On a generalization of a theorem of S. Bernstein

Abstract

In this paper we obtain a solution to the second order boundary value problem of the form $\frac{d}{dt}{\Phi }^{\prime }(\dot{u})=f(t,u,\dot{u}),t\in [0,1],u:\mathbb{R}\to \mathbb{R}$ with Dirichlet and Sturm-Liouville boundary conditions, where $\Phi :\mathbb{R}\to \mathbb{R}$ is strictly convex, differentiable function and $f:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ 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.