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Variational Inequality
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Variational Inequality

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

Deadline for manuscript submissions: closed (31 July 2020) | Viewed by 32928

Special Issue Editor

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

Special Issue Information

Dear Colleagues,

It is well known that variational inequality was introduced by Hartman and Stampacchia in 1966, and was later expanded by Stampacchia. Since then, variational inequality and its various generalizations have become very effective and quite powerful tools in the study of the many problems arising from differential equations, mechanics, contact problems, optimization and control problems, management science, operations research, general equilibrium problems in economics and transportation, unilateral, obstacle, moving, and so on. The motivation of this Special Issue is to report and collect the recent developments of variational inequality in both theory and applications. This Special Issue deals mainly with the theory and applications of variational inequalities and related problems; in particular, any work including new ideas, novelty techniques, and/or results on variational inequalities and its relevant problems are welcome. We are accepting high-quality research or review papers. The purpose of this Special Issue is to connect the efforts of various scientists (particularly mathematicians and engineers), for whom various variational inequality problems are important in their research activity.

Prof. Dr. Jen-Chih Yao
Guest Editor

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Keywords

  • Variational inequality
  • Complementarity problem
  • Variational analysis
  • Optimization
  • Optimal control

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Related Special Issue

Published Papers (14 papers)

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Research

11 pages, 302 KiB  
Article
Interpolative Reich–Rus–Ćirić and Hardy–Rogers Contraction on Quasi-Partial b-Metric Space and Related Fixed Point Results
by Vishnu Narayan Mishra, Luis Manuel Sánchez Ruiz, Pragati Gautam and Swapnil Verma
Mathematics 2020, 8(9), 1598; https://doi.org/10.3390/math8091598 - 17 Sep 2020
Cited by 28 | Viewed by 2421
Abstract
The aim of this paper was to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we have redefined the Reich–Rus–Ćirić type contraction and Hardy–Rogers type contraction [...] Read more.
The aim of this paper was to obtain common fixed point results by using an interpolative contraction condition given by Karapinar in the setting of complete metric space. Here in this paper, we have redefined the Reich–Rus–Ćirić type contraction and Hardy–Rogers type contraction in the framework of quasi-partial b-metric space and proved the corresponding common fixed point theorem by adopting the notion of interpolation. The results are further validated with the application based on them. Full article
(This article belongs to the Special Issue Variational Inequality)
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15 pages, 1102 KiB  
Article
Strong Convergent Theorems Governed by Pseudo-Monotone Mappings
by Liya Liu, Xiaolong Qin and Jen-Chih Yao
Mathematics 2020, 8(8), 1256; https://doi.org/10.3390/math8081256 - 31 Jul 2020
Viewed by 1844
Abstract
The purpose of this paper is to introduce two different kinds of iterative algorithms with inertial effects for solving variational inequalities. The iterative processes are based on the extragradient method, the Mann-type method and the viscosity method. Convergence theorems of strong convergence are [...] Read more.
The purpose of this paper is to introduce two different kinds of iterative algorithms with inertial effects for solving variational inequalities. The iterative processes are based on the extragradient method, the Mann-type method and the viscosity method. Convergence theorems of strong convergence are established in Hilbert spaces under mild assumption that the associated mapping is Lipschitz continuous, pseudo-monotone and sequentially weakly continuous. Numerical experiments are performed to illustrate the behaviors of our proposed methods, as well as comparing them with the existing one in literature. Full article
(This article belongs to the Special Issue Variational Inequality)
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13 pages, 790 KiB  
Article
The Bregman–Opial Property and Bregman Generalized Hybrid Maps of Reflexive Banach Spaces
by Eskandar Naraghirad, Luoyi Shi and Ngai-Ching Wong
Mathematics 2020, 8(6), 1022; https://doi.org/10.3390/math8061022 - 22 Jun 2020
Cited by 1 | Viewed by 2297
Abstract
The Opial property of Hilbert spaces is essential in many fixed point theorems of non-expansive maps. While the Opial property does not hold in every Banach space, the Bregman–Opial property does. This suggests to study fixed point theorems for various Bregman non-expansive like [...] Read more.
The Opial property of Hilbert spaces is essential in many fixed point theorems of non-expansive maps. While the Opial property does not hold in every Banach space, the Bregman–Opial property does. This suggests to study fixed point theorems for various Bregman non-expansive like maps in the general Banach space setting. In this paper, after introducing the notion of Bregman generalized hybrid sequences in a reflexive Banach space, we prove (with using the Bregman–Opial property instead of the Opial property) convergence theorems for such sequences. We also provide new fixed point theorems for Bregman generalized hybrid maps defined on an arbitrary but not necessarily convex subset of a reflexive Banach space. We end this paper with a brief discussion of the existence of Bregman absolute fixed points of such maps. Full article
