUM
cana-327	99	27	)	)	PUNCT
cana-327	99	28	and	and	CCONJ
cana-327	99	29	from	from	ADP
cana-327	99	30	(	(	PUNCT
cana-327	99	31	7	7	NUM
cana-327	99	32	)	)	PUNCT
cana-327	99	33	,	,	PUNCT
cana-327	99	34	we	we	PRON
cana-327	99	35	claim	claim	VERB
cana-327	99	36	lim	lim	PROPN
cana-327	99	37	℘→∞	℘→∞	PROPN
cana-327	99	38	 	 	SPACE
cana-327	99	39	sup	sup	PROPN
cana-327	99	40	∑	∑	PROPN
cana-327	99	41	  	  	SPACE
cana-327	99	42	∞	∞	NUM
cana-327	99	43	𝑗=℘1	𝑗=℘1	NOUN
cana-327	99	44	1	1	NUM
cana-327	99	45	𝑟(𝑗	𝑟(𝑗	PROPN
cana-327	99	46	)	)	PUNCT
cana-327	99	47	∑	∑	ADP
cana-327	99	48	  	  	SPACE
cana-327	99	49	∞	∞	PROPN
cana-327	99	50	𝑚=𝑗1	𝑚=𝑗1	PROPN
cana-327	99	51	𝑞(𝑚	𝑞(𝑚	PROPN
cana-327	99	52	)	)	PUNCT
cana-327	99	53	𝑝	𝑝	NOUN
cana-327	99	54	𝛾2	𝛾2	VERB
cana-327	99	55	𝛾1(𝜏1	𝛾1(𝜏1	PRON
cana-327	99	56	−1(𝜎1(𝑚	−1(𝜎1(𝑚	NOUN
cana-327	99	57	)	)	PUNCT
cana-327	99	58	)	)	PUNCT
cana-327	99	59	)	)	PUNCT
cana-327	100	1	=	=	SYM
cana-327	100	2	∞.	∞.	PROPN
cana-327	100	3	thus	thus	ADV
cana-327	100	4	,	,	PUNCT
cana-327	100	5	lim℘→∞	lim℘→∞	ADJ
cana-327	100	6	 	 	SPACE
cana-327	100	7	𝑠(℘	𝑠(℘	NOUN
cana-327	100	8	)	)	PUNCT
cana-327	100	9	=	=	SYM
cana-327	100	10	0	0	NUM
cana-327	100	11	and	and	CCONJ
cana-327	100	12	s(℘	s(℘	NUM
cana-327	100	13	)	)	PUNCT
cana-327	100	14	is	be	AUX
cana-327	100	15	negative	negative	ADJ
cana-327	100	16	and	and	CCONJ
cana-327	100	17	increasing	increase	VERB
cana-327	100	18	.	.	PUNCT
cana-327	101	1	summing	sum	VERB
cana-327	101	2	(	(	PUNCT
cana-327	101	3	11	11	NUM
cana-327	101	4	)	)	PUNCT
cana-327	101	5	from	from	ADP
cana-327	101	6	℘2	℘2	NOUN
cana-327	101	7	to	to	ADP
cana-327	101	8	℘	℘	PROPN
cana-327	101	9	−	−	PROPN
cana-327	101	10	1	1	NUM
cana-327	101	11	for	for	ADP
cana-327	101	12	℘	℘	PROPN
cana-327	101	13	>	>	SYM
cana-327	101	14	𝑖	𝑖	SYM
cana-327	101	15	,	,	PUNCT
cana-327	101	16	we	we	PRON
cana-327	101	17	get	get	VERB
cana-327	101	18	−𝑟(℘2)(δ𝑠(℘2	−𝑟(℘2)(δ𝑠(℘2	PROPN
cana-327	101	19	)	)	PUNCT
cana-327	101	20	)	)	PUNCT
cana-327	102	1	≤	≤	ADV
cana-327	102	2	∑	∑	PUNCT
cana-327	102	3	  	  	SPACE
cana-327	102	4	℘−1	℘−1	PROPN
cana-327	102	5	𝑠=℘2	𝑠=℘2	PROPN
cana-327	102	6	  	  	SPACE
cana-327	102	7	𝑞(𝑖	𝑞(𝑖	PROPN
cana-327	102	8	)	)	PUNCT
cana-327	102	9	𝑝	𝑝	NOUN
cana-327	102	10	𝛾2	𝛾2	VERB
cana-327	102	11	𝛾1(𝜏1	𝛾1(𝜏1	PRON
cana-327	102	12	−1(𝜎1(𝑖	−1(𝜎1(𝑖	PROPN
cana-327	102	13	)	)	PUNCT
cana-327	102	14	)	)	PUNCT
cana-327	102	15	)	)	PUNCT
cana-327	103	1	𝑠	𝑠	X
cana-327	103	2	𝛾2	𝛾2	VERB
cana-327	103	3	𝛾1(𝜏1	𝛾1(𝜏1	NUM
cana-327	103	4	−1(𝜎1(𝑖	−1(𝜎1(𝑖	PROPN
cana-327	103	5	)	)	PUNCT
cana-327	103	6	)	)	PUNCT
cana-327	103	7	)	)	PUNCT
cana-327	103	8	again	again	ADV
cana-327	103	9	summing	sum	VERB
cana-327	103	10	from	from	ADP
cana-327	103	11	𝜏1	𝜏1	NOUN
cana-327	103	12	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	103	13	)	)	PUNCT
cana-327	103	14	)	)	PUNCT
cana-327	103	15	to	to	ADP
cana-327	103	16	℘	℘	VERB
cana-327	103	17	−	−	PROPN
cana-327	103	18	1	1	NUM
cana-327	103	19	and	and	CCONJ
cana-327	103	20	using	use	VERB
cana-327	103	21	𝑠(℘	𝑠(℘	NOUN
cana-327	103	22	)	)	PUNCT
cana-327	103	23	is	be	AUX
cana-327	103	24	increasing	increase	VERB
cana-327	103	25	and	and	CCONJ
cana-327	103	26	we	we	PRON
cana-327	103	27	have	have	VERB
cana-327	103	28	𝑠(𝜏1	𝑠(𝜏1	ADJ
cana-327	103	29	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	103	30	)	)	PUNCT
cana-327	103	31	)	)	PUNCT
cana-327	103	32	)	)	PUNCT
cana-327	104	1	−	−	PROPN
cana-327	104	2	𝑠(℘	𝑠(℘	NOUN
cana-327	104	3	)	)	PUNCT
cana-327	104	4	≤	≤	NOUN
cana-327	104	5	𝑠	𝑠	AUX
cana-327	104	6	𝛾2	𝛾2	VERB
cana-327	104	7	𝛾1(𝜏1	𝛾1(𝜏1	VERB
cana-327	104	8	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	104	9	)	)	PUNCT
cana-327	104	10	)	)	PUNCT
cana-327	104	11	)	)	PUNCT
cana-327	105	1	∑	∑	ADP
cana-327	105	2	  	  	SPACE
cana-327	105	3	℘−1	℘−1	PROPN
cana-327	105	4	𝑗=𝜏1	𝑗=𝜏1	PUNCT
cana-327	105	5	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	105	6	)	)	PUNCT
cana-327	105	7	)	)	PUNCT
cana-327	105	8	1	1	NUM
cana-327	105	9	𝑟(𝑗	𝑟(𝑗	PROPN
cana-327	105	10	)	)	PUNCT
cana-327	105	11	∑	∑	ADP
cana-327	105	12	  	  	SPACE
cana-327	105	13	𝑗−1	𝑗−1	PROPN
cana-327	105	14	𝑚=𝑗3	𝑚=𝑗3	VERB
cana-327	105	15	𝑞(𝑚	𝑞(𝑚	PROPN
cana-327	105	16	)	)	PUNCT
cana-327	105	17	𝑝	𝑝	NOUN
cana-327	105	18	𝛾2	𝛾2	VERB
cana-327	105	19	𝛾1(𝜏1	𝛾1(𝜏1	PRON
cana-327	105	20	−1(𝜎1(𝑚	−1(𝜎1(𝑚	NOUN
cana-327	105	21	)	)	PUNCT
cana-327	105	22	)	)	PUNCT
cana-327	105	23	)	)	PUNCT
cana-327	105	24	or	or	CCONJ
cana-327	105	25	𝑠(𝜏1	𝑠(𝜏1	ADJ
cana-327	105	26	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	105	27	)	)	PUNCT
cana-327	105	28	)	)	PUNCT
cana-327	105	29	)	)	PUNCT
cana-327	106	1	𝑠	𝑠	X
cana-327	106	2	𝛾2	𝛾2	VERB
cana-327	106	3	𝛾1(𝜏1	𝛾1(𝜏1	ADV
cana-327	106	4	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	106	5	)	)	PUNCT
cana-327	106	6	)	)	PUNCT
cana-327	106	7	)	)	PUNCT
cana-327	107	1	≥	≥	X
cana-327	107	2	∑	∑	PUNCT
cana-327	107	3	  	  	SPACE
cana-327	107	4	℘−1	℘−1	PROPN
cana-327	107	5	𝑗=𝜏1	𝑗=𝜏1	PUNCT
cana-327	107	6	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	107	7	)	)	PUNCT
cana-327	107	8	)	)	PUNCT
cana-327	107	9	1	1	NUM
cana-327	107	10	𝑟(𝑗	𝑟(𝑗	PROPN
cana-327	107	11	)	)	PUNCT
cana-327	107	12	∑	∑	ADP
cana-327	107	13	  	  	SPACE
cana-327	107	14	𝑗−1	𝑗−1	PROPN
cana-327	107	15	𝑚=𝑗3	𝑚=𝑗3	VERB
cana-327	107	16	𝑞(𝑚	𝑞(𝑚	PROPN
cana-327	107	17	)	)	PUNCT
cana-327	107	18	𝑝	𝑝	NOUN
cana-327	107	19	𝛾2	𝛾2	VERB
cana-327	107	20	𝛾1(𝜏1	𝛾1(𝜏1	PRON
cana-327	107	21	−1(𝜎1(𝑚	−1(𝜎1(𝑚	NOUN
cana-327	107	22	)	)	PUNCT
cana-327	107	23	)	)	PUNCT
cana-327	107	24	)	)	PUNCT
cana-327	107	25	.	.	PUNCT
cana-327	108	1	(	(	PUNCT
cana-327	108	2	13	13	NUM
cana-327	108	3	)	)	PUNCT
cana-327	108	4	since	since	SCONJ
cana-327	108	5	𝑠(𝜏1	𝑠(𝜏1	ADJ
cana-327	108	6	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	108	7	)	)	PUNCT
cana-327	108	8	)	)	PUNCT
cana-327	108	9	)	)	PUNCT
cana-327	109	1	𝑠	𝑠	X
cana-327	109	2	𝛾2	𝛾2	VERB
