与渐近非扩张半群不动点问题相关的分裂等式混合均衡问题及其应用

研究了与渐近非扩张半群不动点问题相关的分裂等式混合均衡问题.在等式约束下,为同时逼近两个空间中混合均衡问题和渐近非扩张半群不动点问题的公共解,借助收缩投影方法引出了一种迭代程序.在适当条件下,该迭代算法的强收敛性被证明.文末还把所得结果应用于分裂等式混合变分不等式问题和分裂等式凸极小化问题.     

On iterative methods for equilibrium problems

In this paper, we introduce a hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of finitely many nonexpansive mappings. We prove that the approximate solution converges strongly to a solution of a class of variational inequalities under some mild conditions, which is the optimality condition for some minimization problem. We also give some comments on the results of Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007) 455–469]. Results obtained in this paper may be viewed as an improvement and refinement of the previously known results in this area.     

New iterative scheme with nonexpansive mappings for equilibrium problems and variational inequality problems in Hilbert spaces

Recently, Ceng, Guu and Yao introduced an iterative scheme by viscosity-like approximation method to approximate the fixed point of nonexpansive mappings and solve some variational inequalities in Hilbert space (see Ceng et al. (2009) [9]). Takahashi and Takahashi proposed an iteration scheme to solve an equilibrium problem and approximate the fixed point of nonexpansive mapping by viscosity approximation method in Hilbert space (see Takahashi and Takahashi (2007) [12]). In this paper, we introduce an iterative scheme by viscosity approximation method for finding a common element of the set of a countable family of nonexpansive mappings and the set of an equilibrium problem in a Hilbert space. We prove the strong convergence of the proposed iteration to the unique solution of a variational inequality.     

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.     

Iterative approximation of a common solution of a split equilibrium problem,a variational inequality problem and a fixed point problem

In this paper, we introduce an iterative method to approximate a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem for a nonexpansive mapping in real Hilbert spaces. We prove that the sequences generated by the iterative scheme converge strongly to a common solution of the split equilibrium problem, the variational inequality problem and the fixed point problem for a nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area.     

A relaxed extragradient-like method for a generalized mixed equilibrium problem,a general system of generalized equilibria and a fixed point problem

Very recently, Takahashi and Takahashi [S. Takahashi, W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Anal. 69 (2008) 1025–1033] suggested and analyzed an iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem of a nonexpansive mapping in a Hilbert space. In this paper, based on Takahashi–Takahashi’s iterative method and well-known extragradient method we introduce a relaxed extragradient-like method for finding a common solution of a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem of a strictly pseudocontractive mapping in a Hilbert space and then obtain a strong convergence theorem. Utilizing this theorem, we establish some new strong convergence results in fixed point problems, variational inequalities, mixed equilibrium problems and systems of generalized equilibria.     

A method for the approximate solution of quasi-static problems for hardening bodies

A method for the approximate solution of quasi-static problems for hardening elastoplastic bodies is proposed. The constitutive relation of the model is taken in the form of a variational inequality. An approximate solution of the initial problem is constructed in time steps and, by means of the finite element method, is reduced to the solution of a system of two variational inequalities in corresponding finite-dimensional space. It is shown that the solution of this system is equivalent to finding the saddle point of the corresponding quadratic functional. To find the saddle point, Udzawa's algorithm is used, by means of which the process of finding the velocity vector and stress tensor reduces to the successive calculation of these quantities: the velocity vector is determined from the variational inequality corresponding to the equilibrium equation, and the stress tensor is determined from the variational inequality corresponding to the constitutive relation. The latter inequality is reduced to a certain non-linear equation containing the operation of projection onto a closed convex set corresponding to the elastic strains of the medium. In turn, the solution of the non-linear equation is constructed using the method of successive approximations. To illustrate the use of the proposed method, the one-dimensional problem of the quasi-static deformation of a cylindrical tube under a load applied to its internal surface is considered.     

Halpern- and Mann-Type Algorithms for Fixed Points and Inclusion Problems on Hadamard Manifolds

In this article, we consider an inclusion problem which is defined by means of a sum of a single-valued vector field and a set-valued vector field defined on a Hadamard manifold. We propose Halpern-type and Mann-type algorithms for finding a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. Some particular cases of our problem and algorithm are also discussed. We study the convergence of the proposed algorithm to a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. As applications of our results and algorithms, we derive the solution methods and their convergence results for the optimization problems, variational inequality problems and equilibrium problems in the setting of Hadamard manifolds.     

