Mixed Equilibrium Problems Research Articles

Iterative Algorithm for extended mixed equilibrium problem

In this paper, we introduce and study an extended mixed equilibrium problem by using auxiliary principle technique. A generalized predictor-corrector iterative algorithm is defined for solving extended mixed equilibrium problem. The convergence of the method mentioned requires some condition (∗), g-relatively relaxed Lipschitz continuity and relatively g-relaxed monotonicity of the mappings.

System of generalized mixed equilibrium problems, variational inequality, and fixed point problems

AbstractThe purpose of this paper is to introduce a new iterative algorithm for finding a common element of the set of solutions of a system of generalized mixed equilibrium problems, the set of common fixed points of a finite family of pseudo contraction mappings, and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a real Hilbert space. We establish results on the strong convergence of the sequence by the proposed scheme to a common element of the above three solution sets. These results extend and improve some corresponding results in this area. Finally, we give a numerical example which supports our main theorem.

Existence of Solutions for Nonlinear Implicit Differential Equations: An Equilibrium Problem Approach

ABSTRACTIn this article, we study the existence of solutions for nonlinear implicit differential equations associated to a time-dependent pseudomonotone (respectively, quasimonotone) operator by using a new approach based on the theory of equilibrium \\hboxproblems. More precisely, we first establish some existence results of solutions for mixed equilibrium problems where the \\hboxbifunctions are, respectively, maximal monotone and \\hboxpseudomonotone (quasimonotone) in the topological sense. Then, we write the nonlinear implicit differential equations in the form of a mixed equilibrium problem. By using the existence results for mixed equilibrium problems, we establish the existence of solutions of nonlinear implicit differential equations. This new approach provides some new and nice results which improve and unify most of the recent results obtained in this direction.

Hybrid shrinking iterative solutions to convex feasibility problems and a system of generalized mixed equilibrium problems

Hybrid shrinking iterative solutions to convex feasibility problems and a system of generalized mixed equilibrium problems

Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

Algorithms for common solutions of generalized mixed equilibrium problems and system of variational inclusion problems

Hybrid Projection Methods for Bregman Totally Quasi-D-Asymptotically Nonexpansive Mappings

In this paper, a new iterative scheme by hybrid projection method is proposed for a finite family of Bregman totally quasi-D-asymptotically nonexpansive mappings. Conditions ensuring strong convergence are imposed to common elements of set of common fixed points of the mappings and set of common solutions to a system of generalized mixed equilibrium problems in a reflexive Banach space. These results extend many important recent ones in the literature.

Generalized viscosity approximation methods for mixed equilibrium problems and fixed point problems

In this paper, we present a new iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of common fixed points of a finite family of nonexpansive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative method has strong convergence under some mild conditions imposed on algorithm parameters. The results presented in this paper improve and extend the corresponding results reported by some authors recently.

A strong convergence theorem for finding a common fixed point of a finite family of Bregman nonexpansive mappings in Banach spaces which solves a generalized mixed equilibrium problem

In this paper, we study an iterative method for finding a common fixed point of a finite family of Bregman strongly nonexpansive mappings in real reflexive Banach spaces. Moreover, we prove a strong convergence theorem for finding common fixed points which solve a generalized mixed equilibrium problem. As a special case, we solve mixed variational inequality problems.

Convergence Theorems for BregmanK-Mappings and Mixed Equilibrium Problems in Reflexive Banach Spaces

We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banach space and prove the existence of solution of the equilibrium problem. Using Bregman distance, we introduce the concept of BregmanK-mapping for a finite family of Bregman quasiasymptotically nonexpansive mappings and show the fixed point set of the BregmanK-mapping is the set of common fixed points of{Ti}i=1N. Using the BregmanK-mapping, we introduce an iterative sequence for finding a common point in the set of a common fixed points of the finite family of Bregman quasiasymptotically nonexpansive mappings and the set of solutions of some mixed equilibrium problems. Strong convergence of the iterative sequence is proved. Our results generalise and improve many recent results in the literature.

Mixed Vector Equilibrium Problem Involving Multi-Valued Mapping

Mixed Vector Equilibrium Problem Involving Multi-Valued Mapping

COMMON SOLUTION TO GENERALIZED MIXED EQUILIBRIUM PROBLEM AND FIXED POINT PROBLEM FOR A NONEXPANSIVE SEMIGROUP IN HILBERT SPACE

In this paper, we introduce and study an explicit hybrid re- laxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solu- tion is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.

Strong convergence theorem for generalized mixed equilibrium problems and bregman nonexpansive mapping in Banach spaces

Strong convergence theorem for generalized mixed equilibrium problems and bregman nonexpansive mapping in Banach spaces

Strong convergence theorem for a system of generalized mixed equilibrium problems and finite family of Bregman nonexpansive mappings in Banach spaces

In this paper we introduce a general iterative algorithm for finding the common element of the set of common fixed points of a finite family of Bregman nonexpansive mappings and the set of solutions of systems of generalized mixed equilibrium problems. As an application, we find a solution for system of mixed variational inequality.

