Proximal normal structure and relatively nonexpansive mappings

A. Anthony Eldred¹ ; W. A. Kirk² ; P. Veeramani³

¹Department of Mathematics Indian Institute of Technology, Madras Chennai, India
²Department of Mathematics The University of Iowa Iowa City, IA 52242-1419, U.S.A.
³Department of Mathematics Indian Institute of Technology-Madras Chennai, India

Studia Mathematica, Tome 171 (2005) no. 3, pp. 283-293

Résumé

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 $\Vert x_{0}-Tx_{0}\Vert =\mathop{\rm dist}( A,B) $. If in addition the norm of $X$ is strictly convex, and if (i) is replaced with (i)$^{\prime}$ $T( A) \subseteq A$ 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 $\Vert x_{0} -y_{0}\Vert =\mathop{\rm 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'ski{\u\i} type iteration method for approximating the fixed points of relatively nonexpansive mappings is also given, and some related Hilbert space results are discussed.

Détail

DOI : 10.4064/sm171-3-5

Keywords: notion proximal normal structure introduced study mappings relatively nonexpansive sense defined union subsets banach space satisfy vert tx ty vert leq vert x y vert shown weakly compact convex pair has proximal normal structure relatively nonexpansive mapping cup rightarrow cup satisfying subseteq subseteq has proximal point sense there exists cup vert tx vert mathop dist addition norm strictly convex replaced prime subseteq subseteq conclusion there exist fixed points vert y vert mathop dist because every bounded closed convex pair uniformly convex banach space has proximal normal structure these results uniformly convex spaces krasnoselski type iteration method approximating fixed points relatively nonexpansive mappings given related hilbert space results discussed