IJPAM: Volume 15, No. 4 (2004)


Man Kam Kwong
Lucent Technologies
2000 N. Naperville Road
Naperville, IL 60566, USA
e-mail: mkkwong@lucent.com

Abstract.We present several results related to the classical Lyapunov inequality for the oscillation of second-order linear equations. The first is an improved Lyapunov inequality given in terms of the downswing of the functions $ \int_{a}^{t} (t-a)q(t)\,dt $ and $ \int_{t}^{b} (b-t)q(t)\,dt $, extending earlier results of Kwong and Harris and Kong. Nonoscillation criteria are derived as corollaries. A Lyapunov-type inequality for two consecutive zeros of the derivative of a solution is then established and a nonoscillation criterion given as a corollary. An oscillation criterion for positive $ q(t) $ is also proved. It extends the known condition $ \int_{}^{} t^\gamma q(t)\,dt=\infty $, $ \gamma \in[0,1) $.

Received: July 13, 2004

AMS Subject Classification: 34C10

Key Words and Phrases: linear second-order differential equations, oscillation, integral criteria

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 15
Issue: 4