Hybrid extragradient method with regularization for triple hierarchical variational inequalities with general mixed equilibrium and split feasibility constraints

In this paper, we introduce a hybrid extragradient iterative algorithm with regularization for solving the triple hierarchical variational inequality problem (THVIP) (defined over the common fixed point set of finitely many nonexpansive mappings and a strictly pseudocontraction) with constraints of a general mixed equilibrium problem (GMEP), a split feasibility problem (SFP) and a general system of variational inequalities (GSVI). The iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, Mann's iteration method, hybrid steepest descent method and gradient-projection method (GPM) with regularization. It is proven that, under very mild conditions, the sequences generated by the proposed algorithm converge strongly to a unique solution of the THVIP. We also give the applications of our results for solving some special cases of the THVIP. The results presented in this paper improve and extend some corresponding ones in the earlier and recent literature.

Convergence theorems for split equality generalized mixed equilibrium problems for demi-contractive mappings

In this paper, we introduce a new iterative algorithm for solving the split equality generalized mixed equilibrium problems. The weak and strong convergence theorems are proved for demi-contractive mappings in real Hilbert spaces. Several special cases are also discussed. As applications, we employ our results to get the convergence results for the split equality convex differentiable optimization problem, the split equality convex minimization problem, and the split equality mixed equilibrium problem. The results in this paper generalize, extend, and unify some recent results in the literature.

Mixed equilibrium problems on Hadamard manifolds

In this paper we study the existence of solutions of mixed equilibrium problems on Hadamard manifolds. We also introduce the implicit and explicit algorithms to solve these problems. Under reasonable assumptions, we show that the sequence generated by both implicit and explicit algorithms converges to a solution of mixed equilibrium problems, whenever it exists. Moreover our results generalize some corresponding results, existing in the literature.

Strong convergence theorem for totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings under relaxed conditions

We construct a relaxed hybrid shrinking iteration algorithm for approximating common fixed points of a countable family of totally quasi-ϕ-asymptotically nonexpansive multi-valued mappings. A strong convergence theorem for solving generalized mixed equilibrium problems is established in the framework of Banach spaces under relaxed conditions. Since there is no need to impose a uniformity assumption on the involved mappings and no need to compute complex series in the iteration process, the results improve those of the authors with related interests.

WEAK CONVERGENCE THEOREMS FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS, MONOTONE MAPPINGS AND PSEUDOCONTRACTIVE MAPPINGS

In this paper, we introduce a new iterative algorithm for finding a common element of the set of solutions of a generalized mixed equilibrium problem related to a continuous monotone mapping, the set of solutions of a variational inequality problem for a continuous monotone mapping, and the set of fixed points of a continuous pseudocontractive mapping in Hilbert spaces. Weak convergence for the proposed iterative algorithm is proved. Our results improve and extend some recent results in the literature.

Scalar gap functions and error bounds for generalized mixed vector equilibrium problems with applications

It is well known that equilibrium problems are very important mathematical models and are closely related with fixed point problems, variational inequalities, and Nash equilibrium problems. Gap functions and error bounds which play a vital role in algorithms design, are two much-addressed topics of vector equilibrium problems. This paper is devoted to studying the scalar-valued gap functions and error bounds for the generalized mixed vector equilibrium problem (GMVE). First, a scalar gap function for (GMVE) is proposed without any scalarization methods, and then error bounds of (GMVE) are established in terms of the gap function. As applications, error bounds for generalized vector variational inequalities and vector variational inequalities are derived, respectively. The main results obtained are new and improve corresponding results of Charitha and Dutta (Pac. J. Optim. 6:497-510, 2010) and Sun and Chai (Optim. Lett. 8:1663-1673, 2014).

Implicit and explicit iterative methods for mixed equilibria with constraints of system of generalized equilibria and hierarchical fixed point problem

In this paper, we introduce one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEPs) with the constraints of a system of generalized equilibrium problems (SGEP) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping in a real Hilbert space. We establish the strong convergence of these two composite relaxed extragradient-like schemes to the same common solution of finitely many GMEPs and the SGEP, which is the unique solution of the HFPP for a strictly pseudocontractive mapping. In particular, we make use of weaker control conditions than previous ones for the sake of proving strong convergence. Utilizing these results, we first propose the composite implicit and explicit relaxed extragradient-like schemes for finding a common fixed point of a finite family of strictly pseudocontractive mappings, and then we derive their strong convergence to the unique common solution of the SGEP and some HFPP. Our results complement, develop, improve, and extend the corresponding ones given by some authors recently in this area.