【作者】 田淑环；

【导师】 王培光；

【作者基本信息】 河北大学 ， 基础数学， 2007， 硕士

【摘要】 本硕士论文主要讨论了差分方程的稳定性和边值问题,包含四部分内容.第一章主要介绍了课题的研究背景、现状和本文的主要工作及其意义.第二章通过引入向量函数上拟单调增的概念,建立了一个新的比较定理,揭示了两个向量差分系统与一个纯量差分系统解的关系,并结合向量李雅普诺夫函数,得到了差分系统的两度量稳定性的判据,最后给出例子加以验证.第三章考虑了差分方程的非线性边值问题,首先将非线性边值问题转化为初值问题,再利用耦合上下解方法,从而得到了一种广义的单调迭代方法.第四章包含两部分内容,第一部分考察了只含有滞后项的差分方程的非线性边值问题,主要借助了极大值原理和单调迭代方法,并给出例子加以验证；第二部分利用极大值原理和耦合上下解方法,证明了同时含有滞后和超前项的泛函差分方程解的存在性问题.更多还原

【Abstract】 This dissertation mainly deals with stability and boundary value problems for dif-ference equations and is composed of four chapters.In chapter one, the history of the study about stability and boundary value problems of difference equation is introduced accompanied with some prints of present work as well as the practical and theoretical values of this paper.In chapter two, by employing the concept of the upper quasi-monotone increasing of functions, we get a new comparison theorem, which connects the solutions of two finite difference equations and a scalar one. Combing vector Lyapunov functions, we establish new stability criteria in terms of two measures for difference equations and an example is given to illustrate the results obtained.In chapter three, using coupled upper and lower solutions, translating nonlinear functional boundary value problems into initial value problems, generalized monotone iterative method for difference equations with nonlinear functional boundary value con-ditions are discussed.Chapter four is composed of two sections. In the first section, we mainly consider nonlinear functional boundary value problems of functional difference equations only with retardation through a maximum principle and monotone iterative technique and an example is given to validate the results. In the second section, following the first section using the maximum principle and coupled upper and lower solutions, we prove existence results for functional difference equations involving retardation and anticipation via constructing monotone iterative sequences.更多还原