Positive solutions of second order difference inequalities (Q1374835)

The paper deals with the inequality \(\Delta(r_{n-1} \Delta(x_{n-1} -p_{n-1} x_{n-1- \tau})) +q_nx_{n- \sigma} \leq 0\), where \(n,\tau, \sigma+ 1\in\mathbb{N}\), \(r_n>0\), \(q_n\geq 0\), but \(q_{n_i} >0\) for a certain subsequence. Under the condition \(\sum^\infty_{n=0} r_n^{-1} =\infty\) or \(\sum^\infty_{n=0} r_n<\infty\), several nonexistence criteria are derived for eventually positive solutions \(x_n\) in the three cases \(0\leq p_n<1\), \(p_n\leq 0\) resp. \(p_n>1\) generalizing the case \(-1\leq p_n \leq 0\) of \textit{B. G. Zhang} and \textit{S. S. Cheng} [Fasc. Math. 25, 13-32 (1995; Zbl 0830.39005)].