Về một số định lý điểm bất động cho một vài lớp ánh xạ co trên không gian đều

✶ ▼Ư❈ ▲Ư❈ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❑❤æ♥❣ ❣✐❛♥ ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✷ ✹ ✹ ✻ ởt số ỵ t ❧ỵ♣ →♥❤ ①↕ ❝♦ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ✶✹ ✷✳✶✳ ỵ t ố ợ Φ− ❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ✤è✐ ✈ỵ✐ ❝→❝ →♥❤ ①↕ ❝♦ ❦✐➸✉ ▼❡✐r✲❑❡❡❧❡r tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷ ▼Ð ✣❺❯ ❑❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ✤÷đ❝ ①➙② ❞ü♥❣ ❜ð✐ ❆✳ ❲❡✐❧ ✈➔♦ ♥➠♠ ✶✾✸✽✳ ❑❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ❧➔ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ q✉❛♥ trå♥❣✱ ♥â ❧➔ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ tê♥❣ q✉→t ❝õ❛ ♥❤✐➲✉ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ q✉❡♥ t❤✉ë❝ ♥❤÷ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ✳✳✳◆❤ú♥❣ ❦➳t q✉↔ q trồ tr ợ ổ ữủ ự ❜ð✐ ❏✳ ❑❡❧❧②✱ ◆✳ ❇♦✉r❜❛❦✐✱ ❆✳ ▲✳ ❘♦❜❡rts♦♥ ✭①❡♠ ❬✽❪✮✳ ✣✐➲✉ t❤ó ✈à ❧➔ tỉ♣ỉ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ✤÷đ❝ s✐♥❤ ❜ð✐ ❤å ❝→❝ ❣✐↔ ♠➯tr✐❝✱ ✤➙② ❧➔ ❝ì sð q✉❛♥ trå♥❣ ✤➸ ♥❣÷í✐ t❛ ①➙② ❞ü♥❣ ♥❤✐➲✉ t➼♥❤ ❝❤➜t tữỡ tỹ ữ ổ tr ợ ổ ỵ t ố ợ ❝→❝ →♥❤ ①↕ ❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ ❧➔ ♠ët ❦➳t q✉↔ q✉❛♥ trå♥❣ ❝õ❛ t♦→♥ ❤å❝✳ ❙❛✉ ữủ ự ỵ trð t❤➔♥❤ ♠ët tr♦♥❣ ♥❤ú♥❣ ✈➜♥ ✤➲ t❤✉ ❤ót ✤÷đ❝ r➜t ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ❈→❝ ỵ t ố ợ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ♣❤♦♥❣ ♣❤ó ❝❤♦ ♥❤✐➲✉ ❦✐➸✉ →♥❤ ①↕✱ tr ổ ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ rë♥❣ r➣✐ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❝õ❛ t♦→♥ ❤å❝ ♥❤÷ ●✐↔✐ t➼❝❤✱ P❤÷ì♥❣ tr➻♥❤ ✈✐ t➼❝❤ ♣❤➙♥✳✳✳ ▼ët ✈➜♥ ✤➲ tü ♥❤✐➯♥ s❛✉ ❦❤✐ ①✉➜t ❤✐➺♥ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ỵ ợ ổ ự tr õ ỵ ❦❤✐ ❝→❝ →♣ ❞ö♥❣ ❝õ❛ ♥â tr↔✐ rë♥❣ tr➯♥ ♥❤✐➲✉ ợ ổ q trồ t ữ ợ ổ ỗ ữỡ ổ ♠➯tr✐❝✳ ◆❣÷í✐ ✤➛✉ t✐➯♥ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ tr➯♥ ❧➔ ■✳ ❆✳ ❘✉s ✈ỵ✐ ❝→❝ ❦➳t q✉↔ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ỗ ữỡ ữợ ự ữủ ởt số ♥❤➔ t♦→♥ ❤å❝ t❤ü❝ ❤✐➺♥ ✈➔ ✤↕t ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔ q✉❛♥ ✸ trå♥❣ ✈➔♦ t❤➟♣ ♥✐➯♥ ✽✵ ✈➔ ✾✵ t trữợ ỳ ữớ t ữủ t q tốt t tr ữợ ự ❆♥❣❡❧♦✈✱ ❏✳ ▼❛t❦♦✇s❦✐✱✳✳✳✭①❡♠ ❬✷✱ ✸✱ ✹❪✮✳ ✣➦❝ ❜✐➺t ❝→❝ ✤à♥❤ ỵ t ợ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ✤➣ ❝❤♦ ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ ♥❤✐➲✉ ✈➜♥ ✤➲ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ t➼❝❤ ♣❤➙♥✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳✳✳ ❈→❝ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ ✤à♥❤ ỵ t ợ s✉② rë♥❣ tr➯♥ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ✈➔ ù♥❣ ❞ư♥❣ tú tữỡ ố ợ tr ữợ ợ t ổ ởt t q ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ♠ët sè ❧ỵ♣ →♥❤ ①↕ ❝♦ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ✈➔ ù♥❣ ❞ư♥❣✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ t s ởt số ỵ ❜➜t ✤ë♥❣ ❝❤♦ ♠ët ✈➔✐ ❧ỵ♣ →♥❤ ①↕ ❝♦ tr➯♥ ❦❤æ♥❣ ✤➲✉✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ❝❤♦ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤ ❧➔✿ ♥✐➺♠✱ ✈➼ ❞ö ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉✱ ♠ët ỵ t ①↕ ❝♦ ♣❤✐ t✉②➳♥ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ✈➔ ✤÷❛ r ởt ỵ t ①↕ ❝♦ ❦✐➸✉ ▼❡✐r✲❑❡❡❧❡r tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉✳ ▲✉➟♥ ✈➠♥ ữủ tỹ t trữớ ữợ sỹ ữợ t Pữỡ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ t❤➛②✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ❤å❝✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ❈✉è✐ ❝ị♥❣ ①✐♥ ỡ ỗ t ❧➔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ ❈❛♦ ❤å❝ ✶✼ ●✐↔✐ t➼❝❤ ✤➣ ❝ë♥❣ t→❝✱ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦t ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ t❤✐➳✉ sât✳ ❈❤ó♥❣ tỉ✐ r➜t ữủ ỳ ỵ õ õ t❤➛②✱ ❝ỉ ❣✐→♦ ✈➔ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❱✐♥❤✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶ ❚→❝ ❣✐↔ ✹ ❈❍×❒◆● ✶ ❑❍➷◆● ●■❆◆ ✣➋❯ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❝➛♥ ❞ị♥❣ ✈➲ s❛✉✳ ◆❤ú♥❣ ♥ë✐ ❞✉♥❣ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② tø ❬✶❪✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X = ∅✱ ❤å τ ❝→❝ t➟♣ ❤đ♣ ❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ♠ët tæ♣æ tr➯♥ X ♥➳✉ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✶✮ ∅, X ∈ τ ❀ ✷✮ ◆➳✉ U1 , U2 ∈ τ t❤➻ U1 ∩ U2 ∈ τ ❀ ✸✮ ❱ỵ✐ ♠å✐ ❤å {Uα }α∈I ⊂ τ ✤➲✉ ❝â ∪α∈I Uα ∈ τ ✳ ❑❤✐ ✤â (X, τ ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ ♠é✐ U ∈ τ ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ♠ð✱ t➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ✤â♥❣ ♥➳✉ X \ A ❧➔ ♠ð✳ ❈❤♦ (X, τ ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ x ∈ X ✳ ▼é✐ t➟♣ ❤đ♣ V ✤÷đ❝ ❣å✐ ❧➔ ❧➙♥ X tỗ t t U s ❝❤♦ x ∈ U ⊂ V ✳ ❚❛ ❣å✐ Vx ❧➔ ❤å t➜t ❝↔ ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ x✳ ❑❤✐ ✤â✱ Vx ❝â t➼♥❤ ❝❤➜t s❛✉✿ N1 ✮ V ∈ Vx ⇒ x ∈ V ; N2 ✮ V ∈ Vx ✈➔ V ⊂ U ⇒ U ∈ Vx ❀ N3 ✮ U, V ∈ Vx ⇒ U ∩ V ∈ Vx ❀ N4 ✮ V ∈ Vx ⇒ ∃W ⊆ V s❛♦ ❝❤♦ x ∈ W ✈➔ V ∈ Vy ✱ ✈ỵ✐ ♠å✐ y ∈ W ❍å Wx ⊂ Ux ✤÷đ❝ ❣å✐ ❧➔ ❝ì sð ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ x ợ V Vx tỗ t W ∈ Wx s❛♦ ❝❤♦ W ⊂ V ✳ ✣à♥❤ ỵ s r ộ x X ❝â ❤å ❝♦♥ Ux t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ tr➯♥ t tr X tỗ t tổổ tữỡ t ợ Ux ✱ tù❝ ❧➔ Ux ✺ ❧➔ ❤å ❝→❝ ❧➙♥ ❝➟♥ t x ỵ sỷ X = ✈➔ ✈ỵ✐ ♠é✐ x ∈ X ❝â ❝→❝ t➟♣ ❝♦♥ Ux t÷ì♥❣ ù♥❣ t❤♦↔ ♠➣♥ N1, N2, N3 ✈➔ N4✳ ❑❤✐ ✤â ❤å t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ D ⊆ X s❛♦ ❝❤♦ D ∈ Ux ✈ỵ✐ x ♥➔♦ ✤â ❧➟♣ ♥➯♥ ♠ët tỉ♣ỉ τ tr➯♥ X ✳ ✣è✐ ✈ỵ✐ tæ♣æ τ ✱ ♠é✐ U ∈ Ux ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ x✳ ✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ❝ị♥❣ ✈ỵ✐ ♠ët tỉ♣ỉ tr➯♥ ✤â s t ổ ữợ ❧➔ ❧✐➯♥ tư❝✳ ❚➟♣ ❝♦♥ U tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì X ữủ U U ợ ♠å✐ α ∈ K ✈➔ |α| < 1❀ t➟♣ U ữủ út ợ x X tỗ t > s x U ✈ỵ✐ ♠å✐ |α| < δ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ ổ tỗ t ỡ s U t ỗ t út ợ U U tỗ t V U s V + V ⊂ U ✳ ✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X = d : X ì X R ữủ ❣å✐ ❧➔ ♠ët ♠➯tr✐❝ tr➯♥ X ♥➳✉ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✶✮ d(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X ✱ d(x, y) = ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x = y ❀ ✷✮ d(x, y) = d(y, x) ✈ỵ✐ ♠å✐ x, y ∈ X ❀ ✸✮ d(x, y) d(x, z) + d(z, y) ✈ỵ✐ ♠å✐ x, y, z ∈ X ✳ ❑❤✐ ✤â (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❚➟♣ U tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ữủ ợ a U tỗ t r > s B(a, r) := {x ∈ X : d(x, a) < r} ⊂ U ❚❛ ✤➣ ❜✐➳t τ = {U ⊂ X : U ♠ð} ❧➔ ♠ët tæ♣æ tr➯♥ X ✳ ✻ ✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ✤➲✉ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉✳ ◆❤ú♥❣ ❦➳t q✉↔ ❝õ❛ ♠ư❝ ♥➔② ❝ì ❜↔♥ ✤÷đ❝ tr➻♥❤ ❜➔② tø ❬✶❪✱ ❬✽❪✳ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ t➟♣ ❤đ♣ ❦❤ỉ♥❣ trè♥❣✳ ✶✮ ▼é✐ t➟♣ ❤đ♣ ❝♦♥ U = {(x, y) : x, y ∈ X} ⊂ X × X ✤÷đ❝ ❣å✐ ❧➔ ♠ët q✉❛♥ ❤➺ ✭q✉❛♥ ❤➺ ❤❛✐ ♥❣ỉ✐✮ tr➯♥ X ✳ ✷✮ ❱ỵ✐ ♠é✐ q✉❛♥ ❤➺ U tr➯♥ X ✱ U −1 = {(x, y) ∈ X × X : (y, x) ⊂ U } ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥ ❤➺ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ U ✳ ✸✮ ◗✉❛♥ ❤➺ ❤ñ♣ t❤➔♥❤ ❝õ❛ ❤❛✐ q✉❛♥ ❤➺ U ✱ V ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ U ◦ V = {(x, z) : ∃y ∈ X, (x, y) ∈ U, (y, z) ∈ V } ❚❛ ✈✐➳t U t❤❛② ❝❤♦ U ◦ U ✳ ✹✮ ◗✉❛♥ ❤➺ ∆(X) = {(x, x) : x X} ữủ ữớ ợ ♠é✐ q✉❛♥ ❤➺ U ✈➔ t➟♣ ❤ñ♣ A ⊂ X ✱ ✤➦t U [A] = {y : (x, y) ∈ U ✈ỵ✐ x ♥➔♦ ✤â t❤✉ë❝ A}✳ ❚❛ ✈✐➳t U [x] t❤❛② ❝❤♦ U [{x}]✳ ✶✳✷✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❍å U t X ì X ữủ ♠ët ❝→✐ ✤➲✉ ❤❛② ❝➜✉ tró❝ ✤➲✉ tr➯♥ X ✱ ♥➳✉ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✮ ∆(X) ⊂ U ✈ỵ✐ ♠å✐ U ⊂ U ❀ ✷✮ ◆➳✉ U ∈ U t❤➻ U −1 ⊂ U ❀ ✸✮ ◆➳✉ U U t tỗ t V U s ❝❤♦ V ◦ V ⊂ U ❀ ✹✮ ◆➳✉ U, V ∈ U t❤➻ U ∩ V ∈ U; ✺✮ ◆➳✉ U ∈ U ✈➔ U ⊂ V ⊂ X × X t❤➻ V ⊂ U ✳ ❈➦♣ (X, U ✮ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉✳ ❈→❝ t➼♥❤ ❝❤➜t ✹✮ ✈➔ ✺✮ ❧➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❧å❝✳ ❚r♦♥❣ t❤ü❝ ❤➔♥❤ ♥❣÷í✐ t❛ ❝â t❤➸ t❤❛② ✹✮ ✈➔ ✺✮ ❜ð✐ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ♠ët ❝ì sð ❧å❝✱ tự ợ ộ U, V U tỗ t↕✐ W ∈ U s❛♦ ❝❤♦ W ⊆ U ∩ V ✳ ▼ët ❝ì sð ❝õ❛ ❝➜✉ tró❝ ✤➲✉ tr➯♥ X ❧➔ ♠ët ❝ì sð ❧å❝ tr♦♥❣ X × X t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✶✮✱ ✷✮ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ tr ợ U U tỗ t V ∈ U s❛♦ ❝❤♦ V ⊂ U −1 ✳ ❚➟♣ ❝♦♥ A ∈ U ✤÷đ❝ ❣å✐ ❧➔ ✤è✐ ①ù♥❣ ♥➳✉ A = A−1 ✳ ❘ã r➔♥❣ ❝→❝ t➟♣ ❝â ❞↕♥❣ A ∩ A−1 ❧✉æ♥ ✤è✐ ①ù♥❣✳ ❉♦ ✤â✱ ♠é✐ ❝➜✉ tró❝ ✤➲✉ ❧✉ỉ♥ ❝â ♠ët ❝ì sð ✤è✐ ①ù♥❣✳ ▼é✐ U ∈ U ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✈ị♥❣ ❧➙♥ ❝➟♥ ✤è✐ ①ù♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉ X ✳ ◆➳✉ Y ⊂ X ✱ t❤➻ tr➯♥ Y ❝â ❝➜✉ tró❝ ✤➲✉ ❝↔♠ s✐♥❤ UY = {U ∩ (Y × Y ) : U ∈ U} ✶✳✷✳✸ ❱➼ ❞ö✳ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❑❤✐ ✤â✱ ❤å U = {Un : n = 1, 2, }✱ tr♦♥❣ ✤â Un = {(x, y) ∈ X × X : d(x, y) < } n ❧➔ ❝→✐ ✤➲✉ tr➯♥ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❦✐➸♠ tr❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❝➜✉ tró❝ ✤➲✉ ✤è✐ ✈ỵ✐ U ✳ ❱➻ d(x, x) = ✈ỵ✐ ♠å✐ x ∈ X ♥➯♥ ∆(X) ⊂ Un ✈ỵ✐ ♠å✐ n✳ ❱ỵ✐ ♠å✐ Un ∈ U tø d(x, y) = d(y, x) ✈ỵ✐ ♠å✐ x, y ∈ X s✉② r❛ Un−1 ∈ U ✳ ❱ỵ✐ ♠å✐ U = Un ∈ U ✱ ①➨t } 2n ❑❤✐ ✤â✱ ợ (x, y) V V tỗ t z ∈ X s❛♦ ❝❤♦ (x, z) ∈ V ✈➔ V = U2n = {(x, y) ∈ X × X : d(x, y) < (z, y) ∈ V ✳ ❑❤✐ ✤â d(x, z) < 1 ✈➔ d(z, y) < 2n 2n ❙✉② r❛ 1 + = 2n 2n n ❱➟② (x, y) ∈ U ✱ ❤❛② V ◦ V ⊂ U ✳ ◆➳✉ U, V ∈ U t❤➻ U = Um ✈➔ V = Un ✳ d(x, y) ❚❛ ❝â t❤➸ ❣✐↔ t❤✐➳t m d(x, z) + d(z, y) < n✱ ❞♦ ✤â U ∩ V = Um ∈ U ✽ ❱➟② U ❧➔ ❝➜✉ tró❝ ✤➲✉ tr➯♥ X ✱ ❤❛② ♠å✐ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉✳ ✶✳✷✳✹ ❱➼ ❞ư✳ ▼å✐ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ V ❧➔ ❝ì sð ❧➙♥ ❝➟♥ ❝➙♥ t↕✐ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ X ✳ ❱ỵ✐ ♠é✐ V ∈ V ✱ ✤➦t UV = {(x, y) ∈ X × X : x − y ∈ V } ✈➔ U = {UV }V ∈V ❚❛ ❝❤ù♥❣ ♠✐♥❤ U ❧➔ ❝➜✉ tró❝ ✤➲✉ tr➯♥ X ✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ (x, x) ∈ ∆(X)✱ ✈➻ = x − x ∈ UV ✈ỵ✐ ♠å✐ V ∈ V ♥➯♥ ∆(X) ∈ UV ✳ ❱ỵ✐ ♠å✐ UV ∈ U ✱ tø t➼♥❤ ❝➙♥ ❝õ❛ V s✉② r❛ ✤÷đ❝ UV−1 ∈ U ✳ ❱ỵ✐ ♠é✐ U = UV ∈ U ✳ ❈❤å♥ W ∈ U s❛♦ ❝❤♦ W + W ⊂ V ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ (x, y) ⊂ UW ◦ UW tỗ t z X s (x, z) ∈ UW ✈➔ (z, y) ∈ UW ✳ ❚❛ ❝â x − z ∈ W ✈➔ z − y ∈ W ✳ ❙✉② r❛ x − y = x − z + z − y ∈ W + W ⊂ V, ❤❛② (x, y) ∈ UV = U ✳ ❱➟② UW ◦ UW ⊂ V ❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❦✐➸♠ tr❛ ✤÷đ❝ U ❧➔ ❝ì sð ❧å❝✳ ❱➟② ♠å✐ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉✳ ❚✐➳♣ t❤❡♦ t❛ tr➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ tæ♣æ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉✳ ●✐↔ sû (X, U) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉✳ ❱ỵ✐ ♠é✐ A ∈ U ✱ t❛ ❣å✐ UA (x) = {y ∈ X : (x, y) ∈ A} ❧➔ ❧➙♥ x ỵ ợ ộ x X ✱ ❤å {UA(x) : A ∈ U} t❤♦↔ ♠➣♥ ❝→❝ t✐➯♥ ✤➲ N1) ✤➳♥ N4) ✈➲ ❧➙♥ ❝➟♥✳ ✾ ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ ♠å✐ A ∈ U ❝❤ù❛ ✤÷í♥❣ ❝❤➨♦ ∆(X) ♥➯♥ x ∈ UA(x)✳ ❚❛ ♥❤➟♥ ✤÷đ❝ N1 )✳ ❱ỵ✐ ♠å✐ V = UA (x) ✈➔ V ⊆ U ✳ ✣➦t B = {(x, y) : x ∈ X, y ∈ U }✳ ❑❤✐ ✤â A ⊂ B ✱ ❞♦ ✤â tø t➼♥❤ ❝❤➜t ❝õ❛ ❝➜✉ tró❝ ✤➲✉ s✉② r❛ B ∈ U ✳ ❘ã r➔♥❣ U = UB (x) ∈ U ✳ ❱➟② t✐➯♥ ✤➲ N2 ) ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚✐➳♣ t❤❡♦✱ t❛ ❝❤ù♥❣ ♠✐♥❤ N3 )✳ ❱ỵ✐ ♠å✐ U = UA (x), V = UB (x) t❛ ❝â A ∩ B ∈ U ✳ ❑❤✐ ✤â W = UA∩B (x) = {y ∈ X : (x, y) ∈ A ∩ B} ⊂ {y ∈ X : (x, y) ∈ A} ∩ {y ∈ X : (x, y) ∈ B} = U ∩ V ❈✉è✐ ❝ò♥❣✱ t❛ ❦✐➸♠ tr❛ N4 )✳ ❱ỵ✐ ♠é✐ U = UA (x) t❛ ❧➜② B ∈ U s❛♦ ❝❤♦ B ◦ B ⊂ A✳ ❑❤✐ ✤â V = VB (x)✳ ❘ã r➔♥❣ V ⊂ U ✳ ❚❛ ❝á♥ ♣❤↔✐ ❝❤➾ r❛ V ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ y ✈ỵ✐ ♠å✐ y ∈ V ✳ ◆➳✉ y ∈ UB (x) t❤➻ ✈ỵ✐ ♠å✐ z ∈ UB (y) t❛ ❝â (x, y) ∈ B ✈➔ (y, z) ∈ B ✳ ❙✉② r❛ (x, z) ∈ B ◦ B ⊂ A✳ ❙✉② r❛ UA (x) ❧➔ ❧➙♥ z ỵ ữủ ự ỵ = {UA (x) : x ∈ X, A ∈ U} ❧➔ ♠ët tæ♣æ tr➯♥ X ✈➔ ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ✭tỉ♣ỉ s✐♥❤ ❜ð✐ ❝➜✉ tró❝ ✤➲✉✮✳ ◆➳✉ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❤♦➦❝ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì tỉ♣ỉ t❤➻ tỉ♣ỉ s✐♥❤ ❜ð✐ ❝➜✉ tró❝ ✤➲✉ trị♥❣ ✈ỵ✐ tỉ♣ỉ ①✉➜t ♣❤→t✳ ✶✳✷✳✻ ▼➺♥❤ ✤➲✳ ▼é✐ ❝➜✉ tró❝ ✤➲✉ ❧✉ỉ♥ ❝â ♠ët ❝ì sð ✤è✐ ①ù♥❣ ❝→❝ t➟♣ ✤â♥❣ tr♦♥❣ X × X ✈ỵ✐ tỉ♣ỉ t➼❝❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ A ∈ U ✳ ✣➦t B = A ∩ A−1✳ ❑❤✐ ✤â B ❧➔ ✤è✐ ①ù♥❣ ✈➔ B ◦ B ◦ B ⊂ A✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ B ⊂ A ❚❤➟t ✈➟②✱ ✈ỵ✐ (x0 , y0 ) B tỗ t (x, y) ∈ B s❛♦ ❝❤♦ (x, x0 ) ∈ B ✈➔ (y, y0 ) ∈ B ✳ ❑❤✐ ✤â✱ tø t➼♥❤ ✤è✐ ①ù♥❣ ✈➔ B ◦ B ◦ B ⊂ A s✉② r❛ (x0 , y0 ) ∈ A ✶✵ ❚ø ❝❤ù♥❣ ♠✐♥❤ tr➯♥ ❝ô♥❣ s✉② r❛ B ✤è✐ ①ù♥❣✳ ❉♦ ✤â✱ ❤å ❝→❝ t➟♣ B ❧➔ ♠ët ❝ì sð ✤è✐ ①ù♥❣ ❝→❝ t➟♣ ✤â♥❣✳ ✶✳✷✳✼ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X = ∅✱ ❤➔♠ d : X × X → R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❣✐↔ ♠➯tr✐❝ tr➯♥ X ♥➳✉ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✶✮ d(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X ❀ ♥➳✉ x = y t❤➻ d(x, y) = 0✳ ✷✮ d(x, y) = d(y, x) ✈ỵ✐ ♠å✐ x, y ∈ X ❀ ✸✮ d(x, y) d(x, z) + d(z, y) ✈ỵ✐ ♠å✐ x, y ∈ X ✳ ❘ã r➔♥❣ ♠å✐ ♠➯tr✐❝ ❧➔ ❣✐↔ ♠➯tr✐❝✱ ♥❣÷đ❝ ❧↕✐ ❧➔ ❦❤ỉ♥❣ ✤ó♥❣✳ ❈❤➥♥❣ ❤↕♥ tr♦♥❣ R ①➨t d(x, y) = |x2 − y |✳ ❑❤✐ ✤â d ❧➔ ❣✐↔ ♠➯tr✐❝ ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♠➯tr✐❝✳ ❈❤♦ X = ∅ ✈➔ {dα }α∈I ❧➔ ♠ët ❤å ❝→❝ ❣✐↔ ♠➯tr✐❝ tr➯♥ X ✳ ❱ỵ✐ ♠é✐ α ∈ I ✱ ✤➦t Vε (α) = {(x, y) ∈ X × X : dα (x, y) < ε}, tr♦♥❣ ✤â > ữủ trữợ t U = {U ⊂ X × X : U ❧➔ ❣✐❛♦ ❤ú✉ ❤↕♥ ❝→❝ t➟♣ ❝â ❞↕♥❣ Vε (α)} ❚❛ ❝â ♠➺♥❤ ✤➲ s❛✉✳ ✶✳✷✳✽ ▼➺♥❤ ✤➲✳ ❍å U ❧➔ ❝ì sð ❝õ❛ ♠ët ❝➜✉ tró❝ ✤➲✉ tr➯♥ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø dα(x, x) = ✈ỵ✐ ♠å✐ x ∈ X ✈➔ α ∈ I ✱ s✉② r❛ ∆(X) ⊂ U ✈ỵ✐ ♠å✐ U ∈ U ✳ ❚ø t➼♥❤ ✤è✐ ①ù♥❣ ❝õ❛ ❝→❝ dα s✉② r❛ ♥➳✉ (x, y) ∈ Vε (α) t❤➻ (y, x) ∈ Vε (α)✳ ❉♦ ✤â U −1 ∈ U ✈ỵ✐ ♠å✐ U ∈ U ✳ ❱ỵ✐ ♠å✐ U ∈ U ✳ ❑❤✐ ✤â n U= Vεi (αi ) i=1 ✣➦t n V = V εi (αi ) i=1 ✶✽ ❑❤✐ ✤â✱ ❤å {(d(n,r) : r > 0, n = 1, } s✐♥❤ r❛ ❝➜✉ tró❝ ✤➲✉ tr➯♥ l∞ ✳ ❚r♦♥❣ t❤ü❝ t➳✱ X = l∞ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣ ổ ữợ ởt số ợ ởt ❞➣② t❤ỉ♥❣ t❤÷í♥❣✱ ✈➔ {d(n,r) } ❧➔ ❤å ❝→❝ ♥û❛ ❝❤✉➞♥ tr➯♥ X ✳ ❉♦ ✤â✱ tæ♣æ ✤➲✉ ❝õ❛ X tổổ ỗ ữỡ ró r tổổ ❤ì♥ tỉ♣ỉ s✐♥❤ ❜ð✐ ❝❤✉➞♥ x = sup |xn | ∀x = {xn } ∈ X n ❑❤✐ ✤â✱ t➟♣ ❝❤➾ sè I = {(n, r) : r > 0, n = 1, 2, } ❳➨t →♥❤ ①↕ j : I → I ①→❝ ✤à♥❤ ❜ð✐ ) 2n ✈ỵ✐ ♠å✐ (n, r) ∈ I ✳ ❳➨t ❤å {Φ} = {Φ(n,r) } ①→❝ ✤à♥❤ ❜ð✐ j(n, r) = n, r(1 − Φ(n,r) (t) = 2(n − 1) t, ∀t 2n − ❑❤✐ ✤â✱ ❞➵ t❤➜② {Φ}(n,r) t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ♠ët ❤å Φ−❝♦✳ ❇➙② ❣✐í✱ ①➨t t➟♣ M = {x = {xn } ∈ l∞ : x = sup |xn | = 1, xn n ∀n ✈➔ lim xn = 0} n→∞ ❑❤✐ ✤â M ❧➔ t➟♣ ✤â♥❣ ✈➔ ❜à ❝❤➦♥✳ ❚✉② ♥❤✐➯♥ M ❦❤æ♥❣ ✤➛② ✤õ✳ ❚❤➟② ✈➟②✱ ①➨t ❞➣② xk = (1, 1, , 1, 0, 0, 0, ), k = 1, 2, ❚❛ ❝â {xk } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ n ∈ N✱ ♥➳✉ k, l n t❛ ❝â d(n,r) (xk , xl ) = r|Pn xk − Pn xl | = r|1 − 1| = ú ỵ r ổ ✈ỵ✐ tỉ♣ỉ s✐♥❤ ❜ð✐ ❝❤✉➞♥✮✳ ❚❛ ❝❤➾ r❛ xk ❤ë✐ tư tỵ✐ x = (1, 1, , 1, )✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ n ∈ N ✈➔ ✈ỵ✐ ♠å✐ r > 0✱ ♥➳✉ k n t❛ ❝â d(n,r) (xk , x) = r|Pn xk − Pn x| = r|1 − 1| = ✶✾ ❱➟② d(n,r) (xk , x) → ❦❤✐ k → ∞✳ ❍❛② xk ❤ë✐ tö tỵ✐ x = (1, 1, , 1, ) ∈ / M✳ ❇➙② ❣✐í✱ ①➨t →♥❤ ①↕ T : M → M ①→❝ ✤à♥❤ ❜ð✐ 1 Pn (T x) = − (1 − )(1 − xn ) = + (1 − )xn n n n ✈ỵ✐ ♠å✐ x = {xn } ∈ M ❚❛ ❝❤ù♥❣ ♠✐♥❤ T ❧➔ Φ−❝♦ ✈ỵ✐ ❤å Φ ✈➔ →♥❤ ①↕ j ①→❝ ✤à♥❤ ð tr➯♥✳ ❚❤➟t ✈➟②✱ t❛ ❝â d(n,r) (T x, T y) = r|Pn (T x) − Pn (T y)| = r(1 − )|xn − yn |, n dj(n,r) (x, y) = r(1 − )|xn − yn | 2n ✈➔ )|xn − yn | 2n 2(n − 1) n−1 = r(1 − )|xn − yn | = r |xn − yn | 2n − 2n n Φ(n,r) dj(n,r) (x, y) = Φ(n,r) r(1 − ❚❛ ♥❤➟♥ ✤÷đ❝ d(n,r) (T x, T y) Φ(n,r) dj(n,r) (x, y) ✈ỵ✐ ♠å✐ x, y ∈ M ❱➟② T t❤♦↔ ♠➣♥ tt ỵ trứ tt X ❧➔ ❦❤æ♥❣ tü❛ ✤➛② ✤õ✮✳ ❚✉② ♥❤✐➯♥✱ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ x = (1, 1, , 1, ) ∈ / M ✣à♥❤ ♥❣❤➽❛ →♥❤ ①↕ ❝♦ ❝õ❛ ❇❛♥❛❝❤ ✤÷đ❝ ♣❤→t ❜✐➸✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ♥❤÷ s❛✉ dα (T x, T y) kα dα (x, y) ✈ỵ✐ ♠å✐ x, y ∈ X ✈➔ ❤➡♥❣ sè kα ∈ (0, 1) ✈ỵ✐ ♠å✐ α ∈ (0, 1)✳ ✣à♥❤ ữủ tứ ợ {Φα } ①→❝ ✤à♥❤ ❜ð✐ Φα (t) = kα t✳ ❱➼ ❞ư s❛✉ ❝❤ù♥❣ tä ❧ỵ♣ Φ−❝♦ t❤ü❝ sü rë♥❣ ❤ì♥ ❧ỵ♣ →♥❤ ①↕ ❝♦ ❝õ❛ ❇❛♥❛❝❤✳ ✷✵ ✷✳✶✳✻ ❱➼ ❞ư✳ ❳➨t X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ❝→❝ ❤➔♠ t❤ü❝ ①→❝ ✤à♥❤ tr➯♥ [0, 1]✳ ❚❛ ✤➣ ❜✐➳t X ❧➔ ổ ỗ ữỡ ợ ỷ ✤à♥❤ ❜ð✐✿ pα (f ) = |f (α)|, ∀f ∈ X, α ∈ [0, 1] ❍å ♥û❛ ❝❤✉➞♥ ♥➔② s✐♥❤ r❛ ❤å ❝→❝ ❣✐↔ ♠➯tr✐❝ dα (f, g) = pα (f, g) = |f (α) − g(α)| ✈ỵ✐ ♠å✐ f, g ∈ X ●å✐ Y = {f ∈ X : f (t) 0, ∀t ∈ [0, 1]} ❑❤✐ ✤â Y ❧➔ t➟♣ ❝♦♥ ✤â♥❣ ❝õ❛ X ✳ ❇➙② ❣✐í ①➨t T : Y → Y ①→❝ ✤à♥❤ ❜ð✐ (T f )(t) = f (t) + f (t) t ✈ỵ✐ ♠å✐ 1+t ✈➔ ✈ỵ✐ ♠å✐ α ∈ [0, 1]✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ α ∈ [0, 1]✱ f, g ∈ Y t❛ ❝â ✈ỵ✐ ♠å✐ t ∈ [0, 1]✳ ❚❛ ❝❤➾ r❛ T ❧➔ Φ−❝♦ ✤è✐ ✈ỵ✐ ❤å Φα (t) = t g(α) f (α) − + f (α) + g(α) |f (α) − g(α)| |f (α) − g(α)| = + f (α) + g(α) + f (α)g(α) + |f (α) − g(α)| = Φα (|f (α) − g(α)|) = Φα (dα (f, g)) dα (T f, T g) = |(T f )(α) − (T g)(α)| = ❚r♦♥❣ ❦❤✐ ✤â✱ T ❦❤æ♥❣ ♣❤↔✐ ❧➔ Φ−❝♦ ✈ỵ✐ ❤å Φα (t) = kα t✳ ❚❤➟t ✈➟②✱ ❣✐↔ sỷ ợ [0, 1] tỗ t k (0, 1) s❛♦ ❝❤♦ dα (T f, T g) kα dα (f, g) ✈ỵ✐ ♠å✐ f, g ∈ X ✳ ❚❛ ♥❤➟♥ ✤÷đ❝ < kα < 1, + f (α) + g(α) + f (α)g(α) tr♦♥❣ ✤â kα ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ f, g ✳ ❉♦ ✤â✱ ♥➳✉ ❝❤å♥ f (α)✱ g(α) ✤õ ❜➨ t❤➻ t❛ ♥❤➟♥ ✤÷đ❝ ♠➙✉ t❤✉➝♥✳ ❱➟② T ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ Φ−❝♦ ✈ỵ✐ ❤å (t) = k t ỵ ♠ët ❦➳t q✉↔ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ tr t õ ỵ s ♣❤→t ❜✐➸✉ ❝❤♦ →♥❤ ①↕ Φ−❝♦ tr➯♥ t♦➔♥ ❜ë ❦❤æ♥❣ ỵ X ổ ✤➲✉ ✈➔ →♥❤ ①↕ T : X → X ✳ ●✐↔ sû r➡♥❣✿ ✶✮ T : X → X ❧➔ ởt ợ ộ I tỗ t↕✐ ❤➔♠ Φα t❤♦↔ ♠➣♥ sup{Φj n (α) (t) : n = 1, 2, } Φα (t) ✈➔ Φαt(t) ✤ì♥ t tr (0, +) ỗ t x0 X s ợ I tỗ t q(α) s❛♦ ❝❤♦ dj (α) (x0 , T x0 ) < q(α) ✈ỵ✐ ♠å✐ n = 1, 2, ❑❤✐ ✤â✱ T ❝â ➼t ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ x0 tr♦♥❣ ❣✐↔ t❤✐➳t✱ t❛ ①➙② ❞ü♥❣ ❞➣② xn = T xn−1✱ n = n 1, 2, ✈➔ ✤➦t cαn = dα (xn , xn+1 ) = dα (T n x0 , T n+1 x0 ), n = 1, 2, ❑❤✐ ✤â cαn = dα (T n x0 , T n+1 x0 ) Φα dj(α) (T n−1 x0 , T n x0 ) Φα Φj(α) Φj n−1 (α) dj n (α) (x0 , T x0 ) (❞♦ ✸✮) Φα Φj(α) Φj n−1 (α) dj n (α) (q(α)) n Φα q(α) (❞♦ ✷✮) ❙✉② r❛✱ ✈ỵ✐ ♠å✐ m, n ∈ N p dα (xm+p , xm ) p dα (xm+p−i+1 , xm+p−i ) = i=1 p m+p−i cαm+p−i i=1 Φα i=1 ❚ø q(α) > s✉② r❛ n Φα (q(α)) = Φα (Φn−1 α (q(α))) n−1 Φα (q(α)) Φα (q(α)) q(α) (q(α)) ✷✷ ❱➻ ❤➔♠ Φα (t) ✤ì♥ ✤✐➺✉ t➠♥❣ ♥➯♥ t n n+1 Φα (q(α)) Φα Φα (q(α)) = n n Φα (q(α)) Φ (q(α)) ❚❤❡♦ ❞➜✉ ❤✐➺✉ ❉❛❧❛♠❜❡rt✱ t❛ ❝â ❝❤✉é✐ Φα (q(α)) < q(α) m ∞ m=1 Φα (q(α)) ❤ë✐ tö✳ ❙✉② r❛ p m+p−i Φα (q(α)) → 0, m → ∞, ∀p i=1 ❱➻ ✈➟② dα (xm+p , xm ) → ❦❤✐ m → ∞ ✈ỵ✐ ♠å✐ p✳ ❍❛② {xn } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❱➻ X ✤➛② ✤õ ❞➣② ♥➯♥ xn → x ∈ X ❦❤✐ n → ∞✱ tù❝ ❧➔ ✈ỵ✐ ♠å✐ α✱ dα (xn , x) → ❦❤✐ n → ∞ ❚❛ ❝❤ù♥❣ ♠✐♥❤ x ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ α ∈ I ♥➳✉ ❝➛♥ t❤✐➳t t❤❛② ❜ð✐ ❞➣② ❝♦♥ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t {dα (xn , x)} ✈➔ {dj(α) (xn , x)} ❣✐↔♠ ❞➛♥ tỵ✐ 0✳ ❚❛ ❝â dα (T x, x) dα (x, xn+1 )+dα (T xn , T x) dα (x, xn+1 )+Φα dj(α) (xn , x) ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦ n → ∞ ✈➔ sû ❞ö♥❣ t➼♥❤ ❧✐➯♥ tö❝ ♣❤↔✐ ❝õ❛ Φα t❛ ♥❤➟♥ ✤÷đ❝ dα (T x, x) = 0✳ ❱➟② T x = x✳ ❱ỵ✐ ❝→❝ ❣✐↔ t❤✐➳t ❝õ❛ ỵ tr ữ t t t ❝õ❛ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✷✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ ✤➲✉ X ữủ j ợ x, y X I tỗ t q(x, y, α) s❛♦ ❝❤♦ dj n (α) (x, y) q(x, y, α) ✈ỵ✐ ♠å✐ n = 1, 2, t r ỵ s ✤✐➲✉ ❦✐➺♥ X ❧➔ j−❜à ❝❤➦♥ t❤➻ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû T x = x ✈➔ ✷✸ T y = y ✳ ❑❤✐ ✤â dα (x, y) = dα (T x, T y) Φα dj(α) (x, y) = Φα dj(α) (T x, T y) Φα Φj(α) Φj n−1 (α) dj n (α) (x, y) Φα Φj(α) Φj n−1 (α) q(x, y, α)) n Φα (q(x, y, α)) ❈❤ù♥❣ ♠✐♥❤ tữỡ tỹ ữ tr ỵ t ữủ n n ∞ n=1 Φα (q(x, y, α)) ❤ë✐ tö✳ ❙✉② r❛ lim Φα (q(x, y, α)) = 0✳ ❚❛ ♥❤➟♥ ữủ dj() (x, y) = ợ n ∈ I ✳ ❱➟② x = y ◆➳✉ ❜ê s✉♥❣ t❤➯♠ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ →♥❤ ①↕ j t❤➻ ✤✐➲✉ ỵ õ t tt ỡ ỵ X ổ ✤➲✉ ✈➔ →♥❤ ①↕ T : X → X ✳ ●✐↔ sû r➡♥❣✿ ✶✮ T : X → X ❧➔ ♠ët Φ−❝♦❀ ✷✮ ❱ỵ✐ ♠é✐ α ∈ I ✈➔ ✈ỵ✐ ♠å✐ t > lim Φα Φj(α) Φj n (α) t