O desenvolvimento da pesquisa em equações diferenciais no ICMC-USP: contribuição de Nelson Onuchic. -

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

❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

❖ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ P❡sq✉✐s❛ ❡♠ ❊q✉❛çõ❡s

❉✐❢❡r❡♥❝✐❛✐s ♥♦ ■❈▼❈✲❯❙P✿ ❈♦♥tr✐❜✉✐çã♦ ❞❡

◆❡❧s♦♥ ❖♥✉❝❤✐❝✳

P❛trí❝✐❛ ❞❡ ❙♦✉③❛

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

❖r✐❡♥t❛❞♦r

Pr♦❢✳ ❉r✳ ❙ér❣✐♦ ❘♦❜❡rt♦ ◆♦❜r❡ ❈♦✲♦r✐❡♥t❛❞♦r❛

✶✶✶ ❳✶✶✶①

❙♦✉③❛✱ P❛trí❝✐❛

❖ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ P❡sq✉✐s❛ ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ♥♦ ■❈▼❈✲❯❙P✿ ❈♦♥tr✐❜✉✐çã♦ ❞❡ ◆❡❧s♦♥ ❖♥✉❝❤✐❝✳✴ P❛trí❝✐❛ ❞❡ ❙♦✉③❛✲ ❘✐♦ ❈❧❛r♦✿ ❬s✳♥✳❪✱ ✷✵✶✶✳

✹✹ ❢✳✿ ✜❣✳✱ t❛❜✳

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

❖r✐❡♥t❛❞♦r✿ ❙ér❣✐♦ ❘♦❜❡rt♦ ◆♦❜r❡ ❈♦✲♦r✐❡♥t❛❞♦r❛✿ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐

✶✳ ❆♥á❧✐s❡✳ ✷✳ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ✸✳ ❊q✉❛çã♦ ❉✐❢❡r❡♥❝✐❛❧✳ ✹✳ ▼❛t❡♠át✐❝❛✳ ■✳ ❚ít✉❧♦

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

P❛trí❝✐❛ ❞❡ ❙♦✉③❛

❖ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ P❡sq✉✐s❛ ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s

♥♦ ■❈▼❈✲❯❙P✿ ❈♦♥tr✐❜✉✐çã♦ ❞❡ ◆❡❧s♦♥ ❖♥✉❝❤✐❝✳

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

Pr♦❢✳ ❉r✳ ❙ér❣✐♦ ❘♦❜❡rt♦ ◆♦❜r❡ ❖r✐❡♥t❛❞♦r

Pr♦❢❛✳ ❉r❛✳ ▼❛r✐❛ ❞♦ ❈❛r♠♦ ❈❛r❜✐♥❛tt♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ■❈▼❈ ✲ ❯❙P

Pr♦❢❛✳ ❉r❛✳ ❊❧✐r✐s ❈r✐st✐♥❛ ❘✐③③✐♦❧❧✐

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ■●❈❊ ✲ ❯♥❡s♣

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛ ❡ ♠❡✉s ❛♠✐❣♦s q✉❡ ♠❡ ❛♣♦✐❛r❛♠ ❞✉r❛♥t❡ t♦❞♦ ♦ ♣❡r❝✉rs♦ ❞♦ ♠❡str❛❞♦✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ♣r♦❢❡ss♦r ❙ér❣✐♦ ❘♦❜❡rt♦ ◆♦❜r❡✳

➚ ♠✐♥❤❛ ❝♦✲♦r✐❡♥t❛❞♦r❛ ♣r♦❢❡ss♦r❛ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐✱ q✉❡ ❡♠ ♠✉✐t♦s ♠♦♠❡♥t♦s ♠❡ ✐♥❝❡♥t✐✈♦✉ ❡ ♦r✐❡♥t♦✉✳

❆♦s ♣r♦❢❡ss♦r❡s q✉❡ ♠❡ ❛❥✉❞❛r❛♠ t❛♥t♦✱ ❡♠ ♣❛rt✐❝✉❧❛r à ♣r♦❢❡ss♦r❛ ▼❛r✐❛ ❞♦ ❈❛r♠♦ ❈❛r❜✐♥❛tt♦✳

❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ tr❛❜❛❧❤♦ é ❛♣r❡s❡♥t❛r ✉♠ ✐♠♣♦rt❛♥t❡ ❡ ❝♦♥❤❡❝✐❞♦ t❡♦r❡♠❛ s♦❜r❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s✱ ❞❡✈✐❞♦ ❛♦ Pr♦❢❡ss♦r ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❡ Pr♦❢❡ss♦r P❤✐❧✐♣ ❍❛rt♠❛♥ ♣✉❜❧✐❝❛❞♦ ❡♠ ✶✾✻✸✳ P❛r❛ ✐st♦ é ♥❡❝❡ssár✐♦ ❞❡s❝r❡✈❡r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡ ❝♦♥❝❡✐t♦s ❜ás✐✲ ❝♦s ❞❡ ❆♥á❧✐s❡ ❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r✳ ❆❧é♠ ❞✐ss♦✱ ❛♥❛❧✐s❛r ❝♦♠♦ ❡ss❡ r❡s✉❧t❛❞♦ ❝♦♥tr✐❜✉✐✉ ❡♠ tr❛❜❛❧❤♦s ♣♦st❡r✐♦r❡s ❞❡ ❞✐✈❡rs♦s ♣❡sq✉✐s❛❞♦r❡s ❡ ♦r✐❡♥t❛♥❞♦s ❞♦ Pr♦❢❡ss♦r ◆❡❧s♦♥ ♥❛s ❞é❝❛❞❛s q✉❡ s❡ s❡❣✉✐r❛♠ ❛té ♦s t❡♠♣♦s ❛t✉❛✐s✱ ❞❡st❛❝❛♥❞♦ ❛ s✉❛ ✐♥✢✉ê♥❝✐❛ ❡ ❝♦❧❛❜♦✲ r❛çã♦ ❝♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛s ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ♥❛ r❡❣✐ã♦ ❞❡ ❙ã♦ ❈❛r❧♦s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❛ ✉t✐❧✐③❛çã♦ ❞♦ r❡s✉❧t❛❞♦ ♣♦r ❞♦✐s ❞❡ s❡✉s ♦r✐❡♥t❛♥❞♦s✱ ❍✐❧❞❡❜r❛♥❞♦ ▼✳ ❘♦❞r✐❣✉❡s ❡ ❆❞❛❧❜❡rt♦ ❙♣❡③❛♠✐❣❧✐♦✳ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❢♦✐ ✉♠ ✐♠♣♦r✲ t❛♥t❡ ♣❡sq✉✐s❛❞♦r ♥❛ ár❡❛ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s✳ ❊♠ ✶✾✻✼✱ ❝♦♠❡ç♦✉ ❛ tr❛❜❛❧❤❛r ♥❛ ❯❙P ❝❛♠♣✉s ❙ã♦ ❈❛r❧♦s✱ ♦♥❞❡ ♦r✐❡♥t♦✉ ✈ár✐♦s ❡st✉❞❛♥t❡s ❞❡ ♣ós✲❣r❛❞✉❛çã♦ ❡ t❡✈❡ ❞❡♥s❛ ♣✉❜❧✐❝❛çã♦✳ ❊❧❡ ❢❛❧❡❝❡✉ ❡♠ ✸ ❞❡ s❡t❡♠❜r♦ ❞❡ ✶✾✾✾✳

❆❜str❛❝t

❚❤❡ ❛✐♠ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ ♣r❡s❡♥t ❛♥ ✐♠♣♦rt❛♥t ❛♥❞ ✇❡❧❧ ❦♥♦✇♥ t❤❡♦r❡♠ ♦♥ ❉✐✛❡r✲ ❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❞❡✈❡❧♦♣❡❞ ❜② Pr♦❢❡ss♦r ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❛♥❞ Pr♦❢❡ss♦r P❤✐❧✐♣ ❍❛rt♠❛♥ ❛♥❞ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✾✻✸✳ ❋♦r t❤✐s r❡❛s♦♥ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❞❡s❝r✐❜❡ s♦♠❡ r❡s✉❧ts ❛♥❞ ❝♦♥✲ ❝❡♣ts ♦❢ ❆♥❛❧②s✐s ❛♥❞ ▲✐♥❡❛r ❆❧❣❡❜r❛✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❛✐♠ ❛t ❡①❛♠✐♥✐♥❣ ❤♦✇ t❤❡s❡ r❡s✉❧ts ❝♦♥tr✐❜✉t❡❞ t♦ ❧❛t❡r ♣❛♣❡rs ♦❢ s❡✈❡r❛❧ r❡s❡❛r❝❤❡rs ❛♥❞ st✉❞❡♥ts ❛❞✈✐s❡❞ ❜② Pr♦✲ ❢❡ss♦r ◆❡❧s♦♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✉♥t✐❧ t♦❞❛②✱ ❤✐❣❤❧✐❣❤t✐♥❣ ❤✐s ✐♥✢✉❡♥❝❡ ❛♥❞ ❝♦❧❧❛❜♦r❛t✐♦♥ ♦♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✐♥ t❤❡ ❙ã♦ ❈❛r❧♦s r❡❣✐♦♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ✇✐❧❧ ♣r❡s❡♥t t❤❡ ✉s❡ ♦❢ t❤❡s❡ r❡s✉❧ts ❜② t✇♦ ♦❢ ❤✐s ❛❞✈✐s❡❡s✱ ❍✐❧❞❡❜r❛♥❞♦ ▼✳ ❘♦❞r✐❣✉❡s ❛♥❞ ❆❞❛❧❜❡rt♦ ❙♣❡③❛♠✐❣❧✐♦✳ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ✇❛s ❛♥ ✐♠♣♦rt❛♥t r❡s❡❛r❝❤❡r ✐♥ t❤❡ ✜❡❧❞ ♦❢ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ■♥ ✶✾✻✼✱ ❤❡ st❛rt❡❞ ✇♦r❦ ✇❛s ❛ ❢❛❝✉❧t② ♠❡♠❜❡rs ♦❢ ❯❙P✱ ❙ã♦ ❈❛r❧♦s ❝❛♠♣✉s✱ ✇❤❡r❡ ❤❡ s✉♣❡r✈✐s❡❞ s❡✈❡r❛❧ ❣r❛❞✉❛t❡ st✉❞❡♥ts ❛♥❞ ✇r♦t❡ ❛ ❞❡♥s❡ ❧✐st ♦❢ ♣✉❜❧✐❝❛t✐♦♥s✳ ❍❡ ❞✐❡❞ ♦♥ ❙❡♣t❡♠❜❡r ✸✱ ✶✾✾✾✳

❙✉♠ár✐♦

✶ ■♥tr♦❞✉çã♦ ✼

✷ ❍✐stór✐❝♦ ✾

✸ Pr❡❧✐♠✐♥❛r❡s ✶✷

✸✳✶ ❘❡s✉❧t❛❞♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✸✳✷ ❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✹ ❖ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲❖♥✉❝❤✐❝ ✷✾

✹✳✶ ❖ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲❖♥✉❝❤✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✷ ❯t✐❧✐③❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✸ ❈✐t❛çõ❡s ❞❡ ❛❧❣✉♥s tr❛❜❛❧❤♦s ❞❡ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✺ P✉❜❧✐❝❛çõ❡s ❞❡ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ✸✽

