On the Continuous Dependence on Parameter of the Solution Set of Differential Inclusions

In recent years the theory of differential inclusions in infinite-dimensional Banach spaces has attracted much attention due to its application to optimal control problems described by partial differential equations. An important question that arises in the study of such inclusions is the continuous dependence of the solution set with respect to a parameter contained in the right-hand side. For finite-dimensional spaces a theorem on continuous dependence of solutions of differential inclusions on parameters has already been established (see [1, 101). The aim of this note is to extend this theorem to Banach spaces and to derive from it the first fundamental theorem of Bogoliubov on averaging in finite intervals [2] for differential inclusions in standard form in Banach spaces (see [6, 10]). Throughout the sequel solutions of differential inclusions will always be taken in Carathéodory sense. Let X be a Banach space with a strictly convex norm IIII and the associated metric By Comp X(ConvX) we denote the collection of all non-empty compact (convex and compact, respectively) subsets of Xendowed with the l-lausdorff metric cx(,), by p(x,A) the distance from a point x c X to a set A C X, and by I = [0, TI a segment of the real positive axis R= [0, + ), 0< T R. By C(I,X)we mean the Banach space of continuous mappings from I to X, equipped with the standard norm. We define the modulus of a set A t Comp X to be the number I Al = a(A,I0}). Measurability, strong measurability of mappings and integrals of multi valued mappings are understood as in [8]. Consider the differential inclusion