Mixed Equilibrium Problems Research Articles

A hybrid iterative algorithm for a split mixed equilibrium problem and a hierarchical fixed point problem

A hybrid iterative algorithm for a split mixed equilibrium problem and a hierarchical fixed point problem

Generalized monotone mixed equilibrium problems in FC-spaces: Existence and approximation

Generalized monotone mixed equilibrium problems in FC-spaces: Existence and approximation

Algorithms for Common Solutions to Generalized Mixed Equilibrium Problems and Fixed Point Problems under Nonlinear Transformations in Banach Spaces

The purpose of this paper is to present a new iterative scheme for finding a common solution of the generalized mixed equilibrium problems with an infinite family of inverse strongly monotone mappings and the fixed point problems of demimetric mappings under nonlinear transformations in Banach spaces. Applications are also included. The results in this paper are the extension and improvement of the recent results in the literature.

A new algorithm for solving mixed equilibrium problem and finding common fixed points of Bregman strongly nonexpansive mappings

In this paper, we study a new iterative method for solving mixed equilibrium problem and a common fixed point of a finite family of Bregman strongly nonexpansive mappings in the framework of reflexive real Banach spaces. Moreover, we prove a strong convergence theorem for finding common fixed points which also are solutions of a mixed equilibrium problem.

Optimal Control of Systems Governed by Mixed Equilibrium Problems Under Monotonicity-Type Conditions with Applications

In this paper, we introduce and study an optimal control problem governed by mixed equilibrium problems described by the sum of a maximal monotone bifunction and a bifunction which is pseudomonotone in the sense of Brezis/quasimonotone. Our motivation comes from the fact that many control problems, whose state system is a variational inequality problem or a nonlinear evolution equation or a hemivariational inequality problem, can be formulated as a control problem governed by a mixed equilibrium problem. There are different techniques to study optimal control problems governed by nonlinear evolution equations, variational inequalities or hemivariational inequalities in the literature. However, our technique is completely different from existing ones. It is based on the Mosco convergence and recent results in the theory of equilibrium problems. As an application, we study optimal control problems governed by elliptic variational inequalities with additional state constraints.

Approximation of common solutions of nonlinear problems involving various classes of mappings

A new iterative scheme is proposed which approximates a common solution of split equality fixed point problems involving $$\eta -$$ demimetric mappings, finite family of $$\gamma $$ -inverse strongly monotone mappings, finite family of relatively quasi-nonexpansive mappings and finite family of system of generalized mixed equilibrium problems in real Banach spaces which are 2-uniformly convex and uniformly smooth. Our theorems extend and complement several existing results in this area of research.

A vector auxiliary principle technique approach to systems of mixed vector equilibrium problems

ABSTRACTIn this paper, we introduce and study a system of generalized mixed vector equilibrium problems involving set-valued variational-like inequalities (SGMVEP, in short) in the general setting of Banach spaces. First, we construct an associated vector auxiliary problem and establish existence results for it. This leads to the generation of an iterative algorithm for solving SGMVEP. Then, under weaker assumptions, the strong convergence of the proposed algorithm is established. The method adopted here is based on the vector auxiliary principle and the obtained results improve some existing results in the literature.

General iterative methods for systems of variational inequalities with the constraints of generalized mixed equilibria and fixed point problem of pseudocontractions

In this paper, we introduce two general iterative methods (one implicit method and one explicit method) for finding a solution of a general system of variational inequalities (GSVI) with the constraints of finitely many generalized mixed equilibrium problems and a fixed point problem of a continuous pseudocontractive mapping in a Hilbert space. Then we establish strong convergence of the proposed implicit and explicit iterative methods to a solution of the GSVI with the above constraints, which is the unique solution of a certain variational inequality. The results presented in this paper improve, extend, and develop the corresponding results in the earlier and recent literature.

New algorithms for solving system of generalized mixed equilibrium problems and some nonlinear problems, with applications

ABSTRACTIn this paper, new iterative algorithms of Krasnoselkii-type and Halpern-type for approximating common fixed points of a countable family of equally continuous and totally quasi-φ-asymptotically nonexpansive maps, common zeros of a countable family of -inverse strongly monotone maps and a common element of a set of solutions to a system of generalized mixed equilibrium problems in a 2-uniformly convex and uniformly smooth real Banach space are introduced and studied. Our theorems are significant improvements of some recent important results, in particular, the results of Deng (J Fixed Point Theory appl. 2016); Ofoedu and Malonza (Appl Math Comput. 2011;217:6019–6030), and Wang et al. (Fixed Point Theory Appl. 2012;34).

Algorithms for a system of generalized mixed equilibrium problems and a countable family of some nonlinear multi-valued nonexpansive-type maps

In this paper, relaxed iterative algorithms of Krasnoselskii-type and Halpern-type that approximate a solution of a system of a generalized mixed equilibrium problem and a common fixed point of a countable family of totally quasi-ϕ-asymptotically nonexpansive multi-valued maps are constructed. Strong convergence of the sequence generated by these algorithms is proved in uniformly smooth and strictly convex real Banach spaces with Kadec-Klee property. Furthermore, several applications of our theorems are also presented. Finally, our theorems are significant improvements on several important recent results for this class of nonlinear problems.

