Positive solutions of second order nonlinear difference equations (Q1353660)

Necessary and sufficient conditions are obtained for the existence of positive solutions of the difference equation \(\Delta(r_n(\Delta x_n)^\sigma)+ g_nx_{n+\tau}^\sigma=0\), where \(\sigma\) is a quotient of positive odd integers, \(g_n\geq 0\) but \(g_n>0\) for infinitely many \(n\), and \(r_n>0\) with divergent \(\sum_{k=0}^\infty r_k^{-1/\sigma}\). In particular, the case \(\tau=0\) is studied.