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16 pages, 301 KB

Open AccessArticle

On Higher-Order Nonlinear Fractional Elastic Equations with Dependence on Lower Order Derivatives in Nonlinearity

by Yujun Cui, Chunyu Liang and Yumei Zou

Fractal Fract. 2024, 8(7), 398; https://doi.org/10.3390/fractalfract8070398 - 2 Jul 2024

Viewed by 921

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given [...] Read more.

The paper studied high-order nonlinear fractional elastic equations that depend on low-order derivatives in nonlinearity and established the existence and uniqueness results by using the Leray–Schauder alternative theorem and Perov’s fixed point theorem on an appropriate space under mild assumptions. Examples are given to illustrate the key results. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

24 pages, 365 KB

Open AccessArticle

Solving Fractional Random Differential Equations by Using Fixed Point Methodologies under Mild Boundary Conditions

by Hasanen A. Hammad and Saleh Fahad Aljurbua

Fractal Fract. 2024, 8(7), 384; https://doi.org/10.3390/fractalfract8070384 - 28 Jun 2024

Cited by 5 | Viewed by 1311

Abstract

This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we [...] Read more.

This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we used the fixed-point method to examine the existence of a solution to the proposed system, mainly focusing on the theorems of Leray, Schauder, and Perov in generalized metric spaces. Finally, an example has been provided to support and underscore our results. Full article

(This article belongs to the Special Issue Metric Spaces with Its Application to Fractional Differential Equations)

15 pages, 327 KB

Open AccessArticle

Single and Multi-Valued Ordered-Theoretic Perov Fixed-Point Results for θ-Contraction with Application to Nonlinear System of Matrix Equations

by Fahim Ud Din, Salha Alshaikey, Umar Ishtiaq, Muhammad Din and Salvatore Sessa

Mathematics 2024, 12(9), 1302; https://doi.org/10.3390/math12091302 - 25 Apr 2024

Cited by 9 | Viewed by 1050

Abstract

This paper combines the concept of an arbitrary binary connection with the widely recognized principle of

θ

-contraction to investigate the innovative features of vector-valued metric spaces. This methodology demonstrates the existence of fixed points for both single- and multi-valued mappings within complete [...] Read more.

This paper combines the concept of an arbitrary binary connection with the widely recognized principle of

θ

-contraction to investigate the innovative features of vector-valued metric spaces. This methodology demonstrates the existence of fixed points for both single- and multi-valued mappings within complete vector-valued metric spaces. Through the utilization of binary relations and

θ

-contraction, this study advances and refines the Perov-type fixed-point results in the literature. Furthermore, this article furnishes examples to substantiate the validity of the presented results. Additionally, we establish an application for finding the existence of solutions to a system of matrix equations. Full article

14 pages, 289 KB

Open AccessArticle

Czerwik Vector-Valued Metric Space with an Equivalence Relation and Extended Forms of Perov Fixed-Point Theorem

by Monairah Alansari, Yahya Almalki and Muhammad Usman Ali

Mathematics 2023, 11(16), 3583; https://doi.org/10.3390/math11163583 - 18 Aug 2023

Cited by 2 | Viewed by 1433

Abstract

In this article, we shall generalize the idea of vector-valued metric space and Perov fixed-point theorem. We shall introduce the notion of Czerwik vector-valued

R

-metric space by involving an equivalence relation. A few basic concepts and properties related to Czerwik vector-valued

R

[...] Read more.

In this article, we shall generalize the idea of vector-valued metric space and Perov fixed-point theorem. We shall introduce the notion of Czerwik vector-valued

R

-metric space by involving an equivalence relation. A few basic concepts and properties related to Czerwik vector-valued

R

-metric space shall also be discussed that are required to obtain a few extended types of Perov fixed-point theorem. Full article

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

13 pages, 274 KB

Open AccessArticle

Banach Fixed Point Theorems in Generalized Metric Space Endowed with the Hadamard Product

by Saleh Omran, Ibtisam Masmali and Ghaliah Alhamzi

Symmetry 2023, 15(7), 1325; https://doi.org/10.3390/sym15071325 - 28 Jun 2023

Cited by 5 | Viewed by 2528

Abstract

In this paper, we prove some Banach fixed point theorems in generalized metric space where the contractive conditions are endowed with the Hadamard product of real symmetric positive definite matrices. Since the condition that a matrix A converges to zero is not needed, [...] Read more.

