Fixed-point Acceleration Algorithm of Total Asymptotically Nonexpansive Mappings

Rana Fadhil; Salwa Salman

doi:10.12688/f1000research.172972.1

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Research Article

Fixed-point Acceleration Algorithm of Total Asymptotically Nonexpansive Mappings

[version 1; peer review: awaiting peer review]

Rana Fadhil ¹, Salwa Salman¹

PUBLISHED 22 Dec 2025

Author details Author details

¹ Mathematics, University of Baghdad, Baghdad, Baghdad Governorate, 10011, Iraq

Rana Fadhil
Roles: Writing – Original Draft Preparation

Salwa Salman
Roles: Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS AWAITING PEER REVIEW

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

The fixed-point iteration method is a powerful tool in analysis and its extensions in other branches. It provides a unified structure for proving existence and uniqueness results for various nonlinear problems. This study, conducted within the scope of a real Banach space enriched by uniform convexity, presents an accelerated four-step iterative scheme designed to approximate elements of the fixed-point set corresponding to a mapping called a total asymptotically nonexpansive (briefly, TAN) mapping. The results of weak and strong convergence for the proposed method are established. The results relating to weak convergence require Opial’s condition of the normed space, while the results relating to strong convergence require the addition of the semi-compactness assumption of the mapping under consideration to the previous assumptions. Additionally, numerical simulations are conducted to evaluate the algorithm’s effectiveness and compare it with previously established approaches. These findings extend and refine several recent contributions available in the literature. Lastly, the work includes an application to 2- dimensional equations.

Keywords

Banach Spaces, Iterative Schemes, Total Asymptotically Non-Expansive Mappings, Fixed Points, Strong Convergence, Weak Convergence, Stability.

Corresponding author: Rana Fadhil

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2025 Fadhil R and Salman S. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Fadhil R and Salman S. Fixed-point Acceleration Algorithm of Total Asymptotically Nonexpansive Mappings [version 1; peer review: awaiting peer review]. F1000Research 2025, 14:1430 (https://doi.org/10.12688/f1000research.172972.1) First published: 22 Dec 2025, 14:1430 (https://doi.org/10.12688/f1000research.172972.1) Latest published: 22 Dec 2025, 14:1430 (https://doi.org/10.12688/f1000research.172972.1)

Introduction

In Banach spaces, fixed-point theory has evolved through a series of extensions of contractive-type mappings. One of the earliest key contributions was made by Browder (1965) and Kirk (1965) in,^1,2 who initiated the study of nonexpansive mappings and established essential fixed-point results in uniformly convex Banach spaces. This was extended later by Goebel and Kirk (1972) in,³ who introduced asymptotically nonexpansive mappings, there by widening the scope of fixed-point theory by considering operators whose iterates approach the nonexpansive property in an asymptotic sense. Building on these developments, Alber, Chidume, and Zegeye (2005) in⁴ proposed the class of totally asymptotically nonexpansive TAN-mappings. A mapping $Ψ : X \to X$ is defined as TAN-mapping if $\exists$ nonnegative real sequences $, {κ_{v}}$ where $μ_{v} \to 0, κ_{v} \to 0 as v \to \infty$ , together with a strictly increasing continuous function $\emptyset : ℝ^{+} \to ℝ^{+}$ satisfying $\emptyset (0) = 0$ such that for all $x, y \in X$ .

(1)

‖ Ψ^{v} x - Ψ^{v} y ‖ \leq ‖ x - y ‖ + μ_{v} \emptyset (‖ x - y ‖) + κ_{v} holds for all v \geq 1

This generalization incorporates both nonexpansive and asymptotically nonexpansive mappings as particular cases. Since its introduction, the concept of TAN-mapping has primarily been explored through iterative schemes in Banach spaces. Among the earliest adaptations were the classical Mann and Ishikawa iterations, later followed by the Halpern iteration in uniformly convex Banach spaces.¹⁰ More recently, Abed and Abed (2022) in⁵ investigated the Fibonacci–Halpern iteration for monotone TAN-mappings in partially ordered Banach spaces, establishing its stability and showing faster convergence compared with Mann and Halpern iterations. In 2023, Salman and Abed²⁴ proposed a five-step iterative scheme for (λ,ρ)-firmly nonexpansive multivalued mappings in modular spaces, demonstrating that these satisfy both the (UUC1) property and the Δ2-condition. Khomphungson and Nammanee (2024) further analyzed modified Mann–Ishikawa algorithms, proving strong convergence under Condition (A) in uniformly convex Banach spaces. Moreover, Balooee and Al-Homidan (2024) in^4,7 developed hybrid resolvent-based methods that connect fixed-point problems of TAN-mappings with generalized variational-like inclusions, ensuring the existence of a common element belonging simultaneously to the solution set of such problems and to the fixed-point set of a TAN-mapping. These formulations are closely related to integral and differential equations, which are frequently reformulated as operator inclusions within Banach spaces. Moreover, recent studies have enriched and broadened the theoretical perspective. For instance, Galisu et al. (2024)⁸ introduced enriched asymptotically nonexpansive mappings with center zero in a reflexive and strictly convex Banach space, there by providing a broader framework that incorporates TAN-mappings. Similarly, Sun (2025)⁹ extended the theory to random asymptotically nonexpansive mappings in uniformly convex modules, emphasizing stochastic extensions still tied to Banach-type structures. In summary, as reported in,^1,2 the research direction has progressed from nonexpansive mappings³ to asymptotically nonexpansive mappings, culminating in the more general class of TAN-mappings.⁴ A wide variety of iterative schemes such as Mann, Ishikawa, Halpern, Fibonacci-Halpern, modified Mann-Ishikawa, and hybrid resolvent-based algorithms have played a crucial role in establishing convergence, stability, and practical applications. More recent findings in^5,6 reaffirm the central significance of TAN-mappings in contemporary fixed-point theory within Banach spaces, particularly in relation to stability outcomes and their applications to variational inclusions and integral-differential models.

The primary objective of this work is to examine the convergence and stability characteristics of an inertial accelerated iterative scheme within the framework of TAN-mappings in Banach spaces. Building upon the inertial accelerated approach proposed by Harbau et al. (2022),¹⁰ we establish and prove weak and strong convergence results for the introduced process and further confirm its stability when subjected to perturbations. Moreover, a comparative study with two alternative iterative methods is conducted, demonstrating that the inertial accelerated scheme attains faster convergence. Finally, the obtained convergence results are applied to integral equations, highlighting both the practical importance and the applicability of the theoretical outcomes.

Preliminaries

This part includes requirements for definitions, lemmas, and facts for proving the main results.

Definition 1.

¹¹ If $X$ is a Banach space named uniformly convex if for any $ϵ \in (0, 2),$ $\exists δ (ϵ) > 0 ∋ ‖ \frac{p + q}{2} ‖ \leq 1 - δ (ϵ)$ holds, for all $p, q \in X$ with $‖ p ‖ \leq 1, ‖ q ‖ \leq 1$ , and $‖ p - q ‖ \geq ϵ$ .

Remark 1.

¹⁰ Note that every Hilbert space satisfies Definition (1)

Symbols

i) $X$ denotes the real Banach Space.
ii) $X^{*}$ denotes the dual of a space $X$ .
iii) ⇀ is used for weak convergence, while ⟶ is used for strong convergence.
iv) The set of $ω$ - $weak$ cluster limits of a sequence ${q_{v}}$ is $ω_{c} (q_{v}) = {z : \exists q_{vk} ⇀ z}$ represents.
v) $Fix (Ψ) ≕ {p$ in $X : Ψ (p) = p}$ , all fixed points of $Ψ$ .

Definition 2.

¹⁰ Let $⟨ ., . ⟩$ be a duality pairing on $X \times X^{*}$ . A mapping $J_{φ} : X \to 2^{X^{*}}$ defined as $J_{φ} (p) = {p^{*} \in X^{*} : ⟨ p, p^{*} ⟩ = ‖ p ‖ ‖ p^{*} ‖, ‖ p^{*} ‖ = φ (‖ p ‖)}$ and $φ$ be a gauge function is a generalized duality mapping.

Note, if $φ (t) = t$ for all $t \geq 0$ then $J_{φ} = J_{2}$ is a normalized duality mapping and can be expressed as $J (p) = {p^{*} \in X^{*} : ⟨ p, p^{*} ⟩ = {‖ p ‖}^{2}, ‖ p^{*} ‖ = ‖ p ‖}$ .

Definition 3.

¹² A mapping $Ψ : C \subseteq X \to C$ is called weakly sequentially continuous if, for all sequences ${q_{v}}$ in $X$ with $q_{v} ⇀ z$ , it follows that $(q_{v}) ⇀ Ψ (z)$ .

Definition 4.

¹³ A mapping Ψ: $Ψ : C \subseteq X \to C is called$

(1) Demi-closed at $y_{0} \in C$ , if ${q_{v}}$ in C such that $q_{v} ⇀ z \in C$ and $Ψ q_{v} \to y_{0}$ , then $Ψ q_{0} = y_{0}$ .
(2) Semi-compact if $lim_{v \to \infty} ‖ q_{v} - Ψ q_{v} ‖ = 0$ , for any bounded sequence ${q_{v}} \subset C$ then $\exists$ a subsequence ${q_{vk}} \subset {q_{v}}$ and $q_{vk} \to z \in C .$

Suppose a fixed-point iteration scheme is defined by a general form

(2)

q_{v + 1} = f (Ψ, q_{v})

for

v = 0, 1, 2, \dots,

for a mapping

Ψ

and

q_{0} \in X

. This ensures the convergence of

{q_{v}}_{v = 0}^{\infty}

to

z \in Fix (Ψ)

. Where

f (Ψ, q_{v})

does contain all parameters that define the given fixed-point iteration scheme.

Definition 5.

¹⁴ Let ${p_{v}}_{v = 0}^{\infty}$ be an arbitrary sequence in $X$ and fix $ε_{v} = ‖ p_{v + 1} - f (Ψ, p_{v}) ‖$ for $= 0, 1, 2, . .$ . Then $q_{v + 1} = f (Ψ, p_{v})$ is $Ψ$ -stable (or stable) with respect to $Ψ$ if and only if.

lim_{v \to \infty} ε_{v} = 0 ⬄ lim_{v \to \infty} p_{v} = z .

Definition 6.

¹⁵ Suppose ${q_{v}}_{v = 0}^{\infty}$ and ${s_{v}}_{v = 1}^{\infty}$ are two sequences converging to the same fixed point $z$ with $‖ q_{v} - z ‖ \leq a_{v},$ and $‖ s_{v} - z ‖ \leq b_{v}$ for all $v \geq 0$ . If the sequence $a_{v} \to a$ and $b_{v} \to b$ and additionally $lim_{v \to \infty} \frac{‖ a_{v} - a ‖}{‖ b_{v} - b ‖} = 0,$ then ${q_{v}}_{v = 1}^{\infty}$ is said to converge to $z$ faster than ${s_{v}}_{v = 1}^{\infty}$ .

Definition 7.

¹⁶ A space $X$ has Opial’s condition if for all ${q_{v}} \subset X$ with $q_{v} ⇀ z_{1}$ satisfied $lim_{v \to \infty} sup ‖ q_{v} - z_{1} ‖ < lim_{v \to \infty} sup ‖ q_{v} - z_{2} ‖$ $\forall$ $z_{2} \in X$ and $z_{2} \neq z_{1}$ .

Definition 8.

²³ A set $C$ is convex if $λq + (1 - λ) s \in C,$ for any $q, s$ in $C$ and $λ \in [0, 1]$ .

The Lemma below is known as Opial’s property.

Lemma 1.

¹⁷ If a duality mapping $J$ is weakly continuous on $X$ , and a sequence ${q_{v}} \subset X$ with $q_{v} ⇀ z$ , then for any

(3)

p \in X . lim_{v \to \infty} inf ‖ q_{v} - p ‖ \geq lim_{v \to \infty} inf ‖ q_{v} - z ‖

Moreover, if $X$ is uniformly convex, equality occurs ⬄ $p = z$ .

Lemma 2.

¹⁸ Suppose $X$ is a real uniformly convex Banach space and $\emptyset \neq C \subset X$ is a closed convex. If $Ψ : C \to C$ TAN-mapping, then ( $I - Ψ$ ) is demiclosed at zero, whenever ${q_{v}}$ is a sequence in $C$ such that $q_{v} ⇀ z$ .

Lemma 3.

¹⁹ Assume ${a_{v}}, {b_{v}},$ and ${c_{v}}$ are nonnegative sequences such that $a_{v + 1} \leq (1 + c_{v}) a_{v} + b_{v}$ , $\sum_{v = 0}^{\infty} b_{v} < \infty$ and $\sum_{v = 0}^{\infty} c_{v} < \infty, \forall v \geq 0$ then $:$

(i) ${a_{v}}$ is convergent.
(ii) If $lim_{v \to \infty} inf a_{v} = 0$ then $lim_{v \to \infty} a_{v} = 0$ .

Lemma 4.

²⁰ A Banach space $X$ is uniformly convex if and only if there exists a continuous, strictly increasing function $g : [0, \infty) \to [0, \infty)]$ with $g (0) = 0,$ such that: ${‖ λ p + (1 - λ) q ‖}^{2} \leq λ {‖ p ‖}^{2} + (1 - λ) {‖ q ‖}^{p} - λ (1 - λ) g (‖ p - q ‖)$ , holds for all $p, q in B_{r} = {p \in X : ‖ p ‖ \leq r} and every λ \in [0, 1]$ and a fixed number $r > 0$ .

Lemma 5.

²² A Banach space $X$ is uniformly convex and $0 < b < c < 1$ , which is constant. Let ${a_{v}}$ sequence in $[b, c]$ such that $lim_{v \to \infty} ‖ a_{v} q_{v} + (1 - a_{v}) s_{v} ‖ = d, lim_{v \to \infty} sup ‖ q_{v} ‖ \leq d, lim_{v \to \infty} sup ‖ s_{v} ‖ \leq d$ , satisfy for some $d \geq 0 .$ Then $lim_{v \to \infty} ‖ q_{v} - s_{v} ‖ = 0$ , where ${q_{v}}, {s_{v}} \in X .$

Main results

This section is devoted to presenting results concerning weak and strong convergence, stability, and comparative analysis, as well as an illustrative example and application. Below, $X$ will be a Banach space with uniformly convexity.

Weak convergence

The following Assumptions appeared in²¹

Assumption 1.

(i) Choose sequences ${α_{v}} \subset (0, 1), {σ_{v}}, {δ_{v}} \subset [0, \infty)$ and $\sum_{v = 1}^{\infty} δ_{v} < \infty$ with $δ_{v} = o (σ_{v})$ which means $lim_{n \to \infty} \frac{δ_{v}}{σ_{v}} = 0 .$
(ii) Let $q_{0}, q_{1} \in X$ given, for the terms $q_{v - 1}$ and $q_{v}$ for each $v \geq 1$ choose $ω_{v}$ , such that $0 \leq ω_{v} \leq \bar{ω_{v}}$ . For any $η \geq 3$ define

(4)

\bar{ω_{v}} ≔ {\begin{matrix} min {\frac{v - 1}{v + η - 1}, \frac{δ_{v}}{‖ q_{v} - q_{v - 1} ‖}}, & if q_{v} \neq q_{v - 1} \\ \frac{v - 1}{v + η - 1}, & otherwise \end{matrix}

Remark 2.

From the given Assumption, it directly follows that for every $v \geq 1,$ the relation.

ω_{v} ‖ q_{v} - q_{v - 1} ‖ \leq δ_{v}

Which together with $\sum_{v = 1}^{\infty} δ_{v} < \infty$ and $lim_{v \to \infty} \frac{δ_{v}}{σ_{v}} = 0$ , holds

(5)

\sum_{v = 1}^{\infty} ω_{v} ‖ q_{v} - q_{v - 1} ‖ < \infty

and

(6)

lim_{v \to \infty} \frac{ω_{v}}{σ_{v}} ‖ q_{v} - q_{v - 1} ‖ \leq lim_{n \to \infty} \frac{δ_{v}}{σ_{v}} = 0

Theorem 1.

Let $X$ satisfy Opial’s property and $Ψ : X \to X$ TAN-mapping associated with the sequences ${μ_{v}}, {κ_{v}} \subset (0, \infty)$ and a strictly increasing continuous function $\emptyset : ℝ^{+} \to ℝ^{+}$ such that $\emptyset (0) = 0$ and $\sum_{v = 1}^{\infty} (μ_{v} + κ_{v}) < \infty$ . Define the sequence ${q_{v}}$ as: let.

(7)

{\begin{matrix} q_{0}, q_{1} \in X \\ a_{v} = q_{v} + ω_{v} (q_{v} - q_{v - 1}) \\ b_{v + 1} = \frac{1}{ρ} (Ψ^{v} (a_{v}) - a_{v}) + σ_{v} b_{v} \\ p_{v} = a_{v} + ρ b_{v + 1} \\ q_{v + 1} = λ α_{v} a_{v} + (1 - λ α_{v}) p_{v}, \forall v \geq 1, \end{matrix}

where

λ \in (0, 1], ρ > 0

and

$(C_{1}) \sum_{v = 0}^{\infty} σ_{v} < \infty$

(C_{2}) lim_{v \to \infty} inf λ α_{v} (1 - λ α_{v}) > 0

$(C_{3}) {Ψ^{v} a_{v} - a_{v}}$ is bounded.

