Asymptotic behavior of solutions of second order selfadjoint difference equations with disturbing terms (Q2719907)

Sufficient conditions which guarantee that the solutions of the difference equation NEWLINE\[NEWLINE{\Delta (c_{n}\Delta z_{n})+a_{n}z_{n+1}=f(n,z_{n},z_{n+1})}NEWLINE\]NEWLINE have the asymptotic behavior \(\lim_{n\rightarrow \infty }z_{n}=\alpha \) or \(\lim_{n\rightarrow \infty }\frac{z_{n}}{C_{n}}=\beta ,\) where \(\alpha ,\beta \) are real constants and \(C_{n}=\sum^{n}_{j=1}c_{j}^{-1}.\) Two examples illustrating the results are given.