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"> 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"> 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)) 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 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 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) 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 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"> 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))≠∅. 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) 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)} 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) 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.