F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.172972.2 Research Article Articles Fixed-point Acceleration Algorithm of Total Asymptotically Nonexpansive Mappings

[version 2; peer review: 1 not approved]

 Fadhil Rana Writing – Original Draft Preparation https://orcid.org/0009-0006-6349-4257 a 1 Salman Salwa Writing – Review & Editing 1 Mathematics, University of Baghdad, Baghdad, Baghdad Governorate, 10011, Iraq Rana.Fadel2203@ihcoedu.uobaghdad.edu.iq

No competing interests were disclosed.

 9 2 2026 2025 14 1430 5 2 2026 Copyright: © 2026 Fadhil R and Salman S 2026 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Fixed-point iterative methods play a fundamental role in nonlinear analysis and its applications. They provides a unified framework for establishing existence and uniqueness results for a wide class of nonlinear problems. In this paper, working in a real uniformly convex Banach space, we propose an accelerated four-step iterative scheme for approximating fixed-point of a class of mapping known as total asymptotically nonexpansive TAN-mappings. Weak and strong convergence results for proposed algorithm are established. The weak convergence analysis is obtained under Opial’s condition, while the strong convergence results require in addition a semi-compactness assumption on the underlying mapping. Furthermore, numerical simulations are presented to demonstrate the effectiveness of the proposed scheme and to compare its performance with existing iterative methods. The obtained results extend and refine several recent contributions in the literature. Finally, an application to two-dimensional equations is provided.

 Banach Spaces Iterative Schemes Total Asymptotically Non-Expansive Mappings Fixed Points Strong Convergence Weak Convergence Stability. University of Baghdad The author(s) declared that no grants were involved in supporting this work. Amendments from Version 1

In this revised version of the article, we have addressed the comments and suggestions received during the initial assessment with a focus on improving clarity, language quality, and overall academic presentation, while preserving the original mathematical content and results. Specifically, the manuscript has undergone a comprehensive English language revision across all sections, including the abstract, introduction, preliminaries, main results, proofs, stability analysis, numerical examples, applications, and conclusion. Non-standard expressions and informal wording were replaced with standard academic English, grammatical inconsistencies were corrected, and sentence structures were refined to enhance readability and coherence. Several definitions, lemmas, theorems, and proofs were rewritten stylistically to improve clarity and logical flow, without modifying any assumptions, arguments, or conclusions. The notation and presentation were made more consistent throughout the manuscript, and minor typographical issues were corrected. No changes were made to the title, author list, figures, tables, or underlying data. All mathematical results, convergence analyses, numerical experiments, and applications remain exactly as presented in Version 1. The revisions are limited strictly to language, structure, and presentation, ensuring that the scientific content and originality of the work are fully preserved.

 Introduction

Fixed point theory in Banach spaces has evolved through a sequence of generalizations of contractive-type mappings. One of the earliest fundamental contributions was made by Browder (1965) and Kirk (1965) in, 1, 2 who initiated the systematic study of nonexpansive mappings and established essential fixed-point results in uniformly convex Banach spaces. This line of research was further extended by Goebel and Kirk (1972) in, 3 who introduced the concept of asymptotically nonexpansive mappings, thereby enlarging the scope of fixed-point theory to operators whose iterates approach nonexpansive in an asymptotic sense. Building upon these developments, Alber, Chidume, and Zegeye (2005) in 4 a introduced the class of totally asymptotically nonexpansive TAN-mappings. A mapping Ψ : X → X is defined as TAN-mapping if ∃ nonnegative real sequences , { κ v } where μ v → 0 , κ v → 0 as v → ∞ , together with a strictly increasing continuous function ∅ : ℝ + → ℝ + satisfying ∅ ( 0 ) = 0 such that for all x , y ∈ X . ‖ Ψ v x − Ψ v y ‖ ≤ ‖ x − y ‖ + μ v ∅ ( ‖ x − y ‖ ) + κ v holds for all v ≥ 1

This framework unifies nonexpansive and asymptotically nonexpansive mappings as special cases. Since its introduction, the class of TAN-mapping has primarily been extensively investigated through various iterative schemes in Banach spaces. Classical methods such as the Mann and Ishikawa iterations were among the earliest approaches, followed by the Halpern iteration in uniformly convex Banach spaces. 10 More recent studies have focused on accelerated and hybrid iterative techniques. For instance, Abed and Abed (2022) in 5 studied the Fibonacci–Halpern iteration for monotone TAN-mappings in partially ordered Banach spaces, establishing its stability results and demonstrating faster convergence compared with Mann and Halpern iterations. In 2023, Salman and Abed (2023) 24 proposed a five-step iterative scheme for (λ,ρ)-firmly nonexpansive multivalued mappings in modular spaces, showing that such mapping satisfy both the (UUC1) property and the Δ2-condition. Khomphungson and Nammanee (2024) further analyzed modified Mann–Ishikawa algorithms, and proved strong convergence under Condition (A) in uniformly convex Banach spaces. In addition, Balooee and Al-Homidan (2024) in 4, 7 developed hybrid resolvent-based methods linking fixed-point problems of TAN-mappings with generalized variational-like inclusions, ensuring the existence of a common solution belonging simultaneously to the solution set of the inclusion problems and to the fixed-point set of a TAN-mapping. Such formulations are closely related to integral and differential equations, which are frequently recast as operator inclusions problem in Banach spaces. Further extensions of the theory have been proposed in recent years. Galisu et al. (2024) 8 introduced enriched asymptotically nonexpansive mappings with center zero in a reflexive and strictly convex Banach space, providing a broader framework that includes TAN-mappings as a subclass. Similarly, Sun (2025) 9 extended the analysis to random asymptotically nonexpansive mappings in uniformly convex modules, emphasizing stochastic generalization within Banach-type structures. In summary, fixed point theory has progressed from nonexpansive mappings, 1, 2 to asymptotically nonexpansive mappings, 3 culminating in the general class of TAN-mappings. 4 A wide range of iterative schemes, including Mann, Ishikawa, Halpern, Fibonacci-Halpern, modified Mann-Ishikawa, and hybrid resolvent-based algorithms have played a central role in establishing convergence, stability, and applications to variational inclusions and integral-differential models in. 5, 6

The main objective of the present work is to investigate the convergence and stability properties of an inertial accelerated iterative scheme for TAN-mappings in Banach spaces. Motivated by the inertial accelerated framework proposed by Harbau et al. (2022), 10 we establish weak and strong convergence results for the proposed algorithm and analyze its stability under perturbations. Furthermore, a comparative study with existing iterative methods is provide, demonstrating that superior convergence performance of the inertial accelerated scheme. Finally, the theoretical results are applied to integral equations, illustrating the practical relevance and the applicability of the proposed approach.

