Asymptotic formulas for fundamental system of solutions of high order ordinary differential equations with coefficients -- distributions

Abstract

This paper deals with differential equations of the form $$\tau (y)-{\lambda }^{2m}\varrho (x)y=0,\tau (y)=\sum _{k,s=0}^m({\tau }_{k,s}(x)y^{(m-k)}(x){)}^{(m-s)},$$ where $n=2m⩾2$, $\lambda$ is the large complex parameter, the positive functions\ $\varrho$\ and\ ${\tau }_{0,0}$ \ belong to $W^{1,1}[0,1]$ and the complex valued coefficients ${\tau }_{k,s}$ are such that the anti-derivatives ${\tau }_{k,s}^{(-l)}$ belong to $L_2[0,1]$, provided that $l=\min \{k,s\}$. Here the anti-derivatives are understood in the sense of distributions. The above equation can be reduced to the $n$-th order system of differential equations of the form $${\mathbf{y}}^{\prime }=\lambda \rho (x)\mathrm{B}\mathbf{y}+\mathrm{A}(x)\mathbf{y}+\mathrm{C}(x,\lambda )\mathbf{y}$$ with constant matrix $\mathrm{B}$ and summable matrices $\mathrm{A}(x)$ and $\mathrm{C}(x,\lambda )$. The first objective of the paper is obtain new results on asymptotic representation for the matrix of fundamental solutions of the last equation with respect to $\lambda \to \infty$ in certain sectors of the complex plane. The second objective is to apply the obtained results for analyzing the asymptotic representation of fundamental solutions of the first scalar equation with distribution coefficients.

In Russian, The paper is accepted for publishing in "Sbornik: Mathematics"