Lyapunov and Comparison Methods for Nonlinear Distributed-Delay Equations with Variable Delay and Perturbations

We study boundedness, stability, asymptotic stability, and robustness for the nonlinear distributed-delay equation x′(t)=−∫t−r(t)ta(t,s)g(x(s))ds+p(t,x(t))+f(t), where the delay is bounded and may vary with time. The analysis is based on a direct dissipative decomposition of the distributed-delay term together with Lyapunov and Halanay-type comparison estimates. Using a standard model transformation, the method separates the instantaneous dissipative contribution from a memory remainder controlled by the first absolute moment of the kernel. As in Halanay and Razumikhin arguments, no derivative of the delay is used. We first obtain a priori bounds under an integrable growth majorant and then establish exponential asymptotic stability under a uniform dissipation-versus-memory gap. A variable-coefficient Halanay inequality is recorded to treat nonuniform dissipation and yields explicit decay estimates of the form |x(t)|≤Cs∥ϕ∥∞exp−η∫r0max{t,r0}λ(s)ds. State-dependent perturbations and external forcing are also included, yielding ultimate bounds and convergence under vanishing forcing. For the elementary equation x′(t)=−c∫t−rtx(s)ds, our sufficient criterion gives exponential asymptotic stability whenever cr2<2, whereas the condition in Burton’s linear contraction theorem requires cr2<1. Thus the Lyapunov/comparison method applies on the nonempty parameter range 1≤cr2<2 where that contraction criterion fails. The exact characteristic stability boundary for this example is cr2=π2/2, indicating the remaining gap between the present sufficient condition and the sharp linear threshold.