Điểm bất động và điểm bất động chung của các ánh xạ kiểu () co trong không gian meetric suy rộng

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ P❤➵♠ ❱➝♥ ▲➢➡♥❣ ➜✐Ó♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✺ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ P❤➵♠ ❱➝♥ ▲➢➡♥❣ ➜✐Ó♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✺ ▼ơ❝ ▲ơ❝ ❚r❛♥❣ ▼ơ❝ ❧ơ❝ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ✷ ❈❤➢➡♥❣ ✶✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✶✳✶ ❈➳❝ ❦❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤➢➡♥❣ ✷✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ✺ ✽ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✶✹ ✷✳✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ✶✹ (α, ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ❑Õt ❧✉❐♥ ✸✻ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✼ ✶ ❧ê✐ ♥ã✐ ➤➬✉ ▲ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ❝❤đ ➤Ị ♥❣❤✐➟♥ ❝ø✉ q✉❛♥ trä♥❣ ❝đ❛ ❣✐➯✐ tÝ❝❤✳ ◆ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ tr♦♥❣ t♦➳♥ ❤ä❝ ✈➭ ❝➳❝ ♥❣➭♥❤ ❦ü t❤✉❐t✳ ❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧Ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝đ❛ ❇❛♥❛❝❤✳ ◆➝♠ ✶✾✻✽✱ ❑❛♥❛♥ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ♠ét ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ♠➭ ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❧✐➟♥ tơ❝ ❝ñ❛ ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ◆➝♠ ✷✵✵✵✱ ❇r❛♥❝✐❛r✐ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ➤ã ❧➭ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➭ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➢ỵ❝ t❤❛② t❤Õ ❜ë✐ ❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝✳ ↕♥❣ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣✱ ♠ä✐ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤Ò✉ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ♥❤➢♥❣ ➤✐Ị✉ ♥❣➢ỵ❝ ❧➵✐ ❧➭ ❦❤➠♥❣ ➤ó♥❣✳ ↕♥❣ ❝ị♥❣ ➤➲ t×♠ ❝➳❝❤ ♠ë ré♥❣ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❇❛♥❛❝❤ tr♦♥❣ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❚❤❡♦ ❤➢í♥❣ ♠ë ré♥❣ ♥➭②✱ ♥❤✐Ò✉ ♥❤➭ t♦➳♥ ❤ä❝ ❦❤➳❝ ➤➲ t✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤✉ ➤➢ỵ❝ ♥❤✐Ị✉ ❦Õt q✉➯ t❤ó ✈Þ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ♥➭②✳ ◆➝♠ ✷✵✵✽✱ ❉✉tt❛ ✈➭ ❈❤♦✉❞❤✉r② ➤➲ ♠ë ré♥❣ ➤➢ỵ❝ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛✲ ♥❛❝❤ ❝❤♦ ❧í♣ ➳♥❤ ①➵ (ψ, φ)✲❝♦ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ➜Õ♥ ♥➝♠ ✷✵✶✷✱ ❍✳ ▲❛❦③✐❛♥ ✈➭ ❇✳ ❙❛♠❡t ➤➲ ♠ë ré♥❣ ❦Õt q✉➯ ❝ñ❛ P✳◆✳❉✉tt❛ ✈➭ ❈❤♦✉❞✲ ❤✉r② ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✱ ❝❤ó♥❣ t➠✐ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥➭② ♥❤➺♠ t×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ (ψ, ϕ)✲ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳❚❙✳❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ✷ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✧✳ ▼ô❝ ➤Ý❝❤ ❝ñ❛ ❧✉❐♥ ✈➝♥ ♥➭② ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ➤✐Ó♠ ❜✃t ➤é♥❣✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ ②Õ✉✱ ➳♥❤ ①➵ ❦✐Ó✉ (ψ, ϕ)✲❝♦✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ①➵ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉✱ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị (ψ, ϕ)✲❝♦✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ (ψ, ϕ)✲❝♦ ②Õ✉✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ (α, ψ, ϕ)✲❝♦✱✳✳✳ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ▼ơ❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝ñ❛ ❧✉❐♥ ✈➝♥✳ ❇❛♦ ❣å♠ ❝➳❝ ♥é✐ ❞✉♥❣✿ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❦❤➳✐ ♥✐Ö♠ ❞➲② ❤é✐ tô✱ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ t❤ø tù✱ ❦❤➳✐ ♥✐Ư♠ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ể t ộ rì ột số ị ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ tù ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s ụ ú t trì ột số ị ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❑❛♥❛♥ ✈➭ ♠ë ré♥❣ ❝đ❛ ❝❤ó♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ tết ề ị ý ó r ò trì ệ q ợ rút r từ ị ❧ý ♥➭②✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ (ψ, ϕ)✲ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ▼ô❝ ✶ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❤❛✐ ➳♥❤ ①➵ ✈➭ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ị ❝➳❝ ❦Õt q✉➯ ➤ã ✈➭ tr×♥❤ ❜➭② ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ö♠ ➳♥❤ ①➵ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ❦✐Ư♥ (α, ψ, ϕ)✲❝♦ α✲❝❤Ý♥❤ f ✲α✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ q✉②✱ ❦❤➳✐ ♥✐Ư♠ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ✈➭ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ (α, ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ✈Ò ❝➳❝ ❦Õt ✸ q✉➯ ➤ã ✈➭ tr×♥❤ ❜➭② ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ▲✉❐♥ ✈➝♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤ ❝❤✉ ➤➳♦ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❡♠ ①✐♥ ❜➭② tá sù ❜✐Õt ➡♥ s➞✉ s➽❝ tí✐ ❚❤➬②✳ ◆❤➞♥ ❞Þ♣ ♥➭② ❡♠ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❦❤♦❛ ❚♦➳♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý t❤➬②✱ ❝➠ ❣✐➳♦ tr♦♥❣ tæ ●✐➯✐ ❚Ý❝❤ ❦❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ t❐♥ t×♥❤ ❣✐ó♣ ➤ì ❡♠ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❇❛♥ ❣✐➳♠ ❤✐Ö✉ tr➢ê♥❣ ❚❍P❚ ▲➟ ❍å♥❣ P❤♦♥❣✱ tØ♥❤ ◗✉➯♥❣ ❇×♥❤✱ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤♦➳ ✷✶ ●✐➯✐ ❚Ý❝❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ✈➭ ❣✐❛ ➤×♥❤✱ ➤å♥❣ ♥❣❤✐Ư♣ ➤➲ t➵♦ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ♥❤✃t ➤Ĩ ❣✐ó♣ t➠✐ ❤♦➭♥ t❤➭♥❤ tèt ♥❤✐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♥✳ ❱✐♥❤✱ ♥❣➭② ✷✺ t❤➳♥❣ ✼ ♥➝♠ ✷✵✶✺ P❤➵♠ ❱➝♥ ▲➢➡♥❣ ✹ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦✐Õ♥ t❤ø❝ ❝➡ së ✈➭ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ tù ➳♥❤ ①➵ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❝➬♥ ❞ï♥❣ ❝❤♦ ❝➳❝ tr×♥❤ ❜➭② ✈Ị s❛✉✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ ♠➟tr✐❝ tr➟♥ ❍➭♠ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ ✈➭ d(x, y) ≤ d(x, z) + d(z, y) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ✭✐✐✐✮ X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ ✈➭ ❦Ý ❤✐Ư✉ ❧➭ ➤Õ♥ ➤✐Ĩ♠ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤✐Ĩ♠ x y✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✶❪✮ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ f : (X, d) → (Y, ρ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ ρ[f (x) , f (y)] ≤ αd (x, y) , ✶✳✶✳✸ ➜Þ♥❤ ❧ý✳ ➤➬② ➤đ✱ ➤✐Ĩ♠ d : X ì X R ợ ọ ột d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✐✐✮ ✶✳✶✳✷ X✳ ♥Õ✉ t❤á❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥✿ d(x, y) ≥ ✈í✐ ♠ä✐ x, y ∈ X ✭✐✮ ❚❐♣ X ✭❬✶❪✮ ❈❤♦ t❐♣ ❤ỵ♣ ✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✮ ●✐➯ sö f :X→X x∗ ∈ X ➜✐Ĩ♠ ✈í✐ ♠ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ tõ s❛♦ ❝❤♦ x∗ ∈ X X (X, d) ✈➭ (Y, ρ)✳ ➳♥❤ ①➵ α ∈ [0, 1) s❛♦ ❝❤♦ 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❛ ➳♥❤ ①➵ f✳ ✺ ▼ë ré♥❣ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤✱ P✳ ◆✳ ❉✉tt❛✱ ❇✳ ❙✳ ❈❤♦✉❞❤✉r② ➤➲ t❤✉ ➤➢ỵ❝ ❦Õt q✉➯ s❛✉✳ ➜Þ♥❤ ❧ý✳ ✶✳✶✳✹ ✭❬✻❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ T : X → X X ❧➭ ♠ét tù ➳♥❤ ①➵ tr➟♥ t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝ ψ (d (T x, T y)) ≤ ψ (d (x, y)) − ϕ (d (x, y)) , tr♦♥❣ ➤ã ✈➭ ψ, ϕ : [0, +∞) → [0, +∞) ψ(t) = ϕ(t) = ✈í✐ ♠ä✐ x, y ∈ X, ✭✶✳✶✮ ❧➭ ❝➳❝ ❤➭♠ ❧✐➟♥ tơ❝✱ ➤➡♥ ➤✐Ư✉ ❦❤➠♥❣ ❣✐➯♠ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t ❂ ✵✳ ❑❤✐ ➤ã T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✶✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ (ψ, ϕ) − co ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✺ ①➵ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ✭❬✸❪✮ ❈❤♦ t❐♣ ❤ỵ♣ x, y ∈ X X = φ ✈➭ d : X × X → [0, +∞) ❧➭ ♠ét ➳♥❤ u, v ∈ X \ {x, y}✱ t❛ ✈➭ ✈í✐ ♠ä✐ ❝➷♣ ➤✐Ĩ♠ ♣❤➞♥ ❜✐Öt ❝ã ✭✐✮ ✭✐✐✮ ✭✐✐✐✮ ❑❤✐ ➤ã d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ d(x, y) = d(y, x) d(x, y) ≤ d(x, u) + d(u, v) + d(v, y)✳ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ➜✐Ị✉ ❦✐Ư♥ ✭✐✐✐✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝✳ ✶✳✶✳✻ ➜Þ♥❤ ♥❣❤Ü❛✳ {xn } ⊂ X ✭❬✸❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❞➲② ➤➢ỵ❝ ❣ä✐ ❧➭ ❤é✐ tơ ✈Ị ➤✐Ĩ♠ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n ≥ n0 t❛ ❝ã x∈X d (xn , x) < ε✳ ♥Õ✉ ✈í✐ ♠ä✐ ε>0 ▲ó❝ ➤ã t❛ ❦Ý ❤✐Ư✉ ❧➭ tå♥ t➵✐ n ∈ N∗ lim xn = x ❤❛② n→+∞ xn → x ❦❤✐ n → +∞✳ ✶✳✶✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳ {xn } ⊂ X ✳ (X, d) ✭❬✸❪✮ ❈❤♦ ❚❛ ♥ã✐ r➺♥❣ ♥Õ✉ ✈í✐ ♠ä✐ {xn } ε > 0✱ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❞➲② ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ tå♥ t➵✐ n ε ∈ N∗ d(xn , xm ) < ε✳ ✻ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n > m ≥ nε ✱ t❛ ❝ã ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✽ ✭❬✸❪✮ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ ị ĩ (X, d) ợ ọ ủ (X, d) ➤Ị✉ ❤é✐ tơ tr♦♥❣ ♥ã✳ ✭❬✹❪✮ ●✐➯ sư T, f : X → X ❧➭ ❝➳❝ ➳♥❤ ①➵ tõ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ➜✐Ó♠ ➳♥❤ ①➵ x∈X T y ∈X ✈➭ f ➤➢ỵ❝ ❣ä✐ trị trù ợ t ủ tr➟♥ X ♥Õ✉ tå♥ t➵✐ x∈X s❛♦ ❝❤♦ y = f (x) = T (x)✳ ❑❤✐ ➤ã ➤✐Ĩ♠ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ✭❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t✮ ❝đ❛ ❤❛✐ ➳♥❤ ①➵ T ✈➭ f✳ ➜✐Ĩ♠ x∈X ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❤❛✐ ➳♥❤ ①➵ T ✈➭ f ♥Õ✉ x = T (x) = f (x)✳ (T, f ) ❈➷♣ ➳♥❤ ①➵ ➤➢ỵ❝ ❣ä✐ ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ ♥Õ✉ ♥❤❛✉ t➵✐ ❝➳❝ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝❤ó♥❣✱ ♥❣❤Ü❛ ❧➭ ➤✐Ó♠ ✶✳✶✳✶✵ x∈X ♠➭ T ✈➭ f ❣✐❛♦ ❤♦➳♥ ✈í✐ T f (x) = f T (x) t➵✐ ❝➳❝ T (x) = f (x)✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✭❬✽❪✮ ●✐➯ sư X = φ✳ ◆Õ✉ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ (X, ) (X, d, ) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ t❤ø tù✳ ❑❤✐ ➤ã ❤❛✐ ♣❤➬♥ tư x, y ∈ X ➤➢ỵ❝ ❣ä✐ ❧➭ s♦ s➳♥❤ ➤➢ỵ❝ ♥Õ✉ ✶✳✶✳✶✶ ①➵ ❧➭ ♠ét t❐♣ ❤ỵ♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈í✐ q✉❛♥ ❤Ư t❤ø tù ị ĩ x y y tì x✳ (X, ) ❧➭ t❐♣ ❤ỵ♣ s➽♣ t❤ø tù ❜é ♣❤❐♥ ✈➭ ❤❛✐ ➳♥❤ T, f : X → X ✳ ➳♥❤ ①➵ T ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ➳♥❤ ①➵ f ✲❦❤➠♥❣ ❣✐➯♠ ♥Õ✉ T x f (x) f (y) , ✈í✐ ♠ä✐ x, y ∈ X ✳ ✶✳✶✳✶✷ ➜Þ♥❤ ♥❣❤Ü❛✳ t❐♣ X ●✐➯ sö X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ t➠♣➠✱ ✈➭♦ t❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝ ✭✐✮ ❍➭♠ f α ∈ R f x0 < α f : X → R ❧➭ ➳♥❤ ①➵ tõ R✳ ❚❛ ♥ã✐ r➺♥❣✿ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ t➵✐ ♠➭ T y ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x0 ∈ X t❤× tå♥ t➵✐ ❧➞♥ ❝❐♥ ♠ë f x < α ✈í✐ ♠ä✐ x ∈ U ✱ ✼ U ♥Õ✉ ✈í✐ ♠ä✐ sè t❤ù❝ ❝ñ❛ x0 tr♦♥❣ X s❛♦ ❝❤♦ ✭✐✐✮ ❍➭♠ f α ∈ R ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ t➵✐ ♠➭ f x0 > α x0 ∈ X t❤× tå♥ t➵✐ ❧➞♥ ❝❐♥ ♠ë V ♥Õ✉ ✈í✐ ♠ä✐ sè t❤ù❝ ❝ñ❛ x0 tr♦♥❣ X s❛♦ ❝❤♦ f x > α ✈í✐ ♠ä✐ x ∈ V ✱ ✭✐✐✐✮ ❍➭♠ tr➟♥ f X ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ ✭t➢➡♥❣ ø♥❣ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✮ ♥Õ✉ f ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ ✭t➢➡♥❣ ø♥❣ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐✮ t➵✐ ♠ä✐ ➤✐Ĩ♠ t❤✉é❝ X✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ✶✳✷ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ r ú t trì ột số ị ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❑❛♥❛♥ ✈➭ ♠ë ré♥❣ ❝đ❛ ❝❤ó♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ❚r➢í❝ ❤Õt t❛ ❦ý ❤✐Ö✉ Ψ = {ψ : [0, +∞) → [0, +∞) | ψ ❧➭ ❤➭♠ ❧✐➟♥ tô❝✱ ❦❤➠♥❣ ❣✐➯♠ ✈➭ ψ(t) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = 0}✱ Φ = {ϕ : [0, +∞) → [0, +∞) | ϕ ❧➭ ❤➭♠ ❧✐➟♥ tô❝ ✈➭ ϕ(t) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ t = 0}✳ ❚õ ➤➞② ✈Ò s❛✉✱ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✶✮ ➤➢ỵ❝ ❣ä✐ ❝❤✉♥❣ ❧➭ ➳♥❤ ①➵ ❦✐Ó✉ (ψ, ϕ) − co✳ ◆➝♠ ✷✵✶✷✱ ❍✳ ▲❛❦③✐❛♥✱ ❇✳ ❙❛♠❡t ➤➲ t❤✉ ➤➢ỵ❝ ♠ét ❦Õt q t tự ị ý trờ ợ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ✶✳✷✳✶ ➜Þ♥❤ ❧ý✳ ❍❛✉s❞♦r❢❢✱ ✭❬✼❪✮ ❈❤♦ T :X→X (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ ✈➭ ❧➭ ♠ét tù ➳♥❤ ①➵ tr➟♥ X t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ψ (d (T x, T y)) ≤ ψ (d (x, y)) − ϕ (d (x, y)) , tr♦♥❣ ➤ã ψ ∈ Ψ✱ ϕ ∈ Φ✳ ❑❤✐ ➤ã T x, y ∈ X, ✭✶✳✷✮ ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ♠ét ➤✐Ĩ♠ tï② ý ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ s❛✉ ✈í✐ ♠ä✐ xo ∈ X ✳ ❚❛ ①➳❝ ➤Þ♥❤ ♠ét ❞➲② {xn } ⊂ X xn+1 = T xn , n = 1, 2, ✈➭ t✐Õ♥ ❤➭♥❤ t❤❡♦ ❝➳❝ ❜➢í❝ s❛✉✳ ✽ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ (α, ψ, ϕ)✲❝♦ tr♦♥❣ ✷✳✷ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ➳♥❤ ①➵ ①➵ f ✲α✲ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ ➳♥❤ α✲ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ α✲❝❤Ý♥❤ q✉✐ ✈➭ ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ (α, ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✷✳✷✳✶ ➳♥❤ ①➵ T ✭❬✽❪✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ❈❤♦ ❝➳❝ ➳♥❤ ①➵ f ✲α✲❝❤✃♣ T, f : X → X ✈➭ α : X ì X [0, +) ợ ế ♠ä✐ x, y ∈ X s❛♦ ❝❤♦ α(f x, f y) ≥ 1✱ t❛ ❝ã α(T x, T y) ≥ ế f t tì ị ♥❣❤Ü❛✳ ✷✳✷✳✷ ✭❬✽❪✮ ❈❤♦ X × X → [0, +∞)✳ X ❝❤♦ α(xn , xn+1 ) ≥ 1✱ s❛♦ ❝❤♦ X ✭❬✽❪✮ ❈❤♦ ❧➭ ❝➳❝ tù ➳♥❤ ①➵ tr➟♥ r➺♥❣ ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ♥❤❐♥ ➤➢ỵ❝✳ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ α✲❝❤Ý♥❤ n∈ N α✲❝❤✃♣ ✈➭ q✉✐ ♥Õ✉ ✈í✐ ♠ä✐ ❞➲② xn → x✱ {xn } ⊂ X t❤× tå♥ t➵✐ ❞➲② ❝♦♥ α: s❛♦ {xnk } ⊂ {xn } α(xnk , x) ≥ ✈í✐ ọ k N ị ý (X, d) ợ ❣ä✐ ❧➭ ✈í✐ ♠ä✐ T (f X, d) (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ T, f : X → X s❛♦ ❝❤♦ TX ⊆ fX ✈➭ α : X × X → [0, +∞)✳ ●✐➯ sư ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ ✈➭ ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥ ψ (α(f x, f y)d (T x, T y)) ≤ ψ (M (x, y)) − ϕ (M (x, y)) , tr♦♥❣ ➤ã ψ ∈ Ψ, ϕ ∈ Φ ✈í✐ ♠ä✐ ✈➭ M (x, y) = max{d(f x, f y), d(f x, T x), d(f y, T y)} ❈ị♥❣ ❣✐➯ t❤✐Õt r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥ ✭✐✮ T ❧➭ ➳♥❤ ①➵ ✭✐✐✮ tå♥ t➵✐ f ✲α✲❝❤✃♣ x0 ∈ X ✱ ♥❤❐♥ ➤➢ỵ❝❀ s❛♦ ❝❤♦✿ α(f x0 , T x0 ) ≥ 1; ✷✹ x, y ∈ X ✭✷✳✷✸✮ ✭✐✐✐✮ X ❧➭ α✲❝❤Ý♥❤ ❝ã α(xm , xn ) ≥ 1✱ ✭✐✈✮ ❤♦➷❝ ❑❤✐ ➤ã ✈➭ f T ✈í✐ ♠ä✐ α(f u, f v) ≥ f ✈➭ {xn } ⊂ X q✉✐ ✈➭ ✈í✐ ♠ä✐ ❞➲② ❤♦➷❝ m, n ∈ N ♠➭ α(f v, f u) ≥ T ✈➭ f t❛ m < n❀ ❦❤✐ ♠➭ fu = Tu ó ột trị trù ợ ♥❤✃t tr♦♥❣ ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ t❤× α(xn , xn+1 ) ≥ 1✱ s❛♦ ❝❤♦ X✳ ✈➭ f v = T v✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ T ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✷✸✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ (α, ψ, ϕ) − co ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② ♠ét ➤✐Ĩ♠ tï② ý ➤Þ♥❤ ❝➳❝ ❞➲② ♠ä✐ n ≥ 0✳ {xn } ⊂ X ✈➭ {yn } ⊂ X x0 ∈ X α(f x0 , T x0 ) ≥ 1✳ ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝✿ ◆❣♦➭✐ r❛✱ t❛ ❣✐➯ sö r➺♥❣✱ ♥Õ✉ xn+p+1 = xn+1 ✳ s❛♦ ❝❤♦ yn = f xn+1 = T xn ✱ yn = T xn = T xn+p = yn+p ❙ë ❞Ü ➤✐Ị✉ ♥➭② ❧➭♠ ➤➢ỵ❝ ❧➭ ♥❤ê ❣✐➯ t❤✐Õt tr➢ê♥❣ ợ ệt ế yn = yn+1 tì yn+1 ①➳❝ ✈í✐ t❤× ❝❤ä♥ T X ⊆ f X✳ ❚r♦♥❣ ột trị trù ợ ủ T f ✳ ❇ë✐ ✈❐②✱ t❛ ❝ã t❤Ĩ ❣✐➯ sư r➺♥❣ yn = yn+1 ✱ ✈í✐ ♠ä✐ n ∈ N✳ ❚õ ➤✐Ị✉ ❦✐Ö♥ ✭✐✐✮ t❛ ❝ã ①➵ α(f x0 , T x0 ) = α(f x0 , f x1 ) ≥ 1✳ ❉♦ ❣✐➯ t❤✐Õt T ❧➭ ➳♥❤ f ✲α✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ♥➟♥ t❛ ❝ã α(T x0 , T x1 ) = α(f x1 , f x2 ) ≥ ✈➭ α(T x1 , T x2 ) = α(f x2 , f x3 ) ≥ ❇➺♥❣ q✉✐ ♥➵♣ t❛ s✉② r❛ α(f xn , f xn+1 ) ≥ 1✱ ✈í✐ ♠ä✐ n ≥ 0✳ ❇➞② ❣✐ê✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✸✮ t❛ ❝ã ψ (d (T xn , T xn+1 )) ≤ ψ (α(f xn , f xn+1 )d (T xn , T xn+1 )) ≤ ψ (M (xn , xn+1 )) − ϕ (M (xn , xn+1 )) , ✈í✐ M (xn , xn+1 ) = max{d(f xn , f xn+1 ), d(f xn , T xn ), d(f xn+1 , T xn+1 )} = max{d(yn−1 , yn ), d(yn , yn+1 )} ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ ψ (d (T xn , T xn+1 )) ≤ ψ (M (xn , xn+1 )) − ϕ (M (xn , xn+1 )) , ✷✺ ✈í✐ ♠ä✐ n ∈ N ✭✷✳✷✹✮ ❇➞② ❣✐ê✱ ♥Õ✉ M (xn , xn+1 ) = d (yn , yn+1 ) t❤× tõ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✹✮ t❛ ❝ã ψ (d (yn , yn+1 )) ≤ ψ (d (yn , yn+1 )) − ϕ (d (yn , yn+1 )) ❚õ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ t❤✐Õt yn = yn+1 ✱ ϕ ✈í✐ ♠ä✐ s✉② r❛ n ∈ N✳ d (yn , yn+1 ) = 0✳ ❉♦ ➤ã ✭✷✳✷✺✮ ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ M (xn , xn+1 ) = d (yn−1 , yn ) > 0✱ ✈í✐ ♠ä✐ n ∈ N✳ ❑❤✐ ➤ã tõ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✹✮ t❛ ♥❤❐♥ ➤➢ỵ❝ ψ (d (yn , yn+1 )) ≤ ψ (d (yn−1 , yn )) − ϕ (d (yn−1 , yn )) < ψ (d (yn−1 , yn )) ❉♦ ψ ❧➭ ❤➭♠ ❦❤➠♥❣ ❣✐➯♠ ♥➟♥ {d (yn , yn+1 )} t❤ù❝ d (yn , yn+1 ) < d (yn−1 , yn )✱ ✈í✐ ♠ä✐ n ∈ N✳ ❙✉② r❛ ❧➭ ❞➲② ❝➳❝ sè t❤ù❝ ❦❤➠♥❣ ➞♠✱ ❣✐➯♠✳ ❉♦ ➤ã ♥ã ❤é✐ tơ ✈Ị ♠ét sè s ≥ 0✳ ◆Õ✉ s > 0✱ ❦❤✐ ➤ã ❝❤♦ n → +∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✺✮ t❛ ♥❤❐♥ ➤➢ỵ❝ ψ(s) ≤ ψ(s) − ϕ(s)✳ ❚õ tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ ϕ t❛ s✉② r❛ s = 0✳ ❱× ✈❐②✱ t❛ ❝ã lim d(yn , yn+1 ) = ✭✷✳✷✻✮ n→+∞ ❇➞② ❣✐ê ❣✐➯ sư r➺♥❣ yn = ym ✱ ✈í✐ ♠ä✐ m = n✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ {yn } ❧➭ ❞➲② ❈❛✉❝❤②✳ ➜➬✉ t✐➟♥✱ t❛ ❝❤ø♥❣ tá r➺♥❣ ❞➲② ✈❐②✱ ❞♦ d(yn , yn+1 ) → ❉♦ ➤ã✱ ♥Õ✉ ♥➟♥ tå♥ t➵✐ d(yn , yn+2 ) > L✱ {d(yn , yn+2 )} ❧➭ ❞➲② ❜Þ ❝❤➷♥✳ ❚❤❐t L > s❛♦ ❝❤♦ d(yn , yn+1 ) ≤ L✱ ✈í✐ ♠ä✐ n ∈ N✳ ✈í✐ ♠ä✐ n ∈ N t❤× tõ M (xn , xn+2 ) = max{d(f xn , f xn+2 ), d(f xn , T xn ), d(f xn+2 , T xn+2 )} = d(yn−1 , yn+1 ), ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✐✮ t❛ ❝ã ψ (d (yn , yn+2 )) = ψ (d (T xn , T xn+2 )) ≤ ψ (α(f xn , f xn+2 )d (T xn , T xn+2 )) ≤ ψ (M (xn , xn+2 )) − ϕ (M (xn , xn+2 )) < ψ (d (yn−1 , yn+1 )) ◆❤ê tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ ψ t❛ s✉② r❛ ❞➲② ❜Þ ❝❤➷♥✳ ✷✻ {d(yn , yn+2 )} ❧➭ ❞➲② ❣✐➯♠ ✈➭ ✈× ✈❐② ♥ã ●✐➯ sö tå♥ t➵✐ ♠ét sè n ∈ N ♠➭ d(yn−1 , yn+1 ) ≤ L ✈➭ d(yn , yn+2 ) > L✱ ❦❤✐ ➤ã tõ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ ψ (d (yn , yn+2 )) = ψ (d (T xn , T xn+2 )) ≤ ψ (α(f xn , f xn+2 )d (T xn , T xn+2 )) ≤ ψ (M (xn , xn+2 )) − ϕ (M (xn , xn+2 )) < ψ (M (xn , xn+2 )) ≤ ψ(L) ❱➭ ❞♦ ψ ❧➭ ❤➭♠ ❦❤➠♥❣ ❣✐➯♠ ♥➟♥ t❛ s✉② r❛ t❤✐Õt✳ ❱❐② d(yn , yn+2 ) > L trờ ợ tì d(yn , yn+2 ) < L✳ d(yn , yn+2 ) ≤ L✱ ✈í✐ ♠ä✐ ▼➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ n∈N ✈➭ tr♦♥❣ ❝➯ ❤❛✐ {d(yn , yn+2 )} ❧➭ ❞➲② ❜Þ ❝❤➷♥✳ ❇➞② ❣✐ê t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ lim d(yn , yn+2 ) = ✭✷✳✷✼✮ n→+∞ ●✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ lim d(yn , yn+2 ) = 0✳ n→+∞ ❑❤✐ ➤ã sÏ tå♥ t➵✐ ♠ét ❞➲② ❝♦♥ {ynk } ⊂ {yn } s❛♦ ❝❤♦ lim d(ynk , ynk +2 ) → s > n→+∞ ❚õ ❝➳❝ ❜✃t ➤➻♥❣ t❤ø❝ d(ynk −1 , ynk +1 ) ≤ d(ynk −1 , ynk ) + d(ynk , ynk +2 ) + d(ynk +1 , ynk +2 ), ✈➭ d(ynk , ynk +2 ) ≤ d(ynk −1 , ynk ) + d(ynk −1 , ynk +1 ) + d(ynk +1 , ynk +2 ), ❝❤♦ k → ∞ t❛ ♥❤❐♥ ➤➢ỵ❝ lim d(ynk −1 , ynk +1 ) = s k→+∞ ❇➞② ❣✐ê✱ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✸✮ ❝❤♦ x = xnk ✈➭ y = xnk +2 t❛ ❝ã ψ (d (T xnk , T xnk +2 )) ≤ ψ (α(f xnk , f xnk +2 )d (T xnk , T xnk +2 )) ≤ ψ (M (xnk , xnk +2 )) − ϕ (M (xnk , xnk +2 )) , ✷✼ ✭✷✳✷✽✮ ✈í✐ M (xnk , xnk +2 ) = max{d(f xnk , f xnk +2 ), d(f xnk , T xnk ), d(f xnk +2 , T xnk +2 )} = max{d(ynk −1 , ynk +1 ), d(ynk −1 , ynk ), d(ynk +1 , ynk +2 )} ❱í✐ ❝➳❝ ❧❐♣ ❧✉❐♥ tr➟♥ tõ ➤✐Ò✉ ♥➭② t❛ s✉② r❛ ❈❤♦ k → +∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✽✮✱ t❛ ♥❤❐♥ ➤➢ỵ❝ ◆❤ê tÝ♥❤ ❝❤✃t ❝ñ❛ ❤➭♠ ❱❐② ψ(s) ≤ ψ(s) − ϕ(s)✳ ϕ s✉② r❛ s = 0✳ ➜✐Ị✉ ♥➭② ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ t❤✐Õt s > 0✳ lim d(yn , yn+2 ) = n→+∞ ❚✐Õ♣ t❤❡♦✱ ♥Õ✉ ❣✐➯ sư ♥❣➢ỵ❝ ❧➵✐ r➺♥❣ ➤ã tå♥ t➵✐ sè ✈➭ lim M (xnk , xnk +2 ) = s k→+∞ {yn } ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❞➲② ❈❛✉❝❤②✳ ❑❤✐ ε > s❛♦ ❝❤♦ ✈í✐ ♥ã✱ t ó tể tì ợ {ymk } ⊂ {yn } {ynk } ⊂ {yn }✱ ✈í✐ nk > mk ≥ k t❤á❛ ♠➲♥ d(ymk , ynk ) ≥ ε ◆❣♦➭✐ r❛ ✈í✐ mk t➢➡♥❣ ø♥❣✱ t❛ ❝ã t❤Ó ❝❤ä♥ nk ✭✷✳✷✾✮ ❧➭ sè tù ♥❤✐➟♥ ♥❤á ♥❤✃t s❛♦ ❝❤♦ nk − mk ≥ ✈➭ t❤á❛ ♠➲♥ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✾✮✳ ❑❤✐ ➤ã d(ymk , ynk −1 ) < ε ✭✷✳✸✵✮ ❇➞② ❣✐ê sư ❞ơ♥❣ ✭✷✳✷✾✮✱ ✭✷✳✸✵✮ ✈➭ ❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝ t❛ t❤✉ ➤➢ỵ❝ ε ≤ d(ymk , ynk ) ≤ d(ynk , ynk −2 ) + d(ynk −2 , ynk −1 ) + d(ynk −1 , ymk ) < d(ynk , ynk −2 ) + d(ynk −2 , ynk −1 ) + ε ❈❤♦ k → +∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ ✈➭ sö ❞ơ♥❣ ✭✷✳✷✻✮✱ ✭✷✳✷✼✮ t❛ ♥❤❐♥ ➤➢ỵ❝ d(ymk , ynk ) → ε ▼➷t ❦❤➳❝✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ d(ymk , ynk ) − d(ymk −1 , ymk ) − d(ynk −1 , ynk ) ≤ d(ynk −1 , ymk −1 ) ≤ d(ynk −1 , ynk ) + d(ymk , ynk ) + d(ymk −1 , ymk ), ✷✽ ✭✷✳✸✶✮ ❝❤♦ k → +∞✱ t❛ t❤✉ ➤➢ỵ❝ d(ymk −1 , ynk −1 ) → ε ❚r♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✸✮✱ ❝❤♦ x = xnk ✈➭ ✭✷✳✸✷✮ y = xmk t❛ ➤➢ỵ❝ ψ (d (T xmk , T xnk )) ≤ ψ (α(f xmk , f xnk )d (T xmk , T xnk )) ≤ ψ (M (f xmk , f xnk )) − ϕ (M (f xmk , f xnk )) , ✈í✐ M (f xmk , f xnk ) = max{d(f xmk , f xnk ), d(f xnk , T xnk ), d(f xmk , T xmk )} = max{d(ynk −1 , ymk −1 ), d(ynk −1 , ynk ), d(ymk −1 , ymk )} ◆❤ê tÝ♥❤ ❧✐➟♥ tô❝ ❝đ❛ ❤➭♠ ψ ✈➭ tÝ♥❤ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ ❝đ❛ ❤➭♠ ϕ✱ ♥➟♥ ❦❤✐ ❝❤♦ k → +∞✱ t❛ ♥❤❐♥ ➤➢ỵ❝ ψ(ε) ≤ ψ(ε) − ϕ(ε) ❚õ tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ t❤✐Õt ❧➭ ε > 0✳ ❉♦ ➤ã ϕ t❛ s✉② r❛ {yn } X ❧➭ ✈í✐ ♠ä✐ ➜✐Ị✉ ♥➭② ❞➱♥ tí✐ ♠➞✉ t❤✉➱♥ ✈í✐ ❣✐➯ ❧➭ ❞➲② ❈❛✉❝❤②✳ ▲➵✐ ❞♦ s✉② ré♥❣ ➤➬② ➤ñ✱ ♥➟♥ tå♥ t➵✐ ❉♦ ε = 0✳ z ∈ fX s❛♦ ❝❤♦ (f X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ yn → z ✳ ▲✃② y ∈ X s❛♦ ❝❤♦ f y = z✳ α✲❝❤Ý♥❤ q✉✐✱ ♥➟♥ tå♥ t➵✐ ❞➲② ❝♦♥ {ynk } ⊂ {yn } s❛♦ ❝❤♦ α(ynk −1 , f y) ≥ 1✱ k ∈ N✳ ◆Õ✉ f y = T y✱ t❤× tõ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✸✮✱ ✈í✐ x = xnk ➤➢ỵ❝ ψ (d (T xnk , T y)) ≤ ψ (α(f xnk , f y)d (T xnk , T y)) ≤ ψ (M (f xnk , f y)) − ϕ (M (f xnk , f y)) , tr♦♥❣ ➤ã M (f xnk , f y) = max{d(f xnk , f y), d(f xnk , T xnk ), d(f y, T y)} = max{d(ynk −1 , f y), d(ynk −1 , ynk ), d(f y, T y)} ❇➞② ❣✐ê ❞♦ d(ynk −1 , f y) → ✈➭ d(ynk −1 , ynk ) → 0, ✷✾ ❦❤✐ k → +∞, t❛ ♥❤❐♥ ♥➟♥ ✈í✐ k ➤đ ❧í♥ t❛ ❝ã M (f xnk , f y) = d(f y, T y)✳ ▼➷t ❦❤➳❝✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ d(f y, T y) ≤ d(f y, ynk −1 ) + d(ynk −1 , ynk ) + d(T xnk , T y), t❛ s✉② r❛ ➤➢ỵ❝ d(f y, T y) ≤ lim inf d(Tnk , T y) k→+∞ ❉♦ ψ ❧➭ ❤➭♠ ❧✐➟♥ tô❝ ✈➭ ❦❤➠♥❣ ❣✐➯♠ ♥➟♥ ✈í✐ k ➤đ ❧í♥ t❛ ❝ã ψ (d (f y, T y)) ≤ lim inf ψ(d(Tnk , T y)) ≤ ψ (d (f y, T y)) − ϕ (d (f y, T y)) k→+∞ ❙✉② r❛ d(f y, T y) = 0✱ ♥❣❤Ü❛ ❧➭ z = f y = T y✳ z ❉♦ ➤ã ❧➭ ❣✐➳ trÞ trï♥❣ ❤ỵ♣ ❝đ❛ T f✳ ✈➭ ❚✐Õ♣ t❤❡♦✱ ❣✐➯ sư r➺♥❣ tå♥ t➵✐ r➺♥❣ p = 1✳ ❚❤❐t ✈❐②✱ ♥Õ✉ p>1 n, p ∈ N s❛♦ ❝❤♦ ❦❤✐ ➤ã t❛ ❝ã yn = yn+p ✳ ❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ d(yn+p−1 , yn+p ) > 0✳ ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✺✮ t❛ ♥❤❐♥ ➤➢ỵ❝ ψ (d (yn , yn+1 )) = ψ (d (yn+p , yn+p+1 )) ≤ ψ (d (yn+p−1 , yn+p )) − ϕ (d (yn+p−1 , yn+p )) < ψ (d (yn+p−1 , yn+p )) ❉♦ {d(yn , yn+1 )} ❧➭ ❞➲② ❣✐➯♠ ♥➟♥ t❛ s✉② r❛ ➤➢ỵ❝ ψ (d (yn , yn+1 )) < ψ (d (yn , yn+1 )) ➜✐Ò✉ ♥➭② ♠➞✉ t❤✉➮♥✳ ❉♦ ➤ã ♥➟♥ yn+1 p = 1✳ ❑❤✐ ➤ã✱ t❛ ❝ã ũ trị trù ợ ủ T f f xn+1 = T xn = T xn+1 = yn+1 ❚õ ➤ã s✉② r❛ T ✈➭ f ❝ã ♠ét ❣✐➳ trị trù ợ sử u, v X z w s trị trù ợ ❝ñ❛ T ✈➭ z = f (u) = T (u) ✈➭ w = f (v) = T (v)✳ f✳ ❑❤✐ ➤ã tå♥ t➵✐ ❝➳❝ ➤✐Ĩ♠ ◆❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✐✈✮✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✸✮ t❛ ❝ã ψ(d(z, w)) = ψ(d(T u, T v)) ≤ ψ(α(f u, f v)d(T u, T v)) ≤ ψ(M (u, v)) − ϕ(M (u, v)) ≤ ψ(d(f u, f v)) − ϕ(d(f u, f v)) = ψ(d(z, w)) − ϕ(d(z, w)) ❚õ ❜✃t ➤➻♥❣ t❤ø❝ ♥➭②✱ ♥❤ê tÝ♥❤ ❝❤✃t ❝đ❛ ❤➭♠ z=w ✈➭ ❣✐➳ trÞ trï♥❣ ❤ỵ♣ ❝đ❛ T ✈➭ f ϕ t❛ s✉② r❛ ❧➭ ❞✉② ♥❤✃t✳ ✸✵ d(z, w) = 0✳ ❙✉② r❛ ❇➞② ế z trị trù ợ ủ tí ②Õ✉✱ ❦❤✐ ➤ã t❛ ❝ã ➤✐Ò✉ ♥➭② t❛ s✉② r❛ T ❝ñ❛ ✷✳✷✳✹ ✈➭ T ✈➭ f✱ ➤å♥❣ t❤ê✐ f z = T z ì trị trù ợ ❝ñ❛ T z = f z = T z✱ ♥❣❤Ü❛ ❧➭ z ✈➭ T f ✈➭ f ❧➭ t➢➡♥❣ ❧➭ ❞✉② ♥❤✃t✱ tõ ❧➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t f✳ ❍Ö q✉➯✳ T :X→X (X, d) ✭❬✽❪✮ ❈❤♦ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ✱ ❧➭ ♠ét tù ➳♥❤ ①➵ tr➟♥ X ✈➭ α : X × X → [0, +∞)✳ ●✐➯ sư ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥ ψ (α(x, y)d (T x, T y)) ≤ ψ (M (x, y)) − ϕ (M (x, y)) , tr♦♥❣ ➤ã ψ ∈ Ψ, ϕ ∈ Φ ✈í✐ ♠ä✐ x, y ∈ X, ✭✷✳✸✸✮ ✈➭ M (x, y) = max{d(x, y), d(x, T x), d(y, T y)} ❈ò♥❣ ❣✐➯ t❤✐Õt r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥ ✭✐✮ T ❧➭ α✲❝❤✃♣ ✭✐✐✮ tå♥ t➵✐ ✭✐✐✐✮ x0 ∈ X ✱ ❧➭ ❝ã α(xm , xn ) ≥ 1✱ T α✲❝❤Ý♥❤ ✷✳✷✳✺ q✉✐ ✈➭ ✈í✐ ♠ä✐ ❞➲② α(u, v) ≥ ✈í✐ ♠ä✐ ❤♦➷❝ m, n ∈ N α(v, u) ≥ {xn } ⊂ X ♠➭ s❛♦ ❝❤♦ α(xn , xn+1 ) ≥ 1✱ t❛ m < n❀ ❦❤✐ ♠➭ u = Tu ✈➭ v = T v✳ ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ tr➟♥ α(x0 , T x0 ) ≥ 1; s❛♦ ❝❤♦✿ X ✭✐✈✮ ❤♦➷❝ ❑❤✐ ó ợ r ị ý ế ọ f = IX ❧➭ ➳♥❤ ①➵ ➤å♥❣ ♥❤✃t X ✱ t❤× t❛ s✉② r❛ ♥❣❛② ➤✐Ị✉ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ➜Þ♥❤ ❧ý✳ X → X ✭❬✽❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❧➭ ❝➳❝ tù ➳♥❤ ①➵ tr➟♥ ❳ s❛♦ ❝❤♦ T X ⊆ f X✳ ●✐➯ sö r➺♥❣ T, f : (f X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ ✈➭ ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥ ψ (d (T x, T y)) ≤ ψ (M (x, y)) − ϕ (M (x, y)) , ✸✶ ✈í✐ ♠ä✐ x, y ∈ X, ✭✷✳✸✹✮ ψ ∈ Ψ, ϕ ∈ Φ tr♦♥❣ ➤ã ✈➭ M (x, y) = max{d(f x, f y), d(f x, T x), d(f y, T y)} ❑❤✐ ➤ã ✈➭ f T ✈➭ f ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ t❤× T f ✈➭ x, y ∈ X ✱ T ❍➡♥ ♥÷❛✱ ♥Õ✉ ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❇➺♥❣ ❝➳❝❤ ❝❤ä♥ ❤➭♠ ọ X ó ột trị trù ợ ♥❤✃t tr♦♥❣ α : X × X → [0, +∞) s❛♦ ❝❤♦ α(x, y) = 1✱ rå✐ ➳♣ ❞ô♥❣ trù❝ t✐Õ♣ ➜Þ♥❤ ❧ý ✷✳✷✳✸ t❛ s✉② r❛ ♥❣❛② ➤✐Ị✉ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷✳✻ ✈➭ ✭❬✽❪✮ ➜Þ♥❤ ❧ý✳ T, f : X → X ❈❤♦ (X, d, ) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ t❤ø tù ❧➭ ❝➳❝ tù ➳♥❤ ①➵ tr➟♥ ❳ s❛♦ ❝❤♦ T X ⊆ f X✳ ●✐➯ sö (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ ✈➭ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ψ(d (T x, T y)) ≤ ψ (M (x, y)) − ϕ (M (x, y)) , ✈í✐ ♠ä✐ x, y ∈ X ψ(t) − ϕ(t) ≥ 0✱ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ fx t≥0 f y✱ tr♦♥❣ ➤ã ψ ∈ Ψ ✈➭ ✭✷✳✸✺✮ ϕ ∈ Φ✱ t❤á❛ ♠➲♥ ✈➭ M (x, y) = max{d(f x, f y), d(f x, T x), d(f y, T y)} ❈ị♥❣ ❣✐➯ t❤✐Õt r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥ ✭✐✮ T ❧➭ ➳♥❤ ①➵ ✭✐✐✮ tå♥ t➵✐ ✭✐✐✐✮ ♥Õ✉ f ✲❦❤➠♥❣ x0 ∈ X ✱ {xn } ⊂ X ❣✐➯♠❀ s❛♦ ❝❤♦✿ f x0 T x0 ❀ ❧➭ ❞➲② s❛♦ ❝❤♦ tå♥ t➵✐ ❞➲② ❝♦♥ u, v ∈ X {xnk } ⊂ {xn } xn xn+1 s❛♦ ❝❤♦ xnk x✱ xn → x✱ t❤× k ∈ N❀ fu ó ột trị trù ợ t tr X✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ s❛♦ ❝❤♦ ✈➭ f v = T v✱ ✈í✐ ♠ä✐ ✈➭ t❤× ✭✐✈✮ ✈í✐ ♠ä✐ fu = Tu n∈N ✈í✐ ♠ä✐ fv ❧➭ s♦ s➳♥❤ ➤➢ỵ❝✳ ❑❤✐ ➤ã ✈➭ f T ✈➭ f ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ t❤× T ❈❤ø♥❣ ♠✐♥❤✳ ❳Ðt ➳♥❤ ①➵ α(x, y) = ✈➭ f T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ α : X × X → [0, +∞) ①➳❝ ➤Þ♥❤ ❜ë✐ x, y ∈ f X ♥Õ✉ ✈í✐ ❝➳❝ ❣✐➳ trÞ ❦❤➳❝ ❝đ❛ ✸✷ ✈➭ x y x, y ❑❤✐ ➤ã ❞Ô ❞➭♥❣ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ r➺♥❣ {xn } ⊂ X ❣✐ê ❣✐➯ sư ❦❤✐ ❧➭ ❞➲② s❛♦ ❝❤♦ ❧➭ ➳♥❤ ①➵ f ✲α✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✳ ❇➞② α(xn , xn+1 ) ≥ 1✱ ✈í✐ ♠ä✐ n ∈ N ✈➭ xn → x ∈ X n → +∞✳ ◆❤ê ❝➳❝❤ ①➳❝ ➤Þ♥❤ ❝đ❛ ❤➭♠ α✱ t❛ ❝ã xn , xn+1 ∈ f X ❱× T fX ✈➭ xn ✈í✐ ♠ä✐ n ∈ N ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ✱ ♥➟♥ t❛ s✉② r❛ ➤➢ỵ❝ ♥❤ê ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✐✮✱ tå♥ t➵✐ ❞➲② ❝♦♥ ➤Þ♥❤ ❝đ❛ ♥÷❛ xn+1 , {xnk } ⊂ {xn } xnk s❛♦ ❝❤♦ α s✉② r❛ α(xnk , x) ≥ 1✱ ✈í✐ ♠ä✐ k ∈ N ✈➭ ❞♦ ➤ã X α(xn , xm ) ≥ 1✱ ✈í✐ ♠ä✐ m, n ∈ N ✈➭ m < n✳ ❧➭ x✳ x ∈ f X✳ ▲➵✐ ❚õ ❝➳❝❤ ①➳❝ α✲❝❤Ý♥❤ q✉✐✳ ❍➡♥ ❱× t❤Õ✱ ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✐✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✸ ❧➭ ➤ó♥❣✳ ▲❐♣ ❧✉❐♥ t➢➡♥❣ tù t❛ s✉② r❛ r➺♥❣ tõ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✮ ✈➭ ✭✐✈✮ ❝đ❛ ➤Þ♥❤ ❧ý ♥➭② s✉② r❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✐✐✮ ✈➭ ✭✐✈✮ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✸ ❝ị♥❣ t❤á❛ ♠➲♥✳ ◆❤➢ ✈❐② ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✸ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ ▲➵✐ ❞♦ x, y ∈ X s❛♦ ❝❤♦ fx fy ✈➭ ψ (α(f x, f y)d (T x, T y)) = α(f x, f y) = 1✱ ✈í✐ ♠ä✐ ♥➟♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ tr♦♥❣ ❜✃t tứ ũ ợ tỏ ì tế ❞ơ♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✸ t❛ s✉② r❛ T ✈➭ f ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ ❚õ ➜Þ♥❤ ❧ý ✷✳✷✳✸ t❛ ❝ã t❤Ó s✉② r❛ ♠ét sè ❦Õt q✉➯ s❛✉ ➤➞②✳ ◆❤➽❝ ❧➵✐ r➺♥❣ t❛ ❦ý ❤✐Ö✉ s❛♦ ❝❤♦ γ Λ ❧➭ t❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ ❤➭♠ γ : [0, +∞) → [0, +∞) ❦❤➯ tÝ❝❤ ▲❡❜❡s❣✉❡ tr ỗ t ủ [0, +) ỗ số > t ó (s)ds > 0✳ ❍Ö q✉➯✳ ✷✳✷✳✼ X ✭❬✽❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ T, f : X → ❧➭ ❝➳❝ tù ➳♥❤ ①➵ tr➟♥ ❳ s❛♦ ❝❤♦ (f X, d) r➺♥❣ TX ⊆ fX ✈➭ α : X × X → [0, +∞)✳ ●✐➯ sư ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ ✈➭ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ α(f x,f y)d(T x,T y) M (x,y) γ(s)ds ≤ tr♦♥❣ ➤ã M (x,y) γ(s)ds − γ, δ ∈ Λ δ(s)ds, ✈í✐ ♠ä✐ ✈➭ M (x, y) = max{d(f x, f y), d(f x, T x), d(f y, T y)} ❈ò♥❣ ❣✐➯ t❤✐Õt r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥ ✸✸ x, y ∈ X, ✭✐✮ T f ✲α✲❝❤✃♣ ❧➭ x0 ∈ X ✱ ✭✐✐✮ tå♥ t➵✐ ✭✐✐✐✮ α✲❝❤Ý♥❤ ✈➭ f s❛♦ ❝❤♦ X ❧➭ ❝ã α(xm , xn ) ≥ 1✱ ✭✐✈✮ ❤♦➷❝ ❑❤✐ ➤ã ♥❤❐♥ ➤➢ỵ❝❀ T q✉✐ ✈➭ ✈í✐ ♠ä✐ ❞➲② ✈í✐ ♠ä✐ α(f u, f v) ≥ 1✱ f ✈➭ α(f x0 , T x0 ) ≥ 1❀ ❤♦➷❝ m, n ∈ N {xn } ⊂ X s❛♦ ❝❤♦ T α(f v, f u) ≥ ✈➭ f t❛ m < n❀ ♠➭ fu = Tu ❦❤✐ ♠➭ ❝ã ♠ét trị trù ợ t tr t tí ②Õ✉✱ t❤× α(xn , xn+1 ) ≥ 1✱ X✳ ✈➭ f v = T v✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ t ψ : [0, +∞) → [0, +∞) ❈❤ø♥❣ ♠✐♥❤✳ ❳Ðt ❤➭♠ ❝❤♦ ❜ë✐ ψ(t) = γ(s)ds✱ ✈í✐ t ♠ä✐ t ∈ [0, +∞) ✈➭ ❤➭♠ ϕ : [0, +∞) → [0, +∞) ❝❤♦ ❜ë✐ δ(s)ds✱ ϕ(t) = ✈í✐ ♠ä✐ t ∈ [0, +∞)✳ ❉Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ r➺♥❣ ψ ∈ Ψ ✈➭ ϕ ∈ Φ✳ ❱× t❤Õ✱ ➳♣ ❞ơ♥❣ trự tế ị ý t ó ợ ết q ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷✳✽ ❍Ö q✉➯✳ X →X ✭❬✽❪✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❧➭ ❝➳❝ tù ➳♥❤ ①➵ tr➟♥ ❳ s❛♦ ❝❤♦ (f X, d) ●✐➯ sö r➺♥❣ TX ⊆ fX ✈➭ T, f : α : X × X → [0, +∞)✳ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ ✈➭ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ α(f x,f y)d(T x,T y) M (x,y) γ(s)ds ≤ k tr♦♥❣ ➤ã ✭✐✮ k ∈ [0, 1)✳ T ❧➭ x, y ∈ X ✱ x0 ∈ X ✱ α✲❝❤Ý♥❤ ♥❤❐♥ ➤➢ỵ❝❀ s❛♦ ❝❤♦ X ❧➭ ❝ã α(xm , xn ) ≥ 1✱ ✭✐✈✮ ❤♦➷❝ ✈í✐ ♠ä✐ ❈ị♥❣ ❣✐➯ t❤✐Õt r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➢ỵ❝ t❤á❛ ♠➲♥ f ✲α✲❝❤✃♣ ✭✐✐✮ tå♥ t➵✐ ✭✐✐✐✮ γ(s)ds, α(f x0 , T x0 ) ≥ 1❀ q✉✐ ✈➭ ✈í✐ ♠ä✐ ❞➲② ✈í✐ ♠ä✐ α(f u, f v) ≥ 1✱ ❤♦➷❝ m, n ∈ N {xn } ⊂ X ♠➭ α(f v, f u) ≥ ✸✹ s❛♦ ❝❤♦ α(xn , xn+1 ) ≥ 1✱ m < n❀ ❦❤✐ ♠➭ fu = Tu ✈➭ f v = T v✳ t❛ ❑❤✐ ➤ã ✈➭ f T ✈➭ f ❝ã ♠ét trị trù ợ t tr t tí ②Õ✉✱ t❤× ❈❤ø♥❣ ♠✐♥❤✳ δ(s) = (1 − k)γ(s)✱ T ✈➭ f ❍➡♥ ♥÷❛✱ ♥Õ✉ T ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ ❚õ ❍Ö q✉➯ ✷✳✷✳✼✱ ❜➺♥❣ ❝➳❝❤ ❧✃② ✈í✐ ♠ä✐ X✳ s ∈ [0, +∞)✱ ✈í✐ k δ ∈ Λ ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ ❧➭ ❤➺♥❣ sè s❛♦ ❝❤♦ k ∈ [0, 1) t❤× t❛ ❝ã ♥❣❛② ❦Õt q✉➯ ❝➬♥ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷✳✾ ❱Ý ❞ô✳ tr♦♥❣ X ❈❤♦ ✳ ❚❛ ①➳❝ ➤Þ♥❤ ♠➟tr✐❝ s✉② ré♥❣ d 1 1 , =d , = ✱ 5 d d 1 1 , =d , = ✱ 5 d(x, y) = |x − y| ✈í✐ ❝➳❝ ❣✐➳ trÞ ❦❤➳❝ ❝đ❛ x, y ✳ T :X→X (X, d) 1 1 , =d , = ✱ 5 ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ✳ ❳Ðt ➳♥❤ ❝❤♦ ❜ë✐   Tx =  1−x ✈➭ ❝➳❝ ❤➭♠ ♠ä✐ d ♥❤➢ s❛✉ ❉Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ r➺♥❣ ①➵ 1 1 , , , X = [0, 1] ✈➭ A = ψ, ϕ : [0, +∞) → [0, +∞) ♥Õ✉ x∈A ♥Õ✉ x ∈ [0, 1] \ A, ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ ψ(t) = t, ϕ(t) = t , ✈í✐ t ∈ [0, +∞)✳ ❈✉è✐ ❝ï♥❣✱ ①Ðt ❤➭♠ α : X × X → [0, +∞) ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ α(x, y) = ♥Õ✉ ✈í✐ ❝➳❝ ❣✐➳ trÞ ❦❤➳❝ ❝đ❛ x, y α t❤á❛ ♠➲♥ t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ❍Ư q✉➯ ✷✳✷✳✹✳ ❉♦ ➤ã T ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr➟♥ X ✱ ➤ã ❧➭ x = ✳ ❑❤✐ ➤ã T x, y ∈ A ❤♦➷❝ x = y ✈➭ ✸✺ ❝ã ♠ét ❑Õt ❧✉❐♥ ❙❛✉ t❤ê✐ ❣✐❛♥ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉ ✈Ị ➤Ị t➭✐✿ ✧➜✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ (ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✧✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ t❐♥ t×♥❤✱ ❝❤✉ ➤➳♦ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉✿ ✶✳ ❍Ư t❤è♥❣ ❤ã❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ t❤ø tù✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ α✲❝❤Ý♥❤ q✉✐❀ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❤❛✐ ➳♥❤ ①➵✱ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ ②Õ✉✱ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ ➳♥❤ ①➵ ➤➢ỵ❝✱ ➳♥❤ ①➵ α✲ f ✲❦❤➠♥❣ ❣✐➯♠✱ ➳♥❤ ①➵ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ f ✲α✲ ❝❤✃♣ ♥❤❐♥ (α, ψ, ϕ)✲❝♦❀ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❞➲② ❤é✐ tô✱ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ✷✳ rì ó ệ tố ột số ị ý ề ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❑❛♥❛♥ ✈➭ ❝➳❝ ♠ë ré♥❣ ❝đ❛ ❝❤ó♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣❀ ❣✐í✐ t❤✐Ư✉ ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ (ψ, ϕ)✲❝♦ ②Õ✉✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ (α, ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ♠ét sè ➤Þ♥❤ ❧ý ✈➭ ❤Ư q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ (ψ, ϕ)✲❝♦✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉✱ (α, ψ, ϕ)✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❝❤➻♥❣ ❤➵♥ ♥❤➢ ➜Þ♥❤ ❧ý ✶✳✷✳✶✱ ➜Þ♥❤ ý ị ý ị ý trì ợ ý ứ ột số ị ý ệ q✉➯ ❦❤➳❝ ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ♠✐♥❤ ❤♦➷❝ ♠➠ t➯ ♥❣➽♥ ❣ä♥✳ ✹✳ ●✐í✐ t❤✐Ư✉ ❝❤✐ t✐Õt ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛✳ ❱Ý ❞ô ✷✳✶✳✻ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✶✳✷ ✈➭ ❱Ý ❞ơ ✷✳✷✳✾ ♠✐♥❤ ❤ä❛ ❝❤♦ ➜Þ♥❤ ❧ý ✷✳✷✳✸✳ ✸✻ t➭✐ ❧✐Ư✉ t ỗ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ❆✳ ❆③❛♠✱ ▼✳ ❆rs❤❛❞ ✭✷✵✵✽✮✱ ❑❛♥❛♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✶ ✭✶✮✱ ✹✺✲✹✽✳ ❬✸❪ ❆✳ ❇r❛♥❝✐❛r✐ ✭✷✵✵✵✮✱ ❆ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠ ♦❢ ❇❛♥❛❝❤✲❈❛❝❝✐♣♣♦❧✐ t②♣❡ ♦♥ ❛ ❝❧❛sss ♦❢ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ P✉❜❧✳ ▼❛t❤✳ ❉❡❜r❡❝❡♥✱ ✺✼ ✭✶✲ ✷✮✱ ✸✶✲✸✼✳ ❬✹❪ ❈✳ ❉✐ ❇❛r✐ ❛♥❞ P✳ ❱❡tr♦ ✭✷✵✶✷✮✱ ❈♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❆♣♣❧✳ ▼❛t❤✳ ❈♦♠♣✉t✳✱ ✷✶✽✱ ✼✸✷✷✲✼✸✷✺✳ ❬✺❪ ◆✳ ❈❛❦✐❝ ✭✷✵✶✸✮✱ ❈♦✐♥❝✐❞❡♥❝❡ ❛♥❞ ❝♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r (ψ, ϕ)✲ ✇❡❛❦❧② ❝♦tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❋✐❧♦✲ ♠❛t✱ ✷✼ ✭✽✮✱ ✶✹✶✺ ✲ ✶✹✷✸✳ ❬✻❪ P✳ ◆✳ ❉✉tt❛✱ ❇✳ ❙✳ ❈❤♦✉❞❤✉r② ✭✷✵✵✽✮✱ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ✐♥ ♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✷✵✵✽✱ ✽ ♣❛❣❡s✱ ■❉ ✹✵✻✸✻✽✳ ❬✼❪ ❍✳ ▲❛❦③✐❛♥✱ ❇✳ ❙❛♠❡t ✭✷✵✶✷✮✱ ❋✐①❡❞ ♣♦✐♥ts ❢♦r (ψ, ϕ)✲✇❡❛❦❧② ❝♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ❆♣♣❧✳ ▼❛t❤✳ ▲❡tt❡rs✱ ✷✺✱ ✾✵✷✲ ✾✵✻✳ ❬✽❪ ❱✳ ▲✳ ❘♦s❛✱ P✳ ❱❡tr♦ ✭✷✵✶✹✮✱ ❈♦♠♠♦♥ ❢✐①❡❞ ♣♦✐♥ts ❢♦r (α, ψ, ϕ)✲ ❝♦♥tr❛❝t✐♦♥s ✐♥ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✿ ▼♦❞❡❧❧✐♥❣ ❛♥❞ ❈♦♥tr♦❧✱ ✾ ✭✶✮✱ ✹✸ ✲ ✺✹✳ ❬✾❪ ❇✳ ❙❛♠❡t ✭✷✵✵✾✮✱ ❆ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ❛ ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡s ❢♦r ♠❛♣♣✐♥❣s s❛t✐s❢②✐♥❣ ❛ ❝♦♥tr❛❝t✐✈❡ ❝♦♥❞✐t✐♦♥ ♦❢ ✐♥t❡❣r❛❧ t②♣❡✱ ■♥t✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳✱ ✸ ✭✷✻✮✱ ✶✷✻✺✲✶✷✼✶✳ ✸✼ ❬✶✵❪ ❇✳ ❙❛♠❡t✱ ❈✳ ❱❡tr♦✱ P✳ ❱❡tr♦ ✭✷✵✶✷✮✱ ❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r (α, ϕ)✲ ❝♦♥tr❛❝t✐✈❡ t②♣❡ ♠❛♣♣✐♥❣s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✺✱ ✷✶✺✹ ✲ ✷✶✻✺✳ ❬✶✶❪ ■✳ ❘✳ ❙❛r♠❛✱ ❏✳ ▼✳ ❘❛♦✱ ❙✳ ❙✳ ❘❛♦ ✭✷✵✵✾✮✱❈♦♥tr❛❝t✐♦♥s ♦✈❡r ❣❡♥❡r❛❧✐③❡❞ ♠❡tr✐❝ s♣❛❝❡✱ ❏✳ ◆♦♥❧✐♥❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✷ ✭✸✮✱ ✶✽✵✲✶✽✷✳ ✸✽