Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
FASCICULI
MATHEMATICI
Nr43
2010
A.Acosta,W.Aziz,J.MatkowskiandN.Merentes
UNIFORMLYCONTINUOUSCOMPOSITION
OPERATORINTHESPACEOFϕ–VARIATION
FUNCTIONSINTHESENSEOFRIESZ
Abstract.AssumingthataNemytskiioperatormapsasubset
ofthespaceofboundedvariationfunctionsinthesenseofRiesz
intoanotherspaceofthesametype,andisuniformlycontinuous,
weprovethatthegeneratoroftheoperatorisanaffinefunction.
Keywords:o–variationinthesenseofRiesz,uniformlycontin-
uouscompositionoperator,Jensenequation.
AMSMathematicsSubjectClassification:26A45,47H30.
1.Introduction
LetIbeanintervalofR,(Xj|·|)arealnormedspace,Caclosedsubset
ofX,(Yj|·|)arealBanachspaceandh:I×C→Yagivenfunction.
DenotebyXIthesetofallfunctionsf:I→XandbyH:XI→YI
the(Nemytskiiorsuperposition)compositionoperatorgeneratedbythe
functionhdefinedby
H(f)(·)1h(·jf(·)).
Inthispaper,weprovethatifHmapssomesubsetsofthespaceRVł(IjC)
offunctionsofboundedo–variationinthesenseofRieszintospaceRVψ(IjY),
andisuniformlycontinuous,thenh,thegeneratorfunctionoftheoperator
H,isanaffinefunctionofthesecondvariable.
ThisgeneralizestheresultsofChistyakov[3],andMerentes[12],where
itisassumedthatHisLipschizian.Theuniformlycontinuouscomposition
operatorswerefirstconsideredin[7]forthespaceofdifferentiablefunctions
andabsolutelycontinuousfunctions,laterin[8]forthespaceofH¨
older
function,andin[9]forthespaceofboundedvariationfunctions.
