Sự tồn tại điểm bất động của ánh xạ co trong không gian b mêtric nón

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❚r➬♥ ❚❤Þ ❍➵♥❤ ❙ù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥ ✲ ✷✵✶✻ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❚r➬♥ ❚❤Þ ❍➵♥❤ ❙ù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ▲✉❐♥ ❱➝♥ ❚❤➵❝ ❙ü ❚♦➳♥ ❍ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ➜✐♥❤ ❍✉② ❍♦➭♥❣ ◆❣❤Ư ❆♥ ✲ ✷✵✶✻ ▼ơ❝ ▲ơ❝ ❚r❛♥❣ ▼ơ❝ ❧ô❝ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ✷ ❈❤➢➡♥❣ ✶✳ ❑❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✺ ✶✳✶✳ ▼ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳ ◆ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸✳ ❑❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ❈❤➢➡♥❣ ✷✳ ❙ù tå♥ t➵✐ ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✶✼ ✷✳✶✳ ▼ét sè ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✷✳ ▼ét sè ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ❑Õt ❧✉❐♥ ✸✾ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✹✵ ❧ê✐ ♥ã✐ ➤➬✉ ▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ột tr ữ ề ợ ề t ❤ä❝ q✉❛♥ t➞♠ ✈➭ ♥❣❤✐➟♥ ❝ø✉✳ ◆❣➢ê✐ t❛ ➤➲ t×♠ t❤✃② sù ø♥❣ ❞ơ♥❣ ➤❛ ❞➵♥❣ ❝đ❛ ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ t♦➳♥ ❤ä❝ ✈➭ ♥❤✐Ò✉ ♥❣➭♥❤ ❦ü t❤✉❐t ❦❤➳❝✳ ❙ù ♣❤➳t tr✐Ĩ♥ ❝đ❛ ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❣➽♥ ❧✐Ị♥ ✈í✐ t➟♥ t✉ỉ✐ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❧í♥ ♥❤➢ ❇r♦✉✇❡r✱ ❇❛♥❛❝❤✱ ❙❝❤❛✉❞❡r✱ ❑❛❦✉t❛♥✐✱ ✳✳✳ ❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❇❛♥❛❝❤ ✭✶✾✾✷✮✳ ◆❣➢ê✐ t❛ ➤➲ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ♥➭② ❝❤♦ ♥❤✐Ò✉ ❧♦➵✐ ➳♥❤ ①➵ ✈➭ ♥❤✐Ị✉ ❧♦➵✐ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✳ ▼ét tr♦♥❣ ♥❤÷♥❣ ❤➢í♥❣ ♠ë ré♥❣ ➤ã ❧➭ t❤❛② ➤ỉ✐ ➤✐Ị✉ ❦✐Ư♥ tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ♠➟tr✐❝✱ tõ ➤ã t❤✉ ➤➢ỵ❝ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ré♥❣ ❤➡♥ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❙❛✉ ➤ã✱ ♥❣➢ê✐ t❛ ♥❣❤✐➟♥ ❝ø✉ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❧í♣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ✈õ❛ ➤Þ♥❤ ♥❣❤Ü❛✳✳✳ ◆➝♠ ✷✵✵✼✱ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❚r✉♥❣ ◗✉è❝✿ ❍✉❛♥❣ ▲♦♥❣ ✲ ●✉❛♥❣ ✈➭ ❩❤❛♥❣ ❳✐❛♥ ✭❬✻❪✮ ➤➲ t❤❛② ❣✐➯ t❤✐Õt ❤➭♠ ♠➟tr✐❝ ♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ t❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝ ❦❤➠♥❣ ➞♠ ❜ë✐ ♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ ♠ét ♥ã♥ ➤Þ♥❤ ❤➢í♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛✲ ♥❛❝❤ ✈➭ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❙❛✉ ➤ã✱ ♥❤✐Ò✉ ♥❤➭ t♦➳♥ ❤ä❝ ➤➲ ♥❣❤✐➟♥ ❝ø✉ ✈➭ ➤➵t ➤➢ỵ❝ ♥❤✐Ị✉ ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ◆❤÷♥❣ ♥❣➢ê✐ t❤✉ ➤➢ỵ❝ ♥❤✐Ị✉ ❦Õt q✉➯ t❤❡♦ ❤➢í♥❣ ♥➭② ❧➭✿ ❏✳ ❙✳ ❯♠❡✱ ❘✳ ❆✳ ❙t♦❧t❡♥❜❡❣✱ ❈✳ ❙✳ ❲♦♥❣✱ ❍✳ ▲✳ ●✉❛♥❣ ✈➭ ❩✳ ❳✐❛♥ ✳✳✳ ❑❤➳✐ ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ➤➢ỵ❝ ➤➢❛ r❛ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ❜ë✐ ❙✳ ❈③❡r✇✐❦ ✭❬✺❪✮✳ ❚r♦♥❣ ❬✼❪✱ ◆✳ ❍✉ss❛✐♥ ✈➭ ❝➳❝ ❝é♥❣ sù ➤➲ ♠ë ré♥❣ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ✈➭ ♠➟tr✐❝ ♥ã♥ ❜➺♥❣ ❝➳❝❤ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè tÝ♥❤ ❝❤✃t t➠♣➠ ✈➭ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♥➭②✳ ➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ ✈➭ ❧Ü♥❤ ❤é✐ ✈Ị ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ❝❤ó♥❣ t➠✐ t×♠ ❤✐Ĩ✉✱ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ❝ñ❛ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ✈➭ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✳ ❱× t❤Õ ❝❤ó♥❣ t➠✐ ❝❤ä♥ ➤Ò t➭✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭✿ ✧❙ù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✧✳ ❱í✐ ♠ơ❝ ➤Ý❝❤ ➤ã✱ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❝❤✐❛ ❧➭♠ ❤❛✐ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ❑❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✳ ❚r♦♥❣ ♠ơ❝ ✶✱ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ❝➡ ❜➯♥ ✈Ị ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ ➳♥❤ ①➵ ❧✐➟♥ tơ❝✳✳✳ ❝➬♥ tr♦♥❣ ❧✉❐♥ ✈➝♥✳ ▼ơ❝ ✷ tr×♥❤ ❜➭② ➤Þ♥❤ ♥❣❤Ü❛✱ ✈Ý ❞ơ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ♥ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ▼ơ❝ ✸ tr×♥❤ ❜➭② ➤Þ♥❤ ♥❣❤Ü❛✱ ✈Ý ❞ơ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ❙ù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲ ♠➟tr✐❝ ♥ã♥✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t trì ột số ị ý ề tồ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❇❛♥❛❝❤✱ ❈❤❛tt❡r❥❡❛✱ ❑❛♥♥❛♥✱✳✳✳ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ➤➲ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ tr♦♥❣ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✾❪✳ ▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ ➤➢❛ r❛ ♠ét ✈➭✐ ❦Õt q✉➯ ♠í✐ ✈Ị sù tå♥ t➵✐ ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲ ♠➟tr✐❝ ♥ã♥✱ ➤ã ❧➭ ❝➳❝ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ ✷✳✷✳✸✱ ✷✳✷✳✽✱ ✷✳✷✳✶✵ ✈➭ ❝➳❝ ❍Ö q✉➯ ✷✳✷✳✹✱ ✷✳✷✳✺✱ ✷✳✷✳✼✱ ✷✳✷✳✾ ✈➭ ✷✳✷✳✶✷✳ ❈➳❝ ❦Õt q✉➯ ♥➭② ❧➭ ♠ë ré♥❣ ❝ñ❛ ♠ét sè ❦Õt q✉➯ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✹✱ ✽✱ ✾✱ ✶✵❪✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ✈➭ ♥❣❤✐➟♠ ❦❤➽❝ ❝ñ❛ P●❙✳❚❙✳ ➜✐♥❤ ❍✉② ❍♦➭♥❣✳ ❚➳❝ ❣✐➯ ①✐♥ ❜➭② tỏ ò ết s s ủ ì ế ❚➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥ ❈❤đ ♥❤✐Ư♠ ❑❤♦❛ ❙❛✉ ➤➵✐ ❤ä❝✱ ❇❛♥ ❈❤đ ♥❤✐Ư♠ ❑❤♦❛ ❚♦➳♥ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✳ ❚➳❝ ❣✐➯ ①✐♥ ➤➢ỵ❝ ❝➯♠ ➡♥ q✉ý ❚❤➬② ❣✐➳♦✱ ❈➠ ❣✐➳♦ ❚æ ●✐➯✐ tÝ❝❤ tr♦♥❣ ❑❤♦❛ ❚♦➳♥ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ ♥❤✐Ưt t×♥❤ ❣✐➯♥❣ ❞➵② ✈➭ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ s✉èt t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣✳ ❈✉è✐ ❝ï♥❣ ①✐♥ ❝➯♠ ➡♥ ❣✐❛ ➤×♥❤✱ ➤å♥❣ ♥❣❤✐Ö♣✱ ❜➵♥ ❜❒✱ ➤➷❝ ❜✐Öt ❧➭ ❝➳❝ ❜➵♥ tr♦♥❣ ❧í♣ ❈❛♦ ❤ä❝ ✷✷ ✲ ❈❤✉②➟♥ ♥❣➭♥❤ ●✐➯✐ tÝ❝❤ ➤➲ ❝é♥❣ t➳❝✱ ❣✐ó♣ ➤ì ✈➭ ➤é♥❣ ✈✐➟♥ t➳❝ ❣✐➯ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉✳ ▼➷❝ ❞ï ➤➲ ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣ ♥❤➢♥❣ ❞♦ ❝ß♥ ❤➵♥ ❝❤Õ ✈Ò ♠➷t ❦✐Õ♥ t❤ø❝ ✈➭ t❤ê✐ ❣✐❛♥ ♥➟♥ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ t❤✐Õ✉ sãt✳ ❑Ý♥❤ ♠♦♥❣ q✉ý ❚❤➬② ❈➠ ✈➭ ❜➵♥ ❜❒ ➤ã♥❣ ❣ã♣ ý ❦✐Õ♥ ➤Ó ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ❱✐♥❤✱ t❤➳♥❣ ✽ ♥➝♠ ✷✵✶✻ ❚r➬♥ ❚❤Þ ❍➵♥❤ ❝❤➢➡♥❣ ✶ ❑❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ trì ị ĩ í ụ ột sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝ñ❛ ♥ã♥ ✈➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✱ ❧➭♠ ❝➡ së ❝❤♦ ❈❤➢➡♥❣ ✷✳ 1.1 ▼ét số ế tứ ị ụ trì ột sè ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ❝➡ ❜➯♥ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✱✳✳✳ ❝➬♥ ❞ï♥❣ tr♦♥❣ ❧✉❐♥ ✈➝♥✳ ❈➳❝ ❦Õt q✉➯ ♥➭② ➤➢ỵ❝ trÝ❝❤ r❛ tõ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ✈➭ ❬✸❪✳ 1.1.1 ị ĩ X sử d ợ ọ tr tr X t rỗ ♥Õ✉ ✈í✐ ♠ä✐ d : X × X → R✳ x, y, z ∈ X ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ d(x, y) ≥ ✈➭ d(x, y) = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x = y ❀ d(x, y) = d(y, x)❀ d(x, z) ≤ d(x, y) + d(y, z)✳ ❚❐♣ X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❜ë✐ 1.1.2 d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ (X, d) ❤♦➷❝ X ✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✺❪✮✳ ●✐➯ sư X ❧➭ t❐♣ ❦❤➳❝ rỗ s X ì X [0, +∞) ➤➢ỵ❝ ❣ä✐ ❧➭ b✲♠➟tr✐❝ ♥Õ✉ ✈í✐ ♠ä✐ x, y, z ∈ X ✱ t❛ ❝ã ✭✐✮ ✭✐✐✮ d(x, y) = ⇔ x = y ❀ d(x, y) = d(y, x)❀ d : d(x, y) ≤ s[d(x, z) + d(z, y)] ✭✐✐✐✮ X ❚❐♣ ❝ï♥❣ ✈í✐ ♠ét ✭❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✮✳ b✲♠➟tr✐❝ tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ✈í✐ t❤❛♠ sè s✱ ♥ã✐ ❣ä♥ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❣✐❛♥ b✲♠➟tr✐❝ b✲♠➟tr✐❝ ✈➭ ➤➢ỵ❝ ❦Ý ❤✐Ư✉ ❜ë✐ (X, d) ❤♦➷❝ X✳ 1.1.3 ❱Ý ❞ơ✳ ✶✮ ●✐➯ sư (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ d : X×X → [0, +∞) ❧➭ ❤➭♠ ➤➢ỵ❝ ❝❤♦ ❜ë✐ d(x, y) = (p(x, y))2 , ∀x, y ∈ X ❑❤✐ ➤ã✱ d ❧➭ b✲♠➟tr✐❝ ✈í✐ s = 2✳ ✷✮ ●✐➯ sư X = R ✈➭ tr➟♥ R t❛ ①Ðt ♠➟tr✐❝ t❤➠♥❣ t❤➢ê♥❣✳ ị d : R ì R [0, +∞) ❜ë✐ d(x, y) = |x − y|2 , ∀x, y ∈ R ❑❤✐ ➤ã✱ d ❧➭ ♠➟tr✐❝ ✈í✐ s = ✭t❤❡♦ ✶✮✮ ♥❤➢♥❣ d ❦❤➠♥❣ ❧➭ ♠➟tr✐❝ tr➟♥ R ✈× d(1, −2) = > = d(1, 0) + d(0, −2) 1.1.4 ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✺❪✮✳ ●✐➯ sö {xn } ❧➭ ❞➲② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ (X, d)✳ ❉➲② ❤✐Ư✉ ❜ë✐ n0 {xn } ➤➢ỵ❝ ❣ä✐ ❧➭ b✲❤é✐ tơ ✭♥ã✐ ❣ä♥ ❧➭ ❤é✐ tơ✮ tí✐ x ∈ X xn → x ❤♦➷❝ limn→∞ xn = x ♥Õ✉ ✈í✐ ♠ä✐ > 0✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ s❛♦ ❝❤♦ ❦❤✐ ✈➭ ➤➢ỵ❝ ❦Ý d(xn , x) < ✈í✐ ♠ä✐ n ≥ n0 ✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ xn → x ❦❤✐ ✈➭ ❝❤Ø d(xn , x) → ❦❤✐ n → ∞✳ ❉➲② {xn } ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ > 0✱ tå♥ t➵✐ sè tù ♥❤✐➟♥ n0 s❛♦ ❝❤♦ d(xn , xm ) < ❑❤➠♥❣ ❣✐❛♥ ➤Ị✉ ❤é✐ tơ✳ ✈í✐ ♠ä✐ n, m ≥ n0 ✳ b✲♠➟tr✐❝ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤đ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ ♥ã 1.1.5 ➜Þ♥❤ ♥❣❤Ü❛✳ K = C✳ ❍➭♠ ●✐➯ sö E ❧➭ ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ tr➟♥ tr➢ê♥❣ p : E → R ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤✉➮♥ tr➟♥ E K = R ❤♦➷❝ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ư♥ s❛✉ ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ ❙è ❝đ❛ p(x) ≥ 0, ∀x ∈ E ✈➭ p(x) = ⇔ x = 0❀ p(λx) = |λ|p(x), ∀x ∈ E, ∀λ ∈ K❀ p(x + y) ≤ p(x) + p(y), ∀x, y ∈ E ✳ p(x) ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤✉➮♥ ❝đ❛ ✈❡❝t➡ x ∈ E ✳ x ❧➭ x ✳ ❑❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ❚❛ t❤➢ê♥❣ ❦Ý ❤✐Ư✉ ❝❤✉➮♥ E ❝ï♥❣ ✈í✐ ♠ét ❝❤✉➮♥ ị tr ó ợ ọ ị ❝❤✉➮♥✳ 1.1.6 ▼Ư♥❤ ➤Ị✳ ◆Õ✉ E ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥ t❤× ❝➠♥❣ t❤ø❝ d(x, y) = x − y , ∀x, y ∈ E, ①➳❝ ➤Þ♥❤ ♠ét ♠➟tr✐❝ tr➟♥ E✳ ❚❛ ❣ä✐ ♠➟tr✐❝ ♥➭② ❧➭ ♠➟tr✐❝ s✐♥❤ ❜ë✐ ❝❤✉➮♥ ❤❛② ♠➟tr✐❝ ❝❤✉➮♥✳ ▼ét ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥ ✈➭ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ t❤❡♦ ♠➟tr✐❝ s✐♥❤ ❜ë✐ ❝❤✉➮♥ ợ ọ 1.1.7 ị ý ế ➳♥❤ ①➵ ❝❤✉➮♥✿ ♣❤Ð♣ ❝é♥❣✿ E ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥ t❤× x → x , ∀x ∈ E ❀ (x, y) → x + y, ∀(x, y) ∈ E × E ❀ ✈➭ ♣❤Ð♣ ♥❤➞♥ ✈í✐ ✈➠ ❤➢í♥❣✿ (λ, x) → λx, ∀(λ, x) ∈ K × E ❧➭ ❝➳❝ ➳♥❤ ①➵ ❧✐➟♥ tơ❝✳ E ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ó ỗ a E 1.1.8 ị ý ỗ K, = ➳♥❤ ①➵ ●✐➯ sö x → x + a, x → λx, ∀x ∈ E ❧➭ ❝➳❝ ♣❤Ð♣ ➤å♥❣ ♣❤➠✐ E E 1.1.9 t ợ ị ĩ ≤ ◗✉❛♥ ❤Ö X ✈➭ ≤ ❧➭ ♠ét q✉❛♥ ❤Ö ❤❛✐ ♥❣➠✐ tr➟♥ ➤➢ỵ❝ ❣ä✐ ❧➭ q✉❛♥ ❤Ư t❤ø tù ❜é ♣❤❐♥ tr➟♥ X X✳ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ư♥ s❛✉ ✭✐✮ x ≤ x ✈í✐ ♠ä✐ x ∈ X ❀ ✭✐✐✮ ❚õ ✭✐✐✐✮ x ≤ y ✈➭ y ≤ x s✉② r❛ x = y ✈í✐ ♠ä✐ x, y ∈ X ❀ x ≤ y ❀ y ≤ z s✉② r❛ x ≤ z ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ❚❐♣ ❤ỵ♣ X ❝ï♥❣ ✈í✐ ♠ét t❤ø tù ❜é ♣❤❐♥ tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ ❦Ý ❤✐Ö✉ (X, ≤) ❤♦➷❝ X ✳ 1.1.10 ●✐➯ sö ✧≤✧ ❧➭ ♠ét q✉❛♥ ❤Ư ❤❛✐ ♥❣➠✐ tr➟♥ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭✐✮ P❤➬♥ tư X ✈➭ A ⊆ X ✳ x ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❐♥ tr➟♥ ✭t➢➡♥❣ ø♥❣ ❝❐♥ ❞➢í✐✮ ❝đ❛ A ♥Õ✉ a ≤ x ✭t➢➡♥❣ ø♥❣ x ≤ a✮ ✈í✐ ♠ä✐ ♣❤➬♥ tư a ∈ A❀ ✭✐✐✮ P❤➬♥ tư ❝đ❛ A x ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❐♥ tr➟♥ ➤ó♥❣ ✭t➢➡♥❣ ø♥❣ ❝❐♥ ❞➢í✐ ➤ó♥❣✮ ♥Õ✉ x ❧➭ ♠ét ❝❐♥ tr➟♥ ✭t➢➡♥❣ ø♥❣ ❝❐♥ ❞➢í✐✮ ❝đ❛ ❝ị♥❣ ❧➭ ♠ét ❝❐♥ tr➟♥ ✭t➢➡♥❣ ø♥❣ ❝❐♥ ❞➢í✐✮ ❝đ❛ ✈➭ ♥Õ✉ A t❤× x ≤ y y ✭t➢➡♥❣ y ≤ x✮✳ ❑❤✐ ➤ã✱ t❛ ❦Ý ❤✐Ö✉ x = sup A ✭t➢➡♥❣ ø♥❣ x = inf A✮✳ ø♥❣ 1.2 A ◆ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ 1.2.1 ➜Þ♥❤ ♥❣❤Ü❛✳ R✳ ▼ét t❐♣ ❝♦♥ P ✭✐✮ P ❝đ❛ ❧➭ ➤ã♥❣✱ ✭✐✐✮ ❱í✐ ✭❬✹❪✮✳ ❈❤♦ E ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ tr➟♥ tr➢ê♥❣ sè t❤ù❝ E ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ã♥ tr♦♥❣ E ♥Õ✉ P = ∅✱ P = {0}❀ a, b ∈ R✱ a, b ≥ ✈➭ x, y ∈ P ✭✐✐✐✮ ◆Õ✉ x∈P ✈➭ −x ∈ P t❤× x = 0✳ t❤× ax + by ∈ P ❀ 26 2.2.2 ◆❤❐♥ ①Ðt✳ ✶✮ ❱× ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ❧➭ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ị ý trờ ợ ệt ủ ➜Þ♥❤ ❧ý ✷✳✷✳✶✳ ✷✮ ❚r♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ ♥Õ✉ ❧✃② s = ✭tø❝ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ủ tì t ợ ý ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❇❛♥❛❝❤✳ 2.2.3 ✈➭ ➜Þ♥❤ ❧ý✳ ●✐➯ sư (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟ tr✐❝ ♥ã♥ ➤➬② ➤đ ✈í✐ s ≥ f : X → X ✳ ❑❤✐ ➤ã✱ ♥Õ✉ tå♥ t➵✐ ❝➳❝ ❤➺♥❣ sè ❦❤➠♥❣ ➞♠ a1 , a2 , , a5 s❛♦ ❝❤♦ a1 + a2 + + a5 < (2.14) a1 s + a3 + a4 ≤ 1 a3 + a5 < s (2.15) (2.16) ✈➭ d(f x, f y) ≤ a1 d(x, y) + a2 d(x, f x) + a3 d(x, f y)+ s (2.17) + a4 d(y, f x) + a5 d(y, f y) ∀x, y ∈ X s n t❤× f ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ x ∈ X ✈➭ ✈í✐ ♠ä✐ x0 ∈ X ✱ limn→∞ f x0 = x✳ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❜✃t ❦ú x0 ∈ X ✈➭ ➤➷t xn = f xn−1 ∀n = 1, 2, ❚❛ ❝ã xn = f n x0 ✈í✐ ♠ä✐ n = 1, 2, ❙ư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✼✮ t❛ ❝ã d(xn+1 , xn ) = d(f xn , f xn−1 ) ≤ a1 d(xn , xn−1 ) + a2 d(xn , f xn ) a3 a4 + d(xn , f xn−1 ) + d(xn−1 , f xn ) + a5 d(xn−1 , f xn−1 ) s s a4 = (a1 + a5 )d(xn , xn−1 ) + a2 d(xn , xn+1 ) + d(xn−1 , xn+1 ) s ≤ (a1 + a5 )d(xn , xn−1 ) + a2 d(xn , xn+1 ) + a4 [d(xn−1 , xn ) + d(xn , xn+1 )] = (a1 + a5 + a4 )d(xn , xn−1 ) + (a2 + a4 )d(xn , xn+1 ) (2.18) 27 ✈í✐ ♠ä✐ n = 1, 2, ✳ ❚➢➡♥❣ tù✱ t❛ ❝ã d(xn , xn+1 ) = d(f xn−1 , f xn ) ≤ a1 d(xn−1 , xn ) + a2 d(xn−1 , f xn−1 ) a3 a4 + d(xn−1 , f xn ) + d(xn , f xn−1 ) + a5 d(xn , f xn ) s s ≤ (a1 + a2 + a3 )d(xn−1 , xn ) + (a3 + a5 )d(xn , xn+1 ) (2.19) ✈í✐ ♠ä✐ n = 1, 2, ✳ ❚õ ✭✷✳✶✽✮ ✈➭ ✭✷✳✶✾✮ s✉② r❛ d(xn , xn+1 ) = 2a1 + a2 + a3 + a4 + a5 d(xn−1 , xn ) ∀n = 1, 2, (2.20) − a2 − a3 − a4 − a5 ➜➷t λ= ❚õ ✭✷✳✶✹✮ s✉② r❛ 2a1 + a2 + a3 + a4 + a5 − a2 − a3 − a4 − a5 λ ∈ [0, 1)✳ ➳♣ ❞ô♥❣ ♥❤✐Ò✉ ❧➬♥ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✵✮✱ t❛ ❝ã d(xn , xn+1 ) ≤ λd(xn−1 , xn ) ≤ ≤ λn d(x0 , x1 ) ∀n = 1, 2, ❱× λ ∈ [0, 1) ♥➟♥ λn d(x0 , x1 ) n ỗ c ∈ intP tå♥ t➵✐ sè tù ♥❤✐➟♥ ❱í✐ ♠ä✐ ❉♦ ➤ã tõ ✭✷✳✷✶✮ s✉② r❛ r➺♥❣✱ nc s❛♦ ❝❤♦ lim d(xn , xn+1 ) n→∞ (2.21) c ∀n ≥ nc (2.22) n ✈➭ m ∈ N∗ ✱ sư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✻✮ t❛ ❝ã d(xn , xm ) = d(f xn−1 , f xm−1 ) ≤ a1 d(xn−1 , xm−1 ) + a2 d(xn−1 , xn ) a3 a4 + d(xn−1 , xm ) + d(xm−1 , xn ) + a5 d(xm−1 , xm ) s s ≤ sa1 d(xn−1 , xn ) + a1 s d(xn , xm ) + a1 s2 d(xm , xm−1 ) + a2 d(xn−1 , xn ) + a3 d(xn−1 , xn ) + a3 d(xn , xm ) + a4 d(xm−1 , xm ) + a4 d(xm , xn ) + a5 d(xm−1 , xm ) ❉♦ ➤ã (1 − a1 s2 − a3 − a4 )d(xn , xm ) ≤ (a1 s + a2 + a3 )d(xn , xn−1 ) + (a1 s + a4 + a5 )d(xm , xm−1 ) (2.23) 28 ✈í✐ ♠ä✐ n ✈➭ m ∈ N s r ỗ c intP tồ t➵✐ sè tù ♥❤✐➟♥ (a1 s + a2 + a3 )d(xn , xn−1 ) + (a1 s2 + a4 + a5 )d(xm , xm−1 ) ✈í✐ ♠ä✐ nc s❛♦ ❝❤♦ c n ✈➭ m ≥ nc ✳ ❑Õt ❤ỵ♣ ✈í✐ ✭✷✳✷✸✮ s✉② r❛ (1 − a1 s2 − a3 − a4 )d(xn , xm ) ▼➷t ❦❤➳❝✱ tõ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✺✮ s✉② r❛ c ∀n, m ≥ nc ≤ − a1 s − a3 − a4 ≤ ✳ ❉♦ ➤ã d(xn , xm ) ◆❤➢ ✈❐② c ∀n, m ≥ nc {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ❱× (X, d) ➤➬② ➤đ ♥➟♥ tå♥ t➵✐ x ∈ X s❛♦ ❝❤♦ xn → x✱ tø❝ ❧➭ limn→∞ f n x0 = x✳ ❚✐Õ♣ t❤❡♦✱ t❛ ❝❤ø♥❣ ♠✐♥❤ x ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ✳ ❙ư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✼✮ t❛ ❝ã d(x, f x) ≤ sd(x, xn+1 ) + sd(f xn , f x) ≤ sd(x, xn+1 ) + s[a1 d(xn , x) + a2 d(xn , xn+1 ) + a3 d(xn , f x) s + a4 d(x, xn+1 ) + a5 d(x, f x)] s ≤ sd(x, xn+1 ) + s[a1 d(xn , x) + a2 d(xn , xn+1 ) + a3 d(xn , x) a4 + a3 d(x, f x) + d(x, xn+1 ) + a5 d(x, f x)] ∀n = 1, 2, s ❉♦ ➤ã a4 )d(x, xn+1 ) s + s(a1 + a3 )d(xn , x) + sa2 d(xn , xn+1 ) ∀n = 1, 2, (2.24) (1 − sa3 − sa5 )d(x, f x) ≤ (s + xn x s r ỗ c ∈ intP (s + tå♥ t➵✐ sè tù ♥❤✐➟♥ nc s❛♦ ❝❤♦ a4 )d(x, xn+1 ) + s(a1 + a3 )d(xn , x) + sa2 d(xn , xn+1 ) s ❑Õt ❤ỵ♣ ✈í✐ ✭✷✳✷✹✮ s✉② r❛ (1 − sa3 − sa5 )d(x, f x) c ∀c ∈ intP c ∀n ≥ nc 29 ❚❤❡♦ ❇ỉ ➤Ị ✶✳✷✳✹ ✭✈✐✐✐✮ t❤× ❦✐Ư♥ ✭✷✳✶✻✮ t❤× ✈❐② (1 − sa3 − sa5 )d(x, f x) = 0✳ ▼➷t ❦❤➳❝✱ t❤❡♦ ➤✐Ò✉ (1 − sa3 − sa5 ) > 0✱ ❞♦ ➤ã d(x, f x) = 0✱ tø❝ ❧➭ x = f x✳ ◆❤➢ x ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ✳ ❈✉è✐ ❝ï♥❣ t❛ ❝❤ø♥❣ ♠✐♥❤ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ❧➭ ❞✉② ♥❤✃t✳ ●✐➯ sư y ❝ị♥❣ ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ∈X f ✱ tø❝ y = f y ✳ ❑❤✐ ➤ã✱ d(x, y) = d(f x, f y) ≤ a1 d(x, y) + a2 d(x, x) + a3 d(x, y) s + a4 d(y, x) + a5 d(y, y) s 