❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
▼♦❞❡❧♦s ▼❛t❡♠át✐❝♦s
❡ ❆♣❧✐❝❛çõ❡s ❛♦ ❊♥s✐♥♦ ▼é❞✐♦
▲✉❝✐❛♥♦ ❆♣❛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❛✳ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐
✺✶✼✳✸✽ ▼✷✶✷♠
▼❛❣r✐♥✐✱ ▲✉❝✐❛♥♦ ❆♣❛r❡❝✐❞♦ ▼♦❞❡❧♦s ▼❛t❡♠át✐❝♦s
❡ ❆♣❧✐❝❛çõ❡s ❛♦ ❊♥s✐♥♦ ▼é❞✐♦✴ ▲✉❝✐❛♥♦ ❆♣❛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❛✿ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐
✶✳ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s✳ ✷✳ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ❖r❞✐♥ár✐❛s✳ ✸✳ ▼♦❞❡❧❛❣❡♠ ▼❛t❡♠át✐❝❛✳ ✹✳ ❊st❛❜✐❧✐❞❛❞❡✳ ■✳ ❚ít✉❧♦
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
▲✉❝✐❛♥♦ ❆♣❛r❡❝✐❞♦ ▼❛❣r✐♥✐
▼♦❞❡❧♦s ▼❛t❡♠át✐❝♦s
❡ ❆♣❧✐❝❛çõ❡s ❛♦ ❊♥s✐♥♦ ▼é❞✐♦
❉✐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❛✳ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐ ❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦ ■●❈❊ ✲ ❯◆❊❙P✴❘✐♦ ❈❧❛r♦
Pr♦❢✳ ❉r✳ ▲✉✐③ ❆✉❣✉st♦ ❞❛ ❈♦st❛ ▲❛❞❡✐r❛ ■❈▼❈ ✲ ❯❙P✴❙ã♦ ❈❛r❧♦s
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r t❡r ♣❡r♠✐t✐❞♦ q✉❡ ❡✉ ❝♦♥❝❧✉íss❡ ♠❛✐s ❡st❛ ❡t❛♣❛ ❞❡ ♠✐♥❤❛ ✈✐❞❛ ❡ ❛ss✐♠ r❡❛❧✐③❛ss❡ ✉♠ s♦♥❤♦ ❛♥t✐❣♦✳
❙♦✉ ❣r❛t♦ à ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛✱ Pr♦❢❛✳ ❉r❛✳ ▼❛rt❛ ❈✐❧❡♥❡ ●❛❞♦tt✐✱ ♣❡❧❛ ❛❝♦❧❤✐❞❛ ❡ ♦r✐❡♥t❛çã♦✱ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛✱ ❝❛r✐♥❤♦ ❡ ❞❡❞✐❝❛çã♦ ❝♦♠ q✉❡ ♠❡ ♦r✐❡♥t♦✉ ❡ t♦r♥♦✉ ❡st❡ tr❛❜❛❧❤♦ ♣♦ssí✈❡❧❀ t❛♠❜é♠ ♣❡❧❛s ♣❛❧❛✈r❛s ❞❡ â♥✐♠♦ ❡♠ ❝❛❞❛ ❡✲♠❛✐❧ tr♦❝❛❞♦ ♦✉ ❝♦♥✈❡rs❛ ❡ ♣♦r s✉❛s ✈❛❧✐♦s❛s s✉❣❡stõ❡s ❞✐❛♥t❡ ❞❛s ♠✐♥❤❛s ✐♥ú♠❡r❛s ❞❡s❛t❡♥çõ❡s ❛♦ ❡❧❛❜♦r❛r ❡st❡ t❡①t♦✳
➚ Pr♦❢❛✳ ❉r❛✳ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦ ♣♦r ❛❝r❡❞✐t❛r ♥♦ P❘❖❋▼❆❚✱ ❡♠ ♠✐♠ ❡ ❡♠ ❝❛❞❛ ✉♠ ❞❡ ♠❡✉s ❝♦❧❡❣❛s ❞❡ t✉r♠❛✳ P❡❧❛ ♣r❡s❡♥ç❛ ❝♦♥st❛♥t❡ ❛♦ ♥♦ss♦ ❧❛❞♦✱ ❡✈✐t❛♥❞♦ q✉❡ ❞❡s❛♥✐♠áss❡♠♦s ❛ ❝❛❞❛ ♥♦✈♦ tr♦♣❡ç♦ ♦✉ ❞✐✜❝✉❧❞❛❞❡❀ t❛♠❜é♠ ♣❡❧♦ ❝❛r✐♥❤♦✱ ❞❡❞✐❝❛çã♦ ❡ r❡s♣❡✐t♦ ❝♦♠ ♥♦ss❛s ❧✐♠✐t❛çõ❡s ♠❛t❡♠át✐❝❛s✳
➚ Pr♦❢❛✳ ❉r❛✳ ❆❧✐❝❡ ❑✐♠✐❡ ▼✐✇❛ ▲✐❜❛r❞✐✱ ♣♦r s✉❛ ❝♦♠♣❡tê♥❝✐❛ ❡ ❝♦♥❤❡❝✐♠❡♥t♦❀ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ t❡r ❛♣r❡♥❞✐❞♦ ♠✉✐t♦ ❝♦♠ s✉❛s ❛✉❧❛s ❞✉r❛♥t❡ ♦ ❱❡rã♦ ❞❡ ✷✵✶✷✳
➚s ♠✐♥❤❛s q✉❡r✐❞❛s ❆♥❛ ❈❡❝í❧✐❛✱ ●❧á✉❝✐❛✱ ▼❛r✐❛♥❛✱ ❙✐❜❡❧✐ ❡ P❛trí❝✐❛✱ q✉❡ ❞✐✈✐❞✐r❛♠ s✉❛s ✈✐❞❛s ❝♦♠✐❣♦ ❛♦ ❧♦♥❣♦ ❞♦s ú❧t✐♠♦s ❞♦✐s ❛♥♦s✱ ♦❢❡r❡❝❡♥❞♦ ❛♠✐③❛❞❡✱ â♥✐♠♦✱ ❤♦r❛s ❞❡ ❡st✉❞♦ ❥✉♥t♦s ❡ ♠✉✐t❛s ✈❡③❡s s✉❛s ❝❛s❛s ♣❛r❛ q✉❡ ❡✉ ❡st✉❞❛ss❡ ❛♦s ✜♥s ❞❡ s❡♠❛♥❛✳ ❏✉♥t♦s ❡st✉❞❛♠♦s✱ r✐♠♦s ✭♠✉✐t♦✦✮ ❡ ❞✐✈✐❞✐♠♦s ♠✉✐t❛s ❡①♣❡r✐ê♥❝✐❛s ❡ ❛♥❣úst✐❛s ❛♦ ❧♦♥❣♦ ❞❡st❛ ❥♦r♥❛❞❛✳
❆♦ ♠❡✉ q✉❡r✐❞♦ ❛♠✐❣♦ ❆r✐♦s✈❛❧❞♦ ❚r✐♥❞❛❞❡ ♣❡❧❛s ❝❛r♦♥❛s ❞❡ ❙ã♦ P❛✉❧♦ ❛ ❘✐♦ ❈❧❛r♦ t♦❞❛ s❡♠❛♥❛❀ ♣♦r s✉❛ ❛♠✐③❛❞❡✱ ❝❛r✐♥❤♦ ❡ ❧♦♥❣❛s ❝♦♥✈❡rs❛s ♣❡❧♦ ❝❛♠✐♥❤♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ♦s ❢✉♥❞❛♠❡♥t♦s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥✲ ❝✐❛✐s ♦r❞✐♥ár✐❛s✱ ✐♥str✉♠❡♥t♦s ✐♥❞✐s♣❡♥sá✈❡✐s ♥❛ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ ❞❡ ♣r♦❜❧❡♠❛s ♥❛s ♠❛✐s ✈❛r✐❛❞❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❤✉♠❛♥♦✳ ❚❛♠❜é♠ ❞✐s❝✉t✐♠♦s ❛❧❣✉♥s ♠♦❞❡❧♦s ❝❧áss✐❝♦s ♥❛ ❧✐t❡r❛t✉r❛ ❡ ❛♣r❡s❡♥t❛♠♦s ❝r✐tér✐♦s q✉❡ ❛ss❡❣✉r❛♠ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛s s♦❧✉✲ çõ❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✱ ❛ss✉♥t♦ ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♣❛r❛ t♦❞♦s q✉❡ ❡st✉❞❛♠ ❡st❡ t❡♠❛✳ ❆❧é♠ ❞✐ss♦✱ ❛♣r❡s❡♥t❛♠♦s ❞✉❛s ♣r♦♣♦st❛s ❞✐❞át✐❝❛s ♣❛r❛ ♦s ♣r♦❢❡ss♦r❡s ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ q✉❡ ❞❡s❡❥❛♠ ✐♥❝❧✉✐r ❛ ♠♦❞❡❧❛❣❡♠ ❡♠ s✉❛s ❛✉❧❛s ❝♦♠♦ ❡❧❡♠❡♥t♦ ♠♦t✐✈❛❞♦r ♣❛r❛ ✉♠ ❡st✉❞♦ s✐❣♥✐✜❝❛t✐✈♦ ❞♦s t❡♠❛s ❞✐s❝✉t✐❞♦s ♥♦s ❝✉rrí❝✉❧♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
❆❜str❛❝t
❲❡ ♣r❡s❡♥t t❤❡ ❢✉♥❞❛♠❡♥t❛❧s ♦❢ t❤❡ t❤❡♦r② ♦❢ ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ✐♥✲ ❞✐s♣❡♥s❛❜❧❡ t♦♦❧s ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧✐♥❣ ♦❢ ♣r♦❜❧❡♠s ✐♥ ✈❛r✐♦✉s ❛r❡❛s ♦❢ ❤✉♠❛♥ ❦♥♦✇❧❡❞❣❡✳ ❲❡ ❛❧s♦ ❞✐s❝✉ss s♦♠❡ ❝❧❛ss✐❝ ♠♦❞❡❧s ✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ❧✐t❡r❛t✉r❡ ❛♥❞ ♣r❡✲ s❡♥t ❝r✐t❡r✐❛ t♦ ❡♥s✉r❡ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛♥ ❡q✉❛t✐♦♥ ❞✐✛❡r❡♥t✐❛t❡s s✉❜❥❡❝t ♦❢ ❣r❡❛t ✐♥t❡r❡st t♦ ❛❧❧ ✇❤♦ st✉❞② t❤❡ s✉❜❥❡❝t✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ♣r❡s❡♥t t✇♦ ♣r♦♣♦s❛❧s ❢♦r t❤❡ t❡❛❝❤✐♥❣ ♦❢ ♠❛t❤❡♠❛t✐❝s t❡❛❝❤❡rs ♦❢ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥ ✇❤♦ ✇❛♥t t♦ ✐♥❝❧✉❞❡ ♠♦✲ ❞❡❧✐♥❣ ✐♥ t❤❡✐r ❝❧❛ss❡s ❛s ♠♦t✐✈❛t♦r ❢♦r ♠♦t✐✈❛t✐♥❣ ❛♥❞ ♠❡❛♥✐♥❣❢✉❧ st✉❞② ♦❢ t❤❡ t♦♣✐❝s ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ❝✉rr✐❝✉❧❛ ♦❢ s❡❝♦♥❞❛r② ❡❞✉❝❛t✐♦♥✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❈✉r✈❛s ✐♥t❡❣r❛✐s ❡ ❝❛♠♣♦ ❞❡ ❞✐r❡çõ❡s ❞❡ ✭✷✳✻✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ●rá✜❝♦ ❞❛s s♦❧✉çõ❡s ❞♦ ♠♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✸ ●rá✜❝♦ ❞❛s s♦❧✉çõ❡s ❞♦ ♠♦❞❡❧♦ ❞❡ ❱❡r❤✉❧st ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✹ ❘❡tr❛t♦ ❞❡ ❢❛s❡ ❞❡ ✭✷✳✻✸✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✺ ❘❡tr❛t♦ ❞❡ ❢❛s❡ ❞❡ ✭✷✳✻✹✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✻ ❘❡tr❛t♦ ❞❡ ❢❛s❡ ❞❡ ✭✷✳✻✺✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✶ ◆ó ❊stá✈❡❧✿ λ1 < λ2 <0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
