Mục đích của luận án là mở rộng các kết quả về sự tồn tại điểm bất động cho một số lớp ánh xạ trên các lớp không gian như không gian b-mêtric sắp thứ tự bộ phận không gian b-mêtric nón trên các đại số Banach;...
❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ▲➊ ❚❍❆◆❍ ◗❯❹◆ ✣➚◆❍ ▲Þ ✣■➎▼ ❇❻❚ ✣❐◆● ❈❍❖ ▼❐❚ ❙➮ ▲❰P ⑩◆❍ ❳❸ ❚❘➊◆ ❑❍➷◆● ●■❆◆ b✲▼➊❚❘■❈ ❱⑨ Ù◆● ❉Ö◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ t➼❝❤ ▼➣ sè✿ ✾ ✹✻ ✵✶ ✵✷ ❚➶▼ ❚➁❚ ▲❯❾◆ ⑩◆ ❚■➌◆ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●❍➏ ❆◆ ✲ ✷✵✶✽ ▲✉➟♥ →♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ rữớ t ữợ ✶✳ P●❙✳ ❚❙✳ ❚r➛♥ ❱➠♥ ❹♥ ✷✳ ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ❉ô♥❣ P❤↔♥ ❜✐➺♥ ✶✿ ●❙✳❚❙❑❍ ◆❣✉②➵♥ ◗✉❛♥❣ ❉✐➺✉ P❤↔♥ ❜✐➺♥ ✷✿ ●❙✳❚❙❑❍ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ P❤↔♥ ❜✐➺♥ ✸✿ P●❙✳❚❙ ◆❣✉②➵♥ s ữủ trữợ ỗ ❝❤➜♠ ❧✉➟♥ →♥ ❝➜♣ ❚r÷í♥❣ ❤å♣ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ỗ t ❈â t❤➸ t➻♠ ❤✐➸✉ ❧✉➟♥ →♥ t↕✐✿ ✶✳ ❚❤÷ ✈✐➺♥ ◆❣✉②➵♥ ❚❤ó❝ ❍➔♦✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ✷✳ ❚❤÷ ❱✐➺♥ ố t é ỵ ❝❤å♥ ✤➲ t➔✐ ◆❣✉②➯♥ ❧➼ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝ỉ♥❣ ❝ư ❤ú✉ ➼❝❤ ❝õ❛ t♦→♥ ❤å❝ ❤✐➺♥ ✤↕✐✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♥❤ú♥❣ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ❦➳t q✉↔ ♥➔② ❧➔ ♠ët ♥ë✐ ❞✉♥❣ ❝èt ❧ã✐ ❝õ❛ ❣✐↔✐ t➼❝❤ ♣❤✐ t✉②➳♥✳ ❱➜♥ ✤➲ ♠ð rë♥❣ ◆❣✉②➯♥ ❧➼ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ tr➯♥ ❝→❝ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➣ ✈➔ ✤❛♥❣ ✤÷đ❝ ♥❤✐➲✉ t→❝ ❣✐↔ q✉❛♥ t➙♠ ự t ỳ ữợ t ữủ ♥❤✐➲✉ ❦➳t q✉↔ ✤→♥❣ ❦➸✱ t✐➯✉ ❜✐➸✉ ✈ỵ✐ ♥❤ú♥❣ ❝ỉ♥❣ tr➻♥❤ ♥ê✐ ❜➟t ❝õ❛ ❑❛♥♥❛♥ ✭✶✾✻✽✮✱ ❇♦②❞ ✈➔ ❲♦♥❣ ✭✶✾✻✾✮✱ ❈✐r✐❝ ✭✶✾✼✹✮✱ ❘❤♦❛❞❡s ✭✶✾✼✼✮✱ ❘❛♥ ✈➔ ❘❡✉r✐♥❣s ✭✷✵✵✹✮✱ ❘❛♥ ✈➔ ❘❡✉r✐♥❣s ✭✷✵✵✹✮✱ ❘✉s ✈➔ ❙❡r❜❛♥ ✭✷✵✵✽✮✱ ❙❤❛t❛♥❛✇✐♠ ✈➔ ❆❧✲❘❛✇❛s❤❞❡❤ ✭✷✵✶✷✮✱ rs Pr ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ rë♥❣ r➣✐ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❝õ❛ t♦→♥ ❤å❝ ✈➔ ❦❤♦❛ ❤å❝ ❦❤→❝ ♥❤÷ ❣✐↔✐ t➼❝❤✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐✲t➼❝❤ ♣❤➙♥✱ ❦✐♥❤ t➳ ✈➔ tt t ữợ ự ỵ t❤✉②➳t ✤✐➸♠ ❜➜t ✤ë♥❣ ♠➯tr✐❝ ♣❤→t tr✐➸♥ ❝❤õ ②➳✉ t❤❡♦ s ự ỵ t ✤ë♥❣ ❝❤♦ ❝→❝ →♥❤ ①↕ ❝♦ ✈➔ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr➯♥ ❧ỵ♣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ◆❣❤✐➯♥ ❝ù✉ ỵ t ✈➔ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr➯♥ ❝→❝ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤→❝ ♥❤❛✉✿ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ r✐➯♥❣✱ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✱✳✳✳✳ ◆❣❤✐➯♥ ❝ù✉ ❝→❝ ự ỵ t tr ♠ët sè ❧➽♥❤ ✈ü❝ ❝õ❛ t♦→♥ ❤å❝ ♥❤÷✿ ❝❤ù♥❣ ♠✐♥❤ sỹ tỗ t t ợ ♣❤÷ì♥❣ tr➻♥❤ ✈✐✲t➼❝❤ ♣❤➙♥✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✱✳✳✳ ❚r➯♥ ❝ì sð ✤â ✤➲ t➔✐ ✤➦t ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ s❛✉✿ ✲ ◆❣❤✐➯♥ ❝ù✉ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠ët sè ❧ỵ♣ ❝→❝ →♥❤ ①↕ tr➯♥ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ✲ ❳➙② ❞ü♥❣ ♠ët sè ❧ỵ♣ →♥❤ ①↕ ❝♦ s✉② rë♥❣ tr➯♥ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ✲ Ù♥❣ ❞ö♥❣ ❝→❝ ❦➳t q✉↔ t❤✉ ✤÷đ❝ ✈➔♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ sü tỗ t ởt số ợ ữỡ tr t ợ ỵ tr ú tổ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ ❝❤♦ ❧✉➟♥ →♥ ❝õ❛ ♠➻♥❤ ❧➔✿ ỵ t ởt số ợ →♥❤ ①↕ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈➔ ù♥❣ ❞ư♥❣✧✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ →♥ ❧➔ rở t q sỹ tỗ t ❜➜t ✤ë♥❣ ❝❤♦ ♠ët sè ❧ỵ♣ →♥❤ ①↕ tr➯♥ ❝→❝ ợ ổ ữ ổ btr s tự tỹ ❜ë ♣❤➟♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈➔ ù♥❣ ❞ư♥❣ ❝→❝ ❦➳t q✉↔ t❤✉ ✤÷đ❝ ự sỹ tỗ t ởt số ợ ữỡ tr t ố tữủ ự ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❧✉➟♥ →♥ ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✱ ❝→❝ →♥❤ ①↕ ❝♦ s✉② rë♥❣✱ ✤✐➸♠ ❜➜t ✤ë♥❣✱ ✤✐➸♠ trò♥❣ ♥❤❛✉ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✱ ♠ët sè ợ ữỡ tr t P ự ự ỵ trũ ỵ t tr ổ btr s t❤ù tü ❜ë ♣❤➟♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈➔ ù♥❣ ❞ö♥❣ ❝→❝ ❦➳t q✉↔ t ữủ ự t tỗ t ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ự ú tổ sỷ ữỡ ự ỵ tt t ỵ tt ữỡ tr ữỡ tr t ỵ tt t tr q tr tỹ t ỵ ♥❣❤➽❛ ❦❤♦❛ ❤å❝ ✈➔ t❤ü❝ t✐➵♥ ▲✉➟♥ →♥ ✤➣ ❧➔♠ ú t t q sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ số ỗ tớ ự t q t ữủ ự sỹ tỗ t ởt số ợ ữỡ tr t q ✈➔ ❝➜✉ tró❝ ❧✉➟♥ →♥ ◆ë✐ ❞✉♥❣ ❧✉➟♥ →♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ✸ ❝❤÷ì♥❣✳ ◆❣♦➔✐ r❛✱ ❧✉➟♥ →♥ ❝á♥ ❝â ▲í✐ ❝❛♠ ✤♦❛♥✱ ▲í✐ ❝↔♠ ì♥✱ ▼ư❝ ❧ư❝✱ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❑✐➳♥ ♥❣❤à✱ ❉❛♥❤ ♠ư❝ ❝ỉ♥❣ tr➻♥❤ ❦❤♦❛ ❤å❝ ❝õ❛ ♥❣❤✐➯♥ ❝ù✉ s✐♥❤ ❧✐➯♥ q✉❛♥ trü❝ t✐➳♣ ✤➳♥ ❧✉➟♥ →♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ✈➔ ù♥❣ ❞ö♥❣✳ ▼ö❝ ✶✳✶✱ ú tổ ự sỹ tỗ t trũ ♥❤❛✉ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ T ✲❝♦ ✸ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ▼ư❝ ✶✳✷✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ sỹ tỗ t trũ ợ →♥❤ ①↕ (ψ, L)✲T ✲❤➛✉ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ▼ư❝ ✶✳✸✱ ❝❤ó♥❣ tỉ✐ ✤➣ ù♥❣ ❞ư♥❣ ❦➳t q✉↔ t➻♠ ✤÷đ❝ ✤➸ ♥❣❤✐➯♥ ự sỹ tỗ t ởt ợ ữỡ tr t➼❝❤ ♣❤➙♥✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤➣ ✤÷đ❝ ✤➠♥❣ tr➯♥ t↕♣ ❝❤➼ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✿ ▼♦❞❡❧❧✐♥❣ ❛♥❞ ❈♦♥tr♦❧✳ ❈❤÷ì♥❣ ✷ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ♥❣❤✐➯♥ ❝ù✉ ✈➲ sü tỗ t t tr ổ btr õ ✤➛② ✤õ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ ù♥❣ ❞ö♥❣✳ ▼ư❝ ✷✳✶ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✳ ▼ư❝ ✷✳✷✱ ❝❤ó♥❣ tỉ✐ ♥❣❤✐➯♥ ❝ù✉ sỹ tỗ t t ợ →♥❤ ①↕ ϕ✲❝♦ ②➳✉ s✉② rë♥❣ tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✳ ▼ư❝ ✷✳✸✱ ❝❤ó♥❣ tổ ự sỹ tỗ t t ❜ë ✤æ✐ ❝❤♦ ♠ët sè →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✳ ▼ö❝ ✷✳✹✱ ❝❤ó♥❣ tỉ✐ ✤➣ ù♥❣ ❞ư♥❣ ❦➳t q✉↔ t➻♠ ✤÷đ❝ ự sỹ tỗ t ởt ợ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ ❈→❝ ❦➳t q✉↔ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤❛♥❣ ✤÷đ❝ ❣û✐ ✤➠♥❣ tr➯♥ ♠ët sè t↕♣ ❝❤➼ ❚♦→♥ ❤å❝ q✉è❝ t➳✳ ❈❤÷ì♥❣ ✸ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ự sỹ tỗ t t tr ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈➔ ù♥❣ ❞ư♥❣✳ ▼ư❝ ✸✳✶ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ❚r♦♥❣ ▼ư❝ ✸✳✷✱ ❝❤ó♥❣ tỉ✐ t❤✐➳t ❧➟♣ ♠ët sè ✤à♥❤ ỵ t ợ s✉② rë♥❣ ✈➔ →♥❤ ①↕ ϕ✲❝♦ ❝❤✉➞♥ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ▼ư❝ ✸✳✸✱ ❝❤ó♥❣ tỉ✐ t❤✐➳t ❧➟♣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ởt số ỵ t ở ổ ♠ët ❧ỵ♣ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ▼ư❝ ✸✳✹✱ ❝❤ó♥❣ tổ ự t sỹ tỗ t ởt ợ ữỡ tr t t q ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤➣ ✤÷đ❝ ✤➠♥❣ tr➯♥ t↕♣ ❝❤➼ ❙❝✐❡♥t✐❢✐❝ ♣✉❜❧✐❝❛t✐♦♥s ♦❢ t❤❡ st❛t❡ ✉♥✐✈❡rs✐t② ♦❢ ◆♦✈✐ P❛③❛r ✈➔ t↕♣ ❝❤➼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ▼❛t❤❡♠❛t✐❝❛❧ ❙t✉❞✐❡s✳ ✹ ❈❍×❒◆● ✶ ✣■➎▼ ❚❘Ò◆● ◆❍❆❯ ❈❍❖ ▼❐❚ ❙➮ ▲❰P ⑩◆❍ ❳❸ ❚❘➊◆ ❑❍➷◆● ●■❆◆ b✲▼➊❚❘■❈ ❙➁P ❚❍Ù ❚Ü ❇❐ P❍❾◆ ❱⑨ Ù◆● ❉Ö◆● ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ T ✲❝♦ ✈➔ →♥❤ ①↕ (ψ, L)✲T ✲❤➛✉ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✱ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❦➳t q✉↔ ✈➲ ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ T ✲❝♦ s✉② rë♥❣ ✈➔ →♥❤ ①↕ (ψ, L)✲T ✲❤➛✉ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ tỉ✐ ù♥❣ ❞ư♥❣ ❦➳t q✉↔ t ữủ ự sỹ tỗ t ởt ợ ữỡ tr t trũ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ T ✲❝♦ s✉② rë♥❣ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈➔ t❤✐➳t ❧➟♣ ♠ët sè ❦➳t q✉↔ ✈➲ ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ❈❤♦ X ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ s ≥ ❧➔ ♠ët sè t❤ü❝✳ ❍➔♠ d : X ìX R ữủ btr tr➯♥ X ✱ ♥➳✉ ✈ỵ✐ ♠å✐ x, y, z ∈ X ✱ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✶✳ ≤ d(x, y) ✈➔ d(x, y) = ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x = y ✳ ✷✳ d(x, y) = d(y, x)✳ ✸✳ d(x, z) ≤ s[d(x, y) + d(y, z)]✳ ❑❤✐ ✤â✱ (X, d, s) ✤÷đ❝ ❣å✐ ❧➔ ổ btr ợ số s rữớ ủ s = t❤➻ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✽ ❈❤♦ X ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ ❝→❝ →♥❤ ①↕ f, g : X → X ✳ ❑❤✐ ✤â f ✈➔ g ✤÷đ❝ ❣å✐ ❧➔ ❣✐❛♦ ❤♦→♥ ♥➳✉ f gx = gf x ✈ỵ✐ ♠å✐ x ∈ X ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✾ ❈❤♦ X ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ ❝→❝ →♥❤ ①↕ f, g : X → X ✳ ◆➳✉ w = f x = gx ✈ỵ✐ x ∈ X t❤➻ x ✤÷đ❝ ❣å✐ ❧➔ ♠ët ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝õ❛ f ✈➔ g ✈➔ w ✤÷đ❝ ❣å✐ ❧➔ ❣✐→ trà trò♥❣ ♥❤❛✉ ❝õ❛ f ✈➔ g ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✵ ❈❤♦ (X, d, s, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ✈➔ ❝→❝ →♥❤ ①↕ f, g, h, k : X → X ✳ ❑❤✐ ✤â ✺ ✶✳ ❈➦♣ (f, g) ✤÷đ❝ ❣å✐ ❧➔ h✲t÷ì♥❣ t❤➼❝❤ ♥➳✉ lim d(f hgxn , ghf xn ) = 0✱ ✈ỵ✐ ♠å✐ ❞➣② n→∞ {xn } tr♦♥❣ X s❛♦ ❝❤♦ lim hf xn = lim hgxn = t ✈ỵ✐ t ♥➔♦ ✤â t❤✉ë❝ X ✳ ◆➳✉ ❧➜② n→∞ n→∞ hx = x ✈ỵ✐ ♠å✐ x ∈ X t❤➻ ❝➦♣ (f, g) ✤÷đ❝ ❣å✐ ❧➔ t÷ì♥❣ t❤➼❝❤ ✳ ◆➳✉ t❛ ❧➜② gx = x ✈ỵ✐ ♠å✐ x ∈ X ✱ t❤➻ f ✤÷đ❝ ❣å✐ ❧➔ h✲t÷ì♥❣ t❤➼❝❤✳ ✷✳ ❈➦♣ (f, g) ✤÷đ❝ ❣å✐ ❧➔ h✲t÷ì♥❣ t❤➼❝❤ ②➳✉ ♥➳✉ f hgx = ghf x ✈ỵ✐ ♠é✐ hgx = hf x✳ ◆➳✉ t❛ ❧➜② hx = x ✈ỵ✐ ♠å✐ x ∈ X t❤➻ ❝➦♣ (f, g) ✤÷đ❝ ❣å✐ ❧➔ t÷ì♥❣ t❤➼❝❤ ②➳✉✳ ◆➳✉ t❛ ❧➜② gx = x ✈ỵ✐ ♠å✐ x ∈ X t❤➻ f ✤÷đ❝ ❣å✐ ❧➔ h✲t÷ì♥❣ t❤➼❝❤ ②➳✉✳ ✸✳ ❈➦♣ (f, g) ✤÷đ❝ ❣å✐ ❧➔ h✲t➠♥❣ ②➳✉ ✤è✐ ✈ỵ✐ k ♥➳✉ hf (X) hg(X) ⊆ hk(X) ✈➔ ✈ỵ✐ ♠å✐ x ∈ X ✱ t❛ ❝â hf x hgy ✈ỵ✐ ♠å✐ y ∈ (hk)−1 (hf x) ✈➔ hgx hf y ✈ỵ✐ ♠å✐ y ∈ (hk)−1 (hgx)✳ ◆➳✉ t❛ ❧➜② hx = x ✈ỵ✐ ♠å✐ x ∈ X t❤➻ ❝➦♣ (f, g) ữủ t ố ợ k ✳ ◆➳✉ ❧➜② kx = x ✈ỵ✐ ♠å✐ x ∈ X t❤➻ ❝➦♣ (f, g) ✤÷đ❝ ❣å✐ ❧➔ h✲t➠♥❣ ②➳✉✳ ✹✳ ❈➦♣ (f, g) ✤÷đ❝ ❣å✐ ❧➔ h✲t➠♥❣ ②➳✉ ❜ë ♣❤➟♥ ✤è✐ ✈ỵ✐ k ♥➳✉ hf (X) ⊆ hk(X) ✈➔ ✈ỵ✐ ♠å✐ x ∈ X ✱ t❛ ❝â hf x hgy ✈ỵ✐ ♠å✐ y ∈ (hk)−1 (hf x)✳ ◆➳✉ t❛ ❧➜② hx = x ✈ỵ✐ ♠å✐ x ∈ X t❤➻ ❝➦♣ (f, g) ✤÷đ❝ ❣å✐ ❧➔ t➠♥❣ ②➳✉ ❜ë ♣❤➟♥ ✤è✐ ✈ỵ✐ k ✳ ◆➳✉ t❛ ❧➜② kx = x ợ x X t (f, g) ữủ ❣å✐ ❧➔ h✲t➠♥❣ ②➳✉ ❜ë ♣❤➟♥✳ ✺✳ f ✤÷đ❝ ❣å✐ ❧➔ g ✲✤ì♥ ✤✐➺✉ ❦❤ỉ♥❣ ❣✐↔♠ ✤è✐ ✈ỵ✐ (h, ) ♥➳✉ hgx hgy ✱ t❤➻ t❛ ❝â hf x hf y ✳ ◆➳✉ t❛ ❧➜② hx = x ✈ỵ✐ ♠å✐ x ∈ X ✱ t❤➻ f ✤÷đ❝ ❣å✐ ❧➔ g ✲✤ì♥ ✤✐➺✉ ❦❤ỉ♥❣ ❣✐↔♠ ✤è✐ ✈ỵ✐ “ ”✳ ◆➳✉ t❛ ❧➜② gx = x ✈ỵ✐ ♠å✐ x ∈ X ✱ t❤➻ f ✤÷đ❝ ❣å✐ ❧➔ ✤ì♥ ✤✐➺✉ ❦❤ỉ♥❣ ❣✐↔♠ ✤è✐ ✈ỵ✐ (h, )✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✶ ❈❤♦ (X, d, s) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ s > ✈➔ ❝→❝ →♥❤ ①↕ T, S : X → X ✱ →♥❤ S ữủ T tỗ t k ∈ [0, 1) s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ X ✱ t❛ ❝â d(T Sx, T Sy) ≤ kd(T x, T y) ✭✶✳✶✮ ◆➳✉ T x = x ✈ỵ✐ ♠å✐ x ∈ X t❤➻ →♥❤ ①↕ T ✲❝♦ ❧➔ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤✳ ❱➼ ❞ö ✶✳✶✳✶✷ ▲➜② X = [1, ∞)✱ ①➨t b✲♠➯tr✐❝ ✤÷đ❝ ❝❤♦ ❜ð✐ d(x, y) = |x − y|2 ✈ỵ✐ ♠å✐ x, y ∈ X ✳ ❳➨t ❝→❝ →♥❤ ①↕ T x = − ✈➔ Sx = 4x ✈ỵ✐ ♠å✐ x ∈ X ✳ ❑❤✐ ✤â S ❧➔ →♥❤ ①↕ x T ✲❝♦ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ →♥❤ ①↕ ❝♦ t❤ỉ♥❣ t❤÷í♥❣✳ ◆➠♠ ✷✵✶✺✱ ❍✉❛♥❣✱ ❘❛❞❡♥♦✈➼❝ ✈➔ ❱✉❥❛❦♦✈➼❝ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ s❛✉ ✈➲ ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝❤♦ ❝→❝ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐❛♥ btr s tự tỹ ỵ (X, d, s, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ✈ỵ✐ ❤➺ sè s > ✈➔ f, g, S, R : X → X ❧➔ ❝→❝ →♥❤ ①↕ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ f (X) ⊆ R(X) ✈➔ g(X) ⊆ S(X)✳ ✷✳ ❱ỵ✐ ♠é✐ ❝➦♣ x, y ∈ X s❛♦ ❝❤♦ Sx, Ry ❧➔ s♦ s→♥❤ ✤÷đ❝✱ t❛ ❝â si d(f x, gy) ≤ Ms (x, y), ✭✶✳✷✮ ✻ ð ✤➙② i > ❧➔ ♠ët ❤➡♥❣ sè ✈➔ Ms (x, y) = max d(Sx, Ry), d(Sx, f x), d(Ry, gy), d(Sx, gy) + d(Ry, f x) 2s ✸✳ ❈→❝ →♥❤ ①↕ f, g, R ✈➔ S ❧➔ ❧✐➯♥ tö❝✳ ✹✳ ❈→❝ ❝➦♣ (f, S) ✈➔ (g, R) ❧➔ t÷ì♥❣ t❤➼❝❤✳ ✺✳ ❈→❝ ❝➦♣ (f, g) ✈➔ (g, f ) ❧➔ t➠♥❣ ②➳✉ ❜ë ♣❤➟♥ ✤è✐ ✈ỵ✐ R ✈➔ S ✱ t÷ì♥❣ ù♥❣✳ ❑❤✐ ✤â✱ ❝→❝ ❝➦♣ (f, S) ✈➔ (g, R) ❝â ♠ët ✤✐➸♠ trò♥❣ ♥❤❛✉ z tr♦♥❣ X ✳ ❍ì♥ ♥ú❛✱ ♥➳✉ Rz ✈➔ Sz ❧➔ s♦ s→♥❤ ✤÷đ❝✱ t❤➻ z ❧➔ ♠ët ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝õ❛ f, g, R ✈➔ S ✳ ❙❛✉ ✤➙② ❧➔ ♠ët ✈➼ ❞ö ♠➔ ❝❤ó♥❣ t❛ ❞➵ ❞➔♥❣ ♥❤➟♥ t❤➜② ❝→❝ ❝➦♣ (f, S) ✈➔ (g, R) ❝â ♠ët ✤✐➸♠ trò♥❣ ♥❤❛✉ ❧➔ ✤✐➸♠ tr♦♥❣ X ✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ỵ ỵ ổ ✤÷đ❝ ❝❤♦ ❝→❝ →♥❤ ①↕ f, g, R, S ✳ ❱➼ ❞ö ✶✳✶✳✶✹ ❈❤♦ t➟♣ X = [0, 1]✱ ①➨t b✲♠➯tr✐❝ d ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ d(x, y) = |x−y|2 ✈ỵ✐ ♠å✐ x, y ∈ X ✈➔ q✉❛♥ ❤➺ t❤ù tü “ ” tr➯♥ X ①→❝ ✤à♥❤ ❜ð✐✳ x y ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x ≥ y ✈ỵ✐ ♠å✐ x, y ∈ X ❚❛ ①→❝ ✤à♥❤ ❝→❝ →♥❤ ①↕ g, S, f, R : X → X ❝❤♦ ❜ð✐ x Sx = Rx = √ ✈➔ f x = gx = x 2i+ ✈ỵ✐ ♠å✐ x ∈ X, i > ❇➙② ❣✐í✱ ❝❤ó♥❣ tỉ✐ s➩ tt ởt ỵ trũ ❧ỵ♣ ❝→❝ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ T ✲❝♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ỡ ỳ sỷ ỵ t r ữủ sỹ tỗ t trũ tr ỵ ✶✳✶✳✶✺ ❈❤♦ (X, d, s, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ✈ỵ✐ s > ✈➔ T, f, g, S, R : X → X ❧➔ ❝→❝ →♥❤ ①↕ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ f (X) ⊆ R(X) ✈➔ g(X) ⊆ S(X)✳ ✷✳ T ❧➔ →♥❤ ①↕ ♠ët✲♠ët✳ ✸✳ ▼é✐ ❝➦♣ x, y ∈ X s❛♦ ❝❤♦ T Sx, T Ry ❧➔ s♦ s→♥❤ ✤÷đ❝✱ t❛ ❝â si d(T f x, T gy)) ≤ MsT (x, y), ✭✶✳✸✮ ð ✤➙② i > ❧➔ ♠ët ❤➡♥❣ sè ✈➔ MsT (x, y) = max d(T Sx, T Ry), d(T Sx, T f x), d(T Ry, T gy), d(T Sx, T gy) + d(T Ry, T f x) 2s ✭✶✳✹✮ ✼ ✹✳ ❈→❝ →♥❤ ①↕ f, g, R ✈➔ S ❧➔ ❧✐➯♥ tö❝✳ ✺✳ ❈→❝ ❝➦♣ (f, S) ✈➔ (g, R) ❧➔ T ✲t÷ì♥❣ t❤➼❝❤✳ ✻✳ ❈→❝ ❝➦♣ (f, g) ✈➔ (g, f ) ❧➔ T ✲t➠♥❣ ②➳✉ ❜ë ♣❤➟♥ ✤è✐ ✈ỵ✐ R ✈➔ S ✱ t÷ì♥❣ ù♥❣✳ ❑❤✐ ✤â✱ ❝→❝ ❝➦♣ (f, S) ✈➔ (g, R) ❝â ♠ët ✤✐➸♠ trò♥❣ ♥❤❛✉ z tr♦♥❣ X ✳ ❍ì♥ ♥ú❛✱ ♥➳✉ T Rz ✈➔ T Sz ❧➔ s♦ s→♥❤ ✤÷đ❝✱ t❤➻ z ❧➔ ♠ët ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝õ❛ f, g, R ✈➔ S ✳ ◆❤➟♥ t r ỵ t T ỗ t t t t ữủ ỵ sỷ ỵ t r ữủ sỹ tỗ t ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❱➼ ❞ư ✶✳✶✳✶✹✳ ❚ø ❝→❝ ❦➳t q✉↔ tr➯♥ t❛ s✉② r❛ r➡♥❣ ỵ ởt rở tỹ sỹ ỵ (X, d, s, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ✈ỵ✐ ❤➺ sè s > ✈➔ ❝→❝ →♥❤ ①↕ f, g, T, S, R : X → X ✳ ✶✳ ◆➳✉ ❝→❝ →♥❤ ①↕ f ✈➔ g t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✶✳✸✮ ✈➔ ✭✶✳✹✮ tr♦♥❣ ✣à♥❤ ỵ t (f, g) ữủ tọ ✤✐➲✉ ❦✐➺♥ T ✲❝♦ s✉② rë♥❣✳ ✷✳ ◆➳✉ ❝➦♣ (f, f ) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ T ✲❝♦ s✉② rë♥❣✱ t❤➻ →♥❤ ①↕ f ✤÷đ❝ ❣å✐ ❧➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ T ✲❝♦ s✉② rë♥❣✳ ❍➺ q✉↔ ✶✳✶✳✷✵ ❈❤♦ (X, d, s, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ✈ỵ✐ s > ✈➔ T, g : X → X ❧➔ ❤❛✐ →♥❤ ①↕ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ T ❧➔ →♥❤ ①↕ ♠ët✲♠ët✳ ✷✳ ❱ỵ✐ ♠é✐ ❝➦♣ x, y ∈ X s❛♦ ❝❤♦ T x, T y ❧➔ s♦ s→♥❤ ✤÷đ❝✱ t❛ ❝â si d(T gx, T gy) ≤ d(T x, T y), ð ✤➙② i > ❧➔ ♠ët ❤➡♥❣ sè✳ ✸✳ g ❧➔ ❧✐➯♥ tö❝✳ ✹✳ g ✈➔ T ❧➔ ❣✐❛♦ ❤♦→♥✳ ✺✳ g ❧➔ ✤ì♥ ✤✐➺✉ ❦❤ỉ♥❣ ❣✐↔♠ ✤è✐ ợ (T, ) ỗ t x0 X s ❝❤♦ T x0 T gx0 ✳ ❑❤✐ ✤â✱ g ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ z tr♦♥❣ X ✳ ✭✶✳✺✮ ✽ ✶✳✷ ✣✐➸♠ trò♥❣ ♥❤❛✉ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ (ψ, L)✲T ✲❤➛✉ ❝♦ s✉② rë♥❣ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ (ψ, L)✲T ✲❤➛✉ ❝♦ s✉② rë♥❣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❦➳t q✉↔ ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ (ψ, L)✲T ✲❤➛✉ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❑➼ ❤✐➺✉ Ψ ❧➔ ❤å t➜t ❝↔ ❝→❝ ❤➔♠ ψ : R+ → R+ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ ψ ❧➔ ❤➔♠ ❧✐➯♥ tư❝ ✈➔ ❦❤ỉ♥❣ ❣✐↔♠✳ ✷✳ ψ(t) = ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ t = 0✳ ❑❤✐ ✤â✱ ψ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❤➔♠ t❤❛② ✤ê✐ ❦❤♦↔♥❣ ❝→❝❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳ ❈❤♦ (X, d, s, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ✈ỵ✐ s > ✈➔ ❝→❝ →♥❤ ①↕ S, T, g : X → X ✳ ❑❤✐ ✤â✱ →♥❤ ①↕ S ✤÷đ❝ ❣♦✐ ❧➔ (ψ, L)✲T ✲❤➛✉ ❝♦ s✉② rë♥❣ ✤è✐ ✈ỵ✐ g ♥➳✉ ✈ỵ✐ ♠å✐ x, y ∈ X ♠➔ T gx T gy t❛ ❝â ψ(si d(T Sx, T Sy)) ≤ ψ(MsT (x, y)) + Lψ(NsT (x, y)), ✭✶✳✻✮ ✈ỵ✐ ψ ∈ Ψ✱ i > 1✱ L ≥ ♥➔♦ ✤â✱ tr♦♥❣ ✤â MsT (x, y) = max d(T gx, T gy), d(T gx, T Sx), d(T gy, T Sy), d(T gx, T Sy) + d(T gy, T Sx) , 2s ✈➔ NsT (x, y) = d(T gx, T Sx), d(T gy, T Sy), d(T gx, T Sy), d(T gy, T Sx) ❚r♦♥❣ ✣à♥❤ ♥❣❤➽❛ tr➯♥ ♥➳✉ t❛ ❧➜② T ❧➔ ỗ t tr X t S ①↕ (ψ, L)✲❤➛✉ ❝♦ s✉② rë♥❣ ✤è✐ ✈ỵ✐ g ✳ ◆➠♠ ✷✵✶✺✱ ❍✉❛♥❣✱ ❘❛❞❡♥♦✈➼❝ ✈➔ ❱✉❥❛❦♦✈➼❝ ✤➣ t❤✐➳t ❧➟♣ ❦➳t q✉↔ ✈➲ ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ (ψ, L)✲❤➛✉ ❝♦ s✉② rë♥❣ ✤è✐ ✈ỵ✐ g tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ✣à♥❤ ỵ (X, d, , s) ổ btr ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥ ✈ỵ✐ s > ✈➔ g, S : X → X ❧➔ ❤❛✐ →♥❤ ①↕ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ S(X) ⊆ g(X)✳ ✷✳ ❱ỵ✐ ψ ∈ Ψ✱ i > ✈➔ L ≥ ♥➔♦ ✤â✱ t❛ ❝â ψ(si d(Sx, Sy)) ≤ ψ(Ms (x, y)) + Lψ(Ns (x, y)) ✭✶✳✼✮ tỗ t x0 C(I) s ợ ♠å✐ t ∈ I ✱ t❛ ❝â e−t K(t, r)f (r, x0 (r))dr ≤ T (x0 (t)) T η(t) + λ ✭✶✳✶✵✮ ✈ỵ✐ i > ✈➔ p > số ỵ ợ tt ❝↔ ❝→❝ ❣✐↔ t❤✐➳t (a1 ) − (a6 )✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✽✮ ❝â ♠ët ♥❣❤✐➺♠ tr♦♥❣ X ✳ (a6 ) (λAL)p ≤ 2i(p−1) ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ s r r tỗ t ❤➔♠ sè T, K, g ✈➔ f t❤ä❛ ♠➣♥ t➜t tt ỵ ✶✳✸✳✹✳ ❈❤♦ X = C(I) ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❤➔♠ t❤ü❝ ❧✐➯♥ tö❝ tr➯♥ t➟♣ I = [0, 1]✳ ❚r➯♥ X t❛ ①➨t b✲♠➯tr✐❝ ❝❤♦ ❜ð✐ d(x, y) = supt∈I | x(t) − y(t) |2 ✈ỵ✐ ♠å✐ x, y ∈ X ✈➔ q✉❛♥ ❤➺ t❤ù tü “ ” ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ x y ♥➳✉ x(t) ≥ y(t) ✈ỵ✐ ♠å✐ t ∈ I ✳ ❑❤✐ ✤â✱ (X, d, s, ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tỹ ợ s = t ữỡ tr t➼❝❤ ♣❤➙♥ t2 e−4t x(t) = t − + 5.2i 2i+1 e−t t2 et r2 x(r) + r2 dr ỷ ỵ t r ữủ x(t) = t2 ợ t [0, 1] ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥✳ ❑➳t ❧✉➟♥ ❈❤÷ì♥❣ ✶ ❚r♦♥❣ ❈❤÷ì♥❣ ✶ ❝õ❛ ❧✉➟♥ →♥✱ ❝❤ó♥❣ tỉ✐ ✤➣ t ữủ t q s ữ r ỵ ỵ sỹ tỗ t ✤✐➸♠ trò♥❣ ♥❤❛✉ ❝õ❛ ❧ỵ♣ ❝→❝ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ T ✲❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ btr s tự tỹ ỗ tớ ù♥❣ ❞ö♥❣ ❝→❝ ❦➳t q✉↔ ♥➔② ✈➔♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ sỹ tỗ t ởt ợ ữỡ tr t ♣❤➙♥✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷đ❝ tr➼❝❤ tø ❜➔✐ ❜→♦✿ ❙✳ ❘❛❞❡♥♦✈➼❝✱ ❚✳ ❱✳ ❆♥ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✼✮✱ ✏❙♦♠❡ ❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t r❡s✉❧ts ❢♦r T ✲❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s ♦♥ ♣❛rt✐❛❧❧② ♦r✲ ❞❡r❡❞ b✲♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✑✱ ◆♦♥❧✐♥❡❛r ss tr ữ r ỵ ỵ sỹ tỗ t trũ ợ (, L)✲T ✲❤➛✉ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷đ❝ tr➼❝❤ tø ❜➔✐ ❜→♦✿ ❙✳ ❘❛❞❡♥♦✈➼❝✱ ◆✳ ❉❡❞♦✈➼❝✱ ❚✳ ❱✳ ❆♥ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✼✮✱ ✏❙♦♠❡ ❝♦✐♥❝✐❞❡♥❝❡ t❤❡♦r❡♠ ❢♦r ❛❧♠♦st ❣❡♥❡r❛❧✐③❡❞ (ϕ, L)✲T ✲❝♦♥tr❛❝t✐♦♥s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ b✲♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✑ ✭✤❛♥❣ ❣û✐ ✤➠♥❣✮✳ ✶✷ ❈❍×❒◆● ✷ ✣■➎▼ ❇❻❚ ✣❐◆● ❈❍❖ ▼❐❚ ❙➮ ▲❰P ⑩◆❍ ❳❸ ❚❘➊◆ ❑❍➷◆● ●■❆◆ b✲▼➊❚❘■❈ ◆➶◆ ✣❺❨ ✣Õ ❚❘➊◆ ✣❸■ ❙➮ ❇❆◆❆❈❍ ❱⑨ Ù◆● ❉Ư◆● ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ♠ð rë♥❣ ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ϕ✲❝♦ ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ s❛♥❣ ❝➜✉ tró❝ ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❦➳t q✉↔ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤ỉ✐ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ ♥➔② tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✳ ◆❣♦➔✐ r❛✱ ❝❤ó♥❣ tỉ✐ ù♥❣ ❞ư♥❣ ❦➳t q✉↔ t ữủ ự sỹ tỗ t ởt ợ ữỡ tr t ổ btr ♥â♥ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✣➛✉ t✐➯♥✱ t❛ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ s❛✉✿ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✻✳ ❈❤♦ X ❧➔ t➟♣ ❦❤→❝ ré♥❣✱ s ≥ ❧➔ ♠ët sè t❤ü❝ ✈➔ A ❧➔ ✤↕✐ sè ❇❛♥❛❝❤✳ ⑩♥❤ ①↕ d : X ì X A ữủ btr ♥â♥ tr➯♥ X ✱ ♥➳✉ ✈ỵ✐ ♠å✐ x, y, z ∈ X ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✶✳ θ d(x, y) ✈➔ d(x, y) = θ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x = y ✳ ✷✳ d(x, y) = d(y, x)✳ ✸✳ d(x, z) s[d(x, y) + d(y, z)]✳ ❑❤✐ ✤â✱ (X, A, d, s) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❤➺ sè s✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✽✳ ❈❤♦ A ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ P ❧➔ ♥â♥ tr♦♥❣ A✳ ⑩♥❤ ①↕ ϕ : P → P ✤÷đ❝ ❣å✐ ❧➔ s♦ s→♥❤ ②➳✉ ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ ϕ ❧➔ ❦❤æ♥❣ ❣✐↔♠ ✤è✐ ✈ỵ✐ “ ϕ(t1 ) ϕ(t2 )✳ ”✱ ♥❣❤➽❛ ❧➔ ✈ỵ✐ ♠å✐ t1 , t2 ∈ P ♠➔ t1 ✷✳ {ϕn (t)} ❧➔ c✲❞➣② tr♦♥❣ P ✈ỵ✐ ♠å✐ t ∈ P ✳ ✸✳ ◆➳✉ {un } ❧➔ c✲❞➣② tr♦♥❣ P t❤➻ {ϕ(un )} ❝ô♥❣ ❧➔ c✲❞➣② tr♦♥❣ P ✳ t2 ✱ ❦➨♦ t❤❡♦ ✶✸ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✾✳ ❈❤♦ (X, A, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ P ❧➔ ♥â♥ tr♦♥❣ A✳ ⑩♥❤ ①↕ f : X → X ✤÷đ❝ ❣å✐ ❧➔ ϕ✲❝♦ ②➳✉ ♥➳✉ tỗ t s s : P → P s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ X, t❛ ❝â d f (x), f (y) ϕ d(x, y) ỵ (X, A, d) ổ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ f : X → X ❧➔ →♥❤ ①↕ ϕ✲❝♦ ②➳✉✳ ❑❤✐ ✤â✱ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ u ∈ X ✈➔ lim f n (x) = u ✈ỵ✐ ♠é✐ x ∈ X ✳ n→∞ ✷✳✷ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ ②➳✉ s✉② rë♥❣ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ❳✉➜t ♣❤→t tø ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ϕ✲❝♦ ②➳✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤✱ ❝❤ó♥❣ tỉ✐ ①➙② ❞ü♥❣ ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ϕ✲❝♦ ②➳✉ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❈❤♦ (X, A, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ P ❧➔ ♥â♥ tr♦♥❣ A✳ ⑩♥❤ ①↕ f : X → X ✤÷đ❝ s rở tỗ t ①↕ s♦ s→♥❤ ②➳✉ ϕ : P → P s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ X, t❛ ❝â d f (x), f (y) ✭✷✳✶✮ ϕ d(x, y) ❇ê ✤➲ ✷✳✷✳✷✳ ●✐↔ sû (X, A, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤✱ P ❧➔ ♥â♥ tr♦♥❣ A ✈➔ f : X → X ❧➔ →♥❤ ①↕ ϕ✲❝♦ ②➳✉ s✉② rë♥❣✳ ❑❤✐ ✤â t❛ ❝â ✶✳ ❱ỵ✐ ♠å✐ t1 , t2 ∈ P ♠➔ t1 t2 ✈➔ ✈ỵ✐ ♠å✐ n ∈ N✱ t❛ ❝â ϕn (t1 ) ϕn (t2 )✳ ✷✳ ❱ỵ✐ ♠å✐ x, y ∈ X ✈➔ ♠å✐ n ∈ N✱ t❛ ❝â d f n (x), f n (y) ϕn d(x, y) ◆➠♠ ✷✵✶✼✱ ❍✳ ❍✉❛♥❣✱ ❙✳ ❘❛❞❡♥♦✈✐✁❝ ✈➔ ●✳ ❉❡♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❦➳t q✉↔ s❛✉ ✤➙② ❝❤♦ ✤✐➸♠ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤✳ ✣à♥❤ ỵ (X, A, d, s) ổ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤✱ k ∈ P s❛♦ ❝❤♦ ρ(k) < ✈➔ f : X → X ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d f (x), f (y) kd(x, y), ✈ỵ✐ ♠å✐ x, y ∈ X ✭✷✳✷✮ ❑❤✐ ✤â✱ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ u ∈ X ✈➔ ✈ỵ✐ ♠é✐ x ∈ X t❛ ❝â lim f n (x) = u✳ n→∞ ❙❛✉ ✤➙② ❧➔ ♠ët ✈➼ ❞ư ♠➔ ❝❤ó♥❣ t❛ ❞➵ ❞➔♥❣ ♥❤➟♥ t❤➜② →♥❤ ①↕ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❧➔ ✤✐➸♠ tr♦♥❣ X ✳ ❚✉② ♥❤✐➯♥✱ tr trữớ ủ ỵ ổ ❞ư♥❣ ✤÷đ❝ ❝❤♦ →♥❤ ①↕ f ✳ ✶✹ ❱➼ ❞ư ✷✳✷✳✹✳ ❈❤♦ A = R2 ✱ P = {(u, v) ∈ A : u, v ≥ 0}✳ ❱ỵ✐ x = (x1 , x2 )✱ y = (y1 , y2 ) ∈ A✱ t❛ ①→❝ ✤à♥❤ ✶✳ ❈❤✉➞♥ tr♦♥❣ A ❝❤♦ ❜ð✐ (x1 , x2 ) = |x1 | + |x2 |✳ ✷✳ P❤➨♣ ♥❤➙♥ tr♦♥❣ A ❝❤♦ ❜ð✐ xy = (x1 , x2 )(y1 , y2 ) = (x1 y1 , x1 y2 + x2 y1 ) ✸✳ X = R+ ✈➔ ①→❝ ✤à♥❤ d : X × X → A ❝❤♦ ❜ð✐ d(x, y) = |x − y|2 , ✈ỵ✐ ♠å✐ x, y ∈ X ✹✳ f : X → X ❝❤♦ ❜ð✐ f (x) = x ✈ỵ✐ ♠å✐ x ∈ X x+1 ✺✳ ϕ : P → P ❝❤♦ ❜ð✐ ϕ(z1 , z2 ) = z1 , ✈ỵ✐ ♠å✐ (z1 , z2 ) ∈ P z1 + ❑❤✐ ✤â✱ (X, A, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❤➺ sè s = 2✳ ❉➵ ❞➔♥❣ ♥❤➟♥ t❤➜② →♥❤ ①↕ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❧➔ ✤✐➸♠ tr trữớ ủ ỵ ❧↕✐ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝ ❝❤♦ →♥❤ ①↕ f ✳ ú tổ s tt ởt ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ ϕ✲❝♦ ②➳✉ s✉② rë♥❣ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ỡ ỳ sỷ ỵ t õ t r sỹ tỗ t t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ f tr♦♥❣ ❱➼ ❞ö ✷✳✷✳✹✳ ✣à♥❤ ỵ (X, A, d, s) ổ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ f : X → X ❧➔ →♥❤ ①↕ ϕ✲❝♦ ②➳✉ s✉② rë♥❣✳ ❑❤✐ ✤â✱ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ u ∈ X ✈➔ ✈ỵ✐ ♠é✐ x ∈ X ✱ lim f n (x) = u✳ n→∞ ◆❤➟♥ ①➨t ✷✳✷✳✻✳ r ỵ s = t t t ữủ ỵ ϕ(t) = kt ✈ỵ✐ ♠å✐ t ∈ A tr♦♥❣ ✤â k ∈ P ♠➔ ρ(k) < t❤➻ t❛ t❤✉ ữủ ỵ sỷ ỵ t r ữủ sỹ tỗ t ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ f tr♦♥❣ ❱➼ ❞ö ✷✳✷✳✹✳ ❚ø ❝→❝ ❦➳t q✉↔ tr➯♥ t❛ s✉② r❛ r➡♥❣ ✣à♥❤ ỵ ởt rở tỹ sỹ ỵ t ở ổ ởt sè →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ t❤✐➳t ❧➟♣ ♠ët sè ❦➳t q✉↔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳ P❤➛♥ tû (x, y) ∈ X × X ✤÷ì❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤ỉ✐ ❝õ❛ →♥❤ ①↕ T : X × X → X ♥➳✉ T (x, y) = x ✈➔ T (y, x) = y ✳ ❇ê ✤➲ ✷✳✸✳✷✳ ❈❤♦ (X, A, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❤➺ sè s ≥ ✈➔ →♥❤ ①↕ T : X × X → X ✳ ❚❛ ❦➼ ❤✐➺✉ ✶✺ ✶✳ ρ : X × X → R ❝❤♦ ❜ð✐ ρ (x, y), (u, v) = d(x, u) + d(y, v) ✈ỵ✐ ♠å✐ (x, y)✱ (u, v) ∈ X × X ✱ ð ✤➙② X = X × X ✳ ✷✳ GT : X × X → X × X ❝❤♦ ❜ð✐ GT (x, y) = T (x, y), T (y, x) ✈ỵ✐ ♠å✐ (x, y) ∈ X × X ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ❛✳ (X × X, A, ρ, s) ❝ơ♥❣ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❤➺ sè s✳ ❜✳ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ GT ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❇➙② ❣✐í✱ t❛ t❤✐➳t ❧➟♣ ♠ët sè ❦➳t q✉↔ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐ ❝❤♦ ♠ët sè →♥❤ ①↕ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤✳ ✣à♥❤ ỵ (X, A, d, s) ổ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❤➺ sè s ≥ 1✱ P ❧➔ ♥â♥ tr♦♥❣ A✱ ϕ : P → P ❧➔ →♥❤ ①↕ s♦ s→♥❤ ②➳✉ ✈➔ T : X × X → X ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d T (x, y), T (u, v) + d T (y, x), T (v, u) ϕ d(x, u) + d(y, v) ✭✷✳✸✮ ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✳ ❑❤✐ ✤â✱ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤ỉ✐✳ ✁ ✷✳✹ ×♥❣ ởt ợ ữỡ tr t r ú tổ ự ỵ ự t sỹ tỗ t ởt ợ ữỡ tr t C[a, b] ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❤➔♠ ❣✐→ trà t❤ü❝ ❧✐➯♥ tö❝ tr➯♥ [a, b] ⊂ R✳ ❑➼ ❤✐➺✉ A = R2 ✈➔ P = {(x, y) ∈ A : x, y ≥ 0} ✈ỵ✐ ❝❤✉➞♥✱ ♣❤➨♣ ♥❤➙♥ ✈➔ q✉❛♥ ❤➺ t❤ù tü t÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❱➼ ❞ư ✷✳✷✳✹✳ ❚❛ ①→❝ ✤à♥❤ →♥❤ ①↕ d : C[a, b] × C[a, b] → A ❝❤♦ ❜ð✐ d(x, y) = sup |x(t) − y(t)|2 , sup |x(t) − y(t)|2 t∈[a,b] t∈[a,b] ✈ỵ✐ ♠å✐ x, y ∈ C[a, b]✳ ❑❤✐ ✤â✱ C[a, b], A, d, s ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✈ỵ✐ ❤➺ sè s = ỵ C[a, b], A, d, s ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤ ✤÷đ❝ ♥â✐ tr♦♥❣ ❇ê ✤➲ ✷✳✹✳✶✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ s❛✉✳ b x(t) = η(t) + K t, x(r) dr, t ∈ [a, b], ✭✷✳✹✮ a ð ✤➙② x, η ∈ C[a, b] ✈➔ K : [a, b] × R → R✳ ●✐↔ sû r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② t❤ä❛ ♠➣♥✿ ✶✳ ❱ỵ✐ ♠é✐ t ∈ [a, b]✱ ❤➔♠ K t, x(r) ❧➔ ❦❤↔ t➼❝❤ t❤❡♦ r tr [a, b] ỗ t ởt ❤➔♠ ❧✐➯♥ tư❝ ψ : [a, b] × [a, b] → R s❛♦ ❝❤♦ b ψ(t, r) dr ≤ 1, sup t[a,b] a tỗ t ởt s s γ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ t, r ∈ [a, b] ✈➔ ✈ỵ✐ ♠å✐ x, y ∈ C[a, b], t❛ ❝â K t, x(r) − K t, y(r) ≤ ψ(t, r) γ |x(r) − y(r)| ❑❤✐ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✮ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ u ∈ C[a, b] ❱➼ ❞ö s r tỗ t K ψ ✱ γ ✈➔ η t❤ä❛ ♠➣♥ t➜t ❝↔ ❝→❝ tt ỵ C[0, 1] ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ❤➔♠ t❤ü❝ ❧✐➯♥ tư❝ tr➯♥ [0, 1]✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ √ sin t + + ln x(t) = t − 1 r sin t ln + |x(r)| dr, ✭✷✳✺✮ ợ t [0, 1] ỷ ỵ ✷✳✹✳✷ t❛ ❝❤➾ r❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ u ∈ C[0, 1] ❑➳t ❧✉➟♥ ❈❤÷ì♥❣ ✷ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ t❤✉ ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ ❝❤➼♥❤ s ữ r ỵ sỹ tỗ t t ợ ϕ✲❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ✤↕✐ số ỵ ự sỹ tỗ t t ởt ợ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷đ❝ tr➼❝❤ tø ❜➔✐ ❜→♦✿ ❚✳ ❱✳ ❆♥ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✽✮✱ ✏❖♥ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ ❝❧❛ss ♦❢ ♥♦♥❧✐♥❡❛r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✐♥ ❝♦♥❡ b✲♠❡tr✐❝ s♣❛❝❡s ♦✈❡r ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✑ ỷ ữ r ỵ sỹ tỗ t t ở ổ tr ổ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷đ❝ tr➼❝❤ tø ❜➔✐ ❜→♦✿ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✽✮✱ ✏❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡✲ ♦r❡♠s ✐♥ ❝♦♥❡ b✲♠❡tr✐❝ s♣❛❝❡s ♦✈❡r ❇❛♥❛❝❤ ❛❧❣❡❜r❛s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✑ ✭✤❛♥❣ ❣û✐ ✤➠♥❣✮✳ ✶✼ ❈❍×❒◆● ✸ ✣■➎▼ ❇❻❚ ✣❐◆● ❈❍❖ ▼❐❚ ▲❰P ⑩◆❍ ❳❸ ❚❘❖◆● ❑❍➷◆● ●■❆◆ b✲▼➊❚❘■❈ ❱❰■ ●■⑩ ❚❘➚ ❚❘❖◆● C ∗ ✲✣❸■ ❙➮ ❱⑨ Ù◆● ❉Ư◆● ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tt ởt số ỵ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤ỉ✐ ❝❤♦ ♠ët sè ❧ỵ♣ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ❈❤ó♥❣ tỉ✐ ❝ơ♥❣ ù♥❣ ❞ư♥❣ ❝→❝ ❦➳t q✉↔ t➻♠ ✤÷đ❝ ✈➔♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❜➔✐ t♦→♥ sỹ tỗ t ởt ợ ữỡ tr t➼❝❤ ♣❤➙♥✳ ✸✳✶ ❑❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✣➛✉ t✐➯♥✱ t❛ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ s❛✉✿ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✶✳ ❈❤♦ A ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ ∗ : A → A ❧➔ →♥❤ ①↕ ❝❤♦ ❜ð✐ a → a∗ ✱ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ (a + b)∗ = a∗ + b∗ ✈➔ (λa)∗ = λa∗ ✈ỵ✐ ♠å✐ a, b ∈ A ✈➔ ♠å✐ sè t❤ü❝ λ✳ ❑❤✐ ✤â ✶✳ A ✤÷đ❝ ❣å✐ ❧➔ ∗✲✤↕✐ sè ♥➳✉ A ❧➔ ✤↕✐ sè ❇❛♥❛❝❤ ✈➔ (ab)∗ = b∗ a∗ ✈➔ (a∗ )∗ = a ợ a, b A A ữủ ❧➔ ∗✲✤↕✐ sè ❇❛♥❛❝❤ ♥➳✉ A ❧➔ ∗✲✤↕✐ sè ✤➛② ✤õ s❛♦ ❝❤♦ ab ≤ a b ✈➔ a = a ợ a, b A A ữủ ❣å✐ ❧➔ C ∗ ✲✤↕✐ sè ♥➳✉ A ❧➔ ∗✲✤↕✐ sè ❇❛♥❛❝❤ ✈➔ a∗ a = a ✈ỵ✐ ♠å✐ a ∈ A✳ ✹✳ ❑➼ ❤✐➺✉ 1A ❧➔ ♣❤➛♥ tû ✤ì♥ ✈à ✈➔ 0A ❧➔ ♣❤➛♥ tû ❦❤ỉ♥❣ ❝õ❛ A✳ P❤➛♥ tû a ∈ A ✤÷đ❝ ❣å✐ ❧➔ ❞÷ì♥❣✱ ✈✐➳t ❧➔ 0A a ❤❛② a 0A ✱ ♥➳✉ a∗ = a ✈➔ σ(a) ⊂ R+ ✱ ð ✤➙② σ(a) = {λ ∈ R : λ1A − a ❧➔ ❦❤æ♥❣ ❦❤↔ ♥❣❤à❝❤⑥ ❧➔ ♣❤ê ❝õ❛ a✳ ✺✳ ❱ỵ✐ a, b ∈ A b ữủ ợ ỡ a a ✤÷đ❝ ❣å✐ ❧➔ ♥❤ä ❤ì♥ b✱ ✈✐➳t ❧➔ a ❤❛② b a✱ ♥➳✉ b − a 0A ✳ ❚❛ ❝ô♥❣ ✈✐➳t a ≺ b ♥➳✉ a b ✈➔ a = b✳ b ✻✳ ❚➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ♣❤➛♥ tû ❞÷ì♥❣ ❝õ❛ A ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ A+ ✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✸✳ ❈❤♦ X ❧➔ t➟♣ ❦❤→❝ ré♥❣✱ A ❧➔ C ∗ ✲✤↕✐ sè✱ s ∈ A ♠➔ s d : X × X → A ❧➔ →♥❤ ①↕ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y, z ∈ X ✱ t❛ ❝â ✶✳ 0A d(x, y) ✈➔ d(x, y) = 0A ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x = y ✳ 1A ✈➔ ✶✽ ✷✳ d(x, y) = d(y, x)✳ ✸✳ d(x, z) s[d(x, y) + d(y, z)❪✳ ❑❤✐ ✤â✱ d ✤÷đ❝ ❣å✐ ❧➔ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè tr➯♥ X ✈➔ (X, A, d, s) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈ỵ✐ ❤➺ sè s✳ ❇ê ✤➲ ✸✳✶✳✻✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ⑩♥❤ ①↕ f : X X ữủ tỗ t ởt ❤➔♠ s♦ s→♥❤ ϕ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ X, t❛ ❝â d f (x), f (y) ≤ ϕ d(x, y) ✸✳✷ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ s✉② rë♥❣ ✈➔ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ ❝❤✉➞♥ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ tæ✐ ①➙② ❞ü♥❣ ❦❤→✐ ♥✐➺♠ →♥❤ ①↕ ϕ✲❝♦ s✉② rë♥❣ ✈➔ →♥❤ ①↕ ϕ✲❝♦ ❝❤✉➞♥ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ✤à♥❤ ❧➼ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ❧ỵ♣ →♥❤ ①↕ ♥➔②✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳✶✳ ❈❤♦ (X, A, d, s) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ⑩♥❤ ①↕ ϕ : A+ → A+ ✤÷đ❝ ❣å✐ ❧➔ s♦ s→♥❤ s✉② rë♥❣ ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ ❱ỵ✐ ♠å✐ t1 , t2 ∈ A+ ♠➔ t1 t2 ❦➨♦ t❤❡♦ ϕ(t1 ) ϕ(t2 )✳ ✷✳ lim ϕn (t) = 0A ✈ỵ✐ ♠å✐ t ∈ A+ ✳ n→∞ ✸✳ {ϕ(un )} ❧➔ ❞➣② ❤ë✐ tö ✈➲ 0A tr♦♥❣ A+ ✱ ♥➳✉ {un } ❧➔ ❞➣② ❤ë✐ tö ✈➲ 0A tr♦♥❣ A+ ✳ ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳✷✳ ❈❤♦ (X, A, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ⑩♥❤ ①↕ f : X → X ✤÷đ❝ ❣å✐ ❧➔ ϕ✲❝♦ s rở tỗ t s s s rë♥❣ ϕ : A+ → A+ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ X, t❛ ❝â d f (x), f (y) ϕ d(x, y) ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳✸✳ ❈❤♦ (X, A, d, s) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ⑩♥❤ ①↕ ϕ : A+ → A+ ✤÷đ❝ ❣å✐ ❧➔ s♦ s→♥❤ ❝❤✉➞♥ s✉② rë♥❣ ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ ❱ỵ✐ ♠å✐ t1 , t2 ∈ A+ ♠➔ t1 ≤ t2 ❦➨♦ t❤❡♦ ϕ(t1 ) ≤ ϕ(t2 ) ✳ ✷✳ lim ϕn (t) = 0A ✈ỵ✐ ♠å✐ t ∈ A+ ✳ n→∞ ✶✾ ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳✹✳ ❈❤♦ (X, A, d, s) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ⑩♥❤ ①↕ f : X → X ✤÷đ❝ ❣å✐ s rở tỗ t s♦ s→♥❤ ❝❤✉➞♥ s✉② rë♥❣ ϕ : A+ → A+ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y ∈ X, t❛ ❝â d f (x), f (y) ≤ ϕ d(x, y) ❇➙② ❣✐í t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ♠ët ❦➳t q✉↔ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ ❝❤✉➞♥ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ tr tr C số ỵ (X, A, d, s) ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈➔ f : X → X ❧➔ →♥❤ ①↕ ϕ✲❝♦ ❝❤✉➞♥ s✉② rë♥❣✳ ❑❤✐ ✤â✱ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ u ∈ X ✈➔ ♠é✐ x ∈ X ✱ lim f n (x) = u✳ n→∞ ◆➠♠ ✷✵✶✺✱ ▼❛ ✈➔ ❏✐❛♥❣ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❦➳t q✉↔ s❛✉ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C số ỵ (X, A, d, s) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✱ a ∈ A s❛♦ ❝❤♦ a < ✈➔ f : X → X ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ a∗ d(x, y)a, d f (x), f (y) ✈ỵ✐ ♠å✐ x, y ∈ X ✭✸✳✶✮ ❑❤✐ ✤â✱ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ u ∈ X ✈➔ ♠é✐ x ∈ X, lim f n (x) = u✳ n→∞ ❱➼ ❞ö s❛✉ ✤➙② t❛ ❞➵ ❞➔♥❣ ♥❤➟♥ t❤➜② →♥❤ ①↕ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❧➔ ✤✐➸♠ tr♦♥❣ X ✱ ữ tr trữớ ủ ỵ ổ →♣ ❞ư♥❣ ✤÷đ❝ ❝❤♦ →♥❤ ①↕ f ✳ ❱➼ ❞ư ✸✳✷✳✶✵✳ ❈❤♦ A = C ❧➔ t➟♣ ❤ñ♣ ❝→❝ sè ♣❤ù❝✱ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ✈➔ ♣❤➨♣ ❝ë♥❣ sè ♣❤ù❝ t❤ỉ♥❣ tữớ a = |a|, a = a ợ a ∈ C✳ ❚❛ ①→❝ ✤à♥❤ ✶✳ X = R+ ✈➔ d : X × X → A+ ❝❤♦ ❜ð✐ d(x, y) = |x − y|2 ✈ỵ✐ ♠å✐ x, y ∈ X ✷✳ f : X → X ❝❤♦ ❜ð✐ f (x) = x ✈ỵ✐ ♠å✐ x ∈ X x+1 ✸✳ ϕ : A+ → A+ ❝❤♦ ❜ð✐ ϕ(z) = |z| ✈ỵ✐ ♠å✐ z ∈ A+ |z| + ❑❤✐ ✤â (X, A, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈ỵ✐ ❤➺ sè s = 2✳ ❉➵ ♥❤➟♥ t❤➜② →♥❤ ①↕ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❧➔ ✤✐➸♠ ổ tỗ t a A ợ a < s❛♦ ❝❤♦ ✤✐➲✉ ❦✐➺♥ ✭✸✳✶✮ tr♦♥❣ ✣à♥❤ ỵ tọ ú tổ ự ởt ỵ t ợ ①↕ ϕ✲❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ❍ì♥ ♥ú❛✱ sỷ ỵ t õ t r sỹ tỗ t t f tr ỵ (X, A, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈➔ f : X → X ❧➔ →♥❤ ①↕ ϕ✲❝♦ s✉② rë♥❣✳ ❑❤✐ ✤â✱ f ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ u ∈ X ✈➔ ✈ỵ✐ ♠é✐ x ∈ X ✱ lim f n (x) = u✳ n→∞ ✷✵ ◆❤➟♥ ①➨t ✸✳✷✳✶✷✳ r ỵ : A+ → A+ ❝❤♦ ❜ð✐ ❝æ♥❣ t❤ù❝ ϕ(t) = a∗ ta ✈ỵ✐ ♠å✐ t 0A ✱ tr♦♥❣ ✤â a ∈ A trữợ tọ a < t t t ữủ ỵ sỷ ỵ t r ữủ sỹ tỗ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ f tr♦♥❣ ❱➼ ❞ö ✸✳✷✳✶✵✳ ✸✳✸ ✣✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐ ❝❤♦ ♠ët sè ❧ỵ♣ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ tæ✐ t❤✐➳t ❧➟♣ ♠ët sè ❦➳t q✉↔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤ỉ✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C số ỵ s t q✉↔ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ số q ự ỵ ✸✳✸✳✹✳ ❈❤♦ (X, A, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✱ a ∈ A s❛♦ ❝❤♦ a < ✈➔ F : X × X → X ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d(F (x, y), F (u, v)) a∗ d(x, u)a + a∗ d(y, v)a ✭✸✳✷✮ ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✳ ❑❤✐ ✤â✱ F ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤ỉ✐✳ ❱➼ ❞ư s❛✉ ✤➙② ❞➵ ❞➔♥❣ ♥❤➟♥ t❤➜② →♥❤ ①↕ F ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐ ❧➔ ✤✐➸♠ (0, 0) tr♦♥❣ X ỵ ổ ✤÷đ❝ ❝❤♦ →♥❤ ①↕ F tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✳ ❱➼ ❞ö ✸✳✸✳✺✳ ❈❤♦ A = A = α0 β0 : α, β ∈ R ✈ỵ✐ ❝→❝ ♣❤➨♣ t♦→♥ ❝ë♥❣✱ ♣❤➨♣ ổ ữợ tr tổ tữớ ❧✐➯♥ ❤ñ♣ ❝❤♦ ❜ð✐ A∗ = AT ✭ð ✤➙② AT ❧➔ ♠❛ tr➟♥ ❝❤✉②➸♥ ✈à ❝õ❛ ♠❛ tr➟♥ A✮ ✈➔ ❝❤✉➞♥ A = max{|α|, |β|}✳ ❚❛ ❧➜② X = R ✈➔ ①→❝ ✤à♥❤ ✶✳ d : X × X → A ❝❤♦ ❜ð✐ d(u, v) = |u − v| 0 |u − v| = |u − v| = |u − v|1A ✈ỵ✐ ♠å✐ u, v ∈ X ✷✳ F : X × X → X ❝❤♦ ❜ð✐ F (x, y) = x − 2y ✈ỵ✐ ♠å✐ x, y ∈ X 1/2 0 1/2 ∈ A t❤➻ F ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✸✳✷✮✳ õ ỵ ổ ữủ F tr trữớ ủ ợ a = ❇➙② ❣✐í✱ ❝❤ó♥❣ tỉ✐ t❤✐➳t ❧➟♣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧➼ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ❍ì♥ ♥ú❛✱ sû ❞ư♥❣ ❦➳t q✉↔ ♥➔② ❝❤ó♥❣ tỉ✐ ❝❤➾ r❛ ✤÷đ❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤ỉ✐ ❝õ❛ F tr ỵ (X, A, d, s) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈ỵ✐ ❤➺ sè s 1A ✱ a ∈ A s❛♦ ❝❤♦ a < ✈➔ T : X × X → X ❧➔ →♥❤ ✷✶ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d(T (x, y), T (u, v)) + d(T (y, x), T (v, u)) 2[a∗ d(x, u)a + a∗ d(y, v)a] ✭✸✳✸✮ ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✳ ❑❤✐ ✤â✱ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐✳ r ỵ s = 1A t t❤✉ ✤÷đ❝ ❦➳t q✉↔ s❛✉✳ ❍➺ q✉↔ ✸✳✸✳✼✳ ❈❤♦ (X, A, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✱ a ∈ A s❛♦ ❝❤♦ a < ✈➔ T : X × X → X ❧➔ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d(T (x, y), T (u, v)) + d(T (y, x), T (v, u)) 2[a∗ d(x, u)a + a∗ d(y, v)a] ✭✸✳✹✮ ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✳ ❑❤✐ ✤â✱ T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐✳ ◆❤➟♥ ①➨t ✸✳✸✳✽✳ ❚ø ❍➺ q✉↔ ✸✳✸✳✼ t❛ ❝â ❦➳t ❧✉➟♥ ❝õ❛ ❍➺ q✉↔ ✸✳✸✳✹✳ ❍ì♥ ♥ú❛✱ →♥❤ ①↕ F t❤ä❛ ♠➣♥ t➜t ❝↔ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ❍➺ q✉↔ ✸✳✸✳✼ ✈➔ F ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐ ❧➔ (0, 0)✳ ❚ø ❝→❝ ❦➳t q✉↔ tr➯♥ t❛ t❤➜② r➡♥❣ ❍➺ q✉↔ ✸✳✸✳✼ ❧➔ ♠ët ♠ð rë♥❣ t❤ü❝ sü ❝õ❛ ❍➺ q✉↔ ✸✳✸✳✹✳ ✁ ✸✳✹ ×♥❣ ởt ợ ữỡ tr t r t ự ỵ ự sỹ tỗ t t ởt ợ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥✳ ❈❤♦ E ❧➔ t➟♣ ✤â♥❣ ❦❤→❝ ré♥❣ ❜à ❝❤➦♥ ✤♦ ✤÷đ❝ ▲ì❜❡ tr♦♥❣ Rn ✈➔ µ ❧➔ ✤ë ✤♦ ▲ì❜❡ tr➯♥ E ✳ ❑➼ ❤✐➺✉ |x(t)|2 dµ < +∞} L2 (E) = {x : E → R| E ❑❤✐ ✤â✱ t➟♣ ❤ñ♣ L2 (E) ũ ợ t ổ ữợ ữủ ❜ð✐ x, y = x(t)y(t)dµ E ✈➔ x = x, x 2 |x(t)| dµ = E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ❑➼ ❤✐➺✉ B L2 (E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ t➜t ❝↔ ❝→❝ t♦→♥ tû t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tr➯♥ L2 (E) ✈➔ C(E) ❧➔ ❦❤æ♥❣ ❣✐❛♥ t➜t ❝↔ ❝→❝ ❤➔♠ t❤ü❝ ❧✐➯♥ tư❝ tr➯♥ E ✈ỵ✐ x = sup |x(t)| x C(E) rữợ t t❛ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ B L2 (E) ✳ t∈E ❇ê ✤➲ ✸✳✹✳✶✳ ❈❤♦ a ∈ C(E)✱ f, g ∈ B L2 (E) ✈➔ u, v ∈ L2 (E)✳ ❚❛ ①→❝ ✤à♥❤ ✶✳ P❤➨♣ ♥❤➙♥ ✈➔ ❝❤✉➞♥ tr➯♥ B L2 (E) ❝❤♦ ❜ð✐ (f g)(v) = f g(v) ✈ỵ✐ ♠å✐ v ∈ L2 (E) ✈➔ f = sup f (v) ✳ v =1 ✷✳ ❚♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ f ❧➔ t♦→♥ tû f ∗ : B L2 (E) → B L2 (E) ❝❤♦ ❜ð✐ f u, v = u, f ∗ v ✈ỵ✐ ♠å✐ u, v ∈ L2 (E)✱ ð ✤➙② , ❧➔ t➼❝❤ ✈æ ữợ tr L2 (E) a : L2 (E) → L2 (E) ❧➔ t♦→♥ tû t➼❝❤ ❝❤♦ ❜ð✐ πa (v) = av ✈ỵ✐ ♠å✐ v ∈ L2 (E)✳ ✷✷ ✹✳ ⑩♥❤ ①↕ d : C(E) × C(E) → B L2 (E) ❝❤♦ ❜ð✐ d(x, y) = π|x−y|p ✈ỵ✐ p > ✈➔ ♠å✐ x, y ∈ C(E)✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ❛✳ B L2 (E) ❧➔ ♠ët C ∗ ✲✤↕✐ sè ✈ỵ✐ ♣❤➛♥ tû ✤ì♥ ✈à ❧➔ 1B(L2 (E)) ①→❝ ✤à♥❤ ❜ỵ✐ 1B(L2 (E)) (u) = ✈ỵ✐ ♠å✐ u ∈ L2 (E)✳ ❜✳ B L2 (E) + = g ∈ B L2 (E) : gv, v ≥ ✈ỵ✐ ♠å✐ v ∈ L2 (E) ✳ ❝✳ ❱ỵ✐ ♠å✐ f, g ∈ B L2 (E) ✱ f g ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ (f −g)v, v ≥ ✈ỵ✐ ♠å✐ v ∈ L2 (E)✱ ð ✤➙② “ ” ❧➔ q✉❛♥ ❤➺ t❤ù tü s✐♥❤ ❜ð✐ ♥â♥ B L2 (E) + ✳ ❞✳ πa ∈ B(L2 (E))✱ πa = a ✈➔ πa = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a = ∈ C(E)✳ ❡✳ C(E), B L2 (E) , d, s ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈ỵ✐ s = 2p 1B(L2 (E)) ỵ C(E), B L2 (E) , d, s ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè tr♦♥❣ ❇ê ✤➲ ✸✳✹✳✶✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ s❛✉✳ ✭✸✳✺✮ K t, x(r) dr, t ∈ E x(t) = η(t) + E ð ✤➙② x, η ∈ C(E)✳ ●✐↔ sû r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✶✳ K : E × R → R s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ t ∈ E ❤➔♠ K t, x(r) ❧➔ ❦❤↔ t➼❝❤ t❤❡♦ ❜✐➳♥ r tr➯♥ E ✈➔ K t, x(r) ❧➔ ❜à ❝❤➦♥ ✤➲✉ tr➯♥ E × E ỗ t tử : E × E → R ✈ỵ✐ sup t∈E ✈ỵ✐ ♠å✐ t, r ∈ E ✱ ♠å✐ x, y ∈ C(E)✱ t❛ ❝â K t, x(r) − K t, y(r) ψ(t, r) dr ≤ s❛♦ ❝❤♦ ✈ỵ✐ p > ✈➔ E ≤ ψ(t, r) x(r) − y(r) p p + x(r) − y(r) p ❑❤✐ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✸✳✺✮ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ u ∈ C(E)✳ ❱➼ ❞ö s❛✉ ự tọ r tỗ t K ψ ✈➔ η t❤ä❛ ♠➣♥ t➜t ❝↔ ❝→❝ ❣✐↔ t❤✐➳t ỵ t ữỡ tr t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ x(t) = t + − ln √ sin t + e r sin t x(r) dr + x(r) [0,1] ✈ỵ✐ t ∈ [0, 1]✳ ✣➦t η(t) = t + − ln √ sin t, ψ(t, r) = r sin t ✈ỵ✐ t, r ∈ [0, 1], e ✭✸✳✻✮ ✷✸ K t, x(r) = r sin t x(r) + x(r) ✈ỵ✐ x ∈ C[0, 1] ✈➔ t, r ∈ [0, 1] ❑❤✐ ✤â✱ K ✱ ψ ✈➔ tọ tt ỵ ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✭✸✳✻✮ ❝â ❞✉② ♥❤➜t ♠ët ♥❣❤✐➺♠ u ∈ C[0, 1] ❑➳t ❧✉➟♥ ❈❤÷ì♥❣ ✸ ❚r♦♥❣ ❈❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ t❤✉ ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ ❝❤➼♥❤ s ữ r ỵ ỵ sỹ tỗ t t ợ →♥❤ ①↕ ϕ✲❝♦ s✉② rë♥❣ ✈➔ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ ❝❤✉➞♥ s✉② rë♥❣ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ tr tr C số ỗ tớ ự t q ự sỹ tỗ t ởt ợ ữỡ tr t ❦➳t q✉↔ ♥➔② ✤÷đ❝ tr➼❝❤ tø ❜➔✐ ❜→♦✿ ❚✳ ❱✳ ❆♥ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✽✮✱ ✏●❡♥❡r✲ ❛❧✐③❛t✐♦♥s ♦❢ ϕ✲❝♦♥tr❛❝t✐♦♥s ♦♥ C ∗ ✲❛❧❣❡❜r❛✲✈❛❧✉❡❞ b✲ ♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✑ ❏✳ t t ữ r ỵ sỹ tỗ t t ở ✤æ✐ ❝❤♦ ♠ët sè →♥❤ ①↕ tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷đ❝ tr➼❝❤ tø ❜➔✐ ❜→♦✿ ❙✳ ❘❛❞❡♥♦✈➼❝✱ P✳ ❱❡tr♦✱ ❆✳ ◆❛st❛s✐ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✼✮✱ ✏❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ C ∗ ✲❛❧❣❡❜r❛✲✈❛❧✉❡❞ b✲♠❡tr✐❝ s♣❛❝❡s✑✱ ❙❝✐❡♥t✐❢✐❝ P✉❜❧✐❝❛t✐♦♥s ♦❢ ❚❤❡ ❙t❛t❡ ❯♥✐✈❡rs✐t② ♦❢ ◆♦✈✐ P❛③❛r✱ ❙❡r✳ ❆✿ ❆♣♣❧✳ ▼❛t❤✳ ■♥❢♦r♠✳ ❆♥❞ ▼❡❝❤✳✱ ✾ ✭✶✮✱ ✽✶✲✾✵✳ ✷✹ ❑➌❚ ▲❯❾◆ ❱⑨ ❑■➌◆ ◆●❍➚ ✶ ❑➳t ❧✉➟♥ ▲✉➟♥ →♥ ♥❣❤✐➯♥ ự sỹ tỗ t t ởt sè ❧ỵ♣ ❝→❝ →♥❤ ①↕ tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ s➢♣ t❤ù tü ❜ë ♣❤➟♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè ✈➔ ù♥❣ ❞ö♥❣ ❝→❝ ❦➳t q✉↔ t ữủ sỹ tỗ t ởt ợ ữỡ tr t t q ữ r ỵ sỹ tỗ t trũ ợ →♥❤ ①↕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ T ✲❝♦ s✉② rë♥❣ ✈➔ →♥❤ ①↕ (ψ, L)✲T ✲❤➛✉ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ữ r ỵ sỹ tỗ t tỗ t t t ❜➜t ✤ë♥❣ ❜ë ✤ỉ✐ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ ②➳✉ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ ✤➛② ✤õ tr số ữ r ỵ sỹ tỗ t tỗ t t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ s✉② rë♥❣✱ ϕ✲❝♦ ❝❤✉➞♥ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ✣÷❛ r❛ ❝→❝ ỵ sỹ tỗ t tỗ t ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❜ë ✤æ✐ ❝❤♦ ♠ët sè →♥❤ ①↕ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ✹✮ Ù♥❣ ❞ö♥❣ ❝→❝ ❦➳t q✉↔ t ữủ ự sỹ tỗ t ởt ợ ữỡ tr t ỹ ♠ët sè ❝→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ❝❤♦ ❝→❝ ❦➳t q ỗ tớ ự tọ t q t ữủ ❧➔ sü ♠ð rë♥❣ ❝õ❛ ♥❤ú♥❣ ❦➳t q✉↔ ✤➣ ❝â✳ ✷ ❑✐➳♥ ♥❣❤à ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐ ❝❤ó♥❣ tỉ✐ s➩ t✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ s❛✉✿ ◆❣❤✐➯♥ ❝ù✉ sỹ tỗ t t ởt số ợ ❝→❝ →♥❤ ①↕ ✤❛ trà tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈➔ ự t sỹ tỗ t ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❉❆◆❍ ▼Ö❈ ❈➷◆● ❚❘➐◆❍ ❈Õ❆ ◆●❍■➊◆ ❈Ù❯ ❙■◆❍ ▲■➊◆ ◗❯❆◆ ✣➌◆ ▲❯❾◆ ⑩◆ ✶✳ ❙✳ ❘❛❞❡♥♦✈➼❝✱ ❚✳ ❱✳ ❆♥ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✼✮✱ ✏❙♦♠❡ ❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t r❡s✉❧ts ❢♦r T ✲❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s ♦♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ b✲♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛✲ t✐♦♥s t♦ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✑✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✿ ▼♦❞❡❧❧✐♥❣ ❛♥❞ ❈♦♥tr♦❧✳✱ ✷✷ ✭✹✮✱ ✺✹✺✲✺✻✺✳ ✷✳ ❙✳ ❘❛❞❡♥♦✈➼❝✱ P✳ ❱❡tr♦✱ ❆✳ ◆❛st❛s✐ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✼✮✱ ✏❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ C ∗ ✲❛❧❣❡❜r❛✲✈❛❧✉❡❞ b✲♠❡tr✐❝ s♣❛❝❡s✑✱ ❙❝✐❡♥t✐❢✐❝ ♣✉❜❧✐❝❛t✐♦♥s ♦❢ t❤❡ st❛t❡ ✉♥✐✈❡rs✐t② ♦❢ ◆♦✈✐ P❛③❛r✱ ❙❡r✳ ❆✿ ❆♣♣❧✳ ▼❛t❤✳ ■♥❢♦r♠✳ ❆♥❞ ▼❡❝❤✳✱ ✾ ✭✶✮✱ ✽✶✲✾✵✳ ✸✳ ❚✳ ❱✳ ❆♥ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✱ ✏●❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ϕ✲❝♦♥tr❛❝t✐♦♥s ♦♥ C ∗ ✲❛❧❣❡❜r❛✲✈❛❧✉❡❞ b✲ ♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✑ ❏✳ ❆❞✈✳ ▼❛t❤✳ ❙t✉❞✳✱ ✶✶ ✭✸✮✱ ✺✺✽✲✺✼✺✳ ✹✳ ❙✳ ❘❛❞❡♥♦✈➼❝✱ ◆✳ ❉❡❞♦✈➼❝✱ ❚✳ ❱✳ ❆♥ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✼✮✱ ✏❙♦♠❡ ❝♦✐♥❝✐❞❡♥❝❡ t❤❡♦r❡♠ ❢♦r ❛❧♠♦st ❣❡♥❡r❛❧✐③❡❞ (ϕ, L)✲T ✲❝♦♥tr❛❝t✐♦♥s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ b✲♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✑ ✭✤❛♥❣ ❣û✐ ✤➠♥❣✮✳ ✺✳ ❚✳ ❱✳ ❆♥ ❛♥❞ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✽✮✱ ✏❖♥ t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ ❝❧❛ss ♦❢ ♥♦♥❧✐♥❡❛r ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✐♥ ❝♦♥❡ b✲♠❡tr✐❝ s♣❛❝❡s ♦✈❡r ❇❛♥❛❝❤ ❛❧❣❡❜r❛s✑ ✭✤❛♥❣ ❣û✐ ✤➠♥❣✮✳ ✻✳ ▲✳ ❚✳ ◗✉❛♥✳ ✭✷✵✶✽✮✱ ✏❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ❝♦♥❡ b✲♠❡tr✐❝ s♣❛❝❡s ♦✈❡r ❇❛♥❛❝❤ ❛❧❣❡❜r❛s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✑ ✭✤❛♥❣ ❣û✐ ✤➠♥❣✮✳ ❈⑩❈ ❑➌❚ ◗❯❷ ❈Õ❆ ▲❯❾◆ ìẹ r ●✐↔✐ t➼❝❤ t❤✉ë❝ ❱✐➺♥ s÷ ♣❤↕♠ ❚ü ♥❤✐➯♥ ❚r÷í♥❣ ✣↕✐ rữớ ❤å❝ ❱✐♥❤ ✭✷✵✶✹ ✲ ✷✵✶✽✮✳ • ✣↕✐ ❤ë✐ ❚♦→♥ ❤å❝ t♦➔♥ q✉è❝ t↕✐ ◆❤❛ ❚r❛♥❣ ♥➠♠ ✷✵✶✽✳ ... ♣❤➟♥✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C số ởt số ợ ữỡ tr t ♣❤➙♥✳ ✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ▲✉➟♥ →♥ ♥❣❤✐➯♥ ❝ù✉ ỵ trũ ỵ t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣... ♥â♥ tr➯♥ ❝→❝ ✤↕✐ sè ❇❛♥❛❝❤✱ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ợ tr tr C số ỗ tớ ù♥❣ ❞ư♥❣ ❝→❝ ❦➳t q✉↔ t❤✉ ✤÷đ❝ ✈➔♦ ✈✐➺❝ ♥❣❤✐➯♥ ự sỹ tỗ t ởt số ợ ữỡ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ ✼✳ ❚ê♥❣ q✉❛♥ ✈➔ ❝➜✉ tró❝ ❧✉➟♥ →♥ ◆ë✐... ❝❤ó♥❣ tổ tt ởt số ỵ t ❝❤♦ ❧ỵ♣ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ s✉② rë♥❣ ✈➔ →♥❤ ①↕ ϕ✲❝♦ ❝❤✉➞♥ s✉② rë♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ ❣✐→ trà tr♦♥❣ C ∗ ✲✤↕✐ sè✳ ▼ư❝ ✸✳✸✱ ❝❤ó♥❣ tổ tt ự ởt số ỵ ✤✐➸♠ ❜➜t ✤ë♥❣