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Oscillation of nonlinear difference equations with several coefficients. (English) Zbl 1163.39004

Summary: We provide some sufficient conditions for the oscillation of every solution of the following linear difference equation:
\[ x_{n+1}-x_n +\sum^m_{i=1} p_{in} x_{n-k_i}=0, \]
where \(k_i\in\{\dots,-3,-2\}\) and \(p_{in}\leq 0\) for \(i =1,2,\dots,m\). Furthermore, we consider the oscillation properties of the following nonlinear difference equations
\[ \begin{aligned} &x_{n+1}-x_n+(1+x_n)\sum^m_{i=1} p_{in}x_{n-k_i}=0,\\ & x_{n+1}-x_n+\sum^m_{i=1} p_{in} f_i(x_n-k_i)=0,\end{aligned} \] in the cases \(k_i\in\{0,1,\dots\}\) and \(p_{in}\geq 0\) for \(i=1,2,\dots,m\).

MSC:

39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
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