Sự tồn tại điểm bất động trong không gian mêtric với t khoảng cách

▼Ö❈ ▲Ö❈ ❚r❛♥❣ ✶ ▲❮■ ◆➶■ ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❤÷ì♥❣ ✶✳ ▼ët số ỵ sỹ tỗ t t ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ τ ✲❦❤♦↔♥❣ ❝→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ▼ët sè ỵ sỹ tỗ t t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ✤❛ trà ✈➔ sü tỗ t t ✤ì♥ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ τ ✲❦❤♦↔♥❣ ❝→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ▼Ö❈ ▲Ö❈ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t❤✉②➳t ✤✐➸♠ ❜➜t ✤ë♥❣ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝❤õ ✤➲ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❝❤✉②➯♥ ❣✐❛ ❣✐↔✐ t➼❝❤ ❤➔♠✳ ▼ët sè ❦➳t q✉↔ sỹ tỗ t t t ①✉➜t ❤✐➺♥ tø ✤➛✉ t❤➳ ❦✛ ❳❳✱ tr♦♥❣ ✤â ♣❤↔✐ ỵ tr ổ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✱ ♥➠♠ ✶✾✷✷✳ ❈→❝ ♥❤➔ t♦→♥ ❤å❝ rở ỵ ①↕ ✈➔ ♥❤✐➲✉ ❧♦↕✐ ❦❤ỉ♥❣ ❣✐❛♥✳ ◆❣÷í✐ t❛ t❤÷í♥❣ ❞ü❛ ✈➔♦ ❦❤♦↔♥❣ ❝→❝❤ ✤➸ ✤÷❛ r❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝❤♦ sỹ tỗ t t ởt tr ỳ ữợ rở õ ự sỹ tỗ t t ỡ tr sỹ tỗ t t ỡ tr sỹ tỗ t t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤❛ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tr ợ ỳ ữớ t ữủ ỳ t q q trồ ữợ s ❈✐r➼❝✱ ❚❛t❛r✉✱ ❈❛r✐st✐✳✳✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ❞ü❛ t t ự ỵ tt t sỹ tỗ t t tr ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ τ ✲❦❤♦↔♥❣ ❝→❝❤✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ✤â✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣ ❈❤÷ì♥❣ ✶ ợ ởt số ỵ sỹ tỗ t↕✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà tr ổ tr ợ r ữỡ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ sỹ tỗ t t ✤ì♥ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ τ ✲❦❤♦↔♥❣ ❝→❝❤✳ ▼ư❝ ✤➛✉ t✐➯♥ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❞➔♥❤ ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥ ✈➲ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ →♥❤ ①↕ ❧✐➯♥ tư❝✳✳✳ ♠➔ ❝❤ó♥❣ ❝➛♥ ❞ị♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ▼ư❝ t❤ù ❤❛✐ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ ✈➼ ❞ö ✈➲ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ τ ✲❦❤♦↔♥❣ ❝→❝❤✳ tự tr ởt số ỵ sỹ tỗ t t ✤ì♥ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ τ ữỡ ợ ỹ tỗ t↕✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ✤❛ trà ✈➔ sỹ tỗ t t ①↕ ✤ì♥ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ τ ✲❦❤♦↔♥❣ ❝→❝❤✳ ▼ư❝ ✤➛✉ t✐➯♥ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ✤❛ trà✱ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ tr sỹ tỗ t t →♥❤ ①↕ ✤❛ trà p✲❝♦✳ ▼ö❝ t✐➳♣ t❤❡♦ tr➻♥❤ ❜➔② ởt số ỵ sỹ tỗ t t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ τ ✲❦❤♦↔♥❣ ❝→❝❤✳ ❈→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤õ ②➳✉ ❧➔ ✤➣ ❝â tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ◆❣♦➔✐ ✈✐➺❝ ❤➺ t❤è♥❣✱ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ♠➔ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t ❤♦➦❝ ❜ä q✉❛ ❝❤ù♥❣ ♠✐♥❤✱ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ✤÷❛ r❛ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❦➳t q✉↔ ♠ỵ✐ ✤â ❧➔ ◆❤➟♥ ①➨t ✶✳✷✳✷✱ ▼➺♥❤ ✤➲ ✶✳✷✳✸✱ ❱➼ ❞ư ✶✳✷✳✹✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ t rữớ ữợ sỹ ữợ t t➻♥❤ ✈➔ ♥❣❤✐➯♠ ❦❤➢❝ ❝õ❛ P●❙✳❚❙✳ ✣✐♥❤ ❍✉② ❍♦➔♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ ❚❤➛②✳ ◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❇❛♥ ❈❤õ ♥❤✐➺♠ ❑❤♦❛ qỵ t❤➛② ❝ỉ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ▼➦❝ ❞ị ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❞♦ ❝á♥ ❤↕♥ ❝❤➳ ✈➲ ♠➦t ❦✐➳♥ t❤ù❝ ✈➔ t❤í✐ ❣✐❛♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr ọ ỳ t sõt qỵ t ổ õ õ ỵ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❱✐♥❤✱ t❤→♥❣ ✶✷ ♥➠♠ ✷✵✶✶ ❚→❝ ì ị ĩ ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ✣❒◆ ❚❘➚ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ❱❰■ τ ✲❑❍❖❷◆● ❈⑩❈❍ ✶✳✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ ✤➣ ❝â ❝➛♥ ❞ò♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ t➟♣ ❤đ♣ X ✈➔ ❤➔♠ d : X × X −→ R✳ ❍➔♠ d ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠➯tr✐❝ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✮ d(x, y) ≥ ✈➔ d(x, y) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y ❀ ✷✮ d(x, y) = d(y, x)❀ ✸✮ d(x, z) ≤ d(x, y) + d(y, z) ✈ỵ✐ ♠å✐ x, y, z ∈ X ✳ ❚➟♣ ❤đ♣ X ❝ị♥❣ ✈ỵ✐ ♠ët ♠➯tr✐❝ d tr➯♥ ♥â ữủ ổ tr ỵ (X, d) ❤❛② X ✳ ✶✳✶✳✷✳ ✣à♥❤ ♥❣❤➽❛✳ ❉➣② {xn} tr♦♥❣ ổ tr X ữủ tử tợ x X ợ ộ > tỗ t↕✐ n0 ∈ N s❛♦ ❝❤♦ d(x, xn) < ε ✈ỵ✐ ♠å✐ n ≥ n0 ✳ ❑❤✐ ✤â t❛ ✈✐➳t xn → x ❤♦➦❝ lim xn = x✳ n→∞ ✶✳✶✳✸✳ ◆❤➟♥ ①➨t✳ ✶✮ ❉➣② {xn} tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ❤ë✐ tư tỵ✐ x ∈ X ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ d(x, xn ) → 0✳ ✷✮ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ♠ët ❞➣② ❝❤➾ ❤ë✐ tö ✈➲ ♠ët ✤✐➸♠ ❞✉② ♥❤➜t✳ ✸✮ ◆➳✉ xn → x ✈➔ yn → y t❤➻ d(xn , yn ) → d(x, y)✳ ✶✳✶✳✹✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû (X, d) ✈➔ (Y, p) ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ →♥❤ ①↕ f : X −→ Y ✳ ⑩♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ t↕✐ ✤✐➸♠ x ∈ X ♥➳✉ ✈ỵ✐ ♠å✐ > tỗ t > s ✈ỵ✐ ♠å✐ t ∈ X ♠➔ d(x, t) < δ t❤➻ d(f (x), f (t)) < ε✳ ✹ ✶✳✶✳✺✳ ✣à♥❤ ỵ (X, d) (Y, ) ổ ❣✐❛♥ ♠➯tr✐❝ ✈➔ →♥❤ ①↕ f : X −→ Y ✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ t÷ì♥❣ ✤÷ì♥❣ ✶✮ f ❧✐➯♥ tö❝ t↕✐ x ∈ X ❀ ✷✮ ▼å✐ ❞➣② {xn } ⊂ X s❛♦ ❝❤♦ xn → x t❤➻ f (xn )→ f (x)✳ ✶✳✶✳✻✳ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ▼ët ❞➣② {xn} tr♦♥❣ X ❣å✐ ❧➔ ❞➣② ❈❛✉❝❤② ♥➳✉ ✈ỵ✐ ♠å✐ ε > 0, tỗ t n0 N ợ n m ≥ n0 t❤➻ d(xn , xm ) < ε✳ ▼å✐ ❞➣② ❤ë✐ tư ❧➔ ❞➣② ❈❛✉❝❤②✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ❣å✐ ❧➔ ✤➛② ✤õ ♥➳✉ ♠å✐ ❞➣② ❈❛✉❝❤② tr♦♥❣ 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 X, Y ❧➔ ❤❛✐ t➟♣ ❦❤→❝ ré♥❣✳ ỵ 2Y P (Y ) tt ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ Y ✳ ❚❛ ❣å✐ ♠é✐ →♥❤ ①↕ tø X ✈➔♦ Y ❧➔ ♠ët →♥❤ ①↕ ✤ì♥ trà ❤❛② ❤➔♠ ✤ì♥ trà ✈➔ ❣å✐ ♠é✐ →♥❤ ①↕ tø X ✈➔♦ ♠ët ❤å ❝♦♥ ❝õ❛ 2Y ❧➔ ♠ët →♥❤ ①↕ ✤❛ trà ❤❛② ❤➔♠ ✤❛ trà✳ ✶✳✶✳✽✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû f : X−→X ✈➔ T : X−→U ✈ỵ✐ U ⊂ 2X ✳ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ♥➳✉ f (x) = x✳ ✣✐➸♠ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ♥➳✉ x ∈ T (x)✳ ✣✐➸♠ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ T ♥➳✉ x = f (x) ∈ ✣✐➸♠ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ T (x)✳ ✶✳✷✳ τ ✲❑❍❖❷◆● ❈⑩❈❍ ❚❘➊◆ ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ ✈➼ ❞ö ✈➲ τ ❦❤♦↔♥❣ ❝→❝❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠➯tr✐❝ ✈ỵ✐ τ ❦❤♦↔♥❣ ❝→❝❤✳ ✶✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ợ tr d p : X ì X [0, ∞) ❣å✐ ❧➔ τ ✲❦❤♦↔♥❣ ✺ ❝→❝❤ tr➯♥ X tỗ t : X ì [0, ) −→ [0, ∞) t❤ä❛ ♠➣♥✿ (τ1 ) p(x, z) ≤ p(x, y) + p(y, z) ✈ỵ✐ ♠å✐ x, y, z ∈ X ❀ (τ2 ) inf{η(x, t) : t > 0} = ✈ỵ✐ ♠å✐ x ∈ X ✈➔ η ❦❤æ♥❣ ❣✐↔♠ t❤❡♦ ❜✐➳♥ t❤ù ❤❛✐❀ (τ3 ) lim xn = x ✈➔ lim (sup{η(zn , p(zn , xm )) : m ≥ n}) = ❦➨♦ t❤❡♦ n→ ∞ n→∞ p(w, x) ≤ lim inf p(w, xn ) ✈ỵ✐ ♠å✐ w ∈ X; n→∞ (τ4 ) lim (sup{p(xn , ym ) : m ≥ n}) = ✈➔ lim η(xn , tn ) = ❦➨♦ t❤❡♦ n→∞ n→∞ lim η(yn , tn ) = 0; n→∞ (τ5 ) lim η(zn , p(zn , xn )) = ✈➔ lim η(zn , p(zn , yn )) = ❦➨♦ t❤❡♦ n→∞ n→∞ lim d(xn , yn ) = n→∞ ✶✳✷✳✷✳ ◆❤➟♥ ①➨t✳ ◆➳✉ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ t❤➻ d ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ①→❝ ✤à♥❤ ❤➔♠ η : X × [0, ∞)→[0, ∞) ❜ð✐ ❝æ♥❣ t❤ù❝ η(x, t) = t ợ (x, t) X ì [0, ) ♥❤✐➯♥ d t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (τ1 ) ✈➔ η t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (τ2 ) tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❱➻ d ❧➔ ❤➔♠ ❧✐➯♥ tư❝ tø X × X ✈➔♦ R ♥➯♥ tø lim xn = x ∈ X s✉② r❛ n→∞ d(w, x) = lim d(w, xn ) = lim inf d(w, xn ) n→∞ n→∞ ✈ỵ✐ ♠å✐ w ∈ X ✱ tù❝ ❧➔ ✤✐➲✉ ❦✐➺♥ (τ3 ) ✤÷đ❝ t❤ä❛ ♠➣♥✳ ●✐↔ sû lim η(xn , tn ) = ✈➔ {ηn } ⊂ X ✳ ❑❤✐ ✤â✱ t❛ ❝â n→∞ lim η(yn , tn ) = lim tn = lim η(xn , tn ) = n→∞ n→∞ n→∞ ❞♦ ✤â ✤✐➲✉ ❦✐➺♥ (τ4 ) ✤÷đ❝ t❤ä❛ ♠➣♥✳ ✻ ●✐↔ sû {xn }, {yn }, {zn } ❧➔ ❝→❝ ❞➣② tr♦♥❣ X s❛♦ ❝❤♦ lim η(zn , d(zn , xn )) = lim η(zn , d(zn , yn )) = n→∞ n→∞ ❑❤✐ ✤â✱ tø η(x, t) = t ợ (x, t) X ì [0, ∞) s✉② r❛ lim d(zn , xn ) = lim d(zn , yn ) = n→∞ n→∞ ▼➦t ❦❤→❝✱ t❛ ❝â d(xn , yn ) ≤ d(xn , zn ) + d(zn , yn ) ✈ỵ✐ ♠å✐ n ❉♦ ✤â lim d(xn , yn ) = 0, tù❝ ❧➔ ✤✐➲✉ ❦✐➺♥ (τ5 ) ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❱➟② d ❧➔ n→∞ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ✶✳✷✳✸✳ ▼➺♥❤ ✤➲✳ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ p : X × X −→ R ❧➔ ❤➔♠ ❧✐➯♥ tö❝ t❤❡♦ ❜✐➳♥ t❤ù ❤❛✐✳ ❑❤✐ ✤â✱ ♥➳✉ ♣ t❤ä❛ (1) tỗ t > s❛♦ ❝❤♦ d(x, y) ≤ αp(x, y) ✈ỵ✐ ♠å✐ (x, y) ∈ X × X t❤➻ p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ①→❝ ✤à♥❤ ❤➔♠ η : X × [0, ∞) −→ [0, ∞) ❜ð✐ ❝ỉ♥❣ t❤ù❝ η(x, t) = t ✈ỵ✐ ♠å✐ (x, t) ∈ X × [0, ∞) ❍✐➸♥ ♥❤✐➯♥ η t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (τ2 )✳ ●✐↔ sû {xn } ⊂ X, xn →x✳ ❑❤✐ ✤â✱ ✈➻ p ❧✐➯♥ tö❝ t❤❡♦ ❜✐➳♥ t❤ù ✷ ♥➯♥ ✈ỵ✐ ♠å✐ w ∈ X t❛ ❝â p(w, x) = lim p(w, xn ) = lim inf p(w, xn ) n→∞ n→∞ ❉♦ ✤â ✤✐➲✉ ❦✐➺♥ (τ3 ) t❤ä❛ ♠➣♥✳ ❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ◆❤➟♥ ①➨t ✶✳✷✳✷✱ ✤✐➲✉ ❦✐➺♥ (τ4 ) ✤÷đ❝ t❤ä❛ ♠➣♥✳ ✼ ●✐↔ sû {xn }, {yn }, {zn } ❧➔ ❝→❝ ❞➣② tr♦♥❣ X s❛♦ ❝❤♦ lim η(zn , p(zn , xn )) = lim η(zn , p(zn , yn )) = n→∞ n→∞ ❑❤✐ ✤â✱ tø η(x, t) = t ợ (x, t) X ì [0, ) t❛ ❝â lim p(zn , xn ) = lim p(zn , yn ) = n→∞ n→∞ ❱➻ α > ♥➯♥ lim αp(zn , xn ) = lim αp(zn , yn ) = n→∞ n→∞ ✣✐➲✉ ♥➔② ❝ò♥❣ ✈ỵ✐ ❜➜t ✤➥♥❣ t❤ù❝ d(xn , yn ) ≤ d(xn , zn ) + d(zn , yn ) ≤ αp(xn , zn ) + αp(zn , yn ) ✈ỵ✐ ♠å✐ n ❦➨♦ t❤❡♦ lim d(xn , yn ) = 0✳ ❉♦ ✤â ✤✐➲✉ ❦✐➺♥ (τ5 ) ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❱➟② p ❧➔ n→∞ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ✶✳✷✳✹✳ ❱➼ ❞ö✳ ✶✮ ●✐↔ sû X = [1, M ] ✭✈ỵ✐ < M < ∞✮ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➯tr✐❝ d ✤÷đ❝ ❝↔♠ s✐♥❤ tø ♠➯tr✐❝ t❤ỉ♥❣ t❤÷í♥❣ tr➯♥ R ✈➔ p : X × X −→ R ❧➔ ❤➔♠ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ p(x, y) = y ợ (x, y) X ì X ✤â✱ p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ❚❤➟t ✈➟②✱ ❤✐➸♥ ♥❤✐➯♥ p ❧➔ ❤➔♠ ❧✐➯♥ tö❝ t❤❡♦ ❜✐➳♥ t❤ù ❤❛✐ ✈➔ z = p(x, z) ≤ y + z = p(x, y) + p(y, z) ✈ỵ✐ ♠å✐ x, y, z ∈ X ▼➦t ❦❤→❝ d(x, y) ≤ M p(x, y) ợ (x, y) X ì X ❉♦ ✤â t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✸✱ p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ✷✮ ●✐↔ sû X = {0, 2} ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈ỵ✐ ♠➯tr✐❝ ❝↔♠ s✐♥❤ ❜ð✐ ♠➯tr✐❝ tổ tữớ tr R p : X ì X −→ R ❧➔ ❤➔♠ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ p(x, y) = y ợ (x, y) X ì X ✽ ❑❤✐ ✤â p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ❚❤➟t ✈➟②✱ ❞➵ t❤➜② p t❤ä❛ ♠➣♥ (τ1 ) ✈➔ ❧✐➯♥ tö❝ t❤❡♦ ❜✐➳♥ t❤ù ✷✳ ❉♦ ✤â ♥➳✉ ①→❝ ✤à♥❤ ❤➔♠ η : X × [0, ∞) −→ [0, ∞) ❜ð✐ ❝æ♥❣ t❤ù❝ η(x, t) = t ợ (x, t) X ì [0, ) t ❝→❝ ✤✐➲✉ ❦✐➺♥ (τ2 ), (τ3 ), (τ4 ) ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ p ✈➔ η t❤ä❛ ♠➣♥ (τ5 )✳ ●✐↔ sû {xn }, {yn }, {zn } ❧➔ ❝→❝ ❞➣② tr♦♥❣ X s❛♦ ❝❤♦ lim η(zn , p(zn , xn )) = lim η(zn , p(zn , yn )) = n→∞ n→∞ ❑❤✐ ✤â✱ lim p(zn , xn ) = lim p(zn , yn ) = n→∞ ✭✯✮ ❚❤❡♦ ❝→❝❤ ①→❝ ✤à♥❤ p t❤➻ p(x, y) = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ y = 0✳ ❍ì♥ ♥ú❛✱ ♥➳✉ y ∈ X ♠➔ y = t❤➻ p(x, y) = 2✳ ❑➳t ❤đ♣ ✈ỵ✐ () s r tỗ t n0 N s xn = yn = ✈ỵ✐ ♠å✐ n ≥ n0 ✳ ❉♦ ✤â d(xn , yn ) = 0✳ ◆❤÷ ✈➟② (τ5 ) ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❉♦ ✤â p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✶✳✷✳✺✳ ▼➺♥❤ ✤➲✳ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X, η ❧➔ ❤➔♠ t❤ä❛ ♠➣♥ (τ2), (τ3), (τ4) ✈➔ (τ5)✳ ❑❤✐ ✤â✱ ♥➳✉ q ❧➔ ❤➔♠ tø X × X ✈➔♦ [0, ∞) t❤ä❛ ♠➣♥ (τ1 )q ỗ t c > s min{p(x, y), c} ≤ q(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X ✱ ✐✐✮ n→∞ lim xn = x ✈➔ lim sup{η(zn , q(zn , xm )) : m ≥ n} = ❦➨♦ t❤❡♦ n→∞ q(w, z) ≤ lim inf q(w, xn ) ✈ỵ✐ ♠å✐ w ∈ X n→∞ t❤➻ q ❝ô♥❣ ❧➔ ♠ët τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t θ(x, t) = t + η(x, t) ✭✶✮ ✈ỵ✐ x ∈ X ✈➔ t ∈ [0, ∞)✳ ú ỵ r (x, t) (x, t), ợ x ∈ X ✈➔ t ∈ [0, ∞)✳ ❑❤✐ ✤â tø ❣✐↔ t❤✐➳t s✉② r❛ (τ1 )q , (τ2 )θ ✈➔ (τ3 )q,θ ✳ Ð ✤➙②✱ t❛ ✈✐➳t (τ1 )q ✱ (τ2 )θ , (τ3 )q,θ ❝â ♥❣❤➽❛ ❧➔ q, θ ✈➔ ❝➦♣ q, θ ❧➛♥ ❧÷đt t❤ä❛ ♠➣♥ (τ1 ), (τ2 ), (τ3 )✳ ✾ ❚❛ ❣✐↔ sû r➡♥❣ lim sup{q(xn , ym ) : m ≥ n} = ✈➔ lim θ(xn , tn ) = n→∞ n→∞ ❑❤✐ ✤â lim sup{p(xn , ym ) : m ≥ n} = ✈➔ lim tn = lim η(xn , tn ) = n→∞ n→∞ n→∞ ❚ø (τ4 ) t❛ ❝â lim η(yn , tn ) = ✈➔ ❞♦ ✤â lim θ(yn , tn ) = 0✳ ❉♦ ✤â (τ4 )q,θ n→∞ n→∞ ✤ó♥❣✳ ❚❛ ❣✐↔ sû r➡♥❣ lim θ(zn , q(zn , xn )) = ✈➔ lim θ(zn , q(zn , yn )) = n→∞ n→∞ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ θ t❛ ❝â lim η(zn , q(zn , xn )) = ✈➔ lim q(zn , xn ) = n→∞ n→∞ ❉♦ ✤â tø ❣✐↔ t❤✐➳t s✉② r❛ lim η(zn , p(zn , xn )) = ❱➻ t❤➳ t❛ ❝â n→∞ lim η(zn , p(zn , yn )) = n→∞ ❚ø (τ5 ) s✉② r❛ lim d(xn , yn ) = 0✳ ❚❛ ❝â (τ5 )q,θ ✤ó♥❣✳ ❱➟② t❛ ❝â ✤✐➲✉ ❝➛♥ n→∞ ❝❤ù♥❣ ♠✐♥❤✳ ❈→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ❤➺ q✉↔ trü❝ t✐➳♣ ❝õ❛ ▼➺♥❤ ✤➲ ✶✳✷✳✺✳ ✶✳✷✳✻✳ ▼➺♥❤ ✤➲✳ ●✐↔ sû ♣ ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❳ ✈➔ q ❧➔ ❤➔♠ tø X × X ✈➔♦ [0, ∞) t❤ä❛ ♠➣♥ (τ1)q ✳ ❑❤✐ ✤â✱ ♥➳✉ ỗ t c > tọ min{p(x, y), c} ≤ q(x, y) ✈ỵ✐ x, y ∈ X ❀ ✐✐✮ n→∞ lim xn = x ✈➔ p(w, x) ≤ lim sup p(w, xn ) ✈ỵ✐ ♠å✐ w ∈ X ❦➨♦ t❤❡♦ n→∞ q(w, x) ≤ lim inf q(w, xn ) n→∞ ✈ỵ✐ ♠å✐ w ∈ X t❤➻ q ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû n→∞ lim xn = x ✈➔ lim sup{η(zn , q(zn , xm )) : m ≥ n} = n→∞ ✶✵ ❈❍×❒◆● ✷ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ✣❆ ❚❘➚ ❱⑨ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈❍❯◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ✣❒◆ ❚❘➚ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ❱❰■ τ ✲❑❍❖❷◆● ❈⑩❈❍ ✷✳✶✳ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ✣❆ ❚❘➚ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ sü tỗ t tr t tr sỹ tỗ t t ❝õ❛ →♥❤ ①↕ ✤❛ trà p✲❝♦✳ ✷✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ◆➳✉ T ❧➔ →♥❤ ①↕ tø X ✈➔♦ 2X t❤➻ t❛ ♥â✐ T ❧➔ →♥❤ ①↕ ✤❛ trà tø X ✈➔♦ ❝❤➼♥❤ ♥â✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✱ p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✈➔ T : X −→ 2X ✳ ⑩♥❤ ①↕ T ✤÷đ❝ ❣å✐ ❧➔ p✲❝♦ ♥➳✉ T x = ∅ ✈ỵ✐ ♠å✐ x ∈ X tỗ t r [0, 1) s Q(T x, T y) ≤ rp(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X, tr♦♥❣ ✤â Q(A, B) = sup inf p(a, b) aA bB ỵ ổ tr ✤➛② ✤õ (X, d) ✈➔ p ❧➔ ♠ët τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ●✐↔ sû T ❧➔ ♠ët →♥❤ ①↕ ✤❛ trà p✲❝♦ tø X ✈➔♦ ❝❤➼♥❤ ♥â s❛♦ ❝❤♦ ✈ỵ✐ x ❜➜t ❦ý t❤✉ë❝ X, T x ❧➔ t õ X õ tỗ t x0 ∈ X s❛♦ ❝❤♦ x0 ∈ T x0 ✈➔ p(x0, x0) = ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❣✐↔ t❤✐➳t s✉② r tỗ t r [0, 1) s Q(T x, T y) ≤ r p(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X ✷✼ 1+r ∈ [0, 1) ✈➔ ❝è ✤à♥❤ x, y ∈ X ✈➔ u ∈ T x✳ ❑❤✐ ✤â✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ p(x, y) > 0✱ tỗ t v T y s p(u, v) ≤ rp(x, y)✳ ❚r♦♥❣ tr÷í♥❣ ✣➦t r = ❤đ♣ p(x, y) = t❛ ❝â Q(T x, T y) = õ tỗ t {vn } tr T y s❛♦ ❝❤♦ lim p(u, ) = 0✳ ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✷✱ {vn } ❧➔ p✲❈❛✉❝❤② ✈➔ ❞♦ ✤â {vn } ❧➔ n→∞ ❈❛✉❝❤②✳ ❚ø X ✲✤➛② ✤õ ✈➔ T y ✤â♥❣✱ s✉② r❛ {vn } ❤ë✐ tư tỵ✐ ✤✐➸♠ v ∈ T y ✳ ❉♦ ✤â t❛ ❝â p(u, v) ≤ lim p(u, ) = = rp(x, y) n→∞ ◆❤÷ ✈➟② t❛ ✤➣ ❝❤➾ r❛ ❜➜t ❦ý x, y ∈ X ✈➔ u ∈ T x t❤➻ v ∈ T y s❛♦ ❝❤♦ p(u, v) ≤ rp(x, y)✳ ❈è ✤à♥❤ u0 ∈ X ✈➔ u1 ∈ T u0 t tỗ t u2 T u1 s ❝❤♦ p(u1 , u2 ) ≤ rp(u0 , u1 )✳ ❉♦ ✤â t❛ ❝â ❞➣② {un } tr♦♥❣ X s❛♦ ❝❤♦ Un+1 ∈ T un ✈➔ p(un , un+1 ) ≤ rp(un−1 , un ) ✈ỵ✐ ♠å✐ n ∈ N✳ ❱ỵ✐ ❜➜t ❦ý n ∈ N t❛ ❝â p(un , un+1 ) ≤ rp(un−1 , un ) ≤ r2 p(un−2 , un−1 ) ≤ · · · ≤ rn p(u0 , u1 ) ✈➔ ❞♦ ✤â✱ ✈ỵ✐ ❜➜t ❦ý m, n ∈ N ✈➔ m > n✱ p(un , um ) ≤ p(un , un+1 ) + p(un+1 , un+2 ) + · · · + p(um−1 , um ) ≤ rn p(u0 , u1 ) + rn+1 p(u0 , u1 ) + · · · + rm−1 p(u0 , u1 ) rn ≤ p(u0 , u1 ) 1−r ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✸✱ {un } ❧➔ ❞➣② p✲❈❛✉❝❤②✳ ❉♦ ✤â✱ t❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✶✱ {un } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❉♦ ✤â✱ {un } ❤ë✐ tư tỵ✐ ✤✐➸♠ v0 ∈ X ✳ ❱ỵ✐ n ∈ N✱ tø (τ3 ) t❛ ❝â p(un , v0 ) ≤ lim inf p(un , um ) ≤ n→∞ rn p(u0 , u1 ) 1−r ❚ø tt s r tỗ t wn T v0 s❛♦ ❝❤♦ p(un , wn ) ≤ rp(un−1 , v0 ) ✈ỵ✐ n ∈ N✳ ❱➻ t❤➳ t❛ ❝â lim sup p(un , wn ) ≤ lim sup rp(un−1 , v0 ) n→∞ n→∞ rn ≤ lim p(u0 , v1 ) = n→∞ − r ✷✽ ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✸✱ {wn } ❤ë✐ tư tỵ✐ v0 ✳ ❚ø T v0 ✤â♥❣✱ t❛ ❝â v0 ∈ T v0 ợ v0 tỗ t v1 T v0 s ❝❤♦ p(v0 , v1 ) ≤ rp(v0 , v0 )✳ ❉♦ ✤â t❛ ❝â ❞➣② {vn } tr♦♥❣ X s❛♦ ❝❤♦ vn+1 ∈ T ✈➔ p(v0 , vn+1 ) ≤ rp(v0 , ) ✈ỵ✐ ♠å✐ n ∈ N✳ ❚ø ✤â t❛ ❝â p(v0 , ) ≤ rp(v0 , vn−1 ) ≤ · · · ≤ rn p(v0 , v0 ) ❉♦ ✤â lim sup(un , ) ≤ lim (p(un , v0 ) + p(v0 , )) = n→∞ n→∞ ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✸✱ {vn } ❧➔ ❞➣② p✲❈❛✉❝❤② ✈➔ ❤ë✐ tư tỵ✐ v0 ✳ ❱➟② t❛ ❝â p(v0 , v0 ) ≤ lim p(v0 , ) = n→∞ ✷✳✶✳✸✳ ❱➼ ❞ö✳ ✣➦t X = {0, 1} ✈➔ ①→❝ ✤à♥❤ ♠ët τ ✲❦❤♦↔♥❣ ❝→❝❤ p tr➯♥ X ❜ð✐ p(x, y) = y ♠å✐ x, y ∈ X ✈➔ ♠ët →♥❤ ①↕ ✤❛ trà T tø X ✈➔♦ ❝❤➼♥❤ ♥â ①→❝ ✤à♥❤ ❜ð✐ T (x) = X ✈ỵ✐ ♠å✐ x ∈ X t❤➻ ∈ X ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✈➔ p(1, 1) = 0✳ ✷✳✷✳ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈❍❯◆● ❈Õ❆ ❈⑩❈ ⑩◆❍ ❳❸ ✣❒◆ ❚❘➚ ▼ö❝ ♥➔② tr➻♥❤ ởt số ỵ sỹ tỗ t ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ τ ✲❦❤♦↔♥❣ ❝→❝❤✳ ✷✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ f, g ❧➔ ❤❛✐ →♥❤ ①↕ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ X ✱ p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✈➔ g(X) ⊆ f (X)✳ ❚❛ ♥â✐ g s rở tữỡ ự ợ (p, f ), (0, 1) tỗ t ❝→❝ ❤➔♠ ❦❤æ♥❣ ➙♠ q, r, s, t t❤ä❛ ♠➣♥ sup {q(x, y) + r(x, y) + s(x, y) + 4t(x, y)} ≤ λ < x,y∈X ✭✹✮ ✈➔ max{p(f (x), g(y)), p(g(y), f (x))} ≤ q(x, y)p(x, y) + r(x, y)p(x, f (x)) +s(x, y)p(y, g(y)) + t(x, y)[p(x, g(y)) + p(y, f (x))] ✷✾ ✭✺✮ ✈ỵ✐ ♠å✐ x, y ∈ X ✳ ✷✳✷✳✷✳ ❱➼ ❞ö✳ ❛✮ ◆➳✉ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈➔ p(x, y) = d(x, y) t❤➻ ♠é✐ →♥❤ ①↕ ❝♦ f tr➯♥ X ❧➔ λ✲❝♦ s✉② rë♥❣ ✤è✐ ✈ỵ✐ (p, f )✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚r♦♥❣ ◆❤➟♥ ①➨t ✶✳✷✳✷✳ t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ p = d t❤➻ p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ❱➻ f ❧➔ →♥❤ ①↕ ❝♦ tr➯♥ X ♥➯♥ tỗ t [0, 1) s d(f (x), f (y)) ≤ αd(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X ❚❛ s➩ ❝❤ù♥❣ tä f ❧➔ λ✲❝♦ s✉② rë♥❣ tữỡ ự ợ (p, f ) (0, 1)✳ ◆➳✉ α = t❤➻ f ❧➔ →♥❤ ①↕ ỗ t õ ự ú sû α ∈ (0, 1)✳ ▲➜② λ = α, q(x, y) = α, r(x, y) = s(x, y) = t(x, y) = ✈ỵ✐ ♠å✐ x, y ∈ X ✳ ❑❤✐ ✤â✱ t❛ ❝â sup {q(x, y) + r(x, y) + s(x, y) + 4t(x, y)} = α = λ x,y∈X ✈➔ ✈ỵ✐ ♠å✐ x, y ∈ X t❤➻ max{p(f (x), f (y), p(f (y)), f (x))} = d(f (x), f (y)) ≤ αd(x, y) = q(x, y)d(x, y) + r(x, y)p(x, f (x)) + s(x, y)p(y, f (y)) + t(x, y)[p(x, f (y)) + p(y, f (x))] ❱➟② f s rở tữỡ ự ợ (p, f ) ❜✮ ▲➜② X = [0, 2] ⊆ R ✈➔ x  ✈ỵ✐ x ∈ [0, 1] f (x) = g(x) = 9x  ✈ỵ✐ x ∈ [1; 2] 10 1 q(x, y) = , r(x, y) = s(x, y) = , t(x, y) = 10 11 ✸✵ ✈➔ p(x, y) = |x − y| t❤➻ q ❧➔ s rở ợ (p, f ) ữ õ ổ ♣❤↔✐ ❧➔ →♥❤ ①↕ ❝♦ ✈➻ d(g(1), g(x)) = x 1 − → 10 9 ❦❤✐ x → 1+ ♥❤÷♥❣ d(1, x) → ❦❤✐ x → 1+ ✳ ❇ê ✤➲ s❛✉ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❝→❝ ✤à♥❤ ỵ t t x0 X ✳ ❳→❝ ✤à♥❤ {xn} ❜ð✐ x2n+1 = f (x2n ), x2n+2 = g(x2n+1 ), ✭✻✮ tr♦♥❣ ✤â f, g ❧➔ ❝→❝ →♥❤ ①↕ tr➯♥ X t❤ä❛ ♠➣♥ g ❧➔ λ✲❝♦ s✉② rë♥❣ ✤è✐ ✈ỵ✐ (p, f )✳ ❑❤✐ ✤â✱ {xn} ❧➔ ♠ët ❞➣② ❈❛✉❝❤②✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t M1 = max{p(x2n+1 , x2n+2 ), p(x2n+2 , x2n+1 )}, M2 = max{p(x2n , x2n+1 ), p(x2n+1 , x2n )}✳ ❚ø ✭✹✮✱ ✭✺✮ ✈➔ ✭✻✮ t❛ ❝â M1 = max{p(f (x2n ), g(x2n+1 )), p(g(x2n+1 ), f (x2n ))} ≤ λ max{p(x2n , x2n+1 ), p(x2n , f (x2n )), p(x2n+1 , g(x2n+1 )), [p(x2n , g(x2n+1 )) + p(x2n+1 , f (x2n ))]} ≤ λ max{p(x2n , x2n+1 ), p(x2n , x2n+1 ), p(x2n+1 , x2n+2 ), [p(x2n , x2n+2 ) + p(x2n+1 , x2n+1 )]} = λM (x2n , x2n+1 ), tr♦♥❣ ✤â M (x2n , x2n+1 ) = max{p(x2n , x2n+1 ), p(x2n+1 , x2n+2 ), [p(x2n , x2n+2 ) + p(x2n+1 , x2n+1 )]} ❇➙② ❣✐í ♥➳✉ M (x2n , x2n+1 ) = p(x2n+1 , x2n+2 ) ✸✶ t❤➻ t❛ ❝â p(x2n+1 , x2n+2 ) ≤ λp(x2n+1 , x2n+2 ) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ p(x2n+1 , x2n+2 ) = 0✳ ◆➳✉ M (x2n , x2n+1 ) = [p(x2n , x2n+2 ) + p(x2n+1 , x2n+1 )] t❤➻ p(x2n+1 , x2n+2 ) ≤ λ [p(x2n , x2n+2 ) + p(x2n + 1, x2n+1 )] ❉♦ ✤â p(x2n+1 , x2n+2 ) ≤ λ p(x2n , x2n+2 ) ❤♦➦❝ p(x2n+1 , x2n+2 ) ≤ λ p(x2n+1 , x2n+1 ) ◆➳✉ p(x2n+1 , x2n+2 ) ≤ t❤➻ tø λ p(x2n , x2n+2 ) λ λ p(x2n , x2n+2 ) ≤ [p(x2n , x2n+1 ) + p(x2n+1 , x2n+2 )] 2 λ ≤ p(x2n , x2n+1 ) + p(x2n+1 , x2n+2 ), 2 t❛ ❝â p(x2n+1 , x2n+2 ) ≤ λp(x2n , x2n+1 ) ◆➳✉ p(x2n+1 , x2n+2 ) ≤ λ p(x2n+1 , x2n+1 ) t❤➻ tø λ λ p(x2n+1 , x2n+1 ) ≤ [p(x2n+1 , x2n ) + p(x2n , x2n+1 )], 2 t❛ ❝â p(x2n+1 , x2n+2 ) ≤ λp(x2n+1 , x2n ) ✸✷ ❤♦➦❝ p(x2n+1 , x2n+2 ) ≤ λp(x2n , x2n+1 ) ❉♦ ✤â tr♦♥❣ ❜➜t ❦ý tr÷í♥❣ ❤đ♣ ♥➔♦ t❛ ❝ơ♥❣ ❝â M1 ≤ λp(x2n+1 , x2n ) ❤♦➦❝ M1 ≤ λp(x2n , x2n+1 ) ✭✼✮ ❚÷ì♥❣ tü t❛ ❝â M2 ≤ λp(x2n−1 , x2n ) ❤♦➦❝ M2 ≤ λp(x2n , x2n−1 ) tử ỵ ữ tr t õ p(xn , xn+1 ) ≤ λ max{p(xn−1 , xn ), p(xn , xn−1 )} ≤ ≤ λn max{p(x0 , x1 ), p(x1 , x0 )} ✣➦t r(x0 ) = max{p(x0 , x1 ), p(x1 , x0 )} t❤➻ ✈ỵ✐ ♠é✐ m > n t❛ ❝â m−n−1 p(xm , xn ) ≤ m−n−1 λ(n+k) r(x0 ) ≤ λn r(x0 )(1 − λ)−1 p(xn+k+1 , xn+k ) ≤ k=0 k=0 ❚ø ✤â s✉② r❛ lim (sup{p(xm , xn ) : m ≥ n}) = n→∞ ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✶ ✈➔ ❇ê ✤➲ ✶✳✷✳✶✸ t❤➻ {xn } ❧➔ ỵ (X, d) ổ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✱ ♣ ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ ❳ ✈➔ x0 ∈ X ✱ ❢ ✈➔ ❣ ❧➔ ❤❛✐ →♥❤ ①↕ tr➯♥ ❳ t❤ä❛ ♠➣♥ ❣ ❧➔ λ✲❝♦ s✉② rë♥❣ ✤è✐ ✈ỵ✐ (p, f )✳ ❍ì♥ ♥ú❛✱ ❣✐↔ sû r➡♥❣ ♥➳✉ lim sup{p(xn , xm ) : m > n} = ✈➔ lim sup p(xn , y) = n→∞ n→∞ t❤➻ tø lim sup p(xn, f (xn)) = ❦➨♦ t❤❡♦ f (y) = y ✈➔ tø n→∞ lim p(xn , g(xn )) = n→∞ ❦➨♦ t❤❡♦ g(y) = y ❑❤✐ ✤â✱ f ✈➔ g ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❞✉② ♥❤➜t z s❛♦ ❝❤♦ p(z, z) = ✈➔ (f g)n(x0) → z ✈➔ (gf )n(x0) → z ✳ ✸✸ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x0 ∈ X ✱ ①→❝ ✤à♥❤ ❞➣② {xn } ❜ð✐ x2n+1 = f (x2n ) ✈➔ x2n+2 = g(x2n+1 )✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✸ t❤➻ {xn } ❧➔ ❞➣② ❈❛✉❝❤② ✈➔ ❤ë✐ tư tỵ✐ ✤✐➸♠ z ∈ X ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ f (z) = z, g(z) = z ✳ ❚ø (τ3 ) t❛ ❝â lim sup(p(x2n , f (x2n )) + p(x2n , z)) ≤ lim sup(p(x2n , x2n+1 )) n→∞ n→∞ + lim inf p(x2n , xm ) ≤ lim sup p(x2n , xm ) = m→∞ m≥2n ❚÷ì♥❣ tü t❛ ❝â lim sup(p(x2n+1 , g(x2n+1 )) + p(x2n+1 , z)) = n→∞ ❉♦ ✤â lim (sup{p(xn , xm ) : m > n}) = ✈➔ lim (xn , z) = n→∞ n→∞ ❱➟② t❛ ❝â lim (p(x2n , f (x2n )) = n→∞ ✈➔ lim p(x2n , z) = n→∞ ✣➦t xn = x2n ✳ ❚ø ❣✐↔ t❤✐➳t s✉② r❛ f (z) = z ✳ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â g(z) = z ✳ ❇➙② ❣✐í✱ tr♦♥❣ ✭✺✮ ✤➦t x = y = z t❛ ❝â p(z, z) ≤ λp(z, z) ❉♦ ✤â p(z, z) = ✭✈➻ λ ∈ (0, 1)✮✳ ◆➳✉ u ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ f ✈➔ g t❤➻ t❤❡♦ ✭✺✮ t❛ ❝â max{p(z, u), p(u, z)} ≤ q(z, u), p(z, u) + r(z, u)p(z, z) + s(z, u)p(u, u) + t(z, u)[p(z, u) + p(u, z)] ≤ λ max{p(z, u), p(z, z), p(u, u), [p(z, u) + p(u, z)]} = λ max{p(z, u), [p(z, u) + p(u, z)]} ✣➥♥❣ t❤ù❝ ❝✉è✐ ❝ị♥❣ ✤ó♥❣ ❜ð✐ ✈➻ p(z, z) = p(u, u) = ❚ø ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ s✉② r❛ p(z, u) = p(u, z) = ❈✉è✐ ❝ò♥❣✱ ✈➻ p(z, z) = p(z, u) = ♥➯♥ t❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✸ t❛ ❝â z = u ✸✹ ✷✳✷✳✺✳ ❈❤ó þ✳ ◆➳✉ f ❧✐➯♥ tö❝ t❤➻ tø {xn} ✈➔ {f (zn)} ❤ë✐ tư tỵ✐ y s✉② r❛ f (y) = y ✳ ◆➳✉ lim (sup{p(xn , xm ) : m > n}) = 0, lim p(xn , y) = n→∞ n→∞ ✈➔ lim p(xn , f (xn )) = t❤➻ t❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✸ t❛ ❝â n→∞ lim xn = lim f (xn ) = y n→∞ n→∞ ♥❤÷♥❣ ❦❤ỉ♥❣ ❦➳t ❧✉➟♥ ✤÷đ❝ f (y) = y ✳ n−1 ❈❤➥♥❣ ❤↕♥✱ ❧➜② X = R, xn = , p = d, y = ✈➔ f : R−→ R ①→❝ ✤à♥❤ n ❜ð✐ t ✈ỵ✐ t = f (t) = ✈ỵ✐ t = ❑❤✐ ✤â✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ ♥â✐ tr➯♥ ✤÷đ❝ t❤ä❛ ♠➣♥ ♥❤÷♥❣ f (1) = 1✳ ❈â t❤➸ g k ❧➔ λ✲❝♦ s✉② rë♥❣ ✤è✐ ✈ỵ✐ (p, f ) ✈ỵ✐ k ∈ N ✈➔ k > ♥❤÷♥❣ g ❦❤ỉ♥❣ ♥❤÷ ✈➟②✳ ❱➼ ❞ö s❛✉ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥➔②✳ ✷✳✷✳✻✳ ❱➼ ❞ö✳ ❈❤♦ X = {a, b, c} ✈ỵ✐ a, b, c ∈ R✱ f (x) = a ✈ỵ✐ ♠å✐ x ⊂ X ✱ ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ✤ê✐ tr➯♥ X ✈➔ g : X−→ X ①→❝ ✤à♥❤ ❜ð✐ g(a) = a; g(b) = c, g(c) = a✳ ✣➦t p = d t❛ ❝â g = f ✈➔ g ❧➔ λ✲❝♦ s rở ố ợ (p, f ) ữ tứ g(X) f (X) ♥➯♥ g ❦❤ỉ♥❣ λ✲❝♦ s✉② rë♥❣ ✤è✐ ✈ỵ✐ (p, f )✳ ✷✳✷✳✼✳ ❇ê ✤➲✳ ❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d)✱ p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✳ ●✐↔ sû f, f0 ❧➔ tü ❤❛✐ →♥❤ ①↕ tr➯♥ X t❤ä❛ ♠➣♥ max{p(f0 (x), f (y)),p(f (y), f0 (x))} ≤ λ max{p(x, y), p(x, f0 (x)), p(y, f (y)), p(x, f (y)), p(y, f0 (x))} ✭✾✮ ✈ỵ✐ λ ∈ [0, 1) ✈➔ ♠å✐ x, y ∈ X ✳ ◆➳✉ tỗ t z X s f0(z) = z ✈➔ p(z, z) = t❤➻ f (z) = z ✈➔ z ❧➔ ❞✉② ♥❤➜t✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø f0(z) = z ✈➔ tø ✭✽✮ t❛ ❝â ✸✺ max{p(z, f (z)), p(f (z), z)} = max{p(f0 (z), f (z)), p(f (z), f0 (z))} ≤ λ max{p(z, z), p(z, f (z))} = λp(z, f (z)) ❱➻ λ ∈ (0, 1) s✉② r❛ p(z, f (z)) = 0✳ ❉♦ ✤â tø ❇ê ✤➲ ✶✳✷✳✶✷ t❛ ❝â z = f (z)✳ ◆➳✉ v ∈ X t❤ä❛ ♠➣♥ f0 (v) = v ✈➔ p(v, v) = t❤➻ t❛ ❝â f (v) = v ✈➔ p(z, v) = p(f0 (z), f (v)) ≤ λ max{p(z, v), p(z, z), p(v, v), p(v, z)} = λ max{p(z, v), p(v, z)} ❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝â p(v, z) ≤ λ max{p(z, v), p(v, z)} ❉♦ ✤â p(z, v) = p(v, z) = 0✳ ❚ø p(z, z) = ✈➔ ❇ê ✤➲ ✶✳✷✳✶✷ t❛ ❝â v = z ỵ sỷ (X, d) ổ ♠➯tr✐❝ ✤➛② ✤õ✱ ♣ ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X ✈➔ {fn} ❧➔ ❞➣② ❝→❝ tü →♥❤ ①↕ tr➯♥ X t❤ä❛ ♠➣♥ f0 ❧✐➯♥ tư❝ ✈➔ ✈ỵ✐ ♠é✐ x, y ∈ X t❛ ❝â max{p(f0 (x), fn (y)),p(fn (y), f0 (x))} ≤ λ max{p(x, y), p(x, f0 (x)), p(y, fn (y), [p(x, fn (y)) + p(y, f0 (x))} ✭✶✵✮ ✈ỵ✐ λ ∈ (0, 1) ✈➔ n = 0, 1, 2, ❑❤✐ ✤â✱ {fn} ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❣å✐ ❧➔ z s❛♦ ❝❤♦ p(z, z) = 0✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x0 ∈ X ✱ ①→❝ ✤à♥❤ ❞➣② {xn} ❜ð✐ x1 = f0 (x0 ), x2 = f0 (x1 ) = f02 (x0 ), , xn = f0n (x0 ), ✭✶✶✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ {xn } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❚ø ✭✾✮ t❛ ❝â max{p(xn , xn−1 ), p(xn+1 , xn )} = max{p(f0 (xn−1 ), f0 (xn−2 )), p(f0 (xn−2 ), f0 (xn−1 ))} ≤ λ max{p(xn−2 , xn−1 ), p(xn−1 , xn ), p(xn−2 , xn ) + p(xn−1 , xn−1 )} ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ max{p(xn , xn−1 ), p(xn−1 , xn )} ≤ λ max{p(xn−1 , xn−2 ), p(xn−2 , xn−1 )} ✭✶✷✮ ✸✻ ❚❛ ✤➦t M = max{p(xn−2 , xn−1 ), p(xn−1 , xn ), p(xn−2 , xn ) + p(xn−1 , xn−1 )} ◆➳✉ M = p(xn−1 , xn ) t❤➻ p(xn−1 , xn ) = ✈➔ ✭✶✷✮ ✤ó♥❣✳ ◆➳✉ M = p(xn−2 , xn−1 ) t❤➻ max{p(xn , xn−1 ), p(xn−1 , xn )} ≤ λp(xn−2 , xn−1 ) ✈➔ ✭✶✷✮ ✤ó♥❣✳ ◆➳✉ M = p(xn−2 , xn ) + p(xn−1 , xn−1 ) t❤➻ 4 max{p(xn , xn−1 ), p(xn−1 , xn ) ≤ λp(xn−2 , xn ) + p(xn−1 , xn−1 ) ❉♦ ✤â max{p(xn , xn−1 ), p(xn−1 , xn )} ≤ λp(xn−2 , xn ) ≤ λp(xn−2 , xn−1 ) + p(xn−1 , xn ) ❤♦➦❝ max{p(xn , xn−1 ), p(xn−1 , xn )} ≤ λp(xn−1 , xn−1 ) ≤ λp(xn−1 , xn−2 ) + p(xn−2 , xn−1 ) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ max{p(xn , xn−1 ), p(xn−1 , xn )} ≤ λ max{p(xn−1 , xn−2 ), p(xn−2 , xn−1 )} ❱➟② tr♦♥❣ ♠å✐ tr÷í♥❣ ❤đ♣ ✭✶✷✮ ✤➲✉ ✤ó♥❣✳ ❚✐➳♣ tử ỵ tr t õ p(xn1 , xn ) ≤ λ max{p(xn−2 , xn−1 ), p(xn−1 , xn−2 )} ≤ ≤ λn max{p(x0 , x1 ), p(x1 , x0 )} ✣➦t r(x0 ) = max{p(x0 , x1 ), p(x1 , x0 )}✳ ❱ỵ✐ ♠é✐ m > n t❛ ❝â m−n−1 p(xn , xm ) ≤ m−n−1 λn+k r(x0 ) ≤ λn r(x0 )(1 − λ)−1 p(xn+k , xn+k+1 ) ≤ k=0 k=0 ✸✼ ❱➻ lim (sup{p(xn , xm ) : m > n}) = 0✳ ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✸✱ {xn } ❧➔ ❞➣② n→∞ ❈❛✉❝❤②✳ X ổ tr tỗ t↕✐ ✤✐➸♠ z ∈ X s❛♦ ❝❤♦ lim xn = z ✳ ▼➦t ❦❤→❝✱ ✈➻ f0 ❧✐➯♥ tö❝ ♥➯♥ n→∞ f0 (z) = f0 lim xn = lim (f0 (xn )) = lim (xn+1 ) = z n→∞ n→∞ n→∞ ❉♦ ✤â f0 (z) = z ✳ ❚ø ✭✶✵✮ t❛ ❝â p(z, z) = p(f0 (z), z) = p(z, f0 (z)) = p(f0 (z), f0 (z)) ≤ λp(z, z), ✈➟② p(z, z) = 0✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✷✳✼✱ z ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ f0 ✈➔ fn ✈ỵ✐ n = 1, 2, ú ỵ ◆➳✉ ✤✐➲✉ ❦✐➺♥ f0 ❧✐➯♥ tư❝ ✤÷đ❝ t❤❛② t❤➳ ❜➡♥❣ ỷ tử ữợ p t tự t t ỵ ú p ỷ tử ữợ t tự t t tứ ✭✶✵✮ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ t❛ ❝â✳ p(z, f0 (z)) ≤ p(z, xn ) + p(f0 (xn−1 ), f0 (z)) ≤ p(z, xn ) + λ max{p(z, xn−1 ), p(z, f0 (z)), p(xn−1 , fn (xn−1 )), [p(z, fn (xn−1 )) + p(xn−1 , f0 (z))]} ≤ p(z, xn ) + λ max{p(x, xn−1 ), p(z, f0 (z)), p(xn−1 , xn ), [p(z, xn ) + p(xn−1 , f0 (z))]} ≤ p(z, xn ) + λ[p(z, xn−1 ) + p(z, f0 (z)) + p(xn−1 , xn ) + p(x, z)] ❉♦ ✤â p(z, f0 (z)) ≤ [p(z, xn ) + λ[p(z, xn−1 ) + p(xn−1 , xn ) + p(xn , z)]] 1−λ ❚ø (τ3 ) s✉② r❛ p(xn , z) ≤ lim inf (p(xn , xm ) ≤ λn r(x0 )(1 − λ)−1 n→∞ ❙✉② r❛ lim (p(xn , z)) = 0✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ❝→❝❤ ①➙② ❞ü♥❣ {xn } t❤➻ n→∞ lim (p(xn−1 , xn )) = n→∞ ✸✽ ứ p ỷ tử ữợ t tự ♥❤➜t t❛ ❝â lim p(z, xn ) = lim p(z, xn−1 ) = n→∞ n→∞ ❉♦ ✤â✱ p(z, f0 (z)) = 0✳ ▼➦t ❦❤→❝ p(f0 (z), z) ≤ p(xn , z) + p(f0 (z), f0 (xn−1 )) ≤ p(xn , z) + λ max{p(z, xn−1 ), p(xn−1 , f0 (xn−1 )), p(z, f0 (z)), [p(xn−1 , f0 (z)) + p(z, f0 (xn−1 ))]} ≤ p(xn , z) + λ max{p(z, xn−1 ), p(xn−1 , xn ), p(z, f0 (z)), [p(xn−1 , f0 (z)) + p(z, xn )]} ≤ p(xn , z) + λ[p(z, xn−1 ) + p(xn−1 , xn ) + p(xn , z) + p(z, f0 (z))] ❉♦ ✤â p(f0 (z), z) = 0✳ ❚ø ✤â t❛ ❝â f0 (z) = z ✳ ❚ø ự tr t õ ỵ s ỵ sỷ (X, d) ổ tr ✤õ✱ p ❧➔ τ ✲❦❤♦↔♥❣ ❝→❝❤ tr➯♥ X t❤ä❛ ♠➣♥ p ♥û❛ ❧✐➯♥ tö❝ t❤❡♦ ❜✐➳♥ t❤ù ♥❤➜t ✈➔ {fn} ❞➣② →♥❤ ①↕ tr♦♥❣ X t❤ä❛ ♠➣♥ max{p(f0 (x), fn (y)), p(fn (y), f0 (x))} ≤λ max{p(x, y), p(x, f0 (x), p(y, fn (y)), [p(x, fn (y)) + p(y, f0 (x))]} ✈ỵ✐ ♠é✐ x, y ∈ X, λ ∈ (0, 1) ✈➔ n = 0, 1, 2, t❤➻ {fn} ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❧➔ z t❤ä❛ ♠➣♥ p(z, z) = 0✳ ✸✾ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ởt số t q sỹ tỗ t t ✤ë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ τ ✲❦❤♦↔♥❣ ❝→❝❤✳ ✷✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ♠➔ tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➾ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t✳ ✸✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ♠➔ tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❦❤æ♥❣ ❝❤ù♥❣ ♠✐♥❤✱ ✤â ❧➔ ▼➺♥❤ ✤➲ ✶✳✷✳✻✱ ❱➼ ❞ư ✷✳✷✳✷✭❛✮✳ ✹✳ ✣÷❛ r❛ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❦➳t q✉↔ ♠ỵ✐ ✤â ❧➔ ◆❤➟♥ ①➨t ✶✳✷✳✷✱ ▼➺♥❤ ✤➲ ✶✳✷✳✸ ✈➔ ❱➼ ❞ö ✶✳✷✳✹✳ ✹✵ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ▲➙♠ ❚❤à ❚❤❛♥❤ ▲♦❛♥ ✭✷✵✶✵✮✱ ❙ü tỗ t t tr ổ Otr ❦❤æ♥❣ ❣✐❛♥ ❝â t❤ù tü✱ ▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❚♦→♥ ❤å❝✱ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❬✷❪ ❍♦➔♥❣ ❚❤à ❚❤✉ ✭✷✵✶✵✮✱ ▼ët số ỵ sỹ tỗ t t ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔♦ O✲♠➯tr✐❝✱ ▲✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❚♦→♥ ❤å❝✱ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❬✸❪ ❘✳ ❑❛♥♥❛♥ ✭✶✾✻✾✮✱ ❙♦♠❡ r❡s✉❧ts ♦♥ ❢✐①❡❞ ♣♦✐♥ts✳ ■■✱ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧② ✼✻✱ ✹✵✺ ✲ ✹✵✽✳ ❬✹❪ ❆✳ ▼❡✐r ❛♥❞ ❊✳ ❑❡❡❧❡r ✭✶✾✻✾✮✱ ❆ t❤❡♦r❡♠ ♦♥ ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✷✽✱ ✸✷✻ ✲ ✸✷✾✳ ❬✺❪ ❚✳ ❙✉③✉❦✐ ✭✷✵✵✶✮✱ ●❡♥❡r❛❧✐③❡❞ ❞✐st❛♥❝❡ ❛♥❞ ❡①✐st❡♥❝❡ t❤❡♦r❡♠s ✐♥ ❝♦♠✲ ♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✷✺✸✱ ♥♦✳✷✱ ✹✹✵ ✲ ✹✺✽✳ ❙❡✈❡r❛❧ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❝♦♥❝❡r♥✐♥❣ τ ✲❞✐st❛♥❝❡✱ ✷✵✵✹ ❍✐♥❞❛✇✐ ♣✉❜❧✐s❤✐♥❣ ❝♦r♣♦r❛t✐♦♥✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❬✻❪ ❚✳ ❙✉③✉❦✐ ✭✷✵✵✹✮✱ ❝♦♦②r✐❣❤t ❝ ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✿ ✸✱ ✶✾✺ ✲ ✷✵✾✳ ❬✼❪ ❇✳ ❱❛❦✐❧❛❜❛❞ ❛♥❞ ❙✳ ▼❛♥s♦✉✈ ❱❛❡③♣♦✉r ✭✷✵✶✵✮✱ ●❡♥❡r❛❧✐③❡❞ ❝♦♥tr❛❝t✐♦♥s ❛♥❞ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❝♦♥❝❡r♥✐♥❣ τ ✲❞✐st❛♥❝❡✱ ❏✳ ◆♦♥❧✐♥❡❛r✳ ❙❝✐✳ ❆♣♣❧✳✸✳ ♥♦✳✷✱ ✼✽ ✲ ✽✻✳ ✹✶ ... t t tr sỹ t? ?? t t ❝❤✉♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ tr ợ t ữỡ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ✤❛ trà✱ ✤✐➸♠ ❜? ?t tr sỹ t? ?? t ✤✐➸♠ ❜? ?t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ✤❛ trà p✲❝♦✳ ▼ö❝ t t tr... ✷✳✷✳✼✱ z ❧➔ ✤✐➸♠ ❜? ?t ✤ë♥❣ ❞✉② ♥❤? ?t ❝õ❛ f0 ✈➔ fn ✈ỵ✐ ♠å✐ n = 1, 2, ú ỵ f0 t? ?? ữủ t t ỷ t? ?? ữợ p t t? ?? t t ỵ ú p ỷ t? ?? ữợ t t? ?? t t❤➻ t? ? ✭✶✵✮ ✈➔ ❜? ?t ✤➥♥❣ t? ??ù❝ t? ??♠ ❣✐→❝ t? ?? ❝â✳ p(z, f0 (z))... ✤? ?t r = α ∈ [0, 1)✳ ❚ø 1−α p (T x, T x2 ) ≤ αp(x, T x) + αp (T x, T x) ✈➔ p (T x, T x) ≤ αp (T x, T x) + αp (T x, x), t? ?? ❝â p (T x, T x) + p (T x, T x) ≤ rp (T x, x) + rp(x, T x) ✷✷ ♠å✐ x ∈ X ỵ t