Open Uma introdução às equações funcionais

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛

❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚

❯♠❛ ■♥tr♦❞✉çã♦ às ❊q✉❛çõ❡s

❋✉♥❝✐♦♥❛✐s

♣♦r

❆❧❡① P❡r❡✐r❛ ❇❡③❡rr❛

s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

❆❜r✐❧✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇

†_{❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡}

B574u Bezerra, Alex Pereira.

Uma introdução às equações funcionais / Alex Pereira Bezerra.-- João Pessoa, 2014.

49f.

Orientador: Napoleón Caro Tuesta

Dissertação (Mestrado) – UFPB/CCEN

1. Matemática. 2. Equações funcionais. 3. Equaçoes de Cauchy.

❋✉♥❝✐♦♥❛✐s

♣♦r

❆❧❡① P❡r❡✐r❛ ❇❡③❡rr❛

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ ❉r✳ ◆❛♣♦❧é♦♥ ❈❛r♦ ❚✉❡st❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ❘♦❜❡rt♦ ❈❛❧❧❡❥❛s ❇❡❞r❡❣❛❧ ✲ ❯❋P❇

Pr♦❢✳ ❉r✳ ▼✐❣✉❡❧ ❋✐❞❡♥❝✐♦ ▲♦❛②③❛ ▲♦③❛♥♦ ✲ ❯❋P❊

❆❣r❛❞❡ç♦✱ ♣r✐♠❡✐r♦ ❛ ❉❡✉s ✲ q✉❡ ♥ã♦ t❡♠ ♠❡ ❢❛❧t❛❞♦✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ◆❛♣♦❧❡ó♥ ❈❛r♦✱ ♣❡❧❛s ❤♦r❛s ❞❡❞✐❝❛❞❛s ❛♦ ♠❡✉ ❡♥✲ s✐♥❛♠❡♥t♦✱ ♣❡❧❛s ❝♦♥✈❡rs❛s✱ ❡♠ s✉❛ s❛❧❛✱ ♦✉ ♣❡❧♦ ❢❛❝❡❜♦♦❦ ✭s❡♠♣r❡ é ❝❧❛r♦✱ ❝♦♥✈❡rs❛♠♦s s♦❜r❡ ♠❛t❡♠át✐❝❛✮✱ ✈✐❛ ❡♠❛✐❧ ❡ q✉❡ ♠❡s♠♦ ❡st❛♥❞♦ ❧♦♥❣❡✱ s❡♠♣r❡ ♠❡ ❡s❝❧❛r❡❝✐❛♠✱ ✐♥❝❡♥t✐✈❛♥❞♦✲♠❡ ❛ss✐♠ ❛ ❞❛r ❝♦♥t✐♥✉✐❞❛❞❡ ❛♦s ♠❡✉s ❡st✉❞♦s✱ t♦r♥❛♥❞♦✲s❡ ♣❛r❛ ♠✐♠ ✉♠❛ r❡❢❡rê♥❝✐❛ ❝♦♠♦ ♣r♦❢❡ss♦r✱ ♠❛t❡♠át✐❝♦ ❡ s❡r ❤✉♠❛♥♦✳ ❆❣r❛❞❡ç♦✱ t❛♠❜é♠✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❞❡s♣r❡♥✲ ❞✐❞❛✱ ❛❝r❡❞✐t❛♥❞♦ s❡♠♣r❡ ♥♦ ♠❡✉ tr❛❜❛❧❤♦ ❡ ❝♦♥❤❡❝✐♠❡♥t♦✳ P❡❧❛ ❡♥♦r♠❡ ❝♦♥tr✐❜✉✐çã♦✱ s❡♠ ❛ q✉❛❧ ❡st❡ tr❛❜❛❧❤♦ ♥ã♦ t❡r✐❛ s✐❞♦ r❡❛❧✐③❛❞♦✳

❆ ♠✐♥❤❛ ❡s♣♦s❛✱ ●✐sé❧✐❛ ❞❡ ❙❛♥t❛♥❛ ▼✉♥✐③✱ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ♠❛✐s q✉❡ ❡s♣❡❝✐❛❧✱ ♣♦✐s é ❛ss✐♠ ♦ ❛♠♦r q✉❡ s❡♥t✐♠♦s ✉♠ ♣❡❧♦ ♦✉tr♦✳ ❋♦✐ ❝♦♠ ❛ s✉❛ ❧✉t❛ ❡ ❡s❢♦rç♦ q✉❡ ❝♦♥s❡❣✉✐ ❝❤❡❣❛r ♥❡st❡ ♠♦♠❡♥t♦✱ ♣♦✐s s❡♠ ✈♦❝ê s❡✐ q✉❡ t✉❞♦ q✉❡ ❝♦♥q✉✐st❡✐ ❡ ♣r❡t❡♥❞♦ ❝♦♥q✉✐st❛r ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧✱ ❛♦ t❡✉ ❧❛❞♦ ♦s s♦♥❤♦s s❡ t♦r♥❛♠ r❡❛✐s ❡ ♦ ✐♠♣♦ssí✈❡❧ s❡ t♦r♥❛ ♣♦ssí✈❡❧✳ ▼✐♥❤❛ ♠❡❧❤♦r ❛♠✐❣❛✱ ♠❡✉ ❛♠♦r✱ ❝♦♠♦ ❢♦✐ ❞✐t♦ ❡♠ t✉❛ ❞✐ss❡rt❛çã♦ ❡ ❛ss✐♠ t✉❞♦ ❝♦♠❡ç♦✉✱ a=

r

ax+ate

mo ✳

❆♦s ❝♦❧❡❣❛s ❞❡ ▼❡str❛❞♦✱ ♣r✐❝✐♣❛❧♠❡♥t❡ ❛ ❣❛❧❡r❛ ❞❛ ✈✐❛❣❡♠ ❘❡❝✐❢❡✲ ❏♦ã♦ P❡ss♦❛✱ ❉❡♠✐❧s♦♥✱ ❈❛r❡❝❛ ❡ ❆♥tô♥✐♦✳

❯♠ ❛❣r❛❞❡❝✐♠❡♥t♦ t♦❞♦ ❡s♣❡❝✐❛❧ ❛♦s ♠❡✉s P❛✐s✱ ❏♦sé P❡❞r♦ ✭■♥ ♠❡♠♦✲ r✐❛♠✮ ❡ ●❡r❝✐♥❛ P❡r❡✐r❛✱ q✉❡ s♦✉❜❡r❛♠ ♠❡ ❡❞✉❝❛r✱ ❡ ❛♣❡s❛r ❞❡ t♦❞❛s ❛s ❞✐✜❝✉❧❞❛❞❡s✱ ♠❡ ❞❡r❛♠ ♦ ♣r✐♥❝✐♣✐❛❧ q✉❡ ❢♦✐ ♦ s❡✉ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧

❆♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❞♦ Pr♦❢♠❛t ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s✳

❆♦s ♠❡✉s ✐r♠ã♦s✱ ❆❧♠✐r ❡ ❆♥❞ré❛✱ ♣♦r t♦❞♦s ♦s ♠♦♠❡♥t♦s ✈✐✈✐❞♦s ❥✉♥t♦s✳

❡①♣❧✐❝❛r ❝♦♠ ♠❛❡str✐❛✱ ♠❡ ❞✐③✐❛ ❛ ✈❡❧❤❛ ❢r❛s❡✿ ✏❊st❛ ✈❛✐ ♣❛r❛ ♦ ❧✐✈r♦✑✳ ❉❡✈♦ ♦ q✉❡ s❡✐ ❡♠ ♠❛t❡♠át✐❝❛✱ ❤♦❥❡ ❡♠ ❞✐❛✱ ❛ ✈♦❝ê ♠❡✉ ❛♠✐❣♦✳

❆♦ ❛♠✐❣♦ ❏♦sé ●✐♥❛❧❞♦✱ q✉❡ s❡♠♣r❡ ♠❡✉ ❞❡✉ ❛♣♦✐♦✱ ❡ ♠♦r❛❞✐❛✱ ❞❡s❞❡ ♦ ♣r✐♠❡✐r♦ ❞✐❛ ❡♠ ❏♦ã♦ P❡ss♦❛✱ s❡❥❛ ❡st✉❞❛♥❞♦ ❥✉♥t♦✱ ♦✉ r✐♥❞♦ ❞❡ ♥♦ss❛s ❜❡st❡✐r❛s✱ ✈♦❝ê é ✉♠❛ ❛♠✐③❛❞❡ q✉❡ ❝✉❧t✐✈❛r❡✐ s❡♠♣r❡ ❝♦♠✐❣♦✳

❆♦s ♠❡✉s ♣❛✐s✱ ❏♦sé P❡❞r♦✭✐♥ ♠❡♠♦✲ r✐❛♠✮ ❡ ●❡r❝✐♥❛ P❡r❡✐r❛✱ s❡♠ ♦s q✉❛✐s ❡st❡ tr❛❜❛❧❤♦ ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧✳

❊st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛ ✉♠ ❡st✉❞♦ s♦❜r❡ ❡q✉❛çõ❡s ❢✉♥❝✐♦♥❛✐s✱ ❝♦♥s✐❞❡r❛♥❞♦ s✉❛ r❡❧❡✈â♥❝✐❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛✱ t❡♥❞♦ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✜♥❛❧ ❛♣r❡s❡♥t❛r ✉♠❛ ♣r♦♣♦st❛ q✉❡ ❝♦♥tr✐❜✉❛ ♣❛r❛ ❛ ♠❡❧❤♦r✐❛ ❞♦ ❡♥s✐♥♦ ❞❡st❡ tó♣✐❝♦✳ ➱ ❛♣r❡s❡♥t❛❞♦ ✉♠ r❡s✉♠♦ s♦❜r❡ ❛ ❤✐stór✐❛ ❞❛s ❡q✉❛çõ❡s ❢✉♥❝✐♦♥❛✐s✳ ❊♠ s❡❣✉✐❞❛✱ ♦ ❝❛♣ít✉❧♦ ✶ é ❝♦♥st✐t✉í❞♦ ♣❡❧♦ ❡st✉❞♦ ❞❛s ❡q✉❛çõ❡s ❋✉♥❝✐♦♥❛✐s ❞❡ ❈❛✉❝❤②✱ ♦ ❝❛♣ít✉❧♦ ✷ tr❛t❛ ❞❛s ❡q✉❛çõ❡s ❞❡ ❏❡♥s❡♥✱ P❡①✐❞❡r ❡ ❞✬❆❧❡♠❜❡rt✳ ◆♦ ❝❛♣✐t✉❧♦ ✸ ♠♦str❛♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❞❛s ❡q✉❛çõ❡s ❢✉♥❝✐♦♥❛✐s✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❊q✉❛çõ❡s ❋✉♥❝✐♦♥❛✐s✱ ❈❛✉❝❤②✱ ❏❡♥s❡♥✱ P❡①✐❞❡r✱ ❞✬❆❧❡♠❜❡rt✳

❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts ❛ st✉❞② ♦♥ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥s✱ ❝♦♥s✐❞❡r✐♥❣ ✐ts r❡❧❡✈❛♥❝❡ t♦ t❤❡ t❡❛❝❤✐♥❣ ♦❢ ♠❛t❤❡♠❛t✐❝s✱ ✇✐t❤ t❤❡ ✉❧t✐♠❛t❡ ❣♦❛❧ ♦❢ ♣r❡s❡♥t✐♥❣ ❛ ♣r♦♣♦s❛❧ t♦ ❝♦♥tr✐❜✉t❡ t♦ t❤❡ ✐♠♣r♦✈❡♠❡♥t ♦❢ t❡❛❝❤✐♥❣ t❤✐s t♦♣✐❝✳ ❆ s✉♠♠❛r② ♦❢ t❤❡ ❤✐st♦r② ♦❢ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥s ✐s ♣r❡s❡♥t❡❞✳ ❚❤❡♥✱ ❈❤❛♣t❡r ✶ ❝♦♥s✐sts ♦❢ t❤❡ st✉❞② ♦❢ t❤❡ ❈❛✉❝❤② ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s✱ ❈❤❛♣t❡r ✷ ❞❡❛❧s ✇✐t❤ ❡q✉❛t✐♦♥s ❏❡♥s❡♥✱ P❡①✐❞❡r✱ ❞ ✬❆❧❡♠❜❡rt✳ ■♥ ❝❤❛♣t❡r ✸ ✇❡ s❤♦✇ s♦♠❡ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥s✳ ❑❡②✇♦r❞s✿ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s✱ ❈❛✉❝❤②✱ ❏❡♥s❡♥✱ P❡①✐❞❡r✱ ❞✬❆❧❡♠❜❡rt✳

