Quasi-Solutions and Monotone Method for Systems of Nonlinear Boundary Value Problems

Abstract

**Please note that the full text is embargoed** ABSTRACT: In the study of comparison theorems, existence of extremal solutions and monotone iterative techniques for initial and boundary value problems of ordinary differential systems, it becomes necessary to impose a condition generally known as quasi-monotone property [1,3,5]. In systems which represent physical situations such as a model governing the combustion of a material, quasi-monotonicity is not satisfied, see [4]. However a kind of mixed monotone property holds. To deal with such situations the notion of quasi-solutions was systematically developed in [4]. In this paper, we investigate monotone iterative method for systems of nonlinear boundary value problems when the system possesses a mixed quasi-monotone property. This appears a natural setup for considering quasi-solutions and quasi-extremal solutions in view of the fact extremal solutions need not exist when quasi-monotone property does not hold. Furthermore, the results obtained include as special cases the known results corresponding to quasi-monotone property.