Systems of Set-Valued Quasivariational Inclusion Problems

We propose four kinds of systems of set-valued quasivariational inclusion problems in product spaces, which include many known systems of equilibrium problems and systems of variational inequalities as well as inclusion problems. Sufficient conditions for the solution existence are established. When applied to special cases, these conditions improve many existing results in the literature. To ensure the generality of our problem setting and results, applications in fixed-point theory and quasioptimization problems are included. This work was partially supported by the Vietnam national program on basic research in natural sciences. A part of the work of the first author was completed during his stay at the Department Mathematics of Namur University and its hospitality is gratefully acknowledged. The authors are indebted to an anonymous referee for valuable remarks.     

General implicit variational inclusion problems based on A-maximal (m)-relaxed monotonicity (AMRM) frameworks

A general framework for an algorithmic procedure based on the variational convergence of operator sequences involving A-maximal (m)-relaxed monotone (AMRM) mappings in a Hilbert space setting is developed, and then it is applied to approximating the solution of a general class of nonlinear implicit inclusion problems involving A-maximal (m)-relaxed monotone mappings. Furthermore, some specializations of interest on existence theorems and corresponding approximation solvability theorems on H-maximal monotone mappings are included that may include several other results for general variational inclusion problems on general maximal monotonicity in the literature.     

Complementarity Problems and Variational Inequalities. A Unified Approach of Solvability by an Implicit Leray-Schauder Type Alternative

In several recent papers we obtained existence theorems for complementarity problems and variational inequalities using for each of them a particular notion of exceptional family of elements. Now, in this paper we introduce a new notion of exceptional family of elements. This notion is based on an Implicit Leray-Schauder Alternative. By this new notion we obtain a unification of the study of solvability of complementarity problems and of variational inequalities. The paper is finished with a section dedicated to variational inequalities with δ-pseudomonotone operators.     

Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint

In this paper, we study the well-posedness in the generalized sense for variational inclusion problems and variational disclusion problems, the well-posedness for optimization problems with variational inclusion problems, variational disclusion problems and scalar equilibrium problems as constraint.     

Equilibrium Problems with Applications to Eigenvalue Problems

In this paper, we consider equilibrium problems and introduce the concept of (S)₊ condition for bifunctions. Existence results for equilibrium problems with the (S)₊ condition are derived. As special cases, we obtain several existence results for the generalized nonlinear variational inequality studied by Ding and Tarafdar (Ref. 1) and the generalized variational inequality studied by Cubiotti and Yao (Ref. 2). Finally, applications to a class of eigenvalue problems are given.     

On implicit vector equilibrium problems

In this paper, we introduce and study a class of implicit vector equilibrium problems, which includes a number of (scalar) implicit equilibrium problems, implicit variational inequalities, and implicit complementarity problems as special cases. By using KKM-Fan theorem, we prove some new existence theorems of solutions for this kind of implicit vector equilibrium problems in Hausdorff topological vector spaces. Our results extend and unify some corresponding results of several authors.     

On quasivariational inclusion problems of type I and related problems

The quasivariational inclusion problems are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of a general vector ideal (proper, Pareto, weak) quasi-optimization problems, of quasivariational inequalities, and of vector quasi-equilibrium problems. Further, we prove theorems on the existence for solutions of the sum of these inclusions. As corollaries, we shall show several results on the existence of solutions to another problems in the vector optimization problems concerning multivalued mappings. This work was supported by the National Science Council of the Republic of China and the Academy of Sciences and Technologies of Vietnam. The authors wish to express their gratitude to the referees for their valuable suggestions.     

Existence Criteria for the Solutions of Two Types of Variational Relation Problems

The variational relation problems have been introduced in 2008, by Dinh The Luc, as general models for a large class of problems in nonlinear analysis and applied mathematics. Since this manner of approach proved to be a powerful tool for studying a wide class of problems in nonlinear analysis and applied mathematics, several types of variational relation problems or systems of variational relation problems have been investigated in many recent papers. The present paper fits into this interesting group of works, establishing existence criteria for the solutions of two very general types of variational relation problems.     