✶ ❊q✉❛çõ❡s ❋✉♥❝✐♦♥❛✐s ❞❡ ❈❛✉❝❤② ✶ ✶✳✶ ❊q✉❛çã♦ ❆❞✐t✐✈❛ ❞❡ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❊q✉❛çã♦ ❊①♣♦♥❡♥❝✐❛❧ ❞❡ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❊q✉❛çã♦ ▲♦❣❛rít♠✐❝❛ ❞❡ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✹ ❊q✉❛çã♦ ▼✉❧t✐♣❧✐❝❛t✐✈❛ ❞❡ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✷ ❊q✉❛çõ❡s ❞❡ ❏❡♥s❡♥✱ P❡①✐❞❡r ❡ ❞✬❆❧❡♠❜❡rt ✶✷ ✷✳✶ ❋✉♥çã♦ ❝♦♥✈❡①❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❞❡ ❏❡♥s❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❞❡ ❞✬❆❧❡♠❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸✳✶ ❙♦❧✉çõ❡s ❈♦♥tí♥✉❛s ❞❛ ❊q✉❛çã♦ ❞❡ ❞✬ ❆❧❡♠❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✸✳✷ ❙♦❧✉çã♦ ●❡r❛❧ ❞❛ ❊q✉❛çã♦ ❞❡ ❉✬❆❧❡♠❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳✸ ❈❛r❛❝t❡r✐③❛çã♦ ❞❛ ❋✉♥çã♦ ❈♦✲s❡♥♦ ❛tr❛✈és ❞❡ ✉♠❛ ❊q✉❛çã♦

❋✉♥❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✹ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❞❡ P❡①✐❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽

✸ ❆♣❧✐❝❛çõ❡s ❞❛s ❊q✉❛çõ❡s ❋✉♥❝✐♦♥❛✐s ✸✷ ✸✳✶ ➪r❡❛ ❞♦ r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❉❡s✐♥t❡❣r❛çã♦ r❛❞✐♦❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✸ ❉❡✜♥✐çã♦ ❞❡ ▲♦❣❛r✐t♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✹ ❏✉r♦s ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✺ ❏✉r♦s ❈♦♠♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✻ ❙♦♠❛ ❞❡ P♦tê♥❝✐❛ ❞❡ ■♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸✳✻✳✶ ❙♦♠❛ ❞♦s Pr✐♠❡✐r♦s ♥ ♥ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✻✳✷ ❙♦♠❛ ❞♦s ◗✉❛❞r❛❞♦s ❞♦s ♥ Pr✐♠❡✐r♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s ✳ ✳ ✳ ✸✾ ✸✳✻✳✸ ❙♦♠❛ ❞❛s ❦✲és✐♠❛s P♦tê♥❝✐❛s ❞♦s ♥ Pr✐♠❡✐r♦s ◆ú♠❡r♦s ◆❛t✉r❛✐s ✹✵ ✸✳✼ Pr♦❜❧❡♠❛s ❖❧í♠♣✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

❆ ❇❛s❡ ❞❡ ❍❛♠❡❧ ✹✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✾

❖s ♠❛t❡♠át✐❝♦s ✈❡♠ tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ❡q✉❛çõ❡s ❢✉♥❝✐♦♥❛✐s ❜❡♠ ❛♥t❡s ❞❛ ❢♦r♠❛✲ ❧✐③❛çã♦ ❞❛s ♠❡s♠❛s✳ ❊①❡♠♣❧♦s ❞❡ ❡q✉❛çõ❡s ❢✉♥❝✐♦♥❛✐s ✐♥✐❝✐❛✐s ♣♦❞❡♠ s❡r r❛str❡❛❞♦s ♥♦ tr❛❜❛❧❤♦ ❞♦ ♠❛t❡♠át✐❝♦ ◆✐❝♦❧❡ ❖r❡s♠❡ ❞♦ sé❝✉❧♦ ❳■❱✳ ◆✐❝♦❧❡ ❖r❡s♠❡ ❢♦r♥❡✲ ❝❡✉ ✉♠❛ ❞❡✜♥✐çã♦ ✐♥❞✐r❡t❛ ❞❡ ❢✉♥çõ❡s ❧✐♥❡❛r❡s ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧✳ ❖r❡s♠❡ ♥❛s❝❡✉ ❡♠ ✶✸✷✸✱ ♥❛ ◆♦r♠❛♥❞✐❛ ❡ ♠♦rr❡✉ ❡♠ ✶✸✽✷✱ ❡♠ ▲✐s✐❡✉①✱ ♥❛ ❋r❛♥ç❛✳ ❊♠ ✶✸✺✷✱ ❖r❡s♠❡ ❡s❝r❡✈❡✉ ✉♠ ❣r❛♥❞❡ tr❛t❛❞♦ s♦❜r❡ ❛ ✉♥✐❢♦r♠✐❞❛❞❡ ❡ ❞❡❢♦r♠✐❞❛❞❡ ❞❡ ✐♥t❡♥s✐❞❛❞❡s✱ ✐♥t✐t✉❧❛❞♦ ❚r❛❝t❛t✉s ❞❡ ❝♦♥✜❣✉r❛t✐♦♥✐❜✉s q✉❛❧✐t❛t✉♠ ❡t ♠♦t✉✉♠✳ ◆❡st❡ ✐♠♣♦rt❛♥t❡ tr❛❜❛❧❤♦✱ ❖r❡s♠❡ ❡st❛❜❡❧❡❝✐❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ✉♠ r❡❧❛çã♦ ❢✉♥❝✐♦♥❛❧ ❡♥tr❡ ❞✉❛s ✈❛r✐á✈❡✐s✱ ❡ ❛ ✐❞❡✐❛ ✭❜❡♠ à ❢r❡♥t❡ ❞❡ ❘❡♥é ❉❡s❝❛rt❡s✮ ❞❡ q✉❡ s❡ ♣♦❞❡ ❡①♣r❡ss❛r ❡st❛ r❡❧❛çã♦ ❣❡♦♠❡tr✐❝❛♠❡♥t❡✱ ♣❡❧♦ q✉❡ ❤♦❥❡ ❝❤❛♠❛rí❛♠♦s ✉♠ ❣rá✜❝♦✳ ❆♦ ❧♦♥❣♦ ❞♦s ♣ró①✐♠♦s ❝❡♠ ❛♥♦s✱ ❛s ❡q✉❛çõ❡s ❢✉♥❝✐♦♥❛✐s ❢♦r❛♠ ✉s❛❞❛s✱ ♠❛s ♥❡♥❤✉♠❛ t❡♦r✐❛ ❣❡r❛❧ ❞❡ t❛✐s ❡q✉❛çõ❡s s✉r❣✐✉✳

❉❡♥tr❡ ❡ss❡s ♠❛t❡♠át✐❝♦s ❢♦✐ ●r❡❣ór✐♦ ❞❡ ❙❛✐♥t✲❱✐❝❡♥t ✭✶✺✽✹✲✶✻✻✼✮✱ ❝✉❥♦ tr❛❜❛✲ ❧❤♦ s♦❜r❡ ❛ ❤✐♣ér❜♦❧❡ ❢❡③ ✉s♦ ✐♠♣❧í❝✐t♦ ❞❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ f(xy) = f(x) +f(y)✱

❡ ❢♦✐ ♣✐♦♥❡✐r♦ ♥❛ t❡♦r✐❛ ❞♦ ▲♦❣❛rít♠♦✳ ❊st❡ ❘❡s✉❧t❛❞♦ ❞❡ ❙❛✐♥t✲❱✐❝❡♥t ❛♣❛r❡❝❡✉ ❡♠ ✶✻✹✼ ♥♦ s❡✉ ❣r❛♥❞❡ tr❛t❛❞♦ ✐♥t✐t✉❧❛❞♦ ❖♣✉s ●❡♦♠❡tr✐❝✉♠ q✉❛❞r❛t✉r❛❡ ❝✐r❝✉❧✐ ❡t s❡❝t✐♦♥✉♠ ❝♦♥✐✳ ❙❡ ♦ tít✉❧♦ ❞❡st❡ tr❛❜❛❧❤♦ ♣❛r❡❝❡ ❧♦♥❣♦✱ ♦ ♣ró♣r✐♦ tr❛t❛❞♦✱ ❡♠ ❝❡r❝❛ ❞❡ ✶✷✺✵ ♣á❣✐♥❛s✱ ❡r❛ ♠✉✐t♦ ♠❛✐s ❧♦♥❣♦✳

❊♠❜♦r❛ ❛ ❞❡✜♥✐çã♦ ❞❡ ❧✐♥❡❛r✐❞❛❞❡ ❞❡ ◆✐❝♦❧❡ ❖r❡s♠❡ ♣♦ss❛ s❡r ✐♥t❡r♣r❡t❛❞♦ ❝♦♠♦ ✉♠ ❞♦s ♣r✐♠❡✐r♦s ❡①❡♠♣❧♦s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧✳ ❊❧❛ ♥ã♦ r❡♣r❡s❡♥t❛ ✉♠ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ❛ t❡♦r✐❛ ❞❛s ❡q✉❛çõ❡s ❢✉♥❝✐♦♥❛✐s✳ ❖ t❡♠❛ ❞❡ ❡q✉❛çõ❡s ❢✉♥❝✐♦♥❛✐s é ♠❛✐s ♣r♦♣r✐❛♠❡♥t❡ ❞❛t❛❞♦ ❛ ♣❛rt✐r ❞♦ tr❛❜❛❧❤♦ ❞❡ ❆✉❣✉st✐♥ ▲♦✉✐s ❈❛✉❝❤②✱ ♥❛s❝✐❞♦ ❡♠ ✶✼✽✾✱ ❡♠ P❛r✐s✱ ♥❛ ❋r❛♥ç❛✳ ❯♠ ♠❛t❡♠át✐❝♦ ❜r✐❧❤❛♥t❡✱ ❈❛✉❝❤② tr❛❜❛❧❤♦✉ ❡♠ ✈ár✐❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡❧❡ é ❝♦♥❤❡❝✐❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r s❡✉ tr❛❜❛❧❤♦ s♦❜r❡ ❝á❧❝✉❧♦✱ ❡ é r❡❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✉♠ ❞♦s ❢✉♥❞❛❞♦r❡s ❞❛ ♠♦❞❡r♥❛ t❡♦r✐❛ ❞❛ ❛♥á❧✐s❡ ♠❛t❡♠át✐❝❛✳

❆ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧✱ q✉❡ é ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❛ss♦❝✐❛❞❛ ❝♦♠ ❈❛✉❝❤② é

f(x+y) =f(x) +f(y), ♣❛r❛ t♦❞♦x, y _∈R. (⋆)

❊✱ ❛❣♦r❛✱ é ❝❤❛♠❛❞❛ ❞❡ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❞❡ ❈❛✉❝❤②✳ ❖ q✉❡ s❡ q✉❡r ❞❡t❡r♠✐♥❛r sã♦ t♦❞❛s ❛s ❢✉♥çõ❡s r❡❛✐sf q✉❡ s❛t✐s❢❛③❡♠ (⋆)✳ P♦❞❡♠♦s ✐♠❡❞✐❛t❛♠❡♥t❡ ♥♦t❛r q✉❡

