Điểm bất động bộ đôi của các ánh xạ co suy rộng trong không gian meetric thứ tự

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❍♦➭♥❣ ❉ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥ ✲ ✷✵✶✺ ❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❍♦➭♥❣ ❉ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ▲✉❐♥ ❱➝♥ ❚❤➵❝ ❙ü ❚♦➳♥ ❍ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ö ❆♥ ✲ ✷✵✶✺ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ✷ ❈❤➢➡♥❣ ✶✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✺ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤➢➡♥❣ ✷✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ❄❄ ✶✶ ✷✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❄❄ ✷✳✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ❑Õt ❧✉❐♥ ✸✸ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✹ ✶ ❧ê✐ ♥ã✐ ➤➬✉ ▲ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝❤đ ➤Ị ♥❣❤✐➟♥ ❝ø✉ q✉❛♥ trä♥❣ ❝đ❛ ❣✐➯✐ tÝ❝❤✳ ◆ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ t♦➳♥ ❤ä❝ ✈➭ ❝➳❝ ♥❣➭♥❤ ❦ü t❤✉❐t✳ ❈➳❝ ❦Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❇r♦✇❡r ✈➭♦ ♥➝♠ ✶✾✶✷ ✈➭ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❇❛♥❛❝❤ ✈➭♦ ♥➝♠ ✶✾✷✷✳ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ➤➲ trë t❤➭♥❤ ♠ét ❝➠♥❣ ❝ơ ♣❤ỉ ❞ơ♥❣ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ❜➭✐ t♦➳♥ ✈Ị sù tå♥ t➵✐ tr♦♥❣ ♥❤✐Ị✉ ❝❤✉②➟♥ ♥❣➭♥❤ ❝đ❛ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝ ✈➭ ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ❝➳❝ ♥❣➭♥❤ ❦❤♦❛ ❤ä❝ ❦❤➳❝✳ ❱× t❤Õ ➤➲ ❝ã ♠ét sè ❧í♥ ❝➳❝ ♠ë ré♥❣ ❝đ❛ ♥❣✉②➟♥ ❧ý ❝➡ ❜➯♥ ♥➭② ❝❤♦ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✱ ❜➺♥❣ ❝➳❝❤ ➤✐Ò✉ ❝❤Ø♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝➡ ❜➯♥ ❤♦➷❝ t❤❛② ➤ỉ✐ ❦❤➠♥❣ ❣✐❛♥✳ ❑❤➳✐ ♥✐Ư♠ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ➤➢ỵ❝ ❙✳ ❙✳ ❈❤❛♥❣ ✈➭ ❨✳ ❍✳ ▼❛ ❣✐í✐ t❤✐Ư✉ ♥➝♠ ✶✾✾✶ ✈➭ s❛✉ ➤ã ➤➲ t❤✉ ❤ót sù q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝✳ ◆➝♠ ✷✵✵✻✱ ❚✳ ●✳ ❇❤❛s❦❛r ✈➭ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ➤➲ t❤✐Õt ❧❐♣ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ tù ❜é ♣❤❐♥ ✧≤✧ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ F : X × X → X ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ✈➭ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦✿ ❚å♥ t➵✐ sè x ≥ u, y ≤ v (X, d) ❝ã tr❛♥❣ ❜Þ t❤ø k ∈ (0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y, u, v ∈ X ♠➭ t❛ ❝ã d(T (x, y), T (u, v)) ≤ k d(x, u) + d(y, v) ◆➝♠ ✷✵✵✾✱ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ✈➭ ▲✳ ❈✐r✐❝ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ư✉ tré♥✱ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ♠➭ ❝❤ó♥❣ ❧➭ ♠ë ré♥❣ ❝đ❛ ❝➳❝ ❦Õt q✉➯ ➤➲ t❤✉ ➤➢ỵ❝ ❝đ❛ ❚✳ ●✳ ❇❤❛s❦❛r ✈➭ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠✱✳✳✳ ➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✱ ❝❤ó♥❣ t➠✐ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥➭② ♥❤➺♠ t×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ✷ ✧ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✧ ▼ô❝ ➤Ý❝❤ ❝ñ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❣✐❛♦ ❤♦➳♥✱ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥✱ tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ư✉ tré♥✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû✱ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵ ✈Ị ❝➳❝ ➳♥❤ ①➵ ➤ã✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû ✈➭ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✱✳✳✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ❈➳❝ ♥é✐ ❞✉♥❣ ❣å♠✿ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❣✐❛♦ ❤♦➳♥✱ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥✱ tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ư✉ tré♥✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ết q ó ụ trì ột số ị ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ♠➭ ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ị ý ó r ò trì ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ô❝ ú t trì ột số ị ý ề ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✱ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ➤➢ỵ❝ tr×♥❤ ❜➭②✳ ▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✈➭ ✸ ❝➳❝ ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣✱ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ❝➳❝ ❦Õt q✉➯ ➤ã ✈➭ tr×♥❤ ❜➭② ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤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 t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý t❤➬②✱ ❝➠ tr♦♥❣ tæ ●✐➯✐ ❚Ý❝❤ ❦❤♦❛ ❚♦➳♥ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ Pò ọ ợ t qố tế rờ ọ ò ú ỡ tr q trì ọ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ◆❤➞♥ ➤➞② t➳❝ ❣✐➯ ①✐♥ ❝➳♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤♦➳ ✷✶ ●✐➯✐ ❚Ý❝❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➭✐ ●ß♥✳ ❈✉è✐ ❝ï♥❣ ❝➳♠ ➡♥ ❣✐❛ ➤×♥❤ ✈➭ ❇❛✱ ▼Đ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❣✐ó♣ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ▼➷❝ ❞ï ➤➲ tÝ❝❤ ❝ù❝ ➤➬✉ t➢ ✈➭ ❝ã ♥❤✐Ò✉ ❝è ❣➽♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ỏ ữ s sót ợ ữ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ ❝ñ❛ q✉ý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ❱✐♥❤✱ ♥❣➭② ✷✼ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✺ ◆❣✉②Ơ♥ ❍♦➭♥❣ ❉ị ✹ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ♣❤✐ t✉②Õ♥ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ ♠ét ♠➟tr✐❝ tr➟♥ ✭❬✶❪✮ ❈❤♦ t❐♣ ❤ỵ♣ X = φ✱ ➳♥❤ ①➵ d : X ì X R ợ ọ X ế t❤á❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✶✮ d(x, y) ≥ ✈í✐ ♠ä✐ x, y ∈ X ✈➭ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ 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 ✳ ❚❐♣ X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦Ý ❤✐Ö✉ ❧➭ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤✐Ó♠ x ➤Õ♥ ➤✐Ó♠ y ✳ ✶✳✶✳✷ ❱Ý ❞ô✳ ✶✮ ❳Ðt X = R✱ d : R × R → R ❝❤♦ ❜ë✐ d (x, y) = |x − y|✱ ✈í✐ ♠ä✐ x, y ∈ R✳ ❑❤✐ ➤ã d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ R✳ ✷✮ ❳Ðt X = Rn ✳ ❱í✐ ❜✃t ❦ú x = (x1 , , xn ), y = (y1 , , yn ) ∈ Rn t❛ ➤➷t n |xi − yi | d1 (x, y) = n ✈➭ i=1 i=1 tr➟♥ ✶✳✶✳✸ X |xi − yi |✳ ❑❤✐ ➤ã d1 , d2 ❧➭ ❝➳❝ ♠➟tr✐❝ d2 (x, y) = Rn ✳ ▼Ö♥❤ ➤Ò✳ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d)✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ x, y, u, v ∈ ✱ t❛ ❝ã |d (x, y) − d (u, v)| ≤ d (x, u) + d (y, v) ✺ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✹ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ d(x, A) = inf d (x, y) ✈➭ ❣ä✐ d(x, A) ❧➭ ❦❤♦➯♥❣ y∈A ▼Ö♥❤ ➤Ò✳ ✶✳✶✳✺ x, y ∈ X (X, d)✱ A ⊂ X ✱ x ∈ X ✱ ❦Ý ❤✐Ö✉ ❝➳❝❤ tõ ➤✐Ĩ♠ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ x ➤Õ♥ t❐♣ ❤ỵ♣ (X, d)✱ A ⊂ X ✳ A✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ t❛ ❝ã |d (x, A) − d (y, A)| ≤ d (x, y) ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✻ ❧➭ ❤é✐ tơ ✈Ị ➤✐Ĩ♠ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d)✱ ❞➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐ x ∈ X ♥Õ✉ ✈í✐ ♠ä✐ ε > tå♥ t➵✐ n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n ≥ n0 t❛ ❝ã d (xn , x) < ε✳ ▲ó❝ ➤ã t❛ ❦Ý ❤✐Ö✉ lim xn = x ❤❛② xn → x ❦❤✐ n → ∞✳ n→∞ ▼Ư♥❤ ➤Ị✳ ✶✳✶✳✼ ✶✮ ❚❐♣ ✷✮ E x∈E ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤ã♥❣ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✈í✐ ♠ä✐ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ tå♥ t➵✐ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✽ {xn } ⊂ E {xn } ⊂ E ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ ❝ã (X, d)✱ E ⊂ X ✱ x ∈ X ✳ ♠➭ s❛♦ ❝❤♦ ➜Þ♥❤ ♥❣❤Ü❛✳ ❚❐♣ ❝♦♥ ❣✐❛♥ ❝♦♥ ✶✳✶✳✶✵ xn → x✳ ε > 0✱ tå♥ t➵✐ n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n, m ≥ n0 t❛ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ xn → x t❛ ❝ã x ∈ E ✳ (X, d)✳ ❉➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐ d(xn , xm ) < ε✱ ❤❛② {xn } ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✶✳✶✳✾ ❑❤✐ ➤ã lim n,m→+∞ d(xn , xm ) = 0✳ (X, d)✳ ❚❛ ♥ã✐ (X, d) ❧➭ ➤➬② ➤đ X ➤Ị✉ ❤é✐ tơ✳ M ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤đ ♥Õ✉ ❦❤➠♥❣ M ✈í✐ ♠➟tr✐❝ ❝➯♠ s✐♥❤ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ✳ ❱Ý ❞ơ✳ ✶✮ ❚❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝ R ✈í✐ ♠➟tr✐❝ d (x, y) = |x − y| ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ✷✮ ❚❐♣ ❤ỵ♣ Rn ❣å♠ t✃t ❝➯ ❝➳❝ ❜é n sè t❤ù❝✱ ✈í✐ ♠➟tr✐❝ d1 (x, y)✱ d2 (x, y) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ✻ ✶✳✶✳✶✶ ▼Ư♥❤ ➤Ị✳ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✶✮ ◆Õ✉ M ➤➬② ➤ñ tì ế M t ó ị ♥❣❤Ü❛✳ M (X, d)✱ M ⊂ X ✳ ❑❤✐ ➤ã ❧➭ t❐♣ ➤ã♥❣✳ X ➤➬② ➤đ t❤× M ➤➬② ➤đ✳ ✭❬✶❪✮ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ f : (X, d) → (Y, ρ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ (X, d) ✈➭ (Y, ρ)✳ ➳♥❤ ①➵ α ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X t❛ ❝ã ρ[f (x) , f (y)] ≤ αd (x, y) ❙è t❤ù❝ ✶✳✶✳✶✸ α ∈ [0, 1) ợ ọ ệ ị ý f :XX x X t➵✐ ❞✉② ♥❤✃t ➤✐Ó♠ ①➵ f tr➟♥ X ✳ ✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✮ ●✐➯ sö ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ➜✐Ĩ♠ sè ❝♦ ❝đ❛ ❧➭ ➳♥❤ ①➵ ❝♦ tõ s❛♦ ❝❤♦ X (X, d) ❧➭ ❦❤➠♥❣ ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑❤✐ ➤ã tå♥ f (x∗ ) = x∗ ✳ x∗ ∈ X ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ f✳ ✶✳✶✳✶✹ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✷❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ✈➭ ❝➳❝ ➳♥❤ ①➵ F : X × X → X ✈➭ g : X → X ✳ ❍❛✐ ➳♥❤ ①➵ F ✈➭ g ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐❛♦ ❤♦➳♥ ✈í✐ ♥❤❛✉ ♥Õ✉ F (gx, gy) = g(F (x, y)) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✶✳✶✳✶✺ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✷❪✮ ❈❤♦ t❐♣ X = φ✳ (X, d, ≤) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✭✶✮ (X, ≤) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥✳ ✭✷✮ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ✶✳✶✳✶✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭✶✮ ❈➳❝ ♣❤➬♥ tư ✭❬✷❪✮ ❈❤♦ t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ (X, ≤)✳ x, y ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ ♥❤❛✉ ➤è✐ ✈í✐ t❤ø tù ≤✱ ♥Õ✉ ❤♦➷❝ x ≤ y ✱ ❤♦➷❝ y ≤ x✳ ✼ ✭✷✮ ➳♥❤ ①➵ f : X → X ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐➯♠ ♥Õ✉ ✈í✐ x, y ∈ X ♠➭ x ≤ y t❛ ❝ã ✭✸✮ f (x) ≤ f (y)✳ ➳♥❤ ①➵ f : X → X ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ t➝♥❣ ♥Õ✉ ✈í✐ x, y ∈ X ♠➭ x ≤ y t❛ ❝ã f (y) ≤ f (x)✳ ✭❬✷❪✮ ❈❤♦ t ợ ợ s tứ tự từ ị ♥❣❤Ü❛✳ ➳♥❤ ①➵ F : X × X → X ✈➭ g : X → X ✳ ➳♥❤ ①➵ F ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã ➤✐Ư✉ tré♥ ♥❣➷t ♥Õ✉ ♥❣❤Ü❛ ❧➭ ✈í✐ ❜✃t ❦ú ♥Õ✉ (X, ≤) ✈➭ ✷ tÝ♥❤ ❝❤✃t g ✲➤➡♥ F (x, y) t➝♥❣ ♥❣➷t t❤❡♦ ❜✐Õ♥ x✱ ✈➭ ❣✐➯♠ ♥❣➷t t❤❡♦ ❜✐Õ♥ y ✱ x, y ∈ X x1 , x2 ∈ X, ♠➭ gx1 < gx2 , t❤× t❛ ❝ã F (x1 , y) < F (x2 , y), ✈➭ ♥Õ✉ y1 , y2 ∈ X, ♠➭ gy1 < gy2 , t❤× t❛ ❝ã F (x, y1 ) > F (x, y2 ) ❚r♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ tr➟♥✱ ♥Õ✉ g ❧➭ ➳♥❤ ①➵ ➤å♥❣ ♥❤✃t✱ t❤× ➳♥❤ ①➵ F ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➡♥ ➤✐Ư✉ tré♥ ♥❣➷t✳ ✶✳✶✳✶✽ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ✶✳✶✳✶✾ ✭❬✷❪✮ P❤➬♥ tư ➜Þ♥❤ ♥❣❤Ü❛✳ (x, y) X ì X ợ ọ ể trï♥❣ ❜é ➤➠✐ F : X × X → X ✈➭ g : X → X ♥Õ✉ F (x, y) = gx ✈➭ F (y, x) = gy ✳ ✭❬✷❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➜Þ♥❤ ♥❣❤Ü❛✳ ➤➢➡❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ➳♥❤ ①➵ ✈➭ (X, d)✳ P❤➬♥ tư (x, y) ∈ X × X T : X × X → X ♥Õ✉ T (x, y) = x T (y, x) = y ✳ ✶✳✶✳✷✵ ❱Ý ❞ô✳ ❝➠♥❣ t❤ø❝ X = [0; +∞) ✈➭ ➳♥❤ ①➵ T : X ì X X ợ ị T (x; y) = x + y ✈í✐ ♠ä✐ x, y ∈ X ✳ ❉Ô t❤✃② r➺♥❣ T ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❧➭ ✶✳✶✳✷✶ ❈❤♦ ❱Ý ❞ơ✳ (0, 0)✳ ❈❤♦ X = P([0; 1)) ❧➭ ❤ä t✃t ❝➯ ❝➳❝ t❐♣ ❝♦♥ ❝ñ❛ t❐♣ [0, 1) ✈➭ T : X ì X X ợ ị T (A; B) = A − B ✈í✐ ♠ä✐ A, B ∈ X ✳ ❑❤✐ ➤ã✱ t❛ ❝ã t❤Ó t❤✃② r➺♥❣ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❤❛✐ t❐♣ ❤ỵ♣ rê✐ ♥❤❛✉✳ ✽ T ❧➭ ❝➷♣ (A, B)✱ tr♦♥❣ ➤ã A ✈➭ B ❧➭ ✭❬✷❪✮ ❈❤♦ t❐♣ ❤ỵ♣ ➤➢ỵ❝ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✶✳✶✳✷✷ ➜Þ♥❤ ♥❣❤Ü❛✳ ➳♥❤ ①➵ T : X × X → X✳ (X, ≤) ✈➭ ➳♥❤ ①➵ T ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ♥Õ✉ T (x, y) ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ ❣✐➯♠ t❤❡♦ ❜✐Õ♥ x✱ ✈➭ ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ t➝♥❣ t❤❡♦ ❜✐Õ♥ y ✱ ♥❣❤Ü❛ ❧➭ ✈í✐ ❜✃t ❦ú ♥Õ✉ x, y ∈ X x1 , x2 ∈ X, ♠➭ x1 ≤ x2 , t❤× t❛ ❝ã T (x1 , y) ≤ T (x2 , y), ✈➭ ♥Õ✉ y1 , y2 ✶✳✶✳✷✸ ✭❬✷❪✮ ❈❤♦ t❐♣ ❤ỵ♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ (X, ≤) ✈➭ ❣✐➯ sư r➺♥❣ ➜Þ♥❤ ❧ý✳ ❝ã ♠ét ♠➟tr✐❝ ∈ X, ♠➭ y1 ≤ y2 , t❤× t❛ ❝ã T (x, y1 ) ≥ T (x, y2 ) d tr➟♥ X F :X ×X →X s❛♦ ❝❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ tr➟♥ k ∈ [0, 1) r➺♥❣ tå♥ t➵✐ ♠ét sè d(F (x, y), F (u, v)) ≤ k X✳ ❈❤♦ ●✐➯ sö s❛♦ ❝❤♦ d(x, u) + d(y, v) ✈í✐ ♠ä✐ x, y, u, v ∈ X ♠➭ x ≥ u, y ≤ v ✭✶✳✶✮ ◆Õ✉ tå♥ t➵✐ x, y ∈ X ✶✳✶✳✷✹ x0 , y0 ∈ X x = F (x, y) s❛♦ ❝❤♦ ➜Þ♥❤ ❧ý✳ ✐✮ ◆Õ✉ X ✈➭ x0 ≤ F (x0 , y0 ) ✈➭ y0 ≥ F (y0 , x0 )✱ d tr➟♥ X t❤× tå♥ t➵✐ y = F (y, x)✳ ✭❬✷❪✮ ❈❤♦ t❐♣ ❤ỵ♣ ➤➢ỵ❝ t❤ø tù