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) ≤ E (t , τ) ‖ y τ ‖ + ∫ τ t K 1 (t , s) ‖ y s ‖ d s + ∫ t ∞ K 2 (t , s) ‖ y s ‖ d s + ρ , t ≥ τ ≥ 0. As a simple example of application, the retarded scalar functional differential equation 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 x ˙ = F 0 (t , x) + ∑ i = 1 m F i (t , x (t − r i (t))) on R n with superlinear nonlinearities F i (0 ≤ i ≤ 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 d u d t + A u = F (θ t p , u t) in a Banach space X with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous equilibrium 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. [ABSTRACT FROM AUTHOR]