Oscillatory properties of solutions of higher order differential equations

本文研究了高阶(含二阶)Emden-Fowler型常微分方程解的振动性，并在常微分方程解的基础上研究了椭圆型微分方程解的振动性。两年来得到的新结果和新知识如下：1.对于二阶半线性常微分方程解，得到了Sturm-Liouville线性正则特征值问题的一个推广和类似结果。对于四维Emden-Fowler微分系统，建立了具有特定渐近性质的非振荡解的存在性的完整刻画，如t→∞，并得到了非振荡解的不存在性的一个刻画。对于具有一般非线性的高阶常微分方程解的非振动性，建立了Kiguradze类非振动解存在性的一个刻画。对于高阶线性常微分方程奇特征值问题，证明了存在可数特征值序列，且所考虑的n阶本征值函数在无限区间内正好有n个零点。对于二阶拟线性常微分方程组，完全确定了正解的渐近形式。对于二阶拟线性椭圆型微分方程解的振动性，给出了一个充分条件。对于二维拉普拉斯型半线性椭圆型微分方程组，建立了一个类似的Liouville定理。

The aim of this research is to investigate the oscillatory properties of solutions of higher-order (including second-order) ordinary differential equations of the Emden-Fowler type, and to investigate the oscillatory properties of solutions of elliptic differential equations on the base of the results for ordinary differential equations.The new results and knowledge obtained in the two years are as follows :1. For the second-order half-linear ordinary differential equations, a generalization and an analogue of the Sturm-Liouville linear regular eigenvalue problem are obtained.2. For the four-dimensional Emden-Fowler differential systems, a complete characterization for the existence of nonoscillatory solutions with specific asymptotic properties as t→∞ is established, and a characterization for the nonexistence of nonoscillatory solutions is also obtained.3. For higher-order ordinary differential equations with general nonlinearities, a characterization for the existence of nonoscillatory solutions of the Kiguradze classes is established.4. For a singular eigenvalue value problem to higher-order linear ordinary differential equations, it is shown that there is a countable sequence of eigenvalues and that the n-th eigenfunction has exactly n zeros in an infinite interval under consideration.5. For the second-order quasilinear ordinary differential equations, the asymptotic forms of positive solutions are completely determined.6. For the second-order quasilinear elliptic differential equations, a sufficient condition for the oscillation of all solutions is established.7. For the two-dimensional semilinear elliptic differential systems of the Laplace type, an analogue of the Liouville theorem is established.