不動点定理とα階常微分方程式の解の存在と一意性

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ෆಈ఺ఆཧͱ

_α

֊ৗඍ෼ํఔࣜͷղͷଘࡏͱҰҙੑ

Fixed point theorems and existence and uniqueness of solutions for α-th order ordinary diﬀerential

equations

ɹ๛ాণ࢙

Masashi Toyoda

ۄ઒େֶ޻ֶ෦ϚωδϝϯταΠΤϯεֶՊ, 194–8610 ౦ژ౎ொాࢢۄ઒ֶԂ 6–1–1 Department of Management Science, College of Engineering, Tamagawa University,

6–1–1 Tamagawa-gakuen, Machida-shi, Tokyo 194–8610

Abstract

In this paper, initial value problems of fractional differential equations are discussed. We obtain the existence and uniqueness of solutions for the problems. Two proofs are given. In the first proof, we don’t use any fixed point theorems. In the second proof, we use the fixed point theorem for contraction mappings.

Keywords: Initial value problems, fractional diﬀerential equations, ﬁxed point theorems.

1 ͸͡Ίʹ [4]ʹ͸, ҎԼͷఆཧʹରͯ͠, ॖখࣸ૾ͷ ෆಈ఺ఆཧΛ༻͍Δ৔߹ͱ༻͍ͳ͍৔߹ͷূ໌ ͕ه͞Ε͍ͯΔ. ఆཧ _{1. a, b > 0} ͱ͢Δ. R2 ͷ෦෼ू߹ _D Λ_{D = {(x, y) | x}₀ _{≤ x ≤ x}₀_{+ a, |y − y}₀_{| ≤ b}} ͱ͢Δ_{. f} Λ _D Ͱ࿈ଓͳؔ਺ͱ͠_{, L > 0} ͕ ଘࡏͯ͠, ೚ҙͷ _{(x, y}₁_{), (x, y}₂)∈ D ʹରͯ͠ |f(x, y1)− f(x, y2)| ≤ L|y1− y2| ΛΈͨ͢ͱ͢Δ. ͜ͷͱ͖_{, [x}₀_{, x}₀_{+ h]}͔Β _R ΁ͷ͋Δؔ਺ _y ͕ଘࡏͯ͠ y(x) = f (x, y(x)) y(x0) = y0 ΛΈͨ͢. ͨͩ͠ _h ͸ h < min a, b M, 1 L ΛΈͨ͢ਖ਼ఆ਺Ͱ͋Δ. ͜͜Ͱ M = max (x,y)∈D|f(x, y)| Ͱ͋Δ. ຊ࿦จͰ͸,ҎԼͷఆཧʹରͯ͠,ॖখࣸ૾ ͷෆಈ఺ఆཧΛ༻͍Δ৔߹ͱ༻͍ͳ͍৔߹ͷূ ໌Λه͢. ఆཧ2͸ఆཧ1ͷ֦ுͰ͋Δ. ࣮ࡍ, ఆཧ 2Ͱ _{α = 1}ͷ৔߹͕ఆཧ 1ʹͳΔ. ఆཧ _{2. 0 < α ≤ 1, a, b > 0} ͱ͢Δ. R2 ͷ෦ ෼ू߹ _D Λ D = {(x, y) | x0≤ x ≤ x0+ a, y − y0 Γ(α)x α−1_{≤ b} ͱ͢Δ_{. f} Λ _D Ͱ࿈ଓͳؔ਺ͱ͠_{, L > 0} ͕ ଘࡏͯ͠, ೚ҙͷ _{(x, y}₁_{), (x, y}₂)∈ D ʹରͯ͠ |f(x, y1)− f(x, y2)| ≤ L|y1− y2| ΛΈͨ͢ͱ͢Δ. ͜ͷͱ͖_{, [x}₀_{, x}₀_{+ h]}͔Β _R ΁ͷ͋Δؔ਺ _y ͕ଘࡏͯ͠ Dαy(x) = f (x, y(x)) Dαy(x0) = y0 ΛΈͨ͢. ͨͩ͠ _h ͸ h < min a, Γ(α + 1)b M 1 α , Γ(α + 1) L 1 α