❛ ❡q✉❛çã♦ ❞❡ ❈❛✉❝❤② é s❛t✐s❢❡✐t❛ ♣♦r q✉❛❧q✉❡r ❢✉♥çã♦ ❞❛ ❢♦r♠❛

f(x) = ax✱

♦♥❞❡ ❛ ❝♦♥st❛♥t❡a é r❡❛❧✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ♥♦ss❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦

s✐♠♣❧❡s ♣❛r❛ ❡st❛ ❡q✉❛çã♦ é ❛♣❡♥❛s ✉♠❛ ♣❡q✉❡♥❛ ♣❛rt❡ ❞❛ ❤✐stór✐❛✳ ❉❡✈❡♠♦s t❛♠✲ ❜é♠ ♣❡r❣✉♥t❛r s❡ ❛ ❢❛♠í❧✐❛ ❞❡ ❢✉♥çõ❡s ❞❛ ❢♦r♠❛ f(x) = ax é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s

❛s s♦❧✉çõ❡s ♣❛r❛ ❛ ❡q✉❛çã♦ (⋆)✳ P❛r❡❝❡ r❛③♦á✈❡❧ q✉❡ t❛✐s ❢✉♥çõ❡s ❧✐♥❡❛r❡s s❡❥❛♠ ❛s

ú♥✐❝❛s s♦❧✉çõ❡s ♣❛r❛(⋆)✳ ◆♦ ❡♥t❛♥t♦✱ ✐st♦ ❛❝❛❜❛ ♣♦r s❡r ✈❡r❞❛❞❡ s♦♠❡♥t❡ s❡ ❛❧❣✉♠❛

r❡str✐çã♦ ❧❡✈❡ é ✐♠♣♦st❛ ❛ ❢✉♥çã♦ f✳ P♦r ❡①❡♠♣❧♦✱ ❢✉♥çõ❡s ❞❛ ❢♦r♠❛ f(x) = ax

sã♦ ❛s ú♥✐❝❛s s♦❧✉çõ❡s ♣❛r❛ (⋆) ❡♥tr❡ ❛ ❝❧❛ss❡ ❞❡ ❢✉♥çõ❡s q✉❡ sã♦ ❞❡❧✐♠✐t❛❞❛s ❡♠

❛❧❣✉♠ ✐♥t❡r✈❛❧♦ ❞❛ ❢♦r♠❛(₋c, c)✱ ❡♠ q✉❡ c >0✳ ❆❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ♣♦❞❡✲s❡ ♠♦str❛r

q✉❡ f(x) = ax ❢♦r♠❛ ❛ ú♥✐❝❛ ❝❧❛ss❡ ❞❡ s♦❧✉çõ❡s ❡♥tr❡ ❛s ❢✉♥çõ❡s r❡❛✐s ❝♦♥tí♥✉❛s✳

❍✐st♦r✐❝❛♠❡♥t❡✱ ❏❡❛♥ ❞✬❆❧❡♠❜❡rt ♣r❡❝❡❞❡ ❆✉❣✉st✐♥ ▲♦✉✐s ❈❛✉❝❤②✳ ◆♦ ❡♥t❛♥t♦✱ ♥♦ ❝♦♥t❡①t♦ ❞❡ ❡q✉❛çõ❡s ❢✉♥❝✐♦♥❛✐s✱ ♣❛r❡❝❡ ♠❛✐s ♥❛t✉r❛❧ ❝♦♥s✐❞❡r❛r s✉❛s ❝♦♥tr✐❜✉✐çõ❡s ❛♣ós ❈❛✉❝❤②✳

❏❡❛♥ ❞✬❆❧❡♠❜❡rt ❡r❛ ✉♠ ❤♦♠❡♠ ❞❡ ♠✉✐t♦s ♥♦♠❡s✳ ❖ ✜❧❤♦ ✐❧❡❣ít✐♠♦ ❞❡ ✉♠ ♦✜❝✐❛❧ ❞♦ ❡①ér❝✐t♦✱ ▲♦✉✐s✲❈❛♠✉s ❉❡st♦✉❝❤❡s✱ ❡ ✉♠❛ ❡s❝r✐t♦r❛✱ ❈❧❛✉❞✐♥❡ ●✉ér✐♥ ❞❡ ❚❡♥❝✐♥✱ ❡❧❡ ♥❛s❝❡✉ ❡♠ P❛r✐s ❡♠ ✶✼✶✼✱ ❡♥q✉❛♥t♦ s❡✉ ♣❛✐ ❡st❛✈❛ ♥♦ ❡①t❡r✐♦r✳ ❆♣ós s❡✉ ♥❛s❝✐✲ ♠❡♥t♦✱ s✉❛ ♠ã❡ ♦ ❛❜❛♥❞♦♥♦✉ ♥❛ ✐❣r❡❥❛ ❞❡ ❙❛✐♥t✲❏❡❛♥✲▲❡r♦♥❞✳ ❙❡❣✉✐♥❞♦ ❛ tr❛❞✐çã♦✱ ❢♦✐ ♥♦♠❡❛❞♦ ❏❡❛♥ ▲❡ ❘♦♥❞✱ ❡ ❝♦❧♦❝❛❞♦ ❡♠ ✉♠ ♦r❢❛♥❛t♦✳ ❆♣ós ♦ r❡t♦r♥♦ ❞❡ s❡✉ ♣❛✐✱ ❢♦✐ r❡t✐r❛❞♦ ❞♦ ♦r❢❛♥❛t♦✳ ❊♠❜♦r❛ ❉❡st♦✉❝❤❡s t❡♥❤❛ ❛♣♦✐❛❞♦ s❡✉ ✜❧❤♦ ✜♥❛♥❝❡✐r❛✲ ♠❡♥t❡✱ ❡❧❡ ♦♣t♦✉ ♣♦r ♥ã♦ r❡❝♦♥❤❡❝❡r ♣✉❜❧✐❝❛♠❡♥t❡ ♦ ✜❧❤♦✳ ❊♠ ✶✼✸✽✱ ❏❡❛♥ ▲❡ ❘♦♥❞ ❡♥tr♦✉ ♥❛ ❢❛❝✉❧❞❛❞❡ ❞❡ ❞✐r❡✐t♦✱ ♦♥❞❡ t❡♠ r❡❣✐str♦s ❝♦♠ ♦ ♥♦♠❡ ❉❛r❡♠❜❡r❣✳ ▼❛✐s t❛r❞❡✱ ❡❧❡ ♠✉❞♦✉ s❡✉ ♥♦♠❡ ♣❛r❛ ❞✬❆❧❡♠❜❡rt✳

❆ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧

f(x+y) +f(x₋y) = 2f(x)f(y), ♦♥❞❡ 0_≤y_≤x_≤ π

2,

é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❞❡ ❞✬❆❧❡♠❜❡rt✳

❊q✉❛çõ❡s ❋✉♥❝✐♦♥❛✐s ❞❡ ❈❛✉❝❤②

✶✳✶ ❊q✉❛çã♦ ❆❞✐t✐✈❛ ❞❡ ❈❛✉❝❤②

f(x+y) =f(x) +f(y) ✭✶✳✶✮

é ❝❤❛♠❛❞❛ ❊q✉❛çã♦ ❆❞✐t✐✈❛ ❞❡ ❈❛✉❝❤②✳

❯♠❛ ❢✉♥çã♦ φ : R _−→ R é ❛❞✐t✐✈❛ s❡ s❛t✐s❢❛③ ❛ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❆❞✐t✐✈❛ ❞❡ ❈❛✉❝❤②✱ ✐st♦ é✱

φ(x+y) =φ(x) +φ(y), ♣❛r❛ t♦❞♦ x, y _∈R.

❖ ❡st✉❞♦ ❞❛s ❢✉♥çõ❡s ❛❞✐t✐✈❛ r❡♠♦♥t❛ ❆❞r✐❡♥ ▼❛r✐❡ ▲❡❣❡♥❞r❡ ✭✶✼✾✶✮ ❡ ❏♦❤❛♥♥ ❈❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss ✭✶✽✽✾✮✱ q✉❡ ✜③❡r❛♠ ❛s ♣r✐♠❡✐r❛s t❡♥t❛t✐✈❛s ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✶✳✶✮✳ P♦ré♠✱ ♦ ❡st✉❞♦ s✐st❡♠át✐❝♦ ❞❛ ❡q✉❛çã♦ ✭✶✳✶✮ ❢♦✐ r❡❛❧✐③❛❞♦ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ❢r❛♥❝ês ❆✉❣✉st✐♥ ▲♦✉✐s ❈❛✉❝❤② ❡♠ s❡✉ ❧✐✈r♦ ❈♦✉rs ❞✬❆♥❛❧②s❡ ❡♠ ✶✽✷✶✳

❉❡✜♥✐çã♦ ✶✳✶ ❯♠❛ ❢✉♥çã♦ φ :R_−→R é ❞✐t❛ r❛❝✐♦♥❛❧♠❡♥t❡ ❤♦♠♦❣ê♥❡❛ s❡✱

φ(rx) =rφ(x), ✭✶✳✷✮

♣❛r❛ t♦❞♦ x_∈R ❡ t♦❞♦ r _∈Q.

Pr♦♣♦s✐çã♦ ✶✳✶ ❙❡ φ : R _−→ R é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❛❞✐t✐✈❛ ❞❡ ❈❛✉❝❤②✱ ❡♥tã♦ φ é r❛❝✐♦♥❛❧♠❡♥t❡ ❤♦♠♦❣ê♥❡❛✳ ❆❧é♠ ❞✐ss♦✱ ❡①✐st❡ c _∈ Q t❛❧ q✉❡ φ(r) = cr,

♣❛r❛ t♦❞♦ r_∈Q.

Pr♦✈❛✳ ❙❡❥❛ φ : R _−→ R ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✶✳✶✮✱ ❢❛③❡♥❞♦ x = y = 0 ❡♠

✭✶✳✶✮✱ t❡♠♦s q✉❡ φ(0) =φ(0) +φ(0) ❡ ❞❛í

φ(0) = 0. ✭✶✳✸✮

❙✉❜st✐t✉✐♥❞♦y =₋x ❡♠ ✭✶✳✶✮ ❡ ✉s❛♥❞♦ ✭✶✳✸✮✱ t❡♠♦s q✉❡φ é ✉♠❛ ❢✉♥çã♦ í♠♣❛r✱

✐st♦ é✱

φ(₋x) =₋φ(x), ✭✶✳✹✮

♣❛r❛ t♦❞♦ x _∈ R✳ ❆ss✐♠✱ ❛té ❛❣♦r❛✱ ♠♦str❛♠♦s q✉❡ ❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❛❞✐t✐✈❛ ❞❡ ❈❛✉❝❤② s❡ ❛♥✉❧❛ ♥❛ ♦r✐❣❡♠ ❡ é ✉♠❛ ❢✉♥çã♦ í♠♣❛r✳ ❱❛♠♦s ♣r♦✈❛r ❛❣♦r❛ q✉❡ ❛ s♦❧✉çã♦φ é r❛❝✐♦♥❛❧♠❡♥t❡ ❤♦♠♦❣ê♥❡❛✳ P❛r❛ ✐st♦✱ s❡❥❛x_∈R✳ ❊♥tã♦✱

φ(2x) = φ(x+x) =φ(x) +φ(x) = 2φ(x).

P♦rt❛♥t♦✱

φ(3x) =φ(2x+x) =φ(2x) +φ(x) = 2φ(x) +φ(x) = 3φ(x).

