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Fixed Point, Optimization, and Applications II
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Fixed Point, Optimization, and Applications II

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 34211

Special Issue Editors

Prof. Dr. Mihai Postolache

grade E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, University Politehnica of Bucharest, Bucharest, Romania
Interests: fixed point theory; continuous optimization; numerical algorithms
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jen-Chih Yao

E-Mail Website
Guest Editor
1. Research Center for Interneural Computing, China Medical University Hospital, Taichung City 404332, Taiwan
2. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
Interests: vector optimization; fixed point theory; variational inequalities; complementarity problems; variational analysis; equilibrium problems; optimal control; generalized convexity and generalized monotonicity
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Yonghong Yao

E-Mail Website
Guest Editor
School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
Interests: nonlinear analysis; optimization
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is well known that fixed point theory in suitable spaces is nowadays an active research area. This is due to its versatility in the study of nonlinear phenomena of the real world. Results regarding existence, uniqueness, and numerical reckoning fixed points of nonlinear operators find diverse applications in theoretical and applied sciences.

Optimization plays an important role in the study of some characteristics that describe diverse nonlinear phenomena of the real world, such as efficiency, control, and much more. The research topics in this field include best approximation, numerical algorithms, optimal control, and well-posedness.

The aim of this Special Issue is to report new results in the two research areas recorded above: fixed point and optimization, and their applications. This Special Issue will accept high-quality papers containing original research results, with illustrative applications, and survey articles of exceptional merit.

The research topics include, but are not limited to, the following:

  • The existence and uniqueness of fixed points;
  • Best approximation problems;
  • Iteration processes for fixed points or best proximity points;
  • Nonlinear optimization and applications;
  • Variational inequalities and equilibrium problems;
  • Dynamical systems and special functions;
  • Well-posedness and optimal control.

Prof. Dr. Mihai Postolache
Prof. Dr. Jen-Chih Yao
Prof. Dr. Yonghong Yao
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

21 pages, 334 KiB  
Article
Common Best Proximity Point Theorems for Generalized Dominating with Graphs and Applications in Differential Equations
by Watchareepan Atiponrat, Anchalee Khemphet, Wipawinee Chaiwino, Teeranush Suebcharoen and Phakdi Charoensawan
Mathematics 2024, 12(2), 306; https://doi.org/10.3390/math12020306 - 17 Jan 2024
Viewed by 549
Abstract
In this paper, we initiate a concept of graph-proximal functions. Furthermore, we give a notion of being generalized Geraghty dominating for a pair of mappings. This permits us to establish the existence of and unique results for a common best proximity point of [...] Read more.
In this paper, we initiate a concept of graph-proximal functions. Furthermore, we give a notion of being generalized Geraghty dominating for a pair of mappings. This permits us to establish the existence of and unique results for a common best proximity point of complete metric space. Additionally, we give a concrete example and corollaries related to the main theorem. In particular, we apply our main results to the case of metric spaces equipped with a reflexive binary relation. Finally, we demonstrate the existence of a solution to boundary value problems of particular second-order differential equations. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
11 pages, 258 KiB  
Article
On Enriched Suzuki Mappings in Hadamard Spaces
by Teodor Turcanu and Mihai Postolache
Mathematics 2024, 12(1), 157; https://0-doi-org.brum.beds.ac.uk/10.3390/math12010157 - 03 Jan 2024
Viewed by 633
Abstract
We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. [...] Read more.
We define and study enriched Suzuki mappings in Hadamard spaces. The results obtained here are extending fundamental findings previously established in related research. The extension is realized with respect to at least two different aspects: the setting and the class of involved operators. More accurately, Hilbert spaces are particular Hadamard spaces, while enriched Suzuki nonexpansive mappings are natural generalizations of enriched nonexpansive mappings. Next, enriched Suzuki nonexpansive mappings naturally contain Suzuki nonexpansive mappings in Hadamard spaces. Besides technical lemmas, the results of this paper deal with (1) the existence of fixed points for enriched Suzuki nonexpansive mappings and (2) Δ and strong (metric) convergence of Picard iterates of the α-averaged mapping, which are exactly Krasnoselskij iterates for the original mapping. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
17 pages, 365 KiB  
Article
Fixed Point Results in Controlled Fuzzy Metric Spaces with an Application to the Transformation of Solar Energy to Electric Power
by Umar Ishtiaq, Doha A. Kattan, Khaleel Ahmad, Salvatore Sessa and Farhan Ali
Mathematics 2023, 11(15), 3435; https://0-doi-org.brum.beds.ac.uk/10.3390/math11153435 - 07 Aug 2023
Cited by 1 | Viewed by 587
Abstract
In this manuscript, we give sufficient conditions for a sequence to be Cauchy in the context of controlled fuzzy metric space. Furthermore, we generalize the concept of Banach’s contraction principle by utilizing several new contraction conditions and prove several fixed point results. Furthermore, [...] Read more.
