An Existence Result for a Nonlinear Volterra Integral Equation in a Hilbert Space

We study equations of the form \[u(t) + \int_0^t {a(t - s)gu(s)ds \ni f(t)} ,\quad t \geqq 0\] on a real Hilbert space H. The unknown function is u and a, g, f are given. It is assumed that the kernel a is operator-valued (real-valued as a special case) and g is an arbitrary maximal monotone operator in H. The method can also be applied to time-varying nonlinearity. We prove an existence and uniqueness result that extends earlier results by Londen and Barbu. Finally an application is given.