id	sid	tid	token	lemma	pos
cana-524	1	1	communications	communication	NOUN
cana-524	1	2	on	on	ADP
cana-524	1	3	applied	apply	VERB
cana-524	1	4	nonlinear	nonlinear	ADJ
cana-524	1	5	analysis	analysis	NOUN
cana-524	1	6	issn	issn	NOUN
cana-524	1	7	:	:	PUNCT
cana-524	1	8	1074	1074	NUM
cana-524	1	9	-	-	PUNCT
cana-524	1	10	133x	133x	NUM
cana-524	1	11	vol	vol	NOUN
cana-524	1	12	31	31	NUM
cana-524	1	13	no	no	NOUN
cana-524	1	14	.	.	NOUN
cana-524	1	15	2	2	NUM
cana-524	1	16	(	(	PUNCT
cana-524	1	17	2024	2024	NUM
cana-524	1	18	)	)	PUNCT
cana-524	1	19	119	119	NUM
cana-524	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-524	1	21	oscillatory	oscillatory	ADJ
cana-524	1	22	properties	property	NOUN
cana-524	1	23	of	of	ADP
cana-524	1	24	second	second	ADJ
cana-524	1	25	order	order	NOUN
cana-524	1	26	half	half	ADJ
cana-524	1	27	-	-	PUNCT
cana-524	1	28	linear	linear	ADJ
cana-524	1	29	delay	delay	NOUN
cana-524	1	30	difference	difference	NOUN
cana-524	1	31	equations	equation	NOUN
cana-524	1	32	k.	k.	PROPN
cana-524	1	33	masaniammal	masaniammal	PROPN
cana-524	1	34	1	1	NUM
cana-524	1	35	,	,	PUNCT
cana-524	1	36	i.	i.	PROPN
cana-524	1	37	mohammed	mohammed	PROPN
cana-524	1	38	ali	ali	PROPN
cana-524	1	39	jaffer	jaffer	PROPN
cana-524	1	40	2	2	NUM
cana-524	1	41	1	1	NUM
cana-524	1	42	,	,	PUNCT
cana-524	1	43	2	2	NUM
cana-524	1	44	department	department	NOUN
cana-524	1	45	of	of	ADP
cana-524	1	46	mathematics	mathematic	NOUN
cana-524	1	47	,	,	PUNCT
cana-524	1	48	government	government	NOUN
cana-524	1	49	arts	art	NOUN
cana-524	1	50	college	college	PROPN
cana-524	1	51	,	,	PUNCT
cana-524	1	52	udumalpet-642126	udumalpet-642126	NOUN
cana-524	1	53	,	,	PUNCT
cana-524	1	54	tamilnadu	tamilnadu	ADJ
cana-524	1	55	,	,	PUNCT
cana-524	1	56	india	india	PROPN
cana-524	1	57	.	.	PROPN
cana-524	1	58	1	1	NUM
cana-524	1	59	reka.maths@gmail.com	reka.maths@gmail.com	NOUN
cana-524	1	60	2	2	NUM
cana-524	1	61	jaffermathsgac@gmail.com	jaffermathsgac@gmail.com	NOUN
cana-524	1	62	article	article	NOUN
cana-524	1	63	history	history	NOUN
cana-524	1	64	:	:	PUNCT
cana-524	1	65	received	receive	VERB
cana-524	1	66	:	:	PUNCT
cana-524	1	67	25	25	NUM
cana-524	1	68	-	-	PUNCT
cana-524	1	69	01	01	NUM
cana-524	1	70	-	-	PUNCT
cana-524	1	71	2024	2024	NUM
cana-524	1	72	revised	revise	VERB
cana-524	1	73	:	:	PUNCT
cana-524	1	74	02	02	NUM
cana-524	1	75	-	-	PUNCT
cana-524	1	76	04	04	NUM
cana-524	1	77	-	-	PUNCT
cana-524	1	78	2024	2024	NUM
cana-524	1	79	accepted	accept	VERB
cana-524	1	80	:	:	PUNCT
cana-524	1	81	25	25	NUM
cana-524	1	82	-	-	PUNCT
cana-524	1	83	04	04	NUM
cana-524	1	84	-	-	PUNCT
cana-524	1	85	2024	2024	NUM
cana-524	1	86	abstract	abstract	NOUN
cana-524	1	87	:	:	PUNCT
cana-524	1	88	this	this	DET
cana-524	1	89	study	study	NOUN
cana-524	1	90	explores	explore	NOUN
cana-524	1	91	,	,	PUNCT
cana-524	1	92	some	some	DET
cana-524	1	93	necessary	necessary	ADJ
cana-524	1	94	and	and	CCONJ
cana-524	1	95	sufficient	sufficient	ADJ
cana-524	1	96	conditions	condition	NOUN
cana-524	1	97	that	that	PRON
cana-524	1	98	are	be	AUX
cana-524	1	99	established	establish	VERB
cana-524	1	100	for	for	ADP
cana-524	1	101	oscillatory	oscillatory	ADJ
cana-524	1	102	properties	property	NOUN
cana-524	1	103	of	of	ADP
cana-524	1	104	second	second	ADJ
cana-524	1	105	order	order	NOUN
cana-524	1	106	half	half	ADJ
cana-524	1	107	-	-	PUNCT
cana-524	1	108	linear	linear	ADJ
cana-524	1	109	delay	delay	NOUN
cana-524	1	110	difference	difference	NOUN
cana-524	1	111	equations	equation	NOUN
cana-524	1	112	of	of	ADP
cana-524	1	113	the	the	DET
cana-524	1	114	form	form	NOUN
cana-524	1	115	δ(𝑝(𝜉)(δ𝑥(𝜉))𝑟	δ(𝑝(𝜉)(δ𝑥(𝜉))𝑟	PROPN
cana-524	1	116	)	)	PUNCT
cana-524	1	117	+	+	NUM
cana-524	1	118	𝑞(𝜉)𝑥𝑠(𝜎(𝜉	𝑞(𝜉)𝑥𝑠(𝜎(𝜉	NOUN
cana-524	1	119	)	)	PUNCT
cana-524	1	120	)	)	PUNCT
cana-524	2	1	=	=	PUNCT
cana-524	2	2	0	0	NUM
cana-524	2	3	,	,	PUNCT
cana-524	2	4	for	for	ADP
cana-524	2	5	𝜉	𝜉	X
cana-524	2	6	≥	≥	NOUN
cana-524	2	7	𝜉0	𝜉0	NOUN
cana-524	2	8	.	.	PUNCT
cana-524	3	1	under	under	ADP
cana-524	3	2	the	the	DET
cana-524	3	3	assumption	assumption	NOUN
cana-524	3	4	∑t=𝜉0	∑t=𝜉0	PUNCT
cana-524	3	5	𝜉−1	𝜉−1	NOUN
cana-524	3	6	  	  	SPACE
cana-524	3	7	1	1	NUM
cana-524	3	8	𝑝	𝑝	PROPN
cana-524	3	9	1	1	NUM
cana-524	3	10	𝑟(t	𝑟(t	PROPN
cana-524	3	11	)	)	PUNCT
cana-524	4	1	=	=	SYM
cana-524	4	2	∞.	∞.	PROPN
cana-524	4	3	two	two	NUM
cana-524	4	4	cases	case	NOUN
cana-524	4	5	are	be	AUX
cana-524	4	6	considered	consider	VERB
cana-524	4	7	for	for	ADP
cana-524	4	8	𝑟	𝑟	NOUN
cana-524	4	9	<	<	X
cana-524	4	10	𝑠	𝑠	PROPN
cana-524	4	11	and	and	CCONJ
cana-524	4	12	𝑟	𝑟	X
cana-524	4	13	>	>	X
cana-524	4	14	𝑠	𝑠	PROPN
cana-524	4	15	,	,	PUNCT
cana-524	4	16	where	where	SCONJ
cana-524	4	17	𝑟	𝑟	PRON
cana-524	4	18	and	and	CCONJ
cana-524	4	19	𝑠	𝑠	PROPN
cana-524	4	20	are	be	AUX
cana-524	4	21	the	the	DET
cana-524	4	22	quotients	quotient	NOUN
cana-524	4	23	of	of	ADP
cana-524	4	24	two	two	NUM
cana-524	4	25	positive	positive	ADJ
cana-524	4	26	odd	odd	ADJ
cana-524	4	27	integers	integer	NOUN
cana-524	4	28	.	.	PUNCT
cana-524	5	1	the	the	DET
cana-524	5	2	effectiveness	effectiveness	NOUN
cana-524	5	3	and	and	CCONJ
cana-524	5	4	applicability	applicability	NOUN
cana-524	5	5	of	of	ADP
cana-524	5	6	the	the	DET
cana-524	5	7	result	result	NOUN
cana-524	5	8	are	be	AUX
cana-524	5	9	illustrated	illustrate	VERB
cana-524	5	10	through	through	ADP
cana-524	5	11	few	few	ADJ
cana-524	5	12	examples	example	NOUN
cana-524	5	13	.	.	PUNCT
cana-524	6	1	keywords	keyword	NOUN
cana-524	6	2	:	:	PUNCT
cana-524	6	3	half	half	ADJ
cana-524	6	4	-	-	PUNCT
cana-524	6	5	linear	linear	ADJ
cana-524	6	6	,	,	PUNCT
cana-524	6	7	delay	delay	NOUN
cana-524	6	8	difference	difference	NOUN
cana-524	6	9	equation	equation	NOUN
cana-524	6	10	,	,	PUNCT
cana-524	6	11	oscillation	oscillation	NOUN
cana-524	6	12	.	.	PUNCT
cana-524	7	1	1	1	X
cana-524	7	2	.	.	X
cana-524	7	3	introduction	introduction	NOUN
cana-524	7	4	we	we	PRON
cana-524	7	5	consider	consider	VERB
cana-524	7	6	the	the	DET
cana-524	7	7	second	second	ADJ
cana-524	7	8	order	order	NOUN
cana-524	7	9	half	half	ADJ
cana-524	7	10	-	-	PUNCT
cana-524	7	11	linear	linear	ADJ
cana-524	7	12	delay	delay	NOUN
cana-524	7	13	difference	difference	NOUN
cana-524	7	14	equations	equation	NOUN
cana-524	7	15	of	of	ADP
cana-524	7	16	the	the	DET
cana-524	7	17	form	form	NOUN
cana-524	7	18	δ(𝑝(𝜉)(δ𝑥(𝜉))𝑟	δ(𝑝(𝜉)(δ𝑥(𝜉))𝑟	PROPN
cana-524	7	19	)	)	PUNCT
cana-524	7	20	+	+	NUM
cana-524	7	21	𝑞(𝜉)𝑥𝑠(𝜎(𝜉	𝑞(𝜉)𝑥𝑠(𝜎(𝜉	NOUN
cana-524	7	22	)	)	PUNCT
cana-524	7	23	)	)	PUNCT
cana-524	8	1	=	=	PUNCT
cana-524	8	2	0	0	NUM
cana-524	8	3	,	,	PUNCT
cana-524	8	4	for	for	ADP
cana-524	8	5	𝜉	𝜉	PROPN
cana-524	8	6	≥	≥	NOUN
cana-524	8	7	𝜉0	𝜉0	NOUN
cana-524	8	8	.	.	PUNCT
cana-524	9	1	(	(	PUNCT
cana-524	9	2	1.1	1.1	NUM
cana-524	9	3	)	)	PUNCT
cana-524	9	4	where	where	SCONJ
cana-524	9	5	𝑟	𝑟	PRON
cana-524	9	6	and	and	CCONJ
cana-524	9	7	𝑠	𝑠	PROPN
cana-524	9	8	are	be	AUX
cana-524	9	9	the	the	DET
cana-524	9	10	quotient	quotient	NOUN
cana-524	9	11	of	of	ADP
cana-524	9	12	two	two	NUM
cana-524	9	13	positive	positive	ADJ
cana-524	9	14	odd	odd	ADJ
cana-524	9	15	integers	integer	NOUN
cana-524	9	16	,	,	PUNCT
cana-524	9	17	and	and	CCONJ
cana-524	9	18	δ	δ	PROPN
cana-524	9	19	is	be	AUX
cana-524	9	20	the	the	DET
cana-524	9	21	forward	forward	ADJ
cana-524	9	22	difference	difference	NOUN
cana-524	9	23	operator	operator	NOUN
cana-524	9	24	defined	define	VERB
cana-524	9	25	by	by	ADP
cana-524	9	26	δ𝑥(𝜉	δ𝑥(𝜉	NOUN
cana-524	9	27	)	)	PUNCT
cana-524	9	28	=	=	PUNCT
cana-524	10	1	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	10	2	+	+	NOUN
cana-524	10	3	1	1	X
cana-524	10	4	)	)	PUNCT
cana-524	10	5	−	−	NOUN
cana-524	10	6	𝑥(𝜉	𝑥(𝜉	PROPN
cana-524	10	7	)	)	PUNCT
cana-524	10	8	.	.	PUNCT
cana-524	11	1	the	the	DET
cana-524	11	2	following	follow	VERB
cana-524	11	3	assumptions	assumption	NOUN
cana-524	11	4	are	be	AUX
cana-524	11	5	used	use	VERB
cana-524	11	6	in	in	ADP
cana-524	11	7	this	this	DET
cana-524	11	8	paper	paper	NOUN
cana-524	11	9	to	to	PART
cana-524	11	10	obtain	obtain	VERB
cana-524	11	11	the	the	DET
cana-524	11	12	result	result	NOUN
cana-524	11	13	:	:	PUNCT
cana-524	11	14	h1	h1	PROPN
cana-524	11	15	)	)	PUNCT
cana-524	11	16	{	{	PUNCT
cana-524	11	17	𝑝(𝜉	𝑝(𝜉	PROPN
cana-524	11	18	)	)	PUNCT
cana-524	11	19	}	}	PUNCT
cana-524	11	20	is	be	AUX
cana-524	11	21	sequence	sequence	NOUN
cana-524	11	22	of	of	ADP
cana-524	11	23	positive	positive	ADJ
cana-524	11	24	real	real	ADJ
cana-524	11	25	numbers	number	NOUN
cana-524	11	26	,	,	PUNCT
cana-524	11	27	0	0	PUNCT
cana-524	11	28	<	<	X
cana-524	11	29	𝑝	𝑝	X
cana-524	11	30	<	<	X
cana-524	11	31	1	1	NUM
cana-524	11	32	,	,	PUNCT
cana-524	11	33	𝜎(𝜉	𝜎(𝜉	PROPN
cana-524	11	34	)	)	PUNCT
cana-524	11	35	<	<	X
cana-524	11	36	𝜉	𝜉	X
cana-524	11	37	,	,	PUNCT
cana-524	11	38	lim𝜉→∞	lim𝜉→∞	ADJ
cana-524	11	39	 	 	SPACE
cana-524	11	40	𝜎(𝜉	𝜎(𝜉	PROPN
cana-524	11	41	)	)	PUNCT
cana-524	12	1	=	=	SYM
cana-524	12	2	∞.	∞.	PROPN
cana-524	12	3	h2	h2	NOUN
cana-524	12	4	)	)	PUNCT
cana-524	12	5	{	{	PUNCT
cana-524	12	6	𝑞(𝜉	𝑞(𝜉	PROPN
cana-524	12	7	)	)	PUNCT
cana-524	12	8	}	}	PUNCT
cana-524	12	9	is	be	AUX
cana-524	12	10	a	a	DET
cana-524	12	11	sequence	sequence	NOUN
cana-524	12	12	of	of	ADP
cana-524	12	13	nonnegative	nonnegative	ADJ
cana-524	12	14	real	real	ADJ
cana-524	12	15	numbers	number	NOUN
cana-524	12	16	and	and	CCONJ
cana-524	12	17	𝑞(𝜉	𝑞(𝜉	PRON
cana-524	12	18	)	)	PUNCT
cana-524	12	19	is	be	AUX
cana-524	12	20	not	not	PART
cana-524	12	21	identically	identically	ADV
cana-524	12	22	zero	zero	NUM
cana-524	12	23	for	for	ADP
cana-524	12	24	sufficiently	sufficiently	ADV
cana-524	12	25	large	large	ADJ
cana-524	12	26	values	value	NOUN
cana-524	12	27	of	of	ADP
cana-524	12	28	𝜉.	𝜉.	PROPN
cana-524	12	29	h3	h3	NOUN
cana-524	12	30	)	)	PUNCT
cana-524	12	31	𝑣(𝜉	𝑣(𝜉	NOUN
cana-524	12	32	)	)	PUNCT
cana-524	12	33	=	=	SYM
cana-524	13	1	∑𝑡=𝜉1	∑𝑡=𝜉1	PROPN
cana-524	13	2	𝜉−1	𝜉−1	NOUN
cana-524	13	3	 	 	SPACE
cana-524	13	4	𝑝−	𝑝−	PROPN
cana-524	13	5	1	1	NUM
cana-524	13	6	𝑟(𝑡	𝑟(𝑡	NOUN
cana-524	13	7	)	)	PUNCT
cana-524	13	8	with	with	ADP
cana-524	13	9	lim𝜉→∞	lim𝜉→∞	ADJ
cana-524	13	10	 	 	SPACE
cana-524	13	11	𝑣(𝜉	𝑣(𝜉	NOUN
cana-524	13	12	)	)	PUNCT
cana-524	13	13	=	=	SYM
cana-524	13	14	∞.	∞.	PROPN
cana-524	13	15	h4	h4	PROPN
cana-524	13	16	)	)	PUNCT
cana-524	13	17	0	0	PUNCT
cana-524	14	1	<	<	X
cana-524	14	2	𝜎0(𝜉	𝜎0(𝜉	PROPN
cana-524	14	3	)	)	PUNCT
cana-524	14	4	≤	≤	NOUN
cana-524	14	5	𝜎(𝜉	𝜎(𝜉	NOUN
cana-524	14	6	)	)	PUNCT
cana-524	14	7	,	,	PUNCT
cana-524	14	8	for	for	ADP
cana-524	14	9	δ𝜎0(𝜉	δ𝜎0(𝜉	NUM
cana-524	14	10	)	)	PUNCT
cana-524	14	11	≥	≥	NOUN
cana-524	14	12	𝜎0	𝜎0	NOUN
cana-524	14	13	>	>	X
cana-524	14	14	0	0	NUM
cana-524	14	15	,	,	PUNCT
cana-524	14	16	for	for	ADP
cana-524	14	17	𝜉	𝜉	PROPN
cana-524	14	18	≥	≥	NOUN
cana-524	14	19	𝜉0	𝜉0	NOUN
cana-524	14	20	.	.	PUNCT
cana-524	15	1	2	2	X
cana-524	15	2	.	.	X
cana-524	15	3	preliminary	preliminary	ADJ
cana-524	15	4	results	result	NOUN
cana-524	15	5	in	in	ADP
cana-524	15	6	this	this	DET
cana-524	15	7	section	section	NOUN
cana-524	15	8	,	,	PUNCT
cana-524	15	9	we	we	PRON
cana-524	15	10	provide	provide	VERB
cana-524	15	11	useful	useful	ADJ
cana-524	15	12	lemma	lemma	PROPN
cana-524	15	13	that	that	PRON
cana-524	15	14	will	will	AUX
cana-524	15	15	be	be	AUX
cana-524	15	16	essential	essential	ADJ
cana-524	15	17	in	in	ADP
cana-524	15	18	the	the	DET
cana-524	15	19	analysis	analysis	NOUN
cana-524	15	20	of	of	ADP
cana-524	15	21	the	the	DET
cana-524	15	22	oscillation	oscillation	NOUN
cana-524	15	23	behavior	behavior	NOUN
cana-524	15	24	of	of	ADP
cana-524	15	25	(	(	PUNCT
cana-524	15	26	1.1	1.1	NUM
cana-524	15	27	)	)	PUNCT
cana-524	15	28	.	.	PUNCT
cana-524	16	1	lemma	lemma	PROPN
cana-524	16	2	2.1	2.1	NUM
cana-524	16	3	.	.	PUNCT
cana-524	17	1	assuming	assume	VERB
cana-524	17	2	(	(	PUNCT
cana-524	17	3	𝐻1	𝐻1	NOUN
cana-524	17	4	)	)	PUNCT
cana-524	17	5	−	−	PROPN
cana-524	17	6	(	(	PUNCT
cana-524	17	7	𝐻3	𝐻3	PROPN
cana-524	17	8	)	)	PUNCT
cana-524	17	9	hold	hold	VERB
cana-524	17	10	and	and	CCONJ
cana-524	17	11	that	that	SCONJ
cana-524	17	12	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	17	13	)	)	PUNCT
cana-524	17	14	is	be	AUX
cana-524	17	15	an	an	DET
cana-524	17	16	eventually	eventually	ADV
cana-524	17	17	positive	positive	ADJ
cana-524	17	18	solution	solution	NOUN
cana-524	17	19	of	of	ADP
cana-524	17	20	(	(	PUNCT
cana-524	17	21	1.1	1.1	NUM
cana-524	17	22	)	)	PUNCT
cana-524	17	23	.	.	PUNCT
cana-524	18	1	then	then	ADV
cana-524	18	2	,	,	PUNCT
cana-524	18	3	there	there	PRON
cana-524	18	4	exists	exist	VERB
cana-524	18	5	𝜉1	𝜉1	PROPN
cana-524	18	6	≥	≥	NOUN
cana-524	18	7	𝜉0	𝜉0	NOUN
cana-524	18	8	and	and	CCONJ
cana-524	18	9	𝑑	𝑑	NOUN
cana-524	18	10	>	>	X
cana-524	18	11	0	0	NUM
cana-524	18	12	such	such	ADJ
cana-524	18	13	that	that	SCONJ
cana-524	18	14	mailto:lakshmikanth.mechanical@gmail.com	mailto:lakshmikanth.mechanical@gmail.com	X
cana-524	18	15	mailto:rupaachowdary@gmail.com	mailto:rupaachowdary@gmail.com	X
cana-524	19	1	communications	communication	NOUN
cana-524	19	2	on	on	ADP
cana-524	19	3	applied	apply	VERB
cana-524	19	4	nonlinear	nonlinear	ADJ
cana-524	19	5	analysis	analysis	NOUN
cana-524	19	6	issn	issn	NOUN
cana-524	19	7	:	:	PUNCT
cana-524	19	8	1074	1074	NUM
cana-524	19	9	-	-	PUNCT
cana-524	19	10	133x	133x	NUM
cana-524	19	11	vol	vol	NOUN
cana-524	19	12	31	31	NUM
cana-524	19	13	no	no	NOUN
cana-524	19	14	.	.	NOUN
cana-524	19	15	2	2	NUM
cana-524	19	16	(	(	PUNCT
cana-524	19	17	2024	2024	NUM
cana-524	19	18	)	)	PUNCT
cana-524	19	19	120	120	NUM
cana-524	19	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-524	19	21	0	0	PUNCT
cana-524	19	22	<	<	X
cana-524	19	23	𝑥(𝜉	𝑥(𝜉	PROPN
cana-524	19	24	)	)	PUNCT
cana-524	19	25	≤	≤	NOUN
cana-524	19	26	𝑑𝑣(𝜉	𝑑𝑣(𝜉	NOUN
cana-524	19	27	)	)	PUNCT
cana-524	19	28	,	,	PUNCT
cana-524	19	29	(	(	PUNCT
cana-524	19	30	2.1	2.1	NUM
cana-524	19	31	)	)	PUNCT
cana-524	19	32	𝑣(𝜉	𝑣(𝜉	NOUN
cana-524	19	33	)	)	PUNCT
cana-524	20	1	[	[	X
cana-524	20	2	∑	∑	PUNCT
cana-524	20	3	  	  	SPACE
cana-524	20	4	∞	∞	PROPN
cana-524	20	5	𝜁=𝜉	𝜁=𝜉	PROPN
cana-524	20	6	 	 	SPACE
