Almost automorphic and pseudo-almost automorphic solutions to semilinear evolution equations with nondense domain
This nice short paper concerns the existence and uniqueness of almost automorphic (in the sense of Bochner) and pseudo almost automorphic (in the sense of Liang et al) solutions of the semilinear evolution equation \[ x'(t)=Ax(t)+f(t,t,x(t)),t\in \mathbb R \] where \(A\) is a linear and not necessarily densely defined operator satisfying the Hille-Yosida condition. Assuming that the function \(f\) is pseudo almost automorphic with respect to the first variable and satisfies some Lipschitz type condition, the authors achieve their aims using Banach's fixed point theorem. An application to a partial differential equation is given. This problem goes back to the reviewer's paper [Semigroup Forum 69, No.~1, 80--86 (2004; Zbl 1077.47058)]. The results generalize several recent works in the topics.
scientific article; zbMATH DE number 5624538
scientific article