✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ P❍❸▼ ❚❍➚ ❚❍❯ ❚❘❆◆● ❙Ü ❚➬◆ ❚❸■ ◆●❍■➏▼ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❈❻P ❇❆ ❱❰■ ✣■➋❯ ❑■➏◆ ❇■➊◆ ❉❸◆● ❇❆ ✣■➎▼ ❱⑨ ❉❸◆● ❚➑❈❍ P❍❹◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ P❍❸▼ ❚❍➚ ❚❍❯ ❚❘❆◆● ❙Ü ❚➬◆ ❚❸■ ◆●❍■➏▼ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❈❻P ❇❆ ❱❰■ ✣■➋❯ ❑■➏◆ ❇■➊◆ ❉❸◆● ❇❆ ✣■➎▼ ❱⑨ ❉❸◆● ❚➑❈❍ P❍❹◆ ◆❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✽✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ ✣➐◆❍ ❍Ị◆● ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾ ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trò♥❣ ❧➦♣ ✈ỵ✐ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr ữủ ró ỗ ố ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ P❤↕♠ ❚❤à ❚❤✉ ❚r❛♥❣ ❳→❝ ♥❤➟♥ ❝õ❛ ❦❤♦❛ ❚♦→♥ ❳→❝ ♥❤➟♥ ữớ ữợ r ũ ỡ rữợ tr ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ tæ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ỡ s s tợ r ũ ữớ t t t ữợ tổ tr sốt q tr ❝ù✉ ✤➸ tæ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❦❤♦❛ ❚♦→♥ ❝ò♥❣ t♦➔♥ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ✣❍❙P ❚❤→✐ ◆❣✉②➯♥ ✤➣ tr✉②➲♥ t❤ư ❝❤♦ tỉ✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ q✉❛♥ trå♥❣✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✈➔ ❝❤♦ tỉ✐ ỳ ỵ õ õ qỵ tr sốt q tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❇↔♥ ❧✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❦❤✐➳♠ ❦❤✉②➳t rt ữủ sỹ õ õ ỵ ❦✐➳♥ ❝õ❛ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❈✉è✐ ❝ò♥❣ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❦❤➼❝❤ ❧➺ tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ P❤↕♠ ❚❤à ❚❤✉ ❚r❛♥❣ ✐✐ ▼ö❝ ❧ö❝ ❚r❛♥❣ ❜➻❛ ♣❤ư ▲í✐ ❝❛♠ ✤♦❛♥ ▲í✐ ❝↔♠ ì♥ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✐ ✐✐ ởt số ỵ t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❚♦→♥ tû ❋r❡❞❤♦❧♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❍➔♠ ●r❡❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t ữỡ tr ợ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠ ✈➔ ❞↕♥❣ t➼❝❤ ♣❤➙♥ ✶✷ ✷✳✶ ỹ tỗ t ữỡ tr ❜❛ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ✶✷ ỹ tỗ t ữỡ tr ❜❛ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ❚➔✐ t ởt số ỵ ❤✐➺✉ ✈➔ ✈✐➳t t➢t R t➟♣ ❝→❝ sè t❤ü❝ ∅ t➟♣ ré♥❣ A⊂B A A∪B ❤ñ♣ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B A∩B ❣✐❛♦ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B B t➼❝❤ ❉❡s❝❛rt❡s ❝õ❛ ❤❛✐ t➟♣ ❤đ♣ ker(f ) ❤↕t ♥❤➙♥ ❝õ❛ Coker(f ) ✤è✐ ❤↕t ♥❤➙♥ ❝õ❛ ✷ ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ ❧➔ t➟♣ ❝♦♥ ❝õ❛ ✐✈ B f f A ✈➔ B ▼ð ✤➛✉ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ ✤❛ ❞↕♥❣ tr ỹ t ỵ tt ❈❤➥♥❣ ❤↕♥ ♥❤÷ ❜➔✐ t♦→♥ ①➨t ✤ë ✈ã♥❣ ❝õ❛ ♠ët ợ ữủ t t ợ s s♦♥❣ ❝→❝ ✈➟t ❧✐➺✉ ❦❤→❝ ♥❤❛✉ ❬✽❪✱ ❜➔✐ t♦→♥ ♥❣❤✐➯♥ ❝ù✉ ❞á♥❣ ❝❤↔② ❝õ❛ ♠ët ♠➔♥❣ ♠ä♥❣ ❝❤➜t ❧ä♥❣ ♥❤ỵt tr➯♥ ❜➲ ♠➦t r➢♥✱ ❦❤✐ ♠ët ♠➔♥❣ ♥❤÷ ✈➟② ❝❤↔② ố ởt t t ữợ t ự s ↔♥❤ ❤÷ð♥❣ ❝õ❛ sù❝ ❝➠♥❣ ❜➲ ♠➦t✱ ❧ü❝ ❤➜♣ ❞➝♥ ụ ữ ợt ữỡ tr ❞❛♦ ✤ë♥❣ ❝ơ♥❣ ✤÷đ❝ ✤÷❛ ✈➲ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ❬✶✶❪✳ ❚r♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ✤â✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✤÷đ❝ ❞➝♥ ✤➳♥ ❝â t❤➸ ð ❞↕♥❣ ❜❛ ✤✐➸♠✱ ❞↕♥❣ t➼❝❤ ♣❤➙♥ ❤❛② ❝→❝ ❞↕♥❣ ♣❤✐ t ự sỹ tỗ t t ữỡ tr ợ ❝→❝ ❧♦↕✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❦❤→❝ ♥❤❛✉ t❤✉ ❤ót ✤÷đ❝ ♥❤✐➲✉ sü q✉❛♥ t➙♠ ❝õ❛ ❝→❝ ♥❤➔ t♦→♥ ❤å❝✳ ❑ÿ t❤✉➟t ❦❤→ ♣❤ê ❜✐➳♥ ✤÷đ❝ sû ❞ư♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ tr ữợ ữỡ ♣❤→♣ ❧✐➯♥ tö❝ ❞ü❛ tr➯♥ ✈✐➺❝ ✤→♥❤ ❣✐→ t✐➯♥ ♥❣❤✐➺♠ ❝õ❛ ♠ët ❤å ❝→❝ ❜➔✐ t♦→♥ ✈ỵ✐ ♠ët t❤❛♠ sè t s õ sỷ ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ❬✷❪✱ ❬✸❪✱ ❬✹❪✱ ❬✺❪✳ ❈❤ó♥❣ tỉ✐ ✤➣ ỹ tỗ t ữỡ tr ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠ ✈➔ ❞↕♥❣ t➼❝❤ ♣❤➙♥✑✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦➳t q✉↔ ❝õ❛ ❆❜❞❡❧❦❛❞❡r r sỹ tỗ t ữỡ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ✤➛② ✤õ✿ y (t) = f (t, y(t), y (t), y (t)), ✶ < t < 1, tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣✱ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❉✐r✐❝❤❧❡t ❜❛ ✤✐➸♠ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ t➼❝❤ ♣❤➙♥✳ ỗ ữỡ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ♠ët sè ✤à♥❤ ❧➼ ✤✐➸♠ ❜➜t ✤ë♥❣✱ t♦→♥ tû ❋r❡❞❤♦❧♠ ✈➔ ❤➔♠ ●r❡❡♥✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤↕t ✤÷đ❝ ✤→♥❤ ❣✐→ t✐➯♥ ♥❣❤✐➺♠ ❝õ❛ ♠ët ❤å ❜➔✐ t♦→♥ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ✤➛② ✤õ tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣✿ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ t➼❝❤ ♣❤➙♥✳ ❙❛✉ õ sỷ ỵ t ự ởt số t q sỹ tỗ t ♥❣❤✐➺♠✳ ✷ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ❝➛♥ t❤✐➳t ❝❤♦ ❝❤÷ì♥❣ s❛✉✱ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t ởt số ỵ ❜➜t ✤ë♥❣ ❈❤♦ →♥❤ ①↕ T : A → A✳ ▼é✐ ♥❣❤✐➺♠ ❣å✐ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ x ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ x = Tx ✤÷đ❝ T✳ ởt số ỵ t s ỵ t ỡ ữủ sỷ tr ự sỹ tỗ t t ữỡ tr ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ❇❛♥❛❝❤ ❝❤♦ ❝→❝ t♦→♥ tû ❝♦ ✈ỵ✐ số k ỵ t ❇r♦✉✇❡r ❝❤♦ ❝→❝ t♦→♥ tû ❧✐➯♥ tư❝ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ỳ ỵ t r ❝❤♦ ❝→❝ t♦→♥ tû ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝ tr➯♥ ♠ët t ỗ rộ t tr ổ ❇❛♥❛❝❤ ✭✈æ ❤↕♥ ❝❤✐➲✉✮✳ ✣➙② ❧➔ ♠ët tê♥❣ q✉→t ❤â❛ ỵ t rr ỵ ❜➜t ✤ë♥❣ ❙❝❤❡❛❢❡r ❝❤♦ ❝→❝ t♦→♥ tû ❧✐➯♥ tö❝ ✈➔ ❝♦♠♣❛❝t tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ◆❣♦➔✐ r❛ ♠ët sè ✤à♥❤ ỵ t q trồ ữủ sỷ tr ự sỹ tỗ t ữỡ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤✐ ✸ t✉②➳♥✱ ❝❤➥♥❣ ❤↕♥ ♥❤÷ ✤à♥❤ ỵ r r t tỷ t tr ởt t ỗ rộ ổ ũ ợ ỵ t ❧➼ t❤✉②➳t ❜➟❝ ❇r♦✉✇❡r ✈➔ ❧➼ t❤✉②➳t ❝❤➾ sè ✤✐➸♠ ❜➜t ✤ë♥❣ ❝ơ♥❣ ❧➔ ♥❤ú♥❣ ❝ỉ♥❣ ❝ư q✉❛♥ trå♥❣✱ ✤÷đ❝ ự tr ự sỹ tỗ t ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❧✐➯♥ tư❝ ❝ơ♥❣ ♥❤÷ sỹ tỗ t ữỡ tr t ỵ t t ữỡ tr➻♥❤ ♣❤✐ t✉②➳♥ x = T x ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ♠❡tr✐❝ (X, d) ✭①❡♠ ❬✶✸❪✮ ❚♦→♥ tû ✤÷đ❝ ❣å✐ ❧➔ ❝♦ ✈ỵ✐ ❤➺ sè k T :M ⊆X→X ✭✶✳✶✮ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ d(T x, T y) ≤ kd(x, y) ✈ỵ✐ ♠å✐ x, y ∈ M ✣à♥❤ ỵ k ố k < ỵ t ✭✶✾✷✷✮✮ ●✐↔ sû r➡♥❣ ✭✐✮ T : M ⊆ X → M ❧➔ ♠ët →♥❤ ①↕ tø ▼ ✈➔♦ ❝❤➼♥❤ ♥â❀ ✭✐✐✮ ▼ ❧➔ t➟♣ ✤â♥❣✱ ❦❤→❝ ré♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ (X, d)❀ ✭✐✐✐✮ ❚ ❧➔ ♠ët →♥❤ ①↕ ❝♦ ✈ỵ✐ ❤➺ sè k ✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ x✱ tù❝ ❧➔ T ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ tr➯♥ ▼✳ ỵ t õ ỵ q trå♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤✱ ✤➦❝ ❜✐➺t tr♦♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ sỹ tỗ t t ữỡ tr t ỵ t rr ợ ỵ t ỵ ❜➜t ✤ë♥❣ ❇r♦✉✇❡r ❦❤æ♥❣ ❝❤➾ r❛ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ✤✐➸♠ ❜➜t ✤ë♥❣✱ t✉② ♥❤✐➯♥ ❝→❝ ✹ ❚ø ✤â t❛ ❝â |f1 (t, yn (t), yn (t), yn )| ≤ Mf , ❍ì♥ ♥ú❛ Nf1 (yn ) → Nf1 (y) ✣➸ ❝❤ù♥❣ ♠✐♥❤ Nf1 C (I)✳ tr♦♥❣ ❙✉② r❛ M1 > Nf ❧✐➯♥ tö❝✳ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝✱ ❣å✐ BR = y ∈ C03 (I); y ❑❤✐ õ tỗ t t I (3) R tọ ♠➣♥ Nf1 (y) ≤ M1 , ∀y ∈ BR ✈➔ t1 Nf1 (y)(t1 ) − Nf1 (y)(t2 ) ≤ |f1 (s, y(s), y (s), y (s)|ds ≤ Mf (R)|t1 −t2 |, t2 tr♦♥❣ ✤â Mf (R) = sup {|f (t, y, p, w)|; t ∈ I, |y| ≤ R, |p| ≤ R, |w| ≤ R} ❉♦ ✤â Nf1 ữợ t tử ự C (I)✳ L−1 Nf1 ❝â ✤✐➸♠ ❜➜t ✤ë♥❣✱ t❤➟t tứ ữợ tr t õ t ❜➜t ✤ë♥❣ ❝õ❛ ❤å ♣❤÷ì♥❣ tr➻♥❤ y = λL−1 Nf1 (y), ❜à ❝❤➦♥ tr♦♥❣ 0 0, ✈ỵ✐ ♠å✐ |p| > r1 