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16 pages, 320 KB

Open AccessArticle

Fixed Points of Enriched Mappings with General Real Constants

by Konrawut Khammahawong, Natthaya Boonyam, Sani Salisu and Premyuda Dechboon

Mathematics 2026, 14(6), 937; https://doi.org/10.3390/math14060937 - 10 Mar 2026

Viewed by 754

Building upon classical fixed point theory, the concept of enriched contractions introduces a new class of mappings. For a normed linear space

(X, ∥ \cdot ∥)

, a mapping

T : X \to X

is called an enriched contraction if [...] Read more.

Building upon classical fixed point theory, the concept of enriched contractions introduces a new class of mappings. For a normed linear space

(X, ∥ \cdot ∥)

, a mapping

T : X \to X

is called an enriched contraction if there exist

b \in [0, \infty)

and

θ \in [0, b + 1)

such that

∥ b (x - y) + T x - T y ∥ \leq θ ∥ x - y ∥, \forall x, y \in X .

This class of mappings includes both the well-known Picard–Banach contraction and certain nonexpansive mappings. In this paper, we extend the definition by allowing

b \in R \ {- 1}

instead of

b \in [0, \infty)

. This extension enables the condition to cover both contraction and certain nonexpansive mappings. We establish results on the existence and uniqueness of fixed points and present the Krasnosel’skii iteration for approximating such points. An example is provided to demonstrate mapping that meets the extended condition but not the original. Full article

(This article belongs to the Section C: Mathematical Analysis)

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17 pages, 285 KB

Open AccessArticle

Fixed Point Approximation for Enriched Suzuki Nonexpansive Mappings in Banach Spaces

by Doaa Filali, Fahad Maqbul Alamrani, Esmail Alshaban, Adel Alatawi, Amid Yousef Alanazi and Faizan Ahmad Khan

Axioms 2025, 14(6), 426; https://doi.org/10.3390/axioms14060426 - 30 May 2025

Cited by 2 | Viewed by 1735

Abstract

This paper investigates the approximation of fixed points for mappings that satisfy the enriched (C) condition using a modified iterative process in a Banach space framework. We first establish a weak convergence result and then derive strong convergence theorems under suitable assumptions. To [...] Read more.

This paper investigates the approximation of fixed points for mappings that satisfy the enriched (C) condition using a modified iterative process in a Banach space framework. We first establish a weak convergence result and then derive strong convergence theorems under suitable assumptions. To illustrate the applicability of our findings, we present a numerical example involving mappings that satisfy the enriched (C) condition but not the standard (C) condition. Additionally, numerical computations and graphical representations demonstrate that the proposed iterative process achieves a faster convergence rate compared to several existing methods. As a practical application, we introduce a projection based an iterative process for solving split feasibility problems (SFPs) in a Hilbert space setting. Our findings contribute to the ongoing development of iterative processes for solving optimization and feasibility problems in mathematical and applied sciences. Full article

(This article belongs to the Special Issue Fixed-Point Theory and Its Related Topics, 5th Edition)

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20 pages, 386 KB

Open AccessArticle

Some Fixed Point Results for Novel Contractions with Applications in Fractional Differential Equations for Market Equilibrium and Economic Growth

by Min Wang, Muhammad Din and Mi Zhou

Fractal Fract. 2025, 9(5), 324; https://doi.org/10.3390/fractalfract9050324 - 19 May 2025

Cited by 8 | Viewed by 1475

Abstract

In this study, we introduce two new classes of contractions, namely enriched

(I, ρ, χ)

-contractions and generalized enriched

(I, ρ, χ)

-contractions, within the context of normed spaces. These classes generalize several well-known contraction [...] Read more.

In this study, we introduce two new classes of contractions, namely enriched

(I, ρ, χ)

-contractions and generalized enriched

(I, ρ, χ)

-contractions, within the context of normed spaces. These classes generalize several well-known contraction types, including

χ

-contractions, Banach contractions, enriched contractions, Kannan contractions, Bianchini contractions, Zamfirescu contractions, non-expansive mappings, and

(ρ, χ)

-enriched contractions. We establish related fixed point results for the novel contractions in normed spaces endowed with the binary relations preserving key symmetric properties, ensuring consistency and applicability. The Krasnoselskij iteration method is refined to incorporate symmetric constraints, facilitating fixed point identification within these spaces. By appropriately selecting constants in the definition of enriched

(I, ρ, χ)

-contractions, employing a suitable binary relation, or control function

χ \in Θ

, our framework generalizes and extends classical fixed point theorems. Illustrative examples highlight the significance of our findings in reinforcing fixed point conditions and demonstrating their broader applicability. Additionally, this paper explores how these ideas guarantee the stability of the production–consumption markets equilibrium and the economic growth model. Full article

(This article belongs to the Special Issue Fractional Order Modelling of Dynamical Systems)

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16 pages, 347 KB

Open AccessArticle

Introducing Monotone Enriched Nonexpansive Mappings for Fixed Point Approximation in Ordered CAT(0) Spaces

by Safeer Hussain Khan, Rizwan Anjum and Nimra Ismail

Computation 2025, 13(4), 81; https://doi.org/10.3390/computation13040081 - 21 Mar 2025

Cited by 4 | Viewed by 1603

Abstract

The aim of this paper is twofold: introducing the concept of monotone enriched nonexpansive mappings and a faster iterative process. Our examples illustrate the novelty of our newly introduced concepts. We investigate the iterative estimation of fixed points for such mappings for the [...] Read more.

