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Asymptotic and oscillatory behavior of solutions of nonlinear second order difference equations. (English) Zbl 0784.39003
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. B. Szmanda [J. Math. Anal. Appl. 79, 90-95 (1981; Zbl 0455.39004) or J. W. Hooker and W. T. Patula [ibid. 91, 9-29 (1983; Zbl 0508.39005)].
Reviewer: L.Berg (Rostock)

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)