Special Issue "Fixed Point, Optimization, and Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 31 October 2019

Special Issue Editors

Guest Editor
Prof. Dr. Mihai Postolache

Department of General Education, China Medical University, 40402 Taichung, Taiwan & Department of Mathematics and Computer Science, University Politehnica of Bucharest, 060042 Bucharest, Romania
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Interests: fixed point theory and applications; nalgorithms; continuous optimization
Guest Editor
Prof. Dr. Jen-Chih Yao

Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taiwan
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Interests: vector optimization; fixed point theory; variational inequalities; complementarity problems; variational analysis; equilibrium problems; optimal control; generalized convexity and generalized monotonicity
Guest Editor
Prof. Dr. Yonghong Yao

School of Mathematical Sciences, Tianjin Polytechnic University, China
Website | E-Mail
Interests: nonlinear analysis, optimization

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.

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

Prof. Dr. Mihai Postolache
Prof. Dr. Jen-Chih Yao
Prof. Dr. Yonghong Yao
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics 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 850 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 (17 papers)

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Research

Open AccessArticle
Prešić Type Nonself Operators and Related Best Proximity Results
Mathematics 2019, 7(5), 394; https://doi.org/10.3390/math7050394
Received: 14 March 2019 / Revised: 25 April 2019 / Accepted: 25 April 2019 / Published: 30 April 2019
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Abstract
The purpose of this article is to discuss the existence of best proximity points for Presˇic´-type nonself operators, say T:AkB. We also give several examples to support our results. As a [...] Read more.
The purpose of this article is to discuss the existence of best proximity points for Pre s ˇ i c ´ -type nonself operators, say T : A k B . We also give several examples to support our results. As a consequence of our results, we have provided some interesting formulations of Pre s ˇ i c ´ fixed point results. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessFeature PaperArticle
Numerical Reckoning Fixed Points of (ρE)-Type Mappings in Modular Vector Spaces
Mathematics 2019, 7(5), 390; https://doi.org/10.3390/math7050390
Received: 6 February 2019 / Revised: 15 April 2019 / Accepted: 25 April 2019 / Published: 29 April 2019
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Abstract
In this paper, we study an iteration process introduced by Thakur et al. for Suzuki mappings in Banach spaces, in the new context of modular vector spaces. We establish existence results for a more recent version of Suzuki generalized non-expansive mappings. The stability [...] Read more.
In this paper, we study an iteration process introduced by Thakur et al. for Suzuki mappings in Banach spaces, in the new context of modular vector spaces. We establish existence results for a more recent version of Suzuki generalized non-expansive mappings. The stability and data dependence of the scheme for ρ -contractions is studied as well. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
Generalized Viscosity Implicit Iterative Process for Asymptotically Non-Expansive Mappings in Banach Spaces
Mathematics 2019, 7(5), 379; https://doi.org/10.3390/math7050379
Received: 2 April 2019 / Revised: 21 April 2019 / Accepted: 23 April 2019 / Published: 26 April 2019
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Abstract
In this paper, we propose a generalized viscosity implicit iterative method for asymptotically non-expansive mappings in Banach spaces. The strong convergence theorem of this algorithm is proved, which solves the variational inequality problem. Moreover, we provide some applications to zero-point problems and equilibrium [...] Read more.
In this paper, we propose a generalized viscosity implicit iterative method for asymptotically non-expansive mappings in Banach spaces. The strong convergence theorem of this algorithm is proved, which solves the variational inequality problem. Moreover, we provide some applications to zero-point problems and equilibrium problems. Further, a numerical example is given to illustrate our convergence analysis. The results generalize and improve corresponding results in the literature. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
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Open AccessArticle
Cayley Inclusion Problem Involving XOR-Operation
Mathematics 2019, 7(3), 302; https://doi.org/10.3390/math7030302
Received: 9 February 2019 / Revised: 14 March 2019 / Accepted: 21 March 2019 / Published: 25 March 2019
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Abstract
