Research

19 pages, 334 KiB

Open AccessArticle

Convergence Criteria for Fixed Point Problems and Differential Equations

by Mircea Sofonea and Domingo A. Tarzia

Mathematics 2024, 12(3), 395; https://doi.org/10.3390/math12030395 - 25 Jan 2024

Viewed by 558

We consider a Cauchy problem for differential equations in a Hilbert space X. The problem is stated in a time interval I, which can be finite or infinite. We use a fixed point argument for history-dependent operators to prove the unique [...] Read more.

We consider a Cauchy problem for differential equations in a Hilbert space X. The problem is stated in a time interval I, which can be finite or infinite. We use a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we establish convergence criteria for both a general fixed point problem and the corresponding Cauchy problem. These criteria provide the necessary and sufficient conditions on a sequence

{u_{n}}

, which guarantee its convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result. Full article

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

0 pages, 278 KiB

Open AccessArticle

Fixed Point Results for Hybrid Rational Contractions under a New Compatible Condition with an Application

by Xiaolan Liu, Mi Zhou, Arslan Hojat Ansari, Naeem Saleem and Mukesh Kumar Jain

Mathematics 2024, 12(1), 121; https://doi.org/10.3390/math12010121 - 29 Dec 2023

Viewed by 523

Abstract

In this scholarly discourse, we present proof of the existence of unique fixed points in b-metric spaces for hybrid rational contractions. Moreover, we establish a common fixed point theorem for four self-mappings, assuming S-compatibility for two pairs of self-mappings within the [...] Read more.

In this scholarly discourse, we present proof of the existence of unique fixed points in b-metric spaces for hybrid rational contractions. Moreover, we establish a common fixed point theorem for four self-mappings, assuming S-compatibility for two pairs of self-mappings within the framework of b-metric spaces. As a practical demonstration of the aforementioned results, we apply them to a type of integral equation and derive a theorem that guarantees the existence of solutions. Full article

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

12 pages, 315 KiB

Open AccessArticle

Fixed-Point Approximation of Operators Satisfying (RCSC)—Condition in CAT(0) Spaces

by Naeem Saleem, Kifayat Ullah, Hossam A. Nabwey, Hazrat Bilal, Sharif Ullah and Reny George

Mathematics 2023, 11(22), 4658; https://doi.org/10.3390/math11224658 - 16 Nov 2023

Viewed by 617

Abstract

In this research article, we have proved strong and

Δ

-convergence results for mapping satisfying

(R C S C)

condition via M-iteration process in CAT(0) spaces. Numerical examples are provided to show the superiority of our results over other existing results [...] Read more.

In this research article, we have proved strong and

Δ

-convergence results for mapping satisfying

(R C S C)

condition via M-iteration process in CAT(0) spaces. Numerical examples are provided to show the superiority of our results over other existing results and to illustrate the faster convergence of the M iterative scheme as compared to many well-known iterative schemes. In this process, many results are improved in the current literature of CAT(0) spaces. Full article

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

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15 pages, 303 KiB

Open AccessArticle

Best Approximation of Fixed-Point Results for Branciari Contraction of Integral Type on Generalized Modular Metric Space

by Nesrin Manav Tatar and Ravi P. Agarwal

Mathematics 2023, 11(21), 4455; https://doi.org/10.3390/math11214455 - 27 Oct 2023

Viewed by 789

Abstract

In the realm of generalized modular metric spaces, we substantiate the validity of fixed-point theorems with Branciari contractions. This paper expands and broadens the original theorems in this context. Subsequently, by building upon this foundation, we explore various integral contractions to identify and [...] Read more.

