Approximations of solutions to second order semilinear integrodifferential equations

In the present work we study the approximation of solutions to a class of second order semilinear integrodifferential equations in a Hilbert space. These equations arise in the study of viscoelastic materials with memory. Using a pair of associated nonlinear integral equations and projection operators we consider a pair of approximate nonlinear integral equations. We first show the existence and uniqueness of solutions to this pair of approximate integral equations. We then establish the convergence of the sequences of the approximate solutions and the pair of approximate integral equations to the solution and the pair of associated integral equations, respectively. We finally consider the Faedo-Galerkin approximation of the solution and prove some convergence results.