Treść książki
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6
IshakAltunandDuranTurkoglu
LetdbeasymmetricfunctiononasetX,andforanyε>0andany
x∈X,letS(x,ε)1{y∈X:d(x,y)<ε}.From[6],wecandefinea
topologyTdonXbyU∈Tdifandonlyifforeachx∈U,someS(x,ε)⊂U.
Asymmetricfunctiondisasemi-metricifforeachx∈Xandforeachε>0,
S(x,ε)isaneighborhoodofxinthetopologyTd.AtopologicalspaceXis
saidtobesymmetrizable(resp.semi-metrizable)ifitstopologyisinduced
byasymmetricfunction(resp.semi-metric)onX.Thed-completesym-
metrizablespacesformanimportantclassofd-completetopologicalspaces.
Otherexamplesofd-completetopologicalspacesmaybefoundinHicksand
Rhoades[6].
HicksandRhoades[6]provedthefollowingtheorem.
Theorem1.Let(X,T)beaHausdorffd-completetopologicalspaceand
f,hbew-continuousselfmappingsonXsatisfying
d(hx,hy)≤G(M∗(x,y))
forx,y∈X,where
M
∗(x,y)1max{d(fx,fy),d(fx,hx),d(fy,hy)}
andGisareal-valuedfunctionsatisfyingthefollowing:
(a)0<G(y)<yforeachy>0;G(0)10,
(b)g(y)1
y1G(y)isanon-increasingfunctionon(0,∞),
y
(c)∫
o
y1
g(y)dy<∞foreachy1>0,
(d)G(y)isnon-decreasing.
Supposealsothat
(ź)fandhcommute,
(źź)h(X)⊆f(X).
ThenfandhhaveauniquecommonfixedpointinX.
2.Mainresult
Now,wegiveourmaintheorems.
Theorem2.Letfbeself-mappingofaHausdorffd-completetopological
space(X,T)satisfyingthefollowing
(1)
/
o
d(fx,fy)
o(t)dt≤G(/M(x,y)
o
o(t)dt)
forallx,y∈X,whereo:R+→R+isaLebesgueintegrablemappingwhich
issummableoneachcompactsubsetofR+,non-negativeandsuchthat
(2)
ε≤/
o
ε
o(t)dtforeachε>0,
