Oscillation of a nonlinear difference equation with several delays. (English) Zbl 1055.39015

Luo, X. N.; Zhou, Yong; Li, C. F.

Math. Bohem. 128, No. 3, 309-317 (2003).

Summary: We consider the nonlinear difference equation with several delays \[ (ax_{n + 1} + bx_{n})^k - (cx_{n})^k + \sum _{i = 1}^{m} p_{i} (n) x^k_{n- \sigma _{i}} = 0 \] where \(a, b, c \in (0, \infty )\), \(k = q/r, q, r\) are positive odd integers, \(m\), \(\sigma _{i}\) are positive integers, \(\{p_{i} (n)\}\), \(i = 1, 2, \dots , m\), is a real sequence with \(p_{i} (n) \geq 0\) for all large \(n\), and \(\liminf _{n \to \infty } p_{i} (n) = p_{i} < \infty \), \(i = 1, 2, \dots , m\). Some sufficient conditions for the oscillation of all solutions of the above equation are obtained.

MSC:

39A11

Stability of difference equations (MSC2000)

Keywords:

nonlinear delay difference equtions; oscillation; eventually positive solutions; characteristic equation