Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero
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 o...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2007-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2007/60907 |
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| Summary: | 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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| ISSN: | 0161-1712 1687-0425 |