 Preliminaries

This section presents basic definitions, notations, and auxiliary results that will be used in establishing the main convergence theorems.

11 A Banach space X is said to be uniformly convex if for any ϵ ∈ ( 0 , 2 ) , ∃ δ ( ϵ ) > 0 ∋ ‖ p + q 2 ‖ ≤ 1 − δ ( ϵ ) holds, for all p , q ∈ X with ‖ p ‖ ≤ 1 , ‖ q ‖ ≤ 1 , and ‖ p − q ‖ ≥ ϵ .

10 Every Hilbert space is uniformly convex and hence satisfies Definition (1)

Symbols

X denotes the real Banach Space.

X ∗ denotes the dual of a space X .

The symbol ⇀ denotes weak convergence, while ⟶ denotes strong convergence.

The set of ω - weak cluster limits of a sequence { q v } is ω c ( q v ) = { z : ∃ q vk ⇀ z } represents.

Fix ( Ψ ) ≕ { p in X : Ψ ( p ) = p } , all fixed points of Ψ .

10 Let ⟨ . , . ⟩ be a duality pairing on X × X ∗ . A mapping J φ : X → 2 X ∗ defined as J φ ( p ) = { p ∗ ∈ X ∗ : ⟨ p , p ∗ ⟩ = ‖ p ‖ ‖ p ∗ ‖ , ‖ p ∗ ‖ = φ ( ‖ p ‖ ) } is called a generalized duality mapping, where φ is a gauge function.

Note, if φ ( t ) = t for all t ≥ 0 then J φ = J 2 is a normalized duality mapping and can be expressed as J ( p ) = { p ∗ ∈ X ∗ : ⟨ p , p ∗ ⟩ = ‖ p ‖ 2 , ‖ p ∗ ‖ = ‖ p ‖ } .

12 A mapping Ψ : C ⊆ X → 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 ) .

13 A mapping Ψ: Ψ : C ⊆ X → C is called

Demi-closed at y 0 ∈ C , if { q v } in C such that q v ⇀ z ∈ C and Ψ q v → y 0 , then Ψ q 0 = y 0 .

Semi-compact if lim v → ∞ ‖ q v − Ψ q v ‖ = 0 , for any bounded sequence { q v } ⊂ C with lim v → ∞ ‖ q v − Ψ q v ‖ = 0 , there exists a subsequence { q vk } ⊂ { q v } satisfied q vk → z ∈ C .

Suppose a fixed-point iteration scheme is defined by a general form q v + 1 = f ( Ψ , q v ) for v = 0 , 1 , 2 , … , for a mapping Ψ and q 0 ∈ X . This ensures the convergence of { q v } v = 0 ∞ to z ∈ Fix ( Ψ ) . Where f ( Ψ , q v ) does contain all parameters that define the given fixed-point iteration scheme.

14 Let { p v } v = 0 ∞ 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 → ∞ ε v = 0 ⬄ lim v → ∞ p v = z .

15 Suppose { q v } v = 0 ∞ and { s v } v = 1 ∞ are two sequences converging to the same fixed point z with ‖ q v − z ‖ ≤ a v , and ‖ s v − z ‖ ≤ b v for all v ≥ 0 . If the sequence a v → a and b v → b and additionally lim v → ∞ ‖ a v − a ‖ ‖ b v − b ‖ = 0 , then { q v } v = 1 ∞ is said to converge to z faster than { s v } v = 1 ∞ .

16 A space X has Opial’s condition if for all { q v } ⊂ X with q v ⇀ z 1 satisfied lim v → ∞ sup ‖ q v − z 1 ‖ < lim v → ∞ sup ‖ q v − z 2 ‖ ∀ z 2 ∈ X and z 2 ≠ z 1 .

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

The Lemma below is known as Opial’s property.

17 If a duality mapping J is weakly continuous on X , and a sequence { q v } ⊂ X with q v ⇀ z , then for any p ∈ X . lim v → ∞ inf ‖ q v − p ‖ ≥ lim v → ∞ inf ‖ q v − z ‖

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

18 Suppose X is a real uniformly convex Banach space and ∅ ≠ C ⊂ X is a closed convex. If Ψ : C → C TAN-mapping, then ( I − Ψ ) is demiclosed at zero, whenever { q v } is a sequence in C such that q v ⇀ z .

19 Assume { a v } , { b v } , and { c v } are nonnegative sequences such that a v + 1 ≤ ( 1 + c v ) a v + b v , ∑ v = 0 ∞ b v < ∞ and ∑ v = 0 ∞ c v < ∞ , ∀ v ≥ 0 then :

{ a v } is convergent.

If lim v → ∞ inf a v = 0 then lim v → ∞ a v = 0 .

20 A Banach space X is uniformly convex if and only if there exists a continuous, strictly increasing function g : [ 0 , ∞ ) → [ 0 , ∞ ) ] with g ( 0 ) = 0 , such that: ‖ λ p + ( 1 − λ ) q ‖ 2 ≤ λ ‖ p ‖ 2 + ( 1 − λ ) ‖ q ‖ p − λ ( 1 − λ ) g ( ‖ p − q ‖ ) , holds for all p , q in B r = { p ∈ X : ‖ p ‖ ≤ r } and every λ ∈ [ 0 , 1 ] and a fixed number r > 0 .

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 → ∞ ‖ a v q v + ( 1 − a v ) s v ‖ = d , lim v → ∞ sup ‖ q v ‖ ≤ d , lim v → ∞ sup ‖ s v ‖ ≤ d , satisfy for some d ≥ 0 . Then lim v → ∞ ‖ q v − s v ‖ = 0 , where { q v } , { s v } ∈ X .

Main results

This section is devoted to the investigation of weak and strong convergence properties, stability, analysis and comparative results of the proposed iterative scheme. Throughout this section, X denotes a uniformly convex Banach space.

Weak convergence

The following assumptions are adopted from 21

Let { α v } ⊂ ( 0 , 1 ) and let { σ v } , { δ v } ⊂ [ 0 , ∞ ) be a sequence such that ∑ v = 1 ∞ δ v < ∞ and δ v = o ( σ v ) that is lim n → ∞ δ v σ v = 0 .

