Properties of Fixed Point Set of a Multivalued Map

The study of fixed points for multivalued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin [17]. Later, an interesting and rich fixed point theory for such maps has been developed. The theory of multivalued maps has applications in control theory, convex optimization, differential inclusions, and economics (see, e.g., [3, 8, 16, 22]). The theory of multivalued nonexpansive mappings is harder than the corresponding theory of single-valued nonexpansive mappings. It is natural to expect that the theory of nonself-multivalued noncontinuous functions would be much more complicated. The concept of a ∗-nonexpansive multivalued map has been introduced and studied by Husain and Latif [9] which is a generalization of the usual notion of nonexpansiveness for single-valued maps. In general, ∗-nonexpansive multivalued maps are neither nonexpansive nor continuous (see Example 3.7). Xu [22] has established some fixed point theorems while Beg et al. [2] have recently studied the interplay between best approximation and fixed point results for ∗-nonexpansive maps defined on certain subsets of a Hilbert space and Banach space. For this class of functions, approximating sequences to a fixed point in Hilbert spaces are constructed by Hussain and Khan [10] and its applications to random fixed points and best approximations in Fréchet spaces are given by Khan and Hussain [12]. In this paper, using the best approximation operator, we (i) establish certain properties of the set of fixed points of a ∗-nonexpansive multivalued nonself-map in the setup of a strictly convex Banach space, (ii) prove fixed point results for ∗-nonexpansive random maps in a Banach space under several boundary conditions, and (iii) provide affirmative answers to the questions posed by Ko [14] and Xu and Beg [24] related to the set of fixed points.