❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❍♦➭♥❣ ❚❤Þ ❚❤❛♥❤ ❍✉②Ị♥ ❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❊❞❡❧st❡✐♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥ ✲ ✷✵✶✻ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❍♦➭♥❣ ❚❤Þ ❚❤❛♥❤ ❍✉②Ị♥ ❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❊❞❡❧st❡✐♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✻ ▼ơ❝ ▲ơ❝ ❚r❛♥❣ ▼ơ❝ ❧ơ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▲ê✐ ♥ã✐ ➤➬✉ ❈❤➢➡♥❣ ■✳ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ ❝đ❛ ❑❛r❛♣✐♥❛r ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷✳ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ ❝ñ❛ ❑❛r❛♣✐♥❛r ✳ ✳ ✳ ✺ ✶✳✶✳ ❈➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ së ❈❤➢➡♥❣ ■■✳ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ✶✺ ✷✳✶✳ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ❝đ❛ ❑❛r❛✲ ♣✐♥❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷✳ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ❝đ❛ P♦♣❡s❦✉ ✷✽ ❑Õt ❧✉❐♥ ✹✵ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✹✶ ✐ ▼ë ➤➬✉ ❚r♦♥❣ ❣✐➯✐ tÝ❝❤ ❤➭♠ ♣❤✐ t✉②Õ♥✱ ▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ➤❛♥❣ ♥❣➭② ❝➭♥❣ ➤➢ỵ❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✱ ❜ë✐ ✈× ♥ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ ❦❤➠♥❣ ❝❤Ø tr♦♥❣ ♠ét sè ❝❤✉②➟♥ ♥❣➭♥❤ ❝ñ❛ t♦➳♥ ❤ä❝✱ ❝➳❝ ♥❣➭♥❤ ❦ü t❤✉❐t ♠➭ ❝ß♥ tr♦♥❣ ❝➳❝ ♥❣➭♥❤ ✈Ị ❦✐♥❤ tÕ✳ ❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧Ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❇❛♥❛❝❤✳ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ✭✶✾✷✷✮ ❧➭ ❦Õt q✉➯ ❦❤ë✐ ➤➬✉ ❝❤♦ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❞➵♥❣ ❝♦✱ ♥❤➢♥❣ ♣❤➯✐ ➤Õ♥ ữ ủ tế ỷ ợ t tr✐Ĩ♥ ♠➵♥❤ ♠Ï✳ ◆ã ❝❤♦ ♣❤Ð♣ t❛ ①➞② ❞ù♥❣ ♥❤÷♥❣ t❤✉❐t t♦➳♥ t×♠ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥✳ ◆❤✐Ị✉ ❦Õt q✉➯ t✐➟✉ ❜✐Ĩ✉ t❤❡♦ ❤➢í♥❣ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ❝đ❛ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❝ã t❤Ĩ ❦Ĩ ➤Õ♥ ♥❤➢ ❝ñ❛ ❝➳❝ t➳❝ ❣✐➯ ▼✳ ❊❞❡❧st❡✐♥✱ ❚✳ ❙✉③✉❦✐✱ ❘✳ ❑❛♥♥❛♥✱ ❙✳ ❘❡✐❝❤✱ ●✳ ❊✳ ❍❛r❞②✱ ❚✳ ❉✳ ❘♦❣❡rs✱ ▲✳ ❇✳ ❈✐r✐✬❝✱✳✳✳ ◆➝♠ ✶✾✻✷✱ ▼✳ ❊❞❡❧st❡✐♥ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý ♥ỉ✐ t✐Õ♥❣ s❛✉ ➤➞②✿ ✧❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝ ✈➭ T : X → X ✳ ●✐➯ sö r➺♥❣ d(T x, T y) < d(x, y) ✈í✐ ♠ä✐ x, y ∈ X ♠➭ x = y ✳ ❑❤✐ ➤ã✱ T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣✧✳ ◆➝♠ ✷✵✵✽✱ ❚✳ ❙✉③✉❦✐ ➤➲ ❣✐í✐ t❤✐Ư✉ ♠ét ❧♦➵✐ ➳♥❤ ①➵ ♠í✐ ✈➭ tr×♥❤ ❜➭② ♠ét ♥❣✉②➟♥ ❧ý ♠ë ré♥❣ ❝ñ❛ ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✱ tr♦♥❣ ➤ã tÝ♥❤ ➤➬② ➤đ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ❝ị♥❣ ❝ã t❤Ĩ ➤➢ỵ❝ ➤➷❝ tr➢♥❣ ❜ë✐ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵✳ ❙❛✉ ➤ã ♠ét sè t➳❝ ❣✐➯ ➤➲ t×♠ ❝➳❝❤ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ö♠ ➳♥❤ ①➵ ❙✉③✉❦✐ ✈➭ t❤✐Õt ❧❐♣ ị ý ể t ộ ữ ①➵ ❙✉③✉❦✐ s✉② ré♥❣ ♥➭② tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❚õ ❦Õt q✉➯ ➤➲ t❤✉ ➤➢ỵ❝ ♥➝♠ ✷✵✵✽✱ ♥➝♠ ✷✵✵✾✱ ❚✳ ❙✉③✉❦✐ ➤➲ ➤➢❛ r❛ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝✱ T : X → X ✳ ●✐➯ sö r➺♥❣ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ x = y ✱ ♥Õ✉ 12 d(x, T x) < d(x, y) t❤× t❛ ❝ã d(T x, T y) < d(x, y)✳ ❑❤✐ ➤ã✱ T ❝ã ❦Õt q✉➯ s❛✉ ➤➞② ♠➭ ♥ã ❧➭ ♠ét ♠ë ré♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ❊❞❡❧st❡✐♥✿ ✧❈❤♦ ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣✧✳ ❚õ ➤ã r✃t ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ ➤➲ t×♠ ❝➳❝❤ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ♥➭② ❝đ❛ ▼✳ ❊❞❡❧st❡✐♥ ✈➭ ❚✳ ❙✉③✉❦✐ ♥❤➺♠ ➤➢❛ r❛ ❝➳❝ ❦Õt q✉➯ ♠í✐ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣❛❝t✱ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❦✐Ó✉ ❙✉③✉❦✐✱ ➳♥❤ ①➵ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥✱ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ➤Þ♥❤ ❧ý✱ ❤Ư q✉➯ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ❝❤ø♥❣ ♠✐♥❤ ✧ ❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✧ ❝ß♥ ✈➽♥ t➽t✳ ❱× t❤Õ ❝❤ó♥❣ t➠✐ ❝❤ä♥ ➤Ị t➭✐ ❝❤♦ ❧✉❐♥ ✈➝♥ ❝đ❛ ♠×♥❤ ❧➭✿ ✐✐ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ ❝đ❛ ❑❛r❛♣✐♥❛r✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ▼ơ❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❧✉❐♥ ✈➝♥ ✱ ❜❛♦ ❣å♠ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠➭ ❝❤ó♥❣ ❧➭ ♠ë ré♥❣ ♠ét sè ❦Õt q✉➯ ❝đ❛ ❑❛r❛♣✐♥❛r✱ ❙✉③✉❦✐✳ ▼ơ❝ ✷✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡♥st❡✐♥ ❝đ❛ ❑❛r❛♣✐♥❛r✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ❝➳❝ ➤Þ♥❤ ❧ý ➤ã✳ ◆❣♦➭✐ r ú t ò trì ột số ệ q ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ụ ú t trì ột số ị ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ❝đ❛ ❑❛r❛♣✐♥❛r✳ ▼ơ❝ trì ột số ị ý ể t ộ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ❝đ❛ P♦♣❡s❦✉✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ❝➳❝ ❦Õt q✉➯ ➤ã✳ ❚r×♥❤ ❜➭② ♠ét sè ❤Ư q✉➯ ✈➭ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝❤✉ ➤➳♦ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ t➳❝ ❣✐➯ ①✐♥ ❜➭② tá sù ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ❚❤➬②✳ ◆❤➞♥ ❞Þ♣ ♥➭② ❡♠ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý t❤➬②✱ ❝➠ ❣✐➳♦ tr♦♥❣ tæ ●✐➯✐ ❚Ý❝❤ ❦❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ t❐♥ t×♥❤ ❣✐ó♣ ➤ì ❡♠ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤♦➳ ✷✷ ●✐➯✐ ❚Ý❝❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ✈➭ ❣✐❛ ➤×♥❤✱ ➤å♥❣ ♥❣❤✐Ư♣ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ♥❤✃t ➤Ĩ ❣✐ó♣ t➠✐ ❤♦➭♥ t❤➭♥❤ tèt ♥❤✐Ư♠ ✈ơ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ▼➷❝ ❞ï ➤➲ tÝ❝❤ ❝ù❝ t×♠ ❤✐Ĩ✉ ✈➭ ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✳ ❚➳❝ ❣✐➯ r✃t ♠♦♥❣ ♥❤❐♥ ợ ữ ý ế ó ó ủ qý ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ◆❣❤Ư ❆♥✱ t❤➳♥❣ ✵✽ ♥➝♠ ✷✵✶✻ ❍♦➭♥❣ ❚❤Þ ❚❤❛♥❤ ❍✉②Ị♥ ✐✐✐ ❝❤➢➡♥❣ ✶ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ ❝đ❛ ❑❛r❛♣✐♥❛r ❈➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ së ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ❇❛♦ ❣å♠ ❝➳❝ ♥é✐ ❞✉♥❣✿ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝✱ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❦✐Ó✉ ❙✉③✉❦✐✱ ➳♥❤ ①➵ ❦✐Ó✉ ❊❞❡❧st❡✐♥✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵✱ ❦❤➳✐ ♥✐Ư♠ ➤✐Ĩ♠ ❜✃t ộ ể t ộ rì ột số ị ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ tù ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s❛✉✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ ♠➟tr✐❝ tr➟♥ X ✭❬✶❪✮ ❈❤♦ t❐♣ ợ X d : X ì X R ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮ d(x, y) ≥ ✈í✐ ♠ä✐ x, y ∈ X ✭✷✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✸✮ d(x, y) ≤ d(x, z) + d(z, y) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ❚❐♣ X ✈➭ ❦Ý ❤✐Ư✉ ❧➭ ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ ✈➭ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤♦➯♥❣ ữ x y ị ĩ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ✈➭ (Y, ρ)✳ ➳♥❤ ①➵ f : (X, d) → (Y, ρ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ α ∈ [0, 1) s❛♦ ❝❤♦ ρ(f (x) , f (y)) ≤ αd (x, y) ✶ ✈í✐ ♠ä✐ x, y ∈ X ➜Þ♥❤ ❧ý✳ ✶✳✶✳✸ ➤➬② ➤đ✱ ➤✐Ĩ♠ f :X→X x∗ ∈ X ➜✐Ó♠ ①➵ ❧➭ ➳♥❤ ①➵ ❝♦ tõ s❛♦ ❝❤♦ x∗ ∈ X X (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑❤✐ ➤ã tå♥ t➵✐ ❞✉② ♥❤✃t f (x∗ ) = x∗ ✳ ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ f✳ ➜Þ♥❤ ❧ý✳ ✶✳✶✳✹ X ✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✮ ●✐➯ sö ✭❬✻❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T : X → ❧➭ ♠ét tù ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝ ψ (d (T x, T y)) ≤ ψ (d (x, y)) − ϕ (d (x, y)) tr♦♥❣ ➤ã ψ, ϕ : [0, +∞) → [0, +∞) ✈í✐ ♠ä✐ x, y ∈ X, ❧➭ ❝➳❝ ❤➭♠ ❧✐➟♥ tơ❝✱ ➤➡♥ ➤✐Ư✉ t➝♥❣ ✈➭ ψ(t) = ϕ(t) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t ❂ ✵✳ ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ộ t ị ý ợ ết ế ❧➭ ♠ét ❦Õt q✉➯ ❝ỉ ➤✐Ĩ♥ ❝đ❛ ❣✐➯✐ tÝ❝❤ ♣❤✐ t✉②Õ♥✳ ◆ã ❝ã r✃t ♥❤✐Ị✉ ❞➵♥❣ tỉ♥❣ q✉➳t✳ ◆➝♠ ✷✵✵✽✱ ❙✉③✉❦✐ ❬✾❪✱ ➤➲ ❣✐í✐ t❤✐Ư✉ ♠ét ➳♥❤ ①➵ ❧♦➵✐ ♠í✐ ❧➭ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ➤ã✱ ➤å♥❣ t❤ê✐ ➤➷❝ tr➢♥❣ tÝ♥❤ ➤➬② ➤ñ ❝ñ❛ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❜ë✐ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❧♦➵✐ ♥➭②✳ ✶✳✶✳✺ ➜Þ♥❤ ❧ý✳ ✭❬✾❪✮ ❈❤♦ ❧➭ ♠ét ➳♥❤ ①➵ tr➟♥ ❝❤♦ ❜ë✐ X✳ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ T : X → X ❚❛ ①➳❝ ➤Þ♥❤ ♠ét ❤➭♠ ❦❤➠♥❣ t➝♥❣ 1−r θ(r) = r2 1+r θ : [0, 1) → ( 12 , 1] √ ♥Õ✉ ♥Õ✉ 0≤r≤ √ 5−1 5−1 , ≤r≤ √ , √ ≤ r ≤ ●✐➯ sö r➺♥❣ tå♥ t➵✐ r ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ♥Õ✉ θ(r)d(x, T x) ≤ ♥Õ✉ d(x, y) t❤× t❛ ❝ã d(T x, T y) ≤ rd(x, y)✳ ✷ ❑❤✐ ➤ã✱ tå♥ t➵✐ ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ lim T n x = z n→∞ ➜Þ♥❤ ❧ý✳ ✶✳✶✳✻ X ✈í✐ ♠ä✐ T✳ ❍➡♥ ♥÷❛✱ t❛ ❝ã (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T : X → ✭❬✼❪✮ ❈❤♦ x, y ∈ X ❝ñ❛ x ∈ X✳ ❧➭ ♠ét t♦➳♥ tư ➤➡♥ trÞ ❱í✐ z ♥Õ✉ (s, r)✲❝♦✱ ♥❣❤Ü❛ ❧➭ t♦➳♥ tư t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥✿ d(y, T x) ≤ sd(y, x) t❤× d(T x, T y) ≤ rMT (x, y), r ∈ [0, 1), s > r ✈➭ d(x, T y) + d(y, T x) MT (x, y) = max d(x, y), d(x, T x), d(y, T y), tr♦♥❣ ➤ã ❑❤✐ ➤ã✱ T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ s ≥ t❤× T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ✶✳✶✳✼ ❍Ö q✉➯✳ ❈❤♦ ❧➭ ♠ét ➳♥❤ ①➵ tr➟♥ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T : X → X X✳ ●✐➯ sö r➺♥❣ tå♥ t➵✐ r ∈ [0, 1) ✈➭ s > r s❛♦ ❝❤♦ ♥Õ✉ d(y, T x) ≤ sd(y, x) t❤× t❛ ❝ã d(T x, T y) ≤ rd(x, y) ✈í✐ ♠ä✐ x, y ∈ X ✳ ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ z ✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ s ≥ t❤× T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❉Ô t❤✃② r➺♥❣ t❛ ❧✉➠♥ ❝ã d(x, y) ≤ MT (x, y) ✈í✐ ♠ä✐ x, y ∈ X ✳ ì tế từ ị ý tr t trự tế s r❛ ❦Õt q✉➯ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ✶✳✶✳✽ ➜Þ♥❤ ❧ý✳ ①➵ tr➟♥ T ✭❬✸❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝ ✈➭ T ❧➭ ➳♥❤ X ✳ ●✐➯ sö✱ d(T x, T y) < d(x, y) ✈í✐ ♠ä✐ x, y ∈ X ✱ ✈í✐ x = y ✳ ❑❤✐ ➤ã✱ ❝ã ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ➜Þ♥❤ ❧ý ✶✳✶✳✺✭❬✶✵❪✮ ➤➲ tỉ♥❣ q✉➳t ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥✳ ✶✳✶✳✾ ➜Þ♥❤ ❧ý✳ ➳♥❤ ①➵ tr➟♥ ❑❤✐ ➤ã✱ T ✭❬✶✵❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝ ✈➭ X ✳ ●✐➯ sư✱ ✈í✐ ♠ä✐ x, y ∈ X d(x, T x) < d(x, y) ⇒ d(T x, T y) < d(x, y) ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣✳ ✸ T ❧➭ ✶✳✶✳✶✵ ✭❬✾❪✮ ❑Ý ❤✐Ư✉ ➜Þ♥❤ ❧ý✳ θ✿ [0, 1) → ( 12 , 1] ❧➭ ❤➭♠ ❦❤➠♥❣ t➝♥❣ ❝❤♦ ❜ë✐ √ 1−r θ(r) = r2 1+r ❑❤✐ ➤ã✱ ✈í✐ ✭✶✮ X ♥Õ✉ ♥Õ✉ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❝➳❝ ➤✐Ò✉ s❛✉ ❧➭ t ủ ỗ T X → X tr➟♥ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣✿ tå♥ t➵✐ ♥❣❤Ü❛ ❧➭ ✶✳✶✳✶✶ ♥Õ✉ 5−1 0≤r≤ , √ 5−1 ≤r≤ √ , 2 √ ≤ r ≤ tỏ ữ ề ệ s tì r ∈ [0, 1) s❛♦ ❝❤♦ θ(r)d(x, T x) ≤ d(x, y)✱ ❝ã d(T x, T y) ≤ rd(x, y) ✈í✐ ♠ä✐ x, y ∈ X n ➜Þ♥❤ ❧ý✳ X → X X ✭❬✷❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝ ✈➭ ❧➭ ➳♥❤ ①➵ ❤➬✉ ❝♦✱♥❣❤Ü❛ ❧➭ tå♥ t➵✐ k ∈ [0, 1) ✈➭ L ≥ T : s❛♦ ❝❤♦ d(T x, T y) ≤ kd(x, y) + Ld(y, T x) ✈í✐ ♠ä✐ x, y ∈ X ❑❤✐ ➤ã✱ ✶✳✶✳✶✷ F ix(T ) = {x ∈ X : T x = x} = Ø ➜Þ♥❤ ♥❣❤Ü❛✳ tr♦♥❣ ❳✳ ❉➲② ✭❬✶❪✮ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ {xn }n ❧➭ ❞➲② {xn }n ❧➭ ❞➲② ❝➡ ❜➯♥ ⇔ ∀ > 0, ∃n0 ∈ N : ∀p ∈ N t❤× d(xn+p , xn ) < ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ ♥Õ✉ ♠ä✐ ❞➲② ❝➡ ❜➯♥ ➤Ị✉ ❤é✐ tơ✳ ✶✳✶✳✶✸ ➜Þ♥❤ ♥❣❤Ü❛✳ ♥Õ✉ ♠ä✐ ❞➲② t❤✉é❝ ✭❬✶❪✮ ❈❤♦ {xn } tr♦♥❣ X (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ➤Ò✉ ❝❤ø❛ ♠ét ❞➲② ❝♦♥ X✳ ✹ ❚❛ ♥ã✐ X ❝♦♠♣➽❝ {xnk } ❤é✐ tơ ✈Ị ♠ét ➤✐Ĩ♠ ① ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ ❝đ❛ ✶✳✷ ❑❛r❛♣✐♥❛r ❚r♦♥❣ ♣❤➬♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝đ❛ ❑❛r❛♣✐♥❛r✱ ♠ét sè ❤Ư q✉➯✳ ✶✳✷✳✶ ➜Þ♥❤ ❧ý✳ ✭❬✹❪✮ ❈❤♦ T :X→X ❧➭ ➳♥❤ ①➵ tõ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝ (X, d) ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ●✐➯ sư r➺♥❣✿ ✈í✐ ♠ä✐ x, y ∈ X d(x, T x) < d(x, y) ⇒ d(T x, T y) < M (x, y), tr♦♥❣ ➤ã ❑❤✐ ➤ã✱ T M (x, y) = max{d(x, y), d(T x, x), d(y, T y), 21 d(T x, y), 12 d(x, T y)} ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❈❤ø♥❣ ♠✐♥❤✳ ➜➷t X ✭✶✳✶✮ s❛♦ ❝❤♦ z ∈ X ✱ ♥❣❤Ü❛ ❧➭ T z = z ✳ θ = inf{d(x, T x) : x ∈ X} ✈➭ ❝❤ä♥ ♠ét ❞➲② {xn } ⊂ lim d(xn , Txn ) = θ✳ ❱× X n→∞ ❝♦♠♣➽❝ ♥➟♥ ❦❤➠♥❣ ♠✃t tÝ♥❤ tæ♥❣ q✉➳t t❛ ❝ã t❤Ó ❣✐➯ t❤✐Õt r➺♥❣ {xn } ✈➭ {T xn } ❤é✐ tô ➤Õ♥ z ✈➭ w✱ t➢➡♥❣ ø♥❣ tr♦♥❣ X ✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ θ = 0✳ ●✐➯ sö r➺♥❣ θ > 0✳ ❑❤✐ ➤ã✱ ❧➢✉ ý r➺♥❣ lim d(xn , w) = d(z, w) = lim d(xn , Txn ) = θ n→∞ ❚❛ ❝ã t❤Ó ❝❤ä♥ sè n→∞ k ∈ N s❛♦ ❝❤♦ θ < d(xn , w) ✈➭ d(xn , T xn ) < θ ✈í✐ ♠ä✐ n ≥ k ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ d(xn , T xn ) < d(xn , w) ✈í✐ ♠ä✐ n ≥ k ✳ ◆❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✶✮✱ t❛ ♥❤❐♥ ➤➢ỵ❝ d(T xn , T w) < M (xn , w) ✈í✐ ♠ä✐ n ≥ k ✱ tr♦♥❣ ➤ã 1 M (xn , w) = max{d(xn , w), d(T xn , xn ), d(w, T w), d(T xn , w), d(xn , T w)} 2 ❱× t❤Õ✱ t❛ ❝ã d(w, T w) = lim d(T xn , T w) < lim M (xn , w), n→∞ n→∞ ✺ ✭✶✳✷✮ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ✷✳✷ ré♥❣ ❝đ❛ P♦♣❡s❦✉ ❚r♦♥❣ ♣❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ❝đ❛ P♦♣❡s❦✉ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✶ ✭❬✽❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T ❧➭ ♠ét r ∈ [0, 1), a ∈ [0, 1], b ∈ [0, 1)✱ t❤á❛ ♠➲♥ √ √ (a + b)r2 + r ≤ ♥Õ✉ r ∈ [1/2, 1/ 2) ✈➭ a + (a + b)r ≤ ♥Õ✉ r ∈ [1/ 2, 1) ➳♥❤ ①➵ tr➟♥ X✳ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ●✐➯ sư r➺♥❣ tå♥ t➵✐ x, y ∈ X ad(x, T x) + bd(y, T x) ≤ d(y, x) =⇒ d(T x, T y) ≤ rd(x, y) ❑❤✐ ➤ã✱ ♠ä✐ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t z ∈ X ✳ ❍➡♥ ♥÷❛✱ lim T n x = z ✈í✐ n→∞ x ∈ X✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱× ✈í✐ ♠ä✐ ad(x, T x) + bd(T x, T x) = ad(x, T x) ≤ d(T x, x) ➤ó♥❣ x ∈ X ✱ ♥➟♥ t❤❡♦ ❣✐➯ t❤✐Õt t❛ ❝ã d(T x, T x) ≤ rd(x, T x) ❇➞② ❣✐ê✱ t❛ ❝è ➤Þ♥❤ u∈X ➤ã✱ tõ ✭✷✳✹✷✮ t❛ s✉② r❛ ✈➭ ①➞② ❞ù♥❣ ❞➲② ✈í✐ ♠ä✐ x ∈ X {un } ∈ X ✭✷✳✹✷✮ ❝❤♦ ❜ë✐ ∞ d(un , un+1 ) ≤ rn d(u, T u)✳ ❱× t❤Õ✱ t❛ ❝ã un = T n u✳ ❑❤✐ d(un , un+1 ) < n=1 ∞✳ ❉♦ ➤ã✱ {un } ❧➭ ♠ét ❞➲② ❈❛✉❝❤②✳ ❱× X ❧➭ ➤➬② ➤đ ♥➟♥ {un } ❤é✐ tơ tí✐ z ∈ X ✳ ❚✐Õ♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ❝❤Ø r❛ r➺♥❣ d(T x, z) ≤ rd(x, z) ❱× ✈í✐ ♠ä✐ x ∈ X, x = z ✭✷✳✹✸✮ lim d(un , T un ) = 0✱ lim d(x, T un ) = lim d(x, un ) = d(x, z)✱ ♥➟♥ tå♥ n→∞ n→∞ t➵✐ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ ♠ä✐ n ≥ n0 ✳ ➤➢ỵ❝ n→∞ n0 s❛♦ ❝❤♦ ad(un , T un ) + bd(x, T un ) ≤ d(x, un ) ✈í✐ ❚❤❡♦ ❣✐➯ t❤✐Õt✱ t❛ ❝ã d(T un , T x) ≤ rd(un , x)✳ ❈❤♦ n → ∞✱ t❛ d(z, T x) ≤ rd(z, x)✳ ➜✐Ò✉ ♥➭② ❝❤ø♥❣ ♠✐♥❤ ❝➠♥❣ t❤ø❝ ✭✷✳✹✸✮✳ ❇➞② ❣✐ê✱ t❛ ❣✐➯ sö r➺♥❣ T j z = z ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ j ≥ 1✳ ❑❤✐ ➤ã✱ tõ ✭✷✳✹✸✮ t❛ s✉② r❛ ➤➢ỵ❝ d(T j+1 z, z) ≤ rj d(T z, z) ✷✽ ✭✷✳✹✹✮ ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ j ≥ 1✳ ❚❛ ①Ðt ❜❛ tr➢ê♥❣ ❤ỵ♣ s❛✉ ✭❛✮ 0≤r< 2✱ ✭❜✮ √1 ✱ ≤r< ✭❝✮ √ ≤ r < 1✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ✭❛✮✱ t❛ ❝❤ó ý r➺♥❣ 2r < 1✳ ❑❤✐ ➤ã✱ tõ ✭✷✳✹✷✮ ✈➭ ✭✷✳✹✹✮✱ t❛ ❝ã d(z, T z) ≤ d(z, T z) + d(T z, T z) ≤ rd(z, T z) + rd(z, T z) = 2rd(z, T z) < d(z, T z) ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ✭❜✮✱ t❛ ❝❤ó ý r➺♥❣ 2r2 < 1✳ ◆Õ✉ t❛ ❣✐➯ sö ad(T z, T z) + bd(z, T z) > d(z, T z) t❤× t❤❡♦ ✭✷✳✹✷✮ ✈➭ ✭✷✳✹✹✮✱ t❛ ❝ã d(z, T z) ≤ d(z, T z) + d(T z, T z) < ad(T z, T z) + bd(z, T z) + d(T z, T z) ≤ ar2 d(z, T z) + br2 d(z, T z) + rd(z, T z) = [(a + b)r2 + r]d(z, T z) ≤ d(z, T z) ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❉♦ ➤ã✱ ad(T z, T z) + bd(z, T z) ≤ d(z, T z) ❚õ ❣✐➯ t❤✐Õt ✈➭ ✭✷✳✹✹✮✱ t❛ ❝ã d(z, T z) ≤ d(z, T z) + d(T z, T z) ≤ r2 d(z, T z) + rd(z, T z) ≤ r2 d(z, T z) + r2 d(z, T z) = 2r2 d(z, T z) < d(z, T z) ✷✾ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ✭❝✮✱ t❛ ❣✐➯ sö r➺♥❣ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ ad(un , un+1 ) + bd(z, un+1 ) > d(z, un ) ✈í✐ ♠ä✐ m0 ≥ s❛♦ ❝❤♦ n ≥ m0 ❑❤✐ ➤ã✱ t❛ ❝ã d(z, un ) < ad(un , un+1 ) + b[ad(un+1 , un+2 ) + bd(z, un+2 )] ≤ (a + abr)d(un , un+1 ) + b2 d(z, un+2 ) < (a + abr)d(un , un+1 ) + b2 [ad(un+2 , un+3 ) + bd(z, un+3 )] ≤ (a + abr + ab2 r2 )d(un , un+1 ) + b3 d(z, un+3 ) ❚✐Õ♣ tô❝ q✉➳ trì t ợ d(z, un ) < (a + abr + ab2 r2 + + abp−1 rp−1 )d(un , un+1 ) + bp d(z, un+p ) − (br)p d(un , un+1 ) + bp d(z, un+p ) ≤a − br ✈í✐ ♠ä✐ n ≥ m0 ✈➭ p ≥ 1✳ ❈❤♦ p → ∞✱ t❛ ♥❤❐♥ ➤➢ỵ❝ d(z, un ) ≤ a d(un , un+1 ) − br ✈í✐ ♠ä✐ n ≥ m0 ❉♦ ➤ã✱ d(z, un+1 ) ≤ ✈í✐ ♠ä✐ a ar d(un+1 , un+2 ) ≤ d(un , un+1 ) − br − br n ≥ m0 ✱ ✈× ✈❐② d(un , un+1 ) ≤ d(z, un ) + d(z, un+1 ) a ar < d(un , un+1 ) + d(un , un+1 ) − br − br a + ar d(un , un+1 ) ≤ d(un , un+1 ) = − br ✈í✐ ♠ä✐ ❞➲② n ≥ m0 ✳ ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❉♦ ➤ã✱ tå♥ t➵✐ ♠ét ❞➲② ❝♦♥ {un(k) } ❝ñ❛ {un } s❛♦ ❝❤♦ ad(un(k) , un(k)+1 ) + bd(z, un(k)+1 ) ≤ d(z, un(k) ) ✸✵ ✈í✐ ♠ä✐ k ≥ ◆❤ê ❣✐➯ t❤✐Õt✱ t❛ t❤✉ ➤➢ỵ❝ d(T z, T un(k) ≤ rd(z, un(k) ) k → ∞✱ t❛ ➤➢ỵ❝ d(z, T z) = 0✱ ♥❣❤Ü❛ ❧➭ z = T z ✳ t❤✐Õt k ≥ 1✳ ❈❤♦ ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ T j z = z ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ j ≥ 1✳ ❚õ ➤ã s✉② r❛ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ ❝ã ✈í✐ ♠ä✐ j ≥ s❛♦ ❝❤♦ T j z = z ✳ ◆❤ê ✭✷✳✹✷✮✱ t❛ d(z, T z) = d(T j z, T j+1 z) ≤ rj d(z, T z)✳ ❱× r ∈ [0, 1)✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ d(z, T z) = 0✱ ♥❣❤Ü❛ ❧➭ T z = z ✳ ❇➞② ❣✐ê t❛ ❣✐➯ sư r➺♥❣ y ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦❤➳❝ ❝ñ❛ T ✱ ♥❣❤Ü❛ ❧➭ T y = y ✳ ❑❤✐ ➤ã✱ t❛ ❝ã ad(y, T y) + bd(z, y) ≤ d(z, y), ✈× t❤Õ✱ t❤❡♦ ❣✐➯ t❤✐Õt t❛ s✉② r❛ d(y, z) = d(T y, T z) ≤ rd(y, z)✳ ▲➵✐ ✈× r ∈ [0, 1) tõ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ s✉② r❛ ✷✳✷✳✷ ◆❤❐♥ ①Ðt✳ ❱í✐ d(y, z) = 0✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ r ∈ [0, 12 )✱ ❝❤♦ a = 1, b = 0✱ t❛ t❤✉ ➤➢ỵ❝ ➤✐Ị✉ ❦✐Ư♥ ❙✉③✉❦✐ ủ ị ý ữ từ ề ệ ủ ➜Þ♥❤ ❧ý ✷✳✷✳✶ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ t❛ ➤➢ỵ❝ ad(x, T x) + b[d(x, T x) − d(y, x)] ≤ d(y, x), ♥❣❤Ü❛ ❧➭ a+b d(x, T x) ≤ d(y, x) 1+b ◆Õ✉ r ∈ [ √12 , 1)✱ t❛ ❝ã a+b = = θ(r), 1+b 1+r tõ ➤ã s✉② r❛ r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶ ❦Ð♦ t❤❡♦ ➤✐Ị✉ ❦✐Ư♥ ❙✉③✉❦✐✳ ❈❤ó♥❣ t❛ ❝ị♥❣ ❝❤ó ý r➺♥❣ ♥Õ✉ t❛ ❧✃② a= (1−r) ,b r2 = ✈í✐ r ∈ [ 12 , √12 ) t❤× t ợ ề ệ ó ị ý ❦❤➳✐ q✉➳t ❤ã❛✱ ♠ë ré♥❣ ✈➭ ❤♦➭♥ t❤✐Ư♥ ❤➡♥ ➤Þ♥❤ ❧ý ❙✉③✉❦✐✳ ✷✳✷✳✸ ❱Ý ❞ơ✳ ✭❬✽❪✮ ❑ý ❤✐Ư✉ X = {−1, 0, 1, 2}✳ ♠➟tr✐❝ ➤➬② ➤đ ✈í✐ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣ tr➟♥ ✸✶ ❑❤✐ ➤ã✱ X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ R✳ ❚❛ ①➳❝ ➤Þ♥❤ ➳♥❤ ①➵ T : X → X ❝❤♦ ❜ë✐ Tx = ♥Õ✉ x ∈ {−1, 0, 1}, T = −1 ❑❤✐ ➤ã✱ ♥❤➢♥❣ T T t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶ ✈í✐ ♠ä✐ r ∈ [0, 31 ) ∪ [ 12 , 1)✱ ❦❤➠♥❣ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❙✉③✉❦✐ ❝đ❛ ➜Þ♥❤ ❧ý ✶✳✶✳✺✳ θ(r)d(1, T 1) ≤ = d(1, 2) ✈× t❛ ❝ã ❚❤❐t ✈➞②✱ d(T 1, T 2) = = √ d(1, 2) ♥➟♥ T ◆Õ✉ r ∈ [ 21 , t❛ ➤➢ỵ❝ ✈í✐ ♠ä✐ r ∈ [0, 1)✱ ✈➭ ❦❤➠♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❙✉③✉❦✐✳ ( 5−1) )✱ t❛ ❝ã r2 +r < 1✱ ✈× ✈❐② ♥Õ✉ ❧✃② a, b s❛♦ ❝❤♦ a+b = 1−r ✱ r2 a + b > 1✳ ❉♦ ➤ã✱ ad(1, T 1) + bd(1, T 2) = a + 2b > = d(1, 2) ✈➭ ad(2, T 2) + bd(2, T 1) = 3a + 2b > = d(1, 2) ▲ó❝ ♥➭②✱ râ r➭♥❣ r➺♥❣ T t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶✳ √ ( 5−1) ), 1)✱√t❛ ❧✃② b = 12 ✳ ❑❤✐ ➤ã t❛ ①Ðt ❤❛✐ tr➢ê♥❣ ❤ỵ♣ s❛✉ ◆Õ✉ r ∈ [ 2 ( 5−1) √1 ❚r➢ê♥❣ ❤ỵ♣ ✶✳ r ∈ [ ✳ ), )✳ ❑❤✐ ➤ã✱ t❛ ➤➷t a = 2−2r−r 2r2 2−r ❚r➢ê♥❣ ❤ỵ♣ ✷✳ r ∈ [ √ , 1)✳ ❑❤✐ ➤ã✱ t❛ ➤➷t a = ✳ ▲ó❝ ➤ã✱ tr♦♥❣ ❝➯ ❤❛✐ 2+2r tr➢ê♥❣ ❤ỵ♣ t❛ ❝ã a + 2b = + a > 1✱ ì T tỏ ề ệ ủ ị ý ✷✳✷✳✶✳ ◆Õ✉ r ∈ [0, 1/3) ✈í✐ a = 1, b = 1/2✱ t❤× râ r➭♥❣ T ❝ị♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶✳ ➜Þ♥❤ ❧ý s❛✉ ➤➞② ❧➭ ♠ét ♠ë ré♥❣ ❝đ❛ ➜Þ♥❤ ❧ý ✶✳✶✳✼✳ ✷✳✷✳✹ ➜Þ♥❤ ❧ý✳ ➳♥❤ ①➵ tr➟♥ ✭❬✽❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T ❧➭ ♠ét X ✳ ●✐➯ sö r➺♥❣ tå♥ t➵✐ r ∈ [0, 1) ✈➭ ❝➳❝ sè s > r s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X s−r d(x, T x) + d(y, T x) ≤ sd(y, x) =⇒ d(T x, T y) ≤ rd(x, y) 1+r ✸✷ ❑❤✐ ➤ã✱ T ❝ã ♠ét ể t ộ ữ ế s tì T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ✈í✐ ♠ä✐ ▲✃② u1 ∈ X ✈➭ ①➞② ❞ù♥❣ ❞➲② n ≥ 1✳ ❱× = d(un+1 , T un ) ≤ sd(un+1 , un ) − ♥➟♥ tõ ❣✐➯ t❤✐Õt✱ t❛ ➤➢ỵ❝ t❛ ❝ã {un } ❝❤♦ ❜ë✐ un+1 = T un s−r d(un , T un ), 1+r d(un+1 , un+2 ) ≤ rd(un+1 , un ) ✈í✐ ♠ä✐ n ≥ 1✳ ❉♦ ➤ã d(un+1 , un+2 ) ≤ rn d(u1 , u2 ) ✈í✐ ♠ä✐ n ≥ 1✳ ❱× t❤Õ ∞ ∞ rn−1 d(u1 , u2 ) < ∞ d(un+1 , un ) ≤ n=1 n=1 ➜✐Ò✉ ♥➭② s✉② r❛ {un } ❧➭ ♠ét ❞➲② ❈❛✉❝❤②✳ ❱× X ➤➬② ➤đ ♥➟♥ {un } ❤é✐ tơ tí✐ ➤✐Ĩ♠ z∈X ♥➭♦ ➤ã✳ ❇➞② ❣✐ê✱ t❛ sÏ ❝❤Ø r❛ r➺♥❣ tå♥ t➵✐ ♠ét ❞➲② ❝♦♥ d(z, T un(k) ) ≤ sd(z, un(k) ) − ✈í✐ ♠ä✐ s−r d(un(k) , T un(k) ) 1+r k ≥ 1✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ ♣❤➯♥ ❝❤ø♥❣✱ ❣✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ n0 s❛♦ ❝❤♦ d(z, T un ) > sd(z, un ) − ✈í✐ ♠ä✐ {un(k) } ❝đ❛ ❞➲② {un } s❛♦ ❝❤♦ s−r d(un , T un ) 1+r n ≥ v ✳ ❑❤✐ ➤ã✱ t❛ ❝ã s−r d(un+1 , un+2 ) 1+r s−r s−r > s2 d(z, un ) − s d(un , un+1 ) − d(un+1 , un+2 ) 1+r 1+r s−r ≥ s2 d(z, un ) − [sd(un , un+1 ) + rd(un , un+1 )] 1+r s−r = s2 d(z, un ) − (s + r)d(un , un+1 ) 1+r d(z, un+2 ) > sd(z, un+1 ) − ❇➺♥❣ q✉② ♥➵♣✱ ✈í✐ ♠ä✐ n ≥ n0 ✱ p ≥ 1✱ t❛ t❤✉ ➤➢ỵ❝ d(z, un+p ) > sp d(z, un ) − s − r p−1 (s + sp−2 r + + rp−1 )d(un , un+1 ) 1+r ✸✸ ❑❤✐ ➤ã✱ t❛ ❝ã s − r p−1 − (r/s)p d(z, un+p ) > s d(z, un ) − s d(un , un+1 ) 1+r − r/s s − r − (r/s)p p d(un , un+1 ) = s d(z, un ) − 1+r s−r p ❉♦ ➤ã✱ t❛ ❝ã s p − (r/s)p d(z, un ) − d(un , un+1 ) < d(z, un+p ) 1+r ✭✷✳✹✺✮ ▼➷t ❦❤➳❝✱ t❛ ❝ã d(un+p , un ) ≤ d(un , un+1 ) + d(un+1 , un+2 ) + + d(un+p−1 , un+p ) ≤ (1 + r + + rp−1 )d(un , un+1 ) − rp d(un , un+1 ) = 1−r ❈❤♦ p → ∞✱ ❦❤✐ ➤ã ✈í✐ ♠ä✐ n ≥ 1✱ t❛ ❝ã d(z, un ) ≤ d(z, un+p ) ≤ ❚õ ✭✷✳✹✺✮ ✈➭ ✭✷✳✹✻✮✱ ✈í✐ ♠ä✐ d(un , un+1 ) 1−r ✳ ❉♦ ➤ã✱ t❛ ❝ã d(un+p , un+p+1 ) d(un , un+1 ) ≤ rp 1−r 1−r ✭✷✳✹✻✮ n ≥ n0 ✱ p ≥ 1✱ t❛ ❝ã − ( rs )p rp p d(un , un+1 ) > s d(z, un ) − d(un , un+1 ) 1−r 1+r ✈× ✈❐② ❈❤♦ − ( rs )p ( rs )p d(un , un+1 ) > d(z, un ) − d(un , un+1 ) 1−r 1+r p → ∞✱ t❛ ➤➢ỵ❝ d(z, un ) ≤ d(z, un+1 ) ≤ d(un , un+1 ) 1+r ✈í✐ ♠ä✐ n ≥ n0 ✳ ❑❤✐ ➤ã✱ t❛ ❝ã d(un+1 , un+2 ) d(un , un+1 ) ≤r 1+r 1+r ✈➭ r d(un , un+1 ) d(un , un+1 ) > sd(z, un ) − (s − r) 1+r 1+r ✸✹ ➜✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ ❧➭ d(z, un ) < d(un , un+1 ) 1+r ✈í✐ ♠ä✐ n ≥ n0 ✳ ❉♦ ➤ã✱ d(un , un+1 ) ≤ d(z, un ) + d(z, un+1 ) d(un , un+1 ) d(un , un+1 ) +r 1+r 1+r = d(un , un+1 ) < ➜➞② ❧➭ ♠ét ♠➞✉ t❤✉➱♥✳ ❉♦ ➤ã✱ tå♥ t➵✐ ♠ét ❞➲② ❝♦♥ {un(k) } ❝ñ❛ ❞➲② {un } s❛♦ ❝❤♦ s−r d(un(k) , T un(k) ) 1+r ✈í✐ ♠ä✐ k ≥ 1✳ ❚❤❡♦ ❣✐➯ t❤✐Õt✱ t❛ ❝ã d(T z, T un(k) ) ≤ rd(z, un(k) )✳ ❈❤♦ k → ∞✱ d(z, T un(k) ) ≤ sd(z, un(k) ) − t❛ ➤➢ỵ❝ d(z, T z) = 0✱ ♥❣❤Ü❛ ❧➭ z = T z ✳ ❈✉è✐ ❝ï♥❣✱ ♥Õ✉ s ≥ 1✱ t❛ ❣✐➯ sö r➺♥❣ y ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦❤➳❝ ❝đ❛ T✳ ❑❤✐ ➤ã✱ s−r d(y, T y) 1+r d(z, y) = d(T z, T y) ≤ rd(z, y)✳ ❉♦ r < t❛ d(z, T y) = d(z, y) ≤ sd(z, y) = sd(z, y) − ❱× ✈❐②✱ t❤❡♦ ❣✐➯ t❤✐Õt✱ s✉② r❛ d(z, y) = 0✱ ❤❛② y = z ✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ➜Þ♥❤ ❧ý s❛✉ ➤➞② ♠ë ré♥❣ ➜Þ♥❤ ❧ý ✶✳✶✳✾ ✈➭ ➜Þ♥❤ ❧ý ✶✳✶✳✽✳ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✺ ➳♥❤ ①➵ tr➟♥ ✭❬✽❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝ ✈➭ T ❧➭ ♠ét X ✳ ●✐➯ sư r➺♥❣ ✈í✐ ♠ä✐ x, y ∈ X ad(x, T x) + bd(y, T x) < d(y, x) =⇒ d(T x, T y) ≤ d(x, y), tr♦♥❣ ➤ã a > 0✱ b > 0✱ 2a + b < 1✳ ❑❤✐ ➤ã✱ T ❈❤ø♥❣ ♠✐♥❤✳ tr♦♥❣ X s❛♦ ❝❤♦ ➜➷t ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ β = inf{d(x, T x) : x ∈ X} ✈➭ ❝❤ä♥ ♠ét ❞➲② {xn } lim d(xn , T xn ) = β ✳ ❱× X n→∞ q✉➳t✱ t❛ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣ ❝♦♠♣➽❝ ♥➟♥ ❦❤➠♥❣ ♠✃t tÝ♥❤ tæ♥❣ {xn } ✈➭ {T xn } t➢➡♥❣ ø♥❣ ❤é✐ tô ➤Õ♥ v, w ∈ X ✳ ❑❤✐ ➤ã✱ t❛ ❝ã lim d(xn , w) = lim d(T xn , v) = d(v, w) = β n→∞ n→∞ ✸✺ ❈❤ó♥❣ t❛ sÏ ❝❤Ø r❛ r➺♥❣ ❧➵✐ r➺♥❣ β = 0✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ ♣❤➯♥ ứ sử ợ > ì lim [ad(xn , T xn ) + bd(w, T xn )] = aβ < β = lim d(w, xn ), n→∞ n→∞ ♥➟♥ t❛ ❝ã t❤Ó ❝❤ä♥ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ k0 s❛♦ ❝❤♦ ad(xn , T xn ) + bd(w, T xn ) < d(w, xn ) ✈í✐ ♠ä✐ n ≥ k0 ✳ ❚❤❡♦ ❣✐➯ t❤✐Õt✱ d(T w, T xn ) < d(w, xn ) ➤ó♥❣ ✈í✐ n ≥ k0 ✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ d(w, T w) = lim d(T w, T xn ) ≤ lim d(w, xn ) = β n→∞ ❚õ ❝➳❝❤ ➤Þ♥❤ ♥❣❤Ü❛ n→∞ β ✱ t❛ ợ d(w, T w) = ì ad(w, T w) + bd(T w, T w) < d(T w, w), ♥➟♥ t❛ ❝ã d(T w, T w) < d(w, T w) = β, ➤✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ➤Þ♥❤ ♥❣❤Ü❛ ❝ñ❛ β ✳ ▼➞✉ t❤✉➱♥ ♥➭② ❝❤ø♥❣ tá β = 0✳ ❱× t❤Õ✱ t❛ ❝ã lim d(xn , w) = lim d(T xn , v) = lim d(T xn , xn ) = d(v, w) = n→∞ ♥➟♥ n→∞ n→∞ v = w✳ ❉♦ ➤ã✱ lim xn = lim T xn = w✳ n→∞ n→∞ ❚✐Õ♣ t❤❡♦✱ t❛ sÏ ❝❤Ø r❛ r➺♥❣ T T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ ❦❤➠♥❣ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ♥➭♦ ❝➯✳ ❱× ad(xn , T xn ) + bd(T xn , T xn ) < d(T xn , xn ) ♥➟♥ t❤❡♦ ❣✐➯ t❤✐Õt t❛ s✉② r❛ ✈í✐ ♠ä✐ n ≥ 1, d(T xn , T xn ) < d(T xn , xn )✱ ✈× t❤Õ lim T xn = w✳ n→∞ ❇➺♥❣ ♣❤Ð♣ q✉② ♥➵♣✱ t❛ t❤✉ ➤➢ỵ❝ d(T p xn , T p+1 xn ) < d(T p−1 xn , T p xn ) < < d(xn , T xn ) ✸✻ ✈➭ lim T p xn = w ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ p ≥ 1✳ ◆Õ✉ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ p ≥ n→∞ ✈➭ ♠ét ❞➲② ❝♦♥ {xn(k) } ❝ñ❛ ❞➲② {xn } s❛♦ ❝❤♦ ad(T p−1 xn(k) , T p xn(k) ) + bd(w, T p xn(k) ) < d(w, T p−1 xn(k) ) ✈í✐ ♠ä✐ ❈❤♦ k ≥ tì t tết t ợ d(T w, T p xn(k) < d(w, T p−1 xn(k) )✳ k → ∞✱ t❛ ♥❤❐♥ ➤➢ỵ❝ d(w, T w) = 0✱ ♥❣❤Ü❛ ❧➭ w = T w✱ ➤✐Ị✉ ♥➭② ♠➞✉ t❤✉➮♥ ✈í✐ ❣✐➯ t❤✐Õt ♣❤➯♥ ❝❤ø♥❣ t❤Ĩ ❣✐➯ sư r➺♥❣ ✈í✐ ♠ä✐ T ❦❤➠♥❣ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ♥➭♦ ❝➯✳ ❉♦ ➤ã✱ t❛ ❝ã m ≥ 1✱ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ n(m) ≥ s❛♦ ❝❤♦ ad(T m−1 xn , T m xn ) + bd(w, T m xn ) ≥ d(w, T m−1 