Existence and exponential stability of Stepanov-like almost automorphic mild solutions for semilinear evolution equations

Abstract

This work is concerned with the existence and exponential stability of Stepanov-like almost automorphic mild solutions for the following semilinear evolution equations

x'(t) = Ax(t) + F(t, x(t)),     t ∈ ℝ,

where A is the infinitesimal generator of a C₀-semigroup of bounded linear operator on a Banach space X and F: ℝ × X → X is a Stepanov-like almost automorphic function in t uniformly with respect to the second argument x. By applying the Banach contraction mapping principle (when F satisfies Lipschitz type conditions), and the Schauder's fixed point theorem (when F does not necessarily satisfy Lipschitz type conditions), we obtain the existence and exponential stability of Stepanov-like almost automorphic mild solutions for the semilinear evolution equations. Moreover, as application, two examples are given to illustrate our abstract results.