Some classes of linear operators involved in functional equations

Abstract

Fix N∈N, 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)| K} is of Lebesgue measure zero, fn satisfy Luzin’s condition N, and the set fn−1(A) is of Lebesgue measure zero for every set A⊂R of Lebesgue measure zero. We show that the formula Ph=∑n=1Ngn⋅(h∘fn) 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 φ∈L1([0,1]) of the equation φ=Pφ+g with a given g∈L1([0,1]).