【作者】 颜荣；

【导师】 姚慧丽；

【作者基本信息】 哈尔滨理工大学 ， 基础数学， 2011， 硕士

【摘要】 概周期型函数理论在微分方程的应用这一研究课题备受数学工作者关注。很多数学研究者在讨论具有逐段常变量的微分方程的概周期型解的存在时,通常利用相应的差分方程的概周期型序列解。所以研究差分方程的概周期型序列解的存在性是很有必要的。本文主要讨论三类不同的差分方程的概周期型序列解,并把其中一个方程的遥远概周期序列解的相关结果应用到一类具有逐段常变量微分方程中,考虑其遥远概周期解的唯一存在性。主要内容如下:第一部分讨论了一类离散的Logistic差分方程的渐近概周期序列解的存在性。在已有的结果上,对条件做了一些变动,得到了一些新的结果。第二部分给出了一类常系数的差分方程的渐近概周期序列解存在的充要条件。这一结果的证明是从矩阵的角度来考虑,通过把N阶矩阵转化为一阶来分析问题。第三部分讨论了一类差分方程的遥远概周期序列解的唯一存在性。并把所得到的遥远概周期序列解的相关结果应用到一类逐段常变量微分方程中,得到其遥远概周期解的唯一存在性。本文第一部分和第二部分的结果是已有结果的进一步推广。讨论遥远概周期函数在微分方程中应用的文献不是很多,所以说第三部分的结论是较新的,这对解决其它一些问题有着一定的应用价值。更多还原

【Abstract】 The topic that theories of almost periodic type functions are applied into differential equations has attracted more and more attention among mathematical staffs. The existence of almost periodic type solutions to a differential equation with piecewise constant argument is discussed by mathematical staffs using almost periodic type sequence solutions of the corresponding difference equation. Therefore, studying the existence of almost periodic type solutions is necessary for difference equations.In this article, almost periodic type sequences solutions for three different types of difference equations are researched and the relevant results of remotely almost periodic sequence solution are applied into a differential equation, the existence and uniqueness of remotely almost periodic solution are gotten.Primary contents are as follows:In the first part of paper, existence of asymptotically almost periodic sequence solution for the discrete Logistic differential equation is discussed. In the previous results, some changes on their conditions are made and some new results are obtained.In the second part of writings, the necessary and sufficient conditions of asymptotically almost periodic solution to the difference equation with constant coefficient are given. Results of the conclusion are proved by thinking the N-order matrix into a first-order.In the third part of text, the existence and uniqueness of remotely almost periodic sequence solution for a class of differential equations are discussed. The relevant results of remotely almost periodic sequence solution are applied into the differential equation and the existence and uniqueness of remotely almost periodic solution are gotten.The results of first part and second part are the further extension on the existing results. The literatures on remotely almost periodic solutions in differential equations are not many. So, the results of the third part are relatively new and it has great value to solve the practical problems.更多还原