✻ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✹✷

✶ ■♥tr♦❞✉çã♦

❖s tr❛❜❛❧❤♦s ❞♦ ♣r♦❢❡ss♦r ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❢♦r❛♠ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ▼❛t❡♠át✐❝❛ ♥♦ ❇r❛s✐❧✱ ❡♠ ❡s♣❡❝✐❛❧✱ ◆❡❧s♦♥ t❡✈❡ ❣r❛♥❞❡ ❝♦♥tr✐❜✉✐çã♦ ♣❛r❛ ❛s ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s✱ ár❡❛ ♥❛ q✉❛❧ ❡❧❡ ❢❡③ ❡s❝♦❧❛✳ ❆ ❡①♣❡❝t❛t✐✈❛ ❝♦♠ ❡st❡ tr❛❜❛❧❤♦ é ❛ ❝♦♠♣r♦✲ ✈❛çã♦ ❞❡st❡ ❢❛t♦✱ ♠♦str❛r ❡❢❡t✐✈❛♠❡♥t❡ ❝♦♠♦ s❡ ❞❡✉ ♦ ✐♥í❝✐♦ ❞❡ss❛ ❧✐♥❤❛ ❞❡ ♣❡sq✉✐s❛ ✐♥tr♦❞✉③✐❞❛ ♣♦r ❖♥✉❝❤✐❝ ❡♠ ❙ã♦ ❈❛r❧♦s ❡ ❛ ❝♦♥t✐♥✉❛çã♦ ❞❛ ♠❡s♠❛ ♣♦r s❡✉s ❛❧✉♥♦s ❡ ❛❧✉♥♦s ❞♦s s❡✉s ❛❧✉♥♦s✳ P❛r❛ ✐ss♦ ✈❛♠♦s ❛♣r❡s❡♥t❛r ♦ ✉s♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲ ❖♥✉❝❤✐❝ ❡♠ ❞♦✐s tr❛❜❛❧❤♦s ❞❡ ❉♦✉t♦r❛❞♦✱ ✉♠ ❞❡ ❍✐❧❞❡❜r❛♥❞♦ ▼✉♥❤♦③ ❘♦❞r✐❣✉❡s✶ _{❡ ♦}

♦✉tr♦ ❞❡ ❆❞❛❧❜❡rt♦ ❙♣❡③❛♠✐❣❧✐♦✷_{✱ ❛♠❜♦s ♦r✐❡♥t❛♥❞♦s ❞❡ ◆❡❧s♦♥✳}

❖ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❢♦❝❛ ♦ ❊♥s✐♥♦ ❙✉♣❡r✐♦r✱ ♣♦r ✐ss♦ ❛ ♣r❡♦❝✉♣❛çã♦ ❡♠ ❝♦❧♦❝❛r ❛s ❞❡✜♥✐çõ❡s ❡♥✈♦❧✈✐❞❛s ❞✉r❛♥t❡ ♦ tr❛❜❛❧❤♦ ❡ ❛❧❣✉♠❛s ❜✐❜❧✐♦❣r❛✜❛s✱ ♣♦✐s ❛ss✐♠✱ ✉♠ ❛❧✉♥♦ ❞❡ ❣r❛❞✉❛çã♦ t❡♠ ❝♦♥❞✐çõ❡s ❞❡ ❛❝♦♠♣❛♥❤❛r ❛ ❧❡✐t✉r❛ ❞♦ tr❛❜❛❧❤♦✳

◆♦ ■❈▼❈✲❯❙P✱ ◆❡❧s♦♥ ❖♥✉❝❤✐❝✱ ❢♦✐ ✉♠ ❞♦s ♣r❡❝✉rs♦r❡s ♥♦ tr❛❜❛❧❤♦ ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❡ ♣♦ss✐❜✐❧✐t♦✉ ❛ ❢♦r♠❛çã♦ ❞❡ ❣r✉♣♦s ❞❡ ❡st✉❞♦ ❡ ♣❡sq✉✐s❛ ♥❡st❛ ár❡❛✱ ♦s q✉❛✐s ❛té ❤♦❥❡ ❡stã♦ ❡♠ ❛t✐✈✐❞❛❞❡ ♥❡st❡ ✐♥st✐t✉t♦✳ ❆tr❛✈és ❞❡ s✉❛s ♦r✐❡♥t❛çõ❡s ❡st✐♠✉❧♦✉ s❡✉s ❛❧✉♥♦s ❛ ❝♦♥t✐♥✉❛r ♦s tr❛❜❛❧❤♦s ❡ ❢♦✐ ✉♠ ❣r❛♥❞❡ ❡①❡♠♣❧♦ ♣❛r❛ ❡❧❡s✳

❙❡❣✉♥❞♦ ❇❛❞✐♥ ❬✶❪✱ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ♦r✐❡♥t♦✉ ❝❛t♦r③❡ tr❛❜❛❧❤♦s ❞❡ ▼❡str❛❞♦ ❡ ♥♦✈❡ tr❛❜❛❧❤♦s ❞❡ ❉♦✉t♦r❛❞♦✱ ❡ s❡✉s ♦r✐❡♥t❛♥❞♦s✱ s❡❣✉✐r❛♠ ❡st✉❞❛♥❞♦ ❛s ❊q✉❛çõ❡s ❉✐❢❡✲ r❡♥❝✐❛✐s✱ ♦r✐❡♥t❛♥❞♦ ♦✉tr♦s ❛❧✉♥♦s ❡ ❛ss✐♠ ✐♥✐❝✐♦✉✲s❡ ❛ ❢♦r♠❛çã♦ ❞❡ ❣r✉♣♦ ❞❡ ♣❡sq✉✐s❛ ✐♥✢✉❡♥❝✐❛❞♦ ♣♦r ❡❧❡✳

P❛r❛ ❛❧❝❛♥ç❛r♠♦s ♥♦ss❛ ❡①♣❡❝t❛t✐✈❛✱ ❢♦❝❛♠♦s ❡♠ ✉♠ ❞♦s r❡s✉❧t❛❞♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♣♦r ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❡♠ ♣❛r❝❡r✐❛ ❝♦♠ P❤✐❧✐♣ ❍❛rt♠❛♥✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲❖♥✉❝❤✐❝✱ ♦ q✉❛❧ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❛rt✐❣♦✱ ❖♥ t❤❡ ❆s②♠♣t♦t✐❝ ■♥t❡❣r❛t✐♦♥ ♦❢ ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭P❛❝✐✜❝ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✈♦❧✳ ✶✸✱ ✶✾✻✸✮✳ ❱❡r❡♠♦s ❞✉❛s ❞❡ s✉❛s ❛♣❧✐❝❛çõ❡s ❡ t❛♠❜é♠ ❛❧❣✉♠❛s ❝✐t❛çõ❡s ❞♦ ♠❡s♠♦✳

❖ tr❛❜❛❧❤♦ s❡rá ❡str✉t✉r❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ t❡♠♦s ❛ ■♥tr♦❞✉çã♦ ❞♦ tr❛❜❛❧❤♦✱ ♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡ é ❞❡♥♦♠✐♥❛❞♦ ❍✐stór✐❝♦✱ ♥♦ q✉❛❧ ❡♥❝♦♥tr❛♠♦s ❛❧❣✉♠❛s ✐♥❢♦r♠❛çõ❡s

✶_{Pr♦❢❡ss♦r ❚✐t✉❧❛r ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ▼❛t❡♠át✐❝❛s ❡ ❈♦♠♣✉t❛çã♦✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡}

▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ❈❛♠♣✉s ❙ã♦ ❈❛r❧♦s✳

✷_{Pr♦❢❡ss♦r ▲✐✈r❡✲❉♦❝❡♥t❡ ❞♦ ■♥st✐t✉t♦ ❞❡ ❇✐♦❝✐ê♥❝✐❛s✱ ▲❡tr❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡}

▼❛t❡♠át✐❝❛ ❞❛ ❯♥❡s♣ ❈❛♠♣✉s ❙ã♦ ❏♦sé ❞♦ ❘✐♦ Pr❡t♦✳

✽

❞❛ ✈✐❞❛ ❞❡ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❡ ❛❧❣✉♥s ❞❛❞♦s ❞❡ ❡♥tr❡✈✐st❛ ❢❡✐t❛ ❝♦♠ ♦s ♣r♦❢❡ss♦r❡s P❧á❝✐❞♦ ❩♦❡❣❛ ❚á❜♦❛s✸ _{❡ ❍✐❧❞❡❜r❛♥❞♦ ▼✉♥❤♦③ ❘♦❞r✐❣✉❡s✱ ❛♠❜♦s ❢♦r❛♠ ♦r✐❡♥t❛❞♦s ♣♦r ◆❡❧s♦♥❀}

❛ s❡❣✉✐r ♦ ❝❛♣ít✉❧♦ ❞❡ Pr❡❧✐♠✐♥❛r❡s q✉❡ tr❛③ ❞❡✜♥✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲❖♥✉❝❤✐❝ ❡ ❞❡✜♥✐çõ❡s ✐♠♣❧í❝✐t❛s ❛ ❡❧❡❀ ♣♦st❡r✐♦r♠❡♥t❡ t❡♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ♦ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲❖♥✉❝❤✐❝✱ ❞♦✐s ❡①❡♠♣❧♦s ❞❡ s✉❛ ✉t✐❧✐③❛çã♦✱ ❝♦♠♦ ❥á ❝✐t❛❞♦ ❛❝✐♠❛✱ ♦s ❡①❡♠♣❧♦s ❡stã♦ ♥♦s tr❛❜❛❧❤♦s ❞❡ ❞♦✉t♦r❛❞♦ ❞♦s ♣r♦❢❡ss♦r❡s ❍✐❧❞❡✲ ❜r❛♥❞♦ ▼✳ ❘♦❞r✐❣✉❡s ❡ ❆❞❛❧❜❡rt♦ ❙♣❡③❛♠✐❣❧✐♦ ❡ ❛❧❣✉♠❛s ❝✐t❛çõ❡s ❞♦ ♠❡s♠♦ ❡♠ ❛❧❣✉♥s ❛rt✐❣♦s ❝✐❡♥tí✜❝♦s❀ ✉♠ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠❛ ❧✐st❛ ❞❛s P✉❜❧✐❝❛çõ❡s ❞♦ ♣r♦❢❡ss♦r ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❡ ♣♦r ✜♠ ❛s ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ❡ t❛♠❜é♠ ❛s ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s✳

✸_{Pr♦❢❡ss♦r ❚✐t✉❧❛r ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ▼❛t❡♠át✐❝❛s ❡ ❈♦♠♣✉t❛çã♦✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡}

✷ ❍✐stór✐❝♦

❆s ✐♥❢♦r♠❛çõ❡s ❝♦♥t✐❞❛s ♥❡ss❡ ❝❛♣ít✉❧♦ ❢♦r❛♠ t♦♠❛❞❛s ❝♦♠ ❜❛s❡ ❡♠ ❇❛❞✐♥ ❬✶❪✳ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ♥❛s❝❡✉ ♥♦ ❞✐❛ ✶✶ ❞❡ ♠❛rç♦ ❞❡ ✶✾✷✻✱ ♥❛ ❝✐❞❛❞❡ ❞❡ ❇r♦❞ósq✉✐ ♥♦ ✐♥t❡r✐♦r ❞♦ ❡st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✱ ❇r❛s✐❧✳ ❋✐❧❤♦ ❞❡ ❋r❛♥❝✐s❝♦ ❖♥✉s✐❝ ❡ ▼❛r✐❛ ❉♦❧❡s✳