y R õ tỗ t↕✐ R1 ∈ [r1 , +∞) s❛♦ ❝❤♦ ♥❣❤✐➺♠ y ❝õ❛ ❜➔✐ t♦→♥ t❤ä❛ ♠➣♥ |y (t)| ≤ R1 ✈➔ |y(t)| ≤ R1 , ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ ❝→❝ ❤➔♠ h1 ✈➔ h2 ✭✷✳✶✽✮✱ ✭✷✳✶✾✮ ∀t ∈ I ❧✐➯♥ tö❝ ♥➯♥ tỗ t h0i = max {|hi (u, v)|; u, v ∈ [−r1 ; r1 ]} , i = 1, ●✐↔ sû R1 = max(r1 + h01 , r1 + h02 )✳ ❑❤✐ ✤â R1 > r1 ✷✻ ✈➔ R1 > max(h01 , h02 ) ●✐↔ sû R1 y = ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✽✮✱ ✭✷✳✶✾✮✳ ❚❛ s➩ ❝❤➾ r❛ |y (t)| ≤ ✈ỵ✐ ♠å✐ ✤â ❤♦➦❝ t I sỷ ữủ tỗ t t1 ∈ I y (t1 ) > R1 ❳➨t tr÷í♥❣ ❤ñ♣ ❤♦➦❝ |y (t)| > R1 ✳ ❑❤✐ y (t1 ) < −R1 ✳ y (t1 ) > R1 ✭tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✮✳ max{|y (t)|; t ∈ I} > R1 ✳ ❚❛ ❝â s❛♦ ❝❤♦ ❉♦ y tử tỗ t t2 I s ❝❤♦ y (t2 ) = max{|y (t)|; t ∈ I}✳ ◆➳✉ t2 ∈ (0, 1) t❤➻ y (t2 ) > R1 > r1 , y (t2 ) = y (t2 )y (t2 ) ≤ 0✳ ✈➔ ❚ø ✭❍✸✮ t❛ ❝â y (t2 )f (t2 , y(t2 ), y (t2 ), y (t2 )) = y (t2 )f (t2 , y(t2 ), y (t2 ), 0) > ❉♦ < λ ≤ 1✱ ♥➯♥ tø ✭✷✳✶✽✮ s✉② r❛ ≥ λy (t2 )y (t2 ) > 0✳ ✣✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ◆➳✉ t2 = 0✱ tù❝ ❧➔ y ✤↕t ❣✐→ trà ❝ü❝ ✤↕✐ t↕✐ t = 0✱ ❦❤✐ ✤â y (0) ≤ ✈➔ y (0) > R1 > r1 ✳ ◆➳✉ y (0) = 0✱ t❤❡♦ ✭❍✸✮ t❛ ❝â y (0)y (0) = y (0)f (0, 0, y (0), 0) > s✉② r❛ y (0) > 0✳ y (t) > y (0) = 0✳ ❱➻ ✈➟② y (0) ❉♦ ✤â✱ y ❚÷ì♥❣ tü ✤ì♥ ✤✐➺✉ t➠♥❣ ✈ỵ✐ y t ❣➛♥ ❝ơ♥❣ ✤ì♥ ✤✐➺✉ t➠♥❣ ✈ỵ✐ ❦❤æ♥❣ t❤➸ ❧➔ ❣✐→ trà ❝ü❝ ✤↕✐ ❝õ❛ |y (t)|✳ t ✈➔ t > 0✱ ❣➛♥ s✉② r❛ ✈➔ t > 0✳ ❱➻ ✈➟② ❞➝♥ tỵ✐ ♠➙✉ t❤✉➝♥✳ ◆➳✉ y (0) < 0✳ ❝→❝❤ ①→❝ ✤à♥❤ t❤➻ h01 ≥ y (0) − ay (0) > y (0) > R1 t ợ R1 r trữớ ủ t2 = t tữỡ tỹ ú t ụ tợ ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ◆❤÷ ✈➟②✱ −R1 , y (t) − R1 ≤ 0, ∀t ∈ I ✳ ❚÷ì♥❣ tü✱ ❝ơ♥❣ ❝❤➾ r❛ ✤÷đ❝ y (t) ≥ ∀t ∈ I ✳ ❉♦ ✤â |y (t)| ≤ R1 ❚ø y(t) = t y (s)ds ✈➔ 0≤t≤1 ∀t ∈ I s✉② r❛ |y(t)| ≤ R1 , ◆❤➟♥ ①➨t ∀t ∈ I ✿ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ t❤✉➛♥ ♥❤➜t✱ tù❝ ❧➔ hi (u, v) 0✱ ✈ỵ✐ ♠å✐ u, v ∈ R, i = 1, 2.✱ ❦❤✐ ✤â R1 = r1 ✳ = sỷ tỗ t ❤➡♥❣ sè ❞÷ì♥❣ K1, K2 t❤ä❛ ♠➣♥ |f (t, y, p, w)| ≤ K1 w2 + K2 , ✈ỵ✐ ♠å✐ w ∈ R✱ (t, y, p) ∈ I × [−R1 ; R1 ] × [−R1 ; R1 ]✳ ❑❤✐ ✤â✱ tỗ t R2 > ổ tở s❛♦ ❝❤♦ ✈ỵ✐ ♥❣❤✐➺♠ y ❝õ❛ ❜➔✐ t♦→♥ ♠➔ |y(t)| ≤ R1 ✱ |y (t)| ≤ R1 ✱ ∀t ∈ I t❤➻ |y (t)| ≤ R2 ✱ ∀t ∈ I ✳ ✭✷✳✶✾✮ ❈❤ù♥❣ ♠✐♥❤✳ ♠➣♥ ✭✷✳✶✽✮✱ ●✐↔ sû y = ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✽✮✱ ✭✷✳✶✾✮ t❤ä❛ |y(t)| ≤ R1 , |y (t)| ≤ R1 ✈ỵ✐ ♠å✐ t ∈ I✳ |h01 + R1 | |y (0)| ≤ , a ●å✐ ❚ø ✭✷✳✷✺✮ ✈➔ t∈I s❛♦ ❝❤♦ |h02 + R1 | |y (1)| ≤ b |h01 + R1 | |h02 + R1 | r0 = max , a b ✭✷✳✷✻✮ t❛ ❝â |y (0)| ≤ r0 , ✈➔ |y (1)| ≤ r0 |y (t)| = max{|y (t)|; t ∈ I} > r0 ✳ t↕✐ ♠ët ♥û❛ ❦❤♦↔♥❣ |y (t)| > r0 ❑❤✐ ✤â ✈ỵ✐ ♠å✐ tê♥❣ q✉→t✱ ❣✐↔ sû α, t ⊂ I t ∈ α, t α ≤ t✳ ✭❤♦➦❝ t, α ⊂ I ợ t trữớ ủ y (t) > r0 sỷ tỗ t y C (I) s❛♦ ❝❤♦ t ∈ t, α ✭✷✳✷✺✮ ♥➯♥ tỗ |y ()| = r0 ổ t t ✈ỵ✐ ♠å✐ t ∈ α, t ✳ ❚ø ✭✷✳✶✽✮ s✉② r❛✿ |y (t)| = λ|f (t, y(t), y (t), y (t))| ≤ λ K1 |y (t)|2 + K2 , ❱➻ 0 0✱ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ r0 , r1 , h10 , h20 , K1 , K2 ứ s r tỗ t R2 s |y (t)| ≤ R2 ❉♦ ✤â |y (t)| ≤ R2 t I ỵ sỷ ữủ tọ sỷ tỗ t số ữỡ i, i = 1, ợ (b + 1)β1 + (a + 1)β2 < ✭❍✸✮ ✭❍✹✮ a + b + ✈➔ ❤➔♠ σi : (0, ∞) → (0, ∞) ❧✐➯♥ tư❝✱ ❦❤ỉ♥❣ ❣✐↔♠ t❤ä❛ ♠➣♥ σi (u) ≤ βi u ✈ỵ✐ u > ✈➔ |hi (y1 , y2 ) − hi (z1 , z2 )| ≤ σi (max{|y1 − z1 |, |y2 − z2 |}), ∀y1 , y2 , z1 , z2 ∈ R ❑❤✐ õ t tỗ t t t ởt ♥❣❤✐➺♠✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû y ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✶✽✮ ✈➔ ✭✷✳✶✾✮✳ ❚ø ✤✐➲✉ ❦✐➺♥ ✭❍✸✮ t❛ ❝â |y(t)| ≤ R1 ❚ø ✤✐➲✉ ❦✐➺♥ ✭❍✹✮ t❛ ❝â |y (t)| ≤ R2 ✈➔ |y (t)| ≤ R1 ✈ỵ✐ ♠å✐ ✈ỵ✐ ♠å✐ t ∈ I✳ t ∈ I✳ ●å✐ R3 = max{|f (t, y, p, w)|; t ∈ I, |y| ≤ R1 , |p| ≤ R1 , |w| ≤ R2 } ❑❤✐ ✤â✱ ✤➦t ✣➦t r = max(R1 , R2 , R3 )✱ Ω = {y ∈ C (I); y ❧✐➯♥ tư❝ ❝õ❛ ❤➔♠ f (3) t❛ ✤÷đ❝ y (3) ≤ r < r + 1}✳ ❚ø t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ●r❡❡♥ ✈➔ t➼♥❤ s✉② r❛ t♦→♥ tû G1 : Ω → C (I) ⑩♣ ❞ö♥❣ ✭❍✺✮✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tư❝✳ G2 : Ω → C (I) ❧➔ →♥❤ ①↕ ❝♦✳ ❚❤➟t ✈➟②✱ t❛ ❝â✿ |G2 (y)(t) − G2 (z)(t)| = [ϕ(t, y(s)) − ϕ(t, z(s))] ds ≤ |ϕ(t, y(s)) − ϕ(t, z(s))|ds ❍ì♥ ♥ú❛✱ ✈ỵ✐ i = 1, |Hi (y) − Hi (z)| ≤ σi ( y − z ✷✾ (3) ) ❉♦ ✤â |ϕ(t, y(s)) − ϕ(t, z(s))| ≤ g1 (t)|H1 (y(s)) − H1 (z(s))| + g2 (t)|H2 (y(s)) − H2 (z(s))| ≤ g1 (t)σ1 ( y − z (3) ) + g2 (t)σ2 ( y − z a + 1/2 b + 1/2 β1 + β2 a+b+1 a+b+1 ≤ y−z (3) ) (3) ❚÷ì♥❣ tü ∂ϕ(t, y(s)) ∂ϕ(t, z(s)) ≤ g1 (t)σ1 ( y − z (3) ) + g2 (t)σ2 ( y − z − ∂t ∂t b+1 a+1 ≤ β1 + β2 y − z a+b+1 a+b+1 (3) ) (3) ✈➔ ∂ ϕ(t, y(s)) ∂ ϕ(t, z(s)) − ≤ (β1 + β2 ) y − z ∂t2 ∂t2 a+b+1 (3) ❑❤✐ ✤â G2 (y) − G2 (z) (3) b+1 a+1 β1 + β2 a+b+1 a+b+1 ≤ y−z ❉♦ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✽✮✱ ✭✷✳✶✾✮ ✤➲✉ t❤ä❛ ♠➣♥ ♥➯♥ ổ tỗ t y / (0, 1) s❛♦ ❝❤♦ (3) y (3) ≤ r✱ y = λL−1 F(y)✳ ❉♦ ✤â ❦❤➥♥❣ ✤à♥❤ ✭❆✷✮ ❝õ❛ ỵ ổ r ữ t tỷ L−1 F = G1 + G2 ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✽✮✱ ✭✷✳✶✾✮✳ ❱➼ ❞ö ✷✳✷✳✻✳ ❳➨t ❜➔✐ t♦→♥ ❜✐➯♥     y (t) = ey(t) (y (t) − 1)(1 + y (t)2 ),        y(0) = 