臺灣博碩士論文加值系統

這篇論文是延續之前顏玟檍和蕭婉玲的碩士論文。我們首先研究 Kwong 和 Zettl [8]提出的線性方程的套疊原理。此原理指出，我們可以藉由移除原勢能函數 a(x) 在定義域上的某些區間之後所形成的新勢能函數 a_1 (x) 的研究來確定原線性微分方程是否具有振盪性。此原理有一些應用。特別的是，它能助於確定附有振盪性勢能函數，甚至是週期性勢能函數的微分方程的振盪性。我們更仔細的去研究對於任何的實數 γ， y''+x^γ ϕ(x)y=0 此方程的振盪性。在這裡 ϕ(x) 是一個片段連續、具有 T 週期性並且 ∫_0^T ϕ(x)dx=0 的函數。而這個結果似乎是新的。
另外，套疊原理與其部分應用也可以延伸至半線性微分方程。

This thesis is a follow-up of the studies carried out in two previous theses by W.I. Yen and W.L. Hsiao. We first investigate the telescoping principle introduced by Kwong and Zettl [8]. By the principle, one can determine whether a linear differential equation is oscillatory or not by trimming off some intervals in the domain of the potential function a(x) and studying the new function a_1 (x) .
The principle has a number of implications. In particular, it helps to determine the oscillation of a differential equation with oscillatory potentials, and even periodic potentials. We shall study in detail the oscillation of the equation y''+x^γ ϕ(x)y=0, for any γ∈R . Here ϕ(x) is a piecewise continuous T-periodic function such that ∫_0^T ϕ(x)dx=0. This result seems to be new.
The telescoping principle and some of its implications can also be extended to half-linear equations.

1 Introduction........................1
2 Integral inequalities and linear oscillation....................8
2.1 Integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Telescoping principle and oscillation theorem . . . . . . . . . . . . . . . 10
3 Differential Equations with Oscillatory Potentials...................19
3.1 Willett''s test for class O and class NO . . . . . . . . . . . . . . . . . . . . . .19
3.2 Oscillatory potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
3.3 Periodic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Converse of telescoping principle . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Half-Linear Equations........................................40
4.1 Integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Telescoping principle and oscillation theorem . . . . . . . . . . . . . . . 41

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