n→∞ = 0; ✸✮ →♥❤ ①↕ j : I I t tỗ t xn = T xn−1 ✱ n = 1, 2, t❤♦↔ ♠➣♥ dα (xm , xm+n ) ♠å✐ n ∈ N ✈➔ m = 0, 1, ✳ ❑❤✐ ✤â✱ T ❝â ➼t ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ s❛♦ ❝❤♦ ❞➣② dj(α) (xm , xm+n ) ✈ỵ✐ x0 ∈ X ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❞➣② {xn} ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ✤✐➲✉ ỵ ợ ộ n = 1, 2, ✤➦t cαn = dα (xn , xn+1 )✳ ❑❤✐ ✤â cαn = dα (T n x0 , T n+1 x0 ) Φα dj(α) (T n−1 x0 , T n x0 ) Φα Φj(α) Φj n−1 (α) dj n (α) (x0 , T x0 ) Φα Φj(α) Φj n−1 (α) cα0 ✷✹ ❈❤♦ n → ∞ ✈➔ sû ❞ö♥❣ ❣✐↔ t❤✐➳t ✷✮ ❝õ❛ ✤à♥❤ ỵ t ữủ lim cn = n ợ ♠å✐ α ∈ I ✳ ●✐↔ sû {xn } ❦❤æ♥❣ õ tỗ t > I s ợ tỗ t↕✐ m(ν) > ν ✈➔ p(ν) > t❤♦↔ ♠➣♥ dα0 (xm(ν)+p(ν) , xm(ν) ) ε0 ❱➻ j ❧➔ t tỗ t I s j(α) = α0 ✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ✤✐➲✉ ❦✐➺♥ ✸✮ t❛ ❝â dα (xm(ν) , xm(ν)+p(ν) ) dj(α) (xm(ν) , xm(ν)+p(ν) ) ε0 ●å✐ p ❧➔ sè ♥❤ä ♥❤➜t s❛♦ ❝❤♦ dj(α) (xm(ν) , xm(ν)+p(ν) ) ε0 tr♦♥❣ ❦❤✐ dj() (xm() , xm()+p()1 ) < số p tỗ t↕✐ ❜ð✐ ✈➻ lim cαn = dα (xn , xn+1 ) = ✈ỵ✐ ♠å✐ α ∈ I ✮✳ ✣➦t n→∞ j(α) hν = dj(α) (T m+p x0 , T m x0 ) ❑❤✐ ✤â ε0 j(α) hν = dj(α) (T m+p x0 , T m x0 ) j(α) dj(α) (T m+p x0 , T m+p−1 x0 ) + dj(α) (T m+p−1 x0 , T m x0 ) cm+p−1 + ε0 j(α) ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦ ν → ∞ t❛ ♥❤➟♥ ✤÷đ❝ lim hν ν→∞ ❝â ε0 j(α) hν = dj(α) (T m+p x0 , T m x0 ) = ε0 ❚❛ dα (T m+p x0 , T m x0 ) dα (T m+p x0 , T m+p+1 x0 ) + dα (T m+p+1 x0 , T m+1 x0 ) + dα (T m x0 , T m x0 ) cαm+p + Φα dj(α) (T m+p x0 , T m x0 ) + cαm ✷✺ ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦ ν → ∞ t❛ ❝â ε0 Φα (ε0 ) ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t Φα (t) < t ✈ỵ✐ ♠å✐ t > 0✳ ❱➟② {xn } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❱➻ X ❧➔ tỗ t x X s ❝❤♦ {xn } ❤ë✐ tư tỵ✐ x✱ tù❝ ❧➔ dα (xn , x) → ❦❤✐ n → ∞ ✈ỵ✐ ♠å✐ α ∈ I ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ x ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠å✐ α ∈ I ♥➳✉ ❝➛♥ t❤✐➳t t❤❛② ❜ð✐ ❞➣② ❝♦♥ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t {dα (xn , x)} ✈➔ {dj(α) (xn , x)} ❣✐↔♠ ❞➛♥ tỵ✐ 0✳ ❚❛ ❝â dα (T x, x) dα (x, xn+1 )+dα (T xn , T x) dα (x, xn+1 )+Φα dj(α) (xn , x) ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦ n → ∞ ✈➔ sû ❞ö♥❣ t➼♥❤ ❧✐➯♥ tö❝ ♣❤↔✐ ❝õ❛ Φα t❛ ♥❤➟♥ ✤÷đ❝ dα (T x, x) = 0✳ ❱➟② T x = x✳ ✷✳✶✳✶✶ ◆❤➟♥ ①➨t✳ ✶✮ ❚÷ì♥❣ tü ữ tr ỵ s X j ỵ t T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✷✮ ✣✐➲✉ ❦✐➺♥ lim Φα Φj(α) Φj n (α) t n→∞ = ❦❤æ♥❣ t❤➸ s✉② r❛ tø Φj n (α) t < t✳ ❈❤➥♥❣ ❤↕♥✱ ①➨t I = N ✈➔ ❤å Φ = {Φn (t) = − t : n ∈ N} (1 + n)2 ✈➔ j : N → N ①→❝ ✤à♥❤ ❜ð✐ j(n) = n + 1✳ ❑❤✐ ✤â Φn Φj(n) Φj k (n) t (1 + n)2 − (2 + n)2 − (k + + n)2 − = t (1 + n)2 (2 + n)2 (k + + n)2 n n+k+2 = t n+1 n+k+1 ❙✉② r❛ lim Φn Φj(n) Φj k (n) t k→∞ = n t=0 n+1 ✈ỵ✐ ♠é✐ n ∈ N ✈➔ ✈ỵ✐ ♠å✐ t > tr ỵ t tr ỵ ❚❤➟t ✈➟②✱ ❣✐↔ sû ✈ỵ✐ ♠é✐ α ∈ I ✱ tỗ t t sup{j n () (t) : n = 1, 2, } Φα (t) ✷✻ Φα (t) ✤ì♥ ✤✐➺✉ t➠♥❣ tr➯♥ (0, +∞)✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ t > ❝è ✤à♥❤ ✤➦t t Φ(t) Φα (t) k = < ❙✉② r❛ Φ(t) < kt < t✳ ❱➻ ✤ì♥ ✤✐➺✉ t➠♥❣ ♥➯♥ tø t t Φ(t) < kt < t t❛ ❝â ✈➔ n Φα (t) k n t ▼➦t ❦❤→❝ ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ỵ t õ n (t) lim Φα Φj(α) Φj n (α) t = Φα Φj(α) Φj n (α) t ❉♦ ✤â✱ t❛ ♥❤➟♥ ✤÷đ❝ n→∞ ❙❛✉ ✤➙② ❧➔ ♠ët ❦➳t q✉↔ ❝❤♦ tr÷í♥❣ ❤đ♣ j ♣❤ư t❤✉ë❝ ✈➔♦ t➼♥❤ ❧➦♣ ❧↕✐ ❝õ❛ →♥❤ ①↕ T ✱ t❤❡♦ ♥❣❤➽❛ j : I × N I ỵ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉ ✈➔ →♥❤ ①↕ T : X → X ✳ ●✐↔ sû r➡♥❣✿ ✶✮ ❱ỵ✐ ♠é✐ α I n N tỗ t ,n(t) Φ s❛♦ ❝❤♦ dα (T n x, T n y) ỗ t x0 n = 0, 1, 2, ✈➔ ∈ X Φα,n (dj(α,n) (x, y)), ∀x, y ∈ X s❛♦ ❝❤♦ dj(α,n)(x0, T x0) p(α) < +∞ ∞ Φα,n p(α) < ∞ n=0 ❑❤✐ ✤â T ❝â ➼t ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ x0 t❤♦↔ tt ỵ t õ m d (T m+n n dα T n+i−1 (T x0 ), T n+i−1 x0 x0 , T x0 ) i=1 m Φα,n+i−1 dj(α,n+i−1) x0 , T x0 i=1 m Φα,n+i−1 p(α) i=1 ✈ỵ✐ ♠å✐ ✷✼ ❱➻ ❝❤✉é✐ ∞ n=0 Φα,n p(α) ❤ë✐ tö ♥➯♥ m lim n→∞ Φα,n+i−1 p(α) = i=1 ✈ỵ✐ ♠å✐ m ∈ N✳ ❉♦ ✤â✱ ✈ỵ✐ ♠é✐ α ∈ I t❛ ❝â lim dα (T m+n