Oscillation of positive solution to a nonautonomous nonlinear delay difference equation (Q2719937)

The following main result is obtained: Suppose \(\{r_n\}^\infty_{ n=0}\) is a positive sequence, \(0\leq c<1\), \(k\) is a nonnegative integer, NEWLINE\[NEWLINE\varliminf_{n\to \infty}\sum^{n+k}_{i=n+1}r_i>0,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sum^\infty_{n=0} r_n\left\{{k+1 \over k}\left(\sum^{n+k}_{s=n+1} {r_n\over 1-c}\right)^{1/(k+1)}-1\right\}=\infty.NEWLINE\]NEWLINE Then every positive solution of the difference equation NEWLINE\[NEWLINE\Delta x_n= r_nx_n{1-x_{n-k} \over 1-cx_{n-k}},\;n=0,1,2, \dots,NEWLINE\]NEWLINE is oscillatory about 1.