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Symmetry
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Symmetry in Fixed Point Theory and Optimization: Computations and Applications
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Symmetry in Fixed Point Theory and Optimization: Computations and Applications

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

Deadline for manuscript submissions: closed (20 March 2026) | Viewed by 17517

Special Issue Editor

Dr. Narin Petrot

E-Mail Website
Guest Editor
1. Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2. Center of Excellence in Nonlinear Analysis and Optimization, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Interests: optimization algorithms and its applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Symmetry is a foundational concept that resonates through diverse areas of mathematics and computer science, providing deep insights into the structures and dynamics of many analytical and computational processes. In fixed point theory and optimization, including vector optimization, linear programming, variational inequalities, and equilibrium problems, symmetrical principles often guide iterative schemes, enhance convergence properties, and inspire new approaches to challenging questions in mathematics, engineering, and computational applications.

This Special Issue, titled “Symmetry in Fixed Point Theory and Optimization: Computations and Applications”, will highlight both theoretical breakthroughs and practical advances that capitalize on symmetrical frameworks. We welcome submissions that address, but are not limited to, the following:

  1. Symmetry in Fixed Point Theory
    • Novel fixed point theorems exploiting symmetrical properties in various abstract spaces (metric, Banach, etc.);
    • Convergence analyses of iterative schemes under symmetrical constraints;
    • Fractional calculus approaches to fixed point problems.
  2. Symmetry in Optimization
    • Symmetrical structures and formulations in constrained/unconstrained optimization, including vector optimization and linear programming;
    • Variational inequalities and equilibrium problems that exploit symmetrical properties in their solution methodologies;
    • Approximation algorithms and stochastic systems demonstrating symmetrical advantages.
  3. Computational Approaches
    • Numerical methods, simulations, and software frameworks that utilize symmetry for improved convergence, reduced complexity, or more robust performance;
    • Cross-disciplinary collaborations linking mathematics and computer science to develop or apply symmetrical techniques for computational challenges.
  4. Applications
    • Symmetry-driven solutions in fields such as engineering, physics, finance, and machine learning;
    • Case studies on how symmetry provides deeper insights, reduces computational overheads, or fosters innovative methodological developments;
    • Broader impacts of symmetrical frameworks on modern computational science, algorithmic design, and data analysis.

We request original research articles, review articles, and short communications examining the influence and role of symmetry within the realms of fixed point theory, optimization, and computer science. By uniting experts across these domains, we will show how symmetrical considerations not only deepen theoretical understanding but also spark new applications and innovations.

We look forward to receiving valuable contributions that advance our understanding and utilization of symmetrical properties in fixed point theory and optimization.

Dr. Narin Petrot
Guest Editor

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 250 words) can be sent to the Editorial Office for assessment.

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
  • nonlinear analysis
  • optimization (vector optimization, linear programming)
  • equilibrium and variational inequality problems
  • iterative algorithms
  • fractional calculus
  • stochastic systems
  • approximation algorithms
  • computer science

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (9 papers)

