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Symmetry
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Fixed Point Theory and Computational Analysis with Applications
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Fixed Point Theory and Computational Analysis with Applications

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

Deadline for manuscript submissions: closed (31 May 2021) | Viewed by 70011

Special Issue Editors

Prof. Dr. Wei-Shih Du

E-Mail Website
Guest Editor
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan
Interests: nonlinear analysis; fixed point theory and its applications; variational principles and inequalities; optimization theory; fractional calculus theory
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alicia Cordero

E-Mail Website
Guest Editor
School of Telecommunications Engineering, Universitat Politècnica de València, 46022 Valencia, Spain
Interests: numerical analysis; iterative methods; nonlinear problems; discrete dynamics, real and complex
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Huaping Huang

E-Mail Website
Guest Editor
1. School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404100, China
2. Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Interests: nonlinear functional analysis; complex analysis and differential equation theory; theory of fixed point in metric spaces and abstract metric spaces
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Juan Ramón Torregrosa Sánchez

E-Mail Website
Guest Editor
Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain
Interests: iterative processes; matrix analysis; numerical analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fixed point theory is a fascinating subject that has a wide range of applications in many areas of mathematics. Brouwer’s fixed point theorem and Banach contraction principle are undoubtedly the most important and applicable fixed point theorems. Many authors are dedicated to the generalization of the various directions of well-known fixed point theorems. The rapid development of fixed point theory and its applications has led to many academic papers studying the importance of its promotions and applications in nonlinear analysis, optimization problems, integral and differential equations and inclusions, dynamic system theory, control theory, signal and image processing, economics, game theory, etc. In this special issue, we will accept important research and development on fixed point theory that integrate basic science into the real world.

A plenty of problems caused by the real world can be reduced to solve mathematical models by applying computational analysis. During the previous more than seven decades, computational analysis has made more important contributions to improve our understanding of the real world around us in various fields, such as immunological systems, computational systems, electrical and mechanical structures, financial markets, information and knowledge management, highway transportation networks, telecommunication networks, economics and so on. Due to the significance of the applications of computational analysis, in this special issue, we would like to receive contributions describing applications of mathematics and computation.

We cordially and earnestly invite researchers to contribute their original and high quality research papers which will inspire the advance in fixed point theory, computational analysis and their applications. Potential topics include, but are not limited to:

  • Fixed point theory and best proximity point theory with applications
  • Numerical algorithms for nonlinear problems
  • Variational method and its applications
  • Nonlinear differential and integral equations by fixed point theory
  • Optimization problems by fixed point theory
  • Geometry of Banach spaces
  • Well-posedness and control in fixed point theory and computational analysis
  • Linear and nonlinear dynamical systems
  • Image and signal processing
  • Matrix theory
  • Numerical analysis
  • Computational geometry
  • Computer graphics
  • Finite element method
  • Intelligence computation
  • Scientific and engineering computing

Prof. Dr. Wei-Shih Du
Prof. Dr. Alicia Cordero
Prof. Dr. Huaping Huang
Prof. Dr. Juan R. Torregrosa
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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.

Keywords

  • Fixed point theory
  • Best proximity point theory
  • Optimization problems
  • Algorithms
  • Variational inequality problems
  • Geometry of Banach spaces
  • Dynamical systems
  • Matrix theory
  • Numerical Analysis
  • Image and signal processing
  • Visualization

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Published Papers (25 papers)