❊✱ ❡♠ ❣❡r❛❧✱ ✉s❛♥❞♦ ✐♥❞✉çã♦✱ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡

φ(nx) = n_·φ(x), ✭✶✳✺✮

♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦n✳ ❉❡ ❢❛t♦✱ ♣❛r❛ n= 2 é ✈❡r❞❛❞❡✐r❛ ♣♦r ❞❡✜♥✐çã♦✳ ❆❣♦r❛

❛ss✉♠❛ q✉❡ ✈❛❧❡ ♣r❛ ❛❧❣✉♠ n0✱

φ(x1+x2+...+xn0) =φ(x1) +φ(x2) +...+φ(xn0).

❱❡r✐✜q✉❡♠♦s s❡ ✈❛❧❡ ♣❛r❛ n0+ 1✿

φ(x1+x2 +...+xn0 +xn0+1) = φ((x1+x2+...+xn0) +xn0+1)

= φ(x1+x2+...+xn0) +φ(xn0+1)

= φ(x1) +φ(x2) +...+φ(xn0) +φ(xn0+1),

❢❛③❡♥❞♦ xi =x,∀i✱ ♦❜t❡♠♦s

φ(nx) = n_·φ(x).

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ n é ✉♠ ✐♥t❡✐r♦ ♥❡❣❛t✐✈♦✱ ❡♥tã♦✱ ₋n é ♣♦s✐t✐✈♦ ❡ ♣♦r ✭✶✳✹✮ ❡

✭✶✳✺✮✱ t❡♠♦s

φ(nx) =φ(₋(₋n)x) =₋φ(₋nx) =₋(₋n)φ(x) = nφ(x).

❊♥tã♦✱ ♣r♦✈❛♠♦s q✉❡ φ(nx) = nφ(x), ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ n ❡ ♣❛r❛ t♦❞♦ x _∈ R✳

❆❣♦r❛✱ s❡❥❛ r ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❛r❜✐trár✐♦✱ ❡♥tã♦✱

r= k

l,

♦♥❞❡ k_∈Z ❡ l_∈N∗✳ kx=l(rx)✳ ❯s❛♥❞♦ ♦ q✉❡ ❢♦✐ ♣r♦✈❛❞♦ ❛❝✐♠❛✱ ♦❜t❡♠♦s

kφ(x) = φ(kx) =φ(l(rx)) =lφ(rx),

✐st♦ é✱

φ(rx) = k

lφ(x) = rφ(x).

P♦rt❛♥t♦✱ φ é r❛❝✐♦♥❛❧♠❡♥t❡ ❤♦♠♦❣ê♥❡❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❢❛③❡♥❞♦ x = 1 ♥❛

❡q✉❛çã♦ ❛❝✐♠❛ ❡ ❞❡✜♥✐♥❞♦ c=φ(1)✱ t❡♠♦s q✉❡

φ(r) = cr,

♣❛r❛ t♦❞♦ ♥ú♠❡r♦r _∈Q✳

❚❡♦r❡♠❛ ✶✳✶ ✭❈❛✉❝❤②✮ ❙❡❥❛ φ : R _−→ R ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✳ ❙❡ φ é ❝♦♥tí♥✉❛✱

❡♥tã♦ φ é ❧✐♥❡❛r✱ ✐st♦ é✱ ❡①✐st❡ c_∈R t❛❧ q✉❡ φ(x) = cx✱ ♣❛r❛ t♦❞♦ x_∈R✳

Pr♦✈❛✳ ❙❡❥❛ φ ✉♠❛ s♦❧✉çã♦ ❝♦♥tí♥✉❛ ❞❛ ❡q✉❛çã♦ ✭✶✳✶✮✳ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x✱

❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ rn ❞❡ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s t❛❧ q✉❡ limrn = x✳ P❡❧❛ ♣r♦♣♦s✐çã♦

(1.1)✱ ❡①✐st❡ c_∈R t❛❧ q✉❡ φ(r) =cr✱ ♣❛r❛ t♦❞♦ r _∈Q✳ P♦rt❛♥t♦✱

φ(rn) =crn,

♣❛r❛ t♦❞♦n _∈N.

❆❣♦r❛✱ ✉s❛♥❞♦ ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ φ✱ t❡♠♦s

φ(x) =φ( lim

n→∞rn) = limn→∞φ(rn) = limn→∞crn =cx.

❆ ❝♦♥❞✐çã♦ ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❢♦✐ ❡♥❢r❛q✉❡❝✐❞❛ ♣♦r ❏❡❛♥ ●❛st♦♥ ❉❛r❜♦✉① ❡♠ ✶✽✼✺✱ ❝♦♠♦ ♠♦str❛ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿

Pr♦♣♦s✐çã♦ ✶✳✷ ❙❡❥❛ φ ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✶✳✶✮✳ ❙❡ φ é ❝♦♥tí♥✉❛ ❡♠ ✉♠

♣♦♥t♦✱ ❡♥tã♦✱ ❡❧❛ é ❝♦♥tí♥✉❛ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s✳

Pr♦✈❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡φ é ❝♦♥tí♥✉❛ ❡♠t✱ ♣❛r❛ ❛❧❣✉♠t _∈R✳ ▼♦str❛r❡♠♦s q✉❡φ

é ❝♦♥tí♥✉❛ ❡♠ R✳ P❛r❛ ✐st♦✱ s❡❥❛ x_∈R q✉❛❧q✉❡r✱ ❡♥tã♦✱

lim

h→0φ(x+h) = limh→0φ(t+h+x−t)

= lim

h→0φ(t+h) +φ(x−t),

φ é ❝♦♥tí♥✉❛ ❡♠ t✱ lim

h→0φ(t+h) = φ(t)✳ ▲♦❣♦✱

lim

h→0φ(x+h) = φ(t) +φ(x−t)

= φ(x).

■ss♦ ♣r♦✈❛ q✉❡ φ é ❝♦♥tí♥✉❛ ❡♠ x ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡φ é ❝♦♥tí♥✉❛ ❡♠ R✳

◆♦ ❝❛♣ít✉❧♦ ✺ ❞♦ ❧✐✈r♦ ❈♦✉rs ❞✬❆♥❛❧②s❡✱ ❆✉❣✉st✐♥ ▲♦✉✐s ❈❛✉❝❤② ✭✶✽✷✶✮ ❡st✉❞♦✉ ♦✉tr❛s três ❡q✉❛çõ❡s✱ ❛ s❛❜❡r✱

f(x+y) = f(x)_·f(y), ✭✶✳✻✮ f(xy) = f(x) +f(y), ✭✶✳✼✮ f(xy) = f(x)_·f(y). ✭✶✳✽✮

✶✳✷

❊q✉❛çã♦ ❊①♣♦♥❡♥❝✐❛❧ ❞❡ ❈❛✉❝❤②

❆ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❈❛✉❝❤② é

f(x+y) = f(x)_·f(y), x, y _∈R.

❚❡♦r❡♠❛ ✶✳✷ ❆ s♦❧✉çã♦ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❈❛✉❝❤②✱ t❡♠ ❛ s❡❣✉✐♥t❡ ❢♦r♠❛

φ(x) =eA(x),

♦♥❞❡ A:R_−→Ré ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✱ ♦✉ φ(x) = 0✳

Pr♦✈❛✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ f(x) = 0, _∀x_∈ R é ✉♠❛ s♦❧✉çã♦ ❞❡ (1.6)✳ ❆✜r♠❛♠♦s q✉❡

φ(x) ₆= 0,_∀x_∈ R✳ ❙✉♣♦♥❤❛ q✉❡ ♥ã♦✳ ❊♥tã♦✱ ❡①✐st❡ y0 t❛❧ q✉❡ φ(y0) = 0✳ ❉❡ (1.6)✱ t❡♠♦s

φ(y) =φ((y₋y0) +y0) =φ(y−y0)·φ(y0) = 0✱

♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

❚♦♠❡ x= t

2 ❡♠ (1.6)✱ t❡♠♦s q✉❡

φ(t) = φ(t 2)

2_,

♣❛r❛ t♦❞♦ t _∈ R✳ ❆ss✐♠✱ φ(x) é ❡str✐t❛♠❡♥t❡ ♣♦s✐t✐✈❛✳ ❆❣♦r❛✱ t♦♠❛♥❞♦ ❧♦❣❛r✐t♠♦

♥❛t✉r❛❧ ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡ (1.6)✱ ♦❜t❡♠♦s

Lnφ(x+y) = Lnφ(x) +Lnφ(y).

❉❡✜♥✐♥❞♦ A:R_−→R♣♦r A(x) = Lnφ(x)✱ t❡♠♦s

A(x+y) = A(x) +A(y).

❆ss✐♠✱ t❡♠♦s ❛ s♦❧✉çã♦ φ(x) =eA(x)_✳

❉❡✜♥✐çã♦ ✶✳✷ ❯♠❛ ❢✉♥çã♦φ:R_−→Ré ❝❤❛♠❛❞❛ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ r❡❛❧ s❡ s❛t✐s❢❛③ f(x+y) =f(x)_·f(y)✱ ♣❛r❛ t♦❞♦ x, y _∈R✳

❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧

f(x+y+nxy) =f(x)_·f(y), ✭✶✳✾✮

✈❛❧❡ ♣❛r❛ t♦❞♦s ♦s r❡❛✐s x > ₋1

n ❡ y > −

1

n✳ ◗✉❛♥❞♦ n −→0✱ ❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧

✭✶✳✾✮ s❡ r❡❞✉③ ❛ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❈❛✉❝❤②✳ ❊st❛ ❡q✉❛çã♦ ❢♦✐ ❡st✉❞❛❞❛ ♣♦r ❚❤✐❡❧♠❛♥ ✭✶✾✹✾✮✳

❚❡♦r❡♠❛ ✶✳✸ ❚♦❞❛ s♦❧✉çã♦ φ ❞❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ✭✶✳✾✮ é ❞❛ ❢♦r♠❛

φ(x) = 0 ♦✉ φ(x) =eA(ln(1+nx)),

♦♥❞❡ A:R_−→R é ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✳

Pr♦✈❛✳ ❱❛♠♦s ❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ✭✶✳✾✮ ♥❛ s❡❣✉✐♥t❡ ❢♦r♠❛

φ

(1 +nx)_·(1 +ny)₋1

n

=φ(x)_·φ(y). ✭✶✳✶✵✮

❉❡✜♥❛1 +nx =eu _{❡1 +ny=e}v _{t❛❧ q✉❡u=ln(1 +nx)❡v =ln(1 +ny)✳ ❆❣♦r❛✱} r❡❡s❝r❡✈❡♥❞♦ ✭✶✳✶✵✮✱ ♦❜t❡♠♦s

φ

eu+v

−1 n =φ eu −1 n ·φ ev −1 n

, ♣❛r❛ t♦❞♦u, v _∈R. ✭✶✳✶✶✮

❉❡✜♥✐♥❞♦

ψ(u) =φ

eu

−1

n

✭✶✳✶✷✮

❡♠ ✭✶✳✶✶✮✱ t❡♠♦s

ψ(u+v) = ψ(u)_·ψ(v), ✭✶✳✶✸✮

♣❛r❛ t♦❞♦u, v _∈R✳ ❆ss✐♠✱ ♣❡❧♦ t❡♦r❡♠❛ (1.2)✱ t❡♠♦s

ψ(x) =eA(x) _{♦✉ ψ(x) = 0,}

∀x_∈R. ✭✶✳✶✹✮

P♦rt❛♥t♦✱ ❞❡ ✭✶✳✶✷✮ ❡ ✭✶✳✶✹✮✱ ♦❜t❡♠♦s

φ(x) = 0 ♦✉ φ(x) = eA(ln(1+nx))_,

♦♥❞❡ A:R_−→Ré ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✳

✶✳✸

❊q✉❛çã♦ ▲♦❣❛rít♠✐❝❛ ❞❡ ❈❛✉❝❤②

❆ ❡q✉❛çã♦

f(xy) = f(x) +f(y) _⊙

é ❝❤❛♠❛❞❛ ❞❡ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ▲♦❣❛rít♠✐❝❛ ❞❡ ❈❛✉❝❤②✳

❚❡♦r❡♠❛ ✶✳✹ ❙❡ ❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ❧♦❣❛rít♠✐❝❛ ❞❡ ❈❛✉❝❤② ✈❛❧❡ ♣❛r❛ t♦❞♦ x, y _∈

R∗✱ ❡♥tã♦ ❛ s♦❧✉çã♦ ❣❡r❛❧ é ❞❛❞❛ ♣♦r

φ(x) = A(ln_|x_|), _∀x_∈R∗, ✭✶✳✶✺✮

♦♥❞❡ A é ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✳

Pr♦✈❛✳ ❙✉❜st✐t✉✐♥❞♦x=y=t ❡♠ _⊙✱ t❡♠♦s

φ(t2) = 2φ(t).

