Các bài toán biến phân lồi

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ■■ ❑❍❖❆ ❚❖⑩◆ ✖✖✖✖✖✖✖✖✖✖♦✵♦✖✖✖✖✖✖✖✖✖✖ ✣➄◆● ❚❍➚ ▲❆◆ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ❇■➌◆ P❍❹◆ ▲➬■ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ ữợ ◆❐■✱ ✹✴✷✵✶✹ ▲❮■ ❈❷▼ ❒◆ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ỡ tợ t ổ trữớ ữ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✤➣ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❦❤♦→ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ✣➦❝ ❜✐➺t tæ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ❱➠♥ ❚✉②➯♥ ◆❣✉②➵♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤♦→ ❧✉➟♥ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➔ ❤↕♥ ❝❤➳✳ ❑➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü õ õ ỵ t ổ ✈➔ t♦➔♥ t❤➸ ❜↕♥ ✤å❝ ✤➸ ✤➲ t➔✐ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✹ ❚→❝ ❣✐↔ ✣➦♥❣ ❚❤à ▲❛♥ ✐ ổ ữợ sỹ ữợ ❞➝♥ ❝õ❛ t❤➛② ❣✐→♦ ❚✉②➯♥ ◆❣✉②➵♥ ❱➠♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ tổ ữủ t ổ trũ ợ t t➔✐ ♥➔♦ ❦❤→❝✳ ❚r♦♥❣ ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ♥➔② tỉ✐ ✤➣ sû ❞ư♥❣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ ❧á♥❣ ❜✐➳t ì♥ tr➙♥ trå♥❣✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ✈✐➯♥ ✣➦♥❣ ❚❤à ▲❛♥ ✐✐ ▼ö❝ ❧ö❝ ▲❮■ ▼Ð ✣❺❯ ✶ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✹ ✶✳✶ ✶✳✷ ✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✶ ❈→❝ ❦➼ ❤✐➺✉ ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✷ ❈→❝ ✤➥♥❣ t❤ù❝ ✈➔ ❜➜t ✤➥♥❣ t❤ù❝ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✸ P❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✶✳✹ ❚æ♣æ ♠↕♥❤ ✈➔ ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✺ ❙ü ❤ë✐ tö ②➳✉ ❝õ❛ ❞➣② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❚➟♣ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷✳✷ P❤➨♣ ❝❤✐➳✉ t❤❡♦ ❝❤✉➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✸ ❚➼♥❤ ❝❤➜t tæ♣æ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t t ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ t ổ ỗ ❈→❝ ❜✐➳♥ t❤➸ ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷ ❈⑩❈ ❇⑨■ ❚❖⑩◆ ❇■➌◆ P❍❹◆ ▲➬■ ✹✷ ✷✳✶ ❈➟♥ tr➯♥ ú ữợ ú ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✷ ❈ü❝ t✐➸✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✐✐✐ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✷✳✸ ❚➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ❝ü❝ t✐➸✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t ❝ü❝ t✐➸✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✺ ❉➣② ❝ü❝ t✐➸✉ ❤♦→ ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➌❚ ▲❯❾◆ ✻✵ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✻✶ ✣➦♥❣ ❚❤à ▲❛♥ ✐✈ ❑✸✻❈ ❙P ❚♦→♥ ▲❮■ ▼Ð ✣❺❯ ❝♦♥✈❡① ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠s✮ ❤❛② ❝á♥ ✤÷đ❝ ❣å✐ ❧➔ ❝→❝ t tố ữ ỗ tt rs t ỗ ởt ợ t q✉② ❤♦↕❝❤ t♦→♥ ❤å❝ ❝ü❝ ❦➻ q✉❛♥ trå♥❣✱ ♥â ❜❛♦ ❧❡❛st✲sq✉❛r❡s ♣r♦❜❧❡♠s✮ ✈➔ ❝→❝ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✭❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠s✮✳ ❤➔♠ ❝→❝ ❜➔✐ t♦→♥ ❜➻♥❤ ♣❤÷ì♥❣ tè✐ t❤✐➸✉ ✭ ❈→❝ ❜➔✐ t♦→♥ ♥➔② ❝â r➜t ♥❤✐➲✉ ù♥❣ ỵ tt õ t tr tữỡ ✤è✐ ✤➛② ✤õ✳ ❇➔✐ t♦→♥ ❜➻♥❤ ♣❤÷ì♥❣ tè✐ t❤✐➸✉ ✤➣ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tø t❤➳ ❦➾ t❤ù ✶✺ ✈➔ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ t✉②➳♥ t➼♥❤ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tø ✤➛✉ t❤➳ ❦➾ ✶✾✳ ❚r♦♥❣ ❦❤✐ ✤â✱ ❝→❝ ❜➔✐ t♦→♥ ❜✐➳♥ ỗ ợ ữủ ự tứ ỳ ❝✉è✐ ❝õ❛ t❤➳ ❦➾ ✶✾ ✲ ✤➛✉ t❤➳ ❦➾ ✷✵ ❜ð✐ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ♥❤÷✿ ❈❛r❛t❤❡♦❞♦r②✱ ▼✐♥❦♦✇s❦✐✱ ❙t❡✐♥✐t③✱ ❋❛r❦❛s✳ ❚✐➳♣ t❤❡♦✱ ♥❤ú♥❣ ♥➠♠ ✹✵ ✤➳♥ ♥➠♠ ✺✵ ❝õ❛ t❤➳ ❦➾ ✷✵ ✤➣ ❝â ♥❤ú♥❣ ♥❣❤✐➯♥ ❝ù✉ ♠❛♥❣ t➼♥❤ ✤ët ữ tr ỵ tt trỏ ỡ r tr ỵ tt ố t tố ữ ợ t ỗ ữủ ❝ù✉ ❜ð✐ ❋❡♥❝❤❡❧✳ ❚✐➳♣ s❛✉ ✤â ❧➔ ❝→❝ ❝æ♥❣ tr➻♥❤ ❝õ❛ ❚✳ ❘✳ ❘♦❝❦❛❢❡❧❧❛r ✈➔♦ t❤➟♣ ♥✐➯♥ ✻✵ ✈➔ ✼✵ ❝õ❛ t❤➳ ❦➾ ✷✵✳ ❚✳ ❘✳ ❘♦❝❦❛❢❡❧❧❛r ✤➣ ✤÷❛ r❛ ❝→❝ ❦➳t q✉↔ q✉❛♥ trå♥❣ tr♦♥❣ t❤í✐ ❣✐❛♥ ♥➔② ✈➔ ữớ t tr ỵ tt t ỗ ởt ❝→❝❤ trå♥ ✈➭♥ ♥❤➜t✳ ▼ët ❝➙✉ ❤ä✐ ✤➦t r❛ ❧➔ t s t ỗ õ ởt trỏ q ✶ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ trå♥❣ ♥❤÷ t❤➳ tr♦♥❣ ❧➼ t❤✉②➳t tè✐ ÷✉❄ Ð ✤➙② ❝❤ó♥❣ tỉ✐ ❝â t❤➸ tr➻♥❤ ởt số ỵ ữ s ợ ởt ỗ t ỹ t ữỡ ỹ t t ởt t ỗ tr ❦❤ỉ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❧✉ỉ♥ ❝â ♣❤➛♥ tr♦♥❣ t÷ì♥❣ ố rộ ởt t ỗ t tổ õ ữợ ữủ t ỹ tỗ t ởt ỹ t t ởt ỗ tr ởt t ỗ õ t ✤÷đ❝ ✤➦❝ tr÷♥❣ ❜ð✐ ❝→❝ ♣❤➛♥ tû ❝õ❛ ♥â♥ ❧ò✐ ởt t ỗ õ t tr÷♥❣ ❜ð✐ ❤ú✉ ❤↕♥ ❝→❝ ✤✐➸♠ ❝ü❝ ❜✐➯♥ ✈➔ ❝→❝ ữợ ỹ ởt õ ỗ õ tỹ ố tữỡ ự ợ õ ỹ õ ởt ỗ ỷ tử ữợ tỹ ố tữỡ ự ợ ủ õ ããã ứ ỳ ỵ tr ữủ sỹ ữợ t ữợ tổ t t ỗ õ tốt õ ỗ ữỡ ữỡ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ❣✐↔✐ t ỗ r ữỡ õ tr ởt số t t ỡ ổ rt t ỗ ỗ ởt số tự ỡ ữủ sû ❞ư♥❣ tr♦♥❣ ❈❤÷ì♥❣ ✷✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ t ỗ r ữỡ ▲❛♥ ✷ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❝❤ó♥❣ tổ ợ t ữ tr ú ữợ ú ỹ t ♠ët ❜➔✐ t♦→♥ tè✐ ÷✉✳ ❚✐➳♣ t❤❡♦ ✤â ❝❤ó♥❣ tỉ✐ tr sỹ tỗ t t t ỹ t✐➸✉ ✈➔ ❞➣② ❝ü❝ t✐➸✉ ❤â❛ ❝õ❛ ♠ët ❜➔✐ t♦→♥ ỗ P ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ ❑❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❦❤â❛ ❧✉➟♥✳ ❈❤♦ ổ ữợ d ã|ã ã H ởt ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈ỵ✐ t➼❝❤ ❧➔ ❝❤✉➞♥ s✐♥❤ ❜ð✐ t➼❝❤ ổ ữợ ợ ữủ ữ s (∀x ∈ H), (∀y ∈ H), t❛ ❝â ||x|| = x|x ✈➔ d(x, y) = ||x − y|| ✭✶✳✶✮ ❚♦→♥ tỷ ỗ t tự tr H ✶✳✶✳✶ ❈→❝ ❦➼ ❤✐➺✉ ✈➔ ✈➼ ❞ö ✣à♥❤ ♥❣❤➽❛ ✶✳✶ X ❈❤♦ ❝♦♥ ❝õ❛ X✳ ❱ỵ✐ ♠é✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì t❤ü❝✱ x ∈ R, λC = {λx|x ∈ C}✳ C C ❧➔ t➟♣ ❤đ♣ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ❛❢❢✐♥❡ ♥➳✉ C=∅ ●✐↔ sû X ❝❤ù❛ ✈➔ (∀λ ∈ R) C = λC + (1 − λ)C ✭✶✳✷✮ C = ∅✱ ❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ C✱ tù❝ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ♥❤ä ♥❤➜t ❝õ❛ ✹ X ❝❤ù❛ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ C ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ s♣❛♥C ✱ ❜❛♦ ✤â♥❣ ❝õ❛ ♥â ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ X t➼♥❤ ✤â♥❣ ♥❤ä ♥❤➜t ❝õ❛ ❝❤ù❛ P❤➛♥ ❜ò trü❝ ❣✐❛♦ ❝õ❛ t➟♣ C C⊆H ✈➔ ♥â ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ s♣❛♥C ✳ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ C ⊥ = {x ∈ H | (∀x ∈ C) ▼ët t➟♣ trü❝ ❣✐❛♦ ♥➳✉ s♣❛♥C = H✳ C⊂H I H I H ✤÷đ❝ ❣å✐ ❧➔ t→❝❤ ✤÷đ❝ ✭ x∈H ♠➔ i∈J H ❦❤↔ ❧✐✮ ♥➳✉ (xi )i∈I ❧➔ ❤å ❝→❝ ❧➔ ❧ỵ♣ ❝→❝ t ỳ rộ ữủ ữợ q tỗ t ữủ ởt ❝ì sð trü❝ ❣✐❛♦ ❝õ❛ ❑❤ỉ♥❣ ❣✐❛♥ ✈➔ ❣✐↔ sû tù❝ ❧➔✿ x|u = 0} ❝â ♠ët ❝ì sð trü❝ ❣✐❛♦ ✤➳♠ ✤÷đ❝✳ ❇➙② ❣✐í✱ ❣✐↔ sû ✈❡❝tì tr♦♥❣ C ⊥✱ xi J∈I ⊂✳ ❑❤✐ ✤â ❤ë✐ tö ✤➳♥ (xi )i∈I x ❧➔ ❦❤↔ tê♥❣ ♥➳✉ tù❝ ❧➔✿ (∀ε ∈ R++ ) (∀K ∈ I) (∀J ∈ I) J ⊃ K ⇒ x − xi ε ✭✶✳✹✮ i∈J ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❝❤ó♥❣ t❛ ✈✐➳t tr♦♥❣ [0, +∞]✱ x = i∈I xi ✳ ✣è✐ ✈ỵ✐ (αi )i∈I ❝❤ó♥❣ t❛ ❝â✿ αi = sup i∈I J∈I αi ✭✶✳✺✮ i∈J ✣➙② ❧➔ tr÷í♥❣ ❤đ♣ r✐➯♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈➔ ♥â ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❦❤♦→ ❧✉➟♥ ♥➔②✳ ❱➼ ❞ö ✶✳✶✳ t❤ü❝ ❚ê♥❣ trü❝ t✐➳♣ ❍✐❧❜❡rt ❝õ❛ ♠ët ❤å ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt (Hi , || · ||i )i∈I Hi = ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳ i∈I ✣➦♥❣ ❚❤à ▲❛♥ ||xi ||2i < +∞ x = (xi )i∈I ∈ ×i∈I Hi ✭✶✳✻✮ i∈I ✺ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ tốt ỹ tỗ t ỹ t ỵ f : H [, +] tỹ ỗ ỷ tử ữợ t ủ õ ỗ H s ❝❤♦ ✈ỵ✐ ♠é✐ ξ ∈ R, C ∩ ❧❡✈≤ξ f ❦❤→❝ ré♥❣ ✈➔ ❜à ❝❤➦♥✳ ❚❤➻ f ❝â ♠ët ✤✐➸♠ ❝ü❝ t✐➸✉ tr➯♥ C ✳ C ❈❤ù♥❣ ♠✐♥❤✳ ❚ø f ỷ tử ữợ t t ❤ñ♣ ❉♦ ✤â D ✶✳✺ tr♦♥❣ D = C ∩ C t õ ỗ tứ tt tữỡ ữỡ ợ H f t ❝ü❝ t✐➸✉ tr➯♥ D✳ f ✤↕t ❝ü❝ ❉♦ ✤â tø ❇ê ✤➲ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✹ ❈❤♦ f : H → [−∞, +∞] ✳ ❚❤➻ f ✤÷đ❝ ❣å✐ ❧➔ ✭❝♦❡r❝✐✈❡✮ ♥➳✉ lim f (x) = +∞ ❜ù❝ ✭✷✳✺✮ x →+∞ ✈➔ ❧➔ ✈➔ ❧❡✈≤ξ f õ ỗ t ủ rộ ✈➔ ❝♦♠♣❛❝t ②➳✉✳ ❱➻ ✈➟② C t✐➸✉ tr➯♥ f ❧➔ tỹ ỗ ỷ tử ữợ s r s✐➯✉ ❜ù❝ ✭s✉♣❡r❝♦❡r❝✐✈❡✮ ♥➳✉ f (x) = +∞ →+∞ x lim x ữợ f ự s ự ♥➳✉ ✭✷✳✻✮ H = {0}✳ ▼➺♥❤ ✤➲ ✷✳✻ ❈❤♦ f : H → [−∞, +∞]✳ ❚❤➻ f ❧➔ ❜ù❝ ♥➳✉ õ õ ởt t ủ ự ữợ (❧❡✈≤ξ f )ξ∈R ❧➔ ❜à ❝❤➦♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ✈ỵ✐ ♠é✐ ξ ∈ R, t❛ ❝â t❤➸ ❝❤å♥ ❞➣② ✣➦♥❣ ❚❤à ▲❛♥ (xn )n∈N ❧❡✈≤ξ f ❦❤æ♥❣ ❜à ❝❤➦♥✳ ❑❤✐ ✤â tr♦♥❣ ❧❡✈≤ξ f s❛♦ ❝❤♦ ✹✼ xn → +∞✳ ❙✉② ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ f r❛ ❦❤ỉ♥❣ ❧➔ ❜ù❝✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû t➟♣ ❤đ♣ ự ữợ ợ ộ ξ ∈ R++ ✱ (xn )n∈N t❛ ❧➜② tr♦♥❣ N ∈N H ❧➔ xn → +∞✳ ❑❤✐ ✤â inf n≥N f (xn ) ≥ ξ ✳ ❙✉② r❛ s❛♦ ❝❤♦ s❛♦ ❝❤♦ f f (xn ) → +∞✳ ▼➺♥❤ ✤➲ ✷✳✼ ●✐↔ sû ❦❤æ♥❣ ❣✐❛♥ H ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ f ∈ Γ0(H)✳ ❚❤➻ f ❧➔ ❜ù❝ ♥➳✉ ✈➔ ❝❤➾ tỗ t R s ( f ) ❦❤→❝ ré♥❣ ✈➔ ❜à ❝❤➦♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ f ự t tt t ủ ự ữợ ( f )ξ∈R ❜à ❝❤➦♥ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✻✳ ●✐↔ sû ❧➜② (❧❡✈≤ξ f ) ❦❤→❝ ré♥❣ ✈➔ ❜à ❝❤➦♥ ✈➔ x ∈ ❧❡✈≤ξ f ✳ ❘ã r➔♥❣ t➜t ❝↔ ❝→❝ t ủ ự ữợ [ξ, +∞] ✈➔ ❣✐↔ sû ❧❡✈≤η f ▲➜② ❦❤æ♥❣ ❜à ❝❤➦♥✱ s✉② r❛ r❡❝❧❡✈≤η f y ∈ r❡❝❧❡✈≤η f ✱ tø x ∈ ❧❡✈≤η f ❱ỵ✐ ♠é✐ s✉② r❛ = {0}✳ (∀λ ∈ R++ ) x+λy ∈ ❧❡✈≤η f ✳ λ ∈ [1, +∞]✱ t❛ ❝â x + y = (1 − λ−1 )x + λ−1 (x + λy)✳ ❉♦ ✤â f (x + y) ≤ (1 − λ−1 )f (x) + λ−1 f (x + λy) ≤ (1 − λ−1 )f (x) + λ−1 η)✳ ❚ø ✤â t❛ ❝â ❙✉② r❛ λ(f (x + y) − f (x)) ≤ η − f (x) x + r❡❝❧❡✈≤η f ⊂ ❧❡✈≤ξ f f (x + y) ≤ f (x) ≤ η ✳ ✭❧❡✈≤ξ f ❜à ❝❤➦♥ ✈➔ ❧❡✈≤η f ❦❤æ♥❣ ❜à õ tt t ủ ự ữợ f ✈➔ (❧❡✈≤ξ f )ξ∈R ❧➔ ❜à ❝❤➦♥✳ ❱➻ ✈➟② ❧➔ ❜ù❝ ✭t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✻✮✳ ▼➺♥❤ ✤➲ ✷✳✽ ▲➜② f ∈ Γ0(H) ✈➔ g : H → [−∞, +∞] ❧➔ s✐➯✉ ❜ù❝✳ ❚❤➻ f + g ❧➔ s✐➯✉ ❜ù❝✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â f ✤÷đ❝ ❧➔♠ trë✐ ❜ð✐ ❤➔♠ ❛❢❢✐♥❡ ❧✐➯♥ tö❝✳ ❚❛ ♥â✐ x → x|u + η ✱ ð ✤â u ∈ H, η ∈ R✱ ❞♦ ✤â (∀x ∈ H) f (x) + g(x) ≥ x|u + η + g(x) ≥ − x ✣➦♥❣ ❚❤à ▲❛♥ ✹✽ u + η + g(x) ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❚ø ✤â t❛ ❝â f (x) + g(x) η + g(x) ≥− u + → +∞ x x ❦❤✐ x → +∞✳ ▼➺♥❤ ✤➲ ✷✳✾ ▲➜② f ∈ 0(H) C t ủ õ ỗ ❝õ❛ s❛♦ ❝❤♦ C ∩ ❞♦♠f ✤ó♥❣ H ✭✐✮ f ❧➔ ❜ù❝✳ ✭✐✐✮ C ❜à ❝❤➦♥✳ = ∅✳ ●✐↔ sû ♠ët tr♦♥❣ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ❚❤➻ f ❝â ♠ët ❝ü❝ t✐➸✉ tr➯♥ C ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø C ∩ ❞♦♠f C ∩ ❧❡✈≤f (x) f f = ∅✱ s r x f õ ỗ ré♥❣✳ ❍ì♥ ♥ú❛✱ ❝â ♠ët ❝ü❝ t✐➸✉ tr➯♥ D s❛♦ ❝❤♦ D = ❜à ❝❤➦♥✱ s✉② r❛ C✳ ❍➺ q✉↔ ✷✳✷ f, g ∈ Γ0(H)✳ ●✐↔ sû ❞♦♠f ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ ✤ó♥❣✿ ✭✐✮ f ❧➔ s✐➯✉ ❜ù❝✳ ✭✐✐✮ f ự g ữợ g = ∅ ✈➔ ♠ët tr♦♥❣ ❚❤➻ f + g ❧➔ ❜ù❝ ✈➔ ♥â ❝â ❝ü❝ t✐➸✉ tr➯♥ H✳ ◆➳✉ f g ỗ t t f + g õ ✤ó♥❣ ♠ët ❝ü❝ t✐➸✉ tr➯♥ H✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â f, g ∈ Γ0(H) ♥➯♥ f +g ∈ Γ0(H)✳ ❉♦ ✤â tø ▼➺♥❤ ✤➲ ✷✳✾✭✐✮ t❛ t❤➜② f +g ❧➔ ❜ù❝ tr♦♥❣ ❝↔ ✷ tr÷í♥❣ ❤đ♣✳ ❉♦ f, g ∈ (H) t q ợ ú ỵ f +g ỗ t t õ t t ❝õ❛ ❝ü❝ t✐➸✉✳ ✣➦♥❣ ❚❤à ▲❛♥ ✹✾ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✭✐✮ ❚ø ▼➺♥❤ ✤➲ ✷✳✽ ✈ỵ✐ f, g ∈ Γ0 (H) s✉② r❛ f +g ❧➔ s✐➯✉ ❜ù❝ ♥➯♥ ♥â ❧➔ ❜ù❝ ✳ ✭✐✐✮ ❚➟♣ ❤ñ♣ µ = inf g(H) > −∞ ⇒ (f + g)(x) ≥ f (x) + µ → +∞ x → +∞ ❦❤✐ ✭ t➼♥❤ ❜ù❝ ❝õ❛ f ✮✳ ❍➺ q✉↔ ✷✳✸ f 0(H) ỗ f s✐➯✉ ❜ù❝ ✈➔ ❝â ✤ó♥❣ ♠ët ❝ü❝ t✐➸✉ tr➯♥ H✳ ự ủ q ỗ ợ sè f = βq + (f − βq) f ❙✉② r❛ · ✳ ❚ø ▼➺♥❤ ✤➲ ✶✳✸✸ ❣✐↔ sû f β ∈ R++ t❤➻ f − βq ỗ = tờ s ❜ù❝ βq ✈➔ ❧➔ r❛ f − βq ∈ Γ0 (H)✳ ❧➔ s✐➯✉ ❜ù❝ ✭t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✽✮ ✈➔ t❤❡♦ ❍➺ q✉↔ ✷✳✷ t❤➻ f ❝â ✤ó♥❣ ♠ët ❝ü❝ t✐➸✉✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✺ (xn )n∈N ❈❤♦ C ❧➔ ♠ët ❞➣② tr tữỡ ự ợ C t