Fixed Point Theory and Fractional Calculus with Applications

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (15 April 2019) | Viewed by 74358

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Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan
Interests: nonlinear analysis and its applications; fixed point theory; variational principles and inequalities; optimization theory; equilibrium problems; fractional calculus theory
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College of Mathematics and Statistics, Hubei Normal University, Huangshi, China

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Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL, USA

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Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan

Special Issue Information

Aims and Scope: Fixed point theory is a beautiful mixture of analysis, topology, and geometry. Over the last sixty years or so the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular, fixed point theory has been applied in such diverse fields as biology, chemistry, economics, engineering, game theory, physics and logic programming.

Fractional calculus has played an important role in the study of the real world nonlinear fractional differential equations that arise from the modeling of nonlinear phenomena, optimal control of complex systems and other scientific research. The main research topics in this field include approximation theory and operator theory, numerical computational methods, control for nonlinear fractional differential equations, well-posedness of fractional mathematical models and the development of new technologies.

The aim of this Special Issue is to report new fixed point theorems as well as fractional calculus results and their applications. This Special Issue will accept high quality papers containing original research results and survey articles of exceptional merit.

Topics: The research topics include but are not limited to:

  • Fixed point theory and its applications
  • Best proximity point theory and its applications
  • Algorithms for fixed points and best proximity points
  • Nonlinear problems via fixed point theory approaches
  • Initial value problems of fractional differential equations
  • Boundary value problems of fractional differential equations
  • Singular and impulsive fractional differential and integral equations
  • Well-posedness and control in fixed point theory and fractional differential equations

A limited number of expository and survey articles will also be published.

Prof. Dr. Wei-Shih Du
Prof. Dr. Guoqing Chai
Prof. Dr. Toka Diagana
Prof. Dr. Tomonari Suzuki
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (26 papers)