1 = (a1 + a3 + a4 )d(x, y) s s ▼➷t ❦❤➳❝✱ tõ ✭✷✳✶✹✮ s✉② r❛ t❤× d(x, y) = 0✱ tø❝ ❧➭ x = y ✳ ❱❐② ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f 2.2.4 sè ≤ a1 + a3 + a4 < 1✳ ❉♦ ➤ã t❤❡♦ ❇ỉ ➤Ị ✶✳✷✳✹ ✭✐①✮ ❍Ư q✉➯✳ ●✐➯ sö s ≥ ✈➭ f : X → X ❧➭ ❞✉② ♥❤✃t✳ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ➤➬② ➤đ ✈í✐ t❤❛♠ ❧➭ ➳♥❤ ①➵ s❛♦ ❝❤♦ tå♥ t➵✐ ≤ λ < min{ 21 , 1s } t❤á❛ ♠➲♥ d(f x, f y) ≤ λ[d(x, f x) + d(y, f y)] ∀x, y ∈ X ❑❤✐ ➤ã✱ f ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t x∈X ✈➭ ✈í✐ ♠ä✐ (2.25) x0 ∈ X ✱ lim f n x0 = x n→∞ ❈❤ø♥❣ ♠✐♥❤✳ ➜➷t a1 = a3 = a4 = 0✱ a2 = a5 = λ✳ ❑❤✐ ➤ã✱ t❛ ❝ã a1 + a2 + + a5 = 2λ < 1, a1 s + a3 + a4 = 0, a3 + a5 = λ < 1s ▼➷t ❦❤➳❝✱ tõ ✭✷✳✷✺✮ s✉② r❛ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✼✮ tr♦♥❣ ị ý ợ tỏ ó từ ị ❧ý ✷✳✷✳✸ t❛ ❝ã ➤✐Ò✉ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ 2.2.5 ✈➭ ❍Ư q✉➯✳ ●✐➯ sư (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤➬② ➤đ ✈í✐ s ≥ 1 f : X → X ✳ ❑❤✐ ➤ã✱ ♥Õ✉ tå♥ t➵✐ λ ∈ [0, min{ 2s , s2 }) s❛♦ ❝❤♦ d(f x, f y) ≤ λ[d(x, f y) + d(y, f x)] ∀x, y ∈ X (2.26) 30 t❤× f ❝ã ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣ x∈X ✈➭ ✈í✐ ♠ä✐ x0 ∈ X ✱ limn→∞ f n x0 = x✳ ❈❤ø♥❣ ♠✐♥❤✳ ➜➷t a1 = a2 = a5 = 0✱ a3 = a4 = sλ✳ ❑❤✐ ➤ã✱ t❛ ❞Ô ❞➭♥❣ ể tr ợ ề ệ ủ ị ý ❉♦ ➤ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ s✉② r❛ tõ ị ý 2.2.6 ú ý btr ì ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ❧➭ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt ❝đ❛ ❦❤➠♥❣ ♥ã♥ ♥➟♥ tõ ❍Ö q✉➯ ✷✳✷✳✹ ✈➭ ❍Ö q✉➯ ✷✳✷✳✺ s✉② r❛ ➜Þ♥❤ ❧ý ✷ ✈➭ ➜Þ♥❤ ❧ý ✸ tr♦♥❣ ✭❬✾❪✮ ✈í✐ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❜ỉ s✉♥❣✳ ✷✮ ❚r♦♥❣ ❍Ư q✉➯ ✷✳✷✳✹ ✈➭ ❍Ö q✉➯ ✷✳✷✳✺✱ ♥Õ✉ ❧✃② ❤❛✐ ❦Õt q✉➯ t➢➡♥❣ ø♥❣ s❛✉ ➤➞②✿ ◆Õ✉ f :X→X s = tì t ợ (X, d) ♠➟tr✐❝ ➤➬② ➤đ ✈➭ ❧➭ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❑❛♥♥❛♥ ✭❬✽❪✮ ❤♦➷❝ ❦✐Ó✉ ❈❤❛tt❡r❥❡❛ ✭❬✹❪✮ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣ X✳ 2.2.7 (X, d) ❍Ư q✉➯✳ ●✐➯ sư ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟ tr✐❝ ♥ã♥ ➤➬② ➤ñ ✈➭ f : X → X ✳ ❑❤✐ ➤ã✱ ♥Õ✉ tå♥ t➵✐ ❝➳❝ ❤➺♥❣ sè ❦❤➠♥❣ ➞♠ a1 , a2 , , a5 s❛♦ ❝❤♦ a1 + a2 + + a5 < ✈➭ d(f x, f y) ≤ a1 d(x, y) + a2 d(x, f x) + a3 d(x, f y) + a4 d(y, f x) + a5 d(y, f y) ∀x, y ∈ X t❤× f ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t x ∈ X ✈➭ ✈í✐ ♠ä✐ x0 ∈ X ✱ limn→∞ f n x0 = x✳ ❈❤ø♥❣ ♠✐♥❤✳ ❍Ư q✉➯ ♥➭② ❧➭ tr➢ê♥❣ ❤ỵ♣ ➤➷❝ ❜✐Ưt ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ ❦❤✐ 2.2.8 sè ➜Þ♥❤ ❧ý✳ ●✐➯ sư s = 1✳ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ➤➬② ➤đ ✈í✐ t❤❛♠ s ≥ ✈➭ f : X → X ✳ ❑❤✐ ➤ã✱ ♥Õ✉ tå♥ t➵✐ ❤➺♥❣ sè α s❛♦ ❝❤♦ ≤ α < min{ 1 , } s2 + s3 (2.27) 31 ✈➭ d(f x, f y) ≤ α sup{sd(x, y), d(x, f y) + d(y, f x), s[d(x, f x) + d(y, f y)]} (2.28) ✈í✐ ♠ä✐ x, y ∈ X t❤× f ❝ã ❞✉② ♥❤✃t ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ x∈X ✈➭ ✈í✐ ♠ä✐ x0 ∈ X ✱ limn→∞ f n x0 = x✳ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❜✃t ❦ú x0 ∈ X ✳ ➜➷t xn = f xn−1 ∀n = 1, 2, ❚❛ ❝ã xn = f n x0 ✈í✐ ♠ä✐ n = 1, 2, ✳ ❙ư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✷✽✮ t❛ ❝ã d(xn , xn+1 ) = d(f xn−1 , f xn ) ≤ α sup{sd(xn−1 , xn ), d(xn−1 , xn+1 ) + d(xn , xn ), s[d(xn+1 , xn ) + d(xn , xn+1 )]} = αs[d(xn−1 , xn ) + d(xn , xn+1 )] ∀n = 1, 2, ❉♦ ➤ã αs d(xn−1 , xn ) ∀n = 1, 2, (2.29) − αs 1 ❑❤✐ ➤ã✱ tõ ✭✷✳✷✼✮ s✉② r❛ α < s +1 < 2s ✈í✐ ♠ä✐ s > 1✳ ❉♦ ➤ã d(xn , xn+1 ) ≤ ➜➷t λ= αs 1−αs ✳ λ ∈ [0, 1) ♥Õ✉ s > 1✳ ◆❤➢ ✈❐② ◆Õ✉ s = t❤× α < s2 +1 = ✈➭ ❞♦ ➤ã λ= α 1−α < 1✳ α ∈ [0, 1) ✈í✐ ♠ä✐ s ≥ 1✳ ➳♣ ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✾✮ ♥❤✐Ị✉ ❧➬♥ t❛ ❝ã d(xn , xn+1 ) ≤ λd(xn−1 , xn ) ≤ ≤ λn d(x0 , x1 ) ❱× λ ∈ [0, 1) ♥➟♥ limn→∞ λn d(x0 , x1 ) = số tự ó ỗ c intP tå♥ t➵✐ nc s❛♦ ❝❤♦ d(xn , xn+1 ) ≤ λn d(x0 , x1 ) ❚✐Õ♣ t❤❡♦ t❛ ❝❤ø♥❣ ♠✐♥❤ c ∀n ≥ nc (2.30) {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ❱í✐ ♠ä✐ n ✈➭ m ∈ N∗ t❛ ❝ã d(xn , xm ) = d(f xn−1 , f xm−1 ) ≤ α sup{sd(xn−1 , xm−1 ), d(xn−1 , xm ) + d(xm−1 , xn ), s[d(xn−1 , xn ) + d(xm−1 , xm )]} 32 ≤ α sup{s2 d(xn−1 , xn ) + s3 d(xn , xm ) + s3 