Không gian mêtric nón với thứ tự bộ phận và sự tồn tại điểm bất động

▼Ö❈ ▲Ö❈ ❚r❛♥❣ ▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ◆â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ❈❤÷ì♥❣ ỹ tỗ t t tr ổ ♠➯tr✐❝ ♥â♥ ✈ỵ✐ t❤ù tü ❜ë ♣❤➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ỹ tỗ t t tr ổ ♠➯tr✐❝ ♥â♥ ✈ỵ✐ t❤ù tü ❜ë ♣❤➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t t tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈ỵ✐ t❤ù tü ❜ë ♣❤➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✶ ▼Ð ✣❺❯ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❦❤→✐ ♥✐➺♠ q✉❛♥ trå♥❣ ❝õ❛ t♦→♥ ❤å❝✱ ♥â ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤ ✈➔ ♥❤✐➲✉ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❦❤→❝✳ ◆➠♠ ✷✵✵✼✱ ❍✉❛♥❣ ▲♦♥❣ ✲ ●✐❛♥❣ ✈➔ ❩❤❛♥❣ ❳✐❛♥ ✤➣ ♠ð rë♥❣ ❦❤→✐ ♥✐➺♠ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❜➡♥❣ ❝→❝❤ t❤❛② t➟♣ ủ số tỹ ởt õ ữợ tr ổ t ữủ ợ tờ q✉→t ❤ì♥ ✤â ❧➔ ❦❤→✐ ♥✐➺♠ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✳ ự t t tổổ sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr➯♥ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♥➔② ✤❛♥❣ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ t tr ữợ t ữủt ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✱ ✤➸ ❝â ♥❤ú♥❣ ❤✐➸✉ ❜✐➳t ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ ❝❤ó♥❣ tỉ✐ t➻♠ ❤✐➸✉✱ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈➔ ❝→❝ ỵ sỹ tỗ t t ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ♠➔ tr➯♥ ✤â ❝â tr❛♥❣ ❜à t❤➯♠ ♠ët t❤ù tü ❜ë ♣❤➟♥✳ ▼ð rë♥❣ ♠ët sè ❦➳t q✉↔ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❝❤♦ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈ỵ✐ t❤ù tü ❜ë ♣❤➟♥✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ✤â✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② t❤➔♥❤ ❤❛✐ ❝❤÷ì♥❣ ❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ✤➛✉ t✐➯♥ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ tỉ♣ỉ ✤↕✐ ❝÷ì♥❣✱ ❣✐↔✐ t➼❝❤ ❤➔♠ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ♥ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ✈➼ ❞ư ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❙❛✉ ✤â✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ ✈➼ ❞ư ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ♠➔ ❝❤ó♥❣ ❝➛♥ ❞ị♥❣ ❝❤♦ ❝❤÷ì♥❣ s❛✉✳ ữỡ ỹ tỗ t t tr ổ tr õ ợ tự tỹ r ữỡ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ sỹ tỗ t t t ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ♠➔ tr➯♥ ♥â ❝â ♠ët t❤ù tü ❜ë ♣❤➟♥✳ ◆❣♦➔✐ ✈✐➺❝ ❤➺ t❤è♥❣✱ tr➻♥❤ ❜➔② ❧↕✐✱ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♥❤✐➲✉ ❦➳t q✉↔ ♠➔ tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❦❤æ♥❣ ❝❤ù♥❣ ♠✐♥❤ ❤♦➦❝ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ữ r ự ởt số t q ợ ✤â ❧➔ ❱➼ ❞ö 1.3.2.2)✱ ▼➺♥❤ ✤➲ 1.3.4✱ ❍➺ q✉↔ 1.3.5 1.3.11 ỵ 2.1.4 ỵ 2.1.5✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ữợ sỹ ữợ t t ❝õ❛ P●❙✳❚❙✳ ✣✐♥❤ ❍✉② ❍♦➔♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ ❚❤➛②✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❑❤♦❛ ❙❛✉ ✤↕✐ ❤å❝✱ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ữủ ỡ qỵ ❚❤➛② ❣✐→♦✱ ❈æ ❣✐→♦ ❚ê ●✐↔✐ t➼❝❤ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t ố ũ ỡ ỗ ❜↕♥ ❜➧✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ ❈❛♦ ❤å❝ ✶✽ ✲ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●✐↔✐ t➼❝❤ ✤➣ ❝ë♥❣ t→❝✱ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ ❝á♥ ❤↕♥ ❝❤➳ ✈➲ ♠➦t ❦✐➳♥ t❤ù❝ ✈➔ t❤í✐ ❣✐❛♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ọ ỳ t sõt qỵ ổ õ õ ỵ ữủ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❱✐♥❤✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✷ ❚→❝ ❣✐↔ ✸ ❈❍×❒◆● ✶ ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆ ✶✳✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥ ❝➛♥ ❞ị♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ❈→❝ ❦➳t q✉↔ ð ♠ư❝ ♥➔② ✤÷đ❝ ❧➜② tø ❝→❝ t➔✐ ❧✐➺✉ [1] ❤♦➦❝ [2]✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ t➟♣ ❤ñ♣ X ✳ ❍å τ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❚1✮ ∅, X ∈ τ ❀ ✭❚2✮ ◆➳✉ Gi ∈ τ, i ∈ I t❤➻ Gi ∈ τ ❀ i∈I ✭❚3✮ ◆➳✉ G1, G2 ∈ τ t❤➻ G1 ∩ G2 ∈ τ ✳ ❚➟♣ ❤ñ♣ X ũ ợ tổổ tr õ ữủ ổ tổổ ỵ (X, ) ✤ì♥ ❣✐↔♥ ❤ì♥ ❧➔ X ✳ ❈→❝ ♣❤➛♥ tû ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ✳ ❈→❝ ♣❤➛♥ tû t❤✉ë❝ τ ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ♠ð ✳ ●✐↔ sû A ⊂ X ✳ ❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ✤â♥❣ ♥➳✉ X \ A ❧➔ ♠ð✳ ✶✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ X ✱ t➟♣ ❝♦♥ A ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ x X tỗ t t V X s❛♦ ❝❤♦ x ∈ V ⊆ A✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ X ✱ x ∈ X, U(x) ❧➔ ❤å t➜t ❝↔ ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ x✳ ❍å B(x) ⊂ U(x) ✤÷đ❝ ❣å✐ ❧➔ ❝ì sð ❧➙♥ ❝➟♥ t↕✐ x ♥➳✉ ợ U U(x) tỗ t V B(x) s❛♦ ❝❤♦ V ⊂ U ✳ ✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❉➣② {xn} tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ X ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư tỵ✐ x ∈ X ♥➳✉ ✈ỵ✐ ♠é✐ ❧➙♥ ❝➟♥ U x tỗ t n0 N s xn ∈ U ✈ỵ✐ ♠å✐ n ✹ n0 ❑❤✐ ✤â✱ t❛ ✈✐➳t xn → x ❤♦➦❝ n→∞ lim xn = x✳ ✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ❑❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ X ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ t✐➯♥ ✤➲ ♥➳✉ t↕✐ ♠é✐ ✤✐➸♠ x ∈ X ❝â ♠ët ❝ì sð ❧➙♥ ❝➟♥ B(x) ❝â ❧ü❝ ❧÷đ♥❣ ✤➳♠ ✤÷đ❝✳ ❑❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ X ✤÷đ❝ ❣å✐ ❧➔ T2✲❦❤æ♥❣ ❣✐❛♥ ❤❛② ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ ♥➳✉ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý x, y ∈ X, x = y tỗ t tữỡ ự Ux, Uy x ✈➔ y s❛♦ ❝❤♦ Ux ∩ Uy = ∅✳ ◆➳✉ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ t❤➻ ♠é✐ ❞➣② tr♦♥❣ X ♠➔ ❤ë✐ tư t❤➻ ❤ë✐ tư tỵ✐ ♠ët ✤✐➸♠ ❞✉② ♥❤➜t✳ ✤➳♠ ✤÷đ❝ t❤ù ♥❤➜t ✶✳✶✳✺ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X, Y ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ ✈➔ f : X → Y ✳ ⑩♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ t↕✐ x ∈ X ♥➳✉ ✈ỵ✐ ♠é✐ V f (x) tỗ t U ❝õ❛ x s❛♦ ❝❤♦ f (U ) ⊂ V ✳ ⑩♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ tr➯♥ X ✭♥â✐ ❣å♥ ❧➔ ❧✐➯♥ tö❝✮ ♥➳✉ ♥â ❧✐➯♥ tö❝ t↕✐ ♠å✐ ✤✐➸♠ ❝õ❛ X ✳ ✶✳✶✳✻ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû X ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ d : X × X → R✳ ❍➔♠ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠➯tr✐❝ tr➯♥ X ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥ ✭✐✮ 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 X ũ ợ ởt tr tr õ ữủ ổ tr ỵ (X, d) ❤♦➦❝ X ✳ d ✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ▼ët ❞➣② {xn} tr♦♥❣ ❣å✐ ❧➔ ợ > 0, tỗ t n0 ∈ N s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n ✈➔ m n0 t❤➻ d(xn, xm) < ε✳ ▼å✐ ❞➣② ❤ë✐ tö ❧➔ ❞➣② ❈❛✉❝❤②✳ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ❣å✐ ❧➔ ✤➛② ✤õ ♥➳✉ ♠å✐ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✤➲✉ X ✺ ❤ë✐ tö✳ ❚➟♣ ❝♦♥ A ⊂ X ❣å✐ ❧➔ t➟♣ ✤➛② ✤õ ♥➳✉ ♥â ✤➛② ✤õ ✈ỵ✐ ♠➯tr✐❝ ❝↔♠ s✐♥❤✳ ▼å✐ t➟♣ ❝♦♥ ✤➛② ✤õ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ t➟♣ ✤â♥❣✱ ♠å✐ t➟♣ ❝♦♥ ✤â♥❣ ❝õ❛ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ❧➔ t➟♣ ✤➛② ✤õ✳ ✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ K = R ❤♦➦❝ K = C✳ ❍➔♠ p : E → R ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝❤✉➞♥ tr➯♥ E ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✭✐✮ p(x) 0, ∀x ∈ E ✈➔ p(x) = ⇔ x = 0❀ ✭✐✐✮ p(xλ) = |λ|p(x)✱ ∀x ∈ E ✱ ∀λ ∈ K❀ ✭✐✐✐✮ p(x + y) p(x) + p(y) ∀x, y ∈ E ✳ ❙è p(x) ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉➞♥ ❝õ❛ ✈❡❝tì X ∈ E ✳ ❚❛ t❤÷í♥❣ ❦➼ ❤✐➺✉ ❝❤✉➞♥ ❝õ❛ x ❧➔ x ✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì E ❝ị♥❣ ợ ởt tr õ ữủ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ✶✳✶✳✾ ▼➺♥❤ ✤➲✳ ◆➳✉ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤➻ ❝æ♥❣ t❤ù❝ d(x, y) = x − y , ∀x, y ∈ E, ①→❝ ✤à♥❤ ♠ët ♠➯tr✐❝ tr➯♥ E ✳ ❚❛ ❣å✐ ♠➯tr✐❝ ♥➔② ❧➔ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ ❤❛② ♠➯tr✐❝ ❝❤✉➞♥✳ ▼ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ t❤❡♦ ♠➯tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ỵ E ổ ❝❤✉➞♥ t❤➻ →♥❤ ①↕ ❝❤✉➞♥✿ x → x , ∀x ∈ E ❀ ♣❤➨♣ ❝ë♥❣ ✿(x, y) → x + y, ∀(x, y) ∈ E × E ✈➔ ♣❤➨♣ ♥❤➙♥ ợ ổ ữợ (, x) x, ợ (λ, x) ∈ K × E ❧➔ ❝→❝ →♥❤ ①↕ tử ỵ sỷ E ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ a ∈ E ✈➔ ♠é✐ λ ∈ K, λ = ❝→❝ →♥❤ ①↕ x → x + a, x → λx , x E ỗ ổ E ❧➯♥ E ✳ X✳ ✶✳✶✳✶✷ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ t➟♣ ❤ñ♣ X ✈➔ ❧➔ ♠ët q✉❛♥ ❤➺ ❤❛✐ ♥❣æ✐ tr➯♥ ◗✉❛♥ ❤➺ ✤÷đ❝ ❣å✐ ❧➔ q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭✐✮ x x ✈ỵ✐ ♠å✐ x ∈ X ❀ ✭✐✐✮ ❚ø x y ✈➔ y x s✉② r❛ x = y ✈ỵ✐ ♠å✐ x, y ∈ X ❀ ✭✐✐✐✮ x y❀ y z s✉② r❛ x z ✈ỵ✐ ♠å✐ x, y, z ∈ X ✳ ❚➟♣ ❤đ♣ X ❝ị♥❣ ✈ỵ✐ ♠ët t❤ù tü ❜ë ♣❤➟♥ tr➯♥ ♥â ✤÷đ❝ ❣å✐ ❧➔ t➟♣ s tự tỹ ỵ (X, ) ❤♦➦❝ X ✳ ✶✳✶✳✶✸ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû f : X−→X ✈➔ g : X−→ X ✳ ✣✐➸♠ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ♥➳✉ f (x) = x✳ ✣✐➸♠ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ g ♥➳✉ x f (x) = g(x)✳ = ✶✳✷ ◆➶◆ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ❇❆◆❆❈❍ ▼ư❝ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ✈➜♥ ✤➲ ❝ì ❜↔♥ ✈➲ ♥â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳✭❬✹❪✮ ❈❤♦ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ tr➯♥ tr÷í♥❣ sè t❤ü❝ R✳ ▼ët t➟♣ ❝♦♥ P ❝õ❛ E ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥â♥ tr♦♥❣ E ♥➳✉✿ ✭✐✮ P ❧➔ ✤â♥❣✱ P = ∅✱ P = {0}❀ ✭✐✐✮ ❱ỵ✐ a, b ∈ R✱ a, b ✈➔ x, y ∈ P t❤➻ ax + by ∈ P ❀ ✭✐✐✐✮ ◆➳✉ x ∈ P ✈➔ −x ∈ P t❤➻ x = 0✳ ✶✳✷✳✷ ❱➼ ❞ư✳ ✶✮ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ sè t❤ü❝ R ✈ỵ✐ ❝❤✉➞♥ t❤ỉ♥❣ t❤÷í♥❣✱ ✼ t➟♣ P = {x ∈ R : x 0} ❧➔ ♠ët ♥â♥✳ ✷✮ ●✐↔ sû E = R2, P = {(x, y) ∈ E : x, y 0} ⊂ R2✳ ❑❤✐ ✤â✱ P t❤ä❛ ♠➣♥ ❜❛ ✤✐➲✉ ❦✐➺♥ ✭✐✮ P ❧➔ t➟♣ ✤â♥❣✱ P = ∅, P = {0}❀ ✭✐✐✮ ❱ỵ✐ ♠å✐ (x, y), (u, v) ∈ P ✈➔ ♠å✐ a, b ∈ R, a, b t❛ ❝â a(x, y) + b(u, v) ∈ P ❀ ✭✐✐✐✮ ❱ỵ✐ (x, y) ∈ P ✈➔ (−x, −y) ∈ P t❛ ❝â (x, y) = (0, 0)✳ ❱➟② P ❧➔ ♠ët ♥â♥ tr➯♥ E ✳ ✸✮ ●✐↔ sû C[a,b] ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ sè ♥❤➟♥ ❣✐→ trà t❤ü❝ ❧✐➯♥ tö❝ tr➯♥ [a, b]✳ ❚❛ ✤➣ ❜✐➳t C[a,b] ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ f = sup |f (x)| ∀f ∈ C[a,b] x∈[a,b] ❚r➯♥ C[a,b] ❝â q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ t❤ỉ♥❣ t❤÷í♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✈ỵ✐ f, g ∈ C[a,b]✱ g ⇔ f (x) f g(x) ∀x ∈ [a, b] ✣➦t P = {f ∈ C[a,b] : f } ❑❤✐ ✤â✱ P t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✐✮ P ❧➔ t➟♣ ✤â♥❣✱ P = ∅, P = {0}❀ ✭✐✐✮ ❱ỵ✐ ♠å✐ a, b ∈ R, a, b ✈➔ ♠å✐ f, g ∈ P t❛ ❝â af (x) + bg(x) ∀x ∈ [a, b] ❉♦ ✤â af + bg ∈ P ❀ ✭✐✐✐✮ ❱ỵ✐ f ∈ P ✈➔ −f ∈ P t❛ ❝â f = 0✳ ❱➟② P ❧➔ ♠ët ♥â♥ tr➯♥ E ✳ ✽ ❈❤♦ P ❧➔ ♠ët ♥â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✳ ❚r➯♥ E ✱ t❛ ✤à♥❤ ♥❣❤➽❛ q✉❛♥ ❤➺ t❤ù tü ” ” ①→❝ ✤à♥❤ ❜ð✐ P ♥❤÷ s❛✉ x y ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ y − x ∈ P ✳ ❚❛ ✈✐➳t x < y ♥➳✉ x y ✈➔ x = y ✈➔ ✈✐➳t x y ♥➳✉ y − x ∈ intP ✳ E✳ ✶✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳✭❬✹❪✮ ❈❤♦ P ❧➔ ♠ët ♥â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✶✮ ◆â♥ P ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ tỗ t số tỹ K > s ✈ỵ✐ ♠å✐ x, y ∈ E ✈➔ x y t❛ ❝â x K y ✳ ❙è t❤ü❝ ❞÷ì♥❣ K ♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ ❝õ❛ P ✳ ✷✮ ◆â♥ P ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ❝❤➼♥❤ q✉② ♥➳✉ ♠å✐ ❞➣② t➠♥❣ ✈➔ ❜à ❝❤➦♥ tr➯♥ tr♦♥❣ E ✤➲✉ ❤ë✐ tö ✭♠ët ❝→❝❤ t÷ì♥❣ ✤÷ì♥❣ ❧➔ ♠å✐ ❞➣② ❣✐↔♠ ✈➔ ❜à ❝❤➦♥ ữợ tr E tử {xn} tr E s x1 x2 t tỗ t↕✐ x ∈ E s❛♦ ❝❤♦ ··· ··· xn xn − x → y ✈ỵ✐ y ∈ E, ❦❤✐ n ỵ s ố q✉❛♥ ❤➺ ❣✐ú❛ ♥â♥ ❝❤➼♥❤ q✉② ✈➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ ỵ õ q tr ổ ❇❛♥❛❝❤ ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ ❧➔ ♥â♥ ❝❤➼♥❤ q✉② ♥❤÷♥❣ ❦❤ỉ♥❣ ❝❤✉➞♥ t➢❝✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ n t❛ ❝❤å♥ ✤÷đ❝ tn, sn ∈ P s❛♦ ❝❤♦ tn − sn ∈ P ✈➔ tn sn n2 tn < sn ✳ ❱ỵ✐ ♠é✐ n 1✱ ✤➦t yn = ✈➔ xn = ❚❛ ❝â tn tn xn , yn , yn − xn ∈ P, yn = ✈➔ n2 xn ∞ y ∞ ∞ y n n ❱➻ ❝❤✉é✐ = y = ❤ë✐ tö ♥➯♥ ❤ë✐ tö t✉②➺t ✤è✐ 2 n n n2 ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû n=1 P n=1 tr♦♥❣ E ✳ ❉♦ E ❇❛♥❛❝❤ ♥➯♥ ❝❤✉é✐ ∞ yn n=1 n ✾ n=1 ❤ë✐ tö tr♦♥❣ E ✳ ❚ø P ✤â♥❣ s✉② ∞ r❛ tỗ t y P s nyn2 = y ❇➙② ❣✐í✱ tø xn n=1 ❝õ❛ ❝❤✉é✐ ♥➯♥ t❛ s✉② r❛ x1 x1 + ❱➻ P ❝❤➼♥❤ q✉② ♥➯♥ ❝❤✉é✐ x2 22 x1 + ∞ xn n=1 n n→∞ xn ✈➔ ❝→❝❤ ①→❝ ✤à♥❤ ··· y ❤ë✐ tö✳ ❙✉② r❛ xn = 0, n2 lim ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ n2 x2 x3 + 22 32 yn ✳ ❱➟② P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ ✶✳✷✳✺ ◆❤➟♥ ①➨t✳✭❬✹❪✮ ✣✐➲✉ ữủ ỵ tr õ ổ ✤ó♥❣ ♥❣❤➽❛ ❧➔ ❝â ♥❤ú♥❣ ♥â♥ ❝❤✉➞♥ t➢❝ ♥❤÷♥❣ ❦❤ỉ♥❣ ❝❤➼♥❤ q✉②✳ ❚❤➟t ✈➟②✱ ①➨t ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E = C[0;1] ✈ỵ✐ ❝❤✉➞♥ sup : f = sup |f (x)|✳ ✣➦t x∈[0,1] P = {f ∈ E : f 0}✳ ❑❤✐ ✤â✱ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ K = 1✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû f, g ∈ E ✈➔ f g ✳ ❑❤✐ ✤â✱ f (x) g(x) ✈ỵ✐ ♠å✐ x ∈ [0; 1]✳ ❙✉② r❛ f = sup |f (x)| = sup f (x) x∈[0,1] sup g(x) = sup |g(x)| = g , x∈[0,1] x∈[0,1] x∈[0,1] ❝❤ù♥❣ tä P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ ❇➙② ❣✐í✱ t❛ ❝❤ù♥❣ ♠✐♥❤ P ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♥â♥ ❝❤➼♥❤ q✉②✳ ❚❤➟t ✈➟②✱ ①➨t ❞➣②{fn} ∈ E ①→❝ ✤à♥❤ ữ s fn(x) = xn ợ x [0; 1]✳ ❑❤✐ ✤â✱ ··· xn · · · x2 x, ∀x ∈ [0; 1] ❘ã r➔♥❣ {fn} ❧➔ ❞➣② ữợ ữ ổ tử tr P ✳ ❱➟② P ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♥â♥ ❝❤➼♥❤ q✉②✳ ✶✳✷✳✻ ▼➺♥❤ ✤➲✳✭❬✹❪✮ ◆➳✉ K ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ ❝õ❛ ♥â♥ P t❤➻ K ✶✵ ré♥❣✳ ❑❤✐ ✤â✱ ♥➳✉ ♠å✐ ❞➙② ❝❤✉②➲♥ tr♦♥❣ X ✤➲✉ ❝â ❝➟♥ tr➯♥ t❤➻ tr♦♥❣ X ❝â ♣❤➛♥ tû ❝ü❝ ✤↕✐✳ ✷✳✶✳✸ ❇ê ✤➲✳✭❬✸❪✮ ●✐↔ sû (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ ϕ : X → P ✱ λ ❧➔ ♠ët sè t❤ü❝ ❞÷ì♥❣✱ ✈➔ ϕ ❧➔ ♠ët q✉❛♥ ❤➺ tr➯♥ X ✤÷đ❝ ❝❤♦ ❜ð✐ x ❑❤✐ ✤â✱ ϕ ϕ y ⇔ λd(x, y) ϕ(y) − ϕ(x); x, y ∈ X ❧➔ ♠ët t❤ù tü ❜ë ♣❤➟♥ tr➯♥ X ✳ ❱➻ d(x, x) = ✈ỵ✐ ♠å✐ x ∈ X ♥➯♥ x ϕ x ✈ỵ✐ ♠å✐ x ∈ X ✳ ●✐↔ sû x, y ∈ X s❛♦ ❝❤♦ x ϕ y ✈➔ y ϕ x✳ ❑❤✐ ✤â✱ t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ λd(x, y) ϕ(y) − ϕ(x) ✈➔ λd(x, y) ϕ(x) − ϕ(y) = −(ϕ(y) − ϕ(x)) ❉♦ ✤â d(x, y) = ϕ(y) − ϕ(x) = 0✱ tù❝ ❧➔ x = y✳ ●✐↔ sû x, y, z ∈ X s❛♦ ❝❤♦ x ϕ y ✈➔ y ϕ z✳ ❑❤✐ ✤â✱ t❛ ❝â λd(x, z) λ[d(x, y) + d(y, z)] ϕ(y) − ϕ(x) + ϕ(z) − ϕ(y) = ϕ(z) − ϕ(x) ❉♦ ✤â x ϕ z✳ ❱➟② ϕ ❧➔ ♠ët t❤ù tü tr X r ỵ s t ❣✐↔ t❤✐➳t ♥â♥ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈➔ ▼✐♥✐❤❡❞r❛❧ ♠↕♥❤✱ tù❝ ❧➔ ♠å✐ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ✈➔ ❜à ữợ tr E õ ữợ ú ỵ sỷ (X, d) ổ tr ♥â♥ ✤➛② ✤õ✱ ϕ : X → P ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â✱ ♥➳✉ T : X → X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ λd(x, T (x)) ϕ(x) − ϕ(T (x)), ∀x ∈ X t❤➻ T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣✱ tr♦♥❣ ✤â λ ❧➔ ♠ët sè t❤ü❝ ❞÷ì♥❣ ♥➔♦ ✤â✳ ✷✸ ❚❤❡♦ ❇ê ✤➲ 2.1.2 t❤➻ (X, ϕ) ❧➔ t➟♣ ✤÷đ❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ●✐↔ sû Y ❧➔ ♠ët ❞➙② ❝❤✉②➲♥ tr♦♥❣ X ✳ ❑❤✐ ✤â✱ ✈➻ t➟♣ {ϕ(y) : y ∈ Y } ❦❤→❝ ré♥❣ ✈➔ ❜à ữợ tỗ t ự inf {(y) : y ∈ Y } := c ∈ P ❚❛ s➩ ự tọ Y õ ữợ tr (X, ) tỗ t x Y s (x) = c t❤➻ x = inf Y ✳ ❚❤➟t ✈➟②✱ ♥➳✉ y ∈ Y ♠➔ y ϕ x ✈➔ y = x t❤➻ ϕ(x) − ϕ(y), < λd(x, y) tù❝ ❧➔ ϕ(y) ϕ(x) = c✳ ✣➙② ❧➔ ♠ët ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ●✐↔ sû c < ϕ(y) ✈ỵ✐ ♠å✐ y ∈ Y ✳ ❑❤✐ ✤â ✈➻ c = inf {ϕ(y) : y ∈ Y } ♥➯♥ ✈ỵ✐ d ♥➔♦ ✤â t❤✉ë❝ intP tỗ t y1 Y s c < (y1) < c + d ố d ữỡ tỹ tỗ t↕✐ y2 ∈ Y s❛♦ ❝❤♦ d c < ϕ(y2 ) < inf (c + , ϕ(y1 )) ❱➻ Y ❧➔ ❞➙② ❝❤✉②➲♥ ♥➯♥ y1 ϕ y2 ❤♦➦❝ y2 ϕ y1✳ ❉♦ ✤â✱ tø ϕ(y2) < ϕ(y1) s✉② r❛ y2 ϕ y1✳ ❚✐➳♣ tư❝ s✉② ❧✉➟♥ t÷ì♥❣ tü t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ❞➣② {yn} tr♦♥❣ Y s❛♦ ❝❤♦ yn+1 yn ✈➔ c < ϕ(yn ) < c + d n ∀n = 1, 2, ❚ø P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈➔ nd → ❦❤✐ n → ∞✱ →♣ ❞ö♥❣ ▼➺♥❤ ✤➲ 1.2.10 t❛ s✉② r❛ ϕ(yn) → c ❦❤✐ n → ∞✳ ❱ỵ✐ ♠é✐ n = 1, 2, ✈➔ k ∈ N✱ ✈➻ yn+k ϕ yn ♥➯♥ ϕ(yn ) − ϕ(yn+k ) λd(yn+k , yn ) ▼➦t ❦❤→❝✱ ✈➳ ♣❤↔✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❞➛♥ tỵ✐ ❦❤ỉ♥❣ ❦❤✐ n → ∞ ♥➯♥ s✉② r❛ λd(yn+k , yn) → ❦❤✐ n → ∞ ✈ỵ✐ ♠å✐ k ∈ N✳ ❱➻ λ > ♥➯♥ ✷✹ ❦❤✐ n → ∞ ✈ỵ✐ ♠å✐ k ∈ N✳ ❉♦ ✤â {yn} ❧➔ ❞➣② ❈❛✉❝❤②✳ X tỗ t x X s❛♦ ❝❤♦ yn → x✳ ❑➳t ❤đ♣ ✈ỵ✐ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ϕ t❛ ❝â d(yn+k , yn ) → c = lim ϕ(yn ) = ϕ(x) n→∞ ❇➙② ❣✐í✱ t❛ ❝❤ù♥❣ ♠✐♥❤ x = inf Y ✳ ❱ỵ✐ ♠é✐ k = 1, 2, t❛ ❝â yn ✈ỵ✐ ♠å✐ n > k tù❝ ❧➔ λd(yn , yk ) ϕ(yk ) − ϕ(yn ), ϕ yk ✈ỵ✐ ♠å✐ n > k ❈❤♦ n → ∞✱ tø t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ d ✈➔ ϕ✱ →♣ ❞ö♥❣ ✣à♥❤ ỵ 1.1.8 t õ d(x, yk ) (yk ) − ϕ(x), ∀k = 1, 2, , tù❝ ❧➔ x ϕ yk ✈ỵ✐ ♠å✐ k = 1, 2, ✳ ●✐↔ sû y ∈ Y õ tỗ t n N s ❝❤♦ yn ϕ y t❤➻ x ϕ y ✳ ●✐↔ sû y ϕ yn ✈ỵ✐ ♠å✐ n✱ tù❝ ❧➔ λd(y, yn ) ϕ(yn ) − ϕ(y), ∀n ❚÷ì♥❣ tü ♥❤÷ tr➯♥✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② s✉② r❛ ϕ(x) − ϕ(y) λd(y, x) ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ϕ(x) = c = inf ϕ(Y ) ✈➔ c ∈/ (Y ) ữ x y ợ y ∈ Y ✳ ●✐↔ sû z ϕ yn ✈ỵ✐ ♠å✐ n = 1, 2, ✱ tù❝ ❧➔ λd(z, yn ) ϕ(yn ) − ϕ(z), ∀n ❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝â λd(z, x) ϕ(x) − ϕ(z), tù❝ ❧➔ z ϕ x✳ ❱➟② x = inf Y ✳ ❉♦ ✤â t❤❡♦ ❇ê ✤➲ ❩♦r♥❡ X ❝â ♣❤➛♥ tû ❝ü❝ t✐➸✉✱ ❦➼ ❤✐➺✉ ❧➔ a✳ ❚❤❡♦ ✤✐➲✉ ❦✐➺♥ ❝õ❛ T t❛ ❝â λd(a, T (a)) ϕ(a) − ϕ(T (a)), ✷✺ tù❝ ❧➔ T (a) ϕ a✳ ❱➻ a ❧➔ ♣❤➛♥ tû ❝ü❝ t✐➸✉ ❝õ❛ X ♥➯♥ T (a) = a✳ ❱➟② a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ỵ sỷ P õ q✉② tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✱ t❤ù tü tr➯♥ E ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ P ❧➔ t✉②➳♥ t➼♥❤ tr➯♥ P ✭tù❝ ❧➔ ✈ỵ✐ ♠å✐ a, b ∈ P t❤➻ a b ❤♦➦❝ b ✤õ ✈ỵ✐ t❤ù tü ❜ë ♣❤➟♥ a✮ ✈➔ (X, d, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✳ ❑❤✐ ✤â✱ ♥➳✉ f : X → X ❧➔ ❤➔♠ ❧✐➯♥ tư❝ ❦❤ỉ♥❣ ❣✐↔♠ t❤ä❛ ♠➣♥ ✶✮ ❱ỵ✐ ộ c P, < c tỗ t (c) ∈ intP s❛♦ ❝❤♦ tø c c + δ(c) ✈➔ x d(x, y) < y s✉② r❛ d(f (x), f (y)) < c ỗ t x0 X s ❝❤♦ x0 f (x0 ) t❤➻ f ❝â ✤✐➸♠ ❜➜t x0 = f (x0) t ỵ ữủ ❝❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x0 < f (x0) ✈➔ f ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐↔♠✳ ❇➡♥❣ q✉② ♥↕♣✱ t❛ t❤✉ ✤÷đ❝ ❈❤ù♥❣ ♠✐♥❤✳ x0 < f (x0 ) f (x0 ) f (x0 ) ··· f n (x0 ) f n+1 (x0 ) ✣➦t xn = f n(x0), n = 1, 2, ✳ ❑❤✐ ✤â✱ ❞➣② {xn} ❧➔ ❞➣② ❦❤æ♥❣ ❣✐↔♠✱ tù❝ {xn } ❧➔ ❞➣② t➠♥❣✳ ✣➛✉ t✐➯♥✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ n→∞ lim d(xn , xn+1 ) = 0✳ ◆➳✉ ❞➣② {xn } ổ t t t tỗ t n0 N s❛♦ ❝❤♦ xn = xn +1 = f (xn )✳ ❉♦ ✤â✱ xn ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✈➔ ỵ ữủ ự {xn} t ♥❣➦t t❤➻ ❞➣② {d(xn, xn+1)} ❧➔ ❞➣② ❣✐↔♠ ♥❣➦t✳ ❚❤➟t ✈➟②✱ ✈➻ {xn} ❧➔ ❞➣② t➠♥❣ ♥❣➦t ♥➯♥ xn < xn+1 ✈ỵ✐ ♠å✐ n✳ ❉♦ ✤â✱ < d(xn , xn+1 )✳ ⑩♣ ❞ö♥❣ 1) ❝❤♦ c = d(xn , xn+1 ) t❛ ❝â 0 d(xn+1 , xn+2 ) = d(f (xn ), f (xn+1 )) < d(xn , xn+1 ) ✷✻ ✈ỵ✐ ♠å✐ n ∈ N ❉♦ ✤â {d(xn, xn+1}) ❧➔ ❞➣② ❣✐↔♠ ♥❣➦t✳ ❱➻ P ❧➔ ♥â♥ ❝❤➼♥❤ q✉② ♥➯♥ {d(xn, xn+1)} → c ∈ P ✳ ❚❛ ❝â c < d(xn , xn+1 ) ✈ỵ✐ ♠å✐ n = 0, 1, ✳ ❚❤➟t ✈➟②✱ ✤➦t cn = d(xn , xn+1 ), n = 0, 1, tỗ t n0 ∈ N s❛♦ ❝❤♦ cn c t❤➻ cn < cn c ✈ỵ✐ ♠å✐ n > n0✳ ❉♦ ✤â c − cn < c − cn ✈ỵ✐ ♠å✐ n > n0 ✭✈➻ c − cn − (c − cn ) = cn − cn ∈ P ✈ỵ✐ ♠å✐ n > n0✮✳ ❱➻ P ❧➔ ♥â♥ ❝❤➼♥❤ q✉② ♥➯♥ P ❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ K ✳ ❉♦ ✤â t❛ ❝â 0 0 0 < c − c n0 < K c − c n ✈ỵ✐ ♠å✐ n > n0 ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ cn → c ✳ ❉♦ ✤â c < cn ✈ỵ✐ ♠å✐ n✳ ✣➦t d = inf {cn : n = 0, 1, }✳ ❑❤✐ ✤â✱ d cn ✈ỵ✐ ♠å✐ n = 0, 1, ✳ ❉♦ ✤â✱ t❤❡♦ ❇ê ✤➲ 1.2.7 ①✮ t❛ ❝â d n→∞ lim cn = c ✳ ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ inf imum✱ t❛ ❝â c = d✳ ❱➟② d(xn , xn+1 ) → inf {d(xj , xj+1 ) : j = 0, 1, }✳ ❇➙② ❣✐í✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ c = 0✳ ●✐↔ sû c > 0✳ ⑩♣ ❞ö♥❣ ✤✐➲✉ ❦✐➺♥ 1) ❝❤♦ c = c ✱ ❝â t❤➸ t➻♠ δ(c ) ∈ intP s❛♦ ❝❤♦ tø c ❱➻ c ✈➔ x < y s✉② r❛ d(f (x), f (y)) < c d(x, y) < c + δ(c ) = lim d(xn , xn+1 )✱ n→∞ c õ t ủ ợ xn tỗ t n0 ∈ N s❛♦ ❝❤♦ d(xn , xn+1 ) < c + δ(c ) ∀n < xn0 +1 n0 t❛ ❝â d(xn0 +1 , xn0 +2 ) = d(f (xn0 ), f (xn0 +1 )) < c ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈➻ c = inf {d(xn, xn+1)}✳ ❱➟② c = ❤❛② n→∞ lim d(xn , xn+1 ) = 0✳ ❚✐➳♣ t❤❡♦✱ t❛ ❝❤ù♥❣ tä {xn} ❧➔ ❞➣② ❈❛✉❝❤②✳ ❱ỵ✐ ♠é✐ c ∈ intP ✱ tø ✤✐➲✉ 1) s r tỗ t (c) intP õ t❤➸ ❝❤å♥ δ(c) c✮ s❛♦ ❝❤♦ tø