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Stepanov-like pseudo-almost periodicity and semilinear differential equations with uniform continuity. (English) Zbl 1209.35011
Summary: 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.

MSC:
35B15 Almost and pseudo-almost periodic solutions to PDEs
47D03 Groups and semigroups of linear operators
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