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Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation, II

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  • Sugie, Jitsuro

Abstract

The present paper deals with nonoscillation problem for the second-order linear difference equation cnxn+1+cn−1xn−1=bnxn,n=1,2,…,where {bn} and {cn} are positive sequences. All nontrivial solutions of this equation are nonoscillatory if and only if the Riccati-type difference equation qnzn+1zn−1=1has an eventually positive solution, where qn=cn2/(bnbn+1). Our nonoscillation theorems are proved by using this equivalence relation. In particular, it is focusing on the relation of the triple (q3k−2,q3k−1,q3k) for each k∈N. Our results can also be applied to not only the case that {bn} and {cn} are periodic but also the case that {bn} or {cn} is non-periodic. To compare the obtained results with previous works, we give some concrete examples and those simulations.

Suggested Citation

  • Sugie, Jitsuro, 2017. "Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation, II," Applied Mathematics and Computation, Elsevier, vol. 304(C), pages 142-152.
    • Handle: RePEc:eee:apmaco:v:304:y:2017:i:c:p:142-152
    DOI: 10.1016/j.amc.2017.01.048
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