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


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