id	sid	tid	token	lemma	pos
ejpam-6077	1	1	european	european	PROPN
ejpam-6077	1	2	journal	journal	PROPN
ejpam-6077	1	3	of	of	ADP
ejpam-6077	1	4	pure	pure	ADJ
ejpam-6077	1	5	and	and	CCONJ
ejpam-6077	1	6	applied	applied	ADJ
ejpam-6077	1	7	mathematics	mathematic	NOUN
ejpam-6077	1	8	2025	2025	NUM
ejpam-6077	1	9	,	,	PUNCT
ejpam-6077	1	10	vol	vol	NOUN
ejpam-6077	1	11	.	.	PROPN
ejpam-6077	1	12	18	18	NUM
ejpam-6077	1	13	,	,	PUNCT
ejpam-6077	1	14	issue	issue	NOUN
ejpam-6077	1	15	2	2	NUM
ejpam-6077	1	16	,	,	PUNCT
ejpam-6077	1	17	article	article	NOUN
ejpam-6077	1	18	number	number	NOUN
ejpam-6077	1	19	6077	6077	NUM
ejpam-6077	1	20	issn	issn	PROPN
ejpam-6077	1	21	1307	1307	NUM
ejpam-6077	1	22	-	-	SYM
ejpam-6077	1	23	5543	5543	NUM
ejpam-6077	1	24	–	–	PUNCT
ejpam-6077	1	25	ejpam.com	ejpam.com	X
ejpam-6077	1	26	published	publish	VERB
ejpam-6077	1	27	by	by	ADP
ejpam-6077	1	28	new	new	PROPN
ejpam-6077	1	29	york	york	PROPN
ejpam-6077	1	30	business	business	PROPN
ejpam-6077	1	31	global	global	ADJ
ejpam-6077	1	32	existence	existence	NOUN
ejpam-6077	1	33	and	and	CCONJ
ejpam-6077	1	34	uniqueness	uniqueness	NOUN
ejpam-6077	1	35	of	of	ADP
ejpam-6077	1	36	fixed	fix	VERB
ejpam-6077	1	37	points	point	NOUN
ejpam-6077	1	38	in	in	ADP
ejpam-6077	1	39	mr−metric	mr−metric	ADJ
ejpam-6077	1	40	spaces	space	NOUN
ejpam-6077	1	41	and	and	CCONJ
ejpam-6077	1	42	their	their	PRON
ejpam-6077	1	43	applications	application	NOUN
ejpam-6077	1	44	abed	abe	VERB
ejpam-6077	1	45	al	al	PROPN
ejpam-6077	1	46	-	-	PUNCT
ejpam-6077	1	47	rahman	rahman	PROPN
ejpam-6077	1	48	m.	m.	NOUN
ejpam-6077	1	49	malkawi	malkawi	ADP
ejpam-6077	1	50	department	department	PROPN
ejpam-6077	1	51	of	of	ADP
ejpam-6077	1	52	mathematics	mathematic	NOUN
ejpam-6077	1	53	,	,	PUNCT
ejpam-6077	1	54	faculty	faculty	NOUN
ejpam-6077	1	55	of	of	ADP
ejpam-6077	1	56	arts	art	NOUN
ejpam-6077	1	57	and	and	CCONJ
ejpam-6077	1	58	science	science	NOUN
ejpam-6077	1	59	,	,	PUNCT
ejpam-6077	1	60	amman	amman	PROPN
ejpam-6077	1	61	arab	arab	PROPN
ejpam-6077	1	62	university	university	PROPN
ejpam-6077	1	63	,	,	PUNCT
ejpam-6077	1	64	amman	amman	PROPN
ejpam-6077	1	65	11953	11953	NUM
ejpam-6077	1	66	,	,	PUNCT
ejpam-6077	1	67	jordan	jordan	PROPN
ejpam-6077	1	68	abstract	abstract	PROPN
ejpam-6077	1	69	.	.	PUNCT
ejpam-6077	2	1	this	this	DET
ejpam-6077	2	2	paper	paper	NOUN
ejpam-6077	2	3	investigates	investigate	VERB
ejpam-6077	2	4	fixed	fix	VERB
ejpam-6077	2	5	-	-	PUNCT
ejpam-6077	2	6	point	point	NOUN
ejpam-6077	2	7	theorems	theorem	NOUN
ejpam-6077	2	8	within	within	ADP
ejpam-6077	2	9	mr	mr	PROPN
ejpam-6077	2	10	-	-	PUNCT
ejpam-6077	2	11	metric	metric	ADJ
ejpam-6077	2	12	spaces	space	NOUN
ejpam-6077	2	13	,	,	PUNCT
ejpam-6077	2	14	an	an	DET
ejpam-6077	2	15	extension	extension	NOUN
ejpam-6077	2	16	of	of	ADP
ejpam-6077	2	17	standard	standard	ADJ
ejpam-6077	2	18	metric	metric	ADJ
ejpam-6077	2	19	spaces	space	NOUN
ejpam-6077	2	20	,	,	PUNCT
ejpam-6077	2	21	emphasizing	emphasize	VERB
ejpam-6077	2	22	the	the	DET
ejpam-6077	2	23	existence	existence	NOUN
ejpam-6077	2	24	and	and	CCONJ
ejpam-6077	2	25	uniqueness	uniqueness	NOUN
ejpam-6077	2	26	of	of	ADP
ejpam-6077	2	27	fixed	fix	VERB
ejpam-6077	2	28	points	point	NOUN
ejpam-6077	2	29	for	for	ADP
ejpam-6077	2	30	continuous	continuous	ADJ
ejpam-6077	2	31	mappings	mapping	NOUN
ejpam-6077	2	32	s	s	PART
ejpam-6077	2	33	:	:	PUNCT
ejpam-6077	2	34	x	x	SYM
ejpam-6077	2	35	→	→	SYM
ejpam-6077	2	36	x	x	X
ejpam-6077	2	37	,	,	PUNCT
ejpam-6077	2	38	where	where	SCONJ
ejpam-6077	2	39	x	x	PRON
ejpam-6077	2	40	is	be	AUX
ejpam-6077	2	41	a	a	DET
ejpam-6077	2	42	closed	closed	ADJ
ejpam-6077	2	43	,	,	PUNCT
ejpam-6077	2	44	bounded	bound	VERB
ejpam-6077	2	45	,	,	PUNCT
ejpam-6077	2	46	and	and	CCONJ
ejpam-6077	2	47	convex	convex	PROPN
ejpam-6077	2	48	subset	subset	NOUN
ejpam-6077	2	49	of	of	ADP
ejpam-6077	2	50	a	a	DET
ejpam-6077	2	51	banach	banach	NOUN
ejpam-6077	2	52	space	space	NOUN
ejpam-6077	2	53	(	(	PUNCT
ejpam-6077	2	54	e	e	NOUN
ejpam-6077	2	55	,	,	PUNCT
ejpam-6077	2	56	∥	∥	X
ejpam-6077	2	57	·	·	PUNCT
ejpam-6077	2	58	∥	∥	NUM
ejpam-6077	2	59	)	)	PUNCT
ejpam-6077	2	60	.	.	PUNCT
ejpam-6077	3	1	the	the	DET
ejpam-6077	3	2	study	study	NOUN
ejpam-6077	3	3	establishes	establish	VERB
ejpam-6077	3	4	that	that	SCONJ
ejpam-6077	3	5	if	if	SCONJ
ejpam-6077	3	6	s	s	PRON
ejpam-6077	3	7	satisfies	satisfy	VERB
ejpam-6077	3	8	a	a	DET
ejpam-6077	3	9	contractive	contractive	ADJ
ejpam-6077	3	10	condition	condition	NOUN
ejpam-6077	3	11	involving	involve	VERB
ejpam-6077	3	12	the	the	DET
ejpam-6077	3	13	mr	mr	PROPN
ejpam-6077	3	14	-	-	PUNCT
ejpam-6077	3	15	metric	metric	NOUN
ejpam-6077	3	16	with	with	ADP
ejpam-6077	3	17	a	a	DET
ejpam-6077	3	18	constant	constant	ADJ
ejpam-6077	3	19	k	k	PROPN
ejpam-6077	3	20	∈	∈	PROPN
ejpam-6077	4	1	[	[	X
ejpam-6077	4	2	0	0	NUM
ejpam-6077	4	3	,	,	PUNCT
ejpam-6077	4	4	1	1	NUM
ejpam-6077	4	5	)	)	PUNCT
ejpam-6077	4	6	,	,	PUNCT
ejpam-6077	4	7	and	and	CCONJ
ejpam-6077	4	8	a	a	DET
ejpam-6077	4	9	measure	measure	NOUN
ejpam-6077	4	10	of	of	ADP
ejpam-6077	4	11	noncompactness	noncompactness	ADJ
ejpam-6077	4	12	condition	condition	NOUN
ejpam-6077	4	13	governed	govern	VERB
ejpam-6077	4	14	by	by	ADP
ejpam-6077	4	15	a	a	DET
ejpam-6077	4	16	function	function	NOUN
ejpam-6077	4	17	ϕ	ϕ	NOUN
ejpam-6077	4	18	where	where	SCONJ
ejpam-6077	4	19	ϕ(t	ϕ(t	NUM
ejpam-6077	4	20	)	)	PUNCT
ejpam-6077	4	21	<	<	X
ejpam-6077	4	22	t	t	PROPN
ejpam-6077	4	23	for	for	ADP
ejpam-6077	4	24	t	t	PROPN
ejpam-6077	4	25	>	>	X
ejpam-6077	4	26	0	0	PROPN
ejpam-6077	4	27	,	,	PUNCT
ejpam-6077	4	28	then	then	ADV
ejpam-6077	4	29	s	s	AUX
ejpam-6077	4	30	possesses	possess	VERB
ejpam-6077	4	31	a	a	DET
ejpam-6077	4	32	unique	unique	ADJ
ejpam-6077	4	33	fixed	fix	VERB
ejpam-6077	4	34	point	point	NOUN
ejpam-6077	4	35	υ∗.	υ∗.	VERB
ejpam-6077	4	36	the	the	DET
ejpam-6077	4	37	findings	finding	NOUN
ejpam-6077	4	38	have	have	VERB
ejpam-6077	4	39	significant	significant	ADJ
ejpam-6077	4	40	applications	application	NOUN
ejpam-6077	4	41	in	in	ADP
ejpam-6077	4	42	solving	solve	VERB
ejpam-6077	4	43	nonlinear	nonlinear	ADJ
ejpam-6077	4	44	integral	integral	ADJ
ejpam-6077	4	45	equations	equation	NOUN
ejpam-6077	4	46	,	,	PUNCT
ejpam-6077	4	47	ensuring	ensure	VERB
ejpam-6077	4	48	stability	stability	NOUN
ejpam-6077	4	49	of	of	ADP
ejpam-6077	4	50	iterative	iterative	NOUN
ejpam-6077	4	51	processes	process	NOUN
ejpam-6077	4	52	,	,	PUNCT
ejpam-6077	4	53	optimization	optimization	NOUN
ejpam-6077	4	54	,	,	PUNCT
ejpam-6077	4	55	game	game	NOUN
ejpam-6077	4	56	theory	theory	NOUN
ejpam-6077	4	57	,	,	PUNCT
ejpam-6077	4	58	economic	economic	ADJ
ejpam-6077	4	59	equilibria	equilibrium	NOUN
ejpam-6077	4	60	,	,	PUNCT
ejpam-6077	4	61	and	and	CCONJ
ejpam-6077	4	62	boundary	boundary	ADJ
ejpam-6077	4	63	value	value	NOUN
ejpam-6077	4	64	problems	problem	NOUN
ejpam-6077	4	65	,	,	PUNCT
ejpam-6077	4	66	showcasing	showcase	VERB
ejpam-6077	4	67	the	the	DET
ejpam-6077	4	68	versatility	versatility	NOUN
ejpam-6077	4	69	of	of	ADP
ejpam-6077	4	70	mr	mr	PROPN
ejpam-6077	4	71	-	-	PUNCT
ejpam-6077	4	72	metric	metric	ADJ
ejpam-6077	4	73	spaces	space	NOUN
ejpam-6077	4	74	in	in	ADP
ejpam-6077	4	75	addressing	address	VERB
ejpam-6077	4	76	noncompact	noncompact	ADJ
ejpam-6077	4	77	settings	setting	NOUN
ejpam-6077	4	78	and	and	CCONJ
ejpam-6077	4	79	fixed	fix	VERB
ejpam-6077	4	80	-	-	PUNCT
ejpam-6077	4	81	point	point	NOUN
ejpam-6077	4	82	problems	problem	NOUN
ejpam-6077	4	83	.	.	PUNCT
ejpam-6077	5	1	2020	2020	NUM
ejpam-6077	5	2	mathematics	mathematic	NOUN
ejpam-6077	5	3	subject	subject	NOUN
ejpam-6077	5	4	classifications	classification	NOUN
ejpam-6077	5	5	:	:	PUNCT
ejpam-6077	5	6	47h10	47h10	NUM
ejpam-6077	5	7	,	,	PUNCT
ejpam-6077	5	8	54h25	54h25	NUM
ejpam-6077	5	9	,	,	PUNCT
ejpam-6077	5	10	46t99	46t99	NUM
ejpam-6077	5	11	,	,	PUNCT
ejpam-6077	5	12	47h09	47h09	NUM
ejpam-6077	5	13	key	key	ADJ
ejpam-6077	5	14	words	word	NOUN
ejpam-6077	5	15	and	and	CCONJ
ejpam-6077	5	16	phrases	phrase	NOUN
ejpam-6077	5	17	:	:	PUNCT
ejpam-6077	5	18	mr	mr	PROPN
ejpam-6077	5	19	−metric	−metric	ADV
ejpam-6077	5	20	fixed	fix	VERB
ejpam-6077	5	21	-	-	PUNCT
ejpam-6077	5	22	point	point	NOUN
ejpam-6077	5	23	theorems	theorem	NOUN
ejpam-6077	5	24	,	,	PUNCT
ejpam-6077	5	25	mr	mr	ADJ
ejpam-6077	5	26	-	-	PUNCT
ejpam-6077	5	27	metric	metric	ADJ
ejpam-6077	5	28	spaces	space	NOUN
ejpam-6077	5	29	,	,	PUNCT
ejpam-6077	5	30	measure	measure	NOUN
ejpam-6077	5	31	of	of	ADP
ejpam-6077	5	32	noncompactness	noncompactness	ADJ
ejpam-6077	5	33	,	,	PUNCT
ejpam-6077	5	34	iterative	iterative	NOUN
ejpam-6077	5	35	processes	process	NOUN
ejpam-6077	5	36	,	,	PUNCT
ejpam-6077	5	37	optimization	optimization	NOUN
ejpam-6077	5	38	,	,	PUNCT
ejpam-6077	5	39	game	game	NOUN
ejpam-6077	5	40	theory	theory	NOUN
ejpam-6077	5	41	,	,	PUNCT
ejpam-6077	5	42	nonlinear	nonlinear	ADJ
ejpam-6077	5	43	integral	integral	ADJ
ejpam-6077	5	44	equations	equation	NOUN
ejpam-6077	5	45	,	,	PUNCT
ejpam-6077	5	46	boundary	boundary	ADJ
ejpam-6077	5	47	value	value	NOUN
ejpam-6077	5	48	problems	problem	NOUN
ejpam-6077	5	49	,	,	PUNCT
ejpam-6077	5	50	banach	banach	NOUN
ejpam-6077	5	51	spaces	space	VERB
ejpam-6077	5	52	1	1	NUM
ejpam-6077	5	53	.	.	PUNCT
ejpam-6077	6	1	introduction	introduction	NOUN
ejpam-6077	6	2	fixed	fix	VERB
ejpam-6077	6	3	-	-	PUNCT
ejpam-6077	6	4	point	point	NOUN
ejpam-6077	6	5	theory	theory	NOUN
ejpam-6077	6	6	represents	represent	VERB
ejpam-6077	6	7	a	a	DET
ejpam-6077	6	8	fundamental	fundamental	ADJ
ejpam-6077	6	9	aspect	aspect	NOUN
ejpam-6077	6	10	of	of	ADP
ejpam-6077	6	11	functional	functional	ADJ
ejpam-6077	6	12	analysis	analysis	NOUN
ejpam-6077	6	13	,	,	PUNCT
ejpam-6077	6	14	playing	play	VERB
ejpam-6077	6	15	a	a	DET
ejpam-6077	6	16	pivotal	pivotal	ADJ
ejpam-6077	6	17	role	role	NOUN
ejpam-6077	6	18	in	in	ADP
ejpam-6077	6	19	mathematics	mathematic	NOUN
ejpam-6077	6	20	and	and	CCONJ
ejpam-6077	6	21	numerous	numerous	ADJ
ejpam-6077	6	22	scientific	scientific	ADJ
ejpam-6077	6	23	fields	field	NOUN
ejpam-6077	6	24	.	.	PUNCT
ejpam-6077	7	1	the	the	DET
ejpam-6077	7	2	idea	idea	NOUN
ejpam-6077	7	3	of	of	ADP
ejpam-6077	7	4	a	a	DET
ejpam-6077	7	5	fixed	fix	VERB
ejpam-6077	7	6	point	point	NOUN
ejpam-6077	7	7	,	,	PUNCT
ejpam-6077	7	8	where	where	SCONJ
ejpam-6077	7	9	a	a	DET
ejpam-6077	7	10	function	function	NOUN
ejpam-6077	7	11	maps	map	VERB
ejpam-6077	7	12	a	a	DET
ejpam-6077	7	13	specific	specific	ADJ
ejpam-6077	7	14	point	point	NOUN
ejpam-6077	7	15	to	to	ADP
ejpam-6077	7	16	itself	itself	PRON
ejpam-6077	7	17	,	,	PUNCT
ejpam-6077	7	18	is	be	AUX
ejpam-6077	7	19	essential	essential	ADJ
ejpam-6077	7	20	for	for	ADP
ejpam-6077	7	21	solving	solve	VERB
ejpam-6077	7	22	equations	equation	NOUN
ejpam-6077	7	23	,	,	PUNCT
ejpam-6077	7	24	studying	study	VERB
ejpam-6077	7	25	dynamic	dynamic	ADJ
ejpam-6077	7	26	systems	system	NOUN
ejpam-6077	7	27	,	,	PUNCT
ejpam-6077	7	28	and	and	CCONJ
ejpam-6077	7	29	addressing	address	VERB
ejpam-6077	7	30	optimization	optimization	NOUN
ejpam-6077	7	31	problems	problem	NOUN
ejpam-6077	7	32	.	.	PUNCT
ejpam-6077	8	1	in	in	ADP
ejpam-6077	8	2	this	this	DET
ejpam-6077	8	3	framework	framework	NOUN
ejpam-6077	8	4	,	,	PUNCT
ejpam-6077	8	5	the	the	DET
ejpam-6077	8	6	introduction	introduction	NOUN
ejpam-6077	8	7	of	of	ADP
ejpam-6077	8	8	generalized	generalized	ADJ
ejpam-6077	8	9	metric	metric	ADJ
ejpam-6077	8	10	spaces	space	NOUN
ejpam-6077	8	11	,	,	PUNCT
ejpam-6077	8	12	including	include	VERB
ejpam-6077	8	13	mr	mr	PROPN
ejpam-6077	8	14	-	-	PUNCT
ejpam-6077	8	15	metric	metric	ADJ
ejpam-6077	8	16	spaces	space	NOUN
ejpam-6077	8	17	,	,	PUNCT
ejpam-6077	8	18	offers	offer	VERB
ejpam-6077	8	19	a	a	DET
ejpam-6077	8	20	versatile	versatile	ADJ
ejpam-6077	8	21	structure	structure	NOUN
ejpam-6077	8	22	for	for	ADP
ejpam-6077	8	23	tackling	tackle	VERB
ejpam-6077	8	24	fixed	fix	VERB
ejpam-6077	8	25	-	-	PUNCT
ejpam-6077	8	26	point	point	NOUN
ejpam-6077	8	27	challenges	challenge	NOUN
ejpam-6077	8	28	,	,	PUNCT
ejpam-6077	8	29	particularly	particularly	ADV
ejpam-6077	8	30	in	in	ADP
ejpam-6077	8	31	noncompact	noncompact	ADJ
ejpam-6077	8	32	contexts	contexts	NOUN
ejpam-6077	8	33	.	.	PUNCT
ejpam-6077	9	1	this	this	DET
ejpam-6077	9	2	study	study	NOUN
ejpam-6077	9	3	explores	explore	VERB
ejpam-6077	9	4	mr	mr	ADJ
ejpam-6077	9	5	-	-	PUNCT
ejpam-6077	9	6	metric	metric	ADJ
ejpam-6077	9	7	spaces	space	NOUN
ejpam-6077	9	8	,	,	PUNCT
ejpam-6077	9	9	which	which	PRON
ejpam-6077	9	10	generalize	generalize	VERB
ejpam-6077	9	11	the	the	DET
ejpam-6077	9	12	classical	classical	ADJ
ejpam-6077	9	13	metric	metric	ADJ
ejpam-6077	9	14	space	space	NOUN
ejpam-6077	9	15	framework	framework	NOUN
ejpam-6077	9	16	to	to	PART
ejpam-6077	9	17	encompass	encompass	VERB
ejpam-6077	9	18	a	a	DET
ejpam-6077	9	19	wider	wide	ADJ
ejpam-6077	9	20	array	array	NOUN
ejpam-6077	9	21	of	of	ADP
ejpam-6077	9	22	applications	application	NOUN
ejpam-6077	9	23	.	.	PUNCT
ejpam-6077	10	1	by	by	ADP
ejpam-6077	10	2	defining	define	VERB
ejpam-6077	10	3	a	a	DET
ejpam-6077	10	4	generalized	generalized	ADJ
ejpam-6077	10	5	metric	metric	NOUN
ejpam-6077	10	6	m	m	NOUN
ejpam-6077	10	7	and	and	CCONJ
ejpam-6077	10	8	utilizing	utilize	VERB
ejpam-6077	10	9	the	the	DET
ejpam-6077	10	10	measure	measure	NOUN
ejpam-6077	10	11	of	of	ADP
ejpam-6077	10	12	noncompactness	noncompactness	ADJ
ejpam-6077	10	13	,	,	PUNCT
ejpam-6077	10	14	we	we	PRON
ejpam-6077	10	15	derive	derive	VERB
ejpam-6077	10	16	fixed	fix	VERB
ejpam-6077	10	17	-	-	PUNCT
ejpam-6077	10	18	point	point	NOUN
ejpam-6077	10	19	theorems	theorem	NOUN
ejpam-6077	10	20	for	for	ADP
ejpam-6077	10	21	mappings	mapping	NOUN
ejpam-6077	10	22	s	s	PART
ejpam-6077	10	23	:	:	PUNCT
ejpam-6077	10	24	x	x	SYM
ejpam-6077	10	25	→	→	SYM
ejpam-6077	10	26	x	x	X
ejpam-6077	10	27	,	,	PUNCT
ejpam-6077	10	28	where	where	SCONJ
ejpam-6077	10	29	x	x	PRON
ejpam-6077	10	30	represents	represent	VERB
ejpam-6077	10	31	a	a	DET
ejpam-6077	10	32	closed	closed	ADJ
ejpam-6077	10	33	,	,	PUNCT
ejpam-6077	10	34	bounded	bound	VERB
ejpam-6077	10	35	,	,	PUNCT
ejpam-6077	10	36	and	and	CCONJ
ejpam-6077	10	37	convex	convex	PROPN
ejpam-6077	10	38	subset	subset	NOUN
ejpam-6077	10	39	of	of	ADP
ejpam-6077	10	40	a	a	DET
ejpam-6077	10	41	banach	banach	NOUN
ejpam-6077	10	42	space	space	NOUN
ejpam-6077	10	43	.	.	PUNCT
ejpam-6077	11	1	these	these	DET
ejpam-6077	11	2	results	result	NOUN
ejpam-6077	11	3	ensure	ensure	VERB
ejpam-6077	11	4	the	the	DET
ejpam-6077	11	5	existence	existence	NOUN
ejpam-6077	11	6	and	and	CCONJ
ejpam-6077	11	7	uniqueness	uniqueness	NOUN
ejpam-6077	11	8	of	of	ADP
ejpam-6077	11	9	fixed	fix	VERB
ejpam-6077	11	10	points	point	NOUN
ejpam-6077	11	11	under	under	ADP
ejpam-6077	11	12	certain	certain	ADJ
ejpam-6077	11	13	contractive	contractive	ADJ
ejpam-6077	11	14	conditions	condition	NOUN
ejpam-6077	11	15	and	and	CCONJ
ejpam-6077	11	16	noncompactness	noncompactness	ADJ
ejpam-6077	11	17	criteria	criterion	NOUN
ejpam-6077	11	18	.	.	PUNCT
ejpam-6077	12	1	doi	doi	NOUN
ejpam-6077	12	2	:	:	PUNCT
ejpam-6077	12	3	https://doi.org/10.29020/nybg.ejpam.v18i2.6077	https://doi.org/10.29020/nybg.ejpam.v18i2.6077	PROPN
ejpam-6077	12	4	email	email	NOUN
ejpam-6077	12	5	address	address	NOUN
ejpam-6077	12	6	:	:	PUNCT
ejpam-6077	12	7	a.malkawi@aau.edu.jo	a.malkawi@aau.edu.jo	PROPN
ejpam-6077	12	8	and	and	CCONJ
ejpam-6077	12	9	math.malkawi@gmail.com	math.malkawi@gmail.com	X
ejpam-6077	12	10	(	(	PUNCT
ejpam-6077	12	11	a.	a.	NOUN
ejpam-6077	12	12	malkawi	malkawi	PROPN
ejpam-6077	12	13	)	)	PUNCT
ejpam-6077	12	14	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6077	13	1	1	1	NUM
ejpam-6077	13	2	copyright	copyright	NOUN
ejpam-6077	13	3	:	:	PUNCT
ejpam-6077	13	4	©	©	PROPN
ejpam-6077	13	5	2025	2025	NUM
ejpam-6077	13	6	the	the	DET
ejpam-6077	13	7	author(s	author(s	NOUN
ejpam-6077	13	8	)	)	PUNCT
ejpam-6077	13	9	.	.	PUNCT
ejpam-6077	14	1	(	(	PUNCT
ejpam-6077	14	2	cc	cc	NOUN
ejpam-6077	14	3	by	by	ADP
ejpam-6077	14	4	-	-	PUNCT
ejpam-6077	14	5	nc	nc	PROPN
ejpam-6077	14	6	4.0	4.0	NUM
ejpam-6077	14	7	)	)	PUNCT
ejpam-6077	14	8	a.	a.	NOUN
ejpam-6077	14	9	malkawi	malkawi	ADP
ejpam-6077	14	10	/	/	SYM
ejpam-6077	14	11	eur	eur	PROPN
ejpam-6077	14	12	.	.	PUNCT
ejpam-6077	15	1	j.	j.	PROPN
ejpam-6077	15	2	pure	pure	PROPN
ejpam-6077	15	3	appl	appl	PROPN
ejpam-6077	15	4	.	.	PROPN
ejpam-6077	15	5	math	math	PROPN
ejpam-6077	15	6	,	,	PUNCT
ejpam-6077	15	7	18	18	NUM
ejpam-6077	15	8	(	(	PUNCT
ejpam-6077	15	9	2	2	NUM
ejpam-6077	15	10	)	)	PUNCT
ejpam-6077	15	11	(	(	PUNCT
ejpam-6077	15	12	2025	2025	NUM
ejpam-6077	15	13	)	)	PUNCT
ejpam-6077	15	14	,	,	PUNCT
ejpam-6077	15	15	6077	6077	NUM
ejpam-6077	15	16	2	2	NUM
ejpam-6077	15	17	of	of	ADP
ejpam-6077	15	18	16	16	NUM
ejpam-6077	15	19	the	the	DET
ejpam-6077	15	20	importance	importance	NOUN
ejpam-6077	15	21	of	of	ADP
ejpam-6077	15	22	these	these	DET
ejpam-6077	15	23	findings	finding	NOUN
ejpam-6077	15	24	stems	stem	VERB
ejpam-6077	15	25	from	from	ADP
ejpam-6077	15	26	their	their	PRON
ejpam-6077	15	27	wide	wide	ADV
ejpam-6077	15	28	-	-	PUNCT
ejpam-6077	15	29	ranging	range	VERB
ejpam-6077	15	30	applications	application	NOUN
ejpam-6077	15	31	in	in	ADP
ejpam-6077	15	32	both	both	DET
ejpam-6077	15	33	mathematical	mathematical	ADJ
ejpam-6077	15	34	theory	theory	NOUN
ejpam-6077	15	35	and	and	CCONJ
ejpam-6077	15	36	practical	practical	ADJ
ejpam-6077	15	37	problems	problem	NOUN
ejpam-6077	15	38	,	,	PUNCT
ejpam-6077	15	39	such	such	ADJ
ejpam-6077	15	40	as	as	ADP
ejpam-6077	15	41	nonlinear	nonlinear	ADJ
ejpam-6077	15	42	integral	integral	ADJ
ejpam-6077	15	43	equations	equation	NOUN
ejpam-6077	15	44	,	,	PUNCT
ejpam-6077	15	45	iterative	iterative	NOUN
ejpam-6077	15	46	methods	method	NOUN
ejpam-6077	15	47	,	,	PUNCT
ejpam-6077	15	48	optimization	optimization	NOUN
ejpam-6077	15	49	,	,	PUNCT
ejpam-6077	15	50	game	game	NOUN
ejpam-6077	15	51	theory	theory	NOUN
ejpam-6077	15	52	,	,	PUNCT
ejpam-6077	15	53	and	and	CCONJ
ejpam-6077	15	54	boundary	boundary	ADJ
ejpam-6077	15	55	value	value	NOUN
ejpam-6077	15	56	challenges	challenge	NOUN
ejpam-6077	15	57	.	.	PUNCT
ejpam-6077	16	1	this	this	DET
ejpam-6077	16	2	work	work	NOUN
ejpam-6077	16	3	seeks	seek	VERB
ejpam-6077	16	4	to	to	PART
ejpam-6077	16	5	establish	establish	VERB
ejpam-6077	16	6	a	a	DET
ejpam-6077	16	7	solid	solid	ADJ
ejpam-6077	16	8	theoretical	theoretical	ADJ
ejpam-6077	16	9	framework	framework	NOUN
ejpam-6077	16	10	for	for	ADP
ejpam-6077	16	11	employing	employ	VERB
ejpam-6077	16	12	mr	mr	ADJ
ejpam-6077	16	13	-	-	ADJ
ejpam-6077	16	14	metric	metric	ADJ
ejpam-6077	16	15	spaces	space	NOUN
ejpam-6077	16	16	within	within	ADP
ejpam-6077	16	17	fixedpoint	fixedpoint	NOUN
ejpam-6077	16	18	theory	theory	NOUN
ejpam-6077	16	19	,	,	PUNCT
ejpam-6077	16	20	emphasizing	emphasize	VERB
ejpam-6077	16	21	their	their	PRON
ejpam-6077	16	22	capability	capability	NOUN
ejpam-6077	16	23	to	to	PART
ejpam-6077	16	24	handle	handle	VERB
ejpam-6077	16	25	intricate	intricate	ADJ
ejpam-6077	16	26	and	and	CCONJ
ejpam-6077	16	27	noncompact	noncompact	ADJ
ejpam-6077	16	28	situations	situation	NOUN
ejpam-6077	16	29	effectively	effectively	ADV
ejpam-6077	16	30	.	.	PUNCT
ejpam-6077	17	1	for	for	ADP
ejpam-6077	17	2	additional	additional	ADJ
ejpam-6077	17	3	information	information	NOUN
ejpam-6077	17	4	,	,	PUNCT
ejpam-6077	17	5	we	we	PRON
ejpam-6077	17	6	direct	direct	VERB
ejpam-6077	17	7	readers	reader	NOUN
ejpam-6077	17	8	to	to	ADP
ejpam-6077	17	9	[	[	X
ejpam-6077	17	10	1–27	1–27	PROPN
ejpam-6077	17	11	]	]	X
ejpam-6077	17	12	.	.	PUNCT
ejpam-6077	18	1	definition	definition	NOUN
ejpam-6077	18	2	1	1	NUM
ejpam-6077	18	3	.	.	PUNCT
ejpam-6077	19	1	[	[	X
ejpam-6077	19	2	28	28	NUM
ejpam-6077	19	3	]	]	X
ejpam-6077	19	4	let	let	VERB
ejpam-6077	19	5	x	x	SYM
ejpam-6077	19	6	̸=	̸=	PROPN
ejpam-6077	19	7	∅	∅	NOUN
ejpam-6077	19	8	denote	denote	VERB
ejpam-6077	19	9	a	a	DET
ejpam-6077	19	10	non	non	ADJ
ejpam-6077	19	11	-	-	ADJ
ejpam-6077	19	12	empty	empty	ADJ
ejpam-6077	19	13	set	set	NOUN
ejpam-6077	19	14	,	,	PUNCT
ejpam-6077	19	15	and	and	CCONJ
ejpam-6077	19	16	let	let	VERB
ejpam-6077	19	17	r	r	PRON
ejpam-6077	19	18	>	>	X
ejpam-6077	19	19	1	1	NUM
ejpam-6077	19	20	be	be	AUX
ejpam-6077	19	21	a	a	DET
ejpam-6077	19	22	given	give	VERB
ejpam-6077	19	23	real	real	ADJ
ejpam-6077	19	24	number	number	NOUN
ejpam-6077	19	25	.	.	PUNCT
ejpam-6077	20	1	a	a	DET
ejpam-6077	20	2	function	function	NOUN
ejpam-6077	20	3	m	m	VERB
ejpam-6077	20	4	:	:	PUNCT
ejpam-6077	20	5	x	x	X
ejpam-6077	20	6	×	×	NOUN
ejpam-6077	20	7	x	x	SYM
ejpam-6077	20	8	×	×	NOUN
ejpam-6077	20	9	x	x	INTJ
ejpam-6077	20	10	→	→	X
ejpam-6077	20	11	[	[	X
ejpam-6077	20	12	0,∞	0,∞	NOUN
ejpam-6077	20	13	)	)	PUNCT
ejpam-6077	20	14	is	be	AUX
ejpam-6077	20	15	called	call	VERB
ejpam-6077	20	16	an	an	DET
ejpam-6077	20	17	mr	mr	NOUN
ejpam-6077	20	18	-	-	PUNCT
ejpam-6077	20	19	metric	metric	NOUN
ejpam-6077	20	20	if	if	SCONJ
ejpam-6077	20	21	it	it	PRON
ejpam-6077	20	22	fulfills	fulfill	VERB
ejpam-6077	20	23	the	the	DET
ejpam-6077	20	24	following	follow	VERB
ejpam-6077	20	25	conditions	condition	NOUN
ejpam-6077	20	26	for	for	ADP
ejpam-6077	20	27	all	all	DET
ejpam-6077	20	28	υ	υ	PROPN
ejpam-6077	20	29	,	,	PUNCT
ejpam-6077	20	30	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	20	31	∈	∈	PROPN
ejpam-6077	21	1	x	x	X
ejpam-6077	21	2	:	:	PUNCT
ejpam-6077	21	3	•	•	PRON
ejpam-6077	21	4	(	(	PUNCT
ejpam-6077	21	5	m1	m1	NOUN
ejpam-6077	21	6	)	)	PUNCT
ejpam-6077	21	7	:	:	PUNCT
ejpam-6077	22	1	m(υ	m(υ	PROPN
ejpam-6077	22	2	,	,	PUNCT
ejpam-6077	22	3	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	22	4	)	)	PUNCT
ejpam-6077	22	5	≥	≥	NOUN
ejpam-6077	22	6	0	0	NUM
ejpam-6077	22	7	.	.	NOUN
ejpam-6077	22	8	•	•	NUM
ejpam-6077	22	9	(	(	PUNCT
ejpam-6077	22	10	m2	m2	PROPN
ejpam-6077	22	11	)	)	PUNCT
ejpam-6077	22	12	:	:	PUNCT
ejpam-6077	23	1	m(υ	m(υ	PROPN
ejpam-6077	23	2	,	,	PUNCT
ejpam-6077	23	3	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	23	4	)	)	PUNCT
ejpam-6077	23	5	=	=	SYM
ejpam-6077	23	6	0	0	PUNCT
ejpam-6077	24	1	if	if	SCONJ
ejpam-6077	24	2	and	and	CCONJ
ejpam-6077	24	3	only	only	ADV
ejpam-6077	24	4	if	if	SCONJ
ejpam-6077	24	5	υ	υ	PROPN
ejpam-6077	24	6	=	=	SYM
ejpam-6077	24	7	ξ	ξ	NOUN
ejpam-6077	24	8	=	=	PUNCT
ejpam-6077	24	9	ℑ.	ℑ.	NOUN
ejpam-6077	24	10	•	•	NUM
ejpam-6077	24	11	(	(	PUNCT
ejpam-6077	24	12	m3	m3	PROPN
ejpam-6077	24	13	)	)	PUNCT
ejpam-6077	24	14	:	:	PUNCT
ejpam-6077	24	15	m(υ	m(υ	PROPN
ejpam-6077	24	16	,	,	PUNCT
ejpam-6077	24	17	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	24	18	)	)	PUNCT
ejpam-6077	24	19	=	=	SYM
ejpam-6077	24	20	m(p(υ	m(p(υ	PROPN
ejpam-6077	24	21	,	,	PUNCT
ejpam-6077	24	22	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	24	23	)	)	PUNCT
ejpam-6077	24	24	)	)	PUNCT
ejpam-6077	24	25	,	,	PUNCT
ejpam-6077	24	26	for	for	ADP
ejpam-6077	24	27	any	any	DET
ejpam-6077	24	28	permutation	permutation	NOUN
ejpam-6077	24	29	p(υ	p(υ	NOUN
ejpam-6077	24	30	,	,	PUNCT
ejpam-6077	24	31	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	24	32	)	)	PUNCT
ejpam-6077	24	33	of	of	ADP
ejpam-6077	24	34	υ	υ	PROPN
ejpam-6077	24	35	,	,	PUNCT
ejpam-6077	24	36	ξ,ℑ.	ξ,ℑ.	PROPN
ejpam-6077	24	37	•	•	ADV
ejpam-6077	24	38	(	(	PUNCT
ejpam-6077	24	39	m4	m4	PROPN
ejpam-6077	24	40	)	)	PUNCT
ejpam-6077	24	41	:	:	PUNCT
ejpam-6077	25	1	m(υ	m(υ	PROPN
ejpam-6077	25	2	,	,	PUNCT
ejpam-6077	25	3	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	25	4	)	)	PUNCT
ejpam-6077	25	5	≤	≤	NUM
ejpam-6077	26	1	r	r	NOUN
ejpam-6077	27	1	[	[	X
ejpam-6077	27	2	m(υ	m(υ	PROPN
ejpam-6077	27	3	,	,	PUNCT
ejpam-6077	27	4	ξ	ξ	PROPN
ejpam-6077	27	5	,	,	PUNCT
ejpam-6077	27	6	ℓ1	ℓ1	NOUN
ejpam-6077	27	7	)	)	PUNCT
ejpam-6077	28	1	+	+	SYM
ejpam-6077	28	2	m(υ	m(υ	PROPN
ejpam-6077	28	3	,	,	PUNCT
ejpam-6077	28	4	ℓ1,ℑ	ℓ1,ℑ	NOUN
ejpam-6077	28	5	)	)	PUNCT
ejpam-6077	29	1	+	+	ADJ
ejpam-6077	29	2	m(ℓ1	m(ℓ1	NOUN
ejpam-6077	29	3	,	,	PUNCT
ejpam-6077	29	4	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	29	5	)	)	PUNCT
ejpam-6077	29	6	]	]	PUNCT
ejpam-6077	29	7	.	.	PUNCT
ejpam-6077	30	1	a	a	DET
ejpam-6077	30	2	pair	pair	NOUN
ejpam-6077	30	3	(	(	PUNCT
ejpam-6077	30	4	x	x	X
ejpam-6077	30	5	,	,	PUNCT
ejpam-6077	30	6	m	m	NOUN
ejpam-6077	30	7	)	)	PUNCT
ejpam-6077	30	8	that	that	PRON
ejpam-6077	30	9	satisfies	satisfy	VERB
ejpam-6077	30	10	these	these	DET
ejpam-6077	30	11	properties	property	NOUN
ejpam-6077	30	12	is	be	AUX
ejpam-6077	30	13	called	call	VERB
ejpam-6077	30	14	an	an	DET
ejpam-6077	30	15	mr	mr	PROPN
ejpam-6077	30	16	-	-	PUNCT
ejpam-6077	30	17	metric	metric	ADJ
ejpam-6077	30	18	space	space	NOUN
ejpam-6077	30	19	.	.	PUNCT
ejpam-6077	31	1	definition	definition	NOUN
ejpam-6077	31	2	2	2	NUM
ejpam-6077	31	3	.	.	PUNCT
ejpam-6077	32	1	let	let	VERB
ejpam-6077	32	2	{	{	PUNCT
ejpam-6077	32	3	υin	υin	PROPN
ejpam-6077	32	4	}	}	PUNCT
ejpam-6077	32	5	be	be	AUX
ejpam-6077	32	6	a	a	DET
ejpam-6077	32	7	sequence	sequence	NOUN
ejpam-6077	32	8	in	in	ADP
ejpam-6077	32	9	an	an	DET
ejpam-6077	32	10	mr	mr	PROPN
ejpam-6077	32	11	-	-	PUNCT
ejpam-6077	32	12	metric	metric	ADJ
ejpam-6077	32	13	space	space	NOUN
ejpam-6077	32	14	(	(	PUNCT
ejpam-6077	32	15	y	y	NOUN
ejpam-6077	32	16	,	,	PUNCT
ejpam-6077	32	17	m	m	PROPN
ejpam-6077	32	18	)	)	PUNCT
ejpam-6077	32	19	.	.	PUNCT
ejpam-6077	33	1	the	the	DET
ejpam-6077	33	2	sequence	sequence	NOUN
ejpam-6077	33	3	is	be	AUX
ejpam-6077	33	4	termed	term	VERB
ejpam-6077	33	5	mr	mr	PROPN
ejpam-6077	33	6	-	-	PUNCT
ejpam-6077	33	7	convergent	convergent	NOUN
ejpam-6077	33	8	if	if	SCONJ
ejpam-6077	33	9	there	there	PRON
ejpam-6077	33	10	exists	exist	VERB
ejpam-6077	33	11	an	an	DET
ejpam-6077	33	12	element	element	ADJ
ejpam-6077	33	13	υi1	υi1	NOUN
ejpam-6077	33	14	∈	∈	PROPN
ejpam-6077	33	15	y	y	PROPN
ejpam-6077	33	16	such	such	ADJ
ejpam-6077	33	17	that	that	PRON
ejpam-6077	33	18	for	for	ADP
ejpam-6077	33	19	every	every	DET
ejpam-6077	33	20	ϵ	ϵ	X
ejpam-6077	33	21	>	>	X
ejpam-6077	33	22	0	0	NUM
ejpam-6077	33	23	,	,	PUNCT
ejpam-6077	33	24	there	there	PRON
ejpam-6077	33	25	exists	exist	VERB
ejpam-6077	33	26	a	a	DET
ejpam-6077	33	27	positive	positive	ADJ
ejpam-6077	33	28	integer	integer	NOUN
ejpam-6077	33	29	n	n	AUX
ejpam-6077	33	30	satisfying	satisfy	VERB
ejpam-6077	33	31	m(υin	m(υin	NOUN
ejpam-6077	33	32	,	,	PUNCT
ejpam-6077	33	33	υim	υim	NOUN
ejpam-6077	33	34	,	,	PUNCT
ejpam-6077	33	35	υi1	υi1	PROPN
ejpam-6077	33	36	)	)	PUNCT
ejpam-6077	33	37	<	<	X
ejpam-6077	34	1	ϵ	ϵ	X
ejpam-6077	34	2	for	for	ADP
ejpam-6077	34	3	all	all	DET
ejpam-6077	34	4	m	m	PROPN
ejpam-6077	34	5	,	,	PUNCT
ejpam-6077	34	6	n	n	PRON
ejpam-6077	34	7	≥	≥	NOUN
ejpam-6077	34	8	n	n	ADV
ejpam-6077	34	9	.	.	PUNCT
ejpam-6077	35	1	in	in	ADP
ejpam-6077	35	2	this	this	DET
ejpam-6077	35	3	case	case	NOUN
ejpam-6077	35	4	,	,	PUNCT
ejpam-6077	35	5	the	the	DET
ejpam-6077	35	6	sequence	sequence	NOUN
ejpam-6077	35	7	{	{	PUNCT
ejpam-6077	35	8	υin	υin	NOUN
ejpam-6077	35	9	}	}	PUNCT
ejpam-6077	35	10	is	be	AUX
ejpam-6077	35	11	said	say	VERB
ejpam-6077	35	12	to	to	ADP
ejpam-6077	35	13	mr	mr	PROPN
ejpam-6077	35	14	-	-	PUNCT
ejpam-6077	35	15	converge	converge	NOUN
ejpam-6077	35	16	to	to	ADP
ejpam-6077	35	17	υi1	υi1	VERB
ejpam-6077	35	18	,	,	PUNCT
ejpam-6077	35	19	where	where	SCONJ
ejpam-6077	35	20	υi1	υi1	NOUN
ejpam-6077	35	21	is	be	AUX
ejpam-6077	35	22	considered	consider	VERB
ejpam-6077	35	23	the	the	DET
ejpam-6077	35	24	limit	limit	NOUN
ejpam-6077	35	25	of	of	ADP
ejpam-6077	35	26	the	the	DET
ejpam-6077	35	27	sequence	sequence	NOUN
ejpam-6077	35	28	.	.	PUNCT
ejpam-6077	36	1	definition	definition	NOUN
ejpam-6077	36	2	3	3	NUM
ejpam-6077	36	3	.	.	PUNCT
ejpam-6077	37	1	a	a	DET
ejpam-6077	37	2	sequence	sequence	NOUN
ejpam-6077	37	3	{	{	PUNCT
ejpam-6077	37	4	υin	υin	NOUN
ejpam-6077	37	5	}	}	PUNCT
ejpam-6077	37	6	in	in	ADP
ejpam-6077	37	7	an	an	DET
ejpam-6077	37	8	mr	mr	PROPN
ejpam-6077	37	9	-	-	PUNCT
ejpam-6077	37	10	metric	metric	ADJ
ejpam-6077	37	11	space	space	NOUN
ejpam-6077	37	12	(	(	PUNCT
ejpam-6077	37	13	y	y	PROPN
ejpam-6077	37	14	,	,	PUNCT
ejpam-6077	37	15	m	m	PROPN
ejpam-6077	37	16	)	)	PUNCT
ejpam-6077	37	17	is	be	AUX
ejpam-6077	37	18	called	call	VERB
ejpam-6077	37	19	mr	mr	PROPN
ejpam-6077	37	20	-	-	PUNCT
ejpam-6077	37	21	cauchy	cauchy	PROPN
ejpam-6077	37	22	if	if	SCONJ
ejpam-6077	37	23	for	for	ADP
ejpam-6077	37	24	every	every	DET
ejpam-6077	37	25	ϵ	ϵ	X
ejpam-6077	37	26	>	>	X
ejpam-6077	37	27	0	0	NUM
ejpam-6077	37	28	,	,	PUNCT
ejpam-6077	37	29	there	there	PRON
ejpam-6077	37	30	exists	exist	VERB
ejpam-6077	37	31	a	a	DET
ejpam-6077	37	32	positive	positive	ADJ
ejpam-6077	37	33	integer	integer	NOUN
ejpam-6077	37	34	n	n	CCONJ
ejpam-6077	37	35	such	such	ADJ
ejpam-6077	37	36	that	that	DET
ejpam-6077	37	37	m(υin	m(υin	NOUN
ejpam-6077	37	38	,	,	PUNCT
ejpam-6077	37	39	υim	υim	PROPN
ejpam-6077	37	40	,	,	PUNCT
ejpam-6077	37	41	υip	υip	ADJ
ejpam-6077	37	42	)	)	PUNCT
ejpam-6077	37	43	<	<	X
ejpam-6077	37	44	ϵ	ϵ	X
ejpam-6077	37	45	for	for	ADP
ejpam-6077	37	46	all	all	DET
ejpam-6077	37	47	m	m	PROPN
ejpam-6077	37	48	,	,	PUNCT
ejpam-6077	37	49	n	n	CCONJ
ejpam-6077	37	50	,	,	PUNCT
ejpam-6077	37	51	p	p	NOUN
ejpam-6077	37	52	≥	≥	NOUN
ejpam-6077	37	53	n	n	NOUN
ejpam-6077	37	54	.	.	PUNCT
ejpam-6077	38	1	definition	definition	NOUN
ejpam-6077	38	2	4	4	NUM
ejpam-6077	38	3	.	.	PUNCT
ejpam-6077	39	1	an	an	DET
ejpam-6077	39	2	mr	mr	PROPN
ejpam-6077	39	3	-	-	PUNCT
ejpam-6077	39	4	metric	metric	ADJ
ejpam-6077	39	5	space	space	NOUN
ejpam-6077	39	6	(	(	PUNCT
ejpam-6077	39	7	x	x	X
ejpam-6077	39	8	,	,	PUNCT
ejpam-6077	39	9	m	m	VERB
ejpam-6077	39	10	)	)	PUNCT
ejpam-6077	39	11	is	be	AUX
ejpam-6077	39	12	considered	consider	VERB
ejpam-6077	39	13	bounded	bound	VERB
ejpam-6077	39	14	if	if	SCONJ
ejpam-6077	39	15	there	there	PRON