✸✳✷ ◆ó ■♥stá✈❡❧✿ λ1 > λ2 >0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸
✸✳✸ P♦♥t♦ ❞❡ ❙❡❧❛✿ λ1 <0< λ2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✸✳✹ ◆ó ❊stá✈❡❧✿ λ1 =λ2 =λ <0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✸✳✺ ◆ó ❊stá✈❡❧✿ λ1 =λ2 =λ >0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✸✳✻ ❆✉t♦✈❛❧♦r❡s r❡♣❡t✐❞♦s✿ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❣❡♦♠étr✐❝❛ ♠❡♥♦r q✉❡ ❛ ❛❧❣é❜r✐❝❛✳ ✻✺ ✸✳✼ ❈❡♥tr♦✿ a= 0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✸✳✽ ❋♦❝♦ ■♥stá✈❡❧✿ a >0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✸✳✾ ❋♦❝♦ ❊stá✈❡❧✿ a <0✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻
✹✳✶ ❙♦❧✉çã♦γ(t, t0, x0) ❡stá✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✷✳✶ ❚❛①❛s ❞❡ ♥❛t❛❧✐❞❛❞❡ ❡ ♠♦rt❛❧✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ❡♥tr❡ ✷✵✵✵ ❡ ✷✵✵✺✳ ✳ ✳ ✸✾ ✷✳✷ ❊rr♦ ♥♦ ▼♦❞❡❧♦ ❞❡ ▼❛❧t❤✉s ✲ P♦♣✉❧❛çã♦ ❞♦ ▼✉♥✐❝í♣✐♦ ❞❡ ❙ã♦ P❛✉❧♦
❡♥tr❡ ✷✵✵✵✲✷✵✶✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✶ ❊st❛❜✐❧✐❞❛❞❡ ❞♦s s✐st❡♠❛s ❧✐♥❡❛r❡sX′ =AX ❞❡ ❖r❞❡♠ 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✺✳✶ P♦♣✉❧❛çã♦ ♦✜❝✐❛❧ ❞♦ ♠✉♥✐❝í♣✐♦ ❞❡ ❙ã♦ P❛✉❧♦ ❡♥tr❡ 2000 ❡2005✳ ✳ ✳ ✳ ✳ ✳ ✾✹
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✾
✷ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s ✷✶
✷✳✶ Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ ❊①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✸ ❆❧❣✉♠❛s té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸✳✶ ❱❛r✐á✈❡✐s s❡♣❛rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸✳✷ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ❡s❝❛❧❛r❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ✳ ✳ ✳ ✸✷ ✷✳✸✳✸ ❊q✉❛çõ❡s ❡①❛t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹ ▼♦❞❡❧♦s ❞❡s❝r✐t♦s ♣♦r ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✹✳✶ ▼♦❞❡❧♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❞❡ ▼❛❧t❤✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✹✳✷ ❖ ♠♦❞❡❧♦ ❞❡ ❱❡r❤✉❧st ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✷✳✹✳✸ Pr♦♣❛❣❛çã♦ ❞❛ ♣♦❞r✐❞ã♦ ❡♠ ♠❛çãs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✹✳✹ ❉✐ss❡♠✐♥❛çã♦ ❞❡ ❞♦❡♥ç❛s ❝♦♥t❛❣✐♦s❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✺ ❘❡tr❛t♦s ❞❡ ❢❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✸ ❙✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ✹✾
✸✳✶ ❙♦❧✉çõ❡s ❡ ♣r♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✶✳✶ ❆✉t♦✈❛❧♦r❡s r❡❛✐s ❡ ❞✐st✐♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✶✳✷ ❆✉t♦✈❛❧♦r❡s ❝♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✸✳✶✳✸ ❆✉t♦✈❛❧♦r❡s r❡♣❡t✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✸✳✷ ❙✐st❡♠❛s ♣❧❛♥❛r❡s✿ r❡tr❛t♦s ❞❡ ❢❛s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✸✳✷✳✶ ❆✉t♦✈❛❧♦r❡s r❡❛✐s ❡ ❞✐st✐♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✷✳✷ ❆✉t♦✈❛❧♦r❡s r❡♣❡t✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✸✳✷✳✸ ❆✉t♦✈❛❧♦r❡s ❝♦♠♣❧❡①♦s ❡ ❝♦♥❥✉❣❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✹ ■♥tr♦❞✉çã♦ à ❡st❛❜✐❧✐❞❛❞❡ ✻✼
✺ Pr♦♣♦st❛s ❞✐❞át✐❝❛s ✽✾ ✺✳✶ Pê♥❞✉❧♦ s✐♠♣❧❡s ❡ ♠❛ss❛✲♠♦❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✺✳✷ ❊✈♦❧✉çã♦ ♣♦♣✉❧❛❝✐♦♥❛❧ ❞♦ ♠✉♥✐❝í♣✐♦ ❞❡ ❙ã♦ P❛✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸
❆ ❘❡s✉❧t❛❞♦s ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ✾✼
❆✳✶ ❉✐❛❣♦♥❛❧✐③❛çã♦ ❡ ❢♦r♠❛ ❞❡ ❏♦r❞❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ❆✳✷ ❊①♣♦♥❡♥❝✐❛❧ ❞❡ ♠❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾
✶ ■♥tr♦❞✉çã♦
■s❛❛❝ ◆❡✇t♦♥ ✭✶✻✹✸✲✶✼✷✼✮ ❡ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ✈♦♥ ▲❡✐❜♥✐③ ✭✶✻✹✻✲✶✼✶✻✮ ❢♦r❛♠ r❡s✲ ♣♦♥sá✈❡✐s ♣❡❧❛ ✐♥✈❡♥çã♦ ❞♦ ❈á❧❝✉❧♦ ♥♦ sé❝✉❧♦ ❳❱■■ ❛♦ ❞❡s❝♦❜r✐r❡♠ ❞❡ ♠❛♥❡✐r❛ ✐♥❞❡✲ ♣❡♥❞❡♥t❡ ❛ ❝♦♥❡①ã♦ ❡♥tr❡ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ r❡t❛ t❛♥❣❡♥t❡ ❛♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ f ❡ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❞❡t❡r♠✐♥❛çã♦ ❞❛ ár❡❛ s♦❜ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡r✐✈❛❞❛ ❞❡ f✱ ❡st❛❜❡❝❡♥❞♦ ❞❡st❡ ♠♦❞♦ ♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦✳
◆❡st❡ ♠❡s♠♦ ♣❡rí♦❞♦✱ ❡♠ ❣r❛♥❞❡ ♣❛rt❡ ❞❡✈✐❞♦ ❛♦s tr❛❜❛❧❤♦s ❞❡ ◆❡✇t♦♥✱ ♦s ♣r✐♥❝í✲ ♣✐♦s ❞❛ ▼❡❝â♥✐❝❛ ❈❧áss✐❝❛ ❢♦r❛♠ ❞❡s❝♦❜❡rt♦s ❡ ❡♥❝♦♥tr❛r❛♠ ♥♦ ❈á❧❝✉❧♦ ❛s ❢❡rr❛♠❡♥t❛s ♠❛t❡♠át✐❝❛s ❛♣r♦♣r✐❛❞❛s ♣❛r❛ s❡✉ ❝♦rr❡t♦ tr❛t❛♠❡♥t♦✳ ◆♦ sé❝✉❧♦ ❳❱■■■✱ ❛ ❛❜♦r❞❛❣❡♠ ❞❡ ♣r♦❜❧❡♠❛s ❢ís✐❝♦s ❡ ❣❡♦♠étr✐❝♦s ❝♦♠ ♦s ♠ét♦❞♦s ❞♦ ❈á❧❝✉❧♦ s❡ ♠♦str♦✉ ❜❛st❛♥t❡ ♣r♦❢í❝✉❛✱ s✉r❣✐♥❞♦ ❛ss✐♠ ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✳
◆❡✇t♦♥ ❛♣❧✐❝♦✉ sér✐❡s ✐♥✜♥✐t❛s ❡ ❛ss✐♠ r❡s♦❧✈❡✉ ❛❧❣✉♠❛s ❝❧❛ss❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡✲ r❡♥❝✐❛✐s ❞♦ t✐♣♦ y′ =F(x, y) ♦♥❞❡F(x, y)é ✉♠ ♣♦❧✐♥ô♠✐♦ ♥❛s ✈❛r✐á✈❡✐sx ❡y✳ ▲❡✐❜♥✐③✱
♣♦r ✈♦❧t❛ ❞❡ ✶✻✾✶ ❛♣r❡s❡♥t♦✉ ❛ té❝♥✐❝❛ ❞❛s ✈❛r✐á✈❡✐s s❡♣❛rá✈❡✐s ❡ ♥❛ s❡q✉ê♥❝✐❛ ♦❜t❡✈❡ ❛ s♦❧✉çã♦ ❣❡r❛❧ ❞❛s ❡q✉❛çõ❡s ❞❛ ❢♦r♠❛
dy
dx +p(x)y=q(x), ♦♥❞❡ p ❡ q sã♦ ❢✉♥çõ❡s ❞❡ x✳
