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−i− ▲ê✐ ❝➯♠ ➡♥ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❝➳❝ t❤➭② ❣✐➳♦✱ ❝➠ ❣✐➳♦ ❣✐➯♥❣ ❞➵② ❝❤✉②➟♥ ♥❣➭♥❤ ❚♦➳♥ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ P❤➵♠ ❍➭ ◆é✐ ✷ ➤➲ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✳ ➜➷❝ ❜✐Ưt✱ t➠✐ ①✐♥ ❝➯♠ ➡♥ t❤➬② ❣✐➳♦ ❚❙ ❇ï✐ ❚rä♥❣ ❑✐➟♥ ❣✐➯♥❣ ✈✐➟♥ ❑❤♦❛ ❈➠♥❣ ♥❣❤Ö ❚❤➠♥❣ t✐♥✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❳➞② ❞ù♥❣ ❍➭ ◆é✐ ➤➲ trù❝ t✐Õ♣ ❤➢í♥❣ ❞➱♥ t➠✐ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❧ù❛ ❝❤ä♥ ➤Ò t➭✐ ✈➭ ❤♦➭♥ ❝❤Ø♥❤ ➤Ò t➭✐✳ ❚➠✐ ①✐♥ ❝➯♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❧í♣ ❝❛♦ ❤ä❝ ❑✶✶ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ➤➲ ❣✐ó♣ ➤ì ✈➭ ❝ã ♥❤÷♥❣ ➤ã♥❣ ❣ã♣ q✉Ý ❜➳✉ ❝❤♦ ❜➯♥ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❍➭ ◆é✐✱ t❤➳♥❣ ✶✵ ♥➝♠ ✷✵✵✾ ❚➳❝ ❣✐➯ −1− ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ ▲✉❐♥ ✈➝♥ ❧➭ ❝➠♥❣ trì ứ ủ r t ợ tự ệ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ t❤➬② ❣✐➳♦ ❚❙✳ ❇ï✐ ❚rä♥❣ ❑✐➟♥ ❣✐➯♥❣ ✈✐➟♥ ❑❤♦❛ ❈➠♥❣ ♥❣❤Ö ❚❤➠♥❣ t✐♥✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❳➞② ❞ù♥❣ ❍➭ ◆é✐✳ ◆❣♦➭✐ ❝➳❝ ❦Õt q✉➯ ➤➢ỵ❝ trÝ❝❤ ết q ò ữ ết q ú t t ợ tr q trì ❝ø✉ ✈➭ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✱ ❝➳❝ ❦Õt q✉➯ ♥➭② ❝❤➢❛ ➤➢ỵ❝ ❝➠♥❣ ❜è ë ❜✃t ❦ú t➵♣ ❝❤Ý ♥➭♦✳ ❍➭ ◆é✐✱ t❤➳♥❣ ✶✵ ♥➝♠ ✷✵✵✾ ❚➳❝ ❣✐➯ −2− ▼ô❝ ❧ơ❝ ▼ë ➤➬✉ ✹ ❈❤➢➡♥❣ ✶✳ ❈➳❝ ❦Õt q✉➯ ❜ỉ trỵ ✶✳✶✳ ❈➳❝ ♥❣✉②➟♥ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✶✳✷✳ P❤Ð♣ ❝❤✐Õ✉ ♠❡tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸✳ ❇✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥ ❝❤✐Ò✉ ✳ ✳ ✳ ✳ ✳ ✶✶ ❈❤➢➡♥❣ ✷✳ ❙ù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ t♦➳♥ tư ➤➡♥ ➤✐Ư✉ ❈❤➢➡♥❣ ✸✳ ❙ù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ ✸✳✶✳ ❙ù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❑❛r❛✲ ♠❛r❞✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸✳✷✳ ❙ù tå♥ t➵✐ ♥❣❤✐Ö♠ ❝đ❛ ❱■s ✈í✐ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❇rÐ③✐s ✷✶ ❈❤➢➡♥❣ ✹✳ ❇✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ✈í✐ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✷✺ ✸✼ −3− ❇➯♥❣ ❦ý ❤✐Ö✉ ❱■ ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t② ❱■s ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s ❱■(K, f ) ❜➭✐ t♦➳♥ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ①➳❝ ➤Þ♥❤ ❜ë✐ t❐♣ ❙(K, f ) K ✈➭ ➳♥❤ ①➵ f t❐♣ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❱■(K, f ) −4− ▼ë ➤➬✉ ✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ò t➭✐ ❇✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ✭❱■✱ rt qt ợ ột ì ữ ệ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ❜➭✐ t♦➳♥ ①✉✃t ❤✐Ư♥ tr♦♥❣ ❝➳❝ ❧Ü♥❤ ✈ù❝ ❦❤➳❝ ♥❤❛✉ ❝ñ❛ t♦➳♥ ❤ä❝ ♥❤➢✿ ❧ý t❤✉②Õt tè✐ ➢✉✱ ♣❤➢➡♥❣ tr×♥❤ ➤➵♦ ❤➭♠ r✐➟♥❣✱ ❝➳❝ ❜➭✐ t♦➳♥ ❝➞♥ ❜➺♥❣ ❦✐♥❤ tÕ✱ ❝➡ ❤ä❝✳ ❑Ó tõ ❦❤✐ r❛ ➤ê✐ ❝đ❛ ➤Þ♥❤ ❧ý ❍❛r♠❛♥❞✲❙♣❛♠♣❛❝❝❤✐❛ ♥➝♠ ✶✾✻✻✱ sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈➭ ❝➳❝ ❝❤đ ➤Ị ❧✐➟♥ q✉❛♥ ✈➱♥ ➤❛♥❣ t❤✉ ❤ót sù q✉❛♥ t➞♠ ❝đ❛ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝✳ ◆❤✐Ị✉ ❝➞✉ ❤á✐ ♠ë tr♦♥❣ ❤➢í♥❣ ♥➭② ✈➱♥ ❝ß♥ ➤❛♥❣ tå♥ t➵✐✳ ❈❤ó ý r➺♥❣ ❝➳❝ ❦Õt q✉➯ ❦✐♥❤ ➤✐Ĩ♥ tr➢í❝ ➤➞② ❝❤đ ②Õ✉ ♥❣❤✐➟♥ ❝ø✉ ❝❤♦ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✳ ●➬♥ ➤➞② ♠ét sè ❝➠♥❣ tr×♥❤ ①✉✃t ❤✐Ư♥ tr➟♥ ❝➳❝ t➵♣ ❝❤Ý ❝❤✉②➟♥ ♥❣➭♥❤ ➤➲ ➤➢❛ r❛ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ ♠í✐ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❑❛r❛♠❛❞✐❛♥ ✈➭ t❤❡♦ ♥❣❤Ü❛ ❇rÐ③✐s✳ ◆❣➢ê✐ t❛ ➤➲ ❜✐Õt r➺♥❣ ❤❛✐ ❧í♣ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ö✉ ♥➭② ❧➭ ❤♦➭♥ t♦➭♥ ❦❤➳❝ ♥❤❛✉✳ ◆➝♠ ✷✵✵✵✱ ❉♦♠♦❦♦s ✈➭ ❑♦❧✉♠❜➳♥ ➤➲ ➤➢❛ r❛ ♠ét ➤Þ♥❤ ♥❣❤Ü❛ ♠í✐ ❝❤♦ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉✳ ▲í♣ ❝➳❝ t♦➳♥ tư t❤♦➯ ♠➲♥ ➤Þ♥❤ ♥❣❤Ü❛ ♥➭② ❝❤ø❛ ❝➯ ❤❛✐ ❧í♣ t♦➳♥ tư ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❑❛r❛♠❛❞✐❛♥ ✈➭ ❇rÐ③✐s✳ ❉ù❛ tr➟♥ ❦❤➳✐ ♥✐Ư♠ ♠í✐ ♥➭② ❤ä ➤➲ ➤➵t ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ ❦❤➳ t❤ó ✈Þ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s✳ ▼ơ❝ t✐➟✉ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ t✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s ✈➭ ❑♦❧✉♠❜➳♥ ✈➭ ➤➢❛ r❛ ♠ét ❦Õt q✉➯ ♠í✐ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ tr➢í❝ ➤ã ❝đ❛ ❉♦♠♦❦♦s ✈➭ ❑♦❧✉♠❜➳♥✳ ➜Ĩ ➤➵t ➤➢ỵ❝ ❦Õt q✉➯ ♥➭② tr➢í❝ ❤Õt ❝❤ó♥❣ t❛ ♣❤➯✐ ❦❤➯♦ s➳t ❝➳❝ ❦Õt q✉➯ ✈Ò sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✱ ❝➳❝ ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ −5− ❱■s ❝❤♦ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝ñ❛ ❑❛r❛♠❛❞✐❛♥ ✈➭ ❇rÐ③✐s ✈➭ ❝➳❝ ❦Õt q✉➯ ❣➬♥ ➤➞② ❝đ❛ ❉♦♠♦❦♦s ✈➭ ❑♦❧✉♠❜➳♥✳ ❚r➟♥ ❝➡ së ➤ã ❝❤ó♥❣ t❛ sÏ ➤➷t ❜➭✐ t♦➳♥ ✈➭ ❣✐➯✐ q✉②Õt ✈✃♥ ➤Ò ❜➺♥❣ ✈✐Ư❝ ➤➢❛ r❛ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét ❦Õt q✉➯ ♠í✐ ♥❤♦ ♥❤á✱ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ tr➢í❝ ➤ã ❝đ❛ ❉♦♠♦❦♦s ✈➭ ❑♦❧✉♠❜➳♥✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ♠❛♥❣ t➟♥ ✧❙ù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ✈í✐ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉✧✱ ❜❛♦ ❣å♠ ✹ ❝❤➢➡♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ✶ ❝❤ó♥❣ t❛ sÏ tr×♥❤ ❜➭② ♠ét sè ết q ổ trợ q tớ ị ý ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❈❤➢➡♥❣ ✷ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✳ ❈❤➢➡♥❣ ✸ ❣å♠ ❝➳❝ ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❑❛r❛♠❛❞✐❛♥ ✈➭ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❇rÐ③✐s✳ ❈❤➢➡♥❣ ✹ ➤➢ỵ❝ ❞➭♥❤ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ♠í✐ ♥❤✃t ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ❝❤♦ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ❈✉è✐ ❝❤➢➡♥❣ ♥➭② ❧➭ ♠ét sè ❦Õt q✉➯ ♠í✐ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ✷✳ ▼ơ❝ ➤Ý❝❤ ♥❣❤✐➟♥ ❝ø✉ ❍Ư t❤è♥❣ ❤♦➳ ❧➵✐ ❝➳❝ ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❑❛r❛♠❛❞✐❛♥ ✈➭ ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❝ñ❛ ❇rÐ③✐s✱ tr➟♥ ❝➡ së ➤ã ➤➢❛ r❛ ♠ét ❦Õt q✉➯ ♠í✐ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ✸✳ ◆❤✐Ư♠ ✈ơ ♥❣❤✐➟♥ ❝ø✉ ◆❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉✳ −6− ✹✳ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❙ù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❑❛r❛✲ ♠❛❞✐❛♥✱ ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❇rÐ③✐s ✈➭ ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❉♦♠♦❦♦s✲ ❑♦❧✉♠❜➳♥✳ ✺✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❙ư ❞ơ♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❝ị♥❣ ♥❤➢ ❝➳❝ ❦ü t❤✉❐t ❝đ❛ ❣✐➯✐ tÝ❝❤ ❝ỉ ➤✐Ĩ♥✱ ❣✐➯✐ tÝ❝❤ ❤➭♠✱ ❣✐➯✐ tÝ❝❤ ❧å✐✱ ❣✐➯✐ tÝ❝❤ ❦❤➠♥❣ tr➡♥✱ ❣✐➯✐ tÝ❝❤ ➤❛ trÞ ✈➭ ❧ý t❤✉②Õt tè✐ ➢✉✳ ✻✳ ●✐➯ t❤✉②Õt ❦❤♦❛ ❤ä❝ ➜Ò t➭✐ ➤➲ ➤➢❛ r❛ ➤➢ỵ❝ ❝➳❝ ❦Õt q✉➯ ♠ë ré♥❣ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ✈í✐ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ −7− ❈❤➢➡♥❣ ✶ ❈➳❝ ❦Õt q✉➯ ổ trợ r ú t trì ột số ết q ổ trợ ị ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐♥❤ ❧ý ✈Ị sù tå♥ t➵✐ ì ế ị ý ề tồ t ♥❣❤✐Ư♠ ❝đ❛ ❱■s tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉✳ ✶✳✶✳ ❈➳❝ ♥❣✉②➟♥ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❚r♦♥❣ s✉èt ♠ơ❝ ♥➭② ❝❤ó♥❣ t❛ ❣✐➯ sư r➺♥❣ A⊆X ❧➭ ♠ét t❐♣ ➤ã♥❣✳ ●✐➯ sư ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳ f ♥Õ✉ ➳♥❤ ①➵ f f :A→X X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➭ ❧➭ ♠ét ➳♥❤ ①➵✳ ➜✐Ó♠ x¯ ∈ A ➤➢ỵ❝ ❣ä✐ f (¯ x) = x¯✳ :A→X ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ sè α ∈ [0, 1) s❛♦ ❝❤♦ f (x) − f (y) ≤ α x − y ✈í✐ ♠ä✐ ✭✶✳✶✮ x, y ∈ A✳ ❈❤ó♥❣ t❛ ❝ã ❦Õt q✉➯ q✉❡♥ t❤✉é❝ s❛✉ ➤➞② ✈Ị ❧➭ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✳ ➜Þ♥❤ ❧ý ✶✳✷✳ ●✐➯ sö ➤ã f A⊆X ❧➭ ♠ét t❐♣ ➤ã♥❣ ✈➭ ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ t❤✉é❝ A✳ f : A → A ❧➭ ♠ét ➳♥❤ ①➵ ❝♦✳ ❑❤✐ −8− C⊂X ◆❤➽❝ ❧➵✐ r➺♥❣ ♠ét t❐♣ ➤➢ỵ❝ ❣ä✐ ❧➭ ❧å✐ ♥Õ✉ ✈í✐ ♠ä✐ x, y ∈ C ✈➭ ✈í✐ λ ∈ [0, 1]✱ t❛ ❝ã λx + (1 − λ)y ∈ C ✳ ❚❛ ❝ã ♥❣✉②➟♥ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❇r♦✉✇❡r ♠ä✐ s❛✉ ➤➞②✳ K ⊂X ➜Þ♥❤ ❧ý ✶✳✸✳ ❬✻✱ tr ✽❪ ●✐➯ sö ➳♥❤ ①➵ ❧✐➟♥ tô❝✳ ❑❤✐ ➤ã ✶✳✷✳ h:K →K ❧➭ t❐♣ ❧å✐ ✈➭ ❝♦♠♣❛❝t✱ ❧➭ ♠ét h ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣✳ P❤Ð♣ ❝❤✐Õ✉ ♠❡tr✐❝ ❈❤♦ K⊂X ❧➭ ♠ét t❐♣ ➤ã♥❣ ✈➭ x ∈ X ✳ ➜✐Ó♠ y ∈ K t❤♦➯ ♠➲♥ x − y = inf x − z z∈K ➤➢ỵ❝ ❣ä✐ ❧➭ ì ế tr ủ t ứ ỗ xX x ❧➟♥ t❐♣ K ✈➭ ❦ý ❤✐Ö✉ y = PK (x)✳ P❤Ð♣ y = PK (x) ❣ä✐ ❧➭ ♣❤Ð♣ ❝❤✐Õ✉ ♠❡tr✐❝ ❧➟♥ t❐♣ K ✱ ❦ý ❤✐Ö✉ PK ✳ ❑Õt q✉➯ s❛✉ ➤➞② ❝❤♦ t❛ sù tå♥ t➵✐ ❝ñ❛ ❝❤➞♥ ❤×♥❤ ❝❤✐Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❇ỉ ➤Ị ✶✳✹✳ ❬✻✱ tr ✽✲✾❪ ●✐➯ sö ✈➭ K ❧➭ t❐♣ ❝♦♥ ❧å✐ ➤ã♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt x ∈ H ✳ ❑❤✐ ➤ã tå♥ t➵✐ ❞✉② ♥❤✃t y ∈ K s❛♦ ❝❤♦ x − y = inf x − z ✭✶✳✷✮ z∈K ❈❤ø♥❣ ♠✐♥❤✳ ➜➷t ❞➲② zm ∈ K s❛♦ ❝❤♦ d = inf z∈K x − z ✳ ❚õ ➤Þ♥❤ ♥❣❤Ü❛ ❝ñ❛ ✐♥❢✐♠✉♠✱ tå♥ t➵✐ limm→∞ x − zm = d ❙ư ❞ơ♥❣ q✉✐ t➽❝ ❤×♥❤ ❜×♥❤ ❤➭♥❤ x+y + x−y =2 x + y 2, t❛ ❝ã zm − zk = x − zm H + x − zk − x − (zm + zk ) 2 −9− ❱× K ❧➭ t❐♣ ❧å✐✱ t❛ ❝ã (zm zm − zk ❉♦ ➤ã + zk ) ∈ K ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ♥➟♥ ≤ x − zm limk,m→∞ zm − zk → x − 12 (zm + zk ) ≥ d ❍Ö q✉➯ ❧➭ ✈➭ ✈➭ ✈× ✈❐② + x − zk (zm ) zm → y0 ∈ H ✳ ❱× K − 4d2 ❧➭ ❞➲② ❈❛✉❝❤②✳ ▼➷t ❦❤➳❝ ❧➭ t❐♣ ➤ã♥❣✱ t❛ ❝ã H ❧➭ y0 ∈ K ✳ ❍➡♥ ♥÷❛ x − y0 = lim x − zm = d m→∞ ●✐➯ sö r➺♥❣ tå♥ t➵✐ y1 , y2 ∈ K x − y1 = x − y2 = d✳ ❙ư ❞ơ♥❣ q✉✐ t➽❝ s❛♦ ❝❤♦ ❜×♥❤ ❤➭♥❤ t❛ ❧➵✐ ❝ã y1 − y2 2 = x − y1 ❚õ ➤➞② s✉② r❛ ❝❤ø♥❣ ♠✐♥❤✳ y1 = y2 + x − y2 − x − (y1 + y2 ) ✈➭ ❞♦ ➤ã ❝❤➞♥ ❤×♥❤ ❝❤✐Õ✉ ❝đ❛ x ≤ 4d2 − 4d2 = ❧➭ ❞✉② ♥❤✃t✳ ❇ỉ ➤Ị ➤➢ỵ❝ ✷ ❇ỉ ➤Ị ✶✳✺✳ ❬✻✱ tr ✾✲✶✵❪ ●✐➯ sö ✈➭ K ❧➭ t❐♣ ❝♦♥ ❧å✐ ➤ã♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt H x ∈ H ✳ ❑❤✐ ➤ã y = PK x ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ y, z − y ≥ x, z − y , ∀z ∈ K ❈❤ø♥❣ ♠✐♥❤✿ ●✐➯ y = PK x✳ ❱× K ✭✶✳✸✮ ❧➭ t❐♣ ❧å✐ t❛ ❝ã (1 − t)y + tz = y + t(z − y) ∈ K, ∀z ∈ K, t ∈ [0, 1] ❈è ➤Þ♥❤ t✉ú ý z∈K ✈➭ ①Ðt ❤➭♠ sè Φ(t) = x − ((1 − t)y + tz) ❱× y ❧➭ ❝❤➞♥ ❤×♥❤ ❝❤✐Õ✉ ❝đ❛ x ❧➟♥ t❐♣ K ✱ t❛ ❝ã Φ(t) = x − ((1 − t)y + tz) ➜✐Ò✉ ♥➭② s✉② r❛ ✭✶✳✹✮ ≥ x−y = Φ(0) Φ (0) ≥ 0✳ ▼➷t ❦❤➳❝ t❛ ❧➵✐ ❝ã Φ(t) = x − y − 2t x − y, z − y + t2 z − y − 25 − ❈❤➢➡♥❣ ✹ ❇✃t ➤➻♥❣ t❤ø❝ ❜✐Õ♥ ♣❤➞♥ ✈í✐ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t❛ sÏ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ❝❤♦ ❧í♣ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ ➤➢ỵ❝ ➤➢❛ r❛ ❣➬♥ ➤➞② ❜ë✐ ❉♦♠♦❦♦s✲ ❑♦❧✉♠❜➳♥✳ ▲í♣ ❝➳❝ t♦➳♥ tư ♥➭② ❜❛♦ ❣å♠ ❝➯ ❤❛✐ ❧í♣ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❑❛r❛♠❛❞✐❛♥ ✈➭ ❇rÐ③✐s✳ ◆❤➢ ✈❐② ❝➳❝ ❦Õt q✉➯ ♥➭② ❧➭ sù ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ❝ñ❛ ❝❤➢➡♥❣ ✸✳ P❤➬♥ ❝✉è✐ ❝ñ❛ ❝❤➢➡♥❣ ♥➭② sÏ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ♠í✐ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ❝❤♦ ❧í♣ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ❚r♦♥❣ s✉èt ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t❛ ❣✐➯ sư ➤è✐ ♥❣➱✉ X ∗✱ K ⊂ X ➜Þ♥❤ ♥❣❤Ü❛ ✹✳✶✳ X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈í✐ ❧➭ ♠ét t❐♣ ❧å✐ ➤ã♥❣ ❦❤➳❝ rỗ f : K X ọ ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲ ❑♦❧✉♠❜➳♥ ♥Õ✉ ỗ (xi )iI K ọ x, y ∈ K