❊♠ ✶✾✼✶ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❝♦♠❡ç♦✉ ❛ ❛♣r❡s❡♥t❛r ♦s ♣r✐♠❡✐r♦s s✐♥❛✐s ❞❡ ♠❛❧ ❞❡ P❛r❦✐♥✲ s♦♥ ❡ ✈❡✐♦ ❛ ❢❛❧❡❝❡r ❡♠ ✸ ❞❡ s❡t❡♠❜r♦ ❞❡ ✶✾✾✾✳

❖ tr❛❜❛❧❤♦ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ♥❛ ár❡❛ ❞❡ ♣❡sq✉✐s❛ ❡♠ ▼❛t❡♠át✐❝❛ ❢♦✐ ♠✉✐t♦ ❛♠♣❧♦ ❡ s✐❣♥✐✜❝❛t✐✈♦✳ ❚❡✈❡ ✈ár✐♦s tr❛❜❛❧❤♦s ♣✉❜❧✐❝❛❞♦s ♥♦ ❇r❛s✐❧✱ ❊st❛✲ ❞♦s ❯♥✐❞♦s✱ P♦rt✉❣❛❧✱ ▼é①✐❝♦✱ ■tá❧✐❛✱ ❍♦❧❛♥❞❛✱ ❊s❝ó❝✐❛ ❡ ❏❛♣ã♦✳ ◆♦ ❝❛♣ít✉❧♦ ❝✐♥❝♦ ❡♥❝♦♥tr❛✲s❡ ✉♠❛ ❧✐st❛ ❝♦♠ ❛s ♣✉❜❧✐❝❛çõ❡s ❞❡ ◆❡❧s♦♥ ❖♥✉❝❤✐❝✳

▲✐❝❡♥❝✐♦✉✲s❡ ❡♠ ❋ís✐❝❛ ♣❡❧❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❋✐❧♦s♦✜❛✱ ❈✐ê♥❝✐❛s ❡ ▲❡tr❛s ❞♦ ■♥st✐t✉t♦ ▼❛❝❦❡♥③✐❡✱ ❙ã♦ P❛✉❧♦✳

❚r❛❜❛❧❤♦✉ ❝♦♠♦ Pr♦❢❡ss♦r ❆ss✐st❡♥t❡ ♥♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞♦ ■♥st✐t✉t♦ ❚❡❝♥♦❧ó❣✐❝♦ ❞❛ ❆❡r♦♥á✉t✐❝❛ ✭■❚❆✮ ♥♦ ♣❡rí♦❞♦ ❞❡ ✶✾✺✻ ❛ ✶✾✺✽✳

❊♥tr❡ ✶✾✺✺ ❡ ✶✾✺✻ ❡st✉❞♦✉ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧✱ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧ ❡ ❊str✉t✉r❛s ❯♥✐✲ ❢♦r♠❡s s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞♦ ♣r♦❢❡ss♦r ❈❤❛✐♠ ❙❛♠✉❡❧ ❍ö♥✐❣✳ ❉✉r❛♥t❡ ❡ss❡ ♣❡rí♦❞♦ ♣r♦❞✉③✐✉ s✉❛ t❡s❡ ❞❡ ❞♦✉t♦r❛❞♦ ✐♥t✐t✉❧❛❞❛ ❊str✉t✉r❛s ❯♥✐❢♦r♠❡s s♦❜r❡ P✲❊s♣❛ç♦s ❡ ❆♣❧✐❝❛çõ❡s ❞❛ ❚❡♦r✐❛ ❞❡st❡s ❊s♣❛ç♦s ❡♠ ❚♦♣♦❧♦❣✐❛ ●❡r❛❧ ❛ q✉❛❧ ❛♣r❡s❡♥t♦✉ ♥❛ ❋✳❋✳❈✳▲✳ ❞❛ ❯❙P ❡♠ ✶✾✺✼✳

◆❡❧s♦♥ ❖♥✉❝❤✐❝ ♣❛rt✐❝✐♣♦✉ ❞♦ ✶♦✳ _{❈♦❧óq✉✐♦ ❇r❛s✐❧❡✐r♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ❊s❝r❡✈❡✉ ♦}

❝❛♣ít✉❧♦ s♦❜r❡ ❊s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ❡ ❞❡ ❍✐❧❜❡rt ❞❛ ♣✉❜❧✐❝❛çã♦ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧✱ ❞♦ ♠❡s♠♦ ❝♦❧óq✉✐♦✳ ❊st❛ ♣✉❜❧✐❝❛çã♦ ❢♦✐ ✉t✐❧✐③❛❞❛ ❝♦♠♦ t❡①t♦ ♣❛r❛ ♦ ❝✉rs♦ ♠✐♥✐str❛❞♦ ♣♦r ◆❡❧s♦♥ ♥❡st❡ ♠❡s♠♦ ❝♦❧óq✉✐♦✳

❊♠ ✶✾✺✽✱ ❝♦♥✈✐❞❛❞♦ ♣❡❧♦ ♣r♦❢❡ss♦r ❏♦ã♦ ❉✐❛s ❞❛ ❙✐❧✈❡✐r❛ ♣❛r❛ ❝r✐❛r ♦ ❈✉rs♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❋❛❝✉❧❞❛❞❡ ❞❡ ❋✐❧♦s♦✜❛✱ ❈✐ê♥❝✐❛s ❡ ▲❡tr❛s ❞❡ ❘✐♦ ❈❧❛r♦✱ ❤♦❥❡ ❯♥❡s♣ ❝❛♠♣✉s ❘✐♦ ❈❧❛r♦✱ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❛❝❡✐t♦✉ ♦ ❝♦♥✈✐t❡ ❡ ❢♦✐ r❡❣❡♥t❡ ❞❛ ❝❛❞❡✐r❛ ❞❡ ❆♥á❧✐s❡ ▼❛t❡♠át✐❝❛ ❡ ♣❡r♠❛♥❡❝❡✉ ♥❡st❛ ♣♦s✐çã♦ ❛té ♦ ❛♥♦ ❞❡ ✶✾✻✻✳

❊♠ ✶✾✺✾✱ ◆❡❧s♦♥ ❢♦✐ ♣r♦❢❡ss♦r ✈✐s✐t❛♥t❡ ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ▼♦♥t❡✈✐❞é♦✱ ❯r✉❣✉❛✐✱ ♦♥❞❡ ♠✐♥✐str♦✉ ✉♠ ❝✉rs♦ s♦❜r❡ ♦ ▼ét♦❞♦ ❚♦♣♦❧ó❣✐❝♦ ❞❡ ❲❛③❡✇s❦✐ ❡ ❝♦♥❤❡❝❡✉ ♦s ♠❛t❡♠át✐❝♦s ❏✉❛♥ ❏♦r❣❡ ❙❝❤ä✛❡r ❡ ❏♦sé ▲✉ís ▼❛ss❡r❛✳

❆ ♣❛rt✐r ❞♦ ❝♦♥t❛t♦ ❝♦♠ ▼❛ss❡r❛✱ ❖♥✉❝❤✐❝ t❡✈❡ ✐♥t❡r❡ss❡ ❡♠ ❡st✉❞❛r ❊q✉❛çõ❡s ❉✐❢❡✲

✶✵

r❡♥❝✐❛✐s ❡✱ ♦ ▼ét♦❞♦ ❚♦♣♦❧ó❣✐❝♦ ❞❡ ❲❛③❡✇s❦✐ ❢♦✐ ✉s❛❞♦ ♣❛r❛ r❡s♦❧✈❡r ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ♣r♦♣♦st♦s ♣♦r ▼❛ss❡r❛✳

❊♠ ✶✾✻✶✱ ♥♦ ✸♦✳ _{❈♦❧óq✉✐♦ ❇r❛s✐❧❡✐r♦ ❞❡ ▼❛t❡♠át✐❝❛✱ ❡♠ ❋♦rt❛❧❡③❛✱ ♠✐♥✐str♦✉ ♦ ❝✉rs♦}

❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s✿ ❊st❛❜✐❧✐❞❛❞❡ ❡♠ ❙✐st❡♠❛s ▲✐♥❡❛r❡s✱ ❊q✉✐✈❛❧ê♥❝✐❛ ❆ss✐♥tót✐❝❛ ❡ ▼ét♦❞♦ ❚♦♣♦❧ó❣✐❝♦ ❞❡ ❲❛③❡✇s❦✐✳

❖✉tr♦ ❢❛t♦ ♠✉✐t♦ r❡❧❡✈❛♥t❡ q✉❡ ❢❡③ ❝♦♠ q✉❡ ❖♥✉❝❤✐❝ t✐✈❡ss❡ ✐♥t❡r❡ss❡ ♥❛s ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s✱ ❢♦✐ s✉❛ ✐❞❛ ♣❛r❛ ♦s ❊st❛❞♦s ❯♥✐❞♦s✳ ❉✉r❛♥t❡ ✉♠ ❛♥♦ ◆❡❧s♦♥ ✜❝♦✉ ♥♦ ✧❘❡s❡❛r❝❤ ■♥st✐t✉t❡ ❢♦r ❆❞✈❛♥❝❡❞ ❙t✉❞✐❡s✧ ✭❘■❆❙✮✱ ❇❛❧t✐♠♦r❡✱ ❯❙❆✱ ♦♥❞❡ ❛♣r♦❢✉♥❞♦✉ s❡✉s ❡st✉❞♦s ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s✳ ▲á t❡✈❡ ❣r❛♥❞❡ ✐♥✢✉ê♥❝✐❛ ❞❡ ❏❛❝❦ ❑✳ ❍❛❧❡✱ ♠❛t❡♠át✐❝♦ ❛♠❡r✐❝❛♥♦ ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ q✉❡ t❛♠❜é♠ ❢❛③✐❛ ♣❛rt❡ ❞♦ ❣r✉♣♦✳

❙❡❣✉♥❞♦ ♦ Pr♦❢❡ss♦r P❧á❝✐❞♦✱ ❍❛❧❡ ✐♥✢✉❡♥❝✐♦✉ ♠✉✐t♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛s ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ♥♦ ❇r❛s✐❧✳ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ tr♦✉①❡ ❏❛❦❡ ❍❛❧❡ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣❛r❛ ❛ ❯❙P ❙ã♦ ❈❛r❧♦s ♣❛r❛ q✉❡ ❡❧❡ ❝♦♥❤❡❝❡ss❡ ♦ ❣r✉♣♦ ❞❡ ♣❡sq✉✐s❛ q✉❡ s❡ ❢♦r♠♦✉ ♣♦r ✈♦❧t❛ ❞❡ ✶✾✼✵✳ ❖ ♣r♦❢❡ss♦r P❧á❝✐❞♦ ❞✐③ ❛✐♥❞❛ q✉❡ t❛❧ ❣r✉♣♦ ❝♦♥t❛✈❛✱ ❡♥tr❡ ♦✉tr♦s✱ ❝♦♠ ♦s ♣r♦❢❡ss♦r❡s ❍✐❧❞❡❜r❛♥❞♦ ▼✳ ❘♦❞r✐❣✉❡s✱ ❆♥tô♥✐♦ ❋❡r♥❛♥❞❡s ■③é✱ ❍❡r♠í♥✐♦ ❈❛ss❛❣♦ ❏ú♥✐♦r✳

