ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN NGUYỄN THỊ MINH THÚY TÌM HIỂU VỀ PHƯƠNG TRÌNH FREDHOLM VỚI NHÂN DẠNG CHẬP TRÊN KHOẢNG HỮU HẠN LUẬN VĂN THẠC SĨ TOÁN HỌC Hà Nội - 2018 ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN NGUYỄN THỊ MINH THÚY TÌM HIỂU VỀ PHƯƠNG TRÌNH FREDHOLM VỚI NHÂN DẠNG CHẬP TRÊN KHOẢNG HỮU HẠN Chuyên ngành: Toán giải tích Mã số: 60 46 01 02 LUẬN VĂN THẠC SĨ TOÁN HỌC Cán hướng dẫn: TS Lê Huy Chuẩn Hà Nội - 2018 ▲❮■ ❈❷▼ ❒◆ ✣➸ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✈➔ ❦➳t t❤ó❝ ❦❤â❛ ❤å❝✱ ✈ỵ✐ t➻♥❤ ❝↔♠ ❝❤➙♥ t❤➔♥❤ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ t ỡ s s tợ trữớ ❚ü ♥❤✐➯♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❝â ♠æ✐ tr÷í♥❣ ❤å❝ t➟♣ tèt tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❡♠ ❤å❝ t t trữớ ỷ ỡ tợ t❤➛② ▲➯ ❍✉② ❈❤✉➞♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❡♠ tr sốt q tr ự trỹ t ữợ ❞➝♥ ❡♠ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣ ỗ tớ tọ ỏ ỡ tỵ✐ t❤➛② ❝ỉ ❦❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ ❜↕♥ ❜➧ ✤➣ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ ♥❣➔② ✷✽ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✶✽ ❍å❝ ✈✐➯♥ ◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚❤ó② ✶ ▼ư❝ ❧ư❝ ▲❮■ ❈❷▼ ❒◆ ▲❮■ ▼Ð ✣❺❯ ✶ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ L2(0, w) ✶ ✸ ✹ ✶✳✶ ❳➙② ❞ü♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỹ tỗ t ❝➜✉ tró❝ ❝õ❛ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ Pữỡ tr t ợ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ p L (0, w) ✷✳✶ ❚➼♥❤ ❝❤➜t t♦→♥ tû t➼❝❤ ♣❤➙♥ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr Pữỡ tr t ợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶ Pữỡ tr t ợ t Pữỡ tr t ợ ♣❤↔✐ tr♦♥❣ ❦❤æ♥❣ Wp2 (0, w) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ ❱➼ ❞ö →♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳✳✳✳ ✳✳✳✳ ❣✐❛♥ ✳✳✳✳ ✳✳✳✳ Lp (0, w) ❑➌❚ ▲❯❾◆ ✷✶ ✷✶ ✷✺ ✷✺ ✷✽ ✸✷ ✸✽ ✷ ▲❮■ ▼Ð ✣❺❯ ◆❤✐➲✉ ✈➜♥ ✤➲ tr♦♥❣ t♦→♥ ❤å❝✱ ❝ì ❤å❝✱ ✈➟t ❧➼ ✈➔ ❝→❝ ♥❣➔♥❤ ❦➽ t❤✉➟t ❦❤→❝ ❞➝♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ ▼ư❝ t✐➯✉ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ❤✐➸✉ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ ❝â ❞↕♥❣ w k(x − t)f (t)dt = φ(x), µf (x) + tr♦♥❣ ✤â µ ❧➔ sè ♣❤ù❝ ✈➔ k(x) ∈ L(0, w)✳ ✣➸ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ ♥➔②✱ t❛ s➩ ①➨t t♦→♥ tû t➼❝❤ ♣❤➙♥ ❝â ❞↕♥❣ w d Sf = dx s(x − t)f (t)dt ✈➔ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧♦↕✐ ✶ t÷ì♥❣ ù♥❣ Sf = φ(x) ❇➡♥❣ ❝→❝❤ ❝❤å♥ x s(x) = k(u)du + µ+ (x > 0), k(u)du + µ− (x < 0), x s(x) = µ = à+ + , ữỡ tr tr tr t ữỡ tr r ợ ✤➛✉✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ởt số t q ự ữỡ tr r ợ ố ỗ ữỡ ữỡ tr t➼♥❤ ❦❤↔ ♥❣❤à❝❤ ❝õ❛ t♦→♥ tû S tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L2(0, w)❀ ❝➜✉ tró❝ ❝õ❛ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ✈➔ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ L2(0, w) ợ t ữỡ ✈➠♥ tr➻♥❤ ❜➔② ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ t♦→♥ tû S tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Lp(0, w)✱ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Wp2(0, w) ✈➔ ❝✉è✐ ❝ò♥❣ ❧➔ ♠ët ✈➼ ❞ư ♠✐♥❤ ❤å❛✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② ❞ü❛ t❤❡♦ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪✳ ✸ ❈❤÷ì♥❣ ✶ ❚♦→♥ tû t➼❝❤ 2♣❤➙♥ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ L (0, w) ✶✳✶ ❳➙② ❞ü♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ t❛ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ❦❤↔ ♥❣❤à❝❤ ❝õ❛ t♦→♥ tû S tr♦♥❣ ✈ỵ✐ t♦→♥ tû S ❝â ❞↕♥❣ L2 (0, w) w d Sf = dx s(x − t)f (t)dt, f (x) ∈ L2 (0, w), ✭✶✳✶✮ w tr♦♥❣ ✤â s(x) t❤✉ë❝ L2(−w, w) ✈➔ ❤➔♠ sè g(x) = s(x − t)f (t)dt ❧➔ ♠ët ❤➔♠ sè ❧✐➯♥ tư❝ t✉②➺t ✤è✐✳ ❚♦→♥ tû S ✤÷đ❝ ữ tr t tỷ t ợ ♥❤➙♥ ❞↕♥❣ ❤✐➺✉✳ ✣➸ t➻♠ ✤÷đ❝ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ t♦→♥ tû S t❛ ♣❤↔✐ t➻♠ ❤➔♠ sè N1(x), N2(x) t❤ä❛ ♠➣♥ SN1 (x) = M (x), SN2 (x) = 1, ✈ỵ✐ ❧➔ ❤➔♠ ❤➡♥❣ ❜➡♥❣ ✈➔ M (x) = s(x), ≤ x ≤ w ❑❤✐ ✤â✱ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ T = S −1 ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ q✉❛ ❤➔♠ sè N1 (x) ✈➔ N2 (x)✳ ỵ S t tỷ tr♦♥❣ L2(0, w)✳ ❑❤✐ ✤â✱ t♦→♥ tû S ✤÷đ❝ ❜✐➸✉ ữợ w Sf = d dx s(x, t)f (t)dt, tr♦♥❣ ✤â s(x, t) t❤✉ë❝ L2(0, w) ✈ỵ✐ ♠é✐ x ❝è ✤à♥❤✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ sè ex (t) = 1, ≤ t ≤ x, 0, x < t ≤ w ✹ ❈❤÷ì♥❣ ✶✳ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ L2 (0, w) ◆➳✉ f ∈ L2(0, w) t❤➻ Sf ∈ L2(0, w) t ổ ữợ tr L2(0, w) t❛ ❝â x Sf, ex = (Sf )dt ▲↕✐ ❝â Sf, ex = f, S ∗ ex ✭✶✳✷✮ S ∗ ex = s(x, t), ✭✶✳✸✮ ✈ỵ✐ S ∗ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ t♦→♥ tû S ✳ ✣➦t t❛ ✤÷đ❝ w ∗ s(x, t)f (t)dt f, S ex = ✭✶✳✹✮ ❚ø ✭✶✳✷✮ ✲ ✭✶✳✹✮ t❛ ❝â x w (Sf )dt = Sf, ex = f, S ∗ ex = ❱➟② d Sf = dx s(x, t)f (t)dt w s(x, t)f (t)dt ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❤➔♠ ex ✈➔ ✤➥♥❣ t❤ù❝ ✭✶✳✸✮ t❛ s✉② r❛ ❤➺ q✉↔ s❛✉ ✤➙②✳ ❍➺ q✉↔ ✶✳✶✳2❍➔♠ sè s(x, t) tr♦♥❣ ❝ỉ♥❣ t❤ù❝ ✭✶✳✶✮ ❝â t❤➸ ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ s(x, t) t❤✉ë❝ L (0, w) ✈ỵ✐ ♠é✐ x ✈➔ w |s(x + ∆x, t) − s(x, t)|2 dt ≤ ||S||2 |∆x| s(0, t) = 0; ❚❛ ❦➼ ❤✐➺✉ A ❧➔ t♦→♥ tû t➼❝❤ ♣❤➙♥ tr➯♥ L2(0, w) ①→❝ ✤à♥❤ ❜ð✐ x Af = i f (t)dt ✭✶✳✺✮ ❑❤✐ ✤â✱ t♦→♥ tû ❧✐➯♥ ❤ñ♣ A∗ ❝â ❞↕♥❣ w ∗ A f = −i f (t)dt x ✺ ✭✶✳✻✮ ữỡ tỷ t ợ tr L2 (0, w) ỵ S t♦→♥ tû ❜à ❝❤➦♥ ✈ỵ✐ ♥❤➙♥ ✈✐ ♣❤➙♥ ❞↕♥❣ ✭✶✳✶✮✳ ❑❤✐ ✤â✱ t❛ ❝â ❜✐➸✉ ❞✐➵♥ w (AS − SA∗ )f = i ✭✶✳✼✮ (M (x) + N (t))f (t)dt, tr♦♥❣ ✤â M (x) = s(x), N (x) = −s(−x), ≤ x ≤ w ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ✭✶✳✶✮✱ ✭✶✳✺✮ ✈➔ ✭✶✳✻✮ t❛ ❝â ASf = A w d dx x w SA∗ f = S −i s (x − u) u w f (t)dts(x) − = i − w s(x − t)d f (t)dt u w u u w − f (t)dtdu f (t)dtd(s(x − u)) f (t)dts(x − u) = −i s(x − u) w w f (t)dtdu = i w w u 0w w d f (t)dt = −i dx x w w = −i w f (t)dt = −iS s(−t)f (t)dt 0 x w s(x − t)f (t)dt − i =i 0 w x s(τ − t)f (t)dt s(τ − t)f (t) dτ dτ 0 w d s(x − t)f (t)dt = i w =i w s(x − t)f (t)dt = i s(x) − s(x − t) f (t)dt ❉♦ ✤â (AS − SA∗ )f = ASf − SA∗ f w w s(x − t)f (t)dt − i =i 0 w s(x − t)f (t)dt − i =i 0 0 w s(x − t)f (t)dt = i s(x − t)f (t)dt [s(x) − s(−t)]f (t)dt ✻ w s(x)f (t)dt − i s(−t)f (t)dt + i w s(x)f (t)dt − i s(x − t)f (t)dt w w w s(x)f (t)dt − i s(−t)f (t)dt + i w =i w ữỡ tỷ t ợ ❝❤➟♣ tr♦♥❣ L2 (0, w) w [M (x) + N (t)]f (t)dt (0 ≤ x ≤ w) =i ❱➟② ỵ ữủ ự sỷ t tỷ S ❞↕♥❣ ✭✶✳✶✮ ❝â ♥❣❤à❝❤ ✤↔♦ ❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ ✤➥♥❣ t❤ù❝ ✭✶✳✼✮ ❧➔ ❝ì sð ✤➸ t❛ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ①➙② ❞ü♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ S ✳ ❱ỵ✐ T = S −1 t❛ ❝â (T A − A∗ T )f = T (AS − SA∗ )T f = S −1 (AS − SA∗ )S −1 f w [M (x) + N (t)](S −1 f )(t)dt = S −1 i w = S −1 i M (x)(S −1 f )(t)dt + i N (t)(S −1 f )(t)dt w w (S −1 f )(t)dtS −1 (M (x)) + i =i w N (t)S −1 f (t)dtS −1 (1) =i S −1 f, N1 (x) + i S −1 f, N (t) N2 (x) = i f, (S −1 )∗ N1 (x) + i f, (S −1 )∗ N (t) N2 (x) = i f, M1 (t) N1 (x) + i f, M2 (t) N2 (x) w =i [N1 (x)M1 (t) + N2 (x)M2 (t)]f (t)dt, tr♦♥❣ ✤â S ∗M1 = 1, ❚❛ ❦➼ ❤✐➺✉ ✈➔ t♦→♥ tû S ∗ M2 = N (x), SN1 = M (x), SN2 = Q(x, t) = N1 (x)M1 (t) + N2 (x)M2 (t), w Qf (x) = Q(x, t)f (t)dt ỵ t tỷ T tr♦♥❣ L2(0, w) ✈➔ t❤ä❛ ♠➣♥ T A − A∗ T = iQ t❤➻ ❤➔♠ sè ✭✶✳✽✮ 2w−|x−t| φ(x, t) = Q x+t ✼ s+x−t s−x+t , ds 2 ữỡ tỷ t ợ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ L2 (0, w) ❧✐➯♥ tö❝ t✉②➺t ✤è✐ t❤❡♦ t ✈➔ w ∂ φ(x, t) f (t)dt ∂t d T f (x) = dx ✭✶✳✶✵✮ ❈❤ù♥❣ T t ỵ tỗ t F (x, t) tở L2(0, w) s t tỷ T ữủ ữợ w d Tf = dx F (x, t)f (t)dt ✭✶✳✶✶✮ ❚❤❡♦ ❍➺ q✉↔ ✶✳✶✱ ❤➔♠ sè F (x, t) ❝â t❤➸ ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ w |F (x + ∆x, t) − F (x, t)|2 dt ≤ ||T ||2 |∆x| F (w, t) = 0, ✭✶✳✶✷✮ ❱➻ ✈➟②✱ t➼❝❤ ♣❤➙♥ w F (x, s)ds t ❧✐➯♥ tö❝ t❤❡♦ x ♥➯♥ t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ F1(x, t) ❜ð✐ w w F1 (x, t) = − F (u, s)dsdu x ✭✶✳✶✸✮ t ❚ø ✭✶✳✶✶✮✱ ✭✶✳✶✷✮ t❛ s✉② r❛ t♦→♥ tû T1 ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ w T1 f = F1 (x, t)f (t)dt t❤ä❛ ♠➣♥ ❚❤❡♦ ✭✶✳✽✮✱ t❛ ❝â ✤➥♥❣ t❤ù❝ s❛✉ T1 = iA∗2 T A T1 A − A∗ T1 = iQ1 , tr♦♥❣ ✤â w ✭✶✳✶✹✮ Q1 (x, t)f (t)dt ✭✶✳✶✺✮ Q(u, s)ds(u − x)du ✭✶✳✶✻✮ Q1 f = iA∗2 QAf = w w Q1 (x, t) = x t ✽ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) ❚ø ✭✷✳✻✮ ✈➔ ✭✷✳✼✮ t❛ ❝â |γ(x, t)| ≤ h(x − t) ✈ỵ✐ w−x h(x) = |A (x)B(w)| + |A (w − s)B (s + x)|ds, |A(0)B (w + x)| + w < x < w; w |A (w − s)B (s + x)|ds, + −x −w < x < −x ✷✳✷ P❤÷ì♥❣ tr t ợ t ữỡ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❝â ❞↕♥❣ w d Sf = dx s(x − t)f (t)dt = φ(x) ✷✳✷✳✶ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✈ỵ✐ ✈➳ ♣❤↔✐ ✤➦❝ ❜✐➺t ▼➺♥❤ ✤➲ ✷✳✶✳ ❈❤♦ S ❧➔ t♦→♥ tû ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp(0, w) ❝â ❞↕♥❣ ✭✷✳✶✮✳ ❑❤✐ ✤â✱ ✤➥♥❣ t❤ù❝ ✭✶✳✼✮ ❧✉ỉ♥ ✤ó♥❣ ✈ỵ✐ M (x) = s(x), N (x) = s(x) tự ú ợ Lm ữủ ✭✶✳✸✼✮✳ ❑❤✐ ✤â✱ t♦→♥ tû S ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ r ❣✐❛♥ Lr (0, w)(p ≤ r ≤ q) ✈➔ t♦→♥ tû S ∗ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lr˜(0, w)(˜r = r − ) ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ S ∗ = U SU, U f = f (w − x) ✭✷✳✾✮ ▼➺♥❤ ✤➲ ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ❇ê ✤➲ ✶✳✶ ð ❈❤÷ì♥❣ ✶✳ ❇ê ✤➲ ✷✳✶✳ ◆➳✉ f t❤✉ë❝ Lp(0, w) ✈➔ Sf t❤✉ë❝ Lq (0, w) t❤➻ Sf, U f = f, S ∗ U f ✭✷✳✶✵✮ ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ Sf t❤✉ë❝ Lq (0, w) ♥➯♥ S ∗U f t❤✉ë❝ Lq (0, w)✳ ❉♦ ✈➟②✱ ❤❛✐ ✈➳ ❝õ❛ ✭✷✳✶✵✮ ✤➲✉ ❝â ♥❣❤➽❛✳ ✣➥♥❣ t❤ù❝ ✭✷✳✶✵✮ t❤✉ ✤÷đ❝ tø t➼♥❤ ❝❤➜t s❛✉ f1 , f2 = U f2 , U f1 , f1 ∈ Lq (0, w), f2 Lp (0, w) ỵ ●✐↔ sû t♦→♥ tû S ❝â ❞↕♥❣ ✭✷✳✶✮p ❜à ❝❤➦♥ tr Lp(0, w)(1 p 2) tỗ t ❝→❝ ❤➔♠ sè N1(x) ✈➔ N2(x) t❤✉ë❝ L (0, w) t❤ä❛ ♠➣♥ SN1 = M, SN2 = 1✳ ❑❤✐ ✤â −1 SBγ (x, λ) = eixλ , λ = , ✭✷✳✶✷✮ iγ tr♦♥❣ ✤â γ = SN1 , U N2 − N1 , S ∗ U N2 ; Bγ (x, λ) = + B(x, λ) + λγ ợ B(x, ) tở Lp(0, w) ữủ ❜ð✐ ✭✶✳✺✵✮ ✲ ✭✶✳✺✷✮✳ ✷✺ ✭✷✳✶✸✮ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✱ ✤➥♥❣ t❤ù❝ ✭✶✳✼✮ ✈➔ ❝ỉ♥❣ t❤ù❝ ✭✶✳✸✽✮ ✈➝♥ ✤ó♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ S t❤✉ë❝ Lp(0, w)✳ ❉♦ N (x) t❤✉ë❝ Lq (0, w) tỗ t số C s ||Lm+1 ||p ≤ Cm ||Lm ||p ≤ C m+1 m! ✭✷✳✶✹✮ ❚❛ ❦➼ ❤✐➺✉ ||f ||p ❧➔m❝❤✉➞♥ ❝õ❛ f tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp(0, w)✳ ❚❤❡♦ ✭✷✳✶✹✮✱ ❝❤✉é✐ ∞ (iλ) Bγ (x, λ) = ❤ë✐ tư ✈ỵ✐ |λ| < c−1 ❱➟② SBγ (x, λ) = eiλx, |λ| < c−1✳ m! m=0 ❚❤❡♦ ❇ê ✤➲ ✭✶✳✸✮✱ t❛ ❝â w Bγ (x, λ) = uγ (x, λ) − iλ ✭✷✳✶✺✮ Bγ (t, λ)dt, x tr♦♥❣ ✤â ✭✷✳✶✻✮ uγ (x, λ) = aγ (λ)N1 (x) + bγ (λ)N2 (x) w Bγ (t, x)dt; bγ (x) = + iλ aγ (x) = iλ ❚❤❡♦ ✭✶✳✸✽✮ t❛ ❝â w Bγ (t, λ)N (t)dt w xm i = SA∗ Lm + i m [M (x) + N (t)]Lm (t)dt s✉② r❛ w Lm+1 = − m w Lm (t)dt + x ❉♦ ✤â ∞ (iλ)m Lm+1 = − (m − 1)! m m=1 [N1 (x) + N (t)N2 (x)]Lm (t)dt ∞ (iλ)m (m − 1)! m=1 ∞ + m=1 w Lm (t)dt x w (iλ)m (m − 1)! [N1 (x) + N (t)N2 (x)]Lm (t)dt ❚ø ✤â t❛ ❝â w Bγ (x, λ) − L1 (x) = −iλ w Bγ (t, λ)dt + iλN1 (x) x Bγ (t, λ)dt w + iλN2 (x) N (t)Bγ (t, λ)dt ữỡ Pữỡ tr t r ợ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) w ⇔ Bγ (x, λ) = −iλ w Bγ (t, λ)dt + iλN1 (x) x Bγ (t, λ)dt w N (t)Bγ (t, λ)dt + N2 (x) + iλN2 (x) ❚❛ ✤➦t w aγ (λ) = iλ w Bγ (t, λ)dt; bγ (λ) = + iλ Bγ (t, λ)N (t)dt ✭✷✳✶✼✮ uγ (x, λ) = aγ (λ)N1 (x) + bγ (λ)N2 (x) ❱➟② w w Bγ (t, λ)dt = uγ (x, λ) − iλ Bγ (x, λ) = aγ (λ).N1 (x) + bγ (λ)N2 (x) − iλ x Bγ (t, )dt x t a () a() ữợ ❞↕♥❣ aγ (λ) = iλ Bγ (x, λ), S ∗ U N2 , a(λ) = iλ SBγ (x, λ), U N2 ✭✷✳✶✽✮ ✭✷✳✶✾✮ ❚ø ✭✷✳✶✺✮ ✲ ✭✷✳✶✼✮ t❛ ✤÷đ❝ a(λ) = aγ (λ)(1 + iλγ) ✭✷✳✷✵✮ ✈ỵ✐ γ ①→❝ ✤à♥❤ t b () b() ữợ ❞↕♥❣ bγ (λ) = + iλ Bγ , S ∗ U (1 − N1 ) , b(λ) = + iλ SBγ (x, λ), U (1 − N2 ) ❚÷ì♥❣ tü ✭✷✳✷✵✮✱ t❛ ✤÷đ❝ b(λ) = bγ (1 + iλγ) ❚ø ✭✷✳✷✵✮ ✈➔ ✭✷✳✷✷✮ t❛ t❤✉ ✤÷đ❝ uγ (x, λ) = u(x, λ) B(x, λ) ; Bγ (x, λ) = ; |λ| < c−1 + iλγ + iλγ ❉♦ B(x, λ) ✈➔ eiλx ❣✐↔✐ t➼❝❤ t❤❡♦ ỵ ữủ ự ữỡ Pữỡ tr t r ợ ❝❤➟♣ tr♦♥❣ Lp (0, w) ✷✳✷✳✷ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✈ỵ✐ ✈➳ ♣❤↔✐ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Wp2 (0, w) ❈❤♦ S ❧➔ t♦→♥ tû ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❤✐➺✉ ❜à ❝❤➦♥ tr Lp(0, w) sỷ tỗ t số N1(x) ✈➔ N2(x) t❤ä❛ ♠➣♥ SN1 = M, SN2 = é ỵ tr ú t ữỡ tr ợ (x) = eiλx ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ t❛ sû ❞ư♥❣ ❦➳t q✉↔ ✤â ✤➸ ❣✐↔✐ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✈➳ ♣❤↔✐ ❧➔ ❤➔♠ sè φ(x) t❤✉ë❝ Wp2 ❚❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ sè s❛✉ r(x, t) = N2 (w − t)N1 (x) − N1 (w − t)N2 (x) ❚❛ ❝â r(w − t, w − x) = N2 (w − w + x)N1 (w − t) − N1 (w − w + x)N2 (w − t) = N2 (x)N1 (w − t) − N1 (x − t)N2 (w − t) = −r(x, t), ❱ỵ✐ f (x) ❧➔ ❤➔♠ sè ❜➜t ❦ý t❤✉ë❝ Lq (−w, w) ✤➦t w w−x f (x − t + s)r(t, s)dsdt, I= x ✣ê✐ ❜✐➳♥ t = w − t , s = w − s ✱ ❦❤✐ ✤â w w−x f (x − t + s)r(t, s)dsdt I= x 0 x f (x + t − s )r(w − t , w − s )ds dt = w−x w w−x w f (x + t − s )r(w − t , w − s )ds dt = x w w−x f (x + t − s )r(w − t , w − s )dt ds = x w w−x =− f (x + t − s )r(t , s )dt ds x w w−x =− f (x + t − s)r(t, s)dtds = −I x ✷✽ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) ❙✉② r❛✱ I = ❚❛ ✤à♥❤ ♥❣❤➽❛ t♦→♥ tû T tr♦♥❣ Wp2 ❜ð✐ w w ϕ (t)r(x, t)dt + ϕ(w)N2 (x) − Tϕ = ϕ (x − t + w)N2 (t)dt x w w − ϕ (x + t − s).r(t, s)dsdt ✭✷✳✷✸✮ x w−x ❚❛ t❤➜② T ❧➔ ♠ët →♥❤ ①↕ tø Wp2 ✈➔♦ Lp(0, w)✳ ❚❤❡♦ ✭✶✳✺✵✮ ✲ ✭✶✳✺✷✮ t❛ t❤✉ ✤÷đ❝ B(x, λ) = T eiλx ✭✷✳✷✹✮ ◆➳✉ γ = t❤➻ tø ✭✷✳✶✷✮ t❛ s✉② r❛ ST eiλx = eiλx ; T SB(x, λ) = B(x, λ) ✭✷✳✷✺✮ ❙û ❞ö♥❣ ♠è✐ t ự ữủ ỵ ữợ ỵ sỷ ỵ tọ = õ t♦→♥ tû T ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✷✸✮ ❧➔ ♥❣❤à❝❤ ✤↔♦ ♣❤↔✐ ❝õ❛ S ✱ tù❝ ❧➔✱ ST ϕ = ϕ, ϕ ∈ Wp2 ✭✷✳✷✻✮ ❉♦ ✈➟②✱ ✈ỵ✐ γ = ✈➔ ϕ t❤✉ë❝ Wp2 t❤➻ ❤➔♠ sè f (x) = T ϕ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮✳ ❚❛ ①➨t tr÷í♥❣ ❤đ♣ γ = SN1 , U N2 − N1 , S ∗ U N2 = ❚❛ ✤➦t λ0 = −1 ❚ø ✭✷✳✶✷✮ t❛ ❝â iγ SBγ (x, λ) = eixλ , s✉② r❛ S ❞♦ ✈➟② B(x, λ) iλγ + = eiλx , S(B(x, λ)) = (iλγ + 1)eixλ ❑❤✐ λ t✐➳♥ tỵ✐ λ0, ❞♦ S ❧✐➯♥ tư❝ ✈➔ B ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ t❤❡♦ λ ♥➯♥ SB(x, λ0 ) = lim SB(x, λ) = lim (iλγ + 1)eixλ = λ→λ0 ữỡ Pữỡ tr t r ợ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) ❚❛ ✈✐➳t ❧↕✐ ✭✷✳✶✷✮ ♥❤÷ s❛✉ S(B(x, λ) − B(x, λ0 )) (iλγ + 1)eixλ − = = eixλ iλf + iλγ + ✭✷✳✷✼✮ ❚❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ sè x Cγ ϕ = γ ϕ(t)eiλ0 (x−t) dt, Tγ = T Cγ ✭✷✳✷✽✮ ❚❛ ❞➵ t❤➜② x Cγ e iλx = γ eiλt eiλ0 (x−t) dt x = γ = = = eiλ0 x e(iλ−iλ0 )t dt x iλ e x e(iλ−iλ0 )t iλ − iλ0 γ eiλ0 x (iλ−iλ )x e γ iλ − iλ0 Cγ eiλx = ❚ø ✭✷✳✷✹✮✱ ✭✷✳✷✽✮ ✈➔ ✭✷✳✷✾✮ t❛ ❝â eiλx − eiλ0 x iλγ + eiλx − eiλ0 x iλγ + ❚ø