Proximal normal structure and relatively nonexpansive mappings. (English) Zbl 1078.47013
Summary: The notion of proximal normal structure is introduced and used to study mappings that are “relatively nonexpansive” in the sense that they are defined on the union of two subsets and of a Banach space and satisfy for all , . It is shown that if and are weakly compact and convex, and if the pair has proximal normal structure, then a relatively nonexpansive mapping satisfying (i) and , has a proximal point in the sense that there exists such that . If in addition the norm of is strictly convex, and if (i) is replaced with (i)’ and , then the conclusion is that there exist and such that and are fixed points of and . Because every bounded closed convex pair in a uniformly convex Banach space has proximal normal structure, these results hold in all uniformly convex spaces. A Krasnosel’skiĭ type iteration method for approximating the fixed points of relatively nonexpansive mappings is also given, and some related Hilbert space results are discussed.
|47H09||Mappings defined by “shrinking” properties|
|46B20||Geometry and structure of normed linear spaces|
|47H10||Fixed point theorems for nonlinear operators on topological linear spaces|
|47J25||Iterative procedures (nonlinear operator equations)|