ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - VŨ THỊ THANH NGA MỘT ĐỊNH LÝ HỘI TỤ MẠNH GIẢI BÀI TOÁN CHẤP NHẬN TÁCH VÀ BÀI TỐN ĐIỂM BẤT ĐỘNG TRONG KHƠNG GIAN BANACH Chuyên ngành: Toán ứng dụng Mã số : 46 01 12 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC TS Trương Minh Tuyên TS Li ZhenYang THÁI NGUYÊN - 2019 ✐✐ ▲í✐ ❝↔♠ ì♥ ❚æ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ rữỡ ữớ t t ữợ ❞➝♥✱ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❝→❝ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✕ ❚✐♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✕✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ ❚r÷í♥❣✳ ◆❤➙♥ ❞à♣ ♥➔②✱ tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ỡ t tợ ỳ ữớ t tr ỗ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ởt số ỵ t tt ✤➛✉ ✶ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✸ ✶✳✶ ổ pỗ ổ trỡ ✤➲✉ ✳ ✳ ✳ ✶✳✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ ❙ü ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ỗ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ổ pỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✺ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ⑩♥❤ ①↕ ✤è✐ ♥❣➝✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❑❤♦↔♥❣ ❝→❝❤ ❇r❡❣♠❛♥ ✈➔ ♣❤➨♣ ❝❤✐➳✉ ❇r❡❣♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ❑❤♦↔♥❣ ❝→❝❤ ❇r❡❣♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ P❤➨♣ ❝❤✐➳✉ ❇r❡❣♠❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ ❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✺ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ ♠↕♥❤ tr→✐ ✸ ✸ ✹ ✻ ✾ ✶✶ ✶✸ ✶✻ ✶✻ ✶✼ ữỡ ởt ỵ tử ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✷✻ ✷✳✶ P❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❧❛✐ ❣❤➨♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✸ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ❑➳t ❧✉➟♥ ✹✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✶ ✐✈ ởt số ỵ t tt E ổ ❇❛♥❛❝❤ E∗ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ E R t➟♣ ❤ñ♣ ❝→❝ sè t❤ü❝ ∩ ♣❤➨♣ ❣✐❛♦ inf M ❝➟♥ ữợ ú t ủ số M sup M tr➯♥ ✤ó♥❣ ❝õ❛ t➟♣ ❤đ♣ sè M max M sè ❧ỵ♥ ♥❤➜t tr♦♥❣ t➟♣ ❤đ♣ sè M M sè ♥❤ä ♥❤➜t tr♦♥❣ t➟♣ ❤ñ♣ sè M ❛r❣♠✐♥x∈X F (x) t➟♣ ❝→❝ ✤✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ F tr➯♥ X ∅ t➟♣ ré♥❣ ∀x ✈ỵ✐ ♠å✐ x ❞♦♠(A) ♠✐➲♥ ❤ú✉ t tỷ A I t tỷ ỗ t Lp (Ω) ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❜➟❝ p tr➯♥ Ω lp ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② sè ❦❤↔ tê♥❣ ❜➟❝ p lim sup xn ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② số {xn } lim inf xn ợ ữợ ❞➣② sè {xn } xn → x0 ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ x0 x n ⇀ x0 ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x0 Jp →♥❤ ①↕ ố E () ổ ỗ ổ ❇❛♥❛❝❤ E ρE (τ ) ♠ỉ ✤✉♥ trì♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E F ix(T ) ❤♦➦❝ F (T ) t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T n→∞ n→∞ ✈ ✐♥tM ❡rr PC ♣r♦❥fC iC ♣❤➛♥ tr♦♥❣ ❝õ❛ t➟♣ ủ M s số trữợ tr C r C t ỗ C ✶ ▼ð ✤➛✉ ❈❤♦ C ✈➔ Q ❧➔ ❝→❝ t ỗ õ rộ ổ ❣✐❛♥ ❍✐❧❜❡rt H1 ✈➔ H2 ✱ t÷ì♥❣ ù♥❣✳ ❈❤♦ T : H1 −→ H2 ❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥✳ ❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✭❙❋P✮ ❝â ❞↕♥❣ ♥❤÷ s❛✉✿ ❚➻♠ ♠ët ♣❤➛♥ tû x∗ ∈ C s❛♦ ❝❤♦ T x∗ ∈ Q ✭✵✳✶✮ ❉↕♥❣ tê♥❣ q✉→t ❝õ❛ ❇➔✐ t♦→♥ ✭✵✳✶✮ ❧➔ ❜➔✐ t♦→♥ ✭✵✳✷✮✱ ❜➔✐ t♦→♥ ♥➔② ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ❈❤♦ Ci ✱ i = 1, 2, , N ✈➔ Qj ✱ j = 1, 2, , M t ỗ õ ❝õ❛ H1 ✈➔ H2 t÷ì♥❣ ù♥❣✳ −1 ❚➻♠ ♠ët ♣❤➛♥ tû x∗ ∈ S = ∩N (∩M i=1 Ci ∩ T j=1 Qj ) 6= ∅ ✭✵✳✷✮ ▼æ ❤➻♥❤ ❜➔✐ t P t ữủ ợ t ❝ù✉ ❜ð✐ ❨✳ ❈❡♥s♦r ✈➔ ❚✳ ❊❧❢✈✐♥❣ ❬✻❪ ❝❤♦ ♠æ ❤➻♥❤ ❝→❝ ❜➔✐ t♦→♥ ♥❣÷đ❝✳ ❇➔✐ t♦→♥ ♥➔② ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ❦❤ỉ✐ ♣❤ư❝ ❤➻♥❤ ↔♥❤ tr♦♥❣ ❨ ❤å❝✱ ✤✐➲✉ ❦❤✐➸♥ ❝÷í♥❣ ✤ë ①↕ trà tr♦♥❣ ✤✐➲✉ trà ❜➺♥❤ ✉♥❣ t❤÷✱ ❦❤ỉ✐ ♣❤ư❝ t➼♥ ❤✐➺✉ ✭①❡♠ ❬✸❪✱ ❬✹❪✮ ❤❛② ❝â t❤➸ →♣ ❞ö♥❣ ❝❤♦ ✈✐➺❝ ❣✐↔✐ ❝→❝ ❜➔✐ t tr t ỵ tt trỏ ỡ ❚❛ ❜✐➳t r➡♥❣ C = F (PC )✕t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø H1 ❧➯♥ C ✳ ❉♦ ✤â✱ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✭✵✳✶✮ ❧➔ ♠ët tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ t→❝❤✳ ❉↕♥❣ tê♥❣ q✉→t ❝õ❛ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ t→❝❤ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ❈❤♦ Ti : H1 −→ H1 ✱ i = 1, 2, , N ✈➔ Sj : H2 −→ H2 ✱ j = 1, 2, , M ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ H1 ✈➔ H2 ✱ t÷ì♥❣ ù♥❣✳ −1 ∩M ∅ ❚➻♠ ♣❤➛♥ tû x∗ ∈ S = ∩N j=1 F ix(Sj ) = i=1 F ix(Ti ) ∩ T ✭✵✳✸✮ ❈❤♦ ✤➳♥ ♥❛② ❇➔✐ t♦→♥ ✭✵✳✸✮ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✤➣ ✈➔ ✤❛♥❣ ❧➔ ❝❤õ ✤➲ t❤✉ ❤ót ♥❤✐➲✉ ữớ t tr ữợ q t ❝ù✉✳ ●➛♥ ✤➙②✱ ✤➣ ❝â ♠ët sè t→❝ ❣✐↔ ✤➲ ❝➟♣ ✤➳♥ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ t➻♠ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ♠ỵ✐ t➻♠ ♠ët ♥❣❤✐➺♠ ❝❤✉♥❣ ❝õ❛ ❇➔✐ t♦→♥ ✭✵✳✶✮ ❤❛② ✭✵✳✸✮ ✈➔ ❝→❝ ❧ỵ♣ ❜➔✐ t♦→♥ ❦❤→❝ ✭❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✱ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✳✳✳✮✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❚✉②❡♥ ❚✳▼✳ ✈➔ ❍❛ ✷ ◆✳❙✳ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✶✼❪ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❧❛✐ ❣❤➨♣ t➻♠ ♠ët ♥❣❤✐➺♠ ❝❤✉♥❣ ❝õ❛ ❇➔✐ t♦→♥ ✭✵✳✷✮ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ t♦→♥ tû ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ ♠↕♥❤ tr→✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣ ❝❤➼♥❤✿ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ ✤➲ ❝➟♣ ✤➳♥ ♠ët sè ✈➜♥ ổ ổ pỗ ✤➲✉✱ trì♥ ✤➲✉✱ →♥❤ ①↕ ✤è✐ ♥❣➝✉❀ ❦❤♦↔♥❣ ❝→❝❤ ❇r❡❣♠❛♥✱ ♣❤➨♣ ❝❤✐➳✉ ❇r❡❣♠❛♥❀ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈➔ ❜➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t♦→♥ tû ❇r❡❣♠❛♥ ❦❤æ♥❣ tr ữỡ ởt ỵ tử ♠↕♥❤ ❣✐↔✐ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❧✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ ❝õ❛ ❚✉②❡♥ ❚✳▼✳ ✈➔ ❍❛ ◆✳❙✳ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✶✼❪ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❧❛✐ ❣❤➨♣ t➻♠ ♠ët ♥❣❤✐➺♠ ❝❤✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t♦→♥ tû ❇r❡❣♠❛♥ ổ tr tr ổ pỗ ✈➔ trì♥ ✤➲✉✳ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ữỡ ỗ ử tr ✈➲ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ổ ỗ trỡ ✶✳✷ ❣✐ỵ✐ t❤✐➺✉ ✈➲ →♥❤ ①↕ ✤è✐ ♥❣➝✉ ❝❤✉➞♥ t➢❝✳ ▼ö❝ ✶✳✸ ✤➲ ❝➟♣ ✤➳♥ ❝→❝ ❦❤→✐ ♥✐➺♠ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ✈➔ ♣❤➨♣ ❝❤✐➳✉ tê♥❣ q✉→t ❝ị♥❣ ✈ỵ✐ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝❤ó♥❣✳ ▼ư❝ ✶✳✹ tr➻♥❤ ❜➔② ✈➲ t♦→♥ tû ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ t♦→♥ tû ❣✐↔✐ tê♥❣ q✉→t ✈➔ t♦→♥ tû ❣✐↔✐ ♠➯tr✐❝✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝ t➔✐ ổ pỗ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉ ✶✳✶✳✶ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ ❈❤♦ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ✈➔ X ∗ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ ♥â✳ ✣➸ ❝❤♦ ✤ì♥ ❣✐↔♥ ✈➔ t❤✉➟♥ t✐➺♥ ❤ì♥✱ ❝❤ó♥❣ tỉ✐ t❤è♥❣ ♥❤➜t sû ❞ư♥❣ ❦➼ ❤✐➺✉ k.k ✤➸ ❝❤➾ ❝❤✉➞♥ tr➯♥ X ✈➔ X ∗❀ ❣✐→ trà ❝õ❛ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ x∗ ∈ X ∗ t↕✐ ✤✐➸♠ x X ữủ ỵ hx, xi ♥❣❤➽❛ ✶✳✶✳✶✳ ❑❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✤÷đ❝ ❣å✐ ❧➔ ♣❤↔♥ ợ x E tỗ t↕✐ x ∈ E s❛♦ ❝❤♦ hx, x∗ i = hx∗ , x∗∗ i, ✈ỵ✐ ♠å✐ x∗ ∈ E ∗✳ ❱➼ ❞ư ✶✳✶✳✷✳ ▼å✐ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ lp ❤❛② Lp(Ω)✱ ✈ỵ✐ < p < ∞✱ ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ú ỵ t t ữợ ổ õ t t➻♠ t❤➜② tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✷❪✳ ✐✮ ◆➳✉ ổ X ỗ ổ t t ợ ổ ❣✐❛♥ ♣❤↔♥ ①↕ Y ✱ t❤➻ X ❝ơ♥❣ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ✐✐✮ ▼å✐ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕❀ ✐✐✐✮ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ♣❤↔♥ ①↕ ❦❤✐ ✈➔ ❝❤➾ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ E ∗ ❝õ❛ ♥â ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ✶✳✶✳✷ ❙ü ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ❉➣② {xn} tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ E ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư ②➳✉ ✈➲ ♠ët ♣❤➛♥ tû x ∈ E ✈➔ ✤÷đ❝ ỵ xn x lim hxn , x∗ i = hx, x∗ i, n→∞ ✈ỵ✐ ♠å✐ x∗ ∈ X ∗ ✳ ◆❤➟♥ ①➨t ✶✳✶✳✺✳ ◆➳✉ ❞➣② {xn} ❤ë✐ tö ♠↕♥❤ ✈➲ x✱ tù❝ ❧➔ kxn − xk → 0✱ t❤➻ ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x✳ ❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ❈❤➥♥❣ ❤↕♥✱ ①➨t ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt l2 ✱ ❞➣② {en } ①→❝ ✤à♥❤ ❜ð✐ en = (0, , 0, ✈à tr➼ t❤ù n , 0, ), ✈ỵ✐ ♠å✐ n ≥ 1✱ ❤ë✐ tư ②➳✉ ✈➲ ❦❤ỉ♥❣ ✭①❡♠ ❬✷❪✮✱ ♥❤÷♥❣ ❦❤ỉ♥❣ ❤ë✐ tư ♠↕♥❤ ✈➲ ❦❤ỉ♥❣ ✭✈➻ ken k = ✈ỵ✐ ♠å✐ n ≥ 1✮✳ ▼➺♥❤ ✤➲ ✶✳✶✳✻✳ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✱ ❞➣② {xn} ⊂ E ❤ë✐ tö ②➳✉ ✈➲ x ∈ E ✳ ❑❤✐ ✤â✱ ❞➣② {xn } ❜à ❝❤➦♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ n ≥ 1✱ ①➨t ❞➣② ♣❤✐➳♠ ❤➔♠ {Hxn } ⊂ E ∗∗ ①→❝ ✤à♥❤ ❜ð✐ hx∗ , Hxn i = hxn , x∗ i ✈ỵ✐ ♠å✐ x∗ ∈ E ∗ ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ x∗ ∈ E ∗ ✱ t❛ ❝â hx∗ , Hxn i = hxn , x∗ i → hx, x∗ i õ t q ỵ ợ ✤➲✉ ❇❛♥❛❝❤✲❙t❡♥❤❛✉①✶ ✱ t❛ ❝â sup kxn k = sup kHxn k < ∞ n ✶ ❈❤♦ n X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ Y ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ✈➔ {An } ⊂ L(X, Y )✳ ◆➳✉ ✈ỵ✐ ♠é✐ x ∈ X ✱ ❞➣② {An x} ❤ë✐ tö tr♦♥❣ Y ✱ t❤➻ supn kAn k < ∞ ✺ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✶✳✶✳✼✳ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✱ A ⊂ E ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ✈➔ {xn } ⊂ A t❤ä❛ ♠➣♥ xn ⇀ x✳ ❑❤✐ ✤â✱ xn → x✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû xn x✱ õ tỗ t > ởt ❝♦♥ {xnk } ⊂ {xn } s❛♦ ❝❤♦ kxnk − xk ≥ ε, ✭✶✳✶✮ ✈ỵ✐ ♠å✐ k ≥ 1✳ ❱➻ {xnk } ⊂ A ✈➔ A ❧➔ t➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ố tỗ t {xnkl } {xnk } s❛♦ ❝❤♦ xnkl → y ✳ ❱➻ sü ❤ë✐ tö ♠↕♥❤ ❦➨♦ t❤❡♦ ❤ë✐ tö ②➳✉ ♥➯♥ xnkl ⇀ y ✈➔ ❞♦ ✤â y = x✳ ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✮✱ t❤❛② xnk ❜ð✐ xnkl t❛ ✤÷đ❝ kxnkl − yk ≥ ε, ♠➙✉ t❤✉➝♥ ✈ỵ✐ xnkl ⇀ y ✳ ❱➟② xn → x✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tữớ sỷ t t ữợ ổ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕✳ ▼➺♥❤ ✤➲ ✶✳✶✳✽✳ ✭①❡♠ ❬✷❪ tr❛♥❣ ✹✶✮ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✐✮ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ✐✐✮ ▼å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ E ✱ ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ②➳✉✳ ữợ t ố ỳ t➟♣ ✤â♥❣ ✈➔ t➟♣ ✤â♥❣ ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳ ▼➺♥❤ ✤➲ ✶✳✶✳✾✳ ◆➳✉ C ❧➔ t➟♣ ỗ õ rộ ổ ổ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ X ✱ t❤➻ C ❧➔ t➟♣ ✤â♥❣ ②➳✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤↔♥ ự sỷ tỗ t {xn } C s❛♦ ❝❤♦ xn ⇀ x✱ ♥❤÷♥❣ x ∈ / C ỵ t t ỗ tỗ t x ∈ X ∗ t→❝❤ ♥❣➦t x ✈➔ C ✱ tù❝ tỗ t > s hy, x i ≤ hx, x∗ i − ε, ✷✵ + hz − P iC x, Jp (ΠC x)i + hz − ΠC x, Jp (ΠC x) − Jp (x)i ≥ ⇔ (kzkp − kΠC xkp ) − hz, Jp (x)i + hΠC x, Jp (x)i ≥ p 1 ⇔ (kzkp − kxkp ) − hz − x, Jp (x)i ≥ (kΠC xkp − kxkp ) − hΠC x − x, Jp (x)i p p ⇔Df (z, x) ≥ Df (ΠC x, x) ❙✉② r❛ ΠC x ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❇r❡❣♠❛♥ ❝õ❛ x ❧➯♥ C ✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû ΠC x ❧➔ ❤➻♥❤ ❝❤✐➳✉ ❇r❡❣♠❛♥ ❝õ❛ x ❧➯♥ C ✳ ❑❤✐ ✤â✱ t❛ ❝â Df (ΠC x, x) ≤ Df (z, x) ✈ỵ✐ ♠å✐ z ∈ C C t ỗ z, C x ∈ C ✱ ♥➯♥ zt = tz + (1 − t)ΠC x ∈ C ✈ỵ✐ ♠å✐ t ∈ (0, 1)✳ ❉♦ ✤â Df (ΠC x, x) ≤ Df (zt , x) ợ t (0, 1) tữỡ ữỡ ợ 1 (kC xkp kxkp ) hΠC x − x, Jp (x)i ≤ (kzt kp − kxkp ) − hz − x, Jp (x)i p p ⇔ (kΠC xkp − kzt kp ) + thz − ΠC x, Jp (x)i ≤ p ⇔Df (ΠC x, zt ) + thz − ΠC x, Jp (x) − Jp (zt )i ≤ ❱➻ Df (ΠC x, zt) ≥ ✈➔ t > 0✱ ♥➯♥ t❛ ❝â hz − ΠC x, Jp (x) − Jp (zt )i ≤ ❈❤♦ t → 0+ t❛ ♥❤➟♥ ✤÷đ❝ hz − ΠC x, Jp (x) − Jp (ΠC x)i ≤ ữủ ự ú ỵ ❚ø ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ❇r❡❣♠❛♥✱ t❛ ❝â ∆p (ΠC x, z) ≤ ∆p (x, z) − ∆p (x, ΠC x), ∀z ∈ C ✭✶✳✶✺✮ ✐✐✮ ◆➳✉ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ f (x) = 21 kxk2✱ t❤➻ r tữỡ ự ợ f trũ ợ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝✳ ✷✶ ✶✳✹ ❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ C Q t ỗ õ ✈➔ ❦❤→❝ ré♥❣ ❝õ❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈➔ F ✱ t÷ì♥❣ ù♥❣✳ ❈❤♦ A : E −→ F ❧➔ ♠ët t♦→♥ tû t✉②➳♥ ❜à ❝❤➦♥ A∗ : F ∗ → E ∗ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ A✳ ❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✭❙❋P✮ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ❚➻♠ ♠ët ♣❤➛♥ tû x∗ ∈ S = C ∩ A−1 (Q) 6= ∅ ✭❙❋P✮ ❉↕♥❣ tê♥❣ q✉→t ❝õ❛ ❇➔✐ t♦→♥ ✭❙❋P✮ ❧➔ ❜➔✐ t♦→♥ ✭▼❙❙❋P✮✱ ❜➔✐ t♦→♥ ♥➔② ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ❈❤♦ Ci ✱ i = 1, 2, , N ✈➔ Qj ✱ j = 1, 2, , M ❧➔ ❝→❝ t ỗ õ E F tữỡ ù♥❣✳ −1 M ❚➻♠ ♠ët ♣❤➛♥ tû x∗ ∈ S = ∩N i=1 Ci ∩ A (∩j=1 Qj ) 6= ∅ ✭▼❙❙❋P✮ ▼æ ❤➻♥❤ ❜➔✐ t♦→♥ ✭❙❋P✮ ❧➛♥ ✤➛✉ t✐➯♥ ữủ ợ t ự sr ❚✳ ❊❧❢✈✐♥❣ ❬✻❪ ❝❤♦ ♠ỉ ❤➻♥❤ ❝→❝ ❜➔✐ t♦→♥ ♥❣÷đ❝✳ ❇➔✐ t♦→♥ ♥➔② ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ❦❤æ✐ ♣❤ư❝ ❤➻♥❤ ↔♥❤ tr♦♥❣ ❨ ❤å❝✱ ✤✐➲✉ ❦❤✐➸♥ ❝÷í♥❣ ✤ë ①↕ trà tr♦♥❣ ✤✐➲✉ trà ❜➯♥❤ ✉♥❣ t❤÷✱ ❦❤ỉ✐ ♣❤ư❝ t➼♥ ❤✐➺✉ ❤❛② ❝â t❤➸ →♣ ❞ö♥❣ ❝❤♦ ✈✐➺❝ ❣✐↔✐ t tr t ỵ tt trá ❝❤ì✐✳ ❑❤✐ E ✈➔ F ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ ✤➸ ❣✐↔✐ t P ữỡ ợ ữỡ ❈◗✱ ❇➔✐ t♦→♥ ✭❙❋P✮ ✤÷đ❝ ✤÷❛ ✈➲ ❜➔✐ t♦→♥ t➻♠ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ PC I − γT ∗ (I − PQ )T ✱ tr♦♥❣ ✤â γ > 0✱ PC ✈➔ PQ ❧➛♥ ❧÷đt ❧➔ ❝→❝ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø E ❧➯♥ C ✈➔ tø F ❧➯♥ Q✱ t÷ì♥❣ ù♥❣✳ ❚❛ ❜✐➳t r➡♥❣ ♥➳✉ γ ∈ 0, ∗ I − γT (I − P ✱ t❤➻ P Q T ❧➔ ♠ët →♥❤ ①↕ C kT k2 ❦❤ỉ♥❣ ❣✐➣♥✳ ❉♦ ✤â✱ ♥❣÷í✐ t❛ ❝â t❤➸ ✈➟♥ ❞ư♥❣ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✭♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥✱ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❍❛❧♣❡r♥✱ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❣➢♥ ❦➳t✮ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❇➔✐ t♦→♥ ✭❙❋P✮✳ ❳✉ ❬✷✵❪ ✤➣ ữ r ự t q ữợ rữợ ❤➳t ỉ♥❣ ❝❤➾ r❛ sü ❤ë✐ tư ②➳✉ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❈◗ ✈➲ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❇➔✐ t♦→♥ ✭❙❋P✮✳ ✣à♥❤ ỵ 0, kT k2 t❤➻ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ x1 ∈ E ✈➔ xn+1 = PC I − γT ∗ (I − PQ )T xn ❤ë✐ tö ②➳✉ ✈➲ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭❙❋P✮✳ ✷✷ ❙ü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ▼❛♥♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✤÷đ❝ ỵ ữợ ỵ ❈❤♦ ❞➣② {αn} ⊂ [0, 4/(2 + γkT k2)] t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ∞ X αn n=1 ◆➳✉ γ ∈ 0, kT k2 − αn + γkT k2 = ∞ t❤➻ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ x1 ∈ E ✈➔ xn+1 = (1 − αn )xn + αn PC I − γT ∗ (I − PQ )T xn , ❤ë✐ tö ②➳✉ ✈➲ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭❙❋P✮✳ ◆➠♠ ✷✵✵✻✱ ❳✉ ❬✶✾❪ ✤➣ ✤÷❛ r❛ ❝→❝ t❤✉➟t t♦→♥ ♠ð rë♥❣ ữỡ ữợ t P rữợ t ổ ự sỹ tử ữỡ Pr t P ỵ ✈ỵ✐ βj > ✈ỵ✐ ♠å✐ j = 1, 2, , M ✈➔ ❬✶✾❪ ◆➳✉ γ ∈ 0, L PM L = kT k2 j=1 βj ✱ t❤➻ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ x1 ∈ E ✈➔ xn+1 = PCN (I − γ M X ∗ βj T (I − PQj )T PC1 (I − γ M X βj T ∗ (I − PQj )T )xn j=1 j=1 ❤ë✐ tö ②➳✉ ✈➲ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❇➔✐ t♦→♥ ✭▼❙❙❋P✮✳ ❳✉ ❝ô♥❣ ✤➣ ①➙② ❞ü♥❣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ s♦♥❣ s♦♥❣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ①♦❛② ỏ t P ữợ ỵ 0, ✈ỵ✐ βj > ✈ỵ✐ ♠å✐ j = 1, 2, , M ✱ L PN PM L = kT k2 j=1 βj ✈➔ λi > t❤ä❛ ♠➣♥ i=1 λi = 1✱ t❤➻ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ x1 ∈ E ✈➔ xn+1 = N X i=1 λi PCi (I − γ M X βj T ∗ (I − PQj )T )xn j=1 ❤ë✐ tö ②➳✉ ởt t P ỵ ✶✳✹✳✺✳ ❬✶✾❪ ◆➳✉ γ ∈ 0, ✈ỵ✐ βj > ✈ỵ✐ ♠å✐ j = 1, 2, , M ✈➔ L PM L = kT k2 j=1 βj ✱ t❤➻ ❞➣② {xn } ①→❝ ✤à♥❤ ❜ð✐ x1 ∈ E ✈➔ xn+1 = PC[n+1] (I − γ M X βj T ∗ (I − PQj )T )xn j=1 ❤ë✐ tö ②➳✉ ✈➲ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❇➔✐ t♦→♥ ✭▼❙❙❋P✮✳ ❑❤✐ E ✈➔ F ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ pỗ trỡ ✤➣ ✤÷❛ r❛ ♠ët ❝↔✐ t✐➳♥ ❝❤♦ t❤✉➟t t♦→♥ ❝õ❛ r ự ởt ỵ tử ♠↕♥❤ ❣✐↔✐ ❜➔✐ t♦→♥ ✭▼❙❙❋P✮✳ ❱ỵ✐ ♠é✐ n ∈ N✱ ❲❛♥❣ ✤➣ ①→❝ ✤à♥❤ ❞➣② →♥❤ ①↕ {Tn } ❜ð✐ Tn (x) = Π ≤ i(n) ≤ N, Ci(n) (x) J ∗ [J (x) − t A∗ J (I − P n p Qi(n)−N )A(x)] N + ≤ i(n) ≤ N + M, q p tr♦♥❣ ✤â i : N → {1, 2, , N } ❧➔ →♥❤ ①↕ ✤✐➲✉ ❦❤✐➸♥ ①♦❛② ✈á♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ i(n) = n ♠♦❞ (N + M ) + ✈➔ tn t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ < t ≤ tn ≤ q Cq kAkq 1/(q−1) , ợ Cq ữủ tr ✶✳✷✳✺✳ ❲❛♥❣ ✤➣ ✤➲ ①✉➜t t❤✉➟t t♦→♥ s❛✉✿ ❱ỵ✐ ♠é✐ ♣❤➛♥ tû ❜❛♥ ✤➛✉ x0 = x¯✱ ①→❝ ✤à♥❤ ❞➣② {xn } ❜ð✐ yn = Tn (xn ) D = {w ∈ E : ∆ (y , w) ≤ ∆ (x , w)} n p n p n En = {w ∈ E : hxn − w, Jp (¯ x) − Jp (xn )i ≥ 0} x x), n+1 = ΠDn ∩En (¯ ✭✶✳✶✼✮ p tr♦♥❣ õ p r tữỡ ự ợ sè f (x) = kxkp ✱ ΠC ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❇r❡❣♠❛♥ ✈➔ Jp ❧➔ →♥❤ ①↕ ✤è✐ ♥❣➝✉✳ ❙ü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✭✶✳✶✼✮ ✤÷đ❝ ❝❤♦ ❜ð✐ ỵ ữợ ỵ {xn} ✤à♥❤ ❜ð✐ t❤✉➟t t♦→♥ ✭✶✳✶✼✮ ❤ë✐ tö ♠↕♥❤ ✈➲ ❤➻♥❤ ❝❤✐➳✉ ❇r❡❣♠❛♥ ΠS x¯ ❝õ❛ x¯ ❧➯♥ t➟♣ ♥❣❤✐➺♠ S ✳ ✷✹ ✶✳✺ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ ♠↕♥❤ tr→✐ p ❈❤♦ C ởt t ỗ t f ợ f (x) = kxkp ✱ ≤ p < ∞ ✈➔ T ❧➔ ♠ët →♥❤ ①↕ tø C ✈➔♦ ❝❤➼♥❤ ♥â✳ ▼ët ♣❤➛♥ tû p t❤✉ë❝ ❜❛♦ ✤â♥❣ ❝õ❛ C ✤÷đ❝ ❣♦✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ t✐➺♠ ❝➟♥ ❝õ❛ T ✭①❡♠ ❬✽❪✱ ❬✶✸❪✮ ♥➳✉ C ❝❤ù❛ ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ p s❛♦ ❝❤♦ lim kxn − T (xn )k = 0✳ ❚➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ t✐➺♠ ❝➟♥ T ữủ n ỵ F (T ) ❚♦→♥ tû T ✤÷đ❝ ❣å✐ ❧➔ ❇r❡❣♠❛♥ ❦❤ỉ♥❣ ❣✐➣♥ ♠↕♥❤ tr t tt tữỡ ự ợ t ❜➜t ✤ë♥❣ t✐➺♠ ❝➟♥ Fˆ (T ) ❦❤→❝ ré♥❣✱ ♥➳✉ ∆p (T x, p) ≤ ∆p (x, p), ✭✶✳✶✽✮ ✈ỵ✐ ♠å✐ p ∈ Fˆ (T )✱ x ∈ C ✈➔ ❦❤✐ {xn } ⊂ C ❧➔ ♠ët ❞➣② ❜à ❝❤➦♥✱ p ∈ Fˆ (T ) t❤ä❛ ♠➣♥ lim (∆p (xn , p) − ∆p (T (xn ), p)) = 0, ✭✶✳✶✾✮ n→∞ t❤➻ t❛ ❝â lim ∆p (T (xn ), xn ) = n→∞ ✭✶✳✷✵✮ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ ♠↕♥❤ tr→✐ ❝â ự tr ỵ tt tố ữ õ ❧ỵ♣ ❜➔✐ t♦→♥ ♥➔② ✤➣ t❤✉ ❤ót sü q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐✳ ◆➠♠ ✷✵✶✻✱ ❙❤❡❤✉ ❡t ❛❧✳ ❬✶✻❪ ①➙② ❞ü♥❣ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ♠ỵ✐ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ s❛✉✿ ❚➻♠ ♠ët ♣❤➛♥ tû x∗ ∈ C ∩ A−1 (Q) ∩ F (T ) ✭✶✳✷✶✮ tr♦♥❣ ✤â T ❧➔ ♠ët →♥❤ ①↕ ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ ♠↕♥❤ tr→✐ tø C ✈➔♦ ❝❤➼♥❤ ♥â C T = I ỗ ♥❤➜t✱ t❤➻ F (T ) = C ✈➔ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②✱ ❇➔✐ t♦→♥ ✭✶✳✷✶✮ trð t❤➔♥❤ ❜➔✐ t♦→♥ ✭❙❋P✮✳ ự t q s ỵ E F ổ pỗ ✤➲✉ ✈➔ trì♥ ✤➲✉✳ ❈❤♦ C ✈➔ Q ❧➔ ❝→❝ t ỗ õ rộ E F ✱ t÷ì♥❣ ù♥❣✱ A : E → F ❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ ✈➔ A∗ : F ∗ → E ∗ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ A✳ ●✐↔ sû ❜➔✐ t♦→♥ ❙❋P ✭❙❋P✮ ❝â t➟♣ ♥❣❤✐➺♠ S ❦❤→❝ ré♥❣✳ ❧➔ ♠ët →♥❤ ①↕ ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ ♠↕♥❤ tr→✐ tø C ✈➔♦ ❝❤➼♥❤ ♥â C t❤ä❛ ♠➣♥ F (T ) = Fˆ (T ) ✈➔ F (T ) ∩ S 6= ∅✳ ❈❤♦ {αn } ❧➔ ♠ët ❞➣② sè tr♦♥❣ ❦❤♦↔♥❣ (0, 1)✳ ❱ỵ✐ ♠é✐ u ∈ E1 ❝è ✤à♥❤✱ ❝❤♦ {xn } ✷✺ ❧➔ ❞➣② ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ u1 ∈ E1 x = Π J [J (u ) − t A∗ J (I − P )A(u )] n C q p n n p Q n u n ≥ n+1 = ΠC Jq [αn Jp (u) + (1 − αn )Jp T (xn )], ✭✶✳✷✷✮ ●✐↔ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✐✮ ✐✐✮ lim αn = 0✱ n→∞ ∞ X αn = ∞, n=1 ✐✐✐✮ < t ≤ tn ≤ k < q Cq kAkq 1/(q−1) ❑❤✐ ✤â✱ ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ ♠ët ♣❤➛♥ tû x∗ ∈ F (T ) ∩ S ✱ ð ✤➙② x∗ = ΠF (T )∩S u✳ ✷✻ ❈❤÷ì♥❣ ✷ ởt ỵ tử t ♥❤➟♥ t→❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❧✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ tr➻♥❤ ❜➔② ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❝❤✐➳✉ t➻♠ ♥❣❤✐➺♠ ❝❤✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ t♦→♥ tû ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ tr→✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ tø t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶✼❪✳ ✷✳✶ P❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t❛ ①➨t ❜➔✐ t♦→♥ t➻♠ ♠ët ♣❤➛♥ tû x† s❛♦ ❝❤♦ x† ∈ S = \ N i=1 Ci \ \ M j=1 A−1 (Qj ) \ \ K k=1 F (Tk ) 6= ∅, ✭✷✳✶✮ tr♦♥❣ õ Ci Qj t ỗ õ rộ ổ pỗ trì♥ ✤➲✉ E ✈➔ F ✱ t÷ì♥❣ ù♥❣✱ F (Tk ) ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ ♠↕♥❤ tr→✐ Tk : E −→ E t❤ä❛ ♠➣♥ Fˆ (Tk ) = F (Tk )✱ ✈➔ A : E −→ F ❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥✳ ◆❤➟♥ ①➨t ✷✳✶✳✶✳ ❛✮ ◆➳✉ E = F ✱ Ci = Qj = E ✈➔ A ❧➔ →♥❤ ỗ t tr E t t trð t❤➔♥❤ ❜➔✐ t♦→♥ t➻♠ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ t♦→♥ tû ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ tr Tk ỗ ♥❤➜t tr➯♥ E ✈ỵ✐ ♠å✐ k = 1, 2, , K ✱ t❤➻ ❜➔✐ t♦→♥ ✭✷✳✶✮ trð t❤➔♥❤ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✭▼❙❙❋P✮✳ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❧❛✐ ❣❤➨♣ ✣➸ ❣✐↔✐ ❇➔✐ t♦→♥ ✭✷✳✶✮✱ ❝→❝ t→❝ ❣✐↔ ❚✳▼✳ ❚✉②❡♥ ✈➔ ◆✳❙✳ ❍❛ ✤➣ ✤÷❛ r❛ ♣❤÷ì♥❣ ữợ Pữỡ ợ ộ ♣❤➛♥ tû ❜❛♥ ✤➛✉ x0 = x ∈ E ✱ ①→❝ ✤à♥❤ ❞➣② {xn } ❜ð✐ yi,n = ΠCi xn , i = 1, 2, , N, ❈❤å♥ in s❛♦ ❝❤♦ ∆p(yi ,n, xn) = i=1, ,N max ∆p (yi,n , xn ), ✤➦t yn = yi ,n , n n zj,n = Jq∗ [Jp (yn ) − tn A∗ Jp (I − PQj )A(yn )], j = 1, 2, , M ❈❤å♥ jn s❛♦ ❝❤♦ ∆p(zj ,n, yn) = j=1, ,M max ∆p (zj,n , yn ), ✤➦t zn = zj ,n , n n tk,n = Tk (zn ), k = 1, 2, , K, ❈❤å♥ kn s❛♦ ❝❤♦ ∆p(tk ,n, zn) = k=1, ,K max ∆p (tk,n , zn ), ✤➦t tn = tk ,n , n n Hn = {z ∈ E : ∆p (tn , z) ≤ ∆p (zn , z) ≤ ∆p (yn , z) ≤ ∆p (xn , z)}, Dn = {z ∈ E : hxn − z, Jp (x0 ) − Jp (xn )i ≥ 0}, xn+1 = ΠHn ∩Dn (x0 ), n ≥ 0, tr♦♥❣ ✤â ❞➣② sè {tn} t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✻✮✳ ❚r♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶✼❪✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✷✳✶✱ ❝→❝ t→❝ ❣✐↔ ❚✳▼✳ ❚✉②❡♥ ✈➔ ◆✳❙✳ ❍❛ ✤➣ ❧➛♥ ❧÷đt ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ♠➺♥❤ ✤➲ ữợ r Pữỡ t❛ ❝â S ⊂ Hn ∩ Dn ✈ỵ✐ ♠å✐ n rữợ t t Hn Dn t ỗ õ E u ∈ S ✱ t❛ ❝â ❈❤ù♥❣ ♠✐♥❤✳ ∆p (tn , u) = ∆p (Tkn (zn ), u) ≤ ∆p (zn , u) ✭✷✳✷✮ ❚ø t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ❇r❡❣♠❛♥ ✭✶✳✶✺✮✱ t❛ ❝â ∆p (yn , u) = ∆p (ΠCin (xn ), u) ≤ ∆p (xn , u) ✭✷✳✸✮ ✷✽ ❇➙② ❣✐í✱ t❛ ❝❤➾ r❛ ∆p(zn, u) ≤ ∆p(yn, u)✳ ✣➦t wn = A(yn) − PQ ✤â t❛ ❝â jn A(yn )✳ ❑❤✐ zn = Jq∗ (Jp (yn ) − tn A∗ Jp (wn )) ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ Jp ✈➔ ✭✶✳✶✹✮✱ t❛ ❝â hA(yn ) − A(u), Jp (wn )i = kA(yn ) − PQjn A(yn )kp + hPQjn A(yn ) − A(u), Jp (wn )i ✭✷✳✹✮ ≥ kwn kp ❉♦ ✤â✱ tø ▼➺♥❤ ✤➲ ✶✳✷✳✺ ✈➔ ✭✷✳✹✮✱ t❛ ♥❤➟♥ ✤÷đ❝ ∆p (zn , u) = ∆p (Jq∗ (Jp (yn ) − tn A∗ Jp (wn )), u) = kJp (yn ) − tn A∗ Jp (wn )kq − hu, Jp (yn )i q + tn hA(u), Jp (wn )i + kukp p Cq (tn kAk)q ≤ kJp (yn )kq − tn hAyn , Jp (wn )i + kJp (wn )kq q q − hu, Jp (yn )i + tn hAu, Jp (wn )i + kukp p 1 = kyn kq − hu, Jp (yn )i + kukp + tn hA(u) − A(yn ), Jp (wn )i q p q Cq (tn kAk) kwn kq + q Cq (tn kAk)q = ∆p (yn , u) + tn hA(u) − A(yn ), Jp (wn )i + kwn kq q q Cq (tn kAk) ≤ ∆p (yn , u) − (tn − )kwn kp q ❚ø ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✻✮✱ t❛ t❤✉ ✤÷đ❝ ∆p (zn , u) ≤ ∆p (yn , u) ✭✷✳✺✮ ❉♦ ✈➟②✱ tø ✭✷✳✷✮✱ ✭✷✳✸✮ ✈➔ ✭✷✳✺✮✱ s✉② r❛ u ∈ Hn✳ ❱➻ ✈➟② S ⊂ Hn ✈ỵ✐ ♠å✐ n ≥ 0✳ ❈✉è✐ ❝ị♥❣ t❛ ❝❤➾ r❛ S ⊂ Dn ✈ỵ✐ ♠å✐ n ≥ 0✳❚❤➟t ✈➟②✱ ✈➻ D0 = E ✱ ♥➯♥ S ⊂ D0 ✳ ●✐↔ sû S ⊂ Dn ✈ỵ✐ n ≥ ♥➔♦ ✤â✱ ❦❤✐ ✤â S ⊂ Hn ∩ Dn ✳ ❉♦ ✤â✱ tø xn+1 = ΠH ∩D (x0 ) ✈➔ ✭✶✳✶✹✮✱ t❛ ❝â n n hxn+1 − u, Jp (x0 ) − Jp (xn+1 )i ≥ 0, ✷✾ ✤✐➲✉ ♥➔② s✉② r❛ u ∈ Dn+1✳ ❜➡♥❣ q✉② ♥↕♣ t♦→♥ ❤å❝✱ t❛ ♥❤➟♥ ✤÷đ❝ S ⊂ Dn ợ n ữủ ự ▼➺♥❤ ✤➲ ✷✳✷✳✷✳ ❚r♦♥❣ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✷✳✶✱ t❛ ❝â xn+1 − xn → ❦❤✐ n → ∞✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✶✱ s✉② r❛ ❞➣② {xn} ❧➔ ❤♦➔♥ t♦➔♥ ①→❝ ✤à♥❤✳ ❈è ✤à♥❤ u ∈ S ✳ ❚ø xn+1 = ΠH ∩D (x0) ✈➔ ✭✶✳✶✺✮ s✉② r❛ ❈❤ù♥❣ ♠✐♥❤✳ n n ✭✷✳✻✮ ∆p (xn+1 , u) ≤ ∆p (x0 , u) ❉♦ ✤â✱ ❞➣② {∆p(xn, u)} ❜à ❝❤➦♥✳ ❱➻ ✈➟②✱ tø ✭✶✳✶✷✮✱ s✉② r❛ ❞➣② {xn} ❝ô♥❣ ❜à ❝❤➦♥✳ ❚✐➳♣ t❤❡♦✱ tø xn+1 ∈ Dn ✈➔ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ t➟♣ ❤ñ♣ Dn✱ t❛ ❝â hxn − xn+1 , Jp (x0 ) − Jp (xn )i ≥ ✭✷✳✼✮ hxn − x0 , Jp (x0 ) − Jp (xn )i ≥ hxn+1 − x0 , Jp (x0 ) − Jp (xn )i ✭✷✳✽✮ ❉♦ ✈➟②✱ t❛ ♥❤➟♥ ✤÷đ❝ ❉♦ ✤â✱ tø ✭✶✳✶✷✮✱ t❛ ❝â ✭✷✳✾✮ hxn+1 − x0 , Jp (x0 ) − Jp (xn )i ≥ ∆p (xn , x0 ) + ∆p (x0 , xn ) ❱➻ ✈➟②✱ tø ✭✶✳✶✶✮✱ t❛ ♥❤➟♥ ✤÷đ❝ −∆p (xn , xn+1 ) + ∆p (xn , x0 ) + ∆p (x0 , xn+1 ) ≥ ∆p (xn , x0 ) + ∆p (x0 , xn ) tữỡ ữỡ ợ p (x0 , xn+1 ) ≥ ∆p (x0 , xn ) + ∆p (xn , xn+1 ), ✭✷✳✶✵✮ s✉② r❛ {∆p(x0, xn)} ❧➔ ❞➣② t➠♥❣✳ ❉♦ ✤â✱ tø t➼♥❤ ❜à ❝❤➦♥ {p(x0, xn)} tỗ t ợ ỳ a = lim ∆p (x0 , xn ) n→∞ ❱➻ ✈➟②✱ tø ✭✷✳✶✵✮✱ t❛ t❤✉ ✤÷đ❝ n→∞ lim ∆p (xn , xn+1 ) = 0✳ ❚ø ✭✶✳✶✷✮ s✉② r❛ lim kxn+1 − xn k = n→∞ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✸✵ ▼➺♥❤ ✤➲ ✷✳✷✳✸✳ ❚r♦♥❣ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✷✳✶✱ ❝→❝ ❞➣② {xn − yn }✱ {xn − zn } ✈➔ {xn − tn } ❤ë✐ tö ✈➲ ❦❤✐ n → ∞✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ xn+1 ∈ Hn ✱ ♥➯♥ t❛ ❝â ∆p (tn , xn+1 ) ≤ ∆p (zn , xn+1 ) ≤ ∆(yn , xn+1 ) ≤ ∆(xn , xn+1 ) ❉♦ ✤â✱ tø ▼➺♥❤ ✤➲ ✷✳✷✳✷ ✭∆(xn , xn+1 ) → 0✮✱ t❛ t❤✉ ✤÷đ❝ ∆p (tn , xn+1 ) → 0, ∆p (zn , xn+1 ) → 0, ∆(yn , xn+1 ) → ❚ø ✭✶✳✶✷✮ s✉② r❛ kxn+1 − tn k → 0, kxn+1 − zn k → 0, kxn+1 − yn k → ❦➳t ❤đ♣ ✈ỵ✐ kxn+1 − xn k → 0✱ t❛ ♥❤➟♥ ✤÷đ❝ xn − tn → 0, xn − zn → 0, ✈➔ xn − yn → ▼➺♥❤ ✤➲ ✷✳✷✳✹✳ ❚r♦♥❣ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✷✳✶✱ t❛ ❝â ω✇ (xn ) ⊂ S ✱ ð ✤➙② ω✇ (xn ) ❧➔ t➟♣ ❝→❝ ✤✐➸♠ tö ②➳✉ ❝õ❛ ❞➣② {xn }✳ ❈❤ù♥❣ ♠✐♥❤✳ ❘ã r➔♥❣✱ ωw (xn ) 6= ∅ ✈➻ ❞➣② {xn } ❜à ❝❤➦♥✳ ▲➜② x¯ ∈ ωw (xn ) õ tỗ t ởt {xnk } ❝õ❛ ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ x¯✳ ❚❛ ự t ữợ s ữợ ✶✳ x¯ ∈ K \ F (Tk ) k=1 ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✸✱ t❛ ❝â tn − zn → ✈➔ ❞♦ ✤â ∆p (tn , zn ) → 0✳ ❚ø ❝→❝❤ ①→❝ ✤à♥❤ ♣❤➛♥ tû tn ✱ t❛ ♥❤➟♥ ✤÷đ❝ ∆p (tk,n , zn ) → 0✱ tù❝ ❧➔ ∆p (Tk (zn ), zn ) → ✈ỵ✐ ♠å✐ k = 1, 2, , K ✳ ❙✉② r❛ x¯ ∈ Fˆ (Tk ) = F (Tk ) ✈ỵ✐ ♠å✐ k = 1, 2, , K ✳ ❉♦ ✈➟② x¯ K \ F (Tk ) k=1 ữợ x ∈ N \ Ci i=1 ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✸✱ t❛ ❝â ∆p (yn , xn ) → 0✳ ❉♦ ✤â✱ tø ❝→❝❤ ①→❝ ✤à♥❤ ♣❤➛♥ tû yn s✉② r❛ ∆p (yi,n , xn ) → ✈➔ ✈➻ ✈➟② kyi,n − xn k → 0, ✭✷✳✶✶✮ ✸✶ ✈ỵ✐ ♠å✐ i = 1, 2, , N ✳ ❚❛ ❝➛♥ ❝❤➾ r❛ r➡♥❣ ∆p(¯x, ΠC (¯x)) = ✈ỵ✐ ♠å✐ i = 1, 2, , N ✳ ❚❤➟t ✈➟②✱ tø ✭✶✳✶✶✮✱ ✭✶✳✶✹✮ ✈➔ ✭✶✳✶✷✮✱ t❛ ♥❤➟♥ ✤÷đ❝ ✤→♥❤ ❣✐→ s❛✉ i ∆p (¯ x, ΠCi (¯ x)) ≤ h¯ x − ΠCi x¯, Jp (¯ x) − Jp (ΠCi (¯ x))i = h¯ x − xnk , Jp (¯ x) − Jp (ΠCi (¯ x))i + hxnk − ΠCi (xnk ), Jp (¯ x) − Jp (ΠCi (¯ x))i + hΠCi (xnk ) − ΠCi (¯ x), Jp (¯ x) − Jp (ΠCi (¯ x))i ≤ h¯ x − xnk , Jp (¯ x) − Jp (ΠCi (¯ x))i + hxnk − ΠCi (xnk ), Jp (¯ x) − Jp (ΠCi (¯ x))i = h¯ x − xnk , Jp (¯ x) − Jp (ΠCi (¯ x))i + hxnk − yi,nk , Jp (¯ x) − Jp (ΠCi (¯ x))i ❚ø ✭✷✳✶✶✮✱ ❝❤♦ k → ∞ t❛ ♥❤➟♥ ✤÷đ❝ ∆p(¯x, ΠC (¯x)) = ✈ỵ✐ ♠å✐ i = 1, 2, , N ✱ N \ tù❝ ❧➔ x¯ ∈ Ci ✈ỵ✐ ♠å✐ i = 1, 2, , N ❤❛② x¯ ∈ Ci✳ i i=1 ữợ x M \ A1 Qj j=1 ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✸✱ t❛ ❝â ∆p(zn, yn) → 0✳ ❉♦ ✤â✱ tø ❝→❝ ①→❝ ✤à♥❤ ♣❤➛♥ tû zn✱ t❛ ♥❤➟♥ ✤÷đ❝ ∆p(zj,n, yn) → ✈➔ ✈➻ ✈➟② t❛ t❤✉ ✤÷đ❝ kzj,n − yn k → 0, ✭✷✳✶✷✮ ✈ỵ✐ ♠å✐ j = 1, 2, , M ✳ ❱➻ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉✱ ♥➯♥ →♥❤ ①↕ ✤è✐ ♥❣➝✉ Jp ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ ❝→❝ t➟♣ ❝♦♥ ỵ õ t❛ ❝â tn A∗ Jp (I − PQj )A(yn ) = Jp (yn ) − Jp (zj,n ) → ❱➻ < t ≤ tn ✈ỵ✐ ♠å✐ n✱ ♥➯♥ t❛ ♥❤➟♥ ✤÷đ❝ kA∗ Jp (I − PQj )A(yn )k → ✭✷✳✶✸✮ ❇➙② ❣✐í t❛ ❝è ✤à♥❤ u ∈ S ✱ ❦❤✐ ✤â A(u) ∈ Qj ✈ỵ✐ ♠å✐ j = 1, 2, , M ✳ ❚ø ✭✶✳✶✹✮ s✉② r❛ k(I − PQj )A(ynk )kp = h(I − PQj )A(ynk ), Jp (I − PQj )A(ynk )i ✸✷ = hA(ynk ) − A(u), Jp (I − PQj )A(ynk )i + hA(u) − PQj A(ynk ), Jp (I − PQj )A(ynk )i ≤ hA(ynk ) − A(u), Jp (I − PQj )A(ynk )i ≤ K0 k(I − PQj )A(ynk )kp−1 , t ủ ợ t ữủ k(I − PQj )A(ynk )k → ✈ỵ✐ ♠å✐ j = 1, 2, , M ✱ ð ✤➙② K0 = kAk(supk kynk k + kuk) < ∞✳ ❚ø ✭✶✳✶✹✮✱ t❛ ❝â k(I − PQj )A(¯ x)kp = hA(¯ x) − PQj A(¯ x), Jp (A(¯ x) − PQj A(¯ x))i = hA(¯ x) − A(ynk ), Jp (A(¯ x) − PQj A(¯ x))i + hA(ynk ) − PQj A(¯ x), Jp (A(¯ x) − PQj A(¯ x))i + hPQj A(¯ x) − A(ynk ), Jp (A(¯ x) − PQj A(¯ x))i ≤ hA(¯ x) − A(ynk ), Jp (A(¯ x) − PQj A(¯ x))i + hA(ynk ) − PQj A(¯ x), Jp (A(¯ x) − PQj A(¯ x))i ❚ø t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ A✱ xn − yn → ✈➔ xnk ⇀ x¯✱ s✉② r❛ A(ynk ) ⇀ A(¯ x)✳ ❉♦ ✤â✱ ❝❤♦ k → ∞ ✈➔ sû ❞ö♥❣ ✭✷✳✶✹✮✱ t❛ ♥❤➟♥ ✤÷đ❝ kA(¯ x) − PQj A(¯ x)k = 0, ✈ỵ✐ ♠å✐ j = 1, 2, , M ✱ tù❝ ❧➔ A(¯ x) ∈ M \ A−1 Qj ✳ j=1 õ tứ ữợ ữợ ữợ t ữủ x S x¯ ❧➔ ❜➜t ❦ý✱ ♥➯♥ ωw (xn ) ⊂ S ✳ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❙ü ❤ë✐ tư ♠↕♥❤ Pữỡ ữủ tr ỵ ữợ ỵ r t t ❞➣② {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ x† = ΠS (x0 )✱ n → ∞✳ ●✐↔ sû {xnk } ❧➔ ♠ët ❞➣② ❝♦♥ ❝õ❛ {xn } s❛♦ ❝❤♦ xnk ⇀ x∗ ✳ ❑❤✐ ✤â tø ▼➺♥❤ ✤➲ ✷✳✷✳✹✱ t❛ ❝â x∗ ∈ S ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✸✸ ❱➻ xn+1 = ΠH ∩D n n (x0 )✱ ♥➯♥ xn+1 ∈ Dn✳ ❉♦ ✤â✱ tø ΠS (x0) ∈ S ⊂ Dn✱ t❛ ❝â ∆p (xn+1 , x0 ) ≤ ∆p (ΠS x0 , x0 ), ❦➳t ❤đ♣ ✈ỵ✐ ∆p(xn+1, x0) ≥ ∆p(xn, x0)✱ t❛ ♥❤➟♥ ✤÷đ❝ ✭✷✳✶✺✮ ∆p (xn , x0 ) ≤ ∆p (ΠS x0 , x0 ), ∀n ≥ ❉♦ ✈➟②✱ tø ✭✶✳✶✵✮✱ ✭✶✳✶✶✮ ❛♥❞ ✭✷✳✶✺✮✱ t❛ t❤✉ ✤÷đ❝ ∆p (xnk , ΠS (x0 )) = ∆p (xnk , x0 ) + ∆p (x0 , ΠS (x0 )) + hxnk − x0 , Jp (x0 ) − Jp (ΠS (x0 ))i ≤ ∆p (ΠS (x0 ), x0 ) + ∆p (x0 , ΠS (x0 )) + hΠS (x0 ) − x0 , Jp (x0 ) − Jp (ΠS (x0 ))i + hxnk − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 ))i = hxnk − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 ))i ❱➻ ✈➟②✱ t❛ ❝â lim sup ∆p (xnk , ΠS (x0 )) ≤ lim suphxnk − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 ))i k→∞ k→∞ ≤ hx∗ − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 ))i ≤ 0, s✉② r❛ k→∞ lim ∆p (xn , ΠS (x0 )) = ✈➔ ❞♦ ✤â tø ✭✶✳✶✷✮ t❛ ❝â xn → ΠS (x0 ) ✳ ❚ø t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ❤➻♥❤ ❝❤✐➳✉ ❇r❡❣♠❛♥ ΠS (x0)✱ s✉② r❛ ❞➣② {xn} ❤ë✐ tö ②➳✉ ✈➲ ΠS (x0)✳ ❚ø tỗ t > s k k τ kxn − ΠS (x0 )k ≤ hxn − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 ))i ❈❤♦ n → ∞✱ t❛ ♥❤➟♥ ✤÷đ❝ xn → x† = S (x0) t tứ ỵ t õ q ữợ rữợ t õ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ✭▼❙❙❋P✮ tr♦♥❣ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❍➺ q✉↔ ✷✳✷✳✻✳ ❈❤♦ Ci ✱ i = 1, 2, , N ✈➔ Qj ✱ j = 1, 2, , M t ỗ õ rộ ổ pỗ ✤➲✉ ✈➔ trì♥ ✤➲✉ E ✈➔ F ✱ t÷ì♥❣ ù♥❣✳ ❈❤♦ A : E → F ❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥✳ ●✐↔ sû ✸✹ S = \ N Ci i=1 \ \ M j=1 −1 A (Qj ) 6= ∅✳ ◆➳✉ ❞➣② sè {tn } t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✻✮✱ t❤➻ ❞➣② {xn} ①→❝ ✤à♥❤ ❜ð✐ x0 ∈ E ✈➔ yi,n = ΠCi (xn ), i = 1, 2, , N, ❈❤å♥ in s❛♦ ❝❤♦ ∆p (yin ,n , xn ) = max ∆p (yi,n , xn ), ✤➦t yn = yin ,n , i=1, ,N zj,n = Jq∗ [Jp (yn ) − tn A∗ Jp (I − PQj )A(yn )], j = 1, 2, , M ❈❤å♥ jn s❛♦ ❝❤♦ ∆p (zjn ,n , yn ) = max ∆p (zj,n , yn ), ✤➦t zn = zjn ,n , j=1, ,M Hn = {z ∈ E : ∆p (zn , z) ≤ ∆p (yn , z) ≤ ∆p (xn , z)}, Dn = {z ∈ E : hxn − z, Jp (x0 ) − Jp (xn )i ≥ 0}, xn+1 = ΠHn ∩Dn (x0 ), n ≥ 0, ❤ë✐ tö ♠↕♥❤ ✈➲ x† = ΠS (x0 ) n ỵ ✷✳✷✳✺ ✈ỵ✐ Tk (x) = x ✈ỵ✐ ♠å✐ x ∈ E ✈➔ ♠å✐ k = 1, 2, , K ✱ t❛ ♥❤➟♥ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❈✉è✐ ❝ị♥❣✱ t❛ õ t q ữợ t t ởt ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ t♦→♥ tû ▲✲❇❙◆❊ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ E ởt ổ pỗ ✈➔ trì♥ ✤➲✉✳ ❈❤♦ Tk : E → E ✱ k = 1, 2, , K ❧➔ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ t♦→♥ tû ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ K \ ˆ ♠↕♥❤ tr→✐ s❛♦ ❝❤♦ F (Tk ) = F (Tk ) ✈➔ S = F (Tk ) 6= ∅✳ ❑❤✐ ✤â ❞➣② {xn } ①→❝ k=1 ✤à♥❤ ❜ð✐ x0 ∈ E ✈➔ tk,n = Tk (xn ), k = 1, 2, , K, ❈❤å♥ kn s❛♦ ❝❤♦ ∆p (tkn ,n , xn ) = max ∆p (tk,n , xn ), ✤➦t tn = tkn ,n , k=1, ,K Hn = {z ∈ E : ∆p (tn , z) ≤ ∆p (xn , z)}, Dn = {z ∈ E : hxn − z, Jp (x0 ) − Jp (xn )i ≥ 0}, xn+1 = ΠHn ∩Dn (x0 ), n ≥ 0, ❤ë✐ tö ♠↕♥❤ ✈➲ x† = ΠS (x0 )✱ ❦❤✐ n ỵ ợ E ≡ F ✈➔ Ci = Qj = E ✈ỵ✐ ♠å✐ i = 1, 2, , N ✱ j = 1, 2, , M ✈➔ A = I ✱ t❛ ♥❤➟♥ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