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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

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Tiêu đề Điểm Bất Động Bộ Đôi Và Điểm Trùng Nhau Bộ Đôi Của Các Ánh Xạ Co Kiểu Geraghty Trong Các Không Gian Mê Tríc Có Thứ Tự Và Không Gian S -Mêtric Có Thứ Tự
Tác giả Nguyễn Văn Phúc
Người hướng dẫn PGS. TS. Trần Văn Ân
Trường học Trường Đại học Vinh
Chuyên ngành Toán Giải tích
Thể loại Luận văn Thạc sỹ
Năm xuất bản 2019
Thành phố Nghệ An
Định dạng
Số trang 55
Dung lượng 516,9 KB

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❇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,

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[2] G.V.R. Babu and P. Subhashini (2012), "Coupled fixed point the- orems in partially ordered metric spaces by altering distances via Geraghty's contraction", J. Adv. Resear. Appl. Math., 4 (4), 78-95, doi:10:5373/jaram.1383.040212 Sách, tạp chí
Tiêu đề: Coupled fixed point theorems in partially ordered metric spaces by altering distances via Geraghty's contraction
Tác giả: G.V.R. Babu, P. Subhashini
Nhà XB: J. Adv. Resear. Appl. Math.
Năm: 2012
[3] T. G. Bhaskar and V. Lakshmikantham (2006), "Fixed point theorems in par- tially ordered metric spaces and applications", Nonlinear Anal., 65, 1379- 1393 Sách, tạp chí
Tiêu đề: Fixed point theorems in par-tially ordered metric spaces and applications
Tác giả: T. G. Bhaskar and V. Lakshmikantham
Năm: 2006
[4] B. S. Choudhury, A. Kundu (2010), "A coupled coincidence point result in partially ordered metric spaces for compatible mappings", Nonlinear Anal.TMA, 73, 2524-2531 Sách, tạp chí
Tiêu đề: A coupled coincidence point result inpartially ordered metric spaces for compatible mappings
Tác giả: B. S. Choudhury, A. Kundu
Năm: 2010
[5] B. S. Choudhury and A. Kundu (2012), "On coupled generalized Banach and Kannan type contractions", J. Nonlinear Sci. Appl., 5, 259-270 Sách, tạp chí
Tiêu đề: On coupled generalized Banach and Kannan type contractions
Tác giả: B. S. Choudhury, A. Kundu
Nhà XB: J. Nonlinear Sci. Appl.
Năm: 2012
[6] D. Dukic, Z. Kadelburg, S. Radenovic (2011), "Fixed points of Geraghty- type mappings in various generalized metric spaces", Abstr. Appl. Anal., ID 561245, 13 pages Sách, tạp chí
Tiêu đề: Fixed points of Geraghty-type mappings in various generalized metric spaces
Tác giả: D. Dukic, Z. Kadelburg, S. Radenovic
Năm: 2011
[7] N. V. Dung (2013), "On coupled common fied points for mixed weakly monotone maps in partially ordered S -metric spaces", Fixed Point Thory and Appl., 2013, Article 48, 17 pages Sách, tạp chí
Tiêu đề: On coupled common fied points for mixed weakly monotone maps in partially ordered S -metric spaces
Tác giả: N. V. Dung
Nhà XB: Fixed Point Theory and Applications
Năm: 2013
[8] M. Geraghty (1973), "On contractive mappings", Proc. Amer. Math. Soc., 40, 604-608 Sách, tạp chí
Tiêu đề: On contractive mappings
Tác giả: M. Geraghty
Năm: 1973
[9] M. E. Gordji, E. Akbartabar, Y. J. Cho, and M. Ramezani (2012), "Coupled common fixed point theorems for mixed weakly monotone mappings in partially ordered metric spaces", Fixed Point Theory and Appl., 2012, Ar- ticle 95 Sách, tạp chí
