Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces

In this paper, a general system of nonlinear variational inequality problem in Banach spaces was considered, which includes some existing problems as special cases. For solving this nonlinear variational inequality problem, we construct two methods which were inspired and motivated by Korpelevich’s extragradient method. Furthermore, we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality.     

Levitin-Polyak well-posedness of a generalized mixed variational inequality in Banach spaces

In this paper, we first introduce the concept of Levitin-Polyak well-posedness of a generalized mixed variational inequality in Banach spaces and establish some characterizations of its Levitin-Polyak well-posedness. Under suitable conditions, we prove that the Levitin-Polyak well-posedness of a generalized mixed variational inequality is equivalent to the Levitin-Polyak well-posedness of a corresponding inclusion problem and a corresponding fixed point problem. We also derive some conditions under which a generalized mixed variational inequality in Banach spaces is Levitin-Polyak well-posed.     

Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques

In this paper we consider a semilinear variational inequality with a gradient-dependent nonlinear term. Obviously the nature of this problem is non-variational. Nevertheless we study that problem associating a suitable semilinear variational inequality, variational in nature, with it, and performing an iterative technique used in De Figueiredo et al. (2004) [6] in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity. We prove the existence of a non-trivial non-negative weak solution u for our problem using essentially variational methods, a penalization technique and an iterative scheme. Via Lewy-Stampacchia’s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is α-H?lder continuous on for any α∈(0,1).     

Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems

In this paper, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of the solutions of the variational inequality problem by using a new hybrid method. We obtain a new result for finding a solution of an equilibrium problem and the solutions of the variational inequality problem.     

Optimal control of multivalued quasi variational inequalities

Optimal control of various variational problems has been an area of active research. On the other hand, in recent years many important models in mechanics and economics have been formulated as multi-valued quasi variational inequalities. The primary objective of this work is to study optimal control of the general nonlinear problems of this type. Under suitable conditions, we ensure the existence of an optimal control for a quasi variational inequality with multivalued pseudo-monotone maps. Convergence behavior of the control is studied when the data for the state quasi variational inequality is contaminated by some noise. Some possible applications are discussed.     

Parallel algorithms for variational inequalities over the Cartesian product of the intersections of the fixed point sets of nonexpansive mappings

This paper presents a framework of iterative algorithms for the variational inequality problem over the Cartesian product of the intersections of the fixed point sets of nonexpansive mappings in real Hilbert spaces. Strong convergence theorems are established under a certain contraction assumption with respect to the weighted maximum norm. The proposed framework produces as a simplest example the hybrid steepest descent method, which has been developed for solving the monotone variational inequality problem over the intersection of the fixed point sets of nonexpansive mappings. An application to a generalized power control problem and numerical examples are demonstrated.     

The existence results for optimal control problems governed by a variational inequality

In this paper, we establish the existence of the optimal control for an optimal control problem where the state of the system is defined by a variational inequality problem with monotone type mappings. Moreover, as an application, we get several existence results of an optimal control for the optimal control problem where the system is defined by a quasilinear elliptic variational inequality problem with an obstacle.     

On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form

In this article, we establish the weighted Trudinger–Moser inequality of the scaling invariant form including its best constant and prove the existence of a maximizer for the associated variational problem. The non-singular case was treated by Adachi and Tanaka (1999) [1] and the existence of a maximizer is a new result even for the non-singular case. We also discuss the relation between the best constants of the weighted Trudinger–Moser inequality and the Caffarelli–Kohn–Nirenberg inequality in the asymptotic sense.     

Optimization problems for weighted Sobolev constants

In this paper, we study a variational problem under a constraint on the mass. Using a penalty method we prove the existence of an optimal shape. It will be shown that the minimizers are Hölder continuous and that for a large class they are even Lipschitz continuous. Necessary conditions in form of a variational inequality in the interior of the optimal domain and a condition on the free boundary are derived.     

Variational Inequalities for Combined Control and Stopping Game

Abstract

We study a nonlinear elliptic variational inequality associated with the combined stochastic control problem. By using the dynamic programming principle and the method of penalization, we establish the existence of a unique viscosity solution of the variational inequality and describe it as the value function of the corresponding combined stochastic game problem.     

