## Oscillation of second order nonlinear delay damped difference equations.(English)Zbl 1123.39008

The authors study the second-order nonlinear delay difference equation with damping term $\Delta^2 x_n+p_n\Delta x_n+F(n,x_{n-\tau},\Delta x_{n-\sigma})=0,\quad n=0,1,2,\ldots,\tag{1}$ where $$(p_n)_n$$ is a nonnegative real sequence, $$F:\mathbb N\times \mathbb R^2\to \mathbb R$$ is continuous for each $$n$$, $$\,\tau,\,\sigma\in \mathbb N$$ and $$\Delta$$ denote the forward difference operator $$\Delta x_n=x_{n+1}-x_n$$.
Under some assumptions on $$F$$ and $$(p_n)_n$$, some oscillation criteria for the above equation are proved, which generalize the results of S. R. Grace and H. A. El-Morshedy [Math. Bohemica 125, 421–430 (2000; Zbl 0969.39005)]. Two particular cases of the equation (1), namely
\begin{aligned} \Delta^2x_n+p_n\Delta x_n+q_n x_{n-\sigma}=0, &\quad n=0,1,2,\dots,\\ \Delta^2x_n+p_n\Delta x_n+q_n g(x_{n-\sigma})=e_n, &\quad n=0,1,2,\dots,\end{aligned}
where $$(e_n)_n$$ is a sequence of real numbers, $$xg(x)>0$$ for $$x\not=0$$ and $$g'(x)\geq \eta>0$$, are also investigated.

### MSC:

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

Zbl 0969.39005
Full Text:

### References:

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