Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay

A class of nonlinear fractional order partial differential equations with delay c∂αu(x, t) ∂tα = a(t)△u(x, t) + f (t, u(x, τ1(t)), . . . , u(x, τl(t))), t ∈ [0, T0] be investigated in this paper, where cDα is the standard Caputo’s fractional derivative of order 0 ≤ α ≤ 1, and l is a positive integer number, the function f is defined as f (t, u1, . . . , ul) : R×R×· · · ,×R → R, and x ∈  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.