In the realm of generalized modular metric spaces, we substantiate the validity of fixed-point theorems with Branciari contractions. This paper expands and broadens the original theorems in this context. Subsequently, by building upon this foundation, we explore various integral contractions to identify and characterize fixed points within this context. To highlight the practical implications of our work, we introduce the concept of the best proximity pair, thereby culminating in the best approximation theorem. We apply this theoretical construct to a specific example—one that is guided by the best approximation method described in prior research. Full article

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

18 pages, 321 KiB

Open AccessArticle

Some Common Fixed Point Results in Modular Ultrametric Space Using Various Contractions and Their Application to Well-Posedness

by Yahya Almalki, Balaanandhan Radhakrishnan, Uma Jayaraman and Kandhasamy Tamilvanan

Mathematics 2023, 11(19), 4077; https://doi.org/10.3390/math11194077 - 26 Sep 2023

Viewed by 795

Abstract

The aim of this study is to prove the existence and uniqueness of fixed point and common fixed point theorems for self-mappings in modular ultrametric spaces. These theorems are proved under varying contractive circumstances and without the property of spherical completeness. As a [...] Read more.

The aim of this study is to prove the existence and uniqueness of fixed point and common fixed point theorems for self-mappings in modular ultrametric spaces. These theorems are proved under varying contractive circumstances and without the property of spherical completeness. As a consequence, the examples of fixed point and common fixed point problems are correctly formulated. As an application, the well-posedness of a common fixed point problem is proved. This study expands on prior research in modular ultrametric space to provide a more comprehensive understanding of such spaces using generalized contraction. Full article

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

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20 pages, 458 KiB

Open AccessArticle

Novel Multistep Implicit Iterative Methods for Solving Common Solution Problems with Asymptotically Demicontractive Operators and Applications

by Hai-Yang Xu and Heng-You Lan

Mathematics 2023, 11(18), 3871; https://doi.org/10.3390/math11183871 - 11 Sep 2023

Viewed by 549

Abstract

It is very meaningful and challenging to efficiently seek common solutions to operator systems (CSOSs), which are widespread in pure and applied mathematics, as well as some closely related optimization problems. The purpose of this paper is to introduce a novel class of [...] Read more.

It is very meaningful and challenging to efficiently seek common solutions to operator systems (CSOSs), which are widespread in pure and applied mathematics, as well as some closely related optimization problems. The purpose of this paper is to introduce a novel class of multistep implicit iterative algorithms (MSIIAs) for solving general CSOSs. By using Xu’s lemma and Maingé’s fundamental and important results, we first obtain strong convergence theorems for both one-step and multistep implicit iterative schemes for CSOSs, involving asymptotically demicontractive operators. Finally, for the applications and profits of the main results presented in this paper, we give two numerical examples and present an iterative approximation to solve the general common solution to the variational inequalities and operator equations. Full article

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

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9 pages, 258 KiB

Open AccessArticle

Existence of Best Proximity Point in O-CompleteMetric Spaces

by G. Poonguzali, V. Pragadeeswarar and Manuel De la Sen

Mathematics 2023, 11(16), 3453; https://doi.org/10.3390/math11163453 - 09 Aug 2023

Viewed by 561

Abstract

In this work, we prove the existence of the best proximity point results for ⊥-contraction (orthogonal-contraction) mappings on an O-complete metric space (orthogonal-complete metric space). Subsequently, these existence results are employed to establish the common best proximity point result. Finally, we provide [...] Read more.

In this work, we prove the existence of the best proximity point results for ⊥-contraction (orthogonal-contraction) mappings on an O-complete metric space (orthogonal-complete metric space). Subsequently, these existence results are employed to establish the common best proximity point result. Finally, we provide suitable examples to demonstrate the validity of our results. Full article

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

20 pages, 324 KiB

Open AccessArticle

Common Fixed Point Theorems for Novel Admissible Contraction with Applications in Fractional and Ordinary Differential Equations

by Watchareepan Atiponrat, Pariwate Varnakovida, Pharunyou Chanthorn, Teeranush Suebcharoen and Phakdi Charoensawan

Mathematics 2023, 11(15), 3370; https://doi.org/10.3390/math11153370 - 01 Aug 2023

Viewed by 553

Abstract

In our work, we offer a novel idea of contractions, namely an

{(α, β, γ)}_{P} -

contraction, to prove the existence of a coincidence point and a common fixed point in complete metric spaces. This leads us to [...] Read more.

