Non-autonomous Second Order Differential Inclusions with a Stabilizing Effect

In this paper we prove the existence of mild solutions for a problem governed by a semilinear non-autonomous second order differential inclusion where a stabilization of the solution is expected due to the control of the reaction term. In order to obtain our existence theorem, first we study a more general problem with a differential inclusion which involves a perturbation guided by an operator N : I → C ( C ( I ; X ) ; X ) , where X is a Banach space. Finally we show an illustrative example of application of our results to a problem involving a wave equation.