Unifying semilocal and local convergence of Newton's method on Banach space with a convergence structure

We present a semilocal and local convergence analysis of Newton's method on a Banach space with a convergence structure to locate zeros of operators. P. Meyer introduced the concept of a Banach space with a convergence structure. Using this setting, he presented a finer semilocal convergence analysis for Newton's method than in related studies using the real norm theory. In all these studies the operator involved as well as its Fréchet derivative is bounded above by the same bound-operator. In the present study, we introduce a second bound operator which is a special case of the bound-operator leading to tighter majorizing sequences for Newton's method. Using this more flexible combination of bound-operators, we improve the results in the earlier studies. In the semilocal case, we obtain under the same or weaker sufficient convergence conditions more precise error bounds on the distances involved and in the local case not considered in the earlier studies, we obtain a larger radius of convergence. This way we expand the applicability of Newton's method. Some numerical examples are also provided to show the superiority of the new results over the old results.