id	sid	tid	token	lemma	pos
ejpam-6197	1	1	european	european	PROPN
ejpam-6197	1	2	journal	journal	PROPN
ejpam-6197	1	3	of	of	ADP
ejpam-6197	1	4	pure	pure	ADJ
ejpam-6197	1	5	and	and	CCONJ
ejpam-6197	1	6	applied	applied	ADJ
ejpam-6197	1	7	mathematics	mathematic	NOUN
ejpam-6197	1	8	2025	2025	NUM
ejpam-6197	1	9	,	,	PUNCT
ejpam-6197	1	10	vol	vol	NOUN
ejpam-6197	1	11	.	.	PROPN
ejpam-6197	1	12	18	18	NUM
ejpam-6197	1	13	,	,	PUNCT
ejpam-6197	1	14	issue	issue	NOUN
ejpam-6197	1	15	3	3	NUM
ejpam-6197	1	16	,	,	PUNCT
ejpam-6197	1	17	article	article	NOUN
ejpam-6197	1	18	number	number	NOUN
ejpam-6197	1	19	6197	6197	NUM
ejpam-6197	1	20	issn	issn	VERB
ejpam-6197	1	21	1307	1307	NUM
ejpam-6197	1	22	-	-	SYM
ejpam-6197	1	23	5543	5543	NUM
ejpam-6197	1	24	–	–	PUNCT
ejpam-6197	1	25	ejpam.com	ejpam.com	X
ejpam-6197	1	26	published	publish	VERB
ejpam-6197	1	27	by	by	ADP
ejpam-6197	1	28	new	new	PROPN
ejpam-6197	1	29	york	york	PROPN
ejpam-6197	1	30	business	business	PROPN
ejpam-6197	1	31	global	global	PROPN
ejpam-6197	1	32	the	the	DET
ejpam-6197	1	33	stability	stability	NOUN
ejpam-6197	1	34	of	of	ADP
ejpam-6197	1	35	the	the	DET
ejpam-6197	1	36	generalized	generalized	ADJ
ejpam-6197	1	37	functional	functional	ADJ
ejpam-6197	1	38	equation	equation	NOUN
ejpam-6197	1	39	gang	gang	NOUN
ejpam-6197	1	40	lyu1	lyu1	PROPN
ejpam-6197	1	41	,	,	PUNCT
ejpam-6197	1	42	yingxiu	yingxiu	PROPN
ejpam-6197	1	43	jiang2,∗	jiang2,∗	PROPN
ejpam-6197	1	44	,	,	PUNCT
ejpam-6197	1	45	qi	qi	PROPN
ejpam-6197	1	46	liu3	liu3	PROPN
ejpam-6197	1	47	,	,	PUNCT
ejpam-6197	1	48	choonkil	choonkil	PROPN
ejpam-6197	1	49	park4	park4	PROPN
ejpam-6197	1	50	1	1	NUM
ejpam-6197	1	51	school	school	NOUN
ejpam-6197	1	52	of	of	ADP
ejpam-6197	1	53	general	general	ADJ
ejpam-6197	1	54	education	education	NOUN
ejpam-6197	1	55	,	,	PUNCT
ejpam-6197	1	56	guangzhou	guangzhou	PROPN
ejpam-6197	1	57	college	college	PROPN
ejpam-6197	1	58	of	of	ADP
ejpam-6197	1	59	technology	technology	NOUN
ejpam-6197	1	60	and	and	CCONJ
ejpam-6197	1	61	business	business	NOUN
ejpam-6197	1	62	,	,	PUNCT
ejpam-6197	1	63	guangzhou	guangzhou	PROPN
ejpam-6197	1	64	510850	510850	NUM
ejpam-6197	1	65	,	,	PUNCT
ejpam-6197	1	66	p.r	p.r	PROPN
ejpam-6197	1	67	.	.	PROPN
ejpam-6197	1	68	china	china	PROPN
ejpam-6197	1	69	2	2	PROPN
ejpam-6197	1	70	department	department	NOUN
ejpam-6197	1	71	of	of	ADP
ejpam-6197	1	72	mathematics	mathematic	NOUN
ejpam-6197	1	73	,	,	PUNCT
ejpam-6197	1	74	yanbian	yanbian	ADJ
ejpam-6197	1	75	university	university	NOUN
ejpam-6197	1	76	,	,	PUNCT
ejpam-6197	1	77	yanji	yanji	X
ejpam-6197	1	78	133001	133001	NUM
ejpam-6197	1	79	,	,	PUNCT
ejpam-6197	1	80	p.r	p.r	PROPN
ejpam-6197	1	81	.	.	PROPN
ejpam-6197	1	82	china	china	PROPN
ejpam-6197	1	83	3	3	NUM
ejpam-6197	1	84	school	school	NOUN
ejpam-6197	1	85	of	of	ADP
ejpam-6197	1	86	mathematics	mathematic	NOUN
ejpam-6197	1	87	and	and	CCONJ
ejpam-6197	1	88	physics	physics	NOUN
ejpam-6197	1	89	,	,	PUNCT
ejpam-6197	1	90	anqing	anqe	VERB
ejpam-6197	1	91	normal	normal	ADJ
ejpam-6197	1	92	university	university	NOUN
ejpam-6197	1	93	,	,	PUNCT
ejpam-6197	1	94	anqing	anqe	VERB
ejpam-6197	1	95	246133	246133	NUM
ejpam-6197	1	96	,	,	PUNCT
ejpam-6197	1	97	p.r	p.r	PROPN
ejpam-6197	1	98	.	.	PROPN
ejpam-6197	1	99	china	china	PROPN
ejpam-6197	1	100	4	4	NUM
ejpam-6197	1	101	research	research	PROPN
ejpam-6197	1	102	institute	institute	NOUN
ejpam-6197	1	103	for	for	ADP
ejpam-6197	1	104	convergence	convergence	NOUN
ejpam-6197	1	105	of	of	ADP
ejpam-6197	1	106	basic	basic	ADJ
ejpam-6197	1	107	science	science	NOUN
ejpam-6197	1	108	,	,	PUNCT
ejpam-6197	1	109	hanyang	hanyang	NOUN
ejpam-6197	1	110	university	university	PROPN
ejpam-6197	1	111	,	,	PUNCT
ejpam-6197	1	112	seoul	seoul	PROPN
ejpam-6197	1	113	04763	04763	NUM
ejpam-6197	1	114	,	,	PUNCT
ejpam-6197	1	115	korea	korea	PROPN
ejpam-6197	1	116	abstract	abstract	NOUN
ejpam-6197	1	117	.	.	PUNCT
ejpam-6197	2	1	the	the	DET
ejpam-6197	2	2	aim	aim	NOUN
ejpam-6197	2	3	of	of	ADP
ejpam-6197	2	4	this	this	DET
ejpam-6197	2	5	paper	paper	NOUN
ejpam-6197	2	6	is	be	AUX
ejpam-6197	2	7	to	to	PART
ejpam-6197	2	8	prove	prove	VERB
ejpam-6197	2	9	the	the	DET
ejpam-6197	2	10	stability	stability	NOUN
ejpam-6197	2	11	(	(	PUNCT
ejpam-6197	2	12	in	in	ADP
ejpam-6197	2	13	the	the	DET
ejpam-6197	2	14	sense	sense	NOUN
ejpam-6197	2	15	of	of	ADP
ejpam-6197	2	16	ulam	ulam	NOUN
ejpam-6197	2	17	)	)	PUNCT
ejpam-6197	2	18	of	of	ADP
ejpam-6197	2	19	the	the	DET
ejpam-6197	2	20	functional	functional	ADJ
ejpam-6197	2	21	equation	equation	NOUN
ejpam-6197	2	22	:	:	PUNCT
ejpam-6197	2	23	f(x	f(x	PROPN
ejpam-6197	2	24	)	)	PUNCT
ejpam-6197	2	25	=	=	PUNCT
ejpam-6197	2	26	α(x)f(f1(x	α(x)f(f1(x	NOUN
ejpam-6197	2	27	)	)	PUNCT
ejpam-6197	2	28	)	)	PUNCT
ejpam-6197	3	1	+	+	CCONJ
ejpam-6197	3	2	β(x)f(f2(x	β(x)f(f2(x	NOUN
ejpam-6197	3	3	)	)	PUNCT
ejpam-6197	3	4	)	)	PUNCT
ejpam-6197	3	5	,	,	PUNCT
ejpam-6197	3	6	where	where	SCONJ
ejpam-6197	3	7	α	α	NOUN
ejpam-6197	3	8	and	and	CCONJ
ejpam-6197	3	9	β	β	X
ejpam-6197	3	10	are	be	AUX
ejpam-6197	3	11	given	give	VERB
ejpam-6197	3	12	real	real	ADV
ejpam-6197	3	13	valued	value	VERB
ejpam-6197	3	14	functions	function	NOUN
ejpam-6197	3	15	defined	define	VERB
ejpam-6197	3	16	on	on	ADP
ejpam-6197	3	17	a	a	DET
ejpam-6197	3	18	nonempty	nonempty	ADJ
ejpam-6197	3	19	set	set	VERB
ejpam-6197	3	20	s	s	PRON
ejpam-6197	3	21	such	such	ADJ
ejpam-6197	3	22	that	that	SCONJ
ejpam-6197	3	23	sup{|α(x)|	sup{|α(x)|	NOUN
ejpam-6197	3	24	:	:	PUNCT
ejpam-6197	3	25	x	x	SYM
ejpam-6197	3	26	∈	∈	PROPN
ejpam-6197	3	27	s	s	PART
ejpam-6197	3	28	}	}	PUNCT
ejpam-6197	3	29	<	<	X
ejpam-6197	3	30	1	1	NUM
ejpam-6197	3	31	and	and	CCONJ
ejpam-6197	3	32	fi(x)(i	fi(x)(i	NOUN
ejpam-6197	3	33	=	=	SYM
ejpam-6197	3	34	1	1	NUM
ejpam-6197	3	35	,	,	PUNCT
ejpam-6197	3	36	2	2	NUM
ejpam-6197	3	37	)	)	PUNCT
ejpam-6197	3	38	are	be	AUX
ejpam-6197	3	39	given	give	VERB
ejpam-6197	3	40	mappings	mapping	NOUN
ejpam-6197	3	41	.	.	PUNCT
ejpam-6197	4	1	2020	2020	NUM
ejpam-6197	4	2	mathematics	mathematic	NOUN
ejpam-6197	4	3	subject	subject	NOUN
ejpam-6197	4	4	classifications	classification	NOUN
ejpam-6197	4	5	:	:	PUNCT
ejpam-6197	4	6	39b62	39b62	NUM
ejpam-6197	4	7	,	,	PUNCT
ejpam-6197	4	8	39b52	39b52	NUM
ejpam-6197	4	9	,	,	PUNCT
ejpam-6197	4	10	47h10	47h10	NUM
ejpam-6197	4	11	,	,	PUNCT
ejpam-6197	4	12	46b25	46b25	NUM
ejpam-6197	4	13	key	key	ADJ
ejpam-6197	4	14	words	word	NOUN
ejpam-6197	4	15	and	and	CCONJ
ejpam-6197	4	16	phrases	phrase	NOUN
ejpam-6197	4	17	:	:	PUNCT
ejpam-6197	4	18	functional	functional	ADJ
ejpam-6197	4	19	inequations	inequation	NOUN
ejpam-6197	4	20	,	,	PUNCT
ejpam-6197	4	21	hyers	hyers	PROPN
ejpam-6197	4	22	-	-	PUNCT
ejpam-6197	4	23	ulam	ulam	PROPN
ejpam-6197	4	24	stability	stability	NOUN
ejpam-6197	4	25	,	,	PUNCT
ejpam-6197	4	26	banach	banach	NOUN
ejpam-6197	4	27	space	space	NOUN
ejpam-6197	4	28	1	1	NUM
ejpam-6197	4	29	.	.	PUNCT
ejpam-6197	5	1	introduction	introduction	NOUN
ejpam-6197	5	2	and	and	CCONJ
ejpam-6197	5	3	preliminaries	preliminary	NOUN
ejpam-6197	5	4	the	the	DET
ejpam-6197	5	5	origin	origin	NOUN
ejpam-6197	5	6	of	of	ADP
ejpam-6197	5	7	the	the	DET
ejpam-6197	5	8	stability	stability	NOUN
ejpam-6197	5	9	problem	problem	NOUN
ejpam-6197	5	10	for	for	ADP
ejpam-6197	5	11	functional	functional	ADJ
ejpam-6197	5	12	equations	equation	NOUN
ejpam-6197	5	13	traces	trace	VERB
ejpam-6197	5	14	back	back	ADV
ejpam-6197	5	15	to	to	ADP
ejpam-6197	5	16	a	a	DET
ejpam-6197	5	17	question	question	NOUN
ejpam-6197	5	18	posed	pose	VERB
ejpam-6197	5	19	by	by	ADP
ejpam-6197	5	20	ulam	ulam	PROPN
ejpam-6197	5	21	[	[	X
ejpam-6197	5	22	1	1	NUM
ejpam-6197	5	23	]	]	PUNCT
ejpam-6197	5	24	.	.	PUNCT
ejpam-6197	6	1	hyers	hyer	NOUN
ejpam-6197	7	1	[	[	X
ejpam-6197	7	2	2	2	X
ejpam-6197	7	3	]	]	PUNCT
ejpam-6197	7	4	made	make	VERB
ejpam-6197	7	5	remarkable	remarkable	ADJ
ejpam-6197	7	6	headway	headway	NOUN
ejpam-6197	7	7	in	in	ADP
ejpam-6197	7	8	1941	1941	NUM
ejpam-6197	7	9	.	.	PUNCT
ejpam-6197	8	1	in	in	ADP
ejpam-6197	8	2	the	the	DET
ejpam-6197	8	3	context	context	NOUN
ejpam-6197	8	4	of	of	ADP
ejpam-6197	8	5	banach	banach	NOUN
ejpam-6197	8	6	spaces	space	NOUN
ejpam-6197	8	7	,	,	PUNCT
ejpam-6197	8	8	he	he	PRON
ejpam-6197	8	9	derived	derive	VERB
ejpam-6197	8	10	highly	highly	ADV
ejpam-6197	8	11	renowned	renowned	ADJ
ejpam-6197	8	12	and	and	CCONJ
ejpam-6197	8	13	captivating	captivating	ADJ
ejpam-6197	8	14	results	result	NOUN
ejpam-6197	8	15	regarding	regard	VERB
ejpam-6197	8	16	the	the	DET
ejpam-6197	8	17	cauchy	cauchy	ADJ
ejpam-6197	8	18	functional	functional	ADJ
ejpam-6197	8	19	equation	equation	NOUN
ejpam-6197	8	20	.	.	PUNCT
ejpam-6197	9	1	later	later	ADV
ejpam-6197	9	2	,	,	PUNCT
ejpam-6197	9	3	there	there	PRON
ejpam-6197	9	4	are	be	VERB
ejpam-6197	9	5	also	also	ADV
ejpam-6197	9	6	very	very	ADV
ejpam-6197	9	7	many	many	ADJ
ejpam-6197	9	8	generalizations	generalization	NOUN
ejpam-6197	9	9	of	of	ADP
ejpam-6197	9	10	cauchy	cauchy	ADJ
ejpam-6197	9	11	equations	equation	NOUN
ejpam-6197	9	12	(	(	PUNCT
ejpam-6197	9	13	see	see	VERB
ejpam-6197	9	14	[	[	X
ejpam-6197	9	15	3–5	3–5	NOUN
ejpam-6197	9	16	]	]	PUNCT
ejpam-6197	9	17	)	)	PUNCT
ejpam-6197	9	18	.	.	PUNCT
ejpam-6197	10	1	for	for	ADP
ejpam-6197	10	2	more	more	ADJ
ejpam-6197	10	3	information	information	NOUN
ejpam-6197	10	4	concerning	concern	VERB
ejpam-6197	10	5	various	various	ADJ
ejpam-6197	10	6	functional	functional	ADJ
ejpam-6197	10	7	equations	equation	NOUN
ejpam-6197	10	8	,	,	PUNCT
ejpam-6197	10	9	see	see	VERB
ejpam-6197	10	10	[	[	X
ejpam-6197	10	11	6–16	6–16	PROPN
ejpam-6197	10	12	]	]	PUNCT
ejpam-6197	10	13	.	.	PUNCT
ejpam-6197	11	1	baker	baker	PROPN
ejpam-6197	11	2	was	be	AUX
ejpam-6197	11	3	the	the	DET
ejpam-6197	11	4	first	first	ADJ
ejpam-6197	11	5	to	to	PART
ejpam-6197	11	6	use	use	VERB
ejpam-6197	11	7	the	the	DET
ejpam-6197	11	8	fixed	fix	VERB
ejpam-6197	11	9	point	point	NOUN
ejpam-6197	11	10	method	method	NOUN
ejpam-6197	11	11	to	to	PART
ejpam-6197	11	12	study	study	VERB
ejpam-6197	11	13	the	the	DET
ejpam-6197	11	14	hyers	hyers	PROPN
ejpam-6197	11	15	-	-	PUNCT
ejpam-6197	11	16	ulam	ulam	ADJ
ejpam-6197	11	17	stability	stability	NOUN
ejpam-6197	11	18	of	of	ADP
ejpam-6197	11	19	functional	functional	ADJ
ejpam-6197	11	20	equations	equation	NOUN
ejpam-6197	11	21	in	in	ADP
ejpam-6197	11	22	[	[	X
ejpam-6197	11	23	17	17	NUM
ejpam-6197	11	24	]	]	PUNCT
ejpam-6197	11	25	.	.	PUNCT
ejpam-6197	12	1	in	in	ADP
ejpam-6197	12	2	that	that	DET
ejpam-6197	12	3	work	work	NOUN
ejpam-6197	12	4	,	,	PUNCT
ejpam-6197	12	5	he	he	PRON
ejpam-6197	12	6	actually	actually	ADV
ejpam-6197	12	7	utilized	utilize	VERB
ejpam-6197	12	8	the	the	DET
ejpam-6197	12	9	following	following	ADJ
ejpam-6197	12	10	variant	variant	NOUN
ejpam-6197	12	11	of	of	ADP
ejpam-6197	12	12	banach	banach	NOUN
ejpam-6197	12	13	’s	’s	PART
ejpam-6197	12	14	fixed	fix	VERB
ejpam-6197	12	15	point	point	NOUN
ejpam-6197	12	16	theorem	theorem	VERB
ejpam-6197	12	17	.	.	PROPN
ejpam-6197	13	1	for	for	ADP
ejpam-6197	13	2	the	the	DET
ejpam-6197	13	3	functional	functional	ADJ
ejpam-6197	13	4	equation	equation	NOUN
ejpam-6197	13	5	f(t	f(t	NOUN
ejpam-6197	13	6	)	)	PUNCT
ejpam-6197	13	7	=	=	SYM
ejpam-6197	13	8	(	(	PUNCT
ejpam-6197	13	9	α(t	α(t	PROPN
ejpam-6197	13	10	)	)	PUNCT
ejpam-6197	13	11	+	+	CCONJ
ejpam-6197	13	12	β(t))f(ϕ(t	β(t))f(ϕ(t	NUM
ejpam-6197	13	13	)	)	PUNCT
ejpam-6197	13	14	)	)	PUNCT
ejpam-6197	13	15	,	,	PUNCT
ejpam-6197	13	16	∗corresponding	∗corresponde	VERB
ejpam-6197	13	17	author	author	NOUN
ejpam-6197	13	18	.	.	PUNCT
ejpam-6197	14	1	doi	doi	NOUN
ejpam-6197	14	2	:	:	PUNCT
ejpam-6197	14	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6197	https://doi.org/10.29020/nybg.ejpam.v18i3.6197	PROPN
ejpam-6197	14	4	email	email	NOUN
ejpam-6197	14	5	addresses	address	NOUN
ejpam-6197	14	6	:	:	PUNCT
ejpam-6197	14	7	lvgang@gzgs.edu.cn	lvgang@gzgs.edu.cn	NOUN
ejpam-6197	14	8	(	(	PUNCT
ejpam-6197	14	9	g.	g.	PROPN
ejpam-6197	14	10	lyu	lyu	NOUN
ejpam-6197	14	11	)	)	PUNCT
ejpam-6197	14	12	,	,	PUNCT
ejpam-6197	14	13	yxjiang@ybu.edu.cn	yxjiang@ybu.edu.cn	PROPN
ejpam-6197	14	14	(	(	PUNCT
ejpam-6197	14	15	y.	y.	PROPN
ejpam-6197	14	16	jiang	jiang	PROPN
ejpam-6197	14	17	)	)	PUNCT
ejpam-6197	14	18	,	,	PUNCT
ejpam-6197	14	19	liuq67@aqnu.edu.cn	liuq67@aqnu.edu.cn	PROPN
ejpam-6197	14	20	(	(	PUNCT
ejpam-6197	14	21	q.	q.	PROPN
ejpam-6197	14	22	liu	liu	PROPN
ejpam-6197	14	23	)	)	PUNCT
ejpam-6197	14	24	,	,	PUNCT
ejpam-6197	14	25	baak@hanyang.ac.kr	baak@hanyang.ac.kr	PROPN
ejpam-6197	14	26	(	(	PUNCT
ejpam-6197	14	27	c.	c.	PROPN
ejpam-6197	14	28	park	park	PROPN
ejpam-6197	14	29	)	)	PUNCT
ejpam-6197	14	30	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6197	15	1	1	1	NUM
ejpam-6197	15	2	copyright	copyright	NOUN
ejpam-6197	15	3	:	:	PUNCT
ejpam-6197	15	4	©	©	PROPN
ejpam-6197	15	5	2025	2025	NUM
ejpam-6197	15	6	the	the	DET
ejpam-6197	15	7	author(s	author(s	NOUN
ejpam-6197	15	8	)	)	PUNCT
ejpam-6197	15	9	.	.	PUNCT
ejpam-6197	16	1	(	(	PUNCT
ejpam-6197	16	2	cc	cc	NOUN
ejpam-6197	16	3	by	by	ADP
ejpam-6197	16	4	-	-	PUNCT
ejpam-6197	16	5	nc	nc	PROPN
ejpam-6197	16	6	4.0	4.0	NUM
ejpam-6197	16	7	)	)	PUNCT
ejpam-6197	16	8	g.	g.	PROPN
ejpam-6197	16	9	lyu	lyu	NOUN
ejpam-6197	16	10	et	et	PROPN
ejpam-6197	16	11	al	al	PROPN
ejpam-6197	16	12	.	.	PUNCT
ejpam-6197	16	13	/	/	SYM
ejpam-6197	16	14	eur	eur	PROPN
ejpam-6197	16	15	.	.	PUNCT
ejpam-6197	17	1	j.	j.	PROPN
ejpam-6197	17	2	pure	pure	PROPN
ejpam-6197	17	3	appl	appl	PROPN
ejpam-6197	17	4	.	.	PROPN
ejpam-6197	17	5	math	math	PROPN
ejpam-6197	17	6	,	,	PUNCT
ejpam-6197	17	7	18	18	NUM
ejpam-6197	17	8	(	(	PUNCT
ejpam-6197	17	9	3	3	NUM
ejpam-6197	17	10	)	)	PUNCT
ejpam-6197	17	11	(	(	PUNCT
ejpam-6197	17	12	2025	2025	NUM
ejpam-6197	17	13	)	)	PUNCT
ejpam-6197	17	14	,	,	PUNCT
ejpam-6197	17	15	6197	6197	NUM
ejpam-6197	17	16	2	2	NUM
ejpam-6197	17	17	of	of	ADP
ejpam-6197	17	18	17	17	NUM
ejpam-6197	17	19	if	if	SCONJ
ejpam-6197	17	20	∥g(t)−	∥g(t)−	PROPN
ejpam-6197	17	21	{	{	PUNCT
ejpam-6197	17	22	α(t	α(t	PROPN
ejpam-6197	17	23	)	)	PUNCT
ejpam-6197	17	24	+	+	CCONJ
ejpam-6197	17	25	β(t)}g(ϕ(t))∥	β(t)}g(ϕ(t))∥	ADJ
ejpam-6197	17	26	≤	≤	ADJ
ejpam-6197	17	27	δ	δ	PROPN
ejpam-6197	17	28	,	,	PUNCT
ejpam-6197	17	29	then	then	ADV
ejpam-6197	17	30	there	there	PRON
ejpam-6197	17	31	exists	exist	VERB
ejpam-6197	17	32	a	a	DET
ejpam-6197	17	33	unique	unique	ADJ
ejpam-6197	17	34	mapping	mapping	NOUN
ejpam-6197	17	35	f	f	NOUN
ejpam-6197	17	36	satisfying	satisfy	VERB
ejpam-6197	17	37	this	this	DET
ejpam-6197	17	38	functional	functional	ADJ
ejpam-6197	17	39	equation	equation	NOUN
ejpam-6197	17	40	,	,	PUNCT
ejpam-6197	17	41	and	and	CCONJ
ejpam-6197	17	42	∥f(t)−	∥f(t)−	PROPN
ejpam-6197	17	43	g(t)∥	g(t)∥	NOUN
ejpam-6197	17	44	≤	≤	NUM
ejpam-6197	17	45	δ	δ	PROPN
ejpam-6197	17	46	1−	1−	NUM
ejpam-6197	17	47	λ	λ	X
ejpam-6197	17	48	.	.	PUNCT
ejpam-6197	18	1	sikorska	sikorska	ADJ
ejpam-6197	18	2	[	[	X
ejpam-6197	18	3	18	18	NUM
ejpam-6197	18	4	]	]	PUNCT
ejpam-6197	18	5	built	build	VERB
ejpam-6197	18	6	a	a	DET
ejpam-6197	18	7	direct	direct	ADJ
ejpam-6197	18	8	method	method	NOUN
ejpam-6197	18	9	(	(	PUNCT
ejpam-6197	18	10	see	see	VERB
ejpam-6197	18	11	,	,	PUNCT
ejpam-6197	18	12	e.g.	e.g.	ADV
ejpam-6197	18	13	,	,	PUNCT
ejpam-6197	18	14	[	[	X
ejpam-6197	18	15	19	19	NUM
ejpam-6197	18	16	,	,	PUNCT
ejpam-6197	18	17	20	20	NUM
ejpam-6197	18	18	]	]	PUNCT
ejpam-6197	18	19	)	)	PUNCT
ejpam-6197	18	20	to	to	PART
ejpam-6197	18	21	improve	improve	VERB
ejpam-6197	18	22	the	the	DET
ejpam-6197	18	23	approximating	approximate	VERB
ejpam-6197	18	24	constant	constant	ADJ
ejpam-6197	18	25	for	for	ADP
ejpam-6197	18	26	the	the	DET
ejpam-6197	18	27	following	follow	VERB
ejpam-6197	18	28	mappings	mapping	NOUN
ejpam-6197	18	29	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	19	1	af(h(x))−	af(h(x))−	NOUN
ejpam-6197	19	2	bf(−h(x))∥	bf(−h(x))∥	VERB
ejpam-6197	19	3	≤	≤	NUM
ejpam-6197	19	4	ϵ(x	ϵ(x	NOUN
ejpam-6197	19	5	)	)	PUNCT
ejpam-6197	19	6	,	,	PUNCT
ejpam-6197	19	7	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	19	8	af(hn(x))−	af(hn(x))−	NOUN
ejpam-6197	19	9	bf(hn+1(x))∥	bf(hn+1(x))∥	PROPN
ejpam-6197	19	10	≤	≤	NOUN
ejpam-6197	19	11	ϵ(x	ϵ(x	NOUN
ejpam-6197	19	12	)	)	PUNCT
ejpam-6197	19	13	.	.	PUNCT
ejpam-6197	20	1	it	it	PRON
ejpam-6197	20	2	improves	improve	VERB
ejpam-6197	20	3	the	the	DET
ejpam-6197	20	4	research	research	NOUN
ejpam-6197	20	5	results	result	NOUN
ejpam-6197	20	6	of	of	ADP
ejpam-6197	20	7	sikorska	sikorska	ADJ
ejpam-6197	20	8	[	[	X
ejpam-6197	20	9	18	18	NUM
ejpam-6197	20	10	]	]	PUNCT
ejpam-6197	20	11	,	,	PUNCT
ejpam-6197	20	12	breaking	break	VERB
ejpam-6197	20	13	through	through	ADP
ejpam-6197	20	14	the	the	DET
ejpam-6197	20	15	limitation	limitation	NOUN
ejpam-6197	20	16	that	that	PRON
ejpam-6197	20	17	her	her	PRON
ejpam-6197	20	18	method	method	NOUN
ejpam-6197	20	19	could	could	AUX
ejpam-6197	20	20	only	only	ADV
ejpam-6197	20	21	handle	handle	VERB
ejpam-6197	20	22	cases	case	NOUN
ejpam-6197	20	23	where	where	SCONJ
ejpam-6197	20	24	the	the	DET
ejpam-6197	20	25	intermediate	intermediate	ADJ
ejpam-6197	20	26	function	function	NOUN
ejpam-6197	20	27	is	be	AUX
ejpam-6197	20	28	odd	odd	ADJ
ejpam-6197	20	29	.	.	PUNCT
ejpam-6197	21	1	the	the	DET
ejpam-6197	21	2	paper	paper	NOUN
ejpam-6197	21	3	proposes	propose	VERB
ejpam-6197	21	4	a	a	DET
ejpam-6197	21	5	new	new	ADJ
ejpam-6197	21	6	direct	direct	ADJ
ejpam-6197	21	7	method	method	NOUN
ejpam-6197	21	8	,	,	PUNCT
ejpam-6197	21	9	which	which	PRON
ejpam-6197	21	10	is	be	AUX
ejpam-6197	21	11	not	not	PART
ejpam-6197	21	12	only	only	ADV
ejpam-6197	21	13	used	use	VERB
ejpam-6197	21	14	to	to	PART
ejpam-6197	21	15	solve	solve	VERB
ejpam-6197	21	16	specific	specific	ADJ
ejpam-6197	21	17	functional	functional	ADJ
ejpam-6197	21	18	equations	equation	NOUN
ejpam-6197	21	19	but	but	CCONJ
ejpam-6197	21	20	also	also	ADV
ejpam-6197	21	21	to	to	PART
ejpam-6197	21	22	construct	construct	VERB
ejpam-6197	21	23	and	and	CCONJ
ejpam-6197	21	24	study	study	VERB
ejpam-6197	21	25	a	a	DET
ejpam-6197	21	26	series	series	NOUN
ejpam-6197	21	27	of	of	ADP
ejpam-6197	21	28	functional	functional	ADJ
ejpam-6197	21	29	equations	equation	NOUN
ejpam-6197	21	30	.	.	PUNCT
ejpam-6197	22	1	this	this	PRON
ejpam-6197	22	2	provides	provide	VERB
ejpam-6197	22	3	new	new	ADJ
ejpam-6197	22	4	research	research	NOUN
ejpam-6197	22	5	tools	tool	NOUN
ejpam-6197	22	6	and	and	CCONJ
ejpam-6197	22	7	ideas	idea	NOUN
ejpam-6197	22	8	for	for	ADP
ejpam-6197	22	9	this	this	DET
ejpam-6197	22	10	field	field	NOUN
ejpam-6197	22	11	.	.	PUNCT
ejpam-6197	23	1	the	the	DET
ejpam-6197	23	2	fixed	fix	VERB
ejpam-6197	23	3	point	point	NOUN
ejpam-6197	23	4	theory	theory	NOUN
ejpam-6197	23	5	is	be	AUX
ejpam-6197	23	6	a	a	DET
ejpam-6197	23	7	pivotal	pivotal	ADJ
ejpam-6197	23	8	method	method	NOUN
ejpam-6197	23	9	for	for	ADP
ejpam-6197	23	10	proving	prove	VERB
ejpam-6197	23	11	the	the	DET
ejpam-6197	23	12	stability	stability	NOUN
ejpam-6197	23	13	of	of	ADP
ejpam-6197	23	14	functional	functional	ADJ
ejpam-6197	23	15	equations[21],widely	equations[21],widely	ADV
ejpam-6197	23	16	applied	apply	VERB
ejpam-6197	23	17	in	in	ADP
ejpam-6197	23	18	differential	differential	NOUN
ejpam-6197	23	19	equations[22	equations[22	NOUN
ejpam-6197	23	20	]	]	PUNCT
ejpam-6197	23	21	and	and	CCONJ
ejpam-6197	23	22	computer	computer	NOUN
ejpam-6197	23	23	science[23	science[23	NOUN
ejpam-6197	23	24	]	]	PUNCT
ejpam-6197	23	25	,	,	PUNCT
ejpam-6197	23	26	this	this	DET
ejpam-6197	23	27	theory	theory	NOUN
ejpam-6197	23	28	transforms	transform	VERB
ejpam-6197	23	29	abstract	abstract	ADJ
ejpam-6197	23	30	solvability	solvability	NOUN
ejpam-6197	23	31	problems	problem	NOUN
ejpam-6197	23	32	into	into	ADP
ejpam-6197	23	33	concrete	concrete	ADJ
ejpam-6197	23	34	analyses	analysis	NOUN
ejpam-6197	23	35	of	of	ADP
ejpam-6197	23	36	operator	operator	NOUN
ejpam-6197	23	37	properties	property	NOUN
ejpam-6197	23	38	,	,	PUNCT
ejpam-6197	23	39	remaining	remain	VERB
ejpam-6197	23	40	indispensable	indispensable	ADJ
ejpam-6197	23	41	for	for	ADP
ejpam-6197	23	42	establishing	establish	VERB
ejpam-6197	23	43	existence	existence	NOUN
ejpam-6197	23	44	and	and	CCONJ
ejpam-6197	23	45	uniqueness	uniqueness	NOUN
ejpam-6197	23	46	in	in	ADP
ejpam-6197	23	47	functional	functional	ADJ
ejpam-6197	23	48	equation	equation	NOUN
ejpam-6197	23	49	frameworks	framework	NOUN
ejpam-6197	23	50	.	.	PUNCT
ejpam-6197	24	1	thoroughly	thoroughly	ADV
ejpam-6197	24	2	,	,	PUNCT
ejpam-6197	24	3	we	we	PRON
ejpam-6197	24	4	explore	explore	VERB
ejpam-6197	24	5	the	the	DET
ejpam-6197	24	6	dependence	dependence	NOUN
ejpam-6197	24	7	relationships	relationship	NOUN
ejpam-6197	24	8	and	and	CCONJ
ejpam-6197	24	9	properties	property	NOUN
ejpam-6197	24	10	of	of	ADP
ejpam-6197	24	11	different	different	ADJ
ejpam-6197	24	12	parameters	parameter	NOUN
ejpam-6197	24	13	in	in	ADP
ejpam-6197	24	14	generalized	generalized	ADJ
ejpam-6197	24	15	functional	functional	ADJ
ejpam-6197	24	16	equations	equation	NOUN
ejpam-6197	24	17	.	.	PUNCT
ejpam-6197	25	1	when	when	SCONJ
ejpam-6197	25	2	studying	study	VERB
ejpam-6197	25	3	single	single	ADJ
ejpam-6197	25	4	-	-	PUNCT
ejpam-6197	25	5	variable	variable	ADJ
ejpam-6197	25	6	abstract	abstract	ADJ
ejpam-6197	25	7	equations	equation	NOUN
ejpam-6197	25	8	,	,	PUNCT
ejpam-6197	25	9	we	we	PRON
ejpam-6197	25	10	need	need	VERB
ejpam-6197	25	11	to	to	PART
ejpam-6197	25	12	analyze	analyze	VERB
ejpam-6197	25	13	the	the	DET
ejpam-6197	25	14	influence	influence	NOUN
ejpam-6197	25	15	of	of	ADP
ejpam-6197	25	16	parameters	parameter	NOUN
ejpam-6197	25	17	such	such	ADJ
ejpam-6197	25	18	as	as	ADP
ejpam-6197	25	19	α	α	PROPN
ejpam-6197	25	20	and	and	CCONJ
ejpam-6197	25	21	β	β	X
ejpam-6197	25	22	on	on	ADP
ejpam-6197	25	23	the	the	DET
ejpam-6197	25	24	solutions	solution	NOUN
ejpam-6197	25	25	.	.	PUNCT
ejpam-6197	26	1	also	also	ADV
ejpam-6197	26	2	,	,	PUNCT
ejpam-6197	26	3	expand	expand	VERB
ejpam-6197	26	4	the	the	DET
ejpam-6197	26	5	parameters	parameter	NOUN
ejpam-6197	26	6	from	from	ADP
ejpam-6197	26	7	the	the	DET
ejpam-6197	26	8	real	real	ADJ
ejpam-6197	26	9	number	number	NOUN
ejpam-6197	26	10	field	field	NOUN
ejpam-6197	26	11	to	to	ADP
ejpam-6197	26	12	the	the	DET
ejpam-6197	26	13	complex	complex	ADJ
ejpam-6197	26	14	number	number	NOUN
ejpam-6197	26	15	field	field	NOUN
ejpam-6197	26	16	to	to	PART
ejpam-6197	26	17	reveal	reveal	VERB
ejpam-6197	26	18	the	the	DET
ejpam-6197	26	19	characteristics	characteristic	NOUN
ejpam-6197	26	20	of	of	ADP
ejpam-6197	26	21	the	the	DET
ejpam-6197	26	22	equations	equation	NOUN
ejpam-6197	26	23	more	more	ADV
ejpam-6197	26	24	comprehensively	comprehensively	ADV
ejpam-6197	26	25	.	.	PUNCT
ejpam-6197	27	1	in	in	ADP
ejpam-6197	27	2	this	this	DET
ejpam-6197	27	3	paper	paper	NOUN
ejpam-6197	27	4	,	,	PUNCT
ejpam-6197	27	5	we	we	PRON
ejpam-6197	27	6	discuss	discuss	VERB
ejpam-6197	27	7	the	the	DET
ejpam-6197	27	8	hyers	hyers	PROPN
ejpam-6197	27	9	-	-	PUNCT
ejpam-6197	27	10	ulam	ulam	ADJ
ejpam-6197	27	11	stability	stability	NOUN
ejpam-6197	27	12	of	of	ADP
ejpam-6197	27	13	the	the	DET
ejpam-6197	27	14	following	follow	VERB
ejpam-6197	27	15	functional	functional	ADJ
ejpam-6197	27	16	equation	equation	NOUN
ejpam-6197	27	17	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	27	18	α(x)f(g(x))−	α(x)f(g(x))−	PUNCT
ejpam-6197	27	19	β(x)f(h(x))∥	β(x)f(h(x))∥	PROPN
ejpam-6197	27	20	≤	≤	NUM
ejpam-6197	27	21	ϵ(x	ϵ(x	NOUN
ejpam-6197	27	22	)	)	PUNCT
ejpam-6197	27	23	in	in	ADP
ejpam-6197	27	24	banach	banach	NOUN
ejpam-6197	27	25	spaces	space	NOUN
ejpam-6197	27	26	.	.	PUNCT
ejpam-6197	28	1	in	in	ADP
ejpam-6197	28	2	fact	fact	NOUN
ejpam-6197	28	3	,	,	PUNCT
ejpam-6197	28	4	the	the	DET
ejpam-6197	28	5	above	above	ADJ
ejpam-6197	28	6	mentioned	mention	VERB
ejpam-6197	28	7	inequality	inequality	NOUN
ejpam-6197	28	8	represents	represent	VERB
ejpam-6197	28	9	an	an	DET
ejpam-6197	28	10	even	even	ADV
ejpam-6197	28	11	more	more	ADV
ejpam-6197	28	12	general	general	ADJ
ejpam-6197	28	13	form	form	NOUN
ejpam-6197	28	14	.	.	PUNCT
ejpam-6197	29	1	in	in	ADP
ejpam-6197	29	2	section	section	NOUN
ejpam-6197	29	3	2	2	NUM
ejpam-6197	29	4	,	,	PUNCT
ejpam-6197	29	5	we	we	PRON
ejpam-6197	29	6	introduce	introduce	VERB
ejpam-6197	29	7	certain	certain	ADJ
ejpam-6197	29	8	improvements	improvement	NOUN
ejpam-6197	29	9	to	to	ADP
ejpam-6197	29	10	the	the	DET
ejpam-6197	29	11	existing	exist	VERB
ejpam-6197	29	12	approximations	approximation	NOUN
ejpam-6197	29	13	.	.	PUNCT
ejpam-6197	30	1	specifically	specifically	ADV
ejpam-6197	30	2	,	,	PUNCT
ejpam-6197	30	3	we	we	PRON
ejpam-6197	30	4	enhance	enhance	VERB
ejpam-6197	30	5	the	the	DET
ejpam-6197	30	6	existing	exist	VERB
ejpam-6197	30	7	approximation	approximation	NOUN
ejpam-6197	30	8	methods	method	NOUN
ejpam-6197	30	9	,	,	PUNCT
ejpam-6197	30	10	aiming	aim	VERB
ejpam-6197	30	11	to	to	PART
ejpam-6197	30	12	obtain	obtain	VERB
ejpam-6197	30	13	more	more	ADV
ejpam-6197	30	14	accurate	accurate	ADJ
ejpam-6197	30	15	and	and	CCONJ
ejpam-6197	30	16	comprehensive	comprehensive	ADJ
ejpam-6197	30	17	results	result	NOUN
ejpam-6197	30	18	.	.	PUNCT
ejpam-6197	31	1	these	these	DET
ejpam-6197	31	2	improvements	improvement	NOUN
ejpam-6197	31	3	contribute	contribute	VERB
ejpam-6197	31	4	to	to	ADP
ejpam-6197	31	5	a	a	DET
ejpam-6197	31	6	deeper	deep	ADJ
ejpam-6197	31	7	understanding	understanding	NOUN
ejpam-6197	31	8	of	of	ADP
ejpam-6197	31	9	the	the	DET
ejpam-6197	31	10	problem	problem	NOUN
ejpam-6197	31	11	and	and	CCONJ
ejpam-6197	31	12	provide	provide	VERB
ejpam-6197	31	13	more	more	ADV
ejpam-6197	31	14	effective	effective	ADJ
ejpam-6197	31	15	tools	tool	NOUN
ejpam-6197	31	16	for	for	ADP
ejpam-6197	31	17	subsequent	subsequent	ADJ
ejpam-6197	31	18	research	research	NOUN
ejpam-6197	31	19	.	.	PUNCT
ejpam-6197	32	1	in	in	ADP
ejpam-6197	32	2	section	section	NOUN
ejpam-6197	32	3	3	3	NUM
ejpam-6197	32	4	,	,	PUNCT
ejpam-6197	32	5	we	we	PRON
ejpam-6197	32	6	explore	explore	VERB
ejpam-6197	32	7	several	several	ADJ
ejpam-6197	32	8	applications	application	NOUN
ejpam-6197	32	9	of	of	ADP
ejpam-6197	32	10	the	the	DET
ejpam-6197	32	11	stability	stability	NOUN
ejpam-6197	32	12	results	result	VERB
ejpam-6197	32	13	.	.	PUNCT
ejpam-6197	33	1	we	we	PRON
ejpam-6197	33	2	not	not	PART
ejpam-6197	33	3	only	only	ADV
ejpam-6197	33	4	demonstrate	demonstrate	VERB
ejpam-6197	33	5	the	the	DET
ejpam-6197	33	6	practical	practical	ADJ
ejpam-6197	33	7	value	value	NOUN
ejpam-6197	33	8	of	of	ADP
ejpam-6197	33	9	the	the	DET
ejpam-6197	33	10	stability	stability	NOUN
ejpam-6197	33	11	theory	theory	NOUN
ejpam-6197	33	12	in	in	ADP
ejpam-6197	33	13	different	different	ADJ
ejpam-6197	33	14	scenarios	scenario	NOUN
ejpam-6197	33	15	but	but	CCONJ
ejpam-6197	33	16	also	also	ADV
ejpam-6197	33	17	expand	expand	VERB
ejpam-6197	33	18	its	its	PRON
ejpam-6197	33	19	scope	scope	NOUN
ejpam-6197	33	20	of	of	ADP
ejpam-6197	33	21	application	application	NOUN
ejpam-6197	33	22	,	,	PUNCT
ejpam-6197	33	23	showing	show	VERB
ejpam-6197	33	24	its	its	PRON
ejpam-6197	33	25	potential	potential	NOUN
ejpam-6197	33	26	in	in	ADP
ejpam-6197	33	27	solving	solve	VERB
ejpam-6197	33	28	real	real	ADJ
ejpam-6197	33	29	-	-	PUNCT
ejpam-6197	33	30	world	world	NOUN
ejpam-6197	33	31	problems	problem	NOUN
ejpam-6197	33	32	.	.	PUNCT
ejpam-6197	34	1	for	for	ADP
ejpam-6197	34	2	the	the	DET
ejpam-6197	34	3	sake	sake	NOUN
ejpam-6197	34	4	of	of	ADP
