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✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ▲➊ ✣➐◆❍ ◗❯Ý◆❍ ✣➚◆❍ ▲Þ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ⑩◆❍ ❳❸ ◆Û❆ ❚Ü❆ ❈❖ ❙❯❨ ❘❐◆● ❱⑨ Ù◆● ❉Ö◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾ download by : skknchat@gmail.com ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ▲➊ ✣➐◆❍ ◗❯Ý◆❍ ✣➚◆❍ ▲Þ ✣■➎▼ ❇❻❚ ✣❐◆● ❈Õ❆ ⑩◆❍ ❳❸ ◆Û❆ ❚Ü❆ ❈❖ ❙❯❨ ❘❐◆● ❱⑨ Ù◆● ❉Ö◆● ◆❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✽✹✻✵✶✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ ề ❚❍➌ ❍Ò◆● ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾ download by : skknchat@gmail.com ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷đ❝ ❝❤➾ ró ỗ ố t ữớ ✈✐➳t ❧✉➟♥ ✈➠♥ ▲➯ ✣➻♥❤ ◗✉ý♥❤ ❳→❝ ♥❤➟♥ ❝õ❛ tr÷ð♥❣ ữớ ữợ ❚❙✳ ❇ị✐ ❚❤➳ ❍ị♥❣ ✐ download by : skknchat@gmail.com ▲í✐ ỡ rữợ tr ❧✉➟♥ ✈➠♥✱ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s tợ ũ ũ ữớ t t t ữợ tổ tr sốt q tr ự tæ✐ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚æ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❦❤♦❛ ❚♦→♥ ❝ị♥❣ t♦➔♥ t❤➸ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ✣❍❙P ❚❤→✐ ◆❣✉②➯♥ ✤➣ tr✉②➲♥ t❤ư ❝❤♦ tỉ✐ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ q✉❛♥ trå♥❣✱ t t ủ tổ ỳ ỵ õ õ qỵ tr sốt q tr t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❇↔♥ ❧✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❦❤✐➳♠ ❦❤✉②➳t ✈➻ ✈➟② rt ữủ sỹ õ õ ỵ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❈✉è✐ ❝ị♥❣ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❦❤➼❝❤ ❧➺ tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ▲➯ ✣➻♥❤ ◗✉ý♥❤ ✐✐ download by : skknchat@gmail.com ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▲í✐ ❝↔♠ ì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët số ỵ t tt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ð ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ữỡ ỵ t ①↕ ❝♦ ❇❛♥❛❝❤ ✳ ✳ ✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳ ❙ü ❤ë✐ tư tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳ ❑❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t ❝♦ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ✐✐ ✐✈ ✶ ✸ ✸ ✹ ✻ ✻ ❈❤÷ì♥❣ ✷✳ ỵ t ỷ tỹ ❝♦ s✉② rë♥❣ ✈➔ ù♥❣ ❞ö♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ tü❛ ❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳ ✣à♥❤ ỵ t tỹ s rë♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ t ♥û❛ tü❛ ❝♦ s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺✳ Ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✻ ✷✶ ✷✺ ✸✶ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✐✐✐ download by : skknchat@gmail.com ởt số ỵ t tt N N R R+ C {xn } ∅ A∪B A×B (X, d) O(x; ∞) B(S) ✷ t➟♣ ❝→❝ sè tü ♥❤✐➯♥ t➟♣ ❝→❝ sè tü ♥❤✐➯♥ ❦❤→❝ ❦❤æ♥❣ t➟♣ ❝→❝ sè t❤ü❝ t➟♣ sè t❤ü❝ ❦❤æ♥❣ ➙♠ t➟♣ ❝→❝ sè ♣❤ù❝ ❞➣② sè t➟♣ ré♥❣ ❤ñ♣ ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B t➼❝❤ ❉❡s❝❛rt❡s ❝õ❛ ❤❛✐ t➟♣ ❤ñ♣ A ✈➔ B ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ q✉ÿ ✤↕♦ ❝õ❛ →♥❤ ①↕ T t↕✐ ✤✐➸♠ x t➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ t❤ü❝ ❜à ❝❤➦♥ tr➯♥ S ✈ỵ✐ ❝❤✉➞♥ s✉♣r❡♠✉♠ ❦➳t t❤ó❝ ❝❤ù♥❣ ♠✐♥❤ ✐✈ download by : skknchat@gmail.com ỵ tt t ✤ë♥❣ ✈➔ ù♥❣ ❞ö♥❣ ❧➔ ❧➽♥❤ ✈ü❝ ♥❣❤✐➯♥ ❝ù✉ ❤➜♣ ❞➝♥ ❝õ❛ t♦→♥ ❤å❝ ❤✐➺♥ ✤↕✐✳ ✣➙② ❧➔ ❧➽♥❤ ✈ü❝ ✤➣ ✈➔ ✤❛♥❣ t❤✉ ❤ót ✤÷đ❝ sü q✉❛♥ t➙♠ ❝õ❛ rt t tr ữợ ỵ t❤✉②➳t ✤✐➸♠ ❜➜t ✤ë♥❣ ❧➔ ♠ët ❝ỉ♥❣ ❝ư q✉❛♥ trå♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❤✐➺♥ t÷đ♥❣ ♣❤✐ t✉②➳♥ t➼♥❤✳ ◆â ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ữ sỹ tỗ t ♣❤÷ì♥❣ tr➻♥❤ ✈✐✱ t➼❝❤ ♣❤➙♥✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ q✉ÿ ✤↕♦ ✤â♥❣ ❝õ❛ ❤➺ ✤ë♥❣ ❧ü❝✱ ✳✳✳ ❍ì♥ ♥ú❛✱ ♥â ❝á♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ư♥❣ tr♦♥❣ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❦❤→❝ ♥❤÷ ❦❤♦❛ ❤å❝ ♠→② t➼♥❤✱ ỵ tt ỵ tt trỏ ỡ t ỵ t♦→♥✱ s✐♥❤ ❤å❝✱ ❦✐♥❤ t➳✱ ✳✳✳ ❙ü ♣❤→t tr✐➸♥ ♠↕♥❤ ỵ tt t õ t õ t ỗ tứ ỳ ự rở r õ ỵ tr t ỵ tt t tr ổ tr ỹ r ỵ ũ ✈ỵ✐ ù♥❣ ❞ư♥❣ ❝õ❛ ♥â ✤➣ ♠ð r❛ sü ♣❤→t tr ợ ởt ỵ tt t tr ỵ tt t tr t tr t❤❡♦ ❜❛ ✈➜♥ ✤➲ s❛✉✿ ▼ð rë♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ rở ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ✤➣ ❜✐➳t ❧➯♥ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❝â ❝➜✉ tró❝ t÷ì♥❣ tü ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝❀ ✈➔ t➻♠ ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ❝❤ó♥❣✳ ✣è✐ ✈ỵ✐ ✈➜♥ ✤➲ ♠ð rë♥❣ ✤✐➲✉ ❦✐➺♥ ❝♦ ❝õ❛ →♥❤ ①↕✱ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t ữủ ỳ ợ t ữủ ✤➳♥ ♥❤÷ ❝õ❛ P❛♥t✲ ❙✐♥❣❤✲▼✐s❤r❛ ❬✸❪✱ P♦♣❡s❝✉ ❬✺❪✱ ▼♦t✲ P❡r✉s❡❧ ❬✻❪✱ ❘❤♦❛❞❡s ❬✼❪✱ ❙✐♥❣❤✲ ▼✐s❤r❛ ❬✽❪✱ ❙✉③✉❦✐ ❬✾❪✱ ✳✳✳ ◆➠♠ r ự ỵ t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ tü❛ ❝♦ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦✳ ◆➠♠ ✷✵✶✺✱ ❑✉♠❛♠✲ tttrt ự ỵ t ✤ë♥❣ ❝❤♦ →♥❤ ①↕ ✶ download by : skknchat@gmail.com tü❛ ❝♦ s✉② rë♥❣ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦✳ ❑➳t q✉↔ ♥➔② ❧➔ ♠ð rë♥❣ ❦➳t q✉↔ ❝õ❛ ❈✐r✐❝ ❬✶❪✳ ◆➠♠ ✷✵✶✼✱ P❛♥t ❬✹❪ ✤➣ ự ỵ t ♥û❛ tü❛ ❝♦ s✉② rë♥❣ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦✳ ❑➳t q✉↔ ♥➔② ❧➔ ♠ð rë♥❣ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❈✐r✐❝ ❬✶❪ ✈➔ ❑✉♠❛♠✲ ❉✉♥❣✲ ❙✐tt❤✐t❤❛❦❡r♥❣❦✐❡t ❬✷❪✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ❣✐ỵ✐ t❤✐➺✉ ❧↕✐ ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ❈✐r✐❝ ❬✶❪✱ ❑✉♠❛♠✲ ❉✉♥❣✲ ❙✐tt❤✐t❤❛❦❡r♥❣❦✐❡t ❬✷❪ ✈➔ P❛♥t ỵ t tü❛ ❝♦✱ tü❛ ❝♦ s✉② rë♥❣ ✈➔ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t q ỗ ❝❤÷ì♥❣ ♥ë✐ ❞✉♥❣✱ ♣❤➛♥ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❈❤÷ì♥❣ ✶ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➲ ổ tr ỵ ◆❣♦➔✐ r❛ ❝❤ó♥❣ tỉ✐ ❝á♥ tr➻♥❤ ❜➔② ♠ët sè ♠ð rở ỡ ỵ ❝♦ ❇❛♥❛❝❤✳ ❈❤÷ì♥❣ ✷ ❞➔♥❤ ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦ ✈➔ ởt số ỵ t tü❛ ❝♦✱ tü❛ ❝♦ s✉② rë♥❣ ✈➔ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦✳ ◆❣♦➔✐ r❛✱ ♠ët ù♥❣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ q✉② ❤♦↕❝❤ ✤ë♥❣ ❝ơ♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔②✳ ✷ download by : skknchat@gmail.com ữỡ ỵ t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ❦❤æ♥❣ ❣✐❛♥ tr ỵ t ❝♦ ❇❛♥❛❝❤ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ ✈➔ ♠ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♥â✳ ✶✳✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ✈➼ ❞ư ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ●✐↔ sû X ❧➔ t➟♣ ❤đ♣ ❦❤→❝ ré♥❣✳ ❍➔♠ d : X × X → R ✤÷đ❝ ❣å✐ ❧➔ ♠❡tr✐❝ tr➯♥ X ♥➳✉ t❤ä❛ ♠➣♥ ✭✐✮ d(x, y) ≥ ✈ỵ✐ ♠å✐ x, y ∈ X ✈➔ d(x, y) = ⇔ x = y ✭✐✐✮ d(x, y) = d(y, x) ✈ỵ✐ ♠å✐ x, y ∈ X ✭✐✐✐✮ d(x, y) ≤ d(x, z) + d(z, y) ✈ỵ✐ ♠å✐ x, y, z ∈ X ❑❤✐ ✤â ❝➦♣ (X, d) ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝✳ ❱➼ ❞ư ✶✳✶✳✷✳ ❚r➯♥ C[0,1]✱ ①➨t ❤➔♠ sè d : C[0,1] × C[0,1] → R ❜ð✐ |x(t) − y(t)|dt, d(x, y) = ✈ỵ✐ ♠å✐ x, y ∈ C[0,1] ❚❛ ❝â |x(t) − y(t)|dt ≥ 0, d(x, y) = ●✐↔ sû ✈ỵ✐ ♠å✐ x, y ∈ C[0,1] |x(t) − y(t)|dt = d(x, y) = ✸ download by : skknchat@gmail.com tữỡ ữỡ ợ x(t) = y(t), ✈ỵ✐ ♠å✐ t ∈ [0, 1] ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä x = y✳ ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝â |x(t) − y(t)|dt d(x, y) = |x(t) − z(t) + z(t) − y(t)|dt = ≤ |x(t) − z(t)|dt + |z(t) − y(t)|dt = d(x, z) + d(z, y) ✈ỵ✐ ♠å✐ x, y, z ∈ C[0,1] ❱➟② tr♦♥❣ (C[0,1], d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝✳ ✶✳✷✳ ❙ü ❤ë✐ tư tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝✱ {xn} ❧➔ ♠ët ❞➣② ❝→❝ ♣❤➛♥ tû ❝õ❛ X ✱ t❛ ♥â✐ {xn} ❤ë✐ tö ✤➳♥ z ∈ X ♥➳✉ lim d(xn , z) = n→∞ ❚❛ ❦➼ ❤✐➺✉ n→∞ lim xn = z ❤♦➦❝ xn → z ❦❤✐ n → ∞✳ ỵ sỷ (X, d) ổ ♠❡tr✐❝✳ ❑❤✐ ✤â ✭✐✮ ●✐ỵ✐ ❤↕♥ ❝õ❛ ♠ët ❞➣② ✭♥➳✉ ❝â✮ ❧➔ ❞✉② ♥❤➜t✳ ✭✐✐✮ ◆➳✉ n→∞ lim xn = a; lim yn = b t❤➻ lim d(xn , yn ) = d(a, b)✳ n→∞ n→∞ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚r♦♥❣ X ❣✐↔ sû n→∞ lim xn = a; lim yn = b ✳ ❚❛ ❝â n→∞ d(a, b) ≤ d(a, xn ) + d(xn , b) ✈ỵ✐ ♠å✐ n ❈❤♦ n → ∞ t❛ t❤✉ ✤÷đ❝ d(a, b) = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ a = b ✭✐✐✮ ❱ỵ✐ ♠å✐ n t❛ ✤➲✉ ❝â d(a, b) ≤ d(a, xn ) + d(xn , yn ) + d(yn , b) ✹ download by : skknchat@gmail.com ỵ t ❝õ❛ →♥❤ ①↕ tü❛ ❝♦ s✉② rë♥❣ ❈→❝ ❦➳t q✉↔ ❝õ❛ ♣❤➛♥ ♥➔② ✤÷đ❝ tr➼❝❤ tø ❝ỉ♥❣ tr➻♥❤ ❬✷❪✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳ ⑩♥❤ ①↕ T : X → X ✤÷đ❝ ❣å✐ ❧➔ tü❛ ❝♦ s✉② rë♥❣ tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ X tỗ t số r [0; 1) s ❝❤♦✿ d(T x, T y) ≤ rMG (x, y) ✈ỵ✐ ♠å✐ x, y ∈ X, ✭✷✳✻✮ ð ✤➙② MG (x, y) = max{d(x, y); d(x, T x); d(y, T y); d(x, T y); d(y, T x); d(T x, x); d(T x, T x); d(T x, y); d(T x, T y)} ❙è r > ♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ ✭✷✳✻✮ ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣ sè tü❛ ❝♦ s✉② rë♥❣ ❝õ❛ T ◆❤➟♥ ①➨t✳ ❱➼ ❞ö s❛✉ ❝❤➾ r❛ ♠ët →♥❤ ①↕ ❧➔ tü❛ ❝♦ s✉② rë♥❣ ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ tü❛ ❝♦✳ ❱➼ ❞ư ✷✳✸✳✷✳ ❳➨t t➟♣ X = {1, 2, 3, 4, 5} ✈ỵ✐ ♠❡tr✐❝ d : X × X → R+ ①→❝ ✤à♥❤ ❜ð✐ 0, ♥➳✉ x = y, d(x, y) = 2, ♥➳✉ (x, y) ∈ {(1, 4); (1, 5); (4, 1); (5, 1)}, 1, tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❳➨t →♥❤ ①↕ T : X → X ①→❝ ✤à♥❤ ❜ð✐ T = T = T = 1, T = 2, T = ❑❤✐ ✤â d(T x, T y) = d(1, 1) = ♥➳✉ (x, y) ∈ {1, 2, 3}; d(T 1, T 4) = d(T 2, T 4) = d(T 3, T 4) = d(1, 2) = 1; d(T 1, 4) = d(T 2, 4) = d(T 3, 4) = d(1, 4) = 2; d(T 1, T 5) = d(T 2, T 5) = d(T 3, T 5) = d(1, 3) = 1; d(T 1, 5) = d(T 2, 5) = d(T 3, 5) = d(1, 5) = 2; d(T 4, T 5) = d(2, 3) = 1; d(4, 5) = d(4, T 4) = d(5, T 5) = d(4, T 5) = d(5, T 4) = 1; ✷✶ download by : skknchat@gmail.com d(T 4, 4) = d(T 2, 4) = d(1, 4) = 2; d(T 5, 5) = d(T 3, 5) = d(1, 5) = ❇➡♥❣ ❦✐➸♠ tr❛ trü❝ t✐➳♣✱ ✈ỵ✐ r ∈ [ 21 , 1] t❛ ❝â d(T x, T y) ≤ rMG (x, y) ✈ỵ✐ ♠å✐ x, y ∈ X ❱➟② T ❧➔ ♥û❛ tü❛ ❝♦ s✉② rë♥❣✳ ❇➡♥❣ ❝→❝❤ ❝❤å♥ x = 4, y = t❛ ❝â d(T 4, T 5) > r.M (4, 5) ✈ỵ✐ ♠å✐ r ∈ [0, 1) ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä T ❦❤æ♥❣ ❧➔ →♥❤ ①↕ tü❛ ❝♦✳ ❇ê ✤➲ ✷✳✸✳✸✳ ●✐↔ sû T ♠❡tr✐❝ (X, d) ✈➔ x ∈ X✳ kn (x) ∈ {1, 2, , n} : X → X ❑❤✐ ✤â ♥➳✉ ❧➔ tü❛ ❝♦ s✉② rë♥❣ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ δ[O(x; n)] > ✈ỵ✐ n ∈ N∗ t tỗ t s [O(x; n)] = d(x, T kn (x) x) ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ x ∈ X ✈➔ ≤ i ≤ n − 1, ≤ j ≤ n✱ t❛ ❝â d(T i x, T j x) = d(T T i−1 x, T T j−1 x) ≤ r max{d(T i−1 x, T j−1 x); d(T i−1 x, T T i−1 x); d(T j−1 x, T T j−1 x); d(T i−1 x, T T j−1 x); d(T j−1 x, T T i−1 x); d(T T i−1 x, T i−1 x); d(T T i−1 x, T T i−1 x); d(T T i−1 x, T j−1 x); d(T T i−1 x, T T j−1 x)} = r max{d(T i−1 x, T j−1 x); d(T i−1 x, T i x); d(T j−1 x, T j x); d(T i−1 x, T j x); d(T j−1 x, T i x); d(T i+1 x, T i−1 x); d(T i+1 x, T i x); d(T i+1 x, T j−1 x); d(T i+1 x, T j x)} ✭✷✳✼✮ ≤ rδ[O(x; n)], ð ✤➙② δ[O(x; n)] = max{d(T ix, T j x) : ≤ i, j ≤ n}✳ ❚ø ✭✷✳✼✮ ✈➔ r < tỗ t kn(x) {1, 2, , n} s❛♦ ❝❤♦ d(x, T kn (x) x) = δ[O(x; n)] ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✷ download by : skknchat@gmail.com ỵ sỷ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦ ✈➔ T :X→X ❧➔ →♥❤ ①↕ tü❛ ❝♦ s✉② rë♥❣ ✈ỵ✐ ❤➡♥❣ sè r > 0✳ ❑❤✐ ✤â ✭✐✮ T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t z ∈ X ❀ ✭✐✐✮ n→∞ lim T n x = z ✈ỵ✐ ♠å✐ x ∈ X ❀ r d(x, T x) ✈ỵ✐ ♠å✐ x ∈ X ✈➔ n ∈ N✳ ✭✐✐✐✮ d(T nx, z) ≤ (1−r) ∗ ❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ x ∈ X, tỗ t n N s [O(x; n)] = t❤➻ x = T x ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä x ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T t trữớ ủ [O(x; n)] > ợ ♠å✐ n ∈ N∗✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✸✳✸✱ ✈ỵ✐ ♠é✐ x X n N tỗ t sè kn(x) ∈ N s❛♦ ❝❤♦ n d(x, T kn (x) x) = δ[O(x; n)] ❑❤✐ ✤â t❛ ❝â d(x, T kn (x) x) ≤ d(x, T x) + d(T x, T kn (x) x) ≤ d(x, T x) + rδ[O(x; n)] = d(x, T x) + rd(x, T kn (x) x) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ δ[O(x; n)] = d(x, T kn (x) x) ≤ d(x, T x) 1−r ✭✷✳✽✮ ❱ỵ✐ m > n✱ ✈➻ T ❧➔ tü❛ ❝♦ s✉② rë♥❣ ✈➔ ✭✷✳✽✮ ♥➯♥ d(T n x, T m x) = d(T T n−1 x, T m−n+1 T n−1 x) ≤ rδ[O(T n−1 x; m − n + 1)] = rd(T n−1 x, T km−n+1 (T n−1 = rd(T T n−2 x, T km−n+1 (T x) n−1 T n−1 x) x)+1 T n−2 x) ≤ r2 δ[O(T n−2 x; km−n+1 (T n−1 x) + 1)] ≤ r2 δ[O(T n−2 x; m − n + 2)] ≤ ≤ rn δ[O(x; m] rn d(x, T x) ≤ 1−r ✷✸ download by : skknchat@gmail.com ✭✷✳✾✮ r ❚ø n→∞ lim 1−r d(x, T x) = ♥➯♥ lim d(T n x, T m x) = 0✳ ❱➟② {T n x} ❧➔ m,n→∞ ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ❧➔ T ✲ t q tỗ t z X s❛♦ ❝❤♦ n lim T n x = z n→∞ ❇ð✐ T ❧➔ tü❛ ❝♦ s✉② rë♥❣✱ d(z, T z) ≤ d(z, T n+1 x) + d(T n+1 x, T z) = d(z, T n+1 x) + d(T T n x, T z) ≤ d(z, T n+1 x) + rMG (T n x, T z) ✭✷✳✶✵✮ ▼➦t ❦❤→❝ t❛ ❧↕✐ ❝â MG (T n x, T z) = max{d(T n x, z); d(T n x, T n+1 x); d(z, T z); d(T n x, T z); d(z, T n+1 x); d(T n+2 x; T n x); d(T n+2 x, T n+1 x); d(T n+2 x, z); d(T n+2 x, T z)} ❚ø ✤â s✉② r❛ lim MG (T n x, T z) = d(z, T z) n→∞ ✣✐➲✉ ♥➔② ✈➔ ✭✷✳✶✵✮✱ t❛ s✉② r❛ d(z, T z) ≤ r.d(z, T z) ❱➻ ✈➟② d(z, T z) = 0✳ ❉♦ ✤â z = T z✳ ❱➟② z ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ●✐↔ sû z∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❦❤→❝ ❝õ❛ T ✳ ❱➻ T ❧➔ tü❛ ❝♦ s✉② rë♥❣✱ d(z, z ∗ ) = d(T z, T z ∗ ) ≤ r max{d(z, z ∗ ); d(z, T z); d(z ∗ , T z ∗ ); d(z, T z ∗ ); d(z ∗ , T z) d(T z, z), d(T z, T z), d(T z, z ∗ ), d(T z, T z ∗ ).