ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - PHẠM QUANG DŨNG VỀ PHƯƠNG PHÁP LẶP TÌM ĐIỂM BẤT ĐỘNG CỦA ÁNH XẠ KHƠNG GIÃN TRONG KHƠNG GIAN BANACH LUẬN VĂN THẠC SĨ TỐN HỌC THÁI NGUYÊN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - PHẠM QUANG DŨNG VỀ PHƯƠNG PHÁP LẶP TÌM ĐIỂM BẤT ĐỘNG CỦA ÁNH XẠ KHÔNG GIÃN TRONG KHÔNG GIAN BANACH Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC TS Trần Xuân Quý THÁI NGUYÊN - 2019 ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ❈❤÷ì♥❣ ✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✶✳✶ ✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✹ ✹ ✶✳✶✳✶ ❑❤æ♥❣ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ ỗ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✶ ⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷✳✷ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ❈❤÷ì♥❣ ✷✳ ❱➲ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✶✶ ✷✳✶ ✷✳✷ ✷✳✸ ❈❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✳✶ ❉→♥❣ ✤✐➺✉ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✳✷ ❉→♥❣ ✤✐➺✉ ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❙ü ❤ë✐ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷✳✶ ✣à♥❤ ỵ tử ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tử ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❦✐➸✉ ❍❛❧♣❡r♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸✳✷ ❙ü ❤ë✐ tö ✷✹ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✸✹ ▼ð ✤➛✉ ❇➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ✤➣ ✈➔ ✤❛♥❣ ❧➔ ♠ët ❝❤õ ✤➲ t❤✉ ❤ót sü q✉❛♥ t➙♠ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr♦♥❣ ✈➔ ♥❣♦➔✐ ữợ ởt tr ỳ ữợ ự t ✤✐➸♠ ❜➜t ✤ë♥❣ ❧➔ ①➙② ❞ü♥❣ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✭①➜♣ ①➾✮ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❤♦➦❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ◆❤✐➲✉ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ tợ ữỡ ữủ t r ✈➔ ❣✐↔✐ q✉②➳t ❝❤♦ tø♥❣ ❧ỵ♣ →♥❤ ①↕ ❦❤→❝ ♥❤❛✉✱ ❝❤➥♥❣ ❤↕♥ ❧ỵ♣ →♥❤ ①↕ ❝♦✱ ❧ỵ♣ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱✳ ✳ ✳ ❱ỵ✐ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ t❤↕❝ s➽✱ tæ✐ ❧ü❛ ❝❤å♥ ♠ët ♣❤➛♥ tr♦♥❣ ❜➔✐ t♦→♥ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ❧ỵ♣ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ổ ữợ sỹ ữợ r ỵ tổ t ữỡ ♣❤→♣ ❧➦♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✧✳ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✱ ❝ư t❤➸ ♥❤÷ s❛✉✿ ❈❤÷ì♥❣ r ổ ỗ ỗ ❝❤➦t ✈➔ ❜➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈❤÷ì♥❣ ✷✿ ❚r➻♥❤ ❜➔② ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❝ò♥❣ ỵ tử tử ❝→❝ ♣❤÷ì♥❣ ♣❤→♣✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❡♠ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ ❣✐ó♣ ✤ï ✈➔ ✤ë♥❣ ✈✐➯♥ ❝õ❛ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ♣❤á♥❣ ✣➔♦ t↕♦✱ ❑❤♦❛ ❚♦→♥ ✕❚✐♥✳ ❱ỵ✐ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❡♠ ♠♦♥❣ ♠✉è♥ ✤÷đ❝ ❣â♣ ♠ët ♣❤➛♥ ♥❤ä ❝æ♥❣ sù❝ ❝õ❛ ♠➻♥❤ ✈➔♦ ✈✐➺❝ ❣➻♥ ❣✐ú ✈➔ ♣❤→t sỹ ỳ ỵ t♦→♥ ❤å❝ ✈è♥ ❞➽ ✤➣ r➜t ✤➭♣✳ ✣➙② ❝ô♥❣ ❧➔ ♠ët ❝ì ❤ë✐ ❝❤♦ ❡♠ ❣û✐ ❧í✐ tr✐ ➙♥ tỵ✐ t➟♣ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐↔♥❣ ✈✐➯♥ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ♥â✐ ✷ ✸ ❝❤✉♥❣ ✈➔ ❑❤♦❛ ❚♦→♥ ✕ ❚✐♥ ♥â✐ r✐➯♥❣✱ ✤➣ tr tử tự qỵ ❜→✉ tr♦♥❣ t❤í✐ ❣✐❛♥ ❡♠ ✤÷đ❝ ❧➔ ❤å❝ ✈✐➯♥ ❝õ❛ tr÷í♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ❚❍P❚ ❚❤❛♥❤ ❚❤õ②✱ P❤ó ❚❤å ❝ò♥❣ t♦➔♥ t❤➸ ỗ t tèt ♥❤➜t ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ ✤✐ ❤å❝ ❈❛♦ ❤å❝❀ ❝↔♠ ì♥ ❝→❝ ❛♥❤ ❝❤à ❡♠ ❤å❝ ✈✐➯♥ ợ ỗ ✤➣ tr❛♦ ✤ê✐✱ ✤ë♥❣ ✈✐➯♥ ✈➔ ❦❤➼❝❤ ❧➺ t→❝ ❣✐↔ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s tợ ữợ r ỵ ✤➣ ❧✉æ♥ q✉❛♥ t➙♠ ➙♥ ❝➛♥ ❝❤➾ ❜↔♦✱ ✤ë♥❣ ✈✐➯♥ ú ù t t õ ỵ s s➢❝ ❝❤♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝ô♥❣ ♥❤÷ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ❈❤➦♥❣ ✤÷í♥❣ ✈ø❛ q✉❛ s➩ ỳ ợ ỵ ✤è✐ ✈ỵ✐ ❝→❝ ❛♥❤ ❝❤à ❡♠ ❤å❝ ✈✐➯♥ ❧ỵ♣ ❑✶✶ ♥â✐ ❝❤✉♥❣ ✈➔ ✈ỵ✐ ❜↔♥ t❤➙♥ ❡♠ ♥â✐ r✐➯♥❣✳ ❉➜✉ ➜♥ ➜② ❤✐➸♥ ♥❤✐➯♥ ❦❤ỉ♥❣ t❤➸ t❤✐➳✉ sü ❤é trđ✱ s➫ ❝❤✐❛ ✤➛② ②➯✉ t❤÷ì♥❣ ❝õ❛ ❝❤❛ ♠➭ ❤❛✐ ❜➯♥ ✈➔ ❝→❝ ❛♥❤ ❝❤à ❡♠ ❝♦♥ ❝❤→✉ tr♦♥❣ ❣✐❛ ✤➻♥❤✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t➜t ❝↔ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥ ú ù ỗ ũ tr ✤÷í♥❣ ✈ø❛ q✉❛✳ ▼ët ❧➛♥ ♥ú❛✱ ❡♠ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✷✷ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾ ❍å❝ ✈✐➯♥ P❤↕♠ ◗✉❛♥❣ ❉ơ♥❣ ❈❤÷ì♥❣ ✶ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❤➻♥❤ ❤å❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑✐➳♥ t❤ù❝ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ tê♥❣ ❤đ♣ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪ ✈➔ ❬✹❪✳ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✶✳✶✳✶ ❑❤æ♥❣ ỗ X ổ x0 X trữợ ỵ Sr (x0 ) ♠➦t ❝➛✉ t➙♠ x0 ❜→♥ ❦➼♥❤ r > 0✱ Sr (x0 ) := {x ∈ X : ||x − x0 || = r} ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ữủ ỗ (0, 2] t ý tỗ t = ( ) > s❛♦ ❝❤♦ ♥➳✉ x, y ∈ X ✈ỵ✐ ||x|| = 1, ||y|| = ✈➔ ||x − y|| ≥ ✱ t❤➻ (x + y) ≤ − δ t q ữợ ởt ổ ỗ ỵ ổ Lp[a, b] ợ < p < ổ ỗ ỵ sỷ X ổ ỗ õ ợ t ý d > 0, >0 tỡ tũ ỵ x, y ∈ X ✈ỵ✐ ||x|| ≤ d, ||y|| ≤ d, ||x y|| tỗ t > s❛♦ ❝❤♦ (x + y) ≤ − δ d d ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ❜➜t ❦ý x, y ∈ X ✱ ①➨t z1 = xd , z2 = yd ✱ ✈➔ t➟♣ ¯ = d ✳ ❍✐➸♥ ♥❤✐➯♥ ¯ > ❍ì♥ ♥ú❛✱ ||z1 || ≤ 1, ||z2 || ≤ ✈➔ ||z1 − z2 || = ||x − y|| ≥ = ¯ d d ❚ø t ỗ t õ = d > 0, (z1 + z2 ) − δ(¯), ♥❣❤➽❛ ❧➔ (x + y) ≤ − δ , 2d d s✉② r❛ (x + y) ≤ − δ d d ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ▼➺♥❤ ✤➲ ✶✳✶✳✹✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ỗ sỷ (0, 1) > 0✳ ❑❤✐ ✤â ✈ỵ✐ ❜➜t ❦ý d > 0✱ ♥➳✉ x, y ∈ X t❤ä❛ ♠➣♥ ||x|| ≤ d✱ ||y|| d ||x y|| t tỗ t↕✐ δ = δ > s❛♦ ❝❤♦ d ||αx + (1 − α)|| ≤ − 2δ d min{α, } d ổ ỗ t ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✤÷đ❝ ❣å✐ ❧➔ ỗ t ợ x, y X x = y, ||x|| = ||y|| = 1✱ t❛ ❝â ||λx + (1 − λ)y|| < ∀λ ∈ (0, 1) ỵ ổ ỗ ổ ỗ t ỵ r ởt ợ ổ ỗ t ổ ổ ỗ t ữợ ❧➔ ♠ët ✈➔✐ ✈➼ ❞ư ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❧➔ ỗ t ữ ỗ trữợ > t C[0, 1] ợ ||.||à ữ s x2 (t)dt ||x||à := ||x||0 + ợ ||.||0 ❝❤✉➞♥ s✉♣✳ ❑❤✐ ✤â ||x||0 ≤ ||x||µ (1 + µ)||x||0 , x ∈ C[0, 1], ✈➔ ❤❛✐ ❝❤✉➞♥ ♥➔② t÷ì♥❣ ữỡ ||.||à ||.||0 ợ (C[0, 1], ||.||0 ) ổ ỗ tr ợ t ý > 0, (C[0, 1], ||.||à ) ỗ ▼➦t ❦❤→❝ ✈ỵ✐ ❜➜t ❦ý x+y ✈➔ ∈ (0, 2] tỗ t x, y C[0, 1] ợ ||x||à = ||y||µ = 1, ||x − y|| = ✱ tò② þ ❣➛♥ 1✳ ❱➻ ✈➟② (C[0, 1], ||.||µ ) ❦❤ỉ♥❣ ỗ t à0 c0 = c0(N) ợ ||.||à ợ x = {xn} c0 ữ s ||x||à := ||x||c0 + i=1 xi i 2 tr♦♥❣ ✤â ||.||c0 ❧➔ ❝❤✉➞♥ t❤ỉ♥❣ t❤÷í♥❣✳ ◆❤÷ tr♦♥❣ ✈➼ ❞ư tr➯♥✱ (c0 , ||.||à ) ợ > ỗ t ữ ổ ỗ tr c0 ợ tổ tữớ ổ ỗ t t tr ữủ t ỗ ổ X ỵ X : (0, 2] [0, 1]) ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ♥➔② ▼➺♥❤ ✤➲ ✶✳✶✳✾✳ (a) ❱ỵ✐ ♠å✐ ❦❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❤➔♠ δX ( ) ❦❤æ♥❣ ❣✐↔♠ tr➯♥ (0, 2] (b) ❍➔♠ s t ỗ ổ tử ỗ (c) r ổ ỗ X s t ỗ ổ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ δX ✱ ❧➔ ❤➔♠ t➠♥❣ t❤ü❝ sü✳ ❚r♦♥❣ ♠ư❝ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ❦❤ỉ♥❣ ỗ ỵ ổ X ỗ X ( ) > ✈ỵ✐ ♠å✐ ∈ (0, 2] ✼ ❍➺ q✉↔ r ổ ỗ X t ỗ t t ỵ X ổ ỗ t X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✸✳ ✶✳ ❈❤✉➞♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ♥➳✉ ✈ỵ✐ ♠é✐ y ∈ SX t❤➻ ❣✐ỵ✐ lim t0 x + ty x t tỗ t ợ x SX ỵ y, x ✳ ❑❤✐ ✤â ✭✶✳✶✮ x ✤÷đ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ ●➙t❡❛✉① ❝õ❛ ❝❤✉➞♥✳ ✷✳ ❈❤✉➞♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ♥➳✉ ✈ỵ✐ ♠é✐ y ∈ SX ợ t ữủ ợ x ∈ SX ✳ ✸✳ ❈❤✉➞♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t ♥➳✉ ✈ỵ✐ ♠é✐ x ∈ SX ợ tỗ t ợ y ∈ SX ✳ ✹✳ ❈❤✉➞♥ ❝õ❛ X ✤÷đ❝ ❣å✐ ❧➔ rt ợ tỗ t ✤➲✉ ✈ỵ✐ ♠å✐ x, y ∈ SX ✳ ✶✳✷ ❇➔✐ t t ổ ỵ ❤✐➺✉ 2X ❧➔ t➟♣ t➜t ❝↔ ❝→❝ t➟♣ ❝♦♥ ❝õ❛ X ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ⑩♥❤ ①↕ J s : X → 2X ✱ s > ✭♥â✐ ❝❤✉♥❣ ❧➔ ✤❛ trà✮ ①→❝ ∗ ✤à♥❤ ❜ð✐ J s (x) = x∗ ∈ X ∗ : x∗ , x = x∗ x , x∗ = x s−1 x ∈ X, ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ tê♥❣ q✉→t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✳ ❑❤✐ s = 2✱ →♥❤ ①↕ J ữủ ỵ J ữủ ố t X ỵ ❤✐➺✉ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ✤ì♥ trà ❧➔ j ✳ ✽ ❱➼ ❞ư ✶✳✷✳✷✳ ❚r♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✱ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❧➔ →♥❤ ①↕ ✤ì♥ ✈à I ✳ ❚➼♥❤ ✤ì♥ trà ❝õ❛ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❝â ♠è✐ ❧✐➯♥ ❤➺ ✈ỵ✐ t➼♥❤ ❦❤↔ ✈✐ ❝õ❛ ❝❤✉➞♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♥❤÷ tr ỵ s ỵ ✶✳✷✳✸✳ ❈❤♦ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝ ❑❤✐ ✤â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ (i) X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ trì♥❀ (ii) J ❧➔ ✤ì♥ trà❀ (iii) ❈❤✉➞♥ ❝õ❛ X ❧➔ ❦❤↔ t ợ x = x Jx ỵ ✶✳✷✳✹✳ ●✐↔ sû X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ t❤ü❝✱ ✈➔ →♥❤ ①↕ Jp : X −→ 2X , < p < ∞✱ ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝✳ ❑❤✐ ✤â ✈ỵ✐ ❜➜t ❦ý x, y ∈ X, t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ∗ J : X → 2X ∗ ||x + y||p ≤ ||x||p + p y, jp (x + y) ✭✶✳✷✮ ✈ỵ✐ ♠å✐ jp(x + y) ∈ Jp(x + y) ✣➦❝ ❜✐➺t ♥➳✉ p = t❤➻ ||x + y||2 ≤ ||x||2 + y, j( x + y) ✭✶✳✸✮ ✈ỵ✐ ♠å✐ j(x + y) ∈ J(x + y) ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✺✳ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✳ (i) ⑩♥❤ ①↕ T : C → E ✤÷đ❝ L tử st tỗ t ❤➡♥❣ sè L ≥ s❛♦ ❝❤♦ Tx − Ty ≤ L x − y ∀x, y ∈ C ✭✶✳✹✮ (ii) ❚r♦♥❣ ✭✶✳✹✮✱ ♥➳✉ L ∈ [0, 1) t❤➻ T ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❝♦❀ ♥➳✉ L = t❤➻ T ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ▼å✐ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱ ✈ỵ✐ t➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❦❤→❝ ré♥❣ ❧➔ tü❛ ❦❤æ♥❣ ❣✐➣♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✻✳ ●✐↔ sû K ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ X ✳ ❳➨t T : K → E ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ ❋✐①(T ) = ∅✳ ⑩♥❤ ①↕ T ✤÷đ❝ ❣å✐ ❧➔ tü❛ ❦❤ỉ♥❣ ❣✐➣♥ ♥➳✉ T x − T x∗ ≤ x − x∗ t❤ä❛ ♠➣♥ ✈ỵ✐ ♠å✐ x ∈ K ✈➔ x∗ ∈ F (T )✳ ✷✶ ❈❤å♥ N, K ∈ N + ✤õ ❧ỵ♥✳ ❚ø ✤➥♥❣ t❤ù❝ ✭✷✳✶✶✮ t❛ ❝â✱ ✈ỵ✐ i = N + K ✱ ∆xN +K = ∆f (xN +K ) = β > ❳➨t f ∗ ∈ X ∗ s❛♦ ❝❤♦ = ✈➔ f ∗ (∆xN +K ) = f ✭✷✳✶✷✮ ∆xN +K ✳ ❑❤✐ ✤â ✈ỵ✐ j = 0, 1, 2, ✱ f ∗ (∆f (xN +K−j )) ≤ f ∗ ∆f (xN +K−j ) = ∆f (xN +K−j ) = s ✭✷✳✶✸✮ ❚ø xN +K−j+1 = (1 − CN +K−j )xN +K−j + CN +K−j f (xN +K−j ) ✈➔ Ci = Ci−1 ✈ỵ✐ ♠å✐ i✱ t❛ ❝â ∆xN +K−j+1 = (1 − CN +K−j )∆xN +K−j + CN +K−j ∆f (xN +K−j ) ✭✷✳✶✹✮ ❚✐➳♣ t❤❡♦ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤✿ f ∗ (∆xN +K−j ) ≥ β ✈ỵ✐ j = 0, 1, ✭✷✳✶✺✮ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ t tự q t trữợ ❤➳t t❛ ❝â f ∗ (∆xN +K ) = ∆xN +K = β t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✺✮ ✈ỵ✐ j = 0✳ ❈á♥ ♥➳✉ j = 1✱ tø ✤➥♥❣ t❤ù❝ ✭✷✳✶✹✮ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✸✮ t❛ ❝â f ∗ (∆xN +K−1 ) = ≥ f ∗ (∆xN +K ) − CN +K−1 CN +K−1 − f ∗ (∆f (xN +K−1 )) − CN +K−1 CN +K−1 β− β = β − CN +K−1 − CN +K−1 ●✐↔ sû ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✺✮ ✤ó♥❣ ✈ỵ✐ j = 0, 1, , t✳ ❑❤✐ ✤â →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✶✸✮ ✈➔ ✤➥♥❣ t❤ù❝ ✭✷✳✶✹✮ ✈➔ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❛ ❝â f ∗ (∆xN +K−t−1 ) = ≥ f ∗ (∆xN +K−t ) − CN +K−t−1 CN +K−t−1 − f ∗ (∆f (xN +K−t−1 )) − CN +K−t−1 CN +K−t−1 β− β = β − CN +K−t−1 − CN +K−t−1 ❚÷ì♥❣ tü ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ỵ tờ t tự t❤❡♦ j = 0, 1, , K − t❛ t❤✉ ✤÷đ❝ xN +K − xN ≥ f ∗ (xN +K − xN ) ≥ Kβ, ✭✷✳✶✻✮ ✷✷ ∞ ❤❛② ❞➣② {xi }∞ i=0 ❦❤æ♥❣ ❝â ❞➣② ❝♦♥ ❤ë✐ tư✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t {xn }n=0 ❝â ✤✐➸♠ tö✳ ❉♦ ✤â β = ✈➔ f (q) = q ✳ ◆❣❤➽❛ ❧➔ xn → q ✈➻ f ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ✣➲ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ t✐➳♣ t❤❡♦✱ t❛ ❝➛♥ ❦❤→✐ ♥✐➺♠ s❛✉✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✷✳ ❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ t❤ü❝ X ✳ ⑩♥❤ ①↕ f : C → X ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❝♦♠♣❛❝t ✭❞❡♠✐❝♦♠♣❛❝t✮ t↕✐ h ∈ X ♥➳✉✱ ✈ỵ✐ ❜➜t ❦ý ❞➣② ❜à ❝❤➦♥ {xn }∞ n=0 tr♦♥❣ C s❛♦ ❝❤♦ xn − f (xn ) h n tỗ t↕✐ ❞➣② ❝♦♥ {xnj }∞ j=0 ✈➔ x ∈ C s❛♦ ❝❤♦ xnj → x ❦❤✐ j → ∞ ✈➔ x − f (x) = h✳ ❍➺ q✉↔ ✷✳✷✳✸✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✱ C ❧➔ t➟♣ ❝♦♥ ✤â♥❣ ❜à ❝❤➦♥ ❝õ❛ X ✈➔ f : C → C ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ●✐↔ sû✱ ❤♦➦❝ (i) f ❧➔ ♥û❛ ❝♦♠♣❛❝t t↕✐ 0✱ ❤♦➦❝ (ii) (I − f ) →♥❤ ①↕ t➟♣ ❝♦♥ ✤â♥❣ ❜à ❝❤➦♥ ❝õ❛ X ✈➔♦ t➟♣ ❝♦♥ ✤â♥❣ ❝õ❛ X ✳ ❱ỵ✐ x0 ∈ C ①➨t {xn}∞n=0 ⊆ C ❧➔ ❞➣② ❝❤➜♣ ♥❤➟♥ ữủ ự ợ số tỹ {Cn}n=0 ỗ tớ tọ ♠➣♥ < a ≤ Cn ≤ b < ✈ỵ✐ ♠å✐ n ≥ 1✳ ❑❤✐ ✤â {xn }∞ n=0 ❧➔ ❞➣② ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ f tr♦♥❣ C ✳ ❈❤ù♥❣ ♠✐♥❤✳ (i) ❚ø xn+1 = (1 − Cn)xn + Cnf (xn) t❛ ❝â xn − f (xn ) = {xn − xn−1 } Cn ∞ ❱➻ t➟♣ C ❜à ❝❤➦♥✱ ♥➯♥ ❞➣② {xn }∞ n=0 ❜à ❝❤➦♥ ✈➔ ❞➣② {Cn }n=1 ❝ô♥❣ ❜à ❝❤➦♥ ữợ ỵ s r❛ {xn − f (xn )} ❤ë✐ tư tỵ✐ ✈➻ ✈➟② tø t➼♥❤ ♥û❛ ❝♦♠♣❛❝t ❝õ❛ →♥❤ ①↕ f t↕✐ 0✱ ❞➣② {xn }∞ n=0 ❝â ✤✐➸♠ tö tr♦♥❣ C ỵ t õ ❝❤ù♥❣ ♠✐♥❤✳ (ii) ◆➳✉ q ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✱ t❤➻ ❞➣② { xn − q }∞ n=0 ❦❤æ♥❣ t➠♥❣ t❤❡♦ n✳ ❉♦ ✤â✱ t❛ ❝❤➾ r tỗ t {xn } n=0 ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✳ ❱ỵ✐ x0 ∈ C ✱ ①➨t K ❧➔ ❜❛♦ ✤â♥❣ ♠↕♥❤ ❝õ❛ t➟♣ {xn }∞ n=0 ✳ ❚❤❡♦ ỵ {(I f )(xn )} tư ♠↕♥❤ tỵ✐ ❦❤✐ n → ∞✳ ❱➻ ✈➟②✱ t❤✉ë❝ ✈➔♦ ❜❛♦ ✤â♥❣ ♠↕♥❤ ❝õ❛ t➟♣ (I − f )(K)✱ (I − f )(K) ❧➔ t➟♣ ✤â♥❣ ✭✈➻ t➟♣ K ✤â♥❣ ✈➔ ❜à ❝❤➦♥✮✱ ✈➻ ✈➟② ∈ (I f )(K) tỗ t {xnj }∞ j=0 s❛♦ ❝❤♦ xnj → µ ∈ C ợ tọ (I f )à = õ xn ỵ ❤ë✐ tö ②➳✉ ❚❛ ♥❤➢❝ ❧↕✐ ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ♥û❛ ✤â♥❣✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✹✳ ⑩♥❤ ①↕ T : K → X ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ✤â♥❣ t↕✐ y ♥➳✉ ✈ỵ✐ ∞ ❜➜t ❦ý ❞➣② {xn }∞ n=0 ⊆ K ❤ë✐ tư ②➳✉ tỵ✐ x ∈ K ✱ ❞➣② {T (xn )}n=0 ❤ë✐ tư ♠↕♥❤ tỵ✐ y ∈ K s✉② r T x = y ỵ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❖♣✐❛❧ ✈➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ f : K → K ✱ ✈ỵ✐ K ❧➔ t➟♣ ỗ t tr X t ý x0 K ✱ ①➨t {xn}∞n=0 ⊆ K ❧➔ ❞➣② ❝❤➜♣ ♥❤➟♥ ữủ ự ợ ổ t {Cn}n=1 tọ < a ≤ Cn < ✈ỵ✐ ♠å✐ n ≥ 1✳ ❑❤✐ ✤â {xn}∞n=0 ❤ë✐ tư ②➳✉ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❖♣✐❛❧ ✈➔ f ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ ♥➯♥ (I −f ) ♥û❛ ✤â♥❣✳ ◆❣♦➔✐ r❛✱ t❤❡♦ ỵ f t q ✤â✱ t❤❡♦ ❇r♦✇❞❡r ✈➔ P❡tr②s❤②♥ ✭✶✾✻✻✮✱ ✈ỵ✐ ❜➜t ❦ý ✤✐➸♠ tö ②➳✉ ❝õ❛ {xn }∞ n=0 ⊆ K ✤➲✉ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✳ ❚❛ s➩ ❝❤➾ r❛ {xn }∞ n=0 ⊆ K ❝â ❞✉② ♥❤➜t tử t sỷ tỗ t ✤✐➸♠ tö ②➳✉ ♣❤➙♥ ❜✐➺t ❝õ❛ {xn }∞ n=0 ✱ ❧➔ ∞ ∞ q1 ✈➔ q2 ✱ ✈➔ ❤❛✐ ❞➣② ❝♦♥ {xni }∞ i=1 ❱➔ {xnj }j=1 s❛♦ ❝❤♦ {xni }i=1 ❤ë✐ tư ②➳✉ tỵ✐ q1 ✈➔ {xnj }∞ j=1 ❤ë✐ tư ②➳✉ tỵ✐ q2 ✳ ❳➨t p ∈ ❋✐①(f ) ✈ỵ✐ ❋✐①(f ) ❧➔ t➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ f ✳ ❑❤✐ ✤â t❛ ❞➵ ❞➔♥❣ ❝â xn+1 − p ≤ xn − p ✈ỵ✐ ♠é✐ n s lim xn p tỗ t↕✐ ✈ỵ✐ ♠å✐ p ∈ ❋✐①(f )✳ ❱➻ ✈➟② ✭X ❧➔ ❦❤æ♥❣ n→∞ ❣✐❛♥ ❖♣✐❛❧✮ t❛ ❝â lim xn − q1 = lim xni − q1 < lim xni − q2 = lim xn − q2 n→∞ i→∞ i→∞ n→∞ ✈➔ lim xn − q2 = lim xnj − q2 < lim xnj − q1 = lim xn − q1 , n→∞ i→∞ i→∞ n→∞ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ tt s r tỗ t t ởt tö ②➳✉ ∞ q ❝õ❛ ❞➣② {xn }∞ n=0 ⊆ K ✳ ❚❤❡♦ t➼♥❤ ❝♦♠♣❛❝t ②➳✉ ❝õ❛ K ✱ t❛ ❝â {xn }n=0 ❤ë✐ tư ②➳✉ tỵ✐ q ✳ ✣à♥❤ ỵ K t ỗ õ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ X✱ ✈➔ T : K → X ❧✐➯♥ tö❝ t❤ä❛ ♠➣♥ (i) ❋✐①(T ) = ∅❀ ✷✹ ◆➳✉ T p = p✱ t❤➻ T x − p ≤ x − p ✈ỵ✐ ♠å✐ x ∈ K (iii) ỗ t x0 K tữỡ ù♥❣ ❞➣② ❝❤➜♣ ♥❤➟♥ ✤÷đ❝ {xn }∞ n=0 ⊆ K ❀ (iv) T ❝❤➼♥❤ q✉② t✐➺♠ ❝➟♥ t↕✐ x0 ❀ ∞ ∞ (v) ◆➳✉ {xn }∞ j=1 ❧➔ ❞➣② ❝♦♥ ❝õ❛ {xn }n=0 s❛♦ ❝❤♦ {xn }j=1 ❤ë✐ tö ②➳✉ tỵ✐ x ∈ K ✈➔ {xn − T xn } ❤ë✐ tư ♠↕♥❤ tỵ✐ t❤➻ x − T x = 0❀ (vi) X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❖♣✐❛❧✳ ❑❤✐ ✤â ❞➣② {xn}∞n=0 ❤ë✐ tư ②➳✉ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T tr♦♥❣ K ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❦➳t q✉↔ ❝õ❛ P❡tr②s❤②♥ ❛♥❞ ❲✐❧❧✐❛♠s♦♥ ✭✶✾✼✸✮✱ s✉② r❛ ❞➣② (ii) j j j j {xn }∞ n=0 ❝❤ù❛ ❞➣② ❝♦♥ ❤ë✐ tư ②➳✉ ✈ỵ✐ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ♥â t❤✉ë❝ ❋✐①(T ) ✈➔✱ ♥❣♦➔✐ r❛✱ ✈ỵ✐ ♠å✐ ❞➣② ❝♦♥ ❤ë✐ tư ②➳✉ {xn }∞ n=0 ❝â ❣✐ỵ✐ ❤↕♥ q ∈ ❋✐①(T ) ữỡ tỹ ữ tr ự ỵ ❞➣② {xn }∞ n=0 ❝â ❞✉② ♥❤➜t ✤✐➸♠ tö ②➳✉ q ∈ K ✳ ❚ø t➼♥❤ ❝♦♠♣❛❝t ②➳✉ ❝õ❛ K ✱ s✉② r❛ ❞➣② {xn }∞ n=0 ❤ë✐ tö ②➳✉ tợ q Pữỡ r r ♠ư❝ ♥➔② t❛ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❦✐➸✉ ❍❛❧♣❡r♥ ①➜♣ ①➾ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ✷✳✸✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ❈❤♦ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ tỹ K t ỗ õ X ✈➔ →♥❤ ①↕ T : K → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❈è ✤à♥❤ t ∈ (0, 1)✱ ✈➔ u K tũ ỵ sỷ zt K ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ →♥❤ ①↕ Tt ①→❝ ✤✐♥❤ ❜ð✐ Tt x := tu + (1 − t)T x✱ x ∈ K ✳ ●✐↔ sû ❋✐①(T ) := {x ∈ K : T x = x} = ∅✳ ❱ỵ✐ {αn } ⊂ [0, 1] ✈➔ u ∈ K ✱ ①➨t ❞➣② ❧➦♣ {xn } ∈ K ①→❝ ✤à♥❤ ♥❤÷ s❛✉ x0 ∈ K ✱ ✈➔ xn+1 := α + (1 − αn )T xn , ✷✳✸✳✷ ❙ü ❤ë✐ tö ◆➠♠ ✶✾✽✸✱ ❘❡✐❝❤ ✤➦t r❛ ❝➙✉ ❤ä✐ s❛✉✳ n ≥ ✭✷✳✶✼✮ ✷✺ ❈➙✉ ❤ä✐✳ ❈❤♦ X ❧➔ ổ ỗ t ổ {n} s ợ t ý t ỗ t K ❝õ❛ X ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ t❤➻ ❞➣② {xn } ①→❝ ✤à♥❤ tr♦♥❣ ✭✷✳✶✼✮ ❤ë✐ tư tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✈ỵ✐ ❜➜t ❦ý u ∈ K ✈➔ →♥❤ ①↕ ❦❤ỉ♥❣ T : K K rữợ õ ✶✾✻✼✱ ❍❛❧♣❡r♥ ✤➣ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❤ë✐ tö ❝õ❛ ❞➣② ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✼✮ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt tr♦♥❣ ❦➳t q s ỵ K t ỗ õ ổ rt H ✈➔ T : K → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❈❤♦ ❞➣② sè t❤ü❝ {αn} ⊂ [0, 1] ①→❝ ✤à♥❤ ❜ð✐ αn = n−θ , θ ∈ (0, 1)✳ ợ u K tũ ỵ t {xn} ∈ K ①→❝ ✤à♥❤ ♥❤÷ s❛✉ x1 ∈ K, xn+1 = αn u + (1 − αn )T xn , n ≥ ❑❤✐ ✤â ❞➣② {xn} ❤ë✐ tö ♠↕♥❤ tỵ✐ ♣❤➛♥ tû ❝õ❛ t➟♣ ❋✐①(T ) := {x ∈ K : T x = x} ❣➛♥ u ♥❤➜t✳ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①→❝ ✤à♥❤ ❜ð✐ ❞➣② ❧➦♣ ♥❤÷ tr♦♥❣ ✭✷✳✶✼✮ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❦✐➸✉ ❍❛❧♣❡r♥✳ ◆➠♠ s t ỵ tr tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ❝❤➾ r❛ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ❞➣② {xn } tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ♥➳✉ ❞➣② sè t❤ü❝ {αn } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✭❈✶✮ n→∞ lim αn = 0❀ ✭❈✷✮ αn = ∞❀ n=1 | = 0✳ ✭❈✸✮ n→∞ lim |α −α α ∞ n n−1 n ◆➠♠ ✶✾✽✸✱ ❘❡✐❝❤ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ tr➯♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ X ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ trì♥ ✤➲✉ ✈➔ αn = n−a ✈ỵ✐ < a < 1✳ ❚❛ t❤➜② r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❍❛❧♣❡r♥ ✈➔ ▲✐♦♥s ✤è✐ ✈ỵ✐ ❞➣② sè t❤ü❝ αn := (n + 1)−1 ❜à ❧♦↕✐ trø✳ ✣✐➲✉ ♥➔② ✤➣ ✤÷đ❝ ❦❤➢❝ ♣❤ư❝ ❜ð✐ ❲✐tt♠❛♥♥ ✭✶✾✾✷✮✱ ✈➝♥ ①➨t ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ❞➣② {xn } ❤ë✐ tư ♠↕♥❤ ♥➳✉ {αn } t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❈✹✮ ✭❈✶✮✱ ✭❈✷✮ ✈➔ ∞ |αn+1 − αn | < ∞ ✭✷✳✶✽✮ n=1 ◆➠♠ ✶✾✾✹✱ ❘❡✐❝❤ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❲✐tt♠❛♥♥ ❝❤♦ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉ ✈➔ ❝â ❞➣② →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❧✐➯♥ tö❝ ❞➣② ②➳✉ t❤❡♦ ❞➣②✱ tr♦♥❣ ✷✻ ✤â ❞➣② {αn } ♣❤↔✐ t❤ã❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝ô♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❈✶✮✱ ✭❈✷✮ ✈➔ ❦❤æ♥❣ ❣✐↔♠ ✭tù❝ ❧➔ ✭❈✸✮ tr♦♥❣ ✭✷✳✶✽✮✮✳ ◆➠♠ ✶✾✾✼✱ ❙❤✐♦❥✐ ✈➔ ❚❛❦❛❤❛s❤✐ ✤➣ ♠ð rë♥❣ ❦➳t q✉↔ ♥➔② ❝õ❛ ❲✐tt♠❛♥♥ ❧➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ❦❤↔ ✈✐ ●✂t❡❛✉① tr tr ộ t ỗ õ ❝❤➦♥ ❦❤→❝ ré♥❣ ❝õ❛ K ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ❑➳t q✉↔ ✤÷đ❝ ♣❤→t ❜✐➸✉ ữ s ỵ X ổ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ✈➔ K ❧➔ t ỗ õ X t T : K → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ s❛♦ ❝❤♦ ❋✐①(T ) = ∅✳ ❳➨t ❞➣② {αn} t❤✉ë❝ ✤♦↕♥ [0, 1] t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❈✶✮✱ ✭❈✷✮ ✈➔ ✭❈✸✮✳ ❱ỵ✐ u ∈ K ✱ ❞➣② {xn} ①→❝ ✤à♥❤ ❜ð✐ x0 ∈ K, xn+1 = αn u + (1 − αn )T xn , n ≥ ●✐↔ sû ❞➣② {zt} ❤ë✐ tư ♠↕♥❤ tỵ✐ z ∈ ❋✐①(T ) ❦❤✐ t ⇓ 0✱ tr♦♥❣ ✤â < t < 1, zt ❧➔ ♣❤➛♥ tû ❞✉② ♥❤➜t ❝õ❛ K t❤ä❛ ♠➣♥ zt = tu + (1 − t)T zt✳ ❑❤✐ ✤â ❞➣② {xn} ❤ë✐ tư ♠↕♥❤ tỵ✐ z ✳ ◆➠♠ ✶✾✽✵✱ ❘❡✐❝❤✱ ♥➠♠ ✶✾✽✹ ❚❛❦❛❤❛s❤✐ ✈➔ ❯❡❞❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ♥➳✉ K t❤ä❛ ♠➣♥ t❤➯♠ ♠ët sè ✤✐➲✉ ❦✐➺♥✱ t❤➻ {zt } ①→❝ ✤à♥❤ ♥❤÷ tr➯♥ ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ✣➦❝ ❜✐➺t✱ ❦❤➥♥❣ s ú ỵ sỷ X ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ✈➔ K t ỗ t X T : K → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❱ỵ✐ u ∈ K ✈➔ zt ❧➔ ♣❤➛♥ tû ❞✉② ♥❤➜t ❝õ❛ K t❤ä❛ ♠➣♥ zt = tu + (1 − t)T zt ✈ỵ✐ < t < 1✳ ●✐↔ t❤✐➳t ♠é✐ →♥❤ ①↕ ❜➜t ❜✐➳♥ ✤è✐ ✈ỵ✐ ❝→❝ ❝→❝ t ỗ õ K ự ởt t ✤ë♥❣ ❝õ❛ T ✳ ❑❤✐ ✤â ❞➣② {zt} ❤ë✐ tö ♠❛♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ◆➠♠ ✷✵✵✵✱ ▼♦r❛❧❡s ✈➔ ❏✉♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t q✉↔ s❛✉✳ ỵ K t ỗ õ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ X ✈ỵ✐ ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ✈➔ T : K → K ❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ ❋✐①(T ) = sỷ r t ỗ ✤â♥❣ ❦❤→❝ ré♥❣ ❝õ❛ K ❝â t➼♥❤ ❝❤➜t ✤✐➸♠ ❜➜t ố ợ ổ õ tỗ t↕✐ q✉ÿ ✤↕♦ ❧✐➯♥ tö❝ t → zt, < t < t❤ä❛ ♠➣♥ zt = tu + (1 − t)T zt✱ ✈ỵ✐ ❜➜t ❦ý u ∈ K ✱ ❤ë✐ tö ♠↕♥❤ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳ ✷✼ ◆➠♠ ✷✵✵✷✱ ❳✉ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ ❦➳t q✉↔ ❍❛❧♣❡r♥ ✤ó♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ✭❈✸✮✬ ✭❈✸✮ ❝õ❛ ▲✐♦♥s t❤❛② t❤➳ ❜➡♥❣ ✤✐➲✉ ❦✐➺♥ |αn − αn−1 | = n→∞ αn lim ◆➠♠ ✷✵✵✷ ❳✉ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❦➳t q s ỵ X ổ trỡ K t rộ ỗ ✤â♥❣ ❝õ❛ X, T : K → K ❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ ❋✐①(T ) = ∅✳ ❱ỵ✐ u, x0 K trữợ {n} [0, 1] t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❈✶✮✱ ✭❈✷✮ ✈➔ ✭❈✸✮✬✳ ❑❤✐ ✤â ❞➣② {xn} ①→❝ ✤à♥❤ ❜ð✐ x0 ∈ K, xn+1 := (1 − αn )T xn + αn u, n≥0 ❤ë✐ tư ♠↕♥❤ tỵ✐ x∗ ∈ ❋✐①(T )✳ ◆❤➟♥ ①➨t ✷✳✸✳✻✳ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✭❛✮ ◆➠♠ ✶✾✾✷✱ ❲✐tt♠❛♥ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ữủ ỵ t ❈→❝ ✤✐➲✉ ❦✐➺♥ ✭❈✸✮✬ ✈➔ ✭❈✹✮ ❦❤ỉ♥❣ t❤➸ s♦ s→♥❤✈ỵ✐ ♥❤❛✉✳ ❈❤➥♥❣ ❤↕♥✱ ❞➣② {αn} ①→❝ ✤à♥❤ ❜ð✐ n− 21 , αn := (n− 12 − 1)−1 , ♥➳✉ n ❧➫ ♥➳✉ n ❝❤➤♥, ✭❈✸✮✬ ♥❤÷♥❣ ❦❤ỉ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❈✹✮✳ ✭❜✮ ❍❛❧♣❡r♥ ❝❤➾ r❛ ✤✐➲✉ ❦✐➺♥ ✭❈✶✮ ✈➔ ✤✐➲✉ ❦✐➺♥ ✭❈✷✮ ❧➔ ❝➛♥ t❤✐➳t ❝❤♦ sü ❤ë✐ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ tö ❝õ❛ ❞➣② {xn } ①→❝ ✤à♥❤ tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭✷✳✶✼✮✳ ▼ët sè ❦➳t q✉↔ ✤➣ ❝❤➾ r❛ ♥➳✉ ❞➣② ❧➦♣ ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ ✭✷✳✶✼✮✱ T xn t❤❛② ❜➡♥❣ Tn xn := t❤➻ ❝→❝ ✤✐➲✉ ❦✐➺♥ n n−1 T k xn , k=0 ✭❈✶✮ ✈➔ ✭❈✷✮ ❧➔ ✤✐➲✉ ❦✐➺♥ ✤õ✳ ❇ê ✤➲ ✷✳✸✳✼✳∞ ❈❤♦ {an} ❧➔ ❞➣② sè t❤ü❝ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ an+1 ≤ an +σn, n ≤ s❛♦ ❝❤♦ σn < ∞✳ ❑❤✐ õ limn an tỗ t t {an} n=1 ❝â ❞➣② ❝♦♥ ❤ë✐ tö ✈➲ 0✱ t❤➻ an ❤ë✐ tö ✈➲ ❦❤✐ n → ∞✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ✤→♥❤ ❣✐→ ≤ an+1 ≤ an + σn, n ≥ 0✱ t❛ ❝â 0, ∞ n ≤ an+1 ≤ a1 + σn ≤ a1 + σn < ∞, n ≥ 0, ✷✽ ✈➻ ✈➟② ❞➣② {an } ❜à ❝❤➦♥✳ ◆❣♦➔✐ r❛✱ ❝è ✤à♥❤ m ∈ N✱ t❛ ❝â ≤ an+m ≤ an+m−1 + σn+m−1 ≤ an+m−2 + σn+m−2 + σn+m−1 : n+m−1 ≤ αn + σi i=1 ∞ ▲➜② ✧❧✐♠s✉♣✧ ❦❤✐ m → ∞✱ t❛ t❤✉ ✤÷đ❝ lim sup an ≤ an + m→∞ σi ✳ ❚✐➳♣ t❤❡♦✱ i=n ❧➜② ✧❧✐♠✐♥❢✧ ❦❤✐ n → ∞✱ t❛ ❝â lim sup an ≤ lim inf an ✳ ❱➻ ✈➟② n→∞ lim inf an = lim sup an , n n ợ tỗ t ◆➳✉ t❤➯♠ ✤✐➲✉ ❦✐➺♥ ❞➣② {an } ❝â ❞➣② ❝♦♥ ❤ë✐ tư ✈➲ 0✱ ✈➻ ❣✐ỵ✐ ❤↕♥ ❝õ❛ {an } tỗ t {an } tử n → ∞✳ ❇ê ✤➲ ✷✳✸✳✽✳ ❈❤♦ ❝→❝ ❞➣② {xn} ✈➔ {yn} ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✈➔ {βn} ⊂ [0, 1] ✈ỵ✐ < lim inf βn ≤ lim sup βn < 1✳ ●✐↔ sû n→∞ n→∞ xn+1 = βn yn + (1 − βn )xn ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ n ≥ ✈➔ lim sup( yn+1 − yn − xn+1 − xn ) ≤ n→∞ ❑❤✐ ✤â n→∞ lim yn − xn = 0✳ ❇ê ✤➲ ✷✳✸✳✾✳ ❈❤♦ {an} ❧➔ ❞➣② ❝→❝ sè t❤ü❝ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ an+1 ≤ (1 − αn )an + αn σn + γn , n ≥ 0, tr♦♥❣ ✤â (i) {αn } ⊂ [0, 1], ∞ n=1 αn = ∞❀ (ii) lim sup σn ≤ 0; n→∞ (iii) γn ≥ 0; (n ≥ 0), ∞ n=1 γn ❑❤✐ ✤â an → ❦❤✐ n → ∞✳ < ∞✳ ✷✾ ❚✐➳♣ t❤❡♦✱ ❣✐↔ sû ❞➣② {zt } ❤ë✐ tö ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ z ❝õ❛ T ❦❤✐ t → 0✱ tr♦♥❣ ✤â zt ❧➔ ♣❤➛♥ tû ❞✉② ♥❤➜t ❝õ❛ K t❤ä❛ ♠➣♥ zt = tu + (1 − t)T zt ợ t ý u K ỵ K t ỗ õ rộ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ X ✈ỵ✐ ❝❤✉➞♥ ❦❤↔ ✈✐ ●➙t❡❛✉① ✤➲✉ ✈➔ T : K → K ❧➔ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✈ỵ✐ ❋✐①(T ) = ∅✳ ❱ỵ✐ δ (0, 1) trữợ t S : K K ❝❤♦ ❜ð✐ Sx := (1 − δ)x + δT x, ∀x ∈ K ●✐↔ sû {αn} ⊂ (0, 1) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❈✶✮ ✈➔ ✭❈✷✮✳ ❱ỵ✐ u, x0 ∈ K ✱ ①➨t ❞➣② ❧➦♣ {xn} ①→❝ ✤à♥❤ ♥❤÷ s❛✉ xn+1 = αn u + (1 − αn )Sxn , ✭✷✳✶✾✮ n ≥ ❑❤✐ ✤â ❞➣② {xn} ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ →♥❤ ①↕ S ❦❤æ♥❣ ❣✐➣♥✱ ✈➔ ❝â ❝ò♥❣ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈ỵ✐ T ✳ ✣➦t βn := (1 − δ)αn + δ∀n ≥ 0; yn := xn+1 − xn + βn xn , βn n ≥ ✭✷✳✷✵✮ ❚❛ t❤➜② βn → δ ❦❤✐ n → ∞✱ ✈➔ ♥➳✉ ❞➣② {xn } ❜à ❝❤➦♥✱ t❤➻ ❞➣② {yn } ❝ô♥❣ ❜à ❝❤➦♥✳ ❳➨t x∗ ∈ ❋✐①(T ) = ❋✐①(S)✳ ❇➡♥❣ q✉② ♥↕♣ t❛ ❝â xn − x∗ ≤ max{ x0 − x∗ , u − x∗ } ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ n ≥ 0✱ ✈➻ ✈➟② {xn }, {yn }, {T xn } ✈➔ {Sxn } ❜à ❝❤➦♥✳ ❱➔ ❝â xn+1 − Sxn = αn u − Sxn → 0, ❦❤✐ n → ∞✳ ❚ø ❝→❝❤ ✤➦t βn ✈➔ S ✱ t❛ t❤✉ ✤÷đ❝ yn = (αn u + (1 − αn )δT xn ) βn ✈➻ ✈➟② αn+1 αn − u βn+1 βn (1 − αn+1 ) + δ T xn+1 − T xn βn+1 yn+1 − yn − xn+1 − xn ≤ ✭✷✳✷✶✮ ✸✵ + − αn+1 δ T xn − xn+1 − xn , βn+1 ❞♦ ✤â✱ tø {xn } {T xn } t t ữủ ợ ♠é✐ ❤➡♥❣ sè M1 > 0, ✈➔ M2 > 0, αn+1 αn − ||u|| βn+1 βn n→∞ (1 − αn+1 ) δ − M1 + βn+1 − αn+1 − αn − + δM2 ≤ βn+1 βn lim sup(||yn+1 − yn || − ||xn+1 − xn ||) ≤ lim sup n→∞ ❱➻ ✈➟②✱ t❤❡♦ ❇ê ✤➲ ✷✳✸✳✽ t❛ ❝â ||yn − xn || → ❦❤✐ n → ∞ ❉♦ ✤â✱ lim ||xn+1 − xn || = lim βn ||yn − xn || = n→∞ n→∞ t ủ ợ t t ữủ ||xn Sxn || → ❦❤✐ n → ∞ ✭✷✳✷✷✮ ❚✐➳♣ t❤❡♦✱ t❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ lim sup u − z, j(xn − z) ≤ n→∞ ✭✷✳✷✸✮ ❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ n ≥ 1✱ ①➨t tn ∈ (0, 1) s❛♦ ❝❤♦ tn → 0, ✈➔ ||xn − Sxn || → 0, tn n → ∞ ✭✷✳✷✹✮ ●✐↔ sû ztn ∈ K ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ →♥❤ ①↕ ❝♦ Stn ①→❝ ✤à♥❤ ❜ð✐ Stn x := tn u+(1−tn )Sx, x ∈ K ✳ ❑❤✐ ✤â ztn −xn = tn (u−xn )+(1−tn )(Sztn −xn )✳ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✸✮✱ t❛ ❝â ||ztn − xn ||2 ≤ (1 − tn )2 ||Sztn − xn ||2 + 2tn u − xn , j(ztn − zn ) ≤ (1 − tn )2 (||Sztn − Sxn || + ||Sxn − xn ||)2 + 2tn (||ztn − xn ||2 + u − ztn , j(ztn − xn ) ) ≤ (1 + t2n )||ztn − xn ||2 + ||Sxn − xn ||× (2||ztn − xn || + ||Sxn − xn ||) + 2tn u − ztn , j(ztn − xn ) , ✈➻ ✈➟② u − ztn , j(xn − ztn ) ≤ tn ||Sxn − xn || ||ztn − xn ||2 + × (2||ztn − xn || 2tn ✸✶ + ||Sxn − xn ||) ❱➻ ❝→❝ ❞➣② {xn }, {ztn } ✈➔ {Sxn } ❜à ❝❤➦♥ ✈➔ t❛ ❝â ||Sxn − xn || → 0, n → ∞✱ ♥➯♥ 2tn lim sup u − ztn , j(xn − ztn ) ≤ n→∞ ✭✷✳✷✺✮ ◆❣♦➔✐ r❛✱ u − ztn , j(xn − ztn ) = u − z, j(xn − z) + u − z, j(xn − ztn ) − j(xn − z) + z − ztn , j(xn − ztn ) ✭✷✳✷✻✮ ◆❤÷♥❣✱ t❤❡♦ ❣✐↔ t❤✐➳t ztn → z ∈ ❋✐①(S), n → ∞✳ ❙û ❞ö♥❣ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ {xn } t❛ t❤✉ ✤÷đ❝ z − ztn , j(xn − ztn ) → 0, n → ∞✳ ❚❛ ❝ô♥❣ ❝â✱ u − z, j(xn − ztn ) − j(xn − z) → 0, n → ∞✳ ❱➻ ✈➟②✱ t❛ t❤✉ ✤÷đ❝ ✭✷✳✷✺✮ ✈➔ ✭✷✳✷✻✮ lim sup u − z, j(xn − z) ≤ 0✳ ◆❣♦➔✐ r❛ tø ✭✷✳✶✾✮ t❛ ❝â n→∞ xn+1 − z = αn (u − z) + (1 − αn )(Sxn − z)✳ ❑❤✐ ✤â ||xn+1 − z||2 ≤ (1 − αn )2 ||Sxn − z||2 + 2αn u − z, j(xn+1 − z) ≤ (1 − αn )||xn − z||2 + αn σn , ✈ỵ✐ σn := u − z, j(xn+1 − z) ; γn ≡ ∀n ≥ ❱➻ ✈➟② t❤❡♦ ❇ê ✤➲ ✷✳✸✳✾✱ ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ◆❤➟♥ ①➨t ✷✳✸✳✶✶✳ ▼å✐ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉ ❝â ❝❤✉➞♥ ❦❤↔ t t rộ ỗ ✤â♥❣ ❜à ❝❤➦♥ X ❝â t➼♥❤ ❝❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ✤è✐ ✈ỵ✐ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ❳➨t Sn (x) := n q✉↔ s❛✉✳ n−1 S k x✱ ✈ỵ✐ S : K → K ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❚❛ õ t k=0 ỵ sỷ X ổ tỹ trỡ ỗ ợ u, x0 K trữợ t {xn} s xn+1 = αn u + (1 − αn )Sn xn , n ≥ ✭✷✳✷✼✮ ✸✷ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❈✶✮ ✈➔ ✭❈✷✮ t❤ä❛ ♠➣♥✳ ❑❤✐ ✤â ❞➣② {xn} ❤ë✐ tö ♠↕♥❤ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ S ✳ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♥❤÷♥❣ s r sỡ ữủ ổ ỗ ổ ỗ t t ỗ ỗ t t t t ✤✐➸♠ ❜➜t ✤ë♥❣✳ • ❉→♥❣ ✤✐➺✉ ❝õ❛ t✐➺♠ ❝➟♥ ❝❤➼♥❤ q✉②✱ ❦❤→✐ ♥✐➺♠ t✐➺♠ ❝➟♥ ❝❤➼♥❤ q✉②✱ t➼♥❤ ❝❤➼♥❤ q✉② ❝õ❛ t✐➺♠ ❝➟♥ ✈➔ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ t➼♥❤ ❝❤➼♥❤ q t ợ sỹ tỗ t t ✤ë♥❣ ❝õ❛ →♥❤ ①↕✳ ❙ü ❤ë✐ tö ♠↕♥❤ tr➻♥❤ ❜➔② tử tợ t ❙ü ❤ë✐ tö ②➳✉ tr➻♥❤ ❜➔② ✈➲ sü ❤ë✐ tö tợ t Pữỡ r ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❤ë✐ tư tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ✈➔ ❤ë✐ tư ♠↕♥❤ tỵ✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕✳ ✸✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t t ỗ ố ❬✷❪ ❍♦➔♥❣ ❚ö② ✭✷✵✵✸✮✱ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ❑❤♦❛ ❤å❝ ❑ÿ t❤✉➟t✳ ❬✶❪ ✣é ❱➠♥ ▲÷✉ ✭✷✵✵✵✮✱ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❇r♦✇❞❡r✱ ❋✳❊✳ ❛♥❞ P❡tr②s❤②♥✱ ❲✳❱✳ ✭✶✾✻✻✮✱ ✏❚❤❡ s♦❧✉t✐♦♥ ❜② ✐t❡r❛t✐♦♥ ♦❢ ♥♦♥❧✐♥❡❛r ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❇✉❧❧✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✼✷✱ ♣♣✳ ✺✼✶✕✺✼✺✳ ❬✹❪ ❈❤✐❞✉♠❡✱ ❈✳ ✭✷✵✵✾✮✱ ●❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ ◆♦♥❧✐♥✲ ❡❛r ■t❡r❛t✐♦♥s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✳ ❬✺❪ ●✐✉s❡♣♣❡✱ ▼✳ ✭✷✵✵✻✮✱ ✏❆ ❣❡♥❡r❛❧ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞ ❢♦r ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣✲ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✶✽✱ ♣♣✳ ✹✸✕✺✷✳ ❬✻❪ ●❧♦✇✐♥s❦✐✱ ❘✳ ✭✶✾✽✹✮✱ ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞s ❢♦r ◆♦♥❧✐♥❡❛r ❱❛r✐❛t✐♦♥❛❧ Pr♦❜✲ ❧❡♠s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✳ ♣✐♥❣s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✑✱ ❬✼❪ ■s❤✐❦❛✇❛✱ ❙✳ ✭✶✾✼✻✮✱ ✏❋✐①❡❞ ♣♦✐♥ts ❛♥❞ ✐t❡r❛t✐♦♥ ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣ ✐♥ ❛ ❇❛♥❛❝❤ s♣❛❝❡✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✼✸✱ ♣♣✳✻✶✕✼✶✳ ❬✽❪ ❑r❛s♥♦s❡❧s❦✐✙✱ ▼✳❆✳ ✭✶✾✺✺✮✱ ✏❚✇♦ ♦❜s❡r✈❛t✐♦♥s ❛❜♦✉t t❤❡ ♠❡t❤♦❞ ♦❢ s✉❝✲ ❯s♣❡❤✐ ▼❛t❤✳ ◆❛✉❦✱ ✶✵✱ ♣♣✳✶✷✸✕✶✷✼✳ ❬✾❪ ❑✐♥❞❡r❧❡❤r❡r✱ ❉✳✱ ❙t❛♠♣❛❝❝❤✐❛✱ ●✳ ✭✶✾✽✵✮✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✱ ❆❝❛❞❡♠✐❝ Pr❡ss✱ ■♥❝✳✱ ◆❡✇ ❨♦r❦✲ ❝❡ss✐✈❡ ❛♣♣r♦①✐♠❛t✐♦♥s✧✱ ▲♦♥❞♦♥✳ ✸✹ ✸✺ ❬✶✵❪ ◗✐♥❛✱ ❳✳✱ ❈❤♦✱ ❙✳❨✳✱ ▲✐♥ ❲✳ ✭✷✵✶✽✮✱ ✏❙tr♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❛♥ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠ ✐♥✈♦❧✈✐♥❣ ♥♦♥❧✐♥❡❛r ♠❛♣♣✐♥❣s ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ❛♥❞ ❛❝❝r❡t✐✈❡ t②♣❡✑✱ ❖♣t✐♠✐③❛t✐♦♥✱ ❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✵✽✵✴✵✷✸✸✶✾✸✹✳✷✵✶✽✳✶✹✾✶✾✼✸✳ ❬✶✶❪ ❙✉③✉❦✐✱ ❚✳ ✭✷✵✵✼✮✱ ✏❆ ❙✉❢❢✐❝✐❡♥t ❛♥❞ ◆❡❝❡ss❛r② ❈♦♥❞✐t✐♦♥ ❢♦r ❍❛❧♣❡r♥✲ ❚②♣❡ ❙tr♦♥❣ ❈♦♥✈❡r❣❡♥❝❡ t♦ ❋✐①❡❞ P♦✐♥ts ♦❢ ◆♦♥❡①♣❛♥s✐✈❡ ▼❛♣♣✐♥❣s✑✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✶✸✺✭✶✮✱ ♣♣✳ ✾✾✕✶✵✻✳ ❬✶✷❪ ❩❡✐❞❧❡r✱ ❊✳ ✭✶✾✽✺✮✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✳ ■■■✳ ❱❛r✐❛t✐♦♥❛❧ ▼❡t❤♦❞s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✳ ... TRƯỜNG ĐẠI HỌC KHOA HỌC - PHẠM QUANG DŨNG VỀ PHƯƠNG PHÁP LẶP TÌM ĐIỂM BẤT ĐỘNG CỦA ÁNH XẠ KHƠNG GIÃN TRONG KHƠNG GIAN BANACH Chun ngành: Tốn ứng dụng Mã số : 46 01 12 LUẬN VĂN