ủ ré♥❣ ❝õ❛ H✳ ❚❤➻ (xn )n∈N ✤÷đ❝ ❣å✐ ❧➔ H ✈➔ ❧➜② ✤ì♥ ✤✐➺✉ ❋❡❥➨r ♥➳✉ (∀x ∈ C) (∀n ∈ N) xn+1 − x ≤ xn − x ✭✷✳✼✮ ▼➺♥❤ ✤➲ ✷✳✶✵ ✭❚✐➺♠ ❝➟♥ tr✉♥❣ t➙♠✮ ❈❤♦ (zn)n∈N ❧➔ ❞➣② ❜à ❝❤➦♥ tr♦♥❣ H✱ C ❧➔ t➟♣ ❤ñ♣ õ ỗ rộ H T : C → C ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✱ t➟♣ f : H → R : x → ❧✐♠ x−zn ✳ ❑❤✐ ✤â t❛ ❝â✿ ✭✐✮ f ❧➔ ❤➔♠ số ỗ ợ số f s✐➯✉ ❜ù❝✳ ✣➦♥❣ ❚❤à ▲❛♥ ✺✵ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✭✐✐✐✮ ✭✐✈✮ ✭✈✮ ✭✈✐✮ ✭✈✐✐✮ ❧➔ ❤➔♠ ỗ s ự õ õ t ởt ❝ü❝ t✐➸✉ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ zC ✤÷đ❝ ❣å✐ ❧➔ t✐➺♠ ❝➟♥ tr✉♥❣ t➙♠ ❝õ❛ (zn)n∈N t÷ì♥❣ ✤è✐ ✤➳♥ C ✳ f +ιC ●✐↔ sû z ∈ H ✈➔ zn ✈➔ zC = PC z ✳ z✳ ❚❤➻ ∀x ∈ H f (x) = x−z + f (z) ●✐↔ sû (zn)nN ỡ r tữỡ ự ợ C ❚❤➻ PC zn → zC ✳ ●✐↔ sû (∀n ∈ N)zn+1 = T zn✳ ❚❤➻ zC ∈ ❋✐①T ✳ ●✐↔ sû zn − T zn → 0✳ ❚❤➻ zC ∈ ❋✐①T ✳ ❈❤ù♥❣ ♠✐♥❤✳ r❛ ✭✐✮ ▲➜② x, y ∈ H ✈➔ α ∈ [0, 1]✳ (∀n ∈ N) αx + (1 − α)y − zn zn − α(1 − α) x − y 2 ❚ø ❍➺ q✉↔ ✶✳✶✱ s✉② = α x − zn + (1 − α) y − ✳ ❇➙② ❣✐í t❛ ✤✐ ❧➜② ❣✐ỵ✐ ❤↕♥ tr➯♥✱ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✭✐✐✮ ứ f ỗ ợ số s✉② r❛ f ❧➔ s✐➯✉ ❜ù❝✳ ✭✐✐✐✮ ❉♦ f + C t q ỗ ♠↕♥❤ ✈➔ s✐➯✉ ❜ù❝ ♥➯♥ ❝ô♥❣ t❤❡♦ ❍➺ q✉↔ ✷✳✸ t❤➻ ♥â ❝â ✤ó♥❣ ♠ët ❝ü❝ t✐➸✉ ✈➔ t❛ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✐✈✮ ▲➜② x ∈ H✳ ❑❤✐ ✤â (∀n ∈ N) x − zn x − z|z − zn ❙✉② r❛ ✭✈✮ ❚❛ ❝â PC z ✳ ❱➻ ✈➟② f + ιC ✳ ✈ỵ✐ ♠é✐ ✺✶ + z − zn = x−z ❚ø ✭✐✐✐✮✱ s✉② r❛ n ∈ N, y ∈ C y − PC zn + PC zn − zn ≤ ✣➦♥❣ ❚❤à ▲❛♥ = x−z f (x) = ❧✐♠ x − zn ❝ü❝ t✐➸✉ ❤♦→ y = lim PC zn zC ✳ 2 + + f (z)✳ zC = PC z ✳ t❛ ❝â✿ y − zn ≤ y − PC zn + y − zn ✳ ❉♦ ✤â ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ lim y − zn ≤ inf lim y − zn y∈C zC ∈ C ✈➔ T zC ∈ C ✈ỵ✐ ♠é✐ T zC − T zn ≤ zC − zn ❚ø ✭✐✐✐✮✱ s✉② r❛ zC − zn f + ιC ✳ ❧➔ ❝ü❝ t✐➸✉ ❝õ❛ ❚ø n ∈ N✳ ✳ ❉♦ ✤â ❙✉② r❛ T zC − zn+1 = (f + ιC ) (T zC ) ≤ (f + ιC )(zC )✳ T zC = zC ✳ n ∈ N, ✭✈✐✐✮ ✈ỵ✐ ♠é✐ y y = zC ✳ ✭✐✐✐✮✱ s✉② r❛ ✭✈✐✮ ❚ø ✈➔ T zC − zn ≤ T zC − T zn + T zn − zn ≤ ❉♦ ✤â lim T zC − zn ≤ lim zC − zn ιC )(T zC ) ≤ (f + ιC )(zC )✳ ▲↕✐ tø ✭✐✐✐✮✱ s✉② r❛ ✳ ❱➻ ✈➟② (f + T zC = zC ✳ ❍➺ q✉↔ ✷✳✹ ▲➜② C ❧➔ t➟♣ ❤ñ♣ õ ỗ rộ H →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ ❱ỵ✐ ♠é✐ z0 ∈ C t➟♣ ❤ñ♣ (∀n ∈ N)zn+1 = T zn ✳ ❑❤✐ ✤â ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿ T : C → C ✭✐✮ ❋✐①T = ∅✳ ✭✐✐✮ (zn )n∈N ❜à ❝❤➦♥ ✈ỵ✐ ∀z0 ∈ C ✳ ✭✐✐✐✮ (zn )n∈N ❜à ❝❤➦♥ ✈ỵ✐ z0 ∈ C ♥➔♦ ✤â ✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮⇒ ✭✐✐✮❍✐➸♥ ♥❤✐➯♥✳ ✭✐✐✮⇒ ✭✐✐✐✮ ❍✐➸♥ ♥❤✐➯♥✳ ✭✐✐✐✮⇒ ✭✐✮ ❱ỵ✐ z0 ∈ C ♥➔♦ ✤â (zn )n∈N t❤➻ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✵✭✈✐✮ t❛ ❝â ❜à ❝❤➦♥ ✈➔ zC ∈ ❋✐①T ✳ (∀n ∈ N)zn+1 = T zn ❙✉② r❛ ❋✐①T = ∅✳ ✷✳✺ ❉➣② ❝ü❝ t✐➸✉ ❤♦→ ✣à♥❤ ♥❣❤➽❛ ✷✳✻ ❈❤♦ tr♦♥❣ ❞♦♠f ✳ ❚❤➻ (xn )n∈N ✣➦♥❣ ❚❤à ▲❛♥ f : X → [−∞, +∞] ❧➔ ✈➔ (xn )n∈N ❧➔ ♠ët ❞➣② ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ f ♥➳✉ ✺✷ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ f (xn ) → inf f (X )✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ t❛ ✤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❞➣② ❝ü❝ t✐➸✉ ❤♦→✳ ▼➺♥❤ ✤➲ ✷✳✶✶ ❈❤♦ f : H → [−∞, +∞] ❧➔ ❤➔♠ ❜ù❝✱ ❝❤➼♥❤ t❤÷í♥❣✳ ❚❤➻ ♠é✐ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ f ✤➲✉ ❜à ❝❤➦♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ❍✐➸♥ ♥❤✐➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✻ ✈➔ ✣à♥❤ ♥❣❤➽❛ ✷✳✻✳ f tỹ ỗ tữớ ỷ tử ữợ (xn)nN ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ f ❤ë✐ tö ②➳✉ ✤➳♥ ♠ët ✤✐➸♠ x ∈ H ♥➔♦ ✤â✳ ❚❤➻ f (x) = inf f (H)✳ : H → [−∞, +∞] ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ▼➺♥❤ ✤➲ ✶✳✸✾ t❛ ❧✉æ♥ ❝â f ❧➔ ♥û❛ tử ữợ t õ inf f (H) ≤ f (x) ≤ limf (xn ) = inf f (H) ú ỵ f (H) (xn )n∈N ✭✐✮ ●✐↔ sû (xn )n∈N ✈➔ ❧➜② ❧➔ ❞➣② tr♦♥❣ ❞♦♠f ✳ ❤ë✐ tö ♠↕♥❤ ✤➳♥ ✤✐➸♠ ❝ü❝ t✐➸✉ H = R ❤♦➦❝ x ∈ ✐♥t ❞♦♠f ✳ ❚❤➻ (xn )n∈N f✳ ❚❤➟t ✈➟②✱ t❛ ❧✉æ♥ ❝â (xn )n∈N ✭✐✐✮ ●✐↔ sû (xn )n∈N ❤ñ♣ H ✭✐✐✐✮ ●✐↔ sû t✐➸✉ f ❝õ❛ f ✈➔ ❧➔ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ f (xn ) → f (x)✳ ❤ë✐ tö ♠↕♥❤ ✤➳♥ ✤✐➸♠ ❝ü❝ t✐➸✉ ❦❤æ♥❣ ❧➔ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ f x ❝õ❛ f✳ ❚❤➻ ❦➸ ❝↔ tr♦♥❣ tr÷í♥❣ ❧➔ ♠➦t ♣❤➥♥❣ ❊✉❝❧✐❞❡✳ H ❧➔ ✈æ ❤↕♥ ❝❤✐➲✉ ✈➔ x ❝õ❛ f ✳ ❝❤♦ ❞ò ✣➦♥❣ ❚❤à ▲❛♥ x x∈ ❚❤➻ (xn )n∈N (xn )n∈N ❤ë✐ tö ②➳✉ ✤➳♥ ✤✐➸♠ ❝ü❝ ❝â t❤➸ ❦❤æ♥❣ ❧➔ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ ✐♥t ❞♦♠f ✳ ✺✸ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❱➼ ❞ö ❝❤♦ · (xn )n∈N ❧➔ ❞➣② trü❝ ❝❤✉➞♥ tr♦♥❣ ✳ ❚❤➻ t❤❡♦ ❱➼ ❞ö ✶✳✻ xn ✈➔ H f = ✈➔ t➟♣ ❤ñ♣ f (0) = = inf f (H) ❦❤✐ f (xn ) ≡ 1✳ ❱➼ ❞ö t✐➳♣ t❤❡♦ ♠✐♥❤ ❤♦↕ ♥❤ú♥❣ ✤➦❝ ✤✐➸♠ ❦❤→❝ ❝õ❛ ❞➣② ❝ü❝ t✐➸✉ ❤♦→✳ ❱➼ ❞ö ✷✳✷✳ ●✐↔ sû x ∈ H \ {0} ∀n ∈ N xn = (−1)n x✳ ❚❤➻ ✈➔ t➟♣ ❤ñ♣ (xn )n∈N f = ι[−x,x] ✈➔ ❧➔ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ f ✈➔ ❦❤ỉ♥❣ ❤ë✐ tư ②➳✉✳ ❱➼ ❞ư ✷✳✸✳ s❛♦ ❝❤♦ ●✐↔ sû H ❧➔ t→❝❤ ✤÷đ❝✱ ❧➜② ωk → 0, (xk )k∈N (ωk )k∈N ❧➔ ❞➣② trü❝ ❝❤✉➞♥ ❝õ❛ ❧➔ ❞➣② tr♦♥❣ H ✈➔ ✿ ωk | x|xk |2 f : H → [−∞, +∞] : x → R++ ✭✷✳✽✮ k∈N ❚❤➻ f ❧➔ ❤➔♠ ❣✐→ trà t❤ü❝ tử ỗ t t õ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ f f ❧➔ ✤✐➸♠ ❝ü❝ t✐➸✉ ❞✉② ❦❤ỉ♥❣ ❧➔ ❤➔♠ ❜ù❝✳ ❍ì♥ ♥ú❛✱ (xn )n∈N ❧➔ ❤ë✐ tư ②➳✉ ♥❤÷♥❣ ❦❤ỉ♥❣ ♠↕♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ú ỵ f trt t t t ỡ ♥ú❛ (∀x ∈ H) x | x|xk |2 ≥ = k∈N ❉♦ ✤â f ❧➔ ❤➔♠ ❣✐→ trà t❤ü❝✳ r tứ số ữỡ ỗ tử t❤➻ f f (x) supk∈N ωk f (ωk | ã|xk |2 )kN ỗ ỷ tử ữợ õ tử ự f ỗ t t ❧➜② ✺✹ x, y ∈ H s❛♦ ❝❤♦ x=y ✈➔ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ α ∈ [0, 1]✳ ❝è ✤à♥❤ ❚ø | · |2 ❚❤➻ ∃m ∈ N s❛♦ ❝❤♦ √ ωm x|xm = √ ωm y|xm ỗ t õ m | αx+(1−α)y|xm |2 < αωm | x|xm |2 +(1−α)ωm | y|xm |2 ✈➔ ✈ỵ✐ ♠é✐ ✭✷✳✾✮ k ∈ N \ m✱ ωk | αx + (1 − α)y|xk |2 ≤ αωk | x|xk |2 + (1 − α)ωk | y|xk |2 ✭✷✳✶✵✮ f (αx + (1 − α)y) < αf (x) + (1 − α)f (y)✳ ❇➙② ❣✐í ①➨t xn (∀n ∈ N)yn = √ ✳ ❚❤➻ yn = √ → +∞ ♥❤÷♥❣ f (yn ) ≡ 1✳ ❉♦ ωn ωn ✤â f ❦❤ỉ♥❣ ❧➔ ❜ù❝✳ ❈✉è✐ ❝ò♥❣ f (xn ) = ωn → = f (0) = inf f (H) ❉♦ ✤â ✈➔ tø ❱➼ ❞ö ✶✳✻ ❱➼ ❞ư ✷✳✹✳ xn ●✐↔ sû ♥❤÷♥❣ H = R2 ✳ ✈ỵ✐ ✤✐➸♠ ❝ü❝ t✐➸✉ ❞✉② ♥❤➜t xn 0✳ ❱➼ ❞ö ♥➔② ❝❤♦ t❛ ❤➔♠ ❜ù❝ f ∈ Γ0 (H) x ♠➔ ❝ü❝ t✐➸✉ ❤♦→ ❧✉➙♥ ♣❤✐➯♥ ❝❤♦ t❛ ❞➣② ❤ë✐ tư ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ✈➔ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ♥â ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ x✳ ●✐↔ sû H = R2 ✈➔ H → [−∞, +∞] f: (ξ1 , ξ2 ) → max{2ξ1 − ξ2 , 2ξ2 − ξ1 } + ιR2+ (ξ1 , ξ2 )     2ξ1 − ξ2 , ♥➳✉ ξ1 ≥ ξ2 ≥    = 2ξ2 − ξ1 , ♥➳✉ ξ2 ≥ ξ1 ≥      +∞ , tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❝á♥ ❘ã r f ỷ tử ữợ ỗ ♥❤÷ tê♥❣ ❝õ❛ ❤❛✐ ❤➔♠ sè ✈➔ ♥â ❧➔ ❜ù❝✳ ❍ì♥ ♥ú❛✱ ❝õ❛ ♥â ✈➔ ✭✷✳✶✶✮ f ♥❤➟♥ x = (0, 0) ♥❤÷ ♠ët ❝ü❝ t✐➸✉ ❞✉② ♥❤➜t inf f (H) = 0✳ ✣➦♥❣ ❚❤à ▲❛♥ ✺✺ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❈➠♥ ❝ù ✈➔♦ x0 ∈ R+ × R++ ✱ ❤♦→ ❧✉➙♥ ♣❤✐➯♥ ✈➔ t❛ ①➙② ❞ü♥❣ ❝õ❛ R✳ f (·, ξ2,n+1 ) (xn )n∈N t❛ ❧➦♣ ❧↕✐ ✈✐➺❝ ✤à♥❤ ♥❣❤➽❛ ❞➣② ❝ü❝ t✐➸✉ n, xn = (ξ1,n , ξ2,n ) ❧➔ ✤➣ ❜✐➳t xn+1 = (ξ1,n+1 , ξ2,n+1 )✳ ✣➛✉ t✐➯♥ ❧➜② ξ1,n+1 ❧➔ ❝ü❝ t✐➸✉ tr➯♥ R ♥❤÷ s❛✉✿ ❚↕✐ ✈➔ ❧➜② ξ2,n+1 ❚ø ✭✷✳✶✶✮ ✈ỵ✐ ♠é✐ sè ♥❣✉②➯♥ ❧➔ ❝ü❝ t✐➸✉ ❝õ❛ n≥1 t❛ ❝â f (ξ1,n+1 , ·) tr➯♥ xn = (ξ0,2 , ξ0,2 ) = x ✈➔ f (xn ) = ξ0,2 = inf f (H)✳ ❇➙② ❣✐í ❝❤ó♥❣ t❛ ✤÷❛ r❛ ✤✐➲✉ ❦✐➺♥ ❤ë✐ tö ♠↕♥❤ ✈➔ ②➳✉ ❝❤♦ ❞➣② ❝ü❝ t✐➸✉ ❤♦→✳ ▼➺♥❤ ✤➲ ✷✳✶✸ ❈❤♦ f : H → [, +] tữớ ỷ tử ữợ tỹ ỗ (xn)nN ỹ t f sỷ tỗ t ∈ [inf f (H), +∞] s❛♦ ❝❤♦ C = ❧❡✈≤ξ f ❧➔ ❜à ❝❤➦♥✳ ❑❤✐ ✤â t❛ ❝â✿ ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ ❉➣② (xn)n∈N ❝â ✤✐➸♠ tö ②➳✉ t❤❡♦ ❞➣② ✈➔ ♠é✐ ✤✐➸♠ ♥➔② ✤➲✉ ❧➔ ❝ü❝ t✐➸✉ ❝õ❛ f ✳ sỷ f + C tỹ ỗ t ❚❤➻ f ❝â ✤✐➸♠ ❝ü❝ t✐➸✉ ❞✉② ♥❤➜t ❧➔ x ✈➔ xn x✳ ●✐↔ sû f + ιC ❧➔ ❤➔♠ tỹ ỗ f õ ỹ t ♥❤➜t ❧➔ x ✈➔ xn → x✳ ❈❤ù♥❣ ♠✐♥❤✳ ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ t❛ ❣✐↔ sû ❤♦➔♥ t♦➔♥ tr♦♥❣ t ủ C (xn )nN õ ỗ ỹ tỗ t tử t ❞➣② ❧➔ ✤ó♥❣ t❤❡♦ ❇ê ✤➲ ✶✳✶✶✳ ✣✐➲✉ t❤ù ❤❛✐ ✤÷đ❝ s✉② r❛ tø ▼➺♥❤ ✤➲ ✷✳✶✷✳ ✣➦♥❣ ❚❤à ▲❛♥ ✺✻ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✭✐✐✮ ❚➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ❝ü❝ t✐➸✉ ❧✉ỉ♥ ✤ó♥❣ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✺✭✐✮✳ ❚ø ❇ê ✤➲ ✶✳✶✷✱ s✉② r❛ ✭✐✐✐✮ ❚ø f + C xn x tỹ ỗ t tø ✭✐✐✮ t❤➻ ✤✐➸♠ ❝ü❝ t✐➸✉ x ∈ C✳ ❇➙② ố ổ tỹ ỗ f + ιC ✳ f ❝â ❞✉② ♥❤➜t ♠ët α ∈ [0, 1] ❑❤✐ ✤â ✈ỵ✐ ♠é✐ ✈➔ ❧➜② φ ❧➔ n ∈ N✱ f (x) + α(1 − α)φ( xn − x ) = inf f (H) + α(1 − α)φ( xn − x ) ≤ f (αxn (1 − α)x) + α(1 − α)φ( xn − x ) ≤ max f (xn ), f (x) = f (xn ) ❱➻ ✈➟② tø ✭✷✳✶✷✮ f (xn ) → f (x) s✉② r❛ φ( xn −x ) → ✈➔ xn −x → 0✳ ❍➺ q✉↔ ✷✳✺ ❈❤♦ f ∈ Γ0(H) ❧➔ ❜ù❝ ✈➔ (xn)n∈N ❧➔ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ f ✳ ❑❤✐ ✤â t❛ ❝â✿ ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ ❉➣② (xn)n∈N ❝â ✤✐➸♠ tö ②➳✉ t❤❡♦ ❞➣② ✈➔ ♠é✐ ✤✐➸♠ ♥➔② ✤➲✉ ❧➔ ✤✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ f ✳ ●✐↔ sỷ f tỹ ỗ t f õ ✤✐➸♠ ❝ü❝ t✐➸✉ ❞✉② ♥❤➜t ❧➔ x ✈➔ xn x✳ sỷ f + C tỹ ỗ tr➯♥ ♠é✐ t➟♣ ❤ñ♣ ❝♦♥ ✤â♥❣ ✈➔ ❦❤→❝ ré♥❣ ❝õ❛ ❞♦♠f ✳ ❚❤➻ f ❝â ✤✐➸♠ ❝ü❝ t✐➸✉ ❞✉② ♥❤➜t ❧➔ x ✈➔ xn → x✳ ❱➼ ❞ö ❦❤→❝ ✈➲ t➼♥❤ ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ tr♦♥❣ t ỗ ữ s ✺✼ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣à♥❤ ♥❣❤➽❛ ✷✳✼ ✭❤❛② ♠❡tr✐❝✮ d✳ ❈❤♦ H ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ tr ợ ữớ ởt t ủ ❝♦♥ C ❝õ❛ H ❧➔ ❞✐❛♠C = sup d(x, y) ✭✷✳✶✸✮ (x,y)∈C×C ❑❤♦↔♥❣ ❝→❝❤ tr♦♥❣ t➟♣ C⊂H ❧➔ ❤➔♠ sè dC : H → [0, +∞] : x → inf d(x, C) ✣à♥❤ ♥❣❤➽❛ ✷✳✽ H✳ ❈❤♦ C ✭✷✳✶✹✮ ❧➔ t ủ õ ỗ rộ ❝õ❛ C ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❧➔ ❜❞r②C = C \ (✐♥tC)✳ ▼➺♥❤ ✤➲ ✷✳✶✹ ❈❤♦ f ∈ Γ0(H) ✈➔ C t ủ õ ỗ H s❛♦ ❝❤♦ C∩ ❞♦♠f = ∅✳ ●✐↔ sû C∩ rf = C t ủ ỗ tự tỗ t t : [0, C] → R+ tr✐➺t t✐➯✉ ❞✉② ♥❤➜t t↕✐ s❛♦ ❝❤♦ x+y ; φ( x − y ) (∀x ∈ C) (∀y ∈ C) B ⊂ C, ✭✷✳✶✺✮ ✈➔ (xn)n∈N ❧➔ ❞➣② ❝ü❝ t✐➸✉ ❤♦→ ❝õ❛ f + ιC ✳ ❚❤➻ f ❝â ✤✐➸♠ ❝ü❝ t✐➸✉ ❞✉② ♥❤➜t x tr➯♥ C ✈➔ xn → x✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â t❤➸ ❣✐↔ sû C ❦❤æ♥❣ ❧➔ t➟♣ ♠ët ✤✐➸♠✳ ❚ø C ❧➔ t➟♣ ❤ñ♣ ❜à ❝❤➦♥✱ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✾✭✐✐✮ s✉② r t tỗ t t s r❛ ✤â f f ✈➻ ✈➟② x ❧➔ ✤✐➸♠ tü❛ ❝õ❛ ✣➦♥❣ ❚❤à ▲❛♥ C✳ ❚✐➳♣ t❤❡♦✱ tø ▼➺♥❤ ✤➲ ❆r❣♠✐♥f = ∅✱ ❜❞r②C ✳ ❉♦ ✤â tø ✭✷✳✶✺✮ t❛ ❝â ✐♥tC = ∅✱ ✷✳✶✸✭✐✐✮ ✈➔ ❇ê ✤➲ ✶✳✶✷✱ t❛ ❝â x∈ ♠➔ ❝â ♥❤✐➲✉ ♥❤➜t ♠ët ✤✐➸♠ ❝ü❝ t✐➸✉✳ ❉♦ ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❝ü❝ t✐➸✉ tr➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✸ t❤➻ x C✳ xn x✳ ❍ì♥ ♥ú❛✱ ✣à♥❤ ♥❣❤➽❛ ✺✽ u C∩ ❧➔ ✈❡❝tì ✤à♥❤ ❝❤✉➞♥ ❧✐➯♥ ❑✸✻❈ ❙P ❚♦→♥ ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ xn + x + φ( xn − x )u✳ ❚❤➻ t❤❡♦ ✭✷✳✶✺✮ ✈ỵ✐ zn ∈ C t❛ ❝â✿ φ( xn − x ) = zn − x|u ≤ 0✳ ❙✉② x − xn |u r❛ φ( xn − x ) ≤ → ✈➔ xn → x✳ ❦➳t s❛♦ ❝❤♦ ✣➦♥❣ ❚❤à ▲❛♥ x = 1✳ ▲➜② n ∈ N ✈➔ t➟♣ zn = ✺✾ ❑✸✻❈ ❙P ❚♦→♥ ❑➌❚ ▲❯❾◆ ❑❤♦→ ❧✉➟♥ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ t ỗ tr ổ rt ổ ữ t ỗ ỗ tr t ỗ ởt số ỵ t ứ õ ú tæ✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ỡ ởt t ỗ ữ tr ú ữợ ú ❝ü❝ t✐➸✉✳ ❚r➯♥ ❝ì sð ✤â ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ sỹ tỗ t t t ỹ t ❞➣② ❝ü❝ t✐➸✉ ❤â❛ ❝õ❛ ♠ët ❜➔✐ t♦→♥ ❜✐➳♥ ♣❤➙♥ ỗ t t ❱✐➺t ❬✶❪ ◆❣✉②➵♥ P❤ö ❍②✱ ❬✷❪ ❍♦➔♥❣ ❚✉✢✱ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ❑❍ ✫ ❑❚✱ ♥➠♠ ✷✵✵✺ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ✣❍◗● ❍➔ ◆ë✐✱ ♥➠♠ ✷✵✵✸ ❬❇❪ ❚➔✐ ❧✐➺✉ t✐➳♥❣ ❆♥❤ ❈♦♥✈❡① ❆♥❛❧②s✐s ❛♥❞ ▼♦♥♦t♦♥❡ ❖♣❡r❛t♦r ❚❤❡♦r② ✐♥ ❍✐❧❜❡rt ❙♣❛❝❡s✱ ❙♣r✐♥❣❡r ✷✵✶✵✳ ❬✸❪ ❇❛✉s❝❤❦❡✱ ❍✳ ❍ ❛♥❞ ❈♦♠❜❡tt❡s✱ P✳ ▲✳✱ ❬✹❪ ❘✉s③❝③②♥✬s❦✐✱ ❆✳✱ ◆♦♥❧✐♥❡❛r ❖♣t✐♠✐③❛t✐♦♥✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥ ✷✵✵✻✳ ✻✶