International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 60907, 16 pages
Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero
1PG Department of Mathematics, Khallikote Autonomous College, Berhampur 760001, Orissa, India
2Department of Mathematics, Texas State University, San Marcos 78666, Texas, USA
3Department of Basic Sciences and Humanities, Konark Institute of Science & Technology, Jatni 752050, Bhubaneswar, Orissa, India
4National Institute of Science & Technology, Palur Hills, Berhampur 761008, Orissa, India
Received 20 January 2007; Revised 30 May 2007; Accepted 4 August 2007
Academic Editor: Attila Gilanyi
Copyright © 2007 R. N. Rath 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.
We find necessary conditions for every solution of the neutral delay difference equation to oscillate or to tend to zero as , where is the forward difference operator , and are sequences of real numbers with . Different ranges of , including , are considered in this paper.
We do not assume that is Lipschitzian nor nondecreasing
with for . In this way, the results of this paper improve, generalize, and extend recent results. Also, we provide illustrative examples for our results.