The method of generalized quasilinearization for second-order boundary value problems has been extended when the forcing function is the sum of 2 -hyperconvex and 2 -hyperconcave functions. We develop two sequences under suitable conditions which converge to the unique solution of the boundary value problem. Furthermore, the convergence is of order 3 . Finally, we provide numerical examples to show the application of the generalized quasilinearization method developed here for second-order boundary value problems.