✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ▲➊ ❱❿◆ ▼■◆❍ ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆ ❱⑨ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ✣■➎▼ ❇❻❚ ✣❐◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ư♥❣ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ▲➊ ❱❿◆ ▼■◆❍ ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆ ❱⑨ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ✣■➎▼ ❇❻❚ ✣❐◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ữớ ữợ P Pì ◆●❯❨➊◆ ✲ ✷✵✶✼ ▼ư❝ ❧ư❝ ▼Ð ✣❺❯ ✶ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✶✳✶ ▼ð ✤➛✉ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ◆â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ▼ët sè t➼♥❤ ❝❤➜t ✈➲ ❦❤æ♥❣ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ❙ü ❤ë✐ tö tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✶✳✷✳✷ ◆❣✉②➯♥ ❧➼ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t tr ổ tr õ ởt số rở ỵ →♥❤ ①↕ ❝♦ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶ ◆❣✉②➯♥ ỵ t ✳ ✳ ✳ ✳ ✷✳✶✳✷ ▼ët sè ❞↕♥❣ ♠ð rë♥❣ ❦❤→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ →♥❤ ①↕ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ▼ð ✤➛✉ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ❚r÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸ ✸ ✸ ✼ ✾ ✾ ✶✹ ✶✾ ✶✾ ✶✾ ✷✷ ✸✶ ✸✶ ✸✷ é ỵ t ✤ë♥❣ ❧➔ ♠ët ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ❦❤→ ❝ì ❜↔♥ tr♦♥❣ ❝❤✉②➯♥ ♥❣➔♥❤ t♦→♥ ❣✐↔✐ t➼❝❤ ✈➔ t♦→♥ ù♥❣ ❞ö♥❣✳ ữủ t ợ ổ tr rr ♥❤ú♥❣ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà✱ ✤❛ trà tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤→❝ ♥❤❛✉ ♥❣➔② ❝➔♥❣ t❤✉ ❤ót ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t tr ữợ q t ự ✈➔ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ❝õ❛ t♦→♥ ❤å❝✿ t♦→♥ tè✐ ÷✉✱ ❝→❝ ❜➔✐ t♦→♥ ❦✐♥❤ t➳✳ ❚❛ ♥❤➢❝ ❧↕✐ r➡♥❣✱ ✈ỵ✐ X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✱ f ❧➔ ♠ët →♥❤ ①↕ ỵ ❝❤➾ r❛ r➡♥❣ f ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❱➲ s❛✉ ❝â r➜t ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ✤➣ q t ự t ỵ ❝❤♦ ❝→❝ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤→❝ ♥❤❛✉✳ ◆➠♠ ✷✵✵✼✱ ●✉❛♥❣ ✈➔ ❩❤❛♥❣ ✭❬✸❪✮ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ♠ët ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠ỵ✐✱ ❝→❝ t→❝ ❣✐↔ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ tr♦♥❣ ✤â ❝→❝ t→❝ ❣✐↔ ✤➣ t❤❛② t➟♣ sè t❤ü❝ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ♠➯tr✐❝ ❜ð✐ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ♠➔ tr➯♥ ✤â ✤➣ ✤à♥❤ ♥❣❤➽❛ ♠ët q✉❛♥ ❤➺ t❤ù tü ỹ tr ởt õ ữợ r ổ tr➻♥❤ ♥➔② ❝→❝ t→❝ ❣✐↔ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ♠ët sè t➼♥❤ ❝❤➜t t÷ì♥❣ tü ✈➲ ♠➯tr✐❝ tr➯♥ ♠➯tr✐❝ ♥â♥✱ t ự ỵ ❝♦ ❇❛♥❛❝❤ tr♦♥❣ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ❱➲ s❛✉✱ ❝â ♥❤✐➲✉ t→❝ ❣✐↔ ❦❤→❝ t✐➳♣ tö❝ ♣❤→t tr✐➸♥ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈➔ ỵ t ỳ ổ ❣✐❛♥ ♥➔②✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔ ❣✐ỵ✐ t❤✐➺✉ ❧↕✐ ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❝→❝ t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙② ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ õ ởt số ỵ t ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❣✐ú❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♥➔②✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ✈✐➳t ❞ü❛ tr➯♥ ❝→❝ ❜➔✐ ❜→♦ ❬✸❪✱ ❬✺❪✱ ❬✶❪✱ ❬✹❪✱ ❬✷❪ ✈➔ ❬✻❪✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð t ỗ ữỡ ữỡ ✶ ▼ð ✤➛✉ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ ❝ì ❜↔♥ ✈➲ ♥â♥✱ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❧ỵ♣ ❦❤ỉ♥❣ r ú tổ ợ t ỵ →♥❤ ①↕ ❇❛♥❛❝❤ ❝❤♦ →♥❤ ①↕ ❣✐ú❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ●✉❛♥❣ ✈➔ ❩❤❛♥❣ ♥➠♠ r ữỡ ỵ t tr ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✷ ♥â♥✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ♠ët số t q ỵ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❣✐ú❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤÷đ❝ ❝→❝ t→❝ ❣✐↔ ●✉❛♥❣✱ ❩❤❛♥❣✱ ❘❡③❛♣♦✉r✱ ❍❛♠❧❜❛r❛♥✐✱ ❋✳ ❙❛❜❡t❣❤❛❞❛♠✱ ❍✳ P✳ ▼❛s✐❤❛ ✈➔ ❆✳ ❍✳ ❙❛♥❛t♣♦✉r ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙②✳ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ữợ sỹ ữợ t t t❤➛② ❣✐→♦ P●❙✳ ❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛②✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❑❤♦❛ ❚♦→♥✲❚✐♥ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ q✉❛♥ t➙♠ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ ❚r÷í♥❣✳ ◆❤➙♥ ❞à♣ ♥➔② ❡♠ ụ ữủ ỷ ỡ t tợ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ❜➯♥ ❡♠✱ ❝ê ✈ơ✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ❇↔♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ❝õ❛ t ổ ỗ t❤→♥❣ ✾ ♥➠♠ ✷✵✶✼ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ▲➯ ❱➠♥ ▼✐♥❤ ✸ ❈❤÷ì♥❣ ✶ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✶✳✶ ▼ð ✤➛✉ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✶✳✶✳✶ ◆â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t❛ ❧✉æ♥ ❣✐↔ t❤✐➳t r➡♥❣ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ởt t ỗ P E ữủ ♠ët ♥â♥ tr♦♥❣ E ♥➳✉ ♥â t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✶✳ P ✤â♥❣✱ P = {∅}✱ P = {0} ; ✷✳ ❱ỵ✐ ♠å✐ a, b ∈ R, a, b 0, x, y ∈ P t❤➻ ax + by ∈ P ; ✸✳ ◆➳✉ x ∈ P ✈➔ −x ∈ P t❤➻ x = ❇➙② ❣✐í t❛ ①❡♠ ①➨t ❦❤→✐ ♥✐➺♠ q✉❛♥ ❤➺ t❤ù tü tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ❧✐➯♥ q✉❛♥ ✤➳♥ ♥â♥✳ ❈❤♦ P ⊂ E ❧➔ ♠ët ♥â♥✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ tr➯♥ E ♥❤÷ s❛✉✿ x y ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ y − x ∈ P ; x < y ♥➳✉ x y ✈➔ x = y; x y ♥➳✉ y − x ∈ intP, tr♦♥❣ ✤â intP ❧➔ ❦➼ ❤✐➺✉ ♣❤➛♥ tr♦♥❣ ❝õ❛ ♥â♥ P ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t❛ ❧✉æ♥ ❣✐↔ t❤✐➳t ♥â♥ P ❝â ♣❤➛♥ tr♦♥❣ intP = ∅✳ ✹ ▼➺♥❤ ✤➲ ✶✳✶✳ ❚ø ❦❤→✐ ♥✐➺♠ t❛ ❞➵ ❞➔♥❣ s✉② r❛✿ ◆➳✉ x y t❤➻ x < y ◆➳✉ x y✱ a t❤➻ ax ay ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ◆â♥ P ữủ õ t tỗ t↕✐ ♠ët ❤➡♥❣ sè K > t❤ä❛ ♠➣♥✿ ✤✐➲✉ ❦✐➺♥ x y ❦➨♦ t❤❡♦ x K y ✱ ✈ỵ✐ ♠å✐ x, y ∈ E ❍➡♥❣ sè K > ♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝✳ ✷✳ ◆â♥ ♠✐♥✐❤❡❞r❛❧ ♥➳✉ sup(x, y) tỗ t ợ x, y E ◆â♥ ♠✐♥✐❤❡❞r❛❧ ♠↕♥❤ ♥➳✉ ♠å✐ t➟♣ ❝♦♥ ❜à ❝❤➦♥ tr➯♥ ❝õ❛ E ✤➲✉ ❝â ❝➟♥ tr➯♥ ✤ó♥❣✳ ✹✳ ◆â♥ ✤➦❝ ♥➳✉ intP = ∅ ✺✳ ◆â♥ s✐♥❤ ♥➳✉ E = P − P ✻✳ ◆â♥ ❝❤➼♥❤ q✉② ♥➳✉ ♠å✐ ❞➣② t➠♥❣ ❜à ❝❤➦♥ tr➯♥ ✤➲✉ ❤ë✐ tö✳ ◆❣❤➽❛ ❧➔ ♥➳✉ {xn, n 1} ❧➔ ❞➣② t❤ä❛ ♠➣♥ x1 x2 ··· y ợ y E t tỗ t x E t❤ä❛ ♠➣♥ lim n−→∞ xn − x = ▼➺♥❤ ✤➲ ✶✳✷✳ ▼å✐ ♥â♥ ❝❤➼♥❤ q✉② ✤➲✉ ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû P ❧➔ ♥â♥ ❝❤➼♥❤ q✉② tr♦♥❣ E ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ ❱ỵ✐ ♠é✐ n t❛ ❝❤å♥ tn, sn ∈ P s❛♦ ❝❤♦ tn − sn ∈ P ✈➔ n2 tn < sn ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ n 1, ✤➦t yn = ttn ✈➔ xn = st n ✳ n n ❑❤✐ ✤â xn, yn ✈➔ y∞n − xn ✤➲✉ t❤✉ë❝ P ✱ yn = ✈➔ n < xn ✈ỵ✐ ♠å✐ n ❉♦ ❝❤✉é✐ yn ❧➔ tử P õ tỗ t y P s❛♦ n2 ❝❤♦ n=1 ∞ y= yn n=1 n x1 ú ỵ r x1 + x2 22 x1 + 1 x + x3 22 32 ··· y, ✺ ❞♦ ✤â ❝❤✉é✐ ∞ x n n=1 n ❤ë✐ tö ✈➻ P ❧➔ ♥â♥ ❝❤➼♥❤ q✉②✳ ❉♦ ✤â xn = 0, n−→∞ n2 lim ♠➙✉ t❤✉➝♥✳ ❱➟② P ❧➔ õ t ổ tỗ t õ ❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ K < ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈➔ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ K < ❈❤å♥ ♠ët ♣❤➛♥ tû ❦❤→❝ ổ x P tũ ỵ < < s❛♦ ❝❤♦ K < − ε ❑❤✐ ✤â (1 − ε)x x ♥❤÷♥❣ (1 − ε) x > K x ✣➙② ❝❤➼♥❤ ❧➔ ♠➙✉ t❤✉➝♥✳ ▼➺♥❤ ợ ộ M > ổ tỗ t ♥â♥ ❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ K > M ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû M > ❧➔ ♠ët số tỹ tũ ỵ t E = ax + b : a, b ∈ R, x ∈ [1 − 1/k, 1] , ❦❤✐ ✤â E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ✈ỵ✐ ❝❤✉➞♥ sup✳ ❑➼ ❤✐➺✉ P = {ax + b : a, b ∈ R, a 0, b 0} õ P ởt õ tr E rữợ ❤➳t t❛ ❝❤ù♥❣ ♠✐♥❤ P ❧➔ ♥â♥ ❝❤➼♥❤ q✉②✳ ●å✐ {anx + bn, n 1} ❧➔ ♠ët ❞➣② t➠♥❣✱ ❜à tr tự tỗ t ởt tỷ cx + d ∈ E s❛♦ ❝❤♦ a1 x + b1 a2 x + b ··· cx + d ✻ ✈ỵ✐ ♠å✐ x ∈ [1 − 1/k, 1] ❑❤✐ ✤â {an, n t❤ü❝ t❤ä❛ ♠➣♥ 1}✱ {bn , n b1 b2 ··· d a1 a2 ··· c, 1} ❧➔ ❤❛✐ ❞➣② sè ✈➔ ❞♦ ✤â ❝→❝ ❞➣② {an, n 1}✱ {bn, n 1} ❤ë✐ tö✳ ●✐↔ sû n−→∞ lim an = a, lim bn = b, ❦❤✐ ✤â lim an x + bn = ax + b✳ ❚ø ✤â s✉② r❛ P ❧➔ ♥â♥ n−→∞ n−→∞ ❝❤➼♥❤ q✉②✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷ t❛ s✉② r❛ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳ tỗ t K s ✤✐➲✉ ❦✐➺♥ g f ❦➨♦ t❤❡♦ g K f ✈ỵ✐ ♠å✐ g, f ∈ E ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ K > M ❚❛ t❤➜② f (x) = −M x + M ∈ P, g(x) = M ∈ P ✈➔ f − g ∈ P ✳ ❉♦ ✤â g f ✱ ❦➨♦ t❤❡♦ M= g K f = K ▼➦t ❦❤→❝✱ t❛ ①➨t ❝→❝ ❤➔♠ sè f (x) = −(M + 1/M )x + M, t❤➻ f ∈ P, g ∈ P ✈➔ f − g ∈ P ✳ ❉♦ ✤â M= g ❍ì♥ ♥ú❛ g =M ◆❤÷ ✈➟② ✈➔ g(x) = M g f✱ ❦➨♦ t❤❡♦ K f f = − 1/M + 1/M M = g > M f = M + 1/M − ◆❤÷ ✈➟② M f < g K f , ❦➨♦ t❤❡♦ K > M ▼➺♥❤ ✤➲ ✶✳✺✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E t❛ ❧✉æ♥ ❝â ✐✮ ợ ộ R, > ổ tỗ t↕✐ ♥â♥ ❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➺ sè K > λ ✼ ✐✐✮ ◆â♥ P ❝❤➼♥❤ q✉② ❦❤✐ ✈➔ ❝❤➾ ữợ tử ❞ö ✶✳✶✳ ❈❤♦ E = Rn✱ t❛ ✤➦t P = {(x1 , , xn ) : xi 0, ∀i = 1, , n} ❑❤✐ ✤â P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✱ ♥â♥ s✐♥❤✱ ♠✐♥✐❤❡❞r❛❧✱ ♠✐♥✐❤❡❞r❛❧ ♠↕♥❤ ✈➔ ✤➦❝✳ ❱➼ ❞ö ✶✳✷✳ ❈❤♦ D ⊆ Rn ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t E = C (D) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ sè ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ D✳ ❑➼ ❤✐➺✉ P = {f ∈ E |f (t) 0, ∀x ∈ D } ❑❤✐ ✤â P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✱ ♥â♥ s✐♥❤✱ ✤➦❝ ✈➔ ♠✐♥✐❤❡❞r❛❧ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ ♥â♥ ♠✐♥✐❤❡❞r❛❧ ♠↕♥❤✱ P ❝ơ♥❣ ❦❤ỉ♥❣ ❧➔ ♥â♥ ❝❤➼♥❤ q✉②✳ ❱➼ ❞ö ✶✳✸✳ ❑➼ ❤✐➺✉ E = C[0;1] ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ sè t❤ü❝ ❦❤↔ ✈✐ ❝➜♣ ✶ tr➯♥ ✤♦↕♥ [0; 1] ✈ỵ✐ ❝❤✉➞♥ f = f tr♦♥❣ ✤â ∞ + f ∞ ∞, f ∈ E, ❧➔ ❝❤✉➞♥ max ❑➼ ❤✐➺✉ P = {f ∈ E : f (t) 0} ❉➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ữủ P ởt õ ợ ộ k x, g(x) = x2k ✱ ❦❤✐ ✤â ✈ỵ✐ ♠å✐ t ∈ [0; 1]✱ ❦➨♦ t❤❡♦ g(t) g f✳ ✤➦t f (x) = f (t) ❚❛ t❤➜② f = 2, g = 2k + ❑➨♦ t❤❡♦ f < g ✳ ◆❤÷ ✈➟② k ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ ❝õ❛ P ✈➔ P ❧➔ ♥â♥ ❦❤æ♥❣ ❝❤✉➞♥ t➢❝✳ ✶✳✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❈❤♦ X ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ E ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈ỵ✐ q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ ✤è✐ ✈ỵ✐ ♥â♥ P ✳ ❈❤♦ ❤➔♠ d : ✷✼ ❚❛ ❝❤å♥ N2 ∈ N∗ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n N2 t❛ ❝â c(1 − k) ✈➔ d(xn+1 , x∗ ) d(xn+1 , xn ) 2k ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ n N2 t❛ ❝â d(T x∗ , x∗ ) c(1 − k) d(T xn , T x∗ ) + d(T xn , x∗ ) k(d(T xn , xn ) + d(T x∗ , x∗ )) + d(xn+1 , x∗ ) ❙✉② r❛ d(T x∗ , x∗ ) (kd(T xn+1 , xn ) + d(T xn+1 , x∗ ) 1−k c/2 + c/2 = c ❈❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❛ ữủ d(T x , x ) c m ợ m ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ mc − d(T x∗, x∗) ∈ P ✈ỵ✐ ♠å✐ m ❱➻ mc −→ ❦❤✐ m −→ ∞ ✈➔ P ✤â♥❣ ♥➯♥ −d(T x, x) P ú ỵ r d(T x , x∗ ) ∈ P ✱ ❞♦ ✤â d(T x∗ , x∗ ) = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ T x∗ = x∗ ✱ tø ✤â x∗ ❧➔ ✤✐➸♠ ❜➜t T sỷ tỗ t y T s❛♦ ❝❤♦ T y∗ = y∗✱ ❦❤✐ ✤â t❛ ❝â d(x∗ , y ∗ ) = d(T x∗ , T y ∗ ) k(d(T x∗ , x∗ ) + d(T y ∗ , y ∗ )) = ❙✉② r❛ x∗ = y∗✳ ❱➟② x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ t T ỵ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✈➔ →♥❤ ①↕ T : X −→ X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d(T x, T y) k(d(T