If Assumption (i) holds, $b_{1} = \frac{1}{ρ} (Ψ^{1} a_{0} - a_{0})$ and $Fix (Ψ) \neq \emptyset$ , then $q_{v} ⇀ z \in Fix (Ψ)$ .

Proof:.

The outlines of the proof are the following steps:

1. Proof ${b_{v}}$ is bounded.
2. $lim_{v \to \infty} ‖ q_{v} - z ‖$ exists for any $z \in Fix (Ψ)$
3. $lim_{v \to \infty} ‖ q_{v} - Ψ q_{v} ‖ = 0$
4. $ω_{c} (q_{v}) \subset Fix (Ψ)$
5. ${q_{v}}$ is $ω$ -convergence to a fixed point.

For (1), the condition ( $C_{1}$ ) implies $lim_{v \to \infty} σ_{v} = 0,$ so $\exists v_{0} \in N$ such that $σ_{n} \leq \frac{1}{2}$ , for every $v \geq v_{0}$ . Put $M_{1}$ as follows:

M_{1} ≔ max {max_{1 \leq k \leq v_{0}} ‖ b_{k} ‖, (\frac{2}{ρ}) sup_{v \in N} ‖ Ψ^{v} a_{v} - a_{v} ‖}

Then, by (C₃), the $M_{1} < \infty$ hold. Assume that $‖ b_{n} ‖ \leq M_{1}$ for some $v \geq v_{0}$ , then:

(8)

\begin{matrix} ‖ b_{v + 1} ‖ = ‖ \frac{1}{ρ} (Ψ^{v} a_{v} - a_{v}) + σ_{v} b_{v} ‖ \\ \leq \frac{1}{ρ} ‖ Ψ^{v} a_{v} - a_{v} ‖ + σ_{v} ‖ b_{v} ‖ \\ \leq M_{1} \end{matrix}

This implies to

‖ b_{v} ‖ \leq M_{1}, for all v \geq 0 then {b_{v}} is bounded .

For (2), from the scheme (7), this is obtained by:

(9)

p_{v} = a_{v} + ρ b_{v + 1} = a_{v} + ρ (\frac{1}{ρ} (Ψ^{v} a_{v} - a_{v}) + σ_{v} b_{v}) = Ψ^{v} a_{v} + ρ σ_{v} b_{v}

By (8), (9), and for any $z \in Fix (Ψ),$ getting:

(10)

\begin{matrix} ‖ p_{v} - z ‖ = ‖ Ψ^{v} a_{v} + ρ σ_{v} b_{v} - z ‖ \\ \leq ‖ Ψ^{v} a_{v} - z ‖ + ρ σ_{v} ‖ b_{v} ‖ \\ \leq ‖ a_{v} - z ‖ + μ_{v} \emptyset (‖ a_{v} - z ‖) + κ_{v} + ρ M_{1} σ_{v} \end{matrix}

Let $U = sup {‖ a_{v} ‖ + ‖ a_{v} - z ‖} < \infty,$ $ζ_{v} = μ_{v} \emptyset (U) + κ_{v}$

Additionally,

(11)

\begin{matrix} ‖ a_{v} - z ‖ = ‖ q_{v} - z + ω_{v} (q_{v} - q_{v - 1}) ‖ \\ \leq ‖ q_{v} - z ‖ + ω_{v} ‖ (q_{v} - q_{v - 1}) ‖ \end{matrix}

By (10) and (11), getting:

\begin{matrix} ‖ p_{v} - z ‖ \leq ‖ q_{v} - z ‖ + ω_{v} ‖ (q_{v} - q_{v - 1}) ‖ + κ_{v} + ρ M_{1} σ_{v} \\ = ‖ q_{v} - z ‖ + σ_{v} [\frac{ω_{v}}{σ_{v}} ‖ (q_{v} - q_{v - 1}) ‖ + \frac{κ_{v}}{σ_{v}} + ρ M_{1}] \end{matrix}

By Remark 2, getting that the sequence $\frac{ω_{v}}{σ_{v}} ‖ (q_{v} - q_{v - 1}) ‖$ converges, then $\exists M_{2} > 0$ such that for every $v \geq 1, \frac{ω_{v}}{σ_{v}} ‖ (q_{v} - q_{v - 1}) ‖ + \frac{κ_{v}}{σ_{v}} + ρ M_{1} \leq M_{2}$ .

Thus:

(12)

‖ p_{v} - z ‖ \leq ‖ q_{v} - z ‖ + σ_{v} M_{2}

Now, using (10), (12), and for some $M_{3} > 0$ any $z \in Fix (Ψ)$ getting:

(13)

\begin{matrix} ‖ q_{v + 1} - z ‖ = ‖ λ α_{v} a_{v} + (1 - λ α_{v}) p_{v} - z ‖ \\ \leq λ α_{v} ‖ a_{v} - z ‖ + (1 - λ α_{v}) ‖ p_{v} - z ‖ \\ \leq λ α_{v} [‖ q_{v} - z ‖ + ω_{v} ‖ (q_{v} - q_{v - 1}) ‖] + (1 - λ α_{v}) ‖ p_{v} - z ‖ \\ \leq λ α_{v} ‖ q_{v} - z ‖ + ω_{v} λ α_{v} ‖ (q_{v} - q_{v - 1}) ‖ + (1 - λ α_{v}) ‖ q_{v} - z ‖ \\ + (1 - λ α_{v}) σ_{v} M_{2} \end{matrix}

‖ q_{v + 1} - z ‖ \leq ‖ q_{v} - z ‖ + σ_{v} M_{3},

where

(14)

\frac{ω_{v} λ α_{v}}{σ_{v}} ‖ (q_{v} - q_{v - 1}) ‖ + (1 - λ α_{v}) M_{2} \leq M_{3}

Hence, from Lemma 5 and (C₁), it follows that $lim_{v \to \infty} ‖ q_{v} - z ‖$ exists. As a result, the sequence ${q_{v}}$ is bounded.

For (3) the above step follows that ${a_{v}}$ is bounded, and thus ${Ψ^{v} a_{v}}$ is bounded as well. Define $r = sup_{v \geq 1} {‖ a_{v} ‖, ‖ Ψ^{v} a_{v} ‖}$ and by condition (13), Lemma 4, and hypothesis C1, getting.

\begin{matrix} {‖ q_{v + 1} - z ‖}^{2} = {‖ λ α_{v} a_{v} + (1 - λ α_{v}) p_{v} - z ‖}^{2} \\ \leq {‖ a_{v} - z ‖}^{2} + (1 - λ α_{v}) ζ_{v}^{2} - λ α_{v} (1 - λ α_{v}) g (‖ a_{v} - Ψ^{v} a_{v} ‖) \\ + ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} - 2 λ α_{v} ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} + λ^{2} ρ^{2} α_{v}^{2} σ_{v}^{2} {M_{1}}^{2} \\ + 2 (1 - λ α_{v}) ρ σ_{v} M_{1} (‖ a_{v} - z ‖ + (1 - λ α_{v}) ζ_{v}) \end{matrix}

Since the $lim_{v \to \infty} ‖ q_{v} - z ‖$ exists for any $z \in Fix (Ψ)$ , then by applying (5) and (11), $\exists L > 0$ such that $‖ a_{v} - z ‖ \leq L$

\begin{matrix} {‖ q_{v + 1} - z ‖}^{2} \leq {‖ a_{v} - z ‖}^{2} + (1 - λ α_{v}) ζ_{v}^{2} - λ α_{v} (1 - λ α_{v}) g (‖ a_{v} - Ψ^{v} a_{v} ‖) \\ + ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} - 2 λ α_{v} ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} + λ^{2} ρ^{2} α_{v}^{2} σ_{v}^{2} {M_{1}}^{2} \\ + 2 (1 - λ α_{v}) ρ σ_{v} M_{1} (L + (1 - λ α_{v}) ζ_{v}) \\ \leq {‖ q_{v} - z ‖}^{2} + 2 ω_{v} ‖ q_{v} - q_{v - 1} ‖ ‖ q_{v} - z ‖ + {(ω_{v} ‖ q_{v} - q_{v - 1} ‖)}^{2} \\ - λ α_{v} (1 - λ α_{v}) g (‖ a_{v} - Ψ^{v} a_{v} ‖) + ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} \\ - 2 λ α_{v} ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} + λ^{2} ρ^{2} α_{v}^{2} σ_{v}^{2} {M_{1}}^{2} \\ + 2 (1 - λ α_{v}) ρ σ_{v} M_{1} (L + (1 - λ α_{v}) ζ_{v}) \end{matrix}