Let q 0 , q 1 ∈ X be given, for the terms q v − 1 and q v for each v ≥ 1 choose ω v , such that 0 ≤ ω v ≤ ω v ¯ . For any η ≥ 3 define

 ω v ¯ ≔ { min { v − 1 v + η − 1 , δ v ‖ q v − q v − 1 ‖ } , if q v ≠ q v − 1 v − 1 v + η − 1 , otherwise

Under Assumption, it directly follows immediately that for every v ≥ 1 , the relation. ω v ‖ q v − q v − 1 ‖ ≤ δ v

Which together with ∑ v = 1 ∞ δ v < ∞ and lim v → ∞ δ v σ v = 0 , holds ∑ v = 1 ∞ ω v ‖ q v − q v − 1 ‖ < ∞ Moreover lim v → ∞ ω v σ v ‖ q v − q v − 1 ‖ ≤ lim n → ∞ δ v σ v = 0

Let X satisfy Opial’s property and Ψ : X → X TAN-mapping associated with the sequences { μ v } , { κ v } ⊂ ( 0 , ∞ ) and a strictly increasing continuous function ∅ : ℝ + → ℝ + with ∅ ( 0 ) = 0 such that ∑ v = 1 ∞ ( μ v + κ v ) < ∞ . Let the sequence { q v } as: let. { q 0 , q 1 ∈ X a v = q v + ω v ( q v − q v − 1 ) b v + 1 = 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 , ∀ v ≥ 1 , where λ ∈ ( 0 , 1 ] , ρ > 0 and the following hold:

( C 1 ) ∑ v = 0 ∞ σ v < ∞ ( C 2 ) lim v → ∞ 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 = 1 ρ ( Ψ 1 a 0 − a 0 ) and Fix ( Ψ ) ≠ ∅ , then q v ⇀ z ∈ Fix ( Ψ ) .

The proof is divided in to five steps:

Proof { b v } is bounded.

lim v → ∞ ‖ q v − z ‖ exists for any z ∈ Fix ( Ψ )

lim v → ∞ ‖ q v − Ψ q v ‖ = 0

ω c ( q v ) ⊂ Fix ( Ψ )

{ q v } is ω -convergence to a fixed point.

For (1), the condition ( C 1 ) implies that lim v → ∞ σ v = 0 , so ∃ v 0 ∈ N such that σ n ≤ 1 2 , for every v ≥ v 0 . Put M 1 as follows: M 1 ≔ max { max 1 ≤ k ≤ v 0 ‖ b k ‖ , ( 2 ρ ) sup v ∈ N ‖ Ψ v a v − a v ‖ }

Then, by (C 3), the M 1 < ∞ hold. Assume that ‖ b n ‖ ≤ M 1 for some v ≥ v 0 , then: ‖ b v + 1 ‖ = ‖ 1 ρ ( Ψ v a v − a v ) + σ v b v ‖ ≤ 1 ρ ‖ Ψ v a v − a v ‖ + σ v ‖ b v ‖ ≤ M 1 This implies to ‖ b v ‖ ≤ M 1 , for all v ≥ 0 then { b v } is bounded .

For (2), from the scheme (7), this is obtained by: p v = a v + ρ b v + 1 = a v + ρ ( 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 ∈ Fix ( Ψ ) , we obtain: ‖ p v − z ‖ = ‖ Ψ v a v + ρ σ v b v − z ‖ ≤ ‖ Ψ v a v − z ‖ + ρ σ v ‖ b v ‖ ≤ ‖ a v − z ‖ + μ v ∅ ( ‖ a v − z ‖ ) + κ v + ρ M 1 σ v

Let U = sup { ‖ a v ‖ + ‖ a v − z ‖ } < ∞ , ζ v = μ v ∅ ( U ) + κ v

Additionally, ‖ a v − z ‖ = ‖ q v − z + ω v ( q v − q v − 1 ) ‖ ≤ ‖ q v − z ‖ + ω v ‖ ( q v − q v − 1 ) ‖

By (10) and (11), we obtain: ‖ p v − z ‖ ≤ ‖ q v − z ‖ + ω v ‖ ( q v − q v − 1 ) ‖ + κ v + ρ M 1 σ v = ‖ q v − z ‖ + σ v [ ω v σ v ‖ ( q v − q v − 1 ) ‖ + κ v σ v + ρ M 1 ]

By Remark 2, we obtain that { ω v σ v ‖ ( q v − q v − 1 ) ‖ } is converges, then ∃ M 2 > 0 such that for every v ≥ 1 , ω v σ v ‖ ( q v − q v − 1 ) ‖ + κ v σ v + ρ M 1 ≤ M 2 .

Thus: ‖ p v − z ‖ ≤ ‖ q v − z ‖ + σ v M 2

Next, using (10), (12), and for some M 3 > 0 any z ∈ Fix ( Ψ ) we obtain: ‖ q v + 1 − z ‖ = ‖ λ α v a v + ( 1 − λ α v ) p v − z ‖ ≤ λ α v ‖ a v − z ‖ + ( 1 − λ α v ) ‖ p v − z ‖ ≤ λ α v [ ‖ q v − z ‖ + ω v ‖ ( q v − q v − 1 ) ‖ ] + ( 1 − λ α v ) ‖ p v − z ‖ ≤ 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 ‖ ≤ ‖ q v − z ‖ + σ v M 3 , where ω v λ α v σ v ‖ ( q v − q v − 1 ) ‖ + ( 1 − λ α v ) M 2 ≤ M 3

Hence, from Lemma 5 and (C 1), it follows that lim v → ∞ ‖ 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 ≥ 1 { ‖ a v ‖ , ‖ Ψ v a v ‖ } and by condition (13), Lemma 4, and hypothesis C1, we obtain. ‖ q v + 1 − z ‖ 2 = ‖ λ α v a v + ( 1 − λ α v ) p v − z ‖ 2 ≤ ‖ 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 )

Since the lim v → ∞ ‖ q v − z ‖ exists for any z ∈ Fix ( Ψ ) , then by applying (5) and (11), ∃ L > 0 such that ‖ a v − z ‖ ≤ L ‖ q v + 1 − z ‖ 2 ≤ ‖ 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 ) ≤ ‖ 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 )

