Notes on the basic notions in nonlinear numerical analysis

Abstract

In this paper we investigate the numerical solution of non-linear equations in an abstract (Banach space) setting. The main result is that the convergence can be guaranteed by two, directly checkable conditions (namely, by the consistency and the stability). We show that these conditions together are a sufficient, but not necessary condition for the convergence. Our theoretical results are demonstrated on the numerical solution of a Cauchy problem for ordinary differential equations by means of the explicit Euler method.