Một định lý hội tụ mạnh giải bài toán chấp nhận tách và bài toán điểm bất động trong không gian banach

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 ữỡ tự ổ pỗ ✐✐ ✐✈ ✶ ✸ ✤➲✉ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷ ❙ü ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ỗ ởt số t t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✹ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✺ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ pỗ ố ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸ ❑❤♦↔♥❣ ❝→❝❤ ❇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∗ ❦❤ỉ♥❣ ❣✐❛♥ ✤è✐ ♥❣➝✉ ❝õ❛ R t➟♣ ❤đ♣ số tỹ inf M ữợ ✤ó♥❣ ❝õ❛ t➟♣ ❤đ♣ sè M sup M ❝➟♥ tr➯♥ ✤ó♥❣ ❝õ❛ t➟♣ ❤đ♣ sè M max M sè ❧ỵ♥ ♥❤➜t tr♦♥❣ t➟♣ ❤ñ♣ sè M sè ♥❤ä ♥❤➜t tr♦♥❣ t➟♣ ❤ñ♣ sè ❛r❣♠✐♥x∈X F (x) t➟♣ ❝→❝ ✤✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ ∅ t➟♣ ré♥❣ ∀x ✈ỵ✐ ♠å✐ ❞♦♠(A) ♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ t♦→♥ tû I t♦→♥ tû ỗ t Lp () ổ t ❜➟❝ lp ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❞➣② sè ❦❤↔ tê♥❣ ❜➟❝ E M F tr➯♥ X x A ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❞➣② sè {xn } lim inf xn ❣✐ỵ✐ ❤↕♥ ữợ số {xn } xn x0 {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ xn ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ lim sup xn M p tr➯♥ p n→∞ n→∞ x0 x0 x0 Jp →♥❤ ①↕ ✤è✐ E () ổ ỗ ổ ρE (τ ) ♠ỉ ✤✉♥ trì♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ F ix(T ) ❤♦➦❝ F (T ) t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T E E Ω ✈ ✐♥tM tr t ủ rr s số trữợ PC ♣❤➨♣ ♠➯tr✐❝ ❧➯♥ f M C ♣r♦❥C ♣❤➨♣ ❝❤✐➳✉ r iC t ỗ C C ✶ ▼ð ✤➛✉ ❈❤♦ H1 ✈➔ C ✈➔ H2 ✱ Q t ỗ õ rộ ❝õ❛ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt 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❛✉✿ ❈❤♦ ✤â♥❣ ❝õ❛ H1 ✈➔ H2 Ci ✱ i = 1, 2, , N ✈➔ Qj ✱ j = 1, 2, , M ❧➔ ❝→❝ t➟♣ ỗ tữỡ ự ởt tỷ (∩M x∗ ∈ S = ∩N j=1 Qj ) = ∅ i=1 Ci ∩ T ✭✵✳✷✮ ▼æ ❤➻♥❤ ❜➔✐ t♦→♥ P t ữủ ợ t ự ❜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♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t ❝õ❛ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ t→❝❤✳ ❉↕♥❣ tê♥❣ q✉→t ❝õ❛ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ t→❝❤ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ❈❤♦ j = 1, 2, , M Ti : H1 −→ H1 ✱ i = 1, 2, , N ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ ❚➻♠ ♣❤➛♥ tû H1 ✈➔ H2 ✱ ✈➔ Sj : H2 −→ H2 ✱ t÷ì♥❣ ù♥❣✳ −1 x∗ ∈ S = ∩N ∩M i=1 F ix(Ti ) ∩ T j=1 F ix(Sj ) = ∅ ✭✵✳✸✮ ❈❤♦ ✤➳♥ ♥❛② ❇➔✐ t♦→♥ ✭✵✳✸✮ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✤➣ ✈➔ ✤❛♥❣ ❧➔ ❝❤õ ✤➲ t❤✉ ❤ót ♥❤✐➲✉ ♥❣÷í✐ ❧➔♠ t♦→♥ tr ữợ q t ự ✤➣ ❝â ♠ët sè t→❝ ❣✐↔ ✤➲ ❝➟♣ ✤➳♥ ✈✐➺❝ ự t ữỡ ợ t ởt ♥❣❤✐➺♠ ❝❤✉♥❣ ❝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 ✈➜♥ ✤➲ ✈➲ ❦❤æ♥❣ ổ pỗ trỡ →♥❤ 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❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ ♠↕♥❤ tr tr ổ trỡ pỗ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ♥➔② ❜❛♦ ỗ ử tr