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✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ❍❖⑨◆● ◆●➴❈ ❚❍⑩■ ❇⑨■ ❚❖⑩◆ ✃◆ ✣➚◆❍ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❯❨➌◆ ❚➑◆❍ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ❍❖⑨◆● ◆●➴❈ ❚❍⑩■ ❇⑨■ ❚❖⑩◆ ✃◆ ✣➚◆❍ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❯❨➌◆ ❚➑◆❍ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ữớ ữợ ễ P❍⑩❚ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺ ✐ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷đ❝ ❝❤➾ rã ỗ ố t ữớ t ▲✉➟♥ ✈➠♥ ❍♦➔♥❣ ◆❣å❝ ❚❤→✐ ✐✐ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ữủ t ữợ sỹ ữợ t t➻♥❤ ✈➔ ❝❤➾ ❜↔♦ ♥❣❤✐➯♠ ❦❤➢❝ ❝õ❛ t❤➛② ❣✐→♦ ●❙✳❚❙❑❍ ❱ơ ◆❣å❝ P❤→t✳ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ✤➳♥ t❤➛②✳ ❚æ✐ ①✐♥ ❝❤➙♥ t ỡ t tợ trữớ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❝→❝ t❤➛② ð ❱✐➺♥ ❚♦→♥ ❤å❝ ✲ ❱✐➺♥ ❍➔♥ ❧➙♠ ❑❍❈◆ ❱✐➺t ◆❛♠ ✤➣ t t tr t ỳ tự qỵ ụ ♥❤÷ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ♥❤➜t ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t tỵ✐ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❤é trđ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ❚❤→✐ ♥❣✉②➯♥✱ t❤→♥❣ ✼ ♥➠♠ ✷✵✶✺ ◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥ ❍♦➔♥❣ ◆❣å❝ ❚❤→✐ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✐ ▲í✐ ❝↔♠ ì♥ ✐✐ ▼ư❝ ❧ư❝ ✐✐ ▼ð ✤➛✉ ✶ ❑➼ ❤✐➺✉ t♦→♥ ❤å❝ ✸ ✶ ❈ì sð t♦→♥ ❤å❝ ✹ ✶✳✶ ✣↕✐ sè t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ ❇➔✐ t♦→♥ ê♥ ✤à♥❤ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ P❤÷ì♥❣ ♣❤→♣ ❤➔♠ ▲②❛♣✉♥♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✺ ▼ët sè ❜ê ✤➲ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷ ✃♥ ✤à♥❤ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✶✾ ✷✳✶ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ỉtỉ♥ỉ♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ ætæ♥æ♠ ✳ ✳ ✳ ✳ ✷✼ ❑➳t ❧✉➟♥ ✸✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✽ ✶ ▼ð ✤➛✉ ❇➔✐ t♦→♥ ê♥ ✤à♥❤ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ q✉❛♥ trå♥❣ tr♦♥❣ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ①✉➜t ♣❤→t tø t❤ü❝ t➳✱ ✤á✐ ❤ä✐ ♣❤↔✐ sû ❞ö♥❣ ♥❤✐➲✉ ỵ tt ổ t ỵ t❤✉②➳t ê♥ ✤à♥❤ ❝→❝ ❤➺ ✤ë♥❣ ❧ü❝ ❜➢t ✤➛✉ ✤÷đ❝ ữợ tứ ố t ữớ ỳ þ t÷ð♥❣ ✈➔ ❦➳t q✉↔ q✉❛♥ trå♥❣ ❝õ❛ ♥❤➔ t♦→♥ ❤å❝ ♥❣÷í✐ ◆❣❛ ❆✳▼✳ ▲②❛♣✉♥♦✈✳ ▼ët ❝→❝❤ ❤➻♥❤ t÷đ♥❣✱ ♠ët ❤➺ t❤è♥❣ ✤÷đ❝ ❣å✐ ❧➔ ê♥ ✤à♥❤ t↕✐ tr↕♥❣ t❤→✐ ❝➙♥ ❜➡♥❣ ♥➔♦ ✤â ♥➳✉ ❝→❝ ♥❤✐➵✉ ♥❤ä ❝õ❛ ❝→❝ ❞ú ❦✐➺♥ ❤♦➦❝ ❝→❝ ❝➜✉ tró❝ ❜❛♥ ✤➛✉ ❝õ❛ ❤➺ t❤è♥❣ ❦❤æ♥❣ ❧➔♠ ❝❤♦ ❤➺ t❤è♥❣ t❤❛② ✤ê✐ ♥❤✐➲✉ s♦ ợ tr t õ õ ỵ t❤✉②➳t ê♥ ✤à♥❤ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ①✉➜t ♣❤→t tø t❤ü❝ t✐➵♥ ✈➔ ♥❤✉ ❝➛✉ ♣❤→t tr✐➸♥ ❝õ❛ ♠ët sè ♥❣➔♥❤ ❦❤♦❛ ❤å❝✳ ❚ø ♥❤ú♥❣ ♥➠♠ ✻✵ ❝õ❛ t❤➳ ❦✛ ✷✵✱ ♥❣÷í✐ t❛ ❜➢t ✤➛✉ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ê♥ ✤à♥❤ ❝→❝ ❤➺ ✤✐➲✉ ❦❤✐➸♥ ♥❤÷ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ ✤÷đ❝✱ ❜➔✐ t♦→♥ ê♥ ✤à♥❤ ❤♦→✱ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉✱✳✳✳❚ø ✤â ✤➳♥ ♥❛② t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❝→❝ ❤➺ ✤✐➲✉ ❦❤✐➸♥ t♦→♥ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ sỉ✐ ♥ê✐✱ t❤✉ ✤÷đ❝ ♥❤✐➲✉ t❤➔♥❤ tü✉ rü❝ rï✱ s➙✉ s➢❝ ✈➔ ù♥❣ ❞ö♥❣ rë♥❣ r➣✐ tr♦♥❣ ỹ ữ t ỵ t ❦➽ t❤✉➟t✱ s✐♥❤ t❤→✐ ❤å❝✱ ♠ỉ✐ tr÷í♥❣✳✳✳ ❈â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❈â t❤➸ ❦➸ r❛ ✤➙② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝❤➼♥❤ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ t❤ù ♥❤➜t ▲②❛♣✉♥♦✈ ✭❤❛② ❝á♥ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ♠ơ ✤➦❝ tr÷♥❣✮✱ ♣❤÷ì♥❣ ♣❤→♣ ✷ t❤ù ❤❛✐ ▲②❛♣✉♥♦✈ ✭❤❛② ❝á♥ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ▲②❛♣✉♥♦✈✮✱ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾✱ ♣❤÷ì♥❣ ♣❤→♣ s♦ s→♥❤✳✳✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❜↔♥ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❦❤→✐ ♥✐➺♠ ✈➲ t➼♥❤ ữỡ tr ỗ tớ ợ t ữỡ t t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❈❤÷ì♥❣ ợ t t ữỡ tr ữủ t ữợ sỹ ữợ t t ụ ◆❣å❝ P❤→t✱ ❱✐➺♥ ❚♦→♥ ❤å❝ ✲ ❱✐➺♥ ❍➔♥ ❧➙♠ ❑❤♦❛ ❤å❝ ❈ỉ♥❣ ♥❣❤➺ ❱✐➺t ◆❛♠✳ ❊♠ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❤➛②✳ ❚→❝ ❣✐↔ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❑❤♦❛ ❙❛✉ ✣↕✐ ❤å❝✱ ❑❤♦❛ ❚♦→♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❚✉② ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ s♦♥❣ t❤í✐ ❣✐❛♥ ✈➔ ♥➠♥❣ ❧ü❝ ❜↔♥ t❤➙♥ ❝â ❤↕♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚ỉ✐ r➜t ♠♦♥❣ ❝â ✤÷đ❝ ♥❤ú♥❣ þ ❦✐➳♥ ✤â♥❣ ❣â♣ ❝õ❛ ❝→❝ t❤➛② ❝ỉ ❝ị♥❣ t♦➔♥ t❤➸ ❜↕♥ ✤å❝✳ ✸ ❑➼ ❤✐➺✉ t♦→♥ ❤å❝ Z ❚➟♣ sè ♥❣✉②➯♥✳ Z+ ❚➟♣ sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✳ R ❚➟♣ sè t❤ü❝✳ R+ ❚➟♣ sè t❤ü❝ ❦❤æ♥❣ ➙♠✳ Rn ❑❤æ♥❣ ❣✐❛♥ ✈❡❝tì ❊✉❝❧✐❞❡ n ❝❤✐➲✉✳ Rn×n ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ♠❛ tr➟♥ t❤ü❝ ❝➜♣ n × n✳ I ▼❛ tr➟♥ ✤ì♥ ✈à✳ AT ▼❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ A✳ P > ▼❛ tr➟♥ ①→❝ ✤à♥❤ ❞÷ì♥❣✳ λ(P ) ❈→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ P ✳ C[a,b] ❚➟♣ ❝→❝ ❤➔♠ sè ❧✐➯♥ tö❝ tr♦♥❣ [a, b]✳ ✹ ❈❤÷ì♥❣ ✶ ❈ì sð t♦→♥ ❤å❝ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tỉ✐ tr ỵ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❤➔♠ ▲②❛♣✉♥♦✈✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② t❤❡♦ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✺❪✳ ✶✳✶ ✣↕✐ sè t✉②➳♥ t➼♥❤ • ❱❡❝tì v ∈ Rn , v = ❣å✐ ❧➔ ✈❡❝tì r✐➯♥❣ ❝õ❛ ♠❛ tr➟♥ A ∈ Rn×n ♥➳✉ ❝â ♠ët sè λ ✭❝â t❤➸ ❧➔ sè t❤ü❝ ❤♦➦❝ sè ♣❤ù❝✮ s❛♦ ❝❤♦ Av = λv ❙è λ ❣å✐ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ A ù♥❣ ✈ỵ✐ ✈❡❝tì r✐➯♥❣ v, t➟♣ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ A s➩ ❦➼ ❤✐➺✉ ❧➔ λ(A) ❈→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ A ①→❝ ✤à♥❤ ❜ð✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤❛ t❤ù❝ ✤➦❝ tr÷♥❣ ❝õ❛ A : det(λI − A) = ❤❛② p(λ) = λn + a1 λn−1 + a2 λn−2 + + an1 + an = ỵ ✶✳✶✳✶✳ ✭❈❛②❧❡② ✲ ❍❛♠✐❧t♦♥✮✳ ▼å✐ ♠❛ tr➟♥ A ∈ Rn×n ✤➲✉ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ ✤➦❝ tr÷♥❣ ❝õ❛ ♥â✿ p(A) = An + a1 An−1 + a2 An−2 + + an−1 A + an I = • ❈❤♦ ♠❛ tr➟♥ A ∈ Rn×n , A = (aij ), i, j = 1, 2, , n ❈❤✉➞♥ ❝õ❛ ♠❛ tr➟♥ ✺ A s➩ ①→❝ ✤à♥❤ ❜ð✐ n n 1/2 |aij |2 ||A|| = ✭✶✳✶✮ i=1 j=1 ã tự tũ ỵ n n ck λk , f (λ) = k=0 ♥➳✉ n = ∞ t❤➻ ❝❤✉é✐ ❣✐↔ t❤✐➳t ❧➔ ❤ë✐ tö✳ ❍➔♠ ❝õ❛ ♠❛ tr➟♥ A ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ n ck Ak f (A) = k=0 ỵ ổ tự str A Rnìn ợ tr r✐➯♥❣ λ1 , λ2 , , λn ❦❤→❝ ♥❤❛✉✳ ❈❤♦ f (λ) ❧➔ ❤➔♠ ✤❛ t❤ù❝ ❜➟❝ n ♥➔♦ ✤â ❞↕♥❣ ✭✶✳✶✮✳ ❑❤✐ ✤â n f (A) = Zk f (λk ), k=1 tr♦♥❣ ✤â Zk ①→❝ ✤à♥❤ ❜ð✐ (A − λ1 I)(A − λ2 I) (A − λk−1 I)(A − λk+1 ) (A − λn I) (λk − λ1 )(λk − λ2 ) (λk − λk−1 )(λk − λk+1 ) (λk − λn ) n A − λj I = j=1,j=k λk − λj Zk = • ▼❛ tr➟♥ A ❣å✐ ❧➔ ①→❝ ✤à♥❤ ❞÷ì♥❣ ♥➳✉ ✐✮ Ax, x ≥ 0, ∀x ∈ Rn ✐✐✮ Ax, x > 0, x = tr õ x, y ỵ t ổ ữợ tỡ x = (x1 , x2 , , xn ), y = (y1 , y2 , , yn ) ①→❝ ✤à♥❤ ❜ð✐ n x, y = xi yi i=1 • ◆➳✉ A = AT , t❤➻ A ❣å✐ ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣✳ ❚❛ ❧✉æ♥ ❝â AAT ❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ✈➔ (AB)T = B T AT ◆➳✉ A ❧➔ ❦❤æ♥❣ s✉② ❜✐➳♥✱ tù❝ ❧➔ det A = 0, ✷✹ ❚❤❡♦ ỵ t r t❤ü❝ t➳ t❛ ❝ô♥❣ ❣➦♣ ❜❛➻ t♦→♥ ❝â ❞↕♥❣ ✭✷✳✶✮ ♥❤÷♥❣ ð ✤➙② ♠❛ tr➟♥ A ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♠❛ tr➟♥ ❤➡♥❣ sè ♠➔ ♥â ❝â ❞↕♥❣ A(t) = A + C(t) t❛ ❣å✐ ❤➺ ❝â ❞↕♥❣ ♥❤÷ ✈➟② ❧➔ ữ số ữợ ú tổ s ❣✐ỵ✐ t❤✐➺✉ ♠ët t✐➯✉ ❝❤✉➞♥ ê♥ ✤à♥❤ ❝õ❛ ❤➺ ❤➛✉ ữ số ỵ t tr ✤â A(t) = A + C(t) ●✐↔ sû ❆ ❧➔ ♠❛ tr➟♥ ê♥ ✤à♥❤ ✈➔ ❣✐↔ sû