Về giới hạn của dãy điểm bất động của dãy các ánh xạ trong không gian mêtric riêng

❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❇Ò■ ❍❯❨ ✣Ù❈ ❱➋ ●■❰■ ❍❸◆ ❈Õ❆ ❉❶❨ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❉❶❨ ❈⑩❈ ⑩◆❍ ❳❸ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ❘■➊◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆❣❤➺ ❆♥ ✲ ✷✵✶✻ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ✲ ❇Ò■ ❍❯❨ ✣Ù❈ ❱➋ ●■❰■ ❍❸◆ ❈Õ❆ ❉❶❨ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❉❶❨ ❈⑩❈ ⑩◆❍ ❳❸ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ❘■➊◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳ ✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ P P❍×❒◆● ❈❍■ ◆❣❤➺ ❆♥ ✲ ✷✵✶✻ ▼Ư❈ ▲Ư❈ ▼ư❝ ❧ư❝ ✶ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶ ❙ü tỗ t t →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✺ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t t ♠ët sè ❧ỵ♣ →♥❤ ①↕ ❝♦ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷ ●✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✶✼ ✷✳✶✳ ❙ü ❤ë✐ tö ❝õ❛ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷✳ ❙ü ❤ë✐ tö ❝õ❛ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕ ❝♦ ✈➔ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ▼Ð ✣❺❯ ỵ tt t ỹ ự q✉❛♥ trå♥❣ ❝õ❛ t♦→♥ ❣✐↔✐ t➼❝❤✳ ❙ü ♣❤→t tr✐➸♥ ♠↕♥❤ ỵ tt t t ỗ tứ ♥❤ú♥❣ ù♥❣ ❞ư♥❣ rë♥❣ ❧ỵ♥ ❝õ❛ ♥â tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❝õ❛ t♦→♥ ❤å❝ ✈➔ tr♦♥❣ ❝↔ ❦❤♦❛ ❤å❝ tü tt t ỵ t ✤ë♥❣ ❝õ❛ ❇❛♥❛❝❤ ✭✶✾✷✷✮ ✤è✐ ✈ỵ✐ ❝→❝ →♥❤ ①↕ ❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ❧➔ ♠ët ❦➳t q✉↔ ❦✐♥❤ ✤✐➸♥ ❝õ❛ t♦→♥ ❤å❝✳ ❙❛✉ ❦❤✐ ✤÷đ❝ ❇❛♥❛❝❤ ❝❤ù♥❣ ỵ t ố ợ ①↕ ❝♦ trð t❤➔♥❤ ♠ët tr♦♥❣ ♥❤ú♥❣ ✈➜♥ ✤➲ t❤✉ ❤ót ✤÷đ❝ r➜t ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ự ỵ t ữủ rở ♥❣❤✐➯♥ ❝ù✉ ♣❤♦♥❣ ♣❤ó ❝❤♦ ♥❤✐➲✉ ❦✐➸✉ →♥❤ ①↕ s✉② rë♥❣✱ tr➯♥ ♥❤✐➲✉ ❧♦↕✐ ❦❤æ♥❣ ❣✐❛♥ ❦❤→❝ ♥❤❛✉✳ ▼ët tr♦♥❣ ỳ ữợ ự ỵ tt t tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❧➔ ✈➜♥ ✤➲ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ❝➜✉ tró❝ ❝õ❛ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤å ❝→❝ →♥❤ ①↕✳ ◆❣❤✐➯♥ ❝ù✉ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët ❞➣② ❝→❝ →♥❤ ①↕ ✤÷đ❝ ◆❛❞❧❡r✱ ❋r❛s❡r✳✳✳ t❤ü❝ ❤✐➺♥ ✈➔♦ ❝✉è✐ ♥❤ú♥❣ ♥➠♠ t trữợ ụ t ữủ ♠ët sè ù♥❣ ❞ö♥❣ tr♦♥❣ ✈✐➺❝ ①➜♣ ①➾ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✭①❡♠ ❬✺❪✱ ❬✶✶❪✳✳✳✮✳ ◆➠♠ ✶✾✾✷✱ tr♦♥❣ ❞ü →♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ sü ❤✐➸♥ t❤à ♥❣æ♥ ♥❣ú ✈➔ ❧÷✉ t❤ỉ♥❣ ♠↕♥❣ ♠→② t➼♥❤✱ ❙✳ ●✳ ▼❛tt❤❡✇ ✭❬✾❪✮ ✤➣ ✤➲ ①✉➜t ✈➔ ①➙② ❞ü♥❣ ❦❤→✐ ♥✐➺♠ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ❈→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t tæ♣æ✱ sü ❤ë✐ tư ❝õ❛ ❞➣②✱ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r ổ tr ỵ ❇❛♥❛❝❤ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✤➣ ✤÷đ❝ ❙✳ ●✳ ▼❛tt❤❡✇ ♥❣❤✐➯♥ ❝ù✉ ❦❤→ t❤➜✉ ✤→♦ tr♦♥❣ ✭❬✶✵❪✮✳ ❑❤→✐ ♥✐➺♠ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ♥❤➟♥ ✤÷đ❝ ❜➡♥❣ ❝→❝❤ t❤❛② t❤➳ ✤➥♥❣ t❤ù❝ d(x, x) = tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ♠➯tr✐❝ ❜ð✐ ❜➜t ✤➥♥❣ t❤ù❝ d(x, x) d(x, y) ✈ỵ✐ ♠å✐ x, y ✳ ❱➻ ✈➟②✱ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♥➔② ❧➔ sü ♠ð rë♥❣ t❤ü❝ sü ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ tt ỵ t ố ✈ỵ✐ →♥❤ ①↕ ❝♦ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ❝ơ♥❣ ✤÷đ❝ q✉❛♥ t➙♠ ✤➳♥ ♥❣➔② ♥❛②✳ ❚r♦♥❣ ❦❤✉ỉ♥ ❦❤ê ♠ët ❧✉➟♥ ✈➠♥ t❤↕❝ s➽✱ ❝❤ó♥❣ tỉ✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ sỹ tỗ t t →♥❤ ①↕ ❝♦ s✉② rë♥❣ ✈➔ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ✤â✱ ❝❤ó♥❣ tỉ✐ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ❝❤♦ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤ ❧➔✿ ❱➲ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ự ởt số ỵ sỹ tỗ t t ❝♦✱ →♥❤ ①↕ s✉② rë♥❣ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ❈→❝ ♥ë✐ ❞✉♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ✷ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð❀ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ổ tr r ởt số ỵ sỹ tỗ t t ❝♦✱ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ❝õ❛ ú tổ sỹ tỗ t ợ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷đ❝ ✤➲ ①✉➜t ỹ tr ỵ tữ õ ố ợ ổ ❣✐❛♥ ♠➯tr✐❝ ✈ỵ✐ ♠ët ✈➔✐ ❝↔✐ t✐➳♥ ✈➲ ❦ÿ t❤✉➟t tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❚r÷í♥❣ ữợ sỹ ữợ t P●❙✳ ❚❙✳ ❑✐➲✉ P❤÷ì♥❣ ❈❤✐✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ t❤➛②✳ ◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❙❛✉ ✤↕✐ ❤å❝✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❑❤♦❛ ❚♦→♥ ❤å❝ ✈➔ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❜ë ♠æ♥ ●✐↔✐ t➼❝❤✱ ❑❤♦❛ ❚♦→♥ ❤å❝ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ỗ tr rữớ ❚❍P❚ ❚r➛♥ P❤ó✱ ❍➔ ❚➽♥❤ ✤➣ ❣✐ó♣ ✤ï✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ố ũ ỡ ỗ ❜↕♥ ❜➧✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ ❜↕♥ tr♦♥❣ ❧ỵ♣ ❈❛♦ ❤å❝ ✷✷ ●✐↔✐ t➼❝❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✤➣ ❝ë♥❣ t→❝✱ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ t❤✐➳✉ sât✳ ❈❤ó♥❣ tỉ✐ r➜t ♠♦♥❣ ữủ ỳ ỵ õ õ t ❝ỉ ❣✐→♦ ✈➔ ❜↕♥ ❜➧ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ◆❣❤➺ ❆♥✱ t❤→♥❣ ✼ ♥➠♠ ✷✵✶✻ ❚→❝ ❣✐↔ ❇ị✐ ❍✉② ✣ù❝ ❈❍×❒◆● ✶ ❙Ü ❚➬◆ ❚❸■ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ⑩◆❍ ❳❸ ❈❖ ❱⑨ ⑩◆❍ ❳❸ ❈❖ ❙❯❨ ❘❐◆● ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ❘■➊◆● ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð❀ ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣❀ ♠ët sè ỵ sỹ tỗ t t →♥❤ ①↕ ❝♦✱ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ✶✳✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦➳t q✉↔ ❝ì sð ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷đ❝ tr➼❝❤ r❛ ❝❤õ ②➳✉ tø ❬✶❪✳ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ♠ët t➟♣ ❦❤→❝ ré♥❣✳ ❍➔♠ d : X ì X R ữủ ♠ët ♠➯tr✐❝ tr➯♥ X ♥➳✉ t❤♦↔ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✮ d(x, y) 0✱ ✈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) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ✶✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❉➣② {xn} ⊂ X ✤÷đ❝ ❣å✐ tử tợ x X ỵ xn xx ữủ ợ ❞➣② {xn }✮✱ ♥➳✉ lim d(xn , x) = n→∞ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ♠ët ❞➣② ♥➳✉ ❝â ❧➔ ❞✉② ♥❤➜t✳ ✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✶✮ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ✤÷đ❝ ❣å✐ ❧➔ ❝♦♠♣❛❝t ♥➳✉ ♠å✐ ❞➣② t❤✉ë❝ X ✤➲✉ ❝â ❞➣② ❝♦♥ ❤ë✐ tö tr♦♥❣ X ✳ ✷✮ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ X ✤÷đ❝ ❣å✐ ❧➔ ❝♦♠♣❛❝t ữỡ ợ a X tỗ t r > s❛♦ ❝❤♦ ❜❛♦ ✤â♥❣ ❝õ❛ B(a, r) = {x ∈ X : d(x, a) < r} ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❉➣② {xn } ⊂ X ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ❈❛✉❝❤② ♥➳✉ lim d(xm , xn ) = 0✳ m,n→∞ ❑❤ỉ♥❣ ❣✐❛♥ (X, d) ✤÷đ❝ ❣å✐ ❧➔ ✤➛② ✤õ ♥➳✉ ♠å✐ ❞➣② ❈❛✉❝❤② ❝õ❛ X ✤➲✉ ❤ë✐ tö tr♦♥❣ X ✳ ❈❤♦ B ⊂ X ✳ ❑❤✐ ✤â δ[B] = sup{d(x, y) : x, y ∈ B} ✤÷đ❝ ❣å✐ ❧➔ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ B ✳ ❚➟♣ ❝♦♥ B ✤÷đ❝ ❣å✐ ❧➔ ❜à ❝❤➦♥ ♥➳✉ ♥â ❝â ✤÷í♥❣ ❦➼♥❤ ❤ú✉ ❤↕♥✳ ✶✳✶✳✺ ✣à♥❤ ♥❣❤➽❛✳ ❉➣② ❝→❝ t➟♣ {Bn} ⊂ X ✤÷đ❝ ❣å✐ ❧➔ t❤➢t ❞➛♥ ✤➲✉ ♥➳✉ Bn+1 ⊂ Bn ✈➔ lim δ[Bn ] = 0✳ n→∞ ❑➳t q✉↔ t s ỏ ỵ tr ỵ r ổ tr ❝→❝ t➟♣ ✤â♥❣ t❤➢t ❞➛♥ ✤➲✉ ❝â ✤✐➸♠ ❝❤✉♥❣ ❞✉② ♥❤➜t✳ ✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ (X, d)✱ (Y, ρ) ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ →♥❤ ①↕ f : X → X✳ ✶✮ →♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ ♥➳✉ ✈ỵ✐ ♠å✐ ❞➣② {x } ⊂ X ✈➔ x n n → x t❤➻ f (xn ) → f (x)✳ ✷✮ ❝❤♦✿ →♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ tử ợ > tỗ t↕✐ δ = δ(ε) s❛♦ ρ(f x, f y) < ε, ∀x, y ∈ X, d(x, y) < δ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ♠å✐ →♥❤ ①↕ ❧✐➯♥ tư❝ ✤➲✉ ❧➔ ❧✐➯♥ tư❝✳ ▼➺♥❤ ✤➲ ♥❣÷đ❝ ❧↕✐ ❧➔ ❦❤ỉ♥❣ ✤ó♥❣✳ ❈❤♦ (X, d) ✈➔ (Y, ρ) ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ổ tr t X ì Y ợ tr ✤à♥❤ ❜ð✐ D((x1 , y1 ), (x2 , y2 )) = d(x1 , x2 ) + ρ(y1 , y2 ) ✈ỵ✐ (x1 , y1 ), (x2 , y2 ) ∈ X ì Y ự ữủ ①↕ ❦❤♦↔♥❣ ❝→❝❤ d : X × X → R ❧➔ ❧✐➯♥ tö❝✱ tù❝ ❧➔ ♥➳✉ ❞➣② (xn , yn ) tử tợ tr (x, y) X ì X t❤➻ d(xn , yn ) ❤ë✐ tư tỵ✐ d(x, y) tr♦♥❣ R✳ ✶✳✶✳✽ ✣à♥❤ ♥❣❤➽❛✳ ✷✮ ❍å →♥❤ ①↕ Tα : (X, d) → (Y, ρ)✱ (α ∈ I) ữủ ỗ tử t a X ợ > tỗ t > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ α ∈ I ♠➔ d(x, a) < δ t❤➻ ρ(Tα x, Tα a) < ε ợ x (T ) ữủ ỗ tử tr X ợ > tỗ t > s ợ α ∈ I ✿ ρ(Tα x, Tα y) < ε ✈ỵ✐ ♠å✐ x, y ∈ X ♠➔ d(x, y) < δ ✳ ✶✳✶✳✾ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ d, ρ ❧➔ ❝→❝ ♠➯tr✐❝ tr➯♥ X ✳ ✶✮ d ✈➔ ρ ✤÷đ❝ ❣å✐ tữỡ ữỡ ỗ t id : (X, d) → (X, ρ) ✈➔ →♥❤ ①↕ ♥❣÷đ❝ ❝õ❛ ♥â ❧✐➯♥ tư❝✳ ✷✮ d ✈➔ ρ ✤÷đ❝ ❣å✐ ❧➔ tữỡ ữỡ ỗ t id : (X, d) → (X, ρ) ✈➔ →♥❤ ①↕ ♥❣÷đ❝ ❝õ❛ ♥â ❧✐➯♥ tư❝ ✤➲✉✳ ◆❣÷í✐ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ d ✈➔ ρ ✤÷đ❝ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✤➲✉ ♥➳✉ ✈➔ ❝❤➾ tỗ t a, b > s ad(x, y) ρ(x, y) bd(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X ✳ ✶✳✶✳✶✵ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ (Tn) ❧➔ ❞➣② ❝→❝ →♥❤ ①↕ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ tø ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ (X, d) ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ (Y, ρ)✳ ✶✮ (Tn ) ữủ tử tr X tợ T : X → Y ♥➳✉ ✈ỵ✐ ♠å✐ x ∈ X t❤➻ Tn x ❤ë✐ tư tỵ✐ T x✳ ✷✮ (Tn ) ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư ✤➲✉ tr➯♥ X tỵ✐ T : X → Y ♥➳✉ lim sup ρ(T xn , T x) = n→∞ x∈X ✸✮ (Tn ) ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư ✤➲✉ tr➯♥ ❝→❝ t➟♣ ❝♦♠♣❛❝t ❝õ❛ X tỵ✐ T : X → Y ♥➳✉ ✈ỵ✐ ♠å✐ t➟♣ ❝♦♠♣❛❝t K ❝õ❛ X t❛ ❝â lim sup ρ(T xn , T x) = n→∞ x∈K ❘ã r➔♥❣ sü ❤ë✐ tö ✤➲✉ ❝õ❛ (Tn ) ❦➨♦ t sỹ tử ỵ s ởt ✤✐➲✉ ❦✐➺♥ ✤➸ ❞➣② ❤ë✐ tö ✤✐➸♠ ❧➔ ❤ë✐ tö ✤➲✉✱ ♥â ✤÷đ❝ sû ❞ư♥❣ ♥❤✐➲✉ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t q ữỡ s ỵ (Tn) ỗ tử tr t t K Tn ❤ë✐ tư ✤✐➸♠ tỵ✐ T ❧✐➯♥ tư❝ tr➯♥ K t❤➻ Tn ❤ë✐ tư ✤➲✉ tỵ✐ T tr➯♥ K ✳ ự ợ ộ > trữợ T ❧✐➯♥ tö❝ tr➯♥ K ❝♦♠♣❛❝t ♥➯♥ T ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ K ✳ ❱➻ ✈➟②✱ t❛ t➻♠ ✤÷đ❝ δ1 = δ1 (ε) s❛♦ ❝❤♦ d(T x, T y) < ε ✈ỵ✐ ♠å✐ x, y ∈ K ✈➔ d(x, y) < (Tn ) ỗ tử ợ > tỗ t > s❛♦ ❝❤♦ d(Tn x, Tn y) < ε ✈ỵ✐ ♠å✐ x, y ∈ K ♠➔ d(x, y) < δ2 ✳ ▼➦t ❦❤→❝✱ ✈➻ (Tn ) ❤ë✐ tö ✤✐➸♠ tỵ✐ T tr➯♥ K ♥➯♥ ✈ỵ✐ ♠é✐ a ∈ K tỗ t n0 (a) = n0 (a, ) s ε n0 (a)✳ ✣➦t δ = min{δ1 , δ2 }✳ ❑❤✐ ✤â✱ ❤å ❤➻♥❤ ❝➛✉ {B(a, δ) : a ∈ K} d(Tn , T a) ✈ỵ✐ ♠å✐ n ❧➔ ♣❤õ ♠ð ❝õ❛ K ✳ ❱➻ K ❝♦♠♣❛❝t ♥➯♥ t❛ t➻♠ ✤÷đ❝ a1 , , ak s❛♦ ❝❤♦ k K⊂ B(ai , δ) i=1 ✣➦t N = max{n0 (ai ) : i = 1, k} ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x K tỗ t s x ∈ B(ai , ε)✳ ❱ỵ✐ n d(Tn x, T x) N t❛ ❝â d(Tn x, Tn ) + d(Tn , T ) + d(T , T x) ε ε ε + + = ε 3 ❱➻ ✈➟②✱ t❛ ♥❤➟♥ ✤÷đ❝ sup d(Tn x, T x) ε x∈K ✈ỵ✐ ♠å✐ n T✳ N ✱ tù❝ ❧➔ lim supx∈K d(Tn x, T x) = 0✱ ❤❛② Tn ❤ë✐ tư ✤➲✉ tr➯♥ K tỵ✐ n→∞ ✶✳✷✳ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ◆➠♠ ✶✾✾✷✱ ❙✳ ●✳ ▼❛tt❤❡✇ ✭❬✾❪✮ ✤➣ ✤➲ ①✉➜t ✈➔ ①➙② ❞ü♥❣ ❦❤→✐ ♥✐➺♠ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ❈→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t tỉ♣ỉ✱ sü ❤ë✐ tư ❝õ❛ ❞➣②✱ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❉♦ ✤â d p(xn , xn+1 ) 1−d ❚ø t✐➯♥ ✤➲ (P 2) tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶ t❛ ❝â p(xn+1 , xn+2 ) max{ap(xn , xn+1 ), bp(xn , T xn ), ep(xn , xn ), f p(xn+1 , xn+1 )} rp(xn , xn+1 ), tr♦♥❣ ✤â r = max{a, b, e, f }✳ ❉♦ ✤â✱ tø ♠å✐ tr÷í♥❣ ❤đ♣ ✤➣ ①➨t ð tr➯♥✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✺✮ trð t❤➔♥❤ p(xn+1 , xn+2 ) ✭✶✳✻✮ kp(xn , xn+1 ), d tr♦♥❣ ✤â k ∈ {r, 1−d }✳ ❘ã r➔♥❣ < k < 1✳ ❉♦ ✤â p(xn+1 , xn+2 ) kp(xn , xn+1 ) k p(xn−1 , xn ) ··· k n+1 p(x0 , x1 ) ✭✶✳✼✮ ❚❛ ❝❤ù♥❣ ♠✐♥❤ {xn } ❧➔ ❞➣② ❈❛✉❝❤②✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❝â t❤➸ ❣✐↔ sû n > m✳ ❑❤✐ ✤â✱ tø ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✼✮ ✈➔ (P 2) tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶ t❛ ❝â p(xn , xm ) p(xn , xn−1 ) + p(xn−1 , xn−2 ) + · · · + p(xm+1 , xm ) − [p(xn−1 , xn−1 ) + p(xn−2 , xn−2 ) + · · · p(xm+1 , xm+1 )] p(xn , xn−1 ) + p(xn−1 , xn−2 ) + · · · + p(xm+1 , xm ) [k n−1 + k n−2 + · · · k m ]p(x0 , x1 ) n−m m1 − k =k p(x0 , x1 ) 1−k ❉♦ ✤â✱ lim p(xn , xm ) = 0✳ ✣✐➲✉ ✤â ❝❤ù♥❣ tä {xn } ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ (X, p)✳ n,m→∞ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✾ t❤➻ {xn } ❝ô♥❣ ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ (X, ps )✳ ❍ì♥ ♥ú❛✱ ✈➻ (X, p) ✤➛② ✤õ ♥➯♥ (X, ps ) ❝ô♥❣ ✤➛② ✤õ✳ tỗ t z X s xn z tr♦♥❣ (X, ps )✱ tù❝ ❧➔ ✭✶✳✽✮ lim ps (z, xn ) = n→∞ ❈ô♥❣ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷✳✾ t❛ ❝â p(z, z) = lim p(z, xn ) = n→∞ lim p(xn , xm ) = 0, n,m→∞ ✭✶✳✾✮ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ z ❧➔ ✤✐➸♠ ❜➜t T ú ỵ r tứ tự ✭✶✳✾✮✱ t❛ ❝â p(z, z) = 0✳ ❇➡♥❣ ❝→❝❤ t❤❛② x = xn ✈➔ y = z ✈➔♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ t❛ ♥❤➟♥ ✤÷đ❝ p(xn+1 , T z) = p(T xn , T z) M (xn , z), ✭✶✳✶✵✮ tr♦♥❣ ✤â M (xn , z) = max{ap(xn , z), bp(xn , T xn ), cp(z, T z), d[p(xn , T z) + p(z, T xn )], ep(xn , xn ), f p(z, z)} ❙û ❞ö♥❣ ✤➥♥❣ t❤ù❝ ✭✶✳✾✮ ✈➔ ❇ê ✤➲ ✶✳✷✳✶✵✱ ❝❤♦ n → ∞ ð ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ p(z, T z) M (z, z), ✭✶✳✶✶✮ tr♦♥❣ ✤â M (z, z) = max{ap(z, z), bp(z, z), cp(z, T z), d[p(z, T z) + p(z, z)], ep(z, z), f p(z, z)} ❚❤❡♦ ✤➥♥❣ t❤ù❝ ✭✶✳✾✮✱ t❛ ❝â M (z, z) = cp(z, T z) ❤♦➦❝ M (z, z) = dp(z, T z)✳ ❚r♦♥❣ ❝↔ ❤❛✐ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❧✉ỉ♥ ❝â p(T z, z) = 0✳ ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✶✱ t❛ s✉② r❛ T z = z ✳ ❈✉è✐ ❝ò♥❣✱ t❛ ❝❤ù♥❣ ♠✐♥❤ z ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ T ✳ ●✐↔ sû ữủ tỗ t w X s T w = w✳ ❑❤✐ ✤â t❛ ❝â p(z, w) = p(T z, T w) M (z, w), ✭✶✳✶✷✮ tr♦♥❣ ✤â M (z, w) = max ap(z, w), bp(z, T z), cp(w, T w), d[p(z, T w) + p(w, T z)], ep(z, z), f p(w, w) = max ap(z, w), bp(z, z), cp(w, w), d[p(z, w) + p(w, z)], ep(z, z), f p(w, w) ❱➻ a, b, c, 2d, e, f ∈ (0, 1) ✈➔ t✐➯♥ ✤➲ ✭P✷✮ tr♦♥❣ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶ ♥➯♥ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✷✮ ❦➨♦ t❤❡♦ p(z, w) = 0✳ ❚❤❡♦ ❇ê ✤➲ ✶✳✷✳✶✶ t❛ ❝â z = w ỵ ữủ ự ứ ỵ tr ú t t ữủ q s ✶✳✸✳✺ ❍➺ q✉↔✳ ✭❬✸❪✮ ❈❤♦ (X, p) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✤➛② ✤õ ✈➔ T : X → X ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ p(T x, T y) max ap(x, y), bp(x, T x), cp(y, T y), d[p(x, T y) + p(y, T x)] , ✭✶✳✶✸✮ ✈ỵ✐ ♠å✐ x, y ∈ X ✱ tr♦♥❣ ✤â a, b, c ∈ [0, 1) ✈➔ d ∈ [0, )✳ ❑❤✐ ✤â✱ Xp = x ∈ X : p(x, x) = inf {p(y, y) : y ∈ X} ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✶✳✸✳✻ ❍➺ q✉↔✳ ✭❬✸❪✮ ❈❤♦ (X, p) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✤➛② ✤õ ✈➔ T : X → X ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ p(T x, T y) ap(x, y) + bp(T x, x) + cp(y, T y) + d[p(T x, y) + p(x, T y)] + ep(x, x) + f p(y, y) ✈ỵ✐ ♠å✐ x, y ∈ X ✱ tr♦♥❣ ✤â a + b + c + 2d + e + f < 1✱ ✈➔ a, b, c, d, e, f ✤â✱ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✭✶✳✶✹✮ 0✳ ❑❤✐ ❈❍×❒◆● ✷ ●■❰■ ❍❸◆ ❈Õ❆ ❉❶❨ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ❉❶❨ ❈⑩❈ ⑩◆❍ ❳❸ ❚❘❖◆● ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ❘■➊◆● ❈❤÷ì♥❣ ♥➔② ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ♠➯tr✐❝ r✐➯♥❣ t❤æ♥❣ q✉❛ ♠ët sè ❞↕♥❣ ❤ë✐ tö ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕✳ ❇➔✐ t♦→♥ ♥➔② ✤÷đ❝ ◆❛❞❧❡r✱ ❋r❛s❡r✱✳✳✳ ♥❣❤✐➯♥ ❝ù✉ ❝✉è✐ ♥❤ú♥❣ ♥➠♠ ✻✵ t trữợ ố ợ ổ tr ❦➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❞♦ ❝❤ó♥❣ tỉ✐ ✤➲ ①✉➜t ❞ü❛ tr➯♥ ♠ët sè ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❚✉② ♥❤✐➯♥✱ ❤➛✉ ❤➳t ❝→❝ ❦➳t q✉↔ ♥➔② ✤➲✉ ❝â sü ❦❤â ❦❤➠♥ ❤ì♥ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✶✳ ❙ü ❤ë✐ tö ❝õ❛ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ▼ư❝ ♥➔② ♥❣❤✐➯♥ ❝ù✉ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❧✐➯♥ tư❝ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ (X, p) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✈➔ (Tn) ❧➔ ❞➣② ❝→❝ →♥❤ ①↕ tø X ✈➔♦ X ✳ ❉➣② (an ) ❝→❝ ♣❤➛♥ tû ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ (Tn ) ♥➳✉ Tn (an ) = an ✈ỵ✐ ♠é✐ n = 1, 2, ✷✳✶✳✷ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ (X, p) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✈➔ T ❧➔ →♥❤ ①↕ tø X ✈➔♦ X ✳ T ✤÷đ❝ ❣å✐ ❧➔ ❧✐➯♥ tư❝ ♥➳✉ ✈ỵ✐ ♠å✐ xn → x t❤➻ limn→∞ p(T xn , T xn ) = p(T x, T x) ❚❛ ❝➛♥ ♠ët sè ❦❤→✐ ♥✐➺♠ s❛✉✿ ✷✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ (X, p) ❧➔ ❦❤æ♥❣ ♠➯tr✐❝ r✐➯♥❣ ✈➔ (Tn) ❧➔ ❞➣② ❝→❝ →♥❤ ①↕ tø X ✈➔♦ X ✳ ✶✮ (Tn ) ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư ✤✐➸♠ tỵ✐ →♥❤ ①↕ T : X → X ♥➳✉ Tn x → T x ✈ỵ✐ ♠å✐ x ∈ X ✱ tù❝ ❧➔ p(Tn x, T x) → p(T x, T x), ✈ỵ✐ ♠å✐ a ∈ X ✳ ✷✮ (Tn ) ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư ✤✐➸♠ tr➯♥ X t❤❡♦ ♠➯tr✐❝ ps tỵ✐ →♥❤ ①↕ T : X → X ♥➳✉ ps (Tn x, T x) → ✈ỵ✐ ♠å✐ x ∈ X ✈➔ ✈ỵ✐ n → ∞✳ ✸✮ (Tn ) ữủ tử tợ T : X → X ♥➳✉ sup |p(Tn x, T x) − p(T x, T x)| → x∈X ❦❤✐ n → ∞✳ ✷✳✶✳✹ ❇ê ✤➲✳ ◆➳✉ xn → x ✈➔ yn → y t❤❡♦ ♠➯tr✐❝ ps t❤➻ p(xn, yn) → p(x, y) ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❣✐↔ t❤✐➳t xn → x ✈➔ yn → y t❤❡♦ ♠➯tr✐❝ ps s✉② r❛ ps (xn , x) = max{p(xn , x) − p(x, x), p(xn , x) − p(xn , xn )} → ✈➔ ps (yn , y) = max{p(yn , y) − p(y, y), p(yn , y) − p(yn , yn )} → ❦❤✐ n → ∞✳ ❚ø p(xn , yn ) p(xn , x) + p(x, yn ) − p(x, x) p(xn , x) + p(x, y) + p(y, yn ) − p(y, y) − p(x, x) ❚ø ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â p(xn , yn ) − p(x, y) [p(xn , x) − p(x, x)] + [p(y, yn ) − p(y, y)] → ❦❤✐ n → ∞✳ ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝â p(x, y) p(xn , x) + p(xn , y) − p(xn , xn ) p(xn , x) + p(xn , yn ) + p(y, yn ) − p(yn , yn ) − p(xn , xn ) ❉♦ ✤â✱ t❛ s✉② r❛ p(x, y) − p(xn , yn ) [p(xn , x) − p(xn , xn )] + [p(y, yn ) − p(yn , yn )] → ❦❤✐ n → ∞✳ ❑➳t ❤ñ♣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ |p(xn , yn )−p(x, y)| → ❦❤✐ n → ∞✳ ❱➟②✱ ❜ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✶✳✺ ❇ê ✤➲✳ ❈❤♦ (X, p) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✈➔ T ❧➔ →♥❤ ①↕ tø X ✈➔♦ X ✳ ◆➳✉ (Tn ) ❧➔ ❞➣② →♥❤ ①↕ ❧✐➯♥ tư❝ ✈➔ ❤ë✐ tư ✤➲✉ tỵ✐ →♥❤ ①↕ T t❤➻ T ❧➔ ❧✐➯♥ tö❝✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû (xk ) tử tợ a trữợ > tũ ỵ õ ợ ộ n = 1, 2, tỗ t k0 = k0 (n) s |p(Tn xk , Tn xk ) − p(Tn a, Tn a)| < ε ✈ỵ✐ ♠å✐ k k0 ✳ ❇➙② ❣✐í✱ tø (Tn ) ❤ë✐ tư ✤➲✉ tỵ✐ →♥❤ ①↕ T : X X s r tỗ t n0 = n0 (ε) s❛♦ ❝❤♦ |p(Tn x, T x) − p(T x, T x)| < ε ✈ỵ✐ ♠å✐ x ∈ X ✈➔ ❦❤✐ n n0 ✳ ❱➻ ✈➟②✱ ✈ỵ✐ ♠å✐ n n0 t❛ ❝â |p(Tn xk , T xk ) − p(T xk , T xk )| < ε ✈➔ |p(Tn a, T a) − p(T a, T a)| < ε ❇➙② ❣✐í✱ ✈ỵ✐ k k0 (n0 ) t❛ ❝â |p(T xk , T xk ) − p(T a, T a)| |p(Tn xk , T xk ) − p(T xk , Tk )| + |p(Tn xk , Tn xk ) − p(Tn a, Tn a)| + |p(Tn a, T a) − p(T a, T a)| < 3ε ✈ỵ✐ ♠å✐ n n0 ✳ ❱➻ ✈➟② lim p(T xk , T xk ) = p(T a, T a) ❉♦ ✤â T ❧✐➯♥ tö❝✳ k→∞ ❈❤ó♥❣ tỉ✐ ♥❤➟♥ ✤÷đ❝ ❦➳t q✉↔ s❛✉✳ ✷✳✶✳✻ ✣à♥❤ ỵ (X, d) ổ tr r ✤õ✳ ●✐↔ sû (Tn) ❧➔ ❞➣② ❝→❝ →♥❤ ①↕ ❧✐➯♥ tö❝ tø X ✈➔♦ X ✈➔ (an ) ❧➔ ❞➣② ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ (Tn )✳ ❑❤✐ ✤â✱ ♥➳✉ Tn ❤ë✐ tư ✤➲✉ tr➯♥ X tỵ✐ →♥❤ ①↕ T tỗ t a X s p(a, a) = lim p(an , a) = lim p(an , an ) n→∞ n→∞ t❤➻ a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❣✐↔ t❤✐➳t an ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ Tn ✈➔ lim an = a s✉② r❛ n→∞ lim p(Tn an , Tn an ) = lim p(an , an ) = p(a, a)✳ ❉♦ (Tn ) ❧✐➯♥ tư❝✱ Tn ❤ë✐ tư ✤➲✉ tỵ✐ n→∞ n→∞ T tr➯♥ X t❛ ❝â T ❧✐➯♥ tư❝✳ ◆❤í ❇ê t õ ợ > tỗ t↕✐ n1 s❛♦ ❝❤♦ ε |p(T an , T an ) − p(T a, T a)| < ✭✷✳✶✮ ✈ỵ✐ ♠å✐ n ≥ n1 ✳ ▼➦t ❦❤→❝✱ tø Tn ❤ë✐ tư ✤➲✉ tỵ✐ T ✱ tù❝ ❧➔ lim sup |p(Tn x, T x) − p(T x, T x)| = n xX ợ x X s r tỗ t↕✐ n2 s❛♦ ❝❤♦ |p(Tn x, T x) − p(T x, T x)| < ✈ỵ✐ ♠å✐ n n2 ✈➔ ✈ỵ✐ ♠å✐ x ∈ X ✳ ❱➻ ✈➟②✱ ✈ỵ✐ n ε ✭✷✳✷✮ n0 = max{n1 , n2 } tø ✭✷✳✶✮ ✈➔ ✭✷✳✷✮ t❛ ❝â |p(T a, an ) − p(T a, T a)| = |p(T a, Tn an ) − p(T a, T a)| |p(Tn an , T an ) − p(T an , T an )| + |p(T an , T an ) − p(T a, T a)| < ợ n n0 ữủ lim p(an , T a) = p(T a, T a) ❱➻ ✈➟② lim an = T a✳ ❑➳t ❤đ♣ ✈ỵ✐ ❣✐↔ t❤✐➳t t❛ ❝â n→∞ n→∞ p(a, a) = lim p(an , an ) = p(T a, T a) n→∞ ❙û ❞ö♥❣ ❇ê ✤➲ ✶✳✷✳✶✷ t❛ ❝â a = T a✳ ❍❛② a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ✷✳✶✳✼ ❍➺ q✉↔✳ ✭❬✶✶❪✮ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✳ ●✐↔ sû (Tn) ❧➔ ❞➣② ❝→❝ →♥❤ ①↕ ❧✐➯♥ tö❝ tø X ✈➔♦ X ✈➔ (an ) ❧➔ ❞➣② ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ (Tn )✳ ❑❤✐ ✤â✱ ♥➳✉ Tn ❤ë✐ tư ✤➲✉ tr➯♥ X tỵ✐ →♥❤ ①↕ T ✈➔ lim an = a t❤➻ a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ n→∞ ❱➼ ❞ö s❛✉ ❝❤♦ t❤➜② ♥➳✉ t❤❛② ❤ë✐ tö ✤➲✉ ❜ð✐ ❤ë✐ tö ✤✐➸♠ t❤➻ ❦➳t ❧✉➟♥ ❦❤ỉ♥❣ ❝á♥ ✤ó♥❣✳ ✷✳✶✳✽ ❱➼ ❞ư✳ ❳➨t R ợ tr tổ tữớ sỷ t →♥❤ ①↕ Tn : R → R ①→❝ ✤à♥❤ ❜ð✐    ) x + ♥➳✉ x ∈ / (0,   n  Tn x = (1 − 2n)x + ♥➳✉ < x  2n   1  (2n − 1)x − + ♥➳✉ tũ ỵ (Tn ) ❤ë✐ tư ✤✐➸♠ tỵ✐ T ✈➔ ❤➔♠ ❦❤♦↔♥❣ ❝→❝❤ tử tỗ t n0 s ❝❤♦ p(T x, T y) ✈ỵ✐ ♠å✐ n p(Tn x, Tn y) + ε n0 ✳ ▼➦t ❦❤→❝✱ ✈➻ pn tử tợ p t t ữủ n1 ✭❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ x, y ✮ s❛♦ ❝❤♦ ε max{n0 , n1 } t❛ ❝â |pn (x, y) − p(x, y)| ✈ỵ✐ ♠å✐ n n1 ✳ ❑❤✐ ✤â✱ ✈ỵ✐ n p(T x, T y) p(Tn x, Tn y) + ε 2ε 2ε qpn (T x, T y) + pn (Tn x, Tn y) + > tũ ỵ t ữủ p(T x, T y) qp(x, y) + ε qp(x, y)✳ ❱➟② T ❧➔ →♥❤ ①↕ q−❝♦ t❤❡♦ ♠➯tr✐❝ r✐➯♥❣ p✳ (X, p) tỗ t ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ a ∈ X ❝õ❛ T ✳ ❚ø p(a, a) = p(T a, T a) qp(a, a) s✉② r❛ p(a, a) = 0✳ ❚❛ ❝â pn (a, an ) = pn (a, Tn an ) pn (a, Tn a) + pn (Tn a, Tn an ) pn (a, Tn a) + qpn (a, an ) ✈ỵ✐ ♠å✐ n ✈➔ ✈➻ t❤➳ pn (a, an ) ❱ỵ✐ ♠å✐ n pn (a, Tn a) 1−q n1 t❛ ❝â p(a, an ) − ε p(a, an ) p(a, Tn a) + ε/3 ε + 1−q tù❝ ❧➔ p(a, Tn a) + ε/3 , 1−q ❱➻ lim p(a, Tn a) = p(a, T a) = p(a, a) = ♥➯♥ t❛ t➻♠ ✤÷đ❝ n2 s❛♦ ❝❤♦ p(a, Tn a) n→∞ ε ✈ỵ✐ n n2 ✳ ❑❤✐ ✤â✱ ✈ỵ✐ n max{n1 , n2 } t❛ ❝â p(a, an ) ε 4ε + 3(1 − q) tũ ỵ t õ lim p(a, an ) = = p(a, a)✳ ❚ù❝ ❧➔ an ❤ë✐ tư tỵ✐ a✳ ❱➻ n→∞ p(a, T a) = T a = a ỵ s sỹ rở ỵ ỵ (X, p) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✤➛② ✤õ✳ ❈❤♦ (Tn) ❧➔ ❞➣② ❝→❝ →♥❤ ①↕ tø X ✈➔♦ X õ t (an ) tỗ t < q < s❛♦ ❝❤♦ p(Tn x, Tn y) q max p(x, y), p(x, Tn x), p(y, Tn y), p(x, Tn y) + p(y, Tn x) ✭✷✳✸✮ ✈ỵ✐ ♠å✐ x, y ∈ X ✈➔ Tn ❤ë✐ tư ✤✐➸♠ tr➯♥ X t❤❡♦ ♠➯tr✐❝ ps tỵ✐ →♥❤ ①↕ T t tỗ t lim an = a a ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ n→∞ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ x, y ∈ X ✱ tø ❣✐↔ t❤✐➳t Tn ❤ë✐ tư ✤✐➸♠ tr➯♥ X t❤❡♦ ♠➯tr✐❝ ps tỵ✐ →♥❤ ①↕ T ✱ sû ❞ö♥❣ ❇ê ✤➲ ✷✳✶✳✹ ✈➔ ❝❤♦ n → ∞ t❛ ♥❤➟♥ ✤÷đ❝ p(T x, T y) q max p(x, y), p(x, T x), p(y, T y), p(x, T y) + p(y, T x) ✭✷✳✹✮ ❱➻ t❤➳✱ t❤❡♦ ❍➺ q✉↔ ✶✳✸✳✺ t❛ ❝â T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ a ∈ X ✳ ❚ø p(a, T a) + p(a, T a) } p(a, a) + p(a, a) = p(a, a) q max p(a, a), p(a, a), p(a, a), ✭✷✳✺✮ p(a, a) = p(T a, T a) q max{p(a, a), p(a, T a), p(a, T a), s✉② r❛ p(a, a) = 0✳ ❇➙② ❣✐í✱ ✈ỵ✐ ♠é✐ n = 1, 2, t❛ ❝â p(an , a) ✭✷✳✻✮ p(a, Tn a) + p(Tn a, an ) ▼➦t ❦❤→❝ p(Tn a, an ) = p(Tn a, Tn an ) p(a, Tn an ) + p(an , Tn a) } q max{p(an , a), p(an , Tn an )p(a, Tn a), q max{p(an , a), p(a, Tn a)} q[p(an , a) + p(a, Tn a)] ✭✷✳✼✮ ❚ø ✭✷✳✻✮ ✈➔ ✭✷✳✼✮ t❛ ♥❤➟♥ ✤÷đ❝ (1 − q)p(a, an ) ❤❛② p(a, an ) (1 + q)p(a, Tn a) 1+q p(a, Tn a) 1−q ✈ỵ✐ ♠å✐ n✳ ❚ø (Tn ) ❤ë✐ tư ✤✐➸♠ tỵ✐ T s✉② r❛ lim Tn a = T a = a✳ ❉♦ ✤â n→∞ lim d(a, Tn a) = p(a, a) = 0✳ ❱➻ ✈➟② lim d(an , a) = 0✱ ❤❛② lim an = a tỗ n t a ❜➜t ✤ë♥❣ ❝õ❛ T ✳ n→∞ n→∞ ❚❛ ♥❤➟♥ ✤÷đ❝ ❤➺ q✉↔ s❛✉✳ ✷✳✷✳✻ ❍➺ q✉↔✳ ❈❤♦ (X, p) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✤➛② ✤õ✳ ❈❤♦ (Tn) ❧➔ ❞➣② ❝→❝ →♥❤ ①↕ tø X ✈➔♦ X ❝â ❞➣② ✤✐➸♠ t (an ) tỗ t < a, b, c, 2d, 2e < s❛♦ ❝❤♦ ap(x, y) + bp(x, Tn x) + cp(y, Tn y) + dp(x, Tn y) + ep(y, Tn x)} ✭✷✳✽✮ p(Tn x, Tn y) ✈ỵ✐ ♠å✐ x, y ∈ X ✈➔ Tn ❤ë✐ tư ✤✐➸♠ tr➯♥ X t❤❡♦ ♠➯tr✐❝ ps tỵ✐ →♥❤ ①↕ T t tỗ t lim an = a a ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ n→∞ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t q = max{a, b, c, 2d, 2e}✳ ❑❤✐ ✤â < q < ✈➔ p(Tn x, Tn y) ap(x, y) + bp(x, Tn x) + cp(y, Tn y) + dp(x, Tn y) + ep(y, Tn x)} p(x, Tn y) + p(y, Tn x) q max{p(x, y), p(x, Tn x), p(y, Tn y), } ✭✷✳✾✮ ❉♦ ✤â✱ ❦➳t q ữủ s r tứ ỵ t ▲✉➟♥ ✈➠♥ ✤➣ t❤✉ ✤÷đ❝ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ s❛✉✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ❝ì sð ✈➲ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ✷✳ ❚r➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ tt ởt số t q sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣ ✤➛② ✤õ✳ ✸✳ ❚❤✐➳t ❧➟♣ ✈➔ ự ởt số ỵ sỹ tỗ t ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❞➣② ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❞➣② ❝→❝ →♥❤ ①↕ ❧✐➯♥ tö❝✱ →♥❤ ①↕ ❝♦✱ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✳ ✣÷❛ r❛ ♠ët ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝❤♦ ♠ët ✈➔✐ ❦➳t q✉↔✳ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯✱ ▲➯ ▼➟✉ ❍↔✐ ỡ s ỵ tt t ❚➟♣ ■✱ ■■✱ ◆❳❇ ●✐→♦ ❉ö❝✳ ❬✷❪ ■✳ ❆❧t✉♥✱ ❋✳ ❙♦❧❛ ❛♥❞ ❍✳ ❙✐♠s❡❦ ✭✷✵✶✵✮✱ ●❡♥❡r❛❧✐③❡❞ ❝♦♥tr❛❝t✐♦♥s ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳✱ ✭✶✽✮✱ ✷✼✼✽✲✷✼✽✺✳ ❬✸❪ ❈❤✐ ❑✳P✳✱ ❑❛r❛♣✐♥❛r ❊✳ ❛♥❞ ❚❤❛♥❤ ❚✳❉✳ ✭✷✵✶✷✮ ❆ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ✐♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ▼❛t❤✳ ❈♦♠♣✉t✳ ▼♦❞❡❧❧✐♥❣✱ ✱ ✱ ✶✻✼✸✲✶✻✽✶✳ ❬✹❪ ❊❢❡ ❆✳ ❖✳ ✭✷✵✵✼✮✱ ❘❡❛❧ ❆♥❛❧②s✐s ✇✐t❤ ❊❝♦♥♦♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s✱ Pr✐♥❝❡t♦♥ ❯♥✐✲ ✈❡rs✐t② Pr❡ss✳ ❬✺❪ ❋r❛s❡r ❘✳ ❛♥❞ ◆❛❞❧❡r ❙✳ ❇✳✭✶✾✻✾✮❙❡q✉❡♥❝❡s ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣s ❛♥❞ ❢✐①❡❞ ♣♦✐♥ts P❛❝✐❢✐❝ ❏✳ ▼❛t❤✳ ✸✶✱ ✻✺✾✲✻✻✼✳ ❬✻❪ ❉✳ ■❧✐❝✱ ❱✳ P❛✈❧♦✈✐❝ ❛♥❞ ❱✳ ❘❛❦♦❝❡✈✐❝ ✭✷✵✶✶✮✱ ❙♦♠❡ ♥❡✇ ❡①t❡♥s✐♦♥s ♦❢ ❇❛✲ ♥❛❝❤✬s ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ t♦ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡✱ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt✳✱ ✱ ✶✸✷✻✲✶✸✸✵✳ ❬✼❪ ❉✳ ■❧✐❝✱ ❱✳ P❛✈❧♦✈✐❝ ❛♥❞ ❱✳ ❘❛❦♦❝❡✈✐❝ ✭✷✵✶✷✮✱❊①t❡♥s✐♦♥s ♦❢ t❤❡ ❩❛♠❢✐r❡s❝✉ t❤❡♦r❡♠ t♦ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ▼❛t❤✳ ❈♦♠♣✉t✳ ▼♦❞❡❧❧✐♥❣✱ ✱ ✱ ✽✵✶✲ ✽✵✾✳ ❬✽❪ ■✈❛♥♦✈ ❆✳ ❆✳ ✭✶✾✼✻✮ ❋✐①❡❞ ♣♦✐♥ts ♦❢ ♠❛♣♣✐♥❣s ♦❢ ♠❡tr✐❝ s♣❛❝❡s✱ ✭❘✉ss✐❛♥✮ ❙t✉❞✐❡s ✐♥ t♦♣♦❧♦❣②✱ ■■✳ ▼❛t✳ ■♥st✳ ❙t❡❦❧♦✈✳ ✭▲❖▼■✮ ✻✻✳ ❬✾❪ ▼❛t❤❡✇s ●✳ ❙✳✱✭✶✾✾✷✮✱ P❛rt✐❛❧ ♠❡tr✐❝ t♦♣♦❧♦❣②✱ ❘❡s❡❛❝❤ ❘❡♣♦rt ✷✶✷✱ ❉❡✲ ♣❛rt♠❡♥t ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❯♥✐✈❡rs✐t② ♦❢ ❲❛r✇✐❝❦✳ ✶✺✼ ✺✺ ✭✺✲✻✮ ✷✹ ✺✺ ✭✸✲✹✮ ❬✶✵❪ ▼❛t❤❡✇s ●✳❙✳✱ ✭✶✾✾✹✮✱ P❛rt✐❛❧ ♠❡tr✐❝ t♦♣♦❧♦❣②✱ ❆♥♥✳ ◆❡✇ ❨♦r❦ ❆❝❛❞✳ ❙❝✐✳✱ ✱ ✶✽✸✲✶✾✼✳ ✼✷✽ ❬✶✶❪ ◆❛❞❧❡r✱ ❙✳ ❇✳ ✭✶✾✻✽✮ ❙❡q✉❡♥❝❡s ♦❢ ❝♦♥tr❛❝t✐♦♥s ❛♥❞ ❢✐①❡❞ ♣♦✐♥ts✱ P❛❝✐❢✐❝ ❏✳ ▼❛t❤✳ ✷✼✱ ✺✼✾✲✺✽✺✳ ❬✶✷❪ ❘❤♦❛❞❡s ❇✳ ❊✳ ✭✶✾✼✼✮ ❆ ❝♦♠♣❛r✐s♦♥ ♦❢ ✈❛r✐♦✉s ❞❡❢✐♥✐t✐♦♥s ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✷✷✻ ✭✶✾✼✼✮✱ ✷✺✼✲✷✾✵✳ ❬✶✸❪ ❘❡✐❝❤ ❙✳ ✭✶✾✼✶✮❙♦♠❡ r❡♠❛r❦s ❝♦♥❝❡r♥✐♥❣ ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s✱ ❈❛♥❛❞✳ ▼❛t❤✳ ❇✉❧❧✳ ✶✹✱ ✶✷✶✲✶✷✹✳