cana-327	109	3	𝛾1(𝜏1	𝛾1(𝜏1	ADV
cana-327	109	4	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	109	5	)	)	PUNCT
cana-327	109	6	)	)	PUNCT
cana-327	109	7	)	)	PUNCT
cana-327	110	1	=	=	PUNCT
cana-327	110	2	|𝑠(𝜏1	|𝑠(𝜏1	ADJ
cana-327	110	3	−1(𝜎1(℘)))|	−1(𝜎1(℘)))|	NOUN
cana-327	110	4	1−	1−	NUM
cana-327	110	5	𝛾2	𝛾2	NOUN
cana-327	110	6	𝛾1	𝛾1	NOUN
cana-327	110	7	and	and	CCONJ
cana-327	110	8	1	1	NUM
cana-327	110	9	−	−	NOUN
cana-327	110	10	𝛾2	𝛾2	NOUN
cana-327	110	11	𝛾1	𝛾1	PROPN
cana-327	110	12	>	>	X
cana-327	110	13	0	0	NUM
cana-327	110	14	,	,	PUNCT
cana-327	110	15	we	we	PRON
cana-327	110	16	get	get	VERB
cana-327	110	17	lim	lim	PROPN
cana-327	110	18	℘→∞	℘→∞	PROPN
cana-327	110	19	 	 	SPACE
cana-327	110	20	sup	sup	PROPN
cana-327	110	21	∑	∑	ADP
cana-327	110	22	  	  	SPACE
cana-327	110	23	℘−1	℘−1	PROPN
cana-327	110	24	𝑗=𝜏1	𝑗=𝜏1	PUNCT
cana-327	111	1	−1(𝜎1(℘	−1(𝜎1(℘	NOUN
cana-327	111	2	)	)	PUNCT
cana-327	111	3	)	)	PUNCT
cana-327	112	1	1	1	NUM
cana-327	112	2	𝑟(𝑗	𝑟(𝑗	PROPN
cana-327	112	3	)	)	PUNCT
cana-327	112	4	∑	∑	ADP
cana-327	112	5	  	  	SPACE
cana-327	112	6	𝑗−1	𝑗−1	PROPN
cana-327	112	7	𝑚=𝑗3	𝑚=𝑗3	VERB
cana-327	112	8	𝑞(𝑚	𝑞(𝑚	PROPN
cana-327	112	9	)	)	PUNCT
cana-327	112	10	𝑝	𝑝	NOUN
cana-327	112	11	𝛾2	𝛾2	VERB
cana-327	112	12	𝛾1(𝜏1	𝛾1(𝜏1	PRON
cana-327	112	13	−1(𝜎1(𝑚	−1(𝜎1(𝑚	NOUN
cana-327	112	14	)	)	PUNCT
cana-327	112	15	)	)	PUNCT
cana-327	112	16	)	)	PUNCT
cana-327	113	1	≤	≤	NUM
cana-327	113	2	0	0	NUM
cana-327	113	3	which	which	PRON
cana-327	113	4	contradicts	contradict	VERB
cana-327	113	5	(	(	PUNCT
cana-327	113	6	7	7	NUM
cana-327	113	7	)	)	PUNCT
cana-327	113	8	.	.	PUNCT
cana-327	114	1	theorem	theorem	ADJ
cana-327	114	2	2.4	2.4	NUM
cana-327	114	3	.	.	PUNCT
cana-327	115	1	assume	assume	VERB
cana-327	115	2	𝛾2	𝛾2	NOUN
cana-327	115	3	=	=	SYM
cana-327	115	4	1	1	NUM
cana-327	115	5	and	and	CCONJ
cana-327	115	6	condition	condition	NOUN
cana-327	115	7	(	(	PUNCT
cana-327	115	8	2	2	NUM
cana-327	115	9	)	)	PUNCT
cana-327	115	10	holds	hold	VERB
cana-327	115	11	.	.	PUNCT
cana-327	116	1	if	if	SCONJ
cana-327	116	2	lim	lim	PROPN
cana-327	116	3	𝑡→∞	𝑡→∞	NUM
cana-327	116	4	 	 	SPACE
cana-327	116	5	inf	inf	PROPN
cana-327	116	6	∑	∑	ADP
cana-327	116	7	  	  	SPACE
cana-327	116	8	℘−1	℘−1	PROPN
cana-327	116	9	𝜎1(℘	𝜎1(℘	PROPN
cana-327	116	10	)	)	PUNCT
cana-327	116	11	𝑞(𝑠)𝑌(𝜎1(𝑠	𝑞(𝑠)𝑌(𝜎1(𝑠	PROPN
cana-327	116	12	)	)	PUNCT
cana-327	116	13	)	)	PUNCT
cana-327	116	14	>	>	X
cana-327	116	15	1	1	NUM
cana-327	116	16	𝑒	𝑒	NOUN
cana-327	116	17	,	,	PUNCT
cana-327	116	18	(	(	PUNCT
cana-327	116	19	14	14	NUM
cana-327	116	20	)	)	PUNCT
cana-327	116	21	then	then	ADV
cana-327	116	22	every	every	DET
cana-327	116	23	solution	solution	NOUN
cana-327	116	24	of	of	ADP
cana-327	116	25	(	(	PUNCT
cana-327	116	26	1	1	NUM
cana-327	116	27	)	)	PUNCT
cana-327	116	28	is	be	AUX
cana-327	116	29	either	either	CCONJ
cana-327	116	30	oscillatory	oscillatory	ADJ
cana-327	116	31	or	or	CCONJ
cana-327	116	32	tends	tend	VERB
cana-327	116	33	to	to	ADP
cana-327	116	34	zero	zero	NUM
cana-327	116	35	as	as	ADP
cana-327	116	36	℘	℘	PROPN
cana-327	116	37	→	→	SYM
cana-327	116	38	∞.	∞.	PROPN
cana-327	116	39	proof	proof	NOUN
cana-327	116	40	.	.	PUNCT
cana-327	117	1	we	we	PRON
cana-327	117	2	assume	assume	VERB
cana-327	117	3	that	that	SCONJ
cana-327	117	4	a	a	DET
cana-327	117	5	non	non	ADJ
cana-327	117	6	-	-	ADJ
cana-327	117	7	oscillatory	oscillatory	ADJ
cana-327	117	8	solution	solution	NOUN
cana-327	117	9	z	z	X
cana-327	117	10	of	of	ADP
cana-327	117	11	(	(	PUNCT
cana-327	117	12	1	1	NUM
cana-327	117	13	)	)	PUNCT
cana-327	117	14	,	,	PUNCT
cana-327	117	15	𝑧(℘	𝑧(℘	NUM
cana-327	117	16	)	)	PUNCT
cana-327	117	17	>	>	X
cana-327	117	18	0	0	NUM
cana-327	117	19	,	,	PUNCT
cana-327	117	20	𝑧	𝑧	X
cana-327	117	21	(	(	PUNCT
cana-327	117	22	𝜎1(℘	𝜎1(℘	NOUN
cana-327	117	23	)	)	PUNCT
cana-327	117	24	)	)	PUNCT
cana-327	117	25	>	>	X
cana-327	117	26	0	0	NUM
cana-327	117	27	,	,	PUNCT
cana-327	117	28	𝑧(𝜏1(℘	𝑧(𝜏1(℘	NOUN
cana-327	117	29	)	)	PUNCT
cana-327	117	30	)	)	PUNCT
cana-327	117	31	>	>	X
cana-327	117	32	0	0	NUM
cana-327	117	33	,	,	PUNCT
cana-327	117	34	℘	℘	X
cana-327	117	35	≥	≥	NUM
cana-327	117	36	℘1	℘1	VERB
cana-327	117	37	≥	≥	NOUN
cana-327	117	38	℘0	℘0	NOUN
cana-327	117	39	and	and	CCONJ
cana-327	117	40	that	that	SCONJ
cana-327	117	41	for	for	ADP
cana-327	117	42	s	s	PRON
cana-327	117	43	one	one	NUM
cana-327	117	44	of	of	ADP
cana-327	117	45	the	the	DET
cana-327	117	46	case	case	NOUN
cana-327	117	47	(	(	PUNCT
cana-327	117	48	i	i	NOUN
cana-327	117	49	)	)	PUNCT
cana-327	117	50	and	and	CCONJ
cana-327	117	51	case	case	NOUN
cana-327	117	52	(	(	PUNCT
cana-327	117	53	ii	ii	NOUN
cana-327	117	54	)	)	PUNCT
cana-327	117	55	holds	hold	VERB
cana-327	117	56	.	.	PUNCT
cana-327	118	1	assume	assume	VERB
cana-327	118	2	that	that	SCONJ
cana-327	118	3	𝑠(℘	𝑠(℘	NOUN
cana-327	118	4	)	)	PUNCT
cana-327	118	5	meets	meet	VERB
cana-327	118	6	case	case	NOUN
cana-327	118	7	(	(	PUNCT
cana-327	118	8	i	i	NOUN
cana-327	118	9	)	)	PUNCT
cana-327	118	10	of	of	ADP
cana-327	118	11	lemma	lemma	PROPN
cana-327	118	12	2.1	2.1	NUM
cana-327	118	13	and	and	CCONJ
cana-327	118	14	from	from	ADP
cana-327	118	15	the	the	DET
cana-327	118	16	proof	proof	NOUN
cana-327	118	17	of	of	ADP
cana-327	118	18	case	case	NOUN
cana-327	118	19	(	(	PUNCT
cana-327	118	20	i	i	NOUN
cana-327	118	21	)	)	PUNCT
cana-327	118	22	of	of	ADP
cana-327	118	23	theorem	theorem	NOUN
cana-327	118	24	2.1	2.1	NUM
cana-327	118	25	,	,	PUNCT
cana-327	118	26	we	we	PRON
cana-327	118	27	have	have	VERB
cana-327	118	28	for	for	ADP
cana-327	118	29	𝛾2	𝛾2	NOUN
cana-327	118	30	=	=	SYM
cana-327	118	31	1	1	NUM
cana-327	118	32	that	that	PRON
cana-327	118	33	𝑦(℘	𝑦(℘	X
cana-327	118	34	)	)	PUNCT
cana-327	118	35	=	=	SYM
cana-327	119	1	𝑟(℘)δ𝑠(℘	𝑟(℘)δ𝑠(℘	NOUN
cana-327	119	2	)	)	PUNCT
cana-327	119	3	is	be	AUX
cana-327	119	4	a	a	DET
cana-327	119	5	positive	positive	ADJ
cana-327	119	6	solution	solution	NOUN
