  This paper is concerned with the approximation of solutions to a class of
second order non linear abstract differential equations. The finite-dimensional
approximate solutions of the given system are built with the aid of the
projection operator. We investigate the connection between the approximate
solution and exact solution, and the question of convergence. Moreover, we
define the Faedo-Galerkin(F-G) approximations and prove the existence and
convergence results. The results are obtained by using the theory of cosine
functions, Banach fixed point theorem and fractional power of closed linear
operators. At last, an example of abstract formulation is provided.