ởt số t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✱ ❦❤ỉ♥❣ ỗ trỡ ợ 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 ổ ❇❛♥❛❝❤ 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 ❞ö♥❣ ❦➼ ❤✐➺✉ t↕✐ ✤✐➸♠ ✤➸ ❝❤➾ ❝❤✉➞♥ tr xX tỗ t X tr ❝õ❛ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ x∗ ∈ X ∗ x, x ữủ ỵ x E ∗∗ ✱ X ✳ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ x∈E E ữủ ợ s x, x∗ = x∗ , x∗∗ , ✈ỵ✐ ♠å✐ x∗ ∈ E ∗ ✳ ❱➼ ❞ö ✶✳✶✳✷✳ ❣✐❛♥ lp ❤❛② ▼å✐ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉✱ ❝→❝ ❦❤ỉ♥❣ Lp (Ω)✱ ✈ỵ✐ < p < ∞✱ ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕ ✭①❡♠ ❬✷❪✮✳ ✹ ❈❤ó ỵ t t ữợ ổ ❇❛♥❛❝❤ ♣❤↔♥ ①↕ ❝â t❤➸ t➻♠ t❤➜② tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✷❪✳ ✐✮ ◆➳✉ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ Y✱ t❤➻ X X ỗ ổ t t ợ ổ ①↕ ❝ơ♥❣ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ✐✐✮ ▼å✐ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕❀ ✐✐✐✮ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧➔ ♣❤↔♥ ①↕ ❦❤✐ ✈➔ ❝❤➾ ❦❤ỉ♥❣ ❣✐❛♥ ❧✐➯♥ ❤đ♣ E∗ ❝õ❛ ♥â ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ✶✳✶✳✷ ❙ü ❤ë✐ tư ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ {xn} ❉➣② tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ x∈E ❣å✐ ❧➔ tử ởt tỷ ữủ ỵ ❤✐➺✉ ❧➔ xn x✱ E ✤÷đ❝ ♥➳✉ lim xn , x∗ = x, x∗ , n→∞ ✈ỵ✐ ♠å✐ x∗ ∈ X ∗ ✳ ◆❤➟♥ ①➨t ✶✳✶✳✺✳ {xn } ◆➳✉ ❞➣② ❤ë✐ tư ②➳✉ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt x✳ l2 ✱ {xn } ❤ë✐ tö ♠↕♥❤ ✈➲ ✭✈➻ tù❝ ❧➔ xn − x → 0✱ t❤➻ ❞➣② ❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ❈❤➥♥❣ ❤↕♥✱ ①➨t ❞➣② {en } ①→❝ ✤à♥❤ ❜ð✐ en = (0, , 0, ✈ỵ✐ ♠å✐ x✱ ✈à tr➼ t❤ù n , 0, ), n ≥ 1✱ ❤ë✐ tö ②➳✉ ✈➲ ❦❤ỉ♥❣ ✭①❡♠ ❬✷❪✮✱ ♥❤÷♥❣ ❦❤ỉ♥❣ ❤ë✐ tư ♠↕♥❤ ✈➲ ❦❤ỉ♥❣ en = ✈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✐ x∗ , Hxn = xn , x∗ n ≥ 1✱ ✈ỵ✐ ♠å✐ ①➨t ❞➣② ♣❤✐➳♠ ❤➔♠ x∗ ∈ E ∗ ✳ {Hxn } ⊂ E ∗∗ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ x∗ ∈ E ∗ ✱ ①→❝ ✤à♥❤ ❜ð✐ t❛ ❝â x∗ , Hxn = xn , x∗ → x, x∗ ✶ ❉♦ ✤â✱ t❤❡♦ ❤➺ q ỵ ợ t t ❝â sup xn = sup Hxn < ∞ 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 An < ∞ ✺ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✶✳✶✳✼✳ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✱ A ⊂ E ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ✈➔ {xn } ⊂ A t❤ä❛ ♠➣♥ xn ❈❤ù♥❣ sỷ xn x õ tỗ t >0 x✳ ❑❤✐ ✤â✱ xn → x✳ ✈➔ ♠ët ❞➣② ❝♦♥ {xnk } ⊂ {xn } s❛♦ ❝❤♦ xnk − x ≥ ε, ✈ỵ✐ ♠å✐ ❱➻ ✭✶✳✶✮ k ≥ 1✳ {xnk } ⊂ A {xnk } s❛♦ ❝❤♦ ❞♦ ✤â y = x✳ ✈➔ A ❧➔ t➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐✱ ♥➯♥ tỗ t xnkl y sỹ ❤ë✐ tö ♠↕♥❤ ❦➨♦ t❤❡♦ ❤ë✐ tö ②➳✉ ♥➯♥ ❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✮✱ t❤❛② xnk ❜ð✐ xnkl {xnkl } ⊂ xnkl y ✈➔ t❛ ✤÷đ❝ xnkl − y ≥ ε, ♠➙✉ t❤✉➝♥ ✈ỵ✐ xnkl y✳ xn → x✳ ❱➟② ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ t❤÷í♥❣ ①✉②➯♥ sû ❞ư♥❣ t t ữợ ổ ▼➺♥❤ ✤➲ ✶✳✶✳✽✳ ✭①❡♠ ❬✷❪ tr❛♥❣ ✹✶✮ ❈❤♦ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❑❤✐ ✤â✱ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ ✐✮ ✐✐✮ E ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♣❤↔♥ ①↕✳ ▼å✐ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ E ✱ ✤➲✉ ❝â ởt tử ữợ ❝❤♦ t❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ t➟♣ ✤â♥❣ ✈➔ t➟♣ ✤â♥❣ ②➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳ ▼➺♥❤ C t ỗ õ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥ X ✱ t❤➻ C ❧➔ t➟♣ ✤â♥❣ ②➳✉✳ ❈❤ù♥❣ ♠✐♥❤✳ ❝❤♦ xn ♥❣➦t x ✈➔ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ự sỷ tỗ t x / C x ữ C tự tỗ t ỵ t t ỗ tỗ t >0 s y, x∗ ≤ x, x∗ − ε, {xn } ⊂ C x∗ ∈ X ∗ s❛♦ t→❝❤ ✷✽ ❇➙② ❣✐í✱ t❛ ❝❤➾ r❛ ∆p (zn , u) ≤ ∆p (yn , u)✳ ✣➦t wn = A(yn ) − PQjn A(yn )✳ ❑❤✐ ✤â t❛ ❝â zn = Jq∗ (Jp (yn ) − tn A∗ Jp (wn )) ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ Jp ✈➔ ✭✶✳✶✹✮✱ t❛ ❝â A(yn ) − A(u), Jp (wn ) = A(yn ) − PQjn A(yn ) p + PQjn A(yn ) − A(u), Jp (wn ) ✭✷✳✹✮ ≥ wn p ❉♦ ✤â✱ tø ▼➺♥❤ ✤➲ ✶✳✷✳✺ ✈➔ ✭✷✳✹✮✱ t❛ ♥❤➟♥ ✤÷đ❝ ∆p (zn , u) = ∆p (Jq∗ (Jp (yn ) − tn A∗ Jp (wn )), u) = Jp (yn ) − tn A∗ Jp (wn ) q − u, Jp (yn ) q + tn A(u), Jp (wn ) + u p p Cq (tn A )q ≤ Jp (yn ) q − tn Ayn , Jp (wn ) + Jp (wn ) q q q − u, Jp (yn ) + tn Au, Jp (wn ) + u p p 1 = yn q − u, Jp (yn ) + u p + tn A(u) − A(yn ), Jp (wn ) q p q Cq (tn A ) wn q + q Cq (tn A )q = ∆p (yn , u) + tn A(u) − A(yn ), Jp (wn ) + wn q q q Cq (tn A ) ≤ ∆p (yn , u) − (tn − ) wn p q ❚ø ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✻✮✱ t❛ t❤✉ ✤÷đ❝ ∆p (zn , u) ≤ ∆p (yn , u) ❉♦ ✈➟②✱ tø ✭✷✳✷✮✱ ✭✷✳✸✮ ✈➔ ✭✷✳✺✮✱ s✉② r❛ ❈✉è✐ ❝ò♥❣ t❛ ❝❤➾ r❛ S ⊂ D0 ✳ ●✐↔ sû S ⊂ Dn S ⊂ Dn xn+1 = ΠHn ∩Dn (x0 ) ✈ỵ✐ u ∈ Hn ✳ ✈ỵ✐ ♠å✐ n ≥ ❱➻ ✈➟② ✭✷✳✺✮ S ⊂ Hn n ≥ 0✳❚❤➟t ♥➔♦ ✤â✱ ❦❤✐ ✤â ✈ỵ✐ ♠å✐ ✈➟②✱ ✈➻ D0 = E ✱ S ⊂ Hn ∩ Dn ✳ ✈➔ ✭✶✳✶✹✮✱ t❛ ❝â xn+1 − u, Jp (x0 ) − Jp (xn+1 ) ≥ 0, n ≥ 0✳ ♥➯♥ ❉♦ ✤â✱ tø ✷✾ ✤✐➲✉ ♥➔② s✉② r❛ ♠å✐ u ∈ Dn+1 ✳ ❜➡♥❣ q✉② ♥↕♣ t♦→♥ ❤å❝✱ t❛ ♥❤➟♥ ✤÷đ❝ S Dn ợ n ữủ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✷✳✷✳✷✳ ❚r♦♥❣ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✷✳✶✱ t❛ ❝â xn+1 − xn → ❦❤✐ n → ∞✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈è ✤à♥❤ ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✶✱ s✉② r❛ ❞➣② u ∈ S✳ ❚ø xn+1 = ΠHn ∩Dn (x0 ) {xn } ❧➔ ❤♦➔♥ t♦➔♥ ①→❝ ✤à♥❤✳ ✈➔ ✭✶✳✶✺✮ s✉② r❛ ∆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 xn − xn+1 , Jp (x0 ) − Jp (xn ) ≥ ✭✷✳✼✮ xn − x0 , Jp (x0 ) − Jp (xn ) ≥ xn+1 − x0 , Jp (x0 ) − Jp (xn ) ✭✷✳✽✮ ❉♦ ✈➟②✱ t❛ ♥❤➟♥ ✤÷đ❝ ❉♦ ✤â✱ tø ✭✶✳✶✷✮✱ t❛ ❝â xn+1 − x0 , Jp (x0 ) − Jp (xn ) ≥ ∆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 ❝❤➦♥ ❝õ❛ ✭✷✳✶✵✮ {∆p (x0 , xn )}✱ t↕✐ ❣✐ỵ✐ ❤↕♥ ❤ú✉ ❤↕♥ a = lim ∆p (x0 , xn ) n→∞ ❱➻ ✈➟②✱ tø ✭✷✳✶✵✮✱ t❛ t❤✉ ✤÷đ❝ lim ∆p (xn , xn+1 ) = 0✳ n→∞ lim xn+1 − xn = n→∞ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚ø ✭✶✳✶✷✮ s✉② r tỗ r Pữỡ ✷✳✶✱ ❝→❝ ❞➣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❛ xn+1 − tn → 0, xn+1 − zn → 0, xn+1 − yn → ❦➳t ❤đ♣ ✈ỵ✐ xn+1 − xn → 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 } ự ó r tỗ t