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, tau)parallel to y(tau)parallel to + integral(t)(tau) K-1(t, s)parallel to y(s)parallel to ds + integral(infinity)(t) K-2(t, s)parallel to y(s)parallel to ds + rho, t >= tau >= 0. As a simple example of application, the retarded scalar functional differential equation (x)over dot = -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)over dot = F-0(t, x) + Sigma(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(theta(t) p, u(t)) in a Banach space X with sublinear nonlinearity. In particular, the existence and uniqueness of a nonautonomous equilibrium 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. (C) 2020 Elsevier Inc. All rights reserved.