Phương pháp đạo hàm tăng cường giải bài toán cân bằng trên tập điểm bất động của họ ánh xạ không giãn

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❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷

❍⑨ ✣Ù❈ ✣❸■

P❍×❒◆● P❍⑩P ✣❆❖ ❍⑨▼ ❚❿◆● ❈×❮◆●
●■❷■ ❇⑨■ ❚❖⑩◆ ❈❹◆ ❇➀◆● ❚❘➊◆ ❚❾P ✣■➎▼ ❇❻❚ ✣❐◆●
❈Õ❆ ❍➴ ⑩◆❍ ❳❸ ❑❍➷◆● ●■❶◆

❚♦→♥ ●✐↔✐ t➼❝❤
▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✵✷

❈❤✉②➯♥ ♥❣➔♥❤✿

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿ ●❙✳❚❙❑❍✳ ◆❣✉②➵♥ ❳✉➙♥ ❚➜♥

❍➔ ◆ë✐ ✲ ✷✵✶✻
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▲❮■ ❈❷▼ ❒◆
▲✉➟♥ ✈➠♥ ✤÷ñ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷
❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ t❤➛② ❣✐→♦ ●❙✳ ❚❙❑❍ ◆❣✉②➵♥ ❳✉➙♥ ❚➜♥✳ ❙ü ❣✐ó♣
✤ï ✈➔ ❤÷î♥❣ ❞➝♥ t➟♥ t➻♥❤✱ ♥❣❤✐➯♠ tó❝ ❝õ❛ t❤➛② tr♦♥❣ s✉èt q✉→ tr➻♥❤
t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ❣✐ó♣ t→❝ ❣✐↔ tr÷ð♥❣ t❤➔♥❤ ❤ì♥ r➜t ♥❤✐➲✉ tr♦♥❣

❝→❝❤ t✐➳♣ ❝➟♥ ♠ët ✈➜♥ ✤➲ ♠î✐✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥✱ ❧á♥❣ ❦➼♥❤
trå♥❣ s➙✉ s➢❝ ♥❤➜t ✤è✐ ✈î✐ t❤➛②✳
❚→❝ ❣✐↔ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷
♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ♣❤á♥❣ s❛✉ ✣↕✐ ❤å❝✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ tr♦♥❣ ♥❤➔ tr÷í♥❣✱
❣✐❛ ✤➻♥❤ ❝ò♥❣ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➣ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥
t❤✉➟♥ ❧ñ✐ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝ ❚❤↕❝ s➽ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥
✈➠♥ ♥➔②✦
❍➔ ◆ë✐✱ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✶✻

❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥

❍➔ ✣ù❝ ✣↕✐

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▲❮■ ❈❆▼ ✣❖❆◆
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐
❞÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ ●❙✳ ❚❙❑❍ ◆❣✉②➵♥ ❳✉➙♥ ❚➜♥✳
❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tæ✐ ✤➣ ❦➳ t❤ø❛
♥❤ú♥❣ t❤➔♥❤ q✉↔ ❦❤♦❛ ❤å❝ ❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈➔ ✤ç♥❣ ♥❣❤✐➺♣ ✈î✐ sü
tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣
✤÷ñ❝ ❝❤➾ rã ♥❣✉ç♥ ❣è❝✳
❍➔ ◆ë✐✱ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✶✻

❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥

❍➔ ✣ù❝ ✣↕✐

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▼ö❝ ❧ö❝
❉❛♥❤ ♠ö❝ ❦➼ ❤✐➺✉

✽

✶

❑✐➳♥ t❤ù❝ ❝ì ❜↔♥

✾

✶✳✶

▼ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt

✾

✶✳✷

⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

✶✳✸

❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✶✳✸✳✶

❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➔ ❝→❝ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✶✳✸✳✷

❙ü tç♥ t↕✐ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✳ ✳ ✳ ✳ ✳

✷✵

✶✳✹
✷

▼ët sè ❦➳t q✉↔ ❜ê trñ

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✶

P❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣

tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥

✷✳✶

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣
❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✳✷

✷✳✸

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✷✸

✷✸

▼ët sè ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ tr➯♥ t➟♣ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

P❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣

✸✽

✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✻✸

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

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✹


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ỵ ồ t
C ởt t ỗ õ rộ ừ ởt ổ
rt tỹ H ởt s f : C ì C R tọ f (x, x) =

0, x C t EP (C, f ) t t ởt x C
s f (x , y) 0 ợ ồ y C
t ỡ t tự ữ õ
ữủ ợ t q trồ tở ỹ
ữ t tố ữ t tự t ở t
ỹ s r r x C
ữủ ồ ừ t ừ
t tố ữ ữ ợ ộ x C ởt t ở ừ ởt


S(x) = arg min{f (x, y) : y C}.