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Research

16 pages, 1181 KB  
Article
Inertial Forward–Backward–Forward Algorithm with Moving Point Projection for Monotone Inclusions and Image Restoration
by Purit Thammasiri, Vasile Berinde, Somyot Plubtieng, Kasamsuk Ungchittrakool and Rabian Wangkeeree
Symmetry 2026, 18(5), 782; https://doi.org/10.3390/sym18050782 - 2 May 2026
Viewed by 306
Abstract
This paper introduces a novel inertial forward–backward–forward algorithm driven by a newly conceptualized moving point projection technique for solving monotone inclusion problems in real Hilbert spaces. By leveraging the properties of a Lipschitz continuous, monotone operator and a maximally monotone operator alongside this [...] Read more.
This paper introduces a novel inertial forward–backward–forward algorithm driven by a newly conceptualized moving point projection technique for solving monotone inclusion problems in real Hilbert spaces. By leveraging the properties of a Lipschitz continuous, monotone operator and a maximally monotone operator alongside this innovative projection strategy, we dynamically construct a sequence of nonempty, closed, and convex sets that contain the zeros of the sum of the two operators. This geometric construction ensures that the resulting sequence is well defined and guarantees its weak convergence to a solution. Furthermore, to validate the practical efficacy of the proposed theoretical framework, we evaluate our method on image restoration problems. Numerical experiments measuring the improvement in signal-to-noise ratio (ISNR) and the structural similarity index measure (SSIM) confirm that the proposed algorithm is highly efficient and significantly outperforms existing state-of-the-art methods. Full article
(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)
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38 pages, 7180 KB  
Article
Object-Oriented Geometric Figures with Operations and Transformations for Relational Modeling
by Steven D. P. Moore
Symmetry 2026, 18(5), 725; https://doi.org/10.3390/sym18050725 - 24 Apr 2026
Viewed by 319
Abstract
This article introduces novel methodologies, coordinate systems, and procedures in computational geometry that further develop a Euclidean-based relationalistic framework. The objective is to describe tools using object-oriented relational elements with symmetry, anchored to a fixed point in a relational model, that generate structured [...] Read more.
This article introduces novel methodologies, coordinate systems, and procedures in computational geometry that further develop a Euclidean-based relationalistic framework. The objective is to describe tools using object-oriented relational elements with symmetry, anchored to a fixed point in a relational model, that generate structured point sets serving as blueprints for geometric figures and physical structures representing their source objects. Geometric operations and transformations construct ratio figures and ordered proportional structures. Using discrete N-Euclidean geometry, two relational coordinate systems are introduced—polar-vertex coordinates and radial coordinates—both formed through discrete geometric operations. A relational unit circle of fixed magnitude is defined by a 4::1 proportional equivalence between radius and angular ratios, independent of real-number or arc-length geometry. Euclid’s theory of proportion is extended from static abstract magnitudes to symmetry-driven geometric construction, and a square-pyramid geometric blueprint is produced from an Earth ratio figure with accurate dimensional magnitudes. The findings reveal a novel commensurability between the radius of a circle and the side length of a square using a shared fixed point coupled via a 3:4:5 Pythagorean-triple triangle, introducing the concept of ordered proportions. Full article
(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)
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15 pages, 301 KB  
Article
Existence, Optimal Control, and Numerical Analysis of a Caputo Fractional Model for Oxygen Saturation Regulation
by Nawal Alharbi
Symmetry 2026, 18(3), 482; https://doi.org/10.3390/sym18030482 - 11 Mar 2026
Viewed by 286
Abstract
Fractional-order models are widely recognized for their ability to capture memory and hereditary effects in biological and physiological systems. In this paper, we develop and analyze a Caputo fractional-order dynamical model for the regulation of blood oxygen saturation (SpO2) under bounded [...] Read more.
