A Monotone Method for Infinite System of Nonlinear Boundary Value Problems

Abstract

Monotone iterative methods have been successfully used to generate improvable two-sided point-wise bounds on solutions of nonlinear boundary value problems for both ordinary and partial differential equations. While such procedures take a simple form when the nonlinearities are independent of gradient terms [6,9], the extension of such techniques to fully nonlinear problems has been quite formidable. In the case of scalar ordinary differential equations of the type (1.1) [see PDF for equation] such results have been obtained making use of either a linear maximum principle (3,1] or a nonlinear maximum principle [4]. In either case an essential use is made of a Nagumo-type condition [5] for deriving uniform estimates on the gradient. The lack of similar tools for higher dimensional problems has impeded comparable progress in obtaining similar results for equations of the type.