ụ ỡ s ữ t ự
t tr t t ở ừ ồ ổ

r ữ ự t
tr t t ở ừ ồ ổ ởt
t t tớ sỹ t út ữủ sỹ q t ự ừ
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♥❤✐➲✉ ♥❤➔ ❦❤♦❛ ❤å❝ tr♦♥❣ ✈➔ ♥❣♦➔✐ ♥÷î❝✳ ❱➲ ♠➦t ❧þ t❤✉②➳t✱ ♥❤✐➲✉ ❦➳t
q✉↔ ❝ì ❜↔♥ ✈➔ q✉❛♥ trå♥❣ ✤➣ ✤↕t ✤÷ñ❝ ❝❤♦ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ tê♥❣ q✉→t
tr➯♥ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ trø✉ t÷ñ♥❣✳ ❚✉② ♥❤✐➯♥ ✈➲ ♠➦t t➼♥❤ t♦→♥✱ ❝→❝ ❦➳t q✉↔
❝á♥ ❤↕♥ ❝❤➳✳ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♠î✐ t❤✉ ✤÷ñ❝ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ ❝➙♥
❜➡♥❣ ✈î✐ ❝→❝ s♦♥❣ ❤➔♠ ♥❤➟♥ ❣✐→ trà t❤ü❝ ✈➔ ❝â t❤➯♠ t➼♥❤ ❝❤➜t ✤ì♥ ✤✐➺✉✳
❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝❤♦ ❧î♣ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ tê♥❣ q✉→t ❤ì♥✱ ♥❤➜t ❧➔
❧î♣ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈î✐ s♦♥❣ ❤➔♠ ❝â t➼♥❤ ✤ì♥ ✤✐➺✉ s✉② rë♥❣✱ ♥❤÷ ❣✐↔
✤ì♥ ✤✐➺✉✱ tü❛ ✤ì♥ ✤✐➺✉ ✈✳✈✳ ✳ ✳ ✤❛♥❣ ✤÷ñ❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ù♥❣ ❞ö♥❣ ♥❤✐➲✉✳
❱î✐ ♠♦♥❣ ♠✉è♥ t➻♠ ❤✐➸✉ s➙✉ ✈➲ ✈➜♥ ✤➲ ♥➔②✱ ❝ò♥❣ sü ❣✐ó♣ ✤ï ❤÷î♥❣
❞➝♥ t➟♥ t➻♥❤ ❝õ❛ t❤➛② ●❙✳ ❚❙❑❍✳ ◆❣✉②➵♥ ❳✉➙♥ ❚➜♥✱ tæ✐ ✤➣ ❝❤å♥ ♥❣❤✐➯♥
❝ù✉ ✤➲ t➔✐✿ ✏ P❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥
❜➡♥❣ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✑

❧➔♠

❧✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❝õ❛ ♠➻♥❤✳

✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
◆❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥
❜➡♥❣ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳

✸✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉
❚❤✉ t❤➟♣ t➔✐ ❧✐➺✉ ✈➲ ❧➽♥❤ ✈ü❝ ♥➔② ✈➔ ✈✐➳t t❤➔♥❤ ❜➔✐ tê♥❣ q✉→t ✈➲
♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤å →♥❤
①↕ ❦❤æ♥❣ ❣✐➣♥✳

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✻


✹✳ ✣è✐ t÷ñ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉

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✰ ✣è✐ t÷ñ♥❣ ♥❣❤✐➯♥ ❝ù✉✿

◆❣❤✐➯♥ ❝ù✉ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠

t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤å →♥❤
①↕ ❦❤æ♥❣ ❣✐➣♥✳
✰ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✿

❈→❝ ❜➔✐ ❜→♦✱ ❝→❝ ❝✉è♥ s→❝❤ ✈➔ ❝→❝ t➔✐

❧✐➺✉ ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➔✐ t♦→♥
❝➙♥ ❜➡♥❣ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳

✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
✰ ❙û ❞ö♥❣ ♠ët ✈➔✐ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ ❝➟♥ ♥ê✐ ❜➟t ❣✐↔✐ ❜➔✐ t♦→♥ ♥➔②
tr➯♥ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ❍ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❣➢♥ ❦➳t✱
♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉✱ ♣❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣✱ ♣❤÷ì♥❣ ♣❤→♣
✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ①➜♣ ①➾✳
✰ ❑❤❛✐ t❤→❝ ❝→❝ t➼♥❤ ❝❤➜t ❧✐➯♥ tö❝✱ ♥û❛ ❧✐➯♥ tö❝✱ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛
♠✐➲♥ ❝❤➜♣ ♥❤➟♥ C ✳

✻✳ ✣â♥❣ ❣â♣ ♠î✐
▲✉➟♥ ✈➠♥ s➩ ❧➔ ♠ët tê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥
❜➡♥❣ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳

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✼


Header Page 9 of 132.

❉❛♥❤ ♠ö❝ ❦➼ ❤✐➺✉
R

t➟♣ ❤ñ♣ ❝→❝ sè t❤ü❝❀

Rn

❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ t❤ü❝ ♥✲❝❤✐➲✉❀

H

❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝❀

x

❝❤✉➞♥ ❝õ❛ ✈➨❝ tì❀

x, y

t➼❝❤ ✈æ ❤÷î♥❣ ❝õ❛ ❤❛✐ ✈❡❝tì x ✈➔ y ❀

A⊂B

t➟♣ ❤ñ♣ A ❧➔ ❝♦♥ t❤ü❝ sü ❝õ❛ t➟♣ ❤ñ♣ B ❀

A⊆B

t➟♣ ❤ñ♣ A ❧➔ ❝♦♥ ❝õ❛ t➟♣ ❤ñ♣ B ❀

A×B

t➼❝❤ ✣➲✲❈→❝ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B ❀

arg min {f (x) : x ∈ C} t➟♣ ❝→❝ ✤✐➸♠ ❝ü❝ t✐➸✉ ❝õ❛ ❤➔♠ f tr➯♥ C ❀
∂f (x)

❞÷î✐ ✈✐ ♣❤➙♥ ❝õ❛ f t↕✐ x❀

PrC (x)

❤➻♥❤ ❝❤✐➳✉ ❝õ❛ x ❧➯♥ t➟♣ C ❀

NC

♥â♥ ♣❤→♣ t✉②➳♥ ♥❣♦➔✐ ❝õ❛ C ❀

xn → x

❞➣② {xn } ❤ë✐ tö ♠↕♥❤ tî✐ x❀

V I(C, F )

❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥❀

OP

❜➔✐ t♦→♥ tè✐ ÷✉❀

EP (C, f )

❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣❀

F ix

❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣❀

Sol(C, F )

t➟♣ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ V I(C, F )❀

Sol(C, f )

t➟♣ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ EP (C, f )❀

I

→♥❤ ①↕ ✤ç♥❣ ♥❤➜t❀

F ix(T )

t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ✳

Footer Page 9 of 132.

