Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the stability and instability investigation of a class of nonlinear Volterra integral equations in a Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in a weighted Sobolev space, which has the same stability properties as the Volterra equation.

In this paper we study a linear integral equation $x\left(t\right)=a\left(t\right)-{\int }_0^{t}C(t,s)x\left(s\right)\mathrm{d}s$, its resolvent equation $R(t,s)=C(t,s)-{\int }_s^{t}C(t,u)R(u,s)\mathrm{d}u$, the variation of parameters formula $x\left(t\right)=a\left(t\right)-{\int }_0^{t}R(t,s)a\left(s\right)\mathrm{d}s$, and a perturbed equation. The kernel, $C(t,s)$, satisfies classical smoothness and sign conditions assumed in many real-world problems. We study the effects of perturbations of $C$ and also the limit sets of the resolvent. These results lead us to the study of nonlinear perturbations.