Một số định lý điểm bất động đối với các ánh xạ thỏa mãn điều kiện co cyclic trong không gian mêtric riêng

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❍✉ú♥❤ ❚❤Þ ❑✐♠ ❉✉♥❣ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❈②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✹ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❍✉ú♥❤ ❚❤Þ ❑✐♠ ❉✉♥❣ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❈②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✹ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▲ê✐ ♥ã✐ ➤➬✉ ❈❤➢➡♥❣ ■✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝ ✳ ✳ ✸ ✽ ❈❤➢➡♥❣ ■■✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✷✳✶ ✶✽ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝ ❦✐Ĩ✉ ❤÷✉ tØ ❍❛r❞② ✲ ❘♦❣❡rs tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✷✳✷ ✳ ✳ ✳ ✳ ✶✽ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ❑Õt ❧✉❐♥ ✸✻ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✼ ✶ ▲ê■ ◆ã■ ➜➬❯ ▲ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❧Ü♥❤ ✈ù❝ t♦➳♥ ❤ä❝ ➤➢ỵ❝ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ q✉❛♥ t➞♠✱ ✈➭ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝❤đ ➤Ị ♥❣❤✐➟♥ ❝ø✉ q✉❛♥ trä♥❣ ❝đ❛ ❣✐➯✐ tÝ❝❤✳ ◆ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ ➤❛ ❞➵♥❣ ❝➯ tr♦♥❣ t♦➳♥ ❤ä❝ ❧➱♥ tr♦♥❣ ❝➳❝ ♥❣➭♥❤ ❦❤♦❛ ❤ä❝ ❦❤➳❝✳ ▼ét sè ❦Õt q✉➯ ✈Ị tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ♥ỉ✐ t✐Õ♥❣ ➤➲ ①✉✃t ❤✐Ư♥ tõ ➤➬✉ t❤Õ ❦Ø ❳❳✱ tr♦♥❣ ➤ã ♣❤➯✐ ❦Ó ➤Õ♥ ♥❣✉②➟♥ ❧Ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❇r♦✉✇❡r ✭✶✾✶✷✮ ✈➭ ♥❣✉②➟♥ ❧Ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ✭✶✾✷✷✮✳ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ➤➲ trë t❤➭♥❤ ♠ét ❝➠♥❣ ❝ơ ♣❤ỉ ❞ơ♥❣ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ❜➭✐ t♦➳♥ ✈Ị sù tå♥ t➵✐ tr♦♥❣ ♥❤✐Ị✉ ♥❣➭♥❤ ❝đ❛ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝ ✈➭ ❝➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ♥ã✳ ❱× t❤Õ✱ ➤➲ ❝ã ♠ét sè ❧í♥ ❝➳❝ ♠ë ré♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ❝➡ ❜➯♥ ♥➭② ❝❤♦ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✱ ❜➺♥❣ ❝➳❝❤ ➤✐Ò✉ ❝❤Ø♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝➡ ❜➯♥ ❤♦➷❝ t❤❛② ➤ỉ✐ ❦❤➠♥❣ ❣✐❛♥✳ ❈❤➻♥❣ ❤➵♥ ♥❤➢✿ ♥❤÷♥❣ ➤✐Ị✉ ❦✐Ư♥ tù❛ ❝♦ ❞➵♥❣ ❍❛r❞② ✲ ❘♦❣❡r ✈➭ ❈✐r✐❝✱ ❞➵♥❣ tù❛ ❝♦ ❝ñ❛ ❝♦ ❝②❝❧✐❝✱ ❞➵♥❣ ❝♦ ②Õ✉ ❝ñ❛ ❝♦ ❝②❝❧✐❝✱ ♣❤Ð♣ fψ ✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✈➭ ♠ë ré♥❣ ♥❤÷♥❣ ❦Õt q✉➯ ♠í✐ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤♦ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ❚r➟♥ ❝➡ së ❝➳❝ ❜➭✐ ❜➳♦ tr❛❝t✐✈❡ ❝♦♥❞✐t✐♦♥s ✲➤➬② ➤ñ✱✳✳✳ ❋✐①❡❞ ♣♦✐♥t ❢♦r ♠❛♣♣✐♥❣s s❛t✐s❢②✐♥❣ ❝②❝❧✐❝❛❧ ❝♦♥✲ ❝ñ❛ ❲✳ ❆✳ ❑✐r❦✱ P✳ ❙✳ ❙r✐♥✐✈❛s❛♥✱ P✳ ❱❡❡r❛♠❛♥✐ ✭✷✵✵✸✮✱ ❇❛✲ ♥❛❝❤ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ❢♦r ❝②❝❧✐❝❛❧ ♠❛♣♣✐♥❣s ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s ❝ñ❛ ❚✳ ❆❜❞❡❧❥❛✇❛❞✱ ❏✳ ❖✳ ❆❧③❛❜✉t✱ ❆✳ ▼✉❦❤❡✐♠❡r ✈➭ ❨✳ ❩❛✐❞❛♥ ✭✷✵✶✷✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✈✐❛ ✈❛r✐♦✉s ❝②❝❧✐❝ ❝♦♥tr❛❝t✐✈❡ ❝♦♥❞✐t✐♦♥s ✐♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s ❝ñ❛ ❍✳ ❑✳ ◆❛s❤✐♥❡✱ ❩✳ ❑❛❞❡❧❜✉r❣✱ ✈➭ ❙✳ ❘❛❞❡♥♦✈✐❝ ✭✷✵✶✸✮✱ ❝ï♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❦❤➳❝✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ◆●➛❚✳ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ♥❤÷♥❣ ♠ë ré♥❣ ♥ã✐ tr➟♥ ✈➭ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ▲✉❐♥ ✈➝♥ ♥❤➺♠ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❦❤➳✐ ♥✐Ư♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈Ị ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❝②❝❧✐❝✱ ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ♠è✐ q✉❛♥ ❤Ư ❝đ❛ ❝❤ó♥❣✱ ❝➳❝ ➤Þ♥❤ ❧Ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ❝②❝❧✐❝ tr➟♥ ✶ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ♥❤÷♥❣ ♥❤❐♥ ①Ðt ♠í✐ ❝đ❛ ❝➳❝ ➤è✐ t➢ỵ♥❣ tr➟♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❱í✐ ♠ơ❝ t✐➟✉ tr ợ trì t ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ▼ơ❝ ✶ t➳❝ ❣✐➯ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ▼ơ❝ ✷ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❈❤➢➡♥❣ ✷ ✈í✐ t✐➟✉ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ▼ơ❝ ✶ ❞➭♥❤ ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝②❝❧✐❝ ❦✐Ĩ✉ ❍❛r❞② ❘♦❣❡rs tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ▼ơ❝ ✷ tr×♥❤ ❜➭② ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✳ ❚➳❝ ❣✐➯ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ♥❤✃t ➤Õ♥ t❤➬②✱ ♥❤➞♥ ❞Þ♣ ♥➭② t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❚♦➳♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý t❤➬②✱ ❝➠ tr♦♥❣ tỉ ●✐➯✐ tÝ❝❤ ❦❤♦❛ ❚♦➳♥ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ P❤ß♥❣ tỉ ❝❤ø❝ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❙➭✐ ●ß♥ ➤➲ ❣✐ó♣ ➤ì tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❚➳❝ ❣✐➯ ❝ò♥❣ ①✐♥ ❝➳♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤ã❛ ✷✵ ❚♦➳♥ ✲ ●✐➯✐ tÝ❝❤ t➵✐ ❚r➢ê♥❣ ọ ò t ề ệ t ợ ❣✐ó♣ t➳❝ ❣✐➯ ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ▼➷❝ ❞ï ❝ã ♥❤✐Ị✉ ❝è ❣➽♥❣✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ s❛✐ ①ãt✳ ❘✃t ♠♦♥❣ ợ ữ ý ế ó ó ủ qí t ❝➠ ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥✳ ❱✐♥❤✱ ♥❣➭② ✵✼ t❤➳♥❣ ✺ ♥➝♠ ✷✵✶✹ ❍✉ú♥❤ ❚❤Þ ❑✐♠ ❉✉♥❣ ✷ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❈②❝❧✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ ♠ét ♠➟tr✐❝ tr➟♥ X ✭❬✶❪✮ ❈❤♦ t❐♣ ❤ỵ♣ d(x, y) ≥ ✷✮ d (x, y) = d (y, x) ✸✮ d (x, z) ≤ d (x, y) + d (y, z) ♠➟tr✐❝ X ✈í✐ ♠ä✐ x, y ∈ X ✈í✐ ♠ä✐ ✈➭ ❦ý ❤✐Ư✉ ❧➭ (X, d) ✭❬✶❪✮ ✈➭ d : X ×X → R d (x, y) = x, y ∈ X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ ▼Ư♥❤ ➤Ị✳ ✶✳✶✳✷ ✳ ❍➭♠ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✶✮ ❚❐♣ ❤ỵ♣ X ●✐➯ sö x, y, z ∈ X ❀ ✳ tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ x=y ❀ ✈í✐ ♠ä✐ d ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ X ❦❤➠♥❣ ❣✐❛♥ ✳ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ xi ∈ X, i = 1, 2, , n✳ ❑❤✐ ➤ã✱ t❛ ❝ã d(x1 , xn ) ≤ d(x1 , x2 ) + d(x2 , x3 ) + · · · + d(xn−1 , xn ) ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✸ ❤é✐ tơ ❦❤✐ ✶✳✶✳✹ tí✐ ➤✐Ĩ♠ ✭❬✶❪✮ ❉➲② x∈X {xn } ✈➭ ❦Ý ❤✐Ö✉ ❧➭ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ xn → x ❤❛② (X, d) lim xn = x n→∞ ♥Õ✉ ➤➢ỵ❝ ❣ä✐ ❧➭ d (x, xn ) → n→∞ ✳ ▼Ư♥❤ ➤Ị✳ ✭❬✶❪✮ ●✐➯ sư (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❑❤✐ ➤ã ✶✮ ◆Õ✉ {xn } ⊂ X ✱ xn → x ✈➭ xn → y ✱ t❤× x = y ✳ ✷✮ ◆Õ✉ {xn }, {yn } ❧➭ ❝➳❝ ❞➲② tr♦♥❣ X ✱ xn → x, yn → y ✱ t❤× d (xn , yn ) d (x, y) ị ĩ ợ ❣ä✐ ❧➭ n, m ≥ n0 ✭❬✶❪✮ ❞➲② ❈❛✉❝❤② ◆❤❐♥ ①Ðt✳ ✷✮ ◆Õ✉ ❞➲② {xnk } ε>0 ♥Õ✉ ✈í✐ ♠ä✐ (X, d) tå♥ t➵✐ sè n0 ∈ N {xn } X tr♦♥❣ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ✳ ➤➬② ➤đ ➤➢ỵ❝ ❣ä✐ ❧➭ {xn } ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❝❤♦ ✈í✐ ♠ä✐ ◆❤❐♥ ①Ðt✳ tõ X (X, d) ✈➭ ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ị♥❣ ❤é✐ tơ ✈Ị (Y, ρ) x ✈➭ ❝ã ❞➲② ❝♦♥ ✳ ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ ➳ 0≤k r}✱ t➢➡♥❣ ø♥❣✮ ♠ë tr♦♥❣ X✳ ✶✳✶✳✶✺ f ➜Þ♥❤ ❧Ý✳ ❧✐➟♥ tơ❝ t➵✐ t➵✐ ✭❬✼❪✮ x∈X ●✐➯ sö X ❧➭ ❦❤➠♥❣ ❣✐❛♥ t➠♣➠ ✈➭ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ f f : X → R✳ ❑❤✐ ➤ã✱ ♥ö❛ ❧✐➟♥ tơ❝ tr➟♥ ✈➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ x0 ✳ ị ĩ AR f ợ ❣ä✐ ❧➭ h∈R s❛♦ ❝❤♦ f (x) ≥ h ✶✳✶✳✶✼ ị ĩ ợ ọ ột X ♠➟tr✐❝✱ ❜Þ ❝❤➷♥ ❞➢í✐ ✭❜Þ ❝❤➷♥ tr➟♥✮ ✭t➢➡♥❣ ø♥❣✱ f (x) ≤ h ✮ ✈í✐ ♠ä✐ ✭❬✶✵✱ ✶✹❪✮ ❈❤♦ t❐♣ rỗ tr r = A X tr X X tr➟♥ x∈A ✳ ❍➭♠ A p (x, y) = p (y, x) ✭P✷✮ ◆Õ✉ ✳ p : X × X → R+ ✭P✸✮ p (x, x) ≤ p (x, y) ✭P✹✮ p (x, z) + p (y, y) ≤ p (x, y) + p (y, z) ❀ ≤ p (x, x) = p (x, y) = p (y, y) ✱ t❤× x=y ❀ ❀ ✈í✐ ♠ä✐ x, y, z ∈ X ❑❤✐ ➤ã✱ ❝➷♣ ✳ (X, p) ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ✻ f : ♥Õ✉ tå♥ t➵✐ ♥Õ✉ t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s P ét ỗ tr r p (x, y) < p (x, x) + } ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✮ ❉➲② X tr➟♥ s✐♥❤ r❛ ♠ét t➠♣➠ {Bp (x, ε) : x ∈ X, ε > 0} ❤ä ❝➳❝ ❤×♥❤ p ỗ (X, p) {xn } ⊂ X x∈X ➤➢ỵ❝ ❣ä✐ ❧➭ τp tr♦♥❣ ➤ã tr➟♥ X ❝ã ❝➡ së ❧➭ Bp (x, ε) {y ∈ X ❂ ✿ >0 ✈➭ ✳ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❤é✐ tơ ➤Õ♥ ➤✐Ĩ♠ x∈X lim p(x, xn ) = ♥Õ✉ n→∞ p(x, x) ✳ ✷✮ ❉➲② {xn } tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ lim p(xn , xm ) = p(x, x) ♥Õ✉ tå♥ t➵✐ ✸✮ n,m→∞ ❈❛✉❝❤② {xn } ♥❣❤Ü❛ ❧➭ t❛ ❝ã ✹✮ ➳ ♥❤ ①➵ ε>0 ✺✮ ♥❤ ①➵ x∈X ❉➲② {xn } ❈❛✉❝❤② ✷✮ x∈X ➤➢ỵ❝ ❣ä✐ ❧➭ ➤Ị✉ ❤é✐ tơ t❤❡♦ t➠♣➠ τp ❞➲② ❈❛✉❝❤② ➤➢ỵ❝ ❣ä✐ ❧➭ ✳ ➤➬② ➤đ ♥Õ✉ ♠ä✐ ❞➲② ➤Õ♥ ♠ét ➤✐Ó♠ x∈X ✱ lim p(xn , xm ) = p(x, x) ✳ δ>0 ❧✐➟♥ tơ❝ ➤➢ỵ❝ ❣ä✐ ❧➭ s❛♦ ❝❤♦ t➵✐ x0 ∈ X ♥Õ✉ ✈í✐ ♠ä✐ f (Bp (x0 , δ)) ⊂ Bp (f (x0 ), ) ✳ ➤➢ỵ❝ ❣ä✐ ❧➭ ❧✐➟♥ tơ❝ tr➟♥ X ♥Õ✉ f ❧✐➟♥ tô❝ t➵✐ ✳ ✭❬✶✹❪✮ ❈❤♦ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ♥Õ✉ ❑❤➠♥❣ ❣✐❛♥ X ✈í✐ n,m→∞ f :X → X ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✮ X tr♦♥❣ f : X → X tå♥ t➵✐ ➳ ♠ä✐ ✶✳✶✳✶✾ (X, p) ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ (X, p) (X, p) ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② 0✲ lim p(xn , xm ) = ✳ n,m→∞ (X, p) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤Ị✉ ❤é✐ tơ ✭t❤❡♦ t➠♣➠ τp 0✲➤➬② ➤đ ♥Õ✉ ♠ä✐ ❞➲② ✮ ✈Ị ➤✐Ĩ♠ ♥➭♦ ➤ã x∈X ✲❈❛✉❝❤② tr♦♥❣ s❛♦ ❝❤♦ p(x, x) = ✳ ✶✳✶✳✷✵ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✹❪✮ ●✐➯ sư ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤đ X ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ✭❈✶✮ ❝♦ ❝②❝❧✐❝ f (A) ⊆ B ✈➭ A ✈➭ B (X, p) t ó rỗ ủ s ❝❤♦ X = A∪B ✳ ♥Õ✉ t❤♦➯ ♠➲♥ ❝➳❝ ➤✐Ò✉ ❦✐Ö♥ s❛✉ f (B) ⊆ A ❀ ✼ ➳ ♥❤ ①➵ f :X→ ❈é♥❣ ✈Õ ✈í✐ ✈Õ ✷ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ s✉② r❛ r➺♥❣ p f x, f x ≤ 2A + B + C + D + E p (x, f x) = λp (x, f x) − (B + C + D + E + 2F ) ◆❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✸✳✸✮ t❛ ❝ã 2A+B+C+D+E 2−(B+C+D+E+2F ) ≤ λ = ➤✐Ị✉ ❦✐Ư♥ q✉ü ➤➵♦✳ ❚❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✶✳✷✱ tå♥ t➵✐ p (f z, z) = p (f z, f z) ✳ ●✐➯ sö X (X, p) ➳ ♣ ❞ơ♥❣ ❇ỉ ➤Ị ✷✳✶✳✸✱ f z∈Y < s❛♦ ❝❤♦ ❝ã tÝ♥❤ ❝❤✃t ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✈➭ f ✳❉♦ ➤ã t❤♦➯ p (z, z) = ✈➭ (P ) ✳ f :X→X ❧➭ ♠ét ➳♥❤ ①➵ tõ ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑ý ❤✐Ö✉ Mf5 (x, y) = {p (x, y) , p (x, f x) , p (y, f y) , p (x, f y) , p (y, f x)}; Mf4 (x, y) = {p (x, y) , p (x, f x) , p (y, f y) , 12 (p (x, f y) + p (y, f x))}; Mf3 (x, y) = {p (x, y) , 12 (p (x, f x) + p (y, f y)), 12 (p (x, f y) + p (y, f x))} ✷✳✶✳✶✵ ➜Þ♥❤ ❧Ý✳ ✭❬✶✺❪✮ ●✐➯ sư (X, p) 0✲➤➬② ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤ñ✱ m m ∈ N✱ A1 , A2 , , Am X t ó rỗ ủ Ai ✈➭ i=1 f : Y → Y ✳ ●✐➯ sö r➺♥❣ m ✭❛✮ Y = Ai ❧➭ ♠ét ❜✐Ĩ✉ ❞✐Ơ♥ ❝②❝❧✐❝ ❝đ❛ Y ➤è✐ ✈í✐ f✳ i=1 ✭❜✮ ❚å♥ t➵✐ ✭✈í✐ λ ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ (x, y) ∈ Ai × Ai+1 , i = 1, 2, , m Am+1 = A1 ✮✱ p (f x, f y) ≤ λ max Mfj (x, y) ✈í✐ j = 3, j = ❤♦➷❝ j = 5✳ m ♥÷❛✱ p (z, z) = ✈➭ z ∈ ❑❤✐ ➤ã✱ f (3.7) z✳ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❍➡♥ Ai ✳ i=1 ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ❜✃t ỳ ỗ n N ý ệ rớ ết t ❝❤ø♥❣ ♠✐♥❤ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ x0 ∈ Y t❛ ➤➷t ✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ xn+1 = f xn ✳ sup{p (x, y) : x, y ∈ A} ✳ s❛♦ ❝❤♦ x ∈ Ai ❑❤✐ ➤ã ♥❤ê ❣✐➯ t❤✐Õt t❛ ❝ã Of (x0 ; n) = {x1 , x2 , , xn } Of (x0 ; ∞) = {x1 , x2 , , xn , } i = 1, , m j=5 ❧➭ q✉ü ➤➵♦ t❤ø ❧➭ q✉ü ➤➵♦ ❝đ❛ x0 n ❝đ❛ ✷✹ ❱í✐ {xn } ⊂ Y x0 ✳ ✈➭ ❦ý ❤✐Ư✉ ✳ ❚❛ ❝ị♥❣ ❦ý ệ í ủ t rỗ A ⊂ X ✳ A= ▲➢✉ ý r➺♥❣ ♥Õ✉ ❞✐❛♠ A=0 A t❤× ❧➭ t❐♣ ❣å♠ ♠ét ➤✐Ĩ♠ ❞✉② ♥❤✃t✳ ❚✉② ♥❤✐➟♥ ➤✐Ị✉ ♥❣➢ỵ❝ ❧➵✐ ❧➭ ❦❤➠♥❣ ➤ó♥❣✳ ❚r➢í❝ t✐➟♥✱ t❛ t❤✃② r➺♥❣ ♥Õ✉ t❛ ❝ã f xn = xn+1 = xn p(xn , xn+1 ) = ✱ ♥❣❤Ü❛ ❧➭ p(xn , xn ) = ✳ ❱× t❤Õ t❛ ❣✐➯ sư r➺♥❣ xn ✈í✐ sè ♥➭♦ ➤ã ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ f ≤ i, j ≤ n ✱ t❤× ✈➭ t❤á❛ ♠➲♥ p(xn , xn+1 ) > n∈N Of (x0 ; ∞) ≤ 1−λ p (x1 , x2 ) ✈í✐ ♠ä✐ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❞✐❛♠ ❣✐➯ sö n ∈ N ✳ ✳ ❚❤❐t ✈❐②✱ ✳ ❑❤✐ ➤ã✱ t❛ ❝ã p (xi+1 , xj+1 ) = p (f xi , f xj ) (3.8) = λ max{p (xi , xj ) , p (xi , xi+1 ) , p (xj , xj+1 ) , p (xi , xj+1 ) , p (xj , xi+1 )} ❉♦ xi , xi+1 , xj , xj+1 ∈ Of (x0 ; n) ✱ t❛ s✉② r❛ p (xi+1 , xj+1 ) ≤ λ k≤n ❱× ✈❐②✱ tå♥ t➵✐ t❛ ❝ã ❞✐❛♠ s❛♦ ❝❤♦ ❞✐❛♠ Of (x0 ; n) < ❞✐❛♠ Of (x0 ; n) Of (x0 ; n) = p (x1 , xk ) ✳ ◆❤ê ➤✐Ò✉ ❦✐Ö♥ ✭P✹✮ p (x1 , xk ) ≤ p (x1 , x2 )+p (x2 , xk )−p (x2 , x2 ) ≤ p (x1 , x2 )+p (x2 , xk ) ✱ ♥➟♥ t❛ s✉② r❛ ❞✐❛♠ Of (x0 , n) ≤ p (x1 , x2 ) + λ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ ❞✐❛♠ Of (x0 , n) ≤ ❞✐❛♠ Of (x0 ; n) 1−λ p (x1 , x2 )✳ ▲✃② ❝❐♥ tr➟♥ ➤ó♥❣ ❝đ❛ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭② t❛ ❝ã ➤✐Ò✉ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ❚✐Õ♣ t❤❡♦ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ✈í✐ λn 1−λ p (x1 , x2 )✳ m > n ≥ ✱ t❛ ❝ã p (xm+1 , xn+1 ) ≤ ❚❤❐t ✈❐②✱ t➢➡♥❣ tù ♥❤➢ ➤è✐ ✈í✐ ❝➠♥❣ t❤ø❝ ✭✸✳✽✮ t❛ ❝ã p (xm+1 , xn+1 ) ≤ λ max{p (xm , xn ) , p (xm , xm+1 ) , p (xn , xn+1 ) , p (xm , xn+1 ) , p (xn , xm+1 )} ❱× xm , xm+1 , xn , xn+1 ∈ Of (xn−1 ; m − n + 2) = λp (xn , xk1 ) p (xm+1 , xn+1 ) ≤ λ ✈í✐ sè k1 ♥➭♦ ➤ã ♠➭ ❞✐❛♠ t❛ ❝ã Of (xn−1 ; m − n + 2) = λp(xn , xk1 ), (3.9) n + ≤ k1 ≤ m + ✳ ❚➢➡♥❣ tù✱ t❛ ❝ã p (xn , xk1 ) ≤ λ max{p (xn−1 , xk1 −1 ) , p (xn−1 , xn ) , p (xk1 −1 , xk1 ) , p (xn−1 , xk1 ) , p (xn , xk1 −1 )} ≤ λ ❞✐❛♠ Of (xn−2 ; m − n + 3) ✷✺ ❑Õt ❤ỵ♣ ✈í✐ ✭✸✳✾✮ t❛ ❝ã p (xm+1 , xn+1 ) ≤ λ2 p (xn−1 , xk2 ) ✈í✐ k2 ≤ m + ✳ ❚✐Õ♣ tơ❝ q✉➳ tr×♥❤ ♥➭②✱ t❛ t❤✉ ➤➢ỵ❝ p (xm+1 , xn+1 ) ≤ λn−1 ❞✐❛♠ ≤ λn−1 λ Of (x0 ; m − 1) = λn−1 p x1 , ykn−1 ❞✐❛♠ Of (x0 ; m) ≤ ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ❝ï♥❣ ♥➭② t❛ s✉② r❛ ♥❣❤Ü❛ ❧➭ t➵✐ {xn } z∈Y ❧➭ ❞➲② ✲❈❛✉❝❤②✳ ❉♦ (X, p) λn 1−λ p (x1 , x2 ) p (xn , xm ) → ❧➭ ✲➤➬② ➤ñ ✈➭ Y ❦❤✐ m, n → ∞ ✱ ❧➭ ➤ã♥❣ ♥➟♥ tå♥ s❛♦ ❝❤♦ lim p (xn , z) = = p(z, z) n→∞ m ❍➡♥ ♥÷❛✱ z∈ Ai ✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ fz = z ✳ ❉ï♥❣ tÝ♥❤ ❝❤✃t ✭P✹✮ ❝ñ❛ i=1 ♠➟tr✐❝ r✐➟♥❣ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✭✸✳✼✮ ✈í✐ j=5 ✱ t❛ ❝ã p (f z, z) ≤ p (f z, f xn ) + p (f xn , z) ≤ λ max{p (z, xn ) , p (z, f z) , p (xn , xn+1 ) , p (z, xn+1 ) , p (xn , f z)} +p (xn+1 ; z) ❉♦ p (z, xn ) , p (xn , xn+1 ) ➤Ò ✶✳✶✳✷✸ t❛ ❝ã ❞➬♥ ➤Õ♥ ✵ ❦❤✐ ✱ ♥➟♥ ♥Õ✉ ❣✐➯ sư n→∞ ✱ ➤✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈× ✈➭ ✈× t❤Õ fz = z ✳ ✈➭ ♥❤ê ❇æ p (f z, z) > p (f z, z) ≤ λp (f z, z) p (f z, z) = f z1 = z1 p (z, xn+1 ) p (xn , f z) → p (z, f z) ➤➻♥❣ t❤ø❝ t❛ ❝ã ➤ã✱ ✈➭ ❇➞② ❣✐ê ❣✐➯ sö tå♥ t➵✐ t❤× tõ ❜✃t λ ∈ [0, 1) ✳ ❉♦ z1 ∈ Y s❛♦ ❝❤♦ ✳ ❑❤✐ ➤ã✱ ♥❤ê tÝ♥❤ ❝❤✃t ✭P✷✮ ❝ñ❛ ♠➟tr✐❝ r✐➟♥❣ t❛ ❝ã p (z, z1 ) = p (f z, f z1 ) ≤ λ max{p (z, z1 ) , p (z, f z) , p (f z1 , f z1 ) , p (z, f z1 ) , p (z1 , f z)} = λ max{p (z, z1 ) , p (z, z) , p (z1 , z1 ) , p (z, z1 ) , p (z1 , z)} = λp (z, z1 ) ➜✐Ò✉ ♥➭②✱ ❝❤Ø ①➯② r❛ ❦❤✐ p (z, z1 ) = ✱ ✈× ✈❐② z = z1 ✳ ❉♦ ➤ã✱ f ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈➳❝ tr➢ê♥❣ ❤ỵ♣ t❤ø❝ j=3 ✈➭ j=4 s✉② tõ tr➢ê♥❣ ❤ỵ♣ j=5 max Mf3 (x, y) ≤ max Mf4 (x, y) ≤ max Mf5 (x, y) ✳ ✷✻ ♥❤ê ❝➳❝ ❜✃t ➤➻♥❣ (X, p) ●✐➯ sư ❍Ư q✉➯✳ ✷✳✶✳✶✶ 0✲➤➬② ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ m ∈ ➤ñ✱ m Ai ✈➭ f : Y → N, A1 , A2 , , Am ❧➭ ❝➳❝ t❐♣ ❝♦♥ ➤ã♥❣ rỗ ủ X i=1 Y sử r➺♥❣ m ✭❛✮ Y = Ai ❧➭ ♠ét ❜✐Ĩ✉ ❞✐Ơ♥ ❝②❝❧✐❝ ❝đ❛ Y ➤è✐ ✈í✐ f✳ i=1 ✭❜✮ ❚å♥ t➵✐ ✭✈í✐ λ ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ (x, y) ∈ Ai × Ai+1 , i = 1, 2, , m Am+1 = A1 ✮✱ p (f x, f y) ≤ λ max{p (x, f x) , p (y, f y)} ❤❛② p (f x, f y) ≤ λ max{p (x, y) , p (x, f x) , p (y, f y)} m ❑❤✐ ➤ã✱ f ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ z ❞✉② ♥❤✃t✳ ❍➡♥ ♥÷❛✱ p (z, z) = ✈➭ z ∈ Ai ✳ i=1 ❈❤ø♥❣ ♠✐♥❤✳ ❉Ô t❤✃② r➺♥❣ max{p (x, f x) , p (y, f y)} ≤ max Mf5 (x, y) ✈➭ max{p (x, y) , p (x, f x) , p (y, f y)} ≤ max Mf5 (x, y) ❱× t❤Õ ❦Õt ❧✉❐♥ ❝đ❛ ❤Ư q✉➯ s✉② tõ ➜Þ♥❤ ❧ý ✷✳✶✳✶✵✳ ✷✳✶✳✶✷ ❍Ö q✉➯✳ ✭❬✶✺❪✮ f : X → X ➳♥❤ ①➵ x, y ∈ X xn → z ✈➭ ❣✐➯ sư t❤❡♦ t➠♣➠ ❈❤ø♥❣ ♠✐♥❤✳ {xn } ❱× t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✸✳✼✮ ✈í✐ f z = z✳ f z ✳ ❦❤✐ f ❧✐➟♥ tơ❝ t➵✐ ➤✐Ĩ♠ ❞✉② ♥❤✃t ✈➭ t❤♦➯ ♠➲♥ lim p (xn , z) = p (z, z) τp ❑❤✐ ➤ã✱ λ ∈ [0, 1) t❤❡♦ t➠♣➠ z✱ ✈➭ ✈í✐ ♠ä✐ ♥❣❤Ü❛ ❧➭ ♥Õ✉ τp ✳ f t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✸✳✼✮✱ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✶✳✶✵ ❧➭ ♠ét ❞➲② tr♦♥❣ n→∞ t❤❡♦ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤đ ✈➭ τp ✱ t❤× t❛ ❝ã f xn → f z = z ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ sö ●✐➯ sö X s❛♦ ❝❤♦ p (z, z) = p (f z, f z) = xn → z t❤❡♦ t➠♣➠ τp ❦❤✐ ❚❛ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❦❤✐ ➤ã ❝ã ✳ ●✐➯ n→∞ ✱ ❤❛② f xn → f z = z n→∞ ✱ ♥❣❤Ü❛ ❧➭ lim p (f xn , f z) = p (f z, f z) = p (z, z) = n→∞ ✷✼ (3.10) ❚❤❐t ✈❐②✱ t❛ ❝ã p (f xn , f z) ≤ λ max{p (xn , z) , p (xn , f xn ) , p (z, f z) , p (xn , f z) , p (z, f xn )} ≤ λ max{p (xn , z) , p (xn , z) + p (f z, f xn ) , 0, p (xn , z) , p (f z, f xn )} = λ(p (xn , z) + p (f z, f xn )) ❙✉② r❛ p (f xn , f z) ≤ λ 1−λ p (xn , z) n ì tế tứ ợ ❝❤ø♥❣ ♠✐♥❤✳ ➜Þ♥❤ ❧Ý✳ ✷✳✶✳✶✸ ✭❬✶✺❪✮ ❈❤♦ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧Ý ✷✳✶✳✶✵ ✈í✐ ➜➷❝ ❜✐Ưt✱ tå♥ t➵✐ ♣❤➬♥ tư ✈➭ f ❝ị♥❣ ❝ã tÝ♥❤ ❝❤✃t ❈❤ø♥❣ ♠✐♥❤✳ ❱× f (X, p) ✈➭ ➳♥❤ ①➵ f : Y → Y