Research

20 pages, 375 KiB

Open AccessArticle

Some Fixed Point Theorems for α-Admissible Mappings in Complex-Valued Fuzzy Metric Spaces

by Satish Shukla, Shweta Rai and Rahul Shukla

Symmetry 2023, 15(9), 1797; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15091797 - 20 Sep 2023

Cited by 3 | Viewed by 1021

This paper discusses some properties of complex-valued fuzzy metric spaces and introduces the

α

-admissible mappings in the setting of complex-valued fuzzy metric spaces. We establish fixed point theorems for mappings satisfying symmetric contractive conditions with control functions. The results of this paper [...] Read more.

This paper discusses some properties of complex-valued fuzzy metric spaces and introduces the

α

-admissible mappings in the setting of complex-valued fuzzy metric spaces. We establish fixed point theorems for mappings satisfying symmetric contractive conditions with control functions. The results of this paper generalize, extend, and improve several results from metric, fuzzy metric, and complex-valued fuzzy metric spaces. Several examples are presented that verify and illustrate the new concepts, claims, and results. Full article

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

12 pages, 340 KiB

Open AccessArticle

Fixed Point Results via Orthogonal (α − 𝔶 − 𝔾)-Contraction in Orthogonal Complete Metric Space

by Xiaolan Liu, Gunasekaran Nallaselli, Absar Ul Haq, Arul Joseph Gnanaprakasam and Imran Abbas Baloch

Symmetry 2023, 15(9), 1762; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15091762 - 14 Sep 2023

Viewed by 558

Abstract

In this publication, we establish a suitable symmetry structure for orthogonal

(α - y - G)

-contractive mappings and prove fixed point results for an orthogonal

(α - y - G)

-contractive via orthogonal metric spaces. We give an [...] Read more.

In this publication, we establish a suitable symmetry structure for orthogonal

(α - y - G)

-contractive mappings and prove fixed point results for an orthogonal

(α - y - G)

-contractive via orthogonal metric spaces. We give an application to strengthen our main results from the existing literature to prove the existence of a unique analytical solution to the differential equation by converting it into an integral equation through fixed point analysis. Full article

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

17 pages, 344 KiB

Open AccessArticle

On New Decomposition Theorems for Mixed-Norm Besov Spaces with Ingredient Modulus of Smoothness

by Junjian Zhao, Marko Kostić and Wei-Shih Du

Symmetry 2023, 15(3), 642; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15030642 - 03 Mar 2023

Cited by 1 | Viewed by 763

Abstract

In this paper, we introduce and study the concept of the ingredient modulus of smoothness in component form in

L_{\vec{p}} (R^{d})

and a kind of mixed-norm Sobolev space. We obtain some new properties, inequalities, and auxiliary results in [...] Read more.

In this paper, we introduce and study the concept of the ingredient modulus of smoothness in component form in

L_{\vec{p}} (R^{d})

and a kind of mixed-norm Sobolev space. We obtain some new properties, inequalities, and auxiliary results in mixed-norm spaces

L_{\vec{p}} (R^{d})

. In addition, a new concept of mixed-norm Besov space is presented and a new decomposition theorem for mixed-norm Besov spaces is established. Full article

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

12 pages, 287 KiB

Open AccessArticle

Solving Integral Equations via Hybrid Interpolative ℛℐ-Type Contractions in 𝔟-Metric Spaces

by Ahmad Aloqaily, Dur-e-Shehwar Sagheer, Isma Urooj, Samina Batul and Nabil Mlaiki

Symmetry 2023, 15(2), 465; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15020465 - 09 Feb 2023

Cited by 5 | Viewed by 1116

Abstract

Karapinar et al. established a more general class of contractions, namely, hybrid interpolative

R

iech

I

strǎstescue-type contractions, and presented some results on the platform of metric spaces. This research uses the domain of

b

-metric spaces to modify this class proficiently. Several [...] Read more.

Karapinar et al. established a more general class of contractions, namely, hybrid interpolative

R

iech

I

strǎstescue-type contractions, and presented some results on the platform of metric spaces. This research uses the domain of

b

-metric spaces to modify this class proficiently. Several interesting fixed-point results are presented by using this new class defined on

b

-metric space, where the symmetric condition is preserved in this study. Examples are provided for the authentication of proved results. Eventually, an application is also provided in order to comprehend our extensive effort in a better way. Full article

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

14 pages, 19233 KiB

Open AccessArticle

An Algorithm for Solving Common Points of Convex Minimization Problems with Applications

by Adisak Hanjing, Nattawut Pholasa and Suthep Suantai

Symmetry 2023, 15(1), 7; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15010007 - 20 Dec 2022

Viewed by 910

Abstract

In algorithm development, symmetry plays a vital part in managing optimization problems in scientific models. The aim of this work is to propose a new accelerated method for finding a common point of convex minimization problems and then use the fixed point of [...] Read more.

