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.Mathematics Subject Classification (2010). 35B15, 47D03.