Table of Contents
ISRN Mathematical Analysis
Volume 2013, Article ID 946453, 11 pages
http://dx.doi.org/10.1155/2013/946453
Research Article

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

Kent State University at Stark, 6000 Frank Avenue NW, Canton, OH 44720-7599, USA

Received 6 June 2013; Accepted 25 July 2013

Academic Editors: M. McKibben, A. Peris, and C. Zhu

Copyright © 2013 Gro Hovhannisyan. 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

We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson's asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.