SOBRE A EXISTÊNCIA DE SOLUÇÃO PARA EQUAÇÕES

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❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❙♦❜r❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❡q✉❛çõ❡s

P❡❞r♦ ❆❧✈❛r♦ ❞❛ ❙✐❧✈❛ ❏✉♥✐♦r

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ P❛rr❡✐r❛ ❞❛ ❙✐❧✈❛

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✶✶✶ ❳✶✶✶①

❙✐❧✈❛ ❏✉♥✐♦r✱ P❡❞r♦ ❆❧✈❛r♦

❙♦❜r❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❡q✉❛çõ❡s✴ P❡❞r♦ ❆❧✈❛r♦ ❞❛ ❙✐❧✈❛ ❏✉♥✐♦r✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✸✳

✺✺ ❢✳✿ ✜❣✳✱ t❛❜✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳

❖r✐❡♥t❛❞♦r✿ ❘✐❝❛r❞♦ P❛rr❡✐r❛ ❞❛ ❙✐❧✈❛

✶✳ ❆♥á❧✐s❡✳ ✷✳ ●❡♦♠❡tr✐❛✳ ✸✳ ❚♦♣♦❧♦❣✐❛✳ ✹✳ ▼ét♦❞♦s ✐t❡r❛t✐✈♦s✳ ■✳ ❚ít✉❧♦

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❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

P❡❞r♦ ❆❧✈❛r♦ ❞❛ ❙✐❧✈❛ ❏✉♥✐♦r

❙♦❜r❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ❡q✉❛çõ❡s

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ❘✐❝❛r❞♦ P❛rr❡✐r❛ ❞❛ ❙✐❧✈❛ ❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r❛✳ ❙✉③❡t❡ ▼❛r✐❛ ❙✐❧✈❛ ❆❢♦♥s♦ ■●❈❊ ✲ ❯◆❊❙P

Pr♦❢✳ ❉r❛✳ ❱❡r❛ ▲✉❝✐❛ ❈❛r❜♦♥❡

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ❯❋❙❈❛r

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❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❊❯❙✱ ♣♦r t❡r ♠❡ ❞❛❞♦ ❡st❛ ♦♣♦rt✉♥✐❞❛❞❡✳ ❆ ♠✐♥❤❛ ❢❛♠í❧✐❛ ❡ ❛ ♠✐♥❤❛ ♥❛♠♦r❛❞❛ ♣♦r t❡r ♠❡ ❛♣♦✐❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✱ ❞❡s❞❡ ♦s ♠❛✐s ❛❧❡❣r❡s ❛té ♦s ♠❛✐s ❞❡❧✐❝❛❞♦s✱ ❡♠ ❡s♣❡❝✐❛❧ ♦s ♠❡✉s ♣❛✐s✳ ●♦st❛r✐❛ t❛♠❜é♠ ❞❡ ❞❡✐①❛r ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❡st❡ tr❛❜❛❧❤♦ ❡ ♠❡ ❛❥✉❞❛r❛♠✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❉r✳ ❘✐❝❛r❞♦ P❛rr❡✐r❛ ❞❛ ❙✐❧✈❛ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛✳

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❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❡ ❢✉♥çõ❡s ♣❛r❛ s❡r❡♠ tr❛✲ ❜❛❧❤❛❞❛s ❝♦♠ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ◆♦s ❞♦✐s ♣r✐♠❡✐r♦s ❝❛♣ít✉❧♦s s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ♥♦çõ❡s ❜ás✐❝❛s ❞❡ t♦♣♦❧♦❣✐❛✱ t❛✐s ❝♦♠♦✿ ♠étr✐❝❛✱ ❡s♣❛ç♦s ♠étr✐❝♦s✱ s❡q✉ê♥❝✐❛s ❞❡ ❈❛✉✲ ❝❤②✱ ❡s♣❛ç♦s ♠étr✐❝♦s ❝♦♠♣❧❡t♦s✳ ❊♠ s❡❣✉✐❞❛✱ ✉s❛♥❞♦ ❡st❛s ♥♦çõ❡s✱ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ s✐st❡♠át✐❝♦ s♦❜r❡ ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✱ ❛♣❧✐❝❛♥❞♦✲♦ ❡♠ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ♥✉♠ér✐❝❛s ❡ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▼ét♦❞♦ ❞❡ ◆❡✇t♦♥ ♣❛r❛ ③❡r♦s ❞❡ ❢✉♥çõ❡s✳

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❆❜str❛❝t

❚❤✐s ♣❛♣❡r ❛✐♠s t♦ ♣r❡s❡♥t s♦♠❡ ❡q✉❛t✐♦♥s ❛♥❞ ❢✉♥❝t✐♦♥s t♦ ❜❡ ✇♦r❦❡❞ ✇✐t❤ ❍✐❣❤ ❙❝❤♦♦❧ st✉❞❡♥ts✳ ❚❤❡ ✜rst t✇♦ ❝❤❛♣t❡rs ✇✐❧❧ ♣r❡s❡♥t ❜❛s✐❝s ♦❢ t♦♣♦❧♦❣②✱ s✉❝❤ ❛s✿ ♠❡✲ tr✐❝✱ ♠❡tr✐❝ s♣❛❝❡s✱ ❈❛✉❝❤② s❡q✉❡♥❝❡s ❛♥❞ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡s✳ ❚❤❡♥✱ ✉s✐♥❣ t❤❡s❡ ♥♦t✐♦♥s✱ ✇❡ ✇✐❧❧ ♠❛❦❡ ❛ s②st❡♠❛t✐❝ st✉❞② ♦❢ ❇❛♥❛❝❤✬s ❋✐①❡❞ P♦✐♥t ❚❤❡♦r❡♠✱ ❛♣♣❧②✐♥❣ ✐t t♦ s♦❧✈❡ ♥✉♠❡r✐❝❛❧ ❡q✉❛t✐♦♥s ❛♥❞ ✐♥ t❤❡ ❞❡♠♦♥str❛t✐♦♥ ♦❢ ◆❡✇t♦♥✬s ▼❡t❤♦❞ ❢♦r ③❡r♦s ♦❢ ❢✉♥❝t✐♦♥s✳

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▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ▼étr✐❝❛s d, d′ _❡ _d′′ _❡♠ _R2_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷}

✹✳✶ ●rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) = 1

2cos(x) ❡y =x✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✹✳✷ ●rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) = e−x _❡_y₌_x._{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽}

✹✳✸ ●rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) = 0,75 sen(x) + 1 ❡ y=x. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✹ ●rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) = arctg(x)₋2❡ y=x. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✺ ●rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) = ₋ln(1 +ex₎_❡ _y₌_x. _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸}

✹✳✻ ●rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) = cos(sen(x)) ❡ y=x✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✼ ●rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) = sen(cos(x)) ❡ y=x. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✽ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ▼ét♦❞♦ ❞❡ ◆❡✇t♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✾ ■♥t❡r♣r❡t❛çã♦ ❣❡♦♠étr✐❝❛ ❞♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ■♥t❡r♠❡❞✐ár✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✶✵ ●rá✜❝♦ ❞❡f(x) = x3_{+ 2}_x2₋₅_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷}

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▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✹✳✶ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡f(x) = 1

2cosx. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✷ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡f(x) =e−x_. _{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾}

✹✳✸ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡f(x) = 0,75 sen(x) + 1. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✹ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡f(x) = arctg(x)₋2. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✺ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡f(x) =₋ln(1 +ex₎_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸}

✹✳✻ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡f(x) = cos(sen(x)). ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✼ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡f(x) = sen(cos(x)). ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✽ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡(x, y, z)_→seny

4 , senz

3 + 1, senx

5 + 2

✳ ✹✼ ✹✳✾ ❆♣r♦①✐♠❛çõ❡s ❞❛ r❛✐③ ❞❡f(x) = x3_{+ 2}_x2₋₅_{✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸}

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❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✶✾

✷ ❊s♣❛ç♦s ▼étr✐❝♦s ✷✶

✷✳✶ ❉❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ ❡①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✸ ❊s♣❛ç♦s ▼étr✐❝♦s ❈♦♠♣❧❡t♦s ✷✾

✸✳✶ ❙❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ❊s♣❛ç♦s ❈♦♠♣❧❡t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✹ ▼ét♦❞♦ ❞❛s ❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s ✸✸

✹✳✶ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✷ ❘❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✸ ❖ ▼ét♦❞♦ ❞❡ ◆❡✇t♦♥ ♣❛r❛ ③❡r♦s ❞❡ ❢✉♥çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽

(20)

(21)

✶ ■♥tr♦❞✉çã♦

◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛r❡♠♦s ❡❧❡♠❡♥t♦s ❞❡ t♦♣♦❧♦❣✐❛ q✉❡ s❡rã♦ ✉s❛❞♦s ❝♦♠♦ ❢❡r✲ r❛♠❡♥t❛ ♣❛r❛ ♠♦str❛r ❛ ❡①✐stê♥❝✐❛ ❞❛ s♦❧✉çã♦ ❞❡ ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❡ ❢✉♥çõ❡s✱ ❛♣r❡s❡♥✲ t❛♥❞♦ ❡①❡♠♣❧♦s ❞❡st✐♥❛❞♦s ❛♦ tr❛❜❛❧❤♦ ❞❡ ♣r♦❢❡ss♦r❡s ❝♦♠ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳

◆♦s ♣r✐♠❡✐r♦s ❝❛♣ít✉❧♦s✱ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ❛❧❣✉♠❛s ♥♦çõ❡s ❜ás✐❝❛s ❞❡ t♦♣♦❧♦❣✐❛✱ ❝♦♠♦ ❞❡✜♥✐çã♦ ❞❡ ♠étr✐❝❛✱ ❡s♣❛ç♦s ♠étr✐❝♦s✱ s❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤②✱ ❡s♣❛ç♦s ❝♦♠♣❧❡t♦s✱ s❡♥❞♦ t♦❞♦s ✐❧✉str❛❞♦s ❝♦♠ ❡①❡♠♣❧♦s✳

❊♠ s❡❣✉✐❞❛✱ ❞❡♠♦♥str❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✱ q✉❡ s❡rá ✉s❛❞♦ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ❛❧❣✉♠❛s ❡q✉❛çõ❡s ♥✉♠ér✐❝❛s ❡ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▼ét♦❞♦ ❞❡ ◆❡✇t♦♥ ♣❛r❛ ③❡r♦s ❞❡ ❢✉♥çõ❡s✳

P♦r ✜♠✱ ❛♣r❡s❡♥t❛r❡♠♦s ❡①❡♠♣❧♦s ❞❡ ❡q✉❛çõ❡s ❡ ❢✉♥çõ❡s ♣❛r❛ s❡r❡♠ tr❛❜❛❧❤❛❞❛s ❝♦♠ ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳

(22)

