Taxas de Convergência para Métodos Iterativos Cíclicos em Problemas Mal Postos

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛

▼ét♦❞♦s ■t❡r❛t✐✈♦s ❈í❝❧✐❝♦s ❡♠

Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s

❘✉❜é♥ ❆❧❡① ▼❛rtí♥❡③ ▼✉ñ♦③

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳➸ ❉r✳ ❆♥tô♥✐♦ ▲❡✐tã♦

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❘✉❜é♥ ❆❧❡① ▼❛rtí♥❡③ ▼✉ñ♦③

❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ▼ét♦❞♦s

■t❡r❛t✐✈♦s ❈í❝❧✐❝♦s ❡♠ Pr♦❜❧❡♠❛s ▼❛❧

P♦st♦s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❋í✲ s✐❝❛s ❡ ▼❛t❡♠át✐❝❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ♣❛r❛ ❛ ♦❜✲ t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡✲ ♠át✐❝❛✱ ❝♦♠ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳➸ ❉r✳ ❆♥tô♥✐♦ ▲❡✐✲ tã♦✳

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❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ▼ét♦❞♦s

■t❡r❛t✐✈♦s ❈í❝❧✐❝♦s ❡♠ Pr♦❜❧❡♠❛s ▼❛❧

P♦st♦s

♣♦r

❘✉❜é♥ ❆❧❡① ▼❛rtí♥❡③ ▼✉ñ♦③✶

❊st❛ ❉✐ss❡rt❛çã♦ ❢♦✐ ❥✉❧❣❛❞❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❚ít✉❧♦ ❞❡ ✏▼❡str❡✑✱ ➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛✱ ❡ ❛♣r♦✈❛❞❛ ❡♠ s✉❛ ❢♦r♠❛ ✜♥❛❧ ♣❡❧♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡

❆♣❧✐❝❛❞❛✳

Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ●♦♥ç❛❧✈❡s ❈♦♦r❞❡♥❛❞♦r ❞♦ ❈✉rs♦ ❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛

Pr♦❢✳ ❉r✳ ▼❛r❝❡❧♦ ❙♦❜♦tt❦❛ ✲ ❯❋❙❈ ✭Pr❡s✐❞❡♥t❡ ❞❛ ❇❛♥❝❛✮

Pr♦❢✳ ❉r✳ ❘❛♣❤❛❡❧ ❋❛❧❝ã♦ ❞❛ ❍♦r❛ ✲ ❯❋❙❈

Pr♦❢✳ ❉r✳ ❲❛❣♥❡r ❇❛r❜♦s❛ ▼✉♥✐③ ✲ ❯❋❙❈

Pr♦❢✳ ❉r✳ ❆❞r✐❛♥♦ ❉❡ ❈❡③❛r♦ ✲ ❋❯❘●

❋❧♦r✐❛♥ó♣♦❧✐s✱ ❆❜r✐❧ ❞❡ ✷✵✶✺✳

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✐✐

✧■♣s❡ s❡ ♥✐❤✐❧ s❝✐r❡ ✐❞ ✉♥✉♠ s❝✐❛t✧✳

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦✱ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋❙❈✱ q✉❡ ❞❡✉ ✉♠❛ ♦♣♦rt✉♥✐❞❛❞❡ ♣❛r❛ ♠✐♠✱ ❡ t❛♠❜é♠ à ❈❆P❊❙✱ ♣♦r s❡✉ ✈❛❧✐♦s♦ s✉♣♦rt❡ ✜♥❛♥❝❡✐r♦✳

❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛ ❝❛❞❛ ✉♠ ❞♦s ♣r♦❢❡ss♦r❡s q✉❡ t✐✈❡ ♥❛s ❞✐✈❡rs❛s ❞✐s❝✐♣❧✐♥❛s✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛ ♠❡✉ ♦r✐❡♥t❛❞♦r ❛❝❛❞ê♠✐❝♦ ■❣♦r ▼♦③♦❧❡✈s❦✐ ❡ ❛ ♠❡✉ ♦r✐❡♥t❛❞♦r ❞❡✜♥✐t✐✈♦ ❆♥tô♥✐♦ ▲❡✐tã♦✱ ✜♥❛❧♠❡♥t❡ q✉❡r♦ ❛❣r❛❞❡❝❡r ❛ ♦s ♣r♦❢❡ss♦r❡s q✉❡ ❛❝❡✐t❛r❛♠ s❡r ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❞❛ ♠✐♥❤❛ ❞❡❢❡s❛✱ ♣❛r❛ t♦❞♦s✱ ♠✐s r❡s♣❡t♦s✳

▼❡♥çã♦ ❛♣❛rt❡ é ♣❛r❛ ❊❧✐s❛ ❆♠❛r❛❧✱ ♦❜r✐❣❛❞♦ ♣♦r t✉❞♦✳

❆❣r❛❞❡ç♦✱ ♥ã♦ ♣♦❞✐❛ s❡r ❞❡ ♦✉tr❛ ♠❛♥❡✐r❛✱ ❛ ♠❡✉s ❛♠✐❣♦s ❡ ❝♦♥❤❡✲ ❝✐❞♦s q✉❡ ❣❛♥❤❡✐ ❛q✉✐ ♥❡ss❛ ✐❧❤❛ ♠❛❧✉❝❛✿ ❑❛♠❛❧✱ ❘♦❜❡rt♦✱ ❆❧❡ss❛♥❞r♦✱ ▼❛r❝❡❧♦✱ ❲✐tr♦✱ ▼❛r✐♦✱ ❊❞✉❛r❞♦✱ ❘❛ú❧✱ ❡♥✜♠✳✳✳t✉❞❛ ❛ ❣❡♥t❡ q✉❡ ❝♦✲ ♥❤❡❝✐ ❣r❛ç❛s à ❯❋❙❈✱ ❡ ❡♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡❝❡r à ♠✐♥❤❛ ❛♠✐❣❛ ❏ú✱ ❡❧❛ é ❞❡♠❛✐s✳

❆❣r❛❞❡ç♦✱ ❛ ♠✐♥❤❛ ❧✐♥❞❛ ❢❛♠í❧✐❛✱ ❛ ❢❛♠í❧✐❛ é ♦ ♣r✐♥❝✐♣❛❧ s✉♣♦rt❡ ♥❛ ✈✐❞❛✳

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❘❡s✉♠♦

◆❛ ❝❧❛ss❡ ❞♦s ♠ét♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✐t❡r❛t✐✈♦s✱ ♦s ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③ sã♦ ✉♥s ❞♦s ♠ét♦❞♦s ♠❛✐s ✉t✐❧✐③❛❞♦s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♥❛ ♠❛t❡♠át✐❝❛ ❛♣❧✐❝❛❞❛✳ ◆♦ ❡♥t❛♥t♦✱ ♥❛ ❧✐t❡r❛t✉r❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ r❡✲ s✉❧t❛❞♦s s♦❜r❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❛s r❡s♣❡❝t✐✈❛s t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♥ã♦ é ❛❜✉♥❞❛♥t❡✳ ❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ❛❧❣✉♠❛s ✈❡rsõ❡s ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ ♦❜t❡♥❞♦ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠ét♦❞♦ ♠♦❞✐✜❝❛❞♦ ❝♦♠ ✉♠ ♣❛râ♠❡tr♦ ❞❡ r❡❧❛①❛♠❡♥t♦✱ ❡ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♥♦ ♠ét♦❞♦ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❡♠ ❜❧♦❝♦ ♥❛s ✈❡rsõ❡s s✐♠étr✐❝❛ ❡ ♥ã♦ s✐♠étr✐❝❛✳ ❋✐♥❛❧♠❡♥t❡✱ ❝♦♠♣❛r❛✲s❡ ♠❡✲ ❞✐❛♥t❡ ❡①♣❡r✐♠❡♥t♦s ♥✉♠ér✐❝♦s ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦s ♠ét♦❞♦s ❡st✉❞❛❞♦s ❝♦♠ ♦ ❞❡s❡♠♣❡♥❤♦ ❞❡ ♠ét♦❞♦s ❜❡♠ ❡st❛❜❡❧❡❝✐❞♦s✳

P❛❧❛✈r❛s✲❈❤❛✈❡✿ Pr♦❜❧❡♠❛s ✐♥✈❡rs♦s✱ ♣r♦❜❧❡♠❛s ♠❛❧ ♣♦st♦s✱ ♠é✲ t♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✐t❡r❛t✐✈♦s✱ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ t❛①❛s ❞❡ ❝♦♥✲ ✈❡r❣ê♥❝✐❛✳

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❆❜str❛❝t

■♥ t❤❡ ❝❧❛ss ♦❢ ✐t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞s✱ ❑❛❝③♠❛r③ t②♣❡ ♠❡t❤♦❞s ❛r❡ s♦♠❡ ♦❢ t❤❛t ❛r❡ ♠♦r❡ ♦❢t❡♥ ✉s❡❞ t♦ s♦❧✈❡ ♣r♦❜❧❡♠s ✐♥ ❛♣♣❧✐❡❞ ♠❛t❤❡♠❛t✐❝s✳ ❍♦✇❡✈❡r✱ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ t❤❡ ❛♠♦✉♥t ♦❢ ❝♦♥✲ ✈❡r❣❡♥❝❡ r❡s✉❧ts ❛♥❞ t❤❡✐r ❝♦♥✈❡r❣❡♥❝❡ r❛t❡ ✐s ♥♦t ❛❜✉♥❞❛♥t✳ ❚❤✐s ✇♦r❦ ❞❡❛❧s ✇✐t❤ t❤❡ ❛♥❛❧②s✐s ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s♦♠❡ ✈❡rs✐♦♥s ♦❢ t❤❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♠❡t❤♦❞✱ ♦❜t❛✐♥✐♥❣ ❝♦♥✈❡r❣❡♥❝❡ ❛♥❞ st❛❜✐❧✐t② ♦❢ t❤❡ ♠♦❞✐✜❡❞ ♠❡t❤♦❞ ✇✐t❤ ❛ r❡❧❛①❛t✐♦♥ ♣❛r❛♠❡t❡r✱ ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ r❛✲ t❡s ❢♦r ♠❡t❤♦❞ ❢♦r ❧✐♥❡❛r ❜❧♦❝❦ ♦♣❡r❛t♦rs ✐♥ ✈❡rs✐♦♥s s②♠♠❡tr✐❝❛❧ ❛♥❞ ♥♦♥✲s②♠♠❡tr✐❝❛❧✳ ❋✐♥❛❧❧②✱ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♠❡t❤♦❞s ✐s ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ✇❡❧❧✲❡st❛❜❧✐s❤❡❞ ♠❡t❤♦❞s✳

❑❡②✇♦r❞s✿ ■♥✈❡rs❡ ♣r♦❜❧❡♠s✱ ✐❧❧✲♣♦ss❡❞ ♣r♦❜❧❡♠s✱ ✐t❡r❛t✐✈❡ r❡❣✉✲ ❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞s✱ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✱ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡s✳

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❈♦♥t❡ú❞♦

■♥tr♦❞✉çã♦ ✶

✶ ❘❡❣✉❧❛r✐③❛çã♦ ❞❡ Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s ♣♦r ▼ét♦❞♦s