In this paper, we study an absolutely new problem, namely, the Cayley inclusion problem which involves the Cayley operator and a multi-valued mapping with XOR-operation. We have shown that the Cayley operator is a single-valued comparison and it is Lipschitz-type-continuous. A fixed point [...] Read more.
In this paper, we study an absolutely new problem, namely, the Cayley inclusion problem which involves the Cayley operator and a multi-valued mapping with XOR-operation. We have shown that the Cayley operator is a single-valued comparison and it is Lipschitz-type-continuous. A fixed point formulation of the Cayley inclusion problem is shown by using the concept of a resolvent operator as well as the Yosida approximation operator. Finally, an existence and convergence result is proved. An example is constructed for some of the concepts used in this work. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessFeature PaperArticle
Variational Inequalities Approaches to Minimization Problems with Constraints of Generalized Mixed Equilibria and Variational Inclusions
Mathematics 2019, 7(3), 270; https://doi.org/10.3390/math7030270
Received: 18 February 2019 / Revised: 10 March 2019 / Accepted: 13 March 2019 / Published: 16 March 2019
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Abstract
Multistep composite implicit and explicit extragradient-like schemes are presented for solving the minimization problem with the constraints of variational inclusions and generalized mixed equilibrium problems. Strong convergence results of introduced schemes are given under suitable control conditions. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
A General Algorithm for the Split Common Fixed Point Problem with Its Applications to Signal Processing
Mathematics 2019, 7(3), 226; https://doi.org/10.3390/math7030226
Received: 24 December 2018 / Revised: 18 February 2019 / Accepted: 20 February 2019 / Published: 28 February 2019
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Abstract
In 2014, Cui and Wang constructed an algorithm for demicontractive operators and proved some weak convergence theorems of their proposed algorithm to show the existence of solutions for the split common fixed point problem without using the operator norm. By Cui and Wang’s [...] Read more.
In 2014, Cui and Wang constructed an algorithm for demicontractive operators and proved some weak convergence theorems of their proposed algorithm to show the existence of solutions for the split common fixed point problem without using the operator norm. By Cui and Wang’s motivation, in 2015, Boikanyo constructed also a new algorithm for demicontractive operators and obtained some strong convergence theorems for this problem without using the operator norm. In this paper, we consider a viscosity iterative algorithm in Boikanyo’s algorithm to approximate to a solution of this problem and prove some strong convergence theorems of our proposed algorithm to a solution of this problem. Finally, we apply our main results to some applications, signal processing and others and compare our algorithm with five algorithms such as Cui and Wang’s algorithm, Boikanyo’s algorithm, forward-backward splitting algorithm and the fast iterative shrinkage-thresholding algorithm (FISTA). Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
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Open AccessArticle
Some Mann-Type Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems Variational Inequalities and Fixed Point Problems
Mathematics 2019, 7(3), 218; https://doi.org/10.3390/math7030218
Received: 31 December 2018 / Revised: 19 February 2019 / Accepted: 21 February 2019 / Published: 26 February 2019
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Abstract
This paper discusses a monotone variational inequality problem with a variational inequality constraint over the common solution set of a general system of variational inequalities (GSVI) and a common fixed point (CFP) of a countable family of nonexpansive mappings and an asymptotically nonexpansive [...] Read more.
This paper discusses a monotone variational inequality problem with a variational inequality constraint over the common solution set of a general system of variational inequalities (GSVI) and a common fixed point (CFP) of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping in Hilbert spaces, which is called the triple hierarchical constrained variational inequality (THCVI), and introduces some Mann-type implicit iteration methods for solving it. Norm convergence of the proposed methods of the iteration methods is guaranteed under some suitable assumptions. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
Extension and Application of the Yamada Iteration Algorithm in Hilbert Spaces
Mathematics 2019, 7(3), 215; https://doi.org/10.3390/math7030215
Received: 17 January 2019 / Revised: 19 February 2019 / Accepted: 20 February 2019 / Published: 26 February 2019
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Abstract