❖s ♠ét♦❞♦s ❞♦ ❈á❧❝✉❧♦ ❡ ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❝♦♠❡ç❛r❛♠ ❛ ❣❛♥❤❛r ❢♦rç❛ ❛♦ s❡ ♠♦str❛r❡♠ ❞❡❝✐s✐✈♦s ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳ ❯♠ ❞❡st❡s ♣r♦❜❧❡♠❛s✱ ♣r♦♣♦st♦ ♣♦r ●❛❧✐❧❡✉ ●❛❧✐❧❡✐ ✭✶✺✻✹✲✶✻✹✷✮✱ ❝♦♥s✐st✐❛ ❡♠ ❞❡s❝r❡✈❡r ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ❛ ❢♦r♠❛ ❞❛ ❝✉r✈❛ ❢♦r♠❛❞❛ ♣♦r ✉♠ ✜♦ s✉s♣❡♥s♦ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❡ s✉❥❡✐t♦ ❛♣❡♥❛s à ❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ❡ ❞❡ s❡✉ ♣ró♣r✐♦ ♣❡s♦ ✭❡rr♦♥❡❛♠❡♥t❡ ❛❧❣✉♥s ♣❡♥s❛✈❛♠ s❡r ❛ ♣❛rá❜♦❧❛ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛✮✳ ❊♠ ✶✻✾✵✱ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ✭✶✻✻✼✲✶✼✹✽✮ r❡s♦❧✈❡✉ ♦ ♣r♦❜❧❡♠❛ ♠♦str❛♥❞♦ q✉❡ ❛ ❝✉r✈❛ ♣r♦❝✉r❛❞❛ ✭❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❝❛t❡♥ár✐❛✮ s❛t✐s❢❛③ à ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧
d2y
dx2 =
pg h
s
1 +
dy dx
2 ,
♦♥❞❡ p✐♥❞✐❝❛ ♦ ♣❡s♦ ❞♦ ✜♦✱ g ❛ ❢♦rç❛ ❞❛ ❣r❛✈✐❞❛❞❡ ❡h ❛ ❛❧t✉r❛ ❞♦ ♣♦♥t♦P ♠❛✐s ❜❛✐①♦ ❞❛ ❝✉r✈❛✱ ✐st♦ é✱ ❞♦ ♣♦♥t♦ P ♣❡rt❡♥❝❡♥t❡ à ❝✉r✈❛ q✉❡ ❡stá ♠❛✐s ♣ró①✐♠♦ ❞♦ s♦❧♦✳
✷✵ ■♥tr♦❞✉çã♦
P♦r s✉❛ ✈❡③✱ ▲❡♦♥❤❛r❞ ❊✉❧❡r ✭✶✼✵✼✲✶✼✽✸✮ ❡♠ ✉♠ ❛rt✐❣♦ ❞❡ ✶✼✸✹ ❛♣r❡s❡♥t♦✉ ❛ t❡♦r✐❛ ❞♦s ❢❛t♦r❡s ✐♥t❡❣r❛♥t❡s ❡ ❡①♣❧✐❝✐t♦✉ ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❢♦ss❡ ❡①❛t❛✳ ❆❧é♠ ❞✐ss♦✱ ❡❧❡ r❡s♦❧✈❡✉ ❝♦rr❡t❛♠❡♥t❡ ❛s ❡q✉❛çõ❡s ❞❛ ❢♦r♠❛
a2
d2y
dx2 +a1
dy
dx +a0y(x) = f(x),
♦♥❞❡ a2, a1 ❡ a0 sã♦ ❝♦♥st❛♥t❡s✳ ❈♦✉❜❡ ❛ ❊✉❧❡r t❛♠❜é♠ ❞❡s❡♥✈♦❧✈❡r ✉♠ ❞♦s ♠ét♦❞♦s
♥✉♠ér✐❝♦s ❡♠♣r❡❣❛❞♦s ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞♦ t✐♣♦ dy
dx =F(x, y), s✉❥❡✐t❛ à ❝♦♥❞✐çã♦ y(x0) = y0✳
❆té ❡st❡ ♣♦♥t♦ ❞❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❛ ♣r❡♦❝✉♣❛çã♦ ❡r❛ ❛ ❞❡ r❡s♦❧✈❡r ❡①♣❧✐❝✐✲ t❛♠❡♥t❡ ❛❧❣✉♠❛s ❝❧❛ss❡s ❞❡ ❡q✉❛çõ❡s✱ ❞❡ ♠♦❞♦ q✉❡ ❛s té❝♥✐❝❛s ❞❡s❡♥✈♦❧✈✐❞❛s ♥ã♦ ❡r❛♠ ❣❡r❛✐s❀ ❤❛✈✐❛ ✉♠❛ ♣r♦❢✉sã♦ ❞❡ ♠ét♦❞♦s ♣❛rt✐❝✉❧❛r❡s q✉❡ ♥ã♦ ❡st❛✈❛♠ ❞❡✈✐❞❛♠❡♥t❡ ❢✉♥✲ ❞❛♠❡♥t❛❞♦s✳ ❈♦♠ ♦ ❛✈❛♥ç♦ ❞❛ ❆♥á❧✐s❡✱ ♣♦r ✈♦❧t❛ ❞❛s ú❧t✐♠❛s ❞é❝❛❞❛s ❞❡ ✶✽✵✵ ❤♦✉✈❡ ❛ ♣r❡♦❝✉♣❛çã♦ ❡♠ s❡ ❞❡s❡♥✈♦❧✈❡r ✉♠❛ t❡♦r✐❛ q✉❡ r❡s♣♦♥❞❡ss❡ às q✉❡stõ❡s ❞❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s✳
❊♠ ❣❡r❛❧ ♥ã♦ é ✉♠ ♣r♦❝❡ss♦ s✐♠♣❧❡s ❞❡t❡r♠✐♥❛r ❛s s♦❧✉çõ❡s ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡✲ r❡♥❝✐❛❧ ❡ ❛té ❤♦❥❡ ♥ã♦ ❡①✐st❡♠ ♠ét♦❞♦s ❣❡r❛✐s✳ ◆❛ ✈❡r❞❛❞❡ ❛ ♠❛✐♦r ♣❛rt❡ ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♥ã♦ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛ ❛♥❛❧✐t✐❝❛♠❡♥t❡✱ ✐st♦ é✱ ♥ã♦ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ❡①✲ ♣❧✐❝✐t❛♠❡♥t❡ ❛s s♦❧✉çõ❡s✳ ❏✉❧❡s ❍❡♥r✐ P♦✐♥❝❛ré ✭✶✽✺✹✲✶✾✶✷✮ ♥♦ ✐♥í❝✐♦ ❞♦ sé❝✉❧♦ ❳❳ ❝♦♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❛rt✐❣♦s ♣r♦♣õ❡ ♦ tr❛t❛♠❡♥t♦ q✉❛❧✐t❛t✐✈♦ ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ♦♥❞❡ ♦s ❡s❢♦rç♦s sã♦ ❞❡❞✐❝❛❞♦s ❛ ❡♥t❡♥❞❡r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢❛♠í❧✐❛ ❞❡ s♦❧✉çõ❡s✱ ♦ q✉❡ ❜❛st❛ ♥❛ ♠❛✐♦r✐❛ ❞❛s ❛♣❧✐❝❛çõ❡s✳
❊st❡ ❜r❡✈❡ r❡❧❛t♦ ❤✐stór✐❝♦ é ❜❛s❡❛❞♦ ♥❛ r❡❢❡rê♥❝✐❛ ❬✶❪✳
◗✉❛♥t♦ à ♦r❣❛♥✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ r❡❣✐str❛♠♦s q✉❡ ♥♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❡stã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ t❡♦r✐❛ ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ♦♥❞❡ ✐♥❝❧✉í♠♦s ✭❞❡♥✲ tr❡ ♦✉tr♦s r❡s✉❧t❛❞♦s✮ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s s♦❜ ❤✐♣ó✲ t❡s❡s ❝♦♥✈❡♥✐❡♥t❡s ❡ ❛❧❣✉♥s ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s ❝❧áss✐❝♦s ♥❛ ❧✐t❡r❛t✉r❛✳ ◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ❛ t❡♦r✐❛ ❞♦s s✐st❡♠❛s ❧✐♥❡❛r❡s✱ ❝❛r❛❝t❡r✐③❛♥❞♦ ❛s s♦❧✉çõ❡s ❡♠ t❡r♠♦s ❞❡ ❡①♣♦♥❡♥❝✐❛✐s ❞❡ ♠❛tr✐③❡s ❡ ❛♣r❡s❡♥t❛♥❞♦ ✉♠❛ ❞❡s❝r✐çã♦ ❞♦s r❡tr❛t♦s ❞❡ ❢❛s❡ ❞♦s s✐st❡♠❛s ❡♠ ❞✐♠❡♥sã♦ ❞♦✐s✳
✷ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s
❉❡ ♠❛♥❡✐r❛ ♣♦✉❝♦ ♣r❡❝✐s❛✱ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛ ♣♦❞❡ s❡r ❞❡✜♥✐❞❛ ❝♦♠♦ ✉♠❛ ❡q✉❛çã♦ q✉❡ ❡♥✈♦❧✈❡ ✉♠❛ ✧❢✉♥çã♦ ✐♥❝ó❣♥✐t❛✧✭❞❡ ❛♣❡♥❛s ✉♠❛ ✈❛r✐á✈❡❧✮ ❡ s✉❛s ❞❡r✐✈❛❞❛s✳
❆ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r ❡①♣❧✐❝✐t❛ ❛ ♥♦çã♦ ❞❡ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛ q✉❡ ❡♠♣r❡✲ ❣❛r❡♠♦s ♥❡st❡ t❡①t♦✳
❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛ é ✉♠❛ r❡❧❛çã♦ ❞♦ t✐♣♦
x(d) =F(t, x, x(1), x(2), ..., x(d−1)), ✭✷✳✶✮ ♦♥❞❡ F :U −→R é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❞❡✜♥✐❞❛ ❡♠ ✉♠ ❛❜❡rt♦U ❞❡ R1+d ❝♦♠ t∈R
❡ ♦♥❞❡ x(j) é ❛ j✲és✐♠❛ ❞❡r✐✈❛❞❛ ❞❛ ❢✉♥çã♦ x❞❡ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✱ ❝♦♠ j ∈ {1,2, ..., d}✳
❖ ✐♥t❡✐r♦d q✉❡ ❛♣❛r❡❝❡ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ é ❝❤❛♠❛❞♦ ❞❡ ♦r❞❡♠ ❞❛ ❡q✉❛çã♦ ❞✐❢❡✲ r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛✳
◆♦ r❡st❛♥t❡ ❞❡st❡ tr❛❜❛❧❤♦ ❡s❝r❡✈❡r❡♠♦s s✐♠♣❧❡s♠❡♥t❡ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✱ ❞❡✈❡♥❞♦ ✜❝❛r s✉❜❡♥t❡♥❞✐❞♦ q✉❡ ❡st❛♠♦s ♥♦s r❡❢❡r✐♥❞♦ à ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛✳ ❖ ❡①❡♠✲ ♣❧♦ ❛ s❡❣✉✐r ✐❧✉str❛ ❛ ♠♦❞❡❧❛❣❡♠ ❞❡ ✉♠ ❢❡♥ô♠❡♥♦ ❢ís✐❝♦✳
