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Oscillation theorems for certain class of nonlinear difference equations. (English) Zbl 0906.39008

In recent years there is a growing interest in the study of discrete difference equations with solutions exhibiting oscillatory behavior because they are interesting in many applications. However, most of the oscillatory results are given for homogeneous difference equations with or without delay.
The major goal of this paper is to give new oscillatory results for a class of forced (nonhomogeneous) nonlinear difference equations of the form \[ \Delta \bigl(a_n \Delta (x_n+ p_nx_{n-k}) \bigr) +q_n f(x_{n+1- \ell})= e_n, \quad n=0,1,2, \dots, \] where \(\Delta y_n= y_{n+1} -y_n\), \(\{a_n\}\), \(\{p_n\}\), \(\{e_n\}\), \(\{q_n\}\) are given sequences, \(k\geq 0\) and \(\ell\geq 0\) are given integers and \(f:R\to R\) is a continuous function with \(uf(u) >0\) for \(u\neq 0\). The results are illustrated by examples.

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
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