In this paper, based on the Yamada iteration, we propose an iteration algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping. We obtain a weak convergence [...] Read more.
In this paper, based on the Yamada iteration, we propose an iteration algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly-monotone mapping. We obtain a weak convergence theorem in Hilbert space. In particular, the set of zero points of an inverse strongly-monotone mapping can be transformed into the solution set of the variational inequality problem. Further, based on this result, we also obtain some new weak convergence theorems which are used to solve the equilibrium problem and the split feasibility problem. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
Convergence and Best Proximity Points for Generalized Contraction Pairs
Mathematics 2019, 7(2), 176; https://doi.org/10.3390/math7020176
Received: 12 December 2018 / Revised: 7 February 2019 / Accepted: 8 February 2019 / Published: 15 February 2019
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Abstract
This paper is devoted to studying the existence of best proximity points and convergence for a class of generalized contraction pairs by using the concept of proximally-complete pairs and proximally-complete semi-sharp proximinal pairs. The obtained results are generalizations of the result of Sadiq [...] Read more.
This paper is devoted to studying the existence of best proximity points and convergence for a class of generalized contraction pairs by using the concept of proximally-complete pairs and proximally-complete semi-sharp proximinal pairs. The obtained results are generalizations of the result of Sadiq Basha (Basha, S., Best proximity points: global optimal approximate solutions, J. Glob. Optim. 2011, 49, 15–21) As an application, we give a result for nonexpansive mappings in normed vector spaces. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces
Mathematics 2019, 7(2), 156; https://doi.org/10.3390/math7020156
Received: 6 January 2019 / Revised: 28 January 2019 / Accepted: 30 January 2019 / Published: 8 February 2019
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Abstract
In this article, we study a modified viscosity splitting method combined with inertial extrapolation for accretive operators in Banach spaces and then establish a strong convergence theorem for such iterations under some suitable assumptions on the sequences of parameters. As an application, we [...] Read more.
In this article, we study a modified viscosity splitting method combined with inertial extrapolation for accretive operators in Banach spaces and then establish a strong convergence theorem for such iterations under some suitable assumptions on the sequences of parameters. As an application, we extend our main results to solve the convex minimization problem. Moreover, the numerical experiments are presented to support the feasibility and efficiency of the proposed method. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
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Open AccessArticle
Sehgal Type Contractions on Dislocated Spaces
Mathematics 2019, 7(2), 153; https://doi.org/10.3390/math7020153
Received: 17 December 2018 / Revised: 1 February 2019 / Accepted: 2 February 2019 / Published: 6 February 2019
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Abstract
In this paper, we investigate the contractive type inequalities for the iteration of the mapping at a given point in the setting of dislocated metric space. We consider an example to illustrate the validity of the given result. Further, as an application, we [...] Read more.
In this paper, we investigate the contractive type inequalities for the iteration of the mapping at a given point in the setting of dislocated metric space. We consider an example to illustrate the validity of the given result. Further, as an application, we propose a solution for a boundary value problem of the second order differential equation. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
An Iterative Approach to the Solutions of Proximal Split Feasibility Problems
Mathematics 2019, 7(2), 145; https://doi.org/10.3390/math7020145
Received: 10 January 2019 / Revised: 25 January 2019 / Accepted: 28 January 2019 / Published: 3 February 2019
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Abstract
The proximal split feasibility problem is investigated in Hilbert spaces. An iterative procedure is introduced for finding the solution of the proximal split feasibility problem. Strong convergence analysis of the presented algorithm is proved. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
ON H+Type Multivalued Contractions and Applications in Symmetric and Probabilistic Spaces
Mathematics 2019, 7(2), 144; https://doi.org/10.3390/math7020144
Received: 29 December 2018 / Revised: 30 January 2019 / Accepted: 30 January 2019 / Published: 3 February 2019
Cited by 2 | PDF Full-text (303 KB) | HTML Full-text | XML Full-text
Abstract
