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Approach to the qualitative theory of functional equations by phase plane analysis

通过相平面分析探讨函数方程定性理论

基本信息

批准号：
16540152
负责人：
SUGIE J.
金额：
$ 2.24万
依托单位：
Shimane University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

项目摘要

(i) We considered a nonlinear delay difference equation, and gave sufficient conditions for all nontrivial solutions to be oscillatory and sufficient conditions for the existence of a nonoscillatory solution. Our results were proved by use of phase plane analysis for a system equivalent to this equation.(ii) The prototype of Lienard system was first formulated by the French physicist A. Lienard in 1928. The system played an important role in the development of the qualitative theory of differential equations. In this research, we gave sufficient conditions for the zero solution of nonautonomous Lienard systems to be globally asymptotically stable. We also investigated the existence of homoclinic orbits in generalized Lienard systems and obtained some conditions under which the Lienard system has homoclinic orbits. As a practical application, we gave a necessary and sufficient condition for the the existence of homoclinic orbits in Gause-type predator-prey models which are under the inf … More luence of an Allee effect.(iii) It is well-known that the solution space of linear differential equations has homogeneity and additivity; that is, any multiple of a solution is also a solution and the sum of two solutions is another solution. The investigation of half-linear differential equations has attracted considerable attention in the last two decades. The term half-linear differential equations is derived from the fact that the solution space has just one half of the above properties, namely, homogeneity (but not additivity). Half-linear differential equations are sometimes called differential equations with the one-dimensional p-Laplacian. In this research, we discussed the asymptotic behavior of solutions and the global asymptotical stability of the zero solution of nonautonomous half-linear differential systems.(iv) We dealt with the oscillation problem for various differential equations such as self-adjoint differential equations, half-linear differential equations, nonlinear differential equations with p-Laplacian or with perturbed terms, and elliptic equations. We presented sufficient conditions for all nontrivial solutions to be oscillatory and sufficient conditions for all nontrivial solutions to be nonoscillatory. The obtained theorems extended many previous results on this problem. Our main method is phase plane analysis for a system equivalent to each equation.(v) Combining Lienard system and differential equations with the one-dimensional p-Laplacian, we considered Lienard-type system with p-Laplacian. We gave sufficient conditions under which this new system has at least one stable limit cycle. The main results were proved by means of phase plane analysis with the Poincare-Bendixson theorem. Less

(i)考虑了一类非线性时滞差分方程，给出了所有非平凡解振动的充分条件和存在非振动解的充分条件。我们的结果证明了使用相平面分析的系统等效于这个方程。(ii)Lienard系统的原型是由法国物理学家A. 1928年的Lienard。该系统在微分方程定性理论的发展中发挥了重要作用。本文给出了非自治Lienard系统零解全局渐近稳定的充分条件。我们还研究了广义Lienard系统的同宿轨的存在性，得到了广义Lienard系统存在同宿轨的条件。作为一个实际应用，我们给出了Gause型捕食模型在下推条件下存在同宿轨道的一个充要条件， ...更多信息 Allee效应。(iii)众所周知，线性微分方程的解空间具有齐性和可加性，即一个解的任意倍数也是一个解，两个解的和是另一个解。半线性微分方程的研究在过去的二十年里引起了人们的极大关注。术语半线性微分方程是从解空间只有上述性质的一半，即齐次性（但不是可加性）的事实导出的。半线性微分方程有时被称为一维p-Laplacian微分方程。在本研究中，我们讨论了非自治半线性微分系统解的渐近性态和零解的全局渐近稳定性。(iv)我们研究了自伴微分方程、半线性微分方程、带p-Laplacian项或扰动项的非线性微分方程以及椭圆型方程的振动问题。给出了所有非平凡解振动的充分条件和所有非平凡解非振动的充分条件。所得定理推广了前人关于此问题的许多结果。我们的主要方法是相平面分析的系统相当于每个方程。(v)将Lienard系统与一维p-Laplacian微分方程相结合，研究了具有p-Laplacian算子的Lienard型系统。给出了该系统至少存在一个稳定极限环的充分条件。利用相平面分析的方法，利用Poincare-Bendixson定理对主要结果进行了证明。少