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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 $\Vert Tx-Ty\Vert \leq\Vert x-y\Vert $ for all $x\in A$, $y\in B$. 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 $T:A\cup B\rightarrow A\cup B$ satisfying (i) $T(A) \subseteq B$ and $T(B) \subseteq A$, has a proximal point in the sense that there exists $x_{0}\in A\cup B$ such that $\|x_{0}-Tx_{0}\|=\text{dist}( A,B) $. If in addition the norm of $X$ is strictly convex, and if (i) is replaced with (i)’ $T(A) \subseteq A$ and $T(B) \subseteq B$, then the conclusion is that there exist $x_{0}\in A$ and $y_{0}\in B$ such that $x_{0}$ and $y_{0}$ are fixed points of $T$ and $\|x_{0}-y_{0}\| = \text{dist}(A,B)$. 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.

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)
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