◆❡❧s♦♥ ❖♥✉❝❤✐❝ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ P❤✐❧✐♣ ❍❛rt♠❛♥ ❡s❝r❡✈❡✉ ♦ ❛rt✐❣♦ ✐♥t✐t✉❧❛❞♦ ❖♥ t❤❡ ❆s②♠♣t♦t✐❝ ■♥t❡❣r❛t✐♦♥ ♦❢ ❖r❞✐♥❛r② ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✭P❛❝✐✜❝ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✈♦❧✳ ✶✸✱ ✶✾✻✸✮✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ♦ r❡s✉❧t❛❞♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❢♦✐ ♦ ❚❡♦✲ r❡♠❛ ❞❡ ❍❛rt♠❛♥✲❖♥✉❝❤✐❝✱ ♦ q✉❛❧ é ❝♦♥s✐❞❡r❛❞♦ ✉♠❛ ❞❡ s✉❛s ❣r❛♥❞❡s ❝♦♥tr✐❜✉✐çõ❡s à ▼❛t❡♠át✐❝❛ ❡ q✉❡ tr❛t❛♠♦s ♥❡st❡ tr❛❜❛❧❤♦✳

❆❧é♠ ❞❛s ❛t✐✈✐❞❛❞❡s ❞❡ ♣❡sq✉✐s❛ ❖♥✉❝❤✐❝ ❝♦♥t✐♥✉♦✉ ♠✐♥✐str❛♥❞♦ ❝✉rs♦s ❡ ❛❜r✐♥❞♦ ✉♠❛ ❧✐♥❤❛ ❞❡ ♣❡sq✉✐s❛ ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ♥♦ ❇r❛s✐❧✳ ◆♦ ■♥st✐t✉t♦ ❞❡ P❡sq✉✐s❛s ▼❛t❡♠át✐❝❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ♠✐♥✐str♦✉ ❝✉rs♦s ❞❡ ♣ós✲❣r❛❞✉❛çã♦ ❝♦♠♦✱ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ❡♠ ✶✾✻✸✱ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧ ❡ ❆♣❧✐❝❛çõ❡s ♥❛ ❚❡♦r✐❛ ❞❛s ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ❡♠ ✶✾✻✸ ❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s ❡ ❋✉♥❝✐♦♥❛✐s ❡♠ ✶✾✻✹✳

❈♦♠ ❛ t❡s❡ ✐♥t✐t✉❧❛❞❛ ❈♦♠♣♦rt❛♠❡♥t♦ ❆ss✐♥tót✐❝♦ ❞❛s ❙♦❧✉çõ❡s ❞❡ ✉♠ ❙✐st❡♠❛ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s✱ ◆❡❧s♦♥ ❢♦✐ ❛♣r♦✈❛❞♦ ♥♦ ❝♦♥❝✉rs♦ ♣❛r❛ ❛ ▲✐✈r❡✲ ❉♦❝ê♥❝✐❛ ❥✉♥t♦ ❛ ❈❛❞❡✐r❛ ❞❡ ❈á❧❝✉❧♦ ■♥✜♥✐t❡s✐♠❛❧ ❞❛ ❋✳❋✳❈✳▲✳ ❞❛ ❯❙P ♥♦ ❛♥♦ ❞❡ ✶✾✻✺✳ ❆ ♠✉❞❛♥ç❛ ❞❡ ◆❡❧s♦♥ ♣❛r❛ ❙ã♦ ❈❛r❧♦s ❢♦✐ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠ ❢♦rt❡ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ♥❛ ❊❊❙❈ ❡✱ ♠❛✐s ❛✐♥❞❛✱ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞❡ ✉♠ ❈✉rs♦ ❞❡ ▼❛t❡♠át✐❝❛ ♥❛ ❯❙P ❙ã♦ ❈❛r❧♦s✳

❊♠ ✶✾✻✽✱ ❝♦♠ ❛ t❡s❡ ✐♥t✐t✉❧❛❞❛ ❊st❛❜✐❧✐❞❛❞❡ ❞❡ ❙✐st❡♠❛s P❡rt✉r❜❛❞♦s ❡ ❈♦♠♣♦rt❛✲ ♠❡♥t♦ ♥♦ ■♥✜♥✐t♦ ❞❡ ❙✐st❡♠❛s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❝♦♠ ❘❡t❛r❞❛♠❡♥t♦ ♥♦ ❚❡♠♣♦✱ ◆❡❧s♦♥ ❛♣r❡s❡♥t♦✉ ♦ tr❛❜❛❧❤♦ ♣❛r❛ ♦ ❝♦♥❝✉rs♦ ❞❡ Pr♦❢❡ss♦r ❚✐t✉❧❛r ❞❛ ❊s❝♦❧❛ ❞❡ ❊♥✲ ❣❡♥❤❛r✐❛ ❞❡ ❙ã♦ ❈❛r❧♦s✲❯❙P✳

✶✶

❊♠ ✶✾✼✷✱ ◆❡❧s♦♥ ❖♥✉❝❤✐❝ t♦r♥♦✉✲s❡ ♣r♦❢❡ss♦r t✐t✉❧❛r ❞❡ ▼❛t❡♠át✐❝❛✱ ♥♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ▼❛t❡♠át✐❝❛s ❯❙P ✲ ❙ã♦ ❈❛r❧♦s✱ ♦ q✉❛❧ ❤♦❥❡ é ♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ▼❛t❡♠át✐❝❛s ❡ ❞❡ ❈♦♠♣✉t❛çã♦✳

❙✉❛ ♦❜r❛ ❣❡r♦✉ ♦ ❣r✉♣♦ ❞❡ ♣❡sq✉✐s❛ ❡♠ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❞❡ ❙ã♦ ❈❛r❧♦s✳ ❚❛❧ ❣r✉♣♦ t❡♠ ❝♦♥t✐♥✉✐❞❛❞❡ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡✳

❊♠ ❝♦♥✈❡rs❛ ❝♦♠ ♦ ♣r♦❢❡ss♦r ❍✐❧❞❡❜r❛♥❞♦✱ ♦ ♠❡s♠♦ ❞✐③ q✉❡✱ ❛❧é♠ ❞❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ◆❡❧s♦♥ ♣❛r❛ s✉❛ ❢♦r♠❛çã♦ ♠❛t❡♠át✐❝❛✱ ◆❡❧s♦♥ ❢♦✐ ✐♠♣♦rt❛♥t❡ t❛♠❜é♠ ♣❛r❛ s✉❛ ❢♦r♠❛çã♦ ♣❡ss♦❛❧ ❡ ♣r♦✜ss✐♦♥❛❧✳ ❆♣r❡♥❞❡✉ ❝♦♠ ❡❧❡ ❛ ❢❛③❡r r❡❧❛tór✐♦s ❡ ❛rt✐❣♦s ❝✐❡♥tí✜❝♦s✱ ♣❧❛♥♦s ❞❡ tr❛❜❛❧❤♦✱ ❛t✐t✉❞❡ ❝✐❡♥tí✜❝❛ ❡ s♦❜r❡ ét✐❝❛✳

❈♦♥✈❡rs❛♥❞♦ ❝♦♠ ♦ ♣r♦❢❡ss♦r P❧á❝✐❞♦✱ ❡❧❡ ❝✐t❛ t❛♠❜é♠✱ ❛ s❡r✐❡❞❛❞❡ ❞❡ ◆❡❧s♦♥ q✉❛♥t♦ ❝✐❡♥t✐st❛ ❡ ❝♦♠♦ ❡r❛ ót✐♠♦ ♣r♦❢❡ss♦r✱ q✉❡r✐❞♦ ♣♦r t♦❞♦s✳

❚❛♥t♦ P❧á❝✐❞♦ q✉❛♥t♦ ❍✐❧❞❡❜r❛♥❞♦ r❡ss❛❧t❛r❛♠ ♠✉✐t♦ q✉❡ ❖♥✉❝❤✐❝ ❡r❛ ✉♠❛ ♣❡ss♦❛ ♠✉✐t♦ ❛❝❡ssí✈❡❧✱ ❞❡ ❣r❛♥❞❡ s✐♠♣❛t✐❛✱ q✉❡ ❡st✐♠✉❧❛✈❛ ♦s ❥♦✈❡♥s ❡ ❡st❛✈❛ s❡♠♣r❡ ♣r♦♥t♦ ♣❛r❛ ❛❥✉❞❛r ❡ ❛❝♦❧❤❡r✳

✸ Pr❡❧✐♠✐♥❛r❡s

❊st❡ ❝❛♣ít✉❧♦ é ❝♦♠♣♦st♦ ♥ã♦ s♦♠❡♥t❡ ❞❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❡♥✈♦❧✈❡♥❞♦ ❞✐r❡✲ t❛♠❡♥t❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲❖♥✉❝❤✐❝✱ ♠❛s t❛♠❜é♠ ❝♦♥té♠ ❞❡✜♥✐çõ❡s ✐♠♣❧í❝✐t❛s ❛ ❡ss❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ q✉❡ ♦ ❧❡✐t♦r t❡♥❤❛ ❡♠ ♠ã♦s ❛s ✐♥❢♦r✲ ♠❛çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦ ❚❡♦r❡♠❛✳

P♦r ❡①❡♠♣❧♦✱ ❛s ❞❡✜♥✐çõ❡s ❞❡ ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ❡ ❞❡ ✉♠ ♣❛r ❛❞♠✐ssí✈❡❧ ♣❛r❛ ✉♠❛ ♠❛tr✐③✱ sã♦ ❞✐r❡t❛♠❡♥t❡ ✉t✐❧✐③❛❞❛s ❞❡♥tr♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❍❛rt♠❛♥✲❖♥✉❝❤✐❝✳ ❏á ❛s ❞❡✜♥✐çõ❡s ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❡ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛✐s ♥ã♦ ❛♣❛r❡❝❡♠ ❞✐r❡t❛♠❡♥t❡✱ ♠❛s s✐♠ ❞❡♥tr♦ ❞❡ ♦✉tr❛s ❞❡✜♥✐çõ❡s✳ P♦ré♠ r❡❢♦rç❛♠♦s q✉❡ é ✉♠ ♦❜❥❡t✐✈♦ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦ q✉❡ ♦ ❧❡✐t♦r t❡♥❤❛ ❝♦♥❞✐çõ❡s ❞❡ ❡♥t❡♥❞❡r ♦s t❡r♠♦s ❛♣r❡s❡♥t❛❞♦s ♥❡st❡ t❡①t♦ ❡ ♣♦rt❛♥t♦ ❛♣r❡s❡♥t❛♠♦s ❞❡✜♥✐çõ❡s ♠❛✐s ❜ás✐❝❛s ❡ ♦✉tr❛s ♠❛✐s ❝♦♠♣❧❡①❛s q✉❡ ❛♣❛r❡❝❡♠ ♥♦ ❞❡❝♦rr❡r ❞♦ tr❛❜❛❧❤♦✳

❆s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❡stã♦ ❛♣r❡s❡♥t❛❞♦s ❡♠ ❞✉❛s s❡çõ❡s✱ ✉♠❛ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ❡ ♦✉tr❛ ❝♦♠ ♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r✳ P❛r❛ ♠❛✐s ♣❡sq✉✐s❛s ♣♦❞❡✲s❡ ❝♦♥s✉❧t❛r ❬✷✱ ✸✱ ✹❪✳

❉❡♥tr♦ ❞❛ s❡çã♦ ❞❡ ❆♥á❧✐s❡ t❡♠♦s ❞❡✜♥✐çõ❡s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛s✱ ❊s♣❛ç♦s ▼étr✐❝♦s ❡ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛✳ P❛r❛ ♣❡sq✉✐s❛s ♥❡ss❛s ár❡❛s ♣♦❞❡ ❝♦♥s✉❧t❛r✲s❡ r❡❢❡rê♥❝✐❛s ❝♦♠♦✿ ❬✺✱ ✻✱ ✼✱ ✽❪✳

✸✳✶ ❘❡s✉❧t❛❞♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r

❉❡✜♥✐çã♦ ✸✳✶✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❝♦♥❥✉♥t♦ V 6= ∅✱ ♠✉♥✐❞♦ ❞❡ ✉♠❛ ❛❞✐çã♦ ❡ ❞❡ ✉♠❛

♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠ ❡s❝❛❧❛r✱ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ s❡ sã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

✶✳ u+v =v+u ♣❛r❛ q✉❛✐sq✉❡r u, v ∈V❀

✷✳ u+ (v+w) = (u+v) +w ♣❛r❛ q✉❛✐sq✉❡r u, v, w ∈V❀

✸✳ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ 0∈V t❛❧ q✉❡ 0 +u=u ♣❛r❛ t♦❞♦ u∈V❀ ✹✳ ♣❛r❛ ❝❛❞❛ u∈V ❡①✐st❡v ∈V t❛❧ q✉❡ u+v = 0❀

✺✳ λ(µu) = (λµ)u ♣❛r❛ q✉❛✐sq✉❡r u∈V ❡ λ, µ∈R❀

❘❡s✉❧t❛❞♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ✶✸

✻✳ (λ+µ)u=λu+µu ♣❛r❛ q✉❛✐sq✉❡r u∈V✱ ❡ λ✱ µ∈R❀

✼✳ λ(u+v) =λu+λv ♣❛r❛ q✉❛✐sq✉❡r u, v ∈V ❡ λ∈R❀

✽✳ 1u=u ♣❛r❛ q✉❛❧q✉❡r u∈V✳

❊①❡♠♣❧♦ ✸✳✶✳ ❙❡❥❛♠ A ⊂ R ❡ _ℑ(A;R₎ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s_{f : A −→}R✳

❙❡ f, g ∈ ℑ(A;R₎ ❡ _{λ ∈} R ❞❡✜♥❛ _{f +g : A −→} R ♣♦r _{(f +g)(x) = f(x) + g(x)} ❡

(λf)(x) =λf(x)✱ x ∈A✳ ❊♥tã♦✱ ℑ(A;R₎ ❝♦♠ ❡st❛ ❛❞✐çã♦ ❡ ♣r♦❞✉t♦ ♣♦r ❡s❝❛❧❛r é ✉♠

❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧✳

❉❡✜♥✐çã♦ ✸✳✷✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧✳ ❉✐③❡♠♦s q✉❡ W ⊂V é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ ❞❡ V s❡ ❢♦r❡♠ s❛t✐s❢❡✐t❛s ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s

✶✳ 0∈W❀

✷✳ ❙❡ u, v ∈W ❡♥tã♦ u+v ∈W❀

✸✳ ❙❡ u∈W ❡♥tã♦ λu∈W ♣❛r❛ t♦❞♦ λ∈R✳

❊①❡♠♣❧♦ ✸✳✷✳ ❊♠ R2✱ ❛s r❡t❛s ♣❛ss❛♥❞♦ ♣❡❧❛ ♦r✐❣❡♠ sã♦ s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡R2✳

❉❡✜♥✐çã♦ ✸✳✸✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ✈❡t♦r❡su1, . . . , un ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧

r❡❛❧ V é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ ✭❧✳✐✳✮ s❡ ❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r α1u1+. . .+αnun = 0

só ❢♦r s❛t✐s❢❡✐t❛ q✉❛♥❞♦ α1 =. . .=αn= 0✳

❉❡✜♥✐çã♦ ✸✳✹✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ V é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ s❡ ❡①✐st✐r ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ S ⊂V t❛❧ q✉❡ V = [S]✱ ♦♥❞❡ [S] ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s

❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ❞♦s ❡❧❡♠❡♥t♦s ❞❡ S✱ ♦✉ s❡❥❛✱ u∈[S] s❡ ❡①✐st✐r❡♠ α1, . . . , αn∈R

❡ u1, . . . , un∈S t❛✐s q✉❡ u=α1u1+. . .+αnun✳

❉❡✜♥✐çã♦ ✸✳✺✳ ❉✐♠❡♥sã♦ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ é ♦ ♥ú♠❡r♦ ❞❡ ✈❡t♦r❡s ❞❡ ✉♠❛ ❜❛s❡ q✉❛❧q✉❡r✳ ❇❛s❡ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ ✭❧✳✐✳✮ q✉❡ ❣❡r❛♠ t♦❞♦ ♦ ❡s♣❛ç♦✳

❖❜s❡r✈❛çã♦✿ ❙❡ ♥ã♦ ❡①✐st✐r ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ✈❡t♦r❡s q✉❡ ❣❡r❛♠ ♦ ❡s♣❛ç♦ ❞✐③❡✲ ♠♦s q✉❡ ♦ ❡s♣❛ç♦ t❡♠ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✳

❊①❡♠♣❧♦ ✸✳✸✳ Rn é ✉♠ ❡s♣❛ç♦ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❉❡ ❢❛t♦ ♣♦ss✉✐ ✉♠ ❝♦♥❥✉♥t♦ ❧✐♥❡✲

❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ ❝♦♠ n ✈❡t♦r❡s q✉❡ ❣❡r❛ t♦❞♦ Rn_{✳ P♦rt❛♥t♦✱ dimR}n _{= n✳ P♦r}

♦✉tr♦ ❧❛❞♦✱ C([a, b],R₎♥ã♦ t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛✱ ♣♦✐s ♥ã♦ é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳

❉❡✜♥✐çã♦ ✸✳✻✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧✳ ❯♠ ♣r♦❞✉t♦ ✐♥t❡r♥♦ s♦❜r❡ V é ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ ❛ ❝❛❞❛ ♣❛r (u, v) ∈ V ×V ❛ss♦❝✐❛ ✉♠ ♥ú♠❡r♦ r❡❛❧ ❞❡♥♦t❛❞♦ ♣♦r hu, vi✱

s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

❘❡s✉❧t❛❞♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ✶✹

✷✳ hαu, vi=αhu, vi,∀u, v ∈V ❡ α∈R❀

✸✳ hu, vi=hv, ui,∀u, v ∈V❀ ✹✳ hu, ui>0 s❡ u6= 0,∀u∈V✳

❉❡✜♥✐çã♦ ✸✳✼✳ ❯♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ V ♠✉♥✐❞♦ ❞❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✳

❊①❡♠♣❧♦ ✸✳✹✳ ❙❡ f, g ∈C([a, b];R₎ ❞❡✜♥✐♠♦s✱

hf, gi=

Z b

a

f(x)g(x)dx

♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡ C([a, b];R₎ é ✉♠ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✳

❊①❡♠♣❧♦ ✸✳✺✳ ❈♦♠ r❡❧❛çã♦ ❛♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✱ ❝❛❧✲ ❝✉❧❡♠♦s ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♥tr❡ sen, cos∈C([0,2π];R₎✱

hsen, cosi=

Z 2π

0

senxcosxdx=

sen2_x

2

2π

0

= 0.

❊①❡♠♣❧♦ ✸✳✻✳ ❙❡ x= (x1, . . . , xn), y = (y1, . . . , yn)∈Rn✱ ❞❡✜♥✐♠♦s

hx, yi=x1y1+. . .+xnyn

♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡ Rn é ✉♠ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✳

❊①❡♠♣❧♦ ✸✳✼✳ ❈♦♠ r❡❧❛çã♦ ❛♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✱ ❝❛❧✲ ❝✉❧❡♠♦s ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡♥tr❡ ♦s ✈❡t♦r❡s (1,−1,1),(0,2,4)∈R3✳

h(1,−1,1),(0,2,4)i= 1.0 + (−1).2 + 1.4 = 2.

❉❡✜♥✐çã♦ ✸✳✽✳ ❯♠❛ ♥♦r♠❛ ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ X é ✉♠❛ ❢✉♥çã♦ ❞❡ X ❡♠ R

❞❡✜♥✐❞❛ ♣♦r

x7−→ kxk

❛ q✉❛❧ s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s✿ ✶✳ kxk ≥0❀

✷✳ kxk= 0⇔x= 0❀

✸✳ kαxk=|α| kxk❀

❘❡s✉❧t❛❞♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ✶✺

❉❡✜♥✐çã♦ ✸✳✾✳ ❯♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ X é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ ❝♦♠ ✉♠❛ ♥♦r♠❛ ❞❡✜♥✐❞❛ ♥❡❧❡✳

◆♦t❛çã♦ (X,k.k)

Pr♦♣♦s✐çã♦ ✸✳✶✳ ❙❡❥❛ V ✉♠ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✳ P❛r❛ ❝❛❞❛ u∈V✱ ❞❡✜♥❛

||u||=phu, ui. ❆ ❢✉♥çã♦ u∈V 7→ ||u|| é ✉♠❛ ♥♦r♠❛ ❡♠V✳

❊①❡♠♣❧♦ ✸✳✽✳ ❊♠ Rn_{✱ ❝♦♠ ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❛❞♦ ♥♦ ❡①❡♠♣❧♦ ✸✳✻✱ ❛ ♥♦r♠❛ ❞❡}

x= (x1, . . . , xn)é ❞❛❞❛ ♣♦r

kxk=

q x2

1+. . .+x2n.

❉❡✜♥✐çã♦ ✸✳✶✵✳ ❙❡❥❛♠ U ❡ V ❡s♣❛ç♦s ✈❡t♦r✐❛✐s r❡❛✐s q✉❛✐sq✉❡r✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ T : U −→ V é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r s❡ ❢♦r❡♠ ✈❡r✐✜❝❛❞❛s ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

✶✳ T(u+v) = T(u) +T(v)✱ ∀u, v ∈U❀ ✷✳ T(λu) =λT(u)✱ ∀u∈U✱ ∀λ∈R✳

❊①❡♠♣❧♦ ✸✳✾✳ ❙❡❥❛ T :C([0,1];R_)→R ❢✉♥çã♦ ❞❛❞❛ ♣♦r

T(f) =

Z 1

0

f(x)dx,∀f ∈C([0,1];R₎

T é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ f, g∈C([0,1];R₎✳

Pr♦✈❡♠♦s q✉❡T(f+g) =T(f) +T(g)✳

❉❡ ❢❛t♦✱

T(f+g) =

Z 1

0

(f+g)(x)dx=

Z 1

0

(f(x)+g(x))dx=

Z 1

0

f(x)dx +

Z 1

0

g(x)dx=T(f)+T(g).

