Weighted pseudo almost periodic functions, convolutions and abstract integral equations

Abstract

This paper deals with a systematic study of the convolution operator Kf = f (*) k defined on weighted pseudo almost periodic functions space PAP(X, rho) and with k is an element of L-1(R). Upon making several different assumptions on k, f and rho, we get five main results. The first two main results establish sufficient conditions on k and rho such that the weighted ergodic space PAP(0)(X, rho) is invariant under the operator kappa. The third result specifies a sufficient condition on all functions (k, f and rho) such that the kappa f is an element of PAP(0)(X, rho). The fourth result is a sufficient condition on the weight function p such that PAP(0)(X, rho) is invariant under kappa. The hypothesis of the convolution invariance results allows to establish a fifth result related to the translation invariance of PAP(0)(X, rho). As a consequence of the fifth result, we obtain a new sufficient condition such that the unique decomposition of a weighted pseudo almost periodic function on its periodic and ergodic components is valid and also for the completeness of PAP(X, rho) with the supremum norm. In addition, the results on convolution are applied to general abstract integral and differential equations.