Điểm bất động của các ánh xạ co suy rộng trong không gian meetric suy rộng và không gian meetric nón suy rộng

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❱➝♥ ❚➞♠ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✺ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❱➝♥ ❚➞♠ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ▲✉❐♥ ❱➝♥ ❚❤➵❝ ❙ü ❚♦➳♥ ❍ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ö ❆♥ ✲ ✷✵✶✺ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ✷ ❈❤➢➡♥❣ ✶✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤➢➡♥❣ ✷✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✷✳✶ ✷✽ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ➤Þ❛ ♣❤➢➡♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ✶✵ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ✷✽ ϕ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ❑Õt ❧✉❐♥ ✹✻ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✹✼ ❧ê✐ ♥ã✐ ➤➬✉ ▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ✈✃♥ ➤Ị ♥❣❤✐➟♥ ❝ø✉ q✉❛♥ trä♥❣ ❝đ❛ ❣✐➯✐ tÝ❝❤✳ ◆ã ❝ã r✃t ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ♥❣➭♥❤ t♦➳♥ ❤ä❝ ✈➭ ❝➳❝ ♥❣➭♥❤ ❦ü t❤✉❐t✳ ❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ♥ã✐ ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ➤ã ❧➭ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❇❛♥❛❝❤ ✭♥➝♠ ✶✾✾✷✮✳ ▼ét ❤➢í♥❣ ♥❤×♥ ❦❤➳❝ ✈Ị ♥❣❤✐➟♥ ❝ø✉ ➤✐Ĩ♠ ❜✃t ộ t ò t r ệ tì ể t ➤é♥❣ ❝đ❛ ♠ét ➳♥❤ ①➵ ❧➭ ✈✃♥ ➤Ị ❝ã ♥❤✐Ị✉ ø♥❣ ❞ô♥❣ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ♥❤✃t ❧➭ ❧ý t❤✉②Õt ❝➳❝ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✱ ♣❤➢➡♥❣ tr×♥❤ ➤➵♦ ❤➭♠ r✐➟♥❣✱ ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥✳ ◆❣➢ê✐ t❛ ➤➲ t×♠ ❝➳❝❤ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ♥➭② ❝❤♦ ♥❤✐Ò✉ ➳♥❤ ①➵ ✈➭ ♥❤✐Ò✉ ❧♦➵✐ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✳ ❑❛♥♥❛♥ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ♠ét ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ ❝♦ ♠➭ ♥ã ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ➳♥❤ ①➵ ✭♥➝♠ ✶✾✻✽✮✳ ◆➝♠ ✷✵✵✹✱ ❇❡r✐♥❞❡ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ế ó ũ ò ợ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❤➬✉ ❝♦ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ❙❛✉ ➤ã ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ ❦❤➳❝ t✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ t❤❡♦ ❤➢í♥❣ ♥➭② ✈➭ t❤✉ ➤➢ỵ❝ ♥❤✐Ị✉ ❦Õt q✉➯ t❤ó ✈Þ✳ ➜➷❝ ❜✐Ưt ✈➭♦ ♥➝♠ ✷✵✵✼✱ t❤❡♦ ❤➢í♥❣ ♠ë ré♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❍✉❛♥❣ ▲♦♥❣✲●✉❛♥❣ ✈➭ ❩❤❛♥❣ ❳✐❛♥ ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❜➺♥❣ ❝➳❝❤ t❤❛② ➤ỉ✐ t❐♣ sè t❤ù❝ tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ♠➟tr✐❝ ❜ë✐ ♠ét ♥ã♥ ➤Þ♥❤ ❤➢í♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✳ ❍❛✐ t➳❝ ❣✐➯ ❝ị♥❣ ➤➲ ①➞② ❞ù♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị sù ❤é✐ tơ ❝đ❛ ❞➲②✱ tÝ♥❤ ➤➬② ➤đ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥✱ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ t ợ ữ ết q tú ị tr➟♥ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♥➭②✱ ➤å♥❣ t❤ê✐ ❝ị♥❣ t❤✃② ➤➢ỵ❝ ♠ét sè ø♥❣ ❞ơ♥❣ ❝đ❛ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ tr♦♥❣ ❣✐➯✐ tÝ❝❤ ♣❤✐ t✉②Õ♥✱ tè✐ ➢✉ ✈Ð❝t➡✳ ❍✐Ö♥ ♥❛② ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤❛♥❣ t❤✉ ❤ót sù q✉❛♥ t➞♠ ❝đ❛ ♠ét sè ♥❤➭ t♦➳♥ ❤ä❝ tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝✳ ◆❣➢ê✐ t❛ ❝ị♥❣ ➤➲ t×♠ ❝➳❝❤ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❜➺♥❣ ❝➳❝❤ t❤❛② t❤Õ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ❜➺♥❣ t tú ì ữ t ể r ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr➟♥ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♥➭②✳ ➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✱ ❝❤ó♥❣ t➠✐ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥❤➺♠ t×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✧✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ ❝♦ ➤Þ❛ ♣❤➢➡♥❣✱ ➳♥❤ ①➵ ❦✐Ĩ✉ ϕ✲❝♦✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱✳✳✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ô❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❧✉❐♥ ✈➝♥✱ ❣å♠✿ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ➳♥❤ ①➵ ❝♦✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ♠ét sè ➤Þ♥❤ ❧ý ✈➭ ❤Ư q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦✳ ▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❝➳❝ ➤Þ♥❤ ❧ý✱ ❤Ư q✉➯ ✈➭ ✈Ý ❞ơ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ✈Ị✿ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ➳♥❤ ①➵ ❝♦ ➤Þ❛ ♣❤➢➡♥❣✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✈Ị ❝➳❝ ➳♥❤ ①➵ ❝♦ ➤Þ❛ ♣❤➢➡♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ▼ơ❝ ✷ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ trì ột số ị ý ệ q ề ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ϕ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝❤✉ ➤➳♦ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ t➳❝ ❣✐➯ ①✐♥ ❜➭② tá sù ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ❚❤➬②✳ ◆❤➞♥ ❞Þ♣ ♥➭② t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❚♦➳♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý t❤➬②✱ ❝➠ ❣✐➳♦ tr♦♥❣ ❜é ♠➠♥ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ❦❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ t❐♥ t×♥❤ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ◆❤➞♥ ➤➞②✱ t➳❝ ❣✐➯ ❝ò♥❣ ①✐♥ ❝➯♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤ã❛ ✷✶ ●✐➯✐ tÝ❝❤ t➵✐ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✳ ❈✉è✐ ❝ï♥❣✱ t➳❝ ❣✐➯ ①✐♥ t ì ố ẹ ị ❡♠ ✈➭ t✃t ❝➯ ❜➵♥ ❜❒ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ ❣✐ó♣ ➤ì 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❛✐ sót ợ ữ ý ế ó ❣ã♣ ❝đ❛ q✉ý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥✳ ❱✐♥❤✱ ♥❣➭② ✵✽ t❤➳♥❣ ✵✽ ♥➝♠ ✷✵✶✺ ◆❣✉②Ơ♥ ❱➝♥ ❚➞♠ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ 1.1 ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❧✉❐♥ ✈➝♥✱ ❝➳❝ ♠è✐ q✉❛♥ ❤Ư ❣✐÷❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠✱ ❦Õt q✉➯ tr➟♥ ✈➭ ❝❤♦ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ 1.1.1 ➜Þ♥❤ ♥❣❤Ü❛✳ ♠ét ♠➟tr✐❝ tr➟♥ ✭❬✶❪✮ ❈❤♦ t❐♣ ❤ỵ♣ X = d : X ì X R ợ ❣ä✐ ❧➭ X ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮ d(x, y) ≥ ✈í✐ ♠ä✐ x, y ∈ X ✈➭ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ ✭✷✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✸✮ d(x, y) ≤ d(x, z) + d(z, y) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ❚❐♣ X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❦Ý ❤✐Ö✉ ❧➭ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤✐Ĩ♠ x ➤Õ♥ ➤✐Ĩ♠ y ✳ 1.1.2 ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶❪✮ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ✈➭ (Y, ρ)✳ ➳♥❤ ①➵ f : (X, d) → (Y, ρ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ α ∈ [0, 1) s❛♦ ❝❤♦ ρ[f (x) , f (y)] ≤ αd (x, y) , 1.1.3 ➜Þ♥❤ ❧ý✳ ✈í✐ ♠ä✐ ✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✮ ●✐➯ sö x, y ∈ X (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ f : X → X ❧➭ ➳♥❤ ①➵ ❝♦ tõ X ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑❤✐ ➤ã tå♥ t➵✐ ❞✉② ♥❤✃t ➤✐Ó♠ x∗ ∈ X ✳ ➜✐Ó♠ ①➵ f✳ x∗ ∈ X ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ▼ë ré♥❣ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✱ P✳ ◆✳ ❉✉tt❛✱ ❇✳ ❙✳ r t ợ ết q s 1.1.4 ị ý ✭❬✻❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T :X→X ❧➭ ♠ét tù ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝✿ ψ (d (T x, T y)) ≤ ψ (d (x, y)) − ϕ (d (x, y)) ✈í✐ ♠ä✐ x, y ∈ X, tr♦♥❣ ➤ã ψ, ϕ : [0, +∞) → [0, +∞) ❧➭ ❝➳❝ ❤➭♠ ❧✐➟♥ tơ❝✱ ➤➡♥ ➤✐Ư✉ ❦❤➠♥❣ ❣✐➯♠ ✈➭ ψ(t) = ϕ(t) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t ❂ ✵✳ ❑❤✐ ➤ã T ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ 1.1.5 ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✺❪✮ ❈❤♦ t❐♣ ❤ỵ♣ ♠ét ♠➟tr✐❝ s✉② ré♥❣ tr➟♥ X = φ✳ ❍➭♠ d : X × X → R ➤➢ỵ❝ ❣ä✐ ❧➭ X ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ö♥ ✭✶✮ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ❀ ✭✷✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ❀ ✭✸✮ d(x, y) ≤ d(x, w) + d(w, z) + d(z, y) ✈í✐ ♠ä✐ x, y ∈ X ✈➭ ✈í✐ ♠ä✐ ❝➷♣ ➤✐Ĩ♠ ♣❤➞♥ ❜✐Öt ❚❐♣ w, z ∈ X \ {x, y}✳ X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ s✉② ré♥❣ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦Ý ❤✐Ö✉ ❧➭ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ➜✐Ị✉ ❦✐Ư♥ ✭✸✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝✳ 1.1.6 ◆❤❐♥ ①Ðt✳ ✭❬✷❪✮ ●✐➯ sö (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❱í✐ x ∈ X ✈➭ ε > t❛ ❦ý ❤✐Ö✉ B(x, ε) = {y ∈ X : d(x, y) < ε}✳ ❑❤✐ ➤ã ❤ä B = {B(x, r) : x ∈ X, r > 0} ❧❐♣ t❤➭♥❤ ♠ét ❝➡ së ❝ñ❛ ♠ét t➠♣➠ τd tr➟♥ X ✳ 1.1.7 ❱Ý ❞ô✳ s❛♦ ❝❤♦ ✭❬✶✷❪✮ ❳Ðt X = {t, 2t, 3t, 4t, 5t} ✈í✐ t > ❧➭ ❤➺♥❣ sè✳ ❈❤♦ sè γ ∈ X γ > 0✳ ❚❛ ①➳❝ ➤Þ♥❤ ❤➭♠ d : X × X → R ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ ✭❛✮ d(x, x) = ✈í✐ ♠ä✐ x ∈ X ✳ ✭❜✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭❝✮ d(t, 2t) = 3γ ✳ ✭❞✮ d(t, 3t) = d(2t, 3t) = γ ✳ ✭❡✮ d(t, 4t) = d(2t, 4t) = d(3t, 4t) = 2γ ✳ ✭❢✮ d(t, 5t) = d(2t, 5t) = d(3t, 5t) = d(4t, 5t) = 23 γ ✳ ❑❤✐ ➤ã ❞Ô ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ r➺♥❣ ré♥❣✱ ♥❤➢♥❣ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② (X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ✈× t❛ ❝ã d(t, 2t) = 3γ > γ + γ = d(t, 3t) + d(3t, 2t) 1.1.8 ❱Ý ❞ô✳ ✭❬✶✷❪✮ ❳Ðt X = n : n = 1, 2, ∪ {0, 2}✳ ❚❛ ị d : X ì X R+ ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝   ♥Õ✉     ♥Õ✉ n d(x, y) =  ♥Õ✉  n    ♥Õ✉ x = y, x ∈ {0, 2} ✈➭ y = n1 , x= y ∈ {0, 2} (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ (X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ✈× t❛ ❝ã d 1.1.9 ✈➭ x, y t❤✉é❝ trờ ợ ò ó ễ tử t❤✃② r➺♥❣ ♥❤➢♥❣ n ➜Þ♥❤ ♥❣❤Ü❛✳ 1 1 1 , = > + = d , + d 0, 3 ✭❬✺❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❞➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❤é✐ tơ ✈Ị ➤✐Ĩ♠ x ∈ X ♥Õ✉ ✈í✐ ♠ä✐ ε > tå♥ t➵✐ n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n ≥ n0 t❛ ❝ã d (xn , x) < ε✳ ▲ó❝ ➤ã t❛ ❦Ý ❤✐Ư✉ ❧➭ lim xn = x ❤❛② n→+∞ xn → x ❦❤✐ n → +∞✳ 1.1.10 ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✺❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❞➲② {xn } ⊂ X ✳ ❚❛ ♥ã✐ r➺♥❣ {xn } ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ (X, d) ♥Õ✉ ✈í✐ ♠ä✐ ε > 0✱ tå♥ t➵✐ nε ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n > m ≥ nε ✱ t❛ ❝ã d(xn , xm ) < ε✳ 1.1.11 ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✺❪✮ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤đ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ (X, d) ➤Ị✉ ❤é✐ tơ tr♦♥❣ ♥ã✳ ❚➢➡♥❣ tù ♥❤➢ tr➢ê♥❣ ❤ỵ♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ♥❣➢ê✐ t❛ ➤➲ t❤✉ ➤➢ỵ❝ ❝➳❝ ❦Õt q✉➯ s❛✉✳ 1.1.12 ▼Ư♥❤ ➤Ị✳ ✭❬✶✷❪✮ ◆Õ✉ {xn } ❧➭ ❞➲② ❤é✐ tô tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ t❤× ♥ã ❧➭ ❞➲② ❈❛✉❝❤②✳ 1.1.13 ▼Ư♥❤ ➤Ị✳ ✭❬✶✷❪✮ ◆Õ✉ {xn } ❧➭ ♠ét ❞➲② ❤é✐ tô tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ t❤× ❣✐í✐ ❤➵♥ ❝đ❛ ♥ã ❧➭ ❞✉② ♥❤✃t✳ 1.1.14 ▼Ư♥❤ ➤Ị✳ ✭❬✶✷❪✮ ◆Õ✉ {xn } ❧➭ ❞➲② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ X ♠➭ ♥ã ❤é✐ tơ ✈Ị ➤✐Ĩ♠ x ∈ X ✱ t❤× ♠ä✐ ❞➲② ❝♦♥ {xnk } ❝đ❛ ♥ã ❝ị♥❣ ❤é✐ tơ ✈Ị ➤✐Ĩ♠ x✳ ✭❬✸❪✮ ●✐➯ sư 1.1.15 ➜Þ♥❤ ♥❣❤Ü❛✳ ♠➟tr✐❝ (X, d) ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ➜✐Ó♠ ①➵ T, f : X → X ❧➭ ❝➳❝ ➳♥❤ ①➵ tõ ❦❤➠♥❣ ❣✐❛♥ y X ợ ọ trị t ♦❢ ❝♦✐♥❝✐❞❡♥❝❡✮ ❝ñ❛ ❤❛✐ ➳♥❤ T ✈➭ f tr➟♥ X ♥Õ✉ tå♥ t➵✐ x ∈ X s❛♦ ❝❤♦ y = f (x) = T (x)✳ ❑❤✐ ➤ã ➤✐Ó♠ x ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ✭❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t✮ ❝đ❛ ❤❛✐ ➳♥❤ ①➵ ❈➷♣ ➳♥❤ ①➵ (T, f ) ➤➢ỵ❝ ❣ä✐ ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ ♥Õ✉ T ✈➭ f ❣✐❛♦ ❤♦➳♥ ✈í✐ ♥❤❛✉ t➵✐ ❝➳❝ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝❤ó♥❣✱ ♥❣❤Ü❛ ❧➭ ➤✐Ĩ♠ T ✈➭ f ✳ T f (x) = f T (x) t➵✐ ❝➳❝ x ∈ X ♠➭ T (x) = f (x)✳ 1.1.16 ➜Þ♥❤ ♥❣❤Ü❛✳ ré♥❣ ✈➭ ✭❬✶✶❪✮ ●✐➯ sö X = φ✳ ◆Õ✉ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② (X, ) ❧➭ ♠ét t❐♣ ❤ỵ♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈í✐ q✉❛♥ ❤Ư t❤ø tự tì (X, d, ) ợ ọ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ t❤ø tù✳ ❑❤✐ ➤ã✱ ❤❛✐ ♣❤➬♥ tư x, y ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ s♦ s➳♥❤ ợ ế x 1.1.17 ị ĩ T, f : X → X ✳ ✭❬✶✶❪✮ ❈❤♦ y ❤❛② y x✳ (X, ) ❧➭ t❐♣ ❤ỵ♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ ❤❛✐ 34 ❇➢í❝ ✶✳ ❈❤♦ x ∈ X ✳ ❱× X ❧➭ ❦❤➠♥❣ ❣✐❛♥ 2c ✲❦❤➯ ①Ý❝ ♥➟♥ ú t ó tể tì ợ ột số ữ ❝➳❝ ➤✐Ó♠ ✭n(x) ➤✐Ó♠✮ x = x0 , x1 , x2 , , xn−1 , xn(x) = T x, s❛♦ ❝❤♦ c ✈í✐ ♠ä✐ i = 1, 2, , n(x)✳ ❑❤➠♥❣ ♠✃t tÝ♥❤ tỉ♥❣ q✉➳t✱ t❛ ❣✐➯ sư r➺♥❣ ❝➳❝ ➤✐Ó♠ x1 , x2 , , xn(x) ❧➭ ❦❤➳❝ ♥❤❛✉ ✭✈➭ ❝❤ó♥❣ ❦❤➳❝ ✈í✐ x ✈➭ T x ♥Õ✉ d(xi−1 , xi ) n(x) > 2✮✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ d(x, T x) ❚❤❐t ✈❐②✱ tõ ❑❤✐ ♠➭ n(x).c (2.2) (2.1) t❛ t❤✃② r➺♥❣ ➤✐Ò✉ ♥➭② ❧➭ ❤✐Ó♥ ♥❤✐➟♥ ♥Õ✉ n(x) = 1, 2✳ n(x) > 2✱ t❛ ①Ðt ❤❛✐ tr➢ê♥❣ ❤ỵ♣ s❛✉✳ ❚r➢ê♥❣ ❤ỵ♣ ✶✳ ❱í✐ n(x) ❧➭ sè ❧❰✱ ❦❤✐ ➤ã ➤➷t n(x) = 2l + ✈í✐ ♠ä✐ l ≥ 1✳ ▲ó❝ ➤ã t❛ ❝ã d(x, T x) ≤ d(x, x1 ) + d(x1 , x2 ) + + d(x2l , T x) n(x).c c (2l + 1) = 2 ❚r➢ê♥❣ ❤ỵ♣ ✷✳ ❱í✐ n(x) ❧➭ sè ❝❤➼♥✱ ❦❤✐ ➤ã ➤➷t n(x) = 2l ✈í✐ ♠ä✐ l ≥ 2✳ ▲ó❝ ➤ã✱ ♥❤ê ✭✷✳✶✮✱ t❛ ❝ã d(x, T x) ≤ d(x, x2 ) + d(x2 , x3 ) + + d(x2l−1 , T x) c n(x).c c + (2l − 2) = 2 ▲➵✐ ✈× T ❧➭ (c, λ)✲❝♦ ➤Þ❛ ♣❤➢➡♥❣ ➤Ị✉✱ ♥➟♥ t❛ ❝ã d(T xi−1 , T xi ) ≤ λd(xi−1 , xi ) c λ , ✈í✐ ♠ä✐ i = 1, 2, , n(x) ❉♦ ➤ã✱ ♥❤ê ♣❤Ð♣ q✉② ♥➵♣ ❦❤➠♥❣ ❤♦➭♥ t♦➭♥ t❛ ❝ã d(T m xi−1 , T m xi ) c λm , ✈í✐ ♠ä✐ m ∈ N ❱× t❤Õ✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✮ t❛ ❝ã d(T m x0 , T m x2 ) λm c 35 ❇➞② ❣✐ê ❧➭♠ t➢➡♥❣ tù ♥❤➢ tr➟♥✱ ❝❤ó♥❣ t❛ ❝ã t❤Ó t❤✃② r➺♥❣ d(T m x, T m+1 x) λm n(x).c , ✈í✐ ♠ä✐ m ∈ N (2.3) T m x0 , , T m xn ❧➭ ❜➺♥❣ ❈❤ó ý r➺♥❣ ♥❣❛② ❝➯ ❦❤✐ ♠ét sè ➤✐Ĩ♠ tr♦♥❣ ❝➳❝ ➤✐Ĩ♠ ♥❤❛✉✱ t❤× ❦Õt q✉➯ tr➟♥ ✈➱♥ ➤ó♥❣✳ ❇➢í❝ ✷✳ ➜➬✉ t✐➟♥✱ ❝❤ó♥❣ t❛ ❧➢✉ ý r➺♥❣ ♥Õ✉ T m x = T n x ✈í✐ ❝➳❝ sè ♥➭♦ ➤ã m, n ∈ N, m > n✱ t❤× ❜➺♥❣ ❝➳❝❤ ➤➷t p = m − n ✈➭ u = T n x✱ t❛ ❝ã T p u = u ✈➭ ♥❤➢ ✈❐② T kp u = u ✈í✐ ♠ä✐ k ∈ N✳ ❇➞② ❣✐ê ❧✃② ➤✐Ĩ♠ u ✈➭ T u ✈➭ t✐Õ♥ ❤➭♥❤ ♥❤➢ tr♦♥❣ ❇➢í❝ ✶✱ t❛ ❝ã t❤Ó t❤✃② r➺♥❣ m d(T u, T ✈➭ ✈í✐ sè ❝è ➤Þ♥❤ m+1 u) λm n(u).c , ✈í✐ ♠ä✐ m ∈ N n(u) ∈ N ♥➭♦ ➤ã ✭n(u) ♣❤ô t❤✉é❝ ✈➭♦ u✮✳ ❑❤✐ ➤ã✱ t❛ ❝ã λkp n(u).c d(u, T u) = d(T kp u, T kp+1 u) ❱× P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝✱ tõ ➤ã t❛ ❝ã d(u, T u) = d(T kp u, T kp+1 u) ≤ K ❱× λ ∈ [0, 1)✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ s✉② r❛ K λ λkp n(u) c kp n(u) c → ❦❤✐ k → ∞✳ ❙✉② r❛ d(u, T u) = 0✱ ♥❣❤Ü❛ ❧➭ t❛ ❝ã T u = u✳ ❇➞② ❣✐ê t❛ ❣✐➯ sö r➺♥❣ r➺♥❣ T p x = T q x✱ ✈í✐ ♠ä✐ p, q ∈ N✳ ❑❤✐ ➤ã t❛ sÏ ❝❤Ø r❛ {T m x} ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ X ✳ ❚r➢í❝ ❤Õt t❛ ❝ã ♥❤❐♥ ①Ðt s❛✉✿ ❣✐➯ sư ➤➲ ❝❤ä♥ ➤➢ỵ❝ sè k ∈ N ♠➭ k ≥ 2✱ s❛♦ ❝❤♦ λk < d(T k x, T k+1 x) ✱ n(x) λk n(x).c ❦❤✐ ➤ã ♥❤ê ✭✷✳✸✮ t❛ ❝ã c ✈➭ d(T k+1 x, T k+2 x) λk+1 n(x).c c ❱× t❤Õ✱ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✮ t❛ s✉② r❛ d(T k x, T k+2 x) c ❇➞② ❣✐ê ✈í✐ ❜✃t ❦ú sè ♥❣✉②➟♥ ❞➢➡♥❣ ♠➭ tr➢ê♥❣ ❤ỵ♣✳ (2.4) m > k ✱ ♠ét ❧➬♥ ♥÷❛ t❛ ①Ðt ❤❛✐ 36 ❚r➢ê♥❣ ❤ỵ♣ ✶✳ ◆Õ✉ n ❧➭ sè ❧❰✱ t❤× t❛ ➤➷t n = 2l + 1✱ l ≥ 0✳ ❑❤✐ ➤ã t❛ ❝ã d(T m x, T m+n x) ≤ d(T m x, T m+1 x) + d(T m+1 x, T m+2 x) + + d(T m+2l x, T m+2l+1 x) n.c (λm + λm+1 + + λm+2l ) λm n.c 1−λ rờ ợ ế n số tì t❛ ➤➷t n = 2l✱ l ≥ 1✳ ❑❤✐ ➤ã tõ ❧❐♣ ❧✉❐♥ tr➢í❝ ❝➠♥❣ t❤ø❝ ✭✷✳✹✮ ✈➭ ♥❤ê ✭✷✳✶✮ t❛ ❝ã d(T m x, T m+n x) ≤ d(T m x, T m+2 x) + d(T m+2 x, T m+3 x) + + d(T m+2l−1 x, T m+2l x) n.c λm n(x).c + (λm+2 + + λm+2l−1 ) m+2 λ n.c λm+2 n.c λm−k λk n(x).c + λm−k c + 1−λ 1−λ λm−k c = [2 − 2λ + nλk−2 ] 2(1 − λ) ❑Õt ❤ỵ♣ ❝➯ ❤❛✐ tr➢ê♥❣ ❤ỵ♣✱ ❝❤ó♥❣ t❛ ❝ã d(T m x, T m+n x) tr♦♥❣ ➤ã λm−k c β, 2(1 − λ) β = max{nλk , − 2λ + nλk−2 }✳ ❱× P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ❝ã d(T m x, T m+n x) ≤ K ❉♦ ❣✐➯ t❤✐Õt λm−k c β 2(1 − λ) λ ∈ [0, 1) ì k ố ị tỏ λk < n(x) ♥➟♥ λm−k → ❦❤✐ m−k c m → ∞✳ ❙✉② r❛ t❛ ❝ã K λ2(1−λ) → ❦❤✐ m → ∞✳ ❱× t❤Õ t❛ ❝ã d(T m x, T m+n x) → ❦❤✐ m → ∞✳ ➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá r➺♥❣ {T m x} ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱× sư r➺♥❣ ♥➟♥ X ❧➭ ❦❤➠♥❣ ❣✐❛♥ T ✲q✉ü ➤➵♦ ➤➬② ➤đ✱ ♥➟♥ {T m x} ❧➭ ❤é✐ tơ tr♦♥❣ X ✱ ❣✐➯ lim T m x = u✳ ❍➡♥ ♥÷❛✱ ì ỗ ị ề tô❝✱ m→∞ T ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝✳ ❉♦ ➤ã t❛ ❝ã T (u) = T ( lim T m x) = lim T m+1 x = u m→∞ ➜✐Ò✉ ♥➭② ❦❤➻♥❣ ➤Þ♥❤ r➺♥❣ m→∞ u ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ❇➢í❝ ✸✳ ➜Ĩ ❦✐Ĩ♠ tr❛ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ t❛ ❣✐➯ sư r➺♥❣ v ❝ị♥❣ ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✱ ♥❣❤Ü❛ ❧➭ T v = v ✳ ❱× X ❧➭ ❦❤➠♥❣ 2c 37 í t ó tể tì ợ ♠ét 2c ✲①Ý❝ u = x0 , x1 , , xn = v ✳ ❑❤✐ ➤ã✱ ❜➺♥❣ ❝➳❝❤ t✐Õ♥ ❤➭♥❤ ♥❤➢ tr♦♥❣ ❇➢í❝ ✶✱ ❝❤ó♥❣ t❛ ❝ã t❤Ĩ t❤✃② r➺♥❣ d(T m u, T m v) λm n.c , ✈í✐ ♠ä✐ m ∈ N ❱× t❤Õ✱ t❛ ❝ã d(u, v) = d(T m u, T m v) ❉♦ λm n.c P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ❝ã d(u, v) = d(T m u, T m v) ▲➵✐ ✈× λ ∈ [0, 1)✱ ♥➟♥ tr➟♥ t❛ s✉② r❛ λm n c λm n c → ❦❤✐ m → ∞✳ ❱× t❤Õ tõ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ d(u, v) = 0✳ ❉♦ ➤ã u = v ✳ ◆❤➢ ✈❐② ị ý ợ ứ 2.1.19 í ụ X = {a, b, c, e} ✈➭ P = {x ∈ R : x ≥ 0}✱ ❚❛ ①➳❝ ➤Þ♥❤ ❤➭♠ d : X → R ➤➢ỵ❝ ❝❤♦ ❜ë✐ d(a, b) = 25, d(a, c) = d(b, c) = 1, d(a, e) = d(b, e) = d(c, e) = 2, d(x, x) = 0, ✈í✐ ♠ä✐ x ∈ X ❑❤✐ ➤ã ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✈➭ P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝✱ ♥❤➢♥❣ (X, d) ❦❤➠♥❣ ♣❤➯✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✈× d(a, b) = 25 > d(a, c) + d(b, c) = 2✳ ❍➡♥ ♥÷❛✱ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ 2ε ✲❦❤➯ ①Ý❝ ✈í✐ ε = 4✳ ❇➞② ❣✐ê t❛ ①Ðt ➳♥❤ ①➵ T : X → X ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ Tx = c ♥Õ✉ x ∈ {a, b, c}, a ♥Õ✉ x = e ❇➺♥❣ ❝➳❝ tÝ♥❤ t♦➳♥ trù❝ t✐Õ♣ t❛ t❤✃② r➺♥❣ ➤đ ✈➭ ➳♥❤ ①➵ ♥÷❛✱ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ T ✲q✉ü ➤➵♦ ➤➬② T : X → X ❧➭ ♠ét ➳♥❤ ①➵ ❝♦ ➤Þ❛ ♣❤➢➡♥❣ ✈í✐ ε = ✈➭ λ = 21 ✳ ❍➡♥ T tỏ ề ệ ì ụ ị ❧ý ✷✳✶✳✶✽ t❛ s✉② r❛ r➺♥❣ T ✈➭ ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❧➭ c✳ 38 2.2 ➜✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ϕ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ 2.2.1 ❇ỉ ➤Ị✳ ❈❤♦ P ❧➭ ♠ét ♥ã♥ tr♦♥❣ E ✈➭ {xn }, {yn } ❧➭ ❤❛✐ ❞➲② tr♦♥❣ E ✳ ◆Õ✉ xn → x, yn → y ❦❤✐ n → ∞ ✈➭ xn ≤ yn ✈í✐ ♠ä✐ n t❤× x ≤ y ✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚õ xn ≤ yn t❛ ❝ã yn − xn ∈ P ✳ ❱× P ➤ã♥❣ ✈➭ (yn − xn ) → y − x ♥➟♥ 2.2.2 y − x ∈ P ✳ ❉♦ ➤ã x ≤ y ➜Þ♥❤ ❧ý✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ ❍❛✉s❞♦r❢❢✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ E ❧➭ t❐♣ s➽♣ tèt t❤❡♦ q✉❛♥ ❤Ö t❤ø tù ❜é ♣❤❐♥ ✧ ✧ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✭♥❣❤Ü❛ ❧➭ ❤❛✐ ♣❤➬♥ tư ❜✃t ❦ú ❝đ❛ E ❜❛♦ ❣✐ê ❝ị♥❣ s♦ s➳♥❤ ➤➢ỵ❝ t❤❡♦ q✉❛♥ ❤Ư ✧ ✧ ỗ t rỗ ủ E ị ❝❤➷♥ ❞➢í✐ ➤Ị✉ ❝ã ❝❐♥ ❞➢í✐ ➤ó♥❣✮✳ ●✐➯ sư T : X → X ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ✈í✐ ♠ä✐ x, y ∈ X t❛ ❝ã d(T x, T y) ≤ d(x, T x) + d(y, T y − ϕ d(x, T x), d(y, T y) , (2.5) tr♦♥❣ ➤ã ϕ : P × P → P ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝ ✈➭ ϕ(a, b) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ a = b = 0✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t u ∈ X ✳ ❈❤ø♥❣ ♠✐♥❤✳ ❈❤♦ x0 ∈ X ❧➭ ♠ét ➤✐Ĩ♠ tï② ý✳ ❇➺♥❣ q✉② ♥➵♣ t❛ ❞Ơ ❞➭♥❣ ①➞② ❞ù♥❣ ➤➢ỵ❝ ♠ét ❞➲② {xn } s❛♦ ❝❤♦ xn+1 = T xn = T n+1 x0 ✈í✐ ♠ä✐ n ≥ ◆Õ✉ tå♥ t➵✐ sè ❣✐ê ❣✐➯ sö (2.6) n0 ∈ N✱ xn0 = xn0 +1 = T xn0 ✱ t❤× t❛ ❝ã xn0 ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❇➞② xn = xn+1 ✱ ✈í✐ ♠ä✐ n ∈ N✳ ❑❤✐ ➤ã t❛ t✐Õ♥ ❤➭♥❤ t❤❡♦ ❝➳❝ ❜➢í❝ s❛✉✿ ❇➢í❝ ✶✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ lim d(xn , xn+1 ) = n→∞ (2.7) 39 ❚❤❛② x = xn ✈➭ y = xn−1 ✈➭♦ ✭✷✳✺✮ ✈➭ sư ❞ơ♥❣ ❝➳❝ tÝ♥❤ ❝❤✃t ❝đ❛ ϕ✱ t❛ ➤➢ỵ❝ d(xn+1 , xn ) = d(T xn , T xn−1 ) ≤ d(xn , T xn ) + d(xn−1 , T xn−1 ) − ϕ d(xn , T xn ), d(xn−1 , T xn−1 ) = d(xn , xn+1 ) + d(xn−1 , xn ) − ϕ d(xn , xn+1 ), d(xn−1 , xn ) (2.8) ≤ d(xn , xn+1 ) + d(xn−1 , xn ) ❚õ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ s✉② r❛ d(xn+1 , xn ) ≤ d(xn , xn−1 ) ✈í✐ ♠ä✐ n ≥ ❱× ✈❐② ❝➳❝ ❞➲② {d(xn , xn+1 )} ❧➭ ệ t ị ì E ❧➭ t❐♣ s➽♣ t❤ø tù t♦➭♥ ♣❤➬♥ t❤❡♦ q✉❛♥ ❤Ö t❤ø tù ❜é ♣❤❐♥ ✧≤✧✱ ♥➟♥ tå♥ t➵✐ r ≥ s❛♦ ❝❤♦ lim d(xn , xn+1 ) = r✳ ❈❤♦ n → ∞ tr♦♥❣ ✭✷✳✽✮ ✈➭ sư ❞ơ♥❣ tÝ♥❤ ❧✐➟♥ n→∞ tơ❝ ❝đ❛ ϕ✱ t❛ ❝ã r ≤ 21 (r + r) − ϕ(r, r)✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ ϕ(r, r) ≤ 0✳ ❱× P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝ tõ ➤➞② t❛ ❝ã ❝ñ❛ ❤➭♠ ϕ(r, r) ≤ K = 0✳ ❙✉② r❛ ϕ(r, r) = 0✳ ◆❤ê tÝ♥❤ ❝❤✃t ϕ t❛ s✉② r❛ r = 0✳ ◆❤➢ ✈❐②✱ ✭✷✳✼✮ ➤➲ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ❇➢í❝ ✷✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ lim d(xn , xn+2 ) = n→∞ (2.9) ❚õ ✭✷✳✺✮✱ t❛ ❝ã d(xn+2 , xn ) = d(T xn+1 , T xn−1 ) ≤ = ≤ 2 d(xn+1 , T xn+1 ) + d(xn−1 , T xn−1 ) − ϕ d(xn+1 , T xn+1 ), d(xn−1 , T xn−1 ) d(xn+1 , xn+2 ) + d(xn−1 , xn ) − ϕ d(xn+1 , xn+2 ), d(xn−1 , xn ) d(xn+1 , xn+2 ) + d(xn−1 , xn ) (2.10) ❱× P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝✱ tõ (2.10) t❛ s✉② r❛ d(xn+2 , xn ) ≤ 21 K.