Controllability Results for Evolution Inclusions with Non-Local Conditions
Zeitschrift für Analysis und ihre Anwendungen
In this paper we prove controllability results for mild solutions defined on a compact real interval for first order differential evolution inclusions in Banach spaces with non-local conditions. By using suitable fixed point theorems we study the case when the multi-valued map has convex as well as non-convex values. determine the evolution t → y(t) of the location of a physical object for which we do not know the positions y(0) and y(t k ) for all k, but we know that the non-local condition
... -local condition (2) holds. Consequently, to describe some physical phenomena, the non-local condition can be more useful than the standard initial condition y(0) = y 0 . From (2) it is clear that, when c k = 0 for all k, we have the classical initial condition. Existence and controllability results were proved by Benchohra and Ntouyas in  for equation (1) with non-local conditions of the form y(0) + f (y) = y 0 , where f ∈ C C ([0, b], E), E under the assumption that f was bounded and the multi-valued map F has convex values. Here, we consider the non-local condition (2) and we prove controllability results in the cases when the multi-valued map F has convex or nonconvex values. In the first case a fixed point theorem for condensing maps due to Martelli  is used. In the later we shall present two results. In the first one we rely on a fixed point theorem for contraction multi-valued maps, due to Covitz and Nadler  , and for the second one on Schaefer's fixed point theorem combined with a selection theorem due to Bressan and Colombo  for lower semicontinuous multi-valued operators with non-empty closed and decomposable values. For recent controllability results in the convex case we refer to the papers by Benchohra and Ntouyas [3 -5] and the references cited therein. Other results for the particular case B ≡ 0 can be found in the paper  .