cana-524	20	7	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	NOUN
cana-524	20	8	)	)	PUNCT
cana-524	20	9	)	)	PUNCT
cana-524	20	10	]	]	PUNCT
cana-524	20	11	1	1	NUM
cana-524	20	12	𝑟	𝑟	X
cana-524	20	13	≤	≤	NOUN
cana-524	20	14	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	20	15	)	)	PUNCT
cana-524	20	16	,	,	PUNCT
cana-524	20	17	for	for	ADP
cana-524	20	18	𝜉	𝜉	PROPN
cana-524	20	19	≥	≥	PROPN
cana-524	20	20	𝜉1	𝜉1	PROPN
cana-524	20	21	.	.	PUNCT
cana-524	21	1	(	(	PUNCT
cana-524	21	2	2.2	2.2	NUM
cana-524	21	3	)	)	PUNCT
cana-524	21	4	proof	proof	NOUN
cana-524	21	5	.	.	PUNCT
cana-524	22	1	assume	assume	VERB
cana-524	22	2	that	that	SCONJ
cana-524	22	3	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	22	4	)	)	PUNCT
cana-524	22	5	be	be	VERB
cana-524	22	6	an	an	DET
cana-524	22	7	eventually	eventually	ADV
cana-524	22	8	positive	positive	ADJ
cana-524	22	9	solution	solution	NOUN
cana-524	22	10	of	of	ADP
cana-524	22	11	(	(	PUNCT
cana-524	22	12	1.1	1.1	NUM
cana-524	22	13	)	)	PUNCT
cana-524	22	14	.	.	PUNCT
cana-524	23	1	then	then	ADV
cana-524	23	2	,	,	PUNCT
cana-524	23	3	by	by	ADP
cana-524	23	4	(	(	PUNCT
cana-524	23	5	h1	h1	PROPN
cana-524	23	6	)	)	PUNCT
cana-524	23	7	,	,	PUNCT
cana-524	23	8	there	there	PRON
cana-524	23	9	exists	exist	VERB
cana-524	23	10	a	a	DET
cana-524	23	11	𝜉∗	𝜉∗	NOUN
cana-524	23	12	such	such	ADJ
cana-524	23	13	that	that	SCONJ
cana-524	23	14	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	23	15	)	)	PUNCT
cana-524	23	16	>	>	X
cana-524	23	17	0	0	NUM
cana-524	23	18	and	and	CCONJ
cana-524	23	19	𝑥(𝜎(𝜉	𝑥(𝜎(𝜉	NOUN
cana-524	23	20	)	)	PUNCT
cana-524	23	21	)	)	PUNCT
cana-524	24	1	>	>	X
cana-524	24	2	0	0	PUNCT
cana-524	25	1	for	for	ADP
cana-524	25	2	all	all	DET
cana-524	25	3	𝜉	𝜉	ADP
cana-524	25	4	≥	≥	AUX
cana-524	25	5	𝜉∗	𝜉∗	NOUN
cana-524	25	6	it	it	PRON
cana-524	25	7	follows	follow	VERB
cana-524	25	8	from	from	ADP
cana-524	25	9	(	(	PUNCT
cana-524	25	10	1.1	1.1	NUM
cana-524	25	11	)	)	PUNCT
cana-524	25	12	that	that	PRON
cana-524	25	13	δ(𝑝(𝜉)(δ𝑥(𝜉))𝑟	δ(𝑝(𝜉)(δ𝑥(𝜉))𝑟	VERB
cana-524	25	14	)	)	PUNCT
cana-524	26	1	=	=	SYM
cana-524	26	2	−𝑞(𝜉)𝑥𝑠(𝜎(𝜉	−𝑞(𝜉)𝑥𝑠(𝜎(𝜉	PROPN
cana-524	26	3	)	)	PUNCT
cana-524	26	4	)	)	PUNCT
cana-524	27	1	≤	≤	ADV
cana-524	27	2	0	0	NUM
cana-524	27	3	.	.	PUNCT
cana-524	28	1	(	(	PUNCT
cana-524	28	2	2.3	2.3	NUM
cana-524	28	3	)	)	PUNCT
cana-524	28	4	consequently	consequently	ADV
cana-524	28	5	,	,	PUNCT
cana-524	28	6	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	PROPN
cana-524	28	7	is	be	AUX
cana-524	28	8	nonincreasing	nonincrease	VERB
cana-524	28	9	for	for	ADP
cana-524	28	10	𝜉	𝜉	PROPN
cana-524	28	11	≥	≥	NOUN
cana-524	28	12	𝜉∗.	𝜉∗.	NOUN
cana-524	28	13	next	next	ADV
cana-524	28	14	,	,	PUNCT
cana-524	28	15	we	we	PRON
cana-524	28	16	establish	establish	VERB
cana-524	28	17	that	that	SCONJ
cana-524	28	18	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	NOUN
cana-524	28	19	is	be	AUX
cana-524	28	20	positive	positive	ADJ
cana-524	28	21	.	.	PUNCT
cana-524	29	1	by	by	ADP
cana-524	29	2	contradiction	contradiction	NOUN
cana-524	29	3	,	,	PUNCT
cana-524	29	4	let	let	VERB
cana-524	29	5	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	NOUN
cana-524	29	6	≤	≤	NUM
cana-524	29	7	0	0	NUM
cana-524	29	8	at	at	ADP
cana-524	29	9	a	a	DET
cana-524	29	10	certain	certain	ADJ
cana-524	29	11	time	time	NOUN
cana-524	29	12	𝜉	𝜉	ADP
cana-524	29	13	≥	≥	NOUN
cana-524	29	14	𝜉∗.	𝜉∗.	NOUN
cana-524	29	15	in	in	ADP
cana-524	29	16	accordance	accordance	NOUN
cana-524	29	17	to	to	ADP
cana-524	29	18	𝑞	𝑞	PROPN
cana-524	29	19	is	be	AUX
cana-524	29	20	not	not	PART
cana-524	29	21	identically	identically	ADV
cana-524	29	22	zero	zero	NUM
cana-524	29	23	and	and	CCONJ
cana-524	29	24	by	by	ADP
cana-524	29	25	(	(	PUNCT
cana-524	29	26	2.3	2.3	NUM
cana-524	29	27	)	)	PUNCT
cana-524	29	28	,	,	PUNCT
cana-524	29	29	there	there	PRON
cana-524	29	30	exists	exist	VERB
cana-524	29	31	𝜉1	𝜉1	PROPN
cana-524	29	32	≥	≥	NOUN
cana-524	29	33	𝜉∗	𝜉∗	NOUN
cana-524	29	34	such	such	DET
cana-524	29	35	that	that	SCONJ
cana-524	29	36	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	VERB
cana-524	29	37	≤	≤	NUM
cana-524	29	38	𝑝(𝜉1)(δ𝑥(𝜉1	𝑝(𝜉1)(δ𝑥(𝜉1	NOUN
cana-524	29	39	)	)	PUNCT
cana-524	29	40	)	)	PUNCT
cana-524	30	1	𝑟	𝑟	X
cana-524	30	2	<	<	X
cana-524	30	3	0	0	NUM
cana-524	30	4	,	,	PUNCT
cana-524	30	5	𝜉	𝜉	X
cana-524	30	6	≥	≥	NOUN
cana-524	30	7	𝜉1	𝜉1	PROPN
cana-524	30	8	.	.	PUNCT
cana-524	31	1	(	(	PUNCT
cana-524	31	2	2.4	2.4	NUM
cana-524	31	3	)	)	PUNCT
cana-524	31	4	remember	remember	VERB
cana-524	31	5	that	that	SCONJ
cana-524	31	6	𝑟	𝑟	NOUN
cana-524	31	7	is	be	AUX
cana-524	31	8	the	the	DET
cana-524	31	9	quotient	quotient	NOUN
cana-524	31	10	of	of	ADP
cana-524	31	11	two	two	NUM
cana-524	31	12	positive	positive	ADJ
cana-524	31	13	odd	odd	ADJ
cana-524	31	14	integers	integer	NOUN
cana-524	31	15	.	.	PUNCT
cana-524	32	1	then	then	ADV
cana-524	32	2	,	,	PUNCT
cana-524	32	3	δ𝑥(𝜉	δ𝑥(𝜉	NOUN
cana-524	32	4	)	)	PUNCT
cana-524	32	5	≤	≤	NOUN
cana-524	32	6	(	(	PUNCT
cana-524	32	7	𝑝(𝜉1	𝑝(𝜉1	NOUN
cana-524	32	8	)	)	PUNCT
cana-524	32	9	𝑝(𝜉	𝑝(𝜉	NOUN
cana-524	32	10	)	)	PUNCT
cana-524	32	11	)	)	PUNCT
cana-524	32	12	1	1	NUM
cana-524	32	13	𝑟	𝑟	PRON
cana-524	32	14	δ𝑥(𝜉1	δ𝑥(𝜉1	NOUN
cana-524	32	15	)	)	PUNCT
cana-524	32	16	,	,	PUNCT
cana-524	32	17	for	for	ADP
cana-524	32	18	𝜉	𝜉	PROPN
cana-524	32	19	≥	≥	PROPN
cana-524	32	20	𝜉1	𝜉1	PROPN
cana-524	32	21	.	.	PUNCT
cana-524	33	1	(	(	PUNCT
cana-524	33	2	2.5	2.5	NUM
cana-524	33	3	)	)	PUNCT
cana-524	33	4	summing	sum	VERB
cana-524	33	5	(	(	PUNCT
cana-524	33	6	2.5	2.5	NUM
cana-524	33	7	)	)	PUNCT
cana-524	33	8	from	from	ADP
cana-524	33	9	𝜉1	𝜉1	PROPN
cana-524	33	10	to	to	ADP
cana-524	33	11	𝜉	𝜉	PROPN
cana-524	33	12	−	−	PROPN
cana-524	33	13	1	1	NUM
cana-524	33	14	,	,	PUNCT
cana-524	33	15	we	we	PRON
cana-524	33	16	arrive	arrive	VERB
cana-524	33	17	at	at	ADP
cana-524	33	18	the	the	DET
cana-524	33	19	result	result	NOUN
cana-524	33	20	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	33	21	)	)	PUNCT
cana-524	33	22	≤	≤	NUM
cana-524	33	23	𝑥(𝜉1	𝑥(𝜉1	NOUN
cana-524	33	24	)	)	PUNCT
cana-524	34	1	+	+	CCONJ
cana-524	34	2	(	(	PUNCT
cana-524	34	3	𝑝(𝜉1	𝑝(𝜉1	ADJ
cana-524	34	4	)	)	PUNCT
cana-524	34	5	)	)	PUNCT
cana-524	34	6	1	1	NUM
cana-524	34	7	𝑟δ𝑥(𝜉1)𝑣(𝜉	𝑟δ𝑥(𝜉1)𝑣(𝜉	NOUN
cana-524	34	8	)	)	PUNCT
cana-524	34	9	.	.	PUNCT
cana-524	35	1	(	(	PUNCT
cana-524	35	2	2.6	2.6	NUM
cana-524	35	3	)	)	PUNCT
cana-524	35	4	by	by	ADP
cana-524	35	5	(	(	PUNCT
cana-524	35	6	𝐻3	𝐻3	PROPN
cana-524	35	7	)	)	PUNCT
cana-524	35	8	,	,	PUNCT
cana-524	35	9	the	the	DET
cana-524	35	10	approach	approach	NOUN
cana-524	35	11	of	of	ADP
cana-524	35	12	the	the	DET
cana-524	35	13	right	right	ADJ
cana-524	35	14	hand	hand	NOUN
cana-524	35	15	side	side	NOUN
cana-524	35	16	is	be	AUX
cana-524	35	17	−∞	−∞	X
cana-524	35	18	then	then	ADV
cana-524	35	19	,	,	PUNCT
cana-524	35	20	lim𝜉→∞	lim𝜉→∞	ADJ
cana-524	35	21	 	 	SPACE
cana-524	35	22	𝑣(𝜉	𝑣(𝜉	NOUN
cana-524	35	23	)	)	PUNCT
cana-524	35	24	=	=	VERB
cana-524	36	1	−∞.	−∞.	NOUN
cana-524	36	2	this	this	PRON
cana-524	36	3	is	be	AUX
cana-524	36	4	a	a	DET
cana-524	36	5	contradiction	contradiction	NOUN
cana-524	36	6	to	to	ADP
cana-524	36	7	the	the	DET
cana-524	36	8	fact	fact	NOUN
cana-524	36	9	that	that	SCONJ
cana-524	36	10	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	36	11	)	)	PUNCT
cana-524	36	12	>	>	X
cana-524	36	13	0	0	X
cana-524	36	14	.	.	PUNCT
cana-524	37	1	thus	thus	ADV
cana-524	37	2	,	,	PUNCT
cana-524	37	3	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	VERB
cana-524	37	4	>	>	X
cana-524	37	5	0	0	NUM
cana-524	37	6	,	,	PUNCT
cana-524	37	7	for	for	ADP
cana-524	37	8	all	all	DET
cana-524	37	9	𝜉	𝜉	ADP
cana-524	37	10	≥	≥	NOUN
cana-524	37	11	𝜉∗.	𝜉∗.	NOUN
cana-524	37	12	from	from	ADP
cana-524	37	13	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	NOUN
cana-524	37	14	being	be	AUX
cana-524	37	15	nonincreasing	nonincrease	VERB
cana-524	37	16	,	,	PUNCT
cana-524	37	17	we	we	PRON
cana-524	37	18	have	have	VERB
cana-524	37	19	δ𝑥(𝜉	δ𝑥(𝜉	NOUN
cana-524	37	20	)	)	PUNCT
cana-524	37	21	≤	≤	NOUN
cana-524	37	22	(	(	PUNCT
cana-524	37	23	𝑝(𝜉1	𝑝(𝜉1	NOUN
cana-524	37	24	)	)	PUNCT
cana-524	37	25	𝑝(𝜉	𝑝(𝜉	NOUN
cana-524	37	26	)	)	PUNCT
cana-524	37	27	)	)	PUNCT
cana-524	37	28	1	1	NUM
cana-524	37	29	𝑟	𝑟	PRON
cana-524	37	30	δ𝑥(𝜉1	δ𝑥(𝜉1	NOUN
cana-524	37	31	)	)	PUNCT
cana-524	37	32	,	,	PUNCT
cana-524	37	33	for	for	ADP
cana-524	37	34	𝜉	𝜉	PROPN
cana-524	37	35	≥	≥	PROPN
cana-524	37	36	𝜉1	𝜉1	PROPN
cana-524	37	37	.	.	PUNCT
cana-524	38	1	(	(	PUNCT
cana-524	38	2	2.7	2.7	NUM
cana-524	38	3	)	)	PUNCT
cana-524	38	4	summing	sum	VERB
cana-524	38	5	(	(	PUNCT
cana-524	38	6	2.7	2.7	NUM
cana-524	38	7	)	)	PUNCT
cana-524	38	8	from	from	ADP
cana-524	38	9	𝜉1	𝜉1	PROPN
cana-524	38	10	to	to	ADP
cana-524	38	11	𝜉	𝜉	PROPN
cana-524	38	12	−	−	PROPN
cana-524	38	13	1	1	NUM
cana-524	38	14	,	,	PUNCT
cana-524	38	15	we	we	PRON
cana-524	38	16	obtain	obtain	VERB
cana-524	38	17	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	38	18	)	)	PUNCT
cana-524	38	19	≤	≤	NUM
cana-524	38	20	𝑥(𝜉1	𝑥(𝜉1	NOUN
cana-524	38	21	)	)	PUNCT
cana-524	39	1	+	+	CCONJ
cana-524	39	2	(	(	PUNCT
cana-524	39	3	𝑝(𝜉1	𝑝(𝜉1	ADJ
cana-524	39	4	)	)	PUNCT
cana-524	39	5	)	)	PUNCT
cana-524	39	6	1	1	NUM
cana-524	39	7	𝑟δ𝑥(𝜉1)𝑣(𝜉	𝑟δ𝑥(𝜉1)𝑣(𝜉	NOUN
cana-524	39	8	)	)	PUNCT
cana-524	39	9	.	.	PUNCT
cana-524	40	1	(	(	PUNCT
cana-524	40	2	2.8	2.8	NUM
cana-524	40	3	)	)	PUNCT
cana-524	40	4	since	since	SCONJ
cana-524	40	5	lim𝜉→∞	lim𝜉→∞	ADJ
cana-524	40	6	 	 	SPACE
cana-524	40	7	𝑣(𝜉	𝑣(𝜉	NOUN
cana-524	40	8	)	)	PUNCT
cana-524	40	9	=	=	SYM
cana-524	41	1	∞	∞	PROPN
cana-524	41	2	,	,	PUNCT
cana-524	41	3	there	there	PRON
cana-524	41	4	exists	exist	VERB
cana-524	41	5	a	a	DET
cana-524	41	6	positive	positive	ADJ
cana-524	41	7	constant	constant	NOUN
cana-524	41	8	𝑑	𝑑	NOUN
cana-524	41	9	such	such	ADJ
cana-524	41	10	that	that	SCONJ
cana-524	41	11	(	(	PUNCT
cana-524	41	12	2.1	2.1	NUM
cana-524	41	13	)	)	PUNCT
cana-524	41	14	holds	hold	VERB
cana-524	41	15	.	.	PUNCT
cana-524	42	1	communications	communication	NOUN
cana-524	42	2	on	on	ADP
cana-524	42	3	applied	apply	VERB
cana-524	42	4	nonlinear	nonlinear	ADJ
cana-524	42	5	analysis	analysis	NOUN
cana-524	42	6	issn	issn	NOUN
cana-524	42	7	:	:	PUNCT
cana-524	42	8	1074	1074	NUM
cana-524	42	9	-	-	PUNCT
cana-524	42	10	133x	133x	NUM
cana-524	42	11	vol	vol	NOUN
cana-524	42	12	31	31	NUM
cana-524	42	13	no	no	NOUN
cana-524	42	14	.	.	NOUN
cana-524	42	15	2	2	NUM
cana-524	42	16	(	(	PUNCT
cana-524	42	17	2024	2024	NUM
cana-524	42	18	)	)	PUNCT
cana-524	42	19	121	121	NUM
cana-524	42	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-524	42	21	since	since	SCONJ
cana-524	42	22	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	NOUN
cana-524	42	23	is	be	AUX
cana-524	42	24	positive	positive	ADJ
cana-524	42	25	and	and	CCONJ
cana-524	42	26	nonincreasing	nonincreasing	ADJ
cana-524	42	27	,	,	PUNCT
cana-524	42	28	lim𝜉→∞	lim𝜉→∞	ADJ
cana-524	42	29	 	 	SPACE
cana-524	42	30	𝑝(𝜉)(δ𝑥(𝜉	𝑝(𝜉)(δ𝑥(𝜉	NOUN
cana-524	42	31	)	)	PUNCT
cana-524	42	32	)	)	PUNCT
cana-524	43	1	𝑟	𝑟	NOUN
cana-524	43	2	exists	exist	VERB
cana-524	43	3	and	and	CCONJ
cana-524	43	4	is	be	AUX
cana-524	43	5	nonnegative	nonnegative	ADJ
cana-524	43	6	.	.	PUNCT
cana-524	44	1	summing	sum	VERB
cana-524	44	2	(	(	PUNCT
cana-524	44	3	1.1	1.1	NUM
cana-524	44	4	)	)	PUNCT
cana-524	44	5	from	from	ADP
cana-524	44	6	𝜉	𝜉	PRON
cana-524	44	7	to	to	ADP
cana-524	44	8	𝑏	𝑏	PROPN
cana-524	44	9	−	−	PROPN
cana-524	44	10	1	1	NUM
cana-524	44	11	,	,	PUNCT
cana-524	44	12	we	we	PRON
cana-524	44	13	get	get	VERB
cana-524	44	14	𝑝(𝑏)(δ𝑥(𝑏))𝑟	𝑝(𝑏)(δ𝑥(𝑏))𝑟	PRON
cana-524	44	15	−	−	NOUN
cana-524	45	1	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	NOUN
cana-524	45	2	+	+	CCONJ
cana-524	45	3	∑	∑	PUNCT
cana-524	45	4	 	 	SPACE
cana-524	45	5	𝑏−1	𝑏−1	PROPN
cana-524	45	6	𝑡=𝜉	𝑡=𝜉	PROPN
cana-524	45	7	𝑞(𝑡)𝑥	𝑞(𝑡)𝑥	PROPN
cana-524	45	8	𝑠(𝜎(𝑡	𝑠(𝜎(𝑡	PROPN
cana-524	45	9	)	)	PUNCT
cana-524	45	10	)	)	PUNCT
cana-524	46	1	=	=	PUNCT
cana-524	46	2	0	0	X
cana-524	46	3	.	.	PUNCT
cana-524	47	1	(	(	PUNCT
cana-524	47	2	2.9	2.9	NUM
cana-524	47	3	)	)	PUNCT
cana-524	47	4	letting	let	VERB
cana-524	47	5	limit	limit	NOUN
cana-524	47	6	as	as	ADP
cana-524	47	7	𝑏	𝑏	PROPN
cana-524	47	8	→	→	SYM
cana-524	47	9	∞	∞	PROPN
cana-524	47	10	,	,	PUNCT
cana-524	47	11	we	we	PRON
cana-524	47	12	obtain	obtain	VERB
cana-524	47	13	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	NOUN
cana-524	47	14	≥	≥	NOUN
cana-524	47	15	∑	∑	ADP
cana-524	47	16	 	 	SPACE
cana-524	47	17	∞	∞	PROPN
cana-524	47	18	𝑡=𝜉	𝑡=𝜉	PROPN
cana-524	47	19	𝑞(𝑡)𝑥	𝑞(𝑡)𝑥	PROPN
cana-524	47	20	𝑠(𝜎(𝑡	𝑠(𝜎(𝑡	PROPN
cana-524	47	21	)	)	PUNCT
cana-524	47	22	)	)	PUNCT
cana-524	47	23	.	.	PUNCT
cana-524	48	1	(	(	PUNCT
cana-524	48	2	2.10	2.10	NUM
cana-524	48	3	)	)	PUNCT
cana-524	48	4	then	then	ADV
cana-524	48	5	,	,	PUNCT
cana-524	48	6	δ𝑥(𝜉	δ𝑥(𝜉	NUM
cana-524	48	7	)	)	PUNCT
cana-524	48	8	≥	≥	NOUN
cana-524	48	9	[	[	PUNCT
cana-524	48	10	1	1	NUM
cana-524	48	11	𝑝(𝜉	𝑝(𝜉	NOUN
cana-524	48	12	)	)	PUNCT
cana-524	48	13	∑	∑	ADP
cana-524	48	14	 	 	SPACE
cana-524	48	15	∞	∞	PROPN
cana-524	48	16	𝑡=𝜉	𝑡=𝜉	PRON
cana-524	48	17	 	 	SPACE
cana-524	48	18	𝑞(𝑡)𝑥	𝑞(𝑡)𝑥	PROPN
cana-524	48	19	𝑠(𝜎(𝑡	𝑠(𝜎(𝑡	PROPN
cana-524	48	20	)	)	PUNCT
cana-524	48	21	)	)	PUNCT
cana-524	48	22	]	]	PUNCT
cana-524	48	23	1	1	NUM
cana-524	48	24	𝑟	𝑟	NOUN
cana-524	48	25	.	.	PUNCT
cana-524	49	1	(	(	PUNCT
cana-524	49	2	2.11	2.11	NUM
cana-524	49	3	)	)	PUNCT
cana-524	49	4	since	since	SCONJ
cana-524	49	5	𝑥(𝜉1	𝑥(𝜉1	NOUN
cana-524	49	6	)	)	PUNCT
cana-524	49	7	>	>	X
cana-524	49	8	0	0	NUM
cana-524	49	9	,	,	PUNCT
cana-524	49	10	summing	sum	VERB
cana-524	49	11	(	(	PUNCT
cana-524	49	12	2.11	2.11	NUM
cana-524	49	13	)	)	PUNCT
cana-524	49	14	from	from	ADP
cana-524	49	15	𝜉1	𝜉1	PROPN
cana-524	49	16	to	to	ADP
cana-524	49	17	𝜉	𝜉	PROPN
cana-524	49	18	−	−	PROPN
cana-524	49	19	1	1	NUM
cana-524	49	20	,	,	PUNCT
cana-524	49	21	we	we	PRON
cana-524	49	22	have	have	VERB
cana-524	49	23	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	49	24	)	)	PUNCT
cana-524	49	25	≥	≥	NOUN
cana-524	49	26	∑	∑	ADP
cana-524	49	27	  	  	SPACE
cana-524	49	28	𝑛−1	𝑛−1	PROPN