ejpam-6077	39	16	exists	exist	VERB
ejpam-6077	39	17	a	a	DET
ejpam-6077	39	18	real	real	ADJ
ejpam-6077	39	19	number	number	NOUN
ejpam-6077	39	20	l	l	NOUN
ejpam-6077	39	21	>	>	X
ejpam-6077	39	22	0	0	NUM
ejpam-6077	39	23	such	such	ADJ
ejpam-6077	39	24	that	that	SCONJ
ejpam-6077	39	25	m(υ	m(υ	PROPN
ejpam-6077	39	26	,	,	PUNCT
ejpam-6077	39	27	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	39	28	)	)	PUNCT
ejpam-6077	39	29	≤	≤	NUM
ejpam-6077	39	30	l	l	NOUN
ejpam-6077	39	31	for	for	ADP
ejpam-6077	39	32	all	all	DET
ejpam-6077	39	33	υ	υ	PROPN
ejpam-6077	39	34	,	,	PUNCT
ejpam-6077	39	35	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	39	36	∈	∈	PROPN
ejpam-6077	39	37	x.	x.	NOUN
ejpam-6077	39	38	in	in	ADP
ejpam-6077	39	39	this	this	DET
ejpam-6077	39	40	case	case	NOUN
ejpam-6077	39	41	,	,	PUNCT
ejpam-6077	39	42	m	m	VERB
ejpam-6077	39	43	is	be	AUX
ejpam-6077	39	44	referred	refer	VERB
ejpam-6077	39	45	to	to	ADP
ejpam-6077	39	46	as	as	ADP
ejpam-6077	39	47	an	an	DET
ejpam-6077	39	48	mr	mr	NOUN
ejpam-6077	39	49	-	-	PUNCT
ejpam-6077	39	50	bound	bind	VERB
ejpam-6077	39	51	for	for	ADP
ejpam-6077	39	52	the	the	DET
ejpam-6077	39	53	metric	metric	NOUN
ejpam-6077	39	54	.	.	PUNCT
ejpam-6077	40	1	definition	definition	NOUN
ejpam-6077	40	2	5	5	NUM
ejpam-6077	40	3	.	.	PUNCT
ejpam-6077	41	1	let	let	VERB
ejpam-6077	41	2	e	e	PRON
ejpam-6077	41	3	be	be	AUX
ejpam-6077	41	4	a	a	DET
ejpam-6077	41	5	set	set	NOUN
ejpam-6077	41	6	subset	subset	NOUN
ejpam-6077	41	7	of	of	ADP
ejpam-6077	41	8	an	an	DET
ejpam-6077	41	9	mr	mr	PROPN
ejpam-6077	41	10	-	-	PUNCT
ejpam-6077	41	11	metric	metric	ADJ
ejpam-6077	41	12	space	space	NOUN
ejpam-6077	41	13	(	(	PUNCT
ejpam-6077	41	14	x	x	X
ejpam-6077	41	15	,	,	PUNCT
ejpam-6077	41	16	m	m	VERB
ejpam-6077	41	17	)	)	PUNCT
ejpam-6077	41	18	is	be	AUX
ejpam-6077	41	19	said	say	VERB
ejpam-6077	41	20	to	to	PART
ejpam-6077	41	21	be	be	AUX
ejpam-6077	41	22	m	m	AUX
ejpam-6077	41	23	bounded	bound	VERB
ejpam-6077	41	24	if	if	SCONJ
ejpam-6077	41	25	there	there	PRON
ejpam-6077	41	26	exists	exist	VERB
ejpam-6077	41	27	l	l	NOUN
ejpam-6077	41	28	>	>	X
ejpam-6077	41	29	0	0	NUM
ejpam-6077	41	30	such	such	ADJ
ejpam-6077	41	31	that	that	SCONJ
ejpam-6077	41	32	m(υ	m(υ	PROPN
ejpam-6077	41	33	,	,	PUNCT
ejpam-6077	41	34	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	41	35	)	)	PUNCT
ejpam-6077	41	36	≤	≤	NUM
ejpam-6077	41	37	l	l	NOUN
ejpam-6077	41	38	for	for	ADP
ejpam-6077	41	39	all	all	DET
ejpam-6077	41	40	υ	υ	PROPN
ejpam-6077	41	41	,	,	PUNCT
ejpam-6077	41	42	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	41	43	∈	∈	PROPN
ejpam-6077	41	44	e.	e.	PROPN
ejpam-6077	41	45	definition	definition	NOUN
ejpam-6077	41	46	6	6	NUM
ejpam-6077	41	47	.	.	PUNCT
ejpam-6077	42	1	[	[	X
ejpam-6077	42	2	29	29	NUM
ejpam-6077	42	3	]	]	X
ejpam-6077	42	4	a	a	DET
ejpam-6077	42	5	banach	banach	NOUN
ejpam-6077	42	6	space	space	NOUN
ejpam-6077	42	7	is	be	AUX
ejpam-6077	42	8	a	a	DET
ejpam-6077	42	9	vector	vector	NOUN
ejpam-6077	42	10	space	space	NOUN
ejpam-6077	42	11	x	x	NOUN
ejpam-6077	42	12	over	over	ADP
ejpam-6077	42	13	the	the	DET
ejpam-6077	42	14	field	field	NOUN
ejpam-6077	42	15	r	r	NOUN
ejpam-6077	42	16	or	or	CCONJ
ejpam-6077	42	17	c	c	NOUN
ejpam-6077	42	18	equipped	equip	VERB
ejpam-6077	42	19	with	with	ADP
ejpam-6077	42	20	a	a	DET
ejpam-6077	42	21	norm	norm	NOUN
ejpam-6077	42	22	∥	∥	X
ejpam-6077	42	23	·	·	PUNCT
ejpam-6077	42	24	∥	∥	X
ejpam-6077	42	25	:	:	PUNCT
ejpam-6077	43	1	x	x	X
ejpam-6077	43	2	→	→	PUNCT
ejpam-6077	43	3	r	r	NOUN
ejpam-6077	43	4	that	that	PRON
ejpam-6077	43	5	satisfies	satisfy	VERB
ejpam-6077	43	6	the	the	DET
ejpam-6077	43	7	following	follow	VERB
ejpam-6077	43	8	conditions	condition	NOUN
ejpam-6077	43	9	:	:	PUNCT
ejpam-6077	43	10	(	(	PUNCT
ejpam-6077	43	11	i	i	NOUN
ejpam-6077	43	12	)	)	PUNCT
ejpam-6077	43	13	positivity	positivity	NOUN
ejpam-6077	43	14	:	:	PUNCT
ejpam-6077	43	15	∥υ∥	∥υ∥	X
ejpam-6077	43	16	≥	≥	X
ejpam-6077	43	17	0	0	NUM
ejpam-6077	43	18	for	for	ADP
ejpam-6077	43	19	all	all	PRON
ejpam-6077	43	20	υ	υ	PRON
ejpam-6077	43	21	∈	∈	PROPN
ejpam-6077	43	22	x	x	NOUN
ejpam-6077	43	23	,	,	PUNCT
ejpam-6077	43	24	and	and	CCONJ
ejpam-6077	43	25	∥υ∥	∥υ∥	X
ejpam-6077	43	26	=	=	SYM
ejpam-6077	43	27	0	0	PUNCT
ejpam-6077	43	28	if	if	SCONJ
ejpam-6077	43	29	and	and	CCONJ
ejpam-6077	43	30	only	only	ADV
ejpam-6077	43	31	if	if	SCONJ
ejpam-6077	43	32	υ	υ	PROPN
ejpam-6077	43	33	=	=	NOUN
ejpam-6077	43	34	0	0	PROPN
ejpam-6077	43	35	.	.	PUNCT
ejpam-6077	43	36	(	(	PUNCT
ejpam-6077	43	37	ii	ii	NOUN
ejpam-6077	43	38	)	)	PUNCT
ejpam-6077	43	39	scalar	scalar	ADJ
ejpam-6077	43	40	multiplication	multiplication	NOUN
ejpam-6077	43	41	:	:	PUNCT
ejpam-6077	43	42	∥αυ∥	∥αυ∥	ADJ
ejpam-6077	43	43	=	=	SYM
ejpam-6077	43	44	|α|∥υ∥	|α|∥υ∥	PROPN
ejpam-6077	43	45	for	for	ADP
ejpam-6077	43	46	all	all	DET
ejpam-6077	43	47	α	α	NOUN
ejpam-6077	43	48	∈	∈	NOUN
ejpam-6077	43	49	r	r	NOUN
ejpam-6077	43	50	or	or	CCONJ
ejpam-6077	43	51	c	c	NOUN
ejpam-6077	43	52	,	,	PUNCT
ejpam-6077	43	53	and	and	CCONJ
ejpam-6077	43	54	υ	υ	DET
ejpam-6077	43	55	∈	∈	PROPN
ejpam-6077	43	56	x.	x.	NOUN
ejpam-6077	43	57	(	(	PUNCT
ejpam-6077	43	58	iii	iii	NOUN
ejpam-6077	43	59	)	)	PUNCT
ejpam-6077	43	60	triangle	triangle	NOUN
ejpam-6077	43	61	inequality	inequality	NOUN
ejpam-6077	43	62	:	:	PUNCT
ejpam-6077	43	63	∥υ	∥υ	PROPN
ejpam-6077	43	64	+	+	CCONJ
ejpam-6077	43	65	ξ∥	ξ∥	PROPN
ejpam-6077	43	66	≤	≤	NOUN
ejpam-6077	43	67	∥υ∥+	∥υ∥+	VERB
ejpam-6077	43	68	∥ξ∥	∥ξ∥	PROPN
ejpam-6077	43	69	for	for	ADP
ejpam-6077	43	70	all	all	DET
ejpam-6077	43	71	υ	υ	PROPN
ejpam-6077	43	72	,	,	PUNCT
ejpam-6077	43	73	ξ	ξ	PROPN
ejpam-6077	43	74	∈	∈	PROPN
ejpam-6077	43	75	x.	x.	NOUN
ejpam-6077	43	76	a.	a.	NOUN
ejpam-6077	43	77	malkawi	malkawi	ADP
ejpam-6077	43	78	/	/	SYM
ejpam-6077	43	79	eur	eur	PROPN
ejpam-6077	43	80	.	.	PUNCT
ejpam-6077	44	1	j.	j.	PROPN
ejpam-6077	44	2	pure	pure	PROPN
ejpam-6077	44	3	appl	appl	PROPN
ejpam-6077	44	4	.	.	PROPN
ejpam-6077	44	5	math	math	PROPN
ejpam-6077	44	6	,	,	PUNCT
ejpam-6077	44	7	18	18	NUM
ejpam-6077	44	8	(	(	PUNCT
ejpam-6077	44	9	2	2	NUM
ejpam-6077	44	10	)	)	PUNCT
ejpam-6077	44	11	(	(	PUNCT
ejpam-6077	44	12	2025	2025	NUM
ejpam-6077	44	13	)	)	PUNCT
ejpam-6077	44	14	,	,	PUNCT
ejpam-6077	44	15	6077	6077	NUM
ejpam-6077	44	16	3	3	NUM
ejpam-6077	44	17	of	of	ADP
ejpam-6077	44	18	16	16	NUM
ejpam-6077	44	19	moreover	moreover	ADV
ejpam-6077	44	20	,	,	PUNCT
ejpam-6077	44	21	x	x	PUNCT
ejpam-6077	44	22	is	be	AUX
ejpam-6077	44	23	complete	complete	ADJ
ejpam-6077	44	24	with	with	ADP
ejpam-6077	44	25	respect	respect	NOUN
ejpam-6077	44	26	to	to	ADP
ejpam-6077	44	27	the	the	DET
ejpam-6077	44	28	norm	norm	NOUN
ejpam-6077	44	29	,	,	PUNCT
ejpam-6077	44	30	meaning	mean	VERB
ejpam-6077	44	31	that	that	SCONJ
ejpam-6077	44	32	every	every	DET
ejpam-6077	44	33	cauchy	cauchy	ADJ
ejpam-6077	44	34	sequence	sequence	NOUN
ejpam-6077	44	35	in	in	ADP
ejpam-6077	44	36	x	x	PUNCT
ejpam-6077	44	37	converges	converge	NOUN
ejpam-6077	44	38	to	to	ADP
ejpam-6077	44	39	a	a	DET
ejpam-6077	44	40	limit	limit	NOUN
ejpam-6077	44	41	in	in	ADP
ejpam-6077	44	42	x.	x.	NOUN
ejpam-6077	44	43	definition	definition	NOUN
ejpam-6077	44	44	7	7	NUM
ejpam-6077	44	45	.	.	PUNCT
ejpam-6077	45	1	[	[	X
ejpam-6077	45	2	30	30	NUM
ejpam-6077	45	3	]	]	PUNCT
ejpam-6077	45	4	in	in	ADP
ejpam-6077	45	5	measure	measure	NOUN
ejpam-6077	45	6	theory	theory	NOUN
ejpam-6077	45	7	,	,	PUNCT
ejpam-6077	45	8	a	a	DET
ejpam-6077	45	9	measurable	measurable	ADJ
ejpam-6077	45	10	set	set	NOUN
ejpam-6077	45	11	is	be	AUX
ejpam-6077	45	12	a	a	DET
ejpam-6077	45	13	subset	subset	NOUN
ejpam-6077	45	14	of	of	ADP
ejpam-6077	45	15	a	a	DET
ejpam-6077	45	16	set	set	NOUN
ejpam-6077	45	17	x	x	PUNCT
ejpam-6077	45	18	that	that	PRON
ejpam-6077	45	19	belongs	belong	VERB
ejpam-6077	45	20	to	to	ADP
ejpam-6077	45	21	a	a	DET
ejpam-6077	45	22	σ	σ	NOUN
ejpam-6077	45	23	-	-	PUNCT
ejpam-6077	45	24	algebra	algebra	NOUN
ejpam-6077	45	25	a	a	PRON
ejpam-6077	45	26	over	over	ADP
ejpam-6077	45	27	x.	x.	NOUN
ejpam-6077	45	28	this	this	PRON
ejpam-6077	45	29	means	mean	VERB
ejpam-6077	45	30	:	:	PUNCT
ejpam-6077	45	31	•	•	ADP
ejpam-6077	45	32	the	the	DET
ejpam-6077	45	33	set	set	NOUN
ejpam-6077	45	34	x	x	PUNCT
ejpam-6077	45	35	is	be	AUX
ejpam-6077	45	36	associated	associate	VERB
ejpam-6077	45	37	with	with	ADP
ejpam-6077	45	38	a	a	DET
ejpam-6077	45	39	measure	measure	NOUN
ejpam-6077	45	40	µ	µ	NOUN
ejpam-6077	45	41	,	,	PUNCT
ejpam-6077	45	42	which	which	PRON
ejpam-6077	45	43	is	be	AUX
ejpam-6077	45	44	a	a	DET
ejpam-6077	45	45	function	function	NOUN
ejpam-6077	45	46	defined	define	VERB
ejpam-6077	45	47	on	on	ADP
ejpam-6077	45	48	a	a	DET
ejpam-6077	45	49	collection	collection	NOUN
ejpam-6077	45	50	a.	a.	NOUN
ejpam-6077	45	51	•	•	ADP
ejpam-6077	45	52	a	a	PRON
ejpam-6077	45	53	is	be	AUX
ejpam-6077	45	54	a	a	DET
ejpam-6077	45	55	family	family	NOUN
ejpam-6077	45	56	of	of	ADP
ejpam-6077	45	57	subsets	subset	NOUN
ejpam-6077	45	58	of	of	ADP
ejpam-6077	45	59	x	x	PUNCT
ejpam-6077	45	60	satisfying	satisfy	VERB
ejpam-6077	45	61	the	the	DET
ejpam-6077	45	62	following	follow	VERB
ejpam-6077	45	63	conditions	condition	NOUN
ejpam-6077	45	64	:	:	PUNCT
ejpam-6077	45	65	(	(	PUNCT
ejpam-6077	45	66	i	i	NOUN
ejpam-6077	45	67	)	)	PUNCT
ejpam-6077	45	68	x	x	SYM
ejpam-6077	45	69	∈	∈	NOUN
ejpam-6077	45	70	a.	a.	NOUN
ejpam-6077	45	71	(	(	PUNCT
ejpam-6077	45	72	ii	ii	NOUN
ejpam-6077	45	73	)	)	PUNCT
ejpam-6077	45	74	if	if	SCONJ
ejpam-6077	45	75	a	a	DET
ejpam-6077	45	76	∈	∈	PROPN
ejpam-6077	45	77	a	a	PRON
ejpam-6077	45	78	,	,	PUNCT
ejpam-6077	45	79	then	then	ADV
ejpam-6077	45	80	x	x	PUNCT
ejpam-6077	45	81	\a	\a	ADJ
ejpam-6077	45	82	∈	∈	PROPN
ejpam-6077	45	83	a	a	DET
ejpam-6077	45	84	(	(	PUNCT
ejpam-6077	45	85	closure	closure	NOUN
ejpam-6077	45	86	under	under	ADP
ejpam-6077	45	87	complements	complement	NOUN
ejpam-6077	45	88	)	)	PUNCT
ejpam-6077	45	89	.	.	PUNCT
ejpam-6077	46	1	(	(	PUNCT
ejpam-6077	46	2	iii	iii	X
ejpam-6077	46	3	)	)	PUNCT
ejpam-6077	46	4	if	if	SCONJ
ejpam-6077	46	5	{	{	PUNCT
ejpam-6077	46	6	an}∞n=1	an}∞n=1	X
ejpam-6077	46	7	⊂	⊂	PROPN
ejpam-6077	46	8	a	a	X
ejpam-6077	46	9	,	,	PUNCT
ejpam-6077	46	10	then	then	ADV
ejpam-6077	46	11	⋃∞	⋃∞	PUNCT
ejpam-6077	46	12	n=1an	n=1an	PROPN
ejpam-6077	46	13	∈	∈	PROPN
ejpam-6077	46	14	a	a	DET
ejpam-6077	46	15	(	(	PUNCT
ejpam-6077	46	16	closure	closure	NOUN
ejpam-6077	46	17	under	under	ADP
ejpam-6077	46	18	countable	countable	ADJ
ejpam-6077	46	19	unions	union	NOUN
ejpam-6077	46	20	)	)	PUNCT
ejpam-6077	46	21	.	.	PUNCT
ejpam-6077	47	1	thus	thus	ADV
ejpam-6077	47	2	,	,	PUNCT
ejpam-6077	47	3	a	a	DET
ejpam-6077	47	4	measurable	measurable	ADJ
ejpam-6077	47	5	set	set	NOUN
ejpam-6077	47	6	is	be	AUX
ejpam-6077	47	7	any	any	DET
ejpam-6077	47	8	element	element	NOUN
ejpam-6077	47	9	of	of	ADP
ejpam-6077	47	10	the	the	DET
ejpam-6077	47	11	σ	σ	PROPN
ejpam-6077	47	12	-	-	PUNCT
ejpam-6077	47	13	algebra	algebra	PROPN
ejpam-6077	47	14	a.	a.	NOUN
ejpam-6077	47	15	2	2	NUM
ejpam-6077	47	16	.	.	PUNCT
ejpam-6077	47	17	fixed	fix	VERB
ejpam-6077	47	18	point	point	NOUN
ejpam-6077	47	19	theorem	theorem	VERB
ejpam-6077	47	20	in	in	ADP
ejpam-6077	47	21	mr	mr	PROPN
ejpam-6077	47	22	-	-	PUNCT
ejpam-6077	47	23	metric	metric	ADJ
ejpam-6077	47	24	space	space	NOUN
ejpam-6077	47	25	within	within	ADP
ejpam-6077	47	26	a	a	DET
ejpam-6077	47	27	banach	banach	NOUN
ejpam-6077	47	28	space	space	NOUN
ejpam-6077	47	29	fixed	fix	VERB
ejpam-6077	47	30	-	-	PUNCT
ejpam-6077	47	31	point	point	NOUN
ejpam-6077	47	32	theory	theory	NOUN
ejpam-6077	47	33	within	within	ADP
ejpam-6077	47	34	mr	mr	PROPN
ejpam-6077	47	35	-	-	PUNCT
ejpam-6077	47	36	metric	metric	ADJ
ejpam-6077	47	37	spaces	space	NOUN
ejpam-6077	47	38	broadens	broaden	VERB
ejpam-6077	47	39	traditional	traditional	ADJ
ejpam-6077	47	40	fixed	fix	VERB
ejpam-6077	47	41	-	-	PUNCT
ejpam-6077	47	42	point	point	NOUN
ejpam-6077	47	43	results	result	NOUN
ejpam-6077	47	44	by	by	ADP
ejpam-6077	47	45	addressing	address	VERB
ejpam-6077	47	46	mappings	mapping	NOUN
ejpam-6077	47	47	in	in	ADP
ejpam-6077	47	48	more	more	ADV
ejpam-6077	47	49	generalized	generalized	ADJ
ejpam-6077	47	50	and	and	CCONJ
ejpam-6077	47	51	intricate	intricate	ADJ
ejpam-6077	47	52	settings	setting	NOUN
ejpam-6077	47	53	,	,	PUNCT
ejpam-6077	47	54	including	include	VERB
ejpam-6077	47	55	noncompact	noncompact	ADJ
ejpam-6077	47	56	spaces	space	NOUN
ejpam-6077	47	57	.	.	PUNCT
ejpam-6077	48	1	here	here	ADV
ejpam-6077	48	2	,	,	PUNCT
ejpam-6077	48	3	we	we	PRON
ejpam-6077	48	4	focus	focus	VERB
ejpam-6077	48	5	on	on	ADP
ejpam-6077	48	6	mr	mr	PROPN
ejpam-6077	48	7	-	-	PUNCT
ejpam-6077	48	8	metric	metric	ADJ
ejpam-6077	48	9	spaces	space	NOUN
ejpam-6077	48	10	where	where	SCONJ
ejpam-6077	48	11	x	x	PRON
ejpam-6077	48	12	is	be	AUX
ejpam-6077	48	13	a	a	DET
ejpam-6077	48	14	closed	closed	ADJ
ejpam-6077	48	15	,	,	PUNCT
ejpam-6077	48	16	bounded	bound	VERB
ejpam-6077	48	17	,	,	PUNCT
ejpam-6077	48	18	and	and	CCONJ
ejpam-6077	48	19	convex	convex	PROPN
ejpam-6077	48	20	subset	subset	NOUN
ejpam-6077	48	21	of	of	ADP
ejpam-6077	48	22	a	a	DET
ejpam-6077	48	23	banach	banach	NOUN
ejpam-6077	48	24	space	space	NOUN
ejpam-6077	48	25	(	(	PUNCT
ejpam-6077	48	26	e	e	NOUN
ejpam-6077	48	27	,	,	PUNCT
ejpam-6077	48	28	∥	∥	X
ejpam-6077	48	29	·	·	PUNCT
ejpam-6077	48	30	∥	∥	NUM
ejpam-6077	48	31	)	)	PUNCT
ejpam-6077	48	32	.	.	PUNCT
ejpam-6077	49	1	conditions	condition	NOUN
ejpam-6077	49	2	are	be	AUX
ejpam-6077	49	3	derived	derive	VERB
ejpam-6077	49	4	to	to	PART
ejpam-6077	49	5	ensure	ensure	VERB
ejpam-6077	49	6	that	that	SCONJ
ejpam-6077	49	7	a	a	DET
ejpam-6077	49	8	continuous	continuous	ADJ
ejpam-6077	49	9	mapping	mapping	NOUN
ejpam-6077	49	10	s	s	PART
ejpam-6077	49	11	:	:	PUNCT
ejpam-6077	49	12	x	x	SYM
ejpam-6077	49	13	→	→	SYM
ejpam-6077	49	14	x	x	PART
ejpam-6077	49	15	possesses	possess	VERB
ejpam-6077	49	16	a	a	DET
ejpam-6077	49	17	unique	unique	ADJ
ejpam-6077	49	18	fixed	fix	VERB
ejpam-6077	49	19	point	point	NOUN
ejpam-6077	49	20	.	.	PUNCT
ejpam-6077	50	1	this	this	DET
ejpam-6077	50	2	result	result	NOUN
ejpam-6077	50	3	forms	form	VERB
ejpam-6077	50	4	a	a	DET
ejpam-6077	50	5	crucial	crucial	ADJ
ejpam-6077	50	6	theoretical	theoretical	ADJ
ejpam-6077	50	7	foundation	foundation	NOUN
ejpam-6077	50	8	for	for	ADP
ejpam-6077	50	9	applications	application	NOUN
ejpam-6077	50	10	in	in	ADP
ejpam-6077	50	11	both	both	CCONJ
ejpam-6077	50	12	pure	pure	ADJ
ejpam-6077	50	13	and	and	CCONJ
ejpam-6077	50	14	applied	applied	ADJ
ejpam-6077	50	15	mathematical	mathematical	ADJ
ejpam-6077	50	16	contexts	contexts	NOUN
ejpam-6077	50	17	.	.	PUNCT
ejpam-6077	51	1	theorem	theorem	NOUN
ejpam-6077	51	2	1	1	X
ejpam-6077	51	3	.	.	X
ejpam-6077	51	4	consider	consider	VERB
ejpam-6077	51	5	an	an	DET
ejpam-6077	51	6	mr	mr	ADJ
ejpam-6077	51	7	-	-	PUNCT
ejpam-6077	51	8	metric	metric	ADJ
ejpam-6077	51	9	space	space	NOUN
ejpam-6077	51	10	(	(	PUNCT
ejpam-6077	51	11	x	x	X
ejpam-6077	51	12	,	,	PUNCT
ejpam-6077	51	13	m	m	NOUN
ejpam-6077	51	14	)	)	PUNCT
ejpam-6077	51	15	,	,	PUNCT
ejpam-6077	51	16	where	where	SCONJ
ejpam-6077	51	17	x	x	PRON
ejpam-6077	51	18	is	be	AUX
ejpam-6077	51	19	a	a	DET
ejpam-6077	51	20	closed	closed	ADJ
ejpam-6077	51	21	,	,	PUNCT
ejpam-6077	51	22	bounded	bound	VERB
ejpam-6077	51	23	,	,	PUNCT
ejpam-6077	51	24	and	and	CCONJ
ejpam-6077	51	25	convex	convex	PROPN
ejpam-6077	51	26	subset	subset	NOUN
ejpam-6077	51	27	of	of	ADP
ejpam-6077	51	28	a	a	DET
ejpam-6077	51	29	banach	banach	NOUN
ejpam-6077	51	30	space	space	NOUN
ejpam-6077	51	31	(	(	PUNCT
ejpam-6077	51	32	e	e	NOUN
ejpam-6077	51	33	,	,	PUNCT
ejpam-6077	51	34	∥	∥	X
ejpam-6077	51	35	·	·	PUNCT
ejpam-6077	51	36	∥	∥	NUM
ejpam-6077	51	37	)	)	PUNCT
ejpam-6077	51	38	.	.	PUNCT
ejpam-6077	52	1	let	let	VERB
ejpam-6077	52	2	s	s	PRON
ejpam-6077	52	3	:	:	PUNCT
ejpam-6077	52	4	x	x	SYM
ejpam-6077	52	5	→	→	PUNCT
ejpam-6077	52	6	x	x	PUNCT
ejpam-6077	52	7	satisfy	satisfy	VERB
ejpam-6077	52	8	the	the	DET
ejpam-6077	52	9	following	follow	VERB
ejpam-6077	52	10	conditions	condition	NOUN
ejpam-6077	52	11	:	:	PUNCT
ejpam-6077	52	12	(	(	PUNCT
ejpam-6077	52	13	i	i	NOUN
ejpam-6077	52	14	)	)	PUNCT
ejpam-6077	52	15	s	s	AUX
ejpam-6077	52	16	is	be	AUX
ejpam-6077	52	17	continuous	continuous	ADJ
ejpam-6077	52	18	.	.	PUNCT
ejpam-6077	53	1	(	(	PUNCT
ejpam-6077	53	2	ii	ii	NOUN
ejpam-6077	53	3	)	)	PUNCT
ejpam-6077	53	4	for	for	ADP
ejpam-6077	53	5	all	all	PRON
ejpam-6077	53	6	υ	υ	PROPN
ejpam-6077	53	7	,	,	PUNCT
ejpam-6077	53	8	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	53	9	∈	∈	PROPN
ejpam-6077	53	10	x	x	SYM
ejpam-6077	53	11	,	,	PUNCT
ejpam-6077	53	12	m(s(υ	m(s(υ	PROPN
ejpam-6077	53	13	)	)	PUNCT
ejpam-6077	53	14	,	,	PUNCT
ejpam-6077	53	15	s(ξ	s(ξ	PROPN
ejpam-6077	53	16	)	)	PUNCT
ejpam-6077	53	17	,	,	PUNCT
ejpam-6077	53	18	s(ℑ	s(ℑ	PROPN
ejpam-6077	53	19	)	)	PUNCT
ejpam-6077	53	20	)	)	PUNCT
ejpam-6077	54	1	≤	≤	PUNCT
ejpam-6077	55	1	k	k	X
ejpam-6077	55	2	·	·	PUNCT
ejpam-6077	55	3	m(υ	m(υ	PROPN
ejpam-6077	55	4	,	,	PUNCT
ejpam-6077	55	5	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	55	6	)	)	PUNCT
ejpam-6077	55	7	,	,	PUNCT
ejpam-6077	55	8	where	where	SCONJ
ejpam-6077	55	9	k	k	PROPN
ejpam-6077	55	10	∈	∈	PROPN
ejpam-6077	56	1	[	[	X
ejpam-6077	56	2	0	0	NUM
ejpam-6077	56	3	,	,	PUNCT
ejpam-6077	56	4	1	1	NUM
ejpam-6077	56	5	)	)	PUNCT
ejpam-6077	56	6	is	be	AUX
ejpam-6077	56	7	a	a	DET
ejpam-6077	56	8	constant	constant	ADJ
ejpam-6077	56	9	.	.	PUNCT
ejpam-6077	57	1	under	under	ADP
ejpam-6077	57	2	these	these	DET
ejpam-6077	57	3	conditions	condition	NOUN
ejpam-6077	57	4	,	,	PUNCT
ejpam-6077	57	5	s	s	VERB
ejpam-6077	57	6	has	have	VERB
ejpam-6077	57	7	a	a	DET
ejpam-6077	57	8	unique	unique	ADJ
ejpam-6077	57	9	fixed	fix	VERB
ejpam-6077	57	10	point	point	NOUN
ejpam-6077	57	11	υ∗	υ∗	NOUN
ejpam-6077	57	12	∈	∈	PROPN
ejpam-6077	57	13	x	x	NOUN
ejpam-6077	57	14	,	,	PUNCT
ejpam-6077	57	15	such	such	ADJ
ejpam-6077	57	16	that	that	DET
ejpam-6077	57	17	s(υ∗	s(υ∗	NOUN
ejpam-6077	57	18	)	)	PUNCT
ejpam-6077	57	19	=	=	SYM
ejpam-6077	58	1	υ∗.	υ∗.	VERB
ejpam-6077	58	2	proof	proof	NOUN
ejpam-6077	58	3	.	.	PUNCT
ejpam-6077	59	1	let	let	VERB
ejpam-6077	59	2	υ0	υ0	PROPN
ejpam-6077	59	3	∈	∈	PROPN
ejpam-6077	59	4	x	x	PUNCT
ejpam-6077	59	5	be	be	AUX
ejpam-6077	59	6	an	an	DET
ejpam-6077	59	7	arbitrary	arbitrary	ADJ
ejpam-6077	59	8	element	element	NOUN
ejpam-6077	59	9	,	,	PUNCT
ejpam-6077	59	10	and	and	CCONJ
ejpam-6077	59	11	define	define	VERB
ejpam-6077	59	12	a	a	DET
ejpam-6077	59	13	sequence	sequence	NOUN
ejpam-6077	59	14	{	{	PUNCT
ejpam-6077	59	15	υn	υn	NOUN
ejpam-6077	59	16	}	}	PUNCT
ejpam-6077	59	17	by	by	ADP
ejpam-6077	59	18	υn+1	υn+1	ADJ
ejpam-6077	59	19	=	=	NOUN
ejpam-6077	59	20	s(υn	s(υn	PROPN
ejpam-6077	59	21	)	)	PUNCT
ejpam-6077	59	22	for	for	ADP
ejpam-6077	59	23	all	all	DET
ejpam-6077	59	24	n	n	DET
ejpam-6077	59	25	≥	≥	NOUN
ejpam-6077	59	26	0	0	NUM
ejpam-6077	59	27	.	.	PUNCT
ejpam-6077	60	1	using	use	VERB
ejpam-6077	60	2	the	the	DET
ejpam-6077	60	3	contraction	contraction	NOUN
ejpam-6077	60	4	condition	condition	NOUN
ejpam-6077	60	5	satisfied	satisfy	VERB
ejpam-6077	60	6	by	by	ADP
ejpam-6077	60	7	s	s	PROPN
ejpam-6077	60	8	,	,	PUNCT
ejpam-6077	60	9	we	we	PRON
ejpam-6077	60	10	have	have	VERB
ejpam-6077	60	11	for	for	ADP
ejpam-6077	60	12	all	all	DET
ejpam-6077	60	13	n	n	DET
ejpam-6077	60	14	≥	≥	NOUN
ejpam-6077	60	15	0	0	NUM
ejpam-6077	60	16	:	:	PUNCT
ejpam-6077	60	17	m(υn+1	m(υn+1	NUM
ejpam-6077	60	18	,	,	PUNCT
ejpam-6077	60	19	υn+2	υn+2	PROPN
ejpam-6077	60	20	,	,	PUNCT
ejpam-6077	60	21	υn+3	υn+3	NOUN
ejpam-6077	60	22	)	)	PUNCT
ejpam-6077	60	23	=	=	SYM
ejpam-6077	61	1	m(s(υn	m(s(υn	ADJ
ejpam-6077	61	2	)	)	PUNCT
ejpam-6077	61	3	,	,	PUNCT
ejpam-6077	61	4	s(υn+1	s(υn+1	ADJ
ejpam-6077	61	5	)	)	PUNCT
ejpam-6077	61	6	,	,	PUNCT
ejpam-6077	61	7	s(υn+2	s(υn+2	NUM
ejpam-6077	61	8	)	)	PUNCT
ejpam-6077	61	9	)	)	PUNCT
ejpam-6077	62	1	≤	≤	PUNCT
ejpam-6077	63	1	k	k	X
ejpam-6077	63	2	·	·	SYM
ejpam-6077	63	3	m(υn	m(υn	X
ejpam-6077	63	4	,	,	PUNCT
ejpam-6077	63	5	υn+1	υn+1	NOUN
ejpam-6077	63	6	,	,	PUNCT
ejpam-6077	63	7	υn+2	υn+2	NUM
ejpam-6077	63	8	)	)	PUNCT
ejpam-6077	63	9	.	.	PUNCT
ejpam-6077	64	1	iterating	iterate	VERB
ejpam-6077	64	2	this	this	DET
ejpam-6077	64	3	inequality	inequality	NOUN
ejpam-6077	64	4	yields	yield	NOUN
ejpam-6077	64	5	:	:	PUNCT
ejpam-6077	64	6	m(υn+1	m(υn+1	NUM
ejpam-6077	64	7	,	,	PUNCT
ejpam-6077	64	8	υn+2	υn+2	PROPN
ejpam-6077	64	9	,	,	PUNCT
ejpam-6077	64	10	υn+3	υn+3	NOUN
ejpam-6077	64	11	)	)	PUNCT
ejpam-6077	64	12	≤	≤	NOUN
ejpam-6077	65	1	kn	kn	PROPN
ejpam-6077	65	2	·	·	SYM
ejpam-6077	65	3	m(υ0	m(υ0	PROPN
ejpam-6077	65	4	,	,	PUNCT
ejpam-6077	65	5	υ1	υ1	PROPN
ejpam-6077	65	6	,	,	PUNCT
ejpam-6077	65	7	υ2	υ2	NOUN
ejpam-6077	65	8	)	)	PUNCT
ejpam-6077	65	9	,	,	PUNCT
ejpam-6077	65	10	a.	a.	NOUN
ejpam-6077	65	11	malkawi	malkawi	PROPN
ejpam-6077	65	12	/	/	SYM
ejpam-6077	65	13	eur	eur	PROPN
ejpam-6077	65	14	.	.	PUNCT
ejpam-6077	66	1	j.	j.	PROPN
ejpam-6077	66	2	pure	pure	PROPN
ejpam-6077	66	3	appl	appl	PROPN
ejpam-6077	66	4	.	.	PROPN
ejpam-6077	66	5	math	math	PROPN
ejpam-6077	66	6	,	,	PUNCT
ejpam-6077	66	7	18	18	NUM
ejpam-6077	66	8	(	(	PUNCT
ejpam-6077	66	9	2	2	NUM
ejpam-6077	66	10	)	)	PUNCT
ejpam-6077	66	11	(	(	PUNCT
ejpam-6077	66	12	2025	2025	NUM
ejpam-6077	66	13	)	)	PUNCT
ejpam-6077	66	14	,	,	PUNCT
ejpam-6077	66	15	6077	6077	NUM
ejpam-6077	66	16	4	4	NUM
ejpam-6077	66	17	of	of	ADP
ejpam-6077	66	18	16	16	NUM
ejpam-6077	66	19	where	where	SCONJ
ejpam-6077	66	20	m(υ0	m(υ0	PROPN
ejpam-6077	66	21	,	,	PUNCT
ejpam-6077	66	22	υ1	υ1	PROPN
ejpam-6077	66	23	,	,	PUNCT
ejpam-6077	66	24	υ2	υ2	NOUN
ejpam-6077	66	25	)	)	PUNCT
ejpam-6077	66	26	is	be	AUX
ejpam-6077	66	27	a	a	DET
ejpam-6077	66	28	finite	finite	NOUN
ejpam-6077	66	29	constant	constant	ADJ
ejpam-6077	66	30	since	since	SCONJ
ejpam-6077	66	31	m	m	PROPN
ejpam-6077	66	32	is	be	AUX
ejpam-6077	66	33	non	non	ADJ
ejpam-6077	66	34	-	-	ADJ
ejpam-6077	66	35	negative	negative	ADJ
ejpam-6077	66	36	.	.	PUNCT
ejpam-6077	67	1	as	as	ADP
ejpam-6077	67	2	n	n	NUM
ejpam-6077	67	3	→	→	SYM
ejpam-6077	67	4	∞	∞	PROPN
ejpam-6077	67	5	,	,	PUNCT
ejpam-6077	67	6	the	the	DET
ejpam-6077	67	7	term	term	NOUN
ejpam-6077	67	8	kn	kn	PROPN
ejpam-6077	67	9	→	→	SYM
ejpam-6077	67	10	0	0	PUNCT
ejpam-6077	68	1	because	because	SCONJ
ejpam-6077	68	2	k	k	PROPN
ejpam-6077	68	3	∈	∈	PROPN
ejpam-6077	68	4	[	[	X
ejpam-6077	68	5	0	0	NUM
ejpam-6077	68	6	,	,	PUNCT
ejpam-6077	68	7	1	1	NUM
ejpam-6077	68	8	)	)	PUNCT
ejpam-6077	68	9	.	.	PUNCT
ejpam-6077	69	1	thus	thus	ADV
ejpam-6077	69	2	:	:	PUNCT
ejpam-6077	69	3	lim	lim	PROPN
ejpam-6077	69	4	n→∞	n→∞	NUM
ejpam-6077	69	5	m(υn+1	m(υn+1	PROPN
ejpam-6077	69	6	,	,	PUNCT
ejpam-6077	69	7	υn+2	υn+2	PROPN
ejpam-6077	69	8	,	,	PUNCT
ejpam-6077	69	9	υn+3	υn+3	NOUN
ejpam-6077	69	10	)	)	PUNCT
ejpam-6077	69	11	=	=	SYM
ejpam-6077	69	12	0	0	X
ejpam-6077	69	13	.	.	PUNCT
ejpam-6077	70	1	this	this	PRON
ejpam-6077	70	2	implies	imply	VERB
ejpam-6077	70	3	that	that	SCONJ
ejpam-6077	70	4	the	the	DET
ejpam-6077	70	5	sequence	sequence	NOUN
ejpam-6077	70	6	{	{	PUNCT
ejpam-6077	70	7	υn	υn	NOUN
ejpam-6077	70	8	}	}	PUNCT
ejpam-6077	70	9	is	be	AUX
ejpam-6077	70	10	a	a	DET
ejpam-6077	70	11	cauchy	cauchy	ADJ
ejpam-6077	70	12	sequence	sequence	NOUN
ejpam-6077	70	13	with	with	ADP
ejpam-6077	70	14	respect	respect	NOUN
ejpam-6077	70	15	to	to	ADP
ejpam-6077	70	16	the	the	DET
ejpam-6077	70	17	mr	mr	PROPN
ejpam-6077	70	18	-	-	PUNCT
ejpam-6077	70	19	metric	metric	ADJ
ejpam-6077	70	20	m	m	NOUN
ejpam-6077	70	21	.	.	PUNCT
ejpam-6077	71	1	since	since	SCONJ
ejpam-6077	71	2	(	(	PUNCT
ejpam-6077	71	3	x	x	X
ejpam-6077	71	4	,	,	PUNCT
ejpam-6077	71	5	m	m	VERB
ejpam-6077	71	6	)	)	PUNCT
ejpam-6077	71	7	is	be	AUX
ejpam-6077	71	8	an	an	DET
ejpam-6077	71	9	mr	mr	PROPN
ejpam-6077	71	10	-	-	PUNCT
ejpam-6077	71	11	metric	metric	ADJ
ejpam-6077	71	12	space	space	NOUN
ejpam-6077	71	13	and	and	CCONJ
ejpam-6077	71	14	x	x	NOUN
ejpam-6077	71	15	is	be	AUX
ejpam-6077	71	16	closed	close	VERB
ejpam-6077	71	17	in	in	ADP
ejpam-6077	71	18	the	the	DET
ejpam-6077	71	19	banach	banach	NOUN
ejpam-6077	71	20	space	space	NOUN
ejpam-6077	71	21	e	e	NOUN
ejpam-6077	71	22	,	,	PUNCT
ejpam-6077	71	23	the	the	DET
ejpam-6077	71	24	completeness	completeness	NOUN
ejpam-6077	71	25	of	of	ADP
ejpam-6077	71	26	e	e	PROPN
ejpam-6077	71	27	ensures	ensure	VERB
ejpam-6077	71	28	that	that	SCONJ
ejpam-6077	71	29	the	the	DET
ejpam-6077	71	30	sequence	sequence	NOUN
ejpam-6077	71	31	{	{	PUNCT
ejpam-6077	71	32	υn	υn	NOUN
ejpam-6077	71	33	}	}	PUNCT
ejpam-6077	71	34	converges	converge	NOUN
ejpam-6077	71	35	to	to	ADP
ejpam-6077	71	36	some	some	DET
ejpam-6077	71	37	υ∗	υ∗	NOUN
ejpam-6077	71	38	∈	∈	PROPN
ejpam-6077	71	39	x.	x.	NOUN
ejpam-6077	72	1	that	that	PRON
ejpam-6077	72	2	is	be	AUX
ejpam-6077	72	3	,	,	PUNCT
ejpam-6077	72	4	lim	lim	PROPN
ejpam-6077	72	5	n→∞	n→∞	NUM
ejpam-6077	72	6	υn	υn	PROPN
ejpam-6077	72	7	=	=	SYM
ejpam-6077	72	8	υ∗.	υ∗.	VERB
ejpam-6077	72	9	to	to	PART
ejpam-6077	72	10	show	show	VERB
ejpam-6077	72	11	that	that	DET
ejpam-6077	72	12	υ∗	υ∗	NOUN
ejpam-6077	72	13	is	be	AUX
ejpam-6077	72	14	a	a	DET
ejpam-6077	72	15	fixed	fix	VERB
ejpam-6077	72	16	point	point	NOUN
ejpam-6077	72	17	of	of	ADP
ejpam-6077	72	18	s	s	PROPN
ejpam-6077	72	19	,	,	PUNCT
ejpam-6077	72	20	we	we	PRON
ejpam-6077	72	21	use	use	VERB
ejpam-6077	72	22	the	the	DET
ejpam-6077	72	23	continuity	continuity	NOUN
ejpam-6077	72	24	of	of	ADP
ejpam-6077	72	25	s.	s.	PROPN
ejpam-6077	72	26	by	by	ADP
ejpam-6077	72	27	definition	definition	NOUN
ejpam-6077	72	28	of	of	ADP
ejpam-6077	72	29	s	s	PRON
ejpam-6077	72	30	and	and	CCONJ
ejpam-6077	72	31	the	the	DET
ejpam-6077	72	32	convergence	convergence	NOUN
ejpam-6077	72	33	of	of	ADP
ejpam-6077	72	34	{	{	PUNCT
ejpam-6077	72	35	υn	υn	NOUN
ejpam-6077	72	36	}	}	PUNCT
ejpam-6077	72	37	,	,	PUNCT
ejpam-6077	72	38	we	we	PRON
ejpam-6077	72	39	have	have	VERB
ejpam-6077	72	40	:	:	PUNCT
ejpam-6077	72	41	lim	lim	PROPN
ejpam-6077	72	42	n→∞	n→∞	NUM
ejpam-6077	72	43	s(υn	s(υn	ADJ
ejpam-6077	72	44	)	)	PUNCT
ejpam-6077	72	45	=	=	SYM
ejpam-6077	72	46	s	s	X
ejpam-6077	72	47	(	(	PUNCT
ejpam-6077	72	48	lim	lim	PROPN
ejpam-6077	72	49	n→∞	n→∞	NUM
ejpam-6077	72	50	υn	υn	NOUN
ejpam-6077	72	51	)	)	PUNCT
ejpam-6077	72	52	.	.	PUNCT
ejpam-6077	73	1	substituting	substitute	VERB
ejpam-6077	73	2	υn+1	υn+1	NUM
ejpam-6077	73	3	=	=	SYM
ejpam-6077	73	4	s(υn	s(υn	NOUN
ejpam-6077	73	5	)	)	PUNCT
ejpam-6077	73	6	,	,	PUNCT
ejpam-6077	73	7	we	we	PRON
ejpam-6077	73	8	obtain	obtain	VERB
ejpam-6077	73	9	:	:	PUNCT
ejpam-6077	73	10	υ∗	υ∗	NOUN
ejpam-6077	73	11	=	=	SYM
ejpam-6077	73	12	s(υ∗	s(υ∗	X
ejpam-6077	73	13	)	)	PUNCT
ejpam-6077	73	14	.	.	PUNCT
ejpam-6077	74	1	to	to	PART
ejpam-6077	74	2	prove	prove	VERB
ejpam-6077	74	3	the	the	DET
ejpam-6077	74	4	uniqueness	uniqueness	NOUN
ejpam-6077	74	5	of	of	ADP
ejpam-6077	74	6	the	the	DET
ejpam-6077	74	7	fixed	fix	VERB
ejpam-6077	74	8	point	point	NOUN
ejpam-6077	74	9	,	,	PUNCT
ejpam-6077	74	10	suppose	suppose	VERB
ejpam-6077	74	11	there	there	PRON
ejpam-6077	74	12	exists	exist	VERB
ejpam-6077	74	13	another	another	DET
ejpam-6077	74	14	fixed	fix	VERB
ejpam-6077	74	15	point	point	NOUN
ejpam-6077	74	16	ξ∗	ξ∗	ADJ
ejpam-6077	74	17	̸=	̸=	PROPN
ejpam-6077	74	18	υ∗	υ∗	NOUN
ejpam-6077	74	19	such	such	ADJ
ejpam-6077	74	20	that	that	DET
ejpam-6077	74	21	s(ξ∗	s(ξ∗	NOUN
ejpam-6077	74	22	)	)	PUNCT
ejpam-6077	74	23	=	=	VERB
ejpam-6077	74	24	ξ∗.	ξ∗.	AUX
ejpam-6077	74	25	using	use	VERB
ejpam-6077	74	26	the	the	DET
ejpam-6077	74	27	contraction	contraction	NOUN
ejpam-6077	74	28	condition	condition	NOUN
ejpam-6077	74	29	for	for	ADP
ejpam-6077	74	30	s	s	PROPN
ejpam-6077	74	31	,	,	PUNCT
ejpam-6077	74	32	we	we	PRON
ejpam-6077	74	33	have	have	VERB
ejpam-6077	74	34	:	:	PUNCT
ejpam-6077	74	35	m(υ∗	m(υ∗	X
ejpam-6077	74	36	,	,	PUNCT
ejpam-6077	74	37	υ∗	υ∗	NOUN
ejpam-6077	74	38	,	,	PUNCT
ejpam-6077	74	39	υ∗	υ∗	NOUN
ejpam-6077	74	40	)	)	PUNCT
ejpam-6077	75	1	=	=	SYM
ejpam-6077	75	2	m(s(ξ∗	m(s(ξ∗	PROPN
ejpam-6077	75	3	)	)	PUNCT
ejpam-6077	75	4	,	,	PUNCT
ejpam-6077	75	5	s(ξ∗	s(ξ∗	NUM
ejpam-6077	75	6	)	)	PUNCT
ejpam-6077	75	7	,	,	PUNCT
ejpam-6077	75	8	s(ξ∗	s(ξ∗	NUM
ejpam-6077	75	9	)	)	PUNCT
ejpam-6077	75	10	)	)	PUNCT
ejpam-6077	76	1	≤	≤	PUNCT
ejpam-6077	77	1	k	k	X
ejpam-6077	77	2	·	·	PUNCT
ejpam-6077	77	3	m(ξ∗	m(ξ∗	NOUN
ejpam-6077	77	4	,	,	PUNCT
ejpam-6077	77	5	ξ∗	ξ∗	NOUN
ejpam-6077	77	6	,	,	PUNCT
ejpam-6077	77	7	ξ∗	ξ∗	NOUN
ejpam-6077	77	8	)	)	PUNCT
ejpam-6077	77	9	.	.	PUNCT
ejpam-6077	78	1	since	since	SCONJ
ejpam-6077	78	2	m(υ∗	m(υ∗	PROPN
ejpam-6077	78	3	,	,	PUNCT
ejpam-6077	78	4	υ∗	υ∗	NOUN
ejpam-6077	78	5	,	,	PUNCT
ejpam-6077	78	6	υ∗	υ∗	NOUN
ejpam-6077	78	7	)	)	PUNCT
ejpam-6077	78	8	=	=	SYM
ejpam-6077	78	9	0	0	NUM
ejpam-6077	78	10	by	by	ADP
ejpam-6077	78	11	the	the	DET
ejpam-6077	78	12	properties	property	NOUN
ejpam-6077	78	13	of	of	ADP
ejpam-6077	78	14	the	the	DET
ejpam-6077	78	15	mr	mr	PROPN
ejpam-6077	78	16	-	-	PUNCT
ejpam-6077	78	17	metric	metric	NOUN
ejpam-6077	78	18	,	,	PUNCT
ejpam-6077	78	19	it	it	PRON
ejpam-6077	78	20	follows	follow	VERB
ejpam-6077	78	21	that	that	SCONJ
ejpam-6077	78	22	m(ξ∗	m(ξ∗	NOUN
ejpam-6077	78	23	,	,	PUNCT
ejpam-6077	78	24	ξ∗	ξ∗	NOUN
ejpam-6077	78	25	,	,	PUNCT
ejpam-6077	78	26	ξ∗	ξ∗	ADJ
ejpam-6077	78	27	)	)	PUNCT
ejpam-6077	78	28	=	=	SYM
ejpam-6077	79	1	0	0	X
ejpam-6077	79	2	.	.	PUNCT
ejpam-6077	80	1	the	the	DET
ejpam-6077	80	2	second	second	ADJ
ejpam-6077	80	3	axiom	axiom	NOUN
ejpam-6077	80	4	of	of	ADP
ejpam-6077	80	5	the	the	DET
ejpam-6077	80	6	mr	mr	PROPN
ejpam-6077	80	7	-	-	PUNCT
ejpam-6077	80	8	metric	metric	ADJ
ejpam-6077	80	9	implies	imply	VERB
ejpam-6077	80	10	υ∗	υ∗	NOUN
ejpam-6077	80	11	=	=	SYM
