An Existence Result on a Volterra Equation in a Banach Space

Let W be a real reflexive Banach space, dense in a Hilbert space H and with dual $W’$. Let the injection $W \to H$ be continuous and compact. We consider the nonlinear integral equation \begin{equation}\tag {$1$} u’(t) + \int _0^t {a(t - \tau )Au(\tau )d\tau = f(t),\quad t \geqslant 0,} \end{equation} where a, f, A are given and u is the unknown. The kernel $a(t)$ maps ${R^ + }$ into R and f takes values in H. The nonlinear function A is a maximal monotone mapping $W \to W’$. Making use of the theory of maximal monotone operators we prove an existence result on (1). This result is used to obtain approximate solutions to the related nonlinear hyperbolic differential equation $u''(t) + Au(t) = f’(t),t \geqslant 0$.