Almost periodicity and almost automorphy for some evolution equations using Favard’s theory in uniformly convex Banach spaces

Abstract

In this work, we use an approach due to Favard (Acta Math 51:31-81, 1928) to study the existence of weakly almost periodic and almost automorphic solutions for some evolution equation whose linear part generates a C 0 -group satisfying the Favard condition in uniformly convex Banach spaces. When this C 0 -group is bounded, which is a condition stronger than Favard's condition, we prove the equivalence between almost automorphy and weak almost automorphy of solutions.

Compact almost automorphic solutions for semilinear parabolic evolution equations

“…The concept of almost automorphy introduced by Bochner [7] is not restricted just to continuous functions. One can generalize that notion to measurable functions with some suitable conditions of integrability, namely, Stepanov almost automorphic functions, see [5,7,13]. That is a Stepanov almost automorphic function is neither continuous nor bounded necessarily.…”