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International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 60907, 16 pages
doi:10.1155/2007/60907
Research Article

Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero

R. N. Rath,1 J. G. Dix,2 B. L. S. Barik,3 and B. Dihudi4

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.

Abstract

We find necessary conditions for every solution of the neutral delay difference equation Δ(rnΔ(yn−pnyn−m))+qnG(yn−k)=fn to oscillate or to tend to zero as n→∞, where Δ is the forward difference operator Δxn=xn+1−xn, and pn, qn, rn are sequences of real numbers with qn≥0, rn>0. Different ranges of {pn}, including pn=±1, are considered in this paper. We do not assume that G is Lipschitzian nor nondecreasing with xG(x)>0 for x≠0. In this way, the results of this paper improve, generalize, and extend recent results. Also, we provide illustrative examples for our results.