Dichotomy and μ‐pseudo almost automorphic solutions for delayed partial functional differential equations in admissible spaces

Abstract

We prove the existence and uniqueness of μ‐pseudo almost automorphic solution for a delayed non‐autonomous partial functional differential equation in the exponential dichotomic case, where the nonlinear operator F satisfies the φ‐Lipschitz condition and φ belongs to some admissible spaces. We further prove the existence of an invariant stable manifold around the μ‐pseudo almost automorphic solution in that case. An application is given to illustrate our theory.

Almost Automorphic Solutions in the Sense of Besicovitch to Nonautonomous Semilinear Evolution Equations

“…The rest of this paper is arranged as follows: in Section 2, we study some basic properties of almost automorphic functions in the sense of Besicovitch. In Section 3, we use the results obtained in Section 2 and the Banach fixed point theorem to establish the existence of almost automorphic solutions in the sense of Besicovitch of system (1). In Section 4, we provide an example to illustrate the applicability of our results.…”

Besicovitch Almost Periodic Solutions of Abstract Semi-Linear Differential Equations with Delay

“…On the other hand, it is well known that the existence of periodic solutions and almost periodic solutions is one of the important research contents of the qualitative theory of differential equations, see in [13][14][15][16][17][18][19][20][21][22] and the references therein. Nevertheless, there are still few results on the existence of Besicovitch almost periodic solutions for differential equations [23].…”

Besicovitch almost periodic solutions of abstract semi-linear differential equations with delay

“…On the other hand, the existence of almost periodic solutions in various senses of differential equations is an important content in the qualitative theoretical research and application research of differential equations, see Refs. [13][14][15][16][17][18][19][20][21][22] and the references therein. But the results of the existence of Besicovitch almost periodic solutions of semi-linear differential equations are still very rare.…”