id	sid	tid	token	lemma	pos
ma-255	1	1	2025	2025	NUM
ma-255	1	2	ada	ada	PROPN
ma-255	1	3	academica	academica	PROPN
ma-255	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-255	1	5	.	.	PUNCT
ma-255	2	1	j.	j.	PROPN
ma-255	2	2	math	math	PROPN
ma-255	2	3	.	.	PUNCT
ma-255	3	1	anal	anal	ADJ
ma-255	3	2	.	.	PUNCT
ma-255	4	1	5	5	NUM
ma-255	4	2	(	(	PUNCT
ma-255	4	3	2025	2025	NUM
ma-255	4	4	)	)	PUNCT
ma-255	5	1	5doi	5doi	NOUN
ma-255	5	2	:	:	PUNCT
ma-255	5	3	10.28924	10.28924	NUM
ma-255	5	4	/	/	SYM
ma-255	5	5	ada	ada	PROPN
ma-255	5	6	/	/	SYM
ma-255	5	7	ma.5.5	ma.5.5	PROPN
ma-255	5	8	hybrid	hybrid	ADJ
ma-255	5	9	iterative	iterative	NOUN
ma-255	5	10	methods	method	NOUN
ma-255	5	11	for	for	ADP
ma-255	5	12	solving	solve	VERB
ma-255	5	13	nonlinear	nonlinear	ADJ
ma-255	5	14	equations	equation	NOUN
ma-255	5	15	in	in	ADP
ma-255	5	16	banach	banach	NOUN
ma-255	5	17	spaces	space	NOUN
ma-255	5	18	ioannis	ioannis	PROPN
ma-255	5	19	k.	k.	PROPN
ma-255	5	20	argyros1,∗	argyros1,∗	PROPN
ma-255	5	21	,	,	PUNCT
ma-255	5	22	santhosh	santhosh	PROPN
ma-255	5	23	george2	george2	PROPN
ma-255	5	24	,	,	PUNCT
ma-255	5	25	samundra	samundra	PROPN
ma-255	5	26	regmi3	regmi3	PROPN
ma-255	5	27	,	,	PUNCT
ma-255	5	28	michael	michael	PROPN
ma-255	5	29	i.	i.	PROPN
ma-255	5	30	argyros4	argyros4	PROPN
ma-255	6	1	1department	1department	NUM
ma-255	6	2	of	of	ADP
ma-255	6	3	mathematical	mathematical	ADJ
ma-255	6	4	sciences	sciences	PROPN
ma-255	6	5	,	,	PUNCT
ma-255	6	6	cameron	cameron	PROPN
ma-255	6	7	university	university	PROPN
ma-255	6	8	,	,	PUNCT
ma-255	6	9	lawton	lawton	PROPN
ma-255	6	10	,	,	PUNCT
ma-255	6	11	ok	ok	PROPN
ma-255	6	12	73505	73505	NUM
ma-255	6	13	,	,	PUNCT
ma-255	6	14	usa	usa	PROPN
ma-255	6	15	iargyros@cameron.edu	iargyros@cameron.edu	PROPN
ma-255	7	1	2department	2department	NUM
ma-255	7	2	of	of	ADP
ma-255	7	3	mathematical	mathematical	ADJ
ma-255	7	4	and	and	CCONJ
ma-255	7	5	computational	computational	ADJ
ma-255	7	6	sciences	science	NOUN
ma-255	7	7	,	,	PUNCT
ma-255	7	8	national	national	PROPN
ma-255	7	9	institute	institute	PROPN
ma-255	7	10	of	of	ADP
ma-255	7	11	technology	technology	PROPN
ma-255	7	12	karnataka	karnataka	PROPN
ma-255	7	13	,	,	PUNCT
ma-255	7	14	india-575	india-575	ADJ
ma-255	7	15	025	025	NUM
ma-255	7	16	sgeorge@nitk.edu.in	sgeorge@nitk.edu.in	NOUN
ma-255	7	17	3department	3department	NUM
ma-255	7	18	of	of	ADP
ma-255	7	19	mathematics	mathematic	NOUN
ma-255	7	20	,	,	PUNCT
ma-255	7	21	university	university	PROPN
ma-255	7	22	of	of	ADP
ma-255	7	23	houston	houston	PROPN
ma-255	7	24	,	,	PUNCT
ma-255	7	25	houston	houston	PROPN
ma-255	7	26	,	,	PUNCT
ma-255	7	27	tx	tx	PROPN
ma-255	7	28	77204	77204	NUM
ma-255	7	29	,	,	PUNCT
ma-255	7	30	usa	usa	PROPN
ma-255	7	31	sregmi5@uh.edu	sregmi5@uh.edu	PROPN
ma-255	7	32	4department	4department	NUM
ma-255	7	33	of	of	ADP
ma-255	7	34	computer	computer	NOUN
ma-255	7	35	science	science	NOUN
ma-255	7	36	,	,	PUNCT
ma-255	7	37	university	university	PROPN
ma-255	7	38	of	of	ADP
ma-255	7	39	oklahoma	oklahoma	PROPN
ma-255	7	40	,	,	PUNCT
ma-255	7	41	norman	norman	PROPN
ma-255	7	42	,	,	PUNCT
ma-255	7	43	ok	ok	ADJ
ma-255	7	44	73501	73501	NUM
ma-255	7	45	,	,	PUNCT
ma-255	7	46	usa	usa	PROPN
ma-255	7	47	argyro01@email.franklin.edu	argyro01@email.franklin.edu	PROPN
ma-255	7	48	abstract	abstract	NOUN
ma-255	7	49	.	.	PUNCT
ma-255	8	1	the	the	DET
ma-255	8	2	present	present	ADJ
ma-255	8	3	article	article	NOUN
ma-255	8	4	contributes	contribute	VERB
ma-255	8	5	to	to	ADP
ma-255	8	6	the	the	DET
ma-255	8	7	solution	solution	NOUN
ma-255	8	8	of	of	ADP
ma-255	8	9	equations	equation	NOUN
ma-255	8	10	which	which	PRON
ma-255	8	11	carry	carry	VERB
ma-255	8	12	the	the	DET
ma-255	8	13	symmetryproperty	symmetryproperty	NOUN
ma-255	8	14	of	of	ADP
ma-255	8	15	the	the	DET
ma-255	8	16	problem	problem	NOUN
ma-255	8	17	or	or	CCONJ
ma-255	8	18	not	not	PART
ma-255	8	19	.	.	PUNCT
ma-255	9	1	iterative	iterative	NOUN
ma-255	9	2	methods	method	NOUN
ma-255	9	3	with	with	ADP
ma-255	9	4	inverses	inverse	NOUN
ma-255	9	5	generate	generate	VERB
ma-255	9	6	sequences	sequence	NOUN
ma-255	9	7	convergingfaster	convergingfaster	ADJ
ma-255	9	8	to	to	ADP
ma-255	9	9	a	a	DET
ma-255	9	10	solution	solution	NOUN
ma-255	9	11	of	of	ADP
ma-255	9	12	an	an	DET
ma-255	9	13	equation	equation	NOUN
ma-255	9	14	than	than	ADP
ma-255	9	15	methods	method	NOUN
ma-255	9	16	without	without	ADP
ma-255	9	17	inverses	inverse	NOUN
ma-255	9	18	.	.	PUNCT
ma-255	10	1	however	however	ADV
ma-255	10	2	,	,	PUNCT
ma-255	10	3	the	the	DET
ma-255	10	4	implementationof	implementationof	NOUN
ma-255	10	5	these	these	DET
ma-255	10	6	methods	method	NOUN
ma-255	10	7	has	have	VERB
ma-255	10	8	drawbacks	drawback	NOUN
ma-255	10	9	,	,	PUNCT
ma-255	10	10	since	since	SCONJ
ma-255	10	11	the	the	DET
ma-255	10	12	analytical	analytical	ADJ
ma-255	10	13	form	form	NOUN
ma-255	10	14	of	of	ADP
ma-255	10	15	these	these	DET
ma-255	10	16	inverse	inverse	NOUN
ma-255	10	17	may	may	AUX
ma-255	10	18	be	be	AUX
ma-255	10	19	unavailable	unavailable	ADJ
ma-255	10	20	orcomputationally	orcomputationally	ADV
ma-255	10	21	very	very	ADV
ma-255	10	22	expensive	expensive	ADJ
ma-255	10	23	.	.	PUNCT
ma-255	11	1	this	this	DET
ma-255	11	2	problem	problem	NOUN
ma-255	11	3	is	be	AUX
ma-255	11	4	addressed	address	VERB
ma-255	11	5	in	in	ADP
ma-255	11	6	this	this	DET
ma-255	11	7	paper	paper	NOUN
ma-255	11	8	by	by	ADP
ma-255	11	9	replacing	replace	VERB
ma-255	11	10	the	the	DET
ma-255	11	11	inversewith	inversewith	NOUN
ma-255	11	12	a	a	DET
ma-255	11	13	finite	finite	ADJ
ma-255	11	14	sum	sum	NOUN
ma-255	11	15	of	of	ADP
ma-255	11	16	linear	linear	PROPN
ma-255	11	17	operators	operator	NOUN
ma-255	11	18	.	.	PUNCT
ma-255	12	1	a	a	DET
ma-255	12	2	convergence	convergence	NOUN
ma-255	12	3	analysis	analysis	NOUN
ma-255	12	4	is	be	AUX
ma-255	12	5	developed	develop	VERB
ma-255	12	6	for	for	ADP
ma-255	12	7	the	the	DET
ma-255	12	8	hybrid	hybrid	ADJ
ma-255	12	9	methods.the	methods.the	DET
ma-255	12	10	numerical	numerical	ADJ
ma-255	12	11	examples	example	NOUN
ma-255	12	12	demonstrate	demonstrate	VERB
ma-255	12	13	that	that	SCONJ
ma-255	12	14	the	the	DET
ma-255	12	15	number	number	NOUN
ma-255	12	16	of	of	ADP
ma-255	12	17	iterates	iterate	NOUN
ma-255	12	18	is	be	AUX
ma-255	12	19	essentially	essentially	ADV
ma-255	12	20	the	the	DET
ma-255	12	21	same	same	ADJ
ma-255	12	22	between	between	ADP
ma-255	12	23	thehybrid	thehybrid	NOUN
ma-255	12	24	and	and	CCONJ
ma-255	12	25	the	the	DET
ma-255	12	26	original	original	ADJ
ma-255	12	27	method	method	NOUN
ma-255	12	28	.	.	PUNCT
ma-255	13	1	this	this	DET
ma-255	13	2	technique	technique	NOUN
ma-255	13	3	is	be	AUX
ma-255	13	4	also	also	ADV
ma-255	13	5	extended	extend	VERB
ma-255	13	6	to	to	PART
ma-255	13	7	solve	solve	VERB
ma-255	13	8	generalized	generalized	ADJ
ma-255	13	9	equations	equation	NOUN
ma-255	13	10	.	.	PUNCT
ma-255	14	1	1	1	X
ma-255	14	2	.	.	X
ma-255	14	3	introduction	introduction	NOUN
ma-255	14	4	the	the	DET
ma-255	14	5	letters	letter	NOUN
ma-255	14	6	x	x	PRON
ma-255	14	7	,	,	PUNCT
ma-255	14	8	y	y	PROPN
ma-255	14	9	denote	denote	VERB
ma-255	14	10	banach	banach	NOUN
ma-255	14	11	spaces	space	VERB
ma-255	14	12	;	;	PUNCT
ma-255	14	13	ω	ω	X
ma-255	14	14	⊂	⊂	PROPN
ma-255	14	15	x	x	X
ma-255	14	16	is	be	AUX
ma-255	14	17	a	a	DET
ma-255	14	18	convex	convex	NOUN
ma-255	14	19	and	and	CCONJ
ma-255	14	20	open	open	ADJ
ma-255	14	21	subset	subset	NOUN
ma-255	14	22	of	of	ADP
ma-255	14	23	x	x	X
ma-255	14	24	,	,	PUNCT
ma-255	14	25	and	and	CCONJ
ma-255	14	26	f1	f1	NOUN
ma-255	14	27	:	:	PUNCT
ma-255	15	1	ω	ω	NUM
ma-255	15	2	−→	−→	NOUN
ma-255	15	3	y	y	PROPN
ma-255	15	4	stands	stand	VERB
ma-255	15	5	for	for	ADP
ma-255	15	6	a	a	DET
ma-255	15	7	continuous	continuous	ADJ
ma-255	15	8	operator	operator	NOUN
ma-255	15	9	.	.	PUNCT
ma-255	16	1	numerous	numerous	ADJ
ma-255	16	2	applications	application	NOUN
ma-255	16	3	from	from	ADP
ma-255	16	4	diverse	diverse	ADJ
ma-255	16	5	areas	area	NOUN
ma-255	16	6	of	of	ADP
ma-255	16	7	computationalscience	computationalscience	NOUN
ma-255	16	8	and	and	CCONJ
ma-255	16	9	engineering	engineering	NOUN
ma-255	16	10	can	can	AUX
ma-255	16	11	be	be	AUX
ma-255	16	12	converted	convert	VERB
ma-255	16	13	by	by	ADP
ma-255	16	14	using	use	VERB
ma-255	16	15	mathematical	mathematical	ADJ
ma-255	16	16	modelling	modelling	NOUN
ma-255	16	17	[	[	X
ma-255	16	18	3	3	NUM
ma-255	16	19	,	,	PUNCT
ma-255	16	20	8	8	NUM
ma-255	16	21	,	,	PUNCT
ma-255	16	22	14	14	NUM
ma-255	16	23	,	,	PUNCT
ma-255	16	24	17,19–21,23,26,28,33,35	17,19–21,23,26,28,33,35	NUM
ma-255	16	25	]	]	PUNCT
ma-255	16	26	to	to	ADP
ma-255	16	27	finding	find	VERB
ma-255	16	28	a	a	DET
ma-255	16	29	solution	solution	NOUN
ma-255	16	30	s∗	s∗	PROPN
ma-255	16	31	∈	∈	PROPN
ma-255	16	32	ω	ω	PROPN
ma-255	16	33	of	of	ADP
ma-255	16	34	the	the	DET
ma-255	16	35	nonlinear	nonlinear	NOUN
ma-255	16	36	in	in	ADP
ma-255	16	37	the	the	DET
ma-255	16	38	general	general	ADJ
ma-255	16	39	equation	equation	NOUN
ma-255	16	40	f1(x	f1(x	NOUN
ma-255	16	41	)	)	PUNCT
ma-255	16	42	=	=	SYM
ma-255	17	1	0	0	X
ma-255	17	2	.	.	PUNCT
ma-255	18	1	(	(	PUNCT
ma-255	18	2	1.1	1.1	NUM
ma-255	18	3	)	)	PUNCT
ma-255	18	4	the	the	DET
ma-255	18	5	closed	closed	ADJ
ma-255	18	6	form	form	NOUN
ma-255	18	7	of	of	ADP
ma-255	18	8	the	the	DET
ma-255	18	9	solution	solution	NOUN
ma-255	18	10	s∗	s∗	PROPN
ma-255	18	11	is	be	AUX
ma-255	18	12	attainable	attainable	ADJ
ma-255	18	13	only	only	ADV
ma-255	18	14	in	in	ADP
ma-255	18	15	special	special	ADJ
ma-255	18	16	cases	case	NOUN
ma-255	18	17	.	.	PUNCT
ma-255	19	1	this	this	DET
ma-255	19	2	forces	force	NOUN
ma-255	19	3	researchers	researcher	NOUN
ma-255	19	4	andpractitioners	andpractitioner	NOUN
ma-255	19	5	to	to	PART
ma-255	19	6	solve	solve	VERB
ma-255	19	7	the	the	DET
ma-255	19	8	equation	equation	NOUN
ma-255	19	9	(	(	PUNCT
ma-255	19	10	1.1	1.1	NUM
ma-255	19	11	)	)	PUNCT
ma-255	19	12	iteratively	iteratively	ADV
ma-255	19	13	.	.	PUNCT
ma-255	20	1	single	single	ADJ
ma-255	20	2	-	-	PUNCT
ma-255	20	3	step	step	NOUN
ma-255	20	4	methods	method	NOUN
ma-255	20	5	of	of	ADP
ma-255	20	6	high	high	ADJ
ma-255	20	7	convergence	convergence	NOUN
ma-255	20	8	order	order	NOUN
ma-255	20	9	received	receive	VERB
ma-255	20	10	:	:	PUNCT
ma-255	20	11	3	3	NUM
ma-255	20	12	jul	jul	PROPN
ma-255	20	13	2024	2024	NUM
ma-255	20	14	.	.	PUNCT
ma-255	21	1	key	key	ADJ
ma-255	21	2	words	word	NOUN
ma-255	21	3	and	and	CCONJ
ma-255	21	4	phrases	phrase	NOUN
ma-255	21	5	.	.	PUNCT
ma-255	22	1	inverse	inverse	NOUN
ma-255	22	2	of	of	ADP
ma-255	22	3	an	an	DET
ma-255	22	4	operator	operator	NOUN
ma-255	22	5	,	,	PUNCT
ma-255	22	6	banach	banach	NOUN
ma-255	22	7	space	space	NOUN
ma-255	22	8	,	,	PUNCT
ma-255	22	9	hybrid	hybrid	ADJ
ma-255	22	10	iterative	iterative	NOUN
ma-255	22	11	method	method	NOUN
ma-255	22	12	,	,	PUNCT
ma-255	22	13	generalized	generalized	ADJ
ma-255	22	14	equations	equation	NOUN
ma-255	22	15	,	,	PUNCT
ma-255	22	16	continuous	continuous	ADJ
ma-255	22	17	operator	operator	NOUN
ma-255	22	18	,	,	PUNCT
ma-255	22	19	convergence	convergence	NOUN
ma-255	22	20	.	.	PUNCT
ma-255	23	1	1	1	NUM
ma-255	23	2	https://adac.ee	https://adac.ee	PROPN
ma-255	23	3	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	23	4	eur	eur	PROPN
ma-255	23	5	.	.	PUNCT
ma-255	24	1	j.	j.	PROPN
ma-255	24	2	math	math	PROPN
ma-255	24	3	.	.	PUNCT
ma-255	25	1	anal	anal	PROPN
ma-255	25	2	.	.	PUNCT
ma-255	26	1	10.28924	10.28924	NUM
ma-255	26	2	/	/	SYM
ma-255	26	3	ada	ada	PROPN
ma-255	26	4	/	/	SYM
ma-255	26	5	ma.5.5	ma.5.5	PROPN
ma-255	26	6	2look	2look	NUM
ma-255	26	7	like	like	ADP
ma-255	26	8	the	the	DET
ma-255	26	9	newton	newton	NOUN
ma-255	26	10	-	-	PUNCT
ma-255	26	11	type	type	NOUN
ma-255	26	12	defined	define	VERB
ma-255	26	13	for	for	ADP
ma-255	26	14	each	each	DET
ma-255	26	15	n	n	NOUN
ma-255	26	16	=	=	SYM
ma-255	26	17	0	0	NUM
ma-255	26	18	,	,	PUNCT
ma-255	26	19	1	1	NUM
ma-255	26	20	,	,	PUNCT
ma-255	26	21	2	2	NUM
ma-255	26	22	,	,	PUNCT
ma-255	26	23	...	...	PUNCT
ma-255	26	24	by	by	ADP
ma-255	26	25	x0	x0	PROPN
ma-255	26	26	∈	∈	PROPN
ma-255	26	27	ω	ω	PROPN
ma-255	26	28	,	,	PUNCT
ma-255	26	29	xn+1	xn+1	PROPN
ma-255	26	30	=	=	SYM
ma-255	26	31	xn	xn	PROPN
ma-255	26	32	−	−	NOUN
ma-255	26	33	l−1n	l−1n	NOUN
ma-255	26	34	f1(xn	f1(xn	NUM
ma-255	26	35	)	)	PUNCT
ma-255	26	36	,	,	PUNCT
ma-255	26	37	(	(	PUNCT
ma-255	26	38	1.2	1.2	NUM
ma-255	26	39	)	)	PUNCT
ma-255	26	40	where	where	SCONJ
ma-255	26	41	ln	ln	NOUN
ma-255	26	42	∈	∈	PROPN
ma-255	26	43	l(x	l(x	PROPN
ma-255	26	44	,	,	PUNCT
ma-255	26	45	y	y	PROPN
ma-255	26	46	)	)	PUNCT
ma-255	26	47	which	which	PRON
ma-255	26	48	is	be	AUX
ma-255	26	49	the	the	DET
ma-255	26	50	space	space	NOUN
ma-255	26	51	of	of	ADP
ma-255	26	52	continuous	continuous	ADJ
ma-255	26	53	operators	operator	NOUN
ma-255	26	54	mapping	map	VERB
ma-255	26	55	x	x	PUNCT
ma-255	26	56	into	into	ADP
ma-255	26	57	y	y	PROPN
ma-255	26	58	,	,	PUNCT
ma-255	26	59	and	and	CCONJ
ma-255	26	60	l−1n	l−1n	PROPN
ma-255	26	61	∈	∈	PROPN
ma-255	26	62	l(y	l(y	PROPN
ma-255	26	63	,	,	PUNCT
ma-255	26	64	x	x	NOUN
ma-255	26	65	)	)	PUNCT
ma-255	26	66	for	for	ADP
ma-255	26	67	each	each	DET
ma-255	26	68	n	n	NOUN
ma-255	26	69	=	=	SYM
ma-255	26	70	0	0	NUM
ma-255	26	71	,	,	PUNCT
ma-255	26	72	1	1	NUM
ma-255	26	73	,	,	PUNCT
ma-255	26	74	2	2	NUM
ma-255	26	75	,	,	PUNCT
ma-255	26	76	....	....	PUNCT
ma-255	26	77	by	by	ADP
ma-255	26	78	ln	ln	ADV
ma-255	26	79	,	,	PUNCT
ma-255	26	80	we	we	PRON
ma-255	26	81	denote	denote	VERB
ma-255	26	82	l(xn	l(xn	NOUN
ma-255	26	83	)	)	PUNCT
ma-255	26	84	.	.	PUNCT
ma-255	27	1	some	some	DET
ma-255	27	2	choices	choice	NOUN
ma-255	27	3	for	for	ADP
ma-255	27	4	the	the	DET
ma-255	27	5	operator	operator	NOUN
ma-255	27	6	ln	ln	ADV
ma-255	27	7	can	can	AUX
ma-255	27	8	be	be	AUX
ma-255	27	9	ln	ln	ADJ
ma-255	27	10	=	=	PUNCT
ma-255	27	11	f	f	NOUN
ma-255	27	12	′1(xn	′1(xn	PROPN
ma-255	27	13	)	)	PUNCT
ma-255	27	14	(	(	PUNCT
ma-255	27	15	newton	newton	PROPN
ma-255	27	16	’s	’s	PART
ma-255	27	17	method	method	NOUN
ma-255	27	18	)	)	PUNCT
ma-255	27	19	,	,	PUNCT
ma-255	27	20	ln	ln	NOUN
ma-255	28	1	=	=	PUNCT
ma-255	29	1	[	[	X
ma-255	29	2	xn	xn	X
ma-255	29	3	−	−	PROPN
ma-255	29	4	f1(xn	f1(xn	NUM
ma-255	29	5	)	)	PUNCT
ma-255	29	6	,	,	PUNCT
ma-255	29	7	xn	xn	PROPN
ma-255	30	1	+	+	CCONJ
ma-255	30	2	f1(xn);f	f1(xn);f	PROPN
ma-255	30	3	]	]	PUNCT
ma-255	30	4	(	(	PUNCT
ma-255	30	5	steffensen	steffensen	PROPN
ma-255	30	6	’s	’s	PART
ma-255	30	7	method	method	NOUN
ma-255	30	8	)	)	PUNCT
ma-255	30	9	,	,	PUNCT
ma-255	30	10	ln	ln	NOUN
ma-255	31	1	=	=	PUNCT
ma-255	31	2	i	i	PROPN
ma-255	31	3	(	(	PUNCT
ma-255	31	4	the	the	DET
ma-255	31	5	picard	picard	NOUN
ma-255	31	6	method	method	NOUN
ma-255	31	7	)	)	PUNCT
ma-255	31	8	,	,	PUNCT
ma-255	31	9	the	the	DET
ma-255	31	10	identity	identity	NOUN
ma-255	31	11	operator	operator	NOUN
ma-255	31	12	.	.	PUNCT
ma-255	32	1	here	here	ADV
ma-255	32	2	f	f	PROPN
ma-255	32	3	′1	′1	X
ma-255	32	4	,	,	PUNCT
ma-255	32	5	[	[	X
ma-255	32	6	.	.	PROPN
ma-255	32	7	,	,	PUNCT
ma-255	32	8	.;f1	.;f1	PROPN
ma-255	32	9	]	]	X
ma-255	32	10	denote	denote	VERB
ma-255	32	11	fréchet	fréchet	ADJ
ma-255	32	12	-	-	PUNCT
ma-255	32	13	derivative	derivative	ADJ
ma-255	32	14	and	and	CCONJ
ma-255	32	15	divided	divide	VERB
ma-255	32	16	differences	difference	NOUN
ma-255	32	17	of	of	ADP
ma-255	32	18	order	order	NOUN
ma-255	32	19	one	one	NUM
ma-255	32	20	for	for	ADP
ma-255	32	21	the	the	DET
ma-255	32	22	operator	operator	NOUN
ma-255	32	23	f1	f1	NOUN
ma-255	32	24	,	,	PUNCT
ma-255	32	25	respectively	respectively	ADV
ma-255	32	26	[	[	X
ma-255	32	27	22,25].many	22,25].many	NUM
ma-255	32	28	other	other	ADJ
ma-255	32	29	choices	choice	NOUN
ma-255	32	30	are	be	AUX
ma-255	32	31	possible	possible	ADJ
ma-255	32	32	.	.	PUNCT
ma-255	33	1	it	it	PRON
ma-255	33	2	turns	turn	VERB
ma-255	33	3	out	out	ADP
ma-255	33	4	that	that	SCONJ
ma-255	33	5	the	the	DET
ma-255	33	6	inverse	inverse	NOUN
ma-255	33	7	of	of	ADP
ma-255	33	8	the	the	DET
ma-255	33	9	operator	operator	NOUN
ma-255	33	10	ln	ln	ADV
ma-255	33	11	is	be	AUX
ma-255	33	12	costly	costly	ADJ
ma-255	33	13	orimpossible	orimpossible	ADJ
ma-255	33	14	to	to	PART
ma-255	33	15	find	find	VERB
ma-255	33	16	in	in	ADP
ma-255	33	17	general	general	ADJ
ma-255	33	18	.	.	PUNCT
ma-255	34	1	this	this	DET
ma-255	34	2	concern	concern	NOUN
ma-255	34	3	with	with	ADP
ma-255	34	4	the	the	DET
ma-255	34	5	implementation	implementation	NOUN
ma-255	34	6	of	of	ADP
ma-255	34	7	these	these	DET
ma-255	34	8	methods	method	NOUN
ma-255	34	9	constitutesthe	constitutesthe	VERB
ma-255	34	10	motivation	motivation	NOUN
ma-255	34	11	for	for	ADP
ma-255	34	12	this	this	DET
ma-255	34	13	paper	paper	NOUN
ma-255	34	14	.	.	PUNCT
ma-255	35	1	our	our	PRON
ma-255	35	2	idea	idea	NOUN
ma-255	35	3	is	be	AUX
ma-255	35	4	to	to	PART
ma-255	35	5	replace	replace	VERB
ma-255	35	6	the	the	DET
ma-255	35	7	inverse	inverse	NOUN
ma-255	35	8	with	with	ADP
ma-255	35	9	a	a	DET
ma-255	35	10	finite	finite	ADJ
ma-255	35	11	sum	sum	NOUN
ma-255	35	12	of	of	ADP
ma-255	35	13	linear	linear	PROPN
ma-255	35	14	operatorsconverging	operatorsconverging	NOUN
ma-255	35	15	to	to	ADP
ma-255	35	16	it	it	PRON
ma-255	35	17	.	.	PUNCT
ma-255	36	1	the	the	DET
ma-255	36	2	reasoning	reasoning	NOUN
ma-255	36	3	is	be	AUX
ma-255	36	4	explained	explain	VERB
ma-255	36	5	as	as	SCONJ
ma-255	36	6	follows	follow	VERB
ma-255	36	7	.	.	PUNCT
ma-255	37	1	let	let	VERB
ma-255	37	2	p	p	PRON
ma-255	37	3	∈	∈	PROPN
ma-255	37	4	n	n	PRON
ma-255	37	5	be	be	AUX
ma-255	37	6	fixed.suppose	fixed.suppose	NUM
ma-255	37	7	there	there	ADV
ma-255	37	8	exists	exist	VERB
ma-255	37	9	γ	γ	PROPN
ma-255	37	10	∈	∈	PROPN
ma-255	37	11	l(x	l(x	PROPN
ma-255	37	12	,	,	PUNCT
ma-255	37	13	y	y	PROPN
ma-255	37	14	)	)	PUNCT
ma-255	37	15	such	such	ADJ
ma-255	37	16	that	that	SCONJ
ma-255	37	17	γ−1	γ−1	PROPN
ma-255	37	18	∈	∈	PROPN
ma-255	37	19	l(y	l(y	PROPN
ma-255	37	20	,	,	PUNCT
ma-255	37	21	x	x	NOUN
ma-255	37	22	)	)	PUNCT
ma-255	37	23	and	and	CCONJ
ma-255	37	24	for	for	ADP
ma-255	37	25	a	a	DET
ma-255	37	26	=	=	SYM
ma-255	37	27	a(x	a(x	PROPN
ma-255	37	28	)	)	PUNCT
ma-255	37	29	=	=	NOUN
ma-255	37	30	γ−1(γ−	γ−1(γ−	PUNCT
ma-255	37	31	l(x))the	l(x))the	DET
ma-255	37	32	operator	operator	NOUN
ma-255	37	33	i−a(x	i−a(x	NOUN
ma-255	37	34	)	)	PUNCT
ma-255	37	35	is	be	AUX
ma-255	37	36	also	also	ADV
ma-255	37	37	invertible	invertible	ADJ
ma-255	37	38	,	,	PUNCT
ma-255	37	39	i.e.	i.e.	X
ma-255	37	40	(	(	PUNCT
ma-255	37	41	i−a(x))−1	i−a(x))−1	NOUN
ma-255	37	42	∈	∈	PROPN
ma-255	37	43	l(y	l(y	PROPN
ma-255	37	44	,	,	PUNCT
ma-255	37	45	x	x	NOUN
ma-255	37	46	)	)	PUNCT
ma-255	37	47	.	.	PUNCT
ma-255	38	1	in	in	ADP
ma-255	38	2	this	this	DET
ma-255	38	3	case	case	NOUN
ma-255	38	4	,	,	PUNCT
ma-255	38	5	the	the	DET
ma-255	38	6	newton	newton	PROPN
ma-255	38	7	-	-	PUNCT
ma-255	38	8	typemethod	typemethod	PROPN
ma-255	38	9	can	can	AUX
ma-255	38	10	read	read	VERB
ma-255	38	11	as	as	ADP
ma-255	38	12	x0	x0	PROPN
ma-255	38	13	∈	∈	PROPN
ma-255	38	14	d	d	PROPN
ma-255	38	15	,	,	PUNCT
ma-255	38	16	xn+1	xn+1	PROPN
ma-255	38	17	=	=	SYM
ma-255	38	18	xn	xn	PROPN
ma-255	39	1	−	−	PROPN
ma-255	39	2	(	(	PUNCT
ma-255	39	3	i	i	PRON
ma-255	39	4	−	−	PROPN
ma-255	39	5	a)−1γ−1f1(xn	a)−1γ−1f1(xn	PROPN
ma-255	39	6	)	)	PUNCT
ma-255	39	7	.	.	PUNCT
ma-255	40	1	(	(	PUNCT
ma-255	40	2	1.3	1.3	NUM
ma-255	40	3	)	)	PUNCT
ma-255	40	4	note	note	NOUN
ma-255	40	5	that	that	SCONJ
ma-255	40	6	we	we	PRON
ma-255	40	7	have	have	VERB
ma-255	40	8	(	(	PUNCT
ma-255	40	9	i	i	PRON
ma-255	40	10	−	−	VERB
ma-255	40	11	a)−1γ−1	a)−1γ−1	NOUN
ma-255	41	1	=	=	PUNCT
ma-255	42	1	[	[	X
ma-255	42	2	γ(i	γ(i	NOUN
ma-255	42	3	−	−	NOUN
ma-255	42	4	a)]−1	a)]−1	NOUN
ma-255	42	5	=	=	PUNCT
ma-255	42	6	l−1n	l−1n	X
ma-255	42	7	.	.	PUNCT
ma-255	43	1	(	(	PUNCT
ma-255	43	2	1.4	1.4	NUM
ma-255	43	3	)	)	PUNCT
ma-255	43	4	however	however	ADV
ma-255	43	5	,	,	PUNCT
ma-255	43	6	even	even	ADV
ma-255	43	7	if	if	SCONJ
ma-255	43	8	the	the	DET
ma-255	43	9	linear	linear	ADJ
ma-255	43	10	operator	operator	NOUN
ma-255	43	11	γ−1	γ−1	PROPN
ma-255	43	12	is	be	AUX
ma-255	43	13	known	know	VERB
ma-255	43	14	it	it	PRON
ma-255	43	15	is	be	AUX
ma-255	43	16	still	still	ADV
ma-255	43	17	required	require	VERB
ma-255	43	18	to	to	PART
ma-255	43	19	find	find	VERB
ma-255	43	20	the	the	DET
ma-255	43	21	inverse	inverse	NOUN
ma-255	43	22	of	of	ADP
ma-255	43	23	(	(	PUNCT
ma-255	43	24	i	i	PRON
ma-255	43	25	−a),which	−a),which	PROPN
ma-255	43	26	is	be	AUX
ma-255	43	27	not	not	PART
ma-255	43	28	a	a	DET
ma-255	43	29	fixed	fix	VERB
ma-255	43	30	operator	operator	NOUN
ma-255	43	31	(	(	PUNCT
ma-255	43	32	in	in	ADP
ma-255	43	33	general	general	ADJ
ma-255	43	34	)	)	PUNCT
ma-255	43	35	.	.	PUNCT
ma-255	44	1	but	but	CCONJ
ma-255	44	2	what	what	PRON
ma-255	44	3	if	if	SCONJ
ma-255	44	4	we	we	PRON
ma-255	44	5	replace	replace	VERB
ma-255	44	6	this	this	DET
ma-255	44	7	operator	operator	NOUN
ma-255	44	8	with	with	ADP
ma-255	44	9	m	m	NOUN
ma-255	44	10	=	=	SYM
ma-255	44	11	mp(x	mp(x	ADJ
ma-255	44	12	)	)	PUNCT
ma-255	45	1	=	=	SYM
ma-255	45	2	i	i	PRON
ma-255	45	3	+	+	X
ma-255	45	4	a+	a+	PUNCT
ma-255	45	5	...	...	PUNCT
ma-255	46	1	+	+	CCONJ
ma-255	46	2	ap	ap	PROPN
ma-255	46	3	.	.	PUNCT
ma-255	47	1	then	then	ADV
ma-255	47	2	,	,	PUNCT
ma-255	47	3	method	method	NOUN
ma-255	47	4	(	(	PUNCT
ma-255	47	5	1.3	1.3	NUM
ma-255	47	6	)	)	PUNCT
ma-255	47	7	can	can	AUX
ma-255	47	8	be	be	AUX
ma-255	47	9	written	write	VERB
ma-255	47	10	as	as	ADP
ma-255	47	11	x0	x0	PROPN
ma-255	47	12	∈	∈	PROPN
ma-255	47	13	ω	ω	PROPN
ma-255	47	14	,	,	PUNCT
ma-255	47	15	xn+1	xn+1	PROPN
ma-255	47	16	=	=	SYM
ma-255	47	17	xn	xn	PROPN
ma-255	47	18	−mγ−1f1(xn	−mγ−1f1(xn	NUM
ma-255	47	19	)	)	PUNCT
ma-255	47	20	.	.	PUNCT
ma-255	48	1	(	(	PUNCT
ma-255	48	2	1.5	1.5	NUM
ma-255	48	3	)	)	PUNCT
ma-255	48	4	it	it	PRON
ma-255	48	5	is	be	AUX
ma-255	48	6	clear	clear	ADJ
ma-255	48	7	that	that	SCONJ
ma-255	48	8	(	(	PUNCT
ma-255	48	9	1.5	1.5	NUM
ma-255	48	10	)	)	PUNCT
ma-255	48	11	is	be	AUX
ma-255	48	12	a	a	DET
ma-255	48	13	useful	useful	ADJ
ma-255	48	14	alternative	alternative	NOUN
ma-255	48	15	for	for	ADP
ma-255	48	16	(	(	PUNCT
ma-255	48	17	1.3	1.3	NUM
ma-255	48	18	)	)	PUNCT
ma-255	48	19	because	because	SCONJ
ma-255	48	20	of	of	ADP
ma-255	48	21	(	(	PUNCT
ma-255	48	22	1.4	1.4	NUM
ma-255	48	23	)	)	PUNCT
ma-255	48	24	.	.	PUNCT
ma-255	49	1	by	by	ADP
ma-255	49	2	letting	let	VERB
ma-255	49	3	p	p	PRON
ma-255	49	4	−→	−→	NOUN
ma-255	49	5	+	+	NOUN
ma-255	49	6	∞	∞	NOUN
ma-255	49	7	,	,	PUNCT
ma-255	49	8	weget	weget	VERB
ma-255	49	9	limp→+∞mp	limp→+∞mp	NUM
ma-255	49	10	=	=	PUNCT
ma-255	49	11	l−1n	l−1n	ADJ
ma-255	49	12	if	if	SCONJ
ma-255	49	13	the	the	DET
ma-255	49	14	limit	limit	NOUN
ma-255	49	15	exists	exist	VERB
ma-255	49	16	.	.	PUNCT
ma-255	50	1	the	the	DET
ma-255	50	2	condition	condition	NOUN
ma-255	50	3	‖a‖	‖a‖	PROPN
ma-255	50	4	<	<	X
ma-255	50	5	1	1	NUM
ma-255	50	6	for	for	ADP
ma-255	50	7	each	each	DET
ma-255	50	8	x	x	SYM
ma-255	50	9	∈	∈	PROPN
ma-255	50	10	ω	ω	PROPN
ma-255	50	11	assures	assure	VERB
ma-255	50	12	theexistence	theexistence	NOUN
ma-255	50	13	of	of	ADP
ma-255	50	14	such	such	DET
ma-255	50	15	a	a	DET
ma-255	50	16	limit	limit	NOUN
ma-255	50	17	.	.	PUNCT
ma-255	51	1	if	if	SCONJ
ma-255	51	2	the	the	DET
ma-255	51	3	linear	linear	ADJ
ma-255	51	4	operator	operator	NOUN
ma-255	51	5	m	m	VERB
ma-255	51	6	is	be	AUX
ma-255	51	7	invertible	invertible	ADJ
ma-255	51	8	and	and	CCONJ
ma-255	51	9	the	the	DET
ma-255	51	10	sequence	sequence	NOUN
ma-255	51	11	{	{	PUNCT
ma-255	51	12	xn	xn	NOUN
ma-255	51	13	}	}	PUNCT
ma-255	51	14	given	give	VERB
ma-255	51	15	by(1.5	by(1.5	NOUN
ma-255	51	16	)	)	PUNCT
ma-255	51	17	converges	converge	NOUN
ma-255	51	18	to	to	ADP
ma-255	51	19	some	some	DET
ma-255	51	20	s∗	s∗	PROPN
ma-255	51	21	,	,	PUNCT
ma-255	51	22	then	then	ADV
ma-255	51	23	by	by	ADP
ma-255	51	24	(	(	PUNCT
ma-255	51	25	1.5	1.5	NUM
ma-255	51	26	)	)	PUNCT
ma-255	51	27	we	we	PRON
ma-255	51	28	get	get	VERB
ma-255	51	29	m−1(xn	m−1(xn	PROPN
ma-255	51	30	−	−	NOUN
ma-255	51	31	xn+1	xn+1	NUM
ma-255	51	32	)	)	PUNCT
ma-255	51	33	=	=	SYM
ma-255	51	34	γ−1f1(xn	γ−1f1(xn	X
ma-255	51	35	)	)	PUNCT
ma-255	51	36	leading	lead	VERB
ma-255	51	37	to	to	ADP
ma-255	51	38	0	0	NUM
ma-255	52	1	=	=	SYM
ma-255	53	1	lim	lim	PROPN
ma-255	53	2	n→+∞	n→+∞	VERB
ma-255	53	3	m−1(xn	m−1(xn	PROPN
ma-255	53	4	−	−	NOUN
ma-255	53	5	xn+1	xn+1	NUM
ma-255	53	6	)	)	PUNCT
ma-255	54	1	=	=	VERB
ma-255	54	2	lim	lim	PROPN
ma-255	54	3	n→+∞	n→+∞	VERB
ma-255	54	4	γ−1f1(xn	γ−1f1(xn	PROPN
ma-255	54	5	)	)	PUNCT
ma-255	54	6	,	,	PUNCT
ma-255	54	7	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	54	8	eur	eur	PROPN
ma-255	54	9	.	.	PUNCT
ma-255	55	1	j.	j.	PROPN
ma-255	55	2	math	math	PROPN
ma-255	55	3	.	.	PUNCT
ma-255	56	1	anal	anal	PROPN
ma-255	56	2	.	.	PUNCT
ma-255	57	1	10.28924	10.28924	NUM
ma-255	57	2	/	/	SYM
ma-255	57	3	ada	ada	PROPN
ma-255	57	4	/	/	SYM
ma-255	57	5	ma.5.5	ma.5.5	PROPN
ma-255	57	6	3i.e	3i.e	NUM
ma-255	57	7	.	.	PUNCT
ma-255	58	1	f1(s	f1(s	NUM
ma-255	58	2	∗	∗	NOUN
ma-255	58	3	)	)	PUNCT
ma-255	58	4	=	=	SYM
ma-255	58	5	0	0	X
ma-255	58	6	.	.	PUNCT
ma-255	59	1	thus	thus	ADV
ma-255	59	2	,	,	PUNCT
ma-255	59	3	the	the	DET
ma-255	59	4	point	point	NOUN
ma-255	59	5	s∗	s∗	PROPN
ma-255	59	6	solves	solve	VERB
ma-255	59	7	the	the	DET
ma-255	59	8	equation	equation	NOUN
ma-255	59	9	(	(	PUNCT
ma-255	59	10	1.1	1.1	NUM
ma-255	59	11	)	)	PUNCT
ma-255	59	12	.	.	PUNCT
ma-255	60	1	the	the	DET
ma-255	60	2	same	same	ADJ
ma-255	60	3	reasoning	reasoning	NOUN
ma-255	60	4	leads	lead	VERB
ma-255	60	5	for	for	ADP
ma-255	60	6	a1	a1	NOUN
ma-255	60	7	=	=	PUNCT
ma-255	60	8	(	(	PUNCT
ma-255	60	9	γ−	γ−	PROPN
ma-255	60	10	l(x))γ−1	l(x))γ−1	NUM
ma-255	60	11	and	and	CCONJ
ma-255	60	12	m1	m1	PROPN
ma-255	61	1	=	=	PUNCT
ma-255	62	1	i	i	PRON
ma-255	62	2	+	+	CCONJ
ma-255	62	3	a1	a1	NOUN
ma-255	62	4	+	+	CCONJ
ma-255	62	5	...	...	PUNCT
ma-255	62	6	+	+	CCONJ
ma-255	62	7	ap1	ap1	NOUN
ma-255	62	8	to	to	ADP
ma-255	62	9	the	the	DET
ma-255	62	10	method	method	NOUN
ma-255	62	11	x0	x0	PROPN
ma-255	62	12	∈	∈	PROPN
ma-255	62	13	ω	ω	PROPN
ma-255	62	14	,	,	PUNCT
ma-255	62	15	xn+1	xn+1	PROPN
ma-255	62	16	=	=	SYM
ma-255	62	17	xn	xn	PROPN
ma-255	62	18	−m1γ−1f1(xn	−m1γ−1f1(xn	PROPN
ma-255	62	19	)	)	PUNCT
ma-255	62	20	.	.	PUNCT
ma-255	63	1	(	(	PUNCT
ma-255	63	2	1.6	1.6	NUM
ma-255	63	3	)	)	PUNCT
ma-255	63	4	it	it	PRON
ma-255	63	5	is	be	AUX
ma-255	63	6	clear	clear	ADJ
ma-255	63	7	that	that	SCONJ
ma-255	63	8	the	the	DET
ma-255	63	9	study	study	NOUN
ma-255	63	10	of	of	ADP
ma-255	63	11	the	the	DET
ma-255	63	12	convergence	convergence	NOUN
ma-255	63	13	of	of	ADP
ma-255	63	14	the	the	DET
ma-255	63	15	method	method	NOUN
ma-255	63	16	(	(	PUNCT
ma-255	63	17	1.6	1.6	NUM
ma-255	63	18	)	)	PUNCT
ma-255	63	19	is	be	AUX
ma-255	63	20	analogous	analogous	ADJ
ma-255	63	21	to	to	ADP
ma-255	63	22	(	(	PUNCT
ma-255	63	23	1.5	1.5	NUM
ma-255	63	24	)	)	PUNCT
ma-255	63	25	.	.	PUNCT
ma-255	64	1	that	that	PRON
ma-255	64	2	is	be	AUX
ma-255	64	3	whywe	whywe	PROPN
ma-255	64	4	study	study	NOUN
ma-255	64	5	only	only	ADV
ma-255	64	6	method	method	NOUN
ma-255	64	7	(	(	PUNCT
ma-255	64	8	1.5	1.5	NUM
ma-255	64	9	)	)	PUNCT
ma-255	64	10	in	in	ADP
ma-255	64	11	section	section	NOUN
ma-255	64	12	2	2	NUM
ma-255	64	13	.	.	PUNCT
ma-255	65	1	we	we	PRON
ma-255	65	2	deal	deal	VERB
ma-255	65	3	with	with	ADP
ma-255	65	4	two	two	NUM
ma-255	65	5	kinds	kind	NOUN
ma-255	65	6	of	of	ADP
ma-255	65	7	convergence	convergence	NOUN
ma-255	65	8	:	:	PUNCT
ma-255	65	9	the	the	DET
ma-255	65	10	semi	semi	ADJ
ma-255	65	11	-	-	ADJ
ma-255	65	12	localand	localand	DET
ma-255	65	13	the	the	DET
ma-255	65	14	local	local	NOUN
ma-255	65	15	.	.	PUNCT
ma-255	66	1	the	the	DET
ma-255	66	2	first	first	ADJ
ma-255	66	3	utilizes	utilizes	ADJ
ma-255	66	4	knowledge	knowledge	NOUN
ma-255	66	5	in	in	ADP
ma-255	66	6	a	a	DET
ma-255	66	7	neighborhood	neighborhood	NOUN
ma-255	66	8	of	of	ADP
ma-255	66	9	x0	x0	PROPN
ma-255	66	10	and	and	CCONJ
ma-255	66	11	develops	develop	VERB
ma-255	66	12	estimates	estimate	NOUN
ma-255	66	13	relatedto	relatedto	VERB
ma-255	66	14	‖xn+1−	‖xn+1−	PROPN
ma-255	66	15	xn‖	xn‖	PROPN
ma-255	66	16	and	and	CCONJ
ma-255	66	17	‖s∗−	‖s∗−	PROPN
ma-255	66	18	xn‖	xn‖	PROPN
ma-255	66	19	,	,	PUNCT
ma-255	66	20	and	and	CCONJ
ma-255	66	21	the	the	DET
ma-255	66	22	convergence	convergence	NOUN
ma-255	66	23	conditions	condition	NOUN
ma-255	66	24	assure	assure	VERB
ma-255	66	25	that	that	PRON
ma-255	66	26	limn→+∞	limn→+∞	VERB
ma-255	66	27	xn	xn	PUNCT
ma-255	67	1	=	=	PUNCT
ma-255	67	2	s∗.	s∗.	ADJ
ma-255	67	3	in	in	ADP
ma-255	67	4	thesecond	thesecond	NOUN
ma-255	67	5	kind	kind	NOUN
ma-255	67	6	,	,	PUNCT
ma-255	67	7	knowledge	knowledge	NOUN
ma-255	67	8	about	about	ADP
ma-255	67	9	a	a	DET
ma-255	67	10	neighborhood	neighborhood	NOUN
ma-255	67	11	of	of	ADP
ma-255	67	12	s∗	s∗	PROPN
ma-255	67	13	is	be	AUX
ma-255	67	14	used	use	VERB
ma-255	67	15	to	to	PART
ma-255	67	16	provide	provide	VERB
ma-255	67	17	the	the	DET
ma-255	67	18	same	same	ADJ
ma-255	67	19	estimates	estimate	NOUN
ma-255	67	20	as	as	ADP
ma-255	67	21	inthe	inthe	DET
ma-255	67	22	semi	semi	ADJ
ma-255	67	23	-	-	ADJ
ma-255	67	24	local	local	ADJ
ma-255	67	25	kind	kind	NOUN
ma-255	67	26	and	and	CCONJ
ma-255	67	27	again	again	ADV
ma-255	67	28	limn→+∞	limn→+∞	VERB
ma-255	67	29	xn	xn	NOUN
ma-255	68	1	=	=	PUNCT
ma-255	68	2	s∗.	s∗.	ADJ
ma-255	68	3	it	it	PRON
ma-255	68	4	is	be	AUX
ma-255	68	5	worth	worth	ADJ
ma-255	68	6	noting	note	VERB
ma-255	68	7	that	that	SCONJ
ma-255	68	8	the	the	DET
ma-255	68	9	iterates	iterate	NOUN
ma-255	68	10	generated	generate	VERB
ma-255	68	11	by(1.3),(1.5	by(1.3),(1.5	NOUN
ma-255	68	12	)	)	PUNCT
ma-255	68	13	and	and	CCONJ
ma-255	68	14	(	(	PUNCT
ma-255	68	15	1.6	1.6	NUM
ma-255	68	16	)	)	PUNCT
ma-255	68	17	are	be	AUX
ma-255	68	18	not	not	PART