Because the $lim_{n \to \infty} ‖ q_{v} - z ‖$ exists for any z $\in Fix (Ψ)$ , the sequence ${‖ q_{v} - z ‖}$ is bounded; hence, $\exists H > 0$ such that $‖ q_{v} - z ‖ \leq H$ for all $v \geq 1$ . Then

\begin{matrix} {‖ q_{v + 1} - z ‖}^{2} \leq {‖ q_{v} - z ‖}^{2} + 2 ω_{v} ‖ q_{v} - q_{v - 1} ‖ H + {(ω_{n} ‖ q_{v} - q_{v - 1} ‖)}^{2} \\ - λ α_{v} (1 - λ α_{v}) g (‖ a_{v} - Ψ^{v} a_{v} ‖) + ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} \\ - 2 λ α_{v} ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} + λ^{2} ρ^{2} α_{v}^{2} σ_{v}^{2} {M_{1}}^{2} \\ + 2 (1 - λ α_{v}) ρ σ_{v} M_{1} (L + (1 - λ α_{v}) ζ_{v}) \end{matrix}

Therefore,

\begin{matrix} λ α_{v} (1 - λ α_{v}) g (‖ a_{v} - Ψ^{v} a_{v} ‖) \leq {‖ q_{v} - z ‖}^{2} - {‖ q_{v + 1} - z ‖}^{2} \\ + 2 ω_{v} ‖ q_{v} - q_{v - 1} ‖ H + {(ω_{n} ‖ q_{v} - q_{v - 1} ‖)}^{2} \\ + ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} - 2 λ α_{v} ρ^{2} {σ_{v}}^{2} {M_{1}}^{2} + λ^{2} ρ^{2} α_{v}^{2} σ_{v}^{2} {M_{1}}^{2} \\ + 2 (1 - λ α_{v}) ρ σ_{v} M_{1} (L + (1 - λ α_{v}) ζ_{v}) \end{matrix}

So,

\begin{matrix} \sum_{v = 0}^{\infty} λ α_{v} (1 - λ α_{v}) g (‖ a_{v} - Ψ^{v} a_{v} ‖) \leq {‖ q_{0} - z ‖}^{2} + 2 H \sum_{v = 0}^{\infty} ω_{v} ‖ q_{v} - q_{v - 1} ‖ + \sum_{v = 0}^{\infty} {(ω_{v} ‖ q_{v} - q_{v - 1} ‖)}^{2} \\ + {M_{1}}^{2} ρ^{2} \sum_{v = 0}^{\infty} {σ_{v}}^{2} + 2 λ ρ^{2} {M_{1}}^{2} \sum_{v = 0}^{\infty} {α_{v} σ_{v}}^{2} + λ^{2} ρ^{2} {M_{1}}^{2} \sum_{v = 0}^{\infty} α_{v}^{2} σ_{v}^{2} \\ + \sum_{v = 0}^{\infty} 2 (1 - λ α_{v}) ρ σ_{v} M_{1} (L + (1 - λ α_{v}) ζ_{v}) \end{matrix}

This gives

$\sum_{v = 0}^{\infty} λ α_{v} (1 - λ α_{v}) g (‖ a_{v} - Ψ^{v} a_{v} ‖) < \infty$ . Which implies that: $lim_{v \to \infty} λ α_{v} (1 - λ α_{v}) g (‖ a_{v} - Ψ^{v} a_{v} ‖) = 0 .$ By (C₂), getting:

(15)

lim_{v \to \infty} g (‖ a_{v} - Ψ^{v} a_{v} ‖) = 0 . And lim_{v \to \infty} ‖ a_{v} - Ψ^{v} a_{v} ‖ = 0

Now,

\begin{matrix} ‖ a_{v} - q_{v} ‖ = ‖ q_{v} + ω_{v} (q_{v} - q_{v - 1}) - q_{v} ‖ \\ = ω_{v} ‖ q_{v} - q_{v - 1} ‖ \end{matrix}

Taking the sum over $v$ for the above inequality and by (5), getting: $\sum_{v = 0}^{\infty} ‖ a_{v} - q_{v} ‖ = \sum_{v = 0}^{\infty} ω_{v} ‖ q_{v} - q_{v - 1} ‖ < \infty$ . Which implies that:

(16)

\underset{v \to \infty}{lim ‖ a_{v} - q_{v} ‖} = 0

Since $Ψ$ is TAN- mapping, getting:

(17)

\begin{matrix} ‖ q_{v} - Ψ^{v} q_{v} ‖ = ‖ q_{v} - a_{v} + a_{v} - Ψ^{v} a_{v} + Ψ^{v} a_{v} - Ψ^{v} q_{v} ‖ \\ \leq ‖ q_{v} - a_{v} ‖ + ‖ a_{v} - Ψ^{v} a_{v} ‖ + ‖ Ψ^{v} a_{v} - Ψ^{v} q_{v} ‖ \\ \leq ‖ q_{v} - a_{v} ‖ + ‖ a_{v} - Ψ^{v} a_{v} ‖ + ‖ a_{v} - q_{v} ‖ + μ_{v} \emptyset (‖ a_{v} - q_{v} ‖) + κ_{v} \\ \leq 2 ‖ q_{v} - a_{v} ‖ + ‖ a_{v} - Ψ^{v} a_{v} ‖ + ζ_{v} \end{matrix}

Continuity of (15) and (16) in (17) implies to:

(18)

lim_{v \to \infty} ‖ q_{v} - Ψ^{v} q_{v} ‖ = 0

On the other hand,

\begin{matrix} ‖ a_{v} - p_{v} ‖ = ‖ a_{v} - (a_{v} + ρ b_{v + 1}) ‖ \\ = ρ ‖ b_{v + 1} ‖ \\ \leq ρ (\frac{1}{ρ} ‖ a_{v} - Ψ^{v} a_{v} ‖ + σ_{v} ‖ b_{v} ‖ \\ \leq ‖ a_{v} - Ψ^{v} a_{v} ‖ + ρ σ_{v} M_{1}) \end{matrix}

By (15) and (C₁), getting:

(19)

lim_{v \to \infty} ‖ a_{v} - p_{v} ‖ = 0

‖ q_{v + 1} - Ψ^{v} q_{v + 1} ‖ \leq λ α_{v} ‖ a_{v} - Ψ^{v} a_{v} ‖ + ρ σ_{v} M_{1} + (1 - λ α_{v}) ‖ a_{v} - p_{v} ‖ + ζ_{v}

It follows from (15), (19), and (C₁) that:

(20)

lim_{v \to \infty} ‖ q_{v + 1} - Ψ^{v} q_{v + 1} ‖ = 0

By (18) and (20), getting:

(21)

lim_{v \to \infty} ‖ q_{v} - {Ψ q}_{v} ‖ = 0

For (4) the reflexivity of $X$ makes each bounded sequence has a weakly convergent subsequence.²³ Applying this fact, ${q_{v}}$ has a subsequence ${q_{vk}}$ such that $q_{vk} ⇀ z$ , therefore, from (21), getting $lim_{k \to \infty} ‖ q_{vk} - {Ψ q}_{vk} ‖ = 0$ and consequently by Lemma 2, $Ψ z = z$ . Hence, $ω_{c} (q_{v}) \subset Fix (Ψ)$ . For (5) to establish that the sequence ${q_{v}}$ converges weakly to a fixed point of $Ψ$ , it is enough to demonstrate that $ω_{c} (q_{v})$ consists of a single element. For this purpose, using Lemma 1 and by Definition 7, let $z_{1}, z_{2} \in ω_{c} (q_{v})$ and consider subsequences ${q_{vi}}$ and ${q_{vj}}$ of ${q_{v}}$ such that.

$q_{vi} ⇀ z_{1}$ and $q_{vj} ⇀ z_{2}$ .Then for $z_{1} \neq z_{2}$ we have:

\begin{matrix} lim_{v \to \infty} ‖ q_{v} - z_{1} ‖ < lim_{i \to \infty} ‖ q_{vi} - z_{2} ‖ \\ = lim_{j \to \infty} ‖ q_{vj} - z_{2} ‖ \\ \begin{matrix} < lim_{j \to \infty} ‖ q_{vj} - z_{1} ‖ \\ = lim_{v \to \infty} ‖ q_{v} - z_{1} ‖ \end{matrix} \end{matrix}

This leads to a contradiction, which proves that $ω_{c} (q_{v})$ is a singleton. Hence, the proof is complete.

Strong convergence

Theorem 2.

Let the assumptions of Theorem 1 hold and suppose that $Ψ$ is semi-compact, then the sequence ${q_{v}}$ in (7) converges strongly to a point in $Fix (Ψ)$ .

Proof.

Semi-compactness of $Ψ$ and using steps (2-3) of the proof of Theorem 1, in addition to the facts that the sequence ${q_{v}}$ is bounded and $lim_{v \to \infty} ‖ q_{v} - {Ψ q}_{v} ‖ = 0$ , leads to the existence of a subsequence ${q_{vk}}$ of ${q_{v}}$ with $q_{vk} \to z$ as $k \to \infty .$ Consequently, $q_{vk} ⇀ z$ and thus $z \in ω_{c} (q_{v}) \subseteq Fix (Ψ) .$ Moreover, by proof of Theorem 1- step 2, $lim_{v \to \infty} ‖ q_{v} - z ‖$ exists, then $lim_{v \to \infty} ‖ q_{v} - z ‖ = lim_{k \to \infty} ‖ q_{vk} - z ‖ = 0$ . Hence, $q_{v} \to z \in Fix (Ψ)$ .