d(xm , xm−1 ), sd(xn−1 , xn ) + sd(xn , xm ) + sd(xm−1 , xm ) + sd(xm , xn ), sd(xn−1 , xn ) + sd(xm−1 , xm )} ≤ α[s2 d(xn−1 , xn ) + s3 d(xm−1 , xm ) + max{s3 , 2s}d(xn , xm )] ❉♦ ➤ã ✈í✐ ♠ä✐ n ✈➭ m ∈ N∗ t❛ ❝ã (1 − α max{s3 , 2s})d(xn , xm ) ≤ α[s2 d(xn−1 , xn ) + s3 d(xm−1 , xm )] (2.31) ❚õ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✷✼✮ s✉② r❛ d(xn , xm ) ≤ ✈í✐ ♠ä✐ α [s2 d(xn−1 , xn ) + s3 d(xm−1 , xm )] − α max{s , 2s} n ✈➭ m N s r ỗ d(xn , xm ) ≤ ✈í✐ ♠ä✐ (1 − α max{s3 , 2s}) > 0✳ ❑Õt ❤ỵ♣ ✈í✐ ✭✷✳✸✶✮ s✉② r❛ c ∈ intP tå♥ t➵✐ nc ∈ N∗ s❛♦ ❝❤♦ α [s2 d(xn−1 , xn ) + s3 d(xm−1 , xm )] − α max{s , 2s} n ✈➭ m ≥ nc ✳ ➤➬② ➤ñ ♥➟♥ tå♥ t➵✐ ❉♦ ➤ã x∈X {xn } ❧➭ ❞➲② ❈❛✉❝❤②✳ s❛♦ ❝❤♦ ❱× c (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ xn → x ❦❤✐ n → ∞✳ ❇➞② ❣✐ê✱ t❛ ❝❤ø♥❣ ♠✐♥❤ x ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ✳ ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✷✽✮ t❛ ❝ã d(x, f x) ≤ sd(x, xn ) + sd(f xn−1 , f x) ≤ sd(x, xn ) + sα sup{sd(xn−1 , x), d(xn−1 , f x) + d(x, xn ), s[d(xn−1 , xn ) + d(x, f x)]} ≤ sd(x, xn ) + αs sup{sd(xn−1 , x), sd(xn−1 , x) + sd(x, f x) + d(x, xn ), s[d(xn−1 , xn ) + d(x, f x)]} ≤ sd(x, xn ) + sα[sd(xn−1 , x) + sd(x, f x) + d(x, xn ) + sd(xn−1 , xn )] ✈í✐ ♠ä✐ n = 1, 2, ✳ ❉♦ ➤ã (1 − αs2 )d(x, f x) ≤ s(1 + α)d(x, xn ) + αs2 d(xn−1 , x) + αs2 d(xn−1 , xn ) (2.32) 33 ✈í✐ ♠ä✐ n = 1, 2, ✳ ❱× xn x n ỗ c ∈ intP tå♥ t➵✐ nc ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n ≥ nc t❛ ❝ã s(1 + α)d(x, xn ) + αs2 d(xn−1 , x) + αs2 d(xn−1 , xn ) ❑Õt ❤ỵ♣ ✈í✐ ✭✷✳✸✷✮ t❛ ❝ã ➤Ị ✶✳✷✳✹ t❤× tø❝ ❧➭ (1 − αs2 )d(x, f x) c c ✈í✐ ♠ä✐ c ∈ intP ✳ ❚❤❡♦ ❇ỉ (1 − αs2 )d(x, f x) = 0✳ ▼➷t ❦❤➳❝ − αs2 > ♥➟♥ d(x, f x) = 0✱ x = f x✳ ❱❐② x ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ f ✳ ❈✉è✐ ❝ï♥❣✱ ❣✐➯ sư y∈X ❝ị♥❣ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ✱ tø❝ y = f y ✳ ❉♦ ➤ã t❛ ❝ã d(x, y) = d(f x, f y) ≤ α sup{sd(x, y), d(x, y) + d(y, x), s[d(x, x) + d(y, y)]} = α sup{sd(x, y), 2d(x, y)} = α max{s, 2}d(x, y) ❑Õt ❤ỵ♣ ✈í✐ α max{s, 2} < s✉② r❛ d(x, y) = 0✱ tø❝ x = y ✳ ❱❐② ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ f ❧➭ ❞✉② ♥❤✃t✳ ❚r♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✽✱ ♥Õ✉ ❧✃② 2.2.9 ❍Ư q✉➯✳ ●✐➯ sư s = tì t ợ ệ q s (X, d) ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤➬② ➤ñ ✈➭ f : X → X ✳ ❑❤✐ ➤ã✱ ♥Õ✉ tå♥ t➵✐ a ∈ [0, 21 ) s❛♦ ❝❤♦ ♠ét tr♦♥❣ ❤❛✐ ➤✐Ị✉ ❦✐Ư♥ s ợ tỏ ọ tì x, y X ✶✮ d(f x, f y) ≤ α[d(x, f x) + d(y, f y)] ✷✮ d(f x, f y) ≤ α[d(x, f y) + d(y, f x)] f ❝ã ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣ x ∈ X ✈➭ ✈í✐ ♠ä✐ x0 ∈ X ✱ limn→∞ f n x0 = x✳ 2.2.10 sè ➜Þ♥❤ ❧ý✳ s ≥ 1✱ T ✈➭ ●✐➯ sư (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ➤➬② ➤đ ✈í✐ t❤❛♠ f :X→X ❧➭ ❤❛✐ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ 34 ✐✮ T ➤➡♥ ➳♥❤ ✈➭ ❧✐➟♥ tô❝✱ ✐✐✮ ❚å♥ t➵✐ ❝➳❝ ❤➺♥❣ sè ❦❤➠♥❣ ➞♠ α1 ✱ α2 ✱ α3 s❛♦ ❝❤♦ α2 + α3 < α1 + α2 < 1 , 2s s2 , (2.33) s (2.34) ✈➭ d(T f x, T f y) ≤ α1 d(T x, T f y) + α2 d(T y, T f x) + α3 s[d(T x, T f x) + d(T y, T f y)] ✈í✐ ♠ä✐ (2.35) x, y ∈ X ✳ ❑❤✐ ➤ã✱ ❝➳❝ ❦❤➻♥❣ ➤Þ♥❤ s❛✉ ❧➭ ➤ó♥❣ ✶✮ ❱í✐ ♠ä✐ x0 ∈ X ✱ ❞➲② {T f n x0 } ❤é✐ tô✳ ✷✮ ◆Õ✉ T ❧➭ ➳♥❤ ①➵ ❤é✐ tô ❞➲② ❝♦♥ t❤× ✸✮ ◆Õ✉ T ❧➭ ➳♥❤ ①➵ ❤é✐ tơ tì ỗ tớ ể t ộ ủ ứ ♠✐♥❤✳ ●✐➯ sư f ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ x0 ∈ X ✱ ❞➲② {f n x0 } ❤é✐ tơ f✳ x0 ❧➭ ➤✐Ĩ♠ ❜✃t ❦ú tr♦♥❣ X ✳ ❚❛ ①➞② ❞ù♥❣ ❞➲② {xn } ❜ë✐ xn+1 = f xn = f n+1 x0 ∀n = 0, 1, ➜➷t yn = T xn , n = 0, 1, ➜➬✉ t✐➟♥ t❛ ❝❤ø♥❣ ♠✐♥❤ ❦❤➻♥❣ ➤Þ♥❤ ❛✮✿ ✈í✐ ỗ c intP tồ t số tự nc s❛♦ ❝❤♦ d(xn , xn+1 ) c ∀n ≥ nc ❚❤❐t ✈❐②✱ sư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✺✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ t❛ ❝ã d(yn+1 , yn ) = d(T f xn , T f xn−1 ) ≤ α1 d(yn , yn ) + α2 d(yn−1 , yn+1 ) + α3 s[d(yn , yn+1 ) + d(yn−1 , yn )] ≤ α2 s[d(yn−1 , yn ) + d(yn , yn+1 )] + α3 s[d(yn , yn+1 ) + d(yn−1 , yn )] = (α2 + α3 )s[d(yn , yn+1 ) + d(yn−1 , yn )] 35 ✈í✐ ♠ä✐ n = 1, 2, ❉♦ ➤ã t❛ ❝ã d(yn+1 , yn ) ≤ (α2 + α3 )s d(yn , yn−1 ) − (α2 + α3 )s (2.36) := λd(yn , yn−1 ) ∀n = 1, 2, tr♦♥❣ ➤ã λ= (α2 +α3 )s 1−(α2 +α3 )s ✳ λ ∈ [0, 1)✳ ❙ư ❞ơ♥❣ ✭✷✳✸✻✮ ♥❤✐Ị✉ ❧➬♥ t❛ ➤➢ỵ❝ ❚õ ✭✷✳✸✸✮ s✉② r❛ d(yn+1 , yn ) ≤ λd(yn , yn−1 ) ≤ ≤ λn d(y1 , y0 ) ∀n = 1, 2, ❱× λ ∈ [0, 1) ♥➟♥ λn d(y1 , y0 ) → ❦❤✐ n ó ỗ c intP t➵✐ sè tù ♥❤✐➟♥ tå♥ nc s❛♦ ❝❤♦ d(yn+1 , yn ) ≤ λn d(y1 , y0 ) c ∀n nc ị ợ ứ ♠✐♥❤✳ ❇➞② ❣✐ê✱ t❛ ❝❤ø♥❣ ♠✐♥❤ {yn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ❱í✐ ♠ä✐ n ✈➭ m = 1, 2, t❛ ❝ã d(yn , ym ) = d(T f xn−1 , T f xm−1 ) ≤ α1 d(yn−1 , ym ) + α2 d(ym−1 , yn ) + α3 s[d(yn−1 , yn ) + d(ym−1 , ym )] ≤ α1 sd(yn−1 , yn ) + α1 sd(yn , ym ) + α2 sd(ym−1 , ym ) + α2 sd(ym , yn ) + α3 s[d(xn−1 , xn ) + d(ym−1 , ym )] ❱× α1 + α2 < s ♥➟♥ tõ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② s✉② r❛ ✈í✐ ♠ä✐ d(yn , ym ) ≤ (α1 + α3 )s (α2 + α3 )s d(yn , yn−1 ) + d(ym , ym−1 ) − (α1 + α2 )s − (α1 + α2 )s (2.37) ❚õ ❦❤➻♥❣ ➤Þ♥❤ ❛✮ s✉② r ỗ n m N t ❝ã c ∈ intP tå♥ t➵✐ nc ∈ N s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n m ≥ nc t❛ ❝ã (α1 + α3 )s (α2 + α3 )s d(yn , yn−1 ) + d(ym , ym−1 ) − (α1 + α2 )s − (α1 + α2 )s c 36 ❉♦ ➤ã✱ ❦Õt ❤ỵ♣ ✈í✐ ✭✷✳✸✼✮ t❛ ❝ã c ∀n ✈➭ m ≥ nc d(yn , ym ) ➜✐Ò✉ ♥➭② ❝❤ø♥❣ tá s❛♦ ❝❤♦ {yn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ❱× (X, d) ➤➬② ➤đ ♥➟♥ tå♥ t➵✐ y ∈ E yn → y ✱ tø❝ T f xn = T f n x0 → y ✳ ✷✮ ●✐➯ sư T ❧➭ ➳♥❤ ①➵ ❤é✐ tơ ❞➲② ❝♦♥✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ f ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❱× T ❤é✐ tô ❞➲② ❝♦♥ ✈➭ {T f xn } ❤é✐ tô ♥➟♥ {f xn }∞ n=1 {f xni }∞ i=1 s❛♦ ❝❤♦ f xni → x ∈ X ❤ỵ♣ ✈í✐ T f xn → y s✉② r❛ y = T x✳ ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ✈➭ ➤✐Ị✉ ❦❤✐ ni → ∞✳ ❝ã ❞➲② ❝♦♥ ❉♦ T ❧✐➟♥ tô❝ ♥➟♥ T f xni → T x✳ ❑Õt ❦✐Ö♥ ✭✷✳✸✺✮✱ t❛ ❝ã d(T f x, T x) ≤ sd(T x, T f xn ) + sd(T f x, T f xn ) ≤ sd(T x, yn+1 ) + sα1 d(T x, yn+1 ) + α2 sd(yn , T f x) + α3 s2 [d(T x, T f x) + d(yn , yn+1 )] ≤ s(1 + α)d(T x, yn+1 ) + α2 s2 d(yn , T x) + (α2 + α3 )s2 d(T x, T f x) + α3 s2 d(yn , yn+1 ) ∀n = 1, 2, ❉♦ ➤ã [1 − (α2 + α3 )s2 ]d(T x, T f x) ≤ s(1 + α)d(T x, yn+1 ) + α2 s2 d(yn , T x) + α3 s2 d(yn , yn+1 ) (2.38) ✈í✐ ♠ä✐ ❚õ n = 1, 2, yn T x s r ỗ c ∈ intP tå♥ t➵✐ sè tù ♥❤✐➟♥ nc s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n ≥ nc t❛ ❝ã α(1 + α)d(T x, yn+1 ) + α2 s2 d(T x, yn ) + α3 s2 d(yn , yn+1 ) c ❑Õt ❤ỵ♣ ✈í✐ ✭✷✳✸✽✮ s✉② r❛ [1 − (α2 + α3 )s2 ]d(T x, T f x) ❚❤❡♦ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✸✮ t❤× d(T x, T f x) = 0✱ tø❝ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ f✳ c ∀c ∈ intP − (α2 + α3 )s2 > 0✳ T x = T f x✳ ❱× T (2.39) ❉♦ ➤ã✱ tõ ✭✷✳✸✾✮ s✉② r❛ ➤➡♥ ➳♥❤ ♥➟♥ x = f x✳ ❱❐② x ❧➭ 37 u ❝ị♥❣ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ❈✉è✐ ❝ï♥❣ ❣✐➯ sö tr♦♥❣ X ✱ tø❝ f u = u✳ ❑❤✐ ➤ã✱ t❛ ❝ã d(T x, T u) = d(T f x, T f u) ≤ α1 d(T x, T u) + α2 d(T u, T x) + α3 s[d(T x, T x) + d(T u, T u)] = (α1 + α2 )d(T x, T u) ❑Õt ❤ỵ♣ ✈í✐ ♥➟♥ α1 + α2 < s✉② r❛ d(T x, T u) = 0✱ tø❝ T x = T u✳ ❱× T x = u✳ ❱❐② ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f 2.2.11 ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✵❪✮✳ ●✐➯ sư ➤➡♥ ➳♥❤ ❧➭ ❞✉② ♥❤✃t✳ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ T ✈➭ f : X → X✳ ✶✮ ➳♥❤ ①➵ f ➤➢ỵ❝ ❣ä✐ ❧➭ T ✲❝♦ ❑❛♥♥❛♥ ♥Õ✉ tå♥ t➵✐ ❤➺♥❣ sè α ∈ [0, 21 ) s❛♦ ❝❤♦ d(T f x, T f y) ≤ α[d(T x, T f x) + d(T y, T f y)] ✈í✐ ♠ä✐ ✷✮ x, y ∈ X ✳ ➳♥❤ ①➵ f ➤➢ỵ❝ ❣ä✐ ❧➭ T ✲❝♦ ❈❤❛tt❡r❥❡❛ ♥Õ✉ tå♥ t➵✐ ❤➺♥❣ sè α ∈ [0, 12 ) s❛♦ ❝❤♦ d(T f x, T f y) ≤ α[d(T x, T f y) + d(T y, T f x)] ✈í✐ ♠ä✐ 2.2.12 P x, y ∈ X ✳ ❍Ư q✉➯✳ ✭❬✶✵❪✮✳ ●✐➯ sư ❧➭ ♥ã♥ t ế f :XX ỗ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤➬② ➤đ ✈í✐ T :X →X ❧➭ ➳♥❤ ①➵ ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tô❝ ✈➭ ➤➡♥ ➳♥❤✳ ❑❤✐ ➤ã✱ T ✲❝♦ ❑❛♥♥❛♥ ❤♦➷❝ T ✲❝♦ ❈❤❛tt❡r❥❡❛ t❤× x0 ∈ X lim d(T f n x0 , T f n+1 x0 ) = 0; n→∞ ✷✮ ❚å♥ t➵✐ ✸✮ ◆Õ✉ T v∈X s❛♦ ❝❤♦ limn→∞ T f n x0 = v ❀ ❧➭ ➳♥❤ ①➵ ❤é✐ tụ tì ỗ ộ tụ x0 ∈ X ✱ ❞➲② {f n x0 } ❝ã 38 ✹✮ ❚å♥ t➵✐ ❞✉② ♥❤✃t ✺✮ ◆Õ✉ tí✐ T u∈X s❛♦ ❝❤♦ u = f u❀ ❧➭ ➳♥❤ ①➵ ộ tụ tì ỗ x0 X ❞➲② {f n x0 } ❤é✐ tô u✳ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö f ❧➭ ➳♥❤ ①➵ T ✲❝♦ ❑❛♥♥❛♥✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ α ∈ [0, 21 ) s❛♦ ❝❤♦ d(T f x, T f y) ≤ α[d(T x, T f x) + d(T y, T f y)] ✈í✐ ♠ä✐ ❱× x, y ∈ X ✳ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤➬② ➤đ ♥➟♥ ❝ã t❤Ĩ ①❡♠ ♥ã ❧➭ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥ ➤➬② ➤đ ✈í✐ s = 1✳ ➜➷t α3 = α✱ α1 = α2 = 0✳ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✐✮ ✈➭ ✐✐✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶✵ ợ tỏ ị ủ ➜Þ♥❤ ❧ý ✷✳✷✳✶✵ ❧➭ ➤ó♥❣✳ ❚õ x0 ∈ X s✉② r❛ tå♥ t➵✐ v∈X s❛♦ ❝❤♦ ❑❤✐ ➤ã✱ t❛ t❤✃② s = 1✳ ❉♦ ➤ã ❝➳❝ {T f n x0 } ộ tụ ỗ limn T f n x0 = v ✳ ❉♦ ➤ã✱ t❛ ❝ã lim d(T f n x0 , T f n+1 x0 ) = n→∞ ◆❤➢ ✈❐② ✶✮ ✈➭ ✷✮ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤✳ ❑❤➻♥❣ ị ủ ệ q ợ s r tõ ✈✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ ✷✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶✵✳ ❑❤➻♥❣ ị ủ ệ q ợ s r từ ị ủ ị ý rờ ợ f ➳♥❤ ①➵ T ✲❝♦ ❈❤❛tt❡r❥❡❛ ➤➢ỵ❝ ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù✳ 39 ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ➤➲ ➤➵t ➤➢ỵ❝ ❝➳❝ ❦Õt q í s rì ị ĩ í ❞ơ ✈➭ ♠ét sè tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ❝đ❛ ♥ã♥ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ✲ ❚r×♥❤ ❜➭② ❧➵✐ ♠ét sè ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ➤➲ ❝ã tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦✳ ✲ ➜➢❛ r❛ ♠ét sè ❦Õt q✉➯ ♠í✐ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝ ♥ã♥✱ ➤ã ❧➭ ❝➳❝ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ ✷✳✷✳✸✱ ✷✳✷✳✽✱ ✷✳✷✳✶✵ ✈➭ ❝➳❝ ❍Ö q✉➯ ✷✳✷✳✹✱ ✷✳✷✳✺✱ ✷✳✷✳✼✱ ✷✳✷✳✾ ✈➭ ✷✳✷✳✶✷✳ ❈➳❝ ❦Õt q✉➯ ♥➭② ❧➭ sù ♠ë ré♥❣ ❝ñ❛ ♠ét sè ❦Õt q✉➯ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✹❪✱ ❬✽❪✱ ❬✾❪✱ ❬✶✵❪✳ 40 t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ◆❣✉②Ô♥ ❱➝♥ ❑❤✉➟ ✈➭ ▲➟ ▼❐✉ ❍➯✐ ✭✷✵✵✷✮✱ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝✱ ❚❐♣ ✶✱ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ❙➢ ♣❤➵♠✳ ❬✷❪ ◆❣✉②Ơ♥ P❤ó❝ ✭✷✵✶✺✮✱ ❱Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ b✲♠➟tr✐❝✱ ▲✉❐♥ ✈➝♥ t❤➵❝ sÜ ❚♦➳♥ ❤ä❝✱ ➜➵✐ ❤ä❝ ❱✐♥❤✳ ❬✸❪ ❏✳ ❑❡❧❧❡② ✭✶✾✼✸✮✱ ❚➠♣➠ ➤➵✐ ❝➢➡♥❣✱ ❍➭ ❍✉② ❑❤♦➳✐✱ ❍å ❚❤✉➬♥ ✈➭ ➜✐♥❤ ▼➵♥❤ ❚➢➡♥❣ ✭❞Þ❝❤✮✱ ◆❤➭ ①✉✃t ❜➯♥ ➜➵✐ ❤ä❝ ✈➭ ❚r✉♥❣ ❤ä❝ ❝❤✉②➟♥ ♥❣❤✐Ö♣✱ ❍➭ ◆é✐✳ ❬✹❪ ❙✳ ❑✳ ❈❤❛tt❡r❥❡❛ ✭✶✾✼✷✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✱ ❈✳ ❘✳ ❆❝❛❞✳ ❇✉❧❣❛r❡ ❙❝✐✳ ✷✺✱ ✼✷✼✲✼✸✵ ✳ ❬✺❪ ❙✳ ❈③❡r✇✐❦ ✭✶✾✾✸✮✱ ❈♦♥tr❛❝t✐♦♥ ♠❛♣♣✐♥❣s ✐♥ b✲♠❡tr✐❝ s♣❛❝❡s✱ ❆❝t❛ ▼❛t❤✳ ■♥ ✲ ❢♦r♠✳ ❯♥✐✈✳ ❖str❛✈✳ ✶✱ ✺✲✶✶✳ ❬✻❪ ❍✳ ▲✳ ●✉❛♥❣✱ ❩✳ ❳✐❛♥ ✭✷✵✵✼✮✱ ❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❧♦♣♦❧♦❣✐❝❛❧ ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ♦❢ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✸✸✷✱ ✶✹✻✽✲✶✹✼✻✳ ❬✼❪ ◆✳ ❍✉ss❛✐♥✱ ▼✳ ❍✳ ❙❤❛❤ ✭✷✵✶✵✮✱ ❑❑▼ ♠❛♣♣✐♥❣s ✐♥ ❝♦♠❡ b✲♠❡tr✐❝ s♣❛❝❡s✱ ❈♦♠♣✉t❡rs ❛♥❞ ▼❛t❤✳ ❆♣♣❧✳ ✻✷✱ ✶✻✼✼✲✶✻✽✹✳ ❬✽❪ ❘✳ ❑❛♥♥❛♥ ✭✶✾✻✽✮✱ ❙♦♠❡ r❡s✉❧ts ♦♥ ❢✐①❡❞ ♣♦✐♥ts✱ ❇✉❧❧✳ ❈❛❧❝✉tt❛ ▼❛t❤✳ ❙♦❝✳ ✻✵✱ ✼✶✲✼✻✳ ❬✾❪ ▼✳ ❑✐r✱ ❍✳ ❑✐③✐❧t✉♥❡ ✭✷✵✶✸✮✱ ❖♥ s♦♠❡ ✇❡❧❧ ❦♥♦✇♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ b✲♠❡tr✐❝ s♣❛❝❡s✱ ❚✉r❦✐s❤ ❏♦✉r♥❛❧ ♦❢ ❆♥❛❧②s✐s ❛♥❞ ◆✉♠❜❡r ❚❤❡♦r②✱ ❱♦❧✳ ✶✱ ◆♦✳ ✶✱ ✶✸✲✶✻✳ ❬✶✵❪ ❏✳ ❘✳ ▼♦r❛❧❡s✱ ❊✳ ❘♦❥❛s ✭✷✵✶✵✮✱ ❈♦♥❡ ♠❡tr✐❝ s♣❛❝❡ ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦✲ r❡♠s ♦❢ T ✲❑❛♥♥❛♥ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ ■♥t✳ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤✳ ❆♥❛❧✲ ②s✐s✱ ❱♦❧✳ ✹✱ ✶✼✺✲✶✽✹✳ ... b ✈➭ b ✈➭ E ✱ a, b, c ∈ d ỗ (ii) P intP intP ỗ b b c ỏ t tỏ α ❧➭ sè t❤ù❝ ❞➢➡♥❣✳ ❑❤✐ ➤ã c t❤× a ỗ b b a K ❧➭ ♥ã♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✱ {xn }, {yn } ❧➭ ❝➳❝ ❞➲② tr♦♥❣ E E✳ c t❤× b. .. [a, b] ➜➷t P = {f ∈ C[a ,b] : ≤ f }✳ ❑❤✐ ➤ã P P ✭✐✮ ❧➭ ➤ã♥❣✱ ✭✐✐✮ ❱í✐ ♠ä✐ t❤á❛ ♠➲♥ ❜❛ ➤✐Ị✉ ❦✐Ư♥ P = ∅✱ P = {0}❀ a, b ∈ R✱ a, b ≥ ✈➭ ♠ä✐ f, g ∈ P ✱ t❛ ❝ã ≤ af (x) + bg(y), ∀x ∈ [a, b] ❉♦ ➤ã af + bg... tơ❝ ♥➟♥ intP +intP c t❤× b − a ∈ intP ✈➭ intP + intP ⊂ intP ✳ ❱❐② a ➜Ó ý r➺♥❣ intP +P = ⊂ intP ✳ ◆Õ✉ a c − b ∈ intP ✳ ❙✉② r❛ b c − b ∈ intP ✳ ❙✉② r❛ c − a = c − b + b − a ∈ c✳ x∈P (x + intP )