Phương trình tích phân fredholm loại hai tổng quát

▲❮■ ❈❷▼ ❒◆ ▲í✐ ✤➛✉ t✐➯♥ tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐✿ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ỵ t ổ tr ổ t t t ụ t ũ ữớ ữợ ự ữợ ụ ữ tổ õ t❤➯♠ ♥❣❤à ❧ü❝ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ◆❤➙♥ ❞à♣ ♥➔② tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ t♦➔♥ t❤➸ ợ P õ ỳ ỵ õ ❣â♣✱ ❣✐ó♣ ✤ï✱ ✤ë♥❣ ✈✐➯♥ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❚æ✐ ①✐♥ tr➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❙ì♥ ▲❛✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽✳ ❙✐♥❤ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ❈❤❛♥❣ ♠✉❛ ✶ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✳✶ ❑❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ✈ỵ✐ ♥❤➙♥ t→❝❤ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✹ ✹ ✺ ✽ ✷ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ❋❘❊❉❍❖▲▼ ▲❖❸■ ❍❆■ ❱❰■ ◆❍❹◆ ❚✃◆● ◗❯⑩❚ ✶✷ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ P❤÷ì♥❣ ♣❤→♣ t❤➳ ❧✐➯♥ ❤đ♣ ✳ ✳ P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ❈→❝ ✤à♥❤ ỵ r trú ❣✐↔✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✺ ✶✽ ✷✺ ✸ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ❋❘❊❉❍❖▲▼ ▲❖❸■ ❍❆■ ❱❰■ ◆❍❹◆ ❍❊❘▼■❚■❆◆ ✸✵ ✸✳✶ ✸✳✷ ✸✳✸ ✸✳✹ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥❤➙♥ ❍❡r♠✐t✐❛♥ ❈→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ rt r rt ỵ ❍✐❧❜❡rt✲❙❝❤♠✐❞t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✷ ✸✽ ✹✺ ✺✼ ✺✽ ✶✳ ▲Þ ❉❖ ❈❍➴◆ ❑➶❆ ▲❯❾◆ ▼Ð ✣❺❯ ◆❤✐➲✉ ✈➜♥ ✤➲ t♦→♥ ❤å❝ ữỡ tr ợ ỡ t ỵ ự ữợ t ◆❤ú♥❣ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ ✤â ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✤÷đ❝ ①❡♠ ♥❤÷ ❧➔ ♠ët ❝ỉ♥❣ ❝ư t♦→♥ ❤å❝ ❤ú✉ ➼❝❤ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ♥➯♥ ✤÷đ❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ t❤❡♦ ♥❤✐➲✉ ❦❤➼❛ ❝↕♥❤ ❦❤→❝ ♥❤❛✉✳ ◆â ❝â ù♥❣ ❞ư♥❣ rë♥❣ r➣✐ ❦❤ỉ♥❣ ❝❤➾ tr♦♥❣ t♦→♥ ❤å❝ ♠➔ ❝á♥ tr♦♥❣ ♥❤✐➲✉ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❦❤→❝✱ ✈➼ ❞ư ♥❤÷ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✈ỵ✐ ❝→❝ ✤✐➲✉ ❦✐➺♥ ①→❝ ✤à♥❤ ❤♦➦❝ ✤➸ ❣✐↔✐ qt ởt số t ỵ ữỡ tr ✈✐ ♣❤➙♥ ❦❤ỉ♥❣ t❤➸ ♠ỉ t↔ ✤÷đ❝ ♥❤÷ ❤✐➺♥ t÷đ♥❣ ❦❤✉➳❝❤ t→♥✱ ❤✐➺♥ t÷đ♥❣ tr✉②➲♥✱✳✳✳ ❱➻ ✈➟② ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ỵ tt t ợ ố ự t s ỡ ữỡ tr t ỗ t❤í✐ ✤â♥❣ ❣â♣ t❤➯♠ ♠ët sè ❧í✐ ❣✐↔✐ ❝❤♦ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✱ tæ✐ ♠↕♥❤ ❞↕♥ ❧ü❛ ❝❤å♥ ✤➲ t➔✐ ✧ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ tê♥❣ q✉→t✧ ✤➸ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝✳ ✷✳▼Ö❈ ✣➑❈❍ ◆●❍■➊◆ ❈Ù❯ ❑❤â❛ ❧✉➟♥ t➟♣ tr✉♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ s❛✉✿ ✲ ◆❣❤✐➯♥ ❝ù✉ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝✳ ✲ ❱➟♥ ❞ư♥❣ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t➟♣ ❧✐➯♥ q✉❛♥✳ ✸✳ ✣➮■ ❚×Đ◆● ◆●❍■➊◆ ❈Ù❯ ◆❣❤✐➯♥ ❝ù✉ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝✳ ✹✳ ◆❍■➏▼ ❱Ư ◆●❍■➊◆ ❈Ù❯ ✲ ❚➻♠ ❤✐➸✉ ❦❤→✐ q✉→t ❝→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ❝õ❛ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝✳ ✲ ▲➔♠ rã ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr t õ t ữủ Pì PP ◆●❍■➊◆ ❈Ù❯ ✲ ❙÷✉ t➛♠✱ ✤å❝ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t➔✐ ❧✐➺✉✱ ♣❤➙♥ t➼❝❤ tê♥❣ ❤ñ♣ ❝→❝ ❦✐➳♥ t❤ù❝✳ ✲ ❚r❛♦ t ợ ữợ tr ụ ữ sr ợ tờ ổ ì P ế ợ ♠➫ ❝õ❛ ❦❤â❛ ❧✉➟♥✿ ✣➙② ❧➔ ♠ët ✈➜♥ ✤➲ ❦❤→ ợ ố ợ t tr t ỗ tớ ✤➙② ❝ơ♥❣ ❧➔ ♠ët ✈➜♥ ✤➲ ❝á♥ ❝❤÷❛ ✤÷đ❝ t✐➳♣ ❝➟♥ ♥❤✐➲✉ ✤è✐ ✈ỵ✐ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✣❍❙P ❚♦→♥ ữợ t tr õ ❝ù✉ ✈➔ tê♥❣ ❤đ♣✱ t❤è♥❣ ❦➯ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝✳ ✼✳ ◆❍Ú◆● ✣➶◆● ●➶P ❈Õ❆ ❑❍➶❆ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ ✤➣ ♥➯✉ r❛ ✤÷đ❝ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝❤♦ ♠ët sè ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✈➔ ❝→❝ ❜➔✐ t➟♣ ❧✐➯♥ q✉❛♥✳ ✽✳ ❈❻❯ ❚❘Ó❈ ❑❍➶❆ ợ ữ õ ữủ t ữỡ ợ ỳ s ✤➙②✿ ❈❤÷ì♥❣ ✶✿ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥❀ ❑❤ỉ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ ✈➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt❀ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t→❝❤ ❜✐➳♥✱ ✳ ✳ ✳ ❝➛♥ ❞ò♥❣ ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝❤÷ì♥❣ s❛✉✳ ❈❤÷ì♥❣ ✷✿ ❚r➻♥❤ ❜➔② ✈➲ ❝→❝❤ ❣✐↔✐ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ợ tờ qt ữỡ r ❣✐↔✐ ♠ët sè ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ợ rt ữỡ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❝➛♥ t❤✐➳t ❝❤♦ ❦❤â❛ ❧✉➟♥ ♥❤÷✿ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✱ ✤➦❝ ❜✐➺t ❧➔ ữỡ tr t r ợ t ❜✐➳♥✳ ✶✳✶ ❑❤→✐ ♥✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ♠➔ t t ữợ t t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝â ❞↕♥❣ b λϕ(x) − K(x, t)ϕ(t)dt = f (x), a tr♦♥❣ ✤â ✭✶✳✶✮ f (x) trữợ õ tr ♣❤ù❝ ✈➔ ❧✐➯♥ tư❝ tr➯♥ ✤♦↕♥ [a, b]❀ • K(x, t) trữợ tử tr [a, b] ì [a, b] õ tr ự ữủ ❣å✐ ❧➔ ♥❤➙♥; • λ ❧➔ ❤➡♥❣ sè ♣❤ù❝ ❝❤♦ trữợ (x) t ổ ữủ t❤✐➳t ❧➔ ❦❤↔ t➼❝❤ t❤❡♦ ♥❣❤➽❛ ❘✐❡♠❛♥♥✳ ❚❛ ❝â t❤➸ ♣❤➙♥ ❧♦↕✐ ♥❤÷ s❛✉✿ ✶✳ ◆➳✉ ❤➺ sè λ = t❤➻ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ b K(x, t)ϕ(t)dt = f (x) a P❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦❧♠ ❧♦↕✐ ♠ët ✷✳ ◆➳✉ ❤➺ sè λ = t❤➻ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐✳ ◆➳✉ ♥❤➙♥ K(x, t) ❝â t➼♥❤ ❝❤➜t K(x, t) ≡ ✈ỵ✐ ♠å✐ t > x t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ trð t❤➔♥❤ ♣❤÷ì♥❣ tr➻♥❤ ❱♦❧t❡rr❛✳ ✸✳ ◆➳✉ λ = t❤➻ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ x λϕ(x) − K(x, t)ϕ(t)dt = f (x), a ✈➔ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❱♦❧t❡rr❛ ❧♦↕✐ ❤❛✐✳ ✹ ✹✳ ◆➳✉ λ = t❤➻ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ x K(x, t)ϕ(t)dt = f (x), a ✈➔ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧♦↕✐ ♠ët✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ú tổ t ợ ữỡ tr r ❇➡♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐✱ t❛ ❝â t❤➸ ✈✐➳t ♣❤÷ì♥❣ tr➻♥❤ t r ữợ b (x) = f (x) + λ K(x, t)ϕ(t)dt a ✭✶✳✷✮ ✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤ó♥❣ t❛ ❞ò♥❣ ❝→❝ ❦➼ ❤✐➺✉ Q [a, b] = [a, b] × [a, b] , C [a, b] = {f : [a, b] → C : ❢ ❧✐➯♥ tö❝ tr➯♥ [a, b]}, C(Q [a, b]}) = {f : Q [a, b]} → C : ❢ ❧✐➯♥ tö❝ tr➯♥ Q [a, b]}, R [a, b] ❧➔ t➟♣ ❤ñ♣ ❝→❝ ❤➔♠ ❣✐→ trà ♣❤ù❝ ✈➔ ❦❤↔ t➼❝❤ tr➯♥ [a, b] , R2 [a, b] ❧➔ t➟♣ ❤đ♣ ❝→❝ ❤➔♠ ❜➻♥❤ ♣❤÷ì♥❣ ❦❤↔ t➼❝❤ tr➯♥ [a, b] ❱ỵ✐ ♠é✐ f ∈ C [0, 1] , t❛ ❦➼ ❤✐➺✉ b f |f (x)|dx = a ✈➔ 1/2 b f |f (x)|2 dx = a ❱ỵ✐ ♠é✐ K(x) ∈ C(Q[0, 1]) t❛ ❦➼ ❤✐➺✉ b K 1/2 b |K(x, t)|2 dxdt = a a ❈❤♦ f ✈➔ g ❧➔ ❤❛✐ ❤➔♠ t❤✉ë❝ C [a, b] t t t ổ ữợ b f, g = f (x)g(x)dx a ◆➳✉ f, g = t❤➻ t❛ ♥â✐ f ✈➔ g trö❝ ❣✐❛♦✳ ❚❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ Cauchy − Schwarz b b a b |f (x)|2 dx f (x)g(x)dx ≤ a |g(x)|2 dx a ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ✭❍➺ ❝→❝ ❤➔♠ trü❝ ❝❤✉➞♥✮✳ ❉➣② {ϕn(x)} ❝→❝ ❤➔♠ t❤✉ë❝ C [a, b] ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❤➺ trü❝ ❝❤✉➞♥ ♥➳✉ ϕn , ϕm = ✺ ♥➳✉ n = m, ♥➳✉ n = m ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳ ✭❍➺ ✤➛② ✤õ✮✳ ❈❤♦ Φ = {ϕn(x)}∞n=1 ❧➔ ❤➺ ❝→❝ ❤➔♠ trü❝ ❝❤✉➞♥ ✈➔ f ∈ R2 [a, b] ◆➳✉ f trü❝ ❣✐❛♦ ✈ỵ✐ ♠å✐ ♣❤➛♥ tû ❝õ❛ Φ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f ❤➺ Φ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ✤➛② ✤õ✳ ✣➦t Φm = {ϕ1, , ϕm} ❧➔ t➟♣ ❝♦♥ ❤ú✉ ❤↕♥ ❝õ❛ Φ ◆➳✉ f ❝❤✉é✐ ❋♦✉r✐❡r ❤ë✐ tö ❚r♦♥❣ ✤â ∈ span {Φm } = t❤➻ t❤➻ t❛ ❝â f (x) = f, ϕ1 ϕ1 (x) + · · · + f, ϕm ϕm (x), ✤÷đ❝ ❣å✐ ❧➔ ❤➺ sè ❋♦✉r✐❡r t❤ù n ❝õ❛ f (x) ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳ ✭❙ü ❤ë✐ tö ✤➲✉ ✮✳ ❈❤♦ {fn(x)} ❧➔ ❞➣② ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ [a, b] ❚❛ ♥â✐ ❞➣② {fn(x)} ❤ë✐ tư ✤➲✉ tỵ✐ ❤➔♠ f (x) tr➯♥ [a, b] ợ > 0, tỗ t số ♥❣✉②➯♥ N = N (ε) s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n ≥ N t❤➻ | fn(x) − f (x) |< ε ✈ỵ✐ ♠å✐ x ∈ [a, b] f, ϕn , n = 1, , m ỵ ❈❛✉❝❤②✮✳ ❉➣② ✈æ ❤↕♥ {fn(x)} ❝→❝ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ [a, b] ❤ë✐ tư ✤➲✉ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ > 0, tỗ t số N () s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n, m ≥ N (ε)✱ | fn (x) − fm (x) |< ε ✈ỵ✐ ♠å✐ x [a, b] ỵ {fn(x)}n=1 ❧➔ ❞➣② ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ❤ë✐ tư ✤➲✉ tỵ✐ ❤➔♠ f (x) tr➯♥ [a, b] , t❤➻ f (x) ❝ô♥❣ ❦❤↔ t➼❝❤ tr➯♥ [a, b] ✈➔ b b f (x)dx = lim fn (x)dx n→∞ a a ∞ ❚ø ✤â t❛ s✉② r❛ r➡♥❣ ♥➳✉ ❝❤✉é✐ un(x) ❤ë✐ tö ✤➲✉ ✤➳♥ S(x) tr➯♥ [a, b] ✈➔ ✈ỵ✐ ♠é✐ n=1 n, un (x) ❦❤↔ t➼❝❤ tr➯♥ [a, b] t❤➻ ∞ b b S(x)dx = un (x)dx a a n=1 ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✻✳ ✭❍ë✐ tö tr✉♥❣ ❜➻♥❤✮✳ ❉➣② {fn (x)} tr♦♥❣ R2 [a, b] ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư tr✉♥❣ ❜➻♥❤ tỵ✐ ❤➔♠ ❣✐ỵ✐ ❤↕♥ f (x) tr♦♥❣ R [a, b] ♥➳✉ 1/2 b lim fn − f n→∞ 2 |f (x) − fn (x)| dx = lim n→∞ = a ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✼✳ ✭❚♦→♥ tû ❋r❡❞❤♦❧♠✮✳ ❈❤♦ K(x, t) ❧➔ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥ Q [a, b] ✈➔ ❦❤↔ t➼❝❤ t❤❡♦ tø♥❣ ❜✐➳♥ tr➯♥ [a, b]✳ ❑➼ ❤✐➺✉ t♦→♥ tû K : R2 [a, b] → R2 [a, b] b ϕ(t) → K(x, t)ϕ(t)dt a ✈➔ ❣å✐ t tỷ r tữỡ ự ợ t K(x, t)✳ ✣➦t K1 (x, t) = K(x, t) b K2 (x, t) = K1 (x, s)K(s, t)ds a b Km (x, t) = Km−1 (x, s)K(s, t)ds a ✻ ❚❛ ❣å✐ Km(x, t) ❧➔ ♥❤➙♥ ❧➦♣ t❤ù m ❝õ❛ K(x, t)✳ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â b Km ϕ = ♠ = 1, Km (x, t)ϕ(t)dt, a ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✽✳ ❚♦→♥ tû K ✤÷đ❝ ❣å✐ ❧➔ ❜à tỗ t số C s ❝❤♦ Kϕ ≤ C ϕ , ∀ϕ ∈ R2 [a, b] ◆➳✉ K ❜à ❝❤➦♥ t❤➻ t❛ ✤➦t Kϕ : ϕ ∈ R2 [a, b] , ϕ = ϕ K = sup ✈➔ ❣å✐ ♥â ❧➔ ❝❤✉➞♥ ❝õ❛ t♦→♥ tû K ▼➺♥❤ ✤➲ ✶✳✷✳✾✳ ◆➳✉ ♥❤➙♥ K(x, t) ❝â 2 Kϕ ❈❤ù♥❣ ♠✐♥❤✳ ≤ K K(x, t) 2 ϕ 22 , < ∞ t❤➻ ∀ϕ ∈ R2 [a, b] ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ Cauchy − Schwarz t❛ ❝â b |Kϕ(x)| = K(s, x)ϕ(s)ds a b b |K(x, s)|2 ds ≤ |ϕ(s)|2 ds a a ❉♦ ✈➟② b Kϕ 2 |Kϕ(x)|2 dx = a b ≤ |ϕ(s)|2 ds |K(x, s)| dsdx a = b b K a 2 ϕ a 2 ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✵✳ sỷ K t tỷ r tữỡ ự ợ ♥❤➙♥ K(x, t) ❑➼ ❤✐➺✉ K ∗ (x.t) = K(x, t) ✈➔ t♦→♥ tû K∗ ①→❝ ✤à♥❤ ❜ð✐ b ∗ ∗ K ∗ (x, t)ϕ(t)dt K : ϕ ∈ R [a, b] → (K ϕ) (x) = a ✤÷đ❝ ❣å✐ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ t♦→♥ tû K ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ s✉② r❛✿ ✭✐✮ ❱ỵ✐ ♠å✐ ϕ, ψ t❤✉ë❝ R2 [a, b] t❤➻ Kϕ, ψ ✭✐✐✮ ❱ỵ✐ ♠å✐ m ≥ t❤➻ (Km)∗ = (K∗)m ✼ = ϕ, K∗ ψ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✶✳ ✭▼✐➲♥✮✳ ❚➟♣ Ω ⊂ C ♠ð✱ ❦❤→❝ ré♥❣ ✈➔ ❧✐➯♥ t❤ỉ♥❣ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♠✐➲♥ tr♦♥❣ C ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✷✳ ❈❤♦ ♠✐➲♥ Ω ✈➔ ❤➔♠ f : Ω → C ✭✐✮ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ tr➯♥ Ω ♥➳✉ f ❦❤↔ ✈✐ t↕✐ ♠å✐ ✤✐➸♠ z ∈ Ω ✭✐✐✮ ❍➔♠ f ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♣❤➙♥ ❤➻♥❤ tr➯♥ Ω tỗ t t P s P ❦❤ỉ♥❣ ❝â ✤✐➸♠ ❣✐ỵ✐ ❤↕♥ tr♦♥❣ Ω; ✲ f (z) ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ tr♦♥❣ ♠✐➲♥ Ω\P ; ✲ ▼å✐ ✤✐➸♠ ❝õ❛ P ✤➲✉ ❧➔ ❝ü❝ ✤✐➸♠ ❝õ❛ f (z) ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✸✳ ❍➔♠ f : C → C ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♥❣✉②➯♥ ♥➳✉ f ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ tr➯♥ t♦➔♥ ♠➦t ♣❤➥♥❣ ♣❤ù❝ C ✶✳✸ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ✈ỵ✐ ♥❤➙♥ t→❝❤ ❜✐➳♥ ❚r♦♥❣ ♠ư❝ ♥➔②✱ t❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ❝â ❞↕♥❣ b ϕ(x) = f (x) + λ K(x, t)ϕ(t)dt, a ✭✶✳✸✮ tr♦♥❣ ✤â K(x, t) ❧➔ ♥❤➙♥ t→❝❤ ❜✐➳♥ tr➯♥ Q [a, b] ❝â ❞↕♥❣ n K(x, t) = (x)bi (t), ✭✶✳✹✮ i=1 tr♦♥❣ ✤â ai(x), bi(t) ❧➔ ❝→❝ ❤➔♠ t❤✉ë❝ C [a, b] ❚❤❛② ✭✶✳✹✮ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ t❛ t❤✉ ✤÷đ❝ n ϕ(x) = f (x) + λ b (x) bi (t)ϕ(t)dt a i=1 ❑❤✐ ✤â ♣❤÷ì♥❣ tr✐♥❤ tr➯♥ trð t❤➔♥❤ n ci (x), ϕ(x) = f (x) + λ ✭✶✳✺✮ i=1 tr♦♥❣ ✤â ci = ab bi(t)ϕ(t)dt, i = 1, , n ❚ø ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ s✉② r❛ ♥❣❤✐➺♠ ϕ(x) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ✤÷đ❝ ①→❝ ✤à♥❤ ♥➳✉ ♥❤÷ ①→❝ ✤à♥❤ ✤÷đ❝ ❤➺ sè ci ◆❤➙♥ ❝↔ ❤❛✐ ✈➳ ữỡ tr ợ bi(t) rỗ t t❤❡♦ ❜✐➳♥ t tr➯♥ [a, b] t❛ t❤✉ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ n aij cj , ci = f i + λ i = 1, , n, j=1 tr♦♥❣ b fi = b bi (t)f (t)dt, aij = a aj (t)bi (t)dt a ✽ ❱✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ t÷ì♥❣ ữỡ ợ ởt số t t (■ − λ❆)❝ = ❢, ✭✶✳✻✮ tr♦♥❣ ✤â ■ ❧➔ ♠❛ tr➟♥ ✤♦♥ ✈à ❝➜♣ n × n, ❆ = (aij ) tr n ì n ợ ❝→❝ ♣❤➛♥ tû ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ tr➯♥✱ ❢ = (f1, , fn)T ✈➔ ❝ = (c1, , cn)T ❧➔ ❝→❝ ❤➺ sè ♣❤↔✐ t➻♠✳ ❑➼ ❤✐➺✉ D(λ) = ❞❡t (■ − λ❆) ❱✐➺❝ ❣✐↔✐ ❤➺ t✉②➳♥ t➼♥❤ ✭✶✳✻✮ ♣❤ư t❤✉ë❝ ✈➔♦ ❣✐→ trà ❝õ❛ ❚❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉✿ ❚r÷í♥❣ ❤đ♣ ✶✿ D(λ) = ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② λ ✤÷đ❝ ❣å✐ ❧➔ ❣✐→ trà ❝❤➼♥❤ q✉② ❝õ❛ ♥❤➙♥✳ ❑❤✐ ✤â ❤➺ t✉②➳♥ t➼♥❤ tr➯♥ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t D(λ) ❝ = (■ − λ❆)−1❢, ❤❛② ❝ = D(λ) adj(■ − λ❆❢, tr♦♥❣ ✤â ❛❞❥(■ − λ❆) = (Dji(λ))❧➔ ♠❛ tr➟♥ ♣❤ư ❤đ♣ ❝õ❛ ♠❛ tr➟♥ (■ − λ❆) ❉♦ ✤â ♠é✐ ❤➺ sè ci ❝â ❜✐➸✉ ❞✐➵♥ ci = D(λ) n Dji (λ)fj j=1 ❚❤❛② ❜✐➸✉ ❞✐➵♥ ❝õ❛ ci ✈➔ fi ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ t❛ ✤÷đ❝ b ϕ(x) = f (x) + λ a D(λ) n Dji (λ)ai (x)bj (t) f (x)dt i=1 j=1 ❑➼ ❤✐➺✉ R(x, t; λ) = D(λ) n n n Dji (λ)ai (x)bj (t) i=1 j=1 ✈➔ ❣å✐ ♥â ❧➔ ♥❤➙♥ ❣✐↔✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ ❑❤✐ ✤â ♥❣❤✐➺♠ ϕ(x) ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ b ϕ(x) = f (x) + λ ✭✶✳✼✮ R(x, t; λ)f (t)dt a ✣➦t    ❉(x, t; λ) =     a1 (x) a2 (x) · · · b1 (t) − λa11 −λa12 · · · b2 (t) −λa21 − λa22 · · · ✳✳✳ ✳✳✳ ✳✳✳ bn (t) −λan1 −λan2 ✈➔ D(x, t; λ) = ❞❡t(❉(x, t; λ)) ❑❤✐ ✤â R(x, t; λ) = − ✾ D(x, t; λ) D(λ) ✳✳✳ an (x) −λa1n −λa2n ✳✳✳ · · · − λann        ✭✶✳✽✮ ❚r÷í♥❣ ❤đ♣ ✷✿ D(λ) = ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② λ ✤÷đ❝ ❣å✐ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥✳ ●✐↔ sû λk ❧➔ ♠ët ❣✐→ trà r✐➯♥❣✱ ♥❣❤➽❛ ❧➔ D(λk ) = ❳➨t tr÷í♥❣ ❤đ♣ f = ❑❤✐ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ trð t❤➔♥❤ (■ − λk ❆)❝ = ❱➻ D(λk ) = ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝â pk ♥❣❤✐➺♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐  ❝(j)(λk ) =   (j)  c1 (λk ) (j) ✳✳✳ j = 1, , pk   cn (λk ) ❚❤❛② ❝→❝ ❣✐→ trà ♥➔② ✈➔♦ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ t❛ t❤✉ ✤÷đ❝ ❝→❝ ♥❣❤✐➺♠ n (j) ϕj (x; λk ) = f (x) + λk ci (λk )a)i (x), j = 1, , pk i=1 ◆➳✉ f (x) ≡ tr➯♥ [a, b] t❤➻ ♠é✐ ❤➔♠ n (e) ϕj (x; λk ) (j) = λk ci (λk )a)i (x) i=1 ❧➔ ♠ët ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t❤✉➛♥ ♥❤➜t b K(x, t)ϕ(t)dt ϕ(x) = λk a ❈❤➾ sè tr➯♥ (e) ✤➸ ❦➼ ❤✐➺✉ r➡♥❣ ♥❣❤✐➺♠ ϕ(e) j (x; ) r tữỡ ự ợ trà r✐➯♥❣ λk ▼é✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t s➩ ❝â ❞↕♥❣ pk ϕ (h) (e) (x; λk ) = αj ϕj (x; λk ), j=1 tr♦♥❣ ✤â j số tũ ỵ số tr (h) ❧➔ ✤➸ ❦➼ ❤✐➺✉ r➡♥❣ ϕ(h)(x; λk ) ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t❤✉➛♥ ♥❤➜t tr➯♥✳ ❳➨t tr÷í♥❣ ❤đ♣ ❢ = ❚❛ s➩ sû ❞ư♥❣ ❜ê ✤➲ s❛✉✿ ❇ê ✤➲ ✶✳✸✳✶✳ ❈❤♦ B = (bij )n×n ✈➔ B∗ = (bij )n×n ✳ ❑❤✐ ✤â ♥➳✉ ❞❡t (B) = t❤➻ ❤➺ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t Bx = f ❝â ♥❣❤✐➺♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f trü❝ ợ tt ữỡ tr ❤ñ♣ t❤✉➛♥ ♥❤➜t B∗ y = ❚ø ❜ê ✤➲ ♥➔②✱ t❛ t❤➜② ❤➺ t✉②➳♥ t➼♥❤ (I − λk A)c = f ❝â ♥❣❤✐➺♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f trü❝ ợ tt ữỡ tr (I − λk A)∗ d = ✭✶✳✾✮ ❱➻ ♠❛ tr➟♥ (I − λk A) ✈➔ (I − λk A)∗ ❝â ❝ò♥❣ ❤↕♥❣ ✈➔ sè ❦❤✉②➳t ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮ ❝ơ♥❣ ❝â pk ♥❣❤✐➺♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳ ▲↕✐ ❝â ❚ (I − λk A)∗ d = (I − λk A∗ )d = (I − λk A )d = ✶✵ ❉♦ ✤â b N ϕ(x) = f (x) + λ a n=1 N = f (x) + λ n=1 ϕn (x)ϕn (t) f (t)dt λn − λ f, ϕn ϕn (x) λn − λ ◆➳✉ λ ❧➔ ❣✐→ trà r✐➯♥❣ t t ỵ r tự ố ợ t t ữỡ tr t ✈➔ ❝❤➾ ♥➳✉ f, ϕj = ✈ỵ✐ ♠å✐ j = k, , k + λ − ◆❤÷ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ð ❈❤÷ì♥❣ ✶✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ❝â ❞↕♥❣ ϕ(x) = f (x) + λk ϕ(p) (x, λk ) + βϕ(h) (x, λk ), tr♦♥❣ ✤â f (x) + ϕ(p)(x, λk ) ❧➔ ♥❣❤✐➺♠ ✤➦❝ ❜✐➺t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤✱ β số tũ ỵ (h) (x, ) sỹ ❦➳t ❤đ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ❤➔♠ r✐➯♥❣ t÷ì♥❣ ù♥❣ ✈ỵ✐ λk ❚❤❛② ❜✐➸✉ ❞✐➵♥ N K(x, t) = n=1 ϕn (x)ϕn (t) λn ✈➔ ♣❤÷ì♥❣ tr➻♥❤ t❛ t❤✉ ✤÷đ❝ b N ϕ(x) = f (x) + λk a n=1 ϕn (x)ϕn (t) ϕ(t)dt λn b N = f (x) + λk a n=1 ϕ, ϕn ϕn (t) λn ✭✸✳✹✮ ◆➳✉ j ❧➔ ❝❤➾ sè s❛♦ ❝❤♦ λj = λk t❤➻ t❛ ❝â ϕ, ϕj = f, ϕj + λk ❙✉② r❛ ϕ, ϕj = ϕ, ϕj λj λj f, ϕj λj − λk ✭✸✳✺✮ ❚❤❛② ✭✸✳✺✮ ✈➔♦ ✭✸✳✹✮ t❛ t❤✉ ✤÷đ❝ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ λk f, ϕj ϕj (x) λj − λk ϕ(x) = f (x) + j=k k ❈á♥ ✈ỵ✐ ❝❤➾ sè j ♠➔ λj = λk t❤➻ ϕ(x) = cj ϕj (x) j=1 ❉♦ ✈➟②✱ t❛ ❝â ❜✐➸✉ ❞✐➵♥ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ϕ(x) = f (x) + λk j=k f, ϕj ϕj (x) + λj − λk ✹✹ k+r−1 