In our work, we offer a novel idea of contractions, namely an

{(α, β, γ)}_{P} -

contraction, to prove the existence of a coincidence point and a common fixed point in complete metric spaces. This leads us to an extension of previous results in the literature. Furthermore, we applied our acquired results to prove the existence of a solution for ordinary and fractional differential equations with integral-type boundary conditions. Full article

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

21 pages, 308 KiB

Open AccessArticle

Solving the Fredholm Integral Equation by Common Fixed Point Results in Bicomplex Valued Metric Spaces

by Afrah Ahmad Noman Abdou

Mathematics 2023, 11(14), 3249; https://doi.org/10.3390/math11143249 - 24 Jul 2023

Viewed by 782

Abstract

The purpose of this research work is to explore the solution of the Fredholm integral equation by common fixed point results in bicomplex valued metric spaces. In this way, we develop some common fixed point theorems for generalized contractions containing point-dependent control functions [...] Read more.

The purpose of this research work is to explore the solution of the Fredholm integral equation by common fixed point results in bicomplex valued metric spaces. In this way, we develop some common fixed point theorems for generalized contractions containing point-dependent control functions in the context of bicomplex valued metric spaces. An illustrative and practical example is also given to show the novelty of the most important result. Full article

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

15 pages, 318 KiB

Open AccessArticle

A Novel Inertial Viscosity Algorithm for Bilevel Optimization Problems Applied to Classification Problems

by Kobkoon Janngam, Suthep Suantai, Yeol Je Cho, Attapol Kaewkhao and Rattanakorn Wattanataweekul

Mathematics 2023, 11(14), 3241; https://doi.org/10.3390/math11143241 - 24 Jul 2023

Cited by 2 | Viewed by 772

Abstract

Fixed-point theory plays many important roles in real-world problems, such as image processing, classification problem, etc. This paper introduces and analyzes a new, accelerated common-fixed-point algorithm using the viscosity approximation method and then employs it to solve convex bilevel optimization problems. The proposed [...] Read more.

Fixed-point theory plays many important roles in real-world problems, such as image processing, classification problem, etc. This paper introduces and analyzes a new, accelerated common-fixed-point algorithm using the viscosity approximation method and then employs it to solve convex bilevel optimization problems. The proposed method was applied to data classification with the Diabetes, Heart Disease UCI and Iris datasets. According to the data classification experiment results, the proposed algorithm outperformed the others in the literature. Full article

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

15 pages, 297 KiB

Open AccessArticle

A New Extension of C_J Metric Spaces—Partially Controlled J Metric Spaces

by Suhad Subhi Aiadi, Wan Ainun Mior Othman, Kok Bin Wong and Nabil Mlaiki

Mathematics 2023, 11(13), 2973; https://doi.org/10.3390/math11132973 - 03 Jul 2023

Cited by 1 | Viewed by 611

Abstract

This article introduces the concept of partially controlled J metric spaces; in particular, the J metric space with self-distance is not necessarily zero, which is important in computer science. We prove the existence of a unique fixed point for linear and nonlinear contractions, [...] Read more.

This article introduces the concept of partially controlled J metric spaces; in particular, the J metric space with self-distance is not necessarily zero, which is important in computer science. We prove the existence of a unique fixed point for linear and nonlinear contractions, provide some examples to prove the existence of this metric space, and present some important applications in fractional differential equations, i.e., “Riemann–Liouville derivatives”. Full article

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

11 pages, 277 KiB

Open AccessArticle

Generalized Equilibrium Problems

by Mircea Balaj and Dan Florin Serac

Mathematics 2023, 11(9), 2146; https://doi.org/10.3390/math11092146 - 03 May 2023

Viewed by 921

Abstract

If X is a convex subset of a topological vector space and f is a real bifunction defined on

X \times X

, the problem of finding a point

x_{0} \in X

such that [...] Read more.