c d(x, y) < c + δ(c) ✈➔ x < y s✉② r❛ d(f (x), f (y)) < c ✷✼ ▼➦t ❦❤→❝✱ tø d(xn, xn+1) s r tỗ t n0 N s❛♦ ❝❤♦ < d(xn−1 , xn ) < δ(c) ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ n n0 ✈ỵ✐ ♠å✐ n n0 ✭✷✳✶✮ ❜➡♥❣ q✉② ♥↕♣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ d(xn , xn+p ) c ✈ỵ✐ p = 1, 2, ✭✷✳✷✮ ❚❤➟t ✈➟②✱ tø (2.1)✱ →♣ ❞ư♥❣ ✤✐➲✉ ❦✐➺♥ 1) ✈ỵ✐ c = d(xn−1, xn) ✈➔ sû ❞ö♥❣ δ(c) c t❛ ❝â d(xn , xn+1 ) < c ữ (2.2) ú ợ p = 1✳ ●✐↔ sû (2.2) ✤ó♥❣ ✈ỵ✐ p > 1✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ (2.2) ✤ó♥❣ ✈ỵ✐ p + 1✳ ❙û ❞ư♥❣ (2.1) ✈➔ ❣✐↔ t❤✐➳t q✉② ♥↕♣ ✈ỵ✐ ♠å✐ n n0 ✈➔ ♠å✐ p > t❛ ❝â d(xn−1 , xn+p ) d(xn−1 , xn ) + d(xn , xn+p ) < δ(c) + c ❇➙② ❣✐í✱ t❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ ❚r÷í♥❣ ❤đ♣ ✶✿ c d(xn−1, xn+p)✳ ❑❤✐ ✤â✱ t❛ ❝â c d(xn−1 , xn+p ) < c + δ(c) ❉♦ ✤â tø xn−1 < xn+p s✉② r❛ d(xn , xn+p+1 ) = d(f (xn−1 ), f (xn+p )) < c ữ (2.2) ú ợ p + ❚r÷í♥❣ ❤đ♣ ✷✿ d(xn−1, xn+p) < c✳ ❑❤✐ ✤â✱ ✈➻ {xn} ❧➔ ❞➣② t➠♥❣ ♥❣➦t ♥➯♥ xn−1 < xn < · · · < xn+p ✳ ❉♦ ✤â d(xn−1 , xn+p ) > 0✳ ⑩♣ ❞ư♥❣ ✤✐➲✉ ❦✐➺♥ 1) ✈ỵ✐ c = d(xn−1, xn+p) t❛ ❝â d(xn , xn+p+1 ) = d(f (xn−1 ), f (xn+p )) < d(xn−1 , xn+p ) < c ✷✽ ❱➟② (2.2) ✤ó♥❣ ✈ỵ✐ p + 1✳ ❉♦ ✤â (2.2) ✤ó♥❣ ✈ỵ✐ ♠å✐ n n0 ✈➔ ♠å✐ p ∈ N ♥➯♥ {xn} ❧➔ ❞➣② ❈❛✉❝❤②✳ ứ X ổ tỗ t z ∈ X s❛♦ ❝❤♦ n→∞ lim xn = z ✳ ❉♦ f ❧➔ ❧✐➯♥ tö❝ ♥➯♥ z = n→∞ lim xn+1 = lim f (xn ) = f (z)✳ ❱➟② z ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ n→∞ ❝õ❛ f ✳ ✷✳✷ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈❍❯◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆ ❱❰■ ❚❍Ù ❚Ü ❇❐ P❍❾◆ ❚r♦♥❣ ♠ö❝ ♥➔②✱ t❛ ❣✐↔ t❤✐➳t P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ E ✱ intP = ∅ ✷✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳✭❬✻❪✮ ❈➦♣ tü →♥❤ ①↕ (f, g) tr➯♥ t➟♣ s➢♣ t❤ù tü ❜ë ♣❤➟♥ (X, ) ✤÷đ❝ ❣å✐ ❧➔ t➠♥❣ ②➳✉ ♥➳✉ f (x) g(f (x)) ✈➔ g(x) f (g(x)) ✈ỵ✐ ♠å✐ x ∈ X ✳ ỵ (X, tự tỹ tỹ , d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✈ỵ✐ , (f, g) ❧➔ ❝➦♣ tü →♥❤ ①↕ t➠♥❣ ②➳✉ tr➯♥ X ✈ỵ✐ q✉❛♥ ❤➺ t❤ù ♥â✐ tr➯♥✳ ●✐↔ sû s ữủ tọ ỗ t↕✐ p, q, r, s, t 0✱ t❤ä❛ ♠➣♥ p + q + r + s + t < ✈➔ q = r ❤♦➦❝ s = t s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ X ♠➔ ❝❤ó♥❣ s♦ s→♥❤ ữủ ợ t õ d(f (x), g(y)) pd(x, y)+qd(x, f (x))+rd(y, g(y))+sd(x, g(y))+td(y, f (x)) ✭✷✳✸✮ ✐✐✮ f ❤♦➦❝ g ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ ❤♦➦❝ ✐✐✬✮ ◆➳✉ ❞➣② {xn } ❦❤æ♥❣ ❣✐↔♠ ✈➔ xn → x ∈ X t❤➻ xn n ∈ N✳ ❑❤✐ ✤â✱ f ✈➔ g ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ x∗ ∈ X ✳ x ợ x0 X tũ ỵ ✈➔ ①→❝ ✤à♥❤ ❞➣② {xn} ❜ð✐ x2n+1 = f (x2n) ✈➔ x2n+2 = g(x2n+1) ✈ỵ✐ ♠å✐ n ∈ N✳ ❑❤✐ ✤â✱ sû ❞ö♥❣ ❣✐↔ t❤✐➳t (f, g) ❧➔ ❝➦♣ tü →♥❤ ①↕ t➠♥❣ ②➳✉ tr➯♥ X t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ {xn} ❧➔ ❞➣② ❦❤ỉ♥❣ ❣✐↔♠ ✈ỵ✐ q✉❛♥ ❤➺ t❤ù tü ♥❣❤➽❛ ❧➔ x0 x1 · · · xn xn+1 ✳ ❱➻ x2n x2n+1 ♥➯♥ →♣ ❞ö♥❣ (2.3) t❛ t❤✉ ✤÷đ❝ ❈❤ù♥❣ ♠✐♥❤✳ d(x2n+1 , x2n+2 ) = d(f (x2n ), g(x2n+1 )) pd(x2n , x2n+1 ) + qd(x2n , x2n+1 ) + rd(x2n+1 , x2n+2 ) + sd(x2n , x2n+2 ) + td(x2n+1 , x2n+1 ) pd(x2n , x2n+1 ) + qd(x2n , x2n+1 ) + s[d(x2n , x2n+1 ) + d(x2n+1 , x2n+2 )] + rd(x2n+1 , x2n+2 ) ❉♦ ✤â (1 − r − s)d(x2n+1 , x2n+2 ) ❤❛② d(x2n+1 , x2n+2 ) (p + q + s)d(x2n , x2n+1 ), p+q+s d(x2n , x2n+1 ) − (r + s) ✭✷✳✹✮ ❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ t÷ì♥❣ tü t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ d(x2n+2 , x2n+3 ) p+r+t p+q+s d(x2n , x2n+1 ) − (q + t) − (r + s) ✭✷✳✺✮ ❚ø (2.4) ✈➔ (2.5) ❜➡♥❣ ♣❤➨♣ q✉② ♥↕♣ t❛ t❤✉ ✤÷đ❝ d(x2n+1 , x2n+2 ) p+q+s d(x2n , x2n+1 ) − (r + s) p+q+s p+r+t d(x2n−1 , x2n ) − (r + s) − (q + t) p+q+s p+r+t p+q+s d(x2n−2 , x2n−1 ) − (r + s) − (q + t) − (r + s) p+q+s p+r+t p+q+s n ··· [ ] d(x0 , x1 ) − (r + s) − (q + t) − (r + s) ✸✵ ✈➔ d(x2n+2 , x2n+3 ) ✣➦t A= p+r+t d(x2n+1 , x2n+2 ) − (q + t) p + r + t p + q + s n+1 ··· [ ] d(x0 , x1 ) − (q + t) − (r + s) p+q+s p+r+t ;B = − (r + s) − (q + t) ❚r♦♥❣ tr÷í♥❣ ❤đ♣ q = r t❛ ❝â A.B = p+q+s p+r+t p+q+s p+r+t = < 1.1 = 1 − (r + s) − (q + t) − (q + t) − (r + t) ◆➳✉ s = t t❛ ❝â A.B = p+q+s p+r+t p+q+s p+r+s = < 