This study introduces an optimal fourth-order iterative method derived by combining two established methods, resulting in enhanced convergence when solving nonlinear equations. Through rigorous convergence analysis using both Taylor expansion and the Banach space framework, the fourth-order optimality condition is verified. We demonstrate the superior efficiency and stability of this new method compared to traditional alternatives. Numerical experiments confirm its effectiveness, showing a reduction in the average number of iterations and computational time. Visual analysis with polynomiographs confirms the method's robustness, focusing on convergence area index, iteration count, computational time, fractal dimension, and Wada measure of basins. These findings underscore the potential of this optimal method for tackling complex nonlinear problems in various scientific and engineering fields.

We present monotone convergence results for general iterative methods in order to approximate a solution of a nonlinear equation defined on a partially ordered linear topological space. The main novelty of the paper is that the operators appearing in the iterative method are not necessarily linear. This way we expand of the applicability of iterative methods. Some applications are also provided from fractional calculus using Caputo and Canavati type fractional derivatives and other areas.

We present a weaker convergence analysis of Newton’s method than in Kantorovich and Akilov (1964), Meyer (1987), Potra and Ptak (1984), Rheinboldt (1978), Traub (1964) on a generalized Banach space setting to approximate a locally unique zero of an operator. This way we extend the applicability of Newton’s method. Moreover, we obtain under the same conditions in the semilocal case weaker sufficient convergence criteria; tighter error bounds on the distances involved and an at least as precise information on the location of the solution. In the local case we obtain a larger radius of convergence and higher error estimates on the distances involved. Numerical examples illustrate the theoretical results.