F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.172972.3 Research Article Articles Fixed-point Acceleration Algorithm of Total Asymptotically Nonexpansive Mappings

[version 3; peer review: 1 approved with reservations, 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.

 18 4 2026 2025 14 1430 12 3 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 functional analysis and its applications. In this paper, we consider a real uniformly convex Banach space and introduce a modified accelerated four-step iterative scheme for approximating fixed-point of a class of total asymptotically nonexpansive TAN-mappings. The proposed approach is formulated within a general TAN framework and does not require compactness assumptions at the level of weak convergence. Weak convergence of the algorithm is established under Opial’s condition, whereas strong convergence is obtained by additionally assuming semi-compactness of the underlying mapping. Moreover, a rigorous comparative convergence analysis is provided to examine the rate of convergence of the proposed scheme relative to the HR-type iterative process. The theoretical results are supported by numerical experiments illustrating the convergence dynamics of both methods. Finally, the applicability of the framework is demonstrated by applying it to a nonlinear two-dimensional Volterra integral equation.

 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 2

Dear Professors and Colleagues, Thank you very much for your efforts in supporting research activities. I am uploading a carefully revised version, and I hope it meets your approval. I would like to clarify the following: - The texts have been written in standard format. - The stability theorem has been revised. - The reason for presenting 2D-Volterra integral equations as an application is: They provide powerful mathematical models for systems with two variables, serving as an important theoretical framework in functional analysis, PDEs, and fixed-point theory. Additionally, this type of equation has been rarely used in research papers on approximate fixed points. Regards

 Introduction

Fixed point theory in Banach spaces has developed through successive generalizations of contractive-type mappings. Early foundational contributions by Browder (1965) and Kirk (1965) in, 1 , 2 initiated the systematic study of nonexpansive mappings and established fundamental fixed-point results in uniformly convex Banach spaces. This line of research was later extended by Goebel and Kirk (1972) in, 3 who introduced the asymptotically nonexpansive mappings, thereby broadening the scope of fixed-point analysis to operators whose iterates exhibit asymptotic nonexpansiveness. Building upon these developments, Alber, Chidume, and Zegeye (2005) in 4 a introduced the class of totally asymptotically nonexpansive TAN-mappings. Let X be a Banach space and let Ψ : X → X . The mapping Ψ is said to be totally asymptotically nonexpansive if there exist nonnegative real sequences μ v and { κ v } satisfying μ v → 0 , κ v → 0 as v → ∞ , together with a strictly increasing continuous function ∅ : ℝ + → ℝ + with ∅ ( 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 particular cases and provides a flexible framework for analyzing convergence behavior of iterative schemes in Banach spaces. Since their introduction, TAN-mapping has been extensively investigated through various iterative procedures. Classical schemes such as Mann, Ishikawa and Halpern iterations laid the foundation for subsequent developments. 10 More recently, increasing attention has been devoted to accelerated and hybrid techniques. For instance, Abed and Abed (2022) in 5 analyzed a Fibonacci–Halpern iteration for monotone TAN-mappings in partially ordered Banach spaces, establishing stability and improved convergence properties. Salman and Abed (2023) 24 proposed a five-step iterative process for ( λ,ρ)-firmly nonexpansive multivalued mappings in modular spaces. Further developments were reported by Khomphungson and Nammanee (2024), 6 who analyzed modified Mann–Ishikawa algorithms, and proved strong convergence under Condition (A) in uniformly convex Banach spaces. Similarly, Balooee and Al-Homidan (2024) in 4, 7 introduced hybrid resolvent-based methods for fixed-point problems involving TAN-mappings and 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. These formulations are closely related to integral and differential equations, which frequently arise as operator inclusions problems a Banach spaces. Additional generalizations have been proposed in recent contributions. 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. More recently, Sun (2025) 9 extended the analysis to random asymptotically nonexpansive mappings in uniformly convex modules, emphasizing stochastic generalization within Banach-type structures. Collectively, these studies demonstrate the progressive fixed-point theory from classical nonexpansive mappings, 1, 2 to asymptotically nonexpansive mappings, 3 and ultimately to the general class of TAN-mappings. 4 A wide spectrum of iterative schemes, including Mann, Ishikawa, Halpern, Fibonacci-Halpern, modified Mann-Ishikawa, and hybrid resolvent-based algorithms, has played a central role in establishing convergence, stability, and applications to variational inclusions and integral-differential models in. 5, 6