✽


Header Page 10 of 132.

ữỡ
tự ỡ
ở ừ ữỡ ởt số tự ỡ ổ
rt ổ ỵ t ở t
ợ t t trữớ ủ t ữ t
sỹ tỗ t ừ t ũ ởt số
ỡ ũ ự ỵ ữủ ữỡ s

ởt số t t ỡ ổ
rt
H ởt ổ tỡ tỹ ổ ữợ

tr H ởt


., . : H ì H R , x, y H
(x, y) x, y

ọ
a)

x, y = y, x , x, y H;

b)

x + y, z = x, z + y, z , x, y, z H;

c)

x, y = x, y , x, y H, R;

d)

x, x 0, x H ,

Footer Page 10 of 132.

x, x = 0 x = 0.



Header Page 11 of 132.

❙è x, y ✤÷ñ❝ ❣å✐ ❧➔ t➼❝❤ ✈æ ❤÷î♥❣ ❝õ❛ ❤❛✐ ✈❡❝tì x ✈➔ y ✳ ❈➦♣

(H, ., . ) ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt✳ ❚❛ t❤➜② r➡♥❣ t➼❝❤ ✈æ ❤÷î♥❣
., . ❧➔ ♠ët ❞↕♥❣ s♦♥❣ t✉②➳♥ t➼♥❤ ①→❝ ✤à♥❤ ❞÷ì♥❣ tr➯♥ H ✳ ❑❤✐ ✤â H ✤÷ñ❝
❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt t❤ü❝✳

✭❇➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤② ✲ ❙❝❤✇❛r③✮ ❈❤♦ H ❧➔ ❦❤æ♥❣
❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt t❛ ❧✉æ♥ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉✿
✣à♥❤ ❧þ ✶✳✶✳

([1])

| x, y |2 ≤ x, x y, y , ∀x, y ∈ H.
❚r♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❙❝❤✇❛r③ ❞➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x✱ y
♣❤ö t❤✉ë❝ t✉②➳♥ t➼♥❤✳

❈❤♦ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt✳ ❑❤✐ ✤â
, x ∈ H ①→❝ ✤à♥❤ ♠ët ❝❤✉➞♥ tr➯♥ H ✳

✣à♥❤ ❧þ ✶✳✷✳

x, x

([1])

x =

▼ët ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ❝â t❤➸ ✤➛②
✤õ ❤♦➦❝ ❦❤æ♥❣ ✤➛② ✤õ ✈î✐ ♠❡tr✐❝ ρ(x, y) = x − y .

▼ët ❦❤æ♥❣ ❣✐❛♥ t✐➲♥ ❍✐❧❜❡rt t❤ü❝ ✤➛② ✤õ ✤÷ñ❝ ❣å✐ ❧➔
❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳

❚ø ✤➙② t❛ ❧✉æ♥ ❦þ ❤✐➺✉ H ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝✳
❱➼ ❞ö ✶✳✶✳

a) R ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈î✐ t➼❝❤ ✈æ ❤÷î♥❣ x, y =
n

n

xi y i ,
i=1

tr♦♥❣ ✤â

x = (x1 , x2 , ..., xn ), y = (y1 , y2 , ..., yn ) ∈ Rn .
b) ❑❤æ♥❣ ❣✐❛♥ l2 ❧➟♣ t❤➔♥❤ ❜ð✐ t➜t ❝↔ ❝→❝ ❞➣② sè s❛♦ ❝❤♦
✈î✐ t➼❝❤ ✈æ ❤÷î♥❣ x, y =

∞

∞

x2n < ∞

n=1

xn yn , x, y ∈ l2 ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳

n=1

❉➣② {xn} tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ❤ë✐ tö ♠↕♥❤ ✤➳♥
x ∈ H ✱ ❦þ ❤✐➺✉ xn → x ♥➳✉ lim xn − x = 0.
n→∞
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳

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✶✵


Header Page 12 of 132.

{xn} tr ổ rt H ữủ ồ ở
tử tợ tỷ x H xn, y x, y n , y H


r ổ rt ở tử t ở tử
ữ ữủ ổ ú
ỵ

([6])

H ổ rt

{xn} ở tử x xn x tr H t xn x
ii) ồ tr H ự ở tử

iii) ỹ ở tử ở tử tữỡ ữỡ H ỳ

ỵ r ổ rt H {xn } ở tử
x0 t t tự s ổ ú
i)

lim inf xn x0 < lim inf xn y , y H, y = x0 .

n

n

ự

{xn } ở tử tợ x0 õ ợ
tỗ t
õ y H, y = x0 t õ

xn y

2

= xn x0 + x0 y
= xn x0

2

2

+ x0 y

2

+ 2 xn x0 , x0 y .

t

2 xn x0 , x0 y 0
n

lim inf xn x0 < lim inf xn y .

n

Footer Page 12 of 132.

n




Header Page 13 of 132.

