✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ĩ ữỡ ị P❍×❒◆● ❚❘➐◆❍ ❚■➌◆ ❍➶❆ ◆Û❆ ❚❯❨➌◆ ❚➑◆❍ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ❍➔ ◆ë✐ ✲ ◆➠♠ ✷✵✶✼ ✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ữỡ ị Pì ❚■➌◆ ❍➶❆ ◆Û❆ ❚❯❨➌◆ ❚➑◆❍ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ❚❙✳ ▲➊ ❍❯❨ ❚■➍◆ ❍➔ ◆ë✐ ✲ ◆➠♠ ✷✵✶✼ ▲❮■ ❈❷▼ ❒◆ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t tỵ✐ t ữớ ữợ t t t ữợ t ❧✉➟♥ ✈➠♥ ♥➔②✳ ❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❑❤♦❛ ❚♦→♥ ✲ ❈ì✲ ❚✐♥ ❤å❝✱ P❤á♥❣ ✣➔♦ t↕♦✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✤➣ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ❝❛♦ ❤å❝✳ ❊♠ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ♥❣÷í✐ t❤➙♥ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❝ê ✈ơ✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❍➔ ◆ë✐✱ ♥❣➔② ✵✺ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✼ ❍å❝ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ❍÷ì♥❣ ✐ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✐ ▲í✐ ♥â✐ ✤➛✉ ✶ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✹ ✶✳✶ ✣à♥❤ ỵ rtr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❝♦♥ ê♥ ✤à♥❤ ✈➔ ❦❤æ♥❣ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ỵ rtr ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❇ê ✤➲ ●r♦♥✇❛❧❧✲❇❡❧❧♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸ ❚➼♥❤ ❧✐➯♥ tö❝ ❍☎♦❧❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹ ✣à♥❤ ỵ t ự ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹✳✶ ✣à♥❤ ❧➼ ✤✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹✳✷ Ù♥❣ ❞ư♥❣ ❝õ❛ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ t✐➳♥ ❤â❛ ♥û❛ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❈❤÷ì♥❣ ✷✳ ✣à♥❤ ỵ rtr ữỡ tr t õ ỷ t t ✶✼ ✷✳✶ ⑩♥❤ ①↕ ♥❤à ♣❤➙♥ ✈➔ ♥❤➙♥ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷ ✣➦t ✈➜♥ ✤➲✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Pt ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ự ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✺ ❈❤➼♥❤ ◗✉✐ ❍☎♦❧❞❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ❑➳t ❧✉➟♥ ✸✶ ✐✐ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✷ ✐✐✐ ▲❮■ ◆➶■ ỵ rtr ởt ổ ✈➔ ♣❤ê ❜✐➳♥ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ❤➺ ✤ë♥❣ ❧ü❝✳ ◆â ✤÷❛ r❛ ✤✐➲✉ ❦✐➺♥ ❦❤✐ ♥➔♦ ❝→❝ ♥❣❤✐➺♠ q✉❛♥❤ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❝õ❛ t õ õ trú tữỡ tỹ ợ ♥❣❤✐➺♠ q✉❛♥❤ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❤â❛ t÷ì♥❣ ù♥❣ ♠➔ t❛ ❝â t❤➸ t➻♠ ❜➡♥❣ ❝→❝❤ ①➙② ❞ü♥❣ ❤➺ ♠❛ tr➟♥ ❏❛❝♦❜✐❛♥ t↕✐ ✤✐➸♠ ❝➙♥ ❜➡♥❣✳ ❘ã r➔♥❣✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ s➩ ❞➵ ❞➔♥❣ ♥❣❤✐➯♥ ❝ù✉ ❤ì♥ r➜t ♥❤✐➲✉ s♦ ✈ỵ✐ ❤➺ ♣❤✐ t ỵ rtr ỏ ữủ ỵ t t õ ữỡ tr ♣❤➙♥ ❝â t❤➸ ❞✐➵♥ ✤↕t ♠ët ❝→❝❤ ♥❣➢♥ ❣å♥ ♥❤÷ s❛✉✳ ❚r♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❤②♣❡r❜♦❧✐❝ x∗ = ✱ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥ x = Ax + r(x), x(0) = x0 , tr♦♥❣ ✤â r ∈ C (Rd ), r(x) = o(|x|) t÷ì♥❣ ữỡ ợ ữỡ tr t t õ y = Ay ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ t❛ ❧✉ỉ♥ t➻♠ ✤÷đ❝ ♠ët s♦♥❣ →♥❤ ❧✐➯♥ tö❝ φ : Rd → Rd t❤ä❛ ♠➣♥✿ x(t; φ(x0 )) = φ(eAt x0 ) t ∈ R, x0 ∈ Rd ▼ö❝ ✤➼❝❤ ❝❤➼♥❤ ❝õ❛ t q tứ ỵ tt ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛♥❣ ♣❤÷ì♥❣ tr➻♥❤ t✐➳♥ ❤â❛ ♥û❛ t✉②➳♥ t➼♥❤✳ ❱➜♥ ✤➲ ♥➔② ❦❤æ♥❣ ❤➲ ❞➵ ❞➔♥❣ ợ trữớ ủ ữ ởt số ự ♠✐♥❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ♣❤↔✐ sû ❞ư♥❣ ỵ t rr ỵ ❧↕✐ ❦❤ỉ♥❣ sû ❞ư♥❣ ✤÷đ❝ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉✳ ●å✐ A ❧➔ t♦→♥ tû s✐♥❤ ❝õ❛ C0 ✲♥❤â♠ eAt tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✈➔ r : X X tử st ữỡ tỹ ợ trữớ ❤ñ♣ ❤ú✉ ❤↕♥✱ t❛ ❦✐➸♠ tr❛ t➼♥❤ ♥û❛ t✉②➳♥ t➼♥❤ ✈➔ t➼♥❤ t✉②➳♥ t➼♥❤ ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ❜❛♥ ✤➛✉✳ ❇➔✐ t♦→♥ t✉②➳♥ t➼♥❤ ✤÷đ❝ ①→❝ ✤à♥❤ tr♦♥❣ ❦❤✉ỉ♥ ỵ tt ỷ õ ữ t t ữỡ tr ỷ t t tỗ t ởt ✤õ tèt ❞✉② ♥❤➜t v(t, v0 )✳ ❱➻ ✈➟②✱ ❝➙✉ ❤ä✐ ✤➦t r❛ ❧➔ ♠è✐ ❧✐➯♥ ❦➳t ❣✐ú❛ ♥❣❤✐➺♠ u(t; u0 ) = eAt u0 ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ ✈➔ ♥❣❤✐➺♠ v(t; v0 ) ❝õ❛ ❜➔✐ t♦→♥ ♥û❛ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❣✐→ trà ❜❛♥ ✤➛✉✳ ❈➙✉ ❤ä✐ ♥➔② ❝ơ♥❣ ❧➔ ỵ rtr ữỡ tr t õ ỷ t t ữ ỵ r trứ tr÷í♥❣ ❤đ♣ sè ❝❤✐➲✉ ❧➔ ❤ú✉ ❤↕♥✱ ♠ët ♥❤â♠ C0 ❦❤ỉ♥❣ ❜❛♦ ❣✐í ❝♦♠♣❛❝t✱ ❞♦ ✤â✱ ❦❤ỉ♥❣ ❝â t➼♥❤ ❝♦♠♣❛❝t s➤♥ ❝â✳ ❱➻ ✈➟②✱ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ t❤ỉ♥❣ t❤÷í♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❝â t❤➸ ❦❤ỉ♥❣ ✤÷đ❝ ♠ð rở trữớ ủ ổ ỵ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ t♦→♥ tû s✐♥❤ A ❝õ❛ C ✲♥❤â♠ e At tr➯♥ X ❝â t➼♥❤ ♥❤à ♣❤➙♥ ♠ô✳ ●✐↔ sû r : X → X ❜à ❝❤➦♥ ✈➔ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③✱ ✈ỵ✐ r(0) = ✈➔ ❤➡♥❣ sè ▲✐♣s❝❤✐t③ ✤õ ♥❤ä✳ ❑❤✐ ✤â✱ ❝â ♠ët ỗ ổ : X X s (eAt x) = v(t; φ(x)), t ∈ R, x ∈ X, tr♦♥❣ ✤â v(t; x) ❜✐➸✉ t❤à ♥❣❤✐➺♠ ✤õ tèt ❝õ❛ ❤➺✿ ∂t v = Av + r(v), t R, v(0) = x ỗ ổ φ ❝â ❞↕♥❣ φ = idX + g ✱ ✈ỵ✐ →♥❤ ①↕ g ∈ BU C(X) ✈➔ φ ❧➔ ❞✉② ♥❤➜t✳ ✣➣ ❝â ♠ët sè t➔✐ ❧✐➺✉ tr➻♥❤ ❜➔② ✤à♥❤ ỵ t ự ❦❤→ ♣❤ù❝ t↕♣ ✈➔ t❤✐➳✉ tü ♥❤✐➯♥ ✈➻ ②➯✉ ❝➛✉ ✤✐➲✉ ❦✐➺♥ r(x) = ❦❤✐ x → ∞ ✳ ❉♦ ✤â✱ ❝❤ó♥❣ t❛ s➩ t✐➳♣ ❝➟♥ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ t ởt ữợ rút r ởt ♣❤✐➳♠ ❤➔♠ ❝❤♦ →♥❤ ①↕ φ : X → X, õ ỵ ✤â ❝❤ó♥❣ t❛ ①→❝ ✤à♥❤ ♠ët →♥❤ ①↕ ❦❤→❝ ψ : X → X ✈➔ ❝❤➾ r❛ r➡♥❣ ψ ❧➔ ♠ët ♥❣❤à❝❤ ✤↔♦ tr→✐ ❝õ❛ φ ✈➔ ♥â ❧➔ ♠ët ✤ì♥ →♥❤✳ ✣✐➲✉ ♥➔② ❝❤♦ t❛ t❤➜② r➡♥❣ →♥❤ ①↕ ủ tỹ sỹ ởt ỗ ổ ψ = φ−1 ✳ ▲✉➟♥ ✈➠♥ ❦❤æ♥❣ ❝❤➾ ♠ð rë♥❣ ỵ rtr t ữỡ tr t õ ♥û❛ t✉②➳♥ t➼♥❤✱ ♠➔ ❝á♥ ✤ì♥ ❣✐↔♥ ❤â❛ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ♥❣❛② ❝↔ tr♦♥❣ tr÷í♥❣ ❤đ♣ ❤ú✉ ❤↕♥ ✈➔ ♠❛♥❣ ❧↕✐ ♠ët sè ❦➳t q✉↔ ❝❤➼♥❤ ①→❝ ✈➔ ❤ú✉ ❞ö♥❣ ố ỗ ❤❛✐ ❝❤÷ì♥❣✱ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ ❞❛♥❤ ♠ư❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❡♠ s➩ tr➻♥❤ ❜➔② ❧↕✐ ✤à♥❤ ❧➼ ❍❛rt♠❛♥✲●r♦❜♠❛♥ ❝ê ✤✐➸♥✱ ❝→❝ ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ù♥❣ ❞ö♥❣✱ →♥❤ ①↕ ♥❤à ♣❤➙♥✱ ♥❤➙♥ ●r❡❡♥ ✈➔ ❜ê r ữỡ ỵ rtr ữỡ tr t✐➳♥ ❤â❛ ♥û❛ t✉②➳♥ t➼♥❤✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② tr ỵ tữ t❤ù❝ ❧✐➯♥ q✉❛♥ ♣❤ö❝ ✈ö ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ỵ õ t ỵ ự ❈✉è✐ ❝ò♥❣✱ ❡♠ s➩ tr➻♥❤ ❜➔② t➼♥❤ ❧✐➯♥ tư❝ ❍☎♦❧❞❡r ❝õ❛ →♥❤ ①↕ ♠ô ♥❤à ♣❤➙♥✳ ▲✉➟♥ ✈➠♥ ❧➔ ❝❤✐ t✐➳t ❤â❛ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ▼❛r✐❡✲▲✉✐s❡ ❍❡✐♥ tr♦♥❣ ❜➔✐ ❜→♦ ❬✶❪ ✤÷đ❝ ✈✐➳t ♥➠♠ ✷✵✶✻✳ ❍➔ ◆ë✐✱ ♥❣➔② ✵✺ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✼ ❍å❝ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ❍÷ì♥❣ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❡♠ s➩ tr ỵ rtr ự r ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ♣❤ư❝ ✈ư ❝❤♦ ❝❤÷ì♥❣ ✷✳ ✣✐➲✉ ✤â ❣✐ó♣ t❛ s♦ s→♥❤ ✤÷đ❝ t➼♥❤ ÷✉ ✈✐➺t tr♦♥❣ ❝→❝❤ ự ỵ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ s→❝❤ ❝õ❛ ▲✳ ❇❛r❡✐r❛ ✈➔ ❈✳ ❱❛❧❧s ❬✷❪✳ ✶✳✶ ✣à♥❤ ỵ rtr ổ ✈➔ ❦❤æ♥❣ ê♥ ✤à♥❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ▼❛ tr➟♥ ✈✉æ♥❣ A(n ì n) ữủ r tt ❝→❝ ❣✐→ trà r✐➯♥❣ ❝â ♣❤➛♥ t❤ü❝ ❦❤→❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❈❤♦ f : R n → Rn , ✤✐➸♠ x0 ∈ Rn ✈ỵ✐ f (x0 ) = s❛♦ ❝❤♦ ♠❛ tr➟♥ Df (x0 ) ❧➔ ❤②♣❡r❜♦❧✐❝✱ ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ x = f (x)✳ ❈❤♦ ♠ët ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❤②♣❡r❜♦❧✐❝ x0 ∈ Rn ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ x = f (x)✱ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤✿ x = Ax A = Df (x0 ) ❈❤ó♥❣ t❛ ✤➣ ❜✐➳t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤÷đ❝ ❝❤♦ ❜ð✐✿ x(t) = eA(t−t0 ) x(t0 ), ✹ t ∈ R ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ❈❤♦ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❤②♣❡r❜♦❧✐❝ x ∈ Rn ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ x = f (x)✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❦❤æ♥❣ ❣✐❛♥ ê♥ ✤à♥❤ ✈➔ ❦❤ỉ♥❣ ê♥ ✤à♥❤ ❝õ❛ x0 t÷ì♥❣ ù♥❣ ♥❤÷ s❛✉✿ E s = {x ∈ Rn : eAt x → t → +∞}, E u = {x ∈ Rn : eAt x → t } ỵ x Rn ❧➔ ♠ët ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❤②♣❡r❜♦❧✐❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ x = f (x)✳ ❑❤✐ ✤â✱ ✶✳ E s ✈➔ E u ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❝õ❛ Rn ✈ỵ✐ E s ⊕ E u = Rn ; ✷✳ ∀x ∈ E s , y ∈ E u ✈➔ t ∈ R✱ t❛ ❝â ✈➔ eAt x ∈ E s eAt y ∈ E u ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ ♠❛ tr➟♥ A = Df (x ) ❦❤æ♥❣ ❝â ❣✐→ trà r✐➯♥❣ ♥➔♦ ❝â ♣❤➛♥ t❤ü❝ ❜➡♥❣ ♥➯♥ ❞↕♥❣ ❏♦r❞❛♥ t õ õ t t ữợ A s Au ù♥❣ ✈ỵ✐ sü ♣❤➙♥ t→❝❤ Rn = F s ⊕ F u ✱ tr♦♥❣ ✤â As , Au t÷ì♥❣ ù♥❣ ❧➔ ❝→❝ ❦❤è✐ ❏♦r❞❛♥ ❝õ❛ ❣✐→ trà r✐➯♥❣ ❝â ♣❤➛♥ t❤ü❝ ➙♠ ✈➔ ❦❤è✐ ❏♦r❞❛♥ ❝â ❣✐→ trà r✐➯♥❣ ❝â ♣❤➛♥ t❤ü❝ ữỡ õ t õ ợ x F s t❤➻ eAt → t → +∞, ✈ỵ✐ x ∈ F u t❤➻ eAt → t → −∞ ✺ ✈ỵ✐ ❝❤✉➞♥ |B|B(X) ≤ b1 , r ∈ Lip(X), |r|Lip ≤ m1 ❉♦ ✤â ♥❣❤✐➺♠ ❞✉② ♥❤➜t v(t; x) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (2.1) t❤ä❛ ♠➣♥ |v(t; x) − v(t; x )|X ≤ MA |x − x |X eωA +MA b1 +MA m1 ✭✷✳✷✮ ◆➳✉ t❤➯♠ ✈➔♦ r ❜à ❝❤➦♥ ✈ỵ✐ ❜✐➯♥ |r|∞ ≤ m0 ✱ t❤➻ |x(t; x)|X ≤ MA (|x|X + m0 )e(ωA +MA b1 )|t| , t ≥ ✭✷✳✸✮ ◆➳✉ eAt ❧➔ ♠ët C0 ✲ ♥❤â♠ ✈ỵ✐ ❜✐➯♥ |eAt |B(X) ≤ MA eωA |t| , t R t ữợ ữủ ✤ó♥❣ ✈ỵ✐ ♠å✐ t ∈ R ❍ì♥ ♥ú❛✱ sü ♣❤ư t❤✉ë❝ ❧✐➯♥ tư❝ ❝ơ♥❣ t❤ä❛ ♠➣♥✱ ♥❣❤➽❛ ❧➔ ♥➳✉ Bn → B tr♦♥❣ B(X), ✈➔ rn → r ❤ë✐ tö ✤➲✉ tr➯♥ t➟♣ ❜à ❝❤➦♥ ✭❝♦♠♣❛❝t ✮ ❝õ❛ X ✱ ❦❤✐ ✤â ❝→❝ ♥❣❤✐➺♠ t÷ì♥❣ ù♥❣ ❤ë✐ tư ✤➳♥ v, ❤ë✐ tö ✤➲✉ tr➯♥ ❝→❝ t➟♣ ❝♦♥ ❜à ❝❤➦♥ R ì X t q ữủ s r tø ❜ê ✤➲ ●r♦♥✇❛❧❧✲❇❡❧❧♠❛♥✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✷✳ P❤➨♣ ❝❤✐➳✉ P + ∈ B(X) ✤÷đ❝ ❣å✐ ♣❤➨♣ ❝❤✐➳✉ ♥❤à ♣❤➙♥ ♠ơ ❝õ❛ C0 ✲♥û❛ ♥❤â♠ eAt ❤♦➦❝ ❝õ❛ t♦→♥ tû s✐♥❤ A tr♦♥❣ X ✱ ♥➳✉ ❝â ♠ët ❤➡♥❣ sè M ≥ 1, η > s❛♦ ❝❤♦ ✈ỵ✐ P− = idX − P+ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t❤ä❛ ♠➣♥✿ ✶✳ P+ eAt = eAt P+ ✈ỵ✐ ♠å✐ t ≥ 0, ✷✳ |eAt P+ x|X ≤ M e−ηt |P+ x|X ✈ỵ✐ ♠å✐ x ∈ X ✈➔ t ≥ 0, ✸✳ eAt P− ♠ð rë♥❣ tỵ✐ ♠ët C0 ✲ ♥❤â♠ tr♦♥❣ R(P− ) ❝â t❤➸ ✈✐➳t ❧↕✐ ❜ð✐ eAt P− ✹✳ |eAt P− x|X ≤ M eηt |P− x|X ✈ỵ✐ ♠å✐ x ∈ X ✈➔ t ≤ ◆❤➙♥ ●r❡❡♥ t÷ì♥❣ ù♥❣ ợ ụ ữủ eAt P+ , t ≥ 0, GA (t) = −eAt P− , t < 0, ✶✽ ❍➺ ♠ô ♥❤à ♣❤➙♥ ♥➔② ❞➝♥ ✤➳♥ sü ❦❤❛✐ tr✐➸♥ X = X+ ⊕ X− ❣✐è♥❣ ♥❤÷ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ê♥ ✤à♥❤ X+ = R(P+ ) = N (P− ) ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❦❤æ♥❣ ê♥ ✤à♥❤ X− = N (P+ ) = R(P− )✳ ❚r♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ♥➔②✱ sü ❤↕♥ ❝❤➳ t÷ì♥❣ ù♥❣ ❝õ❛ ♥û❛ ♥❤â♠ t t ụ ỵ ✷✳✶✳✸✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ A ❧➔ t♦→♥ tû s✐♥❤ ❝õ❛ C ✲♥û❛ ♥❤â♠ tr♦♥❣ X ✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t÷ì♥❣ ✤÷ì♥❣✿ ✶✳ ❚♦→♥ tû A ❝â ♥❤à ♣❤➙♥ ♠ơ✳ ✷✳ ❱ỵ✐ ♠é✐ f ∈ BU C(R; X) ❝â ❞✉② ♥❤➜t ♠ët ♥❣❤✐➺♠ tèt u ∈ BU C(R; X) ❝õ❛ u (t) = Au(t) + f (t), t∈R ✭✷✳✹✮ ✸✳ ❈❤♦ ♠é✐ f ∈ Lp (R; X), ≤ p ≤ ∞✱ ❝â ❞✉② ♥❤➜t ♠ët ♥❣❤✐➺♠ ✤õ tèt u ∈ Lp (R; X) ❝õ❛ ✭✶✳✻✮ ✹✳ σ(eA ) S1 = ∅ ◆➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ✤➣ ♥➯✉ tr ụ tữỡ ữỡ ợ iR (A) = ✈➔ Supρ∈R |(iρ − A)−1 |B(X) < ∞ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ tê♥❣ q✉→t (5) ❝❤➾ ❧➔ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥✱ u ∈ BU C(R; X) ❧➔ ♠ët ♥❣❤✐➺♠ ♠↕♥❤ ❝õ❛ ✭✷✳✹✮ ✈ỵ✐ f ∈ BU C(R; X) ♥➳✉ u ∈ BU C(R; X) BU C(R; D(A)) t❤ä❛ ♠➣♥ (2.4) tø♥❣ ✤✐➸♠ tr♦♥❣ R✳ ⑩♥❤ ①↕ u ∈ BU C(R; X) ❧➔ ♠ët ♥❣❤✐➺♠ tèt ❝õ❛ (2.4) ♥➳✉ ❝â ♥❣❤✐➺♠ ♠↕♥❤ uk → u tr♦♥❣ BU C(R; X) ù♥❣ ✈ỵ✐ ✈➳ ♣❤↔✐ fk := uk − Auk → f tr♦♥❣ BU C(R; X)✳ ❚÷ì♥❣ tü✱ ♥❣❤✐➺♠ ♠↕♥❤ ✈➔ ♥❣❤✐➺♠ ✤õ tèt tr♦♥❣ Lp (R; X), ≤ p ≤ ∞✱ ✤÷đ❝ ①→❝ ✤à♥❤ ❜➡♥❣ ✈✐➺❝ t❤❛② ❦➼ ❤✐➺✉ BU C ❜➡♥❣ Lp ✳ ❍ì♥ t❤➳✱ ♣❤➨♣ ❝❤✐➳✉ P+ ❧➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ê♥ ✤à♥❤ ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ P+ = 2πi (z − eA )−1 dz S1 ✶✾ ❍✐➸♥ ♥❤✐➯♥ GA : R → B(X) ❧✐➯♥ tử t = ợ ữợ ữủ |GA (t)|B(X) ≤ M e−η|t| ◆❤➟♥ t❤➜② ♥❣❤✐➺♠ ❝õ❛ (2.4) ✤÷đ❝ ❝❤♦ ❜ð✐ t➼❝❤ ❝❤➟♣ s❛✉✿ u(t) := (GA ∗ f )(t) = GA (s)f (t − s)ds, t ∈ R, R ❑➳t q✉↔ s❛✉ ✤➙② ❦❤→ q✉❛♥ trå♥❣✱ ♥â ❝❤➾ r sỹ tỗ t ụ ởt t t tữỡ ự ợ A ỵ A t tỷ s✐♥❤ ❝õ❛ C ✲ ♥û❛ ♥❤â♠ e At tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❳✱ ✈ỵ✐ ❜✐➯♥ |eAt |B(X ) ≤ MA eωA t , t ≥ ✈➔ ✤➦t B ∈ X ✳ ❑❤✐ ✤â A + B ❝ô♥❣ ❧➔ t♦→♥ tû s✐♥❤ ❝õ❛ C0 ✲♥û❛ ♥❤â♠✱ ✈➔ |e(A+B)t |B(X ) ≤ MA e(ωA +MA |B|B(X ) )t , t ≥ ❍ì♥ ♥ú❛✱ |e(A+B)t − eAt |B(X ) ≤ |t|MA2 |B|B(X) eωA +MA |B|B(X) t , t > A s ởt C0 õ t ỳ ữợ ữủ tr➯♥ ❦❤↔ ❞ö♥❣ tr➯♥ R✳ ✭✐✐✮ ●✐↔ sû r➡♥❣ A ❝â ♥❤à ♣❤➙♥ ♠ơ ✈ỵ✐ ♥❤➙♥ ●r❡❡♥ GA (t) t❤ä❛ ♠➣♥ |GA+B (t)|B(X) ≤ M e−η|t| , t ∈ R✳ õ tỗ t > s ợ ♠å✐ B ∈ B(X) ✈ỵ✐ |B|B(X ) ≤ δ0 ✱ t♦→♥ tû A + B ❝â ♥❤à ♣❤➙♥ ♠ơ ♥❤÷ ✈➟② ✈➔ ♥❤➙♥ ●r❡❡♥ ❝õ❛ ♥â t❤ä❛ ♠➣♥✱ |GA+B (t)|B(X ) ≤ 2M e−η|t|/2 , t∈R GA+B (t) ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ s❛✉ GA (t − s)BGA+B (s)ds, GA+B (t) = GA (t) + t∈R R ❍ì♥ ♥ú❛✱ |GA+B (t) − GA (t)|B(X ) ≤ (8M /η)|B|B(X ) e−η|t|/2 , ❱➻ ✈➟② GA+B ♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝ ✈➔♦ B ✳ ✷✵ t∈R ✷✳✷ ✣➦t t ởt ró r ỵ t÷ð♥❣ ❝õ❛ ❜➔✐ t♦→♥✱ ❝❤ó♥❣ t❛ ❝➛♥ ♥➢♠ ✈ú♥❣ ❝→❝ tự s rữợ t t ①➙② ❞ü♥❣ ♠ët →♥❤ ①↕ φ ●✐↔ sû r➡♥❣ φ C tỗ t t ữợ φ = idX + g, g ∈ C1 (X) ✈➔ g(0) = 0, g (0) = 0✳ ⑩♣ ❞ö♥❣ q✉✐ t➢❝ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ❤ñ♣ ❝❤♦ ✭✶✮ t❛ ❝â✿ ∂t v(t; φ(x)) = φ (eAt x)AeAt x, x ∈ D(A) ✭✷✳✺✮ ▼➦t ❦❤→❝✱ ♥❣❤✐➺♠ v(t; φ(x)) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✭✷✮ ✈➔ ❧✐➯♥ ❤➺ ✈ỵ✐ ✭✶✮ t❛ ❝â ∂t v(t; φ(x)) = Aφ(eAt x) + r(φ(eAt x))✳ ❑➳t ❤đ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ ✭✷✳✶✮ →♥❤ ①↕ φ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ φ (x)Ax = Aφ(x) + r(φ(x)), x ∈ D(A)✳ ❱ỵ✐ φ = idX + g t❛ ❝â✿ g (x)Ax = Ag(x) + r(x + g(x)), x ∈ D(A) ✭✷✳✻✮ ❚✐➳♣ t❤❡♦✱ t❛ ①→❝ ✤à♥❤ →♥❤ ①↕ w : R → X ❜ð✐ w(t) = g(eAt x), x ∈ D(A)✳ ⑩♣ ❞ö♥❣ q✉✐ t➢❝ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ❤ñ♣ ✈➔ (2.6) s✉② r❛✿ ∂t w(t) = Aw(t) + r(eAt x + g(eAt x)) ✭✷✳✼✮ ❇ð✐ ✈➻ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ w t❤➻ ✤÷ì♥❣ ♥❤✐➯♥ ❧➔ w ❜à ❝❤➦♥ ♥➳✉ g ❜à ❝❤➦♥✳ ❍ì♥ ♥ú❛✱ t♦→♥ tû A t❤ø❛ ♥❤➟♥ ♠ët ♣❤➨♣ ❝❤✐➳✉ ♥❤à ♣❤➙♥ ❞♦ ✤â ♥❣❤✐➺♠ ✤õ tèt ❜à ❝❤➦♥ ✈➔ ❞✉② ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t✐➳♥ ❤â❛ ♥û❛ t✉②➳♥ t➼♥❤ (2.7) ✤÷đ❝ ❝❤♦ ❜ð✐✿ GA (s)r(eA(t−s) x + g(eA(t−s) x))ds, w(t) = ✭✷✳✽✮ R ❚r♦♥❣ ✤â GA ❜✐➸✉ t❤à ❝❤♦ ♥❤➙♥ ●r❡❡♥ ù♥❣ ✈ỵ✐ ♥❤à ♣❤➙♥ ❝õ❛ t♦→♥ tû A✳ ❳➨t w(0) = g(x) ✈➔ t = tr♦♥❣ (2.8)✳ ❚❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❝❤♦ →♥❤ ①↕ g ✿ GA (s)r(e−As x + g(e−As x))ds, g(x) = x∈X ✭✷✳✾✮ R P❤÷ì♥❣ tr➻♥❤ ốt ó ỹ ỗ ổ φ ✭✐✐✮ ●✐↔ sû ❝❤ó♥❣ t❛ ❝â ♠ët ♥❣❤✐➺♠ ❧✐➯♥ tö❝ g ∈ C(X) ❝õ❛ (2.9)✳ ❑❤✐ ✤â ♥â ❜à ✷✶ ❝❤➦♥ ❦❤✐ |g∞ | ≤ idX + g ❧➔ →♥❤ ①↕ tr♦♥❣ (1) ♠➔ t❛ ✤❛♥❣ ❝➛♥ t➻♠✳ ❑➳t ❤ñ♣ sü ❧✐➯♥ ❤➺ ❣✐ú❛ φ = idX + g ợ (2.9 x ữủ t t eAt x t❛ ✤÷đ❝ φ(eAt x) = eAt x + GA (s)r(eA(t−s) x + g(eA(t−s) x)ds