這篇論文是配合上解及下解(Upper and Lower Solution) 找出適當的算子，再利用 Schauder 定點定理，探討p-Laplacian算子方程在Sturm-Liouville-Like四點邊界值下對稱正解的存在性。

In this paper, the Sturm–Liouville-like four-point boundary value problem with a p-Laplacian operator. 
By the use of Schauder fixed point theorem and upper and lower solution, we explore the p-Laplacian operator equation in the Sturm-Liouville-Like a four-point boundary value under symmetric Existence of Positive Solutions .

目錄
一、	引言 …………………………………………… 1
二、	背景知識 ……………………………………… 2
三、	主要定理 ……………………………………… 5
四、	附錄  ………………………………………… 11
五、	參考文獻……………………………………… 17

1.	Robert G. Bartle, The Elements of Real Analysis Second Edition, John Wiley and Sons,New York (1978).
2.	Zhanbing Bai and Zengji Du, Positive solutions for some second-order four-point boundary value problems, J. Math. Anal. Appl. (2006).
3.	John B. Conway, A Course in Functional Analysis Second Edition, Springer-Verlag New York Inc. (1985).
4.	Z. Du, C. Xue and W. Ge, Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative, Arch. Math., 84(2005), 341–349.
5.	W. Feng and J.R.L. Webb, Solvability of a three-point nonlinear boundary value problems at resonance, Nonlinear Anal. 30 (1997), 3227–3238.
6.	C. P. Gupta, Solvability of a three-point boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168(1992), 540–551.
7.	V.A. Il’in and E.I. Moiseer, Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differen. Equat. 23 (1987), 803–810.
8.	V.A. Il’in and E.I. Moiseer, Nonlocal boundary value problem of the second kind for a Sturm–Liouville operator, Differen. Equat. 23 (1987), 979–987.
9.	Dehong Ji, Weigao Ge and Yitao Yan, The existence of symmetric positive solutions for Sturm–Liouville-like four-point boundary value problem with a p-Laplacian operator, Applied Mathematics and Computation 189 (2007),1087–1098.
10.	R. A. Khan and J. R. L. Webb, Existence of at least three solutions of a second-order threepoint boundary value problem, Nonlinear Anal., 64(2006), 1356–1366.
11.	Bing Liu, Positive solutions of three-point boundary value problems for the one-dimensional p-Laplacian with infinitely many singularities, Appl. Math. Lett. 17 (2004), 655–661.
12.	FuYI Li and YaJing Zhang, Multiple symmetric nonnegative solutions of second order ordinary differential equations, Appl. Math. Lett. 17 (2004), 261–267.
13.	R. Ma and H. Wang, Positive solutions of a nonlinear three-point boundary value problems, J. Math. Anal. Appl., 279(2003), 216–227.
14.	J. R. L. Webb, Positive solutions of some three point boundary value problems via fixed point index theory, Nonlinear Anal., 47(2001), 4319–4332.
15.	F.H. Wong, T.G. Chen and S.P. Wang, Existence of positive solutions for various boundary value problem, Computers and Mathematics with Applications 56 (2008) 953–958.