xn ) ✈í✐ ♠ä✐ n ≥ n(m)✳ ❱× pbp =0 p→∞ − bp lim ♥➟♥ t❛ ❝ã t❤Ó ❝❤ä♥ ✈➭ 2a d(A, C) t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ❑Õt ❧✉❐♥ ❙❛✉ ♠ét t❤ê✐ ❣✐❛♥ t❐♣ tr✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ò✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉✱ ✈Ị ➤Ị t➭✐✿ ❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❊❞❡❧st❡✐♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉✿ ✶✳ ❍Ö t❤è♥❣ ❤ã❛ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣❛❝t✱ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❦✐Ó✉ ❙✉③✉❦✐✱ ➳♥❤ ①➵ ❦✐Ó✉ ❊❞❡♥st❡✐♥✱ ➤✐Ó♠ trï♥❣ ♥❤❛✉✱ ➤✐Ó♠ ❜✃t ➤é♥❣✱✳✳✳ rì ột số ị ý ể t ộ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ ❝đ❛ ❑❛r❛♣✐♥❛r✱ P♦♣❡s❦✉✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ➤Þ♥❤ ❧ý ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ❝❤ø♥❣ ♠✐♥❤ ❝ß♥ s ợ ị ý ị ý ✶✳✷✳✷✱ ➜Þ♥❤ ❧ý ✷✳✶✳✶✱ ➜Þ♥❤ ❧ý ✷✳✶✳✺✱ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ ➜Þ♥❤ ❧ý ✷✳✷✳✹✱ ➜Þ♥❤ ❧ý ✷✳✷✳✺✱ ❍Ư q✉➯ ✷✳✶✳✷✳ ✹✳ ●✐í✐ t❤✐Ư✉ ❝❤✐ t✐Õt ❱Ý ❞ơ ✷✳✶✳✾✱ ❱Ý ❞ơ ✷✳✷✳✸✳ t ệ t ỗ ❚➠♣➠ ➤➵✐ ❝➢➡♥❣✱ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ❱✳ ❇❡r✐♥❞❡ ✭✷✵✵✹✮✱ ✧❆♣♣r♦①✐♠❛t✐♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ✇❡❛❦ ❝♦♥tr❛❝t✐♦♥s ✉s✲ ✐♥❣ t❤❡ P✐❝❛r❞ ✐t❡r❛t✐♦♥✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ❋♦r✉♠✱ ✾ ✭✶✮✱ ✹✸✲✺✸✳ ❬✸❪ ▼✳ ❊❞❡❧st❡✐♥ ✭✶✾✻✷✮✱ ✧❖♥ ❢✐①❡❞ ❛♥❞ ♣❡r✐♦❞✐❝ ♣♦✐♥ts ✉♥❞❡r ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✧✱ ❏✳ ▲♦❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ✸✼✱ ✼✹✲✼✾✳ ❬✹❪ ❊✳ ❑❛r❛♣✐♥❛r ✭✷✵✶✶✮✱ ✧❊❞❡❧st❡✐♥ t②♣❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✧✱ ❆♥♥✳ ❋✉♥❝t✳ ❆♥❛❧✳✱ ✷ ✭✶✮✱ ✺✶✲✺✽✳ ❬✺❪ ❊✳ ❑❛r❛♣✐♥❛r ✭✷✵✶✷✮✱ ✧❊❞❡❧st❡✐♥ t②♣❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✳✱ ✷✵✶✷✲✶✵✼✱ ✶✷ ♣❛❣❡s✳ ❬✻❪ P✳ ◆✳ ❉✉tt❛✱ ❇✳ ❙✳ ❈❤♦✉❞❤✉r② ✭✷✵✵✽✮✱ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ✐♥ ♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✷✵✵✽✱ ✽ ♣❛❣❡s✱ ■❉ ✹✵✻✸✻✽✳ ❬✼❪ ❖✳ P♦♣❡s❝✉ ✭✷✵✵✾✮✱ ✧❚✇♦ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❣❡♥❡r❛❧✐③❡❞ ❝♦♥tr❛❝✲ t✐♦♥s ✇✐t❤ ❝♦♥st❛♥s ✐♥ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❈❡♥t✳ ❊✉r✳ ❏✳ ▼❛t❤✳✱✱ ✼✱ ✺✷✾✲✺✸✽✳ ❬✽❪ ❖✳ P♦♣❡s❝✉ ✭✷✵✶✸✮✱ ✧❙♦♠❡ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ ❙✉③✉❦✐ ❛♥❞ ❊❞❡❧st❡✐♥ t②♣❡ t❤❡♦r❡♠s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✳✱ ✷✵✶✸✲✸✶✾✱ ✶✶ ♣❛❣❡s✳ ❬✾❪ ❚✳ ❙✉③✉❦✐ ✭✷✵✵✽✮✱ ✧❆ ❣❡♥❡r❛❧✐③❡❞ ❇❛♥❛❝❤ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ❝❤❛r❛❝✲ t❡r✐③❡s ♠❡tr✐❝ ❝♦♠♣❧❡t❡♥❡ss✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✸✻✱ ✶✽✻✶✲✶✽✻✾✳ ❬✶✵❪ ❚✳ ❙✉③✉❦✐ ✭✷✵✵✾✮✱ ✧❆ ♥❡✇ t②♣❡ ♦❢ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ✐♥ ♠❡tr✐❝ s♣❛❝❡s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✶ ✭✶✶✮✱ ✺✸✶✸✲✺✸✶✼✳ ✹✶ ❬✶✶❪ ❚✳ ❇❛♥❛❝❤ ✭✶✾✾✷✮✱ ✧❙✿ ❙✉r ❧❡s ♦♣❡r❛t✐♦♥s ❞❛♥s ❧❡s ❡♥s❡♠❜❧❡s ❛❜str❛✐ts ❡t ❧❡✉r ❛♣♣❧✐❝❛t✐♦♥ ❛✉① ❡q✉❛t✐♦♥s ✐t❡❣r❛❧❡s✧✱ ❋✉♥❞❛♠✳▼❛t❤✳✸✱ ✼✶ ✭✶✶✮✱ ✶✸✸✲ ✶✽✶✳ ✹✷ ... ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡♥st❡✐♥ ❝đ❛ ❑❛r❛♣✐♥❛r✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ❝➳❝ ➤Þ♥❤ ❧ý ó r ú t ò trì ột số ❤Ư q✉➯✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ✳ ❚r♦♥❣ ❝❤➢➡♥❣... ▼ơ❝ ✶✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ❝đ❛ rr ụ trì ột số ị ý ể ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ❝đ❛ P♦♣❡s❦✉✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ❝➳❝ ❦Õt q✉➯ ➤ã✳ ❚r×♥❤... ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ✷✳✶ ré♥❣ ❝đ❛ ❑❛r❛♣✐♥❛r ❚r♦♥❣ ♣❤➬♥ ú t trì ột số ị ý ể ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ s✉② ré♥❣ ❝đ❛ ❑❛r❛♣✐♥❛r ✈➭ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