二阶微分方程边值问题解的存在唯一性.doc

二阶微分方程边值问题解的存在唯一性 2007级信息与计算科学 韩佳彤 指导教师：许国安 摘要：微分不等式理论是处理各类微分方程的边值问题的解的存在性或唯一性及其数值计算的一种简单而有效的理论。 本文主要是通过微分不等式的技巧，研究一类不具备Nagumo条件但满足某种替代性条件的二阶微分方程的两点边值问题的解的存在性。首先，提出了替代Nagumo条件的一些新条件，并且证明了他们也能起着与Nagumo条件的同样作用，再利用Green函数表示出方程解的等价积分方程y(t),利用Schauder不动点定理及在引理的证明基础上用微分不等式理论证明二阶微分方程边值问题解的存在性，最后在附加一定条件下证明解的唯一性。 关键词：Nagumo条件；微分不等式；边值问题；上解；下解；存在性；唯一性 Second Order Differential Equations Existence and Uniqueness Abstract: The theory of differential inequalities to deal with various types of boundary value problems for differential equations Existence and uniqueness of numerical calculation or a simple and effective hand theory. In this paper, through differential inequality techniques to study the conditions for a class but do not have to meet certain alternative Nagumo condition of two second order differential equations the solution of boundary value problems. First, the proposed alternative to some of the new conditions Nagumo condition, and prove they can play the same role with the Nagumo condition, and then expressed by Green function integral equation equivalent to equation y (t), using Schauder fixed point Theorem and the proof of Lemma using differential inequalities on the basis of second-order Differential Equations prove the existence, in the end proved under certain conditions attached uniqueness of solution.Key words: Nagumo’s condition, differential inequality, boundary value problem, upper solution, lower solution, existence, uniqueness. Keywords: Nagumo condition; differential inequality; boundary value problem; on the solution; lower solution; existence; uniqueness 引言..........................................3 引理及其证明 .................................5 主要结论与证明................................7 结论的应用....................................9 参考文献.....................................10 致谢.........................................10 引言： Nagumo于20世纪30年代开创性地提出了二阶微分方程边值问题的微分不等式理论，给出了Nagumo条件和Nagumo定理，奠定了微分不等式理论的基础，其后Howes和Jackso