In this manuscript, we give sufficient conditions for a sequence to be Cauchy in the context of controlled fuzzy metric space. Furthermore, we generalize the concept of Banach’s contraction principle by utilizing several new contraction conditions and prove several fixed point results. Furthermore, we provide a number of non-trivial examples to validate the superiority of main results in the existing literature. At the end, we discuss an important application to the transformation of solar energy to electric power by utilizing differential equations. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
20 pages, 699 KiB  
Article
Hybrid Fixed Point Theorems of Fuzzy Soft Set-Valued Maps with Applications in Integral Inclusions and Decision Making
by Mohammed Shehu Shagari, Maha Noorwali and Akbar Azam
Mathematics 2023, 11(6), 1393; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061393 - 13 Mar 2023
Cited by 1 | Viewed by 1092
Abstract
A lot of work has been completed in efforts to extend the notions of soft set and fuzzy soft set and their applications to other domains. However, neither of the two concepts has been examined in the study of functional equations in b [...] Read more.
A lot of work has been completed in efforts to extend the notions of soft set and fuzzy soft set and their applications to other domains. However, neither of the two concepts has been examined in the study of functional equations in b-metric spaces. Given this background information, this paper proposes the idea of b-hybrid fuzzy soft contraction in b-metric space and investigates new criteria for the existence of fixed points for such mappings. The significance of the obtained principal result lies in the fact that the contractive inequalities can be specialized in various ways, depending on the choice of the parameters, thereby making it possible to unify, deduce, and refine several corresponding results. To motivate further studies in the directions studied herein, two applications regarding decision-making problems and an existence theorem of integral inclusion are considered, using fuzzy soft set-valued maps. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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26 pages, 23051 KiB  
Article
A One-Parameter Memoryless DFP Algorithm for Solving System of Monotone Nonlinear Equations with Application in Image Processing
by Najib Ullah, Abdullah Shah, Jamilu Sabi’u, Xiangmin Jiao, Aliyu Muhammed Awwal, Nuttapol Pakkaranang , Said Karim Shah and Bancha Panyanak
Mathematics 2023, 11(5), 1221; https://0-doi-org.brum.beds.ac.uk/10.3390/math11051221 - 02 Mar 2023
Cited by 2 | Viewed by 1741
Abstract
In matrix analysis, the scaling technique reduces the chances of an ill-conditioning of the matrix. This article proposes a one-parameter scaling memoryless Davidon–Fletcher–Powell (DFP) algorithm for solving a system of monotone nonlinear equations with convex constraints. The measure function that involves all the [...] Read more.
In matrix analysis, the scaling technique reduces the chances of an ill-conditioning of the matrix. This article proposes a one-parameter scaling memoryless Davidon–Fletcher–Powell (DFP) algorithm for solving a system of monotone nonlinear equations with convex constraints. The measure function that involves all the eigenvalues of the memoryless DFP matrix is minimized to obtain the scaling parameter’s optimal value. The resulting algorithm is matrix and derivative-free with low memory requirements and is globally convergent under some mild conditions. A numerical comparison showed that the algorithm is efficient in terms of the number of iterations, function evaluations, and CPU time. The performance of the algorithm is further illustrated by solving problems arising from image restoration. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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15 pages, 357 KiB  
Article
A New Accelerated Algorithm Based on Fixed Point Method for Convex Bilevel Optimization Problems with Applications
by Piti Thongsri, Bancha Panyanak and Suthep Suantai
Mathematics 2023, 11(3), 702; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030702 - 30 Jan 2023
Cited by 4 | Viewed by 1293
Abstract
A new accelerated common fixed point algorithm is introduced and analyzed for a countable family of nonexpansive mappings and then we apply it to solve some convex bilevel optimization problems. Then, under some suitable conditions, we prove a strong convergence result of the [...] Read more.