[ d(xn+1 , xn+2 ) + d(xn−1 , xn ) ]✳ ❱× t❤Õ tõ ✭✷✳✼✮ t❛ t❤✃② r➺♥❣ lim d(xn+2 , xn ) = n→∞ ◆❤➢ ✈❐② ✭✷✳✾✮ ➤➲ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ❇➢í❝ ✸✳ ❈❤ó♥❣ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ T ❝ã ♠ét ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥✳ 40 ❚❤❐t ✈❐②✱ ❣✐➯ sö r➺♥❣ T ❦❤➠♥❣ ❝ã ➤✐Ó♠ t✉➬♥ ❤♦➭♥✱ ❦❤✐ ➤ã {xn } ❧➭ ♠ét ❞➲② ❝➳❝ ➤✐Ĩ♠ ♣❤➞♥ ❜✐Ưt✱ ➤ã ❧➭ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②✱ s✉② ré♥❣ ♥❣✉②➟♥ xn = xm ✈í✐ ♠ä✐ m = n✳ ❈❤ó♥❣ t❛ sÏ ❝❤Ø r❛ r➺♥❣ {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ (X, d)✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ ❝ã ♠ét ♣❤➬♥ tư c k ✱ tå♥ t➵✐ sè ♥❣✉②➟♥ mk > nk > k s d(xnk , xmk ) ỗ s ❝❤♦ ✈í✐ ♠ä✐ sè c (2.11) k ∈ N✱ t❛ ❝ã t❤Ó ❝❤ä♥ mk ∈ N ❧➭ sè tù ♥❤✐➟♥ ❜Ð ♥❤✃t s❛♦ ❝❤♦ mk > nk ✈➭ t❤á❛ ♠➲♥ ✭✷✳✶✶✮✱ ♥❤➢ ✈❐② t❛ ❝ã d(xnk , xmk−1 ) ≤ c (2.12) ❚õ ✭✷✳✶✶✮✱ ✭✷✳✶✷✮ ✈➭ sư ❞ơ♥❣ ❜✃t ➤➻♥❣ tứ ì ữ t t ợ c d(xmk , xnk ) ≤ d(xmk , xmk−2 ) + d(xmk−2 , xmk−1 ) + d(xmk−1 , xnk ) ≤ d(xmk , xmk−2 ) + d(xmk−2 , xmk−1 ) + c ◆❤ê ❇æ ➤Ị ✷✳✷✳✶✱ tõ ✭✷✳✼✮ ✈➭ ✭✷✳✾✮ t❛ ➤➢ỵ❝ lim d(xnk , xmk ) = c k→∞ ❙ư ❞ơ♥❣ ✭✷✳✺✮ ✈í✐ (2.13) x = xmk −1 ✈➭ y = xnk −1 t❛ ❝ã d(xmk , xnk ) = d(T xmk −1 , T xnk −1 ) ≤ d(xmk −1 , xmk ) + d(xnk −1 , xnk ) − ϕ d(xmk −1 , xmk ), d(xnk −1 , xnk ) ❈❤♦ k → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ rå✐ sư ❞ơ♥❣ ❇ỉ ➤Ị ✷✳✷✳✶✱ ✭✷✳✼✮ ✈➭ ✭✷✳✶✸✮ t ợ ề t ì tế ó s✉② ré♥❣✳ ❱× tå♥ t➵✐ c ≤ − ϕ(0, 0) = {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ✱ ♥➟♥ u ∈ X s❛♦ ❝❤♦ xn → u✳ ▲➵✐ ➳♣ ❞ô♥❣ ✭✷✳✺✮ ♠ét ❧➬♥ ♥÷❛ ✈í✐ x = xn ✱ y = u t❛ ➤➢ỵ❝ d(xn+1 , T u) = d(T xn , T u) ≤ d(xn , xn+1 ) + d(u, T u) − ϕ d(xn , xn+1 ), d(u, T u) (2.14) 41 ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② s✉② r❛ d(xn+1 , T u) ≤ d(xn , xn+1 ) + d(u, T u) ❚õ ✭✷✳✼✮ t❛ ❝ã lim sup d(xn+1 , T u) ≤ d(u, T u) n→∞ ❚✐Õ♣ t❤❡♦ t❛ sÏ ❝❤Ø r❛ ❝➳❝ ♠➞✉ t❤✉➮♥ ❣➷♣ ♣❤➯✐ ❦❤✐ (2.15) T ❦❤➠♥❣ ❝ã ể t tr ỗ trờ ợ s ế ✈í✐ ♠ä✐ n ≥ 2✱ t❛ ❝ã xn = u ✈➭ xn = T u✳ ❑❤✐ ➤ã✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ d(u, T u) ≤ d(u, xn ) + d(xn , xn+1 ) + d(xn+1 , T u) ✈➭ sö ❞ô♥❣ ✭✷✳✼✮ t❛ ❝ã d(u, T u) ≤ lim sup d(xn+1 , T u) (2.16) n→∞ ❚õ ✭✷✳✶✺✮ ✈➭ ✭✷✳✶✻✮✱ t❛ ➤➢ỵ❝ d(u, T u) ≤ lim sup d(xn+1 , T u) ≤ d(u, T u) n→∞ ➜✐Ò✉ ♥➭② s✉② r❛ 21 d(u, T u) r❛ (2.17) ≤ 0✳ ❱× t❤Õ t❛ ❝ã d(u, T u) = 0✱ ♥❣❤Ü❛ ❧➭ T u = u✳ ❙✉② u ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ➤✐Ị✉ ❧➭ T ❦❤➠♥❣ ❝ã ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥✳ ❜✮ ◆Õ✉ tå♥ t➵✐ q ≥ s❛♦ ❝❤♦ xq = u ❤♦➷❝ xq = T u✱ t❤× ✈× T ❦❤➠♥❣ ❝ã ➤✐Ó♠ t✉➬♥ ❤♦➭♥ ♥➭♦ ❝➯ ♥➟♥ u = x0 ✳ ❚❤❐t ✈❐② ♥Õ✉ xq = u = x0 t❤× T q x0 = x0 ✱ s✉② r❛ x0 ❧➭ ♠ét ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥ ❝đ❛ T ✳ ▼➷t ❦❤➳❝ ♥Õ✉ xq = T u ✈➭ x0 = u t❤× ❧ó❝ ➤ã t❛ ❝ã T x0 = T u = xq = T q x0 = T q−1 (T x0 )✱ tø❝ ❧➭ T x0 ❧➭ ♠ét ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥ ❝đ❛ T ✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ➤✐Ị✉ ❧➭ ➤ã✱ ✈í✐ ♠ä✐ T ❦❤➠♥❣ ❝ã ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥✳ ❱❐②✱ u = x0 ✳ ❑❤✐ n ≥ 1✱ t❛ ❝ã d(T n u, u) = d(T n xq , u) = d(xn+q , u), ❤♦➷❝ d(T n u, u) = d(T n−1 T u, u) = d(T n−1 xq , u) = d(xn+q−1 , u) ❚r♦♥❣ ✷ ➤➻♥❣ t❤ø❝ tr➟♥✱ sè q ố ị ì ❞➲② {xn+q } ✈➭ {xn+q−1 } ❧➭ ❝➳❝ ❞➲② ❝♦♥ ❝đ❛ {xn }✳ ❱× {xn } ❧➭ ❞➲② ❤é✐ tơ ➤Õ♥ u tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ 42 ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ (X, d) ❍❛✉s❞♦r❢❢✱ ♥➟♥ ❤❛✐ ❞➲② ➤ã ❝ï♥❣ ❤é✐ tô ✈Ị ♠ét ➤✐Ĩ♠ u ❞✉② ♥❤✃t✱ ♥❣❤Ü❛ ❧➭ lim d(xn+q , u) = lim d(xn+q−1 , u) = n→∞ n→∞ ❱× t❤Õ t❛ ❝ã lim d(T n u, u) = n→∞ ▲➵✐ ✈× (2.18) (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❍❛✉s❞♦r❢❢ ♥➟♥ tõ ✭✷✳✶✽✮ t❛ ❝ã lim d(T n+2 u, u) = n→∞ ▼➷t ❦❤➳❝✱ ✈× (2.19) T ❦❤➠♥❣ ❝ã ➤✐Ó♠ t✉➬♥ ❤♦➭♥ ♥➟♥ t❛ ❧✉➠♥ ❝ã T s u = T r u, ✈í✐ ❜✃t ❦ú s, r ∈ N ♠➭ s = r (2.20) ❱× t❤Õ sử ụ t tứ ì ữ t t❛ ➤➢ỵ❝ d(T n+1 u, T u) − d(u, T u) ≤ d(T n+1 u, T n+2 u) + d(T n+2 u, u) ❱× P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ❝ã d(T n+1 u, T u) − d(u, T u) ≤ K.