2018/11/15 by Janusz Morawiec, Morawiec, Janusz, Thomas Zürcher +1
Engineering · Mathematics · #39B12 #47A50 (Primary) 26A24 #47B38 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1811.06275
openalex publication_date 2018/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix N∈\mathbb N and assume that for every n∈\1,…, N\ the functions fn\colon[0,1]→[0,1] and gn\colon[0,1]→\mathbb 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⊂\mathbb R of Lebesgue measure zero. We show that the formula Ph=∑n=1Ngn ⋅ (h∘ fn) defines a linear and continuous operator P\colon 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]).