Nonlinear Volterra equations on a Hilbert space

We consider equations of the form, u(t) = ? ∝₀^tA(t ? τ)g(u(τ)) dτ + ?(t) (I) on a Hubert space $\text{H}$. A(t) is a family of bounded, linear operators on $\text{H}$ while g is a transformation on $\text{D}$_g ?$\text{H}$ which can be nonlinear and unbounded. We give conditions on A and g which yield stability and asymptotic stability of solutions of (I). It is shown, in particular, that linear combinations with positive coefficients of the operators e^Mt and ?e^Mtsin Mt where M is a bounded, negative self-adjoint operator on $\text{H}$ satisfy these conditions. This is shown to yield stability results for differential equations of the form, $\text{Q (}\text{d}\text{dt}\text{) u = ? P (}\text{d}\text{dt}\text{) g(u(t)) + χ(t)}$, on $\text{H}$.