Corollary 1.

In hypotheses of Theorem 1, suppose $Ψ : X \to X$ be a nonexpansive mapping with F $ix (Ψ) \neq \emptyset$ . Then the sequence ${q_{v}}$ is defined by (7) with $λ \in (0, 1], ρ > 0$ and initial value $b_{0} = \frac{1}{ρ} (Ψ a_{0} - a_{0})$ , satisfied $q_{v} ⇀$ $z \in Fix (Ψ)$ , and the following condition hold:

$(C_{1}) \sum_{v = 0}^{\infty} σ_{v} < \infty$ ,

$(C_{2}) lim_{v \to \infty} inf λ α_{v} (1 - λ α_{v}) > 0$ ,

$(C_{3}) {Ψ a_{v} - a_{v}}$ is bounded.

Corollary 2.

Let $X$ be a real Hilbert space and $Ψ : X \to X$ be (TAN-mapping) or a nonexpansive mapping with $Fix (Ψ) \neq \emptyset$ . Then the sequence $q_{v} ⇀$ $z \in Fix (Ψ)$ where $\in (0, 1], ρ > 0, b_{0} = \frac{1}{ρ} (Ψ^{1} a_{0} - a_{0})$ , and the following conditions hold:

$(C_{1}) \sum_{v = 0}^{\infty} σ_{v} < \infty$

$(C_{2}) lim_{v \to \infty} inf λ α_{v} (1 - λ α_{v}) > 0$

$(C_{3}) {Ψ^{v} a_{v} - a_{v}}$ is bounded.

Remark 3.

Theorem 1 covers the wider class of asymptotically nonexpansive mappings, so it provides an extension of Theorem 3.1 of Dong et al., see,²⁰ under the real uniformly convex Banach spaces, which generalize the real Hilbert spaces.

Stability

In this part, discuss the stability of a sequence defined by (7) for a fixed point of TAN-mapping where $f (Ψ, r_{v}) = λ α_{v} a_{v} + (1 - λ α_{v}) p_{v} .$

Theorem 3.

Let $C \subset X$ be a closed convex set and $Ψ : C \to C$ be a TAN-mapping. Suppose that the sequence ${r_{v}}_{v = 1}^{\infty}$ as in (7), $λ \in (0, 1]$ and $ρ > 0$ . If Assumption 1 holds and ${Ψ^{v} a_{v} - a_{v}}$ is bounded, then ${r_{v}}_{v = 1}^{\infty}$ is stable provided that $b_{1} = \frac{1}{ρ} (Ψ^{1} a_{0} - a_{0})$ , with $\sum_{v = 0}^{\infty} σ_{v} < \infty$ and $lim_{v \to \infty} inf λ α_{v} (1 - λ α_{v}) > 0$ .

Proof.

Let ${r_{v}}_{v = 1}^{\infty}$ be a sequence in $C$ defined by (7) such that $r_{v + 1} = f (Ψ, r_{v})$ converging to a unique fixed point $z$ (by Theorem 1).

Set $ε_{v} = ‖ r_{v + 1} - f (Ψ, r_{v}) ‖$ Now to prove that $lim_{v \to \infty} ε_{v} = 0 ⟺ lim_{v \to \infty} r_{v} = z$

Let $lim_{v \to \infty} ε_{v} = 0 .$ Then, getting:

‖ r_{v + 1} - z ‖ \leq ‖ r_{v + 1} - f (Ψ, r_{v}) ‖ + ‖ f (Ψ, r_{v}) - z ‖ = ε_{v} + ‖ r_{v + 1} - z ‖

By Theorem 1, getting:

‖ r_{v + 1} - z ‖ \leq ‖ r_{v} - z ‖ + σ_{v} M_{3}

Since $0 < M_{3} < 1, \sum_{v = 0}^{\infty} σ_{v} < \infty$ and $lim_{v \to \infty} ε_{v} = 0,$ and by using Lemma 3, this gives $lim_{v \to \infty} ‖ r_{v} - z ‖ = 0$ . Hence, $lim_{v \to \infty} r_{v} = z .$

Conversely, let $lim_{v \to \infty} r_{v} = z$ . Then getting:

\begin{matrix} ε_{v} = ‖ r_{v + 1} - f (Ψ, r_{v}) ‖ \leq ‖ r_{v + 1} - z ‖ + ‖ f (Ψ, r_{v}) - z ‖ \\ \leq ‖ r_{v + 1} - z ‖ + ‖ r_{v} - z ‖ + σ_{v} M_{3} \end{matrix}

Continuity of this implies to $lim_{v \to \infty} ε_{v} = 0 .$ Hence, ${r_{v}}_{v = 1}^{\infty}$ is stable.

Now, recall HR-iteration²⁴ which defined by

(22)

\begin{matrix} ρ_{v} = (1 - δ_{v}) d_{v} + δ_{v} Ψ (d_{v}) \\ w_{v} = Ψ ((1 - t_{v}) ρ_{v} + t_{v} Ψ (ρ_{v})) \\ u_{v} = Ψ (Ψ (w_{v})) \\ d_{v + 1} = (1 - ϱ_{v}) u_{v} + ϱ_{v} Ψ (u_{v}) \forall v \geq 1, \\ where δ_{v}, t_{v}, ϱ_{v} \in (0, 1) and \sum_{v = 0}^{\infty} δ_{v} = \sum_{v = 0}^{\infty} t_{v} = \sum_{v = 0}^{\infty} ϱ_{v} = \infty \end{matrix}

Theorem 4:

Let $C \subset X$ be a closed convex and let $Ψ : C \to C$ be a TAN-mapping. If ${q_{v}}_{v = 1}^{\infty}$ is generated by (7), where $λ \in (0, 1], ρ > 0$ . Then ${q_{v}}$ converges faster than ${d_{v}}$ in (22) to the fixed point of $Ψ$ .

Proof:

Recalling inequality (13), we get

\begin{matrix} ‖ q_{v + 1} - z ‖ \leq λ α_{v} ‖ q_{v} - z ‖ + ω_{v} λ α_{v} ‖ (q_{v} - q_{v - 1}) ‖ + (1 - λ α_{v}) ‖ q_{v} - z ‖ + (1 - λ α_{v}) σ_{v} M_{2} \\ ‖ q_{v + 1} - z ‖ \leq ‖ q_{v} - z ‖ + c_{1} \end{matrix}

Where $c_{1} = ω_{v} λ α_{v} ‖ (q_{v} - q_{v - 1}) ‖ + (1 - λ α_{v}) σ_{v} M_{2}$

Now, to proof ${d_{v}}$ converges to $z \in Fix (Ψ)$

\begin{matrix} ‖ d_{v + 1} - z ‖ \leq ‖ u_{v} - z ‖ + ϱ_{v} μ_{v} \emptyset (‖ u_{v} - z ‖) + ϱ_{v} κ_{v} \\ ‖ u_{v} - z ‖ \leq ‖ w_{v} - z ‖ + μ_{v} \emptyset (‖ w_{v} - z ‖) + 2 κ_{v} + μ_{v} \emptyset (‖ {Ψ w}_{v} - z ‖) \\ \begin{matrix} ‖ w_{v} - z ‖ \leq ‖ ρ_{v} - z ‖ + t_{v} μ_{v} \emptyset (‖ ρ_{v} - z ‖) + t_{v} κ_{v} + κ_{v} + μ_{v} \emptyset [(1 - t_{v}) ρ_{v} + t_{v} Ψ ρ_{v}] \\ ‖ ρ_{v} - z ‖ \leq ‖ d_{v} - z ‖ + δ_{v} μ_{v} \emptyset (‖ d_{v} - z ‖) + δ_{v} κ_{v} \end{matrix} \end{matrix}

Then

\begin{matrix} ‖ u_{v} - z ‖ \leq ‖ d_{v} - z ‖ + δ_{v} μ_{v} \emptyset (‖ d_{v} - z ‖) + δ_{v} κ_{v} + t_{v} μ_{v} \emptyset (‖ w_{v} - z ‖) + t_{v} κ_{v} + 3 κ_{v} + μ_{v} \emptyset [(1 - t_{v}) ρ_{v} + t_{v} Ψ ρ_{v}] \\ + μ_{v} \emptyset (‖ Ψ w_{v} - z ‖) ‖ d_{v + 1} - z ‖ \leq ‖ d_{v} - z ‖ \\ + μ_{v} [δ_{v} \emptyset (‖ d_{v} - z ‖) + + t_{v} \emptyset (‖ ρ_{v} - z ‖) + \emptyset [(1 - t_{v}) ρ_{v} + t_{v} Ψ ρ_{v}] + \emptyset (‖ Ψ w_{v} - z ‖) + \emptyset (‖ w_{v} - z ‖) + ϱ_{v} \emptyset (‖ u_{v} - z ‖)] \\ + κ_{v} [ϱ_{v} + δ_{v} + t_{v} + 3] \leq ‖ d_{v} - z ‖ + c_{2} \end{matrix}