❉♦ ♠❡s♠♦ ♠♦❞♦✱ ❢❛③❡♥❞♦ x=₋t ❡y=₋t ❡♠ _⊙✱ t❡♠♦s

φ(t2) = 2φ(₋t).

❆ss✐♠✱ ✈❡♠♦s q✉❡

φ(t) =φ(₋t), _∀t_∈R∗. ✭✶✳✶✻✮

❆❣♦r❛✱ s✉♣♦♥❤❛♠♦s q✉❡ ❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ⊙ ✈❛❧❡ ♣❛r❛ t♦❞♦ x >0❡ y >0✳

❙❡❥❛✱

x=es _{❡ y=e}t_{, ✭✶✳✶✼✮}

❞❡ ♠♦❞♦ q✉❡

s=lnx ❡ t=lny. ✭✶✳✶✽✮

◆♦t❡ q✉❡ s, t _∈ R ✈✐st♦ q✉❡ x, y _∈ R∗ ♦♥❞❡ R+ ={x∈ R/x >0}✳ ❙✉❜st✐t✉✐♥❞♦ ✭✶✳✶✽✮ ❡♠ ⊙✱ t❡♠♦s

φ(es+t) = φ(es) +φ(et).

❉❡✜♥✐♥❞♦

A(s) = φ(es) ✭✶✳✶✾✮

❡ ✉s❛♥❞♦ ❛ ú❧t✐♠❛ ❡q✉❛çã♦✱ t❡♠♦s

A(s+t) = A(s) +A(t),

♣❛r❛ t♦❞♦s, t _∈R✳ P♦r ✐ss♦✱ ❛ ♣❛rt✐r ❞❡_⊙✱ t❡♠♦s

φ(x) = A(lnx), ✭✶✳✷✵✮

♣❛r❛ t♦❞♦x r❡❛❧ ♣♦s✐t✐✈♦✳

❈♦♠♦ φ(t) = φ(₋t)✱ t❡♠♦s q✉❡ ❛ s♦❧✉çã♦ ❣❡r❛❧ ❞❡ (1.7) é

φ(x) = A(ln_|x_|), ✭✶✳✷✶✮

♣❛r❛ t♦❞♦x r❡❛❧ ♥ã♦✲♥✉❧♦✳

❈♦r♦❧ár✐♦ ✶✳✸✳✶ ❆ s♦❧✉çã♦ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ f(xy) = f(x) +f(y)✱ ♣❛r❛

t♦❞♦ x, y _∈R+ é ❞❛❞❛ ♣♦r

φ(x) = A(lnx), ✭✶✳✷✷✮

♦♥❞❡ A:R_−→R é ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✳

❈♦r♦❧ár✐♦ ✶✳✸✳✷ ❆ s♦❧✉çã♦ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ f(xy) =f(x) +f(y) ✱ ♣❛r❛

t♦❞♦ x, y _∈R é ❞❛❞❛ ♣♦r

φ(x) = 0, _∀x_∈R∗. ✭✶✳✷✸✮

Pr♦✈❛✳ ❙✉❜st✐t✉✐♥❞♦ y= 0 ♥❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ❛❝✐♠❛ t❡♠♦s φ(0) = φ(x) +φ(0)✱

❞♦♥❞❡ φ(x) = 0.

❉❡✜♥✐çã♦ ✶✳✸ ❯♠❛ ❢✉♥çã♦f :R+−→R é ❝❤❛♠❛❞❛ ✉♠❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ s❡ s❛t✐s❢❛③ f(xy) =f(x) +f(y)✱ ♣❛r❛ t♦❞♦ x, y _∈R₊✳

✶✳✹

❊q✉❛çã♦ ▼✉❧t✐♣❧✐❝❛t✐✈❛ ❞❡ ❈❛✉❝❤②

❚r❛t❛r❡♠♦s ❛❣♦r❛ ❞❛ ú❧t✐♠❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ❞❡ ❈❛✉❝❤②✱ ❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ♠✉❧t✐♣❧✐❝❛t✐✈❛✱ ✐st♦ é✱

f(xy) =f(x)_·f(y). (_∗)

❊st❛ ❡q✉❛çã♦ é ❛ ♠❛✐s ❝♦♠♣❧✐❝❛❞❛ ❞❛s três ❡q✉❛çõ❡s ❝♦♥s✐❞❡r❛❞❛s ♥❡st❡ ❝❛♣ít✉❧♦✳ P❛r❛ ♦ ♣ró①✐♠♦ t❡♦r❡♠❛ ✈❛♠♦s ♣r❡❝✐s❛r ❞❛ ❞❡✜♥✐çã♦ ❞❛ ❢✉♥çã♦ s✐❣♥❛❧ ❞❡♥♦t❛❞❛ ♣♦r

sgn(x) ❡ ❞❡✜♥✐❞❛ ❝♦♠♦

sgn(x) =

 



1, s❡ x >0,

0, s❡ x= 0,

−1, s❡ x <0.

❚❡♦r❡♠❛ ✶✳✺ ❆ s♦❧✉çã♦ ❣❡r❛❧ ❞❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ♠✉❧t✐♣❧✐❝❛t✐✈❛✱ é ❞❛❞❛ ♣♦r

φ(x) = 0, ✭✶✳✷✹✮

φ(x) = 1, ✭✶✳✷✺✮

φ(x) = eA(Ln|x|)

|sgn(x)_|, ✭✶✳✷✻✮ φ(x) = eA(Ln|x|)_{sgn(x), ✭✶✳✷✼✮}

♦♥❞❡✱ A:R_−→R é ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✳

Pr♦✈❛✳ ❋❛③❡♥❞♦ x=y= 0 ❡♠ (_∗)✱ t❡♠♦s φ(0)_·[1₋φ(0)] = 0✱ ❡ ♣♦rt❛♥t♦✱

φ(0) = 0 ♦✉ φ(0) = 1. ✭✶✳✷✽✮

❉❛ ♠❡s♠❛ ❢♦r♠❛✱ s✉❜st✐t✉✐♥❞♦ x=y= 1 ❡♠ (_∗)✱ t❡♠♦s

φ(1)_·[1₋φ(1)] = 0,

❡ ♣♦rt❛♥t♦✱

φ(1) = 0 ❡ φ(1) = 1. ✭✶✳✷✾✮

❚♦♠❡ x ❝♦♠♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✱ ✐st♦ é x >0✳ ❊♥tã♦✱ (_∗) ✐♠♣❧✐❝❛

φ(x) = φ(√x)2 _{≥0. ✭✶✳✸✵✮}

❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ x0 ∈ R✱ x0 6= 0 t❛❧ q✉❡ φ(x0) = 0✳ ❙❡❥❛ x ∈ R ✉♠ ♥ú♠❡r♦ r❡❛❧ ❛r❜✐trár✐♦✳ ❊♥tã♦✱ ❞❡ (_∗)✱ t❡♠♦s

φ(x) = φ

x0·

x x0

= 0,

♣❛r❛ t♦❞♦x_∈R✱ ❡ ❛ss✐♠✱ ♦❜t❡♠♦s ❛ s♦❧✉çã♦ ❞❡ (1.24)✳

❙✉♣♦♥❤❛♠♦s q✉❡ φ(x)₆= 0♣❛r❛ t♦❞♦ x_∈R∗✳ ❙❡ φ(0) = 1✱ ❡♥tã♦✱ ❢❛③❡♥❞♦ y= 0

❡♠ (_∗)✱ t❡♠♦s

φ(0) =φ(x)_·φ(0).

❆ss✐♠✱

φ(x) = 1,

♣❛r❛ t♦❞♦x_∈R✳ ❆ss✐♠✱ t❡♠♦s ❛ s♦❧✉çã♦ (1.25)✳

❈♦♥s✐❞❡r❡♠♦s ♦ ❝❛s♦φ(0) = 0✳ ◆❡st❡ ❝❛s♦ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡φ é ♥ã♦ ♥✉❧❛ ❡♠

R∗✳ ❙✉♣♦♥❤❛ q✉❡ ♥ã♦✳ ❊♥tã♦✱ ❡①✐st❡ y0 ❡♠ R∗ t❛❧ q✉❡ φ(y0) = 0✳ ❋❛③❡♥❞♦ y = y0 ❡♠ (_∗)✱ t❡♠♦s

φ(xy0) =φ(x)·φ(y0) = 0. ▲♦❣♦✱ φ(x) = 0,_∀x_∈R∗_{✱ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳}

❯s❛♥❞♦ ♦ ❢❛t♦ q✉❡ φ é ♥ã♦ ♥✉❧❛ ❡♠ R∗ ❡ ✭✶✳✸✵✮✱ t❡♠♦s

φ(x)>0, ♣❛r❛ x >0. ✭✶✳✸✶✮

❙❡❥❛♠

x=es ❡ y=et, ✭✶✳✸✷✮

t❡♠♦s q✉❡

s =Lnx ❡ t=Lny ✭✶✳✸✸✮

◆♦t❡ q✉❡ s, t _∈R❡ x, y _∈R+✳ ❙✉❜st✐t✉✐♥❞♦ ✭✶✳✸✸✮ ❡♠ (∗)✱ t❡♠♦s

φ(es+t) =φ(es)_·φ(et).

❈♦♠♦ φ(t) > 0✱ ♣❛r❛ t♦❞♦ t > 0✱ t♦♠❛♥❞♦ ♦ ❧♦❣❛rít♠♦ ♥❛t✉r❛❧ ❡♠ ❛♠❜♦s ♦s

♠❡♠❜r♦s ❞❛ ú❧t✐♠❛ ❡q✉❛çã♦✱ t❡♠♦s q✉❡

A(s+t) = A(s) +A(t)✱

♦♥❞❡✱

A(s) = lnφ(es), _∀s_∈R. ✭✶✳✸✹✮

❊♥tã♦✱ A é ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✳ ❉❡ ✭✶✳✸✸✮ ❡ ✭✶✳✸✹✮✱ t❡♠♦s

φ(x) =eA(Ln|x|), _∀x_∈R+. ✭✶✳✸✺✮

❉❡ ✭✶✳✷✽✮✱ t❡♠♦s q✉❡ φ(1) = 0 ♦✉ φ(1) = 1✳ ❙❡φ(1) = 0✱ ❢❛③❡♥❞♦ y = 1 ❡♠ (_∗)✱

t❡♠♦s

φ(x) = 0, _∀x_∈R∗.

❈♦♥tr❛❞✐③❡♥❞♦ à ❤✐♣ót❡s❡ q✉❡ φ ♥ã♦ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛ ❡♠ x _∈ R∗✳ ❆ss✐♠✱

φ(1) = 1✳ ❆❣♦r❛✱ ❢❛③❡♥❞♦ x=y=₋1 ❡♠ ✭✶✳✽✮✱ t❡♠♦s φ(1) = [φ](₋1)2 _{❡ ❛ss✐♠✱}

φ(₋1) = 1 ♦✉ φ(₋1) = ₋1. ✭✶✳✸✻✮

❙❡ φ(₋1) = 1✱ ❡♥tã♦✱ ❢❛③❡♥❞♦ y=₋1 ❡♠ (_∗)✱ t❡♠♦s

φ(₋x) =φ(x)_·φ(₋1) =φ(x),

♣❛r❛ t♦❞♦x_∈R∗✳

❊♥tã♦✱ ❞❡ ✭✶✳✸✸✮ s❡❣✉❡ q✉❡

φ(x) =eA(Lnx|)_,

♣❛r❛ t♦❞♦x_∈R∗✳ ❆ss✐♠✱ φ(0) = 0✱ t❡♠♦s

φ(x) =

eA(Ln|x|)_{, s❡ x∈R}∗_,

0, s❡ x= 0.

