Abstract

For the differential equation \(y^{\prime \prime \prime }(x)=\lambda y(x) \) on the interval \([0,1] \), we consider the three-point eigenvalue problem \(a_i y^{(i-1)}(0)+y(c)+b_i y^{(i-1)}(1)=0\), \(i=1,2,3 \), where \(\lambda \) is the spectral parameter, the point \(c\in (0,1) \) is fixed, and \(a_i \) and \(b_i \), \(i=1,2,3\), are some complex numbers. Necessary and sufficient conditions that the coefficients \(a_i \) and \(b_i \) must satisfy for the indicated three-point problem to have degenerate boundary conditions are obtained.

中文翻译：

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退化三点边界条件

摘要

对于区间（[[0,1] \）上的微分方程\（y ^ {\ prime \ prime \ prime}（x）= \ lambda y（x）\），我们考虑三点特征值问题 \ （a_i y ^ {（i-1）}（0）+ y（c）+ b_i y ^ {（i-1）}（1）= 0 \）， \（i = 1,2,3 \），其中\（\ lambda \）是光谱参数，点 \（c \ in（0,1）\）是固定的，而\（a_i \）和\（b_i \），\（i = 1,2， 3 \），是一些复数。获得了系数\（a_i \）和\（b_i \）必须满足的条件，以使指示的三点问题具有退化的边界条件。