Điểm bất động của các ánh xạ co suy rộng trong không gian g mêtric

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❍♦➭♥❣ ❍➢♥❣ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✻ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ô♥ ❍♦➭♥❣ ❍➢♥❣ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✻ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✐ ▼ë ➤➬✉ ✐✐ ❈❤➢➡♥❣ ✶✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ Φ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲ ♠➟tr✐❝ ✶ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ❈❤➢➡♥❣ ✷✳ Φ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✶✹ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❣✐❛♥ ✷✳✷ ✻ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✷✳✶ ✶ G✲♠➟tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❣✐❛♥ G✲♠➟tr✐❝ G✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ α✲❝♦ ✶✹ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ❑Õt ❧✉❐♥ ✸✷ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✸ ✐ ▼ë ➤➬✉ ✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ò t➭✐ ❚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♦♥❣ ♥❤÷♥❣ ❝➠♥❣ ❝ô ❝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♦♥❣ ✈✐Ö❝ ①➞② ❞ù♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ tr♦♥❣ t♦➳♥ ❤ä❝ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ✈✃♥ ➤Ị tr♦♥❣ t♦➳♥ ❤ä❝ ø♥❣ ❞ơ♥❣ ✈➭ ❦❤♦❛ ❤ä❝✳ ❱× ✈❐②✱ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ➤➲ ❧➠✐ ❝✉è♥ ♠ét sè ❧➢ỵ♥❣ ❧í♥ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❝ị♥❣ ❧➭ ➤✐Ị✉ ❞Ơ ❤✐Ĩ✉✳ ▼ét sè ♠ë ré♥❣ ❝đ❛ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➲ ➤➢ỵ❝ ➤Ị ①✉✃t ❜ë✐ ♠ét sè t➳❝ ❣✐➯✳ ◆➝♠ ✶✾✾✼✱ ❨✳ ■✳ ❆❧❜❡r ✈➭ ❙✳ ●✉❡rr❡ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ➳♥❤ ①➵ α✲❝♦ ②Õ✉ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➭ t❤✐Õt ❧❐♣ ♠ét ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤♦ ❧í♣ ➳♥❤ ①➵ ➤ã✳ ❙❛✉ ➤ã ♥➝♠ ✷✵✵✶✱ ❇✳ ❊✳ ❘❤♦❛❞❡s ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❦❤➳✐ ệ ế t ợ ột ị ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ❙❛✉ ➤ã✱ ♥❤✐Ị✉ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ♥➭♦ ➤ã ➤➲ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ❜ë✐ ♥❤✐Ị✉ t➳❝ ❣✐➯ ♥❤➢✿ ■✳ ❇❡❣ ✈➭ ▼✳ ❆❜❜❛s ✭✷✵✵✻✮✱ P✳ ◆✳ ❉✉tt❛ ✈➭ ❇✳ ❙✳ ❈❤♦✉❞❤✉r② ✭✷✵✵✽✮✱ ❲✳ ❙❤❛t❛♥❛✇✐ ✭✷✵✶✵✮✱ ❍✳ ❆②❞✐ ✈➭ ❝➳❝ ❝é♥❣ sù ✭✷✵✵✶✮ ✈➭ ❖✳ ❩❤❛♥❣ ✈➭ ❨✳ ❙♦♥❣ ✭✷✵✵✾✮✳ ◆➝♠ ✶✾✻✻✱ ❙✳ ●❛❤❧❡r ➤➲ ❣✐í✐ t❤✐Ư✉ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ❉❤❛❣❡ ➤➲ ❣✐í✐ t❤✐Ư✉ ❝➳❝ ❦❤➳✐ ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ 2✲♠➟tr✐❝ ✈➭ ♥➝♠ ✶✾✾✷ ❇✳ ❈✳ D✲♠➟tr✐❝✳ ❙❛✉ ➤ã✱ ♥➝♠ ✷✵✵✻ ❩✳ ▼✉st❛❢❛ ✈➭ ❇✳ ❙✐♠s ➤➲ ❝❤Ø r❛ r➺♥❣ ❤➬✉ ❤Õt ❝➳❝ ❦Õt q✉➯ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❦❤➠♥❣ ❣✐❛♥ D✲♠➟tr✐❝ ❝ñ❛ ❇✳ ❈✳ ❉❤❛❣❡ ❧➭ ❦❤➠♥❣ ❝ß♥ ➤ó♥❣ ✈➭ ❤ä ➤➲ ❣✐í✐ t❤✐Ư✉ ♠ét ❦❤➳✐ ♥✐Ư♠ ♠í✐ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ♥❤✐Ị✉ ❦Õt q✉➯ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤♦ ❝➳❝ tù ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈í✐ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ♥➭♦ ➤ã✳ ❙❛✉ ➤ã✱ ♥❤✐Ị✉ t➳❝ ❣✐➯ ➤➲ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ❝➳❝ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ✐✐ ré♥❣✳ ●➬♥ ➤➞②✱ ❍✳ ❆②❞✐ ✈➭ ❝é♥❣ sù ➤➲ t❤✐Õt ❧❐♣ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝❤♦ ❤❛✐ tù ➳♥❤ ①➵ ❝➳❝❤ ❣✐➯ sö r➺♥❣ f f ❧➭ ♠ét ➳♥❤ ①➵ ✈➭ g tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ X ❜➺♥❣ G✲❝♦ ②Õ✉ s✉② ré♥❣ ❦✐Ó✉ A ✈➭ B ➤è✐ ✈í✐ g ✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳❚❙ ❚r➬♥ ❱➝♥ ➣♥ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐ ♥❣❤✐➟♥ ❝ø✉ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ G✲♠➟tr✐❝✧✳ ✷✳ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ✲ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ G✲♠➟tr✐❝✱ ❞➲② G✲ ➤➬② ➤đ✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ●✲♠➟tr✐❝✳ ✲ P❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ♠è✐ q ệ ữ ố tợ tr ị ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝➳❝ ➳♥❤ ①➵ φ✲❝♦✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ G✲❝♦ ②Õ✉ s✉② ré♥❣✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ α✲❝♦ ②Õ✉ s✉② ré♥❣ ✈➭ ♠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➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❜➺♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ♣❤➞♥ tÝ❝❤ tổ ợ s s qt ể trì ❜➭② ♠ét ❝➳❝❤ ❤Ö t❤è♥❣ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❝➳❝ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ➳♥❤ ①➵ Φ✲❝♦✱ G✲❝♦ ②Õ✉ s✉② ré♥❣✱ ➳♥❤ ①➵ α✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ✹✳ ▼ơ❝ ➤Ý❝❤ ♥❣❤✐➟♥ ❝ø✉ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❞➲② G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ●✲♠➟tr✐❝✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ✐✐✐ ①➵ ❦✐Ó✉ Φ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ G✲❝♦ α✲②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱✳✳✳ ✈➭ ❝❤♦ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ✺✳ ◆é✐ ❞✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ❦❤➠♥❣ ❣✐❛♥ φ✲❝♦ tr♦♥❣ G✲♠➟tr✐❝✳ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❈➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲❝♦ ②Õ✉ s✉② ré♥❣ α✲❝♦ ②Õ✉ s✉② ré♥❣ G✲♠➟tr✐❝✳ ❈➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ✻✳ ❈✃✉ tró❝ ❧✉❐♥ ✈➝♥ ▲✉❐♥ ✈➝♥ ❣å♠ ✷ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ❦❤➠♥❣ ❣✐❛♥ Φ✲❝♦ tr♦♥❣ G✲♠➟tr✐❝✳ ▼ơ❝ ✶ ♥❤➺♠ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s❛✉✳ ▼ơ❝ ✷ ♥❤➺♠ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ Φ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ②Õ✉ s✉② ré♥❣ Gtr ụ trì ột số ị ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ ✈➭ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ▼ơ❝ ✷ ♥❤➺♠ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ α✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ▲✉❐♥ ✈➝♥ ♥➭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 ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ tØ♥❤ ❚➞② ◆✐♥❤✱ ❇❛♥ ●✐➳♠ ❍✐Ö✉ ❚r➢ê♥❣ ✐✈ ❚❍P❚ ◗✉❛♥❣ ❚r✉♥❣✱ 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♥ ❍♦➭♥❣ ❍➢♥❣ ✈ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ Φ✲❝♦ G✲♠➟tr✐❝ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ◆é✐ ❞✉♥❣ ❣å♠✿ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❞➲② G✲❤é✐ tô✱ ❞➲② G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ❝➬♥ ❞ï♥❣ trì ề s ị ĩ ♠ét ♠➟tr✐❝ tr➟♥ ✭❬✶❪✮ ❈❤♦ t❐♣ ❤ỵ♣ X X = φ✱ ➳♥❤ ①➵ d : X × X → R ➤➢ỵ❝ ❣ä✐ ♥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 ✳ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❦Ý ❤✐Ö✉ ❧➭ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ ❣✐÷❛ ➤✐Ĩ♠ x ✈➭ ➤✐Ĩ♠ y ✳ ❚❐♣ X ✈➭ ✶✳✶✳✷ ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✼❪✮ ❈❤♦ X ột t rỗ G : X ì 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 ➤➻♥❣ ✈í✐ x = y✱ ✈í✐ z = y✱ t❤ø❝ t❛♠ ❣✐➳❝✮✳ G ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♠➟tr✐❝ s✉② ré♥❣✱ ❤❛② ❣ä♥ ❤➡♥ ❧➭ ♠ét G✲♠➟tr✐❝ tr➟♥ X ✱ ✈➭ ❝➷♣ (X, G) ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ❤➭♠ ✶ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ {xn } ❧➭ ♠ét ❞➲② ❝➳❝ ➤✐Ĩ♠ ❝đ❛ X ✳ ➜✐Ĩ♠ x ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐í✐ ❤➵♥ ❝đ❛ ❞➲② {xn } ♥Õ✉ lim G(x, xn , xm ) = 0✳ ▲ó❝ ➤ã t❛ ♥ã✐ r➺♥❣ ❞➲② {xn } ❧➭ G✲❤é✐ tô ✈Ị x✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✸ ✭❬✼❪✮ ❈❤♦ n,m→∞ xn → x tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ (X, G) ♥Õ✉ ✈í✐ ♠ä✐ ε > 0✱ tå♥ t➵✐ k ∈ N s❛♦ ❝❤♦ G(x, xn , xm ) < ε ✈í✐ ♠ä✐ m, n ≥ k ✳ ◆❤➢ ✈❐②✱ ✶✳✶✳✹ ✭❬✼❪✮ ❈❤♦ ▼Ö♥❤ ➤Ị✳ {xn } ⊆ X ✈➭ ➤✐Ĩ♠ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ✈í✐ ❞➲② x ∈ X ✱ ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ➤➞② ❧➭ t➢➡♥❣ ➤➢➡♥❣ {xn } ❧➭ G✲❤é✐ tơ ✈Ị x❀ ✭✶✮ ❉➲② ✭✷✮ G(xn , xn , x) → 0✱ ❦❤✐ n → ∞❀ ✭✸✮ G(xn , x, x) → 0✱ ❦❤✐ n → ∞❀ ✭✹✮ G(xm , xn , x) → 0✱ ❦❤✐ m, n → ∞✳ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❉➲② {xn } ⊆ X ➤➢ỵ❝ ❣ä✐ ❧➭ G✲❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ ε > tå♥ t➵✐ sè tù ♥❤✐➟♥ 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✐❝ (X, G) ➤➢ỵ❝ ❣ä✐ ❧➭ G✲➤➬② ủ Gtr ủ ế ỗ G✲❈❛✉❝❤② tr♦♥❣ (X, G) ❧➭ G✲❤é✐ tô tr♦♥❣ (X, G)✳ ✶✳✶✳✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✼ ▼Ư♥❤ ➤Ị✳ ✭❬✼❪✮ ❑❤➠♥❣ ❣✐❛♥ ✭❬✼❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ {xn } ❧➭ G✲❈❛✉❝❤②❀ ✭✶✮ ❉➲② ✭✷✮ ❱í✐ ♠ä✐ ε > 0✱ tå♥ t➵✐ sè k ∈N s❛♦ ❝❤♦ n, m ≥ k ❀ ✭✸✮ G(gn , gm , gm ) → ❦❤✐ m, n → ∞✳ ✷ G(xn , xm , xm ) < ε ✈í✐ ♠ä✐ (X, G) ✈➭ (X , G ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ f : (X, G) → (X , G )✳ ❑❤✐ ➤ã f ➤➢ỵ❝ ❣ä✐ ❧➭ G✲❧✐➟♥ tơ❝ t➵✐ ➤✐Ĩ♠ a ∈ X ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ✈í✐ ♠ä✐ ε > 0✱ tå♥ t➵✐ sè δ > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ 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, G) ✈➭ (X , G ) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ➳♥❤ ①➵ f : X → X ❧➭ G✲❧✐➟♥ tô❝ t➵✐ x ∈ X ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ♥ã ❧➭ G✲❧✐➟♥ tô❝ t❤❡♦ ❞➲② t➵✐ x✱ ♥❣❤Ü❛ ❧➭ ✈í✐ ♠ä✐ ❞➲② {xn } ⊂ X ❧➭ G✲❤é✐ tơ ➤Õ♥ x✱ t❛ ❝ã ❞➲② ❣✐➳ trÞ {f (xn )} ❧➭ G✲❤é✐ tơ ➤Õ♥ f (x)✳ ▼Ư♥❤ ➤Ị✳ ✶✳✶✳✾ ✭❬✼❪✮ ❈❤♦ ▼Ư♥❤ ➤Ị✳ ✶✳✶✳✶✵ ✭❬✼❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ❤➭♠ G(x, y, z) ❧✐➟♥ tô❝ ➤å♥❣ t❤ê✐ t❤❡♦ t✃t ❝➯ ✸ ❜✐Õ♥ ❝ñ❛ ♥ã✳ (R, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤➠♥❣ tờ ị GS : R ì R ì R → [0, +∞) ❝❤♦ ❜ë✐ ❱Ý ❞ô✳ ✶✳✶✳✶✶ ✭❬✼❪✮ ✶✮ ❈❤♦ Gs (x, y, z) = d (x, y) + d (y, z) + d (x, z) ❚❛ ①➳❝ ✭✶✳✶✮ x, y, z ∈ R✳ ❑❤✐ ➤ã✱ râ r➭♥❣ (R, Gs ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ✷✮ ❈❤♦ X = {a, b}✳ ❍➭♠ G tr➟♥ X × X × X → [0, +∞) ①➳❝ ➤Þ♥❤ ❜ë✐ ✈í✐ ♠ä✐ G (a, a, a) = G (b, b, b) = G (a, a, b) = 1, G (a, b, b) = ✭✶✳✷✮ G ❧➟♥ t♦➭♥ ❜é X × X × X ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ tÝ♥❤ ➤è✐ ①ø♥❣ ❝đ❛ ❝➳❝ ❜✐Õ♥ sè✳ ❑❤✐ ➤ã✱ râ r➭♥❣ r➺♥❣ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❚❛ ♠ë ré♥❣ ✶✳✶✳✶✷ ♠ä✐ ▼Ư♥❤ ➤Ị✳ x, y, z ✈➭ ✭❬✼❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ a ∈ X ✱ t❛ ❝ã G(x, y, z) = 0✱ t❤× 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✲♠➟tr✐❝✳ ❑❤✐ ➤ã✱ ✈í✐ ✈➭ ❞♦ ➤ã ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❣✐➯ t❤✐Õt r➺♥❣ ψ(G(gu, gu, f u)) ≤ ψ ψ t➝♥❣✱ ❝➠♥❣ t❤ø❝ ✭✷✳✶✻✮ trë t❤➭♥❤ G(gu, gu, f u) − φ(0, 2G(gu, gu, f u), 0) G(gu, gu, f u) = ✈➭ ❞♦ ➤ã f u = gu = t✳ ❱× t❤Õ✱ u ❧➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❝đ❛ f ✈➭ g ✱ ✈➭ ✈× ❝➷♣ {f, g} ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ ♥➟♥ t❛ ❝ã f t = gt✳ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ f t = gt = t✳ ❚õ ❝➠♥❣ t❤ø❝ ✭✷✳✶✮✱ t❛ ❝ã ❱× t❤Õ ψ(G(gt, gxn+1 , gxn+1 )) = ψ(G(f t, f xn , f xn )) ≤ ψ( G(gt, f xn , f xn ) + G(gxn , f xn , f xn ) + G(gxn , f t, f t)) − φ(G(gt, f xn , f xn ), G(gxn , f xn , f xn ), G(gxn , f t, f t)) = ψ( G(gt, gxn+1 , gxn+1 ) + G(gxn , gxn+1 , gxn+1 ) + G(gxn , gt, gt))) − φ(G(gt, gxn+1 , gxn+1 ), G(gxn , gxn+1 , gxn+1 ), G(gxn , gt, gt))) ❈❤♦ n → ∞✱ t❛ ❝ã ψ(G(gt, gu, gu)) ≤ ψ( (G(gt, gu, gu)) + + G(gu, gt, gt)) − φ(G(gt, gu, gu), 0, G(gu, gt, gt)) ≤ ψ( G(gt, gu, gu) + G(gt, gu, gu)) − φ(G(gt, gu, gu), 0, G(gu, gt, gt)) 3 = ψ(G(gt, gu, gu)) − φ(G(gt, gu, gu), 0, G(gu, gt, gt)), φ(G(gt, gu, gu), 0, G(gu, gt, gt)) = 0✱ ♥❣❤Ü❛ ❧➭ gt = gu = t✳ ❚õ ➤ã✱ t❛ ❦Õt ❧✉❐♥ r➺♥❣ t = gt = f t, ✈➭ ✈× t❤Õ t ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ f ✈➭ g ✳ ➜Ó ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ t✱ t❛ ❣✐➯ sư t ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ f ✈➭ g ✳ ❚õ ❝➠♥❣ t❤ø❝ ✭✷✳✶✮ t❛ ❝ã ➤✐Ị✉ ♥➭② ❧➭ ➤ó♥❣ ♥Õ✉ ψ(G(t, t, t )) = ψ(G(f t, f t , f t )) ≤ ψ( 31 (G(t, f t , f t ) + G(t , f t , f t ) + G(t , f t, f t))) −φ(G(t, f t , f t ) + G(t , f t , f t ) + G(t , f t, f t)) = ψ( (G(t, t , t ) + G(t , t, t))) − φ(G(t, t , t ), 0, G(t , t, t)) ≤ ψ( 31 (2G(t, t, t ) + G(t , t, t))) − φ(G(t, t , t ), 0, G(t , t, t)) = ψ(G(t, t, t )) − φ(G(t, t , t ), 0, G(t , t, t)) φ(G(t, t , t ), 0, G(t , t, t)) = ◆❤➢ ✈❐② t = t ❱× t❤Õ✱ ✈➭ ❞♦ ➤ã ✶✾ G(t, t , t ) = G(t , t, t) = 0✳ ✷✳✶✳✸ ❱Ý ❞ô✳ ✭❬✷❪✮ ❈❤♦ X = [0, 2] ✈➭ ❤➭♠ G : X × X × X → R ❝❤♦ ❜ë✐✱ t t+s+u G(x, y, z) = max{|x − y|, |y − z|, |z − x|}, ψ(t) = , φ(t, s, u) = k k ≥ 6✱ f x = ✈➭ gx = − x✳ ❉Ô ❞➭♥❣ t❤✃② r➺♥❣ f ré♥❣ ❦✐Ĩ✉ A ➤è✐ ✈í✐ g ✳ ❚❤❐t ✈❐②✱ t❛ ❝ã ψ(G(f x, f y, f z)) = 0✱ ✈í✐ ❧➭ ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② 1 ψ( (G(gx, f y, f y)+G(gy, f z, f z))+G(gz, f x, f x)) = ( (|1−x|+|1−y|+|1−z|)) 3 ✈➭ ψ(G(gx, f y, f y), G(gy, f z, f z), G(gz, f x, f x)) = |1 − x| + |1 − y| + |1 − z| k ❇➞② ❣✐ê✱ ❞Ơ t❤✃② ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✶✮ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ ❘â r➭♥❣✱ f (X) ⊆ g(X)✱ g(X) ❧➭ t❐♣ ❝♦♥ ➤➬② ➤ñ ❝ñ❛ (X, G) ✈➭ ❝➷♣ {f, g} ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❉♦ ➤ã t✃t ❝➯ tết ủ ị ý ợ tỏ ✈× t❤Õ f ✈➭ g ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ➤ã ❧➭ x = 1✳ ✷✳✶✳✹ ❍Ö q✉➯✳ ✭❬✷❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ f, g : X → X t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ G(f x, f y, f z) ≤ α(G(gx, f y, f y) + G(gy, f z, f z) + G(gz, f x, f x))✱ ✈í✐ α ∈ [0, )✳ ●✐➯ sö