❙❡ φ(₋1) =₋1✱ ❢❛③❡♥❞♦y=₋1 ❡♠ (_∗)✱

φ(₋x) =φ(x)_·φ(₋1) =₋φ(x)✱

♣❛r❛ t♦❞♦x_∈R∗✳ ❆ss✐♠✱ ❞❡ ✭✶✳✸✹✮✱ ♦❜t❡♠♦s

φ(x) =

e(ALln|x|)_{, s❡ x >0,}

−eA(Ln|x|)_{, s❡ x <0,}

♣❛r❛ t♦❞♦x_∈R∗_{✳ ❏✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❢❛t♦ q✉❡ φ(0) = 0✱ t❡♠♦s}

φ(x) =

 



eA(Ln|x|)_{, s❡ x >0,}

0, s❡ x= 0,

−eA(Ln|x|)_{, s❡ x <0,}

q✉❡ é ❛ s♦❧✉çã♦ ✭✶✳✷✼✮✳

❈♦r♦❧ár✐♦ ✶✳✹✳✶ ❆ s♦❧✉çã♦ ❣❡r❛❧ ❝♦♥tí♥✉❛ ❞❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧f(xy) =f(x)_·f(y) (_∗)✱ ♣❛r❛ t♦❞♦ x, y _∈R é ❞❛❞❛ ♣♦r

φ(x) = 0, ✭✶✳✸✼✮

φ(x) = 1, ✭✶✳✸✽✮

φ(x) = _|x_|α ✭✶✳✸✾✮

❡

f(x) = _|x_|α

·sgn(x), ✭✶✳✹✵✮

♦♥❞❡ α é ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧ ♣♦s✐t✐✈❛✳

Pr♦✈❛✳ P❡❧♦ t❡♦r❡♠❛ (1.5) ♦✉φ = 0✱ ♦✉ φ= 1✱ ♦✉ φ t❡♠ ❛ ❢♦r♠❛ (1.26) ♦✉(1.27)✱

♦♥❞❡ A:R_−→Ré ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✳ ❈♦♠♦ φ é ❝♦♥tí♥✉❛ ❡✱ A(t) = Lnφ(et_)✱

A é t❛♠❜é♠ ❝♦♥tí♥✉❛ s♦❜r❡ R✳ P♦rt❛♥t♦✱

A(t) =α_·t✱

♦♥❞❡✱ α_∈R é ✉♠❛ ❝♦♥st❛♥t❡ ❛r❜✐trár✐❛✳ ❆ ♣❛rt✐r ❞❡ (1.26) ❡(1.27)✱ t❡♠♦s

f(x) =_|x_|α ❡

f(x) = _|x_|α

·sgn(x)✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❘❡st❛ ♠♦str❛r q✉❡α >0✳ ❙❡α= 0✱ ❡♥tã♦✱(1.39)♣r♦❞✉③φ(x) = 1✱

♣❛r❛x₆= 0✱ ❡ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡f ❞❡✈❡♠♦s t❡rφ(0) = 1✳ ❆ss✐♠✱ t❡♠♦s q✉❡φ= 1✱

❥á ❧✐st❛❞♦ ❡♠ (1.38) ❝♦♠ α= 0✱ t❡♠♦s

φ(x) = 1✱ ♣❛r❛x >0

❡

φ(x) = ₋1✱ ♣❛r❛x <0

❡ ❛ss✐♠✱ φ ♥ã♦ ♣♦❞❡ s❡r ❝♦♥tí♥✉❛✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ s❡ α <0✱ ❡♥tã♦✱ φ s❡❣✉✐❞❛ ❞❡

(1.38) ❡(1.39) s❛t✐s❢❛③

lim

x→ 0+φ(x) = ∞ ❡ ❛ss✐♠✱ ♥ã♦ ♣♦❞❡ s❡r ❝♦♥tí♥✉❛ ❡♠0✳

❉❡✜♥✐çã♦ ✶✳✹ ❯♠❛ ❢✉♥çã♦φ :R_−→Ré ❞✐t❛ ✉♠❛ ❢✉♥çã♦ ♠✉❧t✐♣❧✐❝❛t✐✈❛ s❡ s❛t✐s❢❛③

f(xy) = f(x)_·f(y)✱ ♣❛r❛ t♦❞♦ x, y _∈R✳

❊q✉❛çõ❡s ❞❡ ❏❡♥s❡♥✱ P❡①✐❞❡r ❡

❞✬❆❧❡♠❜❡rt

✷✳✶ ❋✉♥çã♦ ❝♦♥✈❡①❛

❋✉♥çõ❡s ❈♦♥✈❡①❛s ❢♦r❛♠ ♣r✐♠❡✐r❛♠❡♥t❡ ✐♥tr♦❞✉③✐❞❛s ♣♦r ❏♦❤❛♥ ▲✉❞✇✐❣ ❏❡♥s❡♥ ❡♠ ✶✾✵✺✱ ❛♣❡s❛r ❞❡ q✉❡ ❢✉♥çõ❡s ❝♦♥✈❡①❛s ❥á ❤❛✈✐❛♠ s✐❞♦ tr❛t❛❞❛s ♣♦r ❏❛❝q✉❡s ❍❛✲ ❞❛♠❛r❞ ✭✶✽✾✸✮ ❡ ❖tt♦ ❍ö❧❞❡r ✭✶✽✽✾✮✳ ❯♠❛ ❢✉♥çã♦ f : R_−→ R é ❞✐t❛ ❝♦♥✈❡①❛ s❡ s❛t✐s❢❛③ ❛ ❞❡s✐❣✉❛❧❞❛❞❡

f

x+y

2

≤ f(x) +f(y)

2 , x, y ∈R. ✭✷✳✶✮

✷✳✷ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❞❡ ❏❡♥s❡♥

f

x+y

2

= 1

2(f(x) +f(y)), x, y ∈R, (JE)

é ❝❤❛♠❛❞❛ ❞❡ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❞❡ ❏❡♥s❡♥✳

❉❡✜♥✐çã♦ ✷✳✶ ❯♠❛ ❢✉♥çã♦ φ :R_−→R é ❞✐t❛ ❏❡♥s❡♥ s❡ s❛t✐s❢❛③

φ

x+y

2

= φ(x) +φ(y)

2 , ∀x, y ∈R✱

✐st♦ é✱ s❡ é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ❞❡ ❏❡♥s❡♥✳

❖❜s❡r✈❛çã♦ ✶ ▲❡♠❜r❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ f : R _−→ R é ❞✐t❛ ❛✜♠ s❡ ❡❧❛ é ❞❛ ❢♦r♠❛

f(x) = ax+b✱

♦♥❞❡ a ❡ b sã♦ ❝♦♥st❛♥t❡s✳

❚❡♦r❡♠❛ ✷✳✶ ❆ ❢✉♥çã♦ φ : R _−→ R s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ❞❡ ❏❡♥s❡♥ s❡ ❡ s♦♠❡♥t❡ s❡

φ(x) =A(x) +a, ✭✷✳✷✮

♣❛r❛ ❛❧❣✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛ A:R_−→R ❡ ❛❧❣✉♠ a_∈R✳

Pr♦✈❛✳ ❙✉♣♦♥❤❛♠♦s q✉❡ φ(x) =A(x) +a✱ ♦♥❞❡ a_∈R❡A :R_−→R é ✉♠❛ ❢✉♥çã♦ ❛❞✐t✐✈❛✳ ❊♥tã♦✱

φ

x+y

2

= Ax

2 +

y

2

+a

= Ax

2

+Ay

2

+a

=

A(x) +a

2

+

A(y) +a

2

= φ(x) +φ(y)

2 .

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ φ:R_−→R ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❞❡ ❏❡♥s❡♥✱ ❡♥tã♦✱

φ

x+y

2

= φ(x) +φ(y)

2 , (∗)

♣❛r❛ t♦❞♦x, y _∈R✳ ❋❛③❡♥❞♦ y = 0✱ t❡♠♦s q✉❡

φx

2

= φ(x) 2 +

a

2, ♦♥❞❡ a=φ(0). (∗∗)

❙✉❜st✐t✉✐♥❞♦ (_∗∗) ❡♠ (_∗)✱ t❡♠♦s q✉❡

φ(x+y) +a

2 =

φ(x) +φ(y)

2 ✱

q✉❡ ❡q✉✐✈❛❧❡ ❛

φ(x+y) +a=φ(x) +φ(y). ✭✷✳✸✮

❉❡✜♥❛♠♦s A:R_−→R♣♦r A(x) = φ(x)₋a✳ ❉❡ ✭✷✳✸✮✱ t❡♠♦s q✉❡

(x+y) = φ(x+y)₋a

= φ(x) +φ(y)₋2a

= (φ(x)₋a) + (φ(y)₋a) = A(x) +A(y).

P♦rt❛♥t♦✱ A é ❛❞✐t✐✈❛ ❡ φ(x) =A(x) +a✳

❉❡✜♥✐çã♦ ✷✳✷ ❙❡❥❛♠ m ❡ n ❞♦✐s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❞❛ ❢♦r♠❛ m

2n é ❝❤❛♠❛❞♦ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❞✐á❞✐❝♦✳

❚❡♦r❡♠❛ ✷✳✷ ❆ s♦❧✉çã♦ ❝♦♥tí♥✉❛φ ❞❛ ❡q✉❛çã♦ ❞❡ ❏❡♥s❡♥ ♥♦ ✐♥t❡r✈❛❧♦ [a, b]é ✉♠❛

❢✉♥çã♦ ❛✜♠✱ ✐st♦ é✱

f(x) =β+α_·x, ♦♥❞❡ α, β _∈R.

Pr♦✈❛✳ ❉❡✜♥❛♠♦s ✉♠❛ ♥♦✈❛ ❢✉♥çã♦ F : [0,1]_−→R ❝♦♠♦

F(y) =φ((b₋a)y+a), _∀y_∈[0,1]. ✭✷✳✹✮

❙❡❥❛ ψ : [0,1]_−→[a, b] t❛❧ q✉❡ψ(y) = a+ (b₋a)y✱ ❡♥tã♦✱ F =φ_◦ψ✳ P♦rt❛♥t♦✱ F ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❆✜r♠❛♠♦s q✉❡ F s❛t✐s❢❛③ (JE)✳ ❉❡ ❢❛t♦✱

F

x+y

2

= φ

(b₋a)_·

x+y

2

+a

= φ

[(b₋a)_·x+a] + [(b₋a)_·y+a] 2

= φ((b−a)x+a) +φ((b−a)y+a) 2

= F(x) +F(y)

2 , i.e,

F

x+y

2

= F(x) +F(y)

2 , x, y ∈[0,1]. (⋆)

❊♥tã♦✱ F s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧ ❞❡ ❏❡♥s❡♥ ❡♠[0,1]✳ ❋❛③❡♥❞♦x= 0❡y= 1

❡♠ (⋆)✱ t❡♠♦s

F 1

2 =

F(0) +F(1)

2 =

c+d

2 =c+ 1

2(d−c).

♦♥❞❡ c=F(0) ❡ d=F(1)✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ❢❛③❡♥❞♦x= 0 ❡ y= 1

2 ❡♠ (⋆)✱ t❡♠♦s

F 1 4 =

F(0) +F(1 2)

2 =

c+c+1

2(d−c)

2 =c+

1

4(d−c).

