二阶常微分方程边值问题解的存在性

The Existence of Solutions of Second Order Value Problem for Ordinary Differential Equation

设f: [0, 1]×R2 R满足 Carathéodory条件, a∈ L1[0, 1], a(·) ≥ 0 满足 0 ≤∫10a(t)dt < 1. 运用Leray Schauder原理考虑了边值问题x″(t) = f(t, x(t), x′(t)) t∈[0, 1]x′(0) =0 x(1) =∫10a(t)x(t)dt解的存在性.

Let f: [0, 1]×R^2R satisfies Carathéodory condition. a∈L^1[0, 1], a(·)≥0 and 0≤∫~1_0a(t)<1. By means of Leray-Schauder Theorem the following problem is considered:x″(t)=f(t, x(t) x~′(t)), t∈[0, 1]x′(0)=0 x(1)=∫~1_0a(t)x(t)dtThe criteria of admitting solutions for bounday value problem of second order ordinary differential equation is established.