❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❚rÞ♥❤ ❱➝♥ ▲✉➞♥ ❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❙✉③✉❦✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✻ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❚rÞ♥❤ ❱➝♥ ▲✉➞♥ ❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✻ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▲ê✐ ♥ã✐ ➤➬✉ ❈❤➢➡♥❣ ■✳ ❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ ❝ñ❛ ❑✐❦❦❛✇❛✱ ❙✉③✉❦✐ ✈➭ P♦♣❡s❝✉ ✶ ✶✳✶✳ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷✳ ❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ s✉② ré♥❣ ❝đ❛ ❑✐❦❦❛✇❛✱ ❙✉③✉❦✐ ✈➭ P♦♣❡s❝✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤➢➡♥❣ ■■✳ ▼ét sè ♠ë ré♥❣ ❝đ❛ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ ✹ ✶✽ ✷✳✶✳ ▼ét sè ♠ë ré♥❣ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ ❝đ❛ ▼✉r❛❧✲ ✐s❛♥❦❛r ✈➭ ❏❡②❛❜❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷✳ ▼ë ré♥❣ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ❑Õt ❧✉❐♥ ✸✹ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✺ ✐ ▼ë ➤➬✉ ❚r♦♥❣ ✈➭✐ t❤❐♣ ❦û ❣➬♥ ➤➞②✱ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ♠➟tr✐❝ ➤➲ trë t❤➭♥❤ ♠ét ❧Ü♥❤ ✈ù❝ ♥❣❤✐➟♥ ❝ø✉ q✉❛♥ trä♥❣ tr♦♥❣ ❦❤♦❛ ❤ä❝ t❤✉➬♥ tó② ✈➭ ❦❤♦❛ ❤ä❝ ø♥❣ ❞ơ♥❣✳ ❚r♦♥❣ t❤ù❝ tÕ✱ ♥ã ➤➲ trë t❤➭♥❤ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝➠♥❣ ❝ơ ❝èt ②Õ✉ ♥❤✃t tr♦♥❣ ❣✐➯✐ tÝ❝❤ ❤➭♠ ♣❤✐ t✉②Õ♥✱ tè✐ ➢✉ ❤ã❛✱ t♦➳♥ ❤ä❝✱ ❝➳❝ ♠➠ ❤×♥❤ t♦➳♥ ❤ä❝✱ ❦✐♥❤ tÕ ✈➭ ② ❤ä❝✳ ❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ị♥❣ ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ✈✐Ư❝ ①➞② ❞ù♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ tr♦♥❣ t♦➳♥ ❤ä❝ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ✈✃♥ ➤Ị tr♦♥❣ t♦➳♥ ❤ä❝ ø♥❣ ❞ơ♥❣ ✈➭ ❦❤♦❛ ❤ä❝✳ ❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧Ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝đ❛ ❇❛♥❛❝❤✳ ◆ã ➤➲ ➤➢ỵ❝ ✈❐♥ ❞ơ♥❣ r✃t ♣❤ỉ ❜✐Õ♥ ✈➭ t❤➭♥❤ ❝➠♥❣ tr♦♥❣ ✈✐Ö❝ ❝❤ø♥❣ ♠✐♥❤ sù tå♥ t➵✐ ❞✉② ♥❤✃t ♥❣❤✐Ư♠ ✈➭ tÝ♥❤ ①✃♣ ①Ø ♥❣❤✐Ư♠ ❝đ❛ ❝➳❝ ❜➭✐ t♦➳♥ t❤✉é❝ ♥❤✐Ị✉ ❧Ü♥❤ ✈ù❝ ❝đ❛ ❣✐➯✐ tÝ❝❤✳ ❱× t❤Õ ➤➲ ❝ã ♥❤✐Ị✉ ♥❣❤✐➟♥ ❝ø✉✱ t×♠ ❝➳❝❤ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ❝❤♦ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✱ ❜➺♥❣ ❝➳❝❤ ➤✐Ò✉ ❝❤Ø♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝➡ ❜➯♥ ❤♦➷❝ t❤❛② ➤ỉ✐ ❦❤➠♥❣ ❣✐❛♥✳ ◆❤✐Ị✉ ❦Õt q✉➯ t✐➟✉ ❜✐Ĩ✉ t❤❡♦ ❤➢í♥❣ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ❝ñ❛ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❝ã t❤Ĩ ❦Ĩ ➤Õ♥ ♥❤➢ ❝đ❛ ❝➳❝ t➳❝ ❣✐➯ ❚✳ ❙✉③✉❦✐✱ ▼✳ ❊❞❡❧st❡✐♥✱ ❘✳ ❑❛♥♥❛♥✱ ❙✳ ❘❡✐❝❤✱ ▼✳ ❑✐❦❦❛✇❛✱ ❖✳ P♦♣❡s❝✉✱ ❙✳ ❑✳ ❈❤❛tt❡r❥❡❛✱ ▲✳ ❇✳ ❈✐r✐✬❝✱✳✳✳ ◆➝♠ ✶✾✻✷✱ ▼✳ ❊❞❡❧st❡✐♥ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý ♥ỉ✐ t✐Õ♥❣ s❛✉ ➤➞②✿ ✧❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝✱ T : X → X ✳ d(x, y) ✈í✐ ♠ä✐ x, y ∈ X ♠➭ x = y✳ ❑❤✐ ➤ã T ●✐➯ sö r➺♥❣ d(T x, T y) < ❝ã ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣✧✳ ◆➝♠ ✷✵✵✽✱ ❚✳ ❙✉③✉❦✐ ➤➲ ❣✐í✐ t❤✐Ư✉ ♠ét ❧♦➵✐ ➳♥❤ ①➵ ♠í✐ θ : [0, 1) → ( 12 , 1] ✈➭ t❤✉ ➤➢ỵ❝ ♠ét ♠ë ré♥❣ s❛✉ ➤➞② ❝ñ❛ ♥❣✉②➟♥ ❧ý ❝ñ❛ ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✱ tr♦♥❣ ➤ã tÝ♥❤ ➤➬② ➤đ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ❝ị♥❣ ❝ã t❤Ĩ ➤➢ỵ❝ ➤➷❝ tr➢♥❣ ❜ë✐ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ộ ủ ỉ ỗ ①➵ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❑❤✐ ➤ã X T :X →X ❧➭ ➤➬② ➤ñ ❦❤✐ ✈➭ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✯✮ s❛✉ ➤➞② ➤Ị✉ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ (∗) ∃r ∈ [0, 1), ∀x, y ∈ X θ(r)d(x, T x) < d(x, y) ⇒ d(T x, T y) < r.d(x, y) ” ✐✐ ❙❛✉ ➤ã✱ ❨✳ ❊♥❥♦✉❥✐✱ ▼✳ ◆❛❦❛♥✐s❤✐✱ ▼✳ ❑✐❦❦❛✇❛✱ ❖✳ P♦♣❡s❝✉✱ ✳✳✳ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❞➵♥❣ ❜✐Õ♥ t❤Ĩ ❤♦➷❝ ❧➭♠ ♠Þ♥ ❦Õt q✉➯ tr➟♥ ❝ñ❛ ❚✳ ❙✉③✉❦✐✳ ◆➝♠ ✷✵✵✾✱ ❚✳ ❙✉③✉❦✐ ➤➲ ➤➢❛ r❛ ♠ét ❞➵♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ♠í✐ ♠➭ ♥ã ②Õ✉ ❤➡♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝đ❛ ❊❞❡❧st❡✐♥ ✈➭ t❤✉ ➤➢ỵ❝ ♠ét ❞➵♥❣ ➤Þ♥❤ ❧ý ❊❞❡❧st❡✐♥ s✉② ré♥❣ ♥❤➢ s❛✉✿ ✧❈❤♦ ♠ä✐ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝✱ T : X → X ✳ x, y ∈ X ♠➭ d(x, T x) ●✐➯ sư r➺♥❣ ✈í✐ < d(x, y) t❛ ❝ã d(T x, T y) < d(x, y)✳ ❑❤✐ ➤ã✱ T ❝ã ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣✧✳ ●➬♥ ➤➞②✱ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ ❝ị♥❣ ➤➲ t×♠ ❝➳❝❤ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ♥➭② ❝ñ❛ ❚✳ ❙✉③✉❦✐ ♥❤➺♠ ➤➢❛ r❛ ❝➳❝ ❦Õt q✉➯ ♠í✐ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ✧❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✧✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ❙✉③✉❦✐ ♥❤ê ❝➳❝ ❤➭♠ ➮♥ ❝đ❛ ▼✳❑✐❦❦❛✇❛✱ ❚✳❙✉③✉❦✐✱ ❖✳P♦♣❡s❝✉ ✈➭ ♠ë ré♥❣ ♠ét sè ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ❙✉③✉❦✐✱ ➤➢❛ r❛ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ ❝đ❛ ❑✐❦❦❛✇❛✱ ❙✉③✉❦✐ ✈➭ P♦♣❡s❝✉✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥ ♥❤➢✿ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝✱ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ➳♥❤ ể r ò trì ột sè ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ▼ơ❝ trì ột số ị ý ể t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝đ❛ Ps r ò trì ❜➭② ♠ét sè ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ▼ét sè ♠ë ré♥❣ ❝đ❛ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ ❝đ❛ ▼✉r❛❧✐s❛♥❦❛r ✈➭ ❏❡②❛❜❛❧✳ ▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét ✐✐✐ sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ♣❤Ð♣ φ✲❝♦ s✉② ré♥❣✱ ❤➭♠ φ✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢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❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❚➳❝ ❣✐➯ ❝ò♥❣ ①✐♥ ❝➳♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤♦➳ ✷✷ ●✐➯✐ ❚Ý❝❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ✈➭ ❣✐❛ ➤×♥❤ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ♥❤✃t ➤Ĩ ❣✐ó♣ t➠✐ ❤♦➭♥ t❤➭♥❤ tèt ♥❤✐Ư♠ ✈ơ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ▼➷❝ ❞ï ➤➲ tÝ❝❤ ❝ù❝ ➤➬✉ t➢ ✈➭ ❝ã ♥❤✐Ò✉ ❝è ❣➽♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ữ s sót ợ ữ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ ❝ñ❛ q✉ý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥✳ ❱✐♥❤✱ ♥❣➭② ✷✵ t❤➳♥❣ ✼ ♥➝♠ ✷✵✶✻ ❚rÞ♥❤ ❱➝♥ ▲✉➞♥ ✐✈ ❝❤➢➡♥❣ ✶ ➜Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ ❝đ❛ ❑✐❦❦❛✇❛✱ ❙✉③✉❦✐✱ P♦♣❡s❝✉ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❧✉❐♥ ✈➝♥ ✈➭ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ❊❞❡❧st❡✐♥ ❝ï♥❣ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ ♠➟tr✐❝ tr➟♥ X ✭❬✶❪✮ ❈❤♦ t❐♣ ❤ỵ♣ X ✳ d : X ì X R ợ ọ ❧➭ ♠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✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ ❣✐÷❛ x ✈➭ y✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✷ ✭❬✶❪✮ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟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) , ➜Þ♥❤ ❧ý✳ ✶✳✶✳✸ ➤➬② ➤đ✱ ➤✐Ĩ♠ ➜✐Ĩ♠ ①➵ ✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✮ ●✐➯ sö f :X→X x∗ ∈ X ✈í✐ ♠ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ tõ s❛♦ ❝❤♦ x∗ ∈ X X x, y ∈ 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♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✹ ✭❬✼❪✮ ●✐➯ sư X ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ t➵✐ ❍➭♠ ♠ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✳ ❍➭♠ x0 ∈ X ♥Õ✉ ψ :X →R lim sup ψ(x) ≤ ψ(x0 )✳ x→x0 ψ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ tr➟♥ X ♥Õ✉ ♥ã ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ t➵✐ x ∈ X✳ ❍➭♠ ψ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ tr➟♥ X tr➟♥✱ tr♦♥❣ ➤ã ♥Õ✉ ❤➭♠ −ψ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ (−ψ)(x) = −ψ(x) ✈í✐ ♠ä✐ x ∈ X ✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❤➭♠ ψ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ t➵✐ x0 ∈ X ♥Õ✉ lim inf ψ(x) ≥ ψ(x0 )✳ x→x0 ➜➠✐ ❦❤✐✱ t❛ ✈✐Õt lim ψ(x)✱ lim ψ(x) x→x0 ❧➬♥ ❧➢ỵt t❤❛② ❝❤♦ x→x0 lim sup ψ(x) x→x0 ✈➭ lim inf ψ(x)✳ x→x0 ◆➝♠ ✷✵✵✽✱ ❙✉③✉❦✐ tr♦♥❣ ❬✶✼❪✱ ➤➲ ❣✐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✳ ❚❛ ①➳❝ ➤Þ♥❤ ♠ét ❤➭♠ ❦❤➠♥❣ t➝♥❣ 1−r θ(r) = r2 1+r ●✐➯ sư r➺♥❣ tå♥ t➵✐ ♥Õ✉ ✈í✐ ♠ä✐ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ T : X → X θ : [0, 1) → ( 12 , 1] √ ♥Õ✉ 0≤r≤ √ 5−1 ♥Õ✉ ♥Õ✉ 5−1 , ≤r≤ √ , √ ≤ r ≤ r ∈ [0, 1) s❛♦ ❝❤♦ θ(r)d(x, T x) ≤ d(x, y) t❤× t❛ ❝ã d(T x, T y) ≤ rd(x, y), x, y ∈ X ✳ ✷ ✭✶✳✶✮ ❑❤✐ ➤ã✱ tå♥ t➵✐ ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ lim T n x = z n→∞ ✶✳✶✳✻ ➜Þ♥❤ ❧ý✳ ✈í✐ ♠ä✐ X ❝đ❛ T✳ ❍➡♥ ♥÷❛✱ t❛ ❝ã x ∈ X✳ ✭❬✶✼❪✮ ❱í✐ ❤➭♠ sè ✈í✐ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❜✃t ❦ú ✭✐✮ z θ ①➳❝ ➤Þ♥❤ ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý ✶✳✶✳✺✳ ❑❤✐ ➤ã✱ (X, d) ❝➳❝ ❦Õt q✉➯ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ❧➭ ➤➬② ➤ñ ỗ T : X X tỏ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✶✮ ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ◆➝♠ ✷✵✵✶✱ ❚✳ ❈✳ ▲✐♠ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ L✲❤➭♠ ✈➭ ➤➢❛ r❛ ♠ét ➤➷❝ tr➢♥❣ ❝ñ❛ ➳♥❤ ①➵ ❝♦ ▼❡✐r✲❑❡❡❧❡r✱ ♠➭ ♥ã ❧➭ ♠ét ♠ë ré♥❣ r✃t ♠➵♥❤ ❦Õt q✉➯ ❝ñ❛ ❉✳ ❲✳ ❇♦②❞ ✈➭ ❏✳ ❙✳ ❲♦♥❣✳ ✶✳✶✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✷❪✮ ❍➭♠ sè φ : R+ → R+ ➤➢ỵ❝ ❣ä✐ ❧➭ L✲❤➭♠ ♥Õ✉ φ(0) = 0✱ φ(s) > ✈í✐ s > 0✱ ✈➭ ✈í✐ ♠ä✐ s > 0✱ tå♥ t➵✐ δ > s❛♦ ❝❤♦ φ(t) ≤ s✱ ✈í✐ ♠ä✐ t ∈ [s, s + δ]✳ ◆❤❐♥ ét r ỗ ệ t ợ L tỏ ♠➲♥ φ(s) ≤ s✱ ✈í✐ ♠ä✐ s ≥ 0✳ ❚❛ sÏ ❦ý L✲❤➭♠ ❧➭ ▲✳ ◆➝♠ ✶✾✻✷✱ ▼✳ ❊❞❡❧st❡✐♥ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ ♥Õ✉ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝ ✈➭ ➳♥❤ ①➵ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ T :X→X x = y ✱ t❤× T t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ d(T x, T y) < d(x, y) ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣✳ ◆➝♠ ✷✵✵✾✱ ❚✳ ❙✉③✉❦✐ ➤➲ ♠ë ré♥❣ ❦Õt q✉➯ tr➟♥ ❝ñ❛ ▼✳ ❊❞❡❧st❡✐♥ t ợ ết q s ị ý X X T X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝♦♠♣➽❝ ✈➭ ❣✐➯ sư ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✈í✐ ♠ä✐ ❑❤✐ ➤ã ✭❬✶✽❪✮ ❈❤♦ x, y ∈ X, ♥Õ✉ d(x, T x) < d(x, y) t❤× d(T x, T y) < d(x, y)) ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ◆➝♠ ✶✾✼✻✱ ❏✳ ❇♦❣✐♥ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ❦Õt q✉➯ s❛✉ ➤➞②✳ ✸ T : ➜Þ♥❤ ❧ý✳ ✶✳✶✳✾ T :X→ X ✭❬✺❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ ✈➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ö♥ s❛✉ d(T x, T y) ≤ ad(x, y) + b[d(x, T x) + d(y, T y)] + c[d(x, T y) + d(y, T x)], tr♦♥❣ ➤ã a ≥ 0✱ b > 0✱ c > ✈➭ a + 2b + 2c = 1✳ ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ◆➝♠ ✷✵✶✶✱ ❖✳ P♦♣❡s❝✉ ➤➲ t❤✉ ợ ết q s ị ý (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ➳♥❤ ①➵ T : X → X t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ư♥✿ ❱í✐ ❜✃t ❦ú x, y ∈ X ♥Õ✉ 12 d(x, T x) < d(x, y) t❤× t❛ ❝ã d(T x, T y) ≤ ad(x, y) + b[d(x, T x) + d(y, T y)] + c[d(x, T y) + d(y, T x)], tr♦♥❣ ➤ã a ≥ 0✱ b > 0✱ c > ✈➭ a + 2b + 2c = 1✳ ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ✶✳✷ ❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ s✉② ré♥❣ ❝đ❛ ❑✐❦❦❛✇❛✱ ❙✉③✉❦✐✱ P♦♣❡s❝✉ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ trì ột số ị ý ể t ộ ể ❙✉③✉❦✐ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ñ❛ ▼✳ ❑✐❦❦❛✇❛✱ ❚✳ ❙✉③✉❦✐✱ ❖✳ P♦♣❡s❝✉ ✈➭ tr×♥❤ ❜➭② ♠ét sè ❤Ư q✉➯ ❝ï♥❣ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ✶✳✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✸❪✮ ❑ý ❤✐Ư✉ t❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ ❤➭♠ ❧✐➟♥ tô❝ R+ ❧➭ t❐♣ t✃t ❝➯ ❝➳❝ sè t❤ù❝ ❦❤➠♥❣ ➞♠ ✈➭ Ψ ❧➭ F : R6+ → R t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➞② ✭F1 ✮ F (t1 , , t6 ) ❧➭ ❤➭♠ ❦❤➠♥❣ t➝♥❣ t❤❡♦ ❝➳❝ ❜✐Õ♥ t2 , , t6 ✳ ✭F2 ✮ tå♥ t➵✐ sè r ∈ [0, 1) s ỗ ột tr ề ệ F (u, v, v, u, u + v, 0) ≤ ✹ ✭✶✳✷✮ ❙❛✉ ➤➞② t❛ ❧✉➠♥ ❣✐➯ sö r➺♥❣ E ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ P ❧➭ ♠ét ♥ã♥ tr♦♥❣ E ✈í✐ intP = ∅ ✈➭ ✧≤✧ ❧➭ q✉❛♥ ❤Ư t❤ø tù ❜é ♣❤❐♥ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✳ ✷✳✶✳✺ ị ĩ X ột t rỗ E ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ➳♥❤ ①➵ d : X × X → E ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♠➟tr✐❝ ♥ã♥ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ ✭✶✮ ≤ d(x, y) ✈í✐ ♠ä✐ x, y ∈ X ❀ d(x, y) = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x = y ✳ ✭✷✮ 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 ✳ ❑❤✐ ➤ã (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❧➭ sù tæ♥❣ q✉➳t ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❱Ý ❞ơ s❛✉ ❝❤ø♥❣ tá ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❧➭ sù ♠ë ré♥❣ t❤ù❝ sù ❝đ❛ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ✷✳✶✳✻ ❱Ý ❞ơ✳ ✶✮ ❳Ðt ❦❤➠♥❣ ❣✐❛♥ R ✈í✐ ❝❤✉➮♥ t❤➠♥❣ t❤➢ê♥❣✳ ❑❤✐ ➤ã P = {x ∈ R : x ≥ 0} ❧➭ ♠ét ♥ã♥ tr♦♥❣ R✳ ✷✮ ❈❤♦ E = R2 d : X ×X → E tr♦♥❣ ➤ã ✈➭ P = {(x, y) ∈ E : x, y ≥ 0}✳ ①➳❝ ➤Þ♥❤ ❜ë✐ ❳Ðt X = R ✈➭ ➳♥❤ ①➵ d(x, y) = (|x − y|, α|x − y|) ✈í✐ ♠ä✐ x, y ∈ X ✱ α ❧➭ sè t❤ù❝ ❞➢➡♥❣ ❝❤♦ tr➢í❝✳ ❑❤✐ ➤ã ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ✷✳✶✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳ ❞➲② tr♦♥❣ ✈➭ X ✈➭ ✭❬✾❪✮ ❈❤♦ x ∈ X✳ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ {xn } ❧➭ ♠ét ❉➲② {xn } ➤➢ỵ❝ ❣ä✐ ❧➭ ❤é✐ tơ tí✐ x ♥Õ✉ ✈í✐ ♠ä✐ c ∈ E c✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ N s❛♦ ❝❤♦ d(xn , x) ❑❤✐ ➤ã x ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐í✐ c, ✈í✐ ♠ä✐ ❤➵♥ ❝ñ❛ ❞➲② n > N {xn } ✈➭ t❛ ❦ý ❤✐Ö✉ lim xn = x ❤♦➷❝ xn → x ❦❤✐ n → ∞✳ ✷✷ n→∞ ➜Þ♥❤ ❧ý✳ ✷✳✶✳✽ {xn } ✈➭ ✭❬✾❪✮ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✈í✐ ♥ã♥ ❝❤✉➮♥ t➽❝ P ❧➭ ♠ét ❞➲② tr♦♥❣ X✳ ❑❤✐ ➤ã✱ {xn } ❤é✐ tơ tí✐ x∈X ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ d(xn , x) → ❦❤✐ n → ∞✳ ➜Þ♥❤ ❧ý✳ ✷✳✶✳✾ t➽❝ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✈í✐ ♥ã♥ ❝❤✉➮♥ P ✳ ◆Õ✉ {xn } ❧➭ ♠ét ❞➲② tr♦♥❣ X ✷✳✶✳✶✵ X ✭❬✾❪✮ ❈❤♦ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✾❪✮ ❈❤♦ ❤é✐ tơ tí✐ ❝➯ x ✈➭ y t❤× x = y ✳ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❉➲② {xn } tr♦♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ c∈E ♠➭ c✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ n0 ∈ N s❛♦ ❝❤♦ d(xm , xn ) ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ tr♦♥❣ ✷✳✶✳✶✶ P ✈➭ X c (X, d) ✈í✐ ♠ä✐ m, n ≥ n0 ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤đ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② ➤Ị✉ ❤é✐ tơ✳ ➜Þ♥❤ ❧ý✳ {xn } ✭❬✾❪✮ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✈í✐ ♥ã♥ ❝❤✉➮♥ t➽❝ ❧➭ ♠ét ❞➲② tr♦♥❣ X✳ ❑❤✐ ➤ã✱ {xn } ❧➭ ❞➲② ❈❛✉❝❤② ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ d(xn , xm ) → ❦❤✐ m, n → ∞✳ ✷✳✶✳✶✷ P ✈➭ ➜Þ♥❤ ❧ý✳ ✭❬✶✹❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ♠➟tr✐❝ ♥ã♥ ➤➬② ➤đ ✈í✐ ♥ã♥ ❝❤✉➮♥ t➽❝ f : X → X ✳ ◆Õ✉ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ x = y ♠Ư♥❤ ➤Ị s❛✉ ➤➞② d(x, f x) − d(x, y) ∈ / intP ⇒ d(f x, f y) ❧➭ ➤ó♥❣ ✈í✐ sè ♥➭♦ ➤ã α ∈ (0, 1)✱ t❤× f αd(x, y) ✭✷✳✸✮ ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ f ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣✳ ▲✃② ➤✐Ó♠ ❜✃t ❦ú x0 ∈ X ✳ t❤× d(x0 , f x0 ) = 0✳ ❇➺♥❣ ❝➳❝❤ ❧✃② x = x0 , y = f (x0 )✱ ❦❤✐ ➤ã ✈× 21 d(x0 , f x0 ) − ◆Õ✉ d(x0 , f x0 ) ∈ / P✱ ✭✷✳✸✮ t❛ ❝ã f (x0 ) = x0 ✱ t❤× x0 ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ f ✳ ♥➟♥ t❛ ❝ã d(x0 , f x0 ) ◆Õ✉ − d(x0 , f x0 ) ∈ / intP ✳ d(f x0 , f (f x0 )) ≤ rd(x0 , f x0 )✳ ❇➞② ❣✐ê t❛ ➤➷t f (x0 ) = x0 ✱ ❇ë✐ ✈❐②✱ ♥❤ê xn = f xn−1 ✈í✐ ♠ä✐ n ≥ 1✳ ◆Õ✉ ✈í✐ sè ♥➭♦ ➤ã n ∈ N t❛ ❝ã f xn = xn t❤á❛ ♠➲♥✱ t❤× xn ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f✳ ◆Õ✉ ✈í✐ ♠ä✐ n ∈ N t❛ ❝ã xn = f x n ✳ ✷✸ ❑❤✐ ➤ã✱ ❧❐♣ ❧✉❐♥ ♥❤➢ tr➟♥ d(xn , f xn ) t❛ ❝ã − d(xn , f xn ) ∈ / P ✱ ✈× t❤Õ✱ ♥❤ê ✭✷✳✸✮ t❛ ❝ã d(f xn , f (f xn ) ≤ rd(xn , f xn ) ✈í✐ ♠ä✐ n ∈ N✳ ➜✐Ị✉ ♥➭② ❝ị♥❣ ❝ã ♥❣❤Ü❛ ❧➭ d(xn+2 , xn+1 ) ≤ rd(xn+1 , xn ) t❤á❛ ♠➲♥ ✈í✐ ♠ä✐ n ∈ N ❚õ ❣✐➯ t❤✐Õt ❜➺♥❣ q✉② ♥➵♣ t❛ ❝ã d(xn+1 , xn ) ≤ αd(xn , xn−1 ) ≤ α2 d(xn−1 , xn−2 ) ≤ · · · ≤ αn−1 d(x2 , x1 ) m, n ∈ N ✈➭ ❦❤➠♥❣ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t t❛ ❣✐➯ sư m ≥ n✱ ♥❤ê ❚õ ➤ã✱ ✈í✐ ❜✃t ❦ú ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ t❛ ❝ã d(xm , xn ) ≤ d(xm , xm−1 ) + d(xm−1 , xm−2 ) + · · · + d(xn+1 , xn ) ≤ [αm−1 + · · · + αn−1 ]d(x2 , x1 ) αn−1 d(x2 , x1 ) ≤ 1−α ❱× P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ❝ã αn−1 d(xm , xn ) ≤ d(x2 , x1 ) 1−α ❱× α 0✳ ❱❐②✱ tr♦♥❣ ∈ (0, 1)✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ t❤✉ ➤➢ỵ❝ lim d(xm , xn ) = m,n→∞ {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ X s❛♦ ❝❤♦ ❚õ tÝ♥❤ ➤➬② ➤ñ ❝ñ❛ X s✉② r❛ tå♥ t➵✐ ♠ét ➤✐Ó♠ p xn → p ❦❤✐ n → ∞✳ ❇➞② ❣✐ê✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ♣❤➢➡♥❣ ♣❤➳♣ t➢➡♥❣ tù ♥❤➢ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✷ ✈➭ sư ❞ơ♥❣ tÝ♥❤ ❝❤✉➮♥ t➽❝ ❝đ❛ ♥ã♥ ➤✐Ĩ♠ ✷✳✷ P t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ p ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝đ❛ f ✳ ▼ë ré♥❣ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❙✉③✉❦✐ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ♣❤Ð♣ φ✲❝♦ s✉② ré♥❣✱ ❤➭♠ φ✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ✈➭ tr×♥❤ ❜➭② ♠ét sè ❤Ư q✉➯ ❝ï♥❣ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ✷✹ ●✐➯ sư T :X →X ➤✐Ĩ♠ ❝è ➤Þ♥❤✱ t❛ ➤➷t ❧➭ ➳♥❤ ①➵ ❝♦ tr➟♥ ❦❤➠♥❣ tr xn = T n x ỗ n ∈ N✳ ❞ï♥❣ ❝➳❝ ❦ý ❤✐Ư✉ s❛✉✿ ❱í✐ ✷ ❞➲② t ỳ tr ỗ X xX ❧➭ ❚r♦♥❣ ❝➳❝ ♠ô❝ t✐Õ♣ t❤❡♦ t❛ sÏ {xpn } ✈➭ {xqn } ❝ñ❛ ❞➲② {xn } ➤➲ ❝❤♦ n ∈ N t❛ ➤➷t δn = d(xpn , xqn ) ✈➭ ♥Õ✉ δn = 0, 0 ∆n = d(T xpn , T xqn ) ♥Õ✉ δn > δn ◆➝♠ ✶✾✼✸✱ ▼✳ ❆✳ ●❡r❛❣❤t② ➤➲ ❝❤ø♥❣ ợ ết q s ị ý X x ∈ X ✭❬✽❪✮ ❈❤♦ t❛ ➤➷t xn = T n x ỗ n N ó xn ❤é✐ tơ ✈Ị ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝đ❛ ❞➲② T : X → X ❧➭ ♠ét ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ T ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ❤❛✐ ❞➲② ❝♦♥ ❜✃t ❦ú x pn ✈➭ xqn ❝đ❛ {xn } ✈í✐ xpn = xqn ✱ ♥Õ✉ ∆n tì n ị ĩ ❈❤♦ T : X → X ✳ ❚❛ ❣ä✐ T (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ➳♥❤ ①➵ ❧➭ ♠ét ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ ♥Õ✉ ∀ x, y ∈ X, x = y, d(x, T x) ≤ d(x, y) ⇒ d(T x, T y) < d(x, y) ➜Þ♥❤ ❧ý✳ ✷✳✷✳✸ ❣✐❛♥ ♠➟tr✐❝ ✭❬✷❪✮ ❈❤♦ X✳ ❱í✐ T :X →X x∈X ✭✷✳✹✮ ❧➭ ♠ét ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr➟♥ ❦❤➠♥❣ ❝❤♦ tr➢í❝ ✈➭ ❞➲② xn = T n x, n ∈ N✱ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✭✶✮ {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤②✳ ✭✷✮ ❱í✐ ❤❛✐ ❞➲② ❝♦♥ {xpn } ✈➭ {xqn } ♠➭ d(xpn , T xpn ) ≤ d(xpn , xqn ) ✈í✐ ♠ä✐ n ∈ N✱ ♥Õ✉ ∆n → t❤× δn → 0✳ ❈❤ø♥❣ ♠✐♥❤✳ ❉Ơ t❤✃② ❞➲② ❝♦♥ (1) ⇒ (2)✱ ✈× ♥Õ✉ {xn } ❧➭ ❞➲② ❈❛✉❝❤②✱ t❤× ✈í✐ ❤❛✐ {xpn } ✈➭ {xqn }✱ t❛ ❝ã δn → 0✳ ➜Ó ❝❤ø♥❣ ♠✐♥❤ ♣❤Ð♣ ❦Ð♦ t❤❡♦ (2) ⇒ (1)✱ t❛ ❣✐➯ sö {xn } ❧➭ ❞➲② t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ♠Ư♥❤ ➤Ị ✭✷✮✳ ❚r➢í❝ ❤Õt t❛ ❣✐➯ sư ✷✺ xm = xm+1 ✱ ✈í✐ ♠ä✐ m✳ ❑❤✐ ➤ã xn = xm ✱ ✈× t❤Õ {xn } ✈í✐ ♠ä✐ n ≥ m✱ {xn } ♥❣❤Ü❛ ❧➭ ❧➭ ❞➲② ❤➺♥❣ ❦Ó tõ ❝❤Ø sè ❧➭ ♠ét ❞➲② ❈❛✉❝❤②✳ ❇➞② ❣✐ê t❛ ❣✐➯ sö d(xn , T xn ) ≤ d(xn , T xn )✱ xn = xn+1 ♥➟♥ tõ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✹✮ t❛ s✉② r❛ ✈í✐ ♠ä✐ n✳ m✱ ❱× d(xn+1 , xn+2 ) = d(T xn , T (T xn )) < d(xn , T xn ) = d(xn , xn+1 )✱ ♥❣❤Ü❛ ❧➭ ❞➲② {d(xn , xn+1 )} ❧➭ ❣✐➯♠ ♥❣➷t✳ ❉♦ ✈❐② ♠✐♥❤ r➺♥❣ δ = 0✳ d(xn , xn+1 ) → δ δ ✈í✐ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ ❧➭ ♠ét sè ❦❤➠♥❣ ➞♠✳ ❚❛ sÏ ❝❤ø♥❣ δ > 0✱ ❦❤✐ ➤ã ♥Õ✉ t❛ ❧✃② pn = n ✈➭ qn = n + 1✱ t❤× t❛ ❝ã d(xpn , T xpn ) ≤ d(xpn , xqn )✱ ✈í✐ ♠ä✐ n✱ ✈➭ ∆n → tr♦♥❣ ❦❤✐ δn ❝ã = d(xn , xn+1 ) → δ = 0✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ♠Ư♥❤ ➤Ị ✭✷✮✳ ❱× t❤Õ✱ t❛ d(xn , xn+1 ) → 0✳ ❱× d(xn , xn+1 ) → ❦❤✐ n → ∞✱ kn ∈ N s❛♦ ❝❤♦ d(xm , xm+1 ) < n ✈í✐ ♠ä✐ ❞➲② ❈❛✉❝❤②✱ t❤× t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ tå♥ t➵✐ ♥❣✉②➟♥ ❞➢➡♥❣ s❛♦ ❝❤♦ r➺♥❣ n0 q n > p n > kn m ≥ kn ✳ n ∈ N✱ ◆Õ✉ t❛ ❝❤ä♥ ➤➢ỵ❝ sè {xn } ❦❤➠♥❣ ♣❤➯✐ ❧➭ ε > ✈➭ ❤❛✐ ❞➲② {pn } ✈➭ {qn } ❝➳❝ sè ✈➭ d(xpn , xqn ) ≥ ε✳ ❚❛ ❝ã t❤Ó ❣✐➯ t❤✐Õt qn ❧➭ sè ♥❣✉②➟♥ ❞➢➡♥❣ ❜Ð ♥❤✃t s❛♦ ❝❤♦ d(xpn , xqn−1 ) < ε✳ ❉♦ ➤ã✱ t❛ ❝ã ➜✐Ò✉ ỗ < d(xpn , xqn ) ≤ d(xpn , xqn−1 ) + d(xqn−1 , xqn ) < ε + n ∗ ♥➭② ❝❤Ø r❛ r➺♥❣ δn = d(xpn , xqn ) → ε✳ ❑ý ❤✐Ư✉ n0 ∈ N ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ s❛♦ ❝❤♦ n ∈ N ♠➭ n ≥ n0 t❛ ❝ã d(xpn , T xpn ) ≤ d(xpn , xqn )✳ ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ♥➭②✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✹✮ t❛ ♥❤❐♥ ➤➢ỵ❝ d(T xpn , T xqn ) < d(xpn , xqn ) = δn ✈í✐ ♠ä✐ n ≥ n0 ✳▼➷t ❦❤➳❝✱ t❛ ❝ã d(xpn , xqn ) ≤ d(xpn , T xpn )+d(T xpn , T xqn )+d(T xqn , xqn ) ≤ 1 +d(T xpn , T xqn )+ n n ❱× ✈❐②✱ t❛ ❝ã δn − δn n ≤ d(T xpn , T xqn ) = ∆n < δn ✈í✐ ♠ä✐ ❚õ ❜✃t ➤➺♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ s✉② r❛ r➺♥❣ ∆n ➤Ị ✭✷✮ ➤ó♥❣ t❛ ❝ã n ≥ n0 → ✈➭ ✈× ✈❐② ♥❤ê ♠Ư♥❤ δn → ❦❤✐ n → ∞✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ❧❐♣ ❧✉❐♥ tr➟♥ ❧➭ δn = d(xpn , xqn ) → ε > 0✳ ❉♦ ➤ã {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ✷✻ ❑Õt q✉➯ s❛✉ ➤➞② ❧➭ ♠ét ♠ë ré♥❣ ❦✐Ó✉ ❙✉③✉❦✐ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶✳ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✹ ①➵ tr➟♥ X ✭❬✷❪✮ ❈❤♦ X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ ✈➭ T ❧➭ ♠ét ➳♥❤ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ∀ x, y ∈ X, d(x, T x) < d(x, y) ⇒ d(T x, T y) < d(x, y) ❱í✐ x∈X ❝❤♦ tr➢í❝✱ ➤➷t ❞➲② ✭✷✳✺✮ xn = T n x, n ∈ N∗ ✳ ❑❤✐ ➤ã✱ ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✭✶✮ xn → z tr♦♥❣ X ✱ ✈í✐ z ✭✷✮ ❱í✐ ❤❛✐ ❞➲② ❝♦♥ ❜✃t ❦× ✈í✐ ♠ä✐ ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝ñ❛ {xpn } ✈➭ {xqn }✱ ♠➭ z ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ❑❤✐ ➤ã ♥❤ê ➤✐Ị✉ ❦✐Ö♥ ✭✷✳✺✮ t❛ s✉② r❛ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ r➺♥❣ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➤✐Ĩ♠ y∈X ♠➭ T✳ ❱❐② ♥Õ✉ ♠➭ ❝ã ♥❤✐Ò✉ ♥❤✃t ❧➭ ♠ét ➤✐Ĩ♠ z = y t❤× t❛ ❝ã 12 d(z, T z) < d(z, y)✳ d(T z, T y) < d(z, y)✳ T y = y✳ z∈X (1) ⇒ (2) ❱× T z = z✱ ➜✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ ❧➭ ✈➭ z y ♥➟♥ tõ ❦❤➠♥❣ ♣❤➯✐ ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥✱ ✈× ♥Õ✉ ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝ñ❛ {xqn }✱ ✈➭ T T ✱ t❤× ♠ä✐ y = z t❛ ❝ã y ❦❤➠♥❣ ♣❤➯✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ P❤Ð♣ ❦Ð♦ t❤❡♦ ✈➭ d(xpn , T xpn ) ≤ d(xpn , xqn ) n✱ ♥Õ✉ ∆n → t❤× δn → 0✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❜✃t ➤é♥❣✳ ◆Õ✉ T❀ T✳ xn → z tr♦♥❣ X✱ ❑❤✐ ➤ã✱ ✈í✐ ❤❛✐ ❞➲② ❝♦♥ ❜✃t ❦× d(xpn , T xpn ) ≤ d(xpn , xqn ) ✈í✐ ♠ä✐ n✱ ♥Õ✉ ∆n → ✈í✐ z {xpn } t❤× t❛ ❝ã δn = d(xpn , xqn ) → d(z, z) = 0✳ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ t❛ s✉② r❛ ❞➲② ➤đ✱ ♥➟♥ (2) ⇒ (1)✳ ●✐➯ sư ♠Ư♥❤ ➤Ị ✭✷✮ ➤ó♥❣✱ ❦❤✐ ➤ã ♥❤ê {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ❱× X ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② xn → z ✈í✐ z ∈ X ✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ T z = z ✳ ➜➬✉ t✐➟♥✱ t❛ ♥❤❐♥ t❤✃② r➺♥❣ ✈í✐ ♠ä✐ n ≥ t❛ ❝ã d(xn , xn+1 ) < 2d(xn , z) ❤♦➷❝ d(xn+1 , xn+2 ) < 2d(xn+1 , z) ✭✷✳✻✮ ✷✼ ❚❤❐t ✈❐②✱ ❣✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ tå♥ t➵✐ sè n0 ✈➭ 2d(xn0 +1 , z) ≤ d(xn0 +1 , xn0 +2 )✳ d(xn0 , T xn0 )✱ ≥ s❛♦ ❝❤♦ 2d(xn0 , z) ≤ d(xn0 , xn0 +1 ) ❑❤✐ ➤ã✱ ✈× ❤✐Ĩ♥ ♥❤✐➟♥ ♥➟♥ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✮ t❛ ❝ã d(xn0 , T xn0 ) < d(xn0 +1 , xn0 +2 ) < d(xn0 , xn0 +1 )✳ ❉♦ ➤ã✱ t❛ ❝ã 2d(xn0 , xn0 +1 ) ≤ 2d(xn0 , z) + 2d(xn0 +1 , z) ≤ d(xn0 , xn0 +1 ) + d(xn0 +1 , xn0 +2 ) < d(xn0 , xn0 +1 ) + d(xn0 , xn0 +1 ) = 2d(xn0 , xn0 +1 ) ➜✐Ò✉ ♥➭② ❧➭ ✈➠ ❧ý ✈➭ ❞♦ ➤ã t❛ ❝ã ✭✷✳✻✮✳ ❚õ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✮ ✈➭ ✭✷✳✻✮ s✉② r❛ ✈í✐ ♠ä✐ n ≥ t❛ ❝ã d(xn+1 , T z) < d(xn , z) ❤♦➷❝ d(xn+2 , T z) < d(xn+1 , z)) ✭✷✳✼✮ ❱× xn → z ✱ ♥➟♥ tõ ✭✷✳✼✮ s✉② r❛ tå♥ t➵✐ ♠ét ❞➲② ❝♦♥ ◆❤ê tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ❣✐í✐ ❤➵♥ t❛ s✉② r❛ {xn } ❤é✐ tô ➤Õ♥ T z✳ T z = z ✳ ❱❐②✱ z ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ❚õ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❛ ❝ã z ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝đ❛ T ✳ ✷✳✷✳✺ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✷❪✮ ❈❤♦ ❤➭♠ sè s ∈ R+ ✳ ➳♥❤ ①➵ T : X → X φ : R+ → R+ s❛♦ ❝❤♦ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ φ(s) ≤ s ✈í✐ ♠ä✐ X ✱ ➤➢ỵ❝ ❣ä✐ ❧➭ ♣❤Ð♣ φ✲❝♦ s✉② ré♥❣ ♥Õ✉ ∀ x, y ∈ X, x = y, d(x, T x) ≤ d(x, y) ⇒ d(T x, T y) < φ(d(x, y) ❍➭♠ φ : R+ → R+ ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ➤è✐ ✈í✐ ❧➭ ❤➭♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✮✱ ♥Õ✉ ✈í✐ ♠ä✐ ➤✐Ĩ♠ ❜❛♥ ➤➬✉ T x ∈ X✱ ✭✷✳✽✮ ✭❤❛② ♥ã✐ ❣ä♥ ❞➲② ❧➷♣ xn = T n x, n ∈ N✱ ❧❐♣ t❤➭♥❤ ♠ét ❞➲② ❈❛✉❝❤②✳ ✷✳✷✳✻ ❑ý ❤✐Ö✉✳ ❦Ý ❤✐Ö✉ ❧➭ ❆✳ s❛♦ ❝❤♦ ❤➭♠ ✭❬✷❪✮ ❚❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ ❤➭♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ➤è✐ ✈í✐ ➳♥❤ ①➵ ❚❛ sÏ ❦ý ❤✐Ư✉ α(s) = t ợ ữ ợ T φ ∈❆ φ(s) s ➤➡♥ ➤✐Ö✉ ❣✐➯♠ tr♦♥❣ ❧➞♥ ❝❐♥ ❝ñ❛ 0✱ ♥❣❤Ü❛ ❧➭ tå♥ t➵✐ sè δ > s❛♦ ❝❤♦ ♥Õ✉ < s < t < δ t❤× t❛ ❝ã α(t) ≤ α(s) ✷✽ ✭✷✳✾✮ ❚❛ ❦Ý ❤✐Ư✉ ❜ë✐ + ❧➭ t❐♣ ❤ỵ♣ ❝➳❝ ❤➭♠ sè ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❆ α0 = lim inf α(s) = lim inf s→0+ s→0+ φ(s) >0 s ➜Ó ➤➡♥ ❣✐➯♥ tr♦♥❣ ❝➳❝❤ ✈✐Õt✱ ✈í✐ ❤❛✐ ➤✐Ĩ♠ ♣❤➞♥ ❜✐Ưt sÏ ✈✐Õt ✷✳✷✳✼ φ ∈❆0 ❝❤♦ ✭✷✳✶✵✮ x, y ➤➲ ❝❤♦ tr➟♥ X ✱ t❛ α(x, y) t❤❛② ❝❤♦ α(d(x, y))✳ ▼Ư♥❤ ➤Ị✳ ✭❬✷❪✮ ❈❤♦ X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ➳♥❤ ①➵ T : X → X ✳ ●✐➯ sư r➺♥❣ ✈í✐ ♠ét ❤➭♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ♥➭♦ ➤ã φ ∈ ❆✱ t❛ ❝ã ∀ x, y ∈ X, d(x, T x) < d(x, y) =⇒ d(T x, T y) < φ(d(x, y) ❑❤✐ ➤ã✱ T ✭✷✳✶✶✮ ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ➤✐Ó♠ ❜✃t ❦ú x ∈ X ✈➭ ①➞② ❞ù♥❣ ❞➲② ❧➷♣ xn = T n x, n ∈ N ì ợ t❛ ❝ã {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤②✳ ❇➺♥❣ ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ ❝❤ø♥❣ ♠✐♥❤ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✸ t❛ s✉② r❛ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❚õ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✼ ✈➭ ❝➳❝ ❦ý ❤✐Ư✉ tr➟♥ t❛ ❝ã ❦Õt q✉➯ s❛✉✳ ✷✳✷✳✽ ➜Þ♥❤ ❧ý✳ ✭❬✷❪✮ ỗ L ợ ĩ ⊂ ❆✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ d(xm , xm+1 ) = 0✱ ❑❤✐ ➤ã r❛ {xn } X✳ φ ❧➭ ♠ét ❈è ➤Þ♥❤ ✈í✐ ❝❤Ø sè m L✲❤➭♠ x ∈ X ✈➭ T ❧➭ ♠ét ♣❤Ð♣ ✈➭ ➤➷t ♥➭♦ ➤ã✱ t❤× t❛ ❝ã φ ❧➭ ❤➭♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✳ ●✐➯ sư r➺♥❣ s✉② ré♥❣ xn = T n x, n ∈ N✳ xn = xm ❧➭ ❞➲② ❤➺♥❣ tõ ♠ét ❧ó❝ ♥➭♦ ➤ã✱ ➤ã ➤ã φ✲❝♦ {xn } ✈í✐ ♠ä✐ ◆Õ✉ n ≥ m✳ ❧➭ ❞➲② ❈❛✉❝❤②✳ ❙✉② d(xn , xn+1 ) > ✈í✐ ♠ä✐ n ≥ 1✳ ❱× d(xn , T xn ) ≤ d(xn , T xn ) ✈➭ xn = xn+1 ✱ ọ n ề ệ ỗ n ∈ N t❛ ❝ã d(xn+1 , xn+2 ) < φ(d(xn , xn+1 )) ≤ d(xn , xn+1 ) ➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá r➺♥❣ ❞➲② tơ tí✐ sè s ≥ 0✳ ◆Õ✉ {d(xn , xn+1 )} ❧➭ ❞➲② ❣✐➯♠ ♥❣➷t ✈➭ ✈× ✈❐② ♥ã ❤é✐ s > 0✱ t❤× ✈× φ ✷✾ ❧➭ L✲❤➭♠ ♥➟♥ tå♥ t➵✐ δ > s❛♦ ❝❤♦ φ(t) ≤ s ✈í✐ s ≤ t ≤ s + δ ✳ ▲✃② n ∈ N ➤đ ❧í♥ ➤Ó s ≤ d(xn , xn+1 ) ≤ s + δ ✳ ❑❤✐ ➤ã d(xn+1 , xn+2 ) < φ(d(xn , xn+1 )) ≤ s, ➤✐Ò✉ ♥➭② ❧➭ ♠➞✉ t❤✉➱♥✳ ❉♦ ➤ã d(xn , xn+1 ) → 0✳ ❚✐Õ♣ t❤❡♦✱ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ✈➭ ➤➷t s= ε ✳ ❱× {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤②✳ ▲✃② ❜✃t ❦ú ε > φ ❧➭ L✲❤➭♠✱ ♥➟♥ tå♥ t➵✐ δ ∈ (0, s) s❛♦ ❝❤♦ φ(t) ≤ s ✈í✐ s ≤ t ≤ s + δ ✳ ❱× d(xn , xn+1 ) → 0✱ tå♥ t➵✐ sè N ∈ N s❛♦ ❝❤♦ d(xn , xn+1 ) < δ ✈í✐ ♠ä✐ n ≥ N ✳ ❚❛ sÏ ❝❤Ø r❛ r➺♥❣ d(xn , xn+m ) < δ + s ≤ ε, ❚❤❐t ỗ n N ọ n N, m ∈ N ✭✷✳✶✷✮ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ✭✷✳✶✷✮ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣ t❤❡♦ ♠✳ ❘â r➭♥❣ t❛ t❤✃② ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✷✮ t❤á❛ ♠➲♥ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✷✮ t❤á❛ ♠➲♥ ✈í✐ m ∈ N✳ m = 1✳ ●✐➯ sö ❑❤✐ ➤ã tõ ❝➳❝ ❧❐♣ ❧✉❐♥ tr➟♥ t❛ t❤✉ ➤➢ỵ❝ φ(d(xn , xn+m ) ≤ s✳ ❇➞② ❣✐ê✱ ♥Õ✉ d(xn , T xn ) ≤ d(xn , xn+m ) ❦❤✐ ➤ã t❤❡♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✽✮ t❛ ❝ã d(xn+1 , xn+m+1 ) < φ(d(xn , xn+m )) ✈➭ ✈× t❤Õ t❛ ❝ã d(xn , xn+m+1 ) ≤ d(xn , xn+1 ) + d(xn+1 , xn+m+1 ) < δ + s ≤ ε ◆Õ✉ d(xn , xn+m ) < d(xn , T xn ) t❤× ✈× d(xn , xn+1 ) < δ ✱ ♥➟♥ t❛ ❝ã d(xn , xn+m ) < δ ✈➭ ❞♦ ➤ã t❛ ♥❤❐♥ ➤➢ỵ❝ d(xn , xn+m+1 ) ≤ d(xn , xn+m ) + d(xn+m , xn+m+1 ) < δ + δ < δ + s ≤ ε ❉♦ ➤ã ✭✷✳✶✷✮ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ ✈➭ ❞➲② {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ❱× ✈❐②✱ ❤➭♠ φ ❧➭ ❝❤✃♣ ợ t ý ệ t ì sè ❞➢➡♥❣ ❙ ❧➭ t❐♣ t✃t ❝➯ ❝➳❝ ❤➭♠ α : R+ → [0, 1] s❛♦ ❝❤♦✱ ✈í✐ ❞➲② {sn }✱ ♥Õ✉ α(sn ) → t❤× sn → 0✳ ❑❤✐ ➤ã t❛ ❝ã ❦Õt q✉➯ s❛✉✳ ✷✳✷✳✾ ➜Þ♥❤ ❧ý✳ ✭❬✷❪✮ ◆Õ✉ α ∈ ❙✱ t❤× ❤➭♠ sè φ(s) = α(s).s ✈í✐ ♠ä✐ s ∈ R+ ❤➭♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✳ ✸✵ ❧➭ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö α ∈ ❙✳ ❚❛ ①Ðt ❤➭♠ φ(s) = α(s)s ✈í✐ ♠ä✐ s ∈ R+ ✳ ●✐➯ sư T X ✳ ▲✃t ➤✐Ĩ♠ ❜✃t ❦ú x ∈ X ❣✐❛♥ ♠➟tr✐❝ ➜➷t sn = d(xn , xn+1 ), n ∈ N✳ ❝❤Ø ❝➬♥ ❣✐➯ sö sn+1 sn ❧➭ ♠ét ❤➭♠ ①➳❝ ➤Þ♥❤ ❜ë✐ φ✲❝♦ s✉② ré♥❣ tr➟♥ ❦❤➠♥❣ ✈➭ ①➞② ❞ù♥❣ ❞➲② ❧➷♣ xn = T n x, n ∈ N✳ ❚➢➡♥❣ tù ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤ ë ➜Þ♥❤ ❧ý ✷✳✷✳✽✱ t❛ sn > ✈í✐ ♠ä✐ n ∈ N✳ s ✈í✐ s ≥ 0✳ ◆Õ✉ s > t❤× φ : R+ → R + ❑❤✐ ➤ã sn+1 < α(sn )sn ✈➭ ❞♦ ➤ã sn → → ✈➭ ❞♦ ✈❐② α(sn ) → 1✳ ❱× α ∈ ❙✱ ♥➟♥ t❛ ❝ã s = ➤✐Ò✉ ♥➭② ❧➭ ✈➠ ❧ý✳ ❉♦ ✈❐② s = ✈➭ d(xn , xn+1 ) → 0✳ ❱× d(xn , xn+1 ) ỗ n N t ❝❤ä♥ ➤➢ỵ❝ sè kn ∈ N s❛♦ ❝❤♦ d(xm , xm+1 ) < t❤× tå♥ t➵✐ n ✈í✐ ♠ä✐ m ≥ kn ✳ ◆Õ✉ ❞➲② {xn } ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❞➲② ❈❛✉❝❤②✱ ε > ✈➭ {pn }✱ {qn } ❧➭ ❤❛✐ ❞➲② sè ❞➢➡♥❣ s❛♦ ❝❤♦ qn > pn ≥ kn ✈➭ d(xpn , xqn ) ≥ ε✱ d(xpn , xqn−1 ) < ε✳ ❉♦ ➤ã✱ t❛ t❤✉ ➤➢ỵ❝ ε ≤ d(xpn , xqn ) ≤ d(xpn , xqn−1 ) + d(xqn−1 , xqn ) < ε + n ➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá r➺♥❣ d(xpn , xqn ) → ε✳ ❱× d(xpn , T xpn ) ≤ d(xpn , xqn ), ✈í✐ n ∈ N✱ ➤✐Ị✉ ❦✐Ö♥ ✭✷✳✽✮ ❝❤Ø r❛ r➺♥❣ d(xpn+1 , xqn+1 ) < α(sn )sn ✳ ❉♦ ➤ã t❛ ❝ã sn = d(xpn , xqn ) ≤ d(xpn , xpn+1 ) + d(xpn+1 , xqn+1 ) + d(xqn+1 , xqn ) < + α(sn )sn n ❈❤✐❛ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❝❤♦ sn ✱ ✈× ε ❚õ ❣✐➯ t❤✐Õt ✈í✐ ♠ä✐ ≤ α(sn ) ≤ 1✱ t❛ ♥❤❐♥ ➤➢ỵ❝ r➺♥❣ α(sn ) → 1✳ α ∈ ❙ t❛ s✉② r❛ sn → ➤✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ d(xpn , xqn ) ≥ ε n ≥ 1✳ ▼➞✉ t❤✉➱♥ ♥➲② ❝❤ø♥❣ tá {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ❱× ✈❐②✱ φ ❧➭ ❤➭♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✳ ❇➞② ❣✐ê t❛ ♣❤➳t ❜✐Ĩ✉ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ ❝❤♦ ➳♥❤ ①➵ ✷✳✷✳✶✵ φ✲❝♦✳ ➜Þ♥❤ ❧ý✳ ✭❬✷❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ➳♥❤ ①➵ T : X → X ✳ ●✐➯ sư ✈í✐ ❤➭♠ φ ∈ ❆0 ♥➭♦ ➤ã ✈➭ α(s) = ∀ x, y ∈ X, φ(s) s ✱ t❛ ❝ã d(x, T x) < d(x, y) =⇒ d(T x, T y) < φ(d(x, y) + α(x, T x) ✸✶ ✭✷✳✶✸✮ ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ♥Õ✉ tå♥ t➵✐ t❤× ♥ã ❧➭ ❞✉② ♥❤✃t✳ ❚❤❐t ✈❐②✱ ❣✐➯ sư ♠➭ z∈X ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ T ✈➭ y∈X y = z ✱ ❦❤✐ ➤ã t❛ ❝ã (1 + α(z, T z))−1 d(z, T z) = < d(z, y), ✈➭ ✈× t❤Õ ♥❤ê ➤✐Ị✉ ❦✐Ö♥ ✭✷✳✶✸✮ t❛ ❝ã ➤➻♥❣ t❤ø❝ ❝✉è✐ ♥➭② t❛ ❝ã d(T z, T y) < d(z, y)✳ ❱× T z = z ✱ ♥➟♥ tõ ❜✃t d(z, T y) < d(z, y)✳ ➜✐Ò✉ ♥➭② ❝❤Ø r❛ r➺♥❣ t❛ ♣❤➯✐ ❝ã T y = y ✱ ♥❣❤Ü❛ ❧➭ y ❦❤➠♥❣ ♣❤➯✐ ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ❇➞② ❣✐ê✱ t❛ ❝❤ø♥❣ ♠✐♥❤ tå♥ t➵✐ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣✳ ▲✃② ❤❛✐ ➤✐Ĩ♠ X, ✈× x, y ∈ = y ✳ ◆Õ✉ d(x, T x) ≤ d(x, y) t❤× (1 + α(x, T x))−1 d(x, T x) < d(x, y)✱ ✈í✐ x φ ∈ ❆0 ✱ α(x, T x) > ✈➭ d(x, y) > 0✳ ❉♦ ➤ã✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✸✮ t❛ s✉② r❛ T t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✽✮ ✈í✐ xn = T n x, n ∈ N✳ ❱× ❤➭♠ sè {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ❉♦ X t❤❡♦ t❛ sÏ ❝❤Ø r❛ r➺♥❣ ◆Õ✉ ✈➭ ①➳❝ ➤Þ♥❤ ❞➲② ❧➷♣ φ(s) = α(s)s ❧➭ ❤➭♠ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ ♥➟♥ ❞➲② ❧➭ ➤➬② ➤ñ✱ ♥➟♥ tå♥ t➵✐ z∈X xn → z ✳ ❚✐Õ♣ s❛♦ ❝❤♦ T z = z✳ xm = T xm T z = z✳ ✈➭ t❛ ❝ã φ(s) = α(s)s✳ ❈è ➤Þ♥❤ x ∈ X ✈í✐ sè tù ♥❤✐➟♥ ●✐➯ sư r➺♥❣ ❦✐Ư♥ ✭✷✳✾✮ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✈í✐ m ♥➭♦ ➤ã✱ t❤× x n = T xn ✱ δ > ✈í✐ ♠ä✐ xn = z n ≥ 1✳ ✈í✐ ♠ä✐ ❱× φ ∈ ♥➭♦ ➤ã✳ ▲✃② ♠ét sè ❞➢➡♥❣ N n ≥ m ❆0 ✱ ➤✐Ò✉ s❛♦ ❝❤♦ d(xn , T xn ) < δ ✈í✐ n ≥ N ✳ ❑❤✐ ➤ã✱ t❛ ❝ã < d(T xn , T xn ) < φ(d(xn , T xn )) ≤ d(xn , T xn ), ✈➭ tõ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✾✮ t❛ ❝ã α(xn , T xn ) ≤ α(T xn , T xn ), ✈í✐ n ≥ N ✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ α(xn , T xn ) + ≤ 1 + α(xn , T xn ) + α(T xn , T xn ) ✭✷✳✶✹✮ ❚✐Õ♣ t❤❡♦ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ✈í✐ ♠ä✐ n ≥ N, t❛ ❝ã −1 (1 + α(xn , T xn )) d(xn , T xn ) < d(xn , z), ❤♦➷❝ (1 + α(T x , T x ))−1 d(T x , T x ) < d(x , z) n n n n n+1 ✭✷✳✶✺✮ ✸✷ ❚❛ ❝❤ø♥❣ ♠✐♥❤ ❜➺♥❣ ♣❤➯♥ ❝❤ø♥❣✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ ❦Õt ❧✉❐♥ tr➟♥ ❦❤➠♥❣ ➤ó♥❣✱ ❦❤✐ ➤ã ✈í✐ n≥N ♥➭♦ ➤ã✱ t❛ ❝ã d(xn , z) ≤ (1 + α(xn , T xn ))−1 d(xn , T xn ) d(xn+1 , z) ≤ (1 + α(T xn , T xn ))−1 d(T xn , T xn ) ❚õ ✭✷✳✶✹✮ t❛ ❝ã d(xn , T xn ) ≤ d(xn , z) + d(T xn , z) ≤ (1 + α(xn , T xn ))−1 d(xn , T xn )+ ✭✷✳✶✻✮ (1 + α(T xn , T xn ))−1 d(T xn , T xn ) < [(1 + α(xn , T xn ))−1 + ✭✷✳✶✼✮ (1 + α(T xn , T xn ))−1 α(xn , T xN )]d(xn , T xn ) ≤ d(xn , T xn ) ➜✐Ị✉ ♥➭② ❧➭ ✈➠ ❧ý ✈➭ (2.15) ❧➭ ➤ó♥❣✳ ❇➞② ❣✐ê tõ ✭✷✳✶✸✮ ✈➭ (2.15) t❛ s✉② r❛ ∀ n ≥ N, d(xn+1 , T z) < φ(d(xn , z)) ❤♦➷❝ d(xn+2 , T z) < φ(d(xn+1 , z)) ✭✷✳✶✽✮ ❱× xn → z ✈➭ φ(s) ≤ s✱ tõ ✭✷✳✶✽✮ s✉② r❛ tå♥ t➵✐ ♠ét ❞➲② ❝♦♥ ❝ñ❛ {xn } ❤é✐ tơ ➤Õ♥ T z ✳ ➜✐Ị✉ ♥➭② ❝❤ø♥❣ tá T z = z ✳ ✸✸ ❑Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉✱ ✈Ị ➤Ị t➭✐✿ ❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉ ✶✳ ❍Ö t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ♣❤Ð♣ s✉② ré♥❣✱ ❤➭♠ φ✲❝♦ φ✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❙✉③✉❦✐✱ ▲✲❤➭♠✱ ♣❤Ð♣ ❧➷♣ t❤ø n ❝ñ❛ ➳♥❤ ①➵ T ✱✳✳✳ ✷✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ♠Ư♥❤ ➤Ị✱ tÝ♥❤ ❝❤✃t ✈➭ ➤Þ♥❤ ❧ý ❝❤➻♥❣ ❤➵♥ ♥❤➢ ➜Þ♥❤ ❧ý ✶✳✷✳✺✱ ➜Þ♥❤ ❧ý ✶✳✷✳✽✱ ➜Þ♥❤ ❧ý ✷✳✶✳✷✱ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ ➜Þ♥❤ ❧ý ✷✳✷✳✹✱ ➜Þ♥❤ ❧ý ✷✳✷✳✾✱✳✳✳ ✸✳ ●✐í✐ t❤✐Ư✉ ❝❤✐ t✐Õt ❱Ý ❞ơ ✶✳✷✳✸✱ ❱Ý ❞ơ ✶✳✷✳✶✵✱ ❱Ý ❞ơ ✷✳✶✳✻✱ ❱Ý ❞ơ ✷✳✶✳✸✳✳✳ ✸✹ t➭✐ ❧✐Ư✉ t ỗ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ▼✳ ❆❜t❛❤✐ ✭✷✵✶✺✮✱ ✧❙✉③✉❦✐ t②♣❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❣❡♥❡r❛❧✐③❡❞ ❝♦♥✲ tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s t❤❛t ❝❤❛r❛❝t❡r✐③❡ ♠❡tr✐❝ ❝♦♠♣❧❡t❡♥❡ss✧✱ ❇✉❧❧✳ ■r❛♥✐❛♥ ▼❛t❤✳ ❙♦❝✳✱ ✹✶✭✹✮✱ ✾✸✶✲✾✹✸✳ ❬✸❪ ■✳ ❆❧t✉♥✱ ❆✳ ❊r❞✉r❛♥ ✭✷✵✶✶✮✱ ✧❆ ❙✉③✉❦✐ t②♣❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠✧✱ ■♥t❡r✳ ❏✳ ▼❛t❤✳ ▼❛t❤✳ ❙❝✐✳✱ ✷✵✶✶✱ ❆rt✐❝❧❡ ■❉ ✼✸✻✵✻✸✱ ✾ ♣❛❣❡s✳ ❬✹❪ ❉✳ ❲✳ ❇♦②❞ ❛♥❞ ❙✳ ❲✳ ❲♦♥❣ ✭✶✾✻✾✮✱ ✧❖♥ ♥♦♥❧✐♥❡❛r ❝♦♥tr❛❝t✐♦♥s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✷✵✱ ✹✺✽✲✹✻✹✳ ❬✺❪ ❑✳ ❈✳ ❉❡s❤♠✉❦❤✱ ❘✳ ❚✐✇❛r✐✱ ❙✳ ●✉♣t❛ ✭✷✵✶✺✮✱ ✧●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❛ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦❢ ❙✉③✉❦✐ t②♣❡ ✐♥ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡✧✱ ❏✳ Pr♦❣r❡ss✐✈❡ ❘❡s❡❛r❝❤ ✐♥ ▼❛t❤✳✱ ✺ ✭✶✮✱ ✹✽✷ ✲✲ ✹✽✻✳ ❬✻❪ ❉✳ ❉♦r✐❝✱ ❩✳ ❑❛❞❡❧❜❡r❣✱ ❙✱ ❘❛❞❡♥♦✈✐❝ ✭✷✵✶✷✮✱ ✧❊❞❡❧st❡✐♥s✲❙✉③✉❦✐ t②♣❡ ❢✐①❡❞ ♣♦✐♥t r❡s✉❧ts ✐♥ ♠❡tr✐❝ ❛♥❞ ❛❜str❛❝t ♠❡tr✐❝ s♣❛❝❡s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ❚▼❆✱ ✼✺✱ ✶✾✷✼ ✲✲ ✶✾✸✷✳ ❬✼❪ ❘✳ ❊♥❣❡❧❦✐♥❣ ✭✶✾✼✼✮✱ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣②✱ P❲◆✲P♦❧✐s❤✱ ❙❝✐❡♥t✐❢✐❝ P✉❜❧✐s❤✲ ❡rs✱ ❲❛rs③❛✇❛✳ ❬✽❪ ▼✳ ❆✳ ●❡r❛❣❤t② ✭✶✾✼✸✮✱ ✧❖♥ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✹✵✱ ✻✵✹✲✻✵✽✳ ❬✾❪ ❍✳ ▲♦♥❣✲●✉❛♥❣✱ ❩✳ ❳✐❛♥ ✭✷✵✵✼✮✱ ✧❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡✲ ♦r❡♠s ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✸✷✱ ✶✹✻✽ ✲ ✶✹✼✻✳ ❬✶✵❪ ●✳ ❏✉♥❣❝❦ ✭✶✾✼✻✮✱ ✧❈♦♠♠✉t✐♥❣ ♠❛♣♣✐♥❣s ❛♥❞ ❢✐①❡❞ ♣♦✐♥ts✧✱ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧②✱✱ ✽✸✱ ✷✻✶ ✲✲ ✷✻✸✳ ✸✺ ❬✶✶❪ ▼✳ ❑✐❦❦❛✇❛✱ ❚✳ ❙✉③✉❦✐ ✭✷✵✵✽✮✱ ✧❙♦♠❡ s✐♠✐❧❛r ❜❡t✇❡❡♥ ❝♦♥tr❛❝t✐♦♥s ❛♥❞ ❑❛♥♥❛♥ ♠❛♣♣✐♥❣s✧✱ ❙t✉❞✐❛ ▼❛t❤✳✱✼✻✱ ✻✸ ✲✲ ✻✼✳ ❬✶✷❪ ❚✳ ❈✳ ▲✐♠ ✭✷✵✵✶✮✱ ✧❖♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ▼❡✐r✲❑❡❡❧❡r ❝♦♥tr❛❝t✐✈❡ ♠❛♣s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✹✻ ✭✶✮✱ ✶✶✸ ✲✲ ✶✷✵✳ ❬✶✸❪ ❆✳ ▼❡✐r✱ ❊✳ ❆✳ ❑❡❡❧❡r ✭✶✾✻✾✮✱ ✧❆ t❤❡♦r❡♠ ♦♥ ❝♦♥tr❝❛t✐♦♥ ♠❛♣♣✐♥❣s✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✷✽✱ ✸✷✻✲✸✷✾✳ ❬✶✹❪ ❙✳ ▼✉r❛❧✐s❛♥❦❛r✱ ❑✳ ❏❡②❛❜❛❧ ✭✷✵✶✸✮✱ ✧●❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❙✉③✉❦✐ t②♣❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✧✱ ❆❞✈✳ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r②✱ ✸✭✸✮✱ ✺✵✷✲✺✵✾✳ ❬✶✺❪ ❖✳ P♦♣❡s❝✉ ✭✷✵✵✾✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ✐♥ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❇✉❧❧✳ ❚r❛♥✲ s✐❧✈❛♥✐❛ ❯♥✐✈✳ ❇r❛s♦✈✱ ✶ ✭✺✵✮✱ ✹✼✾✲✹✽✷✳ ❬✶✻❪ ❖✳ P♦♣❡s❝✉ ✭✷✵✶✶✮✱ ✧❚✇♦ ❣❡♥❡r❛t✐♦♥s ♦❢ s♦♠❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✧✱ ❈♦♠♣✉t✳ ▼❛t❤✳ ❆♣♣❧✳✱ ✻✷✱ ✸✾✶✷✲✸✾✶✾✳ ❬✶✼❪ ❚✳ ❙✉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 ❆♥❛❧✳✱ ✼✶ ✭✶✶✮✱ ✺✸✶✸✲✺✸✶✼✳ ✸✻ ... ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ọ ụ trì ột số ị ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❦✐Ó✉ ❙✉③✉❦✐ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ñ❛ ▼✳ ❑✐❦❦❛✇❛✱ ❚✳ ❙✉③✉❦✐✱ ❖✳ P♦♣❡s❝✉✳ ◆❣♦➭✐ r ò trì ột số ệ q ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❈❤➢➡♥❣ ✷... x, T y) < r.d(x, y) ” ✐✐ ❙❛✉ ➤ã✱ ❨✳ ❊♥❥♦✉❥✐✱ ▼✳ ◆❛❦❛♥✐s❤✐✱ ▼✳ ❑✐❦❦❛✇❛✱ ❖✳ P♦♣❡s❝✉✱ ✳✳✳ ➤➲ t ợ ột số ế tể ị ❦Õt q✉➯ tr➟♥ ❝ñ❛ ❚✳ ❙✉③✉❦✐✳ ◆➝♠ ✷✵✵✾✱ ❚✳ ❙✉③✉❦✐ ➤➲ ➤➢❛ r❛ ♠ét ❞➵♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ♠í✐... ✶✳✷ ❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦✐Ĩ✉ ❙✉③✉❦✐ s✉② ré♥❣ ❝đ❛ ❑✐❦❦❛✇❛✱ ❙✉③✉❦✐✱ P♦♣❡s❝✉ P❤➬♥ ú t trì ột số ị ý ể ❜✃t ➤é♥❣ ❦✐Ó✉ ❙✉③✉❦✐ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝đ❛ ▼✳ ❑✐❦❦❛✇❛✱ ❚✳ ❙✉③✉❦✐✱ ❖✳ P♦♣❡s❝✉