Điểm bất động bộ đôi và điểm trùng nhau bộ đôi của các ánh xạ co kiểu geraghty trong các không gian mê tric có thứ tự và khong gian s mêtric có thứ tự

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❱➝♥ P❤ó❝ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✾ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❱➝♥ P❤ó❝ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✽✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✾ ✐ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ ➤➞② ❧➭ ❝➠♥❣ tr×♥❤ ❦❤♦❛ ❤ä❝ ❝đ❛ r✐➟♥❣ t➠✐✳ ❈➳❝ ❦Õt q✉➯ tr♦♥❣ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ sư ❞ơ♥❣ ✈➭ trÝ❝❤ ❞➱♥ ❝❤Ý♥❤ ①➳❝✱ râ r➭♥❣✳ ◆❣❤Ö ❆♥✱ ♥❣➭② ✷✷ t❤➳♥❣ ✺ ♥➝♠ ✷✵✶✾ ◆❣➢ê✐ ❝❛♠ ➤♦❛♥ ◆❣✉②Ơ♥ ❱➝♥ P❤ó❝ ✐✐ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ✈➝♥ ♥➭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✳ ❳✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ q✉ý t❤➬② ❝➠ ë ❇é ♠➠♥ ●✐➯✐ ❚Ý❝❤✱ ❱✐Ư♥ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ✱ ❚❙✳ ◆❣✉②Ô♥ ❱➝♥ ➜ø❝ ❚r➢ë♥❣ ❜é ♠➠♥ ✈➭ ❝➳❝ ❚❤➬②✱ ❈➠ ❣✐➳♦ tr♦♥❣ ❇é ♠➠♥ ●✐➯✐ ❚Ý❝❤✱ ❱✐Ö♥ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❙ë ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ tØ♥❤ ◆❣❤Ö ❆♥✱ ❇❛♥ ●✐➳♠ ❤✐Ö✉ ❚r➢ê♥❣ ❚❍P❚ ◗✉ú♥❤ ▲➢✉ ■❱ ➤➲ ❣✐ó♣ ➤ì✱ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❝❤♦ t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ◆❤➞♥ ➤➞② t➠✐ ①✐♥ ❝➳♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ●✐➯✐ ❚Ý❝❤ ❦❤♦➳ ✷✺ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❣ë✐ ❧ê✐ ❝➳♠ ➡♥ ➤Õ♥ ❇è✱ ▼Đ✱ ❱ỵ ✈➭ ❝➳❝ ❛♥❤ ❡♠ tr♦♥❣ ì t ề ệ t ợ ú t ❤♦➭♥ t❤➭♥❤ ♥❤✐Ư♠ ✈ơ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ▼➷❝ ❞ï ➤➲ tÝ❝❤ ❝ù❝ ➤➬✉ t➢ ✈➭ ❝ã ♥❤✐Ò✉ ❝è ❣➽♥❣ tr♦♥❣ ♥❣❤✐➟♥ ❝ø✉✱ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✱ s♦♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ s❛✐ sãt✳ ❚➠✐ ♠♦♥❣ ♥❤❐♥ ợ ữ ý ế ó ó ủ qý ✈➭ ❜➵♥ ➤ä❝ ➤Ĩ ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥✳ ◆❣❤Ư ❆♥✱ ♥❣➭② ✷✷ t❤➳♥❣ ✺ ♥➝♠ ✷✵✶✾ ◆❣✉②Ơ♥ ❱➝♥ P❤ó❝ ✐✐✐ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✈ ▲ê✐ ♥ã✐ ➤➬✉ ❈❤➢➡♥❣ ■✳ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✶✳✶✳ ✶ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷✳ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤➢➡♥❣ ■■✳ ✶✸ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù ✷✺ ✷✳✶✳ ➜✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ②Õ✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ❑Õt ❧✉❐♥ ✹✸ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✹✹ ✐✈ ❉❛♥❤ ♠ơ❝ ❝➳❝ ❦Ý ❤✐Ư✉ {0, 1, 2, } N : ❚❐♣ ❤ỵ♣ ❝➳❝ sè tù ♥❤✐➟♥✱ ❤❛② N∗ : ❚❐♣ ❤ỵ♣ ❝➳❝ sè tù ♥❤✐➟♥ ❦❤➳❝ ✵✱ ❤❛② R : ❚❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝ R+ : ❚❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝ ❦❤➠♥❣ ➞♠✱ ❤❛② [a, b] : ➜♦➵♥ (a, b) : ❑❤♦➯♥❣ [a, b) : ◆ư❛ ❦❤♦➯♥❣ [a, b), ❤❛② t❐♣ ❤ỵ♣ {x ∈ R : a ≤ x < b} Φ : ❚❐♣ ❝➳❝ ❤➭♠ ϕ : R+ → R+ S : ❚❐♣ ❝➳❝ ❤➭♠ β : R+ → [0; 1) Θ : : [0, +∞) [a, b], ❤❛② t❐♣ ❤ỵ♣ {x ∈ R|a ≤ x ≤ b} (x, y) (a, b), ❤❛② t❐♣ ❤ỵ♣ {x ∈ R : a < x < b} (u, v) ❚❐♣ ❝➳❝ ❤➭♠ ♥Õ✉ x u ❦❤➠♥❣ ❣✐➯♠ ✈➭ : {sn }, {tn } s❛♦ ❝❤♦ ✈➭ y t❛ ❝ã v ❚❐♣ ❝➳❝ ❤➭♠ : ❦❤➠♥❣ ➞♠ ❜✃t ❦× t❤á❛ ♠➲♥ ✈➭ ✈í✐ ❤❛✐ ❞➲② sè ❦❤➠♥❣ ➞♠ ❜✃t ❦× θ(sn , tn ) → ⇒ sn , tn → θ : [0; ∞) × [0; ∞) → [0; 1) : ϕ(t) = ⇔ t = β(tn ) → ⇒ tn → θ : [0; ∞) × [0; ∞) → [0; 1) : θ(s, t) = θ(t, s) ∀s, t ∈ [0; ∞) Θ1 {1, 2, 3, } {sn }, {tn } t❛ ❝ã s❛♦ ❝❤♦ ✈í✐ ❤❛✐ ❞➲② sè θ(sn , tn ) → ⇒ sn , tn → ✈ ▼ë ➤➬✉ ■✳ ▲ý ❞♦ ❝❤ä♥ ➤Ị t➭✐ ▲ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝❤đ ➤Ị ♥❣❤✐➟♥ ❝ø✉ r✃t q✉❛♥ trä♥❣ ❝đ❛ ●✐➯✐ tÝ❝❤ ❚♦➳♥ ❤ä❝✳ ➜➞② ❝ị♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝➠♥❣ ❝ơ q✉❛♥ trä♥❣ ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❤✐Ư♥ t➢ỵ♥❣ ♣❤✐ t✉②Õ♥✳ ◆ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ ❚♦➳♥ ❤ä❝ ❝ò♥❣ ♥❤➢ ❝➳❝ ♥❣➭♥❤ ❦❤♦❛ ❤ä❝ ❦ü t❤✉❐t✳ ▼ét tr♦♥❣ ♥❤÷♥❣ ❦Õt q✉➯ q✉❛♥ trä♥❣ ❝đ❛ ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ➤➢ỵ❝ ➤Ị ①✉✃t ❜ë✐ ❇❛♥❛❝❤ ♥➝♠ ✶✾✷✷✳ ◆❣✉②➟♥ ❧ý ♥➭② ➤➲ trë t❤➭♥❤ ♠ét ❝➠♥❣ ❝ơ ♣❤ỉ ❞ơ♥❣ ➤Ĩ ❝❤ø♥❣ ♠✐♥❤ sù tå♥ t➵✐ ❞✉② ♥❤✃t ♥❣❤✐Ư♠ ❝đ❛ ❝➳❝ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥ ✈➭ ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ ❝ị♥❣ ♥❤➢ ❣✐➯✐ q✉②Õt ❝➳❝ ❜➭✐ t♦➳♥ ✈Ò sù tå♥ t➵✐ tr♦♥❣ ♥❤✐Ò✉ ♥❣➭♥❤ ❝đ❛ ●✐➯✐ tÝ❝❤ ❚♦➳♥ ❤ä❝ ✈➭ ➤➢ỵ❝ ø♥❣ ❞ơ♥❣ ✈➭♦ ❝➳❝ ♥❣➭♥❤ ❦❤♦❛ ❤ä❝ ❦❤➳❝✳ ❱× t❤Õ s❛✉ ➤ã ➤➲ ❝ã ♥❤✐Ò✉ ♥❤➭ t♦➳♥ ❤ä❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ t❤❡♦ ♥❤✐Ị✉ ❤➢í♥❣ ợ ề ết q tú ị ❈➳❝ ♠ë ré♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ❝➡ ❜➯♥ ♥➭② ❝❤♦ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✱ t❤➢ê♥❣ ➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ ❜➺♥❣ ❝➳❝❤ ➤✐Ị✉ ❝❤Ø♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝➡ ❜➯♥ ❤♦➷❝ t❤❛② ➤æ✐ ❦❤➠♥❣ ❣✐❛♥✳ ◆➝♠ ✶✾✼✸✱ ●❡r❛❣❤t② ➤➢❛ r❛ ♠ét ❦Õt q✉➯ ♥æ✐ t✐Õ♥❣ ♠➭ ♥ã ❧➭ sù ♠ë ré♥❣ ❝ñ❛ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✿ ❈❤♦ (X, d) f : X → X✳ β : [0, +∞) → [0, 1) ❦✐Ư♥ ✧♥Õ✉ ●✐➯ sư r➺♥❣ tå♥ t➵✐ ❤➭♠ f t❤á❛ ♠➲♥ ➤✐Ò✉ β(tn ) → 1✱ t❤× tn → 0✧ s❛♦ ❝❤♦ d(f (x), f (y)) ≤ β(d(x, y))d(x, y), ❑❤✐ ➤ã✱ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t {f n (x)} ❤é✐ tơ ✈Ị z ❦❤✐ z∈X ✈í✐ ♠ä✐ ✈➭ ✈í✐ ♠ä✐ x, y ∈ X x∈X ❞➲② ❧➷♣ P✐❝❛r❞ n → ∞✳ ◆➝♠ ✷✵✵✻✱ ❇❤❛s❦❛r ✈➭ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤Þ♥❤ ❧ý t❤ó ✈Þ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ tí t ệ ỗ ợ ó st ❈✐r✐❝ ♥➝♠ ✷✵✵✾ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ➳♥❤ ①➵ g ệ ỗ ợ t ợ ị ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ✈✐ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ♠➭ ❝❤ó♥❣ ❧➭ ♠ë ré♥❣ ❝đ❛ ❝➳❝ ➤Þ♥❤ ❧ý ❝đ❛ ❇❤❛s❦❛r ✈➭ ▲❛❦s❤♠✐❦❛♥t❤❛♠✳ ❙❛✉ ➤ã ♥❤✐Ò✉ ♥❤➭ t♦➳♥ ❤ä❝ ➤➲ t❤✐Õt ❧❐♣ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ư✉ t ì tí t g ệ ỗ ợ ❦Õt q✉➯ ♥➭② ❝ã t❤Ĩ ➤➢ỵ❝ ➳♣ ❞ơ♥❣ tr♦♥❣ ♥❤✐Ị✉ t×♥❤ ❤✉è♥❣ ❦❤➳❝ ♥❤❛✉ ✈➭ ♠ë r❛ ♥❤✐Ị✉ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈➭ ø♥❣ ❞ơ♥❣ tr♦♥❣ ♥❤✐Ị✉ ❧Ü♥❤ ✈ù❝✳ ▼➷t ❦❤➳❝✱ ♠ét sè ♥❤➭ t♦➳♥ ❤ä❝ ❦❤➳❝ ❝ò♥❣ ➤➲ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ◆➝♠ ✷✵✶✷✱ ❙❡❞❣❤✐ ✈➭ ❝é♥❣ sù ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❦❤➳✐ ♥✐Ư♠ ♥➭② ❧➭ ♠ë ré♥❣ ❝đ❛ ❦❤➠♥❣ tr r ọ ò ứ ợ ột sè tÝ♥❤ ❝❤✃t ✈➭ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤♦ tù ➳♥❤ ①➵ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝✳ ❙❛✉ ➤ã✱ ❙❡❞❣❤✐ ✈➭ ◆❣✉②Ơ♥ ❱➝♥ ❉ị♥❣ ❝ị♥❣ ➤➲ ứ ợ ột số ị ý ể t ộ tỉ♥❣ q✉➳t tr➟♥ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝✳ ◆➝♠ ✷✵✶✸✱ ◆❣✉②Ơ♥ ❱➝♥ ❉ị♥❣ ➤➲ sư ❞ơ♥❣ ❦❤➳✐ ♥✐Ư♠ ❝➷♣ ➳♥❤ ①➵ ệ ỗ ợ ế ể t ể ị ❧ý ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù ❜é ♣❤❐♥ ✈➭ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ❝❤Ý♥❤ tr♦♥❣ ❬✸✱ ✾❪ t❤➭♥❤ ❝✃✉ tró❝ ❝đ❛ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝✳ ➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✱ ❝❤ó♥❣ t➠✐ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥➭② ♥❤➺♠ t×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ✧➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✧✳ ■■✳ ▼ô❝ ➤Ý❝❤ ♥❣❤✐➟♥ ❝ø✉ ✲ ❚×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t②✱ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù ✲ ❚r×♥❤ ❜➭② ♠ét ❝➳❝❤ ❝ã ❤Ö t❤è♥❣✱ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ✈✐✐ ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✱ ❝❤♦ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ■■■✳ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ✲ ➜è✐ t➢ỵ♥❣ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù✱ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✱ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù✱ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ✈➭ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✳ ✲ P❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ♠è✐ q ệ ữ ố tợ tr ột số ị ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ■❱✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ✲ ❉ï♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❣✐➯✐ tÝ❝❤✱ t➠♣➠✱ ❣✐➯✐ tÝ❝❤ ❤➭♠✳ ✲ ❙ư ❞ơ♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❧ý t❤✉②Õt✱ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ t➭✐ ❧✐Ư✉ ✈➭ sư ❞ơ♥❣ ♠ét sè ❦ü t❤✉❐t ❝❤ø♥❣ ♠✐♥❤ ♠í✐ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ✈✃♥ ➤Ò ➤➷t r❛✳ ❱✳ ◆é✐ ❞✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ✲ ◆❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ●❡r✲ ❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤♦➵✳ ✲ ◆❣❤✐➟♥ ❝ø✉ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤é✐ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ✲ ◆❣❤✐➟♥ ❝ø✉ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ❝➳❝ ❤Ư q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤é✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ✲ ◆❣❤✐➟♥ ❝ø✉ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ✈✐✐✐ ❱■✳ ❈✃✉ tró❝ ❧✉❐♥ ✈➝♥ ▲✉❐♥ ✈➝♥ ❣å♠ ✷ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù✳ ▼ơ❝ ✶✳✶✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù P❤➬♥ ♥➭② ❝❤ó♥❣ t trì ột số ị ý ể t ộ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù✱ ❝➳❝ ❤Ö q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ▼ơ❝ ✶✳✷✳ ➜✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤é✐ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉✱ ❝➳❝ ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✳ ▼ơ❝ ✷✳✶✳ ➜✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❝❤♦ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ▼ô❝ ✷✳✷✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ②Õ✉ P ú t trì ột số ị ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❤Ư q✉➯ ✈➭ ❝➳❝ ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ S ✲♠➟tr✐❝ ❝ã t❤ø tù✱ ❝➳❝ ✸✶ dn = max{S(g(xn , yn ), g(xn , yn ), g(xn+1 , yn+1 )), S(g(yn , xn ), g(yn , xn ), g(yn+1 , xn+1 ))} dn → d ❦❤✐ n → ∞✱ ✈í✐ d ≥ 0✳ ❚✐Õ♣ t❤❡♦✱ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ d = 0✳ ❚❤❐t ✈❐②✱ ❣✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ d > 0✳ ❑❤✐ ➤ã✱ tõ ✭✷✳✶✵✮ ❧➭ ❞➲② ❦❤➠♥❣ t➝♥❣✳ ❚õ ➤ã s✉② r❛ t❛ ♥❤❐♥ ➤➢ỵ❝ max{S(g(xn+1 , yn+1 ), g(xn+1 , yn+1 ), g(xn+2 , yn+2 )), S(g(yn+1 , xn+1 ), g(yn+1 , xn+1 ), g(yn+2 , xn+2 ))} max{S(g(xn , yn ), g(xn , yn ), g(xn+1 , yn+1 )), S(g(yn , xn ), g(yn , xn ), g(yn+1 , xn+1 ))} ≤ θ(S(g(xn , yn ), g(xn , yn ), g(xn+1 , yn+1 )), S(g(yn , xn ), g(yn , xn ), g(yn+1 , xn+1 ))) < ▲✃② ❣✐í✐ ❤➵♥ tr♦♥❣ ❝➳❝ ✈Õ ❝ñ❛ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❦❤✐ n → ∞✱ t❛ ➤➢ỵ❝ θ(S(g(xn , yn ), g(xn , yn ), g(xn+1 , yn+1 )), S(g(yn , xn ), g(yn , xn ), g(yn+1 , xn+1 ))) → ❱× θ ∈ Θ✱ ♥➟♥ t❛ ❝ã S(g(xn , yn ), g(xn , yn ), g(xn+1 , yn+1 ) → ✈➭ ❦❤✐ ❉♦ ➤ã✱ ➤ã✱ S(g(yn , xn ), g(yn , xn ), g(yn+1 , xn+1 ) → n → ∞✳ dn → ❦❤✐ n → ∞✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ t❤✐Õt d > 0✳ ❉♦ d = 0✱ tø❝ ❧➭ dn = max {S(g(xn , yn ), g(xn , yn ), g(xn+1 , yn+1 )), S(g(yn , xn ), g(yn , xn ), g(yn+1 , xn+1 ))} → 0, ❦❤✐ n → ∞✳ ✭✷✳✶✶✮ {g(xn , yn ), (yn , xn )} ❧➭ ❞➲② tr X ì max S tr Ds ợ ①➳❝ ➤Þ♥❤ tr♦♥❣ ❇ỉ ➤Ị ✷✳✶✳✻✳ max )✱ ◆Õ✉ {g(xn , yn ), g(yn , xn )} ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ (X×X, Ds ❇➞② ❣✐ê t❛ sÏ ❝❤ø♥❣ tá r➺♥❣ X ε > 0✱ s❛♦ ❝❤♦ ✈í✐ ♥ã t ó tế tì ợ số {mk } ✈➭ {nk } ➤Ĩ ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ k t❤á❛ ♠➲♥ nk > mk > k ✱ t❛ ❝ã t❤× tå♥ t➵✐ sè Dsmax ((g(xnk , ynk ), g(ynk , xnk )), (g(xnk , ynk ), g(ynk , xnk )), (g(xmk , ymk ), g(ymk , xmk ))) ≥ ε ✈➭ Dsmax ((g(xnk −1 , ynk −1 ), g(ynk −1 , xnk −1 )), (g(xnk −1 , ynk −1 ), g(ynk −1 , xnk −1 )), ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ Dsmax ✱ t❛ ❝ã (g(xmk , ymk ), g(ymk , xmk ))) < ε rk := max {S(g(xnk , ynk ), g(xnk , ynk ), g(xmk , ymk )), S(g(ynk , xnk ), g(ynk , xnk ), g(ymk , xmk ))} ≥ ε ✭✷✳✶✷✮ ✸✷ ✈➭ max{S(g(xnk −1 , ynk −1 ), g(xnk −1 , ynk −1 ), g(xmk , ymk )), ✭✷✳✶✸✮ S(g(ynk −1 , xnk −1 ), g(ynk −1 , xnk −1 ), g(ymk , xmk ))} < ε ❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ✭✷✳✶✷✮✱ ✭✷✳✶✸✮ ✈➭ ❇ỉ ➤Ị ✷✳✶✳✸✱ t❛ ❝ã ε ≤ rk := max{S(g(xnk , ynk ), g(xnk , ynk ), g(xmk , ymk )), S(g(ynk , xnk ), g(ynk , xnk ), g(ymk , xmk ))} ≤ max{S(g(xnk −1 , ynk −1 ), g(xnk −1 , ynk −1 ), g(xmk , ymk )), S(g(ynk −1 , xnk −1 ), g(ynk −1 , xnk −1 ), g(ymk , xmk ))} + max{2S(g(xnk −1 , ynk −1 ), g(xnk −1 , ynk −1 ), g(xnk , ynk )), 2S(g(ynk −1 , xnk −1 ), g(ynk −1 , xnk −1 ), g(ynk , xnk ))} < max{S(g(xnk −1 , ynk −1 ), g(xnk −1 , ynk −1 ), g(xnk , ynk )), S(g(ynk −1 , xnk −1 ), g(ynk −1 , xnk −1 ), g(ynk , xnk ))} + ε ▲✃② ❣✐í✐ ❤➵♥ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❦❤✐ k → ∞✱ t❛ ❝ã rk = max{S(g(xnk , ynk ), g(xnk , ynk ), g(xmk , ymk )), ✭✷✳✶✹✮ S(g(ynk , xnk ), g(ynk , xnk ), g(ymk , xmk ))} → ε ◆❤ê ❇ỉ ➤Ị ✷✳✶✳✸✱ t❛ ❝ã ε ≤ rk := max{S(g(xnk , ynk ), g(xnk , ynk ), g(xmk , ymk )), S(g(ynk , xnk ), g(ynk , xnk ), g(ymk , xmk ))} ≤ max{2S(g(xnk , ynk ), g(xnk , ynk ), g(xnk +1 , ynk +1 )), 2S(g(ynk , xnk ), g(ynk , xnk ), g(ynk +1 , xnk +1 ))} + max{S(g(xnk +1 , ynk +1 ), g(xnk +1 , ynk +1 ), g(xmk , ymk )), S(g(ynk +1 , xnk +1 ), g(ynk +1 , xnk +1 ), g(ymk , xmk ))} < 2dnk + max{2S(g(xmk , ymk ), g(xmk , ymk ), g(xmk +1 , ymk +1 )), 2S(g(ymk , xmk ), g(ymk , xmk ), g(ymk +1 , xmk +1 ))} + max{S(g(xnk +1 , ynk +1 ), g(xnk +1 , ynk +1 ), g(xmk +1 , ymk +1 )), S(g(ynk +1 , xnk +1 ), g(ynk +1 , xnk +1 ), g(ymk +1 , xmk +1 ))} ✸✸ = 2dnk + 2dmk + max{S(f (xnk , ynk ), f (xnk , ynk ), f (xmk , ymk )), S(f (ynk , xnk ), f (ynk , xnk ), f (ymk , xmk ))} < 2dnk + 2dmk + θ(S(g(xnk , ynk ), g(xnk , ynk ), g(xmk , ymk )), S(g(ynk , xnk ), g(ynk , xnk ), g(ymk , xmk ))) × max{S(g(xnk , ynk ), g(xnk , ynk ), g(xmk , ymk )), S(g(ynk , xnk ), g(ynk , xnk ), g(ymk , xmk ))} ≤ 2dnk + 2dmk + rk ▲✃② ❣✐í✐ ❤➵♥ tr♦♥❣ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❦❤✐ k → ∞ ✈➭ sư ❞ơ♥❣ ✭✷✳✶✶✮ ✈➭ ✭✷✳✶✹✮✱ t❛ ➤➢ỵ❝ θ(S(g(xnk , ynk ), g(xnk , ynk ), g(xmk , ymk )), S(g(ynk , xnk ), g(ynk , xnk ), g(ymk , xmk ))) → ❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ θ✱ t❛ ♥❤❐♥ ➤➢ỵ❝ S(g(xnk , ynk ), g(xnk , ynk ), g(xmk , ymk )) → 0, ✈➭ S(g(ynk , xnk ), g(ynk , xnk ), g(ymk , xmk )) → ❦❤✐ k → ∞✳ ❚õ ➤ã✱ t❛ s✉② r❛ lim rk = 0✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ε > 0✳ k→∞ ❱× t❤Õ✱ ❞➲② {g(xn , yn ), (yn , xn )} ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ (X × X, Dsmax )✳ ❱× X ❧➭ ❦❤➠♥❣ ❣✐❛♥ S ✲➤➬② ➤đ✱ ♥➟♥ t❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✾✱ tå♥ t➵✐ (u, v) ∈ X × X s❛♦ ❝❤♦ lim g(xn , yn ) = lim f (xn , yn ) = u n→∞ ▲➵✐ ✈× n→∞ ✈➭ lim g(yn , xn ) = lim f (yn , xn ) = v n→∞ n→∞ ✭✷✳✶✺✮ (f, g) ❧➭ ❝➷♣ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ s✉② ré♥❣✱ ♥➟♥ tõ ✭✷✳✶✺✮ t❛ ♥❤❐♥ ➤➢ỵ❝ lim S(f (g(xn , yn ), g(yn , xn ), f (g(xn , yn ), g(yn , xn ), g(f (xn , yn ), f (yn , xn ))) = 0, n→∞ ✭✷✳✶✻✮ ✈➭ lim S(f (g(yn , xn ), g(xn , yn ), f (g(yn , xn ), g(xn , yn ), g(f (yn , xn ), f (xn , yn ))) = n→∞ ❇➞② ❣✐ê t❛ ❣✐➯ sö r➺♥❣ ❣✐➯ t❤✐Õt ✭✺✮✭❛✮ t❤á❛ ♠➲♥✱ ♥❣❤Ü❛ ❧➭ f, g ❧➭ ❝➳❝ ➳♥❤ ①➵ ❧✐➟♥ tơ❝✳ ❑❤✐ ➤ã✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❇ỉ ➤Ị ✷✳✶✳✸ t❛ ♥❤❐♥ ➤➢ỵ❝ S(g(u, v),g(u, v), f (g(xn , yn ), g(yn , xn ))) ≤ 2S(g(u, v), g(u, v), g(f (xn , yn ), f (yn , xn ))) + S(f (g(xn , yn ), g(yn , xn )), f (g(xn , yn ), g(yn , xn )), g(f (xn , yn ), f (yn , xn ))) ✸✹ ❇➺♥❣ ❝➳❝❤ ❝❤✉②Ĩ♥ q✉❛ ❣✐í✐ ❤➵♥ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ❦❤✐ ❞ô♥❣ ✭✷✳✶✺✮✱ ✭✷✳✶✻✮ ✈➭ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ n → ∞ ✈➭ sư f ✱ t❛ ➤➢ỵ❝ S(g(u, v), g(u, v), f (u, v)) = ❉♦ ➤ã✱ t❛ ❝ã g(u, v) = f (u, v)✳ ❚➢➡♥❣ tù t❛ ❝ò♥❣ ❝ã g(v, u) = f (v, u)✳ ❇➞② ❣✐ê t❛ ❣✐➯ sö r➺♥❣ ❣✐➯ t❤✐Õt ✭✺✮✭❛✮ t❤á❛ ♠➲♥✳ ❑❤✐ ➤ã✱ ♥❤ê ✭✷✳✼✮ ✈➭ ✭✷✳✶✸✮✱ {g(xn , yn )} ✈➭ {g(yn , xn )} ❧➭ ❤❛✐ ❞➲② ❦❤➠♥❣ ❣✐➯♠✱ g(xn , yn ) → u g(yn , xn ) → v ❦❤✐ n → ∞✳ ❉♦ ➤ã ✈í✐ ♠ä✐ n ∈ N✱ t❛ ❝ã t❛ ❝ã g(xn , yn ) u, g(yn , xn ) ✈➭ v ❇➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ✭✷✳✹✮ ✈➭ ❇ỉ ➤Ị ✷✳✶✳✸ t❛ ♥❤❐♥ ➤➢ỵ❝ S(f (u, v),f (u, v), g(u, v)) ≤ 2S(f (u, v), f (u, v), g(xn+1 , yn+1 )) + S(g(xn+1 , yn+1 ), g(xn+1 , yn+1 ), g(u, v)) = 2S(f (u, v), f (u, v), f (xn , yn )) + S(g(xn+1 , yn+1 ), g(xn+1 , yn+1 ), g(u, v)) → 0, n → ∞✳ f (v, u)✳ ❦❤✐ ❉♦ ➤ã✱ t❛ ❝ã g(u, v) = f (u, v)✳ ❚➢➡♥❣ tù t❛ ❝ò♥❣ ❝ã g(v, u) = ❈❤ó ý r➺♥❣ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ✭✺✮✭❜✮ ❣✐➯ t❤✐Õt ✈Ị tÝ♥❤ ❧✐➟♥ tơ❝ ✈➭ tÝ♥❤ t➢➡♥❣ t❤Ý❝❤ s✉② ré♥❣ ❧➭ ❦❤➠♥❣ ❝➬♥ t❤✐Õt tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ②Õ✉ P❤➬♥ ú t trì ột số ị ý ể ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù ✈➭ tr×♥❤ ❜➭② ❝➳❝ ❤Ư q✉➯ ❝ï♥❣ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤♦➵✳ (X, ) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ ❤❛✐ ➳♥❤ ①➵ f, g : X × X → X ✳ ❚❛ ♥ã✐ r➺♥❣ ❝➷♣ (f, g) ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ②Õ✉ tr➟♥ X ♥Õ✉ ✈í✐ ♠ä✐ x, y ∈ X ✱ t❛ ❝ã ✷✳✷✳✶ ✭❛✮ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✾❪✮ ❈❤♦ x ≤ f (x, y), f (y, x) ≤ y f (x, y) ✭❜✮ g(f (x, y), f (y, x)), g(f (y, x), f (x, y)) x ≤ g(x, y), g(y, x) ≤ y g(x, y) ❦Ð♦ t❤❡♦ f (y, x), ❦Ð♦ t❤❡♦ f (g(x, y), g(y, x)), f (g(y, x), g(x, y)) g(y, x) ✸✺ ❱Ý ❞ơ✳ ✷✳✷✳✷ ✭❬✾❪✮ ●✐➯ sư f, g : R × R → R ❧➭ ❤❛✐ ➳♥❤ ①➵ ①➳❝ ➤Þ♥❤ ❜ë✐ f (x, y) = x − 2y, g(x, y) = x − y ❑❤✐ ➤ã✱ ❝➷♣ (f, g) ó tí t ệ ỗ ợ ế ✳ ) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ f : X ×X → X ❧➭ ♠ét ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr➟♥ X ✳ ❑❤✐ ➤ã✱ ✈í✐ ♠ä✐ n ∈ N✱ ❝➷♣ (f n , f n ) ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ②Õ✉ tr➟♥ X ✳ ✭❬✼❪✮ ❈❤♦ (X, ❈❤ó ý✳ ✷✳✷✳✸ X t rỗ f, g : X × X → X ❧➭ ❤❛✐ ➳♥❤ ①➵✳ P❤➬♥ tö (x, y) X ì X ợ ọ ể ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ f ✈➭ g ♥Õ✉ x = f (x, y) = g(x, y) ✈➭ y = f (y, x) = g(y, x)✳ ✷✳✷✳✹ ➜Þ♥❤ ♥❣❤Ü❛✳ ✷✳✷✳✺ ❑Ý ❤✐Ư✉✳ ✭❬✶✻❪✮ ❈❤♦ Θ1 ✭❬✶✻❪✮ ❑Ý ❤✐Ư✉ θ : [0; ∞) × [0; ∞) → [0; 1) ❦❤➠♥❣ ➞♠ ❜✃t ❦× {sn } ✈➭ {tn } ♥Õ✉ ❧➭ ❧í♣ ❝➳❝ ❤➭♠ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥✿ ✈í✐ ❤❛✐ ❞➲② sè t❤ù❝ θ(sn , tn ) → tì sn , tn ị ý ✈➭ ✭❬✶✻❪✮ ●✐➯ sư f, g : X × X → X ✭✶✮ X (X, S) ❧➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ s➽♣ t❤ø tù ❜é ♣❤❐♥ ❧➭ ❤❛✐ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ❀ ✭✷✮ ❈➷♣ (f, g) ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ②Õ✉ tr➟♥ X ✈➭ tå♥ t➵✐ x , y0 ∈ X s❛♦ ❝❤♦ x0 f (x0 , y0 ), f (y0 , x0 ) ✭✸✮ ●✐➯ sö r➺♥❣ tå♥ t➵✐ y0 ❤♦➷❝ x0 g(x0 , y0 ), g(y0 , x0 ) y0 ; θ ∈ Θ1 s❛♦ ❝❤♦ S(f (x, y), f (x, y),g(u, v)) + S(f (y, x), f (y, x), g(v, u)) ✭✷✳✶✼✮ ≤ θ(S(x, x, u), S(y, y, v))[S(x, x, u) + S(y, y, v)], ✈í✐ ♠ä✐ ✭✹✮ f ❤♦➷❝ x, y, u, v ∈ X x u, y v❀ g ❧✐➟♥ tô❝✳ ❑❤✐ ➤ã✱ f ✈➭ ❈❤ø♥❣ ♠✐♥❤✳ sö x0 , y0 ♠➭ g ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ tr♦♥❣ X ✳ ❇➢í❝ ✶✳ ∈ X s❛♦ ❝❤♦ x0 X ✳ ❚❤❐t ✈❐②✱ ❣✐➯ f (y0 , x0 )✳ ➜➷t x1 = f (x0 , y0 ), y1 = ❚❛ ①➞② ❞ù♥❣ ❤❛✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ f (x0 , y0 ), y0 ✸✻ f (y0 , x0 ), x2 = g(x1 , y1 ), y2 = g(y1 , x1 )✳ ❑❤✐ ➤ã✱ tõ ❣✐➯ t❤✐Õt ✈Ị sù tå♥ t➵✐ ❝đ❛ x0 , y0 ∈ X ✈➭ ❝➷♣ (f, g) ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ②Õ✉ t❛ ❝ã x1 = f (x0 , y0 ) g(f (x0 , y0 ), f (y0 , x0 )) = g(x1 , y1 ) = x2 ❉♦ ➤ã t❛ ❝ã x1 x2 ✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ x2 = g(x1 , y1 ) ❉♦ ➤ã✱ t❛ ❝ã x2 x3 ✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ y1 = f (y0 , x0 ) ❉♦ ➤ã✱ t❛ ❝ã y1 ✈í✐ ♠ä✐ • y2 g(f (y0 , x0 ), f (x0 , y0 )) = g(y1 , x1 ) = y2 y2 ✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ y2 = g(y1 , x1 ) ❉♦ ➤ã✱ t❛ ❝ã f (g(x1 , y1 ), g(y1 , x1 )) = f (x2 , y2 ) = x3 f (g(y1 , x1 ), g(x1 , y1 )) = f (y2 , x2 ) = y3 y3 ✳ ❚✐Õ♣ tô❝ q✉➳ trì t ợ x2k+1 = f (x2k , y2k ), y2k+1 = f (y2k , x2k ), x2k+2 = g(x2k+1 , y2k+1 ), y2k+2 = g(y2k+1 , x2k+1 ) ✭✷✳✶✽✮ k ∈ N ✈➭ ❝➳❝ ❞➲② {xn } ✈➭ {yn } ❧➭ ❝➳❝ ❞➲② ➤➡♥ ➤✐Ö✉ x0 x1 xn ., y0 y1 ≤ yn ●✐➯ sö r➺♥❣ tå♥ t➵✐ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣ n0 ∈ N s❛♦ ❝❤♦ S(xn0 +1 , xn0 +1 , xn0 ) + S(yn0 +1 , yn0 +1 , yn0 ) = ❑❤✐ ➤ã✱ t❛ s✉② r❛ r➺♥❣ S(xn0 +1 , xn0 +1 , xn0 ) = S(yn0 +1 , yn0 +1 , yn0 ) = S ✲♠➟tr✐❝✱ t❛ ♥❤❐♥ ➤➢ỵ❝ xn0 +1 = xn0 , yn0 +1 = yn0 ✳ ❱× t❤Õ✱ tõ ✭✷✳✶✽✮ t❛ s✉② r❛ (xn0 , yn0 ) ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ủ f g ị ĩ ủ ã ❇➞② ❣✐ê✱ t❛ ❣✐➯ sư r➺♥❣ ✈í✐ ♠ä✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ n ∈ N t❛ ❝ã S(xn+1 , xn+1 , xn ) + S(yn+1 , yn+1 , yn ) = ✸✼ ❑❤✐ ➤ã✱ sư ❞ơ♥❣ ✭✷✳✶✼✮ ✈➭ ✭✷✳✶✽✮✱ ✈í✐ n = 2k + 1✱ t❛ ❝ã S(x2k+1 , x2k+1 , x2k+2 ) + S(y2k+1 , y2k+1 , y2k+2 ) = S(f (x2k , y2k ), f (x2k , y2k ), g(x2k+1 , y2k+1 ) + S(f (y2k , x2k ), f (y2k , x2k ), g(y2k+1 , x2k+1 ) ≤ θ(S(x2k , x2k , x2k+1 ), S(y2k , y2k , y2k+1 )) ✭✷✳✶✾✮ [S(x2k , x2k , x2k+1 ) + S(y2k , y2k , y2k+1 )] ❚õ ➤✐Ò✉ ♥➭② t❛ t❤✉ ➤➢ỵ❝ S(x2k+1 , x2k+1 , x2k+2 ) + S(y2k+1 , y2k+1 , y2k+2 ) ✭✷✳✷✵✮ < S(x2k , x2k , x2k+1 ) + S(y2k , y2k , y2k+1 ) ❱í✐ ♠ä✐ k ∈ N✱ t❛ ❦Ý ❤✐Ư✉ γ2k+1 = S(x2k+1 , x2k+1 , x2k+2 ) + S(y2k+1 , y2k+1 , y2k+2 ), ❑❤✐ ➤ã✱ tõ ✭✷✳✷✵✮ t❛ s✉② r❛ ❞➲② {γ2k+1 } ❧➭ ❞➲② ➤➡♥ ➤✐Ö✉ ❣✐➯♠✳ ❉♦ ➤ã✱ tå♥ t➵✐ γ ≥ s❛♦ ❝❤♦ lim γ2k+1 = lim [S(x2k+1 , x2k+1 , x2k+2 ) + S(y2k+1 , y2k+1 , y2k+2 )] = γ k→∞ k→∞ ❚❛ sÏ ❝❤ø♥❣ tá r➺♥❣ γ = 0✳ ●✐➯ sö ♥❣➢ỵ❝ ❧➵✐ γ > 0✱ ❦❤✐ ➤ã tõ ✭✷✳✶✾✮ t❛ ❝ã S(x2k+1 , x2k+1 , x2k+2 ) + S(y2k+1 , y2k+1 , y2k+2 ) S(x2k , x2k , x2k+1 ) + S(y2k , y2k , y2k+1 ) ✭✷✳✷✶✮ ≤ θ(S(x2k , x2k , x2k+1 ), S(y2k , y2k , y2k+1 )) < ❈❤♦ k → ∞ tr♦♥❣ ✭✷✳✷✶✮✱ t❛ ➤➢ỵ❝ θ(S(x2k , x2k , x2k+1 ), S(y2k , y2k , y2k+1 )) → ❙ư ❞ơ♥❣ tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ θ✱ t❛ ❝ã S(x2k , x2k , x2k+1 ), S(y2k , y2k , y2k+1 ) → ❦❤✐ k → ∞ ❦❤✐ k → ∞ ❱× ✈❐②✱ t❛ ♥❤❐♥ ➤➢ỵ❝ S(x2k , x2k , x2k+1 ) + S(y2k , y2k , y2k+1 ) → ✸✽ ➜✐Ò✉ ♥➭② tr➳✐ ✈í✐ ❣✐➯ t❤✐Õt γ > 0✳ ❉♦ ➤ã γ = 0✳ ❚➢➡♥❣ tù✱ ✈í✐ n = 2k + 2✱ t❛ ❝ò♥❣ ❝ã lim [S(x2k+2 , x2k+2 , x2k+3 ) + S(y2k+2 , y2k+2 , y2k+3 )] = k→∞ ❉♦ ➤ã✱ t❛ ❝ã lim [S(xn , xn , xn+1 ) + S(yn , yn , yn+1 )] = k→∞ ❇➞② ❣✐ê✱ t❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ (X, S)✳ ❚❤❐t ✈❐②✱ ✭✷✳✷✷✮ {xn } ✈➭ {yn } ❧➭ ❤❛✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ✈í✐ ♠ä✐ n, m ∈ N ✈í✐ n ≤ m ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❇ỉ ➤Ò ✷✳✶✳✸ t❛ ❝ã S(x2n+1 ,x2n+1 , x2m+1 ) + S(y2n+1 , y2n+1 , y2m+1 ) ≤ (2S(x2n+1 , x2n+1 , x2n+2 ) + 2S(y2n+1 , y2n+1 , y2n+2 )) + (S(x2n+2 , x2n+2 , x2m+1 ) + S(y2n+2 , y2n+2 , y2m+1 )) ≤ (2S(x2n+1 , x2n+1 , x2n+2 ) + 2S(y2n+1 , y2n+1 , y2n+2 )) + (2S(x2n+2 , x2n+2 , x2n+3 ) + 2S(y2n+2 , y2n+2 , y2n+3 )) + + (2S(x2m−1 , x2m−1 , x2m ) + 2S(y2m−1 , y2m−1 , y2m )) + (S(x2m , x2m , x2m+1 ) + S(y2m , y2m , y2m+1 )) = 2γ2n+1 + 2γ2n+2 + + 2γ2m−1 + γ2m ▲✃② ❣✐í✐ ❤➵♥ ❦❤✐ ✭✷✳✷✸✮ n, m → ∞ tr♦♥❣ ✭✷✳✷✸✮ ✈➭ sư ❞ơ♥❣ ✭✷✳✷✷✮ t❛ ♥❤❐♥ ➤➢ỵ❝ S(x2n+1 , x2n+1 , x2m+1 ) + S(y2n+1 , y2n+1 , y2m+1 ) → ❉♦ ➤ã✱ t❛ ➤➢ỵ❝ S(x2n+1 , x2n+1 , x2m+1 ), S(y2n+1 , y2n+1 , y2m+1 ) → ❇➺♥❣ ❝➳❝❤ t❤❛② ➤ỉ✐ ✈❛✐ trß ❝đ❛ f ✈➭ g ✈➭ t✐Õ♣ tơ❝ ❝➳❝ ❧❐♣ ❧✉❐♥ ♥❤➢ ➤➲ t❤ù❝ ❤✐Ư♥ ë tr➟♥✱ t❛ ❝ị♥❣ ♥❤❐♥ ➤➢ỵ❝ S(x2n , x2n , x2m+1 ), S(y2n , y2n , y2m+1 ) → 0, S(x2n , x2n , x2m ), S(y2n , y2n , y2m ) → 0, S(x2n+1 , x2n+1 , x2m ), S(y2n+1 , y2n+1 , y2m ) → ✸✾ ❱× t❤Õ✱ ✈í✐ ♠ä✐ n, m ∈ N ✈í✐ n ≤ m✱ t❛ ❝ã lim [S(xn , xn , xm ) + S(yn , yn , ym )] = n,m→∞ ❚õ ➤ã✱ t❛ s✉② r❛ lim S(xn , xn , xm ) = n,m→∞ lim S(yn , yn , ym ) = n,m→∞ {xn } ✈➭ {yn } ❧➭ ❤❛✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ (X, S)✳ ❱× (X, S) ❧➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ➤➬② ➤ñ✱ ♥➟♥ {xn } ✈➭ {yn } ❧➭ ❤❛✐ ❞➲② S ✲❤é✐ tô✳ ❉♦ ➤ã✱ tå♥ t➵✐ x, y ∈ X s❛♦ ❝❤♦ xn → x ✈➭ yn → y ✳ ❇➢í❝ ✷✳ ❚❛ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ (x, y) ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝ñ❛ f ✈➭ g ✳ ❚❤❐t ✈❐②✱ t❛ ①Ðt ❤❛✐ tr➢ê♥❣ ❤ỵ♣ s❛✉✳ ❚r➢ê♥❣ ❤ỵ♣ ✶✳ ●✐➯ sư f ❧✐➟♥ tơ❝✳ ❑❤✐ ➤ã✱ t❛ ❝ã ❱× ✈❐②✱ x = lim xn = lim f (xn , yn ) = f ( lim xn , lim yn ) = f (x, y), n→∞ n→∞ n→∞ n→∞ y = lim yn = lim f (yn , xn ) = f ( lim yn , lim xn ) = f (y, x) n→∞ n→∞ n→∞ n→∞ ❇➞② ❣✐ê✱ sư ❞ơ♥❣ ✭✷✳✶✼✮✱ t❛ ❝ã S(f (x, y), f (x, y), g(x, y)) + S(f (y, x), f (y, x), g(y, x)) ≤ θ(S(x, x, x), S(y, y, y))[S(x, x, x) + S(y, y, y)] ♥❣❤Ü❛ ❧➭ t❛ ❝ã S(x, x, g(x, y)) + S(y, y, g(y, x)) ≤ θ(S(x, x, x), S(y, y, y))[S(x, x, x) + S(y, y, y)] ❱× S(x, x, x) = S(y, y, y) = 0✱ ♥➟♥ t❛ ❞➢ỵ❝ S(x, x, g(x, y)) = S(y, y, g(y, x)) = 0✱ ♥❣❤Ü❛ ❧➭ t❛ ❝ã g(x, y) = x, g(y, x) = y ✳ ❉♦ ➤ã✱ (x, y) ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝ñ❛ f ✈➭ g ✳ ❚r➢ê♥❣ ❤ỵ♣ ✷✳ ●✐➯ sư g ❧✐➟♥ tơ❝✳ ❑❤✐ ➤ã✱ t➢➡♥❣ tù ♥❤➢ ❚r➢ê♥❣ ❤ỵ♣ ✶✱ t❛ ❝ị♥❣ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ (x, y) ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ f ✈➭ g ✳ ✷✳✷✳✼ ➜Þ♥❤ ❧ý✳ ✭❬✶✻❪✮ ●✐➯ sư (X, S) ❧➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ s➽♣ t❤ø tù ❜é ♣❤❐♥✱ f, g : X × X → X ✶✳ X ❧➭ ❤❛✐ ➳♥❤ ①➵ s❛♦ ❝❤♦ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ❀ ✹✵ (f, g) ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ ②Õ✉ tr➟♥ X ✷✳ ❈➷♣ ✈➭ tå♥ t➵✐ x , y0 ∈ X s❛♦ ❝❤♦ x0 ≤ f (x0 , y0 ), f (y0 , x0 ) ≤ y0 ❤♦➷❝ x0 ≤ g(x0 , y0 ), g(y0 , x0 ) ≤ y0 ; ✸✳ ●✐➯ sö r➺♥❣ tå♥ t➵✐ θ ∈ Θ1 s❛♦ ❝❤♦ S(f (x, y), f (x, y), g(u, v)) + S(f (y, x), f (y, x), g(v, u)) ≤ θ(S(x, x, u), S(y, y, v))[S(x, x, u) + S(y, y, v)], ✭✷✳✷✹✮ ✈í✐ ♠ä✐ ✹✳ X x, y, u, v ∈ X s❛♦ ❝❤♦ x ≤ u, y ≥ v ❀ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ {xn } ❧➭ ♠ét ❞➲② t➝♥❣ ✈➭ xn → x✱ t❤× xn ◆Õ✉ {xn } ❧➭ ♠ét ❞➲② ❣✐➯♠ ✈➭ xn → x✱ t❤× x ❛✮ ◆Õ✉ ❜✮ ❑❤✐ ➤ã✱ f ✈➭ x ✈í✐ ♠ä✐ n ∈ N✳ xn ✈í✐ ♠ä✐ n ∈ N✳ g ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ tr♦♥❣ X ✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚✐Õ♣ tơ❝ ❝➳❝ ❜➢í❝ ❝❤ø♥❣ ♠✐♥❤ ❣✐è♥❣ ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ {xn } ❤é✐ tô ➤Õ♥ x ✈➭ ♠ét ❞➲②{yn } ❦❤➠♥❣ t➝♥❣ ❤é✐ tô ➤Õ♥ y ✱ ✈í✐ x, y ∈ X ✳ ◆Õ✉ xn = x, yn = y ✱ ✈í✐ ♠ä✐ n ≥ 0✱ t❤× tõ ❝➳❝❤ ①➞② ❞ù♥❣ ❝➳❝ ❞➲② {xn }✱ {yn } ✈➭ xn+1 = x, yn+1 = y t❛ s✉② r❛ (x, y) ❧➭ ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ f ✈➭ g ✳ ❱× ✈❐② ❝❤ó♥❣ t❛ ❣✐➯ sư r➺♥❣ ❤♦➷❝ xn = x ❤♦➷❝ yn = y ✈í✐ n ≥ 0✳ ❑❤✐ ➤ã✱ ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ✭✷✳✷✹✮ ✈➭ ➜Þ♥❤ ❧ý ✷✳✷✳✻✱ t❛ ♥❤❐♥ ➤➢ỵ❝ ♠ét ❞➲② ❦❤➠♥❣ ❣✐➯♠ ❇ỉ ➤Ị ✷✳✶✳✸ t❛ ❝ã S(x, x,f (x, y)) + S(y, y, f (y, x)) ≤ 2S(x, x, x2k+2 ) + S(x2k+2 , x2k+2 , f (x, y)) + 2S(y, y, y2k+2 ) + S(y2k+2 , y2k+2 , f (y, x)) = 2S(x, x, x2k+2 ) + S(g(x2k+1 , y2k+1 ), g(x2k+1 , y2k+1 ), f (x, y)) + 2S(y, y, y2k+2 ) + S(g(y2k+1 , x2k+1 ), g(y2k+1 , x2k+1 ), f (y, x)) ≤ 2S(x, x, x2k+2 ) + 2S(y, y, y2k+2 ) + θ(S(x2k+1 , x2k+1 , x), S(y2k+1 , y2k+1 , y)) .