Fixed points of locally nonexpansive mappings in geodesic spaces

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• 作者：W.A. Kirk^a ; ^{william-kirk@uiowa.edu" class="auth_mail" title="E-mail the corresponding author} ; Naseer Shahzad^b ; ^{nshahzad@kau.edu.sa" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Geodesic spaces ; Locally nonexpansive mappings ; Fixed point sets ; Approximation of fixed points
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：705-715
• 全文大小：343 K

文摘

This is an examination of the structure of fixed point sets of locally nonexpansive mappings in various geodesic spaces. Among other things, it is shown that if G is a bounded connected open subset of a complete CAT(0) space X   and if $T:\bar{G}\to X$ is continuous on a057905a297ac">$\bar{G}$ and locally nonexpansive on G  , then the condition d(u,T(u))<d(x,T(x))$d(u,T(u))<d(x,T(x))$ for all 34a19be6a7b0368495a3d15" title="Click to view the MathML source">u∈G$u\in G$ and x∈∂G$x\in \partial G$ implies that the fixed point set of T is a nonempty closed convex subset of G. The following theorem is also consequence of one of our main results. Theorem.   Let (X,d)$(X,d)$ be a complete CAT(0) space which has the geodesic extension property and whose Alexandrov curvature is bounded below. Suppose G is a connected open subset of X  , and suppose T:G→G$T:G\to G$ is a locally nonexpansive mapping for which $\overline{Fix\left(T\right)}\subset G$ and for which int(Fix(T))≠∅$int\left(Fix\left(T\right)\right)\ne \varnothing$. Then Fix(T)$Fix\left(T\right)$ is a closed convex subset of G  , and moreover the sequence {Tⁿ(x)}$\left\{T^n\left(x\right)\right\}$ converges to a point of Fix(T)$Fix\left(T\right)$ for each x∈G$x\in G$.