❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ơ♥ ❳✉➞♥ ◗✉ý ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ư ❆♥ ✲ ✷✵✶✼ ❇é ●✐➳♦ ❞ơ❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ◆❣✉②Ô♥ ❳✉➞♥ ◗✉ý ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ P●❙✳ ❚❙✳ ❚r➬♥ ❱➝♥ ➣♥ ◆❣❤Ư ❆♥ ✲ ✷✵✶✼ ▲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✳ ◆❤➞♥ ❞Þ♣ ♥➭② ❡♠ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❱✐Ư♥ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✱ P❤ß♥❣ ➤➭♦ t➵♦ ❙❛✉ ➤➵✐ ❤ä❝✱ q✉ý t❤➬②✱ ❝➠ ❣✐➳♦ tr♦♥❣ tæ ●✐➯✐ ❚Ý❝❤ ✲ ❱✐Ö♥ ❙➢ ♣❤➵♠ ❚ù ♥❤✐➟♥✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ t❐♥ t×♥❤ ❣✐ó♣ ➤ì ❡♠ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ✈➝♥✳ ❈✉è✐ ❝ï♥❣ t➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➳♠ ➡♥ ❣✐❛ ➤×♥❤ ✈➭ ❝➳❝ ❛♥❤ ❝❤Þ ❤ä❝ ✈✐➟♥ ❝❛♦ ❤ä❝ ❦❤♦➳ ✷✸ ●✐➯✐ ❚Ý❝❤ t➵✐ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ✈➭ ❣✐❛ ➤×♥❤ ➤➲ 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♥✳ ❚➳❝ ❣✐➯ ✐ ▼ô❝ ▲ô❝ ❚r❛♥❣ ▼ô❝ ❧ô❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ ▲ê✐ ♥ã✐ ➤➬✉ ❈❤➢➡♥❣ ■✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✶✳✶✳ ❈➳❝ ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ✶ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ❈❤➢➡♥❣ ■■✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✷✳✶✳ ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ✷✸ α✲ψ ✲❝♦ s✉② ré♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳ ➜✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ α✲ψ ✲❝♦ ✷✸ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ❑Õt ❧✉❐♥ ✹✷ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✹✸ ✐✐ ▼ë ➤➬✉ ▲ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❧➭ sù ❦Õt ❤ỵ♣ ữ tí ì ọ ý tết ể ❜✃t ➤é♥❣ ➤➲ ➤ã♥❣ ❣ã♣ ✈❛✐ trß q✉❛♥ trä♥❣ tr♦♥❣ ✈✐Ư❝ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❤✐Ư♥ t➢ỵ♥❣ ♣❤✐ t✉②Õ♥✳ ➜➷❝ ❜✐Ưt✱ ❧ý t❤✉②Õt ➤✐Ĩ♠ ❜✃t ➤é♥❣ ➤➲ ➤➢ỵ❝ ➳♣ ❞ơ♥❣ tr♦♥❣ ❝➳❝ ❧Ü♥❤ ✈ù❝ ➤❛ ❞➵♥❣ ♥❤➢ ❙✐♥❤ ❤ä❝✱ ❍ã❛ ❤ä❝✱ ❑✐♥❤ tÕ✱ ❑ü t❤✉❐t✱ ▲ý t❤✉②Õt trß ❝❤➡✐ ✈➭ ❱❐t ❧ý✳ ❙ù ❤÷✉ Ý❝❤ ❝đ❛ ❝➳❝ ø♥❣ ❞ơ♥❣ ❝➭♥❣ t➝♥❣ ❧➟♥ ♥❤ê sù ♣❤➳t tr✐Ĩ♥ ❝đ❛ ❦❤♦❛ ❤ä❝ ❦ü t❤✉❐t✱ ❦ü t❤✉❐t ♠➳② tÝ♥❤ ➤Ó tÝ♥❤ t♦➳♥ ❝❤Ý♥❤ ①➳❝ ➤✐Ó♠ ❜✃t ➤é♥❣✳ ❑Õt q✉➯ q✉❛♥ trä♥❣ ➤➬✉ t✐➟♥ ♣❤➯✐ ❦Ó ➤Õ♥ tr♦♥❣ ❧ý t❤✉②Õt ➤✐Ó♠ ❜✃t ➤é♥❣ ❧➭ ♥❣✉②➟♥ ❧Ý ➳♥❤ ①➵ ❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ❝ñ❛ ❇❛♥❛❝❤✳ ❙❛✉ ➤ã✱ ◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ ➤➲ trë t❤➭♥❤ ♠ét ❝➠♥❣ ❝ơ ♣❤ỉ ❞ơ♥❣ ➤Ĩ ❣✐➯✐ q✉②Õt ❝➳❝ ❜➭✐ t♦➳♥ ✈Ò sù tå♥ t➵✐ tr♦♥❣ ♥❤✐Ò✉ ❝❤✉②➟♥ ♥❣➭♥❤ ❝đ❛ ●✐➯✐ tÝ❝❤ t♦➳♥ ❤ä❝ ✈➭ ❝ã ♥❤✐Ị✉ ø♥❣ ❞ô♥❣ q✉❛♥ trä♥❣ tr♦♥❣ ♣❤➢➡♥❣ ♣❤➳♣ sè ♥❤➢ P❤➢➡♥❣ ♣❤➳♣ ◆❡✇t♦♥✲ ❘❛♣❤s♦♥✱ t❤✐Õt ❧❐♣ ❝➳❝ ➤Þ♥❤ ❧ý ❧✐➟♥ q✉❛♥ ➤Õ♥ sù tå♥ t➵✐ ✈➭ tÝ♥❤ ❞✉② ♥❤✃t ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ ✈✐ ♣❤➞♥✱ sù tå♥ t➵✐ ♥❣❤✐Ư♠ ❝đ❛ ♣❤➢➡♥❣ tr×♥❤ tÝ❝❤ ♣❤➞♥ ✈➭ ❤Ư ♣❤➢➡♥❣ tr×♥❤ t✉②Õ♥ tÝ♥❤✳ ❱× t❤Õ ➤➲ ❝ã ♠ét sè ❧í♥ ❝➳❝ ♠ë ré♥❣ ❝đ❛ ➤Þ♥❤ ❧ý ♥➭② ❝❤♦ ❝➳❝ ❧í♣ ➳♥❤ ①➵ ✈➭ ❦❤➠♥❣ ❣✐❛♥ ❦❤➳❝ ♥❤❛✉✱ ❜➺♥❣ ❝➳❝❤ ➤✐Ị✉ ❝❤Ø♥❤ ➤✐Ị✉ ❦✐Ư♥ ❝♦ ❝➡ ❜➯♥ ❤♦➷❝ t❤❛② ➤æ✐ ❦❤➠♥❣ ❣✐❛♥✳ ◆➝♠ ✷✵✵✻✱ ❇❤❛s❦❛r ✈➭ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ➤➲ ❣✐í✐ t❤✐Ư✉ ❦❤➳✐ ♥✐Ư♠ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t❤á❛ ♠➲♥ tÝ♥❤ ➤➡♥ ➤✐Ư✉ tré♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✈➭ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ ✈Ò sù tå♥ t➵✐ ✈➭ ❞✉② ♥❤✃t ❝đ❛ ♥❣❤✐Ư♠ ❝đ❛ ❜➭✐ t♦➳♥ ❣✐➳ trÞ ❜✐➟♥ t✉➬♥ ❤♦➭♥✳ ◆➝♠ ✷✵✵✾✱ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ✈➭ ❈✐r✐✬❝ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦ ♣❤✐ t✉②Õ♥ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ t❤ø tù ❜é ♣❤❐♥✳ ◆➝♠ ✷✵✶✵✱ ❈❤✉❞❤✉r② ✈➭ ❑✉♥❞✉ ➤➲ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ tr♦♥❣ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ◆➝♠ ✷✵✶✷✱ ❙❛♠❡t ✈➭ ❝é♥❣ sù ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ➳♥❤ ①➵ α✲ψ ✲❝♦✱ ➳♥❤ ①➵ α✲❝❤✃♣ ♥❤❐♥ ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝➳❝ ➤Þ♥❤ ❧ý ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤♦ ❝➳❝ ➳♥❤ ①➵ ♥➭② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ❙❛✉ ➤ã✱ ❑❛r❛♣✐♥❛r ✈➭ ❙❛♠❡t ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ➳♥❤ ể t ợ ữ ết q ♠í✐ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝❤♦ ❧í♣ ❝➳❝ ➳♥❤ ①➵ ♥➭②✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦✱ ❝➳❝ ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦ s✉② ré♥❣✱ ♠ét sè ➤Þ♥❤ ❧ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ▲✉❐♥ ✈➝♥ ❣å♠ ✷ ❝❤➢➡♥❣✿ ❈❤➢➡♥❣ ✶ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭②✱ ë ♠ô❝ ✶ ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❧✉❐♥ ✈➝♥ ❜❛♦ ❣å♠ ♠ét sè ❦❤➳✐ ♥✐Ö♠✿ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✱ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ✐✐✐ α✲ψ ✲❝♦✱ ➳♥❤ ①➵ α✲❝❤✃♣ ♥❤❐♥✱ ➳♥❤ ①➵ α✲ψ ✲❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ α✲ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■✱ ➳♥❤ ①➵ α✲ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■■✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥✱ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ α✲ψ ✲❝♦ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ▼ô❝ ✷ ể trì ột số ị ý ể t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ❦✐Ĩ✉ α✲ψ ✲❝♦ tr➟♥ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❈❤➢➡♥❣ ✷ ✈í✐ ♥❤❛♥ ➤Ị ➜✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❚r♦♥❣ ♠ô❝ ú t trì ột số ị ý ề ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ α✲ψ ✲❝♦✱ ➳♥❤ ①➵ α✲❝❤✃♣ ♥❤❐♥✱ ➳♥❤ ①➵ α✲ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■✱ ➳♥❤ ①➵ α✲ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ ■■ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ✈➭ ❝➳❝ ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣✳ ▼ơ❝ ✷ ể trì ột số ị ý ể trù ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ➤➡♥ ➤✐Ư✉ tré♥✱ ➳♥❤ ①➵ ❦✐Ó✉ α✲ψ ✲❝♦✱ ➳♥❤ ①➵ α✲❝❤✃♣ ♥❤❐♥ ❝ï♥❣ ✈í✐ ❝➳❝ ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣ ✈➭ ♠ét sè ✈Ý ❞ơ ♠✐♥❤ ❤ä❛✳ ◆❣❤Ư ❆♥✱ ♥❣➭② ✸✵ t❤➳♥❣ ✼ ♥➝♠ ✷✵✶✼ ◆❣✉②Ơ♥ ❳✉➞♥ ◗✉ý ✐✈ ❝❤➢➡♥❣ ✶ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù ❈➳❝ ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶✳✶ P❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ ❣✐í✐ t❤✐Ư✉ q✉❛ ♠ét sè ❦✐Õ♥ t❤ø❝ ❧➭♠ ❝➡ së ❝❤♦ ✈✐Ư❝ tr×♥❤ ❜➭② ❝đ❛ ❧✉❐♥ ✈➝♥✳ ❈➳❝ ♥é✐ ❞✉♥❣ ❣å♠✿ ❑❤➠♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✱ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❦✐Ó✉ ①➵ α✲ψ ✲❝♦✱ ➳♥❤ α✲❝❤✃♣ ♥❤❐♥✱ ➳♥❤ ①➵ α✲ψ ✲❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ α✲ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■✱ ➳♥❤ ①➵ α✲ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■■✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✱ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ ①➵✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝ñ❛ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ♠ét sè ❦Õt q✉➯ ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ α✲ψ ✲❝♦ ✈➭ ♠ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ❝❤♦ ❝➳❝ ❦Õt q✉➯ ➤ã✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✶ ♠➟tr✐❝ tr➟♥ X ✭❬✶❪✮ ❈❤♦ t ợ X d : X ì X → R ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥✿ ✭✶✮ d(x, y) ≥ ✈í✐ ♠ä✐ x, y ∈ X ✭✷✮ d(x, y) = d(y, x) ✈í✐ ♠ä✐ x, y ∈ X ✳ ✭✸✮ d(x, y) ≤ d(x, z) + d(z, y) ✈í✐ ♠ä✐ x, y, z ∈ X ✳ ❚❐♣ X ❝ï♥❣ ✈í✐ ♠ét ♠➟tr✐❝ ✈➭ ❦Ý ❤✐Ö✉ ❧➭ d(x, y) = ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ x = y ✳ ✈➭ d tr➟♥ ♥ã ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ❤❛② ➤➡♥ ❣✐➯♥ ❧➭ X ✳ ❙è d (x, y) ❣ä✐ ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ tõ ➤✐Ó♠ x ➤Õ♥ ➤✐Ó♠ y ✳ ✶✳✶✳✷ ❱Ý ❞ơ✳ ✈í✐ ♠ä✐ ✶✮ ❳Ðt X = R✳ ❍➭♠ d : R × R → R ❝❤♦ ❜ë✐ d (x, y) = |x − y|✱ x, y ∈ R✳ ❑❤✐ ➤ã d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ R✳ ✶ ✷✮ ❳Ðt X = Rn ✳ n ➤➷t ❱í✐ ❜✃t ❦ú |xi − yi |2 d1 (x, y) = x = (x1 , , xn ), y = (y1 , , yn ) ∈ Rn n ✈➭ ✶✳✶✳✸ i=1 Rn ✳ ➜Þ♥❤ ♥❣❤Ü❛✳ n ≥ n0 t❛ ❝ã ✭❬✶❪✮ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ x∈X ❧➭ ❤é✐ tơ ✈Ị ➤✐Ĩ♠ |xi − yi |✳ ❑❤✐ ➤ã d1 , d2 ❧➭ d2 (x, y) = i=1 ❝➳❝ ♠➟tr✐❝ tr➟♥ t❛ ♥Õ✉ ✈í✐ ♠ä✐ d (xn , x) < ε✳ (X, d)✱ ❞➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐ ε > tå♥ t➵✐ n0 ∈ N∗ ▲ó❝ ➤ã t❛ ❦Ý ❤✐Ư✉ ❧➭ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ lim xn = x ❤❛② xn → x ❦❤✐ n→∞ n → ∞✳ ❈❤♦ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✭❞➲② ❝➡ ❜➯♥✮ ♥Õ✉ ✈í✐ ♠ä✐ ❝ã (X, d)✳ ❉➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ε > 0✱ tå♥ t➵✐ n0 ∈ N∗ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n, m ≥ n0 t❛ d(xn , xm ) < ε✱ ❤❛② {xn } ➤➢ỵ❝ ❣ä✐ ❧➭ ❞➲② ❈❛✉❝❤② ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ lim n,m→+∞ ✶✳✶✳✹ ◆❤❐♥ ①Ðt✳ ✷✮ ◆Õ✉ ❞➲② ✶✮ ◆Õ✉ ❞➲② {xn } ❤é✐ tơ t❤× ♥ã ❧➭ ❞➲② ❈❛✉❝❤②✳ {xn } ❧➭ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ X {xnk } ❤é✐ tơ ✈Ị ➤✐Ĩ♠ x ∈ X ✶✳✶✳✺ d(xn , xm ) = ➜Þ♥❤ ♥❣❤Ü❛✳ t❤× ❞➲② ✈➭ ❝ã ❞➲② ❝♦♥ {xn } ❝ị♥❣ ❤é✐ tơ ✈Ị x✳ ✭❬✶❪✮ ❑❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤ñ✱ ♥Õ✉ ♠ä✐ ❞➲② ❈❛✉❝❤② tr♦♥❣ ♥ã ➤Ị✉ ❤é✐ tơ✳ ❚❐♣ ❝♦♥ ❣✐❛♥ ❝♦♥ ✶✳✶✳✻ M M ❝đ❛ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ (X, d) ➤➢ỵ❝ ❣ä✐ ❧➭ ➤➬② ➤đ✱ ♥Õ✉ ❦❤➠♥❣ ✈í✐ ♠➟tr✐❝ ❝➯♠ s✐♥❤ ❧➭ ❦❤➠♥❣ ❣✐❛♥ ➤➬② ➤đ✳ ❱Ý ❞ơ✳ ✶✮ ❚❐♣ ❤ỵ♣ ❝➳❝ sè t❤ù❝ R ✈í✐ ♠➟tr✐❝ d (x, y) = |x − y|, x, y ∈ R ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ✷✮ ❚❐♣ ❤ỵ♣ Rn ❣å♠ t✃t ❝➯ ❝➳❝ ❜é n sè t❤ù❝✱ ✈í✐ ♠➟tr✐❝ d1 (x, y)✱ d2 (x, y) ❧➭ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ✸✮ ♠➟tr✐❝ X = C[a, b] ❧➭ t❐♣ ❤ỵ♣ t✃t ❝➯ ❝➳❝ ❤➭♠ sè t❤ù❝ ❧✐➟♥ tơ❝ tr➟♥ [a, b] ✈í✐ d(x, y) = max {|x(t) − y(t)| : t ∈ [a, b]} ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ✷ ✹✮ lp ∞ p |xn | = {x = (xn )n : < ∞}, p ≥ 1, ✈í✐ ♠➟tr✐❝ ➤Þ♥❤ ❜ë✐✿ x = (xn )n , y = (yn )n tr♦♥❣ lp t❛ ➤Þ♥❤ ♥❣❤Ü❛ p ∞ |xn − yn |p d(x, y) = (lp , d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ✭❬✶❪✮ ❈❤♦ ❝➳❝ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✼ (X, d) ✈➭ (Y, ρ)✳ ➳♥❤ ①➵ f : (X, d) → (Y, ρ) ➤➢ỵ❝ ❣ä✐ ❧➭ ➳♥❤ ①➵ ❝♦ ♥Õ✉ tå♥ t➵✐ k ∈ [0, 1) s❛♦ ❝❤♦ ρ[f (x) , f (y)] ≤ k d (x, y) , ➜Þ♥❤ ❧ý✳ ✶✳✶✳✽ ➤➬② ➤đ✱ ➤✐Ĩ♠ ➜✐Ĩ♠ ①➵ ✭❬✶❪✮ ✭◆❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦✮ ●✐➯ sö f :X→X x∗ ∈ X ❧➭ ➳♥❤ ①➵ ❝♦ tõ s❛♦ ❝❤♦ x∗ ∈ X X x, y ∈ X (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ✈➭♦ ❝❤Ý♥❤ ♥ã✳ ❑❤✐ ➤ã tå♥ t➵✐ ❞✉② ♥❤✃t f (x∗ ) = x∗ ✳ ❝ã tÝ♥❤ ❝❤✃t f (x∗ ) = x∗ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❝đ❛ ➳♥❤ f✳ ➜Þ♥❤ ♥❣❤Ü❛✳ ✶✳✶✳✾ ✭❬✶✽❪✮ ❈❤♦ X ❧➭ ♠ét ➳♥❤ ①➵✳ ➳♥❤ ①➵ F y ✈í✐ ♠ä✐ ❦Ð♦ t❤❡♦ F (x) ≤ F (y)✳ (X, ≤) ❧➭ ♠ét t❐♣ ➤➢ỵ❝ s➽♣ t❤ø tù ✈➭ F : X −→ ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐➯♠✱ ♥Õ✉ ✈í✐ ♠ä✐ ❚➢➡♥❣ tù✱ F x, y ∈ X, x ≤ ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ t➝♥❣✱ ♥Õ✉ ✈í✐ ♠ä✐ x, y ∈ X, x ≤ y ❦Ð♦ t❤❡♦ F (x) ≥ F (y)✳ ❚r♦♥❣ ❬✽❪✱ ❚✳ ●✳ ❇❤❛s❦❛r ✈➭ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ➤➲ ➤➢❛ r❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠ ❞➢í✐ ➤➞② ✈Ị ➳♥❤ ①➵ ➤➡♥ ➤✐Ư✉ tré♥ ✈➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉✳ ✶✳✶✳✶✵ ị ĩ X ì X X ➳♥❤ ①➵ F (X, ≤) ❧➭ ♠ét t❐♣ ❤ỵ♣ ➤➢ỵ❝ s➽♣ t❤ø tù ✈➭ F : ➤➢ỵ❝ ❣ä✐ ❧➭ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥✱ ♥Õ✉ F ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ ❣✐➯♠ t❤❡♦ ❜✐Õ♥ t❤ø ♥❤✃t ✈➭ ➤➡♥ ➤✐Ö✉ ❦❤➠♥❣ t➝♥❣ t❤❡♦ ❜✐Õ♥ t❤ø ❤❛✐ ❝đ❛ ♥ã✱ ♥❣❤Ü❛ ❧➭ ✈í✐ ♠ä✐ x1 , x2 ∈ X, x1 ≤ x2 ✸ s✉② r❛ F (x1 , y) ≤ F (x2 , y) ✈í✐ ❜✃t ❦ú y∈X ❜✃t ❦ú x ∈ X✳ ✶✳✶✳✶✶ ➜Þ♥❤ ♥❣❤Ü❛✳ ✈➭ ✈í✐ ♠ä✐ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ ✶✳✶✳✶✷ y1 , y2 ∈ X, y1 ≤ y2 ✭❬✶✽❪✮ P❤➬♥ tư (x, y) ∈ X × X F : X ì X X ị ĩ s r ế F (x, y1 ) ≥ F (x, y2 ) ✈í✐ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤✐Ĩ♠ ❜✃t ➤é♥❣ F (x, y) = x ✈➭ F (y, x) = y ✳ ✭❬✶✸❪✮ ❈❤♦ (X, ≤) ❧➭ ♠ét t❐♣ ❝ã t❤ø tù✳ ❉➲② {xn } ❣ä✐ ❧➭ ❦❤➠♥❣ ❣✐➯♠ ➤è✐ ✈í✐ q✉❛♥ ❤Ư t❤ø tù ≤ ♥Õ✉ xn ≤ xn+1 ✱ ❞➲② {xn } ⊂ X ➤➢ỵ❝ ❣ä✐ ❧➭ ❦❤➠♥❣ t➝♥❣ ➤è✐ ✈í✐ q✉❛♥ ❤Ư t❤ø tù ≤ ♥Õ✉ xn ✶✳✶✳✶✸ X ➜Þ♥❤ ❧ý✳ ✭❬✽❪✮ ❈❤♦ ⊂ X ➤➢ỵ❝ ≥ xn+1 ✈í✐ ♠ä✐ n ∈ N✳ (X, ≤) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù ✈➭ d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ (X, d) ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư F : X × X −→ X s❛♦ ❝❤♦ ❧➭ ➳♥❤ ①➵ ❧✐➟♥ tơ❝ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ tr➟♥ X ✈➭ ❣✐➯ sö tå♥ t➵✐ k ∈ [0, 1) s❛♦ ❝❤♦ d (F (x, y), F (u, v)) ≤ ◆Õ✉ tå♥ t➵✐ x , y0 ∈ X k [d(x, u) + d(y, v)] , ✈í✐ ♠ä✐ x ≥ u ✈➭ y ≤ v s❛♦ ❝❤♦ x0 ≤ F (x0 , y0 ) ✈➭ y0 ≥ F (y0 , x0 ), x, y ∈ X t❤× tå♥ t➵✐ ✶✳✶✳✶✹ X ➜Þ♥❤ ❧ý✳ s❛♦ ❝❤♦ s❛♦ ❝❤♦ ✭❬✽❪✮ ❈❤♦ (X, d) F (x, y) = x ✈➭ F (y, x) = y ✳ (X, ≤) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù ✈➭ d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ ❧➭ ♠ét ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ ●✐➯ sư r➺♥❣ X ❝ã tÝ♥❤ ❝❤✃t ❞➢í✐ ➤➞②✿ ✭✐✮ ♥Õ✉ (xn ) ❧➭ ❞➲② ❦❤➠♥❣ ❣✐➯♠ ✈➭ {xn } −→ x✱ t❤× xn ≤ x ✈í✐ ♠ä✐ n, ✭✐✐✮ ♥Õ✉ (yn ) ❧➭ ❞➲② ❦❤➠♥❣ t➝♥❣ ✈➭ {yn } −→ y ✱ t❤× y ≤ yn ✈í✐ ♠ä✐ n ●✐➯ sư F : X × X −→ X sö tå♥ t➵✐ ❧➭ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉ tré♥ tr➟♥ X ✈➭ ❣✐➯ k ∈ [0, 1) s❛♦ ❝❤♦ d (F (x, y), F (u, v)) ≤ k [d(x, u) + d(y, v)] , ✹ ✈í✐ ♠ä✐ x ≥ u ✈➭ y ≤ v ✭✐✐✮ ✭✐✐✐✮ tå♥ t➵✐ ♥Õ✉ x , y0 ∈ X s❛♦ ❝❤♦ α∗ ((x0 , y0 ), (F (x0 , y0 ), F (y0 , x0 ))) ≥ 1❀ {xn }, {yn } ❧➭ ❝➳❝ ❞➲② tr♦♥❣ X s❛♦ ❝❤♦ ✈í✐ ♠ä✐ n ∈ N t❤á❛ ♠➲♥ α∗ ((xn , yn ), (xn+1 , yn+1 )) ≥ ✈➭ lim xn = x, lim yn = y, n→∞ α∗ ((xn , yn ), (x, y)) ≥ 1✳ t❤× t❛ ❝ã ❑❤✐ ➤ã✱ tå♥ t➵✐ ➜Þ♥❤ ❧ý✳ ✷✳✶✳✶✸ n→∞ x, y ∈ X