ma-255	68	19	the	the	DET
ma-255	68	20	same	same	ADJ
ma-255	68	21	in	in	ADP
ma-255	68	22	general	general	ADJ
ma-255	68	23	.	.	PUNCT
ma-255	69	1	but	but	CCONJ
ma-255	69	2	we	we	PRON
ma-255	69	3	use	use	VERB
ma-255	69	4	the	the	DET
ma-255	69	5	same	same	ADJ
ma-255	69	6	notation	notation	NOUN
ma-255	69	7	for	for	ADP
ma-255	69	8	simplicity	simplicity	NOUN
ma-255	69	9	.	.	PUNCT
ma-255	70	1	theconvergence	theconvergence	NOUN
ma-255	70	2	for	for	ADP
ma-255	70	3	both	both	DET
ma-255	70	4	kinds	kind	NOUN
ma-255	70	5	relies	rely	VERB
ma-255	70	6	on	on	ADP
ma-255	70	7	generalized	generalized	ADJ
ma-255	70	8	continuity	continuity	NOUN
ma-255	70	9	conditions	condition	NOUN
ma-255	70	10	controlling	control	VERB
ma-255	70	11	the	the	DET
ma-255	70	12	operatorsinvolved	operatorsinvolved	ADJ
ma-255	70	13	[	[	X
ma-255	70	14	5	5	NUM
ma-255	70	15	,	,	PUNCT
ma-255	70	16	6	6	NUM
ma-255	70	17	,	,	PUNCT
ma-255	70	18	9	9	NUM
ma-255	70	19	,	,	PUNCT
ma-255	70	20	18	18	NUM
ma-255	70	21	]	]	PUNCT
ma-255	70	22	.	.	PUNCT
ma-255	71	1	in	in	ADP
ma-255	71	2	particular	particular	ADJ
ma-255	71	3	,	,	PUNCT
ma-255	71	4	our	our	PRON
ma-255	71	5	semi	semi	ADJ
ma-255	71	6	-	-	ADJ
ma-255	71	7	local	local	ADJ
ma-255	71	8	convergence	convergence	NOUN
ma-255	71	9	analysis	analysis	NOUN
ma-255	71	10	depends	depend	VERB
ma-255	71	11	on	on	ADP
ma-255	71	12	the	the	DET
ma-255	71	13	usage	usage	NOUN
ma-255	71	14	ofmajorizing	ofmajorize	VERB
ma-255	71	15	sequences	sequence	NOUN
ma-255	71	16	[	[	X
ma-255	71	17	29,30,34	29,30,34	NUM
ma-255	71	18	]	]	X
ma-255	71	19	.	.	PUNCT
ma-255	72	1	notice	notice	VERB
ma-255	72	2	that	that	SCONJ
ma-255	72	3	the	the	DET
ma-255	72	4	method	method	NOUN
ma-255	72	5	(	(	PUNCT
ma-255	72	6	1.5	1.5	NUM
ma-255	72	7	)	)	PUNCT
ma-255	72	8	can	can	AUX
ma-255	72	9	also	also	ADV
ma-255	72	10	be	be	AUX
ma-255	72	11	written	write	VERB
ma-255	72	12	for	for	ADP
ma-255	72	13	dn	dn	NOUN
ma-255	72	14	=	=	PUNCT
ma-255	72	15	γm−1as	γm−1as	NUM
ma-255	72	16	x0	x0	PROPN
ma-255	72	17	∈	∈	PROPN
ma-255	72	18	ω	ω	PROPN
ma-255	72	19	,	,	PUNCT
ma-255	72	20	f1(xn	f1(xn	NUM
ma-255	72	21	)	)	PUNCT
ma-255	72	22	+	+	NOUN
ma-255	72	23	dn(xn+1	dn(xn+1	NOUN
ma-255	72	24	−	−	NOUN
ma-255	72	25	xn	xn	NUM
ma-255	72	26	)	)	PUNCT
ma-255	72	27	=	=	SYM
ma-255	73	1	0	0	X
ma-255	73	2	.	.	PUNCT
ma-255	74	1	(	(	PUNCT
ma-255	74	2	1.7	1.7	NUM
ma-255	74	3	)	)	PUNCT
ma-255	74	4	in	in	ADP
ma-255	74	5	section	section	NOUN
ma-255	74	6	3	3	NUM
ma-255	74	7	we	we	PRON
ma-255	74	8	also	also	ADV
ma-255	74	9	use	use	VERB
ma-255	74	10	the	the	DET
ma-255	74	11	developed	develop	VERB
ma-255	74	12	methodology	methodology	NOUN
ma-255	74	13	for	for	ADP
ma-255	74	14	solving	solve	VERB
ma-255	74	15	nonlinear	nonlinear	ADJ
ma-255	74	16	equations	equation	NOUN
ma-255	74	17	to	to	ADP
ma-255	74	18	solvegeneralized	solvegeneralize	VERB
ma-255	74	19	equations	equation	NOUN
ma-255	74	20	.	.	PUNCT
ma-255	75	1	that	that	PRON
ma-255	75	2	is	be	AUX
ma-255	75	3	find	find	VERB
ma-255	75	4	x	x	X
ma-255	75	5	∈	∈	NOUN
ma-255	75	6	x	x	X
ma-255	75	7	such	such	ADJ
ma-255	75	8	that	that	DET
ma-255	75	9	f1(x	f1(x	NOUN
ma-255	75	10	)	)	PUNCT
ma-255	76	1	+	+	SYM
ma-255	77	1	f2(x	f2(x	X
ma-255	77	2	)	)	PUNCT
ma-255	77	3	3	3	NUM
ma-255	77	4	0	0	NUM
ma-255	77	5	.	.	PUNCT
ma-255	78	1	(	(	PUNCT
ma-255	78	2	1.8	1.8	NUM
ma-255	78	3	)	)	PUNCT
ma-255	78	4	here	here	ADV
ma-255	78	5	f2	f2	ADV
ma-255	78	6	:	:	PUNCT
ma-255	78	7	x	x	SYM
ma-255	78	8	⇒	⇒	PROPN
ma-255	78	9	y	y	PROPN
ma-255	78	10	is	be	AUX
ma-255	78	11	a	a	DET
ma-255	78	12	set	set	NOUN
ma-255	78	13	-	-	PUNCT
ma-255	78	14	valued	value	VERB
ma-255	78	15	operator	operator	NOUN
ma-255	78	16	mapping	mapping	NOUN
ma-255	78	17	x	x	PUNCT
ma-255	78	18	into	into	ADP
ma-255	78	19	y	y	PROPN
ma-255	78	20	with	with	ADP
ma-255	78	21	closed	closed	ADJ
ma-255	78	22	graph	graph	NOUN
ma-255	78	23	[	[	X
ma-255	78	24	1–4	1–4	PROPN
ma-255	78	25	,	,	PUNCT
ma-255	78	26	13	13	NUM
ma-255	78	27	,	,	PUNCT
ma-255	78	28	17–19	17–19	NUM
ma-255	78	29	,	,	PUNCT
ma-255	78	30	21–23	21–23	NUM
ma-255	78	31	,	,	PUNCT
ma-255	78	32	26	26	NUM
ma-255	78	33	,	,	PUNCT
ma-255	78	34	28	28	NUM
ma-255	78	35	,	,	PUNCT
ma-255	78	36	32	32	NUM
ma-255	78	37	,	,	PUNCT
ma-255	78	38	35	35	NUM
ma-255	78	39	]	]	PUNCT
ma-255	78	40	and	and	CCONJ
ma-255	78	41	operator	operator	NOUN
ma-255	78	42	f	f	PROPN
ma-255	78	43	is	be	AUX
ma-255	78	44	as	as	ADV
ma-255	78	45	previously	previously	ADV
ma-255	78	46	defined	define	VERB
ma-255	78	47	.	.	PUNCT
ma-255	79	1	a	a	DET
ma-255	79	2	plethora	plethora	NOUN
ma-255	79	3	of	of	ADP
ma-255	79	4	applications	application	NOUN
ma-255	79	5	frommathematical	frommathematical	ADJ
ma-255	79	6	programming	programming	NOUN
ma-255	79	7	,	,	PUNCT
ma-255	79	8	variational	variational	ADJ
ma-255	79	9	inequalities	inequality	NOUN
ma-255	79	10	,	,	PUNCT
ma-255	79	11	optimal	optimal	ADJ
ma-255	79	12	control	control	NOUN
ma-255	79	13	,	,	PUNCT
ma-255	79	14	or	or	CCONJ
ma-255	79	15	constrained	constrain	VERB
ma-255	79	16	systems	system	NOUN
ma-255	79	17	arewritten	arewritten	VERB
ma-255	79	18	in	in	ADP
ma-255	79	19	the	the	DET
ma-255	79	20	form	form	NOUN
ma-255	79	21	(	(	PUNCT
ma-255	79	22	1.8	1.8	NUM
ma-255	79	23	)	)	PUNCT
ma-255	79	24	.	.	PUNCT
ma-255	80	1	there	there	PRON
ma-255	80	2	is	be	VERB
ma-255	80	3	extensive	extensive	ADJ
ma-255	80	4	literature	literature	NOUN
ma-255	80	5	on	on	ADP
ma-255	80	6	iterative	iterative	ADJ
ma-255	80	7	methods	method	NOUN
ma-255	80	8	solving	solve	VERB
ma-255	80	9	the	the	DET
ma-255	80	10	generalizedequation	generalizedequation	NOUN
ma-255	80	11	(	(	PUNCT
ma-255	80	12	1.8	1.8	NUM
ma-255	80	13	)	)	PUNCT
ma-255	81	1	[	[	X
ma-255	81	2	1–4,13,17–19,21–23,26,28,32,35	1–4,13,17–19,21–23,26,28,32,35	NUM
ma-255	81	3	]	]	PUNCT
ma-255	81	4	.	.	PUNCT
ma-255	82	1	notice	notice	VERB
ma-255	82	2	that	that	SCONJ
ma-255	82	3	the	the	DET
ma-255	82	4	method	method	NOUN
ma-255	82	5	used	use	VERB
ma-255	82	6	in	in	ADP
ma-255	82	7	the	the	DET
ma-255	82	8	literature	literature	NOUN
ma-255	82	9	tosolve	tosolve	VERB
ma-255	82	10	(	(	PUNCT
ma-255	82	11	1.8	1.8	NUM
ma-255	82	12	)	)	PUNCT
ma-255	82	13	is	be	AUX
ma-255	82	14	defined	define	VERB
ma-255	82	15	by	by	ADP
ma-255	82	16	f1(xn	f1(xn	NUM
ma-255	82	17	)	)	PUNCT
ma-255	83	1	+	+	CCONJ
ma-255	83	2	d̄n(xn+1	d̄n(xn+1	NOUN
ma-255	83	3	−	−	PROPN
ma-255	83	4	xn	xn	NUM
ma-255	83	5	)	)	PUNCT
ma-255	83	6	+	+	NUM
ma-255	83	7	f2(xn+1	f2(xn+1	NOUN
ma-255	83	8	)	)	PUNCT
ma-255	83	9	3	3	NUM
ma-255	83	10	0	0	NUM
ma-255	83	11	,	,	PUNCT
ma-255	83	12	(	(	PUNCT
ma-255	83	13	1.9	1.9	NUM
ma-255	83	14	)	)	PUNCT
ma-255	83	15	where	where	SCONJ
ma-255	83	16	d̄n	d̄n	NOUN
ma-255	83	17	is	be	AUX
ma-255	83	18	a	a	DET
ma-255	83	19	linear	linear	ADJ
ma-255	83	20	operator	operator	NOUN
ma-255	83	21	.	.	PUNCT
ma-255	84	1	it	it	PRON
ma-255	84	2	can	can	AUX
ma-255	84	3	be	be	AUX
ma-255	84	4	chosen	choose	VERB
ma-255	84	5	as	as	ADP
ma-255	84	6	d̄n	d̄n	NOUN
ma-255	84	7	=	=	SYM
ma-255	84	8	ln	ln	ADJ
ma-255	84	9	,	,	PUNCT
ma-255	84	10	d̄n	d̄n	NOUN
ma-255	84	11	=	=	SYM
ma-255	84	12	f	f	PROPN
ma-255	84	13	′1(xn	′1(xn	PROPN
ma-255	84	14	)	)	PUNCT
ma-255	84	15	or	or	CCONJ
ma-255	84	16	d̄n	d̄n	PROPN
ma-255	84	17	∈	∈	PROPN
ma-255	84	18	∂f1(xn	∂f1(xn	PROPN
ma-255	84	19	)	)	PUNCT
ma-255	84	20	or	or	CCONJ
ma-255	84	21	otherchoices	otherchoice	NOUN
ma-255	84	22	[	[	X
ma-255	84	23	9	9	NUM
ma-255	84	24	,	,	PUNCT
ma-255	84	25	11	11	NUM
ma-255	84	26	,	,	PUNCT
ma-255	84	27	23	23	NUM
ma-255	84	28	,	,	PUNCT
ma-255	84	29	24	24	NUM
ma-255	84	30	]	]	PUNCT
ma-255	84	31	.	.	PUNCT
ma-255	85	1	these	these	DET
ma-255	85	2	methods	method	NOUN
ma-255	85	3	have	have	VERB
ma-255	85	4	the	the	DET
ma-255	85	5	same	same	ADJ
ma-255	85	6	problems	problem	NOUN
ma-255	85	7	as	as	ADP
ma-255	85	8	the	the	DET
ma-255	85	9	ones	one	NOUN
ma-255	85	10	for	for	ADP
ma-255	85	11	solving	solve	VERB
ma-255	85	12	nonlinearequations	nonlinearequation	NOUN
ma-255	85	13	.	.	PUNCT
ma-255	86	1	that	that	PRON
ma-255	86	2	is	be	AUX
ma-255	86	3	why	why	SCONJ
ma-255	86	4	it	it	PRON
ma-255	86	5	is	be	AUX
ma-255	86	6	justified	justified	ADJ
ma-255	86	7	to	to	PART
ma-255	86	8	consider	consider	VERB
ma-255	86	9	the	the	DET
ma-255	86	10	analog	analog	NOUN
ma-255	86	11	of	of	ADP
ma-255	86	12	(	(	PUNCT
ma-255	86	13	1.7	1.7	NUM
ma-255	86	14	)	)	PUNCT
ma-255	86	15	defined	define	VERB
ma-255	86	16	by	by	ADP
ma-255	86	17	f1(xn	f1(xn	NOUN
ma-255	86	18	)	)	PUNCT
ma-255	86	19	+	+	NOUN
ma-255	86	20	dn(xn+1	dn(xn+1	NOUN
ma-255	86	21	−	−	NOUN
ma-255	86	22	xn	xn	NUM
ma-255	86	23	)	)	PUNCT
ma-255	86	24	+	+	NUM
ma-255	86	25	f2(xn+1	f2(xn+1	NOUN
ma-255	86	26	)	)	PUNCT
ma-255	86	27	3	3	NUM
ma-255	86	28	0	0	NUM
ma-255	86	29	(	(	PUNCT
ma-255	86	30	1.10	1.10	NUM
ma-255	86	31	)	)	PUNCT
ma-255	86	32	the	the	DET
ma-255	86	33	semi	semi	ADJ
ma-255	86	34	-	-	ADJ
ma-255	86	35	local	local	ADJ
ma-255	86	36	and	and	CCONJ
ma-255	86	37	local	local	ADJ
ma-255	86	38	convergence	convergence	NOUN
ma-255	86	39	of	of	ADP
ma-255	86	40	the	the	DET
ma-255	86	41	method	method	NOUN
ma-255	86	42	(	(	PUNCT
ma-255	86	43	1.10	1.10	NUM
ma-255	86	44	)	)	PUNCT
ma-255	86	45	is	be	AUX
ma-255	86	46	developed	develop	VERB
ma-255	86	47	in	in	ADP
ma-255	86	48	section	section	NOUN
ma-255	86	49	3	3	NUM
ma-255	86	50	in	in	ADP
ma-255	86	51	an	an	DET
ma-255	86	52	analogousway	analogousway	NOUN
ma-255	86	53	to	to	PART
ma-255	86	54	section	section	NOUN
ma-255	86	55	2	2	NUM
ma-255	86	56	for	for	ADP
ma-255	86	57	the	the	DET
ma-255	86	58	method	method	NOUN
ma-255	86	59	(	(	PUNCT
ma-255	86	60	1.5	1.5	NUM
ma-255	86	61	)	)	PUNCT
ma-255	86	62	or	or	CCONJ
ma-255	86	63	(	(	PUNCT
ma-255	86	64	1.7	1.7	NUM
ma-255	86	65	)	)	PUNCT
ma-255	86	66	.	.	PUNCT
ma-255	87	1	in	in	ADP
ma-255	87	2	numerical	numerical	ADJ
ma-255	87	3	section	section	PROPN
ma-255	87	4	4	4	NUM
ma-255	87	5	,	,	PUNCT
ma-255	87	6	the	the	DET
ma-255	87	7	examples	example	NOUN
ma-255	87	8	demonstratethat	demonstratethat	VERB
ma-255	87	9	the	the	DET
ma-255	87	10	number	number	NOUN
ma-255	87	11	of	of	ADP
ma-255	87	12	iterations	iteration	NOUN
ma-255	87	13	of	of	ADP
ma-255	87	14	the	the	DET
ma-255	87	15	hybrid	hybrid	ADJ
ma-255	87	16	methods	method	NOUN
ma-255	87	17	to	to	PART
ma-255	87	18	arrive	arrive	VERB
ma-255	87	19	at	at	ADP
ma-255	87	20	a	a	DET
ma-255	87	21	predetermined	predetermine	VERB
ma-255	87	22	error	error	NOUN
ma-255	87	23	tolerance	tolerance	NOUN
ma-255	87	24	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	87	25	eur	eur	PROPN
ma-255	87	26	.	.	PUNCT
ma-255	88	1	j.	j.	PROPN
ma-255	88	2	math	math	PROPN
ma-255	88	3	.	.	PUNCT
ma-255	89	1	anal	anal	PROPN
ma-255	89	2	.	.	PUNCT
ma-255	90	1	10.28924	10.28924	NUM
ma-255	90	2	/	/	SYM
ma-255	90	3	ada	ada	PROPN
ma-255	90	4	/	/	SYM
ma-255	90	5	ma.5.5	ma.5.5	PROPN
ma-255	90	6	4is	4is	NOUN
ma-255	90	7	essentially	essentially	ADV
ma-255	90	8	the	the	DET
ma-255	90	9	same	same	ADJ
ma-255	90	10	as	as	ADP
ma-255	90	11	with	with	ADP
ma-255	90	12	the	the	DET
ma-255	90	13	original	original	ADJ
ma-255	90	14	methods	method	NOUN
ma-255	90	15	.	.	PUNCT
ma-255	91	1	moreover	moreover	ADV
ma-255	91	2	,	,	PUNCT
ma-255	91	3	the	the	DET
ma-255	91	4	convergence	convergence	NOUN
ma-255	91	5	order	order	NOUN
ma-255	91	6	is	be	AUX
ma-255	91	7	also	also	ADV
ma-255	91	8	thesame.in	thesame.in	X
ma-255	91	9	order	order	NOUN
ma-255	91	10	to	to	PART
ma-255	91	11	achieve	achieve	VERB
ma-255	91	12	all	all	DET
ma-255	91	13	this	this	PRON
ma-255	91	14	we	we	PRON
ma-255	91	15	redevelop	redevelop	VERB
ma-255	91	16	some	some	DET
ma-255	91	17	standard	standard	ADJ
ma-255	91	18	terminology	terminology	NOUN
ma-255	91	19	to	to	PART
ma-255	91	20	make	make	VERB
ma-255	91	21	the	the	DET
ma-255	91	22	paper	paper	NOUN
ma-255	91	23	as	as	SCONJ
ma-255	91	24	self	self	NOUN
ma-255	91	25	-	-	PUNCT
ma-255	91	26	contained	contain	VERB
ma-255	91	27	as	as	ADP
ma-255	91	28	possible	possible	ADJ
ma-255	91	29	.	.	PUNCT
ma-255	92	1	more	more	ADJ
ma-255	92	2	details	detail	NOUN
ma-255	92	3	can	can	AUX
ma-255	92	4	be	be	AUX
ma-255	92	5	found	find	VERB
ma-255	92	6	in	in	ADP
ma-255	92	7	[	[	X
ma-255	92	8	8	8	NUM
ma-255	92	9	,	,	PUNCT
ma-255	92	10	11	11	NUM
ma-255	92	11	,	,	PUNCT
ma-255	92	12	19	19	NUM
ma-255	92	13	]	]	PUNCT
ma-255	92	14	.	.	PUNCT
ma-255	93	1	let	let	VERB
ma-255	93	2	s(z	s(z	PROPN
ma-255	93	3	,	,	PUNCT
ma-255	93	4	ρ	ρ	PROPN
ma-255	93	5	)	)	PUNCT
ma-255	93	6	and	and	CCONJ
ma-255	93	7	its	its	PRON
ma-255	93	8	closure	closure	NOUN
ma-255	93	9	s[z	s[z	PROPN
ma-255	93	10	,	,	PUNCT
ma-255	93	11	ρ]denote	ρ]denote	PROPN
ma-255	93	12	open	open	ADJ
ma-255	93	13	and	and	CCONJ
ma-255	93	14	closed	closed	ADJ
ma-255	93	15	balls	ball	NOUN
ma-255	93	16	,	,	PUNCT
ma-255	93	17	respectively	respectively	ADV
ma-255	93	18	of	of	ADP
ma-255	93	19	center	center	NOUN
ma-255	93	20	z	z	NOUN
ma-255	93	21	∈	∈	PROPN
ma-255	93	22	x	x	X
ma-255	93	23	and	and	CCONJ
ma-255	93	24	radius	radius	PROPN
ma-255	93	25	ρ	ρ	PROPN
ma-255	93	26	>	>	X
ma-255	93	27	0	0	X
ma-255	93	28	.	.	PUNCT
ma-255	94	1	let	let	VERB
ma-255	94	2	c	c	PRON
ma-255	94	3	be	be	AUX
ma-255	94	4	a	a	DET
ma-255	94	5	set	set	NOUN
ma-255	94	6	in	in	ADP
ma-255	94	7	x	x	PROPN
ma-255	94	8	.define	.define	NOUN
ma-255	94	9	the	the	DET
ma-255	94	10	distance	distance	NOUN
ma-255	94	11	for	for	ADP
ma-255	94	12	x	x	SYM
ma-255	94	13	∈	∈	PROPN
ma-255	94	14	x	x	X
ma-255	94	15	to	to	ADP
ma-255	94	16	c	c	NOUN
ma-255	94	17	by	by	ADP
ma-255	94	18	dist(x	dist(x	PROPN
ma-255	94	19	,	,	PUNCT
ma-255	94	20	c	c	NOUN
ma-255	94	21	)	)	PUNCT
ma-255	94	22	=	=	VERB
ma-255	95	1	infx∈c	infx∈c	ADJ
ma-255	95	2	‖x	‖x	NOUN
ma-255	95	3	−	−	NOUN
ma-255	95	4	y‖.	y‖.	NUM
ma-255	95	5	the	the	DET
ma-255	95	6	generalized	generalize	VERB
ma-255	95	7	set	set	NOUN
ma-255	95	8	-	-	PUNCT
ma-255	95	9	valuedoperator	valuedoperator	NOUN
ma-255	95	10	g	g	PROPN
ma-255	95	11	relates	relate	VERB
ma-255	95	12	with	with	ADP
ma-255	95	13	its	its	PRON
ma-255	95	14	graph	graph	NOUN
ma-255	95	15	given	give	VERB
ma-255	95	16	by	by	ADP
ma-255	95	17	gph(g	gph(g	NOUN
ma-255	95	18	)	)	PUNCT
ma-255	95	19	=	=	PRON
ma-255	95	20	{	{	PUNCT
ma-255	95	21	(	(	PUNCT
ma-255	95	22	x	x	NOUN
ma-255	95	23	,	,	PUNCT
ma-255	95	24	y	y	NOUN
ma-255	95	25	)	)	PUNCT
ma-255	95	26	∈	∈	PROPN
ma-255	95	27	x	x	SYM
ma-255	95	28	×	×	PROPN
ma-255	95	29	y	y	PROPN
ma-255	95	30	,	,	PUNCT
ma-255	95	31	y	y	PROPN
ma-255	95	32	∈	∈	PROPN
ma-255	95	33	f2(x	f2(x	PROPN
ma-255	95	34	)	)	PUNCT
ma-255	95	35	}	}	PUNCT
ma-255	95	36	,	,	PUNCT
ma-255	95	37	and	and	CCONJ
ma-255	95	38	its	its	PRON
ma-255	95	39	domain	domain	NOUN
ma-255	95	40	dom(g	dom(g	ADV
ma-255	95	41	)	)	PUNCT
ma-255	95	42	=	=	PRON
ma-255	96	1	{	{	PUNCT
ma-255	96	2	x	x	PROPN
ma-255	96	3	∈	∈	PROPN
ma-255	96	4	x|f2(x	x|f2(x	PROPN
ma-255	96	5	)	)	PUNCT
ma-255	97	1	6=	6=	ADP
ma-255	97	2	0	0	NUM
ma-255	97	3	}	}	PUNCT
ma-255	97	4	.	.	PUNCT
ma-255	98	1	the	the	DET
ma-255	98	2	inverse	inverse	NOUN
ma-255	98	3	of	of	ADP
ma-255	98	4	g	g	PROPN
ma-255	98	5	is	be	AUX
ma-255	98	6	given	give	VERB
ma-255	98	7	as	as	ADP
ma-255	98	8	g−1(y	g−1(y	PROPN
ma-255	98	9	)	)	PUNCT
ma-255	99	1	=	=	PRON
ma-255	99	2	{	{	PUNCT
ma-255	99	3	x	x	PUNCT
ma-255	99	4	∈	∈	PROPN
ma-255	99	5	x	x	NOUN
ma-255	99	6	,	,	PUNCT
ma-255	99	7	y	y	PROPN
ma-255	99	8	∈	∈	PROPN
ma-255	99	9	f2(x	f2(x	PROPN
ma-255	99	10	)	)	PUNCT
ma-255	99	11	}	}	PUNCT
ma-255	99	12	.	.	PUNCT
ma-255	100	1	notethat	notethat	PROPN
ma-255	100	2	a	a	DET
ma-255	100	3	set	set	NOUN
ma-255	100	4	-	-	PUNCT
ma-255	100	5	valued	value	VERB
ma-255	100	6	operator	operator	NOUN
ma-255	100	7	h	h	NOUN
ma-255	100	8	:	:	PUNCT
ma-255	100	9	x	x	SYM
ma-255	100	10	⇒	⇒	PROPN
ma-255	100	11	y	y	PROPN
ma-255	100	12	is	be	AUX
ma-255	100	13	said	say	VERB
ma-255	100	14	to	to	PART
ma-255	100	15	be	be	AUX
ma-255	100	16	metrically	metrically	ADV
ma-255	100	17	regular	regular	ADJ
ma-255	100	18	at	at	ADP
ma-255	100	19	x0	x0	PROPN
ma-255	100	20	for	for	ADP
ma-255	100	21	y0	y0	PRON
ma-255	100	22	if	if	SCONJ
ma-255	100	23	y0	y0	PROPN
ma-255	100	24	∈	∈	NOUN
ma-255	100	25	h(x0)and	h(x0)and	NOUN
ma-255	100	26	there	there	ADV
ma-255	100	27	exists	exist	VERB
ma-255	100	28	neighbourhoods	neighbourhood	NOUN
ma-255	100	29	v1	v1	VERB
ma-255	100	30	of	of	ADP
ma-255	100	31	x0	x0	PROPN
ma-255	100	32	and	and	CCONJ
ma-255	100	33	v2	v2	NOUN
ma-255	100	34	of	of	ADP
ma-255	100	35	y0	y0	PROPN
ma-255	100	36	and	and	CCONJ
ma-255	100	37	β	β	ADJ
ma-255	100	38	>	>	X
ma-255	100	39	0	0	NUM
ma-255	100	40	such	such	ADJ
ma-255	100	41	that	that	DET
ma-255	100	42	gph(h	gph(h	PROPN
ma-255	100	43	∩	∩	NOUN
ma-255	100	44	(	(	PUNCT
ma-255	100	45	v1	v1	VERB
ma-255	100	46	×	×	NOUN
ma-255	100	47	v2	v2	NOUN
ma-255	100	48	)	)	PUNCT
ma-255	100	49	)	)	PUNCT
ma-255	100	50	isclosed	isclose	VERB
ma-255	100	51	and	and	CCONJ
ma-255	100	52	for	for	ADP
ma-255	100	53	each	each	DET
ma-255	100	54	(	(	PUNCT
ma-255	100	55	x	x	NOUN
ma-255	100	56	,	,	PUNCT
ma-255	100	57	y	y	NOUN
ma-255	100	58	)	)	PUNCT
ma-255	100	59	∈	∈	NOUN
ma-255	100	60	v1	v1	NOUN
ma-255	100	61	×	×	NOUN
ma-255	100	62	v2	v2	PROPN
ma-255	100	63	dist(x	dist(x	PROPN
ma-255	100	64	,	,	PUNCT
ma-255	100	65	h−1(y	h−1(y	PROPN
ma-255	100	66	)	)	PUNCT
ma-255	100	67	)	)	PUNCT
ma-255	101	1	≤	≤	NUM
ma-255	101	2	βdist(y	βdist(y	NUM
ma-255	101	3	,	,	PUNCT
ma-255	101	4	h(x	h(x	PROPN
ma-255	101	5	)	)	PUNCT
ma-255	101	6	)	)	PUNCT
ma-255	101	7	.	.	PUNCT
ma-255	102	1	(	(	PUNCT
ma-255	102	2	1.11	1.11	NUM
ma-255	102	3	)	)	PUNCT
ma-255	102	4	the	the	DET
ma-255	102	5	regularity	regularity	NOUN
ma-255	102	6	modulus	modulus	NOUN
ma-255	102	7	of	of	ADP
ma-255	102	8	h	h	NOUN
ma-255	102	9	at	at	ADP
ma-255	102	10	x0	x0	PROPN
ma-255	102	11	for	for	ADP
ma-255	102	12	y0	y0	PROPN
ma-255	102	13	is	be	AUX
ma-255	102	14	the	the	DET
ma-255	102	15	infimum	infimum	NOUN
ma-255	102	16	over	over	ADP
ma-255	102	17	all	all	PRON
ma-255	102	18	β	β	X
ma-255	102	19	>	>	X
ma-255	102	20	0	0	PUNCT
ma-255	103	1	and	and	CCONJ
ma-255	103	2	is	be	AUX
ma-255	103	3	denoted	denote	VERB
ma-255	103	4	by	by	ADP
ma-255	103	5	reg(h	reg(h	PROPN
ma-255	103	6	;	;	PUNCT
ma-255	103	7	x0	x0	PROPN
ma-255	103	8	/	/	SYM
ma-255	103	9	y0	y0	NOUN
ma-255	103	10	)	)	PUNCT
ma-255	103	11	.	.	PUNCT
ma-255	104	1	additionally	additionally	ADV
ma-255	104	2	if	if	SCONJ
ma-255	104	3	the	the	DET
ma-255	104	4	operator	operator	NOUN
ma-255	104	5	∆	∆	PROPN
ma-255	104	6	:	:	PUNCT
ma-255	104	7	v2	v2	PROPN
ma-255	104	8	→	→	SYM
ma-255	104	9	y	y	PROPN
ma-255	104	10	→	→	SYM
ma-255	104	11	h−1(y	h−1(y	PROPN
ma-255	104	12	)	)	PUNCT
ma-255	104	13	∩	∩	NOUN
ma-255	104	14	v1	v1	NOUN
ma-255	104	15	is	be	AUX
ma-255	104	16	not	not	PART
ma-255	104	17	multivalued	multivalue	VERB
ma-255	104	18	on	on	ADP
ma-255	104	19	v2	v2	NOUN
ma-255	104	20	,	,	PUNCT
ma-255	104	21	then	then	ADV
ma-255	104	22	we	we	PRON
ma-255	104	23	say	say	VERB
ma-255	104	24	that	that	SCONJ
ma-255	104	25	h	h	NOUN
ma-255	104	26	is	be	AUX
ma-255	104	27	strongly	strongly	ADV
ma-255	104	28	metrically	metrically	ADV
ma-255	104	29	regular	regular	ADJ
ma-255	104	30	.	.	PUNCT
ma-255	105	1	in	in	ADP
ma-255	105	2	this	this	DET
ma-255	105	3	case	case	NOUN
ma-255	105	4	,	,	PUNCT
ma-255	105	5	∆	∆	PROPN
ma-255	105	6	is	be	AUX
ma-255	105	7	lipchitz	lipchitz	NOUN
ma-255	105	8	continuous	continuous	ADJ
ma-255	105	9	on	on	ADP
ma-255	105	10	v2.finally	v2.finally	ADV
ma-255	105	11	,	,	PUNCT
ma-255	105	12	section	section	NOUN
ma-255	105	13	5	5	NUM
ma-255	105	14	contains	contain	VERB
ma-255	105	15	concluding	conclude	VERB
ma-255	105	16	remarks	remark	NOUN
ma-255	105	17	and	and	CCONJ
ma-255	105	18	directions	direction	NOUN
ma-255	105	19	for	for	ADP
ma-255	105	20	research	research	NOUN
ma-255	105	21	.	.	PUNCT
ma-255	106	1	2	2	X
ma-255	106	2	.	.	X
ma-255	106	3	convergence	convergence	NOUN
ma-255	106	4	for	for	ADP
ma-255	106	5	the	the	DET
ma-255	106	6	method	method	NOUN
ma-255	106	7	(	(	PUNCT
ma-255	106	8	1.5	1.5	NUM
ma-255	106	9	)	)	PUNCT
ma-255	106	10	we	we	PRON
ma-255	106	11	start	start	VERB
ma-255	106	12	with	with	ADP
ma-255	106	13	the	the	DET
ma-255	106	14	study	study	NOUN
ma-255	106	15	of	of	ADP
ma-255	106	16	the	the	DET
ma-255	106	17	semi	semi	ADJ
ma-255	106	18	-	-	ADJ
ma-255	106	19	local	local	ADJ
ma-255	106	20	analysis	analysis	NOUN
ma-255	106	21	in	in	ADP
ma-255	106	22	this	this	DET
ma-255	106	23	section	section	NOUN
ma-255	106	24	.	.	PUNCT
ma-255	107	1	some	some	DET
ma-255	107	2	auxiliary	auxiliary	ADJ
ma-255	107	3	results	result	NOUN
ma-255	107	4	anddefinitions	anddefinition	NOUN
ma-255	107	5	are	be	AUX
ma-255	107	6	useful	useful	ADJ
ma-255	107	7	.	.	PUNCT
ma-255	108	1	lemma	lemma	PROPN
ma-255	108	2	2.1	2.1	NUM
ma-255	108	3	.	.	PUNCT
ma-255	109	1	(	(	PUNCT
ma-255	109	2	banach	banach	ADV
ma-255	109	3	lemma	lemma	PROPN
ma-255	109	4	on	on	ADP
ma-255	109	5	invertible	invertible	ADJ
ma-255	109	6	operators	operator	NOUN
ma-255	109	7	)	)	PUNCT
ma-255	109	8	(	(	PUNCT
ma-255	109	9	[	[	X
ma-255	109	10	14	14	NUM
ma-255	109	11	,	,	PUNCT
ma-255	109	12	22	22	NUM
ma-255	109	13	,	,	PUNCT
ma-255	109	14	30	30	NUM
ma-255	109	15	,	,	PUNCT
ma-255	109	16	34	34	NUM
ma-255	109	17	]	]	PUNCT
ma-255	109	18	)	)	PUNCT
ma-255	109	19	if	if	SCONJ
ma-255	109	20	p	p	NOUN
ma-255	109	21	is	be	AUX
ma-255	109	22	a	a	DET
ma-255	109	23	bounded	bounded	ADJ
ma-255	109	24	linear	linear	ADJ
ma-255	109	25	operator	operator	NOUN
ma-255	109	26	in	in	ADP
ma-255	109	27	x	x	X
ma-255	109	28	,	,	PUNCT
ma-255	109	29	p−1	p−1	PROPN
ma-255	109	30	exists	exist	VERB
ma-255	109	31	if	if	SCONJ
ma-255	109	32	and	and	CCONJ
ma-255	109	33	only	only	ADV
ma-255	109	34	if	if	SCONJ
ma-255	109	35	there	there	PRON
ma-255	109	36	is	be	VERB
ma-255	109	37	a	a	DET
ma-255	109	38	bounded	bounded	ADJ
ma-255	109	39	linear	linear	ADJ
ma-255	109	40	operator	operator	NOUN
ma-255	109	41	p1	p1	NOUN
ma-255	109	42	in	in	ADP
ma-255	109	43	x	x	PUNCT
ma-255	109	44	such	such	ADJ
ma-255	109	45	that	that	SCONJ
ma-255	109	46	p−11	p−11	NOUN
ma-255	109	47	exists	exist	VERB
ma-255	109	48	and	and	CCONJ
ma-255	109	49	‖i	‖i	NOUN
ma-255	109	50	−	−	NOUN
ma-255	109	51	p1p‖	p1p‖	X
ma-255	109	52	<	<	X
ma-255	109	53	1	1	NUM
ma-255	109	54	.	.	PUNCT
ma-255	110	1	if	if	SCONJ
ma-255	110	2	p−1	p−1	PROPN
ma-255	110	3	exists	exist	VERB
ma-255	110	4	,	,	PUNCT
ma-255	110	5	then	then	ADV
ma-255	110	6	p−1	p−1	PROPN
ma-255	110	7	=	=	PROPN
ma-255	111	1	∞∑	∞∑	PROPN
ma-255	111	2	n=0	n=0	NUM
ma-255	111	3	(	(	PUNCT
ma-255	111	4	i	i	PRON
ma-255	111	5	−	−	PROPN
ma-255	111	6	p1p	p1p	VERB
ma-255	111	7	)	)	PUNCT
ma-255	111	8	np1	np1	PROPN
ma-255	111	9	and	and	CCONJ
ma-255	111	10	‖p−1‖	‖p−1‖	PROPN
ma-255	111	11	≤	≤	PROPN
ma-255	111	12	‖p1‖	‖p1‖	NOUN
ma-255	111	13	1−	1−	NUM
ma-255	111	14	‖i	‖i	NOUN
ma-255	111	15	−	−	NOUN
ma-255	111	16	p1p‖	p1p‖	NOUN
ma-255	111	17	.	.	PUNCT
ma-255	112	1	further	far	ADV
ma-255	112	2	,	,	PUNCT
ma-255	112	3	we	we	PRON
ma-255	112	4	use	use	VERB
ma-255	112	5	majorizing	majorize	VERB
ma-255	112	6	sequences	sequence	NOUN
ma-255	112	7	to	to	PART
ma-255	112	8	prove	prove	VERB
ma-255	112	9	the	the	DET
ma-255	112	10	semi	semi	ADJ
ma-255	112	11	-	-	ADJ
ma-255	112	12	local	local	ADJ
ma-255	112	13	convergence	convergence	NOUN
ma-255	112	14	.	.	PUNCT
ma-255	113	1	recall	recall	VERB
ma-255	113	2	the	the	DET
ma-255	113	3	definitionof	definitionof	NOUN
ma-255	113	4	majorizing	majorize	VERB
ma-255	113	5	sequence	sequence	NOUN
ma-255	113	6	.	.	PUNCT
ma-255	114	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	114	2	eur	eur	PROPN
ma-255	114	3	.	.	PUNCT
ma-255	115	1	j.	j.	PROPN
ma-255	115	2	math	math	PROPN
ma-255	115	3	.	.	PUNCT
ma-255	116	1	anal	anal	PROPN
ma-255	116	2	.	.	PUNCT
ma-255	117	1	10.28924	10.28924	NUM
ma-255	117	2	/	/	SYM
ma-255	117	3	ada	ada	PROPN
ma-255	117	4	/	/	SYM
ma-255	117	5	ma.5.5	ma.5.5	PROPN
ma-255	117	6	5	5	NUM
ma-255	117	7	definition	definition	NOUN
ma-255	117	8	2.1	2.1	NUM
ma-255	117	9	.	.	PUNCT
ma-255	118	1	(	(	PUNCT
ma-255	118	2	[	[	X
ma-255	118	3	14,22,30,34	14,22,30,34	NUM
ma-255	118	4	]	]	PUNCT
ma-255	118	5	)	)	PUNCT
ma-255	118	6	let	let	VERB
ma-255	118	7	{	{	PUNCT
ma-255	118	8	xn	xn	VERB
ma-255	118	9	}	}	PUNCT
ma-255	118	10	be	be	AUX
ma-255	118	11	a	a	DET
ma-255	118	12	sequence	sequence	NOUN
ma-255	118	13	in	in	ADP
ma-255	118	14	a	a	DET
ma-255	118	15	normed	normed	ADJ
ma-255	118	16	space	space	NOUN
ma-255	118	17	x	x	X
ma-255	118	18	.	.	PUNCT
ma-255	119	1	then	then	ADV
ma-255	119	2	a	a	DET
ma-255	119	3	nonnegative	nonnegative	ADJ
ma-255	119	4	scalar	scalar	ADJ
ma-255	119	5	sequence	sequence	NOUN
ma-255	119	6	{	{	PUNCT
ma-255	119	7	un	un	PROPN
ma-255	119	8	}	}	PUNCT
ma-255	119	9	for	for	ADP
ma-255	119	10	which	which	PRON
ma-255	119	11	‖xn+1	‖xn+1	NUM
ma-255	119	12	−	−	NOUN
ma-255	119	13	xn‖	xn‖	PROPN
ma-255	119	14	≤	≤	X
ma-255	119	15	un+1	un+1	ADV
ma-255	119	16	−	−	PROPN
ma-255	119	17	un	un	PROPN
ma-255	119	18	∀n	∀n	NUM
ma-255	119	19	≥	≥	NOUN
ma-255	119	20	0	0	NUM
ma-255	119	21	(	(	PUNCT
ma-255	119	22	2.1	2.1	NUM
ma-255	119	23	)	)	PUNCT
ma-255	119	24	holds	hold	NOUN
ma-255	119	25	,	,	PUNCT
ma-255	119	26	is	be	AUX
ma-255	119	27	a	a	DET
ma-255	119	28	majorizing	majorize	VERB
ma-255	119	29	sequence	sequence	NOUN
ma-255	119	30	for	for	ADP
ma-255	119	31	{	{	PUNCT
ma-255	119	32	xn	xn	NUM
ma-255	119	33	}	}	PUNCT
ma-255	119	34	.	.	PUNCT
ma-255	120	1	note	note	VERB
ma-255	120	2	that	that	SCONJ
ma-255	120	3	any	any	DET
ma-255	120	4	majorizing	majorize	VERB
ma-255	120	5	sequence	sequence	NOUN
ma-255	120	6	is	be	AUX
ma-255	120	7	necessarily	necessarily	ADV
ma-255	120	8	nondecreasing	nondecrease	VERB
ma-255	120	9	.	.	PUNCT
ma-255	121	1	moreover	moreover	ADV
ma-255	121	2	,	,	PUNCT
ma-255	121	3	if	if	SCONJ
ma-255	121	4	the	the	DET
ma-255	121	5	sequence	sequence	NOUN
ma-255	121	6	{	{	PUNCT
ma-255	121	7	un	un	PROPN
ma-255	121	8	}	}	PUNCT
ma-255	121	9	converges	converge	NOUN
ma-255	121	10	,	,	PUNCT
ma-255	121	11	then	then	ADV
ma-255	121	12	{	{	PUNCT
ma-255	121	13	xn	xn	X
ma-255	121	14	}	}	PUNCT
ma-255	121	15	converges	converge	NOUN
ma-255	121	16	too	too	ADV
ma-255	121	17	,	,	PUNCT
ma-255	121	18	and	and	CCONJ
ma-255	121	19	for	for	ADP
ma-255	121	20	u∗	u∗	NOUN
ma-255	121	21	=	=	SYM
ma-255	121	22	limn−→∞	limn−→∞	NOUN
ma-255	121	23	un	un	PROPN
ma-255	121	24	‖s∗	‖s∗	PROPN
ma-255	122	1	−	−	PROPN
ma-255	122	2	xn‖	xn‖	PROPN
ma-255	122	3	≤	≤	PROPN
ma-255	122	4	u∗	u∗	PROPN
ma-255	122	5	−	−	PROPN
ma-255	122	6	un	un	PROPN
ma-255	122	7	.	.	PROPN
ma-255	122	8	hence	hence	ADV
ma-255	122	9	,	,	PUNCT
ma-255	122	10	the	the	DET
ma-255	122	11	study	study	NOUN
ma-255	122	12	of	of	ADP
ma-255	122	13	the	the	DET
ma-255	122	14	convergence	convergence	NOUN
ma-255	122	15	of	of	ADP
ma-255	122	16	the	the	DET
ma-255	122	17	sequence	sequence	NOUN
ma-255	122	18	{	{	PUNCT
ma-255	122	19	xn	xn	NOUN
ma-255	122	20	}	}	PUNCT
ma-255	122	21	reduces	reduce	VERB
ma-255	122	22	to	to	ADP
ma-255	122	23	that	that	PRON
ma-255	122	24	of	of	ADP
ma-255	122	25	{	{	PUNCT
ma-255	122	26	un	un	PROPN
ma-255	122	27	}	}	PUNCT
ma-255	122	28	.	.	PUNCT
ma-255	123	1	the	the	DET
ma-255	123	2	analysis	analysis	NOUN
ma-255	123	3	requires	require	VERB
ma-255	123	4	some	some	DET
ma-255	123	5	conditions	condition	NOUN
ma-255	123	6	.	.	PUNCT
ma-255	124	1	let	let	VERB
ma-255	125	1	e	e	NOUN
ma-255	125	2	=	=	PUNCT
ma-255	126	1	[	[	X
ma-255	126	2	0,+∞).suppose(h1	0,+∞).suppose(h1	NOUN
ma-255	126	3	)	)	PUNCT
ma-255	126	4	there	there	PRON
ma-255	126	5	exists	exist	VERB
ma-255	126	6	a	a	DET
ma-255	126	7	function	function	NOUN
ma-255	126	8	φ	φ	NOUN
ma-255	126	9	:	:	PUNCT
ma-255	127	1	e	e	X
ma-255	127	2	×	×	NOUN
ma-255	127	3	e	e	X
ma-255	127	4	×	×	NOUN
ma-255	127	5	e	e	X
ma-255	127	6	→	→	SYM
ma-255	127	7	[	[	X
ma-255	127	8	0,+∞	0,+∞	NUM
ma-255	127	9	)	)	PUNCT
ma-255	127	10	continuous	continuous	ADJ
ma-255	127	11	as	as	ADP
ma-255	127	12	well	well	ADV
ma-255	127	13	nondecreasing	nondecrease	VERB
ma-255	127	14	in	in	ADP
ma-255	127	15	allthree	allthree	ADJ
ma-255	127	16	variables	variable	NOUN
ma-255	127	17	and	and	CCONJ
ma-255	127	18	invertible	invertible	ADJ
ma-255	127	19	operators	operator	NOUN
ma-255	127	20	m	m	PROPN
ma-255	127	21	(	(	PUNCT
ma-255	127	22	.	.	PUNCT
ma-255	127	23	)	)	PUNCT
ma-255	127	24	and	and	CCONJ
ma-255	127	25	γ	γ	NOUN
ma-255	127	26	such	such	ADJ
ma-255	127	27	that	that	DET
ma-255	127	28	for	for	ADP
ma-255	127	29	some	some	DET
ma-255	127	30	x0	x0	PROPN
ma-255	127	31	∈	∈	PROPN
ma-255	127	32	ω	ω	PROPN
ma-255	127	33	,	,	PUNCT
ma-255	127	34	and	and	CCONJ
ma-255	127	35	each	each	DET
ma-255	127	36	x	x	NOUN
ma-255	127	37	,	,	PUNCT
ma-255	127	38	y	y	PROPN
ma-255	127	39	∈	∈	PROPN
ma-255	127	40	ω	ω	PROPN
ma-255	127	41	the	the	DET
ma-255	127	42	following	follow	VERB
ma-255	127	43	mysovskii	mysovskii	ADJ
ma-255	127	44	-	-	PUNCT
ma-255	127	45	like	like	ADJ
ma-255	127	46	condition	condition	NOUN
ma-255	127	47	holds	hold	VERB
ma-255	127	48	‖m(x)γ−1(f1(y)−	‖m(x)γ−1(f1(y)−	PRON
ma-255	127	49	f1(x)−	f1(x)−	PROPN
ma-255	127	50	γm−1(x))‖	γm−1(x))‖	PROPN
ma-255	127	51	≤	≤	PUNCT
ma-255	127	52	φ(‖x	φ(‖x	PROPN
ma-255	127	53	−	−	PROPN
ma-255	127	54	x0‖	x0‖	PROPN
ma-255	127	55	,	,	PUNCT
ma-255	127	56	‖y	‖y	PUNCT
ma-255	128	1	−	−	PROPN
ma-255	128	2	x0‖	x0‖	PROPN
ma-255	128	3	,	,	PUNCT
ma-255	128	4	‖y	‖y	PUNCT
ma-255	129	1	−	−	PROPN
ma-255	129	2	x‖)‖y	x‖)‖y	PROPN
ma-255	129	3	−	−	PROPN
ma-255	129	4	x‖	x‖	PROPN
ma-255	129	5	define	define	VERB
ma-255	129	6	the	the	DET
ma-255	129	7	real	real	ADJ
ma-255	129	8	real	real	ADJ
ma-255	129	9	sequence	sequence	NOUN
ma-255	129	10	{	{	PUNCT
ma-255	129	11	αn	αn	NOUN
ma-255	129	12	}	}	PUNCT
ma-255	129	13	for	for	ADP
ma-255	129	14	α0	α0	ADJ
ma-255	129	15	=	=	SYM
ma-255	129	16	0	0	NUM
ma-255	129	17	,	,	PUNCT
ma-255	129	18	α1	α1	PROPN
ma-255	129	19	≥	≥	PROPN
ma-255	129	20	η	η	PROPN
ma-255	129	21	:	:	PUNCT
ma-255	129	22	=	=	SYM
ma-255	129	23	‖m(x0)γ−1f1(x0)‖	‖m(x0)γ−1f1(x0)‖	PROPN
