It is known that the study of boundary value and mixed problems for integrable linear equations encounters significant difficulties of a fundamental nature. Exceptions are problems with boundary conditions of a special type, which are often called integrable or linearizable. The purpose of this article is to study the asymptotic behaviors of solutions of singularly perturbed general boundary value problems with boundary jumps for higher‐order equations. Using the Schlesinger–Birghof theorem, we constructed a fundamental system of solutions of a homogeneous perturbed equation of conditionally stable type in the critical case. Initial boundary functions are constructed based on the fundamental system of solutions. An analytical representation is found, the existence and uniqueness of a solution to this boundary value problem are proved. Asymptotic estimates of the solution and its derivatives are derived from the analytical representation of the solution of the given boundary value problem. The limit passage of solution of the perturbed problem to the solution of the unperturbed problem is proved. The conditions of the existence of jumps are found. The values of boundary jumps are determined. As a result, a class of boundary value problems is highlighted that has possessing of phenomenon of boundary jumps.

中文翻译：

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具有边界跳跃的奇摄动广义边值问题解的渐近行为

众所周知，对可积线性方程的边值问题和混合问题的研究遇到了根本性的重大困难。例外是特殊类型的边界条件的问题，通常称为可积或线性化的。本文的目的是研究高阶方程带边界跳跃的奇摄动广义边值问题解的渐近行为。使用Schlesinger-Birghof定理，在临界情况下，我们构造了条件稳定类型的齐次摄动方程的基础解的基本系统。初始边界函数是基于解决方案的基本系统构建的。找到了解析表示，证明了该边值问题解的存在性和唯一性。从给定边值问题的解的解析表示中可以得出该解及其衍生物的渐近估计。证明了摄动问题解到无摄动问题解的极限通过。找到存在跳跃的条件。确定边界跳跃的值。结果，突出了具有边界跳跃现象的一类边界值问题。