Table of Contents
Chinese Journal of Mathematics
Volume 2015, Article ID 358936, 8 pages
http://dx.doi.org/10.1155/2015/358936
Research Article

On the Stochastic Stability and Boundedness of Solutions for Stochastic Delay Differential Equation of the Second Order

1Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt
2Department of Mathematics, Faculty of Science, Assiut University, New Valley Branch, Al Khargah, New Valley 72111, Egypt

Received 23 November 2014; Revised 18 March 2015; Accepted 19 March 2015

Academic Editor: Zhen-Lai Han

Copyright © 2015 A. M. A. Abou-El-Ela et al. 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 present two qualitative results concerning the solutions of the following equation: ; the first result covers the stochastic asymptotic stability of the zero solution for the above equation in case , while the second one discusses the uniform stochastic boundedness of all solutions in case . Sufficient conditions for the stability and boundedness of solutions for the considered equation are obtained by constructing a Lyapunov functional. Two examples are also discussed to illustrate the efficiency of the obtained results.