這篇論文的目的是要對二階線性微分方程的比較定理和振盪定理做整理後的概論。我們將會詳細的討論四個比較定理：Sturm-Picone，Levin，Reid 和 Leighton 的比較定理。而在振盪性質方面，我們將研究 Hille-Kneser 的定理及Wintner和Leighton的振盪準則，當中涉及一個黎卡提方程的分析。1969年，J.S.W Wong對微分方程的振盪與非振盪做了更深層次的研究，我們會解釋這些結果並用一些例子做詳細的驗證。
我們論文的結果，主要建基在 Swanson [12] 和鄧宗琦 [20] 等兩本專著，再外加一篇 Wong [18] 的論文。而在論文中，我們將會將一些結果做更簡化的證明討論和延伸。

This thesis is intended to be a survey on the comparison theorems and oscillation theorems for second order linear differential equations. We shall discuss four comparison theorems in detail: Sturm-Picone, Levin, Reid and Leighton comparison theorems. For oscillation properties, we shall study Hille-Kneser theorems, and Wintner and Leighton oscillation criteria, which involves analysis of a Riccati equation. In 1969, J.S.W. Wong had some deep results about oscillatory and nonoscillatory differential equations. We shall explain these results and some of the examples in detail.
This survey is mainly based on the monographs of Swanson [12], and Deng [20], plus a paper of Wong [18]. In some places, we give simplifications and extensions.

1 Introduction . . ... . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 1
2 Comparison Theorems . . . . . .. . . . . . . . . . . . . . . . . .6
2.1 Sturm comparison theorem . . . . . . . . . . . . . . . . . . 6
2.2 Levin's comparison theorem . . . . . . . . . . . . . . . . 10
2.3 Reid's comparison theorem . . . . . . . . . . . . . . . . 14
2.4 Leighton's comparison theorem . . . . . . . . . . . . . 15
3 Oscillation Theorems for Equations with Continuous Coecients. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . 18
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Kneser's criteria and Hille's renement . . . . . . . . 19
3.3 Oscillation criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Hille-Wintner comparison theorem . . . . . . . . . . . 27
4 Oscillation Theorems for Equations with Integrable Coecients . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . .31
4.1 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Appendix. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . 47
6 Bibliography . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. ..51

G.D. Birkhoff and G.C. Rota, Ordinary Differential Equations, John Wiley and Sons, 1989.

E. Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234-252.

P. Hartman, Differential equations with non-oscillatory eigenfunctions, Duke Math. J. 15 (1948), 697-709.

P. Hartman, Ordinary Differential Equations, S.M. Hartman, Baltimore, 1973.

E. Kamke, A new proof of Sturm's comparison theorems, The Amer. Math. Mon. 46 (1939), 417-421.

A. Kneser, Untersuchungen uber die reelen nullstellen der integrale linearer differentialgleichungen, Math. Ann. 42 (1893), 409-435.

W. Leighton, Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc. 13 (1962), 603-610.

A.Ju. Levin, A comparison principle for second-order differential equations, Soviet Math. Dokl. 1 (1960), 1313-1316.

H.J. Li and C.C. Yeh, Sturm comparison theorem for a half-linear second order differential equation, Proc. Royal Society Edinburgh, 125A (1995), 1193-1204.

Z. Nehari, Oscillation criteria for second-order linear differential equations, Trans. Amer. Math.Soc.85 (1957), 428-445.

W.T. Reid, A comparison theorem for self-adjoint differential equations of second order, Ann. of Math. [2] 65 (1957), 197-202.

C.A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1968.

C. Sturm, Sur les equations differentielles lineaires du second ordre, J. Math. Pures Appl. 1 (1836), 106-186.

C.-T. Taam, Non-oscillatory differential equations, Duke Math. J. 19 (1952), 493-497.

A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115-117.

A. Wintner, On the non-existence of conjugate points, Amer. J. Math. 73 (1951), 368-380.

A. Wintner, On the comparison theorem of Kneser-Hille, Math. Scand. 5 (1957), 255-260.

J.S.W. Wong, Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients, Trans. Amer. Math. Soc. 144 (1969), 197-215.

A. Zettl, Sturm-Liouville Theory, American Mathematical Society, Providence, 2005.

鄧宗琦, 常微分方程邊值問題和Sturm比較理論引論, 華中師範大學出版社, 1990.