On a generalization of a theorem of S. Bernstein

TL;DR

This paper extends a classical theorem by providing existence results for a class of second order boundary value problems involving nonlinear operators, using topological degree theory and a priori bounds.

Contribution

It generalizes Bernstein's theorem to nonlinear differential equations with convex energy functions and establishes existence via Leray-Schauder degree methods.

Findings

Existence of solutions under growth conditions on f

Application of Leray-Schauder degree theory

A priori bounds for solutions

Abstract

In this paper we obtain a solution to the second order boundary value problem of the form $\frac{d}{dt}\Phi'(\dot{u})=f(t,u,\dot{u}),\ t\in[0,1],\ u\colon\mathbb{R} \to\mathbb{R}$ with Dirichlet and Sturm-Liouville boundary conditions, where $\Phi\colon\mathbb{R}\to\mathbb{R}$ is strictly convex, differentiable function and $f\colon[0,1]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is continuous and satisfies a suitable growth condition. Our result is based on a priori bounds for the solution and homotopical invariance of the Leray-Schauder degree.