ඍํఔࣜͷڥքͷॱংू߹ʹ͓͚Δෆಈఆཧ
ͷ
2
ͭͷద༻ྫ
Two examples obtained by using a fixed point theorem in partial ordered sets to boundary value problems for differential equations
ɹ๛ాণ࢙‡ ɹลढ़Ұ∗
ɹɹMasashi Toyoda Toshikazu Watanabe
‡ۄେֶֶ෦ϚωδϝϯταΠΤϯεֶՊ, 194–8610 ౦ژொాࢢۄֶԂ 6–1–1 College of Engineering, Tamagawa University,
6–1–1 Tamagawa-gakuen, Machida-shi, Tokyo 194–8610 ∗ຊେֶཧֶ෦, 101–8308 ౦ژઍా۠ਆాॣՏ 1–8–14
College of Science and Technology, Nihon University, 1–8–14 Kanda-Surugadai, Chiyoda-ku, Tokyo 101–8308
Abstract
In this paper, we apply a fixed point theorem in partial ordered set [17] to two boundary value problems for differential equations. First problem is a boundary value problem for fourth order differential equations. Second problem is a boundary value problem for α order differential equations with 3 < α≤ 4.
Keywords: Fixed point theorem, partially ordered set, boundary value problems.
ୈ1અ ͡Ίʹ (X,≤)Λॱংू߹ͱ͢Δ. ॱংू߹ͷ ྻ{xn}͕୯ௐඇݮগͰ͋Δͱ, x1 ≤ x2≤ x3 ≤ · · · ͕Γཱͭͱ͖Λ͍͏. X͔ΒX ͷࣸ૾T ͕୯ௐඇݮগͰ͋Δͱ, ҙͷ x, y∈ Xʹରͯ͠x≤ yͳΒT x≤ T y͕ Γཱͭͱ͖Λ͍͏. [17]ʹ͓͍ͯ,࣍ͷఆཧ͕ࣔ͞Ε͍ͯΔ. ఆཧ 1. (X,≤)Λॱংू߹ͱ͢Δ. ڑd ͕ଘࡏͯ͠ (X, d)͕උڑۭؒͱ͢Δ. X ͷ୯ௐඇݮগͳྻ{xn} ͕xn → x ∈ X (n→ ∞)ΛΈͨ͢ͳΒ,ҙͷxn∈ Xʹର ͯ͠xn ≤ xΛΈͨ͢ͱ͢Δ. T Λ X͔ΒX ͷࣸ૾Ͱ୯ௐඇݮগͱ͢Δ. ͋Δk ∈ [0, 1) ͕ଘࡏͯ͠, ҙͷx, y ∈ Xʹରͯ͠, x ≤ y ͳΒ d(T x, T y)≤ kd(x, y) ΛΈͨ͢ͱ͢Δ. ͋Δ x0 ∈ X ͕ଘࡏͯ͠ x0 ≤ T x0 ΛΈͨ͢ͱ͢Δ. ͜ͷͱ͖T ෆ ಈΛͭ. ҙͷ x, y ∈ X ʹରͯ͠, ͋Δ z∈ X ͕ଘࡏͯ͠, x≤ z ͔ͭy ≤ z ΛΈͨ͢ ͳΒ, ෆಈҰҙͰ͋Δ. ఆཧ 1 ͷΑ͏ʹ, ॱংΛԾఆͨ͠ڑۭؒ ʹ͓͚Δෆಈఆཧͷݚڀ͕ݱࡏਐΊΒΕ͍ͯ Δ. 80ʹ, [26]ʹΑΔݚڀ͕͋Δ. [19]Ͱ ,ఆཧ1ͷT ʹ࿈ଓΛԾఆͨ͠߹ͷ݁Ռ͕ ಘΒΕ͍ͯΔ. ͦͷ߹,ۭؒX ʹʮXͷ୯ௐ ඇݮগͳྻ{xn}͕xn→ x ∈ X (n → ∞)Λ Έͨ͢ͳΒ,ҙͷxn∈ Xʹରͯ͠xn≤ x ΛΈͨ͢ͱ͢Δʯͱ͍͏ԾఆෆཁʹͳΔ. [19] [17]ͷ݁ՌΛ֦ு͢ΔࢼΈͳ͞Ε͍ͯΔ. ྫ͑, [9]Λࢀর͞Ε͍ͨ. ॱংΛԾఆͨ͠ڑۭؒʹ͓͚Δෆಈఆ ཧ, ͞·͟·ͳʹద༻͞Ε͍ͯΔ. [19] Ͱ,Τϧϛʔτߦྻશମ͔ΒͳΔۭؒʹ͓͍ ͯ,ߦྻʹؔ͢ΔํఔࣜͷղͷଘࡏͱҰҙੑΛ ѻ͍ͬͯΔ. [17]Ͱ, 1֊ඍํఔࣜͷपظղ ͷղͷଘࡏͱҰҙੑΛѻ͍ͬͯΔ. ͜ΕΒͷ݁
