In this paper, the results of Lakshmikantham et al. (Ref. 1) for first-order periodic boundary-value problems are extended, by using the extended method of quaislinearization and rapid convergence for initial-value problems of Mohapatra et al. (Ref. 2). Also, it is shown that monotone sequences converge cubically to the unique solution when the forcing function in the differential equation is 2-hyperconvex and converge quartically when the forcing function is 3-hyperconvex. Several other generalizations of the problem are also presented.