(LUẬN văn THẠC sĩ) tính ổn định của một số lớp phương trình sai phân và áp dụng

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ❚❘×❒◆● ❚❍➚ ◆●➴❈ ❚❘❹▼ ❚➑◆❍ ✃◆ ✣➚◆❍ ❈Õ❆ ▼❐❚ ❙➮ ▲❰P P❍×❒◆● ❚❘➐◆❍ ❙❆■ P❍❹◆ ❱⑨ ⑩P ❉Ư◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❇➐◆❍ ✣➚◆❍ ✲ ◆❿▼ ✷✵✷✵ download by : skknchat@gmail.com ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ◗❯❨ ◆❍❒◆ ❚❘×❒◆● ❚❍➚ ◆●➴❈ ❚❘❹▼ ❚➑◆❍ ✃◆ ✣➚◆❍ ❈Õ❆ ▼❐❚ ❙➮ ▲❰P P❍×❒◆● ❚❘➐◆❍ ❙❆■ P❍❹◆ ❱⑨ ⑩P ❉Ư◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P số ữớ ữợ P ❈➷◆● ❍×❰◆● download by : skknchat@gmail.com ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❝→❝ sè ❧✐➺✉ ✈➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ t ữủ t ữợ sỹ ữợ P ổ ữợ ổ ụ ①✐♥ ❝❛♠ ✤♦❛♥ ♠å✐ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ró ỗ ố t❤→♥❣ ✵✼ ♥➠♠ ✷✵✷✵ ❍å❝ ✈✐➯♥ ❚r÷ì♥❣ ❚❤à ◆❣å❝ ❚r➙♠ download by : skknchat@gmail.com ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ❇➜t ✤➥♥❣ t❤ù❝ ●r♦♥✇❛❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ●✐ỵ✐ t❤✐➺✉ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥❣❤✐➺♠ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥❣❤✐➺♠ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ t ỵ tt ữỡ tr➻♥❤ s❛✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸ ✸ ✹ ✶✶ ✶✾ ✷✷ ✷ ❚➑◆❍ ✃◆ ✣➚◆❍ ❈Õ❆ ▼❐❚ ❙➮ ▲❰P P❍×❒◆● ❚❘➐◆❍ ❙❆■ P❍❹◆ ✷✼ ✷✳✶ ❚➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❚➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♣❤✐ t✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❚➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♣❤✐ t✉②➳♥ ✳ ✳ ✳ ✷✳✷✳✷ ❚➼♥❤ ê♥ ✤à♥❤ ởt ợ ữỡ tr s t ổtổổ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ởt ợ ữỡ tr s ♣❤➙♥ ♣❤✐ t✉②➳♥ ❝â tr➵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✸✻ ✸✻ ✹✸ ✹✹ ✸ ▼❐❚ ❙➮ ❱➑ ❉Ö ⑩P ❉Ö◆● ✹✽ ❑➌❚ ▲❯❾◆ ❱⑨ ❑■➌◆ ◆●❍➚ ✺✽ ✸✳✶ ▼ët sè ✈➼ ❞ö ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ ❞➣② sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✷ ▼ët sè ✈➼ ❞ö ✈➲ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♠æ ❤➻♥❤ q✉➛♥ t❤➸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ download by : skknchat@gmail.com ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ download by : skknchat@gmail.com ✺✾ ❇❷◆● ❈⑩❈ ❑Þ ❍■➏❯ R R+ Z Z− Z+ AT x = max (|x(1)|, , |x(k)|) ∆x(n) = x(n + 1) − x(n) N(n0 ) ✿ t➟♣ sè t❤ü❝ ✿ t➟♣ sè t❤ü❝ ❞÷ì♥❣ ✿ t➟♣ sè ♥❣✉②➯♥ ✿ t➟♣ sè ♥❣✉②➯♥ ➙♠ ✿ t➟♣ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✿ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ A ✿ ❝❤✉➞♥ ❝õ❛ ✈❡❝tì x ✿ s❛✐ ♣❤➙♥ ❝õ❛ ❞➣② x(n) ✿ t➟♣ ỗ số tỹ n n0 download by : skknchat@gmail.com ✶ ▼ð ✤➛✉ ❚➼♥❤ ❝❤➜t ❝õ❛ ♥❣❤✐➺♠ ❝→❝ ữỡ tr s ởt ữợ ự q trồ ỵ tt t r ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❝õ❛ ❚♦→♥ ❤å❝ ❝ơ♥❣ ♥❤÷ ❝→❝ ❦❤♦❛ ❤å❝ ❦❤→❝ ♥❤÷ ●✐↔✐ t➼❝❤ sè✱ ỵ tt ỵ tt ữợ ữủ tr ❤å❝✱ ❙✐♥❤ t❤→✐ ❤å❝✱ ❱➻ ✈➟②✱ ✈✐➺❝ ♥❣❤✐➯♥ ự ỵ tt ởt tớ sỹ ❝õ❛ ❚♦→♥ ❤å❝✱ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ q✉❛♥ t➙♠✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙②✱ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ❇✉rt♦♥✱ ❈♦♦❦❡✱ ❨♦r❦❡✱ ❩❤❛♥❣✱ ❘❛❢♦✉❧✱ ■s❧❛♠✱ ❆r❞❥♦✉♥✐✱ ❍✉♦♥❣✱ ▼❛✉ ✈➔ ♠ët sè ♥❤➔ t♦→♥ ❤å❝ ❦❤→❝ ✤➣ ♥❤➟♥ ✤÷đ❝ ♥❤✐➲✉ ❦➳t q t t t ợ ữỡ tr➻♥❤ s❛✐ ♣❤➙♥ ❝â tr➵ ❤♦➦❝ ❦❤æ♥❣ ❝â tr➵✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷✿ ❚r♦♥❣ ❬✹❪✱ ❍✉♦♥❣ ✈➔ ▼❛✉ ✤➣ ✤➲ ①✉➜t ♠ët sè ❦➳t q✉↔ ✈➲ t➼♥❤ ❜à ❝❤➦♥ ♥❣➦t ❝õ❛ t ổ sỹ tỗ t↕✐ ♥❣❤✐➺♠ t✉➛♥ ❤♦➔♥ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♣❤✐ t✉②➳♥ ✈ỵ✐ tr➵ ❜✐➳♥ t❤✐➯♥ x(n + 1) = λ(n)x(n) + α(n)F (x(n − m(n)), n = 0, ❚r♦♥❣ ❬✶✵❪✱ ■s❧❛♠ ✈➔ ❨❛♥❦s♦♥ ✤➣ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤à♥❤ ❧➼ ✤✐➸♠ ❜➜t ✤ë♥❣ ✤➸ ❝❤➾ r❛ t➼♥❤ ❜à ❝❤➦♥ ✈➔ ê♥ ✤à♥❤ ❝õ❛ ♥❣❤✐➺♠ ❦❤ỉ♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♣❤✐ t✉②➳♥ x(n + 1) = a(n)x(n) + c(n)∆x(n − g(n)) + q(x(n), x(n − g(n))) ❚r♦♥❣ ❬✶✶❪✱ ❍✉♦♥❣ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤à♥❤ ❧➼ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ t➼♥❤ t♦→♥ ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ s❛✐ ♣❤➙♥ ✤➣ ❝❤➾ r❛ sü ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ✈➔ t➼♥❤ ❜à ❝❤➦♥ ♥❣➦t ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♣❤✐ t✉②➳♥ ❦❤æ♥❣ æ✲tæ✲♥æ♠ x(n + 1) = λ(n)x(n) + α(n)F (n, x(n − ω(n)), n ≥ download by : skknchat@gmail.com ✷ ❚r♦♥❣ ❬✽❪✱ ●✐❛♥❣ ✈➔ ❍✉♦♥❣ ♥❣❤✐➯♥ ❝ù✉ sü ê♥ ✤à♥❤ ❝õ❛ ♠æ ❤➻♥❤ ❞➙♥ sè t❤æ♥❣ q✉❛ t➼♥❤ ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ x(n + 1) = λx(n) + F (x(n − m)) ◆â✐ r✐➯♥❣✱ t➼♥❤ ❝❤➜t ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ t➼♥❤ ❝❤➜t t❤ó ✈à ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ♥â ❣➢♥ ✈ỵ✐ ❝→❝ ❝→❝ ✈➜♥ ✤➲ tr♦♥❣ s✐♥❤ ❤å❝✱ ② ❤å❝✱ ❝ì ❤å❝✱ ❦ÿ t❤✉➟t✱ ❦✐♥❤ t➳ ❱➻ ✈➟②✱ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ s❛✐ ♣❤➙♥ ❧➔ ♠ët ✈➜♥ ✤➲ ✤❛♥❣ t❤✉ ❤ót sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ tr♦♥❣ ữợ t tr ự t ởt số ợ ữỡ tr s ỗ t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥ ♥❤÷ s❛✉✿ ❈❤÷ì♥❣ ✶✿ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ s➩ ✤÷đ❝ ❞ị♥❣ tr♦♥❣ ❝→❝ ❝❤÷ì♥❣ t✐➳♣ t❤❡♦ ❝õ❛ ❧✉➟♥ ữỡ ởt số ợ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ ♣❤✐ t✉②➳♥✳ ❈❤÷ì♥❣ ✸✿ ▼ët sè ✈➼ ❞ư →♣ ❞ư♥❣ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ✈➼ ❞ö ✈➲ t➼♥❤ ❝❤➜t ❞➣② sè ✈➔ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❝→❝ ♠æ ❤➻♥❤ q✉➛♥ t❤➸✳ ▲✉➟♥ ữủ t ữợ sỹ ữợ trỹ t P ổ ữợ tổ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➲ sü ❝❤➾ ữợ t t t t t tr s✉èt q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ▼➦❝ ❞ò r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ✈➔ tr➻♥❤ ✤ë ♥➯♥ ❜➯♥ ❝↕♥❤ ♥❤ú♥❣ ❦➳t q✉↔ ✤➣ ✤↕t ✤÷đ❝✱ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳ t sõt t ữủ sỹ õ ỵ t t t qỵ t ổ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ download by : skknchat@gmail.com ✸ ❈❤÷ì♥❣ ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ r ữỡ ú tổ tr ỵ tt ữỡ tr s ỵ t❤✉②➳t ê♥ ✤à♥❤ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✳ ❈→❝ ❦✐➳♥ t❤ù❝ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❝❤õ ②➳✉ t❤❛♠ ❦❤↔♦ tr♦♥❣ t tự r ỵ ✶✳✶✳ ✭❳❡♠ ❬✶✶❪✮ ●✐↔ sû n−1 f ( )x( ), n ∈ N(n0 ) x(n) ≤ p(n) + q(n) ✭✶✳✶✮ =n0 ❑❤✐ ✤â n−1 x(n) ≤ p(n) + q(n) p( )f ( ) =n0 ❈❤ù♥❣ ♠✐♥❤ ✣➦t ❤➔♠ y(n) = n−1 n−1 =n0 f ( )x( ) (1 + q(r)f (r)) ✭✶✳✷✮ r= +1 ❚❛ ❝â y(n) = f (n)x(n), y(n0 ) = ✭✶✳✸✮ ❚ø x(n) ≤ p(n) + q(n)y(n) ✈➔ f (n) ≥ 0✱ t❛ ♥❤➟♥ ✤÷đ❝ y(n + 1) − (1 + q(n)f (n))y(n) ≤ p(n)f (n) ❱➻ + q(n)f (n) > ✈ỵ✐ ♠å✐ n ∈ N(n0), t❛ ♥❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✶✳✹✮ ✈ỵ✐ download by : skknchat@gmail.com ✭✶✳✹✮ n =n0 (1 + ✹ q( )f ( ))−1 ✱ t❛ ✤÷đ❝ n n−1 −1 (1 + q( )f ( )) (1 + q( )f ( ))−1 y(n) ≤ p(n)f (n) =n0 =n0 ▲➜② tê♥❣ tø n0 ✤➳♥ n − ✈➔ ❞ò♥❣ y(n0) = t❛ t❤✉ ✤÷đ❝ n−1 n−1 −1 (1 + q( )f ( )) y(n) ≤ r=n0 =n0 =n0 ❚ø ✤â t❛ ❝â n−1 n−1 y(n) ≤ (1 + q(r)f (r))−1 p( )f ( ) (1 + q(r)f (r)) p( )f ( ) ✭✶✳✺✮ r= +1 =n0 ❉♦ ✭✶✳✷✮✱ t❛ ❝â x(n) ≤ p(n) + q(n)y(n)✳ ❚❛ ❝â ✤✐➲✉ ự q r ỵ ❧➜② p(n) = p ✈➔ q(n) = q✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ n ∈ N(n0 )✱ t❛ ❝â n−1 x(n) ≤ p (1 + qf ( )) =n0 ❍➺ q✉↔ ✶✳✷✳ ❚r♦♥❣ ✣à♥❤ ❧➼ ✶✳✶✱ p(n) ❦❤æ♥❣ ❣✐↔♠ ✈➔ q(n) ≥ ✈ỵ✐ ♠å✐ n ∈ N(n0)✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ n ∈ N(n0)✱ t❛ ❝â n−1 x(n) ≤ p(n)q(n) (1 + q( )f ( )) =n0 ✶✳✷ ●✐ỵ✐ t❤✐➺✉ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤✉➛♥ ♥❤➜t ❝â ❞↕♥❣ x(n + 1) = A(n)x(n) ✭✶✳✻✮ ✈ỵ✐ A(n) = (aij (n)) ❧➔ ♠ët ♠❛ tr➟♥ ❝➜♣ k ❦❤æ♥❣ s✉② ❜✐➳♥✳ ◆➳✉ A ❧➔ ♠❛ tr➟♥ ❤➡♥❣ t❤➻ t❛ ❝â ❤➺ x(n + 1) = Ax(n) ✭✶✳✼✮ ❚❛ ①➨t sü tỗ t t ỵ ✶✳✷✳ ✭❳❡♠ ❬✶✻❪✮ ❱ỵ✐ ♠é✐ x0 ∈ Rk ✈➔ n0 ∈ Z+ t❤➻ ❝â ❞✉② ♥❤➜t ♠ët ♥❣❤✐➺♠ x(n, n0, x0) ữỡ tr ợ x(n0, n0, x0) = x0✳ download by : skknchat@gmail.com ✹✻ ●✐↔ sû x ❧➔ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✻✮ ✈➔ ♣❤÷ì♥❣ tr trữ t ợ ữỡ tr õ ❞↕♥❣ λ2 − rλ − s = ✭✷✳✷✼✮ ❑❤✐ ✤â ✭✐✮ ◆➳✉ ❤❛✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✼✮ ♥➡♠ tr♦♥❣ ❤➻♥❤ trá♥ ✤ì♥ ✈à |λ| < t❤➻ ✤✐➸♠ ❝➙♥ ❜➡♥❣ x ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ✤à❛ ♣❤÷ì♥❣✳ ✭✐✐✮ ◆➳✉ ❝â ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✼✮ ❝â ❣✐→ trà t✉②➺t ✤è✐ ❧ỵ♥ ❤ì♥ ✶ t❤➻ x ❦❤æ♥❣ ê♥ ✤à♥❤✳ ✭✐✐✐✮ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ❤❛✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✼✮ ♥➡♠ tr♦♥❣ ❤➻♥❤ trá♥ ✤ì♥ ✈à |λ| < ❧➔ |r| < − s < 2✳ ✭✐✈✮ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ ✤➸ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✼✮ ❝â ❣✐→ trà t✉②➺t ✤è✐ ❜➨ ❤ì♥ ✶ ✈➔ ♥❣❤✐➺♠ ❝á♥ ❧↕✐ ❝â ❣✐→ trà t✉②➺t ✤è✐ ❧ỵ♥ ❤ì♥ ✶ ❧➔ r2 > −4s ✈➔ |r| > |1 − s|✳ ✭✈✮ ✣✐➲✉ ❦✐➺♥ ✤õ ✤➸ ❤❛✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✼✮ ♥➡♠ tr♦♥❣ ❤➻♥❤ trá♥ ✤ì♥ ✈à |λ < ❧➔ |r| + |s| < 1✳ ❱➼ ❞ư ✷✳✻✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✽✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ❧➔ x0, x1✳ ❑❤✐ ✤â✱ ✤✐➸♠ ❝➙♥ ❜➡♥❣ x = α + ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ✤à❛ ♣❤÷ì♥❣ ♥➳✉ α > 1✱ ❦❤æ♥❣ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ♥➳✉ < α < 1✳ ❚❤➟t ✈➟②✱ t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ trữ t ợ ữỡ tr q ❝➙♥ ❜➡♥❣ x = α + ❧➔ 1 λ2 + λ− = ✭✷✳✷✾✮ α+1 α+1 ❚❛ ❝â xn+1 = α + xn−1 , n = 1, 2, xn |r| + |s| = α+1 ◆➳✉ α > t❤➻ |r| + |s| < 1✳ ❉♦ ✤â✱ ❤❛✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✾✮ ♥➡♠ tr♦♥❣ ❤➻♥❤ trá♥ ✤ì♥ ✈à |λ| < ✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶ t❤➻ ✤✐➸♠ ❝➙♥ ❜➡♥❣ x = α + ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ✤à❛ ♣❤÷ì♥❣✳ download by : skknchat@gmail.com ✹✼ ◆➳✉ α < t❤➻ |r| + |s| > 1✳ ❉♦ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✾✮ ❝â ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠ ❝â ❣✐→ trà t✉②➺t ✤è✐ ❧ỵ♥ ❤ì♥ ✶✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶ t❤➻ ✤✐➸♠ ❝➙♥ ❜➡♥❣ x = α + ❦❤ỉ♥❣ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ❱➼ ❞ư ✷✳✼✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ xn+1 = α − βxn , n = 1, , η − xn−1 ✭✷✳✸✵✮ tr♦♥❣ ✤â ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ❧➔ x0, x1❀ α, β, η ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ s❛♦ ❝❤♦ α = (β+η) ✈➔ η > β ✳ ❑❤✐ ✤â✱ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❧➔ ❦❤ỉ♥❣ ê♥ ✤à♥❤✳ ❚❤➟t ✈➟②✱ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ①→❝ ✤à♥❤ ✤✐➸♠ ❝➙♥ ❜➡♥❣ x= α − βx η−x ❍❛② (x)2 − (β + η)x + α = ◆➯♥ (x − ❉♦ ✤â β+η ) = x= β+η Pữỡ tr trữ t ợ ữỡ tr➻♥❤ ✭✷✳✸✵✮ ①✉♥❣ q✉❛♥❤ ✤✐➸♠ ❝➙♥ ❜➡♥❣ x = β+η ❧➔ λ2 + 2β η+β λ− = η−β η−β 2β 2β η+β 2β ✣➦t r = − η−β , s = η+β η−β ✱ ❦❤✐ ✤â |r| + |s| = η−β + η−β > + η−β > ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶ t❤➻ ✤✐➸♠ ❝➙♥ ❜➡♥❣ x = β+η ❦❤æ♥❣ ê♥ ✤à♥❤✳ download by : skknchat@gmail.com ✹✽ ❈❤÷ì♥❣ ✸ ▼❐❚ ❙➮ ❱➑ ❉Ư ⑩P ❉Ư◆● ✸✳✶ ▼ët sè ✈➼ ❞ö ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ ❞➣② sè ❱➼ ❞ö ✸✳✶✳ ❈❤♦ ❞➣② sè x(n) ①→❝ ✤à♥❤ ❜ð✐ x(n + 1) = e−1 x(n), n ∈ N ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ ✈ỵ✐ ❜➜t ❦➻ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x(0) ∈ R ❞➣② x(n) ❤ë✐ tö ✈➲ ✵✳ ❨➯✉ ❝➛✉ ❝õ❛ ❜➔✐ t♦→♥ q✉② ✈➲ ❝❤ù♥❣ ♠✐♥❤ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ tr➯♥ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ❙û ❞ö♥❣ ❦➳t q✉↔ ✈➲ t➼♥❤ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ tr♦♥❣ ❝❤÷ì♥❣ ✷✱ t❛ ❝â t❤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ♥➔② ♥❤÷ s❛✉✳ ❉➵ t❤➜② ♥❣❤✐➺♠ ❝õ❛ ữỡ tr s tr x(n) = en.x(0) ợ n ≥ 1✳ ❉♦ ✤â✱ |x(n0)| = |e−n x(0)| < δ = ❦➨♦ t❤❡♦ |x(n)| = |e−n.x(0)| = |e−n+n |.|e−n x(0)| < |e−n+n | → ❦❤✐ n → ∞ ✈➔ ❞♦ ✤â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ ❧➔ ❤ót✳ ❍ì♥ ♥ú❛✱ |x(n0)| = |e−n x(0)| < δ = {1, ε} s✉② r❛ |x(n)| = |e−n.x(0)| = |e−n+n |.|e−n x(0)| < δ.|e−n+n | < ε ✱ ✈ỵ✐ ♠å✐ n ≥ n0 ✈➔ ❞♦ ✤â ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ tr➯♥ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ❱➟② ❞➣② x(n) ❤ë✐ tö ✈➲ ✵✳ 0 0 0 ❱➼ ❞ö ✸✳✷✳ ❈❤♦ ❞➣② sè x(n) ①→❝ ✤à♥❤ ❜ð✐ x(n + 1) = x2 (n), n ∈ N download by : skknchat@gmail.com ✹✾ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✱ ✈ỵ✐ ❜➜t ❦➻ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ x(0) ∈ R ❞➣② x(n) ❤ë✐ tö ✈➲ 0✳ ❨➯✉ ❝➛✉ ❝õ❛ ❜➔✐ t♦→♥ q✉② ✈➲ ❝❤ù♥❣ ♠✐♥❤ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ tr➯♥ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ❙û ❞ö♥❣ ❦➳t q✉↔ ✈➲ t➼♥❤ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ tr♦♥❣ ❝❤÷ì♥❣ ✷✱ t❛ ❝â t❤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ♥➔② ữ s ợ n0 N c R✱ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❧➔ x(n) = x(n, n0, c) = n−n0 c2 ❉♦ ✤â✱ ✈ỵ✐ n1 ≥ n0, |x(n1) − x(n1)| = |c|2 |x(n) − x(n)| = |c|2 n−n1 n1 −n0 −2 n1 −n0 |c|2 n1 −n0 ✱ tù❝ ❧➔ |c| < ❦➨♦ t❤❡♦ → ❦❤✐ n → ∞ ✈➔ ❞♦ 1, c22 > t❤➻ ✤➦t (u(n), v(n)) ∈ Sr ⊂ R2✱ tr♦♥❣ ✤â r ❧➔ sè ✤õ ♥❤ä c c ✤➸ V (u(n), v(n)) ≥ 1+r − u2 (n) + 1+r − vn2 > ❱➻ t❤➳✱ ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ ❝õ❛ ❤➺ ❧➔ ❦❤ỉ♥❣ ê♥ ✤à♥❤✳ ❉♦ ✤â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ ❝õ❛ ❤➺ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❦❤✐ c21 ≤ 1, c22 ≤ 1✳ ❱➟② ❞➣② u(n), v(n) ❤ë✐ tö ✈➲ ✵ ❦❤✐ c21 ≤ 1, c22 ≤ 1✳ 2 2 2 2 2 2 ✸✳✷ ▼ët sè ✈➼ ❞ö ✈➲ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♠ỉ ❤➻♥❤ q✉➛♥ t❤➸ ❱➼ ❞ư ✸✳✺✳ ❳➨t ♠ỉ ❤➻♥❤ ❧♦➔✐ ✤ì♥ ❝❤♦ ❜ð✐ x(n + 1) = λ x(n) , n ∈ N (1 + αx(n))β ❚r↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ ❝õ❛ ♠ỉ ❤➻♥❤ ♥➔② ❧➔ x = 1−θ αθ ✱ tr♦♥❣ ✤â θ=λ −1 β (0 < θ < 1) download by : skknchat@gmail.com ✭✸✳✶✮ ✺✶ ✣➦t y = xx ✳ ❑❤✐ ✤â ✭✸✳✶✮ trð t❤➔♥❤ y(n + 1) = y(n) (θ + (1 − θ)y(n))β ✭✸✳✷✮ ✣➦t V (y) = (ln y)2✳ ❚❛ ❝â V (y) =[ln y − β ln(θ + (1 − θ)y)]2 − [ln y]2 = −β ln(θ + (1 − θ)y)[2 ln y − β ln(θ + (1 − θ)y)] ❍➔♠ ln(θ + (1 − θ)y) ➙♠ ✈ỵ✐ y ∈ (0, 1) ữỡ ợ y (1, ) t ❤➔♠ h(y) = ln y − β ln(θ + (1 − θ)y) ❇➙② ❣✐í✱ h(1) = 0, h(y) < ❦❤✐ y y → 0+ , h(y) ∼ (1−θ) ❦❤✐ y → ∞ ✈➔ h (y) = [2θ+y(1−θ)(2−β)] y(θ+(1−θ)y) ◆➳✉ < β ≤ t❤➻ h(y) > ❦❤✐ y → ∞ ✈➔ h(y) > ✈ỵ✐ ♠å✐ y ∈ (1, ∞)✱ ❝❤➥♥❣ ❤↕♥✱ V (y) < ✈ỵ✐ ♠å✐ y > ✈➔ y = 1✳ ❉♦ ✤â✱ tø ✣à♥❤ ❧➼ ✷✳✶✵ tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ x = ❝õ❛ ✭✸✳✷✮ ✭❤♦➦❝ t÷ì♥❣ ✤÷ì♥❣ x = 1−θ αθ ❝õ❛ ✭✸✳✶✮✮ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝ ♥➳✉ β ∈ (0, 2] 2−β β ❱➼ ❞ö ✸✳✻✳ ❳➨t ♠æ ❤➻♥❤ ❝↕♥❤ tr❛♥❤ ❤❛✐ ❧♦➔✐ ❝❤♦ ❜ð✐ x(n + 1) = x(n)[θ1 + (1 − θ1 )(x(n) + d1 y(n))]−β1 , y(n + 1) = y(n)[θ2 + (1 − θ2 )(y(n) + d2 y(n))]−β2 tr♦♥❣ ✤â θ1, θ2, d1, d2, β1, β2 ❧➔ ❝→❝ ❤➡♥❣ sè ❞÷ì♥❣✳ ❚r↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤÷đ❝ ❝❤♦ ❜ð✐ x= − d1 − d1 ,y = , − d1 d2 − d1 d2 tr♦♥❣ ✤â di ∈ (0, 1) ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣✱ tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝ ♥➳✉ θ1 = θ2 = θ ✈➔ βi ∈ (0, 1], i = 1, 2✳ ❚❛ ❝â ln(1 − t) ≤ −t, ✭✸✳✸✮ ✈ỵ✐ ♠å✐ t ∈ (−∞, 1) ✱ ✈ỵ✐ ❞➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ t = 0, (1 − t)−p − ≤ pt(1 − t)−1 ✈ỵ✐ ♠å✐ t ∈ (∞, 1) ✈➔ p ∈ (0, 1] download by : skknchat@gmail.com ✭✸✳✹✮ ✺✷ x ▲➜② Vi = x −1−ln( i i xi ) x i ✳ ❚❛ ❝â , i = 1, x V1 =( ) [θ + (1 − θ)(x + d1 y)]−β1 − x + β1 ln[θ + (1 − θ)(x + d1 y)] ❙û ❞ö♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✸✮ ✈➔ ✭✸✳✹✮ ✈ỵ✐ p = β1 ✈➔ t = (1 − θ)(1 − x + d1y)✱ t❛ t❤✉ ✤÷đ❝ β1 ( xx )(1 − θ)(1 − x − d1 y) − β1 (1 − θ)(1 − x − d1 y) V1 ≤ θ + (1 − θ)(x + d1 y) β1 (1 − θ)(1 − x − d1 y) d1 = { (xy − yx) − θ(1 − x − d1 y)} θ + (1 − θ)(x + d1 y) x ❚✉② ♥❤✐➯♥✱ ❦❤✐ 1−x d = y ✱ t❛ ❝â β1 θ(1 − θ)(1 − x − d1 y)2 V1 ≤ θ + (1 − θ)(x + d1 y) β1 d1 (1 − θ)(1 − x − d1 y)(xy − yx) + , x[θ + (1 − θ)(x + d1 y)] ✈ỵ✐ β1 ∈ (0, 1]✱ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ v = v ữỡ tỹ ợ (0, 1] t ụ õ V2 ≤ ✣➦t V = c1 V1 + c − 2V2 V ≤ β2 θ(1 − θ)(1 − y − d2 x)2 θ + (1 − θ)(y + d2 x) β2 d2 (1 − θ)(1 − y − d2 x)(yx − xy) + y[θ + (1 − θ)(y + d1 x)] ✳ ❚❛ ❝â✿ β1 ( xx )(1 − θ)(1 − x − d1 y) − β1 (1 − θ)(1 − x − d1 y) θ + (1 − θ)(x + d1 y) β1 (1 − θ)(1 − x − d1 y) d1 { (xy − yx) − θ(1 − x − d1 y)}, = θ + (1 − θ)(x + d1 y) x tr♦♥❣ ✤â c1β1d1y = c2β2d2x ∈ R ❉♦ ✤â✱ V ≤ ✈ỵ✐ ♠å✐ x, y > ✈➔ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = x, y = y✳ ❱➻ t❤➳ tø ✣à♥❤ ❧➼ ✷✳✶✵✱ tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ ❞÷ì♥❣ x, y ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝✱ ♥➳✉ θ1 = θ2 ✈➔ β1✱ β2 ∈ (0, 1]✳ ❱➼ ❞ö ✸✳✼✳ download by : skknchat@gmail.com ✺✸ ❳➨t ♠æ ❤➻♥❤ ❝❤♦ ❜ð✐ x(n + 1) = x(n)er(1− x(n) ) λ , n ∈ N ✭✸✳✺✮ ❚❛ s➩ ❝❤ù♥❣ tä r➡♥❣ tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ x ❝õ❛ ❤➺ ✭✸✳✺✮ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝ ♥➳✉ r ∈ (0, 2]✳ ✣➦t y = λx ✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✺✮ trð t❤➔♥❤ y(n + 1) = y(n)er(1−y(n)) ✭✸✳✻✮ ✣➦t V (y) = (y − 1)2✱ t❤➻ V (y) = −yh(y)[1 − er(1−y) ] tr♦♥❣ ✤â h(y) = y.er(1−y) + y − 2✳ ❇➙② ❣✐í h(0) < 0, h(1) = ✈➔ h(y) > ✈ỵ✐ y ≤ 2✳ ❳➨t y ∈ (0, 2)✈ỵ✐ y = 1✳ ❘ã r➔♥❣✱ h(y) = ♥➳✉ r = ( 1−y ) ln( 2−y y )✳ ◆➳✉ y ∈ (0, 1)✱ t❤➻ ✤➦t w = 1−y > s❛♦ ❝❤♦ 1 ) − ln(1 − )} w w ∞ −p w−p p+1 w (−1) + } p p r = w{ln(1 + ∞ = w{ p=1 ∞ =2 p=0 p=1 w2p > 2p + 1 ❚÷ì♥❣ tü✱ ♥➳✉ y ∈ (1, 2) t❤➻ ✤➦t w = y−1 > s❛♦ ❝❤♦ r = w{ln(1 + 1 ) − ln(1 − )} > w w ❉♦ ✤â✱ ✈ỵ✐ r ∈ (0, 2] t❛ ❝â h(y) < ✈ỵ✐ y ∈ (0, 1) ✈➔ h(y) > ✈ỵ✐ y ∈ (1, ∞)✳ ❱➻ ✈➟② ❤➔♠ V (y) ❧➔ ❤➔♠ ▲②❛♣✉♥♦✈ ❝õ❛ ✭✸✳✻✮ tr♦♥❣ R+✳ ❚➟♣ ❤ñ♣ ✤✐➸♠ tr♦♥❣ R+ ❦❤✐ V (y) = ❝❤➾ ❝❤ù❛ ✵ ✈➔ ✶✱ ✈➔ ❦❤æ♥❣ ♠ët ♥❣❤✐➺♠ ♥➔♦ ♥➡♠ tr♦♥❣ R+ ❝â t❤➸ t✐➳♥ tỵ✐ ✵ ❦❤✐ n → ∞✳ ❉♦ ✤â✱ tø ✣à♥❤ ❧➼ ✷✳✶✵ tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ y = ❝õ❛ ✭✸✳✻✮ ✭❤♦➦❝ t÷ì♥❣ ✤÷ì♥❣ x = λ ❝õ❛ ✭✸✳✺✮✮ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ t♦➔♥ ❝ö❝ ♥➳✉ r ∈ (0, 2]✳ download by : skknchat@gmail.com ✺✹ ❱➼ ❞ö ✸✳✽✳ ❳➨t ♠ët ❧♦➔✐ ❞✉② ♥❤➜t ✈ỵ✐ ❤➺ ❤❛✐ ❧ỵ♣ t✉ê✐✳ X(n) ❧➔ sè ❧÷đ♥❣ ❝♦♥ ❝á♥ ❜➨✱ Y (n) ❧➔ sè ❧÷đ♥❣ ❝♦♥ ✤➣ tr÷ð♥❣ t❤➔♥❤ ð t❤í✐ ✤✐➸♠ t❤ù n✳ ❑❤✐ ✤â✱ t❛ ❝â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ X(n + 1) = bY (n) Y (n + 1) = cX(n) + sY (n) − DY (n) ,Y ✣➦t X = DX(n) b = DX(n) ✭✸✳✼✮ t❛ ❝â X(n + 1) = Y (n) ✭✸✳✽✮ ✈ỵ✐ a = cb > 0✳ ✣✐➸♠ ❝è ✤à♥❤ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❧➔ (X ∗, Y ∗) ✈ỵ✐ X ∗ = Y ∗ ✈➔ Y ∗ = a + s − 1✳ ▼➦t ❦❤→❝✱ ✤✐➸♠ ❝è ✤à♥❤ X ∗ ✈➔ Y ∗ ♣❤↔✐ ❞÷ì♥❣ ✤➸ ổ t õ ỵ s ✤â a + s − > 0✳ ✣➸ ❞➵ ❞➔♥❣ ①➨t t➼♥❤ ê♥ ✤à♥❤ t❛ ✤➦t x(n) = X(n) − X ∗, y(n) = Y (n) − Y ∗✱ t❛ ❝â ❤➺ Y (n + 1) = aX(n) + sY (n) − Y (n), x(n + 1) = y(n) ✭✸✳✾✮ ✣✐➸♠ ❝è ✤à♥❤ (0, 0) ❝õ❛ ✭✸✳✾✮ t÷ì♥❣ ù♥❣ ✈ỵ✐ ✤✐➸♠ ❝è ✤à♥❤ (X ∗, Y ∗) ❝õ❛ ✭✸✳✽✮✳ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ y(n + 1) = ax(n) + ry(n) − y (n) ✳  x(n + 1) = y(n) y(n + = ax(n) + ry(n) ✭✸✳✶✵✮ ❍➺ tr➯♥ ❝â ❞↕♥❣ x(n + 1) y(n + 1) =A x(n) y(n) , ✈ỵ✐ A = ✱ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ❆ ❧➔ λ2 − rλ − a = 0✳ ◆❣❤✐➺♠ t➛♠ a r t❤÷í♥❣ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ |λ| < 1✳ ❉♦ ✤â ✭✐✮ − r − a > ⇔ − (2 − 2a − s) − a > ⇔ a + s > ✭✐✐✮ + r − a > ⇔ + (2 − 2a − s) > ⇔ 3a + s < download by : skknchat@gmail.com ✺✺ ✣➸ t➻♠ ♠✐➲♥ ê♥ ✤à♥❤ ❝õ❛ ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣✱ t❛ ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ▲②❛♣✉♥♦✈ ❝❤♦ V (x, y) = a2 x2 + 2ar xy + y 1−a P❤÷ì♥❣ tr➻♥❤ Ax2+2Bxy+Cy2 = D ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ♠ët ❡❧✐♣ ♥➳✉ AC−B > ❤❛② 2 a2 − a r > ⇔ a − < r < − a (1 − a)2 ▼➦t ❦❤→❝✱ ❜➡♥❣ ❝→❝❤ ♥❤â♠ sè ❤↕♥❣ ❝❤ù❛ x, y t❛ t❤✉ ✤÷đ❝ A x2 + C y = D ✈ỵ✐ A + C = a2 + > 0, A C > ❉♦ ✤â A C > ❤❛② D > tø ✤â V (x, y) > 0✳ ❍ì♥ ♥ú❛✱ V (x, y) = y2w(x, y) ✈ỵ✐ w(x, y) = (y − r)2 − 2ax − ❉♦ ✤â V (x, y) ≤ 2ar(r − y) + a2 − 1−a ❦❤✐ w(x, y) < 0, (x, y) ∈ G✱ ✈ỵ✐ G = {(x, y) : (y − r)2 − 2ax − 2ar(r − y) + a2 − < 0} 1−a ▼✐➲♥ G ❜à ❝❤➦♥ ❜ð✐ ♣❛r❛❜♦❧ w(x, y) = 0✳ ❍ì♥ ♥ú❛✱ t❛ ✤➣ ❜✐➳t ♠å✐ ♥❣❤✐➺♠ ❜à ❝❤➦♥ tr♦♥❣ G s➩ ❤ë✐ tư tỵ✐ ❣è❝✳ ❳➨t t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ✤✐➸♠ tr♦♥❣ G ♠➔ ❤ë✐ tư tỵ✐ ❣è❝ Vmin = min{V (x0 , y0 ) : (x0 , y0 ) ∈ ∂G} Jm = {(X ∗ , Y ∗ )}, X = x0 + X ∗ , Y = x0 + Y ∗ V (x(m), y(m)) < Vmin , m = 0, 1, ◆➳✉ (x0, y0) ∈ J0 t❤➻ V (x(1), y(1)) ≤ V (x0, y0) < Vmin✱ ❞♦ ✤â (x(1), y(1)) ∈ J0 ❚÷ì♥❣ tü ♥❤÷ ✈➟② t❛ ❝â t❤➸ ❝❤➾ r❛ r➡♥❣ (x(n), y(n)) ∈ J0 ✈ỵ✐ n = 1, 2, ✳ ❉♦ ✤â (x(n), y(n)) → (0, 0) ❦❤✐ → ∞✳ ◆➳✉ (x0, y0) ∈ Jm t❤➻ V (x(m + 1), y(m + 1)) ≤ V (x(m), y(m)) < Vmin ❑❤✐ ✤â t❛ ❝ô♥❣ ❝â (x(n), y(n)) → (0, 0) ❦❤✐ n → ∞✳ ◆❤÷ ✈➟② Jm ❧➔ ♠✐➲♥ ê♥ ✤à♥❤ ❝õ❛ ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣✳ download by : skknchat@gmail.com ✺✻ ❱➼ ❞ư ✸✳✾✳ ✭▼ỉ ❤➻♥❤ ❝õ❛ ◆✐❝❤♦❧s♦♥ ✲❇❛✐❧❡②✮ ●✐↔ sû H(n) ❧➔ ♠➟t ✤ë ❝õ❛ ❧♦➔✐ ✈➟t ❝❤õ tr♦♥❣ t❤➳ ❤➺ t❤ù n, P (n) ❧➔ ♠➟t ✤ë ❦➼ s✐♥❤ t❤➳ ❤➺ t❤ù n✱ f (H(n), P (n)) ❧➔ ♣❤➛♥ ❝õ❛ ❧♦➔✐ ✈➟t ❝❤õ ❦❤æ♥❣ ❝â ❦➼ s✐♥❤✱ λ ❧➔ t➾ ❧➺ s✐♥❤ s↔♥✱ c ❧➔ sè tr✉♥❣ ❜➻♥❤ ❝õ❛ trù♥❣ ❜➡♥❣ ❝→❝❤ ✤➦t ♠ët ❦➼ s✐♥❤ tr➯♥ ✈➟t ❝❤õ✳ ❚❛ ❝â H(n + 1) = λH(n)f (H(n), P (n)) P (n + 1) = cH(n)[1 − f (H(n), P (n))] ❍➺ sè ❣➦♣ ❣ï ❣✐ú❛ ✈➟t ❝❤õ ✈➔ ✈➟t ❦➼ s✐♥❤ ❧➔ He = aH(n)P (n) ✭✸✳✶✶✮ ◆➳✉ µ ❧➔ sè ❝✉ë❝ ❣➦♣ ù t t ỵ s t s✉➜t ❝õ❛ r ❝✉ë❝ ❣➦♣ ❣ï ❧➔ −µ p(r) = ứ ữỡ tr t õ e He ;à = r! H(n) = aP (n) ợ f (H(n), P (n)) = e−aP (n) t❛ ❝â ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ H(n + 1) = λH(n)e−aP (n) , P (n + 1) = cH(n) − e−aP (n) ✭✸✳✶✸✮ ✭✸✳✶✹✮ ✣✐➸♠ ❝➙♥ ❜➡♥❣ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❧➔ H∗ = λl nλ , P ∗ = lnλ (1 − λ)ac a ❇➡♥❣ t✉②➳♥ t➼♥❤ ❤♦→ t❛ ❝â t❤➸ t❤➜② r➡♥❣ (H ∗, P ∗) ❧➔ ❦❤æ♥❣ ê♥ ✤à♥❤✳ ❉♦ ✤â✱ t❛ ①➨t ♠ỉ ❤➻♥❤ t❤ü❝ t➳ ❤ì♥ H(n + 1) = H(n) exp r(1 − H(n) ) − aP (n) , r > k P (n + 1) = cH(n)(1 − exp(−aP (n))) ❈→❝ ✤✐➸♠ ❝➙♥ ❜➡♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ = exp r(1 − H∗ ) − aP ∗ , P ∗ = cH ∗ (1 − exp(−aP ∗ )) K download by : skknchat@gmail.com ✺✼ ❉♦ ✤â P∗ = ❙✉② r❛ P∗ r H∗ r 1− = (1 − q), H ∗ = ∗ a K a (1 − eaP ) ∗ r(1 − HK ) H∗ ) = − exp −r(1 − acH ∗ K ❘ã r➔♥❣ H1∗ = K, P1∗ = ❧➔ ♠ët tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣✳ ✣➸ t❤ü❝ ❤✐➺♥ ♣❤➙♥ t➼❝❤ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ✤✐➸♠ ❝➙♥ ❜➡♥❣ (H2∗, P2∗) t❛ ✤➦t H(n) = x(n) + H2∗, P (n) = y(n) + P2∗✳ ❚ø ✤â✱ t❛ ❝â x(n)+H2∗ ) − a(y(n)) + P2∗ K P2∗ + c(x(n) + H2∗ )[1 − exp(−a(y(n) + P2∗ ))] x(n + 1) = −H2∗ + (x(n) + H2∗ ) exp r(1 − y(n + 1) = , ❇➡♥❣ t✉②➳♥ t➼♥❤ ❤â❛ q✉❛♥❤ ✤✐➸♠ (0, 0) t❛ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ x(n + 1) y(n + 1) =A x(n) y(n) , − rq −arq r(1−q) ✈ỵ✐ A = ✱ tr♦♥❣ ✤â q = HK ✈➔ χ = 1−exp(−r(1−q)) c(1 − exp(−r(1 − q))) χ − r(1 − q) P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ A ❧➔ ∗ λ2 − λ(1 − r + χ) + (1 − rq)χ + rq(1 − q) = ❙û ❞ö♥❣ t✐➯✉ ❝❤✉➞♥ ê♥ ✤à♥❤ ✤➣ ❜✐➳t |λ| < ⇔ |1−r+χ| < 1+(1−rq)χ+r2q(1−q) < ❉♦ ✤â (1 − rq)χ + r2 q(1 − q) < 1, + (1 − rq)χ + r2 q(1 − q) > |1 − r + χ| ❧➔ ♠✐➲♥ ♥❣❤✐➺♠ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ ✤✐➸♠ ❝➙♥ ❜➡♥❣✳ download by : skknchat@gmail.com ✺✽ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ s❛✉✿ ✶✳ ❍➺ tố ró ởt số t q sỹ tỗ t↕✐ ♥❣❤✐➺♠✱ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ♥❣❤✐➺♠ ❝õ❛ ♠ët sè ợ ữỡ tr s t t t✉②➳♥✳ ✷✳ ❍➺ t❤è♥❣✱ ❧➔♠ rã ♠ët sè ❦➳t q✉↔ ✈➲ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♥❣❤✐➺♠ ♠ët sè ❧ỵ♣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✳ ✸✳ ❚r➻♥❤ ❜➔② ♠ët sè ✈➼ ỵ tt ữỡ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✳ download by : skknchat@gmail.com ✺✾ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❉✳❈✳ ❍✉♦♥❣✱ P❡rs✐st❡♥❝❡ ❛♥❞ ❣❧♦❜❛❧ ❛ttr❛❝t✐✈✐t② ❢♦r ❛ ❞✐s❝r❡t✐③❡❞ ✈❡r✲ s✐♦♥ ♦❢ ❛ ❣❡♥❡r❛❧ ♠♦❞❡❧ ♦❢ ❣❧✉❝♦s❡✲✐♥s✉❧✐♥ ✐♥t❡r❛❝t✐♦♥✱ ❉❡♠♦♥str❛t✐♦ ▼❛t❤❡♠❛t✐❝❛✱ ✹✾✭✸✮✭✷✵✶✻✮✱ ✸✵✷✲✸✶✽✳ ❬✷❪ ❉✳❈✳ ❍✉♦♥❣✱ ❖♥ ❆s②♠♣t♦t✐❝ st❛❜✐❧✐t② ❛♥❞ str✐❝t ❜♦✉♥❞❡❞♥❡ss ❢♦r ♥♦♥✲ ❛✉t♦♥♦♠♦✉s ♥♦♥❧✐♥❡❛r ❞✐❢❢❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ t✐♠❡✲✈❛r②✐♥❣ ❞❡❧❛②✱ ❱✐❡t♥❛♠ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✹✹✭✹✮ ✭✷✵✶✻✮✱ ✼✽✾✲✽✵✵✳ ❬✸❪ ❉✳❱✳ ●✐❛♥❣✱ ❉✳❈✳ ❍✉♦♥❣✱ ❊①t✐♥❝t✐♦♥✱ P❡rs✐st❡♥❝❡ ❛♥❞ ●❧♦❜❛❧ st❛❜✐❧✐t② ✐♥ ♠♦❞❡❧ ♦❢ P♦♣✉❧❛t✐♦♥ ●r♦✇t❤✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✸✵✽✭✷✵✵✺✮✱ ✶✾✺✲✷✵✼✳ ❬✹❪ ❉✳❈✳ ❍✉♦♥❣✱ ◆✳❱✳ ▼❛✉✱ ❖♥ ❛ ♥♦♥❧✐♥❡❛r ❞✐❢❢❡r❡♥❝❡ ❡q✉❛t✐♦♥ ✇✐t❤ ✈❛r✐✲ ❛❜❧❡ ❞❡❧❛②✱ ❉❡♠♦♥str❛t✐♦ ♠❛t❤❡♠❛t✐❝❛✱ ❱♦❧✳ ❳▲❱■ ✱ ◆♦ ✶✱ ✷✵✶✸✳ ❬✺❪ ❉✳❈✳ ❍✉♦♥❣✱ ❖♥ t❤❡ ❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦r ♦❢ ❙♦❧✉t✐♦♥s ♦❢ ❛ ◆♦♥❧✐♥❡❛r ❉✐❢❢❡r❡♥❝❡ ❊q✉❛t✐♦♥ ✇✐t❤ ❇♦✉♥❞❡❞ ▼✉❧t✐♣❧❡ ❉❡❧❛②✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤ ✸✹✿✷ ✭✷✵✵✻✮✱ ✶✻✸✲✶✼✵✳ ❬✻❪ ❉✳❈✳ ❍✉♦♥❣✱ ❆s②♠♣t♦t✐❝ st❛❜✐❧✐t② ❛♥❞ str✐❝t ❜♦✉♥❞❡❞♥❡ss ❢♦r ♥♦♥✲❛✉t♦♥♦♠♦✉s ♥♦♥❧✐♥❡❛r ❞✐❢❢❡r❡♥❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ t✐♠❡✲✈❛r②✐♥❣ ❞❡✲ ❧❛②✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳ ✹✹ ✭ ✷✵✶✻✮✱ ✼✽✾✲✽✵✵✳ ❬✼❪ ❉✳❱✳ ●✐❛♥❣✱ ❉✳❈✳ ❍✉♦♥❣✱ ❊①t✐♥❝t✐♦♥✱ ♣❡rs✐st❡♥❝❡ ❛♥❞ ❣❧♦❜❛❧ st❛❜✐❧✐t② ✐♥ ♠♦❞❡❧s ♦❢ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✵✽✭✷✵✵✺✮✱ ✶✾✺✲ ✷✵✼✳ ❬✽❪ ❉✳❱✳ ●✐❛♥❣✱ ❉✳❈✳ ❍✉♦♥❣✱ ◆♦♥tr✐✈✐❛❧ ♣❡r✐♦❞✐❝✐t② ✐♥ ❞✐s❝r❡t❡ ❞❡❧❛② ♠♦❞✲ ❡❧s ♦❢ ♣♦♣✉❧❛t✐♦♥ ❣r♦✇t❤✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✱ ✸✵✺ ✭✷✵✵✺✮✱ ✷✾✶✲✷✾✺✳ download by : skknchat@gmail.com ✻✵ ❬✾❪ ❉✳▲✳ ❏❛❣❡r♠❛♥ ✱ ❉✐❢❢❡r❡♥❝❡ ❊q✉❛t✐♦♥s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s t♦ ◗✉❡✉❡s✱▼❛r❝❡❧ ❉❡❦❦❡r ■♥❡ ✭✷✵✵✵✮✳ ❬✶✵❪ ▼✳◆✳ ■s❧❛♠✱ ❊✳ ❨❛♥❦s♦♥✱ ❇♦✉♥❞❡❞♥❡ss ❛♥❞ st❛❜✐❧✐t② ✐♥ ♥♦♥❧✐♥❡❛r ❞❡❧❛② ❞✐❢❢❡r❡♥❝❡ ❡q✉❛t✐♦♥s ❡♠♣❧♦②✐♥❣ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r②✱ ❊❧❡❝tr♦♥✐❝ ❏♦✉r♥❛❧ ♦❢ ◗✉❛❧✐t❛t✐✈❡ ❚❤❡♦r② ♦❢ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ◆♦✳ ✷✻ ✭✷✵✵✺✮✱ ✶✲✶✽✳ ❬✶✶❪ ❘✳P✳ ❆❣❛r✇❛❧✱ ❉✐❢❢❡r❡♥❝❡ ❊q✉❛t✐♦♥s ❛♥❞ ■♥❡q✉❛❧✐t✐❡s✳ ❚❤❡♦r②✱ ▼❡t❤♦❞s✱ ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ▼❛r❝❡❧ ❉❡❦❦❡r ■♥❡ ✭✷✵✵✵✮✳ ❬✶✷❪ ❙✳◆✳ ❊❧❛②❞✐✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❞✐❢❢❡r❡♥❝❡ ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r ❱❡r❜❣✱ t❤✐r❞ ❡❞✐t✐♦♥ ✭✷✵✵✺✮✳ ❬✶✸❪ ❨✳◆✳ ❘❛❢❢♦✉❧✱ ❈✳ ❚✐s❞❡❧❧✱ P♦s✐t✐✈❡ ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❞✐s✲ ❝r❡t❡ s②st❡♠s ❛♥❞ ♣♦♣✉❧❛t✐♦♥ ♠♦❞❡❧s✱ ✷✵✵✺ ❍✐♥❞❛✇✐✱ P✉❜❧✐s❤✐♥❣ ❈♦r✲ ♣♦r❛t✐♦♥ ✭✷✵✵✺✮✱ ✸✻✾✲✸✽✵✳ ❬✶✹❪ ❨✳◆✳ ❘❛❢❢♦✉❧✱ ❙t❛❜✐❧✐t② ❛♥❞ ♣❡r✐♦❞✐❝✐t② ✐♥ ❞✐s❝r❡t❡ ❞❡❧❛② ❡q✉❛t✐♦♥s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✷✹ ✭✷✵✵✻✮✱ ✶✸✺✻✲✶✸✻✷✳ ❬✶✺❪ ❨✳◆✳ ❘❛❢❢♦✉❧✱ P❡r✐♦❞✐❝✐t② ✐♥ ❣❡♥❡r❛❧ ❞❡❧❛② ♥♦♥✲❧✐♥❡❛r ❞✐❢❢❡r❡♥❝❡ ❡q✉❛✲ t✐♦♥s ✉s✐♥❣ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r②✱ ❏♦✉r♥❛❧ ♦❢ ❉✐❢❢❡r❡♥❝❡ ❊q✉❛t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✈♦❧✳ ✶✵✱ ◆♦✳ ✶✸✲✶✺ ữợ ✤à♥❤ ❧➼ ✈➔ →♣ ❞ö♥❣✱ ◆❳❇ ✣↕✐ ❍å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐ ✭✷✵✶✹✮✳ download by : skknchat@gmail.com ... x(n + 1) = F (x(n), x(n − 1), , x(n − k)), n = 0, 1, tr♦♥❣ ✤â x(−k), x(−k + 1), x(0) số trữợ số tỹ k số ữỡ trữợ F ∈ C[I k+1 , I] ✭✷✳✷✸✮ ✱ ✈ỵ✐ I ❧➔ ✣à♥❤ ♥❣❤➽❛ ✷✳✽✳ ✭❳❡♠ ❬✶✻❪✮ ▼ët ❞➣② sè... ♥➳✉ tỗ t số C s ❝❤♦ Φ(n, n0 ) ≤ C, n ∈ N(n0 ) tỗ t↕✐ ❤➡♥❣ sè ❞÷ì♥❣ C s❛♦ ❝❤♦ Φ(n, m) = Φ(n, n0 )Φ(m, n0 ) ≤ C, n0 ≤ m ≤ n ∈ N(n0 ) ✭✷✳✷✮ ✭✐✐✐✮ ✃♥ ✤à♥❤ ♠↕♥❤ tỗ t số ữỡ C s❛♦... ) ✈➔ x(n1 ) − x(n1 ) < δ s✉② r❛ x(n) − x(n) < ε ✈ỵ✐ ♠å✐ n ∈ N(n0 ) ✭①✮ ✃♥ t ụ tỗ t số > t > tỗ t số δ = δ(ε) > s❛♦ ❝❤♦ ✈ỵ✐ ❜➜t ❦➻ ♥❣❤✐➺♠ x(n) = x(n, n0 , x0 ) ❝õ❛ ✭✶✳✹✹✮ t❤♦↔ n1 ∈ N(n0