ՌΛࢀߟʹ, චऀΒ, [22]Ͱ 4֊ඍํఔࣜ ͷڥք ·ͨ, [25]Ͱα֊ඍํఔࣜͷڥ քͷղͷଘࡏͱҰҙੑΛѻͬͨ. ͜͜Ͱ 3 < α ≤ 4 Ͱ͋Δ. ͔͠͠, [22]͓Αͼ[25]ʹ , Ұ෦, ܭࢉʹෆඋ͕͋Δ͜ͱ͕ͦͷޙͷݚ ڀͰΘ͔ͬͨ. ͦ͜Ͱ,मਖ਼͞ΕͨܭࢉΛຊ จͰࣔ͢. ·ͨ,मਖ਼͢Δ͚ͩͰͳ͘,৽͘͠ಘ ͨ݁Ռड़Δ. ۩ମతʹ, ୈ2અͰ[22]Λ मਖ਼ͨ݁͠ՌΛࣔ͢(ఆཧ 2). ·ͨ, [22]ͷͱ ͖ʹಘΒΕͳ͔ͬͨ݁Ռ(ఆཧ3ͱྫ1, 2) ه͢. ୈ3અͰ[25]Λमਖ਼ͨ݁͠ՌΛࣔ͢(ఆ ཧ9). ఆཧ9 [25]Λ֦ுͨ݁͠Ռʹͳͬͯ ͍Δ. ୈ2અ ద༻ྫͦͷ1 ຊઅͰ,ఆཧ 1 Λڥք { y′′′′(t) + f (t, y(t), y′′(t)) = 0, y(0) = y(1) = y′′(0) = y′′(1) = 0 ʹద༻͢Δ. ͨͩ͠ f [0, 1]× R × R ͔Β Rͷ࿈ଓؔͰ͋Δ. y0 ∈ C4(I, R) ͕͜ͷ ڥքͷԼղͰ͋Δͱ { y′′′′0 (t) + f (t, y0(t), y0′′(t))≤ 0, y0(0) = y0(1) = y′′0(0) = y′′0(1) = 0 ΛΈͨ͢ͱ͖Λ͍͏. ఆཧ 1ΑΓ,࣍ΛಘΔ. ఆཧ 2. f Λ [0, 1]× R × R͔Β R ͷ࿈ଓ ؔͱ͢Δ. ͋Δ µ∈ (0, 8) ͕ଘࡏͯ͠, ҙ ͷ y1 ≤ y2, u1 ≥ u2 ͱͳΔ y1, y2, u1, u2 ∈ R ͓Αͼ t∈ I ʹରͯ͠ 0≤ f(t, y1, u1)− f(t, y2, u2)≤ µ(u1− u2) ͕Γཱͭͱ͢Δ. y′′′0(0)≤ ∫ 1 0 (∫ t 0 f (s, y0(s), y′′0(s))ds ) dt ΛΈͨ͢Լղy0 ͕ଘࡏ͢Δͱ͢Δ. ͜ͷͱ͖, ڥք { y′′′′(t) + f (t, y(t), y′′(t)) = 0, y(0) = y(1) = y′′(0) = y′′(1) = 0 ͷղ͕Ұҙʹଘࡏ͢Δ. ূ໌. X = C([0, 1], R) ͱ͢Δ. ҙͷ x, y ∈ X ʹରͯ͠x≤ y Λ x(t)≤ y(t) (t ∈ [0, 1]) ͰఆΊΔ. ͜ͷͱ͖ (X,≤) ॱংू߹Ͱ͋Δ. X ͷڑ dΛ d(x, y) = sup t∈[0,1]|x(t) − y(y)| ͰఆΊΔͱ͖, (X, d) උڑۭؒͰ͋Δ. {xn} ΛX ͷ୯ௐඇݮগྻͰ xn→ x ΛΈ ͨ͢ͷͱ͢Δ. ͜ͷͱ͖x(t) = supn∈Nxn(t) Ͱ͋Δ͔Βxn(t) ≤ x(t) Ͱ͋Δ. ͕ͨͬͯ͠ xn≤ xͰ͋Δ. X ͔Β X ͷࣸ૾T Λ T u(t) = ∫ 1 0 G(t, s)f (s, y(s), u(s))ds ͰఆΊΔ. ͜͜Ͱ y(t) =− ∫ 1 0 G(t, s)u(s)ds ͔ͭ G(t, s) = { (1− t)s, 0 ≤ s ≤ t ≤ 1, (1− s)t, 0 ≤ t ≤ s ≤ 1 Ͱ͋Δ. ͜ͷͱ͖ T ୯ௐඇݮগͰ͋Δ. ࣮ ࡍ, u1 ≥ u2 ͱͳΔu1, u2∈ X ʹରͯ͠ y1(t) =− ∫ 1 0 G(t, s)u1(s)ds ≤ − ∫ 1 0 G(t, s)u2(s)ds = y2(t) ͕ҙͷ t ∈ [0, 1] ʹରͯ͠Γཱͭ. ͜͜Ͱ G(t, s) ≥ 0 ͕ҙͷ (t, s) ∈ [0, 1] × [0, 1] ʹ ରͯ͠Γཱͭ͜ͱʹҙ͞Ε͍ͨ. ͞Βʹ f (t, y1(t), u1(t))≥ f(t, y2(t), u2(t))Ͱ͋Δ. ͠ ͕ͨͬͯT u1(t)≥ T u2(t)ΛಘΔ. ͋Δk∈ [0, 1)͕ଘࡏͯ͠,ҙͷu1, u2 ∈ X ʹରͯ͠u1≥ u2 ͳΒ d(T u1(t), T u2(t))≤ kd(u1(t), u2(t))
͕Γཱͭ͜ͱΛࣔ͢. u1 ≥ u2 ͱ͢Δ. ͜ͷ ͱ͖ y1≤ y2 Ͱ͋Δ. ͞Βʹ d(T u2(t), T u1(t)) = sup 0≤t≤1|T u2(t)− T u1(t)| ≤ sup 0≤t≤1 ∫ 1 0 G(t, s)× |f(s, y2(s), u2(s))− f(s, y1(s), u1(s))|ds ≤ µ sup 0≤t≤1 ∫ 1 0 G(t, s)|u2(s)− u1(s)|ds ≤ µd(u2, u1) sup 0≤t≤1 ∫ 1 0 G(t, s)ds = µd(u2, u1)× sup 0≤t≤1 (∫ t 0 (1− t)sds + ∫ 1 t (1− s)tds ) = µ 2d(u2, u1) sup0≤t≤1 t(1− t) ≤ µ 8d(u2, u1) ͕Γཱͭ. α = y0′′ ͱ͓͘. α≤ T αͱͳΔ͜ ͱΛࣔ͢. α′′(s)≤ −f(s, y0(s), α(s))͕ҙͷ s∈ [0, 1] ʹରͯ͠ΓཱͭͷͰ α′(t)≤ α′(0)− ∫ t 0 f (s, y0(s), α(s))ds ΛಘΔ. ͞Βʹ α(x)≤ α′(0)x− ∫ x 0 (∫ t 0 f (s, y0(s), α(s))ds ) dt Ͱ͋Δ. Լղ y0 ͕Έͨ͢Ծఆ͔Β α′(0)≤ ∫ 1 0 (∫ t 0 f (s, y0(s), α(s))ds ) dt Ͱ͋Δ. ͜ͷͱ͖ α(x)≤ x ∫ 1 0 (∫ t 0 f (s, y0(s), α(s))ds ) dt − ∫ x 0 (∫ t 0 f (s, y0(s), α(s))ds ) dt = ∫ 1 0 G(x, t)f (t, y0(t), α(t))dt = T α(x) ͕ҙͷ x∈ [0, 1] ʹରͯ͠Γཱͭ. Αͬͯ α≤ T αͰ͋Δ. ఆཧ 1ΑΓղ͕Ұҙʹଘࡏ͢ Δ. □ ఆཧ 2 ͷԼղy0 ͷΈ͖ͨ݅͢ y′′′0(0)≤ ∫ 1 0 (∫ t 0 f (s, y0(s), y′′0(s))ds ) dt ͲͷΑ͏ͳͱ͖ʹΓཱͭͷ͔. ࣸ૾ f Λ [0, 1]× R × [0, ∞)͔Β [0,∞)Ͱߟ͑ΔͱΓ ཱͭ. ࣮ࡍ,࣍ΛಘΔ. ఆཧ 3. f Λ[0, 1]× R × [0, ∞)͔Β[0,∞) ͷ࿈ଓؔͱ͢Δ. ͋Δµ∈ (0, 8)͕ଘࡏͯ͠, ҙͷ y1, y2, u1, u2∈ R ʹରͯ͠, y1 ≤ y2 ͔ ͭu1≥ u2 ͳΒ, ҙͷ t∈ I ʹରͯ͠ 0≤ f(t, y1, u1)− f(t, y2, u2)≤ µ(u1− u2) ΛΈͨ͢. ͜ͷͱ͖, ڥք { y′′′′(t) + f (t, y(t), y′′(t)) = 0, y(0) = y(1) = y′′(0) = y′′(1) = 0 ͷղ͕Ұҙʹଘࡏ͢Δ. ূ໌. X = C([0, 1], R) ͱ͢Δ. ͜ͷͱ͖ (X,≤) ఆཧ 2 ͱಉ༷, උڑۭؒͰ͋Γ, ॱংू߹Ͱ͋Δ. X ͔Β X ͷࣸ૾T Λ T u(t) = ∫ 1 0 G(t, s)f (s, y(s), u(s))ds ͰఆΊΔ. ͜͜Ͱ y(t) =− ∫ 1 0 G(t, s)u(s)ds ͔ͭ G(t, s) = { (1− t)s, 0 ≤ s ≤ t ≤ 1, (1− s)t, 0 ≤ t ≤ s ≤ 1 Ͱ͋Δ. ͜ͷͱ͖,ఆཧ 2 ͱಉ༷ʹͯ͠, T ୯ௐඇݮগͰ͋Γ,͋Δ k∈ [0, 1) ͕ଘࡏͯ͠, u1 ≥ u2 ͳΒ d(T u1(t), T u2(t))≤ kd(u1(t), u2(t)) Ͱ͋Δ. f ͱG ඇෛͳͷͰ T 0 = ∫ 1 0 G(t, s)f (s, y(s), 0)ds≥ 0 Ͱ͋Δ. ఆཧ 1ΑΓղ͕Ұҙʹଘࡏ͢Δ. □
ྫ 1. λ > 0ͱ͢Δ. ڥք { y′′′′(t) + λ log(y′′+ 2) = 0, y(0) = y(1) = y′′(0) = y′′(1) = 0 Λߟ͑Δ. ఆཧ 3 ʹ͓͍ͯ f (t, y, u) = λ log(u + 2) ͱ͢Δͱ y1 ≤ y2 ͓Αͼ u1 ≥ u2 ≥ 0, t ∈ [0, 1]ʹରͯ͠ f (t, y1, u1)− f(t, y2, u2) = λ(log(u1+ 2)− log(u2+ 2)) = λ log ( u1+ 2 u2+ 2 ) = λ log ( 1 +u1− u2 u2+ 2 ) ≤ λ log (1 + u1− u2) ≤ λ(u1− u2) ͕Γཱͭ. ͜ͷͱ͖, ఆཧ3ΑΓղ͕Ұҙʹ ଘࡏ͢Δ. ྫ 2. λ > 0ͱ͢Δ. ڥք { y′′′′(t) + λ2(log(y′′+ 2) + y′′) = 0, y(0) = y(1) = y′′(0) = y′′(1) = 0 Λߟ͑Δ. ఆཧ 3 ʹ͓͍ͯ f (t, y, u) = λ 2(log(u + 2) + u) ͱ͢Δͱy1 ≤ y2 ͓Αͼ u1 ≥ u2 ≥ 0, t ∈ [0, 1] ʹରͯ͠ f (t, y1, u1)− f(t, y2, u2) = λ 2 (log(u1+ 2)− log(u2+ 2) + u1− u2) = λ 2 ( log ( u1+ 2 u2+ 2 ) + u1− u2 ) = λ 2 ( log ( 1 +u1− u2 u2+ 2 ) + u1− u2 ) ≤ λ 2 (log (1 + u1− u2) + u1− u2) ≤ λ2(u1− u2+ u1− u2) = λ(u1− u2) ͕Γཱͭ. ͜ͷͱ͖, ఆཧ3ΑΓղ͕Ұҙʹ ଘࡏ͢Δ. ୈ3અ ద༻ྫͦͷ2 1 < α≤ 2, 1 < β ≤ 2, 2 < α + β ≤ 4ͱ͢ Δ. ڥք { Dβ(Dαu(t)) + f (t, u(t)) = 0,