强不定问题的变分方法

简要回顾近年来关于强不定问题的变分方法某些研究方面的发展．首先介绍强不定问题,接着叙述建立强不定问题的变分框架的基本思路,进而给出局部凸拓扑线性空间的形变理论,最后陈述几个基于此形变理论的处理强不定问题的临界点定理．这些理论的应用将在后续文章中介绍．     

Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces

Multivalued equilibrium problems in general metric spaces are considered. Uniqueness and Hölder continuity of the solution are established under Hölder continuity and relaxed Hölder-related monotonicity assumptions. The assumptions appear to be weaker and the inclusion to be properly stronger than that of the recent results in the literature. Furthermore, our theorems include completely some known results for variational inequalities in Hilbert spaces, which were demonstrated via geometrical techniques based on the orthogonal projection in Hilbert spaces and the linearity of the canonical pair\(\langle .,.\rangle\).     

Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities

This paper attempts to generalize and unify several new results that have been obtained in the ongoing research area of existence of solutions for equilibrium problems. First, we propose sufficient conditions, which include generalized monotonicity and weak coercivity conditions, for existence of equilibrium points. As consequences, we generalize various recent theorems on the existence of such solutions. For applications, we treat some generalized variational inequalities and complementarity problems. In addition, considering penalty functions, we study the position of a selected solution by relying on the viscosity principle.     

On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems

We consider parametric multivalued vector equilibrium problems of both weak and strong types in metric linear spaces. Sufficient conditions for the local uniqueness and Hölder continuity of the solutions are established. As consequences some new results for variational inequalities are derived and compared with recent papers on the subject.     

New approximation-solvability of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.     

Exceptional Families of Elements for Variational Inequalities in Banach Spaces

In keeping with very recent efforts to establish a useful concept of an exceptional family of elements for variational inequality problems rather than complementarity problems as in the past, we propose such a concept. It generalizes previous ones to multivalued variational inequalities in general normed spaces and allows us to obtain several existence results for variational inequalities corresponding to earlier ones for complementarity problems. Compared with the existing literature, we consider problems in more general spaces and under considerably weaker assumptions on the defining map.     

Systems of variational inclusion problems and differential inclusion problems with applications

In this paper, we study the existence theorems of systems of variational inclusion problems. From these existence results, we study the existence theorems of systems of variational differential inclusion problems, mathematical program with systems of variational inclusion constraints, and mathematical program with systems of equilibrium constraints.     

Mehar’s method for solving fuzzy sensitivity analysis problems with LR flat fuzzy numbers

In published works on fuzzy linear programming there are only few papers dealing with stability or sensitivity analysis in fuzzy mathematical programming. To the best of our knowledge, till now there is no method in the literature to deal with the sensitivity analysis of such fuzzy linear programming problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method, named as Mehar’s method, is proposed for the same. To show the advantages of proposed method over existing methods, some fuzzy sensitivity analysis problems which may or may not be solved by the existing methods are solved by using the proposed method.     

Three-step iterative methods for general variational inclusions in L P spaces

In this paper, we show that the general variational inclusions are equivalent to the fixed point problem. We use this equivalence to discuss the existence of the variational inclusions in L ^p spaces. Using the technique of the updating solution, we suggest some three-step iterative methods for solving the general variational inclusion. We also consider the convergence analysis of the proposed iterative methods under some mild conditions. Since the general variational inclusions include several classes of variational inequalities and optimization problems as special cases, results proved in this paper continue to hold for these problems.     

A theorem of variational inclusion problems and various nonlinear mappings

In this paper, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of nonexpansive mappings and the set of solutions of a finite family of variational inclusion problems in Hilbert spaces. Moreover, we utilize our main result to fixed point problems of various nonlinear mappings and the set of solutions of variational inclusion problems.     

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

Existence and multiplicity of positive solutions for singular quasilinear problems

In this work we combine perturbation arguments and variational methods to study the existence and multiplicity of positive solutions for a class of singular p-Laplacian problems. In the first two theorems we prove the existence of solutions in the sense of distributions. By strengthening the hypotheses, in the third and last result, we establish the existence of two ordered positive weak solutions.