■t❡r❛t✐✈♦s ✸

✶✳✶ Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❘❡❣✉❧❛r✐③❛çã♦ ♣♦r ▼ét♦❞♦s ■t❡r❛t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✶ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳✷ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❈❧áss✐❝♦ ❡ ♦ ▼é✲

t♦❞♦ ▲♦♣✐♥❣✲▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✭▲▲❑✮ ❡♠ ❊s✲

♣❛ç♦s ❞❡ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✸ ❘❡❧❛çã♦ ❍✐stór✐❝❛ ❞❡ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷ ❈♦♥✈❡r❣ê♥❝✐❛ ❞♦ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✶✶

✸ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ✷✼

✸✳✶ Pr♦❜❧❡♠❛s ■♥✈❡rs♦s ❡♠ ❆♥á❧✐s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✳✷ ▼ét♦❞♦s ❚✐♣♦ ❑❛❝③♠❛r③ ❡ ❖♣❡r❛❞♦r❡s ❡♠ ❇❧♦❝♦ ✳ ✳ ✳ ✳ ✷✾

✸✳✷✳✶ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♥ã♦ ❙✐♠étr✐❝♦ ❝♦♠ Pré✲❈♦♥❞✐❝✐♦♥❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷✳✷ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❙✐♠étr✐❝♦ ❝♦♠ Pré✲

❈♦♥❞✐❝✐♦♥❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③

♥ã♦ ❙✐♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✹ ❚❛①❛s ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ▼ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③

❙✐♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✹ ❊①♣❡r✐ê♥❝✐❛s ◆✉♠ér✐❝❛s ✺✶

✹✳✶ ▼ét♦❞♦s ■t❡r❛t✐✈♦s ❛ ❙❡r❡♠ ❈♦♠♣❛r❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✹✳✷ ❉❡s❝r✐çã♦ ❞♦ Pr♦❜❧❡♠❛ ❚❡st❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✷✳✶ ●❡r❛♥❞♦ ❉❛❞♦s ❊①❛t♦s ❡ ❝♦♠ ❘✉í❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

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✹✳✸ ❘❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✹✳✸✳✶ ❉❛❞♦s ❊①❛t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✹✳✸✳✷ ❉❛❞♦s ❝♦♠ ❘✉í❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✹ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

❈♦♥❝❧✉sõ❡s ●❡r❛✐s ✺✾

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■♥tr♦❞✉çã♦

❯♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ♣r♦❜❧❡♠❛s ❡♠ ❡♥❣❡♥❤❛r✐❛ ❡ ❝✐ê♥❝✐❛s ❛♣❧✐✲ ❝❛❞❛s ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❢ís✐❝❛

x❛ ♣❛rt✐r ❞❡ ❞❛❞♦s ♦❜s❡r✈❛❞♦s ❞❡ ✉♠❛ ♦✉tr❛ q✉❛♥t✐❞❛❞❡ ❢ís✐❝❛y✳ ❊ss❛s

q✉❛♥t✐❞❛❞❡s ❡stã♦ ❝♦♥❡❝t❛❞❛s ♠❡❞✐❛♥t❡ ✉♠❛ r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧ ❞❛❞❛ ❞❡ ❢♦r♠❛ ❡①♣❧í❝✐t❛ ♦✉ ✐♠♣❧í❝✐t❛✳ ❊♠❜♦r❛ ❛ r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧ s❡❥❛ ❝♦♥❤❡✲ ❝✐❞❛ ❞❡ ❢♦r♠❛ ❡①♣❧í❝✐t❛✱ ♥❡♠ s❡♠♣r❡ é ♣♦ssí✈❡❧ ♦❜t❡r ❞❡ ❢♦r♠❛ ❛♥❛❧ít✐❝❛ ❛ q✉❛♥t✐❞❛❞❡ x❡♠ ❢✉♥çã♦ ❞❡ y✳ P❛r❛ r❡s♦❧✈❡r ❡ss❡ ♣r♦❜❧❡♠❛✱ ✉s❛♠✲s❡

❢❡rr❛♠❡♥t❛s ♥✉♠ér✐❝❛s✱ ♠❛s q✉❛♥❞♦ ❛ r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧ ✐♥✈❡rs❛ q✉❡ ❝♦✲ ♥❡❝t❛y ❝♦♠x♥ã♦ t❡♠ ✏❜♦❛s ♣r♦♣r✐❡❞❛❞❡s✑ ❛s ❛♣r♦①✐♠❛çõ❡s ♥✉♠ér✐❝❛s

t♦r♥❛♠✲s❡ ✐♥stá✈❡✐s✳ ❆q✉✐ ❡♥tr❛♠ ♥♦ ❥♦❣♦ ♦s ♠ét♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ♣❛r❛ t♦r♥❛r ❡stá✈❡✐s ♣r♦❜❧❡♠❛s ❞❡ ❛♣r♦①✐♠❛çã♦ ✐♥stá✈❡✐s✳

◆♦ ❈❛♣ít✉❧♦ ✶✱ ❞❡✜♥❡♠✲s❡ ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♠❛❧ ♣♦st♦✱ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❡ ❛s ♥♦çõ❡s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ss❡s ♠ét♦❞♦s✳ ❉❡♣♦✐s✱ ❡①♣õ❡♠✲s❡ ♦s ♠ét♦❞♦s ♥♦s q✉❛✐s ❡stá ❜❛s❡❛❞♦ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✿ ♦s ♠ét♦❞♦s ❞❡ ▲❛♥❞✇❡❜❡r ❡ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✱ y✱ ❡ ❞❛❞♦s ❝♦♠ r✉í❞♦ yδ

❆q✉✐ ❛ t❡♦r✐❛ ♥ã♦ é tr❛t❛❞❛ ❞❡t❛❧❤❛❞❛♠❡♥t❡✱ ❥á q✉❡ ♦ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛✲ ❜❛❧❤♦ é ❡st❛❜❡❧❡❝❡r t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦s ♠ét♦❞♦s ❡st✉❞❛❞♦s✳ ◆♦ ❡♥t❛♥t♦✱ ✉♠❛ ❛❜♦r❞❛❣❡♠ r✐❣♦r♦s❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛s r❡❢❡rê♥✲ ❝✐❛s✳ ❋✐♥❛❧♠❡♥t❡✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠ r❡s✉♠♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♥✈❡r❣ê♥✲ ❝✐❛✴t❛①❛s ♣❛r❛ ♦s ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③ ♦❜t✐❞♦s ♥♦s ú❧t✐♠♦s ❛♥♦s✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛✲s❡ ❡♠ ❞❡t❛❧❤❡ ❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♣❛r❛ ♦♣❡r❛❞♦r❡s ♥ã♦ ❧✐♥❡❛r❡s✱ t❛♥t♦ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s ❝♦♠♦ ♣❛r❛ ❞❛❞♦s ❝♦♠ r✉í❞♦✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ❛♣r❡s❡♥t❛♠✲s❡ ❛s ✈❡rsõ❡s ♥ã♦ s✐♠étr✐❝❛ ❡ s✐♠étr✐❝❛ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❡♠ ❜❧♦❝♦s ❧✐♥❡❛r❡s ❡ s❡ ❡st❛❜❡❧❡❝❡♠ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❛♠❜❛s ✈❛r✐❛♥t❡s ❞♦ ♠ét♦❞♦✳ ◆♦ ❈❛♣ít✉❧♦ ✹✱ ❝♦♠♣❛r❛✲s❡ ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦s ♠ét♦❞♦s ❡st✉❞❛❞♦s ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸ ❝♦♠ ♠ét♦❞♦s t✐♣♦ ▲❛♥❞✇❡❜❡r ❡ ♦ ♠ét♦❞♦ ❙t❡❡♣❡st ❉❡s❝❡♥t✳

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❈❛♣ít✉❧♦ ✶

❘❡❣✉❧❛r✐③❛çã♦ ❞❡

Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s ♣♦r

▼ét♦❞♦s ■t❡r❛t✐✈♦s

◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❞❡✜♥✐❞♦s ❝♦♥❝❡✐t♦s t❛✐s ❝♦♠♦ ✏Pr♦❜❧❡♠❛s ✐♥✲ ✈❡rs♦s✑✱ ✏Pr♦❜❧❡♠❛s ■♥✈❡rs♦s ▼❛❧ P♦st♦s✑ ❡ ♦ q✉❡ s❡ ❡♥t❡♥❞❡ ♣♦r ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ♣❛r❛ ❡ss❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s ❥✉♥t♦ às ♥♦✲ çõ❡s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♠ét♦❞♦s ❞❡ r❡❣✉❧❛✲ r✐③❛çã♦✳ ❉❡✜♥✐❞♦ ✐ss♦✱ ❛♣r❡s❡♥t❛♠✲s❡ ❝♦♠♦ ❡str❛té❣✐❛s ❞❡ r❡❣✉❧❛r✐③❛✲ çã♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ✐♥✈❡rs♦s ♠❛❧ ♣♦st♦s✱ ♦s ♠ét♦❞♦s ❞❡ ▲❛♥❞✇❡❜❡r ❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rt ❣❡r❛✐s✳

✶✳✶ Pr♦❜❧❡♠❛s ▼❛❧ P♦st♦s

❖s ♣r♦❜❧❡♠❛s ✐♥✈❡rs♦s ♣❡rt❡♥❝❡♠ ❛ ✉♠❛ ❞❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛ ❛♣❧✐❝❛❞❛ ❝♦♠ ♠❛✐♦r ❝r❡s❝✐♠❡♥t♦ ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦rq✉❡ ♠✉✐t♦s ♣r♦❜❧❡♠❛s ♣ró♣r✐♦s ❞❛ ❡♥❣❡♥❤❛r✐❛ ❡ ❝✐ê♥❝✐❛s ♣♦❞❡♠ s❡ ❡♥q✉❛❞r❛r ♥❡ss❛ ❝❧❛ss❡ ❞❡ ♣r♦❜❧❡♠❛s✳

❙❡ ❡♥t❡♥❞❡ ♣♦r Pr♦❜❧❡♠❛ ■♥✈❡rs♦ ❛q✉❡❧❡ q✉❡ ❝♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ❝❛✉s❛s ❡♠ ❢✉♥çã♦ ❞❡ ❡❢❡✐t♦s ♦❜s❡r✈❛❞♦s✳ ❙❡♥❞♦ ♠❛✐s ♣r❡❝✐s♦✱ ❛♦ ❢❛❧❛r ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ✐♠♣❧✐❝✐t❛♠❡♥t❡ s❡ ❡stá ❢❛❧❛♥❞♦ ❞❡ ✉♠ ♦✉tr♦ ♣r♦❜❧❡♠❛✱ q✉❡ s❡r✐❛ ♦ ♣r♦❜❧❡♠❛ ❞✐r❡t♦✳ ❆♠❜♦s ♣r♦❜❧❡♠❛s sã♦ ✐♥✈❡rs♦s ♠✉t✉❛♠❡♥t❡ ❡ ♣♦rt❛♥t♦ ❛ s♦❧✉çã♦ ❞❡ ✉♠ ❡♥✈♦❧✈❡ ❛ s♦❧✉çã♦ ❞♦ ♦✉tr♦✳

