Asymptotics of solutions to linear differential equations of odd order

K. A. Mirzoev ; N. N. Konechnaja

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2020), pp. 23-28

Résumé

Asymptotic formulas are obtained in the paper for $x\to +\infty$ for the fundamental system of solutions to the equation $$ l (y): = i^{2n+1}\{ (qy^{(n+1)})^{(n)}+(qy^{(n)})^{(n+1)}\}+py=\lambda y, \qquad x\in I:=[1,~+\infty), $$ where $\lambda $ is a complex parameter. It is assumed that $q$ is a positive continuously differentiable function, $p$ has the form $p =\sigma^{(k)}$, $0\le k \le n$, where $\sigma$ is a locally integrable on $I$ function, and the derivative is understood in the sense of the theory of distributions. In the case when $k=0$ and $\lambda \ne 0$, and the coefficients $q$ and $p$ of the expression $l (y)$ are such that $q=1/2 +q_1$, and $q_1,\sigma(=p)$ are integrable on $I$, these formulas are well known. It was established in the paper that they are valid under the same restrictions on $q_1$ and $\sigma$ and for any $1\le k \le n-1$. For $k=n$ additional constraints arise on these functions. We consider separately the case when $\lambda= 0 $. Asymptotic formulas were also obtained for solutions to the equation $l (y)=\lambda y$ under the condition $ q(x) = \alpha x^{2n+1+\nu} (1+r(x))^{-2}, $ $ \sigma(x) = x^{k+\nu}(\beta+ s(x)),$ where $\alpha \ne 0$ and $\beta$ are complex numbers, $\nu \geqslant 0$, and the functions $r $ and $s $ satisfy certain conditions of integral decay.