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✷ ❊s♣❛ç♦s ▼étr✐❝♦s

✷✳✶ ❉❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ ❡①❡♠♣❧♦s

❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠❛ ❞✐stâ♥❝✐❛ ♦✉ ♠étr✐❝❛ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ M é ✉♠❛ ❢✉♥çã♦ d✿ M _× M _→ R✱ q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ♣❛r ♦r❞❡♥❛❞♦ (x, y) _∈ M _×M ✉♠ ♥ú♠❡r♦ r❡❛❧ d(x, y)✱ ❝❤❛♠❛❞♦ ❛ ❞✐stâ♥❝✐❛ ❞❡ x ❛ y✱ s❛t✐s❢❛③❡♥❞♦✿

❞1✮ d(x, y)≥0 ❡ d(x, y) = 0⇔x=y❀

❞2✮ d(x, y) = d(y, x), ∀ x, y ∈M❀

❞3✮ d(x, z)≤d(x, y) +d(y, z), ∀ x, y, z ∈M ✭❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✮✳

❯♠ ❡s♣❛ç♦ ♠étr✐❝♦ é ✉♠ ♣❛r(M, d)✱ ♦♥❞❡ M é ✉♠ ❝♦♥❥✉♥t♦ ❡ d é ✉♠❛ ♠étr✐❝❛ ❡♠ M✳ ◗✉❛♥❞♦ ♥ã♦ ❤♦✉✈❡r r✐s❝♦ ❞❡ ❝♦♥❢✉sã♦✱ ♦♠✐t✐r❡♠♦s ❛ ♠étr✐❝❛ ❡ ♥♦s r❡❢❡r✐r❡♠♦s ❛♣❡♥❛s ❛♦ ✧❡s♣❛ç♦ ♠étr✐❝♦ M✧✳

❖❜s❡r✈❛çã♦ ✷✳✶✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ♣♦❞❡♠ s❡r ❞❡ ♥❛t✉r❡③❛ ❜❛st❛♥t❡ ❛r❜✐trár✐❛✿ ♥ú♠❡r♦s✱ ♣♦♥t♦s✱ ✈❡t♦r❡s✱ ♠❛tr✐③❡s✱ ❢✉♥çõ❡s✱ ❝♦♥❥✉♥t♦s ❡t❝✳ P♦ré♠✱ s❡rã♦ s❡♠♣r❡ ❝❤❛♠❛❞♦s ❞❡ ♣♦♥t♦s ❞❡ M✳

❊①❡♠♣❧♦ ✷✳✶✳ ❆ ♠étr✐❝❛ ③❡r♦✲✉♠✳ ❈♦♥s✐❞❡r❡d✿ M_×M _→R✱ ❞❡✜♥✐❞❛ ♣♦rd(x, x) = 0 ❡ d(x, y) = 1 s❡ x₆=y✳ ❯♠ ❡s♣❛ç♦ ♠étr✐❝♦ ♦❜t✐❞♦ ❝♦♠ ❡st❛ ♠étr✐❝❛ é ♠✉✐t♦ út✐❧ ♣❛r❛ ❝♦♥tr❛✲❡①❡♠♣❧♦s✳ ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ dé✱ ❞❡ ❢❛t♦✱ ✉♠❛ ♠étr✐❝❛✿

❞1✮ P❡❧❛ ♣ró♣r✐❛ ❞❡✜♥✐çã♦ ❞❛ ♠étr✐❝❛ ③❡r♦✲✉♠✱ t❡♠♦s✿

d(x, x) = 0 ❡ d(x, y) = 1✱ _∀x, y _∈M✱ ❧♦❣♦ d(x, y)_≥0,_∀x, y _∈M. ❞2✮ P❛r❛ x6=y✱ t❡♠♦s✿ d(x, y) = 1 =d(y, x)✱∀x, y ∈M✳

❞3✮ P❛r❛ ♣r♦✈❛r♠♦s ❛ t❡r❝❡✐r❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ♠étr✐❝❛✱ ✈❛♠♦s ❞✐✈✐❞✐r ❡♠ q✉❛tr♦

❝❛s♦s✿

d(x, y) +d(y, z) = 1 + 1 = 2>1 =d(x, z) s❡x₆=y₆=z✱ d(x, y) +d(y, z) = 0 + 1 = 1 =d(x, z) s❡ x=y, y ₆=z ❡ x₆=z✱ d(x, y) +d(y, z) = 1 + 0 = 1 =d(x, z) s❡ x₆=y, y =z ❡ x₆=z✱ d(x, y) +d(y, z) = 0 + 0 = 0 =d(x, z) s❡ x=y=z✳

❊♠ t♦❞♦s ♦s ❝❛s♦s✱ t❡♠♦s✿ d(x, z)_≤d(x, y) +d(y, z),_∀x, y, z _∈M✳ P♦rt❛♥t♦✱ d é ✉♠❛ ♠étr✐❝❛ ❡♠M✳

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✷✷ ❊s♣❛ç♦s ▼étr✐❝♦s

❊①❡♠♣❧♦ ✷✳✷✳ ❱❡❥❛♠♦s ❛❣♦r❛ ♦ ❡①❡♠♣❧♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦✿ ❛ r❡t❛ r❡❛❧✱ ♦✉ s❡❥❛✱ ♦ ❝♦♥❥✉♥t♦ R ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❙❡❥❛ d : R _×R _→ R ❞❡✜♥✐❞❛ ♣♦r

d(x, y) = _|x₋y_|✱ ❡♥tã♦ d é ✉♠❛ ♠étr✐❝❛ ❡♠ R✳ P♦❞❡♠♦s ❞✐③❡r t❛♠❜é♠ q✉❡ d é ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦sx, y _∈R✳

❞1✮ ❙❡ x 6=y✱ ❡♥tã♦ d(x, y) =|x−y| > 0✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ✈❛❧♦r ❛❜s♦❧✉t♦✳ ❙❡

x=y✱ ❡♥tã♦ d(x, y) =d(x, x) = _|x₋x_|=_|0_|= 0.

❞2✮ d(x, y) =|x−y|=|y−x|=d(y, x)✱ ♣♦✐s|x−y|=|y−x|✳

❞3✮ P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ♠ó❞✉❧♦s✱ s❛❜❡♠♦s q✉❡✿ s❡ a, b∈R✱ ❡♥tã♦

|a+b_{| ≤ |}a_|+_|b_|✱ ♣♦rt❛♥t♦ _|x₋z_|=_|x₋y+y₋z_{| ≤ |}x₋y_|+_|y₋z_|✱ _∀x, y, z _∈R✳

❉❛í✱ ♦❜t❡♠♦s d(x, z)_≤d(x, y) +d(y, z).

P♦rt❛♥t♦✱ dé ✉♠❛ ♠étr✐❝❛ ❡♠ R✱ ♦✉ s❡❥❛✱(R, d)é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❊st❛ ♠étr✐❝❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ♠étr✐❝❛ ✉s✉❛❧ ❞❛ r❡t❛✳

❊①❡♠♣❧♦ ✷✳✸✳ ❈♦♥s✐❞❡r❡ M =Rn✳ ❖s ♣♦♥t♦s ❞❡ Rn sã♦ ❛s ❧✐st❛s x = (x1, x2, ..., xn)

❝♦♠ xi ∈R✳ ❊♠ s❡❣✉✐❞❛✱ ❛♣r❡s❡♥t❛r❡♠♦s três ♠❛♥❡✐r❛s ❞❡ ❞❡✜♥✐r ✉♠❛ ♠étr✐❝❛ ❡♠M✳

❙❡❥❛♠ x= (x1, x2, ..., xn) ❡ y= (y1, y2, ..., yn)✳ ❚❡♠♦s✿

d(x, y) =p(x1−y1)2 +...+ (xn−yn)2 =

v u u t

n

X

i=1

(xi−yi)2;

d′

(x, y) =_|x1−y1|+...+|xn−yn|= n

X

i=1

|xi−yi|;

d′′

(x, y) =max_{|x1−y1|, ...,|xn−yn|}=max1≤i≤n|xi−yi|.

❊①❡♠♣❧♦ ✷✳✹✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ♣❧❛♥♦ R2✳ ❱❛♠♦s ❞❡✜♥✐r ✉♠ ❝♦♥❥✉♥t♦ B ❞❡ ♣♦♥t♦s ❞❡st❡ ♣❧❛♥♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ B[0,1] =_{x_∈R2;d(x,0)_≤1_}✳ ❖s ❝♦♥❥✉♥t♦s B[0,1] r❡❧❛t✐✈❛♠❡♥t❡ às ♠étr✐❝❛s d, d′ ❡ _d′′✱ ♣♦ss✉❡♠ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛s ❢♦r♠❛s ❞❛s ✜❣✉r❛s

❛❜❛✐①♦✳

❋✐❣✉r❛ ✷✳✶✿ ▼étr✐❝❛s d, d′ _❡ _d′′ _❡♠ _R2_✳

❈♦♠ ❡❢❡✐t♦✱ ♣❛r❛ ❛ ♠étr✐❝❛ ❞✱ t❡♠♦s d(x,0)_≤ 1_⇒ p(x1−0)2+ (x2−0)2 ≤ 1⇒

x2

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❉❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ ❡①❡♠♣❧♦s ✷✸

P❛r❛ ❛ ♠étr✐❝❛d′✱ t❡♠♦s q✉❡

|x1−0|+|x2−0| ≤1⇒ |x1|+|x2| ≤1✱ ♦✉ s❡❥❛✱ ✉♠

q✉❛❞r❛❞♦ ❞❡ ❞✐❛❣♦♥❛✐s ♣❛r❛❧❡❧❛s ❛♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ≤2✳ P❛r❛ ❛ ♠étr✐❝❛ d′′✱ t❡♠♦s q✉❡ _max

{|x1 −0|,|x2−0|} ≤ 1 ⇒ max{|x1|,|x2|} ≤ 1✳

❊♥tã♦✱ s❡❣✉❡ q✉❡ |x1| ≤1 ❡|x2| ≤1⇒ −1≤x1 ≤1❡ −1≤x2 ≤1✳

◆♦t❡ q✉❡ ❛ ✜❣✉r❛ ❞❡✜♥✐❞❛ ♣♦r ❡ss❛ ❡①♣r❡ssã♦ s❡r✐❛ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦≤2✳ Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡❥❛♠ d, d′ ❡ _d′′ ❞❡✜♥✐❞❛s ♥♦ ❡①❡♠♣❧♦ ✷✳✸✳ P❛r❛ q✉❛✐sq✉❡r _{x, y}

∈Rn✱

t❡♠♦s✿

d′′

(x, y)_≤d(x, y)_≤d′

(x, y)_≤n.d′′

(x, y).