ởt ωw (xn ) = ∅ {xnk } ❝õ❛ ❞➣② ✈➻ ❞➣② {xn } {xn } ❜à ❝❤➦♥✳ ▲➜② ❤ë✐ tö ②➳✉ ✈➲ x¯ ∈ ωw (xn )✱ ❦❤✐ ✤â x¯✳ ự t ữợ s K ữợ x F (Tk ) k=1 ứ ▼➺♥❤ ✤➲ ✷✳✷✳✸✱ t❛ ❝â tn ✱ ✤à♥❤ ♣❤➛♥ tû ♠å✐ tn − zn → ✈➔ ❞♦ ✤â ∆p (tn , zn ) → 0✳ ❚ø ❝→❝❤ ①→❝ ∆p (tk,n , zn ) → 0✱ tù❝ ❧➔ ∆p (Tk (zn ), zn ) → ✈ỵ✐ x¯ ∈ Fˆ (Tk ) = F (Tk ) ✈ỵ✐ ♠å✐ k = 1, 2, , K ✳ ❉♦ ✈➟② t❛ ♥❤➟♥ ✤÷đ❝ k = 1, 2, , K ✳ ❙✉② r❛ K x F (Tk ) k=1 ữợ x N Ci i=1 ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✸✱ t❛ ❝â s✉② r❛ ∆p (yi,n , xn ) → ∆p (yn , xn ) → 0✳ ❉♦ ✤â✱ tø ❝→❝❤ ①→❝ ✤à♥❤ ♣❤➛♥ tû yn ✈➔ ✈➻ ✈➟② yi,n − xn → 0, ✭✷✳✶✶✮ ✸✶ ✈ỵ✐ ♠å✐ i = 1, 2, , N ✳ ❚❛ ❝➛♥ ❝❤➾ r❛ r➡♥❣ ∆p (¯ x, ΠCi (¯ x)) = ✈ỵ✐ ♠å✐ i = 1, 2, , N ✳ ❚❤➟t ✈➟②✱ tø ✭✶✳✶✶✮✱ ✭✶✳✶✹✮ ✈➔ ✭✶✳✶✷✮✱ t❛ ♥❤➟♥ ✤÷đ❝ ✤→♥❤ ❣✐→ s❛✉ ∆p (¯ x, ΠCi (¯ x)) ≤ x¯ − ΠCi x¯, Jp (¯ x) − Jp (ΠCi (¯ x)) = x¯ − xnk , Jp (¯ x) − Jp (ΠCi (¯ x)) + xnk − ΠCi (xnk ), Jp (¯ x) − Jp (ΠCi (¯ x)) + ΠCi (xnk ) − ΠCi (¯ x), Jp (¯ x) − Jp (ΠCi (¯ x)) ≤ x¯ − xnk , Jp (¯ x) − Jp (ΠCi (¯ x)) + xnk − ΠCi (xnk ), Jp (¯ x) − Jp (ΠCi (¯ x)) x)) = x¯ − xnk , Jp (¯ x) − Jp (ΠCi (¯ x)) + xnk − yi,nk , Jp (¯ x) − Jp (ΠCi (¯ ❚ø ✭✷✳✶✶✮✱ ❝❤♦ k→∞ t❛ ♥❤➟♥ ữủ p ( x, Ci ( x)) = ợ ♠å✐ i = 1, 2, , N ✱ N tù❝ ❧➔ x¯ ∈ Ci ✈ỵ✐ ♠å✐ i = 1, 2, , N x Ci i=1 ữợ x¯ ∈ M A−1 Qj j=1 ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✸✱ t❛ ❝â t❛ ♥❤➟♥ ✤÷đ❝ ∆p (zn , yn ) → 0✳ ∆p (zj,n , yn ) → ❉♦ ✤â✱ tø ❝→❝ ①→❝ ✤à♥❤ ♣❤➛♥ tû ✈➔ ✈➻ ✈➟② t t ữủ zj,n yn 0, ợ ❱➻ E zn ✱ ✭✷✳✶✷✮ j = 1, 2, , M ✳ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ trì♥ ✤➲✉✱ ♥➯♥ →♥❤ ①↕ ✤è✐ ♥❣➝✉ Jp ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ ❝→❝ t➟♣ ỵ ✤â t❛ ❝â tn A∗ Jp (I − PQj )A(yn ) = Jp (yn ) − Jp (zj,n ) → ❱➻ < t ≤ tn ✈ỵ✐ ♠å✐ n✱ ♥➯♥ t❛ ♥❤➟♥ ✤÷đ❝ A∗ Jp (I − PQj )A(yn ) → ❇➙② ❣✐í t❛ ❝è ✤à♥❤ u ∈ S✱ ❦❤✐ ✤â A(u) ∈ Qj ✈ỵ✐ ♠å✐ ✭✷✳✶✸✮ j = 1, 2, , M ✳ s✉② r❛ (I − PQj )A(ynk ) p = (I − PQj )A(ynk ), Jp (I − PQj )A(ynk ) ❚ø ✭✶✳✶✹✮ ✸✷ = A(ynk ) − A(u), Jp (I − PQj )A(ynk ) + A(u) − PQj A(ynk ), Jp (I − PQj )A(ynk ) ≤ A(ynk ) − A(u), Jp (I − PQj )A(ynk ) ≤ K0 (I − PQj )A(ynk ) p−1 , ✤✐➲✉ ♥➔② ❦➳t ❤đ♣ ✈ỵ✐ ✭✷✳✶✸✮✱ t❛ ♥❤➟♥ ữủ (I PQj )A(ynk ) ợ j = 1, 2, , M ✱ ð ✤➙② ✭✷✳✶✹✮ K0 = A (supk ynk + u ) < ∞✳ ❚ø ✭✶✳✶✹✮✱ t❛ ❝â p x) (I − PQj )A(¯ x)) x), Jp (A(¯ x) − PQj A(¯ = A(¯ x) − PQj A(¯ x)) = A(¯ x) − A(ynk ), Jp (A(¯ x) − PQj A(¯ x)) x), Jp (A(¯ x) − PQj A(¯ + A(ynk ) − PQj A(¯ x)) x) − A(ynk ), Jp (A(¯ x) − PQj A(¯ + PQj A(¯ x)) ≤ A(¯ x) − A(ynk ), Jp (A(¯ x) − PQj A(¯ x)) x), Jp (A(¯ x) − PQj A(¯ + A(ynk ) − PQj A(¯ ❚ø t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ❝❤♦ k→∞ A✱ xn − yn → ✈➔ x nk x¯✱ s✉② r❛ A(ynk ) A(¯ x)✳ ❉♦ ✤â✱ ✈➔ sû ❞ư♥❣ ✭✷✳✶✹✮✱ t❛ ♥❤➟♥ ✤÷đ❝ x) = 0, A(¯ x) − PQj A(¯ M ✈ỵ✐ ♠å✐ j = 1, 2, , M ✱ tù❝ ❧➔ A−1 Qj ✳ A(¯ x) j=1 õ tứ ữợ ữợ ữợ t ữủ x S ✳ ❱➻ x¯ ❧➔ ❜➜t ❦ý✱ ωw (xn ) ⊂ S ✳ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❙ü ❤ë✐ tư ♠↕♥❤ ❝õ❛ P❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✷✳✶ ✤÷đ❝ ❝❤♦ tr♦♥❣ ✤à♥❤ ỵ ữợ ỵ r t t ❞➣② {xn} ❤ë✐ tö ♠↕♥❤ ✈➲ x† = ΠS (x0)✱ ❦❤✐ n → ∞✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû {xnk } tø ▼➺♥❤ ✤➲ ✷✳✷✳✹✱ t❛ ❝â ❧➔ ♠ët ❞➣② ❝♦♥ ❝õ❛ x∗ ∈ S ✳ {xn } s❛♦ ❝❤♦ x nk x∗ ✳ ❑❤✐ ✤â ✸✸ ❱➻ xn+1 = ΠHn ∩Dn (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 )) + xnk − x0 , Jp (x0 ) − Jp (ΠS (x0 )) ≤ ∆p (ΠS (x0 ), x0 ) + ∆p (x0 , ΠS (x0 )) + ΠS (x0 ) − x0 , Jp (x0 ) − Jp (ΠS (x0 )) + xnk − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 )) = xnk − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 )) ❱➻ ✈➟②✱ t❛ ❝â lim sup ∆p (xnk , ΠS (x0 )) ≤ lim sup xnk − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 )) k→∞ k→∞ ≤ x∗ − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 )) ≤ 0, s✉② r❛ lim ∆p (xnk , ΠS (x0 )) = ✈➔ ❞♦ ✤â tø ✭✶✳✶✷✮ t❛ ❝â xnk → ΠS (x0 ) ✳ ❚ø t➼♥❤ k→∞ ❞✉② t r ứ tỗ t >0 ΠS (x0 )✱ s✉② r❛ ❞➣② {xn } ❤ë✐ tö ②➳✉ ✈➲ ΠS (x0 )✳ s❛♦ ❝❤♦ τ xn − ΠS (x0 ) ≤ xn − ΠS (x0 ), Jp (x0 ) − Jp (ΠS (x0 )) ❈❤♦ 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 ✸✹ M N S = A−1 (Qj ) Ci j=1 i=1 ✭✶✳✶✻✮✱ = ∅✳ ◆➳✉ ❞➣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 : xn − z, Jp (x0 ) − Jp (xn ) ≥ 0}, xn+1 = ΠHn ∩Dn (x0 ), n ≥ 0, ❤ë✐ tö ♠↕♥❤ ✈➲ x† = ΠS (x0 )✱ ❦❤✐ n → ự ỵ ợ k = 1, 2, , K ✱ Tk (x) = x ợ x E t ữủ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❈✉è✐ ❝ò♥❣✱ t❛ ❝â ❦➳t q✉↔ ữợ t t ởt t ❝❤✉♥❣ ❝õ❛ ♠ët ❤å ❤ú✉ ❤↕♥ t♦→♥ tû ▲✲❇❙◆❊ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❍➺ q✉↔ ✷✳✷✳✼✳ ❈❤♦ E ❧➔ ♠ët ổ pỗ trỡ Tk : E → E ✱ k = 1, 2, , K ❧➔ ♠ët ❤å ❤ú✉ ❤↕♥ ❝→❝ t♦→♥ tû ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ K ♠↕♥❤ tr→✐ s❛♦ ❝❤♦ Fˆ (Tk ) = F (Tk ) ✈➔ S = F (Tk ) = ∅✳ ❑❤✐ ✤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 : xn − z, Jp (x0 ) − Jp (xn ) ≥ 0}, xn+1 = ΠHn ∩Dn (x0 ), n ≥ 0, ❤ë✐ tö ♠↕♥❤ ✈➲ x† = ΠS (x0 )✱ ❦❤✐ n → ∞✳ ❈❤ù♥❣ ♠✐♥❤✳ ⑩♣ ❞ö♥❣ ỵ ợ i = 1, 2, , N ✱ j = 1, 2, , M ✈➔ A = I✱ E ≡ F ✈➔ C i = Qj = E ợ t ữủ ự ✸✺ ✷✳✸ ❱➼ ❞ö ♠✐♥❤ ❤å❛ ❱➼ ❞ö ✷✳✸✳✶✳ ❚❛ ①➨t ❇➔✐ t♦→♥ ✭✷✳✶✮ ✈ỵ✐ Ci ⊂ Rn Qj ⊂ R m ✈➔ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ C Ci = {x ∈ RN : aC i , x ≤ bi }, Q Qj = {x ∈ RM : aQ j , x ≤ bj }, tr♦♥❣ ✤â ✈➔ Tk Q N M aC i ∈ R , aj ∈ R ✈➔ ❧➔ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ tø Q bC i , bj ∈ R RN ❧➯♥ Sk Sk = {x ∈ Rn : ✈ỵ✐ ♠å✐ k = 1, 2, , K ✈➔ ✈ỵ✐ ♠å✐ i = 1, 2, , N ✱ j = 1, 2, , M ✈ỵ✐ x − Ik ≤ Rk2 }, A ❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tø RN ♠❛ tr➟♥ ❝â ❝→❝ ♣❤➛♥ tû ✤÷đ❝ s✐♥❤ ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ✤♦↕♥ ❚✐➳♣ t❤❡♦✱ t❛ ❧➜② ♥❣➝✉ ♥❤✐➯♥ ❣✐→ trà ❝→❝ tå❛ ✤ë ❝õ❛ ✈➔ Q bC i ✱ bj tr♦♥❣ ✤♦↕♥ ❬✷✱✹❪✱ tå❛ ✤ë t➙♠ Sk ❤➻♥❤ ❝➛✉ tr♦♥❣ ✤♦↕♥ [2, 10]✱ ✈ỵ✐ [2, 4]✳ Q aC i ✱ aj [−1, 1] tr♦♥❣ ✤♦↕♥ ✈➔ ❜→♥ ❦➼♥❤ [1, 3] Rk ❝õ❛ K −1 A (Qj ) j=1 i=1 RM t÷ì♥❣ ù♥❣✳ Ci S= tr♦♥❣ ✤♦↕♥ M N ❉➵ t❤➜② Ik ❧➯♥ F (Tk ) = ∅✱ ✈➻ ∈ S✳ k=1 ❇➙② ❣✐í✱ t❛ ❦✐➸♠ tr❛ sü ❤ë✐ tư ❝õ❛ ❚❤✉➟t t♦→♥ ✷✳✶✱ ✈ỵ✐ ♣❤➛♥ tû ❜❛♥ ✤➛✉ x0 ∈ RN ❝â ❝→❝ tå❛ ✤ë ✤÷đ❝ s✐♥❤ ♥❣➝✉ ♥❤✐➯♥ tr♦♥❣ ✤♦↕♥ M = 40✱ N = 50✱ M = 100✱ K = 200 ✈➔ tn = A [−5, 5]✱ N = 20✱ ✳ ❙❛✉ ♥➠♠ ❧➛♥ t❤û✱ t❛ t❤✉ ✤÷đ❝ ❜↔♥❣ t q số ữợ ứ n < 10−5 ✣✐➲✉ ❦✐➺♥ ❞ø♥❣✿ ❚❖▲n < 10−6 ◆♦✳ ❚❖▲n n ◆♦✳ ❚❖▲n n ✶ 9.73191e − 006 525 ✶ 9.82257e − 007 2692 ✷ 9.72380e − 006 382 ✷ 9.88394e − 007 1084 ✸ 9.74093e − 006 594 ✸ 9.99178e − 007 1878 ✹ 9.81788e − 006 793 ✹ 9.82163e − 007 1922 ✺ 9.77395e − 006 250 ✺ 9.98486e − 007 1644 ❇↔♥❣ ✷✳✶✿ ❇↔♥❣ ❦➳t q✉↔ sè ❝❤♦ ú ỵ n = N ❚r♦♥❣ ✈➼ ❞ư tr➯♥✱ ❤➔♠ sè ❚❖▲ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ N xn − PCi xn i=1 + M M Axn − PQj Axn j=1 + K K xn − Tk xn , k=1 ✸✻ ợ xn n ú ỵ r t ữợ tự n n =0 t xn S ✱ tù❝ ❧➔ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ❱➼ ❞ö ✷✳✸✳✸✳ E = F = L2 ([0, 1]) ợ t ổ ữợ f, g = f (t)g(t)dt ✈➔ ❝❤✉➞♥ ①→❝ ✤à♥❤ ❜ð✐ 1/2 f = f (t)dt , ✈ỵ✐ ♠å✐ f, g ∈ L2 ([0, 1]) ✣➦t tr♦♥❣ ✤â (t) = ti−1 ✱ Ci = {x ∈ L2 ([0, 1]) : , x = bi }, ✈ỵ✐ ♠å✐ i = 1, 2, , N ✈➔ t ∈ [0, 1]✱ bi = i+1 Qj = {x ∈ L2 ([0, 1]) : cj , x ≥ dj }, tr♦♥❣ ✤â cj (t) = t + j ✱ dj = ✈ỵ✐ ♠å✐ j = 1, 2, , M ✈➔ t ∈ [0, 1]✱ Tk = P S k , ð ✤➙② Sk = {x ∈ L2 ([0, 1]) : k = 1, 2, , K ✈➔ x − Ik ≤ k + 1}, Ik (t) = t + k ✈ỵ✐ t ∈ [0, 1]✳ ●✐↔ sû A : L2 ([0, 1]) −→ L2 ([0, 1]), (Ax)(t) = ❚❛ ①➨t ❜➔✐ t♦→♥ t➻♠ ♠ët ♣❤➛♥ tû N † x ∈S= S = ∅✱ ✈➻ x† x(t) s❛♦ ❝❤♦ M K −1 Ci i=1 ❉➵ t❤➜② ✈ỵ✐ ♠å✐ A (Qj ) F (Tk ) j=1 k=1 ΠCi (x) = PCi (x) = bi − , x + x, x(t) = t ∈ S ✳ ❚❛ ❝â PQj (x) = max 0, dj − cj , x cj cj + x, ✭✷✳✶✻✮ ✸✼ ✈➔ Tk (x) =   x,  Ik + ♥➳✉ k+1 (x − Ik ), x − Ik tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❦❤→❝ N = 10, M = 20 ❙û ❞ư♥❣ ❚❤✉➟t t♦→♥ ✷✳✶ ✈ỵ✐ x − Ik ≤ k + 1, ✈➔ K = 40✱ t❛ t❤✉ ✤÷đ❝ ❜↔♥❣ ❦➳t q số ữợ ứ xn+1 xn tn = 1✱ x0 (t) = t2 ❡rr xn+1 − xn n < ❡rr −2 10 9.92326e − 003 128 −3 9.90940e − 004 2159 10 −4 10 9.98327e − 005 47840 tn = 1✱ x0 (t) = exp(t) ❡rr xn+1 − xn −2 10 9.06924e − 03 −3 10 9.95338e − 004 −4 10 9.97943e − 005 n 125 1091 11352 ❇↔♥❣ ✷✳✷✿ ❇↔♥❣ ❦➳t q✉↔ sè ❝❤♦ ❱➼ ❞ö ✷✳✸✳✸ ❉→♥❣ ✤✐➺✉ ❝õ❛ xn+1 − xn tr♦♥❣ ❇↔♥❣ ữủ ổ t ỗ t ữợ 10 x0(t)=exp(t) x0(t)=t2 −1 ||xn+1−xn|| 10 −2 10 −3 10 500 1000 1500 2000 2500 Number of interations ❍➻♥❤ ✷✳✶✿ ❉→♥❣ ✤✐➺✉ ❝õ❛ xn+1 − xn ❉→♥❣ ✤✐➺✉ ❝õ❛ ♥❣❤✐➺♠ ①➜♣ ①➾ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❞ø♥❣ xn+1 − xn < 10−3 xn (t) tr♦♥❣ ❝↔ ❤❛✐ tr÷í♥❣ ❤đ♣ xn+1 xn < 103 ữủ tr ữợ ✸✽ 0.9 0.8 The solution x*(t)=t xn(t) with x0(t)=exp(t) 0.7 xn(t) with x0(t)=t2 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 ❍➻♥❤ ✷✳✷✿ ❉→♥❣ ✤✐➺✉ ❝õ❛ xn(t) ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❞ø♥❣ 0.7 0.8 0.9 xn+1 − xn < 10−3 ❚✐➳♣ t❤❡♦✱ ♥❤➡♠ ✤÷❛ r❛ ♠ët s♦ s→♥❤ ✤ì♥ ❣✐↔♥ ❣✐ú❛ ❤❛✐ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✭✶✳✷✷✮ ✈➔ ✭✷✳✶✮✱ t❛ ①➨t ♠ët tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ❇➔✐ t♦→♥ ✭✷✳✶✻✮ ♥❤÷ s❛✉✿ ❚➻♠ ♠ët ♣❤➛♥ tû tr♦♥❣ ✤â C = C ✱ Q = Q2 ✈➔ x† ∈ C ∩ A−1 (Q) ∩ F (T ), ✭✷✳✶✼✮ T = T2 ✳ ⑩♣ ❞ö♥❣ Pữỡ ợ tn = 1✱ αn = n ✈ỵ✐ ♠å✐ n ≥ ✈➔ u(t) = x0 (t) = ❡①♣(t2 + 1) ✈ỵ✐ ♠å✐ t ∈ [0, 1]✱ t❛ ♥❤➟♥ ✤÷đ❝ ❜↔♥❣ ❦➳t q số