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ΛΈͨ͢ਖ਼ఆ਺Ͱ͋Δ. ͜͜Ͱ M = max (x,y)∈D|f(x, y)| Ͱ͋Δ. ఆཧ2 ͷ_Dα ͸_α ֊Riemann-Liouvilleඍ෼ Ͱ͋Δ. ͢ͳΘͪ_{, (0, ∞)} ͔Β _R ΁ͷؔ਺ _y ͷ_α ֊Riemann-Liouvilleඍ෼͸ Dαy(x) = 1 Γ(n − α) dn dtn _x 0 u(s) (x − t)α−n+1dt ͰఆΊΔ. ͜͜Ͱ _{n = [α] + 1}Ͱ͋Γ _[α] ͸_α Λӽ͑ͳ͍࠷େͷࣗવ਺Ͱ͋Δ. ͢ͳΘͪ_{, n} ͸_{n − 1 ≤ α ≤ n} ΛΈͨࣗ͢વ਺Ͱ͋Δ. Γ͸ ΨϯϚؔ਺Ͱ͋Δ. ΨϯϚؔ਺ʹؔͯ͠ _x a (x − t) p−1_{(t − a)}q−1_dt = Γ(p)Γ(q) Γ(p + q)(x − a) p+q−1 ͕੒Γཱͭ(ྫ͑͹, [6, p.70]). ͜͜Ͱ_{p, q > 0} Ͱ͋Δ. 2 ఆཧ ₂ͷূ໌ͦͷ₁ ఆཧ2ͷূ໌Λࣔ͢. ূ໌. ੵ෼ํఔࣜ y(x) = y0 Γ(α)x α−1 + 1 Γ(α) _x x0 (x − t)α−1f (t, y(t))dt ΛΈͨ͢ _y ͕ͨͩͻͱͭଘࡏ͢Δ͜ͱΛࣔ͢. Dͷ෦෼ू߹_D₀ Λ D0 ={(x, y) | x₀ ≤ x ≤ x₀+ h, y − y0 Γ(α)x α−1_{≤ b} ͰఆΊΔ. ؔ਺ྻ _{yn}Λ y1(x) = _Γ(α)y0 xα−1 ͓Αͼ_{n ∈ N}ʹରͯ͠ yn+1(x) = _Γ(α)y0 xα−1 + 1 Γ(α) _x x0 (x − t)α−1f (t, yn(t))dt ͰఆΊΔ. n ∈ N ʹରͯ͠ _{(x, y}_n_{(x)) ∈ D}₀ Ͱ͋Δ. ࣮ࡍ_{, n ∈ N, x ∈ [x}₀_{, x}₀_{+ h]} ʹରͯ͠ yn(x) − y0 Γ(α)x α−1 = 1 Γ(α) _x x0 (x − t)α−1f (t, yn−1(t))dt ≤ M Γ(α) _xx 0 (x − t)α−1dt = M αΓ(α)(x − x0) α ≤ M hα Γ(α + 1) ≤ b Ͱ͋Δ. Αͬͯ_{(x, y}_n_{(x)) ∈ D}₀ Ͱ͋Δ. n ∈ N, x ∈ [x0, x0+ h] ʹରͯ͠ |yn+1(x) − yn(x)| ≤ M L n−1 Γ(nα + 1)(x − x0) nα ͕੒Γཱͭ. ͜ΕΛ਺ֶతؼೲ๏Ͱࣔ͢. n = 1 ͷͱ͖ |y2(x) − y1(x)| =y₂(x) − y0 Γ(α)x α−1 = 1 Γ(α) _xx 0 (x − t)α−1f (t, y1(t))dt ≤ M Γ(α) _xx 0 (x − t)α−1dt = M Γ(α + 1)(x − x0) α ͕੒Γཱͭ_{. n}Ͱ੒Γཱͭͱ͢Δ. ͜ͷͱ͖ |yn+2(x) − yn+1(x)| = 1 Γ(α) _xx 0 (x − t)α−1(f (t, yn+1(t)) −f(t, yn(t))) dt| ≤ L Γ(α) _xx 0 (x − t)α−1|yn+1(t) − yn(t)|dt ≤ M Ln Γ(α)Γ(nα + 1) × _x x0 (x − t)α−1(t − x0)nαdt

