Accelerated Fixed-Point Approximation for Contraction Mappings with Applications to Fractional Models

In this paper, we develop an accelerated three-step iterative scheme for the approximation of fixed points of contraction mappings in Banach spaces, with a particular focus on applications to fractional models. Strong convergence of the proposed iteration is established under standard contraction assumptions, together with stability and data dependence results. A refined rate of convergence analysis shows that the new scheme achieves a smaller effective contraction factor and converges faster than several classical two- and three-step iterative methods, including the Picard, Mann, Ishikawa, and S-iteration processes. The theoretical results are applied to Caputo-type fractional differential equations by reformulating the associated boundary value problems as fixed-point equations. Existence and uniqueness of solutions follow from the Banach contraction principle, while the accelerated convergence of the proposed iteration leads to improved numerical efficiency. Extensive numerical experiments, including fractional differential equations and nonlinear contraction mappings on the real line, are presented to validate the theoretical findings. The results demonstrate that the proposed three-step iteration provides an effective and reliable computational tool for fractional and non-local models.