No abstract

In this paper, we present a new discrete retarded Gronwall–Bellman type inequality. As applications, the dynamics of some delay difference equations are studied. First, the asymptotic behavior of solutions for scalar difference equation normalΔxfalse(nfalse)=−afalse(nfalse)xfalse(nfalse)+Bfalse(n,xnfalse)$$ \Delta x(n)=-a(n)x(n)+B\left(n,{x}_n\right) $$ is discussed, and some new criteria on the asymptotic stability of the zero solution are obtained under weaker assumptions. Then the dissipativity of a nonautonomous delay difference system with superlinear nonlinearities is investigated. By using the inequalities established here, it is shown that the discrete set‐valued process generated by the system possesses a unique global pullback attractor.

No abstract

We establish a discrete Gronwall–Halanay-type inequality with infinite delay, which is not covered in the existing literature. As an application, a new criterion is obtained for the asymptotic stability of the zero solutions for a class of Volterra difference equations. A concrete example is also given to illustrate the efficiency of the general results.