Existence of Antiperiodic Solutions to Semilinear Evolution Equations in Intermediate Banach Spaces

Abstract

We are concerned in this paper with the antiperiodicity of mild solutions for the semilinear evolution equation $x^{\prime}(t) = Ax(t) + f(t,x)$where A is a sectorial operator not necessarily densely defined in X generating an hyperbolic semigroup $(T(t))_{t\geq 0}$in a Banach space X and $f : \mathbb{R} \times X_{\alpha } \rightarrow X$, where X _α is an intermediate space. We prove the existence and uniqueness of an antiperiodic mild solution in X _α , when the function $f : \mathbb{R} \times X_{\alpha }\rightarrow X$is antiperiodic. The result is obtained using the Banach-fixed point theorem.