CONTRACTIVE MAPPINGS WITH MAXIMUM NORMS - COMPARISON OF CONSTANTS OF CONTRACTION AND APPLICATION TO ASYNCHRONOUS ITERATIONS

In this paper, we give two extensions of Stein-Rosenberg's theorem. Th e first, which we name the general result, is an abstract nonlinear ex tension and can be described as follows: Given a first fixed point map ping on a Banach product space, we define a more implicit second fixed point mapping, possibly after a redecomposition of our product space, and we get that the new mapping has a constant of contraction lower o r equal to the constant of contraction of the initial mapping. This re sult allows an efficient use of El Tarazi's theorem [5], about the con vergence of asynchronous iterations. The second extension is close to the linear case and permits us to compare the constants of contraction using strict inequality. We give two applications of these results: t he first, in a context near from the one studied by D.J. Evans and W. Deren, about a diagonal monotone perturbation of linear problems[6]. T he second is a short example in a totally different framework about th e formulation of asynchronous waveform relaxation for a system of ordi nary differential equations with initial conditions. For another point of view concerning Stein-Roseinberg's theorem and asynchronous algori thms, see [7] and [10].