A new approach to convergence analysis of iterative models with optimal error bounds

TL;DR

This work addresses the convergence analysis of Ishikawa-type fixed-point iterations to a common fixed point of two non-expansive mappings $T_1,T_2$ in Banach spaces by introducing optimal error bounds (OEBs). It develops explicit upper and lower bound sequences (OUEB and OLEB) and derives necessary and sufficient conditions for convergence, along with sharp convergence-rate estimates. The paper analyzes both the Ishikawa and modified Ishikawa iterations, providing exact OEB-based rate characterizations and a framework to compare the two schemes. Numerical experiments validate the theory, demonstrating how the sums of step-sizes control convergence and speed, and showing that the modified Ishikawa scheme can outperform the classical one under suitable conditions.

Abstract

In this paper, we study a new approach related to the convergence analysis of Ishikawa-type iterative models to a common fixed point of two non-expansive mappings in Banach spaces. The main novelty of our contribution lies in the so-called \emph{optimal error bounds}, which established some necessary and sufficient conditions for convergence and derived both the error estimates and bounds on the convergence rates for iterative schemes. Although a special interest here is devoted to the Ishikawa and modified Ishikawa iterative sequences, the theory of \emph{optimal error bounds} proposed in this paper can also be favorably applied to various types of iterative models to approximate common fixed points of non-expansive mappings.

Figures (9)

• Figure 1: Errors plotted against $n$ on a log-scale with four test models of the series $\sum{b_n}$ in \ref{['eq:bn']}, where $\alpha_1 = \frac{1}{2}$ and $\alpha_2 = \frac{1}{5}$.
• Figure 2: Errors plotted against $n$ on a log-scale with three test models of the series $\sum{a_n}$ in \ref{['an-fig1b']}, where $\alpha_1 = \frac{1}{2}$ and $\alpha_2 = \frac{1}{5}$.
• Figure 3: Errors plotted against $n$ on a log-scale with four test models of the series $\sum{b_n}$ in \ref{['eq:bn']}, where $\alpha_1 = \frac{1}{2}$ and $\alpha_2 = 1$.
• Figure 4: Errors plotted against $n$ on a log-scale for various choices of $a_n$ and $b_n$ as in \ref{['eq:anbn']}.
• Figure 5: Estimate of convergence rates of Ishikawa iterative process \ref{['xn-I']} with $\alpha_1, \alpha_2$ as in \ref{['alpha12']}.

Theorems & Definitions (16)

• Definition 2.1: Non-expansive mapping
• Definition 2.2: Common fixed point of two mappings
• Definition 2.3: Comparison of convergence rates
• Definition 2.4: Optimal upper error bound (OUEB)
• Example 2.5
• Definition 2.6: Optimal lower error bound (OLEB)
• Lemma 2.7
• Theorem 3.1
• Remark 3.2
• Theorem 3.3