t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ò✉ λ ∈ [0, 21 )✳ ❑❤✐ ➤ã✱ f t❤♦➯ ➤✐Ị✉ ❦✐Ư♥ q✉ü ➤➵♦ ✭✸✳✷✮✳ z ∈ X s❛♦ ❝❤♦ p (z, z) = ✈➭ p (f z, z) = p (f z, f z) (P )✳ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧Ý ✷✳✶✳✶✵ ✈í✐ ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧Ý ✷✳✶✳✶✵ t❛ ❝ã λ ∈ [0, 12 ) ✱ F ix(f ) = φ ✳ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✸✳✷✮ ❝đ❛ ➜Þ♥❤ ❧Ý ✷✳✶✳✷✳ ●✐➯ sư x∈Y s❛♦ ❝❤♦ x = fx f t❤á❛ ✳ ❑❤✐ ➤ã t❛ ❝ã p f x, f x ≤ λ max{p (x, f x) , p (x, f x) , p f x, f x , p x, f x , p (f x, f x)} ≤ λ max{p (x, f x) , p f x, f x , p (x, f x) + p f x, f x , p f x, f x } = λ p (x, f x) + p f x, f x ❙✉② r❛ p f x, f x ≤ λ 1−λ p (x, f x)✱ tr♦♥❣ ➤ã λ 1−λ 0 ✳ ψ(0) = ✈➭ ψ(t) ≤ t ✈í✐ ♠ä✐ ❙❛✉ ➤➞②✱ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ➳♥❤ ①➵ ➜Þ♥❤ ♥❣❤Ü❛✳ ✷✳✷✳✶ ✭❬✶✺❪✮ ●✐➯ sö A1 , A2 , , Am sè ♥❣✉②➟♥ ❞➢➡♥❣✱ m Ai Y = ✳ i=1 ψ∈Ψ (X, p) ➳ f :Y →Y ♥❤ ①➵ t≥0 ✳ fψ ✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣✳ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❧➭ ❝➳❝ t❐♣ ❝♦♥ ➤ã♥❣ rỗ ủ ợ ọ é m X f ✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ ❧➭ ✈➭ ✈í✐ ♥Õ✉ m ✭❛✮ Y = Ai ❧➭ ♠ét ❜✐Ĩ✉ ❞✐Ơ♥ ❝②❝❧✐❝ ❝đ❛ Y ➤è✐ ✈í✐ f ✳ i=1 ✭❜✮ ❚å♥ t➵✐ α, β ∈ [0, 1) ✈í✐ Ai+1 , i = 1, 2, , m α+β < ✭✈í✐ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ Am+1 = A1 (x, y) ∈ Ai × ✮✱ t❛ ❝ã p (f x, f y) ≤ α♠(x, y) + β ▼(x, y) tr♦♥❣ ➤ã (x, y) = ψ p (x, y) ♠ + p (x, f x) + p (x, y) ✈➭ (x, y) = max{ψ(p (x, y)), ψ(p (x, f x)), ψ(p (y, f y)), ψ( (p (x, y) + p (x, f x)))} ▼ ❑Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ♠ơ❝ ♥➭② ❧➭ ➤Þ♥❤ ❧Ý s❛✉✳ ✷✾ ➜Þ♥❤ ❧Ý✳ ✷✳✷✳✷ ✭❬✶✺❪✮ ●✐➯ sö (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤ñ✱ m m ∈ N✱ A1 , A2 , , Am X✱ Y = ❧➭ ❝➳❝ t ó rỗ ủ Ai i=1 sö f :Y →Y ❧➭ ♣❤Ð♣ fψ ✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣ ✈í✐ ψ ∈ Ψ✳ ❑❤✐ ➤ã✱ f ❝ã m z ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❍➡♥ ♥÷❛✱ t❛ ❝ã z∈ Ai ✳ i=1 ❈❤ø♥❣ ♠✐♥❤✳ ✈í✐ ●✐➯ sư x ∈ A1 xn+1 = f xn , n = 0, 1, 2, ✭✈× A1 = φ ✮✳ ❚❛ ①➳❝ ➤Þ♥❤ ❞➲② {xn } ⊂ X ✈➭ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ lim p (xn , xn+1 ) = (4.1) n→∞ ❚❤❐t ✈❐②✱ ♥Õ✉ tå♥ t➵✐ sè ❝♦ ❝②❝❧✐❝ s✉② ré♥❣✱ Ai × Ai+1 s❛♦ ❝❤♦ α, β ∈ [0, 1) ❇➞② ❣✐ê ❣✐➯ sö r➺♥❣ ✈➭ ✱ tå♥ t➵✐ p (xk+1 , xk ) = ✱ t❤× ♥❤ê α+β < ✱ t❛ s✉② r❛ p (xn , xn+1 ) > nN r ỗ k∈N ✈í✐ ♠ä✐ n∈N f ❧➭ ♣❤Ð♣ fψ ✲ lim p (xn , xn+1 ) = ✳ n→∞ ✳ ❚õ ➤✐Ị✉ ❦✐Ư♥ ✭❛✮✱ t❛ t❤✃② i = i(n) ∈ {1, 2, , m} s❛♦ ❝❤♦ (xn , xn+1 ) ∈ ✳ ❑❤✐ ➤ã✱ tõ ➤✐Ị✉ ❦✐Ư♥ ✭❜✮ t❛ ❝ã p (xn , xn+1 ) ≤ α♠(xn−1 , xn ) + β ▼(xn−1 , xn ), n = 1, 2, (4.2) ▼➷t ❦❤➳❝✱ t❛ ❝ã ♠ (xn−1 , xn ) = ψ p (xn , xn+1 ) + p (xn−1 , xn ) = ψ(p (xn , xn+1 )), + p (xn−1 , xn ) ✈➭ (xn−1 , xn ) = max{ψ(p (xn−1 , xn )), ψ(p (xn , xn+1 )) ▼ ψ( 12 (p (xn−1 , xn+1 ) + p (xn , xn )))} (xn−1 , xn ) = ψ(p (xn , xn+1 )) ▼ ✯ ◆Õ✉ ✱ tõ ✭✹✳✷✮ ✈➭ tÝ♥❤ ❝❤✃t ❝ñ❛ ψ t❛ ❝ã p (xn , xn+1 ) ≤ (α + β)ψp (xn , xn+1 ) < p (xn , xn+1 ) ❱× α+β 0 ✮✳ ❑ý ❤✐Ư✉ ✈í✐ ♠ä✐ ✱ ❤❛② ψ : [0, +∞) → [0, +∞) t ∈ [0, +∞) ✳ ❑❤✐ ➤ã✱ ❞Ô t❤✃② ψ∈Ψ ✳ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ tá r➺♥❣ ❝②❝❧✐❝ s✉② ré♥❣✳ ❚❤❐t ✈❐②✱ ✈í✐ ❜✃t ❦ú y≤x Ai ❑❤✐ ➤ã râ r➭♥❣ f ψ ❧➭ ❤➭♠ ❝❤♦ ❧✐➟♥ tô❝ ✈➭ ❧➭ ♣❤Ð♣ fψ ✲❝♦ (x, y) ∈ Ai × Ai+1 i = 1, ✱ ✱ ❣✐➯ sö r➺♥❣ ✱ ❦❤✐ ➤ã t❛ ❝ã p(f x, f y) = max y2 x2 , 2(1 + x) 2(1 + y) ✸✹ x2 = 2(1 + x) (4.8) ▼➷t ❦❤➳❝ ✈× ψ ❧➭ ❤➭♠ t➝♥❣✱ tõ ❣✐➯ t❤✐Õt x x + max y, p x, 2(1+x) ≤x x ≥ y ✈➭ x ≥ x2 ✱ 2(1+x) t❛ s✉② r❛ ✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ (x, y) = max{ψ(p (x, y)), ψ(p (x, f x)), ψ(p (y, f y)), ψ( 21 (p (x, f y) + p (y, f x)))} ▼ 2 y , ψ p y, 2(1+y) x = max ψ(p (x, y)), ψ p x, 2(1+x) ψ = max ψ(x), ψ(x), ψ(y), ψ 2 , y2 x p x, 2(1+y) + p y, 2(1+x) x x + max y, 2(1+x) = ψ(x) (4.9) ❚õ ➤ã ✭✹✳✽✮ ✈➭ ✭✹✳✾✮ t❛ s✉② r❛ p(f x, f y) ≤ 0.♠(x, y) + ▼(x, y) ❉♦ ➤ã✱ tt ề ệ ủ ị í ợ t❤♦➯ ♠➲♥ ✈í✐ ❱❐②✱ f ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t z = ∈ A1 ∩ A2 ✸✺ ✈➭ p (z, z) = ✳ m = ✳ ❑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❛✉ ✶✳ ❍Ư t❤è♥❣ ❤ã❛ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝②❝❧✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✱ ➳♥❤ ①➵ ❦❤➠♥❣ ❣✐➲♥✱ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❞➲② ❤é✐ tô tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ ❞➲② ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ❝②❝❧✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✲❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ✲➤➬② ➤ñ✱ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝✱ ➳♥❤ ①➵ ❝♦ ✲❝♦♠♣➽❝✱ ➳♥❤ ①➵ ❝♦ rót✱ ❞➲② P✐❝❛r❞✱ ➳♥❤ ①➵ t❤♦➯ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ q✉ü ➤➵♦✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ①➵ ❝♦ ❝②❝❧✐❝ ❦✐Ĩ✉ ❤÷✉ tû ❍❛r❞②✲❘♦❣❡rs✱ ♣❤Ð♣ ✷✳ ①➵ α fψ (P ) ✱ ❜✐Ĩ✉ ❞✐Ơ♥ ❝②❝❧✐❝✱ ➳♥❤ ✲❝♦ ❝②❝❧✐❝ s✉② ré♥❣✳ ❚r×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ rót ❝②❝❧✐❝✱ ➳♥❤ ✲❝♦ ❝②❝❧✐❝ ✈í✐ α∈S ❜➟♥ ♣❤➯✐ t❤á❛ ♠➲♥ ϕ ✱ ➳♥❤ ①➵ ✲❝♦ ❝②❝❧✐❝ ✈í✐ ϕ ❧➭ ❤➭♠ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ ≤ ψ(t) < t ✱ ♠ét ♠ë ré♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❈❛r✐st✐ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❝②❝❧✐❝✱ ♠ét ♠ë ré♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❦❤➠♥❣ ❣✐➲♥✳ ●✐í✐ t❤✐Ư✉ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ ❦✐Ĩ✉ ❤÷✉ tû ❍❛r❞②✲❘♦❣❡rs✱ ♠ét sè ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ♣❤Ð♣ fψ s✉② ré♥❣ ✈➭ ♠ét sè ❤Ư q✉➯ ❝ñ❛ ♥ã tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✸✳ ✲❝♦ ❝②❝❧✐❝ ✲➤➬② ➤ñ✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè tÝ♥❤ ❝❤✃t ✈➭ ➤Þ♥❤ ❧ý ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ❝❤ø♥❣ ♠✐♥❤ ❝ß♥ s➡ ợ ị ý ị ý ➜Þ♥❤ ❧ý ✶✳✷✳✻✱ ❍Ư q✉➯ ✷✳✶✳✼✱ ➜Þ♥❤ ❧ý ✷✳✶✳✾✱ ❍Ư q✉➯ ✷✳✶✳✶✶✱ ❍Ö q✉➯ ✷✳✷✳✸✱ ❍Ö q✉➯ ✷✳✷✳✹✱ ❍Ö q✉➯ ✷✳✷✳✺✳ ✹✳ ❚r×♥❤ ❜➭② ❝❤✐ t✐Õt ❱Ý ❞ơ ✷✳✶✳✽ ♠✐♥❤ ❤ä❛ ❝❤♦ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❝②❝❧✐❝ ❦✐Ĩ✉ ❤÷✉ tû ❍❛r❞②✲❘♦❣❡rs✱ ❱Ý ❞ơ ✷✳✷✳✻ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✷✳✷✳ ✸✻ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❚r➬♥ ❱➝♥ ➣♥ ✭✷✵✵✼✮✱ ❇➭✐ ❣✐➯♥❣ ❚➠♣➠ ➤➵✐ ❝➢➡♥❣✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✳ ❬✷❪ ❚r➬♥ ❱➝♥ ➣♥✱ ❇➭✐ ❣✐➯♥❣ ❑❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ t➠♣➠✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✳ ❬✸❪ ❚✳ ❆❜❞❡❧❥❛✇❛❞✱ ❊✳ ❑❛r❛♣✐♥❛r✱ ❑✳ ❚❛s ✭✷✵✶✶✮✱ ❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ❛ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt✳✱ ✷✹ ✭✶✶✮✱ ✶✾✵✵✲✶✾✵✹✳ ❬✹❪ ❚✳ ❆❜❞❡❧❥❛✇❛❞✱ ❏❖✳ ❆❧③❛❜✉t✱ ❆✳ ▼✉❦❤❡✐♠❡r✱ ❨✳ ❩❛✐❞❛♥ ✭✷✵✶✷✮✱ ❇❛♥❛❝❤ t✐❛❧ ❝♦♥tr❛❝t✐♦♥ ♠❡tr✐❝ ♣r✐♥❝✐♣❧❡ s♣❛❝❡s✱❋✐①❡❞ ❢♦r P♦✐♥t ❝②❝❧✐❝❛❧ ❚❤❡♦r② ♠❛♣♣✐♥❣s ❆♣♣❧✳✱ ✶✺✹ ♦♥ ♣❛r✲ ✭✷✵✶✷✮✱ ❞♦✐✿✶✵✳✶✶✽✻✴✶✻✽✼✲✶✽✶✷✲✷✵✶✷✲✶✺✹✳ ❬✺❪ ❏✳ ❈❛r✐st✐ ✭✶✾✼✻✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♠❛♣♣✐♥❣s s❛t✐s❢②✐♥❣ ✐♥✇❛r❞♥❡ss ❝♦♥❞✐t✐♦♥s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✷✶✺✱ ✷✹✶✲✷✺✶✳ ❬✻❪ ▼✳ ❊❞❡❧st❡✐♥ ✭✶✾✻✷✮✱ ❖♥ ❢✐①❡❞ ❛♥❞ ♣❡r✐♦❞✐❝ ♣♦✐♥ts ✉♥❞❡r ❝♦♥tr❛❝✲ t✐✈❡ ♠❛♣♣✐♥❣s✱ ❏✳ ▲♦❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ✸✼✱ ✼✹✲✼✾✳ ❬✼❪ ❘✳ ❊♥❣❡❧❦✐♥❣ ✭✶✾✼✼✮✱ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣②✱ P❲◆✲P♦❧✐s❤✱ ❙❝✐❡♥t✐❢✐❝ P✉❜❧✐s❤❡rs✱ ❲❛rs③❛✇❛✳ ❬✽❪ ▼✳ ❆✳ ●❡r❛❣❤t② ✭✶✾✼✸✮✱ ❖♥ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✹✵✱ ✻✵✹✲✻✵✽✳ ❬✾❪ ❑✳ ●♦❡❜❡❧✱ ❲✳ ❆✳ ❑✐r❦ ✭✷✵✵✵✮✱ ❚♦♣✐❝ ✐♥ ▼❡tr✐❝ ❋✐①❡❞ P♦✐♥t ❚❤❡✲ ♦r② ✭❲✳ ❆✳ ❑✐r❦ ❛♥❞ ❇✳ ❙✐♠s ❡❞s✳✮✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈✳ Pr❡ss✱ ❈❛♠✲ ❜r✐❞❣❡✳ ❬✶✵❪ ❉✳ ■❧✐❝✱ ❱✳ P❛✈❧♦✈✐❝✱ ❱✳ ❘❛❦♦❝❡✈✐❝ ✭✷✵✶✶✮✱ ❙♦♠❡ ♥❡✇ ❡①t❡♥s✐♦♥s ♦❢ ❇❛♥❛❝❤✬s ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ t♦ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt✳✱ ✷✹ ✭✶✶✮✱ ✶✸✷✻✲✶✾✸✵✳ ✸✼ ❬✶✶❪ ●✳❙✳ ❏❡♦♥❣✱ ❇✳ ❊✳ ❘❤♦❛❞❡s ✭✷✵✵✺✮✱ ▼❛♣s ❢♦r ✇❤✐❝❤ F (T ) = F (T n )✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✻✱ ✽✼✲✶✸✶✳ ❬✶✷❪ ❲✳ ❆✳ ❑✐r❦✱ ❲✳ ❉✳ ❘♦②❛❧t② ✭✶✾✼✶✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❝❡rt❛✐♥ ♥♦♥❧✐♠❡❛r ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s✱ ■❧❧✐♥♦✐s ❏✳ ▼❛t❤✳✱ ✹✱ ✻✺✻✲✻✼✸✳ ❬✶✸❪ ❲✳ ❆✳ ❑✐r❦✱ P✳ ❙✳ ❙r✐♥❛✈❛s❛♥✱ P✳ ❱❡❡r❛♠❛♥✐ ✭✷✵✵✸✮✱ ❋✐①❡❞ ♣♦✐♥ts ❢♦r ♠❛♣♣✐♥❣s s❛t✐s❢②✐♥❣ ❝②❝❧✐❝❛❧ ❝♦♥tr❛❝t✐✈❡ ❝♦♥❞✐t✐♦♥s✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r②✱ ✹✱ ✼✾✲✽✾✳ ❬✶✹❪ ●✳❙✳ ▼❛t❤❡✇s ✭✶✾✾✹✮✱ P❛rt✐❛❧ ♠❡tr✐❝ t♦♣♦❧♦❣②✱ Pr♦❝✳ ✽t❤ ❙✉♠♠❡r ❈♦♥❢❡r❡♥❝❡ ♦♥ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❆♥♥❛❧s ♦❢ t❤❡ ◆❡✇ ❨♦r❦ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s✱ ✼✷✽✱ ✶✽✸✲✶✾✼✳ ❬✶✺❪ ❍✳ ❑✳ ◆❛s❤✐♥❡✱ ❩✳ ❑❛❞❡❧❜❡r❣✱ ❙✳ ❘❛❞❡♥♦✈✐❝ ✭✷✵✶✸✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✈✐❛ ❝②❝❧✐❝ ❝♦♥tr❛❝t✐✈❡ ❝♦♥❞✐t✐♦♥s ✐♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ P✉❜❧✳ ■♥st✐t✳▼❛t❤✳✱ ✾✸ ✭✶✵✼✮✱ ✻✾✲✾✸✳ ❬✶✻❪ ❙✳ ❖❧tr❛ ❛♥❞ ❖✳ ❱❛❧❡r♦ ✭✷✵✵✹✮✱ ❇❛♥❛❝❤✬s ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❢♦r ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❘❡♥❞✳ ■st✐t✳ ▼❛t✳ ❯♥✐✈✳ ❚r✐❡st❡✳✱ ✸✻ ✭✶✲✷✮✱ ✶✼✲✷✻✳ ❬✶✼❪ ▼✳ P❛❝✉r❛✱ ■✳ ❆✳ ❘✉s ✭✷✵✶✵✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❢♦r ❝②❝❧✐❝ φ✲ ❝♦♥tr❛❝t✐♦♥s✱ ◆♦♥❧✐♥❡❛r✳ ❆♥❛❧✳✱ ✼✷✱ ✶✶✽✶✲✶✶✽✼✳ ❬✶✽❪ ❙✳ ❘❛❞❡♥♦✈✐❝✱ ❩✳ ❑❛❞❡❧❜❡r❣✱ ❉✳ ❏❛♥❞r❧✐❝✱ ❆✳ ❏❛♥❞r❧✐❝ ✭✷✵✶✷✮✱ ❙♦♠❡ r❡s✉❧ts ♦♥ ✇❡❛❦ ❝♦♥tr❛❝t✐♦♥s ♠❛♣s✱ ❇✉❧❧✳ ■r❛♥✐❛♥ ▼❛t❤✳ ❙♦❝✳✱ ✸✽ ✭✸✮✱ ✻✷✺✲✻✹✺✳ ❬✶✾❪ ❇✳ ❊✳ ❘❤♦❛❞❡s ✭✷✵✵✶✮✱ ❙♦♠❡ t❤❡♦r❡♠s ♦♥ ✇❡❛❦❧② ❝♦♥tr❛❝t✐✈❡ ♠❛♣s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✱ ✹✼ ✭✹✮✱ ✷✻✽✸✲✷✻✾✸✳ ❬✷✵❪ ❙✳ ❘♦♠❛❣✉❡r❛ ✭✷✵✶✵✮✱ ❆ ❑✐r❦ t②♣❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❝♦♠♣❧❡t❡✲ ♥❡ss ❢♦r ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ■❉ ✹✾✸✷✾✽✱ ❞♦✐✿✶✵✿✶✶✺✺✴✷✵✶✵✴✹✾✸✷✾✽✳ ✸✽ ... ❝♦ ❈②❝❧✐❝ ❦✐Ĩ✉ ❤÷✉ tû ❍❛r❞②✲❘♦❣❡rs tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ❚r♦♥❣ ♠ơ❝ ♥➭② ❝❤ó♥❣ t trì ột số ị í ề ể ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ tù ➳♥❤ ①➵ ①➳❝ ➤Þ♥❤ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✲➤➬② ➤ñ ✈➭ t❤♦➯ ♠➲♥... tr Ai {xn } ộ tụ ế ỗ m Ai i = 1, , m z ❞➲② ❧➷♣ i = 1, , m tr♦♥❣ t❐♣ m Ai ì ỗ {xn } ó số ❤➵♥❣ ♥➺♠ ❝ã ♠ét ❞➲② ❝♦♥ ❝ñ❛ Ai , i = 1, , m {xink } ❝ñ❛ ❞➲② ❧➭ ➤ã♥❣✱ ♥➟♥ z ∈ Ai = φ ✳ ❉♦ ➤ã... ❝②❝❧✐❝ ❦✐Ĩ✉ ❤÷✉ tû ❍❛r❞②✲❘♦❣❡rs✱ ♣❤Ð♣ ✷✳ ①➵ α fψ (P ) ✱ ❜✐Ĩ✉ ❞✐Ơ♥ ❝②❝❧✐❝✱ ➳♥❤ ✲❝♦ s rộ rì ột số ị ý ể ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ ❝♦ rót ❝②❝❧✐❝✱ ➳♥❤ ✲❝♦ ❝②❝❧✐❝ ✈í✐ α∈S ❜➟♥ ♣❤➯✐ t❤á❛ ♠➲♥ ϕ ✱ ➳♥❤ ①➵