Because the lim n → ∞ ‖ q v − z ‖ exists for any z ∈ Fix ( Ψ ) , the sequence { ‖ q v − z ‖ } is bounded; hence, ∃ H > 0 such that ‖ q v − z ‖ ≤ H for all v ≥ 1 . Then ‖ q v + 1 − z ‖ 2 ≤ ‖ 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 )

Therefore, we have λ α v ( 1 − λ α v ) g ( ‖ a v − Ψ v a v ‖ ) ≤ ‖ 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 )

So, ∑ v = 0 ∞ λ α v ( 1 − λ α v ) g ( ‖ a v − Ψ v a v ‖ ) ≤ ‖ q 0 − z ‖ 2 + 2 H ∑ v = 0 ∞ ω v ‖ q v − q v − 1 ‖ + ∑ v = 0 ∞ ( ω v ‖ q v − q v − 1 ‖ ) 2 + M 1 2 ρ 2 ∑ v = 0 ∞ σ v 2 + 2 λ ρ 2 M 1 2 ∑ v = 0 ∞ α v σ v 2 + λ 2 ρ 2 M 1 2 ∑ v = 0 ∞ α v 2 σ v 2 + ∑ v = 0 ∞ 2 ( 1 − λ α v ) ρ σ v M 1 ( L + ( 1 − λ α v ) ζ v )

This gives

∑ v = 0 ∞ λ α v ( 1 − λ α v ) g ( ‖ a v − Ψ v a v ‖ ) < ∞ . Which implies that: lim v → ∞ λ α v ( 1 − λ α v ) g ( ‖ a v − Ψ v a v ‖ ) = 0 . By (C 2), we obtain: lim v → ∞ g ( ‖ a v − Ψ v a v ‖ ) = 0 . And lim v → ∞ ‖ a v − Ψ v a v ‖ = 0

Next, ‖ a v − q v ‖ = ‖ q v + ω v ( q v − q v − 1 ) − q v ‖ = ω v ‖ q v − q v − 1 ‖

Taking the sum over v for the above inequality and by (5), we obtain: ∑ v = 0 ∞ ‖ a v − q v ‖ = ∑ v = 0 ∞ ω v ‖ q v − q v − 1 ‖ < ∞ . Which implies that: lim ‖ a v − q v ‖ v → ∞ = 0

Since Ψ is TAN-mapping, we obtain: ‖ q v − Ψ v q v ‖ = ‖ q v − a v + a v − Ψ v a v + Ψ v a v − Ψ v q v ‖ ≤ ‖ q v − a v ‖ + ‖ a v − Ψ v a v ‖ + ‖ Ψ v a v − Ψ v q v ‖ ≤ ‖ q v − a v ‖ + ‖ a v − Ψ v a v ‖ + ‖ a v − q v ‖ + μ v ∅ ( ‖ a v − q v ‖ ) + κ v ≤ 2 ‖ q v − a v ‖ + ‖ a v − Ψ v a v ‖ + ζ v

Continuity of (15) and (16) in (17) implies that: lim v → ∞ ‖ q v − Ψ v q v ‖ = 0

On the other hand, ‖ a v − p v ‖ = ‖ a v − ( a v + ρ b v + 1 ) ‖ = ρ ‖ b v + 1 ‖ ≤ ρ ( 1 ρ ‖ a v − Ψ v a v ‖ + σ v ‖ b v ‖ ≤ ‖ a v − Ψ v a v ‖ + ρ σ v M 1 )

By (15) and (C 1), we obtain: lim v → ∞ ‖ a v − p v ‖ = 0 ‖ q v + 1 − Ψ v q v + 1 ‖ ≤ 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 (C 1) that: lim v → ∞ ‖ q v + 1 − Ψ v q v + 1 ‖ = 0

By (18) and (20), we obtain: lim v → ∞ ‖ 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), lim k → ∞ ‖ q vk − Ψ q vk ‖ = 0 and consequently by Lemma 2, Ψ z = z . Hence, ω c ( q v ) ⊂ 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 Definition 7, let z 1 , z 2 ∈ 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 ≠ z 2 we have: lim v → ∞ ‖ q v − z 1 ‖ < lim i → ∞ ‖ q vi − z 2 ‖ = lim j → ∞ ‖ q vj − z 2 ‖ < lim j → ∞ ‖ q vj − z 1 ‖ = lim v → ∞ ‖ q v − z 1 ‖

This leads to a contradiction. Hence ω c ( q v ) is a singleton, and { q v } is ω-convergence to a fixed point.

 Strong convergence

Let the assumptions of Theorem 1, be satisfied and suppose, in addition, that the mapping Ψ is semi-compact. Then the sequence { q v } defined by (7) converges strongly to a point in Fix ( Ψ ) .

Since Ψ is semi-compactness of Ψ and using steps 2 and 3 of the proof of Theorem 1, together with the facts that the sequence { q v } is bounded satisfies lim v → ∞ ‖ q v − Ψ q v ‖ = 0 , it follows that there exists a subsequence { q vk } of { q v } such that q vk → z as k → ∞ . Consequently, q vk ⇀ z and hence z ∈ ω c ( q v ) ⊆ Fix ( Ψ ) . Moreover, by proof of Theorem 1 - step 2, the lim v → ∞ ‖ q v − z ‖ exists. Therefore, lim v → ∞ ‖ q v − z ‖ = lim k → ∞ ‖ q vk − z ‖ = 0 , and hence, q v → z ∈ Fix ( Ψ ) .

Under the hypotheses of Theorem 1, suppose that Ψ : X → X be a nonexpansive mapping with F ix ( Ψ ) ≠ ∅ . Then the sequence { q v } is defined by (7) with λ ∈ ( 0 , 1 ] , ρ > 0 and initial value b 0 = 1 ρ ( Ψ a 0 − a 0 ) , satisfied q v ⇀ z ∈ Fix ( Ψ ) , Provided that ( C 1) − ( C 3) hold.

Let X be a real Hilbert space and Ψ : X → X be either a TAN-mapping or a nonexpansive mapping with Fix ( Ψ ) ≠ ∅ . Then the sequence { q v } is defined by (7) where λ ∈ ( 0 , 1 ] , ρ > 0 , and b 0 = 1 ρ ( Ψ 1 a 0 − a 0 ) satisfies q v ⇀ z ∈ Fix ( Ψ ) , provided that condition ( C 1) − ( C 3) hold.

