editorial theory and algorithms of variational inequality...

EditorialTheory and Algorithms of Variational Inequality andEquilibrium Problems, and Their Applications

Xie-ping Ding,1 Nan-jing Huang,2 Fu-quan Xia,1

Qing-bang Zhang,3 and Qamrul Hasan Ansari4

1 Sichuan Normal University, Chengdu, Sichuan 610066, China2 Sichuan University, Chengdu, Sichuan 610064, China3 Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China4Aligarh Muslim University, Aligarh 202 002, India

Correspondence should be addressed to Xie-ping Ding; [email protected]

Received 17 September 2014; Accepted 17 September 2014; Published 22 December 2014

Copyright © 2014 Xie-ping Ding et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The variational inequality problem is a general problemformulation that encompasses manymathematical problems,among others, including nonlinear equations, optimizationproblems, complementarity problems, and fixed point prob-lems. Variational inequality is developed as a tool for thestudy of certain classes of partial deferential equations,economic equilibriumproblems, and the pricingmodel of theoption.

Equilibrium problems provide a mathematical frame-work which includes optimization, variational inequalities,fixed point and saddle point problems, and noncooperativegames as particular cases. It has received an increasinginterest mainly because many theoretical and algorithmicresults developed for one of these models can be oftenextended to the others by using the unifying language.

Because of the importance and active impact of varia-tional inequality and equilibrium problem in the nonlinearanalysis and optimization, this special issue, focusing onmost recent contributions, includes works on hemivaria-tional inequality and multivalued variational inequality, thewell-posedness and stability of equilibrium problems, thebasic methods for showing the existence of solution ofthe equilibrium problem, the iterative approximations, andsome applications to the abstract economy and portfolioselection, which are based on a strict international peerreview procedure and our original proposal. A brief reviewof the papers is given as the following five topics.

(I) Variational Inequality and Complementarity Problem

(i) In the paper titled “On a class of variational-hemivariational inequalities involving upper semicon-tinuous set-valued mappings,” G. Tang et al. apply thevarious coercivity conditions to show the existenceof solutions and boundedness of the solution set forthe variational-hemivariational inequalities involvingupper semicontinuous set-valued mappings.

(ii) In the paper titled “Anew system ofmultivaluedmixedvariational inequality problem,” Xi Li and Xue-songLi construct a perturbational 𝑓-projection algorithmfor solving a system of multivalued mixed variationalinequality problem and prove the existence of solu-tions and the convergence of iterative sequences.

(iii) In the paper titled “An improvement of global errorbound for the generalized nonlinear complementarityproblem over a polyhedral cone,” H. Sun et al. studyan easier computed global error bound for the gen-eralized nonlinear complementarity problem over apolyhedral cone.

(II) Well-Posedness and Stability of Equilibrium Problems

(i) In the paper titled “Levitin-Polyak well-posedness ofan equilibrium-like problem in Banach spaces,” R.Deng discusses the Levitin-Polyak well-posedness of

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 969303, 2 pageshttp://dx.doi.org/10.1155/2014/969303

2 Abstract and Applied Analysis

an equilibrium-like problem in Banach spaces undersuitable conditions.

(ii) In the paper titled “Existence and well-posednessfor symmetric vector quasi-equilibrium problems,” K.Wang et al. study the existence of the solutionsand sufficient conditions for the generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems.

(iii) In the paper titled “The property of the set of equilibriaof the equilibrium problem with lower and upperbounds on Hadamard manifolds,” Q.-B. Zhang andG. Tang study the existence of solutions and theessential stability of the solution set of the equilibriumproblem with lower and upper bounds on Hadamardmanifolds.

(III) 𝐾𝐾𝑀Theorem, Coincidence Theorem, and Applications

(i) In the paper titled “Class U-𝐾𝐾𝑀(𝑋, 𝑌, 𝑍), general𝐾𝐾𝑀 type theorems, and their applications in topo-logical vector space,” G. Tang and Q. Zhang establishsome generalized 𝐾𝐾𝑀 theorems and apply themto show some fixed point theorems and existencetheorems of solutions for the generalized vector equi-librium problems.

(ii) In the paper titled “Some new coincidence theoremsin product 𝐺𝐹𝐶-spaces with applications,” J. Zhaoobtains a continuous selection theorem and applies toproof some new collective fixed point theorems andcoincidence theorems in product 𝐺𝐹𝐶-spaces.

(IV) Convergence Iterative Process for Mappings

(i) In the paper titled “A strong convergence algorithmfor the two-operator split common fixed point prob-lem in Hilbert spaces,” C.-C. Hong and Y.-Y. Huanginvestigate an iterative scheme to approximate theminimum norm solution of the two-operator splitcommon fixed point problem with firmly nonexpan-sive mappings and compare the performance of thepresented algorithm.

(ii) In the paper titled “Weak and strong convergencetheorems for finite families of asymptotically quasi-nonexpansive mappings in Banach spaces,” L. Dengand J. Xiao construct a finite-step iteration sequencefor two finite families of asymptotically nonexpansivemappings and prove its weak and strong convergencein Banach space.

(iii) In the paper titled “Iterative schemes for convex mini-mization problems with constraints,” L.-C. Ceng et al.analyze the strong convergence of an implicit iterativealgorithm for finding a solution of the minimiza-tion problem for a convex and continuously Frechetdifferentiable functional with constraints of the gen-eralized mixed equilibrium problem, the system ofgeneralized equilibrium problems, and finitely manyvariational inclusions in real Hilbert spaces.

(iv) In the paper titled “A continuous trust-region-typemethod for solving nonlinear semidefinite complemen-tarity problem,” Y. Ji et al. propose a new method tosolve nonlinear semidefinite complementarity prob-lem by combining a continuous method and a trust-region-typemethod and show the well-definedness ofthe method and its global convergence.

(V) Applications to the Abstract Economy and Portfolio Selec-tion

(i) In the paper titled “Bounded rationality of general-ized abstract fuzzy economies,” by using a nonlinearscalarization technique, L. Wang and Y. Fu study thebounded rationality model for generalized abstractfuzzy economies and prove the structural stability androbustness to (𝜆, 𝜀)-equilibria of generalized abstractfuzzy economies in finite continuous spaces.

(ii) In the paper titled “Continuous-time mean-varianceportfolio selection under the CEV process,” H. Maderives an optimal portfolio strategy and the efficientfrontier of continuous-time mean-variance portfolioselection model and gives some numerical examples.

Acknowledgments

The editors of this special issue would like to express theirgratitude to the authors who have submitted their papers forconsideration and to the expert referees who provided themwith invaluable help in the reviewing process.

Xie-ping DingNan-jing Huang

Fu-quan XiaQing-bang Zhang

Qamrul Hasan Ansari

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