Pr♦✈❡♠♦s ❛❣♦r❛ q✉❡ T(λf) = λT(f)✳

❉❡ ❢❛t♦✱

T(λf) =

Z 1

0

λf(x)dx =λ Z 1

0

f(x)dx=λT(f), λ∈R_.

P♦rt❛♥t♦T é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳

❘❡s✉❧t❛❞♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ✶✻

❉❡✜♥✐çã♦ ✸✳✶✷✳ ❉✐③❡♠♦s q✉❡ ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s r❡❛✐sU ❡ V sã♦ ✐s♦♠♦r❢♦s s❡ ❡①✐st✐r ✉♠ ✐s♦♠♦r✜s♠♦ T :U −→V✳

❊①❡♠♣❧♦ ✸✳✶✵✳ T : U −→ U ❞❛❞❛ ♣♦r T(u) = u é ✉♠ ✐s♦♠♦r✜s♠♦ ❡✱ ♣♦rt❛♥t♦✱ ♦ ❡s♣❛ç♦ U é ✐s♦♠♦r❢♦ ❛ ❡❧❡ ♠❡s♠♦✳

❊①❡♠♣❧♦ ✸✳✶✶✳ T :Rn_−→P

n−1(R)❞❛❞❛ ♣♦rT(x1, . . . , xn) =x1+x2t+. . .+xntn−1

é ✉♠ ✐s♦♠♦r✜s♠♦ ❡✱ ♣♦rt❛♥t♦✱ ♦s ❡s♣❛ç♦s Rn ❡ _P

n−1(R) sã♦ ✐s♦♠♦r❢♦s✳

❉❡✜♥✐çã♦ ✸✳✶✸✳ ❙❡❥❛♠ U ❡ W s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ V r❡❛✐s✳ ❉✐③❡♠♦s q✉❡ U+W é ❛ s♦♠❛ ❞✐r❡t❛ ❞❡ U ❡ W s❡ U ∩W ={0}✳

◆♦t❛çã♦ U⊕W✳

❊①❡♠♣❧♦ ✸✳✶✷✳ R3 é s♦♠❛ ❞✐r❡t❛ ❞❡✱

U =

(x, y, z)∈R3_{;x+y+z = 0} ❡ _{W =(x, y, z)∈}R3_{;x=y= 0 .}

❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡ W é ❞❡ ❢❛t♦ ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ ❞❡ R3 ♣♦✐s

W =

(x, y, z)∈R3_{;x= 0 ∩(x, y, z)∈}R3_{;y= 0}

♦✉✱ ❛❧t❡r♥❛t✐✈❛♠❡♥t❡✱ s❡ u1 = (x1, y1, z1), u2 = (x2, y2, z2)∈W ❡♥tã♦✱

x1 =y1 =x2 =y2 = 0

❡ u1+u2 = (0,0, z1+z2)é ❝❧❛r❛♠❡♥t❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ W✳

❙❡λ∈R ❡♥tã♦✱

λu1 =λ(0,0, z1) = (λ0, λ0, λz1) = (0,0, λz1)∈W.

❋✐♥❛❧♠❡♥t❡✱(0,0,0)∈W✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ♣r♦✈❛ ❞❡ q✉❡W é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ r❡❛❧ ❞❡ R3✳

Pr♦✈❡♠♦s ❛❣♦r❛ q✉❡ R3 _=U+W✳

❉❡ ❢❛t♦✱ ❞❛❞♦(x, y, z)∈R3 ♣♦❞❡♠♦s ❡s❝r❡✈❡r

(x, y, z) = (x, y,−x−y) + (0,0, z+x+y)

❡ ❝♦♠♦ (x, y,−x−y)∈U ❡ (0,0, z+x+y)∈W ♦❜t❡♠♦sR3 _{=U +W}✳

P♦r ✜♠✱ ♠♦str❡♠♦s q✉❡U ∩W ={0}✳

❙❡❥❛(x, y, z)∈U∩W✳ ❚❡♠♦s✱ 

 

 

x+y+z = 0

x= 0

y= 0

⇐⇒(x, y, z) = (0,0,0)

P♦rt❛♥t♦✱ R3 _=U⊕W✳

❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ✶✼

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ V =U ⊕W✱ ✐st♦ é✱ V =U +W ❡U ∩W ={0}✳

❊♥tã♦✱ ❞❛❞♦ v ∈ V ❡①✐st❡♠ u ∈ U ❡ w ∈ W s❛t✐s❢❛③❡♥❞♦ v = u+w✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡ t❛❧ ❞❡❝♦♠♣♦s✐çã♦ é ú♥✐❝❛✳

❙✉♣♦♥❤❛ q✉❡ ❡①✐st❛♠u′ _{∈U ❡w}′ _{∈W t❛✐s q✉❡ v =u}′_+w′_{✳ ❊♥tã♦✱u+w=u}′_+w′_✱

♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ u−u′ _=w′_−w✳

▼❛su−u′ _{∈U ❡ w}′_{−w∈W ❡✱ ♣♦rt❛♥t♦✱ u−u}′ _=w′_{−w∈U∩W ={0}✱ ♦✉ s❡❥❛}

u=u′ _{❡ w=w}′_✳

❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ ♣❛r❛ ❝❛❞❛ v ∈ V ❡①✐st❛♠ ✉♠ ú♥✐❝♦ u∈ U ❡ ✉♠ ú♥✐❝♦ w ∈W s❛t✐s❢❛③❡♥❞♦ v =u+w✳ ➱ ❝❧❛r♦ q✉❡ V =U+W✳ ❘❡st❛ ♠♦str❛r q✉❡ U ∩W ={0}✳

❖❜✈✐❛♠❡♥t❡✱0∈U ∩W✳

❙❡❥❛ v ∈ U ∩W✱ ✐st♦ é✱ v ∈ U ⊂ V ❡ v ∈ W ⊂ V✳ ❊♥tã♦✱ ❡①✐st❡♠ ú♥✐❝♦ u∈ U ❡ ✉♠ ú♥✐❝♦ w∈W s❛t✐s❢❛③❡♥❞♦ v =u+w✳

❖❜s❡r✈❡ q✉❡ v = u+w = (u+v) + (w−v) ❝♦♠ u+v ∈ U ❡ w−v ∈ W ❡✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞❛ ❞❡❝♦♠♣♦s✐çã♦✱ ❞❡✈❡♠♦s t❡r u=u+v ❡ w=w−v✱ ✐st♦ é✱v = 0✳

▲♦❣♦✱ U∩W ={0}✳

✸✳✷ ❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡

❊st❛ s❡çã♦ ❝♦♥té♠ ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❡♥✈♦❧✈❡♥❞♦ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s✱ ❆♥á❧✐s❡ ❋✉♥❝✐♦♥❛❧✱ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ ❡ r❡s✉❧t❛❞♦s ❝❧áss✐❝♦s ❞❡ ❆♥á❧✐s❡✳

❉❡✜♥✐çã♦ ✸✳✶✹✳ ❯♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é ✉♠ s✐st❡♠❛ ❞❛ ❢♦r♠❛✿            y′

1 =f1(t, y1, y2, . . . , yn)

y′

2 =f2(t, y1, y2, . . . , yn)

✳✳✳ y′

n=fn(t, y1, y2, . . . , yn)

✭✸✳✶✮

♦♥❞❡✱f1, f2, . . . , fnsã♦n❢✉♥çõ❡s ❞❡✜♥✐❞❛s ❡♠ ✉♠ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦D✱ n+1✲❞✐♠❡♥s✐♦♥❛❧✱

❡ (y1, y2, . . . , yn) sã♦ ❛s n ❢✉♥çõ❡s ✐♥❝ó❣♥✐t❛s✳

❘❡s♦❧✈❡r ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥✲ tr❛r ✉♠ ✐♥t❡r✈❛❧♦ I ♥♦ ❡✐①♦t ❡n❢✉♥çõ❡s r❡❛✐sφ1(t), φ2(t), . . . , φn(t)❞❡✜♥✐❞❛s ❡♠I t❛✐s

q✉❡✿ ✶✳ φ′

1(t), φ′2(t), . . . , φ′n(t) ❡①✐st❡♠ ♣❛r❛ ❝❛❞❛ t ❡♠ I;

✷✳ (t, φ1(t), φ2(t), . . . , φn(t))∈D ♣❛r❛ ❝❛❞❛ t∈I;

✸✳ φ′

j(t) = fj(t, φ1(t), φ2(t), . . . , φn(t)) ♣❛r❛ ❝❛❞❛ t∈I✱ j = 1,2, . . . , n

❆ s♦❧✉çã♦ ❞❡ ✭✸✳✶✮ ♣♦❞❡ s❡r ✈✐s✉❛❧✐③❛❞❛ ❝♦♠♦ ❛ ❝✉r✈❛ ♥❛ r❡❣✐ã♦D✱ ❝♦♠ ❝❛❞❛ ♣♦♥t♦ p ♥❛ ❝✉r✈❛ ❞❛❞♦ ♣❡❧❛s ❝♦♦r❞❡♥❛❞❛s(t, φ1(t), . . . , φn(t)) ❡ ❝♦♠φ′(t)❝♦♠♦ s❡♥❞♦ ❛ ❝♦♠✲

❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ✶✽

❯♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ♥ã♦ ❤♦♠♦❣ê♥❡♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❞❛ s❡❣✉✐♥t❡

❢♦r♠❛✿ _           y′

1 =a11(t)y1 +a12(t)y2+· · ·+a1n(t)yn+g1(t)

y′

2 =a21(t)y1 +a22(t)y2+· · ·+a2n(t)yn+g2(t)

✳✳✳ y′

n =an1(t)y1+an2(t)y2+· · ·+ann(t)yn+gn(t)

✭✸✳✷✮

♦✉ ❛✐♥❞❛✱

y′ =A(t)y+g(t) ✭✸✳✸✮

♦♥❞❡ A(t) =



 

a11(t) a12(t) . . . a1n(t)

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ an1(t) an2(t) . . . ann(t)





 ❡ g(t) = 

 

g1(t)

✳✳✳ gn(t)



 

◆♦ ❝❛s♦ ❤♦♠♦❣ê♥❡♦✱y′ _{=A(t)y✱ ✐st♦ é✱ g(t) = 0✳}

❊ ❢á❝✐❧ ✈❡r q✉❡ t♦❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ♦r❞❡♠n ≥2 ♣♦❞❡ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛

❞❡ s✐st❡♠❛✱ ❝♦♠♦ ❡♠ ✭✸✳✷✮✳

❊①❡♠♣❧♦ ✸✳✶✸✳ ❙❡❥❛ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠x′′_=a

1·x′+a2·y+g(t)✱

q✉❛♥❞♦ s❡ t♦♠❛ y1 =x ❡y2 =x′✱ ♦❜té♠✲s❡ ♦ s✐st❡♠❛✿

( y′

1 =x′

y′

2 =x′′

∼

( y′

1 =y2

y′

2 =x′′= a1·y2+a2·y1+g(t)

P❛r❛n >2 ♦ ♣r♦❝❡❞✐♠❡♥t♦ é ❛♥á❧♦❣♦✳ ❈♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ♦r❞❡♠ n ❞❛❞❛ ♣♦r✿

x(n)_+a

n−1·x(n−1)+. . .+a1 ·x′ +a0·x=b(t).

