Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity
Let \(X\) be a topological space and \(Y\) a real Banach space. For a set-valued operator \(F\), the single-valued operator \(f\) is called a selection of \(F\) if \(fx\in Fx\) for some \(x\in X\). A~single-valued operator \(m: X\to Y\) is called a minimum selection of \(F\) if it satisfies \(m(x)\in F(x)\), \(\| m(x)\|= m_F(x)\) for all \(x\in X\), where \(m_F(x)= \text{inf}\{\| y\|: y\in Fx\}\). In Part~2 of the paper under review, the following theorem of minimum continuous selection is proved. Theorem 2.4. Let \(X\) be a topological space and \(Y\) a Banach real space. Let \(F: X\to Y\) be a continuous set-valued operator. Suppose that one of the following conditions is satisfied: (a) \(Y\) is uniformly convex and \(F\) is with closed convex values, (b) \(Y\) is locally uniformly convex and \(F\) is with weakly compact values. Then the minimum selection \(m\) of \(F\) is continuous. In the last part of the paper, as an applicaton, some problem concerning the minimum right inverses for linear operators and minimum fixed points for condensing set-valued nonlinear operators are discussed. Also, the existence of minimum solutions for an integral inclusion is proved.
scientific article; zbMATH DE number 5862543
scientific article