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Research

14 pages, 322 KiB  
Article
On Maximal Elements with Applications to Abstract Economies, Fixed Point Theory and Eigenvector Problems
by Liang-Ju Chu and Wei–Shih Du
Symmetry 2019, 11(6), 789; https://doi.org/10.3390/sym11060789 - 13 Jun 2019
Viewed by 1758
Abstract
Two existence theorems of maximal elements in H-spaces are obtained without compactness. More accurately, we deal with the correspondence to be of L -majorized mappings in the setting of noncompact strategy sets but merely requiring a milder coercive condition. As applications, we [...] Read more.
Two existence theorems of maximal elements in H-spaces are obtained without compactness. More accurately, we deal with the correspondence to be of L -majorized mappings in the setting of noncompact strategy sets but merely requiring a milder coercive condition. As applications, we obtain an equilibrium existence theorem for general abstract economies, a new fixed point theorem, and give a sufficient condition for the existence of solutions of the eigenvector problem (EIVP). Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
12 pages, 282 KiB  
Article
Best Proximity Points for Alternative Maps
by Yi Chou Chen
Symmetry 2019, 11(6), 750; https://doi.org/10.3390/sym11060750 - 3 Jun 2019
Cited by 1 | Viewed by 1908
Abstract
Let ( X , d ) be a metric space and Ω i , i = 1 , 2 , , m , be a nonempty subset of ( X , d ) . An operator [...] Read more.
Let ( X , d ) be a metric space and Ω i , i = 1 , 2 , , m , be a nonempty subset of ( X , d ) . An operator T : 1 i m Ω i 1 i m Ω i is called an alternative map if T ( Ω j ) i j Ω i , j = 1 , 2 , , m . In addition, if for any x, y 1 i m Ω i , there exists a constant α [ 0 , 1 ) such that d ( T x , T y ) α d ( x , y ) + ( 1 α ) d ( Ω j , Ω k ) for some Ω j and Ω k { Ω i } i = 1 m with x Ω j and y Ω k , then we call T an alternative contraction. Moreover, if ( X , d ) has an alternative UC property and T is an alternative contraction, then the best proximity point of T exists. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
14 pages, 295 KiB  
Article
Terminal Value Problem for Differential Equations with Hilfer–Katugampola Fractional Derivative
by Mouffak Benchohra, Soufyane Bouriah and Juan J. Nieto
Symmetry 2019, 11(5), 672; https://doi.org/10.3390/sym11050672 - 15 May 2019
Cited by 25 | Viewed by 3034
Abstract
We present in this work the existence results and uniqueness of solutions for a class of boundary value problems of terminal type for fractional differential equations with the Hilfer–Katugampola fractional derivative. The reasoning is mainly based upon different types of classical fixed point [...] Read more.
We present in this work the existence results and uniqueness of solutions for a class of boundary value problems of terminal type for fractional differential equations with the Hilfer–Katugampola fractional derivative. The reasoning is mainly based upon different types of classical fixed point theory such as the Banach contraction principle and Krasnoselskii’s fixed point theorem. We illustrate our main findings, with a particular case example included to show the applicability of our outcomes. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
19 pages, 315 KiB  
Article
Generalized Contractive Mappings and Related Results in b-Metric Like Spaces with an Application
by Hasanen A. Hammad and Manuel De la Sen
Symmetry 2019, 11(5), 667; https://doi.org/10.3390/sym11050667 - 14 May 2019
Cited by 25 | Viewed by 2454
Abstract
In this article, a general contractive mapping is presented and some fixed point results in complete b-metric-like spaces are studied. The results obtained here extend and improve some related results in the literature. Also, new common fixed point results for a graphic [...] Read more.
In this article, a general contractive mapping is presented and some fixed point results in complete b-metric-like spaces are studied. The results obtained here extend and improve some related results in the literature. Also, new common fixed point results for a graphic contraction mappings are proved. Some comparative examples are given to support the obtained results. Moreover, an analytical solution of an integral equation has been presented as an application. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
23 pages, 319 KiB  
Article
Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators
by Ashish Nandal, Renu Chugh and Mihai Postolache
Symmetry 2019, 11(5), 655; https://doi.org/10.3390/sym11050655 - 10 May 2019
Cited by 12 | Viewed by 3037
Abstract
We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in [...] Read more.
We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
21 pages, 354 KiB  
Article
Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses
by Ravi Agarwal, Snezhana Hristova and Donal O’Regan
Symmetry 2019, 11(5), 614; https://doi.org/10.3390/sym11050614 - 2 May 2019
Cited by 10 | Viewed by 4021
Abstract
In this paper a nonlinear system of Riemann–Liouville (RL) fractional differential equations with non-instantaneous impulses is studied. The presence of non-instantaneous impulses require appropriate definitions of impulsive conditions and initial conditions. In the paper several types of initial value problems are considered and [...] Read more.
