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22 pages, 298 KB

Open AccessArticle

Perturbed

F

-Metric Spaces: New Fixed Point Results with Applications to Caputo Fractional Initial Value Problems

by Maryam G. Alshehri and Jamshaid Ahmad

Axioms 2026, 15(8), 559; https://doi.org/10.3390/axioms15080559 - 28 Jul 2026

Viewed by 190

Abstract

The primary objective of this study is to establish some new fixed point results in the framework of perturbed

F

-metric spaces. Specifically, we develop fixed point theorems for three classes of contractions: Banach-type, rational-type, and interpolative contractions. To demonstrate the validity and [...] Read more.

The primary objective of this study is to establish some new fixed point results in the framework of perturbed

F

-metric spaces. Specifically, we develop fixed point theorems for three classes of contractions: Banach-type, rational-type, and interpolative contractions. To demonstrate the validity and applicability of the obtained results, some nontrivial examples are provided. Moreover, our findings yield various consequences that extend existing results in both

F

-metric spaces and perturbed metric spaces. As an application of the main result, we examine the existence of solutions for a Caputo fractional initial value problem. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 3rd Edition)

20 pages, 406 KB

Open AccessArticle

Fixed Point Results in Extended ℱ-Metric Spaces with Applications to Caputo Fractional Differential Equations

by Badriah Alamri

Fractal Fract. 2026, 10(4), 261; https://doi.org/10.3390/fractalfract10040261 - 15 Apr 2026

Cited by 1 | Viewed by 586

Abstract

The purpose of this research work is to propose and develop the notion of

(α, ψ)

-contractions in the setting of extended

F

-metric spaces and to establish corresponding fixed point results. Using these results, we derive fixed point results for graphic [...] Read more.

The purpose of this research work is to propose and develop the notion of

(α, ψ)

-contractions in the setting of extended

F

-metric spaces and to establish corresponding fixed point results. Using these results, we derive fixed point results for graphic contractions in extended

F

-metric spaces as well as for mappings in partially ordered extended

F

-metric spaces. To demonstrate the validity and novelty of the proposed results, a non-trivial example is provided. Moreover, the constructed framework serves as a tool to investigate the existence of solutions for Caputo fractional differential equations, thereby highlighting both its effectiveness and practical significance. Full article

(This article belongs to the Section Numerical and Computational Methods)

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28 pages, 583 KB

Open AccessArticle

Applications of Fixed Point Results to Fractional Differential and Nonlinear Mixed Volterra–Fredholm Integral Equations

by Maryam G. Alshehri and Jamshaid Ahmad

Fractal Fract. 2026, 10(4), 220; https://doi.org/10.3390/fractalfract10040220 - 26 Mar 2026

Cited by 1 | Viewed by 770

Abstract

This work aims to introduce the concept of graphic rational contractions in the framework of extended

F

-metric spaces and to establish fixed point theorems related to these mappings. In addition, we define and examine the class of interpolative Ćirić–Reich–Rus-type cyclic contractions in [...] Read more.

This work aims to introduce the concept of graphic rational contractions in the framework of extended

F

-metric spaces and to establish fixed point theorems related to these mappings. In addition, we define and examine the class of interpolative Ćirić–Reich–Rus-type cyclic contractions in the same setting, deriving several new fixed point results that broaden existing theories. To illustrate the validity and originality of the obtained results, appropriate examples are presented. Furthermore, the developed theoretical results are applied to study the existence of solutions for fractional differential equations and nonlinear mixed Volterra–Fredholm integral equations, highlighting their effectiveness and practical importance. Full article

(This article belongs to the Section General Mathematics, Analysis)

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23 pages, 347 KB

Open AccessArticle

On the Existence of Solutions for Atangana–Baleanu Type Fractional Integral Equations via Fixed Point Theorems in Extended

F

-Metric Spaces

by Amer Hassan Albargi and Jamshaid Ahmad

Fractal Fract. 2026, 10(2), 128; https://doi.org/10.3390/fractalfract10020128 - 17 Feb 2026

Cited by 2 | Viewed by 476

Abstract

The primary objective of this research article is to investigate the concept of extended

F

-metric spaces and to establish a series of fixed point theorems for generalized contractions within this framework. We further introduce and analyze the notion of interpolative Kannan-type cyclic [...] Read more.

