Về một số định lý điểm trùng nhau bộ đôi vè điểm bất động bộ đôi của các ánh xạ trong không gian g mêtric

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❱➝♥ ▼é♥❣ ❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✻ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ô♥ ❱➝♥ ▼é♥❣ ❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ▲✉❐♥ ❱➝♥ ❚❤➵❝ ❙ü ❚♦➳♥ ❍ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✻ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✐ ▼ë ➤➬✉ ✐✐ ❈❤➢➡♥❣ ✶✳ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ✶ G✲♠➟tr✐❝ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ❈❤➢➡♥❣ ✷✳ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ✶✸ G✲♠➟tr✐❝ t❤ø tù ❜é ♣❤❐♥ ✷✳✶ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ t❤ø tù ❜é ♣❤❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✷ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ g ✲➤➡♥ ➤✐Ư✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ t❤ø tù ❜é ♣❤❐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ❑Õt ❧✉❐♥ ✸✻ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✼ ✐ ▼ë ➤➬✉ ✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ị t➭✐ ▼ét tr♦♥❣ ♥❤÷♥❣ ❦Õt q✉➯ ❤÷✉ Ý❝❤ ♥❤✃t ✈➭ ➤➡♥ ❣✐➯♥ ♥❤✃t tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ➤➢ỵ❝ ➤➢❛ r❛ ♥➝♠ ✶✾✷✷ ✈➭ ♥ã ➤➲ trë t❤➭♥❤ ♠ét ❝➠♥❣ ❝ơ ♠➵♥❤ tr♦♥❣ ●✐➯✐ tÝ❝❤✳ ◆❣✉②➟♥ ❧ý ♥➭② ➤➲ ➤➢ỵ❝ ♠ë ré♥❣ t❤❡♦ ❝➳❝ ❤➢í♥❣ ❦❤➳❝ ♥❤❛✉ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉ ❜ë✐ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ tr♦♥❣ ♥❤÷♥❣ ♥➝♠ ❣➬♥ ➤➞②✳ ❱✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ♠ét sè ➤✐Ị✉ ❦✐Ư♥ ❝♦ ♥➭♦ ➤ã ➤➲ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ré♥❣ r➲✐ ❜ë✐ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝✱ ✈× ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ t♦➳♥ ❤ä❝ ✈➭ ❦❤♦❛ ❤ä❝ ø♥❣ ❞ô♥❣ ✈➭ ♥ã trë ♥➟♥ ❝➬♥ t❤✐Õt ➤Ĩ ①❡♠ ①Ðt ❝➳❝ ♠ë ré♥❣ ❦❤➳❝ ♥❤❛✉ ❝đ❛ ♠➟tr✐❝ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥❤➺♠ ♠ë ré♥❣ ♣❤➵♠ ✈✐ ø♥❣ ❞ơ♥❣ ❝đ❛ ♥ã✳ ❚r♦♥❣ ❤➢í♥❣ ♥➭②✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠ê✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ tù❛✲♠➟tr✐❝ ✈➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ❝ã t❤Ĩ ➤➢ỵ❝ ➤➢❛ r❛ ♥❤➢ ❧➭ ❝➳❝ ✈Ý ❞ơ ❝❤Ý♥❤✳ ❈➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ♥❤÷♥❣ ♣❤➢➡♥❣ ♣❤➳♣ t✐Õ♣ ❝❐♥ ❦❤➳❝ ♥❤❛✉ ➤Õ♥ ❝➳❝ ♠➟tr✐❝ ✈➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝❤♦ t❛ t❤✃② r➺♥❣ ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦❤➠♥❣ ❝❤Ø q✉❛♥ trä♥❣ ➤è✐ ✈í✐ ❝➳❝ ♥❣➭♥❤ ❝đ❛ t♦➳♥ ❤ä❝ ❝❤Ý♥❤ t❤è♥❣✱ ♠➭ ❝ß♥ ❝❤♦ ♥❤✐Ị✉ ❝➳❝ ♣❤➞♥ ♥❣➭♥❤ ❝đ❛ ❦❤♦❛ ❤ä❝ ø♥❣ ❞ơ♥❣✳ ◆➝♠ ✷✵✵✻✱ ❩✳▼✉st❛❢❛ ✈➭ ❇✳❙✐♠s ➤➲ ❣✐í✐ t❤✐Ư✉ ♠ét ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❩✳▼✉st❛❢❛✱ ❇✳❙✐♠s ✈➭ ♥❤÷♥❣ ♥❣➢ê✐ ❦❤➳❝ ➤➲ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦❤➳❝ ♥❤❛✉✳ ◆➝♠ ✷✵✵✾✱ ▼✳❆❜❜❛s ✈➭ ❇✳❊✳❘❤♦❛❞❡s ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❦❤➠♥❣ ❣✐❛♦ ❤♦➳♥ ♠➭ ❦❤➠♥❣ ❝➬♥ tÝ♥❤ ❧✐➟♥ tô❝✱ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦❤➳❝ ♥❤❛✉ tr♦♥❣ ✈✐Ư❝ ①➞② ❞ù♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ▼➷t ❦❤➳❝✱ ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤➲ ♥❤❐♥ ➤➢ỵ❝ ♥❤✐Ị✉ sù ❝❤ó ý tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❑Õt q✉➯ ➤➬✉ t✐➟♥ t❤❡♦ ❤➢í♥❣ ♥➭② ➤➲ ➤➢ỵ❝ ➤➢❛ r❛ ❜ë✐ ❆✳❘❛♥ ✈➭ ▼✳❘❡✉r✐♥❣s ♥➝♠ ✷✵✵✹ ✈➭ ❤ä ➤➲ ➤➢❛ r❛ ❝➳❝ ø♥❣ ❞ơ♥❣ tr♦♥❣ ♣❤➢➡♥❣ tr×♥❤ ♠❛ tr❐♥✳ ❙❛✉ ➤ã✱ ❏✳◆✐❡t♦ ✈➭ ❘✳▲ã♣❡③✱ ♥➝♠ ✷✵✵✼ ➤➲ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ♥➭② ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❦❤➠♥❣ ❣✐➯♠ ✈➭ ø♥❣ ❞ô♥❣ ♥ã ➤Ĩ t❤✉ ➤➢ỵ❝ ♥❣❤✐Ư♠ ❞✉② ♥❤✃t ❝❤♦ ❝➳❝ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ❜❐❝ ✶ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ✐✐ ❜✐➟♥ t✉➬♥ ❤♦➭♥✳ ◆➝♠ ✷✵✵✻✱ ❚✳●✳❇❤❛s❦❛r ✈➭ ❱✳▲❛❦s❤♠✐❦❛♥t❤❛♠ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ F :X→X ✈➭ ➳♥❤ ①➵ g:X→X ✈➭ ♥❣❤✐➟♥ ❝ø✉ ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ❜é ♣❤❐♥✳ ◆➝♠ ✷✵✶✵✱ ❙✳❙❡❞❣❤✐ ✈➭ ❝é♥❣ sù ➤➲ ❝❤ø♥❣ ♠✐♥❤ ♠ét ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♠ê ➤➬② ➤ñ✳ ●➬♥ ➤➞②✱ ❇✳❙✳❈❤♦✉❞❤✉r② ✈➭ P✳▼❛✐t② ➤➲ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤✐Ò✉ ❦✐Ư♥ ❝➬♥ ❝❤♦ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ●✲♠➟tr✐❝ t❤ø tù ❜é ♣❤❐♥ ✈➭ ọ ũ t ợ ề ết q tú ị ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳❚❙ ❚r➬♥ ❱➝♥ ➣♥ ❝❤ó♥❣ t➠✐ ➤➲ ❝❤ä♥ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐ ♥❣❤✐➟♥ ❝ø✉ ✧❱Ị ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✧✳ ✷✳ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ✲ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ tô✱ ❞➲② G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ tré♥✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t G✲♠➟tr✐❝ g ✲➤➡♥ G✲♠➟tr✐❝✱ ❞➲② G✲❤é✐ ➤➬② ➤ñ✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ ➤✐Ư✉ tré♥✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ✲ P❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ố tợ tr ị ý ể t ộ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ö✉ tré♥ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ✸✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ✲ ❉ï♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❣✐➯✐ tÝ❝❤✱ t➠♣➠✱ ❣✐➯✐ tÝ❝❤ ❤➭♠✳ ✲ ❙ư ❞ơ♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❧ý t❤✉②Õt✱ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ t➭✐ ❧✐Ư✉ ✈➭ sư ❞ơ♥❣ ♠ét sè ❦ü t❤✉❐t ❝❤ø♥❣ ♠✐♥❤ ♠í✐ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ✈✃♥ ➤Ị ➤➷t r❛✳ ✲ ❉ù❛ ✈➭♦ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦✱ ❜➺♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ♣❤➞♥ tÝ❝❤ tỉ♥❣ ❤ỵ♣✱ s♦ s➳♥❤ ✱ ❦❤➳✐ q✉➳t ❤♦➳✳✳✳ ➤Ĩ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❤Ö t❤è♥❣ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✐✐✐ ❜é ➤➠✐ ❝ñ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ư✉ tré♥✳ ✹✳ ▼ơ❝ ➤Ý❝❤ ♥❣❤✐➟♥ ❝ø✉ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ♥❤➺♠ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❤Ư t❤è♥❣ ❝➳❝ ❦Õt q✉➯ ✈Ị ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤✐Ư✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ g ✲➤➡♥ G✲♠➟tr✐❝ ✈➭ ❝❤♦ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ✺✳ ◆é✐ ❞✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❈➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❈➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ✻✳ ❈✃✉ tró❝ ❧✉❐♥ ✈➝♥ ▲✉❐♥ ✈➝♥ ❣å♠ ✷ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ▼ơ❝ ✶ ♥❤➺♠ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ụ trì ột số ị ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤♦➵✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ t❤ø tù ❜é ♣❤❐♥✳ ▼ô❝ ✶ ♥❤➺♠ trì ột số ị ý ề ể trù ❜é ➤➠✐✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ ❝➳❝ ✈Ý ❞ơ ụ trì ột số ị ❧ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t tré♥ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ✐✈ g ✲➤➡♥ ➤✐Ư✉ ▲✉❐♥ ✈➝♥ ♥➭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✳ ❚➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❚♦➳♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý t❤➬② ❝➠ ë ❇é ♠➠♥ ●✐➯✐ ❚Ý❝❤ ❦❤♦❛ ❚♦➳♥ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❙ë ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ tØ♥❤ ❚➞② ◆✐♥❤✱ ❇❛♥ ●✐➳♠ ❍✐Ư✉ ❚r➢ê♥❣ ❚❍P❚ ❇×♥❤ ❚❤➵♥❤✱ tØ♥❤ ❚➞② ◆✐♥❤ ➤➲ ❣✐ó♣ ➤ì✱ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❝❤♦ t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ◆❤➞♥ ➤➞② t➳❝ ❣✐➯ ①✐♥ ❝➳♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ●✐➯✐ ❚Ý❝❤ ❦❤♦➳ ✷✷ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➭✐ ●ß♥✳ ❈✉è✐ ❝ï♥❣ t➳❝ ❣✐➯ ①✐♥ ❣ë✐ ❧ê✐ ❝➳♠ ➡♥ ➤Õ♥ ❇❛ ♠Đ✱ ❝➳❝ ❛♥❤ ❡♠ tr♦♥❣ ❣✐❛ ➤×♥❤ ➤➲ 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 ♥❤÷♥❣ ý ❦✐Õ♥ ➤ã♥❣ ❣ã♣ ❝đ❛ q✉ý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥✳ ❱✐♥❤✱ ♥❣➭② ✷✵ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✻ ◆❣✉②Ô♥ ❱➝♥ ▼é♥❣ ✈ ❝❤➢➡♥❣ ✶ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ◆é✐ ❞✉♥❣ ❣å♠✿ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ G✲❤é✐ tơ✱ ❞➲② G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ ➤✐Ö✉ tré♥✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ G✲♠➟tr✐❝ G✲♠➟tr✐❝✱ ❞➲② ➤➬② ➤ñ✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ✶✳✶✳✶ G✲♠➟tr✐❝ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ❝➬♥ ù trì ề s ị ĩ X ột t rỗ G : X × X × X → R+ ❧➭ ♠ét ❤➭♠ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ s❛✉ ✭●✶✮ G(x, y, z) = ♥Õ✉ x = y = z ✱ ✭●✷✮ < G(x, x, y) ✈í✐ ♠ä✐ x, y ∈ X ✭●✸✮ G(x, x, y) ≤ G(x, y, z)✱ ✈í✐ ♠ä✐ x, y, z ∈ X ✭●✹✮ G(x, y, z) = G(x, z, y) = G(y, z, x) = , ✭➤è✐ ①ø♥❣ ë ❝➯ ✸ ❜✐Õ♥✮✱ ✭●✺✮ G(x, y, z) ≤ G(x, a, a) + G(a, y, z), ✈í✐ ♠ä✐ x, y, z, a ∈ X ✱ ✭❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ✈í✐ x = y✱ ✈í✐ z = y✱ ❣✐➳❝✮✳ ❑❤✐ ➤ã✱ ❤➭♠ tr➟♥ G ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♠➟tr✐❝ s✉② ré♥❣✱ ❤❛② ❣ä♥ ❤➡♥ ❧➭ ♠ét X ✱ ✈➭ ❝➷♣ (X, G) ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❉Ơ t❤✃② r➺♥❣ ỗ Gtr tr X Gtr s r ột ♠➟tr✐❝ ❜ë✐ dG (x, y) = G(x, y, y) + G(y, x, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✶ G✲♠➟tr✐❝ dG tr➟♥ X ❝❤♦ ✶✳✶✳✷ ▼Ư♥❤ ➤Ị✳ x, y, z ✈➭ a ∈ X✱ (X, G) ✭❬✼❪✮ ❈❤♦ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G(x, y, z) = 0✱ x = y = z✱ ◆Õ✉ ✭✷✮ G(x, y, z) ≤ G(x, x, y) + G(x, x, z)✱ ✭✸✮ G(x, y, y) ≤ 2G(y, x, x)✱ ✭✹✮ G(x, y, z) ≤ G(x, a, z) + G(a, y, z)✱ ✭✺✮ G(x, y, z) ≤ [G(x, y, a) + G(x, a, z) + G(a, y, z)]✱ ✭✻✮ G(x, y, z) ≤ G(x, a, a) + G(y, a, a) + G(z, a, a) ị ĩ ể ủ tì X✳ ➜✐Ó♠ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ x ∈ X G✲♠➟tr✐❝ lim G(x, xn , xm ) = 0✳ ▲ó❝ ➤ã t❛ ♥ã✐ r➺♥❣ ❞➲② {xn } ❧➭ G✲❤é✐ t➵✐ k∈N ✶✳✶✳✹ X xn → x s❛♦ ❝❤♦ ▼Ö♥❤ ➤Ị✳ ✈➭ ➤✐Ĩ♠ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ✭❬✼❪✮ ❈❤♦ {xn } ♥Õ✉ ε > 0✱ tå♥ x✳ ♥Õ✉ ✈í✐ ♠ä✐ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ✈í✐ ❞➲② {xn } ⊆ G✲❤é✐ ❉➲② ✭✷✮ G(xn , xn , x) → 0✱ ✭✸✮ G(xn , x, x) → 0✱ ✭✹✮ G(xm , xn , x) → 0✱ ❧➭ ➜Þ♥❤ ♥❣❤Ü❛✳ ➤➢ỵ❝ ❣ä✐ ❧➭ G✲♠➟tr✐❝ (X, G) tơ ✈Ị ❧➭ ♠ét ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ➤➞② ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✭✶✮ ✶✳✶✳✺ {xn } G(x, xn , xm ) < ε ✈í✐ ♠ä✐ m, n ≥ k ✳ x ∈ X✱ {xn } ✈➭ ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐í✐ ❤➵♥ ❝đ❛ ❞➲② n,m→∞ ◆❤➢ ✈❐②✱ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ t❛ ❝ã ✭✶✮ ✶✳✶✳✸ G✲♠➟tr✐❝✳ tơ ✈Ị ❦❤✐ ❦❤✐ n → ∞❀ n → ∞❀ ❦❤✐ m, n → ∞✳ ✭❬✼❪✮ ❈❤♦ G✲❈❛✉❝❤② x❀ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♥Õ✉ ✈í✐ ♠ä✐ ε > G✲♠➟tr✐❝✳ tå♥ t➵✐ sè tù ♥❤✐➟♥ ❉➲② {xn } ⊆ X N ∈ N s❛♦ ❝❤♦ G(xn , xm , xl ) < ε ✈í✐ ♠ä✐ n, m, l ≥ N ✱ ♥❣❤Ü❛ ❧➭ G(xn , xm , xl ) → ❦❤✐ n, m, l → ∞✳ ✷ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✻ ✭❬✼❪✮ ❑❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ❦❤➠♥❣ ❣✐❛♥ tr♦♥❣ (X, G)✳ ✶✳✶✳✼ ▼Ư♥❤ ➤Ị✳ ✭❬✼❪✮ G✲♠➟tr✐❝ (X, G) ợ ọ G ủ ủ ế ỗ ❞➲② ❈❤♦ (X, G) G✲❈❛✉❝❤② tr♦♥❣ (X, G) G✲♠➟tr✐❝✳ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❧➭ G✲❤é✐ tô ❑❤✐ ➤ã✱ ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ {xn } ✭✶✮ ❉➲② ✭✷✮ ❱í✐ ♠ä✐ G✲❈❛✉❝❤②❀ ❧➭ ε > 0✱ tå♥ t➵✐ sè ♥❣✉②➟♥ N ∈ N s❛♦ ❝❤♦ G(xn , xm , xm ) < ε ✈í✐ ♠ä✐ n, m ≥ N ✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✽ ①➵ ✭❬✼❪✮ ❈❤♦ (X, G) ✈➭ (X , G ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ➳♥❤ f : (X, G) → (X , G )✳ ❑❤✐ ➤ã f ❦❤✐ ✈í✐ ♠ä✐ ➤➢ỵ❝ ❣ä✐ ❧➭ G ❧✐➟♥ tơ❝ t➵✐ ➤✐Ĩ♠ a ∈ X ε > 0✱ tå♥ t➵✐ sè δ > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ ❦❤✐ ✈➭ ❝❤Ø G(a, x, y) < δ t❛ ❝ã G (f (a), f (x), f (y)) < ε✳ ➳♥❤ ①➵ f : (X, G) → (X , G ) ➤➢ỵ❝ ❣ä✐ ❧➭ G✲❧✐➟♥ tơ❝ tr➟♥ X ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ♥ã ❧➭ G✲❧✐➟♥ tô❝ t➵✐ ♠ä✐ ➤✐Ĩ♠ a ∈ X ✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✾ X ×X → X ➤Õ♥ ✭❬✼❪✮ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ➤➢ỵ❝ ❣ä✐ ❧➭ ❧✐➟♥ tơ❝ ♥Õ✉ ✈í✐ ❜✃t ❦ú ❤❛✐ ❞➲② G✲♠➟tr✐❝✳ {xn } ✈➭ ➳♥❤ ①➵ F {yn } ❧➭ G✲❤é✐ : tô x ✈➭ y ✱ t➢➡♥❣ ø♥❣✱ t❛ ❝ã {F (xn , yn )} ❧➭ G✲❤é✐ tơ ➤Õ♥ F (x, y)✳ ✶✳✶✳✶✵ ➜Þ♥❤ ♥❣❤Ü❛✳ X ×X ×X → X ✶✳✶✳✶✶ ▼Ư♥❤ ➤Ị✳ G(x, y, z) ✶✳✶✳✶✷ ✭❬✼❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ➤➢ỵ❝ ❣ä✐ ❧➭ ❧✐➟♥ tơ❝ ♥Õ✉ ✈í✐ ❜✃t ❦ú ❜❛ ❞➲② G✲❤é✐ tô ➤Õ♥ x✱ y ✈➭ z t➢➡♥❣ ø♥❣✱ t❛ ❝ã ✭❬✼❪✮ ❈❤♦ (X, G) ➳♥❤ ①➵ F {xn }✱ {yn } ✈➭ {zn } : ❧➭ {F (xn , yn , zn )} ❧➭ G✲❤é✐ tô ➤Õ♥ F (x, y, z)✳ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ❤➭♠ ❧✐➟♥ tô❝ ➤å♥❣ t❤ê✐ t❤❡♦ t✃t ❝➯ ✸ ❜✐Õ♥ ❝đ❛ ♥ã✳ ❱Ý ❞ơ✳ ✭❬✼❪✮ ❈❤♦ Gs : X × X × X → R+ ♠ä✐ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣✳ ❳Ðt ❤➭♠ ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ Gs (x, y, z) = d(x, y) + d(y, z) + d(x, z)✱ x, y, z ∈ X ✳ ❑❤✐ ➤ã✱ (X, Gs ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ✸ ✈í✐ ✭❛✮ F ❧➭ ❧✐➟♥ tơ❝✱ ❤♦➷❝ ✭❜✮ X ❝ã tÝ♥❤ ❝❤✃t ❞➢í✐ ➤➞②✿ ✭✐✮ ◆Õ✉ ♠ét ❞➲② ❦❤➠♥❣ ❣✐➯♠ ✭✐✐✮ ◆Õ✉ ♠ét ❞➲② ❦❤➠♥❣ t➝♥❣ t❤× F {xn } → x✱ {yn } → y ✱ t❤× xn x yn y ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ n ∈ N✱ n ∈ N✱ ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❈❤ø♥❣ ♠✐♥❤✳ ◆❤ê ❝➳❝ ❣✐➯ t❤✐Õt ➤➲ ❝❤♦✱ tå♥ t➵✐ x0 ❚❛ ①➳❝ ➤Þ♥❤ x1 , y1 ∈ X F (x0 , y0 ) ✈➭ y0 x0 x0 , y0 ∈ X s❛♦ ❝❤♦ F (y0 , x0 ) ❝❤♦ ❜ë✐ x1 = F (x0 , y0 ) ❱× t❤× y0 ✱ t❛ ❝ã F (x0 , y0 ) x0 x0 ✈➭ y1 = F (y0 , x0 ) y0 F (y0 , x0 )✳ ❉♦ ➤ã✱ x1 = F (x0 , y0 ) ❚✐Õ♣ tơ❝ q✉➳ tr×♥❤ tr➟♥✱ t❛ ❝ã ❤❛✐ ❞➲② F (y0 , x0 ) = y1 y0 {xn } ✈➭ {yn } s❛♦ ❝❤♦ xn+1 = F (xn , yn ), yn+1 = F (yn , xn ) ✈➭ xn ✈í✐ ♠ä✐ xn+1 = F (xn , yn ) F (yn , xn ) = yn+1 yn n ≥ 0✳ ◆Õ✉ ❝ã k ∈ N s❛♦ ❝❤♦ xk = yk = α✱ t❤× t❛ ❝ã α ♥❣❤Ü❛ ❧➭✱ F (α, α) F (α, α) α, α = F (α, α)✳ ❱× t❤Õ✱ (α, α) ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ F ✳ ❚✐Õ♣ t❤❡♦✱ ❣✐➯ sö r➺♥❣ xn ≺ y n ✈í✐ ♠ä✐ n ∈ N✳ ❑❤✐ ➤ã✱ ❜➺♥❣ ❝➳❝ ❧❐♣ t tự trì tr ị ý ✷✳✶✳✶✱ t❛ ❝ã t❤Ĩ ❣✐➯ t❤✐Õt r➺♥❣ ✈í✐ ♠ä✐ ✭✷✳✷✼✮ n ≥ 0✱ yn , v = yn+1 ✈➭ (xn , yn ) = (xn+1 , yn+1 )✳ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✻✮ ➤ó♥❣ ✈í✐ ❑❤✐ ➤ã✱ ♥❤ê ❝➠♥❣ t❤ø❝ ✭✷✳✷✼✮✱ x = xn+2 , u = xn+1 , w = xn , y = z = yn+2 P❤➬♥ ❝ß♥ ❧➵✐ ❝đ❛ ❝❤ø♥❣ ♠✐♥❤ ❧➭♠ t❤❡♦ ❣✐è♥❣ ❝➳❝ ❜➢í❝ ♥❤➢ tr ị ý ố trờ ợ ố ✈í✐ tr➢ê♥❣ ❤ỵ♣ ✭❜✮✱ ❝❤ó♥❣ t❛ ❧➭♠ t❤❡♦ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✷✳ ✷✸ X = N ∪ {0} ✈➭ G : X   x + y + z ♥Õ✉       x+z ♥Õ✉     y + z + ♥Õ✉ G(x, y, z) =  ♥Õ✉   y+2    z+1 ♥Õ✉      ♥Õ✉ ❱Ý ❞ơ✳ ✷✳✶✳✽ ❑❤✐ ➤ã✱ X ×X ×X →X ❈❤♦ X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ x, y, z ợ ị ệt x=y=z 0, ✈➭ t✃t ❝➯ ➤Ò✉ ❦❤➳❝ x = 0, y = z ✈➭ ②✱ ③ ❦❤➳❝ 0, 0, x = 0, y = z = 0, x = y = 0, z = 0, x = y = z G✲♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư t❤ø tù ❜é ♣❤❐♥ tr➟♥ ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉✿ ❱í✐ ❈❤♦ x, y ∈ X, x F :X ×X →X y t❤á❛ ♥Õ✉ w u ✈➭ ❝❤✐❛ ❤Õt (x − y) ✈➭ ✈➭ ợ ị F (x, y) = ◆Õ✉ x>y x y x ≺ y, ♥Õ✉ tr trờ ợ v z tì t ó w ≥ u ≥ x ≥ y ≥ v ≥ z✳ ❱× t❤Õ✱ F (x, y) = F (u, v) = F (w, z) = ✈➭ F (y, x) = F (v, u) = F (z, w) = 0✳ ❱× t❤Õ✱ ✈Õ tr➳✐ ❝đ❛ ✭✷✳✷✻✮ trë t❤➭♥❤ G(1, 1, 1) + G(0, 0, 0) = ❱× ✈❐②✱ t❛ ❝ã t❤Ĩ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✼ ❝❤♦ ✈Ý ❞ơ ♥➭② ✈í✐ ♥÷❛✱ F ✷✳✶✳✾ ❜ë✐ ❝ã ❤❛✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❧➭ ◆❤❐♥ ①Ðt✳ ▼ét G✲♠➟tr✐❝ x0 = ✈➭ y0 = 81✳ ❍➡♥ (0, 0) ✈➭ (1, 0)✳ ❝➯♠ s✐♥❤ ♠ét ❝➳❝❤ tù ♥❤✐➟♥ ♠ét ♠➟tr✐❝ dG (x, y) = Gx, y, y) + G(x, x, y)✳ θ ∈ Θ✳ ✈➭ ✭✷✳✷✻✮ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✈í✐ ♠ä✐ ❚õ ♠ét tr♦♥❣ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ u=w dG ❤♦➷❝ ❝❤♦ v = z✱ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✶✮✱ ✭✷✳✶✾✮✱ ✭✷✳✷✺✮ ✈➭ ✭✷✳✷✻✮ ❦❤➠♥❣ q✉② ✈Ò ❜✃t ❦ú ❜✃t ➤➻♥❣ t❤ø❝ ♠➟tr✐❝ ♥➭♦ ✈í✐ ♠➟tr✐❝ dG ✳ ❱× ✈❐②✱ ❝➳❝ ❦Õt q✉➯ ✈Ò ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, dG ) t➢➡♥❣ ø♥❣ ❦❤➠♥❣ t❤Ĩ ➳♣ ❞ơ♥❣ ✈➭♦ ❱Ý ❞ơ ✷✳✶✳✽✳ ✷✳✶✳✶✵ ◆❤❐♥ ①Ðt✳ ❱Ý ❞ơ ✷✳✶✳✽ ❦❤➠♥❣ t❤Ĩ ➳♣ ❞ơ♥❣ ➤➢ỵ❝ ❝➳❝ ➜Þ♥❤ ❧ý ✷✳✶✳✶✱ ➜Þ♥❤ ❧ý ✷✳✶✳✷ ✈➭ ❍Ư q✉➯ ✷✳✶✳✺✳ ➜➞② ❧➭ ➤✐Ị✉ ❤✐Ĩ♥ ♥❤✐➟♥ ❜ë✐ t❤ù❝ r❛ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✶✮✱ ✭✷✳✶✾✮ ✈➭ ✭✷✳✷✺✮ ❦❤➠♥❣ ➤➢ỵ❝ t❤á❛ ♠➲♥ ❦❤✐ w = u = x = y = 3✱ v = ♥÷❛✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❧➭ ❦❤➠♥❣ ❞✉② ♥❤✃t✳ ✷✹ ✈➭ z = 1✳ ❍➡♥ ❚r♦♥❣ ♣❤➬♥ t✐Õ♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ♥❣❤✐➟♥ ❝ø✉ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ➤Ĩ t❤✉ ➤➢ỵ❝ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♣❤❐♥✳ ◆Õ✉ X ×X ♠ä✐ (X, ) ❧➭ ♠ét t❐♣ ❤ỵ♣ s➽♣ t❤ø tù ❜é ♣❤❐♥✱ ❦❤✐ ➤ã t❛ tr❛♥❣ ❜Þ ❝❤♦ tÝ❝❤ (x, y), (u, v) ∈ X × X ✱ (x, y) ➜Þ♥❤ ❧ý✳ ✈➭ (z, t)✳ (u, v) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x u ✈➭ y ❑❤✐ ➤ã✱ F tå♥ t➵✐ ♠ét ➤✐Ĩ♠ (u, v) ∈ X × X ♠➭ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ❝➳❝ ❣✐➯ t❤✐Õt tr➟♥✱ ♥❤ê ➜Þ♥❤ ❧ý ✷✳✶✳✶✱ t❛ s✉② r❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❇➞② ❣✐ê✱ ❣✐➯ sư ❝đ❛ F✱ tø❝ ❧➭✱ ❝➳❝ ❞➲② x=z ✈➭ y = t✳ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ F ❝ã ♠ét (x, y) ✈➭ (z, t) ❧➭ ❝➳❝ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ x = F (x, y), y = F (y, x), z = F (z, t) ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ (u, v) ∈ X × X v✳ ✭❬✶❪✮ ❚❤➟♠ ✈➭♦ ❣✐➯ t❤✐Õt tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✶✱ ❣✐➯ sư r➺♥❣ ✈í✐ (x, y), (z, t) ∈ X × X ✱ (x, y) t❤ø tù ❜é ✈í✐ t❤ø tù ❜é ♣❤❐♥ s❛✉✿ ❈❤♦ ✷✳✶✳✶✶ G✲♠➟tr✐❝ ✈➭ t = F (t, z)✳ ◆❤ê ❣✐➯ t❤✐Õt ➤➲ ❝❤♦✱ ✈í✐ ❚✐Õ♣ t❤❡♦✱ t❛ sÏ (x, y) ✈➭ (z, t) tå♥ t➵✐ (x, y) ✈➭ (z, t)✳ ❚❛ ➤➷t u0 = u ✈➭ v0 = v ✈➭ ①➞② ❞ù♥❣ {un } ✈➭ {vn } ❝❤♦ ❜ë✐ un = F (un−1 , vn−1 ) ✈➭ = F (vn−1 , un−1 ) ✈í✐ ♠ä✐ n N ì (u, v) s s ợ (x, y)✱ t❛ ❣✐➯ sö r➺♥❣ (u0 , v0 ) = (u, v) (x, y)✳ ❙ư ❞ơ♥❣ q✉② ♥➵♣ t♦➳♥ ❤ä❝✱ ❞Ơ ❞➭♥❣ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ (un , ) (x, y) ✈í✐ ♠ä✐ n ∈ N ❚õ ✭✷✳✶✮✱ t❛ ❝ã G(x, x, un ) + G(y, y, ) = G(F (x, y), F (x, y), F (un−1 , vn−1 )) + G(F (y, x), F (y, x), F (vn−1 , un−1 )) ≤ θ(G(x, x, un−1 ), G(vn−1 , y, y))[G(x, x, un−1 ) + G(vn−1 , y, y)] ✭✷✳✷✽✮ < G(x, x, un−1 ) + G(vn−1 , y, y) ❱× t❤Õ✱ ❞➲② {G(x, x, un ) + G(y, y, )} ❧➭ ❞➲② ❦❤➠♥❣ ➞♠ ✈➭ ❣✐➯♠✱ ❞♦ ➤ã t❛ ❝ã G(x, x, un ) + G(y, y, ) → g, ✈í✐ g ≥ 0✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ g = 0✳ ❚❤❐t ✈❐②✱ ♥Õ✉ g>0 t❤× ❧❐♣ ❧✉❐♥ t➢➡♥❣ tù ✈í✐ ♣❤➬♥ ❝❤ø♥❣ ♠✐♥❤ ủ ị ý t ết ợ r (G(x, x, un−1 ), G(vn−1 , y, y)) → ✷✺ ì t ợ G(x, x, un1 ) → ✈➭ G(vn−1 , y, y) → 0✳ ❉♦ ➤ã✱ G(x, x, un−1 ) + G(vn−1 , y, y) → ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ g > 0✳ ❉♦ ➤ã G(x, x, un ) + G(vn , y, y) → ✭✷✳✷✾✮ ❚➢➡♥❣ tù✱ t❛ ❝ã t❤Ó ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ G(x, un , un ) + G(vn , , y) → 0, ✭✷✳✸✵✮ G(z, z, un ) + G(vn , t, t) → 0, ✭✷✳✸✶✮ G(z, un , un ) + G(t, , ) → ✭✷✳✸✷✮ ✈➭ ◆❤ê ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ t❛ ❝ã G(z, x, x) ≤ G(z, un , un ) + G(un , x, x) ✭✷✳✸✸✮ G(y, t, t) ≤ G(y, , ) + G(vn , t, t) ✭✷✳✸✹✮ ✈➭ ❑Õt ❤ỵ♣ ✭✷✳✸✸✮ ✈➭ ✭✷✳✸✹✮✱ t❛ ❝ã G(z, x, x) + G(y, t, t) ≤ [G(z, un , un ) + G(un , x, x)] + [G(y, , ) + G(vn , t, t)] ≤ [G(x, x, un ) + G(vn , y, y)] + [G(x, un , un ) + G(vn , , y)]+ + [G(z, z, un ) + G(vn , t, t)] + [G(z, un , un ) + G(t, , )] ❈❤♦ n → ∞✱ ♥❤ê ✭✷✳✷✾✮✱ ✭✷✳✸✵✮✱ ✭✷✳✸✶✮ ✈➭ ✭✷✳✸✷✮✱ t❛ ❝ã G(z, x, x) + G(y, t, t) ≤ 0✳ ❱× t❤Õ✱ G(z, x, x) = ✈➭ G(y, t, t) = 0✱ ♥❣❤Ü❛ ❧➭ z=x ✈➭ y = t✳ ❉♦ ➤ã✱ F ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ✷✳✶✳✶✷ ♠ä✐ ✈➭ ➜Þ♥❤ ❧ý✳ ✭❬✶❪✮ ❚❤➟♠ ✈➭♦ ❣✐➯ t❤✐Õt tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✷✱ ❣✐➯ sư r➺♥❣ ✈í✐ (x, y), (z, t) ∈ X × X ✱ (z, t)✳ ❑❤✐ ➤ã✱ F tå♥ t➵✐ ♠ét ➤✐Ĩ♠ (u, v) ∈ X × X s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ (x, y) ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ë ➜Þ♥❤ ❧ý ✷✳✶✳✶✶ ✈➭ ➜Þ♥❤ ❧ý ✷✳✶✳✷✳ ✷✻ ✷✳✷ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ g ✲➤➡♥ ➤✐Ö✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ t❤ø tù ❜é ♣❤❐♥ P ú t trì ột số ị ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ❦❤➠♥❣ ❣✐❛♥ g ✲➤➡♥ ➤✐Ö✉ tré♥ tr➟♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ❝➳❝ ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❈❤ó♥❣ t❛ sÏ ❜➽t ➤➬✉ ❜➺♥❣ ♠ét ✈Ý ❞ơ ❝❤♦ t❤✃② ➤✐Ĩ♠ ②Õ✉ ❝đ❛ ➜Þ♥❤ ❧ý ✶✳✶✳✷✶✳ ✷✳✷✳✶ ❱Ý ❞ơ✳ ●✐➯ sư X = R✳ ❚❛ ①➳❝ ➤Þ♥❤ G : X × X × X → [0, ∞) ❝❤♦ ❜ë✐ G(x, y, z) = |x − y| + |x − z| + |y − z| x, y, z ∈ X ✳ ●✐➯ sö (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❚❛ ➤Þ♥❤ ♥❣❤Ü❛ ➳♥❤ ①➵ F : X × X → X ❝❤♦ ❜ë✐ F (x, y) = x + y ✈➭ 8 g : X → X ❝❤♦ ❜ë✐ g(x) = x ✈í✐ ♠ä✐ x, y ∈ X ✳ ●✐➯ sö x = u = z ✳ ❑❤✐ ➤ã✱ t❛ ❝ã   5 G(F (x, y), F (u, v), F (z, w)) = G  x + y, u + v, z + w 8 8 8 ✈í✐ ♠ä✐ ❧➭ t❤ø tù t❤➠♥❣ t❤➢ê♥❣✳ ❑❤✐ ➤ã✱ ✭✷✳✸✺✮ = 5 |v − y| + |w − y| + |w − v|, 8 ✈➭  G(gx, gu, gz) + G(gy, gv, gw) = G  = ❘â r➭♥❣ ❧➭ ❦❤➠♥❣ ❝ã ❣✐➳ trÞ k ∈ 0, 21    7x 7u 7z 7y 7v 7w , ,  + G , ,  8 8 8 ✭✷✳✸✻✮ (|y − v| + |y − w| + |v − w|) s❛♦ ❝❤♦ ❝➠♥❣ t❤ø❝ ✭✶✳✺✮ ❝ñ❛ ị ý ợ tỏ ể (0, 0) ❧➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❞✉② ♥❤✃t ❝đ❛ F r❛✱ ♥ã ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ t❤➠♥❣ t❤➢ê♥❣ ❝ñ❛ ✷✼ F ✈➭ ✈➭ g ✳ ❚❤ù❝ g ✱ ♥❣❤Ü❛ ❧➭✱ F (0, 0) = g0 = 0✳ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✷ ✭❬✾❪✮ ❈❤♦ G✲♠➟tr✐❝ ♠ét ❦❤➠♥❣ ❣✐❛♥ ➳♥❤ ①➵ s❛♦ ❝❤♦ (X, ) F ❧➭ ♠ét t❐♣ ❤ỵ♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ ➤➬② ➤đ✳ ●✐➯ sư g ✲➤➡♥ ❝ã tÝ♥❤ ❝❤✃t F : X ×X → X ➤✐Ö✉ tré♥ tr➟♥ X ✈➭ g:X →X g(X)✱ g ❧➭ G✲❧✐➟♥ tô❝✱ F ♠➭ gx ✈➭ g gu gw, gy ❧➭ ❝➳❝ ➳♥❤ ①➵ ❧➭ ❤❛✐ ✭✷✳✸✼✮ ≤ k [G(gx, gu, gw) + G(gy, gv, gz)] x, y, u, v, z, w ∈ X ❧➭ ✈➭ G(F (x, y), F (u, v), F (w, z)) + G(F (y, x), F (v, u), F (z, w)) ✈í✐ ♠ä✐ (X, G) gz ✳ ●✐➯ sư r➺♥❣ F (X × X) ⊂ gv G✲t➢➡♥❣ t❤Ý❝❤✳ ●✐➯ sư t❤➟♠ ♠ét tr♦♥❣ ❤❛✐ ➤✐Ị✉ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✭❛✮ F ✭❜✮ (X, G, ) ❧➭ ❧✐➟♥ tơ❝✱ ❤♦➷❝ ❧➭ g ✲t❤ø tù ➤➬② ➤đ✳ ❚❛ ❝ị♥❣ ❣✐➯ sư r➺♥❣ tå♥ t➵✐ gy0 ✳ ◆Õ✉ k ∈ [0, 1)✱ (x, y) ∈ (X × X) t❤× F s❛♦ ❝❤♦ ✈➭ g x0 , y0 ∈ X gx0 F (x0 , y0 ) ✈➭ F (y0 , x0 ) ❝ã ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ♥❣❤Ü❛ ❧➭✱ tå♥ t➵✐ g(x) = F (x, y) ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö s❛♦ ❝❤♦ x0 , y0 ∈ X ✈➭ g(y) = F (y, x)✳ s❛♦ ❝❤♦ gx0 F (x0 , y0 ) ✈➭ F (y0 , x0 ) F (X × X) ⊂ g(X)✱ t❛ ❝ã t❤Ĩ ①➞② ❞ù♥❣ ❤❛✐ ❞➲② {xn } ✈➭ {yn } tr♦♥❣ X gy0 ✳ ❱× t❤❡♦ ❝➳❝❤ s❛✉ ➤➞②✿ gxn+1 = F (xn , yn ), ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ✈í✐ ♠ä✐ gxn ❱× gx0 gx0 F (x0 , y0 )✱ F (y0 , x0 ) gx1 ✈➭ gy1 sè tù ♥❤✐➟♥ gyn+1 = F (yn , xn ), n ∈ N ✭✷✳✸✽✮ n ≥ 0✱ t❛ ❝ã gxn+1 ✈➭ gyn ✭✷✳✸✾✮ gyn+1 gy0 ✱ gx1 = F (x0 , y0 ) ✈➭ F (y0 , x0 ) = gy1 ✱ ♥➟♥ t❛ ❝ã gy0 ✱ ♥❣❤Ü❛ ❧➭✱ ✭✷✳✸✾✮ ➤ó♥❣ ✈í✐ n = 0✳ ●✐➯ sư r➺♥❣ ✭✷✳✸✾✮ ➤ó♥❣ ✈í✐ n > ♥➭♦ ➤ã✳ ❱× F ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ö✉ tré♥✱ ♥❤ê ➤➻♥❣ t❤ø❝ ✭✷✳✸✽✮ t❛ ❝ã 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 ✭✷✳✹✶✮ ✈➭ ❇➺♥❣ q✉② ♥➵♣ t♦➳♥ ❤ä❝✱ t❛ s✉② r❛ ✭✷✳✸✾✮ ➤ó♥❣ ✈í✐ ♠ä✐ gx0 gx1 gx2 gxn ✷✽ gxn+1 n ≥ 0✱ ♥❣❤Ü❛ ❧➭✱ gxn+2 , ✭✷✳✹✷✮ ✈➭ gy0 ◆Õ✉ tå♥ t➵✐ n0 ∈ N gy1 gy2 s❛♦ ❝❤♦ gyn gyn+1 ✭✷✳✹✸✮ gyn+2 (gxn0 +1 , gyn0 +1 ) = (gxn0 , gyn0 )✱ t❤× F ✈➭ g ❝ã ♠ét ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✳ ❚❤❐t ✈❐②✱ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ➤ã t❛ sÏ ❝ã (gxn0 +1 , gyn0 +1 ) = (F (xn0 , yn0 ), F (yn0 , xn0 )) = (gxn0 , gyn0 ) ⇐⇒ ❚❛ ❣✐➯ sö r➺♥❣ r➺♥❣ ❤♦➷❝ ❱í✐ ♠ä✐ F (xn0 , yn0 ) = gxn0 (gxn+1 , gyn+1 ) = (gxn , gyn ) ✈➭ F (yn0 , xn0 ) = gyn0 ✈í✐ ♠ä✐ n ∈ N✳ ❈❤Ý♥❤ ①➳❝ ❤➡♥✱ t❛ ❣✐➯ sö gxn+1 = F (xn , yn ) = gxn ✱ ❤♦➷❝ gyn+1 = F (yn , xn ) = gyn ✳ n ∈ N✱ t❛ ➤➷t tn = G(gxn+1 , gxn+1 , gxn ) + G(gyn+1 , gyn+1 , gyn ) ❑❤✐ ➤ã✱ ♥❤ê ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✸✼✮ ỗ n N t ó tn = G(gxn+1 , gxn+1 , gxn ) + G(gyn+1 , gyn+1 , gyn ) = G (F (xn , yn ), F (xn , yn ), F (xn−1 , yn−1 )) + G (F (yn , xn ), F (yn , xn ), F (yn−1 , xn−1 )) ≤ k [G(gxn , gxn , gxn−1 ) + G(gyn , gyn , gyn−1 )] = ktn−1 ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ r➺♥❣ tn ≤ k n t0 , ❇➞② ❣✐ê✱ ✈í✐ ♠ä✐ ♥❣❤Ü❛ m, n ∈ N ♠➭ m > n✱ n ∈ N ✭✷✳✹✹✮ ♥❤ê ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ✭●✺✮ ❝đ❛ ➤Þ♥❤ G✲♠➟tr✐❝ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✹✹✮ t❛ t❤✉ ➤➢ỵ❝ G(gxm , gxm , gxn ) + G(gym , gym , gyn ) = G (gxn , gxm , gxm ) + G (gyn , gym , gym ) ≤ G(gxn , gxn+1 , gxn+1 ) + G(gxn+1 , gxm , gxm )+ + G(gyn , gyn+1 , gyn+1 ) + G(gyn+1 , gym , gym ) ≤ G(gxn , gxn+1 , gxn+1 ) + G(gxn+1 , gxn+2 , gxn+2 ) + G(gxn+2 , gxm , gxm )+ + G(gyn , gyn+1 , gyn+1 ) + G(gyn+1 , gyn+2 , gyn+2 ) + G(gyn+2 , gym , gym ) ✳✳ ✳ ≤ G(gxn , gxn+1 , gxn+1 ) + G(gxn+1 , gxn+2 , gxn+2 ) + + G(gxm−1 , gxm , gxm )+ + G(gyn , gyn+1 , gyn+1 ) + G(gyn+1 , gyn+2 , gyn+2 ) + + G(gym−1 , gym , gym ) = tn + tn+1 + + tm−1 ≤ (k n + k n+1 + + k m−1 ) t0 ≤ kn 1−k t0 ✷✾ ➜✐Ò✉ ♥➭② ❞➱♥ ➤Õ♥ r➺♥❣ lim [G(gxn , gxm , gxm ) + G(gyn , gym , gym )] = n,m→∞ ❑❤✐ ➤ã✱ t❤❡♦ ▼Ö♥❤ ➤Ị ✶✳✶✳✼✱ t❛ ❦Õt ❧✉❐♥ ➤➢ỵ❝ r➺♥❣ ❤❛✐ ❞➲② ❞➲② {gxn } ✈➭ {gyn } ❧➭ ❝➳❝ G✲❈❛✉❝❤②✳ ▲➢✉ ý r➺♥❣ ❧➭ ❝➳❝ g(X) ❧➭ G✲➤➬② ➤ñ✱ ♥➟♥ tå♥ t➵✐ x, y ∈ g(X) s❛♦ ❝❤♦ {gxn } ✈➭ {gyn } G✲❤é✐ tơ ✈Ị x ✈➭ y t➢➡♥❣ ø♥❣✱ ♥❣❤Ü❛ ❧➭✱ lim F (xn , yn ) = lim gxn+1 = x, n→∞ n→∞ ✭✷✳✹✺✮ lim F (yn , xn ) = lim gyn+1 = y n→∞ ❱× F ✈➭ g ❧➭ ❤❛✐ ➳♥❤ ①➵ n→∞ G✲t➢➡♥❣ t❤Ý❝❤✱ ♥❤ê ➤➻♥❣ t❤ø❝ ✭✷✳✹✺✮✱ t❛ ❝ã lim G(gF (xn , yn ), F (gxn , gyn ), F (gxn , gyn )) = 0, n→∞ ✭✷✳✹✻✮ lim G(gF (yn , xn ), F (gyn , gxn ), F (gyn , gxn )) = n→∞ ●✐➯ sư r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭❛✮ tr♦♥❣ ❣✐➯ t❤✐Õt ❧➭ ①➯② r❛✳ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ n > 0✱ t❛ ❝ã G (gx, F (gxn , gyn ), F (gxn , gyn )) + G (gy, F (gyn , gxn ), F (gyn , gxn )) ≤ G (gx, gF (xn , yn ), gF (xn , yn )) + G (gF (xn , yn ), F (gxn , gyn ), F (gxn , gyn )) + + G (gy, gF (yn , xn ), gF (yn , xn )) + G (gF (yn , xn ), F (gyn , gxn ), F (gyn , gxn )) ✭✷✳✹✼✮ ❈❤♦ n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥✱ ♥❤ê ❤❛✐ ➤➻♥❣ t❤ø❝ ✭✷✳✹✺✮✱ ✭✷✳✹✻✮ ✈➭ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ F ✈➭ g ✱ t❛ ❝ã lim [G (gx, F (x, y), F (x, y)) + G (gy, F (y, x), F (y, x))] = n→∞ ❉♦ ➤ã✱ t❛ s✉② r❛ r➺♥❣ gx = F (x, y) ✈➭ gy = F (y, x)✱ ♥❣❤Ü❛ ❧➭✱ (x, y) ∈ X × X ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ F ✈➭ ❧➭ ♠ét g✳ ❇➞② ❣✐ê ❣✐➯ sö r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭❜✮ tr♦♥❣ ❣✐➯ t❤✐Õt ❧➭ ①➯② r❛✳ ❑❤✐ ➤ã✱ ♥❤ê ✭✷✳✹✷✮✱ ✭✷✳✹✸✮ ✈➭ ✭✷✳✹✺✮✱ t❛ ❝ã ggx ❉♦ F ✈➭ g ❧➭ ❤❛✐ ➳♥❤ ①➵ G✲t➢➡♥❣ gx ✈➭ ggy t❤Ý❝❤ ✈➭ g gy ✭✷✳✹✽✮ ❧➭ ❧✐➟♥ tô❝✱ ♥❤ê ➤➻♥❣ t❤ø❝ ✭✷✳✹✺✮ ✈➭ ✭✷✳✹✻✮✱ t❛ ❝ã lim ggxn = gx = lim gF (xn , yn ) = lim F (gxn , gyn ), n→∞ n→∞ n→∞ ✸✵ ✭✷✳✹✾✮ lim ggyn = gy = lim gF (yn , xn ) = lim F (gyn , gxn ) n→∞ n→∞ ✭✷✳✺✵✮ n→∞ ❚õ ❝➳❝ ➤➻♥❣ t❤ø❝ ✭✷✳✹✾✮ ✈➭ ✭✷✳✺✵✮✱ ✈➭ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ G(gx, F (x, y), F (x, y)) + G(gy, F (y, x), F (y, x)) ≤ G(gx, ggxn+1 , ggxn+1 ) + G(ggxn+1 , F (x, y), F (x, y))+ ✭✷✳✺✶✮ + G(gy, ggyn+1 , ggyn+1 ) + G(ggyn+1 , F (y, x), F (y, x)) = G(gx, ggxn+1 , ggxn+1 ) + G(gF (xn , yn ), F (x, y), F (x, y))+ + G(gy, ggyn+1 , ggyn+1 ) + G(gF (yn , xn ), F (y, x), F (y, x)), ❜➺♥❣ ❝➳❝❤ ❝❤♦ n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥✱ ♥❤ê tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ g ✈➭ ✭✷✳✹✾✮✱ ✭✷✳✺✵✮✱ t❛ ❦Õt ❧✉❐♥ ➤➢ỵ❝ r➺♥❣ ≤ G (gx, F (x, y), F (x, y)) + G (gy, F (y, x), F (y, x)) ≤ ◆❤ê tÝ♥❤ ❝❤✃t ✭●✶✮ ❝đ❛ ➤Þ♥❤ ♥❣❤Ü❛ ❉♦ ➤ã✱ ♣❤➬♥ tư ✈➭ (x, y) ∈ X × X G✲♠➟tr✐❝✱ t❛ ❝ã gx = F (x, y) ✭✷✳✺✷✮ ✈➭ gy = F (y, x)✳ ❧➭ ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❤❛✐ ➳♥❤ ①➵ g✳ ✭❬✾❪✮ ❈❤♦ ❍Ư q✉➯✳ ✷✳✷✳✸ ❦❤➠♥❣ ❣✐❛♥ g:X→X G✲♠➟tr✐❝ (X, ) ❧➭ s❛♦ ❝❤♦ ♠ét t❐♣ ❤ỵ♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ (X, G) ❧➭ ❤❛✐ ➳♥❤ ①➵ s❛♦ ❝❤♦ F ❧➭ G✲➤➬② ➤đ✳ ●✐➯ sư ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ (X, G) ❧➭ ➤✐Ö✉ tré♥ tr➟♥ X ❧➭ x, y, u, v ∈ X G✲❧✐➟♥ tô❝✱ F ♠➭ ✈➭ g gx gu, gy ❧➭ ❝➳❝ ➳♥❤ ①➵ gv ✳ ●✐➯ sö r➺♥❣ G✲t➢➡♥❣ t❤Ý❝❤✳ ✈➭ ✈➭ ✭✷✳✺✸✮ ≤ k [G(gx, gu, gu) + G(gy, gv, gv)] ✈í✐ ♠ä✐ ♠ét F : X×X → X G(F (x, y), F (u, v), F (u, v)) + G(F (y, x), F (v, u), F (v, u)) g F F (X × X) ⊂ g(X)✱ ➳♥❤ ①➵ ●✐➯ sư ♠ét tr♦♥❣ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✭❛✮ F ✭❜✮ (X, G, ) ❧➭ ❧✐➟♥ tơ❝✱ ❤♦➷❝ ❧➭ g ✲t❤ø tù ➤➬② ➤đ✳ ❚❛ ❝ị♥❣ ❣✐➯ sö r➺♥❣ tå♥ t➵✐ ◆Õ✉ k ∈ [0, 1)✱ t❤× F ✈➭ g x0 , y0 ∈ X s❛♦ ❝❤♦ gx0 F (x0 , y0 ) ✈➭ gy0 F (y0 , x0 )✳ ❝ã ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✷✱ ❧✃② ❝đ❛ ❤Ư q✉➯✳ ✸✶ z =u ✈➭ w =v t❛ ❝ã ♥❣❛② ❦Õt q✉➯ ✭❬✾❪✮ ❈❤♦ ❍Ö q✉➯✳ ✷✳✷✳✹ ❦❤➠♥❣ ❣✐❛♥ g:X→X G✲♠➟tr✐❝ (X, ) ❧➭ s❛♦ ❝❤♦ ♠ét t❐♣ ❤ỵ♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ (X, G) ❧➭ ❤❛✐ ➳♥❤ ①➵ s❛♦ ❝❤♦ F ❧➭ G✲➤➬② ➤đ✳ ●✐➯ sư ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ (X, G) ❧➭ F : X×X → X ➤✐Ö✉ tré♥ tr➟♥ X G(F (x, y), F (u, v), F (w, z)) + G(F (y, x), F (v, u), F (v, u)) g(X)✱ x, y, u, v, z, w ∈ X ➳♥❤ ①➵ g ❧➭ ♠➭ G✲❧✐➟♥ gx gu gw, gy gv F✳ tơ❝ ✈➭ ❣✐❛♦ ❤♦➳♥ ✈í✐ ✈➭ ✈➭ ✭✷✳✺✹✮ ≤ k [G(gx, gu, gw) + G(gy, gv, gz)] ✈í✐ ♠ä✐ ♠ét gz ✳ ●✐➯ sư r➺♥❣ F (X × X) ⊂ ●✐➯ sư r➺♥❣ ♠ét tr♦♥❣ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✭❛✮ F ✭❜✮ (X, G, ) ❧➭ ❧✐➟♥ tô❝✱ ❤♦➷❝ ❧➭ g ✲t❤ø tù ➤➬② ➤đ✳ ●✐➯ sư t❤➟♠ r➺♥❣ tå♥ t➵✐ ◆Õ✉ k ∈ [0, 1)✱ t❤× F ✈➭ g g ❈❤ø♥❣ ♠✐♥❤✳ ❱× x0 , y0 ∈ X s❛♦ ❝❤♦ gx0 F (x0 , y0 ) ✈➭ gy0 F (y0 , x0 )✳ ❝ã ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✳ ❣✐❛♦ ❤♦➳♥ ✈í✐ F✱ ♥➟♥ F ✈➭ g ❧➭ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤✳ ❱× t❤Õ✱ ❦Õt q✉➯ ❝đ❛ ❤Ư q✉➯ ợ s r từ ị ý ệ q✉➯✳ ✷✳✷✳✺ ❦❤➠♥❣ ❣✐❛♥ g:X→X G✲♠➟tr✐❝ (X, ) ❧➭ s❛♦ ❝❤♦ ♠ét t❐♣ ❤ỵ♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ (X, G) ❧➭ ❤❛✐ ➳♥❤ ①➵ s❛♦ ❝❤♦ F ❧➭ G✲➤➬② ➤đ✳ ●✐➯ sư ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ (X, G) ❧➭ F : X×X → X ➤✐Ư✉ tré♥ tr➟♥ X G(F (x, y), F (u, v), F (u, v)) + G(F (y, x), F (v, u), F (v, u)) g ❧➭ x, y, u, v ∈ X G✲❧✐➟♥ ♠➭ gx gu, gy tơ❝ ✈➭ ❣✐❛♦ ❤♦➳♥ ✈í✐ F✳ gv ✳ ●✐➯ sö r➺♥❣ ✈➭ ✈➭ ✭✷✳✺✺✮ ≤ k [G(gx, gu, gu) + G(gy, gv, gv)] ✈í✐ ♠ä✐ ♠ét F (X × X) ⊂ g(X)✱ ➳♥❤ ①➵ ●✐➯ sö r➺♥❣ ♠ét tr♦♥❣ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✭❛✮ F ✭❜✮ (X, G, ) ❧➭ ❧✐➟♥ tô❝✱ ❤♦➷❝ ❧➭ g ✲t❤ø tù ➤➬② ➤đ✳ ❚❛ ❝ị♥❣ ❣✐➯ sư r➺♥❣ tå♥ t➵✐ ◆Õ✉ k ∈ [0, 1)✱ t❤× F ✈➭ g x0 , y0 ∈ X s❛♦ ❝❤♦ gx0 F (x0 , y0 ) ❝ã ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✳ ✸✷ ✈➭ gy0 F (y0 , x0 )✳ g ❈❤ø♥❣ ♠✐♥❤✳ ❱× ❣✐❛♦ ❤♦➳♥ ✈í✐ F✱ ♥➟♥ F ✈➭ g ❧➭ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤✳ ❱× t❤Õ✱ ❦Õt q✉➯ ❝đ❛ ❤Ư q✉➯ ➤➢ỵ❝ s✉② r❛ tõ ❍Ư q✉➯ ✷✳✷✳✸✳ ◆Õ✉ t❛ ❧✃② g=I ✭➳♥❤ ①➵ ➤å♥❣ ♥❤✃t tr➟♥ X✮ tr ị ý ệ q tì t t❤✉ ➤➢ỵ❝ ❦Õt q✉➯ ❞➢í✐ ➤➞②✳ ✭❬✾❪✮ ❈❤♦ ❍Ư q✉➯✳ ✷✳✷✳✻ ❦❤➠♥❣ ❣✐❛♥ g:X→X G✲♠➟tr✐❝ (X, ) ❧➭ s❛♦ ❝❤♦ ♠ét t❐♣ ❤ỵ♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ (X, G) ❧➭ ❤❛✐ ➳♥❤ ①➵ s❛♦ ❝❤♦ F ❧➭ G✲➤➬② ➤ñ✳ ●✐➯ sö ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ (X, G) ❧➭ F : X×X → X ➤✐Ư✉ tré♥ tr➟♥ X G(F (x, y), F (u, v), F (w, z)) + G(F (y, x), F (v, u), F (v, u)) x, y, u, v, z, w ∈ X ♠➭ x u w, y v z✳ ✈➭ ✈➭ ✭✷✳✺✻✮ ≤ k [G(x, u, w) + G(y, v, z)] ✈í✐ ♠ä✐ ♠ét ●✐➯ sư r➺♥❣ ♠ét tr♦♥❣ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✭❛✮ F ✭❜✮ (X, G, ) ❧➭ ❧✐➟♥ tô❝✱ ❤♦➷❝ ❧➭ g ✲t❤ø tù ➤➬② ➤đ✳ ❚❛ ❝ị♥❣ ❣✐➯ sư r➺♥❣ tå♥ t➵✐ ◆Õ✉ k ∈ [0, 1)✱ t❤× ❦❤➠♥❣ ❣✐❛♥ g:X→X ✈➭ g ✭❬✾❪✮ ❈❤♦ ❍Ö q✉➯✳ ✷✳✷✳✼ F G✲♠➟tr✐❝ x0 , y0 ∈ X s❛♦ ❝❤♦ gx0 F (x0 , y0 ) ✈➭ gy0 ❝ã ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✳ (X, ) ❧➭ s❛♦ ❝❤♦ ♠ét t❐♣ ❤ỵ♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ (X, G) ❧➭ ❤❛✐ ➳♥❤ ①➵ s❛♦ ❝❤♦ F ❧➭ G✲➤➬② ➤đ✳ ●✐➯ sư ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ (X, G) ❧➭ ➤✐Ö✉ tré♥ tr➟♥ ♠➭ x u, y v✳ X ✈➭ ✈➭ ✭✷✳✺✼✮ ≤ k [G(x, u, u) + G(y, v, v)] x, y, u, v ∈ X ♠ét F : X×X → X G(F (x, y), F (u, v), F (u, v)) + G(F (y, x), F (v, u), F (v, u)) ✈í✐ ♠ä✐ F (y0 , x0 )✳ ●✐➯ sư r➺♥❣ ♠ét tr♦♥❣ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✭❛✮ F ✭❜✮ (X, G, ) ❧➭ ❧✐➟♥ tô❝ ❤♦➷❝ ❧➭ g ✲t❤ø tù ➤➬② ➤đ✳ ❚❛ ❣✐➯ sư t❤➟♠ r➺♥❣ tå♥ t➵✐ ◆Õ✉ k ∈ [0, 1)✱ t❤× F ✈➭ g x0 , y0 ∈ X s❛♦ ❝❤♦ gx0 F (x0 , y0 ) ❝ã ♠ét ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✳ ✸✸ ✈➭ gy0 F (y0 , x0 )✳ ✷✳✷✳✽ ❱Ý ❞ơ✳ ❉ï♥❣ ❝➳❝ t❐♣ ❤ỵ♣ ✈➭ ➳♥❤ ①➵ tr♦♥❣ ❱Ý ❞ô ✷✳✷✳✶✱ t❛ ❝ã G(F (x, y), F (u, v), F (z, w)) + G(F (y, x), F (v, u), F (w, z))     5 5 5 1 = G  x + y, u + v, z + w + G  y + x, v + u, w + z  8 8 8 8 8 8 ≤ [(|u − x| + |z − x| + |z − u|) + (|v − y| + |w − y| + |w − v|)] ✭✷✳✺✽✮ ✈➭  G(gx, gu, gz) + G(gy, gv, gw) = G     7x 7u 7z 7y 7v w , ,  + G , ,  8 8 8 ✭✷✳✺✾✮ = [(|u − x| + |z − x| + |z − u|)+ + (|v − y| + |w − y| + |w − v|)] ❚õ ➤ã✱ t❛ t❤✃② r➺♥❣ ❜✃t ❦× k ∈ [ 76 , 1) ➤Ị✉ t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✸✼✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✷✳ ❈❤ó ý r➺♥❣ (0, 0) ❧➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❞✉② ♥❤✃t ❝đ❛ F ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ♥❣❤Ü❛ ❧➭✱ ✷✳✷✳✾ ❱Ý ❞ơ✳ ●✐➯ sư ✈➭ g ✈➭ ❝ò♥❣ ❧➭ F (0, 0) = g0 = X = R ị G : X ì X × X → [0, ∞) ❝❤♦ ❜ë✐ G(x, y, z) = |x − y| + |x − z| + |y − z| ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ♠ét ❦❤➠♥❣ ❣✐❛♥ ●✐➯ sư ❧➭ q✉❛♥ ❤Ư t❤ø tù t❤➠♥❣ t❤➢ê♥❣✳ ❑❤✐ ➤ã✱ (X, G) ❧➭ G✲♠➟tr✐❝✳ F (x, y) = x3 + y ✈➭ g : X → X ❝❤♦ 8 ❜ë✐ g(x) = x ✈í✐ ♠ä✐ x, y ∈ X ✳ ❑❤✐ ➤ã✱ F (X × X) = X = g(X)✳ ❚❛ ♥❤❐♥ t❤✃② r➺♥❣ ❳➳❝ ➤Þ♥❤ ➳♥❤ ①➵ F : X ×X → X ❝❤♦ ❜ë✐ G(F (x, y), F (u, v), F (z, w)) + G(F (y, x), F (v, u), F (v, u))     3 3 3 3 3 = G x + y , u + v , z + w  + G y + x , v + u , w + z  8 8 8 8 8 8 5 1 = |v − y | + |w3 − y | + |w3 − v | + |u3 − x3 | + |z − x3 | + |z − u3 |+ 8 8 8 1 5 + |v − y | + |w3 − y | + |w3 − v | + |u3 − x3 | + |z − x3 | + |z − u3 | 8 8 8 = (|v − y | + |w3 − y | + |w3 − v | + |u3 − x3 | + |z − x3 | + |z − u3 |) ✸✹ ✈➭ G(gx, gu, gz) + G(gy, gv, gw) = G(x3 , u3 , z ) + G(y , v , w3 ) = (|x3 − u3 | + |x3 − z | + |u3 − z |)+ ✭✷✳✻✵✮ + (|y − v | + |y − w3 | + |v − w3 |) ❑❤✐ ó t tứ ủ ị í ợ t❤á❛ ♠➲♥ ✈í✐ ❜✃t ❦× ✈➭ k ∈ ( 43 , 1) (0, 0) ❧➭ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✳ ❈❤ó ý r➺♥❣ ♥Õ✉ t❛ t❤❛② ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✼✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✷ ❜➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✺✮ ❝đ❛ ➜Þ♥❤ ❧ý ✶✳✶✳✷✶✱ ♥❣❤Ü❛ ❧➭✱ G(F (x, y), F (u, v), F (w, z)) ≤ k[G(gx, gu, gw) + G(gy, gv, gz)], tr♦♥❣ ➤ã k ∈ [0, 21 )✱ ✭✷✳✻✶✮ t❤× ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ tå♥ t➵✐ ♠➷❝ ❞ï ➤✐Ị✉ ❦✐Ư♥ ❝♦ rót ❦❤➠♥❣ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ x = u = z ✳ ❑❤✐ ➤ã✱ t❛ ❝ã   3 3 G(F (x, y), F (u, v), F (z, w)) = G  x + y , u + v , z + w  8 8 8 ❈❤Ý♥❤ ①➳❝ ❤➡♥✱ ①Ðt = ✭✷✳✻✷✮ 5 |v − y | + |w3 − y | + |w3 − v | 8 ✈➭ G(gx, gu, gz) + G(gy, gv, gw) = G (x3 , u3 , z ) + G (y , v , w3 ) = |y − v | + |y − w3 | + |v − w3 | ❘â r➭♥❣ r➺♥❣ ➤✐Ò✉ ❦✐Ư♥ ✭✷✳✻✶✮ ➤ó♥❣ ✈í✐ k> ✸✺ ✳ ✭✷✳✻✸✮ ❑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❛✉ ✶✳ ❍Ư t❤è♥❣ ❤ã❛ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ị ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ t❤ø tù ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ G✲❧✐➟♥ G✲♠➟tr✐❝ ➤è✐ ①ø♥❣✱ ❞➲② G✲❤é✐ tơ✱ ❞➲② G✲❈❛✉❝❤②✱ ➳♥❤ ①➵ tơ❝✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ❝➳❝ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ❣✐❛♦ ❤♦➳♥✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t tré♥✱ ❝➳❝ ➳♥❤ ①➵ g ✲➤➡♥ ➤✐Ư✉ G✲t➢➡♥❣ t❤Ý❝❤✳ ✷✳ ❚r×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥✱ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ g ✲➤➡♥ ➤✐Ö✉ tré♥ G✲♠➟tr✐❝ t❤ø tù ❜é ♣❤❐♥✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ➤Þ♥❤ ❧ý ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ứ ứ ò s ợ ✹✳ ❚r×♥❤ ❜➭② ❝❤✐ t✐Õt ❱Ý ❞ơ ✶✳✷✳✺ ➤Ĩ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✶✳✷✳✷ ✈Ị ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ❱Ý ❞ơ ✶✳✷✳✻ ➤Ĩ ♠✐♥❤ ❤ä❛ ❝❤♦ ❍Ư q✉➯ ✶✳✷✳✸✱ ❱Ý ❞ơ ✷✳✶✳✽ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✶✳✼ ✈Ị ➤✐Ị✉ ❦✐Ư♥ ➤Ĩ ♠ét ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ❱Ý ❞ơ ✷✳✷✳✽ ➤Ĩ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✷✳✷ ✈Ị ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❤❛✐ ➳♥❤ ①➵✳ ✸✻ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ▼✳ ❆❜❜❛s✱ P✳ ❑✉♠❛♠ ✭✷✵✶✷✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥tr❛❝✲ t✐✈❡ ♠❛♣♣✐♥❣s ♦♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ G✲♠❡tr✐❝ s♣❛❝❡s✱❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ❱♦❧✳ ✷✵✶✷✱ ✸✶✱ ✶✹ ♣❛❣❡s✱ ❞♦✐✿✶✵✳✶✶✽✻✴✶✻✽✼✲✶✽✶✷✲✷✵✶✷✲✸✶✳ ❬✷❪ ❚✳ ●✳ ❇❤❛s❦❛r✱ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ✭✷✵✵✻✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ♣❛r✲ t✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ✻✺✱ ✶✸✼✾✲✶✸✾✸✳ ❬✸❪ ❇✳ ❙✳ ❈❤♦✉❞❤✉r②✱ ❆✳ ❑✉♥❞✉ ✭✷✵✶✵✮✱ ❆ ❝♦✉♣❧❡❞ ❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t r❡s✉❧t ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❢♦r ❝♦♠♦❛t✐❜❧❡ ♠❛♣♣✐♥❣s✱ ◆♦❧✐♥❡❛r ❆♥❛❧✳✱ ✼✸✱ ✷✺✷✹✲✷✺✸✶✳ ❬✹❪ ❇✳❙✳ ❈❤♦✉❞❤✉r②✱ P✳ ▼❛✐t② ✭✷✵✶✶✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t r❡s✉❧ts ✐♥ ❣❡♥❡r❛❧✲ ✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ▼❛t❤✳ ❈♦♠♣✉t✳ ▼♦❞❡❧✳✱ ✺✹✱ ✼✸✲✼✾✳ ❬✺❪ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠✱ ▲✳ ❈✐r✐❝ ✭✷✵✵✾✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♥♦♥❧✐♥❡❛r ❝♦♥tr❛❝t✐♦♥s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ✼✵ ✭✶✷✮✱ ✹✸✹✶✲✹✸✹✾✳ ❬✻❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❇✳ ❙✐♠s ✭✷✵✵✻✮✱ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❏♦✉r♥❛❧ ♦❢ ◆♦♥❧✐♥❡❛r ❛♥❞ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ ✼ ✭✷✮✱ ✷✽✾✲✷✾✼✳ ❬✼❪ ❍✳ ❑✳ ◆❛s❤✐♥❡ ✭✷✵✶✷✮✱ ❈♦✉♣❧❡❞ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t r❡s✉❧ts ✐♥ ♦r❞❡r❡❞ G✲♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✶✱ ✶✲✶✸✳ ❬✽❪ ❲✳ ❙❤❛t❛♥❛✇✐ ✭✷✵✶✶✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❍❛❝❡tt❡♣❡ ❏✳ ▼❛t❤✳ ❙t❛t✐st✐❝s✱ ✹✵ ✭✸✮✱ ✹✹✶✲✹✹✼✳ ❬✾❪ ❊✳ ❑❛r❛♣✐♥❛r ✭✷✵✶✷✮✱ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ♦♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ G✲♠❡tr✐❝ s♣❛❝❡s✱ ❆♣♣❧✳✱ ❱♦❧✳ ✷✵✶✷✱ ✶✼✹✱ ✶✸ ♣❛❣❡s✱ ❞♦✐✿✶✵✳✶✶✽✻✴✶✻✽✼✲✶✽✶✷✲ ✷✵✶✷✲✶✼✹✳ ✸✼ ... ➤➻♥❣ t❤ø❝ ♥➭② t❛ t❤✉ ➤➢ỵ❝ G( gx, gy, gy) + G( gy, gx, gx) ≤ 2k [G( gx, gy, gy) + G( gy, gx, gx)] ❱× 2k < 1✱ ♥➟♥ t❛ ❝ã G( gx, gy, gy) + G( gy, gx, gx) < G( gx, gy, gy) + G( gy, gx, gx), ➤✐Ị✉ ♥➭② ❧➭ ♠ét ♠➞✉... G( gyn , gyn+1 , gyn+1 ) + G( gyn+1 , gym , gym ) ≤ G( gxn , gxn+1 , gxn+1 ) + G( gxn+1 , gxn+2 , gxn+2 ) + G( gxn+2 , gxm , gxm )+ + G( gyn , gyn+1 , gyn+1 ) + G( gyn+1 , gyn+2 , gyn+2 ) + G( gyn+2 , gym... ➤Þ♥❤ G? ??♠➟tr✐❝ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✹✹✮ t❛ t❤✉ ➤➢ỵ❝ G( gxm , gxm , gxn ) + G( gym , gym , gyn ) = G (gxn , gxm , gxm ) + G (gyn , gym , gym ) ≤ G( gxn , gxn+1 , gxn+1 ) + G( gxn+1 , gxm , gxm )+ + G( gyn