Almost Periodic Mild Solutions to a Class of Fractional Delayed Differential Equations

Abstract

In this paper, one studies the existence and uniqueness of almost periodic mild solutions to fractional delayed differential equations of the form D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> x(t) = Ax(t) + D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α-1</sup> f(t, x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> ) where 1 <; α <; 2, A: D(A) ⊂ X → X is a linear densely defined operator of sectional type on a complex Banach space X and f: R × X → X is jointly continuous. Let f(t, x) be almost periodic in t ∈ R uniformly for x. Under some additional assumptions on A and f, the existence and uniqueness of a almost periodic mild solution to above equation is obtained by using the Banach fixed-point principle. The obtaining results extent corresponding results in time delay with respect to almost periodic mild solutions for fractional differential equations.