Tiêu đề: Coupled common fixed point theorems for mixed weakly monotone mappings in partially ordered metric spaces
Tác giả: M. E. Gordji, E. Akbartabar, Y. J. Cho, M. Ramezani
Nhà XB: Fixed Point Theory and Applications
Năm: 2012
[10] J. Harjani, B. Lopez, K. Sadarangani (2011), "Fixed point theorems for mixed monotone operators and applications to integral equations", Nonlinear Analy., 74, 1749-1760 Sách, tạp chí
Tiêu đề: Fixed point theorems for mixed monotone operators and applications to integral equations
Tác giả: J. Harjani, B. Lopez, K. Sadarangani
Nhà XB: Nonlinear Analy.
Năm: 2011
[11] Z. Kadelberg, P. Kumam, S. Radenovic, W. Sintunavarat (2015), "Com- mon coupled fixed point theorems for Geraghty-type contraction map- pings using monotone property", Fixed Point Theory and Appl., 2015, doi Sách, tạp chí
Tiêu đề: Common coupled fixed point theorems for Geraghty-type contraction mappings using monotone property
Tác giả: Z. Kadelberg, P. Kumam, S. Radenovic, W. Sintunavarat
Nhà XB: Fixed Point Theory and Applications
Năm: 2015
[12] V. Lakshmikantham, L. Ciric (2009), "Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces", Nonlinear Anal.TMA, 70, 4341-4349 Sách, tạp chí
Tiêu đề: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces
Tác giả: V. Lakshmikantham, L. Ciric
Nhà XB: Nonlinear Anal.TMA
Năm: 2009
[13] K.P. R. Sastry, Ch. S. Rao, N. A. Rao, S. S. A. Sastri (2014), "A Coupled fixed point theorem for Geraghty contractions in partially ordered metric spaces", J. Engin. Resear. Appl., 4 (3), 300-308 Sách, tạp chí
Tiêu đề: A Coupled fixed point theorem for Geraghty contractions in partially ordered metric spaces
Tác giả: K.P. R. Sastry, Ch. S. Rao, N. A. Rao, S. S. A. Sastri
Nhà XB: J. Engin. Resear. Appl.
Năm: 2014
[14] S. Sedghi, N. Shobe, A. Aliouche (2012), "A generalization of fixed point theorems in S -metric spaces", Mat. Vesnik, 64, 258-266 Sách, tạp chí
Tiêu đề: A generalization of fixed pointtheorems inS-metric spaces
Tác giả: S. Sedghi, N. Shobe, A. Aliouche
Năm: 2012
[15] F. Skof (1977), "Theorema di punti fisso per applicazioni negli spazi metrici", Atti. Accad. Sci. Torino, 111, 323-329 Sách, tạp chí
Tiêu đề: Theorema di punti fisso per applicazioni negli spazimetrici
Tác giả: F. Skof
Năm: 1977
[16] M. Zhou, X-L. Liu (2016), "On coupled common fixed point theorems for nonlinear contractions with the mixed weakly monotone property in par- tially ordered S -metric spaces", J. Func. Spaces., 2016, Article ID 7529523, 9 pages Sách, tạp chí
Tiêu đề: On coupled common fixed point theorems for nonlinear contractions with the mixed weakly monotone property in par- tially ordered S -metric spaces
Tác giả: M. Zhou, X-L. Liu
Nhà XB: J. Func. Spaces.
Năm: 2016
[17] M. Zhou, X-L. Liu, D. D. Dekic, B. Damjanovic (2016), "Coupled coincidence point results for Geraghty-type contraction by using monotone property in partially ordered S -metric spaces", J. Nonlinear Sci. Appl., 9, 5950-5969 Sách, tạp chí
Tiêu đề: Coupled coincidencepoint results for Geraghty-type contraction by using monotone property inpartially orderedS-metric spaces
Tác giả: M. Zhou, X-L. Liu, D. D. Dekic, B. Damjanovic
Năm: 2016

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