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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| Format: | Article |
| Language: | English |
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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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| author | R. N. Rath J. G. Dix B. L. S. Barik B. Dihudi |
| author_facet | R. N. Rath J. G. Dix B. L. S. Barik B. Dihudi |
| author_sort | R. N. Rath |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-5fdb75ada52e4aa4af55613267fc684c |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-5fdb75ada52e4aa4af55613267fc684c2025-02-03T06:00:06ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252007-01-01200710.1155/2007/6090760907Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to ZeroR. N. Rath0J. G. Dix1B. L. S. Barik2B. Dihudi3PG Department of Mathematics, Khallikote Autonomous College, Berhampur, Orissa 760001, IndiaDepartment of Mathematics, Texas State University, San Marcos, Texas 78666, USADepartment of Basic Sciences and Humanities, Konark Institute of Science & Technology, Jatni 752050, Bhubaneswar, Orissa, IndiaNational Institute of Science & Technology, Palur Hills, Berhampur, Orissa 761008, IndiaWe 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.http://dx.doi.org/10.1155/2007/60907 |
| spellingShingle | R. N. Rath J. G. Dix B. L. S. Barik B. Dihudi Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero International Journal of Mathematics and Mathematical Sciences |
| title | Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero |
| title_full | Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero |
| title_fullStr | Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero |
| title_full_unstemmed | Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero |
| title_short | Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero |
| title_sort | necessary conditions for the solutions of second order non linear neutral delay difference equations to be oscillatory or tend to zero |
| url | http://dx.doi.org/10.1155/2007/60907 |
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