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 $\Delta x(n)=-a(n)x(n)+B(n,x_n)$ is discussed, and some new criterion 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.

This paper is concerned with the bounded fractal and Hausdorff dimension of the pullback attractors for 2D non-autonomous incompressible Navier-Stokes equations with constant delay terms. Using the construction of trace formula with two bases for phase spaces of product flow, the upper boundedness of fractal dimension has been achieved.