In this paper, we prove some Banach fixed point theorems in generalized metric space where the contractive conditions are endowed with the Hadamard product of real symmetric positive definite matrices. Since the condition that a matrix A converges to zero is not needed, this produces stronger results than those of Perov. As an application of our results, we study the existence and uniqueness of the solution for a system of matrix equations. Full article

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

9 pages, 277 KB

Open AccessArticle

Remarks on “Perov Fixed-Point Results on F-Contraction Mappings Equipped with Binary Relation”

by Slobodanka Mitrović, Nicola Fabiano, Slobodan Radojević and Stojan Radenović

Axioms 2023, 12(6), 518; https://doi.org/10.3390/axioms12060518 - 25 May 2023

Cited by 1 | Viewed by 1031

Abstract

Since 1964, when I.A. Perov introduced the so-called generalized metric space where

d (x, y)

is an element of the vector space

R^{m}

, many researchers have considered various contractive conditions in this type of space. In this paper, [...] Read more.

Since 1964, when I.A. Perov introduced the so-called generalized metric space where

d (x, y)

is an element of the vector space

R^{m}

, many researchers have considered various contractive conditions in this type of space. In this paper, we generalize, extend and unify some of those established results. We are primarily concerned with examining the existence of a fixed point of some mapping from X to itself, but if

(x, y)

belongs to some relation R on the set X, then the binary relation R and some F contraction defined on the space cone

R^{m}

are combined. We start our consideration with the recently announced results and give them strict, critical remarks. In addition, we improve several announced results by weakening some of the given conditions. Full article

17 pages, 358 KB

Open AccessArticle

A Mathematical Theoretical Study of a Coupled Fully Hybrid (k, Φ)-Fractional Order System of BVPs in Generalized Banach Spaces

by Abdellatif Boutiara, Sina Etemad, Sabri T. M. Thabet, Sotiris K. Ntouyas, Shahram Rezapour and Jessada Tariboon

Symmetry 2023, 15(5), 1041; https://doi.org/10.3390/sym15051041 - 8 May 2023

Cited by 25 | Viewed by 2310

Abstract

In this paper, we study a coupled fully hybrid system of

(k, Φ)

–Hilfer fractional differential equations equipped with non-symmetric

(k, Φ)

–Riemann-Liouville (

RL

) integral conditions. To prove the existence and uniqueness results, we use [...] Read more.

In this paper, we study a coupled fully hybrid system of

(k, Φ)

–Hilfer fractional differential equations equipped with non-symmetric

(k, Φ)

–Riemann-Liouville (

RL

) integral conditions. To prove the existence and uniqueness results, we use the Krasnoselskii and Perov fixed-point theorems with Lipschitzian matrix in the context of a generalized Banach space (

GBS

). Moreover, the Ulam–Hyers (

UH

) stability of the solutions is discussed by using the Urs’s method. Finally, an illustrated example is given to confirm the validity of our results. Full article

(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)

18 pages, 320 KB

Open AccessArticle

Perov Fixed-Point Results on F-Contraction Mappings Equipped with Binary Relation

by Fahim Ud Din, Muhammad Din, Umar Ishtiaq and Salvatore Sessa

Mathematics 2023, 11(1), 238; https://doi.org/10.3390/math11010238 - 3 Jan 2023

Cited by 9 | Viewed by 2076

Abstract

The purpose of this article is to discuss some new aspects of the vector-valued metric space. The idea of an arbitrary binary relation along with the well-known F contraction is used to demonstrate the existence of fixed points in the context of a [...] Read more.