A new accelerated common fixed point algorithm is introduced and analyzed for a countable family of nonexpansive mappings and then we apply it to solve some convex bilevel optimization problems. Then, under some suitable conditions, we prove a strong convergence result of the proposed algorithm. As an application, we employ the proposed algorithm for regression and classification problems. Moreover, we compare the performance of our algorithm with others. By numerical experiments, we found that our algorithm has a better performance than the others. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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11 pages, 266 KiB  
Article
Common Fixed Point of Two L2 Operators with Convergence Analysis and Application
by Cristina Calineata, Cristian Ciobanescu and Teodor Turcanu
Mathematics 2023, 11(3), 577; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030577 - 21 Jan 2023
Viewed by 826
Abstract
This article introduces a new numerical algorithm for approximating the solution of the common fixed point problem for two operators defined on CAT(0) spaces, belonging to the class L2, which was very recently introduced. The main results refer [...] Read more.
This article introduces a new numerical algorithm for approximating the solution of the common fixed point problem for two operators defined on CAT(0) spaces, belonging to the class L2, which was very recently introduced. The main results refer to Δ and strong convergence of the sequence generated by the new algorithm. A distinct feature of the adopted approach is the use of equivalent sequences. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
23 pages, 338 KiB  
Article
Fixed Point Theorems on Almost (φ,θ)-Contractions in Jleli-Samet Generalized Metric Spaces
by Doru Dumitrescu and Ariana Pitea
Mathematics 2022, 10(22), 4239; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224239 - 13 Nov 2022
Cited by 1 | Viewed by 894
Abstract
In this paper, we present extensions of some classic results regarding the existence and uniqueness of fixed points of operators fulfilling generalized contractive conditions defined by sums involving some functions with suitable properties in the setting of Jleli–Samet generalized metric spaces. As a [...] Read more.
In this paper, we present extensions of some classic results regarding the existence and uniqueness of fixed points of operators fulfilling generalized contractive conditions defined by sums involving some functions with suitable properties in the setting of Jleli–Samet generalized metric spaces. As a consequence, some known results in the literature are obtained. Examples are provided to prove the usability of our developments. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
16 pages, 556 KiB  
Article
New Approach to Split Variational Inclusion Issues through a Three-Step Iterative Process
by Andreea Bejenaru and Mihai Postolache
Mathematics 2022, 10(19), 3617; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193617 - 02 Oct 2022
Viewed by 1195
Abstract
Split variational inclusions are revealed as a large class of problems that includes several other pre-existing split-type issues: split feasibility, split zeroes problems, split variational inequalities and so on. This makes them not only a rich direction of theoretical study but also one [...] Read more.
Split variational inclusions are revealed as a large class of problems that includes several other pre-existing split-type issues: split feasibility, split zeroes problems, split variational inequalities and so on. This makes them not only a rich direction of theoretical study but also one with important and varied practical applications: large dimensional linear systems, optimization, signal reconstruction, boundary value problems and others. In this paper, the existing algorithmic tools are complemented by a new procedure based on a three-step iterative process. The resulting approximating sequence is proved to be weakly convergent toward a solution. The operation mode of the new algorithm is tracked in connection with mixed optimization–feasibility and mixed linear–feasibility systems. Standard polynomiographic techniques are applied for a comparative visual analysis of the convergence behavior. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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17 pages, 1132 KiB  
Article
A Derivative-Free MZPRP Projection Method for Convex Constrained Nonlinear Equations and Its Application in Compressive Sensing
by Ibrahim Mohammed Sulaiman, Aliyu Muhammed Awwal, Maulana Malik, Nuttapol Pakkaranang and Bancha Panyanak
Mathematics 2022, 10(16), 2884; https://0-doi-org.brum.beds.ac.uk/10.3390/math10162884 - 12 Aug 2022
Cited by 8 | Viewed by 1348
Abstract
Nonlinear systems of equations are widely used in science and engineering and, therefore, exploring efficient ways to solve them is paramount. In this paper, a new derivative-free approach for solving a nonlinear system of equations with convex constraints is proposed. The search direction [...] Read more.
