  Some uniform decay estimates are established for solutions of the following
type of retarded integral inequalities: $$y(t)\leq
E(t,\tau)||y_\tau||+\int_\tau^t K_1(t,s)||y_s||ds+\int_t^\infty
K_2(t,s)||y_s||ds+\rho, \hspace{0.5cm} t\geq\tau\geq 0.$$
  As a simple example of application, the retarded scalar functional
differential equation $\dot x=-a(t)x+B(t,x_t)$ is considered, and the global
asymptotic stability of the equation is proved under weaker conditions. Another
example is the
  ODE system $\dot x=F_0(t,x)+\sum_{i=1}^m F_i(t,x(t-r_i(t)))$ on $R^n$ with
superlinear nonlinearities $F_i$ ($0\leq i\leq m$). The existence of a global
pullback attractor of the system is established under appropriate dissipation
conditions.
  The third example for application concerns the study of the dynamics of the
functional cocycle system $\frac{du}{dt}+Au=F(\theta_tp,u_t)$ in a Banach space
$X$ with sublinear nonlinearity. In particular, the existence and uniqueness of
a nonautonomous stationary solution $\Gamma$ is obtained under the
hyperbolicity assumption on operator $A$ and some additional hypotheses, and
the global asymptotic stability of $\Gamma$ is also addressed.