In this paper a nonlinear system of Riemann–Liouville (RL) fractional differential equations with non-instantaneous impulses is studied. The presence of non-instantaneous impulses require appropriate definitions of impulsive conditions and initial conditions. In the paper several types of initial value problems are considered and their mild solutions are given via integral representations. In the linear case the equivalence of the solution and mild solutions is established. Conditions for existence and uniqueness of initial value problems are presented. Several examples are provided to illustrate the influence of impulsive functions and the interpretation of impulses in the RL fractional case. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
14 pages, 289 KiB  
Article
Non-Unique Fixed Point Theorems in Modular Metric Spaces
by Sharafat Hussain
Symmetry 2019, 11(4), 549; https://doi.org/10.3390/sym11040549 - 16 Apr 2019
Cited by 3 | Viewed by 2537
Abstract
This paper is devoted to the study of Ćirić-type non-unique fixed point results in modular metric spaces. We obtain various theorems about a fixed point and periodic points for a self-map on modular spaces which are not necessarily continuous and satisfy certain contractive [...] Read more.
This paper is devoted to the study of Ćirić-type non-unique fixed point results in modular metric spaces. We obtain various theorems about a fixed point and periodic points for a self-map on modular spaces which are not necessarily continuous and satisfy certain contractive conditions. Our results extend the results of Ćirić, Pachpatte, and Achari in modular metric spaces. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
18 pages, 282 KiB  
Article
Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings
by Manatchanok Khonchaliew, Ali Farajzadeh and Narin Petrot
Symmetry 2019, 11(4), 480; https://doi.org/10.3390/sym11040480 - 3 Apr 2019
Cited by 2 | Viewed by 2044
Abstract
This paper presents two shrinking extragradient algorithms that can both find the solution sets of equilibrium problems for pseudomonotone bifunctions and find the sets of fixed points of quasi-nonexpansive mappings in a real Hilbert space. Under some constraint qualifications of the scalar sequences, [...] Read more.
This paper presents two shrinking extragradient algorithms that can both find the solution sets of equilibrium problems for pseudomonotone bifunctions and find the sets of fixed points of quasi-nonexpansive mappings in a real Hilbert space. Under some constraint qualifications of the scalar sequences, these two new algorithms show strong convergence. Some numerical experiments are presented to demonstrate the new algorithms. Finally, the two introduced algorithms are compared with a standard, well-known algorithm. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
15 pages, 296 KiB  
Article
Anti-Periodic Boundary Value Problems for Nonlinear Langevin Fractional Differential Equations
by Fang Li, Hongjuan Zeng and Huiwen Wang
Symmetry 2019, 11(4), 443; https://doi.org/10.3390/sym11040443 - 27 Mar 2019
Cited by 1 | Viewed by 2045
Abstract
In this paper, we focus on the existence of solutions of the nonlinear Langevin fractional differential equations involving anti-periodic boundary value conditions. By using some techniques, formulas of solutions for the above problem and some properties of the Mittag-Leffler functions [...] Read more.
In this paper, we focus on the existence of solutions of the nonlinear Langevin fractional differential equations involving anti-periodic boundary value conditions. By using some techniques, formulas of solutions for the above problem and some properties of the Mittag-Leffler functions E α , β ( z ) , α , β ( 1 , 2 ) , z R are presented. Moreover, we utilize the fixed point theorem under the weak assumptions for nonlinear terms to obtain the existence result of solutions and give an example to illustrate the result. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
11 pages, 287 KiB  
Article
On Suzuki Mappings in Modular Spaces
by Andreea Bejenaru and Mihai Postolache
Symmetry 2019, 11(3), 319; https://doi.org/10.3390/sym11030319 - 3 Mar 2019
Cited by 10 | Viewed by 2649
Abstract
Inspired by Suzuki’s generalization for nonexpansive mappings, we define the ( C ) -property on modular spaces, and provide conditions concerning the fixed points of newly introduced class of mappings in this new framework. In addition, Kirk’s Lemma is extended to modular spaces. [...] Read more.
Inspired by Suzuki’s generalization for nonexpansive mappings, we define the ( C ) -property on modular spaces, and provide conditions concerning the fixed points of newly introduced class of mappings in this new framework. In addition, Kirk’s Lemma is extended to modular spaces. The main outcomes extend the classical results on Banach spaces. The major contribution consists of providing inspired arguments to compensate the absence of subadditivity in the case of modulars. The results herein are supported by illustrative examples. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
9 pages, 252 KiB  
Article
