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19 pages, 266 KiB

Open AccessArticle

Fixed-Point Results in Elliptic-Valued Metric Spaces with Applications

by Badriah Alamri

Axioms 2025, 14(1), 46; https://doi.org/10.3390/axioms14010046 - 8 Jan 2025

Cited by 1 | Viewed by 849

The primary objective of this research is to investigate the notion of elliptic-valued metric spaces and prove new fixed-point theorems for diverse generalized contractions. Our contributions extend several existing results in the field. To underscore the novelty of our main result, we provide [...] Read more.

The primary objective of this research is to investigate the notion of elliptic-valued metric spaces and prove new fixed-point theorems for diverse generalized contractions. Our contributions extend several existing results in the field. To underscore the novelty of our main result, we provide a concrete example. Additionally, we showcase the practical relevance of our primary theorem by solving a Urysohn integral equation. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory with Applications)

10 pages, 269 KiB

Open AccessArticle

On the Fixed Point Theorem for Large Contraction Mappings with Applications to Delay Fractional Differential Equations

by Mouataz Billah Mesmouli, Elvan Akın, Loredana Florentina Iambor, Osman Tunç and Taher S. Hassan

Fractal Fract. 2024, 8(12), 703; https://doi.org/10.3390/fractalfract8120703 - 28 Nov 2024

Cited by 4 | Viewed by 1249

Abstract

This paper explores a new class of mappings and presents several fixed-point results for these mappings. We define these mappings by combining well-known mappings in the literature, specifically the large contraction mapping and Chatterjea’s mapping. This combination allows us to achieve significant fixed-point [...] Read more.

This paper explores a new class of mappings and presents several fixed-point results for these mappings. We define these mappings by combining well-known mappings in the literature, specifically the large contraction mapping and Chatterjea’s mapping. This combination allows us to achieve significant fixed-point results in complete metric spaces, both in a continuous and a non-continuous sense. Additionally, we provide an explicit example to validate our findings. Furthermore, we discuss a general model for fractional differential equations using the Caputo derivative. Finally, we outline the benefits of our study and suggest potential areas for future research. Full article

13 pages, 267 KiB

Open AccessArticle

New Fixed Point Theorems for Generalized Meir–Keeler Type Nonlinear Mappings with Applications to Fixed Point Theory

by Shin-Yi Huang and Wei-Shih Du

Symmetry 2024, 16(8), 1088; https://doi.org/10.3390/sym16081088 - 22 Aug 2024

Cited by 2 | Viewed by 1384

Abstract

In this paper, we investigate new fixed point theorems for generalized Meir–Keeler type nonlinear mappings satisfying the condition (DH). As applications, we obtain many new fixed point theorems which generalize and improve several results available in the corresponding literature. An example is [...] Read more.

In this paper, we investigate new fixed point theorems for generalized Meir–Keeler type nonlinear mappings satisfying the condition (DH). As applications, we obtain many new fixed point theorems which generalize and improve several results available in the corresponding literature. An example is provided to illustrate and support our main results. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

19 pages, 378 KiB

Open AccessArticle

Solving Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions Using Covariant JS-Contractions

by Nawab Hussain, Nawal Alharbi and Ghada Basendwah

Symmetry 2024, 16(8), 939; https://doi.org/10.3390/sym16080939 - 23 Jul 2024

Cited by 3 | Viewed by 1356

Abstract

This paper investigates the existence, uniqueness, and symmetry of solutions for

Φ

–Atangana–Baleanu fractional differential equations of order

μ \in (1, 2]

under mixed nonlocal boundary conditions. This is achieved through the use of covariant and contravariant

J S

-contractions [...] Read more.