R = eAt x + GA (t − s)r(φ(eAs x))ds R ❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ❝õ❛ ♥❤➙♥ ●r❡❡♥ GA ✱ t❛ ❝â t At At eA(t−s) r(φ(eAs x))ds φ(e x) = e φ(x) + ❱➻ ✈➟② p(t) = φ(eAt x) ❧➔ ♠ët ♥❣❤✐➺♠ ✤õ tèt ❝õ❛ ❤➺ ∂t p = Ap + r(p), ✭✷✳✶✵✮ p(0) = φ(x), ♥❣❤➽❛ ❧➔ p(t) = v(t; φ(x)), t ∈ R, x ∈ X ✱ ❜ð✐ t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠✳ ❙❛✉ ✤➙② t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ sü tỗ t tr ữợ ự ♠✐♥❤ ✤à♥❤ ❧➼✳ ✭✐✐✐✮ ▼➦t ❦❤→❝✱ ✤➦t φ = Id X + g(x), t❤ä❛ ♠➣♥ (1) ✈ỵ✐ ♠å✐ t ∈ R, x ∈ X ✳ ❑❤✐ ✤â φ(eAt x) ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ (2) ✈ỵ✐ ❣✐→ trà ❜❛♥ ✤➛✉ φ(x)✱ ✈➻ ✈➟② ♥â ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ t At At eA(t−s) r(φ(eAs x))ds φ(e x) = e φ(x) + ❙û ❞ö♥❣ t✐➳♣ t➼♥❤ ❝❤➜t ❝õ❛ ♥❤➙♥ ●r❡❡♥ t❛ ❝â✿ GA (s)r(φ(e−As x))ds + φ(eAt x) = eAt φ(x) − R GA (s)r(φ(eA(t−s) x))ds R ❳➨t →♥❤ ①↕ k : X → X ①→❝ ✤à♥❤ ❜ð✐ k(x) = R ✭✷✳✶✶✮ GA (s)r(φ(e−As x))ds✳ ❚ø (2.11) s✉② r❛ (g − k)(eAt x) = eAt (g(x) − k(x)), ❱➻ ✈➟② f (x) = g(x) − k(x) t❤ä❛ ♠➣♥ q✉✐ ❧✉➟t ❜➜t ❜✐➳♥ f (eAt x) = eAt f (x), t ∈ R, x ∈ X, ✷✷ f (0) = ✭✷✳✶✷✮ ❙✉② r❛ GA (s)r(e−As x + g(e−As x)ds) g(x) = f (x) + R ợ ữỡ ú t õ t ữủ tt ỗ ổ tọ t ỗ ổ t ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ (2.12) ❝❤➾ ❝â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣✳ ◆â✐ ❝❤✉♥❣✱ ✤➙② ❦❤ỉ♥❣ ♣❤↔✐ tr÷í♥❣ ❤đ♣ ❝➛♥ t➻♠✳ ❚✉② ♥❤✐➯♥✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❝❤➾ r❛ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ φ ❜➡♥❣ ❝→❝❤ ❣✐↔ sû r➡♥❣ φ − id ❜à ❝❤➦♥✳ ❚❤ü❝ sü✱ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❝✉è✐ ❝ò♥❣ ❜à ❝❤➦♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f ❜à ❝❤➦♥✳ ◆❤÷♥❣ ♣❤➨♣ ❝❤✐➳✉ tø (2.12) ✤➳♥ P± X, (t → ±∞) ❝❤➾ r❛ r➡♥❣ f ❜à ❝❤➦♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f = ✭✐✈✮ ●✐↔ sû φ t❤ä❛ ♠➣♥ (1) ✈➔ ❦❤↔ ♥❣❤à❝❤ ✈ỵ✐ φ −1 = ψ ✈➔ ψ = idX − h ✤÷ì♥❣ ♥❤✐➯♥ h →♥❤ ①↕ t➟♣ ❜à ❝❤➦♥ ❧➯♥ t➟♣ ❜à ❝❤➦♥✳ ❑❤✐ ✤â eAt x = ψ(v(t; φ(x))) ❞♦ ✤â eAt ψ(x) = ψ(v(t(x))) ✈➔✿ eAt ψ(x) = v(t; x) − h(v(t(x))) ✭✷✳✶✸✮ ❚❤❛② t❤➳ w(t; x) = h(v(t(x)))✱ ✤✐➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ w(t; x) ❧➔ ♠ët ♥❣❤✐➺♠ ✤õ tèt ❜à ❝❤➦♥ ❝õ❛ ❤➺ s❛✉✿ ∂t ω = Aw + r(v), ✭✷✳✶✹✮ w(0; x) = h(x), ✈➔ w(0; x) = h(v(0; x)) = h(x)✳ ❙û ❞ö♥❣ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♥❤➙♥ ●r❡❡♥ GA ✱ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ✤÷đ❝ ❝❤♦ ❜ð✐ ✿ GA (s)r(v(t − s; x))ds w(t; x) = R ❈❤å♥ t = ✈➔ ♥❤➢❝ ❧↕✐ w(0; x) = h(x) t❛ ❝â ✿ h(x) = GA (s)r(v(−s; x))ds, x ∈ X ✭✷✳✶✺✮ R ❱➻ ✈➟②✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ φ ❦❤↔ ♥❣❤à❝❤ t❤➻ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♥â ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ψ = idX − h tr♦♥❣ ✤â h ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ (2.15)✱ ✤➦❝ ❜✐➺t h ❧✐➯♥ tư❝ r ữợ ự ỵ ú t s r ①↕ ψ t❤ü❝ ❝❤➜t ❧➔ ♠ët ♥❣❤à❝❤ ✤↔♦ tr→✐ t♦➔♥ ❝ö❝ ❝õ❛ φ✱ tø ✤â t❛ ❝â t❤➸ s✉② r❛ r➡♥❣ φ ❧➔ ♠ët ✤ì♥ →♥❤ ✈➔ ψ ❧➔ ♠ët t♦➔♥ →♥❤✳ ✭✈✮ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ ①✉➜t ♣❤→t tø ♠ët ❝æ♥❣ t❤ù❝ ♥❣❤✐➺♠ q✉❛♥ trå♥❣ s❛✉✿ v(t; x) = eAt ψ(x) + GA (s)r(v(t − s; x))ds, ✭✷✳✶✻✮ t ∈ R, x ∈ X R ❚❤ü❝ t➳✱ t❛ ❝â ✤✐➲✉ tr➯♥ tø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♥❤➙♥ ●r❡❡♥ GA (t) eAt ψ(x) = eAt x − eAt GA (−s)r(v(s; x))ds R t = eAt x + eA(t−s) r(v(s; x)) − GA (t − s)r(v(s; x))ds R = v(t; x) − GA (t − s)r(v(s; x))ds R P❤➛♥ ♥➔② s➩ ✤÷đ❝ ❞ò♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✤ì♥ →♥❤ ❝õ❛ φ tr ữợ ỵ ứ õ õ t t r ỗ ổ ❧➔ ✈➜♥ ✤➲ q✉❛♥ trå♥❣ ♥❤➜t ♠➔ ❝❤ó♥❣ t❛ ✤❛♥❣ qt Pt ỵ ỵ X ổ A ❧➔ t♦→♥ tû s✐♥❤ ❝õ❛ C ✲♥❤â♠ eAt tr♦♥❣ X ♠➔ ❝â ♣❤➨♣ ❝❤✐➳✉ ♥❤à ♣❤➙♥ ù♥❣ ✈ỵ✐ ♥❤➙♥ ●r❡❡♥ GA t❤ä❛ ♠➣♥ |GA (t)B(X ) | ≤ M e−η|t| , t ∈ R ✈ỵ✐ ❝→❝ ❤➡♥❣ sè M ≥ ✈➔ η > 0✳ ●✐↔ sû r➡♥❣ r : X → X ❜à ❝❤➦♥ ✈➔ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③✱ ✈ỵ✐ r(0) = ∈ X ✈➔ ❤➡♥❣ sè ▲✐♣s❝❤✐t③ ❝õ❛ →♥❤ ①↕ r t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✼✮ (2M/η)|r|Lip < õ õ ởt ỗ ổ : X → X s❛♦ ❝❤♦ φ(eAt x) = v(t; φ(x)), t ∈ R, x ∈ X, (HG) tr♦♥❣ ✤â v(t; x) (2) ỗ ổ õ φ = idX + g, ✈ỵ✐ →♥❤ ①↕ g ∈ BU C(X) tỗ t t ự ỵ P ự ỵ ữủ r ữợ t ú t ự sỹ tỗ t t t g tổ q ỵ ữợ ỹ tỗ t t t g ✣➦t Y = BU C(X) ✈➔ ✤à♥❤ ♥❣❤➽❛ GA (s)r(e−As + g(e−As ))ds = (T g)(.) g ∈ BU C(X) g(.) → ✭✷✳✶✽✮ R ❑❤✐ ✤â T ❧➔ ỗ t õ tr Y, ❣✐↔ sû ✈ỵ✐ ✤✐➲✉ ❦✐➺♥✱ ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t g ∈ Y ❝õ❛ T ❈❤ù♥❣ ♠✐♥❤✳ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Y = BU C(X) ✤õ ✈ỵ✐ ❝❤✉➞♥✲ ❙✉♣ ởt ổ r ữợ t ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ →♥❤ ①↕ T →♥❤ ①↕ Y ❧➯♥ Y ✳ ❱ỵ✐ ❤➔♠ sè ❜à ❝❤➦♥ g ∈ Y, T g ❝ô♥❣ ❜à ❝❤➦♥✱ ❜ð✐ ✈➻ |T g|∞ ≤ (2M/η)|r|∞ < ∞ ✣➦t x, x ¯ ∈ X, N ∈ N ✈➔ ρ > ❜➜t ❦ý✳ ❉♦ r ❜à ❝❤➦♥ ✈➔ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ ✈ỵ✐ ❤➡♥❣ sè ▲✐♣s❝❤✐t③ |r|Lip t❛ ♥❤➟♥ ✤÷đ❝ e−η|s| |r(φ(e−As x¯))|X ds |(T g)(x) − (T g)(¯ x)|X ≤ M R |s|>N ≤ (4M |r|∞/η )e−ηN +2M |r|Lip ❈❤♦ |e−As (x−¯ x)|X +|g(e−As x)−g(e−As x¯)|X ds e−η|s| ds+M |r|Lip ≤ 2M |r|∞ |s|≤N MA ωA N e |x−¯ x|X +N Sup|s|≤N |g(e−As x)−g(e−As x¯)|X ωA > 0✱ ✤➛✉ t✐➯♥ ❝❤ó♥❣ t❛ ❝❤å♥ N ✤õ ❧ỵ♥✱ ✤➸ ❤↕♥❣ tû ✤➛✉ t✐➯♥ ð ✈➳ ♣❤↔✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ò♥❣ ♥❤ä ❤ì♥ /3✳ ❙❛✉ ✤â ❝❤ó♥❣ t❛ ❝❤å♥ x − x ¯ ≤δ ✈ỵ✐ δ > ✤õ ♥❤ä s❛♦ ❝❤♦ ❤↕♥❣ tû t❤ù ❤❛✐ ♥❤ä ❤ì♥ /3✱ ✈➔ ❝✉è✐ ❝ò♥❣✱ ✈➻ g ∈ BU C(X) ✈➔ eAt ❧➔ ♠ët ♥❤â♠ C0 ✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❝❤å♥ > ❤♦➦❝ ♥❤ä ❤ì♥ s❛♦ ❝❤♦ ❦➳t q✉↔ ❝✉è✐ ❝ò♥❣ ♥❤ä ❤ì♥ /3✳ ✣✐➲✉ ♥➔② ❝❤♦ t❤➜② T g ∈ BU C(X) ✷✺ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ T ❧➔ ♠ët →♥❤ ①↕ ❝♦ tr➯♥ Y ❱ỵ✐ ❜➜t ❦ý g, g¯ ∈ Y ✈➔ x X t õ ữợ ữủ s e|s| |r|Lip |g(e−As x) − g¯(e−As x)|X ds, |(T g)(x) − (T g¯)(x)|X ≤ R ❱➻ ✈➟② |T g − T g¯|∞ ≤ (2M/η)|r|Lip |g − g¯|∞ ✳ ✣✐➲✉ ❦✐➺♥ (2.17) s✉② r❛ |T g − T g¯|∞ ≤ k|g − g¯∞ |, ✈ỵ✐ k = (2M/η)|r|Lip < 1✱ tù❝ ❧➔ T ❧➔ →♥❤ ①↕ ❝♦ tr♦♥❣ Y ✳ ❑❤✐ ✤â✱ ỵ ởt t g Y (2.9) r ữợ t s t tr ữợ φ ❝â ♥❣❤à❝❤ ✤↔♦ tr→✐ ψ ✈ỵ✐ ψ = id h ữủ ợ ♠å✐ x ∈ X →♥❤ ①↕ h t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿ X h(x) = GA (s)r(v(−s; x))ds ✭✷✳✶✾✮ R ❈❤ù♥❣ ♠✐♥❤ ✣➸ t❤➜② r➡♥❣ ψ = idX − h ợ h ữủ (2.19) tr→✐ ❝õ❛ φ ✈ỵ✐ ♠å✐ x ∈ X ✱ ❝❤ó♥❣ t❛ ❝â q✉❛♥ s→t tø →♥❤ ①↕ ❤ñ♣ ψoφ✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ψ t❛ ❝â ψ(φ(x)) = φ(x) − h(φ(x))✳ ❉♦ ✤â✿ ψ(φ(x)) = x + g(x) − GA (s)r(v(−s; φ(x)))ds R ❚ø (2.9) t❛ ❝â GA (s)r(v(−s; φ(x)))ds = g(x), x∈X R ❱➻ ✈➟②✱ ψ = idX − h ❧➔ ♥❣❤à❝❤ ✤↔♦ tr→✐ ❝õ❛ φ✳ ✣✐➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ φ ❧➔ ✤ì♥ →♥❤ ✈➔ ψ ❧➔ t♦➔♥ →♥❤✳ ❱➻ ✈➟② ✤➸ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ t♦➔♥ →♥❤ ❝õ❛ →♥❤ ①↕ φ t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ✤ì♥ →♥❤ ữợ ỡ ự ♠✐♥❤ ✷✻ ●✐↔ sû ψ(x) = ψ(¯ x)✱ ✈ỵ✐ x, x¯ ∈ X ✳ ✣➦t v(t) = v(t; x) ✈➔ v¯(t) = v(t; x¯)✱ tø ♣❤÷ì♥❣ tr➻♥❤ (2.16) t❛ ❝â ✿ ¯ GA (t − s)r(v(s))ds, GA (t − s)r(v(s))ds = eAt ψ(¯ x) = v¯(t) − v(t) − R R ❱➻ ✈➟②✱ ¯ = v(t) − v(t) GA (t − s) r(v(s)) − r(¯ v (s)) ds, t ∈ R R ✣✐➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ v(t) − v¯(t) ❜à ❝❤➦♥✱ ✈➔ |v − v¯|∞ ≤ (2M/η)|r|Lip |v − v¯|∞ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ (2M/η)|r|Lip < ✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥ v(t) = v¯(t) ✈ỵ✐ ♠å✐ t ∈ R, ❞♦ ✤â ✤➦t t = t❛ ✤÷đ❝ x = x ữợ ự tr r r ởt ỗ ổ X ự ỵ t ◗✉✐ ❍☎ ♦❧❞❡r✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ s➩ ❦❤↔♦ s→t t➼♥❤ ❝❤➼♥❤ q✉✐ ❝õ❛ φ ✈➔ ψ ❝ö t❤➸ t❛ ①➨t g ✈➔ h✳ ▼ët ❝→❝❤ tê♥❣ q✉→t✱ t❛ ❦❤æ♥❣ t❤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ h ❧➔ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③✱ ♥❤÷♥❣ ♥â ✈➝♥ ❧✐➯♥ tö❝ ❍☎♦❧❞❡r✳ ✣➸ t❤➜② rã ✤✐➲✉ ♥➔②✱ t❛ ❝➛♥ ♥❤➢❝ ❧↕✐ r➡♥❣ |GA (s)|B(X ) ≤ M e−η|s| tr♦♥❣ X ✱ ✈➔ |r(y) − r(¯ y )|X ≤ γ(|y − y¯|X ), y¯ ∈ X ✱ ✈ỵ✐ γ(t) = min2|r|∞ , t|r|Lip ❚❤❡♦ ✭✷✳✶✮ ✈ỵ✐ ωA + MA |r|Lip < η ✈➔ K = MA t❛ ❝â |v(s; x) − v(s; x¯)|X ≤ Keω|s| |x − x¯|, s ∈ R, x, x¯ ∈ X ❑➳t ❤ñ♣ ✈ỵ✐ ✭✷✳✶✺✮ t❛ s✉② r❛ ✿ ∞ e−η|s| γ(|v(−s; x) − v(−s; x¯)|X )ds |h(x) − h(¯ x)|X ≤ M −∞ ∞ e−ηs γ(eωs K|x − x¯|X )ds ≤ 2M ✷✼ ∞ 2M dt η/ω = [K|x − x¯|X ] γ(t) 1+η/ω ω t K|x−¯ x|X ∞ 2M dt ≤ [K|x − x¯|X ]η/ω γ(t) 1+η/ω ω t η/ω 2M K η/ω η/ω [2|r|∞ ]1−η/ω |r|Lip |x − x|X , = (1 /) t ữợ ữủ ♥➔② ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ❤➡♥❣ sè ❝❤♦ C0 ✲♥❤â♠ eAt ✱ ♥❤➙♥ ●r❡❡♥ GA ✈➔ tr➯♥ ❜✐➯♥ ❝❤♦ r ✱ ✈ỵ✐ |r|∞ , |r|Lip ✳ t❤❡♦ ✤à♥❤ ❧➼ ✭✷✳✹✮✱ ÷ỵ❝ ❧÷đ♥❣ ❍☎ ♦❧❞❡r ♥➔② ❝❤♦ h ❝ơ♥❣ ❧➔ ÷ỵ❝ ữủ ự ợ ọ B ❝õ❛ t♦→♥ tû A✳ ❍➺ q✉↔ ✷✳✺✳✶✳ ●✐↔ sû r➡♥❣ A ❧➔ t♦→♥ tû ❝õ❛ C ✲♥❤â♠ ✈ỵ✐ ❜✐➯♥ |e At |B(X ) ≤ MA eωA |t| , t ∈ R✱ ♠➔ ❝â ♥❤à ♣❤➙♥ ♠ơ ✈ỵ✐ ♥❤➙♥ ●r❡❡♥ GA (t) t❤ä❛ ♠➣♥ |GA (t)|B(X ) ≤ M e−η|t| , t ∈ R ✣➦t |r|∞ ≤ m0 ✈➔ |r|Lip ≤ m1 ❍➔♠ ψ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✶✺✮ ❧➔ ❧✐➯♥ tư❝ ❍☎ ♦❧❞❡r✳ ◆â✐ ❝❤➼♥❤ ①→❝ ❤ì♥ t❤➻ η/ω |h(x) − h(¯ x)|X ≤ c|x − x¯|X , x, x¯ ∈ X, ❚r♦♥❣ ✤â ω = ωA + MA m1 , ✈➔ c > ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ MA , ωA , M, η ❝ơ♥❣ ♥❤÷ tr➯♥ m0 , m1 ✳ ❈❤ó♥❣ t❛ ❝ơ♥❣ ❝â t❤➸ ❝❤➾ r❛ r➡♥❣ g ❧➔ ❧✐➯♥ tö❝ ❍☎ ♦❧❞❡r✱ ù♥❣ ✈ỵ✐ t➼♥❤ ♣❤✐ t✉②➳♥ r✳ ❈ư t❤➸ ❤ì♥✱ ✤è✐ ✈ỵ✐ t tỷ s A õ C0 ợ ữợ ữủ |eAt |B(X ) ≤ MA eωA |t| ✱ ✈➔ ❤➔♠ ♠ô ♥❤à ♣❤➙♥ ✈➔ ♥❤➙♥ ●r❡❡♥ GA s❛♦ ❝❤♦ |GA (t)|B(X ) ≤ M e−η|t| ✱ ❝â ♠ët sè m1 > s❛♦ ❝❤♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦ý r ∈ BU C(X) Lip(X)r ✈ỵ✐ |r|∞ ≤ m0 ✱ ✈➔ |r|Lip ≤ m1 ✤è✐ ✈ỵ✐ ❜➜t ❦ý α < η/ωA , ω = ωA + MA m1 ❝â ♠ët ❤➡♥❣ sè a > s❛♦ ❝❤♦ |g(x) − g(¯ x)|X ≤ a|x − x¯|αX , x, x¯ ∈ X, ✭✷✳✷✵✮ ❈❤♦ m1 ✤õ ♥❤ä ✳ ❚r♦♥❣ t❤ü❝ t➳✱ ①→❝ ✤à♥❤ D ⊂ Y := BU C(X) ❜ð✐ D = {g ∈ Y : (3.14)} ❧➔ ❤ñ♣ ✷✽ ❧➼✱ tr♦♥❣ ✤â ♠ët ❤➡♥❣ sè s➩ ✤÷đ❝ ❝❤å♥ s❛✉ ✤â✳ D ❧➔ t➟♣ ✤â♥❣✱ ✈➔ tr♦♥❣ tr÷í♥❣ ❤đ♣ T D ⊂ D t❤➻ ✤✐➸♠ ❝è ✤à♥❤ ❝õ❛ T ∈ D✱ ❧➔ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❱➻ ✈➟② ❝❤ó♥❣ t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ T D ⊂ D ✣➸ ✤↕t ✤÷đ❝ ♠ư❝ ✤➼❝❤ ♥➔② ❝❤ó♥❣ t ữợ ữủ tữỡ tỹ ữ tr ự tr ợ g D ữ ỵ r (t) t ữợ (t) [2|r |]1 |r|Lip t ≤ [2m0 ]1−α mα1 tα =: c1 tα , t > ❉♦ ✤â t❛ ❝â✿ ∞ e−η|s| γ(|φ(e−As x) − φ(e−As x¯)|X )ds |(T g)(x) − (T g)(¯ x)|X ≤ M −∞ ∞ e−ηs γ(eωA s MA |x − x¯|X + a[eωA s MA |x − x¯|X ]α )ds ≤ 2M 2M [MA |x − x¯|X ]η/ωA = ωA 2M ≤ ωA [MA |x − x¯|X ]η/ωA 2M [MA |x − x¯|X ]η/ωA ≤ ωA ∞ γ(τ + aτ α ) dτ τ 1+η/ωA dτ (γ(τ ) + γ(aτ α )) 1+η/ω A τ MA |x−¯ x|X ∞ dτ (c1 + a|r|Lip )τ α 1+η/ω A τ MA |x−¯ x|X MA |x−¯ x|X ∞ 2M (MA |x − x¯|X )α−η/ωA [MA |x − x¯|X ]η/ωA (c1 + a|r|Lip ) ωA η/ωA − α 2M c1 + am1 ≤ [MA |x − x¯|X ]α ωA η/ωA − α = 2M ❈❤å♥ α > s❛♦ ❝❤♦ α < η/ωA ✈➔ m1 η−ω MAα < 1✱ ✈ỵ✐ a ✤õ ❧ỵ♥ t❛ ❝â ✿ Aα |(T g)(x) − (T g)(¯ x)|X ≤ α|x − x¯|αX , x, x¯ ∈ X, ❉♦ ✤â T →♥❤ ①↕ ❧➯♥ ❝❤➼♥❤ ♥â tr♦♥❣ D ❱➼ ❞ư ✷✳✺✳✷✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët ∂t v(t, x) = a(x)∂x v(t, x) + b(x)v(t, x), (t, x) ∈ R2 , ✭✷✳✷✶✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ t✉➛♥ ❤♦➔♥ v(t, x + 1) = v(t, x), ✷✾ (t, x) ∈ R2 , ✭✷✳✷✷✮ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ✭✷✳✷✸✮ x ∈ R v(0, x) = v0 (x), ❚r♦♥❣ ✈➼ ❞ư ♥➔②✱ t❛ ❧✉ỉ♥ ①➨t a(x), b(x) ❧➔ ❤➔♠ ❜à ❝❤➦♥✱ ❧✐➯♥ tö❝✱ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦➻ ✈➔ t❤ä❛ ♠➣♥ inf{a(x) : x ∈ R} > ✈➔ inf{b(x) : x ∈ R} > ✭✷✳✷✹✮ ❚❛ ①➨t ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X = Cper (R) ✈ỵ✐ Cper := v ∈ C(R) : v(x + 1) = v(x) ợ x R ữủ tr ❜à ❜ð✐ ❝❤✉➞♥ ||v|| := sup {|v(x)| : x ∈ R} ●å✐ t♦→♥ tû Av := a∂x v + b(x)v ✈ỵ✐ ♠✐➲♥ ①→❝ ✤à♥❤ D(A) := Cper (R) ∩ C (R) ❉ü❛ ✈➔♦ ❝æ♥❣ t❤ù ✭✶✳✺✮ ✈➔ ✣à♥❤ ❧➼ ✶✳✶ tr♦♥❣ ❬✽❪ t❛ ❝â (A, D(A)) s✐♥❤ r❛ c0 ✲♥❤â♠ ①→❝ ✤à♥❤ ❜ð✐ t At b(ξ(r, x, t))dr v0 (ξ(0, x, t)), e v0 := exp tr♦♥❣ ✤â✱ v0 ∈ X ✈➔ ξ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ∂τ ξ(τ, x, t) = a(ξ(τ, x, t)), ξ(τ, x, t) = x ◆❣♦➔✐ r❛✱ ❞♦ ✤✐➲✉ ❦✐➺♥ ✭✷✳✷✹✮ t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t ❝õ❛ ✣à♥❤ ❧➼ ✶✳✷ tr♦♥❣ ❬✽❪ ♥➯♥ ❜➔✐ t♦→♥ ✭✷✳✷✶✮✲✭✷✳✷✸✮ ❝â ♥❤à ♣❤➙♥ ♠ô✳ ❚ø ✤â✱ ♥➳✉ ❤➔♠ ♣❤✐ t✉②➳♥ r : X → X t❤ä❛ t tỗ t ỗ ổ : X X s ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ✭✷✳✷✶✮✲✭✷✳✷✸✮ t÷ì♥❣ ✤÷ì♥❣ tỉ ♣ỉ ✭✤✐➲✉ ❦✐➺♥ (HG)✮ ✈ỵ✐ ❜➔✐ t♦→♥ ∂t v(t, x) = a(x)∂x v(t, x) + b(x)v(t, x) + r(v), v(t, x + 1) = v(t, x), (t, x) ∈ R2 , v(0, x) = φ(v0 (x)), x ∈ R ✸✵ (t, x) ∈ R2 , ✭✷✳✷✺✮ ❑➌❚ ▲❯❾◆ ✣â♥❣ ❣â♣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ỗ r ỳ ỡ ự ỵ rt r ♥➯✉ r❛ ♠ët sè ✈➼ ❞ö✳ ✷✳ ❈❤✐ t✐➳t ❤â❛ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❜➔✐ ❜→♦ ❝õ❛ ▼✳✲▲✳❍❡✐♥✳ ❉ü❛ ✈➔♦ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ♠ô ♥❤à ♣❤➙♥ ✤➸ ♠ð rë♥❣ ✈➔ ự ỵ rt r ởt trỹ t tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈æ ❤↕♥ ❝❤✐➲✉ ♠➔ ❦❤æ♥❣ ❝➛♥ sû ❞ư♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ t÷ì♥❣ ✤÷ì♥❣ tỉ✲♣ỉ ❤❛② ✤✐➸♠ ❜➜t ✤ë♥❣ ❇r♦✉✇❡r✳ ✸✳ ◆➯✉ r❛ ù♥❣ ❞ö♥❣ ❝õ❛ ✤à♥❤ ỵ tr ự t tử r ▼➦❝ ❞ò ✤➣ ❝è ❣➢♥❣✱ t✉② ♥❤✐➯♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ s❛✐ sât✱ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü õ ỵ qỵ t ổ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▼❛r✐❡✲▲✉✐s❡ ❍❡✐♥✱ ❏❛♥ Pr✉ss✱ ❚❤❡ ❍❛rt♠❛♥✲●r♦❜♠❛♥ t❤❡♦r❡♠ ❢♦r s❡♠✐✲ ❧✐♥❡❛r ❤②♣❡r❜♦❧✐❝ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥s✱ ❏✳ ❉✐❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✵✵✷✷✲✵✸✾✻ ✭✷✵✶✻✮ ❬✷❪ ❇❛r❡✐r❛ ▲✳✱ ❱❛❧❧s ❈✳ ✭✷✵✶✷✮✱ ❖r❞✐♥❛r② ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥✿ ◗✉❛❧✐t❛t✐✈❡ ❚❤❡♦r②✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✳ ❬✸❪ ▲✳ ❇❛rr❡✐r❛✱ ❈✳ ❱❛❧❧s✱ ❙t❛❜✐❧✐t② ♦❢ ♥♦♥❛✉t♦♥♦♠♦✉s ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ ❏✳ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✷✶✼ ✭✷✵✵✺✮ ✷✵✹✕✷✹✽✳ ❬✹❪ ▲✳ ❇❛rr❡✐r❛✱ ❈✳ ❱❛❧❧s✱ ❙t❛❜✐❧✐t② ♦❢ ◆♦♥❛✉t♦♥♦♠♦✉s ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ▲❡❝t✳ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✈♦❧✳ ✶✾✷✻✱ ❙♣r✐♥❣❡r✱ ✷✵✵✽✳ ❬✺❪ ❈✳ ❈♦❢❢♠❛♥✱ ❆s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ s♦❧✉t✐♦♥s ♦❢ ♦r❞✐♥❛r② ❞✐❢❢❡r❡♥❝❡ ❡q✉❛✲ t✐♦♥s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✶✵ ✭✶✾✻✹✮ ✷✷✕✺✶✳ ❬✻❪ ❲✳ ❈♦♣♣❡❧✱ ❙t❛❜✐❧✐t② ❛♥❞ ❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦r ♦❢ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❉✳ ❈✳ ❍❡❛t❤ ❛♥❞ ❈♦✳✱ ✶✾✻✺✳ ❬✼❪ P✳ ❍❛rt♠❛♥✱ ❆✳ ❲✐♥t♥❡r✱ ❆s②♠♣t♦t✐❝ ✐♥t❡❣r❛t✐♦♥s ♦❢ ❧✐♥❡❛r ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❆♠❡r✳ ❏✳ ▼❛t❤✳ ✼✼ ✭✶✾✺✺✮ ✹✺✕✽✻✳ ❬✽❪ ❘✳❑❧②✉❝❤♥②❦✱ ■✳ ❑♠✐t❛✱ ▲✳ ❘❡❝❦❡❛✱ ❊①♣♦♥❡♥t✐❛❧ ❞✐❝❤♦t♦♠② ❢♦r ❤②♣❡r❜♦❧✐❝ s②st❡♠s ✇✐t❤ ♣❡r✐♦❞✐❝ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✱ ❏✳ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✷✻✷ ✭✷✵✶✼✮✷✹✾✸✕✷✺✷✵✳ ✸✷