A Monotone Method for Infinite System of Nonlinear Boundary Value Problems
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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 . In either case an essential use is made of a Nagumo-type condition  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.