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

Abstract

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities:As a simple example of application, the retarded scalar functional differential equation ẋ = −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 ẋ = F 0 (t, x) + m i=1 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 du dt + Au = F (θ t p, 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.