Fixed Point Results Using Ft-Contractions in Ordered Metric Spaces Having t-Property
by Hassen Aydi, Tawseef Rashid, Qamrul Haq Khan, Zead Mustafa and Mohammed M. M. Jaradat
Symmetry 2019, 11(3), 313; https://doi.org/10.3390/sym11030313 - 2 Mar 2019
Cited by 4 | Viewed by 2029
Abstract
In this paper, we prove the existence of fixed points of F t -contraction mappings in partially ordered metric spaces not necessarily complete. We require that the ordered metric space has the t-property, which is a new concept introduced recently by Rashid [...] Read more.
In this paper, we prove the existence of fixed points of F t -contraction mappings in partially ordered metric spaces not necessarily complete. We require that the ordered metric space has the t-property, which is a new concept introduced recently by Rashid et al. We also give some examples to illustrate the new concepts and obtained results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
15 pages, 246 KiB  
Article
Some Results for Split Equality Equilibrium Problems in Banach Spaces
by Zhaoli Ma, Lin Wang and Yeol Je Cho
Symmetry 2019, 11(2), 194; https://doi.org/10.3390/sym11020194 - 10 Feb 2019
Cited by 20 | Viewed by 2157
Abstract
In this paper, a new algorithm for finding a common element of a split equality fixed point problem for nonexpansive mappings and split equality equilibrium problem in three Banach spaces is introduced. Also, some strong and weak convergence theorems for the proposed algorithm [...] Read more.
In this paper, a new algorithm for finding a common element of a split equality fixed point problem for nonexpansive mappings and split equality equilibrium problem in three Banach spaces is introduced. Also, some strong and weak convergence theorems for the proposed algorithm are proved. Finally, the main results obtained in this paper are applied to solve the split equality convex minimization problem. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
13 pages, 759 KiB  
Article
Generalized Random α-ψ-Contractive Mappings with Applications to Stochastic Differential Equation
by Chayut Kongban, Poom Kumam, Juan Martínez-Moreno and Kanokwan Sitthithakerngkiet
Symmetry 2019, 11(2), 143; https://doi.org/10.3390/sym11020143 - 28 Jan 2019
Viewed by 2406
Abstract
The main purpose in this paper is to define the modification form of random α -admissible and random α - ψ -contractive maps. We establish new random fixed point theorems in complete separable metric spaces. The interpretation of our results provide the main [...] Read more.
The main purpose in this paper is to define the modification form of random α -admissible and random α - ψ -contractive maps. We establish new random fixed point theorems in complete separable metric spaces. The interpretation of our results provide the main theorems of Tchier and Vetro (2017) as directed corollaries. In addition, some applications to second order random differential equations are presenred here to interpret the usability of the obtained results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
20 pages, 338 KiB  
Article
Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations
by Zhenhua Ma, Azhar Hussain, Muhammad Adeel, Nawab Hussain and Ekrem Savas
Symmetry 2019, 11(1), 93; https://doi.org/10.3390/sym11010093 - 15 Jan 2019
Cited by 4 | Viewed by 3472
Abstract
In this paper, we introduce the notion of C ´ iri c ´ type α - ψ - Θ -contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially [...] Read more.
In this paper, we introduce the notion of C ´ iri c ´ type α - ψ - Θ -contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially ordered complete metric spaces through our main results. As a consequence, we obtain some fixed point results for such contraction in complete metric and partially ordered complete metric spaces. Examples are given to illustrate the results obtained. Moreover, we present the existence of a positive definite solution of nonlinear matrix equation X = Q + i = 1 m A i γ ( X ) A i and give a numerical example. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
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24 pages, 368 KiB  
Article
Hybrid Multivalued Type Contraction Mappings in αK-Complete Partial b-Metric Spaces and Applications
by Eskandar Ameer, Hassen Aydi, Muhammad Arshad, Habes Alsamir and Mohd Selmi Noorani
Symmetry 2019, 11(1), 86; https://doi.org/10.3390/sym11010086 - 14 Jan 2019
Cited by 72 | Viewed by 3663
Abstract
In this paper, we initiate the notion of generalized multivalued ( α K * , Υ , Λ ) -contractions and provide some new common fixed point results in the class of α K -complete partial b-metric spaces. The obtained results are [...] Read more.
In this paper, we initiate the notion of generalized multivalued ( α K * , Υ , Λ ) -contractions and provide some new common fixed point results in the class of α K -complete partial b-metric spaces. The obtained results are an improvement of several comparable results in the existing literature. We set up an example to elucidate our main result. Moreover, we present applications dealing with the existence of a solution for systems either of functional equations or of nonlinear matrix equations. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
