On a generalization of a theorem of S. Bernstein

2018/04/05 by Jakub Ciesielski, Ciesielski, J., Jakub Maksymiuk +3
Mathematics · #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1804.01802

openalex publication_date 2018/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we obtain a solution to the second order boundary value problem of the form (d)/(dt)Φ'(u)=f(t,u,u), t∈[0,1], u\colonℝ →ℝ with Dirichlet and Sturm-Liouville boundary conditions, where Φ\colonℝ→ℝ is strictly convex, differentiable function and f\colon[0,1]×ℝ×ℝ→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.

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