【作者】 黄毅卿；

【导师】 周展；

【作者基本信息】 湖南大学 ， 应用数学， 2004， 硕士

【摘要】 本文讨论了几类具连续变量的差分方程及差分系统周期解的存在性，获得了一系列新结果，推广了具离散变量的差分方程的相关结论。同时，本文详细研究了具连续变量差分方程周期解的连续性问题，这种性质是具离散变量差分方程所没有的。 本文由四章构成。 第1章简述了问题产生的历史背景和本文的主要工作，并说明本文工作的理论意义和实际价值。 第2章讨论了具连续变量的一阶线性差分方程的周期解和渐近周期解的存在性，分别给出了判断存在唯一周期解、多个周期解和不存在周期解以及每一个解都是渐近周期解的充分条件。然后特别讨论了连续周期解的存在性，并给出了一些判断存在唯一连续周期解和不存在连续周期解的充分条件。 第3章讨论了具连续变量的一阶线性差分系统的周期解和渐近周期解的存在性及连续周期解的存在性，使第2章的许多结果推广到m(m≥2)维系统中，但由于矩阵运算和数的各种运算的差别，第2章中的有些结果并不能推广到m(m≥2)维系统。 第4章利用压缩不动点定理和Leray-Schauder不动点定理对具连续变量的一阶非线性差分方程周期解的存在性进行了讨论，获得了一系列判断周期解存在性的充分条件，然后利用隐函数定理讨论了连续周期解的存在性问题。最后把第2章中具连续变量的一阶线性差分方程(连续)周期解存在性的结果与本章研究的具连续变量的一阶非线性差分方程(连续)周期解存在性的结果进行了比较。更多还原

【Abstract】 In this paper, we study the existence of periodic solutions of several classes of difference equations with continuous variables and obtain a series of new results about the existence of periodic solutions, which generalize the conclusions of the corresponding difference equations with discrete variables. Moreover, we study carefully the continuity of periodic solutions of difference equations with continuous variables, which does not exist in the difference equations with discrete variables.The paper is composed of four chapters.In the first chapter, we introduce the background of problem-researching, the recent development of the research in this field and what we have done in this paper. Furthermore, we illustrate the importance of our work in theory and practice.In the second chapter, we discuss the existence of periodic solutions, asymptotically periodic solutions and continuous periodic solutions of first order linear difference equation with continuous variables and obtain a series of sufficient conditions for the existence of unique periodic solution, infinitely many periodic solutions and conditions for having no periodic solution, respectively.In the third chapter, we discuss the existence of periodic solutions, asymptotically periodic solutions and continuous periodic solutions of first order linear difference equations with continuous variables and obtain some new results, which expand the results of the second chapter, but there are still some distinction because of the difference between the matrix and the number.In the fourth chapter, we discuss the existence of periodic solutions of first order nonlinear difference equation by applying contraction mapping theorem and Leray-Schauder fixed point theorem and the implicit function theorem. Finally, we compare the sufficient conditions of the existence of continuous periodic solutions for first order linear difference equation with continuous variables with that for first order nonlinear difference equation.更多还原