ejpam-6077	80	12	ξ∗	ξ∗	PROPN
ejpam-6077	80	13	,	,	PUNCT
ejpam-6077	80	14	which	which	PRON
ejpam-6077	80	15	contradicts	contradict	VERB
ejpam-6077	80	16	the	the	DET
ejpam-6077	80	17	assumption	assumption	NOUN
ejpam-6077	80	18	that	that	SCONJ
ejpam-6077	80	19	υ∗	υ∗	NOUN
ejpam-6077	80	20	̸=	̸=	PROPN
ejpam-6077	80	21	ξ∗.	ξ∗.	VERB
ejpam-6077	80	22	hence	hence	ADV
ejpam-6077	80	23	,	,	PUNCT
ejpam-6077	80	24	υ∗	υ∗	PROPN
ejpam-6077	80	25	is	be	AUX
ejpam-6077	80	26	the	the	DET
ejpam-6077	80	27	unique	unique	ADJ
ejpam-6077	80	28	fixed	fix	VERB
ejpam-6077	80	29	point	point	NOUN
ejpam-6077	80	30	of	of	ADP
ejpam-6077	80	31	s.	s.	PROPN
ejpam-6077	80	32	theorem	theorem	VERB
ejpam-6077	80	33	2	2	X
ejpam-6077	80	34	.	.	PUNCT
ejpam-6077	81	1	let	let	AUX
ejpam-6077	81	2	(	(	PUNCT
ejpam-6077	81	3	x	x	X
ejpam-6077	81	4	,	,	PUNCT
ejpam-6077	81	5	m	m	VERB
ejpam-6077	81	6	)	)	PUNCT
ejpam-6077	81	7	be	be	VERB
ejpam-6077	81	8	an	an	DET
ejpam-6077	81	9	mr	mr	ADJ
ejpam-6077	81	10	-	-	PUNCT
ejpam-6077	81	11	metric	metric	ADJ
ejpam-6077	81	12	space	space	NOUN
ejpam-6077	81	13	,	,	PUNCT
ejpam-6077	81	14	where	where	SCONJ
ejpam-6077	81	15	x	x	PRON
ejpam-6077	81	16	is	be	AUX
ejpam-6077	81	17	a	a	DET
ejpam-6077	81	18	closed	closed	ADJ
ejpam-6077	81	19	,	,	PUNCT
ejpam-6077	81	20	bounded	bound	VERB
ejpam-6077	81	21	,	,	PUNCT
ejpam-6077	81	22	and	and	CCONJ
ejpam-6077	81	23	convex	convex	PROPN
ejpam-6077	81	24	subset	subset	NOUN
ejpam-6077	81	25	of	of	ADP
ejpam-6077	81	26	a	a	DET
ejpam-6077	81	27	banach	banach	NOUN
ejpam-6077	81	28	space	space	NOUN
ejpam-6077	81	29	(	(	PUNCT
ejpam-6077	81	30	e	e	NOUN
ejpam-6077	81	31	,	,	PUNCT
ejpam-6077	81	32	∥	∥	X
ejpam-6077	81	33	·	·	PUNCT
ejpam-6077	81	34	∥	∥	NUM
ejpam-6077	81	35	)	)	PUNCT
ejpam-6077	81	36	.	.	PUNCT
ejpam-6077	82	1	assume	assume	VERB
ejpam-6077	82	2	s	s	PRON
ejpam-6077	82	3	:	:	PUNCT
ejpam-6077	82	4	x	x	SYM
ejpam-6077	82	5	→	→	PUNCT
ejpam-6077	82	6	x	x	X
ejpam-6077	82	7	is	be	AUX
ejpam-6077	82	8	a	a	DET
ejpam-6077	82	9	continuous	continuous	ADJ
ejpam-6077	82	10	operator	operator	NOUN
ejpam-6077	82	11	satisfying	satisfy	VERB
ejpam-6077	82	12	the	the	DET
ejpam-6077	82	13	following	follow	VERB
ejpam-6077	82	14	conditions	condition	NOUN
ejpam-6077	82	15	:	:	PUNCT
ejpam-6077	82	16	(	(	PUNCT
ejpam-6077	82	17	i	i	NOUN
ejpam-6077	82	18	)	)	PUNCT
ejpam-6077	82	19	for	for	ADP
ejpam-6077	82	20	all	all	DET
ejpam-6077	82	21	υ	υ	PROPN
ejpam-6077	82	22	,	,	PUNCT
ejpam-6077	82	23	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	82	24	∈	∈	PROPN
ejpam-6077	82	25	x	x	SYM
ejpam-6077	82	26	,	,	PUNCT
ejpam-6077	82	27	m(s(υ	m(s(υ	PROPN
ejpam-6077	82	28	)	)	PUNCT
ejpam-6077	82	29	,	,	PUNCT
ejpam-6077	82	30	s(ξ	s(ξ	PROPN
ejpam-6077	82	31	)	)	PUNCT
ejpam-6077	82	32	,	,	PUNCT
ejpam-6077	82	33	s(ℑ	s(ℑ	PROPN
ejpam-6077	82	34	)	)	PUNCT
ejpam-6077	82	35	)	)	PUNCT
ejpam-6077	82	36	≤	≤	PUNCT
ejpam-6077	83	1	k	k	X
ejpam-6077	83	2	·	·	PUNCT
ejpam-6077	83	3	m(υ	m(υ	PROPN
ejpam-6077	83	4	,	,	PUNCT
ejpam-6077	83	5	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	83	6	)	)	PUNCT
ejpam-6077	83	7	,	,	PUNCT
ejpam-6077	83	8	where	where	SCONJ
ejpam-6077	83	9	k	k	PROPN
ejpam-6077	83	10	∈	∈	PROPN
ejpam-6077	84	1	[	[	X
ejpam-6077	84	2	0	0	NUM
ejpam-6077	84	3	,	,	PUNCT
ejpam-6077	84	4	1	1	NUM
ejpam-6077	84	5	)	)	PUNCT
ejpam-6077	84	6	is	be	AUX
ejpam-6077	84	7	a	a	DET
ejpam-6077	84	8	constant	constant	ADJ
ejpam-6077	84	9	.	.	PUNCT
ejpam-6077	85	1	(	(	PUNCT
ejpam-6077	85	2	ii	ii	X
ejpam-6077	85	3	)	)	PUNCT
ejpam-6077	85	4	the	the	DET
ejpam-6077	85	5	measure	measure	NOUN
ejpam-6077	85	6	of	of	ADP
ejpam-6077	85	7	noncompactness	noncompactness	ADJ
ejpam-6077	85	8	µ	µ	PRON
ejpam-6077	85	9	satisfies	satisfie	NOUN
ejpam-6077	85	10	:	:	PUNCT
ejpam-6077	85	11	µ(s(a	µ(s(a	NUM
ejpam-6077	85	12	)	)	PUNCT
ejpam-6077	85	13	)	)	PUNCT
ejpam-6077	85	14	≤	≤	NUM
ejpam-6077	85	15	ϕ(µ(a	ϕ(µ(a	NOUN
ejpam-6077	85	16	)	)	PUNCT
ejpam-6077	85	17	)	)	PUNCT
ejpam-6077	85	18	,	,	PUNCT
ejpam-6077	85	19	for	for	ADP
ejpam-6077	85	20	every	every	DET
ejpam-6077	85	21	bounded	bounded	NOUN
ejpam-6077	85	22	subset	subset	VERB
ejpam-6077	85	23	a	a	DET
ejpam-6077	85	24	⊂	⊂	PROPN
ejpam-6077	85	25	x	x	NOUN
ejpam-6077	85	26	,	,	PUNCT
ejpam-6077	85	27	where	where	SCONJ
ejpam-6077	85	28	ϕ	ϕ	NOUN
ejpam-6077	85	29	:	:	PUNCT
ejpam-6077	85	30	[	[	X
ejpam-6077	85	31	0,∞	0,∞	NOUN
ejpam-6077	85	32	)	)	PUNCT
ejpam-6077	85	33	→	→	PUNCT
ejpam-6077	86	1	[	[	X
ejpam-6077	86	2	0,∞	0,∞	NUM
ejpam-6077	86	3	)	)	PUNCT
ejpam-6077	86	4	is	be	AUX
ejpam-6077	86	5	a	a	DET
ejpam-6077	86	6	non	non	ADJ
ejpam-6077	86	7	-	-	ADJ
ejpam-6077	86	8	decreasing	decrease	VERB
ejpam-6077	86	9	function	function	NOUN
ejpam-6077	86	10	such	such	ADJ
ejpam-6077	86	11	that	that	SCONJ
ejpam-6077	86	12	ϕ(t	ϕ(t	NUM
ejpam-6077	86	13	)	)	PUNCT
ejpam-6077	86	14	<	<	X
ejpam-6077	86	15	t	t	PROPN
ejpam-6077	86	16	for	for	ADP
ejpam-6077	86	17	all	all	DET
ejpam-6077	86	18	t	t	PROPN
ejpam-6077	86	19	>	>	X
ejpam-6077	86	20	0	0	PUNCT
ejpam-6077	86	21	and	and	CCONJ
ejpam-6077	86	22	ϕ(0	ϕ(0	PROPN
ejpam-6077	86	23	)	)	PUNCT
ejpam-6077	87	1	=	=	PUNCT
ejpam-6077	87	2	0	0	X
ejpam-6077	87	3	.	.	PUNCT
ejpam-6077	87	4	a.	a.	NOUN
ejpam-6077	87	5	malkawi	malkawi	ADP
ejpam-6077	87	6	/	/	SYM
ejpam-6077	87	7	eur	eur	PROPN
ejpam-6077	87	8	.	.	PUNCT
ejpam-6077	88	1	j.	j.	PROPN
ejpam-6077	88	2	pure	pure	PROPN
ejpam-6077	88	3	appl	appl	PROPN
ejpam-6077	88	4	.	.	PROPN
ejpam-6077	88	5	math	math	PROPN
ejpam-6077	88	6	,	,	PUNCT
ejpam-6077	88	7	18	18	NUM
ejpam-6077	88	8	(	(	PUNCT
ejpam-6077	88	9	2	2	NUM
ejpam-6077	88	10	)	)	PUNCT
ejpam-6077	88	11	(	(	PUNCT
ejpam-6077	88	12	2025	2025	NUM
ejpam-6077	88	13	)	)	PUNCT
ejpam-6077	88	14	,	,	PUNCT
ejpam-6077	88	15	6077	6077	NUM
ejpam-6077	88	16	5	5	NUM
ejpam-6077	88	17	of	of	ADP
ejpam-6077	88	18	16	16	NUM
ejpam-6077	88	19	then	then	ADV
ejpam-6077	88	20	s	s	AUX
ejpam-6077	88	21	has	have	VERB
ejpam-6077	88	22	a	a	DET
ejpam-6077	88	23	unique	unique	ADJ
ejpam-6077	88	24	fixed	fix	VERB
ejpam-6077	88	25	point	point	NOUN
ejpam-6077	88	26	υ∗	υ∗	NOUN
ejpam-6077	88	27	∈	∈	PROPN
ejpam-6077	88	28	x	x	NOUN
ejpam-6077	88	29	,	,	PUNCT
ejpam-6077	88	30	such	such	ADJ
ejpam-6077	88	31	that	that	DET
ejpam-6077	88	32	s(υ∗	s(υ∗	NOUN
ejpam-6077	88	33	)	)	PUNCT
ejpam-6077	88	34	=	=	SYM
ejpam-6077	88	35	υ∗.	υ∗.	VERB
ejpam-6077	88	36	proof	proof	NOUN
ejpam-6077	88	37	.	.	PUNCT
ejpam-6077	89	1	let	let	VERB
ejpam-6077	89	2	υ0	υ0	PROPN
ejpam-6077	89	3	∈	∈	PROPN
ejpam-6077	89	4	x	x	AUX
ejpam-6077	89	5	be	be	AUX
ejpam-6077	89	6	arbitrary	arbitrary	ADJ
ejpam-6077	89	7	,	,	PUNCT
ejpam-6077	89	8	and	and	CCONJ
ejpam-6077	89	9	define	define	VERB
ejpam-6077	89	10	a	a	DET
ejpam-6077	89	11	sequence	sequence	NOUN
ejpam-6077	89	12	{	{	PUNCT
ejpam-6077	89	13	υn	υn	NOUN
ejpam-6077	89	14	}	}	PUNCT
ejpam-6077	89	15	by	by	ADP
ejpam-6077	89	16	υn+1	υn+1	ADJ
ejpam-6077	89	17	=	=	NOUN
ejpam-6077	89	18	s(υn	s(υn	PROPN
ejpam-6077	89	19	)	)	PUNCT
ejpam-6077	89	20	for	for	ADP
ejpam-6077	89	21	all	all	DET
ejpam-6077	89	22	n	n	DET
ejpam-6077	89	23	≥	≥	NOUN
ejpam-6077	89	24	0	0	NUM
ejpam-6077	89	25	.	.	PUNCT
ejpam-6077	90	1	using	use	VERB
ejpam-6077	90	2	the	the	DET
ejpam-6077	90	3	contraction	contraction	NOUN
ejpam-6077	90	4	condition	condition	NOUN
ejpam-6077	90	5	of	of	ADP
ejpam-6077	90	6	s	s	PRON
ejpam-6077	90	7	with	with	ADP
ejpam-6077	90	8	respect	respect	NOUN
ejpam-6077	90	9	to	to	ADP
ejpam-6077	90	10	the	the	DET
ejpam-6077	90	11	mr	mr	PROPN
ejpam-6077	90	12	-	-	PUNCT
ejpam-6077	90	13	metric	metric	ADJ
ejpam-6077	90	14	m	m	NOUN
ejpam-6077	90	15	,	,	PUNCT
ejpam-6077	90	16	we	we	PRON
ejpam-6077	90	17	have	have	AUX
ejpam-6077	90	18	:	:	PUNCT
ejpam-6077	90	19	m(υn+1	m(υn+1	NUM
ejpam-6077	90	20	,	,	PUNCT
ejpam-6077	90	21	υn+2	υn+2	PROPN
ejpam-6077	90	22	,	,	PUNCT
ejpam-6077	90	23	υn+3	υn+3	NOUN
ejpam-6077	90	24	)	)	PUNCT
ejpam-6077	90	25	=	=	SYM
ejpam-6077	91	1	m(s(υn	m(s(υn	ADJ
ejpam-6077	91	2	)	)	PUNCT
ejpam-6077	91	3	,	,	PUNCT
ejpam-6077	91	4	s(υn+1	s(υn+1	ADJ
ejpam-6077	91	5	)	)	PUNCT
ejpam-6077	91	6	,	,	PUNCT
ejpam-6077	91	7	s(υn+2	s(υn+2	NUM
ejpam-6077	91	8	)	)	PUNCT
ejpam-6077	91	9	)	)	PUNCT
ejpam-6077	92	1	≤	≤	PUNCT
ejpam-6077	93	1	k	k	X
ejpam-6077	93	2	·	·	SYM
ejpam-6077	93	3	m(υn	m(υn	X
ejpam-6077	93	4	,	,	PUNCT
ejpam-6077	93	5	υn+1	υn+1	NOUN
ejpam-6077	93	6	,	,	PUNCT
ejpam-6077	93	7	υn+2	υn+2	NUM
ejpam-6077	93	8	)	)	PUNCT
ejpam-6077	93	9	.	.	PUNCT
ejpam-6077	94	1	by	by	ADP
ejpam-6077	94	2	induction	induction	NOUN
ejpam-6077	94	3	,	,	PUNCT
ejpam-6077	94	4	it	it	PRON
ejpam-6077	94	5	follows	follow	VERB
ejpam-6077	94	6	that	that	SCONJ
ejpam-6077	94	7	:	:	PUNCT
ejpam-6077	94	8	m(υn+1	m(υn+1	NUM
ejpam-6077	94	9	,	,	PUNCT
ejpam-6077	94	10	υn+2	υn+2	PROPN
ejpam-6077	94	11	,	,	PUNCT
ejpam-6077	94	12	υn+3	υn+3	NOUN
ejpam-6077	94	13	)	)	PUNCT
ejpam-6077	94	14	≤	≤	NOUN
ejpam-6077	94	15	kn	kn	PROPN
ejpam-6077	94	16	·	·	SYM
ejpam-6077	94	17	m(υ0	m(υ0	PROPN
ejpam-6077	94	18	,	,	PUNCT
ejpam-6077	94	19	υ1	υ1	PROPN
ejpam-6077	94	20	,	,	PUNCT
ejpam-6077	94	21	υ2	υ2	NOUN
ejpam-6077	94	22	)	)	PUNCT
ejpam-6077	94	23	,	,	PUNCT
ejpam-6077	94	24	where	where	SCONJ
ejpam-6077	94	25	m(υ0	m(υ0	PROPN
ejpam-6077	94	26	,	,	PUNCT
ejpam-6077	94	27	υ1	υ1	PROPN
ejpam-6077	94	28	,	,	PUNCT
ejpam-6077	94	29	υ2	υ2	NOUN
ejpam-6077	94	30	)	)	PUNCT
ejpam-6077	94	31	is	be	AUX
ejpam-6077	94	32	a	a	DET
ejpam-6077	94	33	finite	finite	NOUN
ejpam-6077	94	34	constant	constant	ADJ
ejpam-6077	94	35	since	since	SCONJ
ejpam-6077	94	36	m	m	PROPN
ejpam-6077	94	37	is	be	AUX
ejpam-6077	94	38	non	non	ADJ
ejpam-6077	94	39	-	-	ADJ
ejpam-6077	94	40	negative	negative	ADJ
ejpam-6077	94	41	.	.	PUNCT
ejpam-6077	95	1	as	as	ADP
ejpam-6077	95	2	n	n	NUM
ejpam-6077	95	3	→	→	SYM
ejpam-6077	95	4	∞	∞	PROPN
ejpam-6077	95	5	,	,	PUNCT
ejpam-6077	95	6	kn	kn	PROPN
ejpam-6077	95	7	→	→	X
ejpam-6077	95	8	0	0	NUM
ejpam-6077	95	9	,	,	PUNCT
ejpam-6077	95	10	and	and	CCONJ
ejpam-6077	95	11	hence	hence	ADV
ejpam-6077	95	12	:	:	PUNCT
ejpam-6077	95	13	lim	lim	PROPN
ejpam-6077	95	14	n→∞	n→∞	NUM
ejpam-6077	95	15	m(υn+1	m(υn+1	PROPN
ejpam-6077	95	16	,	,	PUNCT
ejpam-6077	95	17	υn+2	υn+2	PROPN
ejpam-6077	95	18	,	,	PUNCT
ejpam-6077	95	19	υn+3	υn+3	NOUN
ejpam-6077	95	20	)	)	PUNCT
ejpam-6077	95	21	=	=	SYM
ejpam-6077	95	22	0	0	X
ejpam-6077	95	23	.	.	PUNCT
ejpam-6077	96	1	thus	thus	ADV
ejpam-6077	96	2	,	,	PUNCT
ejpam-6077	96	3	the	the	DET
ejpam-6077	96	4	sequence	sequence	NOUN
ejpam-6077	96	5	{	{	PUNCT
ejpam-6077	96	6	υn	υn	NOUN
ejpam-6077	96	7	}	}	PUNCT
ejpam-6077	96	8	is	be	AUX
ejpam-6077	96	9	a	a	DET
ejpam-6077	96	10	cauchy	cauchy	ADJ
ejpam-6077	96	11	sequence	sequence	NOUN
ejpam-6077	96	12	under	under	ADP
ejpam-6077	96	13	the	the	DET
ejpam-6077	96	14	mr	mr	PROPN
ejpam-6077	96	15	-	-	PUNCT
ejpam-6077	96	16	metric	metric	NOUN
ejpam-6077	96	17	.	.	PUNCT
ejpam-6077	97	1	since	since	SCONJ
ejpam-6077	97	2	x	x	PRON
ejpam-6077	97	3	is	be	AUX
ejpam-6077	97	4	closed	closed	ADJ
ejpam-6077	97	5	and	and	CCONJ
ejpam-6077	97	6	(	(	PUNCT
ejpam-6077	97	7	x	x	X
ejpam-6077	97	8	,	,	PUNCT
ejpam-6077	97	9	m	m	VERB
ejpam-6077	97	10	)	)	PUNCT
ejpam-6077	97	11	is	be	AUX
ejpam-6077	97	12	complete	complete	ADJ
ejpam-6077	97	13	,	,	PUNCT
ejpam-6077	97	14	the	the	DET
ejpam-6077	97	15	sequence	sequence	NOUN
ejpam-6077	97	16	{	{	PUNCT
ejpam-6077	97	17	υn	υn	NOUN
ejpam-6077	97	18	}	}	PUNCT
ejpam-6077	97	19	converges	converge	NOUN
ejpam-6077	97	20	to	to	ADP
ejpam-6077	97	21	some	some	DET
ejpam-6077	97	22	υ∗	υ∗	NOUN
ejpam-6077	97	23	∈	∈	PROPN
ejpam-6077	97	24	x.	x.	NOUN
ejpam-6077	97	25	by	by	ADP
ejpam-6077	97	26	the	the	DET
ejpam-6077	97	27	continuity	continuity	NOUN
ejpam-6077	97	28	of	of	ADP
ejpam-6077	97	29	s	s	PROPN
ejpam-6077	97	30	,	,	PUNCT
ejpam-6077	97	31	we	we	PRON
ejpam-6077	97	32	have	have	VERB
ejpam-6077	97	33	:	:	PUNCT
ejpam-6077	98	1	lim	lim	PROPN
ejpam-6077	98	2	n→∞	n→∞	NUM
ejpam-6077	98	3	s(υn	s(υn	ADJ
ejpam-6077	98	4	)	)	PUNCT
ejpam-6077	98	5	=	=	SYM
ejpam-6077	98	6	s	s	X
ejpam-6077	98	7	(	(	PUNCT
ejpam-6077	98	8	lim	lim	PROPN
ejpam-6077	98	9	n→∞	n→∞	NUM
ejpam-6077	98	10	υn	υn	NOUN
ejpam-6077	98	11	)	)	PUNCT
ejpam-6077	98	12	,	,	PUNCT
ejpam-6077	98	13	which	which	PRON
ejpam-6077	98	14	implies	imply	VERB
ejpam-6077	98	15	υ∗	υ∗	NOUN
ejpam-6077	98	16	=	=	SYM
ejpam-6077	98	17	s(υ∗	s(υ∗	X
ejpam-6077	98	18	)	)	PUNCT
ejpam-6077	98	19	.	.	PUNCT
ejpam-6077	99	1	thus	thus	ADV
ejpam-6077	99	2	,	,	PUNCT
ejpam-6077	99	3	υ∗	υ∗	NOUN
ejpam-6077	99	4	is	be	AUX
ejpam-6077	99	5	a	a	DET
ejpam-6077	99	6	fixed	fix	VERB
ejpam-6077	99	7	point	point	NOUN
ejpam-6077	99	8	of	of	ADP
ejpam-6077	99	9	s.	s.	PROPN
ejpam-6077	99	10	now	now	ADV
ejpam-6077	99	11	,	,	PUNCT
ejpam-6077	99	12	for	for	ADP
ejpam-6077	99	13	uniqueness	uniqueness	NOUN
ejpam-6077	99	14	,	,	PUNCT
ejpam-6077	99	15	assume	assume	VERB
ejpam-6077	99	16	there	there	PRON
ejpam-6077	99	17	exists	exist	VERB
ejpam-6077	99	18	another	another	DET
ejpam-6077	99	19	fixed	fix	VERB
ejpam-6077	99	20	point	point	NOUN
ejpam-6077	99	21	ξ∗	ξ∗	PROPN
ejpam-6077	99	22	̸=	̸=	PROPN
ejpam-6077	100	1	υ∗.	υ∗.	VERB
ejpam-6077	100	2	using	use	VERB
ejpam-6077	100	3	the	the	DET
ejpam-6077	100	4	contraction	contraction	NOUN
ejpam-6077	100	5	condition	condition	NOUN
ejpam-6077	100	6	for	for	ADP
ejpam-6077	100	7	s	s	PRON
ejpam-6077	100	8	with	with	ADP
ejpam-6077	100	9	respect	respect	NOUN
ejpam-6077	100	10	to	to	ADP
ejpam-6077	100	11	the	the	DET
ejpam-6077	100	12	mr	mr	PROPN
ejpam-6077	100	13	-	-	PUNCT
ejpam-6077	100	14	metric	metric	NOUN
ejpam-6077	100	15	,	,	PUNCT
ejpam-6077	100	16	we	we	PRON
ejpam-6077	100	17	have	have	VERB
ejpam-6077	100	18	:	:	PUNCT
ejpam-6077	100	19	m(υ∗	m(υ∗	X
ejpam-6077	100	20	,	,	PUNCT
ejpam-6077	100	21	υ∗	υ∗	NOUN
ejpam-6077	100	22	,	,	PUNCT
ejpam-6077	100	23	υ∗	υ∗	NOUN
ejpam-6077	100	24	)	)	PUNCT
ejpam-6077	101	1	=	=	SYM
ejpam-6077	101	2	m(s(ξ∗	m(s(ξ∗	PROPN
ejpam-6077	101	3	)	)	PUNCT
ejpam-6077	101	4	,	,	PUNCT
ejpam-6077	101	5	s(ξ∗	s(ξ∗	NUM
ejpam-6077	101	6	)	)	PUNCT
ejpam-6077	101	7	,	,	PUNCT
ejpam-6077	101	8	s(ξ∗	s(ξ∗	NUM
ejpam-6077	101	9	)	)	PUNCT
ejpam-6077	101	10	)	)	PUNCT
ejpam-6077	102	1	≤	≤	PUNCT
ejpam-6077	103	1	k	k	X
ejpam-6077	103	2	·	·	PUNCT
ejpam-6077	103	3	m(ξ∗	m(ξ∗	NOUN
ejpam-6077	103	4	,	,	PUNCT
ejpam-6077	103	5	ξ∗	ξ∗	NOUN
ejpam-6077	103	6	,	,	PUNCT
ejpam-6077	103	7	ξ∗	ξ∗	NOUN
ejpam-6077	103	8	)	)	PUNCT
ejpam-6077	103	9	.	.	PUNCT
ejpam-6077	104	1	since	since	SCONJ
ejpam-6077	104	2	m(υ∗	m(υ∗	PROPN
ejpam-6077	104	3	,	,	PUNCT
ejpam-6077	104	4	υ∗	υ∗	NOUN
ejpam-6077	104	5	,	,	PUNCT
ejpam-6077	104	6	υ∗	υ∗	NOUN
ejpam-6077	104	7	)	)	PUNCT
ejpam-6077	104	8	=	=	SYM
ejpam-6077	104	9	0	0	NUM
ejpam-6077	105	1	and	and	CCONJ
ejpam-6077	105	2	k	k	PROPN
ejpam-6077	105	3	∈	∈	PROPN
ejpam-6077	106	1	[	[	X
ejpam-6077	106	2	0	0	NUM
ejpam-6077	106	3	,	,	PUNCT
ejpam-6077	106	4	1	1	NUM
ejpam-6077	106	5	)	)	PUNCT
ejpam-6077	106	6	,	,	PUNCT
ejpam-6077	106	7	it	it	PRON
ejpam-6077	106	8	follows	follow	VERB
ejpam-6077	106	9	that	that	SCONJ
ejpam-6077	106	10	m(ξ∗	m(ξ∗	NOUN
ejpam-6077	106	11	,	,	PUNCT
ejpam-6077	106	12	ξ∗	ξ∗	NOUN
ejpam-6077	106	13	,	,	PUNCT
ejpam-6077	106	14	ξ∗	ξ∗	ADJ
ejpam-6077	106	15	)	)	PUNCT
ejpam-6077	106	16	=	=	SYM
ejpam-6077	107	1	0	0	X
ejpam-6077	107	2	.	.	PUNCT
ejpam-6077	108	1	by	by	ADP
ejpam-6077	108	2	the	the	DET
ejpam-6077	108	3	properties	property	NOUN
ejpam-6077	108	4	of	of	ADP
ejpam-6077	108	5	m	m	PRON
ejpam-6077	108	6	,	,	PUNCT
ejpam-6077	108	7	this	this	PRON
ejpam-6077	108	8	implies	imply	VERB
ejpam-6077	108	9	υ∗	υ∗	NOUN
ejpam-6077	108	10	=	=	SYM
ejpam-6077	108	11	ξ∗.	ξ∗.	VERB
ejpam-6077	108	12	hence	hence	ADV
ejpam-6077	108	13	,	,	PUNCT
ejpam-6077	108	14	the	the	DET
ejpam-6077	108	15	fixed	fix	VERB
ejpam-6077	108	16	point	point	NOUN
ejpam-6077	108	17	is	be	AUX
ejpam-6077	108	18	unique	unique	ADJ
ejpam-6077	108	19	.	.	PUNCT
ejpam-6077	109	1	finally	finally	ADV
ejpam-6077	109	2	,	,	PUNCT
ejpam-6077	109	3	consider	consider	VERB
ejpam-6077	109	4	the	the	DET
ejpam-6077	109	5	measure	measure	NOUN
ejpam-6077	109	6	of	of	ADP
ejpam-6077	109	7	noncompactness	noncompactness	ADJ
ejpam-6077	109	8	µ.	µ.	NOUN
ejpam-6077	109	9	since	since	SCONJ
ejpam-6077	109	10	x	x	PRON
ejpam-6077	109	11	is	be	AUX
ejpam-6077	109	12	closed	close	VERB
ejpam-6077	109	13	and	and	CCONJ
ejpam-6077	109	14	bounded	bound	VERB
ejpam-6077	109	15	,	,	PUNCT
ejpam-6077	109	16	we	we	PRON
ejpam-6077	109	17	have	have	VERB
ejpam-6077	109	18	:	:	PUNCT
ejpam-6077	109	19	µ(s(a	µ(s(a	NUM
ejpam-6077	109	20	)	)	PUNCT
ejpam-6077	109	21	)	)	PUNCT
ejpam-6077	110	1	≤	≤	NUM
ejpam-6077	110	2	ϕ(µ(a	ϕ(µ(a	NOUN
ejpam-6077	110	3	)	)	PUNCT
ejpam-6077	110	4	)	)	PUNCT
ejpam-6077	110	5	,	,	PUNCT
ejpam-6077	110	6	for	for	ADP
ejpam-6077	110	7	any	any	DET
ejpam-6077	110	8	bounded	bounded	NOUN
ejpam-6077	110	9	subset	subset	VERB
ejpam-6077	110	10	a	a	DET
ejpam-6077	110	11	⊂	⊂	PROPN
ejpam-6077	110	12	x.	x.	NOUN
ejpam-6077	110	13	since	since	SCONJ
ejpam-6077	110	14	ϕ(t	ϕ(t	NUM
ejpam-6077	110	15	)	)	PUNCT
ejpam-6077	111	1	<	<	X
ejpam-6077	111	2	t	t	PROPN
ejpam-6077	111	3	for	for	ADP
ejpam-6077	111	4	all	all	DET
ejpam-6077	111	5	t	t	PROPN
ejpam-6077	111	6	>	>	X
ejpam-6077	111	7	0	0	NUM
ejpam-6077	111	8	,	,	PUNCT
ejpam-6077	111	9	repeated	repeat	VERB
ejpam-6077	111	10	application	application	NOUN
ejpam-6077	111	11	of	of	ADP
ejpam-6077	111	12	s	s	PRON
ejpam-6077	111	13	reduces	reduce	VERB
ejpam-6077	111	14	the	the	DET
ejpam-6077	111	15	measure	measure	NOUN
ejpam-6077	111	16	of	of	ADP
ejpam-6077	111	17	noncompactness	noncompactness	NOUN
ejpam-6077	111	18	,	,	PUNCT
ejpam-6077	111	19	ensuring	ensure	VERB
ejpam-6077	111	20	the	the	DET
ejpam-6077	111	21	existence	existence	NOUN
ejpam-6077	111	22	of	of	ADP
ejpam-6077	111	23	a	a	DET
ejpam-6077	111	24	fixed	fix	VERB
ejpam-6077	111	25	point	point	NOUN
ejpam-6077	111	26	.	.	PUNCT
ejpam-6077	112	1	the	the	DET
ejpam-6077	112	2	uniqueness	uniqueness	NOUN
ejpam-6077	112	3	follows	follow	VERB
ejpam-6077	112	4	from	from	ADP
ejpam-6077	112	5	the	the	DET
ejpam-6077	112	6	argument	argument	NOUN
ejpam-6077	112	7	above	above	ADV
ejpam-6077	112	8	.	.	PUNCT
ejpam-6077	113	1	example	example	NOUN
ejpam-6077	114	1	1	1	NUM
ejpam-6077	114	2	.	.	PUNCT
ejpam-6077	114	3	fixed	fix	VERB
ejpam-6077	114	4	point	point	NOUN
ejpam-6077	114	5	in	in	ADP
ejpam-6077	114	6	mr	mr	PROPN
ejpam-6077	114	7	-	-	PUNCT
ejpam-6077	114	8	metric	metric	ADJ
ejpam-6077	114	9	space	space	NOUN
ejpam-6077	114	10	let	let	VERB
ejpam-6077	114	11	x	x	PUNCT
ejpam-6077	114	12	=	=	SYM
ejpam-6077	114	13	r	r	NOUN
ejpam-6077	114	14	and	and	CCONJ
ejpam-6077	114	15	define	define	VERB
ejpam-6077	114	16	the	the	DET
ejpam-6077	114	17	mr	mr	PROPN
ejpam-6077	114	18	-	-	PUNCT
ejpam-6077	114	19	metric	metric	ADJ
ejpam-6077	114	20	m	m	NOUN
ejpam-6077	114	21	:	:	PUNCT
ejpam-6077	115	1	x×	x×	PUNCT
ejpam-6077	115	2	x×	x×	PUNCT
ejpam-6077	115	3	x	x	PUNCT
ejpam-6077	115	4	→	→	PUNCT
ejpam-6077	115	5	[	[	X
ejpam-6077	115	6	0,∞	0,∞	NOUN
ejpam-6077	115	7	)	)	PUNCT
ejpam-6077	115	8	by	by	ADP
ejpam-6077	115	9	:	:	PUNCT
ejpam-6077	115	10	m(υ	m(υ	PROPN
ejpam-6077	115	11	,	,	PUNCT
ejpam-6077	115	12	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	115	13	)	)	PUNCT
ejpam-6077	115	14	=	=	SYM
ejpam-6077	115	15	|υ	|υ	NOUN
ejpam-6077	115	16	−	−	PROPN
ejpam-6077	115	17	ξ|+	ξ|+	PROPN
ejpam-6077	115	18	|ξ	|ξ	VERB
ejpam-6077	115	19	−ℑ|+	−ℑ|+	NOUN
ejpam-6077	115	20	|ℑ	|ℑ	NOUN
ejpam-6077	115	21	−	−	PROPN
ejpam-6077	115	22	υ|	υ|	PROPN
ejpam-6077	115	23	.	.	PUNCT
ejpam-6077	116	1	let	let	VERB
ejpam-6077	116	2	s	s	PRON
ejpam-6077	116	3	:	:	PUNCT
ejpam-6077	116	4	r	r	NOUN
ejpam-6077	116	5	→	→	SYM
ejpam-6077	116	6	r	r	NOUN
ejpam-6077	116	7	be	be	VERB
ejpam-6077	116	8	the	the	DET
ejpam-6077	116	9	mapping	mapping	NOUN
ejpam-6077	116	10	defined	define	VERB
ejpam-6077	116	11	by	by	ADP
ejpam-6077	116	12	:	:	PUNCT
ejpam-6077	116	13	s(υ	s(υ	PROPN
ejpam-6077	116	14	)	)	PUNCT
ejpam-6077	117	1	=	=	SYM
ejpam-6077	117	2	υ	υ	PRON
ejpam-6077	117	3	2	2	NUM
ejpam-6077	117	4	.	.	PUNCT
ejpam-6077	118	1	a.	a.	NOUN
ejpam-6077	118	2	malkawi	malkawi	PROPN
ejpam-6077	118	3	/	/	SYM
ejpam-6077	118	4	eur	eur	PROPN
ejpam-6077	118	5	.	.	PUNCT
ejpam-6077	119	1	j.	j.	PROPN
ejpam-6077	119	2	pure	pure	PROPN
ejpam-6077	119	3	appl	appl	PROPN
ejpam-6077	119	4	.	.	PROPN
ejpam-6077	119	5	math	math	PROPN
ejpam-6077	119	6	,	,	PUNCT
ejpam-6077	119	7	18	18	NUM
ejpam-6077	119	8	(	(	PUNCT
ejpam-6077	119	9	2	2	NUM
ejpam-6077	119	10	)	)	PUNCT
ejpam-6077	119	11	(	(	PUNCT
ejpam-6077	119	12	2025	2025	NUM
ejpam-6077	119	13	)	)	PUNCT
ejpam-6077	119	14	,	,	PUNCT
ejpam-6077	119	15	6077	6077	NUM
ejpam-6077	119	16	6	6	NUM
ejpam-6077	119	17	of	of	ADP
ejpam-6077	119	18	16	16	NUM
ejpam-6077	119	19	we	we	PRON
ejpam-6077	119	20	verify	verify	VERB
ejpam-6077	119	21	that	that	SCONJ
ejpam-6077	119	22	s	s	AUX
ejpam-6077	119	23	satisfies	satisfie	NOUN
ejpam-6077	119	24	the	the	DET
ejpam-6077	119	25	contraction	contraction	NOUN
ejpam-6077	119	26	condition	condition	NOUN
ejpam-6077	119	27	:	:	PUNCT
ejpam-6077	119	28	m(s(υ	m(s(υ	PROPN
ejpam-6077	119	29	)	)	PUNCT
ejpam-6077	119	30	,	,	PUNCT
ejpam-6077	119	31	s(ξ	s(ξ	PROPN
ejpam-6077	119	32	)	)	PUNCT
ejpam-6077	119	33	,	,	PUNCT
ejpam-6077	119	34	s(ℑ	s(ℑ	NUM
ejpam-6077	119	35	)	)	PUNCT
ejpam-6077	119	36	)	)	PUNCT
ejpam-6077	120	1	=	=	SYM
ejpam-6077	120	2	1	1	NUM
ejpam-6077	120	3	2	2	NUM
ejpam-6077	120	4	m(υ	m(υ	PROPN
ejpam-6077	120	5	,	,	PUNCT
ejpam-6077	120	6	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	120	7	)	)	PUNCT
ejpam-6077	120	8	,	,	PUNCT
ejpam-6077	120	9	where	where	SCONJ
ejpam-6077	120	10	k	k	PROPN
ejpam-6077	120	11	=	=	NOUN
ejpam-6077	120	12	1	1	NUM
ejpam-6077	120	13	2	2	NUM
ejpam-6077	120	14	<	<	X
ejpam-6077	120	15	1	1	NUM
ejpam-6077	120	16	.	.	PUNCT
ejpam-6077	121	1	hence	hence	ADV
ejpam-6077	121	2	,	,	PUNCT
ejpam-6077	121	3	s	s	PART
ejpam-6077	121	4	satisfies	satisfie	NOUN
ejpam-6077	121	5	the	the	DET
ejpam-6077	121	6	conditions	condition	NOUN
ejpam-6077	121	7	of	of	ADP
ejpam-6077	121	8	the	the	DET
ejpam-6077	121	9	mr	mr	PROPN
ejpam-6077	121	10	-	-	PUNCT
ejpam-6077	121	11	metric	metric	ADJ
ejpam-6077	121	12	space	space	NOUN
ejpam-6077	121	13	fixed	fix	VERB
ejpam-6077	121	14	point	point	NOUN
ejpam-6077	121	15	theorem	theorem	VERB
ejpam-6077	121	16	.	.	PUNCT
ejpam-6077	122	1	let	let	VERB
ejpam-6077	122	2	υ0	υ0	PROPN
ejpam-6077	122	3	∈	∈	PROPN
ejpam-6077	122	4	r.	r.	PROPN
ejpam-6077	122	5	construct	construct	VERB
ejpam-6077	122	6	the	the	DET
ejpam-6077	122	7	sequence	sequence	NOUN
ejpam-6077	122	8	υn+1	υn+1	NOUN
ejpam-6077	122	9	=	=	SYM
ejpam-6077	122	10	s(υn	s(υn	NOUN
ejpam-6077	122	11	)	)	PUNCT
ejpam-6077	122	12	.	.	PUNCT
ejpam-6077	123	1	by	by	ADP
ejpam-6077	123	2	iteration	iteration	NOUN
ejpam-6077	123	3	:	:	PUNCT
ejpam-6077	123	4	υn	υn	NOUN
ejpam-6077	123	5	=	=	SYM
ejpam-6077	123	6	υ0	υ0	PROPN
ejpam-6077	123	7	2n	2n	NUM
ejpam-6077	123	8	.	.	PUNCT
ejpam-6077	124	1	as	as	ADP
ejpam-6077	124	2	n	n	NUM
ejpam-6077	124	3	→	→	SYM
ejpam-6077	124	4	∞	∞	PROPN
ejpam-6077	124	5	,	,	PUNCT
ejpam-6077	124	6	υn	υn	PROPN
ejpam-6077	124	7	→	→	SYM
ejpam-6077	124	8	0	0	NUM
ejpam-6077	124	9	.	.	PUNCT
ejpam-6077	125	1	thus	thus	ADV
ejpam-6077	125	2	,	,	PUNCT
ejpam-6077	125	3	the	the	DET
ejpam-6077	125	4	fixed	fix	VERB
ejpam-6077	125	5	point	point	NOUN
ejpam-6077	125	6	of	of	ADP
ejpam-6077	125	7	s	s	PROPN
ejpam-6077	125	8	is	be	AUX
ejpam-6077	125	9	υ∗	υ∗	NOUN
ejpam-6077	125	10	=	=	SYM
ejpam-6077	125	11	0	0	PROPN
ejpam-6077	125	12	.	.	NOUN
ejpam-6077	125	13	example	example	NOUN
ejpam-6077	125	14	2	2	NUM
ejpam-6077	125	15	.	.	PUNCT
ejpam-6077	125	16	fixed	fix	VERB
ejpam-6077	125	17	point	point	NOUN
ejpam-6077	125	18	in	in	ADP
ejpam-6077	125	19	banach	banach	NOUN
ejpam-6077	125	20	space	space	NOUN
ejpam-6077	125	21	with	with	ADP
ejpam-6077	125	22	measure	measure	NOUN
ejpam-6077	125	23	of	of	ADP
ejpam-6077	125	24	noncompactness	noncompactness	ADV
ejpam-6077	125	25	let	let	NOUN
ejpam-6077	125	26	x	x	SYM
ejpam-6077	125	27	=	=	SYM
ejpam-6077	125	28	c[0	c[0	PROPN
ejpam-6077	125	29	,	,	PUNCT
ejpam-6077	125	30	1	1	NUM
ejpam-6077	125	31	]	]	PUNCT
ejpam-6077	125	32	,	,	PUNCT
ejpam-6077	125	33	the	the	DET
ejpam-6077	125	34	banach	banach	NOUN
ejpam-6077	125	35	space	space	NOUN
ejpam-6077	125	36	of	of	ADP
ejpam-6077	125	37	continuous	continuous	ADJ
ejpam-6077	125	38	functions	function	NOUN
ejpam-6077	125	39	on	on	ADP
ejpam-6077	125	40	[	[	X
ejpam-6077	125	41	0	0	NUM
ejpam-6077	125	42	,	,	PUNCT
ejpam-6077	125	43	1	1	NUM
ejpam-6077	125	44	]	]	PUNCT
ejpam-6077	125	45	with	with	ADP
ejpam-6077	125	46	the	the	DET
ejpam-6077	125	47	norm	norm	NOUN
ejpam-6077	125	48	:	:	PUNCT
ejpam-6077	125	49	∥f∥	∥f∥	PROPN
ejpam-6077	125	50	=	=	SYM
ejpam-6077	125	51	max	max	PROPN
ejpam-6077	125	52	υ∈[0,1	υ∈[0,1	PROPN
ejpam-6077	125	53	]	]	PUNCT
ejpam-6077	125	54	|f(υ)|	|f(υ)|	NOUN
ejpam-6077	125	55	.	.	NOUN
ejpam-6077	125	56	define	define	VERB
ejpam-6077	125	57	the	the	DET
ejpam-6077	125	58	measure	measure	NOUN
ejpam-6077	125	59	of	of	ADP
ejpam-6077	125	60	noncompactness	noncompactness	ADJ
ejpam-6077	125	61	µ	µ	NOUN
ejpam-6077	125	62	for	for	ADP
ejpam-6077	125	63	a	a	DET
ejpam-6077	125	64	bounded	bounded	ADJ
ejpam-6077	125	65	set	set	NOUN
ejpam-6077	125	66	a	a	DET
ejpam-6077	125	67	⊂	⊂	X
ejpam-6077	125	68	x	x	PUNCT
ejpam-6077	125	69	by	by	ADP
ejpam-6077	125	70	:	:	PUNCT
ejpam-6077	125	71	µ(a	µ(a	PROPN
ejpam-6077	125	72	)	)	PUNCT
ejpam-6077	125	73	=	=	SYM
ejpam-6077	125	74	inf{δ	inf{δ	NOUN
ejpam-6077	125	75	>	>	X
ejpam-6077	125	76	0	0	NUM
ejpam-6077	125	77	:	:	PUNCT
ejpam-6077	125	78	a	a	PRON
ejpam-6077	125	79	can	can	AUX
ejpam-6077	125	80	be	be	AUX
ejpam-6077	125	81	covered	cover	VERB
ejpam-6077	125	82	by	by	ADP
ejpam-6077	125	83	a	a	DET
ejpam-6077	125	84	finite	finite	ADJ
ejpam-6077	125	85	number	number	NOUN
ejpam-6077	125	86	of	of	ADP
ejpam-6077	125	87	sets	set	NOUN
ejpam-6077	125	88	of	of	ADP
ejpam-6077	125	89	diameter	diameter	NOUN
ejpam-6077	125	90	δ	δ	PROPN
ejpam-6077	125	91	}	}	PUNCT
ejpam-6077	125	92	.	.	PUNCT
ejpam-6077	126	1	let	let	VERB
ejpam-6077	126	2	s	s	PRON
ejpam-6077	126	3	:	:	PUNCT
ejpam-6077	126	4	c[0	c[0	PROPN
ejpam-6077	126	5	,	,	PUNCT
ejpam-6077	126	6	1	1	NUM
ejpam-6077	126	7	]	]	PUNCT
ejpam-6077	126	8	→	→	X
ejpam-6077	126	9	c[0	c[0	PROPN
ejpam-6077	126	10	,	,	PUNCT
ejpam-6077	126	11	1	1	NUM
ejpam-6077	126	12	]	]	PUNCT
ejpam-6077	126	13	be	be	AUX
ejpam-6077	126	14	defined	define	VERB
ejpam-6077	126	15	by	by	ADP
ejpam-6077	126	16	:	:	PUNCT
ejpam-6077	126	17	(	(	PUNCT
ejpam-6077	126	18	sf)(υ	sf)(υ	PROPN
ejpam-6077	126	19	)	)	PUNCT
ejpam-6077	126	20	=	=	SYM
ejpam-6077	126	21	1	1	NUM
ejpam-6077	126	22	2	2	NUM
ejpam-6077	126	23	f(υ	f(υ	PROPN
ejpam-6077	126	24	)	)	PUNCT
ejpam-6077	126	25	.	.	PUNCT
ejpam-6077	127	1	we	we	PRON
ejpam-6077	127	2	verify	verify	VERB
ejpam-6077	127	3	that	that	SCONJ
ejpam-6077	127	4	s	s	VERB
ejpam-6077	127	5	satisfies	satisfie	NOUN
ejpam-6077	127	6	:	:	PUNCT
ejpam-6077	127	7	µ(s(a	µ(s(a	NUM
ejpam-6077	127	8	)	)	PUNCT
ejpam-6077	127	9	)	)	PUNCT
ejpam-6077	127	10	≤	≤	NUM
ejpam-6077	127	11	ϕ(µ(a	ϕ(µ(a	NOUN
ejpam-6077	127	12	)	)	PUNCT
ejpam-6077	127	13	)	)	PUNCT
ejpam-6077	127	14	,	,	PUNCT
ejpam-6077	127	15	where	where	SCONJ
ejpam-6077	127	16	ϕ(t	ϕ(t	NUM
ejpam-6077	127	17	)	)	PUNCT
ejpam-6077	128	1	=	=	PUNCT
ejpam-6077	128	2	t	t	PROPN
ejpam-6077	128	3	2	2	NUM
ejpam-6077	128	4	is	be	AUX
ejpam-6077	128	5	non	non	ADJ
ejpam-6077	128	6	-	-	ADJ
ejpam-6077	128	7	decreasing	decrease	VERB
ejpam-6077	128	8	with	with	ADP
ejpam-6077	128	9	ϕ(t	ϕ(t	NUM
ejpam-6077	128	10	)	)	PUNCT
ejpam-6077	128	11	<	<	X
ejpam-6077	128	12	t	t	PROPN
ejpam-6077	128	13	for	for	ADP
ejpam-6077	128	14	t	t	PROPN
ejpam-6077	128	15	>	>	X
ejpam-6077	128	16	0	0	PROPN
ejpam-6077	128	17	.	.	PUNCT
ejpam-6077	129	1	for	for	ADP
ejpam-6077	129	2	f	f	PROPN
ejpam-6077	129	3	∈	∈	PROPN
ejpam-6077	129	4	x	x	NOUN
ejpam-6077	129	5	,	,	PUNCT
ejpam-6077	129	6	consider	consider	VERB
ejpam-6077	129	7	the	the	DET
ejpam-6077	129	8	sequence	sequence	NOUN
ejpam-6077	129	9	fn+1(υ	fn+1(υ	ADJ
ejpam-6077	129	10	)	)	PUNCT
ejpam-6077	130	1	=	=	SYM
ejpam-6077	130	2	s(fn)(υ	s(fn)(υ	PROPN
ejpam-6077	130	3	)	)	PUNCT
ejpam-6077	130	4	.	.	PUNCT
ejpam-6077	131	1	iterating	iterate	VERB
ejpam-6077	131	2	s	s	PRON
ejpam-6077	131	3	gives	give	VERB
ejpam-6077	131	4	:	:	PUNCT
ejpam-6077	131	5	fn(υ	fn(υ	NUM
ejpam-6077	131	6	)	)	PUNCT
ejpam-6077	131	7	=	=	SYM
ejpam-6077	131	8	1	1	NUM
ejpam-6077	131	9	2n	2n	NUM
ejpam-6077	131	10	f0(υ	f0(υ	NOUN
ejpam-6077	131	11	)	)	PUNCT
ejpam-6077	131	12	.	.	PUNCT
ejpam-6077	132	1	as	as	ADP
ejpam-6077	132	2	n	n	NUM
ejpam-6077	132	3	→	→	SYM
ejpam-6077	132	4	∞	∞	PROPN
ejpam-6077	132	5	,	,	PUNCT
ejpam-6077	132	6	fn(υ	fn(υ	NUM
ejpam-6077	132	7	)	)	PUNCT
ejpam-6077	132	8	→	→	SYM
ejpam-6077	132	9	0	0	NUM
ejpam-6077	132	10	,	,	PUNCT
ejpam-6077	132	11	showing	show	VERB
ejpam-6077	132	12	that	that	SCONJ
ejpam-6077	132	13	the	the	DET
ejpam-6077	132	14	fixed	fixed	ADJ
ejpam-6077	132	15	point	point	NOUN
ejpam-6077	132	16	of	of	ADP
ejpam-6077	132	17	s	s	NOUN
ejpam-6077	132	18	is	be	AUX
ejpam-6077	132	19	f∗(υ	f∗(υ	NOUN
ejpam-6077	132	20	)	)	PUNCT
ejpam-6077	132	21	=	=	SYM
ejpam-6077	132	22	0	0	X
ejpam-6077	132	23	.	.	NOUN
ejpam-6077	132	24	example	example	NOUN
ejpam-6077	132	25	3	3	NUM
ejpam-6077	132	26	.	.	PUNCT
ejpam-6077	132	27	fixed	fix	VERB
ejpam-6077	132	28	point	point	NOUN
ejpam-6077	132	29	in	in	ADP
ejpam-6077	132	30	mr	mr	PROPN
ejpam-6077	132	31	-	-	PUNCT
ejpam-6077	132	32	metric	metric	ADJ
ejpam-6077	132	33	space	space	NOUN
ejpam-6077	132	34	with	with	ADP
ejpam-6077	132	35	measure	measure	NOUN
ejpam-6077	132	36	of	of	ADP
ejpam-6077	132	37	noncompactness	noncompactness	ADV
