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AnnalesMathematicaeSilesianae24(2010),7-13
PraceNaukoweUniwersytetuŚląskiegonr2874,Katowice
STABILITYOFTHEPEXIDERFUNCTIONALEQUATION
RomanBadora,BarbaraPrzebieracz,andPeterVoLkmann
Abstract.AstabilityresultforthePexiderequationwillbederivedfroma
stabilitytheorempublishedin[9]fortheCauchyfunctionalequation.Then
wediscussthequalityofsomeconstantsoccuringinthiscontext;asamodel
caseweconsiderfunctionsdefinedonthemultiplicativesemigroup{1,0}.
1.Introduction
InTheorem1belowwedescribethestabilityresultfortheCauchyequa-
tion,whichhadbeenmentionedintheAbstract.
LetSbeagroupoid,i.e.,Sisasetandforallxjg∈Swehaveaproduct
xg∈S.Forx∈Sandk=0j1j2j...thepowersx2
karerecursivelydefinedby
x2
0
=x1=xjx2
k+1
=x2
k
x2
k
.
JózefTabor[8]pointedouttheusefulnessofthefollowingconditionforsta-
bilityinvestigations:
(T)Forxjg∈Stherealwaysisanentirek≥1suchthat
(1)
(xg)2
k
=x2
k
g2
k
.
Inthepresentpaper,groupoidsSsatisfying(T)arecalledTaborgroupoids.
Threeexamplesareparticularcasesofthem;theyareorderedindecreasing
generality:
Received:3.12.2010.Revised:22.02.2011.
(2010)MathematicsSubjectClassification:39B82,39B52.
Keywordsandphrases:Pexiderequation,stability,Taborgrupoids.
