Monotone enclosure for elliptic and parabolic sytems with nonmonotone nonlinearities

Abstract

Monotone iteration techniques for weakly coupled systems having nonmonotone nonlinearities have been developed recently by M. Khavanin and V. Lakshmikantham and by S. Carl. In this paper an existence and enclosing theorem is given in terms of upper and lower solutions and a new monotone iteration technique, which in a natural way extends the monotone iteration technique used in the “monotone case” to the “nonmonotone case” considered here, is proposed. The maximum and minimum functions that occur in the iteration scheme allow one to work under low regularity conditions; particularly, no local Lipschitz conditions on the nonlinearities are required for the iteration process. A crucial role in the analysis is played by a measurable selection theorem which is usually known from optimization theory. Furthermore, conditions which ensure that the iteration process converges to the (uniquely) defined solution are given.