In algorithm development, symmetry plays a vital part in managing optimization problems in scientific models. The aim of this work is to propose a new accelerated method for finding a common point of convex minimization problems and then use the fixed point of the forward-backward operator to explain and analyze a weak convergence result of the proposed algorithm in real Hilbert spaces under certain conditions. As applications, we demonstrate the suggested method for solving image inpainting and image restoration problems. Full article

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

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23 pages, 891 KiB

Open AccessArticle

Common Fixed-Points Technique for the Existence of a Solution to Fractional Integro-Differential Equations via Orthogonal Branciari Metric Spaces

by Arul Joseph Gnanaprakasam, Gunasekaran Nallaselli, Absar Ul Haq, Gunaseelan Mani, Imran Abbas Baloch and Kamsing Nonlaopon

Symmetry 2022, 14(9), 1859; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14091859 - 06 Sep 2022

Cited by 8 | Viewed by 1037

Abstract

The idea of symmetry is a built-in feature of the metric function. In this paper, we investigate the existence and uniqueness of a fixed point of certain contraction via orthogonal triangular α-orbital admissible mapping in the context of orthogonal complete Branciari metric [...] Read more.

The idea of symmetry is a built-in feature of the metric function. In this paper, we investigate the existence and uniqueness of a fixed point of certain contraction via orthogonal triangular α-orbital admissible mapping in the context of orthogonal complete Branciari metric spaces endowed with a transitive binary relation. Our results generalize and extend some pioneering results in the literature. Furthermore, the existence criteria of the solutions to fractional integro-differential equations are established to demonstrate the applicability of our results. Full article

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

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17 pages, 347 KiB

Open AccessArticle

A General Picard-Mann Iterative Method for Approximating Fixed Points of Nonexpansive Mappings with Applications

by Rahul Shukla, Rajendra Pant and Winter Sinkala

Symmetry 2022, 14(8), 1741; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081741 - 21 Aug 2022

Cited by 2 | Viewed by 1606

Abstract

Fixed point theory provides an important structure for the study of symmetry in mathematics. In this article, a new iterative method (general Picard–Mann) to approximate fixed points of nonexpansive mappings is introduced and studied. We study the stability of this newly established method [...] Read more.

Fixed point theory provides an important structure for the study of symmetry in mathematics. In this article, a new iterative method (general Picard–Mann) to approximate fixed points of nonexpansive mappings is introduced and studied. We study the stability of this newly established method which we find to be summably almost stable for contractive mappings. A number of weak and strong convergence theorems of such iterative methods are established in the setting of Banach spaces under certain geometrical assumptions. Finally, we present a number of applications to address various important problems (zero of an accretive operator, mixed equilibrium problem, convex optimization problem, split feasibility problem, periodic solution of a nonlinear evolution equation) appearing in the field of nonlinear analysis. Full article

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

13 pages, 311 KiB

Open AccessFeature PaperArticle

Douglas–Rachford Splitting Method with Linearization for the Split Feasibility Problem

by Ziyue Hu, Qiaoli Dong, Yuchao Tang and Michael Th. Rassias

Symmetry 2022, 14(3), 537; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14030537 - 06 Mar 2022

Cited by 2 | Viewed by 2019

Abstract

The aim of this article is to introduce the Douglas–Rachford splitting method with linearization to solve the split feasibility problem (SFP). Our proposed method includes two existing methods in work of Tang et al. and Wang as special cases. The ranges of the [...] Read more.

The aim of this article is to introduce the Douglas–Rachford splitting method with linearization to solve the split feasibility problem (SFP). Our proposed method includes two existing methods in work of Tang et al. and Wang as special cases. The ranges of the parameters in work of Tang et al. are extended from (0,1) to (0,2). Under standard conditions, we prove the weak convergence of proposed algorithms. We also provide two numerical experiments to illustrate the effectiveness of the proposed algorithm by comparing the algorithms in work of Tang et al. and Wang. Full article

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

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31 pages, 399 KiB

Open AccessArticle

A New Study on the Fixed Point Sets of Proinov-Type Contractions via Rational Forms

by Mi Zhou, Xiaolan Liu, Naeem Saleem, Andreea Fulga and Nihal Özgür

Symmetry 2022, 14(1), 93; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14010093 - 06 Jan 2022

Cited by 6 | Viewed by 1321

Abstract

In this paper, we presented some new weaker conditions on the Proinov-type contractions which guarantees that a self-mapping T has a unique fixed point in terms of rational forms. Our main results improved the conclusions provided by Andreea Fulga (On [...] Read more.