If X is a convex subset of a topological vector space and f is a real bifunction defined on

X \times X

, the problem of finding a point

x_{0} \in X

such that

f (x_{0}, y) \geq 0

for all

y \in X

, is called an equilibrium problem. When the bifunction f is defined on the cartesian product of two distinct sets X and Y we will call it a generalized equilibrium problem. In this paper, we study the existence of the solutions, first for generalized equilibrium problems and then for equilibrium problems. In the obtained results, apart from the bifunction f, another bifunction is introduced, the two being linked by a certain compatibility condition. The particularity of the equilibrium theorems established in the last section consists of the fact that the classical equilibrium condition (

f (x, x) = 0

, for all

x \in X

) is missing. The given applications refer to the Minty variational inequality problem and quasi-equilibrium problems. Full article

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

12 pages, 271 KiB

Open AccessArticle

Double-Composed Metric Spaces

by Irshad Ayoob, Ng Zhen Chuan and Nabil Mlaiki

Mathematics 2023, 11(8), 1866; https://doi.org/10.3390/math11081866 - 14 Apr 2023

Viewed by 970

Abstract

The double-controlled metric-type space

(X, D)

is a metric space in which the triangle inequality has the form [...] Read more.

The double-controlled metric-type space

(X, D)

is a metric space in which the triangle inequality has the form

D (η, μ) \leq ζ_{1} (η, θ) D (η, θ) + ζ_{2} (θ, μ) D (θ, μ)

for all

η, θ, μ \in X

. The maps

ζ_{1}, ζ_{2} : X \times X \to [1, \infty)

are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form

D (η, μ) \leq α (D (η, θ)) + β (D (θ, μ))

for all

η, θ, μ \in X

. In our new space, the control functions

α, β : [0, \infty) \to [0, \infty)

are composed of the metric

D

in the triangle inequality, where the control functions

ζ_{1}, ζ_{2} : X \times X \to [1, \infty)

in a double-controlled metric-type space are multiplied with the metric

D

. We establish some fixed-point theorems along with the examples and applications. Full article

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

10 pages, 257 KiB

Open AccessFeature PaperArticle

Stability Estimates for an Arithmetic Functional Equation with Brzdȩk Fixed Point Approaches

by Heejeong Koh

Mathematics 2023, 11(7), 1611; https://doi.org/10.3390/math11071611 - 27 Mar 2023

Viewed by 822

Abstract

We introduce an arithmetic functional equation

f (x^{2} + y^{2}) = f (x^{2}) + f (y^{2})

and then investigate stability estimates of the functional equation by using the Brzdȩk fixed point theorem on [...] Read more.

We introduce an arithmetic functional equation

f (x^{2} + y^{2}) = f (x^{2}) + f (y^{2})

and then investigate stability estimates of the functional equation by using the Brzdȩk fixed point theorem on a non-Archimedean fuzzy metric space and a non-Archimedean fuzzy normed space. To apply the Brzdȩk fixed point theorem, the proof uses the linear relationship between two variables, x and y. Full article

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

11 pages, 285 KiB

Open AccessArticle

Relational Contractions Involving (c)-Comparison Functions with Applications to Boundary Value Problems

by Ebrahem Ateatullah Algehyne, Musaad Sabih Aldhabani and Faizan Ahmad Khan

Mathematics 2023, 11(6), 1277; https://doi.org/10.3390/math11061277 - 07 Mar 2023

Cited by 5 | Viewed by 922

Abstract

After the introduction of Alam–Imdad’s relation-theoretic contraction principle, the field of metric fixed point theory has attracted much attention. A number of fixed point theorems in the context of relational metric space employing various types of contractions has been appeared during the last [...] Read more.

After the introduction of Alam–Imdad’s relation-theoretic contraction principle, the field of metric fixed point theory has attracted much attention. A number of fixed point theorems in the context of relational metric space employing various types of contractions has been appeared during the last seven years. In this manuscript, one proved a metrical fixed point theorem for

ϕ

-contraction involving (c)-comparison functions employing an amorphous relation. The result proved in this paper refines, modifies, unifies and sharpens several existing fixed point results. We also constructed an example in order to attest the credibility of our results. Finally, we applied our result to establish the existence and uniqueness of solution of certain periodic boundary value problem. Full article

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

29 pages, 418 KiB

Open AccessArticle

Graph Convergence, Algorithms, and Approximation of Common Solutions of a System of Generalized Variational Inclusions and Fixed-Point Problems

by Javad Balooee, Shih-Sen Chang, Lin Wang, Yu Zhang and Zhao-Li Ma

Mathematics 2023, 11(4), 832; https://doi.org/10.3390/math11040832 - 06 Feb 2023

Viewed by 1044

Abstract

In this paper, under some new appropriate conditions imposed on the parameters and mappings involved in the proximal mapping associated with a general H-monotone operator, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. The main contribution [...] Read more.

