几类非自伴微分方程周期解的存在性及渐近性态

项目名称： 几类非自伴微分方程周期解的存在性及渐近性态

项目编号： No.10871160

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 李永祥

作者单位： 西北师范大学

项目金额： 15万元

中文摘要： 本项目主要研究高阶非线性常微分方程、抽象空间一阶非线性发展方程及非线性电报方程三类非自伴微分方程周期解的存在性。我们用上下解方法、谱扰动方法、拓扑度方法及锥上的不动点指数理论等非线性分析工具研究了这三类方程的周期问题，获得了一些有意义的结果。首先，我们扩充和发展了这三类方程在周期边界条件下的极大值原理，从而用上下解方法获得了一些新的存在结果。对高阶常微分方程周期边值问题，我们加强了极大值原理的结论，建立了对应的线性方程解的强正性估计，借助于这些估，我们应用锥上的不动点指数理论获得了对应的超线性方程非平凡解的存在性结果，这与用临界点理论获得的自伴方程的结果是类似的。针对这三类方程的非自伴性，我们精确刻划了相应的线性算子的谱特征，在此基础上提出了与自伴方程非共振条件对应的谱隔离条件，建立了谱隔离条件下解的存在性与存在唯一性结果。这些结果多方面地发展了非自伴微分方程周期解的存在性研究。

中文关键词： 周期解；非自伴算子；强正性估计；谱分离条件

英文摘要： The main purpose of this project is to study the existence of periodic solutions for three types of non-selfadjoint differential equations: higher order nonlinear ordinary differential equations, abstract first order nonlinear evolution equations and nonlinear telegraph equations. we use the theory and method of nonlinear analysis such as upper and lower solutions method, spectral disturbance method, continuation method of topological degree and theory of fixed point index in cones to study the periodic problem of the equations of the three types, and obtain some new significant results. Firstly, we extend and develop the maximum principle for the equation of the three types in periodic boundary conditions respectively, and then obtain some new existence results by using upper and lower solutions method. Next, we strengthen the conclusions of the maximum principle for higher order ordinary differential equations, and build the strongly positive estimates for the solutions of associated linear equations. By this estimates, we use theory of fixed point index in cones to obtain the existence results of nontrivial solutions for superlinear equations, which are similar to the results of selfadjoint equations obtained by the theory of critical point. With the non-selfadjointness of the three types of equations, we to accurately characterize the spectrum of the associated linear operators. Then we present the spectral seperation conditions which correspond to the nonresonance conditions of selfadjoint equations and build the existence and uniqueness results in the conditions. These results develop the researches of the existence of periodic solutions of the non-selfadjoint differential equations in multiple directions.

英文关键词： periodic solution; non-selfadjoint operator; strongly positivty estimate; spectral seperation conditions