The primary objective of this research article is to investigate the concept of extended

F

-metric spaces and to establish a series of fixed point theorems for generalized contractions within this framework. We further introduce and analyze the notion of interpolative Kannan-type cyclic contractions in extended

F

-metric spaces, deriving several novel fixed point results associated with these mappings. In addition, we obtain common fixed point theorems for rational contractions, thereby extending and unifying a variety of existing results available in the literature. To highlight the novelty and effectiveness of the proposed results, several illustrative examples are provided. Moreover, the theoretical findings are successfully applied to the solution of Atangana–Baleanu fractional integral equations as well as Volterra integral equation of Hammerstein type, demonstrating their practical significance and wide-ranging applicability. Full article

(This article belongs to the Section Numerical and Computational Methods)

21 pages, 471 KB

Open AccessArticle

Some Results on Multivalued Proximal Contractions with Application to Integral Equation

by Muhammad Zahid, Fahim Ud Din, Mudasir Younis, Haroon Ahmad and Mahpeyker Öztürk

Mathematics 2024, 12(22), 3488; https://doi.org/10.3390/math12223488 - 7 Nov 2024

Cited by 5 | Viewed by 1436

Abstract

In this manuscript, for the purpose of investigating the coincidence best proximity point, best proximity point, and fixed point results via alternating distance

ϕ

, we discuss some multivalued

{(ϕ - F_{τ})}_{C_{P}}

and [...] Read more.

In this manuscript, for the purpose of investigating the coincidence best proximity point, best proximity point, and fixed point results via alternating distance

ϕ

, we discuss some multivalued

{(ϕ - F_{τ})}_{C_{P}}

and

{(ϕ - F_{τ})}_{B_{P}} -

proximal contractions in the context of rectangular metric spaces. To ascertain the coincidence best proximity point, best proximity point, and the fixed point for single-valued mappings, we reduce these findings using

{(F_{τ})}_{C_{P}}

and

{(F_{τ})}_{B_{P}} -

proximal contractions. To make our work more understandable, examples of both single- and multivalued mappings are provided. These examples support our core findings, which rely on coincidence points, as well as the corollaries that address fixed point conclusions. In the final phase of our study, we use the obtained results to verify that a solution to a Fredholm integral equation exists. This application highlights the theoretical framework we built throughout our study. Full article

(This article belongs to the Special Issue Advanced Theories and Novel Methods for Nonlinear Analysis, Optimization and Applications)

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15 pages, 358 KB

Open AccessArticle

General New Results on

(ϕ, F) -

Contractions in

b -

Metric-like-Spaces

by Kastriot Zoto, Milanka Gardašević-Filipović, Ilir Vardhami, Zoran Mitrović and Stojan Radenović

Axioms 2023, 12(7), 672; https://doi.org/10.3390/axioms12070672 - 7 Jul 2023

Cited by 3 | Viewed by 1731

Abstract

Thispaper recognizes a general approach related to recent fixed point results about the classes of interpolative and hybrid contractions in metric space and general metric spaces. Considering auxiliary functions, so called Wardowski functions, and a rich set of implicit relations, we introduce types [...] Read more.

Thispaper recognizes a general approach related to recent fixed point results about the classes of interpolative and hybrid contractions in metric space and general metric spaces. Considering auxiliary functions, so called Wardowski functions, and a rich set of implicit relations, we introduce types of

(α_{v^{q}}, ϕ, F) -

contractions and

r -

order hybrid

(α_{v^{q}}, ϕ, F) -

contractions in the setting of

b -

metric-like spaces. They generate and simplify many forms of contractions widely used in the literature. The resulting theorems significantly extend, generalize, and unify an excellent work on fixed point theory. Full article

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

17 pages, 338 KB

Open AccessArticle

Existence Results for Wardoski-Type Convex Contractions and the Theory of Iterated Function Systems

by Naeem Saleem, Bilal Iqbal, Fady Hasan and Wasfi Shatanawi

Symmetry 2023, 15(6), 1162; https://doi.org/10.3390/sym15061162 - 27 May 2023

Cited by 6 | Viewed by 1477

Abstract

The purpose of this paper is to define the notion of extended convex ℱ contraction by imposing less conditions on the function ℱ satisfying certain contractive conditions. We prove the existence of fixed points for these types of mappings in the setting of [...] Read more.

The purpose of this paper is to define the notion of extended convex ℱ contraction by imposing less conditions on the function ℱ satisfying certain contractive conditions. We prove the existence of fixed points for these types of mappings in the setting of b-metric spaces. In addition, some illustrative examples are provided to show the usability of the obtained results. Lastly, we use the obtained fixed-point results to find the fractals with respect to the iterated function systems in the framework of b-metric spaces. Furthermore, the variables involved in the b-metric space are symmetric, and symmetry plays an important role in solving the nonlinear problems defined in operator theory. Full article

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

17 pages, 336 KB

Open AccessArticle

C*-Algebra-Valued Partial Modular Metric Spaces and Some Fixed Point Results

by Santanu Narzary, Dipankar Das, Yumnam Mahendra Singh, Mohammad Saeed Khan and Salvatore Sessa

Symmetry 2023, 15(6), 1135; https://doi.org/10.3390/sym15061135 - 23 May 2023

Cited by 5 | Viewed by 2626

Abstract

In the present paper, we introduce the notion of

C^{*}

-algebra-valued partial modular metric space satisfying the symmetry property that generalizes partial modular metric space,

C^{*}

-algebra-valued partial metric space, and

C^{*}

-algebra-valued modular metric space and discuss it with [...] Read more.

