Table of Contents
ISRN Mathematical Analysis
Volume 2011, Article ID 485203, 9 pages
http://dx.doi.org/10.5402/2011/485203
Research Article

Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations

1Department of Basic Courses, Qingdao Technological University (Linyi), Feixian 273400, Shandong, China
2Department of Mathematics, Qufu Normal University, Qufu 273165, Shandong, China

Received 8 August 2011; Accepted 22 September 2011

Academic Editor: Z. Dosla

Copyright Β© 2011 Yazhou Tian and Fanwei Meng. 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

The existence of nonoscillatory solutions of the higher-order nonlinear differential equation [π‘Ÿ(𝑑)(π‘₯(𝑑)+𝑃(𝑑)π‘₯(π‘‘βˆ’πœ))(π‘›βˆ’1)]ξ…ž+βˆ‘π‘šπ‘–=1𝑄𝑖(𝑑)𝑓𝑖(π‘₯(π‘‘βˆ’πœŽπ‘–))=0,𝑑β‰₯𝑑0, where π‘šβ‰₯1,𝑛β‰₯2 are integers, 𝜏>0,πœŽπ‘–β‰₯0,π‘Ÿ,𝑃,π‘„π‘–βˆˆπΆ([𝑑0,∞),𝑅),π‘“π‘–βˆˆπΆ(𝑅,𝑅)(𝑖=1,2,…,π‘š), is studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general 𝑄𝑖(𝑑)(𝑖=1,2,…,π‘š) which means that we allow oscillatory 𝑄𝑖(𝑑)(𝑖=1,2,…,π‘š). In particular, our results improve essentially and extend some known results in the recent references.