Motivated by the growing interest in accelerated iterative framework, the main objective of the present work is to investigate the convergence and stability properties of an inertial accelerated four-step iterative scheme for TAN-mappings in Banach spaces. Inspired by the inertial accelerated technique proposed by Harbau et al. (2022), 10 we establish weak convergence under Opial’s condition and strong convergence under an additional semi-compactness assumption. Furthermore, stability under perturbations is analyzed. A comparative numerical study with existing iterative methods is conducted to illustrate the performance of the proposed scheme. Finally, the theoretical results are applied to a nonlinear two-dimensional Volterra integral equations, demonstrating the practical relevance and the applicability of the developed framework.

 Preliminaries

This section presents the fundamental definitions 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 ) , there exists a δ ( ϵ ) > 0 such that for any p , q ∈ X with ‖ p ‖ ≤ 1 , ‖ q ‖ ≤ 1 , and ‖ p − q ‖ ≥ ϵ , then, ‖ p + q 2 ‖ ≤ 1 − δ ( ϵ ) .

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

Symbols

X denotes the real Banach Space.

X ∗ denotes the dual of X .

⇀ for weak convergence and, ⟶ for strong convergence.

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

Fix ( Ψ ) ≕ { p in X : Ψ ( p ) = p } denotes the set of fixed points of Ψ .

The duality pairing on X × X ∗ denoted by ⟨ . , . ⟩ .

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

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

12 The duality mapping Ψ is said to be weakly sequentially continuous if, for any sequences { q v } in X such that q v ⇀ z , implies ( q v ) ⇀ Ψ ( z ) .

It is known that space has a weakly sequentially continuous duality map if J φ is single-valued and sequentially continuous from with the weak topology to with the weak ∗topology. 23

13 A mapping Ψ: Ψ : C ⊆ X → C said to be.

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

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

Suppose a fixed-point iteration is defined by the general form q v + 1 = f ( Ψ , q v ) for v = 0 , 1 , 2 , … , where Ψ is a mapping and q 0 ∈ X . The function f ( Ψ , q v ) contain all parameters that defining the iteration procedures.

14 Let { p v } v = 0 ∞ be a sequence in X . Then an iteration procedure q v + 1 = f ( Ψ , p v ) converging to a fixed point z is said to be Ψ -stable (or stable) with respect to Ψ if and only if for ε v = ‖ p v + 1 − f ( Ψ , p v ‖ , v ∈ N , we have lim v → ∞ ε v = 0 ⬄ lim v → ∞ p v = z .

15 Let { q v } v = 0 ∞ and { s v } v = 1 ∞ are two sequences that converging to the same fixed point z and ‖ q v − z ‖ ≤ a v , and ‖ s v − z ‖ ≤ b v for all v ≥ 0 . If the sequence { a v } v = 1 ∞ and { b v } v = 1 ∞ converge to a and b, respectively, and lim v → ∞ ‖ a v − a ‖ ‖ b v − b ‖ = 0 , then we say that { q v } v = 1 ∞ converge to z faster than { s v } v = 1 ∞ .

16 A Banach space X is said to satisfy Opial’s condition if for each sequence { q v } in X converging weakly to z 1, we have lim v → ∞ sup ‖ q v − z 1 ‖ < lim v → ∞ sup ‖ q v − z 2 ‖ for all z 2 ∈ X such that z 2 ≠ z 1 .