tõ♥❣ ♣❤➬♥ r➺♥❣ ❝ã ♠ét ♠➟tr✐❝ ●✐➯ sö r➺♥❣ s❛♦ ❝❤♦ s❛♦ ❝❤♦ (X, ≤) ✈➭ ❣✐➯ sö (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉ ➤➞② {xn } ❧➭ ♠ét ❞➲② sè ❦❤➠♥❣ ❣✐➯♠ ✈í✐ xn → x✱ t❤× xn ≤ x ✈í✐ ♠ä✐ ❧➭ ♠ét ❞➲② sè ❦❤➠♥❣ t➝♥❣ ✈í✐ yn → y ✱ t❤× yn ≥ y ✈í✐ ♠ä✐ n ≥ 1✳ ✐✐✮ ◆Õ✉ {yn } n ≥ 1✳ ❈❤♦ F : X ×X → X tå♥ t➵✐ ♠ét sè ❧➭ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr➟♥ k ∈ [0, 1) d(F (x, y), F (u, v)) ≤ k X ✳ ●✐➯ sö r➺♥❣ s❛♦ ❝❤♦ d(x, u) + d(y, v) ✈í✐ ♠ä✐ x, y, u, v ∈ X ♠➭ x ≥ u, y ≤ v ✭✶✳✷✮ ✾ ◆Õ✉ tå♥ t➵✐ x, y ∈ X ✶✳✶✳✷✺ x0 , y ∈ X s❛♦ ❝❤♦ s❛♦ ❝❤♦ x = F (x, y) ✈➭ y0 ≥ F (y0 , x0 )✱ t❤× tå♥ t➵✐ y = F (y, x)✳ ✈➭ ✭❬✽❪✮ ❈❤♦ ➜Þ♥❤ ♥❣❤Ü❛✳ x0 ≤ F (x0 , y0 ) (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ F : X ×X → X sử M t ợ rỗ ❝đ❛ X = X ×X ×X ×X ✳ ❚❛ ♥ã✐ r➺♥❣ M ❧➭ t❐♣ F ✲❜✃t ❝♦♥ ❜✐Õ♥ q✉❛ ➳♥❤ ①➵ F ❝đ❛ X ♥Õ✉ ✈í✐ ♠ä✐ x, y, z, w ∈ X t❛ ❝ã ✐✮ (x, y, z, w) ∈ M ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ (w, z, y, x) ∈ M ✳ ✐✐✮ ◆Õ✉ ✶✳✶✳✷✻ (x, y, z, w) ∈ M ✱ t❤× (F (x, y), F (y, x), F (z, w), F (w, z)) ∈ M ị ĩ t ợ ủ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ M ❧➭ ♠ét X ✳ ❚❛ ♥ã✐ M t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉ ♥Õ✉ ✈í✐ ♠ä✐ x, y, z, w, a, b ∈ X ✱ ♠➭ (x, y, z, w) ∈ M ✈➭ (z, w, a, b) ∈ M ✱ t❤× t❛ ❝ã (x, y, a, b) ∈ M ✳ ✶✳✶✳✷✼ ◆❤❐♥ ①Ðt✳ ❉Ơ ❞➭♥❣ t❤✃② r➺♥❣ t❐♣ ❤ỵ♣ M = X ❧➭ t❐♣ F ✲❜✃t ❜✐Õ♥ t➬♠ t❤➢ê♥❣✱ ✈➭ ♥ã ❝ò♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳ ✶✳✶✳✷✽ ❱Ý ❞ơ✳ ❈❤♦ X = {0, 1, 2, 3} ✈í✐ ♠➟r✐❝ t❤➠♥❣ t❤➢ê♥❣ ✈➭ ➳♥❤ ①➵ F : X × X X ợ ị tứ F (x, y) = ❉Ô t❤✃② r➺♥❣ ♥Õ✉ x, y = {1, 2}, tr trờ ợ ò M = {1, 2}4 ⊆ X ❧➭ t❐♣ F ✲❜✃t ❜✐Õ♥✱ ✈➭ ♥ã ❝ò♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳ ✶✳✶✳✷✾ ❱Ý ❞ơ✳ ❈❤♦ X = R ✈í✐ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣ ✈➭ ➳♥❤ ①➵ F : X × X → X ợ ị tứ F (x, y) = ❉Ô t❤✃② r➺♥❣ x ♥Õ✉ x, y ∈ (−∞, −1) ∪ (1, +∞), cos(x + y) sin(x − y) tr trờ ợ ò M = [(, 1) ∪ (1, ∞)]4 ⊆ X ❧➭ t❐♣ F ✲❜✃t ❜✐Õ♥✱ ✈➭ ♥ã ❝ò♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳ ✶✵ ✶✳✶✳✸✵ ♣❤➬♥ ❱Ý ❞ô✳ ❈❤♦ (X, d) ❧➭ ♠ét tr ợ tr ị tứ tự từ ❈❤♦ F : X × X → X ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥✱ ♥❣❤Ü❛ ❧➭ ✈í✐ ♠ä✐ ♥Õ✉ x, y ∈ X t❛ ❝ã x1 , x2 ∈ X, ♠➭ x1 ≤ x2 , t❤× t❛ ❝ã F (x1 , y) ≤ F (x2 , y), ✈➭ ♥Õ✉ y1 , y2 ∈ X, ♠➭ y1 ≤ y2 , t❤× t❛ ❝ã F (x, y1 ) ≥ F (x, y2 ) ❇➞② ❣✐ê t❛ ➤Þ♥❤ ♥❣❤Ü❛ t❐♣ ❤ỵ♣ M ⊆ X ❝❤♦ ❜ë✐ M = {(a, b, c, d) ∈ X : a ≥ c, b ≤ d} ❑❤✐ ➤ã M ❧➭ t❐♣ F ✲❜✃t ❜✐Õ♥✱ ✈➭ ♥ã ❝ò♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ♣❤✐ t✉②Õ♥ ✶✳✷ ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù P❤➬♥ ♥➭② ú t trì ột số ị ý ể t ➤é♥❣ ❜é ➤➠✐ ♠➭ ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✳ ✶✳✷✳✶ ➜Þ♥❤ ❧ý✳ ✭❬✶✵❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ M ❧➭ ♠ét t rỗ ủ X = ϕ(0) < ϕ(t) < t F :X ×X →X ●✐➯ sö r➺♥❣ ❝ã ♠ét ❤➭♠ sè ✈➭ lim ϕ(r) < t ✈í✐ ♠ä✐ r→t+ ϕ : [0, +∞) → [0, +∞) t > 0✱ ❝ị♥❣ ❣✐➯ sư r➺♥❣ ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦ d(F (x, y), F (u, v)) ≤ ϕ d(x, u) + d(y, v) , ✈í✐ ♠ä✐ (x, y, u, v) ∈ M ✭✶✳✸✮ ●✐➯ sö r➺♥❣ ❤♦➷❝ ❛✮ F ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝✱ ❤♦➷❝ ❜✮ ◆Õ✉ ✈í✐ ❤❛✐ ❞➲② ❜✃t ❦ú n≥1 s❛♦ ❝❤♦ xn → x ✈➭ {xn } , {yn } yn → y ✱ t❤× ✶✶ ♠➭ (xn+1 , yn+1 , xn , yn ) ∈ M ✱ (x, y, xn , yn ) ∈ M ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ n ≥ 1✳ ◆Õ✉ tå♥ t➵✐ ❧➭ t❐♣ F ✲❜✃t x = F (x, y) (x0 , y0 ) ∈ X × X s❛♦ ❝❤♦ (F (x0 , y0 ), F (y, x0 ), x0 , y0 ) ∈ M ❜✐Õ♥ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✱ t❤× tå♥ t➵✐ ✈➭ y = F (y, x)✱ ❈❤ø♥❣ ♠✐♥❤✳ ❱× ♥❣❤Ü❛ ❧➭ F x, y ∈ X ✈➭ M s❛♦ ❝❤♦ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ F (X × X) ⊆ X ❚❛ ❝ã t❤Ó ①➞② ❞ù♥❣ ❤❛✐ ❞➲② {xn } ✈➭ {yn }tr♦♥❣ X s❛♦ ❝❤♦ yn = F (yn−1 , xn−1 ), ✈í✐ ♠ä✐ n ∈ N xn = F (xn−1 , yn−1 ), ◆Õ✉ tå♥ t➵✐ n∗ ∈ N s❛♦ ❝❤♦ xn∗ −1 = xn∗ ✈➭ yn∗ −1 = yn∗ ✱ ♥❣❤Ü❛ ❧➭ t❛ ❝ã xn∗ −1 = F (xn∗ −1 , yn∗ −1 ), ❱× t❤Õ✱ ✭✶✳✹✮ yn∗ −1 = F (yn∗ −1 , xn∗ −1 ) (xn∗ −1 , yn∗ −1 ) ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ F ✳ ❱× ✈❐② t❛ ❝ã t❤Ĩ ❣✐➯ t❤✐Õt r➺♥❣ ❱× xn−1 = xn ❤♦➷❝ yn−1 = yn ✈í✐ ♠ä✐ n ∈ N✳ (F (x0 , y0 ), F (y0 , x0 ), x0 , y0 ) = (x1 , y1 , x0 , y0 ) ∈ M ✈➭ M ❧➭ ♠ét t❐♣ ❤ỵ♣ F ✲ ❜✃t ❜✐Õ♥✱ t❛ s✉② r❛ (F (x1 , y1 ), F (y1 , x1 ), F (x0 , y0 ), F (y0 , x0 )) = (x2 , y2 , x1 , y1 ) ∈ M ❚✐Õ♣ tơ❝ sư ❞ơ♥❣ ❣✐➯ t❤✐Õt M ❧➭ ♠ét t❐♣ F ✲❜✃t ❜✐Õ♥✱ t❛ ❝ã (F (x2 , y2 ), F (y2 , x2 ), F (x1 , y1 ), F (y1 , x1 )) = (x3 , y3 , x2 , y2 ) ∈ M ❇ë✐ tÝ♥❤ t✉➬♥ ❤♦➭♥ ❝ñ❛ ❧❐♣ ❧✉❐♥ ♥➭②✱ t❛ ➤➢ỵ❝ (F (xn−1 , yn−1 ), F (yn−1 , xn−1 ), xn−1 , yn−1 ) = (xn , yn , xn−1 , yn−1 ) ∈ M, ✈í✐ ♠ä✐ n ∈ N ❇➞② ❣✐ê ✈í✐ ♠ä✐ n ∈ N t❛ ➤➷t δn−1 := d(xn , xn−1 ) + d(yn , yn−1 ) > 0✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ δn ≤ 2ϕ ❚❤❐t ✈❐②✱ ✈× δn−1 ✈í✐ ♠ä✐ n ∈ N (xn , yn , xn−1 , yn−1 ) ∈ M ✈í✐ ♠ä✐ n ∈ N✱ tõ ✭✶✳✸✮ t❛ s✉② r❛ d(xn+1 , xn ) = d(F (xn , yn ), F (xn−1 , yn−1 )) d(xn , xn−1 ) + d(yn , yn−1 ) ≤ ϕ δn−1 = ϕ ✶✷ ✭✶✳✺✮ ❱× M ❧➭ ♠ét t❐♣ F ✲❜✃t ❜✐Õ♥ ✈➭ (xn , yn , xn−1 , yn−1 ) ∈ M ✈í✐ ♠ä✐ n ∈ N✱ t❛ ❝ã (yn−1 , xn−1 , yn , xn ) ∈ M ✈í✐ ♠ä✐ n ∈ N✳ ❚õ ✭✶✳✸✮ ✈➭ (yn−1 , xn−1 , yn , xn ) ∈ M ✈í✐ ♠ä✐ n ∈ N✱ t❛ ➤➢ỵ❝ d(yn+1 , yn ) = d(F (yn , xn ), F (yn−1 , xn−1 )) = d(F (yn−1 , xn−1 ), F (yn , xn )) d(yn−1 , yn ) + d(xn−1 , xn ) ≤ ϕ δn−1 = ϕ ✭✶✳✻✮ ❑Õt ❤ỵ♣ ✭✶✳✺✮ ✈➭ ✭✶✳✻✮✱ t❛ ➤➢ỵ❝ δn ≤ 2ϕ ❚õ ✭✶✳✺✮✱ sư ❞ơ♥❣ tÝ♥❤ ❝❤✃t sè ✈í✐ ♠ä✐ n ∈ N ✭✶✳✼✮ ϕ(t) < t ✈í✐ ♠ä✐ t > t❛ ❝ã δn−1 δn ≤ 2ϕ ❱× t❤Õ✱ δn−1 < δn−1 , ✈í✐ ♠ä✐ n ∈ N {δn } ❧➭ ♠ét ❞➲② ➤➡♥ ➤✐Ö✉ ❣✐➯♠✳ ❉♦ ➤ã✱ tå♥ t➵✐ ❣✐í✐ ❤➵♥ lim δn = δ ✈í✐ n→∞ δ ≥ ♥➭♦ ➤ã✳ ❇➞② ❣✐ê t❛ ❝❤Ø r❛ r➺♥❣ ➤➻♥❣ t❤ø❝ ✭✶✳✼✮ ❝❤♦ δ = 0✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ δ > 0✳ ❑❤✐ ➤ã tõ ❜✃t n → ∞ ✈➭ sư ❞ơ♥❣ ❣✐➯ t❤✐Õt lim+ ϕ(r) < t ✈í✐ ♠ä✐ t > 0✱ t❛ r→t s✉② r❛ δ = lim δn ≤ lim ϕ n→∞ δn−1 =2 ➜✐Ò✉ ♥➭② ❞➱♥ ➤Õ♥ ♠➞✉ t❤✉➱♥✳ ❱× ✈❐② lim δn−1 →δ ϕ + δn−1 ✈➭ ❤❛✐ ❞➲② ❝♦♥ ❝➳❝ sè ♥❣✉②➟♥ nk ✈➭ mk ✈í✐ nk > mk ≥ k s❛♦ ❝❤♦ rk := d(xmk , xnk ) + d(ymk , ynk ) ≥ ε, ✈í✐ ♠ä✐ k = 1, 2, 3, ❍➡♥ ♥÷❛✱ t➢➡♥❣ ø♥❣ ✈í✐ ✭✶✳✾✮ mk ✱ t❛ ❝ã t❤Ĩ ❝❤ä♥ nk ❧➭ sè ♥❣✉②➟♥ ♥❤á ♥❤✃t ✈í✐ nk > mk ≥ k t❤á❛ ♠➲♥ ✭✶✳✾✮✳ ❑❤✐ ➤ã✱ t❛ ❝ã d(xmk , xnk −1 ) + d(ymk , ynk −1 ) < ε ✶✸ ✭✶✳✶✵✮ ❙ư ❞ơ♥❣ ✭✶✳✾✮✱ ✭✶✳✶✵✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ t❛ ❝ã ε ≤ rk = d(xmk , xnk ) + d(ymk , ynk ) ≤ d(xmk , xnk −1 ) + d(xnk −1 , xnk ) + d(ymk , ynk −1 ) + d(ynk −1 , ynk ) ✭✶✳✶✶✮ = [d(xmk , xnk −1 ) + d(ymk , ynk −1 )] + [d(xnk , xnk −1 ) + d(ynk , ynk −1 ) < ε + δnk −1 ❚r♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❝❤♦ ❱× k → ∞ ✈➭ sư ❞ơ♥❣ ✭✶✳✽✮ t❛ ❝ã