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Uniform Decay Estimates for Solutions of a Class of Retarded Integral Inequalities

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities: $$y(t)\leq E(t,τ)||y_τ||+\int_τ^t K_1(t,s)||y_s||ds+\int_t^\infty K_2(t,s)||y_s||ds+ρ, \hspace{0.5cm} t\geqτ\geq 0.$$ 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 $\frac{du}{dt}+Au=F(θ_tp,u_t)$ in a Banach space $X$ with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous stationary solution $Γ$ is obtained under the hyperbolicity assumption on operator $A$ and some additional hypotheses, and the global asymptotic stability of $Γ$ is also addressed.