Where $c_{2} = + μ_{v} [δ_{v} \emptyset (‖ d_{v} - z ‖) + + t_{v} \emptyset (‖ ρ_{v} - z ‖) + \emptyset [(1 - t_{v}) ρ_{v} + t_{v} Ψ ρ_{v}] + \emptyset (‖ Ψ w_{v} - z ‖) + \emptyset (‖ w_{v} - z ‖) + ϱ_{v} \emptyset (‖ u_{v} - z ‖)] + κ_{v} [ϱ_{v} + δ_{v} + t_{v} + 3]$

lim_{v \to \infty} \frac{‖ q_{v + 1} - z ‖}{‖ d_{v + 1} - z ‖} \leq lim_{v \to \infty} \frac{‖ q_{v} - z ‖ + c_{1}}{‖ d_{v} - z ‖ + c_{2}} \to 0 as v \to \infty

Then the sequence ${q_{v}}$ converges faster than ${d_{v}}$ to the fixed point of $Ψ$ .

Numerical example

Example 1.

Consider $X = R$ and let $Ψ_{v} : R \to R$ be TAN-mapping defined for each $v \in N, q \in R$ by $Ψ_{v} (q) = \frac{1}{2} q + \frac{1}{4} sin (vq)$ with initial values and parameters for ${q_{v}},$ $q_{0} = 1.5, q_{1} = 1.3, b_{0} = 0, ρ = 1, λ = 0.8, w_{v} = 0, 4, α_{v} = 0.8, σ_{v} = 0, 05$ . The initial values and parameters for ${d_{v}}$ , $d_{0} = 1.5, δ_{v} = t_{v} = ϱ_{v} = 0.3$ where $z = 1.0$ , the obtained numerical outcomes are presented in Table 1 and illustrated in Figure 1.

Let AU-iteration ${s_{v}}$ be sequences generated by the following iteration processes²⁵:

(23)

\begin{matrix} w_{v} = Ψ ((1 - a_{v}) s_{v} + a_{v} Ψ (s_{v})) \\ h_{v} = Ψ (w_{v}) \\ \begin{matrix} u_{v} = Ψ (h_{v}) \\ s_{v + 1} = Ψ (u_{v}) \end{matrix} \end{matrix}

with

a_{v} \in [0, 1]

.

Table 1. Convergence difference between two iteration processes in banach space.

n	$q_{v}$ (first iteration)	$‖ q_{v} - z ‖$	$d_{v}$ (second iteration)	$‖ d_{v} - z ‖$
0	1.5	0.5	1.5	0.5
1	1.3	0.30000000000000000	0.4131915244957580	0.5868084755042420
2	1.0849189420687200	0.0849189420687162	0.3421717021662320	0.6578282978337680
3	0.894252864909753	0.10574713509024700	0.4654479063947990	0.5345520936052010
4	0.7226411051916090	0.27735889480839100	0.47385052804551600	0.5261494719544850
5	0.5766662211471710	0.42333377885282900	0.42500864205177500	0.5749913579482250
6	0.4681780936016550	0.5318219063983450	0.3790195207923410	0.6209804792076590
7	0.3961058503874510	0.6038941496125490	0.33979249758830500	0.6602075024116950
8	0.34834967884856500	0.651650321151435	0.30751332108003200	0.6924866789199680
9	0.312784839284355	0.6872151607156450	0.2810096061988630	0.7189903938011380
10	0.28351141953541100	0.7164885804645890	0.2588649995140480	0.7411350004859530
11	0.23648461196167400	0.7635153880383260	0.23988091774148700	0.7601190822585130
12	0.2221863506406470	0.777813649359353	0.22322847148202800	0.7767715285179720
13	0.20668131716233100	0.7933186828376690	0.20858002638555100	0.7914199736144490
14	0.1934829721577170	0.8065170278422830	0.19611958172549700	0.8038804182745030
15	0.18216323534734700	0.8178367646526530	0.18626269285313600	0.8137373071468640
16	0.17190652441410200	0.8280934755858980	0.17779939354768800	0.8222006064523120
17	0.16295588804357600	0.8370441119564240	0.15218421514936300	0.8478157848506370
18	0.15469693555509800	0.8453030644449020	0.1368389430998990	0.8631610569001010
19	0.14749077873627800	0.8525092212637220	0.18749719248117700	0.8125028075188230
20	0.14060001806657000	0.8593999819334300	0.0017337606561274100	0.9982662393438730
21	0.1348251425142060	0.8651748574857940	-0.08313828356067490	1.0831382835606700
22	0.12870148909100100	0.8712985109089990	0.24523837673964200	0.7547616232603580
23	0.12453648894401900	0.8754635110559810	0.009942217347192690	0.9900577826528070
24	0.11790397057680900	0.882096029423191	0.09865281597152940	0.9013471840284710
25	0.11742242033299300	0.8825775796670070	0.13624664018214500	0.8637533598178550
26	0.10465031328707400	0.8953496867129260	-0.138760418621349	1.138760418621350
27	0.12046358860049400	0.8795364113995060	-0.13114341891967600	1.131143418919680
28	0.06926230316173150	0.9307376968382690	0.25780777965080600	0.742192220349194
29	0.1260300954581600	0.8739699045418400	-0.08321106295447480	1.083211062954480
30	0.03863088476970310	0.961369115230297	-0.2907346622703990	1.2907346622704000

Figure 1. A graphical comparison of (7) and (22).

Theorem 5.

Let $C \subset X$ be a closed convex and let $Ψ : C \to C$ be a TAN-mapping with $Fix (Ψ) \neq \emptyset$ . Let ${q_{v}}$ be defined by (7) and ${s_{v}}$ be defined by (23)

Under the above assumptions, the following are equivalent for $q_{0} = s_{0} \in C$ :

1. ${q_{v}}$ converges to $z \in Fix (Ψ)$
2. ${s_{v}}$ converges to $z \in Fix (Ψ)$

Proof.

Since initial points $q_{1}, q_{0,} s_{0} \in C$ and all operations are convex combinations or bounded perturbations, it follows that all sequences remain bounded in the closed convex subset $C$ . Similar, as all coefficients are in [0,1], boundedness holds for ${q_{v}}, {s_{v}}, {ρ_{v}}, {b_{v}}, {w_{v}}, {h_{v}}, {u_{v}}$ . By Theorem 1 ${q_{v}}$ convergence to z. In ordered to prove ${s_{v}}$ convergence to $z$

\begin{matrix} ‖ s_{v + 1} - z ‖ \leq ‖ u_{v} - z ‖ + μ_{v} \emptyset (‖ u_{v} - z ‖) + k_{v} \\ \leq (1 - a_{v}) ‖ s_{v} - z ‖ + a_{v} ‖ s_{v} - z ‖ + a_{v} μ_{v} \emptyset (‖ s_{v} - z ‖) + a_{v} k_{v} \\ + μ_{v} \emptyset (‖ (1 - a_{v}) s_{v} + a_{v} Ψ (s_{v}) - z ‖) + k_{v} \\ ‖ s_{v + 1} - z ‖ \leq ‖ s_{v} - z ‖ + a_{v} μ_{v} \emptyset (‖ s_{v} - z ‖) + a_{v} k_{v} + μ_{v} \emptyset (‖ (1 - a_{v}) s_{v} + a_{v} Ψ (s_{v}) - z ‖) \\ + k_{v} + μ_{v} \emptyset (‖ w_{v} - z ‖) + k_{v} + μ_{v} \emptyset (‖ h_{v} - z ‖) + k_{v} + μ_{v} \emptyset (‖ u_{v} - z ‖) + k_{v} \\ \leq ‖ s_{v} - z ‖ + μ_{v} [a_{v} \emptyset (‖ s_{v} - z ‖) + \emptyset (‖ w_{v} - z ‖) + \emptyset (‖ h_{v} - z ‖) + \emptyset (‖ (1 - a_{v}) s_{v} + a_{v} Ψ (s_{v}) - z ‖) + \emptyset (‖ u_{v} - z ‖)] \\ + k_{v} (a_{v} + 4) \end{matrix}

By Lemma 3 ${s_{v}}$ convergence to $z$ . Now, define $D_{v} = ‖ q_{v} - s_{v} ‖$ , to show $lim_{v \to \infty} D_{v} = 0$ .