cana-327	119	7	of	of	ADP
cana-327	119	8	the	the	DET
cana-327	119	9	inequality	inequality	NOUN
cana-327	119	10	δ𝑦(℘	δ𝑦(℘	NUM
cana-327	119	11	)	)	PUNCT
cana-327	120	1	+	+	NUM
cana-327	120	2	𝑞(℘)𝑌(𝜎1(℘))𝑦(𝜎1(℘	𝑞(℘)𝑌(𝜎1(℘))𝑦(𝜎1(℘	NOUN
cana-327	120	3	)	)	PUNCT
cana-327	120	4	)	)	PUNCT
cana-327	121	1	≤	≤	ADV
cana-327	121	2	0	0	NUM
cana-327	121	3	.	.	PUNCT
cana-327	122	1	(	(	PUNCT
cana-327	122	2	15	15	NUM
cana-327	122	3	)	)	PUNCT
cana-327	122	4	communications	communication	NOUN
cana-327	122	5	on	on	ADP
cana-327	122	6	applied	apply	VERB
cana-327	122	7	nonlinear	nonlinear	ADJ
cana-327	122	8	analysis	analysis	NOUN
cana-327	122	9	issn	issn	NOUN
cana-327	122	10	:	:	PUNCT
cana-327	122	11	1074	1074	NUM
cana-327	122	12	-	-	PUNCT
cana-327	122	13	133x	133x	NUM
cana-327	122	14	vol	vol	NOUN
cana-327	122	15	31	31	NUM
cana-327	122	16	no	no	NOUN
cana-327	122	17	.	.	NOUN
cana-327	122	18	1	1	NUM
cana-327	122	19	(	(	PUNCT
cana-327	122	20	2024	2024	NUM
cana-327	122	21	)	)	PUNCT
cana-327	122	22	87	87	NUM
cana-327	122	23	https://internationalpubls.com	https://internationalpubls.com	X
cana-327	122	24	on	on	ADP
cana-327	122	25	the	the	DET
cana-327	122	26	other	other	ADJ
cana-327	122	27	hand	hand	NOUN
cana-327	122	28	,	,	PUNCT
cana-327	122	29	from	from	ADP
cana-327	122	30	[	[	X
cana-327	122	31	6	6	NUM
cana-327	122	32	]	]	PUNCT
cana-327	122	33	,	,	PUNCT
cana-327	122	34	we	we	PRON
cana-327	122	35	can	can	AUX
cana-327	122	36	notice	notice	VERB
cana-327	122	37	that	that	SCONJ
cana-327	122	38	equation	equation	NOUN
cana-327	122	39	(	(	PUNCT
cana-327	122	40	14	14	NUM
cana-327	122	41	)	)	PUNCT
cana-327	122	42	guarantees	guarantee	VERB
cana-327	122	43	that	that	SCONJ
cana-327	122	44	(	(	PUNCT
cana-327	122	45	15	15	NUM
cana-327	122	46	)	)	PUNCT
cana-327	122	47	has	have	VERB
cana-327	122	48	no	no	DET
cana-327	122	49	positive	positive	ADJ
cana-327	122	50	solution	solution	NOUN
cana-327	122	51	,	,	PUNCT
cana-327	122	52	which	which	PRON
cana-327	122	53	implies	imply	VERB
cana-327	122	54	contradiction	contradiction	NOUN
cana-327	122	55	.	.	PUNCT
cana-327	123	1	let	let	VERB
cana-327	123	2	𝑠(℘	𝑠(℘	NOUN
cana-327	123	3	)	)	PUNCT
cana-327	123	4	meets	meet	VERB
cana-327	123	5	case	case	NOUN
cana-327	123	6	(	(	PUNCT
cana-327	123	7	ii	ii	NOUN
cana-327	123	8	)	)	PUNCT
cana-327	123	9	of	of	ADP
cana-327	123	10	lemma	lemma	PROPN
cana-327	123	11	2.1	2.1	NUM
cana-327	123	12	.	.	PUNCT
cana-327	124	1	from	from	ADP
cana-327	124	2	this	this	DET
cana-327	124	3	𝑠(℘	𝑠(℘	NOUN
cana-327	124	4	)	)	PUNCT
cana-327	124	5	<	<	X
cana-327	124	6	0	0	PROPN
cana-327	124	7	and	and	CCONJ
cana-327	124	8	δ𝑠(℘	δ𝑠(℘	PROPN
cana-327	124	9	)	)	PUNCT
cana-327	124	10	>	>	SYM
cana-327	124	11	0	0	PUNCT
cana-327	125	1	and	and	CCONJ
cana-327	125	2	also	also	ADV
cana-327	125	3	lim℘→∞	lim℘→∞	ADJ
cana-327	125	4	 	 	SPACE
cana-327	125	5	𝑠(℘	𝑠(℘	NOUN
cana-327	125	6	)	)	PUNCT
cana-327	125	7	=	=	PUNCT
cana-327	126	1	𝑐1	𝑐1	NOUN
cana-327	126	2	≤	≤	NUM
cana-327	126	3	0	0	NUM
cana-327	126	4	,	,	PUNCT
cana-327	126	5	where	where	SCONJ
cana-327	126	6	𝑐1	𝑐1	NOUN
cana-327	126	7	is	be	AUX
cana-327	126	8	a	a	DET
cana-327	126	9	constant	constant	ADJ
cana-327	126	10	.	.	PUNCT
cana-327	127	1	i.e.	i.e.	X
cana-327	127	2	s	s	X
cana-327	127	3	is	be	AUX
cana-327	127	4	bounded	bound	VERB
cana-327	127	5	and	and	CCONJ
cana-327	127	6	as	as	ADP
cana-327	127	7	in	in	ADP
cana-327	127	8	the	the	DET
cana-327	127	9	proof	proof	NOUN
cana-327	127	10	of	of	ADP
cana-327	127	11	lemma	lemma	PROPN
cana-327	127	12	2.1	2.1	NUM
cana-327	127	13	,	,	PUNCT
cana-327	127	14	we	we	PRON
cana-327	127	15	can	can	AUX
cana-327	127	16	say	say	VERB
cana-327	127	17	that	that	SCONJ
cana-327	127	18	z	z	PROPN
cana-327	127	19	is	be	AUX
cana-327	127	20	also	also	ADV
cana-327	127	21	bounded	bound	VERB
cana-327	127	22	.	.	PUNCT
cana-327	128	1	therefore	therefore	ADV
cana-327	128	2	,	,	PUNCT
cana-327	128	3	lim℘→∞	lim℘→∞	PROPN
cana-327	128	4	 	 	SPACE
cana-327	128	5	𝑧(℘	𝑧(℘	X
cana-327	128	6	)	)	PUNCT
cana-327	128	7	=	=	SYM
cana-327	128	8	𝑚1	𝑚1	NOUN
cana-327	128	9	,	,	PUNCT
cana-327	128	10	0	0	NUM
cana-327	128	11	≤	≤	NUM
cana-327	128	12	𝑚1	𝑚1	NOUN
cana-327	128	13	<	<	X
cana-327	128	14	∞.	∞.	PROPN
cana-327	129	1	we	we	PRON
cana-327	129	2	claim	claim	VERB
cana-327	129	3	that	that	SCONJ
cana-327	129	4	𝑚1	𝑚1	NOUN
cana-327	129	5	=	=	SYM
cana-327	129	6	0	0	PROPN
cana-327	129	7	.	.	PUNCT
cana-327	129	8	suppose	suppose	VERB
cana-327	129	9	𝑚1	𝑚1	X
cana-327	129	10	>	>	X
cana-327	129	11	0	0	PROPN
cana-327	129	12	,	,	PUNCT
cana-327	129	13	there	there	PRON
cana-327	129	14	is	be	VERB
cana-327	129	15	a	a	DET
cana-327	129	16	sequence	sequence	NOUN
cana-327	129	17	{	{	PUNCT
cana-327	129	18	℘𝑛	℘𝑛	NOUN
cana-327	129	19	}	}	PUNCT
cana-327	129	20	such	such	ADJ
cana-327	129	21	that	that	DET
cana-327	129	22	lim𝑛→∞	lim𝑛→∞	PROPN
cana-327	129	23	 	 	SPACE
cana-327	129	24	℘𝑛	℘𝑛	NOUN
cana-327	129	25	=	=	NOUN
cana-327	129	26	∞	∞	PROPN
cana-327	129	27	and	and	CCONJ
cana-327	129	28	lim𝑛→∞	lim𝑛→∞	NOUN
cana-327	129	29	 	 	SPACE
cana-327	129	30	𝑧(℘𝑛	𝑧(℘𝑛	NOUN
cana-327	129	31	)	)	PUNCT
cana-327	129	32	=	=	SYM
cana-327	129	33	𝑚1	𝑚1	NOUN
cana-327	129	34	.	.	PUNCT
cana-327	130	1	thus	thus	ADV
cana-327	130	2	𝑠(℘𝑛	𝑠(℘𝑛	NOUN
cana-327	130	3	)	)	PUNCT
cana-327	130	4	=	=	SYM
cana-327	130	5	𝑧(℘𝑛	𝑧(℘𝑛	ADJ
cana-327	130	6	)	)	PUNCT
cana-327	130	7	−	−	PROPN
cana-327	130	8	𝑝(℘𝑛)𝑧𝛾1(𝜏1(℘𝑛	𝑝(℘𝑛)𝑧𝛾1(𝜏1(℘𝑛	NOUN
cana-327	130	9	)	)	PUNCT
cana-327	130	10	)	)	PUNCT
cana-327	130	11	,	,	PUNCT
cana-327	130	12	𝑧(𝜏1(℘𝑛	𝑧(𝜏1(℘𝑛	NOUN
cana-327	130	13	)	)	PUNCT
cana-327	130	14	)	)	PUNCT
cana-327	131	1	=	=	PUNCT
cana-327	131	2	(	(	PUNCT
cana-327	131	3	𝑧(℘𝑛	𝑧(℘𝑛	ADJ
cana-327	131	4	)	)	PUNCT
cana-327	131	5	−	−	NOUN
cana-327	131	6	𝑠(℘𝑛	𝑠(℘𝑛	NOUN
cana-327	131	7	)	)	PUNCT
cana-327	131	8	)	)	PUNCT
cana-327	131	9	1	1	NUM
cana-327	131	10	𝛾1	𝛾1	PROPN
cana-327	131	11	𝑝	𝑝	ADP
cana-327	131	12	1	1	NUM
