【作者】 杨浩；

【导师】 李洪旭；

【作者基本信息】 四川大学 ， 基础数学， 2023， 硕士

【摘要】 本文以时标上的概周期函数理论为基础,主要研究了定义在时标上的S~p-概周期函数与实数轴上S~p-概周期函数的关系以及时标上Duffing方程的伪概周期解的问题,主要内容及结果如下:第一章,介绍关于本文的研究背景及意义、概周期函数理论的研究现状、时标上概周期函数的研究现状以及时标上Duffing方程概周期解的相关研究进展.第二章,引入本文需要的概周期函数相关定义以及后文需要的引理.第三章,研究时标上S~p-概周期函数与R上S~p-概周期函数的关系,得到任意R上S~p-概周期函数可延拓定义至R上使之是R上的S~p-概周期函数,并且通过一个例子,说明R上S~p-概周期函数在T上的限制不是T上的S~p-概周期函数.通过定义时标T上的小平移集,给出在一定条件下R上S~p-概周期函数在T上限制是T上的S~p-概周期函数的充分条件,最后通过一个例子说明这个条件是必要的.第四章,研究时标上带时滞的Duffing方程的伪概周期解问题.为了处理时滞项,我们引入T →П的概周期函数及伪概周期函数,并且分析这类函数的性质,利用这些性质给出复合定理.通过降阶手段将原方程转化为方程组,从而考虑具有S~p-概周期系数和S~p-伪概周期强迫项的抽象方程,利用稳定性条件得到抽象方程伪概周期解的存在唯一性,随后,应用抽象方程的结论以及Banach不动点定理,得到Duffing方程的伪概周期解.最后,作为应用,给出定义域在时标上和实数轴上Duffing方程的例子.第五章,总结本文的研究成果并进行展望.更多还原

【Abstract】 In this thesis,based on the theory of almost periodic functions,we investigate the connection between S~p-almost periodic functions defined on time scales and R,as well as pseudo almost periodic solutions for Duffing equations on time scales.In Chapter 1,the background and significance of this thesis are presented,including the research on the theory of almost periodic functions defined on the real line and time scales,and the relevant research progress of the almost periodic solutions of the Duffing equations on time scales.In Chapter 2,we mainly introduce the relevant definitions of almost periodic functions and some lemmas required in this thesis.In Chapter 3,we study the connection between S~p-almost periodic functions defined on time scales and R.We obtain that an S~p-almost periodic function on T can be extended to an S~p-almost periodic function on R,and a counterexample is given to demonstrate that the restriction of an S~p-almost periodic function on R on T is not S~p-almost periodic on T.By introducing a concept of minor translation set,we give a condition to ensure the sufficiency of the restriction that an S~p-almost periodic function on R on T is S~p-almost periodic on T.Moreover,we show the necessity of this condition with a counterexample.In Chapter 4,we study the pseudo almost periodic solutions of the Duffing equations with delay.To deal with the delay,we introduce some concepts of almost periodic functions and pseudo almost periodic functions from T to Ⅱ,and some basic properties for these concepts.Next,by applying these results,we give a composition theorem.By reducing the second-order equation to a system,we study the corresponding abstract equations with S~p-almost coefficients and S~p-pseudo almost forcing terms.Under the stability condition,we present some results on the existence and uniqueness of pseudo almost periodic solutions for equations.Then,applying these results and the Banach fixed point theorem,we infer the existence and uniqueness of pseudo almost periodic solutions for the Duffing equations.Finally,we give two concrete examples to illustrate our main results.In Chapter 5,the main work of this thesis is summarized,and the work of further study in the future is proposed.更多还原