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

TL;DR

The paper introduces the AT two-step iterative algorithm for fixed-point approximation of weak contractions in complete normed spaces, proving strong convergence to the unique fixed point with the key bound $\|s_{m+1}-s\| \le \zeta^{3}\|s_m-s\|$. It also establishes almost $R$-stability and provides a data-dependence bound: if an approximate operator $F$ satisfies $\|Fp-Rp\|\le \epsilon$, then the distance between fixed points obeys $\|s-t\| \le (5\epsilon + 2\zeta\epsilon + \zeta^2\epsilon)/(1-\zeta)$. Through a numerical example, AT is shown to outperform classical schemes like Picard, Mann, Ishikawa, S, normal-S, Varat, and $F^{*}$ in convergence speed, and its applicability extends to solving nonlinear boundary-value problems via contraction-like integral operators. The work highlights practical data-dependent insights and lays groundwork for future improvements and extensions of two-step fixed-point methods.

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

Figures (2)

• Figure 1: Figure 1. Comparisons of iterations.
• Figure 2: Comparisons errors of different iterations with $AT$ iteration

Theorems & Definitions (24)

• Definition 1
• Definition 2
• Theorem 1
• Definition 3
• Definition 4
• Lemma 1
• Lemma 2
• Lemma 3
• Definition 5
• Definition 6