cana-327	131	13	𝛾1(℘𝑛	𝛾1(℘𝑛	NOUN
cana-327	131	14	)	)	PUNCT
cana-327	131	15	.	.	PUNCT
cana-327	132	1	taking	take	VERB
cana-327	132	2	𝑛	𝑛	PRON
cana-327	132	3	→	→	SYM
cana-327	132	4	∞	∞	PROPN
cana-327	132	5	,	,	PUNCT
cana-327	132	6	we	we	PRON
cana-327	132	7	get	get	VERB
cana-327	132	8	𝑚1	𝑚1	PROPN
cana-327	132	9	≥	≥	PROPN
cana-327	132	10	lim	lim	PROPN
cana-327	132	11	𝑛→∞	𝑛→∞	NUM
cana-327	132	12	 	 	SPACE
cana-327	132	13	𝑧(𝜏1(℘𝑛	𝑧(𝜏1(℘𝑛	NOUN
cana-327	132	14	)	)	PUNCT
cana-327	132	15	)	)	PUNCT
cana-327	133	1	≥	≥	PROPN
cana-327	133	2	(	(	PUNCT
cana-327	133	3	𝑚1	𝑚1	PROPN
cana-327	133	4	𝑝	𝑝	PROPN
cana-327	133	5	)	)	PUNCT
cana-327	133	6	1	1	NUM
cana-327	133	7	𝛾1	𝛾1	NOUN
cana-327	133	8	we	we	PRON
cana-327	133	9	conclude	conclude	VERB
cana-327	133	10	that	that	DET
cana-327	133	11	𝑚1	𝑚1	NOUN
cana-327	133	12	=	=	SYM
cana-327	133	13	0	0	PROPN
cana-327	133	14	,	,	PUNCT
cana-327	133	15	because	because	SCONJ
cana-327	133	16	of	of	ADP
cana-327	133	17	𝑝	𝑝	PROPN
cana-327	133	18	∈	∈	PROPN
cana-327	133	19	(	(	PUNCT
cana-327	133	20	0,1	0,1	NOUN
cana-327	133	21	)	)	PUNCT
cana-327	133	22	,	,	PUNCT
cana-327	133	23	that	that	PRON
cana-327	133	24	is	be	AUX
cana-327	133	25	lim𝑛→∞	lim𝑛→∞	PROPN
cana-327	133	26	 	 	SPACE
cana-327	133	27	𝑧(℘	𝑧(℘	NUM
cana-327	133	28	)	)	PUNCT
cana-327	133	29	=	=	PUNCT
cana-327	134	1	0	0	NUM
cana-327	134	2	.	.	NOUN
cana-327	135	1	3	3	X
cana-327	135	2	.	.	X
cana-327	135	3	examples	example	NOUN
cana-327	135	4	example	example	NOUN
cana-327	135	5	3.1	3.1	NUM
cana-327	135	6	.	.	PUNCT
cana-327	136	1	examine	examine	VERB
cana-327	136	2	second	second	ADJ
cana-327	136	3	order	order	NOUN
cana-327	136	4	neutral	neutral	ADJ
cana-327	136	5	delay	delay	NOUN
cana-327	136	6	difference	difference	NOUN
cana-327	136	7	equation	equation	NOUN
cana-327	136	8	δ	δ	PROPN
cana-327	136	9	(	(	PUNCT
cana-327	136	10	℘δ	℘δ	ADJ
cana-327	136	11	(	(	PUNCT
cana-327	136	12	𝑧(℘	𝑧(℘	NOUN
cana-327	136	13	)	)	PUNCT
cana-327	137	1	−	−	PROPN
cana-327	137	2	𝑝𝑧	𝑝𝑧	NOUN
cana-327	137	3	1	1	NUM
cana-327	137	4	3	3	NUM
cana-327	137	5	(	(	PUNCT
cana-327	137	6	℘	℘	PROPN
cana-327	137	7	2	2	NUM
cana-327	137	8	)	)	PUNCT
cana-327	137	9	)	)	PUNCT
cana-327	137	10	)	)	PUNCT
cana-327	138	1	+	+	CCONJ
cana-327	138	2	8℘	8℘	NUM
cana-327	138	3	(	(	PUNCT
cana-327	138	4	℘	℘	PROPN
cana-327	138	5	3	3	NUM
cana-327	138	6	)	)	PUNCT
cana-327	138	7	=	=	SYM
cana-327	138	8	0	0	NUM
cana-327	138	9	,	,	PUNCT
cana-327	138	10	℘	℘	X
cana-327	138	11	≥	≥	NOUN
cana-327	138	12	1	1	NUM
cana-327	138	13	,	,	PUNCT
cana-327	138	14	(	(	PUNCT
cana-327	138	15	16	16	NUM
cana-327	138	16	)	)	PUNCT
cana-327	138	17	where	where	SCONJ
cana-327	138	18	𝑝	𝑝	X
cana-327	138	19	∈	∈	PROPN
cana-327	138	20	(	(	PUNCT
cana-327	138	21	0,1	0,1	NOUN
cana-327	138	22	)	)	PUNCT
cana-327	138	23	which	which	PRON
cana-327	138	24	is	be	AUX
cana-327	138	25	a	a	DET
cana-327	138	26	constant	constant	ADJ
cana-327	138	27	.	.	PUNCT
cana-327	139	1	here	here	ADV
cana-327	139	2	𝑟(℘	𝑟(℘	NUM
cana-327	139	3	)	)	PUNCT
cana-327	139	4	=	=	SYM
cana-327	139	5	℘	℘	PROPN
cana-327	139	6	,	,	PUNCT
cana-327	139	7	𝑝(℘	𝑝(℘	NUM
cana-327	139	8	)	)	PUNCT
cana-327	139	9	=	=	SYM
cana-327	139	10	𝑝	𝑝	PROPN
cana-327	139	11	,	,	PUNCT
cana-327	139	12	𝑞(℘	𝑞(℘	PROPN
cana-327	139	13	)	)	PUNCT
cana-327	139	14	=	=	SYM
cana-327	139	15	8℘	8℘	NUM
cana-327	139	16	,	,	PUNCT
cana-327	139	17	𝜏1(℘	𝜏1(℘	NOUN
cana-327	139	18	)	)	PUNCT
cana-327	139	19	=	=	SYM
cana-327	139	20	℘	℘	PROPN
cana-327	139	21	2	2	NUM
cana-327	139	22	,	,	PUNCT
cana-327	139	23	𝜎1(℘	𝜎1(℘	NOUN
cana-327	139	24	)	)	PUNCT
cana-327	139	25	=	=	PUNCT
cana-327	139	26	℘	℘	PROPN
cana-327	139	27	3	3	NUM
cana-327	139	28	for	for	ADP
cana-327	139	29	℘	℘	NUM
cana-327	139	30	≥	≥	NOUN
cana-327	139	31	℘1	℘1	VERB
cana-327	139	32	=	=	SYM
cana-327	139	33	1	1	NUM
cana-327	139	34	,	,	PUNCT
cana-327	139	35	𝛾1	𝛾1	PROPN
cana-327	139	36	=	=	SYM
cana-327	139	37	1/3	1/3	NUM
cana-327	139	38	,	,	PUNCT
cana-327	139	39	𝛾2	𝛾2	VERB
cana-327	139	40	=	=	NOUN
cana-327	139	41	1/5	1/5	NUM
cana-327	139	42	and	and	CCONJ
cana-327	139	43	𝑌(℘	𝑌(℘	NOUN
cana-327	139	44	)	)	PUNCT
cana-327	139	45	=	=	SYM
cana-327	139	46	1−℘	1−℘	PROPN
cana-327	139	47	℘	℘	PROPN
cana-327	139	48	.	.	PUNCT
cana-327	140	1	clearly	clearly	ADV
cana-327	140	2	,	,	PUNCT
cana-327	140	3	these	these	DET
cana-327	140	4	calculations	calculation	NOUN
cana-327	140	5	shows	show	VERB
cana-327	140	6	that	that	SCONJ
cana-327	140	7	above	above	ADP
cana-327	140	8	conditions	condition	NOUN
cana-327	140	9	(	(	PUNCT
cana-327	140	10	6	6	NUM
cana-327	140	11	)	)	PUNCT
cana-327	140	12	and	and	CCONJ
cana-327	140	13	(	(	PUNCT
cana-327	140	14	7	7	X
cana-327	140	15	)	)	PUNCT
cana-327	140	16	are	be	AUX
cana-327	140	17	fullfilled	fullfille	VERB
cana-327	140	18	.	.	PUNCT
cana-327	141	1	so	so	ADV
cana-327	141	2	that	that	SCONJ
cana-327	141	3	by	by	ADP
cana-327	141	4	theorem	theorem	NOUN
cana-327	141	5	2.3	2.3	NUM
cana-327	141	6	,	,	PUNCT
cana-327	141	7	every	every	DET
cana-327	141	8	solution	solution	NOUN
cana-327	141	9	of	of	ADP
cana-327	141	10	(	(	PUNCT
cana-327	141	11	16	16	NUM
cana-327	141	12	)	)	PUNCT
cana-327	141	13	is	be	AUX
cana-327	141	14	oscillatory	oscillatory	ADJ
cana-327	141	15	.	.	PUNCT
cana-327	141	16	example	example	NOUN
cana-327	141	17	3.2	3.2	NUM
cana-327	141	18	.	.	PUNCT
cana-327	142	1	examine	examine	VERB
cana-327	142	2	second	second	ADJ
cana-327	142	3	order	order	NOUN
cana-327	142	4	neutral	neutral	ADJ
cana-327	142	5	delay	delay	NOUN
cana-327	142	6	difference	difference	NOUN
cana-327	142	7	equation	equation	NOUN
cana-327	142	8	δ	δ	PROPN
cana-327	142	9	(	(	PUNCT
cana-327	142	10	1	1	NUM
cana-327	142	11	℘	℘	PROPN
cana-327	142	12	δ	δ	PROPN
cana-327	142	13	(	(	PUNCT
cana-327	142	14	𝑧(℘	𝑧(℘	NOUN
cana-327	142	15	)	)	PUNCT
cana-327	142	16	−	−	PROPN
cana-327	142	17	𝑝𝑧	𝑝𝑧	NOUN
cana-327	142	18	1	1	NUM
cana-327	142	19	3	3	NUM
cana-327	142	20	(	(	PUNCT