17 pages, 315 KiB  
Article
Common Fixed Point Results for Fuzzy Mappings on Complex-Valued Metric Spaces with Homotopy Results
by Humaira, Muhammad Sarwar and Poom Kumam
Symmetry 2019, 11(1), 61; https://doi.org/10.3390/sym11010061 - 8 Jan 2019
Cited by 21 | Viewed by 3689
Abstract
Owing to the notion of a complex-valued metric space, we prove fixed point results, which generalize some common fixed point results under contractive condition for rational expression in the context of complex-valued metric spaces. In application, we present a homotopy result to apply [...] Read more.
Owing to the notion of a complex-valued metric space, we prove fixed point results, which generalize some common fixed point results under contractive condition for rational expression in the context of complex-valued metric spaces. In application, we present a homotopy result to apply the results obtained herein. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
7 pages, 235 KiB  
Article
On Interpolative Hardy-Rogers Type Contractions
by Erdal Karapınar, Obaid Alqahtani and Hassen Aydi
Symmetry 2019, 11(1), 8; https://doi.org/10.3390/sym11010008 - 22 Dec 2018
Cited by 107 | Viewed by 3841
Abstract
By using an interpolative approach, we recognize the Hardy-Rogers fixed point theorem in the class of metric spaces. The obtained result is supported by some examples. We also give the partial metric case, according to our result. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
14 pages, 267 KiB  
Article
Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability
by Mostafa Bachar, Osvaldo Mendez and Messaoud Bounkhel
Symmetry 2018, 10(12), 708; https://doi.org/10.3390/sym10120708 - 3 Dec 2018
Cited by 10 | Viewed by 2488
Abstract
We analyze the modular geometry of the Lebesgue space with variable exponent, L p ( · ) . Our central result is that L p ( · ) possesses a modular uniform convexity property. Part of the novelty is that the property holds [...] Read more.
We analyze the modular geometry of the Lebesgue space with variable exponent, L p ( · ) . Our central result is that L p ( · ) possesses a modular uniform convexity property. Part of the novelty is that the property holds even in the case sup x Ω p ( x ) = . We present specific applications to fixed point theory. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
15 pages, 285 KiB  
Article
Radial Symmetry for Weak Positive Solutions of Fractional Laplacian with a Singular Nonlinearity
by Xing Wang and Li Zhang
Symmetry 2018, 10(12), 695; https://doi.org/10.3390/sym10120695 - 3 Dec 2018
Viewed by 2293
Abstract
This paper is concerned with the radial symmetry weak positive solutions for a class of singular fractional Laplacian. The main results in the paper demonstrate the existence and multiplicity of radial symmetry weak positive solutions by Schwarz spherical rearrangement, constrained minimization, and Ekeland’s [...] Read more.
This paper is concerned with the radial symmetry weak positive solutions for a class of singular fractional Laplacian. The main results in the paper demonstrate the existence and multiplicity of radial symmetry weak positive solutions by Schwarz spherical rearrangement, constrained minimization, and Ekeland’s variational principle. It is worth pointing out that our results extend the previous works of T. Mukherjee and K. Sreenadh to a setting in which the testing functions need not have a compact support. Moreover, we weakened one of the conditions used in their papers. Our results improve on existing studies on radial symmetry solutions of nonlocal boundary value problems. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
24 pages, 315 KiB  
Article
Some Fixed-Point Theorems in b-Dislocated Metric Space and Applications
by Panda Sumati Kumari, Obaid Alqahtani and Erdal Karapınar
Symmetry 2018, 10(12), 691; https://doi.org/10.3390/sym10120691 - 2 Dec 2018
Cited by 20 | Viewed by 2950
Abstract
In this article, we prove some fixed-point theorems in b-dislocated metric space. Thereafter, we propose a simple and efficient solution for a non-linear integral equation and non-linear fractional differential equations of Caputo type by using the technique of fixed point. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
8 pages, 268 KiB  
Article
Some Inequalities of Čebyšev Type for Conformable k-Fractional Integral Operators
by Feng Qi, Gauhar Rahman, Sardar Muhammad Hussain, Wei-Shih Du and Kottakkaran Sooppy Nisar
Symmetry 2018, 10(11), 614; https://doi.org/10.3390/sym10110614 - 8 Nov 2018
Cited by 39 | Viewed by 3799
Abstract
In the article, the authors present several inequalities of the Čebyšev type for conformable k-fractional integral operators. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
17 pages, 295 KiB  
Article
Bochner-Like Transform and Stepanov Almost Periodicity on Time Scales with Applications
by Chao-Hong Tang and Hong-Xu Li
Symmetry 2018, 10(11), 566; https://doi.org/10.3390/sym10110566 - 1 Nov 2018
Cited by 8 | Viewed by 2057
Abstract
By using Bochner transform, Stepanov almost periodic functions inherit some basic properties directly from almost periodic functions. Recently, this old work was extended to time scales. However, we show that Bochner transform is not valid on time scales. Then we present a revised [...] Read more.
