Qualitative and Quantitative Analysis of Nonlinear Oscillation of Differential Equations

微分方程非线性振动的定性和定量分析

• 批准号: 09640237
• 负责人: KUSANO Takashi
• 金额: $ 1.92万
• 依托单位: Fukuoka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640237/
• 关键词: oscillation theory; nonlinear differential equation; nonlinear Sturm-Liouville operator; oscillatory; zero; singular solution; 振動解.非振動解; 定性的理論; 定量解析

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.

本研究项目致力于研究涉及非线性Sturm-Liouville微分算子的各种类型微分方程的振荡行为。得到的主要结果如下：(1)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研究。我们注意到，是否存在一类非线性微分算子可以建立picone型恒等式一直是未知的。(2)我们发展了一种广义三角函数理论，在此基础上，人们可以成功地构造出与著名的线性微分方程斯图尔米安理论的精确非线性模拟。由此定义的广义普吕弗变换使得计算一类半线性常微分方程的非振荡解的零个数成为可能。(3)详细分析了一类具有奇异点的非线性Sturm-Liouville方程正解的渐近行为。对涉及奇异Sturm-Liouville算子的微分方程也作了类似的分析。作为一个意想不到的副产品，我们发现了一种从未在文献中出现过的新型奇异解。(4)关于泛函微分方程的振动，我们建立了(i)一个新的适用于一类特殊的非中立型方程的比较定理和（ii）一个涉及非线性Sturm-Liouville微分算子的中立型方程振动的有效判据。

项目成果

J.Jaros and T.Kusano: "On a class of doubly singular differential equations of second order" Fukuoka University Science Reports. 印刷中.

J.Jaros 和 T.Kusano：“关于一类二阶双奇异微分方程”福冈大学科学报告出版。

T.Kusano and M.Naito: "A singular eigenvalue problem for Sturm-Liouville equations" Differentsial'nye Uravneniya. 34. 303-312 (1998)

T.Kusano 和 M.Naito：“Sturm-Liouville 方程的奇异特征值问题”Differentsialnye Uravneniya。

T.Kusano(共著): "Positive entire solutions to nonlinear biharmonic equations in the plane" Journal of computational and Applied Mathematics. (to appear).

T. Kusano（合著者）：“平面上非线性双调和方程的正整解”《计算与应用数学杂志》（待发表）。

T.Kusano, J.Jaros and N.Yoshida: "A Picone type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order" Nonlinear Analysis : Theory, Methods and Applications. (to be published).

T.Kusano、J.Jaros 和 N.Yoshida：“一类二阶半线性偏微分方程的 Picone 型恒等式和 Sturmian 比较和振荡定理”非线性分析：理论、方法和应用。

T.Kusano and J.Jaros: "On a class of doubly singular differential equations of second order" Fukuoka University Science Reports. (to be published).

T.Kusano 和 J.Jaros：“关于一类二阶双奇异微分方程”福冈大学科学报告。