Let (xn) and (◯n) be two vector sequences converging to a common limit. First, we shall define nonlinear hybrid procedures which consist of constructing a new vector sequence (yn) with better convergence properties than (xn) and (◯n). Then, this procedure is used for accelerating the convergence of a given sequence and applied to the construction of fixed point methods. New methods are derived. Finally, the connection between fixed point iterations and methods for the numerical integration of differential equations is also exploited. Numerical results are given.