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

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❚r➬♥ ❱➝♥ ❚rä♥❣ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❚♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥ ✲ ✷✵✶✽ ❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❚r➬♥ ❱➝♥ ❚rä♥❣ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✽✹✻✵✶✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ö ❆♥ ✲ ✷✵✶✽ ✐ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤✱ ♥❣❤✐➟♠ tó❝ ❝đ❛ t❤➬② P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✳ ❚➠✐ ①✐♥ ❜➭② tá sù ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ❚❤➬②✳ ❳✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ q✉ý t❤➬② ❝➠ ë ❇é ♠➠♥ ●✐➯✐ ❚Ý❝❤✱ ❱✐Ö♥ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ✱ ❚❙✳ ◆❣✉②Ô♥ ❱➝♥ ➜ø❝ ❣✐➳♦ ✈✐➟♥ ❝❤đ ♥❤✐Ư♠ ❧í♣ ❝❛♦ ❤ä❝ ●✐➯✐ ❚Ý❝❤ ❦❤♦➳ ✷✹ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❑ü t❤✉❐t ❱Ü♥❤ ▲♦♥❣✱ ❙ë ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ tØ♥❤ ❱Ü♥❤ ▲♦♥❣✱ ❇❛♥ ●✐➳♠ ❍✐Ö✉ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❑ü t❤✉❐t ❱Ü♥❤ ▲♦♥❣ ➤➲ ❣✐ó♣ ➤ì✱ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❝❤♦ t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ◆❤➞♥ ➤➞② t➠✐ ①✐♥ ❝➳♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ●✐➯✐ ❚Ý❝❤ ❦❤♦➳ ✷✹ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠ ❑ü t❤✉❐t ❱Ü♥❤ ▲♦♥❣✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❣ë✐ ❧ê✐ ❝➳♠ ➡♥ ➤Õ♥ ❇❛ ♠Đ✱ ✈ỵ✱ ❝➳❝ ❛♥❤ tr ì t ề ệ t ợ ❣✐ó♣ t➠✐ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ▼➷❝ ❞ï ➤➲ tÝ❝❤ ❝ù❝ ➤➬✉ t➢ ✈➭ ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ s❛✐ sãt✳ ❚➠✐ ợ ữ ý ế ó ó ủ qý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ö♥✳ ❱✐♥❤✱ ♥❣➭② ✸✵ t❤➳♥❣ ✵✺ ♥➝♠ ✷✵✶✽ ❚r➬♥ ❱➝♥ ❚rä♥❣ ✐✐ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✐ ▼ë ➤➬✉ ✐✐✐ ❈❤➢➡♥❣ ✶✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✶ ✳ ✶ ♠➟tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ✶✳✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❈❤➢➡♥❣ ✷✳ ❣✐❛♥ ✷✳✶ φ✲❝♦ ②Õ✉ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ✭ ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ G✲♠➟tr✐❝ ✷✹ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✷✳✷ G✲♠➟tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟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♥❣ ❞ô♥❣✳ ❚r♦♥❣ t❤ù❝ tÕ✱ ♥ã ➤➲ trë t❤➭♥❤ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝➠♥❣ ❝ơ ❝èt ②Õ✉ ♥❤✃t tr♦♥❣ ❣✐➯✐ tÝ❝❤ ❤➭♠ ♣❤✐ t✉②Õ♥✱ tè✐ ➢✉ ❤ã❛✱ t♦➳♥ ❤ä❝✱ ❝➳❝ ♠➠ ❤×♥❤ t♦➳♥ ❤ä❝✱ ❦✐♥❤ tÕ ✈➭ ② ❤ä❝✳ ❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ị♥❣ ➤ã♥❣ ♠ét ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ✈✐Ư❝ ①➞② ❞ù♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ tr♦♥❣ t♦➳♥ ❤ä❝ ➤Ó ❣✐➯✐ q✉②Õt ❝➳❝ ✈✃♥ ➤Ị tr♦♥❣ t♦➳♥ ❤ä❝ ø♥❣ ❞ơ♥❣ ✈➭ ❦❤♦❛ ❤ä❝✳ ❈❤Ý♥❤ ✈× t❤Õ ♠➭ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ s❛✉ ♥➭② ➤➲ ♠ë ré♥❣ ❝➳❝ ➤Þ♥❤ ❧ý ❝➡ ❜➯♥ ♥➭② ❝❤♦ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✱ ❜➺♥❣ ❝➳❝❤ ➤✐Ị✉ ❝❤Ø♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝➡ ❜➯♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❑ü t❤✉❐t ➤➬✉ t✐➟♥ ❧➭ ✧t❤❛② t❤Õ✧ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈í✐ ♠ét ❦❤➠♥❣ ❣✐❛♥ tỉ♥❣ q✉➳t ❤➡♥✳ ❑❤➠♥❣ ❣✐❛♥ tù❛ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ❢✉③③②✲♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ D✲ ♠➟tr✐❝ ✈➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ❧➭ ♥❤÷♥❣ tỉ♥❣ q✉➳t ❤ã❛ ❝đ❛ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❝ã t❤Ĩ ➤➢ỵ❝ ❝♦✐ ♥❤➢ ❧➭ ❝➳❝ ✈Ý ❞ơ ✈Ị t tế ột tr ữ Gtr ợ t❤✐Ö✉ ❜ë✐ ❩✳ ▼✉st❛❢❛ ✈➭ ❇✳ ❉♦ ➤ã✱ tr♦♥❣ t❤❐♣ ❦û q✉❛✱ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➲ ❣✐❛♥ t❤ó ✈Þ ♥❤✃t ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❙✐♠s ♥➝♠ ✷✵✵✻✳ t❤✉ ❤ót ➤➢ỵ❝ ♥❤✐Ị✉ sù ❝❤ó ý ❝đ❛ ❝➳❝ ♥❤➭ ♥❣❤✐➟♥ ❝ø✉✱ ➤➷❝ ❜✐Ưt ❧➭ ♥❤÷♥❣ ♥❣➢ê✐ ♥❣❤✐➟♥ ❝ø✉ ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❑ü t❤✉❐t t❤ø ❤❛✐ ❧➭ sư❛ ➤ỉ✐ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❝➳❝ ➳♥❤ ①➵✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ♥ã ➤ß✐ ❤á✐ ♣❤➯✐ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ ♥➭♦ ➤ã ✈➭ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ➤ã ➳♥❤ ①➵ ❝♦ ❝ã ➤✐Ĩ♠ ❜✃t ộ ột tr ữ ết q ợ t r❛ tõ ♣❤➢➡♥❣ ♣❤➳♣ ♥➭② ❧➭ ❝ñ❛ ❲✳ ❆✳ ❑✐r❦ ✈➭ ❝é♥❣ sù ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ✈➭♦ ✷✵✵✸ t❤➠♥❣ q✉❛ ❦❤➳✐ ♥✐Ư♠ ➳♥❤ ①➵ ❝②❝❧✐❝ ✈➭ ➤✐Ĩ♠ ①✃♣ ①Ø✳ ❙❛✉ trì ị ý ề ể ỉ tèt ♥❤✃t ✈➭ ➤➷❝ ❜✐Ưt ❧➭ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝②❝❧✐❝ ➤➲ trë t❤➭♥❤ ❝❤đ ➤Ị ♥❣❤✐➟♥ ❝ø✉ ré♥❣ r➲✐✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ❝❤ó♥❣ t➠✐ ➤➲ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥➭② ✈➭ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐ ✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✧✳ ■■✳ ▼ô❝ ➤Ý❝❤ ♥❣❤✐➟♥ ❝ø✉ ▼ô❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❝ã ❤Ư t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✐✈ ✈➭ tÝ♥❤ ❝❤✃t ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ tr➟♥ ❝➳❝ G✲♠➟tr✐❝ ✈➭ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ◆❣♦➭✐ r❛ ❝ß♥ tr×♥❤ ❜➭② ❝➳❝ ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ■■■✳ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ✲ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❞➲② G✲ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝✱ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ✲ P❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ♠è✐ q✉❛♥ ệ ữ ố tợ tr ị ý ề ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ■❱✳ ◆❤✐➟♠ ✈ô ♥❣❤✐➟♥ ❝ø✉ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❈❤♦ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❈❤♦ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ❱✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ✲ ❉ï♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❣✐➯✐ tÝ❝❤✱ t➠♣➠✱ ❣✐➯✐ tÝ❝❤ ❤➭♠✳ ✲ ❙ư ❞ơ♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❧ý t❤✉②Õt✱ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ t➭✐ ❧✐Ö✉ ✈➭ sư ❞ơ♥❣ ♠ét sè ❦ü t❤✉❐t ❝❤ø♥❣ ♠✐♥❤ ♠í✐ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ✈✃♥ ➤Ị ➤➷t r❛✳ ❱■✳ ❈✃✉ tró❝ ❧✉❐♥ ✈➝♥ ▲✉❐♥ ✈➝♥ ❣å♠ ✷ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ▼ơ❝ ✶✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ 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✱ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s❛✉✳ ▼ơ❝ ✷✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ➳♥❤ ①➵ φ✲❝♦ ❝②❝❧✐❝ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ò ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ❝➳❝ ➤Þ♥❤ ý ó r ò trì ệ q ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ▼ơ❝ ✶✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦✐Õ♥ t❤ø❝ ❝➡ ❜➯♥ ✈Ị ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝✱ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ tr♦♥❣ Gtr ủ r ò trì ❝➳❝ ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ▼ơ❝ ✷✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ P❤➬♥ ♥➭② ❝❤ó♥❣ t trì ột số ị ý ề ể t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ề ị ý ó r ò trì ❝➳❝ ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ✶ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ✶✳✶ G✲♠➟tr✐❝ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ 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✐ tơ✱ ❞➲② G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝đ❛ ❝❤ó♥❣ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s❛✉✳ ✶✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✷❪✮ ❈❤♦ X ❧➭ ♠ét t rỗ G : X ìX ìX 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) = , ✭tÝ♥❤ ❝❤✃t ➤è✐ ①ø♥❣✮✱ ✭●✺✮ 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, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣✱ ①Ðt ➳♥❤ + ①➵ G : X × X × X → R ❝❤♦ ❜ë✐ ✶✳✶✳✷ ❱Ý ❞ô✳ ✭❬✶✷❪✮ ❈❤♦ G(x, y, z) = max{d(x, y), d(y, z), d(x, z)} ✈í✐ ♠ä✐ x, y, z ∈ X (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❚❤❐t ✈❐②✱ t❛ ❦✐Ó♠ tr❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ t❛ ❝ã✿ ❑❤✐ ➤ã ✷ (G1) ❱× d(x, y), d(y, z), d(x, z)✱ ➤Ị✉ ❧➭ ❝➳❝ sè t❤ù❝ ❦❤➠♥❣ ➞♠ ✈í✐ ♠ä✐ x, y, z ∈ X ♥➟♥ G(x, y, z) = max{d(x, y), d(y, z), d(x, z)} ≥ ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ➜➻♥❣ t❤ø❝ ①➯② r❛ ❦❤✐ x = y = z ✳ (G2) G(x, x, y) = max{d(x, x), d(x, y), d(x, y)} = d(x, y) > ✈í✐ ♠ä✐ x, y ∈ X ♠➭ x = y ✳ (G3) G(x, x, y) = max{d(x, x), d(x, y), d(x, y)} = d(x, y)✳ ≤ max{d(x, y), d(y, z), d(x, z)} = G(x, y, z) ✈í✐ ♠ä✐ x, y, z ∈ X ♠➭ z = y✳ ✭G4) ❚Ý♥❤ ❝❤✃t ➤è✐ ①ø♥❣ ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥✳ ✭G5) ❱í✐ ♠ä✐ x, y, z, a ∈ X t❛ ❝ã G(x, y, z) = max{d(x, y), d(y, z), d(x, z)} ≤ max{d(x, a) + d(a, y), d(y, z), d(x, a) + d(a, z) + d(a, a)} ≤ max{d(x, a), d(x, a), d(a, a)} + max{d(a, y), d(a, z), d(y, z)} = G(x, a, a) + G(a, y, z) (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣✱ ①Ðt ➳♥❤ + ①➵ G : X × X × X → R ❝❤♦ ❜ë✐ ✶✳✶✳✸ ❱Ý ❞ô✳ ✭❬✶✷❪✮ ❈❤♦ G(x, y, z) = d(x, y) + d(y, z) + d(x, z) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❚❤❐t ✈❐②✱ ✈× d ❧➭ ♠ét ♠➟tr✐❝✱ ♥➟♥ tõ ➤➻♥❣ t❤ø❝ G(x, y, z) = d(x, y) + d(y, z) + d(x, z) ✈í✐ ♠ä✐ x, y, z ∈ X ❞Ô t❤✃② ➳♥❤ ①➵ G t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭G1 ✮✱✭G2 ✮✱✭G3 ✮✱✭G4 ✮ tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ G✲♠➟tr✐❝✳ ❚❛ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ ✭G5 ✮✳ ❱í✐ ❜✃t ❦ú x, y, z, a ∈ X ✱ ❦❤✐ ➤ã t❛ ❝ã G(x, y, z) = d(x, y) + d(y, z) + d(x, z) ≤ [d(x, a) + d(a, y) + d(y, z) + d(x, a) + d(a, z) + d(a, a)] = [d(x, a) + d(x, a) + d(a, a)] + [d(a, y) + d(y, z) + d(a, z)] = G(x, a, a) + G(a, y, z)✳ ❱❐② (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ✈➭ {xn } ❧➭ ♠ét ❞➲② tr♦♥❣ X ✳ ❚❛ ♥ã✐ ❞➲② {xn } ❧➭ G✲❤é✐ tơ tí✐ x ∈ X ♥Õ✉ lim G(x, xn , xm ) = ✶✳✶✳✹ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✷❪✮ ❈❤♦ n,m→∞ 0✱ ♥❣❤Ü❛ ❧➭ ✈í✐ sè ε > ❝❤♦ tr➢í❝ tå♥ t➵✐ ♠ét sè N ∈ N✱ s❛♦ ❝❤♦ G(x, xn , xm ) < ε ✈í✐ ♠ä✐ n, m ≥ N ✳ ▲ó❝ ➤ã ➤✐Ĩ♠ x ➤➢ỵ❝ ❣ä✐ ❧➭ ❣✐í✐ ❤➵♥ ❝đ❛ ❞➲② {xn } ✈➭ ✈✐Õt ❧➭ xn → x ❤♦➷❝ lim xn = x✳ ✸ ▼Ư♥❤ ➤Ị✳ ✶✳✶✳✺ ✭❬✶✷❪✮ ❈❤♦ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❈➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ➤➞② ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✭✶✮ ❉➲② {xn } ❧➭ G✲❤é✐ tơ ✈Ị x❀ ✭✷✮ G(xn , xn , x) → 0✱ ❦❤✐ n → ∞❀ ✭✸✮ G(xn , x, x) → 0✱ ❦❤✐ n → ∞❀ ✭✹✮ G(xm , xn , x) → 0✱ ❦❤✐ m, n → ∞✳ ⇒ ✭✷✮✳ ❱× {xn } ❧➭ G✲❤é✐ tơ tí✐ x ♥➟♥ G(x, xm , xn ) → ❦❤✐ n, m → ∞✳ ▼➷t ❦❤➳❝ G(xn , xn , x) = G(x, xn , xn ) ≤ G(x, xm , xn ) → ❦❤✐ n, m → ∞✳ ✭✷✮ ⇒ ✭✸✮✳ ◆❤ê ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶ t❛ ❝ã ❈❤ø♥❣ ♠✐♥❤✳ ✭✶✮ G(xn , x, x) ≤ G(x, xn , xn )+G(xn , x, xn ) = 2G(xn , xn , x) → ✭✸✮ ⇒ ✭✹✮✳ ❚❛ ❝ã G(xm , xn , x) ❦❤✐ n, m → ∞ ≤ G(xm , x, x)+G(x, xn , x) = G(xm , x, x)+ G(xn , x, x) → ❦❤✐ n, m → ∞✳ ✭✹✮ ⇒ ✭✶✮✳ ▲➭ ❤✐Ó♥ ♥❤✐➟♥✳ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❉➲② {xn } ❣ä✐ ❧➭ G✲❈❛✉❝❤② ♥Õ✉ ✈í✐ sè ε > ❝❤♦ tr➢í❝ tå♥ t➵✐ ♠ét sè N ∈ N✱ s❛♦ ❝❤♦ G(xm , xn , xl ) < ε ✈í✐ ♠ä✐ n, m, l ≥ N ✱ ♥❣❤Ü❛ ❧➭ G(xm , xn , xl ) → 0✱ ❦❤✐ n, m, l → ∞✳ ✶✳✶✳✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✼ ▼Ư♥❤ ➤Ị✳ ✭❬✶✷❪✮ ❈❤♦ ✭❬✶✷❪✮ ❈❤♦ (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✳ ❑❤✐ ➤ã ❝➳❝ ♠Ư♥❤ ➤Ị s❛✉ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✭✶✮ ❉➲② {xn } ❧➭ G✲❈❛✉❝❤②❀ ε > tå♥ t➵✐ sè tù ♥❤✐➟♥ N ∈ N s❛♦ ❝❤♦ G(xn , xm , xm ) < ε ✈í✐ ♠ä✐ n, m ≥ N ✳ ✭✷✮ ❱í✐ ♠ä✐ ⇒ ✭✷✮✳ ❱× {xn } ❧➭ G✲❈❛✉❝❤② ♥➟♥ ✈í✐ sè ε > ❝❤♦ tr➢í❝ tå♥ t➵✐ ♠ét sè N ∈ N✱ s❛♦ ❝❤♦ G(xm , xn , xl ) < ε ✈í✐ ♠ä✐ n, m, l ≥ N ✳ ▼➷t ❦❤➳❝ G(xn , xm , xm ) < G(xm , xn , xl ) ✈í✐ ♠ä✐ m = l ♥➟♥ G(xn , xm , xm ) < ε ✈í✐ ♠ä✐ n, m ≥ N ✳ ✭✷✮ ⇒ ✭✶✮✳ ❚õ ➤✐Ị✉ ❦✐Ư♥ ✭G5) tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ G✲♠➟tr✐❝ ❝❤♦ t❛ ❦Õt q✉➯ G(xm , xn , xl ) < G(xm , xn , xn ) + G(xn , xn , xl ) < 2ε ✈í✐ ♠ä✐ n, m, l ≥ N ♥➟♥ {xn } ❧➭ G✲❈❛✉❝❤②✳ ❈❤ø♥❣ ♠✐♥❤✳ ✭✶✮ ✸✸ ❈❤♦ n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✹✮ t❛ ➤➢ỵ❝ lim G(xn , xn , xn+1 ) = n→∞ ✭✷✳✷✺✮ {xn } ❦❤➠♥❣ ❧➭ ❞➲② G✲❈❛✉❝❤② t❤× ♥❤ê ▼Ư♥❤ ➤Ị ✶✳✶✳✼ t❛ s✉② r❛ tå♥ t➵✐ sè ε > ✈➭ ❝➳❝ ❞➲② ❝♦♥ {n(k)} ✈➭ {l(k)} ❝đ❛ ❞➲② sè tù ♥❤✐➟♥ ✈í✐ n(k) > l(k) > k ◆Õ✉ ❞➲② t❤á❛ ♠➲♥ G(xl(k) , xn(k) , xn(k) ) ≥ ε, tr♦♥❣ ➤ã ✭✷✳✷✻✮ {n(k)} ➤➢ỵ❝ ❝❤ä♥ ❧➭ sè ♥❤á ♥❤✃t t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✷✻✮ tr➟♥✱ tø❝ ❧➭ G(xl(k) , xn(k)−1 , xn(k)−1 ) < ε ✭✷✳✷✼✮ ✲ ❚õ ❤❛✐ ❝➠♥❣ t❤ø❝ ✭✷✳✷✻✮✱ ✭✷✳✷✼✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ✭G5✮✱ t❛ ➤➢ỵ❝ ε ≤ G(xl(k) , xn(k) , xn(k) ) < G(xl(k) , xn(k)−1 , xn(k)−1 ) + G(xn(k)−1 , xn(k) , xn(k) ) < ε + G(xn(k)−1 , xn(k) , xn(k) ) ❈❤♦ ✭✷✳✷✽✮ k → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✽✮ ✈➭ sö ❞ơ♥❣ ✭✷✳✷✷✮ t❛ ➤➢ỵ❝ lim G(xl(k) , xn(k) , xn(k) ) = ε n→∞ ✭✷✳✷✾✮ ✲ ❍➡♥ ♥÷❛✱ t❛ ❝ã G(xl(k) , xn(k) , xn(k) ) ≤ G(xl(k) , xn(k)+1 , xn(k)+1 ) + G(xn(k)+1 , xn(k) , xn(k) ) ✭✷✳✸✵✮ ✈➭ G(xl(k) , xn(k)+1 , xn(k)+1 ) ≤ G(xl(k) , xn(k) , xn(k) ) + G(xn(k) , xn(k)+1 , xn(k)+1 ) ✭✷✳✸✶✮ ❈❤♦ k → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✸✵✮✱ ✭✷✳✸✶✮ ✈➭ sư ❞ơ♥❣ ✭✷✳✷✷✮✱ ✭✷✳✷✺✮ ✈➭ ✭✷✳✷✾✮✱ t❛ t❤✉ ➤➢ỵ❝ lim G(xl(k) , xn(k)+1 , xn(k)+1 ) = ε n→∞ ✭✷✳✸✷✮ ✲ ❚➢➡♥❣ tù ♥❤➢ ✭✷✳✸✵✮ ✈➭ ✭✷✳✸✶✮✱ t❛ ❝ã G(xl(k)−1 , xn(k) , xn(k) ) ≤ G(xl(k)−1 , xl(k) , xl(k) ) + G(xl(k) , xn(k) , xn(k) ) ✭✷✳✸✸✮ ✸✹ ✈➭ G(xl(k) , xn(k) , xn(k) ) ≤ G(xl(k) , xl(k)−1 , xl(k)−1 ) + G(xl(k)−1 , xn(k) , xn(k) ) ✭✷✳✸✹✮ ❈❤♦ k → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✸✸✮✱ ✭✷✳✸✹✮ ✈➭ sư ❞ơ♥❣ ✭✷✳✷✷✮✱ ✭✷✳✷✺✮ ✈➭ ✭✷✳✷✾✮✱ t❛ ♥❤❐♥ ➤➢ỵ❝ lim G(xl(k)−1 , xn(k) , xn(k) ) = ε n→∞ ✭✷✳✸✺✮ ✲ ❍➡♥ ♥÷❛✱ t❛ ❝ã G(xl(k)−1 , xn(k)+1 , xn(k)+1 ) ≤ G(xl(k)−1 , xl(k) , xl(k) ) + G(xl(k) , xn(k) , xn(k) ) + G(xn(k) , xn(k)+1 , xn(k)+1 ) ✭✷✳✸✻✮ ✈➭ G(xl(k) , xn(k) , xn(k) ) ≤ G(xl(k) , xl(k)−1 , xl(k)−1 ) + G(xl(k)−1 , xn(k)+1 , xn(k)+1 ) + G(xn(k)+1 , xn(k) , xn(k) ) ❈❤♦ ✭✷✳✸✼✮ k → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✸✻✮✱ ✭✷✳✸✼✮ ✈➭ sư ❞ơ♥❣ ✭✷✳✷✷✮✱ ✭✷✳✷✺✮ ✈➭ ✭✷✳✷✾✮✱ t❛ ➤➢ỵ❝ lim G(xl(k)−1 , xn(k)+1 , xn(k)+1 ) = ε n→∞ ✭✷✳✸✽✮ ✲ ❚õ ❝➠♥❣ t❤ø❝ ✭✷✳✷✻✮ sư ❞ơ♥❣ ✭G3✮ ✈➭ ✭G5✮✱ t❛ ➤➢ỵ❝ ε ≤ G(xl(k) , xn(k) , xn(k) ) ≤ G(xn(k) , xl(k) , xl(k)+1 ) ✭✷✳✸✾✮ ≤ G(xn(k) , xl(k) , xl(k)+1 ) ≤ G(xn(k) , xl(k) , xl(k) ) + G(xl(k) , xl(k) , xl(k)+1 ) ❈❤♦ k → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✸✾✮✱ ✈➭ sư ❞ơ♥❣ ✭✷✳✷✷✮✱ ✭✷✳✷✺✮ ✈➭ ✭✷✳✷✾✮✱ t❛ t❤✉ ➤➢ỵ❝ lim G(xn(k) , xl(k) , xl(k)+1 ) = ε k→∞ ✭✷✳✹✵✮ ✸✺ ✷✳✷✳✷ ❑ý ❤✐Ö✉✳ ✲ ❑ý ❤✐Ö✉ ◆❤➽❝ ❧➵✐ r➺♥❣ tr♦♥❣ ♠ơ❝ ♥➭② t❛ ❝ị♥❣ sÏ ❞ï♥❣ ❝➳❝ ❦ý ❤✐Ö✉ s❛✉ Ψ ❧➭ ❤ä t✃t ❝➯ ❝➳❝ ❤➭♠ ψ : [0, ∞) → [0, ∞) t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✭ψ1 ✮ ψ ❧➭ ❤➭♠ ❧✐➟♥ tơ❝❀ ✭ψ2 ✮ ψ ❧➭ ❤➭♠ ❦❤➠♥❣ ❣✐➯♠❀ ✭ψ3 ✮ ψ(t) = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ t = 0✳ ✲ ❑ý ❤✐Ö✉ Λ ❧➭ ❤ä t✃t ❝➯ ❝➳❝ ❤➭♠ φ : [0, ∞) → [0, ∞) t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ư♥ s❛✉ ✭φ1 ✮ φ ❧➭ ❤➭♠ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐❀ ✭φ2 ✮ φ(t) = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ t = 0✳ ✷✳✷✳✸ ➜Þ♥❤ ❧ý✳ ✭❬✻❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ {Ai }m i=1 m ❧➭ ❤ä ❣å♠ ❝➳❝ t❐♣ ❝♦♥ Aj G✲➤ã♥❣ ❦❤➳❝ rỗ ủ X Y = T :Y Y j=1 ❧➭ ➳♥❤ ①➵ s❛♦ ❝❤♦ T (Aj ) ⊆ Aj+1 , j = 1, 2, · · · , m, ●✐➯ sư tå♥ t➵✐ ❤➭♠ ✈í✐ Am+1 = A1 ψ ∈ Ψ ✈➭ φ ∈ Λ s❛♦ ❝❤♦ ψ(G(T x, T y, T y)) ≤ ψ(M (x, y, y)) − φ(M (x, y, y)), ✈í✐ ♠ä✐ ✭✷✳✹✶✮ ✭✷✳✹✷✮ x ∈ Aj , y ∈ Aj+1 , j = 1, 2, · · · , m✱ tr♦♥❣ ➤ã M (x, y, y) = max{G(x, y, y), G(x, T x, T x), G(y, T y, T y), G(x, y, T x), [2G(x, T y, T y) + G(y, T x, T x)], [G(x, T y, T y) + 2G(y, T x, T x)]} ✭✷✳✹✸✮ m ❑❤✐ ➤ã✱ T Aj ✳ ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t t❤✉é❝ j=1 ➳♥❤ ①➵ t❤á❛ ♠➲♥ ✭✷✳✹✶✮✱ ✭✷✳✹✷✮ ✈➭ ✭✷✳✹✸✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ✭ψ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣✳ ✸✻ ❈❤ø♥❣ ♠✐♥❤✳ ❚r➢í❝ ❤Õt t❛ ❝❤Ø r❛ sù tå♥ t➵✐ ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ T ✳ ●✐➯ sö✱ t❛ ❧✃② x0 ∈ A1 ✈➭ ❞➲② {xn } ợ ị xn = T xn1 , ✈í✐ n = 1, 2, ✭✷✳✹✹✮ x ∈ A1 , x = T x ∈ A2 , x = T x ∈ A3 , · · · ✳ ◆Õ✉ xn0 +1 = xn0 ✈í✐ sè n0 ∈ N ♥➭♦ ➤ã✱ t❤× xn0 ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ T ✳ ●✐➯ sư xn+1 = xn ✈í✐ ♠ä✐ n ∈ N✳ ❑❤✐ ➤ã tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✹✷✮ ♥Õ✉ t❛ ❧✃② x = xn ✈➭ y = xn+1 ✱ t❤× t❛ ❝ã ❱× T ❧➭ ➳♥❤ ①➵ ❝②❝❧✐❝✱ ♥➟♥ t❛ ❝ã ψ(G(T xn , T xn+1 , T xn+1 )) = ψ(G(xn+1 , xn+2 , xn+2 )) ≤ ψ(M (xn , xn+1 , xn+1 )) − φ(M (xn , xn+1 , xn+1 )), ✭✷✳✹✺✮ tr♦♥❣ ➤ã M (xn , xn+1 , xn+1 ) = max{G(xn , xn+1 , xn+1 ), G(xn , T xn , T xn ), G(xn+1 , T xn+1 , T xn+1 ), G(xn , xn+1 , T xn ), [2G(xn , T xn+1 , T xn+1 ) + G(xn+1 , T xn , T xn )], [G(xn , T xn+1 , T xn+1 ) + 2G(xn+1 , T xn , T xn )]} = max{G(xn , xn+1 , xn+1 ), G(xn , xn+1 , xn+1 ), G(xn+1 , xn+2 , xn+2 ), G(xn , xn+1 , xn+1 ), [2G(xn , xn+2 , xn+2 ) + G(xn+1 , xn+1 , xn+1 )], [G(xn , xn+2 , xn+2 ) + 2G(xn+1 , xn+1 , xn+1 )]} = max{G(xn , xn+1 , xn+1 ), G(xn+1 , xn+2 , xn+2 ), G(xn , xn+2 , xn+2 )} ≤ max{G(xn , xn+1 , xn+1 ), G(xn+1 , xn+2 , xn+2 )} ✭✷✳✹✻✮ ◆Õ✉ M (xn , xn+1 , xn+1 ) = G(xn+1 , xn+2 , xn+2 ) t❤× tõ ❜✐Ĩ✉ t❤ø❝ ✭✷✳✹✺✮ t❛ ➤➢ỵ❝ ψ(G(xn+1 , xn+2 , xn+2 )) ≤ ψ(G(xn+1 , xn+2 , xn+2 )) − φ(G(xn+1 , xn+2 , xn+2 )) ✭✷✳✹✼✮ ✸✼ φ(G(xn+1 , xn+2 , xn+2 )) = 0✱ tø❝ ❧➭ G(xn+1 , xn+2 , xn+2 ) = 0✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ t❤✐Õt✱ xn+1 = xn ✈í✐ ♠ä✐ n ∈ N✳ ❱× ✈❐②✱ t❛ ❝ã ❉♦ ➤ã✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✹✼✮ t❛ s✉② r❛ M (xn , xn+1 , xn+1 ) = G(xn , xn+1 , xn+1 ) ✭✷✳✹✽✮ ❱× t❤Õ✱ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✹✺✮ trë t❤➭♥❤ ψ(G(xn+1 , xn+2 , xn+2 )) ≤ ψ(G(xn , xn+1 , xn+1 )) − φ(G(xn , xn+1 , xn+1 )) ≤ ψ(G(xn , xn+1 , xn+1 )) ✭✷✳✹✾✮ {G(xn , xn+1 , xn+1 )} ❧➭ ❞➲② ❦❤➠♥❣ ➞♠✱ ❦❤➠♥❣ t➝♥❣ ❤é✐ tơ ✈Ị L ≥ 0✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ L = 0✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ L > 0✳ ❑❤✐ ➤ã✱ ❜➺♥❣ ❝➳❝❤ ❧✃② lim sup tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✹✾✮✱ t❛ ❝ã ❱× ✈❐②✱ ❞➲② n→+∞ lim supψ(G(xn+1 , xn+2 , xn+2 )) ≤ lim supψ(G(xn , xn+1 , xn+1 )) n→+∞ n→+∞ − lim inf φ(G(xn , xn+1 , xn+1 )) n→+∞ ≤ lim supψ(G(xn , xn+1 , xn+1 )) ✭✷✳✺✵✮ n→+∞ ❇➞② ❣✐ê✱ sư ❞ơ♥❣ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ψ ✈➭ tÝ♥❤ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ ❝đ❛ φ✱ t❛ t❤✉ ➤➢ỵ❝ ψ(L) ≤ ψ(L) − φ(L) ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ s✉② r❛ ✭✷✳✺✶✮ φ(L) = 0✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ L = 0✱ tø❝ ❧➭ t❛ ❝ã lim G(xn , xn+1 , xn+1 ) = n→∞ ▼➷t ❦❤➳❝✱ ♥❤ê ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭G4✮ ✈➭ ✭G5✮ ❝đ❛ ✭✷✳✺✷✮ G✲♠➟tr✐❝✱ ✈í✐ n ∈ N t❛ ❝ã G(xn , xn−1 , xn−1 ) = G(xn−1 , xn−1 , xn ) ✭✷✳✺✸✮ ≤ G(xn−1 , xn , xn ) + G(xn , xn−1 , xn ) = 2G(xn−1 , xn , xn ) ì t ợ lim G(xn , xn−1 , xn−1 ) = n→∞ ✭✷✳✺✺✮ ✸✽ ❚✐Õ♣ t❤❡♦✱ t❛ ❝❤ø♥❣ ♠✐♥❤ ❞➲② {xn } ❧➭ ❞➲② G✲❈❛✉❝❤② tr♦♥❣ (X, G)✳ ●✐➯ sö✱ {xn } G✲❈❛✉❝❤②✳ ❑❤✐ ➤ã✱ ♥❤ê ▼Ư♥❤ ➤Ị ✶✳✶✳✼ tå♥ t➵✐ sè ε > ✈➭ ❝➳❝ ❞➲② ❝♦♥ {n(k)} ✈➭ {l(k)} ❝ñ❛ N s❛♦ ❝❤♦ n(k) > l(k) > k t❤á❛ ♠➲♥ ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❞➲② G(xl(k) , xn(k) , xn(k) ) ≥ ε, tr♦♥❣ ➤ã ✭✷✳✺✻✮ n(k) ➤➢ỵ❝ ❝❤ä♥ ❧➭ sè ♥❤á ♥❤✃t t❤á❛ ♠➲♥ ✭✷✳✺✻✮✱ ♥❣❤Ü❛ ❧➭ G(xl(k) , xn(k)−1 , xn(k)−1 ) < ε ✭✷✳✺✼✮ ❚õ ✭✷✳✺✻✮✱ ✭✷✳✺✼✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ✭G5✮✱ t❛ ➤➢ỵ❝ ε ≤ G(xl(k) , xn(k) , xn(k) ) ≤ G(xl(k) , xn(k)−1 , xn(k)−1 ) + G(xn(k)−1 , xn(k) , xn(k) ) < ε + G(xn(k)−1 , xn(k) , xn(k) ) ❈❤♦ k → ∞ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✺✽✮ ✈➭ sư ❞ơ♥❣ ✭✷✳✺✽✮ ✭✷✳✺✼✮✱ t❛ ➤➢ỵ❝ lim