Điểm bất động của các ánh xạ hầu co ciric trong không gian mê tric và không gian mêtric riêng

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❉✉② ▲➞♠ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥ ✲ ✷✵✶✺ ❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❉✉② ▲➞♠ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ▲✉❐♥ ❱➝♥ ❚❤➵❝ ❙ü ❚♦➳♥ ❍ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✺ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ✷ ❈❤➢➡♥❣ ✶✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✺ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❈❤➢➡♥❣ ✷✳ ✺ ✶✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✷✸ ✷✳✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ❑Õt ❧✉❐♥ ✸✾ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✹✵ ✶ ❧ê✐ ♥ã✐ ➤➬✉ ▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝❤đ ➤Ị ♥❣❤✐➟♥ ❝ø✉ q✉❛♥ trä♥❣ ❝đ❛ ❣✐➯✐ tÝ❝❤✳ ◆ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ t♦➳♥ ❤ä❝ ✈➭ ❝➳❝ ♥❣➭♥❤ ❦ü t❤✉❐t✳ ❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧Ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ ❝đ❛ ❇❛♥❛❝❤✳ ❈ị♥❣ tõ ➤ã ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ➤➲ ➤➢ỵ❝ ✈❐♥ ❞ơ♥❣ r✃t ♣❤ỉ ❜✐Õ♥ ✈➭ t❤➭♥❤ ❝➠♥❣ tr♦♥❣ ✈✐Ö❝ ❝❤ø♥❣ ♠✐♥❤ sù tå♥ t➵✐ ❞✉② ♥❤✃t ♥❣❤✐Ư♠ ✈➭ tÝ♥❤ ①✃♣ ①Ø ♥❣❤✐Ư♠ ❝đ❛ ❝➳❝ ❜➭✐ t♦➳♥ t❤✉é❝ ♥❤✐Ị✉ ❧Ü♥❤ ✈ù❝ ❝đ❛ ❣✐➯✐ tÝ❝❤✳ ❱× t❤Õ ➤➲ ❝ã ♥❤✐Ị✉ ♥❣❤✐➟♥ ❝ø✉✱ t×♠ ❝➳❝❤ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ❝❤♦ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✱ ❜➺♥❣ ❝➳❝❤ ➤✐Ò✉ ❝❤Ø♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝➡ ❜➯♥ ❤♦➷❝ t❤❛② ➤ỉ✐ ❦❤➠♥❣ ❣✐❛♥✳ ◆➝♠ ✶✾✻✽✱ ❑❛♥♥❛♥ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ♠ét ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ♠➭ ♥ã ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ➳♥❤ ①➵✳ ◆➝♠ ✷✵✵✹✱ ❇❡r✐♥❞❡ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ö♠ ❝➳❝ ➳♥❤ ①➵ ❝♦ ②Õ✉ ✭❤➬✉ ❝♦✮ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱✳✳✳ ◆➝♠ ✶✾✾✹✱ ▼❛tt❤❡✇s ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ➤ã ❧➭ ♠ét ♠ë ré♥❣ ❝ñ❛ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣✱ ♥❤➢♥❣ ë ➤➞② ❦❤♦➯♥❣ ❝➳❝❤ ❝đ❛ ♠ét ➤✐Ĩ♠ ➤Õ♥ ❝❤Ý♥❤ ♥ã ❦❤➠♥❣ ♥❤✃t t❤✐Õt ♣❤➯✐ ❜➺♥❣ ❦❤➠♥❣✳ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➲ ❝ã ♥❤✐Ò✉ ø♥❣ ❞ơ♥❣ ré♥❣ r➲✐ tr♦♥❣ ♥❤✐Ị✉ ♥❣➭♥❤ ❝đ❛ t♦➳♥ ❤ä❝ ❝ị♥❣ ♥❤➢ ❝ó ♣❤➳♣ ♣❤➬♥ ♠Ị♠ ♠➳② tÝ♥❤ ✈➭ ♥❣÷ ♥❣❤Ü❛ ❤ä❝✳ ❱í✐ ♥❤÷♥❣ ➤ã♥❣ ❣ã♣ ➤➳♥❣ ❦Ĩ ♥➭②✱ ♥❤✐Ị✉ t➳❝ ❣✐➯ ➤➲ t❐♣ tr✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t t➠♣➠ ❝đ❛ ♥ã✳ ➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✱ ❝❤ó♥❣ t➠✐ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥➭② ♥❤➺♠ t×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳❚❙✳❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ✷ ✧ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✧✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ ❞➲② 0✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤ñ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ ②Õ✉✱ ➳♥❤ ①➵ ✭ϕ✱L✮✲❝♦ ②Õ✉✱ ➳♥❤ ①➵ ❤➬✉ ❝♦✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❦✐Ĩ✉ ❈✐r✐❝✱ q✉ü ➤➵♦ ❝đ❛ ➳♥❤ ①➵ T t➵✐ ➤✐Ĩ♠ x✱ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ T ✲q✉ü ➤➵♦✱ t♦➳♥ tö P✐❝❛r❞ ②Õ✉✱ ❤➭♠ s♦ s➳♥❤✱ ❤➭♠ ✭c✮✲s♦ s➳♥❤✱ ✳✳✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝ñ❛ ❧✉❐♥ ✈➝♥✱ ❣å♠✿ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ ❞➲② 0✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ ②Õ✉✱ ➳♥❤ ①➵ ✭ϕ✱L✮✲❝♦ ②Õ✉✱ ➳♥❤ ①➵ ❤➬✉ ❝♦✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❦✐Ĩ✉ ❈✐r✐❝✱ q✉ü ➤➵♦ ❝đ❛ ➳♥❤ ①➵ ➤➬② ➤đ T t➵✐ ➤✐Ĩ♠ x✱ ❦❤➠♥❣ ❣✐❛♥ T ✲q✉ü ➤➵♦✱ t♦➳♥ tö P✐❝❛r❞ ②Õ✉✱ ❤➭♠ s♦ s➳♥❤✱ ❤➭♠ ✭c✮✲s♦ s➳♥❤✳ ▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❈✐r✐❝✱ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ➤Þ♥❤ ❧ý ➤ã✳ ◆❣♦➭✐ r ò trì ệ q í ❞ơ ♠✐♥❤ ❤♦➵ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➷♣ ➳♥❤ ①➵ t➝♥❣ ②Õ✉ ♥❣➷t✳ ❈❤➢➡♥❣ ✷ ✈í✐ t➟♥ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❦✐Ó✉ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❦✐Ĩ✉ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ s✉② ré♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ➤ã✳ ▲✉❐♥ ✈➝♥ ♥➭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♥❤ ❇➭ ❘Þ❛✲❱ị♥❣ ❚➭✉✱ ❇❛♥ ●✐➳♠ ❍✐Ư✉ ❚r➢ê♥❣ ❚❍P❚ ◆❣✉②Ơ♥ ❱➝♥ ❈õ✱ tØ♥❤ ❇➭ ❘Þ❛✲❱ị♥❣ ❚➭✉ ➤➲ ❣✐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♥ ❇❛ ♠Ñ✱ ❝➳❝ ❛♥❤ ❡♠ tr ì t ề ệ t ợ ú 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 q✉ý ❚❤➬②✱ ❈➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥✳ ❱✐♥❤✱ ♥❣➭② ✷✶ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✺ ◆❣✉②Ơ♥ ❉✉② ▲➞♠ ✹ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ö✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥ ♥❤➢✿ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ ❞➲② 0✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ ②Õ✉✱ ➳♥❤ ①➵ ✭ϕ✱L✮✲❝♦ ②Õ✉✱ ➳♥❤ ①➵ ❤➬✉ ❝♦✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❦✐Ó✉ ❈✐r✐❝✱ q✉ü ➤➵♦ ❝đ❛ ➳♥❤ ①➵ T t➵✐ ➤✐Ĩ♠ x✱ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ T ✲q✉ü ➤➵♦✱ t♦➳♥ tư P✐❝❛r❞ ②Õ✉✱ ❤➭♠ s♦ s➳♥❤✱ ❤➭♠ ✭c✮✲s♦ s➳♥❤✱✳ ✳ ✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ ♠ét tr➟♥ ♠➟tr✐❝ ✭❬✶❪✮ ❈❤♦ t❐♣ ❤ỵ♣ X = φ✱ 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 ❦Ý ❤✐Ư✉ ❧➭ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❦❤♦➯♥❣ ❝➳❝❤ ✈➭ tõ ➤✐Ó♠ x ➤Õ♥ ➤✐Ĩ♠ y ✳ ✶✳✶✳✷ ❱Ý ❞ơ✳ ✶✮ ❳Ðt X = R✱ d : R × R → R ❝❤♦ ❜ë✐ d (x, y) = |x − y|✱ ✈í✐ ♠ä✐ x, y ∈ R✳ ❑❤✐ ➤ã d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ R✳ ✷✮ ❳Ðt X = Rn ✳ ❱í✐ ❜✃t ❦ú x = (x1 , , xn ), y = (y1 , , yn ) ∈ Rn t❛ ➤➷t n |xi − yi | d1 (x, y) = n ✈➭ i=1 tr➟♥ |xi − yi |✳ ❑❤✐ ➤ã d1 , d2 ❧➭ ❝➳❝ ♠➟tr✐❝ d2 (x, y) = i=1 Rn ✳ ✺ ▼Ư♥❤ ➤Ị✳ ✶✳✶✳✸ X ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d)✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ x, y, u, v ∈ ✱ t❛ ❝ã |d (x, y) − d (u, v)| ≤ d (x, u) + d (y, v) ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✹ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ d(x, A) = inf d (x, y) ✈➭ ❣ä✐ d(x, A) ❧➭ ❦❤♦➯♥❣ y∈A ✶✳✶✳✺ ▼Ư♥❤ ➤Ị✳ x, y ∈ t❛ ❝ã ✭❬✶❪✮ (X, d)✱ A ⊂ X ✱ x ∈ X ✱ ❦Ý ❤✐Ư✉ ❝➳❝❤ tõ ➤✐Ĩ♠ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ x ➤Õ♥ t❐♣ ❤ỵ♣ (X, d)✱ A ⊂ X ✳ A✳ ❑❤✐ ➤ã ✈í✐ ♠ä✐ |d (x, A) − d (y, A)| ≤ d (x, y) ✶✳✶✳✻ ❧➭ ➜Þ♥❤ ♥❣❤Ü❛✳ ❤é✐ tơ ✈Ị ➤✐Ĩ♠ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d)✱ ❞➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐ x ∈ X ♥Õ✉ ✈í✐ ♠ä✐ ε > tå♥ t➵✐ n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n ≥ n0 t❛ ❝ã d (xn , x) < ε✳ ▲ó❝ ➤ã t❛ ❦Ý ❤✐Ö✉ lim xn = x ❤❛② xn → x ❦❤✐ n → ∞✳ n→∞ ✶✳✶✳✼ ▼Ư♥❤ ➤Ị✳ ✶✮ ❚❐♣ ✷✮ ✶✳✶✳✽ ❧➭ ❝ã E x∈E ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤ã♥❣ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✈í✐ ♠ä✐ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ tå♥ t➵✐ ➜Þ♥❤ ♥❣❤Ü❛✳ ❞➲② ❈❛✉❝❤② (X, d)✱ E ⊂ X ✱ x ∈ X ✳ {xn } ⊂ E {xn } ⊂ E ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥Õ✉ ✈í✐ ♠ä✐ ♠➭ s❛♦ ❝❤♦ ➜Þ♥❤ ♥❣❤Ü❛✳ ❚❐♣ ❝♦♥ ❣✐❛♥ ❝♦♥ y ∈ E✳ xn → x✳ ε > 0✱ tå♥ t➵✐ n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n, m ≥ n0 t❛ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ t❛ ❝ã (X, d)✳ ❉➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐ d(xn , xm ) < ε✱ ❤❛② {xn } ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ ✶✳✶✳✾ xn → y ❑❤✐ ➤ã lim n,m→+∞ d(xn , xm ) = 0✳ (X, d)✳ ❚❛ ♥ã✐ (X, d) ❧➭ ➤➬② ➤đ X ➤Ị✉ ❤é✐ tơ✳ M ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ M ✈í✐ ♠➟tr✐❝ ❝➯♠ s✐♥❤ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ✳ ✻ ➤➬② ➤đ ♥Õ✉ ❦❤➠♥❣ ✶✳✶✳✶✵ ❱Ý ❞ơ✳ ✶✮ ❚❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝ R ✈í✐ ♠➟tr✐❝ d (x, y) = |x − y| ✈í✐ ♠ä✐ x, y ∈ R ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ Rn ❣å♠ t✃t ❝➯ ❝➳❝ ❜é n sè t❤ù❝✱ ✈í✐ ♠➟tr✐❝ d1 (x, y)✱ d2 (x, y) ❧➭ ✷✮ ❚❐♣ ❤ỵ♣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ✶✳✶✳✶✶ ▼Ư♥❤ ➤Ị✳ ✭❬✶❪✮ ✶✮ ◆Õ✉ M ➤➬② ➤đ t❤× ✷✮ ◆Õ✉ M ❧➭ t❐♣ ➤ã♥❣ ✈➭ ✶✳✶✳✶✷ ➜Þ♥❤ ♥❣❤Ü❛✳ (X, d)✱ M ⊂ X ✳ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ M ❑❤✐ ➤ã ❧➭ t❐♣ ➤ã♥❣✳ X ➤➬② ➤đ t❤× M ➤➬② ➤đ✳ ✭❬✶❪✮ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ f : (X, d) → (Y, ρ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ (X, d) ✈➭ (Y, ρ)✳ ➳♥❤ ①➵ α ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X t❛ ❝ã ρ[f (x) , f (y)] ≤ αd (x, y) ❙è t❤ù❝ α ∈ [0, 1) ➤➢ỵ❝ ❣ä✐ ❧➭ ❤Ư ❈❤♦ ➳♥❤ ①➵ sè ❝♦ ❝đ❛ f tr➟♥ X ✳ f : X → X tõ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ➜✐Ó♠ x∗ ∈ X ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ✶✳✶✳✶✸ ➜Þ♥❤ ❧ý✳ ✭❬✶❪✮✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✮ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T :X→X ✶✮ T 0≤a p(xn+1 , xn+2 ) ✈í✐ ♠ä✐ n ∈ N✳ ❑Õt ❤➡♣ ✈í✐ ✭✷✳✹✮ t❛ ♥❤❐♥ ➤➢ỵ❝ p(xn+1 , xn+2 ) ≤ ϕ (p(xn , xn+1 )) ✭✷✳✺✮ ❇➺♥❣ q✉② ♥➵♣✱ t❛ ❝ã p(xn , xn+1 ) ≤ ϕn (p(x0 , x1 )) ✈í✐ ♠ä✐ n ∈ N ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ ✈í✐ m−1 p(xn , xm ) ≤ m > n t❛ ❝ã m−2 p(xk , xk+1 ) − k=n m−1 ≤ p(xk+1 , xk+1 ) k=n ∞ p(xk , xk+1 ) ≤ k=n ∞ p(xk , xk+1 ) ≤ k=n ì s s ỗ k (p(x0 , x1 )) k=n ϕk (p(x0 , x1 )) ❤é✐ tô✳ ❚õ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ k=0 t❛ s✉② r❛ {xn } ❧➭ ♠ét ❞➲② 0✲❈❛✉❝❤② tr♦♥❣ X ✳ ❱× X ❧➭ ✵✲➤➬② ➤đ✱ ♥➟♥ {xn } ❤é✐ tơ t❤❡♦ t➠♣➠ τp tí✐ ♠ét ➤✐Ĩ♠ z ∈ X s❛♦ ❝❤♦ lim p(xn , z) = p(z, z) = n→∞ ❇➞② ❣✐ê✱ t❛ ❝❤Ø r❛ r➺♥❣ p(z, T z) = 0✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐✱ p(z, T z) > 0✳ ❱× φ ❧➭ ♠ét ❤➭♠ ✭❝✮✲s♦ s➳♥❤✱ ♥➟♥ φ(t) < t ✈í✐ t > ✈➭ ✈× lim p(xn+1 , xn ) = ✱ n→∞ lim p(xn , z) = ♥➟♥ ❝ã tå♥ t➵✐ sè n0 ∈ N s❛♦ ❝❤♦ ✈í✐ n > n0 t❛ ❝ã n→∞ p(xn+1 , xn ) < p (z, T z) , ✷✼ ✭✷✳✻✮ ✈➭ tå♥ t➵✐ sè n1 ∈ N s❛♦ ❝❤♦ ✈í✐ n > n1 t❛ ❝ã p(xn , z) < p (z, T z) ◆Õ✉ t❛ ❧✃② ✭✷✳✼✮ n > max {n0 , n1 }✱ ❦❤✐ ➤ã ❦Õt ❤ỵ♣ ✭✷✳✻✮✱ ✭✷✳✼✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ t❛ ❝ã [p(xn , T z) + p(z, T xn )] ≤ ≤ = ❇➞② ❣✐ê ✈í✐ [p(xn , z) + p(z, T z) − p(z, z) + p(z, T xn )] 1 p(z, T z) + p(z, T z) + 31 p(z, T z) p(z, T z) ✭✷✳✽✮ n > max {n0 , n1 }✱ ❦Õt ❤ỵ♣ ✭✭✷✳✻✮✱✭✷✳✼✮ ✈➭ ✭✷✳✽✮✱ t❛ ❝ã p(xn+1 , T z) = p(T xn , T z) ≤ φ (max {p(xn , z), p(xn , T xn ), p(z, T z), ≤ φ (p(z, T z)) + ❈❤♦ [p(xn , T z) + Lpw (z, xn+1 )) p(z, T xn )]}) + Lpw (z, xn+1 ) n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣✱ t❛ ❝ã p(z, T z) ≤ φ (p(z, T z))✱ ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➮♥✳ ❉♦ ➤ã ✷✳✶✳✶✸ ➜Þ♥❤ ❧ý✳ T :X→X ✭❬✷❪✮ p(z, T z) = ✈➭ z = T z ✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬② ➤ñ ✈➭ ❧➭ ➳♥❤ ①➵ s❛♦ ❝❤♦ p(T x, T y) ≤ ϕ (M (x, y)) + Lpw (y, T x) tr♦♥❣ ➤ã x, y ∈ X, ✭✷✳✾✮ L ✈➭ M (x, y) ợ tr ị ý : [0, ∞) → [0, ∞) ❧➭ ❤➭♠ ♥ö❛ ❧✐➟♥ tô❝ ♣❤➯✐ s❛♦ ❝❤♦ ❜✃t ➤é♥❣ tr♦♥❣ ♥❣❤Ü❛ ❜ë✐ ϕ(t) < t ✈í✐ ♠ä✐ t > ❑❤✐ ➤ã T ❝ã ➤✐Ĩ♠ X ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sư x0 ∈ X {xn } ột Pr ợ ị xn = T xn−1 = T n x0 ✈í✐ ♠ä✐ n ∈ N ✳ ◆Õ✉ xn0 = xn0 +1 ✈í✐ n0 ∈ N t❤× xn0 ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❞♦ ➤ã ✈í✐ ♠ä✐ T ✳ ❱× ✈❐② t❛ ❣✐➯ sư r➺♥❣ xn = xn+1 ✈í✐ ♠ä✐ n ∈ N ✈➭ p (xn , xn+1 ) > 0✳ ❱× [p(xn+1 , xn+1 ) + p(xn , xn+2 )] ≤ [p(xn , xn+1 ) + p(xn+1 , xn+2 )] ≤ max {p(xn , xn+1 ), p(xn+1 , xn+2 )} , ✷✽ t❛ ❝ã M (xn , xn+1 ) = max{p(xn , xn+1 ), p(xn , xn+1 ), p(xn+1 , xn+2 ), [p(xn+1 , xn+1 ) + p(xn , xn+2 )]} = max {p(xn , xn+1 ), p(xn+1 , xn+2 )} ❑❤✐ ➤ã✱ ✈× pw ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ X ♥➟♥ tõ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✾✮✱ t❛ ❝ã p(xn+1 , xn+2 ) = p(T xn , T xn+1 ) ≤ ϕ (M (xn , xn+1 )) + Lpw (xn+1 , xn+1 ) ✭✷✳✶✵✮ = ϕ (max {p (xn , xn+1 ) , p (xn+1 , xn+2 )}) ◆Õ✉ p(xn , xn+1 ) ≤ p(xn+1 , xn+2 ) ✈í✐ sè ♥➭♦ ➤ã n ∈ N✱ t❤× tõ ✭✷✳✶✵✮ t❛ ❝ã p(xn+1 , xn+2 ) ≤ ϕ (p(xn+1 , xn+2 )) < p(xn+1 , xn+2 ), ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➮♥✳ ❉♦ ➤ã p(xn , xn+1 ) > p(xn+1 , xn+2 ) ✈í✐ ♠ä✐ n ∈ N✳ ❑❤✐ ➤ã tõ ✭✷✳✶✵✮ t❛ s✉② r❛ p(xn+1 , xn+2 ) ≤ ϕ (p(xn , xn+1 )) < p(xn , xn+1 ) ❱× t❤Õ✱ tå♥ t➵✐ ♠ét sè ✭✷✳✶✶✮ c ≥ s❛♦ ❝❤♦ lim p(xn , xn+1 ) = lim ϕ (p(xn , xn+1 )) = c n→∞ ◆Õ✉ n→∞ c > 0✱ t❛ ❝ã c = lim sup ϕ (p(xn , xn+1 )) ≤ ϕ(c) < c, n→∞ ➤✐Ị✉ ♥➭② ♠➞✉ t❤✉➮♥✳ ❱× ✈❐② lim p(xn , xn+1 ) = 0✳ m,n→∞ ❚✐Õ♣ ➤Õ♥✱ t❛ ❝❤Ø r❛ r➺♥❣ lim p(xn , xm ) = ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐✱ tå♥ t➵✐ sè m,n→∞ ε > ✈➭ ❝➳❝ ❞➲② {nk }k∈N , {mk }k∈N tr♦♥❣ N ✈í✐ mk > nk ≥ k s❛♦ ❝❤♦ p (xnk , xmk ) ≥ ε ✈í✐ ♠ä✐ k ∈ N ✳ ❈❤ó♥❣ t❛ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣ p (xnk , xmk −1 ) ≤ ε ✈➭ ❦❤✐ ➤ã ε ≤ p (xnk , xmk ) ≤ p xnk , xmk−1 + p xmk−1 , xmk < ε + p xmk−1 , xmk , ❞♦ ➤ã lim p(xnk , xmk ) = ỗ k N m,n k0 ∈ N s❛♦ ❝❤♦ p xnk+1 , xnk < ε ✈➭ p xmk+1 , xmk < ε ✈í✐ ♠ä✐ k ≥ k0 ❑❤✐ ➤ã p (xnk , xmk ) ≤ M (xnk , xmk ) ≤ p (xnk , xmk ) + p xmk+1 , xmk + p xnk , xnk+1 ✷✾ , ✈í✐ ♠ä✐ k ≥ k0 ✳ ❉♦ ➤ã t❛ ❝ã lim M (xnk , xmk ) = ε✳ ❱× M (xnk , xmk ) ≥ ε ✈í✐ ♠ä✐ k→∞ k ∈ N ✈➭ ϕ ❧➭ ❤➭♠ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ ❜➟♥ ♣❤➯✐ ♥➟♥ t❛ s✉② r❛ lim sup ϕ (M (xnk , xmk )) () k t ỗ k N t❛ ❝ã ε ≤ p (xnk , xmk ) ≤ p xnk , xnk+1 + p xnk+1 , xmk+1 + p xmk+1 , xmk ≤ p xnk , xmk+1 + ϕ (M (xnk , xmk )) + Lpw xmk , xnk+1 + p xmk+1 , xmk , ✈× t❤Õ s✉② r❛ ε ≤ lim sup ϕ (M (xnk , xmk )) ≤ ϕ(ε) < ε, k→∞ ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× ✈❐② tr♦♥❣ lim p(xn , xm ) = ❙✉② r❛ {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② n,m→∞ (X, p) ❱× X ❧➭ 0✲➤➬② ➤đ✱ ♥➟♥ tå♥ t➵✐ z ∈ X s❛♦ ❝❤♦ lim p(xn , xm ) = lim p(xn , z) = p(z, z) = n,m→∞ n→∞ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ z ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ◆Õ✉ p(z, T z) t❤× t❛ ❝ã p(z, T z) ≤ p(z, xn+1 ) + p(xn+1 , T z) ≤ p(z, xn+1 ) + ϕ (p(xn , z)) + Lpw (z, xn+1 ) ≤ p(z, xn+1 )+ +ϕ max p(xn , z), p(xn , T xn ), p(z, T z), 12 {p(xn , T z) + p(z, T xn )} +Lpw (z, xn+1 ) ❇➺♥❣ ❝➳❝❤ ❧✃② ❣✐í✐ ❤➵♥ lim sup✱ t❛ ➤➢ỵ❝ n→∞ p(z, T z) ≤ ϕ (p(z, T z)) < p(z, T z), ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➮♥✳ ❉♦ ➤ã✱ ❝❤ó♥❣ t❛ ❝ã p(z, T z) = 0✳ ❱× ✈❐② z = T z ✳ ❇➞② ❣✐ê t❛ tr×♥❤ ❜➭② ♠ét ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❦Õt q✉➯ tr➟♥✳ ✷✳✶✳✶✹ ❱Ý ❞ơ✳ ●✐➯ sư X = [0, 1] ✈➭ ❤➭♠ p : X × X → R ❝❤♦ ❜ë✐ p(x, y) = max{x, y} ♥Õ✉ x = y, ✸✵ ♥Õ✉ x = y > ❑❤✐ ➤ã ①➵ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬② ➤ñ✳ ❚❛ ①➳❝ ➤Þ♥❤ ➳♥❤ T : X → X ❝❤♦ ❜ë✐ Tx =     x 1−x    ♥Õ✉ x ∈ 0, 21 , ♥Õ✉ x ∈ 0, 21 , ♥Õ✉ x = ❇➞② ❣✐ê t❛ sÏ ❝❤Ø r❛ r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✸✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶✷✱ ✭t➢➡♥❣ ø♥❣✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ủ ị ý ợ tỏ (t) = t ✈➭ L = ❚❛ ①Ðt ❝➳❝ tr➢ê♥❣ ❤ỵ♣ s❛✉✿ ❚r➢ê♥❣ ❤ỵ♣ ✶✳ ◆Õ✉ x ❉♦ ➤ã✱ t tết r rờ ợ = y tì p(T x, T y) = ✈➭ ✈× ✈❐② ❦Õt q✉➯ ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥✳ x = y tr♦♥❣ ♥❤÷♥❣ tr➢ê♥❣ ❤ỵ♣ s❛✉✳ ●✐➯ sư x, y ∈ 0, 21 ✱ ❦❤✐ ➤ã x2 ♥Õ✉ x > y, p(T x, T y) = y ♥Õ✉ x < y ≤ x y = p(x, y) ♥Õ✉ x > y, ♥Õ✉ x y, 1−x ♥Õ✉ x < y y ♥Õ✉ x > y, x ♥Õ✉ x < y p(x, y) ≤ 12 M (x, y) = w ϕ (M (x, y)) ≤ ϕ (M (x, y)) + Lp (y, T x) ❚r➢ê♥❣ ❤ỵ♣ ✹✳ ●✐➯ sö x ∈ 0, 21 ✈➭ y ∈ p(T x, T y) =      x2             ,1 ♥Õ✉ ✱ ❦❤✐ ➤ã x2 > 1−y , 1−x ♥Õ✉ x2 = , 1−y 1−y ♥Õ✉ x2 < 2 ✸✶   1−y    x ♥Õ✉ x2 > ,     1−x ≤ ♥Õ✉ x2 = ,       1−y   y ♥Õ✉ x2 < 2 1 p(x, y) ≤ M (x, y) = ϕ (M (x, y)) 2 ≤ ϕ (M (x, y)) + Lpw (y, T x) ≤ ❚r➢ê♥❣ ❤ỵ♣ ✺✳ ,1 y ∈ 0, 21 ✳ ❚r➢ê♥❣ ❤ỵ♣ ♥➭② t➢➡♥❣ tù ●✐➯ sö x∈ ●✐➯ sö x = ✈➭ y = 1✱ ❦❤✐ ➤ã t❛ ❝ã ✈➭ tr➢ê♥❣ ❤ỵ♣ ✹✳ ❚r➢ê♥❣ ❤ỵ♣ ✻✳ p(T x, T y) = p(1, T y) = < ≤ p(x, y) +1 + pw (y, T x) ≤ ϕ (M (x, y)) + Lpw (y, T x) ❚r➢ê♥❣ ❤ỵ♣ ✼✳ ●✐➯ sư y = ✈➭ x = rờ ợ t tự trờ ợ ì ✈❐②✱ t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶✷ ị ý ề ợ tỏ ó T ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ X ✳ ❈❤ó ý r➺♥❣ p(T0 , T1 ) = = M (0, 1)✱ ❦❤✐ ➤ã ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✮ ❦❤➠♥❣ ➤➢ỵ❝ t❤á❛ ì ú t tể tì ợ ột ϕ ♥➭♦ ♠➭ t❤á❛ ♠➲♥ p(T0 , T1 ) = ≤ ϕ (M (0, 1)) = ϕ(1) ✈➭ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮ ❤♦➷❝ ✭✷✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✾✳ ❉♦ ➤ã ➜Þ♥❤ ❧ý ✷✳✶✳✾ ❦❤➠♥❣ t❤Ĩ ➳♣ ❞ơ♥❣ ❝❤♦ ✈Ý ❞ơ ♥➭②✳ ❚r♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✶✸✱ ♥Õ✉ t❛ ❧✃② ϕ(t) = αt, α ∈ [0, 1)✱ t❤× ❝❤ó♥❣ t❛ ❝ã ❦Õt q✉➯ s❛✉ ✷✳✶✳✶✺ ❍Ö q✉➯✳ T :X →X ✭❬✷❪✮ ❈❤♦ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬② ➤đ ✈➭ ❧➭ ➳♥❤ ①➵ ♠➭ ✈í✐ ♥ã tå♥ t➵✐ ❤❛✐ ❤➺♥❣ sè α ∈ [0, 1) ❝❤♦ p(T x, T y) ≤ αM (x, y) + Lpw (y, T x) ✸✷ ✈í✐ ♠ä✐ x, y ∈ X, ✈➭ L≥0 s❛♦ tr♦♥❣ ➤ã M (x, y) ❑❤✐ ➤ã ✷✳✶✳✶✻ T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❍Ư q✉➯✳ T :X →X ợ tr ị ý (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬② ➤ñ ✈➭ ❧➭ ♠ét ➳♥❤ ①➵ s❛♦ ❝❤♦ p(T x, T y) ≤ ϕ(p(x, y)) + Lpw (y, T x) tr♦♥❣ ➤ã ❝❤♦ L≥0 ϕ(t) < t ✈➭ ϕ : [0, ∞) → [0, ∞) ✈í✐ ♠ä✐ t > 0✳ ❑❤✐ ➤ã T ✈í✐ ♠ä✐ x, y ∈ X, ❧➭ ❤➭♠ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ ❜➟♥ ♣❤➯✐ s❛♦ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ◆❤❐♥ ①Ðt r➺♥❣ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✶✷✱ ➜Þ♥❤ ❧ý ✷✳✶✳✶✸ ❝❤ó♥❣ t❛ ➤➲ ❝❤Ø r❛ r➺♥❣ ♥Õ✉ T ❧➭ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❦✐Ĩ✉ ❈✐r✐❝✱ t❤× ♥ã ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣✳ ◆❤➢♥❣ ➤Ó ➤➯♠ ❜➯♦ tÝ♥❤ ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✱ ❝❤ó♥❣ t❛ ♣❤➯✐ ①Ðt t ột ề ệ ữ ó ợ trì tr♦♥❣ ➤Þ♥❤ ❧ý s❛✉ ➤➞②✳ ✷✳✶✳✶✼ ➜Þ♥❤ ❧ý✳ T :X →X sö T ✭❬✷❪✮ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬② ➤ñ ✈➭ ❈❤♦ ❧➭ ➳♥❤ ①➵ s❛♦ ❝❤♦ ❤♦➷❝ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✮ ❤♦➷❝ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✾✮✮✳ ●✐➯ ❝ị♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ tå♥ t➵✐ ♠ét ❤➭♠ s♦ s➳♥❤ ϕ1 ✈➭ sè L1 ≥ s❛♦ ❝❤♦ p(T x, T y) ≤ ϕ1 (M (x, y)) + L1 pw (x, T x) , ✈í✐ ♠ä✐ x, y ∈ X ✳ ❑❤✐ ➤ã ❈❤ø♥❣ ♠✐♥❤✳ ❱× T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ T ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✮ ❤♦➷❝ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✾✮✱ ♥➟♥ ♥❤ê ➜Þ♥❤ ❧ý ✷✳✶✳✶✷ ❤♦➷❝ ➜Þ♥❤ ❧ý ✷✳✶✳✶✸ t❛ s✉② r❛ ❇➞② ❣✐ê ❣✐➯ sư r➭♥❣ ✭✷✳✶✷✮ T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ T ❝ã ❤❛✐ ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ z ✈➭ w✳ ◆Õ✉ p(z, w) = 0✱ t❤× râ z = w✳ ●✐➯ sö p(z, w) > 0✳ ❑❤✐ ➤ã ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✷✮ ✈í✐ x = z ✈➭ y = w, t❛ ❝ã < p(z, w) = p(T z, T w) ≤ ϕ1 (M (z, w)) + L1 pw (z, T z) = ϕ1 (M (z, w)) < M (z, w), ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❉♦ ➤ã T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ✸✸ ✷✳✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❑ý ❤✐Ö✉ F = {φ : [0, ∞) → [0, ∞)| φ ❧✐➟♥ tô❝ ✈➭ ❦❤➠♥❣ ❣✐➯♠}✳ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✶ X→X ✭❬✹❪✮ ❈❤♦ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬② ➤ñ ✈➭ ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ φ (p(T x, T y)) ≤ φ (ϕ (M (x, y))) + Lpw (y, T x), tr♦♥❣ ➤ã ✈➭ ✈í✐ ♠ä✐ x, y ∈ X, ✭✷✳✶✸✮ L ≥ 0✱ M (x, y) = max p(x, y), p(x, T x), p(y, T y), φ∈F T : ϕ ❍➡♥ ♥÷❛✱ ❧➭ ♠ét ❤➭♠ ✭❝✮✲s♦ s➳♥❤✳ ❑❤✐ ➤ã✱ [p(x, T y) + p(y, T x)] , T ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ z tr♦♥❣ X✳ p(z, z) = ❈❤ø♥❣ ♠✐♥❤✳ ❜ë✐❝➠♥❣ t❤ø❝ ●✐➯ sö x0 ∈ X ✈➭ {xn } ột Pr ợ ị xn = T xn−1 = T n x0 ✈í✐ ♠ä✐ n ∈ N ✳ ◆Õ✉ xn0 = xn0 +1 ✈í✐ n0 ∈ N t❤× xn0 ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✳ ❱× ✈❐② t❛ ❣✐➯ sư r➺♥❣ xn = xn+1 ✈í✐ ♠ä✐ n ∈ N✳ ❱× [p(xn+1 , xn+1 ) + p(xn , xn+2 )] ≤ [p(xn , xn+1 ) + p(xn+1 , xn+2 )] ≤ max {p(xn , xn+1 ) + p(xn+1 , xn+2 )} , ♥➟♥ t❛ ❝ã M (xn , xn+1 ) = max{p(xn , xn+1 ), p(xn , xn+1 ), p(xn+1 , xn+2 ), [p(xn+1 , xn+1 ) + p(xn , xn+2 )]} = max {p(xn , xn+1 ), p(xn+1 , xn+2 )} ❇➺♥❣ ❝➳❝❤ ➳♣ ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✶✸✮✱ t❛ ➤➢ỵ❝ φ (p(xn+1 , xn+2 )) = φ (p(T xn , T xn+1 )) ≤ φ (ϕ(M (xn , xn+1 ))) + Lpw (xn+1 , xn+1 ) ✭✷✳✶✹✮ = φ (ϕ(M (xn , xn+1 ))) ◆Õ✉ M (xn , xn+1 ) = p(xn+1 , xn+2 ) ✈í✐ sè ♥➭♦ ➤ã n ∈ N✱ t❤× tõ ✭✷✳✶✹✮ t❛ ➤➢ỵ❝ φ (p(xn+1 , xn+2 )) ≤ φ (ϕ(p(xn , xn+1 ))) , ✸✹ ✈➭ ✈× φ ❧➭ ❤➭♠ ❦❤➠♥❣ ❣✐➯♠ ♥➟♥ s✉② r❛ p(xn+1 , xn+2 ) ≤ ϕ(p(xn+1 , xn+2 )) < p(xn+1 , xn+2 ), ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❱× ✈❐② M (xn , xn+1 ) = p(xn , xn+1 ) ✈í✐ ♠ä✐ n ∈ N✳ ❑Õt ❤➡♣ ✈í✐ ✭✷✳✶✹✮✱ t❛ ❝ã φ (p(xn+1 , xn+2 )) ≤ φ (ϕ(p(xn , xn+1 ))) , ✈➭ ✈× φ ❧➭ ❤➭♠ ❦❤➠♥❣ ❣✐➯♠ ♥➟♥ p(xn+1 , xn+2 ) ≤ ϕ(p(xn , xn+1 )) ❇➺♥❣ q✉② ♥➵♣✱ t❛ ❝ã p(xn+1 , xn+2 ) ≤ ϕn+1 (p(x0 , x1 )) ✈í✐ ♠ä✐ n ∈ N ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ ✈í✐ m > n t❛ ❝ã m−2 m−1 p(xk , xk+1 ) − p(xn , xm ) ≤ p(xk+1 , xk+1 ) k=n ∞ k=n m−1 ∞ p(xk , xk+1 ) ≤ p(xk , xk+1 ) ≤ ≤ k=n k=n ∞ ❱× ϕ ❧➭ ❤➭♠ ✭❝✮✲s♦ s➳♥❤ ♥➟♥ ỗ k (p(x0 , x1 )) k=n k (p(x0 , x1 )) ❤é✐ tô✳ ❚õ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ k=0 t❛ s✉② r❛ {xn } ❧➭ ♠ét ❞➲② 0✲❈❛✉❝❤② tr♦♥❣ X ✳ ❱× X ❧➭ ✵✲➤➬② ➤đ✱ ♥➟♥ {xn } ❤é✐ tơ t❤❡♦ t➠♣➠ τp tí✐ ♠ét ➤✐Ĩ♠ z ∈ X s❛♦ ❝❤♦ lim p(xn , z) = p(z, z) = n→∞ ❇➞② ❣✐ê✱ t❛ ❝❤Ø r❛ r➺♥❣ p(z, T z) = 0✳ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐✱ p(z, T z) > 0✳ ❱× φ ❧➭ ♠ét ❤➭♠ ✭❝✮✲s♦ s➳♥❤✱ ♥➟♥ φ(t) < t ✈í✐ t > ✈➭ ✈× lim p(xn+1 , xn ) = ✱ n→∞ lim p(xn , z) = ♥➟♥ ❝ã tå♥ t➵✐ sè n0 ∈ N s❛♦ ❝❤♦ ✈í✐ n > n0 t❛ ❝ã n→∞ p(xn+1 , xn ) < p (z, T z) , ✈➭ tå♥ t➵✐ sè ✭✷✳✶✺✮ n1 ∈ N s❛♦ ❝❤♦ ✈í✐ n > n1 t❛ ❝ã p(xn , z) < p (z, T z) ✸✺ ✭✷✳✶✻✮ ◆Õ✉ t❛ ❧✃② n > max {n0 , n1 }✱ ❦❤✐ ➤ã ❦Õt ❤ỵ♣ ✭✷✳✶✺✮✱ ✭✷✳✶✻✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ t❛ ❝ã [p(xn , z) + p(z, T z) − p(z, z) + p(z, T xn )] 1 p(z, T z) + p(z, T z) + 31 p(z, T z) p(z, T z) [p(xn , T z) + p(z, T xn )] ≤ ≤ = ❇➞② ❣✐ê ✈í✐ ✭✷✳✶✼✮ n > max {n0 , n1 }✳ ❑❤✐ ➤ã✱ ❦Õt ❤ỵ♣ ✭✷✳✶✺✮✱ ✭✷✳✶✻✮ ✈➭ ✭✷✳✶✼✮ t❛ ❝ã φ (p(xn+1 , T z)) = φ (p(T xn T z, )) ≤ φ (ϕ(M (xn , z)) + Lpw (z, xn+1 )) ✭✷✳✶✽✮ = φ (ϕ(p(z, T z)) + Lpw (z, xn+1 )) ❈❤♦ n → ∞ tr♦♥❣ ✭✷✳✶✽✮✱ t❛ ➤➢ỵ❝ φ (p(z, T z)) ≤ φ (ϕ(p(z, T z))) ❱× φ ❧➭ ❤➭♠ ❦❤➠♥❣ ❣✐➯♠ ♥➟♥ ❉♦ ➤ã p(z, T z) ≤ ϕ(p(z, T z)) < p(z, T z), ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ p(T xn , z) = ✈➭ z = T z ✳ ❚õ ➤Þ♥❤ ❧ý tr➟♥ t❛ s✉② r❛ ❦Õt q✉➯ s❛✉ ❝đ❛ ■✳ ❆❧t✉♥ ✈➭ ❖✳ ❆❝❛r✳ ✷✳✷✳✷ ❍Ö q✉➯✳ T :X→X ✭❬✸❪✮ ❈❤♦ ❧➭ ➳♥❤ ①➵ (X, p) (ϕ, L)✲❝♦ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬② ➤đ ✈➭ ②Õ✉ ✈í✐ ϕ ❧➭ ♠ét ✭❝✮✲❤➭♠ s♦ s➳♥❤✳ ❑❤✐ ➤ã T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ❜ë✐ ❚r♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✶ ♥Õ✉ t❛ ❧✃② ❤➭♠ φ : [0, ∞) → [0, ∞) ❝❤♦ φ(t) = t ✈í✐ ♠ä✐ t ∈ [0, ) tì t t ợ ệ q tr ét r➺♥❣ ♥Õ✉ f : [0, ∞) → [0, ∞) ❧➭ ❤➭♠ ❦❤➯ tÝ❝❤ ▲❡❜❡s❣✉❡ ✈➭ ❤➭♠ t φ : [0, ) [0, ) ợ ị (t) = f (s)ds, ✈í✐ t ∈ [0, ∞)✱ t❤× t❛ ❝ã φ ∈ F ✳ ❑❤✐ ➤ã✱ t❛ ❝ã ❤Ö q✉➯ s❛✉✳ ✷✳✷✳✸ ❍Ö q✉➯✳ T :X→X ✭❬✸❪✮ ❈❤♦ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ p(T x,T y) ϕ(M (x,y)) f (s)ds+Lpw (y, T x), f (s)ds ≤ 0 ✸✻ ✈í✐ ♠ä✐ x, y ∈ X, ➤đ ✈➭ tr♦♥❣ ➤ã L ≥ 0✱ M (x, y) = max p(x, y), p(x, T x), p(y, T y), ϕ f : [0, ∞) → [0, ∞) ❧➭ ♠ét ❤➭♠ ✭❝✮✲s♦ s➳♥❤ ✈➭ ❑❤✐ ➤ã✱ T [p(x, T y) + p(y, T x)] , ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❧➭ ➳♥❤ ①➵ ❦❤➯ tÝ❝❤ ▲❡❜❡s❣✉❡✳ X✳ ❚✐Õ♣ t❤❡♦✱ t❛ tr×♥❤ ❜➭② ♠ét ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ♥➭②✳ ✷✳✷✳✹ ❱Ý ❞ô✳ ❈❤♦ n X = A ∪ B ✱ tr♦♥❣ ➤ã A = {0} ∪ : n = 1, 2, , B = {2, 3, 4, } ✈➭ max{x, y} ♥Õ✉ x = y, p(x, y) = ❑❤✐ ➤ã ①➵ ♥Õ✉ x = y (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬② ➤đ✳ ❚❛ ①➳❝ ➤Þ♥❤ ➳♥❤ T : X → X ❝❤♦ ❜ë✐ x3 ♥Õ✉ x ∈ A, Tx = x ♥Õ✉ x ∈ B ❇➞② ❣✐ê t❛ ❝❤Ø r❛ r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✶✷✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶ t❤á❛ ♠➲♥ ✈í✐ φ(t) = t, ϕ(t) = t ✈➭ ❚r➢ê♥❣ ❤ỵ♣ ✶✳ L = ❚❛ ①Ðt ❝➳❝ tr➢ê♥❣ ❤ỵ♣ s❛✉✿ ◆Õ✉ x ❉♦ ➤ã✱ t❛ ❣✐➯ tết r rờ ợ = y tì p(T x, T y) = ✈➭ ✈× ✈❐② ❦Õt q✉➯ ❧➭ ❤✐Ĩ♥ ♥❤✐➟♥✳ x = y tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ s❛✉✳ ●✐➯ sö x, y ∈ A✳ ❑❤✐ ➤ã✱ ♥Õ✉ x = ❤♦➷❝ y = t❤× inf {|y − x3 | : x, y ∈ A ✈➭ x = y} = ✱ ♥Õ✉ x = ✈➭ y = t❤× x3 ≤ x✳ ❉♦ ➤ã t❛ ❝ã p(T x, T y) = max {x3 , y } ≤ max {x, y} + |y − x3 | = 21 p(x, y) + 2pw (y, T x) ≤ ϕ (M (x, y)) + Lpw (y, T x) ❚r➢ê♥❣ ❤ỵ♣ ✸✳ ●✐➯ sư x, y ∈ B ✳ ❑❤✐ ➤ã p(T x, T y) = max 1 , x y = min{x, y} ≤ 21 max {x, y} = 12 p(x, y) ≤ ϕ (M (x, y)) + Lpw (y, T x) ❚r➢ê♥❣ ❤ỵ♣ ✹✳ ●✐➯ sư x ∈ A, y ∈ B ✳ ❑❤✐ ➤ã p(T x, T y) = max x3 , y1 ≤ y = 12 p(x, y) ≤ ϕ (M (x, y)) + Lpw (y, T x) ✸✼ ❚r➢ê♥❣ ❤ỵ♣ ✺✳ ●✐➯ sư x ∈ B, y ∈ A✳ ❚r➢ê♥❣ ❤ỵ♣ ♥➭② t➢➡♥❣ tù ♥❤➢ tr➢ê♥❣ ❤ỵ♣ ✹✳ ❉♦ ➤ã✱ t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ủ ị ý ợ tỏ ì T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ X ✳ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✺ X→X ✭❬✹❪✮ ❈❤♦ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✵✲➤➬② ➤ñ ✈➭ ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ φ (p(T x, T y)) ≤ φ (ϕ (M (x, y))) + Lpw (y, T x), tr♦♥❣ ➤ã ✈➭ ϕ ϕ1 , [p(x, T y) + p(y, T x)] , sè L1 ≥ ✈➭ ❤➭♠ T ❝ò♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ tå♥ t➵✐ ❤➭♠ s♦ φ1 ∈ F ♠➭ φ1 (t) > ✈í✐ ♠ä✐ φ1 (p(T x, T y)) ≤ φ1 (ϕ1 (M (x, y))) + L1 pw (x, T x) ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣ ❱× ❈❤ø♥❣ ♠✐♥❤✳ ❧ý ✷✳✷✳✶ t❛ s✉② r❛ t>0 ✈í✐ ♠ä✐ x, y ∈ X ✭✷✳✶✾✮ X✳ T ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✸✮ ✱ ♥➟♥ ♥❤ê ➜Þ♥❤ T ❝ã ❤❛✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❧➭ z ✈➭ w✳ ◆Õ✉ p(z, w) = 0✱ t❤× râ z = w✳ ●✐➯ sư p(z, w) > 0✳ ❑❤✐ ➤ã ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✾✮ ✈í✐ x = z ✈➭ y = w, t❛ ❝ã < φ1 (p(z, w)) = φ1 (p(T z, T w)) ≤ φ1 (ϕ1 (M (z, w))) + L1 pw (z, T z) = φ1 (ϕ1 (M (z, T ))) = φ1 (ϕ1 (p(z, w))) ❱× s❛♦ ❝❤♦ T ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❇➞② ❣✐ê ❣✐➯ sö r➭♥❣ x, y ∈ X, ❧➭ ♠ét ❤➭♠ ✭❝✮✲s♦ s➳♥❤✳ ●✐➯ t❤✐Õt t❤➟♠ r➺♥❣ s➳♥❤ ✈í✐ ♠ä✐ L ≥ 0✱ M (x, y) = max p(x, y), p(x, T x), p(y, T y), φ∈F T : φ ❧➭ ❤➭♠ ❦❤➠♥❣ ❣✐➯♠ ♥➟♥ < p(z, w) ≤ ϕ1 (p(z, w)) < p(z, w), ➤✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❉♦ ➤ã T ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ X ✳ ✸✽ ❑Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉ ✶✳ ❍Ö t❤è♥❣ ❤ã❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ò ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ ❞➲② 0✲❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ ②Õ✉✱ ➳♥❤ ①➵ ✭ϕ✱L✮✲❝♦ ②Õ✉✱ ➳♥❤ ①➵ ❤➬✉ ❝♦✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❦✐Ĩ✉ ❈✐r✐❝✱ q✉ü ➤➵♦ ❝đ❛ ➳♥❤ ①➵ T t➵✐ ➤✐Ĩ♠ x✱ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ T ✲q✉ü ➤➵♦✱ t♦➳♥ tö P✐❝❛r❞ ②Õ✉✱ ❤➭♠ s♦ s➳♥❤✱ ❤➭♠ ✭c✮✲s♦ s rì ột số ị ý ề ể ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝✱ ➳♥❤ ①➵ ❤➬✉ ❝♦ ♠➵♥❤ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ◆❣♦➭✐ r❛ ❝❤ó♥❣ t ò trì ột số ết q ề ể ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ s✉② ré♥❣ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ➤ã✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ➤Þ♥❤ ❧ý ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ❝❤ø♥❣ ò s ợ ị ý ➜Þ♥❤ ❧ý ✶✳✷✳✸✱ ➜Þ♥❤ ❧ý ✶✳✷✳✻✱ ➜Þ♥❤ ❧ý ✶✳✷✳✼✱ ➜Þ♥❤ ❧ý ✷✳✶✳✶✷✱ ➜Þ♥❤ ❧ý ✷✳✶✳✶✸✱ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ ➜Þ♥❤ ❧ý ✷✳✷✳✺✳ ✹✳ ❚r×♥❤ ❜➭② ❝❤✐ t✐Õt ❱Ý ❞ơ ✶✳✷✳✺ ➤Ĩ ♠✐♥❤ ❤ä❛ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❱Ý ❞ơ ✷✳✶✳✶✹ ♠✐♥❤ ❤ä❛ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✈➭ ❱Ý ❞ô ✷✳✷✳✹ ♠✐♥❤ ❤ä❛ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ ❈✐r✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ✸✾ t➭✐ ệ t ỗ ❝➢➡♥❣✱ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ⑧ ❆❝❛r ❛♥❞ ❱✳ ❇❡r✐♥❞❡ ❛♥❞ ■✳ ❆❧t✉♥ ✭✷✵✶✷✮✱ ❬✷❪ ❖✳ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❈✐r✐❝✲t②♣❡ str♦♥❣ ❛❧♠♦st ❝♦♥tr❛❝t✐♦♥s ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✶✷ ✭✶✲✷✮✱ ✷✹✼✲✷✺✾✳ ❬✸❪ ■✳ ❆❧t✉♥✱ ❖✳ ❆❝❛r ✭✷✵✶✷✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ✇❡❛❦ ❝♦♥tr❛❝t✐♦♥s ✐♥ t❤❡ s❡♥s❡ ♦❢ ❇❡r✐♥❞❡ ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳✱ ✶✺✾✱ ✷✻✹✷✲✷✻✹✽✳ ❬✹❪ ■✳ ❆❧t✉♥ ❛♥❞ ❑✳ ❙❛❞❛r❛♥❣❛♥✐ ✭✷✵✶✹✮✱ ❣❡♥❡r❛❧✐③❡❞ ❛❧♠♦st ❝♦♥tr❛❝t✐♦♥s ✐♥ ❋✐①❡❞ ♣❛rt✐❛❧ ♣♦✐♥t ♠❡tr✐❝ t❤❡♦r❡♠s s♣❛❝❡s✱ ❢♦r ✶✷✷ ✭✽✮✱ ❞♦✐✿✶✵✳✶✵✵✼✴s✹✵✵✾✻✲✵✶✹✲✵✶✷✷✲✾✳ ❬✺❪ ●✳ ❇❛❜✉✱ ▼✳ ❙❛♥❞❞❤②❛✱ ▼✳ ❑❛♠❡s✇❛r✐ ✭✷✵✵✽✮✱ ❆ ♥♦t❡ ♦♥ ❛ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦❢ ❇❡r✐♥❞❡ ♦♥ ✇❡❛❦ ❝♦♥tr❛❝t✐♦♥s✱ ❈❛r♣❛t❤✐❛♥ ❏✳ ▼❛t❤✳✱ ✷✹ ✭✶✮✱ ✽✲✶✷✳ ❬✻❪ ❱✳❇❡r✐♥❞❡ ✭✷✵✵✸✮✱ ❖♥ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ✇❡❛❦ ❝♦♥✲ tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❈❛r♣❛t❤✐❛♥ ❏✳ ▼❛t❤✳✱ ✶✾✱ ✼✲✷✷✳ ❬✼❪ ❱✳❇❡r✐♥❞❡ ✭✷✵✵✹✮✱ ❆♣♣r♦①✐♠❛t✐♦♥ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ✇❡❛❦ ❝♦♥tr❛❝t✐♦♥s ✉s✲ ✐♥❣ t❤❡ P✐❝❛r❞ ✐t❡r❛t✐♦♥✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ❋♦r✉♠✱ ✾ ✭✶✮✱ ✹✸✲✺✸✳ ❬✽❪ ❱✳❇❡r✐♥❞❡ ✭✷✵✵✽✮✱ ●❡♥❡r❛❧ ❝♦♥str✉❝t✐✈❡ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❈✐r✐❝✲ t②♣❡ ❛❧♠♦st ❝♦♥tr❛❝t✐♦♥s ✐♥ ♠❡tr✐❝ s♣❛❝❡s✱ ❈❛r♣❛t❤✐❛♥ ❏✳ ▼❛t❤✳✱ ✷✹ ✭✷✮✱ ✶✵✲✶✾✳ ❬✾❪ ❱✳❇❡r✐♥❞❡ ✭✷✵✵✾✮✱ ❙♦♠❡ r❡♠❛r❦s ♦♥ ❛ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❢♦r ❈✐r✐❝✲ t②♣❡ ❛❧♠♦st ❝♦♥tr❛❝t✐♦♥s✱ ❈❛r♣❛t❤✐❛♥ ❏✳ ▼❛t❤✳✱ ✷✺✱ ✶✺✼✲✶✻✷✳ ❬✶✵❪ ▲✳ ❈✐r✐❝ ✭✶✾✼✹✮✱ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❇❛♥❛❝❤✬s ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✹✺✱ ✷✻✼✲✷✼✸✳ ✹✵ ❬✶✶❪ ❙✳ ❈♦❜❛③❛s ✭✷✵✶✶✮✱ ❈♦♠♣❧❡t❡♥❡ss ✐♥ q✉❛s✐✲♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❊❦❡❧❛♥❞ ✈❛r✐❛t♦♥❛❧ ♣r✐♥❝✐♣❧❡✱ ❚♦♣♦❧♦❣② ❆♣♣❧✳✱ ✶✺✽✱ ✶✵✼✸✲✶✵✽✹✳ ❬✶✷❪ ●✳❙✳ ▼❛t❤❡✇s ✭✶✾✾✹✮✱ P❛rt✐❛❧ ♠❡tr✐❝ t♦♣♦❧♦❣②✱ Pr♦❝✳ ✽t❤ ❙✉♠♠❡r ❈♦♥✲ ❢❡r❡♥❝❡ ♦♥ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❆♥♥❛❧s ♦❢ t❤❡ ◆❡✇ ❨♦r❦ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s✱ ✼✷✽✱ ✶✽✸✲✶✾✼✳ ❬✶✸❪ ❙✳ ❘♦♠❛❣✉❡r❛ ✭✷✵✶✶✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❣❡♥❡r❛❧✐③❡❞ ❝♦♥tr❛❝✲ t✐♦♥s ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❚♦♣♦❧✳ ❆♣♣❧✳✱ ✷✶✽✱ ✷✸✾✽✲✷✹✵✻✳ ❬✶✹❪ ■✳ ❆✳ ❘✉s ✭✶✾✾✻✮✱ P✐❝❛r❞ ♦♣❡r❛t♦rs ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ❇❛❜❡s✲❇♦❧②❛✐ ❯♥✐✲ ✈❡rs✐t②✳ ❬✶✺❪ ❚✳ ❩❛♠❢✐r❡s❝✉ ✭✶✾✼✷✮✱ ▼❛t❤✳ ✭❇❛s❡❧✮✱ ✷✸✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ♠❡tr✐❝ s♣❛❝❡s✱ ❆r❝❤✳ ✷✾✷✲✷✾✽✳ ✹✶