Novel iterative method for the approximation of fixed point of a class of generalized (\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha ,\beta$$\end{document})-nonexpansive mapping with applications to seir epidemic model

Abstract

In this paper, we construct a novel iterative scheme for approximating fixed points of generalized \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,\beta )$$\end{document}-nonexpansive mappings in the setting of a real Banach space. The proposed scheme not only generalizes but also unifies and extends several well-known fixed point iterative processes available in the literature. We establish both weak and strong convergence results under appropriate conditions. Furthermore, a comparative analysis of the rate of convergence is carried out using a carefully chosen numerical example, with the outcomes demonstrated through both tabular and graphical illustrations.In addition to convergence properties, we derive a data dependence result, offering insights into the stability of the proposed scheme with respect to perturbations in the underlying mapping. We further prove that the scheme satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}$$\end{document}-stability and almost \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {G}$$\end{document}-stability criteria, thereby enhancing its robustness in practical applications. To demonstrate the applicability of our results, we provide significant application of the analysis to a SEIR epidemic model governed by a Caputo-type fractional differential equation, showcasing the utility of the proposed method in the context of real-world dynamical systems. Our findings contribute to the advancement of fixed point theory and its applications in mathematical modeling, offering a flexible and powerful tool for analyzing complex nonlinear problems.