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12
Hencetheresultfollows.
MujahidAbbas
.
Corollary4.LetEbeanonemptyclosedq-starshapedsubsetofaconvex
metricspaceX,andTandSbecontinuousselfmappingsonEsuchthat
S(E)=EandT(E−{q})⊂S(E−{q}),q∈F(S).SupposethatTis
uniformlyasymptoticallyregular,asymptoticallyS-nonexpansiveandSis
affineonE.Ifcl(E−{q})iscompactandSandTareCq−commuting
mappingsonE−{q},thenF(T)∩F(S)isasingletoninE.
3.Invariantapproximationresults
Meinardus[16]wasthefirsttoemployafixedpointtheoremtoprovethe
existenceofaninvariantapproximationinBanachspaces.Subsequently,
severalinterestingandvaluableresultshaveappearedintheliteratureof
approximationtheory([1],[18]and[20]).Inthissectionweobtainresultson
bestapproximationasafixedpointofuniformlyCq−commutingmappings
andCq−commutinginaconvexmetricspace.
Definition5.LetXbeametricspaceandMbeaclosedsubsetofX.If
thereexistsago∈Msuchthatd(xjgo)=d(xjM)=inf{d(xjg):g∈M}j
thengoiscalledabestapproximationtoxoutofM.WedenotebyPM(x)j
thesetofallbestapproximationstoxoutofM.
Remark1.LetMbeaclosedconvexsubsetofaconvexmetricspace.
As,W(ujvjA)∈Mfor(ujvjA)∈M×M×[0j1],thedefinitionofconvexity
structureonXfurtherimpliesthatW(ujvjA)∈PM(x).HencePM(x)isa
convexsubsetofX.Also,PM(x)isaclosedsubsetofX.Moreover,itcan
alsobeshownthatPM(x)⊂∂M,where∂MstandsfortheboundaryofM.
Theorem5.LetMbeanonemptysubsetofaconvexmetricspaceX,
TjfandgbeselfmapsonXsuchthatuiscommonfixedpointoffjgand
TandT(∂M∩M)⊂M.Supposethatfandgareaffineandcontinuous
onPM(u)withPM(u)q−starshaped,f(PM(u))=PM(u)=g(PM(u))and,
q∈F(f)∩F(g).Ifthepairs{Tjf}and{Tjg}areCq−commutingand
satisfy,
(
I
d(fxjgu)
if
g=uj
d(TxjTg)≤
4
max{d(fxjgg)jd(fxjYTx
q
)j
I
l
d(ggjY
q
Ty
)j1
2[d(fxjY
q
Ty
)+d(ggjYTx
q
)]}
if
g∈PM(u)
forallx∈PM(u)∪{u},andifcl(PM(u))iscompactandPM(u)iscomplete,
thenPM(u)∩F(T)∩F(f)∩F(g)isnonempty.
