❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ▲➊ ❚❍❆◆❍ ❍➪❆ ❱➋ ❙Ü ❚➬◆ ❚❸■ ❚❾P ❇❻❚ ❇■➌◆ ❈Õ❆ ❍➏ ❍⑨▼ ▲➄P ●➬▼ ❈⑩❈ ⑩◆❍ ❳❸ ✣❒◆ ❚❘➚✱ ✣❆ ❚❘➚ ❑■➎❯ F ✲❈❖ ❚❘➊◆ ❑❍➷◆● ●■❆◆ b ✲ ▼➊❚❘■❈ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈ ◆●❍➏ ❆◆✱ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❱■◆❍ ▲➊ ❚❍❆◆❍ ❍➪❆ ❱➋ ❙Ü ❚➬◆ ❚❸■ ❚❾P ❇❻❚ ❇■➌◆ ❈Õ❆ ❍➏ ❍⑨▼ ▲➄P ●➬▼ ❈⑩❈ ⑩◆❍ ❳❸ ✣❒◆ ❚❘➚✱ ✣❆ ❚❘➚ ❑■➎❯ F ✲❈❖ ❚❘➊◆ ❑❍➷◆● ●■❆◆ b ✲▼➊❚❘■❈ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙ß ❚❖⑩◆ ❍➴❈ ❈❍❯❨➊◆ ◆●⑨◆❍✿ ❚❖⑩◆ ●■❷■ ữớ ữợ ❦❤♦❛ ❤å❝ ❚❙✳ ❱Ô ❚❍➚ ❍➬◆● ❚❍❆◆❍ ◆●❍➏ ❆◆✱ ✷✵✶✽ ▲❮■ ❈❷▼ ❒◆ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ t❤ü❝ ❤✐➺♥ t↕✐ rữớ ữợ sỹ ữợ ổ ụ ỗ ổ tọ sỹ ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ♥❤➜t ✤➳♥ ổ ụ ỗ ữớ t t ữợ ú ù tổ tr sốt q tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ◆❤➙♥ ❞à♣ ♥➔②✱ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ P❤á♥❣ ✤➔♦ t↕♦ ❙❛✉ ✣↕✐ ❤å❝ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ ❝→❝ t❤➛②✱ ❝ỉ ❣✐→♦ tr♦♥❣ tê ●✐↔✐ ❚➼❝❤ ð ♥❣➔♥❤ ❚♦→♥✱ ❱✐➺♥ ❙÷ ♣❤↕♠ ❚ü ♥❤✐➯♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤✱ ✤➣ ❣✐↔♥❣ ❞↕② ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ❝ơ♥❣ ♥❤÷ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣✱ s♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ t❤✐➳✉ sât✳ ú tổ rt ữủ ỵ õ ỵ ❝õ❛ ❚❤➛②✱ ❈æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ✦ ◆❣❤➺ ❆♥✱ t❤→♥❣ ✵✼ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ▲➯ ❚❤❛♥❤ ❍á❛ ✶ ▼Ö❈ ▲Ö❈ ❚r❛♥❣ ▲❮■ ❈❷▼ ❒◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ▼Ð ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷✳ ▼➯tr✐❝ ❍❛✉s❞♦r❢❢ tr➯♥ ❧ỵ♣ ❝→❝ t➟♣ ❝♦♥ ❝♦♠♣❛❝t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❈❍×❒◆● ✷✳ ❱➋ ❙Ü ❚➬◆ ❚❸■ ❚❾P ❇❻❚ ❇■➌◆ ❈Õ❆ ❍➏ ❍⑨▼ ▲➄P ✣❒◆ ❚❘➚ ❱⑨ ✣❆ ❚❘➚ ❑■➎❯ F ✲❈❖ ❚❘➊◆ ❑❍➷◆● ●■❆◆ b✲▼➊❚❘■❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ỹ tỗ t t t t tỷ ❢r❛❝t❛❧ ✤ì♥ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t t t t♦→♥ tû ❢r❛❝t❛❧ ✤❛ trà ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷ ▼Ð ✣❺❯ ■✳ ▲Þ ❉❖ ❈❍➴◆ ✣➋ ❚⑨■ ❚➟♣ ❢r❛❝t❛❧ ❣✐ú ♠ët ✈❛✐ trá q✉❛♥ trồ tr s ỡ ữủ tỷ ỗ t t ỹ ỵ t ❤➔♠ ❧➦♣ ❝â r➜t ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❝õ❛ ❦❤♦❛ ❤å❝ ù♥❣ ❞ö♥❣✳ ❈→❝ ❧♦↕✐ ♠➯tr✐❝ ✤➣ ữủ rở t ữợ ổ b✲♠➯tr✐❝ ♥➡♠ tr♦♥❣ ♥❤ú♥❣ ♠ð rë♥❣ ✤â✳ ❑❤→✐ ♥✐➺♠ ❦❤æ♥❣ btr ữủ ợ t t r ✭❬✻❪✮ ✈➔♦ ♥➠♠ ✶✾✾✸✳ ❚ø ✤â✱ ❝â r➜t ♥❤✐➲✉ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❢r❛❝t❛❧ ✈➔ ❤➺ ❤➔♠ ❧➦♣ ✤ì♥ trà✱ ✤❛ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ◆➠♠ ✶✾✻✾✱ ◆❛❞❧❡r ữớ t t ủ ỵ tữ →♥❤ ①↕ ✤❛ trà ✈➔ →♥❤ ①↕ ❝♦ ✈➔ ♥❣❤✐➯♥ ự sỹ tỗ t t tr s õ ỵ tt t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ ✤❛ trà ❝❤♦ t❛ ♥❤✐➲✉ ❝ỉ♥❣ ❝ư ✤➸ ❣✐↔✐ q✉②➳t ♥❤✐➲✉ ✈➜♥ ✤➲ t❤✉➛♥ tó② t♦→♥ ❤å❝ ❝ơ♥❣ ♥❤÷ ù♥❣ ❞ư♥❣✳ ▼➦t ❦❤→❝✱ ♠ët tr♦♥❣ ỳ ữợ rở ỹ t rt ❧➔ t❤❛② ✤ê✐ ❝→❝ ❧♦↕✐ →♥❤ ①↕ ❝♦✳ ✣➣ ❝â ♠ët sè ❦➳t q✉↔ t❤✉ ✤÷đ❝ ✈➲ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ sỹ tỗ t t rt ỡ tr tr s ỗ F ✲❝♦✱ F ✲❝♦ tê♥❣ q✉→t✱ ❜✲s♦ s→♥❤✱ ϕ✲❝♦✱ G✲❝♦✱ ▼❡✐r✲❑❡❡❧❡r✱✳✳✳ ❜ð✐ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ❈✳ ❈❤✐❢✉✱ ❆✳ P❡tr✉s❡❧✱ ❙✳ ❈③❡r✇✐❦✱✳✳✳ tr♦♥❣ ✤â →♥❤ ①↕ F ✲❝♦ ❧➔ ❧♦↕✐ →♥❤ ①↕ ❝♦ tê♥❣ q✉→t ❝❤♦ ❝→❝ ❧♦↕✐ ❝♦ ♣❤ê ❜✐➳♥ ♥❤÷ ❝♦ ❑❛♥♥❛♥✱ ❝♦ ❘❡✐❝❤✳✳✳ ❱➻ t❤➳✱ ✤➸ t➟♣ ❞✉②➺t ✈ỵ✐ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈➔ t➻♠ ❤✐➸✉ ✈➲ ✈➜♥ ✤➲ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲ t➔✐ ♥❣❤✐➯♥ ❝ù✉ sỹ tỗ t t t ỗ ①↕ ✤ì♥ trà✱ ✤❛ trà ❦✐➸✉ F ✲❝♦ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✧✳ ■■✳ ▼Ư❈ ✣➑❈❍ ◆●❍■➊◆ ❈Ù❯ ✸ ▼ư❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ❤✐➸✉ ✈➲ ✈✐➺❝ ①➙② ❞ü♥❣ ♠➯tr✐❝ ❍❛✉s❞♦r❢❢✱ ❤➺ ❤➔♠ ❧➦♣ ✤ì♥ trà✱ ✤❛ trà ✈➔ sỹ tỗ t t t q t tỷ s ỡ tr tr ỗ →♥❤ ①↕ ❦✐➸✉ F ✲❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ❚ø ✤â tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ ✈➲ ❝→❝ ✈➜♥ ✤➲ ♥â✐ tr➯♥✳ ■■■✳ ✣➮■ ❚×Đ◆● ❱⑨ P❍❸▼ ❱■ ◆●❍■➊◆ ❈Ù❯ ự sỹ tỗ t t t q ❤➔♠ ❧➦♣ ✤ì♥ trà✱ ✤❛ trà ❦✐➸✉ F ✲ ❝♦ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ b ✲ ♠➯tr✐❝✳ ■❱✳ ◆❍■➏▼ ❱Ư ◆●❍■➊◆ ❈Ù❯ ✶✳ ✣å❝ ❤✐➸✉ ♠ët sè t➔✐ ❧✐➺✉ ❧✐➯♥ q✉❛♥ ✤➳♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✱ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ❞➣②✱ t➟♣ ❜à ❝❤➦♥✱ t➟♣ ✤â♥❣✱ t➟♣ ❝♦♠♣❛❝t ✈➔ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ ✈➲ ✈✐➺❝ ①➙② ❞ü♥❣ ♠➯tr✐❝ ❍❛✉s❞♦r❢❢ ✳✳✳ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ✷✳ ✣å❝ ❤✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ t q sỹ tỗ t t ♠ët sè →♥❤ ①↕ ❝♦ ✤ì♥ trà ✈➔ ✤❛ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ✸✳ ✣å❝ ❤✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ tt sỹ tỗ t t t q ỗ ỡ tr trà ❝õ❛ →♥❤ ①↕ F ✲❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ❝→❝ ❦➳t q✉↔ ♥➔②✳ ✹✳ ❚➻♠ ♠ët sè ✈➼ ❞ö ♠✐♥❤ Pì PP Pữỡ ự ỵ tt t s ổ tữỡ tỹ ❤â❛✱ tê♥❣ q✉→t ❤â❛✳ ❱■✳ ❈❻❯ ❚❘Ó❈ ▲❯❾◆ ❱❿◆ ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ♥ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② t❤➔♥❤ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❝ì sð ✹ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ❝➛♥ ũ tr ỗ ổ btr ▼➯tr✐❝ ❍❛✉s❞♦r❢❢ tr➯♥ ❧ỵ♣ ❝→❝ t➟♣ ❝♦♥ ❝♦♠♣❛❝t ❝õ❛ ❦❤ỉ♥❣ btr ữỡ sỹ tỗ t t t ❜✐➳♥ ❝õ❛ ❤➺ ❤➔♠ ❧➦♣ ✤ì♥ trà ✈➔ ✤❛ trà ❦✐➸✉ F ✲❝♦ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ú tổ tr ỗ ỹ tỗ t t t t tỷ rt ỡ tr ỹ tỗ t t t t tû ❢r❛❝t❛❧ ✤❛ trà✳ ❱■■✳ ❈⑩❈ ❑➌❚ ◗❯❷ ✣❸❚ ✣×❒❈ ✶✳ ❚r➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ ♠➯tr✐❝ ❍❛✉s❞♦r❢❢ tr➯♥ ❧ỵ♣ ❝→❝ t➟♣ ❝♦♠♣❛❝t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ✷✳ ❚r➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❝→❝ ❜➔✐ ❜→♦ ♥❤÷♥❣ ❦❤æ♥❣ ❝❤ù♥❣ ♠✐♥❤ ❤❛② ❝❤ù♥❣ ♠✐♥❤ ❝á♥ ✈➢♥ t➢t ✈➲ sỹ tỗ t t F tr ổ tr sỹ tỗ t t➟♣ ❜➜t ❜✐➳♥ ❝õ❛ t♦→♥ tû ❢r❛❝t❛❧ ✤ì♥ trà s✐♥❤ ❜ð✐ ❤å →♥❤ ①↕ F ✲❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ sỹ tỗ t t t t tỷ ❢r❛❝t❛❧ s✐♥❤ ❜ð✐ ❝→❝ →♥❤ ①↕ F ✲❝♦ tê♥❣ q✉→t tr ổ btr sỹ tỗ t t rt ❝õ❛ t♦→♥ tû ❢r❛❝t❛❧ s✐♥❤ ❜ð✐ ❤➺ ❤➔♠ ❧➦♣ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ ✤❛ trà tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b ✲ tr r sỹ tỗ t ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ F ✲❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ♠➔ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ✺ ❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ❑❤ỉ♥❣ ♥❣ø♥❣ ♠ð rë♥❣ ❦❤↔ ♥➠♥❣ ù♥❣ ❞ư♥❣ ❝ơ♥❣ ♥❤÷ ♣❤↕♠ ✈✐ ♣❤→t tr✐➸♥ ❝õ❛ t♦→♥ ❤å❝✱ ♥➠♠ ✶✾✾✸ ❈③❡r✇✐❦ ✤➣ ♠ð rë♥❣ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ t❤➔♥❤ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ✣➣ ❝â r➜t ♥❤✐➲✉ ❦➳t q✉↔ ✈➲ ✈✐➺❝ ♠ð rë♥❣ ỵ ợ ổ b✲♠➯tr✐❝ ♥➔②✳ ✶✳✶ ❑❍➷◆● ●■❆◆ b ✲▼➊❚❘■❈ ✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳✭❬✻❪✮ ❈❤♦ ❳ ❧➔ t➟♣ ❤ñ♣ ❦❤→❝ ré♥❣ ✈➔ ♠ët sè t❤ü❝ s ∈ [1; ∞)✳ ❍➔♠ d : X × X −→ [0; ∞) ✤÷đ❝ ❣å✐ ❧➔ b✲♠➯tr✐❝ tr➯♥ X ♥➳✉ ♥â t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✳ ✐✮ d(x, y) = ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x = y❀ ✐✐✮ d(x, y) = d(y, x) ✈ỵ✐ ∀x, y ∈ X ❀ ✐✐✐✮ d(x, y) ≤ s[d(x, z) + d(z, y)] ✈ỵ✐ ∀x, y, z ∈ X ❑❤✐ ✤â✱ ❝➦♣ (X, d) ❤❛② (X, d, s) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ b tr ợ số s ỵ r➡♥❣ d ❧➔ b✲♠➯tr✐❝ tr➯♥ X ♥❤÷♥❣ d ❝â t❤➸ ❦❤æ♥❣ ❧➔ ♠➯tr✐❝ tr➯♥ X ✳ ✶✳✶✳✷ ◆❤➟♥ ①➨t✳ ✶✮ ❑❤✐ s = t❤➻ ♠ët ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❉♦ ✤â✱ ♠å✐ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➲✉ ổ btr ỗ t ổ btr (X, d) ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ❚❤➟t ✈➟②✱ ❧➜② X = {0, 1, 2} ✈➔ ①➨t →♥❤ ①↕ d : X × X −→ [0; ∞) ❜ð✐ ♥➳✉ (x, y) ∈ {(0, 2), (2, 0), (0, 1), (1, 0)} 1 d(x, y) = α > ♥➳✉ (x, y) ∈ {(1, 2), (2, 1)} ♥➳✉ x = y ❉➵ ❞➔♥❣ ❦✐➸♠ tr❛ ✤÷đ❝ d(x, y) ≤ α2 [d(x, z) + d(z, y)] ✈ỵ✐ ∀x, y, z ∈ X ✳ ❉♦ ✤â (X, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈ỵ✐ s = α2 > ✈➔ rã r➔♥❣ d ❦❤æ♥❣ ♣❤↔✐ ❧➔ ♠➯tr✐❝✳ ✻ ✸✮ ◆➳✉ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ t❤➻ d ❧➔ ❤➔♠ ❧✐➯♥ tö❝✱ ♥❣❤➽❛ ❧➔ d(xn , yn ) → d(x, y) ❦❤✐ n → ∞ ♥➳✉ xn → x, yn → y ❦❤✐ n → ∞✳ ❚✉② ♥❤✐➯♥✱ ✤✐➲✉ ♥➔② ❦❤ỉ♥❣ ✤ó♥❣ ✤è✐ ✈ỵ✐ b ✲ ♠➯tr✐❝ d✳ ✶✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✻❪✮ ❈❤♦ (X, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b ✲ ♠➯tr✐❝✳ ❉➣② (xn) tr♦♥❣ X ✤÷đ❝ ❣å✐ ❧➔ ✐✮ ❤ë✐ tư ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠å✐ > tỗ t k N s ợ n ≥ k t❛ ❝â d(xn, x) < t❛ ✈✐➳t ❧➔ n→∞ lim d(xn , x) = ✐✐✮ ❞➣② ❈❛✉❝❤② ợ > tỗ t k ∈ N s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ m, n ≥ k t❛ ❝â d(xm , xn ) < ❤❛② lim d(xn , xm ) = n,m→∞ ✶✳✶✳✹ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✻❪✮ ❈❤♦ (X, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ❚➟♣ ❝♦♥ Y ⊂ X ✤÷đ❝ ❣å✐ ❧➔ ✐✮ ✤â♥❣ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♠é✐ ❞➣② xn tr♦♥❣ Y ♠➔ ❤ë✐ tö ✈➲ x t❤➻ x ∈ Y ✐✐✮ ❝♦♠♣❛❝t ♥➳✉ ợ tr Y ổ tỗ t↕✐ ❞➣② ❝♦♥ ❤ë✐ tö ✈➲ ♠ët ♣❤➛♥ tû t❤✉ë❝ Y ✳ ✐✐✐✮ ❜à ❝❤➦♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ {sup d(x, y) : x, y ∈ Y } < +∞ ✐✈✮ ❑❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ (X, d, s) ✤÷đ❝ ❣å✐ ❧➔ ✤➛② ✤õ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ♠å✐ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✤➲✉ ❤ë✐ tö ✈➲ ♣❤➛♥ tû t❤✉ë❝ X ✳ ✶✳✶✳✺ ▼➺♥❤ ✤➲✳✭❬✾❪✮ ❈❤♦ (X, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b ✲ ♠➯tr✐❝✳ ❉➣② (xn) ❧➔ ❞➣② ❈❛✉❝❤② ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ n→∞ lim d(xn , xn+p ) = ✈ỵ✐ ♠å✐ p ∈ N ✶✳✶✳✻ ◆❤➟♥ ①➨t✳ ✭❬✸❪✮ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ (X, d, s) ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❧➔ ✤ó♥❣✳ ✐✮ ▼ët ❞➣② ❤ë✐ tư t❤➻ ✤✐➸♠ ❣✐ỵ✐ ❤↕♥ ❧➔ ❞✉② ♥❤➜t✳ ✐✐✮ ▼å✐ ❞➣② ❤ë✐ tö ✤➲✉ ❧➔ ❞➣② ❈❛✉❝❤②✳ ✐✐✐✮ ◆â✐ ❝❤✉♥❣✱ ❤➔♠ b✲♠➯tr✐❝ d ❧➔ ❤➔♠ ❦❤ỉ♥❣ ❧✐➯♥ tư❝✳ ✐✈✮ Y = {x ∈ X : tỗ t xn s n lim xn = x} ✼ ✶✳✷ ▼➊❚❘■❈ ❍❆❯❙❉❖❘❋❋ ❚❘➊◆ ▲❰P ❈⑩❈ ❚❾P ❈❖◆ ❈❖▼P❆❈❚ ❈Õ❆ ❑❍➷◆● ●■❆◆ b ✲ ▼➊❚❘■❈ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ❑➼ ❤✐➺✉ d(x, A) = inf{d(x, a) : a ∈ A} ✈➔ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø x ✤➳♥ A❀ δ(A, B) = sup{d(a, B) : a ∈ A} ✈➔ ❣å✐ ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø A ✤➳♥ B ❀ H(X) = {A : A ⊂ X, A = ∅, A ❝♦♠♣❛❝t}; dH (A, B) = max{δ(A, B), δ(B, A)} = max sup d(a, B), sup d(b, A) a∈A b∈B ◆➠♠ ✷✵✶✵✱ tr♦♥❣ ❬✽❪✱ ❝→❝ t→❝ ❣✐↔ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲ s❛✉✳ ✶✳✷✳✶ ❇ê ✤➲✳ ✭❬✽❪✮ ●✐↔ sû (X, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ❱ỵ✐ ❜➜t ❦ý A, B, C, D ∈ H(X)✱ t❛ ❧✉æ♥ ❝â ✐✮ ◆➳✉ B ⊆ C t❤➻ sup d(a, C) ≤ sup d(a, B)❀ a∈A ✳ a∈A ✐✐✮ sup d(x, C) = max sup d(a, C), sup d(b, C) ❀ x∈A∪B a∈A b∈B ✐✐✐✮ dH (A ∪ B, C ∪ D) ≤ max {dH (A, C), dH (B, D)}✳ ❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❱➻ B ⊆ C ♥➯♥ ✈ỵ✐ ♠å✐ a ∈ A✱ t❛ ❝â d(a, C) = inf{d(a, c) : c ∈ C} ≤ inf{d(a, b) : b ∈ B} = d(a, B) ▲➜② s✉♣ ❝↔ ❤❛✐ ✈➳ t❛ ❝â sup d(a, C) ≤ sup d(a, B) a∈A a∈A ✐✐✮ ❚❛ ❝â sup d(x, C) = sup{d(x, C) : x ∈ A ∪ B} x∈A∪B = {sup{d(x, C) : x ∈ A}, sup{d(x, C) : x ∈ B}} = max sup d(a, C), sup d(b, C) a∈A b∈B ✐✐✐✮ ❚ø ✐✐✮ t❛ ❝â sup d(x, C ∪ D) ≤ max sup d(x, C ∪ D), sup d(x, C ∪ D) xAB xA xB ỵ r F (dH (Am+1 , Am+2 )) ≤ F (dH (Am , Am+1 )) − τ (dH (Am , Am+1 )) ≤ F (dH (Am−1 , Am )) − τ (dH (Am−1 , Am )) − τ (dH (Am , Am+1 )) ≤ ≤ F (dH (A0 , A1 )) − n0 ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ F (dH (Am+1, Am+2)) → −∞ ❦❤✐ m → ∞✳ ❑➳t ❤ñ♣ ✈ỵ✐ (F2 ), t❛ ❝â lim dH (Am+1 , Am+2 ) = m ứ (F3) s r tỗ t h ∈ (0, 1) s❛♦ ❝❤♦ lim ([dH (Am+1 , Am+2 )]h F (dH (Am+1 , Am+2 )) = m→∞ ❉➝♥ ✤➳♥ [dH (Am , Am+1 )]h F (dH (Am , Am+1 )) − [dH (Am , Am+1 )]h F (dH (A0 , A1 )) ≤ [dH (Am , Am+1 )]h (F (dH (A0 , A1 ) − n0 )) − [dH (Am , Am+1 )]h F (dH (A0 , A1 )) ≤ −n0 [dH (Am , Am+1 )]h ≤ 0✳ ❈❤♦ n → ∞✱ t❛ ✤÷đ❝ lim (m[dH (Am+1 , Am+2 )]h ) = m→∞ ❉♦ ✤â✱ m lim m mh ữ tỗ t m ∈ N s❛♦ ❝❤♦ [dH (Am+1 , Am+2 )] ≤ ✈ỵ✐ ♠å✐ m ≥ n1 ♥➯♥ dH (Am+1 , Am+2 ) ≤ ✈ỵ✐ ♠å✐ m h [dH (Am+1 , Am+2 )] = 0✳ h m ≥ n1 ❇➙② ❣✐í✱ ✈ỵ✐ m, n ∈ N✱ m > n ≥ n1✱ t❛ ❝â dH (An , Am ) ≤ sdH (An , An+1 ) + s2 dH (An+1 , An+2 ) + + sm−n−1 dH (Am−1 , Am ) ∞ ≤ i=n si ih ✷✵ ∞ ❉♦ ❝❤✉é✐ is ❤ë✐ tö ♥➯♥ dH (An, Am) → ❦❤✐ n, m → ∞ i=n ❱➟② {An} ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❉♦ (H(X), dH ) ✤➛② ✤õ ♥➯♥ An → U ❦❤✐ n → ∞ ✈ỵ✐ U ∈ H(X) ●✐↔ sû U = T U ❑❤✐ ✤â✱ dH (U, T U ) = ♥➯♥ i h τ (MT (An , U )) + F (dH (An+1 , T U )) = τ + F (dH (T An , T U )) ≤ F (MT (An , U )) ✈ỵ✐ (MT (An, U )) = max{H(An, U ), H(An, T (An)), H(U, T (U )), H(An ,T (U ))+H(U,T (An )) , 2b H(T (An ), T (An )), H(T (An ), U ), H(T (An ), T (U ))} = max{H(An , U ), H(An , An+1 ), H(U, T (U )), H(An ,T (U ))+H(U,An+1 ) , 2b H((An+2 ), An+1 ), H((An+2 ), U ), H((An+2 ), T (U ))}✳ ❇➙② ❣✐í ❝❤ó♥❣ t❛ ①❡♠ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✳ ✐✮ ◆➳✉ (MT (An, U )) = H(An, U ) t❤➻ lim τ (H(An , U )) + F (H(T (U ), U )) ≤ F (H(U, U )), n→∞ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ lim τ (t) = ✈ỵ✐ ♠å✐ t ≥ t→0 ✐✐✮ ❑❤✐ (MT (An, U )) = H(An, An+1)✱ t❛ ❝â lim τ (H(An , An+1 )) + F (H(T (U ), U )) ≤ F (H(U, U )), n→∞ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥✳ ✐✐✐✮ ❑❤✐ (MT (An, U )) = H(U, T (U )) t❛ ❝â τ (H(U, T (U )) + F (H(T (U ), U )) ≤ F (H(U, T (U ))), ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ τ (H(U, T (U ))) > (A )) ✐✈✮ ◆➳✉ (MT (An, U )) = H(A ,T (U ))+H(U,T t❤➻ 2b n n (An )) ))+H(U,U ) lim τ ( H(An ,T (U ))+H(U,T ) + F (H(T (U ), U )) ≤ F ( H(U,T (U 2b ) 2b n→∞ ✷✶ t❤✉➝♥ ✈ỵ✐ →♥❤ ①↕ F ✲ ❝♦✳ ✈✮ ❑❤✐ (MT (An, U )) = H(An+2, An+1) t❤➻ (U )) ), = F ( H(U,T 2b ✤✐➲✉ ♥➔② ♠➙✉ lim τ (H(An+2 , An+1 )) + F (H(T (U ), U )) ≤ F (H(U, U )), n→∞ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥✳ ✈✐✮ ❑❤✐ (MT (An, U )) = H(An+2, U t❤➻ lim τ (H(An+2 , U )) + F (H(T (U ), U )) ≤ F (H(U, U )), n→∞ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥✳ ✈✐✐✮ ❈✉è✐ ❝ò♥❣ ♥➳✉ (MT (An, U )) = H(An+2, T (U )) t❤➻ lim τ (H(An+2 , T (U )) + F (H(T (U ), U )) ≤ F (H(U, U )), n→∞ ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â✱ U ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❇➙② ❣✐í t❛ ❦✐➸♠ tr❛ U ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ T ✳ ●✐↔ sû U ✈➔ V ❧➔ ❤❛✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✈ỵ✐ H(U, V ) = 0✳ ❉♦ T s✐♥❤ ❜ð✐ ỗ F ♥➯♥ t❛ ❝â τ (MT (U, V )) + F (H(U, V )) = τ (MT (U, V )) + F (H(T (U ), T (V ))) ≤ F (MT (U, V )) ✈ỵ✐ (MT (U, V )) = max{H(U, V ), H(U, T (U )), H(V, T (V )), H(U,T (V ))+H(V,T (U )) , 2b H(T (U ), U )), H(T (U ), V ), H(T (U ), T (V ))} = max{H(U, V ), H(U, U ), H(V, V ), H(U, U ), H(U, V ), H(U, V )} = H(U, V ) ✷✷ H(U,V )+H(V,U ) , 2b ◆❤÷ ✈➟② τ (H(U, V )) + F (H(U, V )) ≤ F (H(U, V )), ✤✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ τ (H(U, V )) > ❱➟②✱ T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t U H(X) ỵ sỷ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ✈➔ T : X → X t❤ä❛ ♠➣♥ d(T x, T y) ≤ ϕ(d(x, y)) ✈ỵ✐ ♠å✐ x, y ∈ X tr♦♥❣ ✤â ϕ ❧➔ ❤➔♠ s♦ s→♥❤ ✭❳❡♠ ✤à♥❤ ♥❣❤➽❛ ✷✳✷✳✻✮✳ ❑❤✐ ✤â✱ T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❧➔ u ✈➔ n→∞ lim d(T n x, u) = ✈ỵ✐ ❜➜t ❦ý x ∈ X ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ❜➜t ❦ý x ∈ X ✈➔ > 0✱ ❧➜② n ∈ N s❛♦ ❝❤♦ ϕn( ) < 41 ✣➦t F = T n ✈➔ xk = Fxk ✈ỵ✐ ♠å✐ k ∈ N✳ ❑❤✐ ✤â✱ ✈ỵ✐ x, y ∈ X ✈➔ α = ϕn t❛ ❝â d(F x, F y) ≤ ϕn (d(x, y)) = α[d(x, y)] ✭✯✮ ❉♦ ✤â✱ ✈ỵ✐ k ∈ N t❛ ❝â d(xk+1, xk ) → ❦❤✐ k → ∞ ▲➜② k ❧➔ sè s❛♦ ❝❤♦ d(xk+1 , xk ) < ✳ ❱ỵ✐ ♠é✐ z ∈ k(xk , ) := {y ∈ X : d(xk , y) ≤ }✱ t❛ ❝â d(Fz , Fx ) ≤ αd(xk , z) ≤ α( ) = ϕn(2) < ✈➔ d(Fx , xk ) = d(xk+1, xk ) < ❉♦ ✤â✱ d(Fz , Fx ) ≤ 2[ + ] = ✱ ✤✐➲✉ ♥➔② ❝❤➾ ①↔② r❛ r➡♥❣ F ❜✐➳♥ ❝→❝ ♣❤➛♥ tû tr♦♥❣ k(xk , ) ✈➔♦ k(xk , )✳ ◆❤÷ ✈➟②✱ d(xm, xs) ≤ ✈ỵ✐ ♠å✐ m, s ≥ k s✉② r❛ {xk } õ tỗ t xk u ❦❤✐ k → ∞ ❍ì♥ ♥ú❛✱ t❤❡♦ t✐➯✉ ❝❤✉➞♥ ❧✐➯♥ tö❝ ❝õ❛ F ✭s✉② tø ✭✯✮✮✱ t❛ ❝â k k k u = lim xk+1 = lim F (xk+1 ) = F (u), k→∞ k→∞ tù❝ u ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ F ✳ ❱➻ α(t) = ϕn(t) < t ✈ỵ✐ ♠å✐ t > ❞♦ ✤â✱ F ❝â ✤ó♥❣ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✷✸ ❍ì♥ ♥ú❛✱ ✈➻ T ❧✐➯♥ tö❝ ♥➯♥ t❛ ❝â T (u) = lim T (F k x) = lim F k (T x) = u k→∞ k→∞ tù❝ u ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❱➻ ✈ỵ✐ ♠é✐ x ∈ X ✈➔ r = 0, 1, , n − t❛ ❝â T nk+r (x) = F k (T k (x)) → u ❦❤✐ k → ∞ ❉♦ ✤â T mx → u ❦❤✐ m ợ ộ x X ỵ s❛✉ ❝❤♦ t❛ ✤✐➲✉ ❦✐➺♥ ✤➸ ♠ët →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ❝â ✤✐➸♠ ❜➜t t ỵ (X, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ✱ ✈➔ T : X → X ❧➔ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ ✈ỵ✐ ❤➺ sè ❝♦ k ∈ [0, 1)✱ ♥❣❤➽❛ ❧➔ d(T x, T y) ≤ kd(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X õ sk < t tỗ t ❞✉② ♥❤➜t x∗ ∈ X s❛♦ ❝❤♦ x∗ = T x∗ ✳ ❍ì♥ ♥ú❛✱ ✈ỵ✐ ❜➜t ❦ý x0 ∈ X ❞➣② (xn ) ①→❝ ✤à♥❤ ❜ð✐ xn+1 = T xn ✈ỵ✐ ♠å✐ n ∈ N s➩ ❤ë✐ tư ✈➲ x∗ t❤❡♦ b✲♠➯tr✐❝ d✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x0 ∈ X ✈➔ (xn) ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ xn+1 = T xn , n = 0, 1, , n ❉♦ T : X → X ❧➔ →♥❤ ①↕ ❝♦ ♥➯♥ t❛ ❝â d(T x0 , T x1 ) ≤ kd(T x0 , T x1 ) ≤ k d(x0 , x1 ) ❚✐➳♣ tö❝ q✉→ tr➻♥❤ ♥➔② t❛ ❝â d(T n x0 , T n x1 ) ≤ k n d(x0 , x1 ) ✷✹ ▲➜② m, n > ✈ỵ✐ m > n✱ t❛ ❝â d(xn , xm ) < sd(xn , xn+1 ) + s2 d(xn , xn+1 ) + s3 d(xn , xn+1 ) + ≤ sk n d(x0 , x1 ) + s2 k n+1 d(x0 , x1 ) + s3 k n+2 d(x0 , x1 ) + = d(x0 , x1 )sk n [1 + sk + (sk)2 + ] ❑❤✐ m, n → ∞✱ ❞♦ k ∈ [0, 1) ✈➔ ks < t❛ ❝â d(xn, xm) → ❉♦ ✤â (xn) ❧➔ ❞➣② ❈❛✉❝❤②✳ ❉♦ (X, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b tr tỗ t x X ✤➸ lim xn = x∗ n→∞ ❇➙② ❣✐í✱ t❛ ❝❤➾ r❛ x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳ ❚❤➟t ✈➟②✱ t❛ ❝â T x∗ = lim T xn = lim xn+1 = x∗ n→∞ n→∞ ❉♦ ✤â x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ●✐↔ sû x ❝ô♥❣ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✱ ♥❣❤➽❛ ❧➔ T x =x✳ ❑❤✐ ✤â✱ t❛ ❝â d(x∗ , x ) = d(T x∗ , T x ) ≤ kd(x∗ , x ) ❉♦ k < ♥➯♥ d(x∗, x ) = s✉② r❛ x ≡ x∗ ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ▼➦❝ ❞ị ❝❤÷❛ ❝â ❦➳t q✉↔ ♥â✐ ✈➲ sü tỗ t t F ❝♦ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ b✲ ♠➯tr✐❝ ♥❤÷♥❣ ❝→❝ ❦➳t q✉↔ s tr r sỹ tỗ t t t ❜✐➳♥ ❝õ❛ t♦→♥ tû ❢r❛❝t❛❧ s✐♥❤ ❜ð✐ ❝→❝ →♥❤ ①↕ F ✲ ❝♦ ✤ì♥ trà tê♥❣ q✉→t tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ b✲ ♠➯tr✐❝✳ ❈❤ó♥❣ tỉ✐ ✤÷❛ r❛ ✈➼ ❞ư s❛✉ ✤➸ ự tọ tỗ t F tr ổ ❣✐❛♥ b✲♠➯tr✐❝ ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✷✳✶✳✶✹ ❱➼ ❞ö✳ ▲➜② X = [1; +∞] ✈➔ ①➨t ❤➔♠ d : X × X → R+ (x, y) → d(x, y) = |x − y|2 ✷✺ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ✶✮ d ❧➔ b✲♠➯tr✐❝ tr➯♥ X ✈ỵ✐ ❤➡♥❣ sè s = 2❀ ✷✮ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ❀ ✸✮ d ❦❤æ♥❣ ❧➔ ♠➯tr✐❝ tr➯♥ X ✳ ❳➨t →♥❤ ①↕ T :X→X x → Tx = + 2x ❑❤✐ ✤â✱ t❛ ❝â T ❧➔ →♥❤ ①↕ F ✲❝♦ ✈ỵ✐ F : R+ → R ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ F x = ln x ✈➔ τ = ln > √ ❍ì♥ ♥ú❛✱ T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❧➔ x∗ = 1+2 ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❉➵ ❞➔♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❝→❝ ❦❤➥♥❣ ✤à♥❤ ✶✮✱ ✷✮ ✈➔ ✸✮✱ ✤ó♥❣✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ T ❧➔ →♥❤ ①↕ F ✲❝♦✳ ❚❛ ❝â T x = 2x1 + ∈ X ✱ T y = 2y1 + ∈ X ✱ 1 1 |x − y|2 d(T x, T y) = d( + 1, + 1) = | − | = 2y 2y 2x 2y 4x2 y ❚❛ ❝â |x − y|2 τ + F (d(T x, T y)) = ln + ln( ) 4x2 y = ln + ln(|x − y|2 ) − ln 4x2 y = ln(|x − y|2 ) + ln 2 2x y < ln(|x − y| ) = F (d(x, y)) ❱➻ x, y ∈ X ♥➯♥ x, y ≥ s✉② r❛ 2x1y ≤ 21 s✉② r❛ ln 2x1y < ❱➟② T ❧➔ →♥❤ ①↕ F ✲❝♦ ✈ỵ✐ τ = ln > ✈➔ F x = ln x✳ ❳➨t ♣❤÷♥❣ tr➻♥❤ x = T x ❤❛② x = 2x1 + 1✱ s✉② r❛ 2x2 − 2x − = √ √ ⇔ x1 = 1+2 ∈ X ✱ x2 = 1−2 ∈ / X✳ 2 2 ✷✻ √ ❱➟② T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❧➔ x∗ = 1+2 ✳ ✷✳✷ ❙Ü ❚➬◆ ❚❸■ ❚❾P ❇❻❚ ❇■➌◆ ❈Õ❆ ❚❖⑩◆ ❚Û ❋❘❆❈❚❆▲ ✣❆ ❚❘➚ ❑➳t q✉↔ s❛✉ ✤➙② ❧➔ ♠ð rë♥❣ ✤➛✉ t ỵ ①↕ ✤❛ trà✱ ✤÷đ❝ ◆❛❞❧❡r ❝❤ù♥❣ ♠✐♥❤ ♥➠♠ ✶✾✻✾✳ ✷✳✷✳✶ ỵ (X, d) ổ tr ✤➛② ✤õ ✈➔ T : X → H(X) ❧➔ →♥❤ ①↕ ❝♦ ✤❛ trà✱ ♥❣❤➽❛ ❧➔ dH (T x, T y) ≤ kd(x, y) ✈ỵ✐ ♠å✐ (x, y) ∈ X k (0, 1) số trữợ ❑❤✐ ✤â T ❝â ❞✉② ♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣✳ ✷✳✷✳✷ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✸❪✮ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈➔ F : X → P(X) ❧➔ →♥❤ ①↕ ✤❛ trà✳ ✐✮ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ F ♥➳✉ x ∈ F x❀ ✐✐✮ x ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤➦t ❝õ❛ F ♥➳✉ {x} = F x❀ ✐✐✐✮ F ✤÷đ❝ ❣å✐ ❧➔ ❝♦ ✤❛ trà ♥➳✉ dH (F x, F y) < d(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X, x = y ỵ F ix(F ) = {x X : x ∈ F x}✿ t➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ F ❀ SF ix(F ) = {x ∈ X : {x} = F x}✿ t➟♣ ❝→❝ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤➦t ❝õ❛ F ✳ ✷✳✷✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✹❪✮ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ▼ët →♥❤ ①↕ ✤❛ trà T : X → H(X) ✤÷đ❝ ❣å✐ ❧➔ F tỗ t F F ✈➔ τ ∈ R+ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ x, y X, y T x, ổ tỗ t z ∈ T y, ♠➔ τ + F (d(y, z)) ≤ F (M (x, y)) ✷✼ ❦❤✐ d(y, z) > tr♦♥❣ ✤â y) M (x, y) = max{d(x, y), d(x, T x), d(y, T y), d(x,T x)+d(y,T } ✷✳✷✳✹ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✸❪✮ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈➔ ❧➔ ❝→❝ →♥❤ ①↕ ❝♦ ✤❛ trà✳ ❑❤✐ ✤â✱ ✐✮ ❍å {F1, F2, , Fm} ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ❤➔♠ ❧➦♣ ✤❛ trà✳ ✐✐✮ ⑩♥❤ ①→❝ ✤à♥❤ ❜ð✐ F1 , F2 , , Fm : X → H(X) TF : H(X) → P(X) m A → TF A = m Fi a Fi A = i=1 i=1 a∈A ✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû ✤❛ ❢r❛❝t❛❧ s✐♥❤ ❜ð✐ ❤å {F1, F2, , Fm}, ✈➔ t➟♣ ❜➜t ❜✐➳♥ q✉❛ TF ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ❢r❛❝t❛❧ ✤❛ trà✳ ❙❛✉ ✤➙② ởt số t q sỹ tỗ t ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ✤❛ trà ❝õ❛ →♥❤ ①↕ F ✲ ❝♦ ✈➔ ❝õ❛ ♠ët sè ❧♦↕✐ →♥❤ ①↕ tr ổ tr btr ỵ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈➔ T : X → H(X) ❧➔ →♥❤ ①↕ F ✲❝♦ ✤â♥❣✱ ♥❣❤➽❛ ❧➔ t➟♣ {(x, y) : x ∈ X, y ∈ T x} ❧➔ t➟♣ ❝♦♥ ✤â♥❣ ❝õ❛ X × X ✳ ❑❤✐ ✤â✱ T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣✳ ự x0 X tũ ỵ x1 ∈ T x0 ◆➳✉ x1 = x0 t❤➻ x1 ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✈➔ t❛ ❝â ❧í✐ ❝❤ù♥❣ ♠✐♥❤ ✤ó♥❣✳ ❉♦ ✤â t❛ ❣✐↔ sû x1 = x0 T F tỗ t x2 ∈ T x1 s❛♦ ❝❤♦ τ + F (d(x1 , x2 )) ≤ F (M (x0 , x1 )) ✈➔ x2 = x1 ❚÷ì♥❣ tü✱ t❛ ❝â x3 ∈ T x2 s❛♦ ❝❤♦ τ + F (d(x3 , x2 )) ≤ F (M (x1 , x2 )) ✷✽ ✈➔ x3 = x2 ▲➦♣ ❧↕✐ q✉→ tr➻♥❤ ♥➔② t❛ ữủ xn ợ x0 s xn+1 ∈ T xn , xn+1 = xn ✈➔ τ + F (d(xn , xn+1 )) ≤ F (M (xn−1 , xn )) ✈ỵ✐ ♠å✐ n ∈ N ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ F (d(xn, xn+1)) < F (M (xn−1, xn)) ✈ỵ✐ ♠å✐ n ∈ N ❉♦ ✤â✱ t❛ ❝â d(xn , xn+1 ) < max{d(xn−1 , xn ), d(xn−1 , T xn−1 ), d(xn , T xn ), d(xn−1 ,T xn )+d(xn ,T xn−1 ) } = max{d(xn−1 , xn ), d(xn , T xn ), d(xn−12,T xn ) } n ,T xn ) ≤ max{d(xn−1 , xn ), d(xn , T xn ), d(xn−1 ,xn )+d(x } = max{d(xn−1 , xn ), d(xn , xn+1 )} ❉➵ t❤➜② r➡♥❣✱ ♥➳✉ d(xn, xn+1) = max {d(xn−1, xn), d(xn, xn+1)} t❤➻ s➩ ❣➦♣ ♣❤↔✐ ♠➙✉ t❤✉➝♥ ♥➯♥ t❛ ❝â d(xn−1 , xn ) = max {d(xn−1 , xn ), d(xn , xn+1 )} ❉♦ ✤â✱ τ + F (d(xn , xn+1 )) ≤ F (d(xn−1 , xn )) ✈ỵ✐ ♠å✐ n ∈ N ✭✯✮ ✣➦t dn = d(xn, xn+1) > ✈ỵ✐ ♠å✐ n ∈ N✳ ❚❤❡♦ ✭✯✮✱ t❛ ❝â F (dn ) ≤ F (dn−1 ) − τ ≤ ≤ F (d0 ) − nτ ✈ỵ✐ ♠å✐ n ∈ N ✭✯✯✮ ✈➔ ❞♦ ✤â✱ n→∞ lim F (dn ) = −∞ ❚❤❡♦ t✐➯✉ ❝❤✉➞♥ (F2) t❛ s✉② r❛ dn → ❦❤✐ n → ∞ ❇➙② ❣✐í✱ ❧➜② k ∈ (0, 1) s❛♦ ❝❤♦ n→∞ lim dkn F (dn ) = ❚❤❡♦ ✭✯✯✮ ✈ỵ✐ ♠å✐ n ∈ N✱ t❛ ❝â dkn F (dn ) − dkn F (d0 ) ≤ dkn (F (d0 ) − nτ ) − dkn F (d0 ) = nτ dkn ≤ ✷✾ ✭✯✯✯✮ ❈❤♦ n → ∞✱ tø ✭✯✯✯✮ t❛ ❝â n→∞ lim (ndkn ) = 0✱ ❞♦ ✤â lim n dn = n→∞ ∞ ❱➟② ❝❤✉é✐ dn ❤ë✐ tö✳ n=1 ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ (xn) ❧➔ ❞➣② ❈❛✉❝❤②✳ ❱➻ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ tỗ t u X : xn u ❦❤✐ n → ∞ ❇➙② ❣✐í✱ t❛ ❝❤ù♥❣ ♠✐♥❤ u ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❚❤➟t ✈➟②✱ ✈➻ T ❧➔ →♥❤ ①↕ ✤❛ trà ✤â♥❣ ✈➔ (xn, xn+1) → (u, u) ♥➯♥ u ∈ Tu tù❝ u ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T✳ ✷✳✷✳✻ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✹❪✮ ❈❤♦ ❤➔♠ ϕ : R+ → R+ ✐✮ ❍➔♠ ϕ ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ s♦ s→♥❤ ♥➳✉ ♥â ❧➔ ❤➔♠ t➠♥❣ ✈➔ t❤ä❛ ♠➣♥ lim ϕk (t) = ✈ỵ✐ ♠é✐ t ∈ R+ ✳ k→∞ ✐✐✮ ❍➔♠ ϕ ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ b✲s♦ s→♥❤ ✈ỵ✐ ❤➡♥❣ sè s ≥ t tỗ t k0 N, a ∈ (0, 1) ✈➔ ❝❤✉é✐ ➙♠ ❤ë✐ tö vk s❛♦ ❝❤♦ k k=1 sk+1 ϕk+1 (t) < ask ϕk (t) + vk ✈ỵ✐ ♠å✐ k ≥ k0 t ý t R+ ỵ r ❧➔ ❤➔♠ b✲s♦ s→♥❤ t❤➻ ϕ ❧➔ ❤➔♠ s♦ s→♥❤✮✳ ✐✐✐✮ ❍➔♠ f : X → X ✤÷đ❝ ❣å✐ ❧➔ tỗ t s dH (f x, f y) ≤ ϕ(d(x, y)) ✷✳✷✳✼ ❇ê ✤➲✳ ✭❬✸❪✮ ❈❤♦ (X, d, s) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈➔ F : X → H(X) ❧➔ →♥❤ ①↕ ✤❛ trà✳ ❑❤✐ ✤â✱ ✈ỵ✐ Y ∈ H(X) t❤➻ F y ∈ H(X) F (Y ) = y∈Y ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ t❛ ❧➜② ❞➣② (yn) F Y t tỗ t (xn) Y s❛♦ ❝❤♦ yn ∈ F xn ✈ỵ✐ ♠å✐ n ∈ N✳ ●✐↔ sû r➡♥❣ xn → x ∈ Y ✈➔ xn = x ✈ỵ✐ ♠é✐ ✸✵ n ∈ N✳ õ ợ ổ tỗ t un F x s❛♦ ❝❤♦ d(yn, un) ≤ sdH (F xn , F x) < sd(xn , x) → ❦❤✐ n → ∞✳ ❉♦ ✤â✱ d(yn , un ) → ❦❤✐ n → ∞ ❱➻ F x ❧➔ t➟♣ ❝♦♠♣❛❝t ♥➯♥ tỗ t un tử tỷ u F x ụ ỵ ❝♦♥ ✤â ❧➔ (un)n ⊆ N✳ ❑❤✐ ✤â✱ t❛ ❝â yn ∈ F x n d(yn , y) ≤ s[d(yn , un ) + d(un , y)] → ❦❤✐ n → ∞✳ ◆❤÷ ✈➟②✱ yn → y ∈ F x ⊂ F Y ❚❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ỵ (X, d) ổ b✲♠➯tr✐❝ ✤➛② ✤õ ✭✈ỵ✐ ❤➡♥❣ sè s ≥ 1✮ s❛♦ ❝❤♦ b✲♠➯tr✐❝ d ❧➔ ❤➔♠ ❧✐➯♥ tư❝ tr➯♥ X × X ✳ ●✐↔ sû Fi : X → H(X) ❧➔ ❤➔♠ ϕ✲❝♦ ✤❛ trà ✈ỵ✐ ♠é✐ i ∈ {1, 2, , m} s❛♦ ❝❤♦ ϕ : R+ → R+ ❧➔ ❤➔♠ b✲❝♦ s♦ s→♥❤✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✳ ✐✮ ❚♦→♥ tû ❢r❛❝t❛❧ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ TF : H → H m A → TF A = Fi a ∈ H(X) i=1 a∈A ✐✐✮ TF ❧➔ ỗ t t A H(X) s ❝❤♦ TF (A∗) = A∗ tù❝ t♦→♥ tû ❋r❛❝t❛❧ TF ❝â ❞✉② ♥❤➜t