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ x= (x1, x2, ..., xn), y = (y1, y2, ..., yn)∈Rn✳ ❊♥tã♦✱

d′′

(x, y) = max1≤i≤n|xi−yi|=|xk−yk|

♣❛r❛ ❛❧❣✉♠ k _{∈ {}1,2, ..., n}✳ ❈♦♠♦

|xk−yk|=

p

(xk−yk)2 ≤

p

(x1−y1)2+...+ (xk−yk)2+...+ (xn−yn)2 =d(x, y),

s❡❣✉❡ q✉❡

d′′

(x, y)_≤d(x, y). ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡d(x, y)_≤d′₍_{x, y}₎_.

◆♦t❡ q✉❡✱

[d(x, y)]2 = (x1−y1)2+...+ (xn−yn)2,

❡♥q✉❛♥t♦✱

[d′

(x, y)]2 = [_|x1−y1|+...+|xn−yn|]2

=_|x1−y1|2+ 2.|x1−y1|.

" _n X

i=2

|xi−yi|

#

+

" _n X

i=2

|xi−yi|

#2

=_|x1−y1|2+

" _n X

i=2

|xi−yi|

#2

+ 2._|x1−y1|.

" _n X

i=2

|xi−yi|

# . ❉❡s❡♥✈♦❧✈❡♥❞♦ " _n X i=2

|xi−yi|

#2

.

" _n X

i=2

|xi−yi|

#2

= [_|x2−y2|+...+|xn−yn|]2

=_|x2−y2|2+ 2.|x2−y2|.

" _n X

i=3

|xi −yi|

#

+

" _n X

i=3

|xi −yi|

#2

(26)

✷✹ ❊s♣❛ç♦s ▼étr✐❝♦s

❘❡♣❡t✐♥❞♦ ♦ ♣r♦❝❡ss♦✱ t❡r❡♠♦s [d′

(x, y)]2 =_|x1−y1|2+...+|xn−yn|2+ Ω

♦♥❞❡ Ω_≥0✳ ❉❡st❛ ❢♦r♠❛✱ [d′

(x, y)]2 = [d(x, y)]2+ Ω ❡✱ ♣♦rt❛♥t♦✱

[d(x, y)]2 _≤_[_d′₍_{x, y}_)]2_,

q✉❡ ✐♠♣❧✐❝❛ ❡♠

d(x, y)_≤d′

(x, y) ✉♠❛ ✈❡③ q✉❡ d, d′ sã♦ ♥ã♦ ♥❡❣❛t✐✈♦s✳

❘❡st❛ ♣r♦✈❛r q✉❡ d′₍_{x, y}₎

≤n.d′′₍_{x, y}₎_.

❖❜s❡r✈❡ q✉❡

d′′

(x, y) = max1≤i≤n|xi−yi|=|xk−yk|,

♣❛r❛ ❛❧❣✉♠ k_{∈ {}1,2, ..., n_}.

❊♥tã♦ |xi−yi| ≤ |xk−yk|✱ ♣❛r❛ t♦❞♦ i∈ {1,2, ..., n} ❡

d′

(x, y) =_|x1−y1|+|x2−y2|+...+|xn−yn|

≤ |xk−yk|+|xk−yk|+...+|xk−yk|=n.|xk−yk|

=n.d′′

(x, y). P♦rt❛♥t♦✱

d′

(x, y)_≤n.d′′

(x, y).

❊①❡♠♣❧♦ ✷✳✺✳ ❙❡❥❛ d✿ Rn

×Rn

→R✳ ❉❛❞♦s x = (x1, x2, ..., xn)✱ y= (y1, y2, ..., yn)∈

Rn_{✱ ✈❛♠♦s ♠♦str❛r q✉❡}

d(x, y) =

v u u t " _n X i=1

(xi−yi)2

#

é ✉♠❛ ♠étr✐❝❛ ❡♠Rn_.

❞1✮ ❙❡ x6=y t❡♠♦s q✉❡(xi−yi)2 >0✱ ♣❛r❛ ❛❧❣✉♠ i∈ {1,2, ..., n}✳ ❊♥tã♦✱

d(x, y) =

v u u t " _n X i=1

(xi−yi)2

#

(27)

❉❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ ❡①❡♠♣❧♦s ✷✺

❙❡ x=y t❡♠♦s q✉❡ (xi−yi)2 = (xi−xi)2 = 02 = 0✱ ♣❛r❛ t♦❞♦ i= 1,2, ..., n✳ ❊♥tã♦✱

d(x, y) =

v u u t " _n X i=1

(xi−xi)2

# = v u u t " _n X i=1 (0)2 #

= 0.

❞2✮ ❙❛❜❡♠♦s q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r a, b∈R✱ ✈❛❧❡ (a−b)2 = (b−a)2✳ ❊♥tã♦✱

d(x, y) =

v u u t " _n X i=1

(xi−yi)2

# = v u u t " _n X i=1

(yi−xi)2

#

=d(y, x).

❞3✮ ❱❛♠♦s ♣r♦✈❛r q✉❡

v u u t n X i=1

(xi−zi)2 ≤

v u u t n X i=1

(xi−yi)2+

v u u t n X i=1

(yi−zi)2.

❈♦♥s✐❞❡r❡ ci = xi −yi ❡ di = yi −zi✱ ❝♦♠ i = 1,2, ..., n✱ ❡♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❛

❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ ❝♦♠♦✱

v u u t n X i=1

(ci+di)2 ≤

v u u t n X i=1

(ci)2+

v u u t n X i=1

(di)2.

◆♦t❡ q✉❡

0_≤

n

X

i=1

(ci−λdi)2 = n

X

i=1

ci2−2λ n

X

i=1

cidi+λ2 n

X

i=1

di2,∀λ∈R,

✐st♦ é✱

2λ

n

X

i=1

cidi ≤ n

X

i=1

ci2+λ2 n

X

i=1

di2,∀λ∈R.

❚♦♠❛♥❞♦

λ=

Pn

i=1cidi

Pn i=1d2i

,(d₆= 0),

t❡♠♦s

" _n X

i=1

cidi

#2

≤

" _n X

i=1

ci2

#

.

" _n X

i=1

di2

# ⇒ n X i=1

cidi

≤ v u u t n X i=1

ci2.

v u u t n X i=1

di2.

P♦rt❛♥t♦✱

n

X

i=1

(ci+di)2 = n

X

i=1

ci2+ n

X

i=1

di2+ 2 n

X

i=1

cidi

≤

n

X

i=1

ci2+ n

X

i=1

di2+ 2

v u u t n X i=1

ci2

v u u t n X i=1

(28)

✷✻ ❊s♣❛ç♦s ▼étr✐❝♦s =   v u u t n X i=1

ci2+

v u u t n X i=1

di2

  2 ⇒ v u u t n X i=1

(ci+di)2 ≤

v u u t n X i=1

(ci)2+

v u u t n X i=1

(di)2

❊♥tã♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ d(x, z)_≤d(x, y) +d(y, z) é ✈á❧✐❞❛✳

P♦rt❛♥t♦✱ d é ✉♠❛ ♠étr✐❝❛ ❡♠Rn✱ ❧♦❣♦ (Rn, d)é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❊st❛ ♠étr✐❝❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ▼étr✐❝❛ ❊✉❝❧✐❞✐❛♥❛✳ ❊❧❛ ♥♦s ❢♦r♥❡❝❡ ❛ ❞✐stâ♥❝✐❛ ✉s✉❛❧ ❞❛ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✳

❊①❡♠♣❧♦ ✷✳✻✳ ❙❡❥❛ d :R_×R _→ R✱ ❞❡✜♥✐❞❛ ♣♦r d(x, y) = (x₋y)2_{✳ ❱❛♠♦s ♠♦str❛r}

q✉❡ d ♥ã♦ é ✉♠❛ ♠étr✐❝❛ ❡♠R.

❉❡ ❢❛t♦✱ d s❛t✐s❢❛③ ❛s ❞✉❛s ♣r✐♠❡✐r❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♠étr✐❝❛✱ ♠❛s ♥ã♦ ✭❞3✮✳ ❇❛st❛

♦❜s❡r✈❛r ♦ ❝♦♥tr❛✲❡①❡♠♣❧♦✿ d(1,4) = 9

d(1,3) = 4 d(3,4) = 1

❊ ❛ss✐♠ d(1,4)> d(1,3) +d(3,4).

▲♦❣♦✱ d ♥ã♦ s❛t✐s❢❛③ t♦❞❛s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♠étr✐❝❛✳ P♦rt❛♥t♦✱ d ♥ã♦ é ✉♠❛ ♠étr✐❝❛ ❡♠ R.

❊①❡♠♣❧♦ ✷✳✼✳ ❊s♣❛ç♦s ✈❡t♦r✐❛✐s ♥♦r♠❛❞♦s✳ ❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧✳ ❯♠❛ ♥♦r♠❛ ❡♠ E é ✉♠❛ ❢✉♥çã♦ r❡❛❧ ⑤⑤ ⑤⑤✿E _→R✱ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ✈❡t♦rx_∈E ♦ ♥ú♠❡r♦ r❡❛❧||x_||✱ ❝❤❛♠❛❞♦ ❛ ♥♦r♠❛ ❞❡x✱ ❞❡ ♠♦❞♦ ❛ s❡r❡♠ ❝✉♠♣r✐❞❛s ❛s ❝♦♥❞✐çõ❡s ❛❜❛✐①♦ ♣❛r❛ q✉❛✐sq✉❡r x, y _∈E ❡ λ ❡s❝❛❧❛r✿

♥1✮ ||x|| ≥0❡ ||x||= 0⇔x= 0;

♥2✮ ||λ.x||=|λ|.||x||;

♥3✮ ||x+y|| ≤ ||x||+||y||.

❚♦❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❞♦ (E,⑤⑤ ⑤⑤✮ ♣♦❞❡ s❡ t♦r♥❛r ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✳ ❇❛st❛ ❞❡✜♥✐r♠♦s ✉♠❛ ♠étr✐❝❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

d(x, y) =_||x₋y_||.

❊st❛ ♠étr✐❝❛ é ❞✐t❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ ♥♦r♠❛✳ ❱❡r✐✜q✉❡♠♦s q✉❡✱ ❞❡ ❢❛t♦✱ ||x₋y_|| é ✉♠❛ ♠étr✐❝❛✿

❞1✮ ❙❡ x6=y✱ ❡♥tã♦ x−y6= 0✱ ❧♦❣♦||x−y|| 6= 0 ❡✱ ♣♦rt❛♥t♦✱d(x, y)>0.

❙❡ x=y✱ ❡♥tã♦ d(x, y) =d(x, x) =_||x₋x_||=_||0_||=_||0.0_||=_|0_|._||0_||= 0.