cana-524	49	29	𝑡=𝜉1	𝑡=𝜉1	PROPN
cana-524	49	30	[	[	PUNCT
cana-524	49	31	1	1	NUM
cana-524	49	32	𝑝(𝑡	𝑝(𝑡	PROPN
cana-524	49	33	)	)	PUNCT
cana-524	49	34	∑	∑	ADP
cana-524	49	35	  	  	SPACE
cana-524	49	36	∞	∞	PROPN
cana-524	49	37	𝜁=𝑡	𝜁=𝑡	PROPN
cana-524	49	38	 	 	SPACE
cana-524	49	39	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	NOUN
cana-524	49	40	)	)	PUNCT
cana-524	49	41	)	)	PUNCT
cana-524	49	42	]	]	PUNCT
cana-524	49	43	1	1	NUM
cana-524	49	44	𝑟	𝑟	NOUN
cana-524	49	45	.	.	PUNCT
cana-524	50	1	(	(	PUNCT
cana-524	50	2	2.12	2.12	NUM
cana-524	50	3	)	)	PUNCT
cana-524	50	4	use	use	VERB
cana-524	50	5	the	the	DET
cana-524	50	6	definition	definition	NOUN
cana-524	50	7	of	of	ADP
cana-524	50	8	𝑣(𝜉	𝑣(𝜉	NOUN
cana-524	50	9	)	)	PUNCT
cana-524	50	10	to	to	PART
cana-524	50	11	obtain	obtain	VERB
cana-524	50	12	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	50	13	)	)	PUNCT
cana-524	50	14	≥	≥	NOUN
cana-524	50	15	𝑣(𝜉	𝑣(𝜉	NOUN
cana-524	50	16	)	)	PUNCT
cana-524	51	1	[	[	X
cana-524	51	2	∑	∑	PUNCT
cana-524	51	3	  	  	SPACE
cana-524	51	4	∞	∞	PROPN
cana-524	51	5	𝜁=𝜉	𝜁=𝜉	PROPN
cana-524	51	6	 	 	SPACE
cana-524	51	7	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	NOUN
cana-524	51	8	)	)	PUNCT
cana-524	51	9	)	)	PUNCT
cana-524	51	10	]	]	PUNCT
cana-524	51	11	1	1	NUM
cana-524	51	12	𝑟	𝑟	NOUN
cana-524	51	13	.	.	PUNCT
cana-524	52	1	(	(	PUNCT
cana-524	52	2	2.13	2.13	NUM
cana-524	52	3	)	)	PUNCT
cana-524	52	4	this	this	DET
cana-524	52	5	yields	yield	NOUN
cana-524	52	6	(	(	PUNCT
cana-524	52	7	2.2	2.2	NUM
cana-524	52	8	)	)	PUNCT
cana-524	52	9	.	.	PUNCT
cana-524	53	1	3	3	X
cana-524	53	2	.	.	X
cana-524	53	3	main	main	ADJ
cana-524	53	4	results	result	NOUN
cana-524	53	5	theorem	theorem	VERB
cana-524	53	6	3.1	3.1	NUM
cana-524	53	7	.	.	PUNCT
cana-524	54	1	assume	assume	VERB
cana-524	54	2	that	that	SCONJ
cana-524	54	3	there	there	PRON
cana-524	54	4	exists	exist	VERB
cana-524	54	5	a	a	DET
cana-524	54	6	constant	constant	ADJ
cana-524	54	7	𝛽1	𝛽1	NOUN
cana-524	54	8	,	,	PUNCT
cana-524	54	9	the	the	DET
cana-524	54	10	quotient	quotient	NOUN
cana-524	54	11	of	of	ADP
cana-524	54	12	two	two	NUM
cana-524	54	13	positive	positive	ADJ
cana-524	54	14	odd	odd	ADJ
cana-524	54	15	integers	integer	NOUN
cana-524	54	16	,	,	PUNCT
cana-524	54	17	such	such	ADJ
cana-524	54	18	that	that	SCONJ
cana-524	54	19	0	0	NUM
cana-524	54	20	<	<	X
cana-524	54	21	𝑠	𝑠	X
cana-524	54	22	<	<	X
cana-524	54	23	𝛽1	𝛽1	NOUN
cana-524	54	24	<	<	X
cana-524	54	25	𝑟.	𝑟.	NOUN
cana-524	54	26	if	if	SCONJ
cana-524	54	27	(	(	PUNCT
cana-524	54	28	𝐻1	𝐻1	PROPN
cana-524	54	29	)	)	PUNCT
cana-524	55	1	−	−	PROPN
cana-524	55	2	(	(	PUNCT
cana-524	55	3	𝐻3	𝐻3	PROPN
cana-524	55	4	)	)	PUNCT
cana-524	55	5	hold	hold	VERB
cana-524	55	6	,	,	PUNCT
cana-524	55	7	then	then	ADV
cana-524	55	8	each	each	DET
cana-524	55	9	solution	solution	NOUN
cana-524	55	10	of	of	ADP
cana-524	55	11	(	(	PUNCT
cana-524	55	12	1.1	1.1	NUM
cana-524	55	13	)	)	PUNCT
cana-524	55	14	is	be	AUX
cana-524	55	15	oscillatory	oscillatory	ADJ
cana-524	55	16	if	if	SCONJ
cana-524	55	17	and	and	CCONJ
cana-524	55	18	only	only	ADV
cana-524	55	19	if	if	SCONJ
cana-524	55	20	∑	∑	PUNCT
cana-524	55	21	 	 	SPACE
cana-524	55	22	∞	∞	PROPN
cana-524	55	23	𝜁=0	𝜁=0	PROPN
cana-524	55	24	𝑞(𝜁)𝑣	𝑞(𝜁)𝑣	PROPN
cana-524	55	25	𝑠(𝜎(𝜁	𝑠(𝜎(𝜁	NOUN
cana-524	55	26	)	)	PUNCT
cana-524	55	27	)	)	PUNCT
cana-524	56	1	=	=	SYM
cana-524	56	2	∞	∞	PROPN
cana-524	56	3	.	.	PUNCT
cana-524	57	1	(	(	PUNCT
cana-524	57	2	3.1	3.1	NUM
cana-524	57	3	)	)	PUNCT
cana-524	57	4	proof	proof	NOUN
cana-524	57	5	.	.	PUNCT
cana-524	58	1	on	on	ADP
cana-524	58	2	the	the	DET
cana-524	58	3	contrary	contrary	NOUN
cana-524	58	4	,	,	PUNCT
cana-524	58	5	let	let	VERB
cana-524	58	6	𝑥(𝜉	𝑥(𝜉	NUM
cana-524	58	7	)	)	PUNCT
cana-524	58	8	be	be	AUX
cana-524	58	9	an	an	DET
cana-524	58	10	eventually	eventually	ADV
cana-524	58	11	positive	positive	ADJ
cana-524	58	12	solution	solution	NOUN
cana-524	58	13	.	.	PUNCT
cana-524	59	1	so	so	ADV
cana-524	59	2	lemma	lemma	PROPN
cana-524	59	3	2.1	2.1	NUM
cana-524	59	4	holds	hold	NOUN
cana-524	59	5	,	,	PUNCT
cana-524	59	6	and	and	CCONJ
cana-524	59	7	then	then	ADV
cana-524	59	8	there	there	PRON
cana-524	59	9	exists	exist	VERB
cana-524	59	10	𝜉1	𝜉1	PROPN
cana-524	59	11	≥	≥	NOUN
cana-524	59	12	𝜉0	𝜉0	NOUN
cana-524	59	13	such	such	ADJ
cana-524	59	14	that	that	SCONJ
cana-524	59	15	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	59	16	)	)	PUNCT
cana-524	59	17	≥	≥	NOUN
cana-524	59	18	𝑣(𝜉)𝑤	𝑣(𝜉)𝑤	SYM
cana-524	59	19	1	1	NUM
cana-524	59	20	𝑟(𝜉	𝑟(𝜉	NUM
cana-524	59	21	)	)	PUNCT
cana-524	59	22	≥	≥	NOUN
cana-524	59	23	0	0	NUM
cana-524	59	24	,	,	PUNCT
cana-524	59	25	for	for	ADP
cana-524	59	26	𝜉	𝜉	X
cana-524	59	27	≥	≥	PROPN
cana-524	59	28	𝜉1	𝜉1	PROPN
cana-524	59	29	,	,	PUNCT
cana-524	59	30	(	(	PUNCT
cana-524	59	31	3.2	3.2	NUM
cana-524	59	32	)	)	PUNCT
cana-524	59	33	communications	communication	NOUN
cana-524	59	34	on	on	ADP
cana-524	59	35	applied	apply	VERB
cana-524	59	36	nonlinear	nonlinear	ADJ
cana-524	59	37	analysis	analysis	NOUN
cana-524	59	38	issn	issn	NOUN
cana-524	59	39	:	:	PUNCT
cana-524	59	40	1074	1074	NUM
cana-524	59	41	-	-	PUNCT
cana-524	59	42	133x	133x	NUM
cana-524	59	43	vol	vol	NOUN
cana-524	59	44	31	31	NUM
cana-524	59	45	no	no	NOUN
cana-524	59	46	.	.	NOUN
cana-524	59	47	2	2	NUM
cana-524	59	48	(	(	PUNCT
cana-524	59	49	2024	2024	NUM
cana-524	59	50	)	)	PUNCT
cana-524	59	51	122	122	NUM
cana-524	59	52	https://internationalpubls.com	https://internationalpubls.com	X
cana-524	59	53	where	where	SCONJ
cana-524	59	54	𝑤(𝜉	𝑤(𝜉	NOUN
cana-524	59	55	)	)	PUNCT
cana-524	60	1	=	=	PROPN
cana-524	60	2	∑	∑	PUNCT
cana-524	60	3	  	  	SPACE
cana-524	60	4	∞	∞	PROPN
cana-524	60	5	𝜁=𝜉	𝜁=𝜉	PROPN
cana-524	60	6	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	NOUN
cana-524	60	7	)	)	PUNCT
cana-524	60	8	)	)	PUNCT
cana-524	60	9	.	.	PUNCT
cana-524	61	1	(	(	PUNCT
cana-524	61	2	3.3	3.3	NUM
cana-524	61	3	)	)	PUNCT
cana-524	61	4	computing	computing	NOUN
cana-524	61	5	we	we	PRON
cana-524	61	6	have	have	VERB
cana-524	61	7	,	,	PUNCT
cana-524	61	8	δ𝑤(𝜉	δ𝑤(𝜉	NUM
cana-524	61	9	)	)	PUNCT
cana-524	61	10	=	=	SYM
cana-524	61	11	−𝑞(𝜉)𝑥𝑠(𝜎(𝜉	−𝑞(𝜉)𝑥𝑠(𝜎(𝜉	PROPN
cana-524	61	12	)	)	PUNCT
cana-524	61	13	)	)	PUNCT
cana-524	61	14	.	.	PUNCT
cana-524	62	1	(	(	PUNCT
cana-524	62	2	3.4	3.4	NUM
cana-524	62	3	)	)	PUNCT
cana-524	62	4	thus	thus	ADV
cana-524	62	5	,	,	PUNCT
cana-524	62	6	𝑤	𝑤	PRON
cana-524	62	7	is	be	AUX
cana-524	62	8	nonnegative	nonnegative	ADJ
cana-524	62	9	and	and	CCONJ
cana-524	62	10	nonincreasing	nonincrease	VERB
cana-524	62	11	.	.	PUNCT
cana-524	63	1	since	since	SCONJ
cana-524	63	2	𝑥	𝑥	PROPN
cana-524	63	3	>	>	X
cana-524	63	4	0	0	NUM
cana-524	63	5	,	,	PUNCT
cana-524	63	6	by	by	ADP
cana-524	63	7	(	(	PUNCT
cana-524	63	8	𝐻2	𝐻2	PROPN
cana-524	63	9	)	)	PUNCT
cana-524	63	10	,	,	PUNCT
cana-524	63	11	in	in	ADP
cana-524	63	12	continuation	continuation	NOUN
cana-524	63	13	𝑞(𝜉)𝑥𝑠(𝜎(𝜉	𝑞(𝜉)𝑥𝑠(𝜎(𝜉	NOUN
cana-524	63	14	)	)	PUNCT
cana-524	63	15	)	)	PUNCT
cana-524	63	16	can	can	AUX
cana-524	63	17	not	not	PART
cana-524	63	18	be	be	AUX
cana-524	63	19	identically	identically	ADV
cana-524	63	20	zero	zero	NUM
cana-524	63	21	.	.	PUNCT
cana-524	64	1	thus	thus	ADV
cana-524	64	2	,	,	PUNCT
cana-524	64	3	δ𝑤	δ𝑤	ADP
cana-524	64	4	can	can	AUX
cana-524	64	5	not	not	PART
cana-524	64	6	be	be	AUX
cana-524	64	7	identically	identically	ADV
cana-524	64	8	zero	zero	NUM
cana-524	64	9	,	,	PUNCT
cana-524	64	10	and	and	CCONJ
cana-524	64	11	𝑤	𝑤	AUX
cana-524	64	12	can	can	AUX
cana-524	64	13	not	not	PART
cana-524	64	14	be	be	AUX
cana-524	64	15	constant	constant	ADJ
cana-524	64	16	.	.	PUNCT
cana-524	65	1	therefore	therefore	ADV
cana-524	65	2	,	,	PUNCT
cana-524	65	3	𝑤(𝜉	𝑤(𝜉	PROPN
cana-524	65	4	)	)	PUNCT
cana-524	65	5	>	>	X
cana-524	65	6	0	0	PUNCT
cana-524	66	1	for	for	SCONJ
cana-524	66	2	𝜉	𝜉	PROPN
cana-524	66	3	≥	≥	PROPN
cana-524	66	4	𝜉1	𝜉1	PROPN
cana-524	66	5	.	.	PUNCT
cana-524	66	6	computing	compute	VERB
cana-524	66	7	we	we	PRON
cana-524	66	8	get	get	VERB
cana-524	66	9	,	,	PUNCT
cana-524	66	10	δ𝑤1−	δ𝑤1−	X
cana-524	66	11	𝛽1	𝛽1	PROPN
cana-524	66	12	𝑟	𝑟	X
cana-524	66	13	(	(	PUNCT
cana-524	66	14	𝜉	𝜉	NOUN
cana-524	66	15	)	)	PUNCT
cana-524	66	16	≥	≥	NOUN
cana-524	66	17	(	(	PUNCT
cana-524	66	18	1	1	NUM
cana-524	66	19	−	−	PROPN
cana-524	66	20	𝛽1	𝛽1	NOUN
cana-524	66	21	𝑟	𝑟	NOUN
cana-524	66	22	)	)	PUNCT
cana-524	66	23	𝑤	𝑤	PART
cana-524	66	24	−𝛽1	−𝛽1	PROPN
cana-524	66	25	𝑟	𝑟	X
cana-524	66	26	(	(	PUNCT
cana-524	66	27	𝜉)δ𝑤(𝜉	𝜉)δ𝑤(𝜉	PROPN
cana-524	66	28	)	)	PUNCT
cana-524	66	29	.	.	PUNCT
cana-524	67	1	(	(	PUNCT
cana-524	67	2	3.5	3.5	NUM
cana-524	67	3	)	)	PUNCT
cana-524	67	4	summing	sum	VERB
cana-524	67	5	(	(	PUNCT
cana-524	67	6	3.5	3.5	NUM
cana-524	67	7	)	)	PUNCT
cana-524	67	8	from	from	ADP
cana-524	67	9	𝜉2	𝜉2	PROPN
cana-524	67	10	to	to	ADP
cana-524	67	11	𝜉	𝜉	PROPN
cana-524	67	12	−	−	PROPN
cana-524	67	13	1	1	NUM
cana-524	67	14	and	and	CCONJ
cana-524	67	15	using	use	VERB
cana-524	67	16	that	that	PRON
cana-524	67	17	𝑤	𝑤	ADP
cana-524	67	18	>	>	X
cana-524	67	19	0	0	NUM
cana-524	67	20	,	,	PUNCT
cana-524	67	21	we	we	PRON
cana-524	67	22	have	have	VERB
cana-524	67	23	𝑤1−	𝑤1−	NUM
cana-524	67	24	𝛽1	𝛽1	ADJ
cana-524	67	25	𝑟	𝑟	X
cana-524	67	26	(	(	PUNCT
cana-524	67	27	𝜉2	𝜉2	PROPN
cana-524	67	28	)	)	PUNCT
cana-524	67	29	≥	≥	NOUN
cana-524	67	30	(	(	PUNCT
cana-524	67	31	1	1	NUM
cana-524	67	32	−	−	PROPN
cana-524	67	33	𝛽1	𝛽1	NOUN
cana-524	67	34	𝑟	𝑟	NOUN
cana-524	67	35	)	)	PUNCT
cana-524	68	1	[	[	X
cana-524	68	2	−	−	X
cana-524	68	3	∑	∑	DET
cana-524	68	4	  	  	SPACE
cana-524	68	5	𝜉−1	𝜉−1	PROPN
cana-524	68	6	𝜁=𝜉2	𝜁=𝜉2	PROPN
cana-524	68	7	 	 	SPACE
cana-524	68	8	𝑤	𝑤	ADP
cana-524	68	9	−𝛽1	−𝛽1	PROPN
cana-524	68	10	𝑟	𝑟	X
cana-524	68	11	(	(	PUNCT
cana-524	68	12	𝜁)δ𝑤(𝜁	𝜁)δ𝑤(𝜁	NUM
cana-524	68	13	)	)	PUNCT
cana-524	68	14	]	]	PUNCT
cana-524	68	15	≥	≥	X
cana-524	68	16	(	(	PUNCT
cana-524	68	17	1	1	NUM
cana-524	68	18	−	−	PROPN
cana-524	68	19	𝛽1	𝛽1	NOUN
cana-524	68	20	𝑟	𝑟	NOUN
cana-524	68	21	)	)	PUNCT
cana-524	69	1	[	[	X
cana-524	69	2	∑	∑	PUNCT
cana-524	69	3	  	  	SPACE
cana-524	69	4	𝜉−1	𝜉−1	PROPN
cana-524	69	5	𝜁=𝜉2	𝜁=𝜉2	PROPN
cana-524	69	6	 	 	SPACE
cana-524	69	7	𝑤	𝑤	ADP
cana-524	69	8	−𝛽1	−𝛽1	PROPN
cana-524	69	9	𝑟	𝑟	NOUN
cana-524	69	10	(	(	PUNCT
cana-524	69	11	𝜁)(𝑞(𝜁)𝑥𝑠(𝜎(𝜁	𝜁)(𝑞(𝜁)𝑥𝑠(𝜎(𝜁	NOUN
cana-524	69	12	)	)	PUNCT
cana-524	69	13	)	)	PUNCT
cana-524	69	14	)	)	PUNCT
cana-524	69	15	]	]	PUNCT
cana-524	69	16	.	.	PUNCT
cana-524	70	1	(	(	PUNCT
cana-524	70	2	3.6	3.6	NUM
cana-524	70	3	)	)	PUNCT
cana-524	70	4	by	by	ADP
cana-524	70	5	(	(	PUNCT
cana-524	70	6	2.1	2.1	NUM
cana-524	70	7	)	)	PUNCT
cana-524	70	8	and	and	CCONJ
cana-524	70	9	(	(	PUNCT
cana-524	70	10	3.2	3.2	NUM
cana-524	70	11	)	)	PUNCT
cana-524	70	12	,	,	PUNCT
cana-524	70	13	we	we	PRON
cana-524	70	14	obtain	obtain	VERB
cana-524	70	15	𝑥𝑠(𝜉	𝑥𝑠(𝜉	PUNCT
cana-524	70	16	)	)	PUNCT
cana-524	70	17	=	=	SYM
cana-524	70	18	𝑥𝑠−𝛽1(𝜉)𝑥𝛽1(𝜉	𝑥𝑠−𝛽1(𝜉)𝑥𝛽1(𝜉	PROPN
cana-524	70	19	)	)	PUNCT
cana-524	70	20	≥	≥	PROPN
cana-524	70	21	(	(	PUNCT
cana-524	70	22	𝑑𝑣(𝜉))𝑠−𝛽1𝑥𝛽1(𝜉	𝑑𝑣(𝜉))𝑠−𝛽1𝑥𝛽1(𝜉	PROPN
cana-524	70	23	)	)	PUNCT
cana-524	70	24	≥	≥	X
cana-524	70	25	(	(	PUNCT
cana-524	70	26	𝑑𝑣(𝜉))𝑠−𝛽1	𝑑𝑣(𝜉))𝑠−𝛽1	X
cana-524	70	27	(	(	PUNCT
cana-524	70	28	𝑣(𝜉)𝑤	𝑣(𝜉)𝑤	NOUN
cana-524	70	29	1	1	NUM
cana-524	70	30	𝑟(𝜉	𝑟(𝜉	NUM
cana-524	70	31	)	)	PUNCT
cana-524	70	32	)	)	PUNCT
cana-524	70	33	𝛽1	𝛽1	NOUN
cana-524	70	34	=	=	SYM
cana-524	70	35	𝑑𝑠−𝛽1𝑣𝑠(𝜉)𝑤	𝑑𝑠−𝛽1𝑣𝑠(𝜉)𝑤	NOUN
cana-524	70	36	𝛽1	𝛽1	NOUN
cana-524	70	37	𝑟	𝑟	X
cana-524	70	38	(	(	PUNCT
cana-524	70	39	𝜉	𝜉	NOUN
cana-524	70	40	)	)	PUNCT
cana-524	70	41	,	,	PUNCT
cana-524	70	42	for	for	ADP
cana-524	70	43	𝜉	𝜉	PROPN
cana-524	70	44	≥	≥	X
cana-524	70	45	𝜉2	𝜉2	PROPN
cana-524	70	46	.	.	PUNCT
cana-524	71	1	since	since	SCONJ
cana-524	71	2	𝑤	𝑤	INTJ
cana-524	71	3	is	be	AUX
cana-524	71	4	nonincreasing	nonincrease	VERB
cana-524	71	5	,	,	PUNCT
cana-524	71	6	𝛽1	𝛽1	NOUN
cana-524	71	7	𝑟	𝑟	SYM
cana-524	71	8	>	>	X
cana-524	71	9	0	0	NUM
cana-524	71	10	,	,	PUNCT
cana-524	71	11	and	and	CCONJ
cana-524	71	12	𝜎(𝑡	𝜎(𝑡	NUM
cana-524	71	13	)	)	PUNCT
cana-524	71	14	<	<	X
cana-524	71	15	𝑡	𝑡	PROPN
cana-524	71	16	,	,	PUNCT
cana-524	71	17	it	it	PRON
cana-524	71	18	follows	follow	VERB
cana-524	71	19	that	that	SCONJ
cana-524	71	20	𝑥𝑠(𝜎(t	𝑥𝑠(𝜎(t	NOUN
cana-524	71	21	)	)	PUNCT
cana-524	71	22	)	)	PUNCT
cana-524	71	23	≥	≥	PROPN
cana-524	71	24	𝑑𝑠−𝛽1𝑣𝑠(𝜎(𝑡))𝑤	𝑑𝑠−𝛽1𝑣𝑠(𝜎(𝑡))𝑤	NUM
cana-524	71	25	𝛽1	𝛽1	PROPN
cana-524	71	26	𝑟	𝑟	NOUN
cana-524	71	27	𝜎((𝑡	𝜎((𝑡	PROPN
cana-524	71	28	)	)	PUNCT
cana-524	71	29	)	)	PUNCT
cana-524	71	30	≥	≥	PROPN
cana-524	71	31	𝑑𝑠−𝛽1𝑣𝑠(𝜎(𝑡))𝑤	𝑑𝑠−𝛽1𝑣𝑠(𝜎(𝑡))𝑤	NUM
cana-524	71	32	𝛽1	𝛽1	PROPN
cana-524	71	33	𝑟	𝑟	X
cana-524	71	34	(	(	PUNCT
cana-524	71	35	𝑡	𝑡	NOUN
cana-524	71	36	)	)	PUNCT
cana-524	71	37	.	.	PUNCT
cana-524	72	1	(	(	PUNCT
cana-524	72	2	3.7	3.7	NUM
cana-524	72	3	)	)	PUNCT
cana-524	72	4	communications	communication	NOUN
cana-524	72	5	on	on	ADP
cana-524	72	6	applied	apply	VERB
cana-524	72	7	nonlinear	nonlinear	ADJ
cana-524	72	8	analysis	analysis	NOUN
cana-524	72	9	issn	issn	NOUN
cana-524	72	10	:	:	PUNCT
cana-524	72	11	1074	1074	NUM
cana-524	72	12	-	-	PUNCT
cana-524	72	13	133x	133x	NUM
cana-524	72	14	vol	vol	NOUN
cana-524	72	15	31	31	NUM
cana-524	72	16	no	no	NOUN
cana-524	72	17	.	.	NOUN
cana-524	72	18	2	2	NUM
cana-524	72	19	(	(	PUNCT
cana-524	72	20	2024	2024	NUM
cana-524	72	21	)	)	PUNCT
cana-524	72	22	123	123	NUM
cana-524	72	23	https://internationalpubls.com	https://internationalpubls.com	X
cana-524	72	24	going	go	VERB
cana-524	72	25	back	back	ADV
cana-524	72	26	to	to	ADP
cana-524	72	27	(	(	PUNCT
cana-524	72	28	3.6	3.6	NUM
cana-524	72	29	)	)	PUNCT
cana-524	72	30	,	,	PUNCT
cana-524	72	31	we	we	PRON
cana-524	72	32	obtain	obtain	VERB
cana-524	72	33	𝑤1−	𝑤1−	PROPN
cana-524	72	34	𝛽1	𝛽1	NOUN
cana-524	72	35	𝑟	𝑟	X
cana-524	72	36	(	(	PUNCT
cana-524	72	37	𝜉2	𝜉2	PROPN
cana-524	72	38	)	)	PUNCT
cana-524	72	39	≥	≥	NOUN
cana-524	72	40	(	(	PUNCT
cana-524	72	41	1	1	NUM
cana-524	72	42	−	−	PROPN
cana-524	72	43	𝛽1	𝛽1	NOUN
cana-524	72	44	𝑟	𝑟	NOUN
cana-524	72	45	)	)	PUNCT
cana-524	72	46	𝑑𝑠−𝛽1	𝑑𝑠−𝛽1	NOUN
cana-524	72	47	[	[	X
cana-524	72	48	∑	∑	NOUN
