A Modified Iterative Scheme for Fixed-Point Approximation in Banach Spaces applied to a Fractional Viscoelastic Model

In this paper, we propose a new three-step iterative scheme to approximate fixed points of contraction operators in Banach spaces. Under standard Lipschitz conditions, we establish the existence, uniqueness, and strong convergence of the iterative sequence. The convergence rate and data dependence of the method are also investigated. A comparative analysis with Noor, Picard–S, Abbas–Nazir, SP, and NIP iterative methods is presented. As an application, the proposed scheme is employed to solve a fractional viscoelastic model involving a Caputo derivative of order 0<α<1, which is reformulated as a Volterra integral equation. The numerical results, including error analysis and graphical illustrations, demonstrate that the proposed method achieves faster convergence and a higher accuracy.