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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 313725, 14 pages
http://dx.doi.org/10.1155/2012/313725
Research Article

Nonoscillatory Solutions of Second-Order Differential Equations without Monotonicity Assumptions

Department of Mathematics and Computer Science, University of Central Missouri, Warrensburg, MO 64093, USA

Received 25 March 2012; Accepted 7 June 2012

Academic Editor: Chong Lin

Copyright © 2012 Lianwen Wang and Rhonda McKee. 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 continuability, boundedness, monotonicity, and asymptotic properties of nonoscillatory solutions for a class of second-order nonlinear differential equations [ 𝑝 ( 𝑡 ) ( 𝑥 ( 𝑡 ) ) 𝑓 ( 𝑥 ( 𝑡 ) ) ] = 𝑞 ( 𝑡 ) 𝑔 ( 𝑥 ( 𝑡 ) ) are discussed without monotonicity assumption for function g. It is proved that all solutions can be extended to infinity, are eventually monotonic, and can be classified into disjoint classes that are fully characterized in terms of several integral conditions. Moreover, necessary and sufficient conditions for the existence of solutions in each class and for the boundedness of all solutions are established.