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Điểm bất động và các phương trình hàm ( Luận văn thạc sĩ)Điểm bất động và các phương trình hàm ( Luận văn thạc sĩ)Điểm bất động và các phương trình hàm ( Luận văn thạc sĩ)Điểm bất động và các phương trình hàm ( Luận văn thạc sĩ)Điểm bất động và các phương trình hàm ( Luận văn thạc sĩ)
✣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 ✈➔ ❝→❝ ❜➔✐ 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 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✸✳✹ ✣à♥❤ 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❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣ ❈❛✉❝❤② ✷✳✶ ✸✵ ◆●❯❨➊◆ ▲Þ ⑩◆❍ ❳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◆● ❈❆❯❈❍❨✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ 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❤ò✱ ♥❣❤➽❛ ❧➔ ❝❤ó♥❣ ♣❤ư t❤✉ë❝ ♥❤✐➲✉ ✈➔♦ ❣✐↔ t❤✐➳t ❝õ❛ tø♥❣ ❜➔✐ t♦→♥ ❝ö 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♥❣ ♣❤→♣ ❝ư t❤➸ ♥➔♦ ✤â t❤✐➳✉ sù❝ t❤✉②➳t ♣❤ư❝✳ ❚r♦♥❣ ❝→❝ ♣❤÷ì♥❣ tr õ ởt ợ ữỡ tr ❤➭♣✱ ❝➠♥ ❝ù tr➯♥ ❝→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝õ❛ ❝→❝ t➔✐ ❧✐➺✉ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✮ ♠➔ ❧í✐ 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 ✸ →♥❤ ①↕ f õ ỗ ♥â✐ ✤➛✉✱ ❤❛✐ ❝❤÷ì♥❣✱ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t ì ị P ✣■➎▼ ❇❻❚ ✣❐◆● ❱⑨ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ❱➋ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛ ✤✐➸♠ ❜➜t ✤ë♥❣✱ 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 ❝→❝ ✈➼ ❞ö t tỹ s t ữợ sỹ ữợ ❞➝♥ ❝õ❛ ❚✳❙ ❍♦➔♥❣ ❱➠♥ ❍ò♥❣✳ ▼ư❝ ✶✳✸ tr➻♥❤ ❜➔② ỵ tr ổ tr ự ỵ ởt sè ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr♦♥❣ tữớ ữủ t tr ợ õ t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t ✭ ✈➼ ❞ư ❧ỵ♣ ❝→❝ ❤➔♠ ❜à ❝❤➦♥✱ ❧ỵ♣ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝✱ ✳✳✳✮✳ ❈→❝ ❧ỵ♣ ❤➔♠ ♥➔② ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ✱ ❝á♥ ữỡ tr ữủ t ữủ t ữợ ❞↕♥❣ ✤â 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 ✈ỉ ♥❣❤✐➺♠ ❝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 t t ì ị ❳❸ ❈❖ ❇❆◆❆❈❍ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼❊❚❘■❈ ❙❯❨ ❘❐◆● ❱⑨ ❙Ü ✃◆ ✣➚◆❍ ◆●❍■➏▼ ❈Õ❆ ❈⑩❈ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❉❸◆● ữỡ tr ỵ tr ổ tr s rở ỵ ❧➔ ❝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❛ ♠ët 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❛ ❚✳❙ ❍♦➔♥❣ ❱➠♥ ❍ò♥❣✱ ❱✐➺♥ ❑❤♦❛ ❤å❝ ❈ì ❜↔♥ ✕ ✣↕✐ ❤å❝ ❍➔♥❣ ❍↔✐ ❱✐➺t ◆❛♠✳ ❚→❝ ❣✐↔ ①✐♥ tọ ỏ t ỡ t tợ t ữợ ❞➝♥ ✈➔ t➟♣ t❤➸ ❝→❝ t❤➔② ❝æ t❤✉ë❝ ❦❤♦❛ ❚♦→♥ ✕ ❚✐♥✱ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❝ơ♥❣ ♥❤÷ t↕♦ ❝→❝ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❝❤÷ì♥❣ tr➻♥❤ ❝❛♦ ❤å❝ ✈➔ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✵ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✸ ◆❣÷í✐ ✈✐➳t r ữỡ ỵ sỡ ❝➜♣ ✈➲ ✤✐➸♠ ❜➜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 õ ❬❛❀❜❪ ✈➔♦ ❝❤➼♥❤ ♥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 ✣à♥❤ ỵ sỷ F (u) ởt tỹ (x, y, s, t) trữợ ✹ ❜✐➳♥ 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 ✶✳✷✳✸ ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ 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❛ ✭✶✳✶✮✳ ❱➼ ❞ö ✶✳ ❚➻♠ 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 ❚❤❡♦ ♥❣✉②➯♥ ỵ q t s r (x )n ợ ❱➟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→∞ ... ✣➦t tr♦♥❣ ✭✶✳✸✮ ② ❂ ① t❛ s✉② r❛✿ F ( (x, x, f (x) , f (x))) = φ (x, x, f (x) , f (x)) ( x ∈ X) ❱➟② ❤➔♠ φ(x, x, f (x), f (x)) ∈ F ix(F ) g(x) = φ(x, x, f (x), f (x)) ✈ỵ✐ ∀x ∈ X ❧✐➯♥ tö❝ tr➯♥ ♠ët... 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... ♣❤↔✐ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤✿ φ (x, x, f (x) , f (x)) = u∗ y=x∈X t❛ ♥❤➟♥ ✤÷đ❝✿ F ( (x, x, f (x) , f (x))) = φ (x, x, f (x) , f (x)) ( x ∈ X) ✭✶✳✷✮ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ f (x) ✣➥♥❣ t❤ù❝ ✭✶✳✷✮ ❝❤ù♥❣ tä