Boundary Value Problems
Volume 2005 (2005), Issue 2, Pages 93-106

Monotone iterative technique for semilinear elliptic systems

A. S. Vatsala and Jie Yang

Department of Mathematics, University of Louisiana at Lafayette, Lafayette 70504-1010, LA, USA

Received 27 September 2004; Revised 23 January 2005

Copyright © 2005 A. S. Vatsala and Jie Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.