ữợ ứ xn+1 xn P❤÷ì♥❣ ♣❤→♣ ✭✶✳✷✷✮ ❡rr xn+1 − xn < ❡rr n 10−6 9.81429e − 07 18 10−7 9.750563778e − 08 56 10−8 9.97665e − 09 174 P❤÷ì♥❣ ♣❤→♣ ✭✷✳✶✮ ❡rr xn+1 − xn n 10−6 8.10708e − 07 17 10−7 4.17743e − 08 41 10−8 9.28195e − 09 831 ❇↔♥❣ ✷✳✸✿ ❇↔♥❣ ❦➳t q✉↔ sè ❝❤♦ ❇➔✐ t♦→♥ ✭✷✳✶✼✮ ✸✾ ❉→♥❣ ✤✐➺✉ ❝õ❛ ♥❣❤✐➺♠ ①➜♣ ①➾ xn (t) ❝❤♦ tr÷í♥❣ ❤đ♣ xn+1 − xn < 10−6 tr♦♥❣ ❇↔♥❣ ✷✳✸ ✤÷đ❝ ♠ỉ t tr ữợ Algorithm (2.1) Algorithm (1.16) 2.5 1.5 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 ❍➻♥❤ ✷✳✸✿ ❉→♥❣ ✤✐➺✉ ❝õ❛ xn(t) ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❞ø♥❣ 0.7 0.8 0.9 xn+1 − xn < 10−6 ✹✵ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❦❤→ ❝❤✐ t✐➳t ✈➔ ❤➺ t❤è♥❣ ✈➲ ❝→❝ ✈➜♥ ✤➲ s❛✉✿ • ▼ët sè t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♣❤↔♥ ①↕✱ ❦❤ỉ♥❣ ỗ pỗ ổ trỡ ✤➲✉✱ q ✲trì♥ ✤➲✉✱ →♥❤ ①↕ ✤è✐ ♥❣➝✉❀ • ❑❤♦↔♥❣ ❝→❝❤ ❇r❡❣♠❛♥✱ ♣❤➨♣ ❝❤✐➳✉ ❇r❡❣♠❛♥❀ • ❇➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤✱ t♦→♥ tû ❇r❡❣♠❛♥ ❦❤ỉ♥❣ ❣✐➣♥ ♠↕♥❤ tr→✐❀ • ❈→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❚✉②❡♥ ❚✳▼✳ ✈➔ ❍❛ ◆✳❙✳ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✶✼❪ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❧❛✐ ❣❤➨♣ t➻♠ ♠ët ♥❣❤✐➺♠ ❝❤✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❝❤➜♣ ♥❤➟♥ t→❝❤ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤✉♥❣ ❝❤♦ ❝→❝ t♦→♥ tû ❇r❡❣♠❛♥ ❦❤æ♥❣ ❣✐➣♥ ♠↕♥❤ tr→✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❨✳■✳ ❆❧❜❡r✱ ▼❡tr✐❝ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ ♣r♦❥❡❝t✐♦♥s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✿ ♣r♦♣❡rt✐❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ✐♥ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ◆♦♥❧✐♥❡❛r ❖♣❡r❛t♦rs ♦❢ ❆❝✲ ❝r❡t✐✈❡ ❛♥❞ ▼♦♥♦t♦♥❡ ❚②♣❡ ✈♦❧ ✶✼✽ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱❯❙❆✱ ❉❡❦❦❡r✱ ◆❡✇ ❨♦r❦✱ ◆❨✱ ♣♣✳ ✶✺✲✺✵ ✭✶✾✾✻✮ ❬✷❪ ❆❣❛r✇❛❧ ❘✳ P✳✱ ❖✬❘❡❣❛♥ ❉✳✱ ❙❛❤✉ ❉✳ ❘✳ ✭✷✵✵✾✮✱ ▲✐♣s❝❤✐t③✐❛♥✲t②♣❡ ▼❛♣♣✐♥❣s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❢♦r ❙♣r✐♥❣❡r✳ ❬✸❪ ❇②r♥❡ ❈✳ ✭✷✵✵✷✮✱ ✏■t❡r❛t✐✈❡ ♦❜❧✐q✉❡ ♣r♦❥❡❝t✐♦♥ ♦♥t♦ ❝♦♥✈❡① s❡ts ❛♥❞ t❤❡ s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠✑✱ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✶✽ ✭✷✮✱ ♣♣✳ ✹✹✶✕✹✺✸✳ ❬✹❪ ❇②r♥❡ ❈✳ ✭✷✵✵✹✮✱ ✏❆ ✉♥✐❢✐❡❞ tr❡❛t♠❡♥t ♦❢ s♦♠❡ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠s ✐♥ s✐❣♥❛❧ ♣r♦❝❡ss✐♥❣ ❛♥❞ ✐♠❛❣❡ r❡❝♦♥str✉❝t✐♦♥✑✱ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✶✽ ✱ ♣♣✳ ✶✵✸✕✶✷✵✳ ❬✺❪ ❇✉t♥❛r✐✉ ❉✳✱ ❘❡s♠❡r✐t❛ ❊✳ ✭✷✵✵✻✮✱ ✏❇r❡❣♠❛♥ ❞✐st❛♥❝❡s✱ t♦t❛❧❧② ❝♦♥✈❡① ❢✉♥❝✲ t✐♦♥s ❛♥❞ ❛ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ♦♣❡r❛t♦r ❡q✉❛t✐♦♥s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✑✱ ❆♣♣❧✳ ❆♥❛❧✳✱ ✷✵✵✻ ❆❜str✳ ✱ ♣♣✳ ✶✕✸✾✳ ❬✻❪ ❈❡♥s♦r ❨✳✱ ❊❧❢✈✐♥❣ ❚✳ ✭✶✾✾✹✮✱ ✏❆ ♠✉❧t✐♣r♦❥❡❝t✐♦♥ ❛❧❣♦r✐t❤♠ ✉s✐♥❣ ❇r❡❣♠❛♥ ♣r♦❥❡❝t✐♦♥s ✐♥ ❛ ♣r♦❞✉❝t s♣❛❝❡✑✱ ◆✉♠❡r✳ ❆❧❣♦r✐t❤♠s✱ ✽ ✭✷✲✹✮✱ ♣♣✳ ✷✷✶✕✷✸✾✳ ❬✼❪ ❈❡♥s♦r ❨✳✱ ▲❡♥t ❆✳ ✭✶✾✽✶✮✱ ✑❆♥ ✐t❡r❛t✐✈❡ r♦✇✲❛❝t✐♦♥ ♠❡t❤♦❞ ❢♦r ✐♥t❡r✈❛❧ ❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣✑✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳✱ ✸✹ ✱ ♣♣✳ ✸✷✶✕✸✺✸✳ ❬✽❪ ❈❡♥s♦r ❨✳✱ ❘❡✐❝❤ ❙✳ ✭✶✾✾✻✮✱ ✏■t❡r❛t✐♦♥s ♦❢ ♣❛r❛❝♦♥tr❛❝t✐♦♥s ❛♥❞ ❢✐r♠❧② ♥♦♥✲ ❡①♣❛♥s✐✈❡ ♦♣❡r❛t♦rs ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❢❡❛s✐❜✐❧✐t② ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥✑✱ t✐♠✐③❛t✐♦♥✱ ✸✼ ❬✾❪ ■✳ ❈✐♦r❛♥❡s❝✉✱ Pr♦❜❧❡♠s✱ ✱ ♣♣✳ ✸✷✸✕✸✸✾✳ ●❡♦♠❡tr② ♦❢ ❇❛♥❛❝❤ ❙♣❛❝❡s✱ ❉✉❛❧✐t② ▼❛♣♣✐♥❣s ❛♥❞ ◆♦♥❧✐♥❡❛r ❑❧✉✇❡r✱ ❉♦r❞r❡❝❤t ✭✶✾✾✵✮✳ ❬✶✵❪ ❉✐❡st❡❧ ❏✳ ✭✶✾✼✵✮✱ ❱❡r❧❛❣✳ ❖♣✲ ●❡♦♠❡tr② ♦❢ ❇❛♥❛❝❤ ❙♣❛❝❡s✲❙❡❧❡❝t❡❞ ❚♦♣✐❝s✱ ❙♣r✐♥❣❡r✲ ✹✷ ❬✶✶❪ ❑✳ ●♦❡❜❡❧✱ ❲✳❆✳ ❑✐r❦✱ ❙t✉❞✳ ❆❞✈✳ ▼❛t❤✳✱ ✷✽ ❚♦♣✐❝s ✐♥ ▼❡tr✐❝ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r②✱ ✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈✳ Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ❯❑✱ ✶✾✾✵✳ ❈❧❛ss✐❝❛❧ ❇❛♥❛❝❤ ❙♣❛❝❡s ■■✿ ❋✉♥❝t✐♦♥ ❬✶✷❪ ▲✐♥❞❡♥str❛✉ss ❏✳✱ ❚③❛❢r✐r✐ ▲✳ ✭✶✾✼✾✮✱ ❙♣❛❝❡s✱ ❈❛♠❜r✐❞❣❡ ❊r❣❡❜♥✐ss❡ ▼❛t❤✳ ●r❡♥③❣❡❜✐❡t❡ ❇❞✳ ✾✼✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ❬✶✸❪ ❘❡✐❝❤ ❙✳ ✭✶✾✾✻✮✱ ✏❆ ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ❢♦r t❤❡ ❛❧t❡r♥❛t✐♥❣ ♠❡t❤♦❞ ✇✐t❤ ❇r❡❣♠❛♥ ❞✐st❛♥❝❡s✑✱ ✐♥✿ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ◆♦♥❧✐♥❡❛r ❖♣✲ ❡r❛t♦rs ♦❢ ❆❝❝r❡t✐✈❡ ❛♥❞ ▼♦♥♦t♦♥❡ ❚②♣❡✱ ▼❛r❝❡❧ ❉❡❦❦❡r✱ ◆❡✇ ❨♦r❦✱ ♣♣✳ ✸✶✸✲✸✶✽✳ ❬✶✹❪ ❘♦❝❦❛❢❡❧❧❛r ❘✳ ❚✳ ✭✶✾✼✵✮✱ ✏❖♥ t❤❡ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥✐❝✐t② ♦❢ s✉❜❞✐❢❢❡r❡♥t✐❛❧ ♠❛♣♣✐♥❣s✑✱ P❛❝✐❢✐❝ ❏✳ ▼❛t❤✳✱ ❱♦❧✳ ✸✸ ✭✶✮✱ ♣♣✳ ✷✵✾✕✷✶✻✳ ❬✶✺❪ ❙❝❤♦♣❢❡r ❋✳✱ ❙❝❤✉st❡r ❚✳✱ ▲♦✉✐s ❆✳❑✳ ✭✷✵✵✽✮✱ ✏❆♥ ✐t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✑✱ ✷✹ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✱ ✵✺✺✵✵✽✳ ❬✶✻❪ ❙❤❡❤✉ ❨✳✱ ■②✐♦❧❛ ❖✳ ❙✳✱ ❊♥②✐ ❈✳ ❉✳ ✭✷✵✶✻✮✱ ✏❆♥ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠ ❢♦r s♦❧✈✲ ✐♥❣ s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t ♣r♦❜❧❡♠s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✑✱ ◆✉♠❡r✳ ❆❧❣♦r✐t❤♠s✱ ✼✷ ✱ ♣♣✳ ✽✸✺✕✽✻✹✳ ❬✶✼❪ ❚✉②❡♥ ❚✳▼✳✱ ❍❛ ◆✳❙✳ ✭✷✵✶✽✮✱ ✏❆ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠ ❢♦r s♦❧✈✐♥❣ t❤❡ s♣❧✐t ❢❡❛s✐❜✐❧✐t② ❛♥❞ ❢✐①❡❞ ♣♦✐♥t ♣r♦❜❧❡♠s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✑✱ ❚❤❡♦r② ❆♣♣❧✳✱ ✷✵ ❏✳ ❋✐①❡❞ P♦✐♥t ✭✶✹✵✮✳ ❬✶✽❪ ❳✉ ❍✳❑✳ ✭✶✾✾✶✮✱ ✏■♥❡q✉❛❧✐t✐❡s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✑✱ ❆♥❛❧✳✱ ❬✶✾❪ ❳✉ ✶✻ ◆♦♥❧✐♥❡❛r ✱ ♣♣✳ ✶✶✷✼✕✶✶✸✽✳ ❍✳❑✳ ✭✷✵✵✻✮✱ ✏❆ ✈❛r✐❛❜❧❡ ❑r❛s♥♦s❡❧✬s❦✐✐✲▼❛♥♥ ♠✉❧t✐♣❧❡✲s❡t s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠✑✱ ❛❧❣♦r✐t❤♠ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✷✷ ❛♥❞ t❤❡ ✱ ♣♣✳ ✷✵✷✶✕✷✵✸✹✳ ❬✷✵❪ ✭✷✵✶✵✮✱ ✏■t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r t❤❡ s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✐♥ ✐♥❢✐♥✐t❡ ❞✐♠❡♥✲ s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡s✑✱ ■♥✈❡rs❡ Pr♦❜❧❡♠s✱ ✷✻ ✱ ✶✵✺✵✶✽✳ ❬✷✶❪ ❲❛♥❣ ❋✳ ✭✷✵✶✹✮✱ ✏❆ ♥❡✇ ❛❧❣♦r✐t❤♠ ❢♦r s♦❧✈✐♥❣ t❤❡ ♠✉❧t✐♣❧❡✲s❡ts s♣❧✐t ❢❡❛✲ s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✑✱ ✾✾✕✶✶✵✳ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✸✺ ✱ ♣♣✳