It is known that:

Any Banach space with a weakly sequentially continuous duality mapping satisfies Opial’s condition, 23 Any Hilbert space satisfies Opial’s condition. 23

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

The following lemma represents Opial’s property.

17 If in a Banach space X having a weakly continuous duality mapping J , the sequence { q v } ⊂ X is weakly convergent to z , then for any p ∈ X . lim v → ∞ inf ‖ q v − p ‖ ≥ lim v → ∞ inf ‖ q v − z ‖

In particular, if the space X is uniformly convex, then equality hold if and only if p = z .

18 Let X be a real uniformly convex Banach space, let C be a nonempty closed convex subset of X. If Ψ : C → C is a TAN-mapping, then ( I − Ψ ) is demiclosed at zero, that is whenever q v ⇀ z and Ψ ( q v ) ⇀ z , it follows that Ψ ( q v ) = z .

19 Let { a v } , { b v } , and { c v } be nonnegative real sequences satisfying the relation a v + 1 ≤ ( 1 + c v ) a v + b v with ∑ v = 0 ∞ b v < ∞ and ∑ v = 0 ∞ c v < ∞ , ∀ v ≥ 0 then :

The sequence { a v } is converges.

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

20 Let r > 0 be a fixed number. Then, a real Banach space X is uniformly convex if and only if there exists a continuous and 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 ‖ ) , for all p , q in B r = { p ∈ X : ‖ p ‖ ≤ r } and λ ∈ [ 0 , 1 ] .

22 Suppose that X is uniformly convex Banach space and let { a v } be real sequence such that 0 < b ≤ a v < c < 1 for all v ≥ 1. Let { q v } and { s v } be sequence in X such that lim v → ∞ ‖ a v q v + ( 1 − a v ) s v ‖ = d , and lim v → ∞ sup ‖ q v ‖ ≤ d , lim v → ∞ sup ‖ s v ‖ ≤ d hold for some d ≥ 0 . Then lim v → ∞ ‖ q v − s v ‖ = 0 .

Main results

This section investigates of weak and strong convergence properties, stability, analysis and comparative performance of the proposed iterative algorithm.

Weak convergence

The following assumptions are adopted from 21

Choose sequence { α v } ⊂ ( 0 , 1 ) , { σ v } , { δ v } ⊂ [ 0 , ∞ ) and ∑ v = 1 ∞ δ v < ∞ with δ v = o ( σ v ) which mean lim n → ∞ δ v σ v = 0 .

Let q 0 , q 1 ∈ X be arbitrary points, for the iterates q v − 1 and q v for each v ≥ 1 choose ω v , such that 0 ≤ ω v ≤ ω v ¯ where, 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 1 that for each v ≥ 1 , we have. ω v ‖ q v − q v − 1 ‖ ≤ δ v which together with ∑ v = 1 ∞ δ v < ∞ and lim v → ∞ δ v σ v = 0 , we respectively obtain. ∑ v = 1 ∞ ω v ‖ q v − q v − 1 ‖ < ∞ and lim v → ∞ ω v σ v ‖ q v − q v − 1 ‖ ≤ lim n → ∞ δ v σ v = 0

Let X be a real uniformly convex Banach space with Opial’s property. Let Ψ : X → X be TAN-mapping with sequences { μ v } , { κ v } ⊂ ( 0 , ∞ ) and a strictly increasing continuous function ∅ : ℝ + → ℝ + with ∅ ( 0 ) = 0 ∑ v = 1 ∞ ( μ v + κ v ) < ∞ and Fix(Ψ) ≠ ∅. Let { q v } be the sequence generated as follows: { 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 , if Assumption (i) holds and set b 1 = 1 ρ ( Ψ 1 a 0 − a 0 ) . Then, the sequence { q v } converges weakly to a point z ∈ Fix(Ψ), provided that the following conditions 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.

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.

