Investigation of Data Dependence and Convergence Behavior of \(\mathcal{M}\)-Iteration on Banach Spaces with an Application

In this article, we re-analyse and improve the stability and data dependence results obtained by Kaur et al. [Math. Probl. Eng., 2022, Article ID 9327527] by eliminating certain restrictions imposed on control sequences by the authors. Additionally, we conceive a few weak and strong convergence results using \(\mathcal{M}\) -iterative technique to approximate the fixed point of generalized \((\alpha,\beta)\) -nonexpansive mappings. However, we provide a couple of nontrivial examples to attest that the class of generalized \((\alpha,\beta)\) -nonexpansive mappings and that of \(C\) - \(\alpha\) nonexpansive mappings are independent. Finally, our findings are applied to approximate the solution of a certain kind of delay nonlinear Volterra integral equation.