G(xl(k) , xn(k) , xn(k) ) = ε k→∞ ▲➢✉ ý r➺♥❣ ỗ k N tồ t s(k) tỏ ♠➲♥ ≤ s(k) ≤ m✱ s❛♦ ❝❤♦ n(k) − l(k) + s(k) ≡ 1(♠♦❞ m) ✭✷✳✻✵✮ k ➤đ ❧í♥✱ t❛ ❝ã r(k) = l(k) − s(k) > 0✱ ✈í✐ xr(k) ✈➭ xn(k) ♥➺♠ tr♦♥❣ ✷ t❐♣ ❧✐➟♥ t✐Õ♣ ♥❤❛✉ Aj ✈➭ Aj+1 t➢➡♥❣ ø♥❣✱ ✈í✐ j ♥➭♦ ➤ã ♠➭ ≤ j ≤ m✳ ❚õ ❝➠♥❣ t❤ø❝ ✭✷✳✹✷✮✱ ❜➺♥❣ ❝➳❝❤ t❤❛② x = xr(k) ✈➭ y = xn(k) ✱ t ợ ì (G(T xr(k) , T xn(k) , T xn(k) )) ≤ ψ(M (xr(k) , xn(k) , xn(k) )) − φ(M (xr(k) , xn(k) , xn(k) )), ✭✷✳✻✶✮ tr♦♥❣ ➤ã M (xr(k) ,xn(k) , xn(k) ) = max{G(xr(k) , xn(k) , xn(k) ), G(xr(k) , xr(k)+1 , xr(k)+1 ), G(xn(k) , xn(k)+1 , xn(k)+1 ), G(xr(k) , xn(k)+1 , xr(k)+1 ), [2G(xr(k) , xn(k)+1 , xn(k)+1 ) + G(xn(k) , xr(k)+1 , xr(k)+1 )], [G(xr(k) , xn(k)+1 , xn(k)+1 ) + 2G(xn(k) , xr(k)+1 , xr(k)+1 )]} ✭✷✳✻✷✮ ✸✾ ❙ư ❞ơ♥❣ ❇ỉ ➤Ị ✷✳✷✳✶✱ t❛ ➤➢ỵ❝ lim [2G(xr(k) , xn(k)+1 , xn(k)+1 ) + G(xn(k) , xr(k)+1 , xr(k)+1 )] = ε, k→∞ ✭✷✳✻✸✮ lim [G(xr(k) , xn(k)+1 , xn(k)+1 ) + 2G(xn(k) , xr(k)+1 , xr(k)+1 )] = ε k ì t t ợ () ψ(max(ε, 0, 0, ε, ε, ε)) − φ(max(ε, 0, 0, ε, ε, ε)) = ψ(ε) − φ(ε) ✭✷✳✻✺✮ φ(ε) = 0✱ s✉② r❛✱ ε = 0✳ ▼➞✉ t❤✉➱♥ ♥➭② ❝❤ø♥❣ tá ❞➲② {xn } ❧➭ ❞➲② G✲❈❛✉❝❤② tr♦♥❣ (X, G)✳ ❱× (X, G) ❧➭ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ♥➟♥ ❞➲② {xn } G✲❤é✐ tơ ✈Ị w ∈ X ✳ ❱× ✈❐②✱ t❛ ❝ã m w∈ Ai ✳ ❚❤❐t ✈❐②✱ ✈× x0 i=1 ∞ ∞ {xm(n−1) }∞ n=1 ⊂ A1 ✱ {xm(n−1)+1 }n=1 ⊂ A2 , · · · , {xm(n−1) }n=1 m Ai ✳ ❞♦ Ai ❧➭ G✲➤ã♥❣✱ ♥➟♥ t✃t ❝➯ ❝➳❝ ❞➲② ❝♦♥ tr➟♥ ❤é✐ tơ ✈Ị w ∈ i=1 ❚✐Õ♣ t❤❡♦ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ∈ A1 ✱ t❛ ❝ã ⊂ Am ✳ ▲➵✐ w ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✱ ♥❣❤Ü❛ ❧➭✱ w = T w✳ ❚õ ❝➠♥❣ t❤ø❝ ✭✷✳✹✷✮✱ ❜➺♥❣ ❝➳❝❤ t❤❛② x = xn ✈➭ y = w ✱ t❛ ➤➢ỵ❝ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ψ(G(T xn , T w, T w)) ≤ ψ(M (xn , w, w)) − φ(M (xn , w, w)), ✭✷✳✻✻✮ tr♦♥❣ ➤ã M (xn , w, w) = max{G(xn , w, w), G(xn , xn+1 , xn+1 ), G(w, T w, T w), G(xn , w, xn+1 ), [2G(xn , T w, T w) + G(w, xn+1 , xn+1 )], [G(xn , T w, T w) + 2G(w, xn+1 , xn+1 )]} ✭✷✳✻✼✮ ❈❤♦ n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✻✻✮✱ t❛ ♥❤❐♥ ➤➢ỵ❝ ψ(G(w, T w, T w)) ≤ ψ(G(w, T w, T w)) − φ(G(w, T w, T w)), φ(G(w, T w, T w)) = 0✳ G(w, T w, T w) = 0✱ ♥❣❤Ü❛ ❧➭ T w = w✳ ➤✐Ị✉ ♥➭② ❦Ð♦ t❤❡♦ ◆❤ê tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ ✭✷✳✻✽✮ φ t❛ s✉② r❛ ✹✵ ❈✉è✐ ❝ï♥❣✱ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ v ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦❤➳❝ ❝đ❛ u ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝đ❛ T ✳ ●✐➯ sö T ✱ tø❝ ❧➭✱ T v = v ✳ m ❱× t❛ ❝ã u, v ∈ Ai ✱ ♥➟♥ ❦❤✐ i=1 ❧✃② x = v ✈➭ y = w tr♦♥❣ ✭✷✳✹✷✮✱ t❛ ➤➢ỵ❝ ψ(G(T v, T w, T w)) ≤ ψ(M (v, w, w)) − φ(M (v, w, w)), ✭✷✳✻✾✮ tr♦♥❣ ➤ã M (v, w, w) = max{G(v, w, w), G(v, T v, T v), G(w, T w, T w), [2G(v, T w, T w) + G(w, T v, T v)], [G(v, T w, T w) + 2G(w, T v, T v)]} ▼➷t ❦❤➳❝✱ ❧✃② x = w ✈➭ y = v tr♦♥❣ ✭✷✳✹✷✮✱ t❛ ➤➢ỵ❝ ψ(G(T w, T v, T v)) ≤ ψ(M (w, v, v)) − φ(M (w, v, v)), ✭✷✳✼✵✮ ✭✷✳✼✶✮ tr♦♥❣ ➤ã M (w, v, v) = max{G(w, v, v), G(w, T w, T w), G(v, T v, T v), G(w, v, T w), [2G(w, T v, T v) + G(v, T w, T w)], [G(w, T v, T v) + 2G(v, T w, T w)]} ✭✷✳✼✷✮ ◆Õ✉ G(v, w, w) = G(w, v, v) t❤× ♥❤ê ✭✷✳✼✵✮ t❛ ❝ã M (v, w, w) = G(v, w, w) = G(w, v, v)✳ ❑❤✐ ➤ã✱ ♥❤ê ✭✷✳✻✾✮ t❛ ❝ã ψ(G(v, w, w)) ≤ ψ(G(v, w, w)) − φ(G(v, w, w)) ✭✷✳✼✸✮ G(w, v, v) = 0✱ s✉② r❛ w = v ✳ ◆Õ✉ G(v, w, w) > G(w, v, v)✱ t❤× ♥❤ê ✭✷✳✼✵✮ t❛ ❝ã M (v, w, w) = G(v, w, w)✳ ❘â r➭♥❣✱ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❦Ð♦ t❤❡♦ ❑❤✐ ➤ã✱ ♥❤ê ✭✷✳✻✾✮ t❛ ❝ã ψ(G(v, w, w)) ≤ ψ(G(v, w, w)) − φ(G(v, w, w)) ✭✷✳✼✹✮ G(w, v, v) = 0✱ s✉② r❛ w = v ✳ G(w, v, v) > G(v, w, w) t❤× ♥❤ê ✭✷✳✼✷✮ ❘â r➭♥❣✱ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❦Ð♦ t❤❡♦ rờ ợ ò ế t ó M (w, v, v) = G(w, v, v)✳ ❑❤✐ ➤ã✱ ♥❤ê ✭✷✳✼✶✮ t❛ ❝ã ψ(G(w, v, v)) ≤ ψ(G(w, v, v)) − φ(G(w, v, v)), ✭✷✳✼✺✮ ✹✶ ❇✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② ❦Ð♦ t❤❡♦ ❱× ✈❐②✱ T G(w, v, v) = 0✱ s✉② r❛ w = v ✳ ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ✳ ❱Ý ❞ô s❛✉ ➤➞② ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✷✳✸✳ −x ✈í✐ ♠ä✐ x ∈ X ✳ ▲✃② A = [−1, 0] ✈➭ B = [0, 1]✳ ❍➭♠ G : X ×X ×X → [0, +∞) ✷✳✷✳✹ ❱Ý ❞ô ✭❬✻❪✮ ❈❤♦ X = [−1, 1] ✈➭ ❤➭♠ T : X → X ➤➢ỵ❝ ❝❤♦ ❜ë✐ T x = ợ ị G (x, y, z) = |x − y| + |y − z| + |z − x|, ❘â r➭♥❣ ❤➭♠ ✈➭ ❤➭♠ x, y, z ∈ X ✭✷✳✼✻✮ φ : [0, ∞) → [0, ∞) ➤➢ỵ❝ ❝❤♦ t ψ : [0, ∞) → [0, ∞) ➤➢ỵ❝ ❝❤♦ ❜ë✐ ψ(t) = ✈í✐ ♠ä✐ G ❧➭ G✲♠➟tr✐❝ tr♦♥❣ X ✳ t ❜ë✐ φ(t) = t ≥ 0✳ ✈í✐ ♠ä✐ ❉Ơ t❤✃② r➺♥❣✱ ➳♥❤ ①➵ T ❳Ðt ❤➭♠ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✹✷✮✳ ❚❤❐t ✈❐②✱ ✈í✐ ❜✃t ❦ú x ∈ A ✈➭ y ∈ B t❛ ❝ã G(T x, T y, T y) = |T x − T y| + |T y − T y| + |T y − T x| = 2|T x − T y| = |y − x| ❉♦ ➤ã✱ t❛ ❝ã ψ(G(T x, T y, T y)) = |y − x| ✭✷✳✼✼✮ ❍➡♥ ♥÷❛✱ t❛ ❝ã M (x, y, y) = max{|x − y| + |y − y| + |y − x|, |x − T x| + |T x − T x| + |T x − x|, |y − T y| + |T y − T y| + |T y − y|, |x − y| + |T x − y| + |T x − x|, [2(|x − T y| + |T y − T y| + |T y − x|) + |y − T x| + |T x − T x| + |T x − y|], [|x − T y| + |T y − T y| + |T y − x| + 2(|y − T x| + |T x − T x| + |T x − y|)]} = max{2|x − y|, 2|T x − x|, 2|T y − y|, 1 [4|T y − x|) + 2|T x − y|], [2|T y − x|) + 4|T x − y|]} ✭✷✳✼✽✮ 3 ✹✷ ❚õ ✭✷✳✼✽✮✱ t❛ t❤✉ ➤➢ỵ❝ 2|x − y| ≤ M (x, y, y) ✭✷✳✼✾✮ ▼➷t ❦❤➳❝✱ t❛ ❝ã ψ(M (x, y, y)) − φ(M (x, y, y)) = M (x, y, y) M (x, y, y) 3M (x, y, y) − = 8 ✭✷✳✽✵✮ ❚õ ✭✷✳✼✾✮ ✈➭ ✭✷✳✽✵✮✱ t❛ s✉② r❛ r➺♥❣ 3|x − y| 3M (x, y, y) ≤ = ψ(M (x, y, y)) − φ(M (x, y, y)) ✭✷✳✽✶✮ ❉♦ ➤ã✱ tõ ✭✷✳✼✼✮ ✈➭ ✭✷✳✽✵✮✱ t❛ ♥❤❐♥ ➤➢ỵ❝ |y − x| 3|x − y| 3M (x, y, y) ≤ ≤ = ψ(M (x, y, y)) − φ(M (x, y, y)) ψ(G(T x, T y, T y)) = ✭✷✳✽✷✮ ❱× t❤Õ✱ ề ệ ủ ị ý ợ tỏ ❚❛ ❞Ơ ❞➭♥❣ t❤✃② r➺♥❣ ➤✐Ĩ♠ x = ∈ A ∩ B ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❝đ❛ ➳♥❤ ①➵ T ✳ ✷✳✷✳✺ ❍Ö q✉➯✳ ✭❬✻❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ {Ai }m i=1 m ❧➭ ❤ä ❣å♠ ❝➳❝ t❐♣ ❝♦♥ Ai ✈➭ T : Y Y Gó rỗ ủ X ✱ Y = ❧➭ i=1 ➳♥❤ ①➵ s❛♦ ❝❤♦ T (Ai ) ⊆ Ai+1 , i = 1, 2, · · · , m, ●✐➯ sư r➺♥❣ tå♥ t➵✐ ✈í✐ k ∈ (0, 1)✱ s❛♦ ❝❤♦ ➳♥❤ ①➵ T Am+1 = A1 t❤á❛ ♠➲♥ G(T x, T y, T z) ≤ kM (x, y, y), ✈í✐ ♠ä✐ ✭✷✳✽✸✮ ✭✷✳✽✹✮ x ∈ Ai , y ∈ Ai+1 , i = 1, 2, · · · , m❀ tr♦♥❣ ➤ã M (x, y, y) = max{G(x, y, y), G(x, T x, T x), G(y, T y, T y), [2G(x, T y, T y) + G(y, T x, T x)] [G(x, T y, T y) + 2G(y, T x, T x)]} ✭✷✳✽✺✮ ✹✸ m ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t Y = Ai ✳ i=1 φ : [0, ∞) → [0, ∞) ➤➢ỵ❝ ❝❤♦ ❜ë✐ φ(t) = (1 − k)t ψ : [0, ∞) → [0, ∞) ➤➢ỵ❝ ❝❤♦ ❜ë✐ ψ(t) = t ✈í✐ ♠ä✐ ❈❤ø♥❣ ♠✐♥❤✳ ❳Ðt ❤➭♠ ✈í✐ ♠ä✐ t ≥ ✈➭ ❤➭♠ t ≥ 0✳ ❑❤✐ ➤ã✱ ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr r ề ệ ủ ị ý ợ tỏ ì tế ụ ị ý t s✉② r❛ ➳♥❤ ①➵ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② m ♥❤✃t Y = Ai ✳ i=1 ✷✳✷✳✻ ❍Ö q✉➯✳ ✭❬✻❪✮ ❈❤♦ (X, G) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ {Ai }m i=1 m ❧➭ ❤ä ❣å♠ ❝➳❝ t❐♣ ❝♦♥ Ai ✈➭ T : Y → Y G✲➤ã♥❣ rỗ ủ X Y = i=1 ①➵ s❛♦ ❝❤♦ T (Ai ) ⊆ Ai+1 , i = 1, 2, · · · , m, ●✐➯ sö r➺♥❣ tå♥ t➵✐ ❝➳❝ ❤➺♥❣ sè tå♥ t➵✐ ψ ∈ Ψ s❛♦ ❝❤♦ ➳♥❤ ①➵ T ✈í✐ Am+1 = A1 ✭✷✳✽✻✮ a, b, c, d, e ✈í✐ < a + b + c + d + e < ✈➭ t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝ ψ(G(T x, T y, T y)) = aG(x, y, y) + bG(x, T x, T x) + cG(y, T y, T y) + d( [2G(x, T y, T y) + G(y, T x, T x)]) + e( [G(x, T y, T y) + 2G(y, T x, T x)]), ✈í✐ ♠ä✐ x ∈ Ai ✈➭ ✈í✐ ♠ä✐ y ∈ Ai+1 , i = 1, 2, · · · , m✳ m ❑❤✐ ➤ã✱ T ✭✷✳✽✼✮ ❝ã ♠ét ➤✐Ó♠ Ai ✳ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣ i=1 ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ❝ã aG(x, y, y) + bG(x, T x, T x) + cG(y, T y, T y) + d( [2G(x, T y, T y) + G(y, T x, T x)]) + e( [G(x, T y, T y) + 2G(y, T x, T x)]) ≤ (a + b + c + d + e)M (x, y, y), ✭✷✳✽✽✮ ✹✹ tr♦♥❣ ➤ã M (x, y, y) = max{G(x, y, y), G(x, T x, T x), G(y, T y, T y), [2G(x, T y, T y) + G(y, T x, T x)] [G(x, T y, T y) + 2G(y, T x, T x)]} ◆❤ê ❍Ö q✉➯ ✷✳✷✳✺✱ t❛ s✉② r❛ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ✳ ✭✷✳✽✾✮ ✹✺ ❑Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉ ✈Ị ➤Ị