0,    y (0) − y (0) =        y (1) + y (1) = y(s)ds, 1 + y (s)2 ds ✸✵ < t < 1, ✭✷✳✷✾✮ ❚r♦♥❣ ✈➼ ❞ư ♥➔②✱ ✈ỵ✐ p > 1✳ f (t, y, p, w) = ey (p−1)(1+w2 )✳ ❚❛ ❝â pf (t, y, p, 0) > √ ✣✐➲✉ ❦✐➺♥ ✭❍✸✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ❦✐➺♥ ✭❍✹✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ♠➣♥ ✈ỵ✐ β1 = β2 = 1/2✳ r1 = 1✱ K1 = K2 = 2e ✈➔ g1 ❦❤✐ ✤â (3) = R1 = + = g2 2 , ✤✐➲✉ (3) ✳ ✭❍✺✮ t❤ä❛ ữ t ỵ t tỗ t↕✐ ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠✳ ❙❛✉ ✤➙② ❝❤ó♥❣ tỉ✐ s➩ tr sỹ tỗ t t ✭✷✳✶✼✮ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣ ❝ư t❤➸ ❤ì♥✳ ❇➔✐ t♦→♥ ✷✳✷✳✼✳ ❳➨t ❜➔✐ t♦→♥ s❛✉     y (t) = f (t, y(t)),        y(0) = 0,    y (0) − ay (0) =        y (1) + by (1) = < t < 1, ✭✷✳✸✵✮ h1 (y(s))ds, h2 (y(s))ds ỵ sỷ tỗ t q ∈ L1(I), Ψ : [0, +∞) → (0, +∞) ❧✐➯♥ tư❝✱ ❦❤ỉ♥❣ ❣✐↔♠ t❤ä❛ ♠➣♥ |f (t, y)| ≤ q(t)Ψ(|y|), ∀(t, y) ∈ I × R ✭❍✼✮ ●✐↔ sû ❤➔♠ hi : R → R ❦❤æ♥❣ ❣✐↔♠✱ hi(0) = tỗ t ci > ợ c1 (b + 1) + c2 (a + 1) < a + b + s❛♦ ❝❤♦ |hi (y) − hi (z)| ≤ ci |y − z|, ∀y, z ∈ R, i = 1, sỷ tỗ t > s❛♦ ❝❤♦ γ > Ψ(γ) ❑❤✐ ✤â ❜➔✐ t♦→♥ ❈❤ù♥❣ ♠✐♥❤✳ ✈ỵ✐ λ = 1✳ c1 (b + 1) + c2 (a + 1) 1− a+b+1 ✭✷✳✸✵✮ −1 G0 q L1 tỗ t t t ởt õ tữỡ ữỡ ợ ữỡ tr t ❍ì♥ ♥ú❛✱ G1 ❧➔ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tư❝ ✈➔ G2 ❧➔ →♥❤ ①↕ ❝♦✳ ❚ø ✭✷✳✷✸✮ ✈➔ ✭❍✻✮✱ t❛ ❝â |G1 (y)(t)| ≤ G(t, s)q(s)Ψ(|y(s)|)ds ✸✶ ✭✷✳✸✶✮ ❚ø ✭✷✳✷✹✮ ✈➔ ✭❍✺✮ t❛ ❝â |G2 (y)(t)| ≤ c1 g1 ❉♦ ✤â |G2 (y)(t)| ≤ (3) + c2 g2 (3) |y(t)| c1 (b + 1) + c2 (a + 1) y a+b+1 ✭✷✳✸✷✮ ❚ø ✭✷✳✷✶✮ t❛ t❤➜② ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✸✵✮ t❤ä❛ ♠➣♥ |y(t)| ≤ G(t, s)q(s)Ψ(|y(s)|)ds + ✣➦t S := {y ∈ C(I); y G(S) ❜à ❝❤➦♥✳ ●✐↔ sû ❑❤✐ ✤â y =γ < γ}✳ y ∈ ∂S ✈➔ c1 (b + 1) + c2 (a + 1) y a+b+1 ❑❤✐ ✤â S ♠ð tr♦♥❣ λ ∈ (0, 1) t❤ä❛ ♠➣♥ ✭✷✳✸✸✮ C = C(I), ∈ S ✈➔ y = λ(G1 + G2 )(y)✳ ✈➔ tø ✭✷✳✸✷✮ t❛ ❝â✿ γ ≤ G0 q L1 Ψ(γ) + c1 (b + 1) + c2 (a + 1) γ, a+b+1 s✉② r❛ γ ≤ Ψ(γ) c1 (b + 1) + c2 (a + 1) 1− a+b+1 −1 G0 q L1 ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ợ ữ ỵ ✷✳✷✳✸ ❦❤æ♥❣ ①↔② r❛✳ ❉♦ ✤â t♦→♥ tû L−1 F = G1 + G2 ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✸✵✮✳ ❇➔✐ t♦→♥ ✷✳✷✳✾✳ 1, 2✱ ❳➨t ❜➔✐ t♦→♥ ✭✷✳✸✵✮ ✈ỵ✐ hi (y(t)) = li (t)y(t)✱ ∀t ∈ I, i = tr♦♥❣ ✤â li ❧✐➯♥ tö❝ ✈➔ ❦❤æ♥❣ ➙♠✳ ✣➦t k(t, s) = g1 (t)l1 (s) + g2 (t)l2 (s)✱ ∀(t, s) ∈ I × I ✳ ✈➔ ❑❤✐ ✤â k(t, s) ≥ G2 (y)(t) = k(t, s)y(s)ds ỵ sỷ M = max {k(t, s); (t, s) ∈ I × I} < tỗ t : [0, ∞) → (0, ∞) ❧✐➯♥ tư❝✱ ❦❤ỉ♥❣ ❣✐↔♠ s❛♦ ❝❤♦ |f (t, y)| (|y|) lim