ejpam-6197	34	5	simplicity	simplicity	NOUN
ejpam-6197	34	6	,	,	PUNCT
ejpam-6197	34	7	we	we	PRON
ejpam-6197	34	8	present	present	VERB
ejpam-6197	34	9	our	our	PRON
ejpam-6197	34	10	results	result	NOUN
ejpam-6197	34	11	for	for	ADP
ejpam-6197	34	12	functions	function	NOUN
ejpam-6197	34	13	with	with	ADP
ejpam-6197	34	14	values	value	NOUN
ejpam-6197	34	15	in	in	ADP
ejpam-6197	34	16	banach	banach	NOUN
ejpam-6197	34	17	spaces	space	NOUN
ejpam-6197	34	18	.	.	PUNCT
ejpam-6197	35	1	however	however	ADV
ejpam-6197	35	2	,	,	PUNCT
ejpam-6197	35	3	with	with	ADP
ejpam-6197	35	4	some	some	DET
ejpam-6197	35	5	minor	minor	ADJ
ejpam-6197	35	6	additional	additional	ADJ
ejpam-6197	35	7	assumptions	assumption	NOUN
ejpam-6197	35	8	,	,	PUNCT
ejpam-6197	35	9	these	these	DET
ejpam-6197	35	10	results	result	NOUN
ejpam-6197	35	11	can	can	AUX
ejpam-6197	35	12	be	be	AUX
ejpam-6197	35	13	extended	extend	VERB
ejpam-6197	35	14	and	and	CCONJ
ejpam-6197	35	15	reformulated	reformulate	VERB
ejpam-6197	35	16	in	in	ADP
ejpam-6197	35	17	more	more	ADV
ejpam-6197	35	18	general	general	ADJ
ejpam-6197	35	19	banach	banach	NOUN
ejpam-6197	35	20	space	space	NOUN
ejpam-6197	35	21	settings	setting	NOUN
ejpam-6197	35	22	.	.	PUNCT
ejpam-6197	36	1	this	this	PRON
ejpam-6197	36	2	indicates	indicate	VERB
ejpam-6197	36	3	that	that	SCONJ
ejpam-6197	36	4	our	our	PRON
ejpam-6197	36	5	research	research	NOUN
ejpam-6197	36	6	has	have	VERB
ejpam-6197	36	7	the	the	DET
ejpam-6197	36	8	potential	potential	NOUN
ejpam-6197	36	9	to	to	PART
ejpam-6197	36	10	be	be	AUX
ejpam-6197	36	11	further	far	ADV
ejpam-6197	36	12	generalized	generalize	VERB
ejpam-6197	36	13	,	,	PUNCT
ejpam-6197	36	14	making	make	VERB
ejpam-6197	36	15	it	it	PRON
ejpam-6197	36	16	applicable	applicable	ADJ
ejpam-6197	36	17	to	to	ADP
ejpam-6197	36	18	a	a	DET
ejpam-6197	36	19	wider	wide	ADJ
ejpam-6197	36	20	range	range	NOUN
ejpam-6197	36	21	of	of	ADP
ejpam-6197	36	22	mathematical	mathematical	ADJ
ejpam-6197	36	23	models	model	NOUN
ejpam-6197	36	24	and	and	CCONJ
ejpam-6197	36	25	real	real	ADJ
ejpam-6197	36	26	world	world	NOUN
ejpam-6197	36	27	applications	application	NOUN
ejpam-6197	36	28	.	.	PUNCT
ejpam-6197	37	1	g.	g.	PROPN
ejpam-6197	37	2	lyu	lyu	VERB
ejpam-6197	37	3	et	et	PROPN
ejpam-6197	37	4	al	al	PROPN
ejpam-6197	37	5	.	.	PUNCT
ejpam-6197	37	6	/	/	SYM
ejpam-6197	37	7	eur	eur	PROPN
ejpam-6197	37	8	.	.	PUNCT
ejpam-6197	38	1	j.	j.	PROPN
ejpam-6197	38	2	pure	pure	PROPN
ejpam-6197	38	3	appl	appl	PROPN
ejpam-6197	38	4	.	.	PROPN
ejpam-6197	38	5	math	math	PROPN
ejpam-6197	38	6	,	,	PUNCT
ejpam-6197	38	7	18	18	NUM
ejpam-6197	38	8	(	(	PUNCT
ejpam-6197	38	9	3	3	NUM
ejpam-6197	38	10	)	)	PUNCT
ejpam-6197	38	11	(	(	PUNCT
ejpam-6197	38	12	2025	2025	NUM
ejpam-6197	38	13	)	)	PUNCT
ejpam-6197	38	14	,	,	PUNCT
ejpam-6197	38	15	6197	6197	NUM
ejpam-6197	38	16	3	3	NUM
ejpam-6197	38	17	of	of	ADP
ejpam-6197	38	18	17	17	NUM
ejpam-6197	38	19	2	2	NUM
ejpam-6197	38	20	.	.	PUNCT
ejpam-6197	38	21	main	main	ADJ
ejpam-6197	38	22	results	result	NOUN
ejpam-6197	38	23	in	in	ADP
ejpam-6197	38	24	this	this	DET
ejpam-6197	38	25	section	section	NOUN
ejpam-6197	38	26	,	,	PUNCT
ejpam-6197	38	27	we	we	PRON
ejpam-6197	38	28	prove	prove	VERB
ejpam-6197	38	29	the	the	DET
ejpam-6197	38	30	main	main	ADJ
ejpam-6197	38	31	theorem	theorem	NOUN
ejpam-6197	38	32	of	of	ADP
ejpam-6197	38	33	the	the	DET
ejpam-6197	38	34	paper	paper	NOUN
ejpam-6197	38	35	.	.	PUNCT
ejpam-6197	39	1	theorem	theorem	NOUN
ejpam-6197	39	2	1	1	NUM
ejpam-6197	39	3	.	.	PUNCT
ejpam-6197	39	4	suppose	suppose	VERB
ejpam-6197	39	5	that	that	SCONJ
ejpam-6197	39	6	x	x	PRON
ejpam-6197	39	7	is	be	AUX
ejpam-6197	39	8	a	a	DET
ejpam-6197	39	9	linear	linear	ADJ
ejpam-6197	39	10	normed	normed	ADJ
ejpam-6197	39	11	space	space	NOUN
ejpam-6197	39	12	and	and	CCONJ
ejpam-6197	39	13	y	y	PROPN
ejpam-6197	39	14	is	be	AUX
ejpam-6197	39	15	a	a	DET
ejpam-6197	39	16	banach	banach	NOUN
ejpam-6197	39	17	space	space	NOUN
ejpam-6197	39	18	.	.	PUNCT
ejpam-6197	40	1	let	let	VERB
ejpam-6197	40	2	f	f	NOUN
ejpam-6197	40	3	:	:	PUNCT
ejpam-6197	40	4	x	x	X
ejpam-6197	40	5	→	→	SYM
ejpam-6197	40	6	y	y	X
ejpam-6197	40	7	be	be	AUX
ejpam-6197	40	8	a	a	DET
ejpam-6197	40	9	mapping	mapping	NOUN
ejpam-6197	40	10	satisfying	satisfy	VERB
ejpam-6197	40	11	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	40	12	α(x)f(g(x))−	α(x)f(g(x))−	NUM
ejpam-6197	40	13	β(x)f(h(x))∥	β(x)f(h(x))∥	PROPN
ejpam-6197	40	14	≤	≤	NUM
ejpam-6197	40	15	ϵ(x	ϵ(x	NOUN
ejpam-6197	40	16	)	)	PUNCT
ejpam-6197	40	17	(	(	PUNCT
ejpam-6197	40	18	1	1	X
ejpam-6197	40	19	)	)	PUNCT
ejpam-6197	40	20	for	for	ADP
ejpam-6197	40	21	all	all	PRON
ejpam-6197	40	22	x	x	SYM
ejpam-6197	40	23	∈	∈	PROPN
ejpam-6197	40	24	x.	x.	NOUN
ejpam-6197	40	25	if	if	SCONJ
ejpam-6197	40	26	α(x	α(x	NOUN
ejpam-6197	40	27	)	)	PUNCT
ejpam-6197	40	28	and	and	CCONJ
ejpam-6197	40	29	β(x	β(x	NOUN
ejpam-6197	40	30	)	)	PUNCT
ejpam-6197	40	31	are	be	AUX
ejpam-6197	40	32	functions	function	NOUN
ejpam-6197	40	33	from	from	ADP
ejpam-6197	40	34	x	x	PUNCT
ejpam-6197	40	35	to	to	ADP
ejpam-6197	40	36	r	r	NOUN
ejpam-6197	40	37	,	,	PUNCT
ejpam-6197	40	38	and	and	CCONJ
ejpam-6197	40	39	g	g	NOUN
ejpam-6197	40	40	,	,	PUNCT
ejpam-6197	40	41	h	h	NOUN
ejpam-6197	40	42	:	:	PUNCT
ejpam-6197	40	43	x	x	X
ejpam-6197	40	44	→	→	PUNCT
ejpam-6197	40	45	x	x	NOUN
ejpam-6197	40	46	are	be	AUX
ejpam-6197	40	47	mappings	mapping	NOUN
ejpam-6197	40	48	and	and	CCONJ
ejpam-6197	40	49	ϵ	ϵ	NOUN
ejpam-6197	40	50	:	:	PUNCT
ejpam-6197	40	51	x	x	X
ejpam-6197	40	52	→	→	SYM
ejpam-6197	40	53	[	[	X
ejpam-6197	40	54	0,∞	0,∞	NUM
ejpam-6197	40	55	)	)	PUNCT
ejpam-6197	40	56	is	be	AUX
ejpam-6197	40	57	a	a	DET
ejpam-6197	40	58	function	function	NOUN
ejpam-6197	40	59	such	such	ADJ
ejpam-6197	40	60	that	that	SCONJ
ejpam-6197	40	61	∞∑	∞∑	NUM
ejpam-6197	40	62	n=0	n=0	NUM
ejpam-6197	40	63	(	(	PUNCT
ejpam-6197	40	64	λnϵ)(x	λnϵ)(x	NOUN
ejpam-6197	40	65	)	)	PUNCT
ejpam-6197	41	1	=	=	NOUN
ejpam-6197	41	2	:	:	PUNCT
ejpam-6197	41	3	ϵ∗(x	ϵ∗(x	NUM
ejpam-6197	41	4	)	)	PUNCT
ejpam-6197	41	5	<	<	X
ejpam-6197	41	6	∞	∞	NUM
ejpam-6197	41	7	holds	hold	NOUN
ejpam-6197	41	8	and	and	CCONJ
ejpam-6197	41	9	λ	λ	NOUN
ejpam-6197	41	10	is	be	AUX
ejpam-6197	41	11	a	a	DET
ejpam-6197	41	12	linear	linear	ADJ
ejpam-6197	41	13	operator	operator	NOUN
ejpam-6197	41	14	defined	define	VERB
ejpam-6197	41	15	by	by	ADP
ejpam-6197	41	16	(	(	PUNCT
ejpam-6197	41	17	λδ)(x	λδ)(x	PROPN
ejpam-6197	41	18	)	)	PUNCT
ejpam-6197	41	19	:	:	PUNCT
ejpam-6197	42	1	=	=	PUNCT
ejpam-6197	42	2	|α(x)|δ(g(x	|α(x)|δ(g(x	NUM
ejpam-6197	42	3	)	)	PUNCT
ejpam-6197	42	4	)	)	PUNCT
ejpam-6197	43	1	+	+	CCONJ
ejpam-6197	43	2	|β(x)|δ(h(x	|β(x)|δ(h(x	PROPN
ejpam-6197	43	3	)	)	PUNCT
ejpam-6197	43	4	)	)	PUNCT
ejpam-6197	43	5	for	for	ADP
ejpam-6197	43	6	a	a	DET
ejpam-6197	43	7	function	function	NOUN
ejpam-6197	43	8	δ	δ	NOUN
ejpam-6197	43	9	:	:	PUNCT
ejpam-6197	43	10	x	x	X
ejpam-6197	43	11	→	→	PUNCT
ejpam-6197	43	12	[	[	X
ejpam-6197	43	13	0,∞	0,∞	NUM
ejpam-6197	43	14	)	)	PUNCT
ejpam-6197	43	15	and	and	CCONJ
ejpam-6197	43	16	x	x	PUNCT
ejpam-6197	43	17	∈	∈	NOUN
ejpam-6197	43	18	x	x	NOUN
ejpam-6197	43	19	,	,	PUNCT
ejpam-6197	43	20	then	then	ADV
ejpam-6197	43	21	there	there	PRON
ejpam-6197	43	22	exists	exist	VERB
ejpam-6197	43	23	a	a	DET
ejpam-6197	43	24	unique	unique	ADJ
ejpam-6197	43	25	determined	determined	ADJ
ejpam-6197	43	26	mapping	mapping	NOUN
ejpam-6197	43	27	k	k	NOUN
ejpam-6197	43	28	:	:	PUNCT
ejpam-6197	43	29	x	x	X
ejpam-6197	43	30	→	→	SYM
ejpam-6197	43	31	y	y	PROPN
ejpam-6197	43	32	,	,	PUNCT
ejpam-6197	43	33	given	give	VERB
ejpam-6197	43	34	by	by	ADP
ejpam-6197	43	35	k(x	k(x	NOUN
ejpam-6197	43	36	)	)	PUNCT
ejpam-6197	43	37	=	=	SYM
ejpam-6197	44	1	α(x)k(g(x	α(x)k(g(x	NUM
ejpam-6197	44	2	)	)	PUNCT
ejpam-6197	44	3	)	)	PUNCT
ejpam-6197	45	1	+	+	CCONJ
ejpam-6197	45	2	β(x)k(h(x	β(x)k(h(x	NUM
ejpam-6197	45	3	)	)	PUNCT
ejpam-6197	45	4	)	)	PUNCT
ejpam-6197	46	1	such	such	ADJ
ejpam-6197	46	2	that	that	DET
ejpam-6197	46	3	∥f(x)−k(x)∥	∥f(x)−k(x)∥	NOUN
ejpam-6197	46	4	≤	≤	ADJ
ejpam-6197	46	5	ϵ∗(x	ϵ∗(x	PROPN
ejpam-6197	46	6	)	)	PUNCT
ejpam-6197	46	7	(	(	PUNCT
ejpam-6197	46	8	2	2	X
ejpam-6197	46	9	)	)	PUNCT
ejpam-6197	46	10	for	for	ADP
ejpam-6197	46	11	all	all	DET
ejpam-6197	46	12	x	x	SYM
ejpam-6197	46	13	∈	∈	ADJ
ejpam-6197	46	14	x.	x.	NOUN
ejpam-6197	46	15	proof	proof	NOUN
ejpam-6197	46	16	.	.	PUNCT
ejpam-6197	47	1	let	let	VERB
ejpam-6197	47	2	t	t	NOUN
ejpam-6197	47	3	:	:	PUNCT
ejpam-6197	47	4	y	y	PROPN
ejpam-6197	47	5	x	x	PUNCT
ejpam-6197	47	6	→	→	SYM
ejpam-6197	47	7	y	y	PROPN
ejpam-6197	47	8	x	x	VERB
ejpam-6197	47	9	be	be	AUX
ejpam-6197	47	10	an	an	DET
ejpam-6197	47	11	operator	operator	NOUN
ejpam-6197	47	12	satisfying	satisfying	NOUN
ejpam-6197	47	13	(	(	PUNCT
ejpam-6197	47	14	tf)(x	tf)(x	PROPN
ejpam-6197	47	15	)	)	PUNCT
ejpam-6197	47	16	)	)	PUNCT
ejpam-6197	48	1	=	=	PUNCT
ejpam-6197	48	2	α(x)f(g(x	α(x)f(g(x	NUM
ejpam-6197	48	3	)	)	PUNCT
ejpam-6197	48	4	)	)	PUNCT
ejpam-6197	49	1	+	+	CCONJ
ejpam-6197	49	2	β(x)f(h(x	β(x)f(h(x	X
ejpam-6197	49	3	)	)	PUNCT
ejpam-6197	49	4	)	)	PUNCT
ejpam-6197	50	1	in	in	ADP
ejpam-6197	50	2	(	(	PUNCT
ejpam-6197	50	3	1	1	NUM
ejpam-6197	50	4	)	)	PUNCT
ejpam-6197	50	5	.	.	PUNCT
ejpam-6197	51	1	then	then	ADV
ejpam-6197	51	2	we	we	PRON
ejpam-6197	51	3	get	get	VERB
ejpam-6197	51	4	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	51	5	(	(	PUNCT
ejpam-6197	51	6	tf)(x))∥	tf)(x))∥	VERB
ejpam-6197	51	7	≤	≤	NUM
ejpam-6197	51	8	ϵ(x	ϵ(x	NOUN
ejpam-6197	51	9	)	)	PUNCT
ejpam-6197	51	10	.	.	PUNCT
ejpam-6197	52	1	(	(	PUNCT
ejpam-6197	52	2	3	3	X
ejpam-6197	52	3	)	)	PUNCT
ejpam-6197	52	4	at	at	ADP
ejpam-6197	52	5	the	the	DET
ejpam-6197	52	6	same	same	ADJ
ejpam-6197	52	7	time	time	NOUN
ejpam-6197	52	8	,	,	PUNCT
ejpam-6197	52	9	we	we	PRON
ejpam-6197	52	10	can	can	AUX
ejpam-6197	52	11	see	see	VERB
ejpam-6197	52	12	that	that	SCONJ
ejpam-6197	52	13	∥(tξ)(x)−	∥(tξ)(x)−	PROPN
ejpam-6197	52	14	(	(	PUNCT
ejpam-6197	52	15	tζ)(x)∥	tζ)(x)∥	NOUN
ejpam-6197	52	16	≤	≤	VERB
ejpam-6197	52	17	|α(x)|∥ξ(g(x))−	|α(x)|∥ξ(g(x))−	ADJ
ejpam-6197	52	18	ζ(g(x))∥+	ζ(g(x))∥+	PUNCT
ejpam-6197	52	19	|β(x)|∥ξ(h(x))−	|β(x)|∥ξ(h(x))−	PROPN
ejpam-6197	52	20	ζ(h(x))∥	ζ(h(x))∥	PROPN
ejpam-6197	52	21	(	(	PUNCT
ejpam-6197	52	22	4	4	NUM
ejpam-6197	52	23	)	)	PUNCT
ejpam-6197	52	24	for	for	ADP
ejpam-6197	52	25	all	all	DET
ejpam-6197	52	26	ξ	ξ	ADJ
ejpam-6197	52	27	,	,	PUNCT
ejpam-6197	52	28	ζ	ζ	PROPN
ejpam-6197	52	29	∈	∈	PROPN
ejpam-6197	52	30	y	y	NOUN
ejpam-6197	52	31	x	x	X
ejpam-6197	52	32	and	and	CCONJ
ejpam-6197	52	33	x	x	SYM
ejpam-6197	52	34	∈	∈	PROPN
ejpam-6197	52	35	x.	x.	NOUN
ejpam-6197	52	36	first	first	ADV
ejpam-6197	52	37	,	,	PUNCT
ejpam-6197	52	38	we	we	PRON
ejpam-6197	52	39	get	get	VERB
ejpam-6197	52	40	by	by	ADP
ejpam-6197	52	41	induction	induction	NOUN
ejpam-6197	52	42	that	that	SCONJ
ejpam-6197	52	43	,	,	PUNCT
ejpam-6197	52	44	for	for	ADP
ejpam-6197	52	45	all	all	DET
ejpam-6197	52	46	n	n	PRON
ejpam-6197	52	47	∈	∈	PROPN
ejpam-6197	52	48	n	n	CCONJ
ejpam-6197	52	49	,	,	PUNCT
ejpam-6197	52	50	∥(tnf)(x))−	∥(tnf)(x))−	PROPN
ejpam-6197	52	51	(	(	PUNCT
ejpam-6197	52	52	tn+1f)(x)∥	tn+1f)(x)∥	NOUN
ejpam-6197	52	53	≤	≤	NOUN
ejpam-6197	52	54	(	(	PUNCT
ejpam-6197	52	55	λnϵ)(x	λnϵ)(x	NOUN
ejpam-6197	52	56	)	)	PUNCT
ejpam-6197	52	57	,	,	PUNCT
ejpam-6197	52	58	x	x	PUNCT
ejpam-6197	52	59	∈	∈	NOUN
ejpam-6197	52	60	x.	x.	NOUN
ejpam-6197	52	61	(	(	PUNCT
ejpam-6197	52	62	5	5	NUM
ejpam-6197	52	63	)	)	PUNCT
ejpam-6197	52	64	obviously	obviously	ADV
ejpam-6197	52	65	,	,	PUNCT
ejpam-6197	52	66	from	from	ADP
ejpam-6197	52	67	(	(	PUNCT
ejpam-6197	52	68	3	3	NUM
ejpam-6197	52	69	)	)	PUNCT
ejpam-6197	52	70	,	,	PUNCT
ejpam-6197	52	71	for	for	ADP
ejpam-6197	52	72	the	the	DET
ejpam-6197	52	73	case	case	NOUN
ejpam-6197	52	74	n	n	NOUN
ejpam-6197	52	75	=	=	SYM
ejpam-6197	52	76	0	0	NUM
ejpam-6197	52	77	,	,	PUNCT
ejpam-6197	52	78	(	(	PUNCT
ejpam-6197	52	79	5	5	X
ejpam-6197	52	80	)	)	PUNCT
ejpam-6197	52	81	holds	hold	VERB
ejpam-6197	52	82	.	.	PUNCT
ejpam-6197	53	1	now	now	ADV
ejpam-6197	53	2	fix	fix	VERB
ejpam-6197	53	3	n	n	PRON
ejpam-6197	53	4	∈	∈	NOUN
ejpam-6197	53	5	n	n	NOUN
ejpam-6197	53	6	and	and	CCONJ
ejpam-6197	53	7	suppose	suppose	VERB
ejpam-6197	53	8	that	that	SCONJ
ejpam-6197	53	9	the	the	DET
ejpam-6197	53	10	inequality	inequality	NOUN
ejpam-6197	53	11	(	(	PUNCT
ejpam-6197	53	12	5	5	NUM
ejpam-6197	53	13	)	)	PUNCT
ejpam-6197	53	14	is	be	AUX
ejpam-6197	53	15	valid	valid	ADJ
ejpam-6197	53	16	.	.	PUNCT
ejpam-6197	54	1	then	then	ADV
ejpam-6197	54	2	,	,	PUNCT
ejpam-6197	54	3	using	use	VERB
ejpam-6197	54	4	(	(	PUNCT
ejpam-6197	54	5	4	4	NUM
ejpam-6197	54	6	)	)	PUNCT
ejpam-6197	54	7	for	for	ADP
ejpam-6197	54	8	all	all	PRON
ejpam-6197	54	9	x	x	SYM
ejpam-6197	54	10	∈	∈	PROPN
ejpam-6197	54	11	x	x	NOUN
ejpam-6197	54	12	,	,	PUNCT
ejpam-6197	54	13	we	we	PRON
ejpam-6197	54	14	have	have	VERB
ejpam-6197	54	15	∥(tn+1f)(x)−	∥(tn+1f)(x)−	NOUN
ejpam-6197	54	16	(	(	PUNCT
ejpam-6197	54	17	tn+2f)(x)∥	tn+2f)(x)∥	NOUN
ejpam-6197	54	18	≤	≤	X
ejpam-6197	54	19	|α(x)|∥(tnf)(g(x))−	|α(x)|∥(tnf)(g(x))−	NUM
ejpam-6197	54	20	(	(	PUNCT
ejpam-6197	54	21	tn+1f)(g(x))∥+	tn+1f)(g(x))∥+	ADJ
ejpam-6197	54	22	|β(x)|∥(tnf)(h(x))−	|β(x)|∥(tnf)(h(x))−	NUM
ejpam-6197	54	23	(	(	PUNCT
ejpam-6197	54	24	tn+1f)(h(x))∥	tn+1f)(h(x))∥	PROPN
ejpam-6197	54	25	≤	≤	PROPN
ejpam-6197	54	26	|α(x)|(λnϵ)(g(x	|α(x)|(λnϵ)(g(x	NOUN
ejpam-6197	54	27	)	)	PUNCT
ejpam-6197	54	28	)	)	PUNCT
ejpam-6197	55	1	+	+	CCONJ
ejpam-6197	55	2	|β(x)|(λnϵ)(h(x	|β(x)|(λnϵ)(h(x	NOUN
ejpam-6197	55	3	)	)	PUNCT
ejpam-6197	55	4	)	)	PUNCT
ejpam-6197	56	1	=	=	SYM
ejpam-6197	56	2	(	(	PUNCT
ejpam-6197	56	3	λn+1ϵ)(x	λn+1ϵ)(x	PROPN
ejpam-6197	56	4	)	)	PUNCT
ejpam-6197	56	5	.	.	PUNCT
ejpam-6197	57	1	g.	g.	PROPN
ejpam-6197	57	2	lyu	lyu	VERB
ejpam-6197	57	3	et	et	PROPN
ejpam-6197	57	4	al	al	PROPN
ejpam-6197	57	5	.	.	PUNCT
ejpam-6197	57	6	/	/	SYM
ejpam-6197	57	7	eur	eur	PROPN
ejpam-6197	57	8	.	.	PUNCT
ejpam-6197	58	1	j.	j.	PROPN
ejpam-6197	58	2	pure	pure	PROPN
ejpam-6197	58	3	appl	appl	PROPN
ejpam-6197	58	4	.	.	PROPN
ejpam-6197	58	5	math	math	PROPN
ejpam-6197	58	6	,	,	PUNCT
ejpam-6197	58	7	18	18	NUM
ejpam-6197	58	8	(	(	PUNCT
ejpam-6197	58	9	3	3	NUM
ejpam-6197	58	10	)	)	PUNCT
ejpam-6197	58	11	(	(	PUNCT
ejpam-6197	58	12	2025	2025	NUM
ejpam-6197	58	13	)	)	PUNCT
ejpam-6197	58	14	,	,	PUNCT
ejpam-6197	58	15	6197	6197	NUM
ejpam-6197	58	16	4	4	NUM
ejpam-6197	58	17	of	of	ADP
ejpam-6197	58	18	17	17	NUM
ejpam-6197	58	19	thus	thus	ADV
ejpam-6197	58	20	we	we	PRON
ejpam-6197	58	21	complete	complete	VERB
ejpam-6197	58	22	the	the	DET
ejpam-6197	58	23	proof	proof	NOUN
ejpam-6197	58	24	of	of	ADP
ejpam-6197	58	25	(	(	PUNCT
ejpam-6197	58	26	5	5	NUM
ejpam-6197	58	27	)	)	PUNCT
ejpam-6197	58	28	.	.	PUNCT
ejpam-6197	59	1	for	for	ADP
ejpam-6197	59	2	n	n	PRON
ejpam-6197	59	3	,	,	PUNCT
ejpam-6197	59	4	k	k	PROPN
ejpam-6197	59	5	∈	∈	PROPN
ejpam-6197	59	6	n	n	PROPN
ejpam-6197	59	7	,	,	PUNCT
ejpam-6197	59	8	k	k	PROPN
ejpam-6197	59	9	>	>	X
ejpam-6197	59	10	0	0	PROPN
ejpam-6197	59	11	,	,	PUNCT
ejpam-6197	59	12	∥(tnf)(x)−	∥(tnf)(x)−	PROPN
ejpam-6197	59	13	(	(	PUNCT
ejpam-6197	59	14	tn+kf)(x)∥	tn+kf)(x)∥	X
ejpam-6197	59	15	≤	≤	X
ejpam-6197	59	16	k−1∑	k−1∑	PROPN
ejpam-6197	59	17	i=0	i=0	PROPN
ejpam-6197	59	18	∥(tnf)(x)−	∥(tnf)(x)−	X
ejpam-6197	59	19	(	(	PUNCT
ejpam-6197	59	20	tn+i+1f)(x)∥	tn+i+1f)(x)∥	X
ejpam-6197	59	21	≤	≤	NUM
ejpam-6197	60	1	n+k−1∑	n+k−1∑	ADP
ejpam-6197	60	2	i	i	PROPN
ejpam-6197	60	3	=	=	NOUN
ejpam-6197	60	4	n	n	X
ejpam-6197	60	5	(	(	PUNCT
ejpam-6197	60	6	λiϵ)(x	λiϵ)(x	NOUN
ejpam-6197	60	7	)	)	PUNCT
ejpam-6197	60	8	≤	≤	NUM
ejpam-6197	60	9	ϵ∗(x	ϵ∗(x	PROPN
ejpam-6197	60	10	)	)	PUNCT
ejpam-6197	61	1	,	,	PUNCT
ejpam-6197	61	2	x	x	PUNCT
ejpam-6197	61	3	∈	∈	NOUN
ejpam-6197	61	4	x.	x.	NOUN
ejpam-6197	61	5	(	(	PUNCT
ejpam-6197	61	6	6	6	NUM
ejpam-6197	61	7	)	)	PUNCT
ejpam-6197	61	8	from	from	ADP
ejpam-6197	61	9	the	the	DET
ejpam-6197	61	10	convergence	convergence	NOUN
ejpam-6197	61	11	of	of	ADP
ejpam-6197	61	12	the	the	DET
ejpam-6197	61	13	series	series	NOUN
ejpam-6197	61	14	∑	∑	PROPN
ejpam-6197	61	15	(	(	PUNCT
ejpam-6197	61	16	λnϵ)(x	λnϵ)(x	NOUN
ejpam-6197	61	17	)	)	PUNCT
ejpam-6197	61	18	,	,	PUNCT
ejpam-6197	61	19	for	for	ADP
ejpam-6197	61	20	every	every	DET
ejpam-6197	61	21	x	x	SYM
ejpam-6197	61	22	∈	∈	PROPN
ejpam-6197	61	23	x	x	NOUN
ejpam-6197	61	24	,	,	PUNCT
ejpam-6197	61	25	{	{	PUNCT
ejpam-6197	61	26	(	(	PUNCT
ejpam-6197	61	27	tnf)(x)}n∈n	tnf)(x)}n∈n	NOUN
ejpam-6197	61	28	is	be	AUX
ejpam-6197	61	29	a	a	DET
ejpam-6197	61	30	cauchy	cauchy	ADJ
ejpam-6197	61	31	sequence	sequence	NOUN
ejpam-6197	61	32	.	.	PUNCT
ejpam-6197	62	1	since	since	SCONJ
ejpam-6197	62	2	y	y	PROPN
ejpam-6197	62	3	is	be	AUX
ejpam-6197	62	4	complete	complete	ADJ
ejpam-6197	62	5	,	,	PUNCT
ejpam-6197	62	6	we	we	PRON
ejpam-6197	62	7	can	can	AUX
ejpam-6197	62	8	define	define	VERB
ejpam-6197	62	9	limn→∞(tnf)(x	limn→∞(tnf)(x	PROPN
ejpam-6197	62	10	)	)	PUNCT
ejpam-6197	62	11	:	:	PUNCT
ejpam-6197	62	12	=	=	SYM
ejpam-6197	62	13	ψ(x	ψ(x	NOUN
ejpam-6197	62	14	)	)	PUNCT
ejpam-6197	62	15	.	.	PUNCT
ejpam-6197	63	1	taking	take	VERB
ejpam-6197	63	2	n	n	NOUN
ejpam-6197	63	3	=	=	SYM
ejpam-6197	63	4	0	0	PROPN
ejpam-6197	63	5	and	and	CCONJ
ejpam-6197	63	6	k	k	PROPN
ejpam-6197	63	7	→	→	SYM
ejpam-6197	63	8	∞	∞	PROPN
ejpam-6197	63	9	in	in	ADP
ejpam-6197	63	10	(	(	PUNCT
ejpam-6197	63	11	6	6	NUM
ejpam-6197	63	12	)	)	PUNCT
ejpam-6197	63	13	,	,	PUNCT
ejpam-6197	63	14	we	we	PRON
ejpam-6197	63	15	know	know	VERB
ejpam-6197	63	16	that	that	SCONJ
ejpam-6197	63	17	(	(	PUNCT
ejpam-6197	63	18	2	2	X
ejpam-6197	63	19	)	)	PUNCT
ejpam-6197	63	20	holds	hold	NOUN
ejpam-6197	63	21	,	,	PUNCT
ejpam-6197	63	22	and	and	CCONJ
ejpam-6197	63	23	∥(tψ)(x)−	∥(tψ)(x)−	PROPN
ejpam-6197	63	24	(	(	PUNCT
ejpam-6197	63	25	tn+1f)(x)∥	tn+1f)(x)∥	ADV
ejpam-6197	63	26	≤	≤	NOUN
ejpam-6197	63	27	(	(	PUNCT
ejpam-6197	63	28	λn+1)ϵ(x	λn+1)ϵ(x	NUM
ejpam-6197	63	29	)	)	PUNCT
ejpam-6197	63	30	,	,	PUNCT
ejpam-6197	64	1	n	n	PROPN
ejpam-6197	64	2	∈	∈	PROPN
ejpam-6197	64	3	n	n	NOUN
ejpam-6197	64	4	,	,	PUNCT
ejpam-6197	64	5	x	x	PUNCT
ejpam-6197	64	6	∈	∈	NOUN
ejpam-6197	64	7	x	x	NOUN
ejpam-6197	64	8	,	,	PUNCT
ejpam-6197	64	9	and	and	CCONJ
ejpam-6197	64	10	thus	thus	ADV
ejpam-6197	64	11	(	(	PUNCT
ejpam-6197	64	12	tψ)(x	tψ)(x	PROPN
ejpam-6197	64	13	)	)	PUNCT
ejpam-6197	64	14	=	=	PROPN
ejpam-6197	65	1	lim	lim	PROPN
ejpam-6197	65	2	n→∞	n→∞	X
ejpam-6197	65	3	(	(	PUNCT
ejpam-6197	65	4	tn+1ψ)(x	tn+1ψ)(x	PROPN
ejpam-6197	65	5	)	)	PUNCT
ejpam-6197	65	6	=	=	SYM
ejpam-6197	65	7	ψ(x	ψ(x	NOUN
ejpam-6197	65	8	)	)	PUNCT
ejpam-6197	65	9	,	,	PUNCT
ejpam-6197	65	10	x	x	PUNCT
ejpam-6197	65	11	∈	∈	PROPN
ejpam-6197	65	12	x.	x.	NOUN
ejpam-6197	65	13	in	in	ADP
ejpam-6197	65	14	order	order	NOUN
ejpam-6197	65	15	to	to	PART
ejpam-6197	65	16	prove	prove	VERB
ejpam-6197	65	17	the	the	DET
ejpam-6197	65	18	uniqueness	uniqueness	NOUN
ejpam-6197	65	19	of	of	ADP
ejpam-6197	65	20	ψ	ψ	NOUN
ejpam-6197	65	21	,	,	PUNCT
ejpam-6197	65	22	suppose	suppose	VERB
ejpam-6197	65	23	that	that	SCONJ
ejpam-6197	65	24	ψ1	ψ1	NOUN
ejpam-6197	65	25	,	,	PUNCT
ejpam-6197	65	26	ψ2	ψ2	NOUN
ejpam-6197	65	27	∈	∈	PROPN
ejpam-6197	65	28	y	y	NOUN
ejpam-6197	65	29	x	x	X
ejpam-6197	65	30	are	be	AUX
ejpam-6197	65	31	two	two	NUM
ejpam-6197	65	32	fixed	fix	VERB
ejpam-6197	65	33	points	point	NOUN
ejpam-6197	65	34	of	of	ADP
ejpam-6197	65	35	t	t	PROPN
ejpam-6197	65	36	with	with	ADP
ejpam-6197	65	37	∥ψi(x)−	∥ψi(x)−	PROPN
ejpam-6197	65	38	f(x)∥	f(x)∥	VERB
ejpam-6197	65	39	≤	≤	NUM
ejpam-6197	65	40	ϵ∗(x	ϵ∗(x	PROPN
ejpam-6197	65	41	)	)	PUNCT
ejpam-6197	65	42	for	for	ADP
ejpam-6197	65	43	all	all	PRON
ejpam-6197	65	44	x	x	SYM
ejpam-6197	65	45	∈	∈	PROPN
ejpam-6197	65	46	x	x	NOUN
ejpam-6197	65	47	,	,	PUNCT
ejpam-6197	65	48	i	i	NOUN
ejpam-6197	65	49	=	=	NOUN
ejpam-6197	65	50	1	1	NUM
ejpam-6197	65	51	,	,	PUNCT
ejpam-6197	65	52	2	2	NUM
ejpam-6197	65	53	.	.	X
ejpam-6197	65	54	we	we	PRON
ejpam-6197	65	55	can	can	AUX
ejpam-6197	65	56	easily	easily	ADV
ejpam-6197	65	57	show	show	VERB
ejpam-6197	65	58	that	that	SCONJ
ejpam-6197	65	59	∥ψ1(x)−	∥ψ1(x)−	PROPN
ejpam-6197	65	60	ψ2(x)∥	ψ2(x)∥	ADP
ejpam-6197	65	61	=	=	SYM
ejpam-6197	65	62	∥(tmψ1)(x)−	∥(tmψ1)(x)−	PROPN
ejpam-6197	65	63	(	(	PUNCT
ejpam-6197	65	64	tmψ2)(x)∥	tmψ2)(x)∥	ADV
ejpam-6197	65	65	≤	≤	ADJ
ejpam-6197	65	66	2	2	NUM
ejpam-6197	65	67	∞∑	∞∑	NUM
ejpam-6197	65	68	i	i	PRON
ejpam-6197	65	69	=	=	NOUN
ejpam-6197	65	70	m	m	PROPN
ejpam-6197	65	71	(	(	PUNCT
ejpam-6197	65	72	λiϵ)(x	λiϵ)(x	NOUN
ejpam-6197	65	73	)	)	PUNCT
ejpam-6197	65	74	,	,	PUNCT
ejpam-6197	65	75	x	x	PUNCT
ejpam-6197	65	76	∈	∈	NOUN
ejpam-6197	65	77	x.	x.	NOUN
ejpam-6197	65	78	thus	thus	ADV
ejpam-6197	65	79	ψ1(x	ψ1(x	NOUN
ejpam-6197	65	80	)	)	PUNCT
ejpam-6197	65	81	=	=	SYM
ejpam-6197	65	82	ψ2(x	ψ2(x	PROPN
ejpam-6197	65	83	)	)	PUNCT
ejpam-6197	65	84	.	.	PUNCT
ejpam-6197	66	1	remark	remark	PROPN
ejpam-6197	66	2	1	1	NUM
ejpam-6197	66	3	.	.	PUNCT
ejpam-6197	67	1	we	we	PRON
ejpam-6197	67	2	can	can	AUX
ejpam-6197	67	3	easily	easily	ADV
ejpam-6197	67	4	prove∥∥∥∥∥f(x)−	prove∥∥∥∥∥f(x)−	PROPN
ejpam-6197	67	5	n∑	n∑	PROPN
ejpam-6197	67	6	i=0	i=0	PROPN
ejpam-6197	67	7	αi(x)(f(fi(x	αi(x)(f(fi(x	NUM
ejpam-6197	67	8	)	)	PUNCT
ejpam-6197	67	9	)	)	PUNCT
ejpam-6197	67	10	)	)	PUNCT
ejpam-6197	68	1	∥∥∥∥∥	∥∥∥∥∥	X
ejpam-6197	68	2	≤	≤	ADJ
ejpam-6197	68	3	ϵ(x	ϵ(x	NOUN
ejpam-6197	68	4	)	)	PUNCT
ejpam-6197	68	5	for	for	ADP
ejpam-6197	68	6	all	all	PRON
ejpam-6197	68	7	x	x	SYM
ejpam-6197	68	8	∈	∈	NOUN
ejpam-6197	68	9	x.	x.	NOUN
ejpam-6197	68	10	thus	thus	ADV
ejpam-6197	68	11	there	there	PRON
ejpam-6197	68	12	exists	exist	VERB
ejpam-6197	68	13	a	a	DET
ejpam-6197	68	14	unique	unique	ADJ
ejpam-6197	68	15	determined	determined	ADJ
ejpam-6197	68	16	mapping	mapping	NOUN
ejpam-6197	68	17	k	k	NOUN
ejpam-6197	68	18	:	:	PUNCT
ejpam-6197	68	19	x	x	X
ejpam-6197	68	20	→	→	SYM
ejpam-6197	68	21	y	y	PROPN
ejpam-6197	68	22	,	,	PUNCT
ejpam-6197	68	23	given	give	VERB
ejpam-6197	68	24	by	by	ADP
ejpam-6197	68	25	k(x	k(x	NOUN
ejpam-6197	68	26	)	)	PUNCT
ejpam-6197	69	1	=	=	SYM
ejpam-6197	69	2	n∑	n∑	PROPN
ejpam-6197	69	3	i=0	i=0	PROPN
ejpam-6197	69	4	αi(x)(k(fi(x	αi(x)(k(fi(x	NUM
ejpam-6197	69	5	)	)	PUNCT
ejpam-6197	69	6	)	)	PUNCT
ejpam-6197	69	7	,	,	PUNCT
ejpam-6197	70	1	x	x	PUNCT
ejpam-6197	70	2	∈	∈	NOUN
ejpam-6197	70	3	x	x	NOUN
ejpam-6197	70	4	,	,	PUNCT
ejpam-6197	70	5	such	such	ADJ
ejpam-6197	70	6	that	that	DET
ejpam-6197	70	7	∥f(x)−k(x)∥	∥f(x)−k(x)∥	NOUN
ejpam-6197	70	8	≤	≤	ADJ
ejpam-6197	70	9	ϵ∗(x	ϵ∗(x	PROPN
ejpam-6197	70	10	)	)	PUNCT
ejpam-6197	70	11	for	for	ADP
ejpam-6197	70	12	all	all	PRON
ejpam-6197	70	13	x	x	SYM
ejpam-6197	70	14	∈	∈	NOUN
ejpam-6197	70	15	x.	x.	NOUN
ejpam-6197	70	16	g.	g.	PROPN
ejpam-6197	70	17	lyu	lyu	PROPN
ejpam-6197	71	1	et	et	PROPN
ejpam-6197	71	2	al	al	PROPN
ejpam-6197	71	3	.	.	PUNCT
ejpam-6197	71	4	/	/	SYM
ejpam-6197	71	5	eur	eur	PROPN
ejpam-6197	71	6	.	.	PUNCT
ejpam-6197	72	1	j.	j.	PROPN
ejpam-6197	72	2	pure	pure	PROPN
ejpam-6197	72	3	appl	appl	PROPN
ejpam-6197	72	4	.	.	PROPN
ejpam-6197	72	5	math	math	PROPN
ejpam-6197	72	6	,	,	PUNCT
ejpam-6197	72	7	18	18	NUM
ejpam-6197	72	8	(	(	PUNCT
ejpam-6197	72	9	3	3	NUM
ejpam-6197	72	10	)	)	PUNCT
ejpam-6197	72	11	(	(	PUNCT
ejpam-6197	72	12	2025	2025	NUM
ejpam-6197	72	13	)	)	PUNCT
ejpam-6197	72	14	,	,	PUNCT
ejpam-6197	72	15	6197	6197	NUM
ejpam-6197	72	16	5	5	NUM
ejpam-6197	72	17	of	of	ADP
ejpam-6197	72	18	17	17	NUM
ejpam-6197	72	19	3	3	NUM
ejpam-6197	72	20	.	.	PUNCT
ejpam-6197	73	1	the	the	DET
ejpam-6197	73	2	stability	stability	NOUN
ejpam-6197	73	3	of	of	ADP
ejpam-6197	73	4	generalized	generalized	ADJ
ejpam-6197	73	5	functional	functional	ADJ
ejpam-6197	73	6	equations	equation	NOUN
ejpam-6197	73	7	in	in	ADP
ejpam-6197	73	8	banach	banach	NOUN
ejpam-6197	73	9	spaces	space	NOUN
ejpam-6197	73	10	in	in	ADP
ejpam-6197	73	11	the	the	DET
ejpam-6197	73	12	section	section	NOUN
ejpam-6197	73	13	,	,	PUNCT
ejpam-6197	73	14	we	we	PRON
ejpam-6197	73	15	use	use	VERB
ejpam-6197	73	16	theorem	theorem	NOUN
ejpam-6197	73	17	1	1	NUM
ejpam-6197	73	18	to	to	PART
ejpam-6197	73	19	prove	prove	VERB
ejpam-6197	73	20	the	the	DET
ejpam-6197	73	21	hyers	hyers	PROPN
ejpam-6197	73	22	-	-	PUNCT
ejpam-6197	73	23	ulam	ulam	ADJ
ejpam-6197	73	24	stability	stability	NOUN
ejpam-6197	73	25	of	of	ADP
ejpam-6197	73	26	functional	functional	ADJ
ejpam-6197	73	27	inequalities	inequality	NOUN
ejpam-6197	73	28	in	in	ADP
ejpam-6197	73	29	banach	banach	NOUN
ejpam-6197	73	30	spaces	space	NOUN
ejpam-6197	73	31	.	.	PUNCT
ejpam-6197	74	1	the	the	DET
ejpam-6197	74	2	several	several	ADJ
ejpam-6197	74	3	mathematicians	mathematician	NOUN
ejpam-6197	74	4	have	have	AUX
ejpam-6197	74	5	investigated	investigate	VERB
ejpam-6197	74	6	the	the	DET
ejpam-6197	74	7	problems	problem	NOUN
ejpam-6197	74	8	of	of	ADP
ejpam-6197	74	9	hyers	hyer	NOUN
ejpam-6197	74	10	-	-	PUNCT
ejpam-6197	74	11	ulam	ulam	PROPN
ejpam-6197	74	12	stability	stability	NOUN
ejpam-6197	74	13	of	of	ADP
ejpam-6197	74	14	functional	functional	ADJ
ejpam-6197	74	15	equations	equation	NOUN
ejpam-6197	74	16	(	(	PUNCT
ejpam-6197	74	17	see	see	VERB
ejpam-6197	74	18	[	[	X
ejpam-6197	74	19	19	19	NUM
ejpam-6197	74	20	,	,	PUNCT
ejpam-6197	74	21	20	20	NUM
ejpam-6197	74	22	,	,	PUNCT
ejpam-6197	74	23	24–35	24–35	NUM
ejpam-6197	74	24	]	]	PUNCT
ejpam-6197	74	25	)	)	PUNCT
ejpam-6197	74	26	.	.	PUNCT
ejpam-6197	75	1	theorem	theorem	NOUN
ejpam-6197	75	2	2	2	NUM
ejpam-6197	75	3	.	.	PUNCT
ejpam-6197	76	1	let	let	VERB
ejpam-6197	76	2	ϵ	ϵ	X
ejpam-6197	76	3	:	:	PUNCT
ejpam-6197	76	4	x3	x3	VERB
ejpam-6197	76	5	→	→	SYM
ejpam-6197	76	6	[	[	X
ejpam-6197	76	7	0,∞	0,∞	X
ejpam-6197	76	8	)	)	PUNCT
ejpam-6197	76	9	be	be	VERB
ejpam-6197	76	10	a	a	DET
ejpam-6197	76	11	function	function	NOUN
ejpam-6197	76	12	with	with	ADP
ejpam-6197	76	13	ϵ(0	ϵ(0	PROPN
ejpam-6197	76	14	,	,	PUNCT
ejpam-6197	76	15	0	0	NUM
ejpam-6197	76	16	,	,	PUNCT
ejpam-6197	76	17	0	0	NUM
ejpam-6197	76	18	)	)	PUNCT
ejpam-6197	77	1	=	=	SYM
ejpam-6197	77	2	0	0	NUM
ejpam-6197	77	3	such	such	ADJ
ejpam-6197	77	4	that	that	SCONJ
ejpam-6197	77	5	there	there	PRON
ejpam-6197	77	6	exists	exist	VERB
ejpam-6197	77	7	an	an	DET
ejpam-6197	77	8	l	l	NOUN
ejpam-6197	77	9	<	<	X
ejpam-6197	77	10	1	1	NUM
ejpam-6197	77	11	with	with	ADP
ejpam-6197	77	12	ϵ(kx	ϵ(kx	PROPN
ejpam-6197	77	13	,	,	PUNCT
ejpam-6197	77	14	ky	ky	PROPN
ejpam-6197	77	15	,	,	PUNCT
ejpam-6197	77	16	kz	kz	PROPN
ejpam-6197	77	17	)	)	PUNCT
ejpam-6197	77	18	≤	≤	NOUN
ejpam-6197	78	1	|k|lϵ(x	|k|lϵ(x	PROPN
ejpam-6197	78	2	,	,	PUNCT
ejpam-6197	78	3	y	y	PROPN
ejpam-6197	78	4	,	,	PUNCT
ejpam-6197	78	5	z	z	NOUN
ejpam-6197	78	6	)	)	PUNCT
ejpam-6197	78	7	for	for	ADP
ejpam-6197	78	8	all	all	DET
ejpam-6197	78	9	x	x	NOUN
ejpam-6197	78	10	,	,	PUNCT
ejpam-6197	78	11	y	y	PROPN
ejpam-6197	78	12	,	,	PUNCT
ejpam-6197	78	13	z	z	NOUN
ejpam-6197	78	14	∈	∈	PROPN
ejpam-6197	78	15	x	x	X
ejpam-6197	78	16	and	and	CCONJ
ejpam-6197	78	17	k	k	PROPN
ejpam-6197	78	18	∈	∈	PROPN
ejpam-6197	78	19	r.	r.	PROPN
ejpam-6197	78	20	suppose	suppose	VERB
ejpam-6197	78	21	that	that	SCONJ
ejpam-6197	78	22	x	x	PRON
ejpam-6197	78	23	is	be	AUX
ejpam-6197	78	24	a	a	DET
ejpam-6197	78	25	linear	linear	ADJ
ejpam-6197	78	26	normed	normed	ADJ
ejpam-6197	78	27	space	space	NOUN
ejpam-6197	78	28	and	and	CCONJ
ejpam-6197	78	29	y	y	PROPN
ejpam-6197	78	30	is	be	AUX
ejpam-6197	78	31	a	a	DET
ejpam-6197	78	32	banach	banach	NOUN
ejpam-6197	78	33	space	space	NOUN
