Stepanov-Like Pseudo-Almost Periodicity and Semilinear Differential Equations with Uniform Continuity

By using the method of the invariant subspaces for unbounded linear operators and Schauder’s fixed point theorem, we give an existence theorem of mild pseudo-almost periodic solutions for some semilinear differential equations with a Stepanov-like pseudo-almost periodic term under some suitable assumptions. For this purpose, we show a new composition theorem of Stepanov-like pseudo-almost periodic functions. As applications, we examine the existence of mild pseudo-almost periodic solutions to some second-order hyperbolic equations. Our work is done under a “uniform continuity” condition instead of the “Lipschitz” condition assumed in the literature.