(18)

❋♦r♠❛❧♠❡♥t❡✱ ♥♦ ❝♦♥t❡①t♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ ❞❛❞♦ ✉♠ ♦♣❡r❛❞♦r

F :X →Y ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rtX ❡Y✱ ❛ ❡q✉❛çã♦

F(x) =y ✭✶✳✶✮

é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❣❡r❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞✐r❡t♦✲✐♥✈❡rs♦❀ ♦ ♣r♦❜❧❡♠❛ ❞✐✲ r❡t♦ é✿ ❞❛❞♦ x X ❛❝❤❛r y Y✱ ❡ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦✿ ❞❛❞♦ y Y

❛❝❤❛r x X✳ ●❡r❛❧♠❡♥t❡✱ ♦s ♣r♦❜❧❡♠❛s ✐♥✈❡rs♦s q✉❡ ❛♣❛r❡❝❡♠ ♥❛

♣rát✐❝❛ ❝♦rr❡s♣♦♥❞❡♠ ❛ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s q✉❡ ♥ã♦ sã♦ ❜❡♠ ♣♦st♦s✱ ♦♥❞❡ s❡❣✉♥❞♦ ❍❛❞❛♠❛r❞✱ ✉♠ ♣r♦❜❧❡♠❛ ♠❛t❡♠át✐❝♦ ❞❡✜♥✐❞♦ ♠❡❞✐❛♥t❡ ❛ ❡q✉❛çã♦ ✭✶✳✶✮ é ❜❡♠ ♣♦st♦ s❡ ♣❛r❛ t♦❞♦s ♦s ❞❛❞♦s ❛❞♠✐ssí✈❡✐s ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦x✱ ❛ q✉❛❧ ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s✳ ❙❡ ✉♠❛

❞❡ss❛s ❝♦♥❞✐çõ❡s é ✈✐♦❧❛❞❛✱ ♦ ♣r♦❜❧❡♠❛ é ❝❤❛♠❛❞♦ ❞❡ ♠❛❧ ♣♦st♦✱ ✐st♦ é✱ ♦ ❝r✐tér✐♦ ❞❡ ❍❛❞❛♠❛r❞ ❞✐③ q✉❡ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♥❛ ❡q✉❛çã♦ ✭✶✳✶✮ é ❜❡♠ ♣♦st♦ s❡ ♦ ♦♣❡r❛❞♦rF é ✉♠❛ ❜✐❥❡çã♦ ❝♦♠ ✐♥✈❡rs❛F−1❝♦♥tí♥✉❛✳

●❡r❛❧♠❡♥t❡✱ ♥❛ ♣rát✐❝❛✱ ♥ã♦ s❡ ❝♦♥❤❡❝❡ ❝♦♠ ♣r❡❝✐sã♦ ♦s ✈❛❧♦r❡s ❞♦s ❞❛❞♦s y ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦✱ ♣♦ré♠✱ só s❡ ❞✐s♣õ❡ ❞❡ ❞❛❞♦s

❛♣r♦①✐♠❛❞♦syδ✱ ♦s q✉❛✐s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❡rt❡♥❝❡♠ à ✐♠❛❣❡♠ ❞♦

♦♣❡r❛❞♦rF✱ ❝✉❥♦ ❡rr♦ ♦✉ ♥í✈❡❧ ❞❡ r✉í❞♦ é ❝♦♠✉♠❡♥t❡ ❣❡r❛❞♦ ♣❡❧♦s ✐♥s✲

tr✉♠❡♥t♦s ❞❡ ♠❡❞✐çã♦ ♦s q✉❛✐s t❡♠ ✉♠❛ ♣r❡❝✐sã♦ ♦❜✈✐❛♠❡♥t❡ ❧✐♠✐t❛❞❛ ✭♥ã♦ ✐♥✜♥✐t❛✮✳ ❖s ❞❛❞♦s ❛♣r♦①✐♠❛❞♦s ✜❝❛♠ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ q✉❡ ♥ã♦ s✉♣❡r❛ ♦ ✈❛❧♦rδ >0✱ ✐st♦ é

kyk ≤δ ✭✶✳✷✮

♦♥❞❡δ é ❝❤❛♠❛❞♦ ♦ ♥í✈❡❧ ❞❡ ❡rr♦ ♦✉ ♥í✈❡❧ ❞❡ r✉í❞♦✳

❋r❡q✉❡♥t❡♠❡♥t❡ ♥❛ ♣rát✐❝❛✱ ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♥❛ ❡q✉❛çã♦ ✭✶✳✶✮ é ♠❛❧ ♣♦st♦ ♦✉ ♠❛❧ ❝♦♥❞✐❝✐♦♥❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❛♥❞♦ ♦ ♦♣❡r❛❞♦rF é

(19)

é ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♠❛❧ ♣♦st♦ ♣♦r ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♣r♦✲ ❜❧❡♠❛s ❜❡♠ ♣♦st♦s✳ ❆ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡ ♥✉♠ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♠❛❧ ♣♦st♦ ♥❛ ❢♦r♠❛ ❞❛ ❡q✉❛çã♦ ✭✶✳✶✮✱ é ♦ ❝á❧❝✉❧♦ ❞❡ ✉♠❛ s♦❧✉çã♦x∗q✉❛♥❞♦ ❞✐s♣õ❡✲s❡ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦yδ✱ ❥á q✉❡ ❝♦♠♦ ♦ ♦♣❡r❛❞♦r ✐♥✈❡rs♦F−1é

❞❡s❝♦♥tí♥✉♦✱ ♣♦❞❡ ❣❡r❛r ✭❡ ❣❡r❛❧♠❡♥t❡ ❣❡r❛✮ s♦❧✉çõ❡s ✐♥❛❞❡q✉❛❞❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳ P♦rt❛♥t♦✱ é ♥❡❝❡ssár✐♦ ♣r♦❝✉r❛r ✉♠❛ ❛♣r♦①✐♠❛çã♦ xδ

α ❞❡

x∗ q✉❡ s❡❥❛ ♣♦ssí✈❡❧ ❞❡ ❝❛❧❝✉❧❛r ❞❡ ♠❛♥❡✐r❛ ❡stá✈❡❧✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ❞❡♣❡♥❞❛ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s ♦❜s❡r✈❛❞♦syδ✱ ❡ q✉❡ ❝♦♥✈✐r❥❛ à s♦✲

❧✉çã♦ x∗ q✉❛♥❞♦ ♦ ♥í✈❡❧ ❞❡ r✉í❞♦ t❡♥❞❡ ♣❛r❛ ③❡r♦✱ ❞❛❞❛ ✉♠❛ ❡s❝♦❧❤❛ ❛❞❡q✉❛❞❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ α✳

P❛r❛ ✈❡r ❡♠ ❞❡t❛❧❤❡ ❛ t❡♦r✐❛ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ✐♥✈❡r✲ s♦s s✉❣❡r❡✲s❡ ❛♦ ❧❡✐t♦r ❝♦♥s✉❧t❛r ❬✽❪✱ ♦♥❞❡ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛❧t❡r♥❛t✐✈❛s ❞❡ ❡s❝♦❧❤❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦α✳ ➱ ✐♠♣♦rt❛♥t❡ r❡ss❛❧t❛r q✉❡

❛ ❡s❝♦❧❤❛ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ r❡q✉❡r ✉♠ ❜❛❧❛♥ç♦ ❡♥tr❡ ❡st❛✲ ❜✐❧✐❞❛❞❡ ❡ ♣r❡❝✐sã♦ ♣❛r❛ ❛ s♦❧✉çã♦ ❛♣r♦①✐♠❛❞❛ r❡❣✉❧❛r✐③❛❞❛ xδ

α✳ P❛r❛

t❛❧✱ ✉♠❛ ❡st✐♠❛t✐✈❛ ❞❡δ♥❛ r❡❧❛çã♦ ✭✶✳✷✮ é ❢✉♥❞❛♠❡♥t❛❧✳

◗✉❛♥❞♦ ✉♠ ♠ét♦❞♦ ❞❡ ❛♣r♦①✐♠❛çã♦ é ❞❡ ❢❛t♦ ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉✲ ❧❛r✐③❛çã♦✱ ✐st♦ é✱ é ❡stá✈❡❧✿

xδα δ→0

−→xα

❡ é ❝♦♥✈❡r❣❡♥t❡✿

xδ α

α,δ→0

−→ x∗,

❝❛❜❡ ❛ ♣❡r❣✉♥t❛ s♦❜r❡ ❛ r❛♣✐❞❡③ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛✳ ■♥❢❡❧✐③♠❡♥t❡✱ ❡♠ ❣❡r❛❧ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ♣❛r❛ r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛ ♠❛❧ ♣♦st♦ ♣♦❞❡ s❡r ❛r❜✐tr❛r✐❛♠❡♥t❡ ❧❡♥t❛✱ ♣♦r ✐ss♦ ❛s t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣♦❞❡♠ s❡r ♦❜t✐❞❛s s♦♠❡♥t❡ ❡♠ s✉❜❝♦♥❥✉♥t♦s ❞❡ X

❡ ❞❡t❡r♠✐♥❛❞❛s ♣♦r ✐♥❢♦r♠❛çõ❡s ❛✲♣r✐♦r✐ s♦❜r❡ ❛ s♦❧✉çã♦ ❡①❛t❛ x∗ ♦✉ s♦❜r❡ ♦s ❞❛❞♦sy✳ ❊st❛s ✐♥❢♦r♠❛çõ❡s ❛✲♣r✐♦r✐ sã♦ ❢♦r♠✉❧❛❞❛s ❡♠ t❡r♠♦s

❞❛s ❝❤❛♠❛❞❛s ❝♦♥❞✐çõ❡s ❞❡ ❢♦♥t❡ ♦✉ s♦✉r❝❡ ❝♦♥❞✐t✐♦♥s ✭♣❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡❥❛ ❬✽❪✮✳ ❋✐♥❛❧♠❡♥t❡✱ ❡♥t❡♥❞❡✲s❡ ♣♦r t❛①❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ ❛ t❛①❛ ♣❛r❛ ❛ q✉❛❧ ❛❝♦♥t❡❝❡

kxδ α−x∗k

δ→0

(20)

✶✳✷ ❘❡❣✉❧❛r✐③❛çã♦ ♣♦r ▼ét♦❞♦s ■t❡r❛t✐✈♦s

◆❛ ♣rát✐❝❛✱ ♦ ♣r♦❜❧❡♠❛ ❞✐r❡t♦ é ♠✉✐t♦ ♠❡❧❤♦r ❡♥t❡♥❞✐❞♦ q✉❡ ♦ ♣r♦✲ ❜❧❡♠❛ ✐♥✈❡rs♦✱ ♣♦r ✐ss♦✱ ♣❛r❛ r❡s♦❧✈❡r ♦s ♣r✐♠❡✐r♦s ❡①✐st❡ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ❜♦♥s ❛❧❣♦r✐t♠♦s ❡ s♦❢t✇❛r❡ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ✐st♦ é✱ s♦❢t✲ ✇❛r❡ ♣❛r❛ ❛✈❛❧✐❛rF(x)❞❛❞♦ ✉♠ ❞❡t❡r♠✐♥❛❞♦xX✳ ❊♥tã♦✱ é ✉♠❛ ❜♦❛