ejpam-6077	132	38	let	let	NOUN
ejpam-6077	132	39	x	x	SYM
ejpam-6077	132	40	=	=	SYM
ejpam-6077	132	41	c[0	c[0	PROPN
ejpam-6077	132	42	,	,	PUNCT
ejpam-6077	132	43	1	1	NUM
ejpam-6077	132	44	]	]	PUNCT
ejpam-6077	132	45	,	,	PUNCT
ejpam-6077	132	46	the	the	DET
ejpam-6077	132	47	banach	banach	NOUN
ejpam-6077	132	48	space	space	NOUN
ejpam-6077	132	49	of	of	ADP
ejpam-6077	132	50	continuous	continuous	ADJ
ejpam-6077	132	51	functions	function	NOUN
ejpam-6077	132	52	on	on	ADP
ejpam-6077	132	53	[	[	X
ejpam-6077	132	54	0	0	NUM
ejpam-6077	132	55	,	,	PUNCT
ejpam-6077	132	56	1	1	NUM
ejpam-6077	132	57	]	]	PUNCT
ejpam-6077	132	58	with	with	ADP
ejpam-6077	132	59	the	the	DET
ejpam-6077	132	60	norm	norm	NOUN
ejpam-6077	132	61	:	:	PUNCT
ejpam-6077	132	62	∥f∥	∥f∥	PROPN
ejpam-6077	132	63	=	=	SYM
ejpam-6077	132	64	max	max	PROPN
ejpam-6077	132	65	υ∈[0,1	υ∈[0,1	PROPN
ejpam-6077	132	66	]	]	PUNCT
ejpam-6077	132	67	|f(υ)|	|f(υ)|	NOUN
ejpam-6077	132	68	.	.	NOUN
ejpam-6077	132	69	define	define	VERB
ejpam-6077	132	70	the	the	DET
ejpam-6077	132	71	mr	mr	PROPN
ejpam-6077	132	72	-	-	PUNCT
ejpam-6077	132	73	metric	metric	ADJ
ejpam-6077	132	74	m	m	NOUN
ejpam-6077	132	75	:	:	PUNCT
ejpam-6077	133	1	x×	x×	PUNCT
ejpam-6077	133	2	x×	x×	PUNCT
ejpam-6077	133	3	x	x	PUNCT
ejpam-6077	133	4	→	→	PUNCT
ejpam-6077	133	5	[	[	X
ejpam-6077	133	6	0,∞	0,∞	NOUN
ejpam-6077	133	7	)	)	PUNCT
ejpam-6077	133	8	by	by	ADP
ejpam-6077	133	9	:	:	PUNCT
ejpam-6077	133	10	m(f	m(f	PROPN
ejpam-6077	133	11	,	,	PUNCT
ejpam-6077	133	12	g	g	PROPN
ejpam-6077	133	13	,	,	PUNCT
ejpam-6077	133	14	h	h	NOUN
ejpam-6077	133	15	)	)	PUNCT
ejpam-6077	133	16	=	=	SYM
ejpam-6077	134	1	∥f	∥f	PROPN
ejpam-6077	134	2	−	−	PROPN
ejpam-6077	134	3	g∥+	g∥+	PROPN
ejpam-6077	134	4	∥g	∥g	PROPN
ejpam-6077	134	5	−	−	PROPN
ejpam-6077	134	6	h∥+	h∥+	NOUN
ejpam-6077	134	7	∥h−	∥h−	NUM
ejpam-6077	134	8	f∥.	f∥.	NOUN
ejpam-6077	134	9	consider	consider	VERB
ejpam-6077	134	10	the	the	DET
ejpam-6077	134	11	operator	operator	NOUN
ejpam-6077	134	12	s	s	PART
ejpam-6077	134	13	:	:	PUNCT
ejpam-6077	134	14	c[0	c[0	PROPN
ejpam-6077	134	15	,	,	PUNCT
ejpam-6077	134	16	1	1	NUM
ejpam-6077	134	17	]	]	PUNCT
ejpam-6077	134	18	→	→	X
ejpam-6077	134	19	c[0	c[0	PROPN
ejpam-6077	134	20	,	,	PUNCT
ejpam-6077	134	21	1	1	NUM
ejpam-6077	134	22	]	]	PUNCT
ejpam-6077	134	23	defined	define	VERB
ejpam-6077	134	24	by	by	ADP
ejpam-6077	134	25	:	:	PUNCT
ejpam-6077	134	26	(	(	PUNCT
ejpam-6077	134	27	sf)(υ	sf)(υ	PROPN
ejpam-6077	134	28	)	)	PUNCT
ejpam-6077	134	29	=	=	SYM
ejpam-6077	134	30	1	1	NUM
ejpam-6077	134	31	2	2	NUM
ejpam-6077	134	32	f(υ	f(υ	PROPN
ejpam-6077	134	33	)	)	PUNCT
ejpam-6077	134	34	.	.	PUNCT
ejpam-6077	135	1	a.	a.	NOUN
ejpam-6077	135	2	malkawi	malkawi	ADP
ejpam-6077	135	3	/	/	SYM
ejpam-6077	135	4	eur	eur	PROPN
ejpam-6077	135	5	.	.	PUNCT
ejpam-6077	136	1	j.	j.	PROPN
ejpam-6077	136	2	pure	pure	PROPN
ejpam-6077	136	3	appl	appl	PROPN
ejpam-6077	136	4	.	.	PROPN
ejpam-6077	136	5	math	math	PROPN
ejpam-6077	136	6	,	,	PUNCT
ejpam-6077	136	7	18	18	NUM
ejpam-6077	136	8	(	(	PUNCT
ejpam-6077	136	9	2	2	NUM
ejpam-6077	136	10	)	)	PUNCT
ejpam-6077	136	11	(	(	PUNCT
ejpam-6077	136	12	2025	2025	NUM
ejpam-6077	136	13	)	)	PUNCT
ejpam-6077	136	14	,	,	PUNCT
ejpam-6077	136	15	6077	6077	NUM
ejpam-6077	136	16	7	7	NUM
ejpam-6077	136	17	of	of	ADP
ejpam-6077	136	18	16	16	NUM
ejpam-6077	136	19	we	we	PRON
ejpam-6077	136	20	verify	verify	VERB
ejpam-6077	136	21	the	the	DET
ejpam-6077	136	22	conditions	condition	NOUN
ejpam-6077	136	23	of	of	ADP
ejpam-6077	136	24	the	the	DET
ejpam-6077	136	25	theorem	theorem	NOUN
ejpam-6077	136	26	:	:	PUNCT
ejpam-6077	136	27	1	1	X
ejpam-6077	136	28	.	.	X
ejpam-6077	136	29	contraction	contraction	NOUN
ejpam-6077	136	30	condition	condition	NOUN
ejpam-6077	136	31	in	in	ADP
ejpam-6077	136	32	mr	mr	PROPN
ejpam-6077	136	33	-	-	PUNCT
ejpam-6077	136	34	metric	metric	NOUN
ejpam-6077	136	35	:	:	PUNCT
ejpam-6077	136	36	for	for	ADP
ejpam-6077	136	37	all	all	PRON
ejpam-6077	136	38	f	f	PROPN
ejpam-6077	136	39	,	,	PUNCT
ejpam-6077	136	40	g	g	PROPN
ejpam-6077	136	41	,	,	PUNCT
ejpam-6077	136	42	h	h	NOUN
ejpam-6077	136	43	∈	∈	PROPN
ejpam-6077	136	44	c[0	c[0	PROPN
ejpam-6077	136	45	,	,	PUNCT
ejpam-6077	136	46	1	1	NUM
ejpam-6077	136	47	]	]	PUNCT
ejpam-6077	136	48	,	,	PUNCT
ejpam-6077	136	49	m(s(f	m(s(f	PROPN
ejpam-6077	136	50	)	)	PUNCT
ejpam-6077	136	51	,	,	PUNCT
ejpam-6077	136	52	s(g	s(g	PROPN
ejpam-6077	136	53	)	)	PUNCT
ejpam-6077	136	54	,	,	PUNCT
ejpam-6077	136	55	s(h	s(h	PROPN
ejpam-6077	136	56	)	)	PUNCT
ejpam-6077	136	57	)	)	PUNCT
ejpam-6077	137	1	=	=	PUNCT
ejpam-6077	137	2	∥s(f)−	∥s(f)−	X
ejpam-6077	137	3	s(g)∥+	s(g)∥+	ADV
ejpam-6077	137	4	∥s(g)−	∥s(g)−	ADJ
ejpam-6077	137	5	s(h)∥+	s(h)∥+	ADJ
ejpam-6077	137	6	∥s(h)−	∥s(h)−	ADV
ejpam-6077	137	7	s(f)∥.	s(f)∥.	ADV
ejpam-6077	137	8	since	since	SCONJ
ejpam-6077	137	9	s(f)(υ	s(f)(υ	NOUN
ejpam-6077	137	10	)	)	PUNCT
ejpam-6077	137	11	=	=	SYM
ejpam-6077	137	12	1	1	NUM
ejpam-6077	137	13	2f(υ	2f(υ	NUM
ejpam-6077	137	14	)	)	PUNCT
ejpam-6077	137	15	,	,	PUNCT
ejpam-6077	137	16	we	we	PRON
ejpam-6077	137	17	have	have	VERB
ejpam-6077	137	18	:	:	PUNCT
ejpam-6077	137	19	∥s(f)−	∥s(f)−	X
ejpam-6077	137	20	s(g)∥	s(g)∥	ADJ
ejpam-6077	137	21	=	=	SYM
ejpam-6077	137	22	1	1	NUM
ejpam-6077	137	23	2	2	NUM
ejpam-6077	137	24	∥f	∥f	NOUN
ejpam-6077	137	25	−	−	NOUN
ejpam-6077	137	26	g∥.	g∥.	NOUN
ejpam-6077	137	27	similarly	similarly	ADV
ejpam-6077	137	28	for	for	ADP
ejpam-6077	137	29	other	other	ADJ
ejpam-6077	137	30	terms	term	NOUN
ejpam-6077	137	31	,	,	PUNCT
ejpam-6077	137	32	giving	give	VERB
ejpam-6077	137	33	:	:	PUNCT
ejpam-6077	137	34	m(s(f	m(s(f	PROPN
ejpam-6077	137	35	)	)	PUNCT
ejpam-6077	137	36	,	,	PUNCT
ejpam-6077	137	37	s(g	s(g	PROPN
ejpam-6077	137	38	)	)	PUNCT
ejpam-6077	137	39	,	,	PUNCT
ejpam-6077	137	40	s(h	s(h	PROPN
ejpam-6077	137	41	)	)	PUNCT
ejpam-6077	137	42	)	)	PUNCT
ejpam-6077	138	1	=	=	SYM
ejpam-6077	138	2	1	1	NUM
ejpam-6077	138	3	2	2	NUM
ejpam-6077	138	4	m(f	m(f	PROPN
ejpam-6077	138	5	,	,	PUNCT
ejpam-6077	138	6	g	g	PROPN
ejpam-6077	138	7	,	,	PUNCT
ejpam-6077	138	8	h	h	NOUN
ejpam-6077	138	9	)	)	PUNCT
ejpam-6077	138	10	.	.	PUNCT
ejpam-6077	139	1	thus	thus	ADV
ejpam-6077	139	2	,	,	PUNCT
ejpam-6077	139	3	the	the	DET
ejpam-6077	139	4	contraction	contraction	NOUN
ejpam-6077	139	5	constant	constant	ADJ
ejpam-6077	139	6	is	be	AUX
ejpam-6077	139	7	k	k	NOUN
ejpam-6077	139	8	=	=	SYM
ejpam-6077	139	9	1	1	NUM
ejpam-6077	139	10	2	2	NUM
ejpam-6077	139	11	<	<	X
ejpam-6077	139	12	1	1	NUM
ejpam-6077	139	13	.	.	NOUN
ejpam-6077	139	14	2	2	NUM
ejpam-6077	139	15	.	.	NOUN
ejpam-6077	139	16	measure	measure	NOUN
ejpam-6077	139	17	of	of	ADP
ejpam-6077	139	18	noncompactness	noncompactness	ADJ
ejpam-6077	139	19	condition	condition	NOUN
ejpam-6077	139	20	:	:	PUNCT
ejpam-6077	139	21	define	define	VERB
ejpam-6077	139	22	the	the	DET
ejpam-6077	139	23	measure	measure	NOUN
ejpam-6077	139	24	of	of	ADP
ejpam-6077	139	25	noncompactness	noncompactness	ADJ
ejpam-6077	139	26	µ	µ	NOUN
ejpam-6077	139	27	on	on	ADP
ejpam-6077	139	28	a	a	DET
ejpam-6077	139	29	bounded	bounded	ADJ
ejpam-6077	139	30	set	set	NOUN
ejpam-6077	139	31	a	a	DET
ejpam-6077	139	32	⊂	⊂	PROPN
ejpam-6077	139	33	c[0	c[0	PROPN
ejpam-6077	139	34	,	,	PUNCT
ejpam-6077	139	35	1	1	NUM
ejpam-6077	139	36	]	]	PUNCT
ejpam-6077	139	37	as	as	ADP
ejpam-6077	139	38	:	:	PUNCT
ejpam-6077	139	39	µ(a	µ(a	PROPN
ejpam-6077	139	40	)	)	PUNCT
ejpam-6077	139	41	=	=	SYM
ejpam-6077	139	42	inf{δ	inf{δ	NOUN
ejpam-6077	139	43	>	>	X
ejpam-6077	139	44	0	0	NUM
ejpam-6077	140	1	:	:	PUNCT
ejpam-6077	140	2	a	a	PRON
ejpam-6077	140	3	can	can	AUX
ejpam-6077	140	4	be	be	AUX
ejpam-6077	140	5	covered	cover	VERB
ejpam-6077	140	6	by	by	ADP
ejpam-6077	140	7	a	a	DET
ejpam-6077	140	8	finite	finite	ADJ
ejpam-6077	140	9	number	number	NOUN
ejpam-6077	140	10	of	of	ADP
ejpam-6077	140	11	sets	set	NOUN
ejpam-6077	140	12	of	of	ADP
ejpam-6077	140	13	diameter	diameter	NOUN
ejpam-6077	140	14	δ	δ	PROPN
ejpam-6077	140	15	}	}	PUNCT
ejpam-6077	140	16	.	.	PUNCT
ejpam-6077	141	1	for	for	ADP
ejpam-6077	141	2	s(a	s(a	PROPN
ejpam-6077	141	3	)	)	PUNCT
ejpam-6077	141	4	⊂	⊂	PROPN
ejpam-6077	141	5	c[0	c[0	PROPN
ejpam-6077	141	6	,	,	PUNCT
ejpam-6077	141	7	1	1	NUM
ejpam-6077	141	8	]	]	PUNCT
ejpam-6077	141	9	,	,	PUNCT
ejpam-6077	141	10	we	we	PRON
ejpam-6077	141	11	compute	compute	VERB
ejpam-6077	141	12	:	:	PUNCT
ejpam-6077	141	13	µ(s(a	µ(s(a	NUM
ejpam-6077	141	14	)	)	PUNCT
ejpam-6077	141	15	)	)	PUNCT
ejpam-6077	142	1	=	=	SYM
ejpam-6077	142	2	µ	µ	X
ejpam-6077	142	3	(	(	PUNCT
ejpam-6077	142	4	{	{	PUNCT
ejpam-6077	142	5	1	1	NUM
ejpam-6077	142	6	2	2	NUM
ejpam-6077	142	7	f	f	NOUN
ejpam-6077	142	8	:	:	PUNCT
ejpam-6077	142	9	f	f	PROPN
ejpam-6077	142	10	∈	∈	PROPN
ejpam-6077	142	11	a	a	PRON
ejpam-6077	142	12	}	}	PUNCT
ejpam-6077	142	13	)	)	PUNCT
ejpam-6077	142	14	.	.	PUNCT
ejpam-6077	143	1	by	by	ADP
ejpam-6077	143	2	the	the	DET
ejpam-6077	143	3	linearity	linearity	NOUN
ejpam-6077	143	4	of	of	ADP
ejpam-6077	143	5	s	s	PRON
ejpam-6077	143	6	and	and	CCONJ
ejpam-6077	143	7	the	the	DET
ejpam-6077	143	8	definition	definition	NOUN
ejpam-6077	143	9	of	of	ADP
ejpam-6077	143	10	µ	µ	NOUN
ejpam-6077	143	11	,	,	PUNCT
ejpam-6077	143	12	we	we	PRON
ejpam-6077	143	13	get	get	VERB
ejpam-6077	143	14	:	:	PUNCT
ejpam-6077	143	15	µ(s(a	µ(s(a	NUM
ejpam-6077	143	16	)	)	PUNCT
ejpam-6077	143	17	)	)	PUNCT
ejpam-6077	144	1	=	=	SYM
ejpam-6077	144	2	1	1	NUM
ejpam-6077	144	3	2	2	NUM
ejpam-6077	144	4	µ(a	µ(a	NOUN
ejpam-6077	144	5	)	)	PUNCT
ejpam-6077	144	6	.	.	PUNCT
ejpam-6077	145	1	define	define	VERB
ejpam-6077	145	2	ϕ(t	ϕ(t	NUM
ejpam-6077	145	3	)	)	PUNCT
ejpam-6077	146	1	=	=	SYM
ejpam-6077	146	2	1	1	NUM
ejpam-6077	146	3	2	2	NUM
ejpam-6077	146	4	t	t	NOUN
ejpam-6077	146	5	,	,	PUNCT
ejpam-6077	146	6	which	which	PRON
ejpam-6077	146	7	satisfies	satisfy	VERB
ejpam-6077	146	8	ϕ(t	ϕ(t	NUM
ejpam-6077	146	9	)	)	PUNCT
ejpam-6077	146	10	<	<	X
ejpam-6077	146	11	t	t	PROPN
ejpam-6077	146	12	for	for	ADP
ejpam-6077	146	13	all	all	DET
ejpam-6077	146	14	t	t	PROPN
ejpam-6077	146	15	>	>	X
ejpam-6077	146	16	0	0	PUNCT
ejpam-6077	146	17	and	and	CCONJ
ejpam-6077	146	18	ϕ(0	ϕ(0	PROPN
ejpam-6077	146	19	)	)	PUNCT
ejpam-6077	147	1	=	=	PUNCT
ejpam-6077	147	2	0	0	X
ejpam-6077	147	3	.	.	PUNCT
ejpam-6077	148	1	thus	thus	ADV
ejpam-6077	148	2	:	:	PUNCT
ejpam-6077	148	3	µ(s(a	µ(s(a	NUM
ejpam-6077	148	4	)	)	PUNCT
ejpam-6077	148	5	)	)	PUNCT
ejpam-6077	149	1	≤	≤	NUM
ejpam-6077	149	2	ϕ(µ(a	ϕ(µ(a	NOUN
ejpam-6077	149	3	)	)	PUNCT
ejpam-6077	149	4	)	)	PUNCT
ejpam-6077	149	5	.	.	PUNCT
ejpam-6077	150	1	since	since	SCONJ
ejpam-6077	150	2	both	both	DET
ejpam-6077	150	3	conditions	condition	NOUN
ejpam-6077	150	4	are	be	AUX
ejpam-6077	150	5	satisfied	satisfied	ADJ
ejpam-6077	150	6	,	,	PUNCT
ejpam-6077	150	7	s	s	AUX
ejpam-6077	150	8	has	have	VERB
ejpam-6077	150	9	a	a	DET
ejpam-6077	150	10	unique	unique	ADJ
ejpam-6077	150	11	fixed	fix	VERB
ejpam-6077	150	12	point	point	NOUN
ejpam-6077	150	13	f∗	f∗	NOUN
ejpam-6077	150	14	∈	∈	PROPN
ejpam-6077	150	15	c[0	c[0	PROPN
ejpam-6077	150	16	,	,	PUNCT
ejpam-6077	150	17	1	1	NUM
ejpam-6077	150	18	]	]	PUNCT
ejpam-6077	150	19	.	.	PUNCT
ejpam-6077	151	1	iteratively	iteratively	ADV
ejpam-6077	151	2	applying	apply	VERB
ejpam-6077	151	3	s	s	PRON
ejpam-6077	151	4	,	,	PUNCT
ejpam-6077	151	5	starting	start	VERB
ejpam-6077	151	6	from	from	ADP
ejpam-6077	151	7	any	any	DET
ejpam-6077	151	8	f0	f0	PROPN
ejpam-6077	151	9	∈	∈	PROPN
ejpam-6077	151	10	c[0	c[0	PROPN
ejpam-6077	151	11	,	,	PUNCT
ejpam-6077	151	12	1	1	NUM
ejpam-6077	151	13	]	]	PUNCT
ejpam-6077	151	14	,	,	PUNCT
ejpam-6077	151	15	we	we	PRON
ejpam-6077	151	16	find	find	VERB
ejpam-6077	151	17	that	that	PRON
ejpam-6077	151	18	:	:	PUNCT
ejpam-6077	151	19	fn(υ	fn(υ	X
ejpam-6077	151	20	)	)	PUNCT
ejpam-6077	151	21	=	=	SYM
ejpam-6077	151	22	1	1	NUM
ejpam-6077	151	23	2n	2n	NUM
ejpam-6077	151	24	f0(υ	f0(υ	NOUN
ejpam-6077	151	25	)	)	PUNCT
ejpam-6077	151	26	.	.	PUNCT
ejpam-6077	152	1	as	as	ADP
ejpam-6077	152	2	n	n	NUM
ejpam-6077	152	3	→	→	SYM
ejpam-6077	152	4	∞	∞	PROPN
ejpam-6077	152	5	,	,	PUNCT
ejpam-6077	152	6	fn(υ	fn(υ	NUM
ejpam-6077	152	7	)	)	PUNCT
ejpam-6077	152	8	→	→	SYM
ejpam-6077	152	9	0	0	X
ejpam-6077	152	10	.	.	PUNCT
ejpam-6077	153	1	therefore	therefore	ADV
ejpam-6077	153	2	,	,	PUNCT
ejpam-6077	153	3	the	the	DET
ejpam-6077	153	4	fixed	fix	VERB
ejpam-6077	153	5	point	point	NOUN
ejpam-6077	153	6	of	of	ADP
ejpam-6077	153	7	s	s	NOUN
ejpam-6077	153	8	is	be	AUX
ejpam-6077	153	9	f∗(υ	f∗(υ	NOUN
ejpam-6077	153	10	)	)	PUNCT
ejpam-6077	153	11	=	=	SYM
ejpam-6077	153	12	0	0	NUM
ejpam-6077	153	13	.	.	NOUN
ejpam-6077	154	1	3	3	X
ejpam-6077	154	2	.	.	X
ejpam-6077	154	3	applications	application	NOUN
ejpam-6077	154	4	of	of	ADP
ejpam-6077	154	5	the	the	DET
ejpam-6077	154	6	theorem	theorem	NOUN
ejpam-6077	154	7	the	the	DET
ejpam-6077	154	8	concept	concept	NOUN
ejpam-6077	154	9	of	of	ADP
ejpam-6077	154	10	anmr	anmr	ADJ
ejpam-6077	154	11	-	-	PUNCT
ejpam-6077	154	12	metric	metric	ADJ
ejpam-6077	154	13	space	space	NOUN
ejpam-6077	154	14	plays	play	VERB
ejpam-6077	154	15	a	a	DET
ejpam-6077	154	16	critical	critical	ADJ
ejpam-6077	154	17	role	role	NOUN
ejpam-6077	154	18	in	in	ADP
ejpam-6077	154	19	the	the	DET
ejpam-6077	154	20	following	follow	VERB
ejpam-6077	154	21	applications	application	NOUN
ejpam-6077	154	22	,	,	PUNCT
ejpam-6077	154	23	as	as	SCONJ
ejpam-6077	154	24	it	it	PRON
ejpam-6077	154	25	provides	provide	VERB
ejpam-6077	154	26	a	a	DET
ejpam-6077	154	27	generalization	generalization	NOUN
ejpam-6077	154	28	of	of	ADP
ejpam-6077	154	29	the	the	DET
ejpam-6077	154	30	standard	standard	ADJ
ejpam-6077	154	31	metric	metric	NOUN
ejpam-6077	154	32	and	and	CCONJ
ejpam-6077	154	33	enables	enable	VERB
ejpam-6077	154	34	the	the	DET
ejpam-6077	154	35	handling	handling	NOUN
ejpam-6077	154	36	of	of	ADP
ejpam-6077	154	37	noncompact	noncompact	ADJ
ejpam-6077	154	38	settings	setting	NOUN
ejpam-6077	154	39	,	,	PUNCT
ejpam-6077	154	40	essential	essential	ADJ
ejpam-6077	154	41	for	for	ADP
ejpam-6077	154	42	various	various	ADJ
ejpam-6077	154	43	fixed	fix	VERB
ejpam-6077	154	44	-	-	PUNCT
ejpam-6077	154	45	point	point	NOUN
ejpam-6077	154	46	problems	problem	NOUN
ejpam-6077	154	47	.	.	PUNCT
ejpam-6077	155	1	a.	a.	NOUN
ejpam-6077	155	2	malkawi	malkawi	ADP
ejpam-6077	155	3	/	/	SYM
ejpam-6077	155	4	eur	eur	PROPN
ejpam-6077	155	5	.	.	PUNCT
ejpam-6077	156	1	j.	j.	PROPN
ejpam-6077	156	2	pure	pure	PROPN
ejpam-6077	156	3	appl	appl	PROPN
ejpam-6077	156	4	.	.	PROPN
ejpam-6077	156	5	math	math	PROPN
ejpam-6077	156	6	,	,	PUNCT
ejpam-6077	156	7	18	18	NUM
ejpam-6077	156	8	(	(	PUNCT
ejpam-6077	156	9	2	2	NUM
ejpam-6077	156	10	)	)	PUNCT
ejpam-6077	156	11	(	(	PUNCT
ejpam-6077	156	12	2025	2025	NUM
ejpam-6077	156	13	)	)	PUNCT
ejpam-6077	156	14	,	,	PUNCT
ejpam-6077	156	15	6077	6077	NUM
ejpam-6077	156	16	8	8	NUM
ejpam-6077	156	17	of	of	ADP
ejpam-6077	156	18	16	16	NUM
ejpam-6077	156	19	1	1	NUM
ejpam-6077	156	20	.	.	PUNCT
ejpam-6077	157	1	solving	solve	VERB
ejpam-6077	157	2	nonlinear	nonlinear	ADJ
ejpam-6077	157	3	integral	integral	ADJ
ejpam-6077	157	4	equations	equation	NOUN
ejpam-6077	157	5	example	example	VERB
ejpam-6077	157	6	4	4	NUM
ejpam-6077	157	7	.	.	PUNCT
ejpam-6077	158	1	consider	consider	VERB
ejpam-6077	158	2	the	the	DET
ejpam-6077	158	3	nonlinear	nonlinear	ADJ
ejpam-6077	158	4	integral	integral	ADJ
ejpam-6077	158	5	equation	equation	NOUN
ejpam-6077	158	6	:	:	PUNCT
ejpam-6077	158	7	υ(t	υ(t	NOUN
ejpam-6077	158	8	)	)	PUNCT
ejpam-6077	158	9	=	=	SYM
ejpam-6077	159	1	∫	∫	PROPN
ejpam-6077	159	2	b	b	PROPN
ejpam-6077	159	3	a	a	DET
ejpam-6077	159	4	k(t	k(t	PROPN
ejpam-6077	159	5	,	,	PUNCT
ejpam-6077	159	6	s	s	X
ejpam-6077	159	7	,	,	PUNCT
ejpam-6077	159	8	υ(s	υ(s	PROPN
ejpam-6077	159	9	)	)	PUNCT
ejpam-6077	159	10	)	)	PUNCT
ejpam-6077	159	11	ds	ds	PROPN
ejpam-6077	159	12	,	,	PUNCT
ejpam-6077	159	13	where	where	SCONJ
ejpam-6077	159	14	k	k	NOUN
ejpam-6077	159	15	:	:	PUNCT
ejpam-6077	160	1	[	[	X
ejpam-6077	160	2	a	a	X
ejpam-6077	160	3	,	,	PUNCT
ejpam-6077	160	4	b	b	NOUN
ejpam-6077	160	5	]	]	X
ejpam-6077	160	6	×	×	NOUN
ejpam-6077	160	7	[	[	X
ejpam-6077	160	8	a	a	X
ejpam-6077	160	9	,	,	PUNCT
ejpam-6077	160	10	b	b	NOUN
ejpam-6077	160	11	]	]	X
ejpam-6077	160	12	×	×	NOUN
ejpam-6077	160	13	r	r	NOUN
ejpam-6077	160	14	→	→	SYM
ejpam-6077	160	15	r	r	NOUN
ejpam-6077	160	16	is	be	AUX
ejpam-6077	160	17	a	a	DET
ejpam-6077	160	18	continuous	continuous	ADJ
ejpam-6077	160	19	function	function	NOUN
ejpam-6077	160	20	that	that	PRON
ejpam-6077	160	21	satisfies	satisfy	VERB
ejpam-6077	160	22	a	a	DET
ejpam-6077	160	23	lipschitz	lipschitz	ADJ
ejpam-6077	160	24	-	-	PUNCT
ejpam-6077	160	25	type	type	NOUN
ejpam-6077	160	26	condition	condition	NOUN
ejpam-6077	160	27	with	with	ADP
ejpam-6077	160	28	respect	respect	NOUN
ejpam-6077	160	29	to	to	ADP
ejpam-6077	160	30	the	the	DET
ejpam-6077	160	31	third	third	ADJ
ejpam-6077	160	32	variable	variable	NOUN
ejpam-6077	160	33	,	,	PUNCT
ejpam-6077	160	34	i.e.	i.e.	X
ejpam-6077	160	35	,	,	PUNCT
ejpam-6077	160	36	there	there	PRON
ejpam-6077	160	37	exists	exist	VERB
ejpam-6077	160	38	a	a	DET
ejpam-6077	160	39	constant	constant	ADJ
ejpam-6077	160	40	l	l	NOUN
ejpam-6077	160	41	>	>	X
ejpam-6077	160	42	0	0	NUM
ejpam-6077	160	43	such	such	ADJ
ejpam-6077	160	44	that	that	PRON
ejpam-6077	160	45	for	for	ADP
ejpam-6077	160	46	all	all	DET
ejpam-6077	160	47	t	t	PROPN
ejpam-6077	160	48	,	,	PUNCT
ejpam-6077	160	49	s	s	PART
ejpam-6077	160	50	∈	∈	PROPN
ejpam-6077	161	1	[	[	X
ejpam-6077	161	2	a	a	X
ejpam-6077	161	3	,	,	PUNCT
ejpam-6077	161	4	b	b	NOUN
ejpam-6077	161	5	]	]	X
ejpam-6077	161	6	and	and	CCONJ
ejpam-6077	161	7	υ1	υ1	PROPN
ejpam-6077	161	8	,	,	PUNCT
ejpam-6077	161	9	υ2	υ2	PROPN
ejpam-6077	161	10	∈	∈	PROPN
ejpam-6077	161	11	r	r	NOUN
ejpam-6077	161	12	,	,	PUNCT
ejpam-6077	161	13	|k(t	|k(t	PROPN
ejpam-6077	161	14	,	,	PUNCT
ejpam-6077	161	15	s	s	NOUN
ejpam-6077	161	16	,	,	PUNCT
ejpam-6077	161	17	υ1)−k(t	υ1)−k(t	NOUN
ejpam-6077	161	18	,	,	PUNCT
ejpam-6077	161	19	s	s	PROPN
ejpam-6077	161	20	,	,	PUNCT
ejpam-6077	161	21	υ2)|	υ2)|	PROPN
ejpam-6077	161	22	≤	≤	PROPN
ejpam-6077	161	23	l|υ1	l|υ1	PROPN
ejpam-6077	161	24	−	−	PROPN
ejpam-6077	161	25	υ2|	υ2|	PROPN
ejpam-6077	161	26	.	.	PUNCT
ejpam-6077	162	1	step	step	NOUN
ejpam-6077	162	2	1	1	NUM
ejpam-6077	162	3	:	:	PUNCT
ejpam-6077	162	4	define	define	VERB
ejpam-6077	162	5	the	the	DET
ejpam-6077	162	6	space	space	NOUN
ejpam-6077	162	7	let	let	VERB
ejpam-6077	162	8	x	x	PUNCT
ejpam-6077	162	9	=	=	SYM
ejpam-6077	162	10	c([a	c([a	PROPN
ejpam-6077	162	11	,	,	PUNCT
ejpam-6077	162	12	b	b	NOUN
ejpam-6077	162	13	]	]	X
ejpam-6077	162	14	)	)	PUNCT
ejpam-6077	162	15	,	,	PUNCT
ejpam-6077	162	16	the	the	DET
ejpam-6077	162	17	space	space	NOUN
ejpam-6077	162	18	of	of	ADP
ejpam-6077	162	19	continuous	continuous	ADJ
ejpam-6077	162	20	functions	function	NOUN
ejpam-6077	162	21	on	on	ADP
ejpam-6077	162	22	the	the	DET
ejpam-6077	162	23	interval	interval	NOUN
ejpam-6077	162	24	[	[	X
ejpam-6077	162	25	a	a	X
ejpam-6077	162	26	,	,	PUNCT
ejpam-6077	162	27	b	b	NOUN
ejpam-6077	162	28	]	]	X
ejpam-6077	162	29	,	,	PUNCT
ejpam-6077	162	30	which	which	PRON
ejpam-6077	162	31	is	be	AUX
ejpam-6077	162	32	a	a	DET
ejpam-6077	162	33	banach	banach	NOUN
ejpam-6077	162	34	space	space	NOUN
ejpam-6077	162	35	when	when	SCONJ
ejpam-6077	162	36	equipped	equip	VERB
ejpam-6077	162	37	with	with	ADP
ejpam-6077	162	38	the	the	DET
ejpam-6077	162	39	supremum	supremum	ADJ
ejpam-6077	162	40	norm	norm	NOUN
ejpam-6077	162	41	:	:	PUNCT
ejpam-6077	162	42	∥υ∥∞	∥υ∥∞	NOUN
ejpam-6077	162	43	=	=	SYM
ejpam-6077	163	1	sup	sup	NOUN
ejpam-6077	163	2	t∈[a	t∈[a	NOUN
ejpam-6077	163	3	,	,	PUNCT
ejpam-6077	163	4	b	b	NOUN
ejpam-6077	163	5	]	]	X
ejpam-6077	163	6	|υ(t)|	|υ(t)|	NOUN
ejpam-6077	163	7	.	.	PUNCT
ejpam-6077	164	1	now	now	ADV
ejpam-6077	164	2	,	,	PUNCT
ejpam-6077	164	3	equip	equip	NOUN
ejpam-6077	164	4	x	x	PUNCT
ejpam-6077	164	5	with	with	ADP
ejpam-6077	164	6	the	the	DET
ejpam-6077	164	7	mr	mr	PROPN
ejpam-6077	164	8	-	-	PUNCT
ejpam-6077	164	9	metric	metric	NOUN
ejpam-6077	164	10	:	:	PUNCT
ejpam-6077	164	11	m(υ	m(υ	PROPN
ejpam-6077	164	12	,	,	PUNCT
ejpam-6077	164	13	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	164	14	)	)	PUNCT
ejpam-6077	164	15	=	=	SYM
ejpam-6077	164	16	sup	sup	NOUN
ejpam-6077	164	17	t∈[a	t∈[a	NOUN
ejpam-6077	164	18	,	,	PUNCT
ejpam-6077	164	19	b	b	NOUN
ejpam-6077	164	20	]	]	PUNCT
ejpam-6077	164	21	|υ(t)−	|υ(t)−	PROPN
ejpam-6077	164	22	ξ(t)|+	ξ(t)|+	PROPN
ejpam-6077	164	23	sup	sup	NOUN
ejpam-6077	164	24	t∈[a	t∈[a	NOUN
ejpam-6077	164	25	,	,	PUNCT
ejpam-6077	164	26	b	b	NOUN
ejpam-6077	164	27	]	]	X
ejpam-6077	164	28	|ξ(t)−ℑ(t)|	|ξ(t)−ℑ(t)|	NUM
ejpam-6077	164	29	.	.	PUNCT
ejpam-6077	165	1	the	the	DET
ejpam-6077	165	2	mr	mr	PROPN
ejpam-6077	165	3	-	-	PUNCT
ejpam-6077	165	4	metric	metric	NOUN
ejpam-6077	165	5	generalizes	generalize	VERB
ejpam-6077	165	6	the	the	DET
ejpam-6077	165	7	usual	usual	ADJ
ejpam-6077	165	8	metric	metric	NOUN
ejpam-6077	165	9	by	by	ADP
ejpam-6077	165	10	considering	consider	VERB
ejpam-6077	165	11	the	the	DET
ejpam-6077	165	12	distance	distance	NOUN
ejpam-6077	165	13	between	between	ADP
ejpam-6077	165	14	three	three	NUM
ejpam-6077	165	15	functions	function	NOUN
ejpam-6077	165	16	simultaneously	simultaneously	ADV
ejpam-6077	165	17	,	,	PUNCT
ejpam-6077	165	18	which	which	PRON
ejpam-6077	165	19	provides	provide	VERB
ejpam-6077	165	20	a	a	DET
ejpam-6077	165	21	richer	rich	ADJ
ejpam-6077	165	22	structure	structure	NOUN
ejpam-6077	165	23	to	to	PART
ejpam-6077	165	24	analyze	analyze	VERB
ejpam-6077	165	25	the	the	DET
ejpam-6077	165	26	operator	operator	NOUN
ejpam-6077	165	27	.	.	PUNCT
ejpam-6077	166	1	step	step	NOUN
ejpam-6077	166	2	2	2	NUM
ejpam-6077	166	3	:	:	PUNCT
ejpam-6077	166	4	define	define	VERB
ejpam-6077	166	5	the	the	DET
ejpam-6077	166	6	operator	operator	NOUN
ejpam-6077	166	7	define	define	VERB
ejpam-6077	166	8	the	the	DET
ejpam-6077	166	9	operator	operator	NOUN
ejpam-6077	166	10	s	s	PART
ejpam-6077	166	11	:	:	PUNCT
ejpam-6077	166	12	x	x	SYM
ejpam-6077	166	13	→	→	SYM
ejpam-6077	166	14	x	x	PUNCT
ejpam-6077	166	15	by	by	ADP
ejpam-6077	166	16	:	:	PUNCT
ejpam-6077	166	17	(	(	PUNCT
ejpam-6077	166	18	sυ)(t	sυ)(t	PROPN
ejpam-6077	166	19	)	)	PUNCT
ejpam-6077	166	20	=	=	SYM
ejpam-6077	167	1	∫	∫	PROPN
ejpam-6077	167	2	b	b	PROPN
ejpam-6077	167	3	a	a	DET
ejpam-6077	167	4	k(t	k(t	PROPN
ejpam-6077	167	5	,	,	PUNCT
ejpam-6077	167	6	s	s	X
ejpam-6077	167	7	,	,	PUNCT
ejpam-6077	167	8	υ(s	υ(s	PROPN
ejpam-6077	167	9	)	)	PUNCT
ejpam-6077	167	10	)	)	PUNCT
ejpam-6077	168	1	ds	ds	PROPN
ejpam-6077	168	2	.	.	PUNCT
ejpam-6077	169	1	the	the	DET
ejpam-6077	169	2	operator	operator	NOUN
ejpam-6077	169	3	s	s	PART
ejpam-6077	169	4	maps	map	NOUN
ejpam-6077	169	5	a	a	DET
ejpam-6077	169	6	function	function	NOUN
ejpam-6077	169	7	υ	υ	X
ejpam-6077	169	8	∈	∈	PROPN
ejpam-6077	169	9	x	x	PUNCT
ejpam-6077	169	10	to	to	ADP
ejpam-6077	169	11	another	another	DET
ejpam-6077	169	12	function	function	NOUN
ejpam-6077	169	13	sυ	sυ	PROPN
ejpam-6077	169	14	,	,	PUNCT
ejpam-6077	169	15	obtained	obtain	VERB
ejpam-6077	169	16	by	by	ADP
ejpam-6077	169	17	evaluating	evaluate	VERB
ejpam-6077	169	18	the	the	DET
ejpam-6077	169	19	integral	integral	ADJ
ejpam-6077	169	20	.	.	PUNCT
ejpam-6077	170	1	step	step	NOUN
ejpam-6077	170	2	3	3	NUM
ejpam-6077	170	3	:	:	PUNCT
ejpam-6077	170	4	verify	verify	VERB
ejpam-6077	170	5	continuity	continuity	NOUN
ejpam-6077	170	6	of	of	ADP
ejpam-6077	170	7	s	s	PRON
ejpam-6077	170	8	since	since	SCONJ
ejpam-6077	170	9	k(t	k(t	PROPN
ejpam-6077	170	10	,	,	PUNCT
ejpam-6077	170	11	s	s	X
ejpam-6077	170	12	,	,	PUNCT
ejpam-6077	170	13	υ	υ	NOUN
ejpam-6077	170	14	)	)	PUNCT
ejpam-6077	170	15	is	be	AUX
ejpam-6077	170	16	continuous	continuous	ADJ
ejpam-6077	170	17	and	and	CCONJ
ejpam-6077	170	18	satisfies	satisfy	VERB
ejpam-6077	170	19	the	the	DET
ejpam-6077	170	20	lipschitz	lipschitz	NOUN
ejpam-6077	170	21	condition	condition	NOUN
ejpam-6077	170	22	in	in	ADP
ejpam-6077	170	23	x	x	PRON
ejpam-6077	170	24	,	,	PUNCT
ejpam-6077	170	25	the	the	DET
ejpam-6077	170	26	operator	operator	NOUN
ejpam-6077	170	27	s	s	VERB
ejpam-6077	170	28	is	be	AUX
ejpam-6077	170	29	well	well	ADV
ejpam-6077	170	30	-	-	PUNCT
ejpam-6077	170	31	defined	define	VERB
ejpam-6077	170	32	and	and	CCONJ
ejpam-6077	170	33	continuous	continuous	ADJ
ejpam-6077	170	34	on	on	ADP
ejpam-6077	170	35	x.	x.	NOUN
ejpam-6077	170	36	specifically	specifically	ADV
ejpam-6077	170	37	,	,	PUNCT
ejpam-6077	170	38	for	for	ADP
ejpam-6077	170	39	any	any	DET
ejpam-6077	170	40	υ	υ	NOUN
ejpam-6077	170	41	,	,	PUNCT
ejpam-6077	170	42	ξ	ξ	PROPN
ejpam-6077	170	43	∈	∈	PROPN
ejpam-6077	170	44	x	x	X
ejpam-6077	170	45	,	,	PUNCT
ejpam-6077	170	46	|(tυ)(t)−	|(tυ)(t)−	PROPN
ejpam-6077	170	47	(	(	PUNCT
ejpam-6077	170	48	tξ)(t)|	tξ)(t)|	PUNCT
ejpam-6077	171	1	=	=	SYM
ejpam-6077	171	2	∣∣∣∣∫	∣∣∣∣∫	PRON
ejpam-6077	171	3	b	b	NOUN
ejpam-6077	171	4	a	a	DET
ejpam-6077	171	5	(	(	PUNCT
ejpam-6077	171	6	k(t	k(t	PROPN
ejpam-6077	171	7	,	,	PUNCT
ejpam-6077	171	8	s	s	NOUN
ejpam-6077	171	9	,	,	PUNCT
ejpam-6077	171	10	υ(s))−k(t	υ(s))−k(t	NOUN
ejpam-6077	171	11	,	,	PUNCT
ejpam-6077	171	12	s	s	PROPN
ejpam-6077	171	13	,	,	PUNCT
ejpam-6077	171	14	ξ(s	ξ(s	PROPN
ejpam-6077	171	15	)	)	PUNCT
ejpam-6077	171	16	)	)	PUNCT
ejpam-6077	171	17	)	)	PUNCT
ejpam-6077	171	18	ds	ds	ADJ
ejpam-6077	171	19	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6077	171	20	.	.	PUNCT
ejpam-6077	172	1	using	use	VERB
ejpam-6077	172	2	the	the	DET
ejpam-6077	172	3	lipschitz	lipschitz	NOUN
ejpam-6077	172	4	condition	condition	NOUN
ejpam-6077	172	5	:	:	PUNCT
ejpam-6077	172	6	|(sυ)(t)−	|(sυ)(t)−	PROPN
ejpam-6077	172	7	(	(	PUNCT
ejpam-6077	172	8	sξ)(t)|	sξ)(t)|	PUNCT
ejpam-6077	172	9	≤	≤	X
ejpam-6077	172	10	∫	∫	PROPN
ejpam-6077	172	11	b	b	PROPN
ejpam-6077	172	12	a	a	PROPN
ejpam-6077	172	13	l|υ(s)−	l|υ(s)−	ADV
ejpam-6077	172	14	ξ(s)|	ξ(s)|	PROPN
ejpam-6077	172	15	ds	ds	PROPN
ejpam-6077	172	16	≤	≤	PROPN
ejpam-6077	172	17	l∥υ	l∥υ	PROPN
ejpam-6077	172	18	−	−	PROPN
ejpam-6077	172	19	ξ∥∞(b−	ξ∥∞(b−	VERB
ejpam-6077	172	20	a	a	PRON
ejpam-6077	172	21	)	)	PUNCT
ejpam-6077	172	22	.	.	PUNCT
ejpam-6077	173	1	thus	thus	ADV
ejpam-6077	173	2	,	,	PUNCT
ejpam-6077	173	3	s	s	VERB
ejpam-6077	173	4	is	be	AUX
ejpam-6077	173	5	a	a	DET
ejpam-6077	173	6	contraction	contraction	NOUN
ejpam-6077	173	7	in	in	ADP
ejpam-6077	173	8	the	the	DET
ejpam-6077	173	9	supremum	supremum	ADJ
ejpam-6077	173	10	norm	norm	NOUN
ejpam-6077	173	11	.	.	PUNCT
ejpam-6077	174	1	step	step	NOUN
ejpam-6077	174	2	4	4	NUM
ejpam-6077	174	3	:	:	PUNCT
ejpam-6077	174	4	verify	verify	VERB
ejpam-6077	174	5	contraction	contraction	NOUN
ejpam-6077	174	6	in	in	ADP
ejpam-6077	174	7	mr	mr	PROPN
ejpam-6077	174	8	-	-	PUNCT
ejpam-6077	174	9	metric	metric	NOUN
ejpam-6077	174	10	for	for	ADP
ejpam-6077	174	11	υ	υ	PROPN
ejpam-6077	174	12	,	,	PUNCT
ejpam-6077	174	13	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	174	14	∈	∈	PROPN
ejpam-6077	174	15	x	x	SYM
ejpam-6077	174	16	,	,	PUNCT
ejpam-6077	174	17	m(sυ	m(sυ	PROPN
ejpam-6077	174	18	,	,	PUNCT
ejpam-6077	174	19	sξ	sξ	INTJ
ejpam-6077	174	20	,	,	PUNCT
ejpam-6077	174	21	sℑ	sℑ	NOUN
ejpam-6077	174	22	)	)	PUNCT
ejpam-6077	174	23	=	=	SYM
ejpam-6077	174	24	sup	sup	NOUN
ejpam-6077	174	25	t∈[a	t∈[a	NOUN
ejpam-6077	174	26	,	,	PUNCT
ejpam-6077	174	27	b	b	NOUN
ejpam-6077	174	28	]	]	X
ejpam-6077	174	29	|(sυ)(t)−	|(sυ)(t)−	PROPN
ejpam-6077	174	30	(	(	PUNCT
ejpam-6077	174	31	sξ)(t)|+	sξ)(t)|+	PROPN
ejpam-6077	174	32	sup	sup	NOUN
ejpam-6077	174	33	t∈[a	t∈[a	NOUN
ejpam-6077	174	34	,	,	PUNCT
ejpam-6077	174	35	b	b	NOUN
ejpam-6077	174	36	]	]	X
ejpam-6077	174	37	|(sξ)(t)−	|(sξ)(t)−	PROPN
ejpam-6077	174	38	(	(	PUNCT
ejpam-6077	174	39	sℑ)(t)|	sℑ)(t)|	ADV
ejpam-6077	174	40	.	.	PUNCT
ejpam-6077	175	1	a.	a.	NOUN
ejpam-6077	175	2	malkawi	malkawi	ADP
ejpam-6077	175	3	/	/	SYM
ejpam-6077	175	4	eur	eur	PROPN
ejpam-6077	175	5	.	.	PUNCT
ejpam-6077	176	1	j.	j.	PROPN
ejpam-6077	176	2	pure	pure	PROPN
ejpam-6077	176	3	appl	appl	PROPN
ejpam-6077	176	4	.	.	PROPN
ejpam-6077	176	5	math	math	PROPN
ejpam-6077	176	6	,	,	PUNCT
ejpam-6077	176	7	18	18	NUM
ejpam-6077	176	8	(	(	PUNCT
ejpam-6077	176	9	2	2	NUM
ejpam-6077	176	10	)	)	PUNCT
ejpam-6077	176	11	(	(	PUNCT
ejpam-6077	176	12	2025	2025	NUM
ejpam-6077	176	13	)	)	PUNCT
ejpam-6077	176	14	,	,	PUNCT
ejpam-6077	176	15	6077	6077	NUM
ejpam-6077	176	16	9	9	NUM
ejpam-6077	176	17	of	of	ADP
ejpam-6077	176	18	16	16	NUM
ejpam-6077	176	19	using	use	VERB
ejpam-6077	176	20	the	the	DET
ejpam-6077	176	21	contraction	contraction	NOUN
ejpam-6077	176	22	property	property	NOUN
ejpam-6077	176	23	derived	derive	VERB
ejpam-6077	176	24	above	above	ADV
ejpam-6077	176	25	:	:	PUNCT
ejpam-6077	176	26	sup	sup	NOUN
ejpam-6077	176	27	t∈[a	t∈[a	NOUN
ejpam-6077	176	28	,	,	PUNCT
ejpam-6077	176	29	b	b	NOUN
ejpam-6077	176	30	]	]	X
ejpam-6077	176	31	|(sυ)(t)−	|(sυ)(t)−	PROPN
ejpam-6077	176	32	(	(	PUNCT
ejpam-6077	176	33	sξ)(t)|	sξ)(t)|	NUM
ejpam-6077	176	34	≤	≤	ADJ
ejpam-6077	176	35	k	k	PROPN
ejpam-6077	176	36	sup	sup	NOUN
ejpam-6077	176	37	t∈[a	t∈[a	NOUN
ejpam-6077	176	38	,	,	PUNCT
ejpam-6077	176	39	b	b	NOUN
ejpam-6077	176	40	]	]	PUNCT
ejpam-6077	176	41	|υ(t)−	|υ(t)−	PROPN
ejpam-6077	176	42	ξ(t)|	ξ(t)|	NOUN
ejpam-6077	176	43	,	,	PUNCT
ejpam-6077	176	44	and	and	CCONJ
ejpam-6077	176	45	similarly	similarly	ADV
ejpam-6077	176	46	:	:	PUNCT
ejpam-6077	176	47	sup	sup	NOUN
ejpam-6077	176	48	t∈[a	t∈[a	NOUN
ejpam-6077	176	49	,	,	PUNCT
ejpam-6077	176	50	b	b	NOUN
ejpam-6077	176	51	]	]	X
ejpam-6077	176	52	|(sξ)(t)−	|(sξ)(t)−	PROPN
ejpam-6077	176	53	(	(	PUNCT
ejpam-6077	176	54	sℑ)(t)|	sℑ)(t)|	ADV
ejpam-6077	176	55	≤	≤	NOUN
