一类二阶三点边值问题变号解的存在性

Existence of Sign-changing Solutions to a Class of Two-order Equations Three-points Boundary Value Problems

研究一类非线性二阶方程三点边值问题变号解的存在性。通过相应的Green函数,将该问题转化为Hammerstein型积分方程,于是此问题的解等价于一个非线性算子的不动点。进一步,利用Green函数的性质,证明了非线性算子所对应的线性算子是强正的,其所有的特征值都是正的,它们的代数重数全为1。最终,根据线性算子的特征值性质以及非线性项所满足的假设条件,借助于一个抽象的理论结果,证明了非线性算子至少有一个变号不动点,从而得到了此类边值问题变号解的存在性。

I order equati transformed on n this paper, the existence with three-point boundary into Hammerstein＇s integral of sign-changing solutions to a c value problems is discussed. Th equations by using the Green＇s solution of this problem is equivalent to a fixed point sing properties of the Green＇s function,we prove that of the nonlinear ope lass of nonlinear second e involved problems are function, so that every rator. Moreover, by uthe involved linear operator of the nonlinear operator is strongly positive and all the eigenvalues are positive, the algebraic multiplicity of all the eigenvalues is one. Finally, applying properties of eigenvalues of the linear operator and the assumption of the nonlinear term and a known result, it is proved that the nonlinear operator has at least one sign-changing fixed point, and thus the existence of sign-changing solutions of this class of boundary value problems is shown.