ma-255	129	24	and	and	CCONJ
ma-255	129	25	each	each	DET
ma-255	129	26	n	n	PROPN
ma-255	129	27	=	=	SYM
ma-255	129	28	0	0	NUM
ma-255	129	29	,	,	PUNCT
ma-255	129	30	1	1	NUM
ma-255	129	31	,	,	PUNCT
ma-255	129	32	2	2	NUM
ma-255	129	33	,	,	PUNCT
ma-255	129	34	...	...	PUNCT
ma-255	129	35	by	by	ADP
ma-255	129	36	αn+1	αn+1	NUM
ma-255	129	37	=	=	SYM
ma-255	129	38	αn	αn	NOUN
ma-255	130	1	+	+	PUNCT
ma-255	130	2	φ(αn−1	φ(αn−1	ADJ
ma-255	130	3	,	,	PUNCT
ma-255	130	4	αn	αn	VERB
ma-255	130	5	,	,	PUNCT
ma-255	130	6	αn	αn	NOUN
ma-255	130	7	−	−	NOUN
ma-255	131	1	αn−1)(αn	αn−1)(αn	PROPN
ma-255	131	2	−	−	PROPN
ma-255	131	3	αn−1	αn−1	PROPN
ma-255	131	4	)	)	PUNCT
ma-255	131	5	,	,	PUNCT
ma-255	131	6	n	n	NOUN
ma-255	131	7	=	=	SYM
ma-255	131	8	1	1	NUM
ma-255	131	9	,	,	PUNCT
ma-255	131	10	2	2	NUM
ma-255	131	11	,	,	PUNCT
ma-255	131	12	...	...	PUNCT
ma-255	131	13	(	(	PUNCT
ma-255	131	14	2.2	2.2	NUM
ma-255	131	15	)	)	PUNCT
ma-255	131	16	notice	notice	VERB
ma-255	131	17	that	that	SCONJ
ma-255	131	18	the	the	DET
ma-255	131	19	constant	constant	ADJ
ma-255	131	20	η	η	PROPN
ma-255	131	21	is	be	AUX
ma-255	131	22	well	well	ADV
ma-255	131	23	defined	define	VERB
ma-255	131	24	since	since	SCONJ
ma-255	131	25	the	the	DET
ma-255	131	26	operator	operator	NOUN
ma-255	131	27	γ	γ	NOUN
ma-255	131	28	is	be	AUX
ma-255	131	29	invertible	invertible	ADJ
ma-255	131	30	.	.	PUNCT
ma-255	132	1	moreover	moreover	ADV
ma-255	132	2	,	,	PUNCT
ma-255	132	3	thesequence	thesequence	NOUN
ma-255	132	4	{	{	PUNCT
ma-255	132	5	αn	αn	NOUN
ma-255	132	6	}	}	PUNCT
ma-255	132	7	defined	define	VERB
ma-255	132	8	by	by	ADP
ma-255	132	9	the	the	DET
ma-255	132	10	formula	formula	NOUN
ma-255	132	11	(	(	PUNCT
ma-255	132	12	2.2	2.2	NUM
ma-255	132	13	)	)	PUNCT
ma-255	132	14	is	be	AUX
ma-255	132	15	proven	prove	VERB
ma-255	132	16	to	to	PART
ma-255	132	17	be	be	AUX
ma-255	132	18	majorizing	majorize	VERB
ma-255	132	19	for	for	ADP
ma-255	132	20	the	the	DET
ma-255	132	21	method	method	NOUN
ma-255	132	22	(	(	PUNCT
ma-255	132	23	1.5)in	1.5)in	NOUN
ma-255	132	24	theorem	theorem	ADJ
ma-255	132	25	2.3	2.3	NUM
ma-255	132	26	.	.	PUNCT
ma-255	133	1	but	but	CCONJ
ma-255	133	2	let	let	VERB
ma-255	133	3	us	we	PRON
ma-255	133	4	present	present	ADJ
ma-255	133	5	convergence	convergence	NOUN
ma-255	133	6	conditions	condition	NOUN
ma-255	133	7	for	for	ADP
ma-255	133	8	it.(h2	it.(h2	NOUN
ma-255	133	9	)	)	PUNCT
ma-255	133	10	there	there	PRON
ma-255	133	11	exists	exist	VERB
ma-255	133	12	a	a	DET
ma-255	133	13	parameter	parameter	NOUN
ma-255	133	14	ρ	ρ	PROPN
ma-255	133	15	≥	≥	PROPN
ma-255	133	16	η	η	PROPN
ma-255	133	17	such	such	ADJ
ma-255	133	18	that	that	PRON
ma-255	133	19	for	for	ADP
ma-255	133	20	each	each	DET
ma-255	133	21	n	n	NOUN
ma-255	133	22	=	=	SYM
ma-255	133	23	0	0	NUM
ma-255	133	24	,	,	PUNCT
ma-255	133	25	1	1	NUM
ma-255	133	26	,	,	PUNCT
ma-255	133	27	2	2	NUM
ma-255	133	28	,	,	PUNCT
ma-255	133	29	...	...	PUNCT
ma-255	133	30	φ(αn−1	φ(αn−1	X
ma-255	133	31	,	,	PUNCT
ma-255	133	32	αn	αn	VERB
ma-255	133	33	,	,	PUNCT
ma-255	133	34	αn	αn	NOUN
ma-255	133	35	−	−	PROPN
ma-255	133	36	αn−1	αn−1	PROPN
ma-255	133	37	)	)	PUNCT
ma-255	133	38	<	<	X
ma-255	133	39	1	1	NUM
ma-255	133	40	and	and	CCONJ
ma-255	133	41	αn	αn	NOUN
ma-255	133	42	≤	≤	NUM
ma-255	133	43	ρ	ρ	NOUN
ma-255	133	44	.	.	PUNCT
ma-255	134	1	it	it	PRON
ma-255	134	2	follows	follow	VERB
ma-255	134	3	by	by	ADP
ma-255	134	4	the	the	DET
ma-255	134	5	condition	condition	NOUN
ma-255	134	6	(	(	PUNCT
ma-255	134	7	h2	h2	NOUN
ma-255	134	8	)	)	PUNCT
ma-255	134	9	and	and	CCONJ
ma-255	134	10	the	the	DET
ma-255	134	11	formula	formula	NOUN
ma-255	134	12	(	(	PUNCT
ma-255	134	13	2.2	2.2	NUM
ma-255	134	14	)	)	PUNCT
ma-255	134	15	that	that	SCONJ
ma-255	134	16	0	0	NUM
ma-255	134	17	≤	≤	X
ma-255	134	18	αn−1	αn−1	ADJ
ma-255	134	19	≤	≤	NUM
ma-255	134	20	αn	αn	NOUN
ma-255	134	21	≤	≤	NUM
ma-255	134	22	ρ	ρ	NOUN
ma-255	134	23	and	and	CCONJ
ma-255	134	24	there	there	PRON
ma-255	134	25	exists	exist	VERB
ma-255	134	26	α∗	α∗	NOUN
ma-255	134	27	∈	∈	PROPN
ma-255	135	1	[	[	X
ma-255	135	2	η	η	PROPN
ma-255	135	3	,	,	PUNCT
ma-255	135	4	ρ	ρ	PROPN
ma-255	135	5	]	]	X
ma-255	136	1	such	such	ADJ
ma-255	136	2	that	that	PRON
ma-255	136	3	limn→+∞	limn→+∞	VERB
ma-255	136	4	αn	αn	NOUN
ma-255	137	1	=	=	SYM
ma-255	138	1	α∗.	α∗.	NOUN
ma-255	138	2	h	h	NOUN
ma-255	138	3	is	be	AUX
ma-255	138	4	well	well	ADV
ma-255	138	5	known	know	VERB
ma-255	138	6	that	that	SCONJ
ma-255	138	7	this	this	DET
ma-255	138	8	limit	limit	NOUN
ma-255	138	9	isthe	isthe	ADJ
ma-255	138	10	unique	unique	ADJ
ma-255	138	11	least	least	ADV
ma-255	138	12	upper	upper	ADJ
ma-255	138	13	bound	bind	VERB
ma-255	138	14	of	of	ADP
ma-255	138	15	the	the	DET
ma-255	138	16	sequence	sequence	NOUN
ma-255	138	17	{	{	PUNCT
ma-255	138	18	αn	αn	NOUN
ma-255	138	19	}	}	PUNCT
ma-255	138	20	and(h3	and(h3	ADJ
ma-255	138	21	)	)	PUNCT
ma-255	138	22	s[x0	s[x0	ADV
ma-255	138	23	,	,	PUNCT
ma-255	138	24	α	α	PROPN
ma-255	138	25	∗	∗	NOUN
ma-255	138	26	]	]	X
ma-255	139	1	⊂	⊂	ADJ
ma-255	139	2	ω.the	ω.the	DET
ma-255	139	3	conditions	condition	NOUN
ma-255	139	4	(	(	PUNCT
ma-255	139	5	h1)-(h3	h1)-(h3	NUM
ma-255	139	6	)	)	PUNCT
ma-255	139	7	combined	combine	VERB
ma-255	139	8	with	with	ADP
ma-255	139	9	the	the	DET
ma-255	139	10	terminology	terminology	NOUN
ma-255	139	11	are	be	AUX
ma-255	139	12	utilized	utilize	VERB
ma-255	139	13	for	for	ADP
ma-255	139	14	the	the	DET
ma-255	139	15	convergence	convergence	NOUN
ma-255	139	16	of	of	ADP
ma-255	139	17	themethod	themethod	NOUN
ma-255	139	18	(	(	PUNCT
ma-255	139	19	1.5	1.5	NUM
ma-255	139	20	)	)	PUNCT
ma-255	139	21	.	.	PUNCT
ma-255	140	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	140	2	eur	eur	PROPN
ma-255	140	3	.	.	PUNCT
ma-255	141	1	j.	j.	PROPN
ma-255	141	2	math	math	PROPN
ma-255	141	3	.	.	PUNCT
ma-255	142	1	anal	anal	PROPN
ma-255	142	2	.	.	PUNCT
ma-255	143	1	10.28924	10.28924	NUM
ma-255	143	2	/	/	SYM
ma-255	143	3	ada	ada	PROPN
ma-255	143	4	/	/	SYM
ma-255	143	5	ma.5.5	ma.5.5	PROPN
ma-255	143	6	6	6	NUM
ma-255	143	7	theorem	theorem	VERB
ma-255	143	8	2.1	2.1	NUM
ma-255	143	9	.	.	PUNCT
ma-255	144	1	suppose	suppose	VERB
ma-255	144	2	that	that	SCONJ
ma-255	144	3	the	the	DET
ma-255	144	4	conditions	condition	NOUN
ma-255	144	5	(	(	PUNCT
ma-255	144	6	h1)-(h3	h1)-(h3	ADJ
ma-255	144	7	)	)	PUNCT
ma-255	144	8	hold	hold	NOUN
ma-255	144	9	.	.	PUNCT
ma-255	145	1	then	then	ADV
ma-255	145	2	,	,	PUNCT
ma-255	145	3	the	the	DET
ma-255	145	4	sequence	sequence	NOUN
ma-255	145	5	{	{	PUNCT
ma-255	145	6	xn	xn	PROPN
ma-255	145	7	}	}	PUNCT
ma-255	145	8	generated	generate	VERB
ma-255	145	9	by	by	ADP
ma-255	145	10	the	the	DET
ma-255	145	11	method	method	NOUN
ma-255	145	12	(	(	PUNCT
ma-255	145	13	1.5	1.5	NUM
ma-255	145	14	)	)	PUNCT
ma-255	145	15	exists	exist	VERB
ma-255	145	16	in	in	ADP
ma-255	145	17	the	the	DET
ma-255	145	18	ball	ball	NOUN
ma-255	145	19	s(x0	s(x0	NOUN
ma-255	145	20	,	,	PUNCT
ma-255	145	21	α	α	PROPN
ma-255	145	22	∗	∗	NOUN
ma-255	145	23	)	)	PUNCT
ma-255	145	24	remains	remain	VERB
ma-255	145	25	in	in	ADP
ma-255	145	26	the	the	DET
ma-255	145	27	same	same	ADJ
ma-255	145	28	ball	ball	NOUN
ma-255	145	29	for	for	ADP
ma-255	145	30	each	each	DET
ma-255	145	31	n	n	NOUN
ma-255	145	32	=	=	SYM
ma-255	145	33	0	0	NUM
ma-255	145	34	,	,	PUNCT
ma-255	145	35	1	1	NUM
ma-255	145	36	,	,	PUNCT
ma-255	145	37	2	2	NUM
ma-255	145	38	,	,	PUNCT
ma-255	145	39	...	...	PUNCT
ma-255	145	40	and	and	CCONJ
ma-255	145	41	is	be	AUX
ma-255	145	42	convergent	convergent	ADJ
ma-255	145	43	to	to	ADP
ma-255	145	44	a	a	DET
ma-255	145	45	unique	unique	ADJ
ma-255	145	46	solution	solution	NOUN
ma-255	145	47	s∗	s∗	PROPN
ma-255	145	48	∈	∈	PROPN
ma-255	145	49	s[x0	s[x0	NOUN
ma-255	145	50	,	,	PUNCT
ma-255	145	51	α	α	PROPN
ma-255	145	52	∗	∗	NOUN
ma-255	145	53	]	]	PUNCT
ma-255	145	54	of	of	ADP
ma-255	145	55	the	the	DET
ma-255	145	56	equation	equation	NOUN
ma-255	145	57	f1(x	f1(x	NOUN
ma-255	145	58	)	)	PUNCT
ma-255	145	59	=	=	SYM
ma-255	145	60	0	0	NUM
ma-255	146	1	such	such	ADJ
ma-255	146	2	that	that	SCONJ
ma-255	146	3	‖s∗	‖s∗	PUNCT
ma-255	146	4	−	−	NOUN
ma-255	146	5	xn‖	xn‖	PROPN
ma-255	146	6	≤	≤	PROPN
ma-255	146	7	α∗	α∗	VERB
ma-255	146	8	−	−	NOUN
ma-255	146	9	αn	αn	NOUN
ma-255	146	10	,	,	PUNCT
ma-255	146	11	n	n	NOUN
ma-255	146	12	=	=	SYM
ma-255	146	13	0	0	NUM
ma-255	146	14	,	,	PUNCT
ma-255	146	15	1	1	NUM
ma-255	146	16	,	,	PUNCT
ma-255	146	17	2	2	NUM
ma-255	146	18	.	.	PUNCT
ma-255	146	19	.	.	PUNCT
ma-255	146	20	.	.	PUNCT
ma-255	147	1	.	.	PUNCT
ma-255	148	1	(	(	PUNCT
ma-255	148	2	2.3	2.3	NUM
ma-255	148	3	)	)	PUNCT
ma-255	148	4	proof	proof	NOUN
ma-255	148	5	.	.	PUNCT
ma-255	149	1	the	the	DET
ma-255	149	2	process	process	NOUN
ma-255	149	3	of	of	ADP
ma-255	149	4	induction	induction	NOUN
ma-255	149	5	is	be	AUX
ma-255	149	6	employed	employ	VERB
ma-255	149	7	to	to	PART
ma-255	149	8	show	show	VERB
ma-255	149	9	the	the	DET
ma-255	149	10	assertion	assertion	NOUN
ma-255	149	11	‖xm+1	‖xm+1	NUM
ma-255	149	12	−	−	PROPN
ma-255	149	13	xm‖	xm‖	PROPN
ma-255	149	14	≤	≤	NUM
ma-255	150	1	αm+1	αm+1	NUM
ma-255	151	1	−	−	NOUN
ma-255	151	2	αm	αm	NOUN
ma-255	151	3	for	for	ADP
ma-255	151	4	m	m	PROPN
ma-255	151	5	=	=	SYM
ma-255	151	6	0	0	NUM
ma-255	151	7	,	,	PUNCT
ma-255	151	8	1	1	NUM
ma-255	151	9	,	,	PUNCT
ma-255	151	10	2	2	NUM
ma-255	151	11	,	,	PUNCT
ma-255	151	12	.	.	PUNCT
ma-255	151	13	.	.	PUNCT
ma-255	151	14	.	.	PUNCT
ma-255	151	15	.	.	PUNCT
ma-255	152	1	(	(	PUNCT
ma-255	152	2	2.4	2.4	NUM
ma-255	152	3	)	)	PUNCT
ma-255	152	4	the	the	DET
ma-255	152	5	choice	choice	NOUN
ma-255	152	6	of	of	ADP
ma-255	152	7	η	η	PROPN
ma-255	152	8	,	,	PUNCT
ma-255	152	9	(	(	PUNCT
ma-255	152	10	2.2	2.2	NUM
ma-255	152	11	)	)	PUNCT
ma-255	152	12	and	and	CCONJ
ma-255	152	13	the	the	DET
ma-255	152	14	method	method	NOUN
ma-255	152	15	(	(	PUNCT
ma-255	152	16	1.5	1.5	NUM
ma-255	152	17	)	)	PUNCT
ma-255	152	18	give	give	VERB
ma-255	152	19	that	that	DET
ma-255	152	20	‖x1	‖x1	NOUN
ma-255	152	21	−	−	PROPN
ma-255	152	22	x0‖	x0‖	PROPN
ma-255	153	1	=	=	PUNCT
ma-255	153	2	‖m(x0)γ−1f1(x0)‖	‖m(x0)γ−1f1(x0)‖	PROPN
ma-255	153	3	≤	≤	PROPN
ma-255	153	4	η	η	PROPN
ma-255	153	5	=	=	PROPN
ma-255	153	6	α1	α1	PROPN
ma-255	153	7	−	−	PROPN
ma-255	153	8	α0	α0	ADJ
ma-255	153	9	<	<	X
ma-255	153	10	α∗	α∗	NOUN
ma-255	153	11	,	,	PUNCT
ma-255	153	12	so	so	CCONJ
ma-255	153	13	the	the	DET
ma-255	153	14	assertion	assertion	NOUN
ma-255	153	15	(	(	PUNCT
ma-255	153	16	2.4	2.4	NUM
ma-255	153	17	)	)	PUNCT
ma-255	153	18	holds	hold	VERB
ma-255	153	19	if	if	SCONJ
ma-255	153	20	m	m	VERB
ma-255	153	21	=	=	SYM
ma-255	153	22	0	0	NUM
ma-255	153	23	,	,	PUNCT
ma-255	153	24	and	and	CCONJ
ma-255	153	25	the	the	DET
ma-255	153	26	iterate	iterate	NOUN
ma-255	153	27	x1	x1	PROPN
ma-255	153	28	∈	∈	PROPN
ma-255	153	29	s(x0	s(x0	NOUN
ma-255	153	30	,	,	PUNCT
ma-255	153	31	α	α	PROPN
ma-255	153	32	∗	∗	NOUN
ma-255	153	33	)	)	PUNCT
ma-255	153	34	.	.	PUNCT
ma-255	154	1	suppose	suppose	VERB
ma-255	154	2	iterates	iterate	VERB
ma-255	154	3	x0	x0	PROPN
ma-255	154	4	,	,	PUNCT
ma-255	154	5	x1	x1	PROPN
ma-255	154	6	,	,	PUNCT
ma-255	154	7	...	...	PUNCT
ma-255	154	8	,	,	PUNCT
ma-255	154	9	xmexist	xmexist	PROPN
ma-255	154	10	and	and	CCONJ
ma-255	154	11	(	(	PUNCT
ma-255	154	12	2.4	2.4	NUM
ma-255	154	13	)	)	PUNCT
ma-255	154	14	holds	hold	VERB
ma-255	154	15	for	for	ADP
ma-255	154	16	all	all	DET
ma-255	154	17	integers	integer	NOUN
ma-255	154	18	smaller	small	ADJ
ma-255	154	19	or	or	CCONJ
ma-255	154	20	equal	equal	ADJ
ma-255	154	21	to	to	ADP
ma-255	154	22	m−	m−	PROPN
ma-255	154	23	1	1	NUM
ma-255	154	24	.	.	PUNCT
ma-255	154	25	notice	notice	VERB
ma-255	154	26	that	that	SCONJ
ma-255	154	27	the	the	DET
ma-255	154	28	iterate	iterate	NOUN
ma-255	154	29	xm+1	xm+1	PROPN
ma-255	154	30	existsby	existsby	NOUN
ma-255	154	31	the	the	DET
ma-255	154	32	method	method	NOUN
ma-255	154	33	(	(	PUNCT
ma-255	154	34	1.5	1.5	NUM
ma-255	154	35	)	)	PUNCT
ma-255	154	36	and	and	CCONJ
ma-255	154	37	the	the	DET
ma-255	154	38	invertability	invertability	NOUN
ma-255	154	39	of	of	ADP
ma-255	154	40	the	the	DET
ma-255	154	41	operators	operator	NOUN
ma-255	154	42	m(xm	m(xm	NUM
ma-255	154	43	)	)	PUNCT
ma-255	154	44	and	and	CCONJ
ma-255	154	45	γ	γ	X
ma-255	154	46	.	.	PROPN
ma-255	154	47	then	then	ADV
ma-255	154	48	,	,	PUNCT
ma-255	154	49	we	we	PRON
ma-255	154	50	can	can	AUX
ma-255	154	51	write	write	VERB
ma-255	154	52	by	by	ADP
ma-255	154	53	themethod	themethod	NOUN
ma-255	154	54	(	(	PUNCT
ma-255	154	55	1.5	1.5	NUM
ma-255	154	56	)	)	PUNCT
ma-255	154	57	the	the	DET
ma-255	154	58	ostrowski	ostrowski	ADJ
ma-255	154	59	-	-	PUNCT
ma-255	154	60	type	type	NOUN
ma-255	154	61	representation	representation	NOUN
ma-255	154	62	for	for	ADP
ma-255	154	63	f1(xm	f1(xm	NUM
ma-255	154	64	)	)	PUNCT
ma-255	154	65	as	as	ADP
ma-255	154	66	f1(xm	f1(xm	NUM
ma-255	154	67	)	)	PUNCT
ma-255	154	68	=	=	PUNCT
ma-255	154	69	f1(xm)−	f1(xm)−	NOUN
ma-255	154	70	f1(xm−1)−	f1(xm−1)−	NOUN
ma-255	154	71	γm−1(xm	γm−1(xm	ADV
ma-255	154	72	−	−	PROPN
ma-255	154	73	xm−1	xm−1	PROPN
ma-255	154	74	)	)	PUNCT
ma-255	154	75	.	.	PUNCT
ma-255	155	1	(	(	PUNCT
ma-255	155	2	2.5	2.5	NUM
ma-255	155	3	)	)	PUNCT
ma-255	155	4	using	use	VERB
ma-255	155	5	the	the	DET
ma-255	155	6	condition	condition	NOUN
ma-255	155	7	(	(	PUNCT
ma-255	155	8	h1	h1	PROPN
ma-255	155	9	)	)	PUNCT
ma-255	155	10	,	,	PUNCT
ma-255	155	11	method	method	NOUN
ma-255	155	12	(	(	PUNCT
ma-255	155	13	1.5	1.5	NUM
ma-255	155	14	)	)	PUNCT
ma-255	155	15	,	,	PUNCT
ma-255	155	16	(	(	PUNCT
ma-255	155	17	2.2	2.2	NUM
ma-255	155	18	)	)	PUNCT
ma-255	155	19	and	and	CCONJ
ma-255	155	20	the	the	DET
ma-255	155	21	induction	induction	NOUN
ma-255	155	22	hypothesis	hypothesis	NOUN
ma-255	155	23	on	on	ADP
ma-255	155	24	(	(	PUNCT
ma-255	155	25	2.5	2.5	NUM
ma-255	155	26	)	)	PUNCT
ma-255	155	27	we	we	PRON
ma-255	155	28	obtain	obtain	VERB
ma-255	155	29	inturn	inturn	NOUN
ma-255	155	30	that	that	PRON
ma-255	155	31	‖xm+1	‖xm+1	VERB
ma-255	155	32	−	−	PROPN
ma-255	155	33	xm‖	xm‖	PROPN
ma-255	155	34	=	=	PUNCT
ma-255	155	35	‖mγ−1(f1(xm))‖	‖mγ−1(f1(xm))‖	PUNCT
ma-255	155	36	=	=	SYM
ma-255	155	37	‖mγ−1(f1(xm)−	‖mγ−1(f1(xm)−	NUM
ma-255	155	38	f1(xm−1)−	f1(xm−1)−	NOUN
ma-255	155	39	γm−1(xm	γm−1(xm	ADP
ma-255	155	40	−	−	NOUN
ma-255	155	41	xm−1))‖	xm−1))‖	X
ma-255	155	42	≤	≤	NOUN
ma-255	155	43	φ(‖xm−1	φ(‖xm−1	ADJ
ma-255	156	1	−	−	PROPN
ma-255	157	1	x0‖	x0‖	PROPN
ma-255	157	2	,	,	PUNCT
ma-255	157	3	‖xm	‖xm	PROPN
ma-255	157	4	−	−	PROPN
ma-255	157	5	x0‖	x0‖	PROPN
ma-255	157	6	,	,	PUNCT
ma-255	157	7	‖xm	‖xm	PROPN
ma-255	157	8	−	−	PROPN
ma-255	158	1	xm−1‖)‖xm	xm−1‖)‖xm	PROPN
ma-255	158	2	−	−	PROPN
ma-255	158	3	xm−1‖	xm−1‖	PROPN
ma-255	158	4	≤	≤	PROPN
ma-255	158	5	φ(αm−1	φ(αm−1	X
ma-255	158	6	,	,	PUNCT
ma-255	158	7	αm	αm	PROPN
ma-255	158	8	,	,	PUNCT
ma-255	158	9	αm	αm	NOUN
ma-255	158	10	−	−	PROPN
ma-255	158	11	αm−1)(αm	αm−1)(αm	PROPN
ma-255	158	12	−	−	NOUN
ma-255	158	13	αm−1	αm−1	NOUN
ma-255	158	14	)	)	PUNCT
ma-255	158	15	=	=	PUNCT
ma-255	159	1	αm+1	αm+1	X
ma-255	159	2	−	−	NUM
ma-255	159	3	αm	αm	INTJ
ma-255	159	4	,	,	PUNCT
ma-255	159	5	(	(	PUNCT
ma-255	159	6	2.6	2.6	NUM
ma-255	159	7	)	)	PUNCT
ma-255	159	8	and	and	CCONJ
ma-255	159	9	‖xm+1	‖xm+1	NUM
ma-255	159	10	−	−	PROPN
ma-255	159	11	x0‖	x0‖	PROPN
ma-255	159	12	≤	≤	PROPN
ma-255	159	13	‖xm+1	‖xm+1	PUNCT
ma-255	160	1	−	−	PROPN
ma-255	160	2	xm‖+	xm‖+	PROPN
ma-255	161	1	‖xm	‖xm	NUM
ma-255	161	2	−	−	PROPN
ma-255	161	3	xm−1‖+	xm−1‖+	NOUN
ma-255	161	4	...	...	PUNCT
ma-255	162	1	+	+	CCONJ
ma-255	162	2	‖x1	‖x1	NOUN
ma-255	162	3	−	−	PROPN
ma-255	162	4	x0‖	x0‖	PROPN
ma-255	162	5	≤	≤	NUM
ma-255	162	6	αm+1	αm+1	NUM
ma-255	162	7	−	−	NOUN
ma-255	163	1	αm	αm	NOUN
ma-255	164	1	+	+	CCONJ
ma-255	164	2	αm	αm	NOUN
ma-255	164	3	−	−	PROPN
ma-255	164	4	αm−1	αm−1	NOUN
ma-255	164	5	+	+	CCONJ
ma-255	164	6	...	...	PUNCT
ma-255	164	7	+	+	CCONJ
ma-255	164	8	α1	α1	PROPN
ma-255	164	9	−	−	PROPN
ma-255	164	10	α0	α0	PROPN
ma-255	164	11	=	=	SYM
ma-255	164	12	αm+1	αm+1	X
ma-255	164	13	<	<	X
ma-255	164	14	α∗	α∗	NOUN
ma-255	164	15	,	,	PUNCT
ma-255	164	16	which	which	PRON
ma-255	164	17	complete	complete	VERB
ma-255	164	18	the	the	DET
ma-255	164	19	induction	induction	NOUN
ma-255	164	20	for	for	ADP
ma-255	164	21	the	the	DET
ma-255	164	22	assertion	assertion	NOUN
ma-255	164	23	(	(	PUNCT
ma-255	164	24	2.4	2.4	NUM
ma-255	164	25	)	)	PUNCT
ma-255	164	26	and	and	CCONJ
ma-255	164	27	show	show	VERB
ma-255	164	28	that	that	SCONJ
ma-255	164	29	all	all	DET
ma-255	164	30	iterates	iterate	NOUN
ma-255	164	31	{	{	PUNCT
ma-255	164	32	xm+1	xm+1	NUM
ma-255	164	33	}	}	PUNCT
ma-255	164	34	⊂	⊂	NOUN
ma-255	164	35	s(x0	s(x0	NOUN
ma-255	164	36	,	,	PUNCT
ma-255	164	37	α	α	PROPN
ma-255	164	38	∗).but	∗).but	NOUN
ma-255	164	39	the	the	DET
ma-255	164	40	sequence	sequence	NOUN
ma-255	164	41	{	{	PUNCT
ma-255	164	42	αm	αm	NOUN
ma-255	164	43	}	}	PUNCT
ma-255	164	44	is	be	AUX
ma-255	164	45	complete	complete	ADJ
ma-255	164	46	by	by	ADP
ma-255	164	47	the	the	DET
ma-255	164	48	condition	condition	NOUN
ma-255	164	49	(	(	PUNCT
ma-255	164	50	h2	h2	NOUN
ma-255	164	51	)	)	PUNCT
ma-255	164	52	as	as	ADP
ma-255	164	53	convergent	convergent	NOUN
ma-255	164	54	.	.	PUNCT
ma-255	165	1	it	it	PRON
ma-255	165	2	follows	follow	VERB
ma-255	165	3	by	by	ADP
ma-255	165	4	the	the	DET
ma-255	165	5	estimate(2.4	estimate(2.4	NOUN
ma-255	165	6	)	)	PUNCT
ma-255	165	7	that	that	SCONJ
ma-255	165	8	{	{	PUNCT
ma-255	165	9	xm	xm	NOUN
ma-255	165	10	}	}	PUNCT
ma-255	165	11	is	be	AUX
ma-255	165	12	also	also	ADV
ma-255	165	13	complete	complete	ADJ
ma-255	165	14	.	.	PUNCT
ma-255	166	1	but	but	CCONJ
ma-255	166	2	the	the	DET
ma-255	166	3	space	space	NOUN
ma-255	166	4	x	x	PUNCT
ma-255	166	5	is	be	AUX
ma-255	166	6	banach	banach	ADV
ma-255	166	7	,	,	PUNCT
ma-255	166	8	so	so	CCONJ
ma-255	166	9	there	there	PRON
ma-255	166	10	exists	exist	VERB
ma-255	166	11	s∗	s∗	PROPN
ma-255	166	12	∈	∈	PROPN
ma-255	166	13	s[x0	s[x0	NOUN
ma-255	166	14	,	,	PUNCT
ma-255	166	15	α	α	NOUN
ma-255	166	16	∗	∗	NOUN
ma-255	166	17	]	]	PUNCT
ma-255	166	18	such	such	ADJ
ma-255	166	19	that	that	SCONJ
ma-255	166	20	lim	lim	PROPN
ma-255	166	21	m→+∞xm	m→+∞xm	PROPN
ma-255	167	1	=	=	PUNCT
ma-255	167	2	s∗.	s∗.	ADJ
ma-255	167	3	next	next	ADV
ma-255	167	4	,	,	PUNCT
ma-255	167	5	by	by	ADP
ma-255	167	6	letting	let	VERB
ma-255	167	7	m	m	PRON
ma-255	167	8	→	→	PUNCT
ma-255	167	9	+	+	ADJ
ma-255	167	10	∞	∞	PROPN
ma-255	167	11	in	in	ADP
ma-255	167	12	(	(	PUNCT
ma-255	167	13	2.6	2.6	NUM
ma-255	167	14	)	)	PUNCT
ma-255	167	15	,	,	PUNCT
ma-255	167	16	the	the	DET
ma-255	167	17	invertibility	invertibility	NOUN
ma-255	167	18	of	of	ADP
ma-255	167	19	the	the	DET
ma-255	167	20	operators	operator	NOUN
ma-255	167	21	m(.),γ	m(.),γ	PROPN
ma-255	167	22	and	and	CCONJ
ma-255	167	23	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	167	24	eur	eur	PROPN
ma-255	167	25	.	.	PUNCT
ma-255	168	1	j.	j.	PROPN
ma-255	168	2	math	math	PROPN
ma-255	168	3	.	.	PUNCT
ma-255	169	1	anal	anal	PROPN
ma-255	169	2	.	.	PUNCT
ma-255	170	1	10.28924	10.28924	NUM
ma-255	170	2	/	/	SYM
ma-255	170	3	ada	ada	PROPN
ma-255	170	4	/	/	SYM
ma-255	170	5	ma.5.5	ma.5.5	PROPN
ma-255	170	6	7the	7the	PROPN
ma-255	170	7	continuity	continuity	NOUN
ma-255	170	8	of	of	ADP
ma-255	170	9	the	the	DET
ma-255	170	10	operator	operator	NOUN
ma-255	170	11	f	f	NOUN
ma-255	170	12	,	,	PUNCT
ma-255	170	13	we	we	PRON
ma-255	170	14	deduce	deduce	VERB
ma-255	170	15	that	that	SCONJ
ma-255	170	16	limm→+∞mγ−1f1(s	limm→+∞mγ−1f1(	VERB
ma-255	170	17	∗	∗	NOUN
ma-255	170	18	)	)	PUNCT
ma-255	171	1	=	=	SYM
ma-255	171	2	0	0	X
ma-255	171	3	.	.	PUNCT
ma-255	172	1	by	by	ADP
ma-255	172	2	the	the	DET
ma-255	172	3	invertibilityof	invertibilityof	PROPN
ma-255	172	4	m	m	PROPN
ma-255	172	5	0	0	PUNCT
ma-255	172	6	=	=	SYM
ma-255	172	7	m−1(0	m−1(0	PROPN
ma-255	172	8	)	)	PUNCT
ma-255	172	9	=	=	SYM
ma-255	173	1	m−1	m−1	PROPN
ma-255	173	2	(	(	PUNCT
ma-255	173	3	lim	lim	PROPN
ma-255	173	4	m→+∞	m→+∞	PROPN
ma-255	173	5	mγ−1f1(xm	mγ−1f1(xm	NOUN
ma-255	173	6	)	)	PUNCT
ma-255	173	7	)	)	PUNCT
ma-255	174	1	=	=	SYM
ma-255	174	2	lim	lim	PROPN
ma-255	174	3	m→+∞	m→+∞	PROPN
ma-255	174	4	mm−1γ−1f1(xm	mm−1γ−1f1(xm	PROPN
ma-255	174	5	)	)	PUNCT
ma-255	175	1	=	=	SYM
ma-255	175	2	lim	lim	PROPN
ma-255	175	3	m→+∞	m→+∞	PROPN
ma-255	175	4	γ−1f1(xm	γ−1f1(xm	PROPN
ma-255	175	5	)	)	PUNCT
ma-255	176	1	=	=	SYM
ma-255	176	2	γ−1	γ−1	PROPN
ma-255	176	3	lim	lim	PROPN
ma-255	176	4	m→+∞	m→+∞	PROPN
ma-255	176	5	f1(xm	f1(xm	PROPN
ma-255	176	6	)	)	PUNCT
ma-255	176	7	=	=	SYM
ma-255	176	8	γ−1f1(s	γ−1f1(	VERB
ma-255	176	9	∗	∗	NOUN
ma-255	176	10	)	)	PUNCT
ma-255	176	11	.	.	PUNCT
ma-255	177	1	so	so	ADV
ma-255	177	2	,	,	PUNCT
ma-255	177	3	f1(s∗	f1(s∗	PUNCT
ma-255	177	4	)	)	PUNCT
ma-255	177	5	=	=	SYM
ma-255	177	6	0	0	NUM
ma-255	177	7	,	,	PUNCT
ma-255	177	8	since	since	SCONJ
ma-255	177	9	γ(0	γ(0	PROPN
ma-255	177	10	)	)	PUNCT
ma-255	177	11	=	=	SYM
ma-255	178	1	0	0	X
ma-255	178	2	.	.	PUNCT
ma-255	178	3	let	let	VERB
ma-255	178	4	i	i	PRON
ma-255	178	5	=	=	NOUN
ma-255	178	6	0	0	NUM
ma-255	178	7	,	,	PUNCT
ma-255	178	8	1	1	NUM
ma-255	178	9	,	,	PUNCT
ma-255	178	10	2	2	NUM
ma-255	178	11	,	,	PUNCT
ma-255	178	12	....	....	PUNCT
ma-255	179	1	then	then	ADV
ma-255	179	2	,	,	PUNCT
ma-255	179	3	the	the	DET
ma-255	179	4	triangle	triangle	NOUN
ma-255	179	5	inequality	inequality	NOUN
ma-255	179	6	and	and	CCONJ
ma-255	179	7	(	(	PUNCT
ma-255	179	8	2.4	2.4	NUM
ma-255	179	9	)	)	PUNCT
ma-255	179	10	give	give	VERB
ma-255	179	11	‖xm+i	‖xm+i	PROPN
ma-255	179	12	−	−	PROPN
ma-255	179	13	xm‖	xm‖	PROPN
ma-255	179	14	≤	≤	PROPN
ma-255	179	15	αm+i	αm+i	NUM
ma-255	180	1	−	−	NOUN
ma-255	180	2	αm	αm	INTJ
ma-255	180	3	.	.	PUNCT
ma-255	181	1	(	(	PUNCT
ma-255	181	2	2.7	2.7	NUM
ma-255	181	3	)	)	PUNCT
ma-255	181	4	hence	hence	ADV
ma-255	181	5	,	,	PUNCT
ma-255	181	6	by	by	ADP
ma-255	181	7	letting	let	VERB
ma-255	181	8	i	i	PRON
ma-255	181	9	→	→	PUNCT
ma-255	181	10	+	+	ADJ
ma-255	181	11	∞	∞	PROPN
ma-255	181	12	in	in	ADP
ma-255	181	13	(	(	PUNCT
ma-255	181	14	2.7	2.7	NUM
ma-255	181	15	)	)	PUNCT
ma-255	181	16	we	we	PRON
ma-255	181	17	prove	prove	VERB
ma-255	181	18	(	(	PUNCT
ma-255	181	19	2.3	2.3	NUM
ma-255	181	20	)	)	PUNCT
ma-255	181	21	.	.	PUNCT
ma-255	182	1	finally	finally	ADV
ma-255	182	2	,	,	PUNCT
ma-255	182	3	to	to	PART
ma-255	182	4	show	show	VERB
ma-255	182	5	the	the	DET
ma-255	182	6	uniqueness	uniqueness	NOUN
ma-255	182	7	part	part	NOUN
ma-255	182	8	,	,	PUNCT
ma-255	182	9	let	let	VERB
ma-255	182	10	w	w	PROPN
ma-255	182	11	∈	∈	PROPN
ma-255	182	12	s[x0	s[x0	NOUN
ma-255	182	13	,	,	PUNCT
ma-255	182	14	α	α	NOUN
ma-255	182	15	∗	∗	NOUN
ma-255	182	16	]	]	PUNCT
ma-255	182	17	with	with	ADP
ma-255	182	18	f1(w	f1(w	NOUN
ma-255	182	19	)	)	PUNCT
ma-255	182	20	=	=	SYM
ma-255	182	21	0	0	NUM
ma-255	182	22	and	and	CCONJ
ma-255	182	23	w	w	PROPN
ma-255	182	24	6=	6=	PROPN
ma-255	182	25	s∗.	s∗.	ADJ
ma-255	182	26	by	by	ADP
ma-255	182	27	using	use	VERB
ma-255	182	28	the	the	DET
ma-255	182	29	conditions	condition	NOUN
ma-255	182	30	(	(	PUNCT
ma-255	182	31	h1	h1	PROPN
ma-255	182	32	)	)	PUNCT
ma-255	182	33	,	,	PUNCT
ma-255	182	34	(	(	PUNCT
ma-255	182	35	h2	h2	NOUN
ma-255	182	36	)	)	PUNCT
ma-255	182	37	we	we	PRON
ma-255	182	38	can	can	AUX
ma-255	182	39	write	write	VERB
ma-255	182	40	inturn	inturn	NOUN
ma-255	182	41	that	that	SCONJ
ma-255	182	42	‖w	‖w	VERB
ma-255	182	43	−	−	PROPN
ma-255	182	44	s∗‖	s∗‖	NOUN
ma-255	182	45	=	=	SYM
ma-255	182	46	‖(mγ−1)(γm−1)(w	‖(mγ−1)(γm−1)(w	PROPN
ma-255	182	47	−	−	NOUN
ma-255	182	48	s∗)‖	s∗)‖	PROPN
ma-255	182	49	=	=	PUNCT
ma-255	182	50	‖mγ−1(f1(w)−	‖mγ−1(f1(w)−	PUNCT
ma-255	182	51	f1(s∗)−	f1(s∗)−	ADJ
ma-255	182	52	γm−1(w	γm−1(w	ADJ
ma-255	182	53	−	−	ADP
ma-255	182	54	s∗))‖	s∗))‖	PROPN
ma-255	182	55	≤	≤	NUM
ma-255	182	56	φ(‖s∗	φ(‖s∗	NOUN
ma-255	182	57	−	−	PROPN
ma-255	182	58	x0‖	x0‖	PROPN
ma-255	182	59	,	,	PUNCT
ma-255	182	60	‖w	‖w	VERB
ma-255	182	61	−	−	PROPN
ma-255	182	62	x0‖	x0‖	PROPN
ma-255	182	63	,	,	PUNCT
ma-255	182	64	‖w	‖w	VERB
ma-255	182	65	−	−	PROPN
ma-255	182	66	s∗‖)‖w	s∗‖)‖w	PROPN
ma-255	182	67	−	−	NOUN
ma-255	182	68	s∗‖	s∗‖	VERB
ma-255	182	69	≤	≤	NUM
ma-255	182	70	φ(α∗	φ(α∗	NOUN
ma-255	182	71	,	,	PUNCT
ma-255	182	72	α∗	α∗	NOUN
ma-255	182	73	,	,	PUNCT
ma-255	182	74	‖w	‖w	VERB
ma-255	182	75	−	−	PROPN
ma-255	182	76	s∗‖)‖w	s∗‖)‖w	PROPN
ma-255	182	77	−	−	PROPN
ma-255	182	78	s∗‖	s∗‖	NOUN
ma-255	182	79	<	<	X
ma-255	182	80	‖w	‖w	NOUN
ma-255	182	81	−	−	NOUN
ma-255	182	82	s∗‖	s∗‖	NOUN
ma-255	182	83	,	,	PUNCT
ma-255	182	84	which	which	PRON
ma-255	182	85	gives	give	VERB
ma-255	182	86	a	a	DET
ma-255	182	87	contradiction	contradiction	NOUN
ma-255	182	88	.	.	PUNCT
ma-255	183	1	therefore	therefore	ADV
ma-255	183	2	,	,	PUNCT
ma-255	183	3	we	we	PRON
ma-255	183	4	conclude	conclude	VERB
ma-255	183	5	that	that	PRON
ma-255	183	6	w	w	PROPN
ma-255	183	7	=	=	PUNCT
ma-255	183	8	s∗.	s∗.	ADJ
ma-255	183	9	�	�	PROPN
ma-255	183	10	remark	remark	VERB
ma-255	183	11	2.1	2.1	NUM
ma-255	183	12	.	.	PUNCT
ma-255	184	1	the	the	DET
ma-255	184	2	condition	condition	NOUN
ma-255	184	3	mysoskii	mysoskii	ADJ
ma-255	184	4	-	-	PUNCT
ma-255	184	5	type	type	NOUN
ma-255	184	6	[	[	X
ma-255	184	7	22	22	NUM
ma-255	184	8	]	]	PUNCT
ma-255	184	9	condition	condition	NOUN
ma-255	184	10	in	in	ADP
ma-255	184	11	(	(	PUNCT
ma-255	184	12	h1	h1	PROPN
ma-255	184	13	)	)	PUNCT
ma-255	184	14	can	can	AUX
ma-255	184	15	be	be	AUX
ma-255	184	16	replaced	replace	VERB
ma-255	184	17	as	as	SCONJ
ma-255	184	18	follows	follow	VERB
ma-255	184	19	:	:	PUNCT
ma-255	184	20	(	(	PUNCT
ma-255	184	21	h1	h1	PROPN
ma-255	184	22	)	)	PUNCT
ma-255	184	23	′	′	PUNCT
ma-255	185	1	with	with	ADP
ma-255	185	2	operator	operator	NOUN
ma-255	185	3	γ	γ	NOUN
ma-255	185	4	as	as	ADP
ma-255	185	5	in	in	ADP
ma-255	185	6	condition	condition	NOUN
ma-255	185	7	(	(	PUNCT
ma-255	185	8	h1	h1	PROPN
ma-255	185	9	)	)	PUNCT
ma-255	185	10	,	,	PUNCT
ma-255	185	11	suppose	suppose	VERB
ma-255	185	12	that	that	SCONJ
ma-255	185	13	there	there	PRON
ma-255	185	14	exists	exist	VERB
ma-255	185	15	a	a	DET
ma-255	185	16	∈	∈	PROPN
ma-255	185	17	(	(	PUNCT
ma-255	185	18	0	0	NUM
ma-255	185	19	,	,	PUNCT
ma-255	185	20	12	12	NUM
ma-255	185	21	)	)	PUNCT
ma-255	185	22	such	such	ADJ
ma-255	185	23	that	that	SCONJ
ma-255	185	24	‖a‖	‖a‖	PROPN
ma-255	185	25	<	<	X
ma-255	185	26	a	a	X
ma-255	185	27	and	and	CCONJ
ma-255	185	28	for	for	ADP
ma-255	185	29	each	each	DET
ma-255	185	30	x	x	NOUN
ma-255	185	31	,	,	PUNCT
ma-255	185	32	y	y	PROPN
ma-255	185	33	∈	∈	PROPN
ma-255	185	34	ω	ω	PROPN
ma-255	186	1	‖γ−1(f1(y)−	‖γ−1(f1(y)−	X
ma-255	186	2	f1(x)−	f1(x)−	PROPN
ma-255	187	1	γm−1(y	γm−1(y	PROPN
ma-255	187	2	−	−	PROPN
ma-255	187	3	x))‖	x))‖	NOUN
ma-255	187	4	≤	≤	ADJ
ma-255	187	5	φ1(‖x	φ1(‖x	NOUN
ma-255	187	6	−	−	PROPN
ma-255	187	7	x0‖	x0‖	PROPN
ma-255	187	8	,	,	PUNCT
ma-255	187	9	‖y	‖y	PUNCT
ma-255	188	1	−	−	PROPN
ma-255	188	2	x0‖	x0‖	PROPN
ma-255	188	3	,	,	PUNCT
ma-255	188	4	‖y	‖y	PUNCT
ma-255	189	1	−	−	PROPN
ma-255	189	2	x‖)‖y	x‖)‖y	PROPN
ma-255	189	3	−	−	PROPN
ma-255	189	4	x‖	x‖	PROPN
ma-255	189	5	,	,	PUNCT
ma-255	189	6	where	where	SCONJ
ma-255	189	7	the	the	DET
ma-255	189	8	function	function	NOUN
ma-255	189	9	φ1	φ1	NOUN
ma-255	189	10	is	be	AUX
ma-255	189	11	as	as	ADP
ma-255	189	12	the	the	DET
ma-255	189	13	function	function	NOUN
ma-255	189	14	φ	φ	NOUN
ma-255	189	15	.	.	PUNCT
ma-255	190	1	then	then	ADV
ma-255	190	2	,	,	PUNCT
ma-255	190	3	the	the	DET
ma-255	190	4	condition	condition	NOUN
ma-255	190	5	(	(	PUNCT
ma-255	190	6	h1	h1	PROPN
ma-255	190	7	)	)	PUNCT
ma-255	190	8	′	′	NUM
ma-255	190	9	implies	imply	VERB
ma-255	190	10	(	(	PUNCT
ma-255	190	11	h1	h1	PROPN
ma-255	190	12	)	)	PUNCT
ma-255	190	13	if	if	SCONJ
ma-255	190	14	we	we	PRON
ma-255	190	15	take	take	VERB
ma-255	190	16	φ	φ	NOUN
ma-255	190	17	=	=	SYM
ma-255	190	18	a0φ1	a0φ1	PROPN
ma-255	190	19	,	,	PUNCT
ma-255	190	20	where	where	SCONJ
ma-255	190	21	a0	a0	PROPN
ma-255	190	22	=	=	PROPN
ma-255	190	23	1	1	NUM
ma-255	190	24	1−a	1−a	NUM
ma-255	190	25	.	.	PUNCT
ma-255	191	1	this	this	PRON
ma-255	191	2	is	be	AUX
ma-255	191	3	the	the	DET
ma-255	191	4	case	case	NOUN
ma-255	191	5	,	,	PUNCT
ma-255	191	6	since	since	SCONJ
ma-255	191	7	by	by	ADP
ma-255	191	8	the	the	DET
ma-255	191	9	definition	definition	NOUN
ma-255	191	10	of	of	ADP
ma-255	191	11	the	the	DET
ma-255	191	12	operator	operator	NOUN
ma-255	191	13	m	m	VERB
ma-255	191	14	,	,	PUNCT
ma-255	191	15	we	we	PRON
ma-255	191	16	have	have	VERB
ma-255	191	17	the	the	DET
ma-255	191	18	estimate	estimate	NOUN
ma-255	191	19	‖m‖	‖m‖	PROPN
ma-255	191	20	=	=	SYM
ma-255	191	21	‖i	‖i	NOUN
ma-255	191	22	+	+	CCONJ
ma-255	191	23	a+	a+	PUNCT
ma-255	191	24	...	...	PUNCT
ma-255	191	25	+	+	NUM
ma-255	191	26	ap‖	ap‖	PROPN
ma-255	191	27	≤	≤	NOUN
ma-255	191	28	1	1	NUM
ma-255	191	29	+	+	CCONJ
ma-255	191	30	a	a	DET
ma-255	191	31	+	+	X
ma-255	191	32	...	...	PUNCT
ma-255	191	33	+	+	CCONJ
ma-255	191	34	ap	ap	PROPN
ma-255	191	35	=	=	SYM
ma-255	191	36	1−	1−	NUM
ma-255	191	37	ap+1	ap+1	NOUN
ma-255	191	38	1−	1−	NUM
ma-255	191	39	a	a	DET
ma-255	191	40	=	=	SYM
ma-255	191	41	a0	a0	PROPN
ma-255	191	42	<	<	X
ma-255	191	43	1	1	NUM
ma-255	191	44	1−	1−	NUM
ma-255	191	45	a	a	PRON
ma-255	191	46	.	.	PUNCT
ma-255	192	1	(	(	PUNCT
ma-255	192	2	2.8	2.8	NUM
ma-255	192	3	)	)	PUNCT
ma-255	192	4	in	in	ADP
ma-255	192	5	this	this	DET
ma-255	192	6	case	case	NOUN
ma-255	192	7	the	the	DET
ma-255	192	8	invertibility	invertibility	NOUN
ma-255	192	9	of	of	ADP
ma-255	192	10	the	the	DET
ma-255	192	11	operator	operator	NOUN
ma-255	192	12	m	m	VERB
ma-255	192	13	is	be	AUX
ma-255	192	14	implied	imply	VERB
ma-255	192	15	by	by	ADP
ma-255	192	16	the	the	DET
ma-255	192	17	banach	banach	NOUN
ma-255	192	18	lemma	lemma	PROPN
ma-255	192	19	2.1	2.1	NUM
ma-255	192	20	,	,	PUNCT
ma-255	192	21	since	since	SCONJ
ma-255	192	22	‖i	‖i	NOUN
ma-255	192	23	−m‖	−m‖	PROPN
ma-255	192	24	≤	≤	NUM
ma-255	192	25	‖a‖+	‖a‖+	NUM