ejpam-6197	78	34	.	.	PUNCT
ejpam-6197	79	1	let	let	VERB
ejpam-6197	79	2	f	f	NOUN
ejpam-6197	79	3	:	:	PUNCT
ejpam-6197	79	4	x	x	X
ejpam-6197	79	5	→	→	SYM
ejpam-6197	79	6	y	y	X
ejpam-6197	79	7	be	be	AUX
ejpam-6197	79	8	a	a	DET
ejpam-6197	79	9	mapping	mapping	NOUN
ejpam-6197	79	10	satisfying	satisfy	VERB
ejpam-6197	79	11	∥f(ax+by	∥f(ax+by	PROPN
ejpam-6197	79	12	+	+	SYM
ejpam-6197	79	13	cz	cz	NOUN
ejpam-6197	79	14	)	)	PUNCT
ejpam-6197	79	15	+	+	NUM
ejpam-6197	79	16	f(ax	f(ax	NOUN
ejpam-6197	79	17	)	)	PUNCT
ejpam-6197	80	1	+	+	NUM
ejpam-6197	80	2	γ(x)f(z	γ(x)f(z	ADJ
ejpam-6197	80	3	)	)	PUNCT
ejpam-6197	80	4	−α(x)f(ax+	−α(x)f(ax+	PROPN
ejpam-6197	80	5	y)−	y)−	PROPN
ejpam-6197	80	6	β(x)f(x+	β(x)f(x+	PUNCT
ejpam-6197	80	7	cz)−	cz)−	NUM
ejpam-6197	80	8	γ(x)f(by	γ(x)f(by	NOUN
ejpam-6197	81	1	+	+	PUNCT
ejpam-6197	82	1	z)∥	z)∥	NUM
ejpam-6197	82	2	≤	≤	NUM
ejpam-6197	83	1	ϵ(x	ϵ(x	PROPN
ejpam-6197	83	2	,	,	PUNCT
ejpam-6197	83	3	y	y	PROPN
ejpam-6197	83	4	,	,	PUNCT
ejpam-6197	83	5	z	z	NOUN
ejpam-6197	83	6	)	)	PUNCT
ejpam-6197	83	7	,	,	PUNCT
ejpam-6197	83	8	(	(	PUNCT
ejpam-6197	83	9	7	7	X
ejpam-6197	83	10	)	)	PUNCT
ejpam-6197	83	11	for	for	ADP
ejpam-6197	83	12	all	all	DET
ejpam-6197	83	13	x	x	NOUN
ejpam-6197	83	14	,	,	PUNCT
ejpam-6197	83	15	y	y	PROPN
ejpam-6197	83	16	,	,	PUNCT
ejpam-6197	83	17	z	z	PROPN
ejpam-6197	83	18	∈	∈	PROPN
ejpam-6197	83	19	x	x	X
ejpam-6197	83	20	,	,	PUNCT
ejpam-6197	83	21	where	where	SCONJ
ejpam-6197	83	22	α	α	X
ejpam-6197	83	23	,	,	PUNCT
ejpam-6197	83	24	β	β	X
ejpam-6197	83	25	,	,	PUNCT
ejpam-6197	83	26	γ	γ	X
ejpam-6197	83	27	:	:	PUNCT
ejpam-6197	83	28	x	x	SYM
ejpam-6197	83	29	→	→	SYM
ejpam-6197	83	30	r	r	NOUN
ejpam-6197	83	31	are	be	AUX
ejpam-6197	83	32	functions	function	NOUN
ejpam-6197	83	33	and	and	CCONJ
ejpam-6197	83	34	a	a	DET
ejpam-6197	83	35	,	,	PUNCT
ejpam-6197	83	36	b	b	NOUN
ejpam-6197	83	37	,	,	PUNCT
ejpam-6197	83	38	c	c	PROPN
ejpam-6197	83	39	∈	∈	PROPN
ejpam-6197	83	40	r	r	NOUN
ejpam-6197	84	1	such	such	ADJ
ejpam-6197	84	2	that	that	SCONJ
ejpam-6197	84	3	ã	ã	PROPN
ejpam-6197	84	4	:	:	PUNCT
ejpam-6197	84	5	=	=	SYM
ejpam-6197	84	6	∣∣a+b+c	∣∣a+b+c	PROPN
ejpam-6197	84	7	a	a	DET
ejpam-6197	84	8	∣∣	∣∣	PROPN
ejpam-6197	84	9	+	+	CCONJ
ejpam-6197	84	10	∣∣γ	∣∣γ	NOUN
ejpam-6197	84	11	(	(	PUNCT
ejpam-6197	84	12	x	x	NOUN
ejpam-6197	84	13	a	a	X
ejpam-6197	84	14	)	)	PUNCT
ejpam-6197	84	15	∣∣	∣∣	NUM
ejpam-6197	84	16	∣∣	∣∣	X
ejpam-6197	84	17	1	1	NUM
ejpam-6197	84	18	a	a	DET
ejpam-6197	84	19	∣∣	∣∣	PROPN
ejpam-6197	84	20	+	+	CCONJ
ejpam-6197	84	21	∣∣α	∣∣α	ADJ
ejpam-6197	84	22	(	(	PUNCT
ejpam-6197	84	23	x	x	NOUN
ejpam-6197	84	24	a	a	X
ejpam-6197	84	25	)	)	PUNCT
ejpam-6197	84	26	∣∣	∣∣	X
ejpam-6197	84	27	∣∣a+1	∣∣a+1	VERB
ejpam-6197	84	28	a	a	DET
ejpam-6197	84	29	∣∣	∣∣	PROPN
ejpam-6197	84	30	+	+	CCONJ
ejpam-6197	84	31	∣∣β	∣∣β	PROPN
ejpam-6197	84	32	(	(	PUNCT
ejpam-6197	84	33	x	x	NOUN
ejpam-6197	84	34	a	a	X
ejpam-6197	84	35	)	)	PUNCT
ejpam-6197	84	36	∣∣	∣∣	X
ejpam-6197	84	37	∣∣c+1	∣∣c+1	ADP
ejpam-6197	84	38	a	a	DET
ejpam-6197	84	39	∣∣	∣∣	ADJ
ejpam-6197	84	40	+	+	CCONJ
ejpam-6197	84	41	∣∣γ	∣∣γ	NOUN
ejpam-6197	84	42	(	(	PUNCT
ejpam-6197	84	43	x	x	NOUN
ejpam-6197	84	44	a	a	X
ejpam-6197	84	45	)	)	PUNCT
ejpam-6197	84	46	∣∣	∣∣	X
ejpam-6197	84	47	∣∣b+1	∣∣b+1	ADP
ejpam-6197	84	48	a	a	DET
ejpam-6197	84	49	∣∣	∣∣	ADJ
ejpam-6197	84	50	<	<	X
ejpam-6197	84	51	1	1	NUM
ejpam-6197	84	52	.	.	PUNCT
ejpam-6197	85	1	then	then	ADV
ejpam-6197	85	2	there	there	PRON
ejpam-6197	85	3	exists	exist	VERB
ejpam-6197	85	4	a	a	DET
ejpam-6197	85	5	unique	unique	ADJ
ejpam-6197	85	6	mapping	mapping	NOUN
ejpam-6197	85	7	k	k	NOUN
ejpam-6197	85	8	:	:	PUNCT
ejpam-6197	85	9	x	x	X
ejpam-6197	85	10	→	→	PUNCT
ejpam-6197	85	11	y	y	PROPN
ejpam-6197	85	12	such	such	ADJ
ejpam-6197	85	13	that	that	SCONJ
ejpam-6197	85	14	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	85	15	k(x)∥	k(x)∥	VERB
ejpam-6197	85	16	≤	≤	ADV
ejpam-6197	85	17	1	1	NUM
ejpam-6197	85	18	1−	1−	NUM
ejpam-6197	85	19	ãl	ãl	NOUN
ejpam-6197	85	20	ϵ(x	ϵ(x	PROPN
ejpam-6197	85	21	,	,	PUNCT
ejpam-6197	85	22	x	x	X
ejpam-6197	85	23	,	,	PUNCT
ejpam-6197	85	24	x	x	X
ejpam-6197	85	25	)	)	PUNCT
ejpam-6197	85	26	for	for	ADP
ejpam-6197	85	27	all	all	DET
ejpam-6197	85	28	x	x	SYM
ejpam-6197	85	29	∈	∈	ADJ
ejpam-6197	85	30	x.	x.	NOUN
ejpam-6197	85	31	proof	proof	NOUN
ejpam-6197	85	32	.	.	PUNCT
ejpam-6197	86	1	letting	let	VERB
ejpam-6197	86	2	x	x	PUNCT
ejpam-6197	86	3	=	=	PUNCT
ejpam-6197	86	4	y	y	PROPN
ejpam-6197	86	5	=	=	SYM
ejpam-6197	86	6	z	z	NOUN
ejpam-6197	86	7	in	in	ADP
ejpam-6197	86	8	(	(	PUNCT
ejpam-6197	86	9	7	7	NUM
ejpam-6197	86	10	)	)	PUNCT
ejpam-6197	86	11	,	,	PUNCT
ejpam-6197	86	12	we	we	PRON
ejpam-6197	86	13	get	get	VERB
ejpam-6197	86	14	∥f((a+b	∥f((a+b	PROPN
ejpam-6197	86	15	+	+	CCONJ
ejpam-6197	86	16	c)x	c)x	X
ejpam-6197	86	17	)	)	PUNCT
ejpam-6197	87	1	+	+	CCONJ
ejpam-6197	88	1	f(ax	f(ax	NOUN
ejpam-6197	88	2	)	)	PUNCT
ejpam-6197	88	3	+	+	NUM
ejpam-6197	88	4	γ(x)f(x	γ(x)f(x	NOUN
ejpam-6197	88	5	)	)	PUNCT
ejpam-6197	88	6	−α(x)f(ax+	−α(x)f(ax+	PROPN
ejpam-6197	88	7	x)−	x)−	PROPN
ejpam-6197	88	8	β(x)f(x+	β(x)f(x+	PUNCT
ejpam-6197	88	9	cx)−	cx)−	NOUN
ejpam-6197	88	10	γ(x)f(bx+	γ(x)f(bx+	X
ejpam-6197	89	1	x)∥	x)∥	PUNCT
ejpam-6197	89	2	≤	≤	PROPN
ejpam-6197	89	3	ϵ(x	ϵ(x	PROPN
ejpam-6197	89	4	,	,	PUNCT
ejpam-6197	89	5	x	x	X
ejpam-6197	89	6	,	,	PUNCT
ejpam-6197	89	7	x	x	NOUN
ejpam-6197	89	8	)	)	PUNCT
ejpam-6197	89	9	.	.	PUNCT
ejpam-6197	90	1	thus	thus	ADV
ejpam-6197	90	2	∥∥∥∥f	∥∥∥∥f	PRON
ejpam-6197	90	3	(	(	PUNCT
ejpam-6197	90	4	(	(	PUNCT
ejpam-6197	90	5	a+b	a+b	X
ejpam-6197	90	6	+	+	CCONJ
ejpam-6197	90	7	c	c	PROPN
ejpam-6197	90	8	a	a	PRON
ejpam-6197	90	9	)	)	PUNCT
ejpam-6197	90	10	x	x	SYM
ejpam-6197	90	11	)	)	PUNCT
ejpam-6197	91	1	+	+	CCONJ
ejpam-6197	91	2	f(x	f(x	PROPN
ejpam-6197	91	3	)	)	PUNCT
ejpam-6197	92	1	+	+	CCONJ
ejpam-6197	92	2	γ	γ	X
ejpam-6197	92	3	(	(	PUNCT
ejpam-6197	92	4	x	x	NOUN
ejpam-6197	92	5	a	a	X
ejpam-6197	92	6	)	)	PUNCT
ejpam-6197	92	7	f	f	NOUN
ejpam-6197	92	8	(	(	PUNCT
ejpam-6197	92	9	x	x	NOUN
ejpam-6197	92	10	a	a	PRON
ejpam-6197	92	11	)	)	PUNCT
ejpam-6197	92	12	−α	−α	NOUN
ejpam-6197	92	13	(	(	PUNCT
ejpam-6197	92	14	x	x	NOUN
ejpam-6197	92	15	a	a	X
ejpam-6197	92	16	)	)	PUNCT
ejpam-6197	92	17	f	f	NOUN
ejpam-6197	92	18	(	(	PUNCT
ejpam-6197	92	19	(	(	PUNCT
ejpam-6197	92	20	a+	a+	X
ejpam-6197	92	21	1)x	1)x	NUM
ejpam-6197	92	22	a	a	NOUN
ejpam-6197	92	23	)	)	PUNCT
ejpam-6197	92	24	−	−	NOUN
ejpam-6197	92	25	β	β	X
ejpam-6197	92	26	(	(	PUNCT
ejpam-6197	92	27	x	x	X
ejpam-6197	92	28	a	a	X
ejpam-6197	92	29	)	)	PUNCT
ejpam-6197	92	30	f	f	NOUN
ejpam-6197	92	31	(	(	PUNCT
ejpam-6197	92	32	(	(	PUNCT
ejpam-6197	92	33	c	c	X
ejpam-6197	92	34	+	+	NOUN
ejpam-6197	92	35	1)x	1)x	NUM
ejpam-6197	92	36	a	a	PRON
ejpam-6197	92	37	)	)	PUNCT
ejpam-6197	92	38	−	−	PROPN
ejpam-6197	93	1	γ	γ	X
ejpam-6197	93	2	(	(	PUNCT
ejpam-6197	93	3	x	x	NOUN
ejpam-6197	93	4	a	a	X
ejpam-6197	93	5	)	)	PUNCT
ejpam-6197	93	6	f	f	NOUN
ejpam-6197	93	7	(	(	PUNCT
ejpam-6197	93	8	(	(	PUNCT
ejpam-6197	93	9	b	b	X
ejpam-6197	93	10	+	+	NOUN
ejpam-6197	93	11	1)x	1)x	NUM
ejpam-6197	93	12	a	a	PRON
ejpam-6197	93	13	)	)	PUNCT
ejpam-6197	93	14	∥∥∥∥	∥∥∥∥	PROPN
ejpam-6197	93	15	≤	≤	NUM
ejpam-6197	94	1	ϵ	ϵ	X
ejpam-6197	94	2	(	(	PUNCT
ejpam-6197	94	3	x	x	X
ejpam-6197	94	4	a	a	PRON
ejpam-6197	94	5	,	,	PUNCT
ejpam-6197	94	6	x	x	X
ejpam-6197	94	7	a	a	PRON
ejpam-6197	94	8	,	,	PUNCT
ejpam-6197	94	9	x	x	X
ejpam-6197	94	10	a	a	PRON
ejpam-6197	94	11	)	)	PUNCT
ejpam-6197	94	12	≤	≤	NUM
ejpam-6197	94	13	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6197	94	14	1a	1a	PROPN
ejpam-6197	94	15	∣∣∣∣lϵ(x	∣∣∣∣lϵ(x	PROPN
ejpam-6197	94	16	,	,	PUNCT
ejpam-6197	94	17	x	x	NOUN
ejpam-6197	94	18	,	,	PUNCT
ejpam-6197	94	19	x	x	NOUN
ejpam-6197	94	20	)	)	PUNCT
ejpam-6197	94	21	.	.	PUNCT
ejpam-6197	95	1	in	in	ADP
ejpam-6197	95	2	remark	remark	NOUN
ejpam-6197	95	3	1	1	NUM
ejpam-6197	95	4	,	,	PUNCT
ejpam-6197	95	5	let	let	VERB
ejpam-6197	95	6	α1(x	α1(x	PRON
ejpam-6197	95	7	)	)	PUNCT
ejpam-6197	96	1	=	=	SYM
ejpam-6197	96	2	−γ	−γ	NOUN
ejpam-6197	96	3	(	(	PUNCT
ejpam-6197	96	4	x	x	NOUN
ejpam-6197	96	5	a	a	X
ejpam-6197	96	6	)	)	PUNCT
ejpam-6197	96	7	,	,	PUNCT
ejpam-6197	96	8	α2(x	α2(x	PROPN
ejpam-6197	96	9	)	)	PUNCT
ejpam-6197	96	10	=	=	SYM
ejpam-6197	96	11	−1	−1	NOUN
ejpam-6197	96	12	,	,	PUNCT
ejpam-6197	96	13	α3(x	α3(x	PROPN
ejpam-6197	96	14	)	)	PUNCT
ejpam-6197	96	15	=	=	SYM
ejpam-6197	96	16	α	α	PROPN
ejpam-6197	96	17	(	(	PUNCT
ejpam-6197	96	18	x	x	NOUN
ejpam-6197	96	19	a	a	NOUN
ejpam-6197	96	20	)	)	PUNCT
ejpam-6197	96	21	,	,	PUNCT
ejpam-6197	96	22	α4	α4	NOUN
ejpam-6197	96	23	=	=	SYM
ejpam-6197	96	24	β	β	X
ejpam-6197	96	25	(	(	PUNCT
ejpam-6197	96	26	x	x	X
ejpam-6197	96	27	a	a	X
ejpam-6197	96	28	)	)	PUNCT
ejpam-6197	96	29	,	,	PUNCT
ejpam-6197	96	30	α5(x	α5(x	X
ejpam-6197	96	31	)	)	PUNCT
ejpam-6197	96	32	=	=	SYM
ejpam-6197	96	33	γ	γ	X
ejpam-6197	96	34	(	(	PUNCT
ejpam-6197	96	35	x	x	NOUN
ejpam-6197	96	36	a	a	NOUN
ejpam-6197	96	37	)	)	PUNCT
ejpam-6197	96	38	,	,	PUNCT
ejpam-6197	96	39	f1(x	f1(x	NOUN
ejpam-6197	96	40	)	)	PUNCT
ejpam-6197	96	41	=	=	PUNCT
ejpam-6197	97	1	x	x	X
ejpam-6197	97	2	a	a	DET
ejpam-6197	97	3	,	,	PUNCT
ejpam-6197	97	4	f2(x	f2(x	PROPN
ejpam-6197	97	5	)	)	PUNCT
ejpam-6197	98	1	=	=	NOUN
ejpam-6197	98	2	a+b+c	a+b+c	ADV
ejpam-6197	98	3	a	a	DET
ejpam-6197	98	4	,	,	PUNCT
ejpam-6197	98	5	f3(x	f3(x	X
ejpam-6197	98	6	)	)	PUNCT
ejpam-6197	98	7	=	=	NOUN
ejpam-6197	98	8	(	(	PUNCT
ejpam-6197	98	9	a+1)x	a+1)x	NOUN
ejpam-6197	98	10	a	a	DET
ejpam-6197	98	11	,	,	PUNCT
ejpam-6197	98	12	f4(x	f4(x	NUM
ejpam-6197	98	13	)	)	PUNCT
ejpam-6197	98	14	=	=	NOUN
ejpam-6197	98	15	(	(	PUNCT
ejpam-6197	98	16	c+1)x	c+1)x	PROPN
ejpam-6197	98	17	a	a	PRON
ejpam-6197	98	18	,	,	PUNCT
ejpam-6197	98	19	f5(x	f5(x	NOUN
ejpam-6197	98	20	)	)	PUNCT
ejpam-6197	98	21	=	=	SYM
ejpam-6197	99	1	(	(	PUNCT
ejpam-6197	99	2	b+1)x	b+1)x	PROPN
ejpam-6197	99	3	a	a	PRON
ejpam-6197	99	4	.	.	PUNCT
ejpam-6197	100	1	g.	g.	PROPN
ejpam-6197	100	2	lyu	lyu	PROPN
ejpam-6197	100	3	et	et	PROPN
ejpam-6197	100	4	al	al	PROPN
ejpam-6197	100	5	.	.	PUNCT
ejpam-6197	100	6	/	/	SYM
ejpam-6197	100	7	eur	eur	PROPN
ejpam-6197	100	8	.	.	PUNCT
ejpam-6197	101	1	j.	j.	PROPN
ejpam-6197	101	2	pure	pure	PROPN
ejpam-6197	101	3	appl	appl	PROPN
ejpam-6197	101	4	.	.	PROPN
ejpam-6197	101	5	math	math	PROPN
ejpam-6197	101	6	,	,	PUNCT
ejpam-6197	101	7	18	18	NUM
ejpam-6197	101	8	(	(	PUNCT
ejpam-6197	101	9	3	3	NUM
ejpam-6197	101	10	)	)	PUNCT
ejpam-6197	101	11	(	(	PUNCT
ejpam-6197	101	12	2025	2025	NUM
ejpam-6197	101	13	)	)	PUNCT
ejpam-6197	101	14	,	,	PUNCT
ejpam-6197	101	15	6197	6197	NUM
ejpam-6197	101	16	6	6	NUM
ejpam-6197	101	17	of	of	ADP
ejpam-6197	101	18	17	17	NUM
ejpam-6197	101	19	consider	consider	VERB
ejpam-6197	101	20	t	t	NOUN
ejpam-6197	101	21	:	:	PUNCT
ejpam-6197	101	22	y	y	PROPN
ejpam-6197	101	23	x	x	PUNCT
ejpam-6197	101	24	→	→	SYM
ejpam-6197	101	25	y	y	PROPN
ejpam-6197	101	26	x	x	PUNCT
ejpam-6197	101	27	given	give	VERB
ejpam-6197	101	28	as	as	ADP
ejpam-6197	101	29	tf(x	tf(x	NUM
ejpam-6197	101	30	)	)	PUNCT
ejpam-6197	101	31	=	=	SYM
ejpam-6197	102	1	−f	−f	NOUN
ejpam-6197	102	2	(	(	PUNCT
ejpam-6197	102	3	(	(	PUNCT
ejpam-6197	102	4	a+b	a+b	X
ejpam-6197	102	5	+	+	CCONJ
ejpam-6197	102	6	c	c	PROPN
ejpam-6197	102	7	a	a	PRON
ejpam-6197	102	8	)	)	PUNCT
ejpam-6197	102	9	x	x	SYM
ejpam-6197	102	10	)	)	PUNCT
ejpam-6197	103	1	−	−	PROPN
ejpam-6197	103	2	γ	γ	X
ejpam-6197	103	3	(	(	PUNCT
ejpam-6197	103	4	x	x	NOUN
ejpam-6197	103	5	a	a	X
ejpam-6197	103	6	)	)	PUNCT
ejpam-6197	103	7	f	f	NOUN
ejpam-6197	103	8	(	(	PUNCT
ejpam-6197	103	9	x	x	X
ejpam-6197	103	10	a	a	X
ejpam-6197	103	11	)	)	PUNCT
ejpam-6197	104	1	+	+	NOUN
ejpam-6197	104	2	α	α	NOUN
ejpam-6197	104	3	(	(	PUNCT
ejpam-6197	104	4	x	x	NOUN
ejpam-6197	104	5	a	a	X
ejpam-6197	104	6	)	)	PUNCT
ejpam-6197	104	7	f	f	NOUN
ejpam-6197	104	8	(	(	PUNCT
ejpam-6197	104	9	(	(	PUNCT
ejpam-6197	104	10	a+	a+	X
ejpam-6197	104	11	1)x	1)x	NUM
ejpam-6197	104	12	a	a	PRON
ejpam-6197	104	13	)	)	PUNCT
ejpam-6197	104	14	+	+	CCONJ
ejpam-6197	104	15	β	β	X
ejpam-6197	104	16	(	(	PUNCT
ejpam-6197	104	17	x	x	X
ejpam-6197	104	18	a	a	X
ejpam-6197	104	19	)	)	PUNCT
ejpam-6197	104	20	f	f	NOUN
ejpam-6197	104	21	(	(	PUNCT
ejpam-6197	104	22	(	(	PUNCT
ejpam-6197	104	23	c	c	X
ejpam-6197	104	24	+	+	NOUN
ejpam-6197	104	25	1)x	1)x	NUM
ejpam-6197	104	26	a	a	PRON
ejpam-6197	104	27	)	)	PUNCT
ejpam-6197	104	28	+	+	CCONJ
ejpam-6197	104	29	γ	γ	X
ejpam-6197	104	30	(	(	PUNCT
ejpam-6197	104	31	x	x	NOUN
ejpam-6197	104	32	a	a	X
ejpam-6197	104	33	)	)	PUNCT
ejpam-6197	104	34	f	f	NOUN
ejpam-6197	104	35	(	(	PUNCT
ejpam-6197	104	36	(	(	PUNCT
ejpam-6197	104	37	b	b	X
ejpam-6197	104	38	+	+	NOUN
ejpam-6197	104	39	1)x	1)x	NUM
ejpam-6197	104	40	a	a	PRON
ejpam-6197	104	41	)	)	PUNCT
ejpam-6197	104	42	for	for	ADP
ejpam-6197	104	43	all	all	DET
ejpam-6197	104	44	x	x	SYM
ejpam-6197	104	45	∈	∈	PROPN
ejpam-6197	104	46	x	x	NOUN
ejpam-6197	104	47	,	,	PUNCT
ejpam-6197	104	48	f	f	PROPN
ejpam-6197	104	49	∈	∈	PROPN
ejpam-6197	104	50	y	y	PROPN
ejpam-6197	104	51	x	x	X
ejpam-6197	104	52	.	.	PUNCT
ejpam-6197	105	1	we	we	PRON
ejpam-6197	105	2	can	can	AUX
ejpam-6197	105	3	define	define	VERB
ejpam-6197	105	4	the	the	DET
ejpam-6197	105	5	operator	operator	NOUN
ejpam-6197	105	6	λ	λ	NOUN
ejpam-6197	105	7	:	:	PUNCT
ejpam-6197	105	8	rx	rx	VERB
ejpam-6197	105	9	+	+	X
ejpam-6197	105	10	→	→	PUNCT
ejpam-6197	105	11	rx	rx	X
ejpam-6197	105	12	+	+	CCONJ
ejpam-6197	105	13	,	,	PUNCT
ejpam-6197	105	14	given	give	VERB
ejpam-6197	105	15	by	by	ADP
ejpam-6197	105	16	λf(x	λf(x	NOUN
ejpam-6197	105	17	)	)	PUNCT
ejpam-6197	106	1	=	=	SYM
ejpam-6197	106	2	f	f	X
ejpam-6197	106	3	(	(	PUNCT
ejpam-6197	106	4	(	(	PUNCT
ejpam-6197	106	5	a+b	a+b	X
ejpam-6197	106	6	+	+	CCONJ
ejpam-6197	106	7	c	c	PROPN
ejpam-6197	106	8	a	a	PRON
ejpam-6197	106	9	)	)	PUNCT
ejpam-6197	106	10	x	x	SYM
ejpam-6197	106	11	)	)	PUNCT
ejpam-6197	107	1	+	+	CCONJ
ejpam-6197	107	2	∣∣∣γ	∣∣∣γ	PROPN
ejpam-6197	107	3	(	(	PUNCT
ejpam-6197	107	4	x	x	X
ejpam-6197	107	5	a	a	X
ejpam-6197	107	6	)	)	PUNCT
ejpam-6197	107	7	∣∣∣	∣∣∣	NOUN
ejpam-6197	107	8	f	f	PROPN
ejpam-6197	107	9	(	(	PUNCT
ejpam-6197	107	10	x	x	X
ejpam-6197	107	11	a	a	X
ejpam-6197	107	12	)	)	PUNCT
ejpam-6197	107	13	+	+	CCONJ
ejpam-6197	107	14	∣∣∣α	∣∣∣α	ADJ
ejpam-6197	107	15	(	(	PUNCT
ejpam-6197	107	16	x	x	NOUN
ejpam-6197	107	17	a	a	X
ejpam-6197	107	18	)	)	PUNCT
ejpam-6197	107	19	∣∣∣	∣∣∣	NOUN
ejpam-6197	107	20	f	f	PROPN
ejpam-6197	107	21	(	(	PUNCT
ejpam-6197	107	22	(	(	PUNCT
ejpam-6197	107	23	a+	a+	X
ejpam-6197	107	24	1)x	1)x	NUM
ejpam-6197	107	25	a	a	PRON
ejpam-6197	107	26	)	)	PUNCT
ejpam-6197	107	27	+	+	NUM
ejpam-6197	107	28	∣∣∣β	∣∣∣β	NOUN
ejpam-6197	107	29	(	(	PUNCT
ejpam-6197	107	30	x	x	NOUN
ejpam-6197	107	31	a	a	X
ejpam-6197	107	32	)	)	PUNCT
ejpam-6197	107	33	∣∣∣	∣∣∣	NOUN
ejpam-6197	107	34	f	f	PROPN
ejpam-6197	107	35	(	(	PUNCT
ejpam-6197	107	36	(	(	PUNCT
ejpam-6197	107	37	c	c	X
ejpam-6197	107	38	+	+	NOUN
ejpam-6197	107	39	1)x	1)x	NUM
ejpam-6197	107	40	a	a	PRON
ejpam-6197	107	41	)	)	PUNCT
ejpam-6197	107	42	+	+	NUM
ejpam-6197	107	43	∣∣∣γ	∣∣∣γ	PROPN
ejpam-6197	107	44	(	(	PUNCT
ejpam-6197	107	45	x	x	X
ejpam-6197	107	46	a	a	X
ejpam-6197	107	47	)	)	PUNCT
ejpam-6197	107	48	∣∣∣	∣∣∣	NOUN
ejpam-6197	107	49	f	f	PROPN
ejpam-6197	107	50	(	(	PUNCT
ejpam-6197	107	51	(	(	PUNCT
ejpam-6197	107	52	b	b	X
ejpam-6197	107	53	+	+	NOUN
ejpam-6197	107	54	1)x	1)x	NUM
ejpam-6197	107	55	a	a	PRON
ejpam-6197	107	56	)	)	PUNCT
ejpam-6197	107	57	for	for	ADP
ejpam-6197	107	58	all	all	PRON
ejpam-6197	107	59	x	x	SYM
ejpam-6197	107	60	∈	∈	ADJ
ejpam-6197	107	61	x.	x.	NOUN
ejpam-6197	107	62	in	in	ADP
ejpam-6197	107	63	particular	particular	ADJ
ejpam-6197	107	64	,	,	PUNCT
ejpam-6197	107	65	λϵ(x	λϵ(x	ADJ
ejpam-6197	107	66	,	,	PUNCT
ejpam-6197	107	67	x	x	X
ejpam-6197	107	68	,	,	PUNCT
ejpam-6197	107	69	x	x	X
ejpam-6197	107	70	)	)	PUNCT
ejpam-6197	107	71	=	=	SYM
ejpam-6197	107	72	ϵ	ϵ	X
ejpam-6197	107	73	(	(	PUNCT
ejpam-6197	107	74	(	(	PUNCT
ejpam-6197	107	75	a+b	a+b	X
ejpam-6197	107	76	+	+	CCONJ
ejpam-6197	107	77	c	c	PROPN
ejpam-6197	107	78	a	a	PRON
ejpam-6197	107	79	)	)	PUNCT
ejpam-6197	107	80	x	x	NOUN
ejpam-6197	107	81	,	,	PUNCT
ejpam-6197	107	82	(	(	PUNCT
ejpam-6197	107	83	a+b	a+b	X
ejpam-6197	107	84	+	+	CCONJ
ejpam-6197	107	85	c	c	PROPN
ejpam-6197	107	86	a	a	PRON
ejpam-6197	107	87	)	)	PUNCT
ejpam-6197	107	88	x	x	NOUN
ejpam-6197	107	89	,	,	PUNCT
ejpam-6197	107	90	(	(	PUNCT
ejpam-6197	107	91	a+b	a+b	X
ejpam-6197	107	92	+	+	CCONJ
ejpam-6197	107	93	c	c	PROPN
ejpam-6197	107	94	a	a	PRON
ejpam-6197	107	95	)	)	PUNCT
ejpam-6197	107	96	x	x	SYM
ejpam-6197	107	97	)	)	PUNCT
ejpam-6197	108	1	+	+	CCONJ
ejpam-6197	108	2	∣∣∣γ	∣∣∣γ	PROPN
ejpam-6197	108	3	(	(	PUNCT
ejpam-6197	108	4	x	x	X
ejpam-6197	108	5	a	a	X
ejpam-6197	108	6	)	)	PUNCT
ejpam-6197	108	7	∣∣∣	∣∣∣	NOUN
ejpam-6197	108	8	ϵ	ϵ	X
ejpam-6197	108	9	(	(	PUNCT
ejpam-6197	108	10	x	x	X
ejpam-6197	108	11	a	a	PRON
ejpam-6197	108	12	,	,	PUNCT
ejpam-6197	108	13	x	x	X
ejpam-6197	108	14	a	a	PRON
ejpam-6197	108	15	,	,	PUNCT
ejpam-6197	108	16	x	x	X
ejpam-6197	108	17	a	a	X
ejpam-6197	108	18	)	)	PUNCT
ejpam-6197	109	1	+	+	CCONJ
ejpam-6197	109	2	∣∣∣α	∣∣∣α	ADJ
ejpam-6197	109	3	(	(	PUNCT
ejpam-6197	109	4	x	x	NOUN
ejpam-6197	109	5	a	a	X
ejpam-6197	109	6	)	)	PUNCT
ejpam-6197	109	7	∣∣∣	∣∣∣	NOUN
ejpam-6197	109	8	ϵ((a+	ϵ((a+	NOUN
ejpam-6197	109	9	1)x	1)x	NUM
ejpam-6197	109	10	a	a	PRON
ejpam-6197	109	11	,	,	PUNCT
ejpam-6197	109	12	(	(	PUNCT
ejpam-6197	109	13	a+	a+	X
ejpam-6197	109	14	1)x	1)x	NUM
ejpam-6197	109	15	a	a	DET
ejpam-6197	109	16	,	,	PUNCT
ejpam-6197	109	17	(	(	PUNCT
ejpam-6197	109	18	a+	a+	X
ejpam-6197	109	19	1)x	1)x	NUM
ejpam-6197	109	20	a	a	PRON
ejpam-6197	109	21	)	)	PUNCT
ejpam-6197	109	22	+	+	NUM
ejpam-6197	109	23	∣∣∣β	∣∣∣β	NOUN
ejpam-6197	109	24	(	(	PUNCT
ejpam-6197	109	25	x	x	NOUN
ejpam-6197	109	26	a	a	X
ejpam-6197	109	27	)	)	PUNCT
ejpam-6197	109	28	∣∣∣	∣∣∣	NOUN
ejpam-6197	109	29	ϵ((c	ϵ((c	PUNCT
ejpam-6197	110	1	+	+	NUM
ejpam-6197	110	2	1)x	1)x	NUM
ejpam-6197	110	3	a	a	PRON
ejpam-6197	110	4	,	,	PUNCT
ejpam-6197	110	5	(	(	PUNCT
ejpam-6197	110	6	c	c	X
ejpam-6197	110	7	+	+	NOUN
ejpam-6197	110	8	1)x	1)x	NUM
ejpam-6197	110	9	a	a	PRON
ejpam-6197	110	10	,	,	PUNCT
ejpam-6197	110	11	(	(	PUNCT
ejpam-6197	110	12	c	c	X
ejpam-6197	110	13	+	+	NOUN
ejpam-6197	110	14	1)x	1)x	NUM
ejpam-6197	110	15	a	a	PRON
ejpam-6197	110	16	)	)	PUNCT
ejpam-6197	110	17	+	+	NUM
ejpam-6197	110	18	∣∣∣γ	∣∣∣γ	PROPN
ejpam-6197	110	19	(	(	PUNCT
ejpam-6197	110	20	x	x	X
ejpam-6197	110	21	a	a	X
ejpam-6197	110	22	)	)	PUNCT
ejpam-6197	110	23	∣∣∣	∣∣∣	NOUN
ejpam-6197	110	24	ϵ((b	ϵ((b	PUNCT
ejpam-6197	111	1	+	+	CCONJ
ejpam-6197	111	2	1)x	1)x	NUM
ejpam-6197	111	3	a	a	PRON
ejpam-6197	111	4	,	,	PUNCT
ejpam-6197	111	5	(	(	PUNCT
ejpam-6197	111	6	b	b	X
ejpam-6197	111	7	+	+	NOUN
ejpam-6197	111	8	1)x	1)x	NUM
ejpam-6197	111	9	a	a	PRON
ejpam-6197	111	10	,	,	PUNCT
ejpam-6197	111	11	(	(	PUNCT
ejpam-6197	111	12	b	b	X
ejpam-6197	111	13	+	+	NOUN
ejpam-6197	111	14	1)x	1)x	NUM
ejpam-6197	111	15	a	a	PRON
ejpam-6197	111	16	)	)	PUNCT
ejpam-6197	111	17	≤	≤	NOUN
ejpam-6197	111	18	∣∣∣∣a+b	∣∣∣∣a+b	NOUN
ejpam-6197	111	19	+	+	CCONJ
ejpam-6197	111	20	c	c	NOUN
ejpam-6197	111	21	a	a	DET
ejpam-6197	111	22	∣∣∣∣lϵ	∣∣∣∣lϵ	ADJ
ejpam-6197	111	23	(	(	PUNCT
ejpam-6197	111	24	x	x	X
ejpam-6197	111	25	,	,	PUNCT
ejpam-6197	111	26	x	x	X
ejpam-6197	111	27	,	,	PUNCT
ejpam-6197	111	28	x	x	X
ejpam-6197	111	29	)	)	PUNCT
ejpam-6197	111	30	+	+	CCONJ
ejpam-6197	111	31	∣∣∣γ	∣∣∣γ	PROPN
ejpam-6197	111	32	(	(	PUNCT
ejpam-6197	111	33	x	x	X
ejpam-6197	111	34	a	a	X
ejpam-6197	111	35	)	)	PUNCT
ejpam-6197	111	36	∣∣∣	∣∣∣	NOUN
ejpam-6197	111	37	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6197	111	38	1a	1a	X
ejpam-6197	111	39	∣∣∣∣lϵ	∣∣∣∣lϵ	ADJ
ejpam-6197	111	40	(	(	PUNCT
ejpam-6197	111	41	x	x	X
ejpam-6197	111	42	,	,	PUNCT
ejpam-6197	111	43	x	x	X
ejpam-6197	111	44	,	,	PUNCT
ejpam-6197	111	45	x	x	X
ejpam-6197	111	46	)	)	PUNCT
ejpam-6197	111	47	+	+	CCONJ
ejpam-6197	111	48	∣∣∣α	∣∣∣α	ADJ
ejpam-6197	111	49	(	(	PUNCT
ejpam-6197	111	50	x	x	NOUN
ejpam-6197	111	51	a	a	X
ejpam-6197	111	52	)	)	PUNCT
ejpam-6197	111	53	∣∣∣	∣∣∣	NOUN
ejpam-6197	111	54	∣∣∣∣a+	∣∣∣∣a+	PROPN
ejpam-6197	111	55	1	1	NUM
ejpam-6197	111	56	a	a	DET
ejpam-6197	111	57	∣∣∣∣lϵ	∣∣∣∣lϵ	ADJ
ejpam-6197	111	58	(	(	PUNCT
ejpam-6197	111	59	x	x	NOUN
ejpam-6197	111	60	,	,	PUNCT
ejpam-6197	111	61	x	x	X
ejpam-6197	111	62	,	,	PUNCT
ejpam-6197	111	63	x	x	PRON
ejpam-6197	111	64	)	)	PUNCT
ejpam-6197	111	65	+	+	CCONJ
ejpam-6197	111	66	∣∣∣β	∣∣∣β	NOUN
ejpam-6197	111	67	(	(	PUNCT
ejpam-6197	111	68	x	x	NOUN
ejpam-6197	111	69	a	a	X
ejpam-6197	111	70	)	)	PUNCT
ejpam-6197	111	71	∣∣∣	∣∣∣	NOUN
ejpam-6197	111	72	∣∣∣∣c	∣∣∣∣c	NOUN
ejpam-6197	112	1	+	+	CCONJ
ejpam-6197	112	2	1	1	NUM
ejpam-6197	112	3	a	a	DET
ejpam-6197	112	4	∣∣∣∣lϵ	∣∣∣∣lϵ	ADJ
ejpam-6197	112	5	(	(	PUNCT
ejpam-6197	112	6	x	x	NOUN
ejpam-6197	112	7	,	,	PUNCT
ejpam-6197	112	8	x	x	X
ejpam-6197	112	9	,	,	PUNCT
ejpam-6197	112	10	x	x	X
ejpam-6197	112	11	)	)	PUNCT
ejpam-6197	113	1	+	+	CCONJ
ejpam-6197	113	2	∣∣∣γ	∣∣∣γ	PROPN
ejpam-6197	113	3	(	(	PUNCT
ejpam-6197	113	4	x	x	X
ejpam-6197	113	5	a	a	X
ejpam-6197	113	6	)	)	PUNCT
ejpam-6197	113	7	∣∣∣	∣∣∣	NOUN
ejpam-6197	113	8	∣∣∣∣b	∣∣∣∣b	NOUN
ejpam-6197	113	9	+	+	CCONJ
ejpam-6197	113	10	1	1	NUM
ejpam-6197	113	11	a	a	DET
ejpam-6197	113	12	∣∣∣∣lϵ	∣∣∣∣lϵ	ADJ
ejpam-6197	113	13	(	(	PUNCT
ejpam-6197	113	14	x	x	NOUN
ejpam-6197	113	15	,	,	PUNCT
ejpam-6197	113	16	x	x	X
ejpam-6197	113	17	,	,	PUNCT
ejpam-6197	113	18	x	x	NOUN
ejpam-6197	113	19	)	)	PUNCT
ejpam-6197	113	20	≤	≤	NOUN
ejpam-6197	113	21	(	(	PUNCT
ejpam-6197	113	22	∣∣∣∣a+b	∣∣∣∣a+b	X
ejpam-6197	113	23	+	+	CCONJ
ejpam-6197	113	24	c	c	PROPN
ejpam-6197	113	25	a	a	DET
ejpam-6197	113	26	∣∣∣∣+	∣∣∣∣+	PROPN
ejpam-6197	113	27	∣∣∣γ	∣∣∣γ	PROPN
ejpam-6197	113	28	(	(	PUNCT
ejpam-6197	113	29	x	x	X
ejpam-6197	113	30	a	a	X
ejpam-6197	113	31	)	)	PUNCT
ejpam-6197	113	32	∣∣∣	∣∣∣	NOUN
ejpam-6197	113	33	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6197	113	34	1a	1a	PROPN
ejpam-6197	113	35	∣∣∣∣+	∣∣∣∣+	PROPN
ejpam-6197	113	36	∣∣∣α	∣∣∣α	PROPN
ejpam-6197	113	37	(	(	PUNCT
ejpam-6197	113	38	x	x	NOUN
ejpam-6197	113	39	a	a	X
ejpam-6197	113	40	)	)	PUNCT
ejpam-6197	113	41	∣∣∣	∣∣∣	NOUN
ejpam-6197	113	42	∣∣∣∣a+	∣∣∣∣a+	PROPN
ejpam-6197	113	43	1	1	NUM
ejpam-6197	113	44	a	a	DET
ejpam-6197	113	45	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6197	113	46	+	+	NUM
ejpam-6197	113	47	∣∣∣β	∣∣∣β	NOUN
ejpam-6197	113	48	(	(	PUNCT
ejpam-6197	113	49	x	x	NOUN
ejpam-6197	113	50	a	a	X
ejpam-6197	113	51	)	)	PUNCT
ejpam-6197	113	52	∣∣∣	∣∣∣	NOUN
ejpam-6197	113	53	∣∣∣∣c	∣∣∣∣c	NOUN
ejpam-6197	114	1	+	+	CCONJ
ejpam-6197	114	2	1	1	NUM
ejpam-6197	114	3	a	a	DET
ejpam-6197	114	4	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-6197	114	5	∣∣∣γ	∣∣∣γ	PROPN
ejpam-6197	114	6	(	(	PUNCT
ejpam-6197	114	7	x	x	X
ejpam-6197	114	8	a	a	X
ejpam-6197	114	9	)	)	PUNCT
ejpam-6197	114	10	∣∣∣	∣∣∣	NOUN
ejpam-6197	114	11	∣∣∣∣b	∣∣∣∣b	NOUN
ejpam-6197	114	12	+	+	CCONJ
ejpam-6197	114	13	1	1	NUM
ejpam-6197	114	14	a	a	DET
ejpam-6197	114	15	∣∣∣∣)lϵ	∣∣∣∣)lϵ	PROPN
ejpam-6197	114	16	(	(	PUNCT
ejpam-6197	114	17	x	x	NOUN
ejpam-6197	114	18	,	,	PUNCT
ejpam-6197	114	19	x	x	X
ejpam-6197	114	20	,	,	PUNCT
ejpam-6197	114	21	x	x	X
ejpam-6197	114	22	)	)	PUNCT
ejpam-6197	114	23	=	=	SYM
ejpam-6197	114	24	ãlϵ	ãlϵ	PROPN
ejpam-6197	114	25	(	(	PUNCT
ejpam-6197	114	26	x	x	X
ejpam-6197	114	27	,	,	PUNCT
ejpam-6197	114	28	x	x	X
ejpam-6197	114	29	,	,	PUNCT
ejpam-6197	114	30	x	x	X
ejpam-6197	114	31	)	)	PUNCT
ejpam-6197	114	32	.	.	PUNCT
ejpam-6197	115	1	g.	g.	PROPN
ejpam-6197	115	2	lyu	lyu	VERB
ejpam-6197	115	3	et	et	PROPN
ejpam-6197	115	4	al	al	PROPN
ejpam-6197	115	5	.	.	PUNCT
ejpam-6197	115	6	/	/	SYM
ejpam-6197	115	7	eur	eur	PROPN
ejpam-6197	115	8	.	.	PUNCT
ejpam-6197	116	1	j.	j.	PROPN
ejpam-6197	116	2	pure	pure	PROPN
ejpam-6197	116	3	appl	appl	PROPN
ejpam-6197	116	4	.	.	PROPN
ejpam-6197	116	5	math	math	PROPN
ejpam-6197	116	6	,	,	PUNCT
ejpam-6197	116	7	18	18	NUM
ejpam-6197	116	8	(	(	PUNCT
ejpam-6197	116	9	3	3	NUM
ejpam-6197	116	10	)	)	PUNCT
ejpam-6197	116	11	(	(	PUNCT
ejpam-6197	116	12	2025	2025	NUM
ejpam-6197	116	13	)	)	PUNCT
ejpam-6197	116	14	,	,	PUNCT
ejpam-6197	116	15	6197	6197	NUM
ejpam-6197	116	16	7	7	NUM
ejpam-6197	116	17	of	of	ADP
ejpam-6197	116	18	17	17	NUM
ejpam-6197	116	19	since	since	SCONJ
ejpam-6197	116	20	λ	λ	PROPN
ejpam-6197	116	21	is	be	AUX
ejpam-6197	116	22	linear	linear	ADJ
ejpam-6197	116	23	,	,	PUNCT
ejpam-6197	116	24	we	we	PRON
ejpam-6197	116	25	can	can	AUX
ejpam-6197	116	26	get	get	VERB
ejpam-6197	116	27	λnϵ(x	λnϵ(x	PROPN
ejpam-6197	116	28	,	,	PUNCT
ejpam-6197	116	29	x	x	X
ejpam-6197	116	30	,	,	PUNCT
ejpam-6197	116	31	x	x	X
ejpam-6197	116	32	)	)	PUNCT
ejpam-6197	116	33	=	=	SYM
ejpam-6197	116	34	(	(	PUNCT
ejpam-6197	116	35	ãl)nϵ(x	ãl)nϵ(x	PROPN
ejpam-6197	116	36	,	,	PUNCT
ejpam-6197	116	37	x	x	NOUN
ejpam-6197	116	38	,	,	PUNCT
ejpam-6197	116	39	x	x	NOUN
ejpam-6197	116	40	)	)	PUNCT
ejpam-6197	116	41	,	,	PUNCT
ejpam-6197	116	42	x	x	PUNCT
ejpam-6197	116	43	∈	∈	PROPN
ejpam-6197	116	44	x	x	X
ejpam-6197	116	45	,	,	PUNCT
ejpam-6197	116	46	n	n	PROPN
ejpam-6197	116	47	∈	∈	PROPN
ejpam-6197	116	48	n0	n0	PROPN
ejpam-6197	116	49	.	.	PUNCT
ejpam-6197	117	1	since	since	SCONJ
ejpam-6197	117	2	ã	ã	PROPN
ejpam-6197	117	3	<	<	X
ejpam-6197	117	4	1	1	NUM
ejpam-6197	117	5	,	,	PUNCT
ejpam-6197	117	6	the	the	DET
ejpam-6197	117	7	series	series	NOUN
ejpam-6197	117	8	∑∞	∑∞	PROPN
ejpam-6197	117	9	n=0	n=0	PROPN
ejpam-6197	117	10	λ	λ	NOUN
ejpam-6197	117	11	nϵ(x	nϵ(x	NUM
ejpam-6197	117	12	,	,	PUNCT
ejpam-6197	117	13	x	x	X
ejpam-6197	117	14	,	,	PUNCT
ejpam-6197	117	15	x	x	X
ejpam-6197	117	16	)	)	PUNCT
ejpam-6197	117	17	is	be	AUX
ejpam-6197	117	18	convergent	convergent	ADJ
ejpam-6197	117	19	for	for	ADP
ejpam-6197	117	20	every	every	DET
ejpam-6197	117	21	x	x	SYM
ejpam-6197	117	22	∈	∈	PROPN
ejpam-6197	117	23	x	x	X
ejpam-6197	117	24	and	and	CCONJ
ejpam-6197	117	25	ϵ∗(x	ϵ∗(x	PROPN
ejpam-6197	117	26	,	,	PUNCT
ejpam-6197	117	27	x	x	X
ejpam-6197	117	28	,	,	PUNCT
ejpam-6197	117	29	x	x	X
ejpam-6197	117	30	)	)	PUNCT
ejpam-6197	117	31	=	=	SYM
ejpam-6197	118	1	∞∑	∞∑	NUM
ejpam-6197	118	2	n=0	n=0	NUM
ejpam-6197	118	3	λnϵ(x	λnϵ(x	NOUN
ejpam-6197	118	4	)	)	PUNCT
ejpam-6197	118	5	=	=	SYM
ejpam-6197	119	1	∞∑	∞∑	NUM
ejpam-6197	119	2	n=0	n=0	NUM
ejpam-6197	119	3	(	(	PUNCT
ejpam-6197	119	4	ãl	ãl	PROPN
ejpam-6197	119	5	)	)	PUNCT
ejpam-6197	119	6	n	n	PRON
ejpam-6197	119	7	ϵ(x	ϵ(x	PROPN
ejpam-6197	119	8	,	,	PUNCT
ejpam-6197	119	9	x	x	X
ejpam-6197	119	10	,	,	PUNCT
ejpam-6197	119	11	x	x	X
ejpam-6197	119	12	)	)	PUNCT
ejpam-6197	119	13	=	=	SYM
ejpam-6197	119	14	1	1	NUM
ejpam-6197	119	15	1−	1−	NUM
ejpam-6197	119	16	ãl	ãl	NOUN
ejpam-6197	119	17	ϵ(x	ϵ(x	PROPN
ejpam-6197	119	18	,	,	PUNCT
ejpam-6197	119	19	x	x	X
ejpam-6197	119	20	,	,	PUNCT
ejpam-6197	119	21	x	x	NOUN
ejpam-6197	119	22	)	)	PUNCT
ejpam-6197	119	23	,	,	PUNCT
ejpam-6197	119	24	x	x	PUNCT
ejpam-6197	119	25	∈	∈	NOUN
ejpam-6197	119	26	x.	x.	NOUN
ejpam-6197	119	27	by	by	ADP
ejpam-6197	119	28	theorem	theorem	NOUN
ejpam-6197	119	29	1	1	NUM
ejpam-6197	119	30	,	,	PUNCT
ejpam-6197	119	31	there	there	PRON
ejpam-6197	119	32	exists	exist	VERB
ejpam-6197	119	33	a	a	DET
ejpam-6197	119	34	mapping	mapping	NOUN
ejpam-6197	119	35	l	l	NOUN
ejpam-6197	119	36	:	:	PUNCT
ejpam-6197	119	37	x	x	X
ejpam-6197	119	38	→	→	SYM
