Eberlein-weakly almost periodic (in Stepanov-like sense) functions and application

Abstract

In this article, we prove a number of properties concerning a (new) class of (Stepanov-like) Eberlein-weakly almost periodic (S ^P -E.w. a. p.) functions with values in a Banach space. We use the results obtained to study the asymptotic behaviour of solutions to the evolution equation:

where A is the generator of an integral resolvent family in a Banach space 𝕏, a ∈ L ¹(ℝ), and f is a given 𝕏-valued function on ℝ. The objective is to deduce Eberlein-weakly almost periodicity (in Stepanov-like sense) of the solution u from corresponding properties on the part f.

Keywords:

• bounded solutions
• almost periodic and Eberlein-weakly almost periodic functions
• linear differential equations
• orbit

AMS Subject Classifications::

• Primary 34K14
• Secondary 47H99

Acknowledgements

The authors were partially supported by the IRD project UMMISCO.