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Asymptotic analysis of solutions of ordinary differential equations with distribution coefficients. (English. Russian original) Zbl 1477.34032

Sb. Math. 211, No. 11, 1623-1659 (2020); translation from Mat. Sb. 211, No. 11, 129-166 (2020).
The differential equation \[ \tau(y)-\lambda^{2m}\varrho(x)y=0,\quad \tau(y)=\sum\limits_{k,s=0}^m\left(\tau_{k,s}(x)y^{(m-k)}(x)\right)^{(m-s)}, \tag{1} \] is concidered in the article. Here \(x\in[0,1]\), \(\lambda\) is a complex parameter, the functions \(\tau_{0,0},\varrho\) are absolutely continuous and positive, \[ \frac{1}{\sqrt{|\tau_{0,0}|}},\ \frac{1}{\sqrt{|\tau_{0,0}|}}\tau_{k,s}^{(-l)}\in L_2[0,1],\quad 0\leq k,s\leq m,\ l=\min\{k,s\} \tag{2} \] hold, where \(f^{(-k)}\) denotes the \(k\)th antiderivative of the function f, understood in the sense of the theory of distributions.
The article consists of 4 sections and references. The first section is devoted to the general introduction into the problem. The aim of the paper is to obtain asymptotic formulas as \(\lambda\rightarrow\infty\) for the solutions of equation (1) in sectors of the complex plane that, in combination, cover the whole plane. The authors restrict Equation (1) to a system of \(2m\) equations of the first order. In the second section some useful results about properties of solutions, that also contain estimations on the fundamental solution matrix of such a systems are formulated and proved.
The first main results of the article are obtained for the introduced system. They are given in Theorem 1. The existence of the fundamental solution matrix and asymptotic formulas as \(\lambda\rightarrow\infty\) in the considered sectors for elements of the matrix are proved. Then in Theorem 3 of Section 4 the precise formulas, that reduce the equation (1) under conditions (2) to such a system are given. With the help of Theorem 1 in Theorem 4 asymptotic formulas as \(\lambda\rightarrow\infty\) in the considered sectors for solutions and their quasi-derivatives are found.
The results presented in the article are important for the further study of properties of solutions to Equation (1).

MSC:

34A30 Linear ordinary differential equations and systems
34A05 Explicit solutions, first integrals of ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
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