cana-524	72	49	  	  	SPACE
cana-524	72	50	𝜉−1	𝜉−1	PROPN
cana-524	72	51	𝑡=𝜉2	𝑡=𝜉2	PROPN
cana-524	72	52	 	 	SPACE
cana-524	72	53	𝑞(𝑡)𝑥𝑠(𝜎(𝑡	𝑞(𝑡)𝑥𝑠(𝜎(𝑡	NOUN
cana-524	72	54	)	)	PUNCT
cana-524	72	55	)	)	PUNCT
cana-524	72	56	]	]	PUNCT
cana-524	72	57	.	.	PUNCT
cana-524	73	1	(	(	PUNCT
cana-524	73	2	3.8	3.8	NUM
cana-524	73	3	)	)	PUNCT
cana-524	73	4	since	since	SCONJ
cana-524	73	5	(	(	PUNCT
cana-524	73	6	1	1	NUM
cana-524	73	7	−	−	PROPN
cana-524	73	8	𝛽1	𝛽1	NOUN
cana-524	73	9	𝑟	𝑟	NOUN
cana-524	73	10	)	)	PUNCT
cana-524	73	11	>	>	X
cana-524	73	12	0	0	NUM
cana-524	73	13	,	,	PUNCT
cana-524	73	14	by	by	ADP
cana-524	73	15	(	(	PUNCT
cana-524	73	16	3.1	3.1	NUM
cana-524	73	17	)	)	PUNCT
cana-524	73	18	the	the	DET
cana-524	73	19	right	right	ADJ
cana-524	73	20	-	-	PUNCT
cana-524	73	21	hand	hand	NOUN
cana-524	73	22	side	side	NOUN
cana-524	73	23	approaches	approach	VERB
cana-524	73	24	+	+	NOUN
cana-524	73	25	∞	∞	NOUN
cana-524	73	26	as	as	ADP
cana-524	73	27	𝜉	𝜉	PROPN
cana-524	73	28	→	→	SYM
cana-524	73	29	∞.	∞.	PROPN
cana-524	73	30	in	in	ADP
cana-524	73	31	contradiction	contradiction	NOUN
cana-524	73	32	with	with	ADP
cana-524	73	33	(	(	PUNCT
cana-524	73	34	3.8	3.8	NUM
cana-524	73	35	)	)	PUNCT
cana-524	73	36	,	,	PUNCT
cana-524	73	37	this	this	PRON
cana-524	73	38	completes	complete	VERB
cana-524	73	39	the	the	DET
cana-524	73	40	sufficiency	sufficiency	NOUN
cana-524	73	41	proof	proof	NOUN
cana-524	73	42	for	for	ADP
cana-524	73	43	eventually	eventually	ADV
cana-524	73	44	positive	positive	ADJ
cana-524	73	45	solutions	solution	NOUN
cana-524	73	46	.	.	PUNCT
cana-524	74	1	similar	similar	ADJ
cana-524	74	2	to	to	ADP
cana-524	74	3	this	this	PRON
cana-524	74	4	the	the	DET
cana-524	74	5	eventually	eventually	ADV
cana-524	74	6	negative	negative	ADJ
cana-524	74	7	solution	solution	NOUN
cana-524	74	8	can	can	AUX
cana-524	74	9	be	be	AUX
cana-524	74	10	dealt	deal	VERB
cana-524	74	11	by	by	ADP
cana-524	74	12	introducing	introduce	VERB
cana-524	74	13	the	the	DET
cana-524	74	14	variables	variable	NOUN
cana-524	74	15	𝜎	𝜎	PROPN
cana-524	75	1	=	=	X
cana-524	75	2	−𝑥.	−𝑥.	PROPN
cana-524	75	3	then	then	ADV
cana-524	75	4	,	,	PUNCT
cana-524	75	5	the	the	DET
cana-524	75	6	necessary	necessary	ADJ
cana-524	75	7	part	part	NOUN
cana-524	75	8	can	can	AUX
cana-524	75	9	be	be	AUX
cana-524	75	10	shown	show	VERB
cana-524	75	11	by	by	ADP
cana-524	75	12	the	the	DET
cana-524	75	13	contrapositive	contrapositive	ADJ
cana-524	75	14	argument	argument	NOUN
cana-524	75	15	.	.	PUNCT
cana-524	76	1	if	if	SCONJ
cana-524	76	2	(	(	PUNCT
cana-524	76	3	3.1	3.1	NUM
cana-524	76	4	)	)	PUNCT
cana-524	76	5	is	be	AUX
cana-524	76	6	not	not	PART
cana-524	76	7	hold	hold	ADJ
cana-524	76	8	,	,	PUNCT
cana-524	76	9	then	then	ADV
cana-524	76	10	for	for	ADP
cana-524	76	11	each	each	DET
cana-524	76	12	𝛼	𝛼	X
cana-524	76	13	>	>	X
cana-524	76	14	0	0	PUNCT
cana-524	76	15	there	there	PRON
cana-524	76	16	exists	exist	VERB
cana-524	76	17	𝜉1	𝜉1	PROPN
cana-524	76	18	≥	≥	NOUN
cana-524	76	19	𝜉0	𝜉0	NOUN
cana-524	76	20	such	such	ADJ
cana-524	76	21	that	that	SCONJ
cana-524	76	22	∑	∑	ADP
cana-524	76	23	 	 	SPACE
cana-524	76	24	∞	∞	PROPN
cana-524	76	25	𝜁=𝑡	𝜁=𝑡	PROPN
cana-524	76	26	𝑞(𝜁)𝑣	𝑞(𝜁)𝑣	PROPN
cana-524	76	27	𝑠(𝜎(𝜁	𝑠(𝜎(𝜁	NOUN
cana-524	76	28	)	)	PUNCT
cana-524	76	29	)	)	PUNCT
cana-524	77	1	≤	≤	NUM
cana-524	77	2	𝛼	𝛼	X
cana-524	77	3	(	(	PUNCT
cana-524	77	4	1	1	NUM
cana-524	77	5	−	−	NUM
cana-524	77	6	𝑠	𝑠	NUM
cana-524	77	7	𝑟	𝑟	NOUN
cana-524	77	8	)	)	PUNCT
cana-524	77	9	2	2	NUM
cana-524	77	10	,	,	PUNCT
cana-524	77	11	for	for	ADP
cana-524	77	12	all	all	DET
cana-524	77	13	𝜉	𝜉	PRON
cana-524	77	14	≥	≥	NOUN
cana-524	77	15	𝜉1	𝜉1	NOUN
cana-524	77	16	.	.	PUNCT
cana-524	78	1	(	(	PUNCT
cana-524	78	2	3.9	3.9	NUM
cana-524	78	3	)	)	PUNCT
cana-524	78	4	we	we	PRON
cana-524	78	5	define	define	VERB
cana-524	78	6	𝑇	𝑇	PROPN
cana-524	78	7	=	=	SYM
cana-524	78	8	{	{	PUNCT
cana-524	78	9	𝑥	𝑥	NOUN
cana-524	78	10	:	:	PUNCT
cana-524	78	11	(	(	PUNCT
cana-524	78	12	𝛼	𝛼	NOUN
cana-524	78	13	2	2	NUM
cana-524	78	14	)	)	PUNCT
cana-524	78	15	1	1	NUM
cana-524	78	16	𝑟	𝑟	NOUN
cana-524	78	17	𝑣(𝜉	𝑣(𝜉	NOUN
cana-524	78	18	)	)	PUNCT
cana-524	78	19	≤	≤	NOUN
cana-524	78	20	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	78	21	)	)	PUNCT
cana-524	78	22	≤	≤	NUM
cana-524	78	23	𝛼	𝛼	DET
cana-524	78	24	1	1	NUM
cana-524	78	25	𝑟𝑣(𝜉	𝑟𝑣(𝜉	NOUN
cana-524	78	26	)	)	PUNCT
cana-524	78	27	,	,	PUNCT
cana-524	78	28	𝜉	𝜉	PROPN
cana-524	78	29	≥	≥	PROPN
cana-524	78	30	𝜉1	𝜉1	PROPN
cana-524	78	31	}	}	PUNCT
cana-524	78	32	.	.	PUNCT
cana-524	79	1	(	(	PUNCT
cana-524	79	2	3.10	3.10	NUM
cana-524	79	3	)	)	PUNCT
cana-524	79	4	an	an	DET
cana-524	79	5	operator	operator	NOUN
cana-524	79	6	𝜙	𝜙	NOUN
cana-524	79	7	is	be	AUX
cana-524	79	8	defined	define	VERB
cana-524	79	9	on	on	ADP
cana-524	79	10	t	t	PROPN
cana-524	79	11	by	by	ADP
cana-524	79	12	(	(	PUNCT
cana-524	79	13	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	79	14	)	)	PUNCT
cana-524	80	1	=	=	PRON
cana-524	80	2	{	{	PUNCT
cana-524	80	3	0	0	NUM
cana-524	80	4	,	,	PUNCT
cana-524	80	5	if	if	SCONJ
cana-524	80	6	𝜉	𝜉	PROPN
cana-524	80	7	≤	≤	PROPN
cana-524	80	8	𝜉1	𝜉1	PROPN
cana-524	80	9	,	,	PUNCT
cana-524	80	10	∑	∑	ADP
cana-524	80	11	  	  	SPACE
cana-524	80	12	𝜉−1	𝜉−1	CCONJ
cana-524	80	13	𝑡=𝜉1	𝑡=𝜉1	PROPN
cana-524	80	14	  	  	SPACE
cana-524	80	15	[	[	PUNCT
cana-524	80	16	1	1	NUM
cana-524	80	17	𝑝(𝑡	𝑝(𝑡	PROPN
cana-524	80	18	)	)	PUNCT
cana-524	80	19	[	[	PUNCT
cana-524	80	20	𝛼	𝛼	X
cana-524	80	21	2	2	NUM
cana-524	80	22	+	+	NOUN
cana-524	80	23	∑	∑	PROPN
cana-524	80	24	  	  	SPACE
cana-524	80	25	∞	∞	PROPN
cana-524	80	26	𝜁=𝑡	𝜁=𝑡	PROPN
cana-524	80	27	 	 	SPACE
cana-524	80	28	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	𝑞(𝜁)𝑥𝑠(𝜎(𝜁	NOUN
cana-524	80	29	)	)	PUNCT
cana-524	80	30	)	)	PUNCT
cana-524	81	1	]	]	PUNCT
cana-524	81	2	]	]	X
cana-524	81	3	1	1	NUM
cana-524	81	4	𝑟	𝑟	NOUN
cana-524	81	5	,	,	PUNCT
cana-524	81	6	if	if	SCONJ
cana-524	81	7	𝜉	𝜉	PROPN
cana-524	81	8	>	>	X
cana-524	81	9	𝜉1	𝜉1	PROPN
cana-524	81	10	.	.	PUNCT
cana-524	82	1	(	(	PUNCT
cana-524	82	2	3.11	3.11	NUM
cana-524	82	3	)	)	PUNCT
cana-524	82	4	if	if	SCONJ
cana-524	82	5	𝑥	𝑥	PROPN
cana-524	82	6	is	be	AUX
cana-524	82	7	a	a	DET
cana-524	82	8	fixed	fix	VERB
cana-524	82	9	point	point	NOUN
cana-524	82	10	of	of	ADP
cana-524	82	11	𝜙	𝜙	NOUN
cana-524	82	12	,	,	PUNCT
cana-524	82	13	i.e.	i.e.	X
cana-524	82	14	,	,	PUNCT
cana-524	82	15	𝜙𝑥	𝜙𝑥	PROPN
cana-524	83	1	=	=	SYM
cana-524	83	2	𝑥	𝑥	PROPN
cana-524	83	3	,	,	PUNCT
cana-524	83	4	then	then	ADV
cana-524	83	5	𝑥	𝑥	PROPN
cana-524	83	6	is	be	AUX
cana-524	83	7	a	a	DET
cana-524	83	8	solution	solution	NOUN
cana-524	83	9	of	of	ADP
cana-524	83	10	(	(	PUNCT
cana-524	83	11	1.1	1.1	NUM
cana-524	83	12	)	)	PUNCT
cana-524	83	13	.	.	PUNCT
cana-524	84	1	first	first	ADV
cana-524	84	2	,	,	PUNCT
cana-524	84	3	we	we	PRON
cana-524	84	4	estimate	estimate	VERB
cana-524	84	5	(	(	PUNCT
cana-524	84	6	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	84	7	)	)	PUNCT
cana-524	84	8	.	.	PUNCT
cana-524	85	1	by	by	ADP
cana-524	85	2	(	(	PUNCT
cana-524	85	3	𝐻3	𝐻3	PROPN
cana-524	85	4	)	)	PUNCT
cana-524	85	5	,	,	PUNCT
cana-524	85	6	we	we	PRON
cana-524	85	7	have	have	VERB
cana-524	85	8	(	(	PUNCT
cana-524	85	9	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	85	10	)	)	PUNCT
cana-524	85	11	≥	≥	NOUN
cana-524	85	12	∑	∑	ADP
cana-524	85	13	  	  	SPACE
cana-524	85	14	𝜉−1	𝜉−1	CCONJ
cana-524	85	15	𝑡=𝜉1	𝑡=𝜉1	PROPN
cana-524	85	16	[	[	PUNCT
cana-524	85	17	1	1	NUM
cana-524	85	18	𝑝(𝑡	𝑝(𝑡	PROPN
cana-524	85	19	)	)	PUNCT
cana-524	85	20	(	(	PUNCT
cana-524	85	21	𝛼	𝛼	NOUN
cana-524	85	22	2	2	NUM
cana-524	85	23	+	+	NUM
cana-524	85	24	0	0	NUM
cana-524	85	25	)	)	PUNCT
cana-524	85	26	]	]	PUNCT
cana-524	86	1	1	1	NUM
cana-524	86	2	𝑟	𝑟	NOUN
cana-524	86	3	=	=	SYM
cana-524	86	4	(	(	PUNCT
cana-524	86	5	𝛼	𝛼	NOUN
cana-524	86	6	2	2	NUM
cana-524	86	7	)	)	PUNCT
cana-524	86	8	1	1	NUM
cana-524	86	9	𝑟	𝑟	NOUN
cana-524	86	10	𝑣(𝜉	𝑣(𝜉	NOUN
cana-524	86	11	)	)	PUNCT
cana-524	86	12	.	.	PUNCT
cana-524	87	1	(	(	PUNCT
cana-524	87	2	3.12	3.12	NUM
cana-524	87	3	)	)	PUNCT
cana-524	87	4	now	now	ADV
cana-524	87	5	,	,	PUNCT
cana-524	87	6	we	we	PRON
cana-524	87	7	establish	establish	VERB
cana-524	87	8	(	(	PUNCT
cana-524	87	9	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	87	10	)	)	PUNCT
cana-524	87	11	from	from	ADP
cana-524	87	12	above	above	ADV
cana-524	87	13	.	.	PUNCT
cana-524	88	1	for	for	ADP
cana-524	88	2	𝑥	𝑥	PROPN
cana-524	88	3	in	in	ADP
cana-524	88	4	𝑇	𝑇	PROPN
cana-524	88	5	,	,	PUNCT
cana-524	88	6	as	as	SCONJ
cana-524	88	7	we	we	PRON
cana-524	88	8	have	have	VERB
cana-524	88	9	𝑥𝑠(𝜎(𝜁	𝑥𝑠(𝜎(𝜁	NOUN
cana-524	88	10	)	)	PUNCT
cana-524	88	11	)	)	PUNCT
cana-524	88	12	≤	≤	NOUN
cana-524	88	13	(	(	PUNCT
cana-524	88	14	𝛼	𝛼	NOUN
cana-524	88	15	1	1	NUM
cana-524	88	16	𝛼𝑣(𝜎(𝜁	𝛼𝑣(𝜎(𝜁	NOUN
cana-524	88	17	)	)	PUNCT
cana-524	88	18	)	)	PUNCT
cana-524	88	19	)	)	PUNCT
cana-524	89	1	𝑠	𝑠	X
cana-524	89	2	.	.	PUNCT
cana-524	90	1	communications	communication	NOUN
cana-524	90	2	on	on	ADP
cana-524	90	3	applied	apply	VERB
cana-524	90	4	nonlinear	nonlinear	ADJ
cana-524	90	5	analysis	analysis	NOUN
cana-524	90	6	issn	issn	NOUN
cana-524	90	7	:	:	PUNCT
cana-524	90	8	1074	1074	NUM
cana-524	90	9	-	-	PUNCT
cana-524	90	10	133x	133x	NUM
cana-524	90	11	vol	vol	NOUN
cana-524	90	12	31	31	NUM
cana-524	90	13	no	no	NOUN
cana-524	90	14	.	.	NOUN
cana-524	90	15	2	2	NUM
cana-524	90	16	(	(	PUNCT
cana-524	90	17	2024	2024	NUM
cana-524	90	18	)	)	PUNCT
cana-524	90	19	124	124	NUM
cana-524	90	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-524	90	21	then	then	ADV
cana-524	90	22	,	,	PUNCT
cana-524	90	23	by	by	ADP
cana-524	90	24	(	(	PUNCT
cana-524	90	25	3.9	3.9	NUM
cana-524	90	26	)	)	PUNCT
cana-524	90	27	,	,	PUNCT
cana-524	90	28	(	(	PUNCT
cana-524	90	29	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	90	30	)	)	PUNCT
cana-524	90	31	≤	≤	NOUN
cana-524	90	32	∑	∑	ADP
cana-524	90	33	  	  	SPACE
cana-524	90	34	𝜉−1	𝜉−1	PRON
cana-524	90	35	𝑡=𝜉1	𝑡=𝜉1	PROPN
cana-524	90	36	  	  	SPACE
cana-524	90	37	[	[	PUNCT
cana-524	90	38	1	1	NUM
cana-524	90	39	𝑝(𝑡	𝑝(𝑡	PROPN
cana-524	90	40	)	)	PUNCT
cana-524	90	41	[	[	PUNCT
cana-524	90	42	𝛼	𝛼	X
cana-524	90	43	2	2	NUM
cana-524	90	44	+	+	NOUN
cana-524	90	45	∑	∑	PROPN
cana-524	90	46	  	  	SPACE
cana-524	90	47	∞	∞	PROPN
cana-524	90	48	𝜁=𝑡	𝜁=𝑡	PROPN
cana-524	90	49	 	 	SPACE
cana-524	90	50	𝑝(𝜁)𝑥𝑠(𝜎(𝜁	𝑝(𝜁)𝑥𝑠(𝜎(𝜁	NOUN
cana-524	90	51	)	)	PUNCT
cana-524	90	52	)	)	PUNCT
cana-524	91	1	]	]	PUNCT
cana-524	91	2	]	]	X
cana-524	91	3	1	1	NUM
cana-524	91	4	𝑟	𝑟	X
cana-524	91	5	≤	≤	NUM
cana-524	91	6	𝛼	𝛼	DET
cana-524	91	7	1	1	NUM
cana-524	91	8	𝑟𝑣(𝜉	𝑟𝑣(𝜉	NOUN
cana-524	91	9	)	)	PUNCT
cana-524	91	10	.	.	PUNCT
cana-524	92	1	(	(	PUNCT
cana-524	92	2	3.13	3.13	NUM
cana-524	92	3	)	)	PUNCT
cana-524	92	4	therefore	therefore	ADV
cana-524	92	5	,	,	PUNCT
cana-524	92	6	𝜙	𝜙	PROPN
cana-524	92	7	maps	map	VERB
cana-524	92	8	𝑇	𝑇	PROPN
cana-524	92	9	to	to	ADP
cana-524	92	10	𝑇	𝑇	PROPN
cana-524	92	11	,	,	PUNCT
cana-524	92	12	next	next	ADV
cana-524	92	13	,	,	PUNCT
cana-524	92	14	we	we	PRON
cana-524	92	15	find	find	VERB
cana-524	92	16	a	a	DET
cana-524	92	17	fixed	fix	VERB
cana-524	92	18	point	point	NOUN
cana-524	92	19	for	for	ADP
cana-524	92	20	𝜙	𝜙	PROPN
cana-524	92	21	in	in	ADP
cana-524	92	22	𝑇.	𝑇.	PROPN
cana-524	92	23	let	let	VERB
cana-524	92	24	us	we	PRON
cana-524	92	25	define	define	VERB
cana-524	92	26	a	a	DET
cana-524	92	27	sequence	sequence	NOUN
cana-524	92	28	of	of	ADP
cana-524	92	29	functions	function	NOUN
cana-524	92	30	in	in	ADP
cana-524	92	31	𝑇	𝑇	PROPN
cana-524	92	32	by	by	ADP
cana-524	92	33	the	the	DET
cana-524	92	34	recurrence	recurrence	NOUN
cana-524	92	35	relation	relation	NOUN
cana-524	92	36	𝜎0(𝜉	𝜎0(𝜉	PROPN
cana-524	92	37	)	)	PUNCT
cana-524	92	38	=	=	SYM
cana-524	92	39	0	0	NUM
cana-524	92	40	,	,	PUNCT
cana-524	92	41	for	for	ADP
cana-524	92	42	𝜉	𝜉	PROPN
cana-524	92	43	≥	≥	NOUN
cana-524	92	44	𝜉0	𝜉0	NOUN
cana-524	92	45	,	,	PUNCT
cana-524	92	46	𝜎1(𝜉	𝜎1(𝜉	NUM
cana-524	92	47	)	)	PUNCT
cana-524	92	48	=	=	SYM
cana-524	92	49	(	(	PUNCT
cana-524	92	50	𝜙𝜎0)(𝜉	𝜙𝜎0)(𝜉	PROPN
cana-524	92	51	)	)	PUNCT
cana-524	92	52	=	=	PRON
cana-524	93	1	{	{	PUNCT
cana-524	93	2	0	0	NUM
cana-524	93	3	,	,	PUNCT
cana-524	93	4	if	if	SCONJ
cana-524	93	5	𝜉	𝜉	X
cana-524	93	6	<	<	X
cana-524	93	7	𝜉1	𝜉1	PROPN
cana-524	93	8	,	,	PUNCT
cana-524	93	9	𝛼	𝛼	PROPN
cana-524	93	10	1	1	NUM
cana-524	93	11	𝑟𝑣(𝜉	𝑟𝑣(𝜉	NOUN
cana-524	93	12	)	)	PUNCT
cana-524	93	13	,	,	PUNCT
cana-524	93	14	if	if	SCONJ
cana-524	93	15	𝜉	𝜉	PROPN
cana-524	93	16	≥	≥	X
cana-524	93	17	𝜉1	𝜉1	PROPN
cana-524	93	18	,	,	PUNCT
cana-524	93	19	𝜎n+1(𝜉	𝜎n+1(𝜉	PROPN
cana-524	93	20	)	)	PUNCT
cana-524	93	21	=	=	PUNCT
cana-524	93	22	(	(	PUNCT
cana-524	93	23	𝜙𝜎n)(𝜉	𝜙𝜎n)(𝜉	NOUN
cana-524	93	24	)	)	PUNCT
cana-524	93	25	,	,	PUNCT
cana-524	93	26	for	for	ADP
cana-524	93	27	n	n	PRON
cana-524	93	28	≥	≥	NUM
cana-524	93	29	1	1	NUM
cana-524	93	30	,	,	PUNCT
cana-524	93	31	𝜉	𝜉	X
cana-524	93	32	≥	≥	PROPN
cana-524	93	33	𝜉1	𝜉1	PROPN
cana-524	93	34	.	.	PUNCT
cana-524	94	1	(	(	PUNCT
cana-524	94	2	3.14	3.14	NUM
cana-524	94	3	)	)	PUNCT
cana-524	94	4	note	note	VERB
cana-524	94	5	that	that	SCONJ
cana-524	94	6	for	for	ADP
cana-524	94	7	each	each	DET
cana-524	94	8	fixed	fix	VERB
cana-524	94	9	𝜉	𝜉	NOUN
cana-524	94	10	,	,	PUNCT
cana-524	94	11	we	we	PRON
cana-524	94	12	have	have	VERB
cana-524	94	13	𝜎1(𝜉	𝜎1(𝜉	NUM
cana-524	94	14	)	)	PUNCT
cana-524	94	15	≥	≥	NOUN
cana-524	94	16	𝜎0(𝜉	𝜎0(𝜉	NUM
cana-524	94	17	)	)	PUNCT
cana-524	94	18	.	.	PUNCT
cana-524	95	1	using	use	VERB
cana-524	95	2	mathematical	mathematical	ADJ
cana-524	95	3	induction	induction	NOUN
cana-524	95	4	,	,	PUNCT
cana-524	95	5	we	we	PRON
cana-524	95	6	can	can	AUX
cana-524	95	7	show	show	VERB
cana-524	95	8	that	that	SCONJ
cana-524	95	9	𝜎n+1(𝜉	𝜎n+1(𝜉	PROPN
cana-524	95	10	)	)	PUNCT
cana-524	95	11	≥	≥	NOUN
cana-524	95	12	𝜎n(𝜉	𝜎n(𝜉	NUM
cana-524	95	13	)	)	PUNCT
cana-524	95	14	.	.	PUNCT
cana-524	96	1	therefore	therefore	ADV
cana-524	96	2	,	,	PUNCT
cana-524	96	3	the	the	DET
cana-524	96	4	sequence	sequence	NOUN
cana-524	96	5	{	{	PUNCT
cana-524	96	6	𝜎n	𝜎n	NOUN