ma-255	192	26	...	...	PUNCT
ma-255	193	1	+	+	NUM
ma-255	193	2	‖a‖p	‖a‖p	NOUN
ma-255	193	3	≤	≤	PUNCT
ma-255	193	4	a	a	DET
ma-255	193	5	+	+	X
ma-255	193	6	...	...	PUNCT
ma-255	193	7	+	+	CCONJ
ma-255	193	8	ap	ap	PROPN
ma-255	194	1	=	=	PUNCT
ma-255	194	2	a	a	DET
ma-255	194	3	1−	1−	NUM
ma-255	194	4	ap	ap	NOUN
ma-255	194	5	1−	1−	NUM
ma-255	194	6	a	a	DET
ma-255	194	7	<	<	X
ma-255	194	8	a	a	DET
ma-255	194	9	1−	1−	NUM
ma-255	194	10	a	a	DET
ma-255	194	11	<	<	X
ma-255	194	12	1	1	NUM
ma-255	194	13	.	.	PUNCT
ma-255	195	1	so	so	ADV
ma-255	195	2	,	,	PUNCT
ma-255	195	3	m	m	VERB
ma-255	195	4	is	be	AUX
ma-255	195	5	invertible	invertible	ADJ
ma-255	195	6	and	and	CCONJ
ma-255	195	7	‖m−1‖	‖m−1‖	PROPN
ma-255	195	8	≤	≤	NUM
ma-255	195	9	b	b	X
ma-255	195	10	=	=	SYM
ma-255	195	11	1−	1−	NUM
ma-255	195	12	a	a	DET
ma-255	195	13	1−	1−	NUM
ma-255	195	14	2a	2a	NUM
ma-255	195	15	.	.	PUNCT
ma-255	196	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	196	2	eur	eur	PROPN
ma-255	196	3	.	.	PUNCT
ma-255	197	1	j.	j.	PROPN
ma-255	197	2	math	math	PROPN
ma-255	197	3	.	.	PUNCT
ma-255	198	1	anal	anal	PROPN
ma-255	198	2	.	.	PUNCT
ma-255	199	1	10.28924	10.28924	NUM
ma-255	199	2	/	/	SYM
ma-255	199	3	ada	ada	PROPN
ma-255	199	4	/	/	SYM
ma-255	199	5	ma.5.5	ma.5.5	PROPN
ma-255	199	6	8	8	NUM
ma-255	199	7	next	next	ADV
ma-255	199	8	,	,	PUNCT
ma-255	199	9	we	we	PRON
ma-255	199	10	develop	develop	VERB
ma-255	199	11	the	the	DET
ma-255	199	12	local	local	ADJ
ma-255	199	13	convergence	convergence	NOUN
ma-255	199	14	.	.	PUNCT
ma-255	200	1	the	the	DET
ma-255	200	2	role	role	NOUN
ma-255	200	3	of	of	ADP
ma-255	200	4	the	the	DET
ma-255	200	5	initial	initial	ADJ
ma-255	200	6	point	point	NOUN
ma-255	200	7	x0	x0	PROPN
ma-255	200	8	is	be	AUX
ma-255	200	9	exchanged	exchange	VERB
ma-255	200	10	by	by	ADP
ma-255	200	11	s∗	s∗	PROPN
ma-255	200	12	and	and	CCONJ
ma-255	200	13	the	the	DET
ma-255	200	14	function	function	NOUN
ma-255	200	15	φ	φ	X
ma-255	200	16	by	by	ADP
ma-255	200	17	ψ	ψ	PROPN
ma-255	200	18	.	.	PUNCT
ma-255	201	1	suppose	suppose	VERB
ma-255	201	2	:	:	PUNCT
ma-255	201	3	(	(	PUNCT
ma-255	201	4	h4	h4	NOUN
ma-255	201	5	)	)	PUNCT
ma-255	201	6	there	there	PRON
ma-255	201	7	exists	exist	VERB
ma-255	201	8	a	a	DET
ma-255	201	9	function	function	NOUN
ma-255	201	10	ψ	ψ	NOUN
ma-255	201	11	:	:	PUNCT
ma-255	201	12	e	e	X
ma-255	201	13	−→	−→	NOUN
ma-255	201	14	[	[	X
ma-255	201	15	0,+∞	0,+∞	NUM
ma-255	201	16	)	)	PUNCT
ma-255	201	17	continuous	continuous	ADJ
ma-255	201	18	and	and	CCONJ
ma-255	201	19	nondecreasing	nondecrease	VERB
ma-255	201	20	such	such	ADJ
ma-255	201	21	that	that	SCONJ
ma-255	201	22	the	the	DET
ma-255	201	23	equation	equation	NOUN
ma-255	201	24	ψ(t)−	ψ(t)−	PROPN
ma-255	201	25	1	1	NUM
ma-255	201	26	=	=	SYM
ma-255	201	27	0	0	PROPN
ma-255	201	28	has	have	AUX
ma-255	201	29	a	a	DET
ma-255	201	30	smallest	small	ADJ
ma-255	201	31	positive	positive	ADJ
ma-255	201	32	solution	solution	NOUN
ma-255	201	33	denoted	denote	VERB
ma-255	201	34	by	by	ADP
ma-255	201	35	r	r	NOUN
ma-255	201	36	.	.	PUNCT
ma-255	202	1	(	(	PUNCT
ma-255	202	2	h5	h5	PROPN
ma-255	202	3	)	)	PUNCT
ma-255	202	4	there	there	PRON
ma-255	202	5	exists	exist	VERB
ma-255	202	6	a	a	DET
ma-255	202	7	solution	solution	NOUN
ma-255	202	8	s∗	s∗	PROPN
ma-255	202	9	∈	∈	PROPN
ma-255	202	10	ω	ω	PROPN
ma-255	202	11	,	,	PUNCT
ma-255	202	12	and	and	CCONJ
ma-255	202	13	invertible	invertible	ADJ
ma-255	202	14	linear	linear	PROPN
ma-255	202	15	operators	operator	NOUN
ma-255	202	16	m	m	PROPN
ma-255	202	17	,	,	PUNCT
ma-255	202	18	γ	γ	PROPN
ma-255	202	19	such	such	ADJ
ma-255	202	20	that	that	PRON
ma-255	202	21	for	for	ADP
ma-255	202	22	each	each	DET
ma-255	202	23	x	x	SYM
ma-255	202	24	∈	∈	PROPN
ma-255	202	25	ω	ω	NUM
ma-255	202	26	‖mγ−1(f1(x)−	‖mγ−1(f1(x)−	PROPN
ma-255	202	27	f1(s∗)−	f1(s∗)−	ADJ
ma-255	202	28	γm−1(x	γm−1(x	NOUN
ma-255	202	29	−	−	ADP
ma-255	202	30	s∗))‖	s∗))‖	PROPN
ma-255	202	31	≤	≤	PUNCT
ma-255	202	32	ψ(‖x	ψ(‖x	PUNCT
ma-255	203	1	−	−	PROPN
ma-255	203	2	s∗‖)‖x	s∗‖)‖x	NOUN
ma-255	203	3	−	−	NOUN
ma-255	203	4	s∗‖	s∗‖	NOUN
ma-255	203	5	and	and	CCONJ
ma-255	203	6	(	(	PUNCT
ma-255	203	7	h6	h6	PROPN
ma-255	203	8	)	)	PUNCT
ma-255	204	1	s[s∗	s[s∗	PROPN
ma-255	204	2	,	,	PUNCT
ma-255	204	3	r	r	NOUN
ma-255	204	4	]	]	PUNCT
ma-255	204	5	⊂	⊂	PROPN
ma-255	204	6	ω	ω	PROPN
ma-255	204	7	.	.	PUNCT
ma-255	205	1	next	next	ADV
ma-255	205	2	,	,	PUNCT
ma-255	205	3	the	the	DET
ma-255	205	4	constant	constant	ADJ
ma-255	205	5	r	r	NOUN
ma-255	205	6	is	be	AUX
ma-255	205	7	shown	show	VERB
ma-255	205	8	to	to	PART
ma-255	205	9	be	be	AUX
ma-255	205	10	a	a	DET
ma-255	205	11	radius	radius	NOUN
ma-255	205	12	of	of	ADP
ma-255	205	13	convergence	convergence	NOUN
ma-255	205	14	for	for	ADP
ma-255	205	15	the	the	DET
ma-255	205	16	method	method	NOUN
ma-255	205	17	(	(	PUNCT
ma-255	205	18	1.5	1.5	NUM
ma-255	205	19	)	)	PUNCT
ma-255	205	20	.	.	PUNCT
ma-255	206	1	theorem	theorem	VERB
ma-255	206	2	2.2	2.2	NUM
ma-255	206	3	.	.	PUNCT
ma-255	207	1	suppose	suppose	VERB
ma-255	207	2	that	that	SCONJ
ma-255	207	3	the	the	DET
ma-255	207	4	conditions	condition	NOUN
ma-255	207	5	(	(	PUNCT
ma-255	207	6	h4	h4	PROPN
ma-255	207	7	)	)	PUNCT
ma-255	207	8	−	−	PROPN
ma-255	207	9	(	(	PUNCT
ma-255	207	10	h6	h6	PROPN
ma-255	207	11	)	)	PUNCT
ma-255	207	12	hold	hold	VERB
ma-255	207	13	.	.	PUNCT
ma-255	208	1	then	then	ADV
ma-255	208	2	,	,	PUNCT
ma-255	208	3	the	the	DET
ma-255	208	4	sequence	sequence	NOUN
ma-255	208	5	{	{	PUNCT
ma-255	208	6	xn	xn	NOUN
ma-255	208	7	}	}	PUNCT
ma-255	208	8	for	for	ADP
ma-255	208	9	x0	x0	PROPN
ma-255	208	10	∈	∈	PROPN
ma-255	208	11	s(s∗	s(s∗	X
ma-255	208	12	,	,	PUNCT
ma-255	208	13	r)−	r)−	PROPN
ma-255	208	14	{	{	PUNCT
ma-255	208	15	s∗	s∗	PROPN
ma-255	208	16	}	}	PUNCT
ma-255	208	17	exists	exist	VERB
ma-255	208	18	in	in	ADP
ma-255	208	19	s(s∗	s(s∗	ADJ
ma-255	208	20	,	,	PUNCT
ma-255	208	21	r	r	NOUN
ma-255	208	22	)	)	PUNCT
ma-255	208	23	,	,	PUNCT
ma-255	208	24	stays	stay	VERB
ma-255	208	25	in	in	ADP
ma-255	208	26	s(s∗	s(s∗	ADJ
ma-255	208	27	,	,	PUNCT
ma-255	208	28	r	r	NOUN
ma-255	208	29	)	)	PUNCT
ma-255	208	30	and	and	CCONJ
ma-255	208	31	converges	converge	VERB
ma-255	208	32	to	to	PART
ma-255	208	33	s∗	s∗	VERB
ma-255	208	34	so	so	SCONJ
ma-255	208	35	that	that	SCONJ
ma-255	208	36	‖xn+1	‖xn+1	X
ma-255	208	37	−	−	NOUN
ma-255	208	38	s∗‖	s∗‖	NOUN
ma-255	208	39	≤	≤	X
ma-255	208	40	ψ(‖xn	ψ(‖xn	PUNCT
ma-255	208	41	−	−	PROPN
ma-255	208	42	s∗‖)‖xn	s∗‖)‖xn	PROPN
ma-255	209	1	−	−	NOUN
ma-255	209	2	s∗‖	s∗‖	VERB
ma-255	209	3	≤	≤	NOUN
ma-255	209	4	‖xn	‖xn	PUNCT
ma-255	209	5	−	−	NOUN
ma-255	210	1	s∗‖	s∗‖	NOUN
ma-255	210	2	<	<	X
ma-255	210	3	r.	r.	X
ma-255	210	4	(	(	PUNCT
ma-255	210	5	2.9	2.9	NUM
ma-255	210	6	)	)	PUNCT
ma-255	210	7	moreover	moreover	ADV
ma-255	210	8	,	,	PUNCT
ma-255	210	9	x∗	x∗	PROPN
ma-255	210	10	is	be	AUX
ma-255	210	11	the	the	DET
ma-255	210	12	only	only	ADJ
ma-255	210	13	solution	solution	NOUN
ma-255	210	14	of	of	ADP
ma-255	210	15	the	the	DET
ma-255	210	16	equation	equation	NOUN
ma-255	210	17	f	f	X
ma-255	210	18	(	(	PUNCT
ma-255	210	19	x	x	X
ma-255	210	20	)	)	PUNCT
ma-255	210	21	=	=	NOUN
ma-255	210	22	0	0	NUM
ma-255	210	23	in	in	ADP
ma-255	210	24	the	the	DET
ma-255	210	25	ball	ball	NOUN
ma-255	210	26	u(s∗	u(s∗	SYM
ma-255	210	27	,	,	PUNCT
ma-255	210	28	r	r	NOUN
ma-255	210	29	)	)	PUNCT
ma-255	210	30	.	.	PUNCT
ma-255	211	1	proof	proof	NOUN
ma-255	211	2	.	.	PUNCT
ma-255	212	1	the	the	DET
ma-255	212	2	iterates	iterate	NOUN
ma-255	212	3	x1	x1	PROPN
ma-255	212	4	,	,	PUNCT
ma-255	212	5	x2	x2	PROPN
ma-255	212	6	,	,	PUNCT
ma-255	212	7	...	...	PUNCT
ma-255	212	8	,	,	PUNCT
ma-255	212	9	xm+1	xm+1	PROPN
ma-255	212	10	are	be	AUX
ma-255	212	11	well	well	ADV
ma-255	212	12	defined	define	VERB
ma-255	212	13	by	by	ADP
ma-255	212	14	the	the	DET
ma-255	212	15	method	method	NOUN
ma-255	212	16	(	(	PUNCT
ma-255	212	17	1.5	1.5	NUM
ma-255	212	18	)	)	PUNCT
ma-255	212	19	,	,	PUNCT
ma-255	212	20	and	and	CCONJ
ma-255	212	21	we	we	PRON
ma-255	212	22	can	can	AUX
ma-255	212	23	write	write	VERB
ma-255	212	24	in	in	ADP
ma-255	212	25	turnthat	turnthat	NOUN
ma-255	213	1	xm+1	xm+1	PROPN
ma-255	213	2	−	−	PROPN
ma-255	213	3	s∗	s∗	PROPN
ma-255	213	4	=	=	SYM
ma-255	213	5	xm	xm	PROPN
ma-255	214	1	−	−	PROPN
ma-255	214	2	s∗	s∗	PROPN
ma-255	214	3	−mγ−1f1(xm	−mγ−1f1(xm	PROPN
ma-255	214	4	)	)	PUNCT
ma-255	215	1	=	=	X
ma-255	215	2	mγ−1(f1(xm)−	mγ−1(f1(xm)−	NOUN
ma-255	215	3	f1(s∗)−	f1(s∗)−	NOUN
ma-255	216	1	γm−1(xm	γm−1(xm	CCONJ
ma-255	216	2	−	−	PROPN
ma-255	216	3	s∗	s∗	PROPN
ma-255	216	4	)	)	PUNCT
ma-255	216	5	)	)	PUNCT
ma-255	216	6	.	.	PUNCT
ma-255	217	1	(	(	PUNCT
ma-255	217	2	2.10	2.10	NUM
ma-255	217	3	)	)	PUNCT
ma-255	217	4	it	it	PRON
ma-255	217	5	follows	follow	VERB
ma-255	217	6	by	by	ADP
ma-255	217	7	the	the	DET
ma-255	217	8	conditions	condition	NOUN
ma-255	217	9	(	(	PUNCT
ma-255	217	10	h4	h4	PROPN
ma-255	217	11	)	)	PUNCT
ma-255	217	12	,	,	PUNCT
ma-255	217	13	(	(	PUNCT
ma-255	217	14	h5	h5	PROPN
ma-255	217	15	)	)	PUNCT
ma-255	217	16	and	and	CCONJ
ma-255	217	17	(	(	PUNCT
ma-255	217	18	2.10	2.10	NUM
ma-255	217	19	)	)	PUNCT
ma-255	217	20	that	that	PRON
ma-255	217	21	‖xm+1	‖xm+1	VERB
ma-255	218	1	−	−	PROPN
ma-255	218	2	s∗‖	s∗‖	VERB
ma-255	218	3	≤	≤	NOUN
ma-255	218	4	ψ(‖xm	ψ(‖xm	PUNCT
ma-255	218	5	−	−	NOUN
ma-255	218	6	s∗‖)‖xm	s∗‖)‖xm	ADP
ma-255	218	7	−	−	PROPN
ma-255	218	8	s∗‖	s∗‖	VERB
ma-255	218	9	≤	≤	NOUN
ma-255	219	1	ξ‖xm	ξ‖xm	NOUN
ma-255	219	2	−	−	NOUN
ma-255	219	3	s∗‖	s∗‖	VERB
ma-255	219	4	≤	≤	NOUN
ma-255	219	5	ξm+1‖x0	ξm+1‖x0	NOUN
ma-255	219	6	−	−	NOUN
ma-255	219	7	s∗‖	s∗‖	AUX
ma-255	219	8	<	<	X
ma-255	219	9	r	r	NOUN
ma-255	219	10	,	,	PUNCT
ma-255	219	11	(	(	PUNCT
ma-255	219	12	2.11	2.11	NUM
ma-255	219	13	)	)	PUNCT
ma-255	219	14	where	where	SCONJ
ma-255	219	15	ξ	ξ	X
ma-255	219	16	=	=	SYM
ma-255	219	17	ψ(‖x0	ψ(‖x0	NOUN
ma-255	219	18	−	−	NOUN
ma-255	219	19	s∗‖	s∗‖	NOUN
ma-255	219	20	)	)	PUNCT
ma-255	219	21	∈	∈	NOUN
ma-255	220	1	[	[	X
ma-255	220	2	0	0	NUM
ma-255	220	3	,	,	PUNCT
ma-255	220	4	1	1	X
ma-255	220	5	)	)	PUNCT
ma-255	220	6	showing	show	VERB
ma-255	220	7	the	the	DET
ma-255	220	8	assertion	assertion	NOUN
ma-255	220	9	(	(	PUNCT
ma-255	220	10	2.9	2.9	NUM
ma-255	220	11	)	)	PUNCT
ma-255	220	12	for	for	ADP
ma-255	220	13	each	each	DET
ma-255	220	14	m	m	NOUN
ma-255	220	15	=	=	NOUN
ma-255	220	16	1	1	NUM
ma-255	220	17	,	,	PUNCT
ma-255	220	18	2	2	NUM
ma-255	220	19	,	,	PUNCT
ma-255	220	20	...	...	PUNCT
ma-255	220	21	,	,	PUNCT
ma-255	220	22	since	since	SCONJ
ma-255	220	23	x0	x0	PROPN
ma-255	220	24	∈	∈	PROPN
ma-255	220	25	s(s∗	s(s∗	X
ma-255	220	26	,	,	PUNCT
ma-255	220	27	r	r	NOUN
ma-255	220	28	)	)	PUNCT
ma-255	220	29	−	−	PROPN
ma-255	220	30	{	{	PUNCT
ma-255	220	31	s∗	s∗	PROPN
ma-255	220	32	}	}	PUNCT
ma-255	220	33	.	.	PUNCT
ma-255	221	1	by	by	ADP
ma-255	221	2	letting	let	VERB
ma-255	221	3	m	m	PRON
ma-255	221	4	→	→	PUNCT
ma-255	221	5	+	+	ADJ
ma-255	221	6	∞	∞	PROPN
ma-255	221	7	in	in	ADP
ma-255	221	8	(	(	PUNCT
ma-255	221	9	2.11	2.11	NUM
ma-255	221	10	)	)	PUNCT
ma-255	221	11	,	,	PUNCT
ma-255	221	12	we	we	PRON
ma-255	221	13	deduce	deduce	VERB
ma-255	221	14	that	that	SCONJ
ma-255	221	15	lim	lim	PROPN
ma-255	221	16	m→+∞xm	m→+∞xm	PROPN
ma-255	221	17	.	.	PUNCT
ma-255	222	1	in	in	ADP
ma-255	222	2	order	order	NOUN
ma-255	222	3	to	to	PART
ma-255	222	4	show	show	VERB
ma-255	222	5	theuniqueness	theuniqueness	ADJ
ma-255	222	6	part	part	NOUN
ma-255	222	7	,	,	PUNCT
ma-255	222	8	suppose	suppose	VERB
ma-255	222	9	there	there	PRON
ma-255	222	10	exists	exist	VERB
ma-255	222	11	a	a	DET
ma-255	222	12	solution	solution	NOUN
ma-255	222	13	w1	w1	NOUN
ma-255	222	14	∈	∈	PROPN
ma-255	222	15	s(s∗	s(s∗	X
ma-255	222	16	,	,	PUNCT
ma-255	222	17	r	r	NOUN
ma-255	222	18	)	)	PUNCT
ma-255	222	19	such	such	ADJ
ma-255	222	20	that	that	DET
ma-255	222	21	w1	w1	NOUN
ma-255	222	22	6=	6=	PUNCT
ma-255	223	1	s∗.	s∗.	ADJ
ma-255	223	2	then	then	ADV
ma-255	223	3	,	,	PUNCT
ma-255	223	4	as	as	ADP
ma-255	223	5	in	in	ADP
ma-255	223	6	thesemi	thesemi	NOUN
ma-255	223	7	-	-	ADJ
ma-255	223	8	local	local	ADJ
ma-255	223	9	case	case	NOUN
ma-255	223	10	,	,	PUNCT
ma-255	223	11	we	we	PRON
ma-255	223	12	can	can	AUX
ma-255	223	13	write	write	VERB
ma-255	223	14	in	in	ADP
ma-255	223	15	turn	turn	NOUN
ma-255	223	16	‖w1	‖w1	CCONJ
ma-255	223	17	−	−	PROPN
ma-255	223	18	s∗‖	s∗‖	NOUN
ma-255	223	19	=	=	SYM
ma-255	223	20	‖(mγ−1)(γm−1(w1	‖(mγ−1)(γm−1(w1	PUNCT
ma-255	223	21	−	−	NOUN
ma-255	223	22	s∗))‖	s∗))‖	NOUN
ma-255	223	23	=	=	SYM
ma-255	223	24	‖mγ−1(f1(w1)−	‖mγ−1(f1(w1)−	NUM
ma-255	223	25	f1(s∗)−	f1(s∗)−	NOUN
ma-255	224	1	γm−1(w1	γm−1(w1	X
ma-255	224	2	−	−	ADP
ma-255	224	3	s∗))‖	s∗))‖	PROPN
ma-255	224	4	≤	≤	NUM
ma-255	224	5	ψ(‖w1	ψ(‖w1	NOUN
ma-255	224	6	−	−	PROPN
ma-255	225	1	s∗‖)‖w1	s∗‖)‖w1	NOUN
ma-255	225	2	−	−	NOUN
ma-255	225	3	s∗‖	s∗‖	NOUN
ma-255	225	4	<	<	X
ma-255	225	5	‖w1	‖w1	PRON
ma-255	225	6	−	−	PROPN
ma-255	225	7	s∗‖	s∗‖	ADJ
ma-255	225	8	(	(	PUNCT
ma-255	225	9	2.12	2.12	NUM
ma-255	225	10	)	)	PUNCT
ma-255	225	11	by	by	ADP
ma-255	225	12	the	the	DET
ma-255	225	13	choice	choice	NOUN
ma-255	225	14	of	of	ADP
ma-255	225	15	r	r	NOUN
ma-255	225	16	.	.	PUNCT
ma-255	226	1	hence	hence	ADV
ma-255	226	2	,	,	PUNCT
ma-255	226	3	we	we	PRON
ma-255	226	4	conclude	conclude	VERB
ma-255	226	5	that	that	DET
ma-255	226	6	w1	w1	NOUN
ma-255	226	7	=	=	PUNCT
ma-255	226	8	s∗	s∗	PROPN
ma-255	226	9	,	,	PUNCT
ma-255	226	10	since	since	SCONJ
ma-255	226	11	(	(	PUNCT
ma-255	226	12	2.12	2.12	NUM
ma-255	226	13	)	)	PUNCT
ma-255	226	14	contradicts	contradict	VERB
ma-255	226	15	the	the	DET
ma-255	226	16	hypothesis	hypothesis	NOUN
ma-255	226	17	w1	w1	NOUN
ma-255	226	18	6=	6=	PROPN
ma-255	226	19	s∗.	s∗.	PROPN
ma-255	226	20	�	�	PROPN
ma-255	226	21	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	226	22	eur	eur	PROPN
ma-255	226	23	.	.	PUNCT
ma-255	227	1	j.	j.	PROPN
ma-255	227	2	math	math	PROPN
ma-255	227	3	.	.	PUNCT
ma-255	228	1	anal	anal	PROPN
ma-255	228	2	.	.	PUNCT
ma-255	229	1	10.28924	10.28924	NUM
ma-255	229	2	/	/	SYM
ma-255	229	3	ada	ada	PROPN
ma-255	229	4	/	/	SYM
ma-255	229	5	ma.5.5	ma.5.5	PROPN
ma-255	229	6	9	9	NUM
ma-255	229	7	remark	remark	NOUN
ma-255	229	8	2.2	2.2	NUM
ma-255	229	9	.	.	PUNCT
ma-255	230	1	comments	comment	NOUN
ma-255	230	2	similar	similar	ADJ
ma-255	230	3	to	to	ADP
ma-255	230	4	the	the	DET
ma-255	230	5	ones	one	NOUN
ma-255	230	6	in	in	ADP
ma-255	230	7	remark	remark	NOUN
ma-255	230	8	2.4	2.4	NUM
ma-255	230	9	can	can	AUX
ma-255	230	10	be	be	AUX
ma-255	230	11	made	make	VERB
ma-255	230	12	provided	provide	VERB
ma-255	230	13	that	that	SCONJ
ma-255	230	14	x0	x0	PROPN
ma-255	230	15	,	,	PUNCT
ma-255	230	16	ψ	ψ	X
ma-255	230	17	are	be	AUX
ma-255	230	18	exchanged	exchange	VERB
ma-255	230	19	by	by	ADP
ma-255	230	20	s∗	s∗	PROPN
ma-255	230	21	,	,	PUNCT
ma-255	230	22	ψ	ψ	NOUN
ma-255	230	23	,	,	PUNCT
ma-255	230	24	respectively	respectively	ADV
ma-255	230	25	.	.	PUNCT
ma-255	231	1	it	it	PRON
ma-255	231	2	is	be	AUX
ma-255	231	3	also	also	ADV
ma-255	231	4	worth	worth	ADJ
ma-255	231	5	noting	note	VERB
ma-255	231	6	that	that	SCONJ
ma-255	231	7	the	the	DET
ma-255	231	8	condition	condition	NOUN
ma-255	231	9	(	(	PUNCT
ma-255	231	10	h4	h4	PROPN
ma-255	231	11	)	)	PUNCT
ma-255	231	12	does	do	AUX
ma-255	231	13	not	not	PART
ma-255	231	14	necessarily	necessarily	ADV
ma-255	231	15	imply	imply	VERB
ma-255	231	16	the	the	DET
ma-255	231	17	usual	usual	ADJ
ma-255	231	18	condition	condition	NOUN
ma-255	231	19	in	in	ADP
ma-255	231	20	local	local	ADJ
ma-255	231	21	convergence	convergence	NOUN
ma-255	231	22	studies	study	NOUN
ma-255	231	23	that	that	SCONJ
ma-255	231	24	the	the	DET
ma-255	231	25	operator	operator	NOUN
ma-255	231	26	f	f	PROPN
ma-255	231	27	′1(s∗	′1(s∗	PROPN
ma-255	231	28	)	)	PUNCT
ma-255	231	29	is	be	AUX
ma-255	231	30	invertible	invertible	ADJ
ma-255	231	31	,	,	PUNCT
ma-255	231	32	i.e.	i.e.	X
ma-255	231	33	that	that	DET
ma-255	231	34	s∗	s∗	PROPN
ma-255	231	35	is	be	AUX
ma-255	231	36	a	a	DET
ma-255	231	37	simple	simple	ADJ
ma-255	231	38	solution	solution	NOUN
ma-255	231	39	of	of	ADP
ma-255	231	40	the	the	DET
ma-255	231	41	equation	equation	NOUN
ma-255	231	42	f1(x	f1(x	NOUN
ma-255	231	43	)	)	PUNCT
ma-255	231	44	=	=	SYM
ma-255	232	1	0	0	X
ma-255	232	2	.	.	PUNCT
ma-255	233	1	consequently	consequently	ADV
ma-255	233	2	the	the	DET
ma-255	233	3	method	method	NOUN
ma-255	233	4	(	(	PUNCT
ma-255	233	5	1.5	1.5	NUM
ma-255	233	6	)	)	PUNCT
ma-255	233	7	can	can	AUX
ma-255	233	8	be	be	AUX
ma-255	233	9	applied	apply	VERB
ma-255	233	10	to	to	PART
ma-255	233	11	find	find	VERB
ma-255	233	12	solutions	solution	NOUN
ma-255	233	13	of	of	ADP
ma-255	233	14	multiplicity	multiplicity	NOUN
ma-255	233	15	greater	great	ADJ
ma-255	233	16	than	than	ADP
ma-255	233	17	one	one	NUM
ma-255	233	18	.	.	PUNCT
ma-255	234	1	3	3	X
ma-255	234	2	.	.	X
ma-255	234	3	convergence	convergence	NOUN
ma-255	234	4	for	for	ADP
ma-255	234	5	the	the	DET
ma-255	234	6	method	method	NOUN
ma-255	234	7	(	(	PUNCT
ma-255	234	8	1.10	1.10	NUM
ma-255	234	9	)	)	PUNCT
ma-255	234	10	let	let	VERB
ma-255	234	11	λ	λ	PROPN
ma-255	234	12	>	>	X
ma-255	234	13	0	0	NUM
ma-255	234	14	,	,	PUNCT
ma-255	234	15	µ	µ	X
ma-255	234	16	>	>	X
ma-255	234	17	0	0	PROPN
ma-255	234	18	,	,	PUNCT
ma-255	234	19	δ	δ	PROPN
ma-255	234	20	≥	≥	NOUN
ma-255	234	21	0	0	NUM
ma-255	234	22	and	and	CCONJ
ma-255	234	23	β	β	X
ma-255	234	24	>	>	X
ma-255	234	25	0	0	PUNCT
ma-255	234	26	be	be	AUX
ma-255	234	27	given	give	VERB
ma-255	234	28	parameters.suppose:(c1	parameters.suppose:(c1	PROPN
ma-255	234	29	)	)	PUNCT
ma-255	234	30	there	there	PRON
ma-255	234	31	exists	exist	VERB
ma-255	234	32	a	a	DET
ma-255	234	33	function	function	NOUN
ma-255	234	34	ϕ1	ϕ1	NOUN
ma-255	234	35	:	:	PUNCT
ma-255	235	1	[	[	X
ma-255	235	2	0	0	NUM
ma-255	235	3	,	,	PUNCT
ma-255	235	4	λ]×	λ]×	ADV
ma-255	235	5	[	[	X
ma-255	235	6	0	0	NUM
ma-255	235	7	,	,	PUNCT
ma-255	235	8	λ]×	λ]×	ADV
ma-255	235	9	[	[	X
ma-255	235	10	0	0	NUM
ma-255	235	11	,	,	PUNCT
ma-255	235	12	λ	λ	X
ma-255	235	13	]	]	X
ma-255	235	14	−→	−→	NOUN
ma-255	235	15	[	[	X
ma-255	235	16	0	0	NUM
ma-255	235	17	,	,	PUNCT
ma-255	235	18	δ	δ	PROPN
ma-255	235	19	]	]	X
ma-255	235	20	which	which	PRON
ma-255	235	21	is	be	AUX
ma-255	235	22	continuous	continuous	ADJ
ma-255	235	23	and	and	CCONJ
ma-255	235	24	nondecreasing	nondecreasing	ADJ
ma-255	235	25	.	.	PUNCT
ma-255	236	1	define	define	VERB
ma-255	236	2	the	the	DET
ma-255	236	3	scalar	scalar	ADJ
ma-255	236	4	sequence	sequence	NOUN
ma-255	236	5	{	{	PUNCT
ma-255	236	6	hn	hn	NOUN
ma-255	236	7	}	}	PUNCT
ma-255	236	8	for	for	ADP
ma-255	236	9	h0	h0	NOUN
ma-255	236	10	=	=	PROPN
ma-255	236	11	0	0	PROPN
ma-255	236	12	,	,	PUNCT
ma-255	236	13	some	some	PRON
ma-255	236	14	h1	h1	ADJ
ma-255	236	15	≥	≥	NOUN
ma-255	236	16	0	0	NUM
ma-255	236	17	,	,	PUNCT
ma-255	236	18	β1	β1	PROPN
ma-255	236	19	>	>	PUNCT
ma-255	236	20	β	β	X
ma-255	236	21	and	and	CCONJ
ma-255	236	22	each	each	DET
ma-255	236	23	n	n	NOUN
ma-255	236	24	=	=	SYM
ma-255	236	25	1	1	NUM
ma-255	236	26	,	,	PUNCT
ma-255	236	27	2	2	NUM
ma-255	236	28	,	,	PUNCT
ma-255	236	29	...	...	PUNCT
ma-255	236	30	by	by	ADP
ma-255	236	31	hn+1	hn+1	PROPN
ma-255	237	1	=	=	NOUN
ma-255	237	2	hn	hn	PROPN
ma-255	237	3	+	+	SYM
ma-255	237	4	β1ϕ1(hn−1	β1ϕ1(hn−1	PROPN
ma-255	237	5	,	,	PUNCT
ma-255	237	6	hn	hn	PROPN
ma-255	237	7	,	,	PUNCT
ma-255	237	8	hn	hn	PROPN
ma-255	237	9	−	−	PROPN
ma-255	237	10	hn−1)(hn	hn−1)(hn	PROPN
ma-255	237	11	−	−	PROPN
ma-255	237	12	hn−1	hn−1	PROPN
ma-255	237	13	)	)	PUNCT
ma-255	237	14	.	.	PUNCT
ma-255	238	1	(	(	PUNCT
ma-255	238	2	3.1	3.1	NUM
ma-255	238	3	)	)	PUNCT
ma-255	238	4	the	the	DET
ma-255	238	5	scalar	scalar	ADJ
ma-255	238	6	sequence	sequence	NOUN
ma-255	238	7	{	{	PUNCT
ma-255	238	8	hn	hn	NOUN
ma-255	238	9	}	}	PUNCT
ma-255	238	10	is	be	AUX
ma-255	238	11	shown	show	VERB
ma-255	238	12	to	to	PART
ma-255	238	13	be	be	AUX
ma-255	238	14	majorizing	majorize	VERB
ma-255	238	15	for	for	ADP
ma-255	238	16	{	{	PUNCT
ma-255	238	17	xn	xn	PRON
ma-255	238	18	}	}	PUNCT
ma-255	238	19	is	be	AUX
ma-255	238	20	generated	generate	VERB
ma-255	238	21	by	by	ADP
ma-255	238	22	the	the	DET
ma-255	238	23	formulain	formulain	NOUN
ma-255	238	24	the	the	DET
ma-255	238	25	theorem	theorem	NOUN
ma-255	238	26	3.2	3.2	NUM
ma-255	238	27	.	.	PUNCT
ma-255	239	1	however	however	ADV
ma-255	239	2	,	,	PUNCT
ma-255	239	3	let	let	VERB
ma-255	239	4	us	we	PRON
ma-255	239	5	present	present	VERB
ma-255	239	6	a	a	DET
ma-255	239	7	convergence	convergence	NOUN
ma-255	239	8	criterion	criterion	NOUN
ma-255	239	9	for	for	ADP
ma-255	239	10	it.(c2	it.(c2	NOUN
ma-255	239	11	)	)	PUNCT
ma-255	239	12	there	there	PRON
ma-255	239	13	exists	exist	VERB
ma-255	239	14	λ0	λ0	NOUN
ma-255	239	15	∈	∈	NOUN
ma-255	240	1	[	[	X
ma-255	240	2	0	0	NUM
ma-255	240	3	,	,	PUNCT
ma-255	240	4	j	j	PROPN
ma-255	240	5	]	]	PUNCT
ma-255	240	6	such	such	ADJ
ma-255	240	7	that	that	SCONJ
ma-255	240	8	for	for	ADP
ma-255	240	9	each	each	DET
ma-255	240	10	n	n	NOUN
ma-255	240	11	=	=	SYM
ma-255	240	12	0	0	NUM
ma-255	240	13	,	,	PUNCT
ma-255	240	14	1	1	NUM
ma-255	240	15	,	,	PUNCT
ma-255	240	16	2	2	NUM
ma-255	240	17	,	,	PUNCT
ma-255	240	18	...	...	PUNCT
ma-255	241	1	hn	hn	PRON
ma-255	241	2	≤	≤	NOUN
ma-255	242	1	λ0.it	λ0.it	PRON
ma-255	242	2	follows	follow	VERB
ma-255	242	3	by	by	ADP
ma-255	242	4	this	this	DET
ma-255	242	5	condition	condition	NOUN
ma-255	242	6	and	and	CCONJ
ma-255	242	7	(	(	PUNCT
ma-255	242	8	3.1	3.1	NUM
ma-255	242	9	)	)	PUNCT
ma-255	242	10	that	that	SCONJ
ma-255	242	11	0	0	NUM
ma-255	242	12	≤	≤	X
ma-255	242	13	hn−1	hn−1	ADJ
ma-255	242	14	≤	≤	PROPN
ma-255	242	15	hn	hn	PROPN
ma-255	242	16	≤	≤	NOUN
ma-255	242	17	λ0	λ0	NOUN
ma-255	243	1	and	and	CCONJ
ma-255	243	2	there	there	PRON
ma-255	243	3	exists	exist	VERB
ma-255	243	4	h∗	h∗	PROPN
ma-255	243	5	∈	∈	PROPN
ma-255	244	1	[	[	X
ma-255	244	2	0	0	NUM
ma-255	244	3	,	,	PUNCT
ma-255	244	4	λ0]such	λ0]such	PROPN
ma-255	244	5	that	that	PRON
ma-255	244	6	l	l	NOUN
ma-255	244	7	imn→∞	imn→∞	X
ma-255	244	8	=	=	PUNCT
ma-255	244	9	h∗.the	h∗.the	DET
ma-255	244	10	limit	limit	NOUN
ma-255	244	11	point	point	NOUN
ma-255	244	12	h∗	h∗	PROPN
ma-255	244	13	is	be	AUX
ma-255	244	14	the	the	DET
ma-255	244	15	unique	unique	ADJ
ma-255	244	16	least	least	ADV
ma-255	244	17	upper	upper	ADJ
ma-255	244	18	bound	bind	VERB
ma-255	244	19	of	of	ADP
ma-255	244	20	the	the	DET
ma-255	244	21	sequence	sequence	NOUN
ma-255	244	22	{	{	PUNCT
ma-255	244	23	hn}.(c3	hn}.(c3	NOUN
ma-255	244	24	)	)	PUNCT
ma-255	244	25	there	there	PRON
ma-255	244	26	exists	exist	VERB
ma-255	244	27	x0	x0	PROPN
ma-255	244	28	∈	∈	PROPN
ma-255	244	29	x	x	X
ma-255	244	30	and	and	CCONJ
ma-255	244	31	y0	y0	PROPN
ma-255	244	32	∈	∈	NOUN
ma-255	244	33	f1(x0	f1(x0	NOUN
ma-255	244	34	)	)	PUNCT
ma-255	244	35	+	+	CCONJ
ma-255	244	36	f2(x0	f2(x0	ADV
ma-255	244	37	)	)	PUNCT
ma-255	244	38	such	such	ADJ
ma-255	244	39	that	that	SCONJ
ma-255	244	40	βδ	βδ	DET
ma-255	244	41	<	<	X
ma-255	244	42	1	1	NUM
ma-255	244	43	and	and	CCONJ
ma-255	244	44	‖y0‖	‖y0‖	PROPN
ma-255	244	45	≤	≤	NOUN
ma-255	244	46	(	(	PUNCT
ma-255	244	47	1−	1−	NUM
ma-255	244	48	βδ	βδ	NOUN
ma-255	244	49	)	)	PUNCT
ma-255	244	50	min{λβ	min{λβ	NOUN
ma-255	244	51	,	,	PUNCT
ma-255	244	52	µ}.choose	µ}.choose	PRON
ma-255	244	53	h1	h1	VERB
ma-255	244	54	≤	≤	NOUN
ma-255	244	55	β1‖y0‖.(c4	β1‖y0‖.(c4	NOUN
ma-255	244	56	)	)	PUNCT
ma-255	244	57	the	the	DET
ma-255	244	58	operator	operator	NOUN
ma-255	244	59	x	x	PUNCT
ma-255	244	60	−→	−→	NOUN
ma-255	244	61	qdn(x	qdn(x	PROPN
ma-255	244	62	)	)	PUNCT
ma-255	244	63	:	:	PUNCT
ma-255	244	64	=	=	SYM
ma-255	244	65	f1(x0	f1(x0	X
ma-255	244	66	)	)	PUNCT
ma-255	244	67	+	+	NOUN
ma-255	244	68	dn(x	dn(x	X
ma-255	244	69	−	−	NOUN
ma-255	244	70	x0	x0	PROPN
ma-255	244	71	)	)	PUNCT
ma-255	245	1	+	+	CCONJ
ma-255	245	2	f2(x	f2(x	X
ma-255	245	3	)	)	PUNCT
ma-255	245	4	(	(	PUNCT
ma-255	245	5	3.2	3.2	NUM
ma-255	245	6	)	)	PUNCT
ma-255	245	7	is	be	AUX
ma-255	245	8	metrically	metrically	ADV
ma-255	245	9	regular	regular	ADJ
ma-255	245	10	at	at	ADP
ma-255	245	11	x0	x0	PROPN
ma-255	245	12	for	for	ADP
ma-255	245	13	y0	y0	PROPN
ma-255	245	14	with	with	ADP
ma-255	245	15	constant	constant	ADJ
ma-255	245	16	β	β	NOUN
ma-255	245	17	and	and	CCONJ
ma-255	245	18	neighborhoods	neighborhood	NOUN
ma-255	245	19	s(x0	s(x0	VERB
ma-255	245	20	,	,	PUNCT
ma-255	245	21	λ	λ	NOUN
ma-255	245	22	)	)	PUNCT
ma-255	245	23	and	and	CCONJ
ma-255	245	24	s(y0	s(y0	NOUN
ma-255	245	25	,	,	PUNCT
ma-255	245	26	µ),respectively.the	µ),respectively.the	DET
ma-255	245	27	mapping	mapping	NOUN
ma-255	245	28	ϕ1	ϕ1	NOUN
ma-255	245	29	relates	relate	VERB
ma-255	245	30	to	to	ADP
ma-255	245	31	the	the	DET
ma-255	245	32	operators	operator	NOUN
ma-255	245	33	on	on	ADP
ma-255	245	34	the	the	DET
ma-255	245	35	method	method	NOUN
ma-255	245	36	(	(	PUNCT
ma-255	245	37	1.10).(c5	1.10).(c5	NUM
ma-255	245	38	)	)	PUNCT
ma-255	245	39	‖f1(x)−	‖f1(x)−	PROPN
ma-255	245	40	f1(xn)−dn(x	f1(xn)−dn(x	ADJ
ma-255	245	41	−	−	PROPN
ma-255	245	42	xn)‖	xn)‖	PROPN
ma-255	245	43	≤	≤	PROPN
ma-255	245	44	ϕ1(‖x	ϕ1(‖x	PROPN
ma-255	246	1	−	−	PROPN
ma-255	246	2	x0‖	x0‖	PROPN
ma-255	246	3	,	,	PUNCT
ma-255	246	4	‖xn	‖xn	PROPN
ma-255	246	5	−	−	PROPN
ma-255	246	6	x0‖	x0‖	PROPN
ma-255	246	7	,	,	PUNCT
ma-255	246	8	‖x	‖x	NOUN
ma-255	246	9	−	−	PROPN
ma-255	247	1	xn‖)‖x	xn‖)‖x	PROPN
ma-255	247	2	−	−	PROPN
ma-255	247	3	xn‖	xn‖	PROPN
ma-255	247	4	,	,	PUNCT
ma-255	247	5	for	for	ADP
ma-255	247	6	each	each	DET
ma-255	247	7	x	x	SYM
ma-255	247	8	∈	∈	PROPN
ma-255	247	9	s(x0	s(x0	NOUN
ma-255	247	10	,	,	PUNCT
ma-255	247	11	λ).next	λ).next	PROPN
ma-255	247	12	,	,	PUNCT
ma-255	247	13	the	the	DET
ma-255	247	14	semi	semi	ADJ
ma-255	247	15	-	-	ADJ
ma-255	247	16	local	local	ADJ
ma-255	247	17	analysis	analysis	NOUN
ma-255	247	18	of	of	ADP
ma-255	247	19	convergence	convergence	NOUN
ma-255	247	20	is	be	AUX
ma-255	247	21	developed	develop	VERB
ma-255	247	22	using	use	VERB
ma-255	247	23	the	the	DET
ma-255	247	24	conditions	condition	NOUN
ma-255	247	25	(	(	PUNCT
ma-255	247	26	c1)−	c1)−	PROPN
ma-255	247	27	(	(	PUNCT
ma-255	247	28	c5	c5	PROPN
ma-255	247	29	)	)	PUNCT
ma-255	247	30	.	.	PUNCT
ma-255	248	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	248	2	eur	eur	PROPN
ma-255	248	3	.	.	PUNCT
ma-255	249	1	j.	j.	PROPN
ma-255	249	2	math	math	PROPN
ma-255	249	3	.	.	PUNCT
ma-255	250	1	anal	anal	PROPN
ma-255	250	2	.	.	PUNCT
ma-255	251	1	10.28924	10.28924	NUM
ma-255	251	2	/	/	SYM
ma-255	251	3	ada	ada	PROPN
ma-255	251	4	/	/	SYM
ma-255	251	5	ma.5.5	ma.5.5	PROPN
ma-255	251	6	10	10	NUM
ma-255	251	7	theorem	theorem	VERB
ma-255	251	8	3.1	3.1	NUM
ma-255	251	9	.	.	PUNCT
ma-255	251	10	suppose	suppose	VERB
ma-255	251	11	that	that	SCONJ
ma-255	251	12	the	the	DET
ma-255	251	13	conditions	condition	NOUN
ma-255	251	14	(	(	PUNCT
ma-255	251	15	c1)−(c5	c1)−(c5	NOUN
ma-255	251	16	)	)	PUNCT
ma-255	251	17	hold	hold	NOUN
ma-255	251	18	.	.	PUNCT
ma-255	252	1	then	then	ADV
ma-255	252	2	,	,	PUNCT
ma-255	252	3	for	for	ADP
ma-255	252	4	each	each	DET
ma-255	252	5	γ	γ	X
ma-255	252	6	∈	∈	PROPN
ma-255	252	7	(	(	PUNCT
ma-255	252	8	βδ	βδ	NOUN
ma-255	252	9	,	,	PUNCT
ma-255	252	10	1	1	NUM
ma-255	252	11	)	)	PUNCT
ma-255	252	12	there	there	PRON
ma-255	252	13	exists	exist	VERB
ma-255	252	14	a	a	DET
ma-255	252	15	sequence	sequence	NOUN
ma-255	252	16	{	{	PUNCT
ma-255	252	17	xn	xn	NOUN
ma-255	252	18	}	}	PUNCT
ma-255	252	19	generated	generate	VERB
ma-255	252	20	by	by	ADP
ma-255	252	21	the	the	DET
ma-255	252	22	method	method	NOUN
ma-255	252	23	(	(	PUNCT
ma-255	252	24	1.10	1.10	NUM
ma-255	252	25	)	)	PUNCT
ma-255	252	26	which	which	PRON
ma-255	252	27	is	be	AUX
ma-255	252	28	well	well	ADV
ma-255	252	29	defined	define	VERB
ma-255	252	30	in	in	ADP
ma-255	252	31	s(x0	s(x0	PROPN
ma-255	252	32	,	,	PUNCT
ma-255	252	33	λ	λ	PROPN
ma-255	252	34	)	)	PUNCT
ma-255	252	35	,	,	PUNCT
ma-255	252	36	remains	remain	VERB
ma-255	252	37	in	in	ADP
ma-255	252	38	s(x0	s(x0	PROPN
ma-255	252	39	,	,	PUNCT
ma-255	252	40	λ	λ	NOUN
ma-255	252	41	)	)	PUNCT
ma-255	252	42	for	for	ADP
ma-255	252	43	each	each	DET
ma-255	252	44	n	n	NOUN
ma-255	252	45	=	=	SYM
ma-255	252	46	0	0	NUM
ma-255	252	47	,	,	PUNCT
ma-255	252	48	1	1	NUM
ma-255	252	49	,	,	PUNCT
ma-255	252	50	2	2	NUM
ma-255	252	51	,	,	PUNCT
ma-255	252	52	...	...	PUNCT
ma-255	252	53	and	and	CCONJ
ma-255	252	54	convergence	convergence	NOUN
ma-255	252	55	to	to	ADP
ma-255	252	56	a	a	DET
ma-255	252	57	solution	solution	NOUN
ma-255	252	58	s∗	s∗	PROPN
ma-255	252	59	∈	∈	PROPN
ma-255	252	60	s[x0	s[x0	NOUN
ma-255	252	61	,	,	PUNCT
ma-255	252	62	λ	λ	X
ma-255	252	63	]	]	X
ma-255	252	64	of	of	ADP
ma-255	252	65	the	the	DET
ma-255	252	66	generated	generate	VERB
ma-255	252	67	equation	equation	NOUN
ma-255	252	68	(	(	PUNCT
ma-255	252	69	1.8	1.8	NUM
ma-255	252	70	)	)	PUNCT
ma-255	252	71	.	.	PUNCT
ma-255	253	1	moreover	moreover	ADV
ma-255	253	2	,	,	PUNCT
ma-255	253	3	the	the	DET
ma-255	253	4	following	follow	VERB
ma-255	253	5	assertion	assertion	NOUN
ma-255	253	6	hold	hold	VERB
ma-255	253	7	‖s∗	‖s∗	PUNCT
ma-255	253	8	−	−	NOUN
ma-255	253	9	xn‖	xn‖	PROPN
ma-255	253	10	≤	≤	PROPN
ma-255	253	11	γnλ	γnλ	NOUN
ma-255	253	12	(	(	PUNCT
ma-255	253	13	3.3	3.3	NUM
ma-255	253	14	)	)	PUNCT
ma-255	253	15	and	and	CCONJ
ma-255	253	16	dist(0	dist(0	PROPN
ma-255	253	17	,	,	PUNCT
ma-255	253	18	f1(xn	f1(xn	NUM
ma-255	253	19	)	)	PUNCT
ma-255	253	20	+	+	NUM
ma-255	253	21	f2(xn	f2(xn	NUM
ma-255	253	22	)	)	PUNCT
ma-255	253	23	)	)	PUNCT
ma-255	253	24	≤	≤	NUM
ma-255	254	1	γn‖y0‖	γn‖y0‖	X
ma-255	254	2	(	(	PUNCT
ma-255	254	3	3.4	3.4	NUM
ma-255	254	4	)	)	PUNCT
ma-255	254	5	for	for	ADP
ma-255	254	6	each	each	DET
ma-255	254	7	n	n	NOUN
ma-255	254	8	=	=	SYM
ma-255	254	9	0	0	NUM
ma-255	254	10	,	,	PUNCT
ma-255	254	11	1	1	NUM
ma-255	254	12	,	,	PUNCT
ma-255	254	13	2	2	NUM
ma-255	254	14	,	,	PUNCT
ma-255	254	15	...	...	PUNCT
ma-255	254	16	thus	thus	ADV
