Tính lồi và tính không giãn

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✯✯✯✯✯✯✯✯✯✯✯✯✯ ◆●❯❨➍◆ ❚❍➚ ❚❍❷❖ ❚➑◆❍ ▲➬■ ❱⑨ ❚➑◆❍ ❑❍➷◆● ●■❶◆ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●✐↔✐ t➼❝❤ ❍➔ ◆ë✐ ✲ ✷✵✶✹ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❑❍❖❆ ❚❖⑩◆ ✯✯✯✯✯✯✯✯✯✯✯✯✯ ◆●❯❨➍◆ ❚❍➚ ❚❍❷❖ ❚➑◆❍ ▲➬■ ❱⑨ ❚➑◆❍ ❑❍➷◆● ●■❶◆ ❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍➴❈ t ữớ ữợ ◆●❯❨➍◆ ❱❿◆ ❚❯❨➊◆ ❍➔ ◆ë✐ ✲ ✷✵✶✹ ▲❮■ ❈❷▼ ❒◆ ữủ ỷ ỡ tợ t ❝ỉ ❣✐→♦ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✤➣ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ✣➦❝ ❜✐➺t ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ❚✉②➯♥ ◆❣✉②➵♥ ❱➠♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✱ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât ✈➔ ❤↕♥ ữủ sỹ õ õ ỵ ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝æ ❣✐→♦ ✈➔ t♦➔♥ t❤➸ ❜↕♥ ✤å❝ ✤➸ ✤➲ t➔✐ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ❍➔ ◆ë✐✱ ♥❣➔② ✶✽ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✐ ▲❮■ ữợ sỹ ữợ ❝õ❛ t❤➛② ❣✐→♦ ◆❣✉②➵♥ ❱➠♥ ❚✉②➯♥ ❦❤â❛ ❧✉➟♥ ❝õ❛ ❡♠ ữủ t ổ trũ ợ t t ♥➔♦ ❦❤→❝✳ ❚r♦♥❣ ❦❤✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐ ❡♠ ✤➣ sû ❞ö♥❣ ✈➔ t❤❛♠ ❦❤↔♦ ❝→❝ t❤➔♥❤ tü✉ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈ỵ✐ ❧á♥❣ ❜✐➳t ì♥ tr➙♥ trå♥❣✳ ❍➔ ◆ë✐✱ ♥❣➔② ✶✽ t❤→♥❣ ✺ ♥➠♠ ✷✵✶✹ ❙✐♥❤ ❱✐➯♥ ◆❣✉②➵♥ ❚❤à ❚❤↔♦ ✐✐ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✶ ▲í✐ ♠ð ✤➛✉ ✶ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✸ ✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✶✳ ❑➼ ❤✐➺✉ ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t tự ỗ t t❤ù❝ ❝ì ❜↔♥ ✳ ✳ ✳ ✺ ✶✳✶✳✸✳ ❚♦→♥ tû t✉②➳♥ t➼♥❤ ✈➔ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✼ ✶✳✶✳✹✳ ❚æ♣æ ♠↕♥❤ ✈➔ ②➳✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✺✳ ❙ü ❤ë✐ tö ②➳✉ ❝õ❛ ❞➣② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✷✳ ❚➼♥❤ ①➜♣ ①➾ tèt ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✷✳✸✳ ❚➼♥❤ ❝❤➜t tæ♣æ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ỵ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỗ t ❦❤æ♥❣ ❣✐➣♥ ✷✾ ✷✳✶✳ ❈→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳ P t ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✸✳ ❈→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹✳ ❈→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ tr✉♥❣ ❜➻♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✺✳ ❈→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ t❤ỉ♥❣ t❤÷í♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ❑➳t ❧✉➟♥ ✺✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✶ ✐✈ ▲í✐ ♠ð ✤➛✉ ❚r♦♥❣ ●✐↔✐ t➼❝❤ t♦→♥ ❤å❝✱ t➼♥❤ ❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③✱ ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❘✉❞♦❧❢ ▲✐♣s❝❤✐t③✱ ❧➔ ♠ët ❞↕♥❣ ♠↕♥❤ ❝õ❛ t➼♥❤ ❧✐➯♥ tö❝ ✤➲✉ ❝õ❛ ❝→❝ ❤➔♠ sè✳ ❈❤♦ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ (X, dX ) ✈➔ (Y, dY )✱ ✈ỵ✐ dX ❧➔ ♠❡tr✐❝ tr➯♥ X ✈➔ dY ❧➔ ♠❡tr✐❝ tr➯♥ Y ✳ ▼ët t♦→♥ tû f : X → Y ✤÷đ❝ ❣å✐ ❧➔ tử st tỗ t ởt số K s❛♦ ❝❤♦ dY (f (x1 ), f (x2 )) ≤ KdX (x1 , x2 ) ∀x1, x2 ∈ X ◆➳✉ ≤ K < t❤➻ f ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠ët t♦→♥ tû ✭→♥❤ ①↕✮ ❝♦✳ ▲ỵ♣ ❤➔♠ ▲✐♣s❝❤✐t③ ✤â♥❣ ♠ët ✈❛✐ trá ✤➦❝ ❜✐➺t q✉❛♥ trå♥❣ tr♦♥❣ ●✐↔✐ t t tr ỵ tt ữỡ tr ✈✐ ♣❤➙♥✱ t➼♥❤ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ ❧➔ ♠ët ✤✐➲✉ ❦✐➺♥ tr t ỵ Pr õ sỹ tỗ t t t t r ỵ tt t t t ❝♦ ❧➔ ♠ët ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ✤↔♠ ❜↔♦ sü tỗ t t t t t tỷ ỵ ▼ët t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❧➔ ♠ët t♦→♥ tû ▲✐♣s❝❤✐t③ ✈ỵ✐ ❤➺ sè ▲✐♣s❝❤✐t③ ✶✳ ▲ỵ♣ t♦→♥ tû ♥➔② ✤â♥❣ ♠ët ✈❛✐ trá tr✉♥❣ t➙♠ tr♦♥❣ t♦→♥ ❤å❝ ù♥❣ ❞ö♥❣✱ ❜ð✐ ✈➻ r➜t ♥❤✐➲✉ ❜➔✐ t♦→♥ tr♦♥❣ ❣✐↔✐ t➼❝❤ ♣❤✐ t✉②➳♥ q✉② ✈➲ ❜➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ởt t tỷ ổ ợ ỵ q trồ tr ữủ sỹ ữợ t tổ t ỗ ✈➔ t➼♥❤ ❦❤æ♥❣ ❣✐➣♥✑ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❑❤â❛ ỗ ữỡ tr ởt số tự ỡ s t ỗ r ✶ ❝❤÷ì♥❣ ♥➔② ❝â tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ỡ ổ rt t ỗ ỗ ✈➔ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✤÷đ❝ sû ❞ư♥❣ tr ữỡ ữỡ tr t ❦❤æ♥❣ ❣✐➣♥ ♠ët t♦→♥ tû ✈➔ ♠ët sè ❜✐➳♥ t❤➸✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ❝á♥ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠ët t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ✤➦❝ ❜✐➺t õ ởt t ỗ tr ổ ❣✐❛♥ ❍✐❧❜❡rt✳ ❚✐➳♣ t❤❡♦ ✤â✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ♠ët t♦→♥ tû ❦❤ỉ♥❣ ❣✐➣♥✳ ❈✉è✐ ❝ò♥❣ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❝→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ tr✉♥❣ ❜➻♥❤✳ ✷ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ◆ë✐ ❞✉♥❣ ❝õ❛ ❦❤♦→ ❧✉➟♥ ♥➔② ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✱ t❛ ❦➼ ❤✐➺✉ ❦❤ỉ♥❣ ❣✐❛♥ ♥➔② ❧➔ H ✈ỵ✐ t➼❝❤ ✈ỉ ữợ ã|ã ã tr H tữỡ ự ợ t ổ ữợ ợ d tự ❧➔✱ x | x ✈➔ d(x, y) = x − y (∀x ∈ H), (∀y ∈ H), t❛ ❝â x = tỷ ỗ t tự tr H ❤✐➺✉ ❧➔ Id✳ Ð ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ✤➥♥❣ t❤ù❝✱ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝ì ❜↔♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ✶✳✶✳✶✳ ❑➼ ❤✐➺✉ ✈➔ ✈➼ ❞ư P❤➛♥ ❜ò trü❝ ❣✐❛♦ ❝õ❛ t➟♣ ❝♦♥ C tr♦♥❣ H✱ ❦➼ ❤✐➺✉ ❧➔ C ⊥ ✱ tù❝ ❧➔ C ⊥ = {u ∈ H | (∀x ∈ C) x | u = 0} ✸ ✭✶✳✷✮ ▼ët t➟♣ trü❝ ❣✐❛♦ C ⊂ H ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❝ì sð trü❝ ❣✐❛♦ ❝õ❛ H ♥➳✉ spanC = H✳ ❑❤ỉ♥❣ ❣✐❛♥ H ❧➔ t→❝❤ ✤÷đ❝ ♥➳✉ ♥â ❝❤ù❛ ♠ët ❝ì sð trü❝ ❣✐❛♦ ✤➳♠ ✤÷đ❝✳ ❈❤♦ (xi )i∈I ❧➔ ♠ët ❤å ❝→❝ ✈❡❝tì t❤✉ë❝ H ✈➔ I ❧➔ ❧ỵ♣ ❝→❝ t➟♣ ❝♦♥ ❤ú✉ ❤↕♥ ❦❤→❝ ré♥❣ ❝õ❛ I ✱ ✤÷đ❝ s➢♣ t❤ù tü ❜ð✐ q✉❛♥ ❤➺ ❜❛♦ ❤➔♠ õ (xi )iI tờ tỗ t↕✐ x ∈ H t❤ä❛ ♠➣♥ i∈J (xi )J∈I ❤ë✐ tư tỵ✐ x✱ tù❝ ❧➔ (∀ε ∈ R++ )(∃K ∈ I)(∀J ∈ I) J ⊃ K ⇒ x− (xi ) ε ✭✶✳✸✮ i∈J ❑❤✐ ✤â✱ t❛ ✈✐➳t x = i∈I (xi ) ❍å (αi )i∈I ①→❝ ✤à♥❤ tr➯♥ [0, +∞]✱ t❛ ❝â (αi ) = sup i∈I J∈I ✭✶✳✹✮ (αi ) i∈J ❱➼ ❞ö ✶✳✶✳ ❚ê♥❣ trü❝ t✐➳♣ ❝õ❛ ♠ët ❤å ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ (H, · i )i∈I ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳ Hi = x = (xi )i∈I ∈ ×i∈I Hi | xi i < + iI iI ữủ tr ợ (x, y) → (xi + yi )i∈I , ♣❤➨♣ ♥❤➙♥ ổ ữợ (, x) (xi )iI , t ổ ữợ (x, y) xi | y i i i∈I ❑❤✐ I ❧➔ ❤ú✉ ❤↕♥ t❛ ✈✐➳t ×i∈I Hi t❤❛② ❝❤♦ i∈I Hi ✳ ●✐↔ sû ✈ỵ✐ ∀i ∈ I ✱ fi : Hi → [−∞, +∞] ✈➔ ♥➳✉ I ❧➔ ❤ú✉ ❤↕♥ t❤➻ inf i∈I fi ≥ 0✳ ❑❤✐ ✤â Hi → [−∞, +∞] : (xi )i∈I → fi : i∈I i∈I fi (xi ) i∈I ỵ rrr D t ỗ õ rộ H ✈➔ T FixT = ∅✳ : D → D ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ t❤➻ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ✣à♥❤ ❧➼ ✶✳✶✷ t❤➜② r➡♥❣ D ❧➔ ✤â♥❣ ②➳✉ t❤❡♦ ❞➣②✱ ✈➔ tø ✣à♥❤ ❧➼ ✶✳✶✸ ♥â ❧➔ ❝♦♠♣❛❝t ②➳✉ t❤❡♦ ❞➣②✳ ▲➜② x0 ∈ D ✈➔ ❞➣② (αn )n∈N t❤✉ë❝ [0, 1] t❤ä❛ ♠➣♥ α0 = ✈➔ αn ↓ 0✳ ❱ỵ✐ ♠å✐ n ∈ N✱ t♦→♥ tû Tn : D → D : x → αn x0 + (1 − αn )T x ❧➔ ♠ët →♥❤ ①↕ ❝♦ ✈ỵ✐ ❤➺ sè ❝♦ (1 − αn ) ✈➔ ♥â ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ xn ∈ D✳ ❍ì♥ ♥ú❛✱ ✈ỵ✐ n ∈ N✱ xn − T xn = Tn xn − T xn = αn x0 − T xn ≤ αn diam(D) ❉♦ ✤â✱ xn − T xn → 0✳ ▼➦t ❦❤→❝✱ (xn )n∈N ♥➡♠ tr♦♥❣ D✱ t❛ ❝â t❤➸ ❧➜② ✤÷đ❝ ♠ët ❞➣② ❝♦♥ ❤ë✐ tư ②➳✉ ❧➔ xkn x ∈ D✳ ❱➻ xkn − T xkn → 0✱ ♥➯♥ ❍➺ q✉↔ ✷✳✻ ❦❤➥♥❣ ✤à♥❤ x ∈ FixT ✳ ▼➺♥❤ ✤➲ ✷✳✼✳ ✭✤÷í♥❣ ❝♦♥❣ ❣➛♥ ✤ó♥❣✮ ❈❤♦ D ❧➔ t➟♣ ỗ õ rộ H T : D → D ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ ε ∈ [0, 1] ✈➔ x ∈ D tỗ t t x D s ❝❤♦ xε = εx + (1 − ε)T xε ✭✷✳✶✷✮ ❱ỵ✐ ♠é✐ ε ∈ [0; 1] ✤➦t Tε : D → D : x → xε ✈➔ ❧➜② x ∈ D✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ♠➺♥❤ ✤➲ s❛✉✿ ✭✐✮ (∀ε ∈ [0; 1]) Tε = εId + (1 − ε)T Tε = (Id − (1 − ε)T )−1 ◦ εId✳ ✭✐✐✮ (∀ε ∈ [0; 1]) Tε ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝✳ ✭✐✐✐✮ (∀ε ∈ [0; 1]) FixTε = FixT ✳ ✭✐✈✮ (∀ε ∈ [0; 1]) ε(x − T xε ) = xε − T xε = (1 − ε)−1 ε(x − xε )✳ ✭✈✮ ●✐↔ sû r➡♥❣ FixT = ∅✳ ❑❤✐ ✤â✱ limε↓0 xε = ∞✳ ✸✼ ✭✈✐✮ (∀ε ∈ [0; 1]) (∀y ∈ FixT ) x − xε + xε − y ≤ x − y 2✳ ✭✈✐✐✮ ●✐↔ sû FixT = ∅✱ ❦❤✐ ✤â limε↓0 xε = PFixT x✳ ✭✈✐✐✐✮ (∀ε ∈ [0; 1]) (δ ∈ [0; 1]) ε−δ 1−ε xε − x + δ(2 − δ) xδ − xε ✭✐①✮ (∀ε ∈ [0; 1]) (δ ∈ [0; 1]) x − xε 2 ≤2 ε−δ xε − x | xδ − xε 1−ε + xε − xδ ≤ x − xδ ✳ ✭①✮ ❍➔♠ [0; 1] → R+ : ε → x − xε ❧➔ ❣✐↔♠✳ ✭①✐✮ ✣÷í♥❣ ❝♦♥❣ [0; 1] → H : ε → xε ❧➔ ❧✐➯♥ tö❝✳ ✭①✐✐✮ ◆➳✉ x ∈ FixT t❤➻ xε ≡ x ❧➔ ❤➡♥❣ sè❀ ♠➦t ❦❤→❝✱ (xε )ε∈[0;1] ❧➔ ✤÷í♥❣ ❝♦♥❣ ♥ë✐ ①↕✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ ε ∈ [0, 1]✱ t♦→♥ tû D → D : z → εx + (1 − ε)T z ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳ ❉♦ ✤â✱ xε ❧➔ ❞✉② ♥❤➜t ✈➔ Tε ❧➔ ✤à♥❤ tốt ỗ t tự ró r ❍ì♥ ♥ú❛✱ εId = Tε − (1 − ε)T Tε = (Id − (1 − ε)T )Tε ✈➔ ✈➻ Id − (1 − ε)T ❧➔ ✤ì♥ →♥❤✱ t❛ t❤✉ ✤÷đ❝ Tε = (Id − (1 − ε)T )−1 ◦ εId✳ ✭✐✐✮✿ ❈❤♦ y ∈ D✳ ❑❤✐ ✤â x − y = ε−1 ((xε − (1 − ε)T xε ) − (yε − (1 − ε)T yε )) = ε−1 ((xε − yε ) − (1 − ε)(T xε − T yε )) ✸✽ ✭✷✳✶✸✮ ❙û ❞ö♥❣ ✭✷✳✶✷✮✱ ✭✷✳✶✸✮ ✈➔ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ t❛ s✉② r❛ r➡♥❣ Tε x − Tε y | (Id − Tε )x − (Id − Tε )y = xε − yε | (1 − ε)(x − T xε ) − (1 − ε)(y − T yε ) = (1 − ε) xε − yε | (x − y) − (T xε − T yε ) = (1 − ε)ε−1 xε − yε | (xε − yε ) − (T xε − T yε ) ≥ (ε−1 − 1)( xε − yε − xε − yε T xε − T y ε ) = (ε−1 − 1) xε − yε ( xε − yε − T xε − T yε ) ✭✷✳✶✹✮ ≥0 ❉♦ ✤â✱ q✉❛ ▼➺♥❤ ✤➲ ✷✳✶✱ Tε ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝✳ ✭✐✐✐✮✿ ❈❤♦ x ∈ D✳ ●✐↔ sû x ∈ FixT ✳ ❑❤✐ ✤â✱ x = εx + (1 − ε)T x ✈➔ ❞♦ ✤â✱ x = xε ❜ð✐ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ xε ✱ s✉② r❛ Tε x = x✳ ❱➻ ✈➟②✱ x ∈ FixTε ✳ ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû x ∈ FixTε ✱ ❦❤✐ ✤â x = xε = εx + (1 − ε)T xε = x + (1 − ε)(T x − x) ✈➔ ❞♦ ✤â x = T x✱ tù❝ ❧➔ x ∈ FixT ✳ ✭✐✈✮✿ ❙✉② r❛ tø ✭✷✳✶✷✮✳ ✭✈✮✿ sỷ tỗ t (n )nN tở [0, 1] t❤ä❛ ♠➣♥ εn ↓ ✈➔ (xεn )n∈N ❜à ❝❤➦♥✳ ❚ø ✭✐✈✮ ✈➔ ❍➺ q✉↔ ✷✳✻ t❤➜② ❝→❝ ✤✐➸♠ ❤ë✐ tö ②➳✉ ❝õ❛ (xεn )n∈N ♥➡♠ tr♦♥❣ FixT ✳ ✭✈✐✮✿ ●✐↔ sû y ∈ FixT ✳ ❇ð✐ ✭✐✐✐✮✱ y = Tε y = yε ✳ ❱➻ Tε ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ ❜ð✐ ✭✐✐✮✱ t❛ ❝â x−y ≥ xε − yε + (x − xε ) − (y − yε ) = xε − y + x − xε ✭✷✳✶✺✮ ✭✈✐✐✮✿ ❈❤♦ (εn )n∈N t❤✉ë❝ [0, 1] t❤ä❛ ♠➣♥ εn ↓ ✈➔ ✤➦t (∀n ∈ N) zn = xεn ✳ ❚ø ✭✈✐✮✱ (zn )n∈N ❧➔ ❜à ❝❤➦♥✳ ❉♦ ✤â✱ sû ❞ö♥❣ ✭✐✈✮✱ t❛ t❤➜② zn − T zn → ✸✾ 0✳ ❈❤♦ z ❧➔ ✤✐➸♠ tö ②➳✉ t❤❡♦ ❞➣② ❝õ❛ (zn )n∈N ✱ t❛ ♥â✐ zkn ✷✳✶ ❦➨♦ t❤❡♦ z ∈ FixT ✳ ❚ø ✭✈✐✮✱ t❛ t❤✉ ✤÷đ❝ lim x − zkn ❱➻ x − zkn z ✳ ✣à♥❤ ❧➼ ≤ x − z 2✳ x − z ✱ ❇ê ✤➲ ✶✳✶✹✭✐✮ ❝❤♦ t❤➜② x − zkn → x − z ✳ ❉♦ ✤â✱ zkn → z ✳ ▼➦t ❦❤→❝✱ sû ❞ö♥❣ ✭✈✐✮ t❛ ✤÷đ❝ (∀n ∈ N) x − zkn x − y ✳ ❑❤✐ n → +∞✱ t❛ ❦➳t ❧✉➟♥ x − z + z−y 2 + zkn − y ≤ ≤ x − y ✳ ❉♦ ✤â✱ (∀y ∈ FixT ) y − z | x − z ≤ 0✳ ❚ø ✣à♥❤ ❧➼ ✶✳✶✶ s✉② r❛ z = PFixT x✳ ❇ð✐ ✈➟②✱ zn → PFixT x ✈➔ ❞♦ ✤â xε → PFixT x ❦❤✐ ε ↓ 0✳ ✭✈✐✐✮✿ ❈❤♦ δ ∈ [0, 1] ✈➔ ✤➦t yε = xε − x✱ yδ = xδ − x✳ ❱➻ yδ = yε + xδ − xε ✱ sû ❞ư♥❣ ✭✷✳✶✷✮ t❛ t❤✉ ✤÷đ❝ xδ − xε ≥ T xδ − T xε = xδ − δx xε − εx − 1−δ 1−ε = yε yδ − 1−δ 1−ε 2 = δ−ε yε + x δ − x ε (1 − δ)2 − ε = (1 − δ)2 δ−ε 1−ε 2 yε +2 δ−ε yε | xδ − xε + xδ − xε 1−ε ✭✷✳✶✻✮ ❇ð✐ ✈➟②✱ ε−δ 1−ε yε + δ(2 − δ) xδ − xε ≤2 ε−δ yε | xδ − xε , ✭✷✳✶✼✮ 1−ε ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ✭✈✐✐✐✮⇒ ✭✐①✮✿ ❈❤♦ δ ∈ [0, ε]✳ ❑❤✐ ✤â✱ xε − x | xδ − xε ≥ 0✳ ❚❛ ✤÷đ❝ ❦➳t q✉↔✱ xδ − x xδ − xε 2 = xδ − xε + xδ − xε | xε − x + xε − x + xε − x ✳ ✭✐①✮⇒ ✭①✮✿ ❉➵ t❤➜②✳ ✹✵ ≥ ✭①✐✮✿ ❚ø ✭✈✐✐✐✮ ✈➔ ❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ s✉② r❛ (∀δ ∈ [0, ε]) xδ − xε ≤ 2(ε − δ) xε − x δ(2 − δ)(1 − ε) ✭✷✳✶✽✮ ❉♦ ✤â✱ xδ − xε ↓ ❦❤✐ δ ↑ ε✱ ❜ð✐ ✈➟② ✤÷í♥❣ ❝♦♥❣ [0, 1] → H : ε → xε ❧➔ ❧✐➯♥ tư❝ tr→✐✳ ❚÷ì♥❣ tü✱ t❛ ❝â (∀δ ∈ [ε, 1]) xδ − xε ≤ 2(δ − ε) xε − x , δ(2 − δ)(1 − ε) ✭✷✳✶✾✮ ✈➻ ✈➟②✱ xδ − xε ↓ ❦❤✐ δ ↓ ε✳ ❉♦ ✤â✱ ✤÷í♥❣ ❝♦♥❣ [0, 1] → H : ε → xε ❧➔ ❧✐➯♥ tö❝ ♣❤↔✐✳ ✭①✐✐✮✿ ◆➳✉ x ∈ FixT t❤➻ x ∈ FixTε ❜ð✐ ✭✐✐✐✮ ✈➔ ❞♦ ✤â x = Tε x = xε ✳ ●✐↔ sû x ∈ / FixT ✱ ♥➳✉ δ ∈ [0, ε] ✈➔ xε = xδ t❤➻ tø ✭✈✐✐✐✮ ❝❤♦ t❤➜② Tε x = xε = x ✈➔ ❞♦ ✤â x ∈ FixTε ✳ ❱➻ ✈➟②✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ✤÷í♥❣ ❝♦♥❣ (xε )ε∈[0,1] ❧➔ ♥ë✐ ①↕✳ ▼➺♥❤ ✤➲ ✷✳✽✳ ❈❤♦ T1 : H → H ✈➔ T2 : H → H ❧➔ ❝→❝ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ ✈➔ T s❛✉✿ = T1 (2T2 − Id) + (Id − T2 )✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ♠➺♥❤ ✤➲ ✭✐✮ 2T − Id = (2T1 − Id)(2T2 − Id)✳ ✭✐✐✮ T ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝✳ ✭✐✐✐✮ FixT = Fix(2T1 − Id)(2T2 − Id)✳ ✭✐✈✮ ●✐↔ sû T1 ❧➔ t♦→♥ tû ❝❤✐➳✉ ❧➯♥ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❛❢❢✐♥❡ ✤â♥❣ t❤➻ FixT = {x ∈ H | T1 x = T2 x}✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮✿ ❑❤❛✐ tr✐➸♥ trü❝ t✐➳♣ t❛ ✤÷đ❝ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ✭✐✐✮✿ ▼➺♥❤ ✤➲ ✷✳✶ ❦❤➥♥❣ ✤à♥❤ 2T1 − Id ✈➔ 2T2 − Id ❧➔ ❦❤æ♥❣ ❣✐➣♥✳ ❱➻ ✈➟②✱ (2T1 − Id)(2T2 − Id) ❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ✈➔ ❝ơ♥❣ ❧➔ 2T − Id ❜ð✐ ✭✐✮✳ ❉♦ ✤â✱ T ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝✳ ✹✶ ✭✐✐✐✮✿ ❚ø ✭✐✮✱ FixT = Fix(2T − Id) = Fix(2T1 − Id)(2T2 − Id)✳ ✭✐✈✮✿ ●✐↔ sû T1 = PC ✱ tr♦♥❣ ✤â C ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❛❢❢✐♥❡ ✤â♥❣ ❝õ❛ H ✈➔ x ∈ H✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✸ s✉② r❛ T1 ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ ✈➔ tø ❍➺ q✉↔ ✶✳✺ t❛ t❤➜② x ∈ FixT ⇔ x = PC (2T2 x + (1 − 2)x) + x − T2 x ⇔ T2 x = 2PC (T2 x) + (1 − 2)PC x ∈ C ⇔ PC (T2 x) = T2 x = 2PC (T2 x) + (1 − 2)PC x ⇔ T2 x = PC x✳ ❍➺ q✉↔ ✷✳✼✳ ❈❤♦ T1 ❧➔ t♦→♥ tû ❝❤✐➳✉ ❧➯♥ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ t✉②➳♥ t➼♥❤ ❝õ❛ H✱ T2 : H → H ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ ✈➔ ✤➦t T T1 )(Id − T2 )✳ ❑❤✐ ✤â✱ T ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ ✈➔ = T1 T2 + (Id − FixT = {x ∈ H | T1 x = T2 x} ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ T = T1(2T2 − Id) + (Id − T2) ♥➯♥ ❦➳t q✉↔ ✤÷đ❝ s✉② r❛ tø ▼➺♥❤ ✤➲ ✷✳✽✳ ✷✳✹✳ ❈→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ tr✉♥❣ ❜➻♥❤ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ T : D → H ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ✈➔ α [0; 1] õ T ữủ ợ ❤➡♥❣ sè α ❤♦➦❝ α ✲ tr✉♥❣ ❜➻♥❤ tr✉♥❣ ❜➻♥❤✱ tỗ t ởt t tỷ ổ R : D → H s❛♦ ❝❤♦ T = (1 − α)Id + αR✳ ◆❤➟♥ ①➨t ✷✳✶✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ T : D → H✳ ✭✐✮ ◆➳✉ T ❧➔ tr✉♥❣ ❜➻♥❤ t❤➻ ♥â ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✳ ✭✐✐✮ ◆➳✉ T ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ t❤➻ ♥â ❦❤æ♥❣ ♥❤➜t t❤✐➳t ♣❤↔✐ ❧➔ tr✉♥❣ ❜➻♥❤✿ ❳➨t T = −Id : H → H ❦❤✐ H = {0}✳ ✹✷ ✭✐✐✐✮ ❚ø ▼➺♥❤ ✤➲ ✷✳✶ t❤➻ T ❧➔ ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♥â ❧➔ 2✲ tr✉♥❣ ❜➻♥❤✳ ✭✐✈✮ ❈❤♦ β ∈ R++ t❤❡♦ t T ỗ ự ♥➳✉ βT ❧➔ 21 ✲ tr✉♥❣ ❜➻♥❤✳ ▼➺♥❤ ✤➲ ✷✳✾✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ T : D → H ❧➔ ❦❤æ♥❣ ❣✐➣♥ ✈➔ α ∈ [0; 1]✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ♠➺♥❤ ✤➲ t÷ì♥❣ ✤÷ì♥❣ s❛✉✿ ✭✐✮ T ❧➔ t♦→♥ tû α✲tr✉♥❣ ❜➻♥❤✳ ✭✐✐✮ − α Id + α1 T ❧➔ ❦❤æ♥❣ ❣✐➣♥✳ ✭✐✐✐✮ (∀x ∈ D) (∀y ∈ D) T x − T y ≤ x−y − 1−α α (Id − T )x − (Id − T )y ✳ ✭✐✈✮ (∀x ∈ D) (∀y ∈ D) T x − T y + (1 − 2α) x − y ≤ 2(1 − α) x − y|T x − T y ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈è ✤à♥❤ x ✈➔ y t❤✉ë❝ D ✈➔ ✤➦t R = (1 − λ)Id + λT ✱ tr♦♥❣ ✤â λ = 1/α ✈➔ ú ỵ T = (1 )Id + R (i) ⇔ (ii) ⇔ (iii)✿ ❍➺ q✉↔ ✶✳✶ ❝❤♦ t❤➜② Rx − Ry = (1 − λ) x − y + λ Tx − Ty − λ(1 − λ) (Id − T )x − (Id − T )y ✭✷✳✷✵✮ ◆â✐ ❝→❝❤ ❦❤→❝✱ α( x − y − Rx − Ry ) = x − y − T x − T y 1−α (Id − T )x − (Id − T )y − α ✭✷✳✷✶✮ ✹✸ ❚❤➜② r➡♥❣ (i) ⇔ (ii) ⇔ R ❧➔ ❦❤æ♥❣ ❣✐➣♥ ⇔ ✈➳ tr→✐ ❝õ❛ ✭✷✳✷✶✮ ❞÷ì♥❣ ⇔ ✭✐✐✐✮✳ (iii) ⇔ (iv)✿ ❙û ❞ư♥❣ (Id − T )x − (Id − T )y = x−y + Tx − Ty −2 x − y|T x − T y tr♦♥❣ ✭✐✐✐✮✳ ◆❤➟♥ ①➨t ✷✳✷✳ ❚ø ✭✐✮⇒✭✐✐✐✮ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✾ ❝❤♦ t❤➜② ❝→❝ t♦→♥ tû tr✉♥❣ ❜➻♥❤ ❧➔ tü❛ ❦❤æ♥❣ ❣✐➣♥✳ ◆❤➟♥ ①➨t ✷✳✸✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✾ ✭✐✐✐✮ ♥➳✉ T : D → H ❧➔ t♦→♥ tû α✲tr✉♥❣ ❜➻♥❤ ✈ỵ✐ α ∈ 0; 12 t❤➻ T ❧➔ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➢❝✳ ❇➙② ❣✐í ❝❤ó♥❣ t❛ tr➻♥❤ ❜➔② ♣❤➨♣ t♦→♥ ❜↔♦ t♦➔♥ t➼♥❤ tr✉♥❣ ❜➻♥❤✳ ▼➺♥❤ ✤➲ ✷✳✶✵✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ T : D → H, α ∈ [0; 1] ✈➔ λ ∈ ✳ ❑❤✐ ✤â✱ T ❧➔ t♦→♥ tû α✲tr✉♥❣ ❜➻♥❤ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ❧➔ t♦→♥ tû λα✲tr✉♥❣ ❜➻♥❤✳ 0; (1 − λ)Id + λT α ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t R = (1 − α−1)Id + α−1T õ t q ữủ s r tứ ỗ ♥❤➜t T = (1−α)Id+αR ✈➔ (1−λ)Id+λT = (1−λα)Id+λαR✳ ❍➺ q✉↔ ✷✳✽✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ T ✈➔ : D → H λ ∈ [0; 2]✳ ❑❤✐ ✤â✱ T ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ (1 − λ)Id + λT ❧➔ λ2 ✲tr✉♥❣ ❜➻♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t α = 1/2 tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✵ ✈➔ sû ❞ö♥❣ ◆❤➟♥ ①➨t ✷✳✶ ✭✐✐✐✮✳ ✹✹ ▼➺♥❤ ✤➲ ✷✳✶✶✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ (Ti)i∈I ❧➔ ❤å ❤ú✉ ❤↕♥ ❝õ❛ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ tø D ✤➳♥ H✱ ❝❤♦ (ωi)i∈I ❧➔ ♥❤ú♥❣ sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0; 1] s❛♦ ❝❤♦ i∈I ωi = ✈➔ (α)i∈I ❧➔ ♥❤ú♥❣ sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0; 1] s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ i ∈ I ✱Ti ❧➔ t♦→♥ tû αi✲tr✉♥❣ ❜➻♥❤ ✈➔ ✤➦t α = maxi∈I αi ❑❤✐ ✤â✱ i∈I ωi Ti ❧➔ t♦→♥ tû α✲tr✉♥❣ ❜➻♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t T = i∈I ωi Ti ✈➔ ❧➜② x ✈➔ y t❤✉ë❝ D✳ ❚ø (i) ⇒ (iii) tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✾ s✉② r❛ (∀i I) Ti x Ti y ứ t ỗ ❝õ❛ · Tx − Ty + 2 + − αi (Id − Ti )x − (Id − Ti )y αi 1−α (Id − T )x − (Id − T )y α ωi Ti x − ≤ ωi T i y i∈I i∈I ωi Ti x − Ti y i∈I ≤ x−y + i∈I ≤ x−y ✭✷✳✷✷✮ ✈➔ (1 − α)/α = mini∈I (1 − αi )/αi ✱ s✉② r❛ = 1−α + α 2 ωi (Id − Ti )x − i∈I ωi Ti y i∈I − αi ωi (Id − Ti )x − (Id − Ti )y αi ✭✷✳✷✸✮ ❚ø (iii) ⇒ (i) tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✾ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✷✳✹✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ ❝❤♦ (Ti)i∈I ❧➔ ❤å ❤ú✉ ❤↕♥ ❝õ❛ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝ tø D ✤➳♥ H✱ (ωi )i∈I ❧➔ ♥❤ú♥❣ sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0; 1] s❛♦ ❝❤♦ i∈I ωi = 1✳ ❑❤✐ ✤â✱ i∈I ωi Ti ❧➔ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ❝❤➢❝✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ◆❤➟♥ ①➨t ✷✳✶ ✭✐✐✐✮✱ ✤➦t αi ≡ 1/2 tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✶✶ t❛ ✤÷đ❝ ❦➳t q✉↔ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ✹✺ ❚✐➳♣ t❤❡♦ t❛ ✤✐ ①➨t ♣❤➨♣ ❤ñ♣ ❝õ❛ ❝→❝ t♦→♥ tû tr✉♥❣ ❜➻♥❤✳ ▼➺♥❤ ✤➲ ✷✳✶✷✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ m ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✱ t➟♣ ❤đ♣ I = {1, , m}, {Ti}i∈I ❧➔ ❤å t♦→♥ tû tø D ✤➳♥ D ✈➔ (αi )i∈I ❧➔ ♥❤ú♥❣ sè t❤ü❝ tr♦♥❣ [0; 1] s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ i ∈ I ✱ Ti ❧➔ t♦→♥ tû αi ✲tr✉♥❣ ❜➻♥❤ ✈➔ ✤➦t T = T1 Tm ✈➔ α = m m−1+ max αi ✭✷✳✷✹✮ i∈I t❤➻ T ❧➔ t♦→♥ tû α✲tr✉♥❣ ❜➻♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t (∀i ∈ I) ki = αi/(1 − αi)✱ k = maxi∈I ki ✈➔ ❝❤♦ x✱ y t❤✉ë❝ D✳ ❚❛ ✤✐ tứ t ỗ ã t (Id − T )x − (Id − T )y 2 ✈➔ sü t÷ì♥❣ ✤÷ì♥❣ (i) ⇔ (iii) tr♦♥❣ /m = ||(x − y) − (Tm x − Tm y) + (Tm x − Tm y) − (Tm−1 Tm x − Tm−1 Tm y) + (Tm−1 Tm x − Tm−1 Tm y) − − (T2 Tm x − T2 Tm y) + (T2 Tm x − T2 Tm y) − (T1 Tm x − T1 Tm y)||2 /m = ||(Id − Tm )x − (Id − Tm )y + (Id − Tm−1 )Tm x − (Id − Tm−1 )Tm y + + (Id − T1 )T2 Tm x − (Id − T1 )T2 Tm y||2 /m ≤ (Id − Tm )x − (Id − Tm )y + (Id − Tm−1 )Tm x − (Id − Tm−1 )Tm y + + (Id − T1 )T2 Tm x − (Id − T1 )T2 Tm y ≤ km ( x − y 2 − Tm x − Tm y ) + km−1 ( Tm x − Tm y − Tm−1 Tm x − Tm−1 Tm y ) + + k1 ( T2 Tm x − T2 Tm y ≤ k( x − y 2 − T1 Tm x − T1 Tm y ) − T x − T y ) ✭✷✳✷✺✮ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✷✳✶✸✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H , β ∈ R++✱ ❧➜② T : D → H ❧➔ ỗ ự [0; 2] ✤â✱ Id − γT ❧➔ t♦→♥ tû γ ✲tr✉♥❣ 2β ❜➻♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ◆❤➟♥ ①➨t ✷✳✶ ✭✐✈✮✱ βT ❧➔ tr õ tỗ t ởt t tỷ ổ ❣✐➣♥ R : D → H t❤ä❛ ♠➣♥ T = (Id + R)/(2β)✳ ❑➳t q✉↔ t❤✉ ✤÷đ❝ ❧➔✱ Id − γT = (1 − γ/(2β))Id + (γ/(2β))(−R)✳ ✷✳✺✳ ❈→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ t❤ỉ♥❣ t❤÷í♥❣ ❑➳t q✉↔ ✤➛✉ t✐➯♥ ✤÷❛ r❛ ❝æ♥❣ t❤ù❝ ①→❝ ✤à♥❤ t➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤đ♣ ❝→❝ t♦→♥ tû tü❛ ❦❤ỉ♥❣ ❣✐➣♥✳ ❑➳t q✉↔ t✐➳♣ t❤❡♦ ✤÷đ❝ ✤÷❛ r❛ ❝❤♦ ❤đ♣ ❝õ❛ ❝→❝ t♦→♥ tû tü❛ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t✳ ▼➺♥❤ ✤➲ ✷✳✶✹✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ (Ti)i∈I ❧➔ ❤å ❤ú✉ ❤↕♥ ❝õ❛ t♦→♥ tû tü❛ ❦❤æ♥❣ ❣✐➣♥ tø D tỵ✐ H s❛♦ ❝❤♦ i∈I FixTi = ∅✱ (ωi )i∈I ❧➔ ♥❤ú♥❣ sè t❤ü❝ ❞÷ì♥❣ s❛♦ ❝❤♦ i∈I ωi = 1✳ ❑❤✐ ✤â FixTi ωi T i = Fix i∈I i∈I ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t T = i∈I ωi Ti ✳ ❘ã r➔♥❣✱ i∈I FixTi ⊂ FixT ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜❛♦ ❤➔♠ t❤ù❝ ♥❣÷đ❝ ❧↕✐✳ ●✐↔ sû y ∈ i∈I FixTi ✳ ❑❤✐ ✤â✱ tø ✭✷✳✸✮ s✉② r❛ (∀i ∈ I)(x ∈ D) Ti x − x | x − y = Ti x − y ≤ − Ti x − x − Ti x − x 2 − x−y ✭✷✳✷✻✮ ❈❤♦ x ∈ FixT ✳ ❑❤✐ ✤â✱ tø ✭✷✳✷✻✮ t❛ ❝â = Tx − x|x − y = ωi T i x − x | x − y ≤ − i∈I ❱➻ ✈➟②✱ i∈I ωi T i x − x ωi Ti x − x i∈I = ✈➔ ❞♦ ✤â x ∈ ✹✼ i∈I FixTi ✳ ✭✷✳✷✼✮ ▼➺♥❤ ✤➲ ✷✳✶✺✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H ✈➔ T1, T2 ❧➔ ♥❤ú♥❣ t♦→♥ tû tü❛ ❦❤æ♥❣ ❣✐➣♥ tø D ✤➳♥ D✳ ●✐↔ sû✱ T1 ❤♦➦❝ T2 ❧➔ t♦→♥ tû tü❛ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t ✈➔ FixT1 ∩ FixT2 = ∅✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ♠➺♥❤ ✤➲ s❛✉✿ ✭✐✮ FixT1 T2 = FixT1 ∩ FixT2 ✳ ✭✐✐✮ T1 T2 ❧➔ t♦→♥ tû tü❛ ❦❤æ♥❣ ❣✐➣♥✳ ✭✐✐✐✮ ●✐↔ sû T1 ✈➔ T2 ❧➔ t♦→♥ tû tü❛ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t✳ ❑❤✐ ✤â✱ T1 T2 ❧➔ t♦→♥ tû tü❛ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮✿ ❉➵ t❤➜② FixT1 ∩ FixT2 ⊂ FixT1 T2 ✳ ❇➙② ❣✐í✱ ❧➜② x ∈ FixT1 T2 ✈➔ y ∈ FixT1 ∩ FixT2 ✳ ❚❛ ①➨t ❜❛ tr÷í♥❣ ❤đ♣ s❛✉ ✭❛✮ T2 x ∈ FixT1 ✳ ❑❤✐ ✤â✱ T2 x = T1 T2 x = x ∈ FixT1 ∩ FixT2 ✳ ✭❜✮ x ∈ FixT2 ✳ ❑❤✐ ✤â✱ T1 x = T1 T2 x = x ∈ FixT1 ∩ FixT2 ✳ ✭❝✮ T2 x ∈ / FixT1 ✈➔ x ∈ / FixT2 ✳ ❚ø ✤â T1 ❤♦➦❝ T2 ❧➔ tü❛ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t✱ ❝â ➼t ♥❤➜t ♠ët tr♦♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ x − y = T1 T2 x − y ≤ T2 x − y ≤ x − y ❧➔ ❝❤➦t✱ ✤✐➲✉ ✤â ❧➔ ❦❤æ♥❣ t❤➸✳ ✭✐✐✮✿ ❈❤♦ x ∈ D ✈➔ y ∈ FixT1 T2 = FixT1 ∩ FixT2 ✳ ❑❤✐ ✤â✱ T1 T2 x − y ≤ T2 x − y ≤ x − y , ✈➔ ❞♦ ✤â T1 T2 ❧➔ tü❛ ❦❤æ♥❣ ❣✐➣♥✳ ✭✐✐✐✮✿ ❈❤♦ x ∈ D \ FixT1 T2 ✈➔ y ∈ FixT1 T2 = FixT1 ∩ FixT2 ✳ ◆➳✉ x ∈ / FixT2 t❤➻ T1 T2 x − y ≤ T2 x − y < x − y ❈✉è✐ ❝ò♥❣✱ ♥➳✉ x ∈ FixT2 t❤➻ x ∈ / FixT1 ✭❤❛② T1 x = x ∈ FixT1 ∩ FixT2 ✱ ✤✐➲✉ ♥➔② ❦❤æ♥❣ t❤➸ ①↔② r❛✮ ✈➔ ✈➻ ✈➟②✱ T1 T2 x − y < x − y ✳ ✹✽ ❍➺ q✉↔ ✷✳✾✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ m ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✣➦t I = {1 m} ✈➔ ❧➜② (Ti)i∈I ❧➔ ❝→❝ t♦→♥ tû tü❛ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t tø D ✤➳♥ D s❛♦ ❝❤♦ i∈I FixTi = ∅ ✈➔ ✤➦t T = T1 Tm✳ ❑❤✐ ✤â✱ T ❧➔ t♦→♥ tû tü❛ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t ✈➔ FixT = i∈I FixTi ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ m✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✶✺ t❛ t❤➜② ❦➳t q✉↔ ❧➔ ❤✐➸♥ ♥❤✐➯♥ ❦❤✐ m = ✈➔ m = 2✳ ●✐↔ sû✱ m ≥ ✈➔ ❦➳t q✉↔ ✈➝♥ ✤ó♥❣ ✈ỵ✐ m t♦→♥ tû✳ ❈❤♦ (Ti )1≤i≤m+1 ❧➔ ♠ët ❤å ❝→❝ t♦→♥ tû tü❛ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t tø D tỵ✐ D s❛♦ ❝❤♦ m+1 i=1 FixTi = ∅✳ ✣➦t R1 = T1 Tm ✈➔ R2 = Tm+1 ✳ ❑❤✐ ✤â R2 ❧➔ tü❛ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t ✈ỵ✐ FixR2 = FixTm+1 ✈➔ ❜➡♥❣ ❣✐↔ t❤✉②➳t q✉② ♥↕♣ R1 ❧➔ tü❛ ❦❤ỉ♥❣ ❣✐➣♥ ❝❤➦t ✈ỵ✐ FixR1 = m i=1 FixTi ✳ ❇ð✐ ✈➟②✱ tø ▼➺♥❤ ✤➲ ✷✳✶✺✭✐✐✐✮ ✈➔ ✭✐✮✱ R1 R2 = T1 Tm+1 ❧➔ tü❛ ❦❤æ♥❣ ❣✐➣♥ ❝❤➦t ✈➔ FixT1 Tm+1 = FixR1 R2 = FixR1 ∩ FixR2 = m+1 i=1 FixTi ✳ ❍➺ q✉↔ ✷✳✶✵✳ ❈❤♦ D ❧➔ t➟♣ ❤ñ♣ ❝♦♥ ❦❤→❝ ré♥❣ ❝õ❛ H✱ m ❧➔ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✣➦t I = {1 m} ✈➔ ❝❤♦ (Ti)i∈I ❧➔ ❤å ❝→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ tr✉♥❣ ❜➻♥❤ tø D ✤➳♥ D s❛♦ ❝❤♦ ∩i∈I FixTi = ∅ ✈➔ ✤➦t T = T1 Tm✳ ❑❤✐ ✤â✱ FixT = ∩i∈I FixTi✳ ❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ ◆❤➟♥ ①➨t ✷✳✷ ✈➔ s✉② r❛ tø ❍➺ q✉↔ ✷✳✾✳ ✹✾ ❑➌❚ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ♥❤➜t ✈➲ ❝→❝ tr t ỗ ữ t ỗ ỗ ỵ t ởt số t t tæ♣æ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈æ ❤↕♥ ❝❤✐➲✉✳ ❚✐➳♣ t❤❡♦ ✤â✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❝→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥✱ ✤➦❝ ❜✐➺t ❧➔ t➼♥❤ t t tỷ tr t ỗ ❈✉è✐ ❝ò♥❣✱ tr♦♥❣ ❦❤â❛ ❧✉➟♥ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ t♦→♥ tû ❦❤æ♥❣ ❣✐➣♥ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ ♥➔② q✉❛ ♠ët sè t♦→♥ tû t→❝ ✤ë♥❣✳ ✺✵ ❚➔✐ ❧✐➺✉ 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❝❤❦❡ ❛♥❞ P✳ ▲✳ ❈♦♠❜❡tt❡s✱ ❬✹❪ ❆✳ ❘✉s③❝③②♥✬s❦✐✱ ◆♦♥❧✐♥❡❛r ❖♣t✐♠✐③❛t✐♦♥✱ Pr❡ss✱ Pr✐♥❝❡t♦♥ ✷✵✵✻✳ ✺✶ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t②