This paper investigates the existence, uniqueness, and symmetry of solutions for

Φ

–Atangana–Baleanu fractional differential equations of order

μ \in (1, 2]

under mixed nonlocal boundary conditions. This is achieved through the use of covariant and contravariant

J S

-contractions within a generalized framework of a sequential extended bipolar parametric metric space. As a consequence, we obtain the results on covariant and contravariant Ćirić, Chatterjea, Kannan, and Reich contractions as corollaries. Additionally, we substantiate our fixed-point findings with specific examples and derive similar results in the setting of sequential extended fuzzy bipolar metric space. Full article

(This article belongs to the Special Issue Symmetry in Metric Spaces and Topology)

17 pages, 312 KiB

Open AccessArticle

Stability of Fixed Points of Partial Contractivities and Fractal Surfaces

by María A. Navascués

Axioms 2024, 13(7), 474; https://doi.org/10.3390/axioms13070474 - 13 Jul 2024

Cited by 2 | Viewed by 955

Abstract

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations [...] Read more.

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces. Full article

(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

17 pages, 338 KiB

Open AccessArticle

On (α,p)-Cyclic Contractions and Related Fixed Point Theorems

by Victory Asem, Yumnam Mahendra Singh, Mohammad Saeed Khan and Salvatore Sessa

Symmetry 2023, 15(10), 1826; https://doi.org/10.3390/sym15101826 - 26 Sep 2023

Cited by 1 | Viewed by 1493

Abstract

Lipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as [...] Read more.

Lipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as

(α, p)

-Liptschitz mappings and studied not only the modified form of the Lipschitz condition, but also the behavior of a finite number of their iterates. The purpose of this paper is to discuss the various types of

(α, p)

-contractions with cyclic representation that extend the results due to Banach, Kannan, and Chatterjea. Moreover, based on such types of contractions and the property of symmetry, we obtain some related fixed-point results in the setting of metric spaces. Some examples are studied to illustrate the validity of our obtained results. As an application of our results, we establish the existence of the solution to a class of Fredholm integral equations. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

10 pages, 264 KiB

Open AccessArticle

On Cyclic Contractive Mappings of Kannan and Chatterjea Type in Generalized Metric Spaces

by Mohammad Al-Khaleel, Sharifa Al-Sharif and Rami AlAhmad

Mathematics 2023, 11(4), 890; https://doi.org/10.3390/math11040890 - 9 Feb 2023

Cited by 10 | Viewed by 1999

Abstract

Novel cyclic contractions of the Kannan and Chatterjea type are presented in this study. With the aid of these brand-new contractions, new results for the existence and uniqueness of fixed points in the setting of complete generalized metric space have been established. Importantly, [...] Read more.

Novel cyclic contractions of the Kannan and Chatterjea type are presented in this study. With the aid of these brand-new contractions, new results for the existence and uniqueness of fixed points in the setting of complete generalized metric space have been established. Importantly, the results are generalizations and extensions of fixed point theorems by Chatterjea and Kannan and their cyclical expansions that are found in the literature. Additionally, several of the existing results on fixed points in generalized metric space will be generalized by the results presented in this work. Interestingly, the findings have a variety of applications in engineering and sciences. Examples have been given at the end to show the reliability of the demonstrated results. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory and Its Applications)

13 pages, 290 KiB

Open AccessArticle

Chatterjea and

\overset{`}{C}

iri

\overset{`}{C}

-Type Fixed-Point Theorems Using (α − ψ) Contraction on C^*-Algebra-Valued Metric Space

by Ibtisam Masmali and Saleh Omran

Mathematics 2022, 10(9), 1615; https://doi.org/10.3390/math10091615 - 9 May 2022

Cited by 3 | Viewed by 3579

Abstract

In the present paper, we provide and verify several results obtained by using the Chatterjea and

\overset{`}{C}

iri

\overset{`}{c}

fixed-point theorems by using

(α - ψ)

-contractive mapping in

C^{*}

-algebra-valued metric space. We provide some examples and [...] Read more.

In the present paper, we provide and verify several results obtained by using the Chatterjea and

\overset{`}{C}

iri

\overset{`}{c}

fixed-point theorems by using

(α - ψ)

-contractive mapping in

C^{*}

-algebra-valued metric space. We provide some examples and an application to illustrate our results. Our study extends and generalizes the results of several studies in the literature. Full article

16 pages, 323 KiB

Open AccessArticle

Some Generalised Fixed Point Theorems Applied to Quantum Operations

by Umar Batsari Yusuf, Poom Kumam and Sikarin Yoo-Kong

Symmetry 2020, 12(5), 759; https://doi.org/10.3390/sym12050759 - 6 May 2020

Cited by 6 | Viewed by 5655

Abstract

In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of [...] Read more.

