Không gian mêtric nón véctơ tôpô và sự tồn tại điểm bất động

▼Ö❈ ▲Ö❈ ❚r❛♥❣ ▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈➨❝tì tỉ♣ỉ ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ◆â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỹ tỗ t t tr ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t↕✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ é ỵ tt t ởt tr♦♥❣ ♥❤ú♥❣ ❝❤õ ✤➲ ✤÷đ❝ ♥❤✐➲✉ ❝❤✉②➯♥ ❣✐❛ ❣✐↔✐ t➼❝❤ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤✉ ✤÷đ❝ ♥❤✐➲✉ ❦➳t q✉↔✳ ❑➳t q✉↔ q✉❛♥ trå♥❣ ✤➛✉ t✐➯♥ ♣❤↔✐ ❦➸ ✤➳♥ ❧➔ ỵ tr ổ tr ✤➛② ✤õ✳ ❉ü❛ ✈➔♦ ❦➳t q✉↔ ♥➔② ♥❣÷í✐ t❛ ✤➣ ♠ð rë♥❣ ♥â ❝❤♦ ♥❤✐➲✉ ❧♦↕✐ →♥❤ ①↕ ✈➔ ♥❤✐➲✉ ổ ởt tr ỳ ữợ rở õ ự sỹ tỗ t t ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ◆❤ú♥❣ ữớ t ữủ t q q trồ ữợ ❧➔ ❈❛r✐st✐✱ ▼✳ ❆❛♠r✐ ❛♥❞ ❉✳ ❊■ ▼♦✉t❛✇❛❦✐❧✱ ❆❜❞✉❧ ▲❛t✐❢✱ ❲❛❢❛❛ ❆✳ ❆❧❜❛r✳✳✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ❞ü❛ t t ự ỵ tt t ự sỹ tỗ t ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ✤â✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣ ❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ ✈➼ ❞ư ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ❈❤÷ì♥❣ ✷✳ ỹ tỗ t t tr ổ tr ♥â♥✲✈❡❝tì tỉ♣ỉ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ❝♦✱ →♥❤ ①↕ ❝♦ s✉② rë♥❣ ✈➔ ởt số t q sỹ tỗ t t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❝♦ ✈➔ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ❈→❝ ❦➳t q✉↔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤õ ②➳✉ ❧➔ ✤➣ ❝â tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤ó♥❣ tỉ✐ t➻♠ ❤✐➸✉ ✈➔ tr➻♥❤ ❜➔② t❤❡♦ ♠ö❝ ✤➼❝❤ ✤➣ ✤➦t r❛✳ ◆❣♦➔✐ ✈✐➺❝ ❤➺ t❤è♥❣✱ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ♠➔ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ✷ ❦❤↔♦ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t ❤♦➦❝ ❜ä q✉❛ ❝❤ù♥❣ ♠✐♥❤ ❝❤ó♥❣ tỉ✐ ❝ơ♥❣ ữ r ự ởt số t q ợ õ ỵ ỵ ỵ ỵ q ỵ ỵ ỵ ỵ ữủ tỹ t rữớ 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♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✳ ❚→❝ ❣✐↔ ữủ ỡ qỵ ổ tr ❚ê ●✐↔✐ t➼❝❤ ✈➔ tr♦♥❣ ❑❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ❈✉è✐ ❝ị♥❣ 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→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sõt qỵ ổ õ õ ỵ ữủ t ỡ ❱✐♥❤✱ t❤→♥❣ ✵✾ ♥➠♠ ✷✵✶✷ ❚→❝ ❣✐↔ ✸ ❈❍×❒◆● ✶ ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆✲❱❊❈❚❒ ❚➷P➷ ❈❤÷ì♥❣ ♥➔②✱ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ ✈➼ ❞ư ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ✶✳✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ❝ì ❜↔♥ ❝➛♥ ❞ị♥❣ tr♦♥❣ ❧✉➟♥ ✈➠♥✳ ✶✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ❈❤♦ t➟♣ ❤ñ♣ X ✈➔ ❤➔♠ d : X ì X R d ữủ ❧➔ ♠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 õ ữủ ổ tr ỵ ❤✐➺✉ (X, d) ❤❛② ✤ì♥ ❣✐↔♥ ❤ì♥ ❧➔ X ✳ ✶✳✶✳✷✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ❈❤♦ t➟♣ ❤ñ♣ X ✳ ❍å τ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ tỉ♣ỉ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❚1✮ ∅ ✈➔ X ∈ τ ❀ ✭❚2✮ ◆➳✉ Gi ∈ τ, ✈ỵ✐ ♠å✐ i ∈ I t❤➻ Gi ∈ τ ❀ i∈I ✭❚3✮ ◆➳✉ G1, G2 ∈ τ t❤➻ G1 ∩ G2 ∈ τ ✳ ❚➟♣ ❤đ♣ X ❝ị♥❣ ✈ỵ✐ tỉ♣ỉ τ tr➯♥ ♥â ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ tỉ♣ỉ ✈➔ ỵ (X, ) ỡ ỡ ❧➔ X ✳ ❈→❝ ♣❤➛♥ tû ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✳ ❈→❝ ♣❤➛♥ ✹ tû ❝õ❛ τ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ t➟♣ ♠ð tr♦♥❣ X ✳ ❚➟♣ ❝♦♥ Y ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ✤â♥❣ tr♦♥❣ X ♥➳✉ X \ Y ∈ τ ✳ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✱ tæ♣æ tr➯♥ ♥â ❧➔ tæ♣æ s✐♥❤ ❜ð✐ ♠➯tr✐❝✳ ✶✳✶✳✸✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tæ♣æ✳ ❚➟♣ ❝♦♥ ❣å✐ ❧➔ ởt X tỗ t↕✐ t➟♣ ♠ð G s❛♦ ❝❤♦ a ∈ G ⊂ U✳ ❍å ϑ ❝→❝ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ a ❣å✐ ❧➔ ❝ì sð ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ a ♥➳✉ ♠å✐ U a tỗ t V ϑ s❛♦ ❝❤♦ V ⊂ U ✳ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ X ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ t✐➯♥ ✤➲ ✤➳♠ ✤÷đ❝ tự t t ộ x X tỗ t ♠ët ❝ì sð ❧➙♥ ❝➟♥ ✤➳♠ ✤÷đ❝✳ U ⊂X ✶✳✶✳✹✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ❉➣② {xn} tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ tæ♣æ X ữủ tử tợ x X ợ ộ U x tỗ t n0 ∈ N s❛♦ ❝❤♦ xn ∈ U ✈ỵ✐ ♠å✐ n ≥ n0 ❑❤✐ ✤â✱ t❛ ✈✐➳t xn → x ❤♦➦❝ n→lim∞ xn = x ✶✳✶✳✺✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ❑❤æ♥❣ ❣✐❛♥ tỉ♣ỉ X ✤÷đ❝ ❣å✐ ❧➔ T1✲❦❤ỉ♥❣ ♥➳✉ ❤❛✐ ✤✐➸♠ t ý x, y X, x = y tỗ t↕✐ ❝→❝ ❧➙♥ ❝➟♥ t÷ì♥❣ ù♥❣ Ux , Uy ❝õ❛ x ✈➔ y s❛♦ ❝❤♦ y ∈ / Ux ✈➔ x ∈ / Uy ✳ ❑❤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✳ ❣✐❛♥ ✺ ✶✳✶✳✻✳ ✣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 ✳ K✳ ✶✳✶✳✼✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ ▼ët tỉ♣ỉ τ tr➯♥ X ❣å✐ ❧➔ t÷ì♥❣ t❤➼❝❤ ✈ỵ✐ ❝➜✉ tró❝ ✤↕✐ sè ❝õ❛ X ♥➳✉ ❝→❝ →♥❤ ①↕ (x, y) −→ x + y : X × X −→ X ✈➔ (λ, x) −→ λx : K × X −→ X ❧➔ ❧✐➯♥ tư❝✳ ▼ët ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ tr➯♥ tr÷í♥❣ K ❧➔ ♠ët ❝➦♣ ✭X ✱τ ✮ tr♦♥❣ ✤â X ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝ tì tr K ỏ ởt tổổ tữỡ t ợ ❝➜✉ tró❝ ✤↕✐ sè ❝õ❛ X ✳ ❙❛✉ ♥➔② τ ♥❤÷ ✈➟② ❣å✐ ❧➔ tỉ♣ỉ ✈❡❝tì ✈➔ t❤❛② ❝❤♦ ✭X ✱τ ✮ t❛ ✈✐➳t X ♥➳✉ ♥❤÷ tỉ♣ỉ τ ✤â ữủ ố ợ a tở ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ X ✈➔ ✈ỵ✐ ♠å✐ α ∈ K✱ α = 0✱ t❛ ❝â ❛✮ P❤➨♣ tà♥❤ t✐➳♥ ❝❤♦ ❜ð✐ f (x) = x + a ✈ỵ✐ ♠å✐ x ∈ X ❀ ❜✮ P❤➨♣ ✈à tü ❝❤♦ ❜ð✐ g(x) = α x ✈ỵ✐ ♠å✐ x ∈ X ❧➔ ỗ ổ tứ X X ✶✳✶✳✾✳ ▼➺♥❤ ✤➲ ✭❬✶❪✮✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ a ∈ X ✈➔ ✈ỵ✐ ♠å✐ α ∈ K✱ α = 0✳ ❑❤✐ ✤â✱ ❛✮ ❚➟♣ V ⊂ X ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ tr♦♥❣ X ❦❤✐ ✈➔ ❝❤✐ ❦❤✐ t➟♣ a + V ❧➔ ✻ ❧➙♥ ❝➟♥ ❝õ❛ a✳ ❜✮ ❚➟♣ V ⊂ X ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ tr♦♥❣ X ❦❤✐ ✈➔ ❝❤✐ ❦❤✐ t➟♣ αV ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ 0✳ ✶✳✶✳✶✵✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ●✐↔ sû A ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ X ✳ ❛✮ ❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ❤ót ♥➳✉ ợ x X tỗ t s❛♦ ❝❤♦ x ∈ λA ✈ỵ✐ ♠å✐ α ∈ K✱ | α | ≥ λ✳ ❜✮ ❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ❝➙♥ ♥➳✉ ✈ỵ✐ ♠å✐ x ∈ A✱ t❛ ❝â αx ∈ A ✈ỵ✐ ♠å✐ α ∈ K✱ | α | ≤ 1✳ ❝✮ ❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ ỗ ợ x y A ợ ♠å✐ t ∈ [0, 1]✱ t❛ ❝â t.x + (1 − t).y ∈ A✳ ❞✮ ❚➟♣ A ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❜à ❝❤➦♥ ♥➳✉ ✈ỵ✐ ♠é✐ ❧➙♥ ❝➟♥ V ❝õ❛ tỗ t số s > s A ⊂ t V ✈ỵ✐ ♠å✐ t > s ✶✳✶✳✶✶✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✶❪✮✳ ●✐↔ sû X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ tr trữớ K X ữủ ổ ỗ ✤à❛ ♣❤÷ì♥❣ ♥➳✉ t↕✐ ✤✐➸♠ ∈ X ❝â ♠ët ❝ì sð ❧➙♥ ❝➟♥ B s❛♦ ❝❤♦ ♠å✐ ♣❤➛♥ tû B t ỗ sỷ E ổ ỗ ữỡ ✤â✱ ∈ E ❝â ❝ì sð ❧➙♥ ❝➟♥ U t❤ä❛ ♠➣♥ ✐✮ α U ∈ U ∀ α ∈ K, α = 0, ∀ U ∈ U ✐✐✮ ▼å✐ U U ỗ ✭❬✶❪✮✳ ●✐↔ sû E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì✳ ❍➔♠ ♣ ①→❝ ✤à♥❤ tr➯♥ E ✈➔ ♥❤➟♥ ❣✐→ trà t❤ü❝ ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❝❤✉➞♥ tr➯♥ E ♥➳✉ ✐✮ p(x) ≥ ∀ x ∈ E ✳ ✐✐✮ p(λ x) = |λ| p(x) ∀ λ ∈ K, ∀ x ∈ E ✼ ✐✐✐✮ p(x + y) ≤ p(x) + p(y) ∀ x, y ∈ E ❚ø ✐✐✮ s✉② r❛ p(0) = p(0.0) = 0.p(0) = 0✳ ◆û❛ ❝❤✉➞♥ p ❣å✐ ❧➔ ❝❤✉➞♥ ♥➳✉ p(x) = s✉② r❛ x = ỵ sỷ Q ♥û❛ ❝❤✉➞♥ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì E ✳ ❑❤✐ ✤â✱ tỗ t tổổ ỗ ữỡ t tr