1.1 = 1 − (r + s) − (q + t) − (r + s) − (q + t) ❇➙② ❣✐í ✈ỵ✐ n < m t❛ ❝â d(x2n+1 , x2m+1 ) d(x2n+1 , x2n+2 ) + · · · + d(x2m , x2m+1 ) m−1 m i (A (AB) + i=n d(x2n , x2m+1 ) d(x2n , x2m ) ✈➔ i=n+1 (AB)n+1 + )d(x0 , x1 ) − AB − AB A(AB)n = (1 + B) d(x0 , x1 ) − AB ( ❚÷ì♥❣ tü t❛ t❤✉ ✤÷đ❝ A(AB)n (AB)i )d(x0 , x1 ) (AB)n (1 + A) d(x0 , x1 ), − AB (AB)n (1 + A) d(x0 , x1 ) − AB d(x2n+1 , x2m ) A(AB)n (1 + B) − AB ✸✶ ❉♦ ✤â✱ ✈ỵ✐ n < m t❛ ❝â (AB)n A(AB)n ; (1 + A) }d(x0 , x1 ) d(xn , xm ) max{(1 + B) − AB − AB = λn d(x0 , x1 ), tr♦♥❣ ✤â λn → ❦❤✐ n → ∞✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ 1.2.7.ii)✱ ❇ê ✤➲ 1.2.8 ✈➔ ❣✐↔ t❤✐➳t intP = ∅ t❛ ❦➳t ❧✉➟♥ ✤÷đ❝ {xn} ❧➔ ❞➣② ❈❛✉❝❤②✳ ▼➦t ❦❤→❝✱ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➛② tỗ t x X s xn → x∗ ❦❤✐ n → ∞✳ ◆➳✉ f ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✱ t❤➻ t❛ ❝â f (xn) → f (x∗)✳ ❱ỵ✐ n ❝❤➤♥ t❛ ❝â f (xn ) = xn+1 → f (x∗ ) ❦❤✐ n → ∞✳ ▼➦t ❦❤→❝ xn → x∗ ❦❤✐ n → ∞ ✈➔ X ❧➔ T2− ❦❤æ♥❣ ❣✐❛♥ ♥➯♥ f (x∗ ) = x∗ ✭✷✳✻✮ ❱➻ x∗ x∗ ♥➯♥ tr♦♥❣ (2.3) ❝❤♦ x = y = x∗✱ t❛ t❤✉ ✤÷đ❝ d(f (x∗ ), g(x∗ )) pd(x∗ , x∗ ) + qd(x∗ , f (x∗ )) + rd(x∗ , g(x∗ )) + sd(x∗ , g(x∗ )) + td(x∗ , f (x∗ )) ❉♦ ✤â t❛ ❝â d(x∗ , g(x∗ ) (r + s)d(x∗ , g(x∗ ) ❚ø < r + s < 1✱ sû ❞ư♥❣ ❇ê ✤➲ 1.2.7.✐①✮ t❛ ✤÷đ❝ d(x∗, g(x∗)) = ❞♦ ✤â g(x∗ ) = x∗ ✭✷✳✼✮ ❚ø (2.6) ✈➔ (2.7) s✉② r❛ x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ g✳ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü tr♦♥❣ tr÷í♥❣ ❤đ♣ g ❧➔ →♥❤ ①↕ ❧✐➯♥ tư❝✳ ❇➙② ❣✐í✱ ①➨t tr÷í♥❣ ❤đ♣ ❦❤✐ ✤✐➲✉ ❦✐➺♥ (ii ) ✤÷đ❝ t❤ä❛ ♠➣♥✱ ✈ỵ✐ ❞➣② {xn} s❛♦ ❝❤♦ xn → x∗ ∈ X ✈➔ xn x∗ (n ∈ N)✳ ❚ø ❝→❝❤ ①➙② ❞ü♥❣ ❞➣② {xn} t❛ ❝â f (xn) → x∗ ✈➔ g(xn) → x∗ (n → ∞)✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ x∗ ❧➔ ✤✐➸♠ ✸✷ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ g✳ ✣➦t x = x∗ ✈➔ y = xn tr♦♥❣ (2.3) t❛ ❝â d(f (x∗ ), g(xn )) pd(x∗ , xn ) + qd(x∗ , f (x∗ )) + rd(xn , g(xn )) + sd(x∗ , g(xn )) + td(xn , f (x∗ )) pd(x∗ , xn ) + q[d(x∗ , xn ) + d(xn , g(xn ) + d(g(xn ), f (x∗ ))] + rd(xn , g(xn )) + sd(x∗ , g(xn )) + [d(xn , g(xn ) + d(g(xn ), f (x∗ ))] ❚ø ✤â t❛ ❝â (1 − q − t)d(f (x∗ ), g(xn )) (p + q)d(x∗ , xn ) + sd(x∗ , g(xn )) + (q + r + t)d(xn , g(xn ) ∀n (2.8) ❱➻ xn → x∗, g(xn) → x∗ ❦❤✐ n → ∞ ♥➯♥ tø ▼➺♥❤ ✤➲ 1.3.7 s✉② r❛ ✈➳ ♣❤↔✐ ❝õ❛ (2.8) ❞➛♥ tỵ✐ ∈ E ✳ ❑➳t ❤đ♣ ✈ỵ✐ (1 − q − t) > t❛ ❦➳t ❧✉➟♥ ✤÷đ❝ d(f (x∗ ), g(xn )) → ❦❤✐ n → ∞✳ ❉♦ ✤â d(f (x∗ ), x∗ ) = 0✱ tù❝ ❧➔ f (x∗ ) = x∗ ✳ ❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✤è✐ ①ù♥❣ ✈➔ sû ❞ư♥❣ x∗ x∗✱ t❛ ❝â g(x∗) = x∗✳ ❱➟② x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ g ✳ ✷✳✷✳✸ ❱➼ ❞ö✳✭❬✻❪✮ ❈❤♦ X = {1; 2; 3}, ❧➔ t❤ù tü ❜ë ♣❤➟♥ tr➯♥ X ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ {(1, 1); (2, 2); (3, 3); (2, 3); (3, 1); (2, 1)} ✈➔ d : X ì X C[0,1] ữủ d(x, y)(t) = ✈ỵ✐ x = y ✈➔ d(1, 2)(t) = d(2, 1)(t) = 30 t 24 t 6et ❀ d(1, 3)(t) = d(3, 1)(t) = e ❀ d(2, 3)(t) = d(3, 2)(t) = e ✳ ●✐↔ sû✱ 7 f (x) = 1, ∀x ∈ X ✈➔ g(1) = g(3) = 1, g(2) = 3✳ ❑❤✐ ✤â✱ ❝→❝ ỵ 2.2.2 ữủ tọ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✱ (f, g) ❧➔ ❝➦♣ →♥❤ ①↕ t➠♥❣ ②➳✉ ✈➔ f ❧✐➯♥ tư❝ tr➯♥ X ✳ ❇➙② ❣✐í✱ t❛ ❦✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ (2.3)✳ ✣➦t p = q = r = s = 0, t = 57 ✳ ❱➻ f (1) = g(1) = f (2) = f (3) = g(3) = ♥➯♥ t❛ ❝❤➾ ❝➛♥ ❦✐➸♠ tr❛ ✤✐➲✉ ❦✐➺♥ (2.3) ❝❤♦ ❝→❝ ❈❤ù♥❣ ♠✐♥❤✳ ✸✸ ❝➦♣ (2, 2), (2, 3)✳ ✣è✐ ✈ỵ✐ ❝➦♣ (2.3) t❛ ❝â d(f (3), g(2)) ❜ð✐ ✈➻ 0d(3, 2)(t) + 0d(3, f (3))(t) + 0d(2, g(2))(t) 0d(3, g(2))(t) + d(2, f (3))(t), 30 d(f (3), g(2))(t) = d(1, 3)(t) = et ✈➔ 5 30 d(2, f (3))(t) = d(2, 1)(t) = 6et = et 7 7 ❚÷ì♥❣ tü t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ✤✐➲✉ ❦✐➺♥ (2.3) ❝ô♥❣ t❤ä❛ ♠➣♥ ❝❤♦ ❝➦♣ (2.2)✳ ❚❤➟t ✈➟②✱ ❞♦ d(f (2), g(2))(t) = d(1, 3)(t) = 307 et ✈➔ 5 30 d(2, f (2))(t) = d(2, 1)(t) = 6et = et , 7 7 ♥➯♥ d(f (2), g(2)) 0d(2, 2)(t) + 0d(2, f (2))(t) + 0d(2, g(2))(t) 0d(2, g(2))(t) + d(2, f (2))(t) ❱➟② f ✱ g t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ỵ 2.2.2 f, g õ ởt t ✤ë♥❣ ❝❤✉♥❣✳ ✷✳✷✳✹ ❍➺ q✉↔✳ ✭❬✻❪✮❈❤♦ (X, t❤ù tü ❜ë ♣❤➟♥ , d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✈ỵ✐ ✱ f : X → X ❧➔ ♠ët tü →♥❤ ①↕ t❤ä❛ ♠➣♥ x f (x) ✈ỵ✐ ♠å✐ x ∈ X ✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ữủ tọ ỗ t p, q, r, s, t t❤ä❛ ♠➣♥ p + q + r + s + t < ✈➔ q = r ❤♦➦❝ s = t s tỗ t n, m N, m d(f m (x), f n (y)) n ✤➸ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤➙② ✤ó♥❣ pd(x, y) + qd(x, f m (x)) + rd(y, f n (y)) + sd(x, f n (y)) + td(y, f m (x)) ✈ỵ✐ ♠å✐ x, y s s ữủ ợ tr X ✐✐✮ f ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â✱ f ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ x∗ ∈ X ✳ ✸✹ ✣➦t f m = f1, f n = g1✳ ❚❤❡♦ ❣✐↔ t❤✐➳t f ❧✐➯♥ tö❝ s✉② r❛ f1 ❧✐➯♥ tö❝✳ ▲↕✐ ❝â f1(x) = f m(x), g1(f1(x)) = f n(f m(x)) = f n+m(x)✳ ❉♦ x f (x) ♥➯♥ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ f m (x) f n (x) (m n) s✉② r❛ f m (x) f m+n (x) ❤❛② f1 (x) g1 (f1 (x))✳ ❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝ơ♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ g1(x) f1(g1(x))✳ ❱➟② (f1, g1) ❧➔ ❝➦♣ tü →♥❤ ①↕ t➠♥❣ ②➳✉ ♥➯♥ (f1, g1) t❤ä❛ ỵ 2.2.2 õ f1, g1 ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ x∗ ∈ X ✳ ❚ø ❣✐↔ t❤✐➳t x f (x) ✈ỵ✐ ♠å✐ x ∈ X s✉② r❛ ❈❤ù♥❣ ♠✐♥❤✳ x∗ f (x∗ ) f (x∗ ) ··· f m (x∗ ) = x∗ ❉♦ ✤â x∗ = f (x∗)✱ tù❝ x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ❚r♦♥❣ ❍➺ q✉↔ 2.2.4✱ ❝❤♦ m = n = t❛ t❤✉ ữủ q ợ tự tỹ ❈❤♦ (X, , d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✱ f : X → X ❧➔ ♠ët tü →♥❤ ①↕ t❤ä❛ ♠➣♥ x f (x) ✈ỵ✐ ♠å✐ x ∈ X ✳ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ữủ tọ ỗ t p, q, r, s, t t❤ä❛ ♠➣♥ p + q + r + s + t < ✈➔ q = r ❤♦➦❝ s = t s tỗ t n, m N, m d(f (x), (y)) n ✤➸ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤➙② ✤ó♥❣ pd(x, y) + qd(x, f (x)) + rd(y, f (y)) + sd(x, (y)) + td(y, f (x)), ợ x, y s s ữủ ợ tr X ❀ ✐✐✮ f ❧➔ →♥❤ ①↕ ❧✐➯♥ tö❝ ❑❤✐ ✤â✱ f ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ x∗ ∈ X ✳ ✸✺ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✷✳ ❚r➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ö ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✳ ❚r➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ tæ♣æ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✳ ✸✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ♠➔ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t ❤♦➦❝ ❜ä q✉❛ ❝❤ù♥❣ ♠✐♥❤✱ ✤â ❧➔ ❇ê ✤➲ 1.2.7✱ ❱➼ ❞ö 2.2.3✱ ❍➺ q✉↔ 2.2.4✳ ✹✳ ✣÷❛ r❛ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❦➳t q✉↔ ♠ỵ✐ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ tr õ sỹ tỗ t t ✤ë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈ỵ✐ t❤ù tü ❜ë ♣❤➟♥✱ ✤â ❧➔ ❱➼ ❞ö 1.3.2.2)✱ ▼➺♥❤ ✤➲ 1.3.4✱ ❍➺ q 1.3.5 1.3.11 ỵ 2.1.4 ỵ 2.1.5 ỡ s ỵ tt ✈➔ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ❚➟♣ ■✱ ◆❤➔ ❳✉➜t ❇↔♥ ●✐→♦ ❉ư❝✳ ❬✷❪ ❏✳ ❑❡❧❧❡② ✭✶✾✼✸✮✱ ❚ỉ♣ỉ ✤↕✐ ❝÷ì♥❣✱ ❍➔ ❍✉② ỗ ữớ ❤å❝ ✈➔ ❚r✉♥❣ ❤å❝ ❝❤✉②➯♥ ♥❣❤✐➺♣✱ ❍➔ ◆ë✐✳ ❬✸❪ ❆✳ ●r❛♥❛s ❛♥❞ ❏✳ ❉✉❣✉♥❞❥✐ ✭✷✵✵✽✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r②✱ ❙♣r✐♥❣❡r ▼♦♥♦✲ ❣r❛♣❤s ✐♥ ▼❛t❤❡♠❛t✐❝s✳ ❬✹❪ ❍✉❛♥❣ ▲♦♥❣ ✲ ●✉❛♥❣ ❛♥❞ ❩❤❛♥❣ ❳✐❛♥ ✭✷✵✵✼✮✱ ❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❏✳▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✸✷✱ ♥♦✳ ✷✱ ✶✹✻✽ ✲ ✶✹✼✻✳ ❬✺❪ ❏❛❝❦✐❡ ❍❛r❥❛♥✐✱ ❇❡❧➨♥ ▲♦♣➨③ ❛♥❞ ❑✐s❤✐♥ ❙❛❞❛r❛♥❣❛♥✐ ✭✷✵✶✶✮✱ ❆ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❢♦r ▼❡✐r✲❑❡❡❧❡r ❝♦♥tr❛❝t✐♦♥s ✐♥ ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❛♥❞ ❆♣♣❧②❝❛t✐♦♥s✱ ✷✵✶✶✿✽✸✳ ❬✻❪ ❩♦r❛♥ ❑❡❞❡❧❜✉r❣❧✱▼✐r❥❛♥❛ P❛✈❧♦✈✐❝✱ ❙t♦❥♦♥ ❘❡❞❡♥♦✈✐❝ ✭✷✵✶✵✮✱ ❈♦♠✲ ♠♦♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♦r❞❡r❡❞ ❝♦♥tr❛❝t✐♦♥s ❛♥❞ q✉❛s✐❝♦♥tr❛❝✲ t✐♦♥s ✐♥ ♦r❞❡r❡❞ ❝♦♥❡ ♠❡tr✐❝ s♣❛❝❡✱ ❈♦♠♣✉t❡rs ❛♥❞ ♠❛t❤❡♠❛t✐❝s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ◆♦✳✺✾✱ ♣♣✳✸✶✹✽ ✲ ✸✶✺✾✳ ✸✼ ... s✉② r❛ x z ✈ỵ✐ ♠å✐ x, y, z ∈ X ✳ ❚➟♣ ❤đ♣ X ❝ị♥❣ ✈ỵ✐ ♠ët t❤ù tü ❜ë ♣❤➟♥ tr➯♥ ♥â ✤÷đ❝ ❣å✐ ❧➔ t s tự tỹ ỵ (X, ) ❤♦➦❝ X ✳ ✶✳✶✳✶✸ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû f : X−→X ✈➔ g : X−→ X ✳ ✣✐➸♠ x ∈ X ✤÷đ❝ ❣å✐