ma-255	254	17	,	,	PUNCT
ma-255	254	18	the	the	DET
ma-255	254	19	convergence	convergence	NOUN
ma-255	254	20	rate	rate	NOUN
ma-255	254	21	is	be	AUX
ma-255	254	22	r	r	NOUN
ma-255	254	23	-	-	PUNCT
ma-255	254	24	linear	linear	NOUN
ma-255	254	25	.	.	PUNCT
ma-255	255	1	furthermore	furthermore	ADV
ma-255	255	2	,	,	PUNCT
ma-255	255	3	if	if	SCONJ
ma-255	255	4	the	the	DET
ma-255	255	5	operator	operator	NOUN
ma-255	255	6	qdn	qdn	NOUN
ma-255	255	7	is	be	AUX
ma-255	255	8	stronglymetrically	stronglymetrically	ADV
ma-255	255	9	regular	regular	ADJ
ma-255	255	10	with	with	ADP
ma-255	255	11	constant	constant	ADJ
ma-255	255	12	β	β	NOUN
ma-255	255	13	and	and	CCONJ
ma-255	255	14	neighbourhoods	neighbourhood	NOUN
ma-255	255	15	s(x0	s(x0	VERB
ma-255	255	16	,	,	PUNCT
ma-255	255	17	λ	λ	NOUN
ma-255	255	18	)	)	PUNCT
ma-255	255	19	and	and	CCONJ
ma-255	255	20	s(y0	s(y0	NOUN
ma-255	255	21	,	,	PUNCT
ma-255	255	22	µ	µ	NOUN
ma-255	255	23	)	)	PUNCT
ma-255	255	24	,	,	PUNCT
ma-255	255	25	respectively	respectively	ADV
ma-255	255	26	,	,	PUNCT
ma-255	255	27	then	then	ADV
ma-255	255	28	the	the	DET
ma-255	255	29	sequence	sequence	NOUN
ma-255	255	30	{	{	PUNCT
ma-255	255	31	xn	xn	PROPN
ma-255	255	32	}	}	PUNCT
ma-255	255	33	is	be	AUX
ma-255	255	34	the	the	DET
ma-255	255	35	only	only	ADJ
ma-255	255	36	one	one	NUM
ma-255	255	37	satisfying	satisfying	ADJ
ma-255	255	38	(	(	PUNCT
ma-255	255	39	1.10	1.10	NUM
ma-255	255	40	)	)	PUNCT
ma-255	255	41	,	,	PUNCT
ma-255	255	42	and	and	CCONJ
ma-255	255	43	staying	stay	VERB
ma-255	255	44	in	in	ADP
ma-255	255	45	s(x0	s(x0	PROPN
ma-255	255	46	,	,	PUNCT
ma-255	255	47	λ	λ	PROPN
ma-255	255	48	)	)	PUNCT
ma-255	255	49	.	.	PUNCT
ma-255	256	1	remark	remark	PROPN
ma-255	256	2	3.1	3.1	NUM
ma-255	256	3	.	.	PUNCT
ma-255	257	1	the	the	DET
ma-255	257	2	proof	proof	NOUN
ma-255	257	3	is	be	AUX
ma-255	257	4	similar	similar	ADJ
ma-255	257	5	to	to	ADP
ma-255	257	6	the	the	DET
ma-255	257	7	one	one	NUM
ma-255	257	8	in	in	ADP
ma-255	257	9	[	[	X
ma-255	257	10	11	11	NUM
ma-255	257	11	,	,	PUNCT
ma-255	257	12	theorem	theorem	VERB
ma-255	257	13	2.2	2.2	NUM
ma-255	257	14	]	]	PUNCT
ma-255	257	15	.	.	PUNCT
ma-255	258	1	but	but	CCONJ
ma-255	258	2	it	it	PRON
ma-255	258	3	uses	use	VERB
ma-255	258	4	(	(	PUNCT
ma-255	258	5	1.10	1.10	NUM
ma-255	258	6	)	)	PUNCT
ma-255	258	7	,	,	PUNCT
ma-255	258	8	(	(	PUNCT
ma-255	258	9	c4	c4	NOUN
ma-255	258	10	)	)	PUNCT
ma-255	258	11	and	and	CCONJ
ma-255	258	12	(	(	PUNCT
ma-255	258	13	c5	c5	PROPN
ma-255	258	14	)	)	PUNCT
ma-255	258	15	,	,	PUNCT
ma-255	258	16	instead	instead	ADV
ma-255	258	17	of	of	ADP
ma-255	258	18	stronger	strong	ADJ
ma-255	258	19	(	(	PUNCT
ma-255	258	20	1.9),(c4	1.9),(c4	NUM
ma-255	258	21	)	)	PUNCT
ma-255	258	22	’	'	PUNCT
ma-255	259	1	x	x	X
ma-255	259	2	−→	−→	NOUN
ma-255	259	3	qan(x	qan(x	PROPN
ma-255	259	4	)	)	PUNCT
ma-255	259	5	:	:	PUNCT
ma-255	259	6	=	=	PUNCT
ma-255	259	7	f1(x0	f1(x0	PROPN
ma-255	259	8	)	)	PUNCT
ma-255	259	9	+	+	X
ma-255	259	10	an(x	an(x	X
ma-255	259	11	−	−	NOUN
ma-255	259	12	x0	x0	PROPN
ma-255	259	13	)	)	PUNCT
ma-255	260	1	+	+	X
ma-255	260	2	f2(x),(c5	f2(x),(c5	ADJ
ma-255	260	3	)	)	PUNCT
ma-255	260	4	’	'	PUNCT
ma-255	261	1	‖f1(x)−	‖f1(x)−	PROPN
ma-255	261	2	f1(xn)−	f1(xn)−	X
ma-255	261	3	ln(x	ln(x	PUNCT
ma-255	261	4	−	−	PROPN
ma-255	261	5	xn)‖	xn)‖	PROPN
ma-255	261	6	≤	≤	PUNCT
ma-255	261	7	w(‖x	w(‖x	PUNCT
ma-255	261	8	−	−	PROPN
ma-255	262	1	xn‖)‖x	xn‖)‖x	PROPN
ma-255	262	2	−	−	PROPN
ma-255	262	3	xn‖	xn‖	PROPN
ma-255	262	4	for	for	ADP
ma-255	262	5	each	each	DET
ma-255	262	6	x	x	SYM
ma-255	262	7	∈	∈	PROPN
ma-255	262	8	s(x0	s(x0	NOUN
ma-255	262	9	,	,	PUNCT
ma-255	262	10	λ	λ	PROPN
ma-255	262	11	)	)	PUNCT
ma-255	262	12	,	,	PUNCT
ma-255	262	13	where	where	SCONJ
ma-255	262	14	w	w	X
ma-255	262	15	:	:	PUNCT
ma-255	263	1	[	[	X
ma-255	263	2	0.λ	0.λ	X
ma-255	263	3	]	]	X
ma-255	263	4	−→	−→	NOUN
ma-255	263	5	[	[	X
ma-255	263	6	0	0	NUM
ma-255	263	7	,	,	PUNCT
ma-255	263	8	δ	δ	PROPN
ma-255	263	9	]	]	PUNCT
ma-255	263	10	satisfies	satisfy	VERB
ma-255	263	11	lim	lim	PROPN
ma-255	263	12	n→+∞w(t	n→+∞w(t	PROPN
ma-255	263	13	)	)	PUNCT
ma-255	263	14	=	=	SYM
ma-255	264	1	0	0	NUM
ma-255	264	2	proof	proof	NOUN
ma-255	264	3	.	.	PUNCT
ma-255	265	1	pick	pick	VERB
ma-255	265	2	γ	γ	PROPN
ma-255	265	3	∈	∈	PROPN
ma-255	265	4	(	(	PUNCT
ma-255	265	5	βδ	βδ	NOUN
ma-255	265	6	,	,	PUNCT
ma-255	265	7	1	1	NUM
ma-255	265	8	)	)	PUNCT
ma-255	265	9	and	and	CCONJ
ma-255	265	10	β1	β1	VERB
ma-255	266	1	so	so	SCONJ
ma-255	266	2	that	that	SCONJ
ma-255	266	3	β	β	X
ma-255	266	4	<	<	X
ma-255	266	5	β1	β1	PROPN
ma-255	266	6	≤	≤	NUM
ma-255	266	7	γ	γ	PROPN
ma-255	266	8	δ	δ	PROPN
ma-255	266	9	and	and	CCONJ
ma-255	266	10	‖y0‖	‖y0‖	PROPN
ma-255	266	11	≤	≤	NOUN
ma-255	266	12	(	(	PUNCT
ma-255	266	13	1−	1−	NUM
ma-255	266	14	γ)min	γ)min	NOUN
ma-255	266	15	{	{	PUNCT
ma-255	266	16	λ	λ	X
ma-255	266	17	β1	β1	PROPN
ma-255	266	18	,	,	PUNCT
ma-255	266	19	µ	µ	X
ma-255	266	20	}	}	PUNCT
ma-255	267	1	.	.	PUNCT
ma-255	268	1	(	(	PUNCT
ma-255	268	2	3.5	3.5	NUM
ma-255	268	3	)	)	PUNCT
ma-255	268	4	the	the	DET
ma-255	268	5	choice	choice	NOUN
ma-255	268	6	of	of	ADP
ma-255	268	7	β1	β1	PROPN
ma-255	268	8	is	be	AUX
ma-255	268	9	certainly	certainly	ADV
ma-255	268	10	possible	possible	ADJ
ma-255	268	11	since	since	SCONJ
ma-255	268	12	there	there	PRON
ma-255	268	13	are	be	VERB
ma-255	268	14	infinitely	infinitely	ADV
ma-255	268	15	many	many	ADJ
ma-255	268	16	numbers	number	NOUN
ma-255	268	17	between	between	ADP
ma-255	268	18	βand	βand	NOUN
ma-255	268	19	γ	γ	PROPN
ma-255	268	20	δ	δ	PROPN
ma-255	268	21	.	.	PUNCT
ma-255	269	1	we	we	PRON
ma-255	269	2	shall	shall	AUX
ma-255	269	3	show	show	VERB
ma-255	269	4	the	the	DET
ma-255	269	5	existence	existence	NOUN
ma-255	269	6	of	of	ADP
ma-255	269	7	the	the	DET
ma-255	269	8	sequence	sequence	NOUN
ma-255	269	9	{	{	PUNCT
ma-255	269	10	xn	xn	VERB
ma-255	269	11	}	}	PUNCT
ma-255	269	12	using	use	VERB
ma-255	269	13	mathematical	mathematical	ADJ
ma-255	269	14	induction	induction	NOUN
ma-255	269	15	for	for	ADP
ma-255	269	16	n	n	NOUN
ma-255	269	17	=	=	SYM
ma-255	269	18	1	1	NUM
ma-255	269	19	,	,	PUNCT
ma-255	269	20	2	2	NUM
ma-255	269	21	,	,	PUNCT
ma-255	269	22	...	...	PUNCT
ma-255	269	23	satisfying(in	satisfying(in	NOUN
ma-255	269	24	)	)	PUNCT
ma-255	269	25	‖xn	‖xn	PROPN
ma-255	269	26	−	−	PROPN
ma-255	269	27	x0‖	x0‖	PROPN
ma-255	269	28	≤	≤	NUM
ma-255	269	29	1−γn	1−γn	NUM
ma-255	269	30	1−γ	1−γ	NUM
ma-255	269	31	β1‖y0‖	β1‖y0‖	PROPN
ma-255	269	32	≤	≤	PROPN
ma-255	269	33	(	(	PUNCT
ma-255	269	34	1−	1−	NUM
ma-255	269	35	γn)λ	γn)λ	NUM
ma-255	269	36	<	<	X
ma-255	269	37	λ.(iin	λ.(iin	PROPN
ma-255	269	38	)	)	PUNCT
ma-255	270	1	‖xn	‖xn	PROPN
ma-255	270	2	−	−	PROPN
ma-255	270	3	xn−1‖	xn−1‖	PROPN
ma-255	270	4	≤	≤	PROPN
ma-255	271	1	hn	hn	PRON
ma-255	271	2	−	−	PROPN
ma-255	271	3	hn−1.(iiin	hn−1.(iiin	PROPN
ma-255	271	4	)	)	PUNCT
ma-255	271	5	0	0	NUM
ma-255	271	6	∈	∈	PROPN
ma-255	271	7	f1(xn−1	f1(xn−1	PRON
ma-255	271	8	)	)	PUNCT
ma-255	271	9	+	+	NOUN
ma-255	271	10	dn−1(xn	dn−1(xn	NOUN
ma-255	271	11	−	−	NOUN
ma-255	271	12	xn−1	xn−1	PROPN
ma-255	271	13	)	)	PUNCT
ma-255	271	14	+	+	CCONJ
ma-255	271	15	f2(xn),where	f2(xn),where	X
ma-255	271	16	dn−1	dn−1	PROPN
ma-255	271	17	=	=	PUNCT
ma-255	271	18	dn−1(x0	dn−1(x0	PROPN
ma-255	271	19	,	,	PUNCT
ma-255	271	20	...	...	PUNCT
ma-255	271	21	,	,	PUNCT
ma-255	271	22	xn−1).by	xn−1).by	PROPN
ma-255	271	23	hypothesis	hypothesis	NOUN
ma-255	271	24	0	0	NUM
ma-255	271	25	∈	∈	PROPN
ma-255	271	26	s(y0	s(y0	NOUN
ma-255	271	27	,	,	PUNCT
ma-255	271	28	µ	µ	NOUN
ma-255	271	29	)	)	PUNCT
ma-255	271	30	and	and	CCONJ
ma-255	271	31	y0	y0	PROPN
ma-255	271	32	∈	∈	NOUN
ma-255	271	33	qd0(x0	qd0(x0	NOUN
ma-255	271	34	)	)	PUNCT
ma-255	271	35	.	.	PUNCT
ma-255	272	1	using	use	VERB
ma-255	272	2	the	the	DET
ma-255	272	3	condition	condition	NOUN
ma-255	272	4	(	(	PUNCT
ma-255	272	5	c4	c4	NOUN
ma-255	272	6	)	)	PUNCT
ma-255	272	7	for	for	ADP
ma-255	272	8	qd0	qd0	NOUN
ma-255	272	9	we	we	PRON
ma-255	272	10	get	get	VERB
ma-255	272	11	dist(x0	dist(x0	ADJ
ma-255	272	12	,	,	PUNCT
ma-255	272	13	q	q	NOUN
ma-255	272	14	−1	−1	NOUN
ma-255	272	15	d0	d0	NOUN
ma-255	272	16	(	(	PUNCT
ma-255	272	17	0	0	NUM
ma-255	272	18	)	)	PUNCT
ma-255	272	19	)	)	PUNCT
ma-255	272	20	≤	≤	NOUN
ma-255	272	21	β	β	X
ma-255	272	22	dist(0	dist(0	PROPN
ma-255	272	23	,	,	PUNCT
ma-255	272	24	qd0(x0	qd0(x0	NOUN
ma-255	272	25	)	)	PUNCT
ma-255	272	26	)	)	PUNCT
ma-255	272	27	≤	≤	NUM
ma-255	273	1	β‖y0‖.	β‖y0‖.	PUNCT
ma-255	274	1	if	if	SCONJ
ma-255	274	2	y0	y0	NOUN
ma-255	274	3	=	=	SYM
ma-255	274	4	0	0	NUM
ma-255	274	5	,	,	PUNCT
ma-255	274	6	pick	pick	VERB
ma-255	274	7	x1	x1	NOUN
ma-255	274	8	=	=	SYM
ma-255	274	9	x0	x0	PROPN
ma-255	274	10	.	.	PUNCT
ma-255	275	1	otherwise	otherwise	ADV
ma-255	275	2	,	,	PUNCT
ma-255	275	3	it	it	PRON
ma-255	275	4	follows	follow	VERB
ma-255	275	5	that	that	SCONJ
ma-255	275	6	dist(x0	dist(x0	NOUN
ma-255	275	7	,	,	PUNCT
ma-255	275	8	q	q	NOUN
ma-255	275	9	−1	−1	NOUN
ma-255	275	10	d0	d0	NOUN
ma-255	275	11	(	(	PUNCT
ma-255	275	12	0	0	NUM
ma-255	275	13	)	)	PUNCT
ma-255	275	14	)	)	PUNCT
ma-255	275	15	≤	≤	NUM
ma-255	275	16	β1‖y0‖.	β1‖y0‖.	PROPN
ma-255	275	17	thus	thus	ADV
ma-255	275	18	,	,	PUNCT
ma-255	275	19	there	there	PRON
ma-255	275	20	exists	exist	VERB
ma-255	275	21	x1	x1	PROPN
ma-255	275	22	∈	∈	PROPN
ma-255	275	23	q−1d0	q−1d0	NOUN
ma-255	275	24	(	(	PUNCT
ma-255	275	25	0	0	NUM
ma-255	275	26	)	)	PUNCT
ma-255	275	27	satisfying	satisfy	VERB
ma-255	275	28	‖x1	‖x1	NOUN
ma-255	275	29	−	−	PROPN
ma-255	276	1	x0‖	x0‖	PROPN
ma-255	277	1	<	<	X
ma-255	278	1	β1‖y0‖	β1‖y0‖	X
ma-255	278	2	<	<	X
ma-255	278	3	(	(	PUNCT
ma-255	278	4	1−	1−	NUM
ma-255	278	5	γ)λ	γ)λ	NOUN
ma-255	278	6	.	.	PUNCT
ma-255	279	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	279	2	eur	eur	PROPN
ma-255	279	3	.	.	PUNCT
ma-255	280	1	j.	j.	PROPN
ma-255	280	2	math	math	PROPN
ma-255	280	3	.	.	PUNCT
ma-255	281	1	anal	anal	PROPN
ma-255	281	2	.	.	PUNCT
ma-255	282	1	10.28924	10.28924	NUM
ma-255	282	2	/	/	SYM
ma-255	282	3	ada	ada	PROPN
ma-255	282	4	/	/	SYM
ma-255	282	5	ma.5.5	ma.5.5	PROPN
ma-255	282	6	11so	11so	NOUN
ma-255	282	7	,	,	PUNCT
ma-255	282	8	(	(	PUNCT
ma-255	282	9	i1	i1	PROPN
ma-255	282	10	)	)	PUNCT
ma-255	282	11	,	,	PUNCT
ma-255	282	12	(	(	PUNCT
ma-255	282	13	ii1	ii1	NOUN
ma-255	282	14	)	)	PUNCT
ma-255	282	15	and	and	CCONJ
ma-255	282	16	(	(	PUNCT
ma-255	282	17	iii1	iii1	PROPN
ma-255	282	18	)	)	PUNCT
ma-255	282	19	hold.suppose	hold.suppose	PROPN
ma-255	282	20	that	that	PRON
ma-255	282	21	for	for	ADP
ma-255	282	22	some	some	DET
ma-255	282	23	natural	natural	ADJ
ma-255	282	24	number	number	NOUN
ma-255	282	25	m	m	VERB
ma-255	282	26	the	the	DET
ma-255	282	27	element	element	NOUN
ma-255	282	28	xm	xm	PROPN
ma-255	282	29	is	be	AUX
ma-255	282	30	defined	define	VERB
ma-255	282	31	so	so	SCONJ
ma-255	282	32	that	that	SCONJ
ma-255	282	33	the	the	DET
ma-255	282	34	inductionhypothesis	inductionhypothesis	NOUN
ma-255	282	35	(	(	PUNCT
ma-255	282	36	i	i	NOUN
ma-255	282	37	m	m	PROPN
ma-255	282	38	)	)	PUNCT
ma-255	282	39	,	,	PUNCT
ma-255	282	40	(	(	PUNCT
ma-255	282	41	iim	iim	NOUN
ma-255	282	42	)	)	PUNCT
ma-255	282	43	and	and	CCONJ
ma-255	282	44	(	(	PUNCT
ma-255	282	45	iiim	iiim	NOUN
ma-255	282	46	)	)	PUNCT
ma-255	282	47	hold	hold	VERB
ma-255	282	48	.	.	PUNCT
ma-255	283	1	we	we	PRON
ma-255	283	2	shall	shall	AUX
ma-255	283	3	show	show	VERB
ma-255	283	4	that	that	SCONJ
ma-255	283	5	the	the	DET
ma-255	283	6	iterate	iterate	NOUN
ma-255	283	7	xm+1	xm+1	PROPN
ma-255	283	8	is	be	AUX
ma-255	283	9	defined	define	VERB
ma-255	283	10	anddthe	anddthe	DET
ma-255	283	11	assertions	assertion	NOUN
ma-255	283	12	(	(	PUNCT
ma-255	283	13	im+1	im+1	NOUN
ma-255	283	14	)	)	PUNCT
ma-255	283	15	,	,	PUNCT
ma-255	283	16	(	(	PUNCT
ma-255	283	17	iim+1	iim+1	NOUN
ma-255	283	18	)	)	PUNCT
ma-255	283	19	and	and	CCONJ
ma-255	283	20	(	(	PUNCT
ma-255	283	21	iiim+1	iiim+1	PROPN
ma-255	283	22	)	)	PUNCT
ma-255	283	23	hold	hold	VERB
ma-255	283	24	.	.	PUNCT
ma-255	284	1	it	it	PRON
ma-255	284	2	follows	follow	VERB
ma-255	284	3	by	by	ADP
ma-255	284	4	(	(	PUNCT
ma-255	284	5	i	i	NOUN
ma-255	284	6	m	m	PROPN
ma-255	284	7	)	)	PUNCT
ma-255	284	8	that	that	PRON
ma-255	284	9	xm	xm	PROPN
ma-255	284	10	∈	∈	PROPN
ma-255	284	11	s(x0	s(x0	NOUN
ma-255	284	12	,	,	PUNCT
ma-255	284	13	λ).set	λ).set	VERB
ma-255	284	14	em	em	PRON
ma-255	284	15	=	=	PUNCT
ma-255	284	16	f1(x0)−	f1(x0)−	NOUN
ma-255	284	17	f1(xm)−dm(x0	f1(xm)−dm(x0	ADV
ma-255	284	18	−	−	PROPN
ma-255	284	19	xm).by	xm).by	PROPN
ma-255	285	1	(	(	PUNCT
ma-255	285	2	c5	c5	PROPN
ma-255	285	3	)	)	PUNCT
ma-255	285	4	for	for	ADP
ma-255	285	5	x	x	X
ma-255	285	6	=	=	SYM
ma-255	285	7	x0	x0	PROPN
ma-255	285	8	,	,	PUNCT
ma-255	285	9	(	(	PUNCT
ma-255	285	10	i	i	NOUN
ma-255	285	11	m	m	PROPN
ma-255	285	12	)	)	PUNCT
ma-255	285	13	,	,	PUNCT
ma-255	285	14	(	(	PUNCT
ma-255	285	15	iim	iim	NOUN
ma-255	285	16	)	)	PUNCT
ma-255	285	17	and	and	CCONJ
ma-255	285	18	the	the	DET
ma-255	285	19	properties	property	NOUN
ma-255	285	20	of	of	ADP
ma-255	285	21	the	the	DET
ma-255	285	22	function	function	NOUN
ma-255	285	23	ϕ1	ϕ1	NOUN
ma-255	285	24	we	we	PRON
ma-255	285	25	can	can	AUX
ma-255	285	26	write	write	VERB
ma-255	285	27	‖em	‖em	PROPN
ma-255	285	28	−	−	PROPN
ma-255	285	29	y0‖	y0‖	PROPN
ma-255	285	30	≤	≤	PROPN
ma-255	285	31	‖y0‖+	‖y0‖+	PROPN
ma-255	285	32	‖f1(x0)−	‖f1(x0)−	PUNCT
ma-255	286	1	f1(xm)−dm(x0	f1(xm)−dm(x0	ADP
ma-255	286	2	−	−	PROPN
ma-255	287	1	xm)‖	xm)‖	PROPN
ma-255	287	2	≤	≤	PROPN
ma-255	287	3	‖y0‖+	‖y0‖+	PROPN
ma-255	288	1	ϕ1(‖x0	ϕ1(‖x0	PROPN
ma-255	288	2	−	−	PROPN
ma-255	289	1	x0‖	x0‖	PROPN
ma-255	289	2	,	,	PUNCT
ma-255	289	3	‖xm	‖xm	PROPN
ma-255	289	4	−	−	PROPN
ma-255	289	5	x0‖	x0‖	PROPN
ma-255	289	6	,	,	PUNCT
ma-255	289	7	‖xm	‖xm	PROPN
ma-255	289	8	−	−	PROPN
ma-255	289	9	x0‖)‖x0	x0‖)‖x0	PROPN
ma-255	289	10	−	−	PROPN
ma-255	289	11	xm‖	xm‖	PROPN
ma-255	289	12	≤	≤	PROPN
ma-255	290	1	‖y0‖+	‖y0‖+	PROPN
ma-255	290	2	ϕ1(0	ϕ1(0	PROPN
ma-255	290	3	,	,	PUNCT
ma-255	290	4	hm	hm	INTJ
ma-255	290	5	,	,	PUNCT
ma-255	290	6	hm)‖x0	hm)‖x0	VERB
ma-255	290	7	−	−	PROPN
ma-255	290	8	xm‖	xm‖	PROPN
ma-255	290	9	≤	≤	PROPN
ma-255	291	1	‖y0‖+	‖y0‖+	PROPN
ma-255	291	2	δ‖x0	δ‖x0	VERB
ma-255	291	3	−	−	PROPN
ma-255	292	1	xm‖	xm‖	PROPN
ma-255	292	2	≤	≤	PROPN
ma-255	292	3	‖y0‖+	‖y0‖+	PROPN
ma-255	292	4	1−	1−	NUM
ma-255	292	5	γm	γm	ADJ
ma-255	292	6	1−	1−	NUM
ma-255	292	7	γ	γ	PROPN
ma-255	292	8	β‖y0‖	β‖y0‖	NOUN
ma-255	292	9	≤	≤	NUM
ma-255	292	10	(	(	PUNCT
ma-255	292	11	1	1	NUM
ma-255	292	12	+	+	SYM
ma-255	292	13	1−	1−	NUM
ma-255	292	14	γm	γm	NUM
ma-255	292	15	1−	1−	NUM
ma-255	292	16	γ	γ	X
ma-255	292	17	γ)‖y0‖	γ)‖y0‖	PUNCT
ma-255	292	18	=	=	SYM
ma-255	292	19	1−	1−	NUM
ma-255	292	20	γm+1	γm+1	NUM
ma-255	292	21	1−	1−	NUM
ma-255	292	22	γ	γ	X
ma-255	292	23	‖y0‖	‖y0‖	PROPN
ma-255	292	24	<	<	X
ma-255	292	25	µ.	µ.	NOUN
ma-255	292	26	if	if	SCONJ
ma-255	292	27	em	em	PRON
ma-255	292	28	∈	∈	PROPN
ma-255	292	29	qdm(xm	qdm(xm	PROPN
ma-255	292	30	)	)	PUNCT
ma-255	292	31	,	,	PUNCT
ma-255	292	32	set	set	VERB
ma-255	292	33	xm+1	xm+1	PROPN
ma-255	292	34	=	=	SYM
ma-255	292	35	xm	xm	PROPN
ma-255	292	36	.	.	PUNCT
ma-255	293	1	otherwise	otherwise	ADV
ma-255	293	2	by	by	ADP
ma-255	293	3	the	the	DET
ma-255	293	4	condition	condition	NOUN
ma-255	293	5	(	(	PUNCT
ma-255	293	6	c4	c4	NOUN
ma-255	293	7	)	)	PUNCT
ma-255	293	8	we	we	PRON
ma-255	293	9	have	have	VERB
ma-255	293	10	dist(xm	dist(xm	NOUN
ma-255	293	11	,	,	PUNCT
ma-255	293	12	q	q	NOUN
ma-255	293	13	−1	−1	NOUN
ma-255	293	14	dm	dm	PROPN
ma-255	293	15	(	(	PUNCT
ma-255	293	16	em	em	NOUN
ma-255	293	17	)	)	PUNCT
ma-255	293	18	)	)	PUNCT
ma-255	293	19	≤	≤	NOUN
ma-255	293	20	β	β	X
ma-255	293	21	dist(em	dist(em	NOUN
ma-255	293	22	,	,	PUNCT
ma-255	293	23	qdm(xn	qdm(xn	NOUN
ma-255	293	24	)	)	PUNCT
ma-255	293	25	)	)	PUNCT
ma-255	293	26	≤	≤	NUM
ma-255	293	27	β1	β1	PROPN
ma-255	293	28	dist(em	dist(em	PROPN
ma-255	293	29	,	,	PUNCT
ma-255	293	30	qdm(xm	qdm(xm	PROPN
ma-255	293	31	)	)	PUNCT
ma-255	293	32	)	)	PUNCT
ma-255	293	33	.	.	PUNCT
ma-255	294	1	next	next	ADV
ma-255	294	2	,	,	PUNCT
ma-255	294	3	there	there	PRON
ma-255	294	4	exists	exist	VERB
ma-255	294	5	an	an	DET
ma-255	294	6	element	element	NOUN
ma-255	294	7	xm+1	xm+1	PROPN
ma-255	294	8	∈	∈	PROPN
ma-255	294	9	q−1dm(em	q−1dm(em	NOUN
ma-255	294	10	)	)	PUNCT
ma-255	294	11	satisfying	satisfy	VERB
ma-255	294	12	‖xm+1	‖xm+1	NUM
ma-255	294	13	−	−	PROPN
ma-255	294	14	xm‖	xm‖	PROPN
ma-255	294	15	≤	≤	PROPN
ma-255	294	16	β1	β1	PROPN
ma-255	294	17	dist(em	dist(em	PROPN
ma-255	294	18	,	,	PUNCT
ma-255	294	19	qdm(xm	qdm(xm	PROPN
ma-255	294	20	)	)	PUNCT
ma-255	294	21	)	)	PUNCT
ma-255	294	22	.	.	PUNCT
ma-255	295	1	by	by	ADP
ma-255	295	2	(	(	PUNCT
ma-255	295	3	iiim	iiim	NOUN
ma-255	295	4	)	)	PUNCT
ma-255	295	5	it	it	PRON
ma-255	295	6	follows	follow	VERB
ma-255	295	7	qdm(xm	qdm(xm	PROPN
ma-255	295	8	)	)	PUNCT
ma-255	296	1	=	=	SYM
ma-255	296	2	f1(x0	f1(x0	PROPN
ma-255	296	3	)	)	PUNCT
ma-255	296	4	+	+	NOUN
ma-255	296	5	dm(xm	dm(xm	PROPN
ma-255	296	6	−	−	PROPN
ma-255	296	7	x0	x0	PROPN
ma-255	296	8	)	)	PUNCT
ma-255	297	1	+	+	CCONJ
ma-255	297	2	f2(xm	f2(xm	X
ma-255	297	3	)	)	PUNCT
ma-255	297	4	3	3	NUM
ma-255	297	5	f1(x0	f1(x0	NOUN
ma-255	297	6	)	)	PUNCT
ma-255	298	1	+	+	PROPN
ma-255	298	2	dm(xm	dm(xm	PROPN
ma-255	298	3	−	−	PROPN
ma-255	298	4	x0)−	x0)−	X
ma-255	298	5	f1(xm−1	f1(xm−1	PROPN
ma-255	299	1	−dm−1(xm	−dm−1(xm	ADP
ma-255	299	2	−	−	PROPN
ma-255	299	3	xm−1	xm−1	PROPN
ma-255	299	4	)	)	PUNCT
ma-255	299	5	.	.	PUNCT
ma-255	300	1	in	in	ADP
ma-255	300	2	view	view	NOUN
ma-255	300	3	of	of	ADP
ma-255	300	4	the	the	DET
ma-255	300	5	condition	condition	NOUN
ma-255	300	6	(	(	PUNCT
ma-255	300	7	c4	c4	NOUN
ma-255	300	8	)	)	PUNCT
ma-255	300	9	with	with	ADP
ma-255	300	10	x	x	X
ma-255	300	11	=	=	SYM
ma-255	300	12	xm	xm	PROPN
ma-255	300	13	,	,	PUNCT
ma-255	300	14	(	(	PUNCT
ma-255	300	15	3.5	3.5	NUM
ma-255	300	16	)	)	PUNCT
ma-255	300	17	and	and	CCONJ
ma-255	300	18	(	(	PUNCT
ma-255	300	19	iim	iim	NOUN
ma-255	300	20	)	)	PUNCT
ma-255	300	21	,	,	PUNCT
ma-255	300	22	we	we	PRON
ma-255	300	23	obtain	obtain	VERB
ma-255	300	24	in	in	ADP
ma-255	300	25	turn	turn	NOUN
ma-255	300	26	that	that	PRON
ma-255	300	27	‖xm+1	‖xm+1	PUNCT
ma-255	301	1	−	−	PROPN
ma-255	301	2	xm‖	xm‖	PROPN
ma-255	301	3	≤	≤	PROPN
ma-255	301	4	β1	β1	NOUN
ma-255	301	5	∥∥em	∥∥em	NOUN
ma-255	301	6	−	−	PROPN
ma-255	302	1	[	[	X
ma-255	302	2	f1(x0)−	f1(x0)−	NOUN
ma-255	302	3	f1(xm−1	f1(xm−1	NOUN
ma-255	302	4	)	)	PUNCT
ma-255	303	1	+	+	NOUN
ma-255	303	2	dm(xm	dm(xm	PROPN
ma-255	303	3	−	−	NOUN
ma-255	303	4	x0	x0	PROPN
ma-255	303	5	)	)	PUNCT
ma-255	303	6	−dm−1(xm	−dm−1(xm	ADP
ma-255	303	7	−	−	PROPN
ma-255	303	8	xm−1	xm−1	PROPN
ma-255	303	9	)	)	PUNCT
ma-255	303	10	]	]	PUNCT
ma-255	303	11	∥∥	∥∥	X
ma-255	303	12	=	=	SYM
ma-255	303	13	β1‖f1(xm)−	β1‖f1(xm)−	PUNCT
ma-255	303	14	f1(xm−1)−dm−1(xm	f1(xm−1)−dm−1(xm	PROPN
ma-255	303	15	−	−	PROPN
ma-255	303	16	xm−1)‖	xm−1)‖	NOUN
ma-255	304	1	≤	≤	ADJ
ma-255	304	2	β1ϕ1(‖xm−1	β1ϕ1(‖xm−1	PROPN
ma-255	304	3	−	−	PROPN
ma-255	304	4	x0‖	x0‖	PROPN
ma-255	304	5	,	,	PUNCT
ma-255	304	6	‖xm	‖xm	PROPN
ma-255	304	7	−	−	PROPN
ma-255	304	8	x0‖	x0‖	PROPN
ma-255	304	9	,	,	PUNCT
ma-255	304	10	‖xm	‖xm	PROPN
ma-255	304	11	−	−	PROPN
ma-255	304	12	xm−1‖	xm−1‖	PROPN
ma-255	304	13	≤	≤	PROPN
ma-255	305	1	β1ϕ1(hm−1	β1ϕ1(hm−1	PROPN
ma-255	305	2	,	,	PUNCT
ma-255	305	3	hm	hm	INTJ
ma-255	305	4	,	,	PUNCT
ma-255	305	5	hm	hm	INTJ
ma-255	305	6	−	−	PROPN
ma-255	305	7	hm−1	hm−1	NOUN
ma-255	305	8	)	)	PUNCT
ma-255	305	9	=	=	PUNCT
ma-255	305	10	hm+1	hm+1	PRON
ma-255	306	1	−	−	NOUN
ma-255	306	2	hm	hm	INTJ
ma-255	306	3	,	,	PUNCT
ma-255	306	4	(	(	PUNCT
ma-255	306	5	3.6	3.6	NUM
ma-255	306	6	)	)	PUNCT
ma-255	306	7	and	and	CCONJ
ma-255	306	8	‖xm+1	‖xm+1	NUM
ma-255	306	9	−	−	PROPN
ma-255	306	10	xm‖	xm‖	PROPN
ma-255	306	11	≤	≤	NOUN
ma-255	307	1	β1δ‖xm	β1δ‖xm	PUNCT
ma-255	307	2	−	−	PROPN
ma-255	307	3	xm−1‖	xm−1‖	PROPN
ma-255	307	4	=	=	PUNCT
ma-255	308	1	γ‖xm	γ‖xm	PROPN
ma-255	308	2	−	−	PROPN
ma-255	308	3	xm−1‖	xm−1‖	PROPN
ma-255	308	4	≤	≤	PROPN
ma-255	308	5	γm‖x1	γm‖x1	NOUN
ma-255	308	6	−	−	PROPN
ma-255	308	7	x0‖	x0‖	PROPN
ma-255	308	8	≤	≤	PROPN
ma-255	309	1	γmβ1‖y0‖.	γmβ1‖y0‖.	PROPN
ma-255	309	2	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	309	3	eur	eur	PROPN
ma-255	309	4	.	.	PUNCT
ma-255	310	1	j.	j.	PROPN
ma-255	310	2	math	math	PROPN
ma-255	310	3	.	.	PUNCT
ma-255	311	1	anal	anal	PROPN
ma-255	311	2	.	.	PUNCT
ma-255	312	1	10.28924	10.28924	NUM
ma-255	312	2	/	/	SYM
ma-255	312	3	ada	ada	PROPN
ma-255	312	4	/	/	SYM
ma-255	312	5	ma.5.5	ma.5.5	PROPN
ma-255	312	6	12thus	12thus	NUM
ma-255	312	7	,	,	PUNCT
ma-255	312	8	the	the	DET
ma-255	312	9	condition	condition	NOUN
ma-255	312	10	(	(	PUNCT
ma-255	312	11	iiim	iiim	NOUN
ma-255	312	12	)	)	PUNCT
ma-255	312	13	holds	hold	VERB
ma-255	312	14	if	if	SCONJ
ma-255	312	15	m	m	PRON
ma-255	312	16	+	+	ADJ
ma-255	312	17	1	1	NUM
ma-255	312	18	replaces	replace	VERB
ma-255	312	19	m.	m.	NOUN
ma-255	312	20	moreover	moreover	ADV
ma-255	312	21	,	,	PUNCT
ma-255	312	22	by	by	ADP
ma-255	312	23	the	the	DET
ma-255	312	24	selection	selection	NOUN
ma-255	312	25	of	of	ADP
ma-255	312	26	xm+1,we	xm+1,we	PROPN
ma-255	312	27	get	get	VERB
ma-255	312	28	em	em	PRON
ma-255	312	29	∈	∈	PROPN
ma-255	312	30	qdm(xm+1	qdm(xm+1	NOUN
ma-255	312	31	)	)	PUNCT
ma-255	313	1	=	=	SYM
ma-255	313	2	f1(x0	f1(x0	X
ma-255	313	3	)	)	PUNCT
ma-255	314	1	+	+	NOUN
ma-255	314	2	dm(xm+1−	dm(xm+1−	ADJ
ma-255	314	3	−	−	NOUN
ma-255	314	4	x0	x0	NUM
ma-255	314	5	)	)	PUNCT
ma-255	314	6	+	+	PUNCT
ma-255	314	7	f2(xm+1	f2(xm+1	NOUN
ma-255	314	8	)	)	PUNCT
ma-255	314	9	.	.	PUNCT
ma-255	315	1	so	so	ADV
ma-255	315	2	,	,	PUNCT
ma-255	315	3	the	the	DET
ma-255	315	4	assertion	assertion	NOUN
ma-255	315	5	(	(	PUNCT
ma-255	315	6	iiin	iiin	PROPN
ma-255	315	7	)	)	PUNCT
ma-255	315	8	holds	hold	VERB
ma-255	315	9	if	if	SCONJ
ma-255	315	10	m	m	PRON
ma-255	315	11	+	+	ADJ
ma-255	315	12	1	1	NUM
ma-255	315	13	replaces	replace	VERB
ma-255	315	14	m.	m.	NOUN
ma-255	315	15	moreover	moreover	ADV
ma-255	315	16	,	,	PUNCT
ma-255	315	17	by	by	ADP
ma-255	315	18	(	(	PUNCT
ma-255	315	19	i	i	NOUN
ma-255	315	20	m	m	VERB
ma-255	315	21	)	)	PUNCT
ma-255	315	22	we	we	PRON
ma-255	315	23	get	get	VERB
ma-255	315	24	in	in	ADP
ma-255	315	25	turn	turn	NOUN
ma-255	315	26	that	that	PRON
ma-255	315	27	‖xm+1	‖xm+1	PUNCT
ma-255	315	28	−	−	PROPN
ma-255	315	29	x0‖	x0‖	PROPN
ma-255	315	30	≤	≤	PROPN
ma-255	315	31	‖xm+1	‖xm+1	PUNCT
ma-255	316	1	−	−	PROPN
ma-255	316	2	xm‖+	xm‖+	PROPN
ma-255	317	1	‖xm	‖xm	PROPN
ma-255	317	2	−	−	PROPN
ma-255	317	3	x0‖	x0‖	PROPN
ma-255	317	4	≤	≤	PROPN
ma-255	317	5	γmβ1‖y0‖+	γmβ1‖y0‖+	X
ma-255	317	6	1−	1−	NUM
ma-255	318	1	γm	γm	NOUN
ma-255	318	2	1−	1−	NUM
ma-255	318	3	γ	γ	X
ma-255	318	4	β1‖y0‖	β1‖y0‖	PROPN
ma-255	319	1	=	=	SYM
ma-255	320	1	1−	1−	NUM
ma-255	321	1	γm+1	γm+1	NUM
ma-255	321	2	1−	1−	NUM
ma-255	321	3	γ	γ	PROPN
ma-255	321	4	β1‖y0‖	β1‖y0‖	PROPN
ma-255	321	5	,	,	PUNCT
ma-255	321	6	which	which	PRON
ma-255	321	7	terminates	terminate	VERB
ma-255	321	8	the	the	DET
ma-255	321	9	induction	induction	NOUN
ma-255	321	10	for	for	ADP
ma-255	321	11	(	(	PUNCT
ma-255	321	12	i	i	NOUN
ma-255	321	13	m	m	PROPN
ma-255	321	14	)	)	PUNCT
ma-255	321	15	.	.	PUNCT
ma-255	322	1	then	then	ADV
ma-255	322	2	,	,	PUNCT
ma-255	322	3	by	by	ADP
ma-255	322	4	(	(	PUNCT
ma-255	322	5	c2	c2	PROPN
ma-255	322	6	)	)	PUNCT
ma-255	322	7	the	the	DET
ma-255	322	8	sequence	sequence	NOUN
ma-255	322	9	is	be	AUX
ma-255	322	10	complete	complete	ADJ
ma-255	322	11	as	as	ADP
ma-255	322	12	con	con	NOUN
ma-255	322	13	-	-	PUNCT
ma-255	322	14	vergent	vergent	NOUN
ma-255	322	15	.	.	PUNCT
ma-255	323	1	it	it	PRON
ma-255	323	2	follows	follow	VERB
ma-255	323	3	by	by	ADP
ma-255	323	4	(	(	PUNCT
ma-255	323	5	iim	iim	NOUN
ma-255	323	6	)	)	PUNCT
ma-255	323	7	that	that	SCONJ
ma-255	323	8	the	the	DET
ma-255	323	9	sequence	sequence	NOUN
ma-255	323	10	{	{	PUNCT
ma-255	323	11	xm	xm	NOUN
ma-255	323	12	}	}	PUNCT
ma-255	323	13	is	be	AUX
ma-255	323	14	also	also	ADV
ma-255	323	15	complete	complete	ADJ
ma-255	323	16	in	in	ADP
ma-255	323	17	a	a	DET
ma-255	323	18	banach	banach	NOUN
ma-255	323	19	space	space	NOUN
ma-255	323	20	xand	xand	ADV
ma-255	323	21	as	as	ADP
ma-255	323	22	such	such	ADJ
ma-255	323	23	it	it	PRON
ma-255	323	24	converges	converge	VERB
ma-255	323	25	to	to	ADP
ma-255	323	26	some	some	PRON
ma-255	323	27	s∗.	s∗.	ADJ
ma-255	323	28	by	by	ADP
ma-255	323	29	(	(	PUNCT
ma-255	323	30	i	i	NOUN
ma-255	323	31	m	m	VERB
ma-255	323	32	)	)	PUNCT
ma-255	324	1	we	we	PRON
ma-255	324	2	have	have	AUX
ma-255	324	3	s∗	s∗	PROPN
ma-255	324	4	∈	∈	PROPN
ma-255	324	5	s(x0	s(x0	NOUN
ma-255	324	6	,	,	PUNCT
ma-255	324	7	λ	λ	PROPN
ma-255	324	8	)	)	PUNCT
ma-255	324	9	.	.	PUNCT
ma-255	325	1	we	we	PRON
ma-255	325	2	must	must	AUX
ma-255	325	3	show	show	VERB
ma-255	325	4	thatthe	thatthe	NOUN
ma-255	325	5	limit	limit	NOUN
ma-255	325	6	point	point	NOUN
ma-255	325	7	s∗	s∗	PROPN
ma-255	325	8	solves	solve	NOUN
ma-255	325	9	(	(	PUNCT
ma-255	325	10	1.8).set	1.8).set	NUM
ma-255	325	11	ym	ym	NOUN
ma-255	325	12	:	:	PUNCT
ma-255	325	13	=	=	PUNCT
ma-255	325	14	f1(xm	f1(xm	PROPN
ma-255	325	15	)	)	PUNCT
ma-255	325	16	−	−	PROPN
ma-255	325	17	f1(xm−1	f1(xm−1	NUM
ma-255	325	18	)	)	PUNCT
ma-255	326	1	−	−	PROPN
ma-255	326	2	dm−1(xm	dm−1(xm	NOUN
ma-255	326	3	−	−	PROPN
ma-255	326	4	xm−1	xm−1	PROPN
ma-255	326	5	)	)	PUNCT
ma-255	326	6	.	.	PUNCT
ma-255	327	1	it	it	PRON
ma-255	327	2	follows	follow	VERB
ma-255	327	3	by	by	ADP
ma-255	327	4	(	(	PUNCT
ma-255	327	5	iiim	iiim	NOUN
ma-255	327	6	)	)	PUNCT
ma-255	327	7	that	that	SCONJ
ma-255	327	8	ym	ym	PROPN
ma-255	327	9	∈	∈	PROPN
ma-255	327	10	f1(xm	f1(xm	PROPN
ma-255	327	11	)	)	PUNCT
ma-255	327	12	+	+	CCONJ
ma-255	327	13	f2(xm	f2(xm	PROPN
ma-255	327	14	)	)	PUNCT
ma-255	327	15	.	.	PUNCT
ma-255	328	1	by	by	ADP
ma-255	328	2	using	use	VERB
ma-255	328	3	(	(	PUNCT
ma-255	328	4	iim	iim	NOUN
ma-255	328	5	)	)	PUNCT
ma-255	328	6	and	and	CCONJ
ma-255	328	7	the	the	DET
ma-255	328	8	condition	condition	NOUN
ma-255	328	9	(	(	PUNCT
ma-255	328	10	c4	c4	NOUN
ma-255	328	11	)	)	PUNCT
ma-255	328	12	for	for	ADP
ma-255	328	13	x	x	X
ma-255	328	14	=	=	SYM
ma-255	328	15	xm	xm	PROPN
ma-255	328	16	,	,	PUNCT
ma-255	328	17	we	we	PRON
ma-255	328	18	get	get	VERB
ma-255	328	19	in	in	ADP
ma-255	328	20	turnthat	turnthat	NOUN
ma-255	328	21	‖ym‖	‖ym‖	PROPN
ma-255	328	22	=	=	PUNCT
ma-255	328	23	‖f1(xm)−	‖f1(xm)−	PROPN
ma-255	329	1	f1(xm−1)−dm−1(xm	f1(xm−1)−dm−1(xm	PROPN
ma-255	329	2	−	−	PROPN
ma-255	329	3	xm−1)‖	xm−1)‖	PROPN
ma-255	329	4	≤	≤	PROPN
ma-255	329	5	ψ1(‖xm−1	ψ1(‖xm−1	PROPN
ma-255	329	6	−	−	PROPN
ma-255	329	7	x0‖	x0‖	PROPN
ma-255	329	8	,	,	PUNCT
ma-255	329	9	‖xm	‖xm	PROPN
ma-255	329	10	−	−	PROPN
ma-255	329	11	x0‖	x0‖	PROPN
ma-255	329	12	,	,	PUNCT
ma-255	329	13	‖xm	‖xm	PROPN
ma-255	329	14	−	−	PROPN
ma-255	330	1	xm−1‖)‖xm	xm−1‖)‖xm	PROPN
ma-255	330	2	−	−	PROPN
ma-255	330	3	xm−1‖	xm−1‖	PROPN
ma-255	330	4	≤	≤	PROPN
ma-255	330	5	ψ1(hm−1	ψ1(hm−1	PROPN
ma-255	330	6	,	,	PUNCT
ma-255	330	7	hm	hm	INTJ
ma-255	330	8	,	,	PUNCT
ma-255	330	9	hm	hm	INTJ
ma-255	330	10	−	−	PROPN
ma-255	330	11	hm−1)(αm	hm−1)(αm	PROPN
ma-255	330	12	−	−	PROPN
ma-255	330	13	αm−1	αm−1	NOUN
ma-255	330	14	)	)	PUNCT
ma-255	330	15	=	=	PUNCT
ma-255	331	1	αm+1	αm+1	X
ma-255	331	2	−	−	NOUN
ma-255	331	3	αm	αm	NOUN
ma-255	331	4	−→	−→	NOUN
ma-255	331	5	0	0	PUNCT
ma-255	332	1	as	as	SCONJ
ma-255	332	2	m	m	PROPN
ma-255	332	3	→	→	SYM
ma-255	332	4	+	+	ADV
ma-255	332	5	∞.consequently	∞.consequently	ADV
ma-255	332	6	,	,	PUNCT
ma-255	332	7	we	we	PRON
ma-255	332	8	deduce	deduce	VERB
ma-255	332	9	that	that	DET
ma-255	332	10	(	(	PUNCT
ma-255	332	11	xm	xm	NOUN
ma-255	332	12	,	,	PUNCT
ma-255	332	13	ym)→	ym)→	NUM
ma-255	332	14	(	(	PUNCT
ma-255	332	15	s∗	s∗	PROPN
ma-255	332	16	,	,	PUNCT
ma-255	332	17	0	0	NUM
ma-255	332	18	)	)	PUNCT
ma-255	332	19	as	as	SCONJ
ma-255	332	20	m	m	PROPN
ma-255	332	21	→	→	SYM
ma-255	332	22	+	+	PROPN
ma-255	332	23	∞.but	∞.but	PROPN
ma-255	332	24	f	f	PROPN
ma-255	332	25	is	be	AUX
ma-255	332	26	continuous	continuous	ADJ
ma-255	332	27	whereas	whereas	SCONJ
ma-255	332	28	g	g	PROPN
ma-255	332	29	has	have	VERB
ma-255	332	30	a	a	DET
ma-255	332	31	closed	closed	ADJ
ma-255	332	32	graph	graph	NOUN
ma-255	332	33	.	.	PUNCT
ma-255	333	1	thus	thus	ADV
ma-255	333	2	,	,	PUNCT
ma-255	333	3	we	we	PRON
ma-255	333	4	conclude	conclude	VERB
ma-255	333	5	0	0	NUM
ma-255	333	6	∈	∈	NOUN
ma-255	333	7	f1(s∗	f1(s∗	PUNCT
ma-255	333	8	)	)	PUNCT
ma-255	334	1	+	+	NOUN
ma-255	334	2	f2(s	f2(s	PROPN
ma-255	334	3	∗).finally	∗).finally	ADV
ma-255	334	4	,	,	PUNCT
ma-255	334	5	if	if	SCONJ
ma-255	334	6	qdm	qdm	NOUN
ma-255	334	7	is	be	AUX
ma-255	334	8	a	a	DET
ma-255	334	9	strongly	strongly	ADV
ma-255	334	10	metrically	metrically	ADV
ma-255	334	11	regular	regular	ADJ
ma-255	334	12	operator	operator	NOUN
ma-255	334	13	.	.	PUNCT
ma-255	335	1	it	it	PRON
ma-255	335	2	follows	follow	VERB
ma-255	335	3	that	that	SCONJ
ma-255	335	4	the	the	DET
ma-255	335	5	iterate	iterate	NOUN
ma-255	335	6	xm+1	xm+1	PROPN
ma-255	335	7	isunique	isunique	NOUN
ma-255	335	8	and	and	CCONJ
ma-255	335	9	is	be	AUX
ma-255	335	10	obtained	obtain	VERB
ma-255	335	11	from	from	ADP
ma-255	335	12	xm	xm	PROPN
ma-255	335	13	.	.	PUNCT
ma-255	336	1	�	�	PROPN
ma-255	336	2	next	next	ADV
ma-255	336	3	,	,	PUNCT
ma-255	336	4	we	we	PRON
ma-255	336	5	develop	develop	VERB
ma-255	336	6	the	the	DET
ma-255	336	7	local	local	ADJ
ma-255	336	8	convergence	convergence	NOUN
ma-255	336	9	analysis	analysis	NOUN
ma-255	336	10	of	of	ADP
ma-255	336	11	the	the	DET
ma-255	336	12	method	method	NOUN
ma-255	336	13	(	(	PUNCT
ma-255	336	14	1.10).suppose	1.10).suppose	NUM
ma-255	336	15	:(	:(	SYM
ma-255	337	1	c6	c6	PROPN
ma-255	337	2	)	)	PUNCT
ma-255	337	3	there	there	PRON
ma-255	337	4	exists	exist	VERB
ma-255	337	5	a	a	DET
