August 2019 Some classes of linear operators involved in functional equations
Janusz Morawiec, Thomas Zürcher
Ann. Funct. Anal. 10(3): 381-394 (August 2019). DOI: 10.1215/20088752-2018-0037

Abstract

Fix NN, and assume that, for every n{1,,N}, the functions fn:[0,1][0,1] and gn:[0,1]R are Lebesgue-measurable, fn is almost everywhere approximately differentiable with |gn(x)|<|f'n(x)| for almost all x[0,1], there exists KN such that the set {x[0,1]:cardfn1(x)>K} is of Lebesgue measure zero, fn satisfy Luzin’s condition N, and the set fn1(A) is of Lebesgue measure zero for every set AR of Lebesgue measure zero. We show that the formula Ph=n=1Ngn(hfn) defines a linear and continuous operator P:L1([0,1])L1([0,1]), and then we obtain results on the existence and uniqueness of solutions φL1([0,1]) of the equation φ=Pφ+g with a given gL1([0,1]).

Citation

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Janusz Morawiec. Thomas Zürcher. "Some classes of linear operators involved in functional equations." Ann. Funct. Anal. 10 (3) 381 - 394, August 2019. https://doi.org/10.1215/20088752-2018-0037

Information

Received: 14 November 2018; Accepted: 12 December 2018; Published: August 2019
First available in Project Euclid: 6 August 2019

zbMATH: 07089125
MathSciNet: MR3989183
Digital Object Identifier: 10.1215/20088752-2018-0037

Subjects:
Primary: 47A50
Secondary: 26A24 , 39B12 , 47B38

Keywords: approximate differentiability , Functional equations , integrable solutions , linear operators , Luzin’s condition N

Rights: Copyright © 2019 Tusi Mathematical Research Group

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Vol.10 • No. 3 • August 2019
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