Faedo-Galerkin approximation of second order nonlinear differential equation with deviated argument

In this manuscript, we consider a second order nonlinear differential equation with deviated argument in a separable Hilbert space X. We used the strongly continuous cosine family of linear operators and fixed point method to study the existence of an approximate solution of the second order differential equation. We define the fractional power of the closed linear operator and used it to prove the convergence of the approximate solution. Also, we prove the existence and convergence of the Faedo-Galerkin approximate solution. Finally, we give an example to illustrate the application of these abstract results.