The Impact of Data Dependence, Convergence and Stability by $AT$ Iterative Algorithms

TL;DR

This paper introduces the $AT$ iterative algorithm for fixed point approximation in normed spaces, demonstrating its faster convergence and stability over existing methods through theoretical proofs and numerical examples.

Contribution

The paper presents a new $AT$ algorithm with proven strong convergence and enhanced stability, outperforming traditional iterative methods for weak contractions.

Findings

$AT$ algorithm converges faster than existing methods.

The algorithm exhibits almost stable behavior for weak contractions.

Numerical examples validate the practical efficiency of $AT$.

Abstract

This article aims to present the $AT$ algorithm, a novel two-step iterative approach for approximating fixed points of weak contractions within complete normed linear spaces. The article demonstrates the convergence of $AT$ algorithm towards fixed points of weak contractions. Notably, it establishes the algorithm's strong convergence properties, highlighting its faster convergence compared to established iterative methods such as $S$, normal-$S$, Varat, Mann, Ishikawa, $F^{*} $, and Picard algorithms. Additionally, the study explores the $AT$ algorithm's almost stable behavior for weak contractions. Emphasizing practical applicability, the paper offers data-dependent results through the $AT$ algorithm and substantiates findings with illustrative numerical examples