Stability of solutions of Volterra integrodifferential equations

Abstract

Systems with past memory (or after-effect), the state of which is given by nonlinear Volterra- type integrodifferential equations with small perturbations, are investigated. The equations depend on functionals in integral form and, in particular, on analytic functionals represented by Fréchet series. The integral kernels can allow for singularities with Abel’s kernel. The stability under persistent disturbances, and the structure of the general solution, are investigated in the neighborhood of zero for an equation with holomorphic nonlinearity assuming asymptotic stability of the trivial solution of the linearized unperturbed equation. Stability in the critical cases (in Lyapunov’s sense) of a single zero root and of pairs of pure imaginary roots for the unperturbed equation is analyzed.