u(0) = u(1) = (Dαu)(0) = (Dαu)(1) = 0
Λߟ͑Δ. ͜͜Ͱ Dα α ֊ͷ Riemann-Liouville ඍͰ͋Δ. f [0, 1]× R ͔Β R ͷؔͰ͋Δ. u Λ (0,∞) ͔ΒR ͷؔͱ͢Δ. uͷ α ֊Riemann-Liouvilleඍ Dαu(t) = 1 Γ(n− α) dn dtn ∫ t 0 u(s) (t− s)α−n+1ds ͰఆΊΔ. ͜͜Ͱ n = [α] + 1 Ͱ͋Γ [α]α Λӽ͑ͳ͍࠷େͷࣗવͰ͋Δ. ͢ͳΘͪ, n n− 1 ≤ α < nΛΈͨࣗ͢વͰ͋Δ. Γ ΨϯϚؔͰ͋Δ. Ұํ, uͷα ֊ Riemman-Liouvilleੵ Iαu(t) = 1 Γ(α) ∫ t 0 (t− s)α−1u(s)ds ͰఆΊΔ. ิॿ໋ 4ิॿ໋ 5Ͱࣜ ∫ x a (x− t)p−1(t− a)q−1dt = Γ(p)Γ(q) Γ(p + q)(x− a) p+q−1 Λ༻͍Δ. ೦ͷͨΊ,ূ໌Λه͢. t− a = sͱ ͓͘ͱ dt = ds Ͱ͋Γ t ͷੵ͕۠ؒ a→ x ͱมԽ͢Δͱ͖ s 0 → x − a ͱมԽ͢Δ. Αͬͯ ∫ x a (x− t)p−1(t− a)q−1dt = ∫ x−a 0 (x− a − s)p−1sq−1ds Ͱ͋Δ. ͞Βʹ s x−a = uͱ͓͘ͱ x−a1 ds = du Ͱ͋Γ sͷੵ͕۠ؒ 0→ x − a ͱมԽ͢Δ
ͱ͖ u 0→ 1 ͱมԽ͢Δ. Αͬͯ ∫ x a (x− t)p−1(t− a)q−1dt = ∫ x−a 0 (x− a − s)p−1sq−1ds = ∫ 1 0 (x− a − (x − a)u)p−1× (x− a)q−1uq−1(x− a)du = (x− a)p+q−1 ∫ 1 0 (1− u)p−1uq−1du = (x− a)p+q−1B(p, q) = (x− a)p+q−1Γ(p)Γ(q) Γ(p + q) ΛಘΔ. α > 0, β > 0ͷͱ͖ Iαtβ = Γ(β + 1) Γ(α + β + 1)t α+β Ͱ͋Δ. ࣮ࡍ Iαtβ = 1 Γ(α) ∫ t 0 (t− s)α−1sβds = 1 Γ(α) · Γ(α)Γ(β + 1) Γ(α + β + 1)t α+β = Γ(β + 1) Γ(α + β + 1)t α+β Ͱ͋Δ. ·ͨ Dαtβ = Γ(β + 1) Γ(β + 1− α)t β−α Ͱ͋Δ. ࣮ࡍ nΛࣗવͱͨ͠߹ Dntβ = Γ(β + 1) Γ(β + 1− n)t β−n Ͱ͋Δ͜ͱʹҙ͢Ε Dαtβ = 1 Γ(n− α) dn dtn ∫ t 0 (t− s)n−α−1sβds = 1 Γ(n− α) dn dtn Γ(n− α)Γ(β + 1) Γ(n− α + β + 1)t n−α+β = Γ(β + 1) Γ(n− α + β + 1) dn dtntn−α+β = Γ(β + 1) Γ(n− α + β + 1) Γ(n− α + β + 1) Γ(β + 1− α) t β−α = Γ(β + 1) Γ(β + 1− α)t β−α Ͱ͋Δ. ิॿ໋ 4. α > 0 ͱ͢Δ. u(t) = tα−n (n = 1, 2, . . . , [α] + 1)ͷͱ͖, Dαu = 0Ͱ͋Δ.ɹٯ ʹ Dαu(t) = 0 ͳΒ, ͋Δ C 1, C2, . . . , Cn ∈ R ͕ଘࡏͯ͠ u(t) = C1tα−1+ C2tα−2+· · · + Cntα−n Ͱ͋Δ. ͜͜Ͱ n = [α] + 1 Ͱ͋Δ. ূ໌. u(t) = tα−n (n = 1, 2, . . . , [α] + 1)ͷͱ ͖, Dαu = 0 Ͱ͋Δ.ɹ࣮ࡍ Dαu(t) = 1 Γ(n− α) dn dtn ∫ t 0 (t− s)n−α−1sα−nds = 1 Γ(n− α) dn dtn Γ(n− α)Γ(α − n + 1) Γ(1) = 0 ΛಘΔ. ٯΛࣔ͢. Dαu(t) = 0ͱ͢Δ. ఆٛΑΓ 1 Γ(n− α) dn dtn ∫ t 0 (t− s)n−α−1u(s)ds = 0 ͢ͳΘͪ dn dtn ∫ t 0 (t− s)n−α−1u(s)ds = 0 Ͱ͋Δ. ͜ΕΑΓ dn−1 dtn−1 ∫ t 0 (t− s)n−α−1u(s)ds = C1