ejpam-6077	176	56	k	k	PROPN
ejpam-6077	176	57	sup	sup	NOUN
ejpam-6077	176	58	t∈[a	t∈[a	NOUN
ejpam-6077	176	59	,	,	PUNCT
ejpam-6077	176	60	b	b	NOUN
ejpam-6077	176	61	]	]	X
ejpam-6077	176	62	|ξ(t)−ℑ(t)|	|ξ(t)−ℑ(t)|	NUM
ejpam-6077	176	63	.	.	PUNCT
ejpam-6077	177	1	therefore	therefore	ADV
ejpam-6077	177	2	:	:	PUNCT
ejpam-6077	177	3	m(sυ	m(sυ	PROPN
ejpam-6077	177	4	,	,	PUNCT
ejpam-6077	177	5	sξ	sξ	INTJ
ejpam-6077	177	6	,	,	PUNCT
ejpam-6077	177	7	sℑ	sℑ	NOUN
ejpam-6077	177	8	)	)	PUNCT
ejpam-6077	177	9	≤	≤	PUNCT
ejpam-6077	178	1	k	k	X
ejpam-6077	178	2	·	·	PUNCT
ejpam-6077	178	3	m(υ	m(υ	PROPN
ejpam-6077	178	4	,	,	PUNCT
ejpam-6077	178	5	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	178	6	)	)	PUNCT
ejpam-6077	178	7	,	,	PUNCT
ejpam-6077	178	8	where	where	SCONJ
ejpam-6077	178	9	k	k	PROPN
ejpam-6077	178	10	=	=	PRON
ejpam-6077	178	11	l(b−	l(b−	PROPN
ejpam-6077	178	12	a	a	NOUN
ejpam-6077	178	13	)	)	PUNCT
ejpam-6077	178	14	<	<	X
ejpam-6077	178	15	1	1	NUM
ejpam-6077	178	16	ensures	ensure	VERB
ejpam-6077	178	17	the	the	DET
ejpam-6077	178	18	contraction	contraction	NOUN
ejpam-6077	178	19	condition	condition	NOUN
ejpam-6077	178	20	in	in	ADP
ejpam-6077	178	21	the	the	DET
ejpam-6077	178	22	mr	mr	PROPN
ejpam-6077	178	23	-	-	PUNCT
ejpam-6077	178	24	metric	metric	NOUN
ejpam-6077	178	25	.	.	PUNCT
ejpam-6077	179	1	step	step	NOUN
ejpam-6077	179	2	5	5	NUM
ejpam-6077	179	3	:	:	PUNCT
ejpam-6077	179	4	existence	existence	NOUN
ejpam-6077	179	5	and	and	CCONJ
ejpam-6077	179	6	uniqueness	uniqueness	NOUN
ejpam-6077	179	7	of	of	ADP
ejpam-6077	179	8	fixed	fix	VERB
ejpam-6077	179	9	point	point	NOUN
ejpam-6077	179	10	by	by	ADP
ejpam-6077	179	11	the	the	DET
ejpam-6077	179	12	theorem	theorem	NOUN
ejpam-6077	179	13	,	,	PUNCT
ejpam-6077	179	14	since	since	SCONJ
ejpam-6077	179	15	s	s	PRON
ejpam-6077	179	16	satisfies	satisfy	VERB
ejpam-6077	179	17	the	the	DET
ejpam-6077	179	18	contraction	contraction	NOUN
ejpam-6077	179	19	condition	condition	NOUN
ejpam-6077	179	20	in	in	ADP
ejpam-6077	179	21	the	the	DET
ejpam-6077	179	22	mr	mr	PROPN
ejpam-6077	179	23	-	-	PUNCT
ejpam-6077	179	24	metric	metric	NOUN
ejpam-6077	179	25	and	and	CCONJ
ejpam-6077	179	26	x	x	NOUN
ejpam-6077	179	27	is	be	AUX
ejpam-6077	179	28	closed	closed	ADJ
ejpam-6077	179	29	,	,	PUNCT
ejpam-6077	179	30	bounded	bound	VERB
ejpam-6077	179	31	,	,	PUNCT
ejpam-6077	179	32	and	and	CCONJ
ejpam-6077	179	33	convex	convex	PROPN
ejpam-6077	179	34	,	,	PUNCT
ejpam-6077	179	35	there	there	PRON
ejpam-6077	179	36	exists	exist	VERB
ejpam-6077	179	37	a	a	DET
ejpam-6077	179	38	unique	unique	ADJ
ejpam-6077	179	39	fixed	fix	VERB
ejpam-6077	179	40	point	point	NOUN
ejpam-6077	179	41	υ∗	υ∗	NOUN
ejpam-6077	179	42	∈	∈	PROPN
ejpam-6077	179	43	x	x	PUNCT
ejpam-6077	179	44	such	such	ADJ
ejpam-6077	179	45	that	that	SCONJ
ejpam-6077	179	46	:	:	PUNCT
ejpam-6077	179	47	υ∗(t	υ∗(t	NOUN
ejpam-6077	179	48	)	)	PUNCT
ejpam-6077	179	49	=	=	PRON
ejpam-6077	179	50	(	(	PUNCT
ejpam-6077	179	51	sυ∗)(t	sυ∗)(t	PROPN
ejpam-6077	179	52	)	)	PUNCT
ejpam-6077	179	53	=	=	SYM
ejpam-6077	180	1	∫	∫	PROPN
ejpam-6077	180	2	b	b	PROPN
ejpam-6077	180	3	a	a	PRON
ejpam-6077	180	4	k(t	k(t	PROPN
ejpam-6077	180	5	,	,	PUNCT
ejpam-6077	180	6	s	s	X
ejpam-6077	180	7	,	,	PUNCT
ejpam-6077	180	8	υ∗(s	υ∗(s	PROPN
ejpam-6077	180	9	)	)	PUNCT
ejpam-6077	180	10	)	)	PUNCT
ejpam-6077	181	1	ds	ds	PROPN
ejpam-6077	181	2	.	.	PUNCT
ejpam-6077	182	1	this	this	DET
ejpam-6077	182	2	fixed	fix	VERB
ejpam-6077	182	3	point	point	NOUN
ejpam-6077	182	4	υ∗	υ∗	NOUN
ejpam-6077	182	5	is	be	AUX
ejpam-6077	182	6	the	the	DET
ejpam-6077	182	7	unique	unique	ADJ
ejpam-6077	182	8	solution	solution	NOUN
ejpam-6077	182	9	to	to	ADP
ejpam-6077	182	10	the	the	DET
ejpam-6077	182	11	nonlinear	nonlinear	ADJ
ejpam-6077	182	12	integral	integral	ADJ
ejpam-6077	182	13	equation	equation	NOUN
ejpam-6077	182	14	.	.	PUNCT
ejpam-6077	183	1	2	2	X
ejpam-6077	183	2	.	.	X
ejpam-6077	183	3	stability	stability	NOUN
ejpam-6077	183	4	of	of	ADP
ejpam-6077	183	5	iterative	iterative	NOUN
ejpam-6077	183	6	processes	process	NOUN
ejpam-6077	183	7	example	example	NOUN
ejpam-6077	183	8	5	5	NUM
ejpam-6077	183	9	.	.	X
ejpam-6077	183	10	consider	consider	VERB
ejpam-6077	183	11	an	an	DET
ejpam-6077	183	12	iterative	iterative	NOUN
ejpam-6077	183	13	scheme	scheme	NOUN
ejpam-6077	183	14	represented	represent	VERB
ejpam-6077	183	15	by	by	ADP
ejpam-6077	183	16	the	the	DET
ejpam-6077	183	17	update	update	NOUN
ejpam-6077	183	18	rule	rule	NOUN
ejpam-6077	183	19	:	:	PUNCT
ejpam-6077	184	1	υn+1	υn+1	NUM
ejpam-6077	184	2	=	=	SYM
ejpam-6077	184	3	s(υn	s(υn	NOUN
ejpam-6077	184	4	)	)	PUNCT
ejpam-6077	184	5	,	,	PUNCT
ejpam-6077	184	6	where	where	SCONJ
ejpam-6077	184	7	s	s	VERB
ejpam-6077	184	8	:	:	PUNCT
ejpam-6077	184	9	x	x	SYM
ejpam-6077	184	10	→	→	PUNCT
ejpam-6077	184	11	x	x	X
ejpam-6077	184	12	is	be	AUX
ejpam-6077	184	13	a	a	DET
ejpam-6077	184	14	mapping	mapping	NOUN
ejpam-6077	184	15	on	on	ADP
ejpam-6077	184	16	a	a	DET
ejpam-6077	184	17	banach	banach	NOUN
ejpam-6077	184	18	space	space	NOUN
ejpam-6077	184	19	x	x	PUNCT
ejpam-6077	184	20	equipped	equip	VERB
ejpam-6077	184	21	with	with	ADP
ejpam-6077	184	22	a	a	DET
ejpam-6077	184	23	norm	norm	NOUN
ejpam-6077	184	24	∥	∥	X
ejpam-6077	184	25	·	·	PUNCT
ejpam-6077	184	26	∥.	∥.	ADP
ejpam-6077	185	1	the	the	DET
ejpam-6077	185	2	space	space	NOUN
ejpam-6077	185	3	x	x	PUNCT
ejpam-6077	185	4	is	be	AUX
ejpam-6077	185	5	further	far	ADV
ejpam-6077	185	6	structured	structure	VERB
ejpam-6077	185	7	with	with	ADP
ejpam-6077	185	8	an	an	DET
ejpam-6077	185	9	mr	mr	PROPN
ejpam-6077	185	10	-	-	PUNCT
ejpam-6077	185	11	metric	metric	NOUN
ejpam-6077	185	12	defined	define	VERB
ejpam-6077	185	13	as	as	ADP
ejpam-6077	185	14	:	:	PUNCT
ejpam-6077	185	15	m(υ	m(υ	PROPN
ejpam-6077	185	16	,	,	PUNCT
ejpam-6077	185	17	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	185	18	)	)	PUNCT
ejpam-6077	186	1	=	=	SYM
ejpam-6077	186	2	∥υ	∥υ	PROPN
ejpam-6077	186	3	−	−	PROPN
ejpam-6077	186	4	ξ∥+	ξ∥+	VERB
ejpam-6077	186	5	∥ξ	∥ξ	PROPN
ejpam-6077	186	6	−ℑ∥.	−ℑ∥.	PROPN
ejpam-6077	186	7	assumptions	assumption	NOUN
ejpam-6077	186	8	:	:	PUNCT
ejpam-6077	186	9	1	1	X
ejpam-6077	186	10	.	.	X
ejpam-6077	186	11	s	s	PART
ejpam-6077	186	12	satisfies	satisfie	NOUN
ejpam-6077	186	13	a	a	DET
ejpam-6077	186	14	contraction	contraction	NOUN
ejpam-6077	186	15	condition	condition	NOUN
ejpam-6077	186	16	in	in	ADP
ejpam-6077	186	17	the	the	DET
ejpam-6077	186	18	mr	mr	PROPN
ejpam-6077	186	19	-	-	PUNCT
ejpam-6077	186	20	metric	metric	ADJ
ejpam-6077	186	21	:	:	PUNCT
ejpam-6077	186	22	m(s(υ	m(s(υ	PROPN
ejpam-6077	186	23	)	)	PUNCT
ejpam-6077	186	24	,	,	PUNCT
ejpam-6077	186	25	s(ξ	s(ξ	PROPN
ejpam-6077	186	26	)	)	PUNCT
ejpam-6077	186	27	,	,	PUNCT
ejpam-6077	186	28	s(ℑ	s(ℑ	PROPN
ejpam-6077	186	29	)	)	PUNCT
ejpam-6077	186	30	)	)	PUNCT
ejpam-6077	186	31	≤	≤	PUNCT
ejpam-6077	187	1	k	k	X
ejpam-6077	187	2	·	·	PUNCT
ejpam-6077	187	3	m(υ	m(υ	PROPN
ejpam-6077	187	4	,	,	PUNCT
ejpam-6077	187	5	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	187	6	)	)	PUNCT
ejpam-6077	187	7	,	,	PUNCT
ejpam-6077	187	8	for	for	ADP
ejpam-6077	187	9	all	all	DET
ejpam-6077	187	10	υ	υ	PROPN
ejpam-6077	187	11	,	,	PUNCT
ejpam-6077	187	12	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	187	13	∈	∈	PROPN
ejpam-6077	187	14	x	x	X
ejpam-6077	187	15	,	,	PUNCT
ejpam-6077	187	16	where	where	SCONJ
ejpam-6077	187	17	k	k	PROPN
ejpam-6077	187	18	∈	∈	PROPN
ejpam-6077	188	1	[	[	X
ejpam-6077	188	2	0	0	NUM
ejpam-6077	188	3	,	,	PUNCT
ejpam-6077	188	4	1	1	NUM
ejpam-6077	188	5	)	)	PUNCT
ejpam-6077	188	6	is	be	AUX
ejpam-6077	188	7	a	a	DET
ejpam-6077	188	8	constant	constant	ADJ
ejpam-6077	188	9	.	.	PUNCT
ejpam-6077	189	1	2	2	X
ejpam-6077	189	2	.	.	X
ejpam-6077	189	3	the	the	DET
ejpam-6077	189	4	norm	norm	NOUN
ejpam-6077	189	5	∥	∥	X
ejpam-6077	189	6	·	·	PUNCT
ejpam-6077	189	7	∥	∥	PRON
ejpam-6077	189	8	ensures	ensure	VERB
ejpam-6077	189	9	that	that	SCONJ
ejpam-6077	189	10	x	x	PRON
ejpam-6077	189	11	is	be	AUX
ejpam-6077	189	12	a	a	DET
ejpam-6077	189	13	complete	complete	ADJ
ejpam-6077	189	14	metric	metric	ADJ
ejpam-6077	189	15	space	space	NOUN
ejpam-6077	189	16	.	.	PUNCT
ejpam-6077	190	1	step	step	NOUN
ejpam-6077	190	2	-	-	PUNCT
ejpam-6077	190	3	by	by	ADP
ejpam-6077	190	4	-	-	PUNCT
ejpam-6077	190	5	step	step	NOUN
ejpam-6077	190	6	analysis	analysis	NOUN
ejpam-6077	190	7	:	:	PUNCT
ejpam-6077	190	8	step	step	NOUN
ejpam-6077	190	9	1	1	NUM
ejpam-6077	190	10	:	:	PUNCT
ejpam-6077	190	11	understanding	understand	VERB
ejpam-6077	190	12	the	the	DET
ejpam-6077	190	13	mr	mr	PROPN
ejpam-6077	190	14	-	-	PUNCT
ejpam-6077	190	15	metric	metric	NOUN
ejpam-6077	190	16	the	the	DET
ejpam-6077	190	17	mr	mr	PROPN
ejpam-6077	190	18	-	-	PUNCT
ejpam-6077	190	19	metric	metric	NOUN
ejpam-6077	190	20	generalizes	generalize	VERB
ejpam-6077	190	21	the	the	DET
ejpam-6077	190	22	standard	standard	NOUN
ejpam-6077	190	23	metric	metric	ADJ
ejpam-6077	190	24	by	by	ADP
ejpam-6077	190	25	simultaneously	simultaneously	ADV
ejpam-6077	190	26	measuring	measure	VERB
ejpam-6077	190	27	the	the	DET
ejpam-6077	190	28	”	"	PUNCT
ejpam-6077	190	29	distances	distance	NOUN
ejpam-6077	190	30	”	"	PUNCT
ejpam-6077	190	31	between	between	ADP
ejpam-6077	190	32	three	three	NUM
ejpam-6077	190	33	elements	element	NOUN
ejpam-6077	190	34	.	.	PUNCT
ejpam-6077	191	1	in	in	ADP
ejpam-6077	191	2	this	this	DET
ejpam-6077	191	3	case	case	NOUN
ejpam-6077	191	4	:	:	PUNCT
ejpam-6077	191	5	m(υ	m(υ	PROPN
ejpam-6077	191	6	,	,	PUNCT
ejpam-6077	191	7	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	191	8	)	)	PUNCT
ejpam-6077	192	1	=	=	SYM
ejpam-6077	192	2	∥υ	∥υ	PROPN
ejpam-6077	192	3	−	−	PROPN
ejpam-6077	192	4	ξ∥+	ξ∥+	VERB
ejpam-6077	192	5	∥ξ	∥ξ	PROPN
ejpam-6077	192	6	−ℑ∥.	−ℑ∥.	PROPN
ejpam-6077	192	7	this	this	DET
ejpam-6077	192	8	metric	metric	NOUN
ejpam-6077	192	9	allows	allow	VERB
ejpam-6077	192	10	the	the	DET
ejpam-6077	192	11	analysis	analysis	NOUN
ejpam-6077	192	12	to	to	PART
ejpam-6077	192	13	track	track	VERB
ejpam-6077	192	14	how	how	SCONJ
ejpam-6077	192	15	the	the	DET
ejpam-6077	192	16	distances	distance	NOUN
ejpam-6077	192	17	between	between	ADP
ejpam-6077	192	18	successive	successive	ADJ
ejpam-6077	192	19	iterations	iteration	NOUN
ejpam-6077	192	20	evolve	evolve	VERB
ejpam-6077	192	21	in	in	ADP
ejpam-6077	192	22	a	a	DET
ejpam-6077	192	23	dynamic	dynamic	ADJ
ejpam-6077	192	24	system	system	NOUN
ejpam-6077	192	25	.	.	PUNCT
ejpam-6077	193	1	step	step	NOUN
ejpam-6077	193	2	2	2	NUM
ejpam-6077	193	3	:	:	PUNCT
ejpam-6077	193	4	contraction	contraction	NOUN
ejpam-6077	193	5	property	property	NOUN
ejpam-6077	193	6	of	of	ADP
ejpam-6077	193	7	s	s	PRON
ejpam-6077	193	8	for	for	ADP
ejpam-6077	193	9	any	any	DET
ejpam-6077	193	10	υ	υ	NOUN
ejpam-6077	193	11	,	,	PUNCT
ejpam-6077	193	12	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	193	13	∈	∈	PROPN
ejpam-6077	193	14	x	x	NOUN
ejpam-6077	193	15	,	,	PUNCT
ejpam-6077	193	16	the	the	DET
ejpam-6077	193	17	contraction	contraction	NOUN
ejpam-6077	193	18	condition	condition	NOUN
ejpam-6077	193	19	ensures	ensure	VERB
ejpam-6077	193	20	:	:	PUNCT
ejpam-6077	193	21	m(s(υ	m(s(υ	PROPN
ejpam-6077	193	22	)	)	PUNCT
ejpam-6077	193	23	,	,	PUNCT
ejpam-6077	193	24	s(ξ	s(ξ	PROPN
ejpam-6077	193	25	)	)	PUNCT
ejpam-6077	193	26	,	,	PUNCT
ejpam-6077	193	27	s(ℑ	s(ℑ	NUM
ejpam-6077	193	28	)	)	PUNCT
ejpam-6077	193	29	)	)	PUNCT
ejpam-6077	194	1	=	=	SYM
ejpam-6077	194	2	∥s(υ)−	∥s(υ)−	X
ejpam-6077	194	3	s(ξ)∥+	s(ξ)∥+	NOUN
ejpam-6077	194	4	∥s(ξ)−	∥s(ξ)−	ADV
ejpam-6077	194	5	s(ℑ)∥	s(ℑ)∥	ADJ
ejpam-6077	194	6	≤	≤	ADJ
ejpam-6077	194	7	k(∥υ	k(∥υ	NOUN
ejpam-6077	194	8	−	−	PROPN
ejpam-6077	194	9	ξ∥+	ξ∥+	PROPN
ejpam-6077	194	10	∥ξ	∥ξ	PROPN
ejpam-6077	194	11	−ℑ∥	−ℑ∥	NOUN
ejpam-6077	194	12	)	)	PUNCT
ejpam-6077	194	13	,	,	PUNCT
ejpam-6077	194	14	a.	a.	NOUN
ejpam-6077	194	15	malkawi	malkawi	PROPN
ejpam-6077	194	16	/	/	SYM
ejpam-6077	194	17	eur	eur	PROPN
ejpam-6077	194	18	.	.	PUNCT
ejpam-6077	195	1	j.	j.	PROPN
ejpam-6077	195	2	pure	pure	PROPN
ejpam-6077	195	3	appl	appl	PROPN
ejpam-6077	195	4	.	.	PROPN
ejpam-6077	195	5	math	math	PROPN
ejpam-6077	195	6	,	,	PUNCT
ejpam-6077	195	7	18	18	NUM
ejpam-6077	195	8	(	(	PUNCT
ejpam-6077	195	9	2	2	NUM
ejpam-6077	195	10	)	)	PUNCT
ejpam-6077	195	11	(	(	PUNCT
ejpam-6077	195	12	2025	2025	NUM
ejpam-6077	195	13	)	)	PUNCT
ejpam-6077	195	14	,	,	PUNCT
ejpam-6077	195	15	6077	6077	NUM
ejpam-6077	195	16	10	10	NUM
ejpam-6077	195	17	of	of	ADP
ejpam-6077	195	18	16	16	NUM
ejpam-6077	195	19	where	where	SCONJ
ejpam-6077	195	20	k	k	PROPN
ejpam-6077	195	21	∈	∈	PROPN
ejpam-6077	196	1	[	[	X
ejpam-6077	196	2	0	0	NUM
ejpam-6077	196	3	,	,	PUNCT
ejpam-6077	196	4	1	1	NUM
ejpam-6077	196	5	)	)	PUNCT
ejpam-6077	196	6	.	.	PUNCT
ejpam-6077	197	1	this	this	PRON
ejpam-6077	197	2	guarantees	guarantee	VERB
ejpam-6077	197	3	that	that	SCONJ
ejpam-6077	197	4	the	the	DET
ejpam-6077	197	5	operator	operator	NOUN
ejpam-6077	197	6	s	s	PART
ejpam-6077	197	7	”	"	PUNCT
ejpam-6077	197	8	brings	bring	VERB
ejpam-6077	197	9	points	point	NOUN
ejpam-6077	197	10	closer	close	ADV
ejpam-6077	197	11	together	together	ADV
ejpam-6077	197	12	”	"	PUNCT
ejpam-6077	197	13	under	under	ADP
ejpam-6077	197	14	the	the	DET
ejpam-6077	197	15	mr	mr	PROPN
ejpam-6077	197	16	-	-	PUNCT
ejpam-6077	197	17	metric	metric	NOUN
ejpam-6077	197	18	.	.	PUNCT
ejpam-6077	198	1	step	step	NOUN
ejpam-6077	198	2	3	3	NUM
ejpam-6077	198	3	:	:	PUNCT
ejpam-6077	198	4	iterative	iterative	NOUN
ejpam-6077	198	5	sequence	sequence	NOUN
ejpam-6077	198	6	start	start	VERB
ejpam-6077	198	7	with	with	ADP
ejpam-6077	198	8	an	an	DET
ejpam-6077	198	9	initial	initial	ADJ
ejpam-6077	198	10	guess	guess	NOUN
ejpam-6077	198	11	υ0	υ0	NOUN
ejpam-6077	198	12	∈	∈	PROPN
ejpam-6077	198	13	x	x	PUNCT
ejpam-6077	198	14	and	and	CCONJ
ejpam-6077	198	15	define	define	VERB
ejpam-6077	198	16	the	the	DET
ejpam-6077	198	17	sequence	sequence	NOUN
ejpam-6077	198	18	:	:	PUNCT
ejpam-6077	198	19	υn+1	υn+1	NUM
ejpam-6077	198	20	=	=	SYM
ejpam-6077	198	21	s(υn	s(υn	PROPN
ejpam-6077	198	22	)	)	PUNCT
ejpam-6077	198	23	,	,	PUNCT
ejpam-6077	198	24	n	n	X
ejpam-6077	198	25	≥	≥	NOUN
ejpam-6077	198	26	0	0	NUM
ejpam-6077	198	27	.	.	PUNCT
ejpam-6077	199	1	for	for	ADP
ejpam-6077	199	2	any	any	DET
ejpam-6077	199	3	n	n	CCONJ
ejpam-6077	199	4	,	,	PUNCT
ejpam-6077	199	5	consider	consider	VERB
ejpam-6077	199	6	three	three	NUM
ejpam-6077	199	7	successive	successive	ADJ
ejpam-6077	199	8	iterates	iterate	NOUN
ejpam-6077	199	9	υn−1	υn−1	PROPN
ejpam-6077	199	10	,	,	PUNCT
ejpam-6077	199	11	υn	υn	NOUN
ejpam-6077	199	12	,	,	PUNCT
ejpam-6077	199	13	υn+1	υn+1	NOUN
ejpam-6077	199	14	:	:	PUNCT
ejpam-6077	199	15	m(υn+1	m(υn+1	NUM
ejpam-6077	199	16	,	,	PUNCT
ejpam-6077	199	17	υn	υn	NOUN
ejpam-6077	199	18	,	,	PUNCT
ejpam-6077	199	19	υn−1	υn−1	PROPN
ejpam-6077	199	20	)	)	PUNCT
ejpam-6077	199	21	=	=	PUNCT
ejpam-6077	199	22	∥υn+1	∥υn+1	NOUN
ejpam-6077	199	23	−	−	PROPN
ejpam-6077	199	24	υn∥+	υn∥+	NOUN
ejpam-6077	199	25	∥υn	∥υn	CCONJ
ejpam-6077	199	26	−	−	NOUN
ejpam-6077	200	1	υn−1∥.	υn−1∥.	NOUN
ejpam-6077	200	2	using	use	VERB
ejpam-6077	200	3	the	the	DET
ejpam-6077	200	4	contraction	contraction	NOUN
ejpam-6077	200	5	property	property	NOUN
ejpam-6077	200	6	of	of	ADP
ejpam-6077	200	7	s	s	NOUN
ejpam-6077	200	8	:	:	PUNCT
ejpam-6077	200	9	m(υn+1	m(υn+1	NUM
ejpam-6077	200	10	,	,	PUNCT
ejpam-6077	200	11	υn	υn	NOUN
ejpam-6077	200	12	,	,	PUNCT
ejpam-6077	200	13	υn−1	υn−1	PROPN
ejpam-6077	200	14	)	)	PUNCT
ejpam-6077	200	15	≤	≤	PUNCT
ejpam-6077	201	1	k	k	X
ejpam-6077	201	2	·	·	SYM
ejpam-6077	201	3	m(υn	m(υn	ADJ
ejpam-6077	201	4	,	,	PUNCT
ejpam-6077	201	5	υn−1	υn−1	ADJ
ejpam-6077	201	6	,	,	PUNCT
ejpam-6077	201	7	υn−2	υn−2	PROPN
ejpam-6077	201	8	)	)	PUNCT
ejpam-6077	201	9	.	.	PUNCT
ejpam-6077	202	1	by	by	ADP
ejpam-6077	202	2	induction	induction	NOUN
ejpam-6077	202	3	:	:	PUNCT
ejpam-6077	202	4	m(υn+1	m(υn+1	NUM
ejpam-6077	202	5	,	,	PUNCT
ejpam-6077	202	6	υn	υn	NOUN
ejpam-6077	202	7	,	,	PUNCT
ejpam-6077	202	8	υn−1	υn−1	PROPN
ejpam-6077	202	9	)	)	PUNCT
ejpam-6077	202	10	≤	≤	NOUN
ejpam-6077	202	11	kn	kn	PROPN
ejpam-6077	202	12	·	·	SYM
ejpam-6077	202	13	m(υ1	m(υ1	ADJ
ejpam-6077	202	14	,	,	PUNCT
ejpam-6077	202	15	υ0	υ0	NOUN
ejpam-6077	202	16	,	,	PUNCT
ejpam-6077	202	17	υ−1	υ−1	PROPN
ejpam-6077	202	18	)	)	PUNCT
ejpam-6077	202	19	,	,	PUNCT
ejpam-6077	202	20	where	where	SCONJ
ejpam-6077	202	21	υ−1	υ−1	PROPN
ejpam-6077	202	22	can	can	AUX
ejpam-6077	202	23	be	be	AUX
ejpam-6077	202	24	taken	take	VERB
ejpam-6077	202	25	as	as	ADP
ejpam-6077	202	26	the	the	DET
ejpam-6077	202	27	initial	initial	ADJ
ejpam-6077	202	28	condition	condition	NOUN
ejpam-6077	202	29	.	.	PUNCT
ejpam-6077	203	1	step	step	NOUN
ejpam-6077	203	2	4	4	NUM
ejpam-6077	203	3	:	:	PUNCT
ejpam-6077	203	4	convergence	convergence	NOUN
ejpam-6077	203	5	to	to	ADP
ejpam-6077	203	6	the	the	DET
ejpam-6077	203	7	fixed	fix	VERB
ejpam-6077	203	8	point	point	NOUN
ejpam-6077	203	9	since	since	SCONJ
ejpam-6077	203	10	k	k	PROPN
ejpam-6077	203	11	∈	∈	PROPN
ejpam-6077	204	1	[	[	X
ejpam-6077	204	2	0	0	NUM
ejpam-6077	204	3	,	,	PUNCT
ejpam-6077	204	4	1	1	NUM
ejpam-6077	204	5	)	)	PUNCT
ejpam-6077	204	6	,	,	PUNCT
ejpam-6077	204	7	kn	kn	PROPN
ejpam-6077	204	8	→	→	SYM
ejpam-6077	204	9	0	0	PROPN
ejpam-6077	204	10	as	as	ADP
ejpam-6077	204	11	n	n	NOUN
ejpam-6077	204	12	→	→	SYM
ejpam-6077	204	13	∞.	∞.	PROPN
ejpam-6077	204	14	this	this	PRON
ejpam-6077	204	15	implies	imply	VERB
ejpam-6077	204	16	:	:	PUNCT
ejpam-6077	204	17	m(υn+1	m(υn+1	NUM
ejpam-6077	204	18	,	,	PUNCT
ejpam-6077	204	19	υn	υn	NOUN
ejpam-6077	204	20	,	,	PUNCT
ejpam-6077	204	21	υn−1	υn−1	PROPN
ejpam-6077	204	22	)	)	PUNCT
ejpam-6077	204	23	→	→	SYM
ejpam-6077	205	1	0	0	X
ejpam-6077	205	2	.	.	X
ejpam-6077	205	3	decomposing	decompose	VERB
ejpam-6077	205	4	m(υn+1	m(υn+1	NUM
ejpam-6077	205	5	,	,	PUNCT
ejpam-6077	205	6	υn	υn	NOUN
ejpam-6077	205	7	,	,	PUNCT
ejpam-6077	205	8	υn−1	υn−1	PROPN
ejpam-6077	205	9	)	)	PUNCT
ejpam-6077	205	10	,	,	PUNCT
ejpam-6077	205	11	we	we	PRON
ejpam-6077	205	12	observe	observe	VERB
ejpam-6077	205	13	that	that	SCONJ
ejpam-6077	205	14	:	:	PUNCT
ejpam-6077	205	15	∥υn+1	∥υn+1	VERB
ejpam-6077	205	16	−	−	PROPN
ejpam-6077	205	17	υn∥	υn∥	PROPN
ejpam-6077	205	18	→	→	SYM
ejpam-6077	205	19	0	0	NUM
ejpam-6077	205	20	and	and	CCONJ
ejpam-6077	205	21	∥υn	∥υn	NOUN
ejpam-6077	205	22	−	−	NOUN
ejpam-6077	205	23	υn−1∥	υn−1∥	NOUN
ejpam-6077	205	24	→	→	SYM
ejpam-6077	205	25	0	0	NUM
ejpam-6077	205	26	.	.	PUNCT
ejpam-6077	206	1	thus	thus	ADV
ejpam-6077	206	2	,	,	PUNCT
ejpam-6077	206	3	the	the	DET
ejpam-6077	206	4	sequence	sequence	NOUN
ejpam-6077	206	5	{	{	PUNCT
ejpam-6077	206	6	υn	υn	NOUN
ejpam-6077	206	7	}	}	PUNCT
ejpam-6077	206	8	is	be	AUX
ejpam-6077	206	9	cauchy	cauchy	ADJ
ejpam-6077	206	10	in	in	ADP
ejpam-6077	206	11	x.	x.	NOUN
ejpam-6077	206	12	since	since	SCONJ
ejpam-6077	206	13	x	x	PRON
ejpam-6077	206	14	is	be	AUX
ejpam-6077	206	15	a	a	DET
ejpam-6077	206	16	banach	banach	NOUN
ejpam-6077	206	17	space	space	NOUN
ejpam-6077	206	18	,	,	PUNCT
ejpam-6077	206	19	{	{	PUNCT
ejpam-6077	206	20	υn	υn	NOUN
ejpam-6077	206	21	}	}	PUNCT
ejpam-6077	206	22	converges	converge	VERB
ejpam-6077	206	23	to	to	ADP
ejpam-6077	206	24	a	a	DET
ejpam-6077	206	25	unique	unique	ADJ
ejpam-6077	206	26	point	point	NOUN
ejpam-6077	206	27	υ∗	υ∗	NOUN
ejpam-6077	206	28	∈	∈	PROPN
ejpam-6077	206	29	x.	x.	NOUN
ejpam-6077	206	30	step	step	NOUN
ejpam-6077	206	31	5	5	NUM
ejpam-6077	206	32	:	:	PUNCT
ejpam-6077	206	33	verification	verification	NOUN
ejpam-6077	206	34	of	of	ADP
ejpam-6077	206	35	the	the	DET
ejpam-6077	206	36	fixed	fix	VERB
ejpam-6077	206	37	point	point	NOUN
ejpam-6077	206	38	from	from	ADP
ejpam-6077	206	39	the	the	DET
ejpam-6077	206	40	continuity	continuity	NOUN
ejpam-6077	206	41	of	of	ADP
ejpam-6077	206	42	s	s	PROPN
ejpam-6077	206	43	,	,	PUNCT
ejpam-6077	206	44	the	the	DET
ejpam-6077	206	45	limit	limit	NOUN
ejpam-6077	206	46	υ∗	υ∗	NOUN
ejpam-6077	206	47	satisfies	satisfy	VERB
ejpam-6077	206	48	:	:	PUNCT
ejpam-6077	207	1	υ∗	υ∗	NOUN
ejpam-6077	207	2	=	=	SYM
ejpam-6077	207	3	lim	lim	PROPN
ejpam-6077	207	4	n→∞	n→∞	NUM
ejpam-6077	207	5	υn	υn	PROPN
ejpam-6077	207	6	=	=	SYM
ejpam-6077	207	7	lim	lim	PROPN
ejpam-6077	207	8	n→∞	n→∞	NUM
ejpam-6077	207	9	s(υn	s(υn	ADJ
ejpam-6077	207	10	)	)	PUNCT
ejpam-6077	207	11	=	=	SYM
ejpam-6077	207	12	s(υ∗	s(υ∗	NOUN
ejpam-6077	207	13	)	)	PUNCT
ejpam-6077	207	14	.	.	PUNCT
ejpam-6077	208	1	hence	hence	ADV
ejpam-6077	208	2	,	,	PUNCT
ejpam-6077	208	3	υ∗	υ∗	PROPN
ejpam-6077	208	4	is	be	AUX
ejpam-6077	208	5	the	the	DET
ejpam-6077	208	6	unique	unique	ADJ
ejpam-6077	208	7	fixed	fix	VERB
ejpam-6077	208	8	point	point	NOUN
ejpam-6077	208	9	of	of	ADP
ejpam-6077	208	10	s.	s.	PROPN
ejpam-6077	208	11	3	3	NUM
ejpam-6077	208	12	.	.	PUNCT
ejpam-6077	209	1	optimization	optimization	NOUN
ejpam-6077	209	2	problems	problem	NOUN
ejpam-6077	209	3	example	example	VERB
ejpam-6077	209	4	6	6	NUM
ejpam-6077	209	5	.	.	PUNCT
ejpam-6077	210	1	consider	consider	VERB
ejpam-6077	210	2	an	an	DET
ejpam-6077	210	3	iterative	iterative	NOUN
ejpam-6077	210	4	scheme	scheme	NOUN
ejpam-6077	210	5	represented	represent	VERB
ejpam-6077	210	6	by	by	ADP
ejpam-6077	210	7	the	the	DET
ejpam-6077	210	8	update	update	NOUN
ejpam-6077	210	9	rule	rule	NOUN
ejpam-6077	210	10	:	:	PUNCT
ejpam-6077	211	1	υn+1	υn+1	NUM
ejpam-6077	211	2	=	=	SYM
ejpam-6077	211	3	s(υn	s(υn	NOUN
ejpam-6077	211	4	)	)	PUNCT
ejpam-6077	211	5	,	,	PUNCT
ejpam-6077	211	6	where	where	SCONJ
ejpam-6077	211	7	s	s	VERB
ejpam-6077	211	8	:	:	PUNCT
ejpam-6077	211	9	x	x	SYM
ejpam-6077	211	10	→	→	PUNCT
ejpam-6077	211	11	x	x	X
ejpam-6077	211	12	is	be	AUX
ejpam-6077	211	13	a	a	DET
ejpam-6077	211	14	mapping	mapping	NOUN
ejpam-6077	211	15	on	on	ADP
ejpam-6077	211	16	a	a	DET
ejpam-6077	211	17	banach	banach	NOUN
ejpam-6077	211	18	space	space	NOUN
ejpam-6077	211	19	x	x	PUNCT
ejpam-6077	211	20	equipped	equip	VERB
ejpam-6077	211	21	with	with	ADP
ejpam-6077	211	22	a	a	DET
ejpam-6077	211	23	norm	norm	NOUN
ejpam-6077	211	24	∥	∥	X
ejpam-6077	211	25	·	·	PUNCT
ejpam-6077	211	26	∥.	∥.	ADP
ejpam-6077	212	1	the	the	DET
ejpam-6077	212	2	space	space	NOUN
ejpam-6077	212	3	x	x	PUNCT
ejpam-6077	212	4	is	be	AUX
ejpam-6077	212	5	further	far	ADV
ejpam-6077	212	6	structured	structure	VERB
ejpam-6077	212	7	with	with	ADP
ejpam-6077	212	8	an	an	DET
ejpam-6077	212	9	mr	mr	PROPN
ejpam-6077	212	10	-	-	PUNCT
ejpam-6077	212	11	metric	metric	NOUN
ejpam-6077	212	12	defined	define	VERB
ejpam-6077	212	13	as	as	ADP
ejpam-6077	212	14	:	:	PUNCT
ejpam-6077	212	15	m(υ	m(υ	PROPN
ejpam-6077	212	16	,	,	PUNCT
ejpam-6077	212	17	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	212	18	)	)	PUNCT
ejpam-6077	213	1	=	=	SYM
ejpam-6077	213	2	∥υ	∥υ	PROPN
ejpam-6077	213	3	−	−	PROPN
ejpam-6077	213	4	ξ∥+	ξ∥+	VERB
ejpam-6077	213	5	∥ξ	∥ξ	PROPN
ejpam-6077	213	6	−ℑ∥.	−ℑ∥.	PROPN
ejpam-6077	213	7	assumptions	assumption	NOUN
ejpam-6077	213	8	:	:	PUNCT
ejpam-6077	213	9	1	1	X
ejpam-6077	213	10	.	.	X
ejpam-6077	213	11	s	s	PART
ejpam-6077	213	12	satisfies	satisfie	NOUN
ejpam-6077	213	13	a	a	DET
ejpam-6077	213	14	contraction	contraction	NOUN
ejpam-6077	213	15	condition	condition	NOUN
ejpam-6077	213	16	in	in	ADP
ejpam-6077	213	17	the	the	DET
ejpam-6077	213	18	mr	mr	PROPN
ejpam-6077	213	19	-	-	PUNCT
ejpam-6077	213	20	metric	metric	ADJ
ejpam-6077	213	21	:	:	PUNCT
ejpam-6077	213	22	m(s(υ	m(s(υ	PROPN
ejpam-6077	213	23	)	)	PUNCT
ejpam-6077	213	24	,	,	PUNCT
ejpam-6077	213	25	s(ξ	s(ξ	PROPN
ejpam-6077	213	26	)	)	PUNCT
ejpam-6077	213	27	,	,	PUNCT
ejpam-6077	213	28	s(ℑ	s(ℑ	PROPN
ejpam-6077	213	29	)	)	PUNCT
ejpam-6077	213	30	)	)	PUNCT
ejpam-6077	213	31	≤	≤	PUNCT
ejpam-6077	214	1	k	k	X
ejpam-6077	214	2	·	·	PUNCT
ejpam-6077	214	3	m(υ	m(υ	PROPN
ejpam-6077	214	4	,	,	PUNCT
ejpam-6077	214	5	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	214	6	)	)	PUNCT
ejpam-6077	214	7	,	,	PUNCT
ejpam-6077	214	8	a.	a.	NOUN
ejpam-6077	214	9	malkawi	malkawi	PROPN
ejpam-6077	214	10	/	/	SYM
ejpam-6077	214	11	eur	eur	PROPN
ejpam-6077	214	12	.	.	PUNCT
ejpam-6077	215	1	j.	j.	PROPN
ejpam-6077	215	2	pure	pure	PROPN
ejpam-6077	215	3	appl	appl	PROPN
ejpam-6077	215	4	.	.	PROPN
ejpam-6077	215	5	math	math	PROPN
ejpam-6077	215	6	,	,	PUNCT
ejpam-6077	215	7	18	18	NUM
ejpam-6077	215	8	(	(	PUNCT
ejpam-6077	215	9	2	2	NUM
ejpam-6077	215	10	)	)	PUNCT
ejpam-6077	215	11	(	(	PUNCT
ejpam-6077	215	12	2025	2025	NUM
ejpam-6077	215	13	)	)	PUNCT
ejpam-6077	215	14	,	,	PUNCT
ejpam-6077	215	15	6077	6077	NUM
ejpam-6077	215	16	11	11	NUM
ejpam-6077	215	17	of	of	ADP
ejpam-6077	215	18	16	16	NUM
ejpam-6077	215	19	for	for	ADP
ejpam-6077	215	20	all	all	DET
ejpam-6077	215	21	υ	υ	PROPN
ejpam-6077	215	22	,	,	PUNCT
ejpam-6077	215	23	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	215	24	∈	∈	PROPN
ejpam-6077	215	25	x	x	X
ejpam-6077	215	26	,	,	PUNCT
ejpam-6077	215	27	where	where	SCONJ
ejpam-6077	215	28	k	k	PROPN
ejpam-6077	215	29	∈	∈	PROPN
ejpam-6077	216	1	[	[	X
ejpam-6077	216	2	0	0	NUM
ejpam-6077	216	3	,	,	PUNCT
ejpam-6077	216	4	1	1	NUM
ejpam-6077	216	5	)	)	PUNCT
ejpam-6077	216	6	is	be	AUX
ejpam-6077	216	7	a	a	DET
ejpam-6077	216	8	constant	constant	ADJ
ejpam-6077	216	9	.	.	PUNCT
ejpam-6077	217	1	2	2	X
ejpam-6077	217	2	.	.	X
ejpam-6077	217	3	the	the	DET
ejpam-6077	217	4	norm	norm	NOUN
ejpam-6077	217	5	∥	∥	X
ejpam-6077	217	6	·	·	PUNCT
ejpam-6077	217	7	∥	∥	PRON
ejpam-6077	217	8	ensures	ensure	VERB
ejpam-6077	217	9	that	that	SCONJ
ejpam-6077	217	10	x	x	PRON
ejpam-6077	217	11	is	be	AUX
ejpam-6077	217	12	a	a	DET
ejpam-6077	217	13	complete	complete	ADJ
ejpam-6077	217	14	metric	metric	ADJ
ejpam-6077	217	15	space	space	NOUN
ejpam-6077	217	16	.	.	PUNCT
ejpam-6077	218	1	step	step	NOUN
ejpam-6077	218	2	-	-	PUNCT
ejpam-6077	218	3	by	by	ADP
ejpam-6077	218	4	-	-	PUNCT
ejpam-6077	218	5	step	step	NOUN
ejpam-6077	218	6	analysis	analysis	NOUN
ejpam-6077	218	7	:	:	PUNCT
ejpam-6077	218	8	step	step	NOUN
ejpam-6077	218	9	1	1	NUM
ejpam-6077	218	10	:	:	PUNCT
ejpam-6077	218	11	understanding	understand	VERB
ejpam-6077	218	12	the	the	DET
ejpam-6077	218	13	mr	mr	PROPN
ejpam-6077	218	14	-	-	PUNCT
ejpam-6077	218	15	metric	metric	NOUN
ejpam-6077	218	16	the	the	DET
ejpam-6077	218	17	mr	mr	PROPN
ejpam-6077	218	18	-	-	PUNCT
ejpam-6077	218	19	metric	metric	NOUN
ejpam-6077	218	20	generalizes	generalize	VERB
ejpam-6077	218	21	the	the	DET
ejpam-6077	218	22	standard	standard	NOUN
ejpam-6077	218	23	metric	metric	ADJ
ejpam-6077	218	24	by	by	ADP
ejpam-6077	218	25	simultaneously	simultaneously	ADV
ejpam-6077	218	26	measuring	measure	VERB
ejpam-6077	218	27	the	the	DET
ejpam-6077	218	28	”	"	PUNCT
ejpam-6077	218	29	distances	distance	NOUN
ejpam-6077	218	30	”	"	PUNCT
ejpam-6077	218	31	between	between	ADP
ejpam-6077	218	32	three	three	NUM
ejpam-6077	218	33	elements	element	NOUN
ejpam-6077	218	34	.	.	PUNCT
ejpam-6077	219	1	in	in	ADP
ejpam-6077	219	2	this	this	DET
ejpam-6077	219	3	case	case	NOUN
ejpam-6077	219	4	:	:	PUNCT
ejpam-6077	219	5	m(υ	m(υ	PROPN
ejpam-6077	219	6	,	,	PUNCT
ejpam-6077	219	7	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	219	8	)	)	PUNCT
ejpam-6077	220	1	=	=	SYM
ejpam-6077	220	2	∥υ	∥υ	PROPN
ejpam-6077	220	3	−	−	PROPN
ejpam-6077	220	4	ξ∥+	ξ∥+	VERB
ejpam-6077	220	5	∥ξ	∥ξ	PROPN
ejpam-6077	220	6	−ℑ∥.	−ℑ∥.	PROPN
ejpam-6077	220	7	this	this	DET
ejpam-6077	220	8	metric	metric	NOUN
ejpam-6077	220	9	allows	allow	VERB
ejpam-6077	220	10	the	the	DET
ejpam-6077	220	11	analysis	analysis	NOUN
ejpam-6077	220	12	to	to	PART
ejpam-6077	220	13	track	track	VERB
ejpam-6077	220	14	how	how	SCONJ
ejpam-6077	220	15	the	the	DET
ejpam-6077	220	16	distances	distance	NOUN
ejpam-6077	220	17	between	between	ADP
ejpam-6077	220	18	successive	successive	ADJ
ejpam-6077	220	19	iterations	iteration	NOUN
ejpam-6077	220	20	evolve	evolve	VERB
ejpam-6077	220	21	in	in	ADP
ejpam-6077	220	22	a	a	DET
ejpam-6077	220	23	dynamic	dynamic	ADJ
ejpam-6077	220	24	system	system	NOUN
ejpam-6077	220	25	.	.	PUNCT
ejpam-6077	221	1	step	step	NOUN
ejpam-6077	221	2	2	2	NUM
ejpam-6077	221	3	:	:	PUNCT
ejpam-6077	221	4	contraction	contraction	NOUN
ejpam-6077	221	5	property	property	NOUN
ejpam-6077	221	6	of	of	ADP
ejpam-6077	221	7	s	s	PRON
ejpam-6077	221	8	for	for	ADP
ejpam-6077	221	9	any	any	DET
ejpam-6077	221	10	υ	υ	NOUN
ejpam-6077	221	11	,	,	PUNCT
ejpam-6077	221	12	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	221	13	∈	∈	PROPN
ejpam-6077	221	14	x	x	NOUN
ejpam-6077	221	15	,	,	PUNCT
ejpam-6077	221	16	the	the	DET
ejpam-6077	221	17	contraction	contraction	NOUN
ejpam-6077	221	18	condition	condition	NOUN
ejpam-6077	221	19	ensures	ensure	VERB
ejpam-6077	221	20	:	:	PUNCT
ejpam-6077	221	21	m(s(υ	m(s(υ	PROPN
ejpam-6077	221	22	)	)	PUNCT
ejpam-6077	221	23	,	,	PUNCT
ejpam-6077	221	24	s(ξ	s(ξ	PROPN
ejpam-6077	221	25	)	)	PUNCT
ejpam-6077	221	26	,	,	PUNCT
ejpam-6077	221	27	s(ℑ	s(ℑ	NUM
ejpam-6077	221	28	)	)	PUNCT
ejpam-6077	221	29	)	)	PUNCT
ejpam-6077	222	1	=	=	SYM
ejpam-6077	222	2	∥s(υ)−	∥s(υ)−	X
ejpam-6077	222	3	s(ξ)∥+	s(ξ)∥+	NOUN
ejpam-6077	222	4	∥s(ξ)−	∥s(ξ)−	ADV
ejpam-6077	222	5	s(ℑ)∥	s(ℑ)∥	ADJ
ejpam-6077	222	6	≤	≤	ADJ
ejpam-6077	222	7	k(∥υ	k(∥υ	NOUN
ejpam-6077	222	8	−	−	PROPN
ejpam-6077	222	9	ξ∥+	ξ∥+	PROPN
ejpam-6077	222	10	∥ξ	∥ξ	PROPN
ejpam-6077	222	11	−ℑ∥	−ℑ∥	NOUN
ejpam-6077	222	12	)	)	PUNCT
ejpam-6077	222	13	,	,	PUNCT
ejpam-6077	222	14	where	where	SCONJ
ejpam-6077	222	15	k	k	PROPN
ejpam-6077	222	16	∈	∈	PROPN
ejpam-6077	222	17	[	[	X
ejpam-6077	222	18	0	0	NUM
ejpam-6077	222	19	,	,	PUNCT
ejpam-6077	222	20	1	1	NUM
ejpam-6077	222	21	)	)	PUNCT
ejpam-6077	222	22	.	.	PUNCT
ejpam-6077	223	1	this	this	PRON
ejpam-6077	223	2	guarantees	guarantee	VERB
ejpam-6077	223	3	that	that	SCONJ
ejpam-6077	223	4	the	the	DET