In the present paper, we introduce the notion of

C^{*}

-algebra-valued partial modular metric space satisfying the symmetry property that generalizes partial modular metric space,

C^{*}

-algebra-valued partial metric space, and

C^{*}

-algebra-valued modular metric space and discuss it with examples. Some fixed point results using

(ϕ, MF)

-contraction mapping are discussed in such space. In addition, we study the stability of obtained results in the spirit of Ulam and Hyers. As an application, we also provide the existence and uniqueness of the solution for a system of Fredholm integral equations. Full article

(This article belongs to the Section B: Mathematics)

20 pages, 338 KB

Open AccessArticle

Best Proximity Point for

Γ_{τ} F

-Fuzzy Proximal Contraction

by Uma Devi Patel, Vesna Todorcevic, Slobodan Radojevic and Stojan Radenović

Axioms 2023, 12(2), 165; https://doi.org/10.3390/axioms12020165 - 6 Feb 2023

Cited by 7 | Viewed by 1888

Abstract

In this writing, first, we disclose the first and second category of a

Γ_{τ} F

-fuzzy proximal contraction for a mapping

O : U \to V

which is nonself and also declare a fuzzy q-property to confirm the existence of the [...] Read more.

In this writing, first, we disclose the first and second category of a

Γ_{τ} F

-fuzzy proximal contraction for a mapping

O : U \to V

which is nonself and also declare a fuzzy q-property to confirm the existence of the best proximity point for nonself function

O

. Then, we discover a few results using the

Γ_{τ} F

-fuzzy proximal contraction of the first category for a continuous and discontinuous nonself function

O

in a non-Archimedean fuzzy metric space. Later, we discuss another result for the

Γ_{τ} F

-fuzzy proximal contraction of the second category as well. In between the fuzzy proximal theorems, many examples are presented in support of the definitions and theorems proved in this writing. Full article

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

21 pages, 357 KB

Open AccessArticle

Common Best Proximity Points and Completeness of ℱ−Metric Spaces

by Mi Zhou, Naeem Saleem, Basit Ali, Misha Mohsin and Antonio Francisco Roldán López de Hierro

Mathematics 2023, 11(2), 281; https://doi.org/10.3390/math11020281 - 5 Jan 2023

Cited by 5 | Viewed by 2547

Abstract

In this paper, we introduce three classes of proximal contractions that are called the proximally

λ - ψ -

dominated contractions, generalized

η_{β}^{γ} -

proximal contractions and Berinde-type weak proximal contractions, and obtain common best proximity points for these proximal contractions [...] Read more.

In this paper, we introduce three classes of proximal contractions that are called the proximally

λ - ψ -

dominated contractions, generalized

η_{β}^{γ} -

proximal contractions and Berinde-type weak proximal contractions, and obtain common best proximity points for these proximal contractions in the setting of

F -

metric spaces. Further, we obtain the best proximity point result for generalized

α - φ -

proximal contractions in

F -

metric spaces. As an application, fixed point and coincidence point results for these contractions are obtained. Some examples are provided to support the validity of our main results. Moreover, we obtain a completeness characterization of the

F -

metric spaces via best proximity points. Full article

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

16 pages, 300 KB

Open AccessArticle

New Contractive Mappings and Solutions to Boundary-Value Problems in Triple Controlled Metric Type Spaces

by Fatima M. Azmi

Symmetry 2022, 14(11), 2270; https://doi.org/10.3390/sym14112270 - 29 Oct 2022

Cited by 11 | Viewed by 2004

Abstract

In this study, we utilize the notion of triple controlled metric type space that preserves the symmetry property, which is a generalization of b-metric-type spaces, to prove new fixed-point results. We introduce (

α

-

F

)-contractive mappings and

Θ

-contractive mappings [...] Read more.

