f-Konvergenz undf-Stetigkeit von Operatorenfolgen auf metrischen Räumen

Summary

Equicontinuity of the operators of a given sequence of nonlinear operators is one of the necessary and sufficient conditions for continuous convergence of this sequence. This lemma, which is due to Rinow, gives a generalization of the Banach-Steinhaus-theorem. Hence it led to some generalizations of the Lax-Richt-myer-theory of difference approximations for initial value problems. The equicontinuity in this case correspondends to numerical stability. But often continuous convergence is a too strong demand in the theory of nonlinear numerical problems (for instance in the case of difference schemes for quasilinear partial differential equations), whereas a restriction to only pointwise convergence possibly leads to numerical instability. Therefore in this paper a set of definitions of convergence is considered lying between pointwise and continuous convergence. Sorts of continuity are described which are as characteristic for these kinds of convergence as equicontinuity for continuous convergence. As an numerical application we study the connection between the solution-depending stability and the sensitiveness to perturbations of difference schemes for quasilinear initial value problems.