A discrete retarded Gronwall–Bellman type inequality and its applications to difference equations

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.