❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❯❇◆❉ tØ♥❤ ❚❤❛♥❤ ❍ã❛ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❍å♥❣ ➜ø❝ ◆❣✉②Ô♥ ❚❤❛♥❤ ❍➯✐ ▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❚ã▼ ❚➽❚ ▲❯❐◆ ❱→◆ ❚❍➵❈ ❙Ü ❚❤❛♥❤ ❍ã❛ ✲ ✷✵✶✻ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ tr➢ê♥❣ ➜➵✐ ❍ä❝ ❍å♥❣ ➜ø❝✳ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥✿ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✳ P❤➯♥ ❜✐Ö♥ ✶✿ P●❙✳ ❚❙✳ ➜✐♥❤ ❍✉② ❍♦➭♥❣ P❤➯♥ ❜✐Ư♥ ✷✿ ❚❙✳ ❍♦➭♥❣ ◆❛♠ ▲✉❐♥ ✈➝♥ ➤➢ỵ❝ ❜➯♦ ✈Ư t➵✐ ❍é✐ ➤å♥❣ ❝❤✃♠ ❧✉❐♥ ✈➝♥ ❚❤➵❝ sÜ ❦❤♦❛ ❤ä❝ ❚➵✐✿ ❱➭♦ ❤å✐✿✾ ❣✐ê ✸✵ ♥❣➭② ✵✺ t❤➳♥❣ ✶✶ ♥➝♠ ✷✵✶✻ ❈ã t❤Ĩ t×♠ ❤✐Ĩ✉ ❧✉❐♥ ✈➝♥ t➵✐ ❚❤➢ ✈✐Ư♥ tr➢ê♥❣ ➜➵✐ ❍ä❝ ❍å♥❣ ➜ø❝✱ ❤♦➷❝ ❇é ♠➠♥ ●✐➯✐ tÝ❝❤✱ tr➢ê♥❣ ➜➵✐ ❍ä❝ ❍å♥❣ ➜ø❝✳ ✶ ▼ë ➤➬✉ ✶✳ ▲ý ❞♦ ❝❤ä♥ ➤Ò t➭✐ ❚r♦♥❣ ❣✐➯✐ tÝ❝❤ ❤➭♠ ♣❤✐ t✉②Õ♥✱ ▲ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤❛♥❣ ♥❣➭② ❝➭♥❣ ➤➢ỵ❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✱ ❜ë✐ ✈× ♥ã ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣ ❦❤➠♥❣ ❝❤Ø tr♦♥❣ ♠ét sè ❝❤✉②➟♥ ♥❣➭♥❤ ❝ñ❛ t♦➳♥ ❤ä❝✱ ❝➳❝ ♥❣➭♥❤ ❦ü t❤✉❐t ♠➭ ❝ß♥ tr♦♥❣ ❝➳❝ ♥❣➭♥❤ ✈Ị ❦✐♥❤ tÕ✳ ❈➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❧➭ ❝➠♥❣ ❝ơ ❤÷✉ Ý❝❤ tr♦♥❣ ✈✐Ư❝ ❦❤➯♦ s➳t sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ❝➳❝ ❜➭✐ t♦➳♥ ❧✐➟♥ q✉❛♥ ➤Õ♥ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✱ ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ ✈➭ ♣❤➢➡♥❣ tr×♥❤ ➤➵♦ ❤➭♠ r✐➟♥❣✳ ❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧Ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❇❛♥❛❝❤✳ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ✭✶✾✷✷✮ ❧➭ ❦Õt q✉➯ ❦❤ë✐ ➤➬✉ ❝❤♦ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❞➵♥❣ ❝♦✱ ♥❤➢♥❣ ♣❤➯✐ ➤Õ♥ ♥❤÷♥❣ ♥➝♠ ✻✵ ❝đ❛ t❤Õ ❦û ❳❳ ♠í✐ ➤➢ỵ❝ ♣❤➳t tr✐Ĩ♥ ♠➵♥❤ ♠Ï✳ ◆ã ➤➲ trë t❤➭♥❤ ♠ét ❝➠♥❣ ❝ơ ♣❤ỉ ❞ơ♥❣ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ❜➭✐ t♦➳♥ ✈Ị sù tå♥ t➵✐ tr♦♥❣ ♥❤✐Ị✉ ♥❣➭♥❤ ❝đ❛ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝ ✈➭ ❝➳❝ ø♥❣ ❞ơ♥❣ ❝đ❛ ♥ã✳ ❱× t❤Õ ➤➲ ❝ã ♠ét sè ❧í♥ ❝➳❝ ♠ë ré♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ❝➡ ❜➯♥ ♥➭② ❝❤♦ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✱ ❜➺♥❣ ❝➳❝❤ ➤✐Ò✉ ❝❤Ø♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝➡ ❜➯♥ ❤♦➷❝ t❤❛② ➤ỉ✐ ❦❤➠♥❣ ❣✐❛♥✳ ◆➝♠ ✶✾✻✽✱ ❑❛♥♥❛♥ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ♠ét ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ ①➵ ❝♦ ♠➭ ♥ã ❦❤➠♥❣ ➤ß✐ ❤á✐ tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ ➳♥❤ ①➵✳ ❚❤❡♦ ❤➢í♥❣ ♠ë ré♥❣ ♥➭②✱ ♥➝♠ ✷✵✵✹✱ ❇❡r✐♥❞❡ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ❝➳❝ ➳♥❤ ①➵ ❝♦ ②Õ✉✱ ♠➭ ♥ã ❝ị♥❣ ❝ß♥ ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❤➬✉ ❝♦ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ❝➳❝ ➳♥❤ ①➵ ❤➬✉ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ❙❛✉ ➤ã ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ ❦❤➳❝ t✐Õ♣ tơ❝ ♥❣❤✐➟♥ ❝ø✉ t❤❡♦ ❤➢í♥❣ ♥➭② ✈➭ t❤✉ ➤➢ỵ❝ ♥❤✐Ị✉ ❦Õt q✉➯ t❤ó ✈Þ✳ ➜➷❝ ❜✐Ưt ♥➝♠ ✷✵✵✼✱ t❤❡♦ ❤➢í♥❣ ♠ë ré♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❍✉❛♥❣ ▲♦♥❣✲ ●✉❛♥❣ ✈➭ ❩❤❛♥❣ ❳✐❛♥ ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ư♠ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❜➺♥❣ ❝➳❝❤ t❤❛② ➤ỉ✐ t❐♣ sè t❤ù❝ tr♦♥❣ ➤Þ♥❤ ♥❣❤Ü❛ ♠➟tr✐❝ ❜ë✐ ♠ét ♥ã♥ ➤Þ♥❤ ❤➢í♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ➤Þ♥❤ ❝❤✉➮♥✳ ❈➳❝ t➳❝ ❣✐➯ ❝ị♥❣ ➤➲ ①➞② ❞ù♥❣ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ✈Ị sù ❤é✐ tơ ❝đ❛ ❞➲②✱ tÝ♥❤ ➤➬② ➤đ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥✱ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤è✐ ✈í✐ ➳♥❤ t ợ ữ ết q s s tr➟♥ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♥➭②✱ ➤å♥❣ t❤ê✐ ❝ị♥❣ t❤✃② ➤➢ỵ❝ ♠ét sè ø♥❣ ❞ơ♥❣ ❝đ❛ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ tr♦♥❣ ✷ ❣✐➯✐ tÝ❝❤ ♣❤✐ t✉②Õ♥✱ tè✐ ➢✉ ✈Ð❝t➡✳ ❍✐Ö♥ ♥❛② ♥❣❤✐➟♥ ❝ø✉ ❝✃✉ tró❝ ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ➤❛♥❣ t❤✉ ❤ót sù q✉❛♥ t➞♠ ❝đ❛ ♠ét sè ♥❤➭ t♦➳♥ ❤ä❝ tr♦♥❣ ✈➭ ♥❣♦➭✐ ♥➢í❝✳ ◆❣➢ê✐ t❛ ❝ị♥❣ ➤➲ t×♠ ❝➳❝❤ ♠ë ré♥❣ ❦❤➳✐ ♥✐Ư♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❜➺♥❣ ❝➳❝❤ t❤❛② t❤Õ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ❜➺♥❣ t tứ ì ữ t ể r ♥✐Ö♠ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ tÝ♥❤ ❝❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣ tr➟♥ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♥➭②✳ ➜Ĩ t❐♣ ❞➢ỵt ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✱ ❝❤ó♥❣ t➠✐ t✐Õ♣ ❝❐♥ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♥❤➺♠ t×♠ ❤✐Ĩ✉ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❚r➟♥ ❝➡ së ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳❚❙✳❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤ù❝ ❤✐Ư♥ ➤Ị t➭✐✿ ✧▼ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✧✳ ✷✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ➤Ị t➭✐ ✲ ❚×♠ ❤✐Ĩ✉ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr➟♥ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ♥➭②✳ ✲❚×♠ ❝➳❝❤ ♠ë ré♥❣ ♠ét ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr➟♥ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ✸✳ ➜è✐ t➢ỵ♥❣ ✈➭ ♣❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ✲ ➜è✐ t➢ỵ♥❣ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉✱✳✳✳ ✲ P❤➵♠ ✈✐ ♥❣❤✐➟♥ ❝ø✉ ❧➭ ❝➳❝ tÝ♥❤ t ố q ệ ữ ố tợ tr ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ✹✳ P❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ✸ ✲ ❙ư ❞ơ♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ ❧ý t❤✉②Õt✱ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ t➭✐ ❧✐Ư✉ ✈➭ sư ❞ơ♥❣ ♠ét sè ❦ü t❤✉❐t ❝❤ø♥❣ ♠✐♥❤ ♠í✐ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ✈✃♥ ➤Ò ➤➷t r❛✳ ✲ ❉ï♥❣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♥❣❤✐➟♥ ❝ø✉ tr♦♥❣ ❣✐➯✐ tÝ❝❤✱ t➠♣➠✱ ❣✐➯✐ tÝ❝❤ ❤➭♠ ❦Õt ❤ỵ♣ ❝➳❝ ♣❤➢➡♥❣ ♣❤➳♣ ♣❤➞♥ tÝ❝❤ tỉ♥❣ ❤ỵ♣✱ s♦ s➳♥❤ ✱ ❦❤➳✐ q✉➳t ❤♦➳✳✳✳ ➤Ĩ tr×♥❤ ❜➭② ♠ét ❝➳❝❤ ❤Ư t❤è♥❣ ❝➳❝ ❦✐Õ♥ t❤ø❝ ❧✐➟♥ q✉❛♥ ➤Õ♥ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ✺✳ ❈✃✉ tró❝ ❧✉❐♥ ✈➝♥ ▲✉❐♥ ✈➝♥ ❣å♠ ✷ ❝❤➢➡♥❣✳ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❑❛♥♥❛♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ❈➳❝ ♥é✐ ❞✉♥❣ ❣å♠✿ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ✈➭ ❦Õt q✉➯ ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ ♥ã♥✱ ♥ã♥ ❝❤✉➮♥ t➽❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ❞➲② ❈❛✉❝❤②✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤đ✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ị✉ ❦✐Ư♥ ❝♦✱ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣✳ ▼è✐ q✉❛♥ ❤Ư ❣✐÷❛ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ tr➟♥ ✈➭ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ụ ú t trì ột số ị ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ t ộ ệ q ợ trì ột sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ➤Þ♥❤ ❧ý ➤➲ ➤➢❛ r❛ ë tr➟♥✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ➤➢ỵ❝ tr×♥❤ ❜➭②✳ ❈❤♦ ♠ét sè ✈Ý ❞ơ ✈➭ tr×♥❤ ❜➭② ♠ét sè ❤Ư q✉➯ ❧✐➟♥ q✉❛♥✳ ▼ơ❝ ✷ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉✱ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✳ ❚r×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ ❦Õt q✉➯ ➤ã ✈➭ tr×♥❤ ❜➭② ♠ét sè ❤Ư q✉➯ ✈➭ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ✹ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❑❛♥♥❛♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ❈➳❝ ❦❤➳✐ ♥✐Ư♠ ❝➡ ❜➯♥ ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ö❝ trì ố q ệ ữ ❦❤➳✐ ♥✐Ö♠✱ ❦Õt q✉➯ tr➟♥ ✈➭ ❝❤♦ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ ♠➟tr✐❝ tr➟♥ X ❈❤♦ t❐♣ ❤ỵ♣ X 6= φ✳ ❍➭♠ d(x, y) ≥ ✈í✐ ♠ä✐ x, y ∈ X ✭✷✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✸✮ d(x, y) ≤ d(x, z) + d(z, y) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ ✈➭ ❦Ý ❤✐Ư✉ ❧➭ ➤Õ♥ ➤✐Ĩ♠ ✶✳✶✳✷ ✈➭ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤✐Ó♠ x y✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) → (Y, ρ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ ρ[f (x) , f (y)] ≤ αd (x, y) , ✶✳✶✳✸ ➤đ✱ f :X→X s❛♦ ❝❤♦ ➜✐Ĩ♠ f✳ ❧➭ ➳♥❤ ①➵ ❝♦ tõ X (X, d) ✈➭ (Y, ρ)✳ ➳♥❤ ①➵ f : α ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✮ ●✐➯ sư ị ý x X ợ ọ ột ế t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮ ❚❐♣ ①➵ d : X ×X →R x, y ∈ X (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑❤✐ ➤ã tå♥ t➵✐ ❞✉② ♥❤✃t ➤✐Ó♠ f (x∗ ) = x∗ ✳ x∗ ∈ X ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ✺ t ợ ị ĩ X 6= d : X ìX R ợ ọ ột tr s✉② ré♥❣ tr➟♥ X ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ ✭✷✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✸✮ d(x, y) ≤ d(x, w) + d(w, z) + d(z, y) ♣❤➞♥ ❜✐Öt ❚❐♣ X ✈í✐ ♠ä✐ x, y ∈ X ✈➭ ✈í✐ ♠ä✐ ❝➷♣ ➤✐Ó♠ w, z ∈ X \ {x, y}✳ ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ s✉② ré♥❣ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ ❦Ý ❤✐Ö✉ ❧➭ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ➜✐Ị✉ ❦✐Ư♥ ✭✸✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ❜✃t ➤➻♥❣ t❤ø❝ tø ❣✐➳❝✳ ✶✳✶✳✺ ✈➭ ●✐➯ sö ◆❤❐♥ ①Ðt✳ ε>0 t❛ ❦ý ❤✐Ö✉ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✳ B(x, ε) = {y ∈ X : d(x, y) < ε}✳ X, r > 0} ❧❐♣ t❤➭♥❤ ♠ét ❝➡ së ❝đ❛ ♠ét t➠♣➠ τd ✶✳✶✳✻ ❝❤♦ ❱Ý ❞ơ✳ ❳Ðt X = {t, 2t, 3t, 4t, 5t} ✈í✐ ❑❤✐ ➤ã ❤ä tr➟♥ t>0 ❱í✐ x∈X B = {B(x, r) : x ∈ X✳ ❧➭ ❤➺♥❣ sè✳ ❈❤♦ sè γ ∈X s❛♦ γ > 0✳ ❚❛ ①➳❝ ➤Þ♥❤ ❤➭♠ d : X × X → R ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝ ✭❛✮ d(x, x) = ✈í✐ ♠ä✐ x ∈ X ✳ ✭❜✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭❝✮ d(t, 2t) = 3γ ✳ ✭❞✮ d(t, 3t) = d(2t, 3t) = γ ✳ ✭❡✮ d(t, 4t) = d(2t, 4t) = d(3t, 4t) = 2γ ✳ ✭❢✮ d(t, 5t) = d(2t, 5t) = d(3t, 5t) = d(4t, 5t) = 23 γ ✳ ❑❤✐ ➤ã ❞Ơ ❞➭♥❣ t❤ư t❤✃② r➺♥❣ ♥❤➢♥❣ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ (X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ✈× t❛ ❝ã d(t, 2t) = 3γ > γ + γ = d(t, 3t) + d(3t, 2t) ✶✳✶✳✼ ➜Þ♥❤ ♥❣❤Ü❛✳ {xn } ⊂ X ❈❤♦ ➤➢ỵ❝ ❣ä✐ ❧➭ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✭❣✳♠✳s✮✳ ❉➲② ❤é✐ tơ ✈Ị ➤✐Ĩ♠ x ∈ X n ≥ n0 xn → x ❦❤✐ n → +∞✳ (X, d) t❛ ❝ã d (xn , x) < ε✳ ♥Õ✉ ✈í✐ ♠ä✐ ε>0 ▲ó❝ ➤ã t❛ ❦Ý ❤✐Ư✉ ❧➭ tå♥ t➵✐ n ∈ N∗ lim xn = x ❤❛② n→+∞ ✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✽ ❞➲② tr♦♥❣ X✳ ❈❤♦ (X, d) ❚❛ ♥ã✐ r➺♥❣ ε > 0✱ tå♥ t➵✐ nε ∈ N∗ {xn } ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✈➭ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❚❛ ❦ý ❤✐Ư✉ ♥Õ✉ ✈í✐ ♠ä✐ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤đ ♥Õ✉ (X, d) ➤Ị✉ ❤é✐ tơ tr♦♥❣ ♥ã✳ Ψ ❧➭ t❐♣ ❝➳❝ ❤➭♠ ψ : [0; +∞) → [0; +∞) t❤á❛ ♠➲♥✿ ✭✶✮ ψ ✭✷✮ ψ (t) = ❦❤✐ ❝❤Ø ❦❤✐ t = 0✳ ❧➭ ❤➭♠ ❧✐➟♥ tô❝ ❦❤➠♥❣ ❣✐➯♠✱ ❚❛ ❦ý ❤✐Ö✉ Φ ❧➭ t❐♣ ❝➳❝ ❤➭♠ ϕ : [0; +∞) → [0; +∞) t❤á❛ ♠➲♥✿ ✭✶✮ ϕ ❧➭ ❤➭♠ ♥đ❛ ❧✐➟♥ tơ❝✱ ✭✷✮ ϕ (t) = ❦❤✐ ❝❤Ø ❦❤✐ t = 0✳ ✶✳✶✳✶✵ ❧➭ ♠ét n > m ≥ nε ✱ t❛ ❝ã d(xn , xm ) < ε✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ✶✳✶✳✾ (X, d) ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❣✳♠✳s {xn } ❈❤♦ ➜Þ♥❤ ❧ý✳ ❞♦r❢❢✳ ●✐➯ sư (X, d) T :X→X ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤ñ ✈➭ ❍❛✉s✲ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ψ (d (T x, T y)) < ψ (d (x, y)) − ϕ (d (x, y)) ✈í✐ ♠ä✐ x, y ∈ X ✱ tr♦♥❣ ➤ã ψ ∈ Ψ ✈➭ ϕ : [0; +∞) → [0; +∞) ❧➭ ❧✐➟♥ tô❝ ✈➭ ϕ (t) = ❦❤✐ ❝❤Ø ❦❤✐ ✶✳✶✳✶✶ ✭✶✮ t = 0✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ➜✐Ĩ♠ y∈X x∈X ➤Ó ➳♥❤ ①➵ ✭✷✮ ❑❤✐ ➤ã✱ T T ❝ã ❞✉② ♥❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❈❤♦ t❐♣ ❤ỵ♣ ❣ä✐ ❧➭ ❣✐➳ trÞ trï♥❣ ♥❤❛✉ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ y = T x = f x✳ ✈➭ ❈➳❝ ➳♥❤ ①➵ X 6= φ ✈➭ ❤❛✐ ➳♥❤ ①➵ T, f : X → X ✳ ❑❤✐ ➤ã✱ ➤✐Ó♠ x T ✈➭ f ♥Õ✉ tå♥ t➵✐ ❣ä✐ ❧➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❝đ❛ ❝➳❝ f✳ T ✈➭ f ❣ä✐ ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ ♥Õ✉ ♥❤➢ t➵✐ ❝➳❝ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❝đ❛ ❝❤ó♥❣✱ ♥❣❤Ü❛ ❧➭ ♥Õ✉ T ✈➭ f ❣✐❛♦ ❤♦➳♥ T x = f x✱ t❤× t❛ ❝ã T f x = f T x✳ ◆➝♠ ✷✵✶✷✱ ❈✳ ❉✐✳ ❇❛r✐ ✈➭ P✳ ❱❡tr♦ ➤➲ ♠ë rộ ị ý t ợ ết q s ✼ ✶✳✶✳✶✷ ➜Þ♥❤ ❧ý✳ X → X (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ❍❛✉s❞♦r❢❢ ✈➭ T, f : ❈❤♦ T X ⊆ f X✳ ❧➭ ❤❛✐ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ●✐➯ sö (f X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ψ (d (T x, T y)) ψ (d (f x, f y)) − ϕ (d (f x, f y)) ✈í✐ ♠ä✐ x, y ∈ X ✱ ❞➢í✐ ✈➭ ϕ (t) = tr♦♥❣ ➤ã ψ ∈Ψ ❦❤✐ ❝❤Ø ❦❤✐ trï♥❣ ♥❤❛✉ tr♦♥❣ X✳ ✈➭ t = 0✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ ϕ : [0 : +∞) → [0 : +∞) ❑❤✐ ➤ã✱ T ✈➭ f T ✈➭ f ❧➭ ♥ö❛ ❧✐➟♥ tơ❝ ❝ã ❞✉② ♥❤✃t ♠ét ❣✐➳ trÞ ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉ t❤× T ✈➭ f ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ ✶✳✶✳✶✸ ➜Þ♥❤ ♥❣❤Ü❛✳ ❝♦♥ ❝đ❛ ❈❤♦ E ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ù❝ ✈➭ P ❧➭ ♠ét t❐♣ E ✳ ❚❐♣ P ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♥ã♥ ♥Õ✉ ỉ ế P t ó rỗ ✈➭ P 6= {0}✳ ✭✷✮ a, b ∈ R, a, b ≥ 0, x, y ∈ P s✉② r❛ ax + by ∈ P ✱ ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✸✮ x ∈ P ✈➭ −x ∈ P s✉② r❛ x = 0✳ ❑❤✐ ➤ã tr➟♥ ✈➭ ❝❤Ø ♥Õ✉ ♥Õ✉ E ①Ðt q✉❛♥ ❤Ö t❤ø tù ✧≤✧ ①➳❝ ➤Þ♥❤ ❜ë✐ P ♥❤➢ s❛✉✿ x ≤ y ♥Õ✉ y − x ∈ P ✳ ❈❤ó♥❣ t❛ q✉② ➢í❝ x < y ♥Õ✉ x ≤ y ✈➭ x 6= y ✱ ❝ß♥ x ≪ y y − x ∈ intP ✈í✐ intP ❧➭ ♣❤➬♥ tr♦♥❣ ❝đ❛ P ✳ ◆ã♥ P ➤➢ỵ❝ ❣ä✐ ❧➭ ❝❤✉➮♥ t➽❝ ♥Õ✉ ❝ã ♠ét sè K > s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ E ✱ t❛ ❝ã ≤ x ≤ y ❦Ð♦ t❤❡♦ kxk ≤ Kkyk, tr♦♥❣ ➤ã k.k ❧➭ ❝❤✉➮♥ tr♦♥❣ E ✳ ❙è K ❞➢➡♥❣ ♥❤á ♥❤✃t t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ tr➟♥ ➤➢ỵ❝ ❣ä✐ ❧➭ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ ❝đ❛ P✳ ❙❛✉ ➤➞② t❛ ❧✉➠♥ ❣✐➯ sö r➺♥❣ tr♦♥❣ ✶✳✶✳✶✹ E ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ P ❧➭ ♠ét ♥ã♥ E ✈í✐ intP 6= ∅ ✈➭ ✧≤✧ ❧➭ q✉❛♥ ❤Ư t❤ø tù ❜é ♣❤❐♥ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✳ ➜Þ♥❤ ♥❣❤Ü❛✳ X ột t rỗ E ❣✐❛♥ ❇❛♥❛❝❤✳ ➳♥❤ ①➵ d : X × X → E ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♠➟tr✐❝ ♥ã♥ ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ ✭✶✮ ≤ d(x, y) ✈í✐ ♠ä✐ x, y ∈ X ❀ d(x, y) = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ x = y ✳ ✭✷✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✸✮ d(x, y) ≤ d(x, z) + d(z, y) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ✽ ❑❤✐ ➤ã (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❧➭ sù tỉ♥❣ q✉➳t ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ❱Ý ❞ơ s❛✉ ❝❤ø♥❣ tá ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❧➭ sù ♠ë ré♥❣ t❤ù❝ sù ❝đ❛ ❧í♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✳ ✶✳✶✳✶✺ ✶✮ ❳Ðt ❦❤➠♥❣ ❣✐❛♥ ❱Ý ❞ô✳ R ✈í✐ ❝❤✉➮♥ t❤➠♥❣ t❤➢ê♥❣✳ ❑❤✐ ➤ã P = {x ∈ R : x ≥ 0} ❧➭ ♠ét ♥ã♥ tr♦♥❣ R✳ E = R2 ✷✮ ❈❤♦ d : X ×X → E α ✈➭ P = {(x, y) ∈ E : x, y ≥ 0}✳ ①➳❝ ➤Þ♥❤ ❜ë✐ ❳Ðt X = R ✈➭ ➳♥❤ ①➵ d(x, y) = (|x − y|, α|x − y|) ✈í✐ ♠ä✐ x, y ∈ X ✱ tr♦♥❣ ➤ã ❧➭ sè t❤ù❝ ❞➢➡♥❣ ❝❤♦ tr➢í❝✳ ❑❤✐ ➤ã ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✳ ✶✳✶✳✶✻ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ X ❧➭ ♠ét t rỗ sử d : X ìX E ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ✭✶✮ < d(x, y)✱ ✈í✐ ♠ä✐ x, y ∈ X, x 6= y ✭✷✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✸✮ d(x, y) ≤ d(x, u) + d(u, v) + d(v, y) ✈í✐ ♠ä✐ x, y ∈ X ♣❤➞♥ ❜✐Öt ✈➭ d(x, y) = ♥Õ✉ x = y ✈➭ ✈í✐ t✃t ❝➯ ❝➳❝ ➤✐Ó♠ u, v ∈ X \ {x, y}✳ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ tr➟♥ X ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❑❤✐ ➤ã d ✳ ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ✈➭ (X, d) ❧➭ ♠ét ◆❤❐♥ ①Ðt r➺♥❣ t➢➡♥❣ tù ♥❤➢ tr➢ê♥❣ ❤ỵ♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣✱ t❛ ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❜✃t ❦ú ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❚✉② ♥❤✐➟♥ ❝❤✐Ị✉ ♥❣➢ỵ❝ ❧➵✐ ❧➭ ❦❤➠♥❣ ➤ó♥❣✳ ✶✳✶✳✶✼ ❱Ý ❞ơ✳ ❈❤♦ E = R2 ✱ P = {(x, y) ∈ E | x, y ≥ 0}✱ X = R ✈➭ d : X × X → E ❧➭ ➳♥❤ ①➵ ①➳❝ ➤Þ♥❤ ❜ë✐ d(x, y) = tr♦♥❣ ➤ã (0, 0) (3α, 3) (α, 1) ♥Õ✉ x = y, ♥Õ✉ x, y ❝ï♥❣ t❤✉é❝ tr♦♥❣ ♥Õ✉ x, y ❦❤➠♥❣ ➤å♥❣ t❤ê✐ t❤✉é❝{1, 2} ✈➭ α > ❧➭ ❤➺♥❣ sè✳ ❑❤✐ ➤ã P ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ♥❤➢♥❣ {1, 2} ✈➭ x 6= y, ❧➭ ♠ét ♥ã♥ tr♦♥❣ E ✈➭ x 6= y, (X, d) ❧➭ ♠ét ❦❤➠♥❣ (X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ✈× d(1, 2) = (3α, 3) > d(1, 3) + d(3, 2) = (2α, 3) ✾ ❧➭ ♠ét ❞➲② tr♦♥❣ ✈í✐ ♠ä✐ n > n0 ✱ ❧➭ ❣✐í✐ ❤➵♥ ❝đ❛ (X, d) ❈❤♦ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶✽ X ✈➭ t❛ ❝ã ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ {xn } x ∈ X ✳ ◆Õ✉ ✈í✐ ♠ä✐ c ∈ E, c ≫ 0✱ tå♥ t➵✐ n0 ∈ N s❛♦ ❝❤♦ d(xn , x) c tì ó {xn } ợ ọ ❧➭ ❤é✐ tơ tí✐ x ✈➭ x {xn }✳ ▲ó❝ ➤ã t❛ ❝ò♥❣ ✈✐Õt ❧➭ xn → x ❦❤✐ n → ∞✳ ❚➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ t❛ ❝ã ❦Õt q✉➯ s❛✉ ❇ỉ ➤Ị✳ ❈❤♦ ✶✳✶✳✶✾ (X, d) ♥ã♥ t❤➠♥❣ t❤➢ê♥❣✱ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ {xn } ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ❧➭ ♠ét ❞➲② tr♦♥❣ X✳ ❑❤✐ ➤ã✱ ❞➲② ❦❤✐ ❧➭ ♠ét n→∞ kd(xn , x)k → ❦❤✐ n → ∞✳ ▲➢✉ ý r➺♥❣ tr♦♥❣ ✭❬❄❪✮ ➤➲ ❝❤ø♥❣ ♠✐♥❤ r➺♥❣ ♥Õ✉ ♠➟tr✐❝ ♥ã♥ ✈➭ xn → x P {xn } X✱ ❧➭ ♠ét ❞➲② ❤é✐ tô tr♦♥❣ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ t❤× ❣✐í✐ ❤➵♥ ❝đ❛ ♥ã ❧➭ ❞✉② ♥❤✃t✳ ❚✉② ♥❤✐➟♥✱ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ❣✐í✐ ❤➵♥ ♥ã✐ ❝❤✉♥❣ ❧➭ ❦❤➠♥❣ t❤á❛ ♠➲♥✳ ❱Ý ❞ô s❛✉ ♠✐♥❤ ❤ä❛ ♥❤❐♥ ①Ðt tr➟♥ ❱Ý ❞ô✳ ✶✳✶✳✷✵ tr♦♥❣ Q ✈➭ ❈❤♦ E = R✱ P = {x ∈ R | x ≥ 0}✳ a, b ∈ R \ Q ♠➭ a 6= b✳ ❚❛ ➤➷t d : X × X → E ❧➭ ➳♥❤ ①➵ ❝❤♦ ❜ë✐ d(x, x) = (0, 0), d(x, y) = d(y, x), d(x , x ) = 1, n m d(xn , b) = n1 , d(xn , a) = n1 , d(a, b) = ❑❤✐ ➤ã {xn } ❧➭ ❞➲② ♥➺♠ X = {x1 , x2 , , xn , } ∪ {a, b} ✈í✐ ♠ä✐ x∈X ✈í✐ ♠ä✐ x, y ∈ X ✈í✐ ♠ä✐ n, m ∈ N∗ , n 6= m ✈í✐ ♠ä✐ n ∈ N∗ ✈í✐ ♠ä✐ n ∈ N∗ ✈➭ ①Ðt (X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ✈× t❛ ❝ã d(x2 , x3 ) = > d(x2 , a) + d(a, x3 ) = ❚✉② ♥❤✐➟♥✱ ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ r➺♥❣ ♥ã♥ s✉② ré♥❣ ✈➭ ✈× ❚➢➡♥❣ tù t❛ ❝ã ✈× ♥❤➢♥❣ ●✐➯ sö d(xn , a) = d(xn , b) = n n →0 →0 ❦❤✐ ❦❤✐ 1 + = (X, d) n → ∞✱ n → ∞✱ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥➟♥ t❛ ❝ã xn → a tr♦♥❣ (X, d)✳ ♥➟♥ t❛ ❝ã xn → b tr♦♥❣ (X, d)✱ a 6= b✳ ◆❤❐♥ ①Ðt r➺♥❣ tõ ✈Ý ❞ô ♥➭② t❛ t❤✃② r➺♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ♠ét ❞➲② ❤é✐ tơ ❝ã t❤Ĩ ❤é✐ tơ ✈Ị ♥❤✐Ị✉ ➤✐Ĩ♠ ❦❤➳❝ ♥❤❛✉✳ ✶✵ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❑❛♥♥❛♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✶✳✷ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❑❛♥♥❛♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤➲ ➤➢❛ r❛ ë tr➟♥✳ ❚r➢í❝ ❤Õt✱ ❝❤ó♥❣ t➠✐ ♥❤➽❝ ❧➵✐ ♠ét sè ❦❤➳✐ ♥✐Ư♠ ✈➭ tÝ♥❤ ❝❤✃t ✈Ị ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤đ✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ ✶✳✷✳✶ ♠ét ❞➲② tr♦♥❣ X✳ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ {xn } ❧➭ ◆Õ✉ ✈í✐ ❜✃t ❦ú c∈E c ≫ 0✱ ♠➭ tå♥ t➵✐ m, n > N, d(xm , xn ) c tì {xn } ợ ❣ä✐ ❧➭ ♠ét ❞➲② N s❛♦ ❝❤♦ ✈í✐ ♠ä✐ ❈❛✉❝❤② tr♦♥❣ X✳ ❚➢➡♥❣ tù ♥❤➢ tr➢ê♥❣ ❤ỵ♣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ t❛ t❤✉ ➤➢ỵ❝ ❦Õt q✉➯ s❛✉✳ ❇ỉ ➤Ị✳ ❈❤♦ ✶✳✷✳✷ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ ♥ã♥ ❝❤✉➮♥ t➽❝✱ {xn } ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ d(xn , xm ) → ◆❤❐♥ ①Ðt✳ ✶✳✷✳✸ {xn } ❧➭ ♠ét ❞➲② tr♦♥❣ ❦❤✐ X✱ ❑❤✐ ➤ã {xn } ❧➭ ♠ét ❧➭ ♠ét ❞➲② ❈❛✉❝❤② m, n → ∞✳ ◆❤➢ t❛ ➤➲ ❜✐Õt ♥Õ✉ ❧➭ ♠ét ❞➲② ❤é✐ tơ tr♦♥❣ X✳ P t❤× (X, d) {xn } ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ✈➭ ❧➭ ♠ét ❞➲② ❈❛✉❝❤② tr♦♥❣ X✳ ❚✉② ♥❤✐➟♥✱ ➤è✐ ✈í✐ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s rộ ề ó ò ú ữ t tr♦♥❣ ❱Ý ❞ô ✶✳✶✳✷✵ t❛ t❤✃② r➺♥❣ ❞➲② ❦❤✐ {xn } ❤é✐ tô✱ ♥❤➢♥❣ d(xn , xm ) → n, m → ∞✱ ✈× t❤Õ {xn } ❦❤➠♥❣ ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ X ✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ ✶✳✷✳✹ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ (X, d) X ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ◆Õ✉ ➤Ị✉ ❤é✐ tơ✱ t❤× X ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ủ r trờ ợ r s tì tí ♥❤✃t ❝đ❛ ❣✐í✐ ❤➵♥ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ ✶✳✷✳✺ ❇ỉ ➤Ị✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤đ✱ P ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ tr♦♥❣ X ✈➭ tå♥ t➵✐ ♠ét sè n0 ∈ N s❛♦ ❝❤♦ k✳ ●✐➯ sö ❧➭ {xn } ❧➭ ♠ét ❞➲② ❈❛✉❝❤② ✶✶ ✭✶✮ xn 6= xm ✭✷✮ xn , x ❧➭ ❝➳❝ ➤✐Ĩ♠ ♣❤➞♥ ❜✐Ưt tr♦♥❣ X ✈í✐ ♠ä✐ n > n0 ✳ ✭✸✮ xn , y ❧➭ ❝➳❝ ➤✐Ĩ♠ ♣❤➞♥ ❜✐Ưt tr♦♥❣ X ✈í✐ ♠ä✐ n > n0 ✳ ✭✹✮ xn → x ✈í✐ ♠ä✐ ✈➭ n, m > n0 ✳ xn → y ❑❤✐ ➤ã t❛ ❝ã ❦❤✐ n → ∞✳ x = y✳ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ❜✃t ❦ú ❈❛✉❝❤②✱ ♥➟♥ tå♥ t➵✐ sè d(xn , x) ≪ c, ❉♦ ➤ã✱ ✈í✐ ♠ä✐ m0 c∈E ♠➭ c ≫ 0✱ ✈× xn → x✱ xn → y ✈➭ {xn } ❧➭ ❞➲② s❛♦ ❝❤♦ d(xn , y) ≪ c ✈➭ d(xn , xm ) ≪ c ✈í✐ ♠ä✐ n, m > m0 n, m > max{n0 , m0 } t❛ ❝ã d(x, y) ≤ d(x, xn ) + d(xn , xm ) + d(xm , y) ≤ 3c ❱× ♥ã♥ P ❝❤✉➮♥ t➽❝✱ ♥➟♥ tõ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ s✉② r❛ ❧✃② tï② ý✱ tõ ❜✃t ➤➻♥❣ t❤ø❝ ❝✉è✐ ♥➭② t❛ s✉② r❛ kd(x, y)k ≤ 3kkck✳ ❱× d(x, y) = 0✱ ❤❛② x = y ✳ c ❇➞② ❣✐ê t❛ ①Ðt tÝ♥❤ ❝❤✃t ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ❑❛♥♥❛♥✳ ✶✳✷✳✻ P ➜Þ♥❤ ♥❣❤Ü❛✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ✱ ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✳ ➳♥❤ ①➵ T : X → X tõ X ✈➭♦ ❝❤Ý♥❤ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ❑❛♥♥❛♥ ♥Õ✉ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ d(T x, T