This paper investigates the fixed points of weak contraction mappings within the framework of an arbitrary Banach space. The study builds on Alam’s iterative procedure [4], aiming to enhance convergence properties and address complex problems in applied mathematics. The convergence behavior of the proposed iterative method is established via a strong convergence theorem. The stability of the iteration process for weak contractions is also examined. Furthermore, a recent strategy that incorporates an operator based on Green’s function into the Alam iterative scheme is analyzed to broaden the applicability of the method. The proposed algorithm demonstrates a higher rate of convergence compared to the analogous methods of Thakur [37], Piri [32], Ali [8], Z [41], Beg [13] and DH [6]. Two numerical examples in \(\mathbb {R}\) and \(\mathbb {R}^2\) validate this claim. The convergence of the modified scheme incorporating Green’s function is also theoretically justified. To illustrate the practical utility of the method, an application to fractional differential equations, specifically a Volterra Fredholm integro type equation, is presented. A supporting numerical example confirms the effectiveness of the proposed approach, underlining its significance in tackling complex problems in applied mathematics.