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Research

12 pages, 638 KiB  
Article
Enhancing Ant-Based Algorithms for Medical Image Edge Detection by Admissible Perturbations of Demicontractive Mappings
by Vasile Berinde and Cristina Ţicală
Symmetry 2021, 13(5), 885; https://doi.org/10.3390/sym13050885 - 17 May 2021
Cited by 8 | Viewed by 2623
Abstract
The aim of this paper is to show analytically and empirically how ant-based algorithms for medical image edge detection can be enhanced by using an admissible perturbation of demicontractive operators. We thus complement the results reported in a recent paper by the second [...] Read more.
The aim of this paper is to show analytically and empirically how ant-based algorithms for medical image edge detection can be enhanced by using an admissible perturbation of demicontractive operators. We thus complement the results reported in a recent paper by the second author and her collaborators, where they used admissible perturbations of demicontractive mappings as test functions. To illustrate this fact, we first consider some typical properties of demicontractive mappings and of their admissible perturbations and then present some appropriate numerical tests to illustrate the improvement brought by the admissible perturbations of demicontractive mappings when they are taken as test functions in ant-based algorithms for medical image edge detection. The edge detection process reported in our study considers both symmetric (Head CT and Brain CT) and asymmetric (Hand X-ray) medical images. The performance of the algorithm was tested visually with various images and empirically with evaluation of parameters. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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14 pages, 973 KiB  
Article
Fixed-Point Iterative Method with Eighth-Order Constructed by Undetermined Parameter Technique for Solving Nonlinear Systems
by Xiaofeng Wang
Symmetry 2021, 13(5), 863; https://doi.org/10.3390/sym13050863 - 12 May 2021
Cited by 12 | Viewed by 2083
Abstract
In this manuscript, by using undetermined parameter method, an efficient iterative method with eighth-order is designed to solve nonlinear systems. The new method requires one matrix inversion per iteration, which means that computational cost of our method is low. The theoretical efficiency of [...] Read more.
In this manuscript, by using undetermined parameter method, an efficient iterative method with eighth-order is designed to solve nonlinear systems. The new method requires one matrix inversion per iteration, which means that computational cost of our method is low. The theoretical efficiency of the proposed method is analyzed, which is superior to other methods. Numerical results show that the proposed method can reduce the computational time, remarkably. New method is applied to solve the numerical solution of nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs). The nonlinear ODEs and PDEs are discretized by finite difference method. The validity of the new method is verified by comparison with analytic solutions. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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19 pages, 334 KiB  
Article
Rational Type Contractions in Extended b-Metric Spaces
by Huaping Huang, Yumnam Mahendra Singh, Mohammad Saeed Khan and Stojan Radenović
Symmetry 2021, 13(4), 614; https://doi.org/10.3390/sym13040614 - 7 Apr 2021
Cited by 21 | Viewed by 3014
Abstract
In this paper, we establish the existence of fixed points of rational type contractions in the setting of extended b-metric spaces. Our results extend considerably several well-known results in the existing literature. We present some nontrivial examples to show the validity of [...] Read more.
In this paper, we establish the existence of fixed points of rational type contractions in the setting of extended b-metric spaces. Our results extend considerably several well-known results in the existing literature. We present some nontrivial examples to show the validity of our results. Furthermore, as applications, we obtain the existence of solution to a class of Fredholm integral equations. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
20 pages, 328 KiB  
Article
Fixed Point Theorems for Nonexpansive Type Mappings in Banach Spaces
by Rajendra Pant, Prashant Patel, Rahul Shukla and Manuel De la Sen
Symmetry 2021, 13(4), 585; https://doi.org/10.3390/sym13040585 - 2 Apr 2021
Cited by 13 | Viewed by 3567
Abstract
In this paper, we present some fixed point results for a class of nonexpansive type and α-Krasnosel’skiĭ mappings. Moreover, we present some convergence results for one parameter nonexpansive type semigroups. Some non-trivial examples have been presented to illustrate facts. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
13 pages, 304 KiB  
Article
New Study of the Existence and Dimension of the Set of Solutions for Nonlocal Impulsive Differential Inclusions with a Sectorial Operator
by Nawal Alsarori, Kirtiwant Ghadle, Salvatore Sessa, Hayel Saleh and Sami Alabiad
