A convergence analysis for a certain family of extended iterative methods: Part I. Theory

Abstract. We present local and semilocal convergence results for some extended methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. In earlier studies the operator involved is assumed to be at least once Fréchet-differentiable. In the present study, we assume that the operator is only continuous. This way we expand the applicability of these methods. In Part II of the study, we present some choices of the operators involved in fractional calculus where the operators satisfy the convergence conditions. Moreover, we present a corrected version of the generalized fractional Taylor's formula given in [14].