cana-524	96	7	}	}	PUNCT
cana-524	96	8	converges	converge	VERB
cana-524	96	9	pointwise	pointwise	VERB
cana-524	96	10	to	to	ADP
cana-524	96	11	a	a	DET
cana-524	96	12	sequence	sequence	NOUN
cana-524	96	13	𝜎.	𝜎.	NOUN
cana-524	96	14	using	use	VERB
cana-524	96	15	the	the	DET
cana-524	96	16	lebesgue	lebesgue	NOUN
cana-524	96	17	dominated	dominate	VERB
cana-524	96	18	convergence	convergence	NOUN
cana-524	96	19	theorem	theorem	VERB
cana-524	96	20	,	,	PUNCT
cana-524	96	21	we	we	PRON
cana-524	96	22	can	can	AUX
cana-524	96	23	show	show	VERB
cana-524	96	24	that	that	SCONJ
cana-524	96	25	𝜎	𝜎	PROPN
cana-524	96	26	is	be	AUX
cana-524	96	27	a	a	DET
cana-524	96	28	fixed	fix	VERB
cana-524	96	29	point	point	NOUN
cana-524	96	30	of	of	ADP
cana-524	96	31	𝜙	𝜙	PROPN
cana-524	96	32	in	in	ADP
cana-524	96	33	𝑇.	𝑇.	PROPN
cana-524	96	34	this	this	PRON
cana-524	96	35	shows	show	VERB
cana-524	96	36	under	under	ADP
cana-524	96	37	assumption	assumption	NOUN
cana-524	96	38	(	(	PUNCT
cana-524	96	39	3.9	3.9	NUM
cana-524	96	40	)	)	PUNCT
cana-524	96	41	,	,	PUNCT
cana-524	96	42	there	there	PRON
cana-524	96	43	is	be	VERB
cana-524	96	44	a	a	DET
cana-524	96	45	nonoscillatory	nonoscillatory	ADJ
cana-524	96	46	solution	solution	NOUN
cana-524	96	47	that	that	PRON
cana-524	96	48	dose	dose	VERB
cana-524	96	49	not	not	PART
cana-524	96	50	converge	converge	VERB
cana-524	96	51	to	to	ADP
cana-524	96	52	zero	zero	NUM
cana-524	96	53	.	.	PUNCT
cana-524	97	1	this	this	PRON
cana-524	97	2	concludes	conclude	VERB
cana-524	97	3	the	the	DET
cana-524	97	4	proof	proof	NOUN
cana-524	97	5	.	.	PUNCT
cana-524	98	1	theorem	theorem	ADJ
cana-524	98	2	3.2	3.2	NUM
cana-524	98	3	.	.	PUNCT
cana-524	99	1	assume	assume	VERB
cana-524	99	2	that	that	SCONJ
cana-524	99	3	there	there	PRON
cana-524	99	4	exists	exist	VERB
cana-524	99	5	a	a	DET
cana-524	99	6	constant	constant	ADJ
cana-524	99	7	𝛽2	𝛽2	NOUN
cana-524	99	8	,	,	PUNCT
cana-524	99	9	the	the	DET
cana-524	99	10	quotient	quotient	NOUN
cana-524	99	11	of	of	ADP
cana-524	99	12	two	two	NUM
cana-524	99	13	positive	positive	ADJ
cana-524	99	14	odd	odd	ADJ
cana-524	99	15	integers	integer	NOUN
cana-524	99	16	,	,	PUNCT
cana-524	99	17	such	such	ADJ
cana-524	99	18	that	that	SCONJ
cana-524	99	19	0	0	NUM
cana-524	99	20	<	<	X
cana-524	99	21	𝑟	𝑟	X
cana-524	99	22	<	<	X
cana-524	99	23	𝛽2	𝛽2	PROPN
cana-524	99	24	<	<	X
cana-524	99	25	𝑠.	𝑠.	NOUN
cana-524	99	26	if	if	SCONJ
cana-524	99	27	(	(	PUNCT
cana-524	99	28	𝐻1	𝐻1	PROPN
cana-524	99	29	)	)	PUNCT
cana-524	100	1	−	−	PROPN
cana-524	100	2	(	(	PUNCT
cana-524	100	3	𝐻4	𝐻4	PROPN
cana-524	100	4	)	)	PUNCT
cana-524	100	5	hold	hold	VERB
cana-524	100	6	and	and	CCONJ
cana-524	100	7	𝑝(𝜉	𝑝(𝜉	NOUN
cana-524	100	8	)	)	PUNCT
cana-524	100	9	is	be	AUX
cana-524	100	10	nondecreasing	nondecrease	VERB
cana-524	100	11	,	,	PUNCT
cana-524	100	12	then	then	ADV
cana-524	100	13	each	each	DET
cana-524	100	14	solution	solution	NOUN
cana-524	100	15	of	of	ADP
cana-524	100	16	(	(	PUNCT
cana-524	100	17	1.1	1.1	NUM
cana-524	100	18	)	)	PUNCT
cana-524	100	19	is	be	AUX
cana-524	100	20	oscillatory	oscillatory	ADJ
cana-524	100	21	if	if	SCONJ
cana-524	100	22	and	and	CCONJ
cana-524	100	23	only	only	ADV
cana-524	100	24	if	if	SCONJ
cana-524	100	25	∑	∑	PUNCT
cana-524	100	26	 	 	SPACE
cana-524	100	27	∞	∞	PROPN
cana-524	100	28	𝑠=𝜉1	𝑠=𝜉1	PROPN
cana-524	100	29	[	[	PUNCT
cana-524	100	30	1	1	NUM
cana-524	100	31	𝑝(𝑠	𝑝(𝑠	NOUN
cana-524	100	32	)	)	PUNCT
cana-524	100	33	∑	∑	ADP
cana-524	100	34	 	 	SPACE
cana-524	100	35	∞	∞	PRON
cana-524	100	36	𝜁=𝑠	𝜁=𝑠	PUNCT
cana-524	100	37	 	 	SPACE
cana-524	100	38	𝑞(𝜁	𝑞(𝜁	PROPN
cana-524	100	39	)	)	PUNCT
cana-524	100	40	]	]	PUNCT
cana-524	100	41	1	1	NUM
cana-524	100	42	𝑟	𝑟	X
cana-524	100	43	=	=	SYM
cana-524	100	44	∞.	∞.	PROPN
cana-524	100	45	(	(	PUNCT
cana-524	100	46	3.15	3.15	NUM
cana-524	100	47	)	)	PUNCT
cana-524	100	48	proof	proof	NOUN
cana-524	100	49	.	.	PUNCT
cana-524	101	1	on	on	ADP
cana-524	101	2	the	the	DET
cana-524	101	3	contrary	contrary	NOUN
cana-524	101	4	,	,	PUNCT
cana-524	101	5	consider	consider	VERB
cana-524	101	6	that	that	SCONJ
cana-524	101	7	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	101	8	)	)	PUNCT
cana-524	101	9	is	be	AUX
cana-524	101	10	an	an	DET
cana-524	101	11	eventually	eventually	ADV
cana-524	101	12	positive	positive	ADJ
cana-524	101	13	solution	solution	NOUN
cana-524	101	14	that	that	PRON
cana-524	101	15	does	do	AUX
cana-524	101	16	not	not	PART
cana-524	101	17	converge	converge	VERB
cana-524	101	18	to	to	ADP
cana-524	101	19	zero	zero	NUM
cana-524	101	20	.	.	PUNCT
cana-524	102	1	using	use	VERB
cana-524	102	2	the	the	DET
cana-524	102	3	same	same	ADJ
cana-524	102	4	argument	argument	NOUN
cana-524	102	5	as	as	ADP
cana-524	102	6	in	in	ADP
cana-524	102	7	lemma	lemma	PROPN
cana-524	102	8	2.1	2.1	NUM
cana-524	102	9	,	,	PUNCT
cana-524	102	10	there	there	PRON
cana-524	102	11	exits	exit	VERB
cana-524	102	12	𝜉1	𝜉1	PROPN
cana-524	102	13	≥	≥	NOUN
cana-524	102	14	𝜉0	𝜉0	NOUN
cana-524	102	15	such	such	ADJ
cana-524	102	16	that	that	DET
cana-524	102	17	𝑥(𝜎(𝜉	𝑥(𝜎(𝜉	NOUN
cana-524	102	18	)	)	PUNCT
cana-524	102	19	)	)	PUNCT
cana-524	103	1	>	>	X
cana-524	103	2	0	0	PUNCT
cana-524	104	1	and	and	CCONJ
cana-524	104	2	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	NOUN
cana-524	104	3	is	be	AUX
cana-524	104	4	positive	positive	ADJ
cana-524	104	5	and	and	CCONJ
cana-524	104	6	nonincreasing	nonincrease	VERB
cana-524	104	7	.	.	PUNCT
cana-524	105	1	since	since	SCONJ
cana-524	105	2	𝑝(𝜉	𝑝(𝜉	NOUN
cana-524	105	3	)	)	PUNCT
cana-524	105	4	>	>	X
cana-524	105	5	0	0	NUM
cana-524	105	6	,	,	PUNCT
cana-524	105	7	𝑥(𝜉	𝑥(𝜉	PROPN
cana-524	105	8	)	)	PUNCT
cana-524	105	9	is	be	AUX
cana-524	105	10	increasing	increase	VERB
cana-524	105	11	for	for	ADP
cana-524	105	12	𝜉	𝜉	PROPN
cana-524	105	13	≥	≥	PROPN
cana-524	105	14	𝜉1	𝜉1	PROPN
cana-524	105	15	.	.	PUNCT
cana-524	106	1	using	use	VERB
cana-524	106	2	𝑥(𝜉	𝑥(𝜉	PROPN
cana-524	106	3	)	)	PUNCT
cana-524	106	4	≥	≥	NOUN
cana-524	106	5	𝑥(𝜉1	𝑥(𝜉1	NOUN
cana-524	106	6	)	)	PUNCT
cana-524	106	7	,	,	PUNCT
cana-524	106	8	we	we	PRON
cana-524	106	9	have	have	VERB
cana-524	106	10	𝑥𝑠(𝜉	𝑥𝑠(𝜉	PUNCT
cana-524	106	11	)	)	PUNCT
cana-524	106	12	≥	≥	NOUN
cana-524	106	13	𝑥𝑠−𝛽2(𝜉)𝑥𝛽2(𝜉	𝑥𝑠−𝛽2(𝜉)𝑥𝛽2(𝜉	NUM
cana-524	106	14	)	)	PUNCT
cana-524	106	15	≥	≥	PROPN
cana-524	106	16	𝑥𝑠−𝛽2(𝜉1)𝑥	𝑥𝑠−𝛽2(𝜉1)𝑥	PROPN
cana-524	106	17	𝛽2(𝜉	𝛽2(𝜉	NOUN
cana-524	106	18	)	)	PUNCT
cana-524	106	19	,	,	PUNCT
cana-524	106	20	(	(	PUNCT
cana-524	106	21	3.16	3.16	NUM
cana-524	106	22	)	)	PUNCT
cana-524	106	23	and	and	CCONJ
cana-524	106	24	hence	hence	ADV
cana-524	106	25	𝑥𝑠	𝑥𝑠	PROPN
cana-524	106	26	(	(	PUNCT
cana-524	106	27	𝜎(𝜉	𝜎(𝜉	PROPN
cana-524	106	28	)	)	PUNCT
cana-524	106	29	)	)	PUNCT
cana-524	107	1	≥	≥	PROPN
cana-524	107	2	𝑥𝑠−𝛽2(𝜉1)𝑥	𝑥𝑠−𝛽2(𝜉1)𝑥	PROPN
cana-524	107	3	𝛽2(𝜎(𝜉	𝛽2(𝜎(𝜉	PROPN
cana-524	107	4	)	)	PUNCT
cana-524	107	5	)	)	PUNCT
cana-524	107	6	,	,	PUNCT
cana-524	107	7	for	for	ADP
cana-524	107	8	𝜉	𝜉	X
cana-524	107	9	≥	≥	NOUN
cana-524	107	10	𝜉2	𝜉2	PROPN
cana-524	107	11	(	(	PUNCT
cana-524	107	12	3.17	3.17	NUM
cana-524	107	13	)	)	PUNCT
cana-524	107	14	communications	communication	NOUN
cana-524	107	15	on	on	ADP
cana-524	107	16	applied	apply	VERB
cana-524	107	17	nonlinear	nonlinear	ADJ
cana-524	107	18	analysis	analysis	NOUN
cana-524	107	19	issn	issn	NOUN
cana-524	107	20	:	:	PUNCT
cana-524	107	21	1074	1074	NUM
cana-524	107	22	-	-	PUNCT
cana-524	107	23	133x	133x	NUM
cana-524	107	24	vol	vol	NOUN
cana-524	107	25	31	31	NUM
cana-524	107	26	no	no	NOUN
cana-524	107	27	.	.	NOUN
cana-524	107	28	2	2	NUM
cana-524	107	29	(	(	PUNCT
cana-524	107	30	2024	2024	NUM
cana-524	107	31	)	)	PUNCT
cana-524	107	32	125	125	NUM
cana-524	107	33	https://internationalpubls.com	https://internationalpubls.com	X
cana-524	107	34	using	use	VERB
cana-524	107	35	(	(	PUNCT
cana-524	107	36	3.17	3.17	NUM
cana-524	107	37	)	)	PUNCT
cana-524	107	38	and	and	CCONJ
cana-524	107	39	𝜎(𝜉	𝜎(𝜉	PROPN
cana-524	107	40	)	)	PUNCT
cana-524	107	41	≥	≥	NOUN
cana-524	108	1	𝜎0(𝜉	𝜎0(𝜉	NUM
cana-524	108	2	)	)	PUNCT
cana-524	108	3	,	,	PUNCT
cana-524	108	4	from	from	ADP
cana-524	108	5	(	(	PUNCT
cana-524	108	6	2.10	2.10	NUM
cana-524	108	7	)	)	PUNCT
cana-524	108	8	,	,	PUNCT
cana-524	108	9	we	we	PRON
cana-524	108	10	have	have	VERB
cana-524	108	11	𝑝(𝜉)(δ𝑥(𝜉))𝑟	𝑝(𝜉)(δ𝑥(𝜉))𝑟	NOUN
cana-524	108	12	≥	≥	PROPN
cana-524	108	13	𝑥𝑠−𝛽2(𝜉1)𝑥	𝑥𝑠−𝛽2(𝜉1)𝑥	PROPN
cana-524	108	14	𝛽2(𝜎0(𝜉))∑	𝛽2(𝜎0(𝜉))∑	PROPN
cana-524	108	15	 	 	SPACE
cana-524	108	16	∞	∞	PROPN
cana-524	108	17	𝑡=𝜉	𝑡=𝜉	PUNCT
cana-524	108	18	𝑞(𝑡	𝑞(𝑡	NUM
cana-524	108	19	)	)	PUNCT
cana-524	108	20	,	,	PUNCT
cana-524	108	21	for	for	ADP
cana-524	108	22	𝜉	𝜉	X
cana-524	108	23	≥	≥	NOUN
cana-524	108	24	𝜉2	𝜉2	PROPN
cana-524	108	25	.	.	PUNCT
cana-524	109	1	(	(	PUNCT
cana-524	109	2	3.18	3.18	NUM
cana-524	109	3	)	)	PUNCT
cana-524	109	4	from	from	ADP
cana-524	109	5	𝑝(𝜉)(δ𝑥(𝜉))r	𝑝(𝜉)(δ𝑥(𝜉))r	NUM
cana-524	109	6	being	be	AUX
cana-524	109	7	nonincreasing	nonincrease	VERB
cana-524	109	8	and	and	CCONJ
cana-524	109	9	𝜎0(𝜉	𝜎0(𝜉	NOUN
cana-524	109	10	)	)	PUNCT
cana-524	109	11	≤	≤	NUM
cana-524	109	12	𝜉	𝜉	PUNCT
cana-524	109	13	,	,	PUNCT
cana-524	109	14	we	we	PRON
cana-524	109	15	have	have	VERB
cana-524	109	16	𝑝(𝜎0(𝜉))(δ𝑥(𝜎0(𝜉	𝑝(𝜎0(𝜉))(δ𝑥(𝜎0(𝜉	NUM
cana-524	109	17	)	)	PUNCT
cana-524	109	18	)	)	PUNCT
cana-524	109	19	)	)	PUNCT
cana-524	110	1	𝑟	𝑟	X
cana-524	110	2	≥	≥	X
cana-524	110	3	𝑝(𝜉)(δ𝑥(𝜉))𝑟.	𝑝(𝜉)(δ𝑥(𝜉))𝑟.	NOUN
cana-524	110	4	(	(	PUNCT
cana-524	110	5	3.19	3.19	NUM
cana-524	110	6	)	)	PUNCT
cana-524	110	7	we	we	PRON
cana-524	110	8	apply	apply	VERB
cana-524	110	9	this	this	PRON
cana-524	110	10	in	in	ADP
cana-524	110	11	the	the	DET
cana-524	110	12	left	left	ADJ
cana-524	110	13	-	-	PUNCT
cana-524	110	14	hand	hand	NOUN
cana-524	110	15	side	side	NOUN
cana-524	110	16	of	of	ADP
cana-524	110	17	(	(	PUNCT
cana-524	110	18	3.18	3.18	NUM
cana-524	110	19	)	)	PUNCT
cana-524	110	20	.	.	PUNCT
cana-524	111	1	then	then	ADV
cana-524	111	2	,	,	PUNCT
cana-524	111	3	dividing	divide	VERB
cana-524	111	4	by	by	ADP
cana-524	111	5	𝑝(𝜎0(𝜉))𝑥	𝑝(𝜎0(𝜉))𝑥	NUM
cana-524	111	6	𝛽2(𝜎0(𝜉	𝛽2(𝜎0(𝜉	NOUN
cana-524	111	7	)	)	PUNCT
cana-524	111	8	)	)	PUNCT
cana-524	111	9	>	>	X
cana-524	111	10	0	0	PUNCT
cana-524	111	11	and	and	CCONJ
cana-524	111	12	raising	raise	VERB
cana-524	111	13	both	both	DET
cana-524	111	14	side	side	NOUN
cana-524	111	15	to	to	ADP
cana-524	111	16	the	the	DET
cana-524	111	17	1	1	NUM
cana-524	111	18	𝑟	𝑟	NOUN
cana-524	111	19	power	power	NOUN
cana-524	111	20	,	,	PUNCT
cana-524	111	21	we	we	PRON
cana-524	111	22	get	get	VERB
cana-524	111	23	δ𝑥(𝜎0(𝜉	δ𝑥(𝜎0(𝜉	NOUN
cana-524	111	24	)	)	PUNCT
cana-524	111	25	)	)	PUNCT
cana-524	112	1	𝑥	𝑥	PROPN
cana-524	112	2	𝛽2	𝛽2	PROPN
cana-524	112	3	𝑟	𝑟	NOUN
cana-524	112	4	(	(	PUNCT
cana-524	112	5	𝜎0(𝜉	𝜎0(𝜉	NOUN
cana-524	112	6	)	)	PUNCT
cana-524	112	7	)	)	PUNCT
cana-524	112	8	≥	≥	NOUN
cana-524	112	9	[	[	PUNCT
cana-524	112	10	𝑥𝑠−𝛽2(𝜉1	𝑥𝑠−𝛽2(𝜉1	X
cana-524	112	11	)	)	PUNCT
cana-524	112	12	𝑝(𝜎0(𝜉	𝑝(𝜎0(𝜉	NOUN
cana-524	112	13	)	)	PUNCT
cana-524	112	14	)	)	PUNCT
cana-524	112	15	∑	∑	ADP
cana-524	112	16	 	 	SPACE
cana-524	112	17	∞	∞	PROPN
cana-524	112	18	𝑡=𝜉	𝑡=𝜉	PRON
cana-524	112	19	 	 	SPACE
cana-524	112	20	𝑞(𝑡	𝑞(𝑡	PROPN
cana-524	112	21	)	)	PUNCT
cana-524	112	22	]	]	PUNCT
cana-524	112	23	1	1	NUM
cana-524	112	24	𝑟	𝑟	NOUN
cana-524	112	25	,	,	PUNCT
cana-524	112	26	for	for	ADP
cana-524	112	27	𝜉	𝜉	X
cana-524	112	28	≥	≥	NOUN
cana-524	112	29	𝜉2	𝜉2	PROPN
cana-524	112	30	.	.	PUNCT
cana-524	113	1	(	(	PUNCT
cana-524	113	2	3.20	3.20	NUM
cana-524	113	3	)	)	PUNCT
cana-524	113	4	multiplying	multiply	VERB
cana-524	113	5	the	the	DET
cana-524	113	6	left	left	ADJ
cana-524	113	7	hand	hand	NOUN
cana-524	113	8	side	side	NOUN
cana-524	113	9	by	by	ADP
cana-524	113	10	δ𝜎0(𝜉	δ𝜎0(𝜉	PROPN
cana-524	113	11	)	)	PUNCT
cana-524	113	12	𝜎0	𝜎0	NOUN
cana-524	113	13	≥	≥	NOUN
cana-524	113	14	1	1	NUM
cana-524	113	15	and	and	CCONJ
cana-524	113	16	summing	sum	VERB
cana-524	113	17	from	from	ADP
cana-524	113	18	𝜉2	𝜉2	PROPN
cana-524	113	19	to	to	ADP
cana-524	113	20	𝜉	𝜉	PROPN
cana-524	113	21	−	−	PROPN
cana-524	113	22	1	1	NUM
cana-524	113	23	,	,	PUNCT
cana-524	113	24	we	we	PRON
cana-524	113	25	have	have	VERB
cana-524	113	26	1	1	NUM
cana-524	113	27	𝜎0	𝜎0	NUM
cana-524	113	28	∑	∑	PUNCT
cana-524	113	29	δ𝑥(𝜎0(t))δ𝜎0(t	δ𝑥(𝜎0(t))δ𝜎0(t	PROPN
cana-524	113	30	)	)	PUNCT
cana-524	113	31	𝑥	𝑥	PROPN
cana-524	113	32	𝛽2	𝛽2	PROPN
cana-524	113	33	𝑟	𝑟	NOUN
cana-524	113	34	(	(	PUNCT
cana-524	113	35	𝜎0(t	𝜎0(t	NOUN
cana-524	113	36	)	)	PUNCT
cana-524	113	37	)	)	PUNCT
cana-524	114	1	𝜉−1	𝜉−1	PROPN
cana-524	114	2	𝑡=𝜉2	𝑡=𝜉2	PROPN
cana-524	114	3	≥	≥	NUM
cana-524	114	4	𝑥𝑠−𝛽2(𝜉1	𝑥𝑠−𝛽2(𝜉1	ADV
cana-524	114	5	)	)	PUNCT
cana-524	115	1	[	[	X
cana-524	115	2	∑	∑	INTJ
cana-524	115	3	1	1	NUM
cana-524	115	4	𝑝(𝜎0(t	𝑝(𝜎0(t	NOUN
cana-524	115	5	)	)	PUNCT
cana-524	115	6	)	)	PUNCT
cana-524	116	1	𝜉−1	𝜉−1	PROPN
cana-524	116	2	𝑡=𝜉2	𝑡=𝜉2	PROPN
cana-524	116	3	∑𝑞(𝜁	∑𝑞(𝜁	PROPN
cana-524	116	4	)	)	PUNCT
cana-524	116	5	∞	∞	PROPN
cana-524	117	1	𝜁=𝑡	𝜁=𝑡	PROPN
cana-524	117	2	]	]	PUNCT
cana-524	117	3	1	1	NUM
cana-524	117	4	r	r	NOUN
cana-524	117	5	.	.	PUNCT
cana-524	118	1	(	(	PUNCT
cana-524	118	2	3.21	3.21	NUM
cana-524	118	3	)	)	PUNCT
cana-524	118	4	on	on	ADP
cana-524	118	5	the	the	DET
cana-524	118	6	left	left	ADJ
cana-524	118	7	-	-	PUNCT
cana-524	118	8	hand	hand	NOUN
cana-524	118	9	side	side	NOUN
cana-524	118	10	,	,	PUNCT
cana-524	118	11	since	since	SCONJ
cana-524	118	12	𝑟	𝑟	NOUN
cana-524	118	13	<	<	X
cana-524	118	14	𝛽2	𝛽2	PROPN
cana-524	118	15	,	,	PUNCT
cana-524	118	16	using	use	VERB
cana-524	118	17	summation	summation	NOUN
cana-524	118	18	by	by	ADP
cana-524	118	19	parts	part	NOUN
cana-524	118	20	,	,	PUNCT
cana-524	118	21	we	we	PRON
cana-524	118	22	have	have	VERB
cana-524	118	23	𝑥	𝑥	NOUN
cana-524	118	24	−𝛽2	−𝛽2	VERB
cana-524	118	25	𝑟	𝑟	NOUN
cana-524	118	26	𝜎0(𝜉)𝑥(𝜎0(𝜉	𝜎0(𝜉)𝑥(𝜎0(𝜉	NOUN
cana-524	118	27	)	)	PUNCT
cana-524	118	28	)	)	PUNCT
cana-524	119	1	−	−	NOUN
cana-524	120	1	x	x	PUNCT
cana-524	120	2	−𝛽2	−𝛽2	VERB
cana-524	120	3	𝑟	𝑟	NOUN
cana-524	120	4	𝜎0(𝜉2)𝑥(𝜎0(𝜉2	𝜎0(𝜉2)𝑥(𝜎0(𝜉2	ADV
cana-524	120	5	)	)	PUNCT
cana-524	120	6	)	)	PUNCT