Symmetry 2021, 13(3), 491; https://doi.org/10.3390/sym13030491 - 17 Mar 2021
Cited by 6 | Viewed by 1529
Abstract
In this article, we are interested in a new generic class of nonlocal fractional impulsive differential inclusions with linear sectorial operator and Lipschitz multivalued function in the setting of finite dimensional Banach spaces. By modifying the definition of PC-mild solutions initiated by Shu, [...] Read more.
In this article, we are interested in a new generic class of nonlocal fractional impulsive differential inclusions with linear sectorial operator and Lipschitz multivalued function in the setting of finite dimensional Banach spaces. By modifying the definition of PC-mild solutions initiated by Shu, we succeeded to determine new conditions that sufficiently guarantee the existence of the solutions. The results are obtained by combining techniques of fractional calculus and the fixed point theorem for contraction maps. We also characterize the topological structure of the set of solutions. Finally, we provide a demonstration to address the applicability of our theoretical results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
11 pages, 270 KiB  
Article
Existence of the Solutions of Nonlinear Fractional Differential Equations Using the Fixed Point Technique in Extended b-Metric Spaces
by Liliana Guran and Monica-Felicia Bota
Symmetry 2021, 13(2), 158; https://doi.org/10.3390/sym13020158 - 20 Jan 2021
Cited by 4 | Viewed by 2232
Abstract
The purpose of this paper is to prove fixed point theorems for cyclic-type operators in extended b-metric spaces. The well-posedness of the fixed point problem and limit shadowing property are also discussed. Some examples are given in order to support our results, [...] Read more.
The purpose of this paper is to prove fixed point theorems for cyclic-type operators in extended b-metric spaces. The well-posedness of the fixed point problem and limit shadowing property are also discussed. Some examples are given in order to support our results, and the last part of the paper considers some applications of the main results. The first part of this section is devoted to the study of the existence of a solution to the boundary value problem. In the second part of this section, we study the existence of solutions to fractional boundary value problems with integral-type boundary conditions in the frame of some Caputo-type fractional operators. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
10 pages, 269 KiB  
Article
On Some New Jungck–Fisher–Wardowski Type Fixed Point Results
by Jelena Vujaković, Eugen Ljajko, Slobodan Radojević and Stojan Radenović
Symmetry 2020, 12(12), 2048; https://doi.org/10.3390/sym12122048 - 10 Dec 2020
Cited by 9 | Viewed by 1682
Abstract
Many authors used the concept of F−contraction introduced by Wardowski in 2012 in order to define and prove new results on fixed points in complete metric spaces. In some later papers (for example Proinov P.D., J. Fixed Point Theory Appl. (2020)22:21, [...] Read more.
Many authors used the concept of F−contraction introduced by Wardowski in 2012 in order to define and prove new results on fixed points in complete metric spaces. In some later papers (for example Proinov P.D., J. Fixed Point Theory Appl. (2020)22:21, doi:10.1007/s11784-020-0756-1) it is shown that conditions (F2) and (F3) are not necessary to prove Wardowski’s results. In this article we use a new approach in proving that the Picard–Jungck sequence is a Cauchy one. It helps us obtain new Jungck–Fisher–Wardowski type results using Wardowski’s condition (F1) only, but in a way that differs from the previous approaches. Along with that, we came to several new contractive conditions not known in the fixed point theory so far. With the new results presented in the article, we generalize, extend, unify and enrich methods presented in the literature that we cite. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
12 pages, 282 KiB  
Article
Some Fixed Point Theorems for (a−p)-Quasicontractions
by Ovidiu Popescu and Gabriel Stan
Symmetry 2020, 12(12), 1973; https://doi.org/10.3390/sym12121973 - 29 Nov 2020
Cited by 4 | Viewed by 1378
Abstract
In this paper, we introduced the notion of (a−p)-quasicontraction and proved two generalizations of some classical fixed point theorems. Furthermore, we present some examples to support our results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
7 pages, 748 KiB  
Article
Existence and Uniqueness of the Solution for an Integral Equation with Supremum, via w-Distances
by Veronica Ilea and Diana Otrocol
Symmetry 2020, 12(9), 1554; https://doi.org/10.3390/sym12091554 - 20 Sep 2020
Cited by 13 | Viewed by 2559
Abstract
Following the idea of T. Wongyat and W. Sintunavarat, we obtain some existence and uniqueness results for the solution of an integral equation with supremum. The paper ends with the study of Gronwall-type theorems, comparison theorems and a result regarding a Ulam–Hyers stability [...] Read more.