❞2✮ d(x, y) = ||x−y|| = ||(−1).(y−x)|| = | −1|.||y−x|| = ||y−x|| = d(y, x)✱

(29)

❉❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ ❡①❡♠♣❧♦s ✷✼

❞3✮d(x, z) =||x−z||=||x−y+y−z||=||(x−y) + (y−z)|| ≤ ||x−y||+||y−z||=

d(x, y) +d(y, z).

❊①❡♠♣❧♦ ✷✳✽✳ ❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧✳ ❯♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♠ E é ✉♠❛ ❢✉♥çã♦ h,_i✿ E_×E _→R✱ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣❛r ♦r❞❡♥❛❞♦ ❞❡ ✈❡t♦r❡s(x, y)_∈E_×E ✉♠ ♥ú♠❡r♦ r❡❛❧ hx, y_i✱ ❝❤❛♠❛❞♦ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❡x♣♦ry✱ ❞❡ ♠♦❞♦ ❛ s❡r❡♠ ❝✉♠♣r✐❞❛s ❛s ❝♦♥❞✐çõ❡s ❛❜❛✐①♦✱ ♣❛r❛ x, y, z _∈E ❡ λ_∈R ❛r❜✐trár✐♦s✿

♣1✮ hx+z, yi=hx, yi+hz, yi;

♣2✮ hλx, yi=λ.hx, yi;

♣3✮ hx, yi=hy, xi;

♣4✮ ❙❡ x6= 0✱ ❡♥tã♦ hx, xi>0.

❉❡ss❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❞❡❝♦rr❡♠✿

hx, y+z_i=_hx, y_i+_hx, z_i;

hx, λy_i=λ._hx, y_i; h0, x_i= 0.

❆ ♣❛rt✐r ❞❡ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠❛ ♥♦r♠❛ ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ (E,_h,_i)✳ ❇❛st❛ ❞❡✜♥✐r✿

||x_||=p_hx, x_i.

◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ❛ ♥♦r♠❛ ♣r♦✈é♠ ❞❡ ✉♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦✳ ❉❡ ❢❛t♦✱

♥1✮ ❙❡ x6= 0✱ ❡♥tã♦ ||x||=

p

hx, x_i>0♣♦r ✭♣4✮✳

♥2✮ ||λ.x||=

p

hλx, λx_i=pλ2_h_{x, x}_i₌_|_λ_|_.p

hx, x_i=_|λ_|._||x_||. P❛r❛ ♣r♦✈❛r ✭♥3✮✱ ❛♥t❡s ♣r❡❝✐s❛♠♦s ❞♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

Pr♦♣♦s✐çã♦ ✷✳✷✳ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✮ ❙❡❥❛ E ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧✱ ♣❛r❛ q✉❛✐sq✉❡r ❞♦✐s ✈❡t♦r❡s x, y _∈E✱ t❡♠✲s❡✿

|hx, y_{i| ≤ ||}x_||._||y_||.

❉❡♠♦♥str❛çã♦✳ ❙❡ x = 0✱ ❡♥tã♦ _|hx, y_i| = 0 ❡ _||x_|| = 0✱ ♦ q✉❡ t♦r♥❛ ó❜✈✐❛ ❛ ❞❡s✐❣✉❛❧✲ ❞❛❞❡✳

❆❣♦r❛✱ ✈❛♠♦s s✉♣♦rx₆= 0✳ ❊♥tã♦✱ _||x_||>0 ❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ s❡❣✉✐♥t❡ ♥ú♠❡r♦ λ= hx, yi

||x_||2.

❉❡st❛ ❢♦r♠❛✱ s❡ t♦♠❛r♠♦s ♦ ❡❧❡♠❡♥t♦z =y₋λx✱ t❡r❡♠♦s hz, x_i=_hy, x_{i −} hx, yi

||x_||2hx, xi=hy, xi −

hx, y_i ||x_||2.||x||

2 _{= 0}_.

❙❡♥❞♦z =y₋λx✱ t❡♠♦s y=z+λx ❡

(30)

✷✽ ❊s♣❛ç♦s ▼étr✐❝♦s

✐st♦ é✱

||y_||2 =_||z_||2+λ2_||x_||2+ 2λ_hx, z_i=_||z_||2+λ2_||x_||2 ✉♠❛ ✈❡③ q✉❡ hx, z_i=_hz, x_i= 0✳

❉❛í✱

||y_||2 ₌_||_z_||2₊_λ2_||_x_||2 _{⇒ ||}_y_||2 _≥_λ2_||_x_||2_.

▼❛s

λ2_||x_||2 =

hx, y_i ||x_||2

2

._||x_||2 =

hx, y_i ||x_||

2

.

❊♥tã♦✱

||y_||2 _≥

hx, y_i ||x_||

2

,

✐st♦ é✱

hx, y_i2 _{≤ ||}x_||2_||y_||2 ❡ ❡①tr❛✐♥❞♦ ❛ r❛✐③ q✉❛❞r❛❞❛ ❞♦s ❞♦✐s ♠❡♠❜r♦s✱ t❡♠♦s

|hx, y_{i| ≤ ||}x_||._||y_||.

❱♦❧t❛♥❞♦ ❛ ♣r♦✈❛ ❞❡ ✭♥3)✿

||x+y_||2 =_hx+y, x+y_i =_||x_||2+_||y_||2 + 2_hx, y_i ≤ ||x_||2+_||y_||2+ 2_|hx, y_i|

(31)

✸ ❊s♣❛ç♦s ▼étr✐❝♦s ❈♦♠♣❧❡t♦s

✸✳✶ ❙❡q✉ê♥❝✐❛s ❞❡ ❈❛✉❝❤②

❉❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ (xn)n∈N ♥✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ M✱ ♠♦str❛r s✉❛ ❝♦♥✈❡r❣ê♥❝✐❛

❝♦♥s✐st❡ ❡♠ ❡①✐❜✐r x = limxn✳ ◆♦ ❡♥t❛♥t♦✱ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❛♣❡♥❛s ❡♠ s❛❜❡r s❡

❡❧❡ ❡①✐st❡✱ ♥ã♦ s❡♥❞♦ ♥❡❝❡sssár✐♦ ❝♦♥❤❡❝❡r t❛❧ ❧✐♠✐t❡✳ P❛r❛ ✐ss♦✱ t❡♠♦s ❛❧❣✉♥s t❡st❡s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛✳ ❖ ♠❛✐s ❝♦♥❤❡❝✐❞♦ é ♦ ❝r✐tér✐♦ ❞❡ ❈❛✉❝❤②✱ s❡❣✉♥❞♦ ♦ q✉❛❧ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s xn é ❝♦♥✈❡r❣❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ limm,n→∞|xm−xn|= 0.

◆❡st❡ ❝❛♣ít✉❧♦✱ s❡rã♦ ❡st✉❞❛❞♦s ♦s ❡s♣❛ç♦s ♠étr✐❝♦sM ♦♥❞❡ ♦ ❝r✐tér✐♦ ❞❡ ❈❛✉❝❤② s❡ ❛♣❧✐❝❛✱ ♦✉ s❡❥❛✱ ✉♠❛ s❡q✉ê♥❝✐❛(xn)n∈N❡♠M ❝♦♥✈❡r❣❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱limm,n→∞d(xm, xn) =

0.

❉❡✜♥✐çã♦ ✸✳✶✳ ❉✐③✲s❡ q✉❡✱ ✉♠❛ s❡q✉ê♥❝✐❛ (xn)✱ ♥✉♠ ❡s♣❛ç♦ ♠étr✐❝♦M✱ é ❞❡ ❈❛✉❝❤②

q✉❛♥❞♦✱ ♣❛r❛ t♦❞♦ ǫ >0✱ é ♣♦ssí✈❡❧ ♦❜t❡r n0 ∈N t❛❧ q✉❡ m, n > n0 ⇒d(xm, xn)< ǫ.

Pr♦♣♦s✐çã♦ ✸✳✶✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ é ❞❡ ❈❛✉❝❤②✳

❉❡♠♦♥str❛çã♦✳ ❙❡ limxn = a ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦ M✱ ❡♥tã♦ ❞❛❞♦ ǫ > 0 ❡①✐st❡ n0 ∈ N

t❛❧ q✉❡ n > n0 ⇒d(xn, a)<

ǫ

2✳ ❙❡ t♦♠❛r♠♦s m, n > n0✱ t❡r❡♠♦s d(xm, xn)≤d(xm, a) +d(xn, a)<

ǫ 2+

ǫ 2 =ǫ. ▲♦❣♦✱ xn é ❞❡ ❈❛✉❝❤②✳

❖❜s❡r✈❛çã♦ ✸✳✶✳ ◆❡♠ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② é ❝♦♥✈❡r❣❡♥t❡✱ ❝♦♠♦ ✈❡r❡♠♦s ♥♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r✳

❊①❡♠♣❧♦ ✸✳✶✳ ❉❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ xn ❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s ❝♦♥✈❡r❣✐♥❞♦ ♣❛r❛ ✉♠

♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ✭♣♦r ❡①❡♠♣❧♦x1 = 1, x2 = 1,7, x3 = 1,73, x4 = 1,732...❝♦♠limxn =

√

3✮✱ s❡♥❞♦ ❝♦♥✈❡r❣❡♥t❡ ❡♠ R s❡❣✉❡ ❞❛ ♣r♦♣♦s✐çã♦ ✸✳✶ q✉❡ (xn) é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡

❈❛✉❝❤② ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦ Q ❞♦s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳ ▼❛s ❡✈✐❞❡♥t❡♠❡♥t❡ (xn) ♥ã♦ é

❝♦♥✈❡r❣❡♥t❡ ❡♠ Q.

Pr♦♣♦s✐çã♦ ✸✳✷✳ ❚♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② é ❧✐♠✐t❛❞❛✳

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✸✵ ❊s♣❛ç♦s ▼étr✐❝♦s ❈♦♠♣❧❡t♦s

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛(xn)✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦M✳ ❉❛❞♦ǫ= 1✱

❡①✐st❡ n0 ∈N t❛❧ q✉❡ m, n > n0 ⇒d(xm, xn)<1.