Step (1), the condition ( C 1 ) implies that lim v → ∞ σ v = 0 thus, there exists v 0 ∈ N such that σ n ≤ 1 2 , for all v ≥ v 0 . Let M 1 be define as follows: M 1 ≔ max { max 1 ≤ k ≤ v 0 ‖ b k ‖ , ( 2 ρ ) sup v ∈ N ‖ Ψ v a v − a v ‖ }

Then, the hypothesis (C 3), implies that M 1 < ∞ hold. Presume 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 means that ‖ b v ‖ ≤ M 1 , for all v ≥ 0 therefore { b v } is bounded .

Step (2), from the algorithm (4), 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 (5), (6), 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 ) ‖

Combining (7) and (8), 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 know that the sequence { ω v σ v ‖ ( q v − q v − 1 ) ‖ } converges, so there exists some constant say M 2 > 0 such that for all 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 (7), (9), 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, using Lemma 5 and condition (C 1), it follows that lim v → ∞ ‖ q v − z ‖ exists. Consequently, the sequence { q v } is bounded.

Step (3) since the sequence { a v } is bounded, it follows that { Ψ v a v } is bounded. Let r = sup v ≥ 1 { ‖ a v ‖ , ‖ Ψ v a v ‖ } and by (10), Lemma 4, and condition 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 using (2) it follows from (8) that there exists 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 )

Since lim n → ∞ ‖ q v − z ‖ exists for any z ∈ Fix ( Ψ ) , then the sequence { ‖ q v − z ‖ } is bounded; therefore, there exists H > 0 such that ‖ q v − z ‖ ≤ H for all v ≥ 1 . Hence, ‖ 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, 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 )

Hence, ∑ 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 )

Using ( C 1), we obtain

∑ 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 condition ( 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 of both sides and considering (2), we obtain: ∑ v = 0 ∞ ‖ a v − q v ‖ = ∑ v = 0 ∞ ω v ‖ q v − q v − 1 ‖ < ∞ . Which implies that: lim v → ∞ ‖ a v − q 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 (12) and (13) in (14) 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 (12) 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 (12), (16), and ( C 1) that: lim v → ∞ ‖ q v + 1 − Ψ v q v + 1 ‖ = 0

By (15) and (17), we obtain: lim v → ∞ ‖ q v − Ψ q v ‖ = 0

Step (4) Since every uniformly convex Banach space is reflexive, 23 so 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 (18), lim k → ∞ ‖ q vk − Ψ q vk ‖ = 0 and consequently by Lemma 2, Ψ z = z . Hence, ω c ( q v ) ⊂ Fix ( Ψ ) .

Step (5) To prove 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

If in addition to all the hypothese of Theorem 1, the mapping Ψ is semi-compact. Then the sequence { q v } generated by (4) converges strongly to a fixed point of Ψ .

Suppose that Ψ is semi-compactness. Starting from steps 2 and 3 in the proof of Theorem 1, and the boundedness of the sequence { q v } with lim v → ∞ ‖ q v − Ψ q v ‖ = 0 , then there exists a subsequence { q vk } of { q v } such that q vk → z as k → ∞ . Therefore q vk ⇀ z and so z ∈ ω c ( q v ) ⊆ Fix ( Ψ ) . From step 2 in the proof of Theorem 1, lim v → ∞ ‖ q v − z ‖ exists, then lim v → ∞ ‖ q v − z ‖ = lim k → ∞ ‖ q vk − z ‖ = 0 , which means that q v → z ∈ Fix ( Ψ ) .

Under the assumptions of Theorem 1, assume that Ψ : X → X be a nonexpansive mapping with F ix ( Ψ ) ≠ ∅ . Then the sequence { q v } generated by (4) with λ ∈ ( 0 , 1 ] , ρ > 0 and initial value b 0 = 1 ρ ( Ψ a 0 − a 0 ) , converges weakly to a point in Fix ( Ψ ) , provided that conditions ( C 1) − ( C 3) hold.

