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

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: u(t) = integral(t)(-infinity) a(t-s)[Au(s) + f(s)]ds, t is an element of R, where A is the generator of an integral resolvent family in a Banach space X, a is an element of 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.