Oscillatory properties of solutions of higher order differential equations

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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.