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= M L n Γ(α)Γ(nα + 1) × Γ(α)Γ(nα + 1) Γ ((n + 1)α + 1)(x − x0) (n+1)α = M L n Γ ((n + 1)α + 1)(x − x0) (n+1)α Ͱ͋Δ. Αͬͯ_{n + 1} Ͱ΋੒Γཱͭ. ·ͨ ∞ n=1 M Ln−1hnα Γ(nα + 1) < ∞ Ͱ͋Δ. ࣮ࡍ_{, n ∈ N}ʹରͯ͠ an= M L n−1_hnα Γ(nα + 1) ͱ͓͘ͱ an+1 an = M Lnh(n+1)α Γ ((n + 1)α + 1)· Γ(nα + 1) M Ln−1hnα = Lhα Γ(nα + 1) Γ ((n + 1)α + 1) = Lhα Γ(nα + 1) Γ(nα + α + 1) = Lh α (nα)α · (nα)α+1Γ(nα) Γ(nα + α + 1)· Γ(nα + 1) (nα)Γ(nα) Ͱ͋Δ. Αͬͯ_{n → ∞}ͷͱ͖ an+1 an → 0 Ͱ͋Δ. ͜͜Ͱ lim x→∞ Γ(x + s) xsΓ(x) = 1 ʹ஫ҙ͞Ε͍ͨ (ྫ͑͹, [3, p.341]). ͠ ͨ ͕ͬͯ, ਖ਼ ߲ ڃ ਺ ∞_n=1_a_n ʹ ର ͠ ͯ lim_n→∞ an+1_a n = 0 Ͱ͋Δ͔Β, μϥϯϕʔϧ ͷ൑ఆ๏(ྫ͑͹, [5, p.4])ΑΓ ∞_n=1an ͸ ऩଋ͢Δ. ͢ͳΘͪ ∞_n=1MLn−1hnα Γ(nα+1) < ∞Ͱ ͋Δ. M൑ఆ๏(ྫ͑͹, [5, p.19])ΑΓ, ؔ਺ྻ {yn} ͸[x0, x0+ h]Ͱؔ਺ y ʹҰ༷ऩଋ͢Δ. ͜ͷͱ͖ y(x) = y0 Γ(α)x α−1 + 1 Γ(α) _x x0 (x − t)α−1f (t, y(t))dt Ͱ͋Δ_{. y} ͸ղͱͳΔ. ࣍ʹҰҙੑΛࣔ͢. ˆy ΋ղͱ͢Δ. ͜ͷͱ ͖,೚ҙͷ _{x ∈ [x}₀_{, x}₀_{+ h], n ∈ N}ʹରͯ͠ |y(x) − ˆy(x)| ≤ N Ln Γ(nα + 1)(x − x0) nα ͕੒Γཱͭ. ͜͜Ͱ N = sup x∈[x0,x0+h] |y(x) − ˆy(x)| Ͱ͋Δ. ͜ΕΛ਺ֶతؼೲ๏Ͱࣔ͢_{. n = 1}ͷ ͱ͖ |y(x) − ˆy(x)| = 1 Γ(α) x x0 (x − t)α−1(f (t, y(t)) −f(t, ˆy(t))) dt| ≤ L Γ(α) _xx 0 (x − t)α−1|y(t) − ˆy(t)|dt ≤ N L Γ(α) _xx 0 (x − t)α−1dt = N L Γ(α + 1)(x − x0) α Ͱ͋Δ_{. n}Ͱ੒Γཱͭͱ͢Δͱ |y(x) − ˆy(x)| ≤ L Γ(α) _xx 0 (x − t)α−1|y(t) − ˆy(t)|dt ≤ M Ln+1 Γ(α)Γ(nα + 1) × _x x0 (x − t)α−1(t − x0)nαdt = M L n+1 Γ(α)Γ(nα + 1) × Γ(α)Γ(nα + 1) Γ ((n + 1)α + 1)(x − x0) (n+1)α = M Ln+1 Γ ((n + 1)α + 1)(x − x0) (n+1)α Ͱ͋Δ. Αͬͯn + 1 Ͱ΋੒Γཱͭ. n → ∞ ͱͯ͠ _{y = ˆ}_y ͕੒Γཱͭ. Αͬͯ, ղ͸ͨͩҰͭଘࡏ͢Δ.