r➺♥❣ f (X) ⊆ g(X)✱ g(X) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ➤➬② ➤ñ ❝ñ❛ (X, G) ✈➭ ❝➷♣ {f, g} ❧➭ ♠ét t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ❝❤✉♥❣ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✷✱ t❛ ❝❤Ø ❝➬♥ ❧✃② ψ(t) = t ✈➭ φ(t, s, u) = ( 13 − α)(t + s + u) ✈í✐ ♠ä✐ t, s, u ∈ [0, ∞)✳ ❑❤✐ ➤ã ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✷ t❛ ❝ã ❦Õt ❧✉❐♥ ❝đ❛ ❤Ư q✉➯ tr➟♥✳ ✷✳✶✳✺ ❍Ö q✉➯✳ f, g : X → X ✭❬✷❪✮ ●✐➯ sö (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ψ(G(f x, f y, f z)) ≤ ψ (G(x, f y, f y) + G(y, f z, f z) + G(z, f x, f x)) −φ(G(x, f y, f y), G(y, f z, f z), G(z, f x, f x)), ✭✷✳✶✼✮ tr♦♥❣ ➤ã ✷✵ ✶✮ ✷✮ ψ ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤✳ φ : [0, ∞)3 → [0, ∞) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳ ❑❤✐ ➤ã✱ f ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ ❝➳❝❤ ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✷ ❦❤✐ ❧✃② g = IdX ❧➭ ➳♥❤ ①➵ ➤å♥❣ ♥❤✃t✱ ❦❤✐ ➤ã t❛ ❝ã ❦Õt ❧✉❐♥ ❝đ❛ ❤Ư q✉➯ tr➟♥✳ ✷✳✶✳✻ ❍Ư q✉➯✳ X→X ✭❬✷❪✮ ●✐➯ sư (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ f : t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ G(f x, f y, f z) ≤ (G(x, f y, f y) + G(y, f z, f z) + G(z, f x, f x)) −φ(G(x, f y, f y), G(y, f z, f z), G(z, f x, f x)), ✭✷✳✶✽✮ φ : [0, ∞)3 → [0, ∞) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝✱ ✈í✐ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳ ❑❤✐ ➤ã✱ f ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ❙✉② trù❝ t✐Õ♣ tõ ❍Ö q✉➯ ✷✳✶✳✺ ❜➺♥❣ ❝➳❝❤ ❧✃② ψ(t) = t ✈í✐ ♠ä✐ t ∈ [0, ∞)✳ tr♦♥❣ ➤ã (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ f, g : X → X ✳ ❚❛ ♥ã✐ r➺♥❣ f ❧➭ ♠ét ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② ré♥❣ ❦✐Ó✉ B ➤è✐ ✈í✐ g ✈í✐ ♠ä✐ x, y, z ∈ X ❜✃t ➤➻♥❣ t❤ø❝ s❛✉ ➤➞② ➤ó♥❣ ✷✳✶✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✷❪✮ ❈❤♦ ψ(G(f x, f y, f z)) ≤ ψ [G(gx, gx, f y) + G(gy, gy, f z) + G(gz, gz, f x)] −φ(G(gx, gx, f y), G(gy, gy, f z), G(gz, gz, f x)), ✭✷✳✶✾✮ tr♦♥❣ ➤ã ✶✮ ✷✮ ψ ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤✳ φ : [0, ∞)3 → [0, ∞) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳ ❙ö ❞ơ♥❣ ❧❐♣ ❧✉❐♥ t➢➡♥❣ tù ♥❤➢ ➜Þ♥❤ ❧ý ✷✳✶✳✷✱ ❝❤ó♥❣ t❛ ❝ã t❤Ĩ ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý s❛✉ ➤➞②✿ ✷✶ (X, G) G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ f, g : X → X s❛♦ ❝❤♦ f ❧➭ ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② ré♥❣ ❦✐Ĩ✉ B ➤è✐ ✈í✐ g ✳ ●✐➯ sö f (X) ⊆ g(X)✱ g(X) ❧➭ t❐♣ ❝♦♥ ➤➬② ❝ñ❛ (X, G) ✈➭ ❝➷♣ f, g t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣✳ ✭❬✷❪✮ ❈❤♦ ➜Þ♥❤ ❧ý✳ ✷✳✶✳✽ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❚➢➡♥❣ tù ♥❤➢ ➤è✐ ✈í✐ ➜Þ♥❤ ❧ý ✷✳✶✳✷✱ t❛ ❝ã t❤Ĩ s✉② r❛ ♥❤÷♥❣ ❤Ư q✉➯ ❦❤➳❝ ♥❤❛✉ tõ ➜Þ♥❤ ❧ý ✷✳✶✳✽✳ ✷✳✶✳✾ ❍Ư q✉➯✳ ✭❬✷❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ f, g : X → X t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ G(f x, f y, f z) ≤ α(G(gx, gx, f y) + G(gy, gy, f z) + G(gz, gz, f x))✱ ✈í✐ α ∈ [0, )✳ ●✐➯ sư f (X) ⊆ g(X)✱ g(X) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❝♦♥ ➤➬② ➤ñ ❝ñ❛ (X, G) ✈➭ ❝➷♣ {f, g} ❧➭ ♠ét t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ ✷✳✶✳✶✵ ❍Ư q✉➯✳ f, g : X → X ✭❬✷❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ [G(x, x, f y) + G(y, y, f z) + G(z, z, f x)] −φ(G(x, x, f y), G(y, y, f z), G(z, z, f x)), ψ(G(f x, f y, f z)) ≤ ψ ✭✷✳✷✵✮ tr♦♥❣ ➤ã ✶✮ ✷✮ ψ ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤✳ φ : [0, ∞)3 → [0, ∞) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳ ❑❤✐ ➤ã✱ ✷✳✶✳✶✶ f ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❍Ư q✉➯✳ X→X ✭❬✷❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ f : t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ψ(G(f x, f y, f z)) ≤ G(x, x, f y) + G(y, y, f z) + G(z, z, f x) −φ(G(x, x, f y), G(y, y, f z), G(z, z, f x)) ✭✷✳✷✶✮ φ : [0, ∞)3 → [0, ∞) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳ ❑❤✐ ➤ã✱ f ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ t➵✐ ✷✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ✷✳✷ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ α✲❝♦ ②Õ✉ s✉② ré♥❣ G✲♠➟tr✐❝ P❤➬♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ α✲❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ ♠ét sè ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ f, g : X → X ✳ ❚❛ ♥ã✐ r➺♥❣ f ❧➭ ♠ét ➳♥❤ ①➵ G✲❝♦ α✲②Õ✉ s✉② ré♥❣ ❦✐Ó✉ A ➤è✐ ✈í✐ g ✈í✐ ♠ä✐ x, y, z ∈ X ❜✃t ➤➻♥❣ t❤ø❝ s❛✉ ➤➞② ➤ó♥❣ ➜Þ♥❤ ♥❣❤Ü❛✳ ✷✳✷✳✶ ✭❬✺❪✮ ❈❤♦ ψ(G(f x, f y, f z)) ≤ ψ(α(G(gx, f y, f y) + G(gy, f z, f z) + G(gz, f x, f x))) −φ(G(gx, f y, f y), G(gy, f z, f z), G(gz, f x, f x)), ✭✷✳✷✷✮ ✈í✐ α ∈ 0, 31 ✶✮ ✷✮ ✱ tr♦♥❣ ➤ã ψ ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤✳ φ : [0, ∞)3 → [0, ∞) ❧➭ ♠ét ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳ ❇➞② ❣✐ê ú t trì ị ý í Gtr ❝➳❝ ➳♥❤ ①➵ f, g : X → X s❛♦ ❝❤♦ f ❧➭ ➳♥❤ ①➵ G✲❝♦ α✲②Õ✉ s✉② ré♥❣ ❦✐Ó✉ A ➤è✐ ✈í✐ g ✱ ✈í✐ α ∈ [0, 31 )✳ ●✐➯ sö r➺♥❣ f (X) ⊆ g(X)✱ g(X) ❧➭ ➤➬② ➤ñ ❝ñ❛ (X, G) ✈➭ ❝➷♣ {f, g} ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ➜Þ♥❤ ❧ý✳ ✷✳✷✳✷ ✭❬✺❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♥❤✃t✳ x1 = x✳ ❑❤✐ ➤ã sư ❞ơ♥❣ f (X) ⊆ g(X)✱ ❝❤ó♥❣ t❛ ❝ã t❤Ó ①➞② ❞ù♥❣ ♠ét ❞➲② {xn } s❛♦ ❝❤♦ g(xn+1 ) = f (xn ) ✈í✐ ♠ä✐ n ∈ N✳ ➜Ĩ ➤➡♥ ❣✐➯♥ tr♦♥❣ ❦ý ❤✐Ư✉ t❛ sÏ ✈✐Õt gn+1 = gxn+1 ✈➭ fn = f xn ✳ ❚❛ ❣✐➯ sư r➺♥❣ gn+1 = gn ✈í✐ ♠ä✐ n ∈ N ì ế ợ tồ t ột số tự ♥❤✐➟♥ n ≥ 1✱ t❤× t➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✶✳✷ t❛ s✉② r❛ f ✈➭ g sÏ ❝ã ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ x∈X ✈➭ ➤➷t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣✳ ✷✸ ❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❝➠♥❣ t❤ø❝ ✭✷✳✷✷✮ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✭✺✮ ❝đ❛ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✷ t❛ ♥❤❐♥ ➤➢ỵ❝ ψ(G(gn , gn+1 , gn+1 )) ≤ ψ(G(fn−1 , fn , fn )) ✭✷✳✷✸✮ ≤ ψ(α(G(gn−1 , fn , fn ) + G(gn , fn , fn )+ + G(gn , fn−1 , fn−1 ))) − φ(G(gn−1 , fn , fn ), G(gn , fn , fn ), G(gn , fn−1 , fn−1 )) ≤ ψ(α(G(gn−1 , gn+1 , gn+1 ) + G(gn , gn+1 , gn+1 )+ + G(gn , gn , gn ))) ≤ ψ(α(G(gn−1 , gn+1 , gn+1 ) + G(gn , gn+1 , gn+1 ))), ✈í✐ ♠ä✐ ❱× n ∈ N✳ ψ ❧➭ ❤➭♠ t➝♥❣✱ ♥➟♥ tõ ❝➠♥❣ t❤ø❝ ✭✷✳✷✸✮ t❛ ♥❤❐♥ ➤➢ỵ❝ G(gn , gn+1 , gn+1 ) ≤ α(G(gn−1 , gn+1 , gn+1 ) + G(gn , gn+1 , gn+1 )) ≤ α(G(gn−1 , gn , gn )) + 2α(G(gn , gn+1 , gn+1 )) ✭✷✳✷✹✮ n ∈ N✱ tõ ❝➠♥❣ t❤ø❝ ✭✷✳✷✹✮ t❛ t❤✉ ➤➢ỵ❝ ❝➠♥❣ t❤ø❝ s❛✉ α G(gn , gn+1 , gn+1 ) ≤ G(gn−1 , gn , gn ) ✭✷✳✷✺✮ − 2α α ≤ G(gn−2 , gn−1 , gn−1 ) − 2α n−1 α ≤ ≤ G(g1 , g2 , g2 ) − 2α ❇➞② ❣✐ê✱ t❛ sÏ ❝❤Ø r❛ r➺♥❣ {gn } ❧➭ ❞➲② G✲❈❛✉❝❤②✳ ●✐➯ sö r➺♥❣ m > n✱ ❦❤✐ ➤ã sư ❱í✐ ♠ä✐ ❞ơ♥❣ ❝➠♥❣ t❤ø❝ ✭✺✮ tr♦♥❣ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✷ ✈➭ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✹✱ t❛ ❝ã < G(gn , gm , gm ) ≤ G(gn , gn+1 , gn+1 ) + G(gn+1 , gn+2 , gn+2 )+ ≤ ≤ ≤ + G(gn+2 , gn+3 , gn+3 ) + + G(gm−1 , gm , gm ) n−1 m−2 α α ) + ··· + G(g1 , g2 , g2 ) − 2α − 2α n−1 α α 1+ + − 2α − 2α α + + · · · G(g1 , g2 , g2 ) − 2α n−1 − 2α α G(g1 , g2 , g2 ) ✭✷✳✷✻✮ − 3α − 2α ✷✹ ❙ư ❞ơ♥❣ ❣✐➯ t❤✐Õt r➺♥❣ α ∈ [0, 13 ) ✈➭ ❝❤♦ q✉❛ ❣✐í✐ ❤➵♥ ❦❤✐ m → ∞✱ t❛ s✉② r❛ α − 2α n−1 → ❦❤✐ n → ∞ G(gn , gm , gm ) → ❦❤✐ m, n → ∞✳ ❑❤✐ ➤ã✱ ♥❤ê ▼Ư♥❤ ➤Ị ✶✳✶✳✼✱ t❛ ❝ã {gn } = {gxn } ❧➭ G✲❈❛✉❝❤② tr♦♥❣ g(X)✳ ❱× g(X) ❧➭ t❐♣ ❝♦♥ ➤➬② ➤ñ ❝ñ❛ (X, G)✱ ♥➟♥ tå♥ t➵✐ z ∈ X s❛♦ ❝❤♦ gxn ❧➭ G✲❤é✐ tô ➤Õ♥ gz ❦❤✐ n → ∞✳ ➜✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ ❧➭ ❱× t❤Õ✱ t❛ ❝ã G(gn , gn , gz) → ❦❤✐ n → ∞ ✭✷✳✷✼✮ ❈ị♥❣ t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✳✹✱ t❛ ♥❤❐♥ ➤➢ỵ❝ G(gn , gz, gz) → ❦❤✐ n → ∞ ✭✷✳✷✽✮ ❇➞② ❣✐ê✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ❣✐í✐ ❤➵♥ s❛✉ (a) G(gn , gn , f z) → G(gz, gz, f z) ❦❤✐ n → ∞, ✭✷✳✷✾✮ (b) G(gn , f z, f z) → G(gz, gz, f z) ❦❤✐ n → ∞ ✭✷✳✸✵✮ ✈➭ ➜Ĩ ❝❤ø♥❣ ♠✐♥❤ ✭❛✮✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❝➠♥❣ t❤ø❝ ✭✸✮ ✈➭ ❝➠♥❣ t❤ø❝ ✭✻✮ tõ ▼Ư♥❤ ➤Ị ✶✳✶✳✶✷✱ t❛ ❝ã G(gn , gn , f z) − G(gz, gz, f z) ≤ 2G(gn , gz, gz) ≤ 4G(gn , gn , gz) ❈ò♥❣ t❤Õ✱ t❛ ❝ã G(gz, gz, f z) − G(gn , gn , f z) ≤ 2G(gn , gn , gz) ❉♦ ➤ã✱ |G(gn , gn , f z) − G(gz, gz, f z)| ≤ 4G(gn , gn , gz) → ❦❤✐ n → ∞ ➜Ĩ ❝❤ø♥❣ ♠✐♥❤ ✭❜✮✳ ❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❧ý ❧✉❐♥ t➢➡♥❣ tù ❝đ❛ ❝➠♥❣ t❤ø❝ ✭✷✳✷✾✮✱ t❛ sÏ ♥❤❐♥ ➤➢ỵ❝ ❦Õt q✉➯ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ✷✺ ❇➞② ❣✐ê✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ f z = gz ✳ ❙ư ❞ơ♥❣ ❝➠♥❣ t❤ø❝ ✭✷✳✶✮✱ t❛ ❝ã ψ(G(gn+1 ,gn+1 , f z)) = ψ(G(fn , fn , f z)) ≤ ψ(α(Ggn , fn , fn ) + G(gn , f z, f z) + G(gz, fn , fn )) − φ(G(gn , fn , fn ) + G(gn , f z, f z) + G(gz, fn , fn )) ✭✷✳✸✶✮ = ψ(α(G(gn , gn+1 , gn+1 ) + G(gn , f z, f z) + G(gz, gn+1 , gn+1 ))) − φ(G(gn , gn+1 , gn+1 ) + G(gn , f z, f z) + G(gz, gn+1 , gn+1 )) ❈❤♦ n → ∞✱ sư ❞ơ♥❣ ❝➠♥❣ t❤ø❝ ✭✷✳✷✾✮✱ ✭✷✳✸✵✮ ✈➭ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ψ ✈➭ φ✱ ❝➠♥❣ t❤ø❝ ✭✷✳✸✶✮ trë t❤➭♥❤ ψ(G(gz, gz, f z)) ≤ ψ(α(G(gz, f z, f z) − φ(0, G(gz, f z, f z), 0) ✭✷✳✸✷✮ ≤ ψ(2αG(gz, gz, f z) − φ(0, 2G(gz, gz, f z), 0) ≤ ψ(2αG(gz, gz, f z) ψ ❧➭ ❤➭♠ t➝♥❣ ✈➭ α ∈ 0, 31 ✱ tõ ❝➠♥❣ t❤ø❝ ✭✷✳✸✷✮ t❛ s✉② r❛ r➺♥❣ G (gz, gz, f z) = 0✳ ❱× t❤Õ✱ t❛ ❝ã f z = gz ✈➭ z ❧➭ ♠ét ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❝đ❛ f ✈➭ g ✳ ❱× ❝➷♣ {f, g} ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ ♥➟♥ t❛ ❝ã f u = gu ✈í✐ u = f z = gz ✳ ❇➞② ❣✐ê✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ f u = gu = u✳ ➜Ó ❝❤Ø r❛ ➤✐Ị✉ ♥➭②✱ ♥❤ê ❝➠♥❣ ◆❤ê ▼Ư♥❤ ➤Ị ✶✳✶✳✶✷✱ ❣✐➯ t❤✐Õt r➺♥❣ t❤ø❝ ✭✷✳✷✷✮ t❛ ❝ã ❜✃t ➤➻♥❣ t❤ø❝ s❛✉ ψ(G(gu, gn+1 ,gn+1 )) = ψ(G(f u, fn , fn )) ≤ ψ(α(G(gu , fn , fn ) + G(gn , fn , fn ) + G(gn , f u, f u)) − φ(G(gu, fn , fn ), G(gn , fn , fn ), G(gn , fn−1 , fn−1 )) = ψ(α(G(gu, gn+1 , gn+1 ) + G(gn , gn+1 , gn+1 ) + G(gn , gu, gu))) − φ(G(gu, gn+1 , gn+1 ) + G(gn , gn+1 , gn+1 ) + G(gn , gu, gu)) ❈❤♦ n → ∞✱ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ❝ã ψ(G(gu, gz, gz)) ≤ ψ(α(G(gu, gz, gz) + G(gz, gu, gu))) − φ(G(gu, gz, gz), 0, G(gz, gu, gu)) ≤ ψ(2αG(gu, gz, gz) − φ(G(gu, gz, gz), 0, G(gz, gu, gu))) ≤ ψ(G(gu, gz, gz) − φ(G(gu, gz, gz), 0, G(gz, gu, gu))), ✷✻ φ(G(gu, gz, gz), 0, G(gz, gu, gu)) = 0✱ ♥❣❤Ü❛ ❧➭ g(u) = u = g(u) = f (u)✳ ❉♦ ➤ã✱ u ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ f ✈➭ ➤✐Ị✉ ♥➭② ➤ó♥❣ ♥Õ✉ g(z) = u✳ ❱× t❤Õ g✳ u✱ t❛ ❣✐➯ sư t ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ f ✈➭ g ♠➭ t ❦❤➳❝ ✈í✐ u ✭♥❣❤Ü❛ ❧➭ t = g(t) = f (t)✮✳ ◆❤ê ❝➠♥❣ t❤ø❝ ✭✷✳✷✷✮✱ t❛ ➜Ĩ ❝❤ø♥❣ ♠✐♥❤ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ❝ã ψ(G(u, t, t)) = ψ(G(f u, f t, f t)) ≤ ψ(α[G(gu, f t, f t) + G(gt, f t, f t) + G(gt, f u, f u)]) − φ(G(gu, f t, f t), G(gt, f t, f t), G(gt, f u, f u)) ≤ ψ(α[G(u, t, t) + G(t, u, u)]) − φ(G(u, t, t), 0, G(t, u, u)) ≤ ψ(3αG(u, t, t)) − φ(G(u, t, t), 0, G(t, u, u)) ≤ ψ(G(u, t, t)) − φ(G(u, t, t), 0, G(t, u, u)) φ(G(u, t, t), 0, G(t, u, u)) = 0✳ ❇ë✐ ✈❐②✱ ♥❤ê tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ φ t❛ ❝ã G(u, t, t) = G(t, u, u) = 0✳ ❱× t❤Õ✱ t = u✳ ❉♦ ➤ã✱ t❛ ♥❤❐♥ ➤➢ỵ❝ ❍Ư q✉➯✳ ✷✳✷✳✸ f, g : X → X ✭❬✺❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ G(f x, f y, f z) ≤ α(G(gx, f y, f y) + G(gy, f z, f z) + G(gz, f x, f x)), α ∈ [0, 31 ) ●✐➯ sö f (X) ⊆ g(X)✱ g(X) ❧➭ t❐♣ ❝♦♥ ➤➬② ➤ñ ❝ñ❛ (X, G) ✈➭ ❝➷♣ {f, g} ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ✈í✐ ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ ❝➳❝❤ ❧✃② ψ(t) = t ✈➭ φ(t, s, u) = 0✱ s❛✉ ó ụ ị ý t ợ ết q ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷✳✹ ❍Ö q✉➯✳ X→X ✭❬✺❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ f : t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ψ(G(f x, f y, f z)) ≤ ψ(α(G(x, f y, f y) + G(y, f z, f z) + G(z, f x, f x))) − φ(G(x, f y, f y), G(y, f z, f z), G(z, f x, f x)), ✈í✐ α ∈ [0, 31 ), ψ ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ : [0, ∞)3 → [0, ∞) ❧➭ t = s = u = 0✳ ❑❤✐ ➤ã✱ f ❝ã ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤ ✈➭ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ✷✼ ❈❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ ❝➳❝❤ ❝❤♦ g = IdX ✱ ✭➳♥❤ ①➵ ➤å♥❣ ♥❤✃t tr➟♥ X ✮✱ s❛✉ ➤ã ➳♣ ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✷ t❛ ❝ã ➤➢ỵ❝ ❦Õt q✉➯ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ❍Ư q✉➯✳ ✷✳✷✳✺ X→X ✭❬✺❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ f : t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ G(f x, f y, f z) ≤ α(G(x, f y, f y) + G(y, f z, f z) + G(z, f x, f x)) − φ(G(x, f y, f y), G(y, f z, f z), G(z, f x, f x)), ✈í✐ α ∈ [0, 31 ), ψ ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ : [0, ∞)3 → [0, ∞) ❧➭ t = s = u = 0✳ ❑❤✐ ➤ã✱ f ❝ã ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤ ✈➭ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ ❝➳❝❤ t❤❛② ψ(t) = t tr♦♥❣ ❍Ư q✉➯ ✷✳✷✳✹ t❛ ❝ã ➤➢ỵ❝ ❦Õt q✉➯ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ f, g : X → X ✳ ❚❛ ♥ã✐ r➺♥❣ f ❧➭ ➳♥❤ ①➵ G✲❝♦ α✲②Õ✉ s✉② ré♥❣ ❦✐Ĩ✉ B ➤è✐ ✈í✐ g ✱ ♥Õ✉ ✈í✐ ♠ä✐ x, y, z ∈ X ✱ t❛ ❝ã ❜✃t ➤➻♥❣ t❤ø❝ s❛✉ ➤➞② ➤ó♥❣ ➜Þ♥❤ ♥❣❤Ü❛✳ ✷✳✷✳✻ ✭❬✺❪ ❈❤♦ ψ(G(f x, f y, f z)) ≤ ψ(α(G(gx, gx, f y) + G(gy, gy, f z) + G(gz, gz, f x))) −φ(G(gx, gx, f y), G(gy, gy, f z), G(gz, gz, f x)), α ∈ [0, 31 )✱ ψ ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤ ✈➭ φ : [0, ∞)3 → [0, ∞) ❧➭ ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳ tr♦♥❣ ➤ã G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ f, g : X → X s❛♦ ❝❤♦ f ❧➭ ➳♥❤ ①➵ G✲❝♦ α✲②Õ✉ s✉② ré♥❣ ❦✐Ĩ✉ B ➤è✐ ✈í✐ g ✱ ✈í✐ α ∈ [0, 31 )✳ ●✐➯ sư f (X) ⊆ g(X)✱ g(X) ❧➭ t❐♣ ❝♦♥ ➤➬② ➤ñ ❝ñ❛ (X, G) ✈➭ ❝➷♣ {f, g} ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ➜Þ♥❤ ❧ý✳ ✷✳✷✳✼ ✭❬✺❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ➜Þ♥❤ ❧ý ♥➭② ó tể ợ ứ sử ụ ữ ❧❐♣ ❧✉❐♥ t➢➡♥❣ tù ♥❤➢ ❦❤✐ ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✷✳✷✳ ❚➢➡♥❣ tù ♥❤➢ ➤è✐ ✈í✐ ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② ré♥❣ ❦✐Ĩ✉ B ✱ t❛ t❤✉ ➤➢ỵ❝ ❝➳❝ ❤Ư q✉➯ s❛✉✳ ✷✽ ✭❬✺❪✮ ❈❤♦ ❍Ö q✉➯✳ ✷✳✷✳✽ f, g : X → X (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ G(f x, f y, f z) ≤ α(G(gx, gx, f y) + G(gy, gy, f z) + G(gz, gz, f x)), α ∈ [0, 13 )✳ ●✐➯ sö r➺♥❣ f (X) ⊆ g(X)✱ g(X) ❧➭ t❐♣ ❝♦♥ ➤➬② ➤ñ ❝ñ❛ (X, G) ✈➭ ❝➷♣ {f, g} ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ➤ã ❝❤✉♥❣ ❞✉② ♥❤✃t✳ ❍Ö q✉➯✳ ✷✳✷✳✾ X→X ✭❬✺❪✮ ●✐➯ sö (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ f : t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ψ(G(f x, f y, f z)) ≤ ψ(α(G(x, x, f y) + G(y, y, f z) + G(z, z, f x))) − φ(G(x, f y, f y), G(y, f z, f z), G(z, f x, f x)), α ∈ [0, 31 ), ψ ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤ ✈➭ φ : [0, ∞)3 → [0, ∞) ❧➭ ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = s = u = 0✳ ❑❤✐ ➤ã✱ f ❝ã tr♦♥❣ ➤ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ✷✳✷✳✶✵ ①➵ ❍Ư q✉➯✳ f :X→X ✭❬✺❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ●✐➯ sư r➺♥❣ ➳♥❤ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ G(f x, f y, f z)) ≤ α(G(x, x, f y) + G(y, y, f z) + G(z, z, f x))) − φ(G(x, x, f y), G(y, y, f z), G(z, z, f x)), ✈í✐ α ∈ [0, 31 ), ψ ❤➭♠ ❧✐➟♥ tơ❝ ✈í✐ φ : [0, ∞)3 → [0, ∞) ❧➭ t = s = u = 0✳ ❑❤✐ ➤ã✱ f ❝ã ❧➭ ❤➭♠ t❤❛② ➤æ✐ ❦❤♦➯♥❣ ❝➳❝❤ ✈➭ φ(t, s, u) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ❞✉② ♥❤✃t ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❙❛✉ ➤➞②✱ ❝❤ó♥❣ t➠✐ sÏ tr×♥❤ ❜➭② ♠ét sè ✈Ý ❞ơ ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ tí ợ ý ủ tết ủ ị ý ✷✳✷✳✷✳ ✷✳✷✳✶✶ ❱Ý ❞ô✳ ❈❤♦ X = [0, 2] ✈➭ G✲♠➟tr✐❝ ①➳❝ ➤Þ♥❤ ❜ë✐ G(x, y, z) = max{|x − y|, |y − z|, |z − x|}, ψ, φ ❝❤♦ ❜ë✐ t t+s+u ψ(t) = , φ(t, s, u) = , k ✈í✐ ♠ä✐ x, y, z ∈ X, ✈➭ ❝➳❝ ❤➭♠ ✷✾ ✈í✐ k≥ , α ∈ [0, ) α f, g tr➟♥ X ❝❤♦ ❜ë✐ f x = ✈➭ g(x) = − x ✈í✐ ♠ä✐ x ∈ X ✳ ❉Ơ ❞➭♥❣ t❤✃② r➺♥❣ f ❧➭ ➳♥❤ ①➵ G✲❝♦ α✲②Õ✉ s✉② ré♥❣ ❦✐Ó✉ A ➤è✐ ✈í✐ g ✱ ✈➭ f (X) ⊆ g(X)✱ g(X) ❧➭ ♠ét t❐♣ ❝♦♥ ➤➬② ➤ñ ❝ñ❛ (X, G)✱ ✈➭ ❝➷♣ {f, g} ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❍➡♥ ♥÷❛✱ t❛ ❝ã G(f x, f y, f z) = 0✱ ❚❛ ①➳❝ ➤Þ♥❤ ❝➳❝ ➳♥❤ ①➵ ψ(α(G(gx, f y, f y) + G(gy, f z, f z) + G(gz, f x, f x))) α = (|1 − x|, |1 − y|, |1 − z|) ✈➭ φ(G(gx, f y, f y) + G(gy, f z, f z) + G(gz, f x, f x)) = |1−x|+|1−y|+|1−z| k ❉♦ ➤ã✱ t✃t ữ ề ệ ủ ị ý ợ tỏ ♠➲♥ ➤è✐ ✈í✐ ✈Ý ❞ơ g ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✱ ❝ơ t❤Ĩ x = ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t ❝đ❛ f ✈➭ g ✳ ♥➭②✳ ❱× t❤Õ✱ f ✈➭ ❇➞② ❣✐ê t❛ ①➞② ❞ù♥❣ ♠ét ✈Ý ❞ơ ✈Ị ♠ét G✲♠➟tr✐❝ ❦❤➠♥❣ ➤è✐ ①ø♥❣✱ t❤á❛ ♠➲♥ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✷✳ ✷✳✷✳✶✷ ❱Ý ❞ơ✳ ❈❤♦ t❐♣ ❤ỵ♣ X = {a, b, c} ị Gtr G : X ì X × X → R ❝❤♦ ❜ë✐ G(x, x, x) = 0, ✈í✐ ♠ä✐ x ∈ X, G(a, a, b) = 2, G(a, a, c) = G(a, b, b) = G(b, b, c) = 3, G(a, b, c) = G(a, c, c) = G(b, c, c) = 4, G ➤è✐ ①ø♥❣ t❤❡♦ t✃t ❝➯ ❝➳❝ ❜✐Õ♥✳ ❑❤✐ ➤ã✱ ❝❤ó ý r➺♥❣ G ❧➭ ❦❤➠♥❣ ➤è✐ ①ø♥❣ ✈× G(x, x, y) = G(x, y, y) ✈í✐ x = y✳ ❳Ðt ❝➳❝ ➳♥❤ ①➵ f, g : X → X ➤➢ỵ❝ ❝❤♦ ❜ë✐ f (a) = f (b) = f (c) = a✱ g(x) = x ✈í✐ ♠ä✐ x ∈ X ✳ ▲✃② ψ(t) = 2t ✈➭ φ(s, t, r) = s+t+r k ✱ ✈í✐ ♠ä✐ t, s, r ∈ 1 [0, ∞)✱ tr♦♥❣ ➤ã k ≥ 2α ✱ ✈í✐ α ∈ [0, )✳ ❑ý ❤✐Ö✉ L = ψ(G(f x, f y, f z))✱ A = G(gx, f y, f y)✱ B = G(gy, f z, f z)✱ C = G(gz, f x, f x)✱ K = ψ(α(A + B + C)) − φ(A, B, C)✳ ❇➞② ❣✐ê t❛ ❧❐♣ ❜➯♥❣ ❝➳❝ ❣✐➳ trÞ ❝đ❛ L, A, B, C ✈➭ K t➢➡♥❣ ø♥❣ ✈í✐ ❝➳❝ ❣✐➳ ✈➭ ✸✵ trÞ ❝đ❛ ❜✐Õ♥ x, y, z ♥❤➢ s❛✉✳ (x, y, z) (a, a, a) (a, a, b) (a, a, c) (a, b, b) (a, b, c) (a, c, c) (b, b, c) (b, c, c) A 0 0 0 0 B 0 2 3 C 3 3 L 0 0 0 0 K 4α − k2 6α − k3 8α − k4 10α − k5 12α − k6 10α − k5 16α − k8 ❑❤✐ ➤ã✱ ❞Ô t❤✃② r➺♥❣ t✃t ❝➯ ❝➳❝ ề ệ ủ ị ý ợ tỏ ✈× t❤Õ f, g ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✱ ➤ã ❧➭ x = a✳ ✸✶ ❑Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ò✉ 