A Unified Algorithm for Solving Split Generalized Mixed Equilibrium Problem, and for Finding Fixed Point of Nonspreading Mapping in Hilbert Spaces

AbstractThe purpose of this paper is to study a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces.We introduce a new iterative algorithm and prove its strong convergence for approximating a common solution of a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. Our algorithm is developed by combining a modified accelerated Mann algorithm and a viscosity approximation method to obtain a new faster iterative algorithm for finding a common solution of these problems in real Hilbert spaces. Also, our algorithm does not require any prior knowledge of the bounded linear operator norm. We further give a numerical example to show the efficiency and consistency of our algorithm. Our result improves and compliments many recent results previously obtained in this direction in the literature.

Well-posedness for a class of generalized variational-hemivariational inequalities involving set-valued operators

The aim of present work is to study some kinds of well-posedness for a class of generalized variational-hemivariational inequality problems involving set-valued operators. Some systematic approaches are presented to establish some equivalence theorems between several classes of well-posedness for the inequality problems and some corresponding metric characterizations, which generalize many known results. Finally, the well-posedness for a class of generalized mixed equilibrium problems is also considered.

J-variational inequalities and zeroes of a family of maximal monotone operators by sunny generalized nonexpansive retraction

In this paper, using generalized resolvents and retractions onto shrinking closed subsets, we present new extragradient algorithms for finding the solution of a J-variational-like inequality, the zero points of a family of maximal monotone operators, and the solution set of mixed equilibrium problems for a J- $$\alpha $$ -inverse-strongly monotone-like operator in a 2-uniformly convex and 2-uniformly smooth Banach space. By introducing the new definitions, we prove strong convergence of generated iterates in the extragradient methods. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in the literature to illustrate the usability of our results.

Approximating solution of split equality and equilibrium problems by viscosity approximation algorithms

In this paper, we employ viscosity approximation methods to solve split equality and generalized mixed equilibrium problems in Hilbert spaces. The iterative method considered herein does not require the prior estimations of spectral radii. Applications of obtained results have also been discussed. Finally, some numerical computations are presented.

Existence theorems via duality for equilibrium problems with trifunctions

We present an existence result for an equilibrium problem formulated with trifunctions, which is motivated by variational inequalities governed by quasimonotone operators. To prove the existence result, we define the dual problem, and some monotonicity notions for trifunctions. From the main result follow, among others, the Browder–Minty theorem for variational inequalities and Ky Fan’s Minimax theorem. Some applications for mixed equilibrium problems and variational inequalities are given.

A strong convergence theorem for split mixed equilibrium and fixed point problems for nonexpansive mappings

The aim of this work is twofold; first, to give an extension to split mixed equilibrium problem. Secondly, we suggest and analyze an iterative method based on Mann iterative method and Halpern iterative method to find a common solution of split mixed equilibrium problem and fixed point problem for a nonexpansive mapping in the framework of real Hilbert spaces. Further, under some mild conditions, strong convergence theorem is obtained by the sequences generated by the proposed iterative method, which solves the variational inequality problem. Furthermore, we derived some consequences from our main result and provide some numerical experiments. Our results can be viewed as significant extension and generalization of the previously known results for solving equilibrium problems.

Equilibrium and mixed equilibrium problems under weak monotonicity on Hadamard manifolds

Equilibrium and mixed equilibrium problems under weak monotonicity on Hadamard manifolds

Convergence and stability analysis of iterative algorithm for a generalized set-valued mixed equilibrium problem

In this paper, we consider a generalized set-valued mixed equilibrium problem (in short, GSMEP) in real Hilbert space. Related to GSMEP, we consider a generalized Wiener-Hopf equation problem (in short, GWHEP) and show an equivalence relation between them. Further, we give a fixed-point formulation of GWHEP and construct an iterative algorithm for GWHEP. Furthermore, we extend the notion of stability given by Harder and Hick [3] and prove the existence of a solution of GWHEP and discuss the convergence and stability analysis of the iterative algorithm. Our results can be viewed as a refinement and improvement of some known results in the literature.

A new Halpern-type algorithm for a generalized mixed equilibrium problem and a countable family of generalized nonexpansive-type maps

Let K be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space with dual space E∗. In this paper, a new iterative algorithm of Halpern-type is constructed and used to approximate a common element of a generalized mixed equilibrium problem and a common fixed points for a countable family of generalized nonexpansive-type maps. Application of our theorem, in the case of real Hilbert spaces, complements, extends and improves several important recent results. Finally, we give numerical experiments to illustrate the convergence of our sequence.

Noncoercive mixed equilibrium problems under pseudomonotone perturbations and applications to nonlinear evolution equations with lack of coercivity

ABSTRACTIn this paper, we study the existence of solutions for noncoercive mixed equilibrium problems which are described by the sum of a maximal monotone bifunction and a pseudomonotone (or quasimonotone) bifunction in the sense of Brézis. Our approach is based on recession analysis and on recent results established by the authors for the existence of solutions of mixed equilibrium problems under pseudomonotone perturbations. As an application, we study the existence of solutions for nonlinear evolution equations associated with a noncoercive time-dependent pseudomonotone (or quasimonotone) operator.