❆❣♦r❛ ❢❛③❡♥❞♦ x= 1

2 ❡ y= 1 ❡♠ (⋆)✱ t❡♠♦s

F 3 4 = F 1 2

+F(1)

2 =c+

3

4(d−c).

❊♠ s❡❣✉✐❞❛✱ ♣r♦✈❡♠♦s q✉❡ s❡ x é ✉♠ ♥ú♠❡r♦ r❡❛❧ ❞❛ ❢♦r♠❛ m

2k✱ ♦♥❞❡ m ❡ k ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s s❛t✐s❢❛③❡♥❞♦ 06m62k_{✱ ❡♥tã♦✱}

F(x) =c+x(d₋c).

❆ ♣r♦✈❛ é ❢❡✐t❛ ♣♦r ✐♥❞✉çã♦ s♦❜r❡k✳ ❏á ♠♦str❛♠♦s q✉❡ ❛ ❛✜r♠❛çã♦ é ✈❡r❞❛❞❡✐r❛

♣❛r❛ k = 1,2✳ ❆ss✉♠❛ q✉❡(2.4) é ✈á❧✐❞❛ ♣❛r❛ n=k ❡ ❝♦♥s✐❞❡r❡♠♦s ❞♦✐s ❝❛s♦s

caso 1 : x= 2m

2n+1,

caso 2 : x= 2m+ 1

2n+1 . ◆♦ ❝❛s♦ ✶✱ t❡♠♦s

F

2m

2n+1

=Fm

2n

=c+ m

2n(d−c) =c+

2m

2n+1(d−c).

❊ ♥♦ ❝❛s♦ ✷✱

F

2m+ 1 2n+1

= F 1 2 m

2n +

m+ 1

2n

=

F m

2n

+F

m+ 1

2n

2

F

2m+ 1 2n+1

= 1 2

c+ m

2n(d−c) +c+

m+ 1

2n (d−c)

= c+2m+ 1

2n+1 (d−c).

❆ss✐♠✱ (2.4) é s❛t✐s❢❡✐t❛ ♣❛r❛ t♦❞♦s r❛❝✐♦♥❛✐s ❞✐á❞✐❝♦s x ❡♠ [0,1]✳ ❯♠❛ ✈❡③ q✉❡

F é ❝♦♥tí♥✉❛ ❡ ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s r❛❝✐♦♥❛✐s ❞✐á❞✐❝♦s ❡♠ [0,1] é ❞❡♥s♦ ❡♠ [0,1]✱ t❡♠♦s

F(x) = c+x_·(d₋c),

♣❛r❛ t♦❞♦x_∈[0,1]✳ ❈♦♠♦ F =φ_◦ψ✱ ❡♥tã♦ φ=F_◦ψ−1_{✳ P♦rt❛♥t♦✱φ(x) = β+α·x✱} ♣❛r❛ ❛❧❣✉♠α, β _∈R✳

✷✳✸ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❞❡ ❞✬❆❧❡♠❜❡rt

f(x+y) +f(x₋y) = 2f(x)_·f(y), (DE)

♣❛r❛ t♦❞♦x, y _∈R✱ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❊q✉❛çã♦ ❋✉♥❝✐♦♥❛❧ ❞❡ ❞✬❆❧❡♠❜❡rt✳

✷✳✸✳✶ ❙♦❧✉çõ❡s ❈♦♥tí♥✉❛s ❞❛ ❊q✉❛çã♦ ❞❡ ❞✬ ❆❧❡♠❜❡rt

❚❡♦r❡♠❛ ✷✳✸ ❙❡❥❛✱ φ : R _−→ R ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✱ q✉❡ s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ❞❡ ❞✬❆❧❡♠❜❡rt✳ ❊♥tã♦✱ φ t❡♠ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❢♦r♠❛s

φ(x) = 0, ✭✷✳✺✮

φ(x) = 1, ✭✷✳✻✮

φ(x) = cosh(α_·x), α_∈R, ✭✷✳✼✮ φ(x) = cos(β_·x), β _∈R. ✭✷✳✽✮

❈♦♠♦ φ é s♦❧✉çã♦✱

φ(x+y) +φ(x₋y) = 2φ(x)_·φ(y). (_⊗)

Pr♦✈❛✳ ❋❛③❡♥❞♦x=y= 0 ❡♠ (_⊗)✱ ♦❜t❡♠♦s

2φ(0) = 2[φ(0)]2.

❆ss✐♠✱

φ(0) = 0 ♦✉ φ(0) = 1.

❙❡ φ(0) = 0✱ ❡♥tã♦✱ t♦♠❛♥❞♦ y= 0 ❡♠ (_⊗)✱ t❡♠♦s

2φ(x) = 2φ(x)_·φ(0).

❉❛í✱

2φ(x) = 0,

s❡❣✉❡ q✉❡

φ(x) = 0, _∀x_∈R.

❚❡♠♦s ❛ s♦❧✉çã♦ ✭✷✳✻✮✳ ❆ss✉♠✐♥❞♦ ❛❣♦r❛ q✉❡ φ é ♥ã♦ ♥✉❧❛✳ ❱❛♠♦s ♠♦str❛r q✉❡ φ é ✉♠❛ ❢✉♥çã♦ ♣❛r✳ ❋❛ç❛ x= 0 ❡♠ (_⊗)✳ ❊♥tã♦✱

φ(y) +φ(₋y) = 2φ(0)_·φ(y)✱

❝♦♠♦ φ é ♥ã♦ ♥✉❧❛✱ φ(0)₆= 0 ❡ φ(0) = 1✳ ❉❛ ❡q✉❛çã♦ ❛❝✐♠❛

φ(y) +φ(₋y) = 2φ(y),

✐st♦ é✱

φ(₋y) =φ(y),

♣❛r❛ t♦❞♦y_∈R✳ ❊♥tã♦✱ φ é ✉♠❛ ❢✉♥çã♦ ♣❛r✳ ❈♦♠♦ φ é ❝♦♥tí♥✉❛ ❡♠R✱ φ t❛♠❜é♠

é ✐♥t❡❣rá✈❡❧ ❡♠ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦ ✜♥✐t♦✳ ❆ss✐♠✱ ♣❛r❛ t >0✱ t❡♠♦s

Z t

−t

φ(x+y)dy+

Z t

−t

φ(x₋y)dy= 2φ(x)

Z t

−t

φ(y)dy. ✭✷✳✾✮

❆❣♦r❛✱

Z t

−t

φ(x+y)dy =

Z x+t

x−t

φ(z)dz =

Z x+t

x−t

φ(y)dy.

❆♥❛❧♦❣❛♠❡♥t❡✱

Z t

−t

φ(x₋y)dy=

Z x−t

x+t

φ(w)(₋dw) =

Z x+t

x−t

φ(w)dw=

Z x+t

x−t

φ(y)dy.

P♦rt❛♥t♦ ✭✷✳✾✮✱ t♦r♥❛✲s❡

Z x+t

x−t

f(y)dy+

Z x+t

x−t

f(y)dy= 2f(x)

Z t

−t

f(y)dy.

♦✉ s❡❥❛✱

Z x+t

x−t

φ(y)dy =φ(x)

Z t

−t

φ(y)dy. ✭✷✳✶✵✮

❈♦♠♦φé ♥ã♦ ♥✉❧❛✱φ(0) = 1✳ ❆❧é♠ ❞✐ss♦✱ ✉♠❛ ✈❡③ q✉❡φ é ❝♦♥tí♥✉❛✱ ❡①✐st❡t >0

t❛❧ q✉❡

Z t

−t

φ(y)dy >0.

◆♦t❡ q✉❡ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❡ ✭✷✳✶✵✮ é ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ r❡s♣❡✐t♦ ❛x♣❡❧♦ ❚❡♦r❡♠❛

❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ✭✈❡❥❛ ❆♥t♦♥✱ ♣❛❣✳ ✸✾✻✲✷✵✵✶✮✳ ❉❛í✱ ♦ ❧❛❞♦ ❞✐r❡✐t♦ t❛♠❜é♠ é ❞✐❢❡r❡♥❝✐á✈❡❧ ❝♦♠ r❡s♣❡✐t♦ ❛ x✳ ❊♥tã♦✱ t♦♠❛♥❞♦ ❛ ❞❡r✐✈❛❞❛ ❝♦♠ r❡s♣❡✐t♦ ❛ x ❡♠

✭✷✳✶✵✮✱ t❡♠♦s

d dx

Z x+t

x−t

φ(y)dy= d

dx[φ(x)

Z t

−t

φ(y)dy].

❆ss✐♠✱ t❡♠♦s

φ(x+t)₋φ(x₋t) = φ′(x)

Z t

−t

f(y)dy. ✭✷✳✶✶✮

■st♦ ♣r♦✈❛ q✉❡ φ é ❞✉❛s ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧ ❡ ❛ss✐♠✱

φ′(x+t)₋φ′(x₋t) =φ′′(x)

Z t

−t

φ(y)dy.

❊♥tã♦✱ φ é ✸ ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧✳ Pr♦❝❡❞❡♥❞♦ ♣❛ss♦ ❛ ♣❛ss♦✱ ✈❡♠♦s q✉❡ t♦❞❛

s♦❧✉çã♦ ❝♦♥tí♥✉❛ ❞❡ (DE) é ✐♥✜♥✐t❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧✳

❚♦♠❛♥❞♦ x= 0 ❡♠ ✭✷✳✶✶✮✱ t❡♠♦s

φ(t)₋φ(₋t) = φ′(0)

Z t

−t

φ(y)dy. ✭✷✳✶✷✮

❈♦♠♦ φ é ♣❛r✱ t❡♠♦s φ(t) = φ(₋t) ❡ ✭✷✳✶✷✮ t♦r♥❛✲s❡

φ′(0)

Z t

−t

φ(y)dy= 0, ✭✷✳✶✸✮

♣♦ré♠✱ Rt

−tφ(y)dy >0✱ ❡ ❞❡ ✭✷✳✶✸✮✱ t❡♠♦s q✉❡

φ′_{(0) = 0. ✭✷✳✶✹✮}

❈♦♠♦ φ_∈C∞_{(R)✱ ❞✐❢❡r❡♥❝✐❛♠♦s (⊗) ❝♦♠ r❡s♣❡✐t♦ ❛ y ❞✉❛s ✈❡③❡s ♣❛r❛ ♦❜t❡r}

φ′(x+y)₋φ′(x+y) = 2φ(x)_·φ′(y), φ′′(x+y) +φ′′(x₋y) = 2φ(x)_·φ′′(y),

♣❛r❛ t♦❞♦x, y _∈R✳ ❋❛③❡♥❞♦ y = 0✱ t❡♠♦s

2φ′′(x) = 2φ(x)_·φ′′(0).

❙❡❥❛ k=φ′′_{(0)✱ ❡♥tã♦✱}

φ′′_{(x) =kφ(x)✱}

q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛♦ s❡❣✉✐♥t❡ Pr♦❜❧❡♠❛ ❞❡ ❱❛❧♦r ■♥✐❝✐❛❧ ✭P✳❱✳■✳✮

d2y

dx2 = ky,

y(0) = 1, y′(0) = 0.

P❛r❛ r❡s♦❧✈❡r ❡st❡ ✭P❱■✮ ✐r❡♠♦s ❝♦♥s✐❞❡r❛r três ❝❛s♦s✿ k = 0, k > 0❡ k <0.

❈❛s♦ ✶✳ ❙✉♣♦♥❤❛ k= 0✳ ❊♥tã♦ ♦ ✭P❱■✮ s❡ r❡❞✉③ ❛

d2_y

dx2 = 0✱ ❞❛í✱

y(x) =c1x+c2.