Nonlinear systems of equations are widely used in science and engineering and, therefore, exploring efficient ways to solve them is paramount. In this paper, a new derivative-free approach for solving a nonlinear system of equations with convex constraints is proposed. The search direction of the proposed method is derived based on a modified conjugate gradient method, in such a way that it is sufficiently descent. It is worth noting that, unlike many existing methods that require a monotonicity assumption to prove the convergence result, our new method needs the underlying function to be pseudomonotone, which is a weaker assumption. The performance of the proposed algorithm is demonstrated on a set of some test problems and applications arising from compressive sensing. The obtained results confirm that the proposed method is effective compared to some existing algorithms in the literature. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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29 pages, 2513 KiB  
Article
On Some Properties of a Class of Eventually Locally Mixed Cyclic/Acyclic Multivalued Self-Mappings with Application Examples
by Manuel De la Sen and Asier Ibeas
Mathematics 2022, 10(14), 2415; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142415 - 11 Jul 2022
Cited by 1 | Viewed by 910
Abstract
In this paper, a multivalued self-mapping is defined on the union of a finite number of subsets p≥2 of a metric space which is, in general, of a mixed cyclic and acyclic nature in the sense that it can perform some [...] Read more.
In this paper, a multivalued self-mapping is defined on the union of a finite number of subsets p≥2 of a metric space which is, in general, of a mixed cyclic and acyclic nature in the sense that it can perform some iterations within each of the subsets before executing a switching action to its right adjacent one when generating orbits. The self-mapping can have combinations of locally contractive, non-contractive/non-expansive and locally expansive properties for some of the switching between different pairs of adjacent subsets. The properties of the asymptotic boundedness of the distances associated with the elements of the orbits are achieved under certain conditions of the global dominance of the contractivity of groups of consecutive iterations of the self-mapping, with each of those groups being of non-necessarily fixed size. If the metric space is a uniformly convex Banach one and the subsets are closed and convex, then some particular results on the convergence of the sequences of iterates to the best proximity points of the adjacent subsets are obtained in the absence of eventual local expansivity for switches between all the pairs of adjacent subsets. An application of the stabilization of a discrete dynamic system subject to impulsive effects in its dynamics due to finite discontinuity jumps in its state is also discussed. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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11 pages, 309 KiB  
Article
Fixed Point Results for Perov–Ćirić–Prešić-Type Θ-Contractions with Applications
by Jamshaid Ahmad, Saleh Abdullah Al-Mezel and Ravi P. Agarwal
Mathematics 2022, 10(12), 2062; https://0-doi-org.brum.beds.ac.uk/10.3390/math10122062 - 15 Jun 2022
Cited by 2 | Viewed by 1104
Abstract
The aim of this paper is to introduce the notion of Perov–Ćirić–Prešić-type Θ-contractions and to obtain some generalized fixed point theorems in the setting of vector-valued metric spaces. We derive some fixed point results as consequences of our main results. A nontrivial [...] Read more.
The aim of this paper is to introduce the notion of Perov–Ćirić–Prešić-type Θ-contractions and to obtain some generalized fixed point theorems in the setting of vector-valued metric spaces. We derive some fixed point results as consequences of our main results. A nontrivial example is also provided to support the validity of our established results. As an application, we investigate the solution of a semilinear operator system in Banach space. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
9 pages, 266 KiB  
Article
Ćirić-Type Operators and Common Fixed Point Theorems
by Claudia Luminiţa Mihiţ, Ghiocel Moţ and Gabriela Petruşel
Mathematics 2022, 10(11), 1947; https://0-doi-org.brum.beds.ac.uk/10.3390/math10111947 - 06 Jun 2022
Cited by 1 | Viewed by 1125
Abstract
In the context of a complete metric space, we will consider the common fixed point problem for two self operators. The operators are assumed to satisfy a general contraction type condition inspired by the Ćirić fixed point theorems. Under some appropriate conditions we [...] Read more.