cana-524	120	7	≤	≤	ADV
cana-524	120	8	∑	∑	ADP
cana-524	120	9	  	  	SPACE
cana-524	120	10	𝜉−1	𝜉−1	PROPN
cana-524	120	11	𝑠=𝜉2	𝑠=𝜉2	PROPN
cana-524	120	12	 	 	SPACE
cana-524	120	13	𝑥(𝜎0(𝑠	𝑥(𝜎0(𝑠	PROPN
cana-524	120	14	+	+	NOUN
cana-524	120	15	1	1	NUM
cana-524	120	16	)	)	PUNCT
cana-524	120	17	)	)	PUNCT
cana-524	121	1	[	[	PUNCT
cana-524	121	2	−	−	PROPN
cana-524	121	3	(	(	PUNCT
cana-524	121	4	𝛽2	𝛽2	PROPN
cana-524	121	5	𝑟	𝑟	NOUN
cana-524	121	6	)	)	PUNCT
cana-524	121	7	𝑥	𝑥	PROPN
cana-524	121	8	𝛽2	𝛽2	PROPN
cana-524	121	9	𝑟	𝑟	X
cana-524	121	10	−1	−1	NOUN
cana-524	121	11	𝜎0(𝑠	𝜎0(𝑠	X
cana-524	121	12	)	)	PUNCT
cana-524	121	13	𝑥	𝑥	PROPN
cana-524	121	14	𝛽2	𝛽2	PROPN
cana-524	121	15	𝑟	𝑟	NOUN
cana-524	121	16	(	(	PUNCT
cana-524	121	17	𝜎0(𝑠))𝑥	𝜎0(𝑠))𝑥	PROPN
cana-524	121	18	𝛽2	𝛽2	PROPN
cana-524	121	19	𝑟	𝑟	NOUN
cana-524	121	20	(	(	PUNCT
cana-524	121	21	𝜎0(𝑠+1	𝜎0(𝑠+1	NUM
cana-524	121	22	)	)	PUNCT
cana-524	121	23	)	)	PUNCT
cana-524	121	24	]	]	PUNCT
cana-524	122	1	<	<	X
cana-524	122	2	∞.	∞.	PROPN
cana-524	122	3	(	(	PUNCT
cana-524	122	4	3.22	3.22	NUM
cana-524	122	5	)	)	PUNCT
cana-524	122	6	on	on	ADP
cana-524	122	7	the	the	DET
cana-524	122	8	right	right	ADJ
cana-524	122	9	-	-	PUNCT
cana-524	122	10	hand	hand	NOUN
cana-524	122	11	side	side	NOUN
cana-524	122	12	of	of	ADP
cana-524	122	13	(	(	PUNCT
cana-524	122	14	3.21	3.21	NUM
cana-524	122	15	)	)	PUNCT
cana-524	122	16	,	,	PUNCT
cana-524	122	17	we	we	PRON
cana-524	122	18	use	use	VERB
cana-524	122	19	that	that	PRON
cana-524	122	20	𝑝(𝜎0(𝑡	𝑝(𝜎0(𝑡	PROPN
cana-524	122	21	)	)	PUNCT
cana-524	122	22	)	)	PUNCT
cana-524	122	23	≤	≤	NUM
cana-524	122	24	𝑝(𝑡	𝑝(𝑡	PROPN
cana-524	122	25	)	)	PUNCT
cana-524	122	26	to	to	PART
cana-524	122	27	conclude	conclude	VERB
cana-524	122	28	that	that	PRON
cana-524	122	29	(	(	PUNCT
cana-524	122	30	3.15	3.15	NUM
cana-524	122	31	)	)	PUNCT
cana-524	122	32	implies	imply	VERB
cana-524	122	33	the	the	DET
cana-524	122	34	right	right	ADJ
cana-524	122	35	hand	hand	NOUN
cana-524	122	36	side	side	NOUN
cana-524	122	37	approaching	approach	VERB
cana-524	122	38	+	+	ADV
cana-524	122	39	∞	∞	PROPN
cana-524	122	40	as	as	ADP
cana-524	122	41	𝑦	𝑦	NOUN
cana-524	122	42	⟶	⟶	NOUN
cana-524	122	43	∞	∞	PROPN
cana-524	122	44	,	,	PUNCT
cana-524	122	45	which	which	PRON
cana-524	122	46	is	be	AUX
cana-524	122	47	a	a	DET
cana-524	122	48	contradiction	contradiction	NOUN
cana-524	122	49	.	.	PUNCT
cana-524	123	1	hence	hence	ADV
cana-524	123	2	,	,	PUNCT
cana-524	123	3	the	the	DET
cana-524	123	4	solution	solution	NOUN
cana-524	123	5	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	123	6	)	)	PUNCT
cana-524	123	7	can	can	AUX
cana-524	123	8	not	not	PART
cana-524	123	9	be	be	AUX
cana-524	123	10	eventually	eventually	ADV
cana-524	123	11	positive	positive	ADJ
cana-524	123	12	.	.	PUNCT
cana-524	124	1	for	for	ADP
cana-524	124	2	eventually	eventually	ADV
cana-524	124	3	negative	negative	ADJ
cana-524	124	4	solutions	solution	NOUN
cana-524	124	5	,	,	PUNCT
cana-524	124	6	the	the	DET
cana-524	124	7	same	same	ADJ
cana-524	124	8	change	change	NOUN
cana-524	124	9	of	of	ADP
cana-524	124	10	variables	variable	NOUN
cana-524	124	11	is	be	AUX
cana-524	124	12	used	use	VERB
cana-524	124	13	as	as	ADP
cana-524	124	14	in	in	ADP
cana-524	124	15	theorem	theorem	ADJ
cana-524	124	16	3.1	3.1	NUM
cana-524	124	17	and	and	CCONJ
cana-524	124	18	is	be	AUX
cana-524	124	19	proceed	procee	VERB
cana-524	124	20	above	above	ADV
cana-524	124	21	.	.	PUNCT
cana-524	125	1	in	in	ADP
cana-524	125	2	order	order	NOUN
cana-524	125	3	to	to	PART
cana-524	125	4	prove	prove	VERB
cana-524	125	5	the	the	DET
cana-524	125	6	necessity	necessity	NOUN
cana-524	125	7	part	part	NOUN
cana-524	125	8	,	,	PUNCT
cana-524	125	9	we	we	PRON
cana-524	125	10	assume	assume	VERB
cana-524	125	11	that	that	SCONJ
cana-524	125	12	(	(	PUNCT
cana-524	125	13	3.15	3.15	NUM
cana-524	125	14	)	)	PUNCT
cana-524	125	15	does	do	AUX
cana-524	125	16	not	not	PART
cana-524	125	17	hold	hold	VERB
cana-524	125	18	and	and	CCONJ
cana-524	125	19	obtain	obtain	VERB
cana-524	125	20	an	an	DET
cana-524	125	21	eventually	eventually	ADV
cana-524	125	22	positive	positive	ADJ
cana-524	125	23	solution	solution	NOUN
cana-524	125	24	that	that	PRON
cana-524	125	25	does	do	AUX
cana-524	125	26	not	not	PART
cana-524	125	27	converge	converge	VERB
cana-524	125	28	to	to	ADP
cana-524	125	29	zero	zero	NUM
cana-524	125	30	.	.	PUNCT
cana-524	126	1	if	if	SCONJ
cana-524	126	2	(	(	PUNCT
cana-524	126	3	3.15	3.15	NUM
cana-524	126	4	)	)	PUNCT
cana-524	126	5	does	do	AUX
cana-524	126	6	not	not	PART
cana-524	126	7	hold	hold	VERB
cana-524	126	8	,	,	PUNCT
cana-524	126	9	then	then	ADV
cana-524	126	10	for	for	ADP
cana-524	126	11	each	each	DET
cana-524	126	12	𝛼	𝛼	X
cana-524	126	13	>	>	X
cana-524	126	14	0	0	PUNCT
cana-524	126	15	there	there	PRON
cana-524	126	16	exists	exist	VERB
cana-524	126	17	𝜉1	𝜉1	PROPN
cana-524	126	18	≥	≥	NOUN
cana-524	126	19	𝜉0	𝜉0	NOUN
cana-524	126	20	such	such	ADJ
cana-524	126	21	that	that	SCONJ
cana-524	126	22	∑	∑	ADP
cana-524	126	23	 	 	SPACE
cana-524	126	24	∞	∞	PROPN
cana-524	126	25	𝑡=𝜉1	𝑡=𝜉1	PROPN
cana-524	126	26	[	[	PUNCT
cana-524	126	27	1	1	NUM
cana-524	126	28	𝑝(𝑡	𝑝(𝑡	PROPN
cana-524	126	29	)	)	PUNCT
cana-524	126	30	∑	∑	ADP
cana-524	126	31	 	 	SPACE
cana-524	126	32	∞	∞	PROPN
cana-524	126	33	𝜁=𝑡	𝜁=𝑡	PROPN
cana-524	126	34	 	 	SPACE
cana-524	126	35	𝑞(𝜁	𝑞(𝜁	PROPN
cana-524	126	36	)	)	PUNCT
cana-524	126	37	]	]	PUNCT
cana-524	126	38	1	1	NUM
cana-524	126	39	𝑟	𝑟	X
cana-524	126	40	≤	≤	NUM
cana-524	126	41	𝛼	𝛼	X
cana-524	126	42	(	(	PUNCT
cana-524	126	43	1−	1−	NUM
cana-524	126	44	𝑠	𝑠	NUM
cana-524	126	45	𝑟	𝑟	NOUN
cana-524	126	46	)	)	PUNCT
cana-524	126	47	2	2	NUM
cana-524	126	48	,	,	PUNCT
cana-524	126	49	for	for	ADP
cana-524	126	50	all	all	DET
cana-524	126	51	𝜉	𝜉	PROPN
cana-524	126	52	≥	≥	PROPN
cana-524	126	53	𝜉1	𝜉1	PROPN
cana-524	126	54	.	.	PUNCT
cana-524	127	1	(	(	PUNCT
cana-524	127	2	3.23	3.23	NUM
cana-524	127	3	)	)	PUNCT
cana-524	127	4	we	we	PRON
cana-524	127	5	define	define	VERB
cana-524	127	6	𝑇	𝑇	PROPN
cana-524	127	7	=	=	SYM
cana-524	127	8	{	{	PUNCT
cana-524	127	9	𝑥	𝑥	NOUN
cana-524	127	10	:	:	PUNCT
cana-524	127	11	𝛼	𝛼	PROPN
cana-524	127	12	2	2	NUM
cana-524	127	13	≤	≤	NOUN
cana-524	127	14	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	127	15	)	)	PUNCT
cana-524	127	16	≤	≤	NUM
cana-524	127	17	𝛼	𝛼	X
cana-524	127	18	,	,	PUNCT
cana-524	127	19	for	for	ADP
cana-524	127	20	𝜉	𝜉	PROPN
cana-524	127	21	≥	≥	PROPN
cana-524	127	22	𝜉1	𝜉1	PROPN
cana-524	127	23	}	}	PUNCT
cana-524	127	24	.	.	PUNCT
cana-524	128	1	(	(	PUNCT
cana-524	128	2	3.24	3.24	NUM
cana-524	128	3	)	)	PUNCT
cana-524	128	4	communications	communication	NOUN
cana-524	128	5	on	on	ADP
cana-524	128	6	applied	apply	VERB
cana-524	128	7	nonlinear	nonlinear	ADJ
cana-524	128	8	analysis	analysis	NOUN
cana-524	128	9	issn	issn	NOUN
cana-524	128	10	:	:	PUNCT
cana-524	128	11	1074	1074	NUM
cana-524	128	12	-	-	PUNCT
cana-524	128	13	133x	133x	NUM
cana-524	128	14	vol	vol	NOUN
cana-524	128	15	31	31	NUM
cana-524	128	16	no	no	NOUN
cana-524	128	17	.	.	NOUN
cana-524	128	18	2	2	NUM
cana-524	128	19	(	(	PUNCT
cana-524	128	20	2024	2024	NUM
cana-524	128	21	)	)	PUNCT
cana-524	128	22	126	126	NUM
cana-524	128	23	https://internationalpubls.com	https://internationalpubls.com	X
cana-524	128	24	we	we	PRON
cana-524	128	25	define	define	VERB
cana-524	128	26	an	an	DET
cana-524	128	27	operator	operator	NOUN
cana-524	128	28	𝜙	𝜙	NOUN
cana-524	128	29	on	on	ADP
cana-524	128	30	𝑇	𝑇	PROPN
cana-524	128	31	by	by	ADP
cana-524	128	32	(	(	PUNCT
cana-524	128	33	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	128	34	)	)	PUNCT
cana-524	128	35	=	=	PRON
cana-524	128	36	{	{	PUNCT
cana-524	129	1	0	0	NUM
cana-524	129	2	,	,	PUNCT
cana-524	129	3	if	if	SCONJ
cana-524	129	4	𝜉	𝜉	PROPN
cana-524	129	5	≤	≤	PROPN
cana-524	129	6	𝜉1	𝜉1	PROPN
cana-524	129	7	,	,	PUNCT
cana-524	129	8	𝛼	𝛼	PROPN
cana-524	129	9	2	2	NUM
cana-524	129	10	+	+	CCONJ
cana-524	129	11	∑	∑	PUNCT
cana-524	129	12	  	  	SPACE
cana-524	129	13	𝜉−1	𝜉−1	PROPN
cana-524	129	14	𝑡=𝜉1	𝑡=𝜉1	ADJ
cana-524	129	15	1	1	NUM
cana-524	129	16	𝑝(𝑡	𝑝(𝑡	PROPN
cana-524	129	17	)	)	PUNCT
cana-524	130	1	[	[	X
cana-524	130	2	∑	∑	PUNCT
cana-524	130	3	 	 	SPACE
cana-524	130	4	∞	∞	PROPN
cana-524	130	5	𝜁=𝑡	𝜁=𝑡	PROPN
cana-524	130	6	 	 	SPACE
cana-524	130	7	𝑞(𝜁)𝑥	𝑞(𝜁)𝑥	PROPN
cana-524	130	8	𝑠(𝜎(𝜁	𝑠(𝜎(𝜁	NOUN
cana-524	130	9	)	)	PUNCT
cana-524	130	10	)	)	PUNCT
cana-524	130	11	]	]	PUNCT
cana-524	130	12	1	1	NUM
cana-524	130	13	r	r	NOUN
cana-524	130	14	  	  	SPACE
cana-524	130	15	,	,	PUNCT
cana-524	130	16	if	if	SCONJ
cana-524	130	17	𝜉	𝜉	PROPN
cana-524	130	18	>	>	X
cana-524	130	19	𝜉1	𝜉1	PROPN
cana-524	130	20	.	.	PUNCT
cana-524	131	1	(	(	PUNCT
cana-524	131	2	3.25	3.25	NUM
cana-524	131	3	)	)	PUNCT
cana-524	131	4	if	if	SCONJ
cana-524	131	5	𝑥	𝑥	PROPN
cana-524	131	6	is	be	AUX
cana-524	131	7	a	a	DET
cana-524	131	8	fixed	fix	VERB
cana-524	131	9	point	point	NOUN
cana-524	131	10	of	of	ADP
cana-524	131	11	𝜙	𝜙	NOUN
cana-524	131	12	,	,	PUNCT
cana-524	131	13	i.e.	i.e.	X
cana-524	131	14	,	,	PUNCT
cana-524	131	15	𝜙𝑥	𝜙𝑥	PROPN
cana-524	132	1	=	=	SYM
cana-524	132	2	𝑥	𝑥	PROPN
cana-524	132	3	,	,	PUNCT
cana-524	132	4	then	then	ADV
cana-524	132	5	,	,	PUNCT
cana-524	132	6	𝑥	𝑥	PRON
cana-524	132	7	is	be	AUX
cana-524	132	8	a	a	DET
cana-524	132	9	solution	solution	NOUN
cana-524	132	10	of	of	ADP
cana-524	132	11	(	(	PUNCT
cana-524	132	12	1.1	1.1	NUM
cana-524	132	13	)	)	PUNCT
cana-524	132	14	.	.	PUNCT
cana-524	133	1	first	first	ADV
cana-524	133	2	,	,	PUNCT
cana-524	133	3	we	we	PRON
cana-524	133	4	estimate	estimate	VERB
cana-524	133	5	(	(	PUNCT
cana-524	133	6	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	133	7	)	)	PUNCT
cana-524	133	8	.	.	PUNCT
cana-524	134	1	let	let	VERB
cana-524	134	2	𝑥	𝑥	PRON
cana-524	134	3	∈	∈	PROPN
cana-524	134	4	𝑀	𝑀	PROPN
cana-524	134	5	,	,	PUNCT
cana-524	134	6	we	we	PRON
cana-524	134	7	have	have	VERB
cana-524	134	8	(	(	PUNCT
cana-524	134	9	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	134	10	)	)	PUNCT
cana-524	134	11	≥	≥	NOUN
cana-524	134	12	𝛼	𝛼	NOUN
cana-524	134	13	2	2	NUM
cana-524	134	14	+	+	CCONJ
cana-524	134	15	0	0	NUM
cana-524	134	16	,	,	PUNCT
cana-524	134	17	now	now	ADV
cana-524	134	18	,	,	PUNCT
cana-524	134	19	we	we	PRON
cana-524	134	20	estimate	estimate	VERB
cana-524	134	21	(	(	PUNCT
cana-524	134	22	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	134	23	)	)	PUNCT
cana-524	134	24	from	from	ADP
cana-524	134	25	above	above	ADV
cana-524	134	26	.	.	PUNCT
cana-524	135	1	let	let	VERB
cana-524	135	2	𝑥	𝑥	PRON
cana-524	135	3	∈	∈	VERB
cana-524	135	4	𝑀.	𝑀.	PROPN
cana-524	135	5	then	then	ADV
cana-524	135	6	𝑥	𝑥	X
cana-524	135	7	≤	≤	X
cana-524	135	8	𝛼	𝛼	NOUN
cana-524	135	9	and	and	CCONJ
cana-524	135	10	by	by	ADP
cana-524	135	11	(	(	PUNCT
cana-524	135	12	3.23	3.23	NUM
cana-524	135	13	)	)	PUNCT
cana-524	135	14	,	,	PUNCT
cana-524	135	15	we	we	PRON
cana-524	135	16	have	have	VERB
cana-524	135	17	(	(	PUNCT
cana-524	135	18	𝜙𝑥)(𝜉	𝜙𝑥)(𝜉	PROPN
cana-524	135	19	)	)	PUNCT
cana-524	135	20	≤	≤	NUM
cana-524	136	1	𝛼	𝛼	X
cana-524	136	2	2	2	NUM
cana-524	136	3	+	+	SYM
cana-524	136	4	𝛼	𝛼	VERB
cana-524	136	5	𝑠	𝑠	NUM
cana-524	136	6	𝑟	𝑟	X
cana-524	136	7	∑	∑	ADP
cana-524	136	8	  	  	SPACE
cana-524	136	9	𝜉−1	𝜉−1	CCONJ
cana-524	136	10	𝑡=𝜉1	𝑡=𝜉1	PROPN
cana-524	136	11	  	  	SPACE
cana-524	136	12	[	[	PUNCT
cana-524	136	13	1	1	NUM
cana-524	136	14	𝑝(𝑡	𝑝(𝑡	PROPN
cana-524	136	15	)	)	PUNCT
cana-524	136	16	∑	∑	ADP
cana-524	136	17	  	  	SPACE
cana-524	136	18	∞	∞	PROPN
cana-524	136	19	𝜁=𝑡	𝜁=𝑡	PROPN
cana-524	136	20	 	 	SPACE
cana-524	136	21	𝑞(𝜁	𝑞(𝜁	PROPN
cana-524	136	22	)	)	PUNCT
cana-524	136	23	]	]	PUNCT
cana-524	136	24	1	1	NUM
cana-524	136	25	𝑟	𝑟	X
cana-524	136	26	≤	≤	NUM
cana-524	136	27	𝛼	𝛼	DET
cana-524	136	28	2	2	NUM
cana-524	136	29	+	+	NUM
cana-524	136	30	𝛼	𝛼	SYM
cana-524	136	31	2	2	NUM
cana-524	136	32	=	=	SYM
cana-524	136	33	𝛼.	𝛼.	NOUN
cana-524	136	34	(	(	PUNCT
cana-524	136	35	3.26	3.26	NUM
cana-524	136	36	)	)	PUNCT
cana-524	136	37	therefore	therefore	ADV
cana-524	136	38	,	,	PUNCT
cana-524	136	39	𝜙	𝜙	PROPN
cana-524	136	40	maps	map	VERB
cana-524	136	41	𝑇	𝑇	PROPN
cana-524	136	42	to	to	ADP
cana-524	136	43	𝑇	𝑇	PROPN
cana-524	136	44	,	,	PUNCT
cana-524	136	45	we	we	PRON
cana-524	136	46	find	find	VERB
cana-524	136	47	a	a	DET
cana-524	136	48	fixed	fix	VERB
cana-524	136	49	point	point	NOUN
cana-524	136	50	for	for	ADP
cana-524	136	51	𝜙	𝜙	PROPN
cana-524	136	52	in	in	ADP
cana-524	136	53	𝑇.	𝑇.	PROPN
cana-524	136	54	let	let	VERB
cana-524	136	55	us	we	PRON
cana-524	136	56	define	define	VERB
cana-524	136	57	a	a	DET
cana-524	136	58	sequence	sequence	NOUN
cana-524	136	59	of	of	ADP
cana-524	136	60	functions	function	NOUN
cana-524	136	61	in	in	ADP
cana-524	136	62	t	t	PROPN
cana-524	136	63	by	by	ADP
cana-524	136	64	the	the	DET
cana-524	136	65	recurrence	recurrence	NOUN
cana-524	136	66	relation	relation	NOUN
cana-524	136	67	𝜎0(𝜉	𝜎0(𝜉	PROPN
cana-524	136	68	)	)	PUNCT
cana-524	136	69	=	=	SYM
cana-524	136	70	0	0	NUM
cana-524	136	71	,	,	PUNCT
cana-524	136	72	for	for	ADP
cana-524	136	73	𝜉	𝜉	PROPN
cana-524	136	74	≥	≥	NOUN
cana-524	136	75	𝜉0	𝜉0	NOUN
cana-524	136	76	,	,	PUNCT
cana-524	136	77	𝜎1(𝜉	𝜎1(𝜉	NUM
cana-524	136	78	)	)	PUNCT
cana-524	136	79	=	=	SYM
cana-524	136	80	(	(	PUNCT
cana-524	136	81	𝜙𝜎0)(𝜉	𝜙𝜎0)(𝜉	PROPN
cana-524	136	82	)	)	PUNCT
cana-524	136	83	=	=	NOUN
cana-524	137	1	1	1	NUM
cana-524	137	2	,	,	PUNCT
cana-524	137	3	for	for	ADP
cana-524	137	4	𝜉	𝜉	PROPN
cana-524	137	5	≥	≥	NOUN
cana-524	137	6	𝜉0	𝜉0	NOUN
cana-524	137	7	,	,	PUNCT
cana-524	137	8	𝜎n+1(𝜉	𝜎n+1(𝜉	PROPN
cana-524	137	9	)	)	PUNCT
cana-524	137	10	=	=	PUNCT
cana-524	137	11	(	(	PUNCT
cana-524	137	12	𝜙𝜎n)(𝜉	𝜙𝜎n)(𝜉	PROPN
cana-524	137	13	)	)	PUNCT
cana-524	137	14	,	,	PUNCT
cana-524	137	15	for	for	ADP
cana-524	137	16	n≥	n≥	PROPN
cana-524	137	17	1	1	NUM
cana-524	137	18	,	,	PUNCT
cana-524	137	19	𝜉	𝜉	X
cana-524	137	20	≥	≥	PROPN
cana-524	137	21	𝜉1	𝜉1	PROPN
cana-524	137	22	.	.	PUNCT
cana-524	138	1	(	(	PUNCT
cana-524	138	2	3.27	3.27	NUM
cana-524	138	3	)	)	PUNCT
cana-524	138	4	note	note	VERB
cana-524	138	5	that	that	SCONJ
