Oscillatory and asymptotic behavior of fourth order nonlinear delay difference equations. (English) Zbl 1009.39007

Graef, John R.; Thandapani, E.

Fasc. Math. 31, 23-36 (2001).

From the summary: The authors consider the nonlinear difference equation \[ \Delta^2 \bigl(a_n\Delta (b_n\Delta y_n)\bigr) +f(n,y_{n-l})=0, \quad n\in N(n_0)= \{n_0,n_0+1, \dots\}, \] where \(\{a_n\}\) and \(\{b_n\}\) are positive real sequences, \(l\) is a nonnegative integer, \(f:N(n_0)\times \mathbb{R} \to\mathbb{R}\) is a continuous function with \(uf(n,u)>0\) for all \(u\neq 0\). They obtain necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior. They also obtain sufficient conditions for all solutions to be oscillatory if \(f\) is either strongly sublinear or strongly superlinear.

Reviewer: Mihaly Pituk (Veszprem)

MSC:

39A11

Stability of difference equations (MSC2000)

Keywords:

fourth-order nonlinear delay difference equations; oscillatory solution; nonoscillatory solutions; asymptotic behavior