ma-255	337	6	parameter	parameter	NOUN
ma-255	337	7	β	β	X
ma-255	337	8	>	>	X
ma-255	337	9	1	1	NUM
ma-255	337	10	and	and	CCONJ
ma-255	337	11	a	a	DET
ma-255	337	12	function	function	NOUN
ma-255	337	13	ϕ2	ϕ2	ADV
ma-255	337	14	:	:	PUNCT
ma-255	337	15	r+	r+	NOUN
ma-255	337	16	→	→	PUNCT
ma-255	337	17	r+	r+	NOUN
ma-255	337	18	which	which	PRON
ma-255	337	19	is	be	AUX
ma-255	337	20	continuous	continuous	ADJ
ma-255	337	21	andnondecreasing	andnondecrease	VERB
ma-255	337	22	such	such	ADJ
ma-255	337	23	that	that	SCONJ
ma-255	337	24	the	the	DET
ma-255	337	25	equation	equation	NOUN
ma-255	337	26	β1ϕ2(t	β1ϕ2(t	PRON
ma-255	337	27	)	)	PUNCT
ma-255	337	28	−	−	PROPN
ma-255	338	1	1	1	NUM
ma-255	338	2	=	=	SYM
ma-255	338	3	0	0	PROPN
ma-255	338	4	has	have	VERB
ma-255	338	5	a	a	DET
ma-255	338	6	smallest	small	ADJ
ma-255	338	7	positive	positive	ADJ
ma-255	338	8	solution.denote	solution.denote	NOUN
ma-255	338	9	such	such	DET
ma-255	338	10	a	a	DET
ma-255	338	11	solution	solution	NOUN
ma-255	338	12	by	by	ADP
ma-255	338	13	r0.(c7	r0.(c7	PROPN
ma-255	338	14	)	)	PUNCT
ma-255	338	15	there	there	PRON
ma-255	338	16	exists	exist	VERB
ma-255	338	17	a	a	DET
ma-255	338	18	solution	solution	NOUN
ma-255	338	19	s∗	s∗	PROPN
ma-255	338	20	∈	∈	PROPN
ma-255	338	21	x	x	PUNCT
ma-255	338	22	of	of	ADP
ma-255	338	23	the	the	DET
ma-255	338	24	generalized	generalized	ADJ
ma-255	338	25	equation	equation	NOUN
ma-255	338	26	(	(	PUNCT
ma-255	338	27	1.8	1.8	NUM
ma-255	338	28	)	)	PUNCT
ma-255	338	29	and	and	CCONJ
ma-255	338	30	parameters	parameter	NOUN
ma-255	338	31	β	β	VERB
ma-255	338	32	>	>	X
ma-255	338	33	0	0	PUNCT
ma-255	338	34	and	and	CCONJ
ma-255	338	35	δ	δ	PROPN
ma-255	338	36	≥	≥	X
ma-255	338	37	0	0	NUM
ma-255	338	38	such	such	ADJ
ma-255	338	39	that	that	SCONJ
ma-255	338	40	βδ	βδ	ADP
ma-255	338	41	<	<	X
ma-255	338	42	1.(c8	1.(c8	NUM
ma-255	338	43	)	)	PUNCT
ma-255	338	44	the	the	DET
ma-255	338	45	operator	operator	NOUN
ma-255	338	46	x	x	PUNCT
ma-255	338	47	−→	−→	NOUN
ma-255	338	48	tdn(x	tdn(x	PROPN
ma-255	338	49	)	)	PUNCT
ma-255	338	50	:	:	PUNCT
ma-255	339	1	=	=	PUNCT
ma-255	339	2	f1(s	f1(s	NUM
ma-255	339	3	∗	∗	NOUN
ma-255	339	4	)	)	PUNCT
ma-255	339	5	+	+	NUM
ma-255	339	6	dn(x	dn(x	X
ma-255	339	7	−	−	PROPN
ma-255	339	8	s∗	s∗	PROPN
ma-255	339	9	)	)	PUNCT
ma-255	339	10	+	+	CCONJ
ma-255	340	1	f2(x	f2(x	X
ma-255	340	2	)	)	PUNCT
ma-255	340	3	is	be	AUX
ma-255	340	4	a	a	DET
ma-255	340	5	metrically	metrically	ADV
ma-255	340	6	regular	regular	ADJ
ma-255	340	7	at	at	ADP
ma-255	340	8	s∗for	s∗for	PROPN
ma-255	340	9	0	0	NUM
ma-255	340	10	with	with	ADP
ma-255	340	11	constant	constant	ADJ
ma-255	340	12	β	β	NOUN
ma-255	340	13	and	and	CCONJ
ma-255	340	14	neighbourhoods	neighbourhood	NOUN
ma-255	340	15	s(s∗	s(s∗	NUM
ma-255	340	16	,	,	PUNCT
ma-255	340	17	λ	λ	NOUN
ma-255	340	18	)	)	PUNCT
ma-255	340	19	and	and	CCONJ
ma-255	340	20	(	(	PUNCT
ma-255	340	21	0	0	NUM
ma-255	340	22	,	,	PUNCT
ma-255	340	23	µ	µ	NOUN
ma-255	340	24	)	)	PUNCT
ma-255	340	25	for	for	ADP
ma-255	340	26	some	some	DET
ma-255	340	27	µ	µ	X
ma-255	340	28	>	>	X
ma-255	340	29	0	0	PROPN
ma-255	340	30	,	,	PUNCT
ma-255	340	31	respectively.the	respectively.the	DET
ma-255	340	32	function	function	NOUN
ma-255	340	33	ϕ2	ϕ2	ADV
ma-255	340	34	relates	relate	VERB
ma-255	340	35	to	to	ADP
ma-255	340	36	the	the	DET
ma-255	340	37	operators	operator	NOUN
ma-255	340	38	on	on	ADP
ma-255	340	39	the	the	DET
ma-255	340	40	method	method	NOUN
ma-255	340	41	(	(	PUNCT
ma-255	340	42	1.10	1.10	NUM
ma-255	340	43	)	)	PUNCT
ma-255	340	44	.	.	PUNCT
ma-255	341	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	341	2	eur	eur	PROPN
ma-255	341	3	.	.	PUNCT
ma-255	342	1	j.	j.	PROPN
ma-255	342	2	math	math	PROPN
ma-255	342	3	.	.	PUNCT
ma-255	343	1	anal	anal	PROPN
ma-255	343	2	.	.	PUNCT
ma-255	344	1	10.28924	10.28924	NUM
ma-255	344	2	/	/	SYM
ma-255	344	3	ada	ada	PROPN
ma-255	344	4	/	/	SYM
ma-255	344	5	ma.5.5	ma.5.5	PROPN
ma-255	344	6	13(c9	13(c9	NUM
ma-255	344	7	)	)	PUNCT
ma-255	344	8	‖f1(s∗)−	‖f1(s∗)−	NOUN
ma-255	344	9	f1(xn)−dn(s∗	f1(xn)−dn(s∗	ADJ
ma-255	344	10	−	−	PUNCT
ma-255	344	11	xn)‖	xn)‖	PROPN
ma-255	344	12	≤	≤	PROPN
ma-255	344	13	ϕ2(‖s∗	ϕ2(‖s∗	PROPN
ma-255	345	1	−	−	PROPN
ma-255	345	2	xn‖)‖s∗	xn‖)‖s∗	PROPN
ma-255	346	1	−	−	PROPN
ma-255	346	2	xn‖	xn‖	PROPN
ma-255	346	3	(	(	PUNCT
ma-255	346	4	3.7	3.7	NUM
ma-255	346	5	)	)	PUNCT
ma-255	346	6	for	for	ADP
ma-255	346	7	x0	x0	PROPN
ma-255	346	8	∈	∈	PROPN
ma-255	346	9	s(s∗	s(s∗	X
ma-255	346	10	,	,	PUNCT
ma-255	346	11	λ	λ	NOUN
ma-255	346	12	)	)	PUNCT
ma-255	346	13	.	.	PUNCT
ma-255	347	1	next	next	ADV
ma-255	347	2	,	,	PUNCT
ma-255	347	3	the	the	DET
ma-255	347	4	local	local	ADJ
ma-255	347	5	convergence	convergence	NOUN
ma-255	347	6	analysis	analysis	NOUN
ma-255	347	7	of	of	ADP
ma-255	347	8	the	the	DET
ma-255	347	9	method	method	NOUN
ma-255	347	10	(	(	PUNCT
ma-255	347	11	1.10	1.10	NUM
ma-255	347	12	)	)	PUNCT
ma-255	347	13	is	be	AUX
ma-255	347	14	presentedusing	presenteduse	VERB
ma-255	347	15	the	the	DET
ma-255	347	16	conditions	condition	NOUN
ma-255	347	17	(	(	PUNCT
ma-255	347	18	c6)−	c6)−	NOUN
ma-255	347	19	(	(	PUNCT
ma-255	347	20	c4	c4	NOUN
ma-255	347	21	)	)	PUNCT
ma-255	347	22	.	.	PUNCT
ma-255	348	1	theorem	theorem	ADJ
ma-255	348	2	3.2	3.2	NUM
ma-255	348	3	.	.	PUNCT
ma-255	349	1	suppose	suppose	VERB
ma-255	349	2	that	that	SCONJ
ma-255	349	3	the	the	DET
ma-255	349	4	conditions	condition	NOUN
ma-255	349	5	(	(	PUNCT
ma-255	349	6	c6)−	c6)−	NOUN
ma-255	349	7	(	(	PUNCT
ma-255	349	8	c9	c9	NOUN
ma-255	349	9	)	)	PUNCT
ma-255	349	10	hold	hold	VERB
ma-255	349	11	.	.	PUNCT
ma-255	350	1	then	then	ADV
ma-255	350	2	,	,	PUNCT
ma-255	350	3	for	for	ADP
ma-255	350	4	each	each	DET
ma-255	350	5	γ	γ	X
ma-255	350	6	∈	∈	PROPN
ma-255	350	7	(	(	PUNCT
ma-255	350	8	βδ	βδ	NOUN
ma-255	350	9	,	,	PUNCT
ma-255	350	10	1	1	NUM
ma-255	350	11	)	)	PUNCT
ma-255	350	12	the	the	DET
ma-255	350	13	re	re	NOUN
ma-255	350	14	exists	exist	VERB
ma-255	350	15	a	a	DET
ma-255	350	16	sequence	sequence	NOUN
ma-255	350	17	{	{	PUNCT
ma-255	350	18	xn	xn	NOUN
ma-255	350	19	}	}	PUNCT
ma-255	350	20	generated	generate	VERB
ma-255	350	21	by	by	ADP
ma-255	350	22	the	the	DET
ma-255	350	23	method	method	NOUN
ma-255	350	24	(	(	PUNCT
ma-255	350	25	1.10	1.10	NUM
ma-255	350	26	)	)	PUNCT
ma-255	350	27	which	which	PRON
ma-255	350	28	is	be	AUX
ma-255	350	29	well	well	ADV
ma-255	350	30	defined	define	VERB
ma-255	350	31	in	in	ADP
ma-255	350	32	s(s∗	s(s∗	ADJ
ma-255	350	33	,	,	PUNCT
ma-255	350	34	λ	λ	NOUN
ma-255	350	35	)	)	PUNCT
ma-255	350	36	,	,	PUNCT
ma-255	350	37	remains	remain	VERB
ma-255	350	38	in	in	ADP
ma-255	350	39	s(s∗	s(s∗	ADJ
ma-255	350	40	,	,	PUNCT
ma-255	350	41	λ	λ	NOUN
ma-255	350	42	)	)	PUNCT
ma-255	350	43	for	for	ADP
ma-255	350	44	each	each	DET
ma-255	350	45	n	n	NOUN
ma-255	350	46	=	=	SYM
ma-255	350	47	0	0	NUM
ma-255	350	48	,	,	PUNCT
ma-255	350	49	1	1	NUM
ma-255	350	50	,	,	PUNCT
ma-255	350	51	2	2	NUM
ma-255	350	52	,	,	PUNCT
ma-255	350	53	..	..	PUNCT
ma-255	350	54	and	and	CCONJ
ma-255	350	55	converges	converge	VERB
ma-255	350	56	to	to	PART
ma-255	350	57	s∗	s∗	VERB
ma-255	350	58	so	so	SCONJ
ma-255	350	59	that	that	SCONJ
ma-255	350	60	‖s∗	‖s∗	PUNCT
ma-255	351	1	−	−	NOUN
ma-255	351	2	xn‖	xn‖	PROPN
ma-255	351	3	≤	≤	PROPN
ma-255	351	4	d‖s∗	d‖s∗	PROPN
ma-255	351	5	−	−	PROPN
ma-255	351	6	xn−1‖	xn−1‖	PROPN
ma-255	351	7	≤	≤	PROPN
ma-255	351	8	dn‖s∗	dn‖s∗	VERB
ma-255	352	1	−	−	PROPN
ma-255	352	2	x0‖	x0‖	PROPN
ma-255	352	3	<	<	X
ma-255	352	4	λ	λ	PROPN
ma-255	352	5	,	,	PUNCT
ma-255	352	6	where	where	SCONJ
ma-255	352	7	d	d	NOUN
ma-255	352	8	=	=	SYM
ma-255	352	9	β1ϕ2(‖s∗−x0)‖	β1ϕ2(‖s∗−x0)‖	NOUN
ma-255	352	10	∈	∈	PROPN
ma-255	353	1	[	[	X
ma-255	353	2	0	0	NUM
ma-255	353	3	,	,	PUNCT
ma-255	353	4	1	1	NUM
ma-255	353	5	)	)	PUNCT
ma-255	353	6	.	.	PUNCT
ma-255	354	1	additionally	additionally	ADV
ma-255	354	2	,	,	PUNCT
ma-255	354	3	if	if	SCONJ
ma-255	354	4	the	the	DET
ma-255	354	5	operator	operator	NOUN
ma-255	354	6	tdn	tdn	NOUN
ma-255	354	7	is	be	AUX
ma-255	354	8	strongly	strongly	ADV
ma-255	354	9	metrically	metrically	ADV
ma-255	354	10	regular	regular	ADJ
ma-255	354	11	with	with	ADP
ma-255	354	12	constant	constant	ADJ
ma-255	354	13	β	β	NOUN
ma-255	354	14	and	and	CCONJ
ma-255	354	15	neighbourhoods	neighbourhood	NOUN
ma-255	354	16	s(s∗	s(s∗	NUM
ma-255	354	17	,	,	PUNCT
ma-255	354	18	λ	λ	NOUN
ma-255	354	19	)	)	PUNCT
ma-255	354	20	and	and	CCONJ
ma-255	354	21	s(0	s(0	PROPN
ma-255	354	22	,	,	PUNCT
ma-255	354	23	µ	µ	NOUN
ma-255	354	24	)	)	PUNCT
ma-255	354	25	,	,	PUNCT
ma-255	354	26	respectively	respectively	ADV
ma-255	354	27	,	,	PUNCT
ma-255	354	28	then	then	ADV
ma-255	354	29	the	the	DET
ma-255	354	30	sequence	sequence	NOUN
ma-255	354	31	{	{	PUNCT
ma-255	354	32	xn	xn	PROPN
ma-255	354	33	}	}	PUNCT
ma-255	354	34	is	be	AUX
ma-255	354	35	the	the	DET
ma-255	354	36	only	only	ADJ
ma-255	354	37	one	one	NUM
ma-255	354	38	satisfying	satisfying	ADJ
ma-255	354	39	(	(	PUNCT
ma-255	354	40	1.8	1.8	NUM
ma-255	354	41	)	)	PUNCT
ma-255	354	42	,	,	PUNCT
ma-255	354	43	and	and	CCONJ
ma-255	354	44	satisfying	satisfy	VERB
ma-255	354	45	in	in	ADP
ma-255	354	46	(	(	PUNCT
ma-255	354	47	s∗	s∗	PROPN
ma-255	354	48	,	,	PUNCT
ma-255	354	49	λ	λ	NOUN
ma-255	354	50	)	)	PUNCT
ma-255	354	51	.	.	PUNCT
ma-255	355	1	proof	proof	NOUN
ma-255	355	2	.	.	PUNCT
ma-255	356	1	simply	simply	ADV
ma-255	356	2	follow	follow	VERB
ma-255	356	3	the	the	DET
ma-255	356	4	proof	proof	NOUN
ma-255	356	5	of	of	ADP
ma-255	356	6	theorem	theorem	ADJ
ma-255	356	7	3.1	3.1	NUM
ma-255	356	8	for	for	ADP
ma-255	356	9	γ	γ	X
ma-255	356	10	∈	∈	PROPN
ma-255	356	11	(	(	PUNCT
ma-255	356	12	βδ	βδ	NOUN
ma-255	356	13	,	,	PUNCT
ma-255	356	14	1)x0	1)x0	NUM
ma-255	356	15	=	=	SYM
ma-255	356	16	s∗	s∗	PROPN
ma-255	356	17	and	and	CCONJ
ma-255	356	18	β	β	X
ma-255	356	19	<	<	X
ma-255	356	20	β1	β1	PROPN
ma-255	356	21	≤	≤	NUM
ma-255	356	22	γ	γ	PROPN
ma-255	356	23	δ	δ	PROPN
ma-255	356	24	toobtain	toobtain	NOUN
ma-255	356	25	as	as	ADP
ma-255	356	26	in	in	ADP
ma-255	356	27	(	(	PUNCT
ma-255	356	28	3.6	3.6	NUM
ma-255	356	29	)	)	PUNCT
ma-255	356	30	but	but	CCONJ
ma-255	356	31	using	use	VERB
ma-255	356	32	(	(	PUNCT
ma-255	356	33	c8	c8	PROPN
ma-255	356	34	)	)	PUNCT
ma-255	356	35	and	and	CCONJ
ma-255	356	36	(	(	PUNCT
ma-255	356	37	3.7	3.7	NUM
ma-255	356	38	)	)	PUNCT
ma-255	356	39	instead	instead	ADV
ma-255	356	40	of	of	ADP
ma-255	356	41	(	(	PUNCT
ma-255	356	42	c4	c4	NOUN
ma-255	356	43	)	)	PUNCT
ma-255	356	44	and	and	CCONJ
ma-255	356	45	(	(	PUNCT
ma-255	356	46	3.6	3.6	NUM
ma-255	356	47	)	)	PUNCT
ma-255	356	48	,	,	PUNCT
ma-255	356	49	respectively	respectively	ADV
ma-255	356	50	to	to	PART
ma-255	356	51	obtain	obtain	VERB
ma-255	356	52	‖s∗	‖s∗	PUNCT
ma-255	356	53	−	−	PUNCT
ma-255	356	54	xm‖	xm‖	PROPN
ma-255	356	55	≤	≤	PROPN
ma-255	356	56	β1ϕ2(‖s∗	β1ϕ2(‖s∗	PUNCT
ma-255	357	1	−	−	PROPN
ma-255	358	1	xm−1‖)‖s∗	xm−1‖)‖s∗	PROPN
ma-255	358	2	−	−	PROPN
ma-255	358	3	xm−1‖	xm−1‖	PROPN
ma-255	358	4	≤	≤	PROPN
ma-255	358	5	d‖s∗	d‖s∗	PROPN
ma-255	359	1	−	−	PROPN
ma-255	359	2	xm−1‖	xm−1‖	PROPN
ma-255	359	3	≤	≤	PROPN
ma-255	359	4	...	...	PUNCT
ma-255	360	1	≤	≤	NUM
ma-255	360	2	dm‖s∗	dm‖s∗	VERB
ma-255	361	1	−	−	PROPN
ma-255	361	2	x0‖	x0‖	PROPN
ma-255	361	3	<	<	X
ma-255	361	4	λ	λ	PROPN
ma-255	361	5	.	.	PUNCT
ma-255	361	6	therefore	therefore	ADV
ma-255	361	7	,	,	PUNCT
ma-255	361	8	we	we	PRON
ma-255	361	9	conclude	conclude	VERB
ma-255	361	10	that	that	SCONJ
ma-255	361	11	lim	lim	PROPN
ma-255	361	12	m−→+∞xm	m−→+∞xm	PROPN
ma-255	361	13	=	=	PRON
ma-255	361	14	s∗	s∗	PROPN
ma-255	361	15	and	and	CCONJ
ma-255	361	16	the	the	DET
ma-255	361	17	iterate	iterate	NOUN
ma-255	361	18	xm	xm	PROPN
ma-255	361	19	∈	∈	PROPN
ma-255	361	20	s(s∗	s(s∗	X
ma-255	361	21	,	,	PUNCT
ma-255	361	22	λ).finally	λ).finally	ADV
ma-255	361	23	,	,	PUNCT
ma-255	361	24	if	if	SCONJ
ma-255	361	25	the	the	DET
ma-255	361	26	operator	operator	NOUN
ma-255	361	27	tdm	tdm	NOUN
ma-255	361	28	is	be	AUX
ma-255	361	29	strongly	strongly	ADV
ma-255	361	30	metrically	metrically	ADV
ma-255	361	31	regular	regular	ADJ
ma-255	361	32	,	,	PUNCT
ma-255	361	33	it	it	PRON
ma-255	361	34	follows	follow	VERB
ma-255	361	35	that	that	SCONJ
ma-255	361	36	the	the	DET
ma-255	361	37	iterate	iterate	NOUN
ma-255	361	38	xmis	xmis	PROPN
ma-255	361	39	unique	unique	ADJ
ma-255	361	40	in	in	ADP
ma-255	361	41	s(s∗	s(s∗	ADJ
ma-255	361	42	,	,	PUNCT
ma-255	361	43	λ	λ	NOUN
ma-255	361	44	)	)	PUNCT
ma-255	361	45	by	by	ADP
ma-255	361	46	the	the	DET
ma-255	361	47	way	way	NOUN
ma-255	361	48	the	the	DET
ma-255	361	49	iterate	iterate	NOUN
ma-255	361	50	xm	xm	PROPN
ma-255	361	51	is	be	AUX
ma-255	361	52	derived	derive	VERB
ma-255	361	53	from	from	ADP
ma-255	361	54	xm−1	xm−1	PROPN
ma-255	361	55	.	.	PUNCT
ma-255	362	1	�	�	PROPN
ma-255	362	2	4	4	NUM
ma-255	362	3	.	.	PUNCT
ma-255	362	4	numerical	numerical	ADJ
ma-255	362	5	examples	example	NOUN
ma-255	362	6	the	the	DET
ma-255	362	7	examples	example	NOUN
ma-255	362	8	use	use	VERB
ma-255	362	9	ln	ln	NOUN
ma-255	362	10	=	=	PUNCT
ma-255	362	11	f	f	NOUN
ma-255	362	12	′1(xn	′1(xn	NOUN
ma-255	362	13	)	)	PUNCT
ma-255	362	14	,	,	PUNCT
ma-255	362	15	γ	γ	X
ma-255	362	16	=	=	VERB
ma-255	362	17	i	i	PROPN
ma-255	362	18	which	which	PRON
ma-255	362	19	is	be	AUX
ma-255	362	20	independent	independent	ADJ
ma-255	362	21	of	of	ADP
ma-255	362	22	x0	x0	PROPN
ma-255	362	23	and	and	CCONJ
ma-255	362	24	s∗.	s∗.	ADJ
ma-255	362	25	example	example	NOUN
ma-255	362	26	4.1	4.1	NUM
ma-255	362	27	.	.	PUNCT
ma-255	363	1	the	the	DET
ma-255	363	2	solution	solution	NOUN
ma-255	363	3	sought	seek	VERB
ma-255	363	4	for	for	ADP
ma-255	363	5	the	the	DET
ma-255	363	6	nonlinear	nonlinear	ADJ
ma-255	363	7	system	system	NOUN
ma-255	363	8	f1	f1	NOUN
ma-255	363	9	=	=	PUNCT
ma-255	363	10	x	x	PUNCT
ma-255	363	11	−	−	NOUN
ma-255	363	12	0.1	0.1	NUM
ma-255	363	13	sin	sin	NOUN
ma-255	363	14	x	x	NOUN
ma-255	363	15	−	−	PROPN
ma-255	363	16	0.3	0.3	NUM
ma-255	363	17	cos	cos	PROPN
ma-255	363	18	y	y	PROPN
ma-255	364	1	+	+	NUM
ma-255	364	2	0.4	0.4	NUM
ma-255	364	3	f2	f2	PROPN
ma-255	364	4	=	=	SYM
ma-255	364	5	y	y	PROPN
ma-255	364	6	−	−	PROPN
ma-255	364	7	0.2	0.2	NUM
ma-255	364	8	cos	cos	NOUN
ma-255	364	9	x	x	PROPN
ma-255	364	10	+	+	NUM
ma-255	364	11	0.1	0.1	NUM
ma-255	364	12	sin	sin	NOUN
ma-255	364	13	y	y	PROPN
ma-255	364	14	+	+	NOUN
ma-255	364	15	0.3	0.3	NUM
ma-255	364	16	let	let	VERB
ma-255	364	17	f1	f1	NOUN
ma-255	364	18	=	=	SYM
ma-255	364	19	(	(	PUNCT
ma-255	364	20	f1	f1	PROPN
ma-255	364	21	,	,	PUNCT
ma-255	364	22	f2	f2	PROPN
ma-255	364	23	)	)	PUNCT
ma-255	364	24	.	.	PUNCT
ma-255	365	1	then	then	ADV
ma-255	365	2	,	,	PUNCT
ma-255	365	3	the	the	DET
ma-255	365	4	system	system	NOUN
ma-255	365	5	becomes	become	VERB
ma-255	365	6	f1(s	f1(	NOUN
ma-255	365	7	)	)	PUNCT
ma-255	365	8	=	=	SYM
ma-255	366	1	0	0	NUM
ma-255	366	2	f	f	PROPN
ma-255	366	3	or	or	CCONJ
ma-255	366	4	s	s	NOUN
ma-255	366	5	=	=	PUNCT
ma-255	366	6	(	(	PUNCT
ma-255	366	7	x	x	X
ma-255	366	8	,	,	PUNCT
ma-255	366	9	y)t	y)t	PROPN
ma-255	366	10	.	.	PUNCT
ma-255	367	1	then	then	ADV
ma-255	367	2	f	f	PROPN
ma-255	367	3	′1((x	′1((x	NOUN
ma-255	367	4	,	,	PUNCT
ma-255	367	5	y	y	NOUN
ma-255	367	6	)	)	PUNCT
ma-255	367	7	)	)	PUNCT
ma-255	368	1	=	=	PUNCT
ma-255	368	2	[	[	PUNCT
ma-255	368	3	1−	1−	NUM
ma-255	368	4	0.1	0.1	NUM
ma-255	368	5	cos(x	cos(x	PROPN
ma-255	368	6	)	)	PUNCT
ma-255	368	7	0.3	0.3	NUM
ma-255	368	8	sin(y	sin(y	NOUN
ma-255	368	9	)	)	PUNCT
ma-255	368	10	0.2	0.2	NUM
ma-255	368	11	sin(x	sin(x	PROPN
ma-255	368	12	)	)	PUNCT
ma-255	369	1	0.1	0.1	NUM
ma-255	369	2	cos(y	cos(y	NOUN
ma-255	369	3	)	)	PUNCT
ma-255	370	1	+	+	CCONJ
ma-255	370	2	1	1	NUM
ma-255	370	3	]	]	PUNCT
ma-255	370	4	.	.	PUNCT
ma-255	371	1	method	method	NOUN
ma-255	371	2	(	(	PUNCT
ma-255	371	3	1.2	1.2	NUM
ma-255	371	4	)	)	PUNCT
ma-255	371	5	xp+1	xp+1	NOUN
ma-255	372	1	=	=	SYM
ma-255	372	2	xp	xp	PROPN
ma-255	373	1	−	−	PROPN
ma-255	373	2	f	f	PROPN
ma-255	373	3	′1(xp)−1f1(xp	′1(xp)−1f1(xp	PROPN
ma-255	373	4	)	)	PUNCT
ma-255	373	5	.	.	PUNCT
ma-255	374	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	374	2	eur	eur	PROPN
ma-255	374	3	.	.	PUNCT
ma-255	375	1	j.	j.	PROPN
ma-255	375	2	math	math	PROPN
ma-255	375	3	.	.	PUNCT
ma-255	376	1	anal	anal	PROPN
ma-255	376	2	.	.	PUNCT
ma-255	377	1	10.28924	10.28924	NUM
ma-255	377	2	/	/	SYM
ma-255	377	3	ada	ada	PROPN
ma-255	377	4	/	/	SYM
ma-255	377	5	ma.5.5	ma.5.5	PROPN
ma-255	377	6	14	14	NUM
ma-255	377	7	method	method	NOUN
ma-255	377	8	(	(	PUNCT
ma-255	377	9	1.5	1.5	NUM
ma-255	377	10	)	)	PUNCT
ma-255	377	11	,	,	PUNCT
ma-255	377	12	p	p	NOUN
ma-255	377	13	=	=	NOUN
ma-255	377	14	1	1	NUM
ma-255	377	15	,	,	PUNCT
ma-255	377	16	γ	γ	X
ma-255	377	17	=	=	SYM
ma-255	377	18	i	i	PROPN
ma-255	377	19	,	,	PUNCT
ma-255	377	20	m1(x	m1(x	NOUN
ma-255	377	21	)	)	PUNCT
ma-255	377	22	=	=	SYM
ma-255	378	1	i	i	PRON
ma-255	378	2	+	+	CCONJ
ma-255	379	1	(	(	PUNCT
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ma-255	383	11	=	=	SYM
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ma-255	385	7	)	)	PUNCT
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ma-255	389	7	)	)	PUNCT
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ma-255	390	19	)	)	PUNCT
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ma-255	390	21	(	(	PUNCT
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ma-255	390	27	)	)	PUNCT
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ma-255	391	15	)	)	PUNCT
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ma-255	393	7	)	)	PUNCT
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ma-255	396	11	=	=	SYM
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ma-255	399	1	+	+	CCONJ
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ma-255	401	1	+	+	CCONJ
ma-255	401	2	(	(	PUNCT
ma-255	401	3	i	i	PRON
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ma-255	402	1	+	+	CCONJ
ma-255	402	2	(	(	PUNCT
ma-255	402	3	i	i	PRON
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ma-255	402	7	,	,	PUNCT
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ma-255	402	10	=	=	SYM
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ma-255	402	15	(	(	PUNCT
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ma-255	403	11	,	,	PUNCT
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ma-255	403	13	=	=	SYM
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ma-255	403	16	)	)	PUNCT
ma-255	403	17	,	,	PUNCT
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ma-255	403	19	=	=	SYM
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ma-255	403	22	)	)	PUNCT
ma-255	403	23	,	,	PUNCT
ma-255	403	24	a	a	DET
ma-255	403	25	=	=	X
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ma-255	403	29	′1(x	′1(x	NOUN
ma-255	403	30	)	)	PUNCT
ma-255	403	31	)	)	PUNCT
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ma-255	403	33	(	(	PUNCT
ma-255	403	34	4.6	4.6	X
ma-255	403	35	)	)	PUNCT
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ma-255	404	1	=	=	SYM
ma-255	404	2	i	i	PRON
ma-255	405	1	+	+	NUM
ma-255	405	2	p∑	p∑	NOUN
ma-255	405	3	i=1	i=1	PROPN
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ma-255	406	2	,	,	PUNCT
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ma-255	406	4	comparison	comparison	NOUN
ma-255	406	5	shows	show	VERB
ma-255	406	6	that	that	SCONJ
ma-255	406	7	the	the	DET
ma-255	406	8	behavior	behavior	NOUN
ma-255	406	9	of	of	ADP
ma-255	406	10	the	the	DET
ma-255	406	11	method	method	NOUN
ma-255	406	12	(	(	PUNCT
ma-255	406	13	1.5	1.5	NUM
ma-255	406	14	)	)	PUNCT
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ma-255	406	19	as	as	ADP
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ma-255	406	22	method	method	NOUN
ma-255	406	23	(	(	PUNCT
ma-255	406	24	1.2	1.2	NUM
ma-255	406	25	)	)	PUNCT
ma-255	406	26	.	.	PUNCT
ma-255	407	1	however	however	ADV
ma-255	407	2	,	,	PUNCT
ma-255	407	3	the	the	DET
ma-255	407	4	iterates	iterate	NOUN
ma-255	407	5	of	of	ADP
ma-255	407	6	the	the	DET
ma-255	407	7	method	method	NOUN
ma-255	407	8	(	(	PUNCT
ma-255	407	9	1.5	1.5	NUM
ma-255	407	10	)	)	PUNCT
ma-255	407	11	are	be	AUX
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ma-255	407	13	to	to	PART
ma-255	407	14	obtain	obtain	VERB
ma-255	407	15	than	than	ADP
ma-255	407	16	newton	newton	PROPN
ma-255	407	17	’s	’s	PART
ma-255	407	18	.	.	PUNCT
ma-255	408	1	as	as	SCONJ
ma-255	408	2	observed	observe	VERB
ma-255	408	3	in	in	ADP
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ma-255	408	5	1	1	NUM
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ma-255	408	7	4	4	NUM
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ma-255	408	10	number	number	NOUN
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ma-255	408	12	iterations	iteration	NOUN
ma-255	408	13	required	require	VERB
ma-255	408	14	for	for	ADP
ma-255	408	15	the	the	DET
ma-255	408	16	proposed	propose	VERB
ma-255	408	17	methods	method	NOUN
ma-255	408	18	with	with	ADP
ma-255	408	19	k	k	PROPN
ma-255	408	20	ranging	range	VERB
ma-255	408	21	from	from	ADP
ma-255	408	22	3	3	NUM
ma-255	408	23	to	to	PART
ma-255	408	24	5	5	NUM
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ma-255	408	26	aligns	align	VERB
ma-255	408	27	with	with	ADP
ma-255	408	28	those	those	PRON
ma-255	408	29	of	of	ADP
ma-255	408	30	newton	newton	PROPN
ma-255	408	31	’s	’s	PART
ma-255	408	32	method	method	NOUN
ma-255	408	33	.	.	PUNCT
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ma-255	409	3	.	.	PUNCT
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ma-255	411	2	.	.	PUNCT
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ma-255	412	11	(	(	PUNCT
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ma-255	412	13	)	)	PUNCT
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ma-255	412	16	(	(	PUNCT
ma-255	412	17	1.2	1.2	NUM
ma-255	412	18	)	)	PUNCT
ma-255	412	19	newton	newton	PROPN
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ma-255	412	36	8	8	NUM
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ma-255	412	39	)	)	PUNCT
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ma-255	412	61	(	(	PUNCT
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ma-255	412	85	(	(	PUNCT
ma-255	412	86	4.5	4.5	NUM
ma-255	412	87	)	)	PUNCT
ma-255	412	88	,	,	PUNCT
ma-255	412	89	p	p	NOUN
ma-255	412	90	=	=	SYM
ma-255	412	91	5	5	NUM
ma-255	412	92	4	4	NUM
ma-255	412	93	(	(	PUNCT
ma-255	412	94	4.6	4.6	NUM
ma-255	412	95	)	)	PUNCT
ma-255	412	96	,	,	PUNCT
ma-255	412	97	p	p	NOUN
ma-255	412	98	=	=	SYM
ma-255	412	99	5	5	NUM
ma-255	412	100	4table	4table	NUM
ma-255	412	101	1	1	NUM
ma-255	412	102	.	.	PUNCT
ma-255	413	1	the	the	DET
ma-255	413	2	number	number	NOUN
ma-255	413	3	of	of	ADP
ma-255	413	4	iterations	iteration	NOUN
ma-255	413	5	to	to	PART
ma-255	413	6	reach	reach	VERB
ma-255	413	7	error	error	NOUN
ma-255	413	8	tolerance	tolerance	NOUN
ma-255	413	9	ε	ε	NOUN
ma-255	413	10	=	=	SYM
ma-255	413	11	10−9	10−9	NUM
ma-255	413	12	with	with	ADP
ma-255	413	13	initialguess	initialguess	NOUN
ma-255	413	14	x0	x0	PROPN
ma-255	413	15	=	=	PUNCT
ma-255	413	16	(	(	PUNCT
ma-255	413	17	1	1	NUM
ma-255	413	18	,	,	PUNCT
ma-255	413	19	1	1	NUM
ma-255	413	20	)	)	PUNCT
ma-255	413	21	and	and	CCONJ
ma-255	413	22	‖i	‖i	NOUN
ma-255	413	23	−	−	NOUN
ma-255	413	24	f	f	PROPN
ma-255	413	25	′1(x0)‖	′1(x0)‖	PROPN
ma-255	413	26	=	=	PUNCT
ma-255	414	1	0.3129	0.3129	NUM
ma-255	414	2	<	<	X
ma-255	414	3	1	1	NUM
ma-255	414	4	.	.	PUNCT
ma-255	414	5	method	method	NOUN
ma-255	414	6	iterations	iteration	NOUN
ma-255	414	7	method	method	NOUN
ma-255	414	8	iterations	iteration	NOUN
ma-255	414	9	(	(	PUNCT
ma-255	414	10	1.2	1.2	NUM
ma-255	414	11	)	)	PUNCT
ma-255	414	12	newton	newton	PROPN
ma-255	414	13	3	3	NUM
ma-255	414	14	(	(	PUNCT
ma-255	414	15	1.2	1.2	NUM
ma-255	414	16	)	)	PUNCT
ma-255	414	17	newton	newton	PROPN
ma-255	414	18	3	3	NUM
ma-255	414	19	(	(	PUNCT
ma-255	414	20	4.1	4.1	NUM
ma-255	414	21	)	)	PUNCT
ma-255	414	22	,	,	PUNCT
ma-255	414	23	p	p	NOUN
ma-255	414	24	=	=	NOUN
ma-255	414	25	1	1	NUM
ma-255	414	26	5	5	NUM
ma-255	414	27	(	(	PUNCT
ma-255	414	28	4.6	4.6	NUM
ma-255	414	29	)	)	PUNCT
ma-255	414	30	,	,	PUNCT
ma-255	414	31	p	p	NOUN
ma-255	414	32	=	=	NOUN
ma-255	414	33	1	1	NUM
ma-255	414	34	3	3	NUM
ma-255	414	35	(	(	PUNCT
ma-255	414	36	4.2	4.2	NUM
ma-255	414	37	)	)	PUNCT
ma-255	414	38	,	,	PUNCT
ma-255	414	39	p	p	NOUN
ma-255	414	40	=	=	NOUN
ma-255	414	41	2	2	NUM
ma-255	414	42	4	4	NUM
ma-255	414	43	(	(	PUNCT
ma-255	414	44	4.6	4.6	NUM
ma-255	414	45	)	)	PUNCT
ma-255	414	46	,	,	PUNCT
ma-255	414	47	p	p	NOUN
ma-255	414	48	=	=	NOUN
ma-255	414	49	2	2	NUM
ma-255	414	50	3	3	NUM
ma-255	414	51	(	(	PUNCT
ma-255	414	52	4.3	4.3	NUM
ma-255	414	53	)	)	PUNCT
ma-255	414	54	,	,	PUNCT
ma-255	414	55	p	p	NOUN
ma-255	414	56	=	=	NOUN
ma-255	414	57	3	3	NUM
ma-255	414	58	3	3	NUM
ma-255	414	59	(	(	PUNCT
ma-255	414	60	4.6	4.6	NUM
ma-255	414	61	)	)	PUNCT
ma-255	414	62	,	,	PUNCT
ma-255	414	63	p	p	NOUN
ma-255	414	64	=	=	NOUN
ma-255	414	65	3	3	NUM
ma-255	414	66	3	3	NUM
ma-255	414	67	(	(	PUNCT
ma-255	414	68	4.4	4.4	NUM
ma-255	414	69	)	)	PUNCT
ma-255	414	70	,	,	PUNCT
ma-255	414	71	p	p	NOUN
ma-255	414	72	=	=	NOUN
ma-255	414	73	4	4	NUM
ma-255	414	74	3	3	NUM
ma-255	414	75	(	(	PUNCT
ma-255	414	76	4.6	4.6	NUM
ma-255	414	77	)	)	PUNCT
ma-255	414	78	,	,	PUNCT
ma-255	414	79	p	p	NOUN
ma-255	414	80	=	=	NOUN
ma-255	414	81	4	4	NUM
ma-255	414	82	3	3	NUM
ma-255	414	83	(	(	PUNCT
ma-255	414	84	4.5	4.5	NUM
ma-255	414	85	)	)	PUNCT
ma-255	414	86	,	,	PUNCT
ma-255	414	87	p	p	NOUN
ma-255	414	88	=	=	SYM
ma-255	414	89	5	5	NUM
ma-255	414	90	3	3	NUM
ma-255	414	91	(	(	PUNCT
ma-255	414	92	4.6	4.6	NUM
ma-255	414	93	)	)	PUNCT
ma-255	414	94	,	,	PUNCT
ma-255	414	95	p	p	NOUN
ma-255	414	96	=	=	SYM
ma-255	414	97	5	5	NUM
ma-255	414	98	3table	3table	NUM
ma-255	414	99	2	2	NUM
ma-255	414	100	.	.	PUNCT
ma-255	415	1	the	the	DET
ma-255	415	2	number	number	NOUN
ma-255	415	3	of	of	ADP
ma-255	415	4	iterations	iteration	NOUN
ma-255	415	5	to	to	PART
ma-255	415	6	reach	reach	VERB
ma-255	415	7	error	error	NOUN
ma-255	415	8	tolerance	tolerance	NOUN
ma-255	415	9	ε	ε	NOUN
ma-255	415	10	=	=	SYM
ma-255	415	11	10−9	10−9	NUM
ma-255	415	12	with	with	ADP
ma-255	415	13	initialguess	initialguess	NOUN
ma-255	415	14	x0	x0	PROPN
ma-255	415	15	=	=	PUNCT
ma-255	415	16	(	(	PUNCT
ma-255	415	17	0	0	NUM
ma-255	415	18	,	,	PUNCT
ma-255	415	19	0	0	NUM
ma-255	415	20	)	)	PUNCT
ma-255	415	21	and	and	CCONJ
ma-255	415	22	‖i	‖i	ADJ
ma-255	415	23	−	−	NOUN
ma-255	415	24	f	f	PROPN
ma-255	415	25	′1(x0)‖	′1(x0)‖	NOUN
ma-255	415	26	=	=	PUNCT
ma-255	415	27	0.1414	0.1414	NUM
ma-255	415	28	<	<	X
ma-255	415	29	1	1	NUM
ma-255	415	30	.	.	PUNCT
ma-255	415	31	method	method	NOUN
ma-255	415	32	iterations	iteration	NOUN
ma-255	415	33	method	method	NOUN
ma-255	415	34	iterations	iteration	NOUN
ma-255	415	35	(	(	PUNCT
ma-255	415	36	1.2	1.2	NUM
ma-255	415	37	)	)	PUNCT
ma-255	415	38	newton	newton	PROPN
ma-255	415	39	5	5	NUM
ma-255	415	40	(	(	PUNCT
ma-255	415	41	1.2	1.2	NUM
ma-255	415	42	)	)	PUNCT
ma-255	415	43	newton	newton	PROPN
ma-255	415	44	5	5	NUM
ma-255	415	45	(	(	PUNCT
ma-255	415	46	4.1	4.1	NUM
ma-255	415	47	)	)	PUNCT
ma-255	415	48	,	,	PUNCT
ma-255	415	49	p	p	NOUN
ma-255	415	50	=	=	NOUN
ma-255	415	51	1	1	NUM
ma-255	415	52	7	7	NUM
ma-255	415	53	(	(	PUNCT
ma-255	415	54	4.6	4.6	NUM
ma-255	415	55	)	)	PUNCT
ma-255	415	56	,	,	PUNCT
ma-255	415	57	p	p	NOUN
ma-255	415	58	=	=	NOUN
ma-255	415	59	1	1	NUM
ma-255	415	60	9	9	NUM
ma-255	415	61	(	(	PUNCT
ma-255	415	62	4.2	4.2	NUM
ma-255	415	63	)	)	PUNCT
ma-255	415	64	,	,	PUNCT
ma-255	415	65	p	p	NOUN
ma-255	415	66	=	=	NOUN
ma-255	415	67	2	2	NUM
ma-255	415	68	5	5	NUM
ma-255	415	69	(	(	PUNCT
ma-255	415	70	4.6	4.6	NUM
ma-255	415	71	)	)	PUNCT
ma-255	415	72	,	,	PUNCT
ma-255	415	73	p	p	NOUN
ma-255	415	74	=	=	NOUN
ma-255	415	75	2	2	NUM
ma-255	415	76	7	7	NUM
ma-255	415	77	(	(	PUNCT
ma-255	415	78	4.3	4.3	NUM
ma-255	415	79	)	)	PUNCT
ma-255	415	80	,	,	PUNCT
ma-255	415	81	p	p	NOUN
ma-255	415	82	=	=	NOUN
ma-255	415	83	3	3	NUM
ma-255	415	84	5	5	NUM
ma-255	415	85	(	(	PUNCT
ma-255	415	86	4.6	4.6	NUM
ma-255	415	87	)	)	PUNCT
ma-255	415	88	,	,	PUNCT
ma-255	415	89	p	p	NOUN
ma-255	415	90	=	=	NOUN
ma-255	415	91	3	3	NUM
ma-255	415	92	6	6	NUM
ma-255	415	93	(	(	PUNCT
ma-255	415	94	4.4	4.4	NUM
ma-255	415	95	)	)	PUNCT
ma-255	415	96	,	,	PUNCT
ma-255	415	97	p	p	NOUN
ma-255	415	98	=	=	NOUN
ma-255	415	99	4	4	NUM
ma-255	415	100	5	5	NUM
ma-255	415	101	(	(	PUNCT
ma-255	415	102	4.6	4.6	NUM
ma-255	415	103	)	)	PUNCT
ma-255	415	104	,	,	PUNCT
ma-255	415	105	p	p	NOUN
ma-255	415	106	=	=	NOUN
ma-255	415	107	4	4	NUM
ma-255	415	108	6	6	NUM
ma-255	415	109	(	(	PUNCT
ma-255	415	110	4.5	4.5	NUM
ma-255	415	111	)	)	PUNCT
ma-255	415	112	,	,	PUNCT
ma-255	415	113	p	p	NOUN
ma-255	415	114	=	=	SYM
ma-255	415	115	5	5	NUM
ma-255	415	116	5	5	NUM
ma-255	415	117	(	(	PUNCT
ma-255	415	118	4.6	4.6	NUM
ma-255	415	119	)	)	PUNCT
ma-255	415	120	,	,	PUNCT
ma-255	415	121	p	p	NOUN
ma-255	415	122	=	=	NOUN
ma-255	415	123	5	5	NUM
ma-255	415	124	5table	5table	NUM
ma-255	415	125	3	3	NUM
ma-255	415	126	.	.	PUNCT
ma-255	416	1	the	the	DET
ma-255	416	2	number	number	NOUN
ma-255	416	3	of	of	ADP
ma-255	416	4	iterations	iteration	NOUN
ma-255	416	5	to	to	PART
ma-255	416	6	reach	reach	VERB
ma-255	416	7	error	error	NOUN
ma-255	416	8	tolerance	tolerance	NOUN
ma-255	416	9	ε	ε	NOUN
ma-255	416	10	=	=	SYM
ma-255	416	11	10−9	10−9	NUM
ma-255	416	12	,	,	PUNCT
ma-255	416	13	where	where	SCONJ
ma-255	416	14	x0	x0	PROPN
ma-255	416	15	=	=	PRON
ma-255	416	16	(	(	PUNCT
ma-255	416	17	−15,−15	−15,−15	PROPN
ma-255	416	18	)	)	PUNCT
ma-255	416	19	and	and	CCONJ
ma-255	416	20	‖i	‖i	ADJ
ma-255	416	21	−	−	NOUN
ma-255	416	22	f	f	PROPN
ma-255	416	23	′1(x0)‖	′1(x0)‖	PROPN
ma-255	416	24	=	=	PUNCT
ma-255	416	25	0.257	0.257	NUM
ma-255	416	26	<	<	SYM
ma-255	416	27	1	1	NUM
ma-255	416	28	.	.	PUNCT
ma-255	416	29	table	table	NOUN
ma-255	416	30	5	5	NUM