Under the assumptions of Theorem 1, assume that Ψ : X → X be a Asymptotically nonexpansive mapping with Fix ( Ψ ) ≠ ∅ . Then the sequence { q v } generated by (4) with λ ∈ ( 0 , 1 ] , ρ > 0 and initial value b 0 = 1 ρ ( Ψ 1 a 0 − a 0 ) , converges weakly to a point in Fix(Ψ) , provided that conditions ( 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 } generated by (4) with λ ∈ ( 0 , 1 ] , ρ > 0 , and b 0 = 1 ρ ( Ψ 1 a 0 − a 0 ) , converges weakly to a point in Fix ( Ψ ) , provided that conditions ( C 1) − ( C 3) hold.

Let X be a real Hilbert space and Ψ : X → X be either a TAN-mapping or a Asymptotically nonexpansive mapping with Fix ( Ψ ) ≠ ∅ . Then the sequence { q v } generated by (4) with λ ∈ ( 0 , 1 ] , ρ > 0 , and b 0 = 1 ρ ( Ψ 1 a 0 − a 0 ) , converges weakly to a point in Fix ( Ψ ) , provided that conditions ( 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. Hence, the present result generalized several known convergence theorems in the literature.

Stability

In this section, we analyze the stability of the iterative sequence defined by (4) with respect to a fixed point of the TAN-mapping. The associated iterative operator is given by f ( Ψ , r v ) = λ α v s v + ( 1 − λ α v ) t v .

Let X be a Banach and Ψ : X → X be a TAN-mapping. Suppose X has a fixed point z. Let the sequence { r v } v = 1 ∞ be a sequence generated by (4) with λ ∈ ( 0 , 1 ] and ρ > 0 . If Assumption 1 holds and the sequence { Ψ v a v − a v } is bounded, then { q v } v = 1 ∞ is stable.

Let { r v } v = 1 ∞ be an arbitrary sequence in X the sequence generated by (4) is r v + 1 = f ( Ψ , r v ) converging to a unique fixed point z and ε v = ‖ r v + 1 − f ( Ψ , r v ) ‖ . We will prove that lim v → ∞ ε v = 0 ⟺ lim v → ∞ r v = z .

Assume lim v → ∞ ε v = 0 and ‖ r v + 1 − z ‖ = ‖ r v + 1 − f ( Ψ , r v ) ‖ + ‖ f ( Ψ , r v ) − z ‖ ≤ ‖ r v + 1 − f ( Ψ , r v ) ‖ + ‖ f ( Ψ , r v ) − z ‖ = ε v + ‖ f ( Ψ r v , r v ) − z ‖ = ε v + ‖ λ α v s v + ( 1 − λ α v ) t v − z ‖ ≤ ε v + λ α v ‖ s v − z ‖ + ( 1 − λ α v ) ‖ t v − z ‖ ≤ ε v + λ α v [ ‖ r v − z ‖ + ω v ‖ ( r v − r v − 1 ) ‖ ] + ( 1 − λ α v ) ‖ t v − z ‖ ≤ ε v + λ α v ‖ r v − z ‖ + ω v λ α v ‖ ( r v − r v − 1 ) ‖ + ( 1 − λ α v ) [ ‖ s v − z ‖ + μ v ∅ ( ‖ s v − z ‖ ) + κ v + ρ σ v ‖ n v ‖ ] ≤ ε v + λ α v ‖ r v − z ‖ + ω v λ α v ‖ ( r v − r v − 1 ) ‖ + ( 1 − λ α v ) [ ‖ r v − z ‖ + ω v ‖ ( r v − r v − 1 ) ‖ + μ v ∅ ( ‖ s v − z ‖ ) + κ v + ρ σ v ‖ n v ‖ ] ‖ r v + 1 − z ‖ ≤ ε v + ‖ r v − z ‖ + ω v ‖ ( r v − r v − 1 ) ‖ + ( 1 − λ α v ) μ v ∅ ( ‖ s v − z ‖ ) + ( 1 − λ α v ) κ v + ( 1 − λ α v ) ρ σ v ‖ n v ‖ ‖ r v + 1 − z ‖ ≤ ‖ r v − z ‖ + δ v ,

