Constrained extremal solutions of multi-valued linear inclusions in Banach spaces

Let X and Y be Banach spaces, L be a linear manifold in X× Y, or, equivalently, the graph of a multi-valued linear operator from X to Y, and let S be a prescribed hyperplane in X, i.e.S= g+ N. A central problem in our general setting is to determine, for a given y∈ Y, a vector w∈ S∩ D(L) such that, for some z∈ L(w) , ∥ z− y∥ = dist (y, L(S∩ D(L))) , such a vector w is called the constrained extremal solution of multi-valued linear inclusions y∈ L(x) in Banach spaces. We establish three equivalent characterizations of constrained extremal solution of linear inclusions in Banach spaces by means of the algebraic operator parts, the metric generalized inverse of multi-valued linear operator L, and the dual mapping of the spaces. As follows from the main results in this paper, we may get the constrained extremal solution of multi-valued linear inclusions, by using the extremal solution of some interrelated multi-valued linear inclusions in the same spaces. The setting in this paper includes large classes of constrained extremal problems and optimal control problems subject to generalized boundary conditions.