关于S.伯恩斯坦
On a generalization of a theorem of S. Bernstein
摘要：
本文给出了一类二阶边值问题的解，其形式为$\frac{d}{dt}Φ'(\dot{u})=f(t,u,\dot{u}),\ t\in[0,1],\ u\colon\mathbb{R} \to\mathbb{R}$，具有Dirichlet和Sturm-Liouville边界条件，其中$Φ\colon\mathbb{R}\to\mathbb{R}$是严格凸的可微函数，$f\colon[0,1]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$是连续的且满足适当的增长条件.我们的结果是基于先验界的解决方案和同伦不变性的勒雷-绍德尔度。
英文摘要：
In this paper we obtain a solution to the second order boundary value problem of the form $\frac{d}{dt}Φ'(\dot{u})=f(t,u,\dot{u}),\ t\in[0,1],\ u\colon\mathbb{R} \to\mathbb{R}$ with Dirichlet and Sturm-Liouville boundary conditions, where $Φ\colon\mathbb{R}\to\mathbb{R}$ 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.
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