Iterative algorithms for variational inclusions,mixed equilibrium and fixed point problems with application to optimization problems

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a nonexpansive mapping, and the the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of a certain optimization problem related to a strongly positive bounded linear operator.     

A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem

In this paper, we give a hybrid extragradient iterative method for finding the approximate element of the common set of solutions of a generalized equilibrium problem, a system of variational inequality problems, a variational inequality problem and a fixed point problem for a strictly pseudocontractive mapping in a real Hilbert space. Further we establish a strong convergence theorem based on this method. The results presented in this paper improves and generalizes the results given in Yao et al. [36] and Ceng et al. [7], and some known corresponding results in the literature.     

Finite difference technique for solving obstacle problems

In this paper, we use the variational inequality theory coupled with finite difference technique to obtain an approximate solution for a class of obstacle problems in elasticity, like those describing the equilibrium configuration of an elastic string stretched over an elastic obstacle. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution of the obstacle problems.     

Fixed point solutions of generalized equilibrium problems for nonexpansive mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a generalized equilibrium problem in a real Hilbert space. Then, strong convergence of the scheme to a common element of the two sets is proved. As an application, problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem is solved. Moreover, solution is given to the problem of finding a common element of fixed points set of nonexpansive mappings and the set of solutions of a variational inequality problem.     

Convergence theorems based on the shrinking projection method for variational inequality and?equilibrium problems

The purpose of this paper is to introduce a hybrid projection algorithm based on the shrinking projection method for two relatively weak nonexpansive mappings. We prove strong convergence theorem which approximate the common element in the fixed point set of two such mappings, the solution set of the variational inequality and the solution set of the equilibrium problem in the framework of Banach spaces. Our results improve and extend previous results.     

On the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings

In this article we consider the problem of finding a common element in the solution set of generalized equilibrium problems, in the solution set of the classical variational inequality and in the fixed point set of strictly pseudocontractive mappings. Weak convergence theorems of common elements are established in real Hilbert spaces.     

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.     

Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings

We introduce iterative algorithms for finding a common element of the set of solutions of a system of equilibrium problems and of the set of fixed points of a finite family and a left amenable semigroup of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Our results extend, for example, the recent result of [V. Colao, G. Marino, H.K. Xu, An Iterative Method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl. 344 (2008) 340–352] to systems of equilibrium problems.     

A wide neighborhood primal-dual predictor-corrector interior-point method for symmetric cone optimization

Our aim in this paper is to introduce a modified viscosity implicit rule for finding a common element of the set of solutions of variational inequalities for two inverse-strongly monotone operators and the set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. Some strong convergence theorems are obtained under some suitable assumptions imposed on the parameters. As an application, we give an algorithm to solve fixed point problems for nonexpansive mappings, variational inequality problems and equilibrium problems in Hilbert spaces. Finally, we give one numerical example to illustrate our convergence analysis.     

An iterative method of solution for equilibrium and optimization problems

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of solutions of an equilibrium problem in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets. The results of this paper extended and improved the results of H. Iiduka and W. Takahashi [Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515]. Therefore, by using the above result, an iterative algorithm for the solution of a optimization problem was obtained.     

Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints

The equilibrium problem with equilibrium constraints (EPEC) can be looked on as a generalization of Nash equilibrium problem (NEP) and the mathematical program with equilibrium constraints (MPEC) whose constraints contain a parametric variational inequality or complementarity system. In this paper, we particularly consider a special class of EPECs where a common parametric P-matrix linear complementarity system is contained in all players?? strategy sets. After reformulating the EPEC as an equivalent nonsmooth NEP, we use a smoothing method to construct a sequence of smoothed NEPs that approximate the original problem. We consider two solution concepts, global Nash equilibrium and stationary Nash equilibrium, and establish some results about the convergence of approximate Nash equilibria. Moreover we show some illustrative numerical examples.     

An iterative method for finding common solutions of equilibrium and fixed point problems

We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.