li NORTH - IRRIAND Generalized Quasilinearization and First-Order Periodic Boundary Value Problem A. S. Vatsala and Donna Stutson Department of Mathematics University of Southwestern Louisiana Lafayette, Louisiana 70504-1010 Transmitted by Melvin Scott ABSTRACT The method of quasilinearization was extended and generalized to include func- tions which are neither convex nor concave. This is known as generalized quasilin- earization. The generalized method yields both-sided bounds as in the monotone method, but the rate of convergence is quadratic as in the quasilinearization method. In this paper, we extend the generalization method to a first order periodic boundary value problem. 1. INTRODUCTION It is well known [1, 2] that the method of quasilinearization offers an approach for obtaining approximate solutions to nonlinear differential equa- tions. Recently this method has been generalized and extended so as to enlarge the class of functions for which it is applicable [3-11]. Currently the method is referred to as the method of generalized quasilinearization. In this paper, we consider the periodic boundary value problem (PBVP for short) of the form x' =f(t, x), x(0) = x(2~r), te J= [0,2~'], (1.1) where f~ C[J × R, R]. We consider the situation when f admits a decomposition of the form f(t, x)= F(t, x)+ G(t, x)+ h(t, x). Here F(t, x) is not convex but F(t, x) + q~(t, x) is convex for some convex function ¢(t, x) and G(t, x) is not concave but G(t, x) + ~b(t, x) is con- APPLIED MATHEMATICS AND COMPUTATION 77:113-129 (1996) © Elsevier Science Inc., 1996 0096-3003//96/$15.00 655 Avenue of the Americas, New York, NY 10010 SSDI 0096-3003(95)00194-M