s❛♦ ❝❤♦ F (x, y) = x ✈➭ F (y, x) = y ✳ ✭❬✶✷❪✮ ➜Þ♥❤ ý ợ s r từ ị ý ứ ♠✐♥❤✳ ❚➢➡♥❣ tù ♥❤➢ ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶✶✱ t❛ ♥❤❐♥ t❤✃② r➺♥❣ TF t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✷✮ ✈➭ ❞♦ ➤ã t✃t ❝➯ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ý ợ tỏ ì TF ó ột ➤✐Ó♠ ❜✃t ➤é♥❣✱ ♥❣❤Ü❛ ❧➭ F ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ➜Þ♥❤ ❧ý✳ ✷✳✶✳✶✹ ♠ä✐ ✭❬✶✷❪✮ ◆❣♦➭✐ ❝➳❝ ❣✐➯ t❤✐Õt ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶✶✱ t❛ ❣✐➯ sư r➺♥❣ ✈í✐ (x, y), (s, t) ∈ X × X tå♥ t➵✐ (u, v) ∈ X × X s❛♦ ❝❤♦ α((x, y), (u, v)) ≥ ✈➭ α((s, t), (u, v)) ≥ 1, ✈➭ ❝ị♥❣ ❣✐➯ sư r➺♥❣ (u, v) ❧➭ s♦ s➳♥❤ ➤➢ỵ❝ ✈í✐ (x, y) ✈➭ (s, t)✳ ❑❤✐ ➤ã✱ F ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❞✉② ♥❤✃t✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚➢➡♥❣ tù ❝❤ø♥❣ ♠✐♥❤ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶✶✱ t❛ ♥❤❐♥ t❤✃② r➺♥❣ TF t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ✭✶✳✷✮ ✈➭ ❞♦ ó tt ề ệ ủ ị ý ợ t❤á❛ ♠➲♥✳ ❱× ✈❐② TF ❝ã ♠ét ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ♥❣❤Ü❛ ❧➭ F ❝ã ♠ét ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ✷✳✶✳✶✺ X ➜Þ♥❤ ❧ý✳ s❛♦ ❝❤♦ ✭❬✶✷❪✮ ❈❤♦ (X, ≤) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù✱ d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ F : X × X −→ X ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥✳ ●✐➯ sư tå♥ t➵✐ ❤➭♠ (x, y), (u, v) ∈ X × X ♠➭ ψ ∈ Ψ ❧➭ ➳♥❤ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ (x, y) ≥2 (u, v) t❛ ❝ã d(x, u) + d(y, v) d(F (x, y), F (u, v)) + d(F (y, x), F (v, u)) ≤ψ 2 ●✐➯ sö r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➞② ❝ị♥❣ t❤á❛ ♠➲♥ ✸✶ ✭✐✮ ✭✐✐✮ tå♥ t➵✐ F (x0 , y0 ) ∈ X × X ❧✐➟♥ tô❝ ❤♦➷❝ s❛♦ ❝❤♦ (x0 , y0 ) ≤2 (F (x0 , y0 ), F (y0 , x0 )); (X , ≤2 , d) ❧➭ ❝❤Ý♥❤ q✉②✳ ❑❤✐ ➤ã✱ F ❝ã ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ ✈í✐ ♠ä✐ (x, y), (u, v) tå♥ t➵✐ (z, w) ∈ X × X s❛♦ ❝❤♦ ∈ X×X (x, y) ≤2 (z, w) ✈➭ (u, v) ≤2 (z, w) t❤× F ❝ã ❞✉② ♥❤✃t ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚❛ ①➳❝ ➤Þ♥❤ ➳♥❤ ①➵ α∗ ((x, y), (u, v)) = ❘â r➭♥❣✱ TF ❧➭ ➳♥❤ ①➵ α∗ : X × X −→ [0, +∞) ❝❤♦ ❜ë✐ (x, y) ≤2 (u, v), ❤♦➷❝(x, y) ≥2 (u, v) ❦❤✐ tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ ❝ß♥ ❧➵✐✳ α✲ψ ✲❝♦ t❤á❛ ♠➲♥ α∗ ((x, y), (u, v))d2 (TF (x, y), TF (u, v)) ≤ ψ(d2 ((x, y), (u, v))), ✈í✐ ♠ä✐ (x, y), (u, v) ∈ X ✳ ❚õ ➤✐Ị✉ ❦✐Ư♥ ✭✐✮ t❛ ❝ã α∗ ((x0 , y0 ), (TF (x0 , y0 ), TF (y0 , x0 ))) ≥ ◆❤ê ❇ỉ ➤Ị ✶✳✶✳✷✶ t❛ s✉② r❛ ❍➡♥ ♥÷❛✱ ✈í✐ ♠ä✐ TF ❧➭ ➳♥❤ ①➵ ❦❤➠♥❣ ❣✐➯♠ t❤❡♦ q✉❛♥ ❤Ö t❤ø tù (x, y), (u, v) ∈ X , tõ tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ư✉ tré♥ ❝đ❛ TF ≤2 ✳ t❛ ❝ã α∗ ((x, y), (u, v)) ≥ ⇒ (x, y) ≥2 (u, v) ❤♦➷❝ (x, y) ≤2 (u, v) ⇒ TF (x, y) ≥ TF (u, v) ❤♦➷❝ TF (x, y) ≤ TF (u, v) ⇒ α∗ (TF (x, y), TF (u, v)) ợ ì t ❇ỉ ➤Ị ✷✳✶✳✽ t❛ s✉② r❛ TF ❉♦ ➤ã✱ F ➳♥❤ ①➵ α✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✳ ❇➞② ❣✐ê✱ ♥Õ✉ F ❧➭ tụ tì ị ý t❛ s✉② r❛ sù tå♥ t➵✐ ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❚r➢ê♥❣ ❤ỵ♣ ♥Õ✉ (X, ≤2 , d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝❤Ý♥❤ q✉② ✈➭ {xn } ❧➭ ❞➲② ❜✃t ❦ú tr♦♥❣ X t❤á❛ ♠➲♥ α ∗ ((x n , yn ), (xn+1 , yn+1 )) ≥ ✈í✐ ♠ä✐ n ∈ N ✈➭ (xn , yn ) −→ (x, y) ∈ X ❦❤✐ n −→ +∞✱ t❤× ♥❤ê ❣✐➯ t❤✐Õt ✸✷ (X , ≤2 , d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝❤Ý♥❤ q✉② t❛ s✉② r❛ tå♥ t➵✐ ♠ét ❞➲② ❝♦♥ {xn(k) } ❝ñ❛ {xn } s❛♦ ❝❤♦ xn(k) ≤ x ✈➭ yn(k) ≥ y ✈í✐ ♠ä✐ k ∈ N✳ ❑❤✐ ➤ã✱ tõ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ α∗ t❛ s✉② r❛ α∗ ((xn(k) , yn(k) ), (x, y)) ≥ ✈í✐ ♠ä✐ k ∈ N✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭② ♥❤ê ➜Þ♥❤ ❧ý ✷✳✶✳✶✷ t❛ s✉② r❛ sù tå♥ t➵✐ ❝đ❛ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ➜Ó ❝❤Ø r❛ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ t❛ ❣✐➯ sư (z, w) ∈ X s❛♦ ❝❤♦ (x, y) ≤2 (z, w) ➤é♥❣ ❜é ➤➠✐✱ ❦❤✐ ➤ã ♥❤ê ❣✐➯ t❤✐Õt tå♥ t➵✐ ✈➭ (u, v) ≤2 (z, w)✳ ❚õ ➤Þ♥❤ ♥❣❤Ü❛ ❝ñ❛ α∗ ((u, v), (z, w)) ≥ 1✳ (x, y), (u, v) ∈ X ❧➭ ❝➳❝ ➤✐Ó♠ ❜✃t α∗ t❛ s✉② r❛ α∗ ((x, y), (z, w)) ≥ ì tế ị ý t s r❛ tÝ♥❤ ❞✉② ♥❤✃t ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❇➞② ❣✐ê✱ t❛ tr×♥❤ ❜➭② ♠ét ❦Õt q✉➯ tr♦♥❣ ❬✻❪ ♠➭ ♥ã ❧➭ ❤Ư q✉➯ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✶✳✶✺✳ ❍Ư q✉➯✳ ✷✳✶✳✶✻ X s❛♦ ❝❤♦ ✭❬✻❪✮ ❈❤♦ (X, ≤) ❧➭ ♠ét t❐♣ s➽♣ t❤ø tù✱ d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ F : X × X −→ X ①➵ ❝ã tÝ♥❤ ➤➡♥ ➤✐Ư✉ tré♥✳ ●✐➯ sư r➺♥❣ tå♥ t➵✐ sè (x, y), (u, v) ∈ X × X ♠➭ k ∈ [0, 1) ❧➭ ➳♥❤ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ (x, y) ≥2 (u, v) t❛ ❝ã [d(F (x, y), F (u, v)) + d(F (y, x), F (v, u))] ≤ k[d(x, u) + d(y, v)] ●✐➯ sư r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ➤➞② ❝ò♥❣ t❤á❛ ♠➲♥ ✭✐✮ ✭✐✐✮ tå♥ t➵✐ F (x0 , y0 ) ∈ X × X ❧✐➟♥ tơ❝ ❤♦➷❝ s❛♦ ❝❤♦ (x0 , y0 ) ≤2 (F (x0 , y0 ), F (y0 , x0 )); (X , ≤2 , d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ❝❤Ý♥❤ q✉②✳ ❑❤✐ ➤ã✱ F ❝ã ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ ✈í✐ ♠ä✐ (x, y), (u, v) tå♥ t➵✐ (z, w) ∈ X × X s❛♦ ❝❤♦ ∈ X×X (x, y) ≤2 (z, w) ✈➭ (u, v) ≤2 (z, w) t❤× F ❞✉② ♥❤✃t ❝đ❛ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✳ ✸✸ ❝ã ✷✳✷ ➜✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù P ú t trì ột số ị ý ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ➤➡♥ ➤✐Ư✉ tré♥✱ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦✱ ➳♥❤ ①➵ α✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝ ❝ï♥❣ ✈í✐ ❝➳❝ ❤Ư q✉➯ ❝đ❛ ❝❤ó♥❣ ✈➭ trì ột số í ụ ọ ị ❧ý✳ t❤á❛ ♠➲♥ ✭❬✶✹❪✮ ❈❤♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ❝➳❝ ➳♥❤ ①➵ F : X × X −→ X, g : X −→ X ❤➭♠ (X, ≤) ❧➭ ♠ét t❐♣ ❝ã t❤ø tù✱ d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ X t❤á❛ ♠➲♥ F ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ư✉✳ ●✐➯ sư tå♥ t➵✐ ❝➳❝ ψ ∈ Ψ ✈➭ α : X × X −→ [0, +∞) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y, u, v ∈ X ♠➭ g(x) ≥ g(u) ✈➭ g(y) ≥ g(v) t❛ ❝ã α ((g(x), g(y)), (g(u), g(u)))d (F (x, y), F (u, v)) ≤ψ d(g(x), g(u)) + d(g(y), g(v)) ✭✷✳✶✶✮ ●✐➯ sư r➺♥❣ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ❝ị♥❣ ➤ó♥❣✿ ✭✶✮ F ✭✷✮ tå♥ t➵✐ ✈➭ g ❧➭ α✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝❀ x , y0 ∈ X s❛♦ ❝❤♦ g(x0 ) ≤ F (x0 , y0 ), g(y0 ) ≥ F (y0 , x0 ), α((g(x0 ), g(y0 )), F (x0 , y0 ), F (y0 , x0 )) ≥ ✈➭ α ((g(y0 ), g(x0 )), F (y0 , x0 ), F (x0 , y0 )) ≥ 1; ✭✸✮ F (X × X) ⊆ g(X)✱ g ❧✐➟♥ tơ❝✱ F ✈➭ g ❧➭ ❤❛✐ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ tr♦♥❣ X; ✭✹✮ F ❑❤✐ ➤ã✱ ❧✐➟♥ tô❝✳ F ✈➭ g ❝ã ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ tø❝ ❧➭ tå♥ t➵✐ F (x, y) = g(x) ✈➭ F (y, x) = g(y) ✸✹ x, y ∈ X ♠➭ ❈❤ø♥❣ ♠✐♥❤✳ ●✐➯ sö x , y0 ∈ X t❤♦➯ ♠➲♥ α((g(x0 ), g(y0 )), F (x0 , y0 ), F (y0 , x0 )) ≥ ✈➭ α ((g(y0 ), g(x0 )), F (y0 , x0 ), F (x0 , y0 )) ≥ 1; ✈➭ g(x0 ) ≤ F (x0 , y0 ) = g(x1 ) ✈➭ g(y0 ) ≥ F (y0 , x0 ) = g(y1 ) ➜➷t x2 , y2 ∈ X s❛♦ ❝❤♦ F (x1 , y1 ) = g(x2 ) ✈➭ F (y1 , x1 ) = g(y2 ) ❚✐Õ♣ tơ❝ q✉➳ tr×♥❤ ♥➭② ❝❤ó♥❣ t❛ ❝ã t❤Ó ①➞② ❞ù♥❣ ❤❛✐ ❞➲② ✈➭ {xn }, {yn } tr♦♥❣ X ♥❤➢ s❛✉ g(xn+1 ) = F (xn , yn ) g(yn+1 ) = F (yn , xn ), ✈í✐ ♠ä✐ n ∈ N ❇➞② ❣✐ê ❝❤ó♥❣ t❛ sÏ ❝❤Ø r❛ r➺♥❣ g(xn ) ≤ g(xn+1 ) ✈➭ g(yn ) ≥ g(yn+1 ), ✈í✐ ♠ä✐ n ∈ N ❱í✐ n = 0, tõ ❣✐➯ t❤✐Õt ✭✷✳✶✷✮ g(x0 ) ≤ F (x0 , y0 ), g(y0 ) ≥ F (y0 , x0 ) ✈➭ g(x1 ) = F (x0 , y0 )✱ g(y1 ) = F (y0 , x0 ) ❝❤ó♥❣ t❛ ❝ã g(x0 ) ≤ g(x1 ), g(y0 ) ≥ g(y1 ) ❉♦ ➤ã✱ ✭✷✳✶✷✮ ➤ó♥❣ ✈í✐ n = 0✳ ❇➞② ❣✐ê✱ t❛ ❣✐➯ sư r➺♥❣ ✭✷✳✶✷✮ ➤ó♥❣ ✈í✐ g(xn ) ≤ g(xn+1 ), g(yn ) ≥ g(yn+1 )✱ ✈× F n ♥➭♦ ➤ã n ≥ ❑❤✐ ➤ã✱ tõ ❣✐➯ t❤✐Õt ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ö✉ ♥➟♥ t❛ ❝ã g(xn+2 ) = F (xn+1 , yn+1 ) ≥ F (xn , yn+1 ) ≥ F (xn , yn ) = g(xn+1 ) ✈➭ g(yn+2 ) = F (yn+1 , xn+1 ) ≤ F (yn , xn+1 ) ≤ F (xn , yn ) = g(yn+1 ) ❚õ tr➟♥✱ ❝❤ó♥❣ t❛ ❝ã t❤Ĩ ❦Õt ❧✉❐♥ r➺♥❣ g(xn+1 ) ≤ g(xn+2 ) ✈➭ g(yn+1 ) ≥ g(yn+2 ), ✈í✐ ♠ä✐ n ∈ N ◆❤➢ ✈❐②✱ ❜➺♥❣ q✉② ♥➵♣ t♦➳♥ ❤ä❝✱ ❝❤ó♥❣ t❛ ❦Õt ❧✉❐♥ r➺♥❣ ✭✷✳✶✷✮ ➤ó♥❣ ✈í✐ ♠ä✐ n ∈ N✳ ◆Õ✉ tå♥ t➵✐ n ∈ N ♥➭♦ ➤ã s❛♦ ❝❤♦ (xn+1 , yn+1 ) = (xn , yn )✳ ❑❤✐ ➤ã t❛ ❝ã ✈➭ (g(xn+1 ), g(yn+1 )) = (g(xn ), g(yn ))✳ ▲ó❝ ➤ã✱ râ r➭♥❣ F (xn , yn ) = g(xn ) F (yn , xn ) = g(yn )✱ ♥❣❤Ü❛ ❧➭ F ✈➭ g ❝ã ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❧➭ (xn , yn )✳ ✸✺ ❇➞② ❣✐ê✱ ❝❤ó♥❣ t❛ ❣✐➯ t❤✐Õt r➺♥❣ (xn+1 , yn+1 ) ❚õ ❣✐➯ t❤✐Õt✱ F ✈➭ = (xn , yn ) ✈í✐ ♠ä✐ n ≥ n(ε) g ❧➭ α✲❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ ♥➟♥ t❛ ❝ã α((g(x0 ), g(y0 )), (g(x1 ), g(y1 ))) = α((g(x0 ), g(y0 )), F (x0 , y0 ), F (y0 , x0 )) ≥ 1, ♥❣❤Ü❛ ❧➭ t❛ ❝ã α((F (x0 , y0 ), F (y0 , x0 )),(F (x1 , y1 ), F (y1 , x1 ))) = α((g(x1 ), g(y1 )), (g(x1 ), g(y1 ))) ≥ ❱× ✈❐②✱ ❜➺♥❣ q✉② ♥➵♣ t♦➳♥ ❤ä❝ t❛ ❝ã α((g(xn ), g(yn )), (g(xn+1 ), g(yn+1 ))) ≥ ✭✷✳✶✸✮ ❚➢➡♥❣ tù ♥❤➢ ✈❐②✱ t❛ ❝ò♥❣ ❝ã α((g(yn ), g(xn )), (g(yn+1 ), g(xn+1 ))) ≥ ✭✷✳✶✹✮ ❚õ ✭✷✳✶✶✮ ✈➭ ❣✐➯ t❤✐Õt ✶✮ ✈➭ ✷✮ ❝đ❛ ➤Þ♥❤ ❧ý t❛ ❝ã d(g(xn ), g(xn+1 ) = d(F (xn−1 , yn−1 , F (xn , yn ) ≤ α((g(xn−1 ), g(yn−1 )), (g(xn ), g(yn )))d(F (xn−1 , yn−1 , F (xn , yn ) ≤ψ ✭✷✳✶✺✮ d(g(xn−1 ), g(xn )) + d(g(yn−1 ), g(yn )) ❚➢➡♥❣ tù✱ t❛ ❝ã d(g(yn ), g(yn+1 ) = d(F (yn−1 , xn−1 , F (yn , xn ) ≤ α((g(yn−1 ), g(xn−1 )), (g(yn ), g(xn )))d(F (yn−1 , xn−1 , F (yn , xn ) ≤ψ ✭✷✳✶✻✮ d(g(yn−1 ), g(yn )) + d(g(xn−1 ), g(xn )) ❚õ ❝➳❝ ❝➠♥❣ t❤ø❝ ✭✷✳✶✺✮ ✈➭ ✭✷✳✶✻✮ t❛ ❝ã✿ d(g(xn ), g(xn+1 )) + d(g(yn ), g(yn+1 )) ≤ψ ▲➷♣ ❧➵✐ ❝❤ø♥❣ ♠✐♥❤ tr➟♥ ✈í✐ ♠ä✐ d(g(xn−1 ), g(xn )) + d(g(yn−1 ), g(yn )) n ∈ N t❛ ❝ã d(g(xn ), g(xn+1 )) + d(g(yn ), g(yn+1 )) ≤ ψn ✸✻ d(g(x0 ), g(x1 )) + d(g(y0 ), g(y1 )) ❱í✐ ε > tå♥ t➵✐ n(ε) ∈ N t❤á❛ ♠➲♥ d (g(x0 ), g(x1 )) + d (g(y0 ), g(y1 )) ψn n≥n(ε) ▲✃② ❝➳❝ sè ε < n, m ∈ N s❛♦ ❝❤♦ m > n > n(ε)✱ ❦❤✐ ➤ã ❜➺♥❣ ❝➳❝❤ sư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ t❛ ❝ã d (g(xn ), g(xm )) + d (g(yn ), g(ym )) m−1 d (g(xk ), g(xk+1 )) + d (g(yk ), g(yk+1 )) ≤ k=n m−1 ψk ≤ k=n ψn ≤ n≥n(ε) d (g(x0 ), g(x1 )) + d (g(y0 ), g(y1 )) d (g(x0 ), g(x1 )) + d (g(y0 ), g(y1 )) ε < ➜✐Ò✉ ♥➭② ❝ã ♥❣❤Ü❛ ❧➭✿ d (g(xn ), g(xm )) + d (g(yn ), g(ym )) < ε ❱× d (g(xn ), g(xm )) < d (g(xn ), g(xm )) + d (g(yn ), g(ym )) < ε ✈➭ d (g(yn ), g(ym )) < d (g(xn ), g(xm )) + d (g(yn ), g(ym )) < ε, t❛ s✉② r❛ {g(xn )} ✈➭ {g(yn )} ❧➭ ❝➳❝ ❞➲② ❈❛✉❝❤② tr♦♥❣ (X, d)✳ ❱× (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✱ ♥➟♥ t➵✐ x, y ∈ X s❛♦ ❝❤♦ {xn } ✈➭ {yn } ❧➭ ❝➳❝ ❞➲② ❤é✐ tô tr♦♥❣ (X, d)✳ ❉♦ ➤ã✱ tå♥ lim F (xn , yn ) = lim g(xn ) = x ✈➭ lim F (yn , xn ) = n→∞ n→∞ n→∞ lim g(yn ) = y n→∞ ❱× F ✈➭ g ❧➭ ❝➳❝ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤✱ ♥➟♥ t❛ ❝ã lim d(g(F (xn , yn ), F (g(xn ), g(yn ))) = n→∞ ✸✼ ✭✷✳✶✼✮ ✈➭ lim d(g(F (yn , xn ), F (g(yn ), g(xn ))) = n→∞ ❚✐Õ♣ t❤❡♦✱ t❛ sÏ ❝❤ø♥❣ tá r➺♥❣ ❱í✐ ✭✷✳✶✽✮ g(x) = F (x, y) ✈➭ g(y) = F (y, x) n ≥ t❛ ❝ã d(g(x),F (g(xn ), g(yn ))) ≤ d(g(x), g(F (xn ; yn ))) + d(g(F (xn ; yn )), F (g(xn ), g(yn ))) ▲✃② ❣✐í✐ ❤➵♥ ❦❤✐ n → +∞ tr♦♥❣ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥✱ ♥❤ê tÝ♥❤ ❧✐➟♥ tơ❝ ❝đ❛ F, g ✈➭ tõ ❝➠♥❣ t❤ø❝ ✭✷✳✶✼✮ t❛ ❝ã d(g(x), F (x, y)) = ❚➢➡♥❣ tù✱ t❛ ❝ã d(g(y), F (y, x)) = ❇ë✐ ✈❐②✱ t❛ ♥❤❐♥ ➤➢ỵ❝ g(x) = F (x, y) ✈➭ g(y) = F (y, x)✳ ❉♦ ➤ã F ✈➭ g ❝ã ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✳ ❇➞② ❣✐ê✱ ❝❤ó♥❣ t❛ t❤❛② ➤✐Ị✉ ❦✐Ư♥ ❧✐➟♥ tơ❝ ❝đ❛ ➳♥❤ ①➵ F tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✶ ❜ë✐ ➤✐Ị✉ ❦✐Ư♥ tr➟♥ ❝➳❝ ❞➲②✳ ➜Þ♥❤ ❧ý✳ ✷✳✷✳✷ ❝❤♦ ✭❬✶✹❪✮ ❈❤♦ (X, ≤) ❧➭ t❐♣ ❝ã t❤ø tù✱ d ❧➭ ♠ét ♠➟tr✐❝ tr➟♥ X s❛♦ (X, d) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ ✈➭ ❤❛✐ ➳♥❤ ①➵ F : X × X −→ X g : X −→ X ✳ ●✐➯ sö F ❝ã tÝ♥❤ ❝❤✃t α : X × X −→ [0, +∞) ✈➭ g ✲➤➡♥ ➤✐Ö✉ ✈➭ tå♥ t➵✐ ❝➳❝ ❤➭♠ ψ ∈ Ψ ✈➭ s❛♦ ❝❤♦ ✈í✐ ♠ä✐ x, y, u, v ∈ X ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ t❤á❛ ♠➲♥ ✭✶✮ ❇✃t ➤➻♥❣ t❤ø❝ ✭✷✳✶✶✮ ✈➭ ➤✐Ị✉ ❦✐Ư♥ ✶✮✱ ✷✮ ✈➭ ✸✮ ❧✉➠♥ ➤ó♥❣✳ ✭✷✮ ◆Õ✉ {xn }, {yn } ❧➭ ❝➳❝ ❞➲② tr♦♥❣ X s❛♦ ❝❤♦ α((g(xn ), g(yn )), g(xn+1 ), g(yn+1 )) ≥ ✸✽ ✈➭ α((g(yn ), g(xn )), g(yn+1 ), g(xn+1 )) ≥ ✈í✐ n ∈ N ✈➭ lim g(xn ) = x ✈➭ lim g(yn ) = y, ✈í✐ ♠ä✐ x, y ∈ X ✱ t❤× n→∞ n→∞ α((g(xn ), g(yn )), g(x), g(y)) ≥ ✈➭ α((g(yn ), g(xn )), g(y), g(x)) ≥ ◆Õ✉ tå♥ t➵✐ t❤× F ✈➭ g x , y0 ∈ X s❛♦ ❝❤♦ g(x0 ) ≤ F (xo , y0 ) ✈➭ g(y0 ) ≥ F (y0 , x0 )✱ ❝ã ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐✱ ♥❣❤Ü❛ ❧➭ tå♥ t➵✐ x, y ∈ X s❛♦ ❝❤♦ F (x, y) = g(x) ✈➭ F (y, x) = g(y)✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚➢➡♥❣ tù ❝❤ø♥❣ ♠✐♥❤ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ t❛ t❤✃② r➺♥❣ {g(xn )} ✈➭ {g(yn )} ❧➭ ❝➳❝ ❞➲② ❈❛✉❝❤② tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ (X, d)✳ ❑❤✐ ➤ã✱ tå♥ t➵✐ x, y ∈ X s❛♦ ❝❤♦ lim g(xn ) = x✱ lim g(yn ) = y ✈➭ ✈í✐ ♠ä✐ n ∈ N n→∞ n→∞ α((g(xn ), g(yn )), g(x), g(y)) ≥ ❚➢➡♥❣ tù✱ ✈í✐ ♠ä✐ ✭✷✳✶✾✮ n ∈ N t❛ ❝ã α((g(yn ), g(xn )), g(y), g(x)) ≥ ❙ư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ t❛♠ ❣✐➳❝✱ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✶✾✮ ✈➭ tÝ♥❤ ❝❤✃t ♠ä✐ ✭✷✳✷✵✮ ψ(t) < t ✈í✐ t > t❛ ❝ã d(F (x, y), g(x)) ≤ d(F (x, y), F (g(xn ), g(yn ))) + d(g(xn+1 , g(x))) ≤ α((g(xn ), g(yn )), (g(x), g(y)))d(F (g(xn ), F (x, y)) + d(g(xn+1 ), g(x)) d(g(x), g(xn )) + d(g(y), g(yn )) + d(g(xn+1 ), g(x)) d(g(xn ), g(x)) + d(g(yn ), g(y)) + d(g(xn+1 ), g(x)) < ≤ψ ✸✾ ❚➢➡♥❣ tù✱ sư ❞ơ♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ✭✷✳✷✵✮ t❛ ❝ã d(F (y, x), g(y)) ≤ d(F (y, x), F (g(yn ), g(xn ))) + d(g(yn+1 , g(y))) ≤ α((g(yn ), g(xn )), (g(y), g(x)))d(F (g(yn ), F (y, x)) + d(g(yn+1 ), g(y)) d(g(y), g(yn )) + d(g(x), g(yn )) + d(g(yn+1 ), g(y)) d(g(yn ), g(y)) + d(g(xn ), g(x)) < + d(g(yn+1 ), g(y)) ≤ψ ▲✃② ❣✐í✐ ❤➵♥ ❦❤✐ n → ∞ tr♦♥❣ ❤❛✐ ❜✃t ➤➻♥❣ t❤ø❝ tr➟♥ t❛ ♥❤❐♥ ➤➢ỵ❝ d(F (x, y), g(x)) = ❇ë✐ ✈❐②✱ t❛ ❝ã ❈❤ó ý✳ ✈➭ d(F (y, x), g(y)) = F (x, y) = g(x)✈➭(F (y, x) = g(y)✱ ❑❤✐ ❧✃② g = I ❧➭ ➳♥❤ ①➵ ➤å♥❣ ♥❤✃t tr➟♥ X ú t ợ ị ý ủ rs ✈➭ ❝é♥❣ sù tr♦♥❣ ❬✶✼❪✳ ✷✳✷✳✸ ❱Ý ❞ô✳ ❈❤♦ X = [0, 1]✳ ❑❤✐ ➤ã (X, ≤) ❧➭ t❐♣ s➽♣ t❤ø tù ✈í✐ t❤ø tù t❤➠♥❣ t❤➢ê♥❣ ❝đ❛ sè t❤ù❝✳ ❈❤♦ d(x, y) = |x − y| ✈í✐ ♠ä✐ x, y ∈ [0, 1] ❑❤✐ ➤ã (X, ≤) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤đ✳ g(x) = x2 ✈í✐ ♠ä✐ x ∈ X ✈➭ x2 − y 2 ➳♥❤ ①➵ F : X ì X X ị F (x, y) = + ✳ ❳Ðt ❤➭♠ sè 3 2t ψ : [0, 1) → [0, 1) ①➳❝ ➤Þ♥❤ ❜ë✐ ψ(t) = ✈í✐ t ∈ [0, 1)✳ ❈❤♦ {xn } ✈➭ {yn } ❧➭ ❤❛✐ ❞➲② tr♦♥❣ X s❛♦ ❝❤♦ lim F (xn , yn ) = a, lim g(xn ) = a ✈➭ lim F (yn , xn ) = n→∞ n→∞ n→∞ 2 b, lim g(yn ) = b✳ ❑❤✐ ➤ã✱ t❛ ❞Ô ❞➭♥❣ t❤✃② r➺♥❣ a = ✈➭ b = ✳ n→∞ 3 2 ❇➞② ❣✐ê✱ ✈í✐ ♠ä✐ n ≥ t❛ ❝ã g(xn ) = xn , g(yn ) = yn ❈❤♦ ➳♥❤ ①➵ g : X → X F (xn , yn ) = ①➳❝ ➤Þ♥❤ ❜ë✐ x2n − yn2 + 3 ✈➭ ✹✵ F (yn , xn ) = yn2 − x2n + 3 ❑❤✐ ➤ã✱ t❛ s✉② r❛ r➺♥❣ lim d(g(F (xn , yn ), F (g(xn ), g(yn ))) = n→∞ ✈➭ lim d(g(F (yn , xn ), F (g(yn ), g(xn ))) = n→∞ ❉♦ ➤ã✱ F ✈➭ g ❧➭ ❤❛✐ ➳♥❤ ①➵ t➢➡♥❣ t❤Ý❝❤ tr♦♥❣ X ✳ ❱í✐ ♠ä✐ x, y, u, v ∈ X ♠➭ g(x) ≥ g(u) ✈➭ g(y) ≥ g(v) t❛ ❝ã x2 − y 2 u2 − v 2 + − + 3 3 = (x2 − u2 ) + (v − y ) ≤ (x2 − u2 ) + (v − y ) ≤ (d(g(x), g(u)) + d(g(y), g(v))) d(F (x, y), F (u, v)) = ❇➞② ❣✐ê ①Ðt ➳♥❤ ①➵✱ α : X × X → [0, +∞) ❝❤♦ ❜ë✐ α ((x, y), (u, v)) = x ≥ y, u ≥ v ❦❤✐ tr trờ ợ ò 2t t > 0✳ ➜å♥❣ t❤ê✐ t❛ ❝ò♥❣ t❤✃② r➺♥❣ F (X × X) ⊆ g(X) ✈➭ F ❝ã tÝ♥❤ ❝❤✃t g ✲➤➡♥ ➤✐Ư✉✳ ❇➺♥❣ ❝➳❝❤ ❉♦ ➤ã✱ ➤✐Ị✉ ❦✐Ư♥ ✭✷✳✶✶✮ ❧➭ ➤ó♥❣ ➤è✐ ✈í✐ ❤➭♠ ❧✃② x0 = 0, ✈➭ y0 = 0, 9✳ ❑❤✐ ➤ã✱ t❛ ❝ã ψ(t) = g(x0 ) = (0, 6)2 = 0, 36 ≤ 0.51 = F (x0 , y0 ) ✈➭ g(y0 ) = (0.9)2 = 0, 81 ≥ 0, 81 = F (y0 , x0 )✳ ◆❤➢ ✈❐②✱ t✃t ❝➯ ❝➳❝ 2 , ❧➭ ➤✐Ĩ♠ trï♥❣ ➤✐Ị✉ ❦✐Ư♥ ❝đ❛ ➜Þ♥❤ ❧ý ✷✳✷✳✶ ➤➢ỵ❝ t❤á❛ ♠➲♥✳ ❉♦ ✈❐② 3 ♥❤❛✉ ❜é ➤➠✐ ❝ñ❛ F ✈➭ g tr♦♥❣ X ✹✶ ❑Õt ❧✉❐♥ ❙❛✉ ♠ét t❤ê✐ ❣✐❛♥ t❐♣ tr✉♥❣ ♥❣❤✐➟♥ ❝ø✉ ✈➭ t❤❛♠ ❦❤➯♦ ♥❤✐Ị✉ t➭✐ ❧✐Ư✉ ❦❤➳❝ ♥❤❛✉ ✈Ị ➤Ị t➭✐✿ ➜✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ ❦✐Ĩ✉ α✲ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ t❤➬② ❣✐➳♦ P●❙✳❚❙✳ ❚r➬♥ ❱➝♥ ➣♥✱ ❝❤ó♥❣ t➠✐ ➤➲ t❤✉ ➤➢ỵ❝ ♠ét sè ❦Õt q✉➯ s❛✉✿ ✶✳ ❍Ö t❤è♥❣ ❤ã❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠✱ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈➭ ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ ✈Ò ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✱ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❦✐Ĩ✉ ➤✐Ư✉ tré♥✱ ➳♥❤ ①➵ α✲ψ ✲❝♦✱ ➳♥❤ ①➵ α✲ ❝❤✃♣ ♥❤❐♥ ➤➢ỵ❝✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ➤➡♥ g ✲ ➤➡♥ ➤✐Ư✉✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣✱ ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐✱ ➤✐Ĩ♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ rì ột số ị ý ể t ➤é♥❣✱ ➤✐Ó♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ✈➭ ➤✐Ó♠ trï♥❣ ♥❤❛✉ ❜é ➤➠✐ ❝đ❛ ❝➳❝ ➳♥❤ ①➵ α✲ψ ✲❝♦ ♠➭ ❝❤ó♥❣ ❧➭ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ❝ñ❛ ▼✉rs❛❧❡❡♥✱ ❆❣❛r✇❛❧✱ ❑❛r❛♣✐♥❛r✳ ✸✳ ❈❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❝➳❝ tÝ♥❤ ❝❤✃t ✈➭ ➤Þ♥❤ ❧ý ♠➭ tr♦♥❣ ❝➳❝ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❝❤➢❛ ❝❤ø♥❣ ứ ò s ợ ị ý ✶✳✷✳✶✱ ➜Þ♥❤ ❧ý ✶✳✷✳✷✱ ➜Þ♥❤ ❧ý ✶✳✷✳✸✱ ➜Þ♥❤ ❧ý ✷✳✶✳✶✶✱ ➜Þ♥❤ ❧ý ✷✳✶✳✶✺✱ ➜Þ♥❤ ❧ý ✷✳✷✳✶✱ ➜Þ♥❤ ❧ý ✷✳✷✳✷✳ ✹✳ ●✐í✐ t❤✐Ư✉ ❝❤✐ t✐Õt ❱Ý ❞ơ ✶✳✷✳✹✱ ❱Ý ❞ơ ✶✳✷✳✺✱ ❱Ý ❞ô ✷✳✶✳✸✱❱Ý ❞ô ✷✳✶✳✹✱ ❱Ý ❞ô ✷✳✶✳✺✱ ❱Ý ❞ô t ệ t ỗ ✭✶✾✾✽✮✱ ❚➠♣➠ ➤➵✐ ❝➢➡♥❣✱ ◆❤➭ ①✉✃t ❜➯♥ ❑❤♦❛ ❤ä❝ ✈➭ ❑ü t❤✉❐t✳ ❬✷❪ ❆✳ ❆❣❤❛❥❛♥✐ ❛♥❞ ▼✳ ❆❜❜❛s ❛♥❞ ❊✳ P✳ ❑❛❧❧❡❤❜❛st✐ ✭✷✵✶✷✮✱ ✧❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥✧✱ ▼❛t❤✳ ❈♦♠♠✉♥✳✱ ✶✼✱ ✹✾✼✲✺✵✾✳ ❬✸❪ ❍✳ ❆②❞✐✱ ❊✳ ❑❛r❛♣✐♥❛r✱ ❇✳ ❙❛♠❡t✱ ❈✳ ❘❛❥✐❝ ✭✷✵✶✸✮✱ ✧❉✐s❝✉ss✐♦♥ ♦♥ s♦♠❡ ❝♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✷✵✶✸✱ ✺✵✱ ❞♦✐✿✶✵✳✶✶✽✻✴✶✻✽✼✲✶✽✶✷✲✷✵✶✸✲✺✵✳ ❬✹❪ ❙✳ ❇❛♥❛❝❤ ✭✶✾✷✷✮✱ ✧❙✉r ❧❡s ♦♣Ðr❛t✐♦♥s ❞❛♥s ❧❡s ❡♥s❡♠❜❧❡s ❛❜str❛✐ts ❡t ❧❡✉r ❛♣♣❧✐❝❛t✐♦♥s ❛✉① Ðq✉❛t✐♦♥s ✐♥tÐ❣r❛❧❡s✧✱ ❋✉♥❞✳ ▼❛t❤✳✱ ✸✱ ✶✸✸✲✶✽✶✳ ❬✺❪ ❱✳ ❇❡r✐♥❞❡ ✭✷✵✵✷✮✱ ■t❡r❛t✐✈❡ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❢✐①❡❞ ♣♦✐♥ts✱ ❊❞✐t✉r❛ ❊❢❡✲ ♠❡r✐❞❡✱ ❇❛✐❛ ▼❛r❡✳ ❬✻❪ ❱✳ ❇❡r✐♥❞❡ ✭✷✵✶✶✮✱ ✧●❡♥❡r❛❧✐③❡❞ ❝♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♠✐①❡❞ ♠♦♥♦t♦♥❡ ♠❛♣♣✐♥❣s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✹✱ ✼✸✹✼✲✼✸✺✺✳ ❬✼❪ ❱✳ ❇❡r✐♥❞❡ ✭✷✵✶✷✮✱ ✧❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r α✲ψ ✲❝♦♥tr❛❝t✐✈❡ ♠✐①❡❞ ♠♦♥♦t♦♥❡ ♠❛♣♣✐♥❣s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✺✱ ✸✷✶✽✲✸✷✷✽✳ ❬✽❪ ❚✳ ●✳ ❇❤❛s❦❛r✱ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠ ✭✷✵✵✻✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✐♥ ♣❛r✲ t✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✻✺✱ ✶✸✼✾✲ ✶✸✾✸✳ ❬✾❪ ❇✳ ❙✳ ❈❤♦✉❞❤✉r②✱ ❑✳ ❉❛s✱ P✳ ❉❛s ✭✷✵✶✷✮✱ ✧❈♦✉♣❧❡❞ ❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t r❡s✉❧ts ❢♦r ❝♦♠♣❛t✐❜❧❡ ♠❛♣♣✐♥❣s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ❢✉③③② ♠❡tr✐❝ s♣❛❝❡s✧✱ ❋✉③③② ❙❡ts ❛♥❞ ❙②st❡♠s✱ ✷✷✷✱ ✽✹✲✾✼✳ ✹✸ ❬✶✵❪ ❇✳ ❙✳ ❈❤♦✉❞❤✉r②✱ ❆✳ ❑✉♥❞✉ ✭✷✵✶✵✮✱ ✧❆ ❈♦✉♣❧❡❞ ❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥t r❡s✉❧t ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❢♦r ❝♦♠♣❛t✐❜❧❡ ♠❛♣♣✐♥❣s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✸ ✭✽✮✱ ✷✺✷✹✲✷✺✸✶✳ ❬✶✶❪ ❆✳ ❍❛r❛♥❞✐ ✭✷✵✵✽✮✱ ✧❆ ❈♦✉♣❧❡❞ ❛♥❞ tr✐♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r② ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠✧✱ ▼❛t❤✳ ❈♦♠♣✉t✳ ▼♦❞❡❧✱ ❞♦✐✿✶✵✳✶✵✶✻✴❥✳♠❝♠✳✷✵✶✶✳✶✷✳✵✵✻✱ ✺✼ ✭✾➊✶✵✮✱ ✷✸✹✸✲✷✸✹✽✳ ❬✶✷❪ ❊✳ ❑❛r❛♣✐♥❛r✱ ❘✳ ❆❣❛r✇❛❧ ✭✷✵✶✸✮✱ ✧❆ ♥♦t❡ ♦♥ ✬✬❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦✲ r❡♠s ❢♦r α✲ψ ✲❝♦♥tr❛❝t✐✈❡✲t②♣❡ ♠❛♣♣✐♥❣s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✧ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❛♥❞ ❆♣♣❧✳✱ ✷✵✶✸✱ ✷✶✻✱ ❞♦✐✿✶✵✳✶✶✽✻✴✶✻✽✼✲✶✽✶✷✲✷✵✶✸✲✷✶✻✳ ❬✶✸❪ ❊✳ ❑❛r❛♣✐♥❛r✱ ❇✳ ❙❛♠❡t ✭✷✵✶✷✮✱ ✧●❡♥❡r❛❧✐③❡❞ α✲ψ ✲❝♦♥tr❛❝t✐✈❡ t②♣❡ ♠❛♣♣✐♥❣s ❛♥❞ r❡❧❛t❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✧✱ ❆❜str✳ ❆♣♣❧✳ ❆♥❛❧✳✱ ✷✵✶✷✱ ❆rt✐❝❧❡ ■❉ ✼✾✸✹✽✻✱ ✶✼ ♣❛❣❡s✳ ❬✶✹❪ P✳ Pr❡❡t✐✱ ❙✳ ❑✉♠❛r ✭✷✵✶✺✮✱ ✧❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t ❢♦r α✲ψ ✲❝♦♥tr❛❝t✐✈❡ ✐♥ ♣❛r✲ t✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ✉s✐♥❣ ❝♦♠♣❛t✐❜❧❡ ♠❛♣♣✐♥❣s✧✱ ❆♣♣❧✳ ▼❛t❤✳✱ ✻✱ ✶✸✽✵✲✶✸✽✽✳ ❬✶✺❪ ❱✳ ▲❛❦s❤♠✐❦❛♥t❤❛♠✱ ▲✳ ❈✐r✐❝ ✭✷✵✵✾✮✱ ✧❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ♥♦♥❧✐♥❡❛r ❝♦♥tr❛❝t✐♦♥s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✵ ✭✶✷✮✱ ✹✸✹✶✲✹✸✹✾✳ ❬✶✻❪ ◆✳ ❱✳ ▲✉♦♥❣✱ ◆✳ ❳✳ ❚❤✉❛♥ ✭✷✵✶✶✮✱ ✧❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥ts ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✹✱ ✾✽✸✲✾✾✷✳ ❬✶✼❪ ▼✳ ▼✉rs❛❧❡❡♥✱ ❙✳ ❆✳ ▼♦❤✐✉❞❞✐♥❡✱ ❘✳ P✳ ❆❣❛r✇❛❧ ✭✷✵✶✷✮✱ ✧❈♦✉♣❧❡❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r α✲ψ ✲❝♦♥tr❛❝t✐✈❡ t②♣❡ ♠❛♣♣✐♥❣s ✐♥ ♣❛rt✐❛❧❧② ♦r❞❡r❡❞ ♠❡tr✐❝ s♣❛❝❡s✧✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✷✵✶✷✱ ✷✷✽✱ ❞♦✐✿✶✵✳✶✶✽✻✴✶✻✽✼✲ ✶✽✶✷✲✷✵✶✷✲✷✷✽✳ ❬✶✽❪ ❍✳ ❑✳ ◆❛s❤✐♥❡✱ ❇✳ ❙❛♠❡t✱ ❛♥❞ ❈✳ ❱❡tr♦ ✭✷✵✶✷✮✱ ✧❈♦✉♣❧❡❞ ❝♦✐♥❝✐❞❡♥❝❡ ♣♦✐♥ts ✹✹ ❢♦r ❝♦♠♣❛t✐❜❧❡ ♠❛♣♣✐♥❣s s❛t✐s❢②✐♥❣ ♠✐①❡❞ ♠♦♥♦t♦♥❡ ♣r♦♣❡rt②✧✱ ❏✳ ◆♦♥❧✐♥✲ ❡❛r ❙❝✐✳ ❆♣♣❧✳✱ ✺✱ ✶✵✹✲✶✶✹✳ ❬✶✾❪ ❏✳ ◆✐❡t♦✱ ❘✳ ▲ã♣❡③ ✭✷✵✵✺✮✱ ✧❈♦♥tr❛❝t✐✈❡ ♠❛♣♣✐♥❣ t❤❡♦r❡♠s ✐♥ ♣❛rt✐❛❧❧② ♦r✲ ❞❡r❡❞ s❡ts ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s t♦ ♦r❞✐♥❛r② ❞✐❢❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✧✱ ❖r❞❡r✱ ✷✷✱ ✷✷✸✲✷✸✾✳ ❬✷✵❪ P✳ ❉✳ Pr♦✐♥♦✈ ✭✷✵✵✼✮✱ ✧❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❇❛♥❛❝❤ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ✇✐t❤ ❤✐❣❤ ♦r❞❡r ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s✉❝❝❡ss✐✈❡ ❛♣♣r♦①✐♠❛t✐♦♥s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✻✼✱ ✷✸✻✶✲✷✸✻✾✳ ❬✷✶❪ ❇✳ ❙❛♠❡t✱ ❈✳❱❡tr♦✱ P✳ ❱❡tr♦ ✭✷✵✶✷✮✱ ✧❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❝♦♥tr❛❝t✐✈❡ t②♣❡ ♠❛♣♣✐♥❣s✧✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧✳✱ ✼✺✱ α✲ψ ✲ ✷✶✺✹✲✷✶✻✺✳ ❬✷✷❪ P✳ ◆✳ ❉✉tt❛✱ ❇✳ ❙✳ ❈❤♦✉❞❤✉r② ✭✷✵✵✽✮✱ ❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ✐♥ ♠❡tr✐❝ s♣❛❝❡s✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳✱ ✹✵✻✸✻✽✳ ✹✺ ✷✵✵✽✱ ✽ ♣❛❣❡s✱ ■❉ ... ➤ñ✱ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✱ ❝➳❝ ➳♥❤ ①➵ ❦✐Ó✉ ✐✐✐ α? ??ψ ✲❝♦✱ ➳♥❤ ①➵ α? ??❝❤✃♣ ♥❤❐♥✱ ➳♥❤ ①➵ α? ??ψ ✲❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ α? ??ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■✱ ➳♥❤ ①➵ α? ??ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■■✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥... α? ??ψ ✲❝♦ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✳ ❚r♦♥❣ ♠ơ❝ ✶ ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ♠ét sè ➤Þ♥❤ ❧ý ✈Ị ➤✐Ĩ♠ ❜✃t ➤é♥❣ ❜é ➤➠✐ ❝đ❛ ➳♥❤ ①➵ α? ??ψ ✲❝♦✱ ➳♥❤ ①➵ α? ??❝❤✃♣ ♥❤❐♥✱ ➳♥❤ ①➵ α? ??ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■✱ ➳♥❤ ①➵ α? ??ψ... ❣✐❛♥ ♠➟tr✐❝ t❤ø tù✱ ❝➳❝ ➳♥❤ ①➵ ❝♦✱ ➳♥❤ ①➵ ❦✐Ó✉ ①➵ α? ??ψ ✲❝♦✱ ➳♥❤ α? ??❝❤✃♣ ♥❤❐♥✱ ➳♥❤ ①➵ α? ??ψ ✲❝♦ s✉② ré♥❣✱ ➳♥❤ ①➵ α? ??ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■✱ ➳♥❤ ①➵ α? ??ψ ✲❝♦ s✉② ré♥❣ ❧♦➵✐ ■■✱ ➳♥❤ ①➵ ❝ã tÝ♥❤ ❝❤✃t ➤➡♥ ➤✐Ö✉