The aim of this paper is twofold: introducing the concept of monotone enriched nonexpansive mappings and a faster iterative process. Our examples illustrate the novelty of our newly introduced concepts. We investigate the iterative estimation of fixed points for such mappings for the first time within an ordered CAT(0) space. It is done by proving some strong and

Δ

-convergence theorems. Additionally, numerical experiments are included to demonstrate the validity of our theoretical results and to establish the superiority of convergence behavior of our iterative process. As an application, we use our newly introduced concepts to find the solution of an integral equation. The outcomes of our study expand upon and enhance certain established findings in the current body of literature. Full article

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19 pages, 321 KB

Open AccessArticle

Certain Fixed-Point Results for (

e

,ψ,Φ)-Enriched Weak Contractions via Theoretic Order with Applications

by Umar Ishtiaq, Muhammad Din, Yumnam Rohen, Khalid A. Alnowibet and Ioan-Lucian Popa

Axioms 2025, 14(2), 135; https://doi.org/10.3390/axioms14020135 - 14 Feb 2025

Cited by 5 | Viewed by 1230

Abstract

This paper aims to establish several fixed-point theorems within the framework of Banach spaces endowed with a binary relation. By utilizing enriched contraction principles involving two classes of altering-distance functions, the study encompasses various types of contractive mappings, including theoretic-order contractions, Picard–Banach contractions, [...] Read more.

This paper aims to establish several fixed-point theorems within the framework of Banach spaces endowed with a binary relation. By utilizing enriched contraction principles involving two classes of altering-distance functions, the study encompasses various types of contractive mappings, including theoretic-order contractions, Picard–Banach contractions, weak contractions, and non-expansive contractions. A suitable Krasnoselskij iterative scheme is employed to derive the results. Many well-known fixed-point theorems (FPTs) can be obtained as special cases of these findings by assigning specific control functions in the main definitions or selecting an appropriate binary relation. To validate the theoretical results, numerous illustrative examples are provided. Furthermore, the paper demonstrates the applicability of the findings through applications to ordinary differential equations. Full article

22 pages, 842 KB

Open AccessEditor’s ChoiceArticle

Fixed Point Results for Fuzzy Enriched Contraction in Fuzzy Banach Spaces with Applications to Fractals and Dynamic Market Equillibrium

by Muhammad Shaheryar, Fahim Ud Din, Aftab Hussain and Hamed Alsulami

Fractal Fract. 2024, 8(10), 609; https://doi.org/10.3390/fractalfract8100609 - 18 Oct 2024

Cited by 9 | Viewed by 2405

Abstract

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for [...] Read more.

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equation and Fixed-Point Theory)

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11 pages, 258 KB

Open AccessArticle

On Enriched Suzuki Mappings in Hadamard Spaces

by Teodor Turcanu and Mihai Postolache

Mathematics 2024, 12(1), 157; https://doi.org/10.3390/math12010157 - 3 Jan 2024

Cited by 4 | Viewed by 2096

Abstract

We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. [...] Read more.

We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. More accurately, Hilbert spaces are particular Hadamard spaces, while enriched Suzuki nonexpansive mappings are natural generalizations of enriched nonexpansive mappings. Next, enriched Suzuki nonexpansive mappings naturally contain Suzuki nonexpansive mappings in Hadamard spaces. Besides technical lemmas, the results of this paper deal with (1) the existence of fixed points for enriched Suzuki nonexpansive mappings and (2)

Δ

and strong (metric) convergence of Picard iterates of the

α

-averaged mapping, which are exactly Krasnoselskij iterates for the original mapping. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)

20 pages, 324 KB

Open AccessArticle

Approximation of the Solution of Delay Fractional Differential Equation Using AA-Iterative Scheme

by Mujahid Abbas, Muhammad Waseem Asghar and Manuel De la Sen

Mathematics 2022, 10(2), 273; https://doi.org/10.3390/math10020273 - 16 Jan 2022

Cited by 20 | Viewed by 4024

Abstract

The aim of this paper is to propose a new faster iterative scheme (called

A A

-iteration) to approximate the fixed point of

(b, η)

-enriched contraction mapping in the framework of Banach spaces. It is also proved that our [...] Read more.

