Phương Trình Tích Phân

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ▲■❊◆P❍❖◆❊ ❈❍❊❯❈❍❖❯❚❍❖❘ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✻ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ▲■❊◆P❍❖◆❊ ❈❍❊❯❈❍❖❯❚❍❖❘ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ữớ ữợ ◆●❯❨➍◆ ❚❍➚ ◆●❹◆ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✻ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ 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 ữủ ró ỗ ố ❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✳✳✳ t❤→♥❣ ✳✳✳ ♥➠♠ ✷✵✶✻ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ▲■❊◆P❍❖◆❊ ❈❍❊❯❈❍❖❯❚❍❖❘ ✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ Lp ✳ ✳ ✳ ✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ổ rt ổ ữợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ❑❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✸ ❑❤æ♥❣ ❣✐❛♥ L2ρ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❚♦→♥ tû ✤è✐ ①ù♥❣ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ P❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❚♦→♥ tû ❧✐➯♥ ❤ñ♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✸ ❚♦→♥ tû ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✹ ❚♦→♥ tû ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✺ ❚♦→♥ tû ✤è✐ ①ù♥❣ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tư❝ ✳ ✳ ✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✷✳✶ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ P❤➙♥ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ ❑❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✷✳✷✳✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦♠ ✳ ✷✳✷✳✸ P❤÷ì♥❣ tr➻♥❤ ❱♦❧t❡rr❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐ ✐✐ ✶ ✹ ✹ ✽ ✽ ✾ ✶✵ ✶✶ ✶✶ ✶✸ ✶✻ ✷✵ ✷✷ ✷✻ ✷✻ ✸✵ ✸✵ ✸✶ ✸✶ ✷✳✸ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ❧♦↕✐ ♠ët ✷✳✸✳✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ❧♦↕✐ ❤❛✐ ✳ ✷✳✹ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ợ ố ự Pữỡ tr t➼❝❤ ♣❤➙♥ ✈ỵ✐ ❤↕❝❤ s✉② ❜✐➳♥ ✳ ✳ ✳ ✷✳✻ ỵ r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ❋r❡❞❤♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✷ ✸✸ ✸✹ ✸✻ ✸✽ ✸✽ ✹✵ ✹✵ ✹✺ ✐✐✐ ▼ð ✤➛✉ ◆❤✐➲✉ ✈➜♥ ✤➲ ❝õ❛ t♦→♥ ỡ t ỵ ỳ ữỡ tr tr õ ữ t ữợ t ♣❤➙♥✳ ◆❤ú♥❣ ♣❤÷ì♥❣ tr➻♥❤ ➜② ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝ỉ♥❣ ❝ư t♦→♥ ❤å❝ ❤ú✉ ➼❝❤ ✤÷đ❝ ❞ị♥❣ tr♦♥❣ t♦→♥ ❤å❝ ỵ tt t ự Pữỡ tr t ♣❤➙♥ ❤♦➦❝ ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦♠ ❧♦↕✐ ♠ët ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿ b f (x) = K (x, y)φ (y) dy, a < x < b, a tr♦♥❣ ✤â f (x)K (x, y) ỳ trữợ (x) ❧➔ ❤➔♠ ❝❤÷❛ ❜✐➳t ❝â ♠➦t ð ❝↔ tr♦♥❣ ✈➔ ♥❣♦➔✐ ❞➜✉ t➼❝❤ ♣❤➙♥ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ➜② ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦♠ ❧♦↕✐ ❤❛✐✿ b φ (x) = K (x, y)φ (y) dy + f (x) , a < x < b a ◆➳✉ ❝➟♥ ữợ t ỳ t ữỡ tr ➜② ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❱♦❧t❡rr❛ ❧♦↕✐ ♠ët ✈➔ ❧♦↕✐ ❤❛✐ t÷ì♥❣ ù♥❣ ❝â ❞↕♥❣✿ x f (x) = K (x, y)φ (y) dy, a < x < b a x φ (x) = K (x, y)φ (y) dy + f (x) , a a < x < b P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝ỉ♥❣ ❝ư t♦→♥ ỳ t ữủ sỷ t ỵ t❤✉②➳t ✈➔ ❣✐↔✐ t➼❝❤ ù♥❣s ❞ö♥❣✳ ✣➦❝ ❜✐➺t ♥â ✶ ❝á♥ ❣✐ó♣ ➼❝❤ ❝❤♦ ✈✐➺❝ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❣✐↔♥❣ ❞↕② ð ❝→❝ tr÷í♥❣ ❝❛♦ ✤➥♥❣ ✈➔ ✤↕✐ ❤å❝✳ ❚r♦♥❣ ❝❤÷ì♥❣ tr➻♥❤ ❚♦→♥ ð ❜➟❝ ✤↕✐ ❤å❝✱ tỉ✐ ✤➣ ữủ t ổ ợ t ữỡ tr t➼❝❤ ♣❤➙♥ ✈➔ ✈❛✐ trá ❝õ❛ ♥â ✤è✐ ✈ỵ✐ ❜ë ♠ỉ♥ t♦→♥ ❤å❝✳ ❙❛✉ ❦❤✐ ✤÷đ❝ ♥❣❤❡ ❝→❝ t❤➛② ❝ỉ ợ t tổ t ữỡ tr t rt q✉❛♥ trå♥❣✳ ❱ỵ✐ t➛♠ q✉❛♥ trå♥❣ ✤â ❝ị♥❣ ✈ỵ✐ sü ữợ ú ù t t t ❝æ ❣✐→♦ tr♦♥❣ ❇ë ♠æ♥ ❣✐↔✐ t➼❝❤ tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐✿ ✧P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✧ ❧➔♠ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣✳ ◗✉❛ ❧✉➟♥ ✈➠♥ ♥➔② tæ✐ ♠✉è♥ ♥❣❤✐➯♥ ❝ù✉ ♠ët số ỵ tt ỡ ữỡ tr t ◆❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ỗ ữỡ ữỡ r ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ Lp✱ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✈➔ ❝→❝ t♦→♥ tû tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ♥❤÷✿ t♦→♥ tû ❧✐➯♥ ❤đ♣✱ t♦→♥ tû ✤è✐ ①ù♥❣✱ t♦→♥ tû ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝✱ t♦→♥ tû ✤è✐ ①ù♥❣ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝✳ ✣➙② ❧➔ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t ❝❤✉➞♥ ❜à ❝❤♦ ❝❤÷ì♥❣ ✷ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❈❤÷ì♥❣ ✷✿ ✣➙② ❧➔ ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♥❤÷✿ t♦→♥ tû t➼❝❤ ♣❤➙♥✱ ♣❤➙♥ ❧♦↕✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✱ ữỡ tr t ợ ố ự ữỡ tr t ợ s ữỡ tr t ợ t ý ỵ r ữỡ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ữợ sỹ ữợ t t ◆❣➙♥✳ ◆❤➙♥ ❞à♣ ♥➔② ❝❤♦ ♣❤➨♣ ✤÷đ❝ ❜➔② tä ❧á♥❣ t ỡ s s tợ ổ ữớ t t ữợ ú ù tổ tr sốt q tr ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚r♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✱ tỉ✐ ♥❤➟♥ ✤÷đ❝ r➜t ♥❤✐➲✉ sü ❣✐ó♣ ✤ï ✤ë♥❣ ✈✐➯♥ ❝õ❛ ❝→❝ t❤➛②✱ ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✈➔ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝✳ ❚ỉ✐ ①✐♥ ✤÷đ❝ ❜➔② tä ✷ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ r➧♥ ❧✉②➺♥ t↕✐ ❑❤♦❛✱ ❚r÷í♥❣✳ ❈✉è✐ ❝ị♥❣ ❞♦ ❦✐♥❤ ♥❣❤✐➺♠ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sõt tổ rt ữủ ỳ ỵ ❦✐➳♥ ❝❤➾ ❜↔♦✱ ✤â♥❣ ❣â♣ ❝õ❛ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ❝❛♦ ❤å❝ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ Lp ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❬✷❪✱❬✸❪ ❈❤♦ (X, M, µ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦✱ tr♦♥❣ ✤â ❳ ❧➔ ❦❤æ♥❣ ❣✐❛♥✱ ▼ ❧➔ ♠ët σ✲✤↕✐ sè ❝→❝ t➟♣ ❝♦♥ ❝õ❛ ❳✱ µ ❧➔ ♠ët ✤ë ✤♦ tr➯♥ ▼✳ ❈❤♦ ♣∈ [1; +∞) ❧➔ ♠ët sè t❤ü❝✳ ❍å t➜t ❝↔ ❝→❝ ❤➔♠ sè ❢✭①✮ ❝â ❧ô② t❤ø❛ ❜➟❝ ♣ ❦❤↔ t tr ổ Lp(X, à) ữ ✈➟② Lp (X, µ) = {f : X −→ R : |f |p dµ < ∞} x ❑❤✐ ❳ ❧➔ t➟♣ ✤♦ ✤÷đ❝ t❤❡♦ ♥❣❤➽❛ ▲❡❜❡s❣✉❡ tr♦♥❣ ▲❡❜❡❣s✉❡ t❤➻ t❛ t Lp(X) t Lp(X, à) ợ p = ỵ Rk L (X) = {f : X −→ R|ess sup|f (x)| < +∞} tr♦♥❣ ✤â ess sup |f (x)| = inf {M > 0|µ{x ∈ X||f (x)| > M } = 0} x∈X [2] , [3] ủ Lp(X, à) ợ t tổ tữớ tr số ợ ①→❝ ✤à♥❤ ❜ð✐ f (x) Lp (X,µ) |f |p dµ = p ợ ộ f Lp (X, à) X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥✳ ✹ ❈❤ù♥❣ ♠✐♥❤✳ ❉➵ t❤➜② r➡♥❣✱ ✈ỵ✐ ♠å✐ f, g ∈ Lp(X, à), ợ k K, t õ |f + g| ≤ 2max{|f |, |g|} ❚ø ✤â✱ s✉② r❛ |f + g|p ≤ 2p max{|f |p , |g|p ≤ 2p (|f |p + |g|p ) ❱➟② f + g Lp(X, à). r kf Lp(X, à) ữ Lp(X, à) õ ố ợ t t❤ỉ♥❣ t❤÷í♥❣ tr➯♥ ❤➔♠ sè ♥➯♥ ♥â ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✳ ❚❛ ❜✐➳t r➡♥❣✱ |f |pdµ = ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f = ❤➛✉ ❦❤➢♣ ♥ì✐ tr➯♥ ❳ X ♥➯♥ ✤✐➲✉ ❦✐➺♥ t❤ù ♥❤➜t ❝õ❛ ❝❤✉➞♥ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ✣✐➲✉ ❦✐➺♥ t❤ù ❤❛✐ ❧➔ ❤✐➸♥ ♥❤✐➯♥✱ ✤✐➲✉ ❦✐➺♥ t❛♠ ❣✐→❝ ✤÷đ❝ s✉② r❛ tø ❜➜t ✤➥♥❣ tự s ữủ ự ỵ [2] , [3] Lp(X, µ) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû {fn} ❧➔ ♠ët ❞➣② ❈❛✉❝❤② tr♦♥❣ Lp(X, µ)✱ tù❝ ❧➔ fn − fm = lim m,n→∞ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ m, n ≥ nk ✱ k N tỗ t ởt số ||fm fn || < ✣➦❝ ❜✐➺t nk ∈ N∗ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ 2k ✈ỵ✐ ♠å✐ n ≥ nk ❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❝â t❤➸ ❣✐↔ t❤✐➳t n1 < n2 < < nk < ❑❤✐ ✤â ||fm − fn || < 2k ||fn+1 − fn || < ❱ỵ✐ ♠å✐ s ∈ N∗✱ ✤➦t 2k s |fnk+1 (x) − fnk (x)| ∈ Lp (X, µ) gs (x) = |fn1 (x)| + k=1 ✺ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ b a f (τ ) dτ = (τ − a) (b − τ ) ✷✳✸✳✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ❧♦↕✐ ❤❛✐ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à s❛✉ ✤➙② [10] b β (t) α (t) ϕ (t) + πi ϕ (t) dτ = f (t), t ∈ (a, b) , τ −t ✭✷✳✶✶✮ a tr♦♥❣ ✤â ❣✐↔ t❤✐➳t r➡♥❣ α2 (t) − β (t) = 1, ∀t ∈ (a, b) ✭✷✳✶✷✮ ◆❤➟♥ ①➨t r➡♥❣ ❦✐➺♥ ❜➡♥❣ ✶ tr♦♥❣ ✭✷✳✶✷✮ ❦❤æ♥❣ q✉❛♥ trå♥❣ ♠➔ q✉❛♥ trå♥❣ ❧➔ ❦❤→❝ ❦❤ỉ♥❣✳ ❈❤ó♥❣ t❛ ❧✉ỉ♥ ❝â t❤➸ ❧➔♠ ❝❤♦ ❝â ✤✐➸✉ ❦✐➺♥ ✭✷✳✶✷✮ ❜➥♥❣ ❝→❝❤ ❝❤✐❛ ❝↔ ❤❛✐ ✈➳ ❝❤♦ α2 (t) − β (t) ●✐↔ sû f (t) ✈➔ G (t) = α(t)−β(t) α(t)+β(t) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③ ❦❤♦↔♥❣ (a, b) ❚❛ ỵ L = (a, b) , L ❧➔ ❝❤✐➲✉ tø a ✤➳♥ b✳ ❚❛ ✤➦t G (a) = ρeiθ , G (b) = ρ ei(θ+∆) , tr♦♥❣ õ ỵ sỹ t rG(t) tr L ●✐→ trà ❝õ❛ argθ t↕✐ ✤➛✉ ♠ót a ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ −2π < θ ≤ 0, ✭✷✳✶✸✮ < θ < 2π, ✭✷✳✶✹✮ ♥➳✉ ♥❣❤✐➺♠ ❜à ❝❤➦♥ t↕✐ a ♥➳✉ ♥❣❤✐➺♠ ❦❤æ♥❣ ❜à t↕✐ a ✣➦t −iθ θ = − 2π + i ln2πρ , 2πi ln ρ e = 2πi ln ρ e−i(θ+∆) = θ+∆ 2π − χ γ= γ ✸✸ − i ln2πρ , tr♦♥❣ ✤â χ ❧➔ sè ♥❣✉②➯♥ ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ θ+∆ , 2π ✭✷✳✶✺✮ θ+∆ + 1, 2π ✭✷✳✶✻✮ χ= ♥➳✉ ♥❣❤✐➺♠ ❜à ❝❤➦♥ t↕✐ b, χ= ổ t b, tr õ ỵ [x] số tỹ x ỵ ❤✐➺✉ G (t) , t ∈ [a; b] , 1, t ∈ (−∞; a) ∪ (b; +∞) G1 (t) = ❚❛ ❝â  iθ   G1 (a − 0) = 1, G1 (a + 0) = G (a) = ρe , G1 (b − 0) = G (b) = ρei(θ+∆) , G1 (b + 0) = 1,   G1 (a−0) = e−iθ , G1 (b−0) = ρ ei(θ+∆) G1 (a+0) ρ G1 (b+0) ✭✷✳✶✼✮ ❈❤➾ sè ❝õ❛ G1(t) t❤❡♦ trö❝ t❤ü❝ 2πi IndG1 (t) = = = = 2πi 2πi 2πi 2πi = = χ [ln G1 (t)]R ln ln ln G1 (a−0) −2πiγ (b−0) −2πiγ + ln G G1 (a+0) e G1 (b+0) e e−iθ e−2πiγ + ln ρ ei(θ+∆) e−2πiγ ρ e−i(θ+2πiγ) + ln ρ ei(θ+∆−2πγ ) ρ 2πiχ ✭✷✳✶✽✮ ln + ln e ✷✳✹ P❤÷ì♥❣ tr t ợ ố ự t ữỡ tr t ữợ b (x) = f (x) + K (x, y)ϕ (y) dy, a tr♦♥❣ ✤â f (x), K(x, y) trữợ tt f (x) ∈ L2(a, b), K (x, y) ✤è✐ ①ù♥❣ ✈➔ ❜➻♥❤ ♣❤÷ì♥❣ ❦❤↔ t➼❝❤✳ ✸✹ P❤÷ì♥❣ tr➻♥❤ ♥➔② ❝â t❤➸ t ữợ trứ tữủ = f + Aϕ tr♦♥❣ ✤â A ❧➔ ♠ët t♦→♥ tû ✤è✐ ①ù♥❣ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tö❝✳ ❚❛ ✤➣ ❜✐➳t t♦→♥ tû A ❝â ♠ët ❤➺ trü❝ ❝❤✉➞♥ ✤➛② ✤õ ✈❡❝tì r✐➯♥❣ {en} ù♥❣ ✈ỵ✐ ❝→❝ ❣✐→ trà r✐➯♥❣ λn ✈➔ λn → ●✐↔ sû t❛ ✤➣ ❜✐➳t {en} ✈➔ λn ❑❤✐ ✤â ♠✉è♥ ①→❝ ✤à♥❤ ϕ t❛ ❝❤➾ ❝➛♥ ①→❝ ✤à♥❤ ❝→❝ ❤➺ sè ❢♦✉r✐❡r (ϕ, ei) ❝õ❛ ♥â ✤è✐ ✈ỵ✐ ❤➺ {ei} ✈➻ ϕ = i (ϕ, ei) ei ữ t ei ợ ộ i t❛ ❝â ϕ, ei = f, ei + Aϕ, ei = f, ei + ϕ, Aei ✭✷✳✷✵✮ = f, e + λ ϕ, e i i i ◆➳✉ λi = t❤➻ ϕ, ei = f, ei (1 − λi ) ❈á♥ ♥➳✉ λi = t❤➻ ✭✷✳✷✵✮ t r f trữợ tọ f, ei = 0, ♥❤÷♥❣ ❦❤✐ ➜② ✭✷✳✶✽✮ ❦❤ỉ♥❣ ✤➦t ✤✐➲✉ ❦✐➺♥ ♥➔♦ ❝❤♦ ϕ, ei ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ❣✐↔ t❤✐➳t f, ei = ❝❤♦ ei ự ợ i = 1, ữỡ tr➻♥❤ ✭✷✳✶✾✮ ❧➔ f, ei ϕ=Σ ei + Σ ξj ej ✭✷✳✷✶✮ (1 − λ) tr♦♥❣ ✤â ξj ❧➔ ♥❤ú♥❣ số tũ ỵ tờ số t ei ❝â λi = 1, ❝á♥ Σ ❝❤➾ tê♥❣ sè ❧➜② t❤❡♦ ❝→❝ ej ❝â λj = ✈➻ λi → 0(i → ∞) ♥➯♥ < ∞, M = supi 1−λ ✈➔ i Σ | f, ei − λi |2 ≤ M Σ | f, ei |2 ≤ M f < ∞ ❉♦ ✤â ❝❤✉é✐ t❤ù ♥❤➜t tr♦♥❣ ✭✷✳✷✶✮ ❤ë✐ tö ✈➲ ϕ ❈á♥ tờ tự tr t ỗ ởt số ❤ú✉ ❤↕♥ ❤↕♥❣ tû ✭✈➻ ❝❤➾ ❝â t❤➸ ❝â ♠ët sè ❤ú✉ ❤↕♥ ei t❤ä❛ ♠➣♥ λi = ❱➟② tê♥❣ ➜② ❜❛♦ ❣✐í ❝ơ♥❣ ①→❝ ✤à♥❤ ✈➔ ❜➡♥❣ ♠ët ♣❤➛♥ tû ϕ ❚❛ ❝â f,ei f,ei Aϕ = Σ 1−λ Ae = Σ λi ei i 1−λ i i f,ei = −Σ f, ei i + Σ 1−λ ei = −f + ϕ i Aϕ = Σ ξj Aei = Σ ξj ei = ϕ ✸✺ ❈❤♦ ♥➯♥ Aϕ = Aϕ + Aϕ = −f + ϕ + ϕ , ✤â ❝❤➼♥❤ ❧➔ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â ϕ t❤✉ë❝ ❦❤♦↔♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ ù♥❣ ✈ỵ✐ tr r ỵ [3] A ❦❤ỉ♥❣ ❝â ❣✐→ trà ♥➔♦ ❜➡♥❣ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✾✮ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t✱ ✈ỵ✐ ♠å✐ f trữợ A õ tr t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✾✮ ❝❤➾ ❝â ♥❣❤✐➺♠ f trü❝ ❣✐❛♦ ✈ỵ✐ ❦❤ỉ♥❣ ❣✐❛♥ ❝♦♥ r✐➯♥❣ ✈ỵ✐ ❣✐→ trà r✐➯♥❣ 1, ✈➔ ❦❤✐ ✤â ♥❣❤✐➺♠ ✤÷đ❝ ①→❝ ✤à♥❤ ①➯ ①➼❝❤ ♠ët tỷ tũ ỵ ổ r Pữỡ tr t ợ s s✉② ❜✐➳♥ ❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧➔ ❤↕❝❤ ❝â ❞↕♥❣ n K(x, y) = ak (x) bk (y) ✭✷✳✷✷✮ k=1 ❈❤ó♥❣ t❛ ❣✐↔ t❤✐➳t r➡♥❣✱ ❝→❝ ❤➔♠ ak (x) ✈➔ bk (y) ❧➔ ❝→❝ ❤➔♠ ❜➻♥❤ ♣❤÷ì♥❣ ❦❤↔ t➼❝❤ tr➯♥ ✭❛✱❜✮✳ ✣➦t ✭✷✳✷✷✮ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ b ϕ (x) − λ t❛ ✤÷đ❝ K (x, y)ϕ (y) dy = f (x) , a < x < b ✭✷✳✷✸✮ a b n ϕ (x) − λ ak (x) k=1 bk (y)ϕ (y) dy = f (x) , ✭✷✳✷✹✮ a sỷ ữỡ tr õ ỵ b bk (y)ϕ (y) dy = Ck ✭✷✳✷✺✮ Ck ak (x) + f (x) , a < x < b ✭✷✳✷✻✮ a ❚ø ✭✷✳✷✹✮ ✈➔ ✭✷✳✷✺✮ s✉② r❛ n ϕ (x) = λ k=1 ✸✻ ◆❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✷✻✮ ✈ỵ✐ bm(x)✱ ❧➜② t➼❝❤ ♣❤➙♥ t❤❡♦ x tr➯♥ ✭❛✱❜✮ sû ỵ t ữủ ữỡ tr số t✉②➳♥ t➼♥❤ n Cm = λ αmk Ck + fm , ✭✷✳✷✼✮ m = 1, 2, , n k=1 tr♦♥❣ ✤â b αmk = b ak (x)bm (x) dx, fm = a f (x)bm (x) dx a ◆➳✉ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✭✷✳✷✼✮ ❦❤æ♥❣ ❝â ♥❣❤✐➺♠ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✸✮ ❝ơ♥❣ ❦❤ỉ♥❣ ❝â ♥❣❤✐➺♠✳ ●✐↔ sû ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✭✷✳✷✼✮ ❝â ♥❣❤✐➺♠ ❧➔ c1, c2, cn ❑❤✐ ✤â ❤➔♠ ϕ(x) ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ ✭✷✳✷✹✮ s➩ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✸✮✳ ❚❤➟t ✈➟②✱ ✤÷❛ ✭✷✳✷✻✮ ✈➔♦ ✈➳ tr→✐ ❝õ❛ ✭✷✳✷✸✮ t❛ ✤÷đ❝ n f (x) + λ m=1   C −  m b n bm (y) f (y) + λ Ck ak (y) dy k=1 a   = f (x)  ❞♦ t❛ ✤➣ ❝â ✭✷✳✷✼✮✳ ✣à♥❤ t❤ù❝ ❝õ❛ ❤➺ ✭✷✳✷✼✮ ❧➔ − λα11 −λα12 −λα21 − λα22 D (λ) = −λαn1 −λαn2 −λα11n −λα2n − λαnn ❘ã r➔♥❣ D(λ) ❧➔ ✤❛ t❤ù❝ ❜➟❝ n ✈➔ D(0) = ❱➟② ♥➳✉ λ ❦❤æ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ D(λ) t❤➻ ❤➺ ✭✷✳✷✼✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳ ❱➼ ❞ö ✷✳✶✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ π ϕ (x) − λ sin (x + y)ϕ (y) dy = f (x) , < x < π ✸✼ ❚❛ ❝â sin (x + y) = sin x cos y + sin y cos x, n = 2, a1 (x) = sin x, b1 (y) = cosy, a2 (x) = cos x, b2 (y) = sin y, π π α11 = π π , α22 = π π , π cos2 xdx = α21 = f1 = sin2 xdx = sin x cos ydx = 0, α12 = sin x cos xdx = 0, π f (x) cos xdx, f2 = f (x) sin xdx ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❤➺ ✭✷✳✷✼✮ ❝â ❞↕♥❣ C1 − λπ C2 = f1 , λπ − C1 + C2 = f2 ✭✷✳✷✽✮ ✣à♥❤ t❤ù❝ ❝õ❛ ❤➺ ✭✷✳✷✽✮ ❧➔ D (λ) − ✈➔ ❝â ♥❣❤✐➺♠ λ1 = − , π λ2 π , λ2 = π ❱➟②✱ ♥➳✉ λ = ± π2 t❤➻ ❤➺ ✭✷✳✷✽✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t C1, C2 ✈➔ ❦❤✐ ✤â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔ ϕ (x) = f (x) + λ (C1 sin x + C2 cosx) ỵ r ✷✳✻✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦♠ b (I − λA) ϕ ≡ ϕ (x) − λ K (x, y)ϕ (y) dy = f (x) , a ✸✽ a