❊①❡♠♣❧♦ ✷✳✶✳ ✭▲❡✐ ❞❡ ❍♦♦❦❡✮ ❙❡❥❛ m❛ ♠❛ss❛ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❛❝♦♣❧❛❞❛ à ✉♠❛ ♠♦❧❛ ❞❡ ❝♦♠♣r✐♠❡♥t♦ l ❡♠ ❡q✉✐❧í❜r✐♦✳ ❈♦♥s✐❞❡r❡ q✉❡ ❛ ♠♦❧❛ s❡❥❛ ❡st✐❝❛❞❛ ✭♦✉ ❝♦♠♣r✐♠✐❞❛✮ ❞❡ ♠♦❞♦ ❛ s♦❢r❡r ✉♠ ❛❢❛st❛♠❡♥t♦ ✭♦✉ ❞❡❢♦r♠❛çã♦✮ x ❡♠ r❡❧❛çã♦ ❛♦ ❡q✉✐❧í❜r✐♦✱ ♥ã♦ ❡st❛♥❞♦ s✉❥❡✐t♦ ❛ q✉❛❧q✉❡r ❢♦rç❛ ❡①t❡r♥❛✳ ❆ ▲❡✐ ❞❡ ❍♦♦❦❡✱ ❛♦ ❛✜r♠❛r q✉❡ ❛ ✐♥t❡♥s✐❞❛❞❡ ❞❛ ❢♦rç❛ ❡❧ást✐❝❛ Ft é ♣r♦♣♦r❝✐♦♥❛❧ à ❞❡❢♦r♠❛çã♦x✱ ♣❡r♠✐t❡✲♥♦s ❡s❝r❡✈❡r
Ft=−Cx, ✭✷✳✷✮
♦♥❞❡ ❛ ❝♦♥st❛♥t❡ C é ✉♠❛ ❝♦♥st❛♥t❡ ❢ís✐❝❛✱ ❞❡♣❡♥❞❡♥❞♦ ♣♦r ❡①❡♠♣❧♦ ❞♦ ♠❛t❡r✐❛❧ ❞❛ ♠♦❧❛✱ ❞❛ ❡str✉t✉r❛ ❞♦s ❛♥é✐s✱ ❡t❝✳ P❡❧❛ ❙❡❣✉♥❞❛ ▲❡✐ ❞❡ ◆❡✇t♦♥ t❡♠♦s
Ft=m
d2x
dt2,
s❡❣✉❡ ❞❡ ✭✷✳✷✮ q✉❡
d2x
dt2 =−kx, ✭✷✳✸✮
✷✷ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s
❞❡s❞❡ q✉❡ ❢❛ç❛♠♦s k = C/m✳ ❆ ❡q✉❛çã♦ ✭✷✳✸✮ ♠♦❞❡❧❛ ♦ s✐st❡♠❛ ♠❛ss❛✲♠♦❧❛ ♥❛s
❝♦♥❞✐çõ❡s ❛❞♠✐t✐❞❛s✳
◆♦t❡ q✉❡ ❡♠ ✭✷✳✸✮ ❛ ❢✉♥çã♦F(t, x) =−kx♥ã♦ ❞❡♣❡♥❞❡ ❡①♣❧✐❝✐t❛♠❡♥❡ ❞❛ ✈❛r✐á✈❡❧t❀ ❞✐③❡♠♦s ♥❡st❡ ❝❛s♦ q✉❡ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ é ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❛✉tô♥♦♠❛ ❡ ❡s❝r❡✈❡♠♦s ♣♦r s✐♠♣❧✐❝✐❞❛❞❡ ❛♣❡♥❛s F(x)✳ ◆❡st❡ ❝❛s♦✱ ✐♥t❡r♣r❡t❛♠♦s ❛ ❛♣❧✐❝❛çã♦
F :U →R ❝♦♠♦ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❞❡✜♥✐❞♦ ♥♦ ❛❜❡rt♦U ⊂Rd✭❡ ♥ã♦ ❡♠ R1+d✮✳ ❖❜s❡r✈❡ q✉❡ ❛ ❢✉♥çã♦x(t)q✉❡ s❛t✐s❢❛③ ❛ ❡q✉❛çã♦ ✭✷✳✸✮ ❞❡✈❡ s❡r t❛❧ q✉❡ s✉❛ ❞❡r✐✈❛❞❛
s❡❣✉♥❞❛ s❡❥❛ ♣r♦♣♦r❝✐♦♥❛❧ à ❡❧❛ ♠❡s♠❛✳ ❖r❛✱ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s s❡♥♦ ❡ ❝♦ss❡♥♦ ♣♦ss✉❡♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡✳ ❆❥✉st❛♥❞♦ ♦s ❛r❣✉♠❡♥t♦s ❞❡ ♠❛♥❡✐r❛ ❝♦♥✈❡♥✐❡♥t❡✱ s♦♠♦s t❡♥t❛❞♦s ❛ ❝♦♥s✐❞❡r❛r ❛s ❢✉♥çõ❡s r❡❛✐s
x1(t) = αcos(t
√
k) ✭✷✳✹✮
❡
x1(t) =αsen(t
√
k) ✭✷✳✺✮
❝♦♠♦ s♦❧✉çõ❡s ♣❛r❛ ✭✷✳✸✮✳ P♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ❞❡ ❢❛t♦ ❡st❛s ❢✉♥çõ❡s sã♦ s♦❧✉çõ❡s✱ ♣♦✐s ❞❡ ✭✷✳✹✮ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
x′1(t) = −α√ksen(t√k)
❡
x′′1(t) =−kαcos(t√k)
❡ ❡♥tã♦ é ❢á❝✐❧ ✈❡r q✉❡ x′′
1(t) = −kx1(t)✳ ❯♠ r❛❝✐♦❝í♥✐♦ ❛♥á❧♦❣♦ ♣❡r♠✐t❡ ♠♦str❛r q✉❡
✭✷✳✺✮ t❛♠❜é♠ é s♦❧✉çã♦ ❞❡ ✭✷✳✸✮✳ ❆❧é♠ ❞✐ss♦✱ ♣❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ♣♦❞❡♠♦s ✈❡r q✉❡ q✉❛❧q✉❡r ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ✭✷✳✹✮ ❡ ✭✷✳✺✮ t❛♠❜é♠ é s♦❧✉çã♦ ❞❡ ✭✷✳✸✮✱ ♦ q✉❡ ❡♠ ❣❡r❛❧ é ✈❡r❞❛❞❡✐r♦ ♣❛r❛ q✉❛❧q✉❡r ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❧✐♥❡❛r✶✳
❆ s❡❣✉✐r ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡✜♥✐çã♦ ❞❡ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ✭✷✳✶✮✳ ❖❜s❡r✈❡ q✉❡ ♦s ✐t❡♥s 1 ❡2 ❞❛ ❞❡✜♥✐çã♦ ❞❛❞❛ sã♦ ❡①✐❣ê♥❝✐❛s ♣❛r❛ q✉❡ ♦ ✐t❡♠ 3❢❛ç❛ s❡♥t✐❞♦✳
❉❡✜♥✐çã♦ ✷✳✷✳ ❯♠❛ s♦❧✉çã♦ ❞❡ ✭✷✳✶✮ é ✉♠❛ ❝✉r✈❛ γ :I →Rd ❞❡ ❝❧❛ss❡ Cd t❛❧ q✉❡✿
✶✳ ❖ ✐♥t❡r✈❛❧♦ I é ✉♠ ❛❜❡rt♦ ❞❡ R❀ ✷✳ ❖ ✈❡t♦r
t, γ(t),dγ dt(t), ...,
dd−1γ
dtd−1(t)
♣❡rt❡♥❝❡ ❛ U ♣❛r❛ t♦❞♦t ∈I❀
✸✳ P❛r❛ t♦❞♦ t∈I✱ t❡♠✲s❡ F
t, γ(t),dγ dt(t), ...,
dk−1γ
dtk−1(t)
= d
dγ
dtd(t)✳
❖s ♣ró①✐♠♦s ❡①❡♠♣❧♦s ✐❧✉str❛♠ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ♣❡rt✐♥❡♥t❡s à ♥♦çã♦ ❞❡ s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✳
✷✸
❊①❡♠♣❧♦ ✷✳✷✳ ❈♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ dx
dt =t, ✭✷✳✻✮
❝✉❥❛s s♦❧✉çõ❡s sã♦ ❞❛❞❛s ❡①♣❧✐❝✐t❛♠❡♥t❡ ♣♦r
x(t) = t
2
2 +k, ✭✷✳✼✮
♦♥❞❡ k é ✉♠❛ ❝♦♥st❛♥t❡✳
❱❛r✐❛♥❞♦ ❛ ❝♦♥st❛♥t❡k ❞❡t❡r♠✐♥❛♠♦s ✐♥✜♥✐t❛s ❝✉r✈❛s ✭❝❤❛♠❛❞❛s ❞❡ ❝✉r✈❛s ✐♥t❡✲ ❣r❛✐s ❞❛ ❡q✉❛çã♦✮ ❡ ♣♦rt❛♥t♦ ✐♥✜♥✐t❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ✭✷✳✻✮✳ ❆ ❡q✉❛çã♦ ✭✷✳✼✮ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ s♦❧✉çã♦ ❣❡r❛❧ ❞❡ ✭✷✳✻✮ ❡ ❛♦ ✜①❛r ❛ ❝♦♥st❛♥t❡k ♦❜t❡♠♦s ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r✳
❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❣❡♦♠étr✐❝♦✱ ❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ y′ =f(x, y)sã♦ t❛✐s q✉❡ ❡♠
❝❛❞❛ ♣♦♥t♦(x, y) ❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ ✐♥t❡❣r❛❧ ♣❛ss❛♥❞♦ ♣❡❧♦ ♣♦♥t♦ t❡♠ ❝♦❡✜❝✐❡♥t❡
❛♥❣✉❧❛r m = f(x, y)✳ ■st♦ s✉❣❡r❡ ✉♠ ♠ét♦❞♦ ❣❡♦♠étr✐❝♦ ♣❛r❛ ❡♥t❡♥❞❡r ♦ ❝♦♠♣♦rt❛✲
♠❡♥t♦ ❞❛s ❝✉r✈❛s ✐♥t❡❣r❛✐s ❞❛ ❡q✉❛çã♦✿ ♣❛r❛ ✐st♦✱ tr❛ç❛♠♦s ✉♠ ♣❡q✉❡♥♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❡♠ ❝❛❞❛ ♣♦♥t♦(x, y)❝♦♠ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛rf(x, y)❀ ❛♦ ❝♦♥❥✉♥t♦ ❞❡st❡s s❡❣♠❡♥t♦s
é ❞❛❞♦ ♦ ♥♦♠❡ ❞❡ ❝❛♠♣♦ ❞❡ ❞✐r❡çõ❡s ❞❛ ❡q✉❛çã♦y′ =f(x, y)✳ ❈♦♠ ♦ ✉s♦ ❞❡ s♦❢t✇❛✲
r❡s ❣r❛t✉✐t♦s ♣♦❞❡♠♦s ❞❡s❡♥❤❛r ♦ ❝❛♠♣♦ ❞❡ ❞✐r❡çõ❡s ❡ ❛ss✐♠ ❛♥❛❧✐s❛r ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s s♦❧✉çõ❡s✳
❆❜❛✐①♦ ❡♥❝♦♥tr❛♠✲s❡ r❡♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ❝✉r✈❛s ✐♥t❡❣r❛✐s ❞❡ ✭✷✳✻✮✳ ❈♦♠♣❛r❡ ❛ s❡♠❡❧❤❛♥ç❛ ❞❡ ❝♦♠♣♦rt❛♠❡♥t♦ ❡♥tr❡ ❛s ❝✉r✈❛s ✐♥t❡❣r❛✐s ❡ ♦ ❝❛♠♣♦ ❞❡ ❞✐r❡çõ❡s ♣❧♦t❛❞♦ ❛❜❛✐①♦✿ ✷
❋✐❣✉r❛ ✷✳✶✿ ❈✉r✈❛s ✐♥t❡❣r❛✐s ❡ ❝❛♠♣♦ ❞❡ ❞✐r❡çõ❡s ❞❡ ✭✷✳✻✮✳
✷✹ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s
✷✳✶ Pr❡❧✐♠✐♥❛r❡s
◆❡st❛ s❡çã♦ ✈❛♠♦s ❛♣r❡s❡♥t❛r ♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ t❡♦r✐❛ ❜ás✐❝❛ ❞❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥✲ ❝✐❛✐s ♦r❞✐♥ár✐❛s ❡ ❛ ❞❡s❡♥✈♦❧✈❡r❡♠♦s ♣r✐✈✐❧❡❣✐❛♥❞♦ ❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❊st❛ ❡s❝♦❧❤❛ ♥ã♦ é ❛r❜✐trár✐❛✱ ♣♦✐s ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ♦r❞❡♠ q✉❛❧q✉❡r ♣♦❞❡ ❝♦♥✈❡♥✐❡♥t❡♠❡♥t❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❛tr❛✈és ❞❡ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ❝♦♥✈❡♥✐❡♥t❡✳
❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ✭✷✳✶✮✳ P❛r❛ tr❛♥s❢♦r♠á✲❧❛ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❜❛st❛ ❞❡✜♥✐r ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿
y1 =x,
y2 =
dx dt, ✳✳✳ yd=
dd−1x
dtd−1,
❞❡ ♠♦❞♦ q✉❡
y1′ =y2,
y2′ =y3,
✳✳✳ yd′−1 =yd,
❡ ❛ ❡q✉❛çã♦ ✭✷✳✶✮ s❡ ❡s❝r❡✈❡ ❝♦♠♦ ♦ s✐st❡♠❛
y′
1 =y2
y′
2 =y3
✳✳✳ y′
d=F(t, y1, ..., yd−1)
✭✷✳✽✮
◆❛ s❡q✉ê♥❝✐❛ ❛♣r❡s❡♥t❛r❡♠♦s ❡ ♣r♦✈❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ ✉s❛r❡♠♦s ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ❡st❡ tr❛❜❛❧❤♦ ❡ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥❛ s❡çã♦ s❡❣✉✐♥t❡✱ q✉❛♥❞♦ ❞❡♠♦♥str❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞❡ ❙♦❧✉çõ❡s✳
❉❡✜♥✐çã♦ ✷✳✸✳ ❯♠❛ ❢✉♥çã♦ f :U ⊂R1+d→Rd é ❧✐♣s❝❤✐t③✐❛♥❛ ♥❛ s❡❣✉♥❞❛ ✈❛r✐á✈❡❧
s❡ ❡①✐st✐r C ≥0 t❛❧ q✉❡
||f(t, x1)−f(t, x2)|| ≤C||x1−x2||,
♣❛r❛ q✉❛✐sq✉❡r ❞♦✐s ♣♦♥t♦s(t, x1)❡ (t, x2)❞❡U ❝✉❥❛s ♣r✐♠❡✐r❛s ❝♦♦r❞❡♥❛❞❛s sã♦ ✐❣✉❛✐s✳
◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ C é ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ▲✐♣s❝❤✐t③ ❡ q✉❡ f(t, x) s❛t✐s❢❛③ ✉♠❛
Pr❡❧✐♠✐♥❛r❡s ✷✺
❉❡✜♥✐çã♦ ✷✳✹✳ ❯♠❛ ♠étr✐❝❛ ♥✉♠ ❝♦♥❥✉♥t♦ M é ✉♠❛ ❢✉♥çã♦ d : M ×M → R q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣❛r ♦r❞❡♥❛❞♦ ❞❡ ❡❧❡♠❡♥t♦s x, y ∈M ✉♠ ♥ú♠❡r♦ r❡❛❧d(x, y)❝❤❛♠❛❞♦ ❞❡
❞✐stâ♥❝✐❛ ❞❡x❛y✱ ❞❡ ♠♦❞♦ q✉❡ sã♦ s❛t✐s❢❡✐t❛s ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ♣❛r❛ q✉❛✐sq✉❡r x, y, z ∈M✿
✶✳ d(x, x) = 0 ❀
✷✳ ❙❡ x6=y ❡♥tã♦ d(x, y)>0❀
✸✳ d(x, y) = d(y, x)❀
✹✳ d(x, z)≤d(x, y) +d(y, z)
❊①❡♠♣❧♦ ✷✳✸✳ ❆ ❢✉♥çã♦ d(x1, x2) =
p
(a1−b1)2+ (a2 −b2)2+...+ (ad−bd)2,
♦♥❞❡ x1 = (a1, a2, ..., ad) ❡ x2 = (b1, b2, ..., bd) sã♦ ❡❧❡♠❡♥t♦s ❞❡ Rd ❞❡✜♥❡ ✉♠❛ ♠étr✐❝❛
❡♠ Rd✱ ❝❤❛♠❛❞❛ ❞❡ ♠étr✐❝❛ ❡✉❝❧✐❞✐❛♥❛✳
❯♠ ❡s♣❛ç♦ ♠étr✐❝♦ é ✉♠ ♣❛r (M, d) ♦♥❞❡ M é ✉♠ ❝♦♥❥✉♥t♦ ❡ d é ✉♠❛ ♠étr✐❝❛ ❡♠ M✳ ❙❡ (M, d) é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ ❡♥tã♦ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ S ⊂ M ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❞❡s❞❡ q✉❡ s❡❥❛ ❝♦♥s✐❞❡r❛❞❛ ❛ r❡str✐çã♦ ❞❡ d ❛ S×S✳ ◗✉❛♥❞♦ s❡ ❢❛③ ✐st♦✱ ❛ ♠étr✐❝❛ ❞❡ S é ❞✐t❛ ✐♥❞✉③✐❞❛ ♣❡❧❛ ❞❡ M✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ M ♦✉ S ♣♦❞❡♠ s❡r ❞❡ ♥❛t✉r❡③❛ ❜❛st❛♥t❡ ❞✐st✐♥t❛✿ ♣♦❞❡♠ s❡r ♥ú♠❡r♦s✱ ❢✉♥çõ❡s✱ ♠❛tr✐③❡s✱ ❝♦♥❥✉♥t♦s ❡ sã♦ ❝❤❛♠❛❞♦s s✐♠♣❧❡s♠❡♥t❡ ❞❡ ♣♦♥t♦s✳
❯♠❛ s❡q✉ê♥❝✐❛(xn)n∈N ❞❡ ♣♦♥t♦s ❡♠ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦M é ❞✐t❛ ❝♦♥✈❡r❣❡♥t❡ s❡x
s❡ ♣❛r❛ t♦❞❛ ❜♦❧❛ ❛❜❡rt❛✸ B t❛❧ q✉❡x∈B ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ í♥❞✐❝❡sj t❛✐s q✉❡ xj ♥ã♦ ♣❡rt❡♥❝❡ ❛B✳ ❯♠❛ s❡q✉ê♥❝✐❛(xn)n∈N❝❤❛♠❛✲s❡ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② s❡ ♣❛r❛
t♦❞♦ ǫ > 0 ❞❛❞♦✱ ❡①✐st✐r n0 ∈ N t❛❧ q✉❡ m, n > n0 ⇒ d(xm, xn) < ǫ✳ ❚♦❞❛ s❡q✉ê♥❝✐❛
❞❡ ❈❛✉❝❤② é ❧✐♠✐t❛❞❛✳ ❯♠ ❡s♣❛ç♦ ♠étr✐❝♦ M é ❝❤❛♠❛❞♦ ❞❡ ❝♦♠♣❧❡t♦ q✉❛♥t♦ t♦❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ M é ❝♦♥✈❡r❣❡♥t❡✳
❊①❡♠♣❧♦ ✷✳✹✳ ❖ ❝♦♥❥✉♥t♦ Rdé ✉♠ ❡s♣❛ç♦ ❝♦♠♣❧❡t♦ ♣❛r❛ q✉❛❧q✉❡r d≥1✳ ❆ ❞❡♠♦♥s✲
tr❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷❪✳
❉❛❞♦s q✉❛✐sq✉❡r s✉❜❝♦♥❥✉♥t♦sE ⊆Rk ❡ F ⊆Rm ❞♦t❛❞♦s ❞❛s r❡s♣❡❝t✐✈❛s ♠étr✐❝❛s
✐♥❞✉③✐❞❛s✱ ❞❡♥♦t❛♠♦s ♣♦r B0(E, F) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❛♣❧✐❝❛çõ❡s f : E → F q✉❡
sã♦ ❧✐♠✐t❛❞❛s✱ ✐st♦ é✱ t❛❧ q✉❡ ❛ ✐♠❛❣❡♠ f(E)⊆F é ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ ❞❡ Rm✳ ❊♠
B0(E, F) ❞❡✜♥✐♠♦s ❛ ♠étr✐❝❛ ❞♦ s✉♣r❡♠♦ ♦✉ ♠étr✐❝❛ ✉♥✐❢♦r♠❡ ♣♦r
d(f, g) = sup
x∈E||
f(x)−g(x)||,
✸P♦r ❞❡✜♥✐çã♦✱ s❡ ♦ ♣❛r (X, d)é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ ❡♥tã♦ ❛ ❜♦❧❛ ❛❜❡rt❛ ❝❡♥tr❛❞❛ ❡♠x∈X ❡
✷✻ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s
♣❛r❛ f, g ∈ B0(E, F)✳ ➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ B0(E, F) ❝♦♠ ❛ ♠étr✐❝❛ ❞♦ s✉♣r❡♠♦ é
✉♠ ❡s♣❛ç♦ ♠étr✐❝♦❀ ♠❛✐s ❛✐♥❞❛✿ tr❛t❛✲s❡ ❞❡ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❝♦♥❢♦r♠❡ ❛ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦✳
Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡ F ⊆ Rm é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ ❡♥tã♦ ♣❛r❛ q✉❛❧q✉❡r E ⊆Rk ♦ ❡s♣❛ç♦ ♠étr✐❝♦ B0(E, F) é ❝♦♠♣❧❡t♦ ❝♦♠ ❛ ♠étr✐❝❛ ✉♥✐❢♦r♠❡✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛(fn)n∈N✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ✭❡ ♣♦rt❛♥t♦ ❧✐♠✐t❛❞❛✮ ❡♠B0(E, F)✳
❋✐①❛♥❞♦ x∈E✱ ❛ s❡q✉ê♥❝✐❛ (xn)❡♠ q✉❡ xn =fn(x)∈F t❛♠❜é♠ é ❞❡ ❈❛✉❝❤② ♣♦✐s
||fn(x)−fm(x)|| ≤d(fn, fm).
❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ F é ❝♦♠♣❧❡t♦✱ ❡①✐st❡ ❡♠ F ♦ ❧✐♠✐t❡ y = limfn(x)✱ q✉❡ ❞❡✜♥❡
✉♠❛ ❢✉♥çã♦ y = f(x) ❞❡ E ❡♠ F✳ ❱❛♠♦s ♠♦str❛r q✉❡ f é ❧✐♠✐t❛❞❛✱ ♦✉ s❡❥❛✱ q✉❡ f é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ B0(E, F) ❡ q✉❡ ❛ s❡q✉ê♥❝✐❛ fn ❝♦♥✈❡r❣❡ ♣❛r❛ f ♥♦ ❡s♣❛ç♦ B0(E, F)✳
❉❡ ❢❛t♦✱ s❡♥❞♦ (fn) ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛✱ ♣♦❞❡♠♦s t♦♠❛r g0 :E →F ❝♦♥st❛♥t❡
q✉❛❧q✉❡r ❡ r >0t❛✐s q✉❡ d(fn, g0)≤r ♣❛r❛ ❝❛❞❛ n∈N✳ ❉❛❞♦ x∈E s❡❣✉❡ q✉❡
||f(x)−g0(x)||= lim||fn(x)−g0(x)|| ≤limd(fn, g0)≤r
❡ ♣♦rt❛♥t♦✱ d(f, g0) ≤ r ❡ f ∈ B0(E, F)✳ P❛r❛ t❡r♠✐♥❛r✱ ❞❛❞♦ ǫ > 0 t♦♠❡♠♦s N t❛❧
q✉❡ d(fk, fn) <
ǫ
2 ♣❛r❛ q✉❛✐sq✉❡r k, n ≥ N ❝♦♠ k, n, N ∈ N✳ ❋✐①❛♥❞♦ ❛❣♦r❛ x ∈ E✱
❞❡❝♦rr❡ q✉❡
||f(x)−fn(x)||= lim
k→+∞||fk(x)−fn(x)|| ≤
ǫ
2 < ǫ.