Fractional-order models are widely recognized for their ability to capture memory and hereditary effects in biological and physiological systems. In this paper, we develop and analyze a Caputo fractional-order dynamical model for the regulation of blood oxygen saturation (SpO2) under bounded control inputs. The model incorporates nonlinear saturation mechanisms and auxiliary state variables to represent delayed oxygen transport and adaptation effects. By reformulating the system as an operator equation in a suitable Banach space, sufficient conditions for existence and uniqueness of solutions are established using fixed-point theory. An optimal control problem is then formulated to steer oxygen saturation toward a prescribed safe target level, and the existence of an optimal control is proved via compactness arguments and the direct method of the calculus of variations. Numerical simulations are provided to illustrate the theoretical findings and to demonstrate the impact of the fractional order on transient oxygen saturation dynamics, including comparison with the classical integer-order case. The results show that fractional modeling offers a mathematically rigorous and physiologically interpretable framework for describing delayed oxygenation responses and achieving stable regulation under bounded control constraints. Full article
(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)
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19 pages, 792 KB  
Article
Generalized Ishikawa Iterative Algorithm with Errors and Variable Generalized Ishikawa Iterative Algorithm for Nonexpansive Mappings in Symmetric Banach Spaces
by Liangjuan Yu, Yuhan Zhu and Wenying Zhao
Symmetry 2026, 18(1), 125; https://doi.org/10.3390/sym18010125 - 9 Jan 2026
Viewed by 392
Abstract
We present a generalized Ishikawa iterative algorithm with an error term and a variable generalized Ishikawa iterative algorithm. Leveraging the geometric symmetry inherent in uniformly convex Banach spaces, we establish their respective weak convergence theorems for nonexpansive mappings. As applications, we extend several [...] Read more.
We present a generalized Ishikawa iterative algorithm with an error term and a variable generalized Ishikawa iterative algorithm. Leveraging the geometric symmetry inherent in uniformly convex Banach spaces, we establish their respective weak convergence theorems for nonexpansive mappings. As applications, we extend several recent results in the literature related to the proximal point algorithm and the split feasibility problem. Consequently, we propose a hyper-generalized proximal point algorithm and a hyper-generalized perturbation CQ algorithm. Our work not only broadens the application scope of these methods but also highlights the foundational role of symmetric space properties in ensuring algorithmic convergence. Full article
(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)
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35 pages, 3250 KB  
Article
On a Novel Iterative Algorithm in CAT(0) Spaces with Qualitative Analysis and Applications
by Muhammad Khan, Mujahid Abbas and Cristian Ciobanescu
Symmetry 2025, 17(10), 1695; https://doi.org/10.3390/sym17101695 - 9 Oct 2025
Cited by 3 | Viewed by 699
Abstract
This study presents a novel and efficient iterative scheme in the setting of CAT(0) spaces and investigates the convergence properties for a generalized class of mappings satisfying the Garcia–Falset property using the proposed iterative scheme. Strong and weak convergence results are established in [...] Read more.
This study presents a novel and efficient iterative scheme in the setting of CAT(0) spaces and investigates the convergence properties for a generalized class of mappings satisfying the Garcia–Falset property using the proposed iterative scheme. Strong and weak convergence results are established in CAT(0) spaces, generalizing many existing results in the literature. Furthermore, we discuss the stability and data dependence of the new iterative process. Numerical experiments include an analysis of error values, the number of iterations, and computational time, providing a comprehensive assessment of the method’s performance. Moreover, graphical comparisons demonstrate the efficiency and reliability of the approach. The obtained results are utilized in solving integral equations. Additionally, the paper concludes with a polynomiographic study of the newly introduced iterative process, in comparison with standard algorithms, such as Newton, Halley, or Kalantari’s B4 iteration, emphasizing symmetry properties. Full article
(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)
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17 pages, 327 KB  
Article
Best Proximity Theory in Metrically Convex Menger PM-Spaces via Cyclic Kannan Maps
by Moosa Gabeleh, Elif Uyanık Ekici and Maggie Aphane
Symmetry 2025, 17(9), 1549; https://doi.org/10.3390/sym17091549 - 16 Sep 2025
Cited by 1 | Viewed by 898
Abstract
A Takahashi convex structure is considered on Menger PM-spaces and used to investigate the existence of best proximity points for weak cyclic Kannan contractions. We then introduce a concept of a probabilistic proximal quasi-normal structure on a convex pair of subsets of Menger [...] Read more.
A Takahashi convex structure is considered on Menger PM-spaces and used to investigate the existence of best proximity points for weak cyclic Kannan contractions. We then introduce a concept of a probabilistic proximal quasi-normal structure on a convex pair of subsets of Menger PM-spaces and prove that every compact and convex pair in metrically convex Menger PM-spaces has the probabilistic proximal quasi-normal structure. By applying this geometric property, we survey the existence of a best proximity point for cyclic relatively Kannan nonexpansive maps which preserves distance. In order to provide more accurate results, we obtain the same conclusions in the framework of CAT(0) spaces. Full article
