臺灣博碩士論文加值系統

我們主要是研究下列型態的差分方程式有界解和無界解的存在性及行為。
a_(n)=a_(n+1)-c_(n){[a_(n+1)]^2-S^2}
，其中{c_(n)}是已知數列，n≧1。
我們得知當正項級數sum_{n=1}^infinity c_(n)收斂，則有有界解存在，且皆為單調。
而正項級數sum_{n=1}^infinity c_(n)發散，則沒有無界解。
最後我們討論當sum_{n=1}^infinity c_(n)不是正項級數時，解的存在及行為。

For sequence , {c_(n)}, we consider the following difference equation.
a_(n)=a_(n+1)-c_(n){[a_(n+1)]^2-S^2}.
We will apply the method of backward induction to establish the existence, the uniqueness and behavior of the solution under certain conditions.
We know that the difference equation has bounded monotone solution if the positive series sum_{n=1}^infinity c_(n) is convergent.
However, the difference equation has no unbounded solution if the positive series sum_{n=1}^infinity c_(n) is divergent.
Finally, we consider the existence, the uniqueness and behavior of the solution of the difference equation under sum_{n=1}^infinity c_(n) is not positive series.

第一章 正線性算子
1.1簡介..................................1
1.2正確的證明............................2
第二章 差分方程式
2.1簡介..................................7
2.2非明顯解的存在性及其行為..............9
2.3無界解的研究.........................18
2.4討論.................................22

參考文獻.........................................28

參考文獻：
[1] Chen, kung-Yu, 第六屆海峽兩岸數學研討會(北京大學 2009).
[2] Chow Y.S. On a Difference Equation with Last Conditions. Differential equations and control theory (Wuhan, 1994), 11-14, Lecture Notes in Pure and Appl. Math.,176,Dekker, New York, (1996).
[3] Earl D. Rainville,Special Functions, Macmillan Company, New York, 1971.
[4] Esra Erkuş, Oktay Duman, A Korvkin type approximation theorem in statistical sense, Stud. Sci. Math. Hungar. 43, 285-294 (2006).
[5] Huang Jei, Some Difference Equations with Last Conditions (2008).
[6] P.P. Korovkin, Linear operators and approximation theory, Hindustan Publ. Corp. (1960) (In Russian).
[7] P.P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR , 90 (1953) pp. 961-964 (In Russian).