Fixed points of nonexpansive type multivalued maps

Abstract

<p>Let (X,d) be a metric space. Husain and Tarafdar call a map J:X→2ˣ (nonempty subsets of X) nonexpansive (here it is called (H∙T)-nonexpansive if for all x ∊X, uᵪ ∊ J(x) there exists vᵧ ∊J(y) for all y∊X such that d(uᵪ,vᵧ) ≤ d(x,y). Clearly this notion generalizes the usual concept of nonexpansive maps and coincides with it for single-valued maps. We study fixed points for such mappings. Further, if in the above inequality we have d(uᵪ,vᵧ) ≤ hd(x,y) for some fixed h, 0≤h<1,. then we generalize the notion of contraction for single valued maps. We proved a fixed point theorem which contains the Banach fixed point theorem as a special case. We introduce and study two more classes of set-valued nonexpansive mappings: s-nonexpansive mappings and f-(H∙T)-nonexpansive mappings. In general, these two types of maps are not related. But both of them contain the class of all single-values nonexpansive maps. However, the class of all f-(H∙T)-nonexpansive mappings contain the class of all (H∙T)-nonexpansive mappings. It is shown that not every (H∙T)-nonexpansive mapping on a nonempty closed convex and bounded subset of a Banach space has a fixed point. Husain and Tarafdar proved that if M is a compact interval of the real line then each (H∙T)-nonexpansive closed convex valued map J:M→2ᴹ has a fixed point. In this thesis, we extend this result which contains some well-known results due to Browder and Karlovitz. Moreover, it includes, as a special case, the result: every single-valued nonexpansive mapping on a nonempty closed convex bounded subset of a reflexive Banach space satisfying Opial's condition has a fixed point. This late result can also be derived from a result due to Kirk, in view of the fact that the Opial's condition implies normal structure. We also prove a fixed point theorem for multivalued s-nonexpansive mapping from which it is derived that every closed convex-values s-nonexpansive mapping on a nonemty closed convex-values s-nonexpansive mapping on a nonempty closed convex bounded subset of a Hilbert space has a fixed point. In addition, we have a fixed point theorem for set-valued (H∙T)-contractive type mappings from which a result due to Kannan can be derived as a special case. Finally, a common fixed point theorem for such mappings is proved.</p>