Topics on Oscillations of Linear Differential Equations

• Main Authors: Shu-jing Tang, 唐淑靜
• Other Authors: Chun-Kong Law
• Format: Others
• Language: en_US
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/70818354599383449916

碩士 === 國立中山大學 === 應用數學系研究所 === 101 === 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.