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Oscillatory second order linear difference equations and Riccati equations. (English) Zbl 0619.39005
A nontrivial solution of the equation $$c_ nx_{n+1}+c_{n-1}x_{n- 1}=b_ nx_ n$$, $$c_ n>0$$, $$b_ n>0$$, is called oscillatory if for every $$N>0$$ there exists $$n\geq N$$ such that $$x_ nx_{n+1}\leq 0$$. Either all nontrivial solutions are oscillatory or none are. Authors’ abstract: Oscillation criteria are established for the equation $$c_ nx_{n+1}+c_{n-1}x_{n-1}x_{n-1}=b_ nx_ n$$, $$c_ n>0$$, involving asymptotic behavior of the quantity $$\alpha_{n,m}=4\prod^{m}_{j=0}(4q_{n+j})^{-1},$$ where $$q_ n=c^ 2_ n/(b_ nb_{n+1})$$. We also show that the given equation is oscillatory if $$y_{n+1}+y_{n-1}=(q_ n^{-1}-1)y_ n$$ is oscillatory. This result is then employed to obtain several new oscillation criteria. Riccati difference equations are used to prove the basic results.
Reviewer: K.Cooke

MSC:
 39A10 Additive difference equations 39A12 Discrete version of topics in analysis
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