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

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.