x, y) + d(T y, x)) ✈ỵ✐ ♠å✐ x, y ∈ X ✱ tr♦♥❣ ✤â k ∈ [0, 21 ) ❧➔ ❤➡♥❣ sè✳ ❑❤✐ ✤â T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t x∗ ∈ X ✳ ❍ì♥ ♥ú❛ ✈ỵ✐ ♠é✐ x ∈ X ✱ lim T n x = x∗ n−→∞ ❈❤ù♥❣ ♠✐♥❤✳ x0 X tũ ỵ t ỹ {xn} ⊆ X ❜ð✐ xn = T n x0 , ợ n ự tữỡ tỹ ữ ỵ t õ d(xn+1 , xn ) k d(xn , xn−1 ) = hd(xn , xn−1 ), 1−k k tr♦♥❣ ✤â h = 1−k ❉♦ ✤â ✈ỵ✐ n > m✱ d(xn , xn−1 ) + d(xn−1 , xn−2 ) + · · · + d(xm+1 , xm ) d(xn , xm ) (hn−1 + hn−2 + · · · + hm )d(x1 , x0 ) hm d(x1 , x0 ) 1−h ❱ỵ✐m♠é✐ c ∈ E, h d(x1 , x0 ) 1−h c✱ t❛ ❝❤å♥ N1 ∈ N∗ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ m c ❑❤✐ ✤â d(xn , xm ) hm d(x1 , x0 ) 1−h N1 t❛ ❝â c ✈ỵ✐ ♠å✐ n > m N1✳ ❙✉② r❛ {xn} ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ♥➯♥ tỗ t x X s n lim xn = x∗ ❚❛ ❝❤å♥ N2 ∈ N∗ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n N2 t❛ ❝â d(xn , x∗ ) ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ n d(T x∗ , x∗ ) N2 t❛ ❝â c(1 − k) d(T xn , T x∗ ) + d(T xn , x∗ ) k(d(T x∗ , xn ) + d(T xn , x∗ )) + d(xn+1 , x∗ ) k(d(T x∗ , x∗ ) + d(T xn , x∗ ) + d(xn+1 , x∗ )) + d(xn+1 , x∗ ) ❙✉② r❛ d(T x∗ , x∗ ) (kd(xn , x∗ ) + d(xn+1 , x∗ ) + d(xn+1 , x∗ ) 1−k c/2 + c/2 + c/3 = c ❈❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❛ ✤÷đ❝ d(T x∗ , x∗ ) c m ✷✾ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ mc − d(T x∗, x∗) ∈ P ✈ỵ✐ ♠å✐ m ❱➻ mc −→ ❦❤✐ m −→ ∞ ✈➔ P ✤â♥❣ ♥➯♥ −d(T x , x ) P ú ỵ r d(T x∗ , x∗ ) ∈ P ✱ ❞♦ ✤â d(T x∗, x∗) = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ T x∗ = x∗✱ tø ✤â x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ T sỷ tỗ t y T s❛♦ ❝❤♦ T y∗ = y∗✳ ❑❤✐ ✤â t❛ ❝â d(x∗ , y ∗ ) = d(T x∗ , T y ∗ ) k(d(T x∗ , y ∗ ) + d(T y ∗ , x∗ )) = 2kd(x∗ , y ∗ ) ❚ø ✤â s✉② r❛ d(x∗, y∗) = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ x∗ = y∗ ❱➟② x∗ ❧➔ t t T ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ◆❣♦➔✐ r❛✱ ❝→❝ t→❝ ❣✐↔ ❝❤ù♥❣ ♠✐♥❤ t ởt t q ợ ữ s ỵ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✈➔ →♥❤ ①↕ T : X −→ X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d(T x, T y) kd(x, y) + ld(y, T x) ✈ỵ✐ ♠å✐ x, y ∈ X ✱ tr♦♥❣ ✤â k, l ∈ [0, 1) ❧➔ ❝→❝ ❤➡♥❣ sè✳ ❑❤✐ ✤â T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ X ✳ ❍ì♥ ♥ú❛ ♥➳✉ k + l < t❤➻ ✤✐➸♠ ❜➜t ✤ë♥❣ ✤â ❧➔ ❞✉② ♥❤➜t✳ ❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x0 X tũ ỵ t ỹ {xn} X ❜ð✐ xn = T n x0 , ✈ỵ✐ ♠å✐ n ≥ ❑❤✐ ✤â d(xn+1 , xn ) = d(T xn , T xn−1 ) = kd(xn , xn−1 ) kd(xn , xn−1 ) + ld(xn , T xn−1 ) k n d(x1 , x0 ) ✸✵ ❉♦ ✤â✱ ✈ỵ✐ n > m t❛ ❝â d(xn , xm ) d(xn , xn−1 ) + d(xn−1 , xn−2 ) + · · · + d(xm+1 , xm ) (hn−1 + hn−2 + · · · + hm )d(x1 , x0 ) km d(x1 , x0 ) 1−k ❱ỵ✐ ♠é✐ c ∈ E, c✱ t❛ ❝❤å♥ N1 ∈ N s❛♦ ❝❤♦ ♠å✐ m N1✳ ❑❤✐ ✤â ∗ d(xn , xm ) km d(x1 , x0 ) 1−k c ✈ỵ✐ c ✈ỵ✐ ♠å✐ n > m N1✳ ❙✉② r❛ {xn} ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ❧➔ ❦❤æ♥❣ tr õ tỗ t x X s❛♦ ❝❤♦ n−→∞ lim xn = x∗ ❚❛ ❝❤å♥ N2 ∈ N∗ s❛♦ ❝❤♦ d(xn, x∗) 3c ✈ỵ✐ ♠å✐ n N2✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ n N2 t❛ ❝â d(T x∗ , x∗ ) d(xn , T x∗ ) + d(xn , x∗ ) d(T xn−1 , T x∗ ) + d(xn , x∗ ) kd(xn−1 , x∗ ) + ld(T xn−1 , x∗ ) + d(xn , x∗ ) d(xn−1 , x∗ ) + d(xn , x∗ ) + d(xn , x∗ ) ❙✉② r❛ d(T x∗ , x∗ ) c/3 + c/3 + c/3 = c ❈❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❛ ✤÷đ❝ d(T x∗ , x∗ ) c m ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ mc − d(T x∗, x∗) ∈ P ✈ỵ✐ ♠å✐ m ❱➻ mc −→ ❦❤✐ m −→ ∞ ✈➔ P ✤â♥❣ ♥➯♥ −d(T x∗ , x∗ ) ∈ P ✳ ❈❤ó þ r➡♥❣ d(T x∗ , x∗ ) ∈ P ✱ ❞♦ ✤â d(T x∗, x∗) = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ T x∗ = x∗✱ tø ✤â x∗ ❧➔ ✤✐➸♠ t T sỷ tỗ t y∗ ∈ T s❛♦ ❝❤♦ T y∗ = y∗ ✈➔ k + l < 1✳ ❑❤✐ ✤â t❛ ❝â d(x∗ , y ∗ ) = d(T x∗ , T y ∗ ) kd(x∗ , y ∗ ) + ld(T x∗ , y ∗ )) = (k + l)d(x∗ , y ∗ ) ❚ø ✤â s✉② r❛ d(x∗, y∗) = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ x∗ = y∗ ❱➟② x∗ ❧➔ t t T ỵ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✳✷ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ →♥❤ ①↕ ✷✳✷✳✶ ▼ð ✤➛✉ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❈❤♦ X ❧➔ ♠ët t➟♣ ❤ñ♣✱ ❧➔ ♠ët q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥ tr➯♥ X ✱ F : X × X → X ❧➔ ♠ët →♥❤ ①↕✳ ❚❛ ♥â✐ F ❝â t➼♥❤ ✤ì♥ ✤✐➺✉ ❝❤ë♥ ♥➳✉ F (x, y) ✤ì♥ ✤✐➺✉ ❦❤ỉ♥❣ ❣✐↔♠ ✤è✐ ✈ỵ✐ x, ✤ì♥ ✤✐➺✉ ❦❤ỉ♥❣ t➠♥❣ ố ợ y ự ợ x1 ợ ♠å✐ y1 • t❛ ❝â F (x1, y) y2 t❛ ❝â F (x, y2 ) x2 F (x2 , y)✱ F (x, y1 )✳ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ ✈➔ (X, ) ❧➔ ♠ët t➟♣ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ❚❛ s➢♣ t❤ù tü tr➯♥ t X ì X ữ s ợ (x, y), (u, v) ∈ X × X ✱ t❛ ✈✐➳t (x, y) (u, v) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x u ✈➔ v y ◆➠♠ ✷✵✵✻✱ ❇❤❛s❦❛r ✈➔ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ✭❬✷❪✮ ✤➣ ự ỵ (X, d) ổ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✱ tr♦♥❣ ✤â (X, ) ❧➔ t➟♣ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ❈❤♦ F : X × X → X ❧➔ →♥❤ ①↕ ❝â t➼♥❤ ❝❤➜t ✤ì♥ sỷ tỗ t k [0, 1) s❛♦ ❝❤♦ d(F (x, y), F (u, v)) k (d(x, u) + d(y, v)) ✸✷ ✈ỵ✐ ♠å✐ u x✱ y v õ tỗ t x0 , y0 ∈ X s❛♦ ❝❤♦ x0 F (x0 , y0 ), F (x0 , y0 ) y0 t tỗ t x, y ∈ X t❤ä❛ ♠➣♥ F (x, y) = x F (y, x) = y ỵ ✷✳✾✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✱ tr♦♥❣ ✤â (X, ) ❧➔ t➟♣ s➢♣ t❤ù tü ❜ë ♣❤➟♥✳ ●✐↔ sû X ❝â t➼♥❤ ❝❤➜t (i) ◆➳✉ ♠ët ❞➣② ❦❤ỉ♥❣ ❣✐↔♠ xn −→ x t❤➻ xn x ✈ỵ✐ ♠å✐ n ∈ N∗ , (ii) ◆➳✉ ♠ët ❞➣② ❦❤æ♥❣ t➠♥❣ yn −→ y t❤➻ y yn ✈ỵ✐ ♠å✐ n ∈ N∗ ❈❤♦ F : X × X → X ❧➔ →♥❤ ①↕ ❝â t➼♥❤ ❝❤➜t ✤ì♥ ✤✐➺✉ ❝❤ë♥✳ sỷ tỗ t k [0, 1) s d(F (x, y), F (u, v)) ✈ỵ✐ ♠å✐ u x✱ y k (d(x, u) + d(y, v)) v ✳ õ tỗ t x0 , y0 X s❛♦ ❝❤♦ x0 F (x0 , y0 ), F (x0 , y0 ) y0 t tỗ t x, y X t❤ä❛ ♠➣♥ F (x, y) = x ✈➔ F (y, x) = y ✳ ✣✐➸♠ (x, y) t❤ä❛ ♠➣♥ F (x, y) = x ✈➔ F (y, x) = y tr ỵ ữủ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ →♥❤ ①↕ F ❱➲ s❛✉ ❝â ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❞↕♥❣ ỵ ỵ ợ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ ❦❤→❝ ♥❤❛✉ ✈➲ ❤➡♥❣ sè ✈➔ ❝→❝ tr÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤→❝ ♥❤❛✉✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ s➩ ❣✐ỵ✐ t❤✐➺✉ ♠ët sè ❦➳t q✉↔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ ❞↕♥❣ ♥➔② ✤÷đ❝ ❋✳ ❙❛❜❡t❣❤❛❞❛♠✱ ❍✳ P✳ ▼❛s✐❤❛ ✈➔ ❆✳ ❍✳ ❙❛♥❛t♣♦✉r ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ❬✻❪✳ ✷✳✷✳✷ ❚r÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✳ P❤➛♥ tû (x, y) ∈ ❣å✐ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ →♥❤ ①↕ F F (x, y) = x ✈➔ F (y, x) = y ✳ X×X : X×X X ỵ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ●✐↔ sû →♥❤ ①↕ F : X × X → X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ d(F (x, y), F (u, v)) kd(x, u) + ld(y, v) ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✱ tr♦♥❣ ✤â k, l ❧➔ ❝→❝ ❤➡♥❣ sè ❦❤ỉ♥❣ ➙♠ ✈ỵ✐ k + < 1✳ ❑❤✐ ✤â F ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❞✉② ♥❤➜t✳ ❈❤ù♥❣ x0, y0 X tũ ỵ t x1 = F (x0 , y0 ), y1 = F (y0 , x0 ), , xn+1 = F (xn , yn ), yn+1 = F (yn , xn ) ❑❤✐ ✤â tø ❣✐↔ t❤✐➳t t❛ ❝â d(xn , xn+1 ) = d(F (xn−1 , yn−1 ), F (xn , yn ) kd(xn−1 , xn ) + ld(yn−1 , yn ) ❚÷ì♥❣ tü t❛ ❝â d(yn , yn+1 ) = d(F (yn−1 , xn−1 ), F (yn , xn ) kd(yn−1 , yn ) + ld(xn−1 , xn ) ✣➦t dn = d(xn , xn+1 ) + d(yn , yn+1 ), t❛ ❝â dn = d(xn , xn+1 ) + d(yn , yn+1 ) kd(xn−1 , xn ) + ld(yn−1 , yn ) + kd(yn−1 , yn ) + ld(xn−1 , xn ) (k + l)(d(xn−1 , xn ) + d(yn−1 , yn )) = (k + l)dn−1 ✣➦t δ = k + l✱ tø ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â dn δdn−1 ··· δ n d0 ✸✹ ◆➳✉ d0 = t❤➻ (x0, y0) ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ F ✳ ❚❛ ①➨t trữớ ủ d0 > ợ ộ n m t ❝â d(xn , xm ) d(xn , xn−1 ) + d(xn−1 n, xn−2 ) + · · · + d(xm+1 