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 data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=c0f7beac0b292c1b0963cef6e199df19">height="14" width="79" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305716-si1.gif">$T:\bar{G}\to X$ is continuous on data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=21622568e4e0d4710fca057905a297ac">height="14" width="15" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305716-si2.gif">$\bar{G}$ and locally nonexpansive on G  , then the condition data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=540c1d14ce2bf953cfeb283aa62bf5a3" title="Click to view the MathML source">d(u,T(u))<d(x,T(x))$d(u,T(u))<d(x,T(x))$ for all data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si4.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=eebd4f06d34a19be6a7b0368495a3d15" title="Click to view the MathML source">u∈G$u\in G$ and data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si203.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=9124d6ba408049897c1b67a786041a94" title="Click to view the MathML source">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 data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si114.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=379580707ca853e4676d9638f8da9bbd" title="Click to view the MathML source">(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 data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si218.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=43f9244d6052a93b4fcde9fa57066821" title="Click to view the MathML source">T:G→G$T:G\to G$ is a locally nonexpansive mapping for which data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si254.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=500ae5edc4337a654758d21a82910b2a">height="19" width="90" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305716-si254.gif">$\overline{Fix\left(T\right)}\subset G$ and for which data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si255.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=e75ad4c33bc1ec25c931ace06905baf2" title="Click to view the MathML source">int(Fix(T))≠∅$int\left(Fix\left(T\right)\right)\ne \varnothing$. Then data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si10.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=535137918363d2de292e2b49886299b2" title="Click to view the MathML source">Fix(T)$Fix\left(T\right)$ is a closed convex subset of G  , and moreover the sequence data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si11.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=cdc8de260c163891b3532185863f4bfb" title="Click to view the MathML source">{Tⁿ(x)}$\left\{T^n\left(x\right)\right\}$ converges to a point of data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si10.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=535137918363d2de292e2b49886299b2" title="Click to view the MathML source">Fix(T)$Fix\left(T\right)$ for each data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305716&_mathId=si12.gif&_user=111111111&_pii=S0022247X16305716&_rdoc=1&_issn=0022247X&md5=49cf3da3d935ef088fa2729c1964890b" title="Click to view the MathML source">x∈G$x\in G$.