cana-524	138	6	for	for	ADP
cana-524	138	7	each	each	DET
cana-524	138	8	fixed	fix	VERB
cana-524	138	9	𝜉	𝜉	NOUN
cana-524	138	10	,	,	PUNCT
cana-524	138	11	we	we	PRON
cana-524	138	12	have	have	VERB
cana-524	138	13	𝜎1(𝜉	𝜎1(𝜉	NUM
cana-524	138	14	)	)	PUNCT
cana-524	138	15	≥	≥	NOUN
cana-524	138	16	𝜎0(𝜉	𝜎0(𝜉	NUM
cana-524	138	17	)	)	PUNCT
cana-524	138	18	.	.	PUNCT
cana-524	139	1	using	use	VERB
cana-524	139	2	mathematical	mathematical	ADJ
cana-524	139	3	induction	induction	NOUN
cana-524	139	4	,	,	PUNCT
cana-524	139	5	we	we	PRON
cana-524	139	6	can	can	AUX
cana-524	139	7	show	show	VERB
cana-524	139	8	that	that	SCONJ
cana-524	139	9	𝜎n+1(𝜉	𝜎n+1(𝜉	PROPN
cana-524	139	10	)	)	PUNCT
cana-524	139	11	≥	≥	NOUN
cana-524	139	12	𝜎n(𝜉	𝜎n(𝜉	NUM
cana-524	139	13	)	)	PUNCT
cana-524	139	14	.	.	PUNCT
cana-524	140	1	therefore	therefore	ADV
cana-524	140	2	,	,	PUNCT
cana-524	140	3	the	the	DET
cana-524	140	4	sequence	sequence	NOUN
cana-524	140	5	{	{	PUNCT
cana-524	140	6	𝜎n	𝜎n	NOUN
cana-524	140	7	}	}	PUNCT
cana-524	140	8	converges	converge	VERB
cana-524	140	9	pointwise	pointwise	VERB
cana-524	140	10	to	to	ADP
cana-524	140	11	a	a	DET
cana-524	140	12	sequence	sequence	NOUN
cana-524	140	13	𝜎	𝜎	NOUN
cana-524	140	14	in	in	ADP
cana-524	140	15	𝑇.	𝑇.	PROPN
cana-524	140	16	then	then	ADV
cana-524	140	17	,	,	PUNCT
cana-524	140	18	𝜎	𝜎	PROPN
cana-524	140	19	is	be	AUX
cana-524	140	20	a	a	DET
cana-524	140	21	fixed	fix	VERB
cana-524	140	22	point	point	NOUN
cana-524	140	23	of	of	ADP
cana-524	140	24	𝜙	𝜙	PROPN
cana-524	140	25	and	and	CCONJ
cana-524	140	26	a	a	DET
cana-524	140	27	positive	positive	ADJ
cana-524	140	28	solution	solution	NOUN
cana-524	140	29	of	of	ADP
cana-524	140	30	(	(	PUNCT
cana-524	140	31	1.1	1.1	NUM
cana-524	140	32	)	)	PUNCT
cana-524	140	33	.	.	PUNCT
cana-524	141	1	this	this	PRON
cana-524	141	2	complete	complete	ADJ
cana-524	141	3	the	the	DET
cana-524	141	4	proof	proof	NOUN
cana-524	141	5	.	.	PUNCT
cana-524	142	1	4	4	X
cana-524	142	2	.	.	NOUN
cana-524	142	3	example	example	NOUN
cana-524	142	4	example	example	NOUN
cana-524	142	5	4.1	4.1	NUM
cana-524	142	6	.	.	PUNCT
cana-524	143	1	consider	consider	VERB
cana-524	143	2	the	the	DET
cana-524	143	3	second	second	ADJ
cana-524	143	4	order	order	NOUN
cana-524	143	5	half	half	ADJ
cana-524	143	6	-	-	PUNCT
cana-524	143	7	linear	linear	ADJ
cana-524	143	8	delay	delay	NOUN
cana-524	143	9	difference	difference	NOUN
cana-524	143	10	equation	equation	NOUN
cana-524	143	11	δ	δ	PROPN
cana-524	143	12	[	[	PUNCT
cana-524	143	13	1	1	NUM
cana-524	143	14	𝜉	𝜉	X
cana-524	143	15	(	(	PUNCT
cana-524	143	16	δ𝑥(𝜉	δ𝑥(𝜉	NOUN
cana-524	143	17	)	)	PUNCT
cana-524	143	18	)	)	PUNCT
cana-524	143	19	7	7	NUM
cana-524	143	20	3	3	NUM
cana-524	143	21	]	]	PUNCT
cana-524	143	22	+	+	CCONJ
cana-524	143	23	2	2	NUM
cana-524	143	24	7	7	NUM
cana-524	143	25	3	3	NUM
cana-524	143	26	[	[	PUNCT
cana-524	143	27	2𝜉+1	2𝜉+1	PROPN
cana-524	143	28	𝜉2+𝜉	𝜉2+𝜉	PROPN
cana-524	143	29	]	]	X
cana-524	143	30	(	(	PUNCT
cana-524	143	31	𝑥(7𝜉	𝑥(7𝜉	NUM
cana-524	143	32	−	−	NOUN
cana-524	143	33	3	3	NUM
cana-524	143	34	)	)	PUNCT
cana-524	143	35	)	)	PUNCT
cana-524	143	36	1	1	NUM
cana-524	143	37	3	3	NUM
cana-524	143	38	=	=	SYM
cana-524	143	39	0	0	NUM
cana-524	143	40	.	.	PUNCT
cana-524	144	1	(	(	PUNCT
cana-524	144	2	4.1	4.1	NUM
cana-524	144	3	)	)	PUNCT
cana-524	144	4	where	where	SCONJ
cana-524	144	5	,	,	PUNCT
cana-524	144	6	𝑝(𝜉	𝑝(𝜉	PROPN
cana-524	144	7	)	)	PUNCT
cana-524	144	8	=	=	SYM
cana-524	144	9	1	1	NUM
cana-524	144	10	𝜉	𝜉	NOUN
cana-524	144	11	,	,	PUNCT
cana-524	144	12	𝑞(𝜉	𝑞(𝜉	PROPN
cana-524	144	13	)	)	PUNCT
cana-524	144	14	=	=	SYM
cana-524	144	15	2	2	NUM
cana-524	144	16	7	7	NUM
cana-524	144	17	3	3	NUM
cana-524	144	18	[	[	PUNCT
cana-524	144	19	2𝜉	2𝜉	NUM
cana-524	144	20	+	+	CCONJ
cana-524	144	21	1	1	NUM
cana-524	144	22	𝜉	𝜉	SYM
cana-524	144	23	2	2	NUM
cana-524	144	24	+	+	NUM
cana-524	144	25	𝜉	𝜉	X
cana-524	144	26	]	]	PUNCT
cana-524	144	27	,	,	PUNCT
cana-524	144	28	𝜎(𝜉	𝜎(𝜉	PROPN
cana-524	144	29	)	)	PUNCT
cana-524	144	30	=	=	VERB
cana-524	145	1	7𝜉	7𝜉	NUM
cana-524	145	2	−	−	NOUN
cana-524	145	3	3	3	NUM
cana-524	145	4	,	,	PUNCT
cana-524	145	5	𝑠	𝑠	PROPN
cana-524	145	6	=	=	SYM
cana-524	145	7	1	1	NUM
cana-524	145	8	3	3	NUM
cana-524	145	9	,	,	PUNCT
cana-524	145	10	𝑟	𝑟	X
cana-524	145	11	=	=	SYM
cana-524	145	12	7	7	NUM
cana-524	145	13	3	3	NUM
cana-524	145	14	,	,	PUNCT
cana-524	145	15	𝛽1	𝛽1	NOUN
cana-524	145	16	=	=	SYM
cana-524	145	17	5	5	NUM
cana-524	145	18	3	3	NUM
cana-524	145	19	we	we	PRON
cana-524	145	20	have	have	VERB
cana-524	145	21	0	0	NUM
cana-524	145	22	<	<	X
cana-524	145	23	𝑠	𝑠	X
cana-524	145	24	<	<	X
cana-524	145	25	𝛽1	𝛽1	NOUN
cana-524	145	26	<	<	X
cana-524	145	27	𝑟.	𝑟.	NOUN
cana-524	145	28	∑	∑	PUNCT
cana-524	145	29	 	 	SPACE
cana-524	145	30	∞	∞	PROPN
cana-524	146	1	𝜁=0	𝜁=0	PROPN
cana-524	146	2	𝑞(𝜁)𝑣	𝑞(𝜁)𝑣	PROPN
cana-524	146	3	𝑠(𝜎(𝜁	𝑠(𝜎(𝜁	NOUN
cana-524	146	4	)	)	PUNCT
cana-524	146	5	)	)	PUNCT
cana-524	147	1	=	=	PUNCT
cana-524	147	2	∑	∑	PUNCT
cana-524	147	3	 	 	SPACE
cana-524	147	4	∞	∞	PROPN
cana-524	147	5	𝜁=0	𝜁=0	PROPN
cana-524	147	6	2	2	NUM
cana-524	147	7	7	7	NUM
cana-524	147	8	3	3	NUM
cana-524	147	9	[	[	PUNCT
cana-524	147	10	2𝜁+1	2𝜁+1	NUM
cana-524	147	11	𝜁	𝜁	PROPN
cana-524	147	12	2+𝜁	2+𝜁	NUM
cana-524	147	13	]	]	PUNCT
cana-524	147	14	∑	∑	PART
cana-524	147	15	  	  	SPACE
cana-524	147	16	𝜉−1	𝜉−1	PRON
cana-524	147	17	𝑡=𝜉	𝑡=𝜉	PROPN
cana-524	147	18	𝑡	𝑡	X
cana-524	147	19	3	3	NUM
cana-524	147	20	7	7	NUM
cana-524	147	21	=	=	SYM
cana-524	147	22	∞.	∞.	PROPN
cana-524	147	23	communications	communication	NOUN
cana-524	147	24	on	on	ADP
cana-524	147	25	applied	apply	VERB
cana-524	147	26	nonlinear	nonlinear	ADJ
cana-524	147	27	analysis	analysis	NOUN
cana-524	147	28	issn	issn	NOUN
cana-524	147	29	:	:	PUNCT
cana-524	147	30	1074	1074	NUM
cana-524	147	31	-	-	PUNCT
cana-524	147	32	133x	133x	NUM
cana-524	147	33	vol	vol	NOUN
cana-524	147	34	31	31	NUM
cana-524	147	35	no	no	NOUN
cana-524	147	36	.	.	NOUN
cana-524	147	37	2	2	NUM
cana-524	147	38	(	(	PUNCT
cana-524	147	39	2024	2024	NUM
cana-524	147	40	)	)	PUNCT
cana-524	147	41	127	127	NUM
cana-524	147	42	https://internationalpubls.com	https://internationalpubls.com	X
cana-524	147	43	hence	hence	ADV
cana-524	147	44	all	all	DET
cana-524	147	45	the	the	DET
cana-524	147	46	conditions	condition	NOUN
cana-524	147	47	of	of	ADP
cana-524	147	48	theorem	theorem	ADJ
cana-524	147	49	3.1	3.1	NUM
cana-524	147	50	are	be	AUX
cana-524	147	51	satisfied	satisfied	ADJ
cana-524	147	52	.	.	PUNCT
cana-524	148	1	hence	hence	ADV
cana-524	148	2	every	every	DET
cana-524	148	3	solution	solution	NOUN
cana-524	148	4	of	of	ADP
cana-524	148	5	(	(	PUNCT
cana-524	148	6	4.1	4.1	NUM
cana-524	148	7	)	)	PUNCT
cana-524	148	8	is	be	AUX
cana-524	148	9	oscillatory	oscillatory	ADJ
cana-524	148	10	.	.	PUNCT
cana-524	149	1	one	one	NUM
cana-524	149	2	of	of	ADP
cana-524	149	3	such	such	ADJ
cana-524	149	4	solution	solution	NOUN
cana-524	149	5	of	of	ADP
cana-524	149	6	equation	equation	NOUN
cana-524	149	7	(	(	PUNCT
cana-524	149	8	1.1	1.1	NUM
cana-524	149	9	)	)	PUNCT
cana-524	149	10	is	be	AUX
cana-524	149	11	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	149	12	)	)	PUNCT
cana-524	149	13	=	=	SYM
cana-524	150	1	(	(	PUNCT
cana-524	150	2	−1)𝜉+1	−1)𝜉+1	PROPN
cana-524	150	3	.	.	PUNCT
cana-524	150	4	example	example	NOUN
cana-524	150	5	4.2	4.2	NUM
cana-524	150	6	.	.	PUNCT
cana-524	151	1	consider	consider	VERB
cana-524	151	2	the	the	DET
cana-524	151	3	second	second	ADJ
cana-524	151	4	order	order	NOUN
cana-524	151	5	half	half	ADJ
cana-524	151	6	-	-	PUNCT
cana-524	151	7	linear	linear	ADJ
cana-524	151	8	delay	delay	NOUN
cana-524	151	9	difference	difference	NOUN
cana-524	151	10	equation	equation	NOUN
cana-524	151	11	δ	δ	PROPN
cana-524	151	12	[	[	PUNCT
cana-524	151	13	1	1	NUM
cana-524	151	14	𝜉2	𝜉2	PROPN
cana-524	151	15	(	(	PUNCT
cana-524	151	16	δ𝑥(𝜉	δ𝑥(𝜉	NOUN
cana-524	151	17	)	)	PUNCT
cana-524	151	18	)	)	PUNCT
cana-524	151	19	1	1	NUM
cana-524	151	20	3	3	NUM
cana-524	151	21	]	]	PUNCT
cana-524	151	22	+	+	CCONJ
cana-524	151	23	2	2	NUM
cana-524	151	24	1	1	NUM
cana-524	151	25	3	3	NUM
cana-524	151	26	[	[	PUNCT
cana-524	151	27	2𝜉2	2𝜉2	NUM
cana-524	151	28	+	+	ADJ
cana-524	151	29	2𝜉+1	2𝜉+1	PROPN
cana-524	151	30	𝜉4	𝜉4	VERB
cana-524	151	31	+	+	NOUN
cana-524	151	32	2𝜉3+𝜉2	2𝜉3+𝜉2	NUM
cana-524	151	33	]	]	PUNCT
cana-524	151	34	(	(	PUNCT
cana-524	151	35	𝑥(𝜉	𝑥(𝜉	PRON
cana-524	151	36	−	−	NOUN
cana-524	151	37	2	2	NUM
cana-524	151	38	)	)	PUNCT
cana-524	151	39	)	)	PUNCT
cana-524	151	40	7	7	NUM
cana-524	151	41	3	3	NUM
cana-524	151	42	=	=	SYM
cana-524	151	43	0	0	NUM
cana-524	151	44	.	.	PUNCT
cana-524	152	1	(	(	PUNCT
cana-524	152	2	4.2	4.2	NUM
cana-524	152	3	)	)	PUNCT
cana-524	152	4	where	where	SCONJ
cana-524	152	5	,	,	PUNCT
cana-524	152	6	𝑝(𝜉	𝑝(𝜉	PROPN
cana-524	152	7	)	)	PUNCT
cana-524	152	8	=	=	SYM
cana-524	152	9	1	1	NUM
cana-524	152	10	𝜉	𝜉	SYM
cana-524	152	11	2	2	NUM
cana-524	152	12	,	,	PUNCT
cana-524	152	13	𝑞(𝜉	𝑞(𝜉	PROPN
cana-524	152	14	)	)	PUNCT
cana-524	152	15	=	=	SYM
cana-524	152	16	2	2	NUM
cana-524	152	17	1	1	NUM
cana-524	152	18	3	3	NUM
cana-524	152	19	[	[	PUNCT
cana-524	152	20	2	2	NUM
cana-524	152	21	𝜉	𝜉	SYM
cana-524	152	22	2	2	NUM
cana-524	152	23	+	+	NUM
cana-524	152	24	2𝜉	2𝜉	NUM
cana-524	152	25	+	+	CCONJ
cana-524	152	26	1	1	NUM
cana-524	152	27	𝜉	𝜉	ADP
cana-524	152	28	4	4	NUM
cana-524	152	29	+	+	NUM
cana-524	152	30	2𝜉	2𝜉	NUM
cana-524	152	31	3+𝜉	3+𝜉	NUM
cana-524	152	32	2	2	NUM
cana-524	152	33	]	]	PUNCT
cana-524	152	34	,	,	PUNCT
cana-524	152	35	𝜎(𝜉	𝜎(𝜉	PROPN
cana-524	152	36	)	)	PUNCT
cana-524	152	37	=	=	SYM
cana-524	152	38	𝜉	𝜉	NOUN
cana-524	152	39	−	−	NOUN
cana-524	152	40	2	2	NUM
cana-524	152	41	,	,	PUNCT
cana-524	152	42	𝑠	𝑠	PROPN
cana-524	152	43	=	=	NUM
cana-524	152	44	7	7	NUM
cana-524	152	45	3	3	NUM
cana-524	152	46	,	,	PUNCT
cana-524	152	47	𝑟	𝑟	NOUN
cana-524	152	48	=	=	SYM
cana-524	152	49	1	1	NUM
cana-524	152	50	3	3	NUM
cana-524	152	51	,	,	PUNCT
cana-524	152	52	𝛽1	𝛽1	NOUN
cana-524	152	53	=	=	SYM
cana-524	152	54	5	5	NUM
cana-524	152	55	3	3	NUM
cana-524	152	56	we	we	PRON
cana-524	152	57	have	have	VERB
cana-524	152	58	𝑠	𝑠	INTJ
cana-524	152	59	>	>	X
cana-524	152	60	𝛽1	𝛽1	PROPN
cana-524	152	61	>	>	PUNCT
cana-524	152	62	𝑟.	𝑟.	NOUN
cana-524	152	63	∑	∑	PUNCT
cana-524	152	64	 	 	SPACE
cana-524	152	65	∞	∞	PROPN
cana-524	152	66	𝑠=𝜉1	𝑠=𝜉1	PROPN
cana-524	152	67	[	[	PUNCT
cana-524	152	68	1	1	NUM
cana-524	152	69	𝑝(𝑠	𝑝(𝑠	NOUN
cana-524	152	70	)	)	PUNCT
cana-524	152	71	∑	∑	ADP
cana-524	152	72	 	 	SPACE
cana-524	152	73	∞	∞	PRON
cana-524	152	74	𝜁=𝑠	𝜁=𝑠	PUNCT
cana-524	152	75	 	 	SPACE
cana-524	152	76	𝑞(𝜁	𝑞(𝜁	PROPN
cana-524	152	77	)	)	PUNCT
cana-524	152	78	]	]	PUNCT
cana-524	152	79	1	1	NUM
cana-524	152	80	𝑟	𝑟	NOUN
cana-524	152	81	=	=	SYM
cana-524	152	82	∑	∑	PUNCT
cana-524	152	83	 	 	SPACE
cana-524	152	84	∞	∞	PROPN
cana-524	152	85	𝑠=𝜉1	𝑠=𝜉1	PROPN
cana-524	153	1	[	[	X
cana-524	153	2	𝑠2∑	𝑠2∑	PROPN
cana-524	153	3	 	 	SPACE
cana-524	153	4	∞	∞	PROPN
cana-524	153	5	𝜁=𝑠	𝜁=𝑠	PUNCT
cana-524	153	6	 	 	SPACE
cana-524	153	7	2	2	NUM
cana-524	153	8	1	1	NUM
cana-524	153	9	3	3	NUM
cana-524	153	10	[	[	PUNCT
cana-524	153	11	2	2	NUM
cana-524	153	12	𝜁	𝜁	PROPN
cana-524	153	13	2	2	NUM
cana-524	153	14	+	+	PROPN
cana-524	153	15	2𝜁+1	2𝜁+1	NOUN
cana-524	153	16	𝜁4	𝜁4	NOUN
cana-524	153	17	+	+	PROPN
cana-524	153	18	2𝜁3+𝜁2	2𝜁3+𝜁2	PROPN
cana-524	153	19	]	]	X
cana-524	153	20	]	]	X
cana-524	153	21	3	3	NUM
cana-524	153	22	=	=	SYM
cana-524	153	23	∞.	∞.	PROPN
cana-524	153	24	hence	hence	ADV
cana-524	153	25	all	all	DET
cana-524	153	26	the	the	DET
cana-524	153	27	conditions	condition	NOUN
cana-524	153	28	of	of	ADP
cana-524	153	29	theorem	theorem	ADJ
cana-524	153	30	3.2	3.2	NUM
cana-524	153	31	are	be	AUX
cana-524	153	32	satisfied	satisfied	ADJ
cana-524	153	33	.	.	PUNCT
cana-524	154	1	hence	hence	ADV
cana-524	154	2	every	every	DET
cana-524	154	3	solution	solution	NOUN
cana-524	154	4	of	of	ADP
cana-524	154	5	(	(	PUNCT
cana-524	154	6	4.2	4.2	NUM
cana-524	154	7	)	)	PUNCT
cana-524	154	8	is	be	AUX
cana-524	154	9	oscillatory	oscillatory	ADJ
cana-524	154	10	.	.	PUNCT
cana-524	155	1	one	one	NUM
cana-524	155	2	of	of	ADP
cana-524	155	3	such	such	ADJ
cana-524	155	4	solution	solution	NOUN
cana-524	155	5	of	of	ADP
cana-524	155	6	equation	equation	NOUN
cana-524	155	7	(	(	PUNCT
cana-524	155	8	1.1	1.1	NUM
cana-524	155	9	)	)	PUNCT
cana-524	155	10	is	be	AUX
cana-524	155	11	𝑥(𝜉	𝑥(𝜉	NOUN
cana-524	155	12	)	)	PUNCT
cana-524	155	13	=	=	SYM
cana-524	155	14	(	(	PUNCT
cana-524	155	15	−1)𝜉+1	−1)𝜉+1	PROPN
cana-524	155	16	.	.	PROPN
cana-524	156	1	5	5	NUM
cana-524	156	2	.	.	X
cana-524	156	3	conclusion	conclusion	NOUN
cana-524	156	4	in	in	ADP
cana-524	156	5	this	this	DET
cana-524	156	6	paper	paper	NOUN
cana-524	156	7	,	,	PUNCT
cana-524	156	8	we	we	PRON
cana-524	156	9	established	establish	VERB
cana-524	156	10	necessary	necessary	ADJ
cana-524	156	11	and	and	CCONJ
cana-524	156	12	sufficient	sufficient	ADJ
cana-524	156	13	conditions	condition	NOUN
cana-524	156	14	for	for	ADP
cana-524	156	15	the	the	DET
cana-524	156	16	oscillation	oscillation	NOUN
cana-524	156	17	of	of	ADP
cana-524	156	18	solution	solution	NOUN
cana-524	156	19	to	to	ADP
cana-524	156	20	second	second	ADJ
cana-524	156	21	order	order	NOUN
cana-524	156	22	half	half	ADJ
cana-524	156	23	-	-	PUNCT
cana-524	156	24	linear	linear	ADJ
cana-524	156	25	delay	delay	NOUN
cana-524	156	26	difference	difference	NOUN
cana-524	156	27	equation	equation	NOUN
cana-524	156	28	.	.	PUNCT
cana-524	157	1	the	the	DET
cana-524	157	2	above	above	ADJ
cana-524	157	3	discussed	discuss	VERB
cana-524	157	4	examples	example	NOUN
cana-524	157	5	illustrate	illustrate	VERB
cana-524	157	6	the	the	DET
cana-524	157	7	significance	significance	NOUN
cana-524	157	8	and	and	CCONJ
cana-524	157	9	relevance	relevance	NOUN
cana-524	157	10	of	of	ADP
cana-524	157	11	the	the	DET
cana-524	157	12	proven	prove	VERB
cana-524	157	13	results	result	NOUN
cana-524	157	14	.	.	PUNCT
cana-524	158	1	references	reference	NOUN
cana-524	158	2	[	[	X
cana-524	158	3	1	1	NUM
cana-524	158	4	]	]	PUNCT
cana-524	158	5	p.	p.	NOUN
