在本篇論文中，我們主要是對下列這一個無限耦合系統之週期解存在性感到興趣 :
x(i)"+cx(i)'=f(i-1)(x(i)-x(i-1))-f(i)(x(i-1)-x(i))+h(i)(t)
其中i是任意整數，而c大於等於零，且對於所有的整數i，h(i)(t)都是T-週期的連續函數。這個系統是用來描述一維度的無限質點耦合運動，而每個質點上都有著週期性的擾動，且最近的兩個質點之間會彼此交互影響。我們主要目的是證明在 的平均值等於零的假設之下會有週期解存在，因為當函數 不同時，整個系統便會隨之變化，所以我們必須對 分類出四種不同的型式，證明系統在這四種型式之下皆會有週期解的存在，因此首先我們將這個無限耦合系統縮減成有限系統，然後再用度數理論來分析此系統，最後，利用極限的觀念便完成了證明。我們也呈現了一些數值分析的圖。

In this paper, we are concerned with the existence of T-eriodic solutions of the following infinite coupled systems :
x(i)"+cx(i)'=f(i-1)(x(i)-x(i-1))-f(i)(x(i-1)-x(i))+h(i)(t)
where i is any integer, c > = 0 and h(i)(t) is a continuous T-periodic function, for each i. This system is used to describe the motions of 1-dimensional lattices of particles with a peri-
odic perturbation and nearest neighbor interaction between par- ticles.Our main propose is to prove the existence of T-periodic solutions under the hypothesis that the mean value of is zero. Since the approach on solving the problem varies when the functions behave differently. It is needed to catalog the func-
tions to four types. We proved the existence of T-periodic sol- utions for each type. To do so,we first truncate the infinitely coupled system into a related finitely coupled system. Then the degree theory is used to analyze such system. Finally, we completed the proof by using a limit argument.

第1章 緒論......................................1
第2章 度數理論................................. 3
2.1 Brouwer 度............................... 3
2.2 Leray-Schauder 度........................10
2.3 Coincidence 度...........................13
第3章 一維度耦合系統之週期解存在性.............21
3.1 預備知識與定義...........................21
3.2 有限系統之連續同倫.......................23
3.3 主要結果.................................34
3.4 數值分析.................................37
參考文獻 .........................................47

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