Following the idea of T. Wongyat and W. Sintunavarat, we obtain some existence and uniqueness results for the solution of an integral equation with supremum. The paper ends with the study of Gronwall-type theorems, comparison theorems and a result regarding a Ulam–Hyers stability result for the corresponding fixed point problem. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
19 pages, 467 KiB  
Article
Convergence Analysis of a Modified Weierstrass Method for the Simultaneous Determination of Polynomial Zeros
by Plamena I. Marcheva and Stoil I. Ivanov
Symmetry 2020, 12(9), 1408; https://doi.org/10.3390/sym12091408 - 24 Aug 2020
Cited by 10 | Viewed by 2271
Abstract
In 2016, Nedzhibov constructed a modification of the Weierstrass method for simultaneous computation of polynomial zeros. In this work, we obtain local and semilocal convergence theorems that improve and complement the previous results about this method. The semilocal result is of significant practical [...] Read more.
In 2016, Nedzhibov constructed a modification of the Weierstrass method for simultaneous computation of polynomial zeros. In this work, we obtain local and semilocal convergence theorems that improve and complement the previous results about this method. The semilocal result is of significant practical importance because of its computationally verifiable initial condition and error estimate. Numerical experiments to show the applicability of our semilocal theorem are also presented. We finish this study with a theoretical and numerical comparison between the modified Weierstrass method and the classical Weierstrass method. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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20 pages, 443 KiB  
Article
A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems
by Aliyu Muhammed Awwal, Lin Wang, Poom Kumam and Hassan Mohammad
Symmetry 2020, 12(6), 874; https://doi.org/10.3390/sym12060874 - 26 May 2020
Cited by 35 | Viewed by 3642
Abstract
In this paper, we propose a two-step iterative algorithm based on projection technique for solving system of monotone nonlinear equations with convex constraints. The proposed two-step algorithm uses two search directions which are defined using the well-known Barzilai and Borwein (BB) spectral parameters.The [...] Read more.
In this paper, we propose a two-step iterative algorithm based on projection technique for solving system of monotone nonlinear equations with convex constraints. The proposed two-step algorithm uses two search directions which are defined using the well-known Barzilai and Borwein (BB) spectral parameters.The BB spectral parameters can be viewed as the approximations of Jacobians with scalar multiple of identity matrices. If the Jacobians are close to symmetric matrices with clustered eigenvalues then the BB parameters are expected to behave nicely. We present a new line search technique for generating the separating hyperplane projection step of Solodov and Svaiter (1998) that generalizes the one used in most of the existing literature. We establish the convergence result of the algorithm under some suitable assumptions. Preliminary numerical experiments demonstrate the efficiency and computational advantage of the algorithm over some existing algorithms designed for solving similar problems. Finally, we apply the proposed algorithm to solve image deblurring problem. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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15 pages, 290 KiB  
Article
New Fixed Point Theorems in Orthogonal F -Metric Spaces with Application to Fractional Differential Equation
by Tanzeela Kanwal, Azhar Hussain, Hamid Baghani and Manuel de la Sen
Symmetry 2020, 12(5), 832; https://doi.org/10.3390/sym12050832 - 19 May 2020
Cited by 23 | Viewed by 2775
Abstract
We present the notion of orthogonal F -metric spaces and prove some fixed and periodic point theorems for orthogonal ⊥ Ω -contraction. We give a nontrivial example to prove the validity of our result. Finally, as application, we prove the existence and uniqueness [...] Read more.
We present the notion of orthogonal F -metric spaces and prove some fixed and periodic point theorems for orthogonal ⊥ Ω -contraction. We give a nontrivial example to prove the validity of our result. Finally, as application, we prove the existence and uniqueness of the solution of a nonlinear fractional differential equation. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
20 pages, 325 KiB  
Article
Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump
by Zhifu Jia, Xinsheng Liu and Cunlin Li
Symmetry 2020, 12(5), 765; https://doi.org/10.3390/sym12050765 - 6 May 2020
Cited by 10 | Viewed by 2474
Abstract
No previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, for [...] Read more.
No previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, for the one-dimensional case, we give two types of uncertain FDEs with jump that are symmetric in terms of form. The next, for the multidimensional case, when the coefficients of the equations satisfy Lipschitz condition and linear growth condition, we establish an existence and uniqueness theorems of uncertain FDEs with jump of Riemann-Liouville type by Banach fixed point theorem. A symmetric proof in terms of form is suitable to the Caputo type. When the coefficients do not satisfy the Lipschitz condition and linear growth condition, we just prove an existence theorem of the Caputo type equation by Schauder fixed point theorem. In the end, we present an application about uncertain interest rate model. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
18 pages, 438 KiB  
Article
Robust Finite-Time Control of Linear System with Non-Differentiable Time-Varying Delay
by Wanwisa Puangmalai, Jirapong Puangmalai and Thaned Rojsiraphisal
Symmetry 2020, 12(4), 680; https://doi.org/10.3390/sym12040680 - 24 Apr 2020
Cited by 6 | Viewed by 2849
Abstract
Practical systems such as hybrid power systems are currently implemented around the world. In order to get the system to work properly, the systems usually require their behavior to be maintained or state values to stay within a certain threshold. However, it is [...] Read more.
Practical systems such as hybrid power systems are currently implemented around the world. In order to get the system to work properly, the systems usually require their behavior to be maintained or state values to stay within a certain threshold. However, it is difficult to form a perfect mathematical model for describing behavior of the practical systems since there may be some information (uncertainties) that is not observed. Thus, in this article, we studied the stability of an uncertain linear system with a non-differentiable time-varying delay. We constructed Lyapunov-Krasovskii functionals (LKFs) containing several symmetric positive definite matrices to obtain robust finite-time stability (RFTS) and stabilization (RFTU) of the uncertain linear system. With the controller and uncertainties in the considered system, there exist nonlinear terms occurring in the formulation process. Past research handled these nonlinear terms as new variables but this led to some difficulty from a computation point of view. Instead, we applied a novel approach via Cauchy-like matrix inequalities to handle these difficulties. In the end, we present three numerical simulations to show the effectiveness of our proposed theory. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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17 pages, 1439 KiB  
Article
Change Point Detection with Mean Shift Based on AUC from Symmetric Sliding Windows
by Yanguang Wang, Guanna Huang, Junjie Yang, Huadong Lai, Shun Liu, Changrun Chen and Weichao Xu
Symmetry 2020, 12(4), 599; https://doi.org/10.3390/sym12040599 - 9 Apr 2020
Cited by 7 | Viewed by 4074
Abstract
Change point detection is widely used in signal detection, industrial engineering, economy, finance, biomedicine and many other fields. The widely used parametric methods require prior knowledge of the noise signal distribution, which are seldom realistic. In practice, when the distribution of noise is [...] Read more.
Change point detection is widely used in signal detection, industrial engineering, economy, finance, biomedicine and many other fields. The widely used parametric methods require prior knowledge of the noise signal distribution, which are seldom realistic. In practice, when the distribution of noise is not known, it is desirable to design algorithms based on non-parametric statistics, which, in the null case (no change point), are completely distribution free. To this end, we propose to use two symmetric sliding windows to compute the Area Under the receiver operating characteristic Curve (AUC) as a test statistic to measure the difference between the distribution of two samples. In the stage of change point detection, a threshold is designed according to hypothesis test which is based on the null distribution of the test statistics. This threshold is used to detect the potential change points in the signal. To reduce the probability of false alarm detection, a key parameter K is set to distinguish and delete the false alarms in potential change points. Comparative studies showed that our proposed method outperforms the classical Relative unconstrained Least-Squares Importance Fitting (RuLSIF) algorithm and is also better than the Hawkins, Qiu, and Kang (HQK) algorithm when the noise follows non-normal distributions. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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11 pages, 253 KiB  
Article
On New Extensions of Darbo’s Fixed Point Theorem with Applications
by Hüseyin Işık, Shahram Banaei, Farhan Golkarmanesh, Vahid Parvaneh, Choonkil Park and Maryam Khorshidi
Symmetry 2020, 12(3), 424; https://doi.org/10.3390/sym12030424 - 6 Mar 2020
Cited by 7 | Viewed by 2473
Abstract
In this paper, we extend Darbo’s fixed point theorem via weak JS-contractions in a Banach space. Our results generalize and extend several well-known comparable results in the literature. The technique of measure of non-compactness is the main tool in carrying out our proof. [...] Read more.
