Constrained extremal solutions of multi-valued linear inclusions in Banach spaces
Zi Wang, Boying Wu, Yuwen Wang · Journal of Inequalities and Applications · 2016
Let X and Y be Banach spaces, L be a linear manifold in $X\times Y$ , 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. $S=g+N$ . A central problem in our general setting is to determine, for a given $y\in Y$ , a vector $w\in S\cap D ( L )$ such that, for some $z\in L ( w )$ , $\| z-y\| =\operatorname{dist} ( y, L ( S\cap D ( L ) ) )$ , such a vector w is called the constrained extremal solution of multi-valued linear inclusions $y\in L ( x )$ 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.