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 \( x'(t) = -\int_{t-r(t)}^{t}a(t,s)g(x(s))\,ds +p(t,x(t))+f(t), \) 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. In contrast with fixed-point approaches based on a uniform contraction condition, the present method separates the instantaneous dissipative contribution from a memory remainder controlled by the first absolute moment of the kernel. 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 developed to treat nonuniform dissipation and yields explicit decay estimates of the form \( |x(t)| \leq \norm{\phi}_\infty \exp\left( -\eta\int_0^t\lambda(s)\,ds \right). \) State-dependent perturbations and external forcing are also included, yielding ultimate bounds and convergence under vanishing forcing. For the elementary equation \( x'(t)=-c\int_{t-r}^{t}x(s)\,ds, \) our criterion gives exponential asymptotic stability whenever \( cr^{2}<2 \), whereas the contraction condition in Burton's fixed-point result requires \( cr^{2}<1 \). Thus the Lyapunov/comparison method applies on the nonempty parameter range \( 1\leq cr^{2}<2 \) where that contraction criterion fails. The exact characteristic stability boundary for this example is \( cr^{2}=\pi^{2}/2 \), indicating the remaining gap between the present sufficient condition and the sharp linear threshold.