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

Abstract

Abstract Let X and Y be Banach spaces, L be a linear manifold in X × Y $X\times Y$ , 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 $S=g+N$ . A central problem in our general setting is to determine, for a given y ∈ Y $y\in Y$ , a vector w ∈ S ∩ D ( L ) $w\in S\cap D ( L )$ such that, for some z ∈ L ( w ) $z\in L ( w )$ , ∥ z − y ∥ = dist ( y , L ( S ∩ D ( L ) ) ) $\| z-y\| =\operatorname{dist} ( y, L ( S\cap D ( L ) ) )$ , such a vector w is called the constrained extremal solution of multi-valued linear inclusions y ∈ L ( x ) $y\in L ( x )$ 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.

Keywords

• Banach space
• linear manifold
• constrained extremal solution
• metric generalized inverse
• algebraic operator part