A convergence analysis for extended iterative algorithms with applications to fractional and vector calculus

We give local and semilocal convergence results for some iterative algorithms 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 extend the applicability of iterative algorithms. We also present some choices of the operators involved in fractional calculus and vector calculus where the operators satisfy the convergence conditions.