cana-327	142	21	℘	℘	PROPN
cana-327	142	22	2	2	NUM
cana-327	142	23	)	)	PUNCT
cana-327	142	24	)	)	PUNCT
cana-327	142	25	)	)	PUNCT
cana-327	143	1	+	+	CCONJ
cana-327	143	2	℘𝑧	℘𝑧	NOUN
cana-327	143	3	(	(	PUNCT
cana-327	143	4	℘	℘	PROPN
cana-327	143	5	3	3	NUM
cana-327	143	6	)	)	PUNCT
cana-327	143	7	=	=	SYM
cana-327	143	8	0	0	NUM
cana-327	143	9	,	,	PUNCT
cana-327	143	10	℘	℘	X
cana-327	143	11	≥	≥	NOUN
cana-327	143	12	1	1	NUM
cana-327	143	13	,	,	PUNCT
cana-327	143	14	(	(	PUNCT
cana-327	143	15	17	17	NUM
cana-327	143	16	)	)	PUNCT
cana-327	143	17	where	where	SCONJ
cana-327	143	18	𝑝	𝑝	X
cana-327	143	19	∈	∈	PROPN
cana-327	143	20	(	(	PUNCT
cana-327	143	21	0,1	0,1	NOUN
cana-327	143	22	)	)	PUNCT
cana-327	143	23	which	which	PRON
cana-327	143	24	is	be	AUX
cana-327	143	25	a	a	DET
cana-327	143	26	constant	constant	ADJ
cana-327	143	27	.	.	PUNCT
cana-327	144	1	here	here	ADV
cana-327	144	2	𝑟(℘	𝑟(℘	NUM
cana-327	144	3	)	)	PUNCT
cana-327	144	4	=	=	SYM
cana-327	144	5	1	1	NUM
cana-327	144	6	℘	℘	PROPN
cana-327	144	7	,	,	PUNCT
cana-327	144	8	𝑝(℘	𝑝(℘	NUM
cana-327	144	9	)	)	PUNCT
cana-327	144	10	=	=	SYM
cana-327	144	11	𝑝	𝑝	PROPN
cana-327	144	12	,	,	PUNCT
cana-327	144	13	𝑞(℘	𝑞(℘	NOUN
cana-327	144	14	)	)	PUNCT
cana-327	144	15	=	=	SYM
cana-327	144	16	℘	℘	PROPN
cana-327	144	17	,	,	PUNCT
cana-327	144	18	𝜏1(℘	𝜏1(℘	NOUN
cana-327	144	19	)	)	PUNCT
cana-327	144	20	=	=	SYM
cana-327	144	21	℘	℘	PROPN
cana-327	144	22	2	2	NUM
cana-327	144	23	,	,	PUNCT
cana-327	144	24	𝜎1(℘	𝜎1(℘	NOUN
cana-327	144	25	)	)	PUNCT
cana-327	144	26	=	=	PUNCT
cana-327	144	27	℘	℘	PROPN
cana-327	144	28	3	3	NUM
cana-327	144	29	for	for	ADP
cana-327	144	30	℘	℘	NUM
cana-327	144	31	≥	≥	NOUN
cana-327	144	32	℘1	℘1	VERB
cana-327	144	33	=	=	SYM
cana-327	144	34	1	1	NUM
cana-327	144	35	,	,	PUNCT
cana-327	144	36	𝛾1	𝛾1	PROPN
cana-327	144	37	=	=	SYM
cana-327	144	38	1/3	1/3	NUM
cana-327	144	39	,	,	PUNCT
cana-327	144	40	𝛾2	𝛾2	NOUN
cana-327	144	41	=	=	SYM
cana-327	144	42	1	1	NUM
cana-327	144	43	and	and	CCONJ
cana-327	144	44	𝑌(℘	𝑌(℘	NOUN
cana-327	144	45	)	)	PUNCT
cana-327	144	46	=	=	SYM
cana-327	145	1	1	1	NUM
cana-327	145	2	−	−	PROPN
cana-327	145	3	℘.	℘.	PROPN
cana-327	145	4	each	each	DET
cana-327	145	5	and	and	CCONJ
cana-327	145	6	every	every	PRON
cana-327	145	7	conditons	conditon	NOUN
cana-327	145	8	of	of	ADP
cana-327	145	9	theorem	theorem	NOUN
cana-327	145	10	2.4	2.4	NUM
cana-327	145	11	with	with	ADP
cana-327	145	12	𝛾2	𝛾2	NOUN
cana-327	145	13	=	=	SYM
cana-327	145	14	1	1	NUM
cana-327	145	15	are	be	AUX
cana-327	145	16	satisfied	satisfied	ADJ
cana-327	145	17	,	,	PUNCT
cana-327	145	18	so	so	ADV
cana-327	145	19	the	the	DET
cana-327	145	20	equation	equation	NOUN
cana-327	145	21	(	(	PUNCT
cana-327	145	22	17	17	NUM
cana-327	145	23	)	)	PUNCT
cana-327	145	24	is	be	AUX
cana-327	145	25	oscillatory	oscillatory	ADJ
cana-327	145	26	.	.	PUNCT
cana-327	146	1	communications	communication	NOUN
cana-327	146	2	on	on	ADP
cana-327	146	3	applied	apply	VERB
cana-327	146	4	nonlinear	nonlinear	ADJ
cana-327	146	5	analysis	analysis	NOUN
cana-327	146	6	issn	issn	NOUN
cana-327	146	7	:	:	PUNCT
cana-327	146	8	1074	1074	NUM
cana-327	146	9	-	-	PUNCT
cana-327	146	10	133x	133x	NUM
cana-327	146	11	vol	vol	NOUN
cana-327	146	12	31	31	NUM
cana-327	146	13	no	no	NOUN
cana-327	146	14	.	.	NOUN
cana-327	146	15	1	1	NUM
cana-327	146	16	(	(	PUNCT
cana-327	146	17	2024	2024	NUM
cana-327	146	18	)	)	PUNCT
cana-327	146	19	88	88	NUM
cana-327	146	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-327	146	21	4	4	NUM
cana-327	146	22	.	.	X
cana-327	146	23	conclusion	conclusion	VERB
cana-327	146	24	the	the	DET
cana-327	146	25	solutions	solution	NOUN
cana-327	146	26	of	of	ADP
cana-327	146	27	nonlinear	nonlinear	ADJ
cana-327	146	28	equations	equation	NOUN
cana-327	146	29	behave	behave	VERB
cana-327	146	30	in	in	ADP
cana-327	146	31	peculiar	peculiar	ADJ
cana-327	146	32	ways	way	NOUN
cana-327	146	33	and	and	CCONJ
cana-327	146	34	these	these	DET
cana-327	146	35	ways	way	NOUN
cana-327	146	36	can	can	AUX
cana-327	146	37	be	be	AUX
cana-327	146	38	developed	develop	VERB
cana-327	146	39	by	by	ADP
cana-327	146	40	means	mean	NOUN
cana-327	146	41	of	of	ADP
cana-327	146	42	different	different	ADJ
cana-327	146	43	strategies	strategy	NOUN
cana-327	146	44	included	include	VERB
cana-327	146	45	in	in	ADP
cana-327	146	46	the	the	DET
cana-327	146	47	method	method	NOUN
cana-327	146	48	.	.	PUNCT
cana-327	147	1	an	an	DET
cana-327	147	2	attempt	attempt	NOUN
cana-327	147	3	was	be	AUX
cana-327	147	4	made	make	VERB
cana-327	147	5	here	here	ADV
cana-327	147	6	to	to	PART
cana-327	147	7	establish	establish	VERB
cana-327	147	8	the	the	DET
cana-327	147	9	sufficient	sufficient	ADJ
cana-327	147	10	conditions	condition	NOUN
cana-327	147	11	with	with	ADP
cana-327	147	12	the	the	DET
cana-327	147	13	fact	fact	NOUN
cana-327	147	14	that	that	SCONJ
cana-327	147	15	the	the	DET
cana-327	147	16	solution	solution	NOUN
cana-327	147	17	space	space	NOUN
cana-327	147	18	of	of	ADP
cana-327	147	19	nonlinear	nonlinear	ADJ
cana-327	147	20	non	non	ADJ
cana-327	147	21	positive	positive	ADJ
cana-327	147	22	neutral	neutral	ADJ
cana-327	147	23	term	term	NOUN
cana-327	147	24	of	of	ADP
cana-327	147	25	difference	difference	NOUN
cana-327	147	26	equation	equation	NOUN
cana-327	147	27	is	be	AUX
cana-327	147	28	reducing	reduce	VERB
cana-327	147	29	to	to	ADP
cana-327	147	30	the	the	DET
cana-327	147	31	solution	solution	NOUN
cana-327	147	32	of	of	ADP
cana-327	147	33	its	its	PRON
cana-327	147	34	limiting	limit	VERB
cana-327	147	35	equation	equation	NOUN
cana-327	147	36	and	and	CCONJ
cana-327	147	37	we	we	PRON
cana-327	147	38	assumed	assume	VERB
cana-327	147	39	with	with	ADP
cana-327	147	40	𝛾2	𝛾2	NOUN
cana-327	147	41	=	=	SYM
cana-327	147	42	1	1	X
cana-327	147	43	.	.	PUNCT
cana-327	147	44	by	by	ADP
cana-327	147	45	these	these	DET
cana-327	147	46	discussions	discussion	NOUN
