The HK-Iteration: A Faster Convergent Scheme for Generalized $(\alpha,\beta)$-Nonexpansive Mappings with Applications

Abstract

In this paper, we introduce and investigate a new three-step iterative process, termed the \emph{HK-iteration}, for approximating fixed points of generalized $(\alpha,\beta)$-nonexpansive mappings in uniformly convex Banach spaces. We establish both weak and strong convergence theorems for the proposed process and prove that the HK-iteration achieves faster convergence than several classical iterative methods, including the Ishikawa \cite{ishikawa},Agarwal\cite{agarwal}, Mann\cite{mann},M\cite{M}and Picard-S\cite{Picard-S} iterations. A comprehensive numerical example is provided to validate the theoretical findings and to illustrate the superior rate of convergence and stability of the HK-iteration compared with the aforementioned schemes. Moreover, the practical applicability of the proposed method is demonstrated through its successful implementation on a nonlinear delay Volterra integral equation, confirming its strong convergence and computational reliability. Consequently, the present study not only unifies but also extends a broad spectrum of results in fixed point approximation theory and underscores the effectiveness of the HK-iteration in both analytical and applied settings.