The aim of this paper is to propose a new faster iterative scheme (called

A A

-iteration) to approximate the fixed point of

(b, η)

-enriched contraction mapping in the framework of Banach spaces. It is also proved that our iteration is stable and converges faster than many iterations existing in the literature. For validity of our proposed scheme, we presented some numerical examples. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in the uniformly convex Banach space. Finally, we approximate the solution of delay fractional differential equations using

A A

-iterative scheme. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

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12 pages, 288 KB

Open AccessArticle

A Modified Krasnosel’skiǐ–Mann Iterative Algorithm for Approximating Fixed Points of Enriched Nonexpansive Mappings

by Vasile Berinde

Symmetry 2022, 14(1), 123; https://doi.org/10.3390/sym14010123 - 10 Jan 2022

Cited by 16 | Viewed by 3793

Abstract

For approximating the fixed points of enriched nonexpansive mappings in Hilbert spaces, we consider a modified Krasnosel’skiǐ–Mann algorithm for which we prove a strong convergence theorem. We also empirically compare the rate of convergence of the modified Krasnosel’skiǐ–Mann algorithm and of the simple [...] Read more.

For approximating the fixed points of enriched nonexpansive mappings in Hilbert spaces, we consider a modified Krasnosel’skiǐ–Mann algorithm for which we prove a strong convergence theorem. We also empirically compare the rate of convergence of the modified Krasnosel’skiǐ–Mann algorithm and of the simple Krasnosel’skiǐ fixed point algorithm. Based on the numerical experiments reported in the paper we conclude that, for the class of enriched nonexpansive mappings, it is more convenient to work with the simple Krasnosel’skiǐ fixed point algorithm than with the modified Krasnosel’skiǐ–Mann algorithm. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry)

12 pages, 277 KB

Open AccessArticle

Approximation of Fixed Points for Enriched Suzuki Nonexpansive Operators with an Application in Hilbert Spaces

by Kifayat Ullah, Junaid Ahmad, Muhammad Arshad and Zhenhua Ma

Axioms 2022, 11(1), 14; https://doi.org/10.3390/axioms11010014 - 29 Dec 2021

Cited by 14 | Viewed by 3309

Abstract

In this article, we introduce the class of enriched Suzuki nonexpansive (ESN) mappings. We show that this new class of mappings properly contains the class of Suzuki nonexpansive as well as the class of enriched nonexpansive mappings. We establish existence of fixed point [...] Read more.

In this article, we introduce the class of enriched Suzuki nonexpansive (ESN) mappings. We show that this new class of mappings properly contains the class of Suzuki nonexpansive as well as the class of enriched nonexpansive mappings. We establish existence of fixed point and convergence of fixed point in a Hilbert space setting under the Krasnoselskii iteration process. One of the our main results is applied to solve a split feasibility problem (SFP) in this new setting of mappings. Our main results are a significant improvement of the corresponding results of the literature. Full article

(This article belongs to the Special Issue Theory and Application of Fixed Point)

16 pages, 437 KB

Open AccessArticle

Approximating Solutions of Matrix Equations via Fixed Point Techniques

by Rahul Shukla, Rajendra Pant, Hemant Kumar Nashine and Manuel De la Sen

Mathematics 2021, 9(21), 2684; https://doi.org/10.3390/math9212684 - 22 Oct 2021

Cited by 4 | Viewed by 2354

Abstract

The principal goal of this work is to investigate new sufficient conditions for the existence and convergence of positive definite solutions to certain classes of matrix equations. Under specific assumptions, the basic tool in our study is a monotone mapping, which admits a [...] Read more.

The principal goal of this work is to investigate new sufficient conditions for the existence and convergence of positive definite solutions to certain classes of matrix equations. Under specific assumptions, the basic tool in our study is a monotone mapping, which admits a unique fixed point in the setting of a partially ordered Banach space. To estimate solutions to these matrix equations, we use the Krasnosel’skiĭ iterative technique. We also discuss some useful examples to illustrate our results. Full article

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15 pages, 322 KB

Open AccessArticle

Fixed Points Theorems for Unsaturated and Saturated Classes of Contractive Mappings in Banach Spaces

by Vasile Berinde and Mădălina Păcurar

Symmetry 2021, 13(4), 713; https://doi.org/10.3390/sym13040713 - 18 Apr 2021

Cited by 22 | Viewed by 3485

Abstract

Based on the technique of enriching contractive type mappings, a technique that has been used successfully in some recent papers, we introduce the concept of a saturated class of contractive mappings. We show that, from this perspective, the contractive type mappings in the [...] Read more.

Based on the technique of enriching contractive type mappings, a technique that has been used successfully in some recent papers, we introduce the concept of a saturated class of contractive mappings. We show that, from this perspective, the contractive type mappings in the metric fixed point theory can be separated into two distinct classes, unsaturated and saturated, and that, for any unsaturated class of mappings, the technique of enriching contractive type mappings provides genuine new fixed-point results. We illustrate the concept by surveying some significant fixed-point results obtained recently for five remarkable unsaturated classes of contractive mappings. In the second part of the paper, we also identify two important classes of saturated contractive mappings, whose main feature is that they cannot be enlarged by enriching the contractive mappings. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Applications, Geometry of Banach Spaces and Symmetry)

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