n, xm ) d(yn , ym ) d(yn , yn−1 ) + d(yn−1 n, yn−2 ) + · · · + d(ym+1 n, ym ) ✈➔ ❉♦ ✤â d(xn , xm ) + d(yn , ym ) dn−1 + dn−2 + · · · + dm (δ n−1 + δ n−2 + · · · + δ m )d0 δm d0 1−δ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ {xn}, {yn} ❧➔ tr X õ tỗ t x∗ , y ∗ ∈ X s❛♦ ❝❤♦ lim xn = x∗ , ❱ỵ✐ c ∈ E, c✱ ✈ỵ✐ m N tỗ t N N s ❝❤♦ d(xn , x∗ ) ✈ỵ✐ ♠å✐ n N lim yn = y ∗ c/2m ✈➔ d(yn, y∗) c/2m ◆❤÷ ✈➟② d(F (x∗ , y ∗ ), x∗ ) d(F (x∗ , y ∗ ), xN +1 ) + d(xN +1 , x∗ ) = d(F (x∗ , y ∗ ), F (xN , yN )) + d(xn+1 , x∗ ) kd(xN , x∗ ) + ld(yN , y ∗ ) + d(xN +1 , x∗ ) c c c (k + l) + 2m 2m m ◆❤÷ ✈➟② d(F (x∗, y∗), x∗) mc ✈ỵ✐ ♠å✐ m ◆❤÷ ✈➟② d(F (x∗, y∗), x∗) = ✈➔ ❞♦ ✤â ✤â F (x∗ , y ∗ ) = x∗ ✳ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â F (y ∗ , x∗ ) = y ∗ , ❞♦ ✤â (x∗, y∗) ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ F ✸✺ ❈✉è✐ ❝ò♥❣ t❛ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ✤✐➸♠ t sỷ tỗ t (x , y ) ❧➔ ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❦❤→❝ ❝õ❛ F, ❦❤✐ ✤â d(x , x∗ ) = d(F (x , y ), F (x∗ , y ∗ )) kd(x , x∗ ) + ld(y , y ∗ ), d(y , y ∗ ) = d(F (y , x ), F (y ∗ , x∗ )) kd(y , y ∗ ) + ld(x , x∗ ) ✈➔ ❉♦ ✤â d(x , x∗ ) + d(y , y ∗ ) (k + l)(d(x , x∗ ) + d(y , y ∗ )) ❚ø ✤✐➲✉ ❦✐➺♥ k + l < t❛ s✉② r❛ d(x , x∗) + d(y , y∗) = ❑➨♦ t❤❡♦ (x , y ) = (x∗ , y ∗ ) ❞♦ ✤â (x∗ , y ∗ ) ❧➔ ✤✐➸♠ ❜➜t t ỵ t t ✤÷đ❝ ❤➺ q✉↔ ❍➺ q✉↔ ✷✳✸✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ●✐↔ sû →♥❤ ①↕ F : X × X → X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ d(F (x, y), F (u, v)) k (d(x, u) + d(y, v)) ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✱ tr♦♥❣ ✤â k ❧➔ ❤➡♥❣ sè ❦❤æ♥❣ ➙♠ ✈➔ k < 1✳ ❑❤✐ ✤â F ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❞✉② ♥❤➜t✳ ❱➼ ❞ö ✷✳✶✳ ❈❤å♥ E = R2✱ ❦➼ ❤✐➺✉ P = {(x, y) ∈ R2 : x, y 0} ⊂ R2 ✈➔ X = [0, 1] ❚❛ ✤à♥❤ ♥❣❤➽❛ d : X × X −→ E ✈ỵ✐ d(x, y) = (|x − y|, |x − y|) ❑❤✐ ✤â (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ❳➨t ❝→❝ →♥❤ ①↕ F, F1 : X × X −→ X ①→❝ ✤à♥❤ ❜ð✐ F (x, y) = x+y , F1 (x, y) = x+y ✸✻ ❑❤✐ ✤â F t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ tr♦♥❣ ❍➺ q✉↔ ✷✳✸ ✈ỵ✐ k = 1/3, tù❝ ❧➔ d(F (x, y), F (u, v)) (d(x, u) + d(y, v)) ❉♦ ✤â t❤❡♦ ❍➺ q✉↔ ✷✳✸✱ F ❝â ❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❧➔ ✤✐➸♠ (0, 0) ⑩♥❤ ①↕ F1 t❤ä❛ ♠➣♥ d(F1 (x, y), F1 (u, v)) (d(x, u) + d(y, v)) ◆â✐ ❝→❝❤ ❦❤→❝ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d(F1 (x, y), F1 (u, v)) k (d(x, u) + d(y, v)) ✈ỵ✐ k = ❚❛ ❞➵ ♥❤➟♥ t❤➜② tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ t❤➜② F1 ❝â ❤❛✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❧➔ (0, 0) ✈➔ (1, 1)✱ tù❝ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❦❤ỉ♥❣ ❧➔ ❞✉② ♥❤➜t✳ ◆❤÷ ✈➟② k + l < tr ỵ ✷✳✶✵ ✈➔ k < tr♦♥❣ ❍➺ q✉↔ ✷✳✸ ❧➔ q✉❛♥ trå♥❣ ✤è✐ ✈ỵ✐ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ✤✐➸♠ ❜➜t ỵ (X, d) ổ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ●✐↔ sû →♥❤ ①↕ F : X × X → X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ d(F (x, y), F (u, v)) kd(F (x, y), x) + ld(F (u, v), u) ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✱ tr♦♥❣ ✤â k, l ❧➔ ❝→❝ ❤➡♥❣ sè ❦❤æ♥❣ ➙♠✱ k + l < 1✳ ❑❤✐ ✤â F ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❞✉② ♥❤➜t✳ ự x0, y0 X tũ ỵ ✤➦t x1 = F (x0 , y0 ), y1 = F (y0 , x0 ), , xn+1 = F (xn , yn ), yn+1 = F (yn , xn ) ❑❤✐ ✤â tø ❣✐↔ t❤✐➳t t❛ ❝â d(xn , xn+1 ) = δd(xn , xn−1 ) d(yn , yn+1 ) = δd(yn , yn−1 ), tr♦♥❣ ✤â δ = k/(1−l) < 1✳ ▲➟♣ ❧✉➟♥ ❣✐è♥❣ ♥❤÷ tr♦♥❣ ỵ t s r {xn}, {yn} tr (X, d) õ tỗ t x, y∗ ∈ X ✸✼ s❛♦ ❝❤♦ lim xn = x∗ , ❱ỵ✐ c ∈ E, lim yn = y ∗ c✱ ✈ỵ✐ m ∈ N∗ ✈➔ ❝❤å♥ ♠ët sè tü ♥❤✐➯♥ N s❛♦ ❝❤♦ d(xn , x∗ ) = ((1 − l)/4m)c ✈ỵ✐ ♠å✐ n N ❑❤✐ ✤â t❛ ❝â d(F (x∗ , y ∗ ), x∗ ) d(xN +1 , F (x∗ , y ∗ )) + d(xN +1 , x∗ ) = d(F (xN , yN ), F (x∗ , y ∗ )) + d(xN +1 , x∗ ) kd(F (xN , yN ), xN ) + ld(F (x∗ , y ∗ ), x∗ ) + d(xN +1 , x∗ ), ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ d(F (x∗ , y ∗ ), x∗ ) k d(xN +1 , xN ) + d(xN +1 , x∗ ) 1l 1l c m m tũ ỵ ♥➯♥ d(F (x∗, y∗), x∗) = ✈➔ ❞♦ ✤â ✤â F (x∗, y∗) = x∗✳ ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü t❛ ❝â F (y∗, x∗) = y∗, ❞♦ ✤â (x∗, y∗) ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ F ❚❛ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ✤✐➸♠ ❜➜t ✤ë♥❣✳ ●✐↔ sû tỗ t (x , y ) ởt t ✤ë♥❣ ❦❤→❝ ❝õ❛ F, ❦❤✐ ✤â d(x , x∗ ) = d(F (x , y ), F (x∗ , y ∗ )) kd(F (x , y ), x ) + ld(F (x∗ , y ∗ ), x∗ ) = 0, ❞♦ ✤â x = x∗✳ ❚÷ì♥❣ tü t❛ ❝â y ❞✉② ♥❤➜t✳ = y∗✱ s✉② r❛ (x∗, y∗) ❧➔ ✤✐➸♠ t ỵ (X, d) ổ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ●✐↔ sû →♥❤ ①↕ F : X × X → X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ d(F (x, y), F (u, v)) kd(F (x, y), u) + ld(F (u, v), x) ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✱ tr♦♥❣ ✤â k, l ❧➔ ❝→❝ ❤➡♥❣ sè ❦❤æ♥❣ ➙♠✱ k + l < 1✳ ❑❤✐ ✤â F ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❞✉② ♥❤➜t✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝ơ♥❣ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ♥❤÷ ✣à♥❤ ỵ x0 , y0 X tũ ỵ ✈➔ ①➙② ❞ü♥❣ ❤❛✐ ❞➣② {xn , yn } ❣✐è♥❣ ữ tr ỵ tự x1 = F (x0 , y0 ), y1 = F (y0 , x0 ), , xn+1 = F (xn , yn ), yn+1 = F (yn , xn ) ❑❤✐ ✤â tø ❣✐↔ t❤✐➳t t❛ ❝â d(xn , xn+1 ) = d(F (xn−1 , yn−1 ), F (xn , yn )) kd(F (xn−1 , yn−1 ), xn ) + ld(F (xn , yn ), xn−1 ) (d(F (xn , yn ), xn ) + d(xn , xn−1 )), ✤✐➲✉ ✤â ❦➨♦ t❤❡♦ d(xn , xn+1 ) l d(xn , xn−1 ) 1−l d(yn , yn+1 ) l d(yn , xy−1 ) 1−l ❚÷ì♥❣ tü t❛ ❝â ✣✐➲✉ ♥➔② s✉② r❛ {xn}, {yn} ❧➔ ❝→❝ ❞➣② ❈❛✉❝❤② tr♦♥❣ (X, d)✱ õ tỗ t x, y X s lim xn = x∗ , ❱ỵ✐ c ∈ E, lim yn = y ∗ c✱ ✈ỵ✐ m ∈ N∗ ✈➔ ❝❤å♥ ♠ët sè tü ♥❤✐➯♥ N s❛♦ ❝❤♦ d(xn , x∗ ) = ((1 − l)/4m)c ✈ỵ✐ ♠å✐ n N ❑❤✐ ✤â t❛ ❝â d(F (x∗ , y ∗ ), x∗ ) d(xN +1 , F (x∗ , y ∗ )) + d(xN +1 , x∗ ) = d(F (xN , yN ), F (x∗ , y ∗ )) + d(xN +1 , x∗ ) kd(F (xN , yN ), x∗ ) + ld(F (x∗ , y ∗ ), xN ) + d(xN +1 , x∗ ), ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ d(F (x∗ , y ∗ ), x∗ ) 1+k c d(xN +1 , x∗ ) + d(xN , x∗ ) 1−l 1−l m d(F (x∗ , y ∗ ), x∗ ) = ✈➔ ❞♦ ✤â ✤â F (x∗ , y ∗ ) = x∗ m tũ ỵ ự tữỡ tü t❛ ❝â F (y∗, x∗) = y∗, ❞♦ ✤â (x∗, y∗) ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ F ❚➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ✤✐➸♠ ❜➜t ✤ë♥❣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tỹ ữ ỵ ợ tt ỵ ứ ỵ ỵ t õ q (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ●✐↔ sû →♥❤ ①↕ F : X × X → X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ k (d(F (x, y), x) + d(F (u, v), u)) ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✱ tr♦♥❣ ✤â k ❧➔ ♠ët ❤➡♥❣ sè ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ d(F (x, y), F (u, v)) k < 1✳ ❑❤✐ ✤â F ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❞✉② ♥❤➜t✳ ❍➺ q✉↔ ✷✳✺✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥ ✤➛② ✤õ✳ ●✐↔ sû →♥❤ ①↕ F : X × X → X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ k (d(F (x, y), u) + d(F (u, v), x)) ✈ỵ✐ ♠å✐ x, y, u, v ∈ X ✱ tr♦♥❣ ✤â k ❧➔ ♠ët ❤➡♥❣ sè ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ d(F (x, y), F (u, v)) k < 1✳ ❑❤✐ ✤â F ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ t ú ỵ F ỵ : XìX X d(F (x, y), F (u, v)) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ ✭tr♦♥❣ kd(F (x, y), x) + ld(F (u, v), u) ✈ỵ✐ ♠å✐ x, y, u, v ∈ X t❤➻ F ❝ô♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ s❛✉ ✤➙② d(F (x, y), F (u, v)) = d(F (u, v), F (x, y)) kd(F (u, v), u) + ld(F (x, y), x) ❉♦ ✤â F t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ d(F (x, y), F (u, v)) ữỡ tỹ F ỵ k+l d(F (x, y), x) + d(F (u, v), u) : X ×X → X d(F (x, y), F (u, v)) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ ✭tr♦♥❣ kd(F (x, y), u) + ld(F (u, v), x) ✈ỵ✐ ♠å✐ x, y, u, v ∈ X t❤➻ F ❝ô♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦ s❛✉ d(F (x, y), F (u, v)) k+l d(F (x, y), u) + d(F (u, v), x) ✹✵ ❑➳t ❧✉➟♥ ❱ỵ✐ ♠ư❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼♥❤ ❝❤➜t ✈➲ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥ ✈➔ ♠ët số ỵ t tr ợ ổ ❣✐❛♥ ♥➔②✱ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥✿ ♥â♥✱ ♥â♥ ❝❤✉➞♥ t➢❝✱ ♥â♥ ❝❤➼♥❤ q✉②✱ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥✱ sü ❤ë✐ tư tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥ ✈➔ ỵ tr ổ tr ♥â♥✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ✤÷đ❝ ①❡♠ ❧➔ ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✱ ❝➛♥ t❤✐➳t ❝❤♦ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ✷✳ P❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ởt số ỵ t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ ❝→❝ →♥❤ ①↕ tr♦♥❣ tr÷í♥❣ ủ ổ tr õ t ỵ ✷✳✶✱ ✷✳✷✱ ✷✳✸✱ ✷✳✹✱ ✷✳✺✱ ✷✳✻✱ ✷✳✼ ❧➔ ❝→❝ ❦➳t q✉↔ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ ❣✐ú❛ ❝→❝ ổ tr õ ỵ ✷✳✶✶✱ ✷✳✶✷ ❧➔ ❝→❝ ❦➳t q✉↔ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝❤♦ →♥❤ ①↕ ❣✐ú❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐ ❝❤ó♥❣ tỉ✐ s➩ t✐➳♣ tư❝ ♣❤→t tr✐➸♥ ✈➜♥ ✤➲ ♥➔② ✤è✐ ✈ỵ✐ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ s✉② rë♥❣ ❦❤→❝✳ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▼✳ ❆s❛❞✐ ❛♥❞ ❍✳ ❙♦❧❡✐♠❛♥✐ ✭✷✵✶✷✮✱ ✧ ❊①❛♠♣❧❡s ✐♥ ❈♦♥❡ ▼❡tr✐❝ ❙♣❛❝❡s✿ ❆ ❙✉r✈❡②✧✱ ▼✐❞❞❧❡ ✲ ❊❛st ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥t✐❢✐❝ ❘❡s❛r❝❤ ✶✶ ✭✶✷✮✿ ✶✻✸✻✲✶✻✹✵✱✷✵✶✷✳ ❬✷❪ ❚✳ ●✳ ❇❤❛s❦❛r ❛♥❞ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ✭✷✵✵✻✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡✲ ♦r❡♠ ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ◆♦♥❧✐♥✲ ❡❛r ❆♥❛❧②s✐s✿ ❚❤❡♦r②✳ ▼❡t❤♦❞s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✈♦❧✳ ✼✵✱ ♥♦✳✶✷✱ ♣♣✳✹✸✹✶ ✲✹✸✹✾✳ ❬✸❪ ▲✲●✳ ❍✉❛♥❣ ❛♥❞ ❳✳ ❩❛♥❣ ✭✷✵✵✼✮✱ ✧❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥ t❤❡♦r❡♠s ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣✧✱ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✈♦❧✳ ✸✸✷✱ ♥♦✳✷✱ ♣♣✳✶✹✻✽ ✲✶✹✼✻✱ ✷✵✵✼✳ ❬✹❪ ❳✳ ❍✉❛♥❣ ✱ ❈✳ ❩❤✉ ❛♥❞ ❳✳ ❲❡♥ ✭✷✵✶✵✮✱ ✧ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❡①♣❛♥❞✐♥❣ ♠❛♣♣✐♥❣s❛♥❞ ❝♦♥❡ ♠❡tr✐❝ s♣❛❝❡✧✱ ❆▼❙ ✷✵✶✵ ❙✉❜❥❡❝t ❈❧❛ss✐❢✐❝❛t✐♦♥✿ ✹✼❍✶✵✱ ✺✹❍✷✺✳ ❬✺❪ ❙❤✳ ❘❡③❛♣♦✉r ❛♥❞ ❘✳ ❍❛♠❧❜❛r❛♥✐ ✭✷✵✵✽✮✱ ✧❙♦♠❡ ♥♦t❡s ♦♥ t❤❡ ♣❛♣❡r ❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ♦❢ ❝♦♥❡ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✱ ✸✹✺✱ ♣♣ ✼✶✾✲✼✷✹✳ ❬✻❪ ❋✳ ❙❛❜❡t❣❤❛❞❛♠✱ ❍✳ P✳ ▼❛s✐❤❛ ✈➔ ❆✳ ❍✳ ❙❛♥❛t♣♦✉r ✭✷✵✵✾✮✱ ✧❙♦♠❡ ❈♦✉♣❧❡❞ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r❡♠s ✐♥ ❈♦♥❡ ▼❡tr✐❝ ❙♣❛❝❡s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❤tt♣s✿✴✴❞♦✐✳♦r❣✴✶✵✳✶✶✺✺✴✷✵✵✾✴✶✷✺✹✷✻✳ ... ▲➊ ❱❿◆ ▼■◆❍ ❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆ ❱⑨ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ✣■➎▼ ❇❻❚ ✣❐◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ư♥❣ ▼➣ số ữớ ữợ ❞➝♥ ❦❤♦❛ ❤å❝✿ P●❙✳❚❙✳ ❍⑨ ❚❘❺◆ P❍×❒◆● ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼ ▼ư❝ ❧ư❝ ▼Ð ✣❺❯ ✶ ❑❤ỉ♥❣ ❣✐❛♥... ①↕ ❝♦ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ỵ t tr ổ tr õ ởt số rở ỵ ỵ ①↕ ❝♦ ❝↔✐ t✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷ ▼ët sè ❞↕♥❣ ♠ð rë♥❣ ❦❤→❝ ✳ ✳ ✳ ✳ ✳ ✳... ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❝→❝ t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙② ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈➔ ởt số ỵ t t ❝➦♣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❣✐ú❛ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ♥➔②✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