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29 pages, 358 KiB

Open AccessArticle

Geraghty–Pata–Suzuki-Type Proximal Contractions and Related Coincidence Best Proximity Point Results

by Naeem Saleem, Maneesha Tur Raazzia, Nawab Hussain and Asim Asiri

Symmetry 2023, 15(8), 1572; https://doi.org/10.3390/sym15081572 - 11 Aug 2023

Cited by 6 | Viewed by 1018

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context [...] Read more.

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an

α - θ -

Geraghty–Pata–Suzuki-type proximal contraction; an

α - θ -

generalized Geraghty–Pata–Suzuki-type proximal contraction; and an

α - θ -

modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches. Full article

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

17 pages, 326 KiB

Open AccessArticle

Fixed Points of (α, β, F*) and (α, β, F**)-Weak Geraghty Contractions with an Application

by Khairul Habib Alam, Yumnam Rohen and Naeem Saleem

Symmetry 2023, 15(1), 243; https://doi.org/10.3390/sym15010243 - 16 Jan 2023

Cited by 19 | Viewed by 3457

Abstract

This study aims to provide some new classes of

(α, β, F^{*})

-weak Geraghty contraction and

(α, β, F^{* *})

-weak Geraghty contraction, which are self-generalized contractions on any metric space. Furthermore, we [...] Read more.

This study aims to provide some new classes of

(α, β, F^{*})

-weak Geraghty contraction and

(α, β, F^{* *})

-weak Geraghty contraction, which are self-generalized contractions on any metric space. Furthermore, we find that the mappings satisfying the definition of such contractions have a unique fixed point if the underlying space is complete. In addition, we provide an application showing the uniqueness of the solution of the two-point boundary value problem. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)

14 pages, 336 KiB

Open AccessArticle

Solution to Integral Equation in an O-Complete Branciari b-Metric Spaces

by Menaha Dhanraj, Arul Joseph Gnanaprakasam, Gunaseelan Mani, Ozgur Ege and Manuel De la Sen

Axioms 2022, 11(12), 728; https://doi.org/10.3390/axioms11120728 - 13 Dec 2022

Cited by 7 | Viewed by 1690

Abstract

In this paper, we prove fixed point theorem via orthogonal Geraghty type

α

-admissible contraction map in an orthogonal complete Branciari b-metric spaces context. An example is presented to strengthen our main result. We provided an application to find the existence and [...] Read more.

In this paper, we prove fixed point theorem via orthogonal Geraghty type

α

-admissible contraction map in an orthogonal complete Branciari b-metric spaces context. An example is presented to strengthen our main result. We provided an application to find the existence and uniqueness of a solution to the Volterra integral equation. We have compared the approximate solution and exact solution numerically. Full article

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

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24 pages, 453 KiB

Open AccessArticle

Survivability of AIDS Patients via Fractional Differential Equations with Fuzzy Rectangular and Fuzzy b-Rectangular Metric like Spaces

by Naeem Saleem, Salman Furqan, Hossam A. Nabwey and Reny George

Symmetry 2022, 14(11), 2450; https://doi.org/10.3390/sym14112450 - 18 Nov 2022

Cited by 3 | Viewed by 1749

Abstract

As it is not always true that the distance between the points in fuzzy rectangular metric spaces is one, so we introduce the notions of rectangular and b-rectangular metric-like spaces in fuzzy set theory that generalize many existing results, which can be [...] Read more.

As it is not always true that the distance between the points in fuzzy rectangular metric spaces is one, so we introduce the notions of rectangular and b-rectangular metric-like spaces in fuzzy set theory that generalize many existing results, which can be regarded as the main advantage of this paper. In these spaces, the symmetry property is preserved, but the self distance may not be equal to one. We discuss topological properties and demonstrate that neither of these spaces is Hausdorff. Using

α - ψ

-contraction and Geraghty contractions, respectively, in our main results, we establish fixed point results in these spaces. We present examples that justify our definitions and results. We use our main results to demonstrate that the solution of a nonlinear fractional differential equation for HIV is unique. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)

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22 pages, 1744 KiB

Open AccessArticle

Fixed Point of (α,β)-Admissible Generalized Geraghty F-Contraction with Application

by Min Wang, Naeem Saleem, Xiaolan Liu, Arslan Hojat Ansari and Mi Zhou

Symmetry 2022, 14(5), 1016; https://doi.org/10.3390/sym14051016 - 17 May 2022

Cited by 4 | Viewed by 2335

Abstract

In this paper, we introduce some new types of extended Geraghty contractions, called

(α, β)

-admissible generalized Geraghty F-contractions, and prove some fixed point results for such contractions in the setting of partial b-metric spaces. Moreover, based on [...] Read more.