In this paper, we presented some new weaker conditions on the Proinov-type contractions which guarantees that a self-mapping T has a unique fixed point in terms of rational forms. Our main results improved the conclusions provided by Andreea Fulga (On

(ψ, φ) -

Rational Contractions) in which the continuity assumption can either be reduced to orbital continuity,

k -

continuity, continuity of

T^{k}

, T-orbital lower semi-continuity or even it can be removed. Meanwhile, the assumption of monotonicity on auxiliary functions is also removed from our main results. Moreover, based on the obtained fixed point results and the property of symmetry, we propose several Proinov-type contractions for a pair of self-mappings

(P, Q)

which will ensure the existence of the unique common fixed point of a pair of self-mappings

(P, Q)

. Finally, we obtained some results related to fixed figures such as fixed circles or fixed discs which are symmetrical under the effect of self mappings on metric spaces, we proposed some new types of

{(ψ, φ)}_{c} -

rational contractions and obtained the corresponding fixed figure theorems on metric spaces. Several examples are provided to indicate the validity of the results presented. Full article

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

23 pages, 3637 KiB

Open AccessArticle

Primal-Dual Splitting Algorithms for Solving Structured Monotone Inclusion with Applications

by Jinjian Chen, Xingyu Luo, Yuchao Tang and Qiaoli Dong

Symmetry 2021, 13(12), 2415; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122415 - 13 Dec 2021

Cited by 2 | Viewed by 1873

Abstract

This work proposes two different primal-dual splitting algorithms for solving structured monotone inclusion containing a cocoercive operator and the parallel-sum of maximally monotone operators. In particular, the parallel-sum is symmetry. The proposed primal-dual splitting algorithms are derived from two approaches: One is the [...] Read more.

This work proposes two different primal-dual splitting algorithms for solving structured monotone inclusion containing a cocoercive operator and the parallel-sum of maximally monotone operators. In particular, the parallel-sum is symmetry. The proposed primal-dual splitting algorithms are derived from two approaches: One is the preconditioned forward–backward splitting algorithm, and the other is the forward–backward–half-forward splitting algorithm. Both algorithms have a simple calculation framework. In particular, the single-valued operators are processed via explicit steps, while the set-valued operators are computed by their resolvents. Numerical experiments on constrained image denoising problems are presented to show the performance of the proposed algorithms. Full article

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

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22 pages, 1101 KiB

Open AccessArticle

New Self-Adaptive Inertial-like Proximal Point Methods for the Split Common Null Point Problem

by Yan Tang, Yeyu Zhang and Aviv Gibali

Symmetry 2021, 13(12), 2316; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122316 - 03 Dec 2021

Cited by 4 | Viewed by 1175

Abstract

Symmetry plays an important role in solving practical problems of applied science, especially in algorithm innovation. In this paper, we propose what we call the self-adaptive inertial-like proximal point algorithms for solving the split common null point problem, which use a new inertial [...] Read more.

Symmetry plays an important role in solving practical problems of applied science, especially in algorithm innovation. In this paper, we propose what we call the self-adaptive inertial-like proximal point algorithms for solving the split common null point problem, which use a new inertial structure to avoid the traditional convergence condition in general inertial methods and avoid computing the norm of the difference between

x_{n}

and

x_{n - 1}

before choosing the inertial parameter. In addition, the selection of the step-sizes in the inertial-like proximal point algorithms does not need prior knowledge of operator norms. Numerical experiments are presented to illustrate the performance of the algorithms. The proposed algorithms provide enlightenment for the further development of applied science in order to dig deep into symmetry under the background of technological innovation. Full article

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

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18 pages, 316 KiB

Open AccessArticle

Completeness of b−Metric Spaces and Best Proximity Points of Nonself Quasi-Contractions

by Arshad Ali Khan and Basit Ali

Symmetry 2021, 13(11), 2206; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112206 - 19 Nov 2021

Cited by 5 | Viewed by 1259

Abstract

The aims of this article are twofold. One is to prove some results regarding the existence of best proximity points of multivalued non-self quasi-contractions of

b -

metric spaces (which are symmetric spaces) and the other is to obtain a characterization of completeness [...] Read more.

The aims of this article are twofold. One is to prove some results regarding the existence of best proximity points of multivalued non-self quasi-contractions of

b -

metric spaces (which are symmetric spaces) and the other is to obtain a characterization of completeness of

b -

metric spaces via the existence of best proximity points of non-self quasi-contractions. Further, we pose some questions related to the findings in the paper. An example is provided to illustrate the main result. The results obtained herein improve some well known results in the literature. Full article

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

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