On Local and Semi-Local Convergence Analysis of A High-Order Iterative Method for Solving Nonlinear Systems Without High Derivatives

In this paper, we study a general high-order iterative method for solving nonlinear systems in Banach spaces without requiring higher-order derivatives. The proposed method constructs each iteration by combining evaluations of the operator and its derivative, together with an adapted correction scheme. A detailed local convergence analysis under majorant conditions is provided, establishing the convergence to the solution. We also show a semi-local convergence by introducing new majorizing sequences. The theoretical results are illustrated with examples, and confirm the theoretical predictions.