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MujahidAbbas
inguniformlyCq−commutingmappingstoasymptoticallyS−nonexpansive
mappings,commonfixedpointtheoremsareproved.Asanapplication,
invariantapproximationresultsforthesemappingsarealsoderived.
Forthesakeofconvenience,wegathersomebasicdefinitionsandsetout
theterminologyneededinthesequel.
Definition1.Let(Xjd)beametricspace.AmappingW:X×X×
[0j1]→XissaidtobeaconvexstructureonX,if,foreach(xjgjA)∈
X×X×[0j1]andu∈X,
d(ujW(xjgjA))≤Ad(ujx)+(1−A)d(ujg).
AmetricspaceXtogetherwithaconvexstructureWiscalledaconvex
metricspace.Obviously,W(xjxjA)=x.
LetXbeaconvexmetricspace.AnonemptysubsetEofXissaid
tobeconvexif,W(xjgjA)∈Ewhenever(xjgjA)∈E×E×[0j1].A
subsetEofaconvexmetricspaceissaidtobeq−starshapedorstarshaped
withrespecttoq,ifthereexistqinEsuchthatW(xjqjA)∈Ejwhenever
(xjA)∈E×[0j1].Obviouslyq-starshapedsubsetsofXcontainallconvex
subsetsofXasapropersubclass.Takahashi[21]hasshownthatopen
spheresB(xjT)={g∈X:d(gjx)<T}andclosedspheresB[xjT]={g∈
X:d(gjx)≤T}areconvexinaconvexmetricspaceX.Aconvexmetric
spaceXissaidtohaveproperty(A)if:d(W(gjxjA)jW(zjxjA))≤Ad(gjz),
forallxjgjz∈XandA∈(0j1).Property(A)isaconvexmetricspace
analogueofcondition(I)forthestarshapedmetricspacesofGuay,Singh
andWhitfield,see,Definition3.2[11].Throughoutthispaper,aconvex
metricspaceXisassumedtohaveaproperty(A).
Alsonotethateverynormedspaceisaconvexmetricspace.Thereare
manyexamplesofconvexmetricspaceswhichcannotbeembeddedinany
normedspace[21].
Example1.LetX={(x1jx2jx3)∈R3:x1jx2jx3>0}.Forx=
(x1jx2jx3),g=(g1jg2jg3)andz=(z1jz2jz3)inX,and0jβjγ∈[0j1]with
0+β+γ=1,defineamappingW:X3×[0j1]3→Xby
W(xjgjzj0jβjγ)=(0x1+βx2+γx3j0g1+βg2+γg3j0z1+βz2+γz3)j
andametricd:X×X→[0j∞)by,d(xjg)=|x1g1+x2g2+x3g3|.Here
Xisaconvexmetricspacebutitisnotanormedspace.
Example2.LetX={(x1jx2)∈R2:x1jx2>0}.Forx=(x1jx2),
g=(g1jg2)inXand0∈[0j1].DefineamappingW:X×X×[0j1]→X
by
W(xjgj0)=(0x1+(1−0)g1j0x1x2
0x1+(1−0)g1)j
+(1−0)g1g2
