Oscillation and nonoscillation of neutral difference equations with positive and negative coefficients (Q1568734)

The authors establish several new sufficient conditions for the oscillation of all solutions of \[ \Delta(x_n-c_n x_{n-\tau}) +p_nx_{n-k}-q_nx_{n-\ell} =0 \] with \(p_n-q_{n-k+\ell}\geq 0\) and not identically zero. Also existence theorems for the positive solutions are obtained. For related results see the papers of \textit{M.-P. Chen} and \textit{B. G. Zhang} [Bull. Inst. Math., Acad. Sin. 22, 295-306 (1994; Zbl 0817.39003)] and \textit{B. G. Zhang} and \textit{H. Wang} [Chin. J. Math. 24, No. 4, 377-393 (1996; Zbl 0868.39005)].