Qualitative and Quantitative Analysis of Nonlinear Oscillation of Differential Equations

本研究计划致力于研究各种类型的非线性Sturm-Liouville微分算子微分方程的振动性。得到的主要结果如下。(1)We建立了非线性Sturm-Liouville算子的Picorie型恒等式，并将其应用于半线性常微分方程和偏微分方程解的比较和振动性的研究。我们观察到，长期以来一直不知道是否存在一类非线性微分算子，其中一个皮康型恒等式可以建立。(2)We已经开发了一个理论的广义三角函数，在此基础上，人们可以成功地构建一个精确的非线性模拟著名的Sturmian理论的线性微分方程。由此定义的广义Prufer变换使得计算一类半线性常微分方程非振动解的零点个数成为可能。(3)We对一类具有奇异性的非线性Sturm-Liouville方程正解的渐近性态进行了详细的分析。一个类似的分析也作出了微分方程涉及奇异Sturm-Liouville算子。作为一个意想不到的副产品，我们发现了一种新的类型的奇异的解决方案，从来没有出现在文献中。（4）关于泛函微分方程的振动性，我们建立了（i）一类特殊的非中立型方程的一个新的比较定理和（ii）含非线性Sturm-Liouville微分算子的中立型方程振动的一个有效判据。

The present research project is devoted to the investigation of the oscillatory behavior of various types of differential equations involving the nonlinear Sturm-Liouville differential operators. The main results obtained are as follows.(1)We have established Picorie-type identities for the nonlinear Sturm-Liouville operators and have applied them to the study of comparison and oscillation of solutions to half-linear ordinary and partial differential equations. We observe that it has long been unknown whether there is a class of nonlinear differential operators for which a Picone-type identity can be established.(2)We have developed a theory of generalized trigonometric functions, on the basis of which one can successfully construct an exact nonlinear analogue of the well-known Sturmian theory for linear differential equations. The generalized Prufer transformations thus defined has made it possible to count the number of zeros of nonoscillatory solutions to a certain class of half-linear ordinary differential equations.(3)We have made a detailed analysis of the asymptotic behavior of positive solutions to some nonlinear Sturm-Liouville equations with singularities. A similar analysis has also been made of differential equations involving singular Sturm-Liouville operators. As an unexpected byproduct we have discovered a new type of singular solution which has never appeared in the literature.(4)Regarding the oscillation of functional differential equations, we have established (i) a new comparison theorem holding for a special class of non-neutral equations and (ii) an effective criterion for oscillation of neutral equations involving the nonlinear Sturm-Liouville differential operators.