C(t) ❧➔ ❤➔♠ ❦❤↔ t➼❝❤ tr➯♥ R+ ✈➔ ||C(t)|| ≤ a, a > ❑❤✐ ✤â ❤➺ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ✈ỵ✐ a > ✤õ ♥❤ä✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â t❤➸ t ữỡ tr ữợ x(t) = Ax(t) + C(t)x(t), t ≥ 0, ❞♦ ✤â ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✈ỵ✐ x(t0 ) = x0 ❝❤♦ ❜ð✐ t A(t−t0 ) x(t) = e eA(t−s) C(s)x(s)ds x0 + t0 ❱➻ A ❧➔ ♠❛ tr➟♥ ê♥ ✤à♥❤ s✉② r❛ ❤➺ x˙ = Ax ❧➔ ê♥ ✤à♥❤ ♠ô✱ ❞♦ ✤â t❤❡♦ ✤à♥❤ ♥❣❤➽❛ s tỗ t số > 0, > s❛♦ ❝❤♦ ||eAt || ≤ µe−δt , ∀t ≥ ❚❛ ❝â ✤→♥❤ ❣✐→ s❛✉ ||x(t)|| ≤ µe−δ(t−t0 ) ||x0 || + inttt0 µe−δ(s−t0 ) a||x(s)||ds ✣➦t u(t) = eδ(t−t0 ) ||x(t)||, C = µ||x0 ||, a(t) = µa ✈➔ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ●r♦♥✇❛❧❧✱ ❇ê ✤➲ ✭✶✳✺✳✶✮✱ t❛ ❝â eδ(t−t0 ) ||x(t)|| ≤ µ||x0 ||eµa(t−t0 ) , ∀t ≥ t0 ✷✺ ❉♦ ✤â ||x(t)|| ≤ µ||x0 ||e(µa−δ)(t−t0 ) , ❈❤➾ ❝➛♥ ❝❤å♥ a < δ µ ∀t ≥ t0 ❦❤✐ ✤â ❤➺ s➩ ê♥ ✤à♥❤ t✐➺♠ ỵ ữủ ự t t➼♥❤ê♥ ✤à♥❤ ❝õ❛ ❤➺ x˙1 = x1 + x2 + cos2 t, x˙2 = − 14 x2 + 31 sin2 t ▲í✐ ❣✐↔✐✳ ❚❛ ❝â A= ✈➔ C(t) = cos t sin t − 15 − 14 , ❱➔ ❆ ❧➔ ♠❛ tr➟♥ ê♥ ✤à♥❤ ✈➻ 1 λ(A) = − , − , µ = 1, δ = ♠➦t ❦❤→❝ ||C(t)|| ≤ 1 =a< ♥➯♥ ❤➺ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ❚ê♥❣ q✉→t ❤ì♥ t❛ ①➨t ❜➔✐ t♦→♥ ê♥ ✤à♥❤ ❝❤♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ỉtỉ♥ỉ♠ ❝â ♥❤✐➵✉ ♣❤✐ t✉②➳♥ x(t) ˙ = Ax(t) + g(t, x) tr♦♥❣ ✤â A = ∂f (0) ∂x , ✭✷✳✸✮ g(t, x) = o(||x||) ỵ sỷ tr ê♥ ✤à♥❤ ✈➔ g(t, x) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ ||g(t, x)|| ≤ ε||x||, ∀x ∈ Rn t❤➻ ❤➺ ✭✷✳✸✮ ❧➔ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ✈ỵ✐ ε > ♥➔♦ ✤â✳ ✷✻ ❈❤ù♥❣ ♠✐♥❤✳ ◆❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ❤➺ ✭✷✳✸✮ ❝❤♦ ❜ð✐ t A(t−t0 ) x(t) = e eA(t−s) g(s, x(s))ds x0 + t0 ❱➻ ❆ ❧➔ ♠❛ tr➟♥ ê♥ ✤à♥❤ ♥➯♥ ❤➺ x˙ = Ax ❧➔ ê♥ ✤à♥❤ ụ õ tỗ t số > 0, > s❛♦ ❝❤♦ ||eAt || ≤ µe−δt , ∀t ≥ ❚❛ ❝â ✤→♥❤ ❣✐→ ♥❣❤✐➺♠ s❛✉ ✤➙② ||x(t)|| = µe −δ(t−t0 ) t ||x0 || + µe−δ(t−s) ||g(s, x(s))||ds t0 ❚❛ ❝â ||x(t)|| = µe −δ(t−t0 ) t ||x0 || + µe−δ(t−s) ε||x(s)||ds t0 ❙û ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ●r♦♥✇❛❧❧✱ ❇ê ✤➲ ✭✶✳✺✳✶✮ t❛ ♥❤➟♥ ✤÷đ❝ ✤→♥❤ ❣✐→ ||x(t)|| à||x0 ||e(tt0 ) e ợ < δ µ t t0 µεds = µ||x0 ||e(µε−δ)(t−t0 ) , ∀t ≥ t0 t❤➻ ||x(t)|| → ❦❤✐ t → +∞ ❤❛② ❤➺ ✤➣ ❝❤♦ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ỵ ữủ ự t t ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ s❛✉ x˙1 = − x1 + x21 cos2 t x˙2 = − + 14 x22 cos2 t 42 ▲í✐ ❣✐↔✐✳ ❚❛ ❝â − A= − , g(t, x) = 2 x1 cos t 2 x2 cos t ✷✼ ❱➻ ❆ ❧➔ ♠❛ tr➟♥ ê♥ ✤à♥❤ ✈➔ 1 ||g(t, x)|| = sin2 t( x41 + x42 ) ≤ ||x||2 , 4 ❞♦ ✤â ❤➺ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ✷✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❦❤æ♥❣ ætæ♥æ♠ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❈❤♦ ❤➔♠ g : R+ → R+, f (x) : Rm → Rn ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ g− ▲✐♣s❝❤✐t③ ♥➳✉ ||f (x1 ) − f (x2 )|| ≤ g(||x1 − x2 ||), ∀x1 , x2 ∈ Rm ❈❤ó þ r➡♥❣ ♠ët ❤➔♠ f (.) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ ♥➳✉ ❧➜② g(t) = L|t|❀ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ t➠♥❣ tr÷ð♥❣ ❍♦❧❞❡r ♥➳✉ ❧➜②g(t) = L|t|α , < α < ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✷✳ ❈❤♦ α, β ❧➔ ❤❛✐ sè ❞÷ì♥❣✳ ❚❛ ♥â✐ r➡♥❣ ♠ët ❤➔♠ g(t) ∈ R+ ❧➔ ❜à ❝❤➦♥ tr♦♥❣ (α, β) ♥➳✉ g(t) < , t ữ ỵ r g(t) = Lt t❤➻ ❦❤✐ ✤â g(t) ❜à ❝❤➦♥ tr♦♥❣ ❦❤♦↔♥❣ (α, β) β ♥➳✉ L < ❑❤✐ g(t) = Ltm , m ∈ (0, 1) t❤➻ g(t) ❜à ❝❤➦♥ tr♦♥❣ (α, β) ♥➳✉ α β L < m α ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ỉtỉ♥ỉ♠✿ x(t) ˙ = A(t)x(t), t ỵ sỷ r ỗ t K > 0, > s❛♦ ❝❤♦✿ ||eA(s)t || ≤ ke−δt , ∀t, s ≥ ✭✷✳✹✮ ✷✽ ✐✐✮ A(t) ❧➔ ♠ët ❤➔♠ g− ▲✐♣s❝❤✐t③✱ tr♦♥❣ ✤â g(t) ❧➔ ❤➔♠ ❜à ❝❤➦♥ tr♦♥❣ ln K δ , δ 2K ❑❤✐ ✤â ❤➺ ✭✷✳✹✮ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✈✐➳t ❧↕✐ ❤➺ ữợ x(t) = A(0)x(t) + [A(t) A(0)]x(t) ✭✷✳✺✮ ❑❤✐ ✤â ♥❣❤✐➺♠ x(t) ❝õ❛ ❤➺ ✭✷✳✺✮ tø x0 tỵ✐ ❧➔ t (t) = eA(0)t x0 + eA(0)(t−s) [A(s) − A(0)]x(s)ds ✭✷✳✻✮ ❚ø i) ✈➔ ii) t❛ ❝â✿ −δt ||x(t)|| ≤ Ke t ||x0 || + K e−δ(t−s) g(s)||x(s)||ds ✣➦t y(t) = eδt ||x(t)|| t❛ ❝â t ||y(t)|| ≤ K||x0 || + g(s)||y(s)||ds ⑩♣ ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ●r♦♥✇❛❧❧✱ ❇ê ✤➲ ✭✶✳✺✳✶✮✱ t❛ ✤÷đ❝ ||y(t)|| ≤ K||x0 ||eKh(t) , tr♦♥❣ ✤â h(t) := t g(s)ds ❇ð✐ ✈➟② ||x(t)|| ≤ K||x0 ||e−δt eKh(t) ỵ r t0 = tù❝ ❧➔ x(t0 ) = x0 t❤➻ ✭✷✳✼✮ trð t❤➔♥❤ ||x(t)|| ≤ K||x0 ||eδ(t−t0 ) eKh(t−t0 ) , ❱➻ t t−t0 g(s − t0 )ds = g(s)ds = h(t − t0 ) ✭✷✳✽✮ ✷✾ ❚❛ ❝â t❤➸ ✤➦t F (t) = δt − Kh(t)− ∈ K ✣➛✉ t✐➯♥ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝â ♠ët T > s❛♦ ❝❤♦ F (T ) > ❚❤➟t ✈➟②✱ ✈➻ g(t) ❧➔ ❤➔♠ ❜à ❝❤➦♥ tr♦♥❣ ln K δ , ♥➯♥ δ 2K δt ln K h(t) < , t 2K tữỡ ữỡ ợ Kh(t) < ❑❤✐ ✤â ∀t ≤ δ ln K = ln K δ 2lnK t❛ ❝â δ F (t) = δt − Kh(t) − ln K > δt − ln K ▲➜② T = 2lnK t❤➻ F (T ) > ❚ø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✼✮ t❛ s✉② r❛ δ ||x(T )|| ≤ ||x0 ||eln K−δT +Kh(T ) ln K −δ− −Kh(T )T T ||x(T )|| ≤ ||x ||e ❉♦ ✤â ||x(T )|| ≤ ||x0 ||e−δ0 T , ✭✷✳✾✮ F (T ) > ❚➼♥❤ ✤➳♥ ✭✷✳✾✮✱ ✈ỵ✐ ♠é✐ k ∈ Z+ ✱ tr♦♥❣ ❜➜t T ✤➥♥❣ t❤ù❝ ✭✷✳✽✮ t❛ ✤➦t t = kT, t0 = (k − 1)T ✱ ❜➡♥❣ q✉② ♥↕♣ ❞➵ t❤➜② r➡♥❣ tr♦♥❣ ✤â δ0 = ✈ỵ✐ ♠é✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ k ∈ Z+ , ||x(T )|| ≤ ||x0 ||eδ0 kT ✭✷✳✶✵✮ ❇➙② ❣✐í t❛ s➩ ❤♦➔♥ t❤➔♥❤ ❝❤ù♥❣ ữ s ợ t t > tỗ t↕✐ k ∈ Z+ ✈➔ τ ∈ [0, T ) s❛♦ ❝❤♦ t = kT + τ ▲➜② t0 = kT, ỵ r t t0 = t − kT = τ ✱ ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✽✮ ✈➔ ✭✷✳✶✵✮ t❛ ✤÷đ❝ ✸✵ ||x(t)|| ≤ ||x(kT )||eln K−δτ +Kh(τ ) , ||x(t)|| ≤ K||x0 ||e−δτ eKh(τ ) e−δ0 kT ❚❤❛② kT = t − τ t❛ ✤÷đ❝ ||x(t)|| ≤ K||x0 ||e−δ0 t eKh(τ ) eτ (δ0 −δ) ❑❤✐ τ < T ✈➔ ln K K − h(T ) < 0, T T ❚❛ ❝â t❤➸ t➻♠ ✤÷đ❝ K1 > s❛♦ ❝❤♦ δ0 − δ = − ||x(t)|| ≤ K1 ||x0 ||e−δ0 t , ∀t ≥ ỵ t ữủ ự t ứ ự ữ ỵ r g(t) ❜à ❝❤➦♥ δ ln K ♥➔♦ ✤â t❤➻ ♥â t❤ä❛ ♠➣♥ ✈ỵ✐ ❜➜t ❦➻ T ≥ ❍ì♥ ♥ú❛✱ t❤❛② K δ − Kc ln K δ , t❛ ❝â t❤➸ t❤❛② t❤➳ ❜ð✐ ✤✐➲✉ ✈➻ ✤✐➲✉ ❦✐➺♥ g(t) ❜à ❝❤➦♥ tr♦♥❣ δ 2K ❦✐➺♥ ln K ln K h < δ K ❜ð✐ c < ❚❛ s➩ ①➨t sü ê♥ ✤à♥❤ t✐➺♠ ❝➟♥ ❝õ❛ ❤➺ ♣❤✐ t✉②➳♥ x(t) ˙ = A(t)x(t) + f (t, x(t)), t ≥ 0, ✭✷✳✶✶✮ tr♦♥❣ ✤â A(t) ∈ Rn×n , f : R+ × Rn → Rn ❧➔ ❤➔♠ ❧✐➯♥ tư❝✱ ❜➡♥❣ ❝→❝❤ t✐➳♣ ❝➟♥ tø ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥ x(k + 1) = A(k)x(k) + g(k, x(k)), k ∈ Z+ ✭✷✳✶✷✮ ❳➨t ❤➺ t❤❡♦ t❤í✐ ❣✐❛♥ rí✐ r↕❝ ✭✷✳✶✷✮ tr♦♥❣ ✤â A(k) ∈ Rn×n , g(k, x) : Z+ × Rn → Rn ❧➔ ♠ët ❤➔♠ ♣❤✐ t✉②➳♥ ✈➔ g(k, 0) = 0, k ∈ Z+ ❱ỵ✐ ♠é✐ ✸✶ x0 ∈ Rn , ♥❣❤✐➺♠ x(k) ❝õ❛ ✭✷✳✶✷✮ ❜➢t ✤➛✉ tø x(0) = x0 ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ k−1 x(k) = G(k, 0)x0 + ✭✷✳✶✸✮ G(k, i + 1)g(i, x(i)), i=0 tr♦♥❣ ✤â k−1 G(k, 0) = A(i), k = 1, 2, i=0 k−1 G(k, k) = E − t♦→♥ tỷ ỗ t, G(k, i) = A(j), k > i j=i ỵ sỷ r ỗ t K > 0, δ > s❛♦ ❝❤♦✿ ||G(k, i)|| ≤ Ke−δ(k−i) , k = 1, 2, , i = 0, 1, 2, , k > i ✐✐✮ ❱ỵ✐ m > 0, a(k) : Z+ → R+ s❛♦ ❝❤♦✿ ||g(k, x)|| ≤ a(k)||x||m , k = 0, 1, 2, ✐✐✐✮ ◆➳✉ < m < : − e−δ a(k) < K ✭✷✳✶✹✮ K −e−δ ✭✷✳✶✺✮ a(k)e−δk(m−1) < +∞ ✭✷✳✶✻✮ δk(1−m) lim sup e k→∞ ◆➳✉ m = : lim sup a(k) < k→∞ ◆➳✉ m > : ∞ k=0 ❑❤✐ ✤â ❤➺ ✭✷✳✶✷✮ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ❜➜t ❦➻ ♥❣❤✐➺♠ x(k) ❜➢t ✤➛✉ tø x0 ✤➲✉ ❝â ❞↕♥❣ ✭✷✳✶✸✮✳ ✸✷ ❉♦ ✤â tø ✐✮ ✈➔ ✐✐✮ t❛ ❝â k−1 Ke−δ(k−i−1) a(i)||x(i)||m δk ||x(k)|| ≤ Ke ||x0 || + i=0 ◆❤➙♥ ❝↔ ❤❛✐ ✈➳ ✈ỵ✐ e−δk ✈➔ ✤➦t z(k) = e−δk ||x(k)||, b(k) = Ke−δ[1+(1−m)k] a(k), t❛ ✤÷đ❝ k−1 b(i)z(i)m x(k) ≤ K||x0 || + ✭✷✳✶✼✮ i=0 ❛✮ ❚r÷í♥❣ ❤đ♣ < m < : ⑩♣ ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ●r♦♥✇❛❧❧✱ ❇ê ✤➲ ✭✶✳✺✳✷✮✱ ❝❤♦ tr÷í♥❣ ❤đ♣ < m < t❛ ✤÷đ❝ k−1 z(k) ≤ (C + 