Approximations of solutions to nonlinear Sobolev type evolution equations

Abstract

In the present work we study the approximations of solutions to a class of nonlinear Sobolev type evolution equations in a Hilbert space. These equations arise in the analysis of the partial neutral functional differential equations with unbounded delay. We consider an associated integral equation and a sequence of approximate integral equations. We establish the existence and uniqueness of the solutions to every approximate integral equation using the fixed point arguments. We then prove the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. Next we consider the Faedo-Galerkin approximations of the solutions and prove some convergence results. Finally we demonstrate some of the applications of the results established.