❉❡st❡ ♠♦❞♦✱ d(f, fn) < ǫ ♣❛r❛ ❝❛❞❛ n ≥ N✱ ♦✉ s❡❥❛✱ limd(fn, f) = 0 q✉❛♥❞♦
n→ ∞✳
❊①❡♠♣❧♦ ✷✳✺✳ ❙❡❥❛ X ♦ ❝♦♥❥✉♥t♦ ❞❛s ❝✉r✈❛s γ : I →Rd ❝♦♥tí♥✉❛s ❡ ❧✐♠✐t❛❞❛s ❝♦♠ ❛
♠étr✐❝❛ ❞♦ s✉♣r❡♠♦✳ ❊♥tã♦(X, d)é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦ ❝♦♥❢♦r♠❡ ❛ Pr♦♣♦s✐çã♦
✷✳✶ ♣♦✐s Rd é ❝♦♠♣❧❡t♦✳
❙❡❥❛ f : M → M ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡✜♥✐❞❛ ♥✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ M q✉❛❧q✉❡r✳ ❯♠ ♣♦♥t♦ ✜①♦ ❞❛ ❛♣❧✐❝❛çã♦f é ✉♠ ♣♦♥t♦x∗ t❛❧ q✉❡ f(x∗) =x∗✳
❉❡✜♥✐çã♦ ✷✳✺✳ ❯♠❛ ❛♣❧✐❝❛çã♦ g : M → M ❝❤❛♠❛✲s❡ ❝♦♥tr❛çã♦ q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ r❡❛❧ 0≤c <1 t❛❧ q✉❡
d(f(x), f(y))≤cd(x, y),
♣❛r❛ q✉❛✐sq✉❡r x, y ∈M✳
Pr❡❧✐♠✐♥❛r❡s ✷✼
▲❡♠❛ ✷✳✶✳ ❙❡❥❛♠ M ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❡ f : M → M ✉♠❛ ❝♦♥tr❛çã♦✳ ❊♥tã♦ f é ✭✉♥✐❢♦r♠❡♠❡♥t❡✮ ❝♦♥tí♥✉❛✳
❚❡♦r❡♠❛ ✷✳✶✳ ✭❞♦ ♣♦♥t♦ ✜①♦ ❞❡ ❇❛♥❛❝❤✮ ❙❡ M é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ t♦❞❛ ❝♦♥tr❛çã♦ f : M → M ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ✜①♦ ❡♠ M✳ Pr❡❝✐s❛♠❡♥t❡✱ ❡s❝♦❧❤❡♥❞♦ ✉♠ ♣♦♥t♦ x0 ∈ M q✉❛❧q✉❡r ❡ ❝♦❧♦❝❛♥❞♦ x1 = f(x0)✱ x2 = f(x1)✱✳✳✳✱xn+1 = f(xn)✱✳✳✳ ❛
s❡q✉ê♥❝✐❛ (xn) ❝♦♥✈❡r❣❡ ❡♠ M ❡ x∗ = limxn é ♦ ú♥✐❝♦ ♣♦♥t♦ ✜①♦ ❞❡ f✳
❉❡♠♦♥str❛çã♦✳ ❆❞♠✐t❛ q✉❡ ❛ s❡q✉ê♥❝✐❛ (xn) ❝♦♥✈✐r❥❛ ♣❛r❛ ✉♠ ♣♦♥t♦ x∗ ∈M ✳ ❈♦♠♦
f é ✉♥✐❢♦r♠❡♠❡♥t❡ ❝♦♥tí♥✉❛ ✭♣❡❧♦ ▲❡♠❛ ✷✳✶✮ ❞❡✈❡♠♦s t❡r
f(x∗) = f(lim(xn)) = limf(xn) = limf(xn) = limxn+1 =x∗,
♦✉ s❡❥❛✱ x∗ ∈M é ✉♠ ♣♦♥t♦ ✜①♦ ❞❡ f✳
P❛r❛ ❛ ♣r♦✈❛ ❞❛ ✉♥✐❝✐❞❛❞❡✱ s✉♣♦♥❤❛ q✉❡ f ❛❞♠✐t❛ ❞♦✐s ♣♦♥t♦s ✜①♦s ❞✐st✐♥t♦s✱ ♦✉ s❡❥❛✱ s✉♣♦♥❤❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ x∗, x∗∗ ∈M t❛✐s q✉❡ f(x∗) =x∗ ❡ f(x∗∗) =x∗∗❀ ❡♥tã♦
d(x∗, x∗∗) =d(f(x∗), f(x∗∗))≤cd(x∗, x∗∗)
❞❡ ♦♥❞❡ (1−c)d(x∗, x∗∗)≤0✳
❈♦♠♦1−c >0✭❞❡ ♦♥❞❡ 0< c <1✮ s❡❣✉❡ q✉❡cd(x∗, x∗∗) = 0 ❡ ♣♦rt❛♥t♦x∗ =x∗∗✳
◗✉❛♥t♦ à ❡①✐stê♥❝✐❛✱ ♥♦t❡ q✉❡(xn)é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ✭❡ ♣♦rt❛♥t♦ ❝♦♥✈❡r✲
❣❡♥t❡ ❡♠ M✱ ♣♦✐sM é ❝♦♠♣❧❡t♦✮❀ ❞❡ ❢❛t♦✱ ♦❜s❡r✈❡ q✉❡
d(x1, x2) = d(f(x0), f(x1))≤cd(x0, x1),
d(x2, x3) =d(f(x1), f(x2))≤cd(x1, x2)≤c2d(x0, x1),
❡ s❡❣✉❡ ✐♥❞✉t✐✈❛♠❡♥t❡ q✉❡ ♣❛r❛ q✉❛❧q✉❡r n∈N ✱ d(xn, xn+1)≤cnd(x0, x1).
P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡rn, p∈N✱ t❡♠✲s❡
d(xn, xn+p)≤d(xn, xn+1) +d(xn+1, xn+2)...+d(xn+p−1, xn+p)
≤[cn+cn+1+...+cn+p−1]d(x0, x1),
♦✉ s❡❥❛✱
d(xn, xn+p)≤
cn
1−cd(x0, x1). ❈♦♠♦ limcn = 0 ✭♣♦✐s 0 ≤ c < 1✮✱ ❛ s❡q✉ê♥❝✐❛ (x
n) é ❞❡ ❈❛✉❝❤② ❡♠ M ❡ ❛
✷✽ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s
✷✳✷ ❊①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s
❯♠ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ✭❛❜r❡✈✐❛❞❛♠❡♥t❡ P❱■✮ ♦✉ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♥s✐st❡ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ s✉❥❡✐t❛ à ✉♠❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✳
❉❡st❡ ♠♦❞♦✱ tr❛t❛✲s❡ ❞❡ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r x(t) t❛❧ q✉❡ ❡♠ t0 s❡✉
✈❛❧♦r s❡❥❛ x0✿
dx
dt =F(t, x),
x(t0) =x0.
✭✷✳✾✮
❉❡♠♦♥str❛r❡♠♦s ❛ s❡❣✉✐r ✉♠ t❡♦r❡♠❛ q✉❡ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦✲ ❧✉çã♦ ♣❛r❛ ✭✷✳✾✮ ♠❡❞✐❛♥t❡ ❤✐♣ót❡s❡s ❝♦♥✈❡♥✐❡♥t❡s✱ s❡❣✉♥❞♦ ❛ r❡❢❡rê♥❝✐❛ ❬✸❪✳
❚❡♦r❡♠❛ ✷✳✷✳ ✭❊①✐stê♥❝✐❛ ❡ ❯♥✐❝✐❞❛❞❡ ❞❡ ❙♦❧✉çõ❡s✮ ❙❡❥❛ F : U → Rd ♦♥❞❡ U é ✉♠
❛❜❡rt♦ ❞❡ R1+d ❝♦♥tí♥✉❛ ❡ ❧♦❝❛❧♠❡♥t❡ ❧✐♣s❝❤✐t③✐❛♥❛ ♥❛ s❡❣✉♥❞❛ ✈❛r✐á✈❡❧✳ ❊♥tã♦✿
✶✳ P❛r❛ t♦❞♦ (t0, x0) ∈ U ❡①✐st❡ γ : I ⊂ R → Rd✱ s♦❧✉çã♦ ❞❡ x′ = F(t, x) ❝♦♠
γ(t0) = x0❀
✷✳ ❙❡ γ1 :I1 →Rd ❡ γ2 : I2 →Rd sã♦ s♦❧✉çõ❡s ❞❡ x′ =F(t, x) ❡ ❡①✐st❡ t0 ∈I1TI2
t❛❧ q✉❡ γ1(t0) =γ2(t0)✱ ❡♥tã♦ γ1(t) = γ2(t) ♣❛r❛ t♦❞♦ t ∈I1TI2✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ δ > 0 t❛❧ q✉❡✹ Q = Bδ(t0)×Bδ(x0) ⊂ U✱ C ❛ ❝♦♥st❛♥t❡ ❞❡
❧✐♣s❝❤✐t③ ❡ Mδ = sup
(t,y)∈Q
||F|| ❡♠ Q✳ ❙❡❥❛ ❛✐♥❞❛ ǫ >0t❛❧ q✉❡
ǫ≤δ, ǫ < 1 C, ǫ <
δ Mδ
. ✭✷✳✶✵✮
❉❡✜♥❛♠♦sX ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝✉r✈❛s ❝♦♥tí♥✉❛s ❞♦ ✐♥t❡r✈❛❧♦(t0−ǫ, t0+ǫ)
❡♠ Rd ❝✉❥❛s ✐♠❛❣❡♥s ♣❡rt❡♥❝❡♠ à Bδ(x0) ♠✉♥✐❞♦ ❞❛ ♠étr✐❝❛ d ❞♦ s✉♣r❡♠♦✱ ✐st♦ é✱ X ={γ(t)∈Bδ(x0)|γ : (t0−ǫ, t0+ǫ)→Rd ❝♦♥tí♥✉❛✱ ❝♦♠ γ(t0) =x0}.
❙❡❣✉❡ ❞♦ ❊①❡♠♣❧♦ ✭✶✳✶✮ q✉❡ ♦ ❝♦♥❥✉♥t♦ X✱ ♠✉♥✐❞♦ ❞❛ ♥♦r♠❛ d ✭ ❞❡♥♦t❛❞♦ ♣♦r
(X, d)✮ é ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✳ ❊♠(X, d)✈❛♠♦s ❞❡✜♥✐r ♦ ♦♣❡r❛❞♦rF :X → C
♣♦r
F(γ(t)) = x0+
Z t
t0
F(s, γ(s))ds ✭✷✳✶✶✮ ❡ ♦❜s❡r✈❛r q✉❡ ♥ã♦ ❡①✐st❡♠ ♣r♦❜❧❡♠❛s ❞❡ ✐♥t❡❣r❛çã♦ ✉♠❛ ✈❡③ q✉❡F(s, γ(s))é ❝♦♥tí♥✉❛
❡ ❧✐♠✐t❛❞❛ ♣❛r❛ s∈[t0, t]✱ ♦ q✉❡ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❛ ✐♥t❡❣r❛❧✳
❆❧é♠ ❞✐ss♦✱ ♦ ♦♣❡r❛❞♦r F ❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ✐st♦ é✱ F(γ)∈ X, ∀γ ∈ X✱ ♣♦✐s
✹P♦rQ❡st❛♠♦s ❞❡♥♦t❛♥❞♦ ♦ ❢❡❝❤♦ ❞♦ ❝♦♥❥✉♥t♦Q✱ ✐st♦ é✱ ❛ ✉♥✐ã♦ ❞❡Q❝♦♠ t♦❞♦s ♦s s❡✉s ♣♦♥t♦s
❊①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ✷✾
✶✳ ❖ ♦♣❡r❛❞♦rF(γ(t)) é ❝♦♥tí♥✉♦ ♣❛r❛t ∈(t0−ǫ, t0+ǫ)❀
✷✳ F(γ(t0)) =x0✱ ♣♦✐s
F(γ(t0)) =x0+
Z t0
t0
F(s, γ(s))ds=x0+ 0 =x0.