(This article belongs to the Special Issue Variational Inequality)
15 pages, 280 KiB  
Article
A Strong Convergence Theorem under a New Shrinking Projection Method for Finite Families of Nonlinear Mappings in a Hilbert Space
by Wataru Takahashi
Mathematics 2020, 8(3), 435; https://doi.org/10.3390/math8030435 - 17 Mar 2020
Cited by 4 | Viewed by 2493
Abstract
In this paper, using a new shrinking projection method, we deal with the strong convergence for finding a common point of the sets of zero points of a maximal monotone mapping, common fixed points of a finite family of demimetric mappings and common [...] Read more.
In this paper, using a new shrinking projection method, we deal with the strong convergence for finding a common point of the sets of zero points of a maximal monotone mapping, common fixed points of a finite family of demimetric mappings and common zero points of a finite family of inverse strongly monotone mappings in a Hilbert space. Using this result, we get well-known and new strong convergence theorems in a Hilbert space. Full article
(This article belongs to the Special Issue Variational Inequality)
7 pages, 206 KiB  
Article
Existence of a Unique Fixed Point for Nonlinear Contractive Mappings
by Simeon Reich and Alexander J. Zaslavski
Mathematics 2020, 8(1), 55; https://doi.org/10.3390/math8010055 - 1 Jan 2020
Cited by 14 | Viewed by 2106
Abstract
In a recent work, we established the existence of a unique fixed point for nonlinear contractive self-mappings of a bounded and closed set in a Banach space. In the present paper we extend this result to the case of unbounded sets. Full article
(This article belongs to the Special Issue Variational Inequality)
18 pages, 793 KiB  
Article
Generalized Mann Viscosity Implicit Rules for Solving Systems of Variational Inequalities with Constraints of Variational Inclusions and Fixed Point Problems
by Lu-Chuan Ceng and Meijuan Shang
Mathematics 2019, 7(10), 933; https://doi.org/10.3390/math7100933 - 10 Oct 2019
Cited by 7 | Viewed by 1750
Abstract
In this work, let X be Banach space with a uniformly convex and q-uniformly smooth structure, where 1 < q ≤ 2 . We introduce and consider a generalized Mann-like viscosity implicit rule for treating a general optimization system of variational inequalities, [...] Read more.
In this work, let X be Banach space with a uniformly convex and q-uniformly smooth structure, where 1 < q ≤ 2 . We introduce and consider a generalized Mann-like viscosity implicit rule for treating a general optimization system of variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonexpansive mappings in X. The generalized Mann-like viscosity implicit rule investigated in this work is based on the Korpelevich’s extragradient technique, the implicit viscosity iterative method and the Mann’s iteration method. We show that the iterative sequences governed by our generalized Mann-like viscosity implicit rule converges strongly to a solution of the general optimization system. Full article
(This article belongs to the Special Issue Variational Inequality)
13 pages, 292 KiB  
Article
Convergence of Two Splitting Projection Algorithms in Hilbert Spaces
by Marwan A. Kutbi, Abdul Latif and Xiaolong Qin
Mathematics 2019, 7(10), 922; https://doi.org/10.3390/math7100922 - 3 Oct 2019
Viewed by 1613
Abstract
The aim of this present paper is to study zero points of the sum of two maximally monotone mappings and fixed points of a non-expansive mapping. Two splitting projection algorithms are introduced and investigated for treating the zero and fixed point problems. Possible [...] Read more.
The aim of this present paper is to study zero points of the sum of two maximally monotone mappings and fixed points of a non-expansive mapping. Two splitting projection algorithms are introduced and investigated for treating the zero and fixed point problems. Possible computational errors are taken into account. Two convergence theorems are obtained and applications are also considered in Hilbert spaces Full article
(This article belongs to the Special Issue Variational Inequality)
16 pages, 305 KiB  
Article
Fixed Point Results for Generalized ℱ-Contractions in Modular b-Metric Spaces with Applications
by Vahid Parvaneh, Nawab Hussain, Maryam Khorshidi, Nabil Mlaiki and Hassen Aydi
Mathematics 2019, 7(10), 887; https://doi.org/10.3390/math7100887 - 23 Sep 2019
Cited by 15 | Viewed by 3242
Abstract
The aim of this paper is to generalize the F -contractive condition in the framework of α − ν-complete modular b-metric spaces. Some results in ordered modular b-metric spaces are also presented. Moreover, an illustrative example and some related applications [...] Read more.
The aim of this paper is to generalize the F -contractive condition in the framework of α − ν-complete modular b-metric spaces. Some results in ordered modular b-metric spaces are also presented. Moreover, an illustrative example and some related applications are presented to support the obtained results. Full article
(This article belongs to the Special Issue Variational Inequality)
19 pages, 294 KiB  
Article
Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings
by Lu-Chuan Ceng, Xiaolong Qin, Yekini Shehu and Jen-Chih Yao