Strict feasibility and stable solvability of bifunction variational inequalities

In this paper, we study strict feasibility of a bifunction variational inequality. It is proved that a monotone bifunction variational inequality has a nonempty and bounded solution set if and only if it is strictly feasible. Stable solvability of the bifunction variational inequality is discussed under strict feasibility assumption when the domain set is perturbed. Our results generalize earlier results on the classical variational inequality to the case of the bifunction variational inequality.     

Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces

In this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth Banach spaces. First, we prove that the S-iteration process recently introduced by Sahu in [14] converges strongly to a unique fixed point of a mapping T, where T is κ-strongly pseudocontractive mapping from a nonempty, closed and convex subset C of a smooth Banach space into itself. It is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of λ_i-strictly pseudocontractive mappings from C into itself. Our results extend and improve some very recent theorems in fixed point theory and variational inequality problems. Particularly, the results presented here extend some theorems of Reich (1980) [1] and Yamada (2001) [15] to a general class of λ-strictly pseudocontractive mappings in uniformly smooth Banach spaces.     

A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications

The purpose of this paper is to prove by using a new hybrid method a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem, the set of solutions for a variational inequality problem and the set of common fixed points for a pair of relatively nonexpansive mappings in a Banach space. As applications, we utilize our results to obtain some new results for finding a solution of an equilibrium problem, a fixed point problem and a common zero-point problem for maximal monotone mappings in Banach spaces.     

Some properties of solutions to the singular quasilinear problems

In this paper, a kind of quasilinear elliptic problem is studied, which involves the critical exponent and singular potentials. By the Caffarelli-Kohn-Nirenberg inequality and variational methods, some important properties of the positive solution to the problem are established.     

The existence results for obstacle optimal control problems

This paper is concerned with the existence of an optimal control problem for a quasi-linear elliptic obstacle variational inequality in which the obstacle is taken as the control. Firstly, we get some existence results under the assumption of the leading operator of the variational inequality with a monotone type mapping in Section 2. In Section 3, as an application, without the assumption of the monotone type mapping for the leading operator of the variational inequality, we prove that the leading operator of the variational inequality is a monotone type mapping. Existence of the optimal obstacle is proved. The method used here is different from [Y.Y. Zhou, X.Q. Yang, K.L. Teo, The existence results for optimal control problems governed by a variational inequality, J. Math. Anal. Appl. 321 (2006) 595-608].     

Feasibility and solvability of vector variational inequalities with moving cones in Banach spaces

In this paper we study the feasibility and solvability of vector variational inequalities with moving cones in Banach spaces. We show that the strict feasibility implies solvability of vector variational inequalities with moving cones under suitable conditions. Further we show that under suitable conditions, the homogeneous vector variational inequality with a moving cone is solvable whenever it is feasible. As consequences, we obtain the solvability of vector variational inequalities with feasibility assumptions in Banach spaces.     

Existence of solutions for generalized variational inequality problems in Banach spaces

In this paper, we prove the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over compact convex subsets in a reflexive Banach space with a Fréchet differentiable norm. Moreover, we give some conditions that guarantee the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over unbounded closed convex subsets. The result obtained in this paper improves and extends the recent ones announced by Yu and Yang [J. Yu, H. Yang, Existence of solutions for generalized variational inequality problems, Nonlinear Anal., 71 (2009) e2327-e2330] and many others.     

Transitivity and variational principles

We apply an order reasoning to mappings satisfying the triangle inequality. This general approach yields the Ekeland’s variational principle as one of the consequences. In addition we obtain an extension of the Brøndsted variational principle and of the Takahashi fixed point theorem.     

优化和均衡的等价性

通过向量优化问题, 向量变分不等式问题以及向量变分原理来分析优化问题及均衡问题的一致性.从而显然, 可以用统一的观点来处理数值优化、向量优化以及博弈论等问题.进而为非线性分析提供了一个新的发展空间.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.