有界連続写像全体からなる空間における不動点定理とそのニューロンモデルに関係する微分方程式への適用例

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༗ք࿈ଓࣸ૾શମ͔ΒͳΔۭؒʹ͓͚Δෆಈ఺ఆཧͱͦͷ

χϡʔϩϯϞσϧʹؔ܎͢Δඍ෼ํఔࣜ΁ͷద༻ྫ

Fixed point theorem in the space of all bounded continuous mappings and its applied example to

differential equations related with a neuron model

ɹ઒࡚හ࣏‡_ɹࠤʑ໦׮∗_{ɹ๛ాণ࢙}‡ Toshiharu KawasakiɹHiroshi SasakiɹMasashi Toyoda

‡_{ۄ઒େֶ޻ֶ෦ϚωδϝϯταΠΤϯεֶՊ,}∗_{ۄ઒େֶ޻ֶ෦ιϑτ΢ΤΞαΠΤϯεֶՊ,}

194–8610౦ژ౎ொాࢢۄ઒ֶԂ 6–1–1 College of Engineering, Tamagawa University, 6–1–1 Tamagawa-gakuen, Machida-shi, Tokyo 194–8610

Abstract

In this paper, we show a fixed point theorem in the space of all bounded continuous mappings. Moreover we consider the existence of a unique solution for fractional differential equations with multiple delays. Using the existence theorem, we discuss a fractional chaos neuron model.

Keywords: Fixed point theorem, fractional differential equation, neuron model.

ୈ1અ ͸͡Ίʹ ੜ෺ͷਆܦܥʹΑΔ৘ใॲཧͷ͘͠ΈΛ޻ ֶతʹԠ༻͢Δ͜ͱΛ໨తͱͯ͠ɼ·ͨਆܦܥ ͷ৘ใॲཧϝΧχζϜͷղ໌Λ໨తͱͯ͠ɼਆ ܦܥͷجຊ୯Ґͱߟ͑ΒΕ͍ͯΔχϡʔϩϯͷ Ϟσϧ͕ఏҊ͞Ε͍ͯΔɽͦͷҰͭʹ[9]ʹΑͬ ͯఏҊ͞Ε͍ͯΔपظܕϑϥΫγϣφϧΧΦε χϡʔϩϯϞσϧ

Dαu(t) =−βu(t) + sin

( πu(t− τ) 2T0 ) ͕͋Δ. ͜͜Ͱ Dα _{͸ඇ੔਺֊ͷඍ෼ԋࢉࢠ}_, u(t) ͸࣌ࠁ t Ͱͷ಺෦ঢ়ଶ, β ͸ࢄҳఆ਺, τ ͸஗Ԇ࣌ؒ, T0 ͸׆ੑԽؔ਺Ͱ͋Δਖ਼ݭؔ਺ͷ पظύϥϝʔλͰ͋Δ. [5] ΋ࢀর͞Ε͍ͨ. ຊ࿦จͰ͸, [9]ʹΑͬͯఏҊ͞Εͨपظܕ ϑϥΫγϣφϧΧΦεχϡʔϩϯϞσϧͷղͷ ଘࡏͱҰҙੑΛѻ͏. [9]ʹ͓͍ͯ,ඇ੔਺֊ඍ ෼ԋࢉࢠDα _͸_Gr¨_{unwald-Letnikv}_ͷఆٛʹΑ Δ΋ͷ͕༻͍ΒΕ͍ͯΔ. ຊ࿦จͰ͸, Caputo ʹΑΔඍ෼ͷఆٛΛ༻͍Δ. ͢ͳΘͪ,஗Εͭ ͖ͷඇ੔਺֊ඍ෼ํఔࣜʹؔ͢Δ໰୊       

c_Dα_{u(t) =}_{−βu(t) + sin}πu(t− τ) 2T0 (t_{∈ [0, T ]),} u(t) = ϕ(t) (t∈ [−τ0, 0]), ͷղͷଘࡏͱҰҙੑΛ, ୈ3અ Ͱࣔ͢. ͜͜ Ͱ, α ∈ (0, 1], β, τ ∈ [0, ∞), T0, τ0 ∈ (0, ∞), ϕ_{∈ C((−∞, 0], R)}Ͱ͋Δ. ͦͷͨΊ,ɹୈ2અ Ͱ,͋Δ༗ք࿈ଓࣸ૾શମ͔ΒͳΔෆಈ఺ఆཧ Λ঺հ͢Δ. ୈ4અͰ͸, [8]͓Αͼ[3]ͷෆಈ ఺ఆཧΛܥͱ͢ΔΑ͏ʹ,ୈ2અͷఆཧΛ֦ு ͢Δ. ୈ2અ ෆಈ఺ఆཧ I Λ۠ؒͱ͢Δ. ༗ݶ۠ؒͰ΋ແݶ۠ؒͰ ΋Α͍. BC(I, R)͸,ͦͷཁૉu ͷϊϧϜΛ ∥u∥ = sup t∈I |u(t)| ͰఆΊͨI ্ͷ༗ք࿈ଓࣸ૾શମ͔ΒͳΔ Ba-nachۭؒͱ͢Δ. J ΛI _{⊂ J} ΛΈͨ۠ؒ͢ͱ

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͢Δ. F Λ BC(I, R) ͷดू߹ͱ͢Δ. ϕ Λ J\ I ͔ΒE ΁ͷࣸ૾ͱ͢Δ. u∈ F ʹରͯ͠ uϕ Λ uϕ= { u (I ্), ϕ (J_{\ I} ্) ͰఆΊΔ. ఆཧ 1. I = [a, b] ͱ͠, J Λ I _{⊂ J} ΛΈͨ͢ ۠ؒͱ͢Δ. F ΛBC(I, R)ͷۭͰͳ͍ดू߹ ͱ͢Δ. ͋Δ J\ I ͔Β R ΁ͷࣸ૾ϕ ͕ଘࡏ ͯ͠, ೚ҙͷ u_{∈ F} ʹରͯ͠ uϕ ∈ BC(J, R) ΛΈͨ͢ͱ͢Δ. A Λ F ͔ΒͦΕࣗ਎΁ͷ ࣸ૾ͱ͠, ͋Δୈ2 ม਺ʹؔͯ͠ੵ෼Մೳͳ I_{× I} ͔Β [0,_∞) ΁ͷؔ਺ G ͕ଘࡏͯ͠, ͋ Δη1, η2 ∈ C(I, J) ͕ଘࡏͯ͠, (H1), (H2)Λ Έͨ͢ͱ͢Δ. (H1) ೚ҙͷ u, v∈ F , t ∈ I ʹରͯ͠ |Au(t) − Av(t)| ≤ ∫ t a G(t, s) (|uϕ(η1(s))− vϕ(η1(s))| +|uϕ(η2(s))− vϕ(η2(s))|) ds ͕੒Γཱͭ. (H2)͋Δ α∈ [ 0,1 2 ) ͕ଘࡏ͠, ͓Αͼ೚ҙͷ t∈ I ʹରͯ͠ ∫ t a G(t, s)y(s)ds≤ αy(t) ΛΈͨ͢Α͏ͳ͋Δy_{∈ BC(J, (0, ∞))}͕ଘࡏ ͯ͠, ೚ҙͷ t_{∈ I} ʹରͯ͠

y(η1(t))≤ y(t), y(η2(t))≤ y(t)

͕੒Γཱͭ. ͜ͷͱ͖A ͸ͨͩͻͱͭͷෆಈ఺Λ΋ͭ. ূ໌. ৚݅ (H2) ΛΈͨ͢ y ∈ BC(J, (0, ∞)) ʹରͯ͠, ͋Δ m, M > 0 ͕ଘࡏͯ͠, ೚ҙ ͷ t ∈ J ʹରͯ͠ m ≤ y(t) ≤ M Ͱ͋Δ. BC(I, R)ͷϊϧϜ∥·∥y Λ ∥u∥y = sup t∈I { 1 y(t)|u(t)| } ͰఆΊΔ. ͜ͷͱ͖ 1 M∥u∥ ≤ ∥u∥y ≤ 1 m∥u∥ Ͱ͋Δ͔Β, _∥·∥y ͸ϊϧϜ ∥·∥ ͱಉ஋Ͱ͋Δ. F ͷڑ཭ dΛ d(u, v) = sup t∈J { 1 y(t)|uϕ(t)− vϕ(t)| } ͰఆΊΔ. d(u, v) = ∥u − v∥y ͓Αͼ ∥ · ∥y ͸ ׬උڑ཭ۭؒΛಋ͘ϊϧϜ ∥ · ∥ͱಉ஋Ͱ͋Δ ͔Β, (F, d)͸׬උڑ཭ۭؒͰ͋Δ. (H1)ΑΓ, ೚ҙͷu, v∈ F , t ∈ I ʹରͯ͠ 1

≤ 2d(u, v)_y(t) αy(t)

= 2αd(u, v)

ΛಘΔ. Αͬͯ

d(Au, Av)_{≤ 2αd(u, v)}

Ͱ͋Δ. ͢ͳΘͪ, A ͸ॖখࣸ૾Ͱ͋Δ. ॖখ ࣸ૾ͷෆಈ఺ఆཧ͔Β, A͸ෆಈ఺Λͨͩͻͱ ͭ΋ͭ. □ ୈ3અ पظܕϑϥΫγϣφϧΧΦεχϡʔϩ ϯϞσϧ΁ͷద༻ α Λਖ਼ͷ࣮਺ͱ͢Δ. ؔ਺ u ͷ α ֊ Ca-putoඍ෼(·ͨ͸Caputo-Rieszඍ෼) cDα Λ c_Dα_u(t) = 1 Γ(n_{− α)} ∫ t 0 1 (t_{− s)}α−n+1 · dn dsnu(s)ds

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ͰఆΊΔ. ͜͜ͰΓ͸Gammaؔ਺Ͱ͋Δ. ͢ ͳΘͪα > 0ʹରͯ͠ Γ(α) = ∫ _∞ 0 tα−1e−tdt Ͱ͋Δ. ·ͨ, n = [α] + 1 Ͱ͋Γ, [α]͸ α Λ ௒͑ͳ͍࠷େͷࣗવ਺Ͱ͋Δ. ͢ͳΘͪ, n͸ n− 1 ≤ α < n ΛΈͨࣗ͢વ਺Ͱ͋Δ. ඇ੔਺ ֊ඍ෼ʹؔͯ͠,ৄ͘͠͸[2, 4] ΛݟΒΕ͍ͨ. ఆཧ 2. α _{∈ (0, 1], τ ∈ C([0, T ], [0, ∞)),} ϕ ∈ C((−∞, 0], R) ͱ͢Δ. f ∈ C([0, T ] × R_{× R, R)} ͱ͠, ࣍ͷ (Hf) ΛΈͨ͢ͱ͢Δ. (Hf) ͋Δ L0, L1 ∈ [0, ∞) ͕ଘࡏͯ͠, ೚ҙͷ t∈ [0, T ], x0, x1, y0, y1∈ R ʹରͯ͠ |f(t, x0, x1)− f(t, y0, y1)| ≤ L0|x0− y0| + L1|x1− y1| ͕੒Γཱͭ. ͜ͷͱ͖,஗Ε͖ͭͷඇ੔਺֊ඍ෼ํఔࣜʹؔ ͢Δ໰୊   

c_Dα_{u(t) = f (t, u(t), u(t}_{− τ(t)))}

(t_{∈ [0, T ]),} u(t) = ϕ(t) (t_{∈ [−τ}0, 0]), ͸ͨͩͻͱͭͷղΛ΋ͭ. ͜͜Ͱ −τ0 = min_{{t − τ(t) | t ∈ I}} Ͱ͋Δ. ূ໌. I = [0, T ], J = [_−τ0, T ]ͱ͢Δ. ͢ͳΘ ͪJ\ I = [−τ0, 0]Ͱ͋Δ. ·ͨF ={u|I | u ∈ C(J, R), u(t) = ϕ(t) (t_{∈ J \ I)}}ͱ͓͘. ͜ͷ ͱ͖, ೚ҙͷ u _{∈ F} ʹରͯ͠ uϕ ∈ BC(J, R) ΛΈͨ͢. ͜ΕΑΓ u(t) = ϕ(0) + 1 Γ(α) ∫ t 0 (t− s)α−1f (s, u(s), uϕ(s− τ(s))ds ΛΈͨ͢ u _{∈ C(I, R)} ͕ͨͩͻͱͭଘࡏ͢Δ ͜ͱΛ͍͏. F ্ͷࣸ૾ A Λ Au(t) = ϕ(0) + 1 Γ(α) ∫ t 0 (t_{− s)}α−1f (s, u(s), uϕ(s− τ(s))ds ͰఆΊΔ. ͜ͷͱ͖, ೚ҙͷ u _{∈ F} ʹରͯ͠ Au∈ F Ͱ͋Δ. ࣮ࡍ, Au(t) ΛI ʹ੍ݶͨ͠ ؔ਺͸࿈ଓؔ਺Ͱ͋Γ,·ͨ, Au(0) = ϕ(0)Ͱ ͋Δ͔Β J_{\ I = [−τ}0, 0] ্ͷؔ਺ ϕͱt = 0 Ͱͷ஋͸Ұக͢Δ. ·ͨ |Au(t) − Av(t)| ≤ 1 Γ(α) ∫ t 0 (t_{− s)}α−1(L0|uϕ(s)− vϕ(s)| +L1|uϕ(s− τ(s)) − vϕ(s− τ(s))|) ds ≤ L Γ(α) ∫ t 0 (t− s)α−1(|uϕ(s)− vϕ(s)| +_|uϕ(s− τ(s)) − vϕ(s− τ(s))|) ds Ͱ͋Δ. ͜͜Ͱ L = max_{L0, L1} Ͱ͋Δ. ؔ ਺GΛ G(t, s) =    L Γ(α)(t− s) α−1 ₍₀_{≤ s < t),} 0 (t≤ s) ͱ͓͘. η1(s) = s, η2(s) = s− τ(s) ͱ͢Δͱ ͖,ఆཧ1 ͷ(H1)͕੒Γཱͭ. ·ͨ, αͱͯ͠ 0 < 2α0 < 1ΛΈͨ͢Α͏ͳ α0 ΛͱΓ, c ͱ ͯ͠ cα _≥ _αL 0 ΛͱΔ. ͞Βʹ y(t) = e ct _ͱ͢ Δ. ͜ͷͱ͖ ∫ t 0 G(t, s)y(s)ds = L Γ(α) ∫ t 0 (t_{− s)}α−1ecsds = Le ct cα_Γ(α) ∫ ct 0 sα−1e−sds ≤ _cL_αect ≤ α0y(t) ΑΓ, ৚݅(H2) ΛΈͨ͢. ͜͜Ͱ 2 ͭΊͷ౳ ߸͸ t− s = 1 cz Ͱஔ׵ੵ෼ͨ͠. ͢ͳΘͪ ds = −1 cdz Ͱ͋Γ s ͕ 0 → t ͷͱ͖ z ͸ ct_{→ 0} Ͱ͋Δ͔Β L Γ(α) ∫ t 0 (t_{− s)}α−1ecsds = L Γ(α) ∫ 0 ct ( 1 cz )α−1 ect−z ( −1 c ) dz = Le ct cα_Γ(α) ∫ ct 0 zα−1e−zdz Ͱ͋Δ. ఆཧ1 ͔Β A ͸ͨͩͻͱͭͷෆಈ఺ Λ΋ͭ. □

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ఆཧ 2 ΑΓ, ΧΦεχϡʔϩϯϑϥΫγ ϣφϧϞσϧ [9] ͷղͷଘࡏͱҰҙੑΛࣔ͢. α _{∈ (0, 1], β, τ ∈ [0, ∞), T}0 ∈ (0, ∞) ͓Αͼ ϕ_{∈ C((−∞, 0], R)} ͱ͢Δ. ஗Ε͖ͭͷඇ੔਺ ֊ඍ෼ํఔࣜͷ໰୊       

c_Dα_{u(t) =}_{−βu(t) + sin}πu(t− τ) 2T0 (t∈ [0, T ]), u(t) = ϕ(t) (t_{∈ (−τ}0, 0]), Λߟ͑Δ. ͜͜Ͱ −τ0 = min{t − τ(t) | t ∈ I} Ͱ͋Δ. f (t, x0, x1) =−βx0+ sinπx_2T1₀ ͱ͓͘ͱ |f(t, x0, x1)− f(t, y0, y1)| ≤ |β||x0− y0| + � � � �sin πx1 2T0 − sin πy1 2T0 � � � � ≤ |β||x0− y0| + π 2T0|x1− y1|, Ͱ͋Δ͔Β, f ͸ (Hf) Λ L0 = |β| ͓Αͼ L1= _2Tπ₀ ʹରͯ͠Έͨ͢. ͕ͨͬͯ͠,ఆཧ 2 ΑΓͨͩͻͱͭͷղΛ΋ͭ. ୈ4અ ෆಈ఺ఆཧͷ֦ு ఆཧ 1ͷΑ͏ʹ, ੵ෼ෆ౳ࣜΛ৚݅ࣜͱ͠ ؚͯΉෆಈ఺ఆཧʹ[8]΍[3]͕͋Δ. ຊઅͰ͸, ͜ΕΒΛܥͱ͢ΔΑ͏ʹ,ఆཧ 1Λ֦ு͢Δ. I Λ۠ؒͱ͢Δ. ༗ݶ۠ؒͰ΋ແݶ۠ؒͰ ΋Α͍. E Λ Banach ۭؒͱ͢Δ. BC(I, E) ͸,ͦͷཁૉ u ͷϊϧϜΛ ∥u∥ = sup t∈I ∥u(t)∥E ͰఆΊͨI ্ͷ༗ք࿈ଓࣸ૾શମ͔ΒͳΔ Ba-nachۭؒͱ͢Δ. J ΛI ⊂ J ΛΈͨ۠ؒ͢ͱ ͢Δ. F Λ BC(I, E) ͷดू߹ͱ͢Δ. ϕ Λ J_{\ I} ͔Β E ΁ͷࣸ૾ͱ͠u_{∈ F} ʹରͯ͠uϕ Λ uϕ= { u (I ্), ϕ (J\ I ্) ͰఆΊΔ. ࣍ΛಘΔ. ఆཧ 3. I Λ۠ؒͱ͢Δ. J0, J ΛI ⊂ J0 ⊂ J ΛΈͨ۠ؒ͢ͱ͢Δ. E Λ Banach ۭؒͱ͢ Δ. F Λ BC(I, E) ͷۭͰͳ͍ดू߹ͱ͢Δ. J_{\ I} ͔Β E ΁ͷ͋Δؔ਺ ϕ ͕ଘࡏͯ͠, ೚ ҙͷ u _{∈ F} ʹରͯ͠ uϕ ∈ BC(J, E) ΛΈͨ ͢ͱ͢Δ. A ΛF ͔ΒͦΕࣗ਎΁ͷࣸ૾ͱ͢ Δ. ͋Δɹβ _{∈ [0, 1)} ͕ଘࡏ͠, ͋Δ I _{× J}0 ͔Β [0,_∞) ΁ͷୈ2ม਺ʹؔͯ͠ੵ෼Մೳͳ ͋Δؔ਺ G ͕ଘࡏ͠, ͋Δ I ͔Β J0 ΁ͷؔ ਺ γ, δ Ͱ γ _{≤ δ} ΛΈͨ͢΋ͷ͕ଘࡏ͠, ͋ Δη1, η2, . . . , ηn∈ C(J0, J) ͕ଘࡏͯ͠, (H1), (H2)ΛΈͨ͢ͱ͢Δ. (H1) ೚ҙͷ u, v∈ F , t ∈ I ʹରͯ͠

∥Au(t) − Av(t)∥E ≤ β∥u(t) − v(t)∥E

+ ∫ δ(t) γ(t) G(t, s) n ∑ i=1 ∥uϕ(ηi(s))− vϕ(ηi(s))∥Eds ΛΈͨ͢. (H2) ͋Δ β + nKα ∈ [0, 1) ΛΈͨ͢ α ∈ [0,∞), K ∈ [0, ∞) ͕ଘࡏ͠, ͞Βʹ, ೚ҙͷ t_{∈ I} ʹରͯ͠ ∫ δ(t) γ(t) G(t, s)y(s)ds_{≤ αy(t)}Λ Έͨ͢ y_{∈ BC(J, (0, ∞))} ͕ଘࡏͯ͠, ೚ҙͷ t_{∈ J}0 ʹରͯ͠ y(ηi(t))≤ Ky(t) (i = 1, 2, . . . , n) ΛΈͨ͢. ͜ͷͱ͖, A ͸ͨͩͻͱͭͷෆಈ఺Λ΋ͭ. ূ໌. ৚݅ (H2) ͷ y ∈ BC(J, (0, ∞)) ʹର ͯ͠, ͋Δ m, M ∈ (0, ∞) ͕ଘࡏͯ͠, ೚ҙ ͷ t ∈ J ʹରͯ͠ m ≤ y(t) ≤ M ΛΈͨ͢. BC(I, E)ͷϊϧϜ∥·∥yΛ ∥u∥y = sup t∈I { 1 y(t)∥u(t)∥E } ͰఆΊΔ. ͜ͷͱ͖ 1 M∥u∥ ≤ ∥u∥y ≤ 1 m∥u∥ Ͱ͋Δ͔Β,_∥·∥y ͸∥·∥ͱಉ஋Ͱ͋Δ. F ͷڑ ཭dΛ d(u, v) = sup t∈J { 1 y(t)∥uϕ(t)− vϕ(t)∥E } ͰఆΊΔ. ͜ͷͱ͖ d(u, v) =∥u − v∥y ͓Αͼ ∥ · ∥y ͸׬උڑ཭ۭؒΛಋ͘ϊϧϜ∥ · ∥ͱಉ஋ Ͱ͋Δ͔Β, (F, d) ͸׬උڑ཭ۭؒͰ͋Δ. · ͨ৚݅(H1)ΑΓ,೚ҙͷ u, v∈ F , t ∈ I ʹର ͯ͠ 1

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≤ β y(t)∥u(t) − v(t)∥E + 1 y(t) ∫ δ(t) γ(t) G(t, s)_× n ∑ i=1 ∥uϕ(ηi(s))− vϕ(ηi(s))∥Eds ≤ βd(u, v) + 1 y(t) ∫ δ(t) γ(t) G(t, s)× n ∑ i=1 d(u, v)y(ηi(s))ds ≤ βd(u, v) + d(u, v) y(t) ∫ δ(t) γ(t) G(t, s) n ∑ i=1 Ky(s)ds = βd(u, v) + nKd(u, v) y(t) ∫ δ(t) γ(t) G(t, s)y(s)ds ≤ (β + nKα)d(u, v) ͕೚ҙͷu, v∈ F , t ∈ I ʹରͯ͠੒Γཱͭ. ͠ ͕ͨͬͯ

d(Au, Av)≤ (β + nKα)d(u, v)

ΛಘΔ. ॖখࣸ૾ͷෆಈ఺ఆཧΑΓ, A ͸ͨͩ ͻͱͭͷෆಈ఺Λ΋ͭ. □ ఆཧ 3ΑΓ,࣍ͷෆಈ఺ఆཧ[8]ΛಘΔ. ܥ 4 (Louͷෆಈ఺ఆཧ). I = [0, T ] ͱ͢Δ. E ΛBanachۭؒͱ͢Δ. F ΛI ͔Β E ΁ͷ ࿈ଓࣸ૾શମ͔ΒͳΔBanachۭؒC(I, E)ͷ ۭͰͳ͍ดू߹ͱ͢Δ. AΛ F ͔ΒͦΕࣗ਎ ΁ͷࣸ૾ͱ͢Δ. ͋Δα, β∈ [0, 1), K ∈ [0, ∞) ͕ଘࡏͯ͠, ೚ҙͷ u, v _{∈ F , t ∈ (0, T ]} ʹର ͯ͠ ∥Au(t) − Av(t)∥E ≤ β∥u(t) − v(t)∥E+ K tα ∫ t 0 ∥u(s) − v(s)∥E ds ΛΈͨ͢. ͜ͷͱ͖, A ͸ͨͩͻͱͭͷෆಈ఺ Λ΋ͭ. ূ໌. J0 = J = [0, T ] ͱ͓͘. ͍·, ͋Δ α, β _{∈ [0, 1), K ∈ [0, ∞)} ͕ଘࡏͯ͠, ೚ҙ ͷu, v∈ F , t ∈ (0, T ]ʹରͯ͠ ∥Au(t) − Av(t)∥E ≤ β∥u(t) − v(t)∥E+ K tα ∫ t 0 ∥u(s) − v(s)∥E ds ΛΈͨ͢. l’Hopital ͷఆཧΑΓ,೚ҙͷ u, v_∈ F ʹରͯ͠ ∥Au(0) − Av(0)∥E ≤ β∥u(0) − v(0)∥E+ lim t→+0 K tα ∫ t 0 ∥u(s) − v(s)∥E ds = β_{∥u(0) − v(0)∥}E+ lim t→+0 K αtα−1∥u(t) − v(t)∥E = β_{∥u(0) − v(0)∥}E ΛΈͨ͢. ؔ਺ GΛ G(t, s) = { K tα (0 < t≤ T ), 0 (t = 0) ͱ͓͘. ·ͨγ(t) = 0, δ(t) = t, n = 1 ͓Αͼ η1(t) = t ͱ͢Δ. ͜ͷͱ͖, ೚ҙͷ u, v ∈ F , t_{∈ I} ʹରͯ͠ ∥Au(t) − Av(t)∥E ≤            β_{∥u(t) − v(t)∥}E +K tα ∫ t 0 ∥u(s) − v(s)∥E ds (0 < t_{≤ T ),} β_{∥u(0) − v(0)∥}E (t = 0) = β∥u(t) − v(t)∥E + ∫ δ(t) γ(t) G(t, s) n ∑ i=1 ∥u(ηi(s))− v(ηi(s))∥Eds ΛΈͨ͢. ͢ͳΘͪ(H1)ΛΈͨ͢. τ ∈ (0, ∞) ΛKτ1−α _{< 1}_{− β} _{ΛΈͨ͢΋ͷͱ͢Δ}_{. α} 0 = Kτ1−α Λ৚݅(H2) ͷ α ͱ͠K0 = 1Λ৚݅ (H2) ͷK ͱ͢Δ. ·ͨ y(t) = { 1 (0_{≤ t ≤ τ),} eτt−1 (τ ≤ t ≤ T ) ͱ͢Δ. ͜ͷͱ͖0≤ t ≤ τ ͳΒ͹ ∫ δ(t) γ(t) G(t, s)y(s)ds = ∫ t 0 K tαds = K tα ∫ t 0 ds = Kt1−α ≤ α0y(t)

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Ͱ͋Δ. τ _{≤ t ≤ T} ͷͱ͖ ∫ δ(t) γ(t) G(t, s)y(s)ds = ∫ τ 0 K tαds + ∫ t τ K tαe s τ−1ds = K tατ + K tατ ( eτt − 1 ) = Kτ tα e t τ−1 ≤ Kτ1−αeτt−1 = α0y(t) Ͱ͋Δ. Αͬͯ,৚݅(H2) ΛΈͨ͢. ͕ͨͬ͠ ͯ, ఆཧ 3 ΑΓ A ͸ͨͩͻͱͭͷෆಈ఺Λ΋ ͭ. □ ఆཧ 3ΑΓ,࣍ͷෆಈ఺ఆཧ[3]ΛಘΔ.

ܥ5 (de Pascale-de Pascaleͷෆಈ఺ఆཧ).

I = [1,∞) ͱ͢Δ. E Λ Banach ۭؒͱ͢ Δ. F Λ BC(I, E) ͷۭͰͳ͍ดू߹ͱ͢Δ. A Λ F ͔ΒͦΕࣗ਎΁ͷࣸ૾ͱ͢Δ. ͋Δ α∈ (1, ∞), β ∈ [0, 1)͓Αͼ K∈ [0, ∞) ͕ଘ ࡏͯ͠, ೚ҙͷ u, v_{∈ F , t ∈ I} ʹରͯ͠ ∥Au(t) − Av(t)∥E ≤ β∥u(t) − v(t)∥E+ K tα ∫ t 1 ∥u(s) − v(s)∥ Eds ΛΈͨ͢. ͜ͷͱ͖, A ͸ͨͩͻͱͭͷෆಈ఺ Λ΋ͭ. ূ໌. I = J0= J = [1,∞)ͱ͓͘. ͍·,͋Δ α∈ (1, ∞), β ∈ [0, 1)͓Αͼ K ∈ [0, ∞) ͕ଘ ࡏͯ͠,೚ҙͷ u, v_{∈ F , t ∈ I} ʹରͯ͠ ∥Au(t) − Av(t)∥E ≤ β∥u(t) − v(t)∥E+K tα ∫ t 1 ∥u(s) − v(s)∥ Eds ΛΈͨ͢. ؔ਺ G Λ G(t, s) = K tα ͱ͓͘. γ(t) = 1, δ(t) = t, n = 1 ͓Αͼ η1(t) = t ͱ͓͘. ͜ͷͱ͖, ೚ҙͷ u, v ∈ F , t_{∈ I} ʹରͯ͠ K tα ∫ t 1 ∥u(s) − v(s)∥E ds = ∫ δ(t) γ(t) G(t, s) n ∑ i=1 ∥u(ηi(s))− v(ηi(s))∥Eds Ͱ͋Δ͔Β ∥Au(t) − Av(t)∥E ≤ β∥u(t) − v(t)∥E+ K tα ∫ t 1 ∥u(s) − v(s)∥ Eds = β∥u(t) − v(t)∥E+ ∫ δ(t) γ(t) G(t, s)× n ∑ i=1 ∥u(ηi(s))− v(ηi(s))∥Eds ͕੒Γཱͭ. ͢ͳΘͪ, ৚݅ (H1) ΛΈͨ͢. c∈ (0, ∞) ͓Αͼ τ ∈ (1, ∞) Λ K(c−1+ τ1−α) < 1_{− β} ΛΈͨ͢΋ͷͱ͢Δ. ৚݅ (H2) ͷα Λ α0 = K(c−1+ τ1−α) Ͱ,৚݅ (H2)ͷ K Λ K0= 1 ͱ͢Δ. ·ͨ y(t) = { ect ₍₁_{≤ t ≤ τ),} ecτ (τ ≤ t) ͱ͢Δ. ͜ͷͱ͖1_{≤ t ≤ τ} ͳΒ͹ ∫ δ(t) γ(t) G(t, s)y(s)ds = ∫ t 1 K tαe cs_ds = K ctα(e ct_{− e}c₎ ≤ Kc−1ect ≤ Kc−1ect+ Kτ1−αect = K(c−1+ τ1−α)ect = α0y(t)

(7)

Ͱ͋Δ. ·ͨ 1_{≤ τ ≤ t} ͷͱ͖ ∫ δ(t) γ(t) G(t, s)y(s)ds = ∫ τ 1 K tαe cs_{ds +}∫ t τ K tαe cτ_ds = K ctα(e cτ_{− e}c_{) +}Kecτ tα (t− τ) ≤ _ctK_αecτ +Ke cτ tα t = K ctαe cτ _{+ Ke}cτ_t1−α ≤ K_c ecτ + Kecττ1−α = K(c−1+ τ1−α)ecτ = α0y(t) Ͱ͋Δ. Αͬͯ,৚݅(H2) ΛΈͨ͢. ͕ͨͬ͠ ͯ, ఆཧ 3ΑΓ A ͸ͨͩͻͱͭͷෆಈ఺Λ΋ ͭ. □ ఆཧ 3Λ࢖͏ͱ, ఆཧ2͸࣍ͷΑ͏ʹ֦ு Ͱ͖Δ. ఆཧ 6. α_{∈ (0, 1]} ͱ͢Δ. ϕ_{∈ C((−∞, 0], E)} ͱ͢Δ. τ0, τ1, τ2, . . . , τm ∈ C([0, T ], [0, ∞)) ͱ͢Δ. E Λ Banach ۭؒͱ͢Δ. f ∈ C([0, T ]_{× E}m+1, E) ͕, ೚ҙͷ t _{∈ [0, T ],} x0, x1, x2, . . . , xm, y0, y1, y2, . . . , ym ∈ E ʹର ͯ͠, (Hf) ΛΈͨ͢ͱ͢Δ. (Hf) ͋Δ L0, L1, L2, . . . , Lm ∈ [0, ∞) ͕ଘࡏ ͯ͠ ∥f(t, x0, x1, . . . , xm)− f(t, y0, y1, . . . , ym)∥E ≤ m ∑ i=0 Li∥xi− yi∥E ΛΈͨ͢. ͜ͷͱ͖,஗Ε͖ͭͷඇ੔਺֊ඍ෼ํఔࣜʹؔ ͢Δ໰୊        c_Dα_u(t)

= f (t, u(t), u(t_{− τ}1(t)), . . . , u(t− τm(t))) (t∈ [0, T ]), u(t) = ϕ(t) (t_{∈ [−τ}0, 0]) ͸ͨͩͻͱͭͷղΛ΋ͭ. ͜͜Ͱ −τ0= min{t − τi(t)| t ∈ I, i = 1, . . . , m} Ͱ͋Δ. ূ໌. I = J0 = [0, T ], J = [−τ0, T ] ͓Αͼ

F = {u|I | u ∈ C(J, E), u(t) = ϕ(t) (t ∈

J_{\ I)}} ͱ͓͘. ͜ͷͱ͖,೚ҙͷ u_{∈ F} ʹର ͯ͠ uϕ∈ BC(J, E) ΛΈͨ͢. ੵ෼ํఔࣜ u(t) = ϕ(0) + 1 Γ(α) ∫ t 0 (t_{− s)}α−1_× f (s, u(s), uϕ(s− τ1(s)), . . . , uϕ(s− τm(s)))ds ΛΈͨ͢u_{∈ C(I, E)}͕ͨͩͻͱͭଘࡏ͢Δ͜ ͱΛࣔ͢. F ্ͷࣸ૾A Λ Au(t) = ϕ(0) + 1 Γ(α) ∫ t 0 (t− s)α−1× f (s, u(s), uϕ(s− τ1(s)), . . . , uϕ(s− τm(s)))ds ͰఆΊΔ. ͜ͷͱ͖, ೚ҙͷ u _{∈ F} ʹରͯ͠ Au∈ F Ͱ͋Δ. ͞Βʹ ∥Au(t) − Av(t)∥E ≤ 1 Γ(α) ∫ t 0 (t− s)α−1× m ∑ i=0 Li∥uϕ(s− τi(s))− vϕ(s− τi(s))∥E ≤ L Γ(α) ∫ t 0 (t_{− s)}α−1_× m ∑ i=0 ∥uϕ(ηi(s))− vϕ(ηi(s))∥Eds Ͱ ͋ Δ. ͜ ͜ Ͱ τ0(t) = 0, L = max{L0, L1, . . . , Lm} ͓Αͼ ηi(t) = t − τi(t) (i = 0, . . . , m) Ͱ͋Δ. β = 0ͱ͓͖ G(t, s) =    L Γ(α)(t− s) α−1 ₍₀_{≤ s < t),} 0 (t≤ s) γ(t) = 0, δ(t) = t ͓Αͼ n = m + 1 ͱ͓͘. ͜ͷͱ͖, ৚݅ (H1) ΛΈͨ͢. ৚݅ (H2) ͷ α Λ0 < (m + 1)α0 < 1ΛΈͨ͢ α0 ͱ͠,৚ ݅(H2) ͷc Λcα ≥ _αL₀ ΛΈͨ͢΋ͷͱ͢Δ.

(8)

K = 1͓Αͼ y(t) = ect _ͱ͢Δ_. _͜ͷͱ͖ ∫ δ(t) γ(t) G(t, s)y(s)ds = L Γ(α) ∫ t 0 (t− s)α−1ecsds = Le ct cα_Γ(α) ∫ ct 0 sα−1e−sds ≤ L cαe ct ≤ α0y(t) Ͱ͋Δ. Αͬͯ, ৚݅ (H2) ΛΈͨ͢. ఆཧ 3 ΑΓ, A ͸ͨͩͻͱͭͷෆಈ఺Λ΋ͭ. □ ୈ5અ ͓ΘΓʹ ຊ࿦จͷલʹ, ඇ੔਺֊ඍ෼ํఔࣜͷݚڀ Λߦͬͨ. [6, 7]Λඇ੔਺֊ඍ෼ํఔࣜʹ֦ு ͢ΔࢼΈͰ͋Δ. ࿦จ͸౤ߘதͰ͋Δ. ൃද ͸ҎԼͷ৔ॴͰߦͬͨ. τϧίͰ։࠵͞Εͨ

ࠃࡍձٞʮInternational Congress in Honour of Professor Ravi P. Agarwalʯ(Uludag Uni-versity, Bursa, 2014೥6݄23೔͔Β26೔·

Ͱ) Ͱ๛ా͕ߨԋͨ͠. ژ౎਺ཧղੳݚڀॴݚ

ڀूձʮThe International Workshop on Non-linear Analysis and Convex Analysisʯ(ژ౎ େֶ, 2014೥8݄19೔͔Β21೔)ʹͯ, ઒࡚

͕ߨԋͨ͠. ΞϝϦΧͰ։࠵͞Εͨࠃࡍձٞ

ʮTenth Mississippi State Conference on Dif-ferential Equations and Computational Simu-lationsʯ(Mississippi State University, 2014೥

10݄23೔͔Β25೔)Ͱ๛ా͕ߨԋͨ͠. ·ͨ, ຊ࿦จͷ಺༰͸,ࠃࡍ਺ཧՊֶڠձʹΑΔγϯ ϙδ΢Ϝ(େࡕࠃࡍେֶ, 2015೥3݄14೔)͓ Αͼ೔ຊ਺ֶձ೥ձ(໌࣏େֶ, 2015೥3݄21 ೔͔Β24೔)ʹͯൃද͢Δ. ͳ͓,ఆཧ 2 ͱಉ༷ͷ໰୊Λѻͬͨ݁Ռʹ [1]͕͋Δ. ͜ͷ࿦จͷ݁Ռͱͷؔ࿈ʹ͍ͭͯ͸ ࠓޙͷ՝୊Ͱ͋Δ. ࢀߟจݙ

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2 ճඍ෼͢Δ—෼਺ႈඍ

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2015೥3݄15೔ݪߘड෇ɹ