Boundary Value Problems
Volume 2005 (2005), Issue 2, Pages 93-106
Monotone iterative technique for semilinear elliptic systems
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.