f (xi ), (1 − t)x + ty − xi ≥ ✱ ∀t ∈ [0, 1] , i ∈ I ➜Þ♥❤ ❧ý ✹✳✷✳ ❬✶✱ tr ✾✼❪ ◆Õ✉ f : K → X∗ t❤× s❛♦ ❝❤♦ xi x ✈➭ f (x), y − x ≥ ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❇rÐ③✐s t❤× ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ f − 26 − ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö ❝ã ❞➲② (xi ) ⊂ K s❛♦ ❝❤♦ f (xi ), (1 − t)x + ty − xi ≥ 0, ❱× ✭✹✳✶✮ ➤ó♥❣ ✈í✐ ♠ä✐ xi → x ✈➭ ∀t ∈ [0, 1], i ∈ I ✭✹✳✶✮ t ∈ [0, 1]✱ ❧➬ ❧➢ỵt ❝❤♦ t = 0, t = t❛ ❝ã f (xi ), x − xi ≥ ✭✹✳✷✮ f (xi ), y − xi ≥ ✭✹✳✸✮ ✈➭ ❚õ ✭✹✳✸✮ t❛ ❝ã lim inf f (xi ), x − xi ≥ ❱× f ❧➭ ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❇rÐ③✐s✱ t❛ ❝ã lim sup f (xi ), y − xi ≤ f (x), y − x ❑Õt ❤ỵ♣ ✈í✐ ✭✹✳✸✮ t❛ ❝ã ≤ f (x), y − x ✳ ❉♦ ➤ã f ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ❉➢í✐ ➤✐Ị✉ ❦✐Ư♥ ❧✐➟♥ tơ❝✱ ❧í♣ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❑❛r❛✲ ♠❛❞✐❛♥ ❝ị♥❣ t❤✉é❝ ✈➭♦ ❧í♣ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲ ❑♦❧✉♠❜➳♥ ♥❤➢ ➤Þ♥❤ ❧ý ❞➢í✐ ➤➞②✳ ➜Þ♥❤ ❧ý ✹✳✸✳ ❬✶✱ tr ✾✽❪ ◆Õ✉ ♠❛❞✐❛♥ ✈➭ f f : K → X∗ ❧➭ ❤❡♠✐ ❧✐➟♥ tơ❝ t❤× f ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❑❛r❛✲ ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲ ❑♦❧✉♠❜➳♥✳ ❱Ý ❞ơ s❛✉ ❝❤Ø r❛ r➺♥❣ ❧í♣ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲ ❑♦❧✉♠❜➳♥ ❧➭ ré♥❣ ❤➡♥ ❤➻♥ ✷ ❧í♣ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❇rÐ③✐s ✈➭ ❝đ❛ ❑❛r❛♠❛❞✐❛♥✳ − 27 − ❱Ý ❞ô ✹✳✹✳ ❬✺✱ tr ✹✽❪ ❳Ðt ➳♥❤ ①➵ ✈➭ f (0) = 0✳ ❑❤✐ ➤ã f f :R→R ➤➢ỵ❝ ❝❤♦ ❜ë✐ f (x) = ♥Õ✉ x=0 ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥ ♥❤➢♥❣ ❦❤➠♥❣ ❣✐➯ ➤➡♥ ➤✐Ư✉ ♥❣❤Ü❛ ❝đ❛ ❇rÐ③✐s ❤♦➷❝ ❝đ❛ ❑❛r❛♠❛❞✐❛♥✳ ❱í✐ ỗ yK t t T1 (y) = {x K : f (x), y − x ≥ 0} ✭✹✳✹✮ T2 (y) = {x ∈ K : f (y), y − x ≥ 0} ✭✹✳✺✮ ✈➭ ❑❤✐ ➤ã ➳♥❤ ①➵ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥ ❝ã t❤Ĩ ➤➢ỵ❝ ➤➷❝ tr➢♥❣ ❜ë✐ tÝ♥❤ ❝❤✃t s❛✉✳ ➜Þ♥❤ ❧ý ✹✳✺✳ ❬✶✱ tr ✾✼❪ ➳♥❤ ①➵ f ❧➭ ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❝ñ❛ ❉♦♠♦❦♦✲❑♦❧✉♠❜➳♥ ❦❤✐ ✈➭ ❝❤✐ ❦❤✐ T1 (z) ∩ [x, y], T1 (z) ∩ [x, y] = ❝❧ ✭✹✳✻✮ z∈K∩[x,y] z∈K∩[x,y] ë ➤ã ❝❧(A) ❧➭ ❜❛♦ ➤ã♥❣ t❤❡♦ t➠ ♣➠ ②Õ✉ ❝ñ❛ t❐♣ A ✈➭ [x, y] ❧➭ ➤♦➵♥ t❤➻♥❣ ♥è✐ x ✈➭ y✳ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ✈Õ tr➳✐ ❧➭ t❐♣ ❝♦♥ ❝ñ❛ ✈Õ ♣❤➯✐✳ ❚r➢í❝ ❤Õt t❛ ❝❤Ø r❛ r➺♥❣ ♥Õ✉ x ∈ ❝❧ T1 (z) ✭✹✳✼✮ z∈K∩[x,y] t❤× x∈ T1 (z) z∈K∩[x,y] ✭✹✳✽✮ − 28 − ❚❤ù❝ ✈❐② tõ ✭✹✳✼✮ s✉② r❛ r➺♥❣ tå♥ t➵✐ ❞➲② xk ∈ z∈K∩[x,y] T1 (z) s❛♦ ❝❤♦ xk → x✳ ➜✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ r➺♥❣ xk ∈ T1 ((1 − t)x + ty) ✈í✐ ♠ä✐ t ∈ [0, 1]✳ ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ t❐♣ T1 (z) t❛ ❝ã f (xk ), (1 − t)x + ty − xk ≥ 0, ∀t ∈ [0, 1] ❉♦ ➤ã t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝ñ❛ ❉♦♠♦❦♦s✲ ❑♦❧✉♠❜➳♥ s✉② r❛ f (x), y − x ≥ ▼➷t ❦❤➳❝ ✈í✐ ♠ä✐ t ∈ [0, 1] t❛ ❝ã f (x), (1 − t)x + ty − x = t f (x), y − x ≥ ❍Ö q✉➯ ❧➭ x ∈ T1 ((1 − t)x + ty) ✈í✐ ♠ä✐ t ∈ [0, 1]✱ ❤❛② x∈ T1 (z) z∈K∩[x,y] ❚❛ ❝ã ✭✹✳✽✮✳ ❇➞② ❣✐ê t❛ ❧✃② u ∈ ❝❧ z∈K∩[x,y] T1 (z) ∩ [x, y]✳ ❑❤✐ ➤ã T1 (z) ∩ [x, u] u ∈ ❝❧ z∈K∩[x,u] ❤♦➷❝ u ∈ ❝❧ T1 (z) ∩ [u, y] z∈K∩[u,y] ❚❤❡♦ ♥❤❐♥ ①Ðt tr➟♥ t❛ ❝ã u∈ T1 (z) ∩ [x, u] z∈K∩[x,u] ❤♦➷❝ u∈ T1 (z) ∩ [u, y] z∈K∩[u,y] − 29 − ❱× ✈❐② u∈ T1 (z) ∩ [x, y] z∈K∩[x,y] ✈➭ ❞♦ ➤ã t❛ ❝ã ✭✹✳✻✮✳ ❇➞② ❣✐ê ❣✐➯ sö r➺♥❣ ❝ã ✭✹✳✻✮ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ♥❣❤Ü❛ ❝ñ❛ ❉♦♠♦❦♦s✲ ❑♦❧✉♠❜➳♥✳ ▲✃② t✉ú ý x, y ∈ K f ❧➭ ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ✈➭ ❞➲② xk → x s❛♦ ❝❤♦ f (xk ), (1 − t)x + ty − xk ≥ 0, ∀t ∈ [0, 1] ❚õ ➤➞② s✉② r❛ r➺♥❣ xk ∈ T1 (z) ✈í✐ ♠ä✐ z ∈ [x, y]✳ ❉♦ ➤ã xk ∈ T1 (z) z∈K∩[x,y] ✈➭ ✈× ✈❐② T1 (z) ∩ [x, y] T1 (z) ∩ [x, y] = x ∈ ❝❧ z∈K∩[x,y] z∈K∩[x,y] ➜✐Ò✉ ♥➭② s✉② r❛ f (x), (1 − t)x + ty − x ≥ 0, ∀t ∈ [0, 1] ❈❤♦ t=1 t❛ ❝ã f (x), y − x ≥ 0✳ ◆❤➢ ✈❐② f ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲ ❑♦❧✉♠❜➳♥✳ ➜Þ♥❤ ❧ý s❛✉ ❧➭ ❦Õt q✉➯ ➤➬✉ t✐➟♥ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ❝❤♦ ❧í♣ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ➜Þ♥❤ ❧ý ✹✳✻✳ ❬✶✱ tr ✾✾✲✶✵✶❪ ❈❤♦ ❳ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❧å✐ ✈➭ f : K → X∗ K⊂X t❐♣ ỗ ột t tử sử r ề ệ s ợ ệ ú Pé ữ ❤➵♥ ❝đ❛ ❑ ✈í✐ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ ❳ ❧➭ ➤ã♥❣✳ ✭✐✐✮ f ❧➭ ❧✐➟♥ tơ❝ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ ❳ ❣✐❛♦ ✈í✐ K✳ − 30 − ✭✐✐✐✮ f ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ✭✐✈✮ f t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❜ø❝ ➤è✐ ✈í✐ t❐♣ ❧å✐ ❝♦♠♣❛❝t ②Õ✉ z0 ∈ B ∩ K B ⊂ X ✱ tø❝ ❧➭ tå♥ t➵✐ s❛♦ ❝❤♦ f (x), z0 − x < 0, ∀x ∈ K \ B ❑❤✐ ➤ã tå♥ t➵✐ x0 ∈ B ∩ K s❛♦ ❝❤♦ f (x0 ), x − x0 ≥ 0, ∀x ∈ K ➜Ó ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý ❝❤ó♥❣ t❛ ❝➬♥ ❜ỉ ➤Ị s❛✉ ❧✐➟♥ q✉❛♥ tí✐ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❇ỉ ➤Ị ✹✳✼✳ ❬✶✱ tr ✾✾❪ T : K → 2X ✭✐✮ ❚å♥ t➵✐ ✭✐✐✮ T X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ K⊂X ❧➭ ♠ét t❐♣ ❧å✐ ➤ã♥❣✱ ❧➭ ♠ét ➳♥❤ ①➵ ➤❛ trÞ t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ z0 ∈ K s❛♦ ❝❤♦ ❜❛♦ ➤ã♥❣ ②Õ✉ ❝❧ ❧➭ ➳♥❤ ①➵ ❑❑▼✱ tø❝ ❧➭ ✈í✐ ♠ä✐ T (z0 ) ❧➭ ❝♦♠♣❛❝t ②Õ✉✳ x1 , x2 , , xn ∈ K t❛ ❝ã n conv{x1 , , xn } ⊂ T (xi ) ✭✹✳✾✮ i=1 ✭✐✐✐✮ ❱í✐ ♠ä✐ X z∈K ❣✐❛♦ ❝đ❛ T (z) ✈í✐ ❜✃t ❦ú ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ ❧➭ ➤ã♥❣✳ ✭✐✈✮ ❱í✐ ♠ä✐ ➤♦➵♥ t❤➻♥❣ D⊂X t❛ ❝ã T (z) ∩ D = ❝❧ T (z) ∩ D z∈K∩D z∈K∩D ❑❤✐ ➤ã T (z) = z∈K − 31 − ❈❤ø♥❣ ♠✐♥❤ ❝đ❛ ➜Þ♥❤ ❧ý ✹✳✻✳ ❈❤ó♥❣ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ T1 (z) = 0, z∈K ó T1 ợ ị tứ ❚õ ➤✐Ị✉ ❦✐Ư♥ (iv) t❛ ❝ã ❝❧ T1 (z0 ) ⊂ B ✳ ❉♦ ➤ã ❝❧ ♠➲♥✳ ❱× T1 (z0 ) f ❧➭ t❐♣ ❝♦♠♣❛❝ ②Õ✉✳ ❱❐② ➤✐Ị✉ ❦✐Ư♥ (i) ❝đ❛ ❇ỉ ➤Ị ✹✳✼ ➤➢ỵ❝ t❤♦➯ ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉✱ t ị ý t ó ề ệ ợ ệ ➤ó♥❣✳ ●✐➯ sư S (iv) ❝đ❛ ❇ỉ ➤Ị ✹✳✼ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ X✳ ❑❤✐ ➤ã t❛ ❝ã T1 (z) ∩ S = {x ∈ K ∩ S : f (x), z − x ≥ 0} ➜➞② ❧➭ t❐♣ ➤ã♥❣ ✈× t❤❡♦ ❣✐➯ t❤✐Õt ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐➟♥ ❧➭ ❧✐➟♥ tơ❝ tr➟♥ K ∩S ✳ ❈ß♥ ❧➵✐ ❝❤ó♥❣ t❛ ♣❤➯✐ (ii) ❝đ❛ ❇ỉ ➤Ị ✹✳✼✳ ▲✃② t✉ú ý x1 , x2 , , xn ∈ K ✱ t❛ ❝❤Ø r❛ r➺♥❣ ✭✹✳✾✮ t❤♦➯ ♠➲♥✳ ▲✃② ë ➤ã ti (ii) f x ∈ ❝♦♥✈{x1 , x2 , , xn }✳ ❑❤✐ ➤ã x = t1 x1 + t2 x2 + · · · tn xn ✱ ≥ ✈➭ t1 + t2 + · · · tn = 1✳ ◆Õ✉ f (x), xi − x < 0, ∀i = 1, 2, , n t❤× t❛ ❝ã ➤✐Ò✉ ✈➠ ❧ý s❛✉✿ n = f (x), x − x = f (x), n ti xi − i=1 ti x i=1 n ti f (x), xi − x < = i=1 ❱❐② ♣❤➯✐ tå♥ t➵✐ i s❛♦ ❝❤♦ f (x), xi − x ≥ ✈➭ ❞♦ ➤ã n x ∈ T1 (xi ) ⊂ T1 (xi ) i=1 ◆❤➢ ✈❐② t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❇ỉ ➤Ị ✹✳✼ ➤➢ỵ❝ t❤♦➯ ♠➲♥✳ ❚❤❡♦ ❜ỉ ➤Ị ♥➭② t❤× tå♥ t➵✐ x0 ∈ T1 (z) z∈K − 32 − x0 ❉♦ ➤ã ❧➭ ♥❣❤✐Ư♠ ❝đ❛ ❱■(K, f )✳ ❚õ ➤Þ♥❤ ❧ý tr➟♥ t❛ t❤✃② ➤✐Ị✉ ❦✐Ư♥ f ❧✐➟♥ tơ❝ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ tr♦♥❣ ❝❤♦ sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝ñ❛ ❱■(K, f )✳ ▼ét ❝➞✉ ❤á✐ ➤➷t r❛ ❧➭ ➤✐Ị✉ ❦✐Ư♥ ♥➭② ❝ã t❤Ĩ ❧➭♠ ②Õ✉ ❤➡♥ ➤➢ỵ❝ ❦❤➠♥❣ ✈➭ ❦ü t❤✉❐t ❝❤ø♥❣ ♠✐♥❤ ❝❤♦ ➜Þ♥❤ ❧ý ✹✳✺ sÏ ♥❤➢ t❤Õ ♥➭♦✳ P❤➬♥ ❝ß♥ ❧➵✐ ❝đ❛ ❝❤➢➡♥❣ ♥➭② sÏ tr➯ ❧ê✐ ❝❤♦ ❝➳❝ ❝➞✉ ❤á✐ ♥ã✐ tr➟♥✳ ●✐➯ sö S ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ ❚❛ ➤Þ♥❤ ♥❣❤Ü❛ ➳♥❤ ①➵ fS : K ∩ S → S X s❛♦ ❝❤♦ S ∩ K = ∅✳ ợ ị tứ fS (x), y = f (x), y , ∀x, y ∈ S ➜Þ♥❤ ♥❣❤Ü❛ ✹✳✽✳ ❚♦➳♥ tư ❤➵♥ ❝❤✐Ị✉ ❝đ❛ X ❍✐Ĩ♥ ♥❤✐➟♥ ♥Õ✉ f ➤➢ỵ❝ ❣ä✐ ❧➭ t➢➡♥❣ t❤Ý❝❤ ✈í✐ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ♥Õ✉ ✈í✐ ♠ä✐ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ề t ó ị tì f f ❧➭ ❧✐➟♥ tơ❝ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ X t❤♦➯ ♠➲♥ ➜Þ♥❤ ♥❣❤Ü❛ ✹✳✽✳ ✈➭ f : K → X∗ ✭✐✮ f f ✈➭ ✈í✐ ♠ä✐ K ⊂ X ✱ ❱■(K ∩ S, fS ) ❝ã ♥❣❤✐Ư♠✳ ➜Þ♥❤ ❧ý ✹✳✾✳ ❈❤♦ ❳ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ✭✐✐✮ S⊂X K ⊂X ❧➭ ♠ét t❐♣ ❧å✐✱ ❝♦♠♣❛❝t ②Õ✉ ❧➭ ♠ét t♦➳♥ tư t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ ❧➭ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ✈í✐ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ X✳ ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝ñ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ❑❤✐ ➤ã tå♥ t➵✐ x0 ∈ K s❛♦ ❝❤♦ f (x0 ), x − x0 ≥ 0, ∀x ∈ K ❈❤ø♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ sÏ sư ❞ơ♥❣ ❦ü t❤✉❐t ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➜Þ♥❤ ❧ý ✸✳✾✳ 33 ý ệ ỗ LL L ọ t✃t ❝➯ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ ❝❤ó♥❣ t❛ ➤➷t KL = K ∩ L✳ (i) ❚õ ➤✐Ị✉ ❦✐Ư♥ X✳ ❱í✐ ❱■(KL , fL ) ❝ã ♥❣❤✐Ư♠ xL KL K ỗ Y L tå♥ t➵✐ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❝❤ó♥❣ t❛ ➤Þ♥❤ ♥❣❤Ü❛ SY L⊇Y x ∈ KL ✈í✐ tÝ♥❤ ❝❤✃t ❧➭ t❐♣ t✃t ❝➯ ❝➳❝ x∈K ✈➭ fL (x), y − x ≥ , ∀y ∈ KL ❉Ô ❞➭♥❣ t❤✃② r➺♥❣ SY = φ ❣✐❛♦ ❤÷✉ ❤➵♥✱ ë ➤➞② SY ❜ë✐ ✈× xY ∈ SY ✳ ❧➭ ➤ã♥❣ ❝đ❛ SY s❛♦ ❝❤♦ ❍➡♥ ♥÷❛ ❤ä ✭✹✳✶✵✮ {S Y }Y ∈L ❝ã tÝ♥❤ ❝❤✃t ✭tr♦♥❣ t➠♣➠ ②Õ✉ ❝ñ❛ ❳✮✳ ❚❤❐t ✈❐② ❧✃② L1 , L2 , , Ln ∈ L ✈➭ ➤➷t M = L1 , L2 , , Ln ❝❤ó♥❣ t❛ ❝ã M ∈ L ✈➭ n SM ⊆ SLi i=1 ❉♦ ➤ã n n SLi ⊆ φ = SM ⊆ S M ⊆ S Li i=1 ❉Ô t❤✃② r➺♥❣ {S Y }Y ∈L SY ⊂ K ✈➭ K i=1 ❧➭ t❐♣ ❝♦♠♣❛❝t✳ ❚Ý♥❤ ❝❤✃t ❣✐❛♦ ❤÷✉ ❤➵♥ ❝đ❛ t❐♣ t❤á❛ ♠➲♥ S Y = φ Y ∈L ◆❤➢ ✈❐② ❝ã ➤✐Ó♠ x0 ∈ S Y y∈K ✈➭ ❝❤ä♥ ▲✃② t✉ú ý ❝❤✃t ❜❛♦ ➤ã♥❣ tå♥ t➵✐ ❞➲② t❐♣ SY ✈í✐ ♠ä✐ Y ∈ L✳ Y ∈ L s❛♦ ❝❤♦ Y ❝❤ø❛ xi ∈ SY xi → x0 ✳ s❛♦ ❝❤♦ y ✈➭ x0 ✳ ❱× x0 ∈ S Y ✱ t❤❡♦ tÝ♥❤ ❚❤❡♦ ❝➳❝❤ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ t❛ ❝ã f (xi ), v − xi ≥ 0, ∀v ∈ KY ➜➷❝ ❜✐Öt f (xi ), (1 − t)x0 + ty − xi ≥ 0, ∀t ∈ [0, 1] − 34 − ❱× f ❧➭ ❣✐➯ ➤➡♥ ➤✐Ö✉ t❤❡♦ ♥❣❤Ü❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥ ♥➟♥ t❛ ❝ã f (x0 ), y − x0 ≥ ◆❤➢ ✈❐② ❝❤ó♥❣ t❛ ➤➲ ❝❤Ø r❛ r➺♥❣ f (x0 ), y − x0 ≥ 0, ➜✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ r➺♥❣ x0 ∀y ∈ K ệ ủ (K, f ) ị ý ợ ứ ♠✐♥❤✳ ✷ ❍Ö q✉➯ ✹✳✶✵✳ ❈❤♦ ❳ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ị rỗ f ❧➭ ♠ét t❐♣ ❧å✐✱ ➤ã♥❣ ❧➭ ♠ét t♦➳♥ tö t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ ❧➭ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ✈í✐ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ f ✭✐✐✮ f : K → X∗ K⊂X X✳ ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ❑❤✐ ➤ã tå♥ t➵✐ x0 ∈ K s❛♦ ❝❤♦ f (x0 ), x − x0 ≥ 0, ∀x ∈ K ❑❤✐ K ❧➭ t❐♣ ❦❤➠♥❣ ❜Þ ❝❤➷♥✱ ❞➢í✐ ➤✐Ị✉ ❦✐Ư♥ ❜ø❝ t❛ ❝ã ❦Õt q✉➯ s❛✉✳ ➜Þ♥❤ ❧ý ✹✳✶✶✳ ❈❤♦ ❳ ❧➭ ❦❤➠♥❣ ❣✐❛♥ rỗ f f : K → X∗ K⊂X ❧➭ ♠ét t❐♣ ❧å✐✱ ➤ã♥❣ ❧➭ ♠ét t♦➳♥ tư t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ ❧➭ t♦➳♥ tư t➢➡♥❣ t❤Ý❝❤ ✈í✐ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ❤÷✉ ❤➵♥ ❝❤✐Ị✉ ❝đ❛ f f X✳ ❧➭ ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❜ø❝ ➤è✐ ✈í✐ t❐♣ ❧å✐ ❝♦♠♣❛❝t ②Õ✉ z0 ∈ C ∩ K s❛♦ ❝❤♦ f (x), z0 − x < 0, ∀x ∈ K \ C C ⊂ X ✱ tø❝ ❧➭ tå♥ t➵✐ − 35 − ❑❤✐ ➤ã tå♥ t➵✐ x0 ∈ K s❛♦ ❝❤♦ f (x0 ), x − x0 ≥ 0, ∀x ∈ K ❈❤ø♥❣ ♠✐♥❤✳ ❈❤ä♥ ë ➤ã ¯ , r) B(z ¯ , r) ❳Ðt ❱■(B(z r > s❛♦ ❝❤♦ C ⊂ B(z0 , r)✳ ❧➭ ❤×♥❤ ❝➬✉ ➤ã♥❣✳ ❱× ②Õ✉✳ ❚❤❡♦ ➜Þ♥❤ ❧ý ✹✳✾✱ tå♥ t➵✐ X ❧➭ ♣❤➯♥ ①➵✱ ¯ , r) ∩ K x0 ∈ B(z ¯ , r) ∩ K B(z ❧➭ t❐♣ ❝♦♠♣❛❝t s❛♦ ❝❤♦ ¯ , r) ∩ K f (x0 ), z − x0 ≥ 0, ∀z ∈ B(z ❚õ (iii) t❛ ó x0 C B(z0 , r) ỗ y ∈ K ✈➭ ❜Ð✳ ❚õ ✭✹✳✶✶✮ t❛ ❝ã f (x0 ), zt − x0 ≥ ❉♦ ➤ã t❛ ❝ã f (x0 ), y − x0 ≥ 0, ∀y ∈ K ✭✹✳✶✶✮ t ∈ (0, 1) ¯ , r) ∩ K x0 + t(y − x0 ) = ty + (1 − t)x0 ✳ ❑❤✐ ➤ã t❛ ❝ã zt B(z ị ý ợ ứ K, f )✱ ✈í✐ t❛ ➤➷t zt = t ∈ (0, 1) ➤ñ − 36 − ❦Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ➤➲ tr×♥❤ ❜➭② ♠ét ❜ø❝ tr❛♥❤ tỉ♥❣ t❤Ĩ ✈Ị sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s ❝❤♦ ♠ét sè ❧í♣ ❝➳❝ t♦➳♥ tư q✉❛♥ trä♥❣ ❜❛♦ ❣å♠✿ ❧í♣ ❝➳❝ t♦➳♥ tư ➤➡♥ ➤✐Ư✉✱ ❧í♣ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❑❛r❛♠❛❞✐❛♥✱ ❧í♣ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❇rÐ③✐s ✈➭ ❝✉è✐ ❝ï♥❣ ❧➭ ❧í♣ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ t❤❡♦ ♥❣❤Ü❛ ❝đ❛ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ❉➢í✐ ♠ét ❣ã❝ ➤é ♥➭♦ ➤ã ❧í♣ ❝➳❝ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ö✉ ❞♦ ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥ ➤➢❛ r❛ ❜❛♦ ❣å♠ ❝➯ ❤❛✐ ❧í♣ t♦➳♥ tư ❣✐➯ ➤➡♥ ➤✐Ư✉ ❝đ❛ ❑❛r❛♠❛❞✐❛♥ ✈➭ ❇rÐ③✐s ➤➲ ➤➢❛ r❛ tr➢í❝ ➤ã✱ ♥ã ❝❤♦ t❛ ♠ét ❝➳✐ ♥❤×♥ t❤è♥❣ ♥❤✃t ❝❤♦ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❱■s✳ ❚r➟♥ ❝➡ së ❤Ư t❤è♥❣ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦Õt q✉➯ ♥ã✐ tr➟♥ ❝❤ó♥❣ t❛ ➤➲ ➤➵t ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ ♠í✐ ♥❤♦ ♥❤á ✈Ị ❝❤đ ➤Ị ♥❣❤✐➟♥ ❝ø✉ ♥➭②✳ ❈➳❝ ❦Õt q✉➯ ♥➭② ➤➢ỵ❝ ứ ự tr ị ý ữ ❦Õt q✉➯ ❝❤Ý♥❤ t❤✉ ➤➢ỵ❝ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭②✳ ❱í✐ ♣❤➵♠ ✈✐ ✈➭ t❤ê✐ ❣✐❛♥ ❝ã ❤➵♥✱ ❝❤➽❝ ❝❤➽♥ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✳ ▼♦♥❣ q✉ý t❤➭② ❝➠ ✈➭ ❝➳❝ ❜➵♥ ➤å♥❣ ❤ä❝ ❣ã♣ ý ➤Ó ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ❳✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥✦ − 37 − ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❆✳ ❉♦♠♦❦♦s ❛♥❞ ❏✳ ❑♦❧✉♠❜❛♥ ✭✷✵✵✵✮✱ ✧❈♦♠♣❛r✐s♦♥ ♦❢ t✇♦ ❞✐❢❢❡r❡♥t t②♣❡s ♦❢ ♣s❡✉❞♦♠♦♥♦t♦♥❡ ♠❛♣♣✐♥❣s✧✱ ❙❡♠✐♥❛✐r❡ ❞❡ ❧❛ t❤❡♦r✐❡ ❞❡ ❧❛ ♠❡✐❧❧❧❡✉r❡ ❛♣♣r♦①✐♠❛t✐♦♥✱❝♦♥✈❡①✐t❡ ❡t ♦♣t✐♠✐s❛t✐♦♥✱ ✾✺✲✶✵✸✳ ❬✷❪ ❇✳ ❚✳ ❑✐❡♥✱ ❏✳ ❈✳ ❨❛♦ ❛♥❞ ◆✳ ❉✳ ❨❡♥ ✭✷✵✵✼✮✱ ✧❖♥ t❤❡ s♦❧✉t✐♦♥ ❡①✐st❡♥❝❡ ♦❢ ♣s❡✉❞♦♠♦♥♦t♦♥❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ ❏✳ ●❧♦❜❛❧ ❖♣t✐♠✳ ❢✐rst ♦♥❧✐♥❡✱ ✹✶✱ ✶✸✺✲✶✹✺✳ ❬✸❪ ❇✳ ❘✐❝❝❡r✐ ✭✶✾✾✺✮✱ ❇❛s✐❝ ❡①✐st❡♥❝❡ t❤❡♦r❡♠s ❢♦r ❣❡♥❡r❛❧✐③❛t✐♦♥ ✈❛r✐❛t✐♦♥❛❧ ❛♥❞ q✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✱ ✐♥ ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s ❛♥❞ ◆❡t✇♦r❦ ❊q✉✐❧✐❜r✐✉♠ Pr♦❜❧❡♠s ✭❋✳ ●✐❛♥♥❡ss✐ ❛♥❞ ❆✳ ▼❛✉❣❡r✐✱ ❊❞s✳✮✱ P❧❡♥✉♠✱ ◆❡✇ ❨♦r❦✱ ✷✺✶✲✷✺✺✳ ❬✹❪ ❉✳ ❆✉ss❡❧ ❛♥❞ ◆✳ ❍❛❞❥✐s❛✈✈❛s ✭✷✵✵✹✮✱ ✧❖♥ q✉❛s✐♠♦♥♦t♦♥❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥✲ ❡q✉❛❧✐t✐❡s✧✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✱ ✶✷✶✱ ✹✹✺✲✹✺✵✳ ❬✺❪ ❉✳ ■♥♦❛♥ ❛♥❞ ❏✳ ❑♦❧✉♠❜➳♥ ✭✷✵✵✻✮✱ ✧❖♥ ♣s❡✉❞♦♠♦♥♦t♦♥❡ s❡t✲✈❛❧✉❡❞ ♠❛♣✲ ♣✐♥❣s✱ Pr❡♣r✐♥t s✉❜♠✐tt❡❞ t♦ ❊s❡❧✈✐❡r ❙❝✐❡♥❝❡✧✱ ❊❧❡s❡✈✐❡r✱ ✻✽✱ ✹✼✲✺✸✳ ❬✻❪ ❉✳ ❑✐♥❞❡r❧❡❤r❡r ❛♥❞ ●✳ ❙t❛♠♣❛❝❝❤✐❛ ✭✶✾✽✵✮✱ ✧❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t✐❡s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✧✱ ❆❝❛❞❡♠✐❝ Pr❡ss✳ ❬✼❪ ❊✳ ❩❡✐❞❧❡r ✭✶✾✽✻✮✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✧✱ ■✳ ❋✐①❡❞✲P♦✐♥t ❚❤❡♦r❡♠s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ − 38 − ❬✽❪ ❊✳ ❩❡✐❞❧❡r ✭✶✾✾✵✮✱ ✧◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ■■✴❇✳ ◆♦♥❧✐♥❡❛r ▼♦♥♦t♦♥❡ ❖♣❡r❛t♦rs✧✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ❬✾❪ ❋✳ ❇r♦✇❞❡r ✭✶✾✻✸✮✱ ✧◆♦♥❧✐♥❡❛r ❡❧❧✐♣t✐❝ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s✧✱ ❇✉❧❧✳❆♠❡r✳ ▼❛t❤✳ ❙♦❝✱ ✻✾✱ ✽✻✷✲✽✼✹✳ ❬✶✵❪ ❋✳ ❇r♦✇❞❡r ✭✶✾✼✷✮✱ ✧◆♦♥❧✐♥❡❛r ♠❛♣♣✐♥❣s ♦❢ ♠♦♥♦t♦♥❡ t②♣❡ ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s✧✱ ❏✳ ❋✉♥❝✳ ❆♥❛❧✱ ✶✶ ✱ ✷✺✶✲✷✾✹✳ ❬✶✶❪ ❋✳ ❋❛❝❝❤✐♥❡✐ ❛♥❞ ❏✳✲❙✳ P❛♥❣ ✭✷✵✵✸✮✱ ✧❋✐♥✐t❡✲❉✐♠❡♥s✐♦♥❛❧ ❱❛r✐❛t✐♦♥❛❧ ■♥✲ ❡q✉❛❧✐t✐❡s ❛♥❞ ❈♦♠♣❧❡♠❡♥t❛r✐t② Pr♦❜❧❡♠s✧✱ ❱♦❧s✳ ■✱ ■■✱ ❙♣r✐♥❣❡r✳ ❬✶✷❪ ●✳ ▼✐♥t② ✭✶✾✻✷✮✱ ✧▼♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✧✱ ❉✉❦❡ ▼❛t❤✱ ✷✾ ✱ ✸✹✶✲✸✹✻✳ Ð ❬✶✸❪ ❍✳ ❇rÐ③✐s ✭✶✾✻✽✮✱ ✧ q✉❛t✐♦♥s ❡t ✐♥Ðq✉❛t✐♦♥s ♥♦♥ ❧✐♥Ð❛✐r❡s ❞❛♥s ❧❡s ❡s♣❛❝❡s ✈❡❝t♦r✐❡❧ ❡♥ ❞✉❛❧✐tÐ✧✱ ❆♥♥✳ ■♥st✳❋♦✉r✐❡r✱ ✶✽✱ ✶✶✺✲✶✼✺✳ ❬✶✹❪ ■✳ ❑♦♥♥♦✈ ✭✷✵✵✺✮✱ ✧●❡♥❡r❛❧✐③❡❞ ♠♦♥♦t♦♥❡ ❡q✉✐❧✐❜r✐✉♠ ♣r♦❜❧❡♠s ❛♥❞ ✈❛r✐✲ ❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✱ ✐♥ ❍❛♥❞❜♦♦❦ ♦❢ ●❡♥❡r❛❧✐③❡❞ ❈♦♥✈❡①✐t② ❛♥❞ ●❡♥❡r❛❧✲ ✐③❡❞ ▼♦♥♦t♦♥✐❝✐t②✱ ✭◆✳ ❍❛❞❥✐s❛✈✈❛s✱ ❙✳ ❑♦♠❧♦s✐ ❛♥❞ ❙✳ ❙❝❤❛✐❜❧❡✱ ❊❞s✮✧✱ ❙♣r✐♥❣❡r✱ ♣♣✳ ✺✺✾✲✻✶✽✳ ❬✶✺❪ ❏✳ ▲✳ ▲✐♦♥s ❛♥❞ ●✳ ❙t❛♠♣❛❝❝❤❛ ✭✶✾✻✼✮✱ ✧❱❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✱ ❈♦♠♠✉♥✧✱ ♣✉r❡ ❆♣♣❧✳ ▼❛t❤✱ ✷✵✱ ✹✾✸✲✺✶✾✳ ❬✶✻❪ ❏✳✲P✳ ❆✉❜✐♥ ❛♥❞ ❆✳ ❈❡❧❧✐♥❛ ✭✶✾✽✹✮✱ ✧❉✐❢❢❡r❡♥t✐❛❧ ■♥❝❧✉s✐♦♥s✧✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✳ ❬✶✼❪ ❏✳✲P✳ ❈r♦✉③❡✐① ✭✶✾✾✼✮✱ ✧Ps❡✉❞♦♠♦♥♦t♦♥❡ ✈❛r❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ♣r♦❜❧❡♠s✿ ❊①✐st❡♥❝❡ ♦❢ s♦❧✉t✐♦♥s✧✱ ▼❛t❤✳ Pr♦❣r❛♠✱ ✼✽ ✱ ✸✵✺✲✸✶✹✳ − 39 − ❬✶✽❪ ❏✳❙✳ ●✉♦ ❛♥❞ ❏✳ ❈✳ ❨❛♦ ✭✶✾✾✹✮✱ ✧❱❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✇✐t❤ ♥♦♥♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs✧✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r✳ ❆♣♣❧✱ ✽✵✱ ✻✸✲✼✹✳ ❬✶✾❪ ❏✳ ❈✳ ❨❛♦ ✭✶✾✾✹✮✱ ✧❱❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✇✐t❤ ❣❡♥❡r❛❧✐③❡❞ ♠♦♥♦t♦♥❡ ♦♣✲ ❡r❛t♦rs✧✱ ▼❛t❤✳ ❖♣❡r✳ ❘❡s✱ ✶✾✱ ✻✾✶✲✼✵✺✳ ❬✷✵❪ ❏✳ ❈✳ ❨❛♦ ✭✶✾✾✹✮✱ ✧▼✉❧t✐✲✈❛❧✉❡❞ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✇✐t❤ ❑✲ ♣s❡✉❞♦♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs✧✱ ❏♦✉r♥❛❧ ♦❢ ❖♣t✐♠✐③❛t✐♦♥ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐✲ ❝❛t✐♦♥s✱ ✽✵✱ ✻✸✲✼✹✳ ❬✷✶❪ ❑✳ ❉❡✐♠❧✐♥❣ ✭✶✾✽✺✮✱ ✧◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥❛❧ ❆♥❛❧②s✐s✧✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✳ ❬✷✷❪ ▼✳ ❙✐♦♥ ✭✶✾✺✽✮✱ ✧❖♥ ❣❡♥❡r❛❧ ♠✐♥✐♠❛① t❤❡♦r❡♠s✧✱ P❛❝✳ ❏✳ ▼❛t❤✱ ✽ ✱ ✶✼✶✲✶✼✻✳ ❬✷✸❪ ◆✳ ❉✳ ❨❡♥ ✭✷✵✵✹✮✱ ✧❖♥ ❛ ♣r♦❜❧❡♠ ♦❢ ❇✳ ❘✐❝❝❡r✐ ♦♥ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✱ ✐♥ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✧ ✱ ◆♦✈❛ ❙❝✐❡♥❝❡ P✉❜❧✐s❤❡rs✱ ◆❡✇ ❨♦r❦✱ ❱♦❧✳ ✺ ✭❨✳ ❏✳ ❈❤♦✱ ❏✳ ❑✳ ❑✐♠ ❛♥❞ ❙✳ ▼✳ ❑❛♥❣✱ ❊❞s✳✮✱ ✶✻✸✲✶✼✸✳ ❬✷✹❪ ◆✳ ❉✳ ❨❡♥ ✭✶✾✾✼✮✱ ✧❆ r❡s✉❧t r❡❧❛t❡❞ t♦ ❘✐❝❝❡r✐✬s ❝♦♥❥❡❝t✉r❡ ♦♥ ❣❡♥❡r❛❧✐③❡❞ q✉s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ ❆r❝❤✳ ▼❛t❤✱ ✻✾✱ ✺✵✼✲✺✶✹✳ ❬✷✺❪ P✳ ❉❛♥✐❡❧❡ ✭✶✾✾✾✮✱ ✧❆✳ ▼❛✉❣❡r✐ ❛♥❞ ❲✳ ❖❡tt❧✐✱ ❚✐♠❡✲❞❡♣❡♥❞❡♥t tr❛❢❢✐❝ ❡q✉✐✲ ❧✐❜r✐❛✧✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r✳ ❆♣♣❧✱ ✶✵✸ ✱ ✺✹✸✲✺✺✺✳ ❬✷✻❪ P✳ ❍❛rt♠❛♥♥ ❛♥❞ ●✳ ❙t❛♠♣❛❝❝❤✐❛ ✭✶✾✻✻✮✱ ✧❖♥ s♦♠❡ ♥♦♥❧✐♥❡❛r ❡❧❧✐♣t✐❝ ❞✐❢✲ ❢❡r❡♥t✐❛❧ ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥s✧✱ ❆❝t❛ ▼❛t❤✱ ✶✶✺✱ ✶✺✸✲✶✽✽✳ ❬✷✼❪ ❙✳ ❑❛r❛♠❛r❞✐❛♥ ✭✶✾✼✻✮✱ ✧❈♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜❧❡♠s ♦✈❡r ❝♦♥❡s ✇✐t❤ ♠♦♥♦✲ t♦♥❡ ❛♥❞ ♣s❡✉❞♦♠♦♥♦t♦♥❡ ♠❛♣s✧✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✱ ✶✽ ✱ ✹✹✺✲✹✺✹✳ ... g(x) ✭✷✳✹✮ ❚➢➡♥❣ tù t❛ ❝ã ❈é♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✸✮ ✈➭ ✭✷✳✹✮ t❛ ❝ã g (x) − g (y), x − y ≥ ị ĩ tử ữ ề ❝đ❛ ❤➵♥ ❝❤Õ t❤× f : K → X∗ X f : E ∩ X → X∗ f (xn ), y → f (x), y ➤➢ỵ❝ ❣ä✐ ❧➭ ❧✐➟♥ tơ❝... ❉♦♠♦❦♦s✲❑♦❧✉♠❜➳♥✳ ➜Þ♥❤ ❧ý ✹✳✻✳ ❬✶✱ tr ✾✾✲✶✵✶❪ ❈❤♦ ❳ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❧å✐ ✈➭ f : K → X∗ KX t ỗ ột t tử sử r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ ♥❣❤✐Ư♠ ➤ó♥❣✿ ✭✐✮ P❤Ð♣ ❣✐❛♦ ❤÷✉ ❤➵♥ ❝đ❛ ❑ ✈í✐ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❝♦♥

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Nhà XB: Math. Oper. Res
Năm: 1994
[20] J. C. Yao (1994), "Multi-valued variational inequalities with K- pseudomonotone operators", Journal of Optimization Theory and Appli- cations, 80, 63-74 Sách, tạp chí
Tiêu đề: Multi-valued variational inequalities with K- pseudomonotone operators
Tác giả: J. C. Yao
Nhà XB: Journal of Optimization Theory and Applications
Năm: 1994

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