Mathematics 2019, 7(10), 881; https://doi.org/10.3390/math7100881 - 22 Sep 2019
Cited by 16 | Viewed by 2143
Abstract
In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms [...] Read more.
In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI). Full article
(This article belongs to the Special Issue Variational Inequality)
14 pages, 279 KiB  
Article
Split Variational Inclusion Problem and Fixed Point Problem for a Class of Multivalued Mappings in CAT(0) Spaces
by Mujahid Abbas, Yusuf Ibrahim, Abdul Rahim Khan and Manuel De la Sen
Mathematics 2019, 7(8), 749; https://doi.org/10.3390/math7080749 - 16 Aug 2019
Cited by 9 | Viewed by 2625
Abstract
The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in [...] Read more.
The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in C A T ( 0 ) spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of C A T ( 0 ) spaces. Full article
(This article belongs to the Special Issue Variational Inequality)
15 pages, 306 KiB  
Article
New Hybrid Contractions on b-Metric Spaces
by Erdal Karapınar and Andreea Fulga
Mathematics 2019, 7(7), 578; https://doi.org/10.3390/math7070578 - 28 Jun 2019
Cited by 27 | Viewed by 2507
Abstract
In this manuscript, we introduce the notion of b-hybrid contraction in the setting of b-metric space. We investigate the existence and uniqueness of a fixed point for this contraction. Our results combine and merge several existing results in the corresponding literature [...] Read more.
In this manuscript, we introduce the notion of b-hybrid contraction in the setting of b-metric space. We investigate the existence and uniqueness of a fixed point for this contraction. Our results combine and merge several existing results in the corresponding literature and we list some of them as corollaries. Finally, we consider an Ulam’s type stability for an application. Full article
(This article belongs to the Special Issue Variational Inequality)
16 pages, 291 KiB  
Article
Strong Convergence Theorems for Variational Inequalities and Common Fixed-Point Problems Using Relaxed Mann Implicit Iteration Methods
by Lu-Chuan Ceng and Meijuan Shang
Mathematics 2019, 7(5), 424; https://doi.org/10.3390/math7050424 - 11 May 2019
Cited by 3 | Viewed by 2166
Abstract
Mann-like iteration methods are significant to deal with convex feasibility problems in Banach spaces. We focus on a relaxed Mann implicit iteration method to solve a general system of accretive variational inequalities with an asymptotically nonexpansive mapping in the intermediate sense and a [...] Read more.
Mann-like iteration methods are significant to deal with convex feasibility problems in Banach spaces. We focus on a relaxed Mann implicit iteration method to solve a general system of accretive variational inequalities with an asymptotically nonexpansive mapping in the intermediate sense and a countable family of uniformly Lipschitzian pseudocontractive mappings. More convergence theorems are proved under some suitable weak condition in both 2-uniformly smooth and uniformly convex Banach spaces. Full article
(This article belongs to the Special Issue Variational Inequality)
14 pages, 283 KiB  
Article
Generalized Implicit Set-Valued Variational Inclusion Problem with ⊕ Operation
by Rais Ahmad, Imran Ali, Saddam Husain, A. Latif and Ching-Feng Wen
Mathematics 2019, 7(5), 421; https://doi.org/10.3390/math7050421 - 10 May 2019
Cited by 2 | Viewed by 1892
Abstract
In this paper, we consider a resolvent operator which depends on the composition of two mappings with ⊕ operation. We prove some of the properties of the resolvent operator, that is, that it is single-valued as well as Lipschitz-type-continuous. An existence and convergence [...] Read more.
In this paper, we consider a resolvent operator which depends on the composition of two mappings with ⊕ operation. We prove some of the properties of the resolvent operator, that is, that it is single-valued as well as Lipschitz-type-continuous. An existence and convergence result is proven for a generalized implicit set-valued variational inclusion problem with ⊕ operation. Some special cases of a generalized implicit set-valued variational inclusion problem with ⊕ operation are discussed. An example is constructed to illustrate some of the concepts used in this paper. Full article
(This article belongs to the Special Issue Variational Inequality)
20 pages, 296 KiB  
Article
Systems of Variational Inequalities with Nonlinear Operators
by Lu-Chuan Ceng and Qing Yuan
Mathematics 2019, 7(4), 338; https://doi.org/10.3390/math7040338 - 9 Apr 2019
Cited by 5 | Viewed by 2538
Abstract
In this work, we concern ourselves with the problem of solving a general system of variational inequalities whose solutions also solve a common fixed-point problem of a family of countably many nonlinear operators via a hybrid viscosity implicit iteration method in 2 uniformly [...] Read more.
In this work, we concern ourselves with the problem of solving a general system of variational inequalities whose solutions also solve a common fixed-point problem of a family of countably many nonlinear operators via a hybrid viscosity implicit iteration method in 2 uniformly smooth and uniformly convex Banach spaces. An application to common fixed-point problems of asymptotically nonexpansive and pseudocontractive mappings and variational inequality problems for strict pseudocontractive mappings is also given in Banach spaces. Full article
(This article belongs to the Special Issue Variational Inequality)
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