ejpam-6077	223	5	operator	operator	NOUN
ejpam-6077	223	6	s	s	PART
ejpam-6077	223	7	”	"	PUNCT
ejpam-6077	223	8	brings	bring	VERB
ejpam-6077	223	9	points	point	NOUN
ejpam-6077	223	10	closer	close	ADV
ejpam-6077	223	11	together	together	ADV
ejpam-6077	223	12	”	"	PUNCT
ejpam-6077	223	13	under	under	ADP
ejpam-6077	223	14	the	the	DET
ejpam-6077	223	15	mr	mr	PROPN
ejpam-6077	223	16	-	-	PUNCT
ejpam-6077	223	17	metric	metric	NOUN
ejpam-6077	223	18	.	.	PUNCT
ejpam-6077	224	1	step	step	NOUN
ejpam-6077	224	2	3	3	NUM
ejpam-6077	224	3	:	:	PUNCT
ejpam-6077	224	4	iterative	iterative	NOUN
ejpam-6077	224	5	sequence	sequence	NOUN
ejpam-6077	224	6	start	start	VERB
ejpam-6077	224	7	with	with	ADP
ejpam-6077	224	8	an	an	DET
ejpam-6077	224	9	initial	initial	ADJ
ejpam-6077	224	10	guess	guess	NOUN
ejpam-6077	224	11	υ0	υ0	NOUN
ejpam-6077	224	12	∈	∈	PROPN
ejpam-6077	224	13	x	x	PUNCT
ejpam-6077	224	14	and	and	CCONJ
ejpam-6077	224	15	define	define	VERB
ejpam-6077	224	16	the	the	DET
ejpam-6077	224	17	sequence	sequence	NOUN
ejpam-6077	224	18	:	:	PUNCT
ejpam-6077	224	19	υn+1	υn+1	NUM
ejpam-6077	224	20	=	=	SYM
ejpam-6077	224	21	s(υn	s(υn	PROPN
ejpam-6077	224	22	)	)	PUNCT
ejpam-6077	224	23	,	,	PUNCT
ejpam-6077	224	24	n	n	X
ejpam-6077	224	25	≥	≥	NOUN
ejpam-6077	224	26	0	0	NUM
ejpam-6077	224	27	.	.	PUNCT
ejpam-6077	225	1	for	for	ADP
ejpam-6077	225	2	any	any	DET
ejpam-6077	225	3	n	n	CCONJ
ejpam-6077	225	4	,	,	PUNCT
ejpam-6077	225	5	consider	consider	VERB
ejpam-6077	225	6	three	three	NUM
ejpam-6077	225	7	successive	successive	ADJ
ejpam-6077	225	8	iterates	iterate	NOUN
ejpam-6077	225	9	υn−1	υn−1	PROPN
ejpam-6077	225	10	,	,	PUNCT
ejpam-6077	225	11	υn	υn	NOUN
ejpam-6077	225	12	,	,	PUNCT
ejpam-6077	225	13	υn+1	υn+1	NOUN
ejpam-6077	225	14	:	:	PUNCT
ejpam-6077	225	15	m(υn+1	m(υn+1	NUM
ejpam-6077	225	16	,	,	PUNCT
ejpam-6077	225	17	υn	υn	NOUN
ejpam-6077	225	18	,	,	PUNCT
ejpam-6077	225	19	υn−1	υn−1	PROPN
ejpam-6077	225	20	)	)	PUNCT
ejpam-6077	225	21	=	=	PUNCT
ejpam-6077	225	22	∥υn+1	∥υn+1	NOUN
ejpam-6077	225	23	−	−	PROPN
ejpam-6077	225	24	υn∥+	υn∥+	NOUN
ejpam-6077	225	25	∥υn	∥υn	CCONJ
ejpam-6077	225	26	−	−	NOUN
ejpam-6077	226	1	υn−1∥.	υn−1∥.	NOUN
ejpam-6077	226	2	using	use	VERB
ejpam-6077	226	3	the	the	DET
ejpam-6077	226	4	contraction	contraction	NOUN
ejpam-6077	226	5	property	property	NOUN
ejpam-6077	226	6	of	of	ADP
ejpam-6077	226	7	s	s	NOUN
ejpam-6077	226	8	:	:	PUNCT
ejpam-6077	226	9	m(υn+1	m(υn+1	NUM
ejpam-6077	226	10	,	,	PUNCT
ejpam-6077	226	11	υn	υn	NOUN
ejpam-6077	226	12	,	,	PUNCT
ejpam-6077	226	13	υn−1	υn−1	PROPN
ejpam-6077	226	14	)	)	PUNCT
ejpam-6077	226	15	≤	≤	PUNCT
ejpam-6077	227	1	k	k	X
ejpam-6077	227	2	·	·	SYM
ejpam-6077	227	3	m(υn	m(υn	ADJ
ejpam-6077	227	4	,	,	PUNCT
ejpam-6077	227	5	υn−1	υn−1	ADJ
ejpam-6077	227	6	,	,	PUNCT
ejpam-6077	227	7	υn−2	υn−2	PROPN
ejpam-6077	227	8	)	)	PUNCT
ejpam-6077	227	9	.	.	PUNCT
ejpam-6077	228	1	by	by	ADP
ejpam-6077	228	2	induction	induction	NOUN
ejpam-6077	228	3	:	:	PUNCT
ejpam-6077	228	4	m(υn+1	m(υn+1	NUM
ejpam-6077	228	5	,	,	PUNCT
ejpam-6077	228	6	υn	υn	NOUN
ejpam-6077	228	7	,	,	PUNCT
ejpam-6077	228	8	υn−1	υn−1	PROPN
ejpam-6077	228	9	)	)	PUNCT
ejpam-6077	228	10	≤	≤	NOUN
ejpam-6077	228	11	kn	kn	PROPN
ejpam-6077	228	12	·	·	SYM
ejpam-6077	228	13	m(υ1	m(υ1	ADJ
ejpam-6077	228	14	,	,	PUNCT
ejpam-6077	228	15	υ0	υ0	NOUN
ejpam-6077	228	16	,	,	PUNCT
ejpam-6077	228	17	υ−1	υ−1	PROPN
ejpam-6077	228	18	)	)	PUNCT
ejpam-6077	228	19	,	,	PUNCT
ejpam-6077	228	20	where	where	SCONJ
ejpam-6077	228	21	υ−1	υ−1	PROPN
ejpam-6077	228	22	can	can	AUX
ejpam-6077	228	23	be	be	AUX
ejpam-6077	228	24	taken	take	VERB
ejpam-6077	228	25	as	as	ADP
ejpam-6077	228	26	the	the	DET
ejpam-6077	228	27	initial	initial	ADJ
ejpam-6077	228	28	condition	condition	NOUN
ejpam-6077	228	29	.	.	PUNCT
ejpam-6077	229	1	step	step	NOUN
ejpam-6077	229	2	4	4	NUM
ejpam-6077	229	3	:	:	PUNCT
ejpam-6077	229	4	convergence	convergence	NOUN
ejpam-6077	229	5	to	to	ADP
ejpam-6077	229	6	the	the	DET
ejpam-6077	229	7	fixed	fix	VERB
ejpam-6077	229	8	point	point	NOUN
ejpam-6077	229	9	since	since	SCONJ
ejpam-6077	229	10	k	k	PROPN
ejpam-6077	229	11	∈	∈	PROPN
ejpam-6077	230	1	[	[	X
ejpam-6077	230	2	0	0	NUM
ejpam-6077	230	3	,	,	PUNCT
ejpam-6077	230	4	1	1	NUM
ejpam-6077	230	5	)	)	PUNCT
ejpam-6077	230	6	,	,	PUNCT
ejpam-6077	230	7	kn	kn	PROPN
ejpam-6077	230	8	→	→	SYM
ejpam-6077	230	9	0	0	PROPN
ejpam-6077	230	10	as	as	ADP
ejpam-6077	230	11	n	n	NOUN
ejpam-6077	230	12	→	→	SYM
ejpam-6077	230	13	∞.	∞.	PROPN
ejpam-6077	230	14	this	this	PRON
ejpam-6077	230	15	implies	imply	VERB
ejpam-6077	230	16	:	:	PUNCT
ejpam-6077	230	17	m(υn+1	m(υn+1	NUM
ejpam-6077	230	18	,	,	PUNCT
ejpam-6077	230	19	υn	υn	NOUN
ejpam-6077	230	20	,	,	PUNCT
ejpam-6077	230	21	υn−1	υn−1	PROPN
ejpam-6077	230	22	)	)	PUNCT
ejpam-6077	230	23	→	→	SYM
ejpam-6077	231	1	0	0	X
ejpam-6077	231	2	.	.	X
ejpam-6077	231	3	decomposing	decompose	VERB
ejpam-6077	231	4	m(υn+1	m(υn+1	NUM
ejpam-6077	231	5	,	,	PUNCT
ejpam-6077	231	6	υn	υn	NOUN
ejpam-6077	231	7	,	,	PUNCT
ejpam-6077	231	8	υn−1	υn−1	PROPN
ejpam-6077	231	9	)	)	PUNCT
ejpam-6077	231	10	,	,	PUNCT
ejpam-6077	231	11	we	we	PRON
ejpam-6077	231	12	observe	observe	VERB
ejpam-6077	231	13	that	that	SCONJ
ejpam-6077	231	14	:	:	PUNCT
ejpam-6077	231	15	∥υn+1	∥υn+1	VERB
ejpam-6077	231	16	−	−	PROPN
ejpam-6077	231	17	υn∥	υn∥	PROPN
ejpam-6077	231	18	→	→	SYM
ejpam-6077	231	19	0	0	NUM
ejpam-6077	231	20	and	and	CCONJ
ejpam-6077	231	21	∥υn	∥υn	NOUN
ejpam-6077	231	22	−	−	NOUN
ejpam-6077	231	23	υn−1∥	υn−1∥	NOUN
ejpam-6077	231	24	→	→	SYM
ejpam-6077	231	25	0	0	NUM
ejpam-6077	231	26	.	.	PUNCT
ejpam-6077	232	1	thus	thus	ADV
ejpam-6077	232	2	,	,	PUNCT
ejpam-6077	232	3	the	the	DET
ejpam-6077	232	4	sequence	sequence	NOUN
ejpam-6077	232	5	{	{	PUNCT
ejpam-6077	232	6	υn	υn	NOUN
ejpam-6077	232	7	}	}	PUNCT
ejpam-6077	232	8	is	be	AUX
ejpam-6077	232	9	cauchy	cauchy	ADJ
ejpam-6077	232	10	in	in	ADP
ejpam-6077	232	11	x.	x.	NOUN
ejpam-6077	232	12	since	since	SCONJ
ejpam-6077	232	13	x	x	PRON
ejpam-6077	232	14	is	be	AUX
ejpam-6077	232	15	a	a	DET
ejpam-6077	232	16	banach	banach	NOUN
ejpam-6077	232	17	space	space	NOUN
ejpam-6077	232	18	,	,	PUNCT
ejpam-6077	232	19	{	{	PUNCT
ejpam-6077	232	20	υn	υn	NOUN
ejpam-6077	232	21	}	}	PUNCT
ejpam-6077	232	22	converges	converge	VERB
ejpam-6077	232	23	to	to	ADP
ejpam-6077	232	24	a	a	DET
ejpam-6077	232	25	unique	unique	ADJ
ejpam-6077	232	26	point	point	NOUN
ejpam-6077	232	27	υ∗	υ∗	NOUN
ejpam-6077	232	28	∈	∈	PROPN
ejpam-6077	232	29	x.	x.	NOUN
ejpam-6077	232	30	step	step	NOUN
ejpam-6077	232	31	5	5	NUM
ejpam-6077	232	32	:	:	PUNCT
ejpam-6077	232	33	verification	verification	NOUN
ejpam-6077	232	34	of	of	ADP
ejpam-6077	232	35	the	the	DET
ejpam-6077	232	36	fixed	fix	VERB
ejpam-6077	232	37	point	point	NOUN
ejpam-6077	232	38	from	from	ADP
ejpam-6077	232	39	the	the	DET
ejpam-6077	232	40	continuity	continuity	NOUN
ejpam-6077	232	41	of	of	ADP
ejpam-6077	232	42	s	s	PROPN
ejpam-6077	232	43	,	,	PUNCT
ejpam-6077	232	44	the	the	DET
ejpam-6077	232	45	limit	limit	NOUN
ejpam-6077	232	46	υ∗	υ∗	NOUN
ejpam-6077	232	47	satisfies	satisfy	VERB
ejpam-6077	232	48	:	:	PUNCT
ejpam-6077	233	1	υ∗	υ∗	NOUN
ejpam-6077	233	2	=	=	SYM
ejpam-6077	233	3	lim	lim	PROPN
ejpam-6077	233	4	n→∞	n→∞	NUM
ejpam-6077	233	5	υn	υn	PROPN
ejpam-6077	233	6	=	=	SYM
ejpam-6077	233	7	lim	lim	PROPN
ejpam-6077	233	8	n→∞	n→∞	NUM
ejpam-6077	233	9	s(υn	s(υn	ADJ
ejpam-6077	233	10	)	)	PUNCT
ejpam-6077	233	11	=	=	SYM
ejpam-6077	233	12	s(υ∗	s(υ∗	NOUN
ejpam-6077	233	13	)	)	PUNCT
ejpam-6077	233	14	.	.	PUNCT
ejpam-6077	234	1	hence	hence	ADV
ejpam-6077	234	2	,	,	PUNCT
ejpam-6077	234	3	υ∗	υ∗	PROPN
ejpam-6077	234	4	is	be	AUX
ejpam-6077	234	5	the	the	DET
ejpam-6077	234	6	unique	unique	ADJ
ejpam-6077	234	7	fixed	fix	VERB
ejpam-6077	234	8	point	point	NOUN
ejpam-6077	234	9	of	of	ADP
ejpam-6077	234	10	s.	s.	PROPN
ejpam-6077	234	11	a.	a.	PROPN
ejpam-6077	234	12	malkawi	malkawi	PROPN
ejpam-6077	234	13	/	/	SYM
ejpam-6077	234	14	eur	eur	PROPN
ejpam-6077	234	15	.	.	PUNCT
ejpam-6077	235	1	j.	j.	PROPN
ejpam-6077	235	2	pure	pure	PROPN
ejpam-6077	235	3	appl	appl	PROPN
ejpam-6077	235	4	.	.	PROPN
ejpam-6077	235	5	math	math	PROPN
ejpam-6077	235	6	,	,	PUNCT
ejpam-6077	235	7	18	18	NUM
ejpam-6077	235	8	(	(	PUNCT
ejpam-6077	235	9	2	2	NUM
ejpam-6077	235	10	)	)	PUNCT
ejpam-6077	235	11	(	(	PUNCT
ejpam-6077	235	12	2025	2025	NUM
ejpam-6077	235	13	)	)	PUNCT
ejpam-6077	235	14	,	,	PUNCT
ejpam-6077	235	15	6077	6077	NUM
ejpam-6077	235	16	12	12	NUM
ejpam-6077	235	17	of	of	ADP
ejpam-6077	235	18	16	16	NUM
ejpam-6077	235	19	4	4	NUM
ejpam-6077	235	20	.	.	PUNCT
ejpam-6077	235	21	game	game	NOUN
ejpam-6077	235	22	theory	theory	NOUN
ejpam-6077	235	23	and	and	CCONJ
ejpam-6077	235	24	economic	economic	ADJ
ejpam-6077	235	25	equilibria	equilibrium	NOUN
ejpam-6077	235	26	example	example	VERB
ejpam-6077	235	27	7	7	NUM
ejpam-6077	235	28	.	.	X
ejpam-6077	235	29	in	in	ADP
ejpam-6077	235	30	game	game	NOUN
ejpam-6077	235	31	theory	theory	NOUN
ejpam-6077	235	32	,	,	PUNCT
ejpam-6077	235	33	consider	consider	VERB
ejpam-6077	235	34	a	a	DET
ejpam-6077	235	35	strategic	strategic	ADJ
ejpam-6077	235	36	game	game	NOUN
ejpam-6077	235	37	where	where	SCONJ
ejpam-6077	235	38	n	n	DET
ejpam-6077	235	39	players	player	NOUN
ejpam-6077	235	40	aim	aim	VERB
ejpam-6077	235	41	to	to	PART
ejpam-6077	235	42	optimize	optimize	VERB
ejpam-6077	235	43	their	their	PRON
ejpam-6077	235	44	individual	individual	ADJ
ejpam-6077	235	45	payoffs	payoff	NOUN
ejpam-6077	235	46	.	.	PUNCT
ejpam-6077	236	1	let	let	VERB
ejpam-6077	236	2	x	x	PUNCT
ejpam-6077	237	1	=	=	SYM
ejpam-6077	238	1	x1	x1	NUM
ejpam-6077	239	1	×	×	NOUN
ejpam-6077	240	1	x2	x2	NOUN
ejpam-6077	240	2	×	×	PROPN
ejpam-6077	240	3	·	·	PUNCT
ejpam-6077	240	4	·	·	PUNCT
ejpam-6077	240	5	·	·	PUNCT
ejpam-6077	241	1	×	×	NOUN
ejpam-6077	241	2	xn	xn	PROPN
ejpam-6077	241	3	represent	represent	VERB
ejpam-6077	241	4	the	the	DET
ejpam-6077	241	5	strategy	strategy	NOUN
ejpam-6077	241	6	space	space	NOUN
ejpam-6077	241	7	,	,	PUNCT
ejpam-6077	241	8	where	where	SCONJ
ejpam-6077	241	9	xi	xi	PROPN
ejpam-6077	241	10	is	be	AUX
ejpam-6077	241	11	the	the	DET
ejpam-6077	241	12	strategy	strategy	NOUN
ejpam-6077	241	13	set	set	NOUN
ejpam-6077	241	14	of	of	ADP
ejpam-6077	241	15	player	player	NOUN
ejpam-6077	241	16	i	i	PRON
ejpam-6077	241	17	,	,	PUNCT
ejpam-6077	241	18	assumed	assume	VERB
ejpam-6077	241	19	to	to	PART
ejpam-6077	241	20	be	be	AUX
ejpam-6077	241	21	a	a	DET
ejpam-6077	241	22	closed	closed	ADJ
ejpam-6077	241	23	,	,	PUNCT
ejpam-6077	241	24	bounded	bound	VERB
ejpam-6077	241	25	,	,	PUNCT
ejpam-6077	241	26	and	and	CCONJ
ejpam-6077	241	27	convex	convex	PROPN
ejpam-6077	241	28	subset	subset	NOUN
ejpam-6077	241	29	of	of	ADP
ejpam-6077	241	30	a	a	DET
ejpam-6077	241	31	banach	banach	NOUN
ejpam-6077	241	32	space	space	NOUN
ejpam-6077	241	33	.	.	PUNCT
ejpam-6077	242	1	the	the	DET
ejpam-6077	242	2	goal	goal	NOUN
ejpam-6077	242	3	is	be	AUX
ejpam-6077	242	4	to	to	PART
ejpam-6077	242	5	find	find	VERB
ejpam-6077	242	6	a	a	DET
ejpam-6077	242	7	nash	nash	ADJ
ejpam-6077	242	8	equilibrium	equilibrium	NOUN
ejpam-6077	242	9	υ∗	υ∗	NOUN
ejpam-6077	242	10	=	=	SYM
ejpam-6077	242	11	(	(	PUNCT
ejpam-6077	242	12	υ∗1	υ∗1	NOUN
ejpam-6077	242	13	,	,	PUNCT
ejpam-6077	242	14	υ	υ	PROPN
ejpam-6077	242	15	∗	∗	NOUN
ejpam-6077	242	16	2	2	NUM
ejpam-6077	242	17	,	,	PUNCT
ejpam-6077	242	18	.	.	PUNCT
ejpam-6077	242	19	.	.	PUNCT
ejpam-6077	243	1	.	.	PUNCT
ejpam-6077	244	1	,	,	PUNCT
ejpam-6077	244	2	υ	υ	NOUN
ejpam-6077	244	3	∗	∗	NOUN
ejpam-6077	244	4	n	n	NOUN
ejpam-6077	244	5	)	)	PUNCT
ejpam-6077	244	6	,	,	PUNCT
ejpam-6077	244	7	where	where	SCONJ
ejpam-6077	244	8	no	no	DET
ejpam-6077	244	9	player	player	NOUN
ejpam-6077	244	10	has	have	VERB
ejpam-6077	244	11	an	an	DET
ejpam-6077	244	12	incentive	incentive	NOUN
ejpam-6077	244	13	to	to	PART
ejpam-6077	244	14	unilaterally	unilaterally	ADV
ejpam-6077	244	15	deviate	deviate	VERB
ejpam-6077	244	16	.	.	PUNCT
ejpam-6077	245	1	best	good	ADJ
ejpam-6077	245	2	-	-	PUNCT
ejpam-6077	245	3	response	response	NOUN
ejpam-6077	245	4	mapping	mapping	NOUN
ejpam-6077	245	5	define	define	VERB
ejpam-6077	245	6	the	the	DET
ejpam-6077	245	7	best	good	ADJ
ejpam-6077	245	8	-	-	PUNCT
ejpam-6077	245	9	response	response	NOUN
ejpam-6077	245	10	operator	operator	NOUN
ejpam-6077	245	11	t	t	PROPN
ejpam-6077	245	12	as	as	ADP
ejpam-6077	245	13	:	:	PUNCT
ejpam-6077	245	14	s(υ	s(υ	PROPN
ejpam-6077	245	15	)	)	PUNCT
ejpam-6077	245	16	=	=	SYM
ejpam-6077	245	17	bestresponse(υ	bestresponse(υ	NOUN
ejpam-6077	245	18	)	)	PUNCT
ejpam-6077	245	19	=	=	SYM
ejpam-6077	245	20	(	(	PUNCT
ejpam-6077	245	21	s1(υ	s1(υ	PROPN
ejpam-6077	245	22	)	)	PUNCT
ejpam-6077	245	23	,	,	PUNCT
ejpam-6077	245	24	s2(υ	s2(υ	PROPN
ejpam-6077	245	25	)	)	PUNCT
ejpam-6077	245	26	,	,	PUNCT
ejpam-6077	245	27	.	.	PUNCT
ejpam-6077	245	28	.	.	PUNCT
ejpam-6077	246	1	.	.	PUNCT
ejpam-6077	247	1	,	,	PUNCT
ejpam-6077	247	2	sn	sn	INTJ
ejpam-6077	247	3	(	(	PUNCT
ejpam-6077	247	4	υ	υ	NOUN
ejpam-6077	247	5	)	)	PUNCT
ejpam-6077	247	6	)	)	PUNCT
ejpam-6077	247	7	,	,	PUNCT
ejpam-6077	247	8	where	where	SCONJ
ejpam-6077	247	9	si(υ	si(υ	NUM
ejpam-6077	247	10	)	)	PUNCT
ejpam-6077	247	11	is	be	AUX
ejpam-6077	247	12	the	the	DET
ejpam-6077	247	13	best	good	ADJ
ejpam-6077	247	14	-	-	PUNCT
ejpam-6077	247	15	response	response	NOUN
ejpam-6077	247	16	of	of	ADP
ejpam-6077	247	17	player	player	NOUN
ejpam-6077	247	18	i	i	PRON
ejpam-6077	247	19	given	give	VERB
ejpam-6077	247	20	the	the	DET
ejpam-6077	247	21	strategies	strategy	NOUN
ejpam-6077	247	22	of	of	ADP
ejpam-6077	247	23	all	all	DET
ejpam-6077	247	24	other	other	ADJ
ejpam-6077	247	25	players	player	NOUN
ejpam-6077	247	26	υ−i	υ−i	VERB
ejpam-6077	247	27	.	.	PUNCT
ejpam-6077	248	1	mathematically	mathematically	ADV
ejpam-6077	248	2	:	:	PUNCT
ejpam-6077	248	3	si(υ	si(υ	NUM
ejpam-6077	248	4	)	)	PUNCT
ejpam-6077	248	5	=	=	PUNCT
ejpam-6077	248	6	argmax	argmax	PROPN
ejpam-6077	248	7	ξ∈xi	ξ∈xi	NOUN
ejpam-6077	248	8	ui(ξ	ui(ξ	PROPN
ejpam-6077	248	9	,	,	PUNCT
ejpam-6077	249	1	υ−i	υ−i	PROPN
ejpam-6077	249	2	)	)	PUNCT
ejpam-6077	249	3	,	,	PUNCT
ejpam-6077	249	4	where	where	SCONJ
ejpam-6077	249	5	ui	ui	PROPN
ejpam-6077	249	6	is	be	AUX
ejpam-6077	249	7	the	the	DET
ejpam-6077	249	8	utility	utility	NOUN
ejpam-6077	249	9	function	function	NOUN
ejpam-6077	249	10	of	of	ADP
ejpam-6077	249	11	player	player	NOUN
ejpam-6077	249	12	i	i	PRON
ejpam-6077	249	13	,	,	PUNCT
ejpam-6077	249	14	and	and	CCONJ
ejpam-6077	249	15	υ−i	υ−i	PROPN
ejpam-6077	249	16	denotes	denote	VERB
ejpam-6077	249	17	the	the	DET
ejpam-6077	249	18	strategy	strategy	NOUN
ejpam-6077	249	19	profile	profile	NOUN
ejpam-6077	249	20	of	of	ADP
ejpam-6077	249	21	all	all	DET
ejpam-6077	249	22	players	player	NOUN
ejpam-6077	249	23	except	except	SCONJ
ejpam-6077	249	24	i.	i.	PROPN
ejpam-6077	249	25	mr	mr	PROPN
ejpam-6077	249	26	-	-	PUNCT
ejpam-6077	249	27	metric	metric	ADJ
ejpam-6077	249	28	equip	equip	NOUN
ejpam-6077	249	29	the	the	DET
ejpam-6077	249	30	strategy	strategy	NOUN
ejpam-6077	249	31	space	space	NOUN
ejpam-6077	249	32	x	x	PUNCT
ejpam-6077	249	33	with	with	ADP
ejpam-6077	249	34	the	the	DET
ejpam-6077	249	35	mr	mr	PROPN
ejpam-6077	249	36	-	-	PUNCT
ejpam-6077	249	37	metric	metric	NOUN
ejpam-6077	249	38	:	:	PUNCT
ejpam-6077	249	39	m(υ	m(υ	PROPN
ejpam-6077	249	40	,	,	PUNCT
ejpam-6077	249	41	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	249	42	)	)	PUNCT
ejpam-6077	250	1	=	=	SYM
ejpam-6077	250	2	∥υ	∥υ	PROPN
ejpam-6077	250	3	−	−	PROPN
ejpam-6077	250	4	ξ∥+	ξ∥+	VERB
ejpam-6077	250	5	∥ξ	∥ξ	PROPN
ejpam-6077	250	6	−ℑ∥	−ℑ∥	NOUN
ejpam-6077	250	7	,	,	PUNCT
ejpam-6077	250	8	where	where	SCONJ
ejpam-6077	250	9	∥	∥	X
ejpam-6077	250	10	·	·	PUNCT
ejpam-6077	251	1	∥	∥	NUM
ejpam-6077	251	2	is	be	AUX
ejpam-6077	251	3	a	a	DET
ejpam-6077	251	4	norm	norm	NOUN
ejpam-6077	251	5	defined	define	VERB
ejpam-6077	251	6	on	on	ADP
ejpam-6077	251	7	x.	x.	NOUN
ejpam-6077	251	8	this	this	DET
ejpam-6077	251	9	metric	metric	NOUN
ejpam-6077	251	10	is	be	AUX
ejpam-6077	251	11	particularly	particularly	ADV
ejpam-6077	251	12	useful	useful	ADJ
ejpam-6077	251	13	for	for	ADP
ejpam-6077	251	14	capturing	capture	VERB
ejpam-6077	251	15	the	the	DET
ejpam-6077	251	16	distances	distance	NOUN
ejpam-6077	251	17	between	between	ADP
ejpam-6077	251	18	successive	successive	ADJ
ejpam-6077	251	19	strategy	strategy	NOUN
ejpam-6077	251	20	profiles	profile	NOUN
ejpam-6077	251	21	υ	υ	PROPN
ejpam-6077	251	22	,	,	PUNCT
ejpam-6077	251	23	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	251	24	during	during	ADP
ejpam-6077	251	25	iterative	iterative	NOUN
ejpam-6077	251	26	updates	update	NOUN
ejpam-6077	251	27	.	.	PUNCT
ejpam-6077	252	1	assumptions	assumption	NOUN
ejpam-6077	252	2	and	and	CCONJ
ejpam-6077	252	3	contraction	contraction	NOUN
ejpam-6077	252	4	property	property	NOUN
ejpam-6077	252	5	1	1	NUM
ejpam-6077	252	6	.	.	PUNCT
ejpam-6077	253	1	convexity	convexity	NOUN
ejpam-6077	253	2	of	of	ADP
ejpam-6077	253	3	strategy	strategy	NOUN
ejpam-6077	253	4	sets	set	NOUN
ejpam-6077	253	5	:	:	PUNCT
ejpam-6077	253	6	the	the	DET
ejpam-6077	253	7	strategy	strategy	NOUN
ejpam-6077	253	8	sets	set	VERB
ejpam-6077	253	9	xi	xi	X
ejpam-6077	253	10	are	be	AUX
ejpam-6077	253	11	convex	convex	PROPN
ejpam-6077	253	12	,	,	PUNCT
ejpam-6077	253	13	ensuring	ensure	VERB
ejpam-6077	253	14	the	the	DET
ejpam-6077	253	15	existence	existence	NOUN
ejpam-6077	253	16	of	of	ADP
ejpam-6077	253	17	well	well	ADV
ejpam-6077	253	18	-	-	PUNCT
ejpam-6077	253	19	defined	define	VERB
ejpam-6077	253	20	best	good	ADJ
ejpam-6077	253	21	responses	response	NOUN
ejpam-6077	253	22	for	for	ADP
ejpam-6077	253	23	each	each	DET
ejpam-6077	253	24	player	player	NOUN
ejpam-6077	253	25	.	.	PUNCT
ejpam-6077	254	1	2	2	X
ejpam-6077	254	2	.	.	X
ejpam-6077	254	3	lipschitz	lipschitz	VERB
ejpam-6077	254	4	continuity	continuity	NOUN
ejpam-6077	254	5	of	of	ADP
ejpam-6077	254	6	s	s	PRON
ejpam-6077	254	7	:	:	PUNCT
ejpam-6077	254	8	assume	assume	VERB
ejpam-6077	254	9	that	that	SCONJ
ejpam-6077	254	10	s	s	AUX
ejpam-6077	254	11	satisfies	satisfie	NOUN
ejpam-6077	254	12	a	a	DET
ejpam-6077	254	13	lipschitz	lipschitz	ADJ
ejpam-6077	254	14	-	-	PUNCT
ejpam-6077	254	15	type	type	NOUN
ejpam-6077	254	16	condition	condition	NOUN
ejpam-6077	254	17	in	in	ADP
ejpam-6077	254	18	the	the	DET
ejpam-6077	254	19	mr	mr	PROPN
ejpam-6077	254	20	-	-	PUNCT
ejpam-6077	254	21	metric	metric	ADJ
ejpam-6077	254	22	:	:	PUNCT
ejpam-6077	254	23	m(s(υ	m(s(υ	PROPN
ejpam-6077	254	24	)	)	PUNCT
ejpam-6077	254	25	,	,	PUNCT
ejpam-6077	254	26	s(ξ	s(ξ	PROPN
ejpam-6077	254	27	)	)	PUNCT
ejpam-6077	254	28	,	,	PUNCT
ejpam-6077	254	29	s(ℑ	s(ℑ	PROPN
ejpam-6077	254	30	)	)	PUNCT
ejpam-6077	254	31	)	)	PUNCT
ejpam-6077	254	32	≤	≤	PUNCT
ejpam-6077	255	1	k	k	X
ejpam-6077	255	2	·	·	PUNCT
ejpam-6077	255	3	m(υ	m(υ	PROPN
ejpam-6077	255	4	,	,	PUNCT
ejpam-6077	255	5	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6077	255	6	)	)	PUNCT
ejpam-6077	255	7	,	,	PUNCT
ejpam-6077	255	8	for	for	ADP
ejpam-6077	255	9	some	some	DET
ejpam-6077	255	10	constant	constant	ADJ
ejpam-6077	255	11	k	k	PROPN
ejpam-6077	255	12	∈	∈	PROPN
ejpam-6077	256	1	[	[	X
ejpam-6077	256	2	0	0	NUM
ejpam-6077	256	3	,	,	PUNCT
ejpam-6077	256	4	1	1	NUM
ejpam-6077	256	5	)	)	PUNCT
ejpam-6077	256	6	.	.	PUNCT
ejpam-6077	257	1	3	3	X
ejpam-6077	257	2	.	.	X
ejpam-6077	257	3	existence	existence	NOUN
ejpam-6077	257	4	and	and	CCONJ
ejpam-6077	257	5	uniqueness	uniqueness	NOUN
ejpam-6077	257	6	:	:	PUNCT
ejpam-6077	257	7	by	by	ADP
ejpam-6077	257	8	the	the	DET
ejpam-6077	257	9	conditions	condition	NOUN
ejpam-6077	257	10	of	of	ADP
ejpam-6077	257	11	the	the	DET
ejpam-6077	257	12	mr	mr	PROPN
ejpam-6077	257	13	-	-	PUNCT
ejpam-6077	257	14	metric	metric	NOUN
ejpam-6077	257	15	theorem	theorem	NOUN
ejpam-6077	257	16	,	,	PUNCT
ejpam-6077	257	17	s	s	PART
ejpam-6077	257	18	has	have	VERB
ejpam-6077	257	19	a	a	DET
ejpam-6077	257	20	unique	unique	ADJ
ejpam-6077	257	21	fixed	fix	VERB
ejpam-6077	257	22	point	point	NOUN
ejpam-6077	257	23	υ∗	υ∗	NOUN
ejpam-6077	257	24	∈	∈	PROPN
ejpam-6077	257	25	x	x	X
ejpam-6077	257	26	,	,	PUNCT
ejpam-6077	257	27	ensuring	ensure	VERB
ejpam-6077	257	28	a	a	DET
ejpam-6077	257	29	unique	unique	ADJ
ejpam-6077	257	30	nash	nash	NOUN
ejpam-6077	257	31	equilibrium	equilibrium	NOUN
ejpam-6077	257	32	.	.	PUNCT
ejpam-6077	258	1	iterative	iterative	NOUN
ejpam-6077	258	2	process	process	NOUN
ejpam-6077	258	3	starting	start	VERB
ejpam-6077	258	4	from	from	ADP
ejpam-6077	258	5	an	an	DET
ejpam-6077	258	6	initial	initial	ADJ
ejpam-6077	258	7	strategy	strategy	NOUN
ejpam-6077	258	8	profile	profile	NOUN
ejpam-6077	258	9	υ0	υ0	PROPN
ejpam-6077	258	10	∈	∈	PROPN
ejpam-6077	258	11	x	x	NOUN
ejpam-6077	258	12	,	,	PUNCT
ejpam-6077	258	13	the	the	DET
ejpam-6077	258	14	sequence	sequence	NOUN
ejpam-6077	258	15	{	{	PUNCT
ejpam-6077	258	16	υn	υn	NOUN
ejpam-6077	258	17	}	}	PUNCT
ejpam-6077	258	18	is	be	AUX
ejpam-6077	258	19	generated	generated	AUX
ejpam-6077	258	20	iteratively	iteratively	ADV
ejpam-6077	258	21	using	use	VERB
ejpam-6077	258	22	:	:	PUNCT
ejpam-6077	258	23	υn+1	υn+1	NUM
ejpam-6077	258	24	=	=	SYM
ejpam-6077	258	25	s(υn	s(υn	NOUN
ejpam-6077	258	26	)	)	PUNCT
ejpam-6077	258	27	.	.	PUNCT
ejpam-6077	259	1	the	the	DET
ejpam-6077	259	2	contraction	contraction	NOUN
ejpam-6077	259	3	property	property	NOUN
ejpam-6077	259	4	of	of	ADP
ejpam-6077	259	5	s	s	NOUN
ejpam-6077	259	6	ensures	ensure	NOUN
ejpam-6077	259	7	that	that	SCONJ
ejpam-6077	259	8	{	{	PUNCT
ejpam-6077	259	9	υn	υn	NOUN
ejpam-6077	259	10	}	}	PUNCT
ejpam-6077	259	11	converges	converge	NOUN
ejpam-6077	259	12	to	to	ADP
ejpam-6077	259	13	the	the	DET
ejpam-6077	259	14	fixed	fixed	ADJ
ejpam-6077	259	15	point	point	NOUN
ejpam-6077	259	16	υ∗	υ∗	NOUN
ejpam-6077	259	17	,	,	PUNCT
ejpam-6077	259	18	which	which	PRON
ejpam-6077	259	19	is	be	AUX
ejpam-6077	259	20	the	the	DET
ejpam-6077	259	21	nash	nash	ADJ
ejpam-6077	259	22	equilibrium	equilibrium	NOUN
ejpam-6077	259	23	:	:	PUNCT
ejpam-6077	259	24	s(υ∗	s(υ∗	X
ejpam-6077	259	25	)	)	PUNCT
ejpam-6077	259	26	=	=	PRON
ejpam-6077	260	1	υ∗.	υ∗.	VERB
ejpam-6077	260	2	interpretation	interpretation	NOUN
ejpam-6077	260	3	of	of	ADP
ejpam-6077	260	4	the	the	DET
ejpam-6077	260	5	fixed	fix	VERB
ejpam-6077	260	6	point	point	NOUN
ejpam-6077	260	7	at	at	ADP
ejpam-6077	260	8	the	the	DET
ejpam-6077	260	9	nash	nash	ADJ
ejpam-6077	260	10	equilibrium	equilibrium	NOUN
ejpam-6077	260	11	υ∗	υ∗	NOUN
ejpam-6077	260	12	=	=	SYM
ejpam-6077	260	13	(	(	PUNCT
ejpam-6077	260	14	υ∗1	υ∗1	NOUN
ejpam-6077	260	15	,	,	PUNCT
ejpam-6077	260	16	υ	υ	PROPN
ejpam-6077	260	17	∗	∗	NOUN
ejpam-6077	260	18	2	2	NUM
ejpam-6077	260	19	,	,	PUNCT
ejpam-6077	260	20	.	.	PUNCT
ejpam-6077	260	21	.	.	PUNCT
ejpam-6077	261	1	.	.	PUNCT
ejpam-6077	262	1	,	,	PUNCT
ejpam-6077	262	2	υ	υ	NOUN
ejpam-6077	262	3	∗	∗	NOUN
ejpam-6077	262	4	n	n	NOUN
ejpam-6077	262	5	)	)	PUNCT
ejpam-6077	262	6	,	,	PUNCT
ejpam-6077	262	7	each	each	DET
ejpam-6077	262	8	player	player	NOUN
ejpam-6077	262	9	’s	’s	PART
ejpam-6077	262	10	strategy	strategy	NOUN
ejpam-6077	262	11	υ∗i	υ∗i	NUM
ejpam-6077	262	12	satisfies	satisfie	NOUN
ejpam-6077	262	13	:	:	PUNCT
ejpam-6077	262	14	ui(υ	ui(υ	NOUN
ejpam-6077	262	15	∗	∗	NOUN
ejpam-6077	263	1	i	i	PRON
ejpam-6077	263	2	,	,	PUNCT
ejpam-6077	263	3	υ	υ	PRON
ejpam-6077	263	4	∗	∗	NOUN
ejpam-6077	263	5	−i	−i	NOUN
ejpam-6077	263	6	)	)	PUNCT
ejpam-6077	263	7	≥	≥	NOUN
ejpam-6077	263	8	ui(ξ	ui(ξ	PROPN
ejpam-6077	263	9	,	,	PUNCT
ejpam-6077	263	10	υ	υ	PROPN
ejpam-6077	263	11	∗	∗	NOUN
ejpam-6077	263	12	−i	−i	NOUN
ejpam-6077	263	13	)	)	PUNCT
ejpam-6077	263	14	,	,	PUNCT
ejpam-6077	263	15	∀ξ	∀ξ	X
ejpam-6077	263	16	∈	∈	NOUN
ejpam-6077	263	17	xi	xi	NOUN
ejpam-6077	263	18	.	.	PUNCT
ejpam-6077	264	1	this	this	PRON
ejpam-6077	264	2	implies	imply	VERB
ejpam-6077	264	3	that	that	SCONJ
ejpam-6077	264	4	no	no	DET
ejpam-6077	264	5	player	player	NOUN
ejpam-6077	264	6	can	can	AUX
ejpam-6077	264	7	improve	improve	VERB
ejpam-6077	264	8	their	their	PRON
ejpam-6077	264	9	utility	utility	NOUN
ejpam-6077	264	10	by	by	ADP
ejpam-6077	264	11	unilaterally	unilaterally	ADV
ejpam-6077	264	12	changing	change	VERB
ejpam-6077	264	13	their	their	PRON
ejpam-6077	264	14	strategy	strategy	NOUN
ejpam-6077	264	15	.	.	PUNCT
ejpam-6077	265	1	a.	a.	NOUN
ejpam-6077	265	2	malkawi	malkawi	ADP
ejpam-6077	265	3	/	/	SYM
ejpam-6077	265	4	eur	eur	PROPN
ejpam-6077	265	5	.	.	PUNCT
ejpam-6077	266	1	j.	j.	PROPN
ejpam-6077	266	2	pure	pure	PROPN
ejpam-6077	266	3	appl	appl	PROPN
ejpam-6077	266	4	.	.	PROPN
ejpam-6077	266	5	math	math	PROPN
ejpam-6077	266	6	,	,	PUNCT
ejpam-6077	266	7	18	18	NUM
ejpam-6077	266	8	(	(	PUNCT
ejpam-6077	266	9	2	2	NUM
ejpam-6077	266	10	)	)	PUNCT
ejpam-6077	266	11	(	(	PUNCT
ejpam-6077	266	12	2025	2025	NUM
ejpam-6077	266	13	)	)	PUNCT
ejpam-6077	266	14	,	,	PUNCT
ejpam-6077	266	15	6077	6077	NUM
ejpam-6077	266	16	13	13	NUM
ejpam-6077	266	17	of	of	ADP
ejpam-6077	266	18	16	16	NUM
ejpam-6077	266	19	5	5	NUM
ejpam-6077	266	20	.	.	PUNCT
ejpam-6077	266	21	boundary	boundary	ADJ
ejpam-6077	266	22	value	value	NOUN
ejpam-6077	266	23	problems	problem	NOUN
ejpam-6077	266	24	example	example	VERB
ejpam-6077	266	25	8	8	NUM
ejpam-6077	266	26	.	.	PUNCT
ejpam-6077	267	1	consider	consider	VERB
ejpam-6077	267	2	the	the	DET
ejpam-6077	267	3	boundary	boundary	ADJ
ejpam-6077	267	4	value	value	NOUN
ejpam-6077	267	5	problem	problem	NOUN
ejpam-6077	267	6	(	(	PUNCT
ejpam-6077	267	7	bvp	bvp	PROPN
ejpam-6077	267	8	):	):	PUNCT
ejpam-6077	267	9	−u′′(υ	−u′′(υ	PROPN
ejpam-6077	267	10	)	)	PUNCT
ejpam-6077	267	11	=	=	SYM
ejpam-6077	267	12	f(υ	f(υ	PROPN
ejpam-6077	267	13	,	,	PUNCT
ejpam-6077	267	14	u(υ	u(υ	NUM
ejpam-6077	267	15	)	)	PUNCT
ejpam-6077	267	16	)	)	PUNCT
ejpam-6077	267	17	,	,	PUNCT
ejpam-6077	267	18	υ	υ	PROPN
ejpam-6077	267	19	∈	∈	PROPN
ejpam-6077	267	20	(	(	PUNCT
ejpam-6077	267	21	0	0	NUM
ejpam-6077	267	22	,	,	PUNCT
ejpam-6077	267	23	1	1	NUM
ejpam-6077	267	24	)	)	PUNCT
ejpam-6077	267	25	,	,	PUNCT
ejpam-6077	267	26	u(0	u(0	PROPN
ejpam-6077	267	27	)	)	PUNCT
ejpam-6077	267	28	=	=	SYM
ejpam-6077	268	1	u(1	u(1	PROPN
ejpam-6077	268	2	)	)	PUNCT
ejpam-6077	268	3	=	=	SYM
ejpam-6077	269	1	0	0	X
ejpam-6077	269	2	.	.	PUNCT
ejpam-6077	270	1	this	this	DET
ejpam-6077	270	2	type	type	NOUN
ejpam-6077	270	3	of	of	ADP
ejpam-6077	270	4	problem	problem	NOUN
ejpam-6077	270	5	often	often	ADV
ejpam-6077	270	6	arises	arise	VERB
ejpam-6077	270	7	in	in	ADP
ejpam-6077	270	8	physics	physics	NOUN
ejpam-6077	270	9	and	and	CCONJ
ejpam-6077	270	10	engineering	engineering	NOUN
ejpam-6077	270	11	,	,	PUNCT
ejpam-6077	270	12	such	such	ADJ
ejpam-6077	270	13	as	as	ADP
ejpam-6077	270	14	in	in	ADP
ejpam-6077	270	15	heat	heat	NOUN
ejpam-6077	270	16	conduction	conduction	NOUN
ejpam-6077	270	17	,	,	PUNCT
ejpam-6077	270	18	elastic	elastic	ADJ
ejpam-6077	270	19	deformation	deformation	NOUN
ejpam-6077	270	20	,	,	PUNCT
ejpam-6077	270	21	and	and	CCONJ
ejpam-6077	270	22	electrostatics	electrostatic	NOUN
ejpam-6077	270	23	.	.	PUNCT
ejpam-6077	271	1	function	function	NOUN
ejpam-6077	271	2	space	space	NOUN
ejpam-6077	271	3	define	define	VERB
ejpam-6077	271	4	the	the	DET
ejpam-6077	271	5	function	function	NOUN
ejpam-6077	271	6	space	space	NOUN
ejpam-6077	271	7	x	x	PUNCT
ejpam-6077	271	8	=	=	SYM
ejpam-6077	271	9	c([0	c([0	PROPN
ejpam-6077	271	10	,	,	PUNCT
ejpam-6077	271	11	1	1	NUM
ejpam-6077	271	12	]	]	NUM
ejpam-6077	271	13	)	)	PUNCT
ejpam-6077	271	14	,	,	PUNCT
ejpam-6077	271	15	the	the	DET
ejpam-6077	271	16	space	space	NOUN
ejpam-6077	271	17	of	of	ADP
ejpam-6077	271	18	continuous	continuous	ADJ
ejpam-6077	271	19	functions	function	NOUN
ejpam-6077	271	20	on	on	ADP
ejpam-6077	271	21	the	the	DET
ejpam-6077	271	22	interval	interval	NOUN
ejpam-6077	271	23	[	[	X
ejpam-6077	271	24	0	0	NUM
ejpam-6077	271	25	,	,	PUNCT
ejpam-6077	271	26	1	1	NUM
ejpam-6077	271	27	]	]	PUNCT
ejpam-6077	271	28	,	,	PUNCT
ejpam-6077	271	29	equipped	equip	VERB
ejpam-6077	271	30	with	with	ADP
ejpam-6077	271	31	the	the	DET
ejpam-6077	271	32	mr	mr	PROPN
ejpam-6077	271	33	-	-	PUNCT
ejpam-6077	271	34	metric	metric	NOUN
ejpam-6077	271	35	:	:	PUNCT
ejpam-6077	271	36	m(u	m(u	PROPN
ejpam-6077	271	37	,	,	PUNCT
ejpam-6077	271	38	v	v	NOUN
ejpam-6077	271	39	,	,	PUNCT
ejpam-6077	271	40	w	w	NOUN
ejpam-6077	271	41	)	)	PUNCT
ejpam-6077	271	42	=	=	SYM
ejpam-6077	271	43	∥u−	∥u−	NUM
ejpam-6077	271	44	v∥∞	v∥∞	NOUN
ejpam-6077	271	45	+	+	PROPN
ejpam-6077	271	46	∥v	∥v	PROPN
ejpam-6077	271	47	−	−	PROPN
ejpam-6077	271	48	w∥∞	w∥∞	PROPN
ejpam-6077	271	49	,	,	PUNCT
ejpam-6077	271	50	where	where	SCONJ
ejpam-6077	271	51	∥u−	∥u−	PROPN
ejpam-6077	271	52	v∥∞	v∥∞	NOUN
ejpam-6077	271	53	=	=	SYM
ejpam-6077	271	54	supυ∈[0,1	supυ∈[0,1	PROPN
ejpam-6077	271	55	]	]	X
ejpam-6077	271	56	|u(υ)−	|u(υ)−	X
ejpam-6077	271	57	v(υ)|	v(υ)|	NOUN
ejpam-6077	271	58	is	be	AUX
ejpam-6077	271	59	the	the	DET
ejpam-6077	271	60	supremum	supremum	ADJ
ejpam-6077	271	61	norm	norm	NOUN
ejpam-6077	271	62	.	.	PUNCT
ejpam-6077	272	1	integral	integral	ADJ
ejpam-6077	272	2	operator	operator	NOUN
ejpam-6077	272	3	representation	representation	NOUN
ejpam-6077	272	4	the	the	DET
ejpam-6077	272	5	solution	solution	NOUN
ejpam-6077	272	6	of	of	ADP
ejpam-6077	272	7	the	the	DET
ejpam-6077	272	8	boundary	boundary	ADJ
ejpam-6077	272	9	value	value	NOUN
ejpam-6077	272	10	problem	problem	NOUN
ejpam-6077	272	11	can	can	AUX
ejpam-6077	272	12	be	be	AUX
ejpam-6077	272	13	represented	represent	VERB
ejpam-6077	272	14	using	use	VERB
ejpam-6077	272	15	an	an	DET
ejpam-6077	272	16	integral	integral	ADJ
ejpam-6077	272	17	operator	operator	NOUN
ejpam-6077	272	18	s	s	NOUN
ejpam-6077	272	19	,	,	PUNCT
ejpam-6077	272	20	defined	define	VERB