cana-524	158	6	gopalakrishnan	gopalakrishnan	NOUN
cana-524	158	7	,	,	PUNCT
cana-524	158	8	a.	a.	NOUN
cana-524	158	9	murugesan	murugesan	PROPN
cana-524	158	10	,	,	PUNCT
cana-524	158	11	c.	c.	PROPN
cana-524	158	12	jayakumar	jayakumar	NOUN
cana-524	158	13	,	,	PUNCT
cana-524	158	14	oscillation	oscillation	NOUN
cana-524	158	15	conditions	condition	NOUN
cana-524	158	16	of	of	ADP
cana-524	158	17	the	the	DET
cana-524	158	18	second	second	ADJ
cana-524	158	19	-	-	PUNCT
cana-524	158	20	order	order	NOUN
cana-524	158	21	noncanonical	noncanonical	ADJ
cana-524	158	22	difference	difference	NOUN
cana-524	158	23	equations	equation	NOUN
cana-524	158	24	,	,	PUNCT
cana-524	158	25	journal	journal	NOUN
cana-524	158	26	of	of	ADP
cana-524	158	27	mathematics	mathematic	NOUN
cana-524	158	28	and	and	CCONJ
cana-524	158	29	computer	computer	NOUN
cana-524	158	30	science	science	NOUN
cana-524	158	31	,	,	PUNCT
cana-524	158	32	25(2022	25(2022	NUM
cana-524	158	33	)	)	PUNCT
cana-524	158	34	,	,	PUNCT
cana-524	158	35	351	351	NUM
cana-524	158	36	-	-	SYM
cana-524	158	37	360	360	NUM
cana-524	158	38	.	.	PUNCT
cana-524	159	1	[	[	X
cana-524	159	2	2	2	NUM
cana-524	159	3	]	]	PUNCT
cana-524	159	4	c.	c.	PROPN
cana-524	159	5	soundara	soundara	PROPN
cana-524	159	6	rajan	rajan	PROPN
cana-524	159	7	and	and	CCONJ
cana-524	159	8	a.	a.	PROPN
cana-524	159	9	murugesan	murugesan	ADJ
cana-524	159	10	,	,	PUNCT
cana-524	159	11	oscillatory	oscillatory	ADJ
cana-524	159	12	and	and	CCONJ
cana-524	159	13	asymptotic	asymptotic	ADJ
cana-524	159	14	behavior	behavior	NOUN
cana-524	159	15	of	of	ADP
cana-524	159	16	solutions	solution	NOUN
cana-524	159	17	to	to	ADP
cana-524	159	18	second	second	ADJ
cana-524	159	19	-	-	PUNCT
cana-524	159	20	order	order	NOUN
cana-524	159	21	non	non	ADJ
cana-524	159	22	-	-	ADJ
cana-524	159	23	linear	linear	ADJ
cana-524	159	24	neutral	neutral	ADJ
cana-524	159	25	difference	difference	NOUN
cana-524	159	26	equations	equation	NOUN
cana-524	159	27	of	of	ADP
cana-524	159	28	advanced	advanced	ADJ
cana-524	159	29	type	type	NOUN
cana-524	159	30	,	,	PUNCT
cana-524	159	31	malaya	malaya	PROPN
cana-524	159	32	journal	journal	PROPN
cana-524	159	33	of	of	ADP
cana-524	159	34	matematik	matematik	PROPN
cana-524	159	35	,	,	PUNCT
cana-524	159	36	vol	vol	NOUN
cana-524	159	37	.	.	PROPN
cana-524	159	38	9	9	NUM
cana-524	159	39	,	,	PUNCT
cana-524	159	40	no	no	INTJ
cana-524	159	41	.	.	NOUN
cana-524	159	42	1	1	NUM
cana-524	159	43	,	,	PUNCT
cana-524	159	44	1160	1160	NUM
cana-524	159	45	1166	1166	NUM
cana-524	159	46	,	,	PUNCT
cana-524	159	47	2021	2021	NUM
cana-524	159	48	.	.	PUNCT
cana-524	160	1	[	[	X
cana-524	160	2	3	3	NUM
cana-524	160	3	]	]	PUNCT
cana-524	160	4	a.	a.	NOUN
cana-524	160	5	murugesan	murugesan	PROPN
cana-524	160	6	and	and	CCONJ
cana-524	160	7	c.	c.	PROPN
cana-524	160	8	jayakumar	jayakumar	PROPN
cana-524	160	9	,	,	PUNCT
cana-524	160	10	oscillation	oscillation	NOUN
cana-524	160	11	condition	condition	NOUN
cana-524	160	12	for	for	ADP
cana-524	160	13	second	second	ADJ
cana-524	160	14	order	order	NOUN
cana-524	160	15	half	half	ADJ
cana-524	160	16	-	-	PUNCT
cana-524	160	17	linear	linear	ADJ
cana-524	160	18	advanced	advanced	ADJ
cana-524	160	19	difference	difference	NOUN
cana-524	160	20	equation	equation	NOUN
cana-524	160	21	with	with	ADP
cana-524	160	22	variable	variable	ADJ
cana-524	160	23	coefficients	coefficient	NOUN
cana-524	160	24	,	,	PUNCT
cana-524	160	25	malaya	malaya	PROPN
cana-524	160	26	journal	journal	PROPN
cana-524	160	27	of	of	ADP
cana-524	160	28	matematik	matematik	PROPN
cana-524	160	29	,	,	PUNCT
cana-524	160	30	vol	vol	NOUN
cana-524	160	31	.	.	PROPN
cana-524	160	32	8	8	NUM
cana-524	160	33	,	,	PUNCT
cana-524	160	34	no	no	INTJ
cana-524	160	35	.	.	NOUN
cana-524	160	36	4	4	NUM
cana-524	160	37	,	,	PUNCT
cana-524	160	38	1872	1872	NUM
cana-524	160	39	1879	1879	NUM
cana-524	160	40	,	,	PUNCT
cana-524	160	41	2020	2020	NUM
cana-524	160	42	.	.	PUNCT
cana-524	161	1	[	[	X
cana-524	161	2	4	4	X
cana-524	161	3	]	]	PUNCT
cana-524	161	4	shyam	shyam	PROPN
cana-524	161	5	sundar	sundar	PROPN
cana-524	161	6	santra	santra	PROPN
cana-524	161	7	,	,	PUNCT
cana-524	161	8	necessary	necessary	ADJ
cana-524	161	9	and	and	CCONJ
cana-524	161	10	sufficient	sufficient	ADJ
cana-524	161	11	conditions	condition	NOUN
cana-524	161	12	for	for	ADP
cana-524	161	13	oscillatory	oscillatory	ADJ
cana-524	161	14	and	and	CCONJ
cana-524	161	15	asymptotic	asymptotic	ADJ
cana-524	161	16	behaviour	behaviour	NOUN
cana-524	161	17	of	of	ADP
cana-524	161	18	solutions	solution	NOUN
cana-524	161	19	to	to	ADP
cana-524	161	20	second	second	ADJ
cana-524	161	21	-order	-order	PROPN
cana-524	161	22	nonlinear	nonlinear	ADJ
cana-524	161	23	neutral	neutral	ADJ
cana-524	161	24	differential	differential	ADJ
cana-524	161	25	equations	equation	NOUN
cana-524	161	26	with	with	ADP
cana-524	161	27	several	several	ADJ
cana-524	161	28	delays	delay	NOUN
cana-524	161	29	,	,	PUNCT
cana-524	161	30	tatra	tatra	PROPN
cana-524	161	31	mountains	mountain	NOUN
cana-524	161	32	mathematical	mathematical	ADJ
cana-524	161	33	publications	publication	NOUN
cana-524	161	34	,	,	PUNCT
cana-524	161	35	vol	vol	NOUN
cana-524	161	36	.	.	PROPN
cana-524	161	37	75	75	NUM
cana-524	161	38	,	,	PUNCT
cana-524	161	39	(	(	PUNCT
cana-524	161	40	2020	2020	NUM
cana-524	161	41	)	)	PUNCT
cana-524	161	42	,	,	PUNCT
cana-524	161	43	121	121	NUM
cana-524	161	44	134	134	NUM
cana-524	161	45	.	.	PUNCT
cana-524	162	1	[	[	X
cana-524	162	2	5	5	X
cana-524	162	3	]	]	PUNCT
cana-524	162	4	shyam	shyam	PROPN
cana-524	162	5	sundar	sundar	PROPN
cana-524	162	6	santra	santra	PROPN
cana-524	162	7	,	,	PUNCT
cana-524	162	8	necessary	necessary	ADJ
cana-524	162	9	and	and	CCONJ
cana-524	162	10	sufficient	sufficient	ADJ
cana-524	162	11	condition	condition	NOUN
cana-524	162	12	for	for	ADP
cana-524	162	13	oscillatory	oscillatory	ADJ
cana-524	162	14	and	and	CCONJ
cana-524	162	15	asymptotic	asymptotic	ADJ
cana-524	162	16	behaviour	behaviour	NOUN
cana-524	162	17	of	of	ADP
cana-524	162	18	second	second	ADJ
cana-524	162	19	-	-	PUNCT
cana-524	162	20	order	order	NOUN
cana-524	162	21	functional	functional	ADJ
cana-524	162	22	differential	differential	NOUN
cana-524	162	23	equations	equation	NOUN
cana-524	162	24	,	,	PUNCT
cana-524	162	25	kragujevac	kragujevac	PROPN
cana-524	162	26	journal	journal	PROPN
cana-524	162	27	of	of	ADP
cana-524	162	28	mathematics	mathematic	NOUN
cana-524	162	29	,	,	PUNCT
cana-524	162	30	vol	vol	NOUN
cana-524	162	31	.	.	PROPN
cana-524	162	32	44	44	NUM
cana-524	162	33	(	(	PUNCT
cana-524	162	34	3	3	NUM
cana-524	162	35	)	)	PUNCT
cana-524	162	36	(	(	PUNCT
cana-524	162	37	2020	2020	NUM
cana-524	162	38	)	)	PUNCT
cana-524	162	39	,	,	PUNCT
cana-524	162	40	pp	pp	ADP
cana-524	162	41	.	.	PUNCT
cana-524	163	1	459	459	NUM
cana-524	163	2	-	-	SYM
cana-524	163	3	473	473	NUM
cana-524	163	4	.	.	PUNCT
cana-524	164	1	[	[	X
cana-524	164	2	6	6	NUM
cana-524	164	3	]	]	PUNCT
cana-524	164	4	basak	basak	PROPN
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cana-524	164	20	equations	equation	NOUN
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cana-524	164	23	type	type	NOUN
cana-524	164	24	,	,	PUNCT
cana-524	164	25	hacettepe	hacettepe	ADJ
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cana-524	164	31	,	,	PUNCT
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cana-524	164	33	.	.	PUNCT
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cana-524	165	2	(	(	PUNCT
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cana-524	165	4	)	)	PUNCT
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cana-524	165	7	)	)	PUNCT
cana-524	165	8	,	,	PUNCT
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cana-524	165	10	-	-	SYM
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cana-524	165	12	.	.	PUNCT
cana-524	166	1	[	[	X
cana-524	166	2	7	7	X
cana-524	166	3	]	]	X
cana-524	166	4	p.	p.	NOUN
cana-524	166	5	dinakar	dinakar	NOUN
cana-524	166	6	,	,	PUNCT
cana-524	166	7	s.	s.	PROPN
cana-524	166	8	selvarangam	selvarangam	PROPN
cana-524	166	9	,	,	PUNCT
cana-524	166	10	e.	e.	PROPN
cana-524	166	11	thandapani	thandapani	PROPN
cana-524	166	12	,	,	PUNCT
cana-524	166	13	new	new	ADJ
cana-524	166	14	oscillation	oscillation	NOUN
cana-524	166	15	conditions	condition	NOUN
cana-524	166	16	for	for	ADP
cana-524	166	17	second	second	ADJ
cana-524	166	18	order	order	NOUN
cana-524	166	19	halflinear	halflinear	VERB
cana-524	166	20	advanced	advanced	ADJ
cana-524	166	21	difference	difference	NOUN
cana-524	166	22	equations	equation	NOUN
cana-524	166	23	,	,	PUNCT
cana-524	166	24	international	international	ADJ
cana-524	166	25	journal	journal	NOUN
cana-524	166	26	of	of	ADP
cana-524	166	27	mathematical	mathematical	ADJ
cana-524	166	28	engineering	engineering	NOUN
cana-524	166	29	and	and	CCONJ
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cana-524	166	31	sciences	science	NOUN
cana-524	166	32	,	,	PUNCT
cana-524	166	33	vol	vol	NOUN
cana-524	166	34	.	.	PROPN
cana-524	167	1	4	4	NUM
cana-524	167	2	,	,	PUNCT
cana-524	167	3	no	no	INTJ
cana-524	167	4	.	.	NOUN
cana-524	167	5	6	6	NUM
cana-524	167	6	,	,	PUNCT
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cana-524	167	8	-	-	SYM
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cana-524	167	10	,	,	PUNCT
cana-524	167	11	2019	2019	NUM
cana-524	167	12	.	.	PUNCT
cana-524	168	1	[	[	X
cana-524	168	2	8	8	NUM
cana-524	168	3	]	]	X
cana-524	168	4	martin	martin	PROPN
cana-524	168	5	bohner	bohner	PROPN
cana-524	168	6	,	,	PUNCT
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cana-524	168	13	,	,	PUNCT
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cana-524	168	15	criteria	criterion	NOUN
cana-524	168	16	for	for	ADP
cana-524	168	17	second	second	ADJ
cana-524	168	18	-	-	PUNCT
cana-524	168	19	order	order	NOUN
cana-524	168	20	neutral	neutral	ADJ
cana-524	168	21	delay	delay	NOUN
cana-524	168	22	differential	differential	PROPN
cana-524	168	23	equations	equation	NOUN
cana-524	168	24	,	,	PUNCT
cana-524	168	25	electronic	electronic	ADJ
cana-524	168	26	journal	journal	NOUN
cana-524	168	27	of	of	ADP
cana-524	168	28	qualitative	qualitative	ADJ
cana-524	168	29	theory	theory	NOUN
cana-524	168	30	of	of	ADP
cana-524	168	31	differential	differential	ADJ
cana-524	168	32	equations	equation	NOUN
cana-524	168	33	,	,	PUNCT
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cana-524	168	35	,	,	PUNCT
cana-524	168	36	no	no	INTJ
cana-524	168	37	.	.	NOUN
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cana-524	168	39	,	,	PUNCT
cana-524	168	40	1	1	NUM
cana-524	168	41	-	-	SYM
cana-524	168	42	12	12	NUM
cana-524	168	43	.	.	PUNCT
cana-524	169	1	[	[	X
cana-524	169	2	9	9	NUM
cana-524	169	3	]	]	PUNCT
cana-524	169	4	hongwu	hongwu	ADJ
cana-524	169	5	wu	wu	PROPN
cana-524	169	6	,	,	PUNCT
cana-524	169	7	lynn	lynn	PROPN
cana-524	169	8	erbe	erbe	PROPN
cana-524	169	9	,	,	PUNCT
cana-524	169	10	allan	allan	PROPN
cana-524	169	11	peterson	peterson	PROPN
cana-524	169	12	,	,	PUNCT
cana-524	169	13	oscillation	oscillation	NOUN
cana-524	169	14	of	of	ADP
cana-524	169	15	solution	solution	NOUN
cana-524	169	16	to	to	ADP
cana-524	169	17	second	second	ADJ
cana-524	169	18	order	order	NOUN
cana-524	169	19	half	half	ADJ
cana-524	169	20	-	-	PUNCT
cana-524	169	21	linear	linear	NOUN
cana-524	169	22	delay	delay	NOUN
cana-524	169	23	dynamic	dynamic	ADJ
cana-524	169	24	equations	equation	NOUN
cana-524	169	25	on	on	ADP
cana-524	169	26	time	time	NOUN
cana-524	169	27	scales	scale	NOUN
cana-524	169	28	,	,	PUNCT
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cana-524	169	34	,	,	PUNCT
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cana-524	169	36	.	.	PROPN
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cana-524	169	38	,	,	PUNCT
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cana-524	169	40	.	.	NOUN
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cana-524	169	42	,	,	PUNCT
cana-524	169	43	pp	pp	ADJ
cana-524	169	44	.	.	PUNCT
cana-524	170	1	1	1	NUM
cana-524	170	2	-	-	SYM
cana-524	170	3	15	15	NUM
cana-524	170	4	.	.	PUNCT
cana-524	171	1	communications	communication	NOUN
cana-524	171	2	on	on	ADP
cana-524	171	3	applied	apply	VERB
cana-524	171	4	nonlinear	nonlinear	ADJ
cana-524	171	5	analysis	analysis	NOUN
cana-524	171	6	issn	issn	NOUN
cana-524	171	7	:	:	PUNCT
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cana-524	171	9	-	-	PUNCT
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cana-524	171	11	vol	vol	NOUN
cana-524	171	12	31	31	NUM
cana-524	171	13	no	no	NOUN
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cana-524	171	16	(	(	PUNCT
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cana-524	171	18	)	)	PUNCT
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cana-524	172	1	[	[	X
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cana-524	172	4	e.	e.	PROPN
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cana-524	172	8	ravi	ravi	PROPN
cana-524	172	9	,	,	PUNCT
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cana-524	172	11	r.	r.	PROPN
cana-524	172	12	graef	graef	PROPN
cana-524	172	13	,	,	PUNCT
cana-524	172	14	oscillation	oscillation	NOUN
cana-524	172	15	and	and	CCONJ
cana-524	172	16	comparison	comparison	NOUN
cana-524	172	17	theorems	theorem	NOUN
cana-524	172	18	for	for	ADP
cana-524	172	19	half	half	ADJ
cana-524	172	20	-	-	PUNCT
cana-524	172	21	linear	linear	NOUN
cana-524	172	22	second	second	ADJ
cana-524	172	23	-	-	PUNCT
cana-524	172	24	order	order	NOUN
cana-524	172	25	difference	difference	NOUN
cana-524	172	26	equations	equation	NOUN
cana-524	172	27	,	,	PUNCT
cana-524	172	28	computers	computer	NOUN
cana-524	172	29	and	and	CCONJ
cana-524	172	30	mathematics	mathematic	NOUN
cana-524	172	31	with	with	ADP
cana-524	172	32	applications	application	NOUN
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cana-524	172	34	(	(	PUNCT
cana-524	172	35	2001	2001	NUM
cana-524	172	36	)	)	PUNCT
cana-524	172	37	,	,	PUNCT
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cana-524	172	39	-	-	SYM
cana-524	172	40	960	960	NUM
cana-524	172	41	.	.	PUNCT