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳

C

✤÷ñ❝ ❣å✐ ❧➔✿

i)

ii)

iii)

iv)

v)

vi)

vii)

❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✳ ❑❤✐ ✤â

❚➟♣ ❧ç✐ ♥➳✉ λx + (1 − λ)y ∈ C,

∀x, y ∈ C

✈➔ ∀λ ∈ [0, 1] ;

❚➟♣ ✤â♥❣ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C t❤ä❛ ♠➣♥ xn → x ❦❤✐ n → ∞ t❛
✤➲✉ ❝â x ∈ C ❀
❚➟♣ ✤â♥❣ ②➳✉ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C t❤ä❛ ♠➣♥ xn ❤ë✐ tö ②➳✉ ✤➳♥
x ❦❤✐ n → ∞ t❛ ✤➲✉ ❝â x ∈ C ❀
❚➟♣ ❝♦♠♣❛❝t ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö ✈➲
♠ët ♣❤➛♥ tû t❤✉ë❝ C ❀
❚➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥
❤ë✐ tö❀
❚➟♣ ❝♦♠♣❛❝t ②➳✉ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ✤➲✉ ❝â ♠ët ❞➣② ❝♦♥ ❤ë✐ tö
②➳✉ ✈➲ ♠ët ♣❤➛♥ tû t❤✉ë❝ C ❀

❚➟♣ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ ②➳✉ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ✤➲✉ ❝â ♠ët ❞➣②
❝♦♥ ❤ë✐ tö ②➳✉✳

▼å✐ t➟♣ ❝♦♠♣❛❝t ✤➲✉ ❧➔ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐✱ ♥❤÷♥❣ ✤✐➲✉ ♥❣÷ñ❝ ❧↕✐ ❦❤æ♥❣
✤ó♥❣✳
▼å✐ t➟♣ ✤â♥❣ ②➳✉ ❧➔ t➟♣ ✤â♥❣✱ ♥❤÷♥❣ ✤✐➲✉ ♥❣÷ñ❝ ❧↕✐ ❦❤æ♥❣ ✤ó♥❣✳

❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H ✈➔ C ❧➔ ♠ët t➟♣ ❝♦♥ ❝õ❛
♥â✳ ❑❤✐ ✤â →♥❤ ①↕ ϕ : C → H ✤÷ñ❝ ❣å✐ ❧➔✿

✣à♥❤ ♥❣❤➽❛ ✶✳✻✳

i)

ii)

◆û❛ ✤â♥❣ t↕✐ ✤✐➸♠ u ♥➳✉ ❞➣②
ϕ(xn ) → u t❤➻ ϕ(x) = u;

{xn } ⊂ C

❤ë✐ tö ②➳✉ ✤➳♥

x

✈➔

▲✐➯♥ tö❝ ②➳✉ ♥➳✉ ❞➣② {xn} ⊂ C ❤ë✐ tö ②➳✉ ✤➳♥ x t❤➻ ϕ(xn) ❤ë✐ tö
②➳✉ ✤➳♥ ϕ(x)✳

Footer Page 13 of 132.

✶✷


Header Page 14 of 132.

C ởt t ỗ õ rộ ừ ổ
rt H x0 C õ t


NC (x0 ) = w H : w, x x0 0 , x C

ữủ ồ õ t ừ C t x0 t NC (x0) ữủ ồ
õ t tr ừ C t x0
t r 0 NC (x0 ) NC (x0 ) ởt õ ỗ õ

C ởt t ỗ õ rộ ừ ổ
rt H ợ ộ y H t ồ


dC (y) = inf x y
xC

tứ y C
tỗ t PrC (y) s
dC (y) = y PrC (y)

t t õ PrC (y) ừ y tr C
ợ ộ y H tỗ t t ởt tỷ PrC (y)

t t r PrC (y) ừ y tr s
ừ t tố ữ

min
xC

1
xy
2

2

.

ứ 1.8 t õ ỵ s
ỵ

rt H
õ

([6])

C t ỗ õ rộ ừ ổ

i) x PrC (x), y PrC (x) 0, x H, y C
ii)

PrC (x) PrC (y) x y , x, y H

Footer Page 14 of 132.

t ổ


Header Page 15 of 132.

iii)

iv)

PrC (x) − PrC (y)

✤ç♥❣ ❜ù❝✮❀

x − PrC (x)

2

2

≤ PrC (x) − PrC (y), x − y , ∀x, y ∈ H

✭t➼♥❤

≤ x − PrC (x), x − y , ∀y ∈ C ✳

❈❤♦ C ❧➔ ♠ët t➟♣ ❧ç✐✱ ✤â♥❣✱ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥
❍✐❧❜❡rt H ✳ ⑩♥❤ ①↕ F : C → R ✤÷ñ❝ ❣å✐ ❧➔

✣à♥❤ ♥❣❤➽❛ ✶✳✾✳

a)

◆û❛ ❧✐➯♥ tö❝ ❞÷î✐ t↕✐ x ∈ C ✱ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ❤ë✐ tö ♠↕♥❤
✤➳♥ x✱ t❤➻
lim inf F (xn ) ≥ F (x);

n→∞

b)

◆û❛ ❧✐➯♥ tö❝ tr➯♥ t↕✐ x ∈ C ✱ ♥➳✉ ♠å✐ ❞➣② {xn} ⊂ C ❤ë✐ tö ♠↕♥❤
✤➳♥ x✱ t❤➻
lim supF (xn ) ≤ F (x);

n→∞

c)

▲ç✐ tr➯♥ C ✱ ♥➳✉ ✈î✐ ♠å✐ x , y

∈C

✈➔ λ ∈ [0, 1]✱ t❤➻

F (λx + (1 − λ)y) ≤ λF (x) + (1 − λ)F (y);
d)