✸✳ F(γ(t))∈Bδ(x0)✱ ♣♦✐s
||F(γ(t))−x0||=||
Z t
t0
F(s, γ(s))ds|| ≤Mδ|t−t0|< Mδǫ < δ.
❖ ♦♣❡r❛❞♦r ❞❛❞♦ ♣♦r ✭✷✳✶✶✮ é ✉♠❛ ❝♦♥tr❛çã♦ ❡♠X✱ ✐st♦ é✱ ❡①✐st❡0≤λ <1t❛❧ q✉❡
d(F(γ1(t),F(γ2(t))≤λd(γ1(t), γ2(t)),
♣❛r❛ q✉❛✐sq✉❡r γ1, γ2 ∈ X ❡t ∈(t0, t0+ǫ)✳ ❉❡ ❢❛t♦✱ ❞❡ ✭✷✳✶✵✮ t❡♠♦s
||F(γ1(t)− F(γ2(t))||=||
Z t
t0
F(s, γ1(s))−F(s, γ2(s))ds|| ≤
Z t
t0
C||γ1(s)−γ2(s)||ds≤C|t−t0|d(γ1(t), γ2(t))≤
Cǫd(γ1(t), γ2(t))
❡ ❜❛st❛ t♦♠❛r λ=Cǫ✳
❖r❛✱ s❡♥❞♦ ✭✷✳✶✶✮ ✉♠❛ ❝♦♥tr❛çã♦ ♥✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠♣❧❡t♦✱ s❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ✭✶✳✶✮ q✉❡ ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❝✉r✈❛ γ∗ ∈ X t❛❧ q✉❡ F(γ∗(t)) =γ∗(t)✱ ♦✉ s❡❥❛✱
γ∗(t) =x0+
Z t
t0
F(s, γ∗(s))ds ∀t∈(t0−ǫ, t0+ǫ).
❋✐♥❛❧♠❡♥t❡✱ ♥♦t❛♥❞♦ q✉❡ ✭✷✳✷✮ é ❞✐❢❡r❡♥❝✐á✈❡❧✱ s❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ q✉❡
(γ∗)′(t) =F(t, γ∗(t)), ∀t∈(t0−ǫ, t0+ǫ). ✭✷✳✶✷✮
■st♦ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ✐t❡♠1✳
P❛r❛ ❛ ♣r♦✈❛ ❞♦ ✐t❡♠2 s❡❥❛♠ γ1 :I1 → Rd ❡ γ2 : I2 →Rd s♦❧✉çõ❡s ❞❡ x′ =F(t, x)
❡ s❡❥❛ t0 ∈I1∩I2 ❝♦♠ γ1(t0) =γ2(t0) =x0✳
❉❡✜♥✐♥❞♦
J ={t ∈I1 ∩I2 :γ1(t) =γ2(t)},
✈❡♠♦s q✉❡✿
✶✳ J é ♥ã♦ ✈❛③✐♦✱ ♣♦✐s t0 ∈J❀
✸✵ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s
✸✳ J é ❛❜❡rt♦❀ ♣❛r❛ ✈❡r ✐ss♦✱ ❜❛st❛ t♦♠❛r s0 ∈J ❡ ❝♦♥s✐❞❡r❛r y0 =γ1(s0) = γ2(s0)✳
P❛r❛ (s0, y0)✱ r❡♣❡t✐♥❞♦ ♦ ❛r❣✉♠❡♥t♦ ✉s❛❞♦ ♥❛ ♣r♦✈❛ ❞♦ ✐t❡♠ 1 ❞❡st❡ t❡♦r❡♠❛✱
♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ǫ > 0 t❛❧ q✉❡ F : X → X é ✉♠❛ ❝♦♥tr❛çã♦ ❡ ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ✜①♦✱ ❞❡ ♦♥❞❡ γ1(t) = γ2(t) ♣❛r❛ t ∈ (s0 − ǫ, s0 +ǫ) ❡ ♣♦rt❛♥t♦
(s0−ǫ, s0+ǫ)∈J✳
P❡❧♦s ✐t❡♥s 1✱ 2❡ 3❛❝✐♠❛✱ J =I1∩I2 ❡ ✐st♦ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✳
❆ ❞❡♠♦♥str❛çã♦ ❛❝✐♠❛ é ❝♦♥str✉t✐✈❛✱ ♣♦✐s ✭✷✳✶✷✮ ❡st❛❜❡❧❡❝❡ q✉❡ ❛ ú♥✐❝❛ s♦❧✉çã♦ γ(t) ❞❛ ❡q✉❛çã♦ x′ = F(t, x) s✉❥❡✐t❛ ❛ γ(t0) = x0 é ♦ ♣♦♥t♦ ✜①♦ ❞♦ ♦♣❡r❛❞♦r F q✉❡
♣♦❞❡ s❡r ♦❜t✐❞♦ ❛tr❛✈és ❞❡ s✉❝❡ss✐✈❛s ✐t❡r❛çõ❡s✱ ❝♦♠♦ ❢❡✐t♦ ♥❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛(1.1)✳
■❧✉str❛♠♦s ❡st❛ ❛✜r♠❛çã♦ ♥♦ ❊①❡♠♣❧♦ ✭✷✳✻✮✳
❊①❡♠♣❧♦ ✷✳✻✳ ❈♦♥s✐❞❡r❡ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ x′ = 2tx s✉❥❡✐t❛ à ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧
(t0, x0) = (0,1)✳
❉❡✜♥✐♥❞♦ ♦ ♦♣❡r❛❞♦r ❝♦♠♦ ❡♠ ✭✷✳✶✶✮ ♣❛r❛ ❡st❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✱ t❡♠♦s
F(γ(t)) = 1 +
Z t
0
2sγ(s)ds.
❚♦♠❛♥❞♦ q✉❛❧q✉❡r ❝✉r✈❛ γ(t) t❛❧ q✉❡ γ(0) = 1 ❝♦♠♦ ♣r✐♠❡✐r❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ❛
s♦❧✉çã♦✱ t❡♠♦s ❣❛r❛♥t✐❞❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞❛s s✉❝❡ss✐✈❛s ✐t❡r❛❞❛s ❞♦ ♦♣❡r❛❞♦r F ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡ x′ = 2tx✳ ❈♦♥s✐❞❡r❛r❡♠♦s ❛ ♠❛✐s s✐♠♣❧❡s ❞❛s ❝✉r✈❛s ❝♦♠γ(0) = 1✱ ❛ ❝✉r✈❛
❝♦♥st❛♥t❡ γ0(t)≡1❡ ❡♥tã♦
γ1(t) = F(γ0(t)) = 1 +
Z t
0
2sds= 1 +t2,
γ2(t) = F(γ1(t)) = 1 +
Z t
0
2s(1 +s2)ds= 1 +t2+t
4
2,
❡ ♣♦r ✐♥❞✉çã♦ t❡♠♦s
γn(t) = 1 +t2+
t4
2 +
t6
6 +...+
t2n
n! +... ✭✷✳✶✸✮
❋✐♥❛❧♠❡♥t❡✱ ❢❛③❡♥❞♦ n→ ∞ ❡♠ ✭✷✳✶✸✮ s❡❣✉❡ q✉❡ γ(t) = et2
é ❛ s♦❧✉çã♦ ❞❡ x′ = 2tx s✉❥❡✐t❛ à (t
0, x0) = (0,1)✱ ❝♦♠♦ é ❢á❝✐❧ ❞❡ ✈❡r✐✜❝❛r✳
❖❜s❡r✈❛çã♦ ✷✳✶✳ ➱ ❝♦♠✉♠ s✉❜st✐t✉✐r♠♦s ❛ ❤✐♣ót❡s❡ ❞❛ ❢✉♥çã♦ F : U → Rd s❡r
❧✐♣s❝❤✐t③✐❛♥❛ ♥❛ s❡❣✉♥❞❛ ✈❛r✐á✈❡❧ ♣❡❧❛ ❤✐♣ót❡s❡ ♠❛✐s ❝❧áss✐❝❛ ❞❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❡s♣❛❝✐❛❧ ∂F
❆❧❣✉♠❛s té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦ ✸✶
❖❜s❡r✈❛çã♦ ✷✳✷✳ ❊①✐❣✐♥❞♦ ♥♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ❚❡♦r❡♠❛ ✭✷✳✷✮ ❛♣❡♥❛s ❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❢✉♥çã♦ F :U →Rd ❡ r❡❧❛①❛♥❞♦ ❛ ❡①✐❣ê♥❝✐❛ ❞❡ s❡r ▲✐♣s❝❤✐t③✱ ❛✐♥❞❛ é ♣♦ssí✈❡❧ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s✱ ♠❛s s❡ ♣❡r❞❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ✉♥✐❝✐❞❛❞❡✳
✷✳✸ ❆❧❣✉♠❛s té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦
❆♣❡s❛r ❞❡ ♥ã♦ ❡①✐st✐r ✉♠ ♠ét♦❞♦ ❣❡r❛❧ ♣❛r❛ s❡ ❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ♣❛r❛ ❛❧❣✉♥s t✐♣♦s ♣❛rt✐❝✉❧❛r❡s é ♣♦ssí✈❡❧ ❡♥❝♦♥trá✲❧❛ s❡❥❛ ❞❡ ❢♦r♠❛ ❡①♣❧í❝✐t❛ ♦✉ ✐♠♣❧í❝✐t❛✳
◆❛ s❡q✉ê♥❝✐❛ ❛♣r❡s❡♥t❛r❡♠♦s ❛ s♦❧✉çã♦ ❞❡ ❛❧❣✉♥s t✐♣♦s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s❀ ❡♠ ♣❛rt✐❝✉❧❛r tr❛t❛r❡♠♦s ❞❡ ❛❧❣✉♥s ❝❛s♦s ❡s❝❛❧❛r❡s✱ ♥♦s ❜❛s❡❛♥❞♦ ♥❛ r❡❢❡rê♥❝✐❛ ❬✹❪✳ ❆♣❡s❛r ❞❛ ✈❛r✐❡❞❛❞❡ ❞❡ té❝♥✐❝❛s ❡♠♣r❡❣❛❞❛s ♦❜s❡r✈❛♠♦s q✉❡ ❛ s♦❧✉çã♦ ❡♥❝♦♥tr❛❞❛ ❡♠ ❝❛❞❛ ❝❛s♦ é ❞❡ ❢❛t♦ ❛ ú♥✐❝❛ s♦❧✉çã♦ ❡①✐st❡♥t❡ ♦ q✉❡ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞♦ ❞✐r❡t❛♠❡♥t❡ ♣♦r ❞❡r✐✈❛çã♦ ❡ ❥✉st✐✜❝❛❞♦ ♣❡❧♦ ❚❡♦r❡♠❛ ✭✷✳✷✮✳
✷✳✸✳✶ ❱❛r✐á✈❡✐s s❡♣❛rá✈❡✐s
❯♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ é ❝❤❛♠❛❞❛ ❞❡ s❡♣❛rá✈❡❧ s❡ ♣♦ss✉✐r ❛ ❢♦r♠❛ x′(t) = f(t)
g(x) ❝♦♠ g(x)6= 0, ✭✷✳✶✹✮
♦♥❞❡ ❛s ❢✉♥çõ❡s f ❡g sã♦ ❝♦♥tí♥✉❛s ❡♠ ❛❧❣✉♠ ✐♥t❡r✈❛❧♦ ❞❡ R✳ ❚r❛t❛♥❞♦ ♦ sí♠❜♦❧♦ dx
dt ❝♦♠♦ ✉♠ q✉♦❝✐❡♥t❡ ❢♦r♠❛❧✺ g(x)dx=f(t)dt, ❡ ♣♦rt❛♥t♦✱ ❛ ❢❛♠í❧✐❛ ❞❡ ❝✉r✈❛s ✐♥t❡❣r❛✐s ❞❡ ✭✷✳✶✹✮ s❛t✐s❢❛③
Z
g(x)dx=
Z
f(t)dt.