In this study, we utilize the notion of triple controlled metric type space that preserves the symmetry property, which is a generalization of b-metric-type spaces, to prove new fixed-point results. We introduce (

α

-

F

)-contractive mappings and

Θ

-contractive mappings on triple controlled metric type space settings. Then, we establish the existence and uniqueness of fixed-point results on complete triple controlled metric type space. Moreover, some examples and applications to boundary-value problems of the fourth-order differential equation are presented to display the usage of the obtained result. Full article

(This article belongs to the Special Issue Symmetry in Fractional Calculus, Fixed Point and Mathematical Control Theory)

15 pages, 290 KB

Open AccessArticle

New Fixed Point Theorems in Orthogonal F -Metric Spaces with Application to Fractional Differential Equation

by Tanzeela Kanwal, Azhar Hussain, Hamid Baghani and Manuel de la Sen

Symmetry 2020, 12(5), 832; https://doi.org/10.3390/sym12050832 - 19 May 2020

Cited by 28 | Viewed by 3511

Abstract

We present the notion of orthogonal

F

-metric spaces and prove some fixed and periodic point theorems for orthogonal

⊥_{Ω}

-contraction. We give a nontrivial example to prove the validity of our result. Finally, as application, we prove the existence and uniqueness [...] Read more.

We present the notion of orthogonal

F

-metric spaces and prove some fixed and periodic point theorems for orthogonal

⊥_{Ω}

-contraction. We give a nontrivial example to prove the validity of our result. Finally, as application, we prove the existence and uniqueness of the solution of a nonlinear fractional differential equation. Full article

(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)

14 pages, 289 KB

Open AccessArticle

Fixed Point Theorems for Generalized (αβ-ψ)-Contractions in F -Metric Spaces with Applications

by Saleh Abdullah Al-Mezel, Jamshaid Ahmad and Giuseppe Marino

Mathematics 2020, 8(4), 584; https://doi.org/10.3390/math8040584 - 14 Apr 2020

Cited by 22 | Viewed by 2707

Abstract

The purpose of this paper is to define generalized (

α β

-

ψ)

-contraction in the context of

F

-metric space and obtain some new fixed point results. As applications, we solve a nonlinear neutral differential equation with an unbounded delay [...] Read more.

The purpose of this paper is to define generalized (

α β

-

ψ)

-contraction in the context of

F

-metric space and obtain some new fixed point results. As applications, we solve a nonlinear neutral differential equation with an unbounded delay

ϑ^{/} (ι) = - ρ_{1} (ι) ϑ (ι) + ρ_{2} (ι) L (ϑ (ι - ς (ι))) + ρ_{3} (ι) ϑ^{/} (ι - ς (ι)),

where

ρ_{1} (ι)

,

ρ_{2} (ι)

are continuous,

ρ_{3} (ι)

is continuously differentiable and

ς (ι) > 0,

for all

ι \in R

and is twice continuously differentiable. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)

16 pages, 305 KB

Open AccessArticle

Fixed Point Results for Generalized ℱ-Contractions in Modular b-Metric Spaces with Applications

by Vahid Parvaneh, Nawab Hussain, Maryam Khorshidi, Nabil Mlaiki and Hassen Aydi

Mathematics 2019, 7(10), 887; https://doi.org/10.3390/math7100887 - 23 Sep 2019

Cited by 23 | Viewed by 4124

Abstract

The aim of this paper is to generalize the

F

-contractive condition in the framework of

α - ν

-complete modular b-metric spaces. Some results in ordered modular b-metric spaces are also presented. Moreover, an illustrative example and some related applications [...] Read more.

The aim of this paper is to generalize the

F

-contractive condition in the framework of

α - ν

-complete modular b-metric spaces. Some results in ordered modular b-metric spaces are also presented. Moreover, an illustrative example and some related applications are presented to support the obtained results. Full article

(This article belongs to the Special Issue Variational Inequality)

10 pages, 257 KB

Open AccessArticle

Reich, Jungck, and Berinde Common Fixed Point Results on ℱ-Metric Spaces and an Application

by Zoran D. Mitrović, Hassen Aydi, Nawab Hussain and Aiman Mukheimer

Mathematics 2019, 7(5), 387; https://doi.org/10.3390/math7050387 - 28 Apr 2019

Cited by 27 | Viewed by 3711

Abstract

Jleli and Samet (2018) introduced a new concept, named an

F

-metric space, as a generalization of the notion of a metric space. In this paper, we prove certain common fixed point theorems in

F

-metric spaces. As consequences of our results, we [...] Read more.

Jleli and Samet (2018) introduced a new concept, named an

F

-metric space, as a generalization of the notion of a metric space. In this paper, we prove certain common fixed point theorems in

F

-metric spaces. As consequences of our results, we obtain results of Banach, Jungck, Reich, and Berinde in these spaces. An application in dynamic programming is also given. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)

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