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 $\parallel Tx-Ty\parallel \le \parallel x-y\parallel $ 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 $T:A\cup B\to A\cup B$ satisfying (i) $T\left(A\right)\subseteq B$ and $T\left(B\right)\subseteq A$, has a proximal point in the sense that there exists ${x}_{0}\in A\cup B$ such that $\parallel {x}_{0}-T{x}_{0}\parallel =\text{dist}(A,B)$. If in addition the norm of $X$ is strictly convex, and if (i) is replaced with (i)’ $T\left(A\right)\subseteq A$ and $T\left(B\right)\subseteq B$, 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 $\parallel {x}_{0}-{y}_{0}\parallel =\text{dist}(A,B)$. 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.

##### MSC:

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) |