Ͱ͋Δ. ·ͨ dn−2 dtn−2 ∫ t 0 (t− s)n−α−1u(s)ds = C1t + C2 Ͱ͋Δ. ಉ༷ʹ dn−3 dtn−3 ∫ t 0 (t−s)n−α−1u(s)ds = C1t2+C2t+C3 Ͱ͋Δ. ͨͩ͠ C1 2 Λ͋ΒͨΊͯ C1 ͱͨ͠. ҎԼ,దٓ,ఆΛஔ͖͑Δ. ܁Γฦ͢ͱ ∫ t 0 (t− s)n−α−1u(s)ds = C1tn−1+ C2tn−2+· · · + Cn Ͱ͋Δ. ͜͜Ͱ྆ลʹ Iα Λࢪ͢ͱ (ࠨล) = Iα ∫ t 0 (t− s)n−α−1u(s)ds = 1 Γ(α) ∫ t 0 (t− s)n−α−1× (∫ s 0 (s− τ)n−α−1u(τ )dτ ) ds = 1 Γ(α) ∫ t 0 u(τ )× (∫ t τ (t− s)α−1(s− τ)n−α−1ds ) dτ = 1 Γ(α) Γ(α)Γ(n− α) Γ(n) (t− τ) n−1 = Γ(n− α) Γ(n) ∫ t 0 (t− τ)n−1u(τ )dτ = Γ(n− α)Inu(t) Ͱ͋Δ. ͜͜Ͱ3ͭΊͷ߸ੵॱংͷަ Λͨ͠. ·ͨ (ӈล) = Iα(C1tn−1+ C2tn−2+· · · + Cn) = C1tα+n−1+ C2tα+n−2+· · · + Cntα Ͱ͋Δ. Αͬͯ Γ(n− α)Inu(t) = C1tα+n−1+ C2tα+n−2+· · · + Cntα ͢ͳΘͪ,ఆΛஔ͖͑ͯ Inu(t) = C1tα+n−1+ C2tα+n−2+· · · + Cntα ΛಘΔ. ͞Βʹ྆ลʹ Dn Λࢪ͢ͱ DnInu(t) = u(t) Ͱ͋Γ Dn(C1tα+n−1+ C2tα+n−2+· · · + Cntα) = C1tα−1+ C2tα−2+· · · + Cntα−n Ͱ͋Δ͔Β u(t) = C1tα−1+ C2tα−2+· · · + Cntα−n ΛಘΔ. □ ࣍, [21, Theorem2.4]ʹ͋Δ. શΛظ͢ ΔͨΊ,ূ໌Λه͢. ิॿ໋ 5. α > 0, u ∈ L(a, b) ͱ͢Δ. ͜ͷ ͱ͖ DαIαu = u Ͱ͋Δ. ূ໌. n = [α] + 1 ͱ͢Δ. Dα, Iα ͷఆ͓ٛΑ ͼੵॱংͷަ͔Β DαIαu = 1 Γ(α)Γ(n− α) dn dxn× ∫ x 0 1 (x− t)α−n+1 (∫ x s (t− s)α−1u(s)ds ) dt = 1 Γ(α)Γ(n− α) dn dxn× ∫ x 0 (∫ x s (x− t)n−α−1(t− s)α−1u(s)dt ) ds = 1 Γ(α)Γ(n− α) dn dxn× ∫ x 0 u(s) (∫ x s (x− t)n−α−1(t− s)α−1dt ) ds ΛಘΔ. ·ͨ ∫ x s (x− t)n−α−1(t− s)α−1dt = Γ(α)Γ(n− α) Γ(n) (x− s) n−1
Ͱ͋Δ͔Β DαIαu = 1 Γ(n) dn dxn ∫ x 0 (x− s)n−1u(s)ds Ͱ͋Δ. ͜͜Ͱ ∫ x 0 (∫ x 0 · · · (∫ x 0 u(t)dt ) · · · dt ) dt = 1 (n− 1)! ∫ x 0 (x− t)n−1u(t)dt Ͱ͋Δ. ূ໌ֶతؼೲ๏ʹΑΓͰ͖Δ. ͠ ͕ͨͬͯ DαIαu = u ΛಘΔ. □ ࣍ [4]ʹ͋Δ. શΛظ͢ΔͨΊʹ,ূ໌ Λه͢. [11]ࢀরͤΑ. ิॿ໋6. α > 0ͱ͢Δ. uΛ(0, 1)Ͱ࿈ଓͰ ੵՄೳͳؔͱ͢Δ,͢ͳΘͪ u∈ C(0, 1) ∩ L(0, 1)ͱ͢Δ. ͞ΒʹDαu∈ C(0, 1)∩L(0, 1) ΛΈͨ͢ͱ͢Δ. n = [α] + 1 ͱ͢Δ. ͜ͷͱ ͖, ͋Δ C1, C2, . . . , Cn∈ R͕ଘࡏͯ͠ IαDαu(t) = u(t) + C1tα−1+ C2tα−2+· · · + Cntα−n ͕Γཱͭ. ূ໌. ิॿ໋ 4ΑΓ Dα(IαDαu− u) = DαIαDαu− Dαu = Dαu− Dαu = 0 Ͱ͋Δ. Αͬͯ, ิॿ໋ 5 ΑΓ, ͋Δ C1, C2, . . . , Cn∈ R͕ଘࡏͯ͠ IαDαu(t)− u(t) = C1tα−1+ C2tα−2+· · · + Cntα−n Ͱ͋Δ. ͜ΕΑΓ༩ࣜΛಘΔ. □ [4, Lemma 2.3]ʹ͕࣍͋Δ. શΛظ͢Δ ͨΊ,ূ໌Λه͢. ิॿ໋ 7. h∈ C[0, 1] ͔ͭ 1 < α ≤ 2 ͱ͢ Δ. ͜ͷͱ͖, ڥք { Dαu(t) + h(t) = 0, u(0) = u(1) = 0 ͷҰҙղ u(t) = ∫ 1 0 Gα(t, s)h(s)ds Ͱ͋Δ. ͜͜Ͱ Gα(t, s) = 1 Γ(α) ( tα−1(1− s)α−1− (t − s)α−1) (0≤ s ≤ t ≤ 1), 1 Γ(α) ( tα−1(1− s)α−1) (0≤ t ≤ s ≤ 1) Ͱ͋Δ. ূ໌. Dαu(t) =−h(t)ΑΓ IαDαu(t) =−Iαh(t) Ͱ͋Δ. ิॿ໋ 6 ΑΓ, ͋ΔC1, C2 ∈ R ͕ ଘࡏͯ͠ u(t) + C1tα−1+ C2tα−2 =− 1 Γ(α) ∫ t 0 (t− s)α−1h(s)ds Ͱ͋Δ. ͜ͷͱ͖ u(t) =− 1 Γ(α) ∫ t 0 (t− s)α−1h(s)ds − C1tα−1− C2tα−2 Ͱ͋Δ. u(0) = 0 ΑΓC2 = 0Ͱ͋Δ. ͞Βʹ u(1) = 0ΑΓ C1 =− 1 Γ(α) ∫ 1 0 (1− s)α−1h(s)ds Ͱ͋Δ. ͕ͨͬͯ͠ u(t) =− 1 Γ(α) ∫ t 0 (t− s)α−1h(s)ds + 1 Γ(α) ∫ 1 0 (1− s)α−1tα−1h(s)ds = ∫ 1 0 Gα(t, s)h(s)ds ΛಘΔ. □
ิॿ໋ 7ͷؔ Gα ࣍ΛΈͨ͢. [18, Lemma 2.8]ΛࢀরͤΑ. ิॿ໋8. ิॿ໋7ʹ͋Δؔ Gα ʹର͠ ͯGα(t, s)≥ 0 ͔ͭ Gα(t, s)≤ Γ(α)1 ͕ҙͷ t, s∈ [0, 1] ʹରͯ͠Γཱͭ. ূ໌. 0≤ s ≤ t ≤ 1ͷͱ͖ Gα(t, s) = 1 Γ(α) ( tα−1(1− s)α−1− (t − s)α−1) ≥ Γ(α)1 (tα−1(1− s)α−1− (t − ts)α−1) = 0 Ͱ͋Δ. ·ͨ 0≤ t ≤ s ≤ 1 ͷͱ͖ Gα(t, s)≥ 0 Ͱ͋Δ. 0≤ s ≤ t ≤ 1 ͷͱ͖ ∂Gα(t, s) ∂t = 1 Γ(α) ( (α− 1)tα−2(1− s)α−1 −(α − 1)(t − s)α−2) ≤ 1 Γ(α) ( (α− 1)tα−2(1− s)α−1 −(α − 1)(t − ts)α−2) = 1 Γ(α)(α− 1)t α−2((1− s)α−1− (1 − s)α−2) ≤ 0 Ͱ͋Δ͔Β Gα(t, s) s≤ t ≤ 1 ʹ͓͍ͯ୯ ௐඇ૿ՃͰ͋Δ. ͕ͨͬͯ͠ Gα(t, s)≤ Gα(s, s) = 1 Γ(α)s α−1(1− s)α−1 ≤ 1 Γ(α) ΛಘΔ. ·ͨ 0≤ t ≤ s ≤ 1 ʹ͓͍ͯ Gα(t, s) ∂t = 1 Γ(α)(α− 1)t α−2(1− s)α−1 ≥ 0 Ͱ͋Δ͔Β Gα(t, s) 0 ≤ t ≤ s ʹ͓͍ͯ୯ ௐඇݮগͰ͋Δ. ͕ͨͬͯ͠ Gα(t, s)≤ Gα(s, s) = 1 Γ(α)s α−1(1− s)α−1 ≤ 1 Γ(α) ΛಘΔ. □ [18] ͷূ໌ํ๏ͱఆཧ1 ΑΓ࣍ΛಘΔ. ఆཧ 9. f Λ [0, 1]× [0, ∞) ͔Β [0,∞) ͷ ࿈ଓͰୈ2มʹؔͯ͠୯ௐඇݮগͳؔͱ͢ Δ. 1 < α≤ 2, 1 < β ≤ 2, 2 < α + β ≤ 4 ͱ ͢Δ. ͋Δ λ ∈ [0, Γ(α)Γ(β)) ͕ଘࡏͯ͠, ҙͷ u, v∈ [0, ∞) ʹରͯ͠ u≥ v ͳΒ, 0≤ f(t, u) − f(t, v) ≤ λ(u − v) ͕ҙͷ t ∈ [0, 1] ʹରͯ͠Γཱͭͱ͢Δ. ͜ͷͱ͖, ڥք { Dβ(Dαu(t)) + f (t, u(t)) = 0,
u(0) = u(1) = (Dαu)(0) = (Dαu)(1) = 0
ͷղ͕Ұҙʹଘࡏ͢Δ.
ূ໌. ڥք
{
Dβ(Dαu(t)) + f (t, u(t)) = 0,
u(0) = u(1) = (Dαu)(0) = (Dαu)(1) = 0
ͷղ u(t) = ∫ 1 0 Gα(t, s) (∫ 1 0 Gβ(s, r)f (r, u(r))dr ) ds Ͱ͋ΒΘ͞ΕΔ. ࣮ࡍ y(t) = Dαu(t) ͱ͓͘. ͜ͷͱ͖ { Dβ(Dαu(t)) + f (t, u(t)) = 0, (Dαu)(0) = (Dαu)(1) = 0 { Dβy(t) + f (t, u(t)) = 0, y(0) = y(1) = 0 ͱಉͰ͋Δ. ิॿ໋7ΑΓ,Ұҙղ y(t) =− ∫ 1 0 Gβ(t, s)f (s, u(s))ds
͕ଘࡏ͢Δ. ͢ͳΘͪ Dαu(t) + ∫ 1 0 Gβ(t, s)f (s, u(s))ds = 0 Ͱ͋Δ. ͞Βʹ,ิॿ໋7ΑΓ, Dαu(t) + ∫ 1 0 Gβ(t, s)f (s, u(s))ds = 0, u(0) = u(1) = 0 ͷҰҙղ u(t) = ∫ 1 0 Gα(t, s) (∫ 1 0 Gβ(s, r)f (r, u(r))dr ) ds ͕ଘࡏ͢Δ. X ={u ∈ C[0, 1] | u(t) ≥ 0} ͱ͓͘. ҙ ͷu, v∈ X ʹରͯ͠ u≤ v Λ u(t)≤ v(t) (t ∈ [0, 1]) ͰఆΊΔ. ͜ͷͱ͖ (X,≤) ॱংू߹Ͱ͋Δ. ·ͨ (X, d) d(u, v) = sup 0≤t≤1|u(t) − v(t)| ͰఆΊΔڑ d ͰඋڑۭؒͰ͋Δ. X ͔ ΒX ͷࣸ૾ T Λ (T u)(t) = ∫ 1 0 Gα(t, s) (∫ 1 0 Gβ(s, r)f (r, u(r))dr ) ds ͰఆΊΔ. ͜ͷͱ͖T X ͔ΒͦΕࣗͷ ࣸ૾Ͱ͋Δ. u≥ v, t ∈ [0, 1]ͱ͢Δ. ิॿ໋ 8ΑΓ (T u)(t) = ∫ 1 0 Gα(t, s) (∫ 1 0 Gβ(s, r)f (r, u(r))dr ) ds ≥ ∫ 1 0 Gα(t, s) (∫ 1 0 Gβ(s, r)f (r, v(r))dr ) ds = (T v)(t) ͕ҙͷ t∈ [0, 1] ʹରͯ͠Γཱͭ. T ୯ ௐඇݮগͰ͋Δ. u ≥ v ͱ͢Δ. ิॿ໋ 8 ΑΓ d(T u, T v) = sup 0≤t≤1|T u(t) − T v(t)| = sup 0≤t≤1 ∫ 1 0 Gα(t, s)× (∫ 1 0 Gβ(s, r)(f (r, u(r))− f(r, v(r)))dr ) ds ≤ sup 0≤t≤1 ∫ 1 0 Gα(t, s)× (∫ 1 0 Gβ(s, r)λ(u(r)− v(r))dr ) ds ≤ λd(u, v) sup 0≤t≤1 ∫ 1 0 Gα(t, s)× (∫ 1 0 Gβ(s, r)dr ) ds ≤ λd(u, v) 1 Γ(β)0sup≤t≤1 ∫ 1 0 Gβ(t, s)ds ≤ λ Γ(α)Γ(β)d(u, v) Ͱ͋Δ. ͕ͨͬͯ͠ d(T u, T v)≤ λ Γ(α)Γ(β)d(u, v) Ͱ͋Δ. ͜͜Ͱ λ Γ(α)Γ(β) < 1 Ͱ͋Δ. ·ͨ T 0≥ 0 Ͱ͋Δ. ͜ͷͱ͖, ఆཧ1 ΑΓղ͕Ұ ҙʹଘࡏ͢Δ. □ ୈ4અ ͓ΘΓʹ ఆཧ 1͕ॖখࣸ૾ʹؔ࿈͢Δ݁ՌͰ͋Δͷ ʹରͯ͠, Kannanࣸ૾ʹؔ͢Δ݁ՌಘΒΕ ͍ͯΔ. [24]Λࢀর͞Ε͍ͨ.
߂લେֶͰߦΘΕͨʮThe eighth interna-tional conference on Nonlinear Analysis and Convex Analysis (NACA2013)ʯʹ͓͍ͯ, ล͕ߨԋൃදΛߦͬͨ. λΠτϧʮKannan mappings in partially ordered sets with met-ricʯͰ͋Δ.
2014ຊֶձձ(ֶशӃେֶ, 2014
3݄15͔Β18·Ͱ)ʹ͓͍ͯ,ล͕ߨ ԋൃදΛߦͬͨ. λΠτϧʮFixed point the-orem for set-valued Kannan mappings with a vector-valued distanceʯͰ͋Δ. ͪ͜Βͷ༰ ݱࡏߘதͰ͋Δ.
ͷͰߦΘΕͨ ICM2014 ͷας ϥΠτࠃࡍձٞʮThe Fourth Asian Confer-ence on Nonlinear Analysis and Optimization (NAOAsia2014)ʯ(ᅳཱࢣൣେֶ, National Taiwan Normal University, 20148݄5͔
