Discrete Convergence of Mappings and Solutions of Equations

Abstract

In this chapter, we establish the fundamental convergence results for solutions of sequences of nonlinear equations with differentiable operators. To prepare the reader for the analysis in this chapter, we examine in Section 6.1 the relationship between the continuity of a mapping on the one hand and the differentiability and boundedness of its derivatives on the other. The most important result in 6.1 is a quantitative formulation of the Inverse Function Theorem (see Theorem 6.7). If we apply this result to sequences of differentiable mappings, then we obtain equivalent characterizations of the concepts of stability and inverse stability (see Section 6.2, Theorem 6.12). In Section 6.3, we introduce the concepts of consistency and of discrete convergence of sequences of mappings. It turns out that discrete convergence is equivalent to stability together with consistency (cf. Theorem 6.13). By virtue of the characterizations of stability to be discussed in Section 6.2, we are able to obtain equivalent conditions for the discrete convergence of differentiable mappings, along with error estimates (cf. Theorem 6.14). The concluding Section 6.4 establishes and characterizes the discrete convergence of solutions by using the concept of inverse discrete convergence. The most important result of this section is Theorem 6.21 which gives equivalent conditions for biconvergence. With an appropriate choice of underlying norms, we are able to state another important result (cf. Theorem 6.23) which allows us to infer from the inverse stability of equicontinuously equidifferentiable mappings, the local solvability, and the convergence of the approximate solutions.