Composition of S^p-weighted pseudo almost automorphic functions and applications

Abstract

Since the background space of $S^p$-weighted pseudo almost automorphic functions (abbr. $S^p$-wpaa functions) is endowed with a norm coming from $L^p$ norm, it is natural to consider the composition of $S^p$-wpaa functions under conditions of $L^p$ norm. However, the known resutls for composition of $S^p$-wpaa functions were always given under conditions of supremum norm. Motivated by this, we establish some new results for the composition of $S^p$-almost automorphic functions and $S^p$-wpaa functions under a ``uniform continuity condition'' with respect to $L^p$ norm. As an application, we prove the existence of mild weighted pseudo almost automorphic solutions for some semilinear differential equations with an $S^p$-wpaa force term. An example of the heat equation illustrates our results.

Keywords

• $S^p$-almost automorphic
• uniform continuity
• $S^p$-weighted pseudo almost automorphic
• semilinear differential equation