In this paper, we extend Darbo’s fixed point theorem via weak JS-contractions in a Banach space. Our results generalize and extend several well-known comparable results in the literature. The technique of measure of non-compactness is the main tool in carrying out our proof. As an application, we study the existence of solutions for a system of integral equations. Finally, we present a concrete example to support the effectiveness of our results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
13 pages, 2956 KiB  
Article
Use of Precise Area Fraction Model for Fine Grid DEM Simulation of ICFB with Large Particles
by Gruorong Wu and Jie Ouyang
Symmetry 2020, 12(3), 399; https://doi.org/10.3390/sym12030399 - 4 Mar 2020
Cited by 2 | Viewed by 2337
Abstract
The heterogeneous structures in a gas–solid fluidized bed can be resolved in discrete element simulation so long as the grid is fine enough. In order to conveniently calculate mean porosity in fine grid simulations, a precise area fraction model is given for two-dimensional [...] Read more.
The heterogeneous structures in a gas–solid fluidized bed can be resolved in discrete element simulation so long as the grid is fine enough. In order to conveniently calculate mean porosity in fine grid simulations, a precise area fraction model is given for two-dimensional simulations. The proposed area fraction model is validated by the discrete element simulation test on a small-scale internal circulation fluidized system of large particles, using a fine grid size of two particle diameters. Simulations show that the discrete element method can perform well in modelling time-varying waveforms for the physical quantities in an internal circulating fluidized bed, employing the precise gas area fraction model. This thought of precise calculation can be generalized to construct a volume fraction porosity model for three-dimensional simulation by use of the similar symmetry of a rectangular grid. Moreover, to construct these area and volume fraction models is to enrich and perfect the underlying model of fine grid simulation. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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13 pages, 678 KiB  
Article
A Novel Numerical Algorithm to Estimate the Subdivision Depth of Binary Subdivision Schemes
by Aamir Shahzad, Faheem Khan, Abdul Ghaffar, Ghulam Mustafa, Kottakkaran Sooppy Nisar and Dumitru Baleanu
Symmetry 2020, 12(1), 66; https://doi.org/10.3390/sym12010066 - 1 Jan 2020
Cited by 14 | Viewed by 2600
Abstract
Subdivision schemes are extensively used in scientific and practical applications to produce continuous geometrical shapes in an iterative manner. We construct a numerical algorithm to estimate subdivision depth between the limit curves/surfaces and their control polygons after k-fold subdivisions. In this paper, [...] Read more.
Subdivision schemes are extensively used in scientific and practical applications to produce continuous geometrical shapes in an iterative manner. We construct a numerical algorithm to estimate subdivision depth between the limit curves/surfaces and their control polygons after k-fold subdivisions. In this paper, the proposed numerical algorithm for subdivision depths of binary subdivision curves and surfaces are obtained after some modification of the results given by Mustafa et al in 2006. This algorithm is very useful for implementation of the parametrization. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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10 pages, 241 KiB  
Article
Two Fixed Point Theorems Concerning F-Contraction in Complete Metric Spaces
by Ovidiu Popescu and Gabriel Stan
Symmetry 2020, 12(1), 58; https://doi.org/10.3390/sym12010058 - 27 Dec 2019
Cited by 34 | Viewed by 3329
Abstract
In this paper, we generalize some results of Wardowski (Fixed Point Theory Appl. 2012:94, 2012), Cosentino and Vetro (Filomat 28:4, 2014), and Piri and Kumam (Fixed Point Theory Appl. 2014:210, 2014) theories by applying some weaker symmetrical conditions on the self map of [...] Read more.
In this paper, we generalize some results of Wardowski (Fixed Point Theory Appl. 2012:94, 2012), Cosentino and Vetro (Filomat 28:4, 2014), and Piri and Kumam (Fixed Point Theory Appl. 2014:210, 2014) theories by applying some weaker symmetrical conditions on the self map of a complete metric space and on the mapping F, concerning the contractions defined by Wardowski. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
15 pages, 280 KiB  
Article
Viscosity Approximation Methods for a General Variational Inequality System and Fixed Point Problems in Banach Spaces
by Yuanheng Wang and Chanjuan Pan
Symmetry 2020, 12(1), 36; https://doi.org/10.3390/sym12010036 - 23 Dec 2019
Cited by 6 | Viewed by 2206
Abstract
In Banach spaces, we study the problem of solving a more general variational inequality system for an asymptotically non-expansive mapping. We give a new viscosity approximation scheme to find a common element. Some strong convergence theorems of the proposed iterative method are obtained. [...] Read more.
