一类高阶线性微分方程解的复振荡

On Complex Oscillation Theory of Solutions of Some Higher Order Linear Differential Equations

中文关键词:  复微分方程  整函数  增长级  零点收敛指数.

英文关键词:complex differential equations  entire function  the growth of order  the exponent of convergence of the zeros.

中文摘要:

      本文利用Nevanlinna 理论研究一类高阶线性微分方程解的复振荡, 得到已知$A(z)$ 是超越整函数且$\rho(A)<\frac{1}{2}$, $k\geq 2$, 如果方程$f^{(k)}+Af=0$ 有一解满足$\lambda(f)<\rho(A)$, 令$A_{1}=A+h$, 其中$h\not\equiv 0$ 是整函数且$\rho(h)<\rho(A)$,则高阶方程$g^{(k)}+A_{1}(z)g=0$ 没有任何一个解满足$\lambda(g)<\infty$.

英文摘要:

      In this paper, we shall use Nevanlinna theory of meromorphic functions to investigate the complex oscillation theory of solutions of some higher order linear differential equation. Suppose that $A$ is a transcendental entire function with $\rho(A)<\frac{1}{2}$. Suppose that $k\geq 2$ and $f^{(k)}+A(z)f=0$ has a solution $f$ with $\lambda(f)<\rho(A)$, and suppose that $A_{1}=A+h$, where $h\not\equiv 0$ is an entire function with $\rho(h)<\rho(A)$. Then $g^{(k)}+A_{1}(z)g=0$ does not have a solution $g$ with $\lambda(g)<\infty$.