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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 A and B of a Banach space X and satisfy Tx-Tyx-y for all xA, yB. It is shown that if A and B are weakly compact and convex, and if the pair (A,B) has proximal normal structure, then a relatively nonexpansive mapping T:ABAB satisfying (i) T(A)B and T(B)A, has a proximal point in the sense that there exists x 0 AB such that x 0 -Tx 0 =dist(A,B). If in addition the norm of X is strictly convex, and if (i) is replaced with (i)’ T(A)A and T(B)B, then the conclusion is that there exist x 0 A and y 0 B such that x 0 and y 0 are fixed points of T and x 0 -y 0 =dist(A,B). 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.

MSC:
47H09Mappings defined by “shrinking” properties
46B20Geometry and structure of normed linear spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)