ejpam-6077	272	21	as	as	ADP
ejpam-6077	272	22	:	:	PUNCT
ejpam-6077	272	23	(	(	PUNCT
ejpam-6077	272	24	su)(υ	su)(υ	PROPN
ejpam-6077	272	25	)	)	PUNCT
ejpam-6077	273	1	=	=	PUNCT
ejpam-6077	273	2	∫	∫	PROPN
ejpam-6077	273	3	1	1	NUM
ejpam-6077	273	4	0	0	NUM
ejpam-6077	274	1	g(υ	g(υ	ADJ
ejpam-6077	274	2	,	,	PUNCT
ejpam-6077	274	3	s)f(s	s)f(	NOUN
ejpam-6077	274	4	,	,	PUNCT
ejpam-6077	274	5	u(s	u(s	NUM
ejpam-6077	274	6	)	)	PUNCT
ejpam-6077	274	7	)	)	PUNCT
ejpam-6077	274	8	ds	ds	PROPN
ejpam-6077	274	9	,	,	PUNCT
ejpam-6077	274	10	where	where	SCONJ
ejpam-6077	274	11	g(υ	g(υ	NOUN
ejpam-6077	274	12	,	,	PUNCT
ejpam-6077	274	13	s	s	PART
ejpam-6077	274	14	)	)	PUNCT
ejpam-6077	274	15	is	be	AUX
ejpam-6077	274	16	the	the	DET
ejpam-6077	274	17	green	green	PROPN
ejpam-6077	274	18	’s	’s	PART
ejpam-6077	274	19	function	function	NOUN
ejpam-6077	274	20	for	for	ADP
ejpam-6077	274	21	the	the	DET
ejpam-6077	274	22	bvp	bvp	NOUN
ejpam-6077	274	23	:	:	PUNCT
ejpam-6077	274	24	g(υ	g(υ	VERB
ejpam-6077	274	25	,	,	PUNCT
ejpam-6077	274	26	s	s	X
ejpam-6077	274	27	)	)	PUNCT
ejpam-6077	274	28	=	=	SYM
ejpam-6077	274	29	{	{	PUNCT
ejpam-6077	274	30	s(1−	s(1−	PROPN
ejpam-6077	274	31	υ	υ	NOUN
ejpam-6077	274	32	)	)	PUNCT
ejpam-6077	274	33	,	,	PUNCT
ejpam-6077	274	34	if	if	SCONJ
ejpam-6077	274	35	s	s	VERB
ejpam-6077	274	36	≤	≤	NUM
ejpam-6077	274	37	υ	υ	NOUN
ejpam-6077	274	38	,	,	PUNCT
ejpam-6077	274	39	υ(1−	υ(1−	PROPN
ejpam-6077	274	40	s	s	PART
ejpam-6077	274	41	)	)	PUNCT
ejpam-6077	274	42	,	,	PUNCT
ejpam-6077	274	43	if	if	SCONJ
ejpam-6077	274	44	s	s	VERB
ejpam-6077	274	45	>	>	X
ejpam-6077	274	46	υ	υ	PROPN
ejpam-6077	274	47	.	.	PUNCT
ejpam-6077	275	1	the	the	DET
ejpam-6077	275	2	green	green	PROPN
ejpam-6077	275	3	’s	’s	PART
ejpam-6077	275	4	function	function	NOUN
ejpam-6077	275	5	satisfies	satisfy	VERB
ejpam-6077	275	6	the	the	DET
ejpam-6077	275	7	boundary	boundary	ADJ
ejpam-6077	275	8	conditions	condition	NOUN
ejpam-6077	275	9	u(0	u(0	NOUN
ejpam-6077	275	10	)	)	PUNCT
ejpam-6077	275	11	=	=	SYM
ejpam-6077	276	1	u(1	u(1	PROPN
ejpam-6077	276	2	)	)	PUNCT
ejpam-6077	276	3	=	=	SYM
ejpam-6077	276	4	0	0	NUM
ejpam-6077	276	5	and	and	CCONJ
ejpam-6077	276	6	accounts	account	VERB
ejpam-6077	276	7	for	for	ADP
ejpam-6077	276	8	the	the	DET
ejpam-6077	276	9	second	second	ADJ
ejpam-6077	276	10	-	-	PUNCT
ejpam-6077	276	11	order	order	NOUN
ejpam-6077	276	12	differential	differential	ADJ
ejpam-6077	276	13	operator	operator	NOUN
ejpam-6077	276	14	.	.	PUNCT
ejpam-6077	277	1	assumptions	assumption	NOUN
ejpam-6077	277	2	for	for	ADP
ejpam-6077	277	3	f(υ	f(υ	PROPN
ejpam-6077	277	4	,	,	PUNCT
ejpam-6077	277	5	u	u	NOUN
ejpam-6077	277	6	)	)	PUNCT
ejpam-6077	277	7	1	1	NUM
ejpam-6077	277	8	.	.	X
ejpam-6077	278	1	continuity	continuity	NOUN
ejpam-6077	278	2	:	:	PUNCT
ejpam-6077	278	3	the	the	DET
ejpam-6077	278	4	function	function	NOUN
ejpam-6077	278	5	f(υ	f(υ	PROPN
ejpam-6077	278	6	,	,	PUNCT
ejpam-6077	278	7	u	u	NOUN
ejpam-6077	278	8	)	)	PUNCT
ejpam-6077	278	9	is	be	AUX
ejpam-6077	278	10	continuous	continuous	ADJ
ejpam-6077	278	11	in	in	ADP
ejpam-6077	278	12	both	both	CCONJ
ejpam-6077	278	13	υ	υ	NOUN
ejpam-6077	278	14	and	and	CCONJ
ejpam-6077	278	15	u	u	NOUN
ejpam-6077	278	16	,	,	PUNCT
ejpam-6077	278	17	ensuring	ensure	VERB
ejpam-6077	278	18	the	the	DET
ejpam-6077	278	19	well	well	NOUN
ejpam-6077	278	20	-	-	PUNCT
ejpam-6077	278	21	posedness	posedness	NOUN
ejpam-6077	278	22	of	of	ADP
ejpam-6077	278	23	the	the	DET
ejpam-6077	278	24	integral	integral	ADJ
ejpam-6077	278	25	operator	operator	NOUN
ejpam-6077	278	26	s.	s.	PROPN
ejpam-6077	278	27	2	2	NUM
ejpam-6077	278	28	.	.	PUNCT
ejpam-6077	278	29	lipschitz	lipschitz	NOUN
ejpam-6077	278	30	condition	condition	NOUN
ejpam-6077	278	31	:	:	PUNCT
ejpam-6077	278	32	there	there	PRON
ejpam-6077	278	33	exists	exist	VERB
ejpam-6077	278	34	a	a	DET
ejpam-6077	278	35	constant	constant	ADJ
ejpam-6077	278	36	l	l	NOUN
ejpam-6077	278	37	>	>	X
ejpam-6077	278	38	0	0	NUM
ejpam-6077	279	1	such	such	ADJ
ejpam-6077	279	2	that	that	SCONJ
ejpam-6077	279	3	:	:	PUNCT
ejpam-6077	279	4	|f(υ	|f(υ	PROPN
ejpam-6077	279	5	,	,	PUNCT
ejpam-6077	279	6	u1)−	u1)−	PROPN
ejpam-6077	279	7	f(υ	f(υ	PROPN
ejpam-6077	279	8	,	,	PUNCT
ejpam-6077	279	9	u2)|	u2)|	NOUN
ejpam-6077	279	10	≤	≤	NOUN
ejpam-6077	279	11	l|u1	l|u1	VERB
ejpam-6077	279	12	−	−	NOUN
ejpam-6077	279	13	u2|	u2|	ADJ
ejpam-6077	279	14	,	,	PUNCT
ejpam-6077	279	15	∀υ	∀υ	X
ejpam-6077	279	16	∈	∈	PROPN
ejpam-6077	279	17	[	[	X
ejpam-6077	279	18	0	0	NUM
ejpam-6077	279	19	,	,	PUNCT
ejpam-6077	279	20	1	1	NUM
ejpam-6077	279	21	]	]	PUNCT
ejpam-6077	279	22	,	,	PUNCT
ejpam-6077	279	23	u1	u1	NOUN
ejpam-6077	279	24	,	,	PUNCT
ejpam-6077	279	25	u2	u2	PROPN
ejpam-6077	279	26	∈	∈	PROPN
ejpam-6077	279	27	r.	r.	NOUN
ejpam-6077	279	28	this	this	DET
ejpam-6077	279	29	condition	condition	NOUN
ejpam-6077	279	30	ensures	ensure	VERB
ejpam-6077	279	31	that	that	SCONJ
ejpam-6077	279	32	s	s	VERB
ejpam-6077	279	33	is	be	AUX
ejpam-6077	279	34	a	a	DET
ejpam-6077	279	35	contraction	contraction	NOUN
ejpam-6077	279	36	mapping	mapping	NOUN
ejpam-6077	279	37	in	in	ADP
ejpam-6077	279	38	the	the	DET
ejpam-6077	279	39	mr	mr	PROPN
ejpam-6077	279	40	-	-	PUNCT
ejpam-6077	279	41	metric	metric	NOUN
ejpam-6077	279	42	.	.	PUNCT
ejpam-6077	280	1	contraction	contraction	NOUN
ejpam-6077	280	2	property	property	NOUN
ejpam-6077	280	3	in	in	ADP
ejpam-6077	280	4	the	the	DET
ejpam-6077	280	5	mr	mr	PROPN
ejpam-6077	280	6	-	-	PUNCT
ejpam-6077	280	7	metric	metric	NOUN
ejpam-6077	280	8	let	let	VERB
ejpam-6077	280	9	u	u	NOUN
ejpam-6077	280	10	,	,	PUNCT
ejpam-6077	280	11	v	v	NOUN
ejpam-6077	280	12	,	,	PUNCT
ejpam-6077	280	13	w	w	PROPN
ejpam-6077	280	14	∈	∈	PROPN
ejpam-6077	280	15	x.	x.	NOUN
ejpam-6077	280	16	for	for	ADP
ejpam-6077	280	17	the	the	DET
ejpam-6077	280	18	operator	operator	NOUN
ejpam-6077	280	19	s	s	PART
ejpam-6077	280	20	,	,	PUNCT
ejpam-6077	280	21	we	we	PRON
ejpam-6077	280	22	have	have	VERB
ejpam-6077	280	23	:	:	PUNCT
ejpam-6077	280	24	∥s(u)−	∥s(u)−	NOUN
ejpam-6077	280	25	s(v)∥∞	s(v)∥∞	NOUN
ejpam-6077	280	26	=	=	SYM
ejpam-6077	280	27	sup	sup	PROPN
ejpam-6077	280	28	υ∈[0,1	υ∈[0,1	PROPN
ejpam-6077	280	29	]	]	X
ejpam-6077	280	30	∣∣∣∣∫	∣∣∣∣∫	PRON
ejpam-6077	280	31	1	1	NUM
ejpam-6077	280	32	0	0	NUM
ejpam-6077	281	1	g(υ	g(υ	VERB
ejpam-6077	281	2	,	,	PUNCT
ejpam-6077	281	3	s	s	AUX
ejpam-6077	281	4	)	)	PUNCT
ejpam-6077	281	5	(	(	PUNCT
ejpam-6077	281	6	f(s	f(s	ADV
ejpam-6077	281	7	,	,	PUNCT
ejpam-6077	281	8	u(s))−	u(s))−	ADJ
ejpam-6077	281	9	f(s	f(	NOUN
ejpam-6077	281	10	,	,	PUNCT
ejpam-6077	281	11	v(s	v(s	PROPN
ejpam-6077	281	12	)	)	PUNCT
ejpam-6077	281	13	)	)	PUNCT
ejpam-6077	281	14	)	)	PUNCT
ejpam-6077	282	1	ds	ds	ADJ
ejpam-6077	282	2	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6077	282	3	.	.	PUNCT
ejpam-6077	283	1	using	use	VERB
ejpam-6077	283	2	the	the	DET
ejpam-6077	283	3	lipschitz	lipschitz	NOUN
ejpam-6077	283	4	condition	condition	NOUN
ejpam-6077	283	5	for	for	ADP
ejpam-6077	283	6	f	f	PROPN
ejpam-6077	283	7	:	:	PUNCT
ejpam-6077	283	8	∥s(u)−	∥s(u)−	NOUN
ejpam-6077	283	9	s(v)∥∞	s(v)∥∞	NOUN
ejpam-6077	283	10	≤	≤	NUM
ejpam-6077	283	11	l	l	NOUN
ejpam-6077	283	12	sup	sup	NOUN
ejpam-6077	283	13	υ∈[0,1	υ∈[0,1	NOUN
ejpam-6077	283	14	]	]	X
ejpam-6077	283	15	∫	∫	PROPN
ejpam-6077	283	16	1	1	NUM
ejpam-6077	283	17	0	0	NUM
ejpam-6077	283	18	|g(υ	|g(υ	PROPN
ejpam-6077	283	19	,	,	PUNCT
ejpam-6077	283	20	s)|	s)|	NOUN
ejpam-6077	283	21	ds	ds	ADJ
ejpam-6077	283	22	·	·	PUNCT
ejpam-6077	283	23	∥u−	∥u−	NUM
ejpam-6077	283	24	v∥∞.	v∥∞.	NOUN
ejpam-6077	283	25	the	the	DET
ejpam-6077	283	26	boundedness	boundedness	NOUN
ejpam-6077	283	27	of	of	ADP
ejpam-6077	283	28	g(υ	g(υ	PROPN
ejpam-6077	283	29	,	,	PUNCT
ejpam-6077	283	30	s	s	PART
ejpam-6077	283	31	)	)	PUNCT
ejpam-6077	283	32	ensures	ensure	VERB
ejpam-6077	283	33	that	that	SCONJ
ejpam-6077	283	34	the	the	DET
ejpam-6077	283	35	contraction	contraction	NOUN
ejpam-6077	283	36	property	property	NOUN
ejpam-6077	283	37	is	be	AUX
ejpam-6077	283	38	satisfied	satisfied	ADJ
ejpam-6077	283	39	.	.	PUNCT
ejpam-6077	284	1	similarly	similarly	ADV
ejpam-6077	284	2	,	,	PUNCT
ejpam-6077	284	3	for	for	ADP
ejpam-6077	284	4	the	the	DET
ejpam-6077	284	5	mr	mr	PROPN
ejpam-6077	284	6	-	-	PUNCT
ejpam-6077	284	7	metric	metric	ADJ
ejpam-6077	284	8	:	:	PUNCT
ejpam-6077	284	9	m(s(u	m(s(u	NOUN
ejpam-6077	284	10	)	)	PUNCT
ejpam-6077	284	11	,	,	PUNCT
ejpam-6077	284	12	s(v	s(v	PROPN
ejpam-6077	284	13	)	)	PUNCT
ejpam-6077	284	14	,	,	PUNCT
ejpam-6077	284	15	s(w	s(w	NOUN
ejpam-6077	284	16	)	)	PUNCT
ejpam-6077	284	17	)	)	PUNCT
ejpam-6077	284	18	=	=	NOUN
ejpam-6077	284	19	∥s(u)−	∥s(u)−	NOUN
ejpam-6077	284	20	s(v)∥∞	s(v)∥∞	AUX
ejpam-6077	284	21	+	+	CCONJ
ejpam-6077	284	22	∥s(v)−	∥s(v)−	NUM
ejpam-6077	284	23	s(w)∥∞.	s(w)∥∞.	PUNCT
ejpam-6077	284	24	a.	a.	NOUN
ejpam-6077	284	25	malkawi	malkawi	PROPN
ejpam-6077	284	26	/	/	SYM
ejpam-6077	284	27	eur	eur	PROPN
ejpam-6077	284	28	.	.	PUNCT
ejpam-6077	285	1	j.	j.	PROPN
ejpam-6077	285	2	pure	pure	PROPN
ejpam-6077	285	3	appl	appl	PROPN
ejpam-6077	285	4	.	.	PROPN
ejpam-6077	285	5	math	math	PROPN
ejpam-6077	285	6	,	,	PUNCT
ejpam-6077	285	7	18	18	NUM
ejpam-6077	285	8	(	(	PUNCT
ejpam-6077	285	9	2	2	NUM
ejpam-6077	285	10	)	)	PUNCT
ejpam-6077	285	11	(	(	PUNCT
ejpam-6077	285	12	2025	2025	NUM
ejpam-6077	285	13	)	)	PUNCT
ejpam-6077	285	14	,	,	PUNCT
ejpam-6077	285	15	6077	6077	NUM
ejpam-6077	285	16	14	14	NUM
ejpam-6077	285	17	of	of	ADP
ejpam-6077	285	18	16	16	NUM
ejpam-6077	285	19	existence	existence	NOUN
ejpam-6077	285	20	and	and	CCONJ
ejpam-6077	285	21	uniqueness	uniqueness	NOUN
ejpam-6077	285	22	by	by	ADP
ejpam-6077	285	23	the	the	DET
ejpam-6077	285	24	mr	mr	PROPN
ejpam-6077	285	25	-	-	PUNCT
ejpam-6077	285	26	metric	metric	ADJ
ejpam-6077	285	27	fixed	fix	VERB
ejpam-6077	285	28	-	-	PUNCT
ejpam-6077	285	29	point	point	NOUN
ejpam-6077	285	30	theorem	theorem	NOUN
ejpam-6077	285	31	,	,	PUNCT
ejpam-6077	285	32	the	the	DET
ejpam-6077	285	33	contraction	contraction	NOUN
ejpam-6077	285	34	property	property	NOUN
ejpam-6077	285	35	of	of	ADP
ejpam-6077	285	36	s	s	NOUN
ejpam-6077	285	37	guarantees	guarantee	NOUN
ejpam-6077	285	38	that	that	SCONJ
ejpam-6077	285	39	s	s	AUX
ejpam-6077	285	40	has	have	VERB
ejpam-6077	285	41	a	a	DET
ejpam-6077	285	42	unique	unique	ADJ
ejpam-6077	285	43	fixed	fix	VERB
ejpam-6077	285	44	point	point	NOUN
ejpam-6077	285	45	u∗	u∗	NOUN
ejpam-6077	285	46	∈	∈	PROPN
ejpam-6077	285	47	x.	x.	NOUN
ejpam-6077	286	1	this	this	DET
ejpam-6077	286	2	fixed	fix	VERB
ejpam-6077	286	3	point	point	NOUN
ejpam-6077	286	4	satisfies	satisfie	NOUN
ejpam-6077	286	5	:	:	PUNCT
ejpam-6077	286	6	u∗(υ	u∗(υ	X
ejpam-6077	286	7	)	)	PUNCT
ejpam-6077	286	8	=	=	SYM
ejpam-6077	287	1	∫	∫	PROPN
ejpam-6077	287	2	1	1	NUM
ejpam-6077	287	3	0	0	NUM
ejpam-6077	288	1	g(υ	g(υ	ADJ
ejpam-6077	288	2	,	,	PUNCT
ejpam-6077	288	3	s)f(s	s)f(	NOUN
ejpam-6077	288	4	,	,	PUNCT
ejpam-6077	288	5	u∗(s	u∗(s	PROPN
ejpam-6077	288	6	)	)	PUNCT
ejpam-6077	288	7	)	)	PUNCT
ejpam-6077	289	1	ds	ds	PROPN
ejpam-6077	289	2	,	,	PUNCT
ejpam-6077	289	3	which	which	PRON
ejpam-6077	289	4	is	be	AUX
ejpam-6077	289	5	the	the	DET
ejpam-6077	289	6	unique	unique	ADJ
ejpam-6077	289	7	solution	solution	NOUN
ejpam-6077	289	8	to	to	ADP
ejpam-6077	289	9	the	the	DET
ejpam-6077	289	10	original	original	ADJ
ejpam-6077	289	11	boundary	boundary	ADJ
ejpam-6077	289	12	value	value	NOUN
ejpam-6077	289	13	problem	problem	NOUN
ejpam-6077	289	14	.	.	PUNCT
ejpam-6077	290	1	iterative	iterative	NOUN
ejpam-6077	290	2	approximation	approximation	NOUN
ejpam-6077	290	3	starting	start	VERB
ejpam-6077	290	4	with	with	ADP
ejpam-6077	290	5	an	an	DET
ejpam-6077	290	6	initial	initial	ADJ
ejpam-6077	290	7	guess	guess	NOUN
ejpam-6077	290	8	u0(υ	u0(υ	NOUN
ejpam-6077	290	9	)	)	PUNCT
ejpam-6077	290	10	∈	∈	PROPN
ejpam-6077	291	1	x	x	NOUN
ejpam-6077	291	2	,	,	PUNCT
ejpam-6077	291	3	the	the	DET
ejpam-6077	291	4	solution	solution	NOUN
ejpam-6077	291	5	can	can	AUX
ejpam-6077	291	6	be	be	AUX
ejpam-6077	291	7	approximated	approximate	VERB
ejpam-6077	291	8	iteratively	iteratively	ADV
ejpam-6077	291	9	using	use	VERB
ejpam-6077	291	10	:	:	PUNCT
ejpam-6077	291	11	un+1(υ	un+1(υ	ADJ
ejpam-6077	291	12	)	)	PUNCT
ejpam-6077	291	13	=	=	SYM
ejpam-6077	291	14	(	(	PUNCT
ejpam-6077	291	15	tun)(υ	tun)(υ	NUM
ejpam-6077	291	16	)	)	PUNCT
ejpam-6077	291	17	=	=	SYM
ejpam-6077	292	1	∫	∫	PROPN
ejpam-6077	292	2	1	1	NUM
ejpam-6077	292	3	0	0	NUM
ejpam-6077	293	1	g(υ	g(υ	ADJ
ejpam-6077	293	2	,	,	PUNCT
ejpam-6077	293	3	s)f(s	s)f(	NOUN
ejpam-6077	293	4	,	,	PUNCT
ejpam-6077	293	5	un(s	un(s	NUM
ejpam-6077	293	6	)	)	PUNCT
ejpam-6077	293	7	)	)	PUNCT
ejpam-6077	294	1	ds	ds	PROPN
ejpam-6077	294	2	.	.	PUNCT
ejpam-6077	295	1	the	the	DET
ejpam-6077	295	2	sequence	sequence	NOUN
ejpam-6077	295	3	{	{	PUNCT
ejpam-6077	295	4	un	un	PROPN
ejpam-6077	295	5	}	}	PUNCT
ejpam-6077	295	6	converges	converge	NOUN
ejpam-6077	295	7	to	to	ADP
ejpam-6077	295	8	the	the	DET
ejpam-6077	295	9	fixed	fixed	ADJ
ejpam-6077	295	10	point	point	NOUN
ejpam-6077	295	11	u∗(υ	u∗(υ	NOUN
ejpam-6077	295	12	)	)	PUNCT
ejpam-6077	295	13	under	under	ADP
ejpam-6077	295	14	the	the	DET
ejpam-6077	295	15	mr	mr	PROPN
ejpam-6077	295	16	-	-	PUNCT
ejpam-6077	295	17	metric	metric	NOUN
ejpam-6077	295	18	.	.	PUNCT
ejpam-6077	296	1	applications	application	NOUN
ejpam-6077	296	2	1	1	NUM
ejpam-6077	296	3	.	.	PUNCT
ejpam-6077	296	4	heat	heat	NOUN
ejpam-6077	296	5	equation	equation	NOUN
ejpam-6077	296	6	:	:	PUNCT
ejpam-6077	296	7	modeling	model	VERB
ejpam-6077	296	8	steady	steady	ADJ
ejpam-6077	296	9	-	-	PUNCT
ejpam-6077	296	10	state	state	NOUN
ejpam-6077	296	11	heat	heat	NOUN
ejpam-6077	296	12	distribution	distribution	NOUN
ejpam-6077	296	13	in	in	ADP
ejpam-6077	296	14	a	a	DET
ejpam-6077	296	15	onedimensional	onedimensional	ADJ
ejpam-6077	296	16	rod	rod	NOUN
ejpam-6077	296	17	.	.	PUNCT
ejpam-6077	297	1	2	2	NUM
ejpam-6077	297	2	.	.	X
ejpam-6077	297	3	elasticity	elasticity	NOUN
ejpam-6077	297	4	:	:	PUNCT
ejpam-6077	297	5	determining	determine	VERB
ejpam-6077	297	6	the	the	DET
ejpam-6077	297	7	deflection	deflection	NOUN
ejpam-6077	297	8	of	of	ADP
ejpam-6077	297	9	a	a	DET
ejpam-6077	297	10	beam	beam	NOUN
ejpam-6077	297	11	under	under	ADP
ejpam-6077	297	12	a	a	DET
ejpam-6077	297	13	distributed	distribute	VERB
ejpam-6077	297	14	load	load	NOUN
ejpam-6077	297	15	.	.	PUNCT
ejpam-6077	298	1	3	3	X
ejpam-6077	298	2	.	.	X
ejpam-6077	299	1	electrostatics	electrostatic	NOUN
ejpam-6077	299	2	:	:	PUNCT
ejpam-6077	299	3	solving	solve	VERB
ejpam-6077	299	4	for	for	ADP
ejpam-6077	299	5	potential	potential	ADJ
ejpam-6077	299	6	distributions	distribution	NOUN
ejpam-6077	299	7	in	in	ADP
ejpam-6077	299	8	one	one	NUM
ejpam-6077	299	9	-	-	PUNCT
ejpam-6077	299	10	dimensional	dimensional	ADJ
ejpam-6077	299	11	domains	domain	NOUN
ejpam-6077	299	12	.	.	PUNCT
ejpam-6077	300	1	the	the	DET
ejpam-6077	300	2	use	use	NOUN
ejpam-6077	300	3	of	of	ADP
ejpam-6077	300	4	the	the	DET
ejpam-6077	300	5	mr	mr	PROPN
ejpam-6077	300	6	-	-	PUNCT
ejpam-6077	300	7	metric	metric	NOUN
ejpam-6077	300	8	provides	provide	VERB
ejpam-6077	300	9	a	a	DET
ejpam-6077	300	10	robust	robust	ADJ
ejpam-6077	300	11	framework	framework	NOUN
ejpam-6077	300	12	for	for	ADP
ejpam-6077	300	13	analyzing	analyze	VERB
ejpam-6077	300	14	the	the	DET
ejpam-6077	300	15	convergence	convergence	NOUN
ejpam-6077	300	16	of	of	ADP
ejpam-6077	300	17	solutions	solution	NOUN
ejpam-6077	300	18	and	and	CCONJ
ejpam-6077	300	19	stability	stability	NOUN
ejpam-6077	300	20	of	of	ADP
ejpam-6077	300	21	the	the	DET
ejpam-6077	300	22	problem	problem	NOUN
ejpam-6077	300	23	.	.	PUNCT
ejpam-6077	301	1	4	4	X
ejpam-6077	301	2	.	.	X
ejpam-6077	301	3	conclusion	conclusion	NOUN
ejpam-6077	301	4	this	this	DET
ejpam-6077	301	5	paper	paper	NOUN
ejpam-6077	301	6	introduces	introduce	VERB
ejpam-6077	301	7	new	new	ADJ
ejpam-6077	301	8	fixed	fix	VERB
ejpam-6077	301	9	-	-	PUNCT
ejpam-6077	301	10	point	point	NOUN
ejpam-6077	301	11	theorems	theorem	NOUN
ejpam-6077	301	12	in	in	ADP
ejpam-6077	301	13	the	the	DET
ejpam-6077	301	14	context	context	NOUN
ejpam-6077	301	15	of	of	ADP
ejpam-6077	301	16	mr	mr	PROPN
ejpam-6077	301	17	-	-	PUNCT
ejpam-6077	301	18	metric	metric	ADJ
ejpam-6077	301	19	spaces	space	NOUN
ejpam-6077	301	20	,	,	PUNCT
ejpam-6077	301	21	which	which	PRON
ejpam-6077	301	22	extend	extend	VERB
ejpam-6077	301	23	classical	classical	ADJ
ejpam-6077	301	24	metric	metric	ADJ
ejpam-6077	301	25	space	space	NOUN
ejpam-6077	301	26	theory	theory	NOUN
ejpam-6077	301	27	.	.	PUNCT
ejpam-6077	302	1	the	the	DET
ejpam-6077	302	2	analysis	analysis	NOUN
ejpam-6077	302	3	focuses	focus	VERB
ejpam-6077	302	4	on	on	ADP
ejpam-6077	302	5	continuous	continuous	ADJ
ejpam-6077	302	6	selfmappings	selfmapping	NOUN
ejpam-6077	302	7	s	s	PART
ejpam-6077	302	8	:	:	PUNCT
ejpam-6077	302	9	x	x	SYM
ejpam-6077	302	10	→	→	SYM
ejpam-6077	302	11	x	x	X
ejpam-6077	302	12	,	,	PUNCT
ejpam-6077	302	13	where	where	SCONJ
ejpam-6077	302	14	x	x	PRON
ejpam-6077	302	15	is	be	AUX
ejpam-6077	302	16	a	a	DET
ejpam-6077	302	17	closed	closed	ADJ
ejpam-6077	302	18	,	,	PUNCT
ejpam-6077	302	19	bounded	bound	VERB
ejpam-6077	302	20	,	,	PUNCT
ejpam-6077	302	21	and	and	CCONJ
ejpam-6077	302	22	convex	convex	PROPN
ejpam-6077	302	23	subset	subset	NOUN
ejpam-6077	302	24	of	of	ADP
ejpam-6077	302	25	a	a	DET
ejpam-6077	302	26	banach	banach	NOUN
ejpam-6077	302	27	space	space	NOUN
ejpam-6077	302	28	.	.	PUNCT
ejpam-6077	303	1	the	the	DET
ejpam-6077	303	2	main	main	ADJ
ejpam-6077	303	3	results	result	NOUN
ejpam-6077	303	4	establish	establish	VERB
ejpam-6077	303	5	the	the	DET
ejpam-6077	303	6	existence	existence	NOUN
ejpam-6077	303	7	and	and	CCONJ
ejpam-6077	303	8	uniqueness	uniqueness	NOUN
ejpam-6077	303	9	of	of	ADP
ejpam-6077	303	10	fixed	fix	VERB
ejpam-6077	303	11	points	point	NOUN
ejpam-6077	303	12	under	under	ADP
ejpam-6077	303	13	two	two	NUM
ejpam-6077	303	14	key	key	ADJ
ejpam-6077	303	15	conditions	condition	NOUN
ejpam-6077	303	16	:	:	PUNCT
ejpam-6077	303	17	(	(	PUNCT
ejpam-6077	303	18	i	i	NOUN
ejpam-6077	303	19	)	)	PUNCT
ejpam-6077	303	20	a	a	DET
ejpam-6077	303	21	contraction	contraction	NOUN
ejpam-6077	303	22	condition	condition	NOUN
ejpam-6077	303	23	with	with	ADP
ejpam-6077	303	24	a	a	DET
ejpam-6077	303	25	constant	constant	ADJ
ejpam-6077	303	26	k	k	PROPN
ejpam-6077	303	27	∈	∈	PROPN
ejpam-6077	304	1	[	[	X
ejpam-6077	304	2	0	0	NUM
ejpam-6077	304	3	,	,	PUNCT
ejpam-6077	304	4	1	1	NUM
ejpam-6077	304	5	)	)	PUNCT
ejpam-6077	304	6	.	.	PUNCT
ejpam-6077	305	1	(	(	PUNCT
ejpam-6077	305	2	ii	ii	X
ejpam-6077	305	3	)	)	PUNCT
ejpam-6077	305	4	a	a	DET
ejpam-6077	305	5	noncompactness	noncompactness	ADJ
ejpam-6077	305	6	condition	condition	NOUN
ejpam-6077	305	7	controlled	control	VERB
ejpam-6077	305	8	by	by	ADP
ejpam-6077	305	9	a	a	DET
ejpam-6077	305	10	function	function	NOUN
ejpam-6077	305	11	φ	φ	PROPN
ejpam-6077	305	12	satisfying	satisfy	VERB
ejpam-6077	305	13	φ(t	φ(t	PROPN
ejpam-6077	305	14	)	)	PUNCT
ejpam-6077	305	15	<	<	X
ejpam-6077	305	16	t	t	PROPN
ejpam-6077	305	17	for	for	ADP
ejpam-6077	305	18	all	all	DET
ejpam-6077	305	19	t	t	PROPN
ejpam-6077	305	20	>	>	X
ejpam-6077	305	21	0	0	X
ejpam-6077	305	22	.	.	PUNCT
ejpam-6077	306	1	these	these	DET
ejpam-6077	306	2	theorems	theorem	NOUN
ejpam-6077	306	3	provide	provide	VERB
ejpam-6077	306	4	a	a	DET
ejpam-6077	306	5	significant	significant	ADJ
ejpam-6077	306	6	generalization	generalization	NOUN
ejpam-6077	306	7	of	of	ADP
ejpam-6077	306	8	classical	classical	ADJ
ejpam-6077	306	9	fixed	fix	VERB
ejpam-6077	306	10	-	-	PUNCT
ejpam-6077	306	11	point	point	NOUN
ejpam-6077	306	12	theory	theory	NOUN
ejpam-6077	306	13	,	,	PUNCT
ejpam-6077	306	14	with	with	ADP
ejpam-6077	306	15	applications	application	NOUN
ejpam-6077	306	16	spanning	span	VERB
ejpam-6077	306	17	multiple	multiple	ADJ
ejpam-6077	306	18	fields	field	NOUN
ejpam-6077	306	19	,	,	PUNCT
ejpam-6077	306	20	including	include	VERB
ejpam-6077	306	21	:	:	PUNCT
ejpam-6077	306	22	•	•	NUM
ejpam-6077	306	23	solutions	solution	NOUN
ejpam-6077	306	24	to	to	PART
ejpam-6077	306	25	nonlinear	nonlinear	ADJ
ejpam-6077	306	26	integral	integral	ADJ
ejpam-6077	306	27	equations	equation	NOUN
ejpam-6077	306	28	•	•	VERB
ejpam-6077	306	29	stability	stability	NOUN
ejpam-6077	306	30	analysis	analysis	NOUN
ejpam-6077	306	31	of	of	ADP
ejpam-6077	306	32	iterative	iterative	ADJ
ejpam-6077	306	33	methods	method	NOUN
ejpam-6077	306	34	•	•	PRON
ejpam-6077	306	35	optimization	optimization	NOUN
ejpam-6077	306	36	problems	problem	NOUN
ejpam-6077	306	37	•	•	ADP
ejpam-6077	306	38	game	game	NOUN
ejpam-6077	306	39	theory	theory	NOUN
ejpam-6077	306	40	and	and	CCONJ
ejpam-6077	306	41	economic	economic	ADJ
ejpam-6077	306	42	equilibria	equilibrium	NOUN
ejpam-6077	306	43	•	•	ADV
ejpam-6077	306	44	boundary	boundary	ADJ
ejpam-6077	306	45	value	value	NOUN
ejpam-6077	306	46	problems	problem	VERB
ejpam-6077	306	47	the	the	DET
ejpam-6077	306	48	framework	framework	NOUN
ejpam-6077	306	49	of	of	ADP
ejpam-6077	306	50	mr	mr	PROPN
ejpam-6077	306	51	-	-	PUNCT
ejpam-6077	306	52	metric	metric	ADJ
ejpam-6077	306	53	spaces	space	NOUN
ejpam-6077	306	54	offers	offer	VERB
ejpam-6077	306	55	a	a	DET
ejpam-6077	306	56	versatile	versatile	ADJ
ejpam-6077	306	57	and	and	CCONJ
ejpam-6077	306	58	powerful	powerful	ADJ
ejpam-6077	306	59	structure	structure	NOUN
ejpam-6077	306	60	for	for	ADP
ejpam-6077	306	61	analyzing	analyze	VERB
ejpam-6077	306	62	noncompact	noncompact	NOUN
ejpam-6077	306	63	and	and	CCONJ
ejpam-6077	306	64	complex	complex	ADJ
ejpam-6077	306	65	systems	system	NOUN
ejpam-6077	306	66	,	,	PUNCT
ejpam-6077	306	67	enhancing	enhance	VERB
ejpam-6077	306	68	both	both	DET
ejpam-6077	306	69	theoretical	theoretical	ADJ
ejpam-6077	306	70	insights	insight	NOUN
ejpam-6077	306	71	and	and	CCONJ
ejpam-6077	306	72	practical	practical	ADJ
ejpam-6077	306	73	applications	application	NOUN
ejpam-6077	306	74	.	.	PUNCT
ejpam-6077	307	1	a.	a.	NOUN
ejpam-6077	307	2	malkawi	malkawi	ADP
ejpam-6077	307	3	/	/	SYM
ejpam-6077	307	4	eur	eur	PROPN
ejpam-6077	307	5	.	.	PUNCT
ejpam-6077	308	1	j.	j.	PROPN
ejpam-6077	308	2	pure	pure	PROPN
ejpam-6077	308	3	appl	appl	PROPN
ejpam-6077	308	4	.	.	PROPN
ejpam-6077	308	5	math	math	PROPN
ejpam-6077	308	6	,	,	PUNCT
ejpam-6077	308	7	18	18	NUM
ejpam-6077	308	8	(	(	PUNCT
ejpam-6077	308	9	2	2	NUM
ejpam-6077	308	10	)	)	PUNCT
ejpam-6077	308	11	(	(	PUNCT
ejpam-6077	308	12	2025	2025	NUM
ejpam-6077	308	13	)	)	PUNCT
ejpam-6077	308	14	,	,	PUNCT
ejpam-6077	308	15	6077	6077	NUM
ejpam-6077	308	16	15	15	NUM
ejpam-6077	308	17	of	of	ADP
ejpam-6077	308	18	16	16	NUM
ejpam-6077	308	19	references	reference	NOUN
ejpam-6077	308	20	[	[	X
ejpam-6077	308	21	1	1	NUM
ejpam-6077	308	22	]	]	PUNCT
ejpam-6077	308	23	m.	m.	NOUN
ejpam-6077	308	24	s.	s.	PROPN
ejpam-6077	308	25	alsauodi	alsauodi	PROPN
ejpam-6077	308	26	,	,	PUNCT
ejpam-6077	308	27	g.	g.	PROPN
ejpam-6077	308	28	m.	m.	PROPN
ejpam-6077	308	29	gharib	gharib	PROPN
ejpam-6077	308	30	,	,	PUNCT
ejpam-6077	308	31	a.	a.	PROPN
ejpam-6077	308	32	malkawi	malkawi	PROPN
ejpam-6077	308	33	,	,	PUNCT
ejpam-6077	308	34	a.	a.	PROPN
ejpam-6077	308	35	m.	m.	PROPN
ejpam-6077	308	36	rabaiah	rabaiah	PROPN
ejpam-6077	308	37	,	,	PUNCT
ejpam-6077	308	38	and	and	CCONJ
ejpam-6077	308	39	w.	w.	PROPN
ejpam-6077	308	40	a.	a.	PROPN
ejpam-6077	308	41	shatanawi	shatanawi	PROPN
ejpam-6077	308	42	.	.	PUNCT
ejpam-6077	309	1	fixed	fix	VERB
ejpam-6077	309	2	point	point	NOUN
ejpam-6077	309	3	theorems	theorem	NOUN
ejpam-6077	309	4	for	for	ADP
ejpam-6077	309	5	monotone	monotone	ADJ
ejpam-6077	309	6	mappings	mapping	NOUN
ejpam-6077	309	7	on	on	ADP
ejpam-6077	309	8	partial	partial	ADJ
ejpam-6077	309	9	m∗-metric	m∗-metric	ADJ
ejpam-6077	309	10	spaces	space	NOUN
ejpam-6077	309	11	.	.	PUNCT
ejpam-6077	310	1	italian	italian	ADJ
ejpam-6077	310	2	journal	journal	NOUN
ejpam-6077	310	3	of	of	ADP
ejpam-6077	310	4	pure	pure	ADJ
ejpam-6077	310	5	and	and	CCONJ
ejpam-6077	310	6	applied	applied	ADJ
ejpam-6077	310	7	mathematics	mathematic	NOUN
ejpam-6077	310	8	,	,	PUNCT
ejpam-6077	310	9	44:154–172	44:154–172	PROPN
ejpam-6077	310	10	,	,	PUNCT
ejpam-6077	310	11	2023	2023	NUM
ejpam-6077	310	12	.	.	PUNCT
ejpam-6077	311	1	[	[	X
ejpam-6077	311	2	2	2	NUM
ejpam-6077	311	3	]	]	PUNCT
ejpam-6077	311	4	i.	i.	PROPN
ejpam-6077	311	5	a.	a.	PROPN
ejpam-6077	311	6	bakhtin	bakhtin	PROPN
ejpam-6077	311	7	.	.	PUNCT
ejpam-6077	312	1	the	the	DET
ejpam-6077	312	2	contraction	contraction	NOUN
ejpam-6077	312	3	mapping	map	VERB
ejpam-6077	312	4	principle	principle	NOUN
ejpam-6077	312	5	in	in	ADP
ejpam-6077	312	6	almost	almost	ADV
ejpam-6077	312	7	metric	metric	ADJ
ejpam-6077	312	8	spaces	space	NOUN
ejpam-6077	312	9	.	.	PUNCT
ejpam-6077	313	1	functional	functional	ADJ
ejpam-6077	313	2	analysis	analysis	NOUN
ejpam-6077	313	3	,	,	PUNCT
ejpam-6077	313	4	30:26–37	30:26–37	PROPN
ejpam-6077	313	5	,	,	PUNCT
ejpam-6077	313	6	1989	1989	NUM
ejpam-6077	313	7	.	.	PUNCT
ejpam-6077	314	1	[	[	X
ejpam-6077	314	2	3	3	X
ejpam-6077	314	3	]	]	X
ejpam-6077	314	4	s.	s.	PROPN
ejpam-6077	314	5	czerwik	czerwik	PROPN
ejpam-6077	314	6	.	.	PUNCT
ejpam-6077	315	1	contraction	contraction	NOUN
ejpam-6077	315	2	mappings	mapping	NOUN
ejpam-6077	315	3	in	in	ADP
ejpam-6077	315	4	b	b	NOUN
ejpam-6077	315	5	-	-	ADJ
ejpam-6077	315	6	metric	metric	ADJ
ejpam-6077	315	7	spaces	space	NOUN
ejpam-6077	315	8	.	.	PUNCT
ejpam-6077	316	1	acta	acta	PROPN
ejpam-6077	316	2	mathematica	mathematica	PROPN
ejpam-6077	316	3	et	et	PROPN
ejpam-6077	316	4	informatica	informatica	PROPN
ejpam-6077	316	5	universitatis	universitatis	PROPN
ejpam-6077	316	6	ostraviensis	ostraviensis	PROPN
ejpam-6077	316	7	,	,	PUNCT
ejpam-6077	316	8	1:5–11	1:5–11	NUM
ejpam-6077	316	9	,	,	PUNCT
ejpam-6077	316	10	1993	1993	NUM
ejpam-6077	316	11	.	.	PUNCT
ejpam-6077	317	1	[	[	X
ejpam-6077	317	2	4	4	X
ejpam-6077	317	3	]	]	X
ejpam-6077	317	4	y.	y.	PROPN
ejpam-6077	317	5	j.	j.	PROPN
ejpam-6077	317	6	cho	cho	PROPN
ejpam-6077	317	7	,	,	PUNCT
ejpam-6077	317	8	p.	p.	NOUN
ejpam-6077	317	9	p.	p.	PROPN
ejpam-6077	318	1	murthy	murthy	ADJ
ejpam-6077	318	2	,	,	PUNCT
ejpam-6077	318	3	and	and	CCONJ
ejpam-6077	318	4	g.	g.	PROPN
ejpam-6077	318	5	jungck	jungck	PROPN
ejpam-6077	318	6	.	.	PUNCT
ejpam-6077	319	1	a	a	DET
ejpam-6077	319	2	common	common	ADJ
ejpam-6077	319	3	fixed	fix	VERB
ejpam-6077	319	4	point	point	NOUN
ejpam-6077	319	5	theorem	theorem	NOUN
ejpam-6077	319	6	of	of	ADP
ejpam-6077	319	7	meir	meir	PROPN
ejpam-6077	319	8	and	and	CCONJ
ejpam-6077	319	9	keeler	keeler	PROPN
ejpam-6077	319	10	type	type	NOUN
ejpam-6077	319	11	.	.	PUNCT
ejpam-6077	320	1	international	international	ADJ
ejpam-6077	320	2	journal	journal	PROPN
ejpam-6077	320	3	of	of	ADP
ejpam-6077	320	4	mathematics	mathematics	PROPN
ejpam-6077	320	5	and	and	CCONJ
ejpam-6077	320	6	mathematical	mathematical	ADJ
ejpam-6077	320	7	sciences	science	NOUN
ejpam-6077	320	8	,	,	PUNCT
ejpam-6077	320	9	16(4):669–674	16(4):669–674	NUM
ejpam-6077	320	10	,	,	PUNCT
ejpam-6077	320	11	1993	1993	NUM
ejpam-6077	320	12	.	.	PUNCT
ejpam-6077	321	1	[	[	X
ejpam-6077	321	2	5	5	X
ejpam-6077	321	3	]	]	PUNCT
ejpam-6077	321	4	r.	r.	PROPN
ejpam-6077	321	5	o.	o.	PROPN
ejpam-6077	321	6	davies	davies	PROPN
ejpam-6077	321	7	and	and	CCONJ
ejpam-6077	321	8	s.	s.	PROPN
ejpam-6077	321	9	sessa	sessa	PROPN
ejpam-6077	321	10	.	.	PUNCT
ejpam-6077	322	1	a	a	DET
ejpam-6077	322	2	common	common	ADJ
ejpam-6077	322	3	fixed	fix	VERB
ejpam-6077	322	4	point	point	NOUN
ejpam-6077	322	5	theorem	theorem	NOUN
ejpam-6077	322	6	of	of	ADP
ejpam-6077	322	7	gregus	gregus	NOUN
ejpam-6077	322	8	type	type	NOUN
ejpam-6077	322	9	for	for	ADP
ejpam-6077	322	10	compatible	compatible	ADJ
ejpam-6077	322	11	mappings	mapping	NOUN
ejpam-6077	322	12	.	.	PUNCT
ejpam-6077	323	1	facta	facta	PROPN
ejpam-6077	323	2	universitatis	universitatis	PROPN
ejpam-6077	323	3	,	,	PUNCT
ejpam-6077	323	4	series	series	NOUN
ejpam-6077	323	5	:	:	PUNCT
ejpam-6077	323	6	mathematics	mathematic	NOUN
ejpam-6077	323	7	and	and	CCONJ
ejpam-6077	323	8	informatics	informatic	NOUN
ejpam-6077	323	9	,	,	PUNCT
ejpam-6077	323	10	7:51–58	7:51–58	NOUN
ejpam-6077	323	11	,	,	PUNCT
ejpam-6077	323	12	1992	1992	NUM
ejpam-6077	323	13	.	.	PUNCT
ejpam-6077	324	1	[	[	X
ejpam-6077	324	2	6	6	NUM
ejpam-6077	324	3	]	]	PUNCT
ejpam-6077	324	4	b.	b.	PROPN
ejpam-6077	324	5	c.	c.	PROPN
ejpam-6077	324	6	dhage	dhage	PROPN
ejpam-6077	324	7	.	.	PUNCT
ejpam-6077	325	1	generalized	generalize	VERB
ejpam-6077	325	2	metric	metric	ADJ
ejpam-6077	325	3	spaces	space	NOUN
ejpam-6077	325	4	and	and	CCONJ
ejpam-6077	325	5	mappings	mapping	NOUN
ejpam-6077	325	6	with	with	ADP
ejpam-6077	325	7	fixed	fix	VERB
ejpam-6077	325	8	points	point	NOUN
ejpam-6077	325	9	.	.	PUNCT
ejpam-6077	326	1	bulletin	bulletin	NOUN
ejpam-6077	326	2	of	of	ADP
ejpam-6077	326	3	the	the	DET
ejpam-6077	326	4	calcutta	calcutta	PROPN
ejpam-6077	326	5	mathematical	mathematical	ADJ
ejpam-6077	326	6	society	society	NOUN
ejpam-6077	326	7	,	,	PUNCT
ejpam-6077	326	8	84:329–336	84:329–336	NUM
ejpam-6077	326	9	,	,	PUNCT
ejpam-6077	326	10	1992	1992	NUM
ejpam-6077	326	11	.	.	PUNCT
ejpam-6077	327	1	[	[	X
ejpam-6077	327	2	7	7	X
ejpam-6077	327	3	]	]	X
ejpam-6077	327	4	g.	g.	PROPN
ejpam-6077	327	5	gharib	gharib	PROPN
ejpam-6077	327	6	,	,	PUNCT
ejpam-6077	327	7	a.	a.	PROPN
ejpam-6077	327	8	malkawi	malkawi	PROPN
ejpam-6077	327	9	,	,	PUNCT
ejpam-6077	327	10	a.	a.	PROPN
ejpam-6077	327	11	rabaiah	rabaiah	PROPN
ejpam-6077	327	12	,	,	PUNCT
ejpam-6077	327	13	w.	w.	PROPN
ejpam-6077	327	14	shatanawi	shatanawi	PROPN
ejpam-6077	327	15	,	,	PUNCT
ejpam-6077	327	16	and	and	CCONJ
ejpam-6077	327	17	m.	m.	NOUN
ejpam-6077	327	18	alsauodi	alsauodi	PROPN