In the context of a complete metric space, we will consider the common fixed point problem for two self operators. The operators are assumed to satisfy a general contraction type condition inspired by the Ćirić fixed point theorems. Under some appropriate conditions we establish existence, uniqueness and approximation results for the common fixed point. In the same framework, the second problem is to study various stability properties. More precisely, we will obtain sufficient conditions assuring that the common fixed point problem is well-posed and has the Ulam–Hyers stability, as well as the Ostrowski property for the considered problem. Some examples and applications are finally given in order to illustrate the abstract theorems proposed in the first part of the paper. Our results extend and complement some theorems in the recent literature. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
15 pages, 336 KiB  
Article
Self-Adaptive Method and Inertial Modification for Solving the Split Feasibility Problem and Fixed-Point Problem of Quasi-Nonexpansive Mapping
by Yuanheng Wang, Tiantian Xu, Jen-Chih Yao and Bingnan Jiang
Mathematics 2022, 10(9), 1612; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091612 - 09 May 2022
Cited by 5 | Viewed by 1142
Abstract
The split feasibility problem (SFP) has many practical applications, which has attracted the attention of many authors. In this paper, we propose a different method to solve the SFP and the fixed-point problem involving quasi-nonexpansive mappings. We relax the conditions of the operator [...] Read more.
The split feasibility problem (SFP) has many practical applications, which has attracted the attention of many authors. In this paper, we propose a different method to solve the SFP and the fixed-point problem involving quasi-nonexpansive mappings. We relax the conditions of the operator as well as consider the inertial iteration and the adaptive step size. For example, the convergence generated by our new method is better than that of other algorithms, and the convergence rate of our algorithm greatly improves that of previous algorithms. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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7 pages, 281 KiB  
Article
Common Fixed-Point and Fixed-Circle Results for a Class of Discontinuous F-Contractive Mappings †
by Pradip Debnath
Mathematics 2022, 10(9), 1605; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091605 - 09 May 2022
Cited by 1 | Viewed by 1525
Abstract
The exploration of contractive inequalities which do not imply the continuity of the mapping at fixed points was an interesting open problem for quite some time. A significant amount of progress was made in the last two decades towards the solution of this [...] Read more.
The exploration of contractive inequalities which do not imply the continuity of the mapping at fixed points was an interesting open problem for quite some time. A significant amount of progress was made in the last two decades towards the solution of this problem. In the current paper, we attempt to address the question of discontinuity at fixed point with the help of F-contractions in a metric space. We establish a common fixed-point (CFP) result for such contractive mappings and investigate its discontinuity at the CFP. A fixed-circle result is also obtained consequently. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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20 pages, 377 KiB  
Article
Modified Mann Subgradient-like Extragradient Rules for Variational Inequalities and Common Fixed Points Involving Asymptotically Nonexpansive Mappings
by Lu-Chuan Ceng, Yekini Shehu and Jen-Chih Yao
Mathematics 2022, 10(5), 779; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050779 - 28 Feb 2022
Cited by 1 | Viewed by 1584
Abstract
In a real Hilbert space, we aim to investigate two modified Mann subgradient-like methods to find a solution to pseudo-monotone variational inequalities, which is also a common fixed point of a finite family of nonexpansive mappings and an asymptotically nonexpansive mapping. We obtain [...] Read more.
In a real Hilbert space, we aim to investigate two modified Mann subgradient-like methods to find a solution to pseudo-monotone variational inequalities, which is also a common fixed point of a finite family of nonexpansive mappings and an asymptotically nonexpansive mapping. We obtain strong convergence results for the sequences constructed by these proposed rules. We give some examples to illustrate our analysis. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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16 pages, 328 KiB  
Article
A New Nonparametric Filled Function Method for Integer Programming Problems with Constraints
by Suxia Ma, Yuelin Gao, Bo Zhang and Wenlu Zuo
Mathematics 2022, 10(5), 734; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050734 - 25 Feb 2022
Cited by 2 | Viewed by 978
Abstract
In this paper, we investigate and develop a new filled function method for solving integer programming problems with constraints. By adopting the appropriate equivalent transformation method, these problems are transformed into a class of box-constrained integer programming problems. Then, an effective nonparametric filled [...] Read more.