ejpam-6197	119	39	y	y	NUM
ejpam-6197	119	40	such	such	ADJ
ejpam-6197	119	41	that	that	SCONJ
ejpam-6197	119	42	l(x	l(x	PROPN
ejpam-6197	119	43	)	)	PUNCT
ejpam-6197	119	44	=	=	PROPN
ejpam-6197	119	45	lim	lim	PROPN
ejpam-6197	119	46	n→∞	n→∞	X
ejpam-6197	119	47	tnf(x	tnf(x	PROPN
ejpam-6197	119	48	)	)	PUNCT
ejpam-6197	119	49	,	,	PUNCT
ejpam-6197	119	50	l(x	l(x	PROPN
ejpam-6197	119	51	)	)	PUNCT
ejpam-6197	120	1	=	=	SYM
ejpam-6197	120	2	−l	−l	NOUN
ejpam-6197	120	3	(	(	PUNCT
ejpam-6197	120	4	(	(	PUNCT
ejpam-6197	120	5	a+b	a+b	X
ejpam-6197	120	6	+	+	CCONJ
ejpam-6197	120	7	c	c	PROPN
ejpam-6197	120	8	a	a	PRON
ejpam-6197	120	9	)	)	PUNCT
ejpam-6197	120	10	x	x	SYM
ejpam-6197	120	11	)	)	PUNCT
ejpam-6197	121	1	−	−	PROPN
ejpam-6197	121	2	γ	γ	X
ejpam-6197	121	3	(	(	PUNCT
ejpam-6197	121	4	x	x	NOUN
ejpam-6197	121	5	a	a	X
ejpam-6197	121	6	)	)	PUNCT
ejpam-6197	121	7	l	l	NOUN
ejpam-6197	121	8	(	(	PUNCT
ejpam-6197	121	9	x	x	NOUN
ejpam-6197	121	10	a	a	X
ejpam-6197	121	11	)	)	PUNCT
ejpam-6197	122	1	+	+	NOUN
ejpam-6197	122	2	α	α	NOUN
ejpam-6197	122	3	(	(	PUNCT
ejpam-6197	122	4	x	x	NOUN
ejpam-6197	122	5	a	a	X
ejpam-6197	122	6	)	)	PUNCT
ejpam-6197	122	7	l	l	NOUN
ejpam-6197	122	8	(	(	PUNCT
ejpam-6197	122	9	(	(	PUNCT
ejpam-6197	122	10	a+	a+	X
ejpam-6197	122	11	1)x	1)x	NUM
ejpam-6197	122	12	a	a	PRON
ejpam-6197	122	13	)	)	PUNCT
ejpam-6197	123	1	+	+	CCONJ
ejpam-6197	123	2	β	β	X
ejpam-6197	123	3	(	(	PUNCT
ejpam-6197	123	4	x	x	X
ejpam-6197	123	5	a	a	X
ejpam-6197	123	6	)	)	PUNCT
ejpam-6197	123	7	l	l	NOUN
ejpam-6197	123	8	(	(	PUNCT
ejpam-6197	123	9	(	(	PUNCT
ejpam-6197	123	10	c	c	X
ejpam-6197	123	11	+	+	NOUN
ejpam-6197	123	12	1)x	1)x	NUM
ejpam-6197	123	13	a	a	PRON
ejpam-6197	123	14	)	)	PUNCT
ejpam-6197	123	15	+	+	CCONJ
ejpam-6197	123	16	γ	γ	X
ejpam-6197	123	17	(	(	PUNCT
ejpam-6197	123	18	x	x	NOUN
ejpam-6197	123	19	a	a	X
ejpam-6197	123	20	)	)	PUNCT
ejpam-6197	123	21	l	l	NOUN
ejpam-6197	123	22	(	(	PUNCT
ejpam-6197	123	23	(	(	PUNCT
ejpam-6197	123	24	b	b	X
ejpam-6197	123	25	+	+	NOUN
ejpam-6197	123	26	1)x	1)x	NUM
ejpam-6197	123	27	a	a	PRON
ejpam-6197	123	28	)	)	PUNCT
ejpam-6197	123	29	,	,	PUNCT
ejpam-6197	123	30	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	123	31	k(x)∥	k(x)∥	VERB
ejpam-6197	123	32	≤	≤	ADV
ejpam-6197	123	33	1	1	NUM
ejpam-6197	123	34	1−	1−	NUM
ejpam-6197	123	35	ãl	ãl	NOUN
ejpam-6197	123	36	ϵ(x	ϵ(x	PROPN
ejpam-6197	123	37	,	,	PUNCT
ejpam-6197	123	38	x	x	X
ejpam-6197	123	39	,	,	PUNCT
ejpam-6197	123	40	x	x	NOUN
ejpam-6197	123	41	)	)	PUNCT
ejpam-6197	123	42	.	.	PUNCT
ejpam-6197	124	1	this	this	PRON
ejpam-6197	124	2	completes	complete	VERB
ejpam-6197	124	3	the	the	DET
ejpam-6197	124	4	proof	proof	NOUN
ejpam-6197	124	5	.	.	PUNCT
ejpam-6197	125	1	theorem	theorem	NOUN
ejpam-6197	125	2	3	3	X
ejpam-6197	125	3	.	.	PUNCT
ejpam-6197	125	4	suppose	suppose	VERB
ejpam-6197	125	5	that	that	SCONJ
ejpam-6197	125	6	x	x	PRON
ejpam-6197	125	7	is	be	AUX
ejpam-6197	125	8	a	a	DET
ejpam-6197	125	9	linear	linear	ADJ
ejpam-6197	125	10	normed	normed	ADJ
ejpam-6197	125	11	space	space	NOUN
ejpam-6197	125	12	,	,	PUNCT
ejpam-6197	125	13	y	y	PROPN
ejpam-6197	125	14	is	be	AUX
ejpam-6197	125	15	a	a	DET
ejpam-6197	125	16	banach	banach	NOUN
ejpam-6197	125	17	space	space	NOUN
ejpam-6197	125	18	and	and	CCONJ
ejpam-6197	125	19	θ	θ	PROPN
ejpam-6197	125	20	≥	≥	NUM
ejpam-6197	125	21	0	0	NUM
ejpam-6197	125	22	.	.	PUNCT
ejpam-6197	126	1	let	let	VERB
ejpam-6197	126	2	f	f	NOUN
ejpam-6197	126	3	:	:	PUNCT
ejpam-6197	126	4	x	x	X
ejpam-6197	126	5	→	→	SYM
ejpam-6197	126	6	y	y	X
ejpam-6197	126	7	be	be	AUX
ejpam-6197	126	8	a	a	DET
ejpam-6197	126	9	mapping	mapping	NOUN
ejpam-6197	126	10	satisfying	satisfy	VERB
ejpam-6197	126	11	∥f(ax+	∥f(ax+	ADV
ejpam-6197	126	12	by	by	ADP
ejpam-6197	126	13	)	)	PUNCT
ejpam-6197	127	1	+	+	CCONJ
ejpam-6197	127	2	f(x−	f(x−	ADP
ejpam-6197	127	3	y)−af(x)−bf(y)−df(−y)∥	y)−af(x)−bf(y)−df(−y)∥	NOUN
ejpam-6197	127	4	≤	≤	NUM
ejpam-6197	127	5	θ(∥x∥+	θ(∥x∥+	ADV
ejpam-6197	127	6	∥y∥	∥y∥	NOUN
ejpam-6197	127	7	)	)	PUNCT
ejpam-6197	128	1	(	(	PUNCT
ejpam-6197	128	2	8)	8)	NUM
ejpam-6197	128	3	for	for	ADP
ejpam-6197	128	4	all	all	DET
ejpam-6197	128	5	x	x	NOUN
ejpam-6197	128	6	,	,	PUNCT
ejpam-6197	128	7	y	y	PROPN
ejpam-6197	128	8	∈	∈	PROPN
ejpam-6197	128	9	x.	x.	NOUN
ejpam-6197	129	1	if	if	SCONJ
ejpam-6197	129	2	a	a	DET
ejpam-6197	129	3	,	,	PUNCT
ejpam-6197	129	4	b	b	NOUN
ejpam-6197	129	5	,	,	PUNCT
ejpam-6197	129	6	a	a	PRON
ejpam-6197	129	7	,	,	PUNCT
ejpam-6197	129	8	b	b	NOUN
ejpam-6197	129	9	and	and	CCONJ
ejpam-6197	129	10	d	d	PROPN
ejpam-6197	129	11	satisfy	satisfy	PROPN
ejpam-6197	129	12	|a+b|+|d|	|a+b|+|d|	PROPN
ejpam-6197	129	13	|a+b|	|a+b|	PROPN
ejpam-6197	129	14	<	<	X
ejpam-6197	129	15	1	1	NUM
ejpam-6197	129	16	and	and	CCONJ
ejpam-6197	129	17	|a+b+d−	|a+b+d−	NOUN
ejpam-6197	129	18	2|	2|	NUM
ejpam-6197	129	19	=	=	SYM
ejpam-6197	129	20	̸	̸	NUM
ejpam-6197	129	21	0	0	NUM
ejpam-6197	129	22	,	,	PUNCT
ejpam-6197	129	23	then	then	ADV
ejpam-6197	129	24	there	there	PRON
ejpam-6197	129	25	exists	exist	VERB
ejpam-6197	129	26	a	a	DET
ejpam-6197	129	27	unique	unique	ADJ
ejpam-6197	129	28	mapping	mapping	NOUN
ejpam-6197	129	29	k	k	NOUN
ejpam-6197	129	30	:	:	PUNCT
ejpam-6197	129	31	x	x	X
ejpam-6197	129	32	→	→	PUNCT
ejpam-6197	129	33	y	y	PROPN
ejpam-6197	129	34	such	such	ADJ
ejpam-6197	129	35	that	that	SCONJ
ejpam-6197	129	36	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	129	37	k(x)∥	k(x)∥	VERB
ejpam-6197	129	38	≤	≤	NOUN
ejpam-6197	129	39	2θ∥x∥	2θ∥x∥	NUM
ejpam-6197	129	40	|a+	|a+	NOUN
ejpam-6197	129	41	b|	b|	PROPN
ejpam-6197	129	42	−	−	PROPN
ejpam-6197	129	43	(	(	PUNCT
ejpam-6197	129	44	|a+b|+	|a+b|+	PROPN
ejpam-6197	129	45	|d|	|d|	PROPN
ejpam-6197	129	46	)	)	PUNCT
ejpam-6197	129	47	for	for	ADP
ejpam-6197	129	48	all	all	DET
ejpam-6197	129	49	x	x	SYM
ejpam-6197	129	50	∈	∈	ADJ
ejpam-6197	129	51	x.	x.	NOUN
ejpam-6197	129	52	proof	proof	NOUN
ejpam-6197	129	53	.	.	PUNCT
ejpam-6197	130	1	letting	let	VERB
ejpam-6197	130	2	x	x	PUNCT
ejpam-6197	130	3	=	=	PUNCT
ejpam-6197	130	4	y	y	PROPN
ejpam-6197	130	5	=	=	SYM
ejpam-6197	130	6	0	0	NUM
ejpam-6197	130	7	in	in	ADP
ejpam-6197	130	8	(	(	PUNCT
ejpam-6197	130	9	8)	8)	NUM
ejpam-6197	130	10	,	,	PUNCT
ejpam-6197	130	11	we	we	PRON
ejpam-6197	130	12	get	get	VERB
ejpam-6197	130	13	f(0	f(0	NOUN
ejpam-6197	130	14	)	)	PUNCT
ejpam-6197	131	1	=	=	SYM
ejpam-6197	131	2	0	0	X
ejpam-6197	131	3	.	.	PUNCT
ejpam-6197	131	4	letting	let	VERB
ejpam-6197	131	5	y	y	PROPN
ejpam-6197	132	1	=	=	PUNCT
ejpam-6197	132	2	x	x	X
ejpam-6197	132	3	in	in	ADP
ejpam-6197	132	4	(	(	PUNCT
ejpam-6197	132	5	8)	8)	NUM
ejpam-6197	132	6	,	,	PUNCT
ejpam-6197	132	7	we	we	PRON
ejpam-6197	132	8	get∥∥∥∥f(x)−	get∥∥∥∥f(x)−	VERB
ejpam-6197	132	9	(	(	PUNCT
ejpam-6197	132	10	a+b)f	a+b)f	PROPN
ejpam-6197	132	11	(	(	PUNCT
ejpam-6197	132	12	1	1	NUM
ejpam-6197	132	13	a+	a+	PRON
ejpam-6197	132	14	b	b	NOUN
ejpam-6197	132	15	x	x	X
ejpam-6197	132	16	)	)	PUNCT
ejpam-6197	132	17	−df	−df	PROPN
ejpam-6197	132	18	(	(	PUNCT
ejpam-6197	132	19	−	−	PROPN
ejpam-6197	132	20	1	1	NUM
ejpam-6197	132	21	a+	a+	SYM
ejpam-6197	132	22	b	b	NOUN
ejpam-6197	132	23	x	x	X
ejpam-6197	132	24	)	)	PUNCT
ejpam-6197	132	25	∥∥∥∥	∥∥∥∥	NUM
ejpam-6197	132	26	≤	≤	NUM
ejpam-6197	132	27	2θ∥x∥	2θ∥x∥	NUM
ejpam-6197	132	28	|a+	|a+	NOUN
ejpam-6197	132	29	b|	b|	ADJ
ejpam-6197	132	30	(	(	PUNCT
ejpam-6197	132	31	9	9	NUM
ejpam-6197	132	32	)	)	PUNCT
ejpam-6197	132	33	for	for	ADP
ejpam-6197	132	34	all	all	DET
ejpam-6197	132	35	x	x	SYM
ejpam-6197	132	36	∈	∈	NOUN
ejpam-6197	132	37	x.	x.	NOUN
ejpam-6197	132	38	consider	consider	VERB
ejpam-6197	132	39	t	t	NOUN
ejpam-6197	132	40	:	:	PUNCT
ejpam-6197	132	41	y	y	PROPN
ejpam-6197	132	42	x	x	PUNCT
ejpam-6197	132	43	→	→	SYM
ejpam-6197	132	44	y	y	PROPN
ejpam-6197	132	45	x	x	X
ejpam-6197	132	46	and	and	CCONJ
ejpam-6197	132	47	ϵ	ϵ	X
ejpam-6197	132	48	:	:	PUNCT
ejpam-6197	132	49	x	x	X
ejpam-6197	132	50	→	→	SYM
ejpam-6197	132	51	r+	r+	X
ejpam-6197	132	52	,	,	PUNCT
ejpam-6197	132	53	jf(x	jf(x	NOUN
ejpam-6197	132	54	)	)	PUNCT
ejpam-6197	132	55	=	=	SYM
ejpam-6197	133	1	(	(	PUNCT
ejpam-6197	133	2	a+b)f	a+b)f	PROPN
ejpam-6197	133	3	(	(	PUNCT
ejpam-6197	133	4	1	1	NUM
ejpam-6197	133	5	a+	a+	PRON
ejpam-6197	133	6	b	b	NOUN
ejpam-6197	133	7	x	x	X
ejpam-6197	133	8	)	)	PUNCT
ejpam-6197	134	1	+	+	NOUN
ejpam-6197	134	2	df	df	NOUN
ejpam-6197	134	3	(	(	PUNCT
ejpam-6197	134	4	−	−	PROPN
ejpam-6197	134	5	1	1	NUM
ejpam-6197	134	6	a+	a+	SYM
ejpam-6197	134	7	b	b	NOUN
ejpam-6197	134	8	x	x	X
ejpam-6197	134	9	)	)	PUNCT
ejpam-6197	135	1	g.	g.	PROPN
ejpam-6197	135	2	lyu	lyu	NOUN
ejpam-6197	135	3	et	et	PROPN
ejpam-6197	135	4	al	al	PROPN
ejpam-6197	135	5	.	.	PUNCT
ejpam-6197	135	6	/	/	SYM
ejpam-6197	135	7	eur	eur	PROPN
ejpam-6197	135	8	.	.	PUNCT
ejpam-6197	136	1	j.	j.	PROPN
ejpam-6197	136	2	pure	pure	PROPN
ejpam-6197	136	3	appl	appl	PROPN
ejpam-6197	136	4	.	.	PROPN
ejpam-6197	136	5	math	math	PROPN
ejpam-6197	136	6	,	,	PUNCT
ejpam-6197	136	7	18	18	NUM
ejpam-6197	136	8	(	(	PUNCT
ejpam-6197	136	9	3	3	NUM
ejpam-6197	136	10	)	)	PUNCT
ejpam-6197	136	11	(	(	PUNCT
ejpam-6197	136	12	2025	2025	NUM
ejpam-6197	136	13	)	)	PUNCT
ejpam-6197	136	14	,	,	PUNCT
ejpam-6197	136	15	6197	6197	NUM
ejpam-6197	136	16	8	8	NUM
ejpam-6197	136	17	of	of	ADP
ejpam-6197	136	18	17	17	NUM
ejpam-6197	136	19	for	for	ADP
ejpam-6197	136	20	all	all	PRON
ejpam-6197	136	21	x	x	SYM
ejpam-6197	136	22	∈	∈	PROPN
ejpam-6197	136	23	x	x	NOUN
ejpam-6197	136	24	,	,	PUNCT
ejpam-6197	136	25	f	f	PROPN
ejpam-6197	136	26	∈	∈	PROPN
ejpam-6197	137	1	y	y	PROPN
ejpam-6197	137	2	x	x	X
ejpam-6197	137	3	and	and	CCONJ
ejpam-6197	137	4	ϵ(x	ϵ(x	PROPN
ejpam-6197	137	5	)	)	PUNCT
ejpam-6197	137	6	=	=	SYM
ejpam-6197	137	7	2θ∥x∥	2θ∥x∥	NUM
ejpam-6197	137	8	|a+	|a+	NOUN
ejpam-6197	137	9	b|	b|	PROPN
ejpam-6197	137	10	,	,	PUNCT
ejpam-6197	137	11	x	x	SYM
ejpam-6197	137	12	∈	∈	NOUN
ejpam-6197	137	13	x.	x.	NOUN
ejpam-6197	137	14	the	the	DET
ejpam-6197	137	15	inequality	inequality	NOUN
ejpam-6197	137	16	(	(	PUNCT
ejpam-6197	137	17	9	9	NUM
ejpam-6197	137	18	)	)	PUNCT
ejpam-6197	137	19	becomes	become	VERB
ejpam-6197	137	20	∥jf(x)−	∥jf(x)−	NOUN
ejpam-6197	138	1	f(x)∥	f(x)∥	VERB
ejpam-6197	138	2	≤	≤	PUNCT
ejpam-6197	138	3	ϵ(x),∀x	ϵ(x),∀x	PUNCT
ejpam-6197	138	4	∈	∈	PROPN
ejpam-6197	138	5	x.	x.	NOUN
ejpam-6197	138	6	for	for	ADP
ejpam-6197	138	7	every	every	DET
ejpam-6197	138	8	g	g	NOUN
ejpam-6197	138	9	,	,	PUNCT
ejpam-6197	139	1	h	h	NOUN
ejpam-6197	139	2	∈	∈	PROPN
ejpam-6197	139	3	y	y	PROPN
ejpam-6197	139	4	x	x	X
ejpam-6197	139	5	and	and	CCONJ
ejpam-6197	139	6	x	x	SYM
ejpam-6197	139	7	∈	∈	PROPN
ejpam-6197	139	8	x	x	NOUN
ejpam-6197	139	9	,	,	PUNCT
ejpam-6197	139	10	∥tg(x)−	∥tg(x)−	PROPN
ejpam-6197	139	11	th(x)∥	th(x)∥	NOUN
ejpam-6197	140	1	=	=	SYM
ejpam-6197	140	2	∥∥∥∥(a+b)g	∥∥∥∥(a+b)g	PROPN
ejpam-6197	140	3	(	(	PUNCT
ejpam-6197	140	4	1	1	NUM
ejpam-6197	140	5	a+	a+	PRON
ejpam-6197	140	6	b	b	NOUN
ejpam-6197	140	7	x	x	X
ejpam-6197	140	8	)	)	PUNCT
ejpam-6197	141	1	+	+	NUM
ejpam-6197	141	2	dg	dg	X
ejpam-6197	141	3	(	(	PUNCT
ejpam-6197	141	4	−	−	PROPN
ejpam-6197	141	5	1	1	NUM
ejpam-6197	141	6	a+	a+	PRON
ejpam-6197	141	7	b	b	NOUN
ejpam-6197	141	8	x	x	X
ejpam-6197	141	9	)	)	PUNCT
ejpam-6197	141	10	−(a+b)h	−(a+b)h	NOUN
ejpam-6197	141	11	(	(	PUNCT
ejpam-6197	141	12	1	1	NUM
ejpam-6197	141	13	a+	a+	PRON
ejpam-6197	141	14	b	b	NOUN
ejpam-6197	141	15	x	x	X
ejpam-6197	141	16	)	)	PUNCT
ejpam-6197	141	17	−dh	−dh	PROPN
ejpam-6197	141	18	(	(	PUNCT
ejpam-6197	141	19	−	−	PROPN
ejpam-6197	141	20	1	1	NUM
ejpam-6197	141	21	a+	a+	SYM
ejpam-6197	141	22	b	b	NOUN
ejpam-6197	141	23	x	x	X
ejpam-6197	141	24	)	)	PUNCT
ejpam-6197	141	25	∥∥∥∥	∥∥∥∥	PROPN
ejpam-6197	141	26	≤	≤	NUM
ejpam-6197	141	27	|a+b|	|a+b|	PROPN
ejpam-6197	141	28	∥∥∥∥g	∥∥∥∥g	PROPN
ejpam-6197	141	29	(	(	PUNCT
ejpam-6197	141	30	1	1	NUM
ejpam-6197	141	31	a+	a+	PRON
ejpam-6197	141	32	b	b	NOUN
ejpam-6197	141	33	x	x	X
ejpam-6197	141	34	)	)	PUNCT
ejpam-6197	141	35	−	−	PROPN
ejpam-6197	141	36	h	h	NOUN
ejpam-6197	141	37	(	(	PUNCT
ejpam-6197	141	38	1	1	NUM
ejpam-6197	141	39	a+	a+	PRON
ejpam-6197	141	40	b	b	NOUN
ejpam-6197	141	41	x	x	PROPN
ejpam-6197	141	42	)	)	PUNCT
ejpam-6197	141	43	∥∥∥∥	∥∥∥∥	PROPN
ejpam-6197	142	1	+	+	CCONJ
ejpam-6197	142	2	|d|	|d|	PROPN
ejpam-6197	142	3	∥∥∥∥g(−	∥∥∥∥g(−	X
ejpam-6197	142	4	1	1	NUM
ejpam-6197	142	5	a+	a+	PRON
ejpam-6197	142	6	b	b	NOUN
ejpam-6197	142	7	x	x	X
ejpam-6197	142	8	)	)	PUNCT
ejpam-6197	142	9	−	−	PROPN
ejpam-6197	142	10	h	h	NOUN
ejpam-6197	142	11	(	(	PUNCT
ejpam-6197	142	12	−	−	PROPN
ejpam-6197	142	13	1	1	NUM
ejpam-6197	142	14	a+	a+	SYM
ejpam-6197	142	15	b	b	NOUN
ejpam-6197	142	16	x	x	PROPN
ejpam-6197	142	17	)	)	PUNCT
ejpam-6197	142	18	∥∥∥∥	∥∥∥∥	NUM
ejpam-6197	142	19	.	.	PUNCT
ejpam-6197	143	1	thus	thus	ADV
ejpam-6197	143	2	we	we	PRON
ejpam-6197	143	3	can	can	AUX
ejpam-6197	143	4	define	define	VERB
ejpam-6197	143	5	f1(x	f1(x	NUM
ejpam-6197	143	6	)	)	PUNCT
ejpam-6197	143	7	=	=	SYM
ejpam-6197	143	8	1	1	NUM
ejpam-6197	143	9	a+bx	a+bx	PROPN
ejpam-6197	143	10	,	,	PUNCT
ejpam-6197	143	11	f2(x	f2(x	PROPN
ejpam-6197	143	12	)	)	PUNCT
ejpam-6197	143	13	=	=	SYM
ejpam-6197	144	1	−	−	PROPN
ejpam-6197	144	2	1	1	NUM
ejpam-6197	144	3	a	a	PRON
ejpam-6197	144	4	=	=	NOUN
ejpam-6197	144	5	bx	bx	X
ejpam-6197	144	6	,	,	PUNCT
ejpam-6197	144	7	l1(x	l1(x	NOUN
ejpam-6197	144	8	)	)	PUNCT
ejpam-6197	144	9	=	=	NOUN
ejpam-6197	144	10	|a	|a	X
ejpam-6197	144	11	+	+	CCONJ
ejpam-6197	144	12	b|	b|	ADJ
ejpam-6197	144	13	,	,	PUNCT
ejpam-6197	144	14	l2(x	l2(x	PROPN
ejpam-6197	144	15	)	)	PUNCT
ejpam-6197	144	16	=	=	SYM
ejpam-6197	144	17	|d|	|d|	PROPN
ejpam-6197	144	18	and	and	CCONJ
ejpam-6197	144	19	the	the	DET
ejpam-6197	144	20	operator	operator	NOUN
ejpam-6197	144	21	λ	λ	PROPN
ejpam-6197	144	22	:	:	PUNCT
ejpam-6197	144	23	rx	rx	VERB
ejpam-6197	144	24	+	+	X
ejpam-6197	144	25	→	→	PUNCT
ejpam-6197	144	26	rx	rx	X
ejpam-6197	144	27	+	+	CCONJ
ejpam-6197	144	28	,	,	PUNCT
ejpam-6197	144	29	given	give	VERB
ejpam-6197	144	30	by	by	ADP
ejpam-6197	144	31	λη(x	λη(x	NOUN
ejpam-6197	144	32	)	)	PUNCT
ejpam-6197	144	33	=	=	SYM
ejpam-6197	145	1	|a+b|η	|a+b|η	NOUN
ejpam-6197	145	2	(	(	PUNCT
ejpam-6197	145	3	1	1	NUM
ejpam-6197	145	4	a+	a+	PRON
ejpam-6197	145	5	b	b	NOUN
ejpam-6197	145	6	x	x	X
ejpam-6197	145	7	)	)	PUNCT
ejpam-6197	146	1	+	+	PUNCT
ejpam-6197	146	2	|d|η	|d|η	NOUN
ejpam-6197	146	3	(	(	PUNCT
ejpam-6197	146	4	−	−	PROPN
ejpam-6197	146	5	1	1	NUM
ejpam-6197	146	6	a+	a+	SYM
ejpam-6197	146	7	b	b	NOUN
ejpam-6197	146	8	x	x	X
ejpam-6197	146	9	)	)	PUNCT
ejpam-6197	146	10	for	for	ADP
ejpam-6197	146	11	all	all	PRON
ejpam-6197	146	12	x	x	SYM
ejpam-6197	146	13	∈	∈	ADJ
ejpam-6197	146	14	x.	x.	NOUN
ejpam-6197	146	15	in	in	ADP
ejpam-6197	146	16	particular	particular	ADJ
ejpam-6197	146	17	,	,	PUNCT
ejpam-6197	146	18	λϵ(x	λϵ(x	ADJ
ejpam-6197	146	19	)	)	PUNCT
ejpam-6197	146	20	=	=	SYM
ejpam-6197	146	21	|a+b|ϵ	|a+b|ϵ	NOUN
ejpam-6197	146	22	(	(	PUNCT
ejpam-6197	146	23	1	1	NUM
ejpam-6197	146	24	a+	a+	PRON
ejpam-6197	146	25	b	b	NOUN
ejpam-6197	146	26	x	x	X
ejpam-6197	146	27	)	)	PUNCT
ejpam-6197	147	1	+	+	CCONJ
ejpam-6197	148	1	|d|ϵ	|d|ϵ	ADJ
ejpam-6197	148	2	(	(	PUNCT
ejpam-6197	148	3	−	−	PROPN
ejpam-6197	148	4	1	1	NUM
ejpam-6197	148	5	a+	a+	SYM
ejpam-6197	148	6	b	b	NOUN
ejpam-6197	148	7	x	x	X
ejpam-6197	148	8	)	)	PUNCT
ejpam-6197	148	9	=	=	SYM
ejpam-6197	148	10	|a+b|θ	|a+b|θ	NOUN
ejpam-6197	148	11	2∥x∥	2∥x∥	NOUN
ejpam-6197	148	12	|a+	|a+	NOUN
ejpam-6197	148	13	b|2	b|2	PROPN
ejpam-6197	148	14	+	+	CCONJ
ejpam-6197	148	15	|d|θ	|d|θ	PROPN
ejpam-6197	148	16	2∥x∥	2∥x∥	NOUN
ejpam-6197	148	17	|a+	|a+	X
ejpam-6197	148	18	b|2	b|2	PROPN
ejpam-6197	148	19	=	=	PUNCT
ejpam-6197	148	20	{	{	PUNCT
ejpam-6197	148	21	|a+b|+	|a+b|+	PROPN
ejpam-6197	148	22	|d|	|d|	PROPN
ejpam-6197	148	23	|a+	|a+	PROPN
ejpam-6197	148	24	b|	b|	PROPN
ejpam-6197	148	25	}	}	PUNCT
ejpam-6197	148	26	ϵ(x	ϵ(x	PROPN
ejpam-6197	148	27	)	)	PUNCT
ejpam-6197	148	28	.	.	PUNCT
ejpam-6197	149	1	since	since	SCONJ
ejpam-6197	149	2	λ	λ	PROPN
ejpam-6197	149	3	is	be	AUX
ejpam-6197	149	4	linear	linear	ADJ
ejpam-6197	149	5	,	,	PUNCT
ejpam-6197	149	6	we	we	PRON
ejpam-6197	149	7	can	can	AUX
ejpam-6197	149	8	get	get	VERB
ejpam-6197	149	9	λnε(x	λnε(x	NOUN
ejpam-6197	149	10	)	)	PUNCT
ejpam-6197	149	11	=	=	NOUN
ejpam-6197	149	12	{	{	PUNCT
ejpam-6197	149	13	|a+b|+	|a+b|+	PROPN
ejpam-6197	149	14	|d|	|d|	PROPN
ejpam-6197	149	15	|a+	|a+	PROPN
ejpam-6197	149	16	b|	b|	PROPN
ejpam-6197	149	17	}	}	PUNCT
ejpam-6197	149	18	n	n	PRON
ejpam-6197	149	19	ϵ(x	ϵ(x	NOUN
ejpam-6197	149	20	)	)	PUNCT
ejpam-6197	149	21	,	,	PUNCT
ejpam-6197	149	22	x	x	PUNCT
ejpam-6197	149	23	∈	∈	PROPN
ejpam-6197	149	24	x	x	X
ejpam-6197	149	25	,	,	PUNCT
ejpam-6197	149	26	n	n	PROPN
ejpam-6197	149	27	∈	∈	PROPN
ejpam-6197	149	28	n0	n0	PROPN
ejpam-6197	149	29	.	.	PUNCT
ejpam-6197	150	1	g.	g.	PROPN
ejpam-6197	150	2	lyu	lyu	VERB
ejpam-6197	150	3	et	et	PROPN
ejpam-6197	150	4	al	al	PROPN
ejpam-6197	150	5	.	.	PUNCT
ejpam-6197	150	6	/	/	SYM
ejpam-6197	150	7	eur	eur	PROPN
ejpam-6197	150	8	.	.	PUNCT
ejpam-6197	151	1	j.	j.	PROPN
ejpam-6197	151	2	pure	pure	PROPN
ejpam-6197	151	3	appl	appl	PROPN
ejpam-6197	151	4	.	.	PROPN
ejpam-6197	151	5	math	math	PROPN
ejpam-6197	151	6	,	,	PUNCT
ejpam-6197	151	7	18	18	NUM
ejpam-6197	151	8	(	(	PUNCT
ejpam-6197	151	9	3	3	NUM
ejpam-6197	151	10	)	)	PUNCT
ejpam-6197	151	11	(	(	PUNCT
ejpam-6197	151	12	2025	2025	NUM
ejpam-6197	151	13	)	)	PUNCT
ejpam-6197	151	14	,	,	PUNCT
ejpam-6197	151	15	6197	6197	NUM
ejpam-6197	151	16	9	9	NUM
ejpam-6197	151	17	of	of	ADP
ejpam-6197	151	18	17	17	NUM
ejpam-6197	151	19	since	since	SCONJ
ejpam-6197	151	20	|a+b|+|d|	|a+b|+|d|	PROPN
ejpam-6197	151	21	|a+b|	|a+b|	PROPN
ejpam-6197	151	22	<	<	X
ejpam-6197	151	23	1	1	NUM
ejpam-6197	151	24	,	,	PUNCT
ejpam-6197	151	25	the	the	DET
ejpam-6197	151	26	series	series	NOUN
ejpam-6197	151	27	∑∞	∑∞	PROPN
ejpam-6197	151	28	n=0	n=0	X
ejpam-6197	151	29	λ	λ	NOUN
ejpam-6197	151	30	nϵ(x	nϵ(x	NUM
ejpam-6197	151	31	)	)	PUNCT
ejpam-6197	151	32	is	be	AUX
ejpam-6197	151	33	convergent	convergent	ADJ
ejpam-6197	151	34	for	for	ADP
ejpam-6197	151	35	every	every	DET
ejpam-6197	151	36	x	x	SYM
ejpam-6197	151	37	∈	∈	PROPN
ejpam-6197	151	38	x	x	X
ejpam-6197	151	39	and	and	CCONJ
ejpam-6197	151	40	ε∗(x	ε∗(x	NOUN
ejpam-6197	151	41	)	)	PUNCT
ejpam-6197	151	42	=	=	PUNCT
ejpam-6197	152	1	∞∑	∞∑	PRON
ejpam-6197	152	2	n=0	n=0	NUM
ejpam-6197	152	3	λnϵ(x	λnϵ(x	NOUN
ejpam-6197	152	4	)	)	PUNCT
ejpam-6197	152	5	=	=	PUNCT
ejpam-6197	153	1	∞∑	∞∑	NUM
ejpam-6197	153	2	n=0	n=0	PUNCT
ejpam-6197	153	3	{	{	PUNCT
ejpam-6197	153	4	|a+b|+	|a+b|+	PROPN
ejpam-6197	153	5	|d|	|d|	PROPN
ejpam-6197	153	6	|a+	|a+	PROPN
ejpam-6197	153	7	b|	b|	PROPN
ejpam-6197	153	8	}	}	PUNCT
ejpam-6197	153	9	n	n	X
ejpam-6197	153	10	ϵ(x	ϵ(x	NOUN
ejpam-6197	153	11	)	)	PUNCT
ejpam-6197	153	12	=	=	SYM
ejpam-6197	153	13	1	1	NUM
ejpam-6197	153	14	1−	1−	NUM
ejpam-6197	153	15	|a+b|+|d|	|a+b|+|d|	PROPN
ejpam-6197	153	16	|a+b|	|a+b|	PROPN
ejpam-6197	153	17	ϵ(x	ϵ(x	PROPN
ejpam-6197	153	18	)	)	PUNCT
ejpam-6197	153	19	=	=	SYM
ejpam-6197	153	20	1	1	NUM
ejpam-6197	153	21	1−	1−	NUM
ejpam-6197	153	22	|a+b|+|d|	|a+b|+|d|	NOUN
ejpam-6197	153	23	|a+b|	|a+b|	NOUN
ejpam-6197	153	24	2θ∥x∥	2θ∥x∥	NUM
ejpam-6197	153	25	|a+	|a+	PROPN
ejpam-6197	153	26	b|	b|	PROPN
ejpam-6197	153	27	=	=	PUNCT
ejpam-6197	153	28	2θ∥x∥	2θ∥x∥	NUM
ejpam-6197	153	29	|a+	|a+	NOUN
ejpam-6197	153	30	b|	b|	PROPN
ejpam-6197	153	31	−	−	PROPN
ejpam-6197	153	32	(	(	PUNCT
ejpam-6197	153	33	|a+b|+	|a+b|+	PROPN
ejpam-6197	153	34	|d|	|d|	PROPN
ejpam-6197	153	35	)	)	PUNCT
ejpam-6197	153	36	,	,	PUNCT
ejpam-6197	153	37	x	x	PUNCT
ejpam-6197	153	38	∈	∈	NOUN
ejpam-6197	153	39	x.	x.	NOUN
ejpam-6197	153	40	by	by	ADP
ejpam-6197	153	41	theorem	theorem	NOUN
ejpam-6197	153	42	1	1	NUM
ejpam-6197	153	43	,	,	PUNCT
ejpam-6197	153	44	there	there	PRON
ejpam-6197	153	45	exists	exist	VERB
ejpam-6197	153	46	a	a	DET
ejpam-6197	153	47	mapping	mapping	NOUN
ejpam-6197	153	48	k	k	NOUN
ejpam-6197	153	49	:	:	PUNCT
ejpam-6197	153	50	x	x	X
ejpam-6197	153	51	→	→	PUNCT
ejpam-6197	153	52	y	y	PROPN
ejpam-6197	153	53	such	such	ADJ
ejpam-6197	153	54	that	that	SCONJ
ejpam-6197	153	55	k(x	k(x	PROPN
ejpam-6197	153	56	)	)	PUNCT
ejpam-6197	154	1	=	=	PROPN
ejpam-6197	154	2	lim	lim	PROPN
ejpam-6197	154	3	n→∞	n→∞	NUM
ejpam-6197	154	4	jnf(x	jnf(x	PROPN
ejpam-6197	154	5	)	)	PUNCT
ejpam-6197	154	6	,	,	PUNCT
ejpam-6197	154	7	k(x	k(x	PROPN
ejpam-6197	154	8	)	)	PUNCT
ejpam-6197	154	9	=	=	SYM
ejpam-6197	155	1	(	(	PUNCT
ejpam-6197	155	2	a+b)k	a+b)k	PRON
ejpam-6197	155	3	(	(	PUNCT
ejpam-6197	155	4	1	1	NUM
ejpam-6197	155	5	a+	a+	PRON
ejpam-6197	155	6	b	b	NOUN
ejpam-6197	155	7	x	x	X
ejpam-6197	155	8	)	)	PUNCT
ejpam-6197	156	1	+	+	ADV
ejpam-6197	156	2	dk	dk	X
ejpam-6197	156	3	(	(	PUNCT
ejpam-6197	156	4	−	−	PROPN
ejpam-6197	156	5	1	1	NUM
ejpam-6197	156	6	a+	a+	SYM
ejpam-6197	156	7	b	b	NOUN
ejpam-6197	156	8	x	x	X
ejpam-6197	156	9	)	)	PUNCT
ejpam-6197	156	10	,	,	PUNCT
ejpam-6197	156	11	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	156	12	k(x)∥	k(x)∥	VERB
ejpam-6197	156	13	≤	≤	NOUN
ejpam-6197	156	14	2θ∥x∥	2θ∥x∥	NUM
ejpam-6197	156	15	|a+	|a+	NOUN
ejpam-6197	156	16	b|	b|	PROPN
ejpam-6197	156	17	−	−	PROPN
ejpam-6197	156	18	(	(	PUNCT
ejpam-6197	156	19	|a+b|+	|a+b|+	PROPN
ejpam-6197	156	20	|d|	|d|	PROPN
ejpam-6197	156	21	)	)	PUNCT
ejpam-6197	156	22	.	.	PUNCT
ejpam-6197	157	1	this	this	PRON
ejpam-6197	157	2	completes	complete	VERB
ejpam-6197	157	3	the	the	DET
ejpam-6197	157	4	proof	proof	NOUN
ejpam-6197	157	5	.	.	PUNCT
ejpam-6197	158	1	4	4	X
ejpam-6197	158	2	.	.	X
ejpam-6197	158	3	the	the	DET
ejpam-6197	158	4	stability	stability	NOUN
ejpam-6197	158	5	of	of	ADP
ejpam-6197	158	6	the	the	DET
ejpam-6197	158	7	generalized	generalize	VERB
ejpam-6197	158	8	function	function	NOUN
ejpam-6197	158	9	inequations	inequation	NOUN
ejpam-6197	158	10	this	this	DET
ejpam-6197	158	11	section	section	NOUN
ejpam-6197	158	12	primarily	primarily	ADV
ejpam-6197	158	13	investigates	investigate	VERB
ejpam-6197	158	14	the	the	DET
ejpam-6197	158	15	stability	stability	NOUN
ejpam-6197	158	16	of	of	ADP
ejpam-6197	158	17	functional	functional	ADJ
ejpam-6197	158	18	equations	equation	NOUN
ejpam-6197	158	19	,	,	PUNCT
ejpam-6197	158	20	improves	improve	VERB
ejpam-6197	158	21	existing	exist	VERB
ejpam-6197	158	22	results	result	NOUN
ejpam-6197	158	23	through	through	ADP
ejpam-6197	158	24	a	a	DET
ejpam-6197	158	25	novel	novel	ADJ
ejpam-6197	158	26	direct	direct	ADJ
ejpam-6197	158	27	method	method	NOUN
ejpam-6197	158	28	,	,	PUNCT
ejpam-6197	158	29	and	and	CCONJ
ejpam-6197	158	30	explores	explore	VERB
ejpam-6197	158	31	the	the	DET
ejpam-6197	158	32	dependency	dependency	NOUN
ejpam-6197	158	33	relationships	relationship	NOUN
ejpam-6197	158	34	among	among	ADP
ejpam-6197	158	35	different	different	ADJ
ejpam-6197	158	36	parameters	parameter	NOUN
ejpam-6197	158	37	in	in	ADP
ejpam-6197	158	38	generalized	generalized	ADJ
ejpam-6197	158	39	functional	functional	ADJ
ejpam-6197	158	40	equations	equation	NOUN
ejpam-6197	158	41	along	along	ADP
ejpam-6197	158	42	with	with	ADP
ejpam-6197	158	43	their	their	PRON
ejpam-6197	158	44	related	related	ADJ
ejpam-6197	158	45	properties	property	NOUN
ejpam-6197	158	46	.	.	PUNCT
ejpam-6197	159	1	it	it	PRON
ejpam-6197	159	2	also	also	ADV
ejpam-6197	159	3	solves	solve	VERB
ejpam-6197	159	4	functional	functional	ADJ
ejpam-6197	159	5	inequalities	inequality	NOUN
ejpam-6197	159	6	and	and	CCONJ
ejpam-6197	159	7	constructs	construct	NOUN
ejpam-6197	159	8	as	as	ADV
ejpam-6197	159	9	well	well	ADV
ejpam-6197	159	10	as	as	ADP
ejpam-6197	159	11	studies	study	NOUN
ejpam-6197	159	12	inequalities	inequality	NOUN
ejpam-6197	159	13	related	relate	VERB
ejpam-6197	159	14	to	to	ADP
ejpam-6197	159	15	functional	functional	ADJ
ejpam-6197	159	16	equations	equation	NOUN
ejpam-6197	159	17	.	.	PUNCT
ejpam-6197	160	1	park	park	NOUN
ejpam-6197	161	1	[	[	X
ejpam-6197	161	2	36	36	NUM
ejpam-6197	161	3	]	]	PUNCT
ejpam-6197	161	4	introduced	introduce	VERB
ejpam-6197	161	5	additive	additive	ADJ
ejpam-6197	161	6	ρ	ρ	ADJ
ejpam-6197	161	7	-	-	ADJ
ejpam-6197	161	8	functional	functional	ADJ
ejpam-6197	161	9	inequalities	inequality	NOUN
ejpam-6197	161	10	and	and	CCONJ
ejpam-6197	161	11	employed	employ	VERB
ejpam-6197	161	12	the	the	DET
ejpam-6197	161	13	direct	direct	ADJ
ejpam-6197	161	14	method	method	NOUN
ejpam-6197	161	15	to	to	PART
ejpam-6197	161	16	establish	establish	VERB
ejpam-6197	161	17	the	the	DET
ejpam-6197	161	18	hyers	hyers	PROPN
ejpam-6197	161	19	-	-	PUNCT
ejpam-6197	161	20	ulam	ulam	ADJ
ejpam-6197	161	21	stability	stability	NOUN
ejpam-6197	161	22	of	of	ADP
ejpam-6197	161	23	these	these	DET
ejpam-6197	161	24	inequalities	inequality	NOUN
ejpam-6197	161	25	within	within	ADP
ejpam-6197	161	26	banach	banach	NOUN
ejpam-6197	161	27	spaces	space	NOUN
ejpam-6197	161	28	.	.	PUNCT
ejpam-6197	162	1	subsequently	subsequently	ADV
ejpam-6197	162	2	,	,	PUNCT
ejpam-6197	162	3	in	in	ADP
ejpam-6197	162	4	2016	2016	NUM
ejpam-6197	162	5	,	,	PUNCT
ejpam-6197	162	6	choi	choi	NOUN
ejpam-6197	162	7	et	et	PROPN
ejpam-6197	162	8	al	al	PROPN
ejpam-6197	162	9	.	.	PUNCT
ejpam-6197	163	1	[	[	X
ejpam-6197	163	2	37	37	NUM
ejpam-6197	163	3	]	]	PUNCT
ejpam-6197	163	4	investigated	investigate	VERB
ejpam-6197	163	5	additive	additive	ADJ
ejpam-6197	163	6	ρ	ρ	ADJ
ejpam-6197	163	7	-	-	ADJ
ejpam-6197	163	8	functional	functional	ADJ
ejpam-6197	163	9	inequalities	inequality	NOUN
ejpam-6197	163	10	in	in	ADP
ejpam-6197	163	11	normed	normed	ADJ
ejpam-6197	163	12	spaces	space	NOUN
ejpam-6197	163	13	.	.	PUNCT
ejpam-6197	164	1	leveraging	leverage	VERB
ejpam-6197	164	2	the	the	DET
ejpam-6197	164	3	fixed	fix	VERB
ejpam-6197	164	4	-	-	PUNCT
ejpam-6197	164	5	point	point	NOUN
ejpam-6197	164	6	method	method	NOUN
ejpam-6197	164	7	,	,	PUNCT
ejpam-6197	164	8	they	they	PRON
ejpam-6197	164	9	demonstrated	demonstrate	VERB
ejpam-6197	164	10	the	the	DET
ejpam-6197	164	11	hyers	hyers	PROPN
ejpam-6197	164	12	-	-	PUNCT
ejpam-6197	164	13	ulam	ulam	ADJ
ejpam-6197	164	14	stability	stability	NOUN
ejpam-6197	164	15	of	of	ADP
ejpam-6197	164	16	two	two	NUM
ejpam-6197	164	17	distinct	distinct	ADJ
ejpam-6197	164	18	additive	additive	ADJ
ejpam-6197	164	19	ρ	ρ	ADJ
ejpam-6197	164	20	-	-	ADJ
ejpam-6197	164	21	functional	functional	ADJ
ejpam-6197	164	22	inequalities	inequality	NOUN
ejpam-6197	164	23	∥f(x+	∥f(x+	VERB
ejpam-6197	164	24	y)−	y)−	PROPN
ejpam-6197	164	25	f(x)−	f(x)−	PROPN
ejpam-6197	164	26	f(y)∥	f(y)∥	NOUN
ejpam-6197	164	27	≤	≤	X
ejpam-6197	164	28	∥∥∥∥ρ(2f	∥∥∥∥ρ(2f	VERB
ejpam-6197	164	29	(	(	PUNCT
ejpam-6197	164	30	x+	x+	PROPN
ejpam-6197	164	31	y	y	PROPN
ejpam-6197	164	32	2	2	NUM
ejpam-6197	164	33	)	)	PUNCT
ejpam-6197	164	34	−	−	PROPN
ejpam-6197	165	1	f(x)−	f(x)−	PROPN
ejpam-6197	165	2	f(y	f(y	PROPN
ejpam-6197	165	3	)	)	PUNCT
ejpam-6197	165	4	)	)	PUNCT
ejpam-6197	166	1	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-6197	166	2	and	and	CCONJ
ejpam-6197	166	3	∥∥∥∥2f	∥∥∥∥2f	PROPN
ejpam-6197	166	4	(	(	PUNCT
ejpam-6197	166	5	x+	x+	PROPN
ejpam-6197	166	6	y	y	PROPN
ejpam-6197	166	7	2	2	NUM
ejpam-6197	166	8	)	)	PUNCT
ejpam-6197	166	9	−	−	PROPN
ejpam-6197	167	1	f(x)−	f(x)−	PROPN
ejpam-6197	167	2	f(y	f(y	PROPN
ejpam-6197	167	3	)	)	PUNCT
ejpam-6197	168	1	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-6197	168	2	≤	≤	NOUN
ejpam-6197	169	1	∥ρ(f(x+	∥ρ(f(x+	PROPN
ejpam-6197	169	2	y)−	y)−	PROPN
ejpam-6197	169	3	f(x)−	f(x)−	NOUN