❈♦♠♦ y(0) = 1✱ c2 = 1✳ ◆♦✈❛♠❡♥t❡ ❝♦♠♦ y′(0) = 0✱ t❡♠♦s c1 = 0✳ P♦rt❛♥t♦✱

y(x) = 1✳

❈❛s♦ ✷✳ ❙✉♣♦♥❤❛ k >0✳ ❚♦♠❛♥❞♦ y=emx _❡♠

d2_y

dx2 =ky, (DE

′₎

♦❜t❡♠♦sm2 _{=k ❡ ❞❛í m=±}√_{k✳ ❊♥tã♦✱}

y(x) = c1eα·x+c2e−α·x, ♦♥❞❡ α=

√

k.

❆❣♦r❛✱

1 = y(0) =c1e0+c2e0 =⇒1 =c1+c2

❡

0 =y′_{(0) =c}

1e0−c2e0 =⇒c1 =c2 ✭❞❡s❞❡ q✉❡α6= 0✮✱

❡♥tã♦✱ c1 =c2 =

1

2✳ P♦rt❛♥t♦✱ ❛ s♦❧✉çã♦ ❞❡(DE

′_{)é ❞❛❞❛ ♣♦r}

y(x) = e

α·x_+e−α·x

2 = cosh(α·x)✳

❊♥tã♦✱ ♥❡st❡ ❝❛s♦✱ f(x) = cosh(α_·x)✱ q✉❡ é ✭2.7✮✳

❈❛s♦ ✸✳ ❙✉♣♦♥❤❛ k <0✳ ❚♦♠❛♥❞♦ y=emx _❡♠ d2y

dx2 =ky✱ ♦❜t❡♠♦s

m2_y=ky✳

❊♥tã♦✱ m =_±iβ✱ ♦♥❞❡ β =√₋k✱ i=√₋1✳ ❆ s♦❧✉çã♦ ❞❡ (DE′_{) é ❞❛❞❛ ♣♦r}

y(x) =c1eiβx+c2e−iβx✳

❆ss✐♠✱ 1 = y(0) =c1+c2 =⇒c2 = 1−c1✳ P♦r ♦✉tr♦ ❧❛❞♦✱

0 = y′_{(0) =iβc}

1 −iβ(1−c1) = 2iβc1−iβ =iβ(2c1−1) = 0 =⇒c1 =c2 =

1 2✳

▲♦❣♦✱

y(x) = e

iβx_+e−iβx

2 = cos(βx)✳

P♦rt❛♥t♦✱ ❛ s♦❧✉çã♦ é ❞❛❞❛ ♣♦r f(x) = cos(βx)✱ q✉❡ é ✭✷✳✽✮✳

✷✳✸✳✷ ❙♦❧✉çã♦ ●❡r❛❧ ❞❛ ❊q✉❛çã♦ ❞❡ ❉✬❆❧❡♠❜❡rt

❯♠❛ s♦❧✉çã♦ E :R_−→C é ❞✐t❛ ❡①♣♦♥❡♥❝✐❛❧ s❡E s❛t✐s❢❛③ ❛ ❡q✉❛çã♦

E(x+y) =E(x)_·E(y), x, y _∈R.

P♦❞❡ s❡r ❞❡♠♦♥str❛❞♦ q✉❡ ✭♣♦r ❡①❡♠♣❧♦✱ ✈❡❥❛✿ ❊❧♦♥ ▲❛❣❡s✱ ❆ ♠❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ✈♦❧✳✶✮

E(x) = eλ·x_{✱ ♦♥❞❡λ∈R.}

❙❡ E :R_−→C é ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ♥ã♦ ♥✉❧❛✱ ❡♥tã♦✱ ❞❡♥♦t❛♠♦s ♣♦r

E∗(y) =E(y)−1. ✭✷✳✶✺✮

Pr♦♣♦s✐çã♦ ✷✳✶ ❙❡ E : R _−→ C é ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❡ E(0) é ③❡r♦✱ ❡♥tã♦✱

E(x)_≡0✱ ♣❛r❛ t♦❞♦ x_∈R✳

Pr♦✈❛✳ ❙❡❥❛E :R_−→C ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❆ss✐♠✱

E(x+y) = E(x)_·E(y), ✭✷✳✶✻✮

♣❛r❛ t♦❞♦x, y _∈R✳ ❋❛③❡♥❞♦ y = 0 ❡♠ ✭✷✳✶✻✮✱ ♦❜t❡♠♦s

E(x) =E(x)_·E(0), ♣❛r❛ x_∈R. ✭✷✳✶✼✮

❈♦♠♦ E(0) = 0✱ ✭✷✳✶✼✮ ✐♠♣❧✐❝❛

E(x) = 0, _∀x_∈R. ✭✷✳✶✽✮

❆ss✐♠✱ E(x) é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✳

Pr♦♣♦s✐çã♦ ✷✳✷ ❙❡❥❛ E : R _−→R ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❙❡ E(x) ₆= 0✱ ❡♥tã♦✱

E(0) = 1✳

Pr♦✈❛✳ ❙❡❥❛ E : R _−→ R ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❆ss✉♠❛ q✉❡ E(x) é ♥ã♦ ♥✉❧❛✳

❋❛ç❛ x=y= 0 ❡♠ ✭✷✳✶✻✮✱ t❡♠♦s

E(0)_·[1₋E(0)] = 0✳

P♦rt❛♥t♦✱

E(0) = 0 ♦✉ E(0) = 1✱

✈❡♠♦s q✉❡ E(0) = 1✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ♥ã♦✱ ❛ss✐♠ E(0) = 0✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ (2.1)✱

E(x) = 0✱ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❆ss✐♠✱E(0) = 1✳

Pr♦♣♦s✐çã♦ ✷✳✸ ❙❡❥❛ E : R _−→ C ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❙❡✱ E(x0) = 0 ♣❛r❛ ❛❧❣✉♠ x0 6= 0✱❡♥tã♦ E(x) = 0✱ ♣❛r❛ t♦❞♦ x∈R

Pr♦✈❛✳ ❙❡❥❛x₆=x0 ∈R✱ ❡♥tã♦✱ ❝♦♠♦ E(x0) = 0✱ t❡♠♦s

E(x) =E((x₋x0) +x0) = E(x−x0)·E(x0) = 0✳

❆ss✐♠✱ E(x)_≡0✳

Pr♦♣♦s✐çã♦ ✷✳✹ ❙❡❥❛ E :R _−→ C ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❙❡ E(x) é ♥ã♦ ♥✉❧♦✱

❡♥tã♦✱

E∗_{(−x) =E(x)✱}

♣❛r❛ t♦❞♦ x_∈R✳

Pr♦✈❛✳ ❙❡❥❛E :R_−→C ❛ ❡①♣♦♥❡♥❝✐❛❧✳ ❆❣♦r❛ ❢❛ç❛ y=₋x ❡♠ ✭✷✳✶✻✮✱ t❡♠♦s

E(0) =E(x)_·E(₋x). ✭✷✳✶✾✮

❈♦♠♦ E(x)₆= 0 ♣❡❧❛ ♣r♦♣♦s✐çã♦ (2.2)✱ E(0) = 1✱ ✐st♦ ✐♠♣❧✐❝❛ q✉❡

E(₋x) = 1

E(x)✱

♦✉ s❡❥❛✱

E(₋x) =E(x)−1 ✭✷✳✷✵✮

♦✉

E∗(₋x) = E(x). ✭✷✳✷✶✮

Pr♦♣♦s✐çã♦ ✷✳✺ ❙❡❥❛E :R_−→C❝♦♠♦ ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳ ❙✉♣♦♥❤❛E(x)₆= 0✱ ❡♥tã♦✱

E∗(x+y) =E∗(x)_·E∗(y), ✭✷✳✷✷✮

♣❛r❛ t♦❞♦ x, y _∈R✳

Pr♦✈❛✳ ❈♦♠♦ E(x) ₆= 0✱ E(x) é ♥ã♦ ♥✉❧❛ s♦❜r❡ R✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ (2.4)✳ ❆❣♦r❛

❝♦♥s✐❞❡r❡

E∗(x+y) = 1

E(x+y)

= 1

E(x)E(y) = E(x)−1_·E(y)−1 = E∗(x)_·E∗(y),

♣❛r❛ t♦❞♦x, y _∈R✳

Pr♦♣♦s✐çã♦ ✷✳✻ ❚♦❞❛ s♦❧✉çã♦ ♥ã♦ ♥✉❧❛ φ : R _−→ C ❞❛ ❡q✉❛çã♦ ❞❡ ❞✬❆❧❡♠❜❡rt é ✉♠❛ ❢✉♥çã♦ ♣❛r✳

Pr♦✈❛✳ ❙✉❜st✐t✉✐♥❞♦y ♣♦r ₋y ♥❛ ❡q✉❛çã♦ (DE)✱ t❡♠♦s

φ(x+y) +φ(x₋y) = 2φ(x)_·φ(₋y). ✭✷✳✷✸✮

❙✉❜tr❛✐♥❞♦ ✭✷✳✷✸✮ ❞❡ (DE)✱ ♦❜t❡♠♦s

φ(y) =φ(₋y)✱

♣❛r❛ t♦❞♦y_∈R✳ ❆ss✐♠✱ φ é ✉♠❛ ❢✉♥çã♦ ♣❛r✳

❚❡♦r❡♠❛ ✷✳✹ ❚♦❞❛ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ φ :R_−→C ❞❛ ❡q✉❛çã♦ ❢✉♥❝✐♦♥❛❧

f(x+y) +f(x₋y) = 2f(x)_·f(y) (DE)

é ❞❛ ❢♦r♠❛

f(x) = E(x) +E

∗_(x)

2 , ✭✷✳✷✹✮

♦♥❞❡✱ E :R_−→C∗ é ✉♠❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧✳

Pr♦✈❛✳ ❙❡❥❛ φ s♦❧✉çã♦ ♥ã♦ tr✐✈✐❛❧ ❞❡ (DE)✱ ✐st♦ é✱ φ ₆= 0✳ ❋❛③❡♥❞♦ x = y = 0 ❡♠ (DE)✱ ♦❜t❡♠♦sφ(0)_·[1₋φ(0)] = 0✳ ❆ss✐♠✱ φ(0) = 0 ♦✉φ(0) = 1✳ ❈♦♠♦φ(x)₆= 0✱

❡♥tã♦✱

φ(0) = 1. ✭✷✳✷✺✮

❚♦♠❛♥❞♦ y=x❡♠ (DE)✱ t❡♠♦s

φ(2x) +φ(0) = 2φ(x)2 =_⇒φ(2x) = 2φ(x)2₋1. ✭✷✳✷✻✮

❚r♦❝❛♥❞♦ ❛❣♦r❛ x ♣♦rx+y ❡ y ♣♦r x₋y ❡♠ (DE)✱ t❡♠♦s

φ(x+y+x₋y) +φ(x+y₋x+y) = 2φ(x+y)_·φ(x₋y)✳

❆ss✐♠✱

φ(2x) +φ(2y) = 2φ(x+y)_·φ(x₋y), ✭✷✳✷✼✮

♣❛r❛ t♦❞♦x, y _∈R✳

❈❛❧❝✉❧❛♥❞♦

[φ(x+y)₋φ(x₋y)]2 = [φ(x+y) +φ(x₋y)]2₋4φ(x+y)_·φ(x₋y) = [2φ(x)_·φ(y)]2₋4φ(x+y)_·φ(x₋y)

= 4φ(x)2_·φ(y)2₋2[φ(2x) +φ(2y)]

= 4φ(x)2φ(y)2₋2[2φ(x)2 ₋1 + 2φ(y)2₋1] = 4φ(x)2_·φ(y)2₋4φ(x)2₋4φ(y)2+ 4 = 4[φ(x)2 ₋1]_·[φ(y)2₋1].

P♦rt❛♥t♦✱

φ(x+y)₋φ(x₋y) =_±2p[φ(x)2_{−1]·[φ(y)}2_−1],

❛❞✐❝✐♦♥❛♥❞♦ ❛ (DE)✱ t❡♠♦s