In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of quantum states was used to establish the existence of a fixed quantum state associated to an order-preserving quantum operation. The method we presented is an alternative in showing the existence of a fixed quantum state associated to quantum operations. Our method does not capitalise on the commutativity of the quantum effects with the fixed quantum state(s) (Luders’s compatibility criteria). The Luders’s compatibility criteria in higher finite dimensional spaces is rather difficult to check for any prospective fixed quantum state. Some part of our results cover the famous contractive fixed point results of Banach, Kannan and Chatterjea. Full article

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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Figure 1

15 pages, 788 KiB

Open AccessArticle

Fundamental Questions and New Counterexamples for b-Metric Spaces and Fatou Property

by Ning Lu, Fei He and Wei-Shih Du

Mathematics 2019, 7(11), 1107; https://doi.org/10.3390/math7111107 - 14 Nov 2019

Cited by 11 | Viewed by 3976

Abstract

In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction [...] Read more.

In this paper, we give new examples to show that the continuity actually strictly stronger than the Fatou property in b-metric spaces. We establish a new fixed point theorem for new essential and fundamental sufficient conditions such that a Ćirić type contraction with contraction constant

λ \in [\frac{1}{s}, 1)

in a complete b-metric space with

s > 1

have a unique fixed point. Many new examples illustrating our results are also given. Our new results extend and improve many recent results and they are completely original and quite different from the well known results on the topic in the literature. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point and Best Proximity Point Problems and Their Applications)

19 pages, 288 KiB

Open AccessArticle

Solutions to Fredholm Integral Inclusions via Generalized Fuzzy Contractions

by Hamed H Al-Sulami, Jamshaid Ahmad, Nawab Hussain and Abdul Latif

Mathematics 2019, 7(9), 808; https://doi.org/10.3390/math7090808 - 2 Sep 2019

Cited by 5 | Viewed by 1851

Abstract

The aim of this study is to investigate the existence of solutions for the following Fredholm integral inclusion [...] Read more.

The aim of this study is to investigate the existence of solutions for the following Fredholm integral inclusion

φ (t) \in [f (t) + \int_{0}^{1} K (t, s, φ (s)) ϱ s]

for

t \in [0, 1],

where

f \in C [0, 1]

is a given real-valued function and

K : [0, 1]

\times [0, 1]

\times R \to K_{c v} (R)

a given multivalued operator, where

K_{c v}

represents the family of non-empty compact and convex subsets of

R

,

φ \in C [0, 1]

is the unknown function and

ϱ

is a metric defined on

C [0, 1]

. To attain this target, we take advantage of fixed point theorems for

α

-fuzzy mappings satisfying a new class of contractive conditions in the context of complete metric spaces. We derive new fixed point results which extend and improve the well-known results of Banach, Kannan, Chatterjea, Reich, Hardy-Rogers, Berinde and Ćirić by means of this new class of contractions. We also give a significantly non-trivial example to support our new results. Full article

13 pages, 256 KiB

Open AccessArticle

On Fixed Point Results for Modified JS-Contractions with Applications

by Vahid Parvaneh, Nawab Hussain, Aiman Mukheimer and Hassen Aydi

Axioms 2019, 8(3), 84; https://doi.org/10.3390/axioms8030084 - 24 Jul 2019

Cited by 15 | Viewed by 3329

Abstract

In [Fixed Point Theory Appl., 2015 (2015):185], the authors introduced a new concept of modified contractive mappings, generalizing Ćirić, Chatterjea, Kannan, and Reich type contractions. They applied the condition

(θ_{4})

(see page 3, ^{Section 2} of the above paper). Later, [...] Read more.