t➟♣ ❜➜t ❜✐➳♥✳ ✐✈✮ dH (TF (A∗, A∗) ≤ sPs(ϕ(dH (A, TF A))) ✈ỵ✐ ♠é✐ A ∈ H(X) ✈✮ dH (A, A∗) ≤ sPs(dH (A, TF A)) ✈ỵ✐ ♠é✐ A ∈ H(X) ❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❱➻ ϕ : R+ → R+ ❧➔ ❤➔♠ b ✲ ❝♦ s♦ s→♥❤ ♥➯♥ ♥â ❝ô♥❣ ❧➔ ❤➔♠ s♦ s→♥❤✳ ❉♦ ✤â✱ Fi ❧➔ ❝♦ ✈ỵ✐ ♠é✐ i ∈ {1, 2, , m} ❞♦ ϕ(t) < t ✈ỵ✐ ♠é✐ t > 0✳ ✸✶ ❉♦ ✤â✱ t❤❡♦ ❜ê ✤➲ ✶✳✶✳✶✹ t❤➻ t♦→♥ tû ❋r❛❝t❛❧ TF : H(X) → H(X) ✤÷đ❝ ①→❝ m ✤à♥❤✱ ♥❣❤➽❛ ❧➔ A ∈ H(X) t❤➻ TF A = Fi a ∈ H(X) i=1 a∈A ✐✐✮ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ dH (TF A, TF B) ≤ ϕ(dH (A, B)) ✈ỵ✐ A, B ∈ H(X)✳ ❚❤➟t ✈➟②✱ ✈ỵ✐ A, B ❜➜t ❦ý t❤✉ë❝ H(X)✱ ❧➜② u ∈ TF A✱ ❦❤✐ ✤â✱ tỗ t i {1, 2, , m} u Fi A ỡ ỳ tỗ t a A ✤➸ u ∈ Fi a✳ ❱ỵ✐ a ∈ A A, B H(X) tỗ t b B s❛♦ ❝❤♦ d(a, b) ≤ dH (A, B)✳ ❉♦ ✤â✱ dH (TF A, TF B) ≤ ϕ(dH (A, B)) ❱➟②✱ TF ❧➔ →♥❤ ①↕ ϕ✲❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ (H(X), dH )✳ ✐✐✐✮ ❉♦ TF ❧➔ ϕ✲❝♦✱ (H(X), dH ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✤➛② ✤õ ♥➯♥ t tỗ t t A H(X) ✤➸ TF (A∗) = A∗ ✳ ✐✈✮ ✈➔ ✈✮ ữủ s trỹ t tứ ỵ ỵ ữủ ự t ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉✳ ✶✳ ❚r➻♥❤ ❜➔② ❝→❝❤ ①➙② ❞ü♥❣ ♠➯tr✐❝ ❍❛✉s❞♦r❢❢ tr➯♥ ❧ỵ♣ ❝→❝ t➟♣ ❝♦♠♣❛❝t ❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ b✲♠➯tr✐❝✳ ✷✳ ❚r➻♥❤ ❜➔② ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝❤✐ t✐➳t ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ❝→❝ ❜➔✐ ❜→♦ ♥❤÷♥❣ ❦❤ỉ♥❣ ❝❤ù♥❣ ♠✐♥❤ ❤❛② ❝❤ù♥❣ ỏ tt sỹ tỗ t t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ F ✲❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ sỹ tỗ t t t t tû ❢r❛❝t❛❧ ✤ì♥ trà s✐♥❤ ❜ð✐ ❤å →♥❤ ①↕ F tr ổ tr sỹ tỗ t t ❜➜t ❜✐➳♥ ❝õ❛ t♦→♥ tû ❢r❛❝t❛❧ s✐♥❤ ❜ð✐ ❝→❝ →♥❤ ①↕ F ✲❝♦ tê♥❣ q✉→t tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b✲♠➯tr✐❝ ✈➔ sỹ tỗ t t rt t tỷ rt s ❜ð✐ ❤➺ ❤➔♠ ❧➦♣ ❝→❝ →♥❤ ①↕ ϕ✲❝♦ ✤❛ trà tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ b ✲ ♠➯tr✐❝✳ ✸✳ ❚➻♠ ✈➼ ❞ư r sỹ tỗ t t t ❝õ❛ →♥❤ ①↕ F ✲❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ b ✲ ♠➯tr✐❝ ♠➔ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝✳ ✸✸ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶❪✳ ❈✳ ❈❤✐❢✉ ❛♥❞ ●✳ P❡tr✉s❡❧ ✭✷✵✶✹✮✱ ✧❋✐①❡❞ ♣♦✐♥ts ❢♦r ♠✉❧t✐✈❛❧✉❡❞ ❝♦♥tr❛❝✲ t✐♦♥s ✐♥ b✲♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❢r❛❝t❛❧s✧✱ ❚❛✐✇❛♥❡s❡✱ ❏✳ ▼❛t❤✳✱ ✶✽✱ ✶✸✻✺✲✸✸✼✺✳ ❬✷❪✳ ❉✳ ❲❛r❞♦✇s❦✐ ✭✷✵✶✷✮✱ ✧❋✐①❡❞ ♣♦✐♥ts ♦❢ ❛ ♥❡✇t②♣❡ ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣✲ ♣✐♥❣s ✐♥ ❝♦♠♣❧❛t❡ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ✾✲✶✶✳ ❬✸❪✳ ▼✳ ❇♦r✐❝❡❛♥✉✱ ▼✳ ❇♦t❛ ❛♥❞ ❆✳ P❡tr✉s❡❧ ✭✷✵✶✵✮✱ ✧▼✉❧t✐✈❛❧✉❡❞ ❢r❛t❛❧s ✐♥ b✲♠❡tr✐❝ s♣❛❝❡s✧✱ ❈❡♥t✳ ❊✉r✳ ❏✳ ▼❛t❤✳✱ ✽✱ ✸✻✼✲✸✼✼✳ ❬✹❪✳ ▼✳ ❙❣r♦✐ ❛♥❞ ❈✳ ❱❡tr♦ ✭✷✵✶✸✮✱ ✧▼✉❧t✐✲✈❛❧✉❡❞ ❋✲❝♦♥tr❛❝t✐♦♥s ❛♥❞ t❤❡ s♦❧✉t✐♦♥ ♦❢ ❝❡rt❛✐♥ ❢✉♥❝t✐♦♥❛❧ ❛♥❞ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✧✱ ❋✐❧♦♠❛t ✷✼ ✭✼✮✱ ✶✷✺✾✲ ✶✷✻✽✳ ❬✺❪✳ ❙✳ ❇✳ ◆❛❞❧❡r ✭✶✾✻✾✮✱ ✧▼✉❧t✐✲✈❛❧✉❡❞ ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s✧✱ P❛❝✳ ❏✳ ▼❛t❤✳✱ ✸✵✱ ✹✼✺✲✹✽✽✳ ❬✻❪✳ ❙✳ ❈③❡r✇✐❦ ✭✶✾✾✸✮✱ ✧❈♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s ✐♥ b✲♠❡tr✐❝ s♣❛❝❡s✧✱ ❆❝t❛ ▼❛t❤ ■♥❢♦r♠✳ ❯♥✐✈✳ ❖sstr❛✈✳✱✶✱ ✺✲✶✶✳ ❬✼❪✳ ❙✳ ❈③❡r✇✐❦ ✭✶✾✾✽✮✱ ✧◆♦♥❧✐♥❡❛r s❡t✲✈❛❧✉❡❞ ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s ✐♥ b✲ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❆tt✐ ❙❡♠✳ ▼❛t✳ ❋✐s✳ ❯♥✐✈✳ ▼♦❞❡♥❛✱ ❱♦❧✳ ✹✻✱ ✷✻✸✲✷✼✻✳ ❬✽❪✳ ❚✳ ◆❛③✐r✱ ❙✳ ❙✐❧✈❡str♦✈ ❛♥❞ ▼✳ ❆❜❜❛s ✭✷✵✶✻✮✱ ✧❋r❛❝t❛❧s ♦❢ ❣❡♥❡r❛❧✐r❡❞ ❋✲❍✉t❝❤✐♥s♦♥ ♦♣❡r❛t♦r✧✱ ❲❛✈❡❧❡ts ❛♥❞ ❋r❛❝t❛❧s ❛❞✈❛♥❝❡❞ ❆♥❛❧✳✱ ✷✱ ✷✾✲✹✵✳ ❬✾❪✳ ❚✳ ◆❛③✐r✱ ❙✳ ❙✐❧✈❡str♦✈ ❛♥❞ ❳✳ ◗✐✭✷✵✶✻✮ ✧❋r❛❝t❛❧s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❋✲ ❍✉t❝❤✐♥s♦♥ ♦♣❡r❛t♦r ✐♥ b✲♠❡tr✐❝ s♣❛❝❡s✧✱ ✶✲✶✺✳ ✸✹