By using Bochner transform, Stepanov almost periodic functions inherit some basic properties directly from almost periodic functions. Recently, this old work was extended to time scales. However, we show that Bochner transform is not valid on time scales. Then we present a revised version, called Bochner-like transform, for time scales, and prove that a function is Stepanov almost periodic if and only if its Bochner-like transform is almost periodic on time scales. Some basic properties including the composition theorem of Stepanov almost periodic functions are obtained by applying Bochner-like transform. Our results correct the recent results where Bochner transform is used on time scales. As an application, we give some results on dynamic equations with Stepanov almost periodic terms. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
15 pages, 269 KiB  
Article
Convergence Analysis of an Inexact Three-Operator Splitting Algorithm
by Chunxiang Zong, Yuchao Tang and Yeol Je Cho
Symmetry 2018, 10(11), 563; https://doi.org/10.3390/sym10110563 - 1 Nov 2018
Cited by 21 | Viewed by 2742
Abstract
The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, [...] Read more.
The three-operator splitting algorithm is a new splitting algorithm for finding monotone inclusion problems of the sum of three maximally monotone operators, where one is cocoercive. As the resolvent operator is not available in a closed form in the original three-operator splitting algorithm, in this paper, we introduce an inexact three-operator splitting algorithm to solve this type of monotone inclusion problem. The theoretical convergence properties of the proposed iterative algorithm are studied in general Hilbert spaces under mild conditions on the iterative parameters. As a corollary, we obtain general convergence results of the inexact forward-backward splitting algorithm and the inexact Douglas-Rachford splitting algorithm, which extend the existing results in the literature. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
11 pages, 264 KiB  
Article
Some Globally Stable Fixed Points in b-Metric Spaces
by Umar Yusuf Batsari, Poom Kumam and Kanokwan Sitthithakerngkiet
Symmetry 2018, 10(11), 555; https://doi.org/10.3390/sym10110555 - 30 Oct 2018
Cited by 3 | Viewed by 2895
Abstract
In this paper, the existence and uniqueness of globally stable fixed points of asymptotically contractive mappings in complete b-metric spaces were studied. Also, we investigated the existence of fixed points under the setting of a continuous mapping. Furthermore, we introduce a contraction [...] Read more.
In this paper, the existence and uniqueness of globally stable fixed points of asymptotically contractive mappings in complete b-metric spaces were studied. Also, we investigated the existence of fixed points under the setting of a continuous mapping. Furthermore, we introduce a contraction mapping that generalizes that of Banach, Kanan, and Chatterjea. Using our new introduced contraction mapping, we establish some results on the existence and uniqueness of fixed points. In obtaining some of our results, we assume that the space is associated with a partial order, and the b-metric function has the regularity property. Our results improve, and generalize some current results in the literature. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
13 pages, 265 KiB  
Article
A New Approach to the Solution of the Fredholm Integral Equation via a Fixed Point on Extended b-Metric Spaces
by Erdal Karapınar, Panda Sumati Kumari and Durdana Lateef
Symmetry 2018, 10(10), 512; https://doi.org/10.3390/sym10100512 - 16 Oct 2018
Cited by 35 | Viewed by 3563
Abstract
It is very well known that real-life applications of fixed point theory are restricted with the transformation of the problem in the form of f ( x ) = x . (1) The Knaster–Tarski fixed point theorem underlies various approaches of checking the [...] Read more.
It is very well known that real-life applications of fixed point theory are restricted with the transformation of the problem in the form of f ( x ) = x . (1) The Knaster–Tarski fixed point theorem underlies various approaches of checking the correctness of programs. (2) The Brouwer fixed point theorem is used to prove the existence of Nash equilibria in games. (3) Dlala et al. proposed a solution for magnetic field problems via the fixed point approach. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
10 pages, 272 KiB  
Article
Fibonacci–Mann Iteration for Monotone Asymptotically Nonexpansive Mappings in Modular Spaces
by Buthinah A. Bin Dehaish and Mohamed A Khamsi
Symmetry 2018, 10(10), 481; https://doi.org/10.3390/sym10100481 - 11 Oct 2018
Cited by 2 | Viewed by 2747
Abstract
In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by [...] Read more.
In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by x n + 1 = t n T ϕ ( n ) ( x n ) + ( 1 t n ) x n , for n N , when T is a monotone asymptotically nonexpansive self-mapping. Full article
(This article belongs to the Special Issue Fixed Point Theory and Fractional Calculus with Applications)
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