▲♦❣♦ ♦ ❝♦♥❥✉♥t♦ {xn0+1, xn0+2, ...} é ❧✐♠✐t❛❞♦ ❡ t❡♠ ❞✐❛♠êtr♦ ♠❡♥♦r q✉❡ ♦✉ ✐❣✉❛❧ ❛

✶✳ ❙❡❣✉❡ q✉❡

{x1, x2, ..., xn, ...}={x1, ..., xn} ∪ {xn0+1, xn0+2, ...}

é ❧✐♠✐t❛❞♦✳

❖❜s❡r✈❛çã♦ ✸✳✷✳ ◆❡♠ t♦❞❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ é ❞❡ ❈❛✉❝❤②✱ ❝♦♠♦ ✈❡r❡♠♦s ♥♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r✳

❊①❡♠♣❧♦ ✸✳✷✳ ❖ ❡①❡♠♣❧♦ ♠❛✐s s✐♠♣❧❡s ♣❛r❛ ♠♦str❛r q✉❡ ❛ r❡❝í♣r♦❝❛ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r é ❢❛❧s❛✱ é ♦ s❡❣✉✐♥t❡✿ ❡♠❜♦r❛ ❧✐♠✐t❛❞❛✱ ❛ s❡q✉ê♥❝✐❛ ❝♦♠ t❡r♠♦s (1,0,1,0, ...) ♥ã♦ é ❞❡ ❈❛✉❝❤②✱ ♣♦✐s d(xn, xn+1) = 1✱ ♣❛r❛ t♦❞♦ n✳

Pr♦♣♦s✐çã♦ ✸✳✸✳ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② q✉❡ ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡ é ❝♦♥✈❡r❣❡♥t❡ ❡ t❡♠ ♦ ♠❡s♠♦ ❧✐♠✐t❡ q✉❡ ❛ s✉❜s❡q✉ê♥❝✐❛✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ♥♦ ❡s♣❛ç♦ ♠étr✐❝♦ M ❡ (xnk)

✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ♦ ♣♦♥t♦ a_∈M.

❆✜r♠❛♠♦s q✉❡ ❧✐♠ xn = a✱ ♣♦✐s ❞❛❞♦ ǫ > 0✱ ❡①✐st❡ p ∈ N t❛❧ q✉❡ nk > p ⇒

d(xnk, a)<

ǫ

2✳ ❊①✐st❡ t❛♠❜é♠q ∈N t❛❧ q✉❡ m, n > q ⇒d(xm, xn)<

ǫ

2✳

❚♦♠❡♠♦s n0 =max{p, q}✳ P❛r❛ t♦❞♦ n > n0 ❡①✐st❡ nk > n0 ❡ ❡♥tã♦✱

d(xn, a)≤d(xn, xnk) +d(xnk, a)<

ǫ 2+

ǫ 2 =ǫ. ▲♦❣♦✱ limxn =a.

❊①❡♠♣❧♦ ✸✳✸✳ ❙❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♣♦ss✉✐ ❞✉❛s s✉❜s❡q✉ê♥❝✐❛s q✉❡ ❝♦♥✈❡r❣❡♠ ♣❛r❛ ❧✐♠✐t❡s ❞✐st✐♥t♦s✱ ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ ♥ã♦ é ❞❡ ❈❛✉❝❤②✳

❊♠ ♣❛rt✐❝✉❧❛r✱ ✉♠❛ s❡q✉ê♥❝✐❛ q✉❡ ❛ss✉♠❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈❛❧♦r❡s ❞✐st✐♥t♦s só ♣♦❞❡ s❡r ❞❡ ❈❛✉❝❤② q✉❛♥❞♦✱ ❛ ♣❛rt✐r ❞❡ ✉♠ ❝❡rt♦ í♥❞✐❝❡✱ ❡❧❛ é ❝♦♥st❛♥t❡✳

✸✳✷ ❊s♣❛ç♦s ❈♦♠♣❧❡t♦s

❉❡✜♥✐çã♦ ✸✳✷✳ ❉✐③✲s❡ q✉❡✱ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ M é ❝♦♠♣❧❡t♦ q✉❛♥❞♦ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ M é ❝♦♥✈❡r❣❡♥t❡✳

❖ ❡①❡♠♣❧♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❡ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ é ❛ r❡t❛ r❡❛❧✳ ❆ ♣r♦♣♦s✐çã♦ q✉❡ s❡ s❡❣✉❡ é ❞❡✈✐❞❛ ❛ ❈❛✉❝❤②✳

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❊s♣❛ç♦s ❈♦♠♣❧❡t♦s ✸✶

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ (xn) ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❞❡ ♥ú♠❡r♦s r❡❛✐s✳ P❛r❛ ❝❛❞❛ n✱

♣♦♥❤❛♠♦s Xn ={xn, xn+1, ...} ❡ an= ✐♥❢ Xn✳ ❈♦♠♦xn é ❧✐♠✐t❛❞❛ ❡X1 ⊃X2 ⊃X3 ⊃

...✱ ♦❜t❡♠♦s ❛ss✐♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❝r❡s❝❡♥t❡ ❧✐♠✐t❛❞❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s a1 ≤ a2 ≤...≤

an ≤ ...✳ ❙❡❥❛ a = ❧✐♠ an✳ ❆✜r♠❛♠♦s q✉❡ a = ❧✐♠ xn✳ P❡❧❛ ♣r♦♣♦s✐çã♦ ✸✳✸✱ ❜❛st❛

♠♦str❛r q✉❡ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ (xnk)❝♦♥✈❡r❣✐♥❞♦ ♣❛r❛ a✳ P❛r❛ ✐ss♦✱ é s✉✜❝✐❡♥t❡

♣r♦✈❛r q✉❡ t♦❞♦ ✐♥t❡r✈❛❧♦ (a₋ǫ, a+ǫ)✱ǫ >0✱ ❝♦♥té♠ ♣♦♥t♦sxn ❝♦♠n s✉✜❝✐❡♥t❡♠❡♥t❡

❣r❛♥❞❡✳ ❖r❛✱ ❞❛❞♦ q✉❛❧q✉❡r n1✱ ❡①✐st❡m > n1 ❝♦♠a−ǫ < am < a+ǫ✳ ❙❡♥❞♦am =✐♥❢

Xm✱am < a+ǫ✐♠♣❧✐❝❛ q✉❡ ❡①✐st❡n > m ✭❡✱ ♣♦rt❛♥t♦✱n > n1✮ t❛❧ q✉❡am ≤xn < a+ǫ✱

✐st♦ é✱ xn∈(a−ǫ, a+ǫ)✱ ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r✳

Pr♦♣♦s✐çã♦ ✸✳✺✳ ❙❡❥❛♠(M1, d1),· · · ,(Mk, dk)❡s♣❛ç♦s ♠étr✐❝♦s✳ ❖ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦

M =M1× · · · ×Mk✱ ♠✉♥✐❞♦ ❞❛ ♠étr✐❝❛

d(x, y) =

v u u t

" _k X

i=1

di(xi, yi)2

#

,

♦♥❞❡ x = (x1,· · · , xn)✱ y = (y1,· · · , yn) ∈ M✱ é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ s❡✱ ❡

s♦♠❡♥t❡ s❡✱ ❝❛❞❛ ✉♠ ❞♦s ❢❛t♦r❡s M1,· · ·, Mk é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳

❉❡♠♦♥str❛çã♦✳ ❙❡ ❝❛❞❛ Mi é ❝♦♠♣❧❡t♦✱ ❞❛❞❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② (xn) ❡♠ M✱

❝❛❞❛ ✉♠❛ ❞❛s s❡q✉ê♥❝✐❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s(xni)n∈N é ❞❡ ❈❛✉❝❤② ❡♠Mi ❡✱ ♣♦rt❛♥t♦✱ ❝♦♥✲

✈❡r❣❡ ❡♠Mi✳ ❙❡❣✉❡✲s❡ q✉❡ (xn)❝♦♥✈❡r❣❡ ❡♠M ❡✱ ♣♦rt❛♥t♦✱M é ❝♦♠♣❧❡t♦✳ ❘❡❝✐♣r♦❝❛✲

♠❡♥t❡✱ s❡ ✉♠ ❞♦s ❢❛t♦r❡s ✭❞✐❣❛♠♦s✱ M1 ♣❛r❛ s✐♠♣❧✐✜❝❛r ❛ ❡s❝r✐t❛✮ ♥ã♦ ❢♦ss❡ ❝♦♠♣❧❡t♦✱

❡①✐st✐r✐❛ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② (yn) ♥ã♦ ❝♦♥✈❡r❣❡♥t❡ ❡♠ M1✳ ❋✐①❡♠♦s ❛r❜✐tr❛r✐❛✲

♠❡♥t❡ ♣♦♥t♦s a2 ∈ M2, ..., ak ∈ Mk✳ ❆ s❡q✉ê♥❝✐❛ ❞❡ ♣♦♥t♦s xn = (yn, a2, ..., ak) ∈ M

s❡r✐❛ ❞❡ ❈❛✉❝❤②✱ ♣♦✐s d(xm, xn) = d1(ym, yn)✱ ❡ ♥ã♦ ❝♦♥✈✐r❣✐r✐❛ ❡♠ M✳ ▲♦❣♦✱ M ♥ã♦

s❡r✐❛ ❝♦♠♣❧❡t♦✳

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(35)

✹ ▼ét♦❞♦ ❞❛s ❛♣r♦①✐♠❛çõ❡s

s✉❝❡ss✐✈❛s

❙✉♣♦♥❤❛♠♦s q✉❡ s❡ ❞❡s❡❥❛ r❡s♦❧✈❡r ✉♠❛ ❡q✉❛çã♦ ❞♦ t✐♣♦f(x) =b✱ ♦♥❞❡ f é ❝♦♥tí✲ ♥✉❛✳

❖ ♠ét♦❞♦ ❞❛s ❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s r❡❛❧✐③❛✲s❡ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳ ■♥tr♦❞✉③✐♠♦s ✉♠❛ ♥♦✈❛ ❢✉♥çã♦ ϕ(x) =f(x) +x₋b✱ ❞❡st❛ ♠❛♥❡✐r❛ ❛ ❡q✉❛çã♦ ♦r✐❣✐♥❛❧ é ❡q✉✐✈❛❧❡♥t❡ ❛ ϕ(x) = x✳ P❛r❛ ♦❜t❡r ✉♠❛ s♦❧✉çã♦ ❞❡st❛ ❡q✉❛çã♦✱ t♦♠❛✲s❡ ✉♠ ✈❛❧♦r ❛r❜✐trár✐♦ x0

❡ ♣õ❡✲s❡✱ s✉❝❡ss✐✈❛♠❡♥t❡✱ x1 = ϕ(x0)✱ x2 = ϕ(x1)...✳ ❙❡ ❛ s❡q✉ê♥❝✐❛ (xn) ❝♦♥✈❡r❣✐r✱

❡♥tã♦ x= limxn s❡rá ✉♠❛ s♦❧✉çã♦ ❞❡ ϕ(x) = x✱ ♣♦✐s ϕ(x) = ϕ(limxn) = limϕ(xn) =

limxn+1 = limxn = x✳ ❊♠ ❝♦♥s❡q✉ê♥❝✐❛✱ x = limxn s❡rá ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦

f(x) =b.