The purpose of this article is to discuss some new aspects of the vector-valued metric space. The idea of an arbitrary binary relation along with the well-known F contraction is used to demonstrate the existence of fixed points in the context of a complete vector-valued metric space for both single- and multi-valued mappings. Utilizing the idea of binary relation, and with the help of F contraction, this work extends and complements some of the very recently established Perov-type fixed-point results in the literature. Furthermore, this work includes examples to justify the validity of the given results. During the discussion, it was found that some of the renowned metrical results proven by several authors using different binary relations, such as partial order, pre-order, transitive relation, tolerance, strict order and symmetric closure, can be weakened by using an arbitrary binary relation. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

9 pages, 275 KB

Open AccessArticle

New Applications of Perov’s Fixed Point Theorem

by Sorin Mureşan, Loredana Florentina Iambor and Omar Bazighifan

Mathematics 2022, 10(23), 4597; https://doi.org/10.3390/math10234597 - 4 Dec 2022

Cited by 4 | Viewed by 1874

Abstract

The goal of this paper is to consider a differential equation system written as an interesting equivalent form that has not been used before. Using Perov’s fixed point theorem in generalized metric spaces, the existence and uniqueness of the solution are obtained for [...] Read more.

The goal of this paper is to consider a differential equation system written as an interesting equivalent form that has not been used before. Using Perov’s fixed point theorem in generalized metric spaces, the existence and uniqueness of the solution are obtained for the proposed system. The approximation of the solution is given, and as a novelty, the approximation of its derivative is also obtained using the same iteration steps. Full article

(This article belongs to the Special Issue New Advances in Mathematical Analysis and Functional Analysis)

11 pages, 309 KB

Open AccessArticle

Fixed Point Results for Perov–Ćirić–Prešić-Type Θ-Contractions with Applications

by Jamshaid Ahmad, Saleh Abdullah Al-Mezel and Ravi P. Agarwal

Mathematics 2022, 10(12), 2062; https://doi.org/10.3390/math10122062 - 15 Jun 2022

Cited by 3 | Viewed by 1826

Abstract

The aim of this paper is to introduce the notion of Perov–Ćirić–Prešić-type

Θ

-contractions and to obtain some generalized fixed point theorems in the setting of vector-valued metric spaces. We derive some fixed point results as consequences of our main results. A nontrivial [...] Read more.

The aim of this paper is to introduce the notion of Perov–Ćirić–Prešić-type

Θ

-contractions and to obtain some generalized fixed point theorems in the setting of vector-valued metric spaces. We derive some fixed point results as consequences of our main results. A nontrivial example is also provided to support the validity of our established results. As an application, we investigate the solution of a semilinear operator system in Banach space. Full article

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

15 pages, 316 KB

Open AccessArticle

A New Class of Coupled Systems of Nonlinear Hyperbolic Partial Fractional Differential Equations in Generalized Banach Spaces Involving the ψ–Caputo Fractional Derivative

by Zidane Baitiche, Choukri Derbazi, Mouffak Benchohra and Yong Zhou

Symmetry 2021, 13(12), 2412; https://doi.org/10.3390/sym13122412 - 13 Dec 2021

Cited by 9 | Viewed by 2765

Abstract

The current study is devoted to investigating the existence and uniqueness of solutions for a new class of symmetrically coupled system of nonlinear hyperbolic partial-fractional differential equations in generalized Banach spaces in the sense of

ψ

–Caputo partial fractional derivative. Our approach is [...] Read more.

The current study is devoted to investigating the existence and uniqueness of solutions for a new class of symmetrically coupled system of nonlinear hyperbolic partial-fractional differential equations in generalized Banach spaces in the sense of

ψ

–Caputo partial fractional derivative. Our approach is based on the Krasnoselskii-type fixed point theorem in generalized Banach spaces and Perov’s fixed point theorem together with the Bielecki norm, while Urs’s approach was used to prove the Ulam–Hyers stability of solutions of our system. Finally, some examples are provided in order to illustrate our theoretical results. Full article

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

11 pages, 313 KB

Open AccessArticle

Existence and Uniqueness Results of Coupled Fractional-Order Differential Systems Involving Riemann–Liouville Derivative in the Space

W_{a^{+}}^{γ_{1}, 1} (a, b) \times W_{a^{+}}^{γ_{2}, 1} (a, b)

with Perov’s Fixed Point Theorem

by Noura Laksaci, Ahmed Boudaoui, Kamaleldin Abodayeh, Wasfi Shatanawi and Taqi A. M. Shatnawi

Fractal Fract. 2021, 5(4), 217; https://doi.org/10.3390/fractalfract5040217 - 13 Nov 2021

Cited by 4 | Viewed by 2050

Abstract

This paper is devoted to studying the existence and uniqueness of a system of coupled fractional differential equations involving a Riemann–Liouville derivative in the Cartesian product of fractional Sobolev spaces [...] Read more.