▲ç✐ ❝❤➦t tr➯♥ C ✱ ♥➳✉ ✈î✐ ♠å✐ x , y

∈ C, x = y

✈➔ λ ∈ (0, 1) t❤➻

F (λx + (1 − λ)y) < λF (x) + (1 − λ)F (y);
e)

▲ç✐ ♠↕♥❤ tr➯♥ C ♥➳✉ tç♥ t↕✐ α > 0 s❛♦ ❝❤♦ ✈î✐ ♠å✐ x , y
λ ∈ [0, 1]✱ t❤➻

∈C

✈➔

1
F (λx + (1 − λ)y) ≤ λF (x) + (1 − λ)F (y) − αλ(1 − λ) x − y 2 ;
2
f)

❚ü❛ ❧ç✐ tr➯♥ C ♥➳✉ ♠å✐ x , y

∈C

✈➔ λ ∈ [0, 1]✱ t❤➻

F (λx + (1 − λ)y) ≤ max {F (x), F (y)} .

❈❤♦ f : H → R ❧➔ ❤➔♠ ❧ç✐ ❝❤➼♥❤ t❤÷í♥❣ tr➯♥ H ✳ ❚❛
♥â✐ ♣❤➛♥ tû w ∈ H ❧➔ ❞÷î✐ ✤↕♦ ❤➔♠ ❝õ❛ f t↕✐ x ∈ H ♥➳✉

✣à♥❤ ♥❣❤➽❛ ✶✳✶✵✳

w, y − x ≤ f (y) − f (x), ∀y ∈ H.
Footer Page 15 of 132.

✶✹


Header Page 16 of 132.

tt ữợ ừ f t x ữủ ồ ữợ ừ f
t x ỵ f (x) f ữủ ồ ữợ t x
f (x) = .

ổ
r ợ t ổ t ở
ừ ổ

C t ỗ õ rộ tr ổ
rt H T : C C ổ



T x T y x y , x, y C.

P tỷ

ữủ ồ ởt t ở ừ
T (x ) = x t ở ừ T ỵ F ix(T )

x C

T



ỹ tỗ t t ở ừ ổ tr ổ
rt ữủ ỵ s

C t ỗ õ ừ ổ
rt H T : C C ởt ổ õ T õ t t
ởt t ở
ỵ

([2])

ứ t t ỗ t ừ ổ rt t tử
ừ ổ t t t t ở F ix(T )
rộ t õ ỗ õ

Footer Page 16 of 132.




✶✳✸ ❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣

Header Page 17 of 132.

✶✳✸✳✶

❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ✈➔ ❝→❝ ✈➼ ❞ö

❚r♦♥❣ ♠ö❝ ♥➔② t❛ ①➨t ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✱ ❝→❝ tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t
✤÷❛ ✈➲ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ♥❤÷✿ ❇➔✐ t♦→♥ tè✐ ÷✉✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝
❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❑❛❦✉t❛♥✐✱ ❜➔✐ t♦→♥ ✤✐➸♠ ②➯♥ ♥❣ü❛✱
❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ◆❛s❤ tr♦♥❣ trá ❝❤ì✐ ❦❤æ♥❣ ❤ñ♣ t→❝✳

❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣✱ ❦❤→❝ ré♥❣ tr♦♥❣ H ✈➔
s♦♥❣ ❤➔♠ f : C × C → R s❛♦ ❝❤♦ f (x, x) = 0✱ ∀x ∈ C ✳ ❇➔✐ t♦→♥ ❝➙♥
❜➡♥❣ ✈✐➳t t➢t EP (C, f ) ✤÷ñ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿
✣à♥❤ ♥❣❤➽❛ ✶✳✶✷✳

❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ f (x∗, x) ≥ 0,

∀x ∈ C.

❙♦♥❣ ❤➔♠ f t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t f (x, x) = 0✱ ∀x ∈ C ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠
❝➙♥ ❜➡♥❣ tr➯♥ C ✳
❚➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ EP (C, f ) ✤÷ñ❝ ❣å✐ ❧➔ Sol(C, f )✳
✣➸ t❤➜② ✤÷ñ❝ t➼♥❤ ✤ì♥ ✤✐➺✉ s✉② rë♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✱ t❛
♥❤➢❝ ❧↕✐ ♠ët sè ✤à♥❤ ♥❣❤➽❛ ❝õ❛ s♦♥❣ ❤➔♠ f tr♦♥❣ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ s❛✉
✤➙②✳
❈❤♦ s♦♥❣ ❤➔♠ f : C × C → R✱ ❦❤✐ ✤â s♦♥❣ ❤➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ✿

a) ✣ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ✈î✐ ❤➺ sè λ > 0✱ ♥➳✉
f (x, y) + f (y, x) ≤ −λ x − y 2 , ∀x, y ∈ C;
b) ✣ì♥ ✤✐➺✉ ❝❤➦t tr➯♥ C ✱ ♥➳✉
f (x, y) + f (y, x) < 0 , ∀x, y ∈ C ✈➔ x = y ;
c) ✣ì♥ ✤✐➺✉ tr➯♥ C ✱ ♥➳✉

f (x, y) + f (y, x) ≤ 0 , ∀x, y ∈ C

Footer Page 17 of 132.

✶✻


Header Page 18 of 132.

d) ●✐↔ ✤ì♥ ✤✐➺✉ tr➯♥ C ✱ ♥➳✉
f (x, y) ≥ 0 ⇒ f (y, x) ≤ 0, ∀x, y ∈ C ;
e) ●✐↔ ✤ì♥ ✤✐➺✉ ❝❤➦t tr➯♥ C ✱ ♥➳✉
f (x, y) ≥ 0 ⇒ f (y, x) < 0, ∀x, y ∈ C ✈➔ x = y;

f ) ▲✐➯♥ tö❝ ❦✐➸✉ ▲✐♣s❝❤✐t③ tr➯♥ C ✈î✐ ❤➡♥❣ sè C1 > 0 ✈➔ C2 > 0✱ ♥➳✉
f (x, y) + f (y, z) ≥ f (x, z) − C1 x − y

2

− C2 y − z 2 , ∀x, y, z ∈ C.