❆ ❞✐s❝✉ssã♦ ❛❝✐♠❛ ♥ã♦ ❛♣r❡s❡♥t❛ ✉♠❛ ❢♦r♠❛❧✐③❛çã♦ ❛❞❡q✉❛❞❛✱ ✈✐st♦ q✉❡ ♥ã♦ ❡①♣❧✐✲ ❝✐t❛♠♦s ♦ ❞♦♠í♥✐♦ ♥❡♠ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❡ f✳ P❛r❛ s✉❜st✐t✉✐r ❛ ❡q✉❛çã♦ f(x) = b✱ ♣♦r ϕ(x) =x✱ é ♥❡❝❡ssár✐♦ s♦♠❛r ❡ s✉❜tr❛✐r ❡❧❡♠❡♥t♦s ♥❡ss❡s ❝♦♥❥✉♥t♦s ❡ t❛♠❜é♠ q✉❡ x ❡ f(x)♣❡rt❡♥ç❛♠ ❛♦ ♠❡s♠♦ ❡s♣❛ç♦✳

❱❛♠♦s ❛❣♦r❛ r❡❛❧✐③❛r ✉♠ tr❛t❛♠❡♥t♦ s✐st❡♠át✐❝♦ ❜❛s❡❛❞♦ ♥♦ ✧❚❡♦r❡♠❛ ❞♦ ♣♦♥t♦ ✜①♦ ❞❛s ❝♦♥tr❛çõ❡s✧ ❞❡✈✐❞♦ ❛ ❙t❡❢❛♥ ❇❛♥❛❝❤✳

✹✳✶ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤

❉❡✜♥✐çã♦ ✹✳✶✳ ❙❡❥❛♠(M, d1)❡ (N, d2)❡s♣❛ç♦s ♠étr✐❝♦s✳ ❯♠❛ ❛♣❧✐❝❛çã♦f :M →N✱

❝❤❛♠❛✲s❡ ✉♠❛ ❝♦♥tr❛çã♦ q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧ k✱ ❝♦♠ 0_≤k < 1 t❛❧ q✉❡ d(f(x), f(y))_≤k.d(x, y),

q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ x, y _∈M.

❉❡✜♥✐çã♦ ✹✳✷✳ ❉❛❞❛ ✉♠❛ ❛♣❧✐❝❛çã♦ f : M _→ M✱ ❞❡ M ❡♠ s✐ ♠❡s♠♦✱ ✉♠ ♣♦♥t♦ x_∈M ❝❤❛♠❛✲s❡ ♣♦♥t♦ ✜①♦ ❞❡ f q✉❛♥❞♦ f(x) =x.

❊①❡♠♣❧♦ ✹✳✶✳ ◆❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ f(x) = x✱ t♦❞♦ ♣♦♥t♦ x_∈M é ♣♦♥t♦ ✜①♦✳ ◆♦ ❡s♣❛ç♦Rn✱ ✵ é ♦ ú♥✐❝♦ ♣♦♥t♦ ✜①♦ ❞❛ ❛♣❧✐❝❛çã♦ f(x) = ₋x.

(36)

✸✹ ▼ét♦❞♦ ❞❛s ❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s

❆ ❛♣❧✐❝❛çã♦ f : R _→ R ❞❡✜♥✐❞❛ ♣♦r f(x) = x2_{✱ t❡♠ ❞♦✐s ♣♦♥t♦s ✜①♦s ✵ ❡ ✶✱ ❝♦♠}

❡❢❡✐t♦ x=x2 _⇒_x2₋_x_{= 0}_⇒_x₍_x₋_{1) = 0}_⇒_x_{= 0} _♦✉_x_{= 1}_.

❙❡ a₆= 0✱ ❛ ❛♣❧✐❝❛çã♦ x_→x+a✱ ❞❡ Rn ❡♠ s✐ ♠❡s♠♦ ♥ã♦ t❡♠ ♣♦♥t♦ ✜①♦✳

❖❜s❡r✈❛çã♦ ✹✳✶✳ ●❡♦♠❡tr✐❝❛♠❡♥t❡✱ q✉❛♥❞♦ tr❛❜❛❧❤❛♠♦s ❝♦♠ ✉♠❛ ❢✉♥çã♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✱ ♦s ♣♦♥t♦s ✜①♦s ❞❛ ❛♣❧✐❝❛çã♦ sã♦ ❛s ❛❜❝✐ss❛s ❞♦s ♣♦♥t♦s ❞♦ ♣❧❛♥♦ ♦♥❞❡ ♦ ❣rá✜❝♦ ❞❡ f ✐♥t❡rs❡❝t❛r ❛ ❞✐❛❣♦♥❛❧ y=x✳

❉❛❞❛ ✉♠❛ ❛♣❧✐❝❛çã♦ f :M _→ M✱ fn₍_x₎ ❞❡♥♦t❛rá ❛ ♥✲és✐♠❛ ✐t❡r❛❞❛ ❞❡ f✱ ❡s❝r❡✈❡✲

r❡♠♦s f2₍_x_{) =} _f₍_f₍_x₎₎_✱ _f3₍_x_{) =} _f₍_f2₍_x₎₎_{, ..., f}n₍_x_{) =}_f₍_fn−1₍_x₎₎_✳

❚❡♦r❡♠❛ ✹✳✶✳ ✭❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✮ ❙❡❥❛ (M, d) ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ ❡ f :M _→M ✉♠❛ ❝♦♥tr❛çã♦✳ ❊♥tã♦✿

✐✮ ❊①✐st❡ ✉♠✱ ❡ s♦♠❡♥t❡ ✉♠✱ x_∈M✱ t❛❧ q✉❡ f(x) =x.

✐✐✮ ◗✉❛❧q✉❡r q✉❡ s❡❥❛ x1 ∈M✱ ❛ s❡q✉ê♥❝✐❛ (xn)n∈N✱ ♦♥❞❡ xn+1 =fn(x1)✱ ❝♦♥✈❡r❣❡

♣❛r❛ x.

✐✐✐✮ P❛r❛ t♦❞♦ n✱ t❡♠♦s q✉❡ d(xn, x) ≤ kn−1.

d(x1, x2)

(1₋k) ✱ ♦♥❞❡ x1, x2 ∈ M, k é ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ❝♦♥tr❛çã♦ ❞❡ f ❡ (xn) é ❛ s❡q✉ê♥❝✐❛ ❞❡✜♥✐❞❛ ❡♠ ✭✐✐✮✳

❉❡♠♦♥str❛çã♦✳ Pr♦✈❛r❡♠♦s ♣r✐♠❡✐r♦ ♦ ✐t❡♠ ✭✐✐✮✱ ♦✉ s❡❥❛✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ t❛❧ ♣♦♥t♦✳ ❙❡❥❛ x1 ∈M q✉❛❧q✉❡r ❡ xn+1 =fn(x)✱ ♦♥❞❡ n = 1,2, ...✳ ❱❛♠♦s ❞❡♠♦♥str❛r q✉❡(xn)n∈N é

✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳ P❛r❛ n >1✱ t❡♠♦s

d(xn, xn+1) = d(f(xn−1), f(xn))≤k.d(xn−1, xn).

P♦r ✐♥❞✉çã♦ s♦❜r❡ ♥✱ ✈❡♠ q✉❡

d(xn, xn+1)≤kn−1.d(x1, x2),

♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♥✳ ❊♥tã♦✱ ♣❛r❛ 1_≤n < m✱ t❡♠♦s d(xn, xm)≤d(xn, xn+1) +d(xn+1, xm)

≤d(xn, xn+1) +d(xn+1, xn+2) +d(xn+2, xm)

.

≤d(xn, xn+1) +d(xn+1, xn+2) +...+d(xm−1, xm).

P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❝♦♥tr❛çã♦✱ t❡♠♦s

d(xn, xm)≤kn−1.d(x1, x2)+...+km−2.d(x1, x2) =kn−1.d(x1, x2)(1+k+...+km−n−1)≤kn−1.

d(x1, x2)

(37)

❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ✸✺

❈♦♠♦ kn

→ 0✱ q✉❛♥❞♦ t♦♠❛♠♦s n s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ s❡❣✉❡ q✉❡ (xn) é ✉♠❛

s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤②✳ ❙❡♥❞♦M ❝♦♠♣❧❡t♦✱ ❡①✐st❡x_∈M t❛❧ q✉❡xn→x.❱❛♠♦s ♠♦str❛r

q✉❡ f(x) = x✳ P❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n✱ t❡♠♦s

d(f(x), xn+1) = d(f(x), f(xn))≤k.d(x, xn)

❡ ❝♦♠♦ d(x, xn)→0✱ s❡❣✉❡ q✉❡ xn→f(x)✳ P❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡✱ t❡♠♦s f(x) =x.

❯♥✐❝✐❞❛❞❡✿ ❙❡❥❛♠ x, y _∈ M✱ t❛❧ q✉❡ x ₆= y✱ ❝♦♠ f(x) = x ❡ f(y) = y✳ ❊♥tã♦ 0 < d(x, y) = d(f(x), f(y)) _≤ k.d(x, y) ❡✱ ♣♦rt❛♥t♦✱ k _≥ 1 ❝♦♥tr❛❞✐③❡♥❞♦ ❛ ❤✐♣ót❡s❡✱ ✐st♦ t❡r♠✐♥❛ ❛ ♣r♦✈❛ ❞❡ ✭✐✮✳

◗✉❛♥t♦ ❛ ✭✐✐✮✱ ❞❛ ❞❡♠♦♥str❛çã♦ ❞❡ ❡①✐stê♥❝✐❛✱ r❡s✉❧t❛ q✉❡ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❛ ❢♦r♠❛ fn₍_x

1), x1 ∈M✱ ❝♦♥✈❡r❣❡ ❛ ✉♠ ♣♦♥t♦ ✜①♦x✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❧✐♠✐t❡✳

P❛r❛ ♣r♦✈❛r ❛ ❛✜r♠❛çã♦ ✭✐✐✐✮ ♦❜s❡r✈❛♠♦s q✉❡ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛ r❡s✉❧t❛✱ ♣❛r❛ 1_≤n < m q✉❡

d(xn, x)≤d(xn, xm) +d(xm, x)≤kn−1.

d(x1, x2)

1₋k +d(xm, x). ❈♦♠♦d(xm, x)→0s❡❣✉❡ ❛ ❛✜r♠❛çã♦ ✭✐✐✐✮✳

❈♦r♦❧ár✐♦ ✹✳✶✳ ❙❡❥❛ f : M _→ M t❛❧ q✉❡ ♣❛r❛ ❛❧❣✉♠ m ❛ ✐t❡r❛❞❛ fm₍_x₎ _{é ✉♠❛}

❝♦♥tr❛çã♦✳ ❊♥tã♦f t❡♠ ✉♠✱ ❡ s♦♠❡♥t❡ ✉♠✱ ♣♦♥t♦ ✜①♦ ❡ ♣❛r❛ t♦❞♦x1 ∈M✱ ❛ s❡q✉ê♥❝✐❛

fn₍_x

1) ❝♦♥✈❡r❣❡ ♣❛r❛ ❡st❡ ♣♦♥t♦ ✜①♦✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ x ♦ ú♥✐❝♦ ♣♦♥t♦ ✜①♦ ❞❡ fm₍_x₎_{✳ Pr♦✈❛r❡♠♦s q✉❡} _x _{é ♦ ♣♦♥t♦ ✜①♦}

❞❡ f✳ ❈♦♠♦ f(fm₍_x_{)) =} _fm₍_f₍_x₎₎_{✱ ♣❛r❛ t♦❞♦} _x

∈M✱ t❡♠♦s f(x) =f(fm(x)) =fm(f(x)).

