On a generalization of a theorem of S. Bernstein

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 $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:\mathbb{R}\to \mathbb{R}$ is strictly convex, differentiable function and $f\text{\hspace{0.17em}⁣}:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ 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.