t➭✐ ✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤✱ ❝❤✉ ➤➳♦ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉✿ G✲♠➟tr✐❝✱ ❞➲② G✲❤é✐ tô✱ ❞➲② G✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤è✐ ①ø♥❣✱ ➳♥❤ ①➵ G✲❧✐➟♥ tơ❝✱ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ φ✲❝♦ ②Õ✉ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ✶✳ ❍Ư t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị✿ ❦❤➠♥❣ ❣✐❛♥ rì ó ệ tố ột số ị ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ φ✲❝♦ ②Õ✉ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ ✈➭ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ✳ ❝➳❝ ➳♥❤ ①➵ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè ➤Þ♥❤ ❧ý✱ ❤Ư q✉➯✱ ❜ỉ ➤Ị ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ G✲♠➟tr✐❝✱ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ φ✲❝♦ ②Õ✉ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤đ✱ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ✭ψ ✲φ✮✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ G✲♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❝❤➻♥❣ ❤➵♥ ♥❤➢ ➜Þ♥❤ ❧ý ✶✳✶✳✶✼✱ ➜Þ♥❤ ❧ý ✶✳✷✳✶✱ ❍Ư q✉➯ ✶✳✷✳✸✱ ➜Þ♥❤ ❧ý ✶✳✷✳✽✱ ➜Þ♥❤ ❧ý ✷✳✶✳✹✱ ❇ỉ ➤Ị ✷✳✷✳✶✱ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ ❍Ư q✉➯ ✷✳✷✳✻✳ ✹✳ ❚r×♥❤ ❜➭② ❝❤✐ t✐Õt ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✿ ❈➳❝ ❱Ý ❞ô ✶✳✶✳✷✱ ❱Ý ❞ô ✶✳✶✳✸ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✶✱ ❱Ý ❞ơ ✶✳✷✳✹ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✶✳✷✳✶✱ ❝➳❝ ❱Ý ❞ô ✶✳✷✳✾✱ ❱Ý ❞ô ✶✳✷✳✶✵ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✶✳✷✳✽✱ ❝➳❝ ❱Ý ❞ơ ✷✳✶✳✶✵✱ ❱Ý ❞ơ ✷✳✶✳✶✶ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✶✳✸✱ ❱Ý ❞ơ ✷✳✷✳✹ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✷✳✸✳ ✹✻ t➭✐ ❧✐Ư✉ t ỗ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ❨✳ ■✳ ❆❧❜❡r ❛♥❞ ❙✳ ●✉❡rr❡✲❉❡❧❛❜r✐❡r❡ ✭✶✾✾✼✮✱ ✧Pr✐♥❝✐♣❧❡ ♦❢ ✇❡❛❦❧② ❝♦♥✲ tr❛❝t✐✈❡ ♠❛♣s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✧✱ ✐♥ ◆❡✇ ❘❡s✉❧ts ✐♥ ❖♣❡r❛t♦r ❚❤❡♦r② ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✈♦❧✳ ✾✽ ♦❢ ❖♣❡r❛t♦r ❚❤❡♦r②✿ ❆❞✈❛♥❝❡s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✼➊✷✷✱ ❇✐r❦❤❛✉s❡r✱ ❇❛s❡❧✱ ❙✇✐t③❡r❧❛♥❞✳ ❬✸❪ ❍✳ ❆②❞✐ ✭✷✵✶✷✮✱ ✧❆ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t ♦❢ ✐♥t❡❣r❛❧ t②♣❡ ❝♦♥tr❛❝t✐♦♥ ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❏✳ ❆❞✈❛♥❝❡❞ ▼❛t❤✳ ❙t✉❞✐❡s✱ ✺ ✭✶✮ ✶✶✶➊✶✶✼✳ ❬✹❪ ❍✳ ❆②❞✐✱ ❆✳ ❋❡❧❤✐✱ ❙✳ ❙❛❤♠✐♠ ✭✷✵✶✼✮✱ ❘❡❧❛t❡❞ ❢✐①❡❞ ♣♦✐♥t r❡s✉❧ts ❢♦r ❝②❝❧✐❝ ❝♦♥tr❛❝t✐♦♥s ♦♥ G✲♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ❋✐❧♦♠❛t✱ ✸✶ ✭✸✮✱ ✽✺✸✲ ✽✻✾✳ ❬✺❪ ❱✳ ❇❡r✐♥❞❡ ✭✶✾✾✼✮✱ ❈♦♥tr❛❝✐✐ ●❡♥❡r❛❧✐③❛t✐✐ ❆♣❧✐❝❛✐✐✱ ✷✷✱ ❊❞✐t✉r❛ ❈✉❜ Pr❡ss✱ ❇❛✐❛ ▼❛r❡✳ ❬✻❪ ◆✳ ❇✐❧❣✐❧✐✱ ❊✳ ❑❛r❛♣✐♥❛r ✭✷✵✶✸✮✱ ✧❈②❝❧✐❝ ❝♦♥tr❛❝t✐♦♥s ✈✐❛ ❛✉①✐❧✐❛r② ❢✉♥❝t✐♦♥s ♦♥ G✲♠❡tr✐❝ s♣❛❝❡s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✷✵✶✸✱ ✭✹✾✮✱ ❞♦✐✿✶✵✳✶✶✽✻✴✶✻✽✼✲✶✽✶✷✲✷✵✶✸✲✹✾✱ ✶✻ ♣❛❣❡s✳ ❬✼❪ ◆✳ ❇✐❧❣✐❧✐✱ ■✳ ▼✳ ❊r❤❛♥✱ ❊✳ ❑❛r❛♣✐♥❛r✱ ❉✳ ❚✉r❦♦❣❧✉ ✭✷✵✶✹✮✱ ✧❈②❝❧✐❝ ❝♦♥✲ tr❛❝t✐♦♥s ❛♥❞ r❡❧❛t❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ♦♥ ▼❛t❤✳ ■♥❢✳ ❙❝✐✳✱ ✽ G✲♠❡tr✐❝ s♣❛❝❡s✧✱ ❆♣♣❧✳ ✭✹✮✱ ✶✺✹✶✲✶✺✺✶✳ ❬✽❪ ❉✳ ❲✳ ❇♦②❞ ❛♥❞ ❏✳ ❙✳ ❲✳ ❲♦♥❣ ✭✶✾✻✾✮✱ ✧❖♥ ♥♦♥❧✐♥❡❛r ❝♦♥tr❛❝t✐♦♥s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✷✵✱ ✹✺✽➊✹✻✹✳ ❬✾❪ ❊✳ ❑❛r❛♣✐♥❛r✱ ❆✳ ❨✐❧❞✐③✲❯❧✉s✱ ■✳ ▼✳ ❊r❤❛♥ ✭✷✵✶✷✮✱ ✧❈②❝❧✐❝ ❝♦♥tr❛❝t✐♦♥s ♦♥ ●✲♠❡tr✐❝ s♣❛❝❡s✧✱ ❆❜str✳ ❆♣♣❧✳ ❆♥❛❧✳✱ ❞♦✐✿✶✵✳✶✶✺✺✴✷✵✶✷✴✶✽✷✾✹✼✳ ✷✵✶✷✱ ❆rt✐❝❧❡ ■❉ ✶✽✷✾✹✼✱ ✶✺ ♣❛❣❡s✳ ✹✼ ❬✶✵❪ ❩✳ ▼✉st❛❢❛ ✭✷✵✵✺✮✱ ❆ ♥❡✇ str✉❝t✉r❡ ❢♦r ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r②✱ P❤✳❉✳ ❚❤❡s✐s✱ ❚❤❡ ❯♥✐✈❡rs✐t② ♦❢ ◆❡✇✲ ❝❛st❧❡✱ ❆✉str❛❧✐❛✳ ❬✶✶❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❇✳ ❙✐♠s ✭✷✵✵✹✮✱ ✧❙♦♠❡ r❡♠❛r❦s ❝♦♥❝❡r♥✐♥❣ ❉✲♠❡tr✐❝ s♣❛❝❡s✧✱ ✐♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✶✽✾➊✶✾✽✱ ❨♦❦♦❤❛♠❛✱ ❏❛♣❛♥✳ ❬✶✷❪ ❩✳ ▼✉st❛❢❛ ❛♥❞ ❇✳ ❙✐♠s ✭✷✵✵✻✮✱ ✧❆ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❏✳ ◆♦♥❧✐♥❡❛r ❛♥❞ ❈♦♥✈❡① ❆♥❛❧✳✱ ✼ ✭✷✮✱ ✷✽✾➊✷✾✼✳