Điểm bất động và các phương trình hàm

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚❘❺◆ ❚❍➚ ❉❯◆● ✣■➎▼ ❇❻❚ ✣❐◆● ❱⑨ ❈⑩❈ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✹ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚❘❺◆ ❚❍➚ ❉❯◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P ▼➣ sè ✿ ✻✵✳✹✻✳✵✶✳✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ ❚❙✿ ❍❖⑨◆● ❱❿◆ ❍Ị◆● ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✹ ▼ư❝ ❧ư❝ ▲í✐ ♥â✐ ✤➛✉ ỵ sỡ t ✤ë♥❣ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✶✳✶ ✶✳✷ ✻ ✣■➎▼ ❇❻❚ ✣❐◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✷ ❱➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ❙❒ ❈❻P ❱➋ ✣■➎▼ ❇❻❚ ✣❐◆● ❱⑨ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✷✳✻ ✣✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ b ✶✳✸ ✶✳✹ f (φ (x)) = af (x)+ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ◆●❯❨➊◆ ▲Þ ⑩◆❍ ❳❸ ❈❖ ❇❆◆❆❈❍ ❱➋ ✣■➎▼ ❇❻❚ ✣❐◆● ❱⑨ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✸✳✶ ✶✾ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✸✳✹ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ▲➄P ❱⑨ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✷✹ ✶✳✹✳✶ ▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✹✳✷ ▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✹✳✸ ▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✹✳✺ ✷ ▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✹✳✻ ▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✹✳✼ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✹✳✽ ▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✹✳✾ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✹✳✶✵ ▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✹✳✶✶ ▼➺♥❤ ✤➲✭ ❜➔✐ t♦→♥ ✶✶✹ ❬✷❪✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✹✳✶✷ ✷✽ ▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tr ổ ▼❡tr✐❝ s✉② rë♥❣ ✈➔ sü ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ữỡ tr ị ❳❸ ❈❖ ❇❆◆❆❈❍ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼❊❚✲ ❘■❈ ❙❯❨ ❘❐◆● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ❇❛♥❛❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣ ✳ ✸✶ ✷✳✶✳✸ ▼➺♥❤ ✤➲ ✭①❡♠ ❙✳✲▼ ❏✉♥❣ ❛♥❞ ❩✳✲❍ ▲❡❡ ❬✻❪✮✳ ✸✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❙Ü ✃◆ ✣➚◆❍ ◆●❍■➏▼ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❆❯❈❍❨✳ ✷✳✸ ❙Ü ✃◆ ✣➚◆❍ ◆●❍■➏▼ ❈Õ❆ ▼❐❚ ▲❰P ❈⑩❈ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❉❸◆● ❈❆❯❈❍❨✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✾ ✷✳✸✳✶ ✣à♥❤ ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✸✳✷ ❍➺ q✉↔ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✸✳✸ ❱➼ ❞ö →♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✸✳✹ ▼➺♥❤ ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ❑➳t ❧✉➟♥ ✹✻ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✽ ✷ ▲í✐ ♥â✐ ✤➛✉ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♠ët ❧➽♥❤ ✈ü❝ ❦❤â tr♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ♥➙♥❣ ❝❛♦ ❝õ❛ t♦→♥ ❝➜♣✳ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ r➜t ✤❛ ❞↕♥❣ ✈➔ t❤÷í♥❣ ♠❛♥❣ t➼♥❤ ✤➦❝ t❤ị✱ ♥❣❤➽❛ ❧➔ ❝❤ó♥❣ ♣❤ö t❤✉ë❝ ♥❤✐➲✉ ✈➔♦ ❣✐↔ t❤✐➳t ❝õ❛ tø♥❣ ❜➔✐ t♦→♥ t rt õ ợ ữỡ tr ❤➔♠ t❤❡♦ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐✳ ▲í✐ ❣✐↔✐ ❝õ❛ ♠ët ❜➔✐ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ t❤÷í♥❣ ✤á✐ ❤ä✐ ♥❤✐➲✉ ❦ÿ ♥➠♥❣ ✈➔ ❦✐➳♥ t❤ù❝ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❤å❝ s✐♥❤✿ ❦ÿ ♥➠♥❣ ❜✐➳♥ ✤ê✐✱ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ❤➔♠ sè✱ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♠ët sè ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝ì ❜↔♥✱✳✳✳ ❍✐➺♥ ❝â ♥❤✐➲✉ t➔✐ ❧✐➺✉ ❝❤✉②➯♥ ❦❤↔♦ ✈➲ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ♥❤÷♥❣ tr♦♥❣ ❤➛✉ ❤➳t ❝→❝ t➔✐ ❧✐➺✉ ✤â ❝â t❤➸ t❤➜② r➡♥❣ sè ❧÷đ♥❣ ❝→❝ ✈➼ ❞ư ♠✐♥❤ ❤å❛ ❝❤♦ ♠é✐ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❧➔ r➜t ➼t✳ ✣✐➲✉ ♥➔② ❝â t❤➸ ❣✐↔✐ t ỵ tự t õ q ✈➼ ❞ư ❝❤♦ ✈✐➺❝ ù♥❣ ❞ư♥❣ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ♥➔♦ ✤â ❝â t❤➸ ❧➔♠ ❝❤♦ ♥❣÷í✐ ✤å❝ ♥❤➔♠ ❝❤→♥✭❝❤➥♥❣ ❤↕♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ f (φ(x)) = a(x)f (x) + b(x), a(x), b(x) ❧➔ ❝→❝ ❤➔♠ trữợ f tr õ (x) ởt ✤➣ ❝❤♦ ❝â ❝❤✉ ❦ý ❧➦♣✱ ❧➔ ❤➔♠ ❝➛♥ t➻♠✮❀ t❤ù ❤❛✐✱ ♥➳✉ ❝â ♠ët ✈➼ ❞ö ♥➔♦ ✤â t❤ü❝ sü ❦❤ỉ♥❣ ❣➙② r❛ ♥❤➔♠ ❝❤→♥ t❤➻ t❤÷í♥❣ ❧í✐ ❣✐↔✐ ❝õ❛ ♥â ❧➔ ♠ët tê ❤đ♣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ✈➔ ①➳♣ ❧í✐ ❣✐↔✐ ✈➼ ❞ư ♥➔② ✈➔♦ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❝ö t❤➸ ♥➔♦ ✤â t❤✐➳✉ sù❝ t❤✉②➳t ♣❤ö❝✳ ❚r♦♥❣ ❝→❝ ữỡ tr õ ởt ợ ữỡ tr ❦❤→ ❤➭♣✱ ❝➠♥ ❝ù tr➯♥ ❝→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝õ❛ ❝→❝ t➔✐ ❧✐➺✉ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✮ ♠➔ õ ỹ sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët →♥❤ ①↕ ♥➔♦ ✤â✳ ❈❤ó♥❣ tỉ✐ ❣å✐ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧♦↕✐ ♥➔② ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❈❤♦ x∗ ∈ X X, Y ❧➔ ❝→❝ t➟♣ ❝â t➼♥❤ ❝❤➜t X ∩Y = ∅ ❣å✐ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ♥➳✉ ✈➔ f :X →Y f (x∗ ) = x∗ ❧➔ ♠ët →♥❤ ①↕✳ ✣✐➸♠ ❇↔♥ ❧✉➟♥ ✈➠♥ ✏✣✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✑ t➟♣ ❤đ♣ ❝→❝ ✈➼ ❞ư ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ♠➔ ❧í✐ ❣✐↔✐ ❝õ❛ ♥â ❝â ❞ò♥❣ ✤➳♥ ❝→❝ t➼♥❤ ❝❤➜t ❦❤→❝ ♥❤❛✉ ❝õ❛ t➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët ✸ →♥❤ f õ ỗ ▲í✐ ♥â✐ ✤➛✉✱ ❤❛✐ ❝❤÷ì♥❣✱ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ t➔✐ t ì ị P ❱➋ ✣■➎▼ ❇❻❚ ✣❐◆● ❱⑨ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ❱➋ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ ✤✐➸♠ ❜➜t ởt số ỵ sỡ t ỵ tr ổ ❣✐❛♥ ♠❡tr✐❝ ✈➔ ♠ët ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ ❬✶❪✳ ❚r♦♥❣ ♠ö❝ ✶✳✷✱ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ✤÷đ❝ ✈➟♥ ❞ư♥❣ ✤➸ t➻♠ ❝→❝ ❤➔♠ ❝â t❤➸ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤÷đ❝ ①➨t✱ ♥❣❤✐➺♠ t❤ü❝ sü ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤÷đ❝ t➻♠ ❜➡♥❣ ❝→❝❤ t❤û trü❝ t✐➳♣ ❝→❝ ❤➔♠ ❦❤↔ ❞➽ ❧➔ ♥❣❤✐➺♠ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤➣ ❝❤♦✳ ▼ët sè tr♦♥❣ ❝→❝ ✈➼ ❞ö ♥➔② ❧➔ ❝→❝ ❜➔✐ t♦→♥ tr♦♥❣ ❝→❝ ❦ý t❤✐ ❖❧②♠♣✐❝ ❚♦→♥ q✉è❝ t➳ ■▼❖✱ ✤➣ trð t❤➔♥❤ ❝→❝ ✈➼ ❞ö ❦✐♥❤ ✤✐➸♥ ❝❤♦ ✈✐➺❝ ù♥❣ ❞ö♥❣ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ♥❤✐➲✉ t➔✐ ❧✐➺✉ ✭ ❝❤➥♥❣ ❤↕♥ ❬✷❪✮✳ ▼ët sè ❝→❝ ✈➼ t tỹ s t ữợ sỹ ữợ ũ tr ỵ tr ổ tr ự ỵ ♠ët sè ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr tữớ ữủ t tr ợ ❝â t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t ✭ ✈➼ ❞ư ❧ỵ♣ ❝→❝ ❤➔♠ ❜à ❝❤➦♥✱ ❧ỵ♣ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝✱ ✳✳✳✮✳ ❈→❝ ❧ỵ♣ ❤➔♠ ♥➔② ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ✱ ❝á♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤÷đ❝ ①➨t ✤÷đ❝ ✈✐➳t ❧↕✐ ữợ õ f t T (T f )(x) = f (x), tr♦♥❣ ❧➔ →♥❤ ①↕ ❝♦ ❝❤➦t tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ t÷ì♥❣ ù♥❣✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr♦♥❣ ♠ư❝ ♥➔② ✤➲✉ ❞✉② ♥❤➜t ♥❣❤✐➺♠✳ ▼ö❝ ✶✳✹ tr➻♥❤ ❜➔② ♠ët ❦➳t q✉↔ ❝õ❛ t→❝ ❣✐↔ ❍♦➔♥❣ ❱➠♥ ❍ò♥❣ tr♦♥❣ ❬✶❪✱ ❦➳t q✉↔ ♥➔② ❝❤♦ ♣❤➨♣ ❦❤➥♥❣ ✤à♥❤ sü ✈æ ♥❣❤✐➺♠ ❝õ❛ ♠ët sè ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞ü❛ tr➯♥ ❝➜✉ tró❝ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❧➦♣ ❝õ❛ ♠ët →♥❤ ①↕ ❣ ♥➔♦ ✤â✳ ❈→❝ ✈➼ ❞ö ❝õ❛ ♠ö❝ ♥➔② ❧➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ①✉➜t ❤✐➺♥ ❝↔ ð tr♦♥❣ ✤↕✐ số t t t ì ị ⑩◆❍ ❳❸ ❈❖ ❇❆◆❆❈❍ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼❊❚❘■❈ ❙❯❨ ❘❐◆● ❱⑨ ❙Ü ✃◆ ✣➚◆❍ ◆●❍■➏▼ ❈Õ❆ ❈⑩❈ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ữỡ tr ỵ tr ổ tr s rở ỵ ♥➔② ❧➔ ❝ì sð ✤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔♦ ✈✐➺❝ ①➨t sü ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤②✳ ▼ư❝ ✷✳✷ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ✹ ❝õ❛ ❈✳P❛r❦ ✈➔ ❚❤✳▼ ❘❛ss✐❛s ✈➲ sü ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ❈→❝ ❦➳t q✉↔ ♥➔② tê♥❣ q✉→t ❦➳t q✉↔ ❝õ❛ ❍②❡rs ❬✹❪ ữủ ự ỵ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣✳ ✣➸ ❝❤➾ r❛ →♣ ❞ö♥❣ ❝õ❛ ❦➳t q✉↔ ✈➔♦ ❧➽♥❤ ✈ü❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝➜♣✱ t→❝ ❣✐↔ ❞➝♥ r❛ ❤❛✐ ✈➼ ❞ö✱ ♠ët ✈➼ ❞ö ❧➜② tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ✈➼ ❞ö ❦❤→❝ t→❝ ❣✐↔ tü s→♥❣ t→❝✳ ▼ö❝ ✷✳✸ tr➻♥❤ ❜➔② ♠ët ❦➳t q✉↔ ❝õ❛ ❙♦♦♥✲▼♦ ❏✉♥❣ ✈➔ ❙❡✉♥❣✇♦♦❦ ▼✐♥ ❬✼❪ ✈➲ sü ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ởt ợ ữỡ tr ự t q ụ ỹ tr ỵ ❝♦ ❇❛♥❛❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣✱ tù❝ ❧➔ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ⑩♣ ❞ư♥❣ ❝õ❛ ❝→❝ ❦➳t q✉↔ ♥➔② ✈➔♦ ❧➽♥❤ ✈ü❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝➜♣ ❧➔ ❝→❝ ❦➳t ❧✉➟♥ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ f (x + y) = Af (x) + Bf (y), tr♦♥❣ ✤â A, B ❧➔ ❝→❝ số t ỗ ữủ t ữợ sỹ ữợ ❞➝♥ ❝õ❛ ❚✳❙ ❍♦➔♥❣ ❱➠♥ ❍ị♥❣✱ ❱✐➺♥ ❑❤♦❛ ❤å❝ ❈ì ❜↔♥ ✕ ✣↕✐ ❤å❝ ❍➔♥❣ ❍↔✐ ❱✐➺t ◆❛♠✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ t❤➔② ữợ t t t ổ tở ❚♦→♥ ✕ ❚✐♥✱ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❝ơ♥❣ ♥❤÷ t↕♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❝❤÷ì♥❣ tr➻♥❤ ❝❛♦ ❤å❝ ✈➔ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✵ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✸ ◆❣÷í✐ t r ữỡ ỵ ❝➜♣ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✶✳✶ ✣■➎▼ ❇❻❚ ✣❐◆● ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛ ❈❤♦ X, Y ❧➔ ❝→❝ t➟♣ ❝â t➼♥❤ ❝❤➜t X ∩ Y = ∅ ✈➔ f : X → Y ❧➔ ♠ët →♥❤ ①↕✳ ✣✐➸♠ x∗ ∈ X ❣å✐ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ♥➳✉ f (x∗ ) = x∗ ❚➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ỵ F ix(f ) ❱➼ ❞ö ✶✮ ⑩♥❤ ①↕ ✷✮ ⑩♥❤ ①↕ f :R→R ❝❤♦ ❜ð✐ g : R → [−1; 1] f (x) = x3 ❝❤♦ ❜ð✐ ❝â ✸ ✤✐➸♠ ❜➜t ✤ë♥❣✱ g(x) = sinx F ix(f ) = {−1, 0, 1} ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ x∗ = 0, F ix (g) = {0} ✸✮ ⑩♥❤ ①↕ h:R→R ❝❤♦ ❜ð✐ h(x) = x + ❦❤æ♥❣ ❝â ✤✐➸♠ ❜➜t ✤ë♥❣✱ F ix(h) = ∅ ✶✳✷ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ❙❒ ❈❻P ❱➋ ✣■➎▼ ❇❻❚ ✣❐◆● ❱⑨ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼✳ ỵ tử tứ ✤â♥❣ ❬❛❀❜❪ ✈➔♦ ❝❤➼♥❤ ♥â ❝â ➼t ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❢ ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ tø ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳ ❱➻ [a; b] f (a) , f (b) ∈ [a; b] ✈➔♦ ❝❤➼♥❤ ♥â✳ ✣➦t ♥➯♥ g(x) = f (x) − x ❑❤✐ ✤â g(x) g (a) g (b) = (f (a) − a) (f (b) − b) ≤ ✻ ❱➟② ♣❤÷ì♥❣ tr➻♥❤ f (x∗ ) = x∗ ✶✳✷✳✷ ❉♦ ✤â ❢ g(x) = ❝â ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠ x∗ ∈ [a; b]✱ tù❝ g(x∗ ) = ❤❛② ❝â t t ởt t ỵ ❢ ❧➔ ❤➔♠ t❤ü❝ sü ❣✐↔♠ tr➯♥ t➟♣ sè t❤ü❝ X t❤➻ ❢ ❦❤æ♥❣ ❝â q✉→ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ tr➯♥ X ✐✐✮ ◆➳✉ ❤➔♠ f (x) x t❤ü❝ sü ✤ì♥ ✤✐➺✉ tr➯♥ t➟♣ sè t❤ü❝ ❳ t❤➻ ❢ ❝â ❦❤æ♥❣ q✉→ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ tr➯♥ X ❈❤ù♥❣ ♠✐♥❤ ✐✮ ❍➔♠ ❣✭①✮ ❂ ❢✭①✮ ✕ ① ❦❤æ♥❣ q✉→ ♠ët ❧➛♥ ❦❤✐ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ◆➳✉ x ∈ X.❉♦ g(X) ✤â ♥➳✉ g(X) ✶✳✷✳✸ ❢ g(x) = ❝õ❛ ♥â ❦❤æ♥❣ q✉→ ♠ët ❧➛♥ tr➯♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr➯♥ X, ♥➳✉ g(X) ♥➯♥ ❣ s➩ ♥❤➟♥ ♠é✐ ❣✐→ trà ❝õ❛ t➟♣ g(X) ❝❤ù❛ ❣✐→ trà ✵ t❤➻ ✐✐✮ ❉♦ t➼♥❤ ✤ì♥ ✤✐➺✉ t❤ü❝ sü✱ ❤➔♠ ❣✐→ trà X t❤ü❝ sü ❣✐↔♠ tr➯♥ ❦❤æ♥❣ ❝❤ù❛ ❣✐→ trà ✵ t❤➻ ❢ g(X) ❦❤æ♥❣ ❝â ❝â ✤ó♥❣ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ f (x) x ,x X ∈X ◆➳✉ ❦❤æ♥❣ ❝❤ù❛ ✶ t❤➻ ❢ ♥❤➟♥ ♠é✐ ❣✐→ trà t❤✉ë❝ ♠✐➲♥ g(X) ❝❤ù❛ ✶ t❤➻ ❢ ❝â ✤ó♥❣ ♠ët ❦❤ỉ♥❣ ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ tr➯♥ X ỵ sỷ F (u) ởt tỹ (x, y, s, t) trữợ ❝õ❛ ✹ ❜✐➳♥ ①✱②✱s✱t ①→❝ ✤à♥❤ tr➯♥ t➟♣ ❞↕♥❣ X × X × R × R ✭ ❳ ❧➔ t➟♣ ❝♦♥ ❝õ❛ t➟♣ sè t❤ü❝ R✮✳ ◆➳✉ ❤➔♠ F (u) ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t u∗ t❤➻ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✿ F (φ (x, y, f (x) , f (y))) = φ (y, x, f (y) , f (x)) (x, y ∈ X ) ✭✶✳✶✮ ✭tr♦♥❣ ✤â ❢ ❧➔ ❤➔♠ ♠ët ❜✐➳♥ ❝➛♥ t➻♠ ❝â t➟♣ ①→❝ ✤à♥❤ ❧➔ X ✮ ♣❤↔✐ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤✿ φ (x, x, f (x) , f (x)) = u∗ y=x∈X t❛ ♥❤➟♥ ✤÷đ❝✿ F (φ (x, x, f (x) , f (x))) = φ (x, x, f (x) , f (x)) (∀x ∈ X) ✭✶✳✷✮ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ f (x) ✣➥♥❣ t❤ù❝ ✭✶✳✷✮ ❝❤ù♥❣ tä ❉♦ F ❧➔ ❤➔♠ t❤ä❛ ♠➣♥ ✭✶✳✶✮ t❤➻ ✤➦t φ (x, x, f (x) , f (x)) ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ F ❝❤➾ ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ u∗ ♥➯♥ t❛ ♣❤↔✐ ❝â✿ φ (x, x, f (x) , f (x)) = u∗ (∀x ∈ X) ✼ ✈ỵ✐ ♠å✐ x ∈ X ỵ ởt ố ợ ữỡ tr ỵ ữỡ tr t❤÷í♥❣ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❣✐→ trà ❝õ❛ φ, F (u) = u ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ u∗ tr♦♥❣ ♠ët ♠✐➲♥ ♥➔♦ ✤â ❝❤ù❛ ♠✐➲♥ s❛✉ ✤â ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ✈➔ t❤û ❝→❝ ♥❣❤✐➺♠ t➻♠ ✤÷đ❝ tø ✭✶✳✷✮ ✈➔♦ ✭✶✳✶✮✳ ❈→❝ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✷✮ t❤ä❛ ♠➣♥ ✭✶✳✶✮ s➩ ❧➔ t➜t ❝↔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✶✮✳ ❱➼ ❞ư ✶✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ f f (x + f (y)) = f (x) + y (∀x, y ∈ R) ●✐↔✐✳ ✣➦t tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ❜➔✐ t♦→♥ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♥❤➜t ❝õ❛ f tr➯♥ R s✉② r❛ ♠å✐ ♥❣❤✐➺♠ ❤❛② f (x) = c − x f y = x = t❛ ❝â f (f (0)) = f (0) ❱➟② f (0) ứ ỵ s r r ỵ f f (0) ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② φ (x, y, f (x) , f (y)) = x + f (y) ❝õ❛ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ ♣❤↔✐ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ f (x) = f (0) − x ❞↕♥❣ R, t❤ü❝ sü ❣✐↔♠ ✈➔ t❤ä❛ ①→❝ ✤à♥❤ tr➯♥ t➟♣ sè t❤ü❝ ✣➦t c ✭ f (0) = c, t❛ x + f (x) = f (0) t❛ s✉② r❛ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ ♣❤↔✐ ❝â ❧➔ ❤➡♥❣ sè t❤ü❝✮✳ ❚❤❛② f (x) = c − x ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ❜➔✐ t♦→♥ t❛ ✤÷ì❝✿ c − (x + c − y) = c − x + y → c = ❱➟② f (x) = −x ♣❤÷ì♥❣ tr➻♥❤ ❢ (x f (x) = −x ❘ã r➔♥❣ ❤➔♠ + f (y)) = f (x) + y ❧➔ t❤ü❝ sü ❣✐↔♠ tr➯♥ ❉♦ ✤â ❤➔♠ f (x) = −x R ✈➔ t❤ä❛ ♠➣♥ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ✤➣ ❝❤♦✳ ❱➼ ❞ö ✷ ✭✶✾✽✸✱■▼❖✮✿ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ ✐✮ ✐✐✮ f (xf (y)) = yf (x) x, y t❤ä❛ ♠➣♥ ✿ ❞÷ì♥❣✳ lim f (x) = x→+∞ ●✐↔✐✳ ✣➦t y=x=1 ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❉♦ ✈ỵ✐ ♠å✐ f : (0, +∞) → (0; +∞) f ▲➜② tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ✐✮ t❛ ✤÷đ❝ x=1 f (1) ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✈➔ y = f (1), x∗ > ♥❤➟♥ ✤÷đ❝ f (1) = ◆➳✉ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f (x∗ )2 = (x∗ )2 (x∗ )n tø ✐✮ t❛ ♥❤➟♥ ✤÷đ❝ t❤➻ ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦ t❛ ❝→❝ ❣✐→ trà ❞÷ì♥❣ ♥➯♥ ✤➥♥❣ t❤ù❝ ♥➔② ❝❤♦ ◆➳✉ f (f (1)) = f (1) ❱➟② (x∗ )2 f ❱➟② u∗ f ❧➔ ♠ët f (f (f (1))) = f (1)2 ✳ ❝❤➾ ♥❤➟♥ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t❤➻ ✤➦t tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ✐✮ t❤➻ ✤➦t tr♦♥❣ ✐✮ f y = x = x∗ ❝ô♥❣ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f (1) f (1) = f (1)2 ❱➻ f t❛ f x = x∗ , y = (x∗ )n f (x∗ )n+1 = (x∗ )n+1 ❚❤❡♦ ỵ q t s r (x )n ợ ♠å✐ ❱➟② t❛ ❝â ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f n ♥❣✉②➯♥ ❞÷ì♥❣✳ ❉♦ ✤â ♥➳✉ x∗ > t❛ s✉② r❛ lim (x∗ )n = +∞, lim f ((x∗ )n ) = n→∞ ✽ n→∞ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♠❡tr✐❝ s✉② rë♥❣ d, ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✺✮ ❝â ♥❣❤➽❛ ❧➔✿ d(fm , f ) ≤ ε (∀m ≥ n0 ) ❱➟② f {fn } ❧➔ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② t tũ ỵ {fn } tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣ t❛ ❦➳t ❧✉➟♥ r➡♥❣ (E, d) (E, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣ ✤➛② ✤õ✳ ✷✳✷ ❙Ü ✃◆ ✣➚◆❍ ◆●❍■➏▼ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❆❯❈❍❨✳ ❱➜♥ ✤➲ ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ỗ tứ t ●✐↔ sû sè ε, (G1 , ∗) ❧➔ ♠ët ♥❤â♠ tỗ t ổ (G1 , , d) = δ (ε) > ❧➔ ♠ët ♥❤â♠ ♠❡tr✐❝ ✈ỵ✐ ♠❡tr✐❝ s❛♦ ❝❤♦ ♥➳✉ h : G1 → G2 d(., ) ❈❤♦ ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝✿ d(h(x ∗ y), h(x) ◦ h(y)) < δ (∀x, y G1 ) t tỗ t ởt ỗ H : G1 → G2 t❤ä❛ ♠➣♥✿ d(h(x), H(x)) < ε (∀x ∈ G1 )? ❍②❡rs ❬✹❪ ✤➣ ❝❤♦ ❝➙✉ tr↔ ❧í✐ ❦❤➥♥❣ ✤à♥❤ ❝õ❛ ❜➔✐ t♦→♥ ❙✳▼✳❯❧❛♠ tr♦♥❣ tr÷í♥❣ ❤đ♣ G1 , G2 ❈❤♦ ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈ư t❤➸✿ (X, X ) ✈➔ (Y, Y ) ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ●✐↔ sû ε>0 ✈➔ f :X →Y t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝✿ f (x + y) − f (x) f (y) Y õ tỗ t ❞✉② ♥❤➜t ♠ët ❤➔♠ ❝ë♥❣ t➼♥❤ f (x) − T (x) Y ≤ ε (∀x, y ∈ X) T :X→Y ✭✷✳✻✮ s❛♦ ❝❤♦✿ ≤ ε (∀x ∈ X) ❚❤✳▼ ❘❛ss✐❛s ✭ ❬✽❪✮ ✤➣ tê♥❣ q✉→t ❦➳t q✉↔ ❝õ❛ ❍②❡rs ❦❤✐ t❤❛② ✭✷✳✻✮ ❜ð✐ ✤✐➲✉ ❦✐➺♥✿ f (x + y) − f (x) − f (y) ≤ ε( x tr♦♥❣ ✤â t➼♥❤ ε>0 L:X→Y ✈➔ ≤ p < 1✳ p X + y P X) (∀x, y ∈ X) ✭✷✳✼✮ t tữỡ ự tỗ t t ởt ❤➔♠ ❝ë♥❣ s❛♦ ❝❤♦ ✿ f (x) − L(x) Y ≤ 2ε x − 2p ✸✹ p X (∀x ∈ X) ✭✷✳✽✮ ◆❣♦➔✐ r❛✱ ♥➳✉ ✈ỵ✐ ♠é✐ x∈X ❤➔♠ f (tx) ❧✐➯♥ tö❝ t❤❡♦ ❜✐➳♥ t∈R t❤➻ L ❧➔ R t t ữợ t tr ❜➔② ♠ët ❦➳t q✉↔ ❝õ❛ ❈✳P❛r❦ ✈➔ ❚❤✳ ▼✳ ❘❛ss✐❛s tê♥❣ q✉→t ❤ì♥ ❦➳t q✉↔ tr➯♥✱ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ỹ tr ỵ t ①↕ ❝♦ ❇❛♥❛❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣ ✭ ỵ t q Pr sss õ ỗ ỳ sè ❇❛♥❛❝❤✳ ❱➻ ❝❤õ ✤➲ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤② ♥➯♥ t→❝ ❣✐↔ ❝❤➾ ①➨t ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ỵ ữợ ởt t q t õ tr ỵ Pr ✈➔ ❚❤✳▼ ❘❛ss✐❛s ❬✸❪✮ ❈❤♦ tr➯♥ tr÷í♥❣ sè K ✭K = R ❤♦➦❝ K = C ✮ ✈➔ (Y, ❳ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì ) ❧➔ ♠ët ❦❤ỉ♥❣ tr K sỷ tỗ t số φ : X × X → [0; +∞) t❤ä❛ ♠➣♥✿ φ(2j x, 2j y) = (∀x, y ∈ X) j→∞ 2j lim ✭✷✳✾✮ f (x + y) − f (x) − f (y) ≤ φ(x, y) (∀x, y ∈ X) tỗ t số L < s ❝❤♦ φ(x, x) ≤ 2Lφ( x2 , x2 ) ✈ỵ✐ x X t tỗ t t ởt ❤➔♠ ❝ë♥❣ t➼♥❤ H : X → Y s❛♦ ❝❤♦✿ f (x) − H(x) ≤ φ(x, x) − 2L (∀x ∈ X) ✭✷✳✶✶✮ ❍ì♥ ♥ú❛✱ ♥➳✉ ✈ỵ✐ ♠é✐ x ∈ X ❝è ✤à♥❤✱ ❤➔♠ f (tx) (t ∈ R) ❜à ❝❤➦♥ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛ ✵ t❤➻ H ❧➔ →♥❤ ①↕ R ✲ t✉②➳♥ t➼♥❤✳ ◆â✐ r✐➯♥❣✱ H ❧➔ →♥❤ ①↕ R ✲ t✉②➳♥ t➼♥❤ ♥➳✉ f (tx) (t ∈ R) ❧✐➯♥ tö❝ t↕✐ t = ✈ỵ✐ ♠é✐ x ❝è ✤à♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t E = {g : X → Y } ❱ỵ✐ h, g tũ ỵ tở d(h, g) = inf {C ∈ [0; +∞] : h(x) − g(x) ≤ Cφ(x, x) ∀x ∈ X} ❚❤❡♦ ♠➺♥❤ ✤➲ ✷✳✶✳✸ t➼♥❤ J :E→E (E, d) ✭✷✳✶✷✮ ❝❤♦ ❜ð✐✿ ❚❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ C≥0 t❛ ✤➦t✿ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣ ✤➛② ✤õ✳ ❳➨t →♥❤ ①↕ t✉②➳♥ (Jg)(x) = g(2x) ✈➔ E J ❧➔ →♥❤ ①↕ ❝♦ ❝❤➦t tr➯♥ số tũ ỵ tọ (Jh)(x) (Jg)(x) = d(h, g) ≤ C (∀x ∈ X) E ❚❤➟t ✈➟②✱ ợ tũ ỵ h, g E t ❝â✿ 1 C (h (2x) − g (2x)) ≤ Cφ(2x, 2x) ≤ 2Lφ(x, x) = LCφ(x, x) 2 ✸✺ ✈ỵ✐ ♠å✐ x ∈ X ❚ø ✭✷✳✶✷✮ t❛ s✉② r❛ d(Jh, Jg) ≤ Ld(h, g) ❱➻ L ❉♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✶✮ t❛ s✉② r❛ ❦❤✐ s❛♦ ❝❤♦ |t| ≤ δ ❝â ❜➜t ✤➥♥❣ t❤ù❝✿ H(tx) ≤ M + ✸✻ φ(x, x) < +∞ − 2L ✭✷✳✶✼✮ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ H(sx) (∀t, s ∈ R), H(θ) = θ H (tx) ❧✐➯♥ tö❝ t❤❡♦ t ∈ R H ((t + s)x) = H(tx) + ❱➻ H ♥➯♥ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ❧✐➯♥ tö❝ t↕✐ t = 0, tù❝ ❧➔ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✿ lim H (tx) = ✭✷✳✶✽✮ t→0 ●✐↔ sû tr→✐ ❧↕✐ r➡♥❣ ✭✷✳✶✽✮ ❦❤ỉ♥❣ ✤ó♥❣✳ ❑❤✐ õ tỗ t số >0 ởt {tk } k=1 s❛♦ ❝❤♦✿ lim tk = 0, tk = (∀k), k→∞ ✣➦t nk = δ |tk | H(tk x) ≥ ε (∀k) {nk }∞ k=1 t❛ ❝â ♠ët ❞➣② ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ lim nk = +∞, |nk tk | ≤ δ k→∞ ✭✷✳✶✾✮ t❤ä❛ ♠➣♥ ❚ø ✭✷✳✶✻✮ ✈➔ ✭✷✳✶✾✮ s✉② r❛✿ lim H(nk tk x) = lim nk H(tk x) = +∞ k→∞ ♥❤÷♥❣ |nk tk | ≤ δ ♥➯♥ ✭✷✳✷✵✮ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ✭✷✳✶✼✮✳ ❱➟② ✤â ❧✐➯♥ tö❝ t↕✐ ♠å✐ ✤✐➸♠ t❤➻ f (tx) t❛ ❝â t✛ t ∈ R ◆➳✉ f (tx) (t ∈ R) H(tx) = tH(x) {qn }∞ n=1 s❛♦ ❝❤♦ H(tx) ❧✐➯♥ tö❝ t↕✐ ♣❤↔✐ ❜à ❝❤➦♥ tr♦♥❣ ♠ët ❧➙♥ ❝➟♥ ♥➔♦ ✤â ❝õ❛ ❧✐➯♥ tö❝ t↕✐ ♠å✐ ✤✐➸♠ ✭✷✳✷✵✮ k→∞ t = 0, ❧✐➯♥ tư❝ t↕✐ t=0 t=0 ✈ỵ✐ ♠é✐ x ✈➔ ❞♦ ❝è ✤à♥❤ ❞♦ ✤â t❛ ❝ô♥❣ ❝â H(tx) (t R ởt số tỹ t tũ ỵ t ❧➔ sè ❤ú✉ t✛ t❤➻ tø ✭✷✳✶✻✮ ✈ỵ✐ ♠å✐ x ∈ X lim qn = t n→∞ ◆➳✉ t số ổ t t tỗ t ởt sè ❤ú✉ ❉ị♥❣ t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ H(tx) t❤❡♦ t t❛ ❝â✿ H (tx) = lim H (qn x) = lim qn H (x) = tH (x) (∀x ∈ X) n→∞ ❱➟② ❤➔♠ R H ❧➔ ❤➔♠ R ✕ t❤✉➛♥ ♥❤➜t✳ ❑➳t ❤đ♣ ✈ỵ✐ t➼♥❤ ❝ë♥❣ t➼♥❤ ❝õ❛ H t❛ s r H t t ỵ ữủ ❝❤ù♥❣ ♠✐♥❤ ❤♦➔♥ t♦➔♥✳ ◆❤➟♥ ①➨t✿ ◆➳✉ ❤➔♠ ❤➔♠ n→∞ φ(x, x) φ : X × X → [0; +∞) t❤ä❛ ♠➣♥ ✭✷✳✾✮ ✈➔ ✤✐➲✉ ❦✐➺♥ ❧➔ ❤➔♠ t❤✉➛♥ ♥❤➜t ❜➟❝ φ(x, x) ≤ 2Lφ( x2 , x2 ) (L < 1) p (p < 1) t❤➻ ❚❤➟t ✈➟②✿ φ(x, x) φ(2j x, 2j x) = lim =0 j→∞ 2j(1−p) j→∞ 2j x x x x x x x x φ(x, x) = φ(2 , ) = 2p φ( , ) = 2.2p−1 φ( , ) = 2.Lφ( , ) (L = 2p−1 < 1) 2 2 2 2 p p ❱➻ ❤➔♠ φ(x, y) = ε x X+ y X (0 ≤ p < 1, ε > 0) ❧➔ ❤➔♠ t❤✉➛♥ ♥❤➜t ❜➟❝ lim p →♥❤ ①↕ X×X ✈➔♦ ❦❤♦↔♥❣ ❚❤✳❘❛ss✐❛s ♥➠♠ ✶✾✼✽✳ ❑❤✐ ✤✐➲✉ ❦✐➺♥ ✈➲ ❤➔♠ φ [0; +∞) p = 0, ỵ tờ qt ỡ t q t õ ữủ tọ ợ (x, y) = L= < ✈➔ φ(x, y) = 12 φ( x2 , y2 ) = ε ✣✐➲✉ ❦✐➺♥ ✭✷✳✶✵✮ trð t❤➔♥❤✿ f (x + y) − f (x) − f (y) ≤ ε ✸✼ ♥➯♥ ✤→♥❤ ❣✐→ ✭✷✳✶✶✮ trð t❤➔♥❤✿ f (x) − H(x) ≤ ε ◆❤÷ ✈➟② ❦➳t q✉↔ ❝õ❛ ❉✳ ❍②❡rs ♥➠♠ ✶✾✹✶ ❝ô♥❣ ❧➔ ♠ët trữớ ủ r ỵ ỵ õ t s r ởt số ✤à♥❤ ❝õ❛ ❝→❝ ❜➔✐ t♦→♥ ❝➜♣✳ ❱➼ ❞ư ✶ ✭ ❇➔✐ t♦→♥ ✺✳✼ ❝❤÷ì♥❣ ■■■ ❬✶✵❪✮✿ ❈❤♦ f ❧➔ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ R t❤ä❛ ♠➣♥✿ |f (x + y) − f (x) − f (y)| ≤ σ ✈ỵ✐ số tỹ R x, y ự r tỗ t↕✐ ❞✉② ♥❤➜t ♠ët ❤➔♠ t✉②➳♥ t➼♥❤ L(x) tr➯♥ s❛♦ ❝❤♦ ❝â ❜✐➸✉ ❞✐➵♥ f (x) = L(x) + ω(x) (x R), r t ỵ ✷✳✷✳✶ ❧➜② trà t✉②➺t ✤è✐✱ K = R ✈➔ φ(x, y) = σ ◆➳✉ f ❧✐➯♥ tư❝ t❤❡♦ t ✈ỵ✐ ♠å✐ sè t❤ü❝ x |ω(x)| ≤ σ (∀x ∈ R) ✈ỵ✐ ❝❤✉➞♥ tr♦♥❣ ❧➔ ❤➔♠ ❧✐➯♥ tư❝ tr➯♥ R ❧➔ ❣✐→ R t❤➻ ❤✐➸♥ ♥❤✐➯♥ f (tx) ❱➟② ♠å✐ ❣✐↔ tt ỵ ữủ tọ |f (x) − L(x)| ≤ σ ω(x) = f (x) − L(x) tr♦♥❣ ✤â X =Y =R ❚❤❡♦ ❦❤➥♥❣ ✤à♥❤ ❝õ❛ ✤à♥❤ ỵ tỗ t t t t ( > 0) L(x) tr➯♥ R t❤ä❛ ♠➣♥✿ (∀x ∈ R ) t❛ ✤÷đ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ◆❤➟♥ ①➨t✿ ▲í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ tr➯♥ tr♦♥❣ s→❝❤ ✤➣ ❞➝♥ ❦❤ỉ♥❣ ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❱➼ ❞ö ✷✿ ❈❤♦ f : (0; +∞) → R ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ t❤ä❛ ♠➣♥✿ |f (xy) − f (x) − f (y)| ≤ |ln x|p + |ln y|p ự r tỗ t ❤➡♥❣ sè C (∀x, y ∈ (0; +∞), ≤ p < 1) s❛♦ ❝❤♦ ❝â ❜✐➸✉ ❞✐➵♥✿ f (x) = C ln x + ω(x) tr♦♥❣ ✤â |ω(x)| ≤ ●✐↔✐✳ ✣➦t 2|ln x|p 2−2p (∀x ∈ (0; +∞) t = ln x, s = ln y ❚ø ❣✐↔ t❤✐➳t ❝õ❛ ❜➔✐ t♦→♥ s✉② r❛✿ f (et+s ) − f (et ) − f (es ) ≤ |t|p + |s|p ✸✽ ✣➦t g(t) = f (et )✱ ❦❤✐ ✤â g ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ tø R ✈➔♦ |g(t + s) − g(t) − g(s)| ≤ |t|p + |s|p ⑩♣ ỵ ợ t t H(t) tứ R r tỗ t số C L(t) tr➯♥ |t|p p 2−2 R ✤➲✉ ❝â ❞↕♥❣ L(t) = Ct ✭ ✈ỵ✐ C = const✮ t❛ s✉② s❛♦ ❝❤♦✿ |g(t) − Ct| ≤ ✣➦t t❛ s✉② r❛ tỗ t t ởt s õ t t❤ù❝✿ |g(t) − H(t)| ≤ ❱➻ ♠å✐ ❤➔♠ t✉②➳♥ t➼♥❤ t❤ä❛ ♠➣♥✿ (∀s, t ∈ R) φ(t, s) = |t|p + |s|p R R α(t) = g(t) − Ct, ω(x) = α(ln x) f (x) = g(ln x) = C ln x + ω(x) ✈➔✿ 2|t|p − 2p t❛ ❝â✿ |ω(x)| ≤ 2|ln x|p 2−2p ✭ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✮ ✷✳✸ ❙Ü ✃◆ ✣➚◆❍ ◆●❍■➏▼ ❈Õ❆ ▼❐❚ ▲❰P ❈⑩❈ Pì ỵ ✈➔ ❙❡✉♥❣✇♦♦❦ ▼✐♥ ❬✼❪✮ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈➨❝ tỡ tr trữớ số K, (Y, K (Y ì Y, 2) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ tr♦♥❣ ✤â ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ tr➯♥ ❧➔ ♠ët ❝❤✉➞♥ t÷ì♥❣ ✤÷ì♥❣ ợ tr Y ì Y õ t t ỗ t ởt số k > s (u, u) − (v, v) ≤k u−v ∀u, v ∈ Y ✭✷✳✷✶✮ ●✐↔ sû ✿ ✐✮ F : Y × Y → Y ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ✈ỵ✐ ❝❤✉➞♥ F ✈➔ t❤ä❛ ♠➣♥✿ F (F (u, u), F (v, v)) = F (F (u, v), F (u, v)), ∀u, v ∈ Y ✭✷✳✷✷✮ ✐✐✮ φ : X × X → [0; +∞) ❧➔ ❤➔♠ sè ❝â t➼♥❤ ❝❤➜t✿ x y φ( , ) ≤ φ(x, y) ∀x, y ∈ X 2 ✭✷✳✷✸✮ ❑❤✐ ✤â✱ ♥➳✉ k F < ✈➔ ❤➔♠ f : X → Y t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝✿ f (x + y) − F (f (x), f (y)) ≤ φ(x, y) ∀x, y X t tỗ t t f ∗ : X → Y ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✿ f (x + y) = F (f (x), f (y)) ✭✷✳✷✺✮ s❛♦ ❝❤♦✿ f (x) − f ∗ (x) ự ỵ ỵ tở E E φ(x, x) ∀x ∈ X 1−k F ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ h : X → Y ✭✷✳✷✻✮ ❱ỵ✐ ❤❛✐ ❤➔♠ h, g tị② t❛ ✤➦t✿ d(h, g) = inf {C ∈ [0; +∞] : h(x) − g(x) ≤ Cφ(x, x) ∀x ∈ X} (E, d) ❚❤❡♦ ♠➺♥❤ ✤➲ ✷✳✸✳✶ ❦❤æ♥❣ ❣✐❛♥ T :E→E ①→❝ ✤à♥❤ ♠ët →♥❤ ①↕ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝✿ x x ,h ) 2 (T h)(x) = F (h ❑❤✐ ✤â t❤ä❛ ♠➣♥ T ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣ ✤➛② ✤õ✳ ❚❛ ∀x ∈ X ❧➔ ♠ët →♥❤ ①↕ ❝♦ ❝❤➦t✳ ❚❤ü❝ ợ tũ ỵ d(g, h) C ✭✷✳✷✼✮ h, g ∈ E ✈➔ C ∈ [0; +∞] t❛ ❝â✿ g(x) − h(x) ≤ Cφ(x, x) ∀x ∈ X ✭✷✳✷✽✮ ❚ø ✭✷✳✷✶✮✱ ✭✷✳✷✸✮✱ ✭✷✳✷✼✮ ✈➔ ✭✷✳✷✽✮ t❛ ❝â✿ (T g) (x) − (T h) (x) = F g ≤ F g ≤ k F Cφ x ,g x x 2, x − h x x x ,g ,h x −F h x ,h ≤ F k g x x −h x ✭✷✳✷✾✮ ≤ k F Cφ (x, x) (∀x ∈ X) ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♠❡tr✐❝ s✉② rë♥❣ d✱ tø ✭✷✳✷✾✮ ✈➔ ❝→❝❤ ❝❤å♥ C s✉② r❛✿ d(T g, T h) ≤ k F d(g, h) ❱➻ k F s❛♦ ❝❤♦✿ (u, u) − (v, v) ≤k u−v ∀u, v ∈ Y ●✐↔ sû F : Y × Y → Y ❧➔ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ ✈ỵ✐ ❝❤✉➞♥ F ✈➔ t❤ä❛ ♠➣♥✿ F (F (u, u), F (v, v)) = F (F (u, v), F (u, v)), ∀u, v ∈ Y ❑❤✐ ✤â✱ ♥➳✉ k F < ✈➔ ❤➔♠ f : X → Y t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝✿ f (x + y) − F (f (x), f (y)) ≤ θ( x ✹✷ p + y p ) ∀x, y ∈ X ✭✷✳✸✹✮ tr♦♥❣ ✤â θ ✈➔ p ❧➔ ❝→❝ sè ❦❤æ♥❣ t tỗ t t f : X → Y t❤ä❛ ♠➣♥✿ f ∗ (x + y) = F (f ∗ (x), f ∗ (y)) s❛♦ ❝❤♦✿ 2θ x p f (x) − f (x) ≤ 1−k F ∗ ✷✳✸✳✸ ✭✷✳✸✺✮ ❱➼ ❞ö →♣ ❞ö♥❣ ▲➜② X = Y = R ✈➔ ①❡♠ X, Y Y × Y = R2 ❚r➯♥ ∀x ∈ X ❑❤✐ ✤â t❛ õ ữ ổ ợ tr t✉②➺t ✤è✐✳ t❛ tr❛♥❣ ❜à ❝❤✉➞♥ ♠❛①✿ (x, x) − (y, y) ❳➨t →♥❤ ①↕ t✉②➳♥ t➼♥❤ (x, y) = max {|x| , |y|} (∀(x, y) ∈ R2 ) = |x − y| ❱➟② ✤✐➲✉ ❦✐➺♥ ✭✷✳✷✶✮ ữủ tọ ợ k = F : R2 → R ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝✿ F (x, y) = A.x + B.y (∀x, y ∈ R) tr♦♥❣ ✤â A, B ❧➔ ❝→❝ ❤➡♥❣ sè t❤ü❝ t❤ä❛ ♠➣♥ |A| + |B| < |F (x, y)| = |A.x + B.y| ≤ (|A| + |B|)max {|x| , |y|} ✈➔ ❚❛ ❝â✿ |A| + |B| = A.sign(A) + B.sign(B) tr♦♥❣ ✤â sign(x) = ♥➳✉ x > 0✱ sign(x) = −1 ♥➳✉ x < 0✳ ❚ø ✤â ❞➵ ❞➔♥❣ s✉② r❛✿ F = sup {|F (x, y)| = |A.x + B.y| : x, y ∈ R&max {|x| , |y|} ≤ 1} = |A| + |B| ❞♦ ✤â k F = |A| + |B| < ▼➦t ❦❤→❝✿ F (F (u, u), F (v, v)) = A(A + B)u + B(A + B)v = A(Au + Bv) + B(Au + Bv) = F (F (u, v), F (u, v)) ❱➟② →♥❤ ①↕ t✉②➳♥ t➼♥❤ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✷✳✷✷✮✳ ◆❤÷ ✈➟② ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➦t ❧➯♥ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ X, Y ✈➔ →♥❤ t t F tr ỵ ữủ tọ ♠➣♥✳ ✣✐➲✉ ♥➔② ❝❤♦ ♣❤➨♣ t❛ ♣❤→t ❜✐➸✉ ♠➺♥❤ ✤➲✿ ✷✳✸✳✹ ▼➺♥❤ ✤➲ ●✐↔ sû ψ : R2 → [0; +∞) ❧➔ ❤➔♠ ❤❛✐ ❜✐➳♥ t❤ü❝ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✷✳✷✸✮✳ ❚❛ ♥â✐ ❤➔♠ f : R → R ❧➔ t ổ ỡ (x, x) tỗ t↕✐ ❤➡♥❣ sè ❞÷ì♥❣ M = M (f ) s❛♦ ❝❤♦ |f (x)| ≤ M ψ(x, x), (∀x ∈ R)✳ ❑❤✐ ✤â ♥➳✉ A, B ❧➔ ❝→❝ ❤➡♥❣ sè ✹✸ t❤ü❝ t❤ä❛ ♠➣♥ |A| + |B| < ✈➔ F (u, v) = Au + Bv (∀u, v ∈ R) t❤➻ ❤➔♠ f ∗ (x) ≡ ❧➔ ♥❣❤✐➺♠ ❞✉② t ữỡ tr tr ợ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ R ✈➔ t➠♥❣ ❦❤æ♥❣ ♥❤❛♥❤ ❤ì♥ ❤➔♠ ψ(x, x) ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû f (x) ①→❝ ✤à♥❤ tr➯♥ R, t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤✿ f (x + y) = A.f (x) + B.f (y) (∀x, y R) tỗ t số (x, y) ✭✷✳✸✻✮ M > ✵ s❛♦ ❝❤♦ |f (x)| ≤ M ψ(x, x) (∀x ∈ R)✳ ✣➦t X = Y = R,❝❤✉➞♥ = max {|x| , |y|} (∀(x, y) ∈ R2 ) ✱ φ(x, y) = (1 − |A| − |B|)M ψ(x, y), f ∗ (x) ≡ rã r➔♥❣ t❤ä❛ ♠➣♥✿ |f ∗ (x + y) − A.f ∗ (x) − B.f ∗ (y)| = ≤ φ(x, y) (x, y R) ỵ t s✉② r❛ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✸✻✮ tr♦♥❣ ❧ỵ♣ ❝→❝ ❤➔♠ h(x) ①→❝ ✤à♥❤ tr➯♥ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝✿ R |f ∗ (x) − h(x)| = |h(x)| ≤ ❱➻ ❝↔ f (x) ✈➔ f ∗ (x) ≡ φ(x, x) = M ψ(x, x) (∀x ∈ R) − |A| − |B| ✤➲✉ t❤ä❛ ♠➣♥ ✭✷✳✸✼✮ ♥➯♥ f (x) = f ∗ (x) ≡ ◆❤➟♥ ①➨t✿ ỹ r ữỡ tr ợ tt t tữớ f ∗ (x) ≡ ✭✷✳✸✼✮ |A| + |B| < tr♦♥❣ t➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ R ❝❤➾ ❝â ♥❣❤✐➺♠ ❙ü ❦✐➺♥ ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ✤ì♥ ❣✐↔♥ ♥❤÷ s❛✉✿ ✣➦t tr♦♥❣ ✭✷✳✸✻✮ ✭✷✳✸✻✮ y=0 x=y=0 t❛ ✤÷đ❝ t❛ s✉② r❛ (1 − A − B)f (0) = → f (0) = f (x) = A.f (x) (∀x ∈ R) ❙✉② r❛ ✣➦t tr♦♥❣ (1 − A)f (x) = (∀x ∈ R) ❱➻ |A| < ♥➯♥ tø ✤➙② s✉② r❛ f (x) = (∀x ∈ R) ❚❤û trü❝ t✐➳♣ t❤➜② ❤➔♠ f ∗ (x) ≡ t❤ä❛ ♠➣♥ ✭✷✳✸✻✮ ♥➯♥ t❛ s✉② r❛ ❦❤➥♥❣ ✤à♥❤ ❝õ❛ ♥❤➟♥ ①➨t ❈❤♦ X, Y ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì tr➯♥ tr÷í♥❣ sè t➼♥❤ tø ❦❤æ♥❣ ❣✐❛♥ ✐✮ A+B ✐✐✮ ❤♦➦❝ Y K ◆➳✉ A, B ❧➔ ❝→❝ →♥❤ ①↕ t✉②➳♥ ✈➔♦ ❝❤➼♥❤ ♥â ❝â t➼♥❤ ❝❤➜t✿ ❦❤æ♥❣ ❝â ❣✐→ trà r✐➯♥❣ ✶ A✱ ❤♦➦❝ B ❦❤ỉ♥❣ ❝â ❣✐→ trà r✐➯♥❣ ✶ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✿ f (x + y) = A(f (x)) + B(f (y)) ❝❤➾ ❝â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t f ≡θ ✭✷✳✸✽✮ tr♦♥❣ ❧ỵ♣ t➜t ❝↔ ❝→❝ →♥❤ ①↕ tø x=y=θ tr♦♥❣ ✭✷✳✸✽✮ t❛ ❝â ✈➔♦ Y (A + B)f (θ) = f (θ) ❦❤æ♥❣ ❝â ❣✐→ trà r✐➯♥❣ ✶ ♥➯♥ tø ✤➥♥❣ t❤ù❝ ♥➔② t❛ s✉② r❛ ✹✹ X f (θ) = θ ❱➻ A+B ❱➻ ✈❛✐ trá ❝õ❛ A, B, x, y ✭✷✳✸✽✮ ❝â ♥❤÷ ♥❤❛✉ tr♦♥❣ ✭✷✳✸✽✮ ♥➯♥ ❝â t❤➸ ❝♦✐ y=θ t❛ ❝â A ❦❤æ♥❣ ❝â ❣✐→ trà r✐➯♥❣ ✶✱ ✤➦t tr♦♥❣ A(f (x)) = f (x) (∀x ∈ X) ❱➻ A ❦❤æ♥❣ ❝â ❣✐→ trà r✐➯♥❣ ✶ ♥➯♥ t❛ ♣❤↔✐ f (x) = θ (∀x ∈ X) ✹✺ ❑➳t ❧✉➟♥ ❇↔♥ ❧✉➟♥ ✈➠♥ ✏✣✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✑ t➟♣ ❤đ♣ ❝→❝ ✈➼ ❞ư ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ♠➔ ❧í✐ ❣✐↔✐ ❝õ❛ ♥â ❝â ❞ò♥❣ ✤➳♥ ❝→❝ t➼♥❤ ❝❤➜t ❦❤→❝ ♥❤❛✉ ❝õ❛ t➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët →♥❤ ①↕ f ♥➔♦ ✤â✳ ❈❤÷ì♥❣ ■ ❝õ❛ ❜↔♥ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② t ởt số ỵ sỡ t ỵ ①↕ ❝♦ ❇❛♥❛❝❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ❝ơ♥❣ ♥❤÷ ♠ët ❦➳t q✉↔ tr♦♥❣ ❜➔✐ ❜→♦ ❬✶❪✳ ❚→❝ ❣✐↔ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❦❤➥♥❣ ✤à♥❤ ♠❛♥❣ t➼♥❤ ❝❤➜t ❦ÿ t❤✉➟t ✤➸ t✐➺♥ sû ❞ư♥❣ tr♦♥❣ ❧í✐ ❣✐↔✐ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr ỵ ✶✳✷ tr➻♥❤ ❜➔② ❝→❝ ✈➼ ❞ư ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ú õ sỷ sỹ tỗ t↕✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët →♥❤ ①↕ ♥➔♦ ✤â✳ ▼ët sè tr♦♥❣ ❝→❝ ✈➼ ❞ö ♥➔② ❧➔ ❝→❝ ❜➔✐ t♦→♥ tr♦♥❣ ❝→❝ ❦ý t❤✐ ❖❧②♠♣✐❝ ❚♦→♥ q✉è❝ t➳ ■▼❖✱ ✤➣ trð t❤➔♥❤ ❝→❝ ✈➼ ❞ö ❦✐♥❤ ✤✐➸♥ ❝❤♦ ✈✐➺❝ ù♥❣ ❞ư♥❣ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ♥❤✐➲✉ t➔✐ ❧✐➺✉✳ ▼ët sè ❝→❝ t tỹ s t ữợ sỹ ữợ ũ tr ỵ tr ổ tr ự ỵ ❣✐↔✐ ♠ët sè ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ tr tữớ ữủ t tr ợ ❤➔♠ ❝â t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t ✭ ✈➼ ❞ư ❧ỵ♣ ❝→❝ ❤➔♠ ❜à ❝❤➦♥✱ ❧ỵ♣ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝✱ ✳✳✳✮✳ ❈→❝ ❧ỵ♣ ❤➔♠ ♥➔② ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ ✱ ❝á♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤÷đ❝ ①➨t ✤÷đ❝ t ữợ õ f t ✈➔ T (T f )(x) = f (x), tr♦♥❣ ❧➔ →♥❤ ①↕ ❝♦ ❝❤➦t tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ t÷ì♥❣ ù♥❣✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr♦♥❣ ♠ư❝ ♥➔② ✤➲✉ ❞✉② ♥❤➜t ♥❣❤✐➺♠✳ ▼ö❝ ✶✳✹ tr➻♥❤ ❜➔② ♠ët ❦➳t q✉↔ ❝õ❛ t→❝ ❣✐↔ ❍♦➔♥❣ ❱➠♥ ❍ò♥❣ tr♦♥❣ ❬✶❪✱ ❦➳t q✉↔ ♥➔② ❝❤♦ ♣❤➨♣ ❦❤➥♥❣ ✤à♥❤ sü ✈æ ♥❣❤✐➺♠ ❝õ❛ ♠ët sè ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞ü❛ tr➯♥ ❝➜✉ tró❝ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❧➦♣ ❝õ❛ ♠ët →♥❤ ①↕ ❣ ♥➔♦ ✤â✳ ❈→❝ ✈➼ ❞ö ❝õ❛ ♠ö❝ ♥➔② ❧➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ①✉➜t ❤✐➺♥ ❝↔ ð tr♦♥❣ ✤↕✐ sè t✉②➳♥ t➼♥❤ ❧➝♥ ❣✐↔✐ t➼❝❤✳ ✹✻ ❈❤÷ì♥❣ tr ỵ tr ổ tr s rở ỵ ỡ sð ✤➸ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔♦ ✈✐➺❝ ①➨t sü ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤②✳ ▼ö❝ ✷✳✷ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝õ❛ ❈✳P❛r❦ ✈➔ ❚❤✳▼ ❘❛ss✐❛s ✈➲ sü ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ❈→❝ ❦➳t q✉↔ ♥➔② tê♥❣ q✉→t ❦➳t q✉↔ ❝õ❛ ❍②❡rs ❬✹❪ ✈➔ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ỵ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣✳ ✣➸ ❝❤➾ r❛ →♣ ❞ư♥❣ ❝õ❛ ❦➳t q✉↔ ✈➔♦ ❧➽♥❤ ✈ü❝ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝➜♣✱ t→❝ ❣✐↔ ❞➝♥ r❛ ❤❛✐ ✈➼ ❞ư✱ ♠ët ✈➼ ❞ö ❧➜② tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ✈➼ ❞ö ❦❤→❝ t→❝ ❣✐↔ tü s→♥❣ t→❝✳ ▼ö❝ ✷✳✸ tr➻♥❤ ❜➔② ♠ët ❦➳t q✉↔ ❝õ❛ ❙♦♦♥✲▼♦ ❏✉♥❣ ✈➔ ❙❡✉♥❣✇♦♦❦ ▼✐♥ ❬✼❪ ✈➲ sü ê♥ ✤à♥❤ ♥❣❤✐➺♠ ❝õ❛ ♠ët ❧ỵ♣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤②✳ ❈❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ ♥➔② ụ ỹ tr ỵ tr ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣✱ tù❝ ❧➔ →♣ ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❑➳t q✉↔ ♥➔② ✤÷đ❝ →♣ ❞ư♥❣ ✈➔♦ ❧➽♥❤ ✈ü❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝➜♣ ✈➔ ❝❤♦ ❝→❝ ❦➳t ❧✉➟♥ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ f (x + y) = Af (x) + Bf (y), tr♦♥❣ ✤â A, B ❧➔ ❝→❝ ❤➡♥❣ sè✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❜↔♥ ❧✉➟♥ ✈➠♥ ✤➣ ❝❤ù♥❣ tä ❦❤→✐ ♥✐➺♠ ✤✐➸♠ ❜➜t ✤ë♥❣✱ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ụ ữ sỹ tỗ t t ởt ❧ỵ♣ ❝→❝ →♥❤ ①↕ ♥➔♦ ✤â ❧➔ r➜t ❤ú✉ ➼❝❤ tr♦♥❣ ✈✐➺❝ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✈➔ ❜➜t ♣❤÷ì♥❣ tr t ỵ ❉②✱ ◆❣✉②➵♥ ❱➠♥ ◆❤♦✱ ❱ô ❱➠♥ ❚❤♦↔✳ ❚✉②➸♥ t➟♣ ✷✵✵ ❜➔✐ t❤✐ ❱æ ✤à❝❤ t♦→♥✳ ❚➟♣ ✸✿ ●✐↔✐ t➼❝❤✳ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ư❝ ✷✵✵✷✳ ❬✷❪ ❍♦➔♥❣ ❱➠♥ ❍ị♥❣✳ ◆❤➟♥ ①➨t ✈➲ ❝→❝ →♥❤ ①↕ ❣✐❛♦ ❤♦→♥ tr➯♥ ♠ët t➟♣ tý ỵ ồổ ố ✶✽ ✭ ✻✴✷✵✵✾✮✱ tr ✾✵✲✾✸✳ ❬✸❪ ❇✳▼✳▼❛❦❛r♦✈✱ ▼✳●✳●♦❧✉③✐♥❛✱ ❆✳❆✳▲♦❞❦✐♥✱ ❆✳◆✳P♦❞❦♦r②t♦✈✳ ❈→❝ ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ✈➲ ❣✐↔✐ t➼❝❤ t❤ü❝✳ ▼♦s❦✈❛✱ ♥❤➔ ①✉➜t ❜↔♥ ✧ ❑❤♦❛ ❤å❝✧✱ ✶✾✾✷ ✭ ❚✐➳♥❣ ◆❣❛✮✳ ❬✹❪ ❈❤♦♦♥❦✐❧ P❛r❦ ✱ ❚❤❡♠✐st♦❝❧❡s ▼✳ ❘❛ss✐❛s✳ ❋✐①❡❞ ♣♦✐♥ts ❛♥❞ st❛❜✐❧✐t② ♦❢ t❤❡ ❈❛✉❝❤② ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥✳ ❚❤❡ ❆✉str❛❧✐❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐✲ ❝❛t✐♦♥s✱ ✈♦❧✉♠❡ ✻✱ ■ss✉❡ ■✱ ❛rt✐❝❧❡ ✶✹✱✶✲✾✱✷✵✵✾✳ ❬✺❪ ❉✳❍✳ ❍②❡rs✳ ❖♥ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥✱ Pr♦❝✳ ◆❛t✳ ❆❝❛❞✳ ❙❝✐✳ ❯❙❆✳ ✷✼ ✭✶✾✹✶✮✱✷✷✷✲✷✷✹✳ ❬✻❪ ❉✳❍ ❍②❡rs✱ ●✳■s❛❝✱ ❛♥❞ ❚❤✳▼ ❘❛ss✐❛s✳ ❚♦♣✐❝s ✐♥ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛✲ t✐♦♥s✱ ❲♦r❧❞ ❙❝✐❡♥t✐❢✐①✱ ❘✐✈❡r ❊❞❣❡✱ ◆❏✱ ❯❙❆✱ ✶✾✾✼✳ ❬✼❪ ❚❤✳▼ ❘❛ss✐❛s✳ ❖♥ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ ❧✐♥❡❛r ♠❛♣♣✐♥❣ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✱ Pr♦✳❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✼✷ ✭✶✾✼✽✮✱✷✾✼✲✸✵✵✳ ❬✽❪ ❙✳▼✳❯❧❛♠✳ ❆ ❝♦❧❧❡❝t✐♦♥ ♦❢ t❤❡ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❜❧❡♠s✱ ■♥t❡rs❝✐❡♥❝❡ P✉❜❧✳ ◆❡✇ ❨♦r❦✱ ✶✾✻✵✳ ❬✾❪ ❙✳✲▼ ❏✉♥❣ ❛♥❞ ❩✳✲❍ ▲❡❡✳ ❆ ❢✐①❡❞ ♣♦✐♥t ❛♣♣r♦❛❝❤ t♦ t❤❡ st❛❜✐❧✐t② ♦❢ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ ✇✐t❤ ✐♥✈♦❧✉t✐♦♥✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱✈♦❧✳ ✷✵✵✽✳ ❬✶✵❪ ❙♦♦♥✲▼♦ ❏✉♥❣ ✈➔ ❙❡✉♥❣✇♦♦❦ ▼✐♥ ✳ ❆ ❢✐①❡❞ ♣♦✐♥t ❛♣♣r♦❛❝❤ t♦ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ ❢✭ ① ✰ ②✮ ❂ ❋✭❢✭①✮✱❢✭②✮✮✳ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✉♠❡ ✷✵✵✾✳ ✹✽