Some new oscillation criteria for second order quasi-linear neutral delay differential equations

Abstract

In this paper, we will establish some new oscillation criteria for second order quasi-linear neutral delay differential equations of the form [ r ( t ) | ( x ( t ) + p ( t ) x ( t - τ ) ) ′ | α - 1 ( x ( t ) + p ( t ) x ( t - τ ) ) ′ ] ′ + q ( t ) f ( x ( σ ( t ) ) ) = 0 . The results obtained essentially improve and extend those of Dzurina and Stavroulakis [J. Dzurina, I.P. Stavroulakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comput. 140 (2003) 445–453], Sun and Meng [Y.G. Sun, F.W. Meng, Note on the paper of Dzurina and Stavroulakis, Appl. Math. Comput. 174 (2006) 1634–1641].