TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* ĐỒN THỊ HÀ HÀM LYAPUNOV LỒI PHÂN THỨ KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích HÀ NỘI – 2018 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TỐN ************* ĐỒN THỊ HÀ HÀM LYAPUNOV LỒI PHÂN THỨ KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích Người hướng dẫn khoa học TS HỒNG THẾ TUẤN HÀ NỘI – 2018 ▲í✐ ❝↔♠ ì♥ ❚r➯♥ t❤ü❝ t➳ ❦❤æ♥❣ ❝â sü t❤➔♥❤ ❝æ♥❣ ♥➔♦ ♠➔ ❦❤æ♥❣ ❣➢♥ ❧✐➲♥ ✈ỵ✐ ♥❤ú♥❣ sü ❤é trđ✱ ❣✐ó♣ ✤ï ❞ò ➼t ❤❛② ♥❤✐➲✉✱ ❞ò trü❝ t✐➳♣ ❤❛② ❣✐→♥ t✐➳♣ ❝õ❛ ♥❣÷í✐ ❦❤→❝✳ ❚r♦♥❣ s✉èt t❤í✐ ❣✐❛♥ tø ❦❤✐ ❜➢t ✤➛✉ ❤å❝ t➟♣ ð ❣✐↔♥❣ ✤÷í♥❣ ✤↕✐ ❤å❝ ✤➳♥ ♥❛②✱ ❡♠ ✤➣ ♥❤➟♥ ✤÷đ❝ r➜t ♥❤✐➲✉ sü q✉❛♥ t➙♠✱ ❣✐ó♣ ✤ï ❝õ❛ qỵ ổ ợ ỏ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t ❡♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ỡ t t ữợ ❝❤➾ ❜↔♦ ❝❤♦ ❡♠ ❦❤æ♥❣ ❝❤➾ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ✈➲ ❝❤✉②➯♥ ♠æ♥ ♠➔ ❝á♥ ❝↔ ✈➲ ❝✉ë❝ sè♥❣✳ ✣➸ ❡♠ t❤➯♠ ②➯✉ ❝♦♥ ✤÷í♥❣ ♠➻♥❤ ✤➣ ❝❤å♥ ✈➔ ✈ú♥❣ t✐♥ ỡ tr ỳ ữớ trữợ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ tê ●✐↔✐ t➼❝❤ ✈➔ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❦❤♦❛ ❚♦→♥✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ✤÷đ❝ ❤å❝ t➟♣ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ✈ø❛ q✉❛✳ ❊♠ ①✐♥ ❝→♠ ì♥ ❜❛♥ ❧➣♥❤ ✤↕♦ ❱✐➺♥ ❚♦→♥ ❤å❝✱ ♣❤á♥❣ ❳→❝ s✉➜t t❤è♥❣ ❦➯ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ õ ố ũ ú qỵ ổ ỗ ọ t ổ tr sỹ qỵ t ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ✣♦➔♥ ❚❤à ❍➔ ▲í✐ ❝❛♠ ✤♦❛♥ õ t ỗ t❤ù✧ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ s❛✉ q✉→ tr➻♥❤ ❤å❝ ❤ä✐✱ ♥❣❤✐➯♥ ự t ợ sỹ ữợ t t➻♥❤ ❝õ❛ ❚❙✳ ❍♦➔♥❣ ❚❤➳ ❚✉➜♥ ♣❤á♥❣ ①→❝ ①✉➜t t❤è♥❣ ❦➯ ❱✐➺♥ ❚♦→♥ ❤å❝✳ ❚r♦♥❣ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤ ❡♠ ❝â t❤❛♠ ❦❤↔♦ ♠ët sè ♥ë✐ ❞✉♥❣✱ ❦➳t q✉↔ ởt ữợ ởt số t ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❦❤→❝✳ ❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤✱ ❦❤æ♥❣ s❛♦ ❝❤➨♣ ❜➜t ❦➻ ❜➔✐ ❦❤â❛ ❧✉➟♥ ♥➔♦ ❦❤→❝✳ ❊♠ ①✐♥ ❝❤à✉ ❤♦➔♥ t♦➔♥ tr→❝❤ ♥❤✐➺♠ ✈ỵ✐ ❧í✐ ❝❛♠ ✤♦❛♥ ❝õ❛ ♠➻♥❤✳ ❍➔ ◆ë✐✱ ♥❣➔② ✵✼ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽ ❙✐♥❤ ✈✐➯♥ ✣♦➔♥ ❚❤à ❍➔ ▼ư❝ ❧ư❝ ▲í✐ ♥â✐ ✤➛✉ ✶ ✶ P❤➨♣ t➼♥❤ ✈✐ t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù✳ ✹ ✶✳✶ ❚➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤➙♥ t❤ù✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ✣↕♦ tự ỗ ìợ ữủ tự ỗ rữớ ❤đ♣ trì♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ P❤➨♣ ①➜♣ ①➾ ♠ët ❤➔♠ ❜à ❝❤➦♥ ✤♦ ✤÷đ❝✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ìợ ữủ tự ỗ rữớ ❤ñ♣ tê♥❣ q✉→t✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❍➺ ✤ë♥❣ ❧ü❝ ♣❤➙♥ t❤ù ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✷✺ ✷✾ ✸✳✶ ◆❤ú♥❣ ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ❚❤✐➳t ❦➳ ✤✐➲✉ ❦✐➸♥ ①➜♣ ①➾✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ❑➳t ❧✉➟♥ ✸✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✽ ✐✐ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❇❷◆● ❑➑ ❍■➏❯ ❑➼ ❤✐➺✉ ❚➯♥ ❣å✐ R ❚➟♣ sè t❤ü❝ Rn ❑❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ t❤ü❝ n ❝❤✐➲✉ C ❚➟♣ ❤ñ♣ ❝→❝ sè ♣❤ù❝ |z| ●✐→ trà t✉②➺t ✤è✐✭♠♦❞✉❧❡✮❝õ❛ sè t❤ü❝ ✭♣❤ù❝✮ ③ · L1 [a, b] ❈❤✉➞♥ ❝õ❛ ♠ët ✈➨❝ tì ❤♦➦❝ ♠❛ tr➟♥ ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b] Lip0 (X, Rk ) ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ C([a, b]; X) ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ♥❤➟♥ ❣✐→ trà tr♦♥❣ ❳ ❧✐➯♥ tö❝ tr➯♥ [a, b] α ❈➜♣ ❝õ❛ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù α ❙è ♥❣✉②➯♥ ♥❤ä ♥❤➜t ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ α Iαα+ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù ❘✐❡♠❛♥♥✕▲✐♦✉✈✐❧❧❡ ❝➜♣ α Dαα+ ❚♦→♥ tû ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❘✐❡♠❛♥♥✕▲✐♦✉✈✐❧❧❡ ❝➜♣ α C ❚♦→♥ tû ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❈❛♣✉t♦ ❝➜♣ α Dαα+ Γ(z) ❍➔♠ ●❛♠♠❛ ∆V (·) ●r❛❞✐❡♥t ❝õ❛ ❤➔♠ V (·) x(j) (α) ✣↕♦ ❤➔♠ ❝➜♣ j ❝õ❛ ❤➔♠ ① t↕✐ α✳ ✶ ▲í✐ ♥â✐ ✤➛✉ P❤➨♣ t➼♥❤ ✈✐✕t➼❝❤ ởt ổ ỵ tữ ổ t ❝→❝ q✉→ tr➻♥❤ t✐➳♥ ❤â❛✳ ❚❤ỉ♥❣ t❤÷í♥❣✱ ♠é✐ q✉→ tr➻♥❤ t✐➳♥ ❤â❛ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ❇➡♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✭✤à♥❤ t➼♥❤ ❤♦➦❝ ✤à♥❤ ❧÷đ♥❣✮ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ ♥❣÷í✐ t❛ ❝â t❤➸ ❜✐➳t tr↕♥❣ t❤→✐ ❤✐➺♥ t❤í✐ ❝ơ♥❣ ♥❤÷ ❞ü ✤♦→♥ ✤÷đ❝ ❞→♥❣ ✤✐➺✉ ð q✉→ ❦❤ù ❤❛② t÷ì♥❣ ❧❛✐ ❝õ❛ q✉→ tr➻♥❤ ✤â✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ ❤✐➺♥ t÷đ♥❣ ❤❛② ❣➦♣ tr♦♥❣ ❝✉ë❝ sè♥❣ ❝â t➼♥❤ ❝❤➜t ♣❤ö t❤✉ë❝ ✈➔♦ ❧à❝❤ sû✳ ✣è✐ ợ tữủ s ❝õ❛ ♥â t↕✐ ♠ët t❤í✐ ✤✐➸♠ t÷ì♥❣ ❧❛✐ tø q✉→ ❦❤ù ♣❤ư t❤✉ë❝ ❝↔ ✈➔♦ q✉❛♥ s→t ✤à❛ ♣❤÷ì♥❣ ❧➝♥ t♦➔♥ ❜ë q✉→ ❦❤ù✳ ❍ì♥ ♥ú❛✱ sü ♣❤ư t❤✉ë❝ ♥â✐ ❝❤✉♥❣ ❝ơ♥❣ ❦❤ỉ♥❣ ❣✐è♥❣ ♥❤❛✉ ð t➜t ❝↔ ❝→❝ t❤í✐ ✤✐➸♠✳ ◆❤ú♥❣ t❤ü❝ t➳ ✈ø❛ ♥➯✉ ❞➝♥ tỵ✐ ♥❤✉ ❝➛✉ ỹ ởt ỵ tt tờ qt t tû ✈✐ ♣❤➙♥ s✐♥❤ r❛ ♥❣❤✐➺♠ ❦❤æ♥❣ ❝â t➼♥❤ ❝❤➜t ữỡ ởt tr ỵ tt ữ ✤÷đ❝ ①➙② ❞ü♥❣ ❧➔ ❣✐↔✐ t➼❝❤ ♣❤➙♥ t❤ù✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❜➔✐ t♦→♥ q✉❛♥ trå♥❣ ❝õ❛ ❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤➙♥ t❤ù ❧➔ ❧➼ t❤✉②➳t ✤à♥❤ t➼♥❤✳ ❚r♦♥❣ ✤â ♥❣÷í✐ t❛ ♠✉è♥ ♥❣❤✐➯♥ ❝ù✉ ❞→♥❣ ✤✐➺✉ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤➙♥ t❤ù✳ ▼ët ✈➜♥ ✤➲ q✉❛♥ trå♥❣ ❦❤→❝ ❧➔ t❤✐➳t ❦➳ ✤✐➲✉ ❦❤✐➸♥ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔②✳ ✷ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❈â ❤❛✐ ♣❤÷ì♥❣ ♣❤→♣ t❤÷í♥❣ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❧➼ t❤✉②➳t ✤à♥❤ t➼♥❤ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤➙♥ t❤ù✿ ✶✳ P❤÷ì♥❣ ♣❤→♣ t✉②➳♥ t➼♥❤ ❤â❛✳ ✷✳ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ▲②❛♣✉♥♦✈✳ ❚r♦♥❣ ❦❤✐ ❧➼ t❤✉②➳t ê♥ ✤à♥❤ t✉②➳♥ t➼♥❤ ❤â❛ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ t÷ì♥❣ ✤è✐ ❝❤✐ t✐➳t t❤➻ ♣❤÷ì♥❣ ♣❤→♣ ❧➼ t❤✉②➳t ❤➔♠ ▲②❛♣✉♥♦✈ ❝á♥ ð ❦❤❛✐✳ ◆❣✉②➯♥ ♥❤➙♥ ❧➔ ✈➻ ✤↕♦ ❤➔♠ ♣❤➙♥ tự ổ õ ỵ t ❤➔♠ ♣❤➙♥ t❤ù ❝õ❛ ❤➔♠ ➙♠ ❦❤æ♥❣ ❦➨♦ t❤❡♦ t➼♥❤ s✉② ❣✐↔♠ ❝õ❛ ❤➔♠ ♥➔② t❤❡♦ t❤í✐ ❣✐❛♥✳ ▼ët ❦❤â ❦❤➠♥ ❦❤→❝ ❧➔ q✉② t➢❝ t➼♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ❤đ♣ ❝ê ✤✐➸♥ ❦❤ỉ♥❣ ❝á♥ ✤ó♥❣ ❝❤♦ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù✳ ✣➸ ❦❤➢❝ ♣❤ö❝ ♥❤ú♥❣ ❤↕♥ ❝❤➳ ♥â✐ tr➯♥ ❣➛♥ ✤➙② t→❝ ❣✐↔ ▼✳■✳ ●♦✲ ♠♦②✉♥♦✈ ✤➣ ✤÷❛ r❛ ♠ët ❝→❝❤ ✤→♥❤ ❣✐→ ❝❤♦ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ❝õ❛ ❝→❝ ỗ tổ q ữớ t õ t❤➸ t❤✐➳t ❧➟♣ ❝→❝ ❤➔♠ ù♥❣ ✈✐➯♥ ▲②❛♣✉♥♦✈ ✤➸ ❣✐↔✐ qt ữợ ữủ tr t ❝ù✉ t➼♥❤ ê♥ ✤à♥❤ ✈➔ t❤✐➳t ❦➳ ✤✐➲✉ ❦❤✐➸♥✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ♣❤➙♥ t➼❝❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ ♥â✐ tr➯♥✳ ◆ë✐ ❞✉♥❣ õ ỗ P t t tự tự ỗ ✸✳ ❍➺ ✤ë♥❣ ❧ü❝ ♣❤➙♥ t❤ù ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✸ ❈❤÷ì♥❣ ✶ P❤➨♣ t➼♥❤ ✈✐ t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù✳ ◆ë✐ tr ữỡ ợ t tờ q ữỡ tr tự ợ t ỳ t ỡ t tự ỗ ✈➲ ❚➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù✳ ✣↕♦ ❤➔♠ ♣❤➙♥ t❤ù✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤➙♥ t❤ù✳ ✶✳✶ ❚➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù✳ ữủ ợ t sỡ ữủ ❦❤→✐ ♥✐➺♠ t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù✳ ❍✐➸✉ t❤❡♦ ♠ët ♥❣❤➽❛ ♥➔♦ ✤â✱ t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù ❧➔ ♠ët ♠ð rë♥❣ tü ♥❤✐➯♥ ❝õ❛ ❦❤→✐ ♥✐➺♠ t➼❝❤ ♣❤➙♥ ❧➦♣ t❤ỉ♥❣ t❤÷í♥❣✳ ❈ö t❤➸✱ ❝❤♦ α > ✈➔ [a, b] ⊂ R✱ ❝❤ó♥❣ t❛ ✤à♥❤ ♥❣❤➽❛ t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù ❘✐❡♠❛♥♥✕▲✐♦✉✈✐❧❧❡ ❝➜♣ α ❝õ❛ ❤➔♠ x : [a, b] → R ❧➔ Iaα+ x(t) t := Γ(α) a ✹ x(τ ) dτ (t − τ )1−α ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ✈ỵ✐ t ∈ [a, b]✱ ð ✤➙② ❤➔♠ ●❛♠♠❛ Γ : (0, ∞) → R>0 ❝â ❜✐➸✉ ❞✐➵♥ ∞ tα−1 exp(−t)dt Γ(α) := ❘ã r➔♥❣✱ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ tr➯♥ ♥➳✉ x ❦❤↔ t➼❝❤ tr➯♥ ✤♦↕♥ [a, b]✱ tù❝ ❧➔ b |x(t)|dt < ∞✱ t❤➻ t➼❝❤ ♣❤➙♥ ♣❤➙♥ t❤ù ❘✐❡♠❛♥♥✕▲✐♦✉✈✐❧❧❡ ❝➜♣ α ❝õ❛ a x tỗ t ỡ tr [a, b] ỡ ỳ ❝❤➼♥❤ ❜↔♥ t❤➙♥ t♦→♥ tû t➼❝❤ ♣❤➙♥ ♥➔② ❝ô♥❣ ❧➔ ♠ët ❤➔♠ ❦❤↔ t➼❝❤✳ ◆❤➟♥ ①➨t ♥➔② ❧➔ ♥ë✐ ❞✉♥❣ ❝õ❛ ❜ê ✤➲ s❛✉ ✤➙②✳ ❇ê ✤➲ ✶✳✶✳ ●✐↔ sû x : [a, b] → R ❧➔ ♠ët ❤➔♠ ❦❤↔ t➼❝❤ tr➯♥ [a, b]✳ ❑❤✐ ✤â✱ t➼❝❤ ♣❤➙♥ Iaα+ x(t) tỗ t t t [a, b] ỡ ỳ Iaα+ x(t) ❝ơ♥❣ ❧➔ ♠ët ❤➔♠ t❤✉ë❝ ❧ỵ♣ L1 [a, b] ự t t ữợ t +∞ α−1 (t − τ ) ð ✤➙② φ1 (u) = φ1 (t − τ )φ2 (τ )dτ x(τ )dτ = α+ −∞ un−1 , 0, ✈➔ φ2 (u) = x(u), 0, 0 tø ✭✷✳✶✽✮ tỗ t J > s t j ∈ N, j ≥ J, ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤ó♥❣✿ ωλV T 1/p sup xk (·) − x(·) ∞ + MV sup ϕk (·) − ϕ(·) k≥j p ≤ ε k≥j ❈❤♦ j ∈ N ✈➔ j ≥ J ✳ ❈❤♦ ❜➜t ❦➻ t ∈ [0, T ], tø ✈✐➺❝ ❝❤å♥ ω, λV ✈➔ MV t❛ ❝â✿ nj |zj (t) − z(t)| ≤ αij V (xkij (t)) − V (x(t)) ϕkij (t) αij V (x(t)) ϕkij (t) − ϕ(t) i=1 nj + i=1 nj nj ≤ωλV αij xkij (·) − x(·) ∞ αij ϕkij (t) − ϕ(t) + MV i=1 i=1 ❚❛ ❝â kij ≥ j ≥ J, i ∈ 1, nj , ✈➔ ❞♦ sü ❧ü❛ ❝❤å♥ J ❝❤ó♥❣ t❛ ❝â✿ zj (·)−z(·) p ≤ ωλV sup xk (·)−x(·) ∞ p +MV k≥j sup ϕk (·)−ϕ(·) p ≤ ε k≥j ❈❤♦ ❜➜t ❦ý t ∈ [0, T ]✱ ✈➔ j ∈ N, tø ✭✷✳✶✾✮ s✉② r❛✿ nj nj αij ψkij (t) ≤ ξj (t) = i=1 ❇ð✐ ✈➻ ξj (·) − ψ(·) αij V (xkij (t)), ϕkij (t) = zj (t) ✭✷✳✷✵✮ i=1 p −→ ✈➔ zj (·) − z(·) p −→ ❦❤✐ j −→ ∞, t❤❡♦ ❬✷✶✱ ❚❤❡♦r❡♠ ✸✳✶✷❪ t❛ ❣✐↔ sû r➡♥❣ |εj (t)−z(t)| −→ ✈➔ |zj (t)−z(t)| −→ 0, ∀t ∈ [0, T ] ❱➻ ✈➟②✱ ❤➛✉ ❤➳t t ∈ [0, T ], ❝❤♦ j −→ ∞ tr♦♥❣ ✭✷✳✷✵✮✳ ❈❤ó♥❣ t❛ t❤✉ ✤÷đ❝ ψ(t) ≤ z(t) ❚➼♥❤ r➡♥❣ ψ(t) = (Dα y)(t), ∀t ∈ [0, T ]✳ ❱➻ ✈➟② ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✮ ✤ó♥❣ ✈ỵ✐ ∀t ∈ [0, T ] ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ✷✼ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ♠✐♥❤✳ ❈❤♦ tr÷í♥❣ ❤đ♣ ❦❤✐ V (x) = x , x ∈ Rn , ❝❤ó♥❣ t❛ ❝â ❤➺ q✉↔ s❛✉✿ ❍➺ q✉↔ ✷✳✷✳ ❈❤♦ x(·) ∈ I0α (L∞([0, T ], Rn)) ✈➔ y(t) = + x(t) , t ∈ [0, T ] ❚❤➻ y(·) ∈ I0α+ (L∞ ([0, T ], R)) ①→❝ ✤à♥❤ ✈➔ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ❜➜t ✤➥♥❣ t❤ù❝ (D0α+ y)(t) ≤ x(t), (D0α+ x)(t) ✈ỵ✐ ∀t ∈ [0, T ] ✷✽ ❈❤÷ì♥❣ ✸ ❍➺ ✤ë♥❣ ❧ü❝ ♣❤➙♥ t❤ù ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✳ ✸✳✶ ◆❤ú♥❣ ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥✳ ❳❡♠ ①➨t ❤➺ ✤ë♥❣ ❧ü❝ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤➙♥ t❤ù s❛✉✿ (C D0α+ x)(t) = g(t, x(t), u(t), v(t)) , t ∈ [0, T ] n nu nv ✭✸✳✶✮ x(t) ∈ R , u(t) ∈ P ⊂ R , v(t) ∈ Q ⊂ R , ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x(0) = x0 , x0 ∈ Rn ✭✸✳✷✮ Ð ✤➙② t ❧➔ ❜✐➳♥ t❤í✐ ❣✐❛♥✱ x ❧➔ ✈➨❝ tì tr↕♥❣ t❤→✐✱ ✉ ❧➔ ✈➨❝ tì ✤✐➲✉ ❦❤✐➸♥ ✈➔ ✈ ❧➔ tr↕♥❣ t❤→✐ ♥❤✐➵✉ ❦❤ỉ♥❣ ①→❝ ✤à♥❤✱ nu , nv ∈ N; P ✈➔ ◗ ❧➔ ❤❛✐ t➟♣ ❝♦♠♣❛❝t❀ x0 ❧➔ ❣✐→ trà ❜❛♥ ✤➛✉ ❝õ❛ ✈➨❝ tì tr↕♥❣ t❤→✐✳ ❍➔♠ g : [0, T ] × Rn × P × Q −→ Rn t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭❣✳✶✮ ❍➔♠ g(·) ❧➔ ❧✐➯♥ tö❝✳ ✷✾ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ t r 0, tỗ t g ≥ s❛♦ ❝❤♦ g(t, x, u, v) − g(t, y, u, v) ≤ λg x − y , t ∈ [0, T ], x, y ∈ B(r), u ∈ P, v Q ỗ t Cg > s❛♦ ❝❤♦ g(t, x, u, v) ≤ (1 + x )Cg , t ∈ [0, T ], x ∈ Rn , u ∈ P, v ∈ Q ✭❣✳✹✮ ❈❤♦ ❜➜t ❦➻ t ∈ [0, T ] ✈➔ x, s ∈ Rn , ữỡ tr ữợ ú max s, g(t, x, u, v) = max s, g(t, x, u, v) u∈P v∈Q v∈Q u∈P ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳ ✣✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ ✈➔ ♥❤✐➵✉ t÷ì♥❣ ù♥❣ ❧➔ ❤➔♠ ✤♦ ✤÷đ❝ u : [0, T ] −→ P ✈➔ v : [0, T ] −→ Q✳ ❚➟♣ t÷ì♥❣ ù♥❣ ❝õ❛ t➜t ❝↔ ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ u(·) v(ã) ữủ ỵ U V ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳ ❈❤✉②➸♥ ✤ë♥❣ ❝õ❛ ❤➺ ✭✸✳✶✮✱ ✭✸✳✷✮ t÷ì♥❣ ù♥❣ ✈ỵ✐ ❣✐→ trà ❜❛♥ ✤➛✉ x0 ∈ Rn t❛ ❝â u(·) ∈ U, v(·) ∈ V ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✭✸✳✶✮✱ ✭✸✳✷✮ ð ✤➙② ❤➔♠ u(·) ✈➔ v(·) ❧➔ ✈➨❝ tì ✤✐➲✉ ❦❤✐➸♥ ✈➔ tr↕♥❣ t❤→✐ ♥❤✐➵✉ ổ ú ỵ r t ♠ët ✤ë♥❣ t❤→✐ ♥❤÷ ✈➟② ❧➔ ♠ët ❤➔♠ x(·) ∈ {x0 } + I α (L∞ ([0, T ], Rn )) ❝ò♥❣ ✈ỵ✐ u(·) ✈➔ v(·) t❤ä❛ ♠➣♥ ✭✸✳✶✮ ❝❤♦ ∀t ∈ [0, T ] ❚➼♥❤ ❝❤➜t ✸✳✶✳✶✳ ❱ỵ✐ ❜➜t ❦➻ ❣✐→ trà ❜❛♥ ✤➛✉ x0 ∈ Rn ✈➔ ❜➜t u(ã) U, v(ã) V, tỗ t ♥❤➜t ❝❤✉②➸♥ ✤ë♥❣ x(·) = x(·; x0 , u(·), v(·)) ✸✵ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ t❤ä❛ ♠➣♥ ❤➺ ✭✸✳✶✮✈➔ ✭✸✳✷✮✳ ❍ì♥ ♥ú❛✱ ❝❤♦ ❜➜t ❦➻ R0 ≥ 0✱ ∃ R ≥ ✈➔ ∃ H ≥ s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦➻ x0 ∈ B(R0 )✱ u(·) ∈ U, v(·) ∈ V ✈➔ x(· ) = x(· , x0 , u(· ), v(· ) t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ x(t) ≤ R, x(t) − x(τ ) ≤ H|t − τ |α , t, τ ∈ [0, T ] ✭✸✳✸✮ ❈❤ù♥❣ ♠✐♥❤✳ ✣➙② ❧➔ ♠➺♥❤ ✤➲ õ ữủ tự tứ ỵ ▼➺♥❤ ✤➲ ✷✳✶ ♥➳✉ ❝❤ó♥❣ t❛ t➼♥❤ r➡♥❣✱ ❝❤♦ ❜➜t ❦➻ u(·) ∈ U ✈➔ v(·) ∈ V, ❞♦ ✭❣✳✶✮✲✭❣✳✸✮ ❤➔♠ f (t, x) = g(t, x, u(t), v(t)), t ∈ [0, T ], x ∈ Rn , t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❢✳✶✮✲✭❢✳✸✮ ✈➔ ❤ì♥ ♥ú❛ ✭❢✳✸✮ ❤♦➔♥ t❤➔♥❤ ✈ỵ✐ ❤➡♥❣ sè Cg ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ u(·) ✈➔ v(·)✳ t ữợ Đ ởt s ❝õ❛ ❤➺ ✭✸✳✶✮✱ ✭✸✳✷✮✳ ❉♦ ✤â tr↕♥❣ t❤→✐ ❝õ❛ ♣❤➨♣ ữợ ữủ ổ t ữỡ tr ♣❤➙♥ t❤ù (C Dα y)(t) = g(t, y(t), u(t), v(t)), t ∈ [0, T ], n ✭✸✳✹✮ y(t) ∈ R , u(t) ∈ P, v(t) ∈ Q, ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ y(0) = y0 , y0 ∈ Rn ✭✸✳✺✮ Ð ✤➙② y ❧➔ ✈➨❝ tì tr↕♥❣ t❤→✐ ữợ u(ã) U, v(ã) V ▲➔ ❝→❝ ✈➨❝ tì ✤✐➲✉ ❦❤✐➸♥ ❝❤➜♣ ♥❤➟♥ ✤÷đ❝✳ ❉♦ ✤â tø ▼➺♥❤ ✤➲ ✸✳✶✳✶ ♥â ❝❤♦ ♠ët ❝❤✉②➸♥ ✤ë♥❣ y(ã) = y(ã; y0 , u(ã), v(ã)) tỗ t t ỡ ỳ õ tọ ữợ t tữỡ tü ✭✸✳✸✮✳ ❚r♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦ ♠ư❝ t✐➯✉ ❝õ❛ ❝❤ó♥❣ t❛ ❧➔ ①→❝ ❧➟♣ ♣❤➨♣ ✤à♥❤ ✸✶ ✣♦➔♥ ❚❤à ❍➔ õ tốt ữợ ❤➺ ❜❛♥ ✤➜✉ ✭✸✳✶✮ ✈➔ ✭✸✳✷✮✳ ✸✳✷ ❚❤✐➳t ❦➳ ✤✐➲✉ ❦✐➸♥ ①➜♣ ①➾✳ ❈❤♦ x0 , y0 ∈ Rn ✈➔ ∆ = {τj }k+1 j=1 ⊂ [0, T ], τ1 = τj+1 > τj , j ∈ 1, k, τk+1 = T, k ∈ N, ❧➔ ♣❤➙♥ ❤♦↕❝❤ ❝õ❛ ✤♦↕♥ [0, T ]✳ ❈❤ó♥❣ t❛ s➩ t❤✐➳t ❧➟♣ ❜✐➳♥ ✤✐➲✉ ❦❤✐➸♥ u(·) ∈ U tr♦♥❣ ❤➺ ✭✸✳✶✮✱ ✭✸✳✷✮ ✈➔ v(ã) V tr ữợ j ∈ 1, k ✈➔ ❣✐→ trà x(τj ), y(τj )✳ ❈❤ó♥❣ t❛ ✤à♥❤ ♥❣❤➽❛✿ u(t) = uj ∈ arg max s(τj ), g(τj , x(τj ), u, v) , u∈P v∈Q t ∈ [τj , τj+1 ) , v(t) = vj ∈ arg max s(τj ), g(τj , x(τj ), u, v) v∈Q u∈P ✭✸✳✻✮ ❑❤✐ ✤â ❝❤ó♥❣ t❛ ❦➼ ❤✐➺✉ s(t) = x(t) − y(t), ∀t [0, T ] ỵ t R0 > > tỗ t k > ✈➔ δ > s❛♦ ❝❤♦ ❣✐→ trà ❜❛♥ ✤➛✉ x0 , y0 ∈ B(R0 ), ❜➜t ợ ữớ diam( ) = maxj1,k (j+1 τj ) ✈➔ ❜➜t ❦➻ v(·) ∈ V, u(·) ∈ U ✈➔ v(·) ∈ V ✤÷đ❝ t❤✐➳t ❧➟♣ t❤❡♦ q✉② t➢❝ ❤ë✐ tư ✭✸✳✻✮✱ t❤➻ ❝❤✉②➸♥ ✤ë♥❣ t÷ì♥❣ ù♥❣ x(·) = x(·; x0 , u(·)v(·)) ❝õ❛ ❤➺ ✭✸✳✶✮✱ ✭✸✳✷✮ ✈➔ y(·) = y(·; y0 , u(·), v(·)) ✸✷ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❝õ❛ ♣❤➨♣ ✤à♥❤ ữợ tọ t tự x(ã) y(·) ∞ ≤ ε + K x0 − y ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ R0 > ✈➔ > 0✳❱ỵ✐ sè R0 ✱ ❝❤å♥ R ✈➔ H ❞♦ ✭❣✳✷✮ √ t❛ ❝❤å♥ λg ❜ð✐ sè R✳ ✣à♥❤ ♥❣❤➽❛ k = Eα (2λg T α )✳ ❈❤♦ η > t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ✿ η ≤ Γ(α + 1)ε2 /(2T α Eα (2λg T α )) ❚❤❡♦ ✭❣✳✶✮✱ ❝❤å♥ δ1 > s❛♦ ❝❤♦✱ ❜➜t ❦➻ t, τ ∈ [0, T ], x ∈ B(R)✱ u ∈ P ✈➔ v ∈ Q✱ ♥➳✉ |t − τ | ≤ δ1 t❤➻ g(t, x, u, v) − g(τ, x, u, v) ≤ η/(16R) ▲➜② δ2 > s❛♦ ❝❤♦ δ2α ≤ min{η/(8H(1 + R)cg ), η/(16Rλg H)} Ð ✤➙② cg ❧➔ ❤➡♥❣ sè tø ✭❣✳✸✮✳ ❈❤♦ δ = min{δ1 , δ2 }✱ ❚❛ ❝❤➾ r❛ r➡♥❣ sè ❑ ✈➔ δ t❤ä❛ ỵ x0 , y0 B(R0 ) ✈➔ ∆ ❝â ✤÷í♥❣ ❦➼♥❤(∆) ≤ δ ✳ ▲➜② v(·) ∈ B, n ˜ ∈ U ✈➔ u(·) ∈ U, v˜(·) ∈ V ✳ ❈❤♦ x(·) = x(·; x0 , u(·), v(·)) ✈➔ y(·) = y(·; y0 , u˜(·), v˜(·)) t❤❛② ✤ê✐ ❜ð✐ ❤➺ ✭✸✳✶✮✱ ✭✸✳✷✮ ✈➔ ✭✸✳✹✮✱ ✭✸✳✺✮✳ ❈❤♦ s(·) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✸✳✼✮✳❚❤➻ t❛ ❝â s(·) ∈ {x0 − y0 } + I α (L∞ ([0, T ], Rn )); s(t) ≤ 2R, s(t) − s(τ ) ≤ 2H|t − τ |α , ✸✸ t, τ ∈ [0, T ]; ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ (C Dα s)(t) = g(t, x(t), u(t), v(t))−g(t, y(t), u(t), v(t)) ✈ỵ✐ ♠å✐ t ∈ [0, T ] ❳➨t ❤➔♠ sè v(t) = s(t) v(t) = s(t) − (x0 − y0 ) − (x0 − y0 ) 2 , t ∈ [0, T ] ❑❤✐ ✤â + x0 − y0 , s(t) − (x0 − y0 ) , , t ∈ [0, T ] t❤➻ tø ❍➺ q✉↔ ✸✳✷✱ t❛ ❝â v(·) ∈ I0α+ (L∞ ([0, T ], R)) ✈➔ (D0α+ v)(t) ≤2 s(t) − (x0 − y0 ), (C D0α+ s)(t) + x0 − y0 , (C D0α+ s)(t) =2 s(t), (C D0α+ s)(t) ✈ỵ✐ ♠å✐ t ∈ [0, T ] ✭✸✳✽✮ ❚ø ✤â s✉② r❛ s(t).(C D0α+ s)(t) ≤ λg s(t) + η ✈ỵ✐ ♠å✐ t ∈ [0, T ], ✭✸✳✾✮ ợ t [0, T ] t ữủ s(t).(C D0α+ s)(t) = s(t), g(t, x(t), u(t), v(t)) − g(t, x(t)u(t), v(t)) + s(t), g(t, x(t), u(t), v(t)) − g(t, y(t), u(t), v(t)) ✭✸✳✶✵✮ ●✐↔ sû j ∈ 1, k ✈➔ t ∈ [τj , τj+1 ) tø ✭❣✳✸✮ ✈➔ ❝❤å♥ δ2 , t❛ ✤÷đ❝ s(t), g(t, x(t), u(t), v(t)) − g(t, x(t), u(t), v(t)) ≤ s(τj ), g(t, x(t), u(t), v(t)) − g(t, x(t), u(t), v(t)) + s(t) − s(τj ) ( g(t, x(t), u(t), v(t)) + g(t, x(t)u(t), v(t)) ) ≤ s(τj )g(t, x(t), u(t), v(t)) − g(t, x(t), u(t), v(t)) + 4H(1 + R)cg δ α ≤ s(τj ), g(t, x(t), u(t), v(t)) − g(t, x(t)u(t), v(t)) + η/2 ✸✹ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❍ì♥ ♥ú❛✱ tø ✈✐➺❝ ❝❤å♥ λg , δ1 ✈➔ δ2 t❛ ✤÷đ❝ s(τj ), g(t, x(t), u(t), v(t)) ≤ s(τj ), g(τj , x(τj ), u(t), v(t)) + s(τj ) g(t, x(t), u(t), v(t)) − g(τj , x(t), u(t), v(t)) + s(τj ) g(τj , x(t), u(t), v(t)) − g(τj , x(τj ), u(t), v(t)) ✭✸✳✶✶✮ ≤ s(τj ), g(τj , x(τj ), u(t), v(t)) +2R g(t, x(t), u(t), v(t)) − g(τj , x(t), u(t), v(t)) + 2Rλg Hδ α ≤ s(τj ), g(τj , x(τj ), u(t), v(t)) + η/4 ❚÷ì♥❣ tü✱ s(τj ), g(t, x(t), u(t), v(t)) ≥ s(τj ), g(τj , x(τj )u(t), v(t)) ❈✉è✐ ❝ò♥❣✱ ✈ỵ✐ ✭❣✳✹✮ ✈➔ ❝❤å♥ uj , vj t❛ ❝â s(τj ), g(τj , x(τj ), u(t), v(t)) − s(τj ), g(τj , x(τj ), u(t), v(t)) = s(τj ), g(τj , x(τj ), uj , v(t)) − s(τj ), g(τj , x(τj ), u(t), vj ) ≤ max s(τj ), g(τj , x(τj ), uj , v) − s(τj ), g(τj , x(τj ), u, vj ) v∈Q u∈P = max s(τj ), g(τj , x(τj ), u, v) − max s(τj ), g(τj , x(τj ), u, v) u∈P v∈Q v∈Q u∈P ❑➳t q✉↔ ❧➔✱ ✈ỵ✐ t ∈ [0, T ]), t❛ ❝â s(t), g(t, x(t), u(t), v(t)) − g(t, x(t), u(t), v(t)) ≤ η ✸✺ ✭✸✳✶✷✮ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❱ỵ✐ t ∈ [0, T ], ❞♦ ✤â ❝❤å♥ λq t❛ ✤÷đ❝ s(t), g(t, x(t), u(t),v(t)) − g(t, y(t), u(t), v(t)) ≤ s(t) g(t, x(t), u(t), v(t))−g(t, y(t), u(t), v(t)) ≤ λg s(t) ✭✸✳✶✸✮ ❚ø ✭✸✳✽✮ ✈➔ ✭✸✳✾✮ t❛ ✤÷đ❝ (D0α+ v)(t) ≤ 2λg s(t) s(t) 2 + 2η ✈ỵ✐ ♠å✐ t ∈ [0, T ] 2ηT α ≤ + x − y0 Γ(α + 1) 2λg + Γ(α) s(τ ) dτ (t − τ )1−α ❉♦ ✤â✱ tø ❇ê ✤➲ ✶✳✶ ✈➔ ❝❤å♥ η ✈➔ K ❱ỵ✐ t ∈ [0, T ], t❛ ✤÷đ❝ s(t) ≤ 2ηT α + x − y0 Γ(α + 1) ✸✻ Eα (2λg T α ) ≤ ε2 + K x0 − y0 ❑➳t ❧✉➟♥ ❚r♦♥❣ ❜➔✐ ❑❤â❛ ❧✉➟♥ ❝õ❛ ♥➔②✱ ❞ü❛ tr➯♥ ♠ët sè t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❡♠ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ s❛✉✿ ✶✳ ❑✐➳♥ t❤ù❝ tê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ♣❤➙♥ t❤ù ỗ õ t t tự ❤➔♠ ♣❤➙♥ t❤ù ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤➙♥ t❤ù✳ tự ỗ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝ỉ♥❣ ❝ư ✤➸ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❝❤➜t ✤à♥❤ t➼♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ ✸✳ ❍➺ ✤ë♥❣ ❧ü❝ ♣❤➙♥ t❤ù ✤✐➲✉ ❦✐➸♥ ✤÷đ❝✳ ❚r➯♥ ✤➙② ❧➔ t♦➔♥ ❜ë ♥ë✐ õ t ỗ t❤ù✧✳ ❙❛✉ q✉→ tr➻♥❤ ❤♦➔♥ t❤✐➺♥ ❦❤â❛ ❧✉➟♥ ❡♠ ✤➣ ❤✐➸✉ ❤ì♥ ✈➲ ✤↕♦ ❤➔♠ ♣❤➙♥ t❤ù ✤➦❝ ❜✐➺t ❝õ❛ ỗ ỳ t t ụ ữ ự ❞ư♥❣ ❝õ❛ ♥â tr♦♥❣ t♦→♥ ❤å❝ ❣✐↔✐ t➼❝❤✳ ▼➦❝ ❞ò ✤➣ ❝è ❣➢♥❣ ❤➳t ♠➻♥❤ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ ❤♦➔♥ ❝❤➾♥❤ ♥❤➜t✱ ♥❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ❝ơ♥❣ ♥❤÷ ✈✐➺❝ ♥➢♠ ❜➢t ❦✐➳♥ t❤ù❝✱ ❦✐♥❤ ♥❣❤✐➺♠ ❝õ❛ ❡♠ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ tr♦♥❣ ❜➔✐ ❑❤â❛ ❧✉➟♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤✐➲✉ s❛✐ sât✳ ❊♠ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü t❤ỉ♥❣ ❝↔♠ ❝ơ♥❣ ♥❤÷ sü ✤â♥❣ ❣â♣ ❝õ❛ qỵ t ổ õ ❝õ❛ ❡♠ ✤÷đ❝ t❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ✦ ✸✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆✳ ❆❣✉✐❧❛✲❝❛♠❛❝❤♦✱ ▼✳❆✳ ❉✉❛rt❡✲▼❡r♦✉❞✱ ❏✳❆ ●❛❧❧❡❣♦s✱ ▲②❛✲ ♣✉♥♦✈ ❋✉♥❝t✐♦♥ ❢♦r ❋r❛❝t✐♦♥ ❖r❞❡r ❙②st❡♠s✳ ❈♦♠♠✉♥✳ ◆♦♥❧✐♥✲ ❡❛r ❙❝✐✳ ◆✉♠❡r✳ ❙✐♠✉❧❛t✳ ✶✾✱ ✐ss✉❡ ✾ ✭✷✵✶✹✮✱ ✷✵✺✶✕✷✾✺✼❀ ❉❖■✿✶✵✳✷✵✶✻✴❥✳❝♥s♥s✳✷✵✶✹✳✵✶✳✵✷✷✳ ❬✷❪ ❆✳❆✳ ❆❧✐❦❤❛♥♦✈ ❆ Pr✐♦r✐ ❊st✐♠❛t❡s ❢♦r ❙♦❧✉t✐♦♥ ♦❢ ❇♦✉♥❞❛r② ❱❛❧✉❡ Pr♦❜❧❡♠s ❢♦r ❋r❛❝t✐♦♥❛❧✲❖r❞❡r ❊q✉❛t✐♦♥s ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛✲ t✐♦♥✳ ✹✻✱ ◆♦ ✺ ✭✷✵✶✵✮✱ ✻✻✵✲✻✻✻❀ ❉❖■✿ ✶✵✳✶✶✸✹✴❙✵✵✶✷✷✻✻✶✶✵✵✺✵✵✺✽✳ ❬✸❪ ❨✉✳ ❆✈❡r❜♦✉❦❤✱ ❊①tr❡♠❛❧ ❙❤✐❢t ❘✉❧❡ ❢♦r ❈♦♥t✐♥✉♦s✲❚✐♠❡ ❩❡r♦✲ ✼ ❙✉♠ ▼❛r❦♦✈ ●❛♠❡s✳ ❉②♥✳ ●❛♠❡s ❆♣♣❧✳ ✱ ✐ss✉❡ ✶ ✭✷✵✶✼✮✱ ✶✕✷✵❀ ❉❖■✿✶✵✳✶✵✵✼✴s✶✸✷✸✺✲✵✶✺✲✵✶✼✸✲③✳ ❬✹❪ ❲✳ ❈❤❡♥✱ ❍✳ ❉❛✐✱ ❨✳ ❙♦♥❣✱ ❩✳ ❩❤❛♥❣✱ ❈♦♥✈❡① ▲②❛♣✉♥♦✈ ❢✉♥❝✲ t✐♦♥s ❢♦r ❙t❛❜✐❧✐t② ❆♥❛❧②s✐s ♦❢ ❋r❛❝t✐♦♥❛❧ ❖r❞❡r ❙②st❡♠s✳ ■❊❚ ❝♦♥✲ tr♦❧ ❚❤❡♦r② ❆♣♣❧✳ ✶✶✱ ◆♦ ✼ ✭✷✵✶✼✮✱ ✶✵✼✵✕✶✵✼✹❀ ❉❖■✿✶✵✳✶✵✹✾✴✐❡t✲ ❝t❛✳✷✵✶✻✳✵✾✺✵✳ ❬✺❪ ❆✳❆✳ ❈❤✐❦r✐✐✱ ■✳■✳ ▼❛t✐❝❤✐♥✱ ●❛♠❡ Pr♦❜❧❡♠s ❢♦r ❋r❛❝t✐♦♥❛❧✲❖r❞❡r ▲✐♥❡❛r ❙②st❡♠s✳ Pr♦❝✳ ❙t❡❦❧♦✈ ■♥st✳ ♦❢ ▼❛t❤✳ ✺✹✕✼✵❀ ❉❖■✿✶✵✳✶✶✸✹✴❙✵✵✽✶✺✹✸✽✶✵✵✺✵✵✺✻✳ ✸✽ ✷✻✽✱ s✉♣♣❧✳ ✶ ✭✷✵✶✵✮✱ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❬✻❪ ❏✳❇✳ ❈♦♥✇❛②✱ ❆ ❈♦✉rs❡ ✐♥ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦ ✭✶✾✽✺✮✳ ❬✼❪ ❑✳ ❉✐❡t❤❡❧♠✱ ❚❤❡ ❆♥❛❧②s✐s ♦❢ ❋r❛❝t✐♦♥❛❧ ❉✐❢❢❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣ ✭✷✵✶✵✮❀ ❉❖■✿ ✶✵✳✶✵✵✼✴✾✼✽✲✸✲ ✻✹✷✲✶✹✺✼✹✲✷✳ ❬✽❪ ❉✳ ■❞❝③❛❦✱ ❘✳ ❑❛♠♦❝❦✐✱ ❖♥ t❤❡ ❊①✐st❡♥❝❡ ❛♥❞ ❯♥✐q✉❡♥❡ss ❛♥❞ ❋♦r✉❧❛ ❢♦r t❤❡ ❙♦❧✉t✐♦♥ ♦❢ ❘✲▲ ❋r❛❝t✐♦♥❛❧ ❈❛✉❝❤② Pr♦❜❧❡♠ ✐♥ Rn ❋r❛❝t✳ ❈❛❧❝✳ ❆♣♣❧✳ ❆♥❛❧✳ ✶✹✱ ◆♦ ✹ ✭✷✵✶✶✮✱ ✺✸✽✕✺✺✸❀ ❉❖■✿ ✶✵✳✷✹✼✽✴s✶✸✺✹✵✲✵✶✶✲✵✵✸✸✲✺✳ ❬✾❪ ▲✳❱✳ ❑❛♥t♦r♦✈✐❝❤✱ ●✳P✳ ❆❦✐❧♦✈✱ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧✐s②s✳ P❡r❣❛♠♦♠ Pr❡ss✱ ❖①❢♦r❞ ✭✶✾✽✷✮✳ ❬✶✵❪ ❆✳❆✳ ❑✐❧❜❛s✱ ❍✳▼✳ ❙r✐✈❛st❛✈❛✱ ❏✳❏✳ ❚r✉❥✐❧❧♦✱ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛✲ t✐♦♥s ♦❢ ❋r❛❝t✐♦♥❛❧ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ❊❧s❡✈✐❡r ✭✷✵✶✻✮✳ ❬✶✶❪ ◆✳◆✳ ❑r❛s✈♦✈s❦✐✐✱ ❆✳◆ ❑♦t❡❧♥✐❦♦✈❛✱ ❙t♦❝❤❛st✐❝ ●✉✐❞❡ ❢♦r ❛ ❚✐♠❡✲❉❡❧❛② ❖❜❥❡❝t ✐♥ ❛ P♦s✐♦♥❛❧ ❉✐❢❢❡r❡♥t✐❛❧ ●❛♠❡✳ Pr♦❝✳ ❙t❡❦❧♦✈ ■♥st✳ ▼❛t❤✳ ✷✼✼✱ s✉♣♣❧✳ ✶✳✭✷✵✶✷✮✱ ✶✹✺✕✶✺✶❀ ❉❖■✿ ✶✵✳✶✶✸✹✴❙✵✵✽✶✺✹✸✽✶✷✵✺✵✶✹✽✳ ❬✶✷❪ ◆✳◆✳ ❑r❛s♦✈s❦✐✐✱ ❆✳◆✳ ❑r❛s♦✈s❦✐✐✱ ❈♦♥tr♦❧ ❯♥❞❡r ▲❛❝❦ ♦❢ ■♥❢♦r✲ ♠❛t✐♦♥✳ ❇✐r❦❤❛✉s❡r✱ ❇❡r❧✐♥ ❡t❝✳ ✭✶✾✾✺✮✳ ❬✶✸❪ ◆✳◆✳ ❑r❛s♦✈s❦✐✐✱ ❆✳■✳ ❙✉❜❜♦t✐♥✱ ●❛♠❡✲ ❚❤❡♦r❡t✐❛❧ ❈♦♥tr♦❧ Pr♦❜✲ ❧❡♠s✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦ ✭✶✾✽✽✮✳ ❬✶✹❪ ❈✳ ▲✐✱ ❋✳ ❩❡♥❣✱ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞s ❢♦r ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s✳ ❈❤❛♣✲ ♠❛♥ ❛♥❞ ❍❛❧❧✱ ◆❡✇ ❨♦r❦✱ ✭✶✵✶✺✮✳ ✸✾ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❬✶✺❪ ◆✳❨✉✳ ▲✉❦♦②❛♥♦✈✱ ◆❡✉tr❛❧✲❚②♣❡ ❆✳❘✳ ❙②♠t❡♠s✿ ❙t❡❦❧♦✈ ■♥st ▼❛t❤✳ ✷✾✶✱ P❧❛❦s✐♥✱ ❆♥ ❉✐❢❢❡r❡♥t✐❛❧ ❆♣♣r♦①✐♠❛t✐♦♥ ●❛♠❡s ❢♦r ▼♦❞❡❧✳Pr♦❝✳ ✐ss✉❡ ✶ ✭✷✵✶✺✮✱ ✶✾✵✲✷✵✷❀ ❉❖■✿ ✶✵✳✶✶✸✹✴❙✵✵✽✶✺✹✸✽✶✺✵✽✵✶✺✺✳ ❬✶✻❪ ❱✳ ▼❛❦s✐♠♦✈✱ ●❛♠❡ ❈♦♥tr♦❧ Pr♦❜❧❡♠ ❢♦r ❛ P❤❛s❡ ❋✐❡❧❞ ❊q✉❛✲t✐♦♥✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✼✵✱ ✐ss✉❡ ✶ ✭✷✵✶✻✮✱ ✷✾✹✲✸✵✼❀ ❉❖■✿✶✵✳✶✵✵✼✴s✶✵✾✺✼✲✵✶✺✲✵✼✷✶✲✵✳ ❬✶✼❪ ❆✳❘ ▼❛t✈✐②❝❤✉❦✱ ❱✳◆✳ ❯s❤❛❦♦✈✱ ❖♥ t❤❡ ❈♦♥str✉❝t✐♦♥ ♦❢ ❘❡s♦❧✈✐♥❣ ❈♦♥tr♦❧s ✐♥ ❈♦♥tr♦❧ Pr♦❜❧❡♠s ✇✐t❤ P❤❛s❡ ❈♦♥✲ str❛✐♥ts✳ ❏✳ ❈♦♠♣✉t✳ ❙②st✳❙❝✐✳ ■♥t✳ ✹✺✱ ◆♦ ✶ ✭✷✵✵✻✮✱ ✶✲✶✻❀ ❉❖■✿ ✶✵✳✶✶✸✹✴s✶✵✻✹✷✸✵✼✵✻✵✶✵✵✶✶✳ ❬✶✽❪ ◆✳◆✳ P❡tr♦✈✱ ❖♥❡ Pr♦♣❧❡♠ ♦❢ ●r♦✉♣ P✉rs✉✐t ✇✐t❤ ❋r❛❝t✐♦♥ ❉❡r✐✈❛✲ t✐✈❡s ❛♥❞ P❤❛s❡ ❈♦♥str❛✐♥ts✳ ❇✉❧❧❡t✐♥ ♦❢ ❯❞♠✉rt ❯♥✐✈❡rs✐t②✳ ▼❛t❤❡✲ ♠❛t✐❝s✱ ▼❡❝❤❛♥✐❝s✱ ❈♦♠♦✉t❡r ❙❝✐❡♥s❡✳ ✷✼✱ ✐ss✉❡ ✶ ✭✷✵✶✼✮✱ ✺✹✕✺✾❀ ❉❖■✿ ✶✵✳✷✵✺✸✼✴✈♠✶✼✵✶✵✺✳ ❬✶✾❪ ❘✳❚✳ ❘♦❝❦❛❢❡❧❧❛r✱ ❈♦♥✈❡r① ❆♥❛❧②s✐s✳ Pr✐♥❝❡s✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡rs❡② ✭✶✾✼✷✮✳ ❬✷✵❪ ❇✳ ❘♦ss✱ ❙✳●✳ ❙❛♠❦♦✱ ❊✳❘✳ ▲♦✈❡✱ ❋✉♥❝t✐♦♥ ❚❤❛t ❍❛✈❡ ◆♦ ❋✐rst ❖r❞❡r ❉❡r✐✈❛t✐✈❡ ▼✐❣❤t ❍❛✈❡ ❋r❛❝t✐♦♥❛❧ ❉❡r✐✈❛t✐✈❡s ♦❢ ❆❧❧ ❖r❞❡rs ▲❡ss ❚❤❛♥ ❖♥❡✳ ❘❡❛❧ ❆♥❛❧✳ ❊①❝❤❛♥❣❡✳ ✷✵✱ ◆♦ ✶ ✭✶✾✾✹✕✶✾✾✺✮✱ ✶✹✵✕ ✶✺✼✳ ❬✷✶❪ ❲✳ ❘✉❞✐♥✱ ❘❡❛❧ ❛♥❞ ❈♦♠♣❧❡① ❆♥❛❧②s✐s✳ ▼❝ ●r❛✇✲❍✐❧❧✱ ◆❡✇ ❨♦r❦ ✭✶✾✾✶✮✳ ✹✵ ✣♦➔♥ ❚❤à ❍➔ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❬✷✷❪ ❲✳ ❘✉❞✐♥✱ ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✳ ▼❝ ●r❛✇✲❍✐❧❧✱ ◆❡✇ ❨♦r❦ ✭✶✾✾✶✮✳ ❬✷✸❪ ❙✳●✳ ❙❛♠❦♦✱ ❆✳❆✳ ❑✐❧❜❛s✱ ❖✳■✳ ▼❛r✐❝❤❡✈✱ ❋r❛❝t✐♦♥❛❧ ■♥t❡❣r❛❧s ❛♥❞ ❉❡r✐✈❛t✐✈❡s✳ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✳●♦r❞♦♥ ❇r❡❛❝❤ ❙❝✐✳ P✉❜✲ ❧✐s❤❡rs ✭✶✾✾✸✮✳ ❬✷✹❪ ❏✳ ❲❛♥❣✱ ❨✳ ❩❤♦✉✱ ❆ ❈❧❛ss ♦❢ ❋r❛❝t✐♦♥ ❊✈♦❧✉t✐♦♥ ❊q✉❛t✐♦♥s ❛♥❞ ❖♣t✐♠❛❧ ❈♦♥tr♦❧s ✳ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✿ ❘❡❛❧ ❲♦r❧❞ ❆♣♣❧✳ ✶✷✱ ◆♦ ✶ ✭✷✵✶✶✮✱ ✷✻✷✲✷✼✷❀ ❉❖■✿ ✶✵✳✶✵✶✻✴❥✳♥♦♥r✇❛✳✷✵✶✵✳✵✻✳✵✶✸✳ ❬✷✺❪ ❊✳ ❩❡✐❞❧❡r✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳ ■✿❋✐①❡❞✲P♦✐♥t ❚❤❡♦r❡♠s✳ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦ ✭✶✾✽✻✮✳ ✹✶ ...TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* ĐỒN THỊ HÀ HÀM LYAPUNOV LỒI PHÂN THỨ KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích Người hướng dẫn khoa học