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3 ఆཧ ₂ͷূ໌ͦͷ₂ ຊઅͰ͸, ఆཧ 2Λॖখࣸ૾ͷෆಈ఺ఆཧ Λ༻͍ͯࣔ͢_{. C[a, b]}Λ _{[a, b]} ͔Β _R ΁ͷ࿈ ଓؔ਺શମ͔ΒͳΔू߹ͱ͢Δ_{. y ∈ C[a, b]} ʹ ରͯ͠ϊϧϜ _· Λ y = max x∈[a,b]|y(x)| ͱ͓͘ͱ _{C[a, b]}͸Banach ۭؒͱͳΔ. ఆཧ 2ͷূ໌Λࣔ͢. ূ໌_{. C[x}₀_{, x}₀ _{+ h]} ͷ෦෼ू߹ _D₀ Λ _D₀ = y |y − y0 Γ(α)xα−1 ≤ b (x ∈ [x0, x0+ h]) Ͱ ఆΊΔ_{. D}₀ ্ͷ࡞༻ૉ_T Λ T y(x) = y0 Γ(α)x α−1 + 1 Γ(α) _x x0 (x − t)α−1f (t, y(t))dt ͰఆΊΔ. y ∈ D0 ͳΒ͹ T y ∈ D0 Ͱ͋Δ. ࣮ࡍ, y ∈ D0 ͱ͢Δ. [x0, x0 + h] ͷ఺ྻ {xn} ͕ xn → x ΛΈͨ͢ͱ͢Δ. n ∈ N ͱ͢Δ. ฏۉ஋ͷఆཧΑΓ, ͋Δ c1 ∈ (xn, x) (·ͨ͸ c1∈ (x, xn)) ͕ଘࡏͯ͠ |xα−1 n − xα−1| = (α − 1)cα−21 |xn− x| Ͱ͋Δ. Αͬͯ,͋Δఆ਺_L₁͕ଘࡏͯ͠ |xα−1 n − xα−1| ≤ L1|xn− x| Ͱ͋Δ. ·ͨ,͋Δ _c₂_{∈ (x}_n_{− t, x − t) (}·ͨ͸ c2∈ (x − t, xn− t))͕ଘࡏͯ͠ |(xn−t)α−1−(x−t)α−1| = (α−1)cα−2₂ |xn−x| Ͱ͋Δ. Αͬͯ,͋Δఆ਺_L₂ ͕ଘࡏͯ͠ |(xn− t)α−1− (x − t)α−1| ≤ L2|xn− x| Ͱ͋Δ. Αͬͯ |T y(xn)− T y(x)| ≤ y0 Γ(α)|x α−1 n − xα−1| + M Γ(α) _xxn|(xn− t)α−1− (x − t)α−1|dt ≤ y0L1 Γ(α)|xn− x| + M L2 Γ(α)|xn− x| 2 ΑΓ_x_n_{→ x} ͷͱ͖ T y(xn)→ T y(x) Ͱ͋Δ. ͕ͨͬͯ͠T y ∈ C[x0, x0+ h]ΛಘΔ. ͞Βʹ_{T y ∈ D}₀ Ͱ͋Δ. ࣮ࡍ_{, x ∈ [x}₀_{, x}₀_{+ h]} ͱ͢Δ. ͜ͷͱ͖ (Ty)(x) − y0 Γ(α)x α−1 = 1 Γ(α) _xx 0 (x − t)α−1f (t, y(t))dt ≤ M Γ(α) _xx 0 (x − t)α−1dt = M αΓ(α)(x − x0) α ≤ M hα Γ(α + 1) ≤ b ΑΓ_{T y ∈ D}₀ Ͱ͋Δ. ·ͨ _T ͸ॖখࣸ૾Ͱ͋Δ. ࣮ࡍ_{, y}₁_{, y}₂ _∈ D0, x ∈ [x0, x0+ h] ͱ͢Δ. |T y1(x) − T y2(x)| = 1 Γ(α) _xx 0 (x − t)α−1(f (t, y1(t)) −f(t, y2(t))) dt| ≤ L Γ(α) _xx 0 (x − t)α−1|y1(t) − y2(t)|dt ≤ L Γ(α) _xx 0 (x − t)α−1dty1− y2 = L αΓ(α)(x − x0) α_y 1− y2 ≤ Lhα Γ(α + 1)y1− y2 Ͱ͋Δ. Αͬͯ T y1− T y2 ≤ Lh α Γ(α + 1)y1− y2 ΛಘΔ. ͜͜Ͱ Lhα Γ(α + 1) < 1 ΑΓ_T ͸ॖখͱͳΔ. ͕ͨͬͯ͠, ॖখࣸ૾ͷෆಈ఺ఆཧΑΓ _T ͸ͨͩͻͱͭͷෆಈ఺_y Λ΋ͭ.

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஫ҙ _{1. [1]}Ͱ͸, ఆཧ 1 Λ _α ֊ඍ෼ํఔ ࣜʹ֦ுͨ݁͠Ռ͕঺հ͞Ε͍ͯΔ. ͦͷࡍ, Weissingerͷෆಈ఺ఆཧ([7])Λ༻͍ͯূ໌͞ Ε͍ͯΔ. Weissingerͷෆಈ఺ఆཧ͸,ॖখࣸ ૾ͷෆಈ఺ఆཧͷͻͱͭͷ֦ுͰ͋Δ. ஫ҙ_{2. [8]}Ͱ͸,͞ΒʹಛҟੑΛ΋ͭํఔࣜͷ ৔߹͕ѻΘΕ͍ͯΔ. ఆཧ2ΛಛҟੑΛ΋ͭ৔ ߹ʹ·Ͱ֦ுͨ͠ఆཧʹରͯ͠,ෆಈ఺ఆཧΛ ༻͍Δ৔߹ͱ༻͍ͳ͍৔߹ͷূ໌Λࣔ͢ͷ΋໘ ന͍. [2] ΋ࢀর͞Ε͍ͨ. ࢀߟจݙ

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2016೥3݄18೔ݪߘड෇ɹ

Received, March 18, 2016

2016 年３月 18 日原稿受付，2016 年３月 30 日採録決定 Received, March 18, 2016; accepted, March 30, 2016