▲♦❣♦✱ f(x) =x.

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ t♦❞♦ ♣♦♥t♦ ✜①♦ ❞❡f é ♣♦♥t♦ ✜①♦ ❞❡fm₍_x₎_✱_x_{é ♦ ú♥✐❝♦ ♣♦♥t♦}

✜①♦ ❞❡ f✳ ❉❡ ❢❛t♦

fk₍_x

1)→x,

♣❡❧♦ ❚❡♦r❡♠❛ ✹✳✶ ✐t❡♠ ✭✐✐✮✱ ❡ ♣❛r❛ t♦❞♦ r ❝♦♠ 1_≤r_≤m₋1, fkm+r₍_x

1) =fkm(fr(x1)) =x

♣❡❧❛ ♠❡s♠❛ r❛③ã♦✳

❊①❡♠♣❧♦ ✹✳✷✳ ❙❡❥❛ f :R_→R ✉♠❛ ❢✉♥çã♦ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ ▲✐♣s❝❤✐t③

|f(x)₋f(y)_{| ≤}c._|x₋y_|,

(38)

✸✻ ▼ét♦❞♦ ❞❛s ❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s

❈♦♠ ❡❢❡✐t♦✱ ❛❞♦t❛♥❞♦ ❛ ♠étr✐❝❛ ✉s✉❛❧✱ R é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❝♦♠♦

0_≤c <1 ❛ ❝♦♥❞✐çã♦ ❞❡ ▲✐♣s❝❤✐t③ é ✉♠❛ ❝♦♥tr❛çã♦ ❡♠ R✱ s❡♥❞♦ ❛ss✐♠✱ ♦ ❚❡♦r❡♠❛ ✹✳✶

❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ✜①♦ x_∈R.

❊①❡♠♣❧♦ ✹✳✸✳ ❙❡❥❛♠F _⊂Rn❢❡❝❤❛❞♦✱f :F _→F s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ ▲✐♣s❝❤✐t③ ||f(x)₋f(y)_{|| ≤}c._||x₋y_||,

♣❛r❛ q✉❛✐sq✉❡r x, y _∈ F✱ ❝♦♠ 0 _≤ c < 1✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ x _∈ F t❛❧ q✉❡ f(x) = x.

❈♦♠ ❡❢❡✐t♦ ❛❞♦t❛♥❞♦ ❛ ♠étr✐❝❛ ♣r♦✈❡♥✐❡♥t❡ ❞❛ ♥♦r♠❛✱Rn_{é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠✲}

♣❧❡t♦✱ ❧♦❣♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦ F _⊂ Rn _{s❡rá ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❝♦♠♦}

0_≤c <1❛ ❝♦♥❞✐çã♦ ❞❡ ▲✐♣s❝❤✐t③ é ✉♠❛ ❝♦♥tr❛çã♦ ❡♠ F✱ s❡♥❞♦ ❛ss✐♠✱ ♦ ❚❡♦r❡♠❛ ✹✳✶ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ✜①♦ x_∈F.

P❛r❛ ♦ ♣ró①✐♠♦ ❡①❡♠♣❧♦ ♣r❡❝✐s❛r❡♠♦s ❞♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✱ q✉❡ s❡rá ❡♥✉♥✲ ❝✐❛❞♦ ❛ s❡❣✉✐r✱ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ♠❡s♠♦ s❡rá ♦♠✐t✐❞❛✳

❚❡♦r❡♠❛ ✹✳✷✳ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♥♦ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦ [a, b] ❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ♥♦ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ (a, b)✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ c❡♠ (a, b) t❛❧ q✉❡✿

f′

(c) = f(b)−f(a) b₋a ♦✉✱ ❞❡ ♠❛♥❡✐r❛ ❡q✉✐✈❛❧❡♥t❡✱

f(b)₋f(a) =f′₍_c₎₍_b

−a).

❊①❡♠♣❧♦ ✹✳✹✳ ❙❡❥❛f ✉♠❛ ❢✉♥çã♦ r❡❛❧ ❞❡ ✈❛r✐á✈❡❧ r❡❛❧ q✉❡ ♣♦ss✉✐ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s x _∈ R ✉♠❛ ❞❡r✐✈❛❞❛ f′₍_x₎ _{s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦}

|f′₍_x₎

| ≤ k < 1✱ ♦♥❞❡ k é ❝♦♥s✲ t❛♥t❡✳ ❊♥tã♦ ♦ ❣rá✜❝♦ ❞❡f ❝♦rt❛ ❛ ❞✐❛❣♦♥❛❧y=x❡①❛t❛♠❡♥t❡ ♥✉♠ ♣♦♥t♦(x, x) = ❧✐♠ (xn, xn)✱ ♦♥❞❡ xn=fn(x0) ❡x0 ∈R é t♦♠❛❞♦ ❛r❜✐tr❛r✐❛♠❡♥t❡✳

❈♦♠ ❡❢❡✐t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✱ f(b)₋f(a) =f′₍_x₎₍_b

−a)✱ ♣❛r❛ ❛❧❣✉♠ x ❡♥tr❡ a ❡ b✱ ❡♥tã♦ _|f(b)₋f(a)_|= _|f′₍_x₎

|._|b₋a|✱ ❝♦♠♦ _|f′₍_x₎

| ≤k < 1✱ s❡❣✉❡ q✉❡ f é ✉♠❛ ❝♦♥tr❛çã♦✱ s❡♥❞♦ R ❝♦♠♣❧❡t♦✱ ♦ ❚❡♦r❡♠❛ ✹✳✶ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ú♥✐❝♦

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❘❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ✸✼

✹✳✷ ❘❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s

❊q✉❛çõ❡s ♥✉♠ér✐❝❛s

❊①❡♠♣❧♦ ✹✳✺✳ ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ x=λcos(x),

♦♥❞❡ 0< λ <1.

❱❡r✐✜q✉❡ s❡ ❛ ❡q✉❛çã♦ t❡♠ s♦❧✉çã♦ ❡ s❡ ❛ s♦❧✉çã♦ é ú♥✐❝❛✳

❈♦♠♦ f(x) = λcos(x) é ✉♠❛ ❢✉♥çã♦ ❞❡ R ❡♠ R✱ ❛❞♦t❛♥❞♦ ❛ ♠étr✐❝❛ ✉s✉❛❧✱ R é

❝♦♠♣❧❡t♦✳ ❚❡♠♦s q✉❡ ♠♦str❛r q✉❡ f é ✉♠❛ ❝♦♥tr❛çã♦ ♣❛r❛ ✉s❛r♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✳

❙❡❥❛d(f(x), f(y)) =d(λ❝♦s✭x), λ❝♦s✭y)) =λ|❝♦s(x) ✲ ❝♦s(y)_{| ≤}λ_|x₋y_|=λd(x, y)✱ ♣♦✐s|❝♦s(x)−❝♦s(y)_{| ≤ |}x₋y|✱ ♣❛r❛ q✉❛✐sq✉❡rx, y _∈R✳ ❆ss✐♠✱f é ✉♠❛ ❝♦♥tr❛çã♦ ❝♦♠ k =λ.

❆ ❞❡s✐❣✉❛❧❞❛❞❡ |cos(x)₋cos(y)_{| ≤ |}x₋y|✱ ✉s❛❞❛ ❛❝✐♠❛ ❞❡❝♦rr❡ ❞♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✳ ❉❡ ❢❛t♦✱ cos(x)₋cos(y) =f′₍_t₎₍_x

−y)✱ ♣❛r❛ ❛❧❣✉♠t ❡♥tr❡x ❡y✳ ❊♥tã♦✱ |cos(x)₋cos(y)_|=_{| −}sen(t)_|._|x₋y_{| ≤ |}x₋y|✱ ♣♦✐ssen(t)_≤1 ♣❛r❛ t♦❞♦t _∈R.

❙❡❣✉❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✱ q✉❡ ♣❛rt✐♥❞♦ ❞❡ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ x1✱ ❛s ✐t❡r❛❞❛s s✉❝❡ss✐✈❛s ❞❡ f ❝♦♥✈❡r❣❡♠ ❛♦ ♥ú♠❡r♦x✱ ♣♦♥t♦ ✜①♦ ❞❡ f✳

xn =λcos(λcos(λcos(...λcos(x1)...))).

❱❛♠♦s r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ t♦♠❛♥❞♦ λ = 1

2✳ ❆❜❛✐①♦✱ t❡♠♦s ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) = 1

2❝♦s(x) ❡ ❛ ❞✐❛❣♦♥❛❧ y=x✳

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✸✽ ▼ét♦❞♦ ❞❛s ❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s

P♦❞❡♠♦s ✈✐s✉❛❧✐③❛r q✉❡ ♦ ♣♦♥t♦ ✜①♦ ❡stá ♣ró①✐♠♦ ❞❡ 0,5✳ ❆ss✐♠✱ ❢❛r❡♠♦s ✉♠❛ t❛❜❡❧❛ ❞❡ ✈❛❧♦r❡s ❝♦♥s✐❞❡r❛♥❞♦ ❝♦♠♦ ♣♦♥t♦ ✐♥✐❝✐❛❧x1 = 0,5✳

i xi cos(xi) xi+1 = 1/2 cos(xi)

✶ ✵✱✺✵✵✵✵✵✵✵ ✵✱✽✼✼✺✽✷✺✻ ✵✱✹✸✽✼✾✶✷✽ ✷ ✵✱✹✸✽✼✾✶✷✽ ✵✱✾✵✺✷✻✺✽✹ ✵✱✹✺✷✻✸✷✾✷ ✸ ✵✱✹✺✷✻✸✷✾✷ ✵✱✽✾✾✷✾✽✼✺ ✵✱✹✹✾✻✹✾✸✽ ✹ ✵✱✹✹✾✻✹✾✸✽ ✵✱✾✵✵✺✾✾✺✻ ✵✱✹✺✵✷✾✾✼✽ ✺ ✵✱✹✺✵✷✾✾✼✽ ✵✱✾✵✵✸✶✻✻✼ ✵✱✹✺✵✶✺✽✸✸ ✻ ✵✱✹✺✵✶✺✽✸✸ ✵✱✾✵✵✸✼✽✷✷ ✵✱✹✺✵✶✽✾✶✶ ✼ ✵✱✹✺✵✶✽✾✶✶ ✵✱✾✵✵✸✻✹✽✸ ✵✱✹✺✵✶✽✷✹✶ ✽ ✵✱✹✺✵✶✽✷✹✶ ✵✱✾✵✵✸✻✼✼✹ ✵✱✹✺✵✶✽✸✽✼ ✾ ✵✱✹✺✵✶✽✸✽✼ ✵✱✾✵✵✸✻✼✶✶ ✵✱✹✺✵✶✽✸✺✺ ✶✵ ✵✱✹✺✵✶✽✸✺✺ ✵✱✾✵✵✸✻✼✷✺ ✵✱✹✺✵✶✽✸✻✷ ❚❛❜❡❧❛ ✹✳✶✿ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡ f(x) = 1

2cosx.

