Oscillation of second order nonlinear delay difference equations. (English) Zbl 1035.39008

Saker, S. H.

Bull. Korean Math. Soc. 40, No. 3, 489-501 (2003).

Consider the nonlinear delay difference equation \[ \Delta (p_n\Delta x_n)+q_nf(x_{n-\sigma })=0,\quad n=0,1,2,\dots \tag{*} \] where \(\Delta u_n=u_{n+1}-u_n\) for any sequence \(\{u_n\}\) of real numbers, \(\sigma\) is a nonnegative integer, \(\{p_n\}_{n=0}^\infty\) and \(\{q_n\}_{n=0}^\infty\) are sequences of real numbers such that \(p_n>0,\) \(\sum^\infty \frac 1{p_n}<\infty,\) \(q_n\geq 0\) and \(q_n\) has a positive subsequence, and \(f\) is a continuous nondecreasing real valued function which satisfies \(uf(u)>0\) for \(u\neq 0\) and \(f(u)/u\geq \gamma >0.\) The author establishes some sufficient conditions which guarantee that every solution of (*) is oscillatory or converges to zero.

Reviewer: Fozi Dannan (Doha)

MSC:

39A11

Stability of difference equations (MSC2000)

Keywords:

oscillation; nonlinear delay difference equations