In this paper, we investigate and develop a new filled function method for solving integer programming problems with constraints. By adopting the appropriate equivalent transformation method, these problems are transformed into a class of box-constrained integer programming problems. Then, an effective nonparametric filled function is constructed, and a new global optimization algorithm is designed using the discrete steepest descent method. Numerical experiments illustrate that this algorithm has effectiveness, feasibility, and better global optimization ability. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
19 pages, 374 KiB  
Article
Optimal Control Problems for Set-Valued Quasivariational Inequalities with Applications
by Shih-Sen Chang, Salahuddin, Lin Wang, Jinfang Tang and Liangcai Zhao
Mathematics 2022, 10(5), 691; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050691 - 23 Feb 2022
Viewed by 995
Abstract
In this paper we investigate the optimal control problem for set-valued quasivariational inequality with unilateral constraints. Under suitable conditions, we prove that the solution to the current optimal control problem converges to a solution to old control problems. By way of application, we [...] Read more.
In this paper we investigate the optimal control problem for set-valued quasivariational inequality with unilateral constraints. Under suitable conditions, we prove that the solution to the current optimal control problem converges to a solution to old control problems. By way of application, we utilize our results presented in the paper to study the optimal control associated with boundary value problems which is described by frictional contact problems and a stationary heat transfer problem with unilateral constraints. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
16 pages, 443 KiB  
Article
Common Fixed Points of Operators with Property (E) in CAT(0) Spaces
by Andreea Bejenaru and Cristian Ciobanescu
Mathematics 2022, 10(3), 433; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030433 - 29 Jan 2022
Cited by 6 | Viewed by 1821
Abstract
This paper features the search for common fixed points of two operators in the nonlinear metric setting provided by CAT(0) spaces. The analysis is performed for the generalized nonexpansivity condition known as condition (E), Garcia-Falset et al., and relies on the three step [...] Read more.
This paper features the search for common fixed points of two operators in the nonlinear metric setting provided by CAT(0) spaces. The analysis is performed for the generalized nonexpansivity condition known as condition (E), Garcia-Falset et al., and relies on the three step iteration procedure Sn by Sintunavarat and Pitea. The convergence analysis reveals the approximate solutions as limit points for an iteration sequence, where both the nonexpansive mappings to be analyzed and the specific curved structure of the framework interfere. To point out properly the meaning of this approach, we provide also examples accompanied by numerical simulations. The Poincaré half-plane is one of the non-positively curved setting to be used. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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18 pages, 303 KiB  
Article
Algorithm for Two Generalized Nonexpansive Mappings in Uniformly Convex Spaces
by Gabriela Ioana Usurelu, Teodor Turcanu and Mihai Postolache
Mathematics 2022, 10(3), 318; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030318 - 20 Jan 2022
Cited by 5 | Viewed by 1281
Abstract
In this paper, we study the common fixed-point problem for a pair of García-Falset mapping and (α,β)-generalized hybrid mapping in uniformly convex Banach spaces. For this purpose, we construct a modified three-step iteration by properly including together these [...] Read more.
In this paper, we study the common fixed-point problem for a pair of García-Falset mapping and (α,β)-generalized hybrid mapping in uniformly convex Banach spaces. For this purpose, we construct a modified three-step iteration by properly including together these two types of mappings into its formula. Under this modified iteration, a necessary and sufficient condition for the existence of a common fixed point as well as weak and strong convergence outcomes are phrased under some additional conditions. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
18 pages, 346 KiB  
Article
A Relaxed and Bound Algorithm Based on Auxiliary Variables for Quadratically Constrained Quadratic Programming Problem
by Chenyang Hu, Yuelin Gao, Fuping Tian and Suxia Ma
Mathematics 2022, 10(2), 270; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020270 - 16 Jan 2022
Viewed by 1275
Abstract
Quadratically constrained quadratic programs (QCQP), which often appear in engineering practice and management science, and other fields, are investigated in this paper. By introducing appropriate auxiliary variables, QCQP can be transformed into its equivalent problem (EP) with non-linear equality constraints. After these equality [...] Read more.