Theorem 1 applies to the broader class of asymptotically nonexpansive mappings and therefore extends Theorem 3.1 of Dong et al., see, 20 to real uniformly convex Banach spaces, which include real Hilbert spaces as special case.

Stability

In this section, we investigate the stability of iterative sequence defined by (7) with respect to a fixed point of TAN-mapping. The iterative operator is given by f ( Ψ , r v ) = λ α v a v + ( 1 − λ α v ) p v .

Let C ⊂ X be a nonempty closed convex set and Ψ : C → C be a TAN-mapping. Suppose that the sequence { r v } v = 1 ∞ is generated by (7) with λ ∈ ( 0 , 1 ] and ρ > 0 . If Assumption 1 holds and the sequence { Ψ v a v − a v } is bounded, then { r v } v = 1 ∞ is stable provided that b 1 = 1 ρ ( Ψ 1 a 0 − a 0 ) , with ∑ v = 0 ∞ σ v < ∞ and lim v → ∞ inf λ α v ( 1 − λ α v ) > 0 .

Let { r v } v = 1 ∞ 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).

Define ε v = ‖ r v + 1 − f ( Ψ , r v ) ‖ We show that lim v → ∞ ε v = 0 ⟺ lim v → ∞ r v = z

Assume first that lim v → ∞ ε v = 0 . Then, ‖ r v + 1 − z ‖ ≤ ‖ r v + 1 − f ( Ψ , r v ) ‖ + ‖ f ( Ψ , r v ) − z ‖ = ε v + ‖ r v + 1 − z ‖

By Theorem 1, we obtain: ‖ r v + 1 − z ‖ ≤ ‖ r v − z ‖ + σ v M 3

Since 0 < M 3 < 1 , ∑ v = 0 ∞ σ v < ∞ and lim v → ∞ ε v = 0 , and by using Lemma 3, yield lim v → ∞ ‖ r v − z ‖ = 0 . Hence, lim v → ∞ r v = z .

Conversely, if lim v → ∞ r v = z . Then getting: ε v = ‖ r v + 1 − f ( Ψ , r v ) ‖ ≤ ‖ r v + 1 − z ‖ + ‖ f ( Ψ , r v ) − z ‖ ≤ ‖ r v + 1 − z ‖ + ‖ r v − z ‖ + σ v M 3

By continuity lim v → ∞ ε v = 0 . Thus, { r v } v = 1 ∞ is stable.

Now, recall HR-iteration 24 which is defined by 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 ) ∀ v ≥ 1 , where δ v , t v , ϱ v ∈ ( 0 , 1 ) and ∑ v = 0 ∞ δ v = ∑ v = 0 ∞ t v = ∑ v = 0 ∞ ϱ v = ∞

Let C ⊂ X be a nonempty closed convex and let Ψ : C → C be a TAN-mapping. If the sequence { q v } v = 1 ∞ is generated by (7), with λ ∈ ( 0 , 1 ] and ρ > 0 , then { q v } converges faster than the HR-iteration sequence { d v } defined by (22) to the fixed point z of Ψ .

From inequality (13), we obtain: ‖ q v + 1 − z ‖ ≤ 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 ‖ ≤ ‖ q v − z ‖ + c 1

Where c 1 = ω v λ α v ‖ ( q v − q v − 1 ) ‖ + ( 1 − λ α v ) σ v M 2

For the HR-iteration sequence { d v } , similar estimates give similar results. ‖ d v + 1 − z ‖ ≤ ‖ u v − z ‖ + ϱ v μ v ∅ ( ‖ u v − z ‖ ) + ϱ v κ v ‖ u v − z ‖ ≤ ‖ w v − z ‖ + μ v ∅ ( ‖ w v − z ‖ ) + 2 κ v + μ v ∅ ( ‖ Ψ w v − z ‖ ) ‖ w v − z ‖ ≤ ‖ ρ v − z ‖ + t v μ v ∅ ( ‖ ρ v − z ‖ ) + t v κ v + κ v + μ v ∅ [ ( 1 − t v ) ρ v + t v Ψ ρ v ] ‖ ρ v − z ‖ ≤ ‖ d v − z ‖ + δ v μ v ∅ ( ‖ d v − z ‖ ) + δ v κ v

Then ‖ u v − z ‖ ≤ ‖ d v − z ‖ + δ v μ v ∅ ( ‖ d v − z ‖ ) + δ v κ v + t v μ v ∅ ( ‖ w v − z ‖ ) + t v κ v + 3 κ v + μ v ∅ [ ( 1 − t v ) ρ v + t v Ψ ρ v ] + μ v ∅ ( ‖ Ψ w v − z ‖ ) ‖ d v + 1 − z ‖ ≤ ‖ d v − z ‖ + μ v [ δ v ∅ ( ‖ d v − z ‖ ) + + t v ∅ ( ‖ ρ v − z ‖ ) + ∅ [ ( 1 − t v ) ρ v + t v Ψ ρ v ] + ∅ ( ‖ Ψ w v − z ‖ ) + ∅ ( ‖ w v − z ‖ ) + ϱ v ∅ ( ‖ u v − z ‖ ) ] + κ v [ ϱ v + δ v + t v + 3 ] ≤ ‖ d v − z ‖ + c 2

Where c 2 = + μ v [ δ v ∅ ( ‖ d v − z ‖ ) + + t v ∅ ( ‖ ρ v − z ‖ ) + ∅ [ ( 1 − t v ) ρ v + t v Ψ ρ v ] + ∅ ( ‖ Ψ w v − z ‖ ) + ∅ ( ‖ w v − z ‖ ) + ϱ v ∅ ( ‖ u v − z ‖ ) ] + κ v [ ϱ v + δ v + t v + 3 ] lim v → ∞ ‖ q v + 1 − z ‖ ‖ d v + 1 − z ‖ ≤ lim v → ∞ ‖ q v − z ‖ + c 1 ‖ d v − z ‖ + c 2 → 0 as v → ∞

Therefore the sequence { q v } converges faster than { d v } to the fixed point z of Ψ .

Numerical example

Let X = R and define the mapping Ψ v : R → R by Ψ v ( q ) = 1 2 q + 1 4 sin ( vq ) for all v ∈ N , q ∈ R . It is straightforward to verify that { Ψ v } is a family of TAN-mappings. For the iterative { q v } defined by (7), the initial values and parameters are chosen as 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 . For the HR-iteration sequence { q v } , the initial values and parameters for are d 0 = 1.5 , δ v = t v = ϱ v = 0.3 where z = 1.0 . The corresponding numerical results are reported in Table 1 and illustrated in Figure 1.