} = rd(z, z ∗ ) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ z = z∗✳ ❱➟② T ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳ ❈❤♦ m → ∞ tr♦♥❣ ✭✷✳✾✮✱ t❛ ✤÷đ❝ rn d(x, T x) d(T x, z) ≤ 1−r n ✷✹ download by : skknchat@gmail.com ❱➼ ❞ö ✷✳✸✳✺✳ ❳➨t t➟♣ X = {1, 2, 3, 4, 5} ✈ỵ✐ ♠❡tr✐❝ d : X × X → R+ ①→❝ ✤à♥❤ ❜ð✐ ♥➳✉ x = y, ♥➳✉ (x, y) ∈ {(1, 4); (1, 5); (4, 1); (5, 1)}, tr♦♥❣ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❳➨t →♥❤ ①↕ T : X → X ①→❝ ✤à♥❤ ❜ð✐ 0, d(x, y) = 2, 1, T = T = T = 1, T = 2, T = ❑❤✐ ✤â (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦✳ ❍ì♥ ♥ú❛✱ t❤❡♦ ❱➼ ❞ư ✷✳✸✳✷✱ T ❧➔ →♥❤ ①↕ tü❛ ❝♦ s✉② rë♥❣ ✈ỵ✐ r ∈ [ 12 , 1] ♥❤÷♥❣ ❦❤ỉ♥❣ ❧➔ →♥❤ ①↕ tü❛ ỵ ữủ ❞➵ t❤➜② z = ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ T ✳ ❚✉② ♥❤✐➯♥ ❦❤ỉ♥❣ →♣ ❞ư♥❣ ✤÷đ❝ ỵ ỵ t →♥❤ ①↕ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ ❈→❝ ❦➳t q✉↔ ❝õ❛ ♣❤➛♥ ♥➔② ✤÷đ❝ tr➼❝❤ ❞➝♥ tø ❝ỉ♥❣ tr➻♥❤ ❬✹❪✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✹✳✶✳ ⑩♥❤ ①↕ T : X → X ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ tr➯♥ ❦❤ỉ♥❣ tr X tỗ t số r [0; 1) s❛♦ ❝❤♦ (1 − r)d(x, T x) ≤ d(x, y) ❦➨♦ t❤❡♦ d(T x, T y) ≤ rMG (x, y) ✭✷✳✶✶✮ ✈ỵ✐ ♠å✐ x, y ∈ X ✱ ð ✤➙② MG (x, y) = max{d(x, y), d(x, T x), d(y, T y), d(x, T y), d(y, T x), d(T x, x), d(T x, T x), d(T x, y), d(T x, T y)} ❙è r > ♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ ✭✷✳✶✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣ sè ♥û❛ tü❛ ❝♦ s✉② rë♥❣ ❝õ❛ T ◆❤➟♥ ①➨t✳ ◆➳✉ T : X → X ❧➔ →♥❤ ①↕ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ ✈ỵ✐ ❤➡♥❣ sè r tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ (X, d) ✈➔ δ[O(x; n)] > ✈ỵ✐ n N t tỗ t j {1, 2, , n} s❛♦ ❝❤♦ δ[O(x; n)] = d(x, T j x) ✷✺ download by : skknchat@gmail.com ❇ê ✤➲ ✷✳✹✳✷✳ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✈➔ T : X → X ❧➔ →♥❤ r✳ ①↕ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ ✈ỵ✐ ❤➡♥❣ sè δ[O(xn+1 ; 1)] > õ tỗ t nN s t δ[O(xn ; 1)] ≤ rn δ[O(x0 ; n + 1)], ð ✤➙② ❞➣② {xn } tr♦♥❣ X ①→❝ ✤à♥❤ ❜ð✐ xn = T n x0 ❈❤ù♥❣ ♠✐♥❤✳ ✈ỵ✐ ♠å✐ n ∈ N ❱ỵ✐ ♠é✐ n ∈ N✱ t❛ ❝â (1 − r)d(xn , T xn ) ≤ d(xn , T xn ) = d(xn , xn+1 ) ❱➻ T ❧➔ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ ♥➯♥ d(xn+1 , xn+2 ) = d(T xn , T xn+1 ) ≤ r max{d(xn , xn+1 ), d(xn+1 , xn+2 ), d(xn , xn+2 )} ❚ø ✤â s✉② r❛ d(xn+1 , xn+2 ) ≤ rδ[O(xn ; 2)] ✈ỵ✐ ♠å✐ n ∈ N ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ δ[O(xn+1 ; 1)] ≤ rδ[O(xn ; 2)] ✈ỵ✐ ♠å✐ n ∈ N ❇➡♥❣ q✉② ♥↕♣✱ t❛ s✉② r❛ d(xn+1 , xn+2 ) ≤ rn δ[O(x0 ; n + 1)] ✈ỵ✐ ♠å✐ n ∈ N ❇ê ✤➲ ✷✳✹✳✸✳ ●✐↔ sû t➜t ❝↔ ❝→❝ ❣✐↔ t❤✐➳t ❝õ❛ ❇ê ✤➲ ✷✳✹✳✷ ✤÷đ❝ t❤♦↔ ♠➣♥✳ ❑❤✐ ✤â δ[O(x0 ; n + 1)] ≤ d(x0 , x1 ) 1−r ✷✻ download by : skknchat@gmail.com ❈❤ù♥❣ ♠✐♥❤✳ s❛♦ ❝❤♦ ❚❤❡♦ ♥❤➟♥ ①➨t ð tr tỗ t số ữỡ k n + δ[O(x0 ; n + 1)] = d(x0 , xk ) ❑❤✐ ✤â δ[O(x0 ; n + 1)] = d(x0 , xk ) ≤ d(x0 , x1 ) + d(x1 , xk ) = d(x0 , x1 ) + δ[O(x1 ; n)] ≤ d(x0 , x1 ) + rδ[O(x0 ; n + 1)] ❉♦ ✤â✱ δ[O(x0 ; n + 1)] ≤ d(x0 , x1 ) 1−r ❇ê ✤➲ ✷✳✹✳✹✳ ●✐↔ sû t➜t ❝↔ ❝→❝ ❣✐↔ t❤✐➳t ❝õ❛ ❇ê ✤➲ ✷✳✹✳✷ ✤÷đ❝ t❤♦↔ ♠➣♥✳ ❑❤✐ ✤â d(T xn , T xn+1 ) ≤ ❈❤ù♥❣ ♠✐♥❤✳ r d(xn , xn+1 ) 1−r ❚ø ❜➜t ✤➥♥❣ t❤ù❝ (1 − r)d(xn , T xn+1 ) ≤ d(xn , T xn ) = d(xn , xn+1 ) ✈➔ T ❧➔ ♥û❛ tü❛ ❝♦ s✉② rë♥❣✱ d(T xn , T xn+1 ) ≤ r max{d(xn , xn+1 ), d(xn+1 , xn+2 ), d(xn , xn+2 )} ≤ r[d(xn , xn+1 ) + d(xn+1 , xn+2 )] = r[d(xn , xn+1 ) + d(T xn , T xn+1 )] ❚ø ✤â ❦➨♦ t❤❡♦ d(T xn , T xn+1 ) ≤ r d(xn , xn+1 ) 1−r download by : skknchat@gmail.com ỵ sỷ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦ ✈➔ T :X→X ❧➔ →♥❤ ①↕ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ ✈ỵ✐ ❤➡♥❣ sè r✳ ❑❤✐ ✤â ✭✐✮ T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t z ∈ X ❀ ✭✐✐✮ n→∞ lim T n x = z ✈ỵ✐ ♠å✐ x ∈ X ❀ r d(x, T x) ✈ỵ✐ ♠å✐ x ∈ X ✈➔ n ∈ N✳ ✭✐✐✐✮ d(T nx, z) ≤ (1−r) ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ▲➜② x0 ∈ X ❝è ✤à♥❤✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ❞➣② {xn } tr♦♥❣ X ①→❝ ✤à♥❤ ❜ð✐ xn = T n x0 ợ n N tỗ t n N s❛♦ ❝❤♦ δ[O(xn; 1)] = t❤➻ n−1 xn = xn+1 = T xn ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä xn ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ T ◆➳✉ δ[O(xn; 1)] > t❤➻ t❤❡♦ ❇ê ✤➲ ✷✳✹✳✷ ✈➔ ❇ê ✤➲ ✷✳✹✳✸ t❛ ❝â rn−1 d(xn , xn+1 ) ≤ d(x0 , x1 ) 1−r ▼➦t ❦❤→❝✱ ✈ỵ✐ m > n✱ t❛ ❝â d(xn , xm ) ≤ d(xn , xn+1 ) + d(xn+1 , xn+2 ) + + d(xm−1 , xm ) ≤ (rn−1 + rn + + rm−1 ) d(x0 , x1 ) 1−r rn−1 ≤ d(x0 , x1 ) (1 − r)2 ❱➻ r ∈ [0, 1) ♥➯♥ ❚ø ✤â s✉② r❛ ✭✷✳✶✷✮ rn−1 d(x0 , x1 ) = lim n→∞ (1 − r)2 lim d(xn , xm ) = n,m→∞ ❱➟② ❞➣② {xn} ❧➔ ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ❧➔ T ✲ ✤➛② ✤õ t❤❡♦ q✉ÿ ✤↕♦ ♥➯♥ tỗ t z X s n lim xn = z ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✈ỵ✐ ♠é✐ n ∈ N✱ ❤♦➦❝ (1 − r)d(xn , T xn ) ≤ d(xn , z) ✷✽ download by : skknchat@gmail.com ❤♦➦❝ (1 − r)d(T xn , T xn+1 ) ≤ d(T xn , z) t sỷ tỗ t n N s❛♦ ❝❤♦ d(xn , z) < (1 − r)d(xn , T xn ) ✈➔ d(T xn, z) < (1 − r)d(T xn, T xn+1) ❚ø ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ ✈➔ ❇ê ✤➲ ✷✳✹✳✹✱ t❛ ❝â d(xn , T xn ) ≤ d(T xn , z) + d(T xn , z) < (1 − r)d(xn , T xn ) + (1 − r)d(T xn , T xn+1 ) r d(xn , T xn )] < (1 − r)[d(xn , T xn ) + 1−r r = (1 − r)[1 + ]d(xn , T xn ) 1−r = d(xn , T xn ) ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥✳ ❱➟② ✈ỵ✐ ♠é✐ n ∈ N✱ ❤♦➦❝ (1 − r)d(xn , T xn ) ≤ d(xn , z) ❤♦➦❝ (1 − r)d(T xn , T xn+1 ) ≤ d(T xn , z) ◆➳✉ (1 − r)d(xn, T xn) ≤ d(xn, z) t❤➻ d(T xn , T z) ≤ r max{d(xn , z), d(xn , T xn ), d(z, T z), d(xn , T z), d(z, T xn ), d(T xn , xn ), d(T xn , T xn ), d(T xn , z), d(T xn , T z)} = r max{d(xn , z), d(xn , xn+1 ), d(z, T z), d(xn , T z), d(z, xn+1 ), d(xn+2 , xn ), d(xn+2 , xn+1 ), d(xn+2 , z), d(xn+2 , T z)} ❈❤♦ n → ∞✱ t❛ t❤✉ ✤÷đ❝ d(z, T z) ≤ rd(z, T z) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ z = T z✳ ❱➟② z ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ◆➳✉ (1 − r)d(T xn, T xn+1) ≤ d(T xn, z) t❤➻ d(T xn+1 , T z) ≤ r max{d(xn+1 , z), d(xn+1 , T xn+1 ), d(z, T z), d(xn+1 , T z), ✷✾ download by : skknchat@gmail.com d(z, T xn+1 ), d(T xn+1 , xn+1 ), d(T xn+1 , T xn+1 ), d(T xn+1 , z), d(T xn+1 , T z)} = r max{d(xn+1 , z), d(xn+1 , xn+2 ), d(z, T z), d(xn+1 , T z), d(z, xn+2 ), d(xn+3 , xn+1 ), d(xn+3 , xn+2 ), d(xn+3 , z), d(xn+3 , T z)} ❈❤♦ n → ∞✱ t❛ t❤✉ ✤÷đ❝ d(z, T z) ≤ rd(z, T z) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ z = T z✳ ❱➟② z ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❇➙② ❣✐í ❣✐↔ sû z ✈➔ w ❧➔ ❤❛✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❇ð✐ (1 − r)d(z, T z) ≤ rd(z, w) ✈➔ T ❧➔ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ ♥➯♥ d(w, z) = d(T w, T z) ≤ r max{d(w, z), d(w, T w), d(z, T z), d(w, T z), d(z, T w), d(T w, z), d(T w, T w), d(T w, z), d(T w, T z)} = rd(w, z) ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ w = z✳ ❱➟② T ❝â ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳ ✭✐✐✮ ❱➻ x0 tý ỵ lim xn = lim T n x = z n→∞ n→∞ ✈ỵ✐ ♠å✐ x ∈ X ✭✐✐✐✮ ❚ø ✭✷✳✶✷✮ t❛ ❝â rn−1 d(T x, T x) = d(xn , xm ) ≤ d(x, T x) (1 − r)2 n m ♠å✐ x ∈ X ✈➔ m, n ∈ N✳ ❈❤♦ m → ∞ t❛ t❤✉ ✤÷đ❝ rn−1 d(T x, z) ≤ d(x, T x) (1 − r)2 n ✈ỵ✐ ♠å✐ x ∈ X ✈➔ n ∈ N ✸✵ download by : skknchat@gmail.com ❱➼ ❞ö ✷✳✹✳✻✳ ●✐↔ sû X = {(0, 0), (4, 0), (0, 4), (4, 5), (5, 4)}✳ ❳➨t ♠❡tr✐❝ d tr➯♥ X ①→❝ ✤à♥❤ ❜ð✐ d((x1 , x2 ), (y1 , y2 )) = |x1 − y1 | + |x2 − y2 | ❑❤✐ ✤â (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ✳ ❳➨t →♥❤ ①↕ T : X → X ❜ð✐ T (0, 0) = (0, 0), T (4, 0) = (0, 4), T (0, 4) = (0, 0), T (4, 5) = (4, 0), T (5, 4) = (0, 4) ❉➵ t❤➜② d(T x, T y) ≤ MG (x, y) ♥➳✉ (x, y) = ((4, 5), (5, 4)) ✈➔ (y, x) = ((4, 5), (5, 4)) 10 ❚r♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐✱ t❛ ❧✉ỉ♥ ❝â (1 − r)d(x, T x) > d(x, y) ✈ỵ✐ ♠å✐ r ∈ [0, 1) ♥➯♥ T t❤♦↔ ♠➣♥ t➜t ❝↔ ❝→❝ ❣✐↔ t❤✐➳t ỵ t z = (0, 0) ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ T ✳ ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝â d((T (5, 4), T (4, 5))) = > rM ((4, 5), (5, 4)) ✈ỵ✐ ♠å✐ r ∈ [0, 1) ✣✐➲✉ ♥➔② ❝❤ù♥❣ tä T ❦❤æ♥❣ ❧➔ →♥❤ ①↕ tü❛ ❝♦ s✉② rë♥❣✳ ❉♦ ✈➟② ổ ữủ ỵ ●✐↔ sû X, Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ●å✐ S ⊂ X ✈➔ D ⊂ Y ❧➔ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ tr↕♥❣ t❤→✐ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ✤✐➲✉ ❦❤✐➸♥✳ ❚❛ ❦➼ ❤✐➺✉ B(S) ❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ t❤ü❝ ❜à ❝❤➦♥ tr➯♥ S ✈ỵ✐ ❝❤✉➞♥ s✉♣r❡♠✉♠ ✳ ❑❤✐ ✤â (B(S), ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ p(x) = sup{η(x, y) + K(x, y, p(ξ(x, y)))}, y∈D ✭✷✳✶✸✮ ð ✤➙② η : S × D → R, ξ : S × D → S, p ∈ B(S) ✈➔ K : S × D × R → R download by : skknchat@gmail.com ỵ sû ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✭✐✮ K ✈➔ η ❧➔ ❜à ❝❤➦♥✳ ✭✐✐✮ ❱ỵ✐ (x, y) ∈ S × D; p, q ∈ B(S) ✈➔ t ∈ S ✿ (1 − r) |p(t) − T p(t)| ≤ |p(t) − q(t)| ❦➨♦ t❤❡♦ |K(x, y, p(t)) − K(x, y, q(t))| ≤ r max{|p(t) − q(t)| ; |p(t) − T p(t)| ; |q(t) − T q(t)| ; |p(t) − T q(t)| ; |q(t) − T p(t)| ; p(t) − T p(t) ; T p(t) − T p(t) ; q(t) − T p(t) ; T q(t) − T p(t) }, ð ✤➙② r ∈ [0, 1) ✈➔ T ①→❝ ✤à♥❤ ❜ð✐ T p(x) = sup{η(x, y) + K(x, y, p(ξ(x, y))}, y∈D ✈ỵ✐ x∈S ✈➔ p ∈ B(S)✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✸✮ ❝â ♥❣❤✐➺♠ ❞✉② t õ ú ỵ r ①↕ T ✤✐ tø B(S) ✈➔♦ ❝❤➼♥❤ ♥â✳ ▲➜② ε > ✈➔ p1 , p2 ∈ B(S) ❱ỵ✐ x S tũ ỵ t y1 , y2 D s❛♦ ❝❤♦ ❈❤ù♥❣ ♠✐♥❤✳ T pi < η(x, yi ) + K(x, yi , pi (xi )) + ε, ✭✷✳✶✹✮ ð ✤➙② xi = ξ(x, yi), i = 1, 2✳ ❇ð✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ T ✱ t❛ ❝â T p1 (x) ≥ η(x, y2 ) + K(x, y2 , p1 (x2 )), ✭✷✳✶✺✮ T p2 (x) ≥ η(x, y1 ) + K(x, y1 , p2 (x1 )) ✭✷✳✶✻✮ ❱ỵ✐ p1, p2 ∈ B(S) ✈➔ x ∈ S ✱ t❛ ❣✐↔ sû |p1 (x) − T p1 (x)| ≤ |p1 (x) − p2 (x)| ð ✤➙② MG (p1 , p2 ) = r max{d(p1 , p2 ); d(p1 , T p1 ); d(p2 , T p2 ); d(p1 , T p2 ); ✸✷ download by : skknchat@gmail.com ✭✷✳✶✼✮ d(p2 , T p1 ); d(p1 , T p1 ); d(T p1 , T p1 ); d(p2 , T p1 ); d(p2 , T p1 )} ✈➔ ❞ ❧➔ ♠❡tr✐❝ s✐♥❤ ❜ð✐ ❝❤✉➞♥ tr➯♥ B(S)✳ ❚ø ✭✷✳✶✹✮✱ ✭✷✳✶✻✮ ✈➔ ✭✷✳✶✼✮ t❛ ❝â T p1 (x) − T p2 (x) < K(x, y1 , p1 (x1 )) − K(x, y1 , p2 (x1 )) + ε ≤ |K(x, y1 , p1 (x1 )) − K(x, y1 , p2 (x1 ))| + ε ≤ MG (p1 , p2 ) + ε ✭✷✳✶✽✮ ❈❤ù♥❣ ♠✐♥❤ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü✱ t❛ ❝ô♥❣ ❝â T p2 (x) − T p1 (x) ≤ MG (p1 , p2 ) + ε ✭✷✳✶✾✮ ❚ø ✭✷✳✶✽✮ ✈➔ ✭✷✳✶✾✮✱ t❛ ❝â✿ |T p1 (x) − T p2 (x)| ≤ MG (p1 , p2 ) + > tũ ỵ |T p1 (x) − T p2 (x)| ≤ MG (p1 , p2 ) ✭✷✳✷✶✮ ❚ø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✷✶✮ ✈➔ ✭✷✳✶✼✮ t❛ ✤÷đ❝ d(p1 , T