(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)
47 pages, 15579 KB  
Article
Geometric Symmetry and Temporal Optimization in Human Pose and Hand Gesture Recognition for Intelligent Elderly Individual Monitoring
by Pongsarun Boonyopakorn and Mahasak Ketcham
Symmetry 2025, 17(9), 1423; https://doi.org/10.3390/sym17091423 - 1 Sep 2025
Cited by 3 | Viewed by 1786
Abstract
This study introduces a real-time, non-intrusive monitoring system designed to support elderly care through vision-based pose estimation and hand gesture recognition. The proposed framework integrates convolutional neural networks (CNNs), temporal modeling using LSTM networks, and symmetry-aware keypoint analysis to enhance the accuracy and [...] Read more.
This study introduces a real-time, non-intrusive monitoring system designed to support elderly care through vision-based pose estimation and hand gesture recognition. The proposed framework integrates convolutional neural networks (CNNs), temporal modeling using LSTM networks, and symmetry-aware keypoint analysis to enhance the accuracy and reliability of behavior detection under varied real-world conditions. By leveraging the bilateral symmetry of human anatomy, the system improves the robustness of posture and gesture classification, even in the presence of partial occlusion or variable lighting. A total of 21 hand landmarks and 33 body pose points are used to recognize predefined actions and communication gestures, enabling seamless interaction without wearable devices. Experimental evaluations across four distinct lighting environments confirm a consistent accuracy above 90%, with real-time alerts triggered via IoT messaging platforms. The system’s modular architecture, interpretability, and adaptability make it a scalable solution for intelligent elderly individual monitoring, offering a novel application of spatial symmetry and optimized deep learning in healthcare technology. Full article
(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)
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36 pages, 544 KB  
Article
Well-Posedness of Cauchy-Type Problems for Nonlinear Implicit Hilfer Fractional Differential Equations with General Order in Weighted Spaces
by Jakgrit Sompong, Samten Choden, Ekkarath Thailert and Sotiris K. Ntouyas
Symmetry 2025, 17(7), 986; https://doi.org/10.3390/sym17070986 - 22 Jun 2025
Viewed by 701
Abstract
This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness [...] Read more.
This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness and Ulam–Hyers stability are then derived using Banach’s contraction principle. By introducing a novel singular-kernel Gronwall inequality, we extend the analysis to Ulam–Hyers–Rassias stability and continuous dependence on initial data. The theoretical framework is unified for general fractional orders and validated through examples, demonstrating its applicability to implicit systems with memory effects. Key contributions include weighted-space analysis and stability criteria for this class of equations. Full article
(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)
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17 pages, 285 KB  
Article
Convergence Analysis of Reinforcement Learning Algorithms Using Generalized Weak Contraction Mappings
by Abdelkader Belhenniche, Roman Chertovskih and Rui Gonçalves
Symmetry 2025, 17(5), 750; https://doi.org/10.3390/sym17050750 - 13 May 2025
Cited by 2 | Viewed by 5073
Abstract
We investigate the convergence properties of policy iteration and value iteration algorithms in reinforcement learning by leveraging fixed-point theory, with a focus on mappings that exhibit weak contractive behavior. Unlike traditional studies that rely on strong contraction properties, such as those defined by [...] Read more.
We investigate the convergence properties of policy iteration and value iteration algorithms in reinforcement learning by leveraging fixed-point theory, with a focus on mappings that exhibit weak contractive behavior. Unlike traditional studies that rely on strong contraction properties, such as those defined by the Banach contraction principle, we consider a more general class of mappings that includes weak contractions. Employing Zamfirscu’s fixed-point theorem, we establish sufficient conditions for norm convergence in infinite-dimensional policy spaces under broad assumptions. Our approach extends the applicability of these algorithms to feedback control problems in reinforcement learning, where standard contraction conditions may not hold. Through illustrative examples, we demonstrate that this framework encompasses a wider range of operators, offering new insights into the robustness and flexibility of iterative methods in dynamic programming. Full article
(This article belongs to the Special Issue Symmetry in Fixed Point Theory and Optimization: Computations and Applications)
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