Next, we consider the AU-iteration { s v } be sequences generated by the following scheme 25 : w v = Ψ ( ( 1 − a v ) s v + a v Ψ ( s v ) ) h v = Ψ ( w v ) u v = Ψ ( h v ) s v + 1 = Ψ ( u v ) with a v ∈ [ 0 , 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
A graphical comparison of (7) and (22).

Let C ⊂ X be a nonempty closed convex and let Ψ : C → C be a TAN-mapping with Fix ( Ψ ) ≠ ∅ . Let { q v } be defined by (7) and { s v } be defined by (23)

Under the above assumptions for q 0 = s 0 ∈ C , the following statements are equivalent:

{ q v } converges to z ∈ Fix ( Ψ )

{ s v } converges to z ∈ Fix ( Ψ )

Since initial points q 1 , q 0 , s 0 ∈ C and all operations involved in the iteration schemes are convex combinations or bounded perturbations, it follows that all generated sequences remain bounded in the closed convex subset C . In particular, the sequence { q v } , { s v } , { ρ v } , { b v } , { w v } , { h v } and { u v } are bounded. By Theorem 1, the sequence { q v } convergence to z. To show that { s v } convergence to the same fixed point z, we observe that. ‖ s v + 1 − z ‖ ≤ ‖ u v − z ‖ + μ v ∅ ( ‖ u v − z ‖ ) + k v ≤ ( 1 − a v ) ‖ s v − z ‖ + a v ‖ s v − z ‖ + a v μ v ∅ ( ‖ s v − z ‖ ) + a v k v + μ v ∅ ( ‖ ( 1 − a v ) s v + a v Ψ ( s v ) − z ‖ ) + k v ‖ s v + 1 − z ‖ ≤ ‖ s v − z ‖ + a v μ v ∅ ( ‖ s v − z ‖ ) + a v k v + μ v ∅ ( ‖ ( 1 − a v ) s v + a v Ψ ( s v ) − z ‖ ) + k v + μ v ∅ ( ‖ w v − z ‖ ) + k v + μ v ∅ ( ‖ h v − z ‖ ) + k v + μ v ∅ ( ‖ u v − z ‖ ) + k v ≤ ‖ s v − z ‖ + μ v [ a v ∅ ( ‖ s v − z ‖ ) + ∅ ( ‖ w v − z ‖ ) + ∅ ( ‖ h v − z ‖ ) + ∅ ( ‖ ( 1 − a v ) s v + a v Ψ ( s v ) − z ‖ ) + ∅ ( ‖ u v − z ‖ ) ] + k v ( a v + 4 )

By Lemma 3 { s v } convergence to z . Finally, define D v = ‖ q v − s v ‖ , to show lim v → ∞ D v = 0 .

Since both sequences converge to the same fixed point z , we have: ‖ q v + 1 − s v + 1 ‖ ≤ ‖ q v + 1 − z ‖ + ‖ s v + 1 − z ‖ → 0 , which implies that lim v → ∞ D v = 0 .

Application

In this section, we apply the proposed TAN-mapping framework to a nonlinear two-dimensional Volterra integral equation of the form. X ( ξ , ρ ) = β ( ξ , ρ ) + ∫ 0 ξ ∫ 0 ρ ℧ 1 ( l , h , X ( l , h ) ) dldh + ζ ∫ 0 ξ ℧ 2 ( ρ , h , X ( ξ , h ) ) dh + γ ∫ 0 ρ ℧ 3 ( ξ , l , X ( ρ , r ) ) dl where , ρ , l , h ∈ [ 0 , 1 ] , X ∈ C [ 0 , 1 ] × C [ 0 , 1 ] , β : [ 0 , 1 ] × [ 0 , 1 ] → R 2 , ℧ j : [ 0 , 1 ] × [ 0 , 1 ] × R 2 → R 2 are continuous, function for j = 1 , 2 , 3 , let X = C [ 0 , 1 ] be endowed with the norm ‖ q − s ‖ ∞ = max σ ∈ [ 0 , 1 ] | q ( σ ) − s ( σ ) | , for all q , s ∈ C [ 0 , 1 ]

Let Z be a nonempty closed convex subset of X and define the mapping Ψ : Z → Z by Ψ X ( ξ , ρ ) = β ( ξ , ρ ) + ∫ 0 ξ ∫ 0 ρ ℧ 1 ( l , h , X ( l , h ) ) dldh + ζ ∫ 0 ξ ℧ 2 ( ρ , h , X ( ξ , h ) ) dh + γ ∫ 0 ρ ℧ 3 ( ξ , l , X ( ρ , l ) ) dl

Assume that:

The function X : X × X → R 2 is continuous.

There exist constants v 1 , v 2 , v 3 > 0 and sequences η v , ε v ≥ 0 with η v → 0 , ε v → 0 , such that for all w 1 , w 2 ∈ R 2 , ‖ ℧ 1 ( l , h , w 1 ) − ℧ 1 ( l , h , w 2 ) ‖ ≤ ( 1 + η v ) v 1 ‖ w 1 − w 1 ‖ + ε v ‖ ℧ 2 ( ρ , h , w 1 ) − ℧ 2 ( ρ , h , w 2 ) ‖ ≤ ( 1 + η v ) v 2 ‖ w 1 − w 1 ‖ + ε v ‖ ℧ 3 ( ξ , l , w 1 ) − ℧ 3 ( ξ , l , w 2 ) ‖ ≤ ( 1 + η v ) v 3 ‖ w 1 − w 1 ‖ + ε v

Then the two-dimensional Volterra integral equation admits a solution in Z × Z provided that Ψ has a fixed point.

For ζ , γ ≥ 0 , v 1 + ζ v 2 + γ v 3 ≤ L , L ∈ ( 0 , 1 )

Let X , X ∗ ∈ X . By direct estimation and using Assumption (2), we obtain: ‖ X − Ψ X ∗ ‖ ∞ = max = max σ ∈ [ 0 , 1 ] | X ( ξ , ρ ) ( σ ) − β ( ξ , ρ ) − ∫ 0 ξ ∫ 0 ρ ℧ 1 ( l , h , X ∗ ( l , u ) ) dldh − ζ ∫ 0 ξ ℧ 2 ( ρ , h , X ∗ ( ξ , h ) ) dh − γ ∫ 0 ρ ℧ 3 ( ξ , l , X ∗ ( ρ , l ) ) dl | ≤ max σ ∈ [ 0 , 1 ] { | X ( ξ , ρ ) ( σ ) − β ( ξ , ρ ) − ∫ 0 ξ ∫ 0 ρ ℧ 1 ( l , h , X ( l , h ) ) dldh − ζ ∫ 0 ξ ℧ 2 ( ρ , h , X ( ξ , h ) ) dh − γ ∫ 0 ρ ℧ 3 ( ξ , l , X ( ρ , l ) ) dl | + | ∫ 0 ξ ∫ 0 ρ ℧ 1 ( l , h , X ( l , h ) ) dldh − ∫ 0 ξ ∫ 0 ρ ℧ 1 ( l , h , X ∗ ( l , h ) ) dldh | + ζ | ∫ 0 ξ ℧ 2 ( ρ , h , X ( ξ , h ) ) dh − ∫ 0 ξ ℧ 2 ( ρ , h , X ∗ ( ξ , h ) ) dh | + γ | ∫ 0 ρ ℧ 3 ( ξ , l , X ( ρ , l ) ) dl − ∫ 0 ρ ℧ 3 ( ξ , l , X ∗ ( ρ , l ) ) dl | }

‖ X − Ψ X ∗ ‖ ∞ ≤ β ‖ X − X ∗ ‖ ∞ + ( 1 + η v ) ( v 1 + ζ v 2 + γ v 3 ) ‖ X − X ∗ ‖ ∞ + ( 1 + ζ + γ ) ε v = [ β + ( 1 + η v ) ( v 1 + ζv 2 + γv 3 ) ] ‖ X − X ∗ ‖ ∞ + ( 1 + ζ + γ ) ε v

Let ∅ ( t ) = t , μ v = η v ( v 1 + ζ v 2 + γ v 3 ) , k v = ( 1 + ζ + γ ) ε v . By Assumption 3, it follows that.

‖ X − Ψ X ∗ ‖ ∞ = ‖ X − X ∗ ‖ ∞ + μ v ∅ ( ‖ X − X ∗ ‖ ∞ ) + k v , which show that Ψ is a TAN-mapping.

Therefore, by

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.

Consider the equation as a special case of Equation 24, given by X ( ξ , ρ ) = cos ( πξρ ) 10 + ∫ 0 ξ ∫ 0 ρ 1 16 cos X ( lh ) dldh + 1 2 ∫ 0 ξ 1 24 cos X ( ξh ) dh + 1 2 ∫ 0 ρ 1 23 cos X ( ρl ) dl

Here, β ( ξ , ρ ) = cos ( πξρ ) 10 , ℧ 1 ( l , h , X ( l , h ) ) = 1 16 cos X ( lh ) , ℧ 2 ( ρ , h , X ( ξ , h ) ) = 1 24 cos X ( ξh ) , ℧ 3 ( ξ , l , X ( ρ , l ) ) = 1 23 cos X ( ρl ) , ζ = 1 2 and γ = 1 2 .

Since all the assumptions of Theorem 6 are satisfied, Equation (25) admits at least one solution.

Conclusion

In this paper, we investigated the convergence properties of the modified iterative algorithm (7) in the Banach space X. The main results establish both weak convergence ( Theorem 1) and strong convergence ( Theorem 2) of the proposed scheme under suitable assumption on the space and on the structure of the iterative process. Moreover, it was shown that, Algorithm 7 is equivalent to Algorithm 23 in the sense that both methods converge to the same fixed point, while the proposed algorithm exhibits a faster rate of convergence. The theoretical findings were supported by a numerical example and illustrated through convergence plots generated using MATLAB. To further demonstrate the applicability and significance of the proposed method, the theoretical results were applied to the solution of a nonlinear two-dimensional Volterra integral equation. Future research directions may include the investigation of a related iterative algorithms and extensions, such as those proposed in Refs. 26 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 Browder FE : Fixed-point theorems for nonexpansive mappings in Banach spaces. Proc. Natl. Acad. Sci. 1965;53(6):1272–1276. 16578602 10.1073/pnas.53.6.1272 PMC219828 Kirk WA : A fixed-point theorem for mappings which do not increase distances. Am. Math. Mon. 1965;72(9):1004–1006. 10.2307/2313345 Goebel K Kirk WA : A fixed-point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972;35(1):171–174. 10.1090/S0002-9939-1972-0298500-3 Alber YI Chidume CE Zegeye H : Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory and Applications. 2006;2006(1):1–18. 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. 10.24996/ijs.2022.63.1.25 Khomphurngson K Nammanee K : Fixed point of total asymptotically nonexpansive mappings in Banach spaces. Science & Technology Asia. 2024;29(2):183–190. 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. 10.1007/s40065-023-00440-1 Salisu S Kumam P Sriwongsa S : Enriched asymptotically nonexpansive mappings with center zero. Filomat. 2024;38(1):343–356. 10.2298/FIL2401343S Sun Y : A fixed-point theorem for random asymptotically nonexpansive mappings. N. Y. J. Math. 2025;31:182–194. Harbau MH Ugwunnadi GC Jolaoso LO : Inertial accelerated algorithm for fixed point of asymptotically nonexpansive mapping in real uniformly convex Banach spaces. Axioms. 2021;10(12):680. 10.3390/axioms10030147 Suzuki T : Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 2008;340:1088–1095. 10.1016/j.jmaa.2007.09.023 Browder FE : Fixed point theorems for nonlinear semicontractive mappings in Banach spaces. Arch. Ration. Mech. Anal. 1966;21:259–269. 10.1007/BF00282247 Kim TH Kim DH : Demiclosedness principles for total asymptotically nonexpansive mappings. Mathematical Analysis Research Institute Reports. 2013;1821:90–106. 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). Berinde V : Iterative Approximation of Fixed Points. Berlin: Springer;2007. O’pial Z : Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967;73:591–597. 10.1090/S0002-9904-1967-11761-0 Gossez JP Dozo EL : Some geometric properties related to the fixed-point theory for nonexpansive mappings. Pac. J. Math. 1972;40:565–573. 10.2140/pjm.1972.40.565 Tan KK Xu HK : Fixed Point Iteration Processes for Asymptotically Nonexpansive Mappings. Proc. Am. Math. Soc. 1994;122:733–739. 10.1090/S0002-9939-1994-1203993-5 Osilike MO Aniagbosor SC : Weak and Strong Convergence Theorems for Fixed Points of Asymptotically Nonexpansive Mappings. Math. Comput. Model. 2000;32:1181–1191. 10.1016/S0895-7177(00)00199-0 Xu HK : Inequalities in Banach Spaces with Applications. Nonlinear Anal. 1991;16:1127–1138. 10.1016/0362-546X(91)90200-K Beck A Teboulle M : A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems. SIAM Journal on Imaging Sciences. 2009;2:183–202. 10.1137/080716542 Schu J : Weak and strong convergence to Fixed Points of Asymptotically Nonexpansive Mappings. Bull. Aust. Mathematical Society. 1991;43:153–159. 10.1017/S0004972700028884 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. 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. 10.3390/math10224257 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. 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. 10.24996/ijs.2024.65.6.28 Salman BB Abed SS : New approximating results by weak convergence of forked sequences. Baghdad Sci. J. 2025;22(2):623–633. 10.5256/f1000research.196056.r456885 Reviewer response for version 2 Thianwan Tanakit 1 Referee Department of Mathematics, University of Phayao (Ringgold ID: 90440), Mueang Phayao District, Phayao, Thailand

