一类时滞积分方程解的存在性

项目名称： 一类时滞积分方程解的存在性

项目编号： No.11271235

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 康淑瑰

作者单位： 山西大同大学

项目金额： 68万元

中文摘要： 具有周期时滞的积分方程、泛函微分方程和分数阶微分方程在许多应用问题中起着重要作用。把泛函微分方程或分数阶微分方程转化为积分方程讨论解的存在性是一种研究方法。本项目主要研究定义在具有周期结构的无界集上变积分区域积分方程和分数阶积分方程解的存在性、唯一性、稳定性等问题。通过应用不动点理论、不动点指数理论、拓扑度理论、临界点理论、Morse理论、重合度理论研究（1）一类定义在具有周期结构的无界集上积分方程周期解的存在唯一性和多解存在性；（2）非线性分数阶积分方程解的存在唯一性和多解存在性；应用插值理论、差分理论和数学分析等方法研究（3）方程的数值解与稳定性等。得到这类方程在非线性项不同条件下依赖参数多个周期解存在性、存在唯一性、稳定性和数值解法问题的结果。这些结果将使得许多泛函微分方程解的存在性成为该方程的特例。本课题的研究在方程定性理论和动力系统研究应用中是重要的。

中文关键词： 积分方程；不动点理论；分数阶微分方程；李雅普诺夫函数；稳定性分析

英文摘要： Integral equations， functional differential equation with periodic delays and fractional differential equation have played a significant role in many problem of application. The existence of solutions for functional differential equation or fractional differential equation is equivalent to the existence of solutions for some integral equations is a kind research method. In this program, we study the existence, uniqueness and stability of solutions for more general integral equations which are defined in a closed but unbounded set with periodic structure and fractional integral equations. By fixed point theory, topological degree theory, fixed point index theory, critical point theory, Morse theory and coincidence degree theory, we mainly discussed (1) The existence, uniqueness and multiplicity of periocic solutions for a class of integral equations defined on a set with periodic structure; (2) Existence, uniqueness and multiplicity of solutions for nonlinear fractional integral equations. By interpolations theory, difference theory and mathematical analysis theory, we mainly discussed (3) Numerical Methods for integral equations, and stability of solutions for integral equation. We obtained the conditions of the existence, uniqueness, stability and numerical methods of solutions for integral equations with diffe

英文关键词： integral equation；fixed point theory；fractional differential equation；Lyapunov function；stability analysis