cj j (x) j=k ỵ rtt é trữợ ✈ỵ✐ sü ❦❤❛✐ tr✐➸♥ s♦♥❣ t✉②➳♥ t➼♥❤ ❤ú✉ ❤↕♥✱ t❛ ✤➣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ✈ỵ✐ ♥❤➙♥ t→❝❤ ❜✐➳♥ ❍❡r✲♠✐t✐❛♥✳ ❱➜♥ ✤➲ ✤➦t r❛ ❧➔ sü ❦❤❛✐ tr✐➸♥ ♥➔② ❧✐➺✉ ❝â ❝á♥ ✤ó♥❣ ♥➳✉ ❝❤➾ sè N tr♦♥❣ tê♥❣ ✤÷đ❝ t❤❛② t❤➳ ❜ð✐ ∞✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ♥â✐ ❝❤✉♥❣ ❧➔ ❝➙✉ tr↔ ❧í✐ ❦❤ỉ♥❣ ♠➜② ❦❤↔ q✉❛♥✱ ✈➻ ❝❤✉é✐ ∞ n=1 ϕn (x)ϕn (t) λn ❝â t❤➸ ❦❤ỉ♥❣ ❤ë✐ tư✳ ❚✉② ♥❤✐➯♥ t❛ ✈➝♥ ❝â ❝á♥ ❣✐↔✐ q✉②➳t ✤÷đ❝ tr♦♥❣ ♠ët sè tr÷í♥❣ ❤đ♣✳ ✣✐➲✉ ♥➔② s➩ ✤÷đ❝ t❤➸ ❤✐➺♥ tr♦♥❣ ✤à♥❤ ỵ rtt ữợ r ỵ t ❣✐↔ sû r➡♥❣ ❝❤✉é✐ ❋♦✉r✐❡r g(x) ∼ g, ϕn ϕn (x) = gn ϕn (x) n n ❝â t❤➸ ①➙② ❞ü♥❣ ✤÷đ❝ ♠➔ ❦❤ỉ♥❣ ♥â✐ ❣➻ ✈➲ ❦✐➸✉ ❤ë✐ tư ❝õ❛ ♥â✳ ❚❛ ❝ơ♥❣ ❦❤ỉ♥❣ ❣✐↔ sû r➡♥❣ t➟♣ ❝→❝ trỹ {n(x)} ỵ ỵ rtt K(x, t) rt t❤✉ë❝ C(Q[a, b]) ●✐↔ sû r➡♥❣ ♥❤➙♥ K(x, t) ❝â ✈æ ❤↕♥ ❣✐→ trà r✐➯♥❣ λ1 , λ2 , ✈➔ ❝→❝ ❤➔♠ r✐➯♥❣ trü❝ ❝❤✉➞♥ t÷ì♥❣ ù♥❣ ϕ1 (x), ϕ2 (x), ❱ỵ✐ g ∈ R2 [a, b] t❛ ①→❝ ✤à♥❤ ❤➔♠ b K(x, t)g(t)dt f (x) = Kg = a ❑❤✐ ✤â f (x) ❝â t❤➸ ✤÷đ❝ ❦❤❛✐ tr✐➸♥ t❤➔♥❤ ❝❤✉é✐ ❋♦✉r✐❡r ❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ✤➲✉ ♥❤÷ s❛✉ ∞ ∞ fn ϕn (x) = f (x) = n=1 n=1 gn ϕn (x), λn tr♦♥❣ ✤â fn = f, ϕn ✈➔ gn = g, ϕn ❧➔ ❝→❝ ❤➺ sè ❋♦✉r✐❡r ❝õ❛ f (x) ✈➔ g(x) ❍ì♥ ♥ú❛ f ❝ô♥❣ t❤✉ë❝ R2 [a, b] ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ K(x, t) ❧➔ ♥❤➙♥ ❍❡r♠✐t✐❛♥✱ ♠é✐ λn ✤➲✉ ❧➔ sè t❤ü❝ ♥➯♥ fn = f, ϕn = Kg, ϕn = g, Kϕn = ❳➨t tê♥❣ r✐➯♥❣ N σN (x) = n=1 ✹✺ g, ϕn λn gn ϕn (x) λn = gn g, ϕn = λn λn ❚❛ ❝â N +p |σN +p (x) − σN (x)|2 = n=N +1 ϕn (x) gn λn N +p N +p gn2 ≤ n=N +1 n=N +1 N +p ∞ gn2 ≤ n=1 n=N +1 ❱➻ ∞ |ϕn (x)|2 λ2n ∞ gn2 | g, φn |2 ≤ g = n=1 ♥➯♥ ✈ỵ✐ N ✤õ ❧ỵ♥ t❤➻ |ϕn (x)|2 λ2n 2 < + n=1 N +p gn2 õ t ọ tũ ỵ n=N +1 ∞ |ϕn (x)|2 λ2n n=1 ❝ô♥❣ ❜à ❝❤➦♥ ✭ t❤❡♦ ❤➺ q✉↔ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❇❡ss❡❧✮✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❱ỵ✐ ♠é✐ x ∈ [a, b] ❝è ✤à♥❤✱ ①➨t ❤➔♠ ❧✐➯♥ tö❝ kx(s) = K(x, s) ❍➺ sè ❋♦✉r✐❡r ❝õ❛ kx(s) tr♦♥❣ ❤➺ trü❝ ❝❤✉➞♥ {ϕn(s)} ✤÷đ❝ ❝❤♦ ❜ð✐ b kx , ϕn = K(x, s)ϕn (s)ds = a ❉♦ ✤â ∞ ❙✉② r❛ ❝❤✉é✐ |ϕn (x)|2 = λ2n n=1 ∞ {σN (x)}N =1 ϕn (x) λn ∞ | kx , ϕn |2 ≤ kx 2 < +∞ n=1 t❤ä❛ ♠➣♥ t✐➯✉ ❝❤✉➞♥ ❈❤❛✉❝❤②✳ ❉♦ õ tỗ t tử s (x) = Nlim σN (x) →∞ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ f (x) = σ(x) ❱➻ σ(x) = Nlim σN (x) ợ > tỗ t số ♥❣✉②➯♥ Nε s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ N ≥ Nε t❤➻ σ ∈ R [a, b] ε |σN (x) − σ(x)| < √ b−a ❙✉② r❛ b ϕN − σ ❤❛② σN − σ < ε 2 = ✈ỵ✐ ♠å✐ N ≥ Nε ε2 |σN (x) − σ(x)| dx < a ✹✻ ▼➦t ❦❤→❝✱ t❛ t❤➜② r➡♥❣ N gn ϕn (x) λn f (x) − σN (x) = Kg − n=1 N b g, ϕn ϕn (x) λn K(x, t)g(t)dt − = a n=1 N b K(x, t)g(t)dt − = a n=1 N b K(x, t) − = a n=1 b λn g(t)ϕn (t)dt ϕn (x) a ϕn (x)ϕn (t) λn g(t)dt b = ∆N +1 (x, t)g(t)dt a ❚❛ ♥❤➟♥ t❤➜② r➡♥❣ ∆N +1(x, t) ❧➔ ♥❤➙♥ ❍❡r♠✐t✐❛♥ ❝➢t ❝öt t❤✉ë❝ C(Q[a, b]) ✣➦t DN +1 : R2 [a, b] → R2 [a, b] b g → DN +1 g = ∆N +1 (x, t)g(t)dt a ❑❤✐ ✤â f − σN 2 = DN +1 g 2 = DN +1 g, DN +1 g = g, D2N +1 g ❚♦→♥ tû D2N +1 ❝ơ♥❣ ❧➔ ❍❡r♠✐t✐❛♥ t÷ì♥❣ ù♥❣ ✈ỵ✐ ♥❤➙♥ ❧➦♣ ∆2N +1(x, t) ❚❤❡♦ ❜ê ✤➲ ✸✳✸✳✺✱ ♥â ❝â ➼t ♥❤➜t ♠ët ❣✐→ trà r✐➯♥❣ λ2N +1 t ỵ t õ 2N +1 = max ϕ ϕ, D2N +1 ϕ ϕ, ϕ ❱ỵ✐ ♠å✐ g ∈ R2[a, b] t❤➻ λ2N +1 ≥ g, D2N +1 g f − σN = g, g g, g 2 ỵ r t❤ù t÷✱ λN → ∞ ❦❤✐ n → +∞ ❉♦ tỗ t M s g, g < , λ2N +1 ❙✉② r❛ f − σN ε < , ∀N ≥ Mε ∀N ≥ M ữ ợ N max{N, M} t❤➻ f −σ ≤ f − σN + σN − σ ✹✼ ≤ ε ε + = ε 2 ❙✉② r❛ f = σ ❱➟② f (x) = Nlim σN (x) ❤❛② →∞ ∞ f (x) = n=1 ▲↕✐ ❝â f = Kg ≤ K g gn ϕn (x) = λn < +∞ ❍➺ q✉↔ ✸✳✹✳✷✳ ∞ fn ϕn (x) n=1 ❉♦ ✤â f ∈ R2[a, b]✳ ✭❈æ♥❣ t❤ù❝ ❍✐❧❜❡rt✮✳ ❈❤♦ K(x, t) õ ỳ t t ữ ỵ g, h ∈ R2 [a, b] t❤➻ ∞ Kg, h = n=1 g, ϕn ϕn , h λn ✣➦❝ ❜✐➺t✱ ♥➳✉ g = h t❤➻ ∞ Kg, g = n=1 ❈❤ù♥❣ ♠✐♥❤✳ | g, ϕn |2 λn ❚❛ ❝â b Kg, h Kg(x)h(x)dx = a b ∞ gn ϕn (x)h(x)dx λn = a n=1 ∞ gn λn = n=1 ∞ b ϕn (x)h(x)dx a g, ϕn ϕn , h λn = n=1 ◆➳✉ g = h t❤➻ ∞ Kg, g = n=1 | g, ϕn |2 λn ❍➺ q✉↔ ✸✳✹✳✸✳ ❈❤♦ K(x, t) õ t t ữ ỵ ✸✳✹✳✶✳ ❑❤✐ ✤â✱ ✈ỵ✐ m ≥ 2, ♥❤➙♥ ❧➦♣ Km (x, t) ❝â ❜✐➸✉ ❞✐➵♥ ❞↕♥❣ s♦♥❣ t✉②➳♥ t➼♥❤ ∞ Km (x, t) = n=1 ϕn (x)ϕn (t) λm n ❉♦ ✈➟②✱ ✈➳t Am ❝õ❛ Km (x, t) ✤÷đ❝ ❝❤♦ ❜ð✐ ∞ b Am = Km (x, x)dx = a n=1 ✹✽ λm n ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣✳ ❈❤♦ s ❝è ✤à♥❤ t❤✉ë❝ [a, b] ✤➦t g(t) = K(t, s) ❱➻ K(x, t) ❧➔ ♥❤➙♥ ❍❡r♠✐t✐❛♥ ✈➔ λn ❧➔ sè t❤ü❝ ♥➯♥ ❈❤ù♥❣ ♠✐♥❤✳ b b K(t, s)ϕn (t)dt = gn = g, ϕn = a a ❚❛ ❝â K(s, t).