1) [1 + b(i)], k = 1, 2, i=0 tr♦♥❣ ✤â C = K||x0 || ❉♦ ✤â k−1 −δk {1 + Keδ[1+(1−m)i] a(i)}, ||x(k)|| ≤ (C + 1)e i=0 tù❝ ❧➔ k−1 {1 + Keδ[1+(1−m)i] a(i)} ||x(k)|| ≤ (K||x0 || + 1) i=0 ❚ø ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✹✮ t❛ ❝â t❤➸ t➻♠ ✤÷đ❝ sè p > s❛♦ ❝❤♦ lim sup Keδk(1−m) a(k) ≤ p < − e−delta ❉♦ ✈➟② s➩ ❝â N ∈ Z+ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ k ≥ N, t❛ ❝â Kek(1−m) a(k) + e−delta < p + e−δ = q < tø ✤â s✉② r❛ ||x(k)|| ≤ (K||x0 || + 1)q k ❑❤✐ q < t❤➻ limk → ∞||x(k)|| = ∀k ≥ N ✸✸ ❜✮ ❚r÷í♥❣ ❤đ♣ m = : ❱ỵ✐ m = t❛ ❝â ♥❣❛② ✭✷✳✶✼✮✱ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ●r♦♥✇❛❧❧✱ ❇ê ✤➲ ✭✶✳✺✳✷✮✱ ❝❤♦ tr÷í♥❣ ❤đ♣ m = t❛ ✤÷đ❝ k−1 z(k) ≤ C [1 + b(i)], k = 1, 2, i=0 ❉♦ ✤â k−1 −δk {1 + Keδ a(i)}, ||x(k)|| ≤ K||x0 ||e i=0 tù❝ ❧➔ k−1 [e−δ + Ka(i)] ||x(k)|| ≤ K||x0 || i=0 ⑩♣ ❞ö♥❣ ❣✐↔ t❤✐➳t ✭✷✳✶✺✮ t❛ t➻♠ ✤÷đ❝ ♠ët sè p > s❛♦ ❝❤♦ lim sup Ka(k) ≤ p < − e−δ ❉♦ tỗ t N Z+ s ợ k ≥ N t❛ ❝â Ka(k) + e−δ < p + e−δ = q < ❙✉② r❛ ||x(k)|| ≤ K||x0 ||q k , tù❝ ❧➔ ❤➺ ✤➣ ❝❤♦ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ❝✮ ❚r÷í♥❣ ❤đ♣ m > : ❱ỵ✐ m > t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✼✮ k−1 b(i)z(i)m x(k) ≤ K||x0 || + i=0 ●✐↔ sû r > ❧➔ ♠ët sè ❜➜t ❦➻ tr♦♥❣ ❦❤♦↔♥❣ (0, 1) ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ●r♦♥✇❛❧❧ ❝❤♦ m > 1, ❇ê ✤➲ ✭✶✳✺✳✸✮✱ t❛ ✤÷đ❝ k−1 z(k) ≤ C − (m − 1)C m−1 b(i) i=1 1−m ✸✹ tr♦♥❣ ✤â C = K||x0 ||, b(i) = Ke−δ[1+i(1−m)] a(k), ♥➳✉ ❝❤➾ ❝â ♠ët k−1 − (m − 1)C m−1 ✭✷✳✶✽✮ b(i) > i=0 ✣➸ þ t❤➜② r➡♥❣ ♥➳✉ ✭✷✳✶✽✮ ❝è ✤à♥❤ ✈ỵ✐ ♠å✐ x0 t❤➻ r (m − 1)K m−1 γ ||x0 || ≤ tr♦♥❣ ✤â γ := k−1 m − := R b(i), ✈➻ ✭✷✳✶✻✮ ❤ú✉ ❤↕♥✳ ❚❤➟t ✈➟②✱ t❛ ❝â i=0 k−1 (m − 1)K m−1 m−1 b(i) ≤ (m − 1)K m−1 γ||x0 ||m−1 ≤ r ||x0 || i=0 ❞♦ ✤â k−1 − (m − 1)K m−1 m−1 ||x0 || b(i) ≥ −r > i=0 ❙✉② r❛✱ ✈ỵ✐ ♠å✐ x0 ♠➔ ||x0 || ≤ R t❛ ❝â ||x(k)|| ≤ K1 e−δk ||x0 ||, tr♦♥❣ ✤â K1 = K 1−r ❱➻ ✈➟② ✈ỵ✐ ❜➜t ❦➻ ε > 0, t❛ ❝❤å♥ ✤÷đ❝ sè δ0 < R, s❛♦ ❝❤♦ ✈ỵ✐ ||x0 || < δ0 ✈➔ k ≥ N t❤➻ ||x(k)|| < ε ε K1 ✈➔ N ∈ Z+ ỵ t ữủ ự t ❚ø ✤✐➲✉ ❦✐➺♥ ✭✷✳✶✹✮ t❛ t❤➜② r➡♥❣ e−δ < tù❝ ❧➔ ❤➺ rí✐ r↕❝ t❤❡♦ t❤í✐ ❣✐❛♥ x(k + 1) = A(k)x(k) ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ ✣✐➲✉ ❦✐➺♥ ✤õ ✸✺ ❝❤♦ ✭✷✳✶✹✮ ❧➔ ∞ ek(1−m) a(k) < +∞ k=0 ◆❤➟♥ ①➨t ✷✳✷✳✼✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ✤õ ❝❤♦ sỷ ỵ (A(k)) ❉ü❛ ✈➔♦ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ✤➣ ❜✐➳t ✈➲ sü ê♥ ✤à♥❤ ❝õ❛ ❤➺ rí✐ r↕❝ t❛ ①➨t sü ê♥ ✤à♥❤ ❝õ❛ ❤➺ ✭✷✳✶✶✮ t❤❡♦ ❝→❝❤ s❛✉✿ ❚❛ ✤➣ ❜✐➳t r➡♥❣ ♥❣❤✐➺♠ x(t) ❝õ❛ ✭✷✳✶✶✮ ❜➢t ✤➛✉ tø x0 tỵ✐ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ t x(t) = Φ(t, 0)x0 + Φ(t, s)f (s, x(s))ds, tr♦♥❣ ✤â Φ(t, s) ❧➔ ♠❛ tr➟♥ ♥❣❤✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❤➺ t✉②➳♥ t➼♥❤ ✭✷✳✹✮✳ ●✐↔ sû k ∈ Z+ ✳ ❱ỵ✐ ♠é✐ t ∈ [k, k + 1] ♥❣❤✐➺♠ x(t) ❦➳t ❤đ♣ ✈ỵ✐ ❤➔♠ xk (t), t ∈ [k, k + 1] ♥➔♦ ✤â✳ ✣➦t A(k) := Φ(k + 1, k), k+1 g(k, x(k)) := Φ(k + 1, k + s)f (k + s, x(k + s))ds, k ❦❤✐ ✤â ❤➺ ✭✷✳✶✶✮ ✤÷đ❝ q✉② ✈➲ ❤➺ rí✐ r↕❝ t❤❡♦ t❤í✐ ❣✐❛♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ C[0,1] ữợ x(k + 1) = A(k)x(k) + g(k, x(k)), k ∈ Z+ , ✭✷✳✶✾✮ tr♦♥❣ ✤â x(k) ∈ C[0,1] ✈ỵ✐ ❝❤✉➞♥ ||x(k)|| = max ||x(k + 1)|| t∈[0,1] ❚❛ t❤➜② r➡♥❣ ♥❣❤✐➺♠ ❝õ❛ ❤➺ rí✐ r↕❝ t❤❡♦ t❤í✐ ❣✐❛♥ ✭✷✳✶✾✮ ❜➢t ✤➛✉ tø x(0) = x0 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✷✳✶✶✮ ❜➢t ✤➛✉ tø x0 tỵ✐ ✵ ữủ ỵ t õ t✐➯✉ ❝❤✉➞♥ ✈➲ sü ê♥ ✤à♥❤ ❝õ❛ ❤➺ ❦❤æ♥❣ ætæ♥æ♠ ỵ sỷ r ỗ t↕✐ K > 0, δ > s❛♦ ❝❤♦ ||Φ(t, s)|| ≤ Ke−δs , ∀t, s ≥ ✐✐✮ ||f (t, x)|| ≤ a(t)||x||m , t ≥ 0, tr♦♥❣ ✤â ◆➳✉ < m < : lim sup eδ(k+s)(1−m) a(k + s)ds < k→∞ ◆➳✉ m = : lim sup k→∞ ◆➳✉ m > : ∞ k=0 − e−δ a(k + s)ds < , K −δ(k+s)(m−1) a(k e + s)ds < +∞, ❑❤✐ ✤â ❤➺ ✭✷✳✶✶✮ ê♥ ✤à♥❤ t✐➺♠ ❝➟♥✳ − e−δ , K ✸✼ ❑➳t ❧✉➟♥ ◆❤ú♥❣ ✈➜♥ ữủ tr tr ã ❚r➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♥❤÷ ♥❣❤✐➺♠ ❝õ❛ ❤➺✱ sü ê♥ ✤à♥❤ t❤❡♦ ▲②❛♣✉♥♦✈✱ ✈➔ ♠ët sè ❦➳t q t ỗ tớ tr ởt số ❦✐➳♥ t❤ù❝ ✈➲ ①➨t sü ê♥ ✤à♥❤ ❝õ❛ ❤➺ t❤❡♦ ▲②❛♣✉♥♦✈✳ • P❤➛♥ trå♥❣ t➙♠ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✤✐➲✉ ❦✐➺♥ ê♥ ✤à♥❤ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤❡♦ t❤í✐ ❣✐❛♥✱ ❤➺ ♣❤✐ t✉②➳♥ t❤❡♦ t❤í✐ ❣✐❛♥ ❧✐➯♥ tư❝✱ ❤➺ ♣❤✐ t✉②➳♥ ✈ỵ✐ t❤í✐ ❣✐❛♥ rí✐ r↕❝✱ ❤➺ ỉtỉ♥ỉ♠✱ ❦❤ỉ♥❣ ỉtỉ♥ỉ♠ ✈➔ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛✳ ✸✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❚❤➳ ❍♦➔♥✱ P❤↕♠ P❤✉ ✭✷✵✵✸✮✱ ❈ì sð ♣❤÷ì♥❣ tr ỵ tt ụ Pt ổ ỵ tt ✤✐➲✉ ❦❤✐➸♥ t♦→♥ ❤å❝✱ ◆❳❇ ✣↕✐ ❍å❝ ◗✉è❝ ●✐❛✳ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❱ô ◆❣å❝ P❤→t ✭✶✾✾✾✮ ❖♥ t❤❡ st❛❜✐❧✐t② ♦❢ t✐♠❡ ✲ ✈❛r✐♥❣ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✱ ❖♣t✐♠✐③❛t✐♦♥✱ ✹✺✿✶✱ ✷✸✼ ✲ ✷✺✹✳ ❬✹❪ ❇❡❧❧♠❛♥✱ ❇✳ ✭✶✾✺✸✮✳ ❙t❛❜✐❧✐t② ❚❤❡♦r② ♦❢ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✳ ▼❛❝✲ ●r❛✇ ✲ ❍✐❧❧✱ ◆❡✇ ❨♦r❦✳ ❬✺❪ ❉❡♠✐❞♦✈✐❝❤✱ ❱✳ ❇✳ ✭✶✾✻✾✮✳ ❖♥ t❤❡ st❛❜✐❧✐t② ❝r✐t❡r✐♦♥ ♦❢ ❞✐❢❢❡r❡♥❝❡ ❡q✉❛✲ t✐♦♥✳ ❉✐❢❢✳ ❊q✉❛t✐♦♥s✱ ❯❙❙❘✱ ❱♦❧✳ ✺✱ ◆♦✳ ✼ ✭✐♥ ❘✉ss✐❛♥✮✳
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