❚♦♠❡ y1 =x✱ y2 =x′✱ ✳ ✳ ✳ ✱ yn=x(n−1) ♣❛r❛ ♦❜t❡r ♦ s✐st❡♠❛✿

               y′

1 = y2

y′

2 = y3

✳✳✳ y′

n−1 = xn−1

y′

n = x(n)

∼                y′

1 =x′ =y2

y′

2 =x′′ =y3

✳✳✳ y′

n−1 =x(n−1) =yn

❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ✶✾

◗✉❡ ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ♠❛tr✐❝✐❛❧ ❝♦♠♦✿

      y′ 1 y′ 2 ✳✳✳ y′ n       =      

0 1 0 . . . 0

0 0 1 . . . 0

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳

−a0 −a1 −a2 . . . −an−1

      ·       y1 y2 ✳✳✳ yn       +       0 0 ✳✳✳ b(t)

      ❖✉ ❛✐♥❞❛✱ ˙

Y =A(t)·Y +g(t) ✭✸✳✹✮

❉❡✜♥✐çã♦ ✸✳✶✺✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛♦ s✐st❡♠❛ ✭✸✳✶✮ ❝♦♥s✐st❡ ♥❛ ❡q✉❛çã♦ ✭✸✳✶✮ ❡ ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧ η∈Rn _{❞❛❞❛ ♥♦ ✐♥st❛♥t❡ ✐♥✐❝✐❛❧ t}

0✳

◆♦t❛çã♦✿ y(t0) = η.

❘❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ P0 = (t0, η1, η2, . . . , ηn)❞❡ D✳

❆ s❡❣✉✐r ❡♥✉♥❝✐❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ♣❛r❛ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❆♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ✐❧✉str❛r ❝♦♠♦ ❛s té❝♥✐❝❛s ❞❡ ❆♥á❧✐s❡ sã♦ ✉t✐❧✐③❛❞❛s ♣❛r❛ ♠♦str❛r r❡s✉❧t❛✲ ❞♦s ♥❛ t❡♦r✐❛ ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s✳

❚❡♦r❡♠❛ ✸✳✶✳ ❙✉♣♦♥❤❛ A(t) ❡ g(t) ❝♦♠♦ ❡♠ ✭✸✳✸✮ ❢✉♥çõ❡s ❘✐❡♠❛♥♥ ✐♥t❡❣rá✈❡✐s ❞❡

t ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ (a, b)✱ ✐st♦ é✱ ❛s ✐♥t❡❣r❛✐s ❞❡ ❘✐❡♠❛♥♥ ❡①✐st❡♠ ❡♠ ❛❧❣✉♠ ✐♥t❡r✈❛❧♦ [c, d] ⊂ (a, b) ♣❛r❛ ❝❛❞❛ aij✱ i, j = 1, . . . , n ❡♠ A(t) ❡ ❝❛❞❛ bj✱ j = 1, . . . , n ❡♠ b(t)✱ ❡

s✉♣♦♥❤❛ t❛♠❜é♠ q✉❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ r❡❛❧ k(t) ❝♦♠ ❞♦♠í♥✐♦ (a, b)✱ t❛❧ q✉❡✿

✶✳ k(t) é ❝♦♥tí♥✉❛ ❡ ❧✐♠✐t❛❞❛ ❡♠ (a, b)❀

✷✳ |A(t)| ≤k(t) ❡ |b(t)| ≤k(t)

❙❡❥❛ t0 ∈ (a, b) ❡ s✉♣♦♥❤❛ q✉❡ y0 é ✉♠ ✈❡t♦r ❞❛❞♦✳ ❊♥tã♦ ❛ ❡q✉❛çã♦ ✭✸✳✹✮ t❡♠

✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ y(t) ❡♠ (a, b) t❛❧ q✉❡ y(t0) =y0 ❡ y s❛t✐s❢❛③✱ ♣❛r❛ t♦❞♦ t∈(a, b)✱ ❛

✐❣✉❛❧❞❛❞❡✿

y(t) =y0+

Z t

t0

A(s)·y(s)ds+

Z t

t0

b(s)ds ✭✸✳✺✮

❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ✷✵

P❛r❛t ∈(a, b) ❞❡✜♥❛✿

y0(t) = y0

y1(t) = y0+

Rt

t0A(s)·y0(s)ds+

Rt

t0b(s)ds

✳✳✳

yn(t) = y0+

Rt

t0A(s)·yn−1(s)ds+

Rt

t0b(s)ds

yn+1(t) =y0+

Rt

t0A(s)·yn(s)ds+

Rt

t0b(s)ds

❝♦♠ n = 1,2, . . . .

yn(t)é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠(a, b)❥á q✉❡ é ❞❡✜♥✐❞❛ ♣❡❧❛ s♦♠❛ ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s✱

❡♥tã♦ A(s)·yn(s)é ✐♥t❡❣rá✈❡❧ ❡♠ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦[c, d]⊂(a, b)❡ ❞❛í✱yn+1 é ❝♦♥tí♥✉❛

❡♠ (a, b)✳

❈♦♥s✐❞❡r❡ t∈(a, b)❀

|y1(t)−y0(t)| ≤

Rt

t0|A(s)| · |y0(s)|+|b(s)|ds≤

≤ Rt

t0(k(s)|y0(s)|+k(s))ds

≤(1 +|y0_|)·Rt

t0|k(s)|ds

❙❡❥❛K(t) =Rt

t0|k(s)|ds✱ ❡♥tã♦

|y1(t)−y0(t)| ≤(1 +|y0|)·K(t)

❚♦♠❛♥❞♦n = 2 t❡♠✲s❡✿

|y2(t)−y1(t)| ≤

Rt

t0|A(s)| · |y−1(s)−y0(s)|ds≤

≤

Rt

t0|A(s)|(1 +|y

0_|)·K(s)ds ≤

≤(1 +|y0_|)·Rt

t0k(s)·K(s)ds

❖❜s❡r✈❡ q✉❡ ✐♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s ♦❜té♠✲s❡

Rt

t0k(s)·K(s)ds= (K(s))

2₋Rt

t0k(s)·K(s)ds⇒

⇒2·Rt

t0k(s)·K(s)ds= (K(t))

2 _⇒

⇒Rt

t0k(s)·K(s)ds=

(K(t))2

2!

❊♥tã♦✱ s❡❣✉❡ q✉❡✿

|y2(t)−y1(t)| ≤(1 +|y0|)·

(K(t))2

❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ✷✶

❯s❛♥❞♦ ✐♥❞✉çã♦ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ é ✈á❧✐❞❛ ♣❛r❛ ✉♠ m q✉❛❧q✉❡r✱ ❜❛st❛ s✉♣♦r q✉❡ ✈❛❧❡ ♣❛r❛ m >0✱ ✐st♦ é✿

|ym(t)−ym−1(t)| ≤(1 +|y0|)·

(K(t))m

m!

❡ ♠♦str❛r q✉❡ ✈❛❧❡ ♣❛r❛ m+ 1✿

|ym+1−ym| ≤

Rt

t0|A(s)| · |ym(s)−ym−1(s)|ds ≤

≤Rt

t0k(s)·(1 +|y

0_|)· (K(s))m

m! ds ≤

≤ (1+_m|y_!0|)Rt

t0k(s)·(K(s))

m_ds

■♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s ♦❜té♠✲s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

|ym+1−ym| ≤

(1 +|y0_|)

m! ·

(K(t))m+1

m+ 1 ≤(1 +|y

0_|)· (K(t))m+1

(m+ 1)!

❇❛st❛ ♠♦str❛r q✉❡ ❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡s(ym(t)) é ❝♦♥✈❡r❣❡♥t❡✿

❉❡ ❢❛t♦✱

|ym(t)−y0(t)|=|(y1(t)−y0(t)) + (y2(t)−y1(t)) +· · ·+ (ym(t)−ym−1(t))| ≤

≤ |(y1(t)−y0(t))|+|(y2(t)−y1(t))|+· · ·+|(ym(t)−ym−1(t))|=

m

X

j=0

|yj+1(t)−yj(t)|

❆ sér✐❡ Pm

j=0|yj+1(t)−yj(t)| é ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ t♦❞♦t ∈[c, d] ♣♦✐s

|yj+1−yj| ≤(1+|y0|)·((k(t))

j+1

j+1 )q✉❡ ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠[c, d]⊂(a, b)♣❛r❛ ✉♠❛

❢✉♥çã♦ ❝♦♥tí♥✉❛y(t)✱ ✐st♦ é✱ ♣❛r❛ t♦❞♦ε >0❡①✐st❡n0 ∈Nt❛❧ q✉❡|yn(t)−y(t)|< _M·(ε_b−a)

♣❛r❛ t♦❞♦ t∈[c, d]✳

❆❣♦r❛ é ♣r❡❝✐s♦ ♠♦str❛r q✉❡

limn→∞

Z t

t0

A(s)·yn(s)ds =

Z t

t0

A(s)·y(s)ds ✭✸✳✻✮ P❛r❛ ✐ss♦✱ ♦❜s❡r✈❡ q✉❡ y(s) é ❝♦♥tí♥✉❛ ♣♦rt❛♥t♦✱ Rt

t0A(s)·y(s)ds ❡①✐st❡ ♣❛r❛ t♦❞♦

t❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❢❡❝❤❛❞♦[c, d]⊂(a, b).❊♥tã♦✱ ❞❛❞♦ ✉♠ε >0✱ t♦♠❛♥❞♦n≤n0t❡♠✲s❡

Z t t0

A(s)·(yn(s)−y(s))ds

≤ Z t t0

❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ✷✷

♦♥❞❡ M é ✉♠ ❧✐♠✐t❛♥t❡ ♣❛r❛k(t) ❡♠ (a, b)✱ s❡❣✉❡ ❡♥tã♦ ✭✸✳✻✮✳

❚❡♦r❡♠❛ ✸✳✷✳ ❙❡ ♣❛r❛ i, j = 1, . . . , n ❝❛❞❛ aij(t) ❡ bj(t) sã♦ ❝♦♥tí♥✉❛s ♣❛r❛ t♦❞♦ t ∈

R✱ ❡♥tã♦✱ s❡ _{(t0, y0)} é ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ ♥♦ ❡s♣❛ç♦ _{(t, y)} ❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦

y(t, t0, y0) ❞❛ ❡q✉❛çã♦ ✭✸✳✶✮ t❛❧ q✉❡ y(t0, t0, y0) = y0 ❡ ❛ s♦❧✉çã♦ y(t, t0, y0) ❡①✐st❡ ♣❛r❛

❡st❡ ❞♦♠í♥✐♦ ♥♦ ❡✐①♦ r❡❛❧ t✳

❉❡♠♦♥str❛çã♦✳ ❆ ♣r♦✈❛ ❞❡st❡ t❡♦r❡♠❛ s❡❣✉❡ ❞❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✶✳ ❖❜s❡r✈❡ q✉❡ s❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ A(t) ❡ b(t) sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♥tã♦✱ ❥á q✉❡ ❛ s♦❧✉çã♦ y(t) é

❝♦♥tí♥✉❛✱ ❛ ❡q✉❛çã♦ ✭✸✳✺✮ ♣♦❞❡ s❡r ❞❡r✐✈❛❞❛ ❝♦♠ r❡s♣❡✐t♦ ❛ t✱ ❡ s❡❣✉❡ q✉❡✿

d

dty(t) = d dt

y0+

Rt

t0A(s)·y(s)ds+

Rt

t0b(s)ds

⇒

⇒y′_{(t) =A(t)·y(t) +b(t).}

s✉❜st✐t✉✐♥❞♦ t0 ♥❛ ❡q✉❛çã♦ ✭✸✳✺✮ ♦❜té♠✲s❡✿

y(t0) =y0+

Z t0

t0

A(s)·y(s)ds+

Z t0

t0

b(s)ds=y0.