sup tỗ t ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠✳ Φ(ρ) 1−M < ❑❤✐ ✤â ❜➔✐ t♦→♥ ρ G0 ✸✷ ✭✷✳✷✳✾✮ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â t tữỡ ữỡ ợ ữỡ tr t ✤ë♥❣ y = G1 y + G2 y, ✈ỵ✐ G1 t tử ữợ ứ t ❝â I − G2 ❦❤↔ ♥❣❤à❝❤✳ ✣➸ t➻♠ ❜✐➸✉ t❤ù❝ (I − G2 )−1 t❛ sû ❞ö♥❣ ❧➼ t❤✉②➳t ❋r❡❞❤♦❧♠ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤✳ y(t) = ψ(t) + k(t, s)y(s)ds ✭✷✳✸✺✮ ❚ø ✤✐➲✉ ❦✐➺♥ M < 1✱ s✉② r❛ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ker [k(t, s)]✳ ❑❤✐ ✤â ✭✷✳✸✺✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝❤♦ ❜ð✐ −1 y(t) = (I − G2 ) ψ(t) = ψ(t) + R(t, s)ψ(s)ds, tr♦♥❣ ✤â R(t, s) t❤ä❛ ♠➣♥ M 1−M R(t, s) ≤ ❉♦ ✤â (I − G2 )−1 ≤ ❚ø ✭✷✳✸✹✮✱ t❛ ❝â y 1−M ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✷✳✾✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t tỷ t tử ữợ y õ tỗ t > y ❧➔ Γ = (I − G2 )−1 G1 s❛♦ ❝❤♦ ♥❣❤✐➺♠ ❝õ❛ y = Γy t❤ä❛ ♠➣♥ ◆❣❤➽❛ ❧➔ −1 y(t) = (I − G2 ) G(t, s)f (s, y(s))ds ✈➔ tø ✭❍✶✵✮ |y(t)| ≤ 1−M ✣➦t ρ0 = y G(t, s)Φ(|y(s)|)ds ✭✷✳✸✼✮ 0 ✳ ❚ø ✭✷✳✸✼✮ s✉② r❛ ρ0 ≤ G0 Φ(ρ0 ) 1−M ✸✸ ✭✷✳✸✽✮ ❚ø ✤✐➲✉ ❦✐➺♥ ❝õ❛ ρ > tr s r tỗ t > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ t❛ ❝â Φ(ρ) − M < , ρ G0 ❤♦➦❝ ❚ø ✭✷✳✸✽✮ ✈➔ ✭✷✳✸✾✮ s✉② G0 Φ(ρ) < ρ 1−M r❛ ρ0 ≤ ρ∗✳ ✣➦t V = {y ∈ C(I); y ❚❤❡♦ ✤à♥❤ ❧➼ ✤✐➸♠ ❜➜t ✤ë♥❣ ❙❝❤❛✉❞❡r✱ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✷✳✾✮✳ ✸✹ Γ ✭✷✳✸✾✮ < ρ∗}✳ ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ V ✈➔ ❑➳t ❧✉➟♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ❱ỵ✐ ự sỹ tỗ t ữỡ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠ ✈➔ ❞↕♥❣ t➼❝❤ ♣❤➙♥✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ✤à♥❤ ❧➼ ✤✐➸♠ ❜➜t ✤ë♥❣✱ t♦→♥ tû ❋r❡❞❤♦❧♠ ✈➔ ❤➔♠ ●r❡❡♥✳ ✷✳ r sỹ tỗ t ữỡ tr ✈✐ ♣❤➙♥ ❝➜♣ ❜❛ ✤➛② ✤õ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠ ✈➔ ❞↕♥❣ t➼❝❤ ♣❤➙♥✳ ❈→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♠ỵ✐ ❝❤➾ ❞ø♥❣ ❧↕✐ ð ✈✐➺❝ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❝→❝ ❦➳t q✉↔ ❝õ❛ ❆❜❞❡❧❦❛❞❡r ❇♦✉❝❤❡r✐❢ tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❞↕♥❣ ❜❛ ✤✐➸♠ ✈➔ ❞↕♥❣ t➼❝❤ ♣❤➙♥✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐✱ ❝❤ó♥❣ tỉ✐ s➩ t✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✈➜♥ ✤➲ ♥➔② ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ♣❤✐ t✉②➳♥ ❦❤→❝ t❤æ♥❣ q✉❛ ✈✐➺❝ ✤÷❛ r❛ ♠ët sè ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤→♥❤ ❣✐→ t sỷ ỵ t ự sỹ tỗ t t❤❛♠ ❦❤↔♦ ❬✶❪ ❆✳ ❆❢t❛❜✐③❛❞❡❤✱ ❈✳P✳ ●✉♣t❛✱ ❏✳ ❳✉✱ ✭✶✾✽✾✮✱ ✧❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡✲ ♥❡ss t❤❡♦r❡♠s ❢♦r t❤r❡❡✲♣♦✐♥t ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s✧✱❙■❆▼ ▼❛t❤✳ ❆♥❛❧✱ ❏✳ ✷✵✭✸✮✱ ✼✶✻✲✼✷✻✳ ❬✷❪ ❆✳ ❇♦✉❝❤❡r✐❢✱ ✭✷✵✵✶✮✱ ✏❚✇♦✲♣♦✐♥t ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r t❤✐r❞ ♦r❞❡r ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ■♥t✑✱ ❏✳ ❉✐❢❢❡r❡♥✳ ❊q✉❛t✳ ❆♣♣❧✱ ✷✭✶✮✱ ✸✾✕✹✺✳ ❬✸❪ ❆✳ ❇♦✉❝❤❡r✐❢✱ ◆✳ ❆❧✲▼❛❧❦✐✱ ✭✷✵✵✼✮✱ ✏◆♦♥❧✐♥❡❛r t❤r❡❡✲♣♦✐♥t t❤✐r❞ ♦r❞❡r ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s✑✱ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✱ ✶✾✵✱ ✶✶✻✽✕✶✶✼✼✳ ❬✹❪ ❉❆✳ ❇♦✉❝❤❡r✐❢✱ ❙✳ ▼✳ ❇♦✉❣✉✐♠❛❜✱ ◆✳ ❆❧✲▼❛❧❦✐❝✱ ❩✳ ❇❡♥❜♦✉③✐❛♥❡✱ ✭✷✵✵✾✮✱ ✧❚❤✐r❞ ♦r❞❡r ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ✐♥t❡❣r❛❧ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ✶✼✸✻✲✶✼✹✸✳ ❬✺❪ ▼✳❘✳ ●r♦ss✐♥❤♦✱ ❋✳▼✳ ▼✐♥❤♦s✱ ✭✷✵✵✶✮✱ ✧❊①✐st❡♥❝❡ r❡s✉❧t ❢♦r s♦♠❡ t❤✐r❞ ♦r❞❡r s❡♣❛r❛t❡❞ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s✧✱ ❚✳▼✳❆✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ✹✼✭✹✮✱ ✷✹✵✼✕✷✹✶✽✳ ❬✻❪ ❉✳❉✳ ❍❛✐✱ ❑✳ ❙❝❤♠✐tt✱ ✭✶✾✾✹✮✱ ✧❇♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r ❤✐❣❤❡r ♦r❞❡r ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ❚✳▼✳❆✱ ✼✭✹✮✱ ✷✾✸✕✸✵✺✳ ❬✼❪ ❨✳❆✳ ❑❧♦❦♦✈✱ ✭✷✵✵✷✮✱ ✧❯♣♣❡r ❛♥❞ ❧♦✇❡r ❢✉♥❝t✐♦♥s ✐♥ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s ❢♦r ❛ t❤✐r❞ ♦r❞❡r ♦r❞✐♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✧✱ ❊q✉❛t✱ ❉✐❢❢❡r❡♥✳ ✸✻✭✶✷✮✱ ✶✼✻✷✕✶✼✻✾✳ ❬✽❪ ❉✳ ❑r❛❥❝✐♥♦✈✐❝✱ ✭✶✾✼✷✮✱ ✧❙❛♥❞✇✐❝❤ ❜❡❛♠ ❛♥❛❧②s✐s✧✱ ✸✾✱ ♣♣✳ ✼✼✸✲✼✼✽✳ ✸✻ ❏✳ ❆♣♣❧✳ ▼❡❝❤✱ ❬✾❪ ❇✳ ▼❡❤r✐✱ ▼✳❆✳ ◆✐❦s✐r❛t✱ ✭✷✵✵✺✮✱ ✧❖♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♣❡r✐♦❞✐❝ s♦❧✉✲ t✐♦♥s ❢♦r ❝❡rt❛✐♥ ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✧✱ ❏✳ ❈♦♠♣✉t✳ ❆♣♣❧✳ ▼❛t❤✱ ✶✼✹✱ ✷✸✾✕✷✹✾✳ ❬✶✵❪ ❨✳ ❆✳ ▼❡❧♥✐❦♦✈✱ ▼✳ ❨ ▼❡❧♥✐❦♦✈ ✭✷✵✶✷✮✱ t✐♦♥ ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❬✶✶❪ ❏✳❏✳ ❙t♦❦❡r✱ ✭✶✾✺✵✮✱ s②st❡♠s✧✱ ●r❡❡♥✬s ❋✉♥❝t✐♦♥s ❈♦♥str✉❝✲ ❉❡ ●r✉②t❡r✳ ✧◆♦♥❧✐♥❡❛r ✈✐❜r❛t✐♦♥ ✐♥ ♠❡❝❤❛♥✐❝❛❧ ❛♥❞ ❡❧❡❝tr✐❝❛❧ ■♥t❡rs❝✐❡♥❝❡ P✉❜❧✐s❤❡r✱ ◆❡✇❨♦r❦✳ ❬✶✷❪ ❊✳ ❖✳ ❚✉❝❦ ❛♥❞ ▲✳ ❲✳ ❙❝❤✇❛rt③✱ ✭✶✾✾✵✮✱ ✧❆ ♥✉♠❡r✐❝❛❧ ❛♥❞ ❛s②♠♣t♦t✐❝ st✉❞② ♦❢ s♦♠❡ t❤✐r❞✲♦r❞❡r ♦r❞✐♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s r❡❧❡✈❛♥t t♦ ❞r❛✐♥✐♥❣ ❛♥❞ ❝♦❛t✐♥❣ ❢❧♦✇s✧✱ ❬✶✸❪ ❊✳ ❩❡✐❞❧❡r ✭✶✾✽✻✮✱ ❙■❆▼ ❘❡✈✱ ✸✷✱ ♣♣✳ ✹✺✸✲✹✻✾✳ ◆♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥✱ ■✿ ❋✐①❡❞ ✲ P♦✐♥t ❚❤❡♦r❡♠s✱ ❙♣✐♥❣❡r✳ ✸✼