[ d(T n+1 u, T n+2 u) + d(T n+2 u, u) ] ❈❤♦ n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ✈➭ sö ❞ơ♥❣ ✭✷✳✶✾✮ ✈➭ ✭✷✳✼✮ t❛ ➤➢ỵ❝ lim d(T n+1 u, T u) = d(u, T u) (2.21) lim d(T n u, T u) = d(u, T u) (2.22) n→∞ ❚➢➡♥❣ tù✱ t❛ ❝ã n→∞ ❇➞② ❣✐ê tõ ✭✷✳✺✮✱ ♥❤ê ❇æ ➤Ò ✷✳✷✳✶ t❛ ❝ã d(T n+1 u, T u) ≤ ❈❤♦ d(T n u, T u) + d(u, T u) − ϕ d(T n u, T u), d(u, T u) (2.23) n → ∞ tr♦♥❣ ✭✷✳✷✸✮ ✈➭ sư ❞ơ♥❣ ✭✷✳✷✶✮✱ ✭✷✳✷✷✮ t❛ ❝ã d(u, T u) ≤ d(u, T u) − ϕ d(T u, u), d(u, T u) ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ s✉② r❛ ❝❤✃t ❝ñ❛ ❤➭♠ ϕ d(T u, u), d(u, T u) = 0✳ ◆❤ê tÝ♥❤ ϕ t❛ ❝ã d(u, T u) = 0✱ ♥❣❤Ü❛ ❧➭ T u = u✳ ❉♦ ➤ã u ❧➭ ♠ét ➤✐Ĩ♠ t✉➬♥ 43 ❤♦➭♥ ❝đ❛ T ✱ ♠➞✉ t❤✉➱♥✳ ❱× ✈❐②✱ tõ ❝➳❝ ♠➞✉ t❤✉➮♥ ♥➭② t❛ s✉② r❛ T ❝ã ♠ét ➤✐Ó♠ t✉➬♥ ❤♦➭♥✱ ♥❣❤Ü❛ ❧➭ tå♥ t➵✐ u ∈ X s❛♦ ❝❤♦ u = Tp u ✈í✐ sè p ≥ ♥➭♦ ➤ã✳ ❇➢í❝ ✹✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ T ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❚❤❐t ✈❐②✱ ♥Õ✉ p = t❤× u = T u✱ ❞♦ ➤ã u ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣✳ ❇➞② ❣✐ê ❣✐➯ sư ➤é♥❣ ❝đ❛ p > 1✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ a = T p−1 u ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t T ✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ T p−1 u = T p u✳ ❑❤✐ ➤ã t❛ ❝ã d(T p−1 u, T p u) = ✈➭ ♥❤➢ ✈❐② ϕ d(T p−1 u, T p u), d(T p−1 u, T p u) = 0✳ ❇➞② ❣✐ê sư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✺✮ t❛ ❝ã d(u, T u) = d(T p u, T p+1 u) = d T (T p−1 u), T (T p u) ≤ d(T p−1 u, T p u) + d(T p u, T (T p u) − ϕ d(T p−1 u, T p u), d(T p u, T p u) (2.24) ❙✉② r❛ d(T p−1 u, T p u)+d(T p u, T (T p u) −ϕ d(T p−1 u, T p u), d(T p u, T p u) −d(u, T u) = v ∈ P ❱× ϕ d(T p−1 u, T p u), d(T p u, T p u) ∈ P ✱ ♥➟♥ tõ ➤Þ♥❤ ♥❣❤Ü❛ ♥ã♥ t❛ s✉② r❛ d(T p−1 u, T p u)+d(T p u, T (T p u) −d(u, T u) = ϕ d(T p−1 u, T p u), d(T p u, T p u) +v ∈ P ❉♦ ➤ã✱ tõ (2.24) ✈× ϕ d(T p−1 u, T p u), d(T p u, T p u) = t❛ ❝ã d(u, T u) < d(T p−1 u, T p u) + d(u, T u) , ♥❣❤Ü❛ ❧➭ d(u, T u) < d(T p−1 u, T p u) (2.25) ▲➵✐ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✮ t❛ ❝ã d(T p−1 u, T p u) = d T (T p−2 u), T (T p−1 u) ≤ d(T p−2 u, T p−1 u) + d(T p−1 u, T p u) − ϕ d(T p−2 u, T p−1 u), d(T p−1 u, T p u) ▲❐♣ ❧✉❐♥ t➢➡♥❣ tù ♥❤➢ tr➟♥✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ d(T p−1 u, T p u) ≤ d(T p−2 u, T p−1 u) ❚✐Õ♣ tơ❝ q✉➳ tr×♥❤ ♥➭② ♥❤➢ tr♦♥❣ ✭✷✳✷✺✮ ✈➭ ✭✷✳✷✻✮ t❛ t❤✉ ➤➢ỵ❝ d(u, T u) < d(T p−1 u, T p u) ≤ d(T p−2 u, T p−1 u) ≤ ≤ d(u, T u) (2.26) 44 ❚❛ ❣➷♣ ♣❤➯✐ ♠➞✉ t❤✉➱♥✳ ❱❐② a = T p−1 u ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ❇➢í❝ ✺✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ❧➭ ❞✉② ♥❤✃t✳ ❚❤❐t ✈❐②✱ ❣✐➯ sư ❝ã ❤❛✐ ➤✐Ĩ♠ ❧➭ T b = b ✈➭ T c = c✳ ❑❤✐ ➤ã ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✮ t❛ ❝ã d(b, c) = d(T b, T c) ≤ ❱× b, c ∈ X ➤Ị✉ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✱ ♥❣❤Ü❛ d(b, b) + d(c, c) − ϕ(d(b, b), d(c, c)) = P ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝ t❛ ❝ã d(b, c) ≤ K.0✳ ❙✉② r❛ d(b, c) = 0✳ ❉♦ ➤ã b = c ị ý ợ ứ ị ý tr t t ợ ết q s❛✉✳ 2.2.3 ❍Ö q✉➯✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ ❍❛✉s❞♦r❢❢✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ E ❧➭ t❐♣ s➽♣ tèt t❤❡♦ q✉❛♥ ❤Ö t❤ø tù ❜é ♣❤❐♥ ✧ ✧ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✭♥❣❤Ü❛ ❧➭ ❤❛✐ ♣❤➬♥ tư ❜✃t ❦ú ❝đ❛ E ❜❛♦ ❣✐ê ❝ị♥❣ s♦ s➳♥❤ ➤➢ỵ❝ t❤❡♦ q ệ ỗ t rỗ ❝đ❛ E ♠➭ ❜Þ ❝❤➷♥ ❞➢í✐ ➤Ị✉ ❝ã ❝❐♥ ❞➢í✐ ➤ó♥❣✮✳ ●✐➯ sư T : X → X ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦ tå♥ t➵✐ k ∈ [0, 1) t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ t❛ ❝ã k ∈ [0, 1) ✈➭ d(T x, T y) ≤ k d(x, T