ejpam-6197	169	4	f(y))∥	f(y))∥	X
ejpam-6197	169	5	(	(	PUNCT
ejpam-6197	169	6	where	where	SCONJ
ejpam-6197	169	7	ρ	ρ	PROPN
ejpam-6197	169	8	<	<	X
ejpam-6197	169	9	1	1	NUM
ejpam-6197	169	10	)	)	PUNCT
ejpam-6197	169	11	in	in	ADP
ejpam-6197	169	12	normed	normed	ADJ
ejpam-6197	169	13	spaces	space	NOUN
ejpam-6197	169	14	.	.	PUNCT
ejpam-6197	170	1	in	in	ADP
ejpam-6197	170	2	2023	2023	NUM
ejpam-6197	170	3	,	,	PUNCT
ejpam-6197	170	4	nawaz	nawaz	NOUN
ejpam-6197	170	5	et	et	PROPN
ejpam-6197	170	6	al	al	PROPN
ejpam-6197	170	7	.	.	PUNCT
ejpam-6197	171	1	[	[	X
ejpam-6197	171	2	38	38	NUM
ejpam-6197	171	3	]	]	PUNCT
ejpam-6197	171	4	analyzed	analyze	VERB
ejpam-6197	171	5	the	the	DET
ejpam-6197	171	6	hyers	hyers	PROPN
ejpam-6197	171	7	-	-	PUNCT
ejpam-6197	171	8	ulam	ulam	ADJ
ejpam-6197	171	9	stability	stability	NOUN
ejpam-6197	171	10	of	of	ADP
ejpam-6197	171	11	cubic	cubic	ADJ
ejpam-6197	171	12	and	and	CCONJ
ejpam-6197	171	13	quartic	quartic	ADJ
ejpam-6197	171	14	ρ	ρ	ADJ
ejpam-6197	171	15	-	-	ADJ
ejpam-6197	171	16	functional	functional	ADJ
ejpam-6197	171	17	inequalities	inequality	NOUN
ejpam-6197	171	18	in	in	ADP
ejpam-6197	171	19	fuzzy	fuzzy	ADJ
ejpam-6197	171	20	matrix	matrix	NOUN
ejpam-6197	171	21	spaces	space	NOUN
ejpam-6197	171	22	.	.	PUNCT
ejpam-6197	172	1	they	they	PRON
ejpam-6197	172	2	used	use	VERB
ejpam-6197	172	3	g.	g.	PROPN
ejpam-6197	172	4	lyu	lyu	PROPN
ejpam-6197	172	5	et	et	PROPN
ejpam-6197	172	6	al	al	PROPN
ejpam-6197	172	7	.	.	PUNCT
ejpam-6197	172	8	/	/	SYM
ejpam-6197	172	9	eur	eur	PROPN
ejpam-6197	172	10	.	.	PUNCT
ejpam-6197	173	1	j.	j.	PROPN
ejpam-6197	173	2	pure	pure	PROPN
ejpam-6197	173	3	appl	appl	PROPN
ejpam-6197	173	4	.	.	PROPN
ejpam-6197	173	5	math	math	PROPN
ejpam-6197	173	6	,	,	PUNCT
ejpam-6197	173	7	18	18	NUM
ejpam-6197	173	8	(	(	PUNCT
ejpam-6197	173	9	3	3	NUM
ejpam-6197	173	10	)	)	PUNCT
ejpam-6197	173	11	(	(	PUNCT
ejpam-6197	173	12	2025	2025	NUM
ejpam-6197	173	13	)	)	PUNCT
ejpam-6197	173	14	,	,	PUNCT
ejpam-6197	173	15	6197	6197	NUM
ejpam-6197	173	16	10	10	NUM
ejpam-6197	173	17	of	of	ADP
ejpam-6197	173	18	17	17	NUM
ejpam-6197	173	19	the	the	DET
ejpam-6197	173	20	fixed	fix	VERB
ejpam-6197	173	21	-	-	PUNCT
ejpam-6197	173	22	point	point	NOUN
ejpam-6197	173	23	method	method	NOUN
ejpam-6197	173	24	to	to	PART
ejpam-6197	173	25	study	study	VERB
ejpam-6197	173	26	the	the	DET
ejpam-6197	173	27	following	follow	VERB
ejpam-6197	173	28	functional	functional	ADJ
ejpam-6197	173	29	inequalities:	inequalities:	NOUN
ejpam-6197	173	30	(	(	PUNCT
ejpam-6197	173	31	f(2x+	f(2x+	PROPN
ejpam-6197	173	32	y	y	PROPN
ejpam-6197	173	33	)	)	PUNCT
ejpam-6197	174	1	+	+	CCONJ
ejpam-6197	174	2	f(2x−	f(2x−	VERB
ejpam-6197	174	3	y)−	y)−	PROPN
ejpam-6197	174	4	2f(x+	2f(x+	NUM
ejpam-6197	174	5	y)−	y)−	PROPN
ejpam-6197	174	6	2f(x−	2f(x−	NUM
ejpam-6197	174	7	y)−	y)−	PROPN
ejpam-6197	174	8	12f(x	12f(x	NUM
ejpam-6197	174	9	)	)	PUNCT
ejpam-6197	174	10	−ρ(4f(x+	−ρ(4f(x+	X
ejpam-6197	174	11	y	y	PROPN
ejpam-6197	174	12	2	2	NUM
ejpam-6197	174	13	)	)	PUNCT
ejpam-6197	174	14	+	+	CCONJ
ejpam-6197	174	15	4(f(x−	4(f(x−	NUM
ejpam-6197	174	16	y	y	NOUN
ejpam-6197	174	17	2	2	NUM
ejpam-6197	174	18	)	)	PUNCT
ejpam-6197	174	19	−	−	NOUN
ejpam-6197	174	20	f(x+	f(x+	NOUN
ejpam-6197	174	21	y)−	y)−	PROPN
ejpam-6197	174	22	f(x−	f(x−	ADP
ejpam-6197	174	23	y))−	y))−	NOUN
ejpam-6197	174	24	6f(x	6f(x	NUM
ejpam-6197	174	25	)	)	PUNCT
ejpam-6197	174	26	,	,	PUNCT
ejpam-6197	174	27	r)|	r)|	ADJ
ejpam-6197	174	28	≥	≥	PROPN
ejpam-6197	174	29	r	r	NOUN
ejpam-6197	174	30	r+φ(x	r+φ(x	NOUN
ejpam-6197	174	31	,	,	PUNCT
ejpam-6197	174	32	y	y	PROPN
ejpam-6197	174	33	)	)	PUNCT
ejpam-6197	174	34	,	,	PUNCT
ejpam-6197	174	35	f(2x+	f(2x+	PROPN
ejpam-6197	174	36	y	y	X
ejpam-6197	174	37	)	)	PUNCT
ejpam-6197	174	38	+	+	CCONJ
ejpam-6197	174	39	f(2x−	f(2x−	VERB
ejpam-6197	174	40	y)−	y)−	PROPN
ejpam-6197	174	41	4f(x+	4f(x+	NUM
ejpam-6197	174	42	y)−	y)−	PROPN
ejpam-6197	174	43	4f(x−	4f(x−	NUM
ejpam-6197	174	44	y)−	y)−	PROPN
ejpam-6197	174	45	24f(x	24f(x	NUM
ejpam-6197	174	46	)	)	PUNCT
ejpam-6197	174	47	+	+	CCONJ
ejpam-6197	174	48	6f(y	6f(y	NUM
ejpam-6197	174	49	)	)	PUNCT
ejpam-6197	174	50	−ρ(8f(x+	−ρ(8f(x+	ADV
ejpam-6197	174	51	y	y	PROPN
ejpam-6197	174	52	2	2	NUM
ejpam-6197	174	53	)	)	PUNCT
ejpam-6197	174	54	+	+	CCONJ
ejpam-6197	174	55	8(f(x−	8(f(x−	NUM
ejpam-6197	174	56	y	y	NOUN
ejpam-6197	174	57	2	2	NUM
ejpam-6197	174	58	)	)	PUNCT
ejpam-6197	174	59	−	−	PROPN
ejpam-6197	175	1	2f(x+	2f(x+	PROPN
ejpam-6197	175	2	y)−	y)−	PROPN
ejpam-6197	175	3	2f(x−	2f(x−	NUM
ejpam-6197	175	4	y))−	y))−	NOUN
ejpam-6197	175	5	12f(x	12f(x	NUM
ejpam-6197	175	6	)	)	PUNCT
ejpam-6197	175	7	+	+	CCONJ
ejpam-6197	175	8	3f(y	3f(y	NUM
ejpam-6197	175	9	)	)	PUNCT
ejpam-6197	175	10	,	,	PUNCT
ejpam-6197	175	11	r	r	X
ejpam-6197	175	12	)	)	PUNCT
ejpam-6197	175	13	≥	≥	NOUN
ejpam-6197	175	14	r	r	NOUN
ejpam-6197	175	15	r+φ(x	r+φ(x	NOUN
ejpam-6197	175	16	,	,	PUNCT
ejpam-6197	175	17	y	y	NOUN
ejpam-6197	175	18	)	)	PUNCT
ejpam-6197	175	19	where	where	SCONJ
ejpam-6197	175	20	ρ	ρ	NOUN
ejpam-6197	175	21	̸=	̸=	PROPN
ejpam-6197	175	22	2	2	NUM
ejpam-6197	175	23	is	be	AUX
ejpam-6197	175	24	a	a	DET
ejpam-6197	175	25	real	real	ADJ
ejpam-6197	175	26	number	number	NOUN
ejpam-6197	175	27	.	.	PUNCT
ejpam-6197	176	1	functional	functional	ADJ
ejpam-6197	176	2	equations	equation	NOUN
ejpam-6197	176	3	play	play	VERB
ejpam-6197	176	4	a	a	DET
ejpam-6197	176	5	fundamental	fundamental	ADJ
ejpam-6197	176	6	role	role	NOUN
ejpam-6197	176	7	in	in	ADP
ejpam-6197	176	8	mathematics	mathematic	NOUN
ejpam-6197	176	9	and	and	CCONJ
ejpam-6197	176	10	its	its	PRON
ejpam-6197	176	11	applications	application	NOUN
ejpam-6197	176	12	,	,	PUNCT
ejpam-6197	176	13	particularly	particularly	ADV
ejpam-6197	176	14	in	in	ADP
ejpam-6197	176	15	areas	area	NOUN
ejpam-6197	176	16	such	such	ADJ
ejpam-6197	176	17	as	as	ADP
ejpam-6197	176	18	information	information	NOUN
ejpam-6197	176	19	theory	theory	NOUN
ejpam-6197	176	20	,	,	PUNCT
ejpam-6197	176	21	economics	economic	NOUN
ejpam-6197	176	22	,	,	PUNCT
ejpam-6197	176	23	and	and	CCONJ
ejpam-6197	176	24	decision	decision	NOUN
ejpam-6197	176	25	sciences	science	NOUN
ejpam-6197	176	26	.	.	PUNCT
ejpam-6197	177	1	one	one	NUM
ejpam-6197	177	2	important	important	ADJ
ejpam-6197	177	3	equation	equation	NOUN
ejpam-6197	177	4	is	be	AUX
ejpam-6197	177	5	the	the	DET
ejpam-6197	177	6	generalized	generalized	ADJ
ejpam-6197	177	7	drygas	drygas	NOUN
ejpam-6197	177	8	equation	equation	NOUN
ejpam-6197	177	9	:	:	PUNCT
ejpam-6197	177	10	f(x+	f(x+	NOUN
ejpam-6197	177	11	y	y	NOUN
ejpam-6197	177	12	)	)	PUNCT
ejpam-6197	178	1	+	+	CCONJ
ejpam-6197	178	2	f(x−	f(x−	PROPN
ejpam-6197	178	3	y	y	NOUN
ejpam-6197	178	4	)	)	PUNCT
ejpam-6197	178	5	=	=	SYM
ejpam-6197	178	6	2f(x	2f(x	PROPN
ejpam-6197	178	7	)	)	PUNCT
ejpam-6197	179	1	+	+	NUM
ejpam-6197	179	2	f(y	f(y	NOUN
ejpam-6197	179	3	)	)	PUNCT
ejpam-6197	179	4	+	+	NUM
ejpam-6197	179	5	f(−y	f(−y	NOUN
ejpam-6197	179	6	)	)	PUNCT
ejpam-6197	179	7	,	,	PUNCT
ejpam-6197	179	8	whose	whose	DET
ejpam-6197	179	9	solution	solution	NOUN
ejpam-6197	179	10	is	be	AUX
ejpam-6197	179	11	called	call	VERB
ejpam-6197	179	12	a	a	DET
ejpam-6197	179	13	drygas	drygas	NOUN
ejpam-6197	179	14	mapping	mapping	NOUN
ejpam-6197	179	15	.	.	PUNCT
ejpam-6197	180	1	the	the	DET
ejpam-6197	180	2	general	general	ADJ
ejpam-6197	180	3	solution	solution	NOUN
ejpam-6197	180	4	of	of	ADP
ejpam-6197	180	5	the	the	DET
ejpam-6197	180	6	above	above	ADJ
ejpam-6197	180	7	functional	functional	ADJ
ejpam-6197	180	8	equation	equation	NOUN
ejpam-6197	180	9	was	be	AUX
ejpam-6197	180	10	given	give	VERB
ejpam-6197	180	11	by	by	ADP
ejpam-6197	180	12	ebanks	ebank	NOUN
ejpam-6197	180	13	,	,	PUNCT
ejpam-6197	180	14	kannappan	kannappan	NOUN
ejpam-6197	180	15	and	and	CCONJ
ejpam-6197	180	16	sahoo	sahoo	NOUN
ejpam-6197	181	1	[	[	X
ejpam-6197	181	2	39	39	NUM
ejpam-6197	181	3	]	]	PUNCT
ejpam-6197	181	4	as	as	ADP
ejpam-6197	181	5	f(x	f(x	PROPN
ejpam-6197	181	6	)	)	PUNCT
ejpam-6197	182	1	=	=	SYM
ejpam-6197	182	2	q(x	q(x	PROPN
ejpam-6197	182	3	)	)	PUNCT
ejpam-6197	183	1	+	+	SYM
ejpam-6197	183	2	a(x	a(x	NOUN
ejpam-6197	183	3	)	)	PUNCT
ejpam-6197	183	4	,	,	PUNCT
ejpam-6197	183	5	where	where	SCONJ
ejpam-6197	183	6	a	a	PRON
ejpam-6197	183	7	is	be	AUX
ejpam-6197	183	8	an	an	DET
ejpam-6197	183	9	additive	additive	ADJ
ejpam-6197	183	10	mapping	mapping	NOUN
ejpam-6197	183	11	and	and	CCONJ
ejpam-6197	183	12	q	q	NOUN
ejpam-6197	183	13	is	be	AUX
ejpam-6197	183	14	a	a	DET
ejpam-6197	183	15	quadratic	quadratic	ADJ
ejpam-6197	183	16	mapping	mapping	NOUN
ejpam-6197	183	17	offering	offer	VERB
ejpam-6197	183	18	insights	insight	NOUN
ejpam-6197	183	19	into	into	ADP
ejpam-6197	183	20	the	the	DET
ejpam-6197	183	21	underlying	underlie	VERB
ejpam-6197	183	22	economic	economic	ADJ
ejpam-6197	183	23	behavior[40].the	behavior[40].the	NOUN
ejpam-6197	183	24	following	following	NOUN
ejpam-6197	183	25	theorem	theorem	ADJ
ejpam-6197	183	26	explores	explore	NOUN
ejpam-6197	183	27	the	the	DET
ejpam-6197	183	28	solutions	solution	NOUN
ejpam-6197	183	29	and	and	CCONJ
ejpam-6197	183	30	stability	stability	NOUN
ejpam-6197	183	31	properties	property	NOUN
ejpam-6197	183	32	of	of	ADP
ejpam-6197	183	33	the	the	DET
ejpam-6197	183	34	generalized	generalized	ADJ
ejpam-6197	183	35	drygas	drygas	NOUN
ejpam-6197	183	36	equation	equation	NOUN
ejpam-6197	183	37	,	,	PUNCT
ejpam-6197	183	38	extending	extend	VERB
ejpam-6197	183	39	its	its	PRON
ejpam-6197	183	40	applications	application	NOUN
ejpam-6197	183	41	to	to	ADP
ejpam-6197	183	42	broader	broad	ADJ
ejpam-6197	183	43	mathematical	mathematical	ADJ
ejpam-6197	183	44	and	and	CCONJ
ejpam-6197	183	45	practical	practical	ADJ
ejpam-6197	183	46	contexts	contexts	NOUN
ejpam-6197	183	47	.	.	PUNCT
ejpam-6197	184	1	theorem	theorem	ADJ
ejpam-6197	184	2	4	4	NUM
ejpam-6197	184	3	.	.	PUNCT
ejpam-6197	185	1	let	let	VERB
ejpam-6197	185	2	ϕ	ϕ	NOUN
ejpam-6197	185	3	:	:	PUNCT
ejpam-6197	185	4	x2	x2	PROPN
ejpam-6197	185	5	→	→	PUNCT
ejpam-6197	186	1	[	[	X
ejpam-6197	186	2	0,∞	0,∞	X
ejpam-6197	186	3	)	)	PUNCT
ejpam-6197	186	4	be	be	VERB
ejpam-6197	186	5	a	a	DET
ejpam-6197	186	6	function	function	NOUN
ejpam-6197	186	7	with	with	ADP
ejpam-6197	186	8	ϕ(0	ϕ(0	PROPN
ejpam-6197	186	9	,	,	PUNCT
ejpam-6197	186	10	0	0	NUM
ejpam-6197	186	11	)	)	PUNCT
ejpam-6197	186	12	=	=	SYM
ejpam-6197	187	1	0	0	NUM
ejpam-6197	187	2	such	such	ADJ
ejpam-6197	187	3	that	that	SCONJ
ejpam-6197	187	4	there	there	PRON
ejpam-6197	187	5	exists	exist	VERB
ejpam-6197	187	6	an	an	DET
ejpam-6197	187	7	l	l	NOUN
ejpam-6197	187	8	<	<	X
ejpam-6197	187	9	1	1	NUM
ejpam-6197	187	10	with	with	ADP
ejpam-6197	187	11	ϕ(ax	ϕ(ax	PROPN
ejpam-6197	187	12	,	,	PUNCT
ejpam-6197	187	13	ay	ay	NOUN
ejpam-6197	187	14	)	)	PUNCT
ejpam-6197	187	15	≤	≤	NOUN
ejpam-6197	187	16	|a|lϕ(x	|a|lϕ(x	PROPN
ejpam-6197	187	17	,	,	PUNCT
ejpam-6197	187	18	y	y	NOUN
ejpam-6197	187	19	)	)	PUNCT
ejpam-6197	187	20	for	for	ADP
ejpam-6197	187	21	all	all	DET
ejpam-6197	187	22	x	x	NOUN
ejpam-6197	187	23	,	,	PUNCT
ejpam-6197	187	24	y	y	PROPN
ejpam-6197	187	25	∈	∈	PROPN
ejpam-6197	187	26	x.	x.	NOUN
ejpam-6197	187	27	suppose	suppose	VERB
ejpam-6197	187	28	that	that	SCONJ
ejpam-6197	187	29	x	x	PRON
ejpam-6197	187	30	is	be	AUX
ejpam-6197	187	31	a	a	DET
ejpam-6197	187	32	linear	linear	ADJ
ejpam-6197	187	33	normed	normed	ADJ
ejpam-6197	187	34	space	space	NOUN
ejpam-6197	187	35	,	,	PUNCT
ejpam-6197	187	36	y	y	PROPN
ejpam-6197	187	37	is	be	AUX
ejpam-6197	187	38	a	a	DET
ejpam-6197	187	39	banach	banach	NOUN
ejpam-6197	187	40	space	space	NOUN
ejpam-6197	187	41	,	,	PUNCT
ejpam-6197	187	42	and	and	CCONJ
ejpam-6197	187	43	ρ	ρ	NOUN
ejpam-6197	187	44	:	:	PUNCT
ejpam-6197	187	45	x	x	X
ejpam-6197	187	46	→	→	X
ejpam-6197	187	47	(	(	PUNCT
ejpam-6197	187	48	0	0	NUM
ejpam-6197	187	49	,	,	PUNCT
ejpam-6197	187	50	1	1	NUM
ejpam-6197	187	51	]	]	PUNCT
ejpam-6197	187	52	is	be	AUX
ejpam-6197	187	53	a	a	DET
ejpam-6197	187	54	function	function	NOUN
ejpam-6197	187	55	.	.	PUNCT
ejpam-6197	188	1	let	let	VERB
ejpam-6197	188	2	f	f	NOUN
ejpam-6197	188	3	:	:	PUNCT
ejpam-6197	188	4	x	x	X
ejpam-6197	188	5	→	→	SYM
ejpam-6197	188	6	y	y	X
ejpam-6197	188	7	be	be	AUX
ejpam-6197	188	8	an	an	DET
ejpam-6197	188	9	odd	odd	ADJ
ejpam-6197	188	10	mapping	mapping	NOUN
ejpam-6197	188	11	satisfying	satisfy	VERB
ejpam-6197	188	12	∥f(ax+	∥f(ax+	ADV
ejpam-6197	188	13	by	by	ADP
ejpam-6197	188	14	)	)	PUNCT
ejpam-6197	189	1	+	+	CCONJ
ejpam-6197	189	2	f(ax−	f(ax−	AUX
ejpam-6197	189	3	by)−af(x)−bf(y)−df(−y)∥	by)−af(x)−bf(y)−df(−y)∥	NOUN
ejpam-6197	189	4	≤	≤	NUM
ejpam-6197	189	5	∥ρ(x)(f(ax+	∥ρ(x)(f(ax+	ADV
ejpam-6197	189	6	by)−	by)−	VERB
ejpam-6197	189	7	f(ax)−	f(ax)−	NOUN
ejpam-6197	189	8	f(by))∥+	f(by))∥+	ADJ
ejpam-6197	189	9	ϕ(x	ϕ(x	PROPN
ejpam-6197	189	10	,	,	PUNCT
ejpam-6197	189	11	y	y	NOUN
ejpam-6197	189	12	)	)	PUNCT
ejpam-6197	189	13	(	(	PUNCT
ejpam-6197	189	14	10	10	NUM
ejpam-6197	189	15	)	)	PUNCT
ejpam-6197	189	16	for	for	ADP
ejpam-6197	189	17	all	all	DET
ejpam-6197	189	18	x	x	NOUN
ejpam-6197	189	19	,	,	PUNCT
ejpam-6197	189	20	y	y	PROPN
ejpam-6197	189	21	∈	∈	PROPN
ejpam-6197	189	22	x.	x.	NOUN
ejpam-6197	190	1	if	if	SCONJ
ejpam-6197	190	2	l	l	PROPN
ejpam-6197	190	3	,	,	PUNCT
ejpam-6197	190	4	a	a	PRON
ejpam-6197	190	5	and	and	CCONJ
ejpam-6197	190	6	a	a	DET
ejpam-6197	190	7	satisfy	satisfy	NOUN
ejpam-6197	190	8	2|a|l	2|a|l	NUM
ejpam-6197	190	9	|a|	|a|	NOUN
ejpam-6197	190	10	<	<	X
ejpam-6197	190	11	1	1	NUM
ejpam-6197	190	12	,	,	PUNCT
ejpam-6197	190	13	then	then	ADV
ejpam-6197	190	14	there	there	PRON
ejpam-6197	190	15	exists	exist	VERB
ejpam-6197	190	16	a	a	DET
ejpam-6197	190	17	unique	unique	ADJ
ejpam-6197	190	18	additive	additive	ADJ
ejpam-6197	190	19	mapping	mapping	NOUN
ejpam-6197	190	20	k	k	NOUN
ejpam-6197	190	21	:	:	PUNCT
ejpam-6197	190	22	x	x	X
ejpam-6197	190	23	→	→	PUNCT
ejpam-6197	190	24	y	y	PROPN
ejpam-6197	190	25	such	such	ADJ
ejpam-6197	190	26	that	that	SCONJ
ejpam-6197	190	27	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	190	28	k(x)∥	k(x)∥	VERB
ejpam-6197	190	29	≤	≤	NUM
ejpam-6197	190	30	|a|	|a|	PROPN
ejpam-6197	190	31	|a|	|a|	PROPN
ejpam-6197	190	32	−	−	PROPN
ejpam-6197	190	33	2l|a|	2l|a|	NUM
ejpam-6197	190	34	ϕ(x	ϕ(x	NOUN
ejpam-6197	190	35	,	,	PUNCT
ejpam-6197	190	36	0	0	NUM
ejpam-6197	190	37	)	)	PUNCT
ejpam-6197	190	38	for	for	ADP
ejpam-6197	190	39	all	all	DET
ejpam-6197	190	40	x	x	SYM
ejpam-6197	190	41	∈	∈	ADJ
ejpam-6197	190	42	x.	x.	NOUN
ejpam-6197	190	43	proof	proof	NOUN
ejpam-6197	190	44	.	.	PUNCT
ejpam-6197	191	1	letting	let	VERB
ejpam-6197	191	2	y	y	PROPN
ejpam-6197	191	3	=	=	PUNCT
ejpam-6197	191	4	0	0	NUM
ejpam-6197	191	5	in	in	ADP
ejpam-6197	191	6	(	(	PUNCT
ejpam-6197	191	7	10	10	NUM
ejpam-6197	191	8	)	)	PUNCT
ejpam-6197	191	9	,	,	PUNCT
ejpam-6197	191	10	we	we	PRON
ejpam-6197	191	11	get	get	VERB
ejpam-6197	191	12	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	191	13	2	2	NUM
ejpam-6197	191	14	a	a	DET
ejpam-6197	191	15	f(ax)∥	f(ax)∥	NOUN
ejpam-6197	191	16	≤	≤	ADV
ejpam-6197	191	17	1	1	NUM
ejpam-6197	191	18	|a|	|a|	PROPN
ejpam-6197	191	19	ϕ(x	ϕ(x	PROPN
ejpam-6197	191	20	,	,	PUNCT
ejpam-6197	191	21	0	0	NUM
ejpam-6197	191	22	)	)	PUNCT
ejpam-6197	191	23	,	,	PUNCT
ejpam-6197	191	24	x	x	AUX
ejpam-6197	191	25	∈	∈	NOUN
ejpam-6197	191	26	x.	x.	NOUN
ejpam-6197	191	27	g.	g.	PROPN
ejpam-6197	191	28	lyu	lyu	PROPN
ejpam-6197	191	29	et	et	PROPN
ejpam-6197	191	30	al	al	PROPN
ejpam-6197	191	31	.	.	PUNCT
ejpam-6197	191	32	/	/	SYM
ejpam-6197	191	33	eur	eur	PROPN
ejpam-6197	191	34	.	.	PUNCT
ejpam-6197	192	1	j.	j.	PROPN
ejpam-6197	192	2	pure	pure	PROPN
ejpam-6197	192	3	appl	appl	PROPN
ejpam-6197	192	4	.	.	PROPN
ejpam-6197	192	5	math	math	PROPN
ejpam-6197	192	6	,	,	PUNCT
ejpam-6197	192	7	18	18	NUM
ejpam-6197	192	8	(	(	PUNCT
ejpam-6197	192	9	3	3	NUM
ejpam-6197	192	10	)	)	PUNCT
ejpam-6197	192	11	(	(	PUNCT
ejpam-6197	192	12	2025	2025	NUM
ejpam-6197	192	13	)	)	PUNCT
ejpam-6197	192	14	,	,	PUNCT
ejpam-6197	192	15	6197	6197	NUM
ejpam-6197	192	16	11	11	NUM
ejpam-6197	192	17	of	of	ADP
ejpam-6197	192	18	17	17	NUM
ejpam-6197	192	19	consider	consider	VERB
ejpam-6197	192	20	the	the	DET
ejpam-6197	192	21	mapping	mapping	NOUN
ejpam-6197	192	22	j	j	NOUN
ejpam-6197	192	23	:	:	PUNCT
ejpam-6197	192	24	y	y	PROPN
ejpam-6197	192	25	x	x	PUNCT
ejpam-6197	192	26	→	→	SYM
ejpam-6197	192	27	y	y	PROPN
ejpam-6197	192	28	x	x	INTJ
ejpam-6197	192	29	such	such	ADJ
ejpam-6197	192	30	that	that	DET
ejpam-6197	192	31	jξ(x	jξ(x	NOUN
ejpam-6197	192	32	)	)	PUNCT
ejpam-6197	192	33	=	=	SYM
ejpam-6197	193	1	2	2	NUM
ejpam-6197	193	2	a	a	DET
ejpam-6197	193	3	ξ(ax	ξ(ax	PROPN
ejpam-6197	193	4	)	)	PUNCT
ejpam-6197	193	5	,	,	PUNCT
ejpam-6197	194	1	x	x	PUNCT
ejpam-6197	194	2	∈	∈	PROPN
ejpam-6197	194	3	x	x	X
ejpam-6197	194	4	,	,	PUNCT
ejpam-6197	194	5	ξ	ξ	PROPN
ejpam-6197	194	6	∈	∈	PROPN
ejpam-6197	194	7	y	y	PROPN
ejpam-6197	194	8	x	x	X
ejpam-6197	194	9	.	.	PUNCT
ejpam-6197	195	1	then	then	ADV
ejpam-6197	195	2	we	we	PRON
ejpam-6197	195	3	get	get	VERB
ejpam-6197	195	4	∥jf(x)−	∥jf(x)−	NOUN
ejpam-6197	195	5	f(x)∥	f(x)∥	NOUN
ejpam-6197	195	6	≤	≤	NUM
ejpam-6197	195	7	1	1	NUM
ejpam-6197	195	8	|a|	|a|	PROPN
ejpam-6197	195	9	ϕ(x	ϕ(x	PROPN
ejpam-6197	195	10	,	,	PUNCT
ejpam-6197	195	11	0	0	NUM
ejpam-6197	195	12	)	)	PUNCT
ejpam-6197	195	13	,	,	PUNCT
ejpam-6197	195	14	x	x	PUNCT
ejpam-6197	195	15	∈	∈	NOUN
ejpam-6197	195	16	x.	x.	NOUN
ejpam-6197	195	17	for	for	ADP
ejpam-6197	195	18	every	every	DET
ejpam-6197	195	19	ξ	ξ	PROPN
ejpam-6197	195	20	,	,	PUNCT
ejpam-6197	195	21	µ	µ	X
ejpam-6197	195	22	∈	∈	X
ejpam-6197	195	23	y	y	PROPN
ejpam-6197	195	24	x	x	PROPN
ejpam-6197	195	25	,	,	PUNCT
ejpam-6197	195	26	∥jg(x)−	∥jg(x)−	PROPN
ejpam-6197	195	27	jh(x)∥	jh(x)∥	NOUN
ejpam-6197	195	28	=	=	SYM
ejpam-6197	195	29	2	2	NUM
ejpam-6197	195	30	|a|	|a|	PROPN
ejpam-6197	195	31	∥g(ax)−	∥g(ax)−	PROPN
ejpam-6197	195	32	h(ax)∥.	h(ax)∥.	NOUN
ejpam-6197	195	33	thus	thus	ADV
ejpam-6197	195	34	j	j	PROPN
ejpam-6197	195	35	satisfies	satisfy	VERB
ejpam-6197	195	36	the	the	DET
ejpam-6197	195	37	inequality	inequality	NOUN
ejpam-6197	195	38	(	(	PUNCT
ejpam-6197	195	39	4	4	NUM
ejpam-6197	195	40	)	)	PUNCT
ejpam-6197	195	41	with	with	ADP
ejpam-6197	195	42	f1(x	f1(x	NOUN
ejpam-6197	195	43	)	)	PUNCT
ejpam-6197	195	44	=	=	SYM
ejpam-6197	195	45	2ax	2ax	NOUN
ejpam-6197	195	46	and	and	CCONJ
ejpam-6197	195	47	l1(x	l1(x	NOUN
ejpam-6197	195	48	)	)	PUNCT
ejpam-6197	196	1	=	=	SYM
ejpam-6197	196	2	2	2	NUM
ejpam-6197	196	3	|a|	|a|	NOUN
ejpam-6197	196	4	.	.	PUNCT
ejpam-6197	197	1	next	next	ADV
ejpam-6197	197	2	,	,	PUNCT
ejpam-6197	197	3	we	we	PRON
ejpam-6197	197	4	define	define	VERB
ejpam-6197	197	5	λ	λ	X
ejpam-6197	197	6	:	:	PUNCT
ejpam-6197	197	7	rx	rx	VERB
ejpam-6197	197	8	+	+	X
ejpam-6197	197	9	→	→	PUNCT
ejpam-6197	197	10	rx	rx	X
ejpam-6197	197	11	+	+	CCONJ
ejpam-6197	197	12	by	by	ADP
ejpam-6197	197	13	λϕ(x	λϕ(x	NOUN
ejpam-6197	197	14	,	,	PUNCT
ejpam-6197	197	15	y	y	NOUN
ejpam-6197	197	16	)	)	PUNCT
ejpam-6197	197	17	=	=	SYM
ejpam-6197	197	18	2	2	NUM
ejpam-6197	197	19	|a|	|a|	PROPN
ejpam-6197	197	20	ϕ(ax	ϕ(ax	PROPN
ejpam-6197	197	21	,	,	PUNCT
ejpam-6197	197	22	ay	ay	PROPN
ejpam-6197	197	23	)	)	PUNCT
ejpam-6197	197	24	<	<	X
ejpam-6197	197	25	2|a|l	2|a|l	NUM
ejpam-6197	197	26	|a|	|a|	PROPN
ejpam-6197	197	27	ϕ(x	ϕ(x	PROPN
ejpam-6197	197	28	,	,	PUNCT
ejpam-6197	197	29	y	y	PROPN
ejpam-6197	197	30	)	)	PUNCT
ejpam-6197	197	31	,	,	PUNCT
ejpam-6197	197	32	x	x	PUNCT
ejpam-6197	197	33	∈	∈	NOUN
ejpam-6197	197	34	x,ϕ	x,ϕ	PRON
ejpam-6197	197	35	∈	∈	PROPN
ejpam-6197	197	36	rx	rx	VERB
ejpam-6197	197	37	+	+	CCONJ
ejpam-6197	197	38	.	.	PUNCT
ejpam-6197	198	1	since	since	SCONJ
ejpam-6197	198	2	λ	λ	PROPN
ejpam-6197	198	3	is	be	AUX
ejpam-6197	198	4	linear	linear	ADJ
ejpam-6197	198	5	,	,	PUNCT
ejpam-6197	198	6	we	we	PRON
ejpam-6197	198	7	get	get	VERB
ejpam-6197	198	8	λnϕ(x	λnϕ(x	PRON
ejpam-6197	198	9	,	,	PUNCT
ejpam-6197	198	10	y	y	NOUN
ejpam-6197	198	11	)	)	PUNCT
ejpam-6197	198	12	=	=	SYM
ejpam-6197	199	1	(	(	PUNCT
ejpam-6197	199	2	2|a|l	2|a|l	NUM
ejpam-6197	199	3	|a|	|a|	NOUN
ejpam-6197	199	4	)	)	PUNCT
ejpam-6197	200	1	n	n	PRON
ejpam-6197	200	2	ϕ(x	ϕ(x	PROPN
ejpam-6197	200	3	,	,	PUNCT
ejpam-6197	200	4	y	y	NOUN
ejpam-6197	200	5	)	)	PUNCT
ejpam-6197	200	6	and	and	CCONJ
ejpam-6197	200	7	ε∗(x	ε∗(x	PROPN
ejpam-6197	200	8	,	,	PUNCT
ejpam-6197	200	9	y	y	NOUN
ejpam-6197	200	10	)	)	PUNCT
ejpam-6197	201	1	=	=	PUNCT
ejpam-6197	202	1	∞∑	∞∑	NUM
ejpam-6197	202	2	n=0	n=0	NUM
ejpam-6197	202	3	(	(	PUNCT
ejpam-6197	202	4	2|a|l	2|a|l	NUM
ejpam-6197	202	5	|a|	|a|	NOUN
ejpam-6197	202	6	)	)	PUNCT
ejpam-6197	202	7	n	n	PRON
ejpam-6197	202	8	ϕ(x	ϕ(x	PROPN
ejpam-6197	202	9	,	,	PUNCT
ejpam-6197	202	10	y	y	NOUN
ejpam-6197	202	11	)	)	PUNCT
ejpam-6197	202	12	=	=	SYM
ejpam-6197	202	13	|a|	|a|	PROPN
ejpam-6197	202	14	|a|	|a|	PROPN
ejpam-6197	202	15	−	−	PROPN
ejpam-6197	202	16	2l|a|	2l|a|	NUM
ejpam-6197	202	17	ϕ(x	ϕ(x	PROPN
ejpam-6197	202	18	,	,	PUNCT
ejpam-6197	202	19	y	y	NOUN
ejpam-6197	202	20	)	)	PUNCT
ejpam-6197	202	21	.	.	PUNCT
ejpam-6197	203	1	by	by	ADP
ejpam-6197	203	2	theorem	theorem	NOUN
ejpam-6197	203	3	1	1	NUM
ejpam-6197	203	4	,	,	PUNCT
ejpam-6197	203	5	there	there	PRON
ejpam-6197	203	6	exists	exist	VERB
ejpam-6197	203	7	a	a	DET
ejpam-6197	203	8	mapping	mapping	NOUN
ejpam-6197	203	9	k	k	NOUN
ejpam-6197	203	10	:	:	PUNCT
ejpam-6197	203	11	x	x	X
ejpam-6197	203	12	→	→	PUNCT
ejpam-6197	203	13	y	y	PROPN
ejpam-6197	203	14	such	such	ADJ
ejpam-6197	203	15	that	that	SCONJ
ejpam-6197	203	16	k(x	k(x	PROPN
ejpam-6197	203	17	)	)	PUNCT
ejpam-6197	204	1	=	=	PROPN
ejpam-6197	204	2	lim	lim	PROPN
ejpam-6197	204	3	n→∞	n→∞	NUM
ejpam-6197	204	4	jnf(x	jnf(x	PROPN
ejpam-6197	204	5	)	)	PUNCT
ejpam-6197	204	6	,	,	PUNCT
ejpam-6197	204	7	k(x	k(x	PROPN
ejpam-6197	204	8	)	)	PUNCT
ejpam-6197	204	9	=	=	SYM
ejpam-6197	204	10	2	2	NUM
ejpam-6197	204	11	a	a	DET
ejpam-6197	204	12	k(ax	k(ax	NOUN
ejpam-6197	204	13	)	)	PUNCT
ejpam-6197	204	14	,	,	PUNCT
ejpam-6197	204	15	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	204	16	k(x)∥	k(x)∥	VERB
ejpam-6197	204	17	≤	≤	NUM
ejpam-6197	204	18	|a|	|a|	PROPN
ejpam-6197	204	19	|a|	|a|	PROPN
ejpam-6197	204	20	−	−	PROPN
ejpam-6197	204	21	2l|a|	2l|a|	NUM
ejpam-6197	204	22	ϕ(x	ϕ(x	NOUN
ejpam-6197	204	23	,	,	PUNCT
ejpam-6197	204	24	0	0	NUM
ejpam-6197	204	25	)	)	PUNCT
ejpam-6197	204	26	.	.	PUNCT
ejpam-6197	205	1	next	next	ADV
ejpam-6197	205	2	,	,	PUNCT
ejpam-6197	205	3	we	we	PRON
ejpam-6197	205	4	prove	prove	VERB
ejpam-6197	205	5	that	that	SCONJ
ejpam-6197	205	6	k	k	PROPN
ejpam-6197	205	7	satisfies	satisfy	VERB
ejpam-6197	205	8	the	the	DET
ejpam-6197	205	9	generalized	generalized	ADJ
ejpam-6197	205	10	drygas	drygas	NOUN
ejpam-6197	205	11	equation	equation	NOUN
ejpam-6197	205	12	.	.	PUNCT
ejpam-6197	206	1	replacing	replace	VERB
ejpam-6197	206	2	x	x	PUNCT
ejpam-6197	206	3	by	by	ADP
ejpam-6197	206	4	x	x	X
ejpam-6197	206	5	a	a	PROPN
ejpam-6197	206	6	and	and	CCONJ
ejpam-6197	206	7	y	y	PROPN
ejpam-6197	206	8	by	by	ADP
ejpam-6197	206	9	y	y	PROPN
ejpam-6197	206	10	b	b	PROPN
ejpam-6197	206	11	in	in	ADP
ejpam-6197	206	12	(	(	PUNCT
ejpam-6197	206	13	10	10	NUM
ejpam-6197	206	14	)	)	PUNCT
ejpam-6197	206	15	,	,	PUNCT
ejpam-6197	206	16	we	we	PRON
ejpam-6197	206	17	have∥∥∥f(x+	have∥∥∥f(x+	VERB
ejpam-6197	206	18	y	y	PROPN
ejpam-6197	206	19	)	)	PUNCT
ejpam-6197	207	1	+	+	CCONJ
ejpam-6197	207	2	f(x−	f(x−	PRON
ejpam-6197	207	3	y)−af	y)−af	PROPN
ejpam-6197	207	4	(	(	PUNCT
ejpam-6197	207	5	x	x	NOUN
ejpam-6197	207	6	a	a	PRON
ejpam-6197	207	7	)	)	PUNCT
ejpam-6197	207	8	−bf	−bf	PROPN
ejpam-6197	207	9	(	(	PUNCT
ejpam-6197	207	10	y	y	PROPN
ejpam-6197	207	11	b	b	PROPN
ejpam-6197	207	12	)	)	PUNCT
ejpam-6197	207	13	−df	−df	PROPN
ejpam-6197	207	14	(	(	PUNCT
ejpam-6197	207	15	−y	−y	PROPN
ejpam-6197	207	16	b	b	NOUN
ejpam-6197	207	17	)	)	PUNCT
ejpam-6197	207	18	∥∥∥	∥∥∥	PROPN
ejpam-6197	207	19	≤	≤	NUM
ejpam-6197	207	20	ρ(x	ρ(x	NOUN
ejpam-6197	207	21	)	)	PUNCT
ejpam-6197	208	1	∥f	∥f	PROPN
ejpam-6197	208	2	(	(	PUNCT
ejpam-6197	208	3	x+	x+	PROPN
ejpam-6197	208	4	y)−	y)−	PROPN
ejpam-6197	208	5	f	f	X
ejpam-6197	208	6	(	(	PUNCT
ejpam-6197	208	7	x)−	x)−	PROPN
ejpam-6197	208	8	f	f	PROPN
ejpam-6197	208	9	(	(	PUNCT
ejpam-6197	208	10	y)∥+	y)∥+	NOUN
ejpam-6197	208	11	ϕ	ϕ	PROPN
ejpam-6197	208	12	(	(	PUNCT
ejpam-6197	208	13	x	x	X
ejpam-6197	208	14	a	a	PRON
ejpam-6197	208	15	,	,	PUNCT
ejpam-6197	208	16	y	y	PROPN
ejpam-6197	208	17	b	b	PROPN
ejpam-6197	208	18	)	)	PUNCT
ejpam-6197	209	1	g.	g.	PROPN
ejpam-6197	209	2	lyu	lyu	PROPN
ejpam-6197	209	3	et	et	PROPN
ejpam-6197	209	4	al	al	PROPN
ejpam-6197	209	5	.	.	PUNCT
ejpam-6197	209	6	/	/	SYM
ejpam-6197	209	7	eur	eur	PROPN
ejpam-6197	209	8	.	.	PUNCT
ejpam-6197	210	1	j.	j.	PROPN
ejpam-6197	210	2	pure	pure	PROPN
ejpam-6197	210	3	appl	appl	PROPN
ejpam-6197	210	4	.	.	PROPN
ejpam-6197	210	5	math	math	PROPN
ejpam-6197	210	6	,	,	PUNCT
ejpam-6197	210	7	18	18	NUM
ejpam-6197	210	8	(	(	PUNCT
ejpam-6197	210	9	3	3	NUM
ejpam-6197	210	10	)	)	PUNCT
ejpam-6197	210	11	(	(	PUNCT
ejpam-6197	210	12	2025	2025	NUM
ejpam-6197	210	13	)	)	PUNCT
ejpam-6197	210	14	,	,	PUNCT
ejpam-6197	210	15	6197	6197	NUM
ejpam-6197	210	16	12	12	NUM
ejpam-6197	210	17	of	of	ADP
ejpam-6197	210	18	17	17	NUM
ejpam-6197	210	19	for	for	ADP
ejpam-6197	210	20	all	all	DET
ejpam-6197	210	21	x	x	NOUN
ejpam-6197	210	22	,	,	PUNCT
ejpam-6197	210	23	y	y	PROPN
ejpam-6197	210	24	∈	∈	PROPN
ejpam-6197	210	25	x.	x.	NOUN
ejpam-6197	211	1	then	then	ADV
ejpam-6197	211	2	,	,	PUNCT
ejpam-6197	211	3	from	from	ADP
ejpam-6197	211	4	the	the	DET
ejpam-6197	211	5	above	above	ADJ
ejpam-6197	211	6	inequality	inequality	NOUN
ejpam-6197	211	7	,	,	PUNCT
ejpam-6197	211	8	we	we	PRON
ejpam-6197	211	9	get	get	VERB
ejpam-6197	211	10	∥f(x+	∥f(x+	NOUN
ejpam-6197	211	11	y	y	NOUN
ejpam-6197	211	12	)	)	PUNCT
ejpam-6197	212	1	+	+	CCONJ
ejpam-6197	212	2	f(x−	f(x−	ADP
ejpam-6197	212	3	y)−	y)−	PROPN
ejpam-6197	212	4	2f(x)∥	2f(x)∥	NUM
ejpam-6197	212	5	≤	≤	ADJ
ejpam-6197	212	6	ρ(x	ρ(x	NOUN
ejpam-6197	212	7	)	)	PUNCT
ejpam-6197	213	1	∥f	∥f	PROPN
ejpam-6197	213	2	(	(	PUNCT
ejpam-6197	213	3	x+	x+	PROPN
ejpam-6197	213	4	y)−	y)−	PROPN
ejpam-6197	213	5	f	f	X
ejpam-6197	213	6	(	(	PUNCT
ejpam-6197	213	7	x)−	x)−	PROPN
ejpam-6197	213	8	f	f	PROPN
ejpam-6197	213	9	(	(	PUNCT
ejpam-6197	213	10	y)∥+	y)∥+	NOUN
ejpam-6197	213	11	ϕ	ϕ	PROPN
ejpam-6197	213	12	(	(	PUNCT
ejpam-6197	213	13	x	x	X
ejpam-6197	213	14	a	a	PRON
ejpam-6197	213	15	,	,	PUNCT
ejpam-6197	213	16	y	y	PROPN
ejpam-6197	213	17	b	b	PROPN
ejpam-6197	213	18	)	)	PUNCT
ejpam-6197	214	1	+	+	CCONJ
ejpam-6197	214	2	ϕ	ϕ	X
ejpam-6197	214	3	(	(	PUNCT
ejpam-6197	214	4	x	x	X
ejpam-6197	214	5	a	a	PRON
ejpam-6197	214	6	,	,	PUNCT
ejpam-6197	214	7	0	0	NUM
ejpam-6197	214	8	)	)	PUNCT
ejpam-6197	215	1	+	+	CCONJ
ejpam-6197	215	2	ϕ	ϕ	X
ejpam-6197	215	3	(	(	PUNCT
ejpam-6197	215	4	0	0	NUM
ejpam-6197	215	5	,	,	PUNCT
ejpam-6197	215	6	y	y	PROPN
ejpam-6197	215	7	b	b	PROPN
ejpam-6197	215	8	)	)	PUNCT
ejpam-6197	215	9	.	.	PUNCT
ejpam-6197	216	1	hence	hence	ADV
ejpam-6197	216	2	∥jf(x+	∥jf(x+	ADV
ejpam-6197	216	3	y	y	X
ejpam-6197	216	4	)	)	PUNCT
ejpam-6197	217	1	+	+	NUM
ejpam-6197	217	2	jf(x−	jf(x−	X
ejpam-6197	217	3	y)−	y)−	PROPN
ejpam-6197	217	4	2jf(x)−	2jf(x)−	PROPN
ejpam-6197	217	5	jf(y)−	jf(y)−	PROPN
ejpam-6197	217	6	jf(−y)∥	jf(−y)∥	X
ejpam-6197	217	7	=	=	SYM
ejpam-6197	217	8	2	2	NUM
ejpam-6197	217	9	|a|	|a|	PROPN
ejpam-6197	217	10	∥f	∥f	PROPN