t❛ t❤➜② T ❧➔ →♥❤ ①↕ tø Wp1 w Tγ ϕ = −1 d γ dx iλ − iλ0 eiλx − eiλ0 x (eiλx − eiλ0 x ) = γi(λ − λ0 ) iλγ + ❱➟② Tγ eiλx = T Cγ eiλx = T − ✈➔♦ = T eiλx − T eiλ0 x B(x, λ) − B(x, λ0 ) = iλγ + iλf + Lp (0, w) ✈➔ ✤÷đ❝ ✈✐➳t ữợ w (u).ei0 (tu) du (x, t)dt (t) + iλ0 ✭✷✳✷✾✮ ❚ø ✭✷✳✷✽✮ ✈➔ ✭✷✳✸✵✮✱ t❛ ❝❤➾ r❛ r➡♥❣ STγ eiλx = S B(x, λ) − B(x, λ0 ) iλγ + ❉♦ ✈➟②✱ t❛ ❝â ỵ s = eix ữỡ Pữỡ tr t r ợ tr Lp (0, w) ỵ sỷ ỵ tọ ✈➔ γ ❦❤→❝ 0✳ ❑❤✐ ✤â t♦→♥ tû Tγ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✸✶✮ ❧➔ ♠ët ♥❣❤à❝❤ ✤↔♦ ♣❤↔✐ ❝õ❛ S ✱ tù❝ ❧➔✱ ✭✷✳✸✷✮ STγ ϕ = ϕ, ϕ(x) Wp1 ỵ t ♣❤÷ì♥❣ ♣❤→♣ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮✳ ✣➸ ♠æ t↔ t➟♣ ♥❣❤✐➺♠ ✈➔ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ♥â✱ t❛ s➩ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t Sf = ❚❛ ❦➼ ❤✐➺✉ HS ❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ Sf = t❤✉ë❝ Lp(0, w)✳ ◆➳✉ dim HS > t❤➻ t õ ỵ s ỵ S ❧➔ t♦→♥ tû ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❤✐➺✉ ❜à ❝❤➦♥ tr♦♥❣ Lp(0, w) (1 ≤ p ≤ 2) ✈➔ < dim HS = n < ∞ ❑❤✐ ✤â✱ HS ❝â ❝ì sð ❧➔ fk (0 ≤ k ≤ n − 1) t❤ä❛ ♠➣♥ fk+1 = A∗ fk , ≤ k ≤ n − 2, ✭✷✳✸✸✮ x w tr♦♥❣ ✤â Af = i f (t)dt ✈➔ A∗f = −i f (t)dt✳ x ❈❤ù♥❣ ♠✐♥❤✳ ❉♦ A∗ ❦❤æ♥❣ ❝â ❦❤æ♥❣ ❣✐❛♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❜➜t ❜✐➳♥ ♥➯♥ A∗HS = HS ❚❤❡♦ ✭✶✳✼✮ t❛ ❝â w (AS − SA∗ )f = i [M (x) + N (t)]f (t)dt ❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ f t❤✉ë❝ HS t❛ ❝â Sf = ♥➯♥ w SA∗ f = −i w [M (x) + N (t)]f (t)dt = −iM (x) w f (t)dt − i f (t).N (t)dt ❱➟② SA∗HS ⊂ s♣❛♥{M (x), 1} ◆➳✉ dim SA∗HS = t tỗ t số N1 N2 tử ❑❤✐ ✤â✱ S ❦❤↔ ♥❣❤à❝❤ ♥➯♥ HS = ✭✈æ ỵ dim HS > dim SAHS = ✣➦t HS1 = A∗HS ∩HS ❱➻ dim SA∗HS ≤ ♥➯♥ dim(A∗HS −∩HS ) ≥ dim A∗HS − = dim HS − = n − ❤❛② dim HS1 = n − ❚÷ì♥❣ tü tr➯♥ t❛ ❝â A∗ HS1 = HS1 ⊂ HS ✈➔ dim S(A∗HS1 ) ≤ dim S(A∗HS ) ≤ ❚÷ì♥❣ tü t❛ ✤➦t HS2 = A∗HS1 ∩HS ❑❤✐ ✤â✱ dim(A∗HS1 ∩HS ) ≤ dim A∗HS1 −1 = n − ❚❛ ❞➵ ❞➔♥❣ t❤➜② ✤÷đ❝ HS2 ⊂ HS1 ❈ù tữỡ tữ ữ ợ t HS = AHSk−1 ∩ HS (2 ≤ k ≤ n − 1) ✈➔ HS ⊃ HS1 ⊃ · · · ⊃ HSn1, dim HSk = n k õ tỗ t↕✐ ❤➔♠ sè f0 ∈ HS t❤ä❛ ♠➣♥ fk = A∗k f0 ∈ HSk (1 ≤ k ≤ n − 1, ||f0||p = 0) ❚❛ ①➨t ❤➺ t❤ù❝ α0 f0 + α1 f1 + α2 f2 + · · · + αn−1 fn−1 = ✸✶ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) ⇐⇒α0 f0 + α1 A∗ f0 + α2 A∗2 f0 + · · · + αn−1 A∗n−1 f0 = ⇐⇒ α0 + α1 A∗ + α2 A∗2 + · · · + αn−1 A∗n−1 f0 = ⇐⇒α0 = α1 = α2 = · · · = αn−1 = ❱➻ ✈➟②✱ fk (0 ≤ k ≤ n − 1) ❧➔ ♠ët ❝ì sð ❝õ❛ HS ✈➔ t❤ä❛ ♠➣♥ fk+1 = A∗fk (0 ≤ k) ✷✳✸ ❱➼ ❞ö →♣ ❞ư♥❣ s❛✉ ❚r♦♥❣ ♠ư❝ ♥➔② ❝❤ó♥❣ t❛ s➩ →♣ ỵ tt tr ữỡ tr s w 1−β (x − t) f (t)dt = φ(x) |x − t|α−1 Sα f = ✭✷✳✸✹✮ ✈ỵ✐ ✭✷✳✸✺✮ ❚❛ s➩ r số L1(x), L2(x) tữỡ ự ợ ữỡ tr➻♥❤ ✭✷✳✸✹✮ ❝â ❞↕♥❣ −1 < β < 1; < α < 2; α = L1 (x) = Dx−ρ (w − x)ρ−µ , L2 (x) = w(ρ − µ) + x L1 (x), 1−µ ✭✷✳✸✻✮ ✭✷✳✸✼✮ tr♦♥❣ ✤â D= sin πρ , π(1 − β) µ = − α, tan πρ = (1 − β) sin πµ , < ρ < (1 + β) + (1 − β) cos πµ ✭✷✳✸✽✮ ✭✷✳✸✾✮ ✭✷✳✹✵✮ ❚❛ ❝ơ♥❣ õ t t ữợ sin = ợ sin (à ) 1+ ỵ ✷✳✽✳ ❈❤♦ t♦→♥ tû Sα ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✸✹✮✱ ✭✷✳✸✺✮ ✈➔ ❝→❝ ❤➔♠ sè L1 (x), L2 (x) ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❜ð✐ ✭✷✳✸✻✮ ✲ ✭✷✳✹✵✮✳ ❑❤✐ ✤â✱ t❛ ❝â Sα Lk = xk−1 , k = 1, ✭✷✳✹✸✮ < < ữỡ Pữỡ tr t r ợ tr Lp (0, w) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✭✷✳✹✸✮ ✈ỵ✐ k = 1✳ ✣ê✐ ❜✐➳♥ x = wu, t = ws t ữợw s(x − t) S L = L (t)dt α 1 |x − t|α−1 s❣♥ 1−β (wu − ws) L1 (ws)ds |wu − ws|α−1 = s❣♥ 1−β (wu − ws) D(ws)−ρ (w − ws)ρ−µ ds |wu − ws|α−1 = s−ρ (1 − s)ρ−µ (1 − β =D s❣♥(u − s))|u − s|µ−1ds u s−ρ (1 − s)ρ−µ (u − s)µ−1 ds(1 − β) + = D s−ρ (1 − s)ρ−µ (u − s)µ−1 ds(1 + β) u ❝â ✣➦t s = zu ð t➼❝❤ ♣❤➙♥ t❤ù ♥❤➜t ✈➔ s = − (1 − u)z ð t➼❝❤ ♣❤➙♥ t❤ù ❤❛✐✱ t❛ u z −ρ (1 − zu)ρ−µ (1 − z)µ−1 dz(1 − β) Sα L1 = D uµ−ρ (1 − z)µ−1 z ρ−µ (1 − (1 − u)z)−ρ dz(1 + ) + (1 u) ỵ Γ(c) F (a, b, c, z) = Γ(b)Γ(c − b) tb−1 (1 − t)c−b−1 dt (1 − tz)a ✭✷✳✹✹✮ tr♣♥❣ ✤â Γ ❧➔ ❤➔♠ ❣❛♠♠❛✳ ❑❤✐ ✤â✱ t❛ ữủ S L1 = D(à)(uà +(1 u) (1 ρ) F (µ − ρ, − ρ, − ρ + µ, u)(1 − β) Γ(1 − ρ + µ) Γ(1 + ρ − µ) F (ρ, + ρ − µ, − ρ, − u)(1 + β)) Γ(1 + ρ) ✣➦t ✸✸ ✭✷✳✹✺✮ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) u ✭✷✳✹✻✮ sp−1 (1 − s)q−1 ds Bu (p, q) = ❚ø ✭✷✳✹✺✮ s✉② r❛ ✭✷✳✹✼✮ F (p, − q, p + 1, u) = pu−p Bu (p, q) ❚ø ❝æ♥❣ t❤ù❝ ✭✷✳✹✼✮ ✈➔ Γ(x + 1) = xΓ(x) t❛ ❝â u Γ(1 − ρ) Sα L1 = DΓ(µ) uµ−ρ (µ − ρ)uρ−µ (µ − ρ)Γ(µ − ρ) tµ−ρ−1 (1 − t)ρ−1 dt(1 − β) 1−u + (1 − u)ρ Γ(1 + ρ − µ) −ρ ρu ρΓ(ρ) tρ−1 (1 − t)µ−ρ−1 dt u =D Γ(1 − ρ)Γ(µ) (1 − β) Γ(µ − ρ) tµ−ρ−1 (1 − t)ρ−1 dt 1−u Γ(1 + ρ − µ)Γ(µ) = (1 + β) Γ(ρ) Γ(z)Γ(1 − z) = ▼➔ π sin πz ✭✷✳✹✾✮ ✭✷✳✺✵✮ Γ(1 − ρ)(1 − β) Γ(1 + ρ − µ)(1 + β) = Γ(µ − ρ) Γ(ρ) 1−u tρ−1 (1 − t)µ−ρ−1 dt = tµ−ρ−1 (1 − t)ρ−1 dt ✭✷✳✺✶✮ u ❚ø ✭✷✳✹✽✮✱ ✭✷✳✺✵✮ ✈➔ ✭✷✳✺✶✮ t❛ ❝â Sα L1 = D ✭✷✳✹✽✮ ❚❤❡♦ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ❣❛♠♠❛ ✈➔ ✭✷✳✹✶✮✱ t❛ ❝â tρ−1 (1 − t)µ−ρ−1 dt u Γ(1 − ρ)Γ(µ) (1 − β) Γ(µ − ρ) tµ−ρ−1 (1 − t)ρ−1 dt + tµ−ρ−1 (1 − t)ρ−1 dt u Γ(1 − ρ)Γ(µ) =D (1 − β) Γ(µ − ρ) tµ−ρ−1 (1 − t)ρ−1 dt ✸✹ ✭✷✳✺✷✮ ❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) ❚ø ✭✷✳✹✽✮✱ ✭✷✳✺✶✮ ✭✷✳✺✷✮ ✈➔ ✤➥♥❣ t❤ù❝ tµ−ρ−1 (1 − t)ρ−1 dt = Γ(µ − ρ)Γ(ρ) Γ(µ) ✭✷✳✺✸✮ t❛ ❝â ✤➥♥❣ t❤ù❝ s❛✉ Γ(1 − ρ)Γ(µ) Γ(µ − ρ)Γ(ρ) (1 − β) Γ(µ − ρ) Γ(µ) = D(Γ(1 − ρ)(1 − β)Γ(ρ)) π =D (1 − β) = sin πρ Sα L1 = D ❱➟② ✭✷✳✹✸✮ ✤ó♥❣ ✈ỵ✐ k = ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✭✷✳✹✸✮ ✈ỵ✐ k = ❚❛ ①➨t x u t1−ρ (w−t)ρ−µ (x−t)µ−1 dt(1−β)+ Sα (xL1 ) = D( t1−ρ (w−t)ρ−µ (t−x)µ−1 dt(1+β)) x ✣ê✐ ❜✐➳♥ x = wu, t = wuz ð t➼❝❤ ♣❤➙♥ t❤ù ♥❤➜t ✈➔ x = wu, t = w[1 − (1 − u)z] ð t➼❝❤ ♣❤➙♥ t❤ù ❤❛✐ t❛ ữủ S (xL1 ) = Dw(uà+1 z (1 − zu)ρ−µ (1 − z)µ−1 dt(1 − β) (1 − z)µ−1 z ρ−µ (1 − (1 − u)z)1−ρ dt(1 + β)) +(1 − u)ρ ✭✷✳✺✹✮ ❚ø ✭✷✳✹✺✮ ✈➔ ✭✷✳✺✹✮ t❛ ❝â Sα L1 = DwΓ(w)(uµ−ρ+1 +(1 − u)ρ Γ(2 − ρ) F (µ − ρ, − ρ, − ρ + µ, u)(1 − β) Γ(2 − ρ + µ) Γ(1 + ρ − µ) F (ρ − 1, + ρ − µ, + ρ, − u)(1 + β)) Γ(1 + ρ) ✭✷✳✺✺✮ ❚❛ ✤à♥❤ ♥❣❤➽❛ ❤➺ t❤ù❝ ●❛✉ss ♥❤÷ s❛✉ F (a, b, c, z) = −a c−1 F (a + 1, b, c, z) + F (a, b, c − 1, z) c−a−1 c−a−1 ✭✷✳✺✻✮ ❚ø ✭✷✳✺✺✮ ✈➔ ✭✷✳✺✻✮ t❛ ❦➳t ❧✉➟♥ Sα L1 = DwΓ(µ)(uµ−ρ+1 Γ(2 − ρ) (ρ − µ)F (µ − ρ + 1, − ρ, − ρ + µ, u) Γ(2 − ρ + µ) ữỡ Pữỡ tr t r ợ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) +(1 − ρ)F (ρ, + ρ − µ, + ρ, − u) + ρF (ρ − 1, + ρ − µ, ρ, − µ)(1 + β)) ✭✷✳✺✼✮ ❚ø ✭✷✳✺✵✮ t❛ ❝â Γ(2 − ρ)(1 − β) Γ(1 + ρ − µ)(1 + β) = Γ(1 + ρ) Γ(µ − ρ)ρ(1 − ρ) ✭✷✳✺✽✮ ❚❛ ✈✐➳t ❧↕✐ ✭✷✳✺✼✮ ♥❤÷ s❛✉ Γ(2 − ρ)(1 − β) µ−ρ+1 −1 (u ( F (µ − ρ + 1, − ρ, − ρ + µ, u) Γ(µ − ρ) µ−ρ+1 1 + F (µ − ρ, − ρ, − ρ + µ, µ)) + (1 − u)ρ ( F (ρ, + ρ − µ, + ρ, − u) µ−ρ ρ Sα L1 = DwΓ(µ) + F (ρ − 1, + ρ − µ, ρ, − u)) 1−ρ ✭✷✳✺✾✮ ❚❛ sû ❞ö♥❣ ❦➼ ❤✐➺✉ φ(ρ, µ, u) ❧➔ ❜✐➸✉ t❤ù❝ tr♦♥❣ ♥❣♦➦❝ ✈✉ỉ♥❣ tr♦♥❣ ✭✷✳✺✾✮✳ ⑩♣ ❞ö♥❣ ✭✷✳✹✻✮ ✈➔ ✭✷✳✹✼✮ t❛ ✈✐➳t ❧↕✐ (, à, u) ữ s tà1 (1 t)2 (u − t)dt φ(ρ, µ, u) = =− Γ(µ − ρ − 1)Γ(ρ − 1) Γ(µ − ρ)Γ(ρ − 1) +u Γ(µ) Γ(µ − 1) ❚❤❡♦ ✭✷✳✸✽✮ ✈➔ ✭✷✳✹✾✮ t❛ ❝â DΓ(2 − ρ).Γ(ρ − 1)(1 − β) = sin πρ sin πρ Γ(2 − ρ)Γ(ρ − 1)(1 − β) = = −1 π(1 − β) sin π(1 − ρ) ❚❤❡♦ ✭✷✳✺✾✮ ✈➔ ✭✷✳✻✵✮ t❛ ❝â Sα(xL1) = x(à 1) + w(à p) ú ợ k = ❚ø ✭✷✳✸✻✮ ✈➔ ✭✷✳✺✸✮ t❛ t❤✉ ✤÷đ❝ w w Dx−ρ (w − x)ρ−µ dx = Dw1−µ L1 (x)dx = R= ✭✷✳✻✵✮ Γ(1 − ρ)Γ(1 + ρ − µ) Γ(2 − µ) ❚❤❡♦ ✭✷✳✸✻✮✱ ✭✷✳✸✼✮ t❛ ❝â Q1 (x, t) = D2 x−ρ (w − x)ρ−µ (w − t)−ρ tρ−µ (x + t − w) R(1 à) ữỡ Pữỡ tr t➼❝❤ ♣❤➙♥ ❋r❡❞❤♦❧♠ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Lp (0, w) ✣➦t s = u + w ✈➔♦ φ1(x, t) ✈➔ tø ✭✷✳✻✶✮ t❛ t❤✉ ✤÷đ❝ 2w−|x−t| φ1 (x, t) = Q1 s+x−t s−x+t , ds 2 x+t w−|x−t| D2 = 2R(1 − µ)2 (w + x − t)2 − u2 t −ρ (w − x + t)2 − u2 ρ−µ udu x+t−w ❚ø ✤â✱ t❛ ỹ ữợ t tỷ S T = , Wp2 T ữ ỵ N2 (x) = L1 (x) ú ỵ t ✤÷đ❝ ❝→❝ ❤➔♠ sè L1 ✈➔ L2 ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✹✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ < α < ✸✼ ❑➌❚ ▲❯❾◆ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♥❤ú♥❣ ✈➜♥ ✤➲ s ữỡ tr r ợ ❳➙② ❞ü♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ t♦→♥ tû t➼❝❤ ợ tr L2(0, w) ❝❤➜t ❝õ❛ t♦→♥ tû t➼❝❤ ♣❤➙♥ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr Lp (0, w) ữỡ tr t ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ Wp2 (0, w)✳ ✸✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ▲❡✈ ❆✳ ❙❛❦❤♥♦✈✐❝❤✱ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ✇✐t❤ ❉✐❢❢❡r❡♥❝❡ ❑❡r♥❡❧s ♦♥ ❋✐♥✐t❡ ■♥t❡r✈❛❧s✱ ❇✐r❦❤❛✉s❡r ✭✷✵✶✺✮✳ ❬✷❪ ❚r➛♥ ✣ù❝ ▲♦♥❣ ✲ P❤↕♠ ❑ý ❆♥❤✱ ●✐→♦ tr➻♥❤ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐ ✭✷✵✵✶✮✳ ❬✸❪ ❚r➛♥ ✣ù❝ ▲♦♥❣ ✲ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣ ✲ ❍♦➔♥❣ ◗✉è❝ ❚♦➔♥✱ ●✐→♦ tr➻♥❤ ●✐↔✐ t➼❝❤✱ ❚➟♣ ✶✱ ✷ ✈➔ ✸✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐ ✭✷✵✵✺✮✳ ✸✾ ... KHOA HỌC TỰ NHIÊN NGUYỄN THỊ MINH THÚY TÌM HIỂU VỀ PHƯƠNG TRÌNH FREDHOLM VỚI NHÂN DẠNG CHẬP TRÊN KHOẢNG HỮU HẠN Chuyên ngành: Tốn giải tích Mã số: 60 46 01 02 LUẬN VĂN THẠC SĨ TOÁN HỌC