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❛ G✲♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ➳♥❤ ①➵ G✲❧✐➟♥ tô❝✱ ➳♥❤ ①➵ C ✲❝♦ ②Õ✉✱ ➳♥❤ ①➵ G✲❝♦ ②Õ✉✱ ➳♥❤ ①➵ ❦✐Ó✉ G✲❝♦ ②Õ✉✱ ➳♥❤ ①➵ ❦✐Ó✉ G✲❝♦ ②Õ✉ s✉② ré♥❣✱ ➳♥❤ ①➵ α✲❝♦ ②Õ✉ s✉② ré♥❣✱ ❤➭♠ t❤❛② ➤ỉ✐ ❦❤♦➯♥❣ ❝➳❝❤✱ G✲♠➟tr✐❝ ➤è✐ ①ø♥❣✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉✱ ❣✐➳ trÞ trï♥❣ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♥❤❛✉✱ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ rì ột số ị ý ề ể t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ➳♥❤ ①➵ G✲ G✲♠➟tr✐❝ ➤➬② ➤đ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ➳♥❤ ①➵ ❦✐Ĩ✉ Φ✲❝♦ tr Gtr r ú t ò trì ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ②Õ✉ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ♠➟tr✐❝✱ ➳♥❤ ①➵ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ➤ã✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤Þ♥❤ ❧ý ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ứ ò s ợ Gtr í ụ ❤ä❛ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ➳♥❤ ①➵ G✲❝♦ ②Õ✉ s✉② ré♥❣ ❦✐Ĩ✉ A ➤è✐ ✈í✐ g ✱ ❱Ý ❞ơ ✷✳✷✳✶✶ ♠✐♥❤ ❤ä❛ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ➳♥❤ ①➵ G✲❝♦ α✲②Õ✉ s✉② ré♥❣ ❦✐Ĩ✉ A ➤è✐ ✈í✐ g ✱ ❱Ý ❞ơ ✷✳✷✳✶✷ ♠✐♥❤ ❤ä❛ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ➳♥❤ ①➵ G✲❝♦ α✲②Õ✉ s✉② ré♥❣ ❦✐Ĩ✉ A ➤è✐ ✈í✐ g tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ G✲♠➟tr✐❝ ❦❤➠♥❣ ➤è✐ ①ø♥❣✳ ✹✳ ❚r×♥❤ ❜➭② ❱Ý ❞ơ ✶✳✶✳✶✶ ➤Ĩ ♠✐♥❤ ❤ä❛ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❦❤➠♥❣ ❣✐❛♥ ✸✷ t➭✐ ❧✐Ư✉ t❤❛♠ ỗ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ❍✳ ❆②❞✐✱ ❲✳ ❙❤❛t❛♥❛✇✐✱ ❈✳ ❱❡tr♦ ✭✷✵✶✶✮✱ ❖♥ ❣❡♥❡r❛❧✐③❡❞ ✇❡❛❦❧② ❝♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣ ✐♥ G✲ G✲♠❡tr✐❝ s♣❛❝❡s✱ ❈♦♠♣✉t✳ ▼❛t❤✳ ❆♣♣❧✱ ✻✷✱ ✹✷✷✷✲ ✹✷✷✾✳ ❬✸❪ ❉❙✳ ❏❛❣❣✐ ✭✷✵✶✶✮✱ ❯♥✐q✉❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✱ ■♥❞✐❛♥ ❏✳ P✉r❡ ❆♣♣❧✳ ▼❛t❤✳✱ ✽✱ ✷✷✸✲✷✸✵✳ ❬✹❪ ▼✳ ❙✳ ❑❤❛♥✱ ▼✳ ❙✇❛❧❡❤✱ ❙✳ ❙❡ss❛ ✭✶✾✽✹✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❜② ❛❧t❡r✐♥❣ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts✱ ❇✉❧❧✳ ❆✉st✳ ▼❛t❤✳ ❙♦❝✱ ❬✺❪ ▼✳ ❑❤❛♥❞❛q❥✐✱ ❙✳ ❆❧✲❙❤❛r✐❢ ✭✷✵✶✸✮✱ ●❡♥❡r❛❧✐③❡❞ ♠❛♣♣✐♥❣s ♦♥ ✸✵✱ ✶✲✾✳ α✲✇❡❛❦❧② ❝♦♥tr❛❝t✐✈❡ G✲♠❡tr✐❝ s♣❛❝❡s✱ ▼❛❧❛②s✐❛♥ ❏✳ ▼❛t❤✳ ❙❝✐✳✱ ✼ ✭✷✮✱ ✶✹✼✲✶✻✹✳ ❬✻❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❇✳❙✐♠s ✭✷✵✵✹✮✱ ❙♦♠❡ r❡♠❛r❦s ❝♦♥❝❡r♥✐♥❣ ❉✲ ♠❡tr✐❝ s♣❛❝❡s✱ ✐♥ Pr♦❝❡❡❡❞✐♥❣s ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❋✐①❡❞ ♣♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐t✐♦♥s✱ ❱❛❧❡♥❝✐❛✱ ❙♣❛✐♥✱ ❏✉❧② ✷✵✵✹✱ ✶✽✾✲✶✾✽✳ ❬✼❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❇✳ ❙✐♠s ✭✷✵✵✻✮✱ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❏♦✉r♥❛❧ ♦❢ ◆♦♥❧✐♥❡❛r ❛♥❞ ❈♦♥✈❡① ❆♥❛❧②s✐s✱ ✼ ✭✷✮✱ ✷✽✾✲✷✾✼✳ ❬✽❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❍✳ ❖❜✐❡❞❛t✱ ❋✳ ❆✇❛✇❞❡❤ ✭✷✵✵✽✮✱ ❙♦♠❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦✲ r❡♠s ❢♦r ♠❛♣♣✐♥❣s ♦♥ ❝♦♠♣❧❡t❡ G✲♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✳ ✷✵✵✽✱ ❆rt✐❝❧❡ ■❉✹✵✶✻✽✹✱✶✷ ♣❛❣❡s✳ ❬✾❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❇✳❙✐♠s ✭✷✵✵✾✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❝♦♥tr❛❝t✐✈❡ ♠❛♣✲ ♣✐♥❣s ✐♥ ❝♦♠♣❧❡t❡ G✲ ♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✳ ✷✵✵✾ ❆rt✐❝❧❡ ■❉ ✾✶✼✶✼✺✱ ✶✵ ♣❛❣❡s✱ ❞♦✐✿ ✶✵✳✶✶✺✺✴✷✵✵✾✴✾✶✼✶✼✺✳ ❬✶✵❪ ❲✳ ❙❤❛t❛♥❛✇✐ ✭✷✵✶✵✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❢♦r ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s s❛t✐s✲ ❢②✐♥❣ Φ✲♠❛♣s ✐♥ G✲♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✳ ✷✵✶✵✱ ❆rt✐❝❧❡ ■❉✶✽✶✻✺✵✱ ✾ ♣❛❣❡s✳ ❬✶✶❪ ❲✳ ❙❤❛t❛♥❛✇✐✱ ▼✳ ❆❜❜❛s✱ ❍✳ ❆②❞✐ ✭✷✵✶✵✮✱ ❖♥ ✇❡❛❦❧② C ✲❝♦♥tr❛❝t✐✈❡ ♠❛♣✲ ♣✐♥❣s ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✳ ✷✵✶✵✱ ❆rt✐❝❧❡ ■❉✶✽✶✻✺✵✱ ✾ ♣❛❣❡s✳ ✸✸ ... φ (G( gt, gxn+1 , gxn+1 ), G( gxn , gxn+1 , gxn+1 ), G( gxn , gt, gt))) ❈❤♦ n → ∞✱ t❛ ❝ã ψ (G( gt, gu, gu)) ≤ ψ( (G( gt, gu, gu)) + + G( gu, gt, gt)) − φ (G( gt, gu, gu), 0, G( gu, gt, gt)) ≤ ψ( G( gt, gu, gu)... G( gt, gu, gu) + G( gt, gu, gu)) − φ (G( gt, gu, gu), 0, G( gu, gt, gt)) 3 = ψ (G( gt, gu, gu)) − φ (G( gt, gu, gu), 0, G( gu, gt, gt)), φ (G( gt, gu, gu), 0, G( gu, gt, gt)) = 0✱ ♥❣❤Ü❛ ❧➭ gt = gu = t✳ ❚õ ➤ã✱... ❝ã G( gn , gn , f z) − G( gz, gz, f z) ≤ 2G( gn , gz, gz) ≤ 4G( gn , gn , gz) ❈ò♥❣ t❤Õ✱ t❛ ❝ã G( gz, gz, f z) − G( gn , gn , f z) ≤ 2G( gn , gn , gz) ❉♦ ➤ã✱ |G( gn , gn , f z) − G( gz, gz, f z)| ≤ 4G( gn