❊①❡♠♣❧♦ ✷✳✼✳ ❙✉♣♦♥❤❛ q✉❡ ✉♠❛ ♣♦♣✉❧❛çã♦ ❞❡ P ❛♥✐♠❛✐s ✈✐✈❛ ✐s♦❧❛❞❛ ❡ q✉❡ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❝♦♥t❛♠✐♥❛❞♦s x(t) ♣♦r ✉♠❛ ❝❡rt❛ ❞♦❡♥ç❛ ♥♦ ✐♥st❛♥t❡ t s❡❥❛ ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s s❛❞✐♦s ❡ ❛♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s ❝♦♥t❛♠✐♥❛❞♦s✳ ➱ ♣♦ssí✈❡❧ ♣r❡✈❡r q✉❡✱ ♥❡st❛s ❝♦♥❞✐çõ❡s✱ t♦❞❛ ❛ ♣♦♣✉❧❛çã♦ s❡ ❝♦♥t❛♠✐♥❛rá ♣❡❧❛ ❞♦❡♥ç❛ ♣❛r❛ t s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✳
❉❡ ❢❛t♦✱ s❡♥❞♦P ♦ t❛♠❛♥❤♦ ❞❛ ♣♦♣✉❧❛çã♦ ✭❝♦♥s✐❞❡r❛❞♦ ❝♦♥st❛♥t❡✮ ❡x(t)♦ ♥ú♠❡r♦
❞❡ ✐♥❞✐✈í❞✉♦s ❞♦❡♥t❡s✱ t❡♠♦s q✉❡ P −x(t) r❡♣r❡s❡♥t❛ ♦ ♥ú♠❡r♦ ❞❡ ✐♥❞✐✈í❞✉♦s s❛❞✐♦s✳
❙✉♣♦♥❞♦ ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ dx
dt ♣r♦♣♦r❝✐♦♥❛❧ ❛ x(t) ❡ ❛ P −x(t) ❡♥tã♦ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ q✉❡ ♠♦❞❡❧❛ ♦ ♣r♦❜❧❡♠❛ ♥❛s ❤✐♣ót❡s❡s ❞❛❞❛s é
dx
dt =αx(P −x), ✭✷✳✶✺✮
✺❆ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ ❛ ✈❛❧✐❞❛❞❡ ❞❡ t❛❧ ♠❛♥✐♣✉❧❛çã♦ ❛❧❣é❜r✐❝❛ é ❛ t❡♦r✐❛ ❞❛s ❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s✳
❯♠❛ ❢♦r♠❛ ❞✐❢❡r❡♥❝✐❛❧ ❡♠R2 é ✉♠❛ ❢✉♥çã♦w: Ω
→(R2)∗ ♦♥❞❡Ωé ✉♠ ❛❜❡rt♦ ❞♦ ♣❧❛♥♦ ❡(R2)∗ é ♦
✸✷ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s
q✉❡ é ❞♦ t✐♣♦ ✭✷✳✶✹✮ ❡ ♣♦rt❛♥t♦ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ♥❛ ❢♦r♠❛ dx
x(P −x) =αdt.
P♦r ✐♥t❡❣r❛çã♦✱ t❡♠✲s❡
Z dx
x(P −x) =
Z
αdt,
❞❡ ♦♥❞❡
Z
1
P
1
x+
1
P −x
dx=αt+k1,
Z
1
x+
1
P −x
dx =P αt+k2,
ln x
P −x =P αt+k2, x
P −x =ke
P αt.
❋✐♥❛❧♠❡♥t❡✱ ✐s♦❧❛♥❞♦ ❛ ❢✉♥çã♦ x(t) ♥❛ ú❧t✐♠❛ ❞❛s ✐❣✉❛❧❞❛❞❡s ❛❝✐♠❛ ❡♥❝♦♥tr❛♠♦s ❛
s♦❧✉çã♦ ❞❡ ✭✷✳✶✺✮✿
x(t) = P 1 + 1
ke
−P αt
. ✭✷✳✶✻✮
❚♦♠❛♥❞♦ ♦ ❧✐♠✐t❡ ❞❡ ✭✷✳✶✻✮ q✉❛♥❞♦ t → ∞t❡♠♦s✿
lim
t→∞x(t) = limt→∞
P
1 + 1
ke
−P αt
=P,
❡ ♣♦rt❛♥t♦ ♦ ♥ú♠❡r♦x(t)❞❡ ❝♦♥t❛♠✐♥❛❞♦s t❡♥❞❡ ❛ P✳
✷✳✸✳✷ ❊q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ❡s❝❛❧❛r❡s ❞❡ ♣r✐♠❡✐r❛ ♦r✲
❞❡♠
❆ ❢♦r♠❛ ❣❡r❛❧ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❧✐♥❡❛r ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ é
x′(t) +p(t)x(t) =q(t), ✭✷✳✶✼✮ ♦♥❞❡ p, q : (a, b) → R sã♦ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❞❡ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ t ❞❡✜♥✐❞❛s ♥♦ ❛❜❡rt♦ (a, b) ❡ x(t) é ♥ã♦ ♥✉❧❛ ❡♠(a, b)✳
❆ s♦❧✉çã♦ ❣❡r❛❧ x(t) ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞❛ ♣❡❧♦ ♠ét♦❞♦ ❞♦ ❢❛t♦r ✐♥t❡❣r❛♥t❡✱
♦♥❞❡ ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ✭✷✳✶✼✮ é ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r ✉♠❛ ❢✉♥çã♦µ=µ(t)❛♣r♦♣r✐❛❞❛✱
❞❡ ♠❛♥❡✐r❛ q✉❡ ❛ ❢✉♥çã♦ ♦❜t✐❞❛ s❡❥❛ ❢❛❝✐❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳ ❙❡❥❛ ♣♦rt❛♥t♦ µ=µ(t) ✉♠
❢❛t♦r ✐♥t❡❣r❛♥t❡ ♣❛r❛ ✭✷✳✶✼✮❀ ❡♥tã♦
❆❧❣✉♠❛s té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦ ✸✸
■♠♣♦♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ q✉❡ ♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ ❞❡ ✭✷✳✶✽✮ s❡❥❛ ❛ ❞❡r✐✈❛❞❛ ❞♦ ♣r♦❞✉t♦ µ(t)x(t) ✭♦✉ s❡❥❛✱ q✉❡ ♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ s❡❥❛ ✐❣✉❛❧ ❛ µ′(t)x(t) +µ(t)x′(t)✮✱ s❡❣✉❡ ❞❡
✭✷✳✶✽✮ q✉❡
µ′(t)x(t) +µ(t)x′(t) = µ(t)x′(t) +µ(t)p(t)x(t), ❞❡ ♦♥❞❡ ✭✉♠❛ ✈❡③ q✉❡ ♣♦r ❤✐♣ót❡s❡ x(t) é ♥ã♦ ♥✉❧❛✮
µ′(t) = µ(t)p(t). ✭✷✳✶✾✮ ◆♦t❡ ❛❣♦r❛ q✉❡ ✭✷✳✶✾✮ é ✉♠❛ ❡q✉❛çã♦ s❡♣❛rá✈❡❧❀ r❡s♦❧✈❡♥❞♦✲❛ ♣❡❧♦ ♠ét♦❞♦ ❞❛ s❡çã♦ ❛♥t❡r✐♦r ❡♥❝♦♥tr❛♠♦s
µ(t) = eR0tp(s)ds. ✭✷✳✷✵✮
❆❣♦r❛✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✶✼✮ ♣♦r ✭✷✳✷✵✮ t❡♠♦s
eR0tp(s)dsx′(t) +e Rt
0p(s)dsp(t)x(t) = e Rt
0p(s)dsq(t),
❝✉❥♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ é ❛ ❞❡r✐✈❛❞❛ ❞♦ ♣r♦❞✉t♦ x(t)eR0tp(s)ds✳ ■♥t❡❣r❛♥❞♦ ❞❡ ❛♠❜♦s ♦s
❧❛❞♦s ♣♦❞❡♠♦s ❡s❝r❡✈❡r
x(t)eR0tp(s)ds =
Z
q(s)eR0sp(ξ)dξds+c.
❡ ✐s♦❧❛♥❞♦ ❛ ❢✉♥çã♦ x(t) ♦❜t❡♠♦s
x(t) = 1
eR0tp(s)ds
Z
q(s)eR0sp(ξ)dξds+c. ✭✷✳✷✶✮
❆ ❡q✉❛çã♦ ✭✷✳✷✶✮ t❡♠ ✐♥t❡r❡ss❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ t❡ór✐❝♦✱ ♣♦✐s ✐♥❞✐❝❛ q✉❡ é ♣♦ssí✈❡❧ ♦❜t❡r ❛s s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ✭✷✳✶✼✮✳ ❈♦♠♦ ❛s ✐♥t❡❣r❛çõ❡s ❡①✐st❡♥t❡s ❡♠ ✭✷✳✷✶✮ ♣♦❞❡♠ ♥ã♦ s❡r ❡①♣r❡ss❛s ❡♠ t❡r♠♦s ❞❡ ❢✉♥çõ❡s ❡❧❡♠❡♥t❛r❡s ❡♠ t♦❞♦s ♦s ❝❛s♦s✱ ♣♦❞❡ ♥ã♦ s❡r ♣♦ssí✈❡❧ ♦❜t❡r ✉♠❛ ❡①♣r❡ssã♦ ❡①♣❧í❝✐t❛ ♣❛r❛ ❛ s♦❧✉çã♦✳
❊①❡♠♣❧♦ ✷✳✽✳ ❆ ❡q✉❛çã♦
dx
dt −2x= 4−t ✭✷✳✷✷✮
é ❧✐♥❡❛r ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♠ p(t) =−2 ❡ q(t) = 4−t✳ P♦r ✭✷✳✶✾✮✱ ♦ ❢❛t♦r ✐♥t❡❣r❛♥t❡ s❡rá
µ(t) = eR0t−2ds =e−2t ✭✷✳✷✸✮
❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✷✳✷✷✮ ♣♦r ✭✷✳✷✸✮ s❡❣✉❡ q✉❡
e−2tdx
dt −2e
−2tx(t) = 4e−2t
−te−2t.
P♦r ✐♥t❡❣r❛çã♦✱ t❡♠♦s
Z
(x(t)e−2t)′dt=
Z