▲♦❣♦✱ y(t0, t0, y0) =y0✳ P♦rt❛♥t♦✱ y(t) é s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✸✳✶✮✳

❉❡✜♥✐çã♦ ✸✳✶✻✳ ❯♠ s✐st❡♠❛ ❛✉tô♥♦♠♦ é ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❡♠ q✉❡ ❛ ❢✉♥çã♦ f1✱ f2✱ . . .✱ fn ♥ã♦ ❞❡♣❡♥❞❡♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❡ t✳ ❘❡♣r❡s❡♥t❛♠♦s t❛❧ s✐st❡♠❛

❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿

           y′

1 =f1(y1, y2, . . . , yn)

y′

2 =f2(y1, y2, . . . , yn)

✳✳✳ y′

n=fn(y1, y2, . . . , yn)

✭✸✳✼✮

♦♥❞❡ fi :D⊂Rn−→R✱ i= 1, . . . , n sã♦ ❢✉♥çõ❡s ❞❡ ❝❧❛ss❡ C1✳

❊①❡♠♣❧♦ ✸✳✶✹✳ ❆ ❡q✉❛çã♦ ❡s❝❛❧❛r ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ y” +y = 0♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦

♦ s✐st❡♠❛✿

˙

x1 =x2

˙

x2 =−x1

.

❉❡ ❢❛t♦✱ q✉❛♥❞♦ s❡ t♦♠❛ x1 =y ❡ x2 =y′ t❡♠✲s❡ q✉❡

˙

x1 =x2

˙

x2 =−x1

❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ✷✸

❈♦♠♦ f ❡ s✉❛s ❢✉♥çõ❡s ❞❡r✐✈❛❞❛s sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠ t♦❞♦ ♦ s❡✉ ❞♦♠í♥✐♦✱ ♣❡❧♦s ❚❡♦r❡♠❛s ✸✳✶ ❡ ✸✳✷ ❞❛❞♦ ❛❧❣✉♠ (t0, η)✱ ❝♦♠ η ♥♦ ❞♦♠í♥✐♦ ❞❡f✱ ❡①✐st❡ ✉♠❛ ú♥✐❝❛

s♦❧✉çã♦ r❡❛❧ φ ❞❛ ❡q✉❛çã♦ ✭✸✳✺✮ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧φ(t0) =η✳ ❊ss❛ s♦❧✉çã♦

❡①✐st❡ ❡♠ ❛❧❣✉♠ ✐♥t❡r✈❛❧♦ I ❡ é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❞❡ (t, t0, η) ❝♦♠ t, t0 ∈I✳

❯♠ ♦✉tr♦ ❡①❡♠♣❧♦ ❞❡ s✐st❡♠❛ ❛✉tô♥♦♠♦ é ♦ s✐st❡♠❛ ❧✐♥❡❛r

˙

y=A·y ✭✸✳✽✮

♦♥❞❡ A é ✉♠❛ ♠❛tr✐③ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s✳

▲❡♠❛ ✸✳✶✳ ❙❡x(t)é ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✸✳✺✮ ♥♦ ✐♥t❡r✈❛❧♦(a, b)❡c✉♠❛ ❝♦♥st❛♥t❡ q✉❛❧q✉❡r✱ ❡♥tã♦ ❛ ❢✉♥çã♦ y(t) = x(t+c) é s♦❧✉çã♦ ❞❡ ✭✸✳✻✮ ♥♦ ✐♥t❡r✈❛❧♦ (a−c, b−c)✳

❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s y′_{(t) = x}′_{(t+c) = f(x(t+c)) = f(y(t)).}

◆♦t❡ q✉❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❣❛r❛♥t✐❞❛ ♣❡❧♦ ▲❡♠❛ ❛❝✐♠❛ ♥ã♦ é ❛ss❡❣✉r❛❞❛ ♣❛r❛ s✐st❡♠❛s ♥ã♦ ❛✉tô♥♦♠♦s✳

❆s ❡q✉❛çõ❡s tr❛t❛❞❛s ♥♦ r❡s✉❧t❛❞♦ ❞❡ ❍❛rt♠❛♥✲❖♥✉❝❤✐❝ sã♦ ❛s s❡❣✉✐♥t❡s✿ ✶✳ x′ _=A(t)x

✷✳ x′ _=A(t)x+g(t)

✸✳ x′ _{=A(t)x+f(t, x)}

❉❡✜♥✐çã♦ ✸✳✶✼✳ ❉✐③✲s❡ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ X✱ ❝✉❥♦s ❡❧❡♠❡♥t♦s ❝❤❛♠❛r❡♠♦s ♣♦♥t♦s✱ é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ s❡ ❛ ❝❛❞❛ ❞♦✐s ♣♦♥t♦s p ❡ q ❞❡ ❳ ❡stá ❛ss♦❝✐❛❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ d(p, q)✱ ❝❤❛♠❛❞♦ ❞✐stâ♥❝✐❛ ❞❡ p ❛ q✱ t❛❧ q✉❡

✶✳ d(p, q)>0 s❡ p6=q❀ d(p, q) = 0❀

✷✳ d(p, q) = d(q, p)❀

✸✳ d(p, q)≤d(p, r) +d(r, q)✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ r ∈X✳

❯♠❛ ♥♦r♠❛ ❡♠ X ❞❡✜♥❡ ✉♠❛ ♠étr✐❝❛ d ❡♠ X ❛ q✉❛❧ é ❞❛❞❛ ♣♦r d(x, y) =

kx−yk,∀x, y ∈X✱ ❞❡♥♦♠✐♥❛❞❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ♣❡❧❛ ♥♦r♠❛✳

❊①❡♠♣❧♦ ✸✳✶✺✳ ❙ã♦ ❡①❡♠♣❧♦s ❞❡ ❡s♣❛ç♦s ♠étr✐❝♦s ♦s ❡s♣❛ç♦s ❡✉❝❧✐❞✐❛♥♦s Rn_{✱ ❝♦♠}

n ≥1✳ ❆ ❞✐stâ♥❝✐❛ ❡♠ Rn _{é ❞❛❞❛ ♣♦r}

❘❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ✷✹

❉❡✜♥✐çã♦ ✸✳✶✽✳ ❉✐③✲s❡ q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ {xn}_n∈N ❡♠ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ X é ✉♠❛

s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② q✉❛♥❞♦ ❞❛❞♦ ✉♠ ǫ >0✱ ❝♦rr❡s♣♦♥❞❡ ✉♠ ✐♥t❡✐r♦ N t❛❧ q✉❡ d(xn, xm)< ǫ

s❡ ♥ ≥N ❡ m ≥N✳

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

❊①❡♠♣❧♦ ✸✳✶✻✳ Rn _{❡ C}n_✳

❉❡✜♥✐çã♦ ✸✳✷✵✳ ❯♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ é ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦ ❝♦♠♣❧❡t♦ ✭❝♦♠♣❧❡t♦ ♥❛ ♠étr✐❝❛ ❞❡✜♥✐❞❛ ♣❡❧❛ ♥♦r♠❛✮✳

❉❡✜♥✐çã♦ ✸✳✷✶✳ ❯♠ ♥✉♠❡r♦ ❛ é ❝❤❛♠❛❞♦ ❞❡ ❧✐♠✐t❛♥t❡ s✉♣❡r✐♦r✭✐♥❢❡r✐♦r✮ ❞❡ ✉♠ s✉❜✲ ❝♦♥❥✉♥t♦ X ❞❛ r❡t❛ s❡ x≤a(x≥a)✱ ♣❛r❛ t♦❞♦ x∈X✳

❉❡✜♥✐çã♦ ✸✳✷✷✳ ❉✐③✲s❡ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ X ⊂ R é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡ q✉❛♥❞♦

❡①✐st❡ ❛❧❣✉♠ b∈R t❛❧ q✉❡ _x≤b ♣❛r❛ t♦❞♦ _x∈X✳

❊①❡♠♣❧♦ ✸✳✶✼✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r♠❡♥t❡ ♥♦ ❝♦♥✲ ❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❝♦♠ ❛ ♦r❞❡♠ ✉s✉❛❧✳

❉❡ ❢❛t♦✱ ✲✶ é ✉♠❛ ❝♦t❛ s✉♣❡r✐♦r ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ Z− ❞♦s ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s✱ ♣♦✐s ✲✶

♥ã♦ é ♠❡♥♦r q✉❡ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ❞❡ Z−✳ ❚♦❞♦ ♥ú♠❡r♦ r❡❛❧ ♠❛✐♦r q✉❡ ✲✶ é t❛♠❜é♠

❝♦t❛ s✉♣❡r✐♦r ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♥❡❣❛t✐✈♦s✳

❉❡✜♥✐çã♦ ✸✳✷✸✳ ❉✐③✲s❡ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ X ⊂ R é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ q✉❛♥❞♦

❡①✐st❡ ❛❧❣✉♠ a∈R t❛❧ q✉❡ _{a ≤x} ♣❛r❛ t♦❞♦ _x∈X✳

❊①❡♠♣❧♦ ✸✳✶✽✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s é ❧✐♠✐t❛❞♦ ✐♥❢❡r✐♦r♠❡♥t❡ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s ❝♦♠ ❛ ♦r❞❡♠ ✉s✉❛❧✳

❉❡ ❢❛t♦✱ ✶ é ✉♠❛ ❝♦t❛ ✐♥❢❡r✐♦r ♣❛r❛ ♦ ❝♦♥❥✉♥t♦Z+ ❞♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✱ ♣♦✐s ✶ ♥ã♦

é ♠❛✐♦r q✉❡ ♥❡♥❤✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❚♦❞♦ ✐♥t❡✐r♦ ♠❡♥♦r q✉❡ ✶ é t❛♠❜é♠ ❝♦t❛ ✐♥❢❡r✐♦r ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✳

❉❡✜♥✐çã♦ ✸✳✷✹✳ ❙❡ ✉♠ ❝♦♥❥✉♥t♦ X é ❧✐♠✐t❛❞♦ s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r♠❡♥t❡✱ ❞✐③✲s❡ q✉❡X é ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦✳

❉❡✜♥✐çã♦ ✸✳✷✺✳ ❈♦♥s✐❞❡r❡ n ✐♥t❡r✈❛❧♦s ❢❡❝❤❛❞♦s I = {x/aj ≤x≤bj, j = 1,2, . . . n}

❡ s❡✉s ✈♦❧✉♠❡s

V(I) =

n

Y

j=1