Competing interests: No competing interests were disclosed.

 19 2 2026 Copyright: © 2026 Thianwan T 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation reject

The manuscript investigates weak and strong convergence results for a proposed four-step accelerated iterative scheme applied to total asymptotically nonexpansive (TAN) mappings in uniformly convex Banach spaces. The topic falls within classical fixed-point theory and nonlinear functional analysis, and the general framework (Opial’s condition for weak convergence and semi-compactness for strong convergence) is standard in the literature.

While the mathematical direction is appropriate and technically aligned with existing fixed-point frameworks, the manuscript in its current form does not meet the academic, linguistic, and structural standards required for indexing or publication in a reputable mathematical journal.

The proposed scheme represents an incremental variation within a well-established class of accelerated/inertial-type iterative methods. The manuscript does not clearly demonstrate a substantial theoretical advancement beyond existing accelerated frameworks.

The comparative convergence claim (“converges faster than HR-iteration”) is not supported by a rigorous theoretical rate analysis. The argument appears mostly inequality-based and heuristic rather than providing explicit rate estimates.

The application to the Volterra integral equation follows a routine operator reformulation approach and does not introduce new analytical techniques.

In addition to the above conceptual and theoretical concerns, the manuscript suffers from serious linguistic and grammatical deficiencies, which significantly affect its academic quality and readability. The manuscript contains numerous grammatical errors throughout the text, including incorrect verb forms, improper tense usage, and frequent subject–verb agreement errors. There are repeated problems with article usage (a, an, the), especially before technical terms. Punctuation and connectors (e.g., “therefore”, “moreover”, “hence”) are frequently misused. Notation and sentence flow in the proof sections are inconsistent, making several derivations difficult to follow.

The manuscript is not suitable for indexing in its current form. I recommend:

Rejection in its current form for indexing consideration.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Yes

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Fixed point theory and optimization

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

 10.5256/f1000research.190743.r446606 Reviewer response for version 1 Thianwan Tanakit 1 Referee Department of Mathematics, University of Phayao (Ringgold ID: 90440), Mueang Phayao District, Phayao, Thailand

Competing interests: No competing interests were disclosed.

 10 1 2026 Copyright: © 2026 Thianwan T 2026 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation reject

The manuscript presents interesting convergence results for accelerated fixed-point algorithms. However, the paper contains numerous grammatical errors, awkward sentence constructions, and inconsistencies in academic style. A comprehensive language revision is required.

Reviewer Comments (Language & Grammar):

The manuscript contains numerous grammatical errors throughout the text.

Many sentences suffer from incomplete or incorrect sentence structure, including sentence fragments and run-on sentences.

Incorrect or missing articles (a, an, the) are frequently observed, particularly before technical terms.

Subject–verb agreement errors appear repeatedly.

The use of verb tense is inconsistent, especially in the Introduction and proof sections.

Several expressions are non-standard in academic English (e.g., “getting”, “to proof”, “there by”).

Prepositions and connectors are often used incorrectly.

Definitions and proofs are not written in standard academic English style and require restructuring for clarity.

Overall readability is reduced, and a comprehensive English language revision is necessary.

 Overall Recommendation:

Major language revision required before the manuscript can be considered for indexing. Applications are also routine, so I do not recommend the paper for indexing.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Yes

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Fixed point theory and optimization

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

Fadhil Rana Mathematics, Not Applicable, Baghdad, Baghdad, Iraq

Competing interests: No competing interests were disclosed

 30 1 2026

Dear Editor,

We would like to thank you and the reviewer for the time and effort devoted to the evaluation of our manuscript entitled

“Fixed-point Acceleration Algorithm of Total Asymptotically Nonexpansive Mappings.”

We greatly appreciate the reviewer’s constructive comments, particularly those concerning language, clarity, and academic style. In response, we have carried out a comprehensive and careful English language revision of the entire manuscript. All sections—including the abstract, introduction, preliminaries, theorems and proofs, stability analysis, numerical examples, applications, and conclusion—have been thoroughly revised to improve grammatical correctness, sentence structure, consistency of tense, and overall academic presentation.

In particular:

• Non-standard expressions and informal wording were replaced with standard academic English.

• Sentence fragments, run-on sentences, and grammatical inconsistencies were corrected.

• Articles, prepositions, and subject–verb agreements were carefully revised throughout the manuscript.

• Definitions and proofs were rewritten stylistically (without altering any mathematical arguments or results) to ensure clarity and conformity with conventional mathematical writing standards.

• The overall readability and coherence of the manuscript were significantly improved.

We emphasize that no mathematical content, assumptions, results, or conclusions were altered during this revision. All convergence results, proofs, numerical experiments, and applications remain exactly the same, with improvements limited strictly to language and presentation. This approach was intentionally adopted to preserve the originality and technical integrity of the work while fully addressing the reviewer’s concerns.

We believe that the revised manuscript now meets the required standards of academic English and clarity, and we respectfully submit it for your reconsideration.

Thank you again for your time and consideration.

Sincerely,

Rana Fadhil