ejpam-6197	217	11	(	(	PUNCT
ejpam-6197	217	12	a(x+	a(x+	ADP
ejpam-6197	217	13	y	y	NOUN
ejpam-6197	217	14	)	)	PUNCT
ejpam-6197	217	15	)	)	PUNCT
ejpam-6197	218	1	+	+	CCONJ
ejpam-6197	218	2	f	f	X
ejpam-6197	218	3	(	(	PUNCT
ejpam-6197	218	4	a(x−	a(x−	NOUN
ejpam-6197	218	5	y))−	y))−	NOUN
ejpam-6197	218	6	2f	2f	NUM
ejpam-6197	218	7	(	(	PUNCT
ejpam-6197	218	8	ax)−	ax)−	PROPN
ejpam-6197	218	9	f	f	PROPN
ejpam-6197	218	10	(	(	PUNCT
ejpam-6197	218	11	ay)−	ay)−	PROPN
ejpam-6197	218	12	f	f	X
ejpam-6197	218	13	(	(	PUNCT
ejpam-6197	218	14	−ay)∥	−ay)∥	NOUN
ejpam-6197	218	15	≤	≤	NOUN
ejpam-6197	218	16	2	2	NUM
ejpam-6197	218	17	|a|	|a|	NOUN
ejpam-6197	218	18	ρ(x)∥f(x+	ρ(x)∥f(x+	ADP
ejpam-6197	218	19	a	a	DET
ejpam-6197	218	20	b	b	NOUN
ejpam-6197	218	21	y)−	y)−	PROPN
ejpam-6197	218	22	f(x)−	f(x)−	PROPN
ejpam-6197	218	23	f	f	X
ejpam-6197	218	24	(	(	PUNCT
ejpam-6197	218	25	a	a	DET
ejpam-6197	218	26	b	b	NOUN
ejpam-6197	218	27	y)∥+	y)∥+	ADJ
ejpam-6197	218	28	2	2	NUM
ejpam-6197	218	29	|a|	|a|	PROPN
ejpam-6197	218	30	ϕ(x	ϕ(x	PROPN
ejpam-6197	218	31	,	,	PUNCT
ejpam-6197	218	32	y	y	NOUN
ejpam-6197	218	33	)	)	PUNCT
ejpam-6197	219	1	+	+	CCONJ
ejpam-6197	219	2	2	2	NUM
ejpam-6197	219	3	|a|	|a|	NOUN
ejpam-6197	219	4	ϕ(x	ϕ(x	NOUN
ejpam-6197	219	5	,	,	PUNCT
ejpam-6197	219	6	0	0	NUM
ejpam-6197	219	7	)	)	PUNCT
ejpam-6197	219	8	+	+	CCONJ
ejpam-6197	219	9	2	2	NUM
ejpam-6197	219	10	|a|	|a|	NUM
ejpam-6197	219	11	ϕ(0	ϕ(0	PROPN
ejpam-6197	219	12	,	,	PUNCT
ejpam-6197	219	13	y	y	PROPN
ejpam-6197	219	14	)	)	PUNCT
ejpam-6197	219	15	and	and	CCONJ
ejpam-6197	219	16	so	so	ADV
ejpam-6197	219	17	∥k(x+	∥k(x+	ADV
ejpam-6197	219	18	y	y	PROPN
ejpam-6197	219	19	)	)	PUNCT
ejpam-6197	219	20	+	+	CCONJ
ejpam-6197	219	21	k(x−	k(x−	PROPN
ejpam-6197	219	22	y)−	y)−	PROPN
ejpam-6197	219	23	k(x)−	k(x)−	PROPN
ejpam-6197	219	24	k(y)−	k(y)−	PROPN
ejpam-6197	219	25	k(−y)∥	k(−y)∥	NOUN
ejpam-6197	219	26	=	=	PROPN
ejpam-6197	219	27	lim	lim	PROPN
ejpam-6197	219	28	n→∞	n→∞	NUM
ejpam-6197	219	29	∥jnf(x+	∥jnf(x+	NUM
ejpam-6197	219	30	y	y	X
ejpam-6197	219	31	)	)	PUNCT
ejpam-6197	220	1	+	+	NUM
ejpam-6197	220	2	jnf(x−	jnf(x−	CCONJ
ejpam-6197	220	3	y)−	y)−	PROPN
ejpam-6197	220	4	2jnf(x)−	2jnf(x)−	NOUN
ejpam-6197	220	5	jnf(y)−	jnf(y)−	PROPN
ejpam-6197	220	6	jnf(−y)∥	jnf(−y)∥	NOUN
ejpam-6197	220	7	≤	≤	PROPN
ejpam-6197	220	8	lim	lim	PROPN
ejpam-6197	220	9	n→∞	n→∞	PRON
ejpam-6197	220	10	∥jnf(x+	∥jnf(x+	PROPN
ejpam-6197	220	11	a	a	DET
ejpam-6197	220	12	b	b	NOUN
ejpam-6197	220	13	y)−	y)−	PROPN
ejpam-6197	220	14	jnf(x)−	jnf(x)−	NOUN
ejpam-6197	220	15	jnf	jnf	PROPN
ejpam-6197	220	16	(	(	PUNCT
ejpam-6197	220	17	a	a	DET
ejpam-6197	220	18	b	b	NOUN
ejpam-6197	220	19	y)∥	y)∥	PUNCT
ejpam-6197	221	1	+	+	CCONJ
ejpam-6197	221	2	lim	lim	PROPN
ejpam-6197	221	3	n→∞	n→∞	X
ejpam-6197	221	4	(	(	PUNCT
ejpam-6197	221	5	jnϕ(x	jnϕ(x	PROPN
ejpam-6197	221	6	,	,	PUNCT
ejpam-6197	221	7	y	y	PROPN
ejpam-6197	221	8	)	)	PUNCT
ejpam-6197	221	9	+	+	CCONJ
ejpam-6197	221	10	jnϕ(x	jnϕ(x	PROPN
ejpam-6197	221	11	,	,	PUNCT
ejpam-6197	221	12	0	0	NUM
ejpam-6197	221	13	)	)	PUNCT
ejpam-6197	222	1	+	+	CCONJ
ejpam-6197	222	2	jnϕ(0	jnϕ(0	ADJ
ejpam-6197	222	3	,	,	PUNCT
ejpam-6197	222	4	y	y	NOUN
ejpam-6197	222	5	)	)	PUNCT
ejpam-6197	222	6	)	)	PUNCT
ejpam-6197	222	7	for	for	ADP
ejpam-6197	222	8	all	all	DET
ejpam-6197	222	9	n	n	PRON
ejpam-6197	222	10	∈	∈	PROPN
ejpam-6197	222	11	n0	n0	NOUN
ejpam-6197	222	12	and	and	CCONJ
ejpam-6197	222	13	x	x	NOUN
ejpam-6197	222	14	,	,	PUNCT
ejpam-6197	222	15	y	y	PROPN
ejpam-6197	222	16	∈	∈	PROPN
ejpam-6197	222	17	x.	x.	NOUN
ejpam-6197	222	18	letting	let	VERB
ejpam-6197	222	19	n→	n→	PROPN
ejpam-6197	222	20	∞	∞	PROPN
ejpam-6197	222	21	,	,	PUNCT
ejpam-6197	222	22	we	we	PRON
ejpam-6197	222	23	obtain	obtain	VERB
ejpam-6197	222	24	∥k(x+	∥k(x+	ADV
ejpam-6197	222	25	y	y	PROPN
ejpam-6197	222	26	)	)	PUNCT
ejpam-6197	223	1	+	+	CCONJ
ejpam-6197	223	2	k(x−	k(x−	PROPN
ejpam-6197	223	3	y)−	y)−	PROPN
ejpam-6197	223	4	2k(x)∥	2k(x)∥	PROPN
ejpam-6197	223	5	≤	≤	ADJ
ejpam-6197	223	6	ρ(x)∥k(x+	ρ(x)∥k(x+	NOUN
ejpam-6197	224	1	y)−	y)−	PROPN
ejpam-6197	224	2	k(x)−	k(x)−	PROPN
ejpam-6197	224	3	k(y)∥	k(y)∥	NOUN
ejpam-6197	224	4	(	(	PUNCT
ejpam-6197	224	5	11	11	NUM
ejpam-6197	224	6	)	)	PUNCT
ejpam-6197	224	7	for	for	ADP
ejpam-6197	224	8	all	all	DET
ejpam-6197	224	9	x	x	NOUN
ejpam-6197	224	10	,	,	PUNCT
ejpam-6197	224	11	y	y	PROPN
ejpam-6197	224	12	∈	∈	PROPN
ejpam-6197	224	13	x.	x.	NOUN
ejpam-6197	224	14	letting	let	VERB
ejpam-6197	224	15	y	y	PROPN
ejpam-6197	224	16	=	=	PUNCT
ejpam-6197	224	17	x	x	X
ejpam-6197	224	18	in	in	ADP
ejpam-6197	224	19	(	(	PUNCT
ejpam-6197	224	20	11	11	NUM
ejpam-6197	224	21	)	)	PUNCT
ejpam-6197	224	22	,	,	PUNCT
ejpam-6197	224	23	we	we	PRON
ejpam-6197	224	24	get	get	VERB
ejpam-6197	224	25	∥k(2x)−	∥k(2x)−	PROPN
ejpam-6197	224	26	2k(x)∥	2k(x)∥	PART
ejpam-6197	224	27	≤	≤	ADJ
ejpam-6197	225	1	ρ(x)∥k(2x)−	ρ(x)∥k(2x)−	PROPN
ejpam-6197	225	2	2k(x)∥	2k(x)∥	PROPN
ejpam-6197	225	3	for	for	ADP
ejpam-6197	225	4	all	all	DET
ejpam-6197	225	5	x	x	SYM
ejpam-6197	225	6	∈	∈	NOUN
ejpam-6197	225	7	x.	x.	NOUN
ejpam-6197	225	8	moreover	moreover	ADV
ejpam-6197	225	9	,	,	PUNCT
ejpam-6197	225	10	k(2x	k(2x	PROPN
ejpam-6197	225	11	)	)	PUNCT
ejpam-6197	225	12	=	=	SYM
ejpam-6197	225	13	2k(x	2k(x	PROPN
ejpam-6197	225	14	)	)	PUNCT
ejpam-6197	225	15	for	for	ADP
ejpam-6197	225	16	all	all	PRON
ejpam-6197	225	17	x	x	SYM
ejpam-6197	225	18	∈	∈	NOUN
ejpam-6197	225	19	x.	x.	NOUN
ejpam-6197	225	20	letting	let	VERB
ejpam-6197	225	21	l	l	NOUN
ejpam-6197	225	22	=	=	PUNCT
ejpam-6197	225	23	x+	x+	PUNCT
ejpam-6197	225	24	y	y	PROPN
ejpam-6197	225	25	,	,	PUNCT
ejpam-6197	225	26	m	m	VERB
ejpam-6197	225	27	=	=	SYM
ejpam-6197	225	28	x−	x−	PROPN
ejpam-6197	225	29	y	y	PROPN
ejpam-6197	225	30	in	in	ADP
ejpam-6197	225	31	(	(	PUNCT
ejpam-6197	225	32	11	11	NUM
ejpam-6197	225	33	)	)	PUNCT
ejpam-6197	225	34	,	,	PUNCT
ejpam-6197	226	1	we	we	PRON
ejpam-6197	226	2	get∥∥∥∥k(l	get∥∥∥∥k(l	PROPN
ejpam-6197	226	3	)	)	PUNCT
ejpam-6197	226	4	+	+	CCONJ
ejpam-6197	226	5	k(m)−	k(m)−	PROPN
ejpam-6197	226	6	2k	2k	PROPN
ejpam-6197	226	7	(	(	PUNCT
ejpam-6197	226	8	l	l	NOUN
ejpam-6197	226	9	+	+	NOUN
ejpam-6197	226	10	m	m	VERB
ejpam-6197	226	11	2	2	NUM
ejpam-6197	226	12	)	)	PUNCT
ejpam-6197	226	13	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-6197	226	14	≤	≤	NUM
ejpam-6197	226	15	ρ(x	ρ(x	NOUN
ejpam-6197	226	16	)	)	PUNCT
ejpam-6197	226	17	∥∥∥∥k(l)−	∥∥∥∥k(l)−	ADP
ejpam-6197	226	18	k	k	X
ejpam-6197	226	19	(	(	PUNCT
ejpam-6197	226	20	l	l	X
ejpam-6197	226	21	+	+	NOUN
ejpam-6197	226	22	m	m	VERB
ejpam-6197	226	23	2	2	NUM
ejpam-6197	226	24	)	)	PUNCT
ejpam-6197	227	1	−	−	PROPN
ejpam-6197	228	1	k	k	NOUN
ejpam-6197	228	2	(	(	PUNCT
ejpam-6197	228	3	l	l	NOUN
ejpam-6197	228	4	−m	−m	NOUN
ejpam-6197	228	5	2	2	NUM
ejpam-6197	228	6	)	)	PUNCT
ejpam-6197	228	7	∥∥∥∥	∥∥∥∥	NUM
ejpam-6197	228	8	(	(	PUNCT
ejpam-6197	228	9	12	12	NUM
ejpam-6197	228	10	)	)	PUNCT
ejpam-6197	228	11	for	for	ADP
ejpam-6197	228	12	all	all	DET
ejpam-6197	228	13	l	l	NOUN
ejpam-6197	228	14	,	,	PUNCT
ejpam-6197	228	15	m	m	VERB
ejpam-6197	228	16	∈	∈	NOUN
ejpam-6197	228	17	x.	x.	NOUN
ejpam-6197	228	18	by	by	ADP
ejpam-6197	228	19	(	(	PUNCT
ejpam-6197	228	20	11	11	NUM
ejpam-6197	228	21	)	)	PUNCT
ejpam-6197	228	22	and	and	CCONJ
ejpam-6197	228	23	(	(	PUNCT
ejpam-6197	228	24	12	12	NUM
ejpam-6197	228	25	)	)	PUNCT
ejpam-6197	228	26	,	,	PUNCT
ejpam-6197	228	27	∥k(l	∥k(l	NOUN
ejpam-6197	228	28	)	)	PUNCT
ejpam-6197	229	1	+	+	CCONJ
ejpam-6197	230	1	k(m)−	k(m)−	PROPN
ejpam-6197	230	2	k(l	k(l	PROPN
ejpam-6197	230	3	+	+	PROPN
ejpam-6197	230	4	m)∥	m)∥	PROPN
ejpam-6197	230	5	≤	≤	NOUN
ejpam-6197	230	6	1	1	NUM
ejpam-6197	230	7	2	2	NUM
ejpam-6197	230	8	(	(	PUNCT
ejpam-6197	230	9	ρ(x))2∥k(l	ρ(x))2∥k(l	X
ejpam-6197	230	10	)	)	PUNCT
ejpam-6197	231	1	+	+	CCONJ
ejpam-6197	231	2	k(m)−	k(m)−	PROPN
ejpam-6197	231	3	k(l	k(l	PROPN
ejpam-6197	231	4	+	+	CCONJ
ejpam-6197	231	5	m)∥	m)∥	PROPN
ejpam-6197	231	6	for	for	ADP
ejpam-6197	231	7	all	all	DET
ejpam-6197	231	8	l	l	NOUN
ejpam-6197	231	9	,	,	PUNCT
ejpam-6197	231	10	m	m	VERB
ejpam-6197	231	11	∈	∈	ADJ
ejpam-6197	231	12	x.	x.	NOUN
ejpam-6197	231	13	thus	thus	ADV
ejpam-6197	231	14	k(l	k(l	PROPN
ejpam-6197	231	15	+	+	PROPN
ejpam-6197	231	16	m	m	NOUN
ejpam-6197	231	17	)	)	PUNCT
ejpam-6197	231	18	=	=	SYM
ejpam-6197	231	19	k(l	k(l	PROPN
ejpam-6197	231	20	)	)	PUNCT
ejpam-6197	232	1	+	+	NUM
ejpam-6197	232	2	k(m	k(m	PROPN
ejpam-6197	232	3	)	)	PUNCT
ejpam-6197	232	4	for	for	ADP
ejpam-6197	232	5	all	all	DET
ejpam-6197	232	6	l	l	NOUN
ejpam-6197	232	7	,	,	PUNCT
ejpam-6197	232	8	m	m	VERB
ejpam-6197	232	9	∈	∈	NOUN
ejpam-6197	232	10	x.	x.	NOUN
ejpam-6197	233	1	so	so	ADV
ejpam-6197	233	2	f	f	PROPN
ejpam-6197	233	3	is	be	AUX
ejpam-6197	233	4	additive	additive	ADJ
ejpam-6197	233	5	.	.	PUNCT
ejpam-6197	234	1	theorem	theorem	NOUN
ejpam-6197	234	2	5	5	NUM
ejpam-6197	234	3	.	.	PUNCT
ejpam-6197	235	1	let	let	VERB
ejpam-6197	235	2	ψ	ψ	X
ejpam-6197	235	3	:	:	PUNCT
ejpam-6197	235	4	x2	x2	X
ejpam-6197	235	5	→	→	PUNCT
ejpam-6197	235	6	[	[	X
ejpam-6197	235	7	0,∞	0,∞	X
ejpam-6197	235	8	)	)	PUNCT
ejpam-6197	235	9	be	be	VERB
ejpam-6197	235	10	a	a	DET
ejpam-6197	235	11	function	function	NOUN
ejpam-6197	235	12	such	such	ADJ
ejpam-6197	235	13	that	that	SCONJ
ejpam-6197	235	14	there	there	PRON
ejpam-6197	235	15	exists	exist	VERB
ejpam-6197	235	16	an	an	DET
ejpam-6197	235	17	l	l	NOUN
ejpam-6197	235	18	<	<	X
ejpam-6197	235	19	1	1	NUM
ejpam-6197	235	20	with	with	ADP
ejpam-6197	235	21	ψ	ψ	X
ejpam-6197	235	22	(	(	PUNCT
ejpam-6197	235	23	ax	ax	NOUN
ejpam-6197	235	24	,	,	PUNCT
ejpam-6197	235	25	ay	ay	NOUN
ejpam-6197	235	26	)	)	PUNCT
ejpam-6197	235	27	≤	≤	NOUN
ejpam-6197	235	28	a2lψ(x	a2lψ(x	NOUN
ejpam-6197	235	29	,	,	PUNCT
ejpam-6197	235	30	y	y	NOUN
ejpam-6197	235	31	)	)	PUNCT
ejpam-6197	236	1	g.	g.	PROPN
ejpam-6197	236	2	lyu	lyu	NOUN
ejpam-6197	236	3	et	et	PROPN
ejpam-6197	236	4	al	al	PROPN
ejpam-6197	236	5	.	.	PUNCT
ejpam-6197	236	6	/	/	SYM
ejpam-6197	236	7	eur	eur	PROPN
ejpam-6197	236	8	.	.	PUNCT
ejpam-6197	237	1	j.	j.	PROPN
ejpam-6197	237	2	pure	pure	PROPN
ejpam-6197	237	3	appl	appl	PROPN
ejpam-6197	237	4	.	.	PROPN
ejpam-6197	237	5	math	math	PROPN
ejpam-6197	237	6	,	,	PUNCT
ejpam-6197	237	7	18	18	NUM
ejpam-6197	237	8	(	(	PUNCT
ejpam-6197	237	9	3	3	NUM
ejpam-6197	237	10	)	)	PUNCT
ejpam-6197	237	11	(	(	PUNCT
ejpam-6197	237	12	2025	2025	NUM
ejpam-6197	237	13	)	)	PUNCT
ejpam-6197	237	14	,	,	PUNCT
ejpam-6197	237	15	6197	6197	NUM
ejpam-6197	237	16	13	13	NUM
ejpam-6197	237	17	of	of	ADP
ejpam-6197	237	18	17	17	NUM
ejpam-6197	237	19	for	for	ADP
ejpam-6197	237	20	all	all	DET
ejpam-6197	237	21	x	x	NOUN
ejpam-6197	237	22	,	,	PUNCT
ejpam-6197	237	23	y	y	PROPN
ejpam-6197	237	24	∈	∈	PROPN
ejpam-6197	237	25	x.	x.	NOUN
ejpam-6197	237	26	let	let	VERB
ejpam-6197	237	27	f	f	NOUN
ejpam-6197	237	28	:	:	PUNCT
ejpam-6197	237	29	x	x	X
ejpam-6197	237	30	→	→	SYM
ejpam-6197	237	31	y	y	X
ejpam-6197	237	32	be	be	AUX
ejpam-6197	237	33	an	an	DET
ejpam-6197	237	34	even	even	ADV
ejpam-6197	237	35	mapping	mapping	NOUN
ejpam-6197	237	36	with	with	ADP
ejpam-6197	237	37	f(0	f(0	NOUN
ejpam-6197	237	38	)	)	PUNCT
ejpam-6197	237	39	=	=	SYM
ejpam-6197	237	40	0	0	PUNCT
ejpam-6197	238	1	satisfying	satisfy	VERB
ejpam-6197	238	2	∥f(ax+	∥f(ax+	INTJ
ejpam-6197	238	3	by	by	ADP
ejpam-6197	238	4	)	)	PUNCT
ejpam-6197	239	1	+	+	CCONJ
ejpam-6197	239	2	f(ax−	f(ax−	NOUN
ejpam-6197	239	3	by)−af(x)−bf(y)−df(−y)∥	by)−af(x)−bf(y)−df(−y)∥	NOUN
ejpam-6197	239	4	≤	≤	NOUN
ejpam-6197	239	5	∥ρ(x)(f(ax+	∥ρ(x)(f(ax+	ADV
ejpam-6197	239	6	by	by	ADP
ejpam-6197	239	7	)	)	PUNCT
ejpam-6197	239	8	+	+	CCONJ
ejpam-6197	239	9	f(ax−	f(ax−	X
ejpam-6197	239	10	by)−	by)−	VERB
ejpam-6197	239	11	2f(ax)−	2f(ax)−	PROPN
ejpam-6197	239	12	2f(by))∥+	2f(by))∥+	PROPN
ejpam-6197	239	13	ψ(x	ψ(x	PROPN
ejpam-6197	239	14	,	,	PUNCT
ejpam-6197	239	15	y	y	NOUN
ejpam-6197	239	16	)	)	PUNCT
ejpam-6197	239	17	(	(	PUNCT
ejpam-6197	239	18	13	13	NUM
ejpam-6197	239	19	)	)	PUNCT
ejpam-6197	239	20	for	for	ADP
ejpam-6197	239	21	all	all	DET
ejpam-6197	239	22	x	x	NOUN
ejpam-6197	239	23	,	,	PUNCT
ejpam-6197	239	24	y	y	PROPN
ejpam-6197	239	25	∈	∈	PROPN
ejpam-6197	239	26	x.	x.	NOUN
ejpam-6197	239	27	assume	assume	VERB
ejpam-6197	239	28	that	that	SCONJ
ejpam-6197	239	29	l	l	NOUN
ejpam-6197	239	30	,	,	PUNCT
ejpam-6197	239	31	a	a	PRON
ejpam-6197	239	32	and	and	CCONJ
ejpam-6197	239	33	a	a	DET
ejpam-6197	239	34	satisfy	satisfy	NOUN
ejpam-6197	239	35	2a2l	2a2l	NUM
ejpam-6197	239	36	|a|	|a|	NOUN
ejpam-6197	239	37	<	<	X
ejpam-6197	239	38	1	1	NUM
ejpam-6197	239	39	,	,	PUNCT
ejpam-6197	239	40	and	and	CCONJ
ejpam-6197	239	41	ρ	ρ	NOUN
ejpam-6197	239	42	:	:	PUNCT
ejpam-6197	239	43	x	x	X
ejpam-6197	239	44	→	→	X
ejpam-6197	239	45	(	(	PUNCT
ejpam-6197	239	46	0	0	NUM
ejpam-6197	239	47	,	,	PUNCT
ejpam-6197	239	48	1	1	NUM
ejpam-6197	239	49	]	]	PUNCT
ejpam-6197	239	50	is	be	AUX
ejpam-6197	239	51	a	a	DET
ejpam-6197	239	52	function	function	NOUN
ejpam-6197	239	53	.	.	PUNCT
ejpam-6197	240	1	then	then	ADV
ejpam-6197	240	2	there	there	PRON
ejpam-6197	240	3	exists	exist	VERB
ejpam-6197	240	4	a	a	DET
ejpam-6197	240	5	unique	unique	ADJ
ejpam-6197	240	6	quadratic	quadratic	ADJ
ejpam-6197	240	7	mapping	mapping	NOUN
ejpam-6197	240	8	h	h	NOUN
ejpam-6197	240	9	:	:	PUNCT
ejpam-6197	240	10	x	x	X
ejpam-6197	240	11	→	→	PUNCT
ejpam-6197	240	12	y	y	PROPN
ejpam-6197	240	13	such	such	ADJ
ejpam-6197	240	14	that	that	SCONJ
ejpam-6197	240	15	∥f(x)−h(x)∥	∥f(x)−h(x)∥	ADP
ejpam-6197	240	16	≤	≤	NUM
ejpam-6197	240	17	|a|	|a|	NUM
ejpam-6197	240	18	|a|	|a|	PROPN
ejpam-6197	240	19	−	−	PROPN
ejpam-6197	240	20	2la2	2la2	NUM
ejpam-6197	240	21	ψ(x	ψ(x	NOUN
ejpam-6197	240	22	,	,	PUNCT
ejpam-6197	240	23	0	0	NUM
ejpam-6197	240	24	)	)	PUNCT
ejpam-6197	240	25	for	for	ADP
ejpam-6197	240	26	all	all	DET
ejpam-6197	240	27	x	x	SYM
ejpam-6197	240	28	∈	∈	ADJ
ejpam-6197	240	29	x.	x.	NOUN
ejpam-6197	240	30	proof	proof	NOUN
ejpam-6197	240	31	.	.	PUNCT
ejpam-6197	241	1	letting	let	VERB
ejpam-6197	241	2	y	y	PROPN
ejpam-6197	241	3	=	=	PUNCT
ejpam-6197	241	4	0	0	NUM
ejpam-6197	241	5	in	in	ADP
ejpam-6197	241	6	(	(	PUNCT
ejpam-6197	241	7	13	13	NUM
ejpam-6197	241	8	)	)	PUNCT
ejpam-6197	241	9	,	,	PUNCT
ejpam-6197	241	10	we	we	PRON
ejpam-6197	241	11	get	get	VERB
ejpam-6197	241	12	∥f(x)−	∥f(x)−	PROPN
ejpam-6197	241	13	2	2	NUM
ejpam-6197	241	14	a	a	DET
ejpam-6197	241	15	f(ax)∥	f(ax)∥	NOUN
ejpam-6197	241	16	≤	≤	ADV
ejpam-6197	241	17	1	1	NUM
ejpam-6197	241	18	|a|	|a|	PROPN
ejpam-6197	241	19	ψ(x	ψ(x	PROPN
ejpam-6197	241	20	,	,	PUNCT
ejpam-6197	241	21	0	0	NUM
ejpam-6197	241	22	)	)	PUNCT
ejpam-6197	241	23	,	,	PUNCT
ejpam-6197	241	24	x	x	PUNCT
ejpam-6197	241	25	∈	∈	NOUN
ejpam-6197	241	26	x.	x.	NOUN
ejpam-6197	241	27	similar	similar	ADJ
ejpam-6197	241	28	to	to	ADP
ejpam-6197	241	29	theorem	theorem	NOUN
ejpam-6197	241	30	4	4	NUM
ejpam-6197	241	31	,	,	PUNCT
ejpam-6197	241	32	we	we	PRON
ejpam-6197	241	33	can	can	AUX
ejpam-6197	241	34	prove	prove	VERB
ejpam-6197	241	35	that	that	SCONJ
ejpam-6197	241	36	h(x	h(x	PROPN
ejpam-6197	241	37	)	)	PUNCT
ejpam-6197	242	1	=	=	PROPN
ejpam-6197	242	2	lim	lim	PROPN
ejpam-6197	242	3	n→∞	n→∞	NUM
ejpam-6197	242	4	jnf(x	jnf(x	PROPN
ejpam-6197	242	5	)	)	PUNCT
ejpam-6197	242	6	,	,	PUNCT
ejpam-6197	242	7	h(x	h(x	PROPN
ejpam-6197	242	8	)	)	PUNCT
ejpam-6197	242	9	=	=	PUNCT
ejpam-6197	243	1	2	2	NUM
ejpam-6197	243	2	a	a	DET
ejpam-6197	243	3	h(ax	h(ax	NOUN
ejpam-6197	243	4	)	)	PUNCT
ejpam-6197	243	5	,	,	PUNCT
ejpam-6197	243	6	∥f(x)−h(x)∥	∥f(x)−h(x)∥	ADP
ejpam-6197	243	7	≤	≤	NUM
ejpam-6197	243	8	|a|	|a|	NUM
ejpam-6197	243	9	|a|	|a|	PROPN
ejpam-6197	243	10	−	−	PROPN
ejpam-6197	243	11	2la2	2la2	NUM
ejpam-6197	243	12	ψ(x	ψ(x	NOUN
ejpam-6197	243	13	,	,	PUNCT
ejpam-6197	243	14	0	0	NUM
ejpam-6197	243	15	)	)	PUNCT
ejpam-6197	243	16	.	.	PUNCT
ejpam-6197	244	1	moreover	moreover	ADV
ejpam-6197	244	2	,	,	PUNCT
ejpam-6197	244	3	∥f(x+	∥f(x+	VERB
ejpam-6197	244	4	y	y	NOUN
ejpam-6197	244	5	)	)	PUNCT
ejpam-6197	245	1	+	+	CCONJ
ejpam-6197	245	2	f(x−	f(x−	ADP
ejpam-6197	245	3	y)−	y)−	PROPN
ejpam-6197	245	4	2f(x)−	2f(x)−	NUM
ejpam-6197	245	5	f(y)−	f(y)−	PROPN
ejpam-6197	245	6	f(−y)∥	f(−y)∥	NOUN
ejpam-6197	245	7	≤	≤	NOUN
ejpam-6197	245	8	∥ρ(x)(f(x+	∥ρ(x)(f(x+	PROPN
ejpam-6197	245	9	y	y	NOUN
ejpam-6197	245	10	)	)	PUNCT
ejpam-6197	246	1	+	+	CCONJ
ejpam-6197	246	2	f(x−	f(x−	ADP
ejpam-6197	246	3	y)−	y)−	PROPN
ejpam-6197	246	4	2f(x)−	2f(x)−	NUM
ejpam-6197	246	5	2f(y))∥	2f(y))∥	NUM
ejpam-6197	246	6	+	+	NUM
ejpam-6197	246	7	ψ(x	ψ(x	PROPN
ejpam-6197	246	8	/	/	SYM
ejpam-6197	246	9	a	a	NOUN
ejpam-6197	246	10	,	,	PUNCT
ejpam-6197	246	11	y	y	PROPN
ejpam-6197	246	12	/	/	SYM
ejpam-6197	246	13	b	b	NOUN
ejpam-6197	246	14	)	)	PUNCT
ejpam-6197	246	15	+	+	SYM
ejpam-6197	246	16	ψ(x	ψ(x	NOUN
ejpam-6197	246	17	/	/	SYM
ejpam-6197	246	18	a	a	NOUN
ejpam-6197	246	19	,	,	PUNCT
ejpam-6197	246	20	0	0	NUM
ejpam-6197	246	21	)	)	PUNCT
ejpam-6197	247	1	+	+	CCONJ
ejpam-6197	248	1	ψ(0	ψ(0	ADJ
ejpam-6197	248	2	,	,	PUNCT
ejpam-6197	248	3	y	y	PROPN
ejpam-6197	248	4	/	/	SYM
ejpam-6197	248	5	b	b	NOUN
ejpam-6197	248	6	)	)	PUNCT
ejpam-6197	248	7	.	.	PUNCT
ejpam-6197	249	1	next	next	ADV
ejpam-6197	249	2	,	,	PUNCT
ejpam-6197	249	3	we	we	PRON
ejpam-6197	249	4	prove	prove	VERB
ejpam-6197	249	5	h(x	h(x	PROPN
ejpam-6197	249	6	)	)	PUNCT
ejpam-6197	249	7	is	be	AUX
ejpam-6197	249	8	quadratic	quadratic	ADJ
ejpam-6197	249	9	.	.	PUNCT
ejpam-6197	250	1	by	by	ADP
ejpam-6197	250	2	the	the	DET
ejpam-6197	250	3	definition	definition	NOUN
ejpam-6197	250	4	of	of	ADP
ejpam-6197	250	5	h(x	h(x	PROPN
ejpam-6197	250	6	)	)	PUNCT
ejpam-6197	250	7	,	,	PUNCT
ejpam-6197	250	8	we	we	PRON
ejpam-6197	250	9	have	have	VERB
ejpam-6197	250	10	∥h(x+	∥h(x+	ADV
ejpam-6197	250	11	y	y	NOUN
ejpam-6197	250	12	)	)	PUNCT
ejpam-6197	251	1	+	+	ADP
ejpam-6197	251	2	h(x−	h(x−	ADP
ejpam-6197	251	3	y)−	y)−	PROPN
ejpam-6197	251	4	2h(x)−	2h(x)−	NUM
ejpam-6197	251	5	2h(y)∥	2h(y)∥	NUM
ejpam-6197	251	6	≤	≤	NOUN
ejpam-6197	251	7	∥ρ(x)(h(x+	∥ρ(x)(h(x+	SCONJ
ejpam-6197	251	8	y	y	X
ejpam-6197	251	9	)	)	PUNCT
ejpam-6197	252	1	+	+	VERB
ejpam-6197	252	2	h(x−	h(x−	ADP
ejpam-6197	252	3	y)−	y)−	PROPN
ejpam-6197	252	4	2h(x)−	2h(x)−	NUM
ejpam-6197	252	5	2h(y))∥	2h(y))∥	NUM
ejpam-6197	252	6	for	for	ADP
ejpam-6197	252	7	all	all	DET
ejpam-6197	252	8	x	x	NOUN
ejpam-6197	252	9	,	,	PUNCT
ejpam-6197	252	10	y	y	PROPN
ejpam-6197	252	11	∈	∈	PROPN
ejpam-6197	252	12	x.	x.	NOUN
ejpam-6197	253	1	so	so	ADV
ejpam-6197	253	2	h(x+y)+h(x−y	h(x+y)+h(x−y	X
ejpam-6197	253	3	)	)	PUNCT
ejpam-6197	253	4	=	=	SYM
ejpam-6197	253	5	2h(x)+2h(y	2h(x)+2h(y	NUM
ejpam-6197	253	6	)	)	PUNCT
ejpam-6197	253	7	for	for	ADP
ejpam-6197	253	8	all	all	DET
ejpam-6197	253	9	x	x	NOUN
ejpam-6197	253	10	,	,	PUNCT
ejpam-6197	253	11	y	y	PROPN
ejpam-6197	253	12	∈	∈	PROPN
ejpam-6197	253	13	x.	x.	NOUN
ejpam-6197	253	14	this	this	PRON
ejpam-6197	253	15	completes	complete	VERB
ejpam-6197	253	16	the	the	DET
ejpam-6197	253	17	proof	proof	NOUN
ejpam-6197	253	18	.	.	PUNCT
ejpam-6197	254	1	5	5	X
ejpam-6197	254	2	.	.	X
ejpam-6197	254	3	conclusion	conclusion	NOUN
ejpam-6197	254	4	this	this	DET
ejpam-6197	254	5	paper	paper	NOUN
ejpam-6197	254	6	focuses	focus	VERB
ejpam-6197	254	7	on	on	ADP
ejpam-6197	254	8	the	the	DET
ejpam-6197	254	9	generalized	generalized	ADJ
ejpam-6197	254	10	drygas	drygas	NOUN
ejpam-6197	254	11	functional	functional	ADJ
ejpam-6197	254	12	equation	equation	NOUN
ejpam-6197	254	13	and	and	CCONJ
ejpam-6197	254	14	deeply	deeply	ADV
ejpam-6197	254	15	explores	explore	VERB
ejpam-6197	254	16	its	its	PRON
ejpam-6197	254	17	hyers	hyer	NOUN
ejpam-6197	254	18	ulam	ulam	PROPN
ejpam-6197	254	19	stability	stability	NOUN
ejpam-6197	254	20	in	in	ADP
ejpam-6197	254	21	the	the	DET
ejpam-6197	254	22	context	context	NOUN
ejpam-6197	254	23	from	from	ADP
ejpam-6197	254	24	a	a	DET
ejpam-6197	254	25	real	real	ADJ
ejpam-6197	254	26	vector	vector	NOUN
ejpam-6197	254	27	space	space	NOUN
ejpam-6197	254	28	to	to	ADP
ejpam-6197	254	29	a	a	DET
ejpam-6197	254	30	banach	banach	NOUN
ejpam-6197	254	31	space	space	NOUN
ejpam-6197	254	32	.	.	PUNCT
ejpam-6197	255	1	a	a	DET
ejpam-6197	255	2	g.	g.	NOUN
ejpam-6197	255	3	lyu	lyu	VERB
ejpam-6197	255	4	et	et	PROPN
ejpam-6197	255	5	al	al	PROPN
ejpam-6197	255	6	.	.	PUNCT
ejpam-6197	255	7	/	/	SYM
ejpam-6197	255	8	eur	eur	PROPN
ejpam-6197	255	9	.	.	PUNCT
ejpam-6197	256	1	j.	j.	PROPN
ejpam-6197	256	2	pure	pure	PROPN
ejpam-6197	256	3	appl	appl	PROPN
ejpam-6197	256	4	.	.	PROPN
ejpam-6197	256	5	math	math	PROPN
ejpam-6197	256	6	,	,	PUNCT
ejpam-6197	256	7	18	18	NUM
ejpam-6197	256	8	(	(	PUNCT
ejpam-6197	256	9	3	3	NUM
ejpam-6197	256	10	)	)	PUNCT
ejpam-6197	256	11	(	(	PUNCT
ejpam-6197	256	12	2025	2025	NUM
ejpam-6197	256	13	)	)	PUNCT
ejpam-6197	256	14	,	,	PUNCT
ejpam-6197	256	15	6197	6197	NUM
ejpam-6197	256	16	14	14	NUM
ejpam-6197	256	17	of	of	ADP
ejpam-6197	256	18	17	17	NUM
ejpam-6197	256	19	series	series	NOUN
ejpam-6197	256	20	of	of	ADP
ejpam-6197	256	21	key	key	ADJ
ejpam-6197	256	22	results	result	NOUN
ejpam-6197	256	23	have	have	AUX
ejpam-6197	256	24	been	be	AUX
ejpam-6197	256	25	obtained	obtain	VERB
ejpam-6197	256	26	by	by	ADP
ejpam-6197	256	27	using	use	VERB
ejpam-6197	256	28	the	the	DET
ejpam-6197	256	29	fixed	fix	VERB
ejpam-6197	256	30	point	point	NOUN
ejpam-6197	256	31	method	method	NOUN
ejpam-6197	256	32	.	.	PUNCT
ejpam-6197	257	1	regarding	regard	VERB
ejpam-6197	257	2	the	the	DET
ejpam-6197	257	3	stability	stability	NOUN
ejpam-6197	257	4	of	of	ADP
ejpam-6197	257	5	the	the	DET
ejpam-6197	257	6	functional	functional	ADJ
ejpam-6197	257	7	equation	equation	NOUN
ejpam-6197	257	8	,	,	PUNCT
ejpam-6197	257	9	under	under	ADP
ejpam-6197	257	10	different	different	ADJ
ejpam-6197	257	11	conditions	condition	NOUN
ejpam-6197	257	12	(	(	PUNCT
ejpam-6197	257	13	such	such	ADJ
ejpam-6197	257	14	as	as	ADP
ejpam-6197	257	15	the	the	DET
ejpam-6197	257	16	specific	specific	ADJ
ejpam-6197	257	17	restrictions	restriction	NOUN
ejpam-6197	257	18	on	on	ADP
ejpam-6197	257	19	a	a	DET
ejpam-6197	257	20	,	,	PUNCT
ejpam-6197	257	21	b	b	PROPN
ejpam-6197	257	22	,	,	PUNCT
ejpam-6197	257	23	a	a	DET
ejpam-6197	257	24	,	,	PUNCT
ejpam-6197	257	25	b	b	NOUN
ejpam-6197	257	26	,	,	PUNCT
ejpam-6197	257	27	d	d	PROPN
ejpam-6197	257	28	)	)	PUNCT
ejpam-6197	257	29	,	,	PUNCT
ejpam-6197	257	30	the	the	DET
ejpam-6197	257	31	close	close	ADJ
ejpam-6197	257	32	relationship	relationship	NOUN
ejpam-6197	257	33	between	between	ADP
ejpam-6197	257	34	the	the	DET
ejpam-6197	257	35	mapping	mapping	NOUN
ejpam-6197	257	36	satisfying	satisfy	VERB
ejpam-6197	257	37	the	the	DET
ejpam-6197	257	38	relevant	relevant	ADJ
ejpam-6197	257	39	inequality	inequality	NOUN
ejpam-6197	257	40	and	and	CCONJ
ejpam-6197	257	41	the	the	DET
ejpam-6197	257	42	unique	unique	ADJ
ejpam-6197	257	43	generalized	generalized	ADJ
ejpam-6197	257	44	drygas	drygas	NOUN
ejpam-6197	257	45	mapping	mapping	NOUN
ejpam-6197	257	46	is	be	AUX
ejpam-6197	257	47	clearly	clearly	ADV
ejpam-6197	257	48	given	give	VERB
ejpam-6197	257	49	,	,	PUNCT
ejpam-6197	257	50	and	and	CCONJ
ejpam-6197	257	51	the	the	DET
ejpam-6197	257	52	error	error	NOUN
ejpam-6197	257	53	estimate	estimate	NOUN
ejpam-6197	257	54	is	be	AUX
ejpam-6197	257	55	accurately	accurately	ADV
ejpam-6197	257	56	provided	provide	VERB
ejpam-6197	257	57	.	.	PUNCT
ejpam-6197	258	1	this	this	PRON
ejpam-6197	258	2	not	not	PART
ejpam-6197	258	3	only	only	ADV
ejpam-6197	258	4	deepens	deepen	VERB
ejpam-6197	258	5	the	the	DET
ejpam-6197	258	6	theoretical	theoretical	ADJ
ejpam-6197	258	7	understanding	understanding	NOUN
ejpam-6197	258	8	of	of	ADP
ejpam-6197	258	9	the	the	DET
ejpam-6197	258	10	properties	property	NOUN
ejpam-6197	258	11	of	of	ADP
ejpam-6197	258	12	this	this	DET
ejpam-6197	258	13	equation	equation	NOUN
ejpam-6197	258	14	but	but	CCONJ
ejpam-6197	258	15	also	also	ADV
ejpam-6197	258	16	provides	provide	VERB
ejpam-6197	258	17	an	an	DET
ejpam-6197	258	18	important	important	ADJ
ejpam-6197	258	19	methodological	methodological	ADJ
ejpam-6197	258	20	reference	reference	NOUN
ejpam-6197	258	21	and	and	CCONJ
ejpam-6197	258	22	theoretical	theoretical	ADJ
ejpam-6197	258	23	basis	basis	NOUN
ejpam-6197	258	24	for	for	ADP
ejpam-6197	258	25	subsequent	subsequent	ADJ
ejpam-6197	258	26	studies	study	NOUN
ejpam-6197	258	27	on	on	ADP
ejpam-6197	258	28	the	the	DET
ejpam-6197	258	29	stability	stability	NOUN
ejpam-6197	258	30	of	of	ADP
ejpam-6197	258	31	such	such	ADJ
ejpam-6197	258	32	equations	equation	NOUN
ejpam-6197	258	33	.	.	PUNCT
ejpam-6197	259	1	in	in	ADP
ejpam-6197	259	2	the	the	DET
ejpam-6197	259	3	study	study	NOUN
ejpam-6197	259	4	of	of	ADP
ejpam-6197	259	5	the	the	DET
ejpam-6197	259	6	functional	functional	ADJ
ejpam-6197	259	7	inequality	inequality	NOUN
ejpam-6197	259	8	,	,	PUNCT
ejpam-6197	259	9	it	it	PRON
ejpam-6197	259	10	is	be	AUX
ejpam-6197	259	11	successfully	successfully	ADV
ejpam-6197	259	12	proven	prove	VERB
ejpam-6197	259	13	that	that	SCONJ
ejpam-6197	259	14	there	there	PRON
ejpam-6197	259	15	exist	exist	VERB
ejpam-6197	259	16	unique	unique	ADJ
ejpam-6197	259	17	additive	additive	ADJ
ejpam-6197	259	18	and	and	CCONJ
ejpam-6197	259	19	quadratic	quadratic	ADJ
ejpam-6197	259	20	mappings	mapping	NOUN
ejpam-6197	259	21	under	under	ADP
ejpam-6197	259	22	specific	specific	ADJ
ejpam-6197	259	23	conditions	condition	NOUN
ejpam-6197	259	24	,	,	PUNCT
ejpam-6197	259	25	and	and	CCONJ
ejpam-6197	259	26	the	the	DET
ejpam-6197	259	27	corresponding	corresponding	ADJ
ejpam-6197	259	28	error	error	NOUN
ejpam-6197	259	29	estimates	estimate	NOUN
ejpam-6197	259	30	are	be	AUX
ejpam-6197	259	31	obtained	obtain	VERB
ejpam-6197	259	32	.	.	PUNCT
ejpam-6197	260	1	these	these	DET
ejpam-6197	260	2	conclusions	conclusion	NOUN
ejpam-6197	260	3	are	be	AUX
ejpam-6197	260	4	of	of	ADP
ejpam-6197	260	5	great	great	ADJ
ejpam-6197	260	6	significance	significance	NOUN
ejpam-6197	260	7	in	in	ADP
ejpam-6197	260	8	theoretical	theoretical	ADJ
ejpam-6197	260	9	research	research	NOUN
ejpam-6197	260	10	fields	field	NOUN
ejpam-6197	260	11	such	such	ADJ
ejpam-6197	260	12	as	as	ADP
ejpam-6197	260	13	mathematical	mathematical	ADJ
ejpam-6197	260	14	analysis	analysis	NOUN
ejpam-6197	260	15	and	and	CCONJ
ejpam-6197	260	16	function	function	NOUN
ejpam-6197	260	17	approximation	approximation	NOUN
ejpam-6197	260	18	.	.	PUNCT
ejpam-6197	261	1	they	they	PRON
ejpam-6197	261	2	provide	provide	VERB
ejpam-6197	261	3	new	new	ADJ
ejpam-6197	261	4	perspectives	perspective	NOUN
ejpam-6197	261	5	and	and	CCONJ
ejpam-6197	261	6	methods	method	NOUN
ejpam-6197	261	7	for	for	ADP
ejpam-6197	261	8	function	function	NOUN
ejpam-6197	261	9	classification	classification	NOUN
ejpam-6197	261	10	and	and	CCONJ
ejpam-6197	261	11	property	property	NOUN
ejpam-6197	261	12	characterization	characterization	NOUN
ejpam-6197	261	13	.	.	PUNCT
ejpam-6197	262	1	in	in	ADP
ejpam-6197	262	2	practical	practical	ADJ
ejpam-6197	262	3	applications	application	NOUN
ejpam-6197	262	4	,	,	PUNCT
ejpam-6197	262	5	such	such	ADJ
ejpam-6197	262	6	as	as	ADP
ejpam-6197	262	7	in	in	ADP
ejpam-6197	262	8	the	the	DET
ejpam-6197	262	9	design	design	NOUN
ejpam-6197	262	10	of	of	ADP
ejpam-6197	262	11	optimization	optimization	NOUN
ejpam-6197	262	12	algorithms	algorithm	NOUN
ejpam-6197	262	13	and	and	CCONJ
ejpam-6197	262	14	signal	signal	NOUN
ejpam-6197	262	15	processing	processing	NOUN
ejpam-6197	262	16	,	,	PUNCT
ejpam-6197	262	17	they	they	PRON
