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

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

International Journal of Mathematics and Mathematical Sciences, 2007, vol. 2007, 1-16

Abstract:

We find necessary conditions for every solution of the neutral delay difference equation Δ ( r n Δ ( y n − p n y n − m ) ) + q n G ( y n − k ) = f n to oscillate or to tend to zero as n → ∞ , where Δ is the forward difference operator Δ x n = x n + 1 − x n , and p n ,   q n ,   r n are sequences of real numbers with q n ≥ 0 ,   r n > 0 . Different ranges of { p n } , including p n = ± 1 , are considered in this paper. We do not assume that G is Lipschitzian nor nondecreasing with x G ( 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.

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:060907

DOI: 10.1155/2007/60907

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