一类高阶周期微分方程的解和小函数的关系

基金项目:

国家自然科学基金资助项目

详细信息

The Relation Between Solutions of a class of HigherOrder Differential Equations With Periodic Coefficients andFunctions of Small Growth

摘要: 研究了微分方程~ $f^{(k)}+[P_{k-1}(\mathrm{e}^{z})+Q_{k-1}(\mathrm{e}^{-z})]f^{(k-1)}+\cdots+[P_{0}(\mathrm{e}^{z})+Q_{0}(\mathrm{e}^{-z})]f=0$ 和 ~ $f^{(k)}+[P_{k-1}(\mathrm{e}^{z})+Q_{k-1}(\mathrm{e}^{-z})]f^{(k-1)}+\cdots+[P_{0}(\mathrm{e}^{z})+Q_{0}(\mathrm{e}^{-z})]f=R_{1}(\mathrm{e}^{z})+R_{2}(\mathrm{e}^{-z})$ ~的解以及它们的一阶导数与小函数的关系, 其中~ $P_{j}(z)$ ~,~ $Q_{j}(z)$ ~ $(j=0，1,2，\cdots，k-1)$ ~和~ $R_{i}(z)(i=1，2)$ ~是关于~z~的多项式.
- 收敛指数.
Abstract: Linear differential equations ~ $f^{(k)}+[P_{k-1}(\mathrm{e}^{z})+Q_{k-1}(\mathrm{e}^{-z})]f^{(k-1)}+\cdots+[P_{0}(\mathrm{e}^{z})+Q_{0}(\mathrm{e}^{-z})]f=0$ ~and ~ $f^{(k)}+[P_{k-1}(\mathrm{e}^{z})+Q_{k-1}(\mathrm{e}^{-z})]f^{(k-1)}+\cdots+[P_{0}(\mathrm{e}^{z})+Q_{0}(\mathrm{e}^{-z})]f=R_{1}(\mathrm{e}^{z})+R_{2}(\mathrm{e}^{-z})$ ~ where ~ $P_{j}(z)$ ~,~ $Q_{j}(z)$ ~ $(j=0，1,2，\cdots，k-1)$ ~and~ $R_{i}(z)(i=1，2)$ ~are polynomials in z are investigated. The relationship between solutions and their 1th derivatives and small functions are studied.
- Exponent of convergence.