✐❞❡✐❛ t❡♥t❛r r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ♠❡❞✐❛♥t❡ ✐t❡r❛çõ❡s s✉❝❡ss✐✈❛s✱ ✉s❛♥❞♦ ❡ss❛ ❜❛t❡r✐❛ ❞❡ s♦❢t✇❛r❡s ❡ ❛❧❣♦r✐t♠♦s✳

❖s ♠ét♦❞♦s ✐t❡r❛t✐✈♦s t❡♠ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❛✉t♦✲r❡❣✉❧❛r✐③❛çã♦✱ ♥♦ s❡♥t✐❞♦ q✉❡ ♦ tér♠✐♥♦ ❛♥t❡❝✐♣❛❞♦ ❞♦ ♣r♦❝❡ss♦ ✐t❡r❛t✐✈♦ t❡♠ ✉♠ ❡❢❡✐t♦ r❡❣✉❧❛r✐③❛❞♦r✱ ♦✉ s❡❥❛✱ ♦ í♥❞✐❝❡ ❞❛ ✐t❡r❛çã♦ ❢❛③ ❛ ❢✉♥çã♦ ❞♦ ♣❛râ♠❡tr♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦ α❡✱ ♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ❢❛③ ❛ ❢✉♥çã♦ ❞♦ ♠ét♦❞♦ ❞❡

s❡❧❡çã♦ ❞♦ ♣❛râ♠❡tr♦✳ P❛r❛ ✈❡r ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ r❡❣✉❧❛r✐③❛çã♦ q✉❡ ♣♦ss✉❡♠ ♦s ♠ét♦❞♦s ✐t❡r❛t✐✈♦s ✈❡❥❛ ❬✽✱✶✻❪✳

✶✳✷✳✶ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r

❆ ❝♦♥❞✐çã♦ ❞❡ ♦t✐♠❛❧✐❞❛❞❡ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ♠í♥✐♠♦s q✉❛❞r❛❞♦s ♥ã♦ ❧✐♥❡❛r

min

x∈X

1

2kF(x)−yk

2,

é ❛ ❡q✉❛çã♦ ♥♦r♠❛❧

F′(x)∗F(x) =F′(x)∗y, ✭✶✳✸✮

♦♥❞❡ F′(x) é ❛ ❞❡r✐✈❛❞❛ ❞❡ ❋ré❝❤❡t ❞❡ F ❡♠ x X F(x)é ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ♦♣❡r❛❞♦r ❛❞❥✉♥t♦✳ ❚r❛♥s❢♦r♠❛♥❞♦ ❛ ❡①♣r❡ssã♦ ✭✶✳✸✮ ❡♠ ✉♠❛ ✐t❡r❛çã♦ ❞❡ ♣♦♥t♦ ✜①♦ ♦❜té♠✲s❡

x=xF′(x)(F(x)y), ✭✶✳✹✮

❧♦❣♦ ❞❡♥♦t❛♥❞♦ ♣♦rxδ

n ❛s ✐t❡r❛çõ❡s ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦ yδ ✭❡

♣♦rxn ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❡①❛t♦sy✮ ♦❜té♠✲s❡ ❛ ✐t❡r❛çã♦ ❞❡ ▲❛♥❞✇❡❜❡r

xδn+1=xδn−F′(xδn)∗(F(xδn)−yδ). ✭✶✳✺✮

❯♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❡st❡ ♠ét♦❞♦ ♣❛r❛ ♦s ❝❛s♦s ❡♠ q✉❡ ♦ ♦♣❡r❛❞♦rF

(21)

✶✳✷✳✷ ▼ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❈❧áss✐❝♦ ❡

♦ ▼ét♦❞♦ ▲♦♣✐♥❣✲▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③✭

▲❑✮

❡♠ ❊s♣❛ç♦s ❞❡ ❍✐❧❜❡rt

❖ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r ❛t✉❛ s♦❜r❡ ♣r♦❜❧❡♠❛s ❞❛ ❢♦r♠❛ ❞❛ ❡①♣r❡s✲ sã♦ ✭✶✳✶✮✱ ♦✉ s❡❥❛✱ ♣❛r❛ ✉♠❛ ❡q✉❛çã♦✳ ◆♦ ❝❛s♦ ❞❡ t❡r ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✱ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ s❡r✐❛ ❛❣r✉♣❛r ❡❧❛s ♥✉♠ ♦♣❡r❛❞♦r ♣♦r ❜❧♦❝♦s ♦✉ ❝♦♠ ❡str✉t✉r❛ ✈❡t♦r✐❛❧ ✭❝♦♠ ✈❛❧♦r❡s ♥✉♠ ❡s♣❛ç♦ ♣r♦❞✉t♦✮✳ ❊ss❛ ❛❧✲ t❡r♥❛t✐✈❛ ✐♠♣❧✐❝❛ q✉❡ ❡♠ ❝❛❞❛ ✐t❡r❛çã♦ t❡♠ q✉❡ ❛✈❛❧✐❛r t♦❞♦s ♦s ♦♣❡r❛✲ ❞♦r❡s ❡♥✈♦❧✈✐❞♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥❛ ❛♣r♦①✐♠❛çã♦ ❛t✉❛❧✱ ❛t✐✈✐❞❛❞❡ q✉❡ ♣❛r❛ ✉♠ ❣r❛♥❞❡ ♥ú♠❡r♦ ❞❡ ❡q✉❛çõ❡s t♦r♥❛ ♦ ♣r♦❝❡ss♦ ♣♦✉❝♦ ❡❝♦♥ô♠✐❝♦ ❝♦♠♣✉t❛❝✐♦♥❛❧♠❡♥t❡✳ ❆ s❡❣✉✐r s❡rá ❞❡✜♥✐❞♦ ♦ ♣r♦❜❧❡♠❛ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s✳

❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞❡t❡r♠✐♥❛r ❛❧❣✉♠❛ q✉❛♥t✐❞❛❞❡ ❢ís✐❝❛x∈ X ❛ ♣❛rt✐r ❞❡p∈N❞❛❞♦s yip−1

i=0 ⊂Y

p♦s q✉❛✐s ❡stã♦ ❢✉♥❝✐♦♥❛❧♠❡♥t❡

r❡❧❛❝✐♦♥❛❞♦s ♣♦r

Fi(x) =yi, i= 0, . . . , p−1 ✭✶✳✻✮

♦♥❞❡Fi:Di⊆X →Y sã♦ ♦♣❡r❛❞♦r❡s ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❍✐❧❜❡rtX ❡Y

❝♦♠ r❡s♣❡❝t✐✈♦s ♣r♦❞✉t♦s ✐♥t❡r♥♦sh·,·iX✱h·,·iY✱ ❡ ❛s ♥♦r♠❛s ❛ss♦❝✐❛❞❛s

k · kX✱ k · kY✳ ◗✉❛♥❞♦ ♦ ❝♦♥t❡①t♦ é ❝❧❛r♦ s❡rã♦ ✉s❛❞♦s h·,·i ❡ k · k

♣♦r s✐♠♣❧✐❝✐❞❛❞❡✳ ❈♦♠♦ ❥á ♦❜s❡r✈❛❞♦✱ ♥❛ ♣rát✐❝❛✱ ♦s ❞❛❞♦s ♥ã♦ sã♦ ❝♦♥❤❡❝✐❞♦s ❞❡ ♠❛♥❡✐r❛ ❡①❛t❛✱ ♦✉ s❡❥❛✱ só sã♦ ❞✐s♣♦♥í✈❡✐s ❛♣r♦①✐♠❛çõ❡s

yδi

❞♦s ❞❛❞♦s ❡①❛t♦s s❛t✐s❢❛③❡♥❞♦✱

kyδiyik< δi, i= 0, . . . , p1. ✭✶✳✼✮

❆♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦ s❡ ❛ss✉♠❡ q✉❡ ♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ✭✶✳✻✮ t❡♠ ♣❡❧♦ ♠❡♥♦s ✉♠❛ s♦❧✉çã♦ x∗ ∈Tpi=0−1Di✳ ❖ ♣r♦♣ós✐t♦ ❞♦ ♣r❡s❡♥t❡

tr❛❜❛❧❤♦ é ❡st✉❞❛r s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞♦ t✐♣♦ ✭✶✳✻✮ ♣❛r❛ ♦s q✉❛✐s ❛ s♦❧✉çã♦ x∗ ♥ã♦ ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞♦s ❞❛❞♦s yi✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦

♦ ♣r♦❜❧❡♠❛ ✐♥✈❡rs♦ ❞❡✜♥✐❞♦ ♣❡❧♦ s✐st❡♠❛ ✭✶✳✻✮ é ♠❛❧ ♣♦st♦ ❡ ♣♦rt❛♥t♦ ♣r❡❝✐s❛ s❡r r❡❣✉❧❛r✐③❛❞♦✳

❊①✐st❡♠ ♣❡❧♦ ♠❡♥♦s ❞♦✐s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♠❛❧ ♣♦st♦s ❞❛ ❢♦r♠❛ ❞♦ s✐st❡♠❛ ✭✶✳✻✮✿ ▼ét♦❞♦s ❞❡ ❘❡❣✉❧❛r✐③❛çã♦ ■t❡✲ r❛t✐✈♦s ❡ ▼ét♦❞♦s ❞❡ ❘❡❣✉❧❛r✐③❛çã♦ ❞❡ ❚✐❦❤♦♥♦✈✱ ♠❛s ❛♠❜♦s ♠ét♦❞♦s sã♦ ✐♥❡✜❝✐❡♥t❡s ♣❛r❛p❣r❛♥❞❡ ♦✉ s❡ ❛s ❛✈❛❧✐❛çõ❡s ❞❡Fi(x)❡Fi′(x)∗ sã♦

(22)

❖ ♠ét♦❞♦ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦ é ❞❡✜♥✐❞♦ ♣♦r

xδn+1=xδn−F[′n](xδn)∗(F[n](xδn)−yδ

[n]

), ✭✶✳✽✮

♦♥❞❡[n] :=n modp∈ {0,1, . . . , p−1}✳ ◆♦t❡ q✉❡ ♣❛r❛p= 1t❡♠✲s❡ ♦

♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✳

❖❜s❡r✈❛çã♦ ✶✳✶✳ ➱ ✐♠❡❞✐❛t♦ ♦❜s❡r✈❛r q✉❡ ❛ ❡str❛té❣✐❛ t✐♣♦ ❑❛❝③♠❛r③ ♣♦❞❡ s❡r ✉s❛❞❛ ❝♦♥❥✉♥t❛♠❡♥t❡ ❝♦♠ q✉❛❧q✉❡r ♦✉tr♦ ♠ét♦❞♦ ✐t❡r❛t✐✈♦ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♠❛❧ ♣♦st♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♠ét♦❞♦s t✐♣♦ ◆❡✇t♦♥✱ ❡t❝✳ ❊ss❡♥❝✐❛❧♠❡♥t❡✱ ❛♣❧✐❝❛✲s❡ ✉♠❛ ✐t❡r❛çã♦ ❞♦ ♠ét♦❞♦ ❡s❝♦❧❤✐❞♦ ♣❛r❛ ❝❛❞❛ ✉♠❛ ❞❛s ❡q✉❛çõ❡s ❞♦ s✐st❡♠❛ ❞❡ ❢♦r♠❛ ❝í❝❧✐❝❛✳

