Generalizing the local convergence analysis of a class of $k$-step iterative algorithms with Hölder continuous derivative in Banach spaces
We present the local convergence analysis of a class of $k$-step iterative algorithms using the Hölder continuity condition for approximating solutions of nonlinear equations in Banach spaces. Our analysis is based on the Hölder continuity of the first-order Fréchet derivative and boosts the applicability of the family when the Lipschitz condition fails. This convergence analysis generalizes the local convergence results with the Lipschitz continuity condition. Also, it produces radii of balls of convergence along with the bounds on the error and uniqueness of the solution. The dynamical properties of the class are also explored using complex dynamics tools. Finally, numerical tests are conducted in support of our new theoretical results.