Về sự tồn tại tập bất biến của toán tử fractal đa trị sinh bởi các họ ánh xạ co kiểu ciric trên không gian mêtric

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ♣❤➵♠ ✈➝♥ ❤ï♥❣ ❱Ò sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤❛ trÞ s✐♥❤ ❜ë✐ ❝➳❝ ❤ä ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ❈✐r✐❝ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❧✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥✱ ✷✵✶✽ ❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ♣❤➵♠ ✈➝♥ ❤ï♥❣ ❱Ị sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤❛ trÞ s✐♥❤ ❜ë✐ ❝➳❝ ❤ä ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ❈✐r✐❝ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❧✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✽✹✻✵✶✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❚❙✳ ❱ị ❚❤Þ ❍å♥❣ ❚❤❛♥❤ ◆❣❤Ö ❆♥✱ ✷✵✶✽ ✐ ❧ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤✱ ♥❣❤✐➟♠ tó❝ ❝đ❛ ❈➠ ❚❙✳ ❱ị ❚❤Þ ❍å♥❣ ❚❤❛♥❤✳ ❚➠✐ ①✐♥ ❜➭② tá sù ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ❈➠✳ ❳✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ q✉ý t❤➬② ❝➠ ë ❇é ♠➠♥ ●✐➯✐ tÝ❝❤✱ ❱✐Ö♥ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❚❙✳ ◆❣✉②Ô♥ ❱➝♥ ➜ø❝ ❣✐➳♦ ✈✐➟♥ ❝❤đ ♥❤✐Ư♠ ❧í♣ ❝❛♦ ❤ä❝ ●✐➯✐ ❚Ý❝❤ ❦❤ã❛ ✷✹ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❑ü t❤✉❐t ❱Ü♥❤ ▲♦♥❣✱ ❙ë ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ tØ♥❤ ❱Ü♥❤ ▲♦♥❣✱ ❇❛♥ ●✐➳♠ ❍✐Ö✉ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❑ü t❤✉❐t ❱Ü♥❤ ▲♦♥❣✱ ệ rờ P ò ì ú ỡ t ề ệ t ợ t tr q trì ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ▲✉❐♥ ✈➝♥✳ ◆❤➞♥ ➤➞② t➠✐ ①✐♥ ❝➳♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ●✐➯✐ ❚Ý❝❤ ❦❤♦➳ ✷✹ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❑ü t❤✉❐t ❱Ü♥❤ ▲♦♥❣✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❣ë✐ ❧ê✐ ❝➳♠ ➡♥ ➤Õ♥ ❣✐❛ ➤×♥❤✱ ❜➵♥ ❜❒✱ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❣✐ó♣ t➠✐ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ▼➷❝ ❞ï ➤➲ ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝ ❤✐Ö♥ ❧✉❐♥ ✈➝♥✱ s♦♥❣ ❧✉❐♥ ✈➝♥ tr ỏ ữ s sót ợ ♥❤÷♥❣ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ ❝đ❛ q✉ý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥✳ ❱Ü♥❤ ▲♦♥❣✱ ♥❣➭② ✷✵ t❤➳♥❣ ✻ ♥➝♠ ✷✵✶✽ P❤➵♠ ❱➝♥ ❍ï♥❣ ✐✐ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✐✐ ▼ë ➤➬✉ ✶ ❈❤➢➡♥❣ ✶✳ ✶✳✶ ❑✐Õ♥ t❤ø❝ ❝➡ së ▼➟tr✐❝ ❍❛✉s❞♦r❢❢ ✈➭ sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤➡♥ trÞ ✶✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❙ù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤❛ trÞ ❈❤➢➡♥❣ ✷✳ ❱Ị sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤❛ trÞ s✐♥❤ ❜ë✐ ❝➳❝ ➳♥❤ ①➵ ➤❛ trÞ ❝♦ ❦✐Ĩ✉ ✷✳✶ C´ ✐r✐c´ ✶✶ ❱Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ➤➡♥ trÞ ✈➭ ➤❛ trÞ ❝♦ ❦✐Ĩ✉ C´ ✐r✐c´ ✷✳✷ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❱Ò sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ➤❛ trÞ ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤ä ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ C´ ✐r✐c´ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❑Õt ❧✉❐♥ ✸✽ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✾ ✶ ♠ë ➤➬✉ ◆➝♠ ✶✾✽✶✱ ❏✳ ❊✳ ❍✉t❝❤✐s♦♥ ❝❤Ø r❛ r➺♥❣ ❝ø ❝ã ♠ét ❤ä ❤÷✉ ❤➵♥ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ tr➟♥ Rn t❤× ❝ã ♠ét t♦➳♥ tư ❢r❛❝t❛❧ ✈➭ t❐♣ ❢r❛❝t❛❧ ❝❤Ý♥❤ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤ã✳ ❚❐♣ ❢r❛❝t❛❧ ❝ã r✃t ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ♥❤✐Ị✉ ❧Ü♥❤ ✈ù❝ ❝đ❛ ❦❤♦❛ ❤ä❝ ❝ị♥❣ ♥❤➢ tr♦♥❣ ➤ê✐ sè♥❣✳ ❈❤Ý♥❤ ✈× t❤Õ✱ ♥❣➢ê✐ t❛ ❧✉➠♥ t×♠ ❝➳❝❤ ♠ë ré♥❣ ✈✐Ö❝ ①➞② ❞ù♥❣ ❝➳❝ t❐♣ ❢r❛❝t❛❧✳ ◆➝♠ ✶✾✻✾✱ ◆❛❞❧❡r ➤➲ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ❝❤♦ ➳♥❤ ①➵ ➤❛ trÞ ✈➭ tõ ➤ã ❧ý t❤✉②Õt ể t ộ trị ũ ợ ♣❤➳t tr✐Ĩ♥ ♥❤❛♥❤ ❝❤ã♥❣ ✈➭ ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ●✐➯✐ tÝ❝❤ ♣❤✐ t✉②Õ♥✱ ➤➷❝ ❜✐Öt tr♦♥❣ ❦✐♥❤ tÕ✱ ❧ý tết tố ý tết trò ì tế ột r✃t tù ♥❤✐➟♥ ❧➭ ♥❣➢ê✐ t❛ ♥❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤ä ❝➳❝ ➳♥❤ ①➵ ❝♦ ➤❛ trÞ✳ ◆➝♠ ✶✾✼✶✱ C´ ✐r✐c´ ➤➲ ➤➢❛ r❛ ♠ét ❧♦➵✐ ➳♥❤ ①➵ ❝♦ ♠➭ ♥ã tỉ♥❣ q✉➳t ❝đ❛ ❝➳❝ ❧♦➵✐ ❝♦ ❦✐♥❤ ➤✐Ĩ♥ ♣❤ỉ ❜✐Õ♥✱ ♥❤➢ ❝♦ ❑❛♥♥❛♥✱ ❝♦ ❘❡✐❝❤✱ ❝♦ ❘✉s✱ ❝♦ ❍❛r❞②✲❘♦❣❡r✳✳✳✈➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤♦ ➳♥❤ ①➵ ➤➡♥ trÞ ✈➭ ➤❛ trÞ ❝♦ ❦✐Ĩ✉ ♥➭②✳ ◆❣❛② s❛✉ ➤ã✱ ❝ã r✃t ♥❤✐Ò✉ ♥❤➭ t♦➳♥ ❤ä❝ ♥❤➢ ❆✳ P❡tr✉s❡❧✱ ■✳ ❆✳ ❘✉s✱ ◆✳ ❱✳ ❉✉♥❣✱ P✳ ❑✉♠❛♠✱✳✳✳✱ ♥❣❤✐➟♥ ứ t ợ ề ết q tú ị ề sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ trị ể t ợt ứ ọ ✈➭ t×♠ ❤✐Ĩ✉ ❝➳❝ ✈✃♥ ➤Ị ♥➭②✱ ❝❤ó♥❣ t➠✐ ❝❤ä♥ ➤Ị t➭✐ ♥❣❤✐➟♥ ❝ø✉ ❝❤♦ ❧✉❐♥ ✈➝♥ ❝đ❛ ♠×♥❤ ❧➭ ✧❱Ị sù tå♥ t➵✐ ´ ✐r✐c´ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤❛ trÞ s✐♥❤ ❜ë✐ ❤ä ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ C tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✧✳ ✷✳ ▼ơ❝ ➤Ý❝❤ ♥❣❤✐➟♥ ❝ø✉ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ tì ể ết q t ợ ề tồ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣✿ ❝đ❛ ➳♥❤ ①➵ ➤➡♥ trÞ✱ ➤❛ trÞ ❝♦ s✐♥❤ ❜ë✐ ➳♥❤ ①➵ ➤➡♥ trÞ✱ ➤❛ trÞ ❝♦ ➳♥❤ ①➵ ➤➡♥ trÞ✱ ➤❛ trÞ ❝♦ C´ ✐r✐c´❀ ❝đ❛ ➳♥❤ ①➵ t❐♣ C´ ✐r✐c´❀ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤ä ❝➳❝ C´ ✐r✐c´ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ✸✳ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ✷ ➜è✐ t➢ỵ♥❣✿ ♥❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ q✉❛ t♦➳♥ tư ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❝➳❝ ➳♥❤ ①➵ ➤❛ trÞ ❝♦ ❦✐Ó✉ C´ ✐r✐c´ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ P❤➵♠ ✈✐✿ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ✹✳ ữ ó ó ủ rì ①➞② ❞ù♥❣✿ ♠➟tr✐❝ ❍❛✉s❞♦r❢❢✱ ➳♥❤ ①➵ t❐♣ s✐♥❤ ❜ë✐ ♠ét ➳♥❤ ①➵ ✈➭ t♦➳♥ tö ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤ä ❝➳❝ ➳♥❤ ①➵ ➤➡♥ trÞ ✈➭ ➤❛ trÞ✱ sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤➡♥ trÞ ✈➭ trị rì ết q t ợ ❝đ❛ ❝➳❝ ♥❤➭ ❚♦➳♥ ❤ä❝ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ t❐♣ ❜✃t ❜✐Õ♥ ❝ñ❛ ❧♦➵✐ ➳♥❤ ①➵ ❝♦ ➤➡♥ trÞ ❞♦ ❣ä✐ ❧➭ ❝♦ C´ ✐r✐c´ ❤❛② C´ ✐r✐c´ ➤➢❛ r❛ ✭t❛ tù❛ ❝♦✮ ✈➭ ❝ñ❛ ➳♥❤ ①➵ tù❛ ❝♦ tỉ♥❣ q✉➳t ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ✲ ❚r×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ C´ ✐r✐c´✳ ✺✳ ◆❤✐Ư♠ ✈ơ ♥❣❤✐➟♥ ❝ø✉ ✺✳✶✳ ➜ä❝ ❤✐Ĩ✉ ♠ét sè t➭✐ ❧✐Ư✉ ❧✐➟♥ q✉❛♥ ➤Õ♥ ♠➟tr✐❝ ❍❛✉s❞♦r❢❢✱ ❤Ư ❤➭♠ ❧➷♣ ➤➡♥ trÞ✱ ➤❛ trÞ✱ sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ q✉❛ ➳♥❤ ①➵ ✭➤➡♥ trÞ✱ ➤❛ trÞ✮ ❣å♠ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ C´ ✐r✐c´✱ ➤➡♥ trÞ ✈➭ ➤❛ trÞ ❝♦ ❦✐Ĩ✉ ❤Ư ❤➭♠ ❧➷♣ ➤➡♥ trÞ✱ ➤❛ trÞ C´ ✐r✐c´ ✈➭ C´ ✐r✐c´✱ t♦➳♥ tư ❢r❛❝t❛❧ sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ ❝➳❝ t♦➳♥ tư ♥➭②✳ ✺✳✷✳ ❚r×♥❤ ❜➭② ♠ét ❝➳❝❤ ❝ã ❤Ư t❤è♥❣ ❝➳❝ ❦Õt q✉➯ ✈Ị ❝➳❝ ✈✃♥ ➤Ị ♥ã✐ tr➟♥✱ ❝❤ó trä♥❣ ❝➳❝ ✈✃♥ ➤Ị ✈Ị sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ q✉❛ t♦➳♥ tư ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤Ư ❤➭♠ ❧➷♣ ❣å♠ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ C´ ✐r✐c´✳ ❚r×♥❤ ❜➭② ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ✺✳✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ♠Ư♥❤ ➤Ị✱ ➤Þ♥❤ ❧ý ✈Ị sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ q✉❛ t♦➳♥ tö ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤Ư ❤➭♠ ❧➷♣ ❣å♠ ❝➳❝ ➳♥❤ ①➵ ➤❛ trÞ ❝♦ ❦✐Ĩ✉ C´ ✐r✐c´ tr×♥❤ ❜➭② tr♦♥❣ ❧✉❐♥ ✈➝♥ ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❦❤➠♥❣ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ❝❤ø♥❣ ♠✐♥❤ ❝ß♥ ✈➽♥ t➽t✳ ✸ ✻✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❧ý t❤✉②Õt✿ ➜ä❝ t➭✐ ❧✐Ö✉✱ s✉② ❞✐Ơ♥ ❧➠❣✐❝✱ t➢➡♥❣ tù ❤ã❛✱ tỉ♥❣ q✉➳t ❤ã❛✳ ✼✳ ❈✃✉ tró❝ ❧✉❐♥ ✈➝♥ ◆❣♦➭✐ ♣❤➬♥ ▼ơ❝ ❧ơ❝✱ ▼ë ➤➬✉✱ ❑Õt ❧✉❐♥ ✈➭ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦✳ ◆é✐ ❞✉♥❣ ❧✉❐♥ ợ trì t ế t❤ø❝ ❝➡ së ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ së ❝➬♥ ❞ï♥❣ tr♦♥❣ ❧✉❐♥ ✈➝♥✱ ❣å♠✿ ✶✳✶✳ ▼➟tr✐❝ ❍❛✉s❞♦r❢❢ ✈➭ sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤➡♥ trÞ✳ ✶✳✷✳ ❙ù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤❛ trÞ✳ ❈❤➢➡♥❣ ✷✳ ❱Ị sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤❛ trÞ s✐♥❤ ❜ë✐ ❝➳❝ ➳♥❤ ①➵ ➤❛ trÞ ❝♦ ❦✐Ĩ✉ C´ ✐r✐c´✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛✿ ➳♥❤ ①➵ ➤➡♥ trÞ✱ ➤❛ trÞ ❝♦ ❦✐Ĩ✉ ①➵ ➤➡♥ ✭➤❛✮ trÞ ❝♦ ❝♦ ❦✐Ĩ✉ C´ ✐r✐c´ ✈➭ C´ ✐r✐c´❀ C´ ✐r✐c´❀ ➳♥❤ ①➵ t❐♣ s✐♥❤ ❜ë✐ ➳♥❤ t♦➳♥ tư ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤Ư ❤➭♠ ❧➷♣ trị trì í ụ ọ ◆é✐ ❞✉♥❣ ❝ơ t❤Ĩ ❣å♠✿ ✷✳✶✳ ❱Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ➤➡♥ trÞ ✈➭ ➤❛ trÞ ❝♦ ❦✐Ĩ✉ C´ ✐r✐c´✳ ✷✳✷✳ ❱Ị sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ➤❛ trÞ ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤ä ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ C´ ✐r✐c´✳ ✶ ❝❤➢➡♥❣ ✶ ❑✐Õ♥ t❤ø❝ ❝➡ së ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ së ❝➬♥ ❞ï♥❣ tr♦♥❣ ✈✐Ư❝ tr×♥❤ ❜➭② ♥é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥✳ ➜ã ❧➭ ①➞② ❞ù♥❣ ♠➟tr✐❝ ❍❛✉s❞♦r❢❢ tr➟♥ ❧í♣ ❝➳❝ t❐♣ ❝♦♥ ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ①➞② ❞ù♥❣ ➳♥❤ ①➵ t❐♣ ✈➭ t♦➳♥ tö ❢r❛❝t❛❧✱ ➤å♥❣ t❤ê✐ ❝❤Ø r❛ sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤➡♥ trÞ ✈➭ ➤❛ trÞ✳ ✶✳✶ ▼➟tr✐❝ ❍❛✉s❞♦r❢❢ ✈➭ sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ➤➡♥ trÞ ✶✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ x ∈ X ✈➭ A, B ⊂ X ✳ ❚❛ ❦ý ❤✐Ö✉ i) D(x, A) = inf{d(x, a) : a ∈ A}❂ inf d(x, a) a∈A ➤✐Ó♠ ✈➭ ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ tõ x ➤Õ♥ t❐♣ A❀ ii) δ(A, B) = sup{d(a, b) : a ∈ A, b ∈ B} ❦❤✐ A = B ❧➭✿ d(A) ⇒ d(A) = sup{d(a, b) : a, b ∈ A} ✈➭ ❣ä✐ ❧➭ ➤➢ê♥❣ t❤× δ(A, B) ❦Ý ❤✐Ư✉ ❦Ý♥❤ ❝đ❛ t❐♣ iii) ρ(A, B) = sup{D(a, B)} ✈➭ ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ tõ A ➤Õ♥ B ❀ a∈A iv) K(X) = {A : A ⊂ X, A ❝♦♠♣❛❝t, A = Ø}✳ ✶✳✶✳✷ ❇ỉ ➤Ị✳ ✭❬✶❪✮ ❱í✐ ♠ä✐ A, B, C ⊂ X t❛ ❧✉➠♥ ❝ã✱ ρ(A, B) ≤ ρ(A, C) + ρ(C, B) ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ❜✃t ❦ú a ∈ A✱ t❛ ❝ã✿ D(a, B) = inf d(a, b) ≤ inf {d(a, c) + d(c, b)} ✈í✐ ❜✃t ❦ú c ∈ C b∈B b∈B = d(a, c) + D(c, B) ✈í✐ ❜✃t ❦ú c ∈ C A❀ ✷ ❉♦ ➤ã✱ t❛ ❝ã D(a, B) ≤ inf d(a, c) + sup D(c, B) c∈C c∈C = D(a, C) + ρ(C, B) ▲✃② s✉♣r✐♠✉♠ ❤❛✐ ✈Õ ❝ñ❛ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❤❡♦ a ∈ A t❛ ❝ã ρ(A, B) ≤ ρ(A, C) + ρ(C, B) ❉Ô t❤✃② r➺♥❣✱ tå♥ t➵✐ ♠➟tr✐❝ tr➟♥ ✶✳✶✳✸ A, B ⊂ X ♠➭ ρ(A, B) = ρ(B, A)✳ ❉♦ ➤ã✱ ρ ❦❤➠♥❣ ❧➭ K(X)✳ ▼Ư♥❤ ➤Ị s❛✉ ➤➞② ❝❤♦ t❛ ❝➳❝❤ ①➳❝ ➤Þ♥❤ ♠➟tr✐❝ tr➟♥ K(X)✳ ▼Ư♥❤ ➤Ị✳ ✭❬✶❪✮ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❳Ðt ➳♥❤ ①➵ h : K(X) × K(X) → [0; +∞) (A, B) → h(A, B) = max{ρ(A, B), ρ(B, A)} ❑❤✐ ➤ã✱ h ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ K(X)✳ ♠➟tr✐❝ ➤➬② ➤đ t❤× ❍➡♥ ♥÷❛✱ ♥Õ✉ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ (K(X), h) ❝ị♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❦✐Ĩ♠ tr❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ♠ét ♠➟tr✐❝ ➤è✐ ✈í✐ i) ❚õ ❝➳❝❤ ①➳❝ ➤Þ♥❤ h t❛ ❝ã h(A, B) ≥ ✈í✐ ♠ä✐ h✳ A, B ∈ K(X) h(A, B) = ❦Ð♦ t❤❡♦ ρ(A, B) = ρ(B, A) = 0✳ ❉♦ ➤ã A = B ✈× A, B ∈ K(X)✳ ii) ❍✐Ó♥ ♥❤✐➟♥ t❛ ❝ã h(A, B) = h(B, A) ✈í✐ ♠ä✐ A, B ∈ K(X)✳ iii) ❚❛ ❝❤ø♥❣ ♠✐♥❤ ✈í✐ ♠ä✐ A, B, C ∈ K(X)✱ t❛ ❝ã h(A, B) ≤ h(A, C) + h(C, B) ❚❤❡♦ ❇ỉ ➤Ị ✶✳✶✳✷✱ t❛ ❝ã ρ(A, B) ≤ ρ(A, C) + ρ(C, B) ✈➭ ρ(B, A) ≤ ρ(B, C) + ρ(C, A)✱ ✈í✐ ♠ä✐ A, B, C ∈ K(X)✳ ❉➱♥ ➤Õ♥ h(A, B) = max{ρ(B, A), ρ(A, B)} ≤ max{ρ(A, C) + ρ(C, B), ρ(B, C) + ρ(C, A)} ≤ max{ρ(A, C), ρ(C, A)} + max{ρ(C, B), ρ(B, C)} = h(A, C) + h(B, C) ❱❐②✱ h ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ K(X)✳ ✈➭ ✸ ✶✳✶✳✹ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶❪✮ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ❤➭♠ ①➳❝ ➤Þ♥❤ ♥❤➢ ë ▼Ư♥❤ ➤Ị ✶✳✶✳✸ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠➟tr✐❝ ❍❛✉s❞♦r❢❢ tr h ợ K(X) ú t trì ❜➭② ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ♠➟tr✐❝ ❍❛✉s❞♦r❢❢ ❝ñ❛ ❤➭♠ ✶✳✶✳✺ D(x, A), ρ(A, B) ❝➬♥ ❞ï♥❣ ❝❤♦ tr×♥❤ ❜➭② ♥é✐ ❞✉♥❣ ❝❤Ý♥❤ ❝đ❛ ❧✉❐♥ ✈➝♥✳ ❇ỉ ➤Ị✳ ✭❬✶❪✮ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ◆Õ✉ A ⊂ B t❤× ρ(C, A) ≥ ρ(C, B) ✈➭ ρ(A, C) ≤ ρ(B, C) ✈í✐ ❜✃t ❦ú C ⊂ X ✳ ❈❤ø♥❣ ♠✐♥❤✳ ❉♦ ❤❛② t❛ ❝ã A ⊂ B ♥➟♥ ỗ c C t ó bB D(c, A) ≥ D(c, B)✳ ▲✃② s✉♣r❡♠✉♠ ❤❛✐ ✈Õ ❝ñ❛ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❤❡♦ sup D(c, A) ≥ sup D(c, B), c∈C c ∈ C ✱ t❛ ❝ã ρ(C, A) ≥ ρ(C, B) ❤❛② c∈C ❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù✱ t❛ t❤✉ ➤➢ỵ❝ ✶✳✶✳✻ inf d(c, a) ≥ inf d(c, b) a∈A ❇ỉ ➤Ị✳ ✭❬✶❪✮ ❱í✐ ♠ä✐ ρ(A, C) ≤ ρ(B, C)✳ A, B, C ⊂ X t❛ ❧✉➠♥ ❝ã h(A, B ∪ C) ≤ max{h(A, B), h(A, C)} ❈❤ø♥❣ ♠✐♥❤✳ ❱× B, C ⊂ B ∪ C ρ(A, B ∪ C) ≤ ρ(A, B) ❙✉② r❛ ♥➟♥ t❤❡♦ ❇ỉ ➤Ị ✶✳✶✳✺ t❛ ❝ã ✈➭ ρ(A, B ∪ C) ≤ ρ(A, C) ρ(A, B ∪ C) ≤ max{ρ(A, B), ρ(A, C)}✳ ▼➷t ❦❤➳❝✱ ρ(B ∪ C, A) = sup D(x, A) = max{sup D(x, A), sup D(x, A)} x∈B∪C x∈B x∈C = max{ρ(B, A), ρ(C, A)} ❉♦ ➤ã✱ t❛ ❝ã h(A, B ∪ C) = max{ρ(A, B ∪ C), ρ(B ∪ C, A)} ≤ max{ρ(A, B), ρ(B, A), ρ(A, C), ρ(C, A)} = max{max{ρ(A, B), ρ(B, A)}, max{ρ(A, C), ρ(C, A)}} = max{h(A, B), h(A, C)} ✷✺ d(xn , x∗ ) + δ(x∗ , T x∗ ), δ(x∗ , T xn )) ❈❤♦ t❛ ➤➢ỵ❝ n→∞ ✭✷✳✷✸✮ tr♦♥❣ ✭✷✳✷✸✮✱ sư ❞ơ♥❣ ✭✷✳✷✵✮✱ ✭✷✳✷✶✮✱ ✭✷✳✷✷✮ ✈➭ ❤➭♠ ϕ ❧✐➟♥ tô❝ δ(x∗ , T x∗ ) = 0✳ ❉➱♥ ➤Õ♥ T x∗ = {x∗ } ✈➭ ❞♦ ➤ã x∗ ∈ SFT ✳ ❚✐Õ♣ t❤❡♦✱ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ FT = SFT = {x∗ }✳ ▲✃② y ∗ ∈ FT ✳ ❚õ ✭✷✳✶✺✮ t❛ ➤➢ỵ❝ δ(y ∗ , T y ∗ ) ≤ δ(T y ∗ , T y ∗ ) ≤ ϕ(d(y ∗ , T y ∗ ), δ(y ∗ , T y ∗ ), δ(y ∗ , T y ∗ ), δ(y ∗ , T y ∗ )) ≤ ϕ(δ(y ∗ , y ∗ ), δ(y ∗ , T y ∗ ), δ(y ∗ , T y ∗ ), δ(y ∗ , T y ∗ ), δ(y ∗ , T y ∗ )) ❚õ ❇ỉ ➤Ị ✷✳✶✳✾ t❛ ❝ã ❱× t❤Õ δ(y ∗ , T y ∗ ) = ❞♦ ➤ã T y ∗ = {y ∗ } ♥❣❤Ü❛ ❧➭ y ∗ ∈ SFT ✳ FT = SFT ✳ ❚õ ✭✷✳✶✹✮✱ t❛ ❝ã d(x∗ , y ∗ ) = δ(T x∗ , T y ∗ ) ≤ ϕ(d(x∗ , y ∗ ), δ(x∗ , T x∗ ), δ(y ∗ , T y ∗ ), δ(x∗ , T y ∗ ), δ(y ∗ , T x∗ )) = ϕ(d(x∗ , y ∗ ), d(x∗ , x∗ ), d(y ∗ , y ∗ ), d(x∗ , y ∗ ), d(y ∗ , x∗ )) = ϕ(d(x∗ , y ∗ ), d(x∗ , y ∗ ), d(x∗ , y ∗ ), d(x∗ , y ∗ ), d(x∗ , y ∗ )) ❚õ ❇ỉ ➤Ị ✷✳✶✳✾ ✈➭ tõ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ s✉② r❛ d(x∗ , y ∗ ) = 0✱ ❞♦ ➤ã x∗ = y ∗ ✳ ❱❐②✱ t❛ ❝ã FT = SFT = {x∗ }✳ (ii)✳ ❝❤♦ ❚õ ✭✷✳✷✵✮ t❛ ❝ã x0 ∈ X ✈➭ tå♥ t➵✐ ♠ét q✉ü ➤➵♦ {xn } ❝ñ❛ T t➵✐ x0 s❛♦ lim xn = x∗ ✳ n→∞ ◆➝♠ ✷✵✶✺✱ P✳ ❑✉♠❛♠✱ ◆✳ ❱✳ ❉✉♥❣ ✈➭ ❑✳ ❙✐tt❤✐t❤❛❦❡r♥❣❦✐❡t ➤➲ ♠ë ré♥❣ ❦Õt q✉➯ ♥➝♠ ✶✾✼✹ ❝ñ❛ C´ ✐r✐c´ t❤Ĩ ❤✐Ư♥ tr♦♥❣ ❝➳❝ ➜Þ♥❤ ❧Ý ✷✳✶✳✻✱ ➜Þ♥❤ ❧Ý ✷✳✶✳✼ ❜➺♥❣ ❝➳❝❤ t❤❛② ➳♥❤ ①➵ tù❛ ❝♦ ❜ë✐ ➳♥❤ ①➵ tù❛ ❝♦ tæ♥❣ q✉➳t✳ ❈➳❝ ❦Õt q✉➯ ➤ã ợ trì s T ị ĩ ✭❬✺❪✮ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ T ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ tù❛ ❝♦ tỉ♥❣ q✉➳t ♥Õ✉ tå♥ t➵✐ q ∈ [0, 1) : X → X✳ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ✱ t❛ ❝ã d(T x, T y) ≤ q.max{d(x, y), d(x, T x), d(y, T y), d(x, T y), d(y, T x), ✷✻ d(T x, x), d(T x, T x), d(T x, y), d(T x, T y)} ✷✳✶✳✶✷ ➜Þ♥❤ ❧Ý✳ ✭❬✺❪✮ ❈❤♦ X →X (X, d) ✭✷✳✷✹✮ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ●✐➯ sö r➺♥❣ ❧➭ ♠ét ➳♥❤ ①➵ tù❛ ❝♦ tæ♥❣ q✉➳t ✈➭ X ❧➭ T ✲q✉ü ➤➵♦ ➤➬② ➤ñ✳ T : ❑❤✐ ➤ã✱ t❛ ❝ã ✐✮ T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ x∗ ∈ X ✳ lim T n x = x∗ ✈í✐ ❜✃t ❦ú x ∈ X ✳ ✐✐✮ n→∞ n ∗ ✐✐✐✮ d(T x, x ) ≤ qn 1−q d(x, T x) ✈í✐ ♠ä✐ x∈X ❈❤ø♥❣ ♠✐♥❤✳ ✐✮✳ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ x∈X ✈➭ T ✈➭ n ∈ N✳ ❝ã ể t ộ ỗ i n − 1❀ ≤ j ≤ n✱ t❛ ❝ã d(T i x, T j x) = d(T T i−1 x, T T j−1 x) ✭✷✳✷✺✮ ≤ q max{d(T i−1 x, T j−1 x), d(T i−1 x, T T i−1 x), d(T j−1 x, T T j−1 x), d(T i−1 x, T T j−1 x), d(T j−1 x, T T i−1 x), d(T T i−1 x, T i−1 x), d(T T i−1 x, T T i−1 x), d(T T i−1 x, T j−1 x), d(T T i−1 x, T T j−1 x)} = q max{d(T i−1 x, T j−1 x), d(T i−1 x, T i x), d(T j−1 x, T j x), d(T i−1 x, T j x), d(T j−1 x, T i x), d(T i+1 x, T i−1 x), d(T i+1 x, T i x), d(T i+1 x, T j−1 x), d(T i+1 x, T j x)} ≤ q.δ[OT (x, n)] tr♦♥❣ ➤ã δ[OT (x, n)] = max{d(T i x, T j x) : ≤ i, j ≤ n}✳ ❚õ ✭✷✳✷✺✮✱ ✈× ≤ q < 1✱ ♥➟♥ tå♥ t➵✐ kn (x) ≤ n s❛♦ ❝❤♦ d(x, T kn (x) x) = δ[OT (x, n)] ❑❤✐ ➤ã✱ t❛ ❝ã d(x, T kn (x) x) ≤ d(x, T x) + d(T x, T kn (x) x) ≤ d(x, T x) + q.δ[OT (x, n)] = d(x, T x) + q.d(x, T kn (x) x) ✭✷✳✷✻✮ ✷✼ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ δ[OT (x, n)] = d(x, T kn (x) x) ≤ ❱í✐ ♠ä✐ d(x, T x) 1−q n, m ≥ ✈➭ n < m✱ tõ ➤✐Ị✉ ❦✐Ư♥ tù❛ ❝♦ tỉ♥❣ q✉➳t ❝đ❛ T d(T n x, T m x) = d(T T n−1 x, T m−n+1 T n−1 x) ✭✷✳✷✼✮ ✈➭ ✭✷✳✷✼✮ t❛ ❝ã ✭✷✳✷✽✮ ≤ q.δ[OT (T n−1 x, m − n + 1)] = q.d(T n−1 x, T km−n+1 (T n−1 x) = q.d(T T n−2 x, T km−n+1 (T T n−1 x) n−1 x)+1 T n−2 x) ≤ q δ[OT (T n−2 x, km−n+1 (T n−1 x) + 1)] ≤ q δ[OT (T n−2 x, m − n + 2)] ≤ ≤ q n δ[OT (x, m)] qn ≤ d(x, T x) 1−q ❱× lim q n = 0✱ ♥➟♥ {T n x} ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱× X n→∞ ♥➟♥ tå♥ t➵✐ x∗ ∈ X ❧➭ T ✲q✉ü ➤➵♦ ➤➬② ➤ñ✱ s❛♦ ❝❤♦ lim T n x = x∗ ✭✷✳✷✾✮ n→∞ ❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ tù❛ ❝♦ tỉ♥❣ q✉➳t ❝đ❛ T ♠ét ❧➬♥ ♥÷❛✱ t❛ ❝ã d(x∗ , T x∗ ) ≤ d(x∗ , T n+1 x) + d(T n+1 x, T x∗ ) ✭✷✳✸✵✮ = d(x∗ , T n+1 x) + d(T T n x, T x∗ ) ≤ d(x∗ , T n+1 x) + q max{d(T n x, x∗ ), d(T n x, T T n x), d(x∗ , T x∗ ), d(T n x, T x∗ ), d(x∗ , T T n x), d(T T n x, T n x), d(T T n x, T T n x), d(T T n x, x∗ ), d(T T n x, T x∗ )} = d(x∗ , T n+1 x) + q max{d(T n x, x∗ ), d(T n x, T n+1 x), d(x∗ , T x∗ ), d(T n x, T x∗ ), d(x∗ , T n+1 x), d(T n+2 x, T n x), d(T n+2 x, T n+1 x), d(T n+2 x, x∗ ), d(T n+2 x, T x∗ )} ▲✃② ❣✐í✐ ❤➵♥ ❦❤✐ q.d(x∗ , T x∗ )✳ n→∞ tr♦♥❣ ✭✷✳✸✵✮✱ ✈➭ sư ❞ơ♥❣ ✭✷✳✷✾✮✱ t❛ ❝ã d(x∗ , T x∗ ) ≤ ✷✽ ❱× q ∈ [0, 1)✱ ♥➟♥ t❛ ➤➢ỵ❝ d(x∗ , T x∗ ) = 0✱ ❞♦ ➤ã✱ ➤✐Ó♠ ❜✃t ➤é♥❣✳ ❇➞② ❣✐ê t❛ ❝❤Ø r❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❧➭ ❤❛✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ❱× T T x ∗ = T x∗ ✳ ❱❐② ❧➭ ❞✉② ♥❤✃t✳ ●✐➯ sö T ❝ã x∗ , y ∗ ❧➭ tù❛ ❝♦ tæ♥❣ q✉➳t✱ ♥➟♥ t❛ ❝ã d(x∗ , y ∗ ) = d(T x∗ , T y ∗ ) ≤ q max{d(x∗ , y ∗ ), d(x∗ , T x∗ ), d(y ∗ , T y ∗ ), d(x∗ , T y ∗ ), d(y ∗ , T x∗ ), d(T x∗ , x∗ ), d(T x∗ , T x∗ ), d(T x∗ , y ∗ ), d(T x∗ , T y ∗ )} = q.d(x∗ , y ∗ ) ❱× q ∈ [0, 1)✱ ➤é♥❣ ❝đ❛ T ♥➟♥ t❛ ➤➢ỵ❝ d(x∗ , y ∗ ) = 0✳ ❉➱♥ ➤Õ♥ x∗ = y ∗ ✳ ❱❐② ➤✐Ó♠ ❜✃t ❧➭ ❞✉② ♥❤✃t✳ ii)✳ ➜➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ ❜ë✐ ✭✷✳✷✾✮✳ iii)✳ ▲✃② ❣✐í✐ ❤➵♥ ❦❤✐ m → ∞ tr♦♥❣ ✭✷✳✷✽✮✱ t❛ ❝ã d(T n x, x∗ ) ≤ ✷✳✶✳✶✸ ❍Ö q✉➯✳ ✭❬✺❪✮ ❈❤♦ qn 1−q d(x, T x)✳ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ T : X → X ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✭✶✮ X ❧➭ T ✲q✉ü ➤➵♦ ➤➬② ➤ñ✳ ✭✷✮ ❚å♥ t➵✐ k ∈ N ✈➭ q ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ✱ d(T k x, T k y) ≤ q max{d(x, y), d(x, T k x), d(y, T k y), d(x, T k y), d(y, T k x)} ✭✷✳✸✶✮ ❑❤✐ ➤ã t❛ ❝ã (i) T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ (ii) d(T n x, x∗ ) ≤ x∈X ✈➭ qm 1−q x∗ ∈ X ❀ max{d(T i x, T i+k x) : i = 0, 1, , k − 1} ✈í✐ ♠ä✐ n ∈ N tr♦♥❣ ➤ã m ❧➭ sè ♥❣✉②➟♥ ❧í♥ ♥❤✃t ❦❤➠♥❣ ✈➢ỵt q✉➳ n k❀ (iii) lim T n x = x∗ ✈í✐ ♠ä✐ x ∈ X ✳ n→∞ ➜Þ♥❤ ❧Ý s❛✉ ➤➞② tr×♥❤ ❜➭② ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ tù❛ ❝♦ tỉ♥❣ q✉➳t ➤❛ trÞ✳ ✷✳✶✳✶✹ ➜Þ♥❤ ❧Ý✳ ✭❬✺❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ F : X → BN (X) ❧➭ ➳♥❤ ①➵ ➤❛ trÞ tù❛ ❝♦ tỉ♥❣ q✉➳t ✈➭ X ❧➭ F ✲q✉ü ➤➵♦ ➤➬② ➤ñ✳ ❑❤✐ ➤ã✱ t❛ ❝ã ✷✾ ✭✶✮ F ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ ✭✷✮ ❱í✐ ♠ä✐ x0 ∈ X ✱ x∗ ∈ X F x∗ = {x∗ }✳ ✈➭ tå♥ t➵✐ ♠ét q✉ü ➤➵♦ {xn }n ❝ñ❛ F t➵✐ x0 s❛♦ ❝❤♦ lim xn = x∗ ✈í✐ ♠ä✐ x ∈ X ✳ n→∞ ✭✸✮ d(xn , x∗ ) ≤ (q 1−a )n d(x0 , x1 ) ✈í✐ ♠ä✐ 1−q 1−a n ∈ N✱ tr♦♥❣ ➤ã a < ❧➭ sè ❞➢➡♥❣ ❜✃t ❦×✳ ❈❤ø♥❣ ỗ a (0, 1) ✈➭ ①➳❝ ➤Þ♥❤ ➳♥❤ ①➵ ➤➡♥ trÞ T :X →X x ∈ X ✱ ❧✃② T x ∈ F x t❤á❛ ♠➲♥ d(x, T x) ≥ q a ρ(x, F x)✳ ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ✈Ị F ✲ q✉ü ➤➵♦ ➤➬② ủ tì ỗ x, y X t ❝ã d(T x, T y) ≤ ρ(F x, F y) ≤ q max{d(x, y), ρ(x, F x), ρ(y, F y), D(x, F y), D(y, F x), D(F x, x), D(F x, F x), D(F x, y), D(F x, F y)} = q.q −a max{q a d(x, y), q a ρ(x, F x), q a ρ(y, F y), q a D(x, F y), q a D(y, F x), q a D(F x, x), q a D(F x, F x), q a D(F x, y), q a D(F x, F y)} ≤ q 1−a max{d(x, y), d(x, T x), d(y, T y), d(x, T y), d(y, T x), d(T x, x), d(T x, T x), d(T x, y), d(T x, T y)} ❚❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✶✳✶✷✱ t❛ ❦Õt ❧✉❐♥ r➺♥❣ ❑❤✐ ➤ã ρ(x∗ , F x∗ ) ≤ q a d(x∗ , T x∗ ) = ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ F ✈➭ T ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝ã ♥❣❤Ü❛ ❧➭ ρ(x∗ , F x∗ ) = 0✳ ❉♦ ➤ã x∗ ✳ x∗ F x∗ = {x } ữ ết q ủ ị í ✷✳✶✳✶✷ tr♦♥❣ ➤ã xn = T n x ✈í✐ ♠ä✐ n ∈ N✱ t❛ ❝ã ✭✷✮ ✈➭ ✭✸✮ ✳ ❱Ý ❞ơ s❛✉ ❝❤Ø r❛ ❝➳❝ ❧í♣ ➳♥❤ ①➵ tù❛ ❝♦ tỉ♥❣ q✉➳t t❤ù❝ sù ré♥❣ ❤➡♥ ❧í♣ ❝➳❝ ➳♥❤ ①➵ tù❛ ❝♦✳ ✷✳✶✳✶✺ X = {1, 2, 3, 4} ✈í✐ d ợ ị ế x = y;    d(x, y) = ♥Õ✉ (x, y) ∈ {(1, 3), (3, 1), (1, 4), (4, 1)};     1 tr♦♥❣ ❝➳❝ tr➢ê♥❣ ợ ò í ụ T : X X ợ ị T = T = 1, T = 2, T = ❚❤❐t ✈❐②✱ t❛ ❝ã 1) d(x, y) ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ♥❤➢ tr➟♥ ❧➭ ♠ét ♠➟tr✐❝ ➤➬② ➤đ✳ 2) ❚❛ ❦✐Ĩ♠ tr❛ tÝ♥❤ tù❛ ❝♦ ✈➭ tù❛ ❝♦ tỉ♥❣ q✉➳t ❝ñ❛ ❚✱ t❛ ❝ã d(T x, T y) = d(1, 1) = ♥Õ✉ x, y ∈ {1, 2}; d(T 1, T 3) = d(T 2, T 3) = d(1, 2) = 1; d(T 1, T 4) = d(T 2, T 4) = d(1, 3) = 2; d(T 1, 3) = d(T 2, 3) = d(1, 3) = 2; d(T 1, 4) = d(T 2, 4) = d(1, 4) = 2; d(T 3, T 4) = d(2, 3) = 1; d(3, 4) = d(3, T 3) = d(4, T 3) = d(4, T 4) = 1; d(T 3, 3) = d(T 2, 3) = d(1, 3) = 2; d(T 4, 4) = d(T 3, 4) = d(2, 4) = 1; d(T 3, T 3) = d(T 2, T 3) = d(1, 2) = 1; d(T 3, 4) = d(T 2, 4) = d(1, 4) = 2; d(T 4, T 4) = d(T 3, T 4) = d(2, 3) = 1; d(T 3, T 4) = d(T 2, T 4) = d(1, 3) = ❈➳❝ tÝ♥❤ t♦➳♥ tr➟♥ ❝❤♦ t❤✃② r➺♥❣ ❦❤➠♥❣ ❝ã sè ❦❤➠♥❣ ❧➭ tù❛ ❝♦ ✈í✐ x=3 ✈➭ y=4 ✈× ≤ q < t❤á❛ ♠➲♥ ♣❤➢➡♥❣ tr×♥❤ ✭✷✳✷✹✮✳ ❚✉② ♥❤✐➟♥✱ ♠ä✐ T T ❧➭ tù❛ ❝♦ tæ♥❣ q✉➳t ✈× ✭✷✳✷✹✮ ➤ó♥❣ ❦❤✐ ❧✃② x, y ∈ X ✳ ❍➡♥ ♥÷❛✱ ❞Ơ ❝❤Ø r❛ X ❧➭ T ✲q✉ü ➤➵♦ ➤➬② ➤đ✳ q ∈ [0.5, 1) ✈➭ ✈í✐ ✸✶ ✷✳✷ ❱Ị sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ➤❛ trÞ ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤ä ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ C´ ✐r✐c´ ❚r➢í❝ ❤Õt✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ t♦➳♥ tư ❢r❛❝t❛❧ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✷✳✷✳✶ ➜Þ♥❤ ❧Ý✳ ✭❬✽❪✮ ❈❤♦ (X, d) C´ ✐r✐c´ ❦❤✉②Õt δ(x, T x) ✈➭ δ(y, T y)✳ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ❝❤♦ T : X → Pcl (X) ❧➭ ♠ét t♦➳♥ tư ➤❛ trÞ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥✳ ●✐➯ sư tå♥ t➵✐ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ✈➭ t t từ ế ợ ị : R3+ → R+ s❛♦ ❝❤♦ ❤➭♠ Ψ : R+ → R+ Ψ(t) := ϕ(t, t, t) t❤á❛ ♠➲♥ ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ (1) Ψn (t) → ❦❤✐ n → +∞✱ ✈í✐ ♠ä✐ t > 0❀ (2) (t − Ψ(t)) → +∞ ❦❤✐ t → +∞✳ ●✐➯ sư r➺♥❣ ✈í✐ ♠ä✐ x, y ∈ X ✱ h(T x, T y) ≤ ϕ(d(x, y), D(x, T y), D(y, T x)).( ) Tˆ : K(X) → K(X) s✐♥❤ r❛ ❜ë✐ T ❝ã t❐♣ ❜✃t ❜✐Õ♥ ❞✉② ∗ ˆ(A∗ ) = A∗ ✳ ♥❤✃t✱ tø❝ ❧➭✱ tå♥ t➵✐ ❞✉② ♥❤✃t A ∈ K(X) s❛♦ ❝❤♦ T ❑❤✐ ➤ã ➳♥❤ ①➵ t❐♣ ❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ Tˆ t❤á❛ ♠➲♥ h(TˆA, TˆB) ≤ ϕ(h(A, B), h(A, T B), h(B, T A)), ✈í✐ ♠ä✐ A, B ∈ K(X)✳ ✸✷ ❚❤❐t ✈❐②✱ t❛ ❝ã ρ(T A, T B) = sup ρ(T a, T B) a∈A = sup{ inf ρ(T a, T b)} a∈A b∈B ≤ sup{ inf h(T a, T b)} a∈A b∈B ≤ sup{ inf ϕ(d(a, b), D(a, T b), D(b, T a))} a∈A b∈B ≤ sup ϕ{ inf (d(a, b), inf D(a, T b), inf D(b, T a)} a∈A b∈B b∈B b∈B = sup ϕ(D(a, B), D(a, T B), D(T a, B)) a∈A = ϕ(sup D(a, B), sup D(a, T B), sup D(T a, B)) a∈A a∈A a∈A = ϕ(ρ(A, B), ρ(A, T B), ρ(T A, B)) ≤ ϕ(h(A, B), h(A, T B), h(B, T A)) ❚➢➡♥❣ tù ❝❤♦ ρ(T A, T B)✳ ❱❐②✱ t❛ ❝ã h(T A, T B) ≤ ϕ(h(A, B), h(A, T B), h(B, T A)) ➳♣ ụ ị í ủ s T tì T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣ K(X)✱ tø❝ ❧➭ tå♥ t➵✐ ❞✉② ♥❤✃t t❐♣ A∗ ∈ K(X) s❛♦ ❝❤♦ Tˆ(A∗ ) = A∗ ✳ ✷✳✷✳✷ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶❪✮ t xF ế ỉ ế ỗ ❝❤♦ ✈í✐ ♠ä✐ X ❝ã t➞♠ F ♠ä✐ ➳♥❤ ①➵ ➤❛ trÞ F : X → Y x, y ∈ BX (x, r) t❤× U F y ⊂ U✱ ❝đ❛ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ F x✱ tå♥ t➵✐ ♠ét sè ❞➢➡♥❣ r tr♦♥❣ ➤ã BX (x, r) s❛♦ ❧➭ ❤×♥❤ ❝➬✉ tr♦♥❣ x ✈➭ ❜➳♥ ❦Ý♥❤ r✳ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ♥ã ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ t➵✐ x ∈ F✳ ▼➷❝ ❞ï t♦➳♥ tö ❢r❛❝t❛❧ s✐♥❤ ❜ë✐ ❝➳❝ ➳♥❤ ①➵ ➤❛ trÞ ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧Ý ✷✳✷✳✶ ❦❤➠♥❣ ❝ã t❐♣ ❜✃t ❜✐Õ♥✱ ♥❤➢♥❣ t❤➟♠ ❣✐➯ t❤✐Õt tÝ♥❤ ❣✐❛♦ ❤♦➳♥ ❣✐÷❛ ❝➳❝ ➳♥❤ ①➵ ✈➭ ❤Ư ❝❤Ø ❣å♠ ❤❛✐ ➳♥❤ ①➵✱ tr♦♥❣ ❬✹❪✱ ❝➳❝ t➳❝ ❣✐➯ ❝❤Ø r❛ ❦Õt ❧✉❐♥ s❛✉✳ ✸✸ ➜Þ♥❤ ❧Ý✳ ✭❬✹❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ✷✳✷✳✸ Pcl (X) ❧➭ ❝➳❝ ➳♥❤ ①➵ ➤❛ trÞ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ ✈➭ ❤➭♠ sè s❛♦ ❝❤♦ ✈í✐ ♠ä✐ F1 , F2 : X → ϕ1 , ϕ2 : R3+ → R+ ❧➭ ❝➳❝ i ∈ {1, 2} t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉❀ (1) ϕi ❧➭ ❝➳❝ ❤➭♠ sè ❧✐➟♥ tô❝ ✈➭ t➝♥❣ t❤❡♦ tõ♥❣ ❜✐Õ♥❀ (2) lim ψin (t) = ✈í✐ ♠ä✐ n→∞ t > ✈➭ lim (t − ψi (t)) = ∞✱ t→∞ tr♦♥❣ ➤ã ψi (t) = ϕi (t, t, t) ✈í✐ ♠ä✐ t ∈ R+ ❀ (3) ❱í✐ ♠ä✐ x, y ∈ X t❤× h(Fi x, Fi y) ≤ ϕi (d(x, y), D(x, Fi y), D(y, Fi x)); ✭✷✳✸✷✮ (4) F1 oF2 x = F2 oF1 x✱ ✈í✐ ♠ä✐ x ∈ X ✳ ❑❤✐ ➤ã✱ t♦➳♥ tư rt trị T : K(X) K(X) ợ ➤Þ♥❤ ❜ë✐ T A = F1 A ∪ F2 A ❝ã Ýt ♥❤✃t ♠ét t❐♣ ❜✃t ❜✐Õ♥✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ♠ä✐ ❞✉② ♥❤✃t i ∈ {1, 2}✱ t❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✷✳✶✱ t❛ t❤✃② r➺♥❣ tå♥ t➵✐ A∗i ∈ K(X) t❤á❛ ♠➲♥ Fi A∗i = A∗i ✳ ❍➡♥ ♥÷❛✱ t❤❡♦ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➜Þ♥❤ ❧Ý ✷✳✷✳✶✱ ✈í✐ ♠ä✐ A ∈ K(X)✱ lim Fin (A) = A∗i ✭✷✳✸✸✮ n→∞ tr♦♥❣ (K(X), h)✳ ➜➷t ∞ ∞ A = n=0 ❚❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ t ỗ A K(X) {xk } A rờ ợ t ột số ỗ F2n (A∗1 ) F1n (A∗2 ) ∪ ∗ n=0 ∗ ✈➭ T (A ) = A∗ ✳ t❛ ①❡♠ ①Ðt ❤❛✐ tr➢ê♥❣ ❤ỵ♣ s❛✉✳ n0 s❛♦ ❝❤♦ {xkl } ⊂ F1n0 (A∗2 ) ❤♦➷❝ {xkl } ⊂ F2n0 (A∗1 ) {xkl } s ủ ỗ {xk } ì F1n0 (A2 ), F2n0 (A∗1 ) ∈ K(X) ♥➟♥ tå♥ t➵✐ ♠ét ỗ ộ tụ tr ủ {xkl } ó ũ ột ỗ ộ tụ ủ {xk } rờ ợ t ột ỗ s kn {nl } ủ {n} ột ỗ kn k nl ủ {k} xknl ∈ F1 l (A∗2 ) ✈í✐ ♠ä✐ knl ✱ ❤♦➷❝ xknl ∈ F2 l (A∗1 ) ✈í✐ ♠ä✐ knl ✳ ❚❛ ❝❤Ø ✸✹ ❝➬♥ ①❡♠ ①Ðt tr➢ê♥❣ ❤ỵ♣ kn xknl ∈ F1 l A∗2 ✈í✐ ♠ä✐ k nl ✱ tr➢ê♥❣ ợ ò s r t tự t ❝ã lim h(F1n A∗2 , A∗1 ) = 0, n→∞ tr♦♥❣ ➤ã h(F1n A∗2 , A∗1 ) = max{ sup an ∈F1n A∗2 d(an , A∗1 ), sup d(a, F1n A∗2 )} a∈A∗1 ❈ã ♥❣❤Ü❛ ❧➭ lim d(xknl , A∗1 ) = knl →∞ ❱× A∗1 ∈ K(X)✱ tå♥ t➵✐ a1kn ∈ A∗1 s❛♦ ❝❤♦ l d(xknl , A∗1 ) = d(xknl , a1kn ) l ✈í✐ ♠ä✐ knl ✳ ❱× t❤Õ lim d(xknl , a1kn ) = knl →∞ ❱× ✭✷✳✸✹✮ l A∗1 ∈ K(X)✱ tå♥ t➵✐ ♠ét ❞➲② {a1kn } ❝ñ❛ {a1kn } s❛♦ ❝❤♦ ls l lim a1kn = a1 ∈ A∗1 knls →∞ ❱í✐ ♠ä✐ k n ls ✭✷✳✸✺✮ ls t❛ t❤✃② r➺♥❣ d(xknl , a1 ) ≤ d(xknl , a1kn ) + d(a1kn , a1 ) s s ls ls ❇➺♥❣ ❝➳❝❤ ➳♣ ❞ô♥❣ ✭✷✳✸✺✮ ✈➭ ✭✷✳✸✻✮ t❛ ❝ã lim d(xknl , a1 ) = knls →∞ ❱× ✈❐② s {xknl } ❧➭ ♠ét ❞➲② ❤é✐ tơ ❝đ❛ {xk }✳ s ❚õ ❤❛✐ tr➢ê♥❣ ❤ỵ♣ tr➟♥ t❛ t❤✃② r➺♥❣ A∗ ∈ K(X)✳ ❇➞② ❣✐ê t❛ ❝❤Ø r❛ r➺♥❣ T (A∗ ) = A∗ ✳ ❱× ✈❐② ✈í✐ ♠ä✐ x ∈ X t❛ ❝ã F1 oF2 x = F2 oF1 x ❞♦ ➤ã F1 oF2 (A) = ✸✺ F2 oF1 (A) ✈í✐ ọ A K(X) ì tế t ợ ∗ T (A ) = T ( ∞ f1n x∗2 n=0 ∞ = = = = = f2n x∗1 ) ∪ n=0 ∞ ∞ ∞ n ∗ n ∗ f1 ( ∪ f2 x1 ) ∪ f2 ( f1 x2 ∪ f2n x∗1 ) n=0 n=0 n=0 n=0 ∞ ∞ ∞ ∞ n ∗ n ∗ n ∗ ( f1 of1 x2 ∪ f1 of2 x1 ) ∪ ( f2 of1 x2 ∪ f2 of2n x∗1 ) n=0 n=0 n=0 n=0 ∞ ∞ ∞ ∞ ( f1n+1 x∗2 ∪ f2n of1 x∗1 ) ∪ ( f1n of2 x∗2 ∪ f2n+1 x∗1 ) n=0 n=0 n=0 n=0 ∞ ∞ ∞ ∞ f1n+1 x∗2 ∪ ( f2n x∗1 ) ∪ ( f1n x∗2 ∪ f2n+1 x∗1 ) n=0 n=0 n=0 n=0 ∞ ∞ f1n x∗2 ∪ f2n x∗1 = A∗ n=0 n=0 f1n x∗2 ❱Ý ❞ô s❛✉ tr♦♥❣ ❬✼❪ ♥❤➺♠ ❝❤Ø r❛ r➺♥❣ tr♦♥❣ ✭✷✳✷✮ ✈➭ tr♦♥❣ ✭✷✳✶✹✮ t❛ ❦❤➠♥❣ t❤Ó t❤❛② δ(T x, T y) ❜ë✐ h(T x, T y) ✈➭ ❝ị♥❣ ❦❤➠♥❣ t❤Ĩ t❤❛② ✈Õ ♣❤➯✐ ❝đ❛ ( ) tr♦♥❣ ➜Þ♥❤ ❧Ý ✷✳✷✳✶ ❜ë✐ q max{d(x, y), δ(x, T x), δ(y, T y), D(x, T y), D(y, T x)} ❤♦➷❝ ❜ë✐ q max{d(x, y), h(x, T x), h(y, T y), D(x, T y), D(y, T x)} ✈í✐ q ✷✳✷✳✹ ❱Ý ❞ô✳ ✭❬✸❪✮ ▲✃② ❧❐♣ ➳♥❤ ①➵ X = N✱ d : X ì X R+ ợ ➤Þ♥❤ ❜ë✐    ♥Õ✉ x = y    d(x, y) = ♥Õ✉ |x − y| =     1 ♥Õ✉ |x − y| ≥ T : X → Pb (X) t❤á❛ T x = {x + 1, x + 2} ✈í✐ ♠ä✐ x ∈ X ✳ ❑❤✐ ➤ã✱ t❛ ❝ã ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉✳ ∈ [0, 1)✳ ✸✻ (1) (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ❀ (2) T x ∈ Pcp (X) ✈í✐ ♠ä✐ x ∈ X ✈➭ T ❧➭ ➳♥❤ ①➵ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥❀ (3) h(T x, T y) = 21 max{δ(x, T x), δ(y, T y)} ✈í✐ ♠ä✐ x, y ∈ X ❀ (4) h(T x, T y) = 12 max{h(x, T x), h(y, T y)} ✈í✐ ♠ä✐ x, y ∈ X ❀ (5) T ✈➭ T ❦❤➠♥❣ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝ã ✭✶✮✱ ✭✷✮ ✈➭ ✭✺✮ ❧➭ râ r➭♥❣✳ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ ✭✸✮ ✈➭ ✭✹✮✳ ❚r➢í❝ ❤Õt✱ ➤Ĩ ý r➺♥❣ ỗ xX t ó h(x, T x) = max{D(x, T x), sup d(b, x)} b∈T x = max{ inf d(b, x), sup d(b, x)} b∈T x b∈T x = sup d(b, x) = δ(x, T x) b∈T x ❉♦ ➤ã✱ ✈í✐ ♠ä✐ x, y ∈ X ✱ t❛ ❝ã max{h(x, T x), h(y, T y)} = max{δ(x, T x), δ(y, T y)} ❱× ✈❐②✱ t❛ ❝❤Ø ❝➬♥ ❝❤ø♥❣ ♠✐♥❤ ✭✸✮✳ ❱í✐ ♠ä✐ x