cana-327	147	47	,	,	PUNCT
cana-327	147	48	(	(	PUNCT
cana-327	147	49	1	1	X
cana-327	147	50	)	)	PUNCT
cana-327	147	51	is	be	AUX
cana-327	147	52	oscillatory	oscillatory	ADJ
cana-327	147	53	or	or	CCONJ
cana-327	147	54	asymptotically	asymptotically	ADV
cana-327	147	55	zero	zero	NUM
cana-327	147	56	as	as	ADP
cana-327	147	57	℘	℘	PROPN
cana-327	147	58	→	→	SYM
cana-327	147	59	∞.	∞.	PROPN
cana-327	147	60	references	reference	NOUN
cana-327	147	61	[	[	X
cana-327	147	62	1	1	NUM
cana-327	147	63	]	]	PUNCT
cana-327	147	64	agarwal	agarwal	PROPN
cana-327	147	65	,	,	PUNCT
cana-327	147	66	r.p	r.p	PROPN
cana-327	147	67	.	.	PROPN
cana-327	147	68	,	,	PUNCT
cana-327	147	69	grace	grace	NOUN
cana-327	147	70	,	,	PUNCT
cana-327	147	71	s.r	s.r	PROPN
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cana-327	147	73	,	,	PUNCT
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cana-327	147	75	.	.	PROPN
cana-327	147	76	,d	,d	PUNCT
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cana-327	147	78	of	of	ADP
cana-327	147	79	higher	high	ADJ
cana-327	147	80	order	order	NOUN
cana-327	147	81	difference	difference	NOUN
cana-327	147	82	equations	equation	NOUN
cana-327	147	83	via	via	ADP
cana-327	147	84	comparision	comparision	NOUN
cana-327	147	85	,	,	PUNCT
cana-327	147	86	glasnik.mat	glasnik.mat	PROPN
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cana-327	147	88	,	,	PUNCT
cana-327	147	89	39	39	NUM
cana-327	147	90	(	(	PUNCT
cana-327	147	91	2004	2004	NUM
cana-327	147	92	)	)	PUNCT
cana-327	147	93	,	,	PUNCT
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cana-327	147	95	-	-	SYM
cana-327	147	96	301	301	NUM
cana-327	147	97	.	.	PUNCT
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cana-327	148	2	2	2	NUM
cana-327	148	3	]	]	PUNCT
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cana-327	148	5	,	,	PUNCT
cana-327	148	6	r.p	r.p	PROPN
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cana-327	148	8	,	,	PUNCT
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cana-327	148	10	,	,	PUNCT
cana-327	148	11	m.	m.	NOUN
cana-327	148	12	,	,	PUNCT
cana-327	148	13	grace	grace	NOUN
cana-327	148	14	,	,	PUNCT
cana-327	148	15	s.r	s.r	PROPN
cana-327	148	16	.	.	PROPN
cana-327	148	17	,	,	PUNCT
cana-327	148	18	o'regan	o'regan	PROPN
cana-327	148	19	,	,	PUNCT
cana-327	148	20	d.	d.	PROPN
cana-327	148	21	,	,	PUNCT
cana-327	148	22	discrete	discrete	VERB
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cana-327	148	24	theory	theory	NOUN
cana-327	148	25	,	,	PUNCT
cana-327	148	26	hindawi	hindawi	ADJ
cana-327	148	27	,	,	PUNCT
cana-327	148	28	new	new	PROPN
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cana-327	148	30	,	,	PUNCT
cana-327	148	31	2005	2005	NUM
cana-327	148	32	.	.	PUNCT
cana-327	149	1	[	[	X
cana-327	149	2	3	3	NUM
cana-327	149	3	]	]	X
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cana-327	149	5	,	,	PUNCT
cana-327	149	6	r.p	r.p	PROPN
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cana-327	149	8	,	,	PUNCT
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cana-327	149	10	equations	equation	NOUN
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cana-327	149	12	inqualities	inqualitie	NOUN
cana-327	149	13	,	,	PUNCT
cana-327	149	14	theory	theory	NOUN
cana-327	149	15	,	,	PUNCT
cana-327	149	16	methods	method	NOUN
cana-327	149	17	and	and	CCONJ
cana-327	149	18	applications	application	NOUN
cana-327	149	19	,	,	PUNCT
cana-327	149	20	second	second	ADJ
cana-327	149	21	edition	edition	NOUN
cana-327	149	22	,	,	PUNCT
cana-327	149	23	revised	revise	VERB
cana-327	149	24	and	and	CCONJ
cana-327	149	25	expanded	expand	VERB
cana-327	149	26	,	,	PUNCT
cana-327	149	27	new	new	PROPN
cana-327	149	28	york	york	PROPN
cana-327	149	29	,	,	PUNCT
cana-327	149	30	marcel	marcel	PROPN
cana-327	149	31	dekker	dekker	PROPN
cana-327	149	32	,	,	PUNCT
cana-327	149	33	2000	2000	NUM
cana-327	149	34	.	.	PUNCT
cana-327	150	1	[	[	X
cana-327	150	2	4	4	X
cana-327	150	3	]	]	SYM
cana-327	150	4	ayyappan	ayyappan	PROPN
cana-327	150	5	,	,	PUNCT
cana-327	150	6	g.	g.	PROPN
cana-327	150	7	,	,	PUNCT
cana-327	150	8	chatzarakis	chatzarakis	PROPN
cana-327	150	9	,	,	PUNCT
cana-327	150	10	g.e	g.e	PROPN
cana-327	150	11	.	.	PROPN
cana-327	150	12	,	,	PUNCT
cana-327	150	13	gopal	gopal	NOUN
cana-327	150	14	,	,	PUNCT
cana-327	150	15	t.	t.	PROPN
cana-327	150	16	,	,	PUNCT
cana-327	150	17	thandapani	thandapani	PROPN
cana-327	150	18	,	,	PUNCT
cana-327	150	19	e.	e.	PROPN
cana-327	150	20	,	,	PUNCT
cana-327	150	21	oscillation	oscillation	NOUN
cana-327	150	22	criteria	criterion	NOUN
cana-327	150	23	of	of	ADP
cana-327	150	24	third	third	ADJ
cana-327	150	25	order	order	NOUN
cana-327	150	26	non	non	ADJ
cana-327	150	27	-	-	ADJ
cana-327	150	28	linear	linear	ADJ
cana-327	150	29	neutral	neutral	ADJ
cana-327	150	30	delay	delay	NOUN
cana-327	150	31	difference	difference	NOUN
cana-327	150	32	equations	equation	NOUN
cana-327	150	33	with	with	ADP
cana-327	150	34	noncanonical	noncanonical	ADJ
cana-327	150	35	operators	operator	NOUN
cana-327	150	36	,	,	PUNCT
cana-327	150	37	appl.anal.discrete	appl.anal.discrete	ADJ
cana-327	150	38	math	math	NOUN
cana-327	150	39	.	.	PUNCT
cana-327	151	1	15	15	NUM
cana-327	151	2	(	(	PUNCT
cana-327	151	3	2021	2021	NUM
cana-327	151	4	)	)	PUNCT
cana-327	151	5	,	,	PUNCT
cana-327	151	6	413	413	NUM
cana-327	151	7	-	-	SYM
cana-327	151	8	425	425	NUM
cana-327	151	9	.	.	PUNCT
cana-327	152	1	[	[	X
cana-327	152	2	5	5	NUM
cana-327	152	3	]	]	X
cana-327	152	4	chatzarakis	chatzaraki	NOUN
cana-327	152	5	,	,	PUNCT
cana-327	152	6	g.e	g.e	PROPN
cana-327	152	7	.	.	PROPN
cana-327	152	8	,	,	PUNCT
cana-327	152	9	kanagasabapathi	kanagasabapathi	PROPN
cana-327	152	10	,	,	PUNCT
cana-327	152	11	r.	r.	PROPN
cana-327	152	12	,	,	PUNCT
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cana-327	152	14	and	and	CCONJ
cana-327	152	15	thandapani	thandapani	PROPN