In Banach spaces, we study the problem of solving a more general variational inequality system for an asymptotically non-expansive mapping. We give a new viscosity approximation scheme to find a common element. Some strong convergence theorems of the proposed iterative method are obtained. A numerical experiment is given to show the implementation and efficiency of our main theorem. Our results presented in this paper generalize and complement many recent ones. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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12 pages, 237 KiB  
Article
Some Fixed Point Theorems for Quadratic Quasicontractive Mappings
by Ovidiu Popescu and Gabriel Stan
Symmetry 2019, 11(11), 1329; https://doi.org/10.3390/sym11111329 - 24 Oct 2019
Cited by 3 | Viewed by 2721
Abstract
In this paper, we introduce the notion of quadratic quasicontractive mapping and prove two generalizations of some classical fixed point theorems. Furthermore, we present some examples to support our main results. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
16 pages, 1628 KiB  
Article
A Unified Proximity Algorithm with Adaptive Penalty for Nuclear Norm Minimization
by Wenyu Hu, Weidong Zheng and Gaohang Yu
Symmetry 2019, 11(10), 1277; https://doi.org/10.3390/sym11101277 - 11 Oct 2019
Cited by 2 | Viewed by 2147
Abstract
The nuclear norm minimization (NNM) problem is to recover a matrix that minimizes the sum of its singular values and satisfies some linear constraints simultaneously. The alternating direction method (ADM) has been used to solve this problem recently. However, the subproblems in ADM [...] Read more.
The nuclear norm minimization (NNM) problem is to recover a matrix that minimizes the sum of its singular values and satisfies some linear constraints simultaneously. The alternating direction method (ADM) has been used to solve this problem recently. However, the subproblems in ADM are usually not easily solvable when the linear mappings in the constraints are not identities. In this paper, we propose a proximity algorithm with adaptive penalty (PA-AP). First, we formulate the nuclear norm minimization problems into a unified model. To solve this model, we improve the ADM by adding a proximal term to the subproblems that are difficult to solve. An adaptive tactic on the proximity parameters is also put forward for acceleration. By employing subdifferentials and proximity operators, an equivalent fixed-point equation system is constructed, and we use this system to further prove the convergence of the proposed algorithm under certain conditions, e.g., the precondition matrix is symmetric positive definite. Finally, experimental results and comparisons with state-of-the-art methods, e.g., ADM, IADM-CG and IADM-BB, show that the proposed algorithm is effective. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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10 pages, 259 KiB  
Article
On Some Iterative Numerical Methods for Mixed Volterra–Fredholm Integral Equations
by Sanda Micula
Symmetry 2019, 11(10), 1200; https://doi.org/10.3390/sym11101200 - 24 Sep 2019
Cited by 18 | Viewed by 2632
Abstract
In this paper, we propose a class of simple numerical methods for approximating solutions of one-dimensional mixed Volterra–Fredholm integral equations of the second kind. These methods are based on fixed point results for the existence and uniqueness of the solution (results which also [...] Read more.
In this paper, we propose a class of simple numerical methods for approximating solutions of one-dimensional mixed Volterra–Fredholm integral equations of the second kind. These methods are based on fixed point results for the existence and uniqueness of the solution (results which also provide successive iterations of the solution) and suitable cubature formulas for the numerical approximations. We discuss in detail a method using Picard iteration and the two-dimensional composite trapezoidal rule, giving convergence conditions and error estimates. The paper concludes with numerical experiments and a discussion of the methods proposed. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
8 pages, 237 KiB  
Article
A Closed Formula for the Horadam Polynomials in Terms of a Tridiagonal Determinant
by Feng Qi, Can Kızılateş and Wei-Shih Du
Symmetry 2019, 11(6), 782; https://doi.org/10.3390/sym11060782 - 12 Jun 2019
Cited by 19 | Viewed by 5090
Abstract
In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the [...] Read more.
In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
15 pages, 1432 KiB  
Article
Fixed Point Root-Finding Methods of Fourth-Order of Convergence
by Alicia Cordero, Lucía Guasp and Juan R. Torregrosa
Symmetry 2019, 11(6), 769; https://doi.org/10.3390/sym11060769 - 6 Jun 2019
Cited by 6 | Viewed by 2767
Abstract
In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of [...] Read more.
In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub’s conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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