ma-255	416	31	shows	show	VERB
ma-255	416	32	the	the	DET
ma-255	416	33	results	result	NOUN
ma-255	416	34	of	of	ADP
ma-255	416	35	calculations	calculation	NOUN
ma-255	416	36	to	to	PART
ma-255	416	37	determine	determine	VERB
ma-255	416	38	the	the	DET
ma-255	416	39	computational	computational	ADJ
ma-255	416	40	order	order	NOUN
ma-255	416	41	of	of	ADP
ma-255	416	42	convergence	convergence	NOUN
ma-255	416	43	(	(	PUNCT
ma-255	416	44	coc	coc	PROPN
ma-255	416	45	)	)	PUNCT
ma-255	416	46	and	and	CCONJ
ma-255	416	47	the	the	DET
ma-255	416	48	approximated	approximated	ADJ
ma-255	416	49	computational	computational	ADJ
ma-255	416	50	order	order	NOUN
ma-255	416	51	of	of	ADP
ma-255	416	52	convergence	convergence	NOUN
ma-255	416	53	(	(	PUNCT
ma-255	416	54	acoc	acoc	ADJ
ma-255	416	55	)	)	PUNCT
ma-255	416	56	aiming	aim	VERB
ma-255	416	57	to	to	PART
ma-255	416	58	compare	compare	VERB
ma-255	416	59	the	the	DET
ma-255	416	60	convergence	convergence	NOUN
ma-255	416	61	order	order	NOUN
ma-255	416	62	of	of	ADP
ma-255	416	63	method	method	NOUN
ma-255	416	64	(	(	PUNCT
ma-255	416	65	1.5	1.5	NUM
ma-255	416	66	)	)	PUNCT
ma-255	416	67	with	with	ADP
ma-255	416	68	the	the	DET
ma-255	416	69	convergence	convergence	NOUN
ma-255	416	70	order	order	NOUN
ma-255	416	71	of	of	ADP
ma-255	416	72	newton	newton	PROPN
ma-255	416	73	’s	’s	PART
ma-255	416	74	method	method	NOUN
ma-255	416	75	(	(	PUNCT
ma-255	416	76	1.2	1.2	NUM
ma-255	416	77	)	)	PUNCT
ma-255	416	78	.	.	PUNCT
ma-255	417	1	definition	definition	NOUN
ma-255	417	2	4.1	4.1	NUM
ma-255	417	3	.	.	PUNCT
ma-255	418	1	computational	computational	ADJ
ma-255	418	2	order	order	NOUN
ma-255	418	3	of	of	ADP
ma-255	418	4	convergence	convergence	NOUN
ma-255	418	5	of	of	ADP
ma-255	418	6	a	a	DET
ma-255	418	7	sequence	sequence	NOUN
ma-255	418	8	{	{	PUNCT
ma-255	418	9	xj}j≥0	xj}j≥0	NOUN
ma-255	418	10	is	be	AUX
ma-255	418	11	defined	define	VERB
ma-255	418	12	by	by	ADP
ma-255	418	13	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	418	14	eur	eur	PROPN
ma-255	418	15	.	.	PUNCT
ma-255	419	1	j.	j.	PROPN
ma-255	419	2	math	math	PROPN
ma-255	419	3	.	.	PUNCT
ma-255	420	1	anal	anal	PROPN
ma-255	420	2	.	.	PUNCT
ma-255	421	1	10.28924	10.28924	NUM
ma-255	421	2	/	/	SYM
ma-255	421	3	ada	ada	PROPN
ma-255	421	4	/	/	SYM
ma-255	421	5	ma.5.5	ma.5.5	PROPN
ma-255	421	6	16	16	NUM
ma-255	421	7	method	method	NOUN
ma-255	421	8	iterations	iteration	NOUN
ma-255	421	9	method	method	NOUN
ma-255	421	10	iterations	iteration	NOUN
ma-255	421	11	(	(	PUNCT
ma-255	421	12	1.2	1.2	NUM
ma-255	421	13	)	)	PUNCT
ma-255	421	14	newton	newton	PROPN
ma-255	421	15	7	7	NUM
ma-255	421	16	(	(	PUNCT
ma-255	421	17	1.2	1.2	NUM
ma-255	421	18	)	)	PUNCT
ma-255	421	19	newton	newton	PROPN
ma-255	421	20	7	7	NUM
ma-255	421	21	(	(	PUNCT
ma-255	421	22	4.1	4.1	NUM
ma-255	421	23	)	)	PUNCT
ma-255	421	24	,	,	PUNCT
ma-255	421	25	p	p	NOUN
ma-255	421	26	=	=	NOUN
ma-255	421	27	1	1	NUM
ma-255	421	28	8	8	NUM
ma-255	421	29	(	(	PUNCT
ma-255	421	30	4.6	4.6	NUM
ma-255	421	31	)	)	PUNCT
ma-255	421	32	,	,	PUNCT
ma-255	421	33	p	p	NOUN
ma-255	421	34	=	=	NOUN
ma-255	421	35	1	1	NUM
ma-255	421	36	12	12	NUM
ma-255	421	37	(	(	PUNCT
ma-255	421	38	4.2	4.2	NUM
ma-255	421	39	)	)	PUNCT
ma-255	421	40	,	,	PUNCT
ma-255	421	41	p	p	NOUN
ma-255	421	42	=	=	NOUN
ma-255	421	43	2	2	NUM
ma-255	421	44	7	7	NUM
ma-255	421	45	(	(	PUNCT
ma-255	421	46	4.6	4.6	NUM
ma-255	421	47	)	)	PUNCT
ma-255	421	48	,	,	PUNCT
ma-255	421	49	p	p	NOUN
ma-255	421	50	=	=	SYM
ma-255	421	51	2	2	NUM
ma-255	421	52	8	8	NUM
ma-255	421	53	(	(	PUNCT
ma-255	421	54	4.3	4.3	NUM
ma-255	421	55	)	)	PUNCT
ma-255	421	56	,	,	PUNCT
ma-255	421	57	p	p	NOUN
ma-255	421	58	=	=	NOUN
ma-255	421	59	3	3	NUM
ma-255	421	60	7	7	NUM
ma-255	421	61	(	(	PUNCT
ma-255	421	62	4.6	4.6	NUM
ma-255	421	63	)	)	PUNCT
ma-255	421	64	,	,	PUNCT
ma-255	421	65	p	p	NOUN
ma-255	421	66	=	=	NOUN
ma-255	421	67	3	3	NUM
ma-255	421	68	8	8	NUM
ma-255	421	69	(	(	PUNCT
ma-255	421	70	4.4	4.4	NUM
ma-255	421	71	)	)	PUNCT
ma-255	421	72	,	,	PUNCT
ma-255	421	73	p	p	NOUN
ma-255	421	74	=	=	NOUN
ma-255	421	75	4	4	NUM
ma-255	421	76	7	7	NUM
ma-255	421	77	(	(	PUNCT
ma-255	421	78	4.6	4.6	NUM
ma-255	421	79	)	)	PUNCT
ma-255	421	80	,	,	PUNCT
ma-255	421	81	p	p	NOUN
ma-255	421	82	=	=	NOUN
ma-255	421	83	4	4	NUM
ma-255	421	84	7	7	NUM
ma-255	421	85	(	(	PUNCT
ma-255	421	86	4.5	4.5	NUM
ma-255	421	87	)	)	PUNCT
ma-255	421	88	,	,	PUNCT
ma-255	421	89	p	p	NOUN
ma-255	421	90	=	=	NOUN
ma-255	421	91	5	5	NUM
ma-255	421	92	7	7	NUM
ma-255	421	93	(	(	PUNCT
ma-255	421	94	4.6	4.6	NUM
ma-255	421	95	)	)	PUNCT
ma-255	421	96	,	,	PUNCT
ma-255	421	97	p	p	NOUN
ma-255	421	98	=	=	SYM
ma-255	421	99	5	5	NUM
ma-255	421	100	7table	7table	NUM
ma-255	421	101	4	4	NUM
ma-255	421	102	.	.	PUNCT
ma-255	422	1	the	the	DET
ma-255	422	2	number	number	NOUN
ma-255	422	3	of	of	ADP
ma-255	422	4	iterations	iteration	NOUN
ma-255	422	5	to	to	PART
ma-255	422	6	reach	reach	VERB
ma-255	422	7	error	error	NOUN
ma-255	422	8	tolerance	tolerance	NOUN
ma-255	422	9	ε	ε	NOUN
ma-255	422	10	=	=	SYM
ma-255	422	11	10−12	10−12	PROPN
ma-255	422	12	,	,	PUNCT
ma-255	422	13	where	where	SCONJ
ma-255	422	14	x0	x0	PROPN
ma-255	422	15	=	=	PRON
ma-255	422	16	(	(	PUNCT
ma-255	422	17	−15,−15	−15,−15	PROPN
ma-255	422	18	)	)	PUNCT
ma-255	422	19	and	and	CCONJ
ma-255	422	20	‖i	‖i	ADJ
ma-255	422	21	−	−	NOUN
ma-255	422	22	f	f	PROPN
ma-255	422	23	′1(x0)‖	′1(x0)‖	PROPN
ma-255	422	24	=	=	PUNCT
ma-255	422	25	0.257	0.257	NUM
ma-255	422	26	<	<	X
ma-255	422	27	1	1	NUM
ma-255	422	28	.	.	PUNCT
ma-255	422	29	υj	υj	VERB
ma-255	422	30	=	=	PUNCT
ma-255	422	31	ln	ln	ADJ
ma-255	422	32	|ej+1	|ej+1	PROPN
ma-255	422	33	/	/	SYM
ma-255	422	34	ej	ej	NOUN
ma-255	423	1	|	|	ADV
ma-255	423	2	ln	ln	ADJ
ma-255	423	3	|ej	|ej	NOUN
ma-255	423	4	/	/	SYM
ma-255	423	5	ej−1|	ej−1|	NOUN
ma-255	423	6	,	,	PUNCT
ma-255	423	7	where	where	SCONJ
ma-255	423	8	xj−1	xj−1	PROPN
ma-255	423	9	,	,	PUNCT
ma-255	423	10	xj	xj	PROPN
ma-255	423	11	,	,	PUNCT
ma-255	423	12	xj+1	xj+1	NUM
ma-255	423	13	are	be	AUX
ma-255	423	14	three	three	NUM
ma-255	423	15	consecutive	consecutive	ADJ
ma-255	423	16	iterations	iteration	NOUN
ma-255	423	17	near	near	ADP
ma-255	423	18	the	the	DET
ma-255	423	19	root	root	NOUN
ma-255	423	20	α	α	NOUN
ma-255	423	21	and	and	CCONJ
ma-255	423	22	ej	ej	PROPN
ma-255	423	23	=	=	PROPN
ma-255	423	24	xj	xj	PROPN
ma-255	423	25	−	−	PROPN
ma-255	423	26	α	α	PRON
ma-255	424	1	[	[	X
ma-255	424	2	6	6	NUM
ma-255	424	3	]	]	PUNCT
ma-255	424	4	.	.	PUNCT
ma-255	425	1	definition	definition	NOUN
ma-255	425	2	4.2	4.2	NUM
ma-255	425	3	.	.	PUNCT
ma-255	426	1	the	the	DET
ma-255	426	2	approximated	approximated	ADJ
ma-255	426	3	computational	computational	ADJ
ma-255	426	4	order	order	NOUN
ma-255	426	5	of	of	ADP
ma-255	426	6	convergence	convergence	NOUN
ma-255	426	7	of	of	ADP
ma-255	426	8	a	a	DET
ma-255	426	9	sequence	sequence	NOUN
ma-255	426	10	{	{	PUNCT
ma-255	426	11	xj}j≥0	xj}j≥0	NOUN
ma-255	426	12	is	be	AUX
ma-255	426	13	defined	define	VERB
ma-255	426	14	by	by	ADP
ma-255	426	15	υ̂n	υ̂n	X
ma-255	426	16	=	=	NOUN
ma-255	426	17	ln	ln	ADJ
ma-255	426	18	|êj+1	|êj+1	PUNCT
ma-255	426	19	/	/	SYM
ma-255	426	20	êj	êj	NOUN
ma-255	426	21	|	|	INTJ
ma-255	426	22	ln	ln	ADJ
ma-255	426	23	|êj	|êj	PROPN
ma-255	426	24	/	/	SYM
ma-255	426	25	êj−1|	êj−1|	PROPN
ma-255	426	26	,	,	PUNCT
ma-255	426	27	where	where	SCONJ
ma-255	426	28	êj	êj	NOUN
ma-255	426	29	=	=	SYM
ma-255	426	30	xj	xj	PROPN
ma-255	427	1	−	−	PROPN
ma-255	427	2	xj−1	xj−1	PROPN
ma-255	427	3	.	.	PUNCT
ma-255	428	1	xj	xj	PROPN
ma-255	428	2	,	,	PUNCT
ma-255	428	3	xj−1	xj−1	PROPN
ma-255	428	4	,	,	PUNCT
ma-255	428	5	xj−2	xj−2	PROPN
ma-255	428	6	are	be	AUX
ma-255	428	7	three	three	NUM
ma-255	428	8	consecutive	consecutive	ADJ
ma-255	428	9	iterates	iterate	NOUN
ma-255	428	10	[	[	X
ma-255	428	11	6	6	NUM
ma-255	428	12	]	]	PUNCT
ma-255	428	13	.	.	PUNCT
ma-255	429	1	method	method	PROPN
ma-255	429	2	coc	coc	PROPN
ma-255	429	3	acoc	acoc	PROPN
ma-255	429	4	(	(	PUNCT
ma-255	429	5	1.2	1.2	NUM
ma-255	429	6	)	)	PUNCT
ma-255	429	7	newton	newton	PROPN
ma-255	429	8	1.8624	1.8624	NUM
ma-255	429	9	1.9697	1.9697	NUM
ma-255	429	10	(	(	PUNCT
ma-255	429	11	4.1	4.1	NUM
ma-255	429	12	)	)	PUNCT
ma-255	429	13	,	,	PUNCT
ma-255	429	14	p	p	NOUN
ma-255	429	15	=	=	NOUN
ma-255	429	16	1	1	NUM
ma-255	429	17	0.863	0.863	NUM
ma-255	429	18	1	1	NUM
ma-255	429	19	(	(	PUNCT
ma-255	429	20	4.2	4.2	NUM
ma-255	429	21	)	)	PUNCT
ma-255	429	22	,	,	PUNCT
ma-255	429	23	p	p	NOUN
ma-255	429	24	=	=	NOUN
ma-255	429	25	2	2	NUM
ma-255	429	26	0.2695	0.2695	NUM
ma-255	429	27	1.0438	1.0438	NUM
ma-255	429	28	(	(	PUNCT
ma-255	429	29	4.3	4.3	NUM
ma-255	429	30	)	)	PUNCT
ma-255	429	31	,	,	PUNCT
ma-255	429	32	p	p	NOUN
ma-255	429	33	=	=	NOUN
ma-255	429	34	3	3	NUM
ma-255	429	35	1.9714	1.9714	NUM
ma-255	429	36	2.3569	2.3569	NUM
ma-255	429	37	(	(	PUNCT
ma-255	429	38	4.4	4.4	NUM
ma-255	429	39	)	)	PUNCT
ma-255	429	40	,	,	PUNCT
ma-255	429	41	p	p	NOUN
ma-255	429	42	=	=	NOUN
ma-255	429	43	4	4	NUM
ma-255	429	44	1.8354	1.8354	NUM
ma-255	429	45	1.9453	1.9453	NUM
ma-255	429	46	(	(	PUNCT
ma-255	429	47	4.5	4.5	NUM
ma-255	429	48	)	)	PUNCT
ma-255	429	49	,	,	PUNCT
ma-255	429	50	p	p	NOUN
ma-255	429	51	=	=	NOUN
ma-255	429	52	5	5	NUM
ma-255	429	53	1.8642	1.8642	NUM
ma-255	429	54	1.9661	1.9661	NUM
ma-255	429	55	(	(	PUNCT
ma-255	429	56	4.6	4.6	NUM
ma-255	429	57	)	)	PUNCT
ma-255	429	58	,	,	PUNCT
ma-255	429	59	p	p	NOUN
ma-255	429	60	=	=	NOUN
ma-255	429	61	1	1	NUM
ma-255	429	62	0.9065	0.9065	NUM
ma-255	429	63	1.0118	1.0118	NUM
ma-255	429	64	(	(	PUNCT
ma-255	429	65	4.6	4.6	NUM
ma-255	429	66	)	)	PUNCT
ma-255	429	67	,	,	PUNCT
ma-255	429	68	p	p	NOUN
ma-255	429	69	=	=	NOUN
ma-255	429	70	2	2	NUM
ma-255	429	71	0.5912	0.5912	NUM
ma-255	429	72	0.999	0.999	NUM
ma-255	429	73	(	(	PUNCT
ma-255	429	74	4.6	4.6	NUM
ma-255	429	75	)	)	PUNCT
ma-255	429	76	,	,	PUNCT
ma-255	429	77	p	p	NOUN
ma-255	429	78	=	=	NOUN
ma-255	429	79	3	3	NUM
ma-255	429	80	0.7321	0.7321	NUM
ma-255	429	81	0.9926	0.9926	NUM
ma-255	429	82	(	(	PUNCT
ma-255	429	83	4.6	4.6	NUM
ma-255	429	84	)	)	PUNCT
ma-255	429	85	,	,	PUNCT
ma-255	429	86	p	p	NOUN
ma-255	429	87	=	=	NOUN
ma-255	429	88	4	4	NUM
ma-255	429	89	1.933	1.933	NUM
ma-255	429	90	2.0151	2.0151	NUM
ma-255	429	91	(	(	PUNCT
ma-255	429	92	4.6	4.6	NUM
ma-255	429	93	)	)	PUNCT
ma-255	429	94	,	,	PUNCT
ma-255	429	95	p	p	NOUN
ma-255	429	96	=	=	SYM
ma-255	429	97	5	5	NUM
ma-255	429	98	1.8679	1.8679	NUM
ma-255	429	99	1.9578table	1.9578table	NUM
ma-255	429	100	5	5	NUM
ma-255	429	101	.	.	PUNCT
ma-255	430	1	the	the	DET
ma-255	430	2	computational	computational	ADJ
ma-255	430	3	order	order	NOUN
ma-255	430	4	of	of	ADP
ma-255	430	5	convergence	convergence	NOUN
ma-255	430	6	and	and	CCONJ
ma-255	430	7	the	the	DET
ma-255	430	8	approximated	approximate	VERB
ma-255	430	9	compu	compu	PROPN
ma-255	430	10	-	-	PUNCT
ma-255	430	11	tational	tational	ADJ
ma-255	430	12	order	order	NOUN
ma-255	430	13	of	of	ADP
ma-255	430	14	convergence	convergence	NOUN
ma-255	430	15	,	,	PUNCT
ma-255	430	16	where	where	SCONJ
ma-255	430	17	x0	x0	PROPN
ma-255	430	18	=	=	PRON
ma-255	430	19	(	(	PUNCT
ma-255	430	20	−15,−15	−15,−15	PROPN
ma-255	430	21	)	)	PUNCT
ma-255	430	22	,	,	PUNCT
ma-255	430	23	ε	ε	PROPN
ma-255	430	24	=	=	SYM
ma-255	430	25	10−12	10−12	PROPN
ma-255	430	26	.	.	PUNCT
ma-255	431	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	431	2	eur	eur	PROPN
ma-255	431	3	.	.	PUNCT
ma-255	432	1	j.	j.	PROPN
ma-255	432	2	math	math	PROPN
ma-255	432	3	.	.	PUNCT
ma-255	433	1	anal	anal	PROPN
ma-255	433	2	.	.	PUNCT
ma-255	434	1	10.28924	10.28924	NUM
ma-255	434	2	/	/	SYM
ma-255	434	3	ada	ada	PROPN
ma-255	434	4	/	/	SYM
ma-255	434	5	ma.5.5	ma.5.5	PROPN
ma-255	434	6	17	17	NUM
ma-255	434	7	table	table	NOUN
ma-255	434	8	5	5	NUM
ma-255	434	9	shows	show	VERB
ma-255	434	10	that	that	SCONJ
ma-255	434	11	the	the	DET
ma-255	434	12	convergence	convergence	NOUN
ma-255	434	13	of	of	ADP
ma-255	434	14	the	the	DET
ma-255	434	15	proposed	propose	VERB
ma-255	434	16	methods	method	NOUN
ma-255	434	17	closely	closely	ADV
ma-255	434	18	corresponds	correspond	VERB
ma-255	434	19	with	with	ADP
ma-255	434	20	the	the	DET
ma-255	434	21	convergence	convergence	NOUN
ma-255	434	22	of	of	ADP
ma-255	434	23	newton	newton	PROPN
ma-255	434	24	’s	’s	PART
ma-255	434	25	method	method	NOUN
ma-255	434	26	,	,	PUNCT
ma-255	434	27	particularly	particularly	ADV
ma-255	434	28	for	for	ADP
ma-255	434	29	values	value	NOUN
ma-255	434	30	of	of	ADP
ma-255	434	31	k	k	NOUN
ma-255	434	32	ranging	range	VERB
ma-255	434	33	from	from	ADP
ma-255	434	34	4	4	NUM
ma-255	434	35	to	to	PART
ma-255	434	36	5	5	NUM
ma-255	434	37	with	with	ADP
ma-255	434	38	the	the	DET
ma-255	434	39	convergence	convergence	NOUN
ma-255	434	40	order	order	NOUN
ma-255	434	41	closely	closely	ADV
ma-255	434	42	approximating	approximate	VERB
ma-255	434	43	2	2	NUM
ma-255	434	44	.	.	PUNCT
ma-255	434	45	example	example	NOUN
ma-255	434	46	4.2	4.2	NUM
ma-255	434	47	.	.	PUNCT
ma-255	435	1	let	let	VERB
ma-255	435	2	x	x	SYM
ma-255	435	3	=	=	SYM
ma-255	435	4	y	y	PROPN
ma-255	435	5	=	=	SYM
ma-255	435	6	r3	r3	PROPN
ma-255	435	7	and	and	CCONJ
ma-255	435	8	ω	ω	NUM
ma-255	435	9	=	=	SYM
ma-255	435	10	s[s∗	s[s∗	PROPN
ma-255	435	11	,	,	PUNCT
ma-255	435	12	1	1	NUM
ma-255	435	13	]	]	PUNCT
ma-255	435	14	.	.	PUNCT
ma-255	436	1	the	the	DET
ma-255	436	2	mapping	mapping	NOUN
ma-255	436	3	f	f	X
ma-255	436	4	is	be	AUX
ma-255	436	5	defined	define	VERB
ma-255	436	6	on	on	ADP
ma-255	436	7	ω	ω	NUM
ma-255	436	8	for	for	ADP
ma-255	436	9	a	a	DET
ma-255	436	10	=	=	SYM
ma-255	436	11	(	(	PUNCT
ma-255	436	12	a1	a1	PROPN
ma-255	436	13	,	,	PUNCT
ma-255	436	14	a2	a2	PROPN
ma-255	436	15	,	,	PUNCT
ma-255	436	16	a3	a3	NOUN
ma-255	436	17	)	)	PUNCT
ma-255	436	18	tr	tr	PROPN
ma-255	436	19	∈	∈	PROPN
ma-255	436	20	r3	r3	PROPN
ma-255	436	21	as	as	ADP
ma-255	436	22	f1(a	f1(a	PROPN
ma-255	436	23	)	)	PUNCT
ma-255	436	24	=	=	PUNCT
ma-255	436	25	(	(	PUNCT
ma-255	436	26	a1	a1	PROPN
ma-255	436	27	,	,	PUNCT
ma-255	436	28	e	e	NOUN
ma-255	436	29	a2	a2	NOUN
ma-255	436	30	−	−	PROPN
ma-255	436	31	1	1	NUM
ma-255	436	32	,	,	PUNCT
ma-255	436	33	e	e	NOUN
ma-255	436	34	−	−	PROPN
ma-255	436	35	1	1	NUM
ma-255	436	36	2	2	NUM
ma-255	436	37	a23	a23	NOUN
ma-255	436	38	+	+	SYM
ma-255	436	39	a3	a3	NOUN
ma-255	436	40	)	)	PUNCT
ma-255	436	41	tr	tr	VERB
ma-255	436	42	.	.	PUNCT
ma-255	437	1	then	then	ADV
ma-255	437	2	,	,	PUNCT
ma-255	437	3	the	the	DET
ma-255	437	4	definition	definition	NOUN
ma-255	437	5	of	of	ADP
ma-255	437	6	the	the	DET
ma-255	437	7	derivative	derivative	NOUN
ma-255	437	8	according	accord	VERB
ma-255	437	9	to	to	ADP
ma-255	437	10	fréchet	fréchet	NOUN
ma-255	437	11	[	[	X
ma-255	437	12	22,25	22,25	X
ma-255	437	13	]	]	PUNCT
ma-255	437	14	is	be	AUX
ma-255	437	15	given	give	VERB
ma-255	437	16	for	for	ADP
ma-255	437	17	the	the	DET
ma-255	437	18	mapping	mapping	NOUN
ma-255	437	19	f1	f1	NOUN
ma-255	437	20	f	f	PROPN
ma-255	437	21	′1(a	′1(a	PROPN
ma-255	437	22	)	)	PUNCT
ma-255	438	1	=	=	SYM
ma-255	438	2			NOUN
ma-255	438	3	1	1	NUM
ma-255	438	4	0	0	NUM
ma-255	438	5	0	0	NUM
ma-255	438	6	0	0	NUM
ma-255	438	7	ea2	ea2	VERB
ma-255	438	8	0	0	NUM
ma-255	438	9	0	0	NUM
ma-255	438	10	0	0	NUM
ma-255	439	1	(	(	PUNCT
ma-255	439	2	e	e	X
ma-255	439	3	−	−	PROPN
ma-255	439	4	1)a3	1)a3	NUM
ma-255	439	5	+	+	CCONJ
ma-255	439	6	1	1	NUM
ma-255	439	7			NOUN
ma-255	439	8	.	.	PUNCT
ma-255	440	1	the	the	DET
ma-255	440	2	point	point	NOUN
ma-255	440	3	s∗	s∗	PROPN
ma-255	440	4	=	=	SYM
ma-255	440	5	(	(	PUNCT
ma-255	440	6	0	0	NUM
ma-255	440	7	,	,	PUNCT
ma-255	440	8	0	0	NUM
ma-255	440	9	,	,	PUNCT
ma-255	440	10	0)tr	0)tr	NUM
ma-255	440	11	solves	solve	VERB
ma-255	440	12	the	the	DET
ma-255	440	13	equation	equation	NOUN
ma-255	440	14	f1(a	f1(a	PROPN
ma-255	440	15	)	)	PUNCT
ma-255	440	16	=	=	SYM
ma-255	441	1	0	0	X
ma-255	441	2	.	.	PUNCT
ma-255	442	1	moreover	moreover	ADV
ma-255	442	2	,	,	PUNCT
ma-255	442	3	f	f	PROPN
ma-255	442	4	′1(s∗	′1(s∗	PROPN
ma-255	442	5	)	)	PUNCT
ma-255	442	6	=	=	VERB
ma-255	442	7	i.	i.	NOUN
ma-255	442	8	the	the	DET
ma-255	442	9	conditions	condition	NOUN
ma-255	442	10	of	of	ADP
ma-255	442	11	the	the	DET
ma-255	442	12	theorem	theorem	ADJ
ma-255	442	13	2.2	2.2	NUM
ma-255	442	14	hold	hold	NOUN
ma-255	442	15	for	for	ADP
ma-255	442	16	p	p	NOUN
ma-255	442	17	=	=	NOUN
ma-255	442	18	1	1	NUM
ma-255	442	19	,	,	PUNCT
ma-255	442	20	if	if	SCONJ
ma-255	442	21	ϕ(t	ϕ(t	NUM
ma-255	442	22	)	)	PUNCT
ma-255	443	1	=	=	SYM
ma-255	443	2	a(t	a(t	NOUN
ma-255	443	3	)	)	PUNCT
ma-255	443	4	=	=	NOUN
ma-255	443	5	(	(	PUNCT
ma-255	443	6	e−1)t	e−1)t	PROPN
ma-255	443	7	.	.	PUNCT
ma-255	444	1	then	then	ADV
ma-255	444	2	,	,	PUNCT
ma-255	444	3	we	we	PRON
ma-255	444	4	can	can	AUX
ma-255	444	5	have	have	VERB
ma-255	444	6	r	r	NOUN
ma-255	444	7	∈	∈	PROPN
ma-255	444	8	(	(	PUNCT
ma-255	444	9	0	0	NUM
ma-255	444	10	,	,	PUNCT
ma-255	444	11	0.2909883534	0.2909883534	NUM
ma-255	444	12	)	)	PUNCT
ma-255	444	13	.	.	PUNCT
ma-255	445	1	example	example	NOUN
ma-255	446	1	4.3	4.3	NUM
ma-255	446	2	.	.	PUNCT
ma-255	447	1	let	let	VERB
ma-255	447	2	h[0	h[0	PROPN
ma-255	447	3	,	,	PUNCT
ma-255	447	4	1	1	NUM
ma-255	447	5	]	]	PUNCT
ma-255	447	6	stand	stand	VERB
ma-255	447	7	for	for	ADP
ma-255	447	8	the	the	DET
ma-255	447	9	space	space	NOUN
ma-255	447	10	of	of	ADP
ma-255	447	11	continuous	continuous	ADJ
ma-255	447	12	functions	function	NOUN
ma-255	447	13	mapping	map	VERB
ma-255	447	14	the	the	DET
ma-255	447	15	interval	interval	NOUN
ma-255	447	16	[	[	X
ma-255	447	17	0	0	NUM
ma-255	447	18	,	,	PUNCT
ma-255	447	19	1	1	NUM
ma-255	447	20	]	]	PUNCT
ma-255	447	21	into	into	ADP
ma-255	447	22	the	the	DET
ma-255	447	23	real	real	ADJ
ma-255	447	24	numbers	number	NOUN
ma-255	447	25	.	.	PUNCT
ma-255	448	1	let	let	VERB
ma-255	448	2	x	x	SYM
ma-255	448	3	=	=	SYM
ma-255	448	4	y	y	PROPN
ma-255	448	5	=	=	SYM
ma-255	448	6	h[0	h[0	PROPN
ma-255	448	7	,	,	PUNCT
ma-255	448	8	1	1	NUM
ma-255	448	9	]	]	PUNCT
ma-255	448	10	and	and	CCONJ
ma-255	448	11	ω	ω	NUM
ma-255	448	12	=	=	SYM
ma-255	448	13	s[s∗	s[s∗	PROPN
ma-255	448	14	,	,	PUNCT
ma-255	448	15	1	1	NUM
ma-255	448	16	]	]	PUNCT
ma-255	448	17	with	with	ADP
ma-255	448	18	s∗(v	s∗(v	NOUN
ma-255	448	19	)	)	PUNCT
ma-255	448	20	=	=	SYM
ma-255	449	1	0	0	X
ma-255	449	2	.	.	PUNCT
ma-255	450	1	the	the	DET
ma-255	450	2	operator	operator	NOUN
ma-255	450	3	f1	f1	NOUN
ma-255	450	4	is	be	AUX
ma-255	450	5	defined	define	VERB
ma-255	450	6	on	on	ADP
ma-255	450	7	h[0	h[0	PROPN
ma-255	450	8	,	,	PUNCT
ma-255	450	9	1	1	NUM
ma-255	450	10	]	]	PUNCT
ma-255	450	11	as	as	ADP
ma-255	450	12	f1(z)(v	f1(z)(v	NOUN
ma-255	450	13	)	)	PUNCT
ma-255	451	1	=	=	SYM
ma-255	451	2	z(v)−	z(v)−	PROPN
ma-255	451	3	4	4	NUM
ma-255	451	4	∫	∫	NOUN
ma-255	451	5	1	1	NUM
ma-255	451	6	0	0	NUM
ma-255	451	7	vz(τ)3dτ	vz(τ)3dτ	NOUN
ma-255	451	8	.	.	NOUN
ma-255	451	9	then	then	ADV
ma-255	451	10	,	,	PUNCT
ma-255	451	11	of	of	ADP
ma-255	451	12	the	the	DET
ma-255	451	13	derivative	derivative	NOUN
ma-255	451	14	according	accord	VERB
ma-255	451	15	to	to	ADP
ma-255	451	16	fréchet	fréchet	NOUN
ma-255	451	17	[	[	X
ma-255	451	18	1	1	NUM
ma-255	451	19	,	,	PUNCT
ma-255	451	20	10,15,22,30	10,15,22,30	PROPN
ma-255	451	21	]	]	PUNCT
ma-255	451	22	is	be	AUX
ma-255	451	23	given	give	VERB
ma-255	451	24	below	below	ADV
ma-255	451	25	for	for	ADP
ma-255	451	26	the	the	DET
ma-255	451	27	operator	operator	NOUN
ma-255	451	28	f1	f1	NOUN
ma-255	451	29	f	f	PROPN
ma-255	451	30	′1(z(w))(v	′1(z(w))(v	X
ma-255	451	31	)	)	PUNCT
ma-255	451	32	=	=	SYM
ma-255	451	33	w(v)−	w(v)−	PROPN
ma-255	451	34	12	12	NUM
ma-255	451	35	∫	∫	NOUN
ma-255	451	36	1	1	NUM
ma-255	451	37	0	0	NUM
ma-255	451	38	vτz(τ)2w(τ)dτ	vτz(τ)2w(τ)dτ	NOUN
ma-255	451	39	for	for	ADP
ma-255	451	40	each	each	DET
ma-255	451	41	w	w	PROPN
ma-255	451	42	∈	∈	PROPN
ma-255	451	43	h[0	h[0	PROPN
ma-255	451	44	,	,	PUNCT
ma-255	451	45	1	1	NUM
ma-255	451	46	]	]	PUNCT
ma-255	451	47	.	.	PUNCT
ma-255	452	1	therefore	therefore	ADV
ma-255	452	2	,	,	PUNCT
ma-255	452	3	the	the	DET
ma-255	452	4	conditions	condition	NOUN
ma-255	452	5	of	of	ADP
ma-255	452	6	the	the	DET
ma-255	452	7	theorem	theorem	ADJ
ma-255	452	8	2.2	2.2	NUM
ma-255	452	9	hold	hold	NOUN
ma-255	452	10	if	if	SCONJ
ma-255	452	11	,	,	PUNCT
ma-255	452	12	since	since	SCONJ
ma-255	452	13	for	for	ADP
ma-255	452	14	s∗	s∗	PROPN
ma-255	452	15	=	=	SYM
ma-255	452	16	0	0	NUM
ma-255	452	17	,	,	PUNCT
ma-255	452	18	f	f	PROPN
ma-255	452	19	′1(x	′1(x	NOUN
ma-255	452	20	∗(v	∗(v	NOUN
ma-255	452	21	)	)	PUNCT
ma-255	452	22	)	)	PUNCT
ma-255	453	1	=	=	PUNCT
ma-255	453	2	i	i	PRON
ma-255	453	3	hold	hold	VERB
ma-255	453	4	for	for	ADP
ma-255	453	5	that	that	DET
ma-255	453	6	p	p	NOUN
ma-255	453	7	=	=	NOUN
ma-255	453	8	1	1	NUM
ma-255	453	9	,	,	PUNCT
ma-255	453	10	if	if	SCONJ
ma-255	453	11	ϕ(t	ϕ(t	NUM
ma-255	453	12	)	)	PUNCT
ma-255	453	13	=	=	SYM
ma-255	454	1	a(t	a(t	NOUN
ma-255	454	2	)	)	PUNCT
ma-255	454	3	=	=	PUNCT
ma-255	454	4	6	6	NUM
ma-255	454	5	t.	t.	NOUN
ma-255	454	6	then	then	ADV
ma-255	454	7	,	,	PUNCT
ma-255	454	8	again	again	ADV
ma-255	454	9	by	by	ADP
ma-255	454	10	the	the	DET
ma-255	454	11	definition	definition	NOUN
ma-255	454	12	of	of	ADP
ma-255	454	13	r	r	NOUN
ma-255	454	14	,	,	PUNCT
ma-255	454	15	we	we	PRON
ma-255	454	16	can	can	AUX
ma-255	454	17	choose	choose	VERB
ma-255	454	18	r	r	NOUN
ma-255	454	19	∈	∈	PROPN
ma-255	454	20	(	(	PUNCT
ma-255	454	21	0	0	NUM
ma-255	454	22	,	,	PUNCT
ma-255	454	23	0.83̄	0.83̄	NOUN
ma-255	454	24	)	)	PUNCT
ma-255	454	25	.	.	PUNCT
ma-255	455	1	5	5	X
ma-255	455	2	.	.	X
ma-255	455	3	concluding	conclude	VERB
ma-255	455	4	remarks	remark	VERB
ma-255	455	5	the	the	DET
ma-255	455	6	paper	paper	NOUN
ma-255	455	7	addresses	address	VERB
ma-255	455	8	the	the	DET
ma-255	455	9	issue	issue	NOUN
ma-255	455	10	with	with	ADP
ma-255	455	11	the	the	DET
ma-255	455	12	inverses	inverse	NOUN
ma-255	455	13	appearing	appear	VERB
ma-255	455	14	in	in	ADP
ma-255	455	15	the	the	DET
ma-255	455	16	study	study	NOUN
ma-255	455	17	of	of	ADP
ma-255	455	18	the	the	DET
ma-255	455	19	convergenceof	convergenceof	ADJ
ma-255	455	20	simple	simple	ADJ
ma-255	455	21	-	-	PUNCT
ma-255	455	22	step	step	NOUN
ma-255	455	23	iterative	iterative	NOUN
ma-255	455	24	methods	method	NOUN
ma-255	455	25	.	.	PUNCT
ma-255	456	1	it	it	PRON
ma-255	456	2	is	be	AUX
ma-255	456	3	shown	show	VERB
ma-255	456	4	that	that	SCONJ
ma-255	456	5	the	the	DET
ma-255	456	6	inverse	inverse	NOUN
ma-255	456	7	can	can	AUX
ma-255	456	8	be	be	AUX
ma-255	456	9	replaced	replace	VERB
ma-255	456	10	by	by	ADP
ma-255	456	11	a	a	DET
ma-255	456	12	finite	finite	ADJ
ma-255	456	13	sumof	sumof	PROPN
ma-255	456	14	linear	linear	PROPN
ma-255	456	15	operators	operator	NOUN
ma-255	456	16	related	relate	VERB
ma-255	456	17	to	to	ADP
ma-255	456	18	the	the	DET
ma-255	456	19	operator	operator	NOUN
ma-255	456	20	involved	involve	VERB
ma-255	456	21	.	.	PUNCT
ma-255	457	1	the	the	DET
ma-255	457	2	resulting	result	VERB
ma-255	457	3	hybrid	hybrid	ADJ
ma-255	457	4	methods	method	NOUN
ma-255	457	5	demonstratethe	demonstratethe	ADP
ma-255	457	6	effectiveness	effectiveness	NOUN
ma-255	457	7	of	of	ADP
ma-255	457	8	these	these	DET
ma-255	457	9	methods	method	NOUN
ma-255	457	10	since	since	SCONJ
ma-255	457	11	the	the	DET
ma-255	457	12	number	number	NOUN
ma-255	457	13	of	of	ADP
ma-255	457	14	iterations	iteration	NOUN
ma-255	457	15	is	be	AUX
ma-255	457	16	essentially	essentially	ADV
ma-255	457	17	the	the	DET
ma-255	457	18	same	same	ADJ
ma-255	457	19	as	as	ADP
ma-255	457	20	wellas	wellas	PRON
ma-255	457	21	the	the	DET
ma-255	457	22	convergence	convergence	NOUN
ma-255	457	23	order	order	NOUN
ma-255	457	24	of	of	ADP
ma-255	457	25	the	the	DET
ma-255	457	26	methods	method	NOUN
ma-255	457	27	.	.	PUNCT
ma-255	458	1	however	however	ADV
ma-255	458	2	,	,	PUNCT
ma-255	458	3	the	the	DET
ma-255	458	4	hybrid	hybrid	NOUN
ma-255	458	5	method	method	NOUN
ma-255	458	6	is	be	AUX
ma-255	458	7	cheaper	cheap	ADJ
ma-255	458	8	to	to	ADP
ma-255	458	9	implement.this	implement.this	PRON
ma-255	458	10	idea	idea	NOUN
ma-255	458	11	can	can	AUX
ma-255	458	12	be	be	AUX
ma-255	458	13	used	use	VERB
ma-255	458	14	for	for	ADP
ma-255	458	15	multiple	multiple	ADJ
ma-255	458	16	steps	step	NOUN
ma-255	458	17	and	and	CCONJ
ma-255	458	18	multiple	multiple	ADJ
ma-255	458	19	point	point	NOUN
ma-255	458	20	methods	method	NOUN
ma-255	458	21	with	with	ADP
ma-255	458	22	the	the	DET
ma-255	458	23	same	same	ADJ
ma-255	458	24	advantages[3	advantages[3	PROPN
ma-255	458	25	,	,	PUNCT
ma-255	458	26	5–7,9	5–7,9	NUM
ma-255	458	27	,	,	PUNCT
ma-255	458	28	14,23,24,31	14,23,24,31	NUM
ma-255	458	29	]	]	PUNCT
ma-255	458	30	.	.	PUNCT
ma-255	459	1	this	this	PRON
ma-255	459	2	is	be	AUX
ma-255	459	3	the	the	DET
ma-255	459	4	direction	direction	NOUN
ma-255	459	5	of	of	ADP
ma-255	459	6	future	future	ADJ
ma-255	459	7	research	research	NOUN
ma-255	459	8	.	.	PUNCT
ma-255	460	1	acknowledgement	acknowledgement	NOUN
ma-255	460	2	:	:	PUNCT
ma-255	460	3	we	we	PRON
ma-255	460	4	would	would	AUX
ma-255	460	5	like	like	VERB
ma-255	460	6	to	to	PART
ma-255	460	7	thank	thank	VERB
ma-255	460	8	graduate	graduate	NOUN
ma-255	460	9	student	student	NOUN
ma-255	460	10	mr	mr	PROPN
ma-255	460	11	.	.	PROPN
ma-255	460	12	mykhailo	mykhailo	PROPN
ma-255	460	13	havdiak	havdiak	PROPN
ma-255	460	14	from	from	ADP
ma-255	460	15	the	the	DET
ma-255	460	16	department	department	NOUN
ma-255	460	17	of	of	ADP
ma-255	460	18	optimalprocesses	optimalprocesse	NOUN
ma-255	460	19	,	,	PUNCT
ma-255	460	20	ivan	ivan	PROPN
ma-255	460	21	franko	franko	PROPN
ma-255	460	22	national	national	PROPN
ma-255	460	23	university	university	PROPN
ma-255	460	24	of	of	ADP
ma-255	460	25	lviv	lviv	PROPN
ma-255	460	26	,	,	PUNCT
ma-255	460	27	lviv	lviv	PROPN
ma-255	460	28	ukraine	ukraine	PROPN
ma-255	460	29	for	for	ADP
ma-255	460	30	providing	provide	VERB
ma-255	460	31	example	example	NOUN
ma-255	460	32	4.1	4.1	NUM
ma-255	460	33	.	.	PUNCT
ma-255	461	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	461	2	eur	eur	PROPN
ma-255	461	3	.	.	PUNCT
ma-255	462	1	j.	j.	PROPN
ma-255	462	2	math	math	PROPN
ma-255	462	3	.	.	PUNCT
ma-255	463	1	anal	anal	PROPN
ma-255	463	2	.	.	PUNCT
ma-255	464	1	10.28924	10.28924	NUM
ma-255	464	2	/	/	SYM
ma-255	464	3	ada	ada	PROPN
ma-255	464	4	/	/	SYM
ma-255	464	5	ma.5.5	ma.5.5	PROPN
ma-255	464	6	18references	18references	X
ma-255	465	1	[	[	X
ma-255	465	2	1	1	X
ma-255	465	3	]	]	PUNCT
ma-255	465	4	s.	s.	PROPN
ma-255	465	5	adly	adly	PROPN
ma-255	465	6	,	,	PUNCT
ma-255	465	7	r.	r.	PROPN
ma-255	465	8	cibulka	cibulka	PROPN
ma-255	465	9	,	,	PUNCT
ma-255	465	10	h.v	h.v	PROPN
ma-255	465	11	.	.	PROPN
ma-255	465	12	ngai	ngai	PROPN
ma-255	465	13	,	,	PUNCT
ma-255	465	14	newton	newton	PROPN
ma-255	465	15	’s	’s	PART
ma-255	465	16	method	method	NOUN
ma-255	465	17	for	for	ADP
ma-255	465	18	solving	solve	VERB
ma-255	465	19	inclusions	inclusion	NOUN
ma-255	465	20	using	use	VERB
ma-255	465	21	set	set	NOUN
ma-255	465	22	-	-	PUNCT
ma-255	465	23	valued	value	VERB
ma-255	465	24	approximations	approximation	NOUN
ma-255	465	25	,	,	PUNCT
ma-255	465	26	siam	siam	PROPN
ma-255	465	27	j.optim	j.optim	NOUN
ma-255	465	28	.	.	PUNCT
ma-255	466	1	25	25	NUM
ma-255	466	2	(	(	PUNCT
ma-255	466	3	2015	2015	NUM
ma-255	466	4	)	)	PUNCT
ma-255	466	5	159	159	NUM
ma-255	466	6	-	-	SYM
ma-255	466	7	184.[2	184.[2	NUM
ma-255	466	8	]	]	PUNCT
ma-255	466	9	s.	s.	PROPN
ma-255	466	10	adly	adly	PROPN
ma-255	466	11	,	,	PUNCT
ma-255	466	12	h.v	h.v	PROPN
ma-255	466	13	.	.	PROPN
ma-255	466	14	ngai	ngai	PROPN
ma-255	466	15	,	,	PUNCT
ma-255	466	16	n.v	n.v	PROPN
ma-255	466	17	.	.	PROPN
ma-255	466	18	vu	vu	PROPN
ma-255	466	19	,	,	PUNCT
ma-255	466	20	newton	newton	PROPN
ma-255	466	21	’s	’s	PART
ma-255	466	22	method	method	NOUN
ma-255	466	23	for	for	ADP
ma-255	466	24	solving	solve	VERB
ma-255	466	25	generalized	generalized	ADJ
ma-255	466	26	equations	equation	NOUN
ma-255	466	27	:	:	PUNCT
ma-255	466	28	kantorovich	kantorovich	PROPN
ma-255	466	29	’s	’s	PART
ma-255	466	30	and	and	CCONJ
ma-255	466	31	smale	smale	PROPN
ma-255	466	32	’s	’s	PART
ma-255	466	33	ap	ap	PROPN
ma-255	466	34	-	-	PUNCT
ma-255	466	35	proaches	proaches	PROPN
ma-255	466	36	,	,	PUNCT
ma-255	466	37	j.	j.	PROPN
ma-255	466	38	math	math	PROPN
ma-255	466	39	.	.	PUNCT
ma-255	467	1	anal	anal	PROPN
ma-255	467	2	.	.	PUNCT
ma-255	468	1	appl	appl	PROPN
ma-255	468	2	.	.	PUNCT
ma-255	469	1	439	439	NUM
ma-255	469	2	(	(	PUNCT
ma-255	469	3	2016	2016	NUM
ma-255	469	4	)	)	PUNCT
ma-255	469	5	396	396	NUM
ma-255	469	6	-	-	SYM
ma-255	469	7	418.[3	418.[3	NUM
ma-255	469	8	]	]	X
ma-255	469	9	e.l	e.l	PROPN
ma-255	469	10	.	.	PROPN
ma-255	469	11	allgower	allgower	PROPN
ma-255	469	12	,	,	PUNCT
ma-255	469	13	k.	k.	PROPN
ma-255	469	14	georg	georg	PROPN
ma-255	469	15	,	,	PUNCT
ma-255	469	16	introduction	introduction	NOUN
ma-255	469	17	to	to	ADP
ma-255	469	18	numerical	numerical	ADJ
ma-255	469	19	continuation	continuation	NOUN
ma-255	469	20	algorithms	algorithm	NOUN
ma-255	469	21	,	,	PUNCT
ma-255	469	22	springer	springer	NOUN
ma-255	469	23	,	,	PUNCT
ma-255	469	24	berlin	berlin	PROPN
ma-255	469	25	,	,	PUNCT
ma-255	469	26	heidelberg	heidelberg	PROPN
ma-255	469	27	new	new	ADJ
ma-255	469	28	york,1989.[4	york,1989.[4	PROPN
ma-255	469	29	]	]	X
ma-255	469	30	f.j	f.j	PROPN
ma-255	469	31	.	.	PROPN
ma-255	469	32	aragón	aragón	PROPN
ma-255	469	33	artacho	artacho	PROPN
ma-255	469	34	,	,	PUNCT
ma-255	469	35	a.	a.	NOUN
ma-255	469	36	melyakov	melyakov	PROPN
ma-255	469	37	,	,	PUNCT
ma-255	469	38	a.l	a.l	PROPN
ma-255	469	39	.	.	PROPN
ma-255	469	40	dontchev	dontchev	PROPN
ma-255	469	41	,	,	PUNCT
ma-255	469	42	m.	m.	NOUN
ma-255	469	43	lopez	lopez	NOUN
ma-255	469	44	,	,	PUNCT
ma-255	469	45	local	local	ADJ
ma-255	469	46	convergence	convergence	NOUN
ma-255	469	47	of	of	ADP
ma-255	469	48	quasi	quasi	ADJ
ma-255	469	49	-	-	ADJ
