Solvability of Dirichlet boundary value problems governed by non-monotone differential operators

TL;DR

This paper establishes existence results for Dirichlet boundary value problems involving non-monotone differential operators, broadening the understanding of solvability under mild conditions and including heteroclinic solutions on the half-line.

Contribution

It provides new existence theorems for boundary value problems with non-monotone operators, extending classical results to more general and less restrictive settings.

Findings

Existence of solutions under mild assumptions

Solutions include heteroclinic solutions on the half-line

Applicable to equations with non-monotone differential operators

Abstract

We prove existence results for Dirichlet boundary value problems for equations of the type \begin{align*} \left( \Phi(k(t) x'(t) ) \right)' = f(t, x(t) , x'(t) ) \qquad \text{for a.e. } t \in I:=[0,T] , \end{align*} where $\Phi : J \to \mathbb{R} $ is a generic possibly non-monotone differential operator defined in a open interval $J\subseteq \mathbb{R}$, $k:I \to \mathbb{R}$, $k$ is measurable with $k(t) >0$ for a.e. $t \in I$ and $f: \mathbb{R}^3 \to \mathbb{R}$ is a Carath\'eodory function. Under very mild assumptions, we prove the existence of solutions for suitably prescribed boundary conditions, and we also address the study of the existence of heteroclinic solutions on the half-line $[0,+\infty)$.