❉÷î✐ ✤➙② ❧➔ ♠ët sè tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t ✤÷❛ ✈➲ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✳
❱➼ ❞ö ✶✳✷✳

❇➔✐ t♦→♥ tè✐ ÷✉

❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣✱ ❦❤→❝ ré♥❣ tr♦♥❣ H ✈➔✿ ϕ : C → R ❧➔
♠ët ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ C ✳ ❇➔✐ t♦→♥ tè✐ ÷✉✱ ✈✐➳t t➢t (OP ) ✤÷ñ❝ ♣❤→t ❜✐➸✉
♥❤÷ s❛✉✿
❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ ϕ(x∗ ) ≤ ϕ(y), ∀y ∈ C.

❱î✐ ♠é✐ x , y ∈ C ✤➦t f (x, y) := ϕ(y) − ϕ(x)✳ ❑❤✐ ✤â✱ ❜➔✐ t♦→♥ (OP ) s➩
✤÷ñ❝ ✤÷❛ ✈➲ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ EP (C, f )✳
❱➼ ❞ö ✶✳✸✳

❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❑❛❦✉t❛♥✐✳

❈❤♦ C ❧➔ ♠ët t➟♣ ❧ç✐✱ ✤â♥❣✱ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H
✈➔ →♥❤ ①↕ ✤ì♥ trà F : C → C ✳ ❇➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ (F ix) t❤æ♥❣
t❤÷í♥❣ ❧➔ ❜➔✐ t♦→♥✳ ❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ x∗ = F (x∗ )✳
✣➦t f (x, y) := x − F (x), y − x ✳ ❑❤✐ ✤â ❜➔✐ t♦→♥ (F ix) trð t❤➔♥❤ ❜➔✐
t♦→♥ EP (C, f )✳
❚ê♥❣ q✉→t ❤ì♥ t❛ ①➨t ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ❑❛❦✉t❛♥✐ ✤❛ trà✳
❈❤♦ F : C → 2C ✳ ✣✐➸♠ x∗ ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ F ♥➳✉

x∗ ∈ F (x∗ )✳ ●✐↔ sû ∀x ∈ C ✱ F (x) ❧ç✐✱ ❝♦♠♣❛❝t✱ ❦❤→❝ ré♥❣✳
❇➡♥❣ ❝→❝❤ ✤➦t f (x, y) := max x − u, y − x , ∀x, y ∈ C.
u∈F (x)

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✶✼


Header Page 19 of 132.

õ t t ởt t ở ừ f õ t ổ t ữợ
t EP (C, f )
ỹ tỗ t t ở t ỹ ỵ t ở
t ỵ ỡ s ự sỹ tỗ t ừ
t

ử

t ỹ

C t ỗ õ rộ ừ H t A, B

C s L : A ì B R t ỹ ữủ t
ữ s

(x , y ) A ì B
.
L(x , y) L(x , y ) L(x, y ), (x, y) A ì B
(x , y ) AìB tọ t tự tr ữủ ồ
ỹ ừ L tr A ì B t K = A ì B t s f : K ì K R
ợ

f (x, y) := L(y1 , x2 ) L(x1 , y2 ), ợ x = (x1 , x2 ), y = (y1 , y2 ).
õ t ỹ tữỡ ữỡ t
ử

t t tự

C ởt t ỗ õ rộ tr H F : C H
õ t t tự t tt V I(C, F ) ữủ t
ữợ
x C s F (x ), x x 0, x C.
ừ t V I(C, F ) ữủ Sol(C, F )
t f (x, y) := F (x), y x , x, y C t ữ ữủ t

V I(C, F ) t EP (C, f ) õ t ỡ ỡ

t ỡ ỡ tử st ừ F tữỡ
ữỡ ợ t ỡ ỡ t ỡ ỡ
tử st ừ s f
Footer Page 19 of 132.




Header Page 20 of 132.

a) ✣ì♥ ✤✐➺✉ ♠↕♥❤ tr➯♥ C ✈î✐ ❤➡♥❣ sè λ > 0✱ ♥➳✉
F (x) − F (y), x − y ≥ λ x − y 2 , ∀x, y ∈ C;
b) ✣ì♥ ✤✐➺✉ ❝❤➦t tr➯♥ C ✱ ♥➳✉
F (x) − F (y), x − y > 0, ∀x, y ∈ C, x = y;
c) ✣ì♥ ✤✐➺✉ tr➯♥ C ✱ ♥➳✉
F (x) − F (y), x − y ≥ 0, ∀x, y ∈ C;
d) ●✐↔ ✤ì♥ ✤✐➺✉ tr➯♥ C ✱ ♥➳✉
F (y), x − y ≥ 0 ⇒ F (x), x − y ≥ 0, ∀x, y ∈ C;
e) ▲✐➯♥ tö❝ ▲✐♣s❝❤✐t③ ✈î✐ ❤➡♥❣ sè L > 0 tr➯♥ C ✱ ♥➳✉
F (x) − F (y) ≤ L x − y , ∀x, y ∈ C.

