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

Francesca Anceschi, Cristina Marcelli, Francesca Papalini

TL;DR

This work develops existence results for Dirichlet boundary value problems governed by potentially non-monotone differential operators of the form $(\Phi(k(t)x'))' = f(t,x,x')$, under mild compatibility conditions between the right-hand side and boundary data. Existence is obtained via a fixed-point approach on an auxiliary problem with controlled right-hand side and a truncation argument that handles non-monotonicity; the framework covers monotone, singular, and bounded $\Phi$, and yields heteroclinic solutions on the half-line. The authors illustrate the approach with several examples, including Perona–Malik diffusion, sine-type diffusions, the $r$-Laplacian and mean curvature-type operators, showing solvability without Nagumo-type conditions. On the half-line, they prove heteroclinic existence through a limiting procedure on finite intervals, leveraging Dunford–Pettis compactness. Overall, the results unify solvability for a broad class of nonlinear ODEs with non-monotone operators and have potential applications in image processing and nonlinear diffusion models.

Abstract

We prove existence results for Dirichlet boundary value problems for equations of the type \begin{align*} \left( Φ(k(t) x'(t) ) \right)' = f(t, x(t) , x'(t) ) \qquad \text{for a.e. } t \in I:=[0,T] , \end{align*} where $Φ: 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 Carathéodory 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 $[0,+\infty)$.

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

TL;DR

This work develops existence results for Dirichlet boundary value problems governed by potentially non-monotone differential operators of the form , under mild compatibility conditions between the right-hand side and boundary data. Existence is obtained via a fixed-point approach on an auxiliary problem with controlled right-hand side and a truncation argument that handles non-monotonicity; the framework covers monotone, singular, and bounded , and yields heteroclinic solutions on the half-line. The authors illustrate the approach with several examples, including Perona–Malik diffusion, sine-type diffusions, the -Laplacian and mean curvature-type operators, showing solvability without Nagumo-type conditions. On the half-line, they prove heteroclinic existence through a limiting procedure on finite intervals, leveraging Dunford–Pettis compactness. Overall, the results unify solvability for a broad class of nonlinear ODEs with non-monotone operators and have potential applications in image processing and nonlinear diffusion models.

Abstract

We prove existence results for Dirichlet boundary value problems for equations of the type \begin{align*} \left( Φ(k(t) x'(t) ) \right)' = f(t, x(t) , x'(t) ) \qquad \text{for a.e. } t \in I:=[0,T] , \end{align*} where is a generic possibly non-monotone differential operator defined in a open interval , , is measurable with for a.e. and is a Carathéodory 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 .
Paper Structure (5 sections, 8 theorems, 140 equations)

This paper contains 5 sections, 8 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Auxiliary problem
  3. Proof of Theorem \ref{['thm:main1']}
  4. Examples
  5. Boundary value problems on a half-line

Key Result

Theorem 1.2

Let assumption ip:kappa hold, and suppose that $\Phi$ is locally strictly monotone at $s^*$, where $s^*$ is the slope defined in eq:notat. Let $J^*\subseteq J$ be an open interval, containing $s^*$, where $\Phi$ is strictly monotone. Assume that there exists a positive function $\psi \in L^1_+(I)$ s and for every $x\in \mathbb R$ and $y\in L^p(I)$ such that: Then, there exists a solution $x \in

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 3.1
  • ...and 12 more