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FASCICULI
MATHEMATICI
Nr42
2009
IshakAltunandDuranTurkoglu
SOMEFIXEDPOINTTHEOREMSFORMAPPINGS
SATISFYINGCONTRACTIVECONDITION
OFINTEGRALTYPEON
d-COMPLETETOPOLOGICALSPACES
Abstract.Inthispaper,weprovetwofixedpointtheorems
formappingssatisfyingcontractiveconditionofintegraltypeon
d-completeHausdorfftopologicalspaces.
Keywords:fixedpoints,d-completetopologicalspacescontrac-
tiveconditionofintegraltype.
AMSMathematicsSubjectClassification:54H25,47H10.
1.Introduction
Branciari[5]obtainedafixedpointresultforasinglemappingsatisfying
ananalogueofBanach’scontractionprincipleforanintegraltypeinequality.
Theauthorsin[2],[3],[4],[13],[14]and[15]provedsomefixedpointthe-
oremsinvolvingmoregeneralcontractiveconditions.Recently([6])some
fixedpointtheoremshavebeenprovedinnon-metricsettingwhereinthe
distancefunctionusedneednotsatisfyingtriangleinequality.Thepurpose
ofthispaperistoinvestigatesomenewresultoffixedpointsinnon-metric
settings.Inthesequel,weusecontractiveconditionofintegraltypeon
d-completeHausdorfftopologicalspaces.
Let(X,T)beatopologicalspaceandd:X×X→[0,∞)besuch
thatd(x,y)10ifandonlyifx1y.ThenXissaidtobed-completeif
Σ
∞
nl1d(xn,xn+1)<∞impliesthatthesequence{xn}isconvergentinX.A
mappingT:X→Xisw-continuousatxifxn→ximpliesTxn→Tx.For
detailsond-completetopologicalspaces,werefertoIseki[7]andKasahara
[9]-[11].
Inthesequel,weshallusethefollowing:
AsymmetricfunctiononasetXisarealvalueddonX×Xsuchthat
forallx,y∈X
(ź)d(x,y)≥0,andd(x,y)10ifandonlyifx1y,
(źź)d(x,y)1d(y,x).
