Asymptotic and oscillatory behavior of solutions of nonlinear second order difference equations (Q686209)

For the difference equation \(\Delta(a_ n\Delta y_ n)+p_ n\Delta y_ n+q_ nf(y_ n)=0\) with \(\Delta y_ n=y_{n+1}-y_ n\), \(a_ n>0\), \(p_ n\geq 0\) and \(q_ n>0\) for \(n\geq n_ 0\) as well as \(uf(u)>0\) for \(u\neq 0\) there are proved 3 theorems that under additional conditions all real nontrivial solutions \(y_ n\) are either oscillatory or monotonically tending to zero as \(n\to\infty\). Three examples show that the theorems do not require the condition \(\sum 1/a_ n=\infty\) as in earlier papers, cf. e.g. \textit{B. Szmanda} [J. Math. Anal. Appl. 79, 90-95 (1981; Zbl 0455.39004) or \textit{J. W. Hooker} and \textit{W. T. Patula} [ibid. 91, 9-29 (1983; Zbl 0508.39005)].