In this paper, under some new appropriate conditions imposed on the parameters and mappings involved in the proximal mapping associated with a general H-monotone operator, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. The main contribution of this work is the establishment of a new equivalence relationship between the graph convergence of a sequence of general strongly H-monotone mappings and their associated proximal mappings, respectively, to a given general strongly H-monotone mapping and its associated proximal mapping by using the notions of graph convergence and proximal mapping concerning a general strongly H-monotone mapping. By employing the concept of proximal mapping relating to general strongly H-monotone mapping, some iterative algorithms are proposed, and as an application of the obtained equivalence relationship mentioned above, a convergence theorem for approximating a common element of the set of solutions of a system of generalized variational inclusions involving general strongly H-monotone mappings and the set of fixed points of an

({a_{n}}, {b_{n}}, ϕ)

-total uniformly L-Lipschitzian mapping is proved. It is significant to emphasize that our results are new and improve and generalize many known corresponding results. Full article

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

21 pages, 357 KiB

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 - 05 Jan 2023

Cited by 3 | Viewed by 1103

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)

13 pages, 313 KiB

Open AccessArticle

On a Class of Multistage Stochastic Hierarchical Problems

by Domenico Scopelliti

Mathematics 2022, 10(21), 4044; https://doi.org/10.3390/math10214044 - 31 Oct 2022

Cited by 1 | Viewed by 839

Abstract

In this paper, following the multistage stochastic approach proposed by Rockafellar and Wets, we analyze a class of multistage stochastic hierarchical problems: the Multistage Stochastic Optimization Problem with Quasi-Variational Inequality Constraints. Such a problem is defined in a suitable functional setting relative to [...] Read more.

In this paper, following the multistage stochastic approach proposed by Rockafellar and Wets, we analyze a class of multistage stochastic hierarchical problems: the Multistage Stochastic Optimization Problem with Quasi-Variational Inequality Constraints. Such a problem is defined in a suitable functional setting relative to a finite set of possible scenarios and certain information fields. The key of this multistage stochastic hierarchical problem turns out to be the nonanticipativity: some constraints have to be included in the formulation to take into account the partial information progressively revealed. In this way, we are able to study real-world problems in which the hierarchical decision processes are characterized by sequential decisions in response to an increasing level of information. As an application of this class of multistage stochastic hierarchical problems, we focus on the study of a suitable Single-Leader-Multi-Follower game. Full article

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

13 pages, 2167 KiB

Open AccessArticle

Bayesian Aerosol Retrieval-Based PM_2.5 Estimation through Hierarchical Gaussian Process Models

by Junbo Zhang, Daoji Li, Yingzhi Xia and Qifeng Liao

Mathematics 2022, 10(16), 2878; https://doi.org/10.3390/math10162878 - 11 Aug 2022

Cited by 2 | Viewed by 1201

Abstract

Satellite-based aerosol optical depth (AOD) data are widely used to estimate land surface

{PM}_{2.5}

concentrations in areas not covered by ground

{PM}_{2.5}

monitoring stations. However, AOD data obtained from satellites are typically at coarse spatial resolutions, limiting their applications on small [...] Read more.

Satellite-based aerosol optical depth (AOD) data are widely used to estimate land surface

{PM}_{2.5}

concentrations in areas not covered by ground

{PM}_{2.5}

monitoring stations. However, AOD data obtained from satellites are typically at coarse spatial resolutions, limiting their applications on small or medium scales. In this paper, we propose a new two-step approach to estimate 1-km-resolution

{PM}_{2.5}

concentrations in Shanghai using high spatial resolution AOD retrievals from MODIS. In the first step, AOD data are refined to a

1 \times 1 {km}^{2}

resolution via a Bayesian AOD retrieval method. In the second step, a hierarchical Gaussian process model is used to estimate

{PM}_{2.5}

concentrations. We evaluate our approach by model fitting and out-of-sample cross-validation. Our results show that the proposed approach enjoys accurate predictive performance in estimating

{PM}_{2.5}

concentrations. Full article

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

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