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

Rath, R. N.; Dix, J. G.; Barik, B. L. S.; Dihudi, B.

doi:https://doi.org/10.1155/2007/60907

International Journal of Mathematics and Mathematical Sciences

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Research Article | Open Access

Volume 2007 | Article ID 060907 | https://doi.org/10.1155/2007/60907

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

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

Academic Editor: Attila Gilanyi

Received20 Jan 2007

Revised30 May 2007

Accepted04 Aug 2007

Published10 Sept 2007

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.

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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.

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