The paper focuses on a nonlinear eigenvalue problem of Sturm–Liouville type with real spectral parameter under first type boundary conditions and additional local condition. The nonlinear term is an arbitrary monotonically increasing function. It is shown that for small nonlinearity the negative eigenvalues can be considered as perturbations of solutions to the corresponding linear eigenvalue problem, whereas big positive eigenvalues cannot be considered in this way. Solvability results are found, asymptotics of negative as well as positive eigenvalues are derived, distribution of zeros of the eigenfunctions is presented. As a by-product, a comparison theorem between eigenvalues of two problems with different data is derived. Applications of the found results in electromagnetic theory are given.

中文翻译：

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非线性Sturm-Liouville问题的渐近分析：线性和非线性解决方案

本文重点研究在第一类边界条件和附加局部条件下具有实谱参数的Sturm-Liouville型非线性特征值问题。非线性项是任意单调递增的函数。结果表明，对于较小的非线性，可以将负特征值视为对相应线性特征值问题解的扰动，而不能以这种方式考虑较大的正特征值。找到可解性结果，导出负特征值和正特征值的渐近性，并给出特征函数零的分布。作为副产品，推导了两个具有不同数据的问题的特征值之间的比较定理。给出了研究结果在电磁理论中的应用。