❱➼ ❞ö ✶✳✻✳

❇➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ ◆❛s❤✳

❈â n ❤➣♥❣ ❝ò♥❣ t❤❛♠ ❣✐❛ s↔♥ ①✉➜t ♠ët ❧♦↕✐ s↔♥ ♣❤➞♠✱ ❝❤♦ I := {1, 2, ...n}✱

pi ❧➔ ❣✐→ ❝õ❛ ❤➣♥❣ i ∈ I ♣❤ö t❤✉ë❝ ✈➔♦ tê♥❣ sè ❧÷ñ♥❣ s↔♥ ♣❤➞♠ ❝õ❛ t➜t
❝↔ ❝→❝ ❤➣♥❣ σx =

n

xi , tr♦♥❣ ✤â xi (i = 1, 2, ..., n) ❧➔ sè ❧÷ñ♥❣ s↔♥ ♣❤➞♠

i=1

♠➔ ❤➣♥❣ t❤ù i ✤à♥❤ s↔♥ ①✉➜t✳ ❑þ ❤✐➺✉ hi (xi ) ❧➔ ❝❤✐ ♣❤➼ ❝õ❛ ❤➣♥❣ i ❦❤✐
s↔♥ ①✉➜t xi s↔♥ ♣❤➞♠✳ ❑❤✐ ✤â✱ ❤➔♠ ❧ñ✐ ♥❤✉➟♥ ❝õ❛ ❤➣♥❣ i ✤÷ñ❝ ❝❤♦ ❜ð✐✿

fi (x1 , x2 , ..., xn ) := xi pi (σx ) − hi (xi ), i = 1, 2, ..., n.
●å✐ Ci := {x ∈ R| x ≥ 0} , i = 1, 2, ..., n ❧➔ t➟♣ ❝❤✐➳♥ ❧÷ñ❝ ❝õ❛ ❤➣♥❣ t❤ù

i✳
✣✐➸♠ x∗ = (x∗1 , x∗2 , ..., x∗n ) ∈ C1 × C2 × ...Cn ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝➙♥
❜➡♥❣ ◆❛s❤ ♥➳✉

fi (x∗1 , ..., x∗i−1 , y1 , x∗i+1 , ..., x∗n ) ≤ fi (x∗1 , x∗2 , ..., x∗n ), ∀yi ∈ Ci , i ∈ I.

Footer Page 20 of 132.

✶✾


Header Page 21 of 132.

ỵ t t s õ r t ý

ồ ữỡ s t r ọ tr
ỏ ỳ ữỡ s t t
t t ờ s tt ỡ t C := C1 ì C2 ì ...Cn

f : C ì C R
n

[fi (x1 , x2 , ..., xn ) fi (x1 , ..., xi1 , y, ..., xi+1 , ..., xn )],

f (x, y) :=
i=1

t t s tữỡ ữỡ ợ t EP (C, f )


ỹ tỗ t ừ t

r t tr ởt số sỹ tỗ t
t t ừ t ừ t EP (C, f )

C ởt t ỗ t rộ s
f : C ì C R tọ
ỵ

i) f (., y)
ii) f (x, .)

x C

([3])

ỷ tử tr ợ ộ y C
ỗ ỷ tử ữợ ữợ tr C ợ ộ

õ t EP (C, f ) õ
C t ỗ õ s f : C ì C R
sỷ ự () s ữủ tọ ỗ t t t
s
q

C B = , x C\B, y C : f (x, y) < 0.

õ t EP (C, f ) õ
C ởt t ỗ õ rộ tr H
f : C ì C R s tr C õ t t
ỵ

Footer Page 21 of 132.

([7])




Header Page 22 of 132.

i) f (., y)

ii) f (x, .)

ỷ tử tr ợ ộ y C
tỹ ỗ tr C ợ ộ x C

õ t EP (C, f ) õ ữ C t
ự () tọ
C ởt t ỗ t rộ s
f : C ì C R õ
ỵ

a)

b)

([8])

f ỡ t tr C t t EP (C, f ) õ
t ởt
f (., y) ỷ tử tr ợ ộ y C f (x, .) ỗ ỷ
tử ữợ ợ ộ x C f ỡ tr C t t
EP (C, f ) ổ õ t ởt

ởt số t q ờ trủ
ử tr ởt số ỡ
ũ ự ỵ ữủ ữỡ s

C ởt t ỗ õ rộ ừ H
ổ T : C C F ix(T ) = t I T ỷ õ


C ởt t ỗ õ rộ ừ H
sỷ {xn} tọ tự xn+1 u xn u , u C, n 0.
õ {PrC (xn)} ở tử tợ x C

{an} {bn} {cn} số tỹ ổ tọ
an+1 (1 + bn ) an + cn , n 0;

tr õ
Footer Page 22 of 132.



bn <
n=1




n=1

cn <

õ n
lim an tỗ t




Header Page 23 of 132.

i)

xy

2

r ổ rt H t õ tự s

= x

ii) tx + (1 t)y
t [0, 1]

2
2

y

2

2 x y, y , x, y H;

= t x 2 + (1 t) y

2

t(1 t) x y

2

x, y H

C ởt t ỗ õ rộ ừ ổ
rt H {Tn} tứ C õ sỷ





sup( Tn+1 (z) Tn (z) : z C) < .
n=1

õ ợ ộ y C {Tn(y)} ở tử tợ ởt ừ C
ỡ ỳ T : C C
T (y) = lim Tn (y), y C,
n

t n
lim sup(

T (z) Tn (z) : z C) = 0

{n} số tỹ s {n} [, ] (0, 1), c > 0
{xn} {yn} tr H tọ

lim sup xn c,


n

lim sup y n c,
n



lim sup n xn + (1 n )y n = c,
n

õ n
lim

Footer Page 23 of 132.

xn y n = 0.