E s❛♦ ❝❤♦ ♠å✐ ♥û❛ ❝❤✉➞♥ t❤✉ë❝ Q ❧➔ ❧✐➯♥ tö❝✳ ✶✳✷✳ ◆➶◆ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ❱❊❈❚❒ ❚➷P➷ ▼ö❝ ♥➔② s➩ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ ✈➼ ❞ö ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ✳ ✶✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ●✐↔ sû E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✈➔ P ❧➔ t➟♣ ❝♦♥ ❝õ❛ E ✳ P ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ tr♦♥❣ E ♥➳✉ ✶✮ P = ∅✱ P = {0} ✈➔ P ✤â♥❣ tr♦♥❣ E ❀ ✷✮ ax + by ∈ P ✱ ∀ x✱ y ∈ P ✈➔ ∀ a✱ b ∈ R✱ a ≥ 0✱ b ≥ 0❀ ✸✮ P ∩ (−P ) ❂ {0}✳ ●✐↔ sû P ❧➔ ♥â♥ tr♦♥❣ E ✳ ❑❤✐ ✤â t❛ ①→❝ ✤à♥❤ ❤❛✐ q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ tr E ữ s ợ x y E x ≤ y ⇔ y − x ∈ P; x y ⇔ y − x ∈ intP ❚❛ ❣å✐ ❤❛✐ q✉❛♥ ❤➺ ♥➔② ❧➔ ❤❛✐ q✉❛♥ ❤➺ t❤ù tü tr➯♥ X ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ♥â♥ P ✳ ❚❛ ✈✐➳t x < y t❤❛② ❝❤♦ x ≤ y ✈➔ x = y✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t❛ ❧✉æ♥ ❣✐↔ t❤✐➳t P ❧➔ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ E ✈ỵ✐ ✐♥tP = ∅❀ ≤ ✈➔ ❧➔ ❤❛✐ q✉❛♥ ❤➺ t❤ù tü tr➯♥ E ✤÷đ❝ ①→❝ ❜ð✐ P✳ ✽ ✶✳✷✳✷✳ ❱➼ ❞ư✳ ✶✮ ❱➻ ♠é✐ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ♥➯♥ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❝ơ♥❣ ❧➔ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ✳ ✷✮ ❚❛ ❦➼ ❤✐➺✉ CR ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ ❧✐➯♥ tử tứ R õ ỵ CR ổ ỗ ữỡ ợ tổổ ✤à♥❤ ❜ð✐ ❤å ❝→❝ ♥û❛ ❝❤✉➞♥ {qn : n = 1, 2, }✱ tr♦♥❣ ✤â qn (f ) = sup{ |f (x)| : x ∈ [−n, n] } , f ∈ CR ✣➦t P = { f ∈ CR : f (x) ≥ ∀ x ∈ R } ❍✐➸♥ ♥❤✐➯♥ P = ∅✱ P = {0}✳ ●✐↔ sû {fα} ❧➔ ♠ët ❞➣② s✉② rë♥❣ tr♦♥❣ P ✈➔ {fα } ❤ë✐ tư tỵ✐ f ∈ CR ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â qn (fα − f ) ❤ë✐ tư tỵ✐ 0✳ ❚ø ✤â s✉② r❛ fα (x) → f (x) ✈ỵ✐ ♠å✐ x ∈ R ▼➦t ❦❤→❝ fα(x) ≥ ✈ỵ✐ ♠å✐ α ♥➯♥ f (x) ≥ ✈ỵ✐ ♠å✐ x ∈ R✳ ❉♦ ✤â P ✤â♥❣ tr♦♥❣ CR✳ ◆❤÷ ✈➟② P t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✶✮ tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❉➵ t❤➜② P ❝ô♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✷✮✱ ✸✮ tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❱➟② P ❧➔ ♥â♥ tr♦♥❣ CR ✳ ❚r♦♥❣ ✈➼ ❞ư ♥➔②✱ t❛ t❤➜② r➡♥❣✱ ✈ỵ✐ f ✈➔ g ∈ CR t❤➻ f ≤ g ⇔ g − f ∈ P ⇔ f (x) − g(x) ≥ ∀ x ∈ CR ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä q✉❛♥ ❤➺ ≤ tr➯♥ E ①→❝ ✤à♥❤ ❜ð✐ P trị♥❣ ✈ỵ✐ q✉❛♥ ❤➺ ≤ t❤ỉ♥❣ t❤÷í♥❣ tr➯♥ t➟♣ ❤đ♣ ❝→❝ ❤➔♠ sè✳ ✶✳✷✳✸✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✺❪✮✳ ●✐↔ sû x ✈➔ y ∈ E ✱ x ≤ y✳ ✣➦t [x, y] = {z ∈ E : x ≤ z ≤ y} ✾ ❚➟♣ A E ữủ ỗ t tự tỹ ♥➳✉ [x, y] ⊂ A ✈ỵ✐ ♠å✐ x✱ y ∈ A ♠➔ x ≤ y✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ E ữủ ỗ t tự tỹ t E õ ởt ỡ s ỗ t ỗ t tự tỹ E ỗ t tự tỹ t õ P ữủ ❝❤✉➞♥ t➢❝✳ ✶✳✷✳✹✳ ▼➺♥❤ ✤➲ ✭❬✺❪✮✳ ●✐↔ sû P ❧➔ ♥â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ E ✳ ❑❤✐ ✤â✱ E ỗ t tự tỹ t tỗ t sè ❦ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x✱ y ∈ E ✱ ≤ x ≤ y t❛ ❝â x ≤❦ y sỷ E ỗ t tự tỹ B = B(0, 1) ❧➔ ❤➻♥❤ ❝➛✉ ♠ð t➙♠ tr E õ tỗ t ❧➙♥ ❝➟♥ U ❝õ❛ tr♦♥❣ E s❛♦ ❝❤♦ U ỗ t tự tỹ U B t B t tỗ t ❦ > s❛♦ ❝❤♦ B ⊂ kU ✳ ●✐↔ sû x✱ y ∈ E ✱ ≤ x ≤ y✳ ◆➳✉ y = t❤➻ x = 0✱ ❞♦ ✤â x ❂ ❦ y ✳ ●✐↔ sû y = 0✳ ❑❤✐ ✤â✱ tø y − x ∈ P s✉② r❛ (y − x) ∈ P ✱ tù❝ ❧➔ y ❈❤ù♥❣ ♠✐♥❤✳ y x≤ k y x≤ 0≤ ❉♦ ✤â t❛ ❝â 0≤ y y ∈ B ⊂ kU k y y ∈ U ⊂ B ứ t ỗ t tự tỹ U s r❛ k y ❉♦ ✤â x ≤ ❦ y x ∈ U ⊂ B ✳ ✶✳✷✳✺✳ ▼➺♥❤ ✤➲ ✭❬✹❪✱✭❬✺❪✮✮✳ ●✐↔ sû P ❧➔ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ E ✳ ❑❤✐ ✤â✱ ✶✮ ◆➳✉ x✱ y ✱ z ∈ E ✱ x ≤ y ✈➔ y z t❤➻ x z tỗ t no N s ❝❤♦ d(xn , x) c ∀ n ≥ no ❉♦ ✤â xn ∈ B(x, c) ⊂ V ✈ỵ✐ ♠å✐ n no xn x r ỵ s❛✉✱ t❛ ❣✐↔ sû (X, d)✱ (Y, d ) ❧➔ ❤❛✐ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ✈ỵ✐ ❝→❝ ♠➯tr✐❝ ♥â♥ ♥❤➟♥ ❣✐→ trà tr♦♥❣ ♥â♥ ❝❤✉➞♥ t➢❝ P ✳ ✶✳✸✳✻✳ ỵ sỷ X Y ổ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✱ f ✿ X −→ Y ✈➔ a ∈ X ✳ ❑❤✐ ✤â✱ f ❧✐➯♥ tö❝ t↕✐ a ❦❤✐ ✈➔ ❝❤✐ ❦❤✐ ♠é✐ ❞➣② {xn } ⊂ X ♠➔ xn → a t❤➻ f (xn ) → f (a)✳ ●✐↔ sû f ❧✐➯♥ tư❝ t↕✐ a ♥❤÷♥❣ tỗ t {xn} tr X xn a ữ f (xn ) ổ tử tợ f (a) õ t ỵ {d (f (a), f (xn))} ❦❤ỉ♥❣ ❤ë✐ tư 0✳ ❉♦ ✤â✱ tø ▼➺♥❤ ✤➲ s r tỗ t yo intP s ợ ộ n N tỗ t mn N mn > n ✈➔ yo − d (f (a), f (xm )) ∈/ intP ✳ ❱➻ f ❧✐➯♥ tö❝ t↕✐ a ✈➔ B(f (a), yo) ❧➔ ♠ð tr♦♥❣ Y ♥➯♥ tỗ t co intP s ự n f (B(a, co )) ⊂ B(f (a), yo ) ❱➻ xn a tỗ t n1 s xn ∈ B(a, co) ✈ỵ✐ ♠å✐ n ≥ n1✳ ❉♦ ✤â✱ f (xn ) ∈ B(f (a), yo ) ✈ỵ✐ ♠å✐ n ≥ n1, ❤❛② ✈ỵ✐ ♠å✐ n ≥ n1 ❉♦ ✤â yo − d (f (a), f (xn)) ∈ intP ✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ d (f (a), f (xn )) yo yo − d (f (a), f (xmn1 )) ∈ / intP ❚ø ✤â s✉② r❛ f (xn) → f (a)✳ ✶✾ ✈ỵ✐ mn > n1 ữủ sỷ r ợ ộ {xn} tr X ♠➔ xn → a t❤➻ f (xn ) → f (a)✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ f ❧✐➯♥ tö❝ t↕✐ a ∈ X ✳ ●✐↔ sû f ❦❤ỉ♥❣ ❧✐➯♥ tư❝ t↕✐ a ∈ X ✳ ❑❤✐ ✤â✱ tø ✣à♥❤ ♥❣❤➽❛ s r tỗ t yo intP s ✈ỵ✐ ♠å✐ c ∈ intP ✤➲✉ ❝â f (B(a, c)) ⊂ B(f (a), yo )) ❚ø ✤â s✉② r❛ r➡♥❣ ợ ộ n = 1, 2, tỗ t xn ∈ B(a, nc ) s❛♦ ❝❤♦ f (xn ) ∈ / B(f (a), yo ) ❚ø xn ∈ B(a, nc ) ✈ỵ✐ ♠å✐ n = 1, 2, ✈➔ nc → 0✱ →♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✶✳✷✳✺✳✻✮ s✉② r❛ d(a, xn) → 0✳ ❉♦ ✤â xn → a✳ ❱➻ t❤➳ t❤❡♦ ❣✐↔ t❤✐➳t ❝õ❛ ✤✐➲✉ ❦✐➺♥ ✤õ t❛ ❝â f (xn) → f (a)✳ ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ f (xn) ∈/ B(f (a), yo) ✈ỵ✐ ♠å✐ n✳ ❱➟② f ❧✐➯♥ tư❝ t↕✐ a✳ ✷✵ ❈❍×❒◆● ✷ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ❱➋ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆✲❱❊❈❚❒ ❚➷P➷ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè t q sỹ tỗ t t →♥❤ ①↕ ❝♦ ✈➔ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ✷✳✶✳ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ⑩◆❍ ❳❸ ❈❖ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆✲❱❊❈❚❒ ❚➷P➷ ✷✳✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✭❬✸❪✮✳ ❈❤♦ (X, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ✈➔ {xn} ⊂ X ✳ ❉➣② {xn} ⊂ X ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ❈❛✉❝❤② ♥➳✉ ợ c intP tỗ t no N s❛♦ ❝❤♦ d(xm , xn ) c ✈ỵ✐ ♠å✐ m, n ≥ no ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ (X, d) ✤÷đ❝ ❣å✐ ❧➔ ✤➛② ✤õ ♥➳✉ ♠å✐ ❞➣② ❈❛✉❝❤② {xn} tr X tử tự tỗ t x ∈ X s❛♦ ❝❤♦ xn → x ✭d(x, xn) → ✮✳ ✷✳✶✳✷✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû (X, d) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ⑩♥❤ ①↕ f ✿ X −→ X ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❝♦ tỗ t [0, 1) s d(f (x), f (xn )) ≤ α d(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X ❚❛ ❣å✐ α ❧➔ ❤➡♥❣ sè f ú ỵ ú t t f x t❤❛② ❝❤♦ f (x)✳ ✷✳✶✳✸✳ ◆❤➟♥ ①➨t✳ ◆➳✉ f ✿ X −→ X ❧➔ →♥❤ ①↕ ❝♦ t❤➻ f ❧✐➯♥ tö❝✳ ✷✶ ●✐↔ sû x ∈ X ✈➔ {xn} ❧➔ ❞➣② tr♦♥❣ X ✱ xn → x✳ ❑❤✐ õ t ỵ d(x, xn) f tỗ t [0, 1) s❛♦ ❝❤♦ ❈❤ù♥❣ ♠✐♥❤✳ ≤ d(f (x), f (xn )) ≤ α d(x, xn ) → ❉♦ ✤â d(f (x), f (xn)) → 0✳ ❉♦ ✤â✱ t ỵ f (xn) f (x) ỵ f tử t x x ❧➔ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ X ♥➯♥ f ❧✐➯♥ tö❝ tr X ỵ (X, d) ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ✤➛② ✤õ ✈➔ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ t❤➻ ♠å✐ →♥❤ ①↕ ❝♦ tr➯♥ X ✤➲✉ ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ●✐↔ sû f ✿ X −→ X ❧➔ →♥❤ ①↕ ❝♦ ✈ỵ✐ ❤➡♥❣ sè ❝♦ α✳ ▲➜② xo ∈ X ✳ ✣➦t f (xo ) = x1 ✱ f (x1 ) = x2 ✱✳✳✳✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ d(xn , xn+1 ) = d(f xn−1 , f xn ) ≤ α d(xn−1 , xn ) = α d(f xn−2 , f xn−1 ) ≤ α2 d(xn−2 , xn−1 ) ≤ ≤ αn d(xo , x1 ) ❉♦ ✤â✱ ✈ỵ✐ ♠é✐ n = 1, 2, ✈➔ ✈ỵ✐ ♠é✐ p ∈ N t❛ ❝â d(xn , xn+p ) ≤ ❞(xn , xn+1 ) + d(xn+1 , xn+2 ) + + d(xn+p−1 , xn+p ) ≤ ✭αn + αn+1 + + αn+p−1 ) d(xo , x1 ) − αp αn d(xo , x1 ) ≤ d(xo , x1 ) =α 1−α 1−α αn α ∈ [0, 1) s✉② r❛ rn := 1−α → ❦❤✐ n → ∞✳ ●✐↔ n ✭✶✮ sû U ❧➔ ❧➙♥ ❝➟♥ ❚ø ❜➜t ❦ý ❝õ❛ tr E s tỗ t no N t❤ä❛ ♠➣♥ n1 d(xo, x1) ∈ U ❱➻ rn → tỗ t n1 N s rn ≤ n1 ✈ỵ✐ ♠å✐ n ≥ n1✳ ❚ø d(xo , x1 ) ∈ P ✈➔ n1 − rn ≥ ✈ỵ✐ ♠å✐ n ≥ n1 s✉② r❛ o o o 1 d(xo , x1 ) − rn d(xo , x1 ) = ( − rn ) d(xo , x1 ) ∈ P, no no ✷✷ tù❝ ❧➔ ✭✷✮ ❱➻ P ❝❤✉➞♥ t➢❝ ♥➯♥ t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t U t ỗ t tự tỹ õ tứ s✉② r❛ rn d(xo , x1 ) ∈ U ✈ỵ✐ ♠å✐ n ≥ n1 ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä rn d(xo, x1) → 0✳ ❑➳t ❤đ♣ ✈ỵ✐ ✭✶✮ ✈➔ →♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✶✳✷✳✺✳✹✮ t❛ ❝â d(xn, xn+p) → ❦❤✐ n → ∞✱ ✈ỵ✐ ♠å✐ p ∈ N✳ ❉♦ ✤â✱ {xn} ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X tỗ t a X s xn → a ❤❛② d(a, xn) → 0✳ ❇➙② ❣✐í✱ t❛ ❝❤ù♥❣ ♠✐♥❤ a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ❚❤➟t ✈➟②✱ t❛ ❝â f (xn ) = xn+1 → a ❱➻ xn → a ✈➔ f ❧✐➯♥ tö❝ ♥➯♥ f (xn ) → f (a)✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✸✳✹✳✹✮ t❤➻ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢✳ ❉♦ ✤â✱ t❛ ❝â a = f (a)✱ tù❝ a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ❈✉è✐ ❝ò♥❣✱ t❛ ❝❤ù♥❣ ♠✐♥❤ a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ f ✳ ●✐↔ sû b ∈ X ❝ô♥❣ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✈➔ a = b✳ ❑❤✐ ✤â✱ tø α ∈ [0, 1) s✉② r❛ ≤ rn d(xo , x1 ) ≤ d(xo , x1 ) ∀ n ≥ n1 no < d(a, b) = d(f (a), f (b)) ≤ α d(a, b) < d(a, b) ❚❛ ❝â ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â a = b✳ ✷✳✶✳✺✳ ú ỵ ứ ự ỵ t t r tt ỵ ✤÷đ❝ t❤ä❛ ♠➣♥ ✈➔ a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f t❤➻ f nx → a ✈ỵ✐ ♠å✐ x ∈ X ✱ tr♦♥❣ ✤â f 2x ❂ f (f x)✱ f 3x ❂ f (f 2x)✱✳✳✳ ✷✳✶✳✻✳ ❍➺ q✉↔✳ ◆➳✉ (X, d) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ✤➛② ✤õ✱ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈➔ g ✿ X −→ X s❛♦ ❝❤♦ g no ❧➔ →♥❤ ①↕ ❝♦ ✈ỵ✐ no ❧➔ sè tü ♥❤✐➯♥ ♥➔♦ ✤â✳ ❑❤✐ ✤â✱ g ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ a ✈➔ g n x −→ a ✈ỵ✐ ♠å✐ x ∈ X ✳ ✷✸ ❱➻ gn ❧➔ →♥❤ ①↕ ❝♦ ♥➯♥ t❤❡♦ ✣à♥❤ ỵ gn õ t t a ❑❤✐ ✤â✱ tø gn a = a s✉② r❛ ❈❤ù♥❣ ♠✐♥❤✳ o o o g no (ga) = g(g no a) = ga ◆❤÷ ✈➟② ga ❝ơ♥❣ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ gn ✳ ❉♦ ✤â a = ga✱ tù❝ a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ g✳ ●✐↔ sû x ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ X ✳ ❑❤✐ ✤â✱ t❤❡♦ ú ỵ t õ (g n )n x a✳ ❚ø ✤â t❛ ❝â o o (g no )n (g j x) → a ❥ = 1, 2, , no lim g no n+j x = a ❥ = 1, 2, , no ❚ù❝ ❧➔ n→∞ ❚ø ✤â s✉② r❛ gnx → a ✈ỵ✐ ♠å✐ x ∈ X ●✐↔ sû b ❝ô♥❣ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ g✳ ❚❤❡♦ ❦➳t q✉↔ ✈ø❛ ❝❤ù♥❣ ♠✐♥❤ t❤➻ gnb → a ❱➻ b ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ g ♥➯♥ gnb = b ✈ỵ✐ ♠å✐ n✳ ❉♦ ✤â b = a✳ ✷✳✶✳✼✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû f ✈➔ ❞➣② {fn} ❧➔ ❝→❝ →♥❤ ①↕ tø ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ X ✈➔♦ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ Y ✳ ❚❛ ♥â✐ {fn } ❤ë✐ tư ✤➲✉ tỵ✐ f tr➯♥ X ♥➳✉ ✈ỵ✐ c intP tỗ t số tỹ nc s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n ≥ nc ✈➔ ✈ỵ✐ ♠å✐ x ∈ X t❛ ❝â d(fn x, f x) c ỵ sỷ X ổ tr ♥â♥✲✈❡❝tì tỉ♣ỉ ✤➛② ✤õ✱ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✱ F ✿ X −→ X ✈➔ Fn ✿ X −→ X ❧➔ ❞➣② ❝→❝ →♥❤ ①↕ ❧✐➯♥ tư❝ s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ Fn ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ xn ✱ n = 1, 2, ✈➔ Fn ❤ë✐ tö ✤➲✉ tỵ✐ F tr➯♥ X ✳ ❑❤✐ ✤â✱ ✷✹ ✶✮ ◆➳✉ xn → xo ❤♦➦❝ F xn → xo t❤➻ xo ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ F ❀ ✷✮ ◆➳✉ F ❧➔ →♥❤ ①↕ ❝♦ t❤➻ {xn } ❤ë✐ tư tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ F✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➛✉ t✐➯♥✱ t❛ ❝❤ù♥❣ ♠✐♥❤ F ❧✐➯♥ tö❝ tr➯♥ X ✳ ❱➻ ❞➣② {Fn} ❤ë✐ tư ✤➲✉ tỵ✐ F tr➯♥ X ợ c intP tỗ t số tỹ ♥❤✐➯♥ nc s❛♦ ❝❤♦ d(F x, Fn x) c ∀ n ≥ nc , ∀ x ∈ X ✭✸✮ ✈ỵ✐ ♠é✐ a ∈ X, ✈➻ Fn ❧✐➯♥ tö❝ t↕✐ a tỗ t Ua a s c ✭✹✮ c ∀ x ∈ Ua d(Fnc a, Fnc x) ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ x ∈ Ua✱ tø ✭✸✮ ✈➔ ✭✹✮ t❛ ❝â d(F a, F x) ≤ d(F a, Fnc a) + d(Fnc a, Fnc x) + d(Fnc x, F x) 3c ❚ø ✤â s✉② r❛ F ❧✐➯♥ tư❝ t↕✐ a✳ ◆❤÷ ✈➟② F ❧✐➯♥ tư❝ tr➯♥ X ✳ ✶✮ ●✐↔ sû xn → xo✭ tù❝ ❧➔ d(xn, xo) → 0✮✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ c ∈ intP tỗ t số tỹ nc s d(xn , xo ) ✭✺✮ c ∀ n ≥ nc tử xn xo tỗ t↕✐ sè tü ♥❤✐➯♥ nc s❛♦ ❝❤♦ ✭✻✮ c ∀ n ≥ nc d(F xo , F xn ) ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✮ t❛ ❝â d(F xn , xn ) = d(F xn , Fn xn ) ✭✼✮ c ∀ n ≥ nc ❚ø ✭✺✮✱✭✻✮✱✭✼✮ s✉② r❛ ✈ỵ✐ ♠å✐ n ≥ max(nc, nc, nc ) t❛ ❝â d(F xo , xo ) d(F xo , F xn ) + d(F xn , Fn xn ) + d(Fn xn , xo ) ✷✺ 3c ❉♦ ✤â tø ▼➺♥❤ ✤➲ ✶✳✷✳✺✳✼✮ s✉② r❛ d(F xo, xo) = 0✱ tù❝ ❧➔ F xo = xo ❱➟② xo ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ F ✳ ●✐↔ sû F xn → xo✭ tù❝ ❧➔ d(xo, F xn) → 0✮✳ ❑❤✐ ✤â✱ tø ✭✼✮ s✉② r❛ d(F xn , xn ) → 0✳ ❑➳t ❤đ♣ ✈ỵ✐ ❜➜t ✤➥♥❣ t❤ù❝ ≤ d(xo , xn ) ≤ d(xo , F xn ) + d(F xn , xn ) ∀ n ❚❛ ❝â d(xo, xn) → 0✱ tù❝ ❧➔ xn → xo✳ ❉♦ ✤â✱ t❤❡♦ ❦➳t q✉↔ ✤➣ ❝❤ù♥❣ ♠✐♥❤ xo ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ F ✳ ✷✮ ●✐↔ sû F ❧➔ →♥❤ õ t ỵ t F ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ a✳ ❚❛ ❝â d(a, xn ) = d(F a, Fn xn ) ≤ d(F a, F xn ) + d(F xn , Fn xn ) ≤ α d(a, xn ) + d(F xn , Fn xn ) ∀ n ❉♦ ✤â✱ ✈ỵ✐ ♠å✐ n t❛ ❝â ≤ d(a, xn ) ≤ 1 d(F xn , Fn xn ) = d(F xn , xn ) 1−α 1−α ❱➻ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈➔ d(F xn, xn) → ❦❤✐ n → ∞ ♥➯♥ tø ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ✈➔ ▼➺♥❤ ✤➲ ✶✳✷✳✺✳✹✮ s✉② r❛ d(a, xn) → 0✱ tù❝ ❧➔ xn → a ✷✳✷✳ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ⑩◆❍ ❳❸ ❈❖ ❙❯❨ ❘❐◆● ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆✲❱❊❈❚❒ ❚➷P➷ ▼ö❝ ♥➔② tr ởt số t q sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ✷✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû (X, d) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲ ✈❡❝tì tỉ♣ỉ✳ ⑩♥❤ ①↕ ❢✿ ❳ −→ ❳ ❧➔ →♥❤ ①↕ ❝♦ s rở tỗ t : P −→ [0, 1) s❛♦ ❝❤♦ d(f (x), f (y)) ≤ ϕ(d(x, y))d(x, y) ∀ x, y ∈ X ✷✳✷✳✷✳ ✣à♥❤ ỵ sỷ X ởt ổ tr õtỡ tæ♣æ ✤➛② ✤õ✱ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈➔ ❢✿ ❳ −→ ❳ ❧➔ →♥❤ ①↕ ❝♦ s✉② rë♥❣✳ ❑❤✐ ✤â✱ ♥➳✉ ϕ ❧➔ ❤➔♠ ❦❤æ♥❣ ❣✐↔♠✱ tù❝ ❧➔ tø t1 ≤ t2 s✉② r❛ ϕ(t1 ) ≤ ϕ(t2 ) t❤➻ f ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ▲➜② xo ∈ X ✳ ✣➦t x1 f (xn ), ✳ ❱ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ = f (xo ), x2 = f (x1 ), , xn+1 = d(xn , xn+1 ) = d(f xn−1 , f xn ) ≤ ϕ(d(xn−1 , xn ))d(xn−1 , xn ) ≤ d(xn−1 , xn ) ❱➻ ϕ ❦❤æ♥❣ ❣✐↔♠ ♥➯♥ ✈ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â ϕ(d(xn , xn+1 )) ≤ ϕ(d(xn−1 , xn )) ❉♦ ✤â✱ ✈ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â d(xn , xn+1 ) ≤ ϕ(d(xn−1 , xn )) d(xn−1 , xn ) ≤ ϕ(d(xn−1 , xn )) ϕ(d(xn−2 , xn−1 )) d(xn−2 , xn−1 ) ≤ [ϕ(d(xn−2 , xn−1 ))]2 d(xn−2 , xn−1 ) ≤ ≤ [ϕ(d(xo , x1 ))]n d(xo , x1 ) ❚ø ✤â ✈➔ sû ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ s✉② r❛ r➡♥❣ ✈ỵ✐ ♠é✐ n = 1, 2, ✈➔ ✈ỵ✐ ♠é✐ p = 1, 2, t❛ ❝â d(xn , xn+p ) ≤ d(xn , xn+1 ) + d(xn+1 , xn+2 ) + + d(xn+p−1 , xn+p ) ≤ {[ϕ(d(xo , x1 ))]n + [ϕ(d(xo , x1 ))]n+1 ✷✼ + + [ϕ(d(xo , x1 ))]n+p−1 }d(xo , x1 ) p n − [ϕ(d(xo , x1 ))] d(xo , x1 ) = [ϕ(d(xo , x1 ))] − ϕ(d(xo , x1 )) [ϕ(d(xo , x1 ))]n ≤ d(xo , x1 ) − ϕ(d(xo , x1 )) ✭✶✮ ❚ø ≤ ϕ(d(xo, x1)) < s✉② r❛ [ϕ(d(xo , x1 ))]n d(xo , x1 ) → − ϕ(d(xo , x1 )) ▼➦t ❦❤→❝✱ ✈➻ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ♥➯♥ tø ✭✶✮ ✈➔ ▼➺♥❤ ✤➲ ✶✳✷✳✺✳✹✮ s✉② r❛ d(xn , xn+p ) → ❦❤✐ n → ∞ ✈ỵ✐ ♠å✐ p ∈ N✳ ❉♦ ✤â {xn } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❱➻ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤➛② ✤õ tỗ t x X s xn x õ t ỵ d(x, xn) ❚ø ✤â ✈➔ ϕ(t) ∈ [0, 1) ✈ỵ✐ ♠å✐ t ∈ [0, +∞) t❛ ❝â ≤ d(f x, f xn ) ≤ ϕ(d(x, xn ))d(x, xn ) → ❉♦ ✤â✱ tø ▼➺♥❤ ✤➲ ✶✳✷✳✺✳✹✮ s✉② r❛ d(f x, f xn) → 0, tù❝ ❧➔ f xn → f x ▼➦t ❦❤→❝✱ ✈➻ f xn = xn+1 ✈ỵ✐ ♠å✐ n = 1, 2, ♥➯♥ xn → f x✳ ❱➻ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ ♥➯♥ x = f x✱ tù❝ x ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ●✐↔ sû y ❝ô♥❣ ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ❑❤✐ ✤â✱ y = f y✳ ❉♦ ✤â ♥➳✉ x = y t❤➻ d(x, y) = d(f x, f y) ≤ ϕ(d(x, y))d(x, y) < d(x, y) ❚❛ ❝â ✤✐➲✉ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â x = y✳ ❱➟② f ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✷✳✷✳✸✳ t r ỵ (t) = k ✈ỵ✐ ♠å✐ ✈➔ k ❧➔ ♠ët ❤➡♥❣ sè t❤✉ë❝ [0, 1) t❤➻ f ❧➔ →♥❤ ①↕ ❝♦✳ ◆❤÷ ỵ ởt trữớ ủ t ỵ t [0, +) ỵ sỹ tỗ t ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ ❦✐➸✉ ❑❛♥❛♥ tr♦♥❣ ❦❤æ♥❣ tr õtỡ tổổ ỵ sỷ (X, d) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ✤➛② ✤õ✱ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈➔ f ❧➔ →♥❤ ①↕ tứ X X õ tỗ t ❤➡♥❣ sè α ∈ [0, 21 ) s❛♦ ❝❤♦ d(f x, f y) ≤ α[d(x, f x) + d(y, f y)] ∀ x, y ∈ X t❤➻ f ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ●✐↔ sû α ∈ [0, 21 ) s❛♦ ❝❤♦ ❈❤ù♥❣ ♠✐♥❤✳ f ❧➔ →♥❤ ①↕ tứ X X tỗ t số d(f x, f y) ≤ α [d(f x, x) + d(f y, y)] ∀ x, y ∈ X ▲➜② xo ∈ X ✈➔ ✤➦t f (xo) = x1✱ f (x1) = x2✱✳✳✳✱f (xn) = xn+1 ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â d(xn+1 , xn ) = d(f xn , f xn−1 ) ≤ α [d(f xn , xn ) + d(f xn−1 , xn−1 )] = α [d(xn+1 , xn ) + d(xn , xn−1 )] ❙✉② r❛ d(xn+1 , xn ) ≤ α d(xn , xn−1 ) = h d(xn , xn−1 ), ∀ n ≥ 1, 1−α α tr♦♥❣ ✤â h = 1−α ❱➻ α ∈ [0, 21 ) ♥➯♥ h ∈ [0, 1) ❱ỵ✐ ♠é✐ n > m t❛ ❝â d(xn , xm ) = d(xn , xn−1 ) + + d(xm+1 , xm ) ≤ (hn−1 + hn−2 + + hm ) d(x1 , xo ) hm ≤ d(xo , x1 ) 1−h ✭✷✮ h ❚ø h ∈ [0, 1) s✉② r❛ 1−h d(xo , x1 ) → m → ∞ ❚ø ✭✷✮ ✈➔ ▼➺♥❤ ✤➲ ✶✳✷✳✺✳✹✮ t❛ ❝â d(xn, xm) → ❦❤✐ m → ∞ ✈ỵ✐ ♠å✐ n > m✳ ❉♦ ✤â {xn} m ✷✾ ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ❧➔ ✤➛② ✤õ ♥➯♥ xn → u ∈ X ✳ ❚❛ ❝â ≤ d(f u, u) ≤ d(f xn , f u) + d(f xn , u) ≤ α(d(f xn , xn ) + d(f u, u)) + d(xn+1 , u) ∀ n ❙✉② r❛ ≤ d(f u, u) ≤ (α d(xn+1 , xn ) + d(xn+1 , u)) ∀ n 1−α ❱➻ xn → u ♥➯♥ ✈➳ ♣❤↔✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❞➛♥ tỵ✐ ❦❤ỉ♥❣ ❦❤✐ n → ∞✳ ❱➻ P ❝❤✉➞♥ t➢❝ ♥➯♥ d(f u, u) = ❤❛② f u = u✳ ❱➟② u ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ◆➳✉ v ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f t❤➻ d(u, v) = d(f u, f v) ≤ α (d(f u, u) + d(f v, v)) = ❙✉② r❛ d(u, v) = 0, ❤❛② u = v✳ ❱➟② f ❝â ❞✉② t ởt t ỵ sỷ (X, d) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ ✤➛② ✤õ✱ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈➔ f ❧➔ →♥❤ tứ X X õ tỗ t↕✐ ❤➡♥❣ sè α ∈ [0, 12 ) s❛♦ ❝❤♦ d(f x, f y) ≤ α[d(f x, y) + d(x, f y)] ∀ x, y ∈ X t❤➻ f ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ●✐↔ sû α ∈ [0, 21 ) s❛♦ ❝❤♦ ❈❤ù♥❣ ♠✐♥❤✳ f ❧➔ →♥❤ tứ X X tỗ t số d(f x, f y) ≤ α [d(f x, y) + d(x, f