一类非线性三点边值问题多个正解的存在性

Existence of Multiple Positive Solution of a Nonlinear Three Point Boundary Value Problem

利用Leggett-Williams不动点定理讨论以下一类二阶三点边值问题y″ +f(y) =0 ,0≤t≤ 1,y′(0 ) =0 ,y(1) =αy(η)多个正解的存在性 .在 0 <α <1,0 <η <1时 ,通过对 f限制适当的增长性条件 ,获得了此类问题至少三个正解的存在性以及它们各自的存在区间 ,并就所得结论给出了实际应用 .在主要定理的证明中 ,利用三点边值问题的核函数首先实现了微分方程向积分方程的转化 ,再通过定义合适的锥及凹函数 ,完整解决了以上三点边值问题的多解性 .不但推广了前人在两点边值问题方面的工作 。

By using Leggett-Williams fixed point theorem ,we discuss the existence of multiple positive solutions of the following boundary value problem y″+f(y)=0,0≤t≤1, y′(0)=0,y(1)=αy(η) Growth conditions imposed on f which yield the existence of at least three positive solutions and existence interval of each one in the case of 0<α<1,0<η<1. An example of application is also present in the paper. For the existence of multiple positive of the above boundary value problem , we make a conversion from differential equation to integral equation by using corresponding kernel function; And by defining suitable cone and concave function, the three_point problem can be done completely. Our results not only extend former work, but also take on some inspiration for those who research the existence of multiple solutions of multiple point problems.