Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 195619, 10 pages
http://dx.doi.org/10.1155/2011/195619
Research Article

Oscillation of Certain Second-Order Sub-Half-Linear Neutral Impulsive Differential Equations

Yuangong Sun

School of Mathematics, University of Jinan, Shandong, Jinan 250022, China

Received 17 May 2011; Accepted 30 June 2011

Academic Editor: Garyfalos Papaschinopoulos

Copyright © 2011 Yuangong Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By introducing auxiliary functions, we investigate the oscillation of a class of second-order sub-half-linear neutral impulsive differential equations of the form [ 𝑟 ( 𝑡 ) 𝜙 𝛽 ( 𝑧 ( 𝑡 ) ) ] + 𝑝 ( 𝑡 ) 𝜙 𝛼 ( 𝑥 ( 𝜎 ( 𝑡 ) ) ) = 0 , 𝑡 ≠ 𝜃 𝑘 , Δ 𝜙 𝛽 ( 𝑧 ( 𝑡 ) ) | 𝑡 = 𝜃 𝑘 + 𝑞 𝑘 𝜙 𝛼 ( 𝑥 ( 𝜎 ( 𝜃 𝑘 ) ) ) = 0 , Δ 𝑥 ( 𝑡 ) | 𝑡 = 𝜃 𝑘 = 0 , where 𝛽 > 𝛼 > 0 , 𝑧 ( 𝑡 ) = 𝑥 ( 𝑡 ) + 𝜆 ( 𝑡 ) 𝑥 ( 𝜏 ( 𝑡 ) ) . Several oscillation criteria for the above equation are established in both the case 0 ≤ 𝜆 ( 𝑡 ) ≤ 1 and the case − 1 < − 𝜇 ≤ 𝜆 ( 𝑡 ) ≤ 0 , which generalize and complement some existing results in the literature. Two examples are also given to illustrate the effect of impulses on the oscillatory behavior of solutions to the equation.