Theoretical ApproximationMethods

Abstract

This chapter introduces the theoretical methods that are constructive. We first prove in Section 3.2, an existence result in a special closed set generated by the lower and upper solutions. Next in Section 3.3, we describe a constructive technique that offers monotone sequences which converge to the extremal solutions. This technique is very important because the iterates are solutions of a certain causal differential equationwhich can be computed explicitly and the method can be applied to many nonlinear problems. In Section 3.4, themonotone iterative technique is extended to causal differential equationswhere the right hand side is the sum of the two functions, one of which is monotone nondecreasing and the other is monotone nonincreasing. The results obtained include several special cases and hence, very valuable.