cana-327	152	16	,	,	PUNCT
cana-327	152	17	e.	e.	PROPN
cana-327	152	18	,	,	PUNCT
cana-327	152	19	oscillations	oscillation	NOUN
cana-327	152	20	in	in	ADP
cana-327	152	21	second	second	ADJ
cana-327	152	22	-	-	PUNCT
cana-327	152	23	order	order	NOUN
cana-327	152	24	damped	damped	NOUN
cana-327	152	25	difference	difference	NOUN
cana-327	152	26	equations	equation	NOUN
cana-327	152	27	with	with	ADP
cana-327	152	28	a	a	DET
cana-327	152	29	superlinear	superlinear	ADJ
cana-327	152	30	neutral	neutral	ADJ
cana-327	152	31	term	term	NOUN
cana-327	152	32	,	,	PUNCT
cana-327	152	33	advances	advance	NOUN
cana-327	152	34	in	in	ADP
cana-327	152	35	mathematics	mathematic	NOUN
cana-327	152	36	:	:	PUNCT
cana-327	152	37	scientific	scientific	ADJ
cana-327	152	38	journal	journal	NOUN
cana-327	152	39	,	,	PUNCT
cana-327	152	40	9	9	NUM
cana-327	152	41	(	(	PUNCT
cana-327	152	42	12	12	NUM
cana-327	152	43	)	)	PUNCT
cana-327	152	44	(	(	PUNCT
cana-327	152	45	2020	2020	NUM
cana-327	152	46	)	)	PUNCT
cana-327	152	47	.	.	PUNCT
cana-327	153	1	[	[	X
cana-327	153	2	6	6	NUM
cana-327	153	3	]	]	PUNCT
cana-327	153	4	erbe	erbe	PROPN
cana-327	153	5	,	,	PUNCT
cana-327	153	6	l.h	l.h	PROPN
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cana-327	153	8	,	,	PUNCT
cana-327	153	9	kong	kong	PROPN
cana-327	153	10	,	,	PUNCT
cana-327	153	11	q.	q.	PROPN
cana-327	153	12	,	,	PUNCT
cana-327	153	13	zhang	zhang	PROPN
cana-327	153	14	,	,	PUNCT
cana-327	153	15	b.g	b.g	PROPN
cana-327	153	16	.	.	PROPN
cana-327	153	17	,	,	PUNCT
cana-327	153	18	oscillation	oscillation	NOUN
cana-327	153	19	of	of	ADP
cana-327	153	20	discrete	discrete	ADJ
cana-327	153	21	analogues	analogue	NOUN
cana-327	153	22	of	of	ADP
cana-327	153	23	delay	delay	NOUN
cana-327	153	24	equations	equation	NOUN
cana-327	153	25	,	,	PUNCT
cana-327	153	26	differential	differential	ADJ
cana-327	153	27	and	and	CCONJ
cana-327	153	28	integral	integral	ADJ
cana-327	153	29	equations	equation	NOUN
cana-327	153	30	,	,	PUNCT
cana-327	153	31	2	2	NUM
cana-327	153	32	(	(	PUNCT
cana-327	153	33	3	3	NUM
cana-327	153	34	)	)	PUNCT
cana-327	153	35	(	(	PUNCT
cana-327	153	36	1989	1989	NUM
cana-327	153	37	)	)	PUNCT
cana-327	153	38	,	,	PUNCT
cana-327	153	39	300	300	NUM
cana-327	153	40	-	-	NUM
cana-327	153	41	309	309	NUM
cana-327	153	42	.	.	PUNCT
cana-327	154	1	[	[	X
cana-327	154	2	7	7	X
cana-327	154	3	]	]	X
cana-327	154	4	erbe	erbe	PROPN
cana-327	154	5	,	,	PUNCT
cana-327	154	6	l.h	l.h	PROPN
cana-327	154	7	.	.	PROPN
cana-327	154	8	,	,	PUNCT
cana-327	154	9	kong	kong	PROPN
cana-327	154	10	,	,	PUNCT
cana-327	154	11	q.	q.	PROPN
cana-327	154	12	,	,	PUNCT
cana-327	154	13	zhang	zhang	PROPN
cana-327	154	14	,	,	PUNCT
cana-327	154	15	b.g	b.g	PROPN
cana-327	154	16	.	.	PROPN
cana-327	154	17	,	,	PUNCT
cana-327	154	18	oscillation	oscillation	NOUN
cana-327	154	19	theory	theory	NOUN
cana-327	154	20	for	for	ADP
cana-327	154	21	functional	functional	ADJ
cana-327	154	22	differential	differential	ADJ
cana-327	154	23	equations	equation	NOUN
cana-327	154	24	,	,	PUNCT
cana-327	154	25	new	new	PROPN
cana-327	154	26	york	york	PROPN
cana-327	154	27	,	,	PUNCT
cana-327	154	28	1995	1995	NUM
cana-327	154	29	.	.	PUNCT
cana-327	155	1	[	[	X
cana-327	155	2	8	8	NUM
cana-327	155	3	]	]	PUNCT
cana-327	155	4	elaydi	elaydi	PROPN
cana-327	155	5	,	,	PUNCT
cana-327	155	6	s.	s.	PROPN
cana-327	155	7	an	an	DET
cana-327	155	8	introduction	introduction	NOUN
cana-327	155	9	to	to	ADP
cana-327	155	10	difference	difference	NOUN
cana-327	155	11	equations	equation	NOUN
cana-327	155	12	,	,	PUNCT
cana-327	155	13	springer	springer	NOUN
cana-327	155	14	-	-	PUNCT
cana-327	155	15	verlag	verlag	PROPN
cana-327	155	16	,	,	PUNCT
cana-327	155	17	new	new	PROPN
cana-327	155	18	york	york	PROPN
cana-327	155	19	,	,	PUNCT
cana-327	155	20	1996	1996	NUM
cana-327	155	21	.	.	PUNCT
cana-327	156	1	[	[	X
cana-327	156	2	9	9	NUM
cana-327	156	3	]	]	SYM
cana-327	156	4	grace	grace	NOUN
cana-327	156	5	,	,	PUNCT
cana-327	156	6	s.r	s.r	PROPN
cana-327	156	7	.	.	PROPN
cana-327	156	8	,	,	PUNCT
cana-327	156	9	oscillatory	oscillatory	ADJ
cana-327	156	10	behavior	behavior	NOUN
cana-327	156	11	of	of	ADP
cana-327	156	12	second	second	ADJ
cana-327	156	13	order	order	NOUN
cana-327	156	14	nonlinear	nonlinear	ADJ
cana-327	156	15	differential	differential	ADJ
cana-327	156	16	equations	equation	NOUN
cana-327	156	17	with	with	ADP
cana-327	156	18	a	a	DET
cana-327	156	19	nonpositive	nonpositive	ADJ
cana-327	156	20	neutral	neutral	ADJ
cana-327	156	21	terms	term	NOUN
cana-327	156	22	,	,	PUNCT
cana-327	156	23	mediterr	mediterr	PROPN
cana-327	156	24	.	.	PUNCT
cana-327	157	1	j.	j.	PROPN
cana-327	157	2	math	math	PROPN
cana-327	157	3	.	.	PUNCT
cana-327	158	1	14(2017	14(2017	X
cana-327	158	2	)	)	PUNCT
cana-327	158	3	,	,	PUNCT
cana-327	159	1	art	art	NOUN
cana-327	159	2	.	.	PUNCT
cana-327	160	1	229	229	NUM
cana-327	160	2	.	.	PUNCT
cana-327	161	1	[	[	X
cana-327	161	2	10	10	NUM
cana-327	161	3	]	]	X
cana-327	161	4	grace	grace	NOUN
cana-327	161	5	,	,	PUNCT
cana-327	161	6	s.r	s.r	PROPN
cana-327	161	7	,	,	PUNCT
cana-327	161	8	kaleeswari	kaleeswari	NOUN
cana-327	161	9	,	,	PUNCT
cana-327	161	10	s.	s.	PROPN
cana-327	161	11	,	,	PUNCT
cana-327	161	12	oscillation	oscillation	NOUN
cana-327	161	13	of	of	ADP
cana-327	161	14	even	even	ADV
cana-327	161	15	order	order	VERB
cana-327	161	16	nonlinear	nonlinear	ADJ
cana-327	161	17	difference	difference	NOUN
cana-327	161	18	equations	equation	NOUN
cana-327	161	19	with	with	ADP
cana-327	161	20	an	an	DET
cana-327	161	21	advanced	advanced	ADJ
cana-327	161	22	argument	argument	NOUN
cana-327	161	23	,	,	PUNCT
cana-327	161	24	30	30	NUM
cana-327	161	25	(	(	PUNCT
cana-327	161	26	3a	3a	NUM
cana-327	161	27	)	)	PUNCT
cana-327	161	28	(	(	PUNCT
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cana-327	161	30	)	)	PUNCT
cana-327	161	31	,	,	PUNCT
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cana-327	161	33	-	-	SYM
cana-327	161	34	251	251	NUM
cana-327	161	35	.	.	PUNCT
cana-327	162	1	[	[	X
cana-327	162	2	11	11	NUM
cana-327	162	3	]	]	PUNCT
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cana-327	162	5	,	,	PUNCT
cana-327	162	6	b.	b.	PROPN
cana-327	162	7	,	,	PUNCT
cana-327	162	8	kaleeswari	kaleeswari	PROPN
cana-327	162	9	,	,	PUNCT
cana-327	162	10	s.	s.	PROPN
cana-327	162	11	,	,	PUNCT
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cana-327	162	15	solutions	solution	NOUN
cana-327	162	16	for	for	ADP
cana-327	162	17	certain	certain	ADJ
cana-327	162	18	third	third	ADJ
cana-327	162	19	order	order	NOUN
cana-327	162	20	non	non	ADJ
cana-327	162	21	-	-	ADJ
cana-327	162	22	linear	linear	ADJ
cana-327	162	23	difference	difference	NOUN
cana-327	162	24	equations	equation	NOUN
cana-327	162	25	,	,	PUNCT
cana-327	162	26	far	far	PROPN
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cana-327	162	28	journal	journal	PROPN
cana-327	162	29	of	of	ADP
cana-327	162	30	mathematical	mathematical	ADJ
cana-327	162	31	science	science	NOUN
cana-327	162	32	,	,	PUNCT
cana-327	162	33	98	98	NUM
cana-327	162	34	(	(	PUNCT
cana-327	162	35	8)	8)	NUM
cana-327	162	36	(	(	PUNCT
cana-327	162	37	2015	2015	NUM
cana-327	162	38	)	)	PUNCT
cana-327	162	39	,	,	PUNCT
cana-327	162	40	963	963	NUM
cana-327	162	41	.	.	PUNCT
cana-327	163	1	[	[	X
cana-327	163	2	12	12	NUM
cana-327	163	3	]	]	X
cana-327	163	4	selvarangam	selvarangam	PROPN
cana-327	163	5	,	,	PUNCT
cana-327	163	6	s.	s.	PROPN
cana-327	163	7	,	,	PUNCT
cana-327	163	8	madhan	madhan	PROPN
cana-327	163	9	,	,	PUNCT
cana-327	163	10	m.	m.	NOUN
cana-327	163	11	,	,	PUNCT
cana-327	163	12	thandapani	thandapani	PROPN
cana-327	163	13	,	,	PUNCT
cana-327	163	14	e.	e.	PROPN
cana-327	163	15	,	,	PUNCT
cana-327	163	16	pinelas	pinelas	PROPN
cana-327	163	17	,	,	PUNCT
cana-327	163	18	s.	s.	PROPN
cana-327	163	19	,	,	PUNCT
cana-327	163	20	improved	improve	VERB
cana-327	163	21	oscillation	oscillation	NOUN
cana-327	163	22	conditions	condition	NOUN
cana-327	163	23	for	for	ADP
cana-327	163	24	third	third	ADJ
cana-327	163	25	order	order	NOUN
cana-327	163	26	neutral	neutral	ADJ
cana-327	163	27	type	type	NOUN
cana-327	163	28	difference	difference	NOUN
cana-327	163	29	equations	equation	NOUN
cana-327	163	30	,	,	PUNCT
cana-327	163	31	electron	electron	PROPN
cana-327	163	32	j.	j.	PROPN
cana-327	163	33	differential	differential	PROPN
cana-327	163	34	equations	equation	NOUN
cana-327	163	35	,	,	PUNCT
cana-327	163	36	90	90	NUM
cana-327	163	37	(	(	PUNCT
cana-327	163	38	2017	2017	NUM
cana-327	163	39	)	)	PUNCT
cana-327	163	40	,	,	PUNCT
cana-327	163	41	1	1	NUM
cana-327	163	42	-	-	SYM
cana-327	163	43	13	13	NUM
cana-327	163	44	.	.	PUNCT
cana-327	164	1	[	[	X
cana-327	164	2	13	13	NUM
cana-327	164	3	]	]	X
cana-327	164	4	thandapani	thandapani	PROPN
cana-327	164	5	,	,	PUNCT
cana-327	164	6	e.	e.	PROPN
cana-327	164	7	,	,	PUNCT
cana-327	164	8	selvarangam	selvarangam	PROPN
cana-327	164	9	,	,	PUNCT
cana-327	164	10	s.	s.	PROPN
cana-327	164	11	,	,	PUNCT
cana-327	164	12	oscillation	oscillation	NOUN
cana-327	164	13	results	result	NOUN
cana-327	164	14	for	for	ADP
cana-327	164	15	third	third	ADJ
cana-327	164	16	order	order	NOUN
cana-327	164	17	half	half	NOUN
cana-327	164	18	linear	linear	ADJ
cana-327	164	19	difference	difference	NOUN
cana-327	164	20	equations	equation	NOUN
cana-327	164	21	,	,	PUNCT
cana-327	164	22	bull	bull	NOUN
cana-327	164	23	.	.	PUNCT
cana-327	165	1	math	math	NOUN
cana-327	165	2	.	.	PUNCT
cana-327	166	1	anal	anal	PROPN
cana-327	166	2	.	.	PUNCT
cana-327	166	3	appl	appl	PROPN
cana-327	166	4	.	.	PROPN
cana-327	166	5	,	,	PUNCT
cana-327	166	6	4(2012	4(2012	NUM
cana-327	166	7	)	)	PUNCT
cana-327	166	8	,	,	PUNCT
cana-327	166	9	891	891	NUM
cana-327	166	10	-	-	SYM
cana-327	166	11	102	102	NUM
cana-327	166	12	.	.	PUNCT
cana-327	167	1	[	[	X
cana-327	167	2	14	14	NUM
cana-327	167	3	]	]	X
cana-327	167	4	ladas	ladas	PROPN
cana-327	167	5	,	,	PUNCT
cana-327	167	6	g.	g.	PROPN
cana-327	167	7	,	,	PUNCT
cana-327	167	8	philos	philo	NOUN
cana-327	167	9	,	,	PUNCT
cana-327	167	10	ch.g	ch.g	PROPN
cana-327	167	11	.	.	PUNCT
cana-327	167	12	,	,	PUNCT
cana-327	167	13	sficas	sfica	NOUN
cana-327	167	14	,	,	PUNCT
cana-327	167	15	sharp	sharp	ADJ
cana-327	167	16	conditions	condition	NOUN
cana-327	167	17	for	for	ADP
cana-327	167	18	the	the	DET
cana-327	167	19	oscillation	oscillation	NOUN
cana-327	167	20	of	of	ADP
cana-327	167	21	delay	delay	NOUN
cana-327	167	22	difference	difference	NOUN
cana-327	167	23	equations	equation	NOUN
cana-327	167	24	,	,	PUNCT
cana-327	167	25	j.	j.	PROPN
cana-327	167	26	appl	appl	PROPN
cana-327	167	27	.	.	PROPN
cana-327	167	28	math	math	PROPN
cana-327	167	29	.	.	PUNCT
cana-327	168	1	simulation	simulation	NOUN
cana-327	168	2	2	2	NUM
cana-327	168	3	,	,	PUNCT
cana-327	168	4	(	(	PUNCT
cana-327	168	5	1989	1989	NUM
cana-327	168	6	)	)	PUNCT
cana-327	168	7	,	,	PUNCT
cana-327	168	8	101	101	NUM
cana-327	168	9	-	-	SYM
cana-327	168	10	111	111	NUM
cana-327	168	11	.	.	PUNCT
cana-327	169	1	[	[	X
cana-327	169	2	15	15	NUM
cana-327	169	3	]	]	X
cana-327	169	4	walter	walter	PROPN
cana-327	169	5	,	,	PUNCT
cana-327	169	6	g.k	g.k	PROPN
cana-327	169	7	.	.	PROPN
cana-327	169	8	,	,	PUNCT
cana-327	169	9	allan	allan	PROPN
cana-327	169	10	,	,	PUNCT
cana-327	169	11	c.p	c.p	PROPN
cana-327	169	12	.	.	PROPN
cana-327	169	13	,	,	PUNCT
cana-327	169	14	difference	difference	NOUN
cana-327	169	15	equations	equation	NOUN
cana-327	169	16	introduction	introduction	NOUN
cana-327	169	17	with	with	ADP
cana-327	169	18	applications	application	NOUN
cana-327	169	19	,	,	PUNCT
cana-327	169	20	second	second	ADJ
cana-327	169	21	edition	edition	NOUN
cana-327	169	22	,	,	PUNCT
cana-327	169	23	academic	academic	ADJ
cana-327	169	24	press	press	NOUN
cana-327	169	25	,	,	PUNCT
cana-327	169	26	1991	1991	NUM
cana-327	169	27	.	.	PUNCT