A Relaxation Theorem for a Banach-Space Integral-Inclusion with Delays and Shifts

We consider the following integral-inclusion x(t) = ∫taƒ(t, s, u(s), u(t − d1), ..., u(t − dk)) ds + Q(t)u(t) ∈ F(t, ξ(x)(t)) a.e. in T in Banach space which, in particular, includes a control system defined by partial differential equations with delayed or shifted controls and an ordinary differential inclusion as special cases. We prove a relaxation theorem for this integral-inclusion and discuss some properties of its relaxed solution set which include conditions under which the set of relaxed solution is closed or compact as well as continuous dependence of the set of relaxed solutions on various parameters.