x) + d(y, T y) ✈í✐ ♠ä✐ x, y ∈ X (2.27) ❑❤✐ ➤ã T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❤➭♠ ϕ : P ×P → P ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ ϕ(u, v) = 1−k (u+ v) ✈í✐ ♠ä✐ u, v ∈ P ✳ ❑❤✐ ➤ã ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ❝♦ (2.28) t❛ s✉② r❛ r➺♥❣ T t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ (2.5) ✈í✐ ❤➭♠ ì ụ ị ý t s r❛ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ 2.2.4 ❍Ö q✉➯✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ ❍❛✉s❞♦r❢❢✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ E ❧➭ t❐♣ s➽♣ tèt t❤❡♦ q✉❛♥ ❤Ư t❤ø tù ❜é ♣❤❐♥ ✧ ✧ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✭♥❣❤Ü❛ ❧➭ ❤❛✐ ♣❤➬♥ tư ❜✃t ❦ú ❝đ❛ E ❜❛♦ ❣✐ê ❝ị♥❣ s♦ s➳♥❤ ➤➢ỵ❝ 45 t❤❡♦ q✉❛♥ ệ ỗ t rỗ ủ E ♠➭ ❜Þ ❝❤➷♥ ❞➢í✐ ➤Ị✉ ❝ã ❝❐♥ ❞➢í✐ ➤ó♥❣✮✳ ●✐➯ sö T : X → X ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ t❛ ❝ã k ∈ [0, 1) ✈➭ d(T x, T y) ≤ 1 d(x, T x) + d(y, T y) − ψ d(x, T x) + d(y, T y) 2 ✈í✐ ♠ä✐ x, y ∈ X, (2.28) tr♦♥❣ ➤ã ψ : P → P ❧➭ ❤➭♠ ❧✐➟♥ tô❝ ✈➭ ψ −1 (0) = {0}✳ ❑❤✐ ➤ã T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❤➭♠ ϕ : P × P → P ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ ϕ(u, v) = ψ u+v ✈í✐ ♠ä✐ u, v ∈ P ❑❤✐ ➤ã✱ t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✷ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ ❉♦ ➤ã✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ T ❝ã ♠ét 46 ❑Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉✱ ✈Ị ➤Ị t➭✐✿ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉ ✶✳ ❍Ư t❤è♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ ❦✐Ó✉ ϕ✲❝♦✱ ♥ã♥✱ ♥ã♥ ❝❤✉➮♥ t➽❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ➳♥❤ ①➵ ❝♦ ➤Þ❛ ♣❤➢➡♥❣✱ ➳♥❤ ①➵ (c, λ)✲❝♦ ➤Þ❛ ♣❤➢➡♥❣ ➤Ị✉✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ c✲❦❤➯ ①✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ T ✲q✉ü ➤➵♦ ➤➬② ➤ñ✱✳✳✳ ✷✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ➤Þ♥❤ ❧ý ❝❤➻♥❣ ❤➵♥ ♥❤➢ ➜Þ♥❤ ❧ý ✶✳✷✳✸✱ ➜Þ♥❤ ❧ý ✶✳✷✳✹✱ ➜Þ♥❤ ❧ý ✶✳✷✳✺✱ ➜Þ♥❤ ❧ý ✶✳✷✳✼✱ ➜Þ♥❤ ❧ý ✶✳✷✳✶✶✳ ✸✳ ➜➢❛ r❛ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ♥❤➢ ➜Þ♥❤ ❧ý ✷✳✶✳✶✽✱ ➜Þ♥❤ ❧ý ✷✳✷✳✷✱ ❍Ư q✉➯ ✷✳✷✳✸ ✈➭ ❍Ư q✉➯ ✷✳✷✳✹✳ ✹✳ ●✐í✐ t❤✐Ư✉ ❝❤✐ t✐Õt ❱Ý ❞ô ✶✳✷✳✽✱ ❱Ý ❞ô ✷✳✶✳✽✱ ❱Ý ❞ô ✷✳✶✳✶✾✳ 47 t ệ t ỗ ➤➵✐ ❝➢➡♥❣✱ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ❆✳ ❆③❛♠✱ ▼✳ ❆rs❤❛❞ ✭✷✵✵✽✮✱ ❑❛♥❛♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✶ ✭✶✮✱ ✹✺✲✹✽✳ ❬✸❪ ❈✳ ❉✐ ❇❛r✐ ❛♥❞ P✳ ❱❡tr♦ ✭✷✵✶✷✮✱ ❈♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳✱ ✷✶✽✱ ✼✸✷✷✲✼✸✷✺✳ ❬✹❪ ❱✳❇❡r✐♥❞❡ ✭✷✵✵✽✮✱ ●❡♥❡r❛❧ ❝♦♥str✉❝t✐✈❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❈✐r✐❝✲ t②♣❡ ❛❧♠♦st ❝♦♥tr❛❝t✐♦♥s ✐♥ ♠❡tr✐❝ s♣❛❝❡s✱ ❈❛r♣❛t❤✐❛♥ ❏✳ ▼❛t❤✳✱ ✷✹ ✭✷✮✱ ✶✵✲✶✾✳ ❬✺❪ ❆✳ ❇r❛♥❝✐❛r✐ ✭✷✵✵✵✮✱ ❆ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦❢ ❇❛♥❛❝❤✲❈❛❝❝✐♣♣♦❧✐ t②♣❡ ♦♥ ❛ ❝❧❛sss ♦❢ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ P✉❜❧✳ ▼❛t❤✳ ❉❡❜r❡❝❡♥✱ ✺✼ ✭✶✲ ✷✮✱ ✸✶✲✸✼✳ ❬✻❪ P✳ ◆✳ ❉✉tt❛✱ ❇✳ ❙✳ ❈❤♦✉❞❤✉r② ✭✷✵✵✽✮✱ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ✐♥ ♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✷✵✵✽✱ ✽ ♣❛❣❡s✱ ■❉ ✹✵✻✸✻✽✳ ❬✼❪ ❆✳ ❋♦rr❛✱ ❆✳ ❇❡❧❧♦✉r✱ ❆✳ ❆❧✲❇s♦✉❧ ✭✷✵✵✾✮✱ ❙♦♠❡ r❡s✉❧ts ✐♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r② ❝♦♥❝❡r♥✐♥❣ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ▼❛t❤✳ ❱❡s♥✐❦✱ ✻✶ ✭✸✮✱ ✷✵✸✲ ✷✵✽✳ ❬✽❪ ❍✳ ▲♦♥❣✲●✉❛♥❣✱ ❩✳ ❳✐❛♥ ✭✷✵✵✼✮✱ ❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳✱ ✸✸✷✱ ✶✹✻✽ ✲ ✶✹✼✻✳ ❬✾❪ ▼✳ ❏❧❡❧✐✱ ❇✳ ❙❛♠❡t ✭✷✵✵✾✮✱ ❚❤❡ ❑❛♥♥❛♥✬s ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ✐♥ ❛ ❝♦♥❡ r❡❝t❛♥❣✉❧❛r ♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✸ ✭✷✮✱ ✶✻✶✲✶✻✼✳ ❬✶✵❪ ❉✳ ▼✐❤❡t ✭✷✵✵✾✮✱ ❑❛♥❛♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✷ ✭✷✮✱ ✾✷✲✾✻✳ 48 ❬✶✶❪ ❲✳ ❙❤❛t❛♥❛✇✐✱ ❆✳ ❆❧✲❘❛✇❛s❤❞❡❤✱ ❍✳ ❆②❞✐✱ ❍✳ ❑✳ ◆❛s❤✐♥❡ ✭✷✵✶✷✮✱ ❖♥ ❛ ❢✐①❡❞ ♣♦✐♥t ❢♦r ❣❡♥❡r❛❧✐③❡❞ ❝♦♥tr❛❝t✐♦♥s ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❆❜str❛✳ ❆♣♣❧✳ ❆♥❛❧✳✱ ❞♦✐✿✶✵✳✶✶✺✺✴✷✵✶✷✴✷✹✻✵✽✺✳ ❬✶✷❪ ❉✳ P✳ ❙❤✉❦❧❛ ✭✷✵✶✹✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ■♥t❡r✳ ❏✳ ▼❛t❤✳ ❆r❝❤✐✈❡✱ ✺ ✭✽✮✱ ✷✶✲✷✹✳