ejpam-6197	262	18	provide	provide	VERB
ejpam-6197	262	19	powerful	powerful	ADJ
ejpam-6197	262	20	mathematical	mathematical	ADJ
ejpam-6197	262	21	tools	tool	NOUN
ejpam-6197	262	22	,	,	PUNCT
ejpam-6197	262	23	which	which	PRON
ejpam-6197	262	24	can	can	AUX
ejpam-6197	262	25	effectively	effectively	ADV
ejpam-6197	262	26	improve	improve	VERB
ejpam-6197	262	27	algorithm	algorithm	NOUN
ejpam-6197	262	28	performance	performance	NOUN
ejpam-6197	262	29	and	and	CCONJ
ejpam-6197	262	30	optimize	optimize	VERB
ejpam-6197	262	31	signal	signal	ADJ
ejpam-6197	262	32	processing	processing	NOUN
ejpam-6197	262	33	effects	effect	NOUN
ejpam-6197	262	34	.	.	PUNCT
ejpam-6197	263	1	the	the	DET
ejpam-6197	263	2	research	research	NOUN
ejpam-6197	263	3	results	result	NOUN
ejpam-6197	263	4	of	of	ADP
ejpam-6197	263	5	this	this	DET
ejpam-6197	263	6	paper	paper	NOUN
ejpam-6197	263	7	expand	expand	VERB
ejpam-6197	263	8	the	the	DET
ejpam-6197	263	9	research	research	NOUN
ejpam-6197	263	10	boundaries	boundary	NOUN
ejpam-6197	263	11	of	of	ADP
ejpam-6197	263	12	the	the	DET
ejpam-6197	263	13	stability	stability	NOUN
ejpam-6197	263	14	theory	theory	NOUN
ejpam-6197	263	15	of	of	ADP
ejpam-6197	263	16	functional	functional	ADJ
ejpam-6197	263	17	equations	equation	NOUN
ejpam-6197	263	18	and	and	CCONJ
ejpam-6197	263	19	open	open	VERB
ejpam-6197	263	20	up	up	ADP
ejpam-6197	263	21	new	new	ADJ
ejpam-6197	263	22	directions	direction	NOUN
ejpam-6197	263	23	for	for	ADP
ejpam-6197	263	24	the	the	DET
ejpam-6197	263	25	future	future	ADJ
ejpam-6197	263	26	development	development	NOUN
ejpam-6197	263	27	of	of	ADP
ejpam-6197	263	28	this	this	DET
ejpam-6197	263	29	field	field	NOUN
ejpam-6197	263	30	.	.	PUNCT
ejpam-6197	264	1	follow	follow	VERB
ejpam-6197	264	2	up	up	ADP
ejpam-6197	264	3	research	research	NOUN
ejpam-6197	264	4	can	can	AUX
ejpam-6197	264	5	further	far	ADV
ejpam-6197	264	6	explore	explore	VERB
ejpam-6197	264	7	the	the	DET
ejpam-6197	264	8	properties	property	NOUN
ejpam-6197	264	9	of	of	ADP
ejpam-6197	264	10	the	the	DET
ejpam-6197	264	11	generalized	generalized	ADJ
ejpam-6197	264	12	drygas	drygas	NOUN
ejpam-6197	264	13	functional	functional	ADJ
ejpam-6197	264	14	equation	equation	NOUN
ejpam-6197	264	15	and	and	CCONJ
ejpam-6197	264	16	inequality	inequality	NOUN
ejpam-6197	264	17	in	in	ADP
ejpam-6197	264	18	more	more	ADJ
ejpam-6197	264	19	complex	complex	ADJ
ejpam-6197	264	20	spaces	space	NOUN
ejpam-6197	264	21	or	or	CCONJ
ejpam-6197	264	22	under	under	ADP
ejpam-6197	264	23	different	different	ADJ
ejpam-6197	264	24	types	type	NOUN
ejpam-6197	264	25	of	of	ADP
ejpam-6197	264	26	operators	operator	NOUN
ejpam-6197	264	27	.	.	PUNCT
ejpam-6197	265	1	at	at	ADP
ejpam-6197	265	2	the	the	DET
ejpam-6197	265	3	same	same	ADJ
ejpam-6197	265	4	time	time	NOUN
ejpam-6197	265	5	,	,	PUNCT
ejpam-6197	265	6	it	it	PRON
ejpam-6197	265	7	is	be	AUX
ejpam-6197	265	8	necessary	necessary	ADJ
ejpam-6197	265	9	to	to	PART
ejpam-6197	265	10	strengthen	strengthen	VERB
ejpam-6197	265	11	the	the	DET
ejpam-6197	265	12	applied	apply	VERB
ejpam-6197	265	13	research	research	NOUN
ejpam-6197	265	14	in	in	ADP
ejpam-6197	265	15	interdisciplinary	interdisciplinary	ADJ
ejpam-6197	265	16	fields	field	NOUN
ejpam-6197	265	17	,	,	PUNCT
ejpam-6197	265	18	fully	fully	ADV
ejpam-6197	265	19	explore	explore	VERB
ejpam-6197	265	20	its	its	PRON
ejpam-6197	265	21	potential	potential	ADJ
ejpam-6197	265	22	value	value	NOUN
ejpam-6197	265	23	,	,	PUNCT
ejpam-6197	265	24	and	and	CCONJ
ejpam-6197	265	25	promote	promote	VERB
ejpam-6197	265	26	the	the	DET
ejpam-6197	265	27	coordinated	coordinated	ADJ
ejpam-6197	265	28	development	development	NOUN
ejpam-6197	265	29	of	of	ADP
ejpam-6197	265	30	related	related	ADJ
ejpam-6197	265	31	fields	field	NOUN
ejpam-6197	265	32	.	.	PUNCT
ejpam-6197	266	1	declarations	declaration	NOUN
ejpam-6197	266	2	conflict	conflict	NOUN
ejpam-6197	266	3	of	of	ADP
ejpam-6197	266	4	interest	interest	NOUN
ejpam-6197	266	5	the	the	DET
ejpam-6197	266	6	authors	author	NOUN
ejpam-6197	266	7	declare	declare	VERB
ejpam-6197	266	8	that	that	SCONJ
ejpam-6197	266	9	they	they	PRON
ejpam-6197	266	10	have	have	VERB
ejpam-6197	266	11	no	no	DET
ejpam-6197	266	12	competing	compete	VERB
ejpam-6197	266	13	interests	interest	NOUN
ejpam-6197	266	14	.	.	PUNCT
ejpam-6197	267	1	fundings	funding	NOUN
ejpam-6197	267	2	this	this	DET
ejpam-6197	267	3	work	work	NOUN
ejpam-6197	267	4	was	be	AUX
ejpam-6197	267	5	supported	support	VERB
ejpam-6197	267	6	by	by	ADP
ejpam-6197	267	7	national	national	ADJ
ejpam-6197	267	8	natural	natural	PROPN
ejpam-6197	267	9	science	science	PROPN
ejpam-6197	267	10	foundation	foundation	PROPN
ejpam-6197	267	11	of	of	ADP
ejpam-6197	267	12	china	china	PROPN
ejpam-6197	267	13	(	(	PUNCT
ejpam-6197	267	14	no	no	INTJ
ejpam-6197	267	15	.	.	NOUN
ejpam-6197	267	16	11761074	11761074	NUM
ejpam-6197	267	17	)	)	PUNCT
ejpam-6197	267	18	,	,	PUNCT
ejpam-6197	267	19	project	project	NOUN
ejpam-6197	267	20	of	of	ADP
ejpam-6197	267	21	jilin	jilin	PROPN
ejpam-6197	267	22	science	science	PROPN
ejpam-6197	267	23	and	and	CCONJ
ejpam-6197	267	24	technology	technology	NOUN
ejpam-6197	267	25	development	development	NOUN
ejpam-6197	267	26	for	for	ADP
ejpam-6197	267	27	leading	lead	VERB
ejpam-6197	267	28	talent	talent	NOUN
ejpam-6197	267	29	of	of	ADP
ejpam-6197	267	30	science	science	NOUN
ejpam-6197	267	31	and	and	CCONJ
ejpam-6197	267	32	technology	technology	NOUN
ejpam-6197	267	33	innovation	innovation	NOUN
ejpam-6197	267	34	in	in	ADP
ejpam-6197	267	35	middle	middle	ADJ
ejpam-6197	267	36	and	and	CCONJ
ejpam-6197	267	37	young	young	ADJ
ejpam-6197	267	38	and	and	CCONJ
ejpam-6197	267	39	team	team	NOUN
ejpam-6197	267	40	project	project	NOUN
ejpam-6197	267	41	(	(	PUNCT
ejpam-6197	267	42	no	no	INTJ
ejpam-6197	267	43	.	.	PUNCT
ejpam-6197	268	1	20200301053rq	20200301053rq	NOUN
ejpam-6197	268	2	)	)	PUNCT
ejpam-6197	268	3	,	,	PUNCT
ejpam-6197	268	4	the	the	DET
ejpam-6197	268	5	natural	natural	ADJ
ejpam-6197	268	6	science	science	NOUN
ejpam-6197	268	7	foundation	foundation	PROPN
ejpam-6197	268	8	of	of	ADP
ejpam-6197	268	9	jilin	jilin	PROPN
ejpam-6197	268	10	province	province	PROPN
ejpam-6197	268	11	(	(	PUNCT
ejpam-6197	268	12	no	no	INTJ
ejpam-6197	268	13	.	.	PUNCT
ejpam-6197	268	14	ydzj202101zyts136	ydzj202101zyts136	NOUN
ejpam-6197	268	15	)	)	PUNCT
ejpam-6197	268	16	and	and	CCONJ
ejpam-6197	268	17	the	the	DET
ejpam-6197	268	18	scientific	scientific	ADJ
ejpam-6197	268	19	research	research	NOUN
ejpam-6197	268	20	project	project	NOUN
ejpam-6197	268	21	of	of	ADP
ejpam-6197	268	22	guangzhou	guangzhou	PROPN
ejpam-6197	268	23	college	college	PROPN
ejpam-6197	268	24	of	of	ADP
ejpam-6197	268	25	technology	technology	NOUN
ejpam-6197	268	26	and	and	CCONJ
ejpam-6197	268	27	business	business	NOUN
ejpam-6197	268	28	in	in	ADP
ejpam-6197	268	29	2024	2024	NUM
ejpam-6197	268	30	(	(	PUNCT
ejpam-6197	268	31	no	no	INTJ
ejpam-6197	268	32	.	.	PUNCT
ejpam-6197	269	1	kyzd202404	kyzd202404	NOUN
ejpam-6197	269	2	)	)	PUNCT
ejpam-6197	269	3	.	.	PUNCT
ejpam-6197	270	1	authors	author	NOUN
ejpam-6197	270	2	’	'	PUNCT
ejpam-6197	270	3	contributions	contribution	NOUN
ejpam-6197	270	4	the	the	DET
ejpam-6197	270	5	authors	author	NOUN
ejpam-6197	270	6	equally	equally	ADV
ejpam-6197	270	7	conceived	conceive	VERB
ejpam-6197	270	8	of	of	ADP
ejpam-6197	270	9	the	the	DET
ejpam-6197	270	10	study	study	NOUN
ejpam-6197	270	11	,	,	PUNCT
ejpam-6197	270	12	participated	participate	VERB
ejpam-6197	270	13	in	in	ADP
ejpam-6197	270	14	its	its	PRON
ejpam-6197	270	15	design	design	NOUN
ejpam-6197	270	16	and	and	CCONJ
ejpam-6197	270	17	coordination	coordination	NOUN
ejpam-6197	270	18	,	,	PUNCT
ejpam-6197	270	19	drafted	draft	VERB
ejpam-6197	270	20	the	the	DET
ejpam-6197	270	21	manuscript	manuscript	NOUN
ejpam-6197	270	22	,	,	PUNCT
ejpam-6197	270	23	participated	participate	VERB
ejpam-6197	270	24	in	in	ADP
ejpam-6197	270	25	the	the	DET
ejpam-6197	270	26	sequence	sequence	NOUN
ejpam-6197	270	27	alignment	alignment	NOUN
ejpam-6197	270	28	,	,	PUNCT
ejpam-6197	270	29	and	and	CCONJ
ejpam-6197	270	30	read	read	VERB
ejpam-6197	270	31	and	and	CCONJ
ejpam-6197	270	32	approved	approve	VERB
ejpam-6197	270	33	the	the	DET
ejpam-6197	270	34	final	final	ADJ
ejpam-6197	270	35	manuscript	manuscript	NOUN
ejpam-6197	270	36	.	.	PUNCT
ejpam-6197	271	1	references	reference	NOUN
ejpam-6197	271	2	[	[	X
ejpam-6197	271	3	1	1	NUM
ejpam-6197	271	4	]	]	X
ejpam-6197	271	5	s	s	NOUN
ejpam-6197	271	6	m	m	NOUN
ejpam-6197	271	7	ulam	ulam	PROPN
ejpam-6197	271	8	.	.	PUNCT
ejpam-6197	272	1	problems	problem	NOUN
ejpam-6197	272	2	in	in	ADP
ejpam-6197	272	3	modern	modern	ADJ
ejpam-6197	272	4	mathematics	mathematic	NOUN
ejpam-6197	272	5	.	.	PUNCT
ejpam-6197	273	1	john	john	PROPN
ejpam-6197	273	2	wiley	wiley	PROPN
ejpam-6197	273	3	&	&	CCONJ
ejpam-6197	273	4	sons	sons	PROPN
ejpam-6197	273	5	,	,	PUNCT
ejpam-6197	273	6	inc	inc	PROPN
ejpam-6197	273	7	.	.	PROPN
ejpam-6197	273	8	,	,	PUNCT
ejpam-6197	273	9	new	new	PROPN
ejpam-6197	273	10	york	york	PROPN
ejpam-6197	273	11	,	,	PUNCT
ejpam-6197	273	12	1964	1964	NUM
ejpam-6197	273	13	.	.	PUNCT
ejpam-6197	274	1	g.	g.	PROPN
ejpam-6197	274	2	lyu	lyu	VERB
ejpam-6197	274	3	et	et	PROPN
ejpam-6197	274	4	al	al	PROPN
ejpam-6197	274	5	.	.	PUNCT
ejpam-6197	274	6	/	/	SYM
ejpam-6197	274	7	eur	eur	PROPN
ejpam-6197	274	8	.	.	PUNCT
ejpam-6197	275	1	j.	j.	PROPN
ejpam-6197	275	2	pure	pure	PROPN
ejpam-6197	275	3	appl	appl	PROPN
ejpam-6197	275	4	.	.	PROPN
ejpam-6197	275	5	math	math	PROPN
ejpam-6197	275	6	,	,	PUNCT
ejpam-6197	275	7	18	18	NUM
ejpam-6197	275	8	(	(	PUNCT
ejpam-6197	275	9	3	3	NUM
ejpam-6197	275	10	)	)	PUNCT
ejpam-6197	275	11	(	(	PUNCT
ejpam-6197	275	12	2025	2025	NUM
ejpam-6197	275	13	)	)	PUNCT
ejpam-6197	275	14	,	,	PUNCT
ejpam-6197	275	15	6197	6197	NUM
ejpam-6197	275	16	15	15	NUM
ejpam-6197	275	17	of	of	ADP
ejpam-6197	275	18	17	17	NUM
ejpam-6197	275	19	[	[	X
ejpam-6197	275	20	2	2	NUM
ejpam-6197	275	21	]	]	PUNCT
ejpam-6197	275	22	d	d	PROPN
ejpam-6197	275	23	h	h	PROPN
ejpam-6197	275	24	hyers	hyer	NOUN
ejpam-6197	275	25	.	.	PUNCT
ejpam-6197	276	1	on	on	ADP
ejpam-6197	276	2	the	the	DET
ejpam-6197	276	3	stability	stability	NOUN
ejpam-6197	276	4	of	of	ADP
ejpam-6197	276	5	the	the	DET
ejpam-6197	276	6	linear	linear	ADJ
ejpam-6197	276	7	functional	functional	ADJ
ejpam-6197	276	8	equation	equation	NOUN
ejpam-6197	276	9	.	.	PUNCT
ejpam-6197	277	1	proc	proc	PROPN
ejpam-6197	277	2	.	.	PUNCT
ejpam-6197	278	1	natl	natl	PROPN
ejpam-6197	278	2	.	.	PUNCT
ejpam-6197	279	1	acad	acad	PROPN
ejpam-6197	279	2	.	.	PUNCT
ejpam-6197	280	1	sci	sci	PROPN
ejpam-6197	280	2	.	.	PUNCT
ejpam-6197	280	3	u.s.a	u.s.a	PROPN
ejpam-6197	280	4	.	.	PROPN
ejpam-6197	280	5	,	,	PUNCT
ejpam-6197	280	6	27:222–224	27:222–224	NUM
ejpam-6197	280	7	,	,	PUNCT
ejpam-6197	280	8	1941	1941	NUM
ejpam-6197	280	9	.	.	PUNCT
ejpam-6197	281	1	[	[	X
ejpam-6197	281	2	3	3	X
ejpam-6197	281	3	]	]	PUNCT
ejpam-6197	281	4	t	t	PROPN
ejpam-6197	281	5	aoki	aoki	PROPN
ejpam-6197	281	6	.	.	PUNCT
ejpam-6197	282	1	on	on	ADP
ejpam-6197	282	2	the	the	DET
ejpam-6197	282	3	stability	stability	NOUN
ejpam-6197	282	4	of	of	ADP
ejpam-6197	282	5	the	the	DET
ejpam-6197	282	6	linear	linear	ADJ
ejpam-6197	282	7	transformation	transformation	NOUN
ejpam-6197	282	8	in	in	ADP
ejpam-6197	282	9	banach	banach	NOUN
ejpam-6197	282	10	spaces	space	NOUN
ejpam-6197	282	11	.	.	PUNCT
ejpam-6197	283	1	j.	j.	PROPN
ejpam-6197	283	2	math	math	PROPN
ejpam-6197	283	3	.	.	PUNCT
ejpam-6197	284	1	soc	soc	PROPN
ejpam-6197	284	2	.	.	PUNCT
ejpam-6197	285	1	japan	japan	PROPN
ejpam-6197	285	2	,	,	PUNCT
ejpam-6197	285	3	2:64–66	2:64–66	NUM
ejpam-6197	285	4	,	,	PUNCT
ejpam-6197	285	5	1950	1950	NUM
ejpam-6197	285	6	.	.	PUNCT
ejpam-6197	286	1	[	[	X
ejpam-6197	286	2	4	4	X
ejpam-6197	286	3	]	]	X
ejpam-6197	286	4	p	p	NOUN
ejpam-6197	286	5	gǎvruţa	gǎvruţa	NOUN
ejpam-6197	286	6	.	.	PUNCT
ejpam-6197	287	1	a	a	DET
ejpam-6197	287	2	generalization	generalization	NOUN
ejpam-6197	287	3	of	of	ADP
ejpam-6197	287	4	the	the	DET
ejpam-6197	287	5	hyers	hyers	PROPN
ejpam-6197	287	6	-	-	PUNCT
ejpam-6197	287	7	ulam	ulam	ADJ
ejpam-6197	287	8	-	-	PUNCT
ejpam-6197	287	9	rassias	rassias	PROPN
ejpam-6197	287	10	stability	stability	NOUN
ejpam-6197	287	11	of	of	ADP
ejpam-6197	287	12	approximately	approximately	ADV
ejpam-6197	287	13	additive	additive	ADJ
ejpam-6197	287	14	mappings	mapping	NOUN
ejpam-6197	287	15	.	.	PUNCT
ejpam-6197	288	1	j.	j.	PROPN
ejpam-6197	288	2	math	math	PROPN
ejpam-6197	288	3	.	.	PUNCT
ejpam-6197	289	1	anal	anal	PROPN
ejpam-6197	289	2	.	.	PUNCT
ejpam-6197	290	1	appl	appl	PROPN
ejpam-6197	290	2	.	.	PROPN
ejpam-6197	290	3	,	,	PUNCT
ejpam-6197	290	4	184:431–436	184:431–436	NUM
ejpam-6197	290	5	,	,	PUNCT
ejpam-6197	290	6	1994	1994	NUM
ejpam-6197	290	7	.	.	PUNCT
ejpam-6197	291	1	[	[	X
ejpam-6197	291	2	5	5	NUM
ejpam-6197	291	3	]	]	PUNCT
ejpam-6197	291	4	t	t	PROPN
ejpam-6197	291	5	m	m	NOUN
ejpam-6197	291	6	rassias	rassias	PROPN
ejpam-6197	291	7	.	.	PUNCT
ejpam-6197	292	1	on	on	ADP
ejpam-6197	292	2	the	the	DET
ejpam-6197	292	3	stability	stability	NOUN
ejpam-6197	292	4	of	of	ADP
ejpam-6197	292	5	the	the	DET
ejpam-6197	292	6	linear	linear	ADJ
ejpam-6197	292	7	mapping	mapping	NOUN
ejpam-6197	292	8	in	in	ADP
ejpam-6197	292	9	banach	banach	NOUN
ejpam-6197	292	10	spaces	space	NOUN
ejpam-6197	292	11	.	.	PUNCT
ejpam-6197	293	1	proc	proc	NOUN
ejpam-6197	293	2	.	.	PUNCT
ejpam-6197	294	1	amer	amer	PROPN
ejpam-6197	294	2	.	.	PUNCT
ejpam-6197	294	3	math	math	PROPN
ejpam-6197	294	4	.	.	PUNCT
ejpam-6197	295	1	soc	soc	PROPN
ejpam-6197	295	2	.	.	PUNCT
ejpam-6197	295	3	,	,	PUNCT
ejpam-6197	295	4	72:297–300	72:297–300	PROPN
ejpam-6197	295	5	,	,	PUNCT
ejpam-6197	295	6	1978	1978	NUM
ejpam-6197	295	7	.	.	PUNCT
ejpam-6197	296	1	[	[	X
ejpam-6197	296	2	6	6	NUM
ejpam-6197	296	3	]	]	PUNCT
ejpam-6197	296	4	l	l	NOUN
ejpam-6197	296	5	aiemsomboon	aiemsomboon	NOUN
ejpam-6197	296	6	and	and	CCONJ
ejpam-6197	296	7	w	w	PROPN
ejpam-6197	296	8	sintunavarat	sintunavarat	NOUN
ejpam-6197	296	9	.	.	PUNCT
ejpam-6197	297	1	stability	stability	NOUN
ejpam-6197	297	2	of	of	ADP
ejpam-6197	297	3	the	the	DET
ejpam-6197	297	4	generalized	generalize	VERB
ejpam-6197	297	5	logarithmic	logarithmic	ADJ
ejpam-6197	297	6	functional	functional	ADJ
ejpam-6197	297	7	equations	equation	NOUN
ejpam-6197	297	8	arising	arise	VERB
ejpam-6197	297	9	from	from	ADP
ejpam-6197	297	10	fixed	fix	VERB
ejpam-6197	297	11	point	point	NOUN
ejpam-6197	297	12	theory	theory	NOUN
ejpam-6197	297	13	.	.	PUNCT
ejpam-6197	298	1	rev	rev	PROPN
ejpam-6197	298	2	.	.	PROPN
ejpam-6197	298	3	r.	r.	PROPN
ejpam-6197	298	4	acad	acad	PROPN
ejpam-6197	298	5	.	.	PUNCT
ejpam-6197	299	1	cienc	cienc	PROPN
ejpam-6197	299	2	.	.	PUNCT
ejpam-6197	300	1	exactas	exactas	PROPN
ejpam-6197	300	2	f́ıs	f́ıs	PROPN
ejpam-6197	300	3	.	.	PUNCT
ejpam-6197	301	1	nat	nat	PROPN
ejpam-6197	301	2	.	.	PUNCT
ejpam-6197	302	1	ser	ser	PROPN
ejpam-6197	302	2	.	.	PUNCT
ejpam-6197	303	1	a	a	DET
ejpam-6197	303	2	mat	mat	NOUN
ejpam-6197	303	3	.	.	PUNCT
ejpam-6197	303	4	racsam	racsam	PROPN
ejpam-6197	303	5	,	,	PUNCT
ejpam-6197	303	6	112(1):229–238	112(1):229–238	NUM
ejpam-6197	303	7	,	,	PUNCT
ejpam-6197	303	8	2018	2018	NUM
ejpam-6197	303	9	.	.	PUNCT
ejpam-6197	304	1	[	[	X
ejpam-6197	304	2	7	7	X
ejpam-6197	304	3	]	]	X
ejpam-6197	304	4	n	n	DET
ejpam-6197	304	5	alessa	alessa	NOUN
ejpam-6197	304	6	,	,	PUNCT
ejpam-6197	304	7	k	k	PROPN
ejpam-6197	304	8	tamilvanan	tamilvanan	PROPN
ejpam-6197	304	9	,	,	PUNCT
ejpam-6197	304	10	g	g	PROPN
ejpam-6197	304	11	balasubramanian	balasubramanian	PROPN
ejpam-6197	304	12	,	,	PUNCT
ejpam-6197	304	13	and	and	CCONJ
ejpam-6197	304	14	k	k	PROPN
ejpam-6197	304	15	loganathan	loganathan	PROPN
ejpam-6197	304	16	.	.	PUNCT
ejpam-6197	305	1	stability	stability	NOUN
ejpam-6197	305	2	results	result	NOUN
ejpam-6197	305	3	of	of	ADP
ejpam-6197	305	4	the	the	DET
ejpam-6197	305	5	functional	functional	ADJ
ejpam-6197	305	6	equation	equation	NOUN
ejpam-6197	305	7	deriving	derive	VERB
ejpam-6197	305	8	from	from	ADP
ejpam-6197	305	9	quadratic	quadratic	ADJ
ejpam-6197	305	10	function	function	NOUN
ejpam-6197	305	11	in	in	ADP
ejpam-6197	305	12	random	random	ADJ
ejpam-6197	305	13	normed	normed	ADJ
ejpam-6197	305	14	spaces	space	NOUN
ejpam-6197	305	15	.	.	PUNCT
ejpam-6197	306	1	aims	aim	VERB
ejpam-6197	306	2	math	math	NOUN
ejpam-6197	306	3	.	.	PUNCT
ejpam-6197	306	4	,	,	PUNCT
ejpam-6197	306	5	6(3):2385–2397	6(3):2385–2397	PROPN
ejpam-6197	306	6	,	,	PUNCT
ejpam-6197	306	7	2021	2021	NUM
ejpam-6197	306	8	.	.	PUNCT
ejpam-6197	307	1	[	[	X
ejpam-6197	307	2	8	8	NUM
ejpam-6197	307	3	]	]	PUNCT
ejpam-6197	307	4	a	a	DET
ejpam-6197	307	5	bahyrycz	bahyrycz	NOUN
ejpam-6197	307	6	and	and	CCONJ
ejpam-6197	307	7	j	j	PROPN
ejpam-6197	307	8	sikorska	sikorska	PROPN
ejpam-6197	307	9	.	.	PUNCT
ejpam-6197	308	1	on	on	ADP
ejpam-6197	308	2	a	a	DET
ejpam-6197	308	3	general	general	ADJ
ejpam-6197	308	4	n	n	CCONJ
ejpam-6197	308	5	-	-	PUNCT
ejpam-6197	308	6	linear	linear	ADJ
ejpam-6197	308	7	functional	functional	ADJ
ejpam-6197	308	8	equation	equation	NOUN
ejpam-6197	308	9	.	.	PUNCT
ejpam-6197	309	1	results	result	VERB
ejpam-6197	309	2	math	math	PROPN
ejpam-6197	309	3	.	.	PUNCT
ejpam-6197	309	4	,	,	PUNCT
ejpam-6197	309	5	77(3(128)):1–14	77(3(128)):1–14	NUM
ejpam-6197	309	6	,	,	PUNCT
ejpam-6197	309	7	2022	2022	NUM
ejpam-6197	309	8	.	.	PUNCT
ejpam-6197	310	1	[	[	X
ejpam-6197	310	2	9	9	NUM
ejpam-6197	310	3	]	]	X
ejpam-6197	310	4	y	y	PROPN
ejpam-6197	310	5	cho	cho	PROPN
ejpam-6197	310	6	,	,	PUNCT
ejpam-6197	310	7	c	c	PROPN
ejpam-6197	310	8	park	park	NOUN
ejpam-6197	310	9	,	,	PUNCT
ejpam-6197	310	10	and	and	CCONJ
ejpam-6197	310	11	r	r	NOUN
ejpam-6197	310	12	saadati	saadati	NOUN
ejpam-6197	310	13	.	.	PUNCT
ejpam-6197	311	1	functional	functional	ADJ
ejpam-6197	311	2	inequalities	inequality	NOUN
ejpam-6197	311	3	in	in	ADP
ejpam-6197	311	4	non	non	ADJ
ejpam-6197	311	5	-	-	ADJ
ejpam-6197	311	6	archimedean	archimedean	ADJ
ejpam-6197	311	7	banach	banach	NOUN
ejpam-6197	311	8	spaces	space	VERB
ejpam-6197	311	9	.	.	PUNCT
ejpam-6197	312	1	appl	appl	PROPN
ejpam-6197	312	2	.	.	PROPN
ejpam-6197	312	3	math	math	PROPN
ejpam-6197	312	4	.	.	PUNCT
ejpam-6197	313	1	lett	lett	PROPN
ejpam-6197	313	2	.	.	PROPN
ejpam-6197	313	3	,	,	PUNCT
ejpam-6197	313	4	23(10):1236–1242	23(10):1236–1242	PROPN
ejpam-6197	313	5	,	,	PUNCT
ejpam-6197	313	6	2010	2010	NUM
ejpam-6197	313	7	.	.	PUNCT
ejpam-6197	314	1	[	[	X
ejpam-6197	314	2	10	10	NUM
ejpam-6197	314	3	]	]	X
ejpam-6197	314	4	y	y	PROPN
ejpam-6197	314	5	cho	cho	PROPN
ejpam-6197	314	6	,	,	PUNCT
ejpam-6197	314	7	r	r	NOUN
ejpam-6197	314	8	saadati	saadati	NOUN
ejpam-6197	314	9	,	,	PUNCT
ejpam-6197	314	10	and	and	CCONJ
ejpam-6197	314	11	y	y	PROPN
ejpam-6197	314	12	yang	yang	PROPN
ejpam-6197	314	13	.	.	PUNCT
ejpam-6197	315	1	approximation	approximation	NOUN
ejpam-6197	315	2	of	of	ADP
ejpam-6197	315	3	homomorphisms	homomorphism	NOUN
ejpam-6197	315	4	and	and	CCONJ
ejpam-6197	315	5	derivations	derivation	NOUN
ejpam-6197	315	6	on	on	ADP
ejpam-6197	315	7	lie	lie	NOUN
ejpam-6197	315	8	c∗-algebras	c∗-algebras	PROPN
ejpam-6197	315	9	via	via	ADP
ejpam-6197	315	10	fixed	fix	VERB
ejpam-6197	315	11	point	point	NOUN
ejpam-6197	315	12	method	method	NOUN
ejpam-6197	315	13	.	.	PUNCT
ejpam-6197	316	1	j.	j.	PROPN
ejpam-6197	316	2	inequal	inequal	PROPN
ejpam-6197	316	3	.	.	PUNCT
ejpam-6197	317	1	appl	appl	PROPN
ejpam-6197	317	2	.	.	PROPN
ejpam-6197	317	3	,	,	PUNCT
ejpam-6197	317	4	2013(415):1–9	2013(415):1–9	NUM
ejpam-6197	317	5	,	,	PUNCT
ejpam-6197	317	6	2013	2013	NUM
ejpam-6197	317	7	.	.	PUNCT
ejpam-6197	318	1	[	[	X
ejpam-6197	318	2	11	11	NUM
ejpam-6197	318	3	]	]	PUNCT
ejpam-6197	318	4	a	a	DET
ejpam-6197	318	5	kumar	kumar	PROPN
ejpam-6197	318	6	and	and	CCONJ
ejpam-6197	318	7	m	m	PROPN
ejpam-6197	318	8	k	k	PROPN
ejpam-6197	318	9	antil	antil	PROPN
ejpam-6197	318	10	.	.	PUNCT
ejpam-6197	319	1	stability	stability	NOUN
ejpam-6197	319	2	of	of	ADP
ejpam-6197	319	3	the	the	DET
ejpam-6197	319	4	functional	functional	ADJ
ejpam-6197	319	5	equation	equation	NOUN
ejpam-6197	319	6	deriving	derive	VERB
ejpam-6197	319	7	from	from	ADP
ejpam-6197	319	8	quadratic	quadratic	ADJ
ejpam-6197	319	9	function	function	NOUN
ejpam-6197	319	10	in	in	ADP
ejpam-6197	319	11	banach	banach	NOUN
ejpam-6197	319	12	space	space	NOUN
ejpam-6197	319	13	.	.	PUNCT
ejpam-6197	320	1	commun	commun	PROPN
ejpam-6197	320	2	.	.	PUNCT
ejpam-6197	321	1	math	math	PROPN
ejpam-6197	321	2	.	.	PUNCT
ejpam-6197	322	1	appl	appl	PROPN
ejpam-6197	322	2	.	.	PROPN
ejpam-6197	322	3	,	,	PUNCT
ejpam-6197	322	4	15(2):557–569	15(2):557–569	NUM
ejpam-6197	322	5	,	,	PUNCT
ejpam-6197	322	6	2024	2024	NUM
ejpam-6197	322	7	.	.	PUNCT
ejpam-6197	323	1	[	[	X
ejpam-6197	323	2	12	12	NUM
ejpam-6197	323	3	]	]	X
ejpam-6197	323	4	g	g	PROPN
ejpam-6197	323	5	lu	lu	PROPN
ejpam-6197	323	6	,	,	PUNCT
ejpam-6197	323	7	q	q	PROPN
ejpam-6197	323	8	liu	liu	PROPN
ejpam-6197	323	9	,	,	PUNCT
ejpam-6197	323	10	y	y	PROPN
ejpam-6197	323	11	jin	jin	NOUN
ejpam-6197	323	12	,	,	PUNCT
ejpam-6197	323	13	and	and	CCONJ
ejpam-6197	323	14	j	j	PROPN
ejpam-6197	323	15	xie	xie	PROPN
ejpam-6197	323	16	.	.	PUNCT
ejpam-6197	324	1	3	3	NUM
ejpam-6197	324	2	-	-	PUNCT
ejpam-6197	324	3	variable	variable	NOUN
ejpam-6197	324	4	jensen	jensen	PROPN
ejpam-6197	324	5	ρ	ρ	PROPN
ejpam-6197	324	6	-	-	ADJ
ejpam-6197	324	7	functional	functional	ADJ
ejpam-6197	324	8	equations	equation	NOUN
ejpam-6197	324	9	.	.	PUNCT
ejpam-6197	325	1	j.	j.	PROPN
ejpam-6197	325	2	nonlinear	nonlinear	PROPN
ejpam-6197	325	3	sci	sci	PROPN
ejpam-6197	325	4	.	.	PUNCT
ejpam-6197	325	5	appl	appl	PROPN
ejpam-6197	325	6	.	.	PROPN
ejpam-6197	325	7	,	,	PUNCT
ejpam-6197	325	8	9(12):5995–6003	9(12):5995–6003	NUM
ejpam-6197	325	9	,	,	PUNCT
ejpam-6197	325	10	2016	2016	NUM
ejpam-6197	325	11	.	.	PUNCT
ejpam-6197	326	1	[	[	X
ejpam-6197	326	2	13	13	NUM
ejpam-6197	326	3	]	]	X
ejpam-6197	326	4	g	g	PROPN
ejpam-6197	326	5	lu	lu	PROPN
ejpam-6197	326	6	and	and	CCONJ
ejpam-6197	326	7	c	c	PROPN
ejpam-6197	326	8	park	park	NOUN
ejpam-6197	326	9	.	.	PUNCT
ejpam-6197	327	1	hyers	hyer	NOUN
ejpam-6197	327	2	-	-	PUNCT
ejpam-6197	327	3	ulam	ulam	PROPN
ejpam-6197	327	4	stability	stability	NOUN
ejpam-6197	327	5	of	of	ADP
ejpam-6197	327	6	additive	additive	ADJ
ejpam-6197	327	7	set	set	NOUN
ejpam-6197	327	8	-	-	PUNCT
ejpam-6197	327	9	valued	value	VERB
ejpam-6197	327	10	functional	functional	ADJ
ejpam-6197	327	11	equations	equation	NOUN
ejpam-6197	327	12	.	.	PUNCT
ejpam-6197	328	1	appl	appl	PROPN
ejpam-6197	328	2	.	.	PROPN
ejpam-6197	328	3	math	math	PROPN
ejpam-6197	328	4	.	.	PUNCT
ejpam-6197	329	1	lett	lett	PROPN
ejpam-6197	329	2	.	.	PROPN
ejpam-6197	329	3	,	,	PUNCT
ejpam-6197	329	4	24(8):1312–1316	24(8):1312–1316	NUM
ejpam-6197	329	5	,	,	PUNCT
ejpam-6197	329	6	2011	2011	NUM
ejpam-6197	329	7	.	.	PUNCT
ejpam-6197	330	1	[	[	X
ejpam-6197	330	2	14	14	NUM
ejpam-6197	330	3	]	]	X
ejpam-6197	330	4	g	g	PROPN
ejpam-6197	330	5	lu	lu	PROPN
ejpam-6197	330	6	and	and	CCONJ
ejpam-6197	330	7	c	c	PROPN
ejpam-6197	330	8	park	park	NOUN
ejpam-6197	330	9	.	.	PUNCT
ejpam-6197	331	1	hyers	hyer	NOUN
ejpam-6197	331	2	-	-	PUNCT
ejpam-6197	331	3	ulam	ulam	PROPN
ejpam-6197	331	4	stability	stability	NOUN
ejpam-6197	331	5	of	of	ADP
ejpam-6197	331	6	general	general	PROPN
ejpam-6197	331	7	jensen	jensen	PROPN
ejpam-6197	331	8	-	-	PUNCT
ejpam-6197	331	9	type	type	NOUN
ejpam-6197	331	10	mappings	mapping	NOUN
ejpam-6197	331	11	in	in	ADP
ejpam-6197	331	12	banach	banach	NOUN
ejpam-6197	331	13	algebras	algebra	NOUN
ejpam-6197	331	14	.	.	PUNCT
ejpam-6197	332	1	results	result	VERB
ejpam-6197	332	2	math	math	PROPN
ejpam-6197	332	3	.	.	PUNCT
ejpam-6197	333	1	,	,	PUNCT
ejpam-6197	333	2	66(3	66(3	NOUN
ejpam-6197	333	3	-	-	SYM
ejpam-6197	333	4	4):385–404	4):385–404	NUM
ejpam-6197	333	5	,	,	PUNCT
ejpam-6197	333	6	2014	2014	NUM
ejpam-6197	333	7	.	.	PUNCT
ejpam-6197	334	1	[	[	X
ejpam-6197	334	2	15	15	NUM
ejpam-6197	334	3	]	]	X
ejpam-6197	334	4	c	c	NOUN
ejpam-6197	334	5	park	park	NOUN
ejpam-6197	334	6	,	,	PUNCT
ejpam-6197	334	7	y	y	PROPN
ejpam-6197	334	8	cho	cho	PROPN
ejpam-6197	334	9	,	,	PUNCT
ejpam-6197	334	10	and	and	CCONJ
ejpam-6197	334	11	m	m	PROPN
ejpam-6197	334	12	han	han	PROPN
ejpam-6197	334	13	.	.	PUNCT
ejpam-6197	334	14	functional	functional	ADJ
ejpam-6197	334	15	inequalities	inequality	NOUN
ejpam-6197	334	16	associated	associate	VERB
ejpam-6197	334	17	with	with	ADP
ejpam-6197	334	18	jordan	jordan	PROPN
ejpam-6197	334	19	-	-	PUNCT
ejpam-6197	334	20	vonneumann	vonneumann	PROPN
ejpam-6197	334	21	-	-	PUNCT
ejpam-6197	334	22	type	type	NOUN
ejpam-6197	334	23	additive	additive	ADJ
ejpam-6197	334	24	functional	functional	ADJ
ejpam-6197	334	25	equations	equation	NOUN
ejpam-6197	334	26	.	.	PUNCT
ejpam-6197	335	1	j.	j.	PROPN
ejpam-6197	335	2	inequal	inequal	PROPN
ejpam-6197	335	3	.	.	PUNCT
ejpam-6197	336	1	appl	appl	PROPN
ejpam-6197	336	2	.	.	PROPN
ejpam-6197	336	3	,	,	PUNCT
ejpam-6197	336	4	2007(41820):1–12	2007(41820):1–12	NUM
ejpam-6197	336	5	,	,	PUNCT
ejpam-6197	336	6	2007	2007	NUM
ejpam-6197	336	7	.	.	PUNCT
ejpam-6197	337	1	[	[	X
ejpam-6197	337	2	16	16	NUM
ejpam-6197	337	3	]	]	X
ejpam-6197	337	4	z	z	PROPN
ejpam-6197	337	5	wang	wang	PROPN
ejpam-6197	337	6	.	.	PUNCT
ejpam-6197	338	1	stability	stability	NOUN
ejpam-6197	338	2	of	of	ADP
ejpam-6197	338	3	two	two	NUM
ejpam-6197	338	4	types	type	NOUN
ejpam-6197	338	5	of	of	ADP
ejpam-6197	338	6	cubic	cubic	ADJ
ejpam-6197	338	7	fuzzy	fuzzy	ADJ
ejpam-6197	338	8	set	set	NOUN
ejpam-6197	338	9	-	-	PUNCT
ejpam-6197	338	10	valued	value	VERB
ejpam-6197	338	11	functional	functional	ADJ
ejpam-6197	338	12	equations	equation	NOUN
ejpam-6197	338	13	.	.	PUNCT
ejpam-6197	339	1	results	result	VERB
ejpam-6197	339	2	math	math	PROPN
ejpam-6197	339	3	.	.	PUNCT
ejpam-6197	340	1	,	,	PUNCT
ejpam-6197	340	2	70(1	70(1	PROPN
ejpam-6197	340	3	-	-	PUNCT
ejpam-6197	340	4	2):1–14	2):1–14	PROPN
ejpam-6197	340	5	,	,	PUNCT
ejpam-6197	340	6	2016	2016	NUM
ejpam-6197	340	7	.	.	PUNCT
ejpam-6197	341	1	[	[	X
ejpam-6197	341	2	17	17	NUM
ejpam-6197	341	3	]	]	PUNCT
ejpam-6197	341	4	a	a	DET
ejpam-6197	341	5	baker	baker	NOUN
ejpam-6197	341	6	.	.	PUNCT
ejpam-6197	342	1	the	the	DET
ejpam-6197	342	2	stability	stability	NOUN
ejpam-6197	342	3	of	of	ADP
ejpam-6197	342	4	certain	certain	ADJ
ejpam-6197	342	5	functional	functional	ADJ
ejpam-6197	342	6	equation	equation	NOUN
ejpam-6197	342	7	.	.	PUNCT
ejpam-6197	343	1	proc	proc	PROPN
ejpam-6197	343	2	.	.	PUNCT
ejpam-6197	344	1	amer	amer	PROPN
ejpam-6197	344	2	.	.	PUNCT
ejpam-6197	344	3	math	math	PROPN
ejpam-6197	344	4	.	.	PUNCT
ejpam-6197	345	1	soc	soc	PROPN
ejpam-6197	345	2	.	.	PUNCT
ejpam-6197	345	3	,	,	PUNCT
ejpam-6197	345	4	112(3):729–732	112(3):729–732	NUM
ejpam-6197	345	5	,	,	PUNCT
ejpam-6197	345	6	1991	1991	NUM
ejpam-6197	345	7	.	.	PUNCT
ejpam-6197	346	1	[	[	X
ejpam-6197	346	2	18	18	NUM
ejpam-6197	346	3	]	]	X
ejpam-6197	346	4	j	j	PROPN
ejpam-6197	346	5	sikorska	sikorska	PROPN
ejpam-6197	346	6	.	.	PUNCT
ejpam-6197	347	1	on	on	ADP
ejpam-6197	347	2	a	a	DET
ejpam-6197	347	3	direct	direct	ADJ
ejpam-6197	347	4	method	method	NOUN
ejpam-6197	347	5	for	for	ADP
ejpam-6197	347	6	proving	prove	VERB
ejpam-6197	347	7	the	the	DET
ejpam-6197	347	8	hyers	hyer	NOUN
ejpam-6197	347	9	-	-	PUNCT
ejpam-6197	347	10	ulam	ulam	ADJ
ejpam-6197	347	11	stability	stability	NOUN
ejpam-6197	347	12	of	of	ADP