ejpam-6077	327	19	.	.	PUNCT
ejpam-6077	328	1	a	a	DET
ejpam-6077	328	2	common	common	ADJ
ejpam-6077	328	3	fixed	fix	VERB
ejpam-6077	328	4	point	point	NOUN
ejpam-6077	328	5	theorem	theorem	VERB
ejpam-6077	328	6	in	in	ADP
ejpam-6077	328	7	m∗-metric	m∗-metric	ADJ
ejpam-6077	328	8	space	space	NOUN
ejpam-6077	328	9	and	and	CCONJ
ejpam-6077	328	10	an	an	DET
ejpam-6077	328	11	application	application	NOUN
ejpam-6077	328	12	.	.	PUNCT
ejpam-6077	329	1	nonlinear	nonlinear	ADJ
ejpam-6077	329	2	functional	functional	ADJ
ejpam-6077	329	3	analysis	analysis	NOUN
ejpam-6077	329	4	and	and	CCONJ
ejpam-6077	329	5	applications	application	NOUN
ejpam-6077	329	6	,	,	PUNCT
ejpam-6077	329	7	27(2):289–308	27(2):289–308	NUM
ejpam-6077	329	8	,	,	PUNCT
ejpam-6077	329	9	2022	2022	NUM
ejpam-6077	329	10	.	.	PUNCT
ejpam-6077	330	1	[	[	X
ejpam-6077	330	2	8	8	NUM
ejpam-6077	330	3	]	]	X
ejpam-6077	330	4	a.	a.	NOUN
ejpam-6077	330	5	malkawi	malkawi	PROPN
ejpam-6077	330	6	,	,	PUNCT
ejpam-6077	330	7	a.	a.	NOUN
ejpam-6077	330	8	tallafha	tallafha	NOUN
ejpam-6077	330	9	,	,	PUNCT
ejpam-6077	330	10	and	and	CCONJ
ejpam-6077	330	11	w.	w.	PROPN
ejpam-6077	330	12	shatanawi	shatanawi	PROPN
ejpam-6077	330	13	.	.	PUNCT
ejpam-6077	331	1	coincidence	coincidence	NOUN
ejpam-6077	331	2	and	and	CCONJ
ejpam-6077	331	3	fixed	fix	VERB
ejpam-6077	331	4	point	point	NOUN
ejpam-6077	331	5	results	result	NOUN
ejpam-6077	331	6	for	for	ADP
ejpam-6077	331	7	(	(	PUNCT
ejpam-6077	331	8	ψ	ψ	NOUN
ejpam-6077	331	9	,	,	PUNCT
ejpam-6077	331	10	l)-m	l)-m	PROPN
ejpam-6077	331	11	-weak	-weak	PROPN
ejpam-6077	331	12	contraction	contraction	NOUN
ejpam-6077	331	13	mapping	mapping	NOUN
ejpam-6077	331	14	on	on	ADP
ejpam-6077	331	15	mb	mb	ADJ
ejpam-6077	331	16	-	-	ADJ
ejpam-6077	331	17	metric	metric	ADJ
ejpam-6077	331	18	spaces	space	NOUN
ejpam-6077	331	19	.	.	PUNCT
ejpam-6077	332	1	italian	italian	ADJ
ejpam-6077	332	2	journal	journal	NOUN
ejpam-6077	332	3	of	of	ADP
ejpam-6077	332	4	pure	pure	ADJ
ejpam-6077	332	5	and	and	CCONJ
ejpam-6077	332	6	applied	applied	ADJ
ejpam-6077	332	7	mathematics	mathematic	NOUN
ejpam-6077	332	8	,	,	PUNCT
ejpam-6077	332	9	47:751–768	47:751–768	NUM
ejpam-6077	332	10	,	,	PUNCT
ejpam-6077	332	11	2022	2022	NUM
ejpam-6077	332	12	.	.	PUNCT
ejpam-6077	333	1	[	[	X
ejpam-6077	333	2	9	9	NUM
ejpam-6077	333	3	]	]	PUNCT
ejpam-6077	333	4	a.	a.	NOUN
ejpam-6077	333	5	malkawi	malkawi	PROPN
ejpam-6077	333	6	,	,	PUNCT
ejpam-6077	333	7	a.	a.	NOUN
ejpam-6077	333	8	tallafha	tallafha	NOUN
ejpam-6077	333	9	,	,	PUNCT
ejpam-6077	333	10	and	and	CCONJ
ejpam-6077	333	11	w.	w.	PROPN
ejpam-6077	333	12	shatanawi	shatanawi	PROPN
ejpam-6077	333	13	.	.	PUNCT
ejpam-6077	334	1	coincidence	coincidence	NOUN
ejpam-6077	334	2	and	and	CCONJ
ejpam-6077	334	3	fixed	fix	VERB
ejpam-6077	334	4	point	point	NOUN
ejpam-6077	334	5	results	result	NOUN
ejpam-6077	334	6	for	for	ADP
ejpam-6077	334	7	generalized	generalized	ADJ
ejpam-6077	334	8	weak	weak	ADJ
ejpam-6077	334	9	contraction	contraction	NOUN
ejpam-6077	334	10	mapping	mapping	NOUN
ejpam-6077	334	11	on	on	ADP
ejpam-6077	334	12	b	b	NOUN
ejpam-6077	334	13	-	-	PUNCT
ejpam-6077	334	14	metric	metric	ADJ
ejpam-6077	334	15	spaces	space	NOUN
ejpam-6077	334	16	.	.	PUNCT
ejpam-6077	335	1	nonlinear	nonlinear	ADJ
ejpam-6077	335	2	functional	functional	ADJ
ejpam-6077	335	3	analysis	analysis	NOUN
ejpam-6077	335	4	and	and	CCONJ
ejpam-6077	335	5	applications	application	NOUN
ejpam-6077	335	6	,	,	PUNCT
ejpam-6077	335	7	26(1):177–195	26(1):177–195	NOUN
ejpam-6077	335	8	,	,	PUNCT
ejpam-6077	335	9	2021	2021	NUM
ejpam-6077	335	10	.	.	PUNCT
ejpam-6077	336	1	[	[	X
ejpam-6077	336	2	10	10	NUM
ejpam-6077	336	3	]	]	PUNCT
ejpam-6077	336	4	t.	t.	NOUN
ejpam-6077	336	5	qawasmeh	qawasmeh	NOUN
ejpam-6077	336	6	,	,	PUNCT
ejpam-6077	336	7	a.	a.	PROPN
ejpam-6077	336	8	bataihah	bataihah	PROPN
ejpam-6077	336	9	,	,	PUNCT
ejpam-6077	336	10	k.	k.	PROPN
ejpam-6077	336	11	bataihah	bataihah	PROPN
ejpam-6077	336	12	,	,	PUNCT
ejpam-6077	336	13	a.	a.	NOUN
ejpam-6077	336	14	qazza	qazza	PROPN
ejpam-6077	336	15	,	,	PUNCT
ejpam-6077	336	16	and	and	CCONJ
ejpam-6077	336	17	r.	r.	PROPN
ejpam-6077	336	18	hatamleh	hatamleh	PROPN
ejpam-6077	336	19	.	.	PUNCT
ejpam-6077	337	1	nth	nth	PROPN
ejpam-6077	337	2	composite	composite	ADJ
ejpam-6077	337	3	iterative	iterative	NOUN
ejpam-6077	337	4	scheme	scheme	NOUN
ejpam-6077	337	5	via	via	ADP
ejpam-6077	337	6	weak	weak	ADJ
ejpam-6077	337	7	contractions	contraction	NOUN
ejpam-6077	337	8	with	with	ADP
ejpam-6077	337	9	application	application	NOUN
ejpam-6077	337	10	.	.	PUNCT
ejpam-6077	338	1	international	international	ADJ
ejpam-6077	338	2	journal	journal	PROPN
ejpam-6077	338	3	of	of	ADP
ejpam-6077	338	4	mathematics	mathematics	PROPN
ejpam-6077	338	5	and	and	CCONJ
ejpam-6077	338	6	mathematical	mathematical	ADJ
ejpam-6077	338	7	sciences	science	NOUN
ejpam-6077	338	8	,	,	PUNCT
ejpam-6077	338	9	2023:6953499	2023:6953499	NUM
ejpam-6077	338	10	,	,	PUNCT
ejpam-6077	338	11	2023	2023	NUM
ejpam-6077	338	12	.	.	PUNCT
ejpam-6077	339	1	[	[	X
ejpam-6077	339	2	11	11	NUM
ejpam-6077	339	3	]	]	PUNCT
ejpam-6077	339	4	a.	a.	NOUN
ejpam-6077	339	5	bataihah	bataihah	PROPN
ejpam-6077	339	6	and	and	CCONJ
ejpam-6077	339	7	t.	t.	NOUN
ejpam-6077	339	8	qawasmeh	qawasmeh	NOUN
ejpam-6077	339	9	.	.	PUNCT
ejpam-6077	340	1	a	a	DET
ejpam-6077	340	2	new	new	ADJ
ejpam-6077	340	3	type	type	NOUN
ejpam-6077	340	4	of	of	ADP
ejpam-6077	340	5	distance	distance	NOUN
ejpam-6077	340	6	spaces	space	NOUN
ejpam-6077	340	7	and	and	CCONJ
ejpam-6077	340	8	fixed	fix	VERB
ejpam-6077	340	9	point	point	NOUN
ejpam-6077	340	10	results	result	NOUN
ejpam-6077	340	11	.	.	PUNCT
ejpam-6077	341	1	journal	journal	NOUN
ejpam-6077	341	2	of	of	ADP
ejpam-6077	341	3	mathematical	mathematical	ADJ
ejpam-6077	341	4	analysis	analysis	NOUN
ejpam-6077	341	5	,	,	PUNCT
ejpam-6077	341	6	15(4):81–90	15(4):81–90	NUM
ejpam-6077	341	7	,	,	PUNCT
ejpam-6077	341	8	2024	2024	NUM
ejpam-6077	341	9	.	.	PUNCT
ejpam-6077	342	1	[	[	X
ejpam-6077	342	2	12	12	NUM
ejpam-6077	342	3	]	]	PUNCT
ejpam-6077	342	4	a.	a.	NOUN
ejpam-6077	342	5	bataihah	bataihah	PROPN
ejpam-6077	342	6	and	and	CCONJ
ejpam-6077	342	7	t.	t.	NOUN
ejpam-6077	342	8	qawasmeh	qawasmeh	NOUN
ejpam-6077	342	9	.	.	PUNCT
ejpam-6077	343	1	a	a	DET
ejpam-6077	343	2	new	new	ADJ
ejpam-6077	343	3	type	type	NOUN
ejpam-6077	343	4	of	of	ADP
ejpam-6077	343	5	distance	distance	NOUN
ejpam-6077	343	6	spaces	space	NOUN
ejpam-6077	343	7	and	and	CCONJ
ejpam-6077	343	8	fixed	fix	VERB
ejpam-6077	343	9	point	point	NOUN
ejpam-6077	343	10	results	result	NOUN
ejpam-6077	343	11	.	.	PUNCT
ejpam-6077	344	1	journal	journal	NOUN
ejpam-6077	344	2	of	of	ADP
ejpam-6077	344	3	mathematical	mathematical	ADJ
ejpam-6077	344	4	analysis	analysis	NOUN
ejpam-6077	344	5	,	,	PUNCT
ejpam-6077	344	6	15(4):81–90	15(4):81–90	NUM
ejpam-6077	344	7	,	,	PUNCT
ejpam-6077	344	8	2024	2024	NUM
ejpam-6077	344	9	.	.	PUNCT
ejpam-6077	345	1	[	[	X
ejpam-6077	345	2	13	13	NUM
ejpam-6077	345	3	]	]	X
ejpam-6077	345	4	w.	w.	PROPN
ejpam-6077	345	5	shatanawi	shatanawi	PROPN
ejpam-6077	345	6	,	,	PUNCT
ejpam-6077	345	7	t.	t.	NOUN
ejpam-6077	345	8	qawasmeh	qawasmeh	NOUN
ejpam-6077	345	9	,	,	PUNCT
ejpam-6077	345	10	a.	a.	NOUN
ejpam-6077	345	11	bataihah	bataihah	PROPN
ejpam-6077	345	12	,	,	PUNCT
ejpam-6077	345	13	and	and	CCONJ
ejpam-6077	345	14	a.	a.	NOUN
ejpam-6077	345	15	tallafha	tallafha	NOUN
ejpam-6077	345	16	.	.	PUNCT
ejpam-6077	346	1	new	new	ADJ
ejpam-6077	346	2	contractions	contraction	NOUN
ejpam-6077	346	3	and	and	CCONJ
ejpam-6077	346	4	some	some	DET
ejpam-6077	346	5	fixed	fix	VERB
ejpam-6077	346	6	point	point	NOUN
ejpam-6077	346	7	results	result	NOUN
ejpam-6077	346	8	with	with	ADP
ejpam-6077	346	9	application	application	NOUN
ejpam-6077	346	10	based	base	VERB
ejpam-6077	346	11	on	on	ADP
ejpam-6077	346	12	extended	extended	ADJ
ejpam-6077	346	13	quasi	quasi	ADJ
ejpam-6077	346	14	b	b	NOUN
ejpam-6077	346	15	-	-	ADJ
ejpam-6077	346	16	metric	metric	ADJ
ejpam-6077	346	17	spaces	space	NOUN
ejpam-6077	346	18	.	.	PUNCT
ejpam-6077	347	1	u.p.b	u.p.b	ADJ
ejpam-6077	347	2	.	.	PUNCT
ejpam-6077	348	1	scientific	scientific	ADJ
ejpam-6077	348	2	bulletin	bulletin	NOUN
ejpam-6077	348	3	,	,	PUNCT
ejpam-6077	348	4	series	series	NOUN
ejpam-6077	348	5	a	a	PRON
ejpam-6077	348	6	,	,	PUNCT
ejpam-6077	348	7	83(2):53–64	83(2):53–64	NUM
ejpam-6077	348	8	,	,	PUNCT
ejpam-6077	348	9	2021	2021	NUM
ejpam-6077	348	10	.	.	PUNCT
ejpam-6077	349	1	[	[	X
ejpam-6077	349	2	14	14	NUM
ejpam-6077	349	3	]	]	PUNCT
ejpam-6077	349	4	t.	t.	NOUN
ejpam-6077	349	5	qawasmeh	qawasmeh	NOUN
ejpam-6077	349	6	,	,	PUNCT
ejpam-6077	349	7	w.	w.	PROPN
ejpam-6077	349	8	shatanawi	shatanawi	PROPN
ejpam-6077	349	9	,	,	PUNCT
ejpam-6077	349	10	a.	a.	NOUN
ejpam-6077	349	11	bataihah	bataihah	PROPN
ejpam-6077	349	12	,	,	PUNCT
ejpam-6077	349	13	and	and	CCONJ
ejpam-6077	349	14	a.	a.	NOUN
ejpam-6077	349	15	tallafha	tallafha	NOUN
ejpam-6077	349	16	.	.	PUNCT
ejpam-6077	350	1	fixed	fix	VERB
ejpam-6077	350	2	point	point	NOUN
ejpam-6077	350	3	results	result	NOUN
ejpam-6077	350	4	and	and	CCONJ
ejpam-6077	350	5	(	(	PUNCT
ejpam-6077	350	6	α	α	NOUN
ejpam-6077	350	7	,	,	PUNCT
ejpam-6077	350	8	β)-triangular	β)-triangular	ADJ
ejpam-6077	350	9	admissibility	admissibility	NOUN
ejpam-6077	350	10	in	in	ADP
ejpam-6077	350	11	the	the	DET
ejpam-6077	350	12	frame	frame	NOUN
ejpam-6077	350	13	of	of	ADP
ejpam-6077	350	14	complete	complete	ADJ
ejpam-6077	350	15	extended	extended	ADJ
ejpam-6077	350	16	b	b	NOUN
ejpam-6077	350	17	-	-	PUNCT
ejpam-6077	350	18	metric	metric	ADJ
ejpam-6077	350	19	spaces	space	NOUN
ejpam-6077	350	20	and	and	CCONJ
ejpam-6077	350	21	application	application	NOUN
ejpam-6077	350	22	.	.	PUNCT
ejpam-6077	351	1	u.p.b	u.p.b	PROPN
ejpam-6077	351	2	.	.	PUNCT
ejpam-6077	352	1	scientific	scientific	ADJ
ejpam-6077	352	2	bulletin	bulletin	NOUN
ejpam-6077	352	3	,	,	PUNCT
ejpam-6077	352	4	series	series	PROPN
ejpam-6077	352	5	a	a	PROPN
ejpam-6077	352	6	,	,	PUNCT
ejpam-6077	352	7	83(1):113–124	83(1):113–124	PROPN
ejpam-6077	352	8	,	,	PUNCT
ejpam-6077	352	9	2021	2021	NUM
ejpam-6077	352	10	.	.	PUNCT
ejpam-6077	353	1	[	[	X
ejpam-6077	353	2	15	15	NUM
ejpam-6077	353	3	]	]	X
ejpam-6077	353	4	a.	a.	NOUN
ejpam-6077	353	5	bataihah	bataihah	PROPN
ejpam-6077	353	6	,	,	PUNCT
ejpam-6077	353	7	w.	w.	PROPN
ejpam-6077	353	8	shatanawi	shatanawi	PROPN
ejpam-6077	353	9	,	,	PUNCT
ejpam-6077	353	10	and	and	CCONJ
ejpam-6077	353	11	a.	a.	NOUN
ejpam-6077	353	12	tallafha	tallafha	NOUN
ejpam-6077	353	13	.	.	PUNCT
ejpam-6077	354	1	fixed	fix	VERB
ejpam-6077	354	2	point	point	NOUN
ejpam-6077	354	3	results	result	NOUN
ejpam-6077	354	4	with	with	ADP
ejpam-6077	354	5	simulation	simulation	NOUN
ejpam-6077	354	6	functions	function	NOUN
ejpam-6077	354	7	.	.	PUNCT
ejpam-6077	355	1	nonlinear	nonlinear	ADJ
ejpam-6077	355	2	functional	functional	ADJ
ejpam-6077	355	3	analysis	analysis	NOUN
ejpam-6077	355	4	and	and	CCONJ
ejpam-6077	355	5	applications	application	NOUN
ejpam-6077	355	6	,	,	PUNCT
ejpam-6077	355	7	25(1):13–23	25(1):13–23	NUM
ejpam-6077	355	8	,	,	PUNCT
ejpam-6077	355	9	2020	2020	NUM
ejpam-6077	355	10	.	.	PUNCT
ejpam-6077	356	1	[	[	X
ejpam-6077	356	2	16	16	NUM
ejpam-6077	356	3	]	]	PUNCT
ejpam-6077	356	4	k.	k.	PROPN
ejpam-6077	356	5	abodayeh	abodayeh	PROPN
ejpam-6077	356	6	,	,	PUNCT
ejpam-6077	356	7	w.	w.	PROPN
ejpam-6077	356	8	shatanawi	shatanawi	PROPN
ejpam-6077	356	9	,	,	PUNCT
ejpam-6077	356	10	a.	a.	NOUN
ejpam-6077	356	11	bataihah	bataihah	PROPN
ejpam-6077	356	12	,	,	PUNCT
ejpam-6077	356	13	and	and	CCONJ
ejpam-6077	356	14	a.	a.	PROPN
ejpam-6077	356	15	h.	h.	PROPN
ejpam-6077	356	16	ansari	ansari	PROPN
ejpam-6077	356	17	.	.	PUNCT
ejpam-6077	357	1	some	some	DET
ejpam-6077	357	2	fixed	fix	VERB
ejpam-6077	357	3	point	point	NOUN
ejpam-6077	357	4	and	and	CCONJ
ejpam-6077	357	5	common	common	ADJ
ejpam-6077	357	6	fixed	fix	VERB
ejpam-6077	357	7	point	point	NOUN
ejpam-6077	357	8	results	result	NOUN
ejpam-6077	357	9	through	through	ADP
ejpam-6077	357	10	ω	ω	NOUN
ejpam-6077	357	11	-	-	PUNCT
ejpam-6077	357	12	distance	distance	NOUN
ejpam-6077	357	13	under	under	ADP
ejpam-6077	357	14	nonlinear	nonlinear	ADJ
ejpam-6077	357	15	contractions	contraction	NOUN
ejpam-6077	357	16	.	.	PUNCT
ejpam-6077	358	1	gazi	gazi	PROPN
ejpam-6077	358	2	a.	a.	PROPN
ejpam-6077	358	3	malkawi	malkawi	PROPN
ejpam-6077	358	4	/	/	SYM
ejpam-6077	358	5	eur	eur	PROPN
ejpam-6077	358	6	.	.	PUNCT
ejpam-6077	359	1	j.	j.	PROPN
ejpam-6077	359	2	pure	pure	PROPN
ejpam-6077	359	3	appl	appl	PROPN
ejpam-6077	359	4	.	.	PROPN
ejpam-6077	359	5	math	math	PROPN
ejpam-6077	359	6	,	,	PUNCT
ejpam-6077	359	7	18	18	NUM
ejpam-6077	359	8	(	(	PUNCT
ejpam-6077	359	9	2	2	NUM
ejpam-6077	359	10	)	)	PUNCT
ejpam-6077	359	11	(	(	PUNCT
ejpam-6077	359	12	2025	2025	NUM
ejpam-6077	359	13	)	)	PUNCT
ejpam-6077	359	14	,	,	PUNCT
ejpam-6077	359	15	6077	6077	NUM
ejpam-6077	359	16	16	16	NUM
ejpam-6077	359	17	of	of	ADP
ejpam-6077	359	18	16	16	NUM
ejpam-6077	359	19	university	university	NOUN
ejpam-6077	359	20	journal	journal	NOUN
ejpam-6077	359	21	of	of	ADP
ejpam-6077	359	22	science	science	NOUN
ejpam-6077	359	23	,	,	PUNCT
ejpam-6077	359	24	30(1):293–302	30(1):293–302	NOUN
ejpam-6077	359	25	,	,	PUNCT
ejpam-6077	359	26	2017	2017	NUM
ejpam-6077	359	27	.	.	PUNCT
ejpam-6077	360	1	[	[	X
ejpam-6077	360	2	17	17	NUM
ejpam-6077	360	3	]	]	PUNCT
ejpam-6077	360	4	a.	a.	NOUN
ejpam-6077	360	5	bataihah	bataihah	PROPN
ejpam-6077	360	6	,	,	PUNCT
ejpam-6077	360	7	a.	a.	NOUN
ejpam-6077	360	8	tallafha	tallafha	NOUN
ejpam-6077	360	9	,	,	PUNCT
ejpam-6077	360	10	and	and	CCONJ
ejpam-6077	360	11	w.	w.	PROPN
ejpam-6077	360	12	shatanawi	shatanawi	PROPN
ejpam-6077	360	13	.	.	PUNCT
ejpam-6077	361	1	fixed	fix	VERB
ejpam-6077	361	2	point	point	NOUN
ejpam-6077	361	3	results	result	NOUN
ejpam-6077	361	4	with	with	ADP
ejpam-6077	361	5	ω	ω	NOUN
ejpam-6077	361	6	-	-	PUNCT
ejpam-6077	361	7	distance	distance	NOUN
ejpam-6077	361	8	by	by	ADP
ejpam-6077	361	9	utilizing	utilize	VERB
ejpam-6077	361	10	simulation	simulation	NOUN
ejpam-6077	361	11	functions	function	NOUN
ejpam-6077	361	12	.	.	PUNCT
ejpam-6077	362	1	italian	italian	ADJ
ejpam-6077	362	2	journal	journal	NOUN
ejpam-6077	362	3	of	of	ADP
ejpam-6077	362	4	pure	pure	ADJ
ejpam-6077	362	5	and	and	CCONJ
ejpam-6077	362	6	applied	applied	ADJ
ejpam-6077	362	7	mathematics	mathematic	NOUN
ejpam-6077	362	8	,	,	PUNCT
ejpam-6077	362	9	43:185–196	43:185–196	NOUN
ejpam-6077	362	10	,	,	PUNCT
ejpam-6077	362	11	2020	2020	NUM
ejpam-6077	362	12	.	.	PUNCT
ejpam-6077	363	1	[	[	X
ejpam-6077	363	2	18	18	NUM
ejpam-6077	363	3	]	]	PUNCT
ejpam-6077	363	4	k.	k.	PROPN
ejpam-6077	363	5	abodayeh	abodayeh	PROPN
ejpam-6077	363	6	,	,	PUNCT
ejpam-6077	363	7	a.	a.	PROPN
ejpam-6077	363	8	bataihah	bataihah	PROPN
ejpam-6077	363	9	,	,	PUNCT
ejpam-6077	363	10	and	and	CCONJ
ejpam-6077	363	11	w.	w.	PROPN
ejpam-6077	363	12	shatanawi	shatanawi	PROPN
ejpam-6077	363	13	.	.	PUNCT
ejpam-6077	364	1	generalized	generalize	VERB
ejpam-6077	364	2	ω	ω	NUM
ejpam-6077	364	3	-	-	PUNCT
ejpam-6077	364	4	distance	distance	NOUN
ejpam-6077	364	5	mappings	mapping	NOUN
ejpam-6077	364	6	and	and	CCONJ
ejpam-6077	364	7	some	some	DET
ejpam-6077	364	8	fixed	fix	VERB
ejpam-6077	364	9	point	point	NOUN
ejpam-6077	364	10	theorems	theorem	NOUN
ejpam-6077	364	11	.	.	PUNCT
ejpam-6077	365	1	u.p.b	u.p.b	PROPN
ejpam-6077	365	2	.	.	PUNCT
ejpam-6077	366	1	scientific	scientific	ADJ
ejpam-6077	366	2	bulletin	bulletin	NOUN
ejpam-6077	366	3	,	,	PUNCT
ejpam-6077	366	4	series	series	NOUN
ejpam-6077	366	5	a	a	NOUN
ejpam-6077	366	6	,	,	PUNCT
ejpam-6077	366	7	79(4):223–232	79(4):223–232	NOUN
ejpam-6077	366	8	,	,	PUNCT
ejpam-6077	366	9	2017	2017	NUM
ejpam-6077	366	10	.	.	PUNCT
ejpam-6077	367	1	[	[	X
ejpam-6077	367	2	19	19	NUM
ejpam-6077	367	3	]	]	PUNCT
ejpam-6077	367	4	t.	t.	NOUN
ejpam-6077	367	5	qawasmeh	qawasmeh	NOUN
ejpam-6077	367	6	,	,	PUNCT
ejpam-6077	367	7	w.	w.	PROPN
ejpam-6077	367	8	shatanawi	shatanawi	PROPN
ejpam-6077	367	9	,	,	PUNCT
ejpam-6077	367	10	and	and	CCONJ
ejpam-6077	367	11	a.	a.	NOUN
ejpam-6077	367	12	bataihah	bataihah	PROPN
ejpam-6077	367	13	.	.	PUNCT
ejpam-6077	368	1	common	common	ADJ
ejpam-6077	368	2	fixed	fix	VERB
ejpam-6077	368	3	point	point	NOUN
ejpam-6077	368	4	results	result	NOUN
ejpam-6077	368	5	for	for	ADP
ejpam-6077	368	6	rational	rational	ADJ
ejpam-6077	368	7	(	(	PUNCT
ejpam-6077	368	8	α	α	NOUN
ejpam-6077	368	9	,	,	PUNCT
ejpam-6077	368	10	β)ϕ-mω	β)ϕ-mω	NOUN
ejpam-6077	368	11	contractions	contraction	NOUN
ejpam-6077	368	12	in	in	ADP
ejpam-6077	368	13	complete	complete	ADJ
ejpam-6077	368	14	quasi	quasi	ADJ
ejpam-6077	368	15	metric	metric	ADJ
ejpam-6077	368	16	spaces	space	NOUN
ejpam-6077	368	17	.	.	PUNCT
ejpam-6077	369	1	mathematics	mathematic	NOUN
ejpam-6077	369	2	,	,	PUNCT
ejpam-6077	369	3	7(5):392	7(5):392	NUM
ejpam-6077	369	4	,	,	PUNCT
ejpam-6077	369	5	2019	2019	NUM
ejpam-6077	369	6	.	.	PUNCT
ejpam-6077	370	1	[	[	X
ejpam-6077	370	2	20	20	NUM
ejpam-6077	370	3	]	]	PUNCT
ejpam-6077	370	4	a.	a.	NOUN
ejpam-6077	370	5	rabaiah	rabaiah	PROPN
ejpam-6077	370	6	,	,	PUNCT
ejpam-6077	370	7	a.	a.	NOUN
ejpam-6077	370	8	tallafha	tallafha	NOUN
ejpam-6077	370	9	,	,	PUNCT
ejpam-6077	370	10	and	and	CCONJ
ejpam-6077	370	11	w.	w.	PROPN
ejpam-6077	370	12	shatanawi	shatanawi	PROPN
ejpam-6077	370	13	.	.	PUNCT
ejpam-6077	371	1	common	common	ADJ
ejpam-6077	371	2	fixed	fix	VERB
ejpam-6077	371	3	point	point	NOUN
ejpam-6077	371	4	results	result	NOUN
ejpam-6077	371	5	for	for	ADP
ejpam-6077	371	6	mappings	mapping	NOUN
ejpam-6077	371	7	under	under	ADP
ejpam-6077	371	8	nonlinear	nonlinear	ADJ
ejpam-6077	371	9	contraction	contraction	NOUN
ejpam-6077	371	10	of	of	ADP
ejpam-6077	371	11	cyclic	cyclic	ADJ
ejpam-6077	371	12	form	form	NOUN
ejpam-6077	371	13	in	in	ADP
ejpam-6077	371	14	b	b	NOUN
ejpam-6077	371	15	-	-	ADJ
ejpam-6077	371	16	metric	metric	ADJ
ejpam-6077	371	17	spaces	space	NOUN
ejpam-6077	371	18	.	.	PUNCT
ejpam-6077	372	1	advances	advance	NOUN
ejpam-6077	372	2	in	in	ADP
ejpam-6077	372	3	mathematics	mathematic	NOUN
ejpam-6077	372	4	:	:	PUNCT
ejpam-6077	372	5	scientific	scientific	ADJ
ejpam-6077	372	6	journal	journal	NOUN
ejpam-6077	372	7	,	,	PUNCT
ejpam-6077	372	8	10(2):289–301	10(2):289–301	PROPN
ejpam-6077	372	9	,	,	PUNCT
ejpam-6077	372	10	2021	2021	NUM
ejpam-6077	372	11	.	.	PUNCT
ejpam-6077	373	1	[	[	X
ejpam-6077	373	2	21	21	NUM
ejpam-6077	373	3	]	]	PUNCT
ejpam-6077	373	4	a.	a.	NOUN
ejpam-6077	373	5	rabaiah	rabaiah	PROPN
ejpam-6077	373	6	,	,	PUNCT
ejpam-6077	373	7	a.	a.	NOUN
ejpam-6077	373	8	malkawi	malkawi	PROPN
ejpam-6077	373	9	,	,	PUNCT
ejpam-6077	373	10	a.	a.	PROPN
ejpam-6077	373	11	al	al	PROPN
ejpam-6077	373	12	-	-	PUNCT
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ejpam-6077	373	14	,	,	PUNCT
ejpam-6077	373	15	d.	d.	PROPN
ejpam-6077	373	16	mahmoud	mahmoud	PROPN
ejpam-6077	373	17	,	,	PUNCT
ejpam-6077	373	18	and	and	CCONJ
ejpam-6077	373	19	m.	m.	NOUN
ejpam-6077	373	20	qousini	qousini	PROPN
ejpam-6077	373	21	.	.	PUNCT
ejpam-6077	374	1	fixed	fix	VERB
ejpam-6077	374	2	point	point	NOUN
ejpam-6077	374	3	theorems	theorem	NOUN
ejpam-6077	374	4	in	in	ADP
ejpam-6077	374	5	mr	mr	PROPN
ejpam-6077	374	6	-	-	PUNCT
ejpam-6077	374	7	metric	metric	ADJ
ejpam-6077	374	8	space	space	NOUN
ejpam-6077	374	9	through	through	ADP
ejpam-6077	374	10	semi	semi	NOUN
ejpam-6077	374	11	-	-	NOUN
ejpam-6077	374	12	compatibility	compatibility	NOUN
ejpam-6077	374	13	.	.	PUNCT
ejpam-6077	375	1	advances	advance	NOUN
ejpam-6077	375	2	in	in	ADP
ejpam-6077	375	3	mathematics	mathematic	NOUN
ejpam-6077	375	4	:	:	PUNCT
ejpam-6077	375	5	scientific	scientific	ADJ
ejpam-6077	375	6	journal	journal	NOUN
ejpam-6077	375	7	,	,	PUNCT
ejpam-6077	375	8	10(6):2831–2845	10(6):2831–2845	NUM
ejpam-6077	375	9	,	,	PUNCT
ejpam-6077	375	10	2021	2021	NUM
ejpam-6077	375	11	.	.	PUNCT
ejpam-6077	376	1	[	[	X
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ejpam-6077	376	3	]	]	X
ejpam-6077	376	4	b.	b.	PROPN
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ejpam-6077	376	7	.	.	PUNCT
ejpam-6077	377	1	a	a	DET
ejpam-6077	377	2	fixed	fix	VERB
ejpam-6077	377	3	point	point	NOUN
ejpam-6077	377	4	theorem	theorem	NOUN
ejpam-6077	377	5	for	for	ADP
ejpam-6077	377	6	generalized	generalized	ADJ
ejpam-6077	377	7	metric	metric	ADJ
ejpam-6077	377	8	spaces	space	NOUN
ejpam-6077	377	9	.	.	PUNCT
ejpam-6077	378	1	international	international	ADJ
ejpam-6077	378	2	journal	journal	PROPN
ejpam-6077	378	3	of	of	ADP
ejpam-6077	378	4	mathematics	mathematics	PROPN
ejpam-6077	378	5	and	and	CCONJ
ejpam-6077	378	6	mathematical	mathematical	ADJ
ejpam-6077	378	7	sciences	science	NOUN
ejpam-6077	378	8	,	,	PUNCT
ejpam-6077	378	9	19(1):145–153	19(1):145–153	NUM
ejpam-6077	378	10	,	,	PUNCT
ejpam-6077	378	11	1996	1996	NUM
ejpam-6077	378	12	.	.	PUNCT
ejpam-6077	379	1	[	[	X
ejpam-6077	379	2	23	23	NUM
ejpam-6077	379	3	]	]	X
ejpam-6077	379	4	s.	s.	PROPN
ejpam-6077	379	5	sedghi	sedghi	PROPN
ejpam-6077	379	6	,	,	PUNCT
ejpam-6077	379	7	d.	d.	PROPN
ejpam-6077	379	8	turkoglu	turkoglu	PROPN
ejpam-6077	379	9	,	,	PUNCT
ejpam-6077	379	10	and	and	CCONJ
ejpam-6077	379	11	n.	n.	PROPN
ejpam-6077	379	12	shobe	shobe	PROPN
ejpam-6077	379	13	.	.	PUNCT
ejpam-6077	380	1	common	common	ADJ
ejpam-6077	380	2	fixed	fix	VERB
ejpam-6077	380	3	point	point	NOUN
ejpam-6077	380	4	theorems	theorem	NOUN
ejpam-6077	380	5	for	for	ADP
ejpam-6077	380	6	six	six	NUM
ejpam-6077	380	7	weakly	weakly	ADJ
ejpam-6077	380	8	compatible	compatible	ADJ
ejpam-6077	380	9	mappings	mapping	NOUN
ejpam-6077	380	10	in	in	ADP
ejpam-6077	380	11	d∗-metric	d∗-metric	ADJ
ejpam-6077	380	12	spaces	space	NOUN
ejpam-6077	380	13	.	.	PUNCT
ejpam-6077	381	1	thai	thai	PROPN
ejpam-6077	381	2	journal	journal	PROPN
ejpam-6077	381	3	of	of	ADP
ejpam-6077	381	4	mathematics	mathematic	NOUN
ejpam-6077	381	5	,	,	PUNCT
ejpam-6077	381	6	7(2):381	7(2):381	NUM
ejpam-6077	381	7	–	–	PUNCT
ejpam-6077	381	8	391	391	NUM
ejpam-6077	381	9	,	,	PUNCT
ejpam-6077	381	10	2009	2009	NUM
ejpam-6077	381	11	.	.	PUNCT
ejpam-6077	382	1	[	[	X
ejpam-6077	382	2	24	24	NUM
ejpam-6077	382	3	]	]	PUNCT
ejpam-6077	382	4	a.	a.	NOUN
ejpam-6077	382	5	branciari	branciari	PROPN
ejpam-6077	382	6	.	.	PUNCT
ejpam-6077	383	1	a	a	DET
ejpam-6077	383	2	fixed	fix	VERB
ejpam-6077	383	3	point	point	NOUN
ejpam-6077	383	4	theorem	theorem	NOUN
ejpam-6077	383	5	for	for	ADP
ejpam-6077	383	6	mappings	mapping	NOUN
ejpam-6077	383	7	satisfying	satisfy	VERB
ejpam-6077	383	8	a	a	DET
ejpam-6077	383	9	general	general	ADJ
ejpam-6077	383	10	contractive	contractive	ADJ
ejpam-6077	383	11	condition	condition	NOUN
ejpam-6077	383	12	of	of	ADP
ejpam-6077	383	13	integral	integral	ADJ
ejpam-6077	383	14	type	type	NOUN
ejpam-6077	383	15	.	.	PUNCT
ejpam-6077	384	1	international	international	ADJ
ejpam-6077	384	2	journal	journal	PROPN
ejpam-6077	384	3	of	of	ADP
ejpam-6077	384	4	mathematics	mathematics	PROPN
ejpam-6077	384	5	and	and	CCONJ
ejpam-6077	384	6	mathematical	mathematical	ADJ
ejpam-6077	384	7	sciences	science	NOUN
ejpam-6077	384	8	,	,	PUNCT
ejpam-6077	384	9	29(9):531–536	29(9):531–536	NUM
ejpam-6077	384	10	,	,	PUNCT
ejpam-6077	384	11	2002	2002	NUM
ejpam-6077	384	12	.	.	PUNCT
ejpam-6077	385	1	[	[	X
ejpam-6077	385	2	25	25	NUM
ejpam-6077	385	3	]	]	PUNCT
ejpam-6077	385	4	t.	t.	NOUN
ejpam-6077	385	5	qawasmeh	qawasmeh	NOUN
ejpam-6077	385	6	.	.	PUNCT
ejpam-6077	386	1	(	(	PUNCT
ejpam-6077	386	2	h	h	NOUN
ejpam-6077	386	3	,	,	PUNCT
ejpam-6077	386	4	ωb)-interpolative	ωb)-interpolative	ADJ
ejpam-6077	386	5	contractions	contraction	NOUN
ejpam-6077	386	6	in	in	ADP
ejpam-6077	386	7	ωb	ωb	NOUN
ejpam-6077	386	8	-	-	PUNCT
ejpam-6077	386	9	distance	distance	NOUN
ejpam-6077	386	10	mappings	mapping	NOUN
ejpam-6077	386	11	with	with	ADP
ejpam-6077	386	12	applications	application	NOUN
ejpam-6077	386	13	.	.	PUNCT
ejpam-6077	387	1	european	european	ADJ
ejpam-6077	387	2	journal	journal	PROPN
ejpam-6077	387	3	of	of	ADP
ejpam-6077	387	4	pure	pure	ADJ
ejpam-6077	387	5	and	and	CCONJ
ejpam-6077	387	6	applied	applied	ADJ
ejpam-6077	387	7	mathematics	mathematic	NOUN
ejpam-6077	387	8	,	,	PUNCT
ejpam-6077	387	9	16(3):1717–1730	16(3):1717–1730	NUM
ejpam-6077	387	10	,	,	PUNCT
ejpam-6077	387	11	2023	2023	NUM
ejpam-6077	387	12	.	.	PUNCT
ejpam-6077	388	1	[	[	X
ejpam-6077	388	2	26	26	NUM
ejpam-6077	388	3	]	]	PUNCT
ejpam-6077	388	4	k.	k.	PROPN
ejpam-6077	388	5	abodayeh	abodayeh	PROPN
ejpam-6077	388	6	,	,	PUNCT
ejpam-6077	388	7	t.	t.	NOUN
ejpam-6077	388	8	qawasmeh	qawasmeh	NOUN
ejpam-6077	388	9	,	,	PUNCT
ejpam-6077	388	10	w.	w.	PROPN
ejpam-6077	388	11	shatanawi	shatanawi	PROPN
ejpam-6077	388	12	,	,	PUNCT
ejpam-6077	388	13	and	and	CCONJ
ejpam-6077	388	14	a.	a.	NOUN
ejpam-6077	388	15	tallafha	tallafha	NOUN
ejpam-6077	388	16	.	.	PUNCT
ejpam-6077	389	1	ϵφ	ϵφ	NOUN
ejpam-6077	389	2	-	-	NOUN
ejpam-6077	389	3	contraction	contraction	NOUN
ejpam-6077	389	4	and	and	CCONJ
ejpam-6077	389	5	some	some	DET
ejpam-6077	389	6	fixed	fix	VERB
ejpam-6077	389	7	point	point	NOUN
ejpam-6077	389	8	results	result	NOUN
ejpam-6077	389	9	via	via	ADP
ejpam-6077	389	10	modified	modify	VERB
ejpam-6077	389	11	ω	ω	VERB
ejpam-6077	389	12	-	-	PUNCT
ejpam-6077	389	13	distance	distance	NOUN
ejpam-6077	389	14	mappings	mapping	NOUN
ejpam-6077	389	15	in	in	ADP
ejpam-6077	389	16	the	the	DET
ejpam-6077	389	17	frame	frame	NOUN
ejpam-6077	389	18	of	of	ADP
ejpam-6077	389	19	complete	complete	ADJ
ejpam-6077	389	20	quasi	quasi	ADJ
ejpam-6077	389	21	metric	metric	ADJ
ejpam-6077	389	22	spaces	space	NOUN
ejpam-6077	389	23	and	and	CCONJ
ejpam-6077	389	24	applications	application	NOUN
ejpam-6077	389	25	.	.	PUNCT
ejpam-6077	390	1	international	international	ADJ
ejpam-6077	390	2	journal	journal	NOUN
ejpam-6077	390	3	of	of	ADP
ejpam-6077	390	4	electrical	electrical	ADJ
ejpam-6077	390	5	and	and	CCONJ
ejpam-6077	390	6	computer	computer	NOUN
ejpam-6077	390	7	engineering	engineering	NOUN
ejpam-6077	390	8	,	,	PUNCT
ejpam-6077	390	9	10(4):3839–3853	10(4):3839–3853	NUM
ejpam-6077	390	10	,	,	PUNCT
ejpam-6077	390	11	2020	2020	NUM
ejpam-6077	390	12	.	.	PUNCT
ejpam-6077	391	1	[	[	X
ejpam-6077	391	2	27	27	NUM
ejpam-6077	391	3	]	]	PUNCT
ejpam-6077	391	4	t.	t.	NOUN
ejpam-6077	391	5	qawasmeh	qawasmeh	NOUN
ejpam-6077	391	6	,	,	PUNCT
ejpam-6077	391	7	a.	a.	NOUN
ejpam-6077	391	8	tallafha	tallafha	NOUN
ejpam-6077	391	9	,	,	PUNCT
ejpam-6077	391	10	and	and	CCONJ
ejpam-6077	391	11	w.	w.	PROPN
ejpam-6077	391	12	shatanawi	shatanawi	PROPN
ejpam-6077	391	13	.	.	PUNCT
ejpam-6077	392	1	fixed	fix	VERB
ejpam-6077	392	2	and	and	CCONJ
ejpam-6077	392	3	common	common	ADJ
ejpam-6077	392	4	fixed	fix	VERB
ejpam-6077	392	5	point	point	NOUN
ejpam-6077	392	6	theorems	theorem	NOUN
ejpam-6077	392	7	through	through	ADP
ejpam-6077	392	8	modified	modify	VERB
ejpam-6077	392	9	ω	ω	NUM
ejpam-6077	392	10	-	-	PUNCT
ejpam-6077	392	11	distance	distance	NOUN
ejpam-6077	392	12	mappings	mapping	NOUN
ejpam-6077	392	13	.	.	PUNCT
ejpam-6077	393	1	nonlinear	nonlinear	ADJ
ejpam-6077	393	2	functional	functional	ADJ
ejpam-6077	393	3	analysis	analysis	NOUN
ejpam-6077	393	4	and	and	CCONJ
ejpam-6077	393	5	applications	application	NOUN
ejpam-6077	393	6	,	,	PUNCT
ejpam-6077	393	7	24(2):221–239	24(2):221–239	NUM
ejpam-6077	393	8	,	,	PUNCT
ejpam-6077	393	9	2019	2019	NUM
ejpam-6077	393	10	.	.	PUNCT
ejpam-6077	394	1	[	[	X
ejpam-6077	394	2	28	28	NUM
ejpam-6077	394	3	]	]	X
ejpam-6077	394	4	a.	a.	NOUN
ejpam-6077	394	5	malkawi	malkawi	PROPN
ejpam-6077	394	6	,	,	PUNCT
ejpam-6077	394	7	a.	a.	PROPN
ejpam-6077	394	8	rabaiah	rabaiah	PROPN
ejpam-6077	394	9	,	,	PUNCT
ejpam-6077	394	10	w.	w.	PROPN
ejpam-6077	394	11	shatanawi	shatanawi	PROPN
ejpam-6077	394	12	,	,	PUNCT
ejpam-6077	394	13	and	and	CCONJ
ejpam-6077	394	14	a.	a.	NOUN
ejpam-6077	394	15	tallafha	tallafha	NOUN
ejpam-6077	394	16	.	.	PUNCT
ejpam-6077	395	1	mr	mr	PROPN
ejpam-6077	395	2	-	-	PUNCT
ejpam-6077	395	3	metric	metric	ADJ
ejpam-6077	395	4	spaces	space	NOUN
ejpam-6077	395	5	and	and	CCONJ
ejpam-6077	395	6	an	an	DET
ejpam-6077	395	7	application	application	NOUN
ejpam-6077	395	8	.	.	PUNCT
ejpam-6077	396	1	preprint	preprint	NOUN
ejpam-6077	396	2	,	,	PUNCT
ejpam-6077	396	3	2021	2021	NUM
ejpam-6077	396	4	.	.	PUNCT
ejpam-6077	397	1	[	[	X
ejpam-6077	397	2	29	29	NUM
ejpam-6077	397	3	]	]	X
ejpam-6077	397	4	w.	w.	PROPN
ejpam-6077	397	5	rudin	rudin	PROPN
ejpam-6077	397	6	.	.	PUNCT
ejpam-6077	398	1	functional	functional	ADJ
ejpam-6077	398	2	analysis	analysis	NOUN
ejpam-6077	398	3	.	.	PUNCT
ejpam-6077	399	1	mcgraw	mcgraw	PROPN
ejpam-6077	399	2	-	-	PUNCT
ejpam-6077	399	3	hill	hill	PROPN
ejpam-6077	399	4	,	,	PUNCT
ejpam-6077	399	5	new	new	PROPN
ejpam-6077	399	6	york	york	PROPN
ejpam-6077	399	7	,	,	PUNCT
ejpam-6077	399	8	2	2	NUM
ejpam-6077	399	9	edition	edition	NOUN
ejpam-6077	399	10	,	,	PUNCT
ejpam-6077	399	11	1991	1991	NUM
ejpam-6077	399	12	.	.	PUNCT
ejpam-6077	400	1	[	[	X
ejpam-6077	400	2	30	30	NUM
ejpam-6077	400	3	]	]	X
ejpam-6077	400	4	h.	h.	PROPN
ejpam-6077	400	5	l.	l.	PROPN
ejpam-6077	400	6	royden	royden	PROPN
ejpam-6077	400	7	and	and	CCONJ
ejpam-6077	400	8	p.	p.	PROPN
ejpam-6077	400	9	m.	m.	PROPN
ejpam-6077	400	10	fitzpatrick	fitzpatrick	PROPN
ejpam-6077	400	11	.	.	PUNCT
ejpam-6077	401	1	real	real	ADJ
ejpam-6077	401	2	analysis	analysis	NOUN
ejpam-6077	401	3	.	.	PUNCT
ejpam-6077	402	1	pearson	pearson	PROPN
ejpam-6077	402	2	,	,	PUNCT
ejpam-6077	402	3	upper	upper	ADJ
ejpam-6077	402	4	saddle	saddle	NOUN
ejpam-6077	402	5	river	river	PROPN
ejpam-6077	402	6	,	,	PUNCT
ejpam-6077	402	7	nj	nj	PROPN
ejpam-6077	402	8	,	,	PUNCT
ejpam-6077	402	9	4	4	NUM
ejpam-6077	402	10	edition	edition	NOUN
ejpam-6077	402	11	,	,	PUNCT
ejpam-6077	402	12	2010	2010	NUM
ejpam-6077	402	13	.	.	PUNCT