In this paper, we introduce some new types of extended Geraghty contractions, called

(α, β)

-admissible generalized Geraghty F-contractions, and prove some fixed point results for such contractions in the setting of partial b-metric spaces. Moreover, based on the obtained fixed point results and the property of symmetry, we inaugurate a fixed point result for graphic generalized Geraghty F-contractions defined on partial metric spaces endowed with a directed graph. As an application, we examine the existence of a unique solution to the first-order periodic boundary value by the obtained fixed point result. Moreover, some examples are presented to illustrate the validity of the new results. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)

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18 pages, 311 KiB

Open AccessArticle

Existence and Well-Posedness of Tripled Fixed Points with Application to a System of Differential Equations

by Rashwan A. Rashwan, Hasanen A. Hammad, Ahmed Nafea and Fahd Jarad

Symmetry 2022, 14(4), 745; https://doi.org/10.3390/sym14040745 - 5 Apr 2022

Cited by 2 | Viewed by 1868

Abstract

The purpose of this manuscript is to demonstrate the existence and uniqueness of triple fixed-point results for Geraghty-type contractions in ordinary metric spaces with binary relations. Moreover, the well-posedness of the tripled fixed point problem is investigated. Consequently,

α

-dominated mapping on such [...] Read more.

The purpose of this manuscript is to demonstrate the existence and uniqueness of triple fixed-point results for Geraghty-type contractions in ordinary metric spaces with binary relations. Moreover, the well-posedness of the tripled fixed point problem is investigated. Consequently,

α

-dominated mapping on such space is discussed. Ultimately, to promote our paper, some illustrative examples are derived, and the existence of the solution to a system of differential equations is obtained as an application. Full article

(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points)

19 pages, 307 KiB

Open AccessArticle

Common Fixed Point Results for Almost ℛ_g-Geraghty Type Contraction Mappings in b₂-Metric Spaces with an Application to Integral Equations

by Samera M. Saleh, Salvatore Sessa, Waleed M. Alfaqih and Fawzia Shaddad

Axioms 2021, 10(2), 101; https://doi.org/10.3390/axioms10020101 - 24 May 2021

Cited by 4 | Viewed by 2437

Abstract

In this paper, we define almost

R_{g}

-Geraghty type contractions and utilize the same to establish some coincidence and common fixed point results in the setting of

b_{2}

-metric spaces endowed with binary relations. As consequences of our newly proved results, [...] Read more.

In this paper, we define almost

R_{g}

-Geraghty type contractions and utilize the same to establish some coincidence and common fixed point results in the setting of

b_{2}

-metric spaces endowed with binary relations. As consequences of our newly proved results, we deduce some coincidence and common fixed point results for almost g-

α

-

η

Geraghty type contraction mappings in

b_{2}

-metric spaces. In addition, we derive some coincidence and common fixed point results in partially ordered

b_{2}

-metric spaces. Moreover, to show the utility of our main results, we provide an example and an application to non-linear integral equations. Full article

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

17 pages, 789 KiB

Open AccessArticle

Common Fixed Point Theorems for Generalized Geraghty (α,ψ,ϕ)-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces

by Haitham Qawaqneh, Mohd Noorani, Wasfi Shatanawi and Habes Alsamir

Axioms 2018, 7(4), 74; https://doi.org/10.3390/axioms7040074 - 25 Oct 2018

Cited by 17 | Viewed by 4052

Abstract

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty

(α, ψ, ϕ)

-quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show [...] Read more.

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty

(α, ψ, ϕ)

-quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show the superiority of our results. The obtained results progress some well-known fixed (common fixed) point results in the literature. Our main results cannot be specifically attained from the corresponding metric space versions. This paper is scientifically novel because we take Geraghty contraction self-mapping in partially ordered metric-like spaces via

α -

admissible mapping. This opens the door to other possible fixed (common fixed) point results for non-self-mapping and in other generalizing metric spaces. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications II)

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