ϕn (t)dt = ∞ Kg = n=1 ∞ gn ϕn (x) = λn n=1 ϕn (s) λn ϕn (x)ϕn (s) λ2n ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝â b Kg = K(x, t)K(t, s)dt = K2 (x, s) a ❙✉② r❛ ∞ K2 (x, t) = n=1 ϕn (x)ϕn (t) λ2n ữ ợ m = t ổ tự tr✐➸♥ s♦♥❣ t✉②➳♥ t➼♥❤ tr➯♥ ❧➔ ✤ó♥❣✳ ●✐↔ sû ❝ỉ♥❣ t❤ù❝ ❦❤❛✐ tr✐➸♥ tr➯♥ ❧➔ ✤ó♥❣ ✈ỵ✐ m = 2, , M ✣➦t g(t) = KM (t, s) ❱ỵ✐ m > 2, t❛ ❝â b gn = KM (t, s)ϕn (t)dt g, ϕn = a b = KM (s, t)ϕn (t)dt = a ∞ ∞ ❉♦ ✈➟② Kg = λg n=1 ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝â n ϕn (x) = n n=1 +1 ϕn (x)ϕn (s) λM n b Kg = ϕn (s) λM n b K(x, t)g(t)dt = a K(x, t)KM (t, s)dt = KM +1 (x, s) a ❉♦ ✈➟② ∞ KM +1 (x, s) = n=1 ϕn (x)ϕn (s) +1 λM n ◆❤÷ ✈➟②✱ sü ❦❤❛✐ tr✐➸♥ s♦♥❣ t✉②➳♥ t➼♥❤ tr➯♥ ❧➔ ✤ó♥❣ ✈ỵ✐ ♠å✐ m ≥ ❙✉② r❛ ∞ b Am = K(x, x)dx = a n=1 λm n ❚❛ ✤÷đ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ ✸✳✹✳✹✳ ✭◆❤➙♥ ❣✐↔✐ t❤ù❝✮✳ ❈❤♦ K(x, t) ❝â ❝→❝ t➼♥❤ ❝❤➜t ♥❤÷ tr♦♥❣ ỵ õ tự R(x, t; ) tữỡ ự ợ K(x, t) õ tr s♦♥❣ t✉②➳♥ ✹✾ t➼♥❤ ✈æ ❤↕♥ ∞ R(x, t; λ) = K(x, t) + λ λn (λn − λ) n=1 ✈➔ sü ❤ë✐ tö ❝õ❛ ❝❤✉é✐ n ≥ ∞ n=1 λn (λn −λ) ϕn (t)ϕn (t) ϕn (t)ϕn (t) ❧➔ t✉②➺t ✤è✐ ✈➔ ✤➲✉ ♥➳✉ λ = λn ✈ỵ✐ ♠å✐ ❱ỵ✐ ♠é✐ x, t ∈ Q[a, b], ♥➳✉ |λ| < |λ1| t❤➻ ❝❤✉é✐ ❣✐↔✐ t❤ù❝ ❤ë✐ tö tt ố t ỵ t ❝❤ù♥❣ ♠✐♥❤ ð ✷✳✷✳ ❑❤✐ ①➙② ❞ü♥❣ ❝❤✉é✐ s♦♥❣ t✉②➳♥ t tự t ú ỵ r ộ s♦♥❣ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❤↕♥❣ tû ✤➛✉ t✐➯♥ K1(x, t) = K(x, t) ❦❤ỉ♥❣ ♥❤➜t t❤✐➳t ♣❤↔✐ ❤ë✐ tư✳ ❚❤❡♦ q trữợ t õ ự m1 Km (x, t) R(x, t; λ) = m=1 ∞ λm−1 Km (x, t) = K(x, t) + m=2 ∞ ∞ = K(x, t) + λ m=2 n=1 ∞ = K(x, t) + λ n=1 m ϕn (x)ϕn (t) (λ/λn )2 ϕn (x)ϕn (t) − (λ/λn ) n=1 ∞ = K(x, t) + λ λ λn λn (λn − λ) ϕn (x)ϕn (t) ❚❤❡♦ q✉② t➢❝ t❤→❝ tr✐➸♥ ❣✐↔✐ t➼❝❤✱ ❞↕♥❣ ❜✐➸✉ ❞✐➵♥ ♥➔② ❧➔ ✤ó♥❣ ✈ỵ✐ ♠å✐ n ≥ λ = λn ✈ỵ✐ ▼➺♥❤ ✤➲ ✸✳✹✳✺✳ ❈❤♦ K(x, t) ❧➔ ♥❤➙♥ ❍❡r♠✐t✐❛♥ t❤✉ë❝ C(Q[a, b]) ✈➔ K(x, t) = ◆➳✉ K(x, t) ❦❤æ♥❣ ➙♠ t❤➻ t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉✿ ✭✐✮ ❚➜t ❝↔ ❝→❝ ❣✐→ trà r✐➯♥❣ λn ❝õ❛ K(x, t) ✤➲✉ ❞÷ì♥❣✳ ✭✐✐✮ ❱ỵ✐ ♠å✐ x ∈ [a, b] t❤➻ K(x, x) ≥ ✭✐✐✐✮ ◆➳✉ ∆N +1 ❧➔ ♥❤➙♥ ❝➢t ❝öt N ∆N +1 (x, t) = K(x, t) − n=1 ϕn (x)ϕn (t) , λn t❤➻ t♦→♥ tû ❋r❡❞❤♦❧♠ ①→❝ ✤à♥❤ ❜ð✐ b DN +1 g(x) = ∆N +1 (x, t)g(t)dt, a t❤ä❛ ♠➣♥ ❝→❝ t➼♥❤ ❝❤➜t s❛✉✿ ∞ DN +1 ϕ(x) = N +1 ϕ, ϕN ϕN (x); λn ✺✵ ✈➔ ∞ DN +1 ϕ, ϕ = N +1 | ϕ, ϕN |2 ; λn ❉♦ ✤â✱ ∆N +1 (x, t) ❧➔ ♠ët ♥❤➙♥ ❦❤ỉ♥❣ ➙♠✳ ✭✐✈✮ ❱ỵ✐ ♠å✐ N ≥ ✈➔ x ∈ [a, b], t❛ ❝â N n=1 |ϕn (x)|2 ≤ K(x, x) λn ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ◆➳✉ λn ❧➔ ♠ët ❣✐→ trà r✐➯♥❣ K(x, t) t tỗ t n s ❝❤♦ λn Kϕn = ϕn ❑❤✐ ✤â λn Kϕn , ϕn = λn Kϕn , ϕn = ϕn , ϕn > ▼➔ Kϕn , ϕn ≥ ✈➻ K(x, t) ❧➔ ♥❤➙♥ ❦❤æ♥❣ ➙♠✳ ❉♦ ✤â λn > 0✳ ✭✐✐✮ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♣❤↔♥ ❝❤ù♥❣✳ ▲➜② c ∈ (a, b) ●✐↔ sû r➡♥❣ K(c, c) < ❱➻ K(x, t) ❧✐➯♥ tö❝ tr➯♥ ♥➯♥ ợ (c, c) Q[a, b], tỗ t ♠ët ❧➙♥ ❝➟♥ ✈➔ c − δ ≤ t ≤ c + δ s❛♦ ❝❤♦ Re {K(x, t)} < tr➯♥ S(c, δ) ✈â✐ δ ✤õ ♥❤ä✳ ❚❛ ①→❝ ✤à♥❤ ❤➔♠ ①✉♥❣ ❧÷đ♥❣ 1, ♥➳✉ c − δ ≤ x ≤ c + δ p(x; c, δ) = 0, ♥➳✉ a ≤ x < c − δ ✈➔ c + δ < x ≤ b ❚❤❡♦ ♠➺♥❤ ✤➲ ✸✳✶✳✶✱ Kp, p ♥❤➟♥ ❣✐→ trà t❤ü❝ ♥➯♥ t❛ ❝â S(c, δ) = (x, t) : c − δ ≤ x ≤ c + δ b Kp, p b = K(x, t)p(t)p(x)dxdt a a = Re{K(x, t)}dxdt < 0, S(c,δ) ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ sû r➡♥❣ K(x, t) ❧➔ ♥❤➙♥ ❦❤ỉ♥❣ ➙♠✳ ❙✉② r❛ K(c, c) ≥ ✈ỵ✐ ♠å✐ (c, c) ∈ Q[a, b] ❚❛ ❝ô♥❣ ❝â K(a, a) ≥ ✈➔ K(b, b) ≥ t❤❡♦ sü ❧➟♣ ❧✉➟♥ ❧✐➯♥ t✐➳♣✳ ❉♦ ✈➟② K(x, x) ≥ ✈ỵ✐ ♠å✐ x ∈ [a, b]✳ ✭✐✐✐✮ ❚❤❡♦ ❜ê ✤➲ ✸✳✸✳✺✱ ∆N +1(x, t) ❧✐➯♥ tö❝ tr➯♥ Q[a, b] ✈➔ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥â ❝ô♥❣ ❧➔ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ K(x, t) ✈ỵ✐ n ≥ N + R2[a, b] t tứ ỵ rtt t❛ ❝â N b K(x, t) − DN +1 ϕ(x) = a n=1 ϕn (x)ϕn (t) ϕ(t)dt λn b N b K(x, t)ϕ(t)dt − = a ∞ = n=1 ∞ a n=1 N ϕn (x)ϕn (t) ϕ(t)dt λn 1 ϕn (x) ϕ, ϕn − ϕn (x) ϕ, ϕn λn λ n n=1 = n=N +1 ϕn (x) ϕ, ϕn λn ✺✶ ❙✉② r❛ b DN +1 ϕ, ϕ = DN +1 ϕ(x)ϕ(x)dx a ∞ b = a n=N +1 ∞ = n=N +1 ∞ = n=N +1 ϕ, ϕn ϕn (x)ϕ(x)dx λn ϕ, ϕn ϕn , ϕ λn | ϕ, ϕn |2 λn ❱➻ tr r n ữỡ ợ n = 1, 2, ♥➯♥ DN +1ϕ, ϕ > ϕ ∈ R[a, b] ❚ø ✤â t❛ ❦➳t ❧✉➟♥ r➡♥❣ ∆N +1 (x, t) ❧➔ ♥❤➙♥ ❦❤æ♥❣ ➙♠✳ ✭✐✈✮ ❱➻ λn > ✈ỵ✐ ♠å✐ n = 1, 2, ♥➯♥ t❛ ❝â N ∆N +1 (x, x) = K(x, x) − n=1 ∞ = n=N +1 ❙✉② r❛ N n=1 ✈ỵ✐ ♠å✐ |ϕn (x)|2 λn |ϕn (x)|2 ≥ λn |ϕn (x)|2 ≤ K(x, x) λn ❚ø ❜ê ✤➲ ♥➔② t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ❦➳t q✉↔ s ỵ ( ỵ rr) K(x, t) ❧➔ ♥❤➙♥ ❍❡r♠✐t✐❛♥ t❤✉ë❝ C(Q[a, b]) ✈➔ K(x, t) = ◆➳✉ K(x, t) ❧➔ ♥❤➙♥ ❦❤æ♥❣ ➙♠ t❤➻ ♥â ✤÷đ❝ ❦❤❛✐ tr✐➸♥ t❤➔♥❤ ❝❤✉é✐ s♦♥❣ t✉②➳♥ t➼♥❤ ✈ỉ ❤↕♥ ∞ K(x, t) = n=1 ϕn (x)ϕn (t) λn ✭✸✳✻✮ ✈➔ ❝❤✉é✐ ♥➔② ❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ✤➲✉ tr➯♥ Q[a, b] ❈❤ù♥❣ ♠✐♥❤✳ ∞ ❚❤❡♦ ♠➺♥❤ ✤➲ ✸✳✹✳✺✱ ✈ỵ✐ ♠é✐ x ∈ [a, b] t❤➻ ❝❤✉é✐ ∞ n=1 λn |ϕn (x)| ❤ë✐ tö ✈➔ ✣➦t B = axb max K(x, x) õ tỗ t số Nx (ε) ♣❤ö t❤✉ë❝ ✈➔♦ x ✈➔ ε s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ p ≥ m ≥ Nx(ε) t❤➻ n=1 λn |ϕn (x)| ≤ K(x, x) p ε |ϕn (x)|2 < λ B n=m n ✺✷ ✭✸✳✼✮ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ t❛ ❝â p ϕn (x)ϕn (t) p |ϕn (x)ϕn (t)| ≤ n=m n=m p p |ϕn (x)|2 λ n=m n ≤ |ϕn (x)|2 λ n=m n p |ϕn (x)|2 K(t, t) λ n=m n ≤ ε B = ε, B ≤ ∀p ≥ m ≥ Nx (ε) ❉♦ ✤â ✈ỵ✐ ♠é✐ x ❝è ✤à♥❤ t❤✉ë❝ [a, b]✱ ❝❤✉é✐ ✭✸✳✻✮ ❤ë✐ tư tỵ✐ ❤➔♠ L(x, t) ❧✐➯♥ tư❝ t❤❡♦ ❜✐➳♥ t ❚✐➳♣ t❤❡♦ t❛ ❝❤➾ r❛ r➡♥❣ L(x, t) ❍✐❧❜❡rt✲❙❝❤♠✐❞t t❛ ❝â = K(x, t) b tr➯♥✳ ▲➜② ϕ b (L(x, t) K(x, t))(t)dt = t ỵ b L(x, t)ϕ(t)dt − a ∈ R2 [a, b], a K(x, t)ϕ(t)dt a ∞ ∞ b = a n=1 ∞ = n=1 ∞ = n=1 1 ϕn (x)ϕn (t) ϕ(t)dt − ϕ, ϕn ϕn (x) λn λ n n=1 ϕn (x) λn ∞ ϕ(t)ϕn (t)dt − n=1 ϕ, ϕn ϕn (x) λn ∞ 1 ϕ, ϕn ϕn (x) − ϕ, ϕn ϕn (x) = λn λ n=1 n ✣➦❝ ❜✐➺t✱ ♥➳✉ t❛ ❝❤å♥ ❤➔♠ ϕ(t) = L(x, t) − K(x, t) t❤➻ t❛ t❤✉ ✤÷đ❝ b L(x, t) − K(x, t) 2 |L(x, t) − K(x, t)|2 dt = = a ❱➻ ❝↔ L(x, t) ✈➔ K(x, t) ✤➲✉ ❧➔ ❝→❝ ❤➔♠ t❤❡♦ ❜✐➳♥ t tr➯♥ ✤♦↕♥ [a, b] ♥➯♥ L(x, t) = K(x, t) ✈ỵ✐ ♠é✐ x ❝è ✤à♥❤ t❤✉ë❝ [a, b] ❉♦ ✤â ∞ L(x, t) = K(x, t) = n=1 ϕn (x)ϕn (t) , λn ∀(x, t) ∈ Q[a, b] ❚✐➳♣ t❤❡♦ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝❤✉é✐ ✭✸✳✻✮ ❤ë✐ tö ✤➲✉ t❤❡♦ ❜✐➳♥ x ❈❤å♥ t = x, t❛ t❤✉ ✤÷đ❝ ∞ K(x, x) = n=1 |ϕn (x)|2 λn ❱➻ ❤➔♠ ❣✐ỵ✐ ❤↕♥ K(x, x) ❧✐➯♥ tư❝ tr➯♥ [a, b] t ỵ t ộ tử [a, b] tỗ t số N (ε) s❛♦ ❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✼✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ p ≥ m ≥ N (ε) ✤ë❝ ❧➟♣ ✈ỵ✐ x t tự ụ ú ữợ ✤✐➲✉ ❦✐➺♥ ♥➔②✳ ❉♦ ✤â ❝❤✉é✐ ✭✸✳✻✮ ❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ✤➲✉ tỵ✐ K(x, t) ✺✸ ❍➺ q✉↔ ✸✳✹✳✼✳ ●✐↔ sû K(x, t) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ tr♦♥❣ ỵ õ t A1 K(x, t) ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ ∞ b A1 = K(x, x)dx = a ❈❤ù♥❣ ♠✐♥❤✳ n=1 λn ❚ø ỵ t õ |n (x)|2 n K(x, x) = n=1 ❉♦ ✈➟② b A1 = K(x, x)dx a ∞ b |λn (x)|2 λn = a n=1 b ∞ |λn (x)|2 dx λn = n=1 ∞ = n=1 dx a n ỵ t ♣❤÷ì♥❣ tr➻♥❤ t✐❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ b ϕ(x) = f (x) + λ K(x, t)ϕ(t)dt, a tr♦♥❣ ✤â f (x) ∈ C[a, b]✱ K(x, t) ∈ C(Q[a, b]) ❧➔ ♥❤➙♥ ❍❡r♠✐t✐❛♥ ✈➔ K(x, t) = ●å✐ λ1 , λ2 , ❧➔ ❝→❝ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ ✤÷đ❝ s➢♣ ①➳♣ t❤❡♦ t❤ó tü t➠♥❣ ❞➛♥ ❝õ❛ ♠♦❞✉❧ ✈➔ ϕ1 , ϕ2 , ❧➔ ❝→❝ ❤➔♠ r✐➯♥❣ trü❝ ❝❤✉➞♥ t÷ì♥❣ ù♥❣✳ ❑❤✐ ✤â✿ ✭✐✮ ◆➳✉ λ ❦❤ỉ♥❣ ❧➔ ❣✐→ trà r✐➯♥❣ ❝õ❛ ♥❤➙♥ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ tr➯♥ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ ∞ ϕ(x) = f (x) + λ n=1 f, ϕn ϕn (x) λn − λ ✭✐✐✮ ◆➳✉ λ = λk = · · · = λk+r−1 ❧➔ ❣✐→ trà r✐➯♥❣ r t ợ r tữỡ ù♥❣ ϕk , , ϕk+r−1 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❝â ♥❣❤✐➺♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f, ϕj = ✈â✐ j = k, , k + r − ❑❤✐ ✤â✱ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ ϕ(x) = f (x) + λk j f, ϕj ϕj (x) + λj − λk k+r−1 cj ϕj (x), j=k tr♦♥❣ ✤â tê♥❣ ✤➛✉ t✐➯♥ ❧➜② ✈ỵ✐ ❝→❝ ❝❤➾ sè j s❛♦ ❝❤♦ λj = λk ✈➔ ❝→❝ số cj tr tờ tự tũ ỵ ✺✹ ✭✐✮ ❱ỵ✐ ♥❤ú♥❣ ❣✐↔ t❤✐➳t ✤➣ ❝❤♦ ✈➔ t❤❡♦ q ỵ rtt ữỡ tr t ❝â t❤➸ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❧↕✐ ♥❤÷ s❛✉ ❈❤ù♥❣ ♠✐♥❤✳ ∞ ϕ, ϕn ϕn (x) λn ϕ(x) = f (x) + λ n=1 ❉♦ ✤â ϕ, ϕn = f, ϕn + λ ❙✉② r❛ f, ϕn λn λn − λ ϕ, ϕn = ❉♦ ✤â ϕ, ϕn λn ∞ ϕ(x) = f (x) + λ n=1 f, ϕn ϕn (x) λn − λ ▼➦t ❦❤→❝✱ t❛ ❧↕✐ ❝â ∞ n=1 f, ϕn ϕn (x) = λn − λ ∞ b f (t)ϕn (t)dt ϕn (x) λn − λ ϕn (x)ϕn (t) λn − λ f (t)dt a n=1 b ∞ = a n=1 b = R(x, t; λ)f (t)dt a ❈❤ù♥❣ tä ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ ❜ð✐ b ϕ(x) = f (x) + λ R(x, t; λ)f (t)dt a ✭✐✐✮ ●✐↔ sû λ = λk = · · · = λk+r−1 ❧➔ ♠ët ❣✐→ trà r✐➯♥❣ ❜ë✐ r ❝õ❛ ♥❤➙♥✳ ❑❤✐ ✤â✱ t❤❡♦ ỵ r tự ữỡ tr t õ ♥❣❤✐➺♠ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ f (x) trü❝ ❣✐❛♦ ✈ỵ✐ t➜t ❝↔ ❝→❝ ❤➔♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t tữỡ ự ợ K(x, t) ♥❤➙♥ ❍❡r♠✐t✐❛♥ ♥➯♥ ❝→❝ ❤➔♠ r✐➯♥❣ ❝õ❛ K(t, x) ❝ô♥❣ ❧➔ ❤➔♠ r✐➯♥❣ ❝õ❛ K(x, t) ❈ô♥❣ t❤❡♦ ♠➺♥❤ ✤➲ ✸✳✷✳✸✱ λ ❧➔ sè t❤ü❝✳ ❉♦ ✤â tr♦♥❣ tr÷í♥❣ ❤đ♣ tỗ t t ✤õ ❧➔ f, ϕj = 0, ✈ỵ✐ ♠å✐ j = k, , k + r − ◆➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ trü❝ ❣✐❛♦ ♥➔② ✤÷đ❝ t❤ä❛ ♠➣♥ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ❝â ♥❣❤✐➺♠ ✈➔ ♥❣❤✐➺♠ ✤â ❝â ❞↕♥❣ ϕ(x) = f (x) + λk ϕ(p) (x; λk ) + βϕ(h) (x; λk ) tr♦♥❣ ✤â f (x) + ϕ(p)(x; λ) ❧➔ ♥❣❤✐➺♠ ✤➦❝ ❜✐➺t ❝õ❛ ♣❤÷ì♥❣ tr t số tũ ỵ ϕ(h)(x; λk ) ❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ r tữỡ ự ợ k j ♠ët ❝❤➾ sè s❛♦ ❝❤♦ λj = λk t❤➻ ϕ, ϕj = f, ϕj + λk ✺✺ ϕ, ϕj λj ❙✉② r❛ ϕ, ϕj = λj f, ϕj λj − λk ❚❤❛② ✈➔♦ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ✤÷đ❝ ❝❤♦ tr♦♥❣ ỵ rtt t ữủ (x) = f (x) + λ j ◆➳✉ ❝❤➾ sè j ❧➔♠ ❝❤♦ λj = λk t❤➻ f, ϕj ϕj (x) λj − λk f, ϕj = ϕ(x) = ❚ø ✤â t❛ t❤✉ ✤÷đ❝ cj ϕj (x) j ❝❤➼♥❤ ❧➔ tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝→❝ ❤➔♠ r✐➯♥❣ tr♦♥❣ tê♥❣ t❤ù ❤❛✐✳ ✺✻ ❑➌❚ ▲❯❾◆ ❑❤â❛ ❧✉➟♥ ✤➣ tr➻♥❤ ❜➔② ✤÷đ❝ ♠ët sè ✈➜♥ ✤➲ ❝❤➼♥❤ s❛✉ ✤➙②✿ ✶✳ ✣à♥❤ ♥❣❤➽❛ ✈➔ ♣❤➙♥ ❧♦↕✐ ❝→❝ ữỡ tr t r sử tỗ t ✈➔ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ✈➔ ✤÷❛ r❛ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ❝ư t❤➸ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥❤➙♥ ❝â ❞↕♥❣ t→❝❤ ❜✐➳♥✳ ✷✳ ❚r➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ♥❣❤✐➺♠ ❧✐➯♥ t✐➳♣ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ t❤➳ ❧✐➯♥ t✐➳♣ ❦➧♠ t❤❡♦ ♠ët sè ✈➼ ❞ư ♠✐♥❤ ❤♦↕❝ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥❤➙♥ tê♥❣ q✉→t tọ ởt số trữợ r sỹ tỗ t t ữỡ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ❧♦↕✐ ❤❛✐ ✈ỵ✐ ♥❤➙♥ tê♥❣ q✉→t✳ ✸✳ ▼ët sè ❦➳t q✉↔ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❋r❡❞❤♦❧♠ ❧❛♦à ❤❛✐ ❝â ♥❤➙♥ ❍❡r♠✐t✐❛♥ ✈➔ ✤÷❛ r❛ ❝ỉ♥❣ t❤ù❝ ❜✐➸✉ ❞✐➵♥ ♥❣❤✐➺♠ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✳ ❚✉② ♥❤✐➯♥ ❦❤â❛ ❧✉➟♥ ❝ơ♥❣ ✤➸ ❧↕✐ ♥❤✐➲✉ ữợ t tr ữủ ự ởt ữù ú tổ ợ ữợ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❝❤♦ ♠ët sè ❜➔✐ t♦→♥ ❝ư t❤➸ ❝❤ù ❝❤÷❛ ❝â ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉♥❣✳ ❈➙✉ tr↔ ❧í✐ ❝á♥ ✤á✐ ❤ä✐ tỉ✐ ♣❤↔✐ t✐➳♣ tư❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐✳ ❈✉è✐ ❝ò♥❣✱ tr♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥✱ ♠➦❝ ❞ò ✤➣ r➜t ❝è ❣➢♥❣✱ ❝➞♥ t❤➟♥✱ t➾ ♠➾ s♦♥❣ ❝á♥ ❝â ♥❤✐➲✉ ❤↕♥ ❝❤➳ t❤✐➳✉ sât ❝❤÷❛ ❦❤➢❝ ♣❤ư❝ ✤÷đ❝✳ ❚ỉ✐ r➜t ♠♦♥❣ ữủ sỹ õ õ ỵ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✤➸ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✺✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❬✷❪ ❬✸❪ ❬✹❪ ❬✺❪ ❬✻❪ ◆❣✉②➵♥ ❱➠♥ ❑❤✉➯ ✭ ✷✵✵✶✮✱ ❈ì sð ❧➼ t❤✉②➳t ❤➔♠ ✈➔ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ◆①❜ ●✐→♦ ❞ö❝✳ ◆❣✉②➵♥ ❳✉➙♥ ▲✐➯♠ ✭✷✵✵✶✮✱ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆①❜ ●✐→♦ ❞ư❝ P❤↕♠ ▼✐♥❤ ❚❤ỉ♥❣ ✭✷✵✵✾✮✱ ●✐↔✐ t➼❝❤ ❤➔♠✱ ●✐→♦ tr➻♥❤ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❚➙② ❇➢❝✳ P❤↕♠ ▼✐♥❤ ❚❤æ♥❣ ✭✷✵✵✼✮✱ ❑❤æ♥❣ ❣✐❛♥ tæ♣æ ✲ ✣ë ✤♦ ✕ ❚➼❝❤ ♣❤➙♥✱ ◆①❜ ●✐→♦ ❞ö❝✳ ❍♦➔♥❣ ❚ö② ✭✷✵✵✺✮✱ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆①❜ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ❆✳ ◆✳ ❑♦❧♠♦❣♦r♦✈✱ ❙✳❱✳ ❋♦♠✐♥❡ ✭✶✾✼✶ ✲ ✶✾✽✶✮✱ ❈ì sð ❧➼ t❤✉②➳t ❤➔♠ ✈➔ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ◆①❜ ●✐→♦ ❞ö❝✳ ❬✼❪ ❈✳ ❙✳ ❑✉❜r✉s❧② ✭✷✵✶✶✮✱ ❚❤❡ ❊❧❡♠❡♥ts ♦❢ ❖♣❡r❛t♦r ❚❤❡♦r②✱ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ ❛♥❞ ❇✉s✐✲ ♥❡ss ▼❡❞✐❛✳ ❬✽❪ ❘ P✳ ❑❛♥✇❛❧ ✭✶✾✾✼✮✱ ▲✐♥❡❛r ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s✱ ❇✐r❦❤➠❛✉s❡r✳ ❬✾❪ P✳ ❑②t❤❡ ❛♥❞ P✳ P✉r✐ ✭✷✵✵✷✮✱ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❡t❤♦❞s ❢♦r ▲✐♥❡❛r ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s✱ ❙♣r✐♥❣❡r✳ ❬✶✵❪ ❆✳ ❉✳ P♦❧②❛♥✐♥ ❛♥❞ ❆✳ ❱✳ ▼❛♥③❤✐r♦✈ ✭✷✵✵✽✮✱ ❍❛♥❞❜♦♦❦ ♦❢ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s✱ ❈❤❛♣✲ ♠❛♥ ❛♥❞ ❍❛❧❧✴ ❈❘❈✳ ❬✶✶❪ ▼✳ ❘❛❤♠❛♥ ✭✷✵✵✼✮✱ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ❛♥❞ t❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✱ ❲■❚ Pr❡ss✳ ✺✽