y)] ∀ x, y ∈ X ▲➜② xo ∈ X ✈➔ ✤➦t f (xo) = x1✱ f (x1) = x2✱✳✳✳✱f (xn) = xn+1 ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â d(xn+1 , xn ) = d(f xn , f xn−1 ) ≤ α [d(f xn , xn−1 ) + d(f xn−1 , xn )] = α d(xn+1 , xn−1 ) ≤ α (d(xn+1 , xn ) + d(xn , xn−1 )) ✸✵ ❙✉② r❛ d(xn+1 , xn ) ≤ α d(xn , xn−1 ) = h d(xn , xn−1 ), ∀ n ≥ 1, 1−α α tr♦♥❣ ✤â h = 1−α ❱➻ α ∈ [0, 21 ) ♥➯♥ h ∈ [0, 1) ❱ỵ✐ ♠é✐ n > m t❛ ❝â d(xn , xm ) = d(xn , xn−1 ) + + d(xm+1 , xm ) ≤ (hn−1 + hn−2 + + hm ) d(x1 , xo ) hm ≤ d(xo , x1 ) 1−h ✭✸✮ h d(xo , x1 ) → m → ∞✳ ❉♦ ✤â✱ tø ✭✸✮ ✈➔ ❚ø h ∈ [0, 1) s✉② r❛ 1−h ▼➺♥❤ ✤➲ ✶✳✷✳✺✳✹✮ t❛ ❝â d(xn, xm) → ❦❤✐ m → ∞ ✈ỵ✐ ♠å✐ n > m✳ ❱➻ t❤➳ {xn } ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ❧➔ ✤➛② ✤õ ♥➯♥ xn → u ∈ X ✳ ❚❛ ❝â m d(f u, u) ≤ d(f xn , f u) + d(f xn , u) ≤ α(d(f xn , u) + d(f u, xn )) + d(xn+1 , u) ≤ α (d(f u, u) + d(u, xn ) + d(xn+1 , u)) + d(xn+1 , u) ❙✉② r❛ d(f u, u) ≤ (α (d(xn+1 , u) + d(xn , u)) + d(xn+1 , u) 1−α ❱➻ ✈➳ ♣❤↔✐ ❝õ❛ ❜➜t ✤➥♥❣ ♥➔② ❞➛♥ tỵ✐ ❦❤ỉ♥❣ ❦❤✐ n → ∞ ♥➯♥ t❛ ❝â d(f u, u) = ❤❛② f u = u✳ ❱➟② u ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f ✳ ◆➳✉ v ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f t❤➻ d(u, v) = d(f u, f v) ≤ α (d(f u, v) + d(f v, u)) = α d(u, v) ❚ø α ∈ [0, 21 ) s✉② r❛ d(u, v) = 0, ❤❛② u = v✳ ❱➟② f ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✸✶ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ ✤↕t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ ✈➼ ❞ư ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥â♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì tỉ♣ỉ ✈➔ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ✷✳ ❈❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ♠ët sè ❦➳t q✉↔ ♠➔ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤ù♥❣ ♠✐♥❤ ✈➢♥ t➢t ❤♦➦❝ ❜ä q✉❛ ❝❤ù♥❣ ♠✐♥❤✱ ✤â ❧➔ ❝→❝ ▼➺♥❤ ✤➲ ✶✳✷✳✹❀ ✶✳✷✳✺❀ ✶✳✷✳✻✳ ✸✳ ✣÷❛ r❛ ❱➼ ❞ư ✶✳✷✳✷ ✈➔ ❱➼ ❞ư ✶✳✸✳✷✱ ♠✐♥❤ ❤å❛ ❝❤♦ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲✈❡❝tì tỉ♣ỉ✳ ✣÷❛ r❛ ♠ët sè t➼♥❤ ❝❤➜t ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ tr õtỡ tổổ õ ỵ ỵ ỵ ữ r ự ởt số t q sỹ tỗ t t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❝♦ ✈➔ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✲ ✈❡❝tì tỉ♣ỉ✱ ✤â ỵ q ỵ ỵ ỵ ỵ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯ ✈➔ ▲➯ ỡ s ỵ tt t➼❝❤ ❤➔♠✱ t➟♣ ✶✲✷✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ö❝✳ ❬✷❪ ❆❦❜❛r ❆③❛♠✱ ■s♠❛t ❇❡❣ ❛♥❞ ▼✉❤❛♠♠❛❞ ❆rs❤❛❞ ✭✷✵✶✵✮✱ ❋✐①❡❞ P♦✐♥t ✐♥ t♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡s✲✈❛❧✉❡❞ ❝♦♥❡ ♠❡tr✐❝ s♣❛❝❡s✱ ❍✐♥❞❛✇✐ P✉❜✲ ❧✐s❤✐♥❣ ❝♦r♣♦r❛t✐♦♥ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ❆rt✐❝❧❡ ■❉ ✻✵✹✵✽✹✱ ✾ ♣❛❣❡s✱ ❞♦✐✿ ✶✵✳✶✶✺✺✴✻✵✹✵✽✹✳ ❬✸❪ ▼✉❤❛♠♠❛❞ ❆rs❤❛❞ ❛♥❞ ❆❦❜❛r ❆③❛♠✭✷✵✶✵✮✱ ❋✐①❡❞ ♣♦✐♥t r❡s✉❧ts ✈✐❛ ✐t❡r❛t❡s ♦❢ ❢♦✉r ♠❛♣s ✐♥ ❚❱❙✲✈❛❧✉❡❞ ❝♦♥❡ ♠❡tr✐❝ s♣❛❝❡✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✇♦r❧❞ ❝♦♥❣r❡ss ♦♥ ❡♥❣✐♥❡❡r✐♥❣ ❱♦❧ ■■■✱ ■❙❇◆✿✾✼✽✲✾✽✽✲✶✽✷✶✵✲✽✲✾✱ ■❙❙◆✱ ■❙❙◆✿✷✵✼✽✲✵✾✺✽✭♣r✐♥t✮❀ ■❙❙◆✿✷✵✼✽✲✵✾✻✻✭♦❧✐♥❡✮✳ ❬✹❪ ▼✉❥❛❤✐❞ ❆❜❜❛s✱ ❨❡♦❧ ❏❡ ❈❤♦ ❛♥❤ ❚❛❧❛t ◆❛③✐r✭✷✵✶✶✮✱ ❈♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❢♦✉r ♠❛♣♣✐♥❣s ✐♥ ❚❱❙✲✈❛❧✉❡❞ ❝♦♥❡ ♠❡tr✐❝ s♣❛❝❡s✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ■♥❡q✉❛❧✐t✐❡s✱ ❱♦❧✉♠❡ ✺ ◆✉♠❜❡r ✷✱ ✷✽✼✲✷✾✾✳ ❬✺❪ ❩♦r❛♥ ❑❛❞❡❧❜✉r❣✱ ❙t♦❥❛♥ ❘❛❞❡♥♦✈✐❝ ❛♥❞ ❱❧❛❞✐♠✐r ❘❛❝♦❝❡✈✐✭✷✵✶✵✮✱ ❚♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡✲✈❛❧✉❡❞ ❝♦♥❡ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✱ ❍✐♥❞❛✇✐ P✉❜❧✐s❤✐♥❣ ❈♦r♣♦r❛t✐♦♥✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❆rt✐❝❧❡ ■❉ ✶✼✵✷✺✸✱ ✶✼ ♣❛❣❡s✱ ❞♦✐✿ ✶✵✳✶✶✺✺✴✶✼✵✷✺✸✳ ✸✸