p1 ) ≤ d(p1 , p1 ) ❦➨♦ t❤❡♦ d(T p1, T p1) ≤ 21 MG(p1, p1) ❱➟② T ❧➔ →♥❤ ①↕ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ ✈ỵ✐ ❤➡♥❣ sè r = 21 ỵ tỗ t t p∗ ∈ B(S) s❛♦ ❝❤♦ p∗ = T p∗✳ ❚ø ✤â s✉② r❛ p∗ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ✈➔ ❜à ữỡ tr ỵ ữủ ự ♠✐♥❤✳ ✸✸ download by : skknchat@gmail.com ❑➳t ❧✉➟♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✈➔ ♠ët số ỵ t ởt số ♠ð rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤➛② ✤õ✳ ✷✳ ❚r➻♥❤ ỵ t tỹ ❝♦ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t❤❡♦ q ỵ r ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ tü❛ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t❤❡♦ q✉ÿ ỵ r ỵ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ♥û❛ tü❛ ❝♦ s✉② rë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ T ✲ ✤➛② ✤õ t❤❡♦ q✉ÿ ỵ r ự t q ỵ download by : skknchat@gmail.com ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❈✐r✐❝✱ ▲❥✳ ❇✳ ✭✶✾✼✹✮✱ ✏❆ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❇❛♥❝❤✬s ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐✲ ♣❧❡✑ ✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✹✺✱ ✷✻✼✲✷✼✸✳ ❬✷❪ ❑✉♠❛♠✱ P✱✳ ❱❛♥ ❉✉♥❣✱ ◆✳✱ ❙✐tt❤✐t❤❛❦❡r♥❣❦✐❡t✱ ❑✳ ✭✷✵✶✺✮✱ ✏❆ ❣❡♥❡r✲ ❛❧✐③❛t✐♦♥ ♦❢ ❝✐r✐❝ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s✑ ✱ ❋✐❧♦♠❛t ✷✾✭✼✮✱ ✶✺✹✾✲ ✶✺✺✻✳ ❬✸❪ P❛♥t✱ ❘✳✱ ❙✐♥❣❤✱ ❙✳▲✳✱ ▼✐s❤r❛✱ ❙✳◆ ✭✷✵✶✻✮✱ ✏❆ ❝♦✐♥❝✐❞❡♥❝❡ ❛♥❞ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r s❡♠✐✲q✉❛s✐ ❝♦♥tr❛❝t✐♦♥s✑ ✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r②✱ ✶✼✱ ◆♦✳✷✱ ✹✹✾✲✹✺✻✳ ❬✹❪ P❛♥t✱ ❘✳✱ ✭✷✵✶✼✮✱ ✏❋✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❣❡♥❡r❛❧✐③❛t✐♦♥ s❡♠✐✲q✉❛s✐ ❝♦♥tr❛❝t✐♦♥s✑ ✱ ❏✳ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❆♣♣❧✳ ✶✾ ✶✺✽✶✲ ✶✺✾✵✳ ❬✺❪ P♦♣❡s❝✉✱ ❖ ✭✷✵✵✾✮✱ ✏❚✇♦ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r❡♠s ❢♦r ❣❡♥❡r❛❧✐③❡❞ ❝♦♥✲ tr❛❝t✐♦♥s ✇✐t❤ ❝♦♥st❛♥ts ✐♥ ❝♦♠♣❧❡t❡ ♠❡tr✐❝ s♣❛❝❡s✑ ✱ ❈❡♥t✳ ❊✉r✳ ❏✳ ▼❛t❤✳✼✭✸✮✱ ✺✷✾✲ ✺✸✽✳ ❬✻❪ ▼♦t✱ ●✳✱ P❡r✉s❡❧✱ ❆ ✭✷✵✵✾✮✱ ✏❋✐①❡❞ ♣♦✐♥t t❤❡♦r② ❢♦r ❛ ♥❡✇ t②♣❡ ♦❢ ❝♦♥tr❛t✐✈❡ ♠✉❧t✐✈❛❧✉❡❞ ♦♣❡r❛t♦rs✑ ✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧ ✼✵✱ ✸✸✼✶✲ ✸✸✼✼✳ ❬✼❪ ❘❤♦❛❞❡s✱ ❇✳❊ ✭✶✾✼✼✮✱ ✏❆ ❝♦♠♣❛r✐s♦♥ ♦❢ ✈❛r✐♦✉s ❞❡❢✐♥✐t✐♦♥s ♦❢ ❝♦♥✲ tr❛❝t✐✈❡ ♠❛♣♣✐♥❣s✑ ✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝ ✷✷✻✱ ✷✺✼✲ ✷✾✵✳ ❬✽❪ ❙✐♥❣❤✱ ❙✳▲✳✱ ▼✐s❤r❛✱ ❙✳◆ ✭✷✵✵✶✮✱ ✏❈♦✐♥❝✐❞❡♥❝❡ ❛♥❞ ❢✐①❡❞ ♣♦✐♥ts ♦❢ ♥♦♥✲ s❡❧❢ ❤②❜r✐❞ ❝♦♥tr❛❝t✐♦♥s✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧ ✷✺✻✱ ✹✽✻✲ ✹✾✼✳ ❬✾❪ ❙✉③✉❦✐✱ ❚ ✭✷✵✵✼✮✱ ✏❆ ❣❡♥❡r❛❧✐③❡❞ ❇❛♥❛❝❤ ❝♦♥tr❛❝t✐♦♥ ♣r✐♥❝✐♣❧❡ t❤❛t ❝❤❛r❛❝t❡r✐③❡s ♠❡tr✐❝ ❝♦♠♣❧❡t❡♥❡ss✑ ✱ ❏Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✶✸✻✭✸✮ ✱ ✶✽✻✶✲ ✶✽✻✾✳ ✸✺ download by : skknchat@gmail.com ... sè✿ ✽✹✻✵✶✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ữớ ữợ ề ề ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾ download by : skknchat@gmail.com ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣... ▲➯ ✣➻♥❤ ◗✉ý♥❤ ❳→❝ ♥❤➟♥ ❝õ❛ tr÷ð♥❣ ❦❤♦❛ ❚♦→♥ ữớ ữợ ũ ❚❤➳ ❍ị♥❣ ✐ download by : skknchat@gmail.com ▲í✐ ❝↔♠ ì♥ rữợ tr tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ũ ũ ữớ t t t ữợ tæ✐ tr♦♥❣... t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ▲➯ ✣➻♥❤ ◗✉ý♥❤ ✐✐ download by : skknchat@gmail.com ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
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