On summable solutions of a class of nonlinear integral equations on the whole line

Abstract

In this paper, we study a class of nonlinear integral equations with a noncompact monotone Hammerstein-Nemytskii operator on the whole real line. This class of equations is widely used in various fields of natural science. In particular, such equations arise in mathematical biology and in the theory of radiative transfer. A constructive existence theorem for a nonnegative nontrivial summable and bounded solution is proved. We also study the asymptotic behavior of the solution at $\pm\infty$. At the end of the paper, specific examples of the indicated equations are given, that satisfy all the conditions of the proved existence theorem. In an important particular case, it is possible to prove a uniqueness theorem in a certain class of essentially bounded functions.