Since both sequences converge to the same fixed point $z$ , we can use: $‖ q_{v + 1} - s_{v + 1} ‖ \leq ‖ q_{v + 1} - z ‖ + ‖ s_{v + 1} - z ‖ \to 0$ . Therefore $lim_{v \to \infty} ‖ q_{v} - s_{v} ‖ = 0$ .

Application

In this section, the TAN-mapping is applied to nonlinear Volterra integral equation over two dimensions, which is:

X (ξ, ρ) = β (ξ, ρ) + \int_{0}^{ξ} \int_{0}^{ρ} ℧_{1} (l, h, X (l, h)) dldh + ζ \int_{0}^{ξ} ℧_{2} (ρ, h, X (ξ, h)) dh + γ \int_{0}^{ρ} ℧_{3} (ξ, l, X (ρ, r)) dl

where

, ρ,

l, h \in [0, 1], X \in C [0, 1] \times C [0, 1], β : [0, 1] \times [0, 1] \to R^{2}

,

℧_{j} : [0, 1] \times [0, 1] \times R^{2} \to R^{2}

is continuous,

j = 1, 2, 3

and

X = C [0, 1]

with

{‖ q - s ‖}_{\infty} = {max}_{σ \in [0, 1]} | q (σ) - s (σ) |,

for all

q, s \in C [0, 1]

Theorem 6.

Suppose that $Z$ is a closed convex subset of $X$ and $Ψ : Z \to Z$ define as

(24)

Ψ X (ξ, ρ) = β (ξ, ρ) + \int_{0}^{ξ} \int_{0}^{ρ} ℧_{1} (l, h, X (l, h)) dldh + ζ \int_{0}^{ξ} ℧_{2} (ρ, h, X (ξ, h)) dh + γ \int_{0}^{ρ} ℧_{3} (ξ, l, X (ρ, l)) dl

Assumptions

1) The function $X : X \times X \to R^{2}$ is continuous.
2) There exist constants $v_{1}, v_{2}, v_{3} > 0$ and sequences $η_{v}, ε_{v} \geq 0 with η_{v} \to 0, ε_{v} \to 0$ , such that for all $w_{1}, w_{2} \in R^{2}$ ,
$\begin{matrix} ‖ ℧_{1} (l, h, w_{1}) - ℧_{1} (l, h, w_{2}) ‖ \leq (1 + η_{v}) v_{1} ‖ w_{1} - w_{1} ‖ + ε_{v} \\ ‖ ℧_{2} (ρ, h, w_{1}) - ℧_{2} (ρ, h, w_{2}) ‖ \leq (1 + η_{v}) v_{2} ‖ w_{1} - w_{1} ‖ + ε_{v} \\ ‖ ℧_{3} (ξ, l, w_{1}) - ℧_{3} (ξ, l, w_{2}) ‖ \leq (1 + η_{v}) v_{3} ‖ w_{1} - w_{1} ‖ + ε_{v} \end{matrix}$

Then 2D Volterra integral equation has a solution in $Z \times Z$ provided that $Ψ$ has a fixed point.

3) For $ζ, γ \geq 0, v_{1} + ζ v_{2} + γ v_{3} \leq L, L \in (0, 1)$

Proof.

Let $X, X^{*} \in X,$ then

{‖ X - {Ψ X}^{*} ‖}_{\infty} = {}_{σ \in [0, 1]}^{max}{| X (ξ, ρ) (σ) - X^{*} (ξ, ρ) (σ) |} = {max}_{σ \in [0, 1]} | X (ξ, ρ) (σ) - β (ξ, ρ) - \int_{0}^{ξ} \int_{0}^{ρ} ℧_{1} (l, h, X^{*} (l, u)) dldh - ζ \int_{0}^{ξ} ℧_{2} (ρ, h, X^{*} (ξ, h)) dh - γ \int_{0}^{ρ} ℧_{3} (ξ, l, X^{*} (ρ, l)) dl | \leq {max}_{σ \in [0, 1]} {| X (ξ, ρ) (σ) - β (ξ, ρ) - \int_{0}^{ξ} \int_{0}^{ρ} ℧_{1} (l, h, X (l, h)) dldh - ζ \int_{0}^{ξ} ℧_{2} (ρ, h, X (ξ, h)) dh - γ \int_{0}^{ρ} ℧_{3} (ξ, l, X (ρ, l)) dl | + | \int_{0}^{ξ} \int_{0}^{ρ} ℧_{1} (l, h, X (l, h)) dldh - \int_{0}^{ξ} \int_{0}^{ρ} ℧_{1} (l, h, X^{*} (l, h)) dldh | + ζ | \int_{0}^{ξ} ℧_{2} (ρ, h, X (ξ, h)) dh - \int_{0}^{ξ} ℧_{2} (ρ, h, X^{*} (ξ, h)) dh | + γ | \int_{0}^{ρ} ℧_{3} (ξ, l, X (ρ, l)) dl - \int_{0}^{ρ} ℧_{3} (ξ, l, X^{*} (ρ, l)) dl |}

Then by hypothesis 2

\begin{matrix} {‖ X - {Ψ X}^{*} ‖}_{\infty} \leq β {‖ X - X^{*} ‖}_{\infty} + (1 + η_{v}) (v_{1} + ζ v_{2} + γ v_{3}) {‖ X - X^{*} ‖}_{\infty} + (1 + ζ + γ) ε_{v} \\ = [β + (1 + η_{v}) (v_{1} + {ζv}_{2} + {γv}_{3})] {‖ X - X^{*} ‖}_{\infty} + (1 + ζ + γ) ε_{v} \end{matrix}

Let $\emptyset (t) = t, μ_{v} = η_{v} (v_{1} + ζ v_{2} + γ v_{3}), k_{v} = (1 + ζ + γ) ε_{v}$ and by hypothesis 3.

Then ${‖ X - {Ψ X}^{*} ‖}_{\infty} = {‖ X - X^{*} ‖}_{\infty} + μ_{v} \emptyset ({‖ X - X^{*} ‖}_{\infty}) + k_{v}$ is a TAN-mapping.

Under all previous hypotheses in this section and Theorem 1, the iterative scheme generated by (7) converges weakly to a solution of the integral Equation (24).

Applying Theorem 6, in the following example.

Example 2.

Consider the equation as a special case of Equation 24

(25)

X (ξ, ρ) = \frac{cos (πξρ)}{10} + \int_{0}^{ξ} \int_{0}^{ρ} \frac{1}{16} cos X (lh) dldh + \frac{1}{2} \int_{0}^{ξ} \frac{1}{24} cos X (ξh) dh + \frac{1}{2} \int_{0}^{ρ} \frac{1}{23} cos X (ρl) dl

where, $β (ξ, ρ) = \frac{cos (πξρ)}{10}$ , $℧_{1} (l, h, X (l, h)) = \frac{1}{16} cos X (lh)$ , $℧_{2} (ρ, h, X (ξ, h)) = \frac{1}{24} cos X (ξh)$ , $℧_{3} (ξ, l, X (ρ, l)) = \frac{1}{23} cos X (ρl)$ , $ζ = \frac{1}{2}$ and $γ = \frac{1}{2}$ .

Since all the assumptions of Theorem 6 hold. Then the Equation (25) has a solution.

Conclusion

This paper presented the convergence of the modified algorithm (7) in space $X$ as discussed throughout the paper. The outputs summarize the proof of weak convergence (Theorem 1) and strong convergence (Theorem 2) by adding some appropriate hypotheses to the space and the structure of the iterative algorithm. Note that, Algorithm 7 is equivalent to Algorithm 23, meaning they converge to the same fixed point, with the 7 convergences being faster. These results are supported by an example and a plot of the convergence processes using MATLAB. To demonstrate the importance of the work, the results were applied to solving an integral equation. For future work related to this area, see Algorithms in²⁶ and.²⁷

Ethical clearance

The project was approved by the local ethical committee at the University of Baghdad.

Data availability

No data associated with this article.