where δ v = ε v + ω v ‖ ( r v − r v − 1 ) ‖ + ( 1 − λ α v ) μ v ∅ ( ‖ s v − z ‖ ) + ( 1 − λ α v ) κ v + ( 1 − λ α v ) ρ σ v ‖ n v ‖

Since , μ v → 0 , κ v → 0 , σ v → 0 , and assumption ∑ v = 0 ∞ ω v ‖ ( r v − r v − 1 ) ‖ < ∞ , it follows that δ v → 0 as v → ∞ . Therefore, by Lemma 3, we have lim v → ∞ ‖ r v − z ‖ = 0 , which gives lim v → ∞ r v = z .

On the other hand, suppose that lim v → ∞ r v = z . Then, ε v = ‖ r v + 1 − f ( Ψ , r v ) ‖ ≤ ‖ r v + 1 − z ‖ + ‖ f ( Ψ , r v ) − z ‖ ≤ ‖ r v + 1 − z ‖ + ‖ λ α v s v + ( 1 − λ α v ) t v − z ‖ ≤ ‖ r v + 1 − z ‖ + λ α v ‖ s v − z ‖ + ( 1 − λ α v ) ‖ t v − z ‖ ≤ ‖ r v + 1 − z ‖ + λ α v [ ‖ r v − z ‖ + ω v ‖ ( r v − r v − 1 ) ‖ ] + ( 1 − λ α v ) ‖ t v − z ‖ ≤ ‖ r v + 1 − z ‖ + λ α v ‖ r v − z ‖ + ω v λ α v ‖ ( r v − r v − 1 ) ‖ + ( 1 − λ α v ) [ ‖ s v − z ‖ + μ v ∅ ( ‖ s v − z ‖ ) + κ v + ρ σ v ‖ n v ‖ ] ≤ ‖ r v + 1 − z ‖ + λ α v ‖ r v − z ‖ + ω v λ α v ‖ ( r v − r v − 1 ) ‖ + ( 1 − λ α v ) [ ‖ r v − z ‖ + ω v ‖ ( r v − r v − 1 ) ‖ + μ v ∅ ( ‖ s v − z ‖ ) + κ v + ρ σ v ‖ n v ‖ ] ≤ ‖ r v + 1 − z ‖ + ‖ r v − z ‖ + ω v ‖ ( r v − r v − 1 ) ‖ + ( 1 − λ α v ) μ v ∅ ( ‖ s v − z ‖ ) + ( 1 − λ α v ) κ v + ( 1 − λ α v ) ρ σ v ‖ n v ‖

Taking the limit as v → ∞ by both sides, we get lim v → ∞ ε v = 0 .

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 be a nonempty closed convex subset of a Banach space X and Ψ : C → C be a TAN-mapping. For a given q 0 = d 0, let { q v } v = 1 ∞ be sequence generated by (4), with λ ∈ ( 0 , 1 ] and ρ > 0 and let, { d v } be sequence generated by (19). Then { q v } converges to z faster than { d v } .

From inequality (10), 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 can be verify that { Ψ v } from a family of TAN-mappings. For the iterative { q v } defined by (4), 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 } defined by (19), the initial values and parameters are selected as d 0 = 1.5 , δ v = t v = ϱ v = 0.3 where the fixed point is z = 1.0 . The numerical results are presented in Table 1 and a graphical comparison is shown 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 (4) and (19).

Let C be a nonempty closed convex subset of a Banach space X and Ψ : C → C be a TAN-mapping with Fix ( Ψ ) ≠ ∅ . Let the sequence { q v } and { s v } be defined by (4) and (20), respectively.

