Approximations of solutions for a nonlinear differential equation with a deviating argument

In this paper, we prove the existence and convergence of approximate solution for a class of nonlinear differential equations with a deviated argument in a Hilbert space. We establish the existence and uniqueness of a solution to every approximate integral equation using the fixed point argument. Then, we prove the convergence of a solution of the approximate integral equation to the solution of the associated integral equation. We also consider the Faedo–Galerkin approximation of a solution and prove some convergence results.