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.