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.