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Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay. (English) Zbl 1217.35206
Summary: A class of nonlinear fractional order partial differential equations with delay \[ \frac{^c\partial^{\alpha}u(x,t)}{\partial t^{\alpha}}=a(t)\Delta u(x,t)+f(t,u(x,\tau_1(t)),\dots,u(x,\tau_l(t))),~t\in[0,T_0] \] be investigated in this paper, where \(^{c}D^{\alpha }\) is the standard Caputo’s fractional derivative of order \(0\leq \alpha \leq 1\), and \(l\) is a positive integer number, the function \(f\) is defined as \(f(t,u_{1},\dots ,u_{l}):\mathbb{R}\times \mathbb{R}\times \dots \mathbb{R}\to\mathbb{R},\), and \(x\in\Omega\) is a \(M\) dimension space. Using Lebesgue dominated convergence theorem, Leray-Schauder fixed point theorem and Banach contraction mapping theorem, we obtain some sufficient conditions for the existence of the solutions of the above fractional order partial differential equations.

MSC:
35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
45K05 Integro-partial differential equations
47N20 Applications of operator theory to differential and integral equations
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