Journal of Applied Mathematics and Simulation, Volume 2, Number 4, 1989 217 MONOTONE METHOD FOR FIRST ORDER SINGULAR SYSTEMS WITH BOUNDARY CONDITIONS * J.A. Uvah and A.S. Vatsala Department of Mathematics University of Southwestern Louisiana P.O. Box 41010, USL Lafayette, Louisiana 70504-1010 ABSTRACT Existence of maximal and minimal solutions are proved for the first order singular systems with boundary conditions by combining the method of upper and lower solutions and the monotone iterative technique. Key words: Singular Systems, Extremal Solutions, Upper and Lower Solutions, Iterative Technique. AMS Subject Classification: 34A34, 34B99 1. INTRODUCTION It is well known [4] that by combining the method of upper and lower solutions with monotone iterative techniques, one can prove the existence of extremal solutions on nonlinear problems in a closed set, namely, the sector defined by means of the upper and lower solutions. Recently the result has been extended [6] to singular systems with initial conditions since singular systems do occur in many physical applications. In this paper, we extend this result to singular systems with boundary conditions. This is achieved by developing the necessary comparison result. The crucial part is the consistency condition. An example is given to illustrate that such a condition is attainable. 2. PRELIMINARY RESULTS Consider the boundary value problem (1.1) A.2 =f(t,x), Ex(O) = a, Fx(T) = b where A ,nxn is singular matrix, E, F are real n x n nonsingular matrices, and f C [J x I", In], J = [0, T]. In this paper we combine the method of upper and lower solutions together with monotone iterative technique to prove the existence of * Received: March, 1989; Revised: October, 1989