Quadratically constrained quadratic programs (QCQP), which often appear in engineering practice and management science, and other fields, are investigated in this paper. By introducing appropriate auxiliary variables, QCQP can be transformed into its equivalent problem (EP) with non-linear equality constraints. After these equality constraints are relaxed, a series of linear relaxation subproblems with auxiliary variables and bound constraints are generated, which can determine the effective lower bound of the global optimal value of QCQP. To enhance the compactness of sub-rectangles and improve the ability to remove sub-rectangles, two rectangle-reduction strategies are employed. Besides, two ϵ-subproblem deletion rules are introduced to improve the convergence speed of the algorithm. Therefore, a relaxation and bound algorithm based on auxiliary variables are proposed to solve QCQP. Numerical experiments show that this algorithm is effective and feasible. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
12 pages, 272 KiB  
Article
On a Periodic Boundary Value Problem for Fractional Quasilinear Differential Equations with a Self-Adjoint Positive Operator in Hilbert Spaces
by Mikhail Kamenskii, Garik Petrosyan, Paul Raynaud de Fitte and Jen-Chih Yao
Mathematics 2022, 10(2), 219; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020219 - 12 Jan 2022
Cited by 1 | Viewed by 1130
Abstract
In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in [...] Read more.
In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in Hilbert space and a nonlinear part is a map obeying Carathéodory type conditions. We find the mild solution of this problem in the form of a series in a Hilbert space. In the space of continuous functions, we construct the corresponding resolving operator, and for it, by using Schauder theorem, we prove the existence of a fixed point. At the end of the paper, we give an example for a boundary value problem for a diffusion type equation. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
17 pages, 304 KiB  
Article
On Mann-Type Subgradient-like Extragradient Method with Linear-Search Process for Hierarchical Variational Inequalities for Asymptotically Nonexpansive Mappings
by Lu-Chuan Ceng, Jen-Chih Yao and Yekini Shehu
Mathematics 2021, 9(24), 3322; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243322 - 20 Dec 2021
Cited by 3 | Viewed by 1682
Abstract
We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include [...] Read more.
We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
10 pages, 260 KiB  
Article
An Improved Alternating CQ Algorithm for Solving Split Equality Problems
by Yan-Juan He, Li-Jun Zhu and Nan-Nan Tan
Mathematics 2021, 9(24), 3313; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243313 - 19 Dec 2021
Viewed by 1727
Abstract
The CQ algorithm is widely used in the scientific field and has a significant impact on phase retrieval, medical image reconstruction, signal processing, etc. Moudafi proposed an alternating CQ algorithm to solve the split equality problem, but he only obtained the result of [...] Read more.
The CQ algorithm is widely used in the scientific field and has a significant impact on phase retrieval, medical image reconstruction, signal processing, etc. Moudafi proposed an alternating CQ algorithm to solve the split equality problem, but he only obtained the result of weak convergence. The work of this paper is to improve his algorithm so that the generated iterative sequence can converge strongly. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
13 pages, 254 KiB  
Article
Existence and Generic Stability of Strong Noncooperative Equilibria of Vector-Valued Games
by Yu Zhang, Shih-Sen Chang and Tao Chen
Mathematics 2021, 9(24), 3158; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243158 - 07 Dec 2021
Cited by 2 | Viewed by 1828
Abstract
In this paper, we obtain an existence theorem of general strong noncooperative equilibrium point of vector-valued games, in which every player maximizes all goals. We also obtain an existence theorem of strong equilibrium point of vector-valued games with single-leader–multi-follower framework by using the [...] Read more.
In this paper, we obtain an existence theorem of general strong noncooperative equilibrium point of vector-valued games, in which every player maximizes all goals. We also obtain an existence theorem of strong equilibrium point of vector-valued games with single-leader–multi-follower framework by using the upper semicontinuous of parametric strong noncooperative equilibrium point set of the followers. Moreover, we obtain some results on the generic stability of general strong noncooperative equilibrium point vector-valued games. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
8 pages, 233 KiB  
Article
Sehgal–Guseman-Type Fixed Point Theorem in b-Rectangular Metric Spaces
by Dingwei Zheng, Guofei Ye and Dawei Liu
Mathematics 2021, 9(24), 3149; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243149 - 07 Dec 2021
Cited by 3 | Viewed by 1856
Abstract
In this paper, we prove a Sehgal–Guseman-type fixed point theorem in b-rectangular metric spaces which provides a complete solution to an open problem raised by Zoran D. Mitrović (A note on a Banach’s fixed point theorem in b-rectangular metric space and b [...] Read more.
In this paper, we prove a Sehgal–Guseman-type fixed point theorem in b-rectangular metric spaces which provides a complete solution to an open problem raised by Zoran D. Mitrović (A note on a Banach’s fixed point theorem in b-rectangular metric space and b-metric space). The result presented in the paper generalizes and unifies some results in fixed point theory. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications II)
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