References

1. Browder FE: Fixed-point theorems for nonexpansive mappings in Banach spaces. Proc. Natl. Acad. Sci. 1965; 53(6): 1272–1276. PubMed Abstract | Publisher Full Text | Free Full Text
2. Kirk WA: A fixed-point theorem for mappings which do not increase distances. Am. Math. Mon. 1965; 72(9): 1004–1006. Publisher Full Text
3. Goebel K, Kirk WA: A fixed-point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972; 35(1): 171–174. Publisher Full Text
4. Alber YI, Chidume CE, Zegeye H: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory and Applications. 2006; 2006(1): 1–18.
5. Abed SS, Abed AN: Convergence and stability of iterative scheme for a monotone total asymptotically non-expansive mapping in partially ordered Banach spaces. Iraqi Journal of Science. 2022; 63(1): 241–250. Publisher Full Text
6. Khomphurngson K, Nammanee K: Fixed point of total asymptotically nonexpansive mappings in Banach spaces. Science & Technology Asia. 2024; 29(2): 183–190.
7. Balooee J, Al-Homidan S: System of generalized variational-like inclusions involving (P,η)-accretive mappings and fixed point problems in real Banach spaces. Arab. J. Math. 2024; 13: 1–33. Publisher Full Text
8. Salisu S, Kumam P, Sriwongsa S, et al.: Enriched asymptotically nonexpansive mappings with center zero. Filomat. 2024; 38(1): 343–356. Publisher Full Text
9. Sun Y: A fixed-point theorem for random asymptotically nonexpansive mappings. N. Y. J. Math. 2025; 31: 182–194.
10. Harbau MH, Ugwunnadi GC, Jolaoso LO, et al.: Inertial accelerated algorithm for fixed point of asymptotically nonexpansive mapping in real uniformly convex Banach spaces. Axioms. 2021; 10(12): 680. Publisher Full Text
11. Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 2008; 340: 1088–1095. Publisher Full Text
12. Browder FE: Fixed point theorems for nonlinear semicontractive mappings in Banach spaces. Arch. Ration. Mech. Anal. 1966; 21: 259–269. Publisher Full Text
13. Kim TH, Kim DH: Demiclosedness principles for total asymptotically nonexpansive mappings. Mathematical Analysis Research Institute Reports. 2013; 1821: 90–106.
14. Harder AM: Fixed point theory and stability results for fixed point iteration procedures. St. Rolla, MO, USA: University of Missouri-Rolla; 1987. (Ph.D. thesis).
15. Berinde V: Iterative Approximation of Fixed Points. Berlin: Springer; 2007.
16. O’pial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967; 73: 591–597. Publisher Full Text
17. Gossez JP, Dozo EL: Some geometric properties related to the fixed-point theory for nonexpansive mappings. Pac. J. Math. 1972; 40: 565–573. Publisher Full Text
18. Tan KK, Xu HK: Fixed Point Iteration Processes for Asymptotically Nonexpansive Mappings. Proc. Am. Math. Soc. 1994; 122: 733–739. Publisher Full Text
19. Osilike MO, Aniagbosor SC: Weak and Strong Convergence Theorems for Fixed Points of Asymptotically Nonexpansive Mappings. Math. Comput. Model. 2000; 32: 1181–1191. Publisher Full Text
20. Xu HK: Inequalities in Banach Spaces with Applications. Nonlinear Anal. 1991; 16: 1127–1138. Publisher Full Text
21. Beck A, Teboulle M: A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences. 2009; 2: 183–202. Publisher Full Text
22. Schu J: Weak and strong convergence to Fixed Points of Asymptotically Nonexpansive Mappings. Bull. Aust. Mathematical Society. 1991; 43: 153–159. Publisher Full Text
23. Agarwal RP, El-Gebeily MA, O’Regan D: Fixed point theory for Lipschitzian-type mappings with applications. Topological Fixed-Point Theory and Its Applications. New York: Springer; 2009; vol. 6. .
24. Hammad HA, Rehman H, De la Sen M: A new four-step iterative procedure for approximating fixed points with application to 2D Volterra integral equations. Mathematics. 2022; 10(24): 4827. Publisher Full Text
25. Udofia UE, Igbokwe DI: Monotone α-nonexpansive mapping in ordered Banach space by AU-iteration algorithm with application to delay differential equation. International Journal of Nonlinear Analysis and Applications. 2022; 13(2): 673–690.
26. Salman BB, Abed SS: Convergence of Forked Sequence to a Fixed Point in Modular Function Spaces. Iraqi Journal of Science. 2024; 65(6): 3291–3301. Publisher Full Text
27. Salman BB, Abed SS: New approximating results by weak convergence of forked sequences. Baghdad Sci. J. 2025; 22(2): 623–633.

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¹ Mathematics, University of Baghdad, Baghdad, Baghdad Governorate, 10011, Iraq

Rana Fadhil
Roles: Writing – Original Draft Preparation

Salwa Salman
Roles: Writing – Review & Editing

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No competing interests were disclosed.

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[1] 1. Browder FE: Fixed-point theorems for nonexpansive mappings in Banach spaces. Proc. Natl. Acad. Sci. 1965; 53(6): 1272–1276. PubMed Abstract | Publisher Full Text | Free Full Text

[2] 2. Kirk WA: A fixed-point theorem for mappings which do not increase distances. Am. Math. Mon. 1965; 72(9): 1004–1006. Publisher Full Text

[3] 3. Goebel K, Kirk WA: A fixed-point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972; 35(1): 171–174. Publisher Full Text

[4] 4. Alber YI, Chidume CE, Zegeye H: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory and Applications. 2006; 2006(1): 1–18.

[5] 5. Abed SS, Abed AN: Convergence and stability of iterative scheme for a monotone total asymptotically non-expansive mapping in partially ordered Banach spaces. Iraqi Journal of Science. 2022; 63(1): 241–250. Publisher Full Text

[6] 6. Khomphurngson K, Nammanee K: Fixed point of total asymptotically nonexpansive mappings in Banach spaces. Science & Technology Asia. 2024; 29(2): 183–190.

[7] 7. Balooee J, Al-Homidan S: System of generalized variational-like inclusions involving (P,η)-accretive mappings and fixed point problems in real Banach spaces. Arab. J. Math. 2024; 13: 1–33. Publisher Full Text

[8] 8. Salisu S, Kumam P, Sriwongsa S, et al.: Enriched asymptotically nonexpansive mappings with center zero. Filomat. 2024; 38(1): 343–356. Publisher Full Text

[9] 9. Sun Y: A fixed-point theorem for random asymptotically nonexpansive mappings. N. Y. J. Math. 2025; 31: 182–194.

[10] 10. Harbau MH, Ugwunnadi GC, Jolaoso LO, et al.: Inertial accelerated algorithm for fixed point of asymptotically nonexpansive mapping in real uniformly convex Banach spaces. Axioms. 2021; 10(12): 680. Publisher Full Text

[11] 11. Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 2008; 340: 1088–1095. Publisher Full Text

[12] 12. Browder FE: Fixed point theorems for nonlinear semicontractive mappings in Banach spaces. Arch. Ration. Mech. Anal. 1966; 21: 259–269. Publisher Full Text

[13] 13. Kim TH, Kim DH: Demiclosedness principles for total asymptotically nonexpansive mappings. Mathematical Analysis Research Institute Reports. 2013; 1821: 90–106.

[14] 14. Harder AM: Fixed point theory and stability results for fixed point iteration procedures. St. Rolla, MO, USA: University of Missouri-Rolla; 1987. (Ph.D. thesis).

[15] 15. Berinde V: Iterative Approximation of Fixed Points. Berlin: Springer; 2007.

[16] 16. O’pial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967; 73: 591–597. Publisher Full Text

[17] 17. Gossez JP, Dozo EL: Some geometric properties related to the fixed-point theory for nonexpansive mappings. Pac. J. Math. 1972; 40: 565–573. Publisher Full Text

[18] 18. Tan KK, Xu HK: Fixed Point Iteration Processes for Asymptotically Nonexpansive Mappings. Proc. Am. Math. Soc. 1994; 122: 733–739. Publisher Full Text

[19] 19. Osilike MO, Aniagbosor SC: Weak and Strong Convergence Theorems for Fixed Points of Asymptotically Nonexpansive Mappings. Math. Comput. Model. 2000; 32: 1181–1191. Publisher Full Text

[20] 20. Xu HK: Inequalities in Banach Spaces with Applications. Nonlinear Anal. 1991; 16: 1127–1138. Publisher Full Text

[21] 21. Beck A, Teboulle M: A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences. 2009; 2: 183–202. Publisher Full Text

[22] 22. Schu J: Weak and strong convergence to Fixed Points of Asymptotically Nonexpansive Mappings. Bull. Aust. Mathematical Society. 1991; 43: 153–159. Publisher Full Text

[23] 23. Agarwal RP, El-Gebeily MA, O’Regan D: Fixed point theory for Lipschitzian-type mappings with applications. Topological Fixed-Point Theory and Its Applications. New York: Springer; 2009; vol. 6. .

[24] 24. Hammad HA, Rehman H, De la Sen M: A new four-step iterative procedure for approximating fixed points with application to 2D Volterra integral equations. Mathematics. 2022; 10(24): 4827. Publisher Full Text

[25] 25. Udofia UE, Igbokwe DI: Monotone α-nonexpansive mapping in ordered Banach space by AU-iteration algorithm with application to delay differential equation. International Journal of Nonlinear Analysis and Applications. 2022; 13(2): 673–690.

[26] 26. Salman BB, Abed SS: Convergence of Forked Sequence to a Fixed Point in Modular Function Spaces. Iraqi Journal of Science. 2024; 65(6): 3291–3301. Publisher Full Text

[27] 27. Salman BB, Abed SS: New approximating results by weak convergence of forked sequences. Baghdad Sci. J. 2025; 22(2): 623–633.