Solutions Of Mixed Equilibrium Problem Research Articles

Equilibrium problems under relaxed α-monotonicity on Hadamard manifolds

In this paper, we study the existence of solutions of both equilibrium problems and mixed equilibrium problems on Hadamard manifolds. Under relaxed $$\alpha $$ -pseudomonotonicity assumption on the underlying bifunction we prove that the solution set of the equilibrium problem is nonempty. We also provide the existence of solution of mixed equilibrium problems with relaxed $$\alpha $$ -monotonicity. The results presented in this paper generalize and improve some known results given in literature, see for example (Colao et al. in J Math Anal Appl 388:61–77, 2012; Jana and Nahak in Rend Circ Mat Palermo(2) 65(1):97–109, 2016; Mahato and Nahak in OPSEARCH, 2013. https://doi.org/10.1007/s12597-013-0142-5 ).

Existence Results for Mixed Equilibrium Problems Involving Set-Valued Operators with Applications to Quasi-Hemivariational Inequalities

In this paper, we study the existence of solutions for mixed equilibrium problems associated with a set-valued operator in the general setting of vector spaces in duality, and in particular in Banach spaces. We use a Galerkin-type method and the notion of pseudomonotonicity in the sense of Brezis for bifunctions. As application, we study the existence of solutions for quasi-hemivariational inequalities governed by a set-valued mapping and perturbed with a nonlinear term. Our main results can be applied to differential inclusions, evolution equations and evolution hemivariational inequalities.

Noncoercive Stationary Navier–Stokes Equations of Heat-Conducting Fluids Modeled by Hemivariational Inequalities: An Equilibrium Problem Approach

In this paper, we study the existence of solutions for noncoercive stationary Navier–Stokes equations of heat-conducting fluids with nonmonotone boundary conditions modeled by hemivariational inequalities. Our method is new and differs from most of the existing techniques developed in the literature. It is based on a recent approach developed on the existence of solutions for mixed equilibrium problems described by the sum of a maximal monotone bifunction and a pseudomonotone bifunction in the sense of Brezis. We introduce a Browder–Tikhonov regularization method for mixed equilibrium problems by means of the duality mapping with gauge function $$\mu (t)$$. By using this regularization procedure and techniques from the recession analysis, we study the existence of solutions for the problem considered in this paper.

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.

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.

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.

Iterative methods for triple hierarchical variational inequalities with mixed equilibrium problems, variational inclusions, and variational inequalities constraints

In this paper, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm and multi-step composite Mann-type viscosity iterative algorithm for finding a solution of triple hierarchical variational inequalities defined over the common set of solutions of mixed equilibrium problems, variational inclusions, variational inequalities, and fixed point problems. Under appropriate assumptions, we prove that the proposed algorithms converge strongly to a common element of the fixed point set of a strict pseudocontractive mapping, a solution set of finitely many generalized mixed equilibrium problems, a solution set of finitely many variational inclusions, and a solution set of a general system of variational inequalities. Such an element is a unique solution of a triple hierarchical variational inequality problem. In addition, we also consider as an application the proposed algorithm to solve a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, and a general system of variational inequalities. The results obtained in this paper improve and extend the corresponding results announced by many other authors.

Demiclosed principle and convergence theorems for asymptotically strictly pseudononspreading mappings and mixed equilibrium problems

In this paper, the demiclosed principle for a k-asymptotically strictly pseudononspreading mapping is shown. Meanwhile, an iterative scheme is introduced to approximate a common element of the set of common fixed points of k-asymptotically strictly pseudononspreading mappings and the set of solutions of mixed equilibrium problems in Hilbert spaces, and some weak and strong convergence theorems are proved. The results presented in this paper improve and extend some recent corresponding results.

An Iterative Method for Equilibrium Problems and Fixed Point Problems

In this paper, we prove the strong convergence of an iterative method for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings under some suitable conditions. Result presented in this paper improves and extends the recent known results in this area.

Iterative Algorithms for Mixed Equilibrium Problems, System of Quasi-Variational Inclusion, and Fixed Point Problem in Hilbert Spaces

We introduce a new iterative algorithm for approximating a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Strong convergence of the proposed iterative algorithm is obtained. Our results generalize, extend, and improve the results of Peng and Yao, 2009, Qin et al. 2010 and many authors.

Strong Convergence Theorems for Mixed Equilibrium Problem and AsymptoticallyI-Nonexpansive Mapping in Banach Spaces

This paper aims to use a hybrid algorithm for finding a common element of a fixed point problem for a finite family of asymptotically nonexpansive mappings and the set solutions of mixed equilibrium problem in uniformly smooth and uniformly convex Banach space. Then, we prove some strong convergence theorems of the proposed hybrid algorithm to a common element of the above two sets under some suitable conditions.

Hybrid methods for a mixed equilibrium problem and fixed points of a countable family of multivalued nonexpansive mappings

In this paper, we prove a strong convergence theorem for a new hybrid method, using shrinking projection method introduced by Takahashi and a fixed point method for finding a common element of the set of solutions of mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We also apply our main result to the convex minimization problem and the fixed point problem of a countable family of multivalued nonexpansive mappings.

An iterative approach to mixed equilibrium problems and fixed points problems

Abstract In the present paper, an iterative algorithm for solving mixed equilibrium problems and fixed points problems has been constructed. It is shown that under some mild conditions, the sequence generated by the presented algorithm converges strongly to the common solution of mixed equilibrium problems and fixed points problems. As an application, we can find the minimum norm element without involving projection. MSC:47J05, 47J25, 47H09.

Mixed Equilibrium Problems with Weakly Relaxedα-Monotone Bifunction in Banach Spaces

We introduce the class of mixed equilibrium problems with the weakly relaxedα-monotone bi-function in Banach spaces. Using the KKM technique, we obtain the existence of solutions for mixed equilibrium problem with weakly relaxedα-monotone bi-function in Banach spaces. The results presented in this paper extend and improve the corresