Existence and Uniqueness of Mild Solutions for Fractional Partial Integro-Differential Equations
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Existence and Uniqueness of Mild Solutions for Fractional Partial Integro-Differential Equations Bo Zhu and Baoyan Han Abstract. In this paper, we study a class of nonlinear time fractional partial integro-differential equations. The main results for the problem are obtained by the measure of noncompactness, solution operator, convexpower condensing operator and general Banach contraction mapping principle. Mathematics Subject Classification. 35R11, 35A01, 35A02, 35A24, 34G20. Keywords. Nonlinear time fractional partial integro-differential equation, Solution operator, Measure of noncompactness, General Banach contraction mapping principle, Convex-power condensing operator.
1. Introduction Over the past few decades, fractional calculus has been successfully applied in various areas, such as engineering, physics, finance, chemistry, etc. Many good applications of the fractional calculus have been obtained; see [1–11] and the references therein. In [1–11], those applications are mentioned to control theory of dynamical systems, physics, fluid flow, etc. In [12,13], Zhang, Cheng et al. studied the multiple positive solutions for a system of fractional boundary value problems. In [14], Zhang and Zhong studied the uniqueness for the fractional differential equations by fixed point theorems. In [15], Bai et al. studied the existence of the mild solutions for the fractional differential equations by monotone iterative method. In [16–18], Chen et al. studied the existence results of the non-autonomous evolution equations with nonlocal conditions or non-instantaneous impulses. In [19–21], Ouyang, Zhu et al. studied the following time fractional partial differential equations: 0123456789().: V,-vol
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⎧ ∂2 ⎪ c β ⎪ D u(x, t) − a(t) u(x, t) = g(t, u(x, τ1 (t)), ⎪ t ⎪ ∂x2 ⎪ ⎨ u(x, τ2 (t)), . . . , u(x, τk (t))), t ∈ [0, T0 ], ⎪ ⎪ u(x, t) = 0, (x, t) ∈ ∂Ω × [0, T0 ], ⎪ ⎪ ⎪ ⎩ u(x, 0) = ϕ(x), x ∈ Ω, where c Dtβ is the Caputo, s fractional derivative of order 0 < β ≤ 1, the function a(t) is diffusion coefficient, k is a nature number, Ω ⊂ Rk is a bounded domain with a sufficiently smooth boundary ∂Ω, ϕ ∈ L2 (Ω). Ouyang [19] proved the existence of the local solutions by Leray–Schauder fixed theorem. In [20,21], Zhu et al. transferred the above time fractional partial differential equations into the abstract form of the time fractional differential equations in Banach space L2 (Ω), and obtained the existence and uniqueness results by Banach contraction mapping principle and strict contraction mapping principle. In this paper, we consider the following initial boundary value problem of nonlinear time fractional partial integro-differential equations: ⎧ t ∂ (t − s)β−2 ∂ 2 ⎪ ⎪ ⎪ (u(x, t) + h(u(x, t))) = (u(x, s) + h(u(x, s)))ds ⎪ 2 ⎪ ⎪ 0 Γ(β − 1) ∂x ⎨ ∂t + f (t, u(x, t), Gu(x, t)), t ∈ [0, b], (1.1) ⎪ ⎪ ⎪ u(0, t) = u(π, t) = 0, t ∈ [0, b], ⎪ ⎪ ⎪ ⎩ u(x, 0) = ϕ(x), x ∈ [0, π], where β ∈ (1, 2), h : R → R and f : [0, b] × R2 → R are continuous, ϕ ∈ L2 ([0, π]), linear operator G defined by Gu(x, t) =
t
K(t, s)
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