ejpam-6197	347	13	functional	functional	ADJ
ejpam-6197	347	14	equations	equation	NOUN
ejpam-6197	347	15	.	.	PUNCT
ejpam-6197	348	1	eur	eur	PROPN
ejpam-6197	348	2	.	.	PUNCT
ejpam-6197	349	1	j.	j.	PROPN
ejpam-6197	349	2	pure	pure	PROPN
ejpam-6197	349	3	appl	appl	PROPN
ejpam-6197	349	4	.	.	PUNCT
ejpam-6197	349	5	math	math	PROPN
ejpam-6197	349	6	.	.	PUNCT
ejpam-6197	349	7	,	,	PUNCT
ejpam-6197	350	1	372(1):99–109	372(1):99–109	NUM
ejpam-6197	350	2	,	,	PUNCT
ejpam-6197	350	3	2010	2010	NUM
ejpam-6197	350	4	.	.	PUNCT
ejpam-6197	351	1	[	[	X
ejpam-6197	351	2	19	19	NUM
ejpam-6197	351	3	]	]	X
ejpam-6197	351	4	d	d	PROPN
ejpam-6197	351	5	zhang	zhang	PROPN
ejpam-6197	351	6	,	,	PUNCT
ejpam-6197	351	7	q	q	PROPN
ejpam-6197	351	8	liu	liu	PROPN
ejpam-6197	351	9	,	,	PUNCT
ejpam-6197	351	10	j	j	PROPN
ejpam-6197	351	11	m	m	PROPN
ejpam-6197	351	12	rassias	rassias	PROPN
ejpam-6197	351	13	,	,	PUNCT
ejpam-6197	351	14	and	and	CCONJ
ejpam-6197	351	15	y	y	PROPN
ejpam-6197	351	16	li	li	PROPN
ejpam-6197	351	17	.	.	PUNCT
ejpam-6197	352	1	the	the	DET
ejpam-6197	352	2	stability	stability	NOUN
ejpam-6197	352	3	of	of	ADP
ejpam-6197	352	4	functional	functional	ADJ
ejpam-6197	352	5	equations	equation	NOUN
ejpam-6197	352	6	with	with	ADP
ejpam-6197	352	7	a	a	DET
ejpam-6197	352	8	new	new	ADJ
ejpam-6197	352	9	direct	direct	ADJ
ejpam-6197	352	10	method	method	NOUN
ejpam-6197	352	11	.	.	PUNCT
ejpam-6197	353	1	math	math	NOUN
ejpam-6197	353	2	.	.	PUNCT
ejpam-6197	353	3	,	,	PUNCT
ejpam-6197	354	1	10(7(1188)):1–14	10(7(1188)):1–14	PROPN
ejpam-6197	354	2	,	,	PUNCT
ejpam-6197	354	3	2022	2022	NUM
ejpam-6197	354	4	.	.	PUNCT
ejpam-6197	355	1	[	[	X
ejpam-6197	355	2	20	20	NUM
ejpam-6197	355	3	]	]	X
ejpam-6197	355	4	d	d	PROPN
ejpam-6197	355	5	zhang	zhang	PROPN
ejpam-6197	355	6	,	,	PUNCT
ejpam-6197	355	7	j	j	PROPN
ejpam-6197	355	8	m	m	PROPN
ejpam-6197	355	9	rassias	rassias	PROPN
ejpam-6197	355	10	,	,	PUNCT
ejpam-6197	355	11	q	q	PROPN
ejpam-6197	355	12	liu	liu	PROPN
ejpam-6197	355	13	,	,	PUNCT
ejpam-6197	355	14	and	and	CCONJ
ejpam-6197	355	15	y	y	PROPN
ejpam-6197	355	16	li	li	PROPN
ejpam-6197	355	17	.	.	PUNCT
ejpam-6197	356	1	a	a	DET
ejpam-6197	356	2	note	note	NOUN
ejpam-6197	356	3	on	on	ADP
ejpam-6197	356	4	the	the	DET
ejpam-6197	356	5	stability	stability	NOUN
ejpam-6197	356	6	of	of	ADP
ejpam-6197	356	7	functional	functional	ADJ
ejpam-6197	356	8	equations	equation	NOUN
ejpam-6197	356	9	via	via	ADP
ejpam-6197	356	10	a	a	DET
ejpam-6197	356	11	celebrated	celebrate	VERB
ejpam-6197	356	12	direct	direct	ADJ
ejpam-6197	356	13	method	method	NOUN
ejpam-6197	356	14	.	.	PUNCT
ejpam-6197	357	1	eur	eur	PROPN
ejpam-6197	357	2	.	.	PUNCT
ejpam-6197	358	1	j.	j.	PROPN
ejpam-6197	358	2	math	math	PROPN
ejpam-6197	358	3	.	.	PUNCT
ejpam-6197	359	1	anal	anal	PROPN
ejpam-6197	359	2	.	.	PUNCT
ejpam-6197	359	3	,	,	PUNCT
ejpam-6197	359	4	3(7):1–19	3(7):1–19	PROPN
ejpam-6197	359	5	,	,	PUNCT
ejpam-6197	359	6	2022	2022	NUM
ejpam-6197	359	7	.	.	PUNCT
ejpam-6197	360	1	[	[	X
ejpam-6197	360	2	21	21	NUM
ejpam-6197	360	3	]	]	PUNCT
ejpam-6197	360	4	a	a	DET
ejpam-6197	360	5	kumar	kumar	PROPN
ejpam-6197	360	6	and	and	CCONJ
ejpam-6197	360	7	m	m	PROPN
ejpam-6197	360	8	k	k	PROPN
ejpam-6197	360	9	antil	antil	PROPN
ejpam-6197	360	10	.	.	PUNCT
ejpam-6197	361	1	stability	stability	NOUN
ejpam-6197	361	2	of	of	ADP
ejpam-6197	361	3	the	the	DET
ejpam-6197	361	4	functional	functional	ADJ
ejpam-6197	361	5	equation	equation	NOUN
ejpam-6197	361	6	deriving	derive	VERB
ejpam-6197	361	7	from	from	ADP
ejpam-6197	361	8	quadratic	quadratic	ADJ
ejpam-6197	361	9	function	function	NOUN
ejpam-6197	361	10	in	in	ADP
ejpam-6197	361	11	banach	banach	NOUN
ejpam-6197	361	12	space	space	NOUN
ejpam-6197	361	13	.	.	PUNCT
ejpam-6197	362	1	communications	communication	NOUN
ejpam-6197	362	2	in	in	ADP
ejpam-6197	362	3	mathematics	mathematic	NOUN
ejpam-6197	362	4	and	and	CCONJ
ejpam-6197	362	5	applig	applig	ADJ
ejpam-6197	362	6	.	.	PUNCT
ejpam-6197	363	1	lyu	lyu	NOUN
ejpam-6197	363	2	et	et	PROPN
ejpam-6197	363	3	al	al	PROPN
ejpam-6197	363	4	.	.	PUNCT
ejpam-6197	363	5	/	/	SYM
ejpam-6197	363	6	eur	eur	PROPN
ejpam-6197	363	7	.	.	PUNCT
ejpam-6197	364	1	j.	j.	PROPN
ejpam-6197	364	2	pure	pure	PROPN
ejpam-6197	364	3	appl	appl	PROPN
ejpam-6197	364	4	.	.	PROPN
ejpam-6197	364	5	math	math	PROPN
ejpam-6197	364	6	,	,	PUNCT
ejpam-6197	364	7	18	18	NUM
ejpam-6197	364	8	(	(	PUNCT
ejpam-6197	364	9	3	3	NUM
ejpam-6197	364	10	)	)	PUNCT
ejpam-6197	364	11	(	(	PUNCT
ejpam-6197	364	12	2025	2025	NUM
ejpam-6197	364	13	)	)	PUNCT
ejpam-6197	364	14	,	,	PUNCT
ejpam-6197	364	15	6197	6197	NUM
ejpam-6197	364	16	16	16	NUM
ejpam-6197	364	17	of	of	ADP
ejpam-6197	364	18	17	17	NUM
ejpam-6197	364	19	cations	cation	NOUN
ejpam-6197	364	20	,	,	PUNCT
ejpam-6197	364	21	15(2):557–569	15(2):557–569	NUM
ejpam-6197	364	22	,	,	PUNCT
ejpam-6197	364	23	2024	2024	NUM
ejpam-6197	364	24	.	.	PUNCT
ejpam-6197	365	1	[	[	X
ejpam-6197	365	2	22	22	NUM
ejpam-6197	365	3	]	]	SYM
ejpam-6197	365	4	l	l	NOUN
ejpam-6197	365	5	shahid	shahid	PROPN
ejpam-6197	365	6	,	,	PUNCT
ejpam-6197	365	7	m	m	PROPN
ejpam-6197	365	8	rashid	rashid	PROPN
ejpam-6197	365	9	,	,	PUNCT
ejpam-6197	365	10	a	a	DET
ejpam-6197	365	11	azam	azam	PROPN
ejpam-6197	365	12	,	,	PUNCT
ejpam-6197	365	13	and	and	CCONJ
ejpam-6197	365	14	f	f	PROPN
ejpam-6197	365	15	ali	ali	PROPN
ejpam-6197	365	16	.	.	PROPN
ejpam-6197	366	1	existence	existence	PROPN
ejpam-6197	366	2	results	result	VERB
ejpam-6197	366	3	for	for	ADP
ejpam-6197	366	4	nonlinear	nonlinear	ADJ
ejpam-6197	366	5	fractional	fractional	ADJ
ejpam-6197	366	6	differential	differential	ADJ
ejpam-6197	366	7	inclusions	inclusion	NOUN
ejpam-6197	366	8	via	via	ADP
ejpam-6197	366	9	q	q	ADJ
ejpam-6197	366	10	-	-	PUNCT
ejpam-6197	366	11	rof	rof	NOUN
ejpam-6197	366	12	fixed	fix	VERB
ejpam-6197	366	13	point	point	NOUN
ejpam-6197	366	14	.	.	PUNCT
ejpam-6197	367	1	fractal	fractal	ADJ
ejpam-6197	367	2	and	and	CCONJ
ejpam-6197	367	3	fractional	fractional	ADJ
ejpam-6197	367	4	,	,	PUNCT
ejpam-6197	367	5	,	,	PUNCT
ejpam-6197	367	6	7(1):1–12	7(1):1–12	NUM
ejpam-6197	367	7	,	,	PUNCT
ejpam-6197	367	8	2023	2023	NUM
ejpam-6197	367	9	.	.	PUNCT
ejpam-6197	368	1	[	[	X
ejpam-6197	368	2	23	23	NUM
ejpam-6197	368	3	]	]	X
ejpam-6197	368	4	m	m	PROPN
ejpam-6197	368	5	rashid	rashid	PROPN
ejpam-6197	368	6	,	,	PUNCT
ejpam-6197	368	7	l	l	PROPN
ejpam-6197	368	8	shahid	shahid	PROPN
ejpam-6197	368	9	,	,	PUNCT
ejpam-6197	368	10	f	f	PROPN
ejpam-6197	368	11	dar	dar	PROPN
ejpam-6197	368	12	,	,	PUNCT
ejpam-6197	368	13	i	i	PRON
ejpam-6197	368	14	ayoob	ayoob	VERB
ejpam-6197	368	15	,	,	PUNCT
ejpam-6197	368	16	and	and	CCONJ
ejpam-6197	368	17	n	n	PRON
ejpam-6197	368	18	mlaiki	mlaiki	PROPN
ejpam-6197	368	19	.	.	PUNCT
ejpam-6197	369	1	existence	existence	NOUN
ejpam-6197	369	2	of	of	ADP
ejpam-6197	369	3	solution	solution	NOUN
ejpam-6197	369	4	of	of	ADP
ejpam-6197	369	5	a	a	DET
ejpam-6197	369	6	system	system	NOUN
ejpam-6197	369	7	of	of	ADP
ejpam-6197	369	8	non	non	ADJ
ejpam-6197	369	9	-	-	ADJ
ejpam-6197	369	10	linear	linear	ADJ
ejpam-6197	369	11	differential	differential	ADJ
ejpam-6197	369	12	inclusions	inclusion	NOUN
ejpam-6197	369	13	with	with	ADP
ejpam-6197	369	14	non	non	ADJ
ejpam-6197	369	15	-	-	ADJ
ejpam-6197	369	16	local	local	ADJ
ejpam-6197	369	17	,	,	PUNCT
ejpam-6197	369	18	integral	integral	ADJ
ejpam-6197	369	19	boundary	boundary	ADJ
ejpam-6197	369	20	conditions	condition	NOUN
ejpam-6197	369	21	via	via	ADP
ejpam-6197	369	22	fixed	fix	VERB
ejpam-6197	369	23	points	point	NOUN
ejpam-6197	369	24	of	of	ADP
ejpam-6197	369	25	hybrid	hybrid	ADJ
ejpam-6197	369	26	contractions	contraction	NOUN
ejpam-6197	369	27	.	.	PUNCT
ejpam-6197	370	1	rashid	rashid	PROPN
ejpam-6197	370	2	et	et	PROPN
ejpam-6197	370	3	al	al	PROPN
ejpam-6197	370	4	.	.	PROPN
ejpam-6197	370	5	boundaryvalueproblems	boundaryvalueproblem	NOUN
ejpam-6197	370	6	,	,	PUNCT
ejpam-6197	370	7	2024(90):1	2024(90):1	NUM
ejpam-6197	370	8	–	–	PUNCT
ejpam-6197	370	9	20	20	NUM
ejpam-6197	370	10	,	,	PUNCT
ejpam-6197	370	11	2024	2024	NUM
ejpam-6197	370	12	.	.	PUNCT
ejpam-6197	371	1	[	[	X
ejpam-6197	371	2	24	24	NUM
ejpam-6197	371	3	]	]	SYM
ejpam-6197	371	4	l	l	NOUN
ejpam-6197	371	5	aiemsomboon	aiemsomboon	NOUN
ejpam-6197	371	6	and	and	CCONJ
ejpam-6197	371	7	w	w	PROPN
ejpam-6197	371	8	sintunavarat	sintunavarat	NOUN
ejpam-6197	371	9	.	.	PUNCT
ejpam-6197	372	1	two	two	NUM
ejpam-6197	372	2	new	new	ADJ
ejpam-6197	372	3	generalised	generalise	VERB
ejpam-6197	372	4	hyperstability	hyperstability	NOUN
ejpam-6197	372	5	results	result	NOUN
ejpam-6197	372	6	for	for	ADP
ejpam-6197	372	7	the	the	DET
ejpam-6197	372	8	drygas	drygas	NOUN
ejpam-6197	372	9	functional	functional	ADJ
ejpam-6197	372	10	equation	equation	NOUN
ejpam-6197	372	11	.	.	PUNCT
ejpam-6197	373	1	bull	bull	NOUN
ejpam-6197	373	2	.	.	PUNCT
ejpam-6197	374	1	aust	aust	PROPN
ejpam-6197	374	2	.	.	PUNCT
ejpam-6197	374	3	math	math	PROPN
ejpam-6197	374	4	.	.	PUNCT
ejpam-6197	375	1	soc	soc	PROPN
ejpam-6197	375	2	.	.	PUNCT
ejpam-6197	375	3	,	,	PUNCT
ejpam-6197	375	4	95(2):269–280	95(2):269–280	PROPN
ejpam-6197	375	5	,	,	PUNCT
ejpam-6197	375	6	2017	2017	NUM
ejpam-6197	375	7	.	.	PUNCT
ejpam-6197	376	1	[	[	X
ejpam-6197	376	2	25	25	NUM
ejpam-6197	376	3	]	]	X
ejpam-6197	376	4	a	a	DET
ejpam-6197	376	5	baza	baza	NOUN
ejpam-6197	376	6	and	and	CCONJ
ejpam-6197	376	7	m	m	PROPN
ejpam-6197	376	8	rossafi	rossafi	NOUN
ejpam-6197	376	9	.	.	PUNCT
ejpam-6197	377	1	generalized	generalize	VERB
ejpam-6197	377	2	hyers	hyers	PROPN
ejpam-6197	377	3	-	-	PUNCT
ejpam-6197	377	4	ulam	ulam	PROPN
ejpam-6197	377	5	stability	stability	NOUN
ejpam-6197	377	6	of	of	ADP
ejpam-6197	377	7	quadratic	quadratic	ADJ
ejpam-6197	377	8	functional	functional	ADJ
ejpam-6197	377	9	inequality	inequality	NOUN
ejpam-6197	377	10	in	in	ADP
ejpam-6197	377	11	modular	modular	ADJ
ejpam-6197	377	12	spaces	space	NOUN
ejpam-6197	377	13	and	and	CCONJ
ejpam-6197	377	14	β	β	NOUN
ejpam-6197	377	15	-	-	ADJ
ejpam-6197	377	16	homogeneous	homogeneous	ADJ
ejpam-6197	377	17	banach	banach	NOUN
ejpam-6197	377	18	spaces	space	VERB
ejpam-6197	377	19	.	.	PUNCT
ejpam-6197	378	1	nonlinear	nonlinear	ADJ
ejpam-6197	378	2	funct	funct	NOUN
ejpam-6197	378	3	.	.	PUNCT
ejpam-6197	379	1	anal	anal	PROPN
ejpam-6197	379	2	.	.	PUNCT
ejpam-6197	379	3	appl	appl	PROPN
ejpam-6197	379	4	.	.	PROPN
ejpam-6197	380	1	,	,	PUNCT
ejpam-6197	380	2	29(1):295–306	29(1):295–306	NUM
ejpam-6197	380	3	,	,	PUNCT
ejpam-6197	380	4	2024	2024	NUM
ejpam-6197	380	5	.	.	PUNCT
ejpam-6197	381	1	[	[	X
ejpam-6197	381	2	26	26	NUM
ejpam-6197	381	3	]	]	PUNCT
ejpam-6197	381	4	s	s	PART
ejpam-6197	381	5	bowmiya	bowmiya	NOUN
ejpam-6197	381	6	,	,	PUNCT
ejpam-6197	381	7	g	g	PROPN
ejpam-6197	381	8	balasubramanian	balasubramanian	PROPN
ejpam-6197	381	9	,	,	PUNCT
ejpam-6197	381	10	v	v	ADJ
ejpam-6197	381	11	govindan	govindan	PROPN
ejpam-6197	381	12	,	,	PUNCT
ejpam-6197	381	13	m	m	VERB
ejpam-6197	381	14	donganont	donganont	NOUN
ejpam-6197	381	15	,	,	PUNCT
ejpam-6197	381	16	and	and	CCONJ
ejpam-6197	381	17	h	h	NOUN
ejpam-6197	381	18	byeon	byeon	NOUN
ejpam-6197	381	19	.	.	PUNCT
ejpam-6197	382	1	generalized	generalize	VERB
ejpam-6197	382	2	linear	linear	PROPN
ejpam-6197	382	3	differential	differential	NOUN
ejpam-6197	382	4	equation	equation	NOUN
ejpam-6197	382	5	using	use	VERB
ejpam-6197	382	6	hyers	hyers	PROPN
ejpam-6197	382	7	-	-	PUNCT
ejpam-6197	382	8	ulam	ulam	PROPN
ejpam-6197	382	9	stability	stability	PROPN
ejpam-6197	382	10	approach	approach	NOUN
ejpam-6197	382	11	.	.	PUNCT
ejpam-6197	383	1	eur	eur	PROPN
ejpam-6197	383	2	.	.	PUNCT
ejpam-6197	384	1	j.	j.	PROPN
ejpam-6197	384	2	pure	pure	PROPN
ejpam-6197	384	3	appl	appl	PROPN
ejpam-6197	384	4	.	.	PUNCT
ejpam-6197	384	5	math	math	PROPN
ejpam-6197	384	6	.	.	PUNCT
ejpam-6197	384	7	,	,	PUNCT
ejpam-6197	384	8	17(4):3415–3435	17(4):3415–3435	NUM
ejpam-6197	384	9	,	,	PUNCT
ejpam-6197	384	10	2024	2024	NUM
ejpam-6197	384	11	.	.	PUNCT
ejpam-6197	385	1	[	[	X
ejpam-6197	385	2	27	27	NUM
ejpam-6197	385	3	]	]	SYM
ejpam-6197	385	4	s	s	PART
ejpam-6197	385	5	bowmiya	bowmiya	NOUN
ejpam-6197	385	6	,	,	PUNCT
ejpam-6197	385	7	g	g	PROPN
ejpam-6197	385	8	balasubramanian	balasubramanian	PROPN
ejpam-6197	385	9	,	,	PUNCT
ejpam-6197	385	10	v	v	ADJ
ejpam-6197	385	11	govindan	govindan	PROPN
ejpam-6197	385	12	,	,	PUNCT
ejpam-6197	385	13	m	m	VERB
ejpam-6197	385	14	donganont	donganont	NOUN
ejpam-6197	385	15	,	,	PUNCT
ejpam-6197	385	16	and	and	CCONJ
ejpam-6197	385	17	h	h	NOUN
ejpam-6197	385	18	byeon	byeon	NOUN
ejpam-6197	385	19	.	.	PUNCT
ejpam-6197	386	1	hyersulam	hyersulam	PROPN
ejpam-6197	386	2	stability	stability	NOUN
ejpam-6197	386	3	of	of	ADP
ejpam-6197	386	4	fifth	fifth	ADJ
ejpam-6197	386	5	order	order	NOUN
ejpam-6197	386	6	linear	linear	PROPN
ejpam-6197	386	7	differential	differential	NOUN
ejpam-6197	386	8	equations	equation	NOUN
ejpam-6197	386	9	.	.	PUNCT
ejpam-6197	387	1	eur	eur	PROPN
ejpam-6197	387	2	.	.	PUNCT
ejpam-6197	388	1	j.	j.	PROPN
ejpam-6197	388	2	pure	pure	PROPN
ejpam-6197	388	3	appl	appl	PROPN
ejpam-6197	388	4	.	.	PUNCT
ejpam-6197	388	5	math	math	PROPN
ejpam-6197	388	6	.	.	PUNCT
ejpam-6197	388	7	,	,	PUNCT
ejpam-6197	388	8	17(4):3585–3609	17(4):3585–3609	NUM
ejpam-6197	388	9	,	,	PUNCT
ejpam-6197	388	10	2024	2024	NUM
ejpam-6197	388	11	.	.	PUNCT
ejpam-6197	389	1	[	[	X
ejpam-6197	389	2	28	28	NUM
ejpam-6197	389	3	]	]	X
ejpam-6197	389	4	i	i	PROPN
ejpam-6197	389	5	el	el	PROPN
ejpam-6197	389	6	-	-	PUNCT
ejpam-6197	389	7	fassii	fassii	NOUN
ejpam-6197	389	8	.	.	PUNCT
ejpam-6197	390	1	generalized	generalized	ADJ
ejpam-6197	390	2	hyperstability	hyperstability	NOUN
ejpam-6197	390	3	of	of	ADP
ejpam-6197	390	4	a	a	DET
ejpam-6197	390	5	drygas	drygas	NOUN
ejpam-6197	390	6	functional	functional	ADJ
ejpam-6197	390	7	equation	equation	NOUN
ejpam-6197	390	8	on	on	ADP
ejpam-6197	390	9	a	a	DET
ejpam-6197	390	10	restricted	restricted	ADJ
ejpam-6197	390	11	domain	domain	NOUN
ejpam-6197	390	12	using	use	VERB
ejpam-6197	390	13	brzdek	brzdek	PROPN
ejpam-6197	390	14	’s	’s	PART
ejpam-6197	390	15	fixed	fix	VERB
ejpam-6197	390	16	point	point	NOUN
ejpam-6197	390	17	theorem	theorem	VERB
ejpam-6197	390	18	.	.	PUNCT
ejpam-6197	391	1	j.	j.	PROPN
ejpam-6197	391	2	fixed	fix	VERB
ejpam-6197	391	3	point	point	PROPN
ejpam-6197	391	4	theory	theory	NOUN
ejpam-6197	391	5	appl	appl	PROPN
ejpam-6197	391	6	.	.	PROPN
ejpam-6197	391	7	,	,	PUNCT
ejpam-6197	391	8	19(4):2529	19(4):2529	NUM
ejpam-6197	391	9	–	–	PUNCT
ejpam-6197	391	10	2540	2540	NUM
ejpam-6197	391	11	,	,	PUNCT
ejpam-6197	391	12	2017	2017	NUM
ejpam-6197	391	13	.	.	PUNCT
ejpam-6197	392	1	[	[	X
ejpam-6197	392	2	29	29	NUM
ejpam-6197	392	3	]	]	SYM
ejpam-6197	392	4	g	g	PROPN
ejpam-6197	392	5	l	l	NOUN
ejpam-6197	392	6	forti	forti	PROPN
ejpam-6197	392	7	.	.	PUNCT
ejpam-6197	393	1	stability	stability	NOUN
ejpam-6197	393	2	of	of	ADP
ejpam-6197	393	3	quadratic	quadratic	ADJ
ejpam-6197	393	4	and	and	CCONJ
ejpam-6197	393	5	drygas	drygas	ADJ
ejpam-6197	393	6	functional	functional	ADJ
ejpam-6197	393	7	equations	equation	NOUN
ejpam-6197	393	8	with	with	ADP
ejpam-6197	393	9	an	an	DET
ejpam-6197	393	10	application	application	NOUN
ejpam-6197	393	11	for	for	ADP
ejpam-6197	393	12	solving	solve	VERB
ejpam-6197	393	13	alternative	alternative	ADJ
ejpam-6197	393	14	quadratic	quadratic	ADJ
ejpam-6197	393	15	equation	equation	NOUN
ejpam-6197	393	16	.	.	PUNCT
ejpam-6197	394	1	handbook	handbook	NOUN
ejpam-6197	394	2	of	of	ADP
ejpam-6197	394	3	functional	functional	ADJ
ejpam-6197	394	4	equations	equation	NOUN
ejpam-6197	394	5	,	,	PUNCT
ejpam-6197	394	6	96:155	96:155	NUM
ejpam-6197	394	7	–	–	PUNCT
ejpam-6197	394	8	179	179	NUM
ejpam-6197	394	9	,	,	PUNCT
ejpam-6197	394	10	2014	2014	NUM
ejpam-6197	394	11	.	.	PUNCT
ejpam-6197	395	1	[	[	X
ejpam-6197	395	2	30	30	NUM
ejpam-6197	395	3	]	]	X
ejpam-6197	395	4	z	z	NOUN
ejpam-6197	395	5	jin	jin	NOUN
ejpam-6197	395	6	and	and	CCONJ
ejpam-6197	395	7	j	j	PROPN
ejpam-6197	395	8	wu	wu	PROPN
ejpam-6197	395	9	.	.	PUNCT
ejpam-6197	396	1	ulam	ulam	PROPN
ejpam-6197	396	2	stability	stability	NOUN
ejpam-6197	396	3	of	of	ADP
ejpam-6197	396	4	some	some	DET
ejpam-6197	396	5	fuzzy	fuzzy	ADJ
ejpam-6197	396	6	number	number	NOUN
ejpam-6197	396	7	-	-	PUNCT
ejpam-6197	396	8	valued	value	VERB
ejpam-6197	396	9	functional	functional	ADJ
ejpam-6197	396	10	equations	equation	NOUN
ejpam-6197	396	11	and	and	CCONJ
ejpam-6197	396	12	drygas	drygas	NOUN
ejpam-6197	396	13	type	type	NOUN
ejpam-6197	396	14	functional	functional	ADJ
ejpam-6197	396	15	equation	equation	NOUN
ejpam-6197	396	16	.	.	PUNCT
ejpam-6197	397	1	j.	j.	PROPN
ejpam-6197	397	2	southwest	southwest	PROPN
ejpam-6197	397	3	univ	univ	PROPN
ejpam-6197	397	4	.	.	PROPN
ejpam-6197	397	5	,	,	PUNCT
ejpam-6197	397	6	4:59–66	4:59–66	NOUN
ejpam-6197	397	7	,	,	PUNCT
ejpam-6197	397	8	2018	2018	NUM
ejpam-6197	397	9	.	.	PUNCT
ejpam-6197	398	1	[	[	X
ejpam-6197	398	2	31	31	NUM
ejpam-6197	398	3	]	]	X
ejpam-6197	398	4	s	s	X
ejpam-6197	398	5	jung	jung	NOUN
ejpam-6197	398	6	and	and	CCONJ
ejpam-6197	398	7	p	p	PROPN
ejpam-6197	398	8	k	k	PROPN
ejpam-6197	398	9	sahoo	sahoo	PROPN
ejpam-6197	398	10	.	.	PUNCT
ejpam-6197	399	1	stability	stability	NOUN
ejpam-6197	399	2	of	of	ADP
ejpam-6197	399	3	a	a	DET
ejpam-6197	399	4	functional	functional	ADJ
ejpam-6197	399	5	equation	equation	NOUN
ejpam-6197	399	6	of	of	ADP
ejpam-6197	399	7	drygas	drygas	NOUN
ejpam-6197	399	8	.	.	PUNCT
ejpam-6197	400	1	aequationes	aequatione	NOUN
ejpam-6197	400	2	math	math	PROPN
ejpam-6197	400	3	.	.	PUNCT
ejpam-6197	400	4	,	,	PUNCT
ejpam-6197	400	5	64(3):263–273	64(3):263–273	PROPN
ejpam-6197	400	6	,	,	PUNCT
ejpam-6197	400	7	2002	2002	NUM
ejpam-6197	400	8	.	.	PUNCT
ejpam-6197	401	1	[	[	X
ejpam-6197	401	2	32	32	NUM
ejpam-6197	401	3	]	]	SYM
ejpam-6197	401	4	b	b	X
ejpam-6197	401	5	v	v	NUM
ejpam-6197	401	6	senthil	senthil	PROPN
ejpam-6197	401	7	kumar	kumar	PROPN
ejpam-6197	401	8	,	,	PUNCT
ejpam-6197	401	9	h	h	PROPN
ejpam-6197	401	10	dutta	dutta	PROPN
ejpam-6197	401	11	,	,	PUNCT
ejpam-6197	401	12	and	and	CCONJ
ejpam-6197	401	13	s	s	NOUN
ejpam-6197	401	14	sabarinathan	sabarinathan	NOUN
ejpam-6197	401	15	.	.	PUNCT
ejpam-6197	402	1	modular	modular	ADJ
ejpam-6197	402	2	stabilities	stability	NOUN
ejpam-6197	402	3	of	of	ADP
ejpam-6197	402	4	a	a	DET
ejpam-6197	402	5	reciprocal	reciprocal	ADJ
ejpam-6197	402	6	second	second	ADJ
ejpam-6197	402	7	power	power	NOUN
ejpam-6197	402	8	functional	functional	ADJ
ejpam-6197	402	9	equation	equation	NOUN
ejpam-6197	402	10	.	.	PUNCT
ejpam-6197	403	1	eur	eur	PROPN
ejpam-6197	403	2	.	.	PUNCT
ejpam-6197	404	1	j.	j.	PROPN
ejpam-6197	404	2	pure	pure	PROPN
ejpam-6197	404	3	appl	appl	PROPN
ejpam-6197	404	4	.	.	PUNCT
ejpam-6197	404	5	math	math	PROPN
ejpam-6197	404	6	.	.	PUNCT
ejpam-6197	404	7	,	,	PUNCT
ejpam-6197	405	1	13(5):1162–1175	13(5):1162–1175	NUM
ejpam-6197	405	2	,	,	PUNCT
ejpam-6197	405	3	2020	2020	NUM
ejpam-6197	405	4	.	.	PUNCT
ejpam-6197	406	1	[	[	X
ejpam-6197	406	2	33	33	NUM
ejpam-6197	406	3	]	]	PUNCT
ejpam-6197	406	4	m	m	VERB
ejpam-6197	406	5	piszczek	piszczek	NOUN
ejpam-6197	406	6	and	and	CCONJ
ejpam-6197	406	7	j.	j.	PROPN
ejpam-6197	406	8	szczawi	szczawi	PROPN
ejpam-6197	406	9	.	.	PUNCT
ejpam-6197	407	1	stability	stability	NOUN
ejpam-6197	407	2	of	of	ADP
ejpam-6197	407	3	the	the	DET
ejpam-6197	407	4	drygas	drygas	NOUN
ejpam-6197	407	5	functional	functional	ADJ
ejpam-6197	407	6	equation	equation	NOUN
ejpam-6197	407	7	on	on	ADP
ejpam-6197	407	8	restricted	restricted	ADJ
ejpam-6197	407	9	domain	domain	NOUN
ejpam-6197	407	10	.	.	PUNCT
ejpam-6197	408	1	results	result	VERB
ejpam-6197	408	2	math	math	PROPN
ejpam-6197	408	3	.	.	PUNCT
ejpam-6197	408	4	,	,	PUNCT
ejpam-6197	408	5	68(1):11–24	68(1):11–24	NUM
ejpam-6197	408	6	,	,	PUNCT
ejpam-6197	408	7	2015	2015	NUM
ejpam-6197	408	8	.	.	PUNCT
ejpam-6197	409	1	[	[	X
ejpam-6197	409	2	34	34	NUM
ejpam-6197	409	3	]	]	X
ejpam-6197	409	4	w	w	NOUN
ejpam-6197	409	5	smajdor	smajdor	NOUN
ejpam-6197	409	6	.	.	PUNCT
ejpam-6197	410	1	on	on	ADP
ejpam-6197	410	2	set	set	NOUN
ejpam-6197	410	3	-	-	PUNCT
ejpam-6197	410	4	valued	value	VERB
ejpam-6197	410	5	solutions	solution	NOUN
ejpam-6197	410	6	of	of	ADP
ejpam-6197	410	7	a	a	DET
ejpam-6197	410	8	functional	functional	ADJ
ejpam-6197	410	9	equation	equation	NOUN
ejpam-6197	410	10	of	of	ADP
ejpam-6197	410	11	drygas	drygas	NOUN
ejpam-6197	410	12	.	.	PUNCT
ejpam-6197	411	1	aequationes	aequatione	NOUN
ejpam-6197	411	2	math	math	PROPN
ejpam-6197	411	3	.	.	PUNCT
ejpam-6197	411	4	,	,	PUNCT
ejpam-6197	412	1	77(1	77(1	NOUN
ejpam-6197	412	2	-	-	SYM
ejpam-6197	412	3	2):89–97	2):89–97	NUM
ejpam-6197	412	4	,	,	PUNCT
ejpam-6197	412	5	2009	2009	NUM
ejpam-6197	412	6	.	.	PUNCT
ejpam-6197	413	1	[	[	X
ejpam-6197	413	2	35	35	NUM
ejpam-6197	413	3	]	]	X
ejpam-6197	413	4	d	d	PROPN
ejpam-6197	413	5	yang	yang	PROPN
ejpam-6197	413	6	.	.	PUNCT
ejpam-6197	414	1	remarks	remark	NOUN
ejpam-6197	414	2	on	on	ADP
ejpam-6197	414	3	the	the	DET
ejpam-6197	414	4	stability	stability	NOUN
ejpam-6197	414	5	of	of	ADP
ejpam-6197	414	6	drygas	drygas	NOUN
ejpam-6197	414	7	equation	equation	NOUN
ejpam-6197	414	8	and	and	CCONJ
ejpam-6197	414	9	the	the	DET
ejpam-6197	414	10	pexider	pexider	NOUN
ejpam-6197	414	11	quadratic	quadratic	ADJ
ejpam-6197	414	12	equation	equation	NOUN
ejpam-6197	414	13	.	.	PUNCT
ejpam-6197	415	1	aequationes	aequatione	NOUN
ejpam-6197	415	2	math	math	PROPN
ejpam-6197	415	3	.	.	PUNCT
ejpam-6197	416	1	,	,	PUNCT
ejpam-6197	416	2	68(1	68(1	NOUN
ejpam-6197	416	3	-	-	SYM
ejpam-6197	416	4	2):108–116	2):108–116	NUM
ejpam-6197	416	5	,	,	PUNCT
ejpam-6197	416	6	2004	2004	NUM
ejpam-6197	416	7	.	.	PUNCT
ejpam-6197	417	1	[	[	X
ejpam-6197	417	2	36	36	NUM
ejpam-6197	417	3	]	]	X
ejpam-6197	417	4	c	c	PROPN
ejpam-6197	417	5	park	park	NOUN
ejpam-6197	417	6	.	.	PUNCT
ejpam-6197	418	1	additive	additive	VERB
ejpam-6197	418	2	ρ	ρ	ADJ
ejpam-6197	418	3	-	-	ADJ
ejpam-6197	418	4	functional	functional	ADJ
ejpam-6197	418	5	inequalities	inequality	NOUN
ejpam-6197	418	6	and	and	CCONJ
ejpam-6197	418	7	equations	equation	NOUN
ejpam-6197	418	8	.	.	PUNCT
ejpam-6197	419	1	j.	j.	PROPN
ejpam-6197	419	2	math	math	PROPN
ejpam-6197	419	3	.	.	PUNCT
ejpam-6197	420	1	inequal	inequal	ADJ
ejpam-6197	420	2	.	.	PUNCT
ejpam-6197	420	3	,	,	PUNCT
ejpam-6197	420	4	9(1):17	9(1):17	NUM
ejpam-6197	420	5	–	–	PUNCT
ejpam-6197	420	6	26	26	NUM
ejpam-6197	420	7	,	,	PUNCT
ejpam-6197	420	8	2015	2015	NUM
ejpam-6197	420	9	.	.	PUNCT
ejpam-6197	421	1	[	[	X
ejpam-6197	421	2	37	37	NUM
ejpam-6197	421	3	]	]	X
ejpam-6197	421	4	j	j	PROPN
ejpam-6197	421	5	choi	choi	PROPN
ejpam-6197	421	6	,	,	PUNCT
ejpam-6197	421	7	j	j	PROPN
ejpam-6197	421	8	seong	seong	PROPN
ejpam-6197	421	9	,	,	PUNCT
ejpam-6197	421	10	and	and	CCONJ
ejpam-6197	421	11	c	c	PROPN
ejpam-6197	421	12	park	park	NOUN
ejpam-6197	421	13	.	.	PUNCT
ejpam-6197	422	1	additive	additive	VERB
ejpam-6197	422	2	ρ	ρ	ADJ
ejpam-6197	422	3	-	-	ADJ
ejpam-6197	422	4	functional	functional	ADJ
ejpam-6197	422	5	inequalities	inequality	NOUN
ejpam-6197	422	6	in	in	ADP
ejpam-6197	422	7	normed	normed	ADJ
ejpam-6197	422	8	spaces	space	NOUN
ejpam-6197	422	9	.	.	PUNCT
ejpam-6197	423	1	j.	j.	PROPN
ejpam-6197	423	2	nonlinear	nonlinear	PROPN
ejpam-6197	423	3	sci	sci	PROPN
ejpam-6197	423	4	.	.	PUNCT
ejpam-6197	423	5	appl	appl	PROPN
ejpam-6197	423	6	.	.	PROPN
ejpam-6197	423	7	,	,	PUNCT
ejpam-6197	423	8	9(1):247–253	9(1):247–253	NUM
ejpam-6197	423	9	,	,	PUNCT
ejpam-6197	423	10	2016	2016	NUM
ejpam-6197	423	11	.	.	PUNCT
ejpam-6197	424	1	[	[	X
ejpam-6197	424	2	38	38	NUM
ejpam-6197	424	3	]	]	SYM
ejpam-6197	424	4	s	s	VERB
ejpam-6197	424	5	nawaz	nawaz	NOUN
ejpam-6197	424	6	,	,	PUNCT
ejpam-6197	424	7	a	a	DET
ejpam-6197	424	8	bariq	bariq	NOUN
ejpam-6197	424	9	,	,	PUNCT
ejpam-6197	424	10	a	a	DET
ejpam-6197	424	11	batool	batool	NOUN
ejpam-6197	424	12	,	,	PUNCT
ejpam-6197	424	13	and	and	CCONJ
ejpam-6197	424	14	a	a	DET
ejpam-6197	424	15	akgül	akgül	PROPN
ejpam-6197	424	16	.	.	PUNCT
ejpam-6197	425	1	generalized	generalize	VERB
ejpam-6197	425	2	hyers	hyers	PROPN
ejpam-6197	425	3	-	-	PUNCT
ejpam-6197	425	4	ulam	ulam	PROPN
ejpam-6197	425	5	stability	stability	NOUN
ejpam-6197	425	6	of	of	ADP
ejpam-6197	425	7	ρ	ρ	ADJ
ejpam-6197	425	8	-	-	ADJ
ejpam-6197	425	9	functional	functional	ADJ
ejpam-6197	425	10	inequalities	inequality	NOUN
ejpam-6197	425	11	.	.	PUNCT
ejpam-6197	426	1	j.	j.	PROPN
ejpam-6197	426	2	inequal	inequal	PROPN
ejpam-6197	426	3	.	.	PUNCT
ejpam-6197	427	1	appl	appl	PROPN
ejpam-6197	427	2	.	.	PROPN
ejpam-6197	427	3	,	,	PUNCT
ejpam-6197	427	4	2023(135):1–18	2023(135):1–18	NUM
ejpam-6197	427	5	,	,	PUNCT
ejpam-6197	427	6	2023	2023	NUM
ejpam-6197	427	7	.	.	PUNCT
ejpam-6197	428	1	[	[	X
ejpam-6197	428	2	39	39	NUM
ejpam-6197	428	3	]	]	X
ejpam-6197	428	4	b	b	X
ejpam-6197	428	5	ebanks	ebank	NOUN
ejpam-6197	428	6	,	,	PUNCT
ejpam-6197	428	7	p	p	NOUN
ejpam-6197	428	8	kannappan	kannappan	NOUN
ejpam-6197	428	9	,	,	PUNCT
ejpam-6197	428	10	and	and	CCONJ
ejpam-6197	428	11	p	p	PROPN
ejpam-6197	428	12	sahoo	sahoo	PROPN
ejpam-6197	428	13	.	.	PUNCT
ejpam-6197	429	1	a	a	DET
ejpam-6197	429	2	common	common	ADJ
ejpam-6197	429	3	generaliztion	generaliztion	NOUN
ejpam-6197	429	4	of	of	ADP
ejpam-6197	429	5	functional	functional	ADJ
ejpam-6197	429	6	g.	g.	PROPN
ejpam-6197	429	7	lyu	lyu	PROPN
ejpam-6197	429	8	et	et	PROPN
ejpam-6197	429	9	al	al	PROPN
ejpam-6197	429	10	.	.	PUNCT
ejpam-6197	429	11	/	/	SYM
ejpam-6197	429	12	eur	eur	PROPN
ejpam-6197	429	13	.	.	PUNCT
ejpam-6197	430	1	j.	j.	PROPN
ejpam-6197	430	2	pure	pure	PROPN
ejpam-6197	430	3	appl	appl	PROPN
ejpam-6197	430	4	.	.	PROPN
ejpam-6197	430	5	math	math	PROPN
ejpam-6197	430	6	,	,	PUNCT
ejpam-6197	430	7	18	18	NUM
ejpam-6197	430	8	(	(	PUNCT
ejpam-6197	430	9	3	3	NUM
ejpam-6197	430	10	)	)	PUNCT
ejpam-6197	430	11	(	(	PUNCT
ejpam-6197	430	12	2025	2025	NUM
ejpam-6197	430	13	)	)	PUNCT
ejpam-6197	430	14	,	,	PUNCT
ejpam-6197	430	15	6197	6197	NUM
ejpam-6197	430	16	17	17	NUM
ejpam-6197	430	17	of	of	ADP
ejpam-6197	430	18	17	17	NUM
ejpam-6197	430	19	equations	equation	NOUN
ejpam-6197	430	20	charactering	character	VERB
ejpam-6197	430	21	normed	normed	ADJ
ejpam-6197	430	22	and	and	CCONJ
ejpam-6197	430	23	quasi	quasi	ADJ
ejpam-6197	430	24	-	-	ADJ
ejpam-6197	430	25	inner	inner	ADJ
ejpam-6197	430	26	-	-	PUNCT
ejpam-6197	430	27	product	product	NOUN
ejpam-6197	430	28	spaces	space	NOUN
ejpam-6197	430	29	.	.	PUNCT
ejpam-6197	431	1	canad	canad	PROPN
ejpam-6197	431	2	.	.	PUNCT
ejpam-6197	432	1	math	math	NOUN
ejpam-6197	432	2	.	.	PUNCT
ejpam-6197	433	1	bull	bull	PROPN
ejpam-6197	433	2	.	.	PUNCT
ejpam-6197	433	3	,	,	PUNCT
ejpam-6197	433	4	35(1):321–327	35(1):321–327	PROPN
ejpam-6197	433	5	,	,	PUNCT
ejpam-6197	433	6	1992	1992	NUM
ejpam-6197	433	7	.	.	PUNCT
ejpam-6197	434	1	[	[	X
ejpam-6197	434	2	40	40	NUM
ejpam-6197	434	3	]	]	X
ejpam-6197	434	4	b	b	X
ejpam-6197	434	5	ebanks	ebanks	PROPN
ejpam-6197	434	6	.	.	PUNCT
ejpam-6197	435	1	generalized	generalize	VERB
ejpam-6197	435	2	cauchy	cauchy	ADJ
ejpam-6197	435	3	difference	difference	NOUN
ejpam-6197	435	4	functional	functional	ADJ
ejpam-6197	435	5	equations	equation	NOUN
ejpam-6197	435	6	.	.	PUNCT
ejpam-6197	436	1	aequ	aequ	PROPN
ejpam-6197	436	2	.	.	PUNCT
ejpam-6197	437	1	math	math	NOUN
ejpam-6197	437	2	.	.	PUNCT
ejpam-6197	437	3	,	,	PUNCT
ejpam-6197	438	1	70(1):154–176	70(1):154–176	PROPN
ejpam-6197	438	2	,	,	PUNCT
ejpam-6197	438	3	2005	2005	NUM
ejpam-6197	438	4	.	.	PUNCT