x0 , T n x0 ) = ✈ỵ✐ ♠å✐ m ∈ N✳ ❱➟② {xn } := n→∞ {T n x 0} ❧➔ ❞➣② ❈❛✉❝❤②✳ ❱➻ X ❧➔ ✤➛② ✤õ ❞➣② ♥➯♥ tỗ t x X s lim xn = x ❚❛ ❝â n→∞ dα (x, T x) dα (T x, xn+1 )+dα (xn+1 , x) Φα,1 dj(α,1) (x, xn ) +dα (xn+1 , x) ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❝❤♦ n → ∞ t❛ ♥❤➟♥ ✤÷đ❝ dα (x, T x) = ✈ỵ✐ ♠å✐ α ∈ I ✳ ❱➟② T x = x ỵ t ✤è✐ ✈ỵ✐ ❝→❝ →♥❤ ①↕ ❝♦ ❦✐➸✉ ▼❡✐r✲ ❑❡❡❧❡r tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ▼ư❝ ♥➔② ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ♠ët t q ỵ t ố ợ ❝→❝ →♥❤ ①↕ ❝♦ ❦✐➸✉ ▼❡✐r✲❑❡❡❧❡r tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉✳ t q ỵ tữ tứ t q rr ổ tr rữợ t t ỵ t ố ợ ①↕ ❝♦ ❦✐➸✉ ▼❡✐r✲❑❡❡❧❡r tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ✷✳✷✳✶ ✣à♥❤ ỵ (X, d) ổ tr ✤õ ❞➣②✳ ◆➳✉ →♥❤ ①↕ S : X → X t ợ > 0, tỗ t > s❛♦ ❝❤♦ ε d(x, y) < ε + δ ❦➨♦ t❤❡♦ d(Sx, Sy) < ε, t❤➻ S ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ z ✈➔ n→∞ lim S n z = z ỵ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉ ✤➛② ✤õ ❞➣②✳ ◆➳✉ →♥❤ ①↕ T : t❤♦↔ ♠➣♥ ✈ỵ✐ ♠é✐ α ∈ I ợ > 0, tỗ t > s❛♦ ❝❤♦ ε dα (x, y) < ε + δ ❦➨♦ t❤❡♦ dα (T x, T y) < ε, t❤➻ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ x ✈➔ n→∞ lim T n x = x✳ X→X ✷✽ ❈❤ù♥❣ ♠✐♥❤✳ ✣➛✉ t✐➯♥ t❛ ❝❤ù♥❣ ♠✐♥❤ T ❧✐➯♥ tö❝✳ ●✐↔ sû {xn} ❧➔ ❞➣② ✭s✉② rë♥❣✮ tr♦♥❣ X ✈➔ ♥â ❤ë✐ tư tỵ✐ x ∈ X ✳ ❚❛ ❝❤➾ r❛ T xn tử tợ T x ữủ t tỗ t I s t❤❛② ❞➣② ❝♦♥ ❝õ❛ {xn } ♥➳✉ ❝➛♥ t❤✐➳t t❤➻ ε0 = inf d(T xn , T x) > 0✳ ợ < < tỗ t n s❛♦ ❝❤♦ ε < dα (xn , x) < ε + 1✳ ❑❤✐ ✤â✱ t❤❡♦ ❣✐↔ t❤✐➳t ❝õ❛ ✤à♥❤ ỵ t õ d (T xn , T x) < ε < ε0 ❚❛ ♥❤➟♥ ✤÷đ❝ sü ♠➙✉ t❤✉➝♥✳ ❱➟② T ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳ ▲➜② x0 ∈ X ✱ ①➨t ❞➣② ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ xn = T n x0 ✈ỵ✐ ♠é✐ n = 1, 2, ✳ tỗ t n0 N s xn0 +1 = xn0 t❤➻ xn0 +1 = T xn0 = xn0 ✳ ❚❛ ♥❤➟♥ ✤÷đ❝ xn0 ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❱➻ ✈➟②✱ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t xn = xn+1 ✈ỵ✐ ♠å✐ n✳ ❉♦ ✤â ✈ỵ✐ ♠é✐ α ∈ I t❛ ❝â dα (xn , xn+1 ) > ợ n ứ tt ỵ s✉② r❛ dα (xn , xn+1 ) = pα (T xn−1 , T xn ) < pα (xn−1 , xn ) ✈ỵ✐ ✈ỵ✐ ♠å✐ n✳ ❉♦ ✤â {dα (xn , xn+1 )} ❧➔ ❞➣② ❣✐↔♠ ♥❣➦t ❝→❝ sè t❤ü❝ ❞÷ì♥❣✳ r tỗ t 0 s lim d (xn , xn+1 ) = εα0 ❍ì♥ ♥ú❛ n→∞ dα (xn , xn+1 ) > εα0 ✈ỵ✐ ♠å✐ n✳ ❚❛ ❝❤➾ r❛ εα0 = 0✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû εα0 > 0✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ δ0α > s❛♦ ❝❤♦ ✈ỵ✐ n ✤õ ❧ỵ♥ t❛ ❝â εα0 < dα (xn+1 , xn+2 ) = dα (T xn , T xn+1 ) < εα0 + δ0α ❚ø ❣✐↔ tt ỵ s r d (xn , xn+1 ) < εα0 ▼➙✉ t❤✉➝♥ ✈ỵ✐ dα (xn , xn+1 ) > εα0 ✈ỵ✐ ♠å✐ n✳ ❱➟② lim dα (xn , xn+1 ) = n→∞ εα0 = ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ {xn } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ α ∈ I ✱ ✈ỵ✐ ♠é✐ ε > ❧➜② δ = δ(ε) > s❛♦ ❝❤♦ δ < ε✳ ❚ø lim dα (xn , xn+1 ) = = s r tỗ t N s ❝❤♦ n→∞ dα (xn−1 , xn ) < δ ✷✾ ✈ỵ✐ ♠å✐ n > N ✳ ❱ỵ✐ n > N t❛ ❝â dα (xn , xn+p ) ε ✈ỵ✐ ♠å✐ p = 1, 2, ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❦❤➥♥❣ ✤à♥❤ tr➯♥ ❜➡♥❣ q✉② ♥↕♣✳ ❱➻ δ < ε ♥➯♥ dα (xn , xn+1 ) dα (xn−1 , xn ) < δ < ε ❱➟② ❦❤➥♥❣ ✤à♥❤ ✤ó♥❣ ✈ỵ✐ p = 1✳ ●✐↔ sû ❦❤➥♥❣ ✤à♥❤ ✤ó♥❣ ✈ỵ✐ p✱ tù❝ ❧➔ d(xn , xn+p ) < ε ❑❤✐ ✤â dα (xn−1 , xn+p ) dα (xn−1 , xn ) + dα (xn , xn+p ) < δ + ε ❚ø ✤✐➲✉ ỵ t õ d (T xn1 , T xn+p ) = dα (xn , xn+p+1 ) < ε ❑❤➥♥❣ ✤à♥❤ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ p + 1✳ ❱➟② {xn } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❱➻ X ✤➛② ✤õ tỗ t x X s {xn } ❤ë✐ tư tỵ✐ x ❚❛ ❝❤ù♥❣ ♠✐♥❤ x ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ T ✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ ♠é✐ α ∈ I ✱ tø t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ T ✈➔ dα t❛ ❝â = lim dα (xn , xn+1 ) = dα (xn , T xn ) = dα (x, T x) n→∞ ❱➟② T x = x✳ ●✐↔ sû y ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❦❤→❝ ❝õ❛ T ✳ ❑❤✐ ✤â✱ t❛ ❝â dα (x, y) = dα (T x, T y) ✈ỵ✐ ♠å✐ α ∈ I ✳ ◆➳✉ ε := dα (x, y) > t tứ tt ỵ t õ d (T x, T y) < ε ❚❛ ♥❤➟♥ ✤÷đ❝ sü ♠➙✉ t❤✉➝♥✳ ❱➟② dα (x, y) = 0✱ tù❝ ❧➔ x = y ỵ ữủ ự ởt ỵ tr➯♥✳ ✷✳✷✳✸ ❱➼ ❞ư✳ ❳➨t ❦❤ỉ♥❣ ❣✐❛♥ R∞ = {x = {xn} : xn ∈ R, n = 1, 2, }✳ ❑❤✐ ✤â R∞ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❝→❝ ♣❤➨♣ t♦→♥ ①→❝ ✤à♥❤ ❜ð✐ x + y = {xn + yn }, λx = {λxn } ✸✵ ✈ỵ✐ ♠å✐ x = {xn }, y = {yn } ∈ R∞ } ✈➔ λ ∈ R✳ ❍ì♥ ♥ú❛✱ R∞ ❧➔ ổ ỗ ữỡ ợ ỷ {pn } ①→❝ ✤à♥❤ ❜ð✐✿ ✈ỵ✐ ♠é✐ n = 1, 2, pn (x) = |xn |, ✈ỵ✐ ♠å✐ x = {xn } ∈ R∞ }✳ ❍å ♥û❛ ❝❤✉➞♥ tr➯♥ s✐♥❤ r❛ ❤å ❝→❝ ❣✐↔ ♠➯tr✐❝ dn (x, y) = pn (x − y) = |xn − yn |, ✈ỵ✐ ♠å✐ x = {xn }, y = {yn } ∈ R∞ }✳ ❑❤✐ ✤â R∞ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉ ✤➛② ✤õ ❞➣②✳ ❳➨t →♥❤ ①↕ T : R∞ → R∞ ❝❤♦ ❜ð✐ Pn (T x) = 1 + (1 − )xn , ✈ỵ✐ ♠å✐ x = {xn } ∈ R∞ , n n tr♦♥❣ ✤â Pn ❧➔ ♣❤➨♣ ❝❤✐➳✉ Pn ({xn }) = xn ✈ỵ✐ ♠é✐ n = 1, 2, ❱ỵ✐ ♠é✐ ε n = 1, 2, ợ > tỗ t = s❛♦ ❝❤♦ ♥➳✉ n e ε dn (x, y) = |xn − yn | < ε + n t❤➻ n2 − dn (T x, T y) = (1 − )|xn − yn | < ε < ε n n2 T t ỵ õ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❉➵ 1 ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ x = { } = (1, , , , ) ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ n n ✸✶ ❦➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉✿ ✶✮ ❚r➻♥❤ ❜➔② ❝â ❤➺ t❤è♥❣ ❦❤→✐ ♥✐➺♠✱ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè t➼♥❤ ❝❤➜t ✈➲ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉✳ ❈❤ù♥❣ ♠✐♥❤ ♠ët sè ❦➳t q✉↔✱ ✈➼ ❞ư ✈➔ ♥❤➟♥ ①➨t ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ♠➔ t➔✐ ❧✐➺✉ ✤÷❛ r❛ ♠➔ ❦❤ỉ♥❣ ❝❤ù♥❣ ♠✐♥❤ ♥❤÷✿ ❱➼ ❞ư ✶✳✷✳✸✱ ❱➼ ❞ư ✶✳✷✳✹✱ ▼➺♥❤ ✤➲ ✶✳✷✳✽✳ ✷✮ ❚r➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝õ❛ ❆♥❣❡❧♦✈ ✈➲ sỹ tỗ t t tr ổ ữ ỵ ỵ ỵ ỵ t ✷✳✶✳✶✶✳✳✳ ❈→❝ ❦➳t q✉↔ tr➯♥ t→❝ ❣✐↔ ✤÷❛ r❛ ♥❤÷♥❣ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t✱ ❤♦➦❝ ❦❤ỉ♥❣ ❝❤ù♥❣ ♠✐♥❤✱ ❝❤ó♥❣ tỉ✐ ự tt ữ r ởt ỵ ỵ sỹ tỗ t t ❝❤♦ →♥❤ ①↕ ❝♦ ❦✐➸✉ ▼❡✐r✲❑❡❡❧❡r tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ✤➲✉ ✈➔ ♠ët ✈➼ ❞ö →♣ ❞ö♥❣✳ ✸✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯ ✈➔ ▲➯ ▼➟✉ ❍↔✐ ✭✷✵✵✷✮✱ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ❚➟♣ ■✱ ■■✱ ◆❳❇ ●✐→♦ ❉ö❝✳ ❬✷❪ ❱✳ ●✳ ❆❧❣❡❧♦✈ ✭✷✵✵✾✮✱ ❋✐①❡❞ ♣♦✐♥ts t✐♦♥s ✱ ❈❧✉❥ ❯♥✐✈❡rs✐t② Prss ỡ s ỵ tt r s ❛♥❞ ❆♣♣❧✐❝❛✲ ❬✸❪ ❱✳ ●✳ ❆♥❣❡❧♦✈ ✭✷✵✵✹✮✱ ❆♥ ❡①t❡♥s✐♦♥ ♦❢ ❑✐r❦✲❈❛r✐st✐ t❤❡♦r❡♠ t♦ ✉♥✐✲ ❢♦r♠ s♣❛❝❡s✱ ❆♥t❛r❝t✳ ❏✳ ▼❛t❤✳ ✶✱ ♥♦✳ ✶✱ ✹✼✲✺✶✳ ❖♥ t❤❡ ✐t❡r❛t✐✈❡ t❡st ❢♦r J ✲❝♦♥tr❛❝t✐✈❡ ♠❛♣✲ ♣✐♥❣s ✐♥ ✉♥✐❢♦r♠ s♣❛❝❡s✱ ❉✐s❝✉ss✳ ▼❛t❤✳ ❉✐❢❢❡r❡♥t✐❛❧ ■♥❝❧✳ ✶✾✱ ♥♦✳ ❬✹❪ ❱✳ ●✳ ❆♥❣❡❧♦✈ ✭✶✾✾✾✮✱ ✶✲✷✱ ✶✵✸✲✶✵✾✳ ❬✺❪ ❱✳●✳ ❆♥❣❡❧♦✈ ✭✶✾✾✽✮✱ ❋✐①❡❞ ♣♦✐♥ts ♦❢ ♠✉❧t✐✲✈❛❧✉❡❞ ♠❛♣♣✐♥❣s ✐♥ ✉♥✐✲ ❢♦r♠ s♣❛❝❡s✱ ▼❛t❤✳ ❇❛❧❦❛♥✐❝❛ ✭◆✳❙✳✮ ✶✷✱ ♥♦✳ ✶✲✷✱ ✷✾✲✸✺✳ ❬✻❪ ❱✳ ●✳ ❆♥❣❡❧♦✈ ✭✶✾✽✼✮✱❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ✐♥ ✉♥✐❢♦r♠ s♣❛❝❡s ❛♣♣❧✐❝❛t✐♦♥s✱ ❈③❡❝❤♦s❧♦✈❛❦ ▼❛t❤✳ ❏✳ ✸✼✭✶✶✷✮✱ ♥♦✳ ✶✱ ✶✾✲✸✸✳ ❬✼❪ ❉✳ ❲✳ ❇♦②❞ ❛♥❞ ❏✳ ❙✳ ❲✳ ❲♦♥❣✱ ❖♥ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✷✵ ✶✾✻✾✱ ✹✺✽✲✹✻✹✳ ❛♥❞ ♥♦♥❧✐♥❡❛r ❝♦♥tr❛❝t✐♦♥s✱ Pr♦❝✳ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣②✱ ■♥❝✳ Pr✐♥❝❡t♦♥✱ ◆❡✇ ❏❡rs❡②✳ ❬✾❪ ❆✳ ❑❡❡❧❡r ❛♥❞ ❆✳ ▼❡✐r ✭✶✾✻✾✮✱ ❆ t❤❡♦r❡♠ ♦♥ ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s✱ ❬✽❪ ❏✳ ❊✳ ❑❡❧❧② ✭✶✾✺✸✮✱ ❏✳ ▼❛t❤✳❆♥❛❧✳ ❆♣♣❧✳✱ ✷✽✱✸✷✻✲✸✷✾✳ ❬✶✵❪ ❆✳ ❲❡✐❧ ✭✶✾✸✽✮✱ ❙✉r ❧❡s ❡s♣❛❝❡s ❛ str✉❝t✉r❡ ✉♥✐❢♦r♠❡ ❡t s✉r ❧❛ t♦♣♦❧♦✲ ❣✐❡ ❣❡♥❡r❛❧❡✱ ❍❡r♠❛♥♥✫❈✲✐❡✱ ❊❞✳ P❛r✐s✳ ... ✤➛✉ t✐➯♥ ♥❣❤✐➯♥ ❝ù✉ ✈➜♥ ✤➲ tr➯♥ ❧➔ ■✳ ❆✳ ❘✉s ✈ỵ✐ ❝→❝ ❦➳t q✉↔ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ỗ ữỡ ữợ ự ữủ ởt số ♥❤➔ t♦→♥ ❤å❝ t❤ü❝ ❤✐➺♥ ✈➔ ✤↕t ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔ q✉❛♥ ✸ trå♥❣ ✈➔♦ t❤➟♣ ♥✐➯♥ ✽✵ ✈➔ ✾✵ t trữợ... ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ♠ët sè ❧ỵ♣ →♥❤ ①↕ ❝♦ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ✈➔ ù♥❣ ❞ư♥❣✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ t s ởt số ỵ ❜➜t ✤ë♥❣ ❝❤♦ ♠ët ✈➔✐ ❧ỵ♣ →♥❤ ①↕ ❝♦ tr➯♥ ❦❤æ♥❣ ✤➲✉✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐... ✤➛② ✤õ ❞➣② ♥➳✉ ♠å✐ t➟♣ ✤â♥❣ ✈➔ ì ị ✣❐◆● ❈❍❖ ❈⑩❈ ▲❰P ⑩◆❍ ❳❸ ❈❖ ❚❘➊◆ ❑❍➷◆● ●■❆◆ ữỡ tr ởt số ỵ ❜➜t ✤ë♥❣ ❝❤♦ ❝→❝ →♥❤ ①↕ ❝♦ ♣❤✐ t✉②➳♥ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✤➲✉ ❝õ❛ ❆♥❣❡❧♦✈ ✈➔ ✤÷❛ r❛ ♠ët ✤à♥❤ ỵ