Stepanov-like pseudo-almost automorphic mild solutions for some abstract differential equations

Abstract

In this work, we give some theorems on pseudo-almost automorphic solutions for some abstract semilinear differential equations with uniform continuity. To facilitate this we give a new composition theorem of Stepanov-like pseudo-almost automorphic functions. Since $S^{p}$-pseudo-almost automorphic functions are more general and complicated than almost automorphic functions and $S^{p}$-pseudo-almost periodic functions, our work improves the known results by making use of a uniform continuity condition instead of the Lipschitz condition.