P❛r❛ r❡s♦❧✈❡r ♦ s✐st❡♠❛ ✭✶✳✻✮ ❝♦♠ ❛ r❡❧❛çã♦ ✭✶✳✼✮ s❡rá ✉s❛❞♦ ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✐♥❝♦r♣♦r❛♥❞♦ ✉♠ ♣❛râ♠❡tr♦ ❞❡ r❡❧❛①❛♠❡♥t♦ ❜❛♥❣✲❜❛♥❣ ❛♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦ ✭✈❡❥❛ ❬✷✵❪✮✱ ❝♦♠✲ ❜✐♥❛❞♦ ❝♦♠ ✉♠ ♥♦✈♦ ♣r✐♥❝í♣✐♦ ❞❛ ❞✐s❝r❡♣â♥❝✐❛✱ ✐st♦ é✱

xδn+1=xδn−ωnF[′n](xδn)∗(F[n](xδn)−yδ

[n]

), ✭✶✳✾✮

❝♦♠

ωn:=ωn(δ, yδ) =

1, kF[n](xδn)−yδ

[n]

k> τ δ[n]

0, ❝❛s♦ ❝♦♥trár✐♦✳ ✭✶✳✶✵✮

♦♥❞❡ τ > 2 é ✉♠❛ ❝♦♥st❛♥t❡ ❡s❝♦❧❤✐❞❛ ❛♣r♦♣r✐❛❞❛♠❡♥t❡✳ ❆ ✐t❡r❛çã♦

❛❝❛❜❛ q✉❛♥❞♦ kFi(xδn)−yδ

i

k ≤τ δi,

∀i ∈ {0,1, . . . , p1}✳ ❊ss❛ ✈❛✲

r✐❛♥t❡ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ ❧♦♣✐♥❣ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✭▲▲❑✮✳ P❛r❛ ❞❛❞♦s ❡①❛t♦s ✭δi = 0,i

{0,1, . . . , p−1}✮ t❡♠✲s❡ ωn = 1,∀n ∈ N✱ ❡ ❛ss✐♠ ♥❡st❡ ❝❛s♦ ♣❛rt✐✲

❝✉❧❛r ♦ ♠ét♦❞♦▲▲❑ é ♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦✳ ❆

s❡❣✉✐r✱ ❡ ♣❛r❛ ❝♦♥t❡①t✉❛❧✐③❛r ♦ tr❛❜❛❧❤♦ ❢❡✐t♦ ♥♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠ ❜r❡✈❡ r❡s✉♠♦ s♦❜r❡ ♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❡ t❛①❛s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♦❜t✐❞♦s ♥♦s ú❧t✐♠♦s ❛♥♦s✳

✶✳✷✳✸ ❘❡❧❛çã♦ ❍✐stór✐❝❛ ❞❡ ❘❡s✉❧t❛❞♦s ❖❜t✐❞♦s ♣❛r❛

▼ét♦❞♦s ❚✐♣♦ ❑❛❝③♠❛r③

(23)

❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛✲s❡ ✉♠ ❜r❡✈❡ r❡s✉♠♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ♠ét♦❞♦s t✐♣♦ ❑❛❝③♠❛r③✱ ♣❛r❛ ♦s ❝❛s♦s ❧✐♥❡❛r ❡ ♥ã♦ ❧✐♥❡❛r✿

❬✷✵✵✻❪ ■t❡r❛t✐✈❡❧②✲❘❡❣✉❧❛r✐③❡❞✲●❛✉ss✲◆❡✇t♦♥✲❑❛❝③♠❛r③ ❬✸❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ ❝♦♠ t❛①❛s❀

❬✷✵✵✼❪ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❬✶✶✱ ✶✸❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀ ❬✷✵✵✽❪ ❙t❡❡♣❡st✲❉❡s❝❡♥t✲❑❛❝③♠❛r③ ❬✻❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀ ❬✷✵✵✾❪ ❊①♣❡❝t❛t✐♦♥✲▼❛①✐♠✐③❛t✐♦♥✲❑❛❝③♠❛r③ ❬✶✷❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀

❬✷✵✵✾❪ ❇❧♦❝❦✲▼❛①✐♠✐③❛t✐♦♥✲❑❛❝③♠❛r③ ❬✶✵❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s ♣❛r❛ s✐st❡♠❛s ❧✐♥❡❛r❡s❀

❬✷✵✶✵❪ ▲❡✈❡♥❜❡r❣✲▼❛rq✉❛r❞t✲❑❛❝③♠❛r③ ❬✷❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀ ❬✷✵✶✶❪ ■t❡r❛t❡❞ ❚✐❦❤♦♥♦✈✲❑❛❝③♠❛r③ ❬✺❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s❀ ❬✷✵✶✶❪ P❛r❛❧❡❧❧✲❘❡❣✉❧❛r✐③❡❞✲◆❡✇t♦♥✲❑❛❝③♠❛r③ ❬✶❪❀ ❝♦♥✈❡r❣ê♥❝✐❛ s❡♠ t❛①❛s✳

(24)
(25)

❈❛♣ít✉❧♦ ✷

❆♥á❧✐s❡ ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛

❞♦ ▼ét♦❞♦

▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③

◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ♠ét♦❞♦▲▲❑✱ ✐st♦ é✱ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s ✭❚❡♦r❡♠❛ ✷✳✶✮ ❡ ♥♦

❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦ ✭❚❡♦r❡♠❛ ✷✳✷✮ ♦ ♠ét♦❞♦▲▲❑ ♠❛✐s ♦ ❝r✐tér✐♦

❞❡ ♣❛r❛❞❛ ✭❛ s❡r ❞❡✜♥✐❞♦✮✱ é ✉♠ ♠ét♦❞♦ ❞❡ r❡❣✉❧❛r✐③❛çã♦✳

❆♦ ❧♦♥❣♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ s❡rã♦ ❛ss✉♠✐❞❛s ❤✐♣ót❡s❡s q✉❡ sã♦ ♣❛✲ ❞rã♦ ♥❛ ❛♥á❧✐s❡ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❡ ♠ét♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✐t❡r❛t✐✈♦s✳ ◆♦ ❝♦♥t❡①t♦ ❞♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ✭✶✳✻✮✱ s✉♣õ❡✲s❡ q✉❡ ♦s ♦♣❡r❛❞♦r❡s

Fi, i = 0,1, . . . , p−1 sã♦ ❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡✐s ❡ q✉❡ ∃ρ > 0 t❛❧ q✉❡

∀i= 0,1, . . . , p1✱ t❡♠✲s❡

kFi′(x)k ≤1, x∈Bρ(x0)⊂D:=

p\−1

i=0

Di ✭✷✳✶✮

❡✱

kFi(x)−Fi(xe)−Fi′(x)(x−xe)kY ≤ ηkFi(x)−Fi(xe)kY

∀x,xeBρ(x0)⊂D, ✭✷✳✷✮

❝♦♠0< η <1/2❡Bρ(x0) ={x∈X :kx−x0k< ρ}✳ ❆❧é♠ ❞♦ ❛♥t❡r✐♦r✱

♦ ♣❛râ♠❡tr♦τ ❞❡ ✭✶✳✶✵✮ s❛t✐s❢❛③✱

τ >2 1 +η

12η >2. ✭✷✳✸✮

(26)

✶✷

❈♦♠♦ ❢♦✐ ❛♣♦♥t❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ♣❛r❛ ❞❛❞♦s ❧✐✈r❡s ❞❡ r✉í❞♦ ♦ ♠ét♦❞♦▲▲❑ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ❝❧áss✐❝♦✱

♠❛s ♥♦ ❝❛s♦ ❞❡ ❞❛❞♦s ❝♦♠ r✉í❞♦✱ ♦s ♠ét♦❞♦s ❞❡ r❡❣✉❧❛r✐③❛çã♦ ✐t❡r❛t✐✈♦s ♣r❡❝✐s❛♠ t❡r♠✐♥❛r ❛s ✐t❡r❛çõ❡s ❛♥t❡❝✐♣❛❞❛♠❡♥t❡✱ ❢❛t♦ q✉❡ s❡ ❝♦♥s❡❣✉❡ ❝♦♠ ✉♠ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ❛❞❡q✉❛❞♦✳ ◆♦ s❡❣✉✐♥t❡ ❧❡♠❛✱ ♦❜té♠✲s❡ ✉♠❛ ❡st✐♠❛t✐✈❛ r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞♦ ❡rr♦✱ kenk:=kxδn−

x∗k✱ r❡s✉❧t❛❞♦ q✉❡ ♠♦t✐✈❛ ❛ ❞❡✜♥✐çã♦ ❞♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦ ♠ét♦❞♦▲▲❑✳

▲❡♠❛ ✷✳✶✳ ❙❡❥❛ x∗ ✉♠❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ✭✶✳✻✮ ♥❛ ❜♦❧❛ B

ρ/2(x0)✱

❝♦♠ ♦s ♦♣❡r❛❞♦r❡s Fi ❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ Bρ(x0) s❛t✐s❢❛③❡♥❞♦

✭✷✳✶✮ ❡ ✭✷✳✷✮✳ ❙❡❥❛ {xδ

n}n∈N ❛ s❡q✉ê♥❝✐❛ ❞❡✜♥✐❞❛ ♣♦r ✭✶✳✾✮✱ ✭✶✳✶✵✮ ❡ ❛✐♥❞❛ s✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❝❡rt♦nN✱xδnBρ/2(x)Bρ(x0)✳ ❊♥tã♦

kxδ

n+1−x∗k2− kxδn−x∗k2

≤ωnkF[n](xδn)−yδ

[n]

k2(1 +η)δ[n]

−(12η)kF[n](xδn)−yδ

[n]

k,

✭✷✳✹✮

♦♥❞❡[n] :=n modp∈ {0,1, . . . , p−1}✳

Pr♦✈❛✿ P❛r❛ ωn = 0 ❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❡ xδn+1 = xδn✱ ❧♦❣♦ ♣❛r❛

ωn6= 0 t❡♠✲s❡

kxδn+1−x∗k2− kxδn−x∗k2=

=hx∗xδ

n+1, x∗−xδn+1i − hx∗−xδn, x∗−xδni

= 2 hx∗, xδni − hx∗, xδn+1i

− kxδnk2+hxδn+1±xδn, xδn+1±xδni

= 2hx∗, xδn−xδn+1i+kxδn+1−xδnk2+ 2hxnδ+1−xδn, xδni

= 2hx∗−xδn, xδn−xnδ+1i+kxδn+1−xδnk2

(1.9)

= 2ωnhx∗−xδn, F[′n](x

δ

n)∗(F[n](xδn)−yδ

[n]

