新型不动点定理及其在时滞分数阶微分方程的应用

项目名称： 新型不动点定理及其在时滞分数阶微分方程的应用

项目编号： No.11301039

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 吴君

作者单位： 长沙理工大学

项目金额： 23万元

中文摘要： 分数阶微分方程是一类允许非整数阶导数和积分的数学模型，是近年来在粘弹性、粘塑性力学理论以及控制论中分数阶控制器等领域涌现出来的一类重要时间演化系统。本项目拟以铸造适合研究特定非线性算子的新工具为出破口，研究以下三个内容：（1）分别在连续函数空间和逐段连续函数空间中，建立适合分数阶微分方程解算子的新型不动点定理；（2）以分数阶双曲型偏泛函微分方程、分数阶双曲型偏泛函微分包含、分数阶脉冲泛函微分方程以及分数阶脉冲泛函微分包含为对象，研究其特定解（周期解等）的存在唯一性；（3）用数值模拟方法，探索脉冲、时滞量等参数对这些微分方程或包含的特定解的影响规律。该研究结果将丰富和拓广经典的非线性算子理论，也是不动点理论的新发展，同时，也将丰富和发展时滞分数阶微分方程的基本理论。探索脉冲、时滞等参数如何影响分数阶偏泛函微分方程的特定解的相关工作，国内外尚未见报道。

中文关键词： 分数阶微分方程；不动点定理；时滞；非线性算子；脉冲

英文摘要： Fractional differential equation is a generalization of the ordinary differentiation to arbitrary non-integer order, which is more appropriate and more efficient in the modeling of Mechanics (theory of viscoelasticity and viscoplasticity), Bio-Chemistry (modeling of polymers and proteins), Electrical Engineering (transmission of ultrasound waves), Control Theory (implementation of fractional order controllers), etc. The project focuses on the following three primary coverages: (1)We will build some new fixed point theorems for some special nonlinear operators in continuous function spaces or piecewise continuous function spaces. The fixed point theorems we construct will be matched with the solution operators for the fractional differential equations;(2) The main research work will cover the exsitence and uniqueness of solution or periodic solution for the fractional order hyperbolic partial functional differential equations、fractional order hyperbolic partial functional differential inclusions、fractional order impulsive partial functional differential equations, and fractional order impulsive partial functional differential inclusions;(3)Also， we will depict the effects of the delay and impulse to the dynamical behaviors by simulations. This research would improve and generalize the theory of fractional order

英文关键词： fractional differential equations；fixed point theorems；delay；nonlinear operator；impulse