Qualitative and Quantitative Analysis of Nonlinear Oscillation of Differential Equations

点击翻译按钮获取中文摘要

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.