)i+kxδn+1−xδnk2

= 2ωnhF[′n](x

δ

n)(x∗−xδn), F[n](xδn)−yδ

[n]

i+kxδn+1−xδnk2

= 2ωnhF[′n](x

δ

n)(x∗−xδn)±(F[n](xδn)−yδ

[n]

), F[n](xδn)−yδ

[n]

i

+kxδn+1−xδnk2

= 2ωnhF[n](xnδ)−F[n](x∗)−F[′n](xδn)(xδn−x∗) +y−yδ

[n]

,

F[n](xδn)−yδ

[n]

i −2ωnkF[n](xδn)−yδ

[n]

k2+

kxδn+1−xδnk2

❝✳s✱(∗)

(2.2)

≤ 2ωn

ηkF[n](xδn)−F[n](x∗)k+ky−yδ

[n]

kkF[n](xδn)−yδ

[n]

(27)

✶✸

−ωnkF[n](xδn)−yδ

[n]

k2 ≤2ωn

ηkF[n](xδn)−yδ

[n]

k+δ[n]+δ[n]kF[n](xδn)−yδ

[n]

k

−ωnkF[n](xδn)−yδ

[n]

k2

=ωnkF[n](xδn)−yδ

[n]

k2(η+ 1)δ[n](12η)kF[n](xδn)−yδ

[n]

k,

♦❜t❡♥❞♦ ❛ss✐♠ ❛ ❡st✐♠❛t✐✈❛ ❞❡s❡❥❛❞❛✳

❊♠()❢♦✐ ✉s❛❞♦ ✭✶✳✾✮ ❡ ✭✷✳✶✮ ♣❛r❛ ♦❜t❡r✱

kxδn+1−xδnk2=ωn2kF[′n](xδn)∗(F[n](xδn)−yδ

[n]

)k2

≤ωnkF[n](xδn)−yδ

[n]

k2.

❆ s❡❣✉✐r✱ ❡ ♠♦t✐✈❛❞♦ ♣❡❧♦ ▲❡♠❛ ✷✳✶✱ ❞❡✜♥❡✲s❡ ♦ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦ ♠ét♦❞♦▲▲❑✳

❉❡✜♥✐çã♦ ✷✳✶✳ ❉❡✜♥❡✲s❡ nδ

∗ := nδ∗(yδ)✱ ♦ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦ ♠ét♦❞♦ ▲▲❑✱ ❝♦♠♦ ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♠ú❧t✐♣❧♦ ❞❡p∈Nt❛❧ q✉❡

xδnδ

∗ =x

δ nδ

∗+1=· · ·=x

δ nδ

∗+p. ✭✷✳✺✮

❆❣♦r❛✱ ✉s❛♥❞♦ ♦ ▲❡♠❛ ✷✳✶ ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ í♥❞✐❝❡ ❞❡ ♣❛r❛❞❛ ♣❛r❛ ♦ ♠ét♦❞♦ ▲▲❑✱ t❡♠✲s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❞❡ ♠♦♥♦t♦♥✐❝✐❞❛❞❡ ❞♦ ❡rr♦

kenk:=kxδn−x∗k✿

▲❡♠❛ ✷✳✷✳ ❙❡❥❛♠x∗Fi, i= 0, . . . , p1 xδ

n ❝♦♠♦ ♥♦ ▲❡♠❛ ✷✳✶ ❡nδ∗ ❝♦♠♦ ❞❡✜♥✐❞♦ ❡♠ ✭✷✳✺✮✳ ❊♥tã♦✱

kxδn+1−x∗k ≤ kxnδ −x∗k, n= 0, . . . , nδ∗, ✭✷✳✻✮ ❡ ✭✷✳✺✮ ✐♠♣❧✐❝❛ ωnδ

∗+i= 0✱ ♣❛r❛ t♦❞♦ i∈ {0, . . . , p−1}✱ ✐st♦ é

kFi(xδnδ

∗)−y

δi

k ≤τ δi, i= 0, . . . , p1. ✭✷✳✼✮

Pr♦✈❛✿ ◆❛s ❤✐♣ót❡s❡s ❞♦ ▲❡♠❛ ✷✳✶✱ s❡ωn= 0✱ ❡♥tã♦ ✭✷✳✻✮ é s❛t✐s❢❡✐t♦

❥á q✉❡ ♥❡ss❡ ❝❛s♦ xδ

n+1 = xδn✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ωn ❡♠

✭✶✳✶✵✮ ❡ ❞❡τ ❡♠ ✭✷✳✸✮ t❡♠✲s❡

kF[n](xδn)−yδ

[n]

k> τ δ[n]>2(1 +η)

12η δ

(28)

✶✹

♦ q✉❡ ✐♠♣❧✐❝❛

2(1 +η)δ[n](12η)kF[n](xδn)−yδ

[n]

k<0,

❡ ❞❡ss❛ ❢♦r♠❛ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✹✮ é ♥ã♦ ♣♦s✐t✐✈♦ ♦❜t❡♥❞♦ ❛ss✐♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✻✮ ♣❛r❛ ❡ss❡n✳ ❆ ♣r♦✈❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✻✮

é ❝♦♠♣❧❡t❛❞❛ ♣♦r ✐♥❞✉çã♦ s♦❜r❡n✳ P❛r❛ n= 0✱ ❝♦♠♦ xδ

0 ∈Bρ/2(x∗)✱

❡♥tã♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✻✮ ✐♠♣❧✐❝❛ xδ

1 ∈ Bρ/2(x∗) ⊂ Bρ(x0)✳ ❆❣♦r❛✱

♣❛r❛ 0 < n < nδ

∗ t❛❧ q✉❡ xδn ∈ Bρ/2(x∗)✱ t❡♠✲s❡ ♥♦✈❛♠❡♥t❡ ♣♦r ✭✷✳✻✮

q✉❡xδ

n+1∈Bρ/2(x∗)⊂Bρ(x0)✱ ❧♦❣♦ é ✈á❧✐❞❛ ♣❛r❛ t♦❞♦n= 0, . . . , nδ

♦ q✉❡ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛✳ P❛r❛ ♣r♦✈❛r ✭✷✳✼✮✱ ✉s❛✲s❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✷✳✹✮ ❝♦♠n =nδ

∗+i✱ i∈ {0, . . . , p−1}✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ nδ∗✱ t❡♠✲s❡ q✉❡

xδ nδ

∗+i=x

δ nδ

∗ ❡[n

δ

∗+i] =i✱ ❛ss✐♠

0≤ωnδ

∗+ikFi(x

δ nδ

∗)−y

δi

k2(1 +η)δi−(1−2η)kFi(xδnδ

∗)−y

δi

k,

♣❛r❛ i∈ {0, . . . , p−1}✳ ❙❡ ωnδ

∗+i 6= 0♣❛r❛ ❛❧❣✉♠ i∈ {0, . . . , p−1}✱

❡♥tã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱

2(1 +η)δi(12η)kFi(xδnδ

∗)−y

δi

k ≥0,

♦ q✉❡ ✐♠♣❧✐❝❛ ♣♦r ✭✷✳✸✮ q✉❡ kFi(xδnδ

∗)−y

δi

k < τ δi ♦ q✉❡ ❝♦♥tr❛❞✐③ ❛

❞❡✜♥✐çã♦ ❞❡ωnδ

∗+i✱ ❧♦❣♦ωnδ∗+i= 0✱ ♣❛r❛ t♦❞♦i∈ {0, . . . , p−1}❡ ❛ss✐♠

♦❜té♠✲s❡ ✭✷✳✼✮✳

◆♦t❡ q✉❡ ♣❛r❛ n > nδ✱ t❡♠✲s❡ ωn ≡0 ❡ ❛ss✐♠ xδn =xδnδ

∗✱ ✐st♦ é✱ ♦

♠ét♦❞♦▲▲❑ t♦r♥❛✲s❡ ❡st❛❝✐♦♥ár✐♦ ❛♣ós ♦ í♥❞✐❝❡nδ

∗✳

❆❣♦r❛✱ ✉s❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ❛♥t❡r✐♦r t❡♠✲s❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳ Pr♦♣♦s✐çã♦ ✷✳✶✳ ◆❛s ❤✐♣ót❡s❡s ❞♦ ▲❡♠❛ ✷✳✶✱ ♦❜té♠✲s❡ ❛s s❡❣✉✐♥t❡s ❡st✐♠❛t✐✈❛s

(τmin

i (δ i))2

∗−1

X

n=0

ωnkyδ

[n]

−F[n](xδn)k2≤

τkx0−x∗k2

(1−2η)τ−2(1 +η), ✭✷✳✽✮ ❡ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✱ ✐st♦ é✱ yδi

=yi,

∀i= 0, . . . , p1✱ t❡♠✲s❡

X

n=0

(29)

✶✺

Pr♦✈❛✿ P❛r❛ n= 0, . . . , nδ

∗−1✱ ❝♦♠♦ ωn = 1✱ ♣♦r ❞❡✜♥✐çã♦ ✐♥❢❡r❡✲s❡

q✉❡δ[n]< τ−1kyδ[n]−F[n](xδn)k✳ ❈❤❛♠❛♥❞♦ ❞❡en=xn−x∗❡ ✉s❛♥❞♦

✭✷✳✹✮ t❡♠✲s❡✱

ken+1k2− kenk2≤ωnkyδ

[n]

−F[n](xδn)k2 2τ−1(1 +η) + 2η−1

,

♦ q✉❡ ✐♠♣❧✐❝❛ ♣♦r ✭✷✳✸✮✱

(τ δ[n])2ωnkyδ

[n]

−F[n](xδn)k2≤ k

enk2− ken+1k2

12τ−1(1 +η),

❡ ❛ss✐♠✱

(τmin

i (δ i))2

≤ωnkyδ

[n]

−F[n](xδn)k2≤

τ(kenk2− ken+1k2)

(12η)τ2(1 +η).

❙♦♠❛♥❞♦ ♥❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠n= 0, . . . , nδ

∗−1✱ t❡♠✲s❡✱

(τmin

i (δ i))2

∗−1

X

n=0

ωnkyδ

[n]

−F[n](xδn)k2≤

τ(ke0k2− kenδ

∗k

2)

(12η)τ2(1 +η),

♦❜t❡♥❞♦ ✜♥❛❧♠❡♥t❡✱

∗(τmini (δi))2≤

∗−1

X

n=0

ωnkyδ

[n]

−F[n](xδn)k2≤

τkx0−x∗k2

(1−2η)τ−2(1 +η).

