A note on the theory of limit processes for solving finite equations

Abstract

In the first of the two paragraphs a limit operator for solving equationG(x)=0 in the bounds of abstract Banach space has been developed. Theorem 1 contains the results obtained, and it is in two respects an extension of a known theorem due to Rheinboldt. The main result appears in the fact that in the assumptions of the later theorem the Lipschitz condition[Figure not available: see fulltext.] has been weakened to the form of the Holder condition[Figure not available: see fulltext.]. Instead of a limitpoint, the existence of a limitoperator, which maps a finite neighbourhood of the root into the root itself, and so guarantees the numerical stability of the process, represents another extension included in Theorem 1.--In the second paragraph it has been shown that the Holder condition is also sufficient for the convergence and validity of the Newton sequences, i.e. of the primary form of the modified Newton sequences and of the original one.