MONOTONE ITERATIVE TECHNIQUE FOR SEMILINEAR ELLIPTIC SYSTEMS A. S. VATSALA AND JIE YANG Received 27 September 2004 and in revised form 23 January 2005 We develop monotone iterative technique for a system of semilinear elliptic boundary value problems when the forcing function is the sum of Caratheodory functions which are nondecreasing and nonincreasing, respectively. The splitting of the forcing function leads to four different types of coupled weak upper and lower solutions. In this paper, relative to two of these coupled upper and lower solutions, we develop monotone iterative technique. We prove that the monotone sequences converge to coupled weak minimal and. | MONOTONE ITERATIVE TECHNIQUE FOR SEMILINEAR ELLIPTIC SYSTEMS A. S. VATSALA AND JIE YANG Received 27 September 2004 and in revised form 23 January 2005 We develop monotone iterative technique for a system of semilinear elliptic boundary value problems when the forcing function is the sum of Caratheodory functions which are nondecreasing and nonincreasing respectively. The splitting of the forcing function leads to four different types of coupled weak upper and lower solutions. In this paper relative to two of these coupled upper and lower solutions we develop monotone iterative technique. We prove that the monotone sequences converge to coupled weak minimal and maximal solutions of the nonlinear elliptic systems. One can develop results for the other two types on the same lines. We further prove that the linear iterates of the monotone iterative technique converge monotonically to the unique solution of the nonlinear BVP under suitable conditions. 1. Introduction Semilinear systems of elliptic equations arise in a variety of physical contexts specially in the study of steady-state solutions of time-dependent problems. See 1 4 5 for example. Existence and uniqueness of classical solutions of such systems by monotone method has been established in 2 4 . Using generalized monotone method the existence and uniqueness of coupled weak minimal and maximal solutions for the scalar semilinear elliptic equation has been established in 3 . They have utilized the existence and uniqueness result of weak solution of the linear equation from 1 . In 3 the authors have considered coupled upper and lower solutions and have obtained natural sequences as well as alternate sequences which converge to coupled weak minimal and maximal solutions of the scalar semilinear elliptic equation. In this paper we develop generalized monotone method combined with the method of upper and lower solutions for the system of semilinear elliptic equations. For this purpose we have developed a comparison
