Global attractivity in a delay logistic difference equation (Q1812218)

Consider the nonautonomous delay logistic difference equation (*) \(y_{n+1}-y_n=p_ny_n(1-y_{\tau (n)})\), \(n=0,1,\dots\), where \(\{p_n\}_{n=0}^\infty\) is a nondecreasing sequence of integers, \(\tau (n)<n\) and \(\lim_{n\rightarrow \infty}\tau (n)=\infty\). The author proves the following: Theorem: Assume that \(\sum_{j=\tau (n)}^np_j\leq \frac 54\) for sufficiently large \(n\), then every solution of (*) that oscillates about \(1\) tends to 1 as \(n\rightarrow\infty\).