In [Fixed Point Theory Appl., 2015 (2015):185], the authors introduced a new concept of modified contractive mappings, generalizing Ćirić, Chatterjea, Kannan, and Reich type contractions. They applied the condition

(θ_{4})

(see page 3, ^{Section 2} of the above paper). Later, in [Fixed Point Theory Appl., 2016 (2016):62], Jiang et al. claimed that the results in [Fixed Point Theory Appl., 2015 (2015):185] are not real generalizations. In this paper, by restricting the conditions of the control functions, we obtain a real generalization of the Banach contraction principle (BCP). At the end, we introduce a weakly JS-contractive condition generalizing the JS-contractive condition. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Topics)

11 pages, 264 KiB

Open AccessArticle

Some Globally Stable Fixed Points in b-Metric Spaces

by Umar Yusuf Batsari, Poom Kumam and Kanokwan Sitthithakerngkiet

Symmetry 2018, 10(11), 555; https://doi.org/10.3390/sym10110555 - 30 Oct 2018

Cited by 4 | Viewed by 3329

Abstract

In this paper, the existence and uniqueness of globally stable fixed points of asymptotically contractive mappings in complete b-metric spaces were studied. Also, we investigated the existence of fixed points under the setting of a continuous mapping. Furthermore, we introduce a contraction [...] Read more.

In this paper, the existence and uniqueness of globally stable fixed points of asymptotically contractive mappings in complete b-metric spaces were studied. Also, we investigated the existence of fixed points under the setting of a continuous mapping. Furthermore, we introduce a contraction mapping that generalizes that of Banach, Kanan, and Chatterjea. Using our new introduced contraction mapping, we establish some results on the existence and uniqueness of fixed points. In obtaining some of our results, we assume that the space is associated with a partial order, and the b-metric function has the regularity property. Our results improve, and generalize some current results in the literature. Full article

(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)

9 pages, 272 KiB

Open AccessArticle

New Types of F_c-Contractions and the Fixed-Circle Problem

by Nihal Taş, Nihal Yılmaz Özgür and Nabil Mlaiki

Mathematics 2018, 6(10), 188; https://doi.org/10.3390/math6100188 - 2 Oct 2018

Cited by 51 | Viewed by 3716

Abstract

In this paper we investigate some fixed-circle theorems using Ćirić’s technique (resp. Hardy-Rogers’ technique, Reich’s technique and Chatterjea’s technique) on a metric space. To do this, we define new types of

F_{c}

-contractions such as Ćirić type, Hardy-Rogers type, Reich type and [...] Read more.

In this paper we investigate some fixed-circle theorems using Ćirić’s technique (resp. Hardy-Rogers’ technique, Reich’s technique and Chatterjea’s technique) on a metric space. To do this, we define new types of

F_{c}

-contractions such as Ćirić type, Hardy-Rogers type, Reich type and Chatterjea type. Two illustrative examples are presented to show the effectiveness of our results. Also, it is given an application of a Ćirić type

F_{c}

-contraction to discontinuous self-mappings which have fixed circles. Full article

11 pages, 240 KiB

Open AccessArticle

Some Simultaneous Generalizations of Well-Known Fixed Point Theorems and Their Applications to Fixed Point Theory

by Wei-Shih Du, Erdal Karapınar and Zhenhua He

Mathematics 2018, 6(7), 117; https://doi.org/10.3390/math6070117 - 9 Jul 2018

Cited by 7 | Viewed by 3560

Abstract

In this paper, we first establish a new fixed point theorem that generalizes and unifies a number of well-known fixed point results, including the Banach contraction principle, Kannan’s fixed point theorem, Chatterjea fixed point theorem, Du-Rassias fixed point theorem and many others. The [...] Read more.

In this paper, we first establish a new fixed point theorem that generalizes and unifies a number of well-known fixed point results, including the Banach contraction principle, Kannan’s fixed point theorem, Chatterjea fixed point theorem, Du-Rassias fixed point theorem and many others. The presented results not only unify and generalize the existing results, but also yield several new fixed point theorems, which are different from the well-known results in the literature. Full article

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

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