Header Page 24 of 132.

❈❤÷ì♥❣ ✷
P❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣
❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ tr➯♥ t➟♣

✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❤å →♥❤ ①↕ ❦❤æ♥❣
❣✐➣♥
◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ♥❤➢❝ ❧↕✐ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ t➻♠
✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❚✐➳♣ t❤❡♦ ❧➔ ❣✐î✐ t❤✐➺✉ ♠ët sè
♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ ❝➟♥ ♥ê✐ ❜➟t ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t
✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❙❛✉ ❝ò♥❣ ♥➯✉ r❛ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ sû
❞ö♥❣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣ tr➯♥ t➟♣ ✤✐➸♠ ❜➜t
✤ë♥❣ ❝õ❛ ♠ët →♥❤ ①↕✱ ❤å →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❈❤õ ②➳✉ ❞ü❛ t❤❡♦ ❝→❝ t➔✐
❧✐➺✉ ❬✹❪ ❬✺❪ ❬✶✶❪ ❬✶✷❪✱ ❝â t❤❛♠ ❦❤↔♦ ❝→❝ t➔✐ ❧✐➺✉ ❦❤→❝✳

✷✳✶ ▼ët sè ♣❤÷ì♥❣ ♣❤→♣ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛
→♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥
❍✐➺♥ ♥❛②✱ ❧þ t❤✉②➳t ✤✐➸♠ ❜➜t ✤ë♥❣ ✤❛♥❣ ♣❤→t tr✐➸♥ ❤➳t sù❝ ♠↕♥❤
♠➩ ✈➔ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ t❤ü❝ t➳✳ ◆â t❤✉ ❤ót sü q✉❛♥ t➙♠ ♥❣❤✐➯♥
❝ù✉ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr♦♥❣ ✈➔ ♥❣♦➔✐ ♥÷î❝✳ ◆❤ú♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲
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Header Page 25 of 132.

❜➔✐ t♦→♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✤➣ trð t❤➔♥❤ ♠ët
tr♦♥❣ ♥❤ú♥❣ ❤÷î♥❣ ♥❣❤✐➯♥ ❝ù✉ ❤➳t sù❝ sæ✐ ✤ë♥❣✳
❉÷î✐ ✤➙② ❧➔ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❝ì ❜↔♥ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛
→♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳
❈❤♦ C ❧➔ t➟♣ ❝♦♥ ❧ç✐✱ ✤â♥❣✱ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt H
✈➔ T : C → C ❧➔ ♠ët →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ◆➠♠ ✶✾✺✸ ❲✳❘✳ ▼❛♥♠ ✤➣
♥❣❤✐➯♥ ❝ù✉ ✈➔ ✤➲ ①✉➜t ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ s❛✉✳
❱î✐ ❣✐→ trà x0 ∈ C ❜➜t ❦ý✱ ❞➣② {xn } ✤÷ñ❝ ①➙② ❞ü♥❣ ♥❤÷ s❛✉✿

xn+1 = αn xn + (1 − αn )T xn , ∀n ≥ 0,
tr♦♥❣ ✤â {αn } ❧➔ ♠ët ❞➣② sè t❤ü❝ t❤ä❛ ♠➣♥ α0 = 1, 0 < αn < 1✱ ∀n ≥ 1✱
∞

αn = ∞.
n=0
∞

❲✳❘✳ ▼❛♥♠ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ r➡♥❣✱ ♥➳✉ ❞➣② {αn } t❤ä❛ ♠➣♥

αn (1 − αn ) = ∞, t❤➻ ❦❤✐ ✤â ❞➣② ❧➦♣ {xn } ❤ë✐ tö ②➳✉ ✤➳♥ z ∈ F ix(T )✳

n=1

❚✐➳♣ t❤❡♦ ✤â ♥➠♠ ✶✾✼✹ ❙✳ ■s❤✐❦❛✇❛ ✤➣ ♠ð rë♥❣ ❞➣② ❧➦♣ ▼❛♥♠

0


 x ∈ C,

y n = βn xn + (1 − βn )T xn ,





xn+1 = αn xn + (1 − αn )T y n , ∀n ≥ 0,

tr♦♥❣ ✤â {αn } ✈➔ {βn } ❧➔ ❞➣② ❝→❝ sè t❤ü❝ tr♦♥❣ ✤♦↕♥ [0.1]✱ t❤ä❛ ♠➣♥
∞

0 ≤ αn ≤ βn ≤ 1, ∀n ≥ 1, lim βn = 0,
n→∞

αn βn = ∞. ❑❤✐ ✤â ❞➣② ❧➦♣

n=1

{xn } ❤ë✐ tö ②➳✉ ✤➳♥ z ∈ F ix(T )✳

❱î✐ ♠♦♥❣ ♠✉è♥ ✤↕t ✤÷ñ❝ sü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❞➣② ❧➦♣ {xn }✳ ◆➠♠
✷✵✵✸ ❑✳ ◆❛❦❛❥♦ ✈➔ ❲✳ ❚❛❦❛❤❛s❤✐ ✤➣ ✤➲ ①✉➜t ♠ët ❝↔✐ ❜✐➯♥ ❝õ❛ ♣❤÷ì♥❣
♣❤→♣ ❧➦♣ ❲✳❘✳ ▼❛♥♠ ❞ü❛ tr➯♥ ♣❤÷ì♥❣ ♣❤→♣ ❧❛✐ ❣❤➨♣ tr♦♥❣ q✉② ❤♦↕❝❤
t♦→♥ ❤å❝✳

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