❆♣ós ✶✵ ✐t❡r❛❞❛s✱ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r q✉❡ ♦ ✈❛❧♦r ❞♦ ♣♦♥t♦ ✜①♦ ❝♦♠ ♣r❡❝✐sã♦ ❞❡ s❡✐s ❝❛s❛s ❞❡❝✐♠❛✐s é x= 0,450183✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❡st❛ é ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✐♥✐❝✐❛❧ x= 1

2 cos(x).

❊①❡♠♣❧♦ ✹✳✻✳ ❯s❡ ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ ♣❛r❛ ♠♦str❛r q✉❡ ❡♠ R ❛

❡q✉❛çã♦ x=e−x _{t❡♠ s♦♠❡♥t❡ ✉♠❛ s♦❧✉çã♦✳ ❉❡t❡r♠✐♥❡ ✉♠ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ ❛♣ós ✷✵}

✐t❡r❛çõ❡s✳

❖❜s❡r✈❡ ♦ ❣rá✜❝♦ ❛❜❛✐①♦✿

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❘❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ✸✾

P♦❞❡♠♦s ✈✐s✉❛❧✐③❛r q✉❡ ❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) = e−x _{❡ ❛}

❞✐❛❣♦♥❛❧ y =x ❝♦♥s✐st❡ ❡♠ ✉♠ ú♥✐❝♦ ♣♦♥t♦✳

P♦ré♠✱ ❛ ❢✉♥çã♦ f(x) = e−x _{♥ã♦ é ✉♠❛ ❝♦♥tr❛çã♦ ❡♠} _R_{✱ ♣♦r ❡①❡♠♣❧♦✱}

|f(₋2)₋ f(0)_{| ≃}6,38>_{| −}2₋0_|✳ ❏á ❛ s❡❣✉♥❞❛ ✐t❡r❛❞❛g(x) = f2₍_x_{) =} _e(−e−x₎

é ✉♠❛ ❝♦♥tr❛çã♦ ❡♠ R✳

❙❡❣✉❡ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦ q✉❡ g(x)₋g(y) = g′

(t)(x₋y)

♣❛r❛ ❛❧❣✉♠ t ❡♥tr❡ x ❡ y✱ ♦♥❞❡ _|g(t)_|=_|e(−e−t₎

| ❡ |g′

(t)_| =_|e−(t+e−t₎

| ≤ e−1 _{✭✈✐st♦ q✉❡}

t+e−t

≥1♣❛r❛ t♦❞♦t_∈R)✱ ♣♦r ✐ss♦f2 _{t❡♠ ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ❝♦♥tr❛çã♦} 1

e <1✳ ❙❡❣✉❡✱

♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤ q✉❡✱ ♣❛rt✐♥❞♦ ❞❡ q✉❛❧q✉❡r ✈❛❧♦r ✐♥✐❝✐❛❧ x1✱ ❛s

✐t❡r❛❞❛s s✉❝❡ss✐✈❛s ❞❡ f✱ ❝♦♥✈❡r❣❡♠ ♣❛r❛ ♦ ♣♦♥t♦ ✜①♦ ❞❡ f✱ q✉❡ t❛♠❜é♠ é ❛ s♦❧✉çã♦ ♣r♦❝✉r❛❞❛ ❞❛ ❡q✉❛çã♦ x=e−x_✳

❚♦♠❛♥❞♦ ❝♦♠♦ ✈❛❧♦r ✐♥✐❝✐❛❧x1 = 0,5✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛ ❞❡ ✈❛❧♦r❡s✿

i xi xi+1 =e−xi

✶ ✵✱✺✵✵✵✵✵✵✵ ✵✱✻✵✻✺✸✵✻✻ ✷ ✵✱✻✵✻✺✸✵✻✻ ✵✱✺✹✺✷✸✾✷✶ ✸ ✵✱✺✹✺✷✸✾✷✶ ✵✱✺✼✾✼✵✸✵✾ ✹ ✵✱✺✼✾✼✵✸✵✾ ✵✱✺✻✵✵✻✹✻✸ ✺ ✵✱✺✻✵✵✻✹✻✸ ✵✱✺✼✶✶✼✷✶✺ ✻ ✵✱✺✼✶✶✼✷✶✺ ✵✱✺✻✹✽✻✷✾✺ ✼ ✵✱✺✻✹✽✻✷✾✺ ✵✱✺✻✽✹✸✽✵✺ ✽ ✵✱✺✻✽✹✸✽✵✺ ✵✱✺✻✻✹✵✾✹✺ ✾ ✵✱✺✻✻✹✵✾✹✺ ✵✱✺✻✼✺✺✾✻✸ ✶✵ ✵✱✺✻✼✺✺✾✻✸ ✵✱✺✻✻✾✵✼✷✶ ✶✶ ✵✱✺✻✻✾✵✼✷✶ ✵✱✺✻✼✷✼✼✷✵ ✶✷ ✵✱✺✻✼✷✼✼✷✵ ✵✱✺✻✼✵✻✼✸✺ ✶✸ ✵✱✺✻✼✵✻✼✸✺ ✵✱✺✻✼✶✽✻✸✻ ✶✹ ✵✱✺✻✼✶✽✻✸✻ ✵✱✺✻✼✶✶✽✽✻ ✶✺ ✵✱✺✻✼✶✶✽✽✻ ✵✱✺✻✼✶✺✼✶✹ ✶✻ ✵✱✺✻✼✶✺✼✶✹ ✵✱✺✻✼✶✸✺✹✸ ✶✼ ✵✱✺✻✼✶✸✺✹✸ ✵✱✺✻✼✶✹✼✼✺ ✶✽ ✵✱✺✻✼✶✹✼✼✺ ✵✱✺✻✼✶✹✵✼✻ ✶✾ ✵✱✺✻✼✶✹✵✼✻ ✵✱✺✻✼✶✹✹✼✷ ✷✵ ✵✱✺✻✼✶✹✹✼✷ ✵✱✺✻✼✶✹✷✹✽

❚❛❜❡❧❛ ✹✳✷✿ ❆♣r♦①✐♠❛çõ❡s ❞♦ ♣♦♥t♦ ✜①♦ ❞❡ f(x) = e−x_.

(42)

✹✵ ▼ét♦❞♦ ❞❛s ❛♣r♦①✐♠❛çõ❡s s✉❝❡ss✐✈❛s

❝❛s❛s ❞❡❝✐♠❛✐s é x = 0,56714✱ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ♦r✐❣✐♥❛❧ x=e−x_.

❊①❡♠♣❧♦ ✹✳✼✳ ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ x=λsen(x) +c, ♦♥❞❡ 0< λ <1 ❡c_∈R.

❱❡r✐✜q✉❡ s❡ ❛ ❡q✉❛çã♦ t❡♠ s♦❧✉çã♦ ❡ s❡ ❛ s♦❧✉çã♦ é ú♥✐❝❛✳

❈♦♠♦ f(x) =λsen(x) +c é ✉♠❛ ❢✉♥çã♦ ❞❡ R ❡♠ R✱ ❛❞♦t❛♥❞♦ ❛ ♠étr✐❝❛ ✉s✉❛❧✱ R

é ❝♦♠♣❧❡t♦✳ ❚❡♠♦s q✉❡ ♠♦str❛r q✉❡ f é ✉♠❛ ❝♦♥tr❛çã♦ ♣❛r❛ ✉s❛r♠♦s ♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✳

❙❡❥❛ d(f(x), f(y)) = d(λsen(x) +c, λsen(y) +c) = _|λsen(x) +c₋λsen(y)₋c_|= λ_|sen(x)₋sen(y)_{| ≤}λ_|x₋y_|=λd(x, y)✱ ♣♦✐s_|sen(x)₋sen(y)_{| ≤ |}x₋y_|✱ ♣❛r❛ q✉❛✐sq✉❡r x, y _∈R✳ ❆ss✐♠ f é ✉♠❛ ❝♦♥tr❛çã♦ ❝♦♠ k=λ✳

❆ ❞❡s✐❣✉❛❧❞❛❞❡ |sen(x)₋sen(y)_{| ≤ |}x₋y_|✱ ✉s❛❞❛ ❛❝✐♠❛ ❞❡❝♦rr❡ ❞♦ ❚❡♦r❡♠❛ ❞♦ ❱❛❧♦r ▼é❞✐♦✳ ❉❡ ❢❛t♦✱ sen(x)₋sen(y) =f′₍_t₎₍_x

−y)✱ ♣❛r❛ ❛❧❣✉♠ t ❡♥tr❡x ❡ y✳ ❊♥tã♦ |sen(x)₋sen(y)_|=_|cos(t)_|._|x₋y_{| ≤ |}x₋y_|✱ ♣♦✐s cos(t)_≤1 ♣❛r❛ t♦❞♦t _∈R.

❙❡❣✉❡✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞♦ P♦♥t♦ ❋✐①♦ ❞❡ ❇❛♥❛❝❤✱ q✉❡ ♣❛rt✐♥❞♦ ❞❡ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ x1✱ ❛s ✐t❡r❛❞❛s s✉❝❡ss✐✈❛s ❞❡ f ❝♦♥✈❡r❣❡♠ ❛♦ ♥ú♠❡r♦x✱ ♣♦♥t♦ ✜①♦ ❞❡ f✳

xn=λsen(λsen(λsen(...[λsen(x1) +c]...))) +c.

❘❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ♣❛r❛ ♦ ❝❛s♦ λ = 0,75 ❡ k = 1✳ ❆❜❛✐①♦✱ t❡♠♦s ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) = 0,75 sen(x) + 1 ❡ ❛ ❞✐❛❣♦♥❛❧ y=x✳

❋✐❣✉r❛ ✹✳✸✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ f(x) = 0,75 sen(x) + 1 ❡ y=x.