An Ambrosetti-Prodi type result for integral equations involving dispersal operator

TL;DR

This paper establishes an Ambrosetti-Prodi type result for nonlocal integral equations involving dispersal operators, providing conditions for existence, non-existence, and multiplicity of solutions using topological methods.

Contribution

It extends Ambrosetti-Prodi theory to nonlocal dispersal operators in integral equations, offering new existence and multiplicity criteria.

Findings

Necessary condition for non-existence of solutions

Existence of at least one solution under certain conditions

Existence of multiple solutions in some cases

Abstract

In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =f(x,u)+g(x) , \ \mbox{in} \ \Omega, \] where $\Omega \subset \mathbb{R}^{N}$, $N\geq 1$, is a bounded connected open, $g \in C(\overline{\Omega})$, $f:\overline{\Omega} \times \mathbb{R} \to \mathbb{R}$ are function, and $L_0 : C(\overline{\Omega}) \to C(\overline{\Omega})$ is a nonlocal dispersal operator. Using a sub-supersolution method and the degree theory for $\gamma$-Condensing maps, we have obtained a result of the Ambrosetti-Prodi type, that is, we obtain a necessary condition on $g$ for the non-existence of solutions, the existence of at least one solution, and the existence of at least two distinct solutions.