Controllability Results for Evolution Inclusions with Non-Local Conditions

Mouffak Benchohra, E. P. Gatsori, L. Gorniewicz, S. K. Ntouyas
2003 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
more » ... -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 [3] 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 [20] 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 [11] , and for the second one on Schaefer's fixed point theorem combined with a selection theorem due to Bressan and Colombo [7] 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 [2] .
doi:10.4171/zaa/1153 fatcat:hcoqikoopzg6lf2aiey3c43nty