ma-255	469	50	newton	newton	PROPN
ma-255	469	51	methods	method	NOUN
ma-255	469	52	undermetric	undermetric	ADJ
ma-255	469	53	regularity	regularity	NOUN
ma-255	469	54	,	,	PUNCT
ma-255	469	55	comput	comput	NOUN
ma-255	469	56	.	.	PUNCT
ma-255	470	1	optim	optim	PROPN
ma-255	470	2	.	.	PUNCT
ma-255	470	3	appl	appl	PROPN
ma-255	470	4	.	.	PUNCT
ma-255	471	1	58	58	NUM
ma-255	471	2	(	(	PUNCT
ma-255	471	3	2014	2014	NUM
ma-255	471	4	)	)	PUNCT
ma-255	471	5	225	225	NUM
ma-255	471	6	-	-	SYM
ma-255	471	7	247.[5	247.[5	NUM
ma-255	471	8	]	]	X
ma-255	471	9	i.k	i.k	PROPN
ma-255	471	10	.	.	PROPN
ma-255	471	11	argyros	argyros	PROPN
ma-255	471	12	,	,	PUNCT
ma-255	471	13	convergence	convergence	NOUN
ma-255	471	14	and	and	CCONJ
ma-255	471	15	applications	application	NOUN
ma-255	471	16	of	of	ADP
ma-255	471	17	newton	newton	NOUN
ma-255	471	18	-	-	PUNCT
ma-255	471	19	type	type	NOUN
ma-255	471	20	iterations	iteration	NOUN
ma-255	471	21	,	,	PUNCT
ma-255	471	22	springer	springer	NOUN
ma-255	471	23	,	,	PUNCT
ma-255	471	24	berlin	berlin	PROPN
ma-255	471	25	,	,	PUNCT
ma-255	471	26	2008.[6	2008.[6	NUM
ma-255	471	27	]	]	X
ma-255	471	28	i.k	i.k	PROPN
ma-255	471	29	.	.	PROPN
ma-255	471	30	argyros	argyros	PROPN
ma-255	471	31	,	,	PUNCT
ma-255	471	32	the	the	DET
ma-255	471	33	theory	theory	NOUN
ma-255	471	34	and	and	CCONJ
ma-255	471	35	application	application	NOUN
ma-255	471	36	of	of	ADP
ma-255	471	37	iterative	iterative	ADJ
ma-255	471	38	methods	method	NOUN
ma-255	471	39	with	with	ADP
ma-255	471	40	applications	application	NOUN
ma-255	471	41	,	,	PUNCT
ma-255	471	42	second	second	ADJ
ma-255	471	43	edition	edition	NOUN
ma-255	471	44	,	,	PUNCT
ma-255	471	45	crc	crc	NOUN
ma-255	471	46	press	press	NOUN
ma-255	471	47	,	,	PUNCT
ma-255	471	48	taylorand	taylorand	NOUN
ma-255	471	49	francis	francis	PROPN
ma-255	471	50	,	,	PUNCT
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ma-255	471	53	,	,	PUNCT
ma-255	471	54	2002.[7	2002.[7	NUM
ma-255	471	55	]	]	X
ma-255	471	56	i.k	i.k	PROPN
ma-255	471	57	.	.	PROPN
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ma-255	471	59	,	,	PUNCT
ma-255	471	60	s.	s.	PROPN
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ma-255	471	62	,	,	PUNCT
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ma-255	471	64	the	the	DET
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ma-255	471	66	convergence	convergence	NOUN
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ma-255	471	68	for	for	ADP
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ma-255	471	70	-	-	PUNCT
ma-255	471	71	type	type	NOUN
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ma-255	471	73	solving	solve	VERB
ma-255	471	74	generalizedequations	generalizedequation	NOUN
ma-255	471	75	with	with	ADP
ma-255	471	76	the	the	DET
ma-255	471	77	aubi	aubi	PROPN
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ma-255	471	79	,	,	PUNCT
ma-255	471	80	j.	j.	PROPN
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ma-255	471	82	.	.	PUNCT
ma-255	472	1	81	81	NUM
ma-255	472	2	(	(	PUNCT
ma-255	472	3	2024	2024	NUM
ma-255	472	4	)	)	PUNCT
ma-255	472	5	101817.[8	101817.[8	NUM
ma-255	472	6	]	]	PUNCT
ma-255	472	7	j.-p	j.-p	PROPN
ma-255	472	8	.	.	PUNCT
ma-255	473	1	aubin	aubin	PROPN
ma-255	473	2	,	,	PUNCT
ma-255	473	3	h.	h.	PROPN
ma-255	473	4	frankowska	frankowska	PROPN
ma-255	473	5	,	,	PUNCT
ma-255	473	6	set	set	NOUN
ma-255	473	7	-	-	PUNCT
ma-255	473	8	valued	value	VERB
ma-255	473	9	analysis	analysis	NOUN
ma-255	473	10	,	,	PUNCT
ma-255	473	11	systems	system	NOUN
ma-255	473	12	and	and	CCONJ
ma-255	473	13	control	control	NOUN
ma-255	473	14	:	:	PUNCT
ma-255	473	15	foundations	foundation	NOUN
ma-255	473	16	and	and	CCONJ
ma-255	473	17	applications	application	NOUN
ma-255	473	18	,	,	PUNCT
ma-255	473	19	birkhäuser	birkhäuser	NOUN
ma-255	473	20	,	,	PUNCT
ma-255	473	21	boston	boston	PROPN
ma-255	473	22	,	,	PUNCT
ma-255	473	23	1990.[9	1990.[9	NUM
ma-255	473	24	]	]	X
ma-255	473	25	e.	e.	PROPN
ma-255	473	26	catinas	catinas	PROPN
ma-255	473	27	,	,	PUNCT
ma-255	473	28	the	the	DET
ma-255	473	29	inexact	inexact	ADJ
ma-255	473	30	,	,	PUNCT
ma-255	473	31	inexact	inexact	ADJ
ma-255	473	32	perturbed	perturb	VERB
ma-255	473	33	,	,	PUNCT
ma-255	473	34	and	and	CCONJ
ma-255	473	35	quasi	quasi	PROPN
ma-255	473	36	-	-	ADJ
ma-255	473	37	newton	newton	PROPN
ma-255	473	38	algorithms	algorithm	NOUN
ma-255	473	39	are	be	AUX
ma-255	473	40	equivalent	equivalent	ADJ
ma-255	473	41	models	model	NOUN
ma-255	473	42	,	,	PUNCT
ma-255	473	43	math	math	NOUN
ma-255	473	44	.	.	PUNCT
ma-255	474	1	comp	comp	PROPN
ma-255	474	2	.	.	PUNCT
ma-255	475	1	74(2005	74(2005	NUM
ma-255	475	2	)	)	PUNCT
ma-255	475	3	291	291	NUM
ma-255	475	4	-	-	SYM
ma-255	475	5	301.[10	301.[10	NUM
ma-255	475	6	]	]	X
ma-255	475	7	p.g	p.g	PROPN
ma-255	475	8	.	.	PROPN
ma-255	475	9	ciarlet	ciarlet	PROPN
ma-255	475	10	,	,	PUNCT
ma-255	475	11	c.	c.	PROPN
ma-255	475	12	mardare	mardare	NOUN
ma-255	475	13	,	,	PUNCT
ma-255	475	14	on	on	ADP
ma-255	475	15	the	the	DET
ma-255	475	16	newton	newton	PROPN
ma-255	475	17	-	-	PUNCT
ma-255	475	18	kantorovich	kantorovich	PROPN
ma-255	475	19	theorem	theorem	PROPN
ma-255	475	20	,	,	PUNCT
ma-255	475	21	anal	anal	NOUN
ma-255	475	22	.	.	PUNCT
ma-255	476	1	appl	appl	PROPN
ma-255	476	2	.	.	PROPN
ma-255	477	1	10	10	NUM
ma-255	477	2	(	(	PUNCT
ma-255	477	3	2012	2012	NUM
ma-255	477	4	)	)	PUNCT
ma-255	477	5	249	249	NUM
ma-255	477	6	-	-	SYM
ma-255	477	7	269.[11	269.[11	NUM
ma-255	477	8	]	]	X
ma-255	477	9	r.	r.	PROPN
ma-255	477	10	cibulka	cibulka	PROPN
ma-255	477	11	,	,	PUNCT
ma-255	477	12	a.l	a.l	PROPN
ma-255	477	13	.	.	PROPN
ma-255	477	14	dontchev	dontchev	PROPN
ma-255	477	15	,	,	PUNCT
ma-255	477	16	j.	j.	PROPN
ma-255	477	17	preininger	preininger	PROPN
ma-255	477	18	,	,	PUNCT
ma-255	477	19	t.	t.	PROPN
ma-255	477	20	roubal	roubal	PROPN
ma-255	477	21	,	,	PUNCT
ma-255	477	22	v.	v.	PROPN
ma-255	477	23	veliov	veliov	PROPN
ma-255	477	24	,	,	PUNCT
ma-255	477	25	kantorovich	kantorovich	NOUN
ma-255	477	26	-	-	PUNCT
ma-255	477	27	type	type	NOUN
ma-255	477	28	theorems	theorem	NOUN
ma-255	477	29	for	for	ADP
ma-255	477	30	generalized	generalized	ADJ
ma-255	477	31	equations	equation	NOUN
ma-255	477	32	,	,	PUNCT
ma-255	477	33	j.	j.	PROPN
ma-255	477	34	convex	convex	PROPN
ma-255	477	35	anal	anal	NOUN
ma-255	477	36	.	.	PUNCT
ma-255	478	1	25	25	NUM
ma-255	478	2	(	(	PUNCT
ma-255	478	3	2018	2018	NUM
ma-255	478	4	)	)	PUNCT
ma-255	478	5	459	459	NUM
ma-255	478	6	-	-	SYM
ma-255	478	7	486.[12	486.[12	NUM
ma-255	478	8	]	]	X
ma-255	478	9	a.l	a.l	PROPN
ma-255	478	10	.	.	PROPN
ma-255	478	11	dontchev	dontchev	PROPN
ma-255	478	12	,	,	PUNCT
ma-255	478	13	local	local	ADJ
ma-255	478	14	analysis	analysis	NOUN
ma-255	478	15	of	of	ADP
ma-255	478	16	a	a	DET
ma-255	478	17	newton	newton	NOUN
ma-255	478	18	-	-	PUNCT
ma-255	478	19	type	type	NOUN
ma-255	478	20	method	method	NOUN
ma-255	478	21	based	base	VERB
ma-255	478	22	on	on	ADP
ma-255	478	23	partial	partial	ADJ
ma-255	478	24	linearization	linearization	NOUN
ma-255	478	25	,	,	PUNCT
ma-255	478	26	in	in	ADP
ma-255	478	27	:	:	PUNCT
ma-255	478	28	the	the	DET
ma-255	478	29	mathematics	mathematic	NOUN
ma-255	478	30	ofnumerical	ofnumerical	ADJ
ma-255	478	31	analysis	analysis	NOUN
ma-255	478	32	,	,	PUNCT
ma-255	478	33	lectures	lecture	VERB
ma-255	478	34	appl	appl	PROPN
ma-255	478	35	.	.	PUNCT
ma-255	479	1	math	math	NOUN
ma-255	479	2	.	.	PUNCT
ma-255	480	1	32	32	NUM
ma-255	480	2	,	,	PUNCT
ma-255	480	3	amer	amer	PROPN
ma-255	480	4	.	.	PROPN
ma-255	480	5	math	math	PROPN
ma-255	480	6	.	.	PUNCT
ma-255	481	1	soc	soc	PROPN
ma-255	481	2	.	.	PUNCT
ma-255	481	3	,	,	PUNCT
ma-255	481	4	providence	providence	NOUN
ma-255	481	5	,	,	PUNCT
ma-255	481	6	1996	1996	NUM
ma-255	481	7	,	,	PUNCT
ma-255	481	8	pp	pp	ADP
ma-255	481	9	.	.	PUNCT
ma-255	482	1	295	295	NUM
ma-255	482	2	-	-	SYM
ma-255	482	3	306.[13	306.[13	NUM
ma-255	482	4	]	]	X
ma-255	482	5	a.l	a.l	PROPN
ma-255	482	6	.	.	PROPN
ma-255	482	7	dontchev	dontchev	PROPN
ma-255	482	8	,	,	PUNCT
ma-255	482	9	r.t	r.t	PROPN
ma-255	482	10	.	.	PROPN
ma-255	482	11	rockafellar	rockafellar	ADJ
ma-255	482	12	,	,	PUNCT
ma-255	482	13	implicit	implicit	ADJ
ma-255	482	14	functions	function	NOUN
ma-255	482	15	and	and	CCONJ
ma-255	482	16	solution	solution	NOUN
ma-255	482	17	mappings	mapping	NOUN
ma-255	482	18	:	:	PUNCT
ma-255	482	19	a	a	DET
ma-255	482	20	view	view	NOUN
ma-255	482	21	from	from	ADP
ma-255	482	22	variational	variational	ADJ
ma-255	482	23	analysis	analysis	NOUN
ma-255	482	24	,	,	PUNCT
ma-255	482	25	2nd	2nd	ADJ
ma-255	482	26	ed	ed	NOUN
ma-255	482	27	.	.	PROPN
ma-255	482	28	,springer	,springer	PUNCT
ma-255	482	29	,	,	PUNCT
ma-255	482	30	berlin	berlin	PROPN
ma-255	482	31	,	,	PUNCT
ma-255	482	32	2014.[14	2014.[14	PROPN
ma-255	482	33	]	]	X
ma-255	482	34	p.	p.	NOUN
ma-255	482	35	deuflhard	deuflhard	PROPN
ma-255	482	36	,	,	PUNCT
ma-255	482	37	newton	newton	PROPN
ma-255	482	38	algorithms	algorithm	VERB
ma-255	482	39	for	for	ADP
ma-255	482	40	nonlinear	nonlinear	ADJ
ma-255	482	41	problems	problem	NOUN
ma-255	482	42	:	:	PUNCT
ma-255	482	43	affine	affine	VERB
ma-255	482	44	invariance	invariance	NOUN
ma-255	482	45	and	and	CCONJ
ma-255	482	46	adaptive	adaptive	ADJ
ma-255	482	47	algorithms	algorithm	NOUN
ma-255	482	48	,	,	PUNCT
ma-255	482	49	springer	springer	NOUN
ma-255	482	50	seriesin	seriesin	PROPN
ma-255	482	51	computational	computational	ADJ
ma-255	482	52	mathematics	mathematic	NOUN
ma-255	482	53	,	,	PUNCT
ma-255	482	54	vol	vol	NOUN
ma-255	482	55	.	.	PROPN
ma-255	482	56	35	35	NUM
ma-255	482	57	,	,	PUNCT
ma-255	482	58	springer	springer	NOUN
ma-255	482	59	-	-	PUNCT
ma-255	482	60	verlag	verlag	PROPN
ma-255	482	61	,	,	PUNCT
ma-255	482	62	berlin	berlin	PROPN
ma-255	482	63	,	,	PUNCT
ma-255	482	64	2004.[15	2004.[15	PROPN
ma-255	482	65	]	]	X
ma-255	482	66	j.a	j.a	PROPN
ma-255	482	67	.	.	PROPN
ma-255	482	68	ezquerro	ezquerro	PROPN
ma-255	482	69	,	,	PUNCT
ma-255	482	70	j.m	j.m	PROPN
ma-255	482	71	.	.	PROPN
ma-255	482	72	gutiérrez	gutiérrez	PROPN
ma-255	482	73	,	,	PUNCT
ma-255	482	74	m.a	m.a	PROPN
ma-255	482	75	.	.	PROPN
ma-255	482	76	hernández	hernández	PROPN
ma-255	482	77	,	,	PUNCT
ma-255	482	78	n.	n.	PROPN
ma-255	482	79	romero	romero	PROPN
ma-255	482	80	,	,	PUNCT
ma-255	482	81	m.j	m.j	PROPN
ma-255	482	82	.	.	PROPN
ma-255	482	83	rubio	rubio	PROPN
ma-255	482	84	,	,	PUNCT
ma-255	482	85	the	the	DET
ma-255	482	86	newton	newton	PROPN
ma-255	482	87	algorithm	algorithm	PROPN
ma-255	482	88	:	:	PUNCT
ma-255	482	89	from	from	ADP
ma-255	482	90	newton	newton	PROPN
ma-255	482	91	tokantorovich	tokantorovich	PROPN
ma-255	482	92	(	(	PUNCT
ma-255	482	93	spanish	spanish	ADJ
ma-255	482	94	)	)	PUNCT
ma-255	482	95	,	,	PUNCT
ma-255	482	96	gac	gac	PROPN
ma-255	482	97	.	.	PUNCT
ma-255	482	98	r.	r.	PROPN
ma-255	482	99	soc	soc	PROPN
ma-255	482	100	.	.	PUNCT
ma-255	483	1	mat	mat	PROPN
ma-255	483	2	.	.	PUNCT
ma-255	484	1	esp	esp	PROPN
ma-255	484	2	.	.	PUNCT
ma-255	485	1	13	13	NUM
ma-255	485	2	(	(	PUNCT
ma-255	485	3	2010	2010	NUM
ma-255	485	4	)	)	PUNCT
ma-255	485	5	53	53	NUM
ma-255	485	6	-	-	SYM
ma-255	485	7	76.[16	76.[16	NUM
ma-255	485	8	]	]	X
ma-255	485	9	j.a	j.a	PROPN
ma-255	485	10	.	.	PROPN
ma-255	485	11	ezquerro	ezquerro	PROPN
ma-255	485	12	,	,	PUNCT
ma-255	485	13	m.a	m.a	PROPN
ma-255	485	14	.	.	PROPN
ma-255	485	15	hernández	hernández	PROPN
ma-255	485	16	-	-	PUNCT
ma-255	485	17	veron	veron	PROPN
ma-255	485	18	,	,	PUNCT
ma-255	485	19	domains	domain	NOUN
ma-255	485	20	of	of	ADP
ma-255	485	21	global	global	ADJ
ma-255	485	22	convergence	convergence	NOUN
ma-255	485	23	for	for	ADP
ma-255	485	24	newton	newton	PROPN
ma-255	485	25	’s	’s	PART
ma-255	485	26	algorithm	algorithm	NOUN
ma-255	485	27	from	from	ADP
ma-255	485	28	auxiliary	auxiliary	ADJ
ma-255	485	29	points	point	NOUN
ma-255	485	30	,	,	PUNCT
ma-255	485	31	appl	appl	PROPN
ma-255	485	32	.	.	PROPN
ma-255	485	33	math	math	PROPN
ma-255	485	34	.	.	PUNCT
ma-255	486	1	lett	lett	PROPN
ma-255	486	2	.	.	PROPN
ma-255	486	3	85	85	NUM
ma-255	486	4	(	(	PUNCT
ma-255	486	5	2018	2018	NUM
ma-255	486	6	)	)	PUNCT
ma-255	486	7	48	48	NUM
ma-255	486	8	-	-	SYM
ma-255	486	9	56.[17	56.[17	PROPN
ma-255	486	10	]	]	X
ma-255	486	11	f.	f.	PROPN
ma-255	486	12	facchinei	facchinei	PROPN
ma-255	486	13	,	,	PUNCT
ma-255	486	14	j.-s	j.-s	PROPN
ma-255	486	15	.	.	PUNCT
ma-255	487	1	pang	pang	NOUN
ma-255	487	2	,	,	PUNCT
ma-255	487	3	finite	finite	ADJ
ma-255	487	4	-	-	ADJ
ma-255	487	5	dimensional	dimensional	ADJ
ma-255	487	6	variational	variational	ADJ
ma-255	487	7	inequalities	inequality	NOUN
ma-255	487	8	and	and	CCONJ
ma-255	487	9	complementarity	complementarity	NOUN
ma-255	487	10	problems	problem	NOUN
ma-255	487	11	,	,	PUNCT
ma-255	487	12	springer	springer	NOUN
ma-255	487	13	,	,	PUNCT
ma-255	487	14	newyork	newyork	NOUN
ma-255	487	15	,	,	PUNCT
ma-255	487	16	2003.[18	2003.[18	NUM
ma-255	487	17	]	]	X
ma-255	487	18	o.p	o.p	PROPN
ma-255	487	19	.	.	PROPN
ma-255	487	20	ferreira	ferreira	PROPN
ma-255	487	21	,	,	PUNCT
ma-255	487	22	g.n	g.n	PROPN
ma-255	487	23	.	.	PROPN
ma-255	487	24	silva	silva	PROPN
ma-255	487	25	,	,	PUNCT
ma-255	487	26	inexact	inexact	PROPN
ma-255	487	27	newton	newton	PROPN
ma-255	487	28	’s	’s	PART
ma-255	487	29	method	method	NOUN
ma-255	487	30	to	to	ADP
ma-255	487	31	nonlinear	nonlinear	ADJ
ma-255	487	32	functions	function	NOUN
ma-255	487	33	with	with	ADP
ma-255	487	34	values	value	NOUN
ma-255	487	35	in	in	ADP
ma-255	487	36	a	a	DET
ma-255	487	37	cone	cone	NOUN
ma-255	487	38	,	,	PUNCT
ma-255	487	39	appl	appl	PROPN
ma-255	487	40	.	.	PROPN
ma-255	487	41	anal	anal	PROPN
ma-255	487	42	.	.	PUNCT
ma-255	488	1	97(2018	97(2018	NOUN
ma-255	488	2	)	)	PUNCT
ma-255	488	3	1461	1461	NUM
ma-255	488	4	-	-	SYM
ma-255	488	5	1477.[19	1477.[19	NUM
ma-255	488	6	]	]	X
ma-255	488	7	a.f	a.f	PROPN
ma-255	488	8	.	.	PROPN
ma-255	488	9	izmailov	izmailov	PROPN
ma-255	488	10	,	,	PUNCT
ma-255	488	11	m.v	m.v	PROPN
ma-255	488	12	.	.	PROPN
ma-255	489	1	solodov	solodov	PROPN
ma-255	489	2	,	,	PUNCT
ma-255	489	3	newton	newton	NOUN
ma-255	489	4	-	-	PUNCT
ma-255	489	5	type	type	NOUN
ma-255	489	6	methods	method	NOUN
ma-255	489	7	for	for	ADP
ma-255	489	8	optimization	optimization	NOUN
ma-255	489	9	and	and	CCONJ
ma-255	489	10	variational	variational	ADJ
ma-255	489	11	problems	problem	NOUN
ma-255	489	12	,	,	PUNCT
ma-255	489	13	springer	springer	NOUN
ma-255	489	14	,	,	PUNCT
ma-255	489	15	berlin	berlin	PROPN
ma-255	489	16	,	,	PUNCT
ma-255	489	17	2014.[20	2014.[20	NUM
ma-255	489	18	]	]	X
ma-255	489	19	c.t	c.t	PROPN
ma-255	489	20	.	.	PROPN
ma-255	489	21	kelley	kelley	PROPN
ma-255	489	22	,	,	PUNCT
ma-255	489	23	solving	solve	VERB
ma-255	489	24	nonlinear	nonlinear	ADJ
ma-255	489	25	equations	equation	NOUN
ma-255	489	26	with	with	ADP
ma-255	489	27	newton	newton	PROPN
ma-255	489	28	’s	’s	PART
ma-255	489	29	method	method	NOUN
ma-255	489	30	,	,	PUNCT
ma-255	489	31	fundamentals	fundamental	NOUN
ma-255	489	32	of	of	ADP
ma-255	489	33	algorithms	algorithm	NOUN
ma-255	489	34	,	,	PUNCT
ma-255	489	35	siam	siam	PROPN
ma-255	489	36	,	,	PUNCT
ma-255	489	37	philadelphia,2003.[21	philadelphia,2003.[21	PROPN
ma-255	489	38	]	]	PUNCT
ma-255	489	39	m.	m.	PROPN
ma-255	489	40	kummer	kummer	PROPN
ma-255	489	41	,	,	PUNCT
ma-255	489	42	newton	newton	PROPN
ma-255	489	43	’s	’s	PART
ma-255	489	44	method	method	NOUN
ma-255	489	45	for	for	ADP
ma-255	489	46	non	non	ADJ
ma-255	489	47	-	-	ADJ
ma-255	489	48	differentiable	differentiable	ADJ
ma-255	489	49	functions	function	NOUN
ma-255	489	50	,	,	PUNCT
ma-255	489	51	in	in	ADP
ma-255	489	52	:	:	PUNCT
ma-255	489	53	j.	j.	PROPN
ma-255	489	54	guddat	guddat	PROPN
ma-255	489	55	et	et	PROPN
ma-255	489	56	al	al	PROPN
ma-255	489	57	.	.	PROPN
ma-255	489	58	(	(	PUNCT
ma-255	489	59	eds	ed	NOUN
ma-255	489	60	.	.	PUNCT
ma-255	489	61	)	)	PUNCT
ma-255	489	62	,	,	PUNCT
ma-255	489	63	advances	advance	NOUN
ma-255	489	64	in	in	ADP
ma-255	489	65	mathematicaloptimization	mathematicaloptimization	NOUN
ma-255	489	66	,	,	PUNCT
ma-255	489	67	ser	ser	NOUN
ma-255	489	68	.	.	PROPN
ma-255	489	69	math	math	PROPN
ma-255	489	70	.	.	PUNCT
ma-255	490	1	res	re	NOUN
ma-255	490	2	.	.	PROPN
ma-255	491	1	45	45	NUM
ma-255	491	2	,	,	PUNCT
ma-255	491	3	akademie	akademie	NOUN
ma-255	491	4	-	-	PUNCT
ma-255	491	5	verlag	verlag	PROPN
ma-255	491	6	,	,	PUNCT
ma-255	491	7	berlin	berlin	PROPN
ma-255	491	8	,	,	PUNCT
ma-255	491	9	1988	1988	NUM
ma-255	491	10	,	,	PUNCT
ma-255	491	11	pp	pp	ADV
ma-255	491	12	.	.	PUNCT
ma-255	492	1	114	114	NUM
ma-255	492	2	-	-	SYM
ma-255	492	3	125.[22	125.[22	NUM
ma-255	492	4	]	]	X
ma-255	492	5	l.v	l.v	PROPN
ma-255	492	6	.	.	PROPN
ma-255	492	7	kantorovich	kantorovich	PROPN
ma-255	492	8	,	,	PUNCT
ma-255	492	9	g.	g.	PROPN
ma-255	492	10	akilov	akilov	PROPN
ma-255	492	11	,	,	PUNCT
ma-255	492	12	functional	functional	ADJ
ma-255	492	13	analysis	analysis	NOUN
ma-255	492	14	in	in	ADP
ma-255	492	15	normed	normed	ADJ
ma-255	492	16	spaces	space	NOUN
ma-255	492	17	,	,	PUNCT
ma-255	492	18	pergamon	pergamon	PROPN
ma-255	492	19	press	press	PROPN
ma-255	492	20	,	,	PUNCT
ma-255	492	21	london	london	PROPN
ma-255	492	22	,	,	PUNCT
ma-255	492	23	1981.[23	1981.[23	NUM
ma-255	492	24	]	]	X
ma-255	492	25	r.h	r.h	PROPN
ma-255	492	26	.	.	PROPN
ma-255	492	27	moore	moore	PROPN
ma-255	492	28	,	,	PUNCT
ma-255	492	29	m.z	m.z	PROPN
ma-255	492	30	.	.	PROPN
ma-255	492	31	nashed	nashe	VERB
ma-255	492	32	,	,	PUNCT
ma-255	492	33	approximations	approximation	NOUN
ma-255	492	34	to	to	ADP
ma-255	492	35	generalized	generalize	VERB
ma-255	492	36	inverses	inverse	NOUN
ma-255	492	37	of	of	ADP
ma-255	492	38	linear	linear	PROPN
ma-255	492	39	operators	operator	NOUN
ma-255	492	40	,	,	PUNCT
ma-255	492	41	siam	siam	PROPN
ma-255	492	42	j.	j.	PROPN
ma-255	492	43	appl	appl	PROPN
ma-255	492	44	.	.	PROPN
ma-255	492	45	math	math	PROPN
ma-255	492	46	.	.	PUNCT
ma-255	493	1	27	27	NUM
ma-255	493	2	(	(	PUNCT
ma-255	493	3	1974)1	1974)1	NUM
ma-255	493	4	-	-	SYM
ma-255	493	5	16.[24	16.[24	NUM
ma-255	493	6	]	]	X
ma-255	493	7	m.z	m.z	PROPN
ma-255	493	8	.	.	PROPN
ma-255	493	9	nashed	nashe	VERB
ma-255	493	10	,	,	PUNCT
ma-255	493	11	generalized	generalize	VERB
ma-255	493	12	inverses	inverse	NOUN
ma-255	493	13	and	and	CCONJ
ma-255	493	14	applications	application	NOUN
ma-255	493	15	,	,	PUNCT
ma-255	493	16	academic	academic	ADJ
ma-255	493	17	press	press	NOUN
ma-255	493	18	,	,	PUNCT
ma-255	493	19	new	new	PROPN
ma-255	493	20	york	york	PROPN
ma-255	493	21	,	,	PUNCT
ma-255	493	22	1976	1976	NUM
ma-255	493	23	.	.	PUNCT
ma-255	494	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	494	2	eur	eur	PROPN
ma-255	494	3	.	.	PUNCT
ma-255	495	1	j.	j.	PROPN
ma-255	495	2	math	math	PROPN
ma-255	495	3	.	.	PUNCT
ma-255	496	1	anal	anal	PROPN
ma-255	496	2	.	.	PUNCT
ma-255	497	1	10.28924	10.28924	NUM
ma-255	497	2	/	/	SYM
ma-255	497	3	ada	ada	PROPN
ma-255	497	4	/	/	SYM
ma-255	497	5	ma.5.5	ma.5.5	PROPN
ma-255	497	6	19	19	NUM
ma-255	497	7	[	[	X
ma-255	497	8	25	25	NUM
ma-255	497	9	]	]	X
ma-255	497	10	f.a	f.a	PROPN
ma-255	497	11	.	.	PROPN
ma-255	497	12	potra	potra	PROPN
ma-255	497	13	,	,	PUNCT
ma-255	497	14	v.	v.	ADP
ma-255	497	15	pták	pták	ADJ
ma-255	497	16	,	,	PUNCT
ma-255	497	17	nondiscrete	nondiscrete	ADJ
ma-255	497	18	induction	induction	NOUN
ma-255	497	19	and	and	CCONJ
ma-255	497	20	iterative	iterative	NOUN
ma-255	497	21	processes	process	NOUN
ma-255	497	22	,	,	PUNCT
ma-255	497	23	research	research	NOUN
ma-255	497	24	notes	note	NOUN
ma-255	497	25	in	in	ADP
ma-255	497	26	mathematics	mathematics	PROPN
ma-255	497	27	103	103	NUM
ma-255	497	28	,	,	PUNCT
ma-255	497	29	pitman	pitman	NOUN
ma-255	497	30	,	,	PUNCT
ma-255	497	31	boston	boston	PROPN
ma-255	497	32	,	,	PUNCT
ma-255	497	33	1984.[26	1984.[26	NUM
ma-255	497	34	]	]	X
ma-255	497	35	a.	a.	NOUN
ma-255	497	36	padcharoen	padcharoen	PROPN
ma-255	497	37	,	,	PUNCT
ma-255	497	38	p.	p.	PROPN
ma-255	497	39	kumam	kumam	PROPN
ma-255	497	40	,	,	PUNCT
ma-255	497	41	p.	p.	PROPN
ma-255	497	42	chaipunya	chaipunya	PROPN
ma-255	497	43	,	,	PUNCT
ma-255	497	44	y.	y.	PROPN
ma-255	497	45	shehu	shehu	PROPN
ma-255	497	46	,	,	PUNCT
ma-255	497	47	convergence	convergence	NOUN
ma-255	497	48	of	of	ADP
ma-255	497	49	inertial	inertial	ADJ
ma-255	497	50	modified	modify	VERB
ma-255	497	51	krasnoselskii	krasnoselskii	PROPN
ma-255	497	52	-	-	PUNCT
ma-255	497	53	mann	mann	PROPN
ma-255	497	54	iterationwith	iterationwith	NOUN
ma-255	497	55	application	application	NOUN
ma-255	497	56	to	to	ADP
ma-255	497	57	image	image	NOUN
ma-255	497	58	recovery	recovery	NOUN
ma-255	497	59	,	,	PUNCT
ma-255	497	60	thai	thai	PROPN
ma-255	497	61	j.	j.	PROPN
ma-255	497	62	math	math	PROPN
ma-255	497	63	.	.	PUNCT
ma-255	498	1	18	18	NUM
ma-255	498	2	(	(	PUNCT
ma-255	498	3	2020	2020	NUM
ma-255	498	4	)	)	PUNCT
ma-255	498	5	126	126	NUM
ma-255	498	6	-	-	SYM
ma-255	498	7	142.[27	142.[27	NUM
ma-255	498	8	]	]	X
ma-255	498	9	p.d	p.d	PROPN
ma-255	498	10	.	.	PROPN
ma-255	498	11	proinov	proinov	PROPN
ma-255	498	12	,	,	PUNCT
ma-255	498	13	new	new	ADJ
ma-255	498	14	general	general	ADJ
ma-255	498	15	convergence	convergence	NOUN
ma-255	498	16	theory	theory	NOUN
ma-255	498	17	for	for	ADP
ma-255	498	18	iterative	iterative	NOUN
ma-255	498	19	processes	process	NOUN
ma-255	498	20	and	and	CCONJ
ma-255	498	21	its	its	PRON
ma-255	498	22	applications	application	NOUN
ma-255	498	23	to	to	ADP
ma-255	498	24	newton	newton	PROPN
ma-255	498	25	-	-	PUNCT
ma-255	498	26	kantarovichtype	kantarovichtype	NOUN
ma-255	498	27	theorems	theorem	NOUN
ma-255	498	28	,	,	PUNCT
ma-255	498	29	j.	j.	PROPN
ma-255	498	30	complex	complex	PROPN
ma-255	498	31	.	.	PUNCT
ma-255	499	1	25	25	NUM
ma-255	499	2	(	(	PUNCT
ma-255	499	3	2010	2010	NUM
ma-255	499	4	)	)	PUNCT
ma-255	499	5	3	3	NUM
ma-255	499	6	-	-	SYM
ma-255	499	7	42.[28	42.[28	PROPN
ma-255	499	8	]	]	PUNCT
ma-255	499	9	l.	l.	PROPN
ma-255	499	10	qi	qi	PROPN
ma-255	499	11	,	,	PUNCT
ma-255	499	12	j.	j.	PROPN
ma-255	499	13	sun	sun	PROPN
ma-255	499	14	,	,	PUNCT
ma-255	499	15	a	a	DET
ma-255	499	16	nonsmooth	nonsmooth	ADJ
ma-255	499	17	version	version	NOUN
ma-255	499	18	of	of	ADP
ma-255	499	19	newton	newton	PROPN
ma-255	499	20	’s	’s	PART
ma-255	499	21	method	method	NOUN
ma-255	499	22	,	,	PUNCT
ma-255	499	23	math	math	NOUN
ma-255	499	24	.	.	PUNCT
ma-255	499	25	program	program	NOUN
ma-255	499	26	.	.	PUNCT
ma-255	500	1	58	58	NUM
ma-255	500	2	(	(	PUNCT
ma-255	500	3	1993	1993	NUM
ma-255	500	4	)	)	PUNCT
ma-255	501	1	353	353	NUM
ma-255	501	2	-	-	SYM
ma-255	501	3	367.[29	367.[29	NUM
ma-255	501	4	]	]	X
ma-255	501	5	s.m	s.m	PROPN
ma-255	501	6	.	.	PROPN
ma-255	501	7	robinson	robinson	PROPN
ma-255	501	8	,	,	PUNCT
ma-255	501	9	extension	extension	NOUN
ma-255	501	10	of	of	ADP
ma-255	501	11	newton	newton	PROPN
ma-255	501	12	’s	’s	PART
ma-255	501	13	method	method	NOUN
ma-255	501	14	to	to	ADP
ma-255	501	15	nonlinear	nonlinear	ADJ
ma-255	501	16	functions	function	NOUN
ma-255	501	17	with	with	ADP
ma-255	501	18	values	value	NOUN
ma-255	501	19	in	in	ADP
ma-255	501	20	a	a	DET
ma-255	501	21	cone	cone	NOUN
ma-255	501	22	,	,	PUNCT
ma-255	501	23	numer	numer	PROPN
ma-255	501	24	.	.	PROPN
ma-255	501	25	math	math	NOUN
ma-255	501	26	.	.	PUNCT
ma-255	502	1	19	19	NUM
ma-255	502	2	(	(	PUNCT
ma-255	502	3	1972)341	1972)341	PROPN
ma-255	502	4	-	-	SYM
ma-255	502	5	347.[30	347.[30	PROPN
ma-255	502	6	]	]	X
ma-255	502	7	w.c	w.c	PROPN
ma-255	502	8	.	.	PROPN
ma-255	502	9	rheinboldt	rheinboldt	PROPN
ma-255	502	10	,	,	PUNCT
ma-255	502	11	a	a	DET
ma-255	502	12	unified	unified	ADJ
ma-255	502	13	convergence	convergence	NOUN
ma-255	502	14	theory	theory	NOUN
ma-255	502	15	for	for	ADP
ma-255	502	16	a	a	DET
ma-255	502	17	class	class	NOUN
ma-255	502	18	of	of	ADP
ma-255	502	19	iterative	iterative	NOUN
ma-255	502	20	process	process	NOUN
ma-255	502	21	,	,	PUNCT
ma-255	502	22	siam	siam	PROPN
ma-255	502	23	j.	j.	PROPN
ma-255	502	24	numer	numer	PROPN
ma-255	502	25	.	.	PUNCT
ma-255	503	1	anal	anal	PROPN
ma-255	503	2	.	.	PUNCT
ma-255	504	1	5	5	NUM
ma-255	504	2	(	(	PUNCT
ma-255	504	3	1968	1968	NUM
ma-255	504	4	)	)	PUNCT
ma-255	504	5	42	42	NUM
ma-255	504	6	-	-	SYM
ma-255	504	7	63.[31	63.[31	NOUN
ma-255	504	8	]	]	X
ma-255	504	9	s.	s.	PROPN
ma-255	504	10	regmi	regmi	PROPN
ma-255	504	11	,	,	PUNCT
ma-255	504	12	i.k	i.k	PROPN
ma-255	504	13	.	.	PROPN
ma-255	504	14	argyros	argyros	PROPN
ma-255	504	15	,	,	PUNCT
ma-255	504	16	s.	s.	PROPN
ma-255	504	17	george	george	PROPN
ma-255	504	18	,	,	PUNCT
ma-255	504	19	c.i	c.i	PROPN
ma-255	504	20	.	.	PROPN
ma-255	504	21	argyros	argyros	PROPN
ma-255	504	22	,	,	PUNCT
ma-255	504	23	extended	extended	ADJ
ma-255	504	24	convergence	convergence	NOUN
ma-255	504	25	of	of	ADP
ma-255	504	26	three	three	NUM
ma-255	504	27	step	step	NOUN
ma-255	504	28	iterative	iterative	NOUN
ma-255	504	29	methods	method	NOUN
ma-255	504	30	for	for	ADP
ma-255	504	31	solvingequations	solvingequation	NOUN
ma-255	504	32	in	in	ADP
ma-255	504	33	banach	banach	NOUN
ma-255	504	34	space	space	NOUN
ma-255	504	35	with	with	ADP
ma-255	504	36	applications	application	NOUN
ma-255	504	37	,	,	PUNCT
ma-255	504	38	symmetry	symmetry	NOUN
ma-255	504	39	14	14	NUM
ma-255	504	40	(	(	PUNCT
ma-255	504	41	2022	2022	NUM
ma-255	504	42	)	)	PUNCT
ma-255	505	1	1484.[32	1484.[32	NUM
ma-255	505	2	]	]	X
ma-255	505	3	g.n	g.n	PROPN
ma-255	505	4	.	.	PROPN
ma-255	505	5	silva	silva	PROPN
ma-255	505	6	,	,	PUNCT
ma-255	505	7	kantorovich	kantorovich	PROPN
ma-255	505	8	’s	’s	PART
ma-255	505	9	theorem	theorem	PROPN
ma-255	505	10	on	on	ADP
ma-255	505	11	newton	newton	PROPN
ma-255	505	12	’s	’s	PART
ma-255	505	13	method	method	NOUN
ma-255	505	14	for	for	ADP
ma-255	505	15	solving	solve	VERB
ma-255	505	16	generalized	generalized	ADJ
ma-255	505	17	equations	equation	NOUN
ma-255	505	18	under	under	ADP
ma-255	505	19	the	the	DET
ma-255	505	20	majorant	majorant	NOUN
ma-255	505	21	condi	condi	NOUN
ma-255	505	22	-	-	PUNCT
ma-255	505	23	tion	tion	NOUN
ma-255	505	24	,	,	PUNCT
ma-255	505	25	appl	appl	PROPN
ma-255	505	26	.	.	PROPN
ma-255	505	27	math	math	PROPN
ma-255	505	28	.	.	PUNCT
ma-255	506	1	comput	comput	NOUN
ma-255	506	2	.	.	PUNCT
ma-255	507	1	286	286	NUM
ma-255	507	2	(	(	PUNCT
ma-255	507	3	2016	2016	NUM
ma-255	507	4	)	)	PUNCT
ma-255	507	5	178	178	NUM
ma-255	507	6	-	-	SYM
ma-255	507	7	188.[33	188.[33	NUM
ma-255	507	8	]	]	X
ma-255	507	9	j.f	j.f	PROPN
ma-255	507	10	.	.	PROPN
ma-255	507	11	traub	traub	PROPN
ma-255	507	12	,	,	PUNCT
ma-255	507	13	h.	h.	PROPN
ma-255	507	14	wozniakowsi	wozniakowsi	PROPN
ma-255	507	15	,	,	PUNCT
ma-255	507	16	convergence	convergence	NOUN
ma-255	507	17	and	and	CCONJ
ma-255	507	18	complexity	complexity	NOUN
ma-255	507	19	of	of	ADP
ma-255	507	20	newton	newton	PROPN
ma-255	507	21	iteration	iteration	PROPN
ma-255	507	22	for	for	ADP
ma-255	507	23	operator	operator	NOUN
ma-255	507	24	equations	equation	NOUN
ma-255	507	25	,	,	PUNCT
ma-255	507	26	j.	j.	PROPN
ma-255	507	27	assoc	assoc	PROPN
ma-255	507	28	.	.	PUNCT
ma-255	508	1	comput.mach	comput.mach	PRON
ma-255	508	2	.	.	PUNCT
ma-255	509	1	26	26	NUM
ma-255	509	2	(	(	PUNCT
ma-255	509	3	1979	1979	NUM
ma-255	509	4	)	)	PUNCT
ma-255	509	5	250	250	NUM
ma-255	509	6	-	-	SYM
ma-255	509	7	258.[34	258.[34	NUM
ma-255	509	8	]	]	PUNCT
ma-255	509	9	t.	t.	PROPN
ma-255	509	10	yamamoto	yamamoto	PROPN
ma-255	509	11	,	,	PUNCT
ma-255	509	12	a	a	DET
ma-255	509	13	convergence	convergence	NOUN
ma-255	509	14	theorem	theorem	NOUN
ma-255	509	15	for	for	ADP
ma-255	509	16	newton	newton	PROPN
ma-255	509	17	-	-	PUNCT
ma-255	509	18	like	like	ADJ
ma-255	509	19	algorithms	algorithm	NOUN
ma-255	509	20	in	in	ADP
ma-255	509	21	banach	banach	NOUN
ma-255	509	22	spaces	space	NOUN
ma-255	509	23	,	,	PUNCT
ma-255	509	24	numer	numer	PROPN
ma-255	509	25	.	.	PROPN
ma-255	509	26	math	math	NOUN
ma-255	509	27	.	.	PUNCT
ma-255	510	1	51	51	NUM
ma-255	510	2	(	(	PUNCT
ma-255	510	3	1987	1987	NUM
ma-255	510	4	)	)	PUNCT
ma-255	510	5	545	545	NUM
ma-255	510	6	-	-	SYM
ma-255	510	7	557.[35	557.[35	NUM
ma-255	510	8	]	]	PUNCT
ma-255	510	9	m.	m.	NOUN
ma-255	510	10	ulbrich	ulbrich	PROPN
ma-255	510	11	,	,	PUNCT
ma-255	510	12	semismooth	semismooth	PROPN
ma-255	510	13	newton	newton	PROPN
ma-255	510	14	methods	method	NOUN
ma-255	510	15	for	for	ADP
ma-255	510	16	variational	variational	ADJ
ma-255	510	17	inequalities	inequality	NOUN
ma-255	510	18	and	and	CCONJ
ma-255	510	19	constrained	constrained	ADJ
ma-255	510	20	optimization	optimization	NOUN
ma-255	510	21	problems	problem	NOUN
ma-255	510	22	infunction	infunction	NOUN
ma-255	510	23	spaces	space	NOUN
ma-255	510	24	,	,	PUNCT
ma-255	510	25	siam	siam	PROPN
ma-255	510	26	,	,	PUNCT
ma-255	510	27	philadelphia	philadelphia	PROPN
ma-255	510	28	,	,	PUNCT
ma-255	510	29	2011	2011	NUM
ma-255	510	30	.	.	PUNCT
ma-255	511	1	https://doi.org/10.28924/ada/ma.5.5	https://doi.org/10.28924/ada/ma.5.5	PROPN
ma-255	511	2	1	1	NUM
ma-255	511	3	.	.	PUNCT
ma-255	511	4	introduction	introduction	NOUN
ma-255	511	5	2	2	NUM
ma-255	511	6	.	.	PUNCT
ma-255	511	7	convergence	convergence	NOUN
ma-255	511	8	for	for	ADP
ma-255	511	9	the	the	DET
ma-255	511	10	method	method	NOUN
ma-255	511	11	(	(	PUNCT
ma-255	511	12	1.5	1.5	NUM
ma-255	511	13	)	)	PUNCT
ma-255	511	14	3	3	NUM
ma-255	511	15	.	.	X
ma-255	511	16	convergence	convergence	NOUN
ma-255	511	17	for	for	ADP
ma-255	511	18	the	the	DET
ma-255	511	19	method	method	NOUN
ma-255	511	20	(	(	PUNCT
ma-255	511	21	1.10	1.10	NUM
ma-255	511	22	)	)	PUNCT
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ma-255	511	24	.	.	PUNCT
ma-255	511	25	numerical	numerical	ADJ
ma-255	511	26	examples	example	NOUN
ma-255	511	27	5	5	NUM
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ma-255	511	29	concluding	conclude	VERB
ma-255	511	30	remarks	remark	NOUN
ma-255	511	31	acknowledgement	acknowledgement	NOUN
ma-255	511	32	:	:	PUNCT
ma-255	511	33	references	reference	NOUN