The main idea in this article is to establish some fixed and common fixed point results for multivalued H+-type contraction mappings in symmetric spaces. New results are accompanied with illustrative examples. An application of the obtained results to probabilistic spaces is [...] Read more.
The main idea in this article is to establish some fixed and common fixed point results for multivalued H + -type contraction mappings in symmetric spaces. New results are accompanied with illustrative examples. An application of the obtained results to probabilistic spaces is presented. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems
Mathematics 2019, 7(2), 142; https://doi.org/10.3390/math7020142
Received: 22 December 2018 / Revised: 26 January 2019 / Accepted: 29 January 2019 / Published: 2 February 2019
Cited by 2 | PDF Full-text (844 KB) | HTML Full-text | XML Full-text
Abstract
In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common [...] Read more.
In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common fixed points of an asymptotically nonexpansive mapping and a countable of uniformly Lipschitzian pseudocontractive mappings in Hilbert spaces, which is called the triple hierarchical constrained variational inequality. Strong convergence of the proposed method to the unique solution of the problem is guaranteed under some suitable assumptions. As a sub-result, we provide an algorithm to solve problem of common fixed points of pseudocontractive, nonexpansive mappings, variational inequality problems and generalized mixed bifunction equilibrium problems in Hilbert spaces. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
An Iterative Algorithm for Solving Generalized Variational Inequalities and Fixed Points Problems
Mathematics 2019, 7(1), 61; https://doi.org/10.3390/math7010061
Received: 20 December 2018 / Revised: 4 January 2019 / Accepted: 6 January 2019 / Published: 8 January 2019
Cited by 6 | PDF Full-text (793 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested [...] Read more.
In this paper, a generalized variational inequality and fixed points problem is presented. An iterative algorithm is introduced for finding a solution of the generalized variational inequalities and fixed point of two quasi-pseudocontractive operators under a nonlinear transformation. Strong convergence of the suggested algorithm is demonstrated. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
A Class of Nonlinear Fuzzy Variational Inequality Problems
Mathematics 2019, 7(1), 54; https://doi.org/10.3390/math7010054
Received: 11 December 2018 / Revised: 1 January 2019 / Accepted: 3 January 2019 / Published: 7 January 2019
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Abstract
In this paper, we consider nonlinear variational inequality problems with fuzzy variables. The fuzzy variables were introduced to deal with the variational inequality containing noise for which historical data is not available. The fuzzy expected residual minimization (FERM) problems were established. We discussed [...] Read more.
In this paper, we consider nonlinear variational inequality problems with fuzzy variables. The fuzzy variables were introduced to deal with the variational inequality containing noise for which historical data is not available. The fuzzy expected residual minimization (FERM) problems were established. We discussed the S C 1 property of the FERM model. Furthermore, results of convergence analysis were obtained based on an approximation model of the FERM model. The convergence of global optimal solutions and the convergence of stationary points were analysed. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
Open AccessArticle
System of Variational Inclusions and Fixed Points of Pseudocontractive Mappings in Banach Spaces
Mathematics 2019, 7(1), 5; https://doi.org/10.3390/math7010005
Received: 9 December 2018 / Accepted: 18 December 2018 / Published: 20 December 2018
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Abstract
The purpose of this paper is to solve the general system of variational inclusions (GSVI) with hierarchical variational inequality (HVI) constraint, for an infinite family of continuous pseudocontractive mappings in Banach spaces. By utilizing the equivalence between the GSVI and the fixed point [...] Read more.
The purpose of this paper is to solve the general system of variational inclusions (GSVI) with hierarchical variational inequality (HVI) constraint, for an infinite family of continuous pseudocontractive mappings in Banach spaces. By utilizing the equivalence between the GSVI and the fixed point problem, we construct an implicit multiple-viscosity approximation method for solving the GSVI. Under very mild conditions, we prove the strong convergence of the proposed method to a solution of the GSVI with the HVI constraint, for infinitely many pseudocontractions. Full article
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)
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