Proximal normal structure and relatively nonexpansive mappings

Volume 171 / 2005

A. Anthony Eldred, W. A. Kirk, P. Veeramani Studia Mathematica 171 (2005), 283-293 MSC: Primary 47H10. DOI: 10.4064/sm171-3-5


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.


  • A. Anthony EldredDepartment of Mathematics
    Indian Institute of Technology, Madras
    Chennai, India
  • W. A. KirkDepartment of Mathematics
    The University of Iowa
    Iowa City, IA 52242-1419, U.S.A.
  • P. VeeramaniDepartment of Mathematics
    Indian Institute of Technology-Madras
    Chennai, India

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