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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)

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