Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 42120, 22 pages
doi:10.1155/JIA/2006/42120

Oscillation and nonoscillation theorems for a class of even-order quasilinear functional differential equations

Jelena Manojlović1 and Tomoyuki Tanigawa2

 1Department of Mathematics and Computer Science, Faculty of Science and Mathematics, University of Niš, Višegradska 33, Niš 18000, Serbia and Montenegro
2Department of Mathematics, Faculty of Science Education, Joetsu University of Education, Niigata 943-8512, Japan
Received 13 November 2005; Accepted 30 January 2006

Abstract

We are concerned with the oscillatory and nonoscillatory behavior of solutions of even-order quasilinear functional differential equations of the type (|y(n)(t)|αsgny(n)(t))(n)+q(t)|y(g(t))|βsgny(g(t))=0, where α and β are positive constants, g(t) and q(t) are positive continuous functions on [0,∞), and g(t) is a continuously differentiable function such that g′(t)>0, limt→∞g(t)=∞. We first give criteria for the existence of nonoscillatory solutions with specific asymptotic behavior, and then derive conditions (sufficient as well as necessary and sufficient) for all solutions to be oscillatory by comparing the above equation with the related differential equation without deviating argument.