Newton-Type Methods on Generalized Banach Spaces and Applications in Fractional Calculus

We present a semilocal convergence study of Newton-type methods on a generalized Banach space setting to approximate a locally unique zero of an operator. Earlier studies require that the operator involved is Fréchet differentiable. In the present study we assume that the operator is only continuous. This way we extend the applicability of Newton-type methods to include fractional calculus and problems from other areas. Moreover, under the same or weaker conditions, we obtain weaker sufficient convergence criteria, tighter error bounds on the distances involved and an at least as precise information on the location of the solution. Special cases are provided where the old convergence criteria cannot apply but the new criteria can apply to locate zeros of operators. Some applications include fractional calculus involving the Riemann-Liouville fractional integral and the Caputo fractional derivative. Fractional calculus is very important for its applications in many applied sciences.