Existence results for a nonlinear transport equation in bounded geometry on L1-spaces

In the paper (Latrach, 2001) [22] the existence of solutions to the boundary value problem (1) and (2) was discussed on LpLp spaces for p∈(1,∞)p∈(1,∞) in bounded geometry. Due to the lack of compactness on L1L1-spaces, the analysis did not cover the case p=1p=1. The purpose of this work is to extend the results of [22] to the case p=1p=1. Our analysis uses the concept of Dunford–Pettis operators and a new version of the Darbo fixed point theorem for the measure of weak noncompactness.