❆❣♦r❛✱ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s δi= 0,

∀i= 0, . . . , p1✱ ❞❡ ✭✷✳✹✮ s❡ ❞❡❞✉③✱

kxn+1−x∗k2− kxn−x∗k2≤(2η−1)kF[n](xn)−y[n]k2,

❡ s♦♠❛♥❞♦ ♦❜té♠✲s❡ ❛ ❡st✐♠❛t✐✈❛✱

N

X

n=0

kF[n](xn)−y[n]k2≤ k

x0−x∗k2− kxN+1−x∗k2

1−2η

≤ kx0−x∗k

2

12η ,∀N ∈N,

❞❡ ♦♥❞❡ ✐♥❢❡r❡✲s❡ q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ s♦♠❛s ♣❛r❝✐❛✐s é ❧✐♠✐t❛❞❛✱ ❡ ❛ss✐♠ ✜♥❛❧♠❡♥t❡ ♣♦❞❡✲s❡ t♦♠❛r ❧✐♠✐t❡ q✉❛♥❞♦N → ∞♣❛r❛ ♦❜t❡r

X

n=0

kF[n](xn)−y[n]k2≤k

x0−x∗k2

(30)

✶✻

❞❡ ♦♥❞❡ ♦❜té♠✲s❡ ♦ r❡s✉❧t❛❞♦✳

❉❛ Pr♦♣♦s✐çã♦ ✷✳✶✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♣♦r ✭✷✳✽✮✱ ✐♥❢❡r❡✲s❡ q✉❡ ♥❡❝❡s✲ s❛r✐❛♠❡♥t❡ nδ

∗ <∞ ❡ ❛ss✐♠ ❛ ✐t❡r❛çã♦ t❡r♠✐♥❛ ♥✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣❛ss♦s✳

◆❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✱ ❛♣r❡s❡♥t❛♠✲s❡ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s s♦❧✉çõ❡s ❞❡ ✭✶✳✻✮ ♥✉♠❛ ❜♦❧❛Bρ(x0)✳ ❆ ♣r♦✈❛ ❢❡✐t❛ ♥♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦

♠❡❧❤♦r❛ ❛ ♣r♦✈❛ ❢❡✐t❛ ❡♠ ❬✷✵❪ ✭Pr♦♣♦s✐çã♦ ✹✳✶✮✱ q✉❡ é ✐♥❝♦♠♣❧❡t❛✳ Pr♦♣♦s✐çã♦ ✷✳✷✳ ❙❡❥❛♠ ♦s ♦♣❡r❛❞♦r❡sFi, i= 0, . . . , p−1s❛t✐s❢❛③❡♥❞♦

✭✷✳✶✮ ❡ ✭✷✳✷✮✳ ❙❡ x∗B

ρ(x0)é ✉♠❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ✭✶✳✻✮✱ ❡♥tã♦

✶✳ t♦❞❛ ♦✉tr❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮✱ ex∗Bρ(x0)s❛t✐s❢❛③✱

x∗−xe∗∈ N(Fi′(x∗)),∀i= 0, . . . , p−1,

♦♥❞❡N(A)é ♦ ♥ú❝❧❡♦ ❞♦ ♦♣❡r❛❞♦r ❧✐♥❡❛rA✱

✷✳ ❡①✐st❡ ✉♠❛ x0✲s♦❧✉çã♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛✱x†∈Bρ(x0)✱ ✐st♦ é✱x†

é ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮ ❡ s❛t✐s❢❛③

kx†x0k= inf

x∈Skx−x0k ♦♥❞❡S:={x∈Bρ(x0) :xs❛t✐s❢❛③ ✭✶✳✻✮}

✸✳ ❛ x0✲s♦❧✉çã♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛ x† é ú♥✐❝❛ ❡ s❛t✐s❢❛③

x†x0∈

p\−1

i=0

N(Fi′(x†))

!⊥

.

Pr♦✈❛✿

✶✳ ❉❡ ✭✷✳✷✮ t❡♠✲s❡ q✉❡ ♣❛r❛ t♦❞♦i∈ {0, . . . , p1}x,xeBρ(x0)

|kFi(x)−Fi(xe)k − kFi′(x)(x−xe)k| ≤ηkFi(x)−Fi(xe)k,

❡ ❧❡♠❜r❛♥❞♦ q✉❡η <1/2✱ ✐♥❢❡r❡✲s❡✱

kFi(x)−Fi(ex)k ≤

1 1ηkF

i(x)(x−xe)k.

❙✐♠✐❧❛r♠❡♥t❡ t❡♠✲s❡✱ 1 1 +ηkF

(31)

✶✼

❡ ❛ss✐♠✱

1 1 +ηkF

i(x)(x−xe)k ≤ kFi(x)−Fi(xe)k ≤ 1

1ηkF

i(x)(x−xe)k,

✭✷✳✶✵✮ ❧♦❣♦ ♣❛r❛x∗,xe∗∈Bρ(x0)s♦❧✉çõ❡s✱ t❡♠✲s❡kFi′(x∗)(x∗−xe∗)k= 0✱

♣❛r❛ t♦❞♦i∈ {0, . . . , p1}✱ ♣r♦✈❛♥❞♦ ❛ss✐♠ ♦ ✐t❡♠ ✶✳

✷✳ ❉❡✜♥❛

m:= inf

x∈Skx−x0k, ❡ ❡s❝♦❧❤❛σ >0t❛❧ q✉❡m≤ kx∗x

0k< σ < ρ✳ ❉❡✜♥❛ ❛❣♦r❛

Sσ:={x∈Bσ(x0) :xs❛t✐s❢❛③ ✭✶✳✻✮} ⊂ S,

❡♥tã♦ ✉s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞♦ í♥✜♠♦ t❡♠✲s❡✱

inf

x∈Sσk

xx0k=m.

❙❡❥❛{xk}k∈N⊂ Sσ t❛❧ q✉❡

kxk−x0k

k→∞

−→ m.

❈♦♠♦{xk}k∈N⊂Bρ(x0)✱ ❡♥tã♦ é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡ ♣♦r✲

t❛♥t♦ ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❢r❛❝❛♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡✱ q✉❡ s❡ ❞❡♥♦t❛rá ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛✱ ❡ ❝✉❥♦ ❧✐♠✐t❡ ❢r❛❝♦ s❡rá ❞❡♥♦t❛❞♦ ♣♦r ex✳ ❈♦♠♦ ♦s ♦♣❡r❛❞♦r❡s Fi sã♦ ❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡✐s✱ ❡♥tã♦

♦s ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡sF′

i(x∗)sã♦ ❧✐♠✐t❛❞♦s ❡ ♣♦rt❛♥t♦

Fi′(x∗)(xk−x∗)⇀ Fi′(x∗)(xe−x∗), ∀i= 0, . . . , p−1.

❉❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r ❞❛ ♥♦r♠❛ ♥❛ t♦♣♦❧♦❣✐❛ ❢r❛❝❛ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt s❡❣✉❡ q✉❡✱

kexx0k ≤ lim inf

k→∞ kxk−x0k, ✭✷✳✶✶✮

kF′

i(x∗)(ex−x∗)k ≤ lim inf k→∞ kF

i(x∗)(xk−x∗)k. ✭✷✳✶✷✮

❈♦♠♦ xk, x∗ ∈ S,∀k ∈ N✱ ♣❡❧♦ ✐t❡♠ ✶ ❞❛ ♣r❡s❡♥t❡ ♣r♦♣♦s✐çã♦✱

t❡♠✲s❡

(32)

✶✽

❡ ❛ss✐♠ ♣♦r ✭✷✳✶✷✮✱ kF′

i(x∗)(xe−x∗)k= 0,∀i= 0, . . . , p−1✳ P♦r

✭✷✳✶✶✮✱ s❡❣✉❡ q✉❡ xe Bσ(x0) ⊂ Bρ(x0)✱ ❧♦❣♦ ♣♦r ✭✷✳✶✵✮ xe é

s♦❧✉çã♦ ❞♦ s✐st❡♠❛ ✭✶✳✻✮✱ ✐st♦ é✱xe∈ Sσ ❡ ❛ss✐♠✱

kexx0k ≥ inf

x∈Sσk

xx0k.

❋✐♥❛❧♠❡♥t❡✱ ❝♦♠♦

inf

x∈Sσk

xx0k= lim

k→∞kxk−x0k= lim infk→∞ kxk−x0k

(2.11)

≥ kexx0k,

✐♥❢❡r❡✲s❡ q✉❡ exé ✉♠❛x0✲s♦❧✉çã♦ ❞❡ ♥♦r♠❛ ♠í♥✐♠❛✳

✸✳ ❉❡✜♥❛ ♦ ♦♣❡r❛❞♦r✱

F:= (F0, . . . , Fp−1) :D=

p\−1

i=0

Di⊂X→Yp,

❡♥tã♦✱ ♦ ♣r♦❜❧❡♠❛ ✭✶✳✻✮ ♣♦❞❡ s❡ ❢♦r♠✉❧❛r ❝♦♠♦ F(x) =y✱ ♦♥❞❡

y= (y0, . . . , yp−1)✳ ◆❡st❡ ❝❛s♦✱

N(F′(x)) =

p\−1

i=0

N(Fi′(x)),

❡ ❝♦♠♦ ♦s ♦♣❡r❛❞♦r❡sFi, i= 0, . . . , p−1s❛t✐s❢❛③❡♠ ✭✷✳✷✮✱ t❡♠✲s❡✱

kF(x)F(xe)F′(x)(xxe)k2Yp

=

p−1

X

i=0

kFi(x)−Fi(xe)−Fi′(x)(x−xe)k2Y

≤η2

p−1

X

i=0

kFi(x)−Fi(xe)k2=η2kF(x)−F(ex)k2Yp,

♣❛r❛ t♦❞♦ x,exBρ(x0)✱ ♦✉ s❡❥❛✱ ♦ ♦♣❡r❛❞♦rF s❛t✐s❢❛③ ❛s ❤✐♣ó✲

t❡s❡s ❞❛ Pr♦♣♦s✐çã♦ ✷✳✶ ❞❡ ❬✶✻❪✱ ❡ ❛ss✐♠ x† é ❛ ú♥✐❝❛ x

0✲s♦❧✉çã♦

❞❡ ♥♦r♠❛ ♠í♥✐♠❛ ♥❛ ❜♦❧❛Bρ(x0)❡ s❛t✐s❢❛③✱

x†x

0∈(N(F′(x)))⊥=

p\−1

i=0

N(F′

i(x))

!⊥

,

(33)

✶✾

❆❣♦r❛ ❛♣r❡s❡♥t❛✲s❡ ❛ ♣r♦✈❛ ❞♦ ♣r✐♠❡✐r♦ ❞♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s ❞♦ ♣r❡s❡♥t❡ ❝❛♣ít✉❧♦✱ ✐st♦ é✱ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❧♦❝❛❧ ❞♦ ♠ét♦❞♦ ❞❡ ▲❛♥❞❡✇❡❜❡r✲ ❑❛❝③♠❛r③ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s✳

❚❡♦r❡♠❛ ✷✳✶✳ ❆ss✉♠❛ q✉❡ ♦s ♦♣❡r❛❞♦r❡s Fi, i = 0, . . . , p−1✱ sã♦

❋ré❝❤❡t ❞✐❢❡r❡♥❝✐á✈❡✐s ♥❛ ❜♦❧❛ Bρ(x0)✱ s❛t✐s❢❛③❡♥❞♦ ✭✷✳✶✮✱ ✭✷✳✷✮ ❡ q✉❡

✭✶✳✻✮ t❡♠ ✉♠❛ s♦❧✉çã♦ ❡♠ Bρ/2(x0)✳ ❊♥tã♦✱ ♣❛r❛ ❞❛❞♦s ❡①❛t♦s yδ

i

=

yi, i= 0, . . . , p

−1✱ ❛ s❡q✉ê♥❝✐❛{xn}n∈N ❣❡r❛❞❛ ♣♦r ✭✶✳✾✮ ❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮✳ ❆✐♥❞❛ ♠❛✐s✱ s❡x† ❞❡♥♦t❛ ❛ ú♥✐❝❛x

0✲s♦❧✉çã♦ ❞❡

♥♦r♠❛ ♠í♥✐♠❛ ❞❡ ✭✶✳✻✮ ❡♠ Bρ(x0)❡✱

N(F′

i(x†))⊂ N(Fi′(x)),∀i= 0, . . . , p−1,∀x∈Bρ(x0), ✭✷✳✶✸✮

❡♥tã♦✱

xnn−→→∞x†.