lim rk = ε > 0✳ k→∞ nk > mk ✈➭ M t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✱ t❛ ♥❤❐♥ ➤➢ỵ❝ (xnk , ynk , xmk , ymk ) ∈ M ✈➭ (ymk , xmk , ynk , xnk ) ∈ M ✭✶✳✶✷✮ ❚õ ✭✶✳✸✮ ✈➭ ✭✶✳✶✷✮✱ t❛ t❤✉ ➤➢ỵ❝ d(xmk +1 , xnk +1 ) = d(F (xmk , ymk ), F (xnk , ynk )) = d(F (xnk , ynk ), F (xmk , ymk ) d(xnk , xmk ) + d(ynk , ymk ) rk =ϕ ≤ϕ ✭✶✳✶✸✮ ✈➭ d(ymk +1 , ynk +1 ) = d(F (ymk , xmk ), F (ynk , xnk )) d(ymk , xnk ) + d(xmk , xnk ) ≤ ϕ rk = ϕ ✭✶✳✶✹✮ ❑Õt ❤ỵ♣ ✭✶✳✶✸✮ ✈➭ ✭✶✳✶✹✮✱ t❛ ➤➢ỵ❝ rk+1 ≤ 2ϕ rk , ✈í✐ ♠ä✐ k = 1, 2, 3, ✭✶✳✶✺✮ ❈❤♦ k → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✶✳✶✺✮ ✈➭ sư ❞ơ♥❣ ❣✐➯ t❤✐Õt lim+ ϕ(r) < t ✈í✐ ♠ä✐ t > t❛ s✉② r❛ r→t ε = lim rk+1 ≤ lim ϕ k→∞ k→∞ rk ➜✐Ò✉ ♥➭② ❞➱♥ ➤Õ♥ ♠➞✉ t❤✉➱♥✳ ❱× t❤Õ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ tå♥ t➵✐ = lim+ ϕ rk →ε rk 0✳ r→t ✶✺ ❇➺♥❣ tÝ♥❤ t♦➳♥ ➤➡♥ ❣✐➯♥✱ t❛ t❤✃② r➺♥❣ ✈í✐ ♠ä✐ x, y, u, v ∈ X t❛ ❝ã x+y+2 u+v+2 − 3 ≤ [d(x, u) + d(y, v)] d(x, u) + d(y, v) = d(x, u) + d(y, v) =ϕ d(F (x, y), F (u, v)) = ❍➡♥ ♥÷❛✱ ♥Õ✉ t❛ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧Ý ✶✳✷✳✶ ✈í✐ ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳ ❉Ơ t❤✃② r➺♥❣ ➤✐Ĩ♠ ♥❤✃t ❝đ❛ ✶✳✷✳✸ M = X ✱ t❤× F ❝ã ➤✐Ĩ♠ ❜✃t (2, 2) ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② F✳ ◆❤❐♥ ①Ðt✳ ▼➷❝ ❞ï tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ❧➭ ❝➠♥❣ ❝ơ ❝èt ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ➤Ó ❝❤Ø r❛ sù tå♥ t➵✐ ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ◆ã✐ ❝❤✉♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❝ã t❤Ó ❦❤➠♥❣ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ♥❤➢ tr♦♥❣ ✈Ý ❞ơ tr➟♥✳ ❱× tế ị í ợ q t ó ♠ét ❝➠♥❣ ❝ơ ❜ỉ trỵ ♠í✐ tr♦♥❣ ✈✐Ư❝ ❝❤Ø r❛ sù tå♥ t➵✐ ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ◆Õ✉ t❛ ❧✃② ➳♥❤ ①➵ ϕ(t) = kt ✈í✐ k ∈ [0; 1) tr ị í tì t t ợ ❦Õt q✉➯ s❛✉ ➤➞②✳ ✶✳✷✳✹ ❍Ö q✉➯✳ ✭❬✶✵❪✮ ❈❤♦ (X, d) ột t ợ rỗ ủ s ❝❤♦ tå♥ t➵✐ k ∈ [0, 1) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ ✈➭ X 4✳ ●✐➯ sư r➺♥❣ ✐✮ (x, y, u, v) ∈ M ✳ F ❧➭ ❧➭ ♠ét ➳♥❤ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ d(F (x, y), F (u, v)) ≤ k ✈í✐ ♠ä✐ F :X ×X →X M d(x, u) + d(y, v) ✭✶✳✷✵✮ ●✐➯ sư r➺♥❣ ❤♦➷❝ ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝✱ ❤♦➷❝ ✐✐✮ ❱í✐ ❤❛✐ ❞➲② ❜✃t ❦ú ♥Õ✉ xn → x ✈➭ {xn } , {yn } ♠➭ (xn+1 , yn+1 , xn , yn ) ∈ M yn → y ❦❤✐ n → ∞✱ ✶✻ t❤× (x, y, xn , yn ) ∈ M ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ n ∈ N✱ n ∈ N✳ ◆Õ✉ tå♥ t➵✐ ❤ỵ♣ F ✲❜✃t x = F (x, y) (x0 , y0 ) ∈ X × X (F (x0 , y0 ), F (y0 , x0 ), x0 , y0 ) ∈ M ❜✐Õ♥ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✱ t❤× tå♥ t➵✐ ✈➭ y = F (y, x)✱ ➜Þ♥❤ ❧ý✳ ✶✳✷✳✺ s❛♦ ❝❤♦ F x, y ∈ X M t❐♣ s❛♦ ❝❤♦ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ✭❬✶✵❪✮ ◆❣♦➭✐ ♥❤÷♥❣ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ❧Ý ✶✳✷✳✶✱ t❛ ❣✐➯ t❤✐Õt t❤➟♠ r➺♥❣ ✈í✐ ♠ä✐ (x, y, u, v) ∈ M ♥❣❤Ü❛ ❧➭ ✈➭ ✈➭ (x, y), (z, t) ∈ X × X ✱ (z, t, u, v) ∈ M ✳ ❑❤✐ ➤ã tå♥ t➵✐ F (u, v) ∈ X × X s❛♦ ❝❤♦ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➤Þ♥❤ ❧ý✱ tõ ➜Þ♥❤ ❧Ý ✶✳✷✳✶✱ t❛ ❜✐Õt r➺♥❣ F ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❇➞② ❣✐ê✱ ❣✐➯ sö r➺♥❣ ♥❣❤Ü❛ ❧➭ (x, y) ✈➭ (z, t) ❧➭ ❤❛✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ F ✱ x = F (x, y), y = F (y, x), z = F (z, t), t = F (t, z)✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ x = z ✈➭ y = t✳ ❚❤❐t ✈❐②✱ ♥❤ê ❣✐➯ t❤✐Õt ✈í✐ (x, y) ✈➭ (z, t) tå♥ t➵✐ (u, v) ∈ X × X s❛♦ ❝❤♦ (x, y, u, v) ∈ M ✈➭ (z, t, u, v) ∈ M ✳ ❚❛ ➤➷t u0 = u ✈➭ v0 = v ✈➭ ①➞② ❞ù♥❣ ❤❛✐ ❞➲② {un } ✈➭ {vn } ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ un = F (un−1 , vn−1 ), ❱× = F (vn−1 , un−1 ), ✈í✐ ♠ä✐ n ∈ N M ❧➭ F ✲❜✃t ❜✐Õ♥ ✈➭ (x, y, u0 , v0 ) = (x, y, u, v) ∈ M ✱ t❛ ❝ã (F (x, y), F (y, x), F (u0 , v0 ), F (v0 , u0 )) ∈ M, ♥❣❤Ü❛ ❧➭ (x, y, u1 , v1 ) ∈ M ❚õ (x, y, u1 , v1 ) ∈ M ✱ ♥Õ✉ t❛ sư ❞ơ♥❣ ♠ét ❧➬♥ ♥÷❛ tÝ♥❤ ❝❤✃t F ✲❜✃t ❜✐Õ♥✱ t❤× t❛ ❝ã (F (x, y), F (y, x), F (u1 , v1 ), F (v1 , u1 )) ∈ M, ✈➭ ✈× t❤Õ (x, y, u2 , v2 ) ∈ M ❇ë✐ tÝ♥❤ t✉➬♥ ❤♦➭♥ ❝đ❛ ❧❐♣ ❧✉❐♥ ♥➭②✱ t❛ ➤➢ỵ❝ (x, y, un , ) ∈ M, ✈í✐ ♠ä✐ n ∈ N ✶✼ ✭✶✳✷✶✮ ❚õ ✭✶✳✸✮ ✈➭ ✭✶✳✷✶✮✱ t❛ ❝ã d(x, un+1 ) = d(F (x, y), F (un , )) ≤ ϕ ❱× d(x, un ) + d(y, ) ✭✶✳✷✷✮ M ❧➭ F ✲❜✃t ❜✐Õ♥ ✈➭ (x, y, un , ) ∈ M ✈í✐ ♠ä✐ n ∈ N✱ t❛ ❝ã (vn , un , y, x) ∈ M ✈í✐ ♠ä✐ n ∈ N ✭✶✳✷✸✮ ❚õ ✭✶✳✸✮ ✈➭ ✭✶✳✷✸✮✱ t❛ t❤✉ ➤➢ỵ❝ d(vn+1 , y) = d(F (vn , un ), F (y, x)) ≤ ϕ d(vn , y) + d(un , x) ✭✶✳✷✹✮ ❉♦ ➤ã✱ tõ ✭✶✳✷✷✮ ✈➭ ✭✶✳✷✸✮✱ t❛ ❝ã d(x, un+1 ) + d(y, vn+1 ) ≤ϕ d(x, un ) + d(y, ) ✈í✐ ♠ä✐ n ∈ N ✭✶✳✷✺✮ ✈í✐ ♠ä✐ n ∈ N ✭✶✳✷✻✮ ❇ë✐ tÝ♥❤ t✉➬♥ ❤♦➭♥ ❝ñ❛ ❧❐♣ ❧✉❐♥ ♥➭②✱ t❛ ➤➢ỵ❝ d(x, un+1 ) + d(y, vn+1 ) ≤ ϕn ❚õ ❣✐➯ t❤✐Õt d(x, u1 ) + d(y, v1 ) ϕ(t) < t ✈➭ lim+ ϕ(r) < t✱ t❛ s✉② r❛ lim ϕn (t) = ✈í✐ ♠ä✐ t > 0✳ ❱× n→∞ r→t t❤Õ✱ tõ ✭✶✳✷✻✮✱ t❛ ❝ã lim [d(x, un+1 ) + d(y, vn+1 )] = ✭✶✳✷✼✮ n→∞ ❚➢➡♥❣ tù✱ t❛ ❝ã t❤Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ lim [d(z, un+1 ) + d(t, vn+1 )] = ✭✶✳✷✽✮ n→∞ ◆❤ê ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ ✈í✐ ♠ä✐ n ∈ N✱ t❛ ❝ã d(x, z) + d(y, t) ≤ [d(x, un+1 ) + d(un+1 , z)] + [d(y, vn+1 ) + d(vn+1 , t)] ≤ [d(x, un+1 ) + d(y, vn+1 )] + [d(z, un+1 ) + d(t, vn+1 )] ❈❤♦ n → ∞ tr♦♥❣ ✭✶✳✷✼✮ rå✐ sư ❞ơ♥❣ ✭✶✳✷✺✮ ✈➭ ✭✶✳✷✻✮✱ t❛ ❝ã ➜✐Ị✉ ♥➭② ①➮② r❛ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❱× t❤Õ✱ ✭✶✳✷✾✮ d(x, z) + d(y, t) = 0✳ d(x, z) = ✈➭ d(y, t) = 0✱ ♥❣❤Ü❛ ❧➭ x = z ✈➭ y = t✳ F ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳ ✶✽ ✶✳✷✳✻ ❍Ö q✉➯✳ ❝ã ♠ét ♠➟tr✐❝ ✭❬✶✵❪✮ ❈❤♦ d tr➟♥ r➺♥❣ ❝ã ♠ét ❤➭♠ sè X (X, ≤) s❛♦ ❝❤♦ ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ ❣✐➯ sö (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư ϕ : [0; ∞) → [0; ∞) ✈í✐ = ϕ(0) < ϕ(t) < t t > ✈➭ ❝ị♥❣ ❣✐➯ sư r➺♥❣ F : X × X → X ✈í✐ ♠ä✐ lim ϕ(r) < t ✈➭ r→t+ ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦ F ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ✈➭ d(x, u) + d(y, v) d(F (x, y), F (u, v)) ≤ ϕ x, y, u, v ∈ X ✈í✐ ♠ä✐ ♠➭ x≥u ❛✮ F ❧✐➟♥ tơ❝✱ ❤♦➷❝ ❜✮ X ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉ ✈➭ y ≤ v✳ ●✐➯ sö r➺♥❣ ❤♦➷❝ ✐✮ ◆Õ✉ xn ❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠ ✈í✐ ✐✐✮ ◆Õ✉ yn ❧➭ ❞➲② ❦❤➠♥❣ t➝♥❣ ✈í✐ ◆Õ✉ tå♥ t➵✐ x0 , y ∈ X x, y ∈ X xn → x✱ yn → y ✱ t❤× t❤× xn ≤ x yn ≥ y ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ n ∈ N✳ n ∈ N✳ s❛♦ ❝❤♦ x0 ≤ F (x0 , y0 ), t❤× tå♥ t➵✐ ✭✶✳✸✵✮ s❛♦ ❝❤♦ y0 ≥ F (y0 , x0 ), x = F (x, y) ✈➭ y = F (y, x)✱ ♥❣❤Ü❛ ❧➭ F ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❈❤ø♥❣ ♠✐♥❤✳ ➜➬✉ t✐➟♥✱ t❛ ①➳❝ ➤Þ♥❤ ♠ét t❐♣ ❝♦♥ M ⊆ X ❜ë✐ M = (a, b, c, d) ∈ X : a ≥ c, b ≤ d ❉Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣ M ❧➭ t❐♣ ❤ỵ♣ F ✲❜✃t ❜✐Õ♥ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳ ◆❤ê ✭✶✳✸✵✮ t❛ ❝ã d(F (x, y), F (u, v)) ≤ ϕ ✈í✐ ♠ä✐ d(x, u) + d(y, v) x, y, u, v ∈ X ♠➭ (x, y, u, v) ∈ M ✳ ❱× x0 , y0 ∈ X s❛♦ ❝❤♦ x0 ≤ F (x0 , y0 ), y0 ≥ F (y0 , x0 ), t❛ t❤✉ ➤➢ỵ❝ (F (x0 , y0 ), F (y0 , x0 ), x0 , y0 ) ∈ M ✶✾ ✭✶✳✸✶✮ ◆Õ✉ ❣✐➯ t❤✐Õt ợ tỏ tì t ỳ ❝❤♦ {xn } ✈➭ {yn } s❛♦ {xn } ❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠ tr♦♥❣ X ♠➭ xn → x ✈➭ {yn } ❧➭ ❞➲② ❦❤➠♥❣ t➝♥❣ tr♦♥❣ X ♠➭ yn → y ✱ t❛ ❝ã x1 ≤ x2 ≤ ≤ xn ≤ ≤ x ✈➭ y1 ≥ y2 ≥ ≥ yn ≥ ≥ y ✈í✐ ♠ä✐ n ∈ N✳ ❉♦ ➤ã✱ t❛ ❝ã (x, y, xn , yn ) ∈ M ✈í✐ ♠ä✐ n ∈ N✳ ì tế tết tr ị í ợ t❤á❛ ♠➲♥✳ ❚õ ❝➳❝ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ t❛ s✉② r❛ tt tết ủ ị í ợ t❤á❛ ♠➲♥✱ ❞♦ ➤ã ➳♣ ❞ơ♥❣ ➤Þ♥❤ ❧ý ♥➭② t❛ s✉② r❛ F ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ✶✳✷✳✼ ❍Ư q✉➯✳ ✭❬✶✵❪✮ ◆❣♦➭✐ ♥❤÷♥❣ ❣✐➯ t❤✐Õt tr♦♥❣ ❍Ư q✉➯ ✶✳✷✳✻✱ t❛ ❣✐➯ t❤✐Õt t❤➟♠ r➺♥❣ ✈í✐ ♠ä✐ u, y ≤ v ✈➭ (x, y), (z, t) ∈ X × X ✱ z ≥ u, t ≤ v ✳ ❑❤✐ ➤ã F tå♥ t➵✐ (u, v) ∈ X × X s❛♦ ❝❤♦ x≥ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ➜➬✉ t✐➟♥✱ t❛ ①➳❝ ➤Þ♥❤ ♠ét t❐♣ ❝♦♥ M ⊆ X ❝❤♦ ❜ë✐ M = (a, b, c, d) ∈ X : a ≥ c, b ≤ d ❉Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣ M ❧➭ t❐♣ ❤ỵ♣ F ✲❜✃t ❜✐Õ♥ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳ ❱× ✈❐②✱ ➳♣ ❞ơ♥❣ ❍Ư q✉➯ ✶✳✷✳✻ t❛ s✉② r❛ ❇➞② ❣✐ê ❣✐➯ sö (x, y), (z, t) ∈ X × X ❧➭ ❝➳❝ ➤✐Ĩ♠ ❜✃t ❦ú tr♦♥❣ X × X ✱ ❦❤✐ ➤ã t❤❡♦ ❣✐➯ t❤✐Õt tå♥ t➵✐ ❝➳❝❤ ①➳❝ ➤Þ♥❤ F ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ (u, v) ∈ X × X s❛♦ ❝❤♦ x ≥ u, y ≤ v ✈➭ z ≥ u, t ≤ v ✳ ❚õ M t❛ s✉② r❛ (x, y, u, v) ∈ M ✈➭ (z, t, u, v) ∈ M ✳ ❱× t❤Õ✱ t✃t ❝➯ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ị í ợ tỏ ụ ị ý ♥➭② t❛ s✉② r❛ ✶✳✷✳✽ ❍Ö q✉➯✳ ♠ét ♠➟tr✐❝ F ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳ ✭❬✷❪✮ ❈❤♦ d tr➟♥ X F : X×X → X (X, ≤) ❧➭ t❐♣ ❤ỵ♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ ❣✐➯ sö ❝ã s❛♦ ❝❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ tr➟♥ ✷✵ ●✐➯ sư X✳ ●✐➯ t❤✐Õt r➺♥❣ tå♥ t➵✐ k ∈ [0, 1) s❛♦ ❝❤♦ d(x, u) + d(y, v) d(F (x, y), F (u, v)) ≤ k ✈í✐ ♠ä✐ x, y, u, v ∈ X ♠➭ x ≥ u, y ≤ v ✳ ◆Õ✉ tå♥ t➵✐ x0 ≤ F (x0 , y0 ), t❤× tå♥ t➵✐ x, y ∈ X s❛♦ ❝❤♦ ✭✶✳✸✷✮ x0 , y0 ∈ X s❛♦ ❝❤♦ y0 ≥ F (y0 , x0 ), x = F (x, y) ✈➭ y = F (y, x)✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ ❍Ö q✉➯ ✶✳✷✳✻✱ ♥Õ✉ t❛ ❧✃② ❤➭♠ ϕ(t) = kt ✈í✐ ♠ä✐ t ∈ [0, +∞)✱ tr♦♥❣ ➤ã k ∈ [0, 1) ❧➭ ❤➺♥❣ sè✱ t❤× ✈í✐ ❝➳❝ ❣✐➯ t❤✐Õt ➤➲ ❝❤♦ ❤➭♠ F t❤á❛ ♠➲♥ ❣✐➯ t❤✐Õt ✭❛✮ ✈í✐ ❤➭♠ ϕ✳ ❉♦ ➤ã ➳♣ ❞ơ♥❣ ❤Ư q✉➯ ♥➭② t❛ t❤✉ ➤➢ỵ❝ ❦Õt ❧✉❐♥ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ✶✳✷✳✾ ❍Ö q✉➯✳ ♠ét ♠➟tr✐❝ r➺♥❣ X ✭❬✷❪✮ ❈❤♦ d tr➟♥ X (X, ≤) ❧➭ t❐♣ ❤ỵ♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ ❣✐➯ sư ❝ã s❛♦ ❝❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ t❤á❛ ♠➲♥ ❤❛✐ tÝ♥❤ ❝❤✃t s❛✉ ✐✮ ◆Õ✉ xn ❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠ ✈➭ ✐✐✮ ◆Õ✉ yn ❧➭ ❞➲② ❦❤➠♥❣ t➝♥❣ ✈➭ ●✐➯ sö ●✐➯ sö F :X ×X →X ●✐➯ t❤✐Õt r➺♥❣ tå♥ t➵✐ xn → x✱ yn → y ✱ x, y, u, v ∈ X k ∈ [0, 1) x, y ∈ X yn ≥ y ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ n ∈ N✳ n ∈ N✳ s❛♦ ❝❤♦ ♠➭ x ≥ u, y ≤ v ✳ d(x, u) + d(y, v) ◆Õ✉ tå♥ t➵✐ x0 ≤ F (x0 , y0 ), t❤× tå♥ t➵✐ t❤× xn ≤ x ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr➟♥ ❳✳ d(F (x, y), F (u, v)) ≤ k ✈í✐ ♠ä✐ t❤× s❛♦ ❝❤♦ x0 , y0 ∈ X ✭✶✳✸✸✮ s❛♦ ❝❤♦ y0 ≥ F (y0 , x0 ), x = F (x, y) ✈➭ y = F (y, x)✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ ❍Ö q✉➯ ✶✳✷✳✻✱ ♥Õ✉ t❛ ❧✃② ❤➭♠ ϕ(t) = kt ✈í✐ ♠ä✐ t ∈ [0, +∞)✱ tr♦♥❣ ➤ã k ∈ [0, 1) ❧➭ ❤➺♥❣ sè✱ t❤× ✈í✐ ❝➳❝ ❣✐➯ t❤✐Õt ➤➲ ❝❤♦ t❛ s✉② r❛ ❤➭♠ F t❤á❛ ♠➲♥ ❣✐➯ t❤✐Õt ✭❜✮ ✈í✐ ❤➭♠ ϕ✳ ❉♦ ➤ã ➳♣ ❞ơ♥❣ ❤Ư q✉➯ ♥➭② t❛ t❤✉ ➤➢ỵ❝ ❦Õt ❧✉❐♥ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ✷✶ ❝❤➢➡♥❣ ✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ✷✳✶ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù P❤➬♥ ú t trì ột số ị ý ề ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❈❤♦ (X, ≤) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ X s❛♦ ❝❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ❚❛ ❝ị♥❣ tr❛♥❣ ❜Þ ❝❤♦ ❦❤➠♥❣ ❣✐❛♥ tÝ❝❤ X × X ♠ét q✉❛♥ ❤Ư t❤ø tù tõ♥❣ ♣❤➬♥ ♥❤➢ s❛✉✿ (x, y), (u, v) ∈ X × X, (u, v) ≤ (x, y) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x ≥ u, y ≤ v ✷✳✶✳✶ ➜Þ♥❤ ❧ý✳ ✭❬✺❪✮ ❈❤♦ r➺♥❣ tå♥ t➵✐ ♠ét ♠➟tr✐❝ x≥u ✈➭ y ≤ v✱ α, β ∈ [0, 1) y1 ✱ t❤× T (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ♠➭ α+β < ✈➭ ✈í✐ ♠ä✐ x, y, u, v ∈ X ✱ t❛ ❝ã d(T (x, y), T (u, v)) ≤ α ◆Õ✉ tå♥ t➵✐ ➤✐Ó♠ s❛♦ ❝❤♦ ❧➭ ♠ét ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ s❛♦ ❝❤♦ ✈í✐ ❝➳❝ sè ♥➭♦ ➤ã ♠➭ ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ ❣✐➯ sö d tr➟♥ X T : X ×X → X ➤đ✳ ❈❤♦ (X, ≤) d(x, T (x, y)).d(u, T (u, v))) + βd(x, u) d(x, u) (x0 , y0 ) ∈ X × X s❛♦ ❝❤♦ x0 ≤ T (x0 , y0 ) = x1 ✈➭ ✭✷✳✶✮ y0 ≥ T (y0 , x0 ) = ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② (x0 , y0 ) ∈ X × X s❛♦ ❝❤♦ x0 ≤ T (x0 , y0 ) = x1 ✈➭ y0 ≥ T (y0 , x0 ) = y1 ✳ ❚❛ ➤➷t x1 = T (x0 , y0 ), y1 = T (y0 , x0 )✱ ✈➭ xn+1 = T (xn , yn ), yn+1 = T (yn , xn ) ✈í✐ ♠ä✐ n ≥ 1✳ ❑❤✐ ➤ã t❛ ❝ã T (x0 , y0 ) = T (T (x0 , y0 ), T (y0 , x0 )) = T (x1 , y1 ) = x2 , ✷✷ ✈➭ T (y0 , x0 ) = T (T (y0 , x0 ), T (x0 , y0 )) = T (y1 , x1 ) = y2 ◆❤ê tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ❝đ❛ T ✱ t❛ t❤✉ ➤➢ỵ❝ x2 = T (x0 , y0 ) = T (x1 , y1 ) ≥ T (x0 , y0 ) = x1 , y2 = T (y0 , x0 ) = T (y1 , x1 ) ≤ T (y0 , x0 ) = y1 ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ tỉ♥❣ q✉➳t✱ ✈í✐ n ∈ N✱ t❛ ❝ã xn+1 = T n+1 (x0 , y0 ) = T (T n (x0 , y0 ), T n (y0 , x0 )), yn+1 = T n+1 (y0 , x0 ) = T (T n (y0 , x0 ), T n (x0 , y0 )) ❘â r➭♥❣✱ t❛ t❤✃② r➺♥❣ x0 ≤ T (x0 , y0 ) = x1 ≤ T (x0 , y0 ) = x2 ≤ ≤ T n (x0 , y0 ) = xn ≤ , ✈➭ y0 ≥ T (y0 , x0 ) = y1 ≥ T (y0 , x0 ) = y2 ≥ ≥ T n (y0 , x0 ) = yn ≥ ❉♦ ➤ã✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✶✮ t❛ ❝ã d(xn+1 , xn ) = d(T (xn , yn ), T (xn−1 , yn−1 )) d(xn , T (xn , yn )).d(xn−1 , T (xn−1 , yn−1 )) + βd(xn , xn−1 ) d(xn , xn−1 ) d(xn , xn+1 ).d(xn−1 , xn ) =α + βd(xn , xn−1 ) d(xn , xn−1 ) ≤α = αd(xn , xn+1 ) + βd(xn , xn−1 ) ❚õ ➤✐Ị✉ ♥➭②✱ t❛ s✉② r❛ ➤➢ỵ❝ d(xn , xn+1 ) ≤ β 1−α d(xn , xn−1 ) ✭✷✳✷✮ ❚➢➡♥❣ tù✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✶✮ t❛ ❧➵✐ ❝ã d(yn+1 , yn ) = d(T (yn , xn ), T (yn−1 , xn−1 )) ≤α d(yn , T (yn , xn )).d(yn−1 , T (yn−1 , xn−1 )) + βd(yn , yn−1 ) d(yn , yn−1 ) = αd(yn , yn+1 ) + βd(yn , yn−1 ), ✈➭ ♥❤ê ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ d(yn , yn+1 ) ≤ β 1−α ✷✸ d(yn , yn−1 ) ✭✷✳✸✮ ❚õ ✭✷✳✷✮ ✈➭ ✭✷✳✸✮ t❛ ❝ã β 1−α d(xn , xn+1 ) + d(yn , yn+1 ) ≤ ➜➷t δn = d(xn , xn+1 ) + d(yn , yn+1 ) ✈➭ λ = β ✳ 1−α [d(xn , xn−1 ) + d(yn , yn−1 )] ❑❤✐ ➤ã✱ tõ ✭✷✳✹✮ t❛ ❝ã δn ≤ λδn−1 ≤ λ2 δn−2 ≤ ≤ λn δ0 ◆Õ✉ δ0 ✭✷✳✹✮ ✭✷✳✺✮ = t❤× (x0 , y0 ) ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ T ✳ ●✐➯ sư δ0 > 0✳ ❑❤✐ ó ỗ r N tứ ✈➭ ➳♣ ❞ô♥❣ ❧✐➟♥ t✐Õ♣ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ t❛ t❤✉ ➤➢ỵ❝ d(xn , xn+r ) + d(yn , yn+r ) ≤ [d(xn , xn+1 ) + d(xn+1 , xn+2 ) + + d(xn+r−1 , xn+r )] +[d(yn , yn+1 ) + d(yn+1 , yn+2 ) + + d(yn+r−1 , yn+r )] = [d(xn , xn+1 ) + d(yn , yn+1 )] + [d(xn+1 , xn+2 ) + d(yn+1 , yn+2 )] + + [d(xn+r−1 , xn+r ) + d(yn+r−1 , yn+r )] ≤ δn + δn+1 + + δn+r−1 ≤ λn (1−λr )δ0 1−λ → ❦❤✐ n → ∞ ✭✷✳✻✮ ❱× ✈❐②✱ {xn }, {yn } ❧➭ ❝➳❝ ❞➲② ❈❛✉❝❤② tr♦♥❣ (X, d)✳ ❱× (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ ♥➟♥ tå♥ t➵✐ x∗ , y ∗ ∈ X s❛♦ ❝❤♦ lim xn = x∗ ✈➭ lim yn = y ∗ ✳ ❇➞② ❣✐ê t❛ sÏ ❝❤Ø r❛ r➺♥❣ ❚❤❐t ✈❐②✱ ✈í✐ ∗ n→∞ ∗ n→∞ (x , y ) ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ T ✳ ε > ❜Ð tï② ý✱ ♥❤ê tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ T t➵✐ (x∗ , y ∗ )✱ ♥➟♥ ✈í✐ ε > tå♥ t➵✐ ♠ét sè δ > s❛♦ ❝❤♦ ✈í✐ u, v ∈ X ♠➭ d(x∗ , u) + d(y ∗ , v) < δ t❛ ❝ã d(T (x∗ , y ∗ ), T (u, v)) < tå♥ t➵✐ ❝➳❝ sè ✳ ▲➵✐ ✈× xn → x∗ , yn → y ∗ ✱ ♥➟♥ ✈í✐ ζ = min( 2ε , 2δ ) > 0✱ n0 , m0 s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n ≥ n0 , m ≥ m0 t❛ ❝ã d(xn , x∗ ) < ζ ✈➭ d(xm , x∗ ) < ζ ✳ ❉♦ ➤ã✱ ✈í✐ n ∈ N ♠➭ n ≥ max{n0 , m0 } t❛ ❝ã d(T (x∗ , y ∗ ), x∗ ) ≤ d(T (x∗ , y ∗ ), xn+1 ) + d(xn+1 , x∗ ) = d(T (x∗ , y ∗ ), T (xn , yn )) + d(xn+1 , x∗ ) < ❱× ε + ζ ≤ ε ε > ❜Ð tï② ý✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ r➺♥❣ T (x∗ , y ∗ ) = x∗ ✳ ▲❐♣ ❧✉❐♥ t➢➡♥❣ tù✱ t❛ ❝ò♥❣ s r ợ ì T (y , x ) = y ∗ ✳ (x∗ , y ∗ ) ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ T ✳ ✷✹ ➜Þ♥❤ ❧ý✳ ✷✳✶✳✷ ✈í✐ ♠ä✐ ✭❬✺❪✮ ❱í✐ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶✱ t❛ ❣✐➯ t❤✐Õt t❤➟♠ r➺♥❣ (x, y), (z, t) ∈ X × X s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ tå♥ t➵✐ ➤✐Ĩ♠ (T (x, y), T (y, x)) ✈➭ (u, v) ∈ X × X (T (z, t), T (t, z))✳ s❛♦ ❝❤♦ (T (u, v), T (v, u)) ❑❤✐ ➤ã T ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶ t❛ s✉② r❛ r➺♥❣ T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❇➞② ❣✐ê t❛ ❣✐➯ sö r➺♥❣ ❝ñ❛ T ✱ ♥❣❤Ü❛ ❧➭ x = T (x, y), y = T (y, x), z = T (z, t), t = T (t, z)✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❝❤♦ (x, y), (z, t) ∈ X × X ❧➭ ❝➳❝ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ x = z ✈➭ y = t✳ ❚❤❐t ✈❐②✱ ♥❤ê ❣✐➯ t❤✐Õt tå♥ t➵✐ (u, v) ∈ X × X s❛♦ (T (u, v), T (v, u)) s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ (T (x, y), T (y, x)) ✈➭ (T (z, t), T (t, z))✳ ➜➷t u0 = u, v0 = v ✈➭ ❝❤ä♥ u1 = T (u0 , v0 ) ✈➭ v1 = T (v0 , u0 )✳ ❑❤✐ ➤ã t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶ t❛ ❝ã t❤Ĩ ①➞② ❞ù♥❣ ➤➢ỵ❝ ❝➳❝ ❞➲② {un } ✈➭ {vn } s❛♦ ❝❤♦ un+1 = T (un , ) ✈➭ vn+1 = T (vn , un ) ✈í✐ ♠ä✐ n ≥ 0✳ ❚➢➡♥❣ tù ♥❤➢ ✈❐②✱ t❛ ➤➷t ❞ù♥❣ tr➟♥ t❛ ①➞② ❞ù♥❣ ❝➳❝ ❞➲② x0 = x, y0 = y, z0 = z, t0 = t ✈➭ t❤❡♦ ❝➳❝❤ ①➞② {xn }, {yn }, {zn }, {tn } ✈➭ ❜➺♥❣ ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✶ t❛ ❝❤Ø r❛ ➤➢ỵ❝ r➺♥❣ xn → x = T (x, y), yn → y = T (y, x), zn → z = T (z, t), tn → t = T (t, z)✳ ❱× (T (x, y), T (y, x)) = (x, y) ✈➭ (T (u, v), T (v, u)) = (u1 , v1 ) ❧➭ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ ♥❤❛✉✱ ♥➟♥ ❦❤➠♥❣ ♠✃t tÝ♥❤ tæ♥❣ q✉➳t t❛ ❝ã x ≥ u1 ✈➭ y ≤ v1 ✳ ❚✐Õ♣ t❤❡♦ t❛ sÏ ❝❤Ø r❛ r➺♥❣ ❝➳❝ ❝➷♣ (x, y) ✈➭ (un , ) ❧➭ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ ♥❤❛✉✱ ♥❣❤Ü❛ ❧➭ x r➺♥❣ x ≥ un ✈➭ y ≤ ✈í✐ ♠ä✐ n ≥ 1✳ ❚❤❐t ✈❐②✱ ❣✐➯ sư ≥ un ✈➭ y ≤ ➤ó♥❣ ✈í✐ sè tù ♥❤✐➟♥ n ≥ 1✱ ❦❤✐ ➤ã ♥❤ê tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ❝ñ❛ T t❛ ❝ã un+1 = T (un , ) ≤ T (x, y) = x ✈➭ vn+1 = T (vn , un ) ≥ T (y, x) = y ✳ ❉♦ ➤ã t❛ ❝ã x ≥ un ✈➭ y ≤ ➤ó♥❣ ✈í✐ ♠ä✐ n ≥ 1✳ ❱× ✈❐②✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✶✮ t❛ ❝ã d(x, un+1 ) = d(T (x, y), T (u, v)) n ,T (un ,vn ))) + βd(x, un ) ≤ α d(x,T (x,y).d(u d(x,un ) n ,T (un ,vn )) = α d(x,x).d(u + βd(x, un ) d(x,un ) ✭✷✳✼✮ = βd(x, un ) ❚➢➡♥❣ tù t❛ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ ➤➢ỵ❝ ✷✺ d(y, vn+1 ) ≤ βd(y, )✳ ❚õ ➤ã t❛ t❤✉ d(x, un+1 ) + d(y, vn+1 ) ≤ β[d(x, un ) + d(y, )] ≤ (β)2 [d(x, un−1 ) + d(y, vn−1 )] ≤ ≤ (β)n+1 [d(x, u0 ) + d(y, v0 )] ❈❤♦ n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ ♥❤❐♥ ➤➢ỵ❝ lim [d(x, un+1 ) + d(y, vn+1 )] = n→∞ tõ ➤ã t❛ ❝ã lim d(x, un+1 ) = ✈➭ lim d(y, vn+1 ) = 0✳ n→∞ n→∞ ❚➢➡♥❣ tù t❛ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ lim d(z, un+1 ) = lim d(t, vn+1 ) = 0✳ n→∞ n→∞ ▼➷t ❦❤➳❝✱ ♥❤ê ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ t❛ ❝ã d(x, z) ≤ d(x, un ) + d(un , z) ✈➭ d(y, t) ≤ d(y, ) + d(vn , t) ❈❤♦ n → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ ❝ã d(x, z) = d(y, t) = 0✱ ♥❣❤Ü❛ ❧➭ x = z ✈➭ y = t✳ ❱× ✈❐②✱ ✷✳✶✳✸ T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✳ ➜Þ♥❤ ❧ý✳ ✭❬✺❪✮ ❱í✐ ❝➳❝ ❣✐➯ tết ủ ị ý ế s ợ t❤× tå♥ t➵✐ ❞✉② ♥❤✃t x∈X s❛♦ ❝❤♦ x0 ✈➭ y0 ❧➭ s♦ x = T (x, x)✳ ❈❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ ❝➳❝❤ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✶✱ t❛ ①➞② ❞ù♥❣ ➤➢ỵ❝ ❝➳❝ ❞➲② {xn } ✈➭ {yn } tr♦♥❣ X s❛♦ ❝❤♦ xn → x = T (x, y), yn → y = T (y, x)✳ ì x0 y0 s s ợ sö x0 ≤ y0 ✳ ❚❛ sÏ ❝❤Ø r❛ r➺♥❣ ❦❤✐ ➤ã xn ≤ yn ✈í✐ ♠ä✐ n ≥ 1✱ tr♦♥❣ ➤ã xn = T (xn−1 , yn−1 ) ✈➭ yn = T (yn−1 , xn−1 )✳ ❚❤❐t ✈❐②✱ ❣✐➯ sư ➤✐Ị✉ ➤ã ➤ó♥❣ ✈í✐ sè tù ♥❤✐➟♥ ➤✐Ư✉ tré♥ ❝ñ❛ n ≥ 1✱ ❦❤✐ ➤ã ♥❤ê tÝ♥❤ ❝❤✃t ➤➡♥ T t❛ ❝ã xn+1 = T (xn , yn ) ≤ T (yn , xn ) = yn+1 ✳ ❉♦ ➤ã ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✶✮ t❛ ❝ã d(xn+1 , yn+1 ) = d(T (xn , yn ), T (yn , xn )) n )).d(yn ,T (yn ,xn )) + βd(xn , yn ) ≤ α d(xn ,T (xn ,yd(x n ,yn ) ).d(yn ,yn+1 ) = α d(xn ,xn+1 + βd(xn , yn ) d(xn ,yn ) ❈❤♦ n → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ t❤✉ ợ d(y, x) .d(y, x) ì < tõ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ s✉② r❛ d(y, x) = 0✳ ❱× t❤Õ t❛ ❝ã T (x, y) = x = y = T (y, x)✱ ♥❣❤Ü❛ t❛ tå♥ t➵✐ ❞✉② ♥❤✃t x ∈ X s❛♦ ❝❤♦ x = T (x, x)✳ ✷✻ ◆Õ✉ y0 ❞✉② ♥❤✃t ✷✳✷ ≤ x0 ✱ t❤× ❧❐♣ ❧✉❐♥ ❤♦➭♥ t♦➭♥ t➢➡♥❣ tù t❛ ❝❤Ø r❛ ➤➢ỵ❝ r➺♥❣ tå♥ t➵✐ x ∈ X s❛♦ ❝❤♦ x = T (x, x)✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ✷✳✷✳✶ ➜Þ♥❤ ❧ý✳ ✭❬✸❪✮ ❈❤♦ r➺♥❣ tå♥ t➵✐ ♠ét ♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư ❝❤✃t (X, ≤) d ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù tõ♥❣ ♣❤➬♥ ✈➭ ❣✐➯ sö tr➟♥ F : X ×X → X X ✈➭ s❛♦ ❝❤♦ (X, d) g:X→X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❧➭ ❝➳❝ ➳♥❤ ①➵ s❛♦ ❝❤♦ F ❝ã tÝ♥❤ g ✲➤➡♥ ➤✐Ư✉ tré♥ ♥❣➷t tr➟♥ X ✳ ●✐➯ sư r➺♥❣ tå♥ t➵✐ ❤❛✐ ♣❤➬♥ tö x0 , y0 ∈ X t❤á❛ ♠➲♥ gx0 < F (x0 , y0 ) gy0 > F (y0 , x0 ) ✈➭ ✈➭ tå♥ t➵✐ α, β ∈ [0, 1) ✈í✐ α+β gyn+1 ✭✷✳✶✶✮ ➜Ó ❧➭♠ ➤✐Ị✉ ➤ã t❛ sÏ sư ❞ơ♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣ t♦➳♥ ❤ä❝✳ ❱í✐ n = 0✱ ✈× gx0 < F (x0 , y0 ) ✈➭ gy0 > F (y0 , x0 ) ✈➭ gx1 = F (x0 , y0 ) ✈➭ gy1 = F (y0 , x0 ) t❤❡♦ ❝➳❝❤ ➤➷t tr➢í❝✱ ♥➟♥ t❛ ❝ã gx0 < gx1 ✈➭ gy0 > gy1 ✱ ♥❣❤Ü❛ ❧➭ ✭✷✳✶✵✮✱ ✭✷✳✶✶✮ ➤ó♥❣ ✈í✐ n = 0✳ ●✐➯ sư r➺♥❣ ✭✷✳✶✵✮ ✈➭ ✭✷✳✶✶✮ ➤ó♥❣ ✈í✐ sè tù ♥❤✐➟♥ n > 0✳ ❱× F ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ö✉ tré♥ ♥❣➷t ✈➭ gxn < gxn+1 , gyn > gyn+1 ✱ ♥➟♥ tõ ✭✷✳✾✮✱ t❛ ♥❤❐♥ ➤➢ỵ❝ gxn+1 = F (xn , yn ) < F (xn+1 , yn ) < F (xn+1 , yn+1 ) = gxn+2 , ✭✷✳✶✷✮ gyn+1 = F (yn , xn ) > F (yn+1 , xn ) > F (yn+1 , xn+1 ) = gyn+2 ✭✷✳✶✸✮ ✈➭ ❇➞② ❣✐ê tõ ✭✷✳✶✷✮ ✈➭ ✭✷✳✶✸✮ t❛ t❤✉ ➤➢ỵ❝ gxn+1 < gxn+2 ✈➭ gyn+1 > gyn+2 ✳ ❉♦ ➤ã✱ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣ t❛ ❦Õt ❧✉❐♥ ➤➢ỵ❝ r➺♥❣ ✭✷✳✶✵✮ ✈➭ ✭✷✳✶✶✮ ❧➭ ➤ó♥❣ ✈í✐ ♠ä✐ n ≥ 0✳ ❱× ✈❐② t❛ ❝ã gx0 < gx1 < gx2 < < gxn < gxn+1 < , ✭✷✳✶✹✮ gy0 > gy1 > gy2 > > gyn > gyn+1 > ✭✷✳✶✺✮ ✈➭ ❱× gxn > gxn−1 ✈➭ gyn < gyn−1 ✱ ♥➟♥ tõ ✭✷✳✽✮ ✈➭ ✭✷✳✾✮✱ t❛ ❝ã d(gxn+1 , gxn ) = d(F (xn , yn ), F (xn−1 , yn−1 )) d(gxn , F (xn , yn )).d(gxn−1 , F (xn−1 , yn−1 )) + β.d(gxn , gxn−1 ) d(gxn , gxn−1 ) d(gxn , gxn+1 ).d(gxn−1 , gxn ) = α + β.d(gxn , gxn−1 ) d(gxn , gxn−1 ) ≤ α = α.d(gxn , gxn+1 ) + β.d(gxn , gxn−1 ) ◆❤ê ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ r➺♥❣ d(gxn+1 , gxn ) ≤ β d(gxn , gxn−1 ) 1−α d(gyn+1 , gyn ) ≤ β d(gyn , gyn−1 ) 1−α ❚➢➡♥❣ tù t❛ ❝ã ✷✽ ❉♦ ➤ã✱ t❛ ❝ã d(gxn+1 , gxn ) + d(gyn+1 , gyn ) ≤ ➜➷t { n := d(gxn+1 , gxn ) + d(gyn+1 , gyn )} ✈➭ δ = 0≤ ❱× β [d(gxn , gxn−1 ) + d(gyn , gyn−1 )] 1−α n ≤δ n−1 ≤ δ2 n−2 β 1−α < 1✱ t❛ ❝ã ≤ ≤ δ n ≤ δ < 1✱ tõ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ lim n→∞ n = lim [d(gxn+1 , gxn ) + d(gyn+1 , gyn )] = ✭✷✳✶✻✮ n→∞ ❱× t❤Õ✱ t❛ ❝ã lim d(gxn+1 , gxn ) = ✈➭ lim d(gyn+1 , gyn )] = n→∞ n→∞ ❇➞② ❣✐ê t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ {gxn } ✈➭ {gyn } ❧➭ ❝➳❝ ❞➲② ỗ số m, n N ♠➭ m ≥ n✱ t❛ ❝ã d(gxm , gxn ) ≤ d(gxm , gxm−1 ) + d(gxm−1 , gxm−2 ) + + d(gxn+1 , gxn ) ✈➭ d(gym , gyn ) ≤ d(gym , gym−1 ) + d(gym−1 , gym−2 ) + + d(gyn+1 , gyn ) ❱× t❤Õ t❛ ❝ã d(gxm , gxn ) + d(gym , gyn ) ≤ m−1 + m−2 + + n ≤ (δ m−1 + δ m−2 + + δ n ) δn ≤ 1−δ ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ lim [d(gxm , gxn ) + d(gym , gyn )] = m,n→∞ ❉♦ ➤ã {gxn } ✈➭ {gyn } ❧➭ ❝➳❝ ❞➲② ❈❛✉❝❤② tr♦♥❣ g(X)✳ ❉♦ X ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ♥➟♥ ❝ã (x, y) ∈ X × X s❛♦ ❝❤♦ gxn → x ✈➭ gyn → y ✳ ▲➵✐ ✈× g ❧✐➟♥ tô❝ ♥➟♥ g(gxn ) → gx ✈➭ g(gyn ) → gy ✳ ▼➷t ❦❤➳❝ ✈× F ❧➭ ❧✐➟♥ tơ❝✱ ♥➟♥ F (gxn , gyn ) → F (x, y) ✈➭ F (gyn , gxn ) → F (y, x)✳ ▲➵✐ ❞♦ F ❣✐❛♦ ❤♦➳♥ ✈í✐ g ✱ ♥➟♥ t❛ ❝ã F (gxn , gyn ) = gF (xn , yn ) = g(gxn+1 ) → gx ✈➭ F (gyn , gxn ) = gF (yn , xn ) = g(gyn+1 ) → gy ✳ ◆❤ê ✷✾ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ❣✐í✐ ❤➵♥✱ t❛ ♥❤❐♥ ➤➢ỵ❝ gx = F (x, y) ✈➭ gy = F (y, x)✳ ❉♦ ➤ã F ✈➭ g ❝ã ➤✐Ó♠ trï♥❣ ❜é ➤➠✐✳ ❇➞② ❣✐ê t❛ sÏ ♥❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ ✈➭ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐✳ ➜Ĩ ❧➭♠ ➤✐Ị✉ ➤ã ❧➢✉ ý r➺♥❣ ♥Õ✉ tù tõ♥❣ ♣❤➬♥ t❤× t❛ tr❛♥❣ ❜Þ ❝❤♦ ❦❤➠♥❣ ❣✐❛♥ tÝ❝❤ (X, ≤) ❧➭ ♠ét t❐♣ s➽♣ t❤ø X × X ♠ét q✉❛♥ ❤Ư t❤ø tù tõ♥❣ ♣❤➬♥ ♥❤➢ s❛✉✿ ✈í✐ ✭❬✸❪✮ ◆❣♦➭✐ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ t❛ ❣✐➯ t❤✐Õt t❤➟♠ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✷ (u, v) ≤ (x, y) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x ≤ u, y ≥ v (x, y), (u, v) ∈ X × X, r➺♥❣ ✈í✐ ♠ä✐ (x, y), (z, t) ∈ X ×X ✱ tå♥ t➵✐ (u, v) ∈ X ×X ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t✱ ♥❣❤Ü❛ ❧➭ tå♥ t➵✐ ❞✉② ♥❤✃t ♠ét (x, y) ∈ X ×X s❛♦ ❝❤♦ x = gx = F (x, y) ✈➭ ✈➭ (F (z, t), F (t, z))✳ (F (u, v), F (v, u)) g ❧➭ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ (F (x, y), F (y, x)) s❛♦ ❝❤♦ ❑❤✐ ➤ã F ✈➭ y = gy = F (y, x)✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ❝➳❝ ❣✐➯ t❤✐Õt ➤➲ ❝❤♦✱ ♥❤ê ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ t❛ s✉② r❛ t❐♣ ❝➳❝ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ F ✈➭ g rỗ sử r (x, y) (z, t) ❧➭ ❝➳❝ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ F ✈➭ g ✱ ♥❣❤Ü❛ ❧➭ gx = F (x, y), gy = F (y, x), gz = F (z, t), gt = F (t, z) ❚❛ sÏ ❝❤Ø r❛ r➺♥❣ gx = gz ✈➭ gy = gt✳ ❚❤❐t ✈❐②✱ ♥❤ê ❣✐➯ t❤✐Õt✱ tå♥ t➵✐ (u, v) ∈ X ×X s❛♦ ❝❤♦ (F (u, v), F (v, u)) ❧➭ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ ❝❤ä♥ (F (x, y), F (y, x)) ✈➭ (F (z, t), F (t, z))✳ ➜➷t u0 = u, v0 = v ✈➭ u1 , v1 ∈ X s❛♦ ❝❤♦ gu1 = F (u0 , v0 ) ✈➭ gv1 = F (v0 , u0 )✳ ❑❤✐ ➤ã t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ q✉② ♥➵♣ t❛ ❝ã t❤Ó ①➞② ❞ù♥❣ ❝➳❝ ❞➲② ♠ä✐ {gun }, {gvn } s❛♦ ❝❤♦ gun+1 = F (un , ) ✈➭ gvn+1 = F (vn , un ) n ữ t ự ợ ❝➳❝ ❞➲② x0 = x, y0 = y, z0 = z, t0 = t ❜➺♥❣ ❝➳❝❤ t➢➡♥❣ tù t❛ ①➞② {gxn }, {gyn } ✈➭ {gzn }, {gtn }✳ ❑❤✐ ➤ã ❝ị♥❣ ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ t❛ ❝ã t❤Ĩ ❝❤Ø r❛ r➺♥❣ gxn → gx = F (x, y)✱ gyn → gy = F (y, x)✱ gzn → gz = F (z, t)✱ gtn → gt = F (t, z)✱ ✈í✐ ♠ä✐ n ≥ 1✳ ❱× (F (x, y), F (y, x)) = (gx, gy) ✈➭ (F (u, v), F (v, u)) = (gu1 , gv1 ) ❧➭ s♦ s➳♥❤ ➤➢ỵ❝✱ ♥➟♥ gx ≥ gu1 ✈➭ gy ≤ gv1 ✳ ✸✵ ❇➞② ❣✐ê t❛ sÏ ❝❤Ø r❛ r➺♥❣ (gx, gy) ✈➭ (gun , gvn ) ❧➭ s♦ s➳♥❤ ➤➢ỵ❝✱ ♥❣❤Ü❛ ❧➭ gx ≥ gun ✈➭ gy ≤ gvn ✈í✐ ♠ä✐ n✳ ●✐➯ sư r➺♥❣ ♥ã ➤ó♥❣ ✈í✐ sè tù ♥❤✐➟♥ ➤➡♥ ➤✐Ư✉ tré♥ ♥❣➷t ❝đ❛ n ≥ 0✱ t❤× ❦❤✐ ➤ã ♥❤ê tÝ♥❤ ❝❤✃t g ✲ F ✱ t❛ ❝ã gun+1 = F (un , ) ≤ F (x, y) = gx ✈➭ gvn+1 = F (vn , un ) ≥ F (y, x) = gy ✳ ❉♦ ➤ã gx ≥ gun ✈➭ gy ≤ gvn ➤ó♥❣ ✈í✐ ♠ä✐ n✳ ❱× ✈❐② tõ ✭✷✳✽✮ t❛ ❝ã d(gx, gun+1 ) = d(F (x, y), F (un , )) n ,F (un ,vn )) + β.d(gx, gun ) ≤ α d(gx,F (x,y)).d(gu d(gx,gun ) ✭✷✳✶✼✮ = β.d(gx, gun ) ❚➢➡♥❣ tù✱ t❛ ❝ã t❤Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ d(gy, gvn+1 ) ≤ βd(gy, gvn )✳ ❉♦ ➤ã d(gx, gun+1 ) + d(gy, gvn+1 ) ≤ β[d(gx, gun ) + d(gy, gvn )] ≤ (β)2 [d(gx, gun−1 ) + d(gy, gvn−1 )] ≤ ≤ (β)n+1 [d(gx, gu0 ) + d(gy, gv0 )] ❈❤♦ n → ∞✱ t❛ ♥❤❐♥ ➤➢ỵ❝ lim [d(gx, gun+1 ) + d(gy, gvn+1 )] = n→∞ ❱× t❤Õ t❛ ❝ã lim d(gx, gun+1 ) = ✈➭ lim d(gy, gvn+1 )] = n→∞ n→∞ ❚➢➡♥❣ tù✱ t❛ ❝ã t❤Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ d(gz, gun ) = = lim d(gt, gvn )✳ ❈✉è✐ ❝ï♥❣✱ ♥❤ê ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ t❛ ❝ã d(gx, gz) ≤ d(gx, gun ) + d(gun , gz) ✈➭ d(gy, gt) ≤ d(gy, gvn ) + d(gvn , gt)✳ ❈❤♦ n → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭②✱ t ợ ì d(gx, gz) = = d(gy, gt)✱ ♥❣❤Ü❛ ❧➭ gx = gz ✈➭ gy = gt✳ ▲➵✐ gx = F (x, y) ✈➭ gy = F (y, x)✱ ♥❤ê tÝ♥❤ ❣✐❛♦ ❤♦➳♥ ❝ñ❛ F ✈➭ g ✱ t❛ ❝ã g(gx) = g(F (x, y)) = F (gx, gy), g(gy) = g(F (y, x)) = F (gy, gx) ➜➷t ✭✷✳✶✽✮ gx = p ✈➭ gy = q ✳ ❑❤✐ ➤ã t❛ ❝ã gp = F (p, q) ✈➭ gq = F (q, p)✳ ❉♦ ➤ã (p, q) ❧➭ ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✳ ❑❤✐ ➤ã tõ ✭✷✳✶✽✮ ✈í✐ ✸✶ z = p ✈➭ t = q t❛ s✉② r❛ ➤➢ỵ❝ gp = gx ✈➭ gq = gy ✱ ♥❣❤Ü❛ ❧➭ gp = p ✈➭ gq = q ✳ ❉♦ ➤ã t❛ ❝ã p = gp = F (p, q) ✈➭ q = gq = F (q, p)✳ ❱× ✈❐② (p, q) ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ F ✈➭ g ✳ ➜Ĩ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❞✉② ♥❤✃t✱ t❛ ❣✐➯ sö r➺♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❦❤➳❝✳ ❑❤✐ ➤ã✱ ♥❤ê ✭✷✳✶✽✮ t❛ ❝ã ❱× ✈❐② r = gr = gp = p ✈➭ s = gs = gq = q ✳ (p, q) ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❞✉② ♥❤✃t ❝đ❛ F ✈➭ g ✳ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✸ (r, s) ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✭❬✸❪✮ ◆❣♦➭✐ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ý ế s s ợ tì F tồ t ♠ét ➤✐Ĩ♠ (x, y) ∈ X × X ✈➭ g gx0 ✈➭ gy0 ❧➭ ❝ã ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❞✉② ♥❤✃t✱ ♥❣❤Ü❛ ❧➭ s❛♦ ❝❤♦ gx = F (x, y) = F (y, x) = gy ❈❤ø♥❣ ♠✐♥❤✳ ❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ t❛ ❝ã t❤Ó ①➞② ❞ù♥❣ ❤❛✐ ❞➲② {gxn } ✈➭ {gyn } tr♦♥❣ X s❛♦ ❝❤♦ gxn tr♦♥❣ ➤ã → gx ✈➭ gyn → gy ✱ (x, y) ❧➭ ♠ét ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❝đ❛ F ✈➭ g ✳ ●✐➯ sö r➺♥❣ gx0 ≤ gy0 ✳ ❚❛ sÏ ❝❤Ø r❛ r➺♥❣ gxn ≤ gyn ✱ tr♦♥❣ ➤ã gxn = F (xn−1 , yn−1 ), gyn = F (yn−1 , xn−1 ) ✈í✐ ♠ä✐ n✳ ❚❤❐t ✈❐②✱ ❣✐➯ sư ➤✐Ị✉ ♥➭② ➤ó♥❣ ✈í✐ sè tù ♥❤✐➟♥ n > 0✳ ❑❤✐ ➤ã ♥❤ê tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ư✉ tré♥ ♥❣➷t ❝đ❛ F ✱ t❛ ❝ã gxn+1 = F (xn , yn ) ≤ F (yn , xn ) = gyn+1 ✳ ◆❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✽✮ t❛ ❝ã d(gxn+1 , gyn+1 ) = d(F (xn , yn ), F (yn , xn )) d(gxn , F (xn , yn )).d(gyn , F (yn , xn )) + β.d(gxn , gyn ) d(gxn , gyn ) d(gxn , gxn+1 ).d(gyn , gyn+1 ) = α + β.d(gxn , gyn ) d(gxn , gyn ) ≤ α ❈❤♦ n ❱× → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭②✱ t❛ ➤➢ỵ❝ d(gy, gx) ≤ βd(gy, gx)✳ β < tõ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ d(gy, gx) = 0✳ ❱× t❤Õ✱ t❛ ❝ã F (x, y) = gx = gy = F (y, x)✳ ❚r➢ê♥❣ ❤ỵ♣ gy0 ≤ gx0 t❛ ❝ị♥❣ ❧❐♣ ❧✉❐♥ ❤♦➭♥ t♦➭♥ t➢➡♥❣ tù✳ ✸✷ ❑Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉ ✶✳ ❍Ö t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ò ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❣✐❛♦ ❤♦➳♥✱ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥✱ tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ö✉ tré♥✱ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû✱ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵ ✈Ị ❝➳❝ ➳♥❤ ①➵ ➤ã✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✱✳✳✳ ✷✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ♠Ư♥❤ ➤Ị✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ứ ò s ợ tệ ❝❤✐ t✐Õt ❱Ý ❞ơ ✶✳✶✳✷✵ ✈➭ ❱Ý ❞ơ ✶✳✶✳✷✶ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ❱Ý ❞ơ ✶✳✷✳✷ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✶✳✷✳✶✳ ✸✸ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ỗ t ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ❚✳ ●✳ ❇❤❛s❦❛r✱ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ♣❛r✲ t✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②✲ s✐s✱ ✻✺ ✭✷✵✵✻✮✱ ✶✸✼✾✲✶✸✾✸✳ ❬✸❪ ❙✳ ❈❤❛♥❞♦❦ ✭✷✵✶✸✮✱ ❈♦✉♣❧❡❞ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❛ ❝♦♥tr❛❝t✐✈❡ ❝♦♥❞✐t✐♦♥ ♦❢ r❛t✐♦♥❛❧ t②♣❡ ♦♥ ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ❆♣♣❧✳ ▼❛t❤✳ ■♥❢♦r♠❛t✐❝s✱ ✸✸ ✭✺✲✻✮✱ ✻✹✸✲✻✹✾✳ ❬✹❪ ▲✳ ❇✳ ❈✐rÝ❝✱ ▼✳ ❖✳ ❖❧❛t✐♥✇✇♦✱ ❉✳ ●♦♣❛❧ ❛♥❞ ●✳ ❆❦✐♥❜♦ ✭✷✵✶✷✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♠❛♣♣✐♥❣s s❛t✐s❢②✐♥❣ ❛ ❝♦♥tr❛❝✲ t✐✈❡ ❝♦♥❞✐t✐♦♥ ♦❢ r❛t✐♦♥❛❧ t②♣❡ ♦♥ ❛ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡✱ ❆❞✈✳ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r②✱ ✷ ✭✶✮✱ ✶✲✽✳ ❬✺❪ ▲✳ ❇✳ ❈✐rÝ❝✱ ▼✳ ❖✳ ❖❧❛t✐♥✇✇♦✱ ❉✳ ●♦♣❛❧ ❛♥❞ ●✳ ❆❦✐♥❜♦ ✭✷✵✶✷✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♠❛♣♣✐♥❣s s❛t✐s❢②✐♥❣ ❛ ❝♦♥tr❛❝✲ t✐✈❡ ❝♦♥❞✐t✐♦♥ ♦❢ r❛t✐♦♥❛❧ t②♣❡ ♦♥ ❛ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡✱ ❆❞✈✳ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r②✱ ✷ ✭✶✮✱ ✶✲✽✳ ❬✻❪ ❏✳ ❍❛r❥❛♥✐✱ ❇✳ ▲♦♣❡③ ❛♥❞ ❑✳ ❙❛❞❛r❛❣❛♥✐ ✭✷✵✶✵✮✱ ❆ ❢✐①❡❞ ♣♦✐♥t t❤❡✲ ♦r❡♠ ❢♦r ♠❛♣♣✐♥❣s s❛t✐s❢②✐♥❣ ❛ ❝♦♥tr❛❝t✐✈❡ ❝♦♥❞✐t✐♦♥ ♦❢ r❛t✐♦♥❛❧ t②♣❡ ♦♥ ❛ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡✱ ❆❜str✳ ❆♣♣❧✳ ❆♥❛❧✳✱ ❆r✲ t✐❝❧❡ ■❉ ✶✾✵✼✵✶✱ ✽ ♣❛❣❡s✳ ❬✼❪ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠✱ ▲✳ ❈✐r✐❝ ✭✷✵✵✾✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡✲ ♦r❡♠s ❢♦r ♥♦♥❧✐♥❡❛r ❝♦♥tr❛❝t✐♦♥s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ✼✵ ✭✶✷✮✱ ✹✸✹✶✲✹✸✹✾✳ ❬✽❪ ❇✳ ❙❛♠❡t✱ ❈✳ ❱❡tr♦ ✭✷✵✶✵✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t ❛♥❞ ❢✐①❡❞ ♣♦✐♥t ♦❢ N ✲♦r❞❡r✱ F ✲✐♥✈❛r✐❛♥t ❆♥♥✳ ❋✉♥❝t✳ ❆♥❛❧✳✱ ✶✱ ✹✺✲✹✻✳ ✸✹ s❡t ❬✾❪ ❲✳ ❙✐♥t✉♥❛✈❛r❛t✱ ❨❏✳ ❈❤♦✱ ❛♥❞ P✳ ❑✉♠❛♠ ✭✷✵✶✷✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ✇❡❛❦ ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s ✉♥❞❡r F ✲✐♥✈❛r✐❛♥t s❡t✱ ❆❜str✳ ❆♣♣❧✳ ❆♥❛❧✳✱ ❆rt✐❝❧❡ ■❉ ✸✷✹✽✼✹✳ ❬✶✵❪ ❲✳ ❙✐♥t✉♥❛✈❛r❛t✱ P✳ ❑✉♠❛♠✱ ❛♥❞ ❨❏✳ ❈❤♦ ✭✷✵✶✷✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♥♦♥❧✐♥❡❛r ❝♦♥tr❛❝t✐♦♥s ✇✐t❤♦✉t ♠✐①❡❞ ♠♦♥♦✲ t♦♥❡ ♣r♦♣❡rt②✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✷✵✶✷✱ ❞♦✐✿✶✵✳✶✶✽✻✴✶✻✽✼✲ ✶✽✶✷✲✷✵✶✷✲✶✼✵✳ ✸✺ ... +∞), cos(x + y) sin(x − y) tr♦♥❣ trờ ợ ò M = [(, 1) (1, ∞)]4 ⊆ X ❧➭ t❐♣ F ✲❜✃t ❜✐Õ♥✱ ✈➭ ♥ã ❝ò♥❣ t❤á❛ ♠➲♥ tÝ♥❤ ❝❤✃t ❜➽❝ ❝➬✉✳ ✶✵ ✶✳✶✳✸✵ ♣❤➬♥ ❱Ý ❞ô✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ tr ợ tr ị tứ tự từ... X ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ t➝♥❣ ♥Õ✉ ✈í✐ x, y ∈ X ♠➭ x ≤ y t❛ ❝ã f (y) ≤ f (x)✳ ✭❬✷❪✮ ❈❤♦ t❐♣ ợ ợ s tứ tự từ ị ĩ ➳♥❤ ①➵ F : X × X → X ✈➭ g : X → X ✳ ➳♥❤ ①➵ F ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã ➤✐Ư✉ tré♥ ♥❣➷t ♥Õ✉ ♥❣❤Ü❛... ị ý ề ể ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❤÷✉ tû tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tự ứ tết ết q ợ trì ụ ú t trì ột số ị ❧ý ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