[S(x2k+1 , x2k+1 , x) + S(y2k+1 , y2k+1 , y)] < 2S(x, x, x2k+2 ) + 2S(y, y, y2k+2 ) + S(x2k+1 , x2k+1 , x) + S(y2k+1 , y2k+1 , y) ✹✶ ❈❤♦ n → ∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥✱ t❛ ➤➢ỵ❝ S(x, x, f (x, y)) + S(y, y, f (y, x)) = ❉♦ ➤ã✱ x = f (x, y), y = f (y, x)✳ ❇➺♥❣ ❝➳❝❤ t❤❛② ➤ỉ✐ ✈❛✐ trß ❝đ❛ f ✈➭ g ✈➭ sư ❞ơ♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❣✐è♥❣ ♥❤➢ ♣❤➢➡♥❣ ♣❤➳♣ ➤➲ ➤➢ỵ❝ ♥❤➽❝ ➤Õ♥ ë tr➟♥✱ t❛ ❝❤ø♥❣ x = g(x, y), y = g(y, x)✳ ❱× t❤Õ✱ (x, y) ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ ❝đ❛ f ♠✐♥❤ ➤➢ỵ❝ ✈➭ g✳ (X, ) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ S ❧➭ ♠ét S ✲♠➟tr✐❝ tr➟♥ X s❛♦ ❝❤♦ (X, S) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư r➺♥❣ f, g : X × X → X ❧➭ ❤❛✐ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ②Õ✉ ✈➭ ❣✐➯ sư r➺♥❣ tå♥ t➵✐ µ ∈ Θ1 s❛♦ ❝❤♦ ✷✳✷✳✽ ❍Ư q✉➯✳ ✭❬✶✻❪✮ ●✐➯ sö S(f (x, y),f (x, y), g(u, v)) ≤ µ(S(x, x, u), S(y, y, v))[S(x, x, u) + S(y, y, v)] ✈í✐ ♠ä✐ x, y, u, v ∈ X s❛♦ ❝❤♦ x u, y v ✳ ✭✷✳✷✺✮ ●✐➯ sö r➺♥❣ ❤♦➷❝ ✶✳ f ✷✳ X ❤♦➷❝ g ❧✐➟♥ tô❝✱ ❤♦➷❝ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ {xn } ❧➭ ♠ét ❞➲② t➝♥❣ ✈➭ xn → x✱ t❤× xn x ✈í✐ ♠ä✐ n ∈ N✳ ◆Õ✉ {xn } ❧➭ ♠ét ❞➲② ❣✐➯♠ ✈➭ xn → x✱ ❦❤✐ ➤ã x xn ✈í✐ ♠ä✐ n ∈ N✳ ✭❛✮ ◆Õ✉ ✭❜✮ x0 , y0 ∈ X s❛♦ ❝❤♦ x0 ≤ f (x0 , y0 ), f (y0 , x0 ) ≤ y0 ❤♦➷❝ x0 ≤ g(x0 , y0 ), g(y0 , x0 ) ≤ y0 ✱ t❤× f ✈➭ g ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ tr♦♥❣ X✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ♠ä✐ x, y, u, v ∈ X s❛♦ ❝❤♦ x u, y v ✱ ♥❤ê ✭✷✳✷✺✮ t❛ ◆Õ✉ tå♥ t➵✐ ❝ã S(f (y, x),f (y, x), g(v, u)) ≤ µ(S(y, y, v), S(x, x, u))[S(y, y, v) + S(x, x, u)] ✭✷✳✷✻✮ ❑Õt ❤ỵ♣ ✈í✐ ✭✷✳✷✺✮ ✈➭ ✭✷✳✷✻✮ t❛ ♥❤❐♥ ➤➢ỵ❝ S(f (x, y),f (x, y), g(u, v)) + S(f (y, x), f (y, x), g(v, u)) ≤ [µ(S(x, x, u), S(y, y, v)) + µ(S(y, y, v), S(x, x, u))] [S(x, x, u) + S(y, y, v)] ✹✷ [µ(β1 , β2 ) + µ(β2 , β1 )]✱ ✈í✐ ♠ä✐ β1 , β2 ∈ [0, ∞)✳ ❉Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣ ì tế ụ ị ý ị ý t ợ ết t (1 , β2 ) = q✉➯ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ ❝➳❝❤ ❧✃② µ(β1 , β2 ) = k tr♦♥❣ ❍Ư q✉➯ ✷✳✷✳✽ ✈í✐ ♠ä✐ β1 , β2 ∈ [0, ∞) ✈➭ k ∈ [0, 1)✱ t❛ ➤➢ỵ❝ ❤Ư q✉➯ s❛✉ ➤➞② tr♦♥❣ ❬✼❪✳ ♠ä✐ ✭❬✼❪✮ ❚❤➟♠ ✈➭♦ ❣✐➯ t❤✐Õt ❝đ❛ ❍Ư q✉➯ ✷✳✷✳✽✱ t❛ ❣✐➯ sư r➺♥❣ ✈í✐ ❍Ư q✉➯✳ ✷✳✷✳✾ x, y, u, v ∈ X ♠➭ x u, y v ✈➭ ✈í✐ sè k ∈ [0, 1) ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✺✮ tr♦♥❣ ❍Ư q✉➯ ✷✳✷✳✽ ➤➢ỵ❝ t❤❛② t❤Õ ❜ë✐ k S(f (x, y), f (x, y), g(u, v)) ≤ [S(x, x, u) + S(y, y, v)] ❑❤✐ ➤ã✱ f ✈➭ g ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❜é ➤➠✐ tr♦♥❣ X ✳ ❇➺♥❣ ❝➳❝❤ ❝❤ä♥ f = g tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✻ ✈➭ ➜Þ♥❤ ❧ý ✷✳✷✳✼ ➤å♥❣ t❤ê✐ sư ụ ú ý t t ợ ị ý ể ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ f ➤➢ỵ❝ ❝❤♦ ❜ë✐ ❤Ư q✉➯ s❛✉✳ ❍Ư q✉➯✳ ✷✳✷✳✶✵ ●✐➯ sư f :X ×X →X ✶✳ X ✷✳ f (X, ) ❧➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ ❧➭ ♠ét ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ❀ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ tr➟♥ X ✈➭ tå♥ t➵✐ x , y0 ∈ X s❛♦ ❝❤♦ x0 ≤ f (x0 , y0 ), f (y0 , x0 ) ≤ y0 ; ✸✳ ●✐➯ sö r➺♥❣ tå♥ t➵✐ θ ∈ Θ1 s❛♦ ❝❤♦ S(f (x, y),f (x, y), f (u, v)) + S(f (y, x), f (y, x), f (v, u)) ≤ θ(S(x, x, u), S(y, y, v)).[S(x, x, u) + S(y, y, v)], ✈í✐ ♠ä✐ ✹✳ f x, y, u, v ∈ X ❧✐➟♥ tô❝ ❤♦➷❝ X ♠➭ x u, y v❀ ❝ã ❝➳❝ tÝ♥❤ ❝❤✃t s❛✉✿ ✭❛✮ ◆Õ✉ {xn } ❧➭ ♠ét ❞➲② t➝♥❣ ✈➭ xn → x✱ t❤× xn x ✈í✐ ♠ä✐ n ∈ N✳ ✭❜✮ ◆Õ✉ {xn } ❧➭ ♠ét ❞➲② ❣✐➯♠ ✈➭ xn → x✱ t❤× x xn ✈í✐ ♠ä✐ n ∈ N✳ ❑❤✐ ➤ã✱ f ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ tr♦♥❣ X✳ ✹✸ ❑Õt ❧✉❐♥ ❙❛✉ ♠ét t❤ê✐ ❣✐❛♥ t❐♣ tr✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ò✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉✱ ✈Ị ➤Ị t➭✐✿ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❦❤➠♥❣ ❣✐❛♥ S✲ ♠➟tr✐❝ ❝ã t❤ø tù✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉✿ ✶✳ ❍Ö t❤è♥❣ ❤ã❛ ❝➳❝ ❦❤➳✐ ♥✐Ö♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù✱ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✱ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù ✈➭ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ó✉ ●❡r❛❣❤t② tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù✱ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✱✳✳✳ ✷✳ rì ột số ị ý ể t ộ ộ ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã t❤ø tù✳ ✸✳ ❚r×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝ã tứ tự rì ột số ị ý ể trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✳ ✺✳ rì ột số ị ý ể t ộ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❦✐Ĩ✉ ●❡r❛❣❤t② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝ ❝ã t❤ø tù✳ ✹✹ t➭✐ ❧✐Ö✉ t ỗ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ●✳❱✳❘✳ ❇❛❜✉ ❛♥❞ P✳ ❙✉❜❤❛s❤✐♥✐ ✭✷✵✶✷✮✱ ✧❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡✲ ♦r❡♠s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❜② ❛❧t❡r✐♥❣ ❞✐st❛♥❝❡s ✈✐❛ ●❡r❛❣❤t②✬s ❝♦♥tr❛❝t✐♦♥✧✱ ❏✳ ❆❞✈✳ ❘❡s❡❛r✳ ❆♣♣❧✳ ▼❛t❤✳✱ ✹ ✭✹✮✱ ✼✽✲✾✺✱ ❞♦✐✿✶✵✿✺✸✼✸✴❥❛r❛♠✳✶✸✽✸✳✵✹✵✷✶✷✳ ❬✸❪ ❚✳ ●✳ ❇❤❛s❦❛r ❛♥❞ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ✭✷✵✵✻✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ♣❛r✲ t✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✻✺✱ ✶✸✼✾✲ ✶✸✾✸✳ ❬✹❪ ❇✳ ❙✳ ❈❤♦✉❞❤✉r②✱ ❆✳ ❑✉♥❞✉ ✭✷✵✶✵✮✱ ✧❆ ❝♦✉♣❧❡❞ ❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t r❡s✉❧t ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❢♦r ❝♦♠♣❛t✐❜❧❡ ♠❛♣♣✐♥❣s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ❚▼❆✱ ✼✸✱ ✷✺✷✹✲✷✺✸✶✳ ❬✺❪ ❇✳ ❙✳ ❈❤♦✉❞❤✉r② ❛♥❞ ❆✳ ❑✉♥❞✉ ✭✷✵✶✷✮✱ ✧❖♥ ❝♦✉♣❧❡❞ ❣❡♥❡r❛❧✐③❡❞ ❇❛♥❛❝❤ ❛♥❞ ❑❛♥♥❛♥ t②♣❡ ❝♦♥tr❛❝t✐♦♥s✧✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✺✱ ✷✺✾✲✷✼✵✳ ❬✻❪ ❉✳ ❉✉❦✐❝✱ ❩✳ ❑❛❞❡❧❜✉r❣✱ ❙✳ ❘❛❞❡♥♦✈✐❝ ✭✷✵✶✶✮✱ ✧❋✐①❡❞ ♣♦✐♥ts ♦❢ ●❡r❛❣❤t②✲ t②♣❡ ♠❛♣♣✐♥❣s ✐♥ ✈❛r✐♦✉s ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❆❜str✳ ❆♣♣❧✳ ❆♥❛❧✳✱ ■❉ ✺✻✶✷✹✺✱ ✶✸ ♣❛❣❡s✳ ❬✼❪ ◆✳ ❱✳ ❉✉♥❣ ✭✷✵✶✸✮✱ ✧❖♥ ❝♦✉♣❧❡❞ ❝♦♠♠♦♥ ❢✐❡❞ ♣♦✐♥ts ❢♦r ♠✐①❡❞ ✇❡❛❦❧② ♠♦♥♦t♦♥❡ ♠❛♣s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ❛♥❞ ❆♣♣❧✳✱ ✷✵✶✸✱ S ✲♠❡tr✐❝ s♣❛❝❡s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤♦r② ❆rt✐❝❧❡ ✹✽✱ ✶✼ ♣❛❣❡s✳ ❬✽❪ ▼✳ ●❡r❛❣❤t② ✭✶✾✼✸✮✱ ✧❖♥ ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✧✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✹✵✱ ✻✵✹✲✻✵✽✳ ❬✾❪ ▼✳ ❊✳ ●♦r❞❥✐✱ ❊✳ ❆❦❜❛rt❛❜❛r✱ ❨✳ ❏✳ ❈❤♦✱ ❛♥❞ ▼✳ ❘❛♠❡③❛♥✐ ✭✷✵✶✷✮✱ ✧❈♦✉♣❧❡❞ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♠✐①❡❞ ✇❡❛❦❧② ♠♦♥♦t♦♥❡ ♠❛♣♣✐♥❣s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✳✱ ✷✵✶✷✱ ❆r✲ t✐❝❧❡ ✾✺✳ ❬✶✵❪ ❏✳ ❍❛r❥❛♥✐✱ ❇✳ ▲♦♣❡③✱ ❑✳ ❙❛❞❛r❛♥❣❛♥✐ ✭✷✵✶✶✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♠✐①❡❞ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②✳✱ ✼✹✱ ✶✼✹✾✲✶✼✻✵✳ ❬✶✶❪ ❩✳ ❑❛❞❡❧❜❡r❣✱ P✳ ❑✉♠❛♠✱ ❙✳ ❘❛❞❡♥♦✈✐❝✱ ❲✳ ❙✐♥t✉♥❛✈❛r❛t ✭✷✵✶✺✮✱ ✧❈♦♠✲ ♠♦♥ ❝♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ●❡r❛❣❤t②✲t②♣❡ ❝♦♥tr❛❝t✐♦♥ ♠❛♣✲ ♣✐♥❣s ✉s✐♥❣ ♠♦♥♦t♦♥❡ ♣r♦♣❡rt②✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✳✱ ✶✵✳✶✶✽✻✴s✶✸✻✻✸✲✵✶✺✲✵✷✼✽✲✺✱ ✶✹ ♣❛❣❡s✳ ✷✵✶✺✱ ❞♦✐✿ ✹✺ ❬✶✷❪ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠✱ ▲✳ ❈✐r✐❝ ✭✷✵✵✾✮✱ ✧❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♥♦♥❧✐♥❡❛r ❝♦♥tr❛❝t✐♦♥s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳ ❚▼❆✱ ✼✵✱ ✹✸✹✶✲✹✸✹✾✳ ❬✶✸❪ ❑✳P✳ ❘✳ ❙❛str②✱ ❈❤✳ ❙✳ ❘❛♦✱ ◆✳ ❆✳ ❘❛♦✱ ❙✳ ❙✳ ❆✳ ❙❛str✐ ✭✷✵✶✹✮✱ ✧❆ ❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ❢♦r ●❡r❛❣❤t② ❝♦♥tr❛❝t✐♦♥s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❏✳ ❊♥❣✐♥✳ ❘❡s❡❛r✳ ❆♣♣❧✳✱ ✹ ✭✸✮✱ ✸✵✵✲✸✵✽✳ ❬✶✹❪ ❙✳ ❙❡❞❣❤✐✱ ◆✳ ❙❤♦❜❡✱ ❆✳ ❆❧✐♦✉❝❤❡ ✭✷✵✶✷✮✱ ✧❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ S ✲♠❡tr✐❝ s♣❛❝❡s✧✱ ▼❛t✳ ❱❡s♥✐❦✱ ✻✹✱ ✷✺✽✲✷✻✻✳ ❬✶✺❪ ❋✳ ❙❦♦❢ ✭✶✾✼✼✮✱ ✧❚❤❡♦r❡♠❛ ❞✐ ♣✉♥t✐ ❢✐ss♦ ♣❡r ❛♣♣❧✐❝❛③✐♦♥✐ ♥❡❣❧✐ s♣❛③✐ ♠❡tr✐❝✐✧✱ ❆tt✐✳ ❆❝❝❛❞✳ ❙❝✐✳ ❚♦r✐♥♦✱ ✶✶✶✱ ✸✷✸✲✸✷✾✳ ❬✶✻❪ ▼✳ ❩❤♦✉✱ ❳✲▲✳ ▲✐✉ ✭✷✵✶✻✮✱ ✧❖♥ ❝♦✉♣❧❡❞ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♥♦♥❧✐♥❡❛r ❝♦♥tr❛❝t✐♦♥s ✇✐t❤ t❤❡ ♠✐①❡❞ ✇❡❛❦❧② ♠♦♥♦t♦♥❡ ♣r♦♣❡rt② ✐♥ ♣❛r✲ t✐❛❧❧② ♦r❞❡r❡❞ S ✲♠❡tr✐❝ s♣❛❝❡s✧✱ ❏✳ ❋✉♥❝✳ ❙♣❛❝❡s✳✱ ✷✵✶✻✱ ❆rt✐❝❧❡ ■❉ ✼✺✷✾✺✷✸✱ ✾ ♣❛❣❡s✳ ❬✶✼❪ ▼✳ ❩❤♦✉✱ ❳✲▲✳ ▲✐✉✱ ❉✳ ❉✳ ❉❡❦✐❝✱ ❇✳ ❉❛♠❥❛♥♦✈✐❝ ✭✷✵✶✻✮✱ ✧❈♦✉♣❧❡❞ ❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t r❡s✉❧ts ❢♦r ●❡r❛❣❤t②✲t②♣❡ ❝♦♥tr❛❝t✐♦♥ ❜② ✉s✐♥❣ ♠♦♥♦t♦♥❡ ♣r♦♣❡rt② ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ S ✲♠❡tr✐❝ s♣❛❝❡s✧✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✾✱ ✺✾✺✵✲✺✾✻✾✳ ... max {S( x, u, w), S( y, v, t)} ≤ max {S( x, x, a), S( u, u, a)} + max {S( w, w, a), S( y, y, b)} + max {S( v, v, b), S( t, t, b)} = Dsmax ((x, y), (x, y), (a, b)) + Dsmax ((u, v), (u, v), (a, b)) + Dsmax... (X, S) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ S ✲♠➟tr✐❝✳ ❑❤✐ ➤ã S( x, x, z) ≤ 2S( x, x, y) + S( y, y, z) ✈➭ S( x, x, z) ≤ 2S( x, x, y) + S( z, z, y) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ (X, S) ❧➭ ♠ét S( x, x, y) = S( y, y, x) ✈í✐ ♠ä✐ x, y... tr❛ Ds t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ t❤ø ❤❛✐ ❝đ❛ S ✲♠➟tr✐❝✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ♠ä✐ ❚❤❐t ✈❐②✱ t❛ ❝ã Ds ((x, y), (u, v), (w, t)) = S( x, u, w) + S( y, v, t) ≤ S( x, x, a) + S( u, u, a) + S( w, w, a) ✭✷✳✸✮ + S( y,