Β9·Ͱ)ʹ͓͍ͯ, ๛ా͕ߨԋൃදΛߦͬ ͨ. λΠτϧʮApplication of a fixed point theorem in partial ordered sets to boundary value problems for 3.5 order differential equa-tionsʯͰ͋Δ. ద༻ྫͦͷ1ͷ༰ͱ, ద༻ྫ ͦͷ2ͷඍํఔࣜͰڥք݅u(0) = u′(0) = u′′(1) = u′′′(1) = 0ͷ߹ͷڥքͷ݁Ռ Λൃදͨ͠. ڥք݅ u(0) = u′(0) = u′′(1) = u′′′(1) = 0 ͔ΓͰͳ͘, u(0) = u(1) = u′(0) = u′(1) = 0 ͷ߹ಘΒΕ͓ͯΓ, ݱ ࡏจͱͯ͠ߘதͰ͋Δ. ڥք݅ u(0) = u′(0) = u′′(0) = u′′(1) = 0 ͷ߹ʹ͍ͭͯ, [3]ͱ [14]Λࢀর͞Ε͍ͨ. 2014ຊֶձळق૯߹Պձ(ౡେ ֶ, 20149݄25͔Β9݄28)ʹ͓͍ͯɺ ล͕ߨԋൃදΛߦͬͨ. λΠτϧʮॱংू ߹ʹ͓͚Δ֦ு͞Εͨॖখࣸ૾λΠϓͷෆಈ ఆཧʹ͍ͭͯʯͰ͋Δ. ͍ΘΏΔWeissingerλ ΠϓͰ͋Δ. ࠷ޙʹ,ࠓޙͷ՝ʹ͍ͭͯड़Δ. 1ͭΊ ͷ՝, [9]ͱ[24]ͷ݁ՌͷൺֱͰ͋Δ. [9]ʹ ͓͍ͯ,࣍ͷఆཧ͕ࣔ͞Ε͍ͯΔ. ఆཧ 10. (X,≤)Λॱংू߹ͱ͢Δ. ڑd ͕ଘࡏͯ͠(X, d)උڑۭؒͱ͢Δ. Xͷ ҙͷ୯ௐඇݮগͳྻ{xn}ʹରͯ͠, ͋Δ ෦ྻ {xnk} ͕ଘࡏͯ͠, ҙͷ k ʹରͯ͠ xnk ≤ x͕Γཱͭͱ͢Δ. T Λ X͔ΒX ͷࣸ૾Ͱ୯ௐඇݮগͱ͢Δ. [0,∞)͔Β [0,∞) ͷ͋Δ୯ௐඇݮগͳؔψ Ͱ,ҙͷ t > 0 ʹରͯ͠ ∞ ∑ n=1 ψn(t) <∞ ΛΈͨ͢ͷ͕ଘࡏͯ͠, ҙͷ x, y ∈ X ʹ ରͯ͠, x≥ y ͳΒ d(T x, T y)≤ ψ(M(x, y)) ΛΈͨ͢ͱ͢Δ. ͜͜Ͱ M (x, y) = max { d(x, y),1 2(d(x, T x) + d(y, T y)), 1 2(d(x, T y) + d(y, T x)) } Ͱ͋Δ. ·ͨ, ͋Δ x0 ∈ X͕ଘࡏͯ͠x0 ≤ T x0ΛΈͨ͢ͱ͢Δ. ͜ͷͱ͖TෆಈΛ ͭ. ҙͷ x, y∈ X ʹରͯ͠, ͋Δ z∈ X ͕ ଘࡏͯ͠, x ≤ z ͔ͭ y ≤ z ΛΈͨ͢ͳΒ, ෆಈҰҙͰ͋Δ. ͜ͷఆཧʹ͓͍ͯ, ψ(t) = kt (0 < k < 1) ͱ͢Ε ψ(M (x, y)) = max { kd(x, y),k 2(d(x, T x) + d(y, T y)), k 2(d(x, T y) + d(y, T x)) } Ͱ͋Γ, t > 0ʹରͯ͠, 0 < k < 1 ΑΓ ∞ ∑ n=1 ψn(t) = ∞ ∑ n=1 knt <∞ Ͱ͋Δ. [9]ͱ[24]ͷఆཧͰ,ҰҙੑΛࣔ͢ͷ ʹඞཁͳ͕݅ҟͳ͓ͬͯΓ,྆ऀͷൺֱݕ౼ ͕ඞཁͰ͋Δ. 2ͭΊͷ՝, [24]ʹଓ͘ݚڀͰ͋Δ. [12] ʹ͓͍ͯ,࣍ͷఆཧ͕ࣔ͞Ε͍ͯΔ. ఆཧ 11. X Λඋڑۭؒͱ͢Δ. T Λ X ͔ΒͦΕࣗͷࣸ૾Ͱ, ͋Δ α ∈ [ 0,1 2 ) ͕ଘࡏͯ͠, ҙͷ x, y ∈ X ʹରͯ͠ φ(α)d(x, T x)≤ d(x, y) ͳΒ d(T x, T y)≤ αd(x, T x) + αd(y, T y) ΛΈͨ͢ͱ͢Δ. ͜͜Ͱ φ(α) = { 1 (0≤ α <√2− 1) 1− α (√2− 1 ≤ α < 12) Ͱ͋Δ. ͜ͷͱ͖, T ͨͩͻͱͭͷෆಈΛ ͭ.
ॱংΛԾఆͨ͠ڑۭؒʹ͓͚Δ͜ͷఆཧ ͷࣸ૾ʹରͯ͠,ΓෆಈఆཧΛಘΒΕΔ ͷ͔Ͳ͏͔͞Εͨ՝Ͱ͋Δ. ͠ಘΒΕ Ε, [24]ͷఆཧͷ֦ுʹͳΔͰ͋Ζ͏. 3ͭΊͷ՝, ద༻ྫͦͷ2ͷඍํఔ ࣜͰڥք݅(IV) u′′(0) = u′′(1) = u′′′(0) = u′′′(1) = 0, (V) u(0) = u′(0) = 0, u′′(1) = 0, u′′′(1) = Cu(1) (͜͜Ͱ C ఆ), (VI) u′′(0) = C1u′(0), u′′′(0) = C2u(0), u′′(1) = C3u′(1), u′′′(1) = C4u(1) (C1, C2, C3, C4 ఆ) ͷ߹Λࣔ͢͜ͱͰ͋Δ. ݱࡏ, (I)
u(0) = u(1) = u′(0) = u′(1) = 0, (II)
u(0) = u′(0) = u′′(1) = u′′′(1) = 0 ͓Αͼ
(III) u(0) = u(1) = u′′(0) = u′′(1) = 0 ͷ߹
ղ໌͞Ε͍ͯΔ. [6]ͷ21՝[20], [13] ࢀ র͍ͨ͠. 4ͭΊͷ՝,ಛҟੑΛͭ߹ͷॳظ ͷղͷଘࡏͱҰҙੑΛࣔ͢͜ͱͰ͋Δ. [5] ʹ͓͍ͯ,ಛҟੑΛͭඍํఔࣜu′′ = f (t, u) ͷڥք͕ߟ͞Ε͍ͯΔ. Ұํ, ஶऀ ͷ͏ͪ๛ా, හ࣏ࢯ (ۄେֶֶ෦) ͱڞʹಛҟੑΛͭ2֊ඍํఔࣜͷॳظ Λߟͨ͠([10]). ͜ͷจͰѻ͍ͬͯ Δॳظ݅ u(0) = 0, u′(0) = λ (λ > 0) ͷ߹Ͳ͏ͳΔͰ͋Ζ͏͔. ࣸ૾ Au(t) = λt +∫0t(t− s)f(s, u(s))dsͰߟ͑ΕΑ͍ͩΖ ͏͕, ৄ͍͠ݚڀࠓޙͷ՝Ͱ͋Δ. [15] ࢀর͍ͨ͠. ࢀߟจݙ
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20153݄13ݪߘडɹ
