Global attractivity in a delay logistic difference equation

Abstract

This paper studies the global attractivity of the positive equilibrium 1 of the delay logistic difference equation

$$\vartriangle y_n = p_n y_n \left( {1 - y_{\tau \left( n \right)} } \right), n = 0, 1, 2, . . .,$$

(*)

where {p _n} is a sequence of positive real numbers, {τ(n)} is a nondecreasing sequence of integers, τ(n)<n and $\mathop {\lim }\limits_{x \to \infty } \tau \left( n \right) = \infty$. It is proved that if

$$\sum\limits_{j = \tau \left( n \right)}^n {p_j \leqslant \frac{5}{4}} for sufficiently large n and \sum\limits_{j = 0}^\infty {p_j = \infty ,} $$

then all positive solutions of Eq. (*) tend to 1 as n → ∞. The results improve the existing results in literature.