y) ≤ α[d(T x, x) + d(T y, y)], tr♦♥❣ ➤ã ✶✳✷✳✼ X x, y ∈ X ✭✶✳✶✮ α ∈ [0, 12 ) ❧➭ sè ♥➭♦ ➤ã ❝❤♦ tr➢í❝✳ ❇ỉ ➤Ị✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤đ✱ P ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ tõ ✈í✐ ♠ä✐ K ✱ α ∈ [0, 12 ) ✈➭ ➳♥❤ ①➵ ✈➭♦ ❝❤Ý♥❤ ♥ã ❧➭ ➳♥❤ ①➵ ❝♦ ❑❛♥♥❛♥✳ ❑❤✐ ➤ã ✈í✐ ❜✃t ❦ú x∈X ❧➭ T :X→X ✈➭ ✈í✐ n ❧➭ ♠ét sè ♥❣✉②➟♥ ❞➢➡♥❣✱ t❛ ❝ã n d(T x, T tr♦♥❣ ➤ã r= α 1−α n+1 ∈ [0, 1) x) ≤ α 1−α n d(x, T x) = rn d(x, T x), ✭✶✳✷✮ ✶✷ ➜Þ♥❤ ❧ý✳ ✶✳✷✳✽ P ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ✱ ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ ➳♥❤ ①➵ T :X→X tõ X ✈➭♦ ❝❤Ý♥❤ ♥ã ❧➭ ➳♥❤ ①➵ ❝♦ ❑❛♥♥❛♥✳ ❑❤✐ ➤ã ✭✶✮ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣ ✭✷✮ ❱í✐ ❜✃t ❦ú x ∈ X✱ ❞➲② ❧➷♣ {T n x} X✳ ❤é✐ tô tí✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ❈✉è✐ ❝ï♥❣ ➤Ĩ ♠✐♥❤ ❤ä❛ ➜Þ♥❤ ❧ý ✶✳✷✳✽✱ t❛ tr×♥❤ ❜➭② ✈Ý ❞ơ s❛✉✳ ✶✳✷✳✾ ❱Ý ❞ô✳ ❈❤♦ t➽❝ tr♦♥❣ E✳ E=C P = {x + iy | x, y ∈ R, x, y ≥ 0} ❑❤✐ ➤ã q✉❛♥ ❤Ư t❤ø tù ❜é ♣❤❐♥ tr➟♥ ❱í✐ ❜✃t ❦ú z1 = x1 + iy1 ∈ E ✱ z2 = x2 + iy2 ∈ E z1 ≤ z2 ❳Ðt t❐♣ ✈➭ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ X = {1, 2, 3, 4} d(x, x) = 0, x1 ≤ x2 ✈➭ ①➳❝ ➤Þ♥❤ ❤➭♠ ✈í✐ ♠ä✐ E ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ ❝❤♦ ❜ë✐✿ t❛ ❝ã ✈➭ y1 ≤ y2 d:X ×X →E ❝❤♦ ❜ë✐ x ∈ X, d(1, 2) = d(2, 1) = + 9i, d(2, 3) = d(3, 2) = d(1, 3) = d(3, 1) = + 3i, d(1, 4) = d(4, 1) = d(2, 4) = d(4, 2) = d(3, 4) = d(4, 3) = + 12i ❑❤✐ ➤ã ❞Ơ ❞➭♥❣ ❦✐Ĩ♠ tr❛ ➤➢ỵ❝ r➺♥❣ ré♥❣ ➤➬② ➤ñ✱ ♥❤➢♥❣ (X, d) (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ❦❤➠♥❣ ♣❤➯✐ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ❜ë✐ ✈× ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝ ❦❤➠♥❣ ➤ó♥❣ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ + 9i = d(1, 2) > d(1, 3) + d(3, 2) = + 6i T : X → X ♥❤➢ s❛✉ ( ♥Õ✉ x 6= 4, Tx = ♥Õ✉ x = ❇➞② ❣✐ê t❛ ➤Þ♥❤ ♥❣❤Ü❛ ➳♥❤ ①➵ ◆❤❐♥ ①Ðt r➺♥❣ t❤➢ê♥❣ tr♦♥❣ X T ❦❤➠♥❣ ♣❤➯✐ ❧➭ ♠ét ➳♥❤ ①➵ ❝♦ ➤è✐ ✈í✐ ♠➟tr✐❝ t❤➠♥❣ ✈× t❛ ❝ã |T − T 2| = = |4 − 2| ❚✉② ♥❤✐➟♥ T ❧➵✐ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❑❛♥♥❛♥ d(T x, T y) ≤ α[d(x, T x) + d(y, T y)], ✈í✐ ♠ä✐ x, y ∈ X, ✶✸ ✈í✐ ì ụ ị ý t s✉② r❛ r➺♥❣ ∗ α = ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣ X ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ➤ã ❧➭ T ♠ét ➤✐Ó♠ ❜✃t x = 3✳ (X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥✱ ♥➟♥ t❛ ❦❤➠♥❣ t❤Ó ▲➢✉ ý r➺♥❣ ✈× ➳♣ ❞ơ♥❣ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤➲ ❜✐Õt tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ ❝❤♦ ✈Ý ❞ơ ♥➭②✳ ❍Ư q✉➯✳ ❈❤♦ ✶✳✷✳✶✵ P (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ✱ ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ ①➵ tõ X ●✐➯ sö T :X→X ❧➭ ➳♥❤ ✈➭♦ ❝❤Ý♥❤ ♥ã t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ d(T x, T y) ≤ λd(x, y), tr♦♥❣ ➤ã ✭✶✮ K✳ T λ ∈ [0, 31 )✳ ✈í✐ ♠ä✐ ❑❤✐ ➤ã ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣ ✭✷✮ ❱í✐ ❜✃t ❦ú x ∈ X✱ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② x, y ∈ X, ❞➲② ❧➷♣ x, y ∈ X ✱ {T n x} X✳ ❤é✐ tơ tí✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ t❛ ❝ã d(T x, T y) ≤ λd(x, y) ≤ λ[d(x, T x) + d(T x, T y) + d(y, T y)], ♥❣❤Ü❛ ❧➭ λ[d(x, T x) + d(T x, T y) + d(y, T y)] − d(T x, T y) ∈ P ❑❤✐ ➤ã t❛ ❝ã λ[d(x, T x) + d(y, T y)] − (1 − λ)d(T x, T y) ∈ P ❙✉② r❛ λ [d(x, T x) + d(y, T y)] − d(T x, T y) ∈ P 1−λ ❱× ✈❐② t❛ ❝ã d(T x, T y) ≤ ➜➷t α= λ 1−λ ∈ [0, 12 )✱ λ [d(x, T x) + d(y, T y)] 1−λ ❦❤✐ ➤ã ➳♥❤ ①➵ T :X→X d(T x, T y) ≤ α[d(x, T x) + d(y, T y)], t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✈í✐ ♠ä✐ x, y ∈ X ❚❛ t❤✃② r➺♥❣ t✃t ❝➯ ❝➳❝ ❣✐➯ t❤✐Õt ❝ñ❛ ị ý ề ợ tỏ T T ì tế ụ ị ý t❛ s✉② r❛ ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t tr♦♥❣ ✭✷✮ ❱í✐ ❜✃t ❦ú x ∈ X✱ ❞➲② ❧➷♣ {T n x} X✳ ❤é✐ tơ tí✐ ➤✐Ĩ♠ ❜✃t ➤é♥❣✳ ✶✹ ❝❤➢➡♥❣ ✷ ➜✐Ó♠ ❜✃t ➤é♥❣ ✈➭ ➤✐Ó♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✷✳✶ ➜✐Ó♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ✈➭ ♠ét sè ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣ ➤å♥❣ t❤ê✐ tr×♥❤ ❜➭② ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ●✐➯ sö P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ➜Þ♥❤ ♥❣❤Ü❛✳ ✷✳✶✳✶ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✱ ✧≤✧ ❧➭ t❤ø tù tr➟♥ E ①➳❝ ➤Þ♥❤ ❜ë✐ P ✳ ❍➭♠ ψ : P → P ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ t➵✐ x0 ∈ P ♥Õ✉ lim sup ψ(x) ≤ x→x0 ψ(x0 )✳ ❍➭♠ ψ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ tr➟♥ P ♥Õ✉ ♥ã ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥ t➵✐ ♠ä✐ x ∈ P ✳ ❍➭♠ ψ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ tr➟♥ P ♥Õ✉ ❤➭♠ −ψ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ tr➟♥✱ tr♦♥❣ ➤ã (−ψ)(x) = −ψ(x) ✈í✐ ♠ä✐ x ∈ P ✳ ◆ã✐ ❝➳❝❤ ❦❤➳❝✱ ❤➭♠ ψ ➤➢ỵ❝ ❣ä✐ ❧➭ ♥ư❛ ❧✐➟♥ tơ❝ ❞➢í✐ t➵✐ x0 ∈ P ♥Õ✉ lim inf ψ(x) ≥ ψ(x0 )✳ x→x0 ➜➠✐ ❦❤✐✱ t❛ ✈✐Õt lim ψ(x)✱ lim ψ(x) ❧➬♥ ❧➢ỵt t❤❛② ❝❤♦ lim sup ψ(x) ✈➭ x→x0 x→x0 x→x0 lim inf ψ(x)✳ x→x0 ❍➭♠ ψ : P → P ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐➯♠ ♥Õ✉ ✈í✐ ❜✃t ❦ú x, y ∈ P ♠➭ x ≤ y t❛ ❝ã ψ(x) ≤ ψ(y)✳ ✷✳✶✳✷ ❑ý ❤✐Ö✉✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✱ X 6= φ✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ T : X → X ❧➭ ➳♥❤ ①➵ tõ X ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❱í✐ ♠ä✐ x, y ∈ X, t❛ ➤➷t M (x, y) = max{d(x, y), d(x, T x), d(y, T y)} ✭✷✳✶✮ ✶✺ ❚❛ ❝ò♥❣ sÏ ❦ý ❤✐Ö✉ Ψ = {ψ : P → P | ❧✐➟♥ tô❝✱ ❦❤➠♥❣ ❣✐➯♠ ✈➭ ψ(t) = ⇔ t = 0}, ✈➭ Φ = {φ : P → P | ◆Õ✉ ψ∈Ψ t❤× , φ(t) > ♥ư❛ ❧✐➟♥ tụ ợ ọ X ị ĩ ❈❤♦ ❣ä✐ ❧➭ ❝ã ➤✐Ó♠ t✉➬♥ ❤♦➭♥ p≥1 ❧➭ ♠ét t ợ rỗ ế tồ t ột ể ➤ã✳ ▲ó❝ ➤ã t❛ ❝ị♥❣ ♥ã✐ ➤✐Ĩ♠ ✷✳✶✳✹ p=1 t❤× ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥ ➜Þ♥❤ ❧ý✳ u t>0 φ(0) = 0} ✈➭ ❤➭♠ t❤❛② ➤ỉ✐ ❦❤♦➯♥❣ ❝➳❝❤✳ u∈X ❉Ơ t❤✃② r➺♥❣ ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ♥Õ✉ ✈í✐ ♠ä✐ ❝đ❛ T ❧➭ T ➳♥❤ ①➵ T : X → X ➤➢ỵ❝ u∈X s❛♦ ❝❤♦ u = T p u ✈í✐ sè ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥ ❝đ❛ T✳ ❧➭ ➤✐Ĩ♠ t✉➬♥ ❤♦➭♥ ❝đ❛ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ T ✈➭ T✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ ❍❛✉s❞♦r❢❢✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ E ❧➭ t❐♣ s➽♣ tèt t❤❡♦ q✉❛♥ ❤Ö t❤ø tù ❜é ♣❤❐♥ ✧≪✧ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✭♥❣❤Ü❛ ❧➭ ❤❛✐ ♣❤➬♥ tư ❜✃t ❦ú ❝đ❛ E ❜❛♦ ❣✐ê ❝ị♥❣ s♦ s ợ t q ệ ỗ t rỗ ủ E ị ề ó ❝❐♥ ❞➢í✐ ➤ó♥❣✮✳ ●✐➯ sư r➺♥❣ T : X → X ❧➭ ➳♥❤ ①➵ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ✱ t❛ ❝ã ψ(d(T x, T y)) ≤ ψ(M (x, y)) − φ(M (x, y)), tr♦♥❣ ➤ã ψ ∈ Ψ✱ φ ∈ Φ ✈➭ M (x, y) ➤➢ỵ❝ ①➳❝ ➤Þ♥❤ ❜ë✐ ❝➠♥❣ t❤ø❝ ✭✷✳✷✮ ✭✷✳✶✮✳ ❑❤✐ ➤ã tå♥ t➵✐ ♠ét ➤✐Ó♠ ❞✉② ♥❤✃t u ∈ X s❛♦ ❝❤♦ u = T u✳ ✷✳✶✳✺ ❍Ö q✉➯✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ ❍❛✉s❞♦r❢❢✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ E ❧➭ t❐♣ s➽♣ tèt t❤❡♦ q✉❛♥ ❤Ö t❤ø tù ❜é ♣❤❐♥ ✧≪✧ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✭♥❣❤Ü❛ ❧➭ ❤❛✐ ♣❤➬♥ tư ❜✃t ❦ú ❝đ❛ E ❜❛♦ ❣✐ê ❝ị♥❣ s♦ s➳♥❤ ợ t q ệ ỗ t rỗ ủ E ị ề ó ❞➢í✐ ➤ó♥❣✮✳ ●✐➯ sư r➺♥❣ T : X → X ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ k ∈ [0, 1) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X t❛ ❝ã d(T x, T y) ≤ k max{d(x, y), d(x, T x), d(y, T y)} ❑❤✐ ➤ã✱ T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ✭✷✳✸✮ ✶✻ ❈❤ø♥❣ ♠✐♥❤✳ ❙ư ❞ơ♥❣ trù❝ t✐Õ♣ ➜Þ♥❤ ❧ý ✷✳✶✳✹ ✈í✐ ❝➳❝ ❤➭♠ ψ : P → P ✈➭ φ : P → P ❝❤♦ ❜ë✐ ψ(t) = t ✈➭ φ(t) = (1 − k)t ✈í✐ ♠ä✐ t ∈ P t❛ t❤✉ ợ ết q tì ệ q (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ ❍❛✉s❞♦r❢❢✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ E ❧➭ t❐♣ s➽♣ tèt t❤❡♦ q✉❛♥ ❤Ö t❤ø tù ❜é ♣❤❐♥ ✧≪✧ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✭♥❣❤Ü❛ ❧➭ ❤❛✐ ♣❤➬♥ tư ❜✃t ❦ú ❝đ❛ E ❜❛♦ ❣✐ê ❝ị♥❣ s♦ s ợ t q ệ ỗ t rỗ ủ E ị ề ó ❝❐♥ ❞➢í✐ ➤ó♥❣✮✳ ●✐➯ sư r➺♥❣ T : X → X ❧➭ ➳♥❤ ①➵ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ tå♥ t➵✐ sè α ∈ [0, 21 ) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X t❛ ❝ã d(T x, T y) ≤ α d(x, T x) + d(y, T y) ✭✷✳✹✮ ❑❤✐ ➤ã T ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ▲✃② k = 2α✳ ❑❤✐ ➤ã ✈× α ∈ [0, 21 ) t❛ ❝ã k ∈ [0, 1)✳ ❉♦ ➤ã ♥Õ✉ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✹✮ t❤á❛ ♠➲♥ t❤× t❛ ❝ã d(x, T x) + d(y, T y) d(T x, T y) ≤ α d(x, T x) + d(y, T y) = k ≤ k max d(x, y), d(x, T x), d(y, T y) ❱× t❤Õ✱ ♥❤ê ❍Ư q✉➯ ✷✳✶✳✺ t❛ s✉② r❛ ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✭✷✳✺✮ ❚õ ➜Þ♥❤ ❧ý ✷✳✶✳✹ t❛ t❤✉ ➤➢ỵ❝ ❦Õt q✉➯ s❛✉ ➤➞② tr♦♥❣ ❬❄❪✳ ✷✳✶✳✼ ❍Ö q✉➯✳ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ s✉② ré♥❣ ➤➬② ➤đ ❍❛✉s✲ ❞♦r❢❢✳ ●✐➯ sư r➺♥❣ T : X → X ❧➭ ➳♥❤ ①➵ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y ∈ X ✱ t❛ ❝ã ψ d(T x, T y) ≤ ψ M (x, y) − φ M (x, y) , tr♦♥❣ ➤ã ψ : [0, +∞) → [0, +∞) ❧➭ ❤➭♠ ❧✐➟♥ tô❝✱ ❦❤➠♥❣ ❣✐➯♠✱ ψ(t) = ⇔ t = 0✱ φ : [0, +∞) → [0, +∞) ❧➭ ❤➭♠ ♥ö❛ ❧✐➟♥ tơ❝ ❞➢í✐✱ φ(t) = ⇔ t = M (x, y) ợ ị tứ ✭✷✳✶✮✳ ❑❤✐ ➤ã tå♥ t➵✐ ♠ét ➤✐Ó♠ ❞✉② ♥❤✃t u ∈ X s❛♦ ❝❤♦ u = T u✳ ❈❤ø♥❣ ♠✐♥❤✳ ❙✉② trù❝ t✐Õ♣ tõ ➜Þ♥❤ ❧ý ✷✳✶✳✹ ✈í✐ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E = R ✈➭ ♥ã♥ P = [0, +∞)✳ ✶✼ ➜✐Ĩ♠ trï♥❣ ♥❤❛✉ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❝➳❝ ✷✳✷ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ❚r♦♥❣ ♣❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ sÏ ❝❤ø♥❣ ♠✐♥❤ ♠ét ✈➭✐ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❝đ❛ ❤❛✐ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣✳ ❈❤♦ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✱ ✧≤✧ ❧➭ t❤ø tù tr➟♥ E ①➳❝ ➤Þ♥❤ ❜ë✐ P ✳ ❚❛ ❦ý ❤✐Ö✉ Φ∗ ❧➭ t❐♣ ❝➳❝ ❤➭♠ ϕ : P → P t❤á❛ ♠➲♥✿ (ϕ1 ) lim inf ϕ (t) > ✈í✐ ♠ä✐ r > 0✱ t→r (ϕ2 ) ϕ (t) = ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ t = 0✳ ✷✳✷✳✶ ➜Þ♥❤ ❧ý✳ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ ❍❛✉s✲ ❞♦r❢❢✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ E ❧➭ t❐♣ s➽♣ tèt t❤❡♦ q✉❛♥ ❤Ö t❤ø tù ❜é ♣❤❐♥ ✧6✧ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✭♥❣❤Ü❛ ❧➭ ❤❛✐ ♣❤➬♥ tư ❜✃t ❦ú ❝đ❛ E ❜❛♦ ❣✐ê ❝ị♥❣ s♦ s➳♥❤ ➤➢ỵ❝ t❤❡♦ q✉❛♥ ệ ỗ t rỗ ủ E ♠➭ ❜Þ ❝❤➷♥ ❞➢í✐ ➤Ị✉ ❝ã ❝❐♥ ❞➢í✐ ➤ó♥❣✮✱ T, f : X → X ❧➭ ❤❛✐ ➳♥❤ ①➵ tr➟♥ X t❤á❛ ♠➲♥ T X ⊆ f X ✳ ●✐➯ sö r➺♥❣ (f X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ ❍❛✉s❞♦r❢❢ ✈➭ T ✱ f t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ n ψ d(T x, T y) ψ max d(f x, f y), 1 d(f x, T x) + d(f y, T y) , d(f y, T x) − ϕ d(f x, f y) o − ✭✷✳✻✮ ✈í✐ ♠ä✐ x, y ∈ X ✱ tr♦♥❣ ➤ã ψ ∈ Ψ ✈➭ ϕ ∈ Φ∗ ❑❤✐ ➤ã✱ T ✈➭ f ❝ã ♠ét ❣✐➳ trÞ trï♥❣ ♥❤❛✉ ❞✉② ♥❤✃t tr♦♥❣ X ✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ T ✈➭ f ❧➭ t➢➡♥❣ t❤Ý❝❤ ②Õ✉✱ t❤× T ✈➭ f ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤✉♥❣ ❞✉② ♥❤✃t✳ ➳ ✭ ♥❤ ①➵ ✷✳✷✳✷ ❍Ö q✉➯✳ T t❤á❛ ♠➲♥ ✭✷✳✻✮ ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ (ψ, ϕ)✲❝♦ ②Õ✉✳✮ ❈❤♦ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤ñ ❍❛✉s❞♦r❢❢✱ P ❧➭ ♠ét ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E ✈í✐ ❤➺♥❣ sè ❝❤✉➮♥ t➽❝ K ✈➭ E ❧➭ t❐♣ s➽♣ tèt t❤❡♦ q✉❛♥ ❤Ư t❤ø tù ❜é ♣❤❐♥ ✧≪✧ ①➳❝ ➤Þ♥❤ ❜ë✐ P ✭♥❣❤Ü❛ ❧➭ ❤❛✐ ♣❤➬♥ tư ❜✃t ❦ú ❝đ❛ E ❜❛♦ ❣✐ê ❝ị♥❣ s♦ s➳♥❤ ➤➢ỵ❝ t❤❡♦ ✶✽ q✉❛♥ ệ ỗ t rỗ ủ E ♠➭ ❜Þ ❝❤➷♥ ❞➢í✐ ➤Ị✉ ❝ã ❝❐♥ ❞➢í✐ ➤ó♥❣✮✱ T : X → X ❧➭ ➳♥❤ ①➵ tr➟♥ X t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ s❛✉✿ 1 ψ d (T x, T y) ψ max d (x, y) , d (x, T x) + d (y, T y) , d (y, T x) − ϕ d (x, y) ✭✷✳✼✮ ✈í✐ ♠ä✐ x, y ∈ X ✱ ë ➤➞② ψ ∈ Ψ ✈➭ ϕ ∈ Φ ✳ ❑❤✐ ➤ã✱ T ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ∗ ➤é♥❣✳ ❈❤ø♥❣ ♠✐♥❤✳ ❙✉② tõ ➜Þ♥❤ ❧ý ✷✳✷✳✶ ❜➺♥❣ ❝➳❝❤ ❝❤ä♥ f = IX ❧➭ ➳♥❤ ①➵ ➤å♥❣ ♥❤✃t tr➟♥ X ✳ ✷✳✷✳✸ ❱Ý ❞ô✳ ❈❤♦ X = {0, 1, 2, 3} ❳➞② ❞ù♥❣ d : X × X → R ♥❤➢ s❛✉✿ d (x, y) = d (y, x) ✈í✐ ♠ä✐ x, y ∈ X ✈➭ d (x, y) = ❦❤✐ ❝❤Ø ❦❤✐ y = x✳ ◆❣♦➭✐ r❛ d (0, 3) = d (2, 3) = 1; d (0, 2) = d (1, 3) = 2; d (0, 1) = 4; d (1, 2) = ❉Ô ❞➭♥❣ t❤✃② r➺♥❣ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ♥ã♥ s✉② ré♥❣ ➤➬② ➤đ✱ ✈í✐ P = {x ∈ R : x ≥ 0} ❧➭ ♥ã♥ ❝❤✉➮♥ t➽❝ tr♦♥❣ R✱ ♥❤➢♥❣ (X, d) ❦❤➠♥❣ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ❜ë✐ ì t tứ t ò ú ♠ä✐ x, y, z ∈ X ✳ ❈❤➻♥❣ ❤➵♥ = d (0, 1) > d (0, 2) + d (2, 1) = + = ❇➞② ❣✐ê t❛ ①➞② ❞ù♥❣ ➳♥❤ ①➵ T, f : X → X ♥❤➢ s❛✉✿ T x = 0, ♥Õ✉ x ∈ {0, 1, 2} ; T x = 2, ♥Õ✉ x = 3; f (0) = 0, f (1) = 2, f (2) = 3, f (3) = ❑❤✐ ➤ã T ✈➭ f t❤á❛ ♠➲♥ ✭✷✳✻✮ ✈í✐ ψ (t) = 2t ✈➭ ϕ (t) = t ✳ ❚❤❐t ✈❐②✱ d (T x, T y) > ❝❤Ø ✈í✐ x ∈ {0, 1, 2} ✈➭ y = 3✳ ❍➡♥ ♥÷❛✱ t❛ ❝ã d (f (0) , f (3)) 4 = ψ d (T (0) , T (3)) 2.d (f (0) , f (3)) − = 10 − = 8; d (f (1) , f (3)) = 10 − = 7, 5; = ψ d (T (1) , T (3)) 2.d (f (1) , f (3)) − 2 d (f (2) , f (3)) d (f (2) , T (2)) + d (f (3) , T (3)) − = = ψ d (T (2) , T (3)) 2 3 = [1 + 5] − = 4, ❱× t❤Õ✱ T ✈➭ f t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ✭✷✳✻✮✳ ❉Ơ t❤✃② r➺♥❣ T (X) ⊂ f (X)✳ ❍➡♥ ♥÷❛✱ T ✈➭ f ❧➭ t➢➡♥❣ tí ế ì tế ụ ị ý t s✉② r❛ T ✈➭ f ❝ã ❞✉② ♥❤✃t ➤✐Ó♠ ❜✃t ➤é♥❣ z = 0✳