Pr♦✈❛✿ ❙❡❥❛xe∗✉♠❛ s♦❧✉çã♦ ❞❡ ✭✶✳✻✮ ❡♠ B

ρ/2(x0)✳ ❉❡✜♥❛

en:=xe∗−xn.

P❡❧♦ ▲❡♠❛ ✷✳✷✱ ❛ s❡q✉ê♥❝✐❛{kenk}n∈Né ♠♦♥ót♦♥❛ ❞❡❝r❡s❝❡♥t❡ ❧✐♠✐t❛❞❛ ✐♥❢❡r✐♦r♠❡♥t❡ ♣♦r ✵✱ ❧♦❣♦ é ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ❛❧❣✉♠ ε 0✳ ❆ ♣r♦✈❛

❝♦♥s✐st❡ ❡♠ ♠♦str❛r q✉❡ {en}n∈N é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✱ ♦ q✉❡ ❛ s✉❛ ✈❡③ ✐♠♣❧✐❝❛ q✉❡ {xn}n∈N é ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✱ ❛ss✐♠ ❢❡✐t♦ ✐ss♦✱ só r❡st❛ ♣r♦✈❛r q✉❡ ♦ ❧✐♠✐t❡ ❞❡ {xn}n∈N é ✉♠❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✭✶✳✻✮✳

❙❡❥❛♠ j, k∈N0 t❛✐s q✉❡kj✳ P❛r❛k=k0·p+k1j =j0·p+j1

❝♦♠k1, j1∈ {0, . . . , p−1}✱ s❡❥❛k0≤l0≤j0 s❛t✐s❢❛③❡♥❞♦✱

p−1

X

s=0

kys−Fs(xl0·p+s)k ≤

p−1

X

s=0

kys−Fs(xm0·p+s)k,♣❛r❛k0≤m0≤j0.

✭✷✳✶✹✮ ❉❡✜♥❛l1:=p−1 ❡l:=l0·p+l1✳ ❊♥tã♦✱

kej−ekk ≤ kej−elk+kel−ekk,

kej−elk2 = hej−el, ej−eli

= kejk2−2hej, eli+kelk2

= kejk2− kelk2−2hej, eli+ 2hel, eli

(34)

✷✵

❆♥❛❧♦❣❛♠❡♥t❡ ♦❜té♠✲s❡✱

kel−ekk2=kekk2− kelk2+ 2hel−ek, eli.

P❛r❛k→ ∞✱ (kejk2− kelk2)❡(kekk2− kelk2)t❡♥❞❡♠ ❛ ③❡r♦✳ ❆ss✐♠✱

♣❛r❛ ♠♦str❛r q✉❡kej−ekkk−→→∞0✱ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡hel−ej, eli❡

hel−ek, elit❡♥❞❡♠ ❛ ③❡r♦ q✉❛♥❞♦k→ ∞✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞♦ ♠ét♦❞♦ ❞❡

▲❛♥❞✇❡❜❡r✲❑❛❝③♠❛r③ ✭✶✳✽✮✱ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③ t❡♠✲s❡✱

|hel−ek, eli| =

*l1 X

i=k

(xi−xi+1),ex∗−xl

+ =

l−1

X

i=k

D

F[′i](xi)∗(F[i](xi)−y[i]),ex∗−xl

E =

l−1

X

i=k

D

F[i](xi)−y[i], F[′i](xi)(ex∗−xl)

E

l−1

X

i=k

kF[i](xi)−y[i]kkF[′i](xi)(ex∗−xl)k.

❆❣♦r❛ é ♣r❡❝✐s♦ ❡st✐♠❛rkF′

[i](xi)(xe∗−xl)k✳ ❯s❛♥❞♦ ✭✷✳✷✮ ❡ ❛ ❞❡s✐❣✉❛❧✲

❞❛❞❡ tr✐❛♥❣✉❧❛r t❡♠✲s❡✱

kF[′i](xi)(xe∗−xl)k=kF[′i](xi)(ex∗−xi+xi−xl)k

=kF[i](ex∗)−(F[i](xe∗)−F[i](xi)−F[′i](xi)(ex∗−xi)

−F[i](xl)−(F[i](xi)−F[i](xl)−F[′i](xi)(xi−xl)k

≤ kF[i](ex∗)−F[i](xl)k+ηkF[i](ex∗)−F[i](xi)k

+ηkF[i](xi)−F[i](ex∗) +F[i](xe∗)−F[i](xl)k

≤(1 +η)ky[i]F[i](xl)k+ 2ηky[i]−F[i](xi)k.

❈♦♠❜✐♥❛♥❞♦ ❛s ❞✉❛s ú❧t✐♠❛s ❡st✐♠❛t✐✈❛s ♦❜té♠✲s❡✱

|hel−ek, eli| ≤(1 +η) l−1

X

i=k

ky[i]F[i](xi)kky[i]−F[i](xl)k

+ 2η

l−1

X

i=k

ky[i]F[i](xi)k2.

(35)

✷✶

■st♦ ♠♦str❛ q✉❡ hel−ek, eli k→∞

−→ 0✱ s❡

l−1

X

i=k

ky[i]−F[i](xi)kky[i]−F[i](xl)kk−→→∞0, ✭✷✳✶✻✮

❥á q✉❡ ♣♦r ✭✷✳✾✮ t❡♠✲s❡✱

l−1

X

i=k

ky[i]−F[i](xi)k2 k→∞

−→ 0.

P❛r❛ ♣r♦✈❛r ✭✷✳✶✻✮✱ é ♥❡❝❡ssár✐♦ ♣r✐♠❡✐r♦ ❛❝❤❛r ✉♠❛ ❡st✐♠❛t✐✈❛ ♣❛r❛

ky[i]F

[i](xl)k✳ ❋❛③❡♥❞♦i=i0·p+i1 ❝♦♠i1∈ {0, . . . , p−1}✱ t❡♠✲s❡✱

ky[i]

−F[i](xl)k=kyi1−Fi1(xl0·p+l1)k

≤ kyi1F

i1(xl0·p+i1)k+kFi1(xl0·p+i1)−Fi1(xl0·p+l1)k,

❡ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ✉s❛♥❞♦ ✭✷✳✶✵✮✱ ✭✷✳✶✮ ❡ ✭✶✳✽✮ ♦❜té♠✲s❡✱

kFi1(xl0·p+i1)−Fi1(xl0·p+l1)k=

lX1−1

s=i1

(Fi1(xl0·p+s)−Fi1(xl0·p+s+1))

lX1−1

s=i1

kFi1(xl0·p+s)−Fi1(xl0·p+s+1)k

1 1 −η

lX1−1

s=i1

kFi′1(xl0·p+s)(xl0·p+s−xl0·p+s+1)k

1 1 −η

lX1−1

s=i1

kxl0·p+s−xl0·p+s+1k

= 1 1−η

lX1−1

s=i1

kF′

s(xl0·p+s)∗(Fs(xl0·p+s)−y

s)

k

1 1 −η

lX1−1

s=i1

kFs(xl0·p+s)−y

s

k

= 1 1η

p−2

X

s=i1

kFs(xl0·p+s)−y

s

k,

❛ss✐♠✱ ❝♦♠♦

kyi1F

i1(xl0·p+i1)k ≤

p−2

X

s=i1

kFs(xl0·p+s)−y

s

(36)

✷✷

t❡♠✲s❡✱

ky[i]F[i](xl)k ≤

2η

1η

p−2

X

s=i1

kFs(xl0·p+s)−y

s

k.

❯s❛♥❞♦ ✭✷✳✶✹✮ ❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ t❡♠✲s❡✱

l−1

X

i=k

ky[i]

−F[i](xi)kky[i]−F[i](xl)k

l0

X

m0=k0

p−1

X

i1=0

kyi1

−Fi1(xm0·p+i1)kky

i1

−Fi1(xl0·p+l1)k

≤21−η −η

l0

X

m0=k0

p−1

X

i1=0

kyi1

−Fi1(xm0·p+i1)k

p−2

X

s=i1

kysFs(xl0·p+s)k

!

≤21−η −η

l0

X

m0=k0

p−1

X

i1=0

kyi1F

i1(xm0·p+i1)k

!2

≤21−η −ηp

l0

X

m0=k0

p−1

X

i1=0

kyi1

−Fi1(xm0·p+i1)k

2

≤21−η −ηp

l

X

i=k−k1

ky[i]−F[i[(xi)k2.

✭✷✳✶✼✮ ❈♦♠♦(k−k1)k−→ ∞→∞ ✱ ✉s❛♥❞♦ ✭✷✳✾✮ ♣r♦✈❛✲s❡ ✭✷✳✶✻✮ ❡ ❛ss✐♠ ♣♦r ✭✷✳✶✺✮

t❡♠✲s❡ q✉❡ hel−ek, eli → 0 q✉❛♥❞♦k → ∞✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ ❡ t❡♥❞♦

❝✉✐❞❛❞♦ ❡♠ tr♦❝❛r ♦s í♥❞✐❝❡s ❞♦s s♦♠❛tór✐♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ♥♦ ❝❛s♦

k ≤ j ≤ l✭q✉❡ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ l0 = j0✮✱ ♣r♦✈❛✲s❡ hel−ej, eli → 0

q✉❛♥❞♦ k → ∞ ❡ ❛ss✐♠ kej−elk k→∞

−→ 0 ❡ kel−ekk k→∞

−→ 0✱ ✐st♦ é✱

kej−ekk k→∞

−→ 0✳ ❈♦♠ ✐st♦ ✐♥❢❡r❡✲s❡ q✉❡{en}n∈N é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ {xn}n∈N é ❝♦♥✈❡r❣❡♥t❡✳ ❙❡❥❛ x∗ ♦ ❧✐♠✐t❡✳ ◆♦✈❛♠❡♥t❡ ♣♦r ✭✷✳✾✮✱ ♦❜té♠✲s❡ q✉❡ky[n]

−F[n](xn)k →0♣❛r❛n→ ∞✱

♦ q✉❡ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐♠♣❧✐❝❛✱

kyi−Fi(x∗)k= lim n→∞ky

i

−Fi(xn·p+i)k= 0, ∀i= 0, . . . , p−1,

Figure

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