This paper is devoted to studying the existence and uniqueness of a system of coupled fractional differential equations involving a Riemann–Liouville derivative in the Cartesian product of fractional Sobolev spaces

E = W_{a^{+}}^{γ_{1}, 1} (a, b) \times W_{a^{+}}^{γ_{2}, 1} (a, b)

. Our strategy is to endow the space

E

with a vector-valued norm and apply the Perov fixed point theorem. An example is given to show the usefulness of our main results. Full article

(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)

12 pages, 308 KB

Open AccessArticle

Some New Observations for F-Contractions in Vector-Valued Metric Spaces of Perov’s Type

by Nikola Mirkov, Stojan Radenović and Slobodan Radojević

Axioms 2021, 10(2), 127; https://doi.org/10.3390/axioms10020127 - 21 Jun 2021

Cited by 8 | Viewed by 2325

Abstract

The main purpose of this article is to improve, generalize and complement some recently established results for Perov’s type F-contractions. In our approach, we use only the property (F1) of Wardowski while other authors employed all three conditions. Working only with the fact [...] Read more.

The main purpose of this article is to improve, generalize and complement some recently established results for Perov’s type F-contractions. In our approach, we use only the property (F1) of Wardowski while other authors employed all three conditions. Working only with the fact that the function F is strictly increasing on

{(0, + \infty)}^{m},

we obtain as a consequence new families of contractive conditions in the realm of vector-valued metric spaces of Perov’s type. At the end of the article, we present an example that supports obtained theoretical results and genuinely generalizes several known results in existing literature. Full article

14 pages, 286 KB

Open AccessArticle

A Perov Version of Fuzzy Metric Spaces and Common Fixed Points for Compatible Mappings

by Juan Martínez-Moreno and Dhananjay Gopal

Mathematics 2021, 9(11), 1290; https://doi.org/10.3390/math9111290 - 4 Jun 2021

Cited by 1 | Viewed by 2480

Abstract

In this paper, we define and study the Perov fuzzy metric space and the topology induced by this space. We prove Banach contraction theorems. Moreover, we devised new results for Kramosil and Michálek fuzzy metric spaces. In the process, some results about multidimensional [...] Read more.

In this paper, we define and study the Perov fuzzy metric space and the topology induced by this space. We prove Banach contraction theorems. Moreover, we devised new results for Kramosil and Michálek fuzzy metric spaces. In the process, some results about multidimensional common fixed points as coupled/tripled common fixed point results are derived from our main results. Full article

(This article belongs to the Special Issue Abstract Metric Spaces: Usefulness in Statistics and Fixed Point Theory)

13 pages, 290 KB

Open AccessArticle

Fixed Point Problems on Generalized Metric Spaces in Perov’s Sense

by Liliana Guran, Monica-Felicia Bota and Asim Naseem

Symmetry 2020, 12(5), 856; https://doi.org/10.3390/sym12050856 - 22 May 2020

Cited by 4 | Viewed by 2605

Abstract

The aim of this paper is to give some fixed point results in generalized metric spaces in Perov’s sense. The generalized metric considered here is the w-distance with a symmetry condition. The operators satisfy a contractive weakly condition of Hardy–Rogers type. The [...] Read more.

The aim of this paper is to give some fixed point results in generalized metric spaces in Perov’s sense. The generalized metric considered here is the w-distance with a symmetry condition. The operators satisfy a contractive weakly condition of Hardy–Rogers type. The second part of the paper is devoted to the study of the data dependence, the well-posedness, and the Ulam–Hyers stability of the fixed point problem. An example is also given to sustain the presented results. Full article

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