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Saker, S. H.; Zhang, B. G.

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

Acta Math. Sin., Engl. Ser. 23, No. 4, 715-722 (2007).
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.
Reviewer: Rodica Luca Tudorache (Iaşi)

MSC:

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

Keywords:

oscillatory solution; Riccati techniques; second-order nonlinear delay difference equation; damping term

Citations:

Zbl 0969.39005
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\textit{S. H. Saker} and \textit{B. G. Zhang}, Acta Math. Sin., Engl. Ser. 23, No. 4, 715--722 (2007; Zbl 1123.39008)
Full Text: DOI

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