<p>The maximal order of convergence of Green’s function method for fifth order two-point boundary value problems with deviating argument is obtained. The involved iterated splines technique uses quartic splines and the corrected Simpson quadrature rule. The convergence of this method is proven by obtaining the error estimates in the discrete and continuous approximation of the solution. Some numerical experiments confirm the obtained theoretical order of convergence and illustrate the accuracy of the method.</p>