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 both iteration algorthim consist of convex combinations and bounded perturbations, it follows that all generated sequences remain 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. Assume first that { q v } converges to z ∈ Fix(Ψ). By Theorem 1, this convergence is ensured under the stated assumption. To prove that { s v } also converges to the same fixed point z, we estimate. ‖ 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 )

Since μ → 0, k v → 0, and all invloved sequences are bounded, Lemma 3 implies that { s v } converges 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 . Therefore, the two statements are equivalent.

Application

In this section, we apply the proposed framework of TAN-mapping 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 For all ξ , ρ , l , h ∈ [ 0 , 1 ] , where 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 ] is a Banach space with the maximum norm ‖ q − s ‖ ∞ = max σ ∈ [ 0 , 1 ] | q ( σ ) − s ( σ ) | , for all q , s ∈ C [ 0 , 1 ] .

Assume that Z is a nonempty closed convex subset of X and Ψ : Z → Z described as 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

Also assume the assertions below are true

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 (21) has 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 , where 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, we obtain ‖ X − Ψ X ∗ ‖ ∞ = ‖ X − X ∗ ‖ ∞ + μ v ∅ ( ‖ X − X ∗ ‖ ∞ ) + k v , it follows that Ψ is a TAN-mapping.

Therefore, by

Theorem 1, the iterative algorthim generated by (4) converges weakly to a solution of the integral Equation (21).

The following example supports Theorem 6.

Consider the following two-dimensional Volterra integral equation.

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

It is clear that problem (22) is a special case of (21) with, 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 the cosine function is Lipschitz continuous on bounded intervals and the constants 1 16 , 1 24 , 1 23 satisfy the contraction type bound required in Assumption (3) all assumptions of Theorem 6 are satisfied, Therefore, Equation (22) admits at least one solution in X.

Conclusion

In this paper, we investigated the convergence properties of the modified iterative algorithm (4) in the Banach space X. The main results establish both weak convergence ( Theorem 1) and strong convergence ( Theorem 2) of the proposed scheme under appropriate assumptions on the space and on the structure of the iterative process. Furthermore, stability of the generated sequence was analyzed, and a comparative study demonstrated that the proposed method converges faster than the HR-iteration scheme. Algorithm 4 was shown to be equivalent to Algorithm 20 in the sense that both methods converge to the same fixed point, while the proposed algorithm exhibits a superior convergence rate. 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. The obtained results demonstrate that the proposed framework provides a unified approach for analyzing convergence, stability, and acceleration of iterative algorithm under TAN-mapping conditions. Future research directions may include the investigation of a related iterative algorithms and their extensions under weaker contractive type assumptions, 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.197668.r486754 Reviewer response for version 3 Shah Rahim 1 Referee https://orcid.org/0009-0001-9044-5470 Kohsar University Murree, Murree, Pakistan

Competing interests: No competing interests were disclosed.

 9 6 2026 Copyright: © 2026 Shah R 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 approve-with-reservations

Recommendation: Major Revision

Comments: 

The manuscript still contains substantial grammatical, formatting, and notation inconsistencies that require professional editing.

The revised stability theorem is not yet mathematically rigorous and needs further clarification.

The convergence-rate comparison with the HR iteration is not convincingly proved.

The numerical example appears inconsistent with the claimed fixed point and convergence behavior. It needs to add some discussions in detail. 

Several steps in the weak convergence proof require more detailed justification.

The application to the 2D Volterra integral equation contains notation inconsistencies and remains relatively superficial.

The authors should carefully verify all mathematical derivations, numerical results, and assumptions before the manuscript can be considered for acceptance.

 Overall evaluation: The topic is interesting and potentially publishable, but significant mathematical and presentation issues remain. A further major revision is required before the manuscript can be recommended for acceptance.

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:

Analysis, Fixed Point Theory, Applied Mathematics

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

 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