Điểm bất động của các ánh xạ co kiểu geraghty trong không gian mêtric riêng

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❚❤Þ ❍➺♥❣ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥ ✲ ✷✵✶✼ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❚❤Þ ❍➺♥❣ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ö ❆♥ ✲ ✷✵✶✼ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ t❤➬② ❣✐➳♦✱ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✳ ❚➠✐ ①✐♥ ❜➭② tá sù ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ❚❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ♥❣➢ê✐ ➤➲ t❐♥ t×♥❤ ❤➢í♥❣ ❞➱♥✱ ❣✐ó♣ ➤ì✱ ❝❤Ø ❜➯♦ t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ t❤ù❝ ❤✐Ư♥ ❧✉❐♥ ✈➝♥✳ ◆❤➞♥ ❞Þ♣ ♥➭② t➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❇❛♥ ❣✐➳♠ ❤✐Ö✉ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ P❤ß♥❣ ➜➭♦ t➵♦ s❛✉ ➜➵✐ ❤ä❝ tr➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❇❛♥ ❧➲♥❤ ➤➵♦ ❱✐Ö♥ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✲❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❝❤♦ t➠✐ tr♦♥❣ s✉èt t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣ ✈➭ t❤ù❝ ❤✐Ö♥ ❧✉❐♥ ✈➝♥✳ ❚➠✐ ①✐♥ tr➞♥ trä♥❣ ❝➯♠ ➡♥ ❝➳❝ t❤➬②✱ ❝➠ ❣✐➳♦ tr♦♥❣ t➠t ●✐➯✐ tÝ❝❤ t❤✉é❝ ❱✐Ö♥ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✲❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ➤➲ ❣✐➯♥❣ ❞➵② ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ❈✉è✐ ❝ï♥❣✱ t➠✐ ố ẹ ị ữ t❤➞♥ ❝ị♥❣ ♥❤➢ ❜➵♥ ❜❒ ➤➲ ❣✐ó♣ ➤ì✱ ➤é♥❣ ✈✐➟♥ t➠✐ tr♦♥❣ s✉èt t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉✳ ✐✐ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ▲ê✐ ♥ã✐ ➤➬✉ ❈❤➢➡♥❣ ■✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✶✳✶✳ ✶ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t②✲❈✐r✐❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤➢➡♥❣ ■■✳ ✽ ➜✐Ó♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✶✺ ✷✳✶✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✷✹ ❑Õt ❧✉❐♥ ✸✷ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✸ ✐ ▼ë ➤➬✉ ❚r♦♥❣ ❣✐➯✐ tÝ❝❤ ❤➭♠ ♣❤✐ t✉②Õ♥✱ ▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ➤❛♥❣ ♥❣➭② ợ q t ứ ì ó ó ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ ❦❤➠♥❣ ❝❤Ø tr♦♥❣ ♠ét sè ❝❤✉②➟♥ ♥❣➭♥❤ ❝đ❛ t♦➳♥ ❤ä❝✱ ❝➳❝ ♥❣➭♥❤ ❦ü t❤✉❐t ♠➭ ❝ß♥ tr♦♥❣ ❝➳❝ ♥❣➭♥❤ ✈Ò ❦✐♥❤ tÕ✳ ▼ét sè ❦Õt q✉➯ ✈Ò 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➵ ❝♦ ❇❛♥❛❝❤ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❦Õt q✉➯ ❝➡ së tr♦♥❣ ❣✐➯✐ tÝ❝❤ ♣❤✐ t✉②Õ♥ ✈➭ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ✈➭ ♥ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ❝➳❝ ♥❣➭♥❤ t♦➳♥ ❤ä❝ ✈➭ tr♦♥❣ ♥❤✐Ị✉ ♥❣➭♥❤ ❦❤♦❛ ❤ä❝ ø♥❣ ❞ơ♥❣✳ ◆❤✐Ị✉ ❦Õt q✉➯ t✐➟✉ ❜✐Ĩ✉ t❤❡♦ ❤➢í♥❣ ♠ë ré♥❣ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ➤➢ỵ❝ ❣✐í✐ t❤✐Ư✉ ❜ë✐ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ ❝ã t❤Ĩ ❦Ó ➤Õ♥ ♥❤➢✿ ▼✳ ❊❞❡❧st❡✐♥✱ ❚✳ ❙✉③✉❦✐✱ ❉✳ ❲✳ ❇♦②❞✱ ❙✳ ❲✳ ❲♦♥❣✱ ▼✳ ●❡r❛❣❤t②✱ ❘✳ ❑❛♥♥❛♥✱ ❙✳ ❘❡✐❝❤✱ ●✳ ❊✳ ❍❛r❞②✱ ❚✳ ❉✳ ❘♦❣❡rs✱ ▲✳ ❇✳ ❈✐r✐✬❝✱✳✳✳ ◆➝♠ ✶✾✼✸✱ ▼✳ ●❡r❛❣❤t② ✭❬✶✷❪✮ ➤➲ ➤➢❛ r❛ ❦Õt q✉➯ s❛✉ ➤➞② ♠➭ ♥ã ❧➭ ♠ét ♠ë (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ f : X → X ✳ ●✐➯ sö r➺♥❣ tå♥ t➵✐ ❤➭♠ β : [0, +∞) → [0, 1) t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✧♥Õ✉ β(tn ) → 1✱ t❤× tn → 0✧ s❛♦ ❝❤♦ ré♥❣ ❝đ❛ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✿ ❈❤♦ d(f (x), f (y)) ≤ β(d(x, y))d(x, y), f ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t n {f (x)} ❤é✐ tơ ✈Ị z ❦❤✐ n → ∞✳ ❑❤✐ ➤ã✱ z ∈X ✈í✐ ♠ä✐ x, y ∈ X ✈➭ ✈í✐ ♠ä✐ x∈X ❞➲② P✐❝❛r❞ ❚❤❡♦ ♠ét ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ❦❤➳❝✱ ❝➳❝ ♥❤➭ t♦➳♥ ❤ä❝ tr♦♥❣ ❧Ü♥❤ ✈ù❝ ♥➭② ➤➲ t×♠ ❝➳❝❤ ♠ë ré♥❣✱ ①➞② ❞ù♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠í✐ tỉ♥❣ q✉➳t ❤➡♥ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ s❛✉ ➤ã ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ➳♥❤ ①➵ ❝♦ ✈➭ ❝➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ♥ã tr➟♥ ❝➳❝ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠í✐ ♥➭②✳ ◆➝♠ ✶✾✾✹✱ tr♦♥❣ ❞ù ➳♥ ♥❣❤✐➟♥ ứ ề ể tị ữ t ❞÷ ❧✐Ư✉ tr➟♥ ♠➵♥❣ ♠➳② tÝ♥❤ ❙✳ ●✳ ▼❛tt❤❡✇s ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✭❬✶✹❪✮ ✈➭ t ợ ột tổ qt ủ ị ý ể ❜✃t ➤é♥❣ ❇❛♥❛❝❤ ❝❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ◆➝♠ ✷✵✶✵✱ ■✳ ❆❧t✉♥ ✈➭ ❝é♥❣ sù ➤➲ ➤➢❛ r❛ ♠ét sè ♣❤✐➟♥ ❜➯♥ tỉ♥❣ q✉➳t ❝đ❛ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ▼❛tt❤❡✇s ✭❬✺❪✮✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ t×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ●❡r❛❣❤t② ✈➭ ❝➳❝ ♠ë ré♥❣ ❝ñ❛ ♥ã tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❚×♠ ❤✐Ĩ✉ ✈➭ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❝ã ❤Ư t❤è♥❣✱ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❝❤♦ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛❀ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ➤Þ♥❤ ❧ý✱ ❤Ư q✉➯ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ❝❤ø♥❣ ♠✐♥❤ ❝ß♥ ✈➽♥ t➽t✳ ✐✐ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❧✉❐♥ ✈➝♥✱ ❜❛♦ ❣å♠ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤đ✳ ▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t②✲❈✐r✐❝ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ò ị ý ó r ò trì ột sè ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ụ ú t trì ột số ị ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ t❤ø tù ✈➭ ❝➳❝ ❤Ö q✉➯ ❝đ❛ ❝❤ó♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ➤ã ✈➭ tr×♥❤ ❜➭② ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ụ ú t trì ột số ị ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤ñ t❤ø tù✱ ❣✐í✐ t❤✐Ư✉ ♠ét sè ♥❤❐♥ ①Ðt ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ s✉② ré♥❣ ✈➭ tr×♥❤ ❜➭② ♠ét sè ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ◆❣❤Ư ❆♥✱ ♥❣➭② ✸✵ t❤➳♥❣ ✵✼ ♥➝♠ ✷✵✶✼ ◆❣✉②Ơ♥ ❚❤Þ ❍➺♥❣ ✐✐✐ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ❇❛♦ ❣å♠ ❝➳❝ ♥é✐ ❞✉♥❣✿ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ủ rì ột số ị ý ể t ộ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤đ✱ ♠ét sè ❤Ư q✉➯✱ tÝ♥❤ ❝❤✃t ❧✐➟♥ q✉❛♥ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s❛✉ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ X ♠➟tr✐❝ tr➟♥ ✭❬✶❪✮ ❈❤♦ t ợ X d : X ì X → R ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮ d(x, y) ≥ ✈í✐ ♠ä✐ x, y ∈ X ✭✷✮ 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✐❝ ✈➭ ❦Ý ❤✐Ö✉ ❧➭ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ ✈➭ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤✐Ó♠ x ➤Õ♥ ➤✐Ó♠ y ✳ ✶✳✶✳✷ ♠ä✐ ❱Ý ❞ô✳ ✶✮ ❳Ðt X = R✱ d : R × R → R ❝❤♦ ❜ë✐ d (x, y) = |x − y|✱ ✈í✐ 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 d1 (x, y) = |xi − yi |2 n ✈➭ i=1 ♠➟tr✐❝ tr➟♥ |xi − yi |✳ ❑❤✐ ➤ã d1 , d2 ❧➭ ❝➳❝ d2 (x, y) = i=1 Rn ✳ ✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✸ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❣ä✐ ❧➭ ❤é✐ tơ ✈Ị ➤✐Ĩ♠ n ≥ n0 t❛ ❝ã x∈X d (xn , x) < ε✳ ♥Õ✉ ✈í✐ ♠ä✐ (X, d)✳ {xn } ⊂ X ❉➲② ➤➢ỵ❝ ε > tå♥ t➵✐ n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ▲ó❝ ➤ã t❛ ❦Ý ❤✐Ư✉ ❧➭ lim xn = x ❤❛② xn → x ❦❤✐ n→∞ n → ∞✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✹ ✶✮ ❉➲② n ∈ N∗ ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ {xn } ⊂ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈í✐ ♠ä✐ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❈❛✉❝❤② ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ (X, d)✱ {xn } ⊂ X ✳ n, m ≥ n0 lim n,m→+∞ t❛ ❝ã d(xn , xm ) < ε✱ ε > 0✱ tå♥ t➵✐ {xn } ❧➭ ❞➲② ❤❛② d(xn , xm ) = 0✳ ✷✮ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤đ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ ♥ã ➤Ị✉ ❤é✐ tơ✳ ➜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) , ➜Þ♥❤ ❧ý✳ ✶✳✶✳✻ ➤➬② ➤đ✱ ➤✐Ĩ♠ ➜✐Ĩ♠ ①➵ ✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✮ ●✐➯ sö f :X→X x∗ ∈ X ✈í✐ ♠ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ tõ s❛♦ ❝❤♦ x∗ ∈ X X x, y ∈ X (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑❤✐ ➤ã tå♥ t➵✐ ❞✉② ♥❤✃t f (x∗ ) = x∗ ✳ ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ f ✳ ❚❛ ❦ý ❤✐Ư✉ F ix(f ) ❧➭ t❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵ f ✳ ✶✳✶✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✵❪✮ ❈❤♦ X = ∅✳ ❑❤✐ ➤ã✱ (X, p, ) ➤➢ỵ❝ ❣ä✐ ❧➭ ♠➟tr✐❝ t❤ø tù ♥Õ✉ ✭✐✮ ✭✐✐✮ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ (X, ) ❧➭ t❐♣ t❤ø tù ❜é ♣❤❐♥✳ ✷ ❦❤➠♥❣ ❣✐❛♥ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✽ ✭❬✶✶❪✮ ❛✳ ●✐➯ sư ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tô❝ tr➟♥ t➵✐ ❍➭♠ ♠ä✐ X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✳ ❍➭♠ x0 ∈ X ♥Õ✉ ψ:X→R lim sup ψ(x) ≤ ψ(x0 )✳ x→x0 ψ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ tr➟♥ X ♥Õ✉ ♥ã ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ t➵✐ x ∈ X✳ ❍➭♠ ψ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ tr➟♥ X tr➟♥✱ tr♦♥❣ ➤ã ♥Õ✉ ❤➭♠ −ψ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ (−ψ)(x) = −ψ(x) ✈í✐ ♠ä✐ x ∈ X ✳ ψ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❤➭♠ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ t➵✐ x0 ∈ X ♥Õ✉ lim inf ψ(x) ≥ ψ(x0 )✳ x→x0 ➜➠✐ ❦❤✐✱ t❛ ✈✐Õt lim ψ(x)✱ lim ψ(x) x→x0 ❧➬♥ ❧➢ỵt t❤❛② ❝❤♦ lim sup ψ(x) ✈➭ x→x0 x→x0 lim inf ψ(x)✳ x→x0 ❜✳ ❍➭♠ x0 ∈ X ♥Õ✉ ψ ✿ [0, +∞) → [0, +∞) ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tô❝ tr➟♥ ❜➟♥ ♣❤➯✐ t➵✐ lim sup ψ(x) ≤ ψ(x0 )✳ x→x0 + ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✾ ✭❬✶✹❪✮ ❈❤♦ t❐♣ ❤ỵ♣ ❧➭ ♠ét ♠➟tr✐❝ r✐➟♥❣ tr➟♥ X X = ∅✳ p : X ì X R+ ợ ọ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮ x = y ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ p(x, x) = p(x, y) = p(y, y) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✷✮ p(x, x) ≤ p(x, y) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✸✮ p(x, y) = p(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✹✮ p(x, y) ≤ p(x, z) + p(z, y) − p(z, z) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ❚❐♣ X = ∅ ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ r✐➟♥❣ p tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ♠➟tr✐❝ r✐➟♥❣ ✈➭ ❦Ý ❤✐Ö✉ ❧➭ ✶✳✶✳✶✵ ❱Ý ❞ô✳ ✶✮ ❈❤♦ ❣✐❛♥ (X, p)✳ X = R+ ✱ p : X ì X R+ ị p (x, y) = max {x, y}✱ ✈í✐ ♠ä✐ x, y ∈ X ✳ ❑❤✐ ➤ã✱ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❚❤❐t ✈❐②✱ t❛ t❤✃② p t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮✱ ✭✷✮✱ ✭✸✮ ❝đ❛ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✾✳ ▲✃② x, y, z ∈ X ❜✃t ❦ú✳ ❚❛ ①Ðt ❝➳❝ tr➢ê♥❣ ❤ỵ♣ s❛✉✳ ✸ ✰ ❚r➢ê♥❣ ❤ỵ♣ ✶✿ ◆Õ✉ x ≤ y ≤ z t❤× t❛ ❝ã✿ p(x, y) = y ≤ z + z − z = p(x, z) + p(z, y) − p(z, z)✳ ✰ ❈➳❝ tr➢ê♥❣ ❤ỵ♣ ❤ỵ♣ tr➟♥✳ ❱❐② x ≤ z ≤ y ✈➭ z ≤ x ≤ y ❝❤ø♥❣ ♠✐♥❤ t➢➡♥❣ tù tr➢ê♥❣ p t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✹✮ ❝đ❛ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✶✳✾✳ ❉♦ ➤ã (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ✷✮ ❈❤♦ X = R− = {x ∈ R : x ≤ 0}✱ ✈í✐ x, y ∈ R− ✱ t❛ ➤Þ♥❤ ♥❣❤Ü❛ p (x, y) = − {x, y}✳ ❑❤✐ ➤ã✱ p ❧➭ ♠ét ♠➟tr✐❝ r✐➟♥❣ tr➟♥ R− ✳ ✸✮ ❈❤♦ X = [0, 1]✱ ✈í✐ x, y ∈ X ✱ t❛ ➤Þ♥❤ ♥❣❤Ü❛ p (x, y) = emax{x,y} − t❤× p ❧➭ ♠ét ♠➟tr✐❝ r✐➟♥❣ tr➟♥ X ✳ ✶✳✶✳✶✶ sè ❇ỉ ➤Ị✳ ✭❬✶✹❪✮ ❈❤♦ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ❑❤✐ ➤ã ❝➳❝ ❤➭♠ ps , pw : X ì X [0, ) ợ ❜ë✐ ps (x, y) = 2p (x, y) − p (x, x) − p (y, y) , pw (x, y) = p (x, y) − min{p (x, x) , p (y, y)}, ✈í✐ ♠ä✐ ✶✳✶✳✶✷ x, y ∈ X ❧➭ ❝➳❝ ♠➟tr✐❝ t➢➡♥❣ ➤➢➡♥❣ tr➟♥ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✹❪✮ ❈❤♦ (X, p) X✳ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ x ∈ X ✈➭ ε > 0✳ ❚❛ ❦Ý ❤✐Ö✉ Bp (x, ε) = {y ∈ X : p (x, y) < p (x, x) + ε} ✈➭ ❣ä✐ Bp (x, ε) ❧➭ ì t r ị ý x ❦Ý♥❤ ε tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ (X, p)✳ ✭❬✶✹❪✮ ợ tt ì tr ♠➟tr✐❝ (X, p) ❧➭ ❝➡ së ❝ñ❛ ♠ét t➠♣➠ τp tr➟♥ X ✳ ❈❤ø♥❣ ♠✐♥❤✳ ❉Ô t❤✃② r➺♥❣ X = Bp (x, ε)✳ ●✐➯ sö Bp (x, ε) , Bp (y, δ) x∈X,ε>0 ❧➭ ❝➳❝ ❤×♥❤ ❝➬✉ ♠ë tï② ý tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ Bp (y, δ) = φ✳ ó ỗ (X, p) Bp (x, ) ∩ z ∈ Bp (x, ε) ∩ Bp (y, δ) t❛ ❝ã Bp (z, η) ⊂ Bp (x, ε) ∩ Bp (y, δ)✱ ✈í✐ η := p (z, z) + {ε − p (x, z) , δ − p (y, z)}✳ ✶✳✶✳✶✹ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶✹❪✮ ❈❤♦ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✈➭ ❞➲② {xn } ⊂ X ✳ ❑❤✐ ➤ã ✹ ❇➞② ❣✐ê✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❧➭ α✲❝❤Ý♥❤ {xnk } ❝ñ❛ {xn } s❛♦ ❝❤♦ α(xnk , z) ≥ 1✱ ✈í✐ ♠ä✐ k ∈ N✳ q✉②✱ ♥➟♥ tå♥ t➵✐ ❞➲② ❝♦♥ ◆Õ✉ z ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ f ✳ ❱× X p(z, f z) > 0✱ sư ❞ơ♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✮✱ ✈í✐ x = xnk ✈➭ y = z ✱ t❛ t❤✉ ➤➢ỵ❝ p(xnk +1 , f z) ≤ α(xnk , z)p(f xnk , f z) ≤ β(M (xnk , z))M (xnk , z) ❇➞② ❣✐ê✱ ✈í✐ k ➤đ ❧í♥✱ t❛ ❝ã M (xnk , z) = p(z, f z)✳ ❱× ✈❐②✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ tr➢í❝ ✈➭ ❇ỉ ➤Ị ✶✳✶✳✶✺✱ t❛ ♥❤❐♥ ➤➢ỵ❝ = lim p(xnk +1 , f z) k→+∞ ❚õ ➤ã t❛ ❝ã ❱× M (xnk , z) = lim β(M (xnk , z)) ≤ k→+∞ lim β(M (xnk , z)) = 1✳ k→+∞ β ∈ S ✱ t❛ s✉② r❛ lim M (xnk , z) = 0✳ ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ k→+∞ ❱× ✈❐②✱ t❛ ❝ã ➤é♥❣ ❝đ❛ p(z, f z) = ✈➭ ✈× t❤Õ t❛ ❝ã f z = z ✳ ❱❐②✱ z ❧➭ ♠ét ➤✐Ó♠ ❜✃t f✳ ❇➞② ❣✐ê ❣✐➯ sö r➺♥❣ u ✈➭ v ✱ ✈í✐ u = v ❧➭ ❝➳❝ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ f ✳ ❑❤✐ ➤ã✱ t❛ ❝ã < p(u, v) ≤ α(u, v)p(f u, f v) ≤ β(M (u, v))M (u, v) < M (u, v), tr♦♥❣ ➤ã M (u, v) = max p(u, v), p(u, f u), p(v, f v)), [p(u, f v)+p(f u, v)] = p(u, v) ❙✉② r❛ < p(u, v) < M (u, v) = p(u, v)✳ ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❉♦ ➤ã✱ t❛ ❝ã ➤é♥❣ ❝đ❛ u = v ✳ ❱× ✈❐② t❛ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ tÝ♥❤ ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t f✳ X = [0, 1]✱ d(x, y) = |x − y|✱ ✈í✐ ♠ä✐ x, y ∈ X ✱ e−t p(x, y) = max{x, y}✱ ✈í✐ ♠ä✐ x, y ∈ X (t) = ỗ t > ✈➭ (t + 1) ✷✳✶✳✻ ❱Ý ❞ô✳ ✭❬✷✵❪✮ ❈❤♦ ✷✶ β(0) = ✳ ➜➷t    α(x, y) =   ➳♥❤ ①➵ f : X → X ♥Õ✉ (x, y) = (0, 0) ♥Õ✉ (x, y) = (0, 0) ①➳❝ ➤Þ♥❤ ❜ë✐ f (x) = x ✈í✐ x ∈ X ❧➭ α✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ ♥❤➢♥❣ ♥ã ❦❤➠♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➤Þ♥❤ ❧ý ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d)✳ ❚❤❐t ✈❐②✱ ❜➺♥❣ ❝➳❝❤ ❧✃② x = ✈➭ y = 0✱ t❛ ❝ã 1 d(f 1, f 0) = d( , 0) = − = 3 ✈➭ β(d(1, 0))d(1, 0) = β(|1 − 0|)|1 − 0| = β(1) = ❱× > 2e 2e ị ý rt ợ sử ❞ơ♥❣ ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f✳ ▼➷t ❦❤➳❝✱ t❛ ➤Ĩ ý t❤✃② r➺♥❣ ➳♥❤ ①➵ f ❦❤➠♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✸✳✶ tr♦♥❣ ❬✶✵❪ ➤è✐ ✈í✐ ♠➟tr✐❝ r✐➟♥❣ ①➳❝ ➤Þ♥❤ ë tr➟♥✱ ❜ë✐ ✈× > = β(p(1, 0))p(1, 0) 2e ▼➷t ❦❤➳❝✱ ❜➺♥❣ ❝➳❝❤ ❧✃② x, y ∈ X ✱ ❝❤➻♥❣ ❤➵♥✱ x ≥ y ✈➭ x > 0✱ ❦❤✐ ➤ã t❛ ❝ã p(f 1, f 0) = M (x, y) = max{p(x, y), p(x, f x), p(y, f y), [p(x, f y) + p(f x, y)]} = x α(x, y)p(f x, f y) = x 12 ✈➭ β(M (x, y))M (x, y) = β(x)x = ❇➞② ❣✐ê✱ ✈× 12 < 2e ≤ e−x ✱ ✈í✐ ♠ä✐ x+1 e−x x x+1 x ∈ [0, 1]✱ t❛ t❤✃② r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✺✮ t❤á❛ ♠➲♥✳ ❉Ơ t❤✃② r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✐✮✲✭✐✈✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✺ t❤á❛ ♠➲♥✱ ✷✷ f ♥➟♥ ➳♣ ❞ơ♥❣ ➤Þ♥❤ ❧ý ♥➭② t❛ s✉② r❛ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✱ ➤ã ❝❤Ý♥❤ ❧➭ z = 0✳ ➜Þ♥❤ ❧ý s❛✉ ❝❤♦ t❛ ♠ét ❦Õt q✉➯ ✈Ị sù tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ tù ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ t❤ø tù✳ ✷✳✶✳✼ ➤đ✱ ✭❬✷✵❪✮ ❈❤♦ ➜Þ♥❤ ❧ý✳ (X, p, ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ t❤ø tù ➤➬② α : X × X → [0; +∞) ❧➭ ♠ét ❤➭♠ ✈➭ f : X → X ❣✐➯♠✳ ●✐➯ sö r➺♥❣ tå♥ t➵✐ β∈S ❧➭ ♠ét ➳♥❤ ①➵ ❦❤➠♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ p(f x, f y) ≤ β(M (x, y))M (x, y) ✈í✐ ♠ä✐ x, y ∈ X ♠➭ x ✭✷✳✼✮ y ✱ tr♦♥❣ ➤ã M (x, y) = max p(x, y), p(x, f x), p(y, f y), [p(x, f y) + p(f x, y)] ●✐➯ t❤✐Õt ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ ✭✐✮ ✭✐✐✮ ❚å♥ t➵✐ x0 ∈ X s❛♦ ❝❤♦ ◆Õ✉ ❞➲② ❦❤➠♥❣ ❣✐➯♠ {xn } s❛♦ ❝❤♦ xnk ✭✐✐✐✮ ❑❤✐ ➤ã✱ x0 f x0 ❀ {xn } ❤é✐ tơ tí✐ x✱ t❤× tå♥ t➵✐ ❞➲② ❝♦♥ {xnk } ❝đ❛ x✱ ✈í✐ ♠ä✐ k ∈ N❀ x✱ y ❧➭ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ ♥❤❛✉ ✈í✐ ❜✃t ❦ú x, y ∈ F ix(f )✳ f ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t ❈❤ø♥❣ ♠✐♥❤✳ ❳➳❝ ➤Þ♥❤ ➳♥❤ ①➵ α(x, y) = z ∈ X✳ α : X × X → [0; +∞) ế x ế tr trờ ợ ò ❚❛ ❝ã t❤Ĩ ❞Ơ ❞➭♥❣ t❤✃② r➺♥❣ f y, ❧➭ ợ ì ề ❦✐Ư♥ ✭✐✮ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✶✳✺ t❤á❛ ♠➲♥✳ ❚õ ➤✐Ị✉ ❦✐Ư♥ ✭✐✮ ë tr➟♥ t❛ s✉② r❛ ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✮ ủ ị ý ợ tỏ ❝❤♦ {xn } ❧➭ ♠ét ❞➲② tr♦♥❣ X s❛♦ α(xn , xn+1 ) ≥ 1✱ ✈í✐ ♠ä✐ n ∈ N ✈➭ xn → x ∈ X ✱ ❦❤✐ n → +∞✳ ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ α✱ t❛ ❝ã xn xn+1 ✱ ✈í✐ ♠ä✐ n ∈ N✳ ◆❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✮✱ tå♥ t➵✐ ❞➲② ✷✸ {xnk } ❝ñ❛ {xn } s❛♦ ❝❤♦ xnk ❝♦♥ ✈í✐ ♠ä✐ k ∈ N ✈➭ ❞♦ ➤ã✱ X m, n ∈ N ♠➭ m < n✳ ❧➭ x✱ ✈í✐ ♠ä✐ k ∈ N ✈➭ ✈× t❤Õ α(xnk , x) ≥ 1✱ α✲ ❝❤Ý♥❤ q✉②✳ ❍➡♥ ♥÷❛✱ α(xm , xn ) ≥ 1✱ ✈í✐ ♠ä✐ ❉♦ ➤ã ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✐✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✺ t❤á❛ ♠➲♥✳ ❇➺♥❣ ❝➳❝❤ ①Ðt t➢➡♥❣ tù t❛ t❤✃② r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✐✮ ❝đ❛ ➤Þ♥❤ ❧ý ❦Ð♦ t❤❡♦ ➤✐Ị✉ ❦✐Ư♥ ✭✐✈✮ ❝đ❛ ➜Þ♥❤ ý ì tế tết ủ ị ❧ý ✷✳✶✳✺ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ ❉Ơ t❤✃② r➺♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✺✮ ❝ị♥❣ t❤á❛ ♠➲♥ ✈× s❛♦ ❝❤♦ x y ✈➭ α(x, y) = ♥Õ✉ x α(x, y) = 1✱ ✈í✐ ♠ä✐ x, y ∈ X y ✳ ❉♦ ➤ã✱ ♥❤ê ➜Þ♥❤ ❧ý ✷✳✶✳✺✱ t❛ s✉② r❛ f ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ✷✳✷ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ 0✲➤➬② ➤đ t❤ø tù✱ ♠ét sè ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣ ✈➭ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ♠➟tr✐❝ r✐➟♥❣ ▼ét sè ♥❤❐♥ ①Ðt ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ s rộ ũ ợ trì ị ĩ ộ ➤➠✐ ❝ñ❛ ➳♥❤ ①➵ ❈❤♦ F ❦Ð♦ t❤❡♦ tn ✭❬✷✹❪✮ P❤➬♥ tư f : X × X −→ X ❧➭ ❧í♣ ❝➳❝ ❤➭♠ → 0✳ (x, y) ∈ X × X ♥Õ✉ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ f (x, y) = x ✈➭ f (y, x) = y ✳ θ : [0, ∞) → [0, 1) t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥ θ(tn ) → ◆➝♠ ✷✵✶✵ ❆✳ ❆♠✐♥✐✲❍❛r❛♥❞✐ ✈➭ trì ết q s ị ❧ý✳ ✈➭ ♠ét ➳♥❤ ①➵ ✭✐✮ ✭✐✐✮ ✭❬✻❪✮ ❈❤♦ (X, ) t s tứ tự ợ tr ị ột ♠➟tr✐❝ d f : X → X ✳ ●✐➯ sö r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥ (X, d) ❧➭ ➤➬② ➤đ✳ ✭✶✮ f ❧✐➟♥ tơ❝ ❤♦➷❝ ✷✹ ✭✷✮ {xn } ♥Õ✉ ❞➲② ❦❤➠♥❣ ❣✐➯♠ tr♦♥❣ X ❤é✐ tơ tí✐ ➤✐Ĩ♠ x ∈ X✱ t❤× x✱ ✈í✐ ♠ä✐ n ∈ N✳ xn ✭✐✐✐✮ f ✭✐✈✮ tå♥ t➵✐ x0 ∈ X ✭✈✮ tå♥ t➵✐ θ∈F ❧➭ ➳♥❤ ①➵ ❦❤➠♥❣ ❣✐➯♠✳ s❛♦ ❝❤♦ x0 f x0 ✳ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ♠➭ x y t❛ ❝ã d(f x, f y) ≤ θ(d(x, y))d(x, y) ❑❤✐ ➤ã f t➵✐ ♠ét ➤✐Ĩ♠ ✭✷✳✽✮ ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❍➡♥ ♥÷❛✱ ế ỗ zX s x z y (x, y) ∈ X tå♥ z ✱ t❤× t❛ ♥❤❐♥ ➤➢ỵ❝ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❚✐Õ♣ t ú t trì ột ị ý ể t ➤é♥❣ ❝❤♦ tr♦♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✸ ❈❤♦ ➳♥❤ ①➵ ✭❬✷✷❪✮ ❈❤♦ f :X→X 0✲ ➤➬② ➤ñ t❤ø tù✳ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤ñ t❤ø tù✳ ❧➭ ➳♥❤ ①➵ ❦❤➠♥❣ ❣✐➯♠ s❛♦ ❝❤♦ p(f x, f y) ≤ θ(p(x, y))p(x, y), ✈í✐ ♠ä✐ x, y ∈ X x0 ∈ X s❛♦ ❝❤♦ ✭✶✮ f ✭✷✮ ♥Õ✉ ❞➲② x0 x y ✭❤♦➷❝ y f x0 ✭❤♦➷❝ x0 x✮ x2 ∈ X θ ∈ F✳ ●✐➯ sö r➺♥❣ tå♥ t➵✐ f x0 ✮✳ ❚❛ ❝ò♥❣ ❣✐➯ t❤✐Õt r➺♥❣ f {xn } ❦❤➠♥❣ ❣✐➯♠ tr♦♥❣ X xn ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ s❛♦ ❝❤♦ ♠➭ ❤é✐ tơ tí✐ ➤✐Ĩ♠ x ∈ X ✱ t❤× xn x x)✱ ✈í✐ ♠ä✐ n ∈ N✳ w✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐➯ t❤✐Õt tå♥ t➵✐ x0 x1 ∈ X ✈➭ ✭✷✳✾✮ ❧✐➟♥ tô❝ ❤♦➷❝ ✭❤♦➷❝ ❑❤✐ ➤ã✱ ♠➭ (X, p) x1 = f x ✳ ❑❤✐ ➤ã✱ t❛ ❝ã✱ ∈ X s❛♦ ❝❤♦ x0 x1 x0 ✳ ▼ét ❝➳❝❤ t➢➡♥❣ tù✱ t❛ ❧✃② x2 = f x1 ✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭②✱ x2 = f x1 t❤❡♦ ❝➳❝❤ ♥➭② t❛ ❝ã ❞➲② {xn } tr♦♥❣ X f x0 ✳ ❚❛ ①➳❝ ➤Þ♥❤ f x0 = x1 ✳ ❚✐Õ♣ tơ❝ s❛♦ ❝❤♦ f xn = xn+1 , n ∈ N ✷✺ ✭✷✳✶✵✮ ◆Õ✉ tå♥ t➵✐ sè ✈➭ ✈× ✈❐② ➤✐Ĩ♠ n0 ∈ N s❛♦ ❝❤♦ xn0 +1 = xn0 ✱ t❤× t❛ ❝ã xn0 +1 = xn0 = f xn0 xn0 ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ f ✳ ❉♦ ➤ã✱ ✈✐Ư❝ ❝❤ø♥❣ ♠✐♥❤ tå♥ t➵✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❦Õt t❤ó❝✳ ●✐➯ sư xn+1 = xn ✱ ✈í✐ ♠ä✐ n ∈ N✳ ❱× ❤➭♠ f ❦❤➠♥❣ ❣✐➯♠✱ ♥➟♥ tõ ❝➳❝❤ ➤➷t tr➟♥ t❛ ♥❤❐♥ ➤➢ỵ❝ x0 f x = x1 f x = x2 xn = f xn−1 xn+1 = f xn ✭✷✳✶✶✮ ❚õ ✭✷✳✾✮✱ ✭✷✳✶✵✮ ✈➭ ✭✷✳✶✶✮✱ ✈í✐ ♠ä✐ n ≥ 1✱ t❛ ♥❤❐♥ ➤➢ỵ❝ p(xn+1 , xn+2 ) = p(f xn , f xn+1 ) ≤ θ(p(xn , xn+1 ))p(xn , xn+1 ) ≤ p(xn , xn+1 ) ❑❤✐ ➤ã✱ ❞➲② ✭✷✳✶✷✮ {p(xn , xn+1 )} ❧➭ ❞➲② ➤➡♥ ➤✐Ö✉ ❣✐➯♠✳ ❉♦ ➤ã✱ p(xn , xn+1 ) → c ≥ n → ∞✳ ●✐➯ sư r➺♥❣ c > 0✳ ❑❤✐ ➤ã✱ tõ ➤✐Ị✉ ❦✐Ö♥ ✭✷✳✾✮✱ t❛ ❝ã p(xn+1 , xn+2 ) ≤ θ(p(xn , xn+1 )) ✭✷✳✶✸✮ p(xn , xn+1 ) ❚õ ❝➠♥❣ t❤ø❝ ✭✷✳✶✸✮ t❛ s✉② r❛ θ(p(xn , xn+1 )) → ❦❤✐ n → ∞✳ ❚õ tÝ♥❤ ❝❤✃t ❝ñ❛ ❦❤✐ θ∈F s✉② r❛ r➺♥❣ p(xn , xn+1 ) → ❦❤✐ n → ∞ ❇➞② ❣✐ê✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❞➲② {xn } ❧➭ ❞➲② ✭✷✳✶✹✮ 0✲❈❛✉❝❤②✳ ●✐➯ sö ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ lim sup p(xn , xm ) > ✭✷✳✶✺✮ n,m→∞ ❚õ ➤✐Ị✉ ❦✐Ư♥ ✭✹✮ ❝đ❛ ♠➟tr✐❝ r✐➟♥❣ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✾✮✱ ✈í✐ n > m t❛ ❝ã p(xn , xm ) ≤ p(xn , xn+1 ) + p(xn+1 , xm+1 ) + p(xm+1 , xm ) − p(xm+1 , xm+1 ) − p(xn+1 , xn+1 ) ≤ p(xn , xn+1 ) + p(xm+1 , xm ) + θ(p(xn , xm ))p(xn , xm ) ≤ p(xn , xn+1 ) + p(xm+1 , xm ) − θ(p(xn , xm )) ✷✻ ✭✷✳✶✻✮ ❚õ ✭✷✳✶✹✮ ✈➭ ✭✷✳✶✺✮✱ t❛ ♥❤❐♥ ➤➢ỵ❝ lim n,m→∞ tõ ➤ã t❛ ❝ã − θ(p(xn , xm )) = ∞, ✭✷✳✶✼✮ lim θ(p(xn , xm )) ≥ 1✱ ➤✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ r➺♥❣ lim θ(p(xn , xm )) = n,m→∞ n,m→∞ 1✳ ì F t ợ lim p(xn , xm ) = 0✳ ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥✳ ❉♦ ➤ã✱ n,m→∞ ❞➲② {xn } ❧➭ ❞➲② 0✲❈❛✉❝❤②✳ ❱× (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤ñ✱ ♥➟♥ tå♥ t➵✐ w∈X s❛♦ ❝❤♦ xn → w tr♦♥❣ (X, p) ✈➭ p(w, w) = 0✳ ❍➡♥ ♥÷❛✱ t❛ ❝ã lim p(xn , w) = p(w, w) = ✭✷✳✶✽✮ n→∞ ❇➞② ❣✐ê t❛ sÏ ❝❤Ø r❛ r➺♥❣ ✰ ❚r➢ê♥❣ ❤ỵ♣ ✶✿ ◆Õ✉ f f w = w✳ ▼✉è♥ t❤Õ t❛ ét trờ ợ s tụ tì t ó f w = lim f xn = lim xn+1 = w n→∞ ❱× t❤Õ✱ t❛ ❝ã n→∞ ✭✷✳✶✾✮ f w = w✳ ✰ ❚r➢ê♥❣ ❤ỵ♣ ✷✿ ◆Õ✉ ✭✷✳✾✮ t❤á❛ ♠➲♥✱ t❤× t❛ ❝ã p(w, f w) ≤ p(w, xn+1 ) + p(xn+1 , f w) − p(xn+1 , xn+1 ) ≤ p(w, xn+1 ) + p(xn+1 , f w) ≤ p(w, xn+1 ) + θ(p(xn , w))p(xn , w) ≤ p(w, xn+1 ) + p(xn , w) ❇ë✐ ✈× ✭✷✳✷✵✮ p(xn , w) → ❦❤✐ n → ∞✱ ♥➟♥ t❛ ♥❤❐♥ ➤➢ỵ❝ f w = w✳ ❙❛✉ ➤➞② ❧➭ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✷✳✸ ✈➭ ❝ị♥❣ ❝❤Ø r❛ r➺♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✸ ❧➭ ♠ë ré♥❣ t❤ù❝ sù ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✷✳ ✷✳✷✳✹ ❱Ý ❞ơ ➤Þ♥❤ ❜ë✐ X X = [0, +∞) ∩ Q✱ ✈➭ ①Ðt p : X ì X R+ ợ p(x, y) = max{x, y}✱ ✈í✐ ♠ä✐ x, y ∈ X ✳ ❑❤✐ ➤ã✱ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ❚r➟♥ ✳ ✭❬✷✷❪✮ ❈❤♦ 0✲➤➬② ➤ñ✳ ◆❤➢♥❣ ♥ã ✈➱♥ ❝❤➢❛ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤đ✳ t❛ ①Ðt q✉❛♥ ❤Ư t❤ø tù ❜é ♣❤❐♥ ❝❤♦ ❜ë✐ x y ⇔ x = y; ❤♦➷❝ x, y ∈ [0, 1], ✷✼ ✈í✐ x ≤ y ✭✷✳✷✶✮ ❈❤♦ ❤➭♠ sè ❳➳❝ ➤Þ♥❤ ➳♥❤ ①➵ ●✐➯ sö r➺♥❣ θ(t) = (t + 1)−1 ✱ ✈í✐ ♠ä✐ t ≥ 0✳ ❑❤✐ ➤ã✱ râ r➭♥❣ r➺♥❣ θ ∈ F ✳ f : X → X ❝❤♦ ❜ë✐   x3    + x3 fx =     x2 ♥Õ✉ x ∈ [0, 1], ✭✷✳✷✷✮ ♥Õ✉ x > y ≤ x✳ ❑❤✐ ➤ã✱ t❛ ①Ðt ❤❛✐ tr➢ê♥❣ ❤ỵ♣ s❛✉✿ ✰ ❚r➢ê♥❣ ❤ỵ♣ ✶✿ ◆Õ✉ x ∈ [0, 1] (y ∈ [0, 1])✱ t❤× t❛ ❝ã p(f x, f y) = max y3 x3 , + x3 + y = x3 , + x3 p(x, y) = max{x, y} = x ✭✷✳✷✸✮ ❱× ✈❐②✱ t❛ ❝ã x x3 θ(p(x, y))p(x, y) − p(f x, f y) = − + x + x3 x − x2 ≥ = + x3 ❉♦ ➤ã✱ ✈í✐ ✭✷✳✷✹✮ x ∈ [0, 1]✱ t❤× p(f x, f y) ≤ θ(p(x, y))p(x, y)✳ ✰ ❚r➢ê♥❣ ❤ỵ♣ ✷✿ ◆Õ✉ x > (y = x)✱ t❤× t❛ ❝ã p(f x, f y) = max 1 , x2 y = , x2 p(x, y) = max{x, y} = x ✭✷✳✷✺✮ ❱× ✈❐②✱ t❛ ❝ã θ(p(x, y))p(x, y) − p(f x, f y) = x − 1+x x x3 − x − = = − k ≥ 0, x3 + x2 ✭✷✳✷✻✮ tr♦♥❣ ➤ã x2 + x + 0 1✱ t❤× p(f x, f y) ≤ θ(p(x, y))p(x, y)✳ ❍➡♥ ♥÷❛✱ tõ ✈✐Ư❝ ①Ðt ❚r➢ê♥❣ ❤ỵ♣ ✶ ✈➭ ❚r➢ê♥❣ ❤ỵ♣ ✷✱ râ r➭♥❣ r➺♥❣ ❝➯ ❤❛✐ ❣✐➯ t❤✐Õt ✭✶✮ ✈➭ ✭✷✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✸ ➤➢ỵ❝ t❤á❛ ♠➲♥✱ ✈➭ ✈í✐ x0 = 0✱ t❛ ❝ã x0 f x0 ✳ ❱× t❤Õ✱ t✃t ❝➯ tết ủ ị ý ề ợ tỏ ♠➲♥✳ ❇ë✐ ✈❐②✱ ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ w = 0✳ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ tr♦♥❣ ❱Ý ❞ơ ✷✳✷✳✹ t❛ ①Ðt ♠➟tr✐❝ t✐➟✉ ❝❤✉➮♥ ✈í✐ ♠ä✐ f x, y ∈ X ✱ t❤× ❜➺♥❣ ❝➳❝❤ ❧✃② x = ✈➭ d(x, y) = |x − y| y = 1✱ t❛ ❝ã (1/3)3 1 = 0.46 , − d(f , f 1) = + (1/3)3 1 d( , 1) = | − 1| = 0.66 3 ✭✷✳✷✽✮ ✈➭ ✈× ✈❐② t❛ ❝ã 1 d(f , f 1) = 0.46 > 0, = θ(d( , 1))d( , 1) 3 ❉♦ ➤ã✱ ❣✐➯ t❤✐Õt ✭✷✳✷✾✮ d(f x, f y) ≤ θ(d(x, y))d(x, y) ❦❤➠♥❣ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ ❑Õt q✉➯ s❛✉ ➤➞② tỉ♥❣ q✉➳t ✈➭ ♠ë ré♥❣ ➜Þ♥❤ ❧ý ✷✳✶ tr♦♥❣ ❬✽❪✳ ✷✳✷✳✺ ➜Þ♥❤ ❧ý✳ ➳♥❤ ①➵ ✭❬✷✷❪✮ ❈❤♦ F : X2 → X✳ (X, p) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤ñ ✈➭ ●✐➯ sư r➺♥❣✱ ✈í✐ ♠ä✐ x, y, a, b ∈ X ✈➭ θ ∈ F ✱ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ p(F (x, y), F (a, b)) ≤ θ((p(x, a) + p(y, b)) × 2−1 ) ((p(x, a) + p(y, b)) × 2−1 ) ❑❤✐ ➤ã✱ F ✭✷✳✸✵✮ ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❈❤ø♥❣ ♠✐♥❤✳ ❳Ðt ♠➟tr✐❝ P : X × X → [0, ∞) ❝❤♦ ❜ë✐ P (A, B) = p(x, a) + p(y, b), ∀A = (x, y), B = (a, b) ∈ X ✷✾ ✭✷✳✸✶✮ ❑❤✐ ➤ã✱ ♥Õ✉ (X , p) ➤➬② ➤đ ✭0✲➤➬② ➤đ✮✱ t❤× (X , P ) ❧➭ ➤➬② ➤ñ ✭t➢➡♥❣ ø♥❣ 0✲➤➬② ➤đ✮✳ ❇➞② ❣✐ê✱ t❛ ①➳❝ ➤Þ♥❤ ➳♥❤ ①➵ T : X → X ❝❤♦ ❜ë✐ T (A) = (F (x, y), F (y, x)), ∀A = (x, y) ∈ X ●✐➯ sö ✭✷✳✸✷✮ P : X × X → [0, ∞) ❧➭ ♠➟tr✐❝ tr➟♥ X ợ ị P (A, B) = P (A, B) , ❚õ ✭✷✳✸✵✮✱ ✈í✐ ♠ä✐ ✈í✐ ♠ä✐ A = (x, y), B = (a, b) ∈ X (x, y), (a, b) ∈ X ♠➭ x a ✈➭ y b✱ t❛ ❝ã p(F (x, y), F (a, b)) ≤ θ((p(x, a) + p(y, b)) × 2−1 ) ((p(x, a) + p(y, b)) × 2−1 ), p(F (b, a), F (y, x)) ≤ θ((p(x, a) + p(y, b)) × 2−1 ) ((p(x, a) + p(y, b)) × 2−1 ) ➜✐Ị✉ ♥➭② ❦Ð♦ t❤❡♦ r➺♥❣ ✈í✐ ♠ä✐ (x, y), (a, b) ∈ X ♠➭ x a ✈➭ y ✭✷✳✸✸✮ b✱ t❛ ❝ã {p(F (x, y), F (a, b)) + p(F (b, a), F (y, x))} × 2−1 ≤ θ((p(x, a) + p(y, b)) × 2−1 ) ((p(x, a) + p(y, b)) × 2−1 ), ✭✷✳✸✹✮ ♥❣❤Ü❛ ❧➭ t❛ ❝ã P (T (A), T (B)) ≤ θ(P (A, B))P (A, B) ❇✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❝❤Ý♥❤ ❧➭ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✾✮✱ ✈í✐ ♠ä✐ A ♠➭ A = (x, y) ✭✷✳✸✺✮ = (x, y), B = (a, b) ∈ X (a, b) = B ✳ ❱× ✈❐②✱ t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✸ ➤Ị✉ tỏ r trờ ợ ụ ị ý ✷✳✷✳✸✱ t❛ s✉② r❛ r➺♥❣ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣✳ ❑❤✐ ➤ã✱ tõ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ T ❝ã T ✱ t❛ ❧➵✐ s✉② r❛ x = F (x, y) ✈➭ y = F (y, x)✳ ➜✐Ò✉ ➤ã ❝ã ♥❣❤Ü❛ ❧➭ (x, y) ❧➭ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ F ✳ ✷✳✷✳✻ ❈❤ó ý ✭❬✷✷❪✮ ◆Õ✉ t❛ ❧✃② θ(t) = k ✱ ✈í✐ ♠ä✐ t ∈ [0, +∞)✱ ë ➤➞② ≤ k < tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✸✵✮✱ tì t ợ ết q ủ sr s t❤❛♠ ✭❬✼❪✮ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ✸✵ 0✲➤➬② ➤ñ✳ í ụ s ọ trờ ợ ị ý ụ ợ ị ý tr tì ụ ợ í ụ t❤ø❝ ✭❬✷✷❪✮ ❈❤♦ X = [0, 1] p : X R+ ợ ị p(x, y) = max{x, y} + |x − y|✱ ✈í✐ ♠ä✐ x, y ∈ X ✳ ❑❤✐ ➤ã✱ (X, p) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ 0✲➤➬② ➤ñ✱ ♥❤➢♥❣ ♥ã ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤đ✳ ❚❛ ①Ðt q✉❛♥ ❤Ư t❤ø tù tr➟♥ x, y ∈ X, x ❳Ðt ❤➭♠ ✈➭ ❤➭♠ X ♥❤➢ s❛✉ y ⇔ x = y ❤♦➷❝ (x, y) ∈ {(0, 0), (0, 1), (1, 1)} θ(t) = (t + 1)−1 ✱ ✈í✐ ♠ä✐ t ≥ 0✳ ➤Þ♥❤ ➳♥❤ ①➵ F : X2 → X ❝❤♦ ❜ë✐ ❑❤✐ ➤ã✱ râ r➭♥❣ r➺♥❣ F (x, y) = (y − x) ✱ ✈í✐ ♠ä✐ ✭✷✳✸✻✮ θ ∈ F ✳ ❚❛ ①➳❝ x, y ∈ X ✳ ❑❤✐ ➤ã t❛ ❝ã ❝➳❝ tr➢ê♥❣ ❤ỵ♣ s❛✉✳ ✰ ❚r➢ê♥❣ ❤ỵ♣ ✶✿ ◆Õ✉ ❤♦➷❝ (x, y) = (a, b) = (0, 0) ❤♦➷❝ (x, y) = (a, b) = (1, 1) (x, y) = (0, 0) ✈➭ (a, b) = (1, 1)✱ t❤× t❛ ❝ã✱ p(F (x, y), F (a, b)) = 0✳ ❱× ✈❐②✱ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✵✮ t❤á❛ ♠➲♥✳ (x, y) = (0, 0) ✈➭ (a, b) = (0, 1)✱ t❤× t❛ ❝ã p(F (0, 0), F (0, 1)) = < = θ((p(0, 0) + p(0, 1)) × 2−1 ) ✰ ❚r➢ê♥❣ ❤ỵ♣ ✷✿ ◆Õ✉ ((p(0, 0) + p(0, 1)) × 2−1 ) ✭✷✳✸✼✮ ❱× ✈❐②✱ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✵✮ t❤á❛ ♠➲♥✳ (x, y) = (0, 1) ✈➭ (a, b) = (1, 1)✱ t❤× t❛ ❝ã p(F (0, 1), F (1, 1)) = < 5 = θ((p(0, 1) + p(1, 1)) × 2−1 ) ✰ ❚r➢ê♥❣ ợ ế ((p(0, 1) + p(1, 1)) ì 21 ) ✭✷✳✸✽✮ ❉♦ ➤ã✱ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✸✵✮ t❤á❛ ♠➲♥✳ ❱× ✈❐②✱ t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✺ ➤Ị✉ t❤á❛ ♠➲♥ ✈➭ (0, 0) ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ F ✳ ❱× (X, p) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤đ✱ ♥➟♥ ➜Þ♥❤ ❧ý ✷✳✶ tr♦♥❣ ❬✽❪ ❦❤➠♥❣ ➳♣ ❞ơ♥❣ ➤➢ỵ❝✳ ✸✶ ❑Õt ❧✉❐♥ ❙❛✉ ♠ét t❤ê✐ ❣✐❛♥ t❐♣ tr✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉✱ ✈Ị ➤Ị t➭✐✿ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝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✐❝✱ ➳♥❤ ①➵ ❝♦✱ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✱ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠❡tr✐❝ r✐➟♥❣ t❤ø tù✱ ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣ ủ rì ột số ị ý ể ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ❈✐r✐❝ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ r✐➟♥❣✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ➤Þ♥❤ ❧ý ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ứ ứ ò s ợ ♥❤➢ ➜Þ♥❤ ❧ý ✶✳✷✳✶✱ ➜Þ♥❤ ❧ý ✷✳✶✳✹✱ ➜Þ♥❤ ❧ý ✷✳✶✳✺✱ ➜Þ♥❤ ❧ý ✷✳✶✳✼✱ ➜Þ♥❤ ❧ý ✷✳✷✳✸✱ ➜Þ♥❤ ❧ý ✷✳✷✳✺✳ ✹✳ ●✐í✐ t❤✐Ư✉ ❝❤✐ t✐Õt ❱Ý ❞ơ ✶✳✷✳✹✱ ❱Ý ❞ơ ✷✳✶✳✻✱❱Ý ❞ơ ✷✳✷✳✹✱❱Ý ❞ơ ✷✳✷✳✼✳ ✸✷ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ỗ t ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ❚✳ ❆❜❞❡❧❥❛✇❛❞✱ ❊✳ ❑❛r❛♣✐♥❛r✱ ❑✳ ❚❛s ✭✶✾✼✸✮✱ ✧❊①✐st❡♥❝❡ ❛♥❞ ✉♥✐q✉❡♥❡ss ♦❢ ❛ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt❡rs✱ ✷✹ ✭✶✶✮✱ ✶✾✵✵✲✶✾✵✹✳ ❬✸❪ ■✳ ❆❧t✉♥✱ ❆✳ ❊r❞✉r❛♥ ✭✷✵✶✶✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♠♦♥♦t♦♥❡ ♠❛♣♣✐♥❣s ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s ✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✳✱ ■❉ ✺✵✽✼✸✵✱ ✶✵ ♣❛❣❡s✳ ❬✹❪ ■✳ ❆❧t✉♥✱ ❑✳ ❙❛❞❛r❛♥❣❛♥✐ ✭✷✵✶✸✮✱ ✧●❡♥❡r❛❧✐③❡❞ ●❡r❛❣❤t② t②♣❡ ♠❛♣♣✐♥❣s ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥t r❡s✉❧ts✧✱ ❆r❛❜ ❏✳ ▼❛t❤✳✱ ✷✱ ✷✹✼✲✷✺✸✳ ❬✺❪ ■✳ ❆❧t✉♥✱ ❋✳ ❙♦❧❛✱ ❍✳ ❙✐♠s❡❦ ✭✷✵✶✵✮✱ ✧●❡♥❡r❛❧✐③❡❞ ❝♦♥tr❛❝t✐♦♥s ♦♥ ♣❛rt✐❛❧ ♠❡t✲ r✐❝ s♣❛❝❡s✧✱ ❚♦♣♦❧✳ ❆♣♣❧✳✱ ✶✺✼✱ ✷✼✼✽✲✷✼✽✺✳ ❬✻❪ ❆✳ ❆♠✐♥✐✲❍❛r❛♥❞✐✱ ❍✱ ❊♠❛♠✐ ✭✷✵✶✵✮✱ ✧❆ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r♠❡ ❢♦r ❝♦♥tr❛❝✲ t✐♦♥ t②♣❡ ♠➳♣ ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ t♦ ♦r❞✐✲ ♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s✿ ❚❤❡♦r②✱ ▼❡t❤♦❞s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✼✷ ✭✺✮✱ ✷✷✸✽✲✷✷✹✷✳ ❬✼❪ ❚✳ ●✳ ❇❤❛s❦❛r✱ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ✭✷✵✵✻✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ♣❛r✲ t✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✻✺ ✭✼✮✱ ✶✸✼✾✲✶✸✾✸✳ ❬✽❪ ❇✳ ❙✳ ❈❤♦✉❞❤✉r②✱ ❆✳ ❑✉♥❞✉ ✭✷✵✶✷✮✱ ✧❖♥ ❝♦✉♣❧❡❞ ❣❡♥❡r❛❧✐s❡❞ ❇❛♥❛❝❤ ❛♥❞ ❑❛♥♥❛♥ t②♣❡ ❝♦♥tr❛❝t✐♦♥s✧✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✺ ✭✹✮✱ ✷✺✾✲✷✼✵✳ ❬✾❪ ❍✳ P✳ ❈♦r♣♦r❛t✐♦♥ ✭✷✵✶✻✮✱ ✧❋✐①❡❞ ♣♦✐♥ts t❤❡♦r❡♠ ❢♦r ●❡r❛❣❤t②✲ ❚②♣❡ ❈♦♥✲ tr❛❝t✐✈❡ ▼❛♣♣✐♥❣s ❛♥❞ ❈♦✉♣❧❡❞ ❋✐①❡❞ P♦✐♥t ❘❡s✉❧ts ✐♥ ✵✲ ❈♦♠♣❧❡t❡ ❖r✲ ❞❡r❡❞ P❛rt✐❛❧ ▼❡tr✐❝ ❙♣❛❝❡s✧✱ ■♥t❡r✳ ❏✳ ❆♥❛❧✳✱ ■❉ ✽✾✹✼✵✷✵✱ ✺ ♣❛❣❡s✳ ✸✸ ❬✶✵❪ ❉✳ ❉✉❦✐❝✱ ❩✳ ❑❛❞❡❧❜✉r❣✱ ❙✳ ❘❛❞❡♥♦✈✐❝ ✭✷✵✶✶✮✱ ✧❋✐①❡❞ ♣♦✐♥ts ♦❢ ●❡r❛❣❤t②✲ t②♣❡ ♠❛♣♣✐♥❣s ✐♥ ✈❛r✐♦✉s ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❆❜str✳ ❆♣♣❧✳ ❆♥❛❧✳✱ ■❉ ✺✻✶✷✹✺✱ ✶✸ ♣❛❣❡s✳ ❬✶✶❪ ❘✳ ❊♥❣❡❧❦✐♥❣ ✭✶✾✼✼✮✱ ●❡♥❡r❛❧ ❚♦♣♦❧♦❣②✱ P❲◆✲P♦❧✐s❤✱ ❙❝✐❡♥t✐❢✐❝ P✉❜❧✐s❤❡rs✱ ❲❛rs③❛✇❛✳ ❬✶✷❪ ▼✳ ●❡r❛❣❤t② ✭✶✾✼✸✮✱ ✧❖♥ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✹✵✱ ✻✵✹✲✻✵✽✳ ❬✶✸❪ ❊✳ ❑❛r❛♣✐♥❛r✱ ■✳ ▼✳ ❊r❤❛♥ ✭✷✵✶✶✮✱ ✧❆ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❢♦r ♦♣❡r❛t♦rs ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt❡rs✱ ✷✹ ✭✶✶✮✱ ✶✽✾✹✲✶✽✾✾✳ ❬✶✹❪ ❙✳ ●✳ ▼❛tt❤❡✇s ✭✶✾✾✹✮✱ ✧P❛rt✐❛❧ ♠❡tr✐❝ t♦♣♦❧♦❣②✧✱ ❆♥♥✳ ◆❡✇ ❨♦r❦ ❆❝❛❞✳ ❙❝✐✳✱ ✼✷✽✱ ✶✽✸✲✶✾✼✳ ❬✶✺❪ ❉✳ P❛❡s❛♥♦✱ P✳ ❱❡tr♦ ✭✷✵✶✷✮✱ ✧❙✉③✉❦✐✬s t②♣❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❝♦♠♣❧❡t❡✲ ♥❡ss ❢♦r ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❢✐①❡❞ ♣♦✐♥ts ❢♦r ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❚♦♣♦❧✳ ❆♣♣❧✳✱ ✶✺✾✱ ✾✶✶✲✾✷✵✳ ❬✶✻❪ ❙✳ ❘❛❞❡♥♦✈✐❝✱ ❩✳ ❑❛❞❡❧❜✉r❣✱ ❉✳ ❏❛♥❞r❧✐❝✱ ❆✳ ❏❛♥❞r❧✐❝ ✭✷✵✶✷✮✱ ✧❙♦♠❡ r❡s✉❧ts ♦♥ ✇❡❛❦ ❝♦♥tr❛❝t✐✈❡ ♠❛♣s✧✱ ❇✉❧❧✳ ■r❛♥✐❛♥ ▼❛t❤✳ ❙♦❝✳✱ ✸✽ ✭✸✮✱ ✻✷✺✲✻✹✺✳ ❬✶✼❪ ❙✳ ❘♦♠❛❣✉❡r❛ ✭✷✵✶✵✮✱ ✧❆ ❑✐r❦ t②♣❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❝♦♠♣❧❡t❡♥❡ss ❢♦r ♣❛r✲ t✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ■❉ ✹✾✸✷✾✽✱ ✻ ♣❛❣❡s✳ ❬✶✽❪ ❙✳ ❘♦♠❛❣✉❡r❛ ✭✷✵✶✶✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❢♦r ❣❡♥❡r❛❧✐❞❡❞ ❝♦♥tr❛❝t✐♦♥s ♦♥ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❚♦♣♦❧✳ ❆♣♣❧✳✱ ✷✶✽✱ ✷✸✾✽✲✷✹✵✻✳ ❬✶✾❪ ❙✳ ❘♦♠❛❣✉❡r❛ ✭✷✵✶✶✮✱ ✧▼❛t❦♦✇s❦✐ ✐s t②♣❡ t❤❡♦r❡♠s ❢♦r ❣❡♥❡r❛❧✐❞❡❞ ❝♦♥tr❛❝✲ t✐♦♥s ♦♥ ✭♦r❞❡r❡❞✮ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❆♣♣❧✳ ●❡♥✳ ❚♦♣♦❧✳✱ ✶✷✱ ✷✶✸✲✷✷✵✳ ❬✷✵❪ ❱✳ ▲✳ ❘♦s❛✱ P✳ ❱❡tr♦ ✭✷✵✶✹✮✱ ✧❋✐①❡❞ ♣♦✐♥ts ❢♦r ●❡r❛❣❤t②✲ ❝♦♥tr❛❝t✐♦♥s ✐♥ ♣❛r✲ t✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✼✱ ✶✲✶✵✳ ✸✹ ❬✷✶❪ ❇✳ ❙❛♠❡t✱ ❈✳ ❱❡tr♦✱ P✳ ❱❡tr♦ ✭✷✵✶✷✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❢♦r α✲ψ ✲ ❝♦♥tr❛❝t✐✈❡ t②♣❡ ♠❛♣♣✐♥❣s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✺✱ ✷✶✺✹✲✷✶✻✺✳ ❬✷✷❪ ❊✳ ❨♦❧❛❝❛♥ ✭✷✵✶✻✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ●❡r❛❣❤t② ❝♦♥tr❛❝t✐✈❡ ♠❛♣✲ ♣✐♥❣s ❛♥❞ ❝♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t r❡s✉❧ts ✐♥ 0✲❝♦♠♣❧❡t❡ ♦r❞❡r❡❞ ♣❛rt✐❛❧ ♠❡tr✐❝ s♣❛❝❡s✧✱ ■♥t❡r✳ ❏✳ ❆♥❛❧✳✱ ■❉ ✽✾✹✼✵✷✵✱ ✺ ♣❛❣❡s✳ ❬✷✸❪ ❆✳ ❆♠✐♥✐✲❍❛r❛♥❞✐✱ ❍✳ ❊♠❛♠✐ ✭✷✵✶✵✮✱ ✧❆ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❢♦r ❝♦♥tr❛❝t✐♦♥ t②♣❡ ♠❛♣s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ t♦ ♦r❞✐♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✷ ✭✺✮✱ ✷✷✸✽✲✷✷✹✷✳ ❬✷✹❪ ❍✳ ❑✳ ◆❛s❤✐♥❡✱ ❇✳ ❙❛♠❡t✱ ❛♥❞ ❈✳ ❱❡tr♦ ✭✷✵✶✷✮✱ ✧❈♦✉♣❧❡❞ ❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥ts ❢♦r ❝♦♠♣❛t✐❜❧❡ ♠❛♣♣✐♥❣s s❛t✐s❢②✐♥❣ ♠✐①❡❞ ♠♦♥♦t♦♥❡ ♣r♦♣❡rt②✧✱ ❏✳ ◆♦♥❧✐♥✲ ❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✺✱ ✶✵✹✲✶✶✹✳ ✸✺