<p>This manuscript provides a local convergence analysis of modified Potra–Pták iteration methods for solving nonlinear equations in Banach spaces. A key aspect of this analysis is the role of the first-order Fréchet derivative, which satisfies both the Lipschitz and Hölder conditions. In the convergence analysis, we provide theorems to determine the radius of convergence and establish error bounds, alongside proving the uniqueness of solutions. We verify both conditions through numerous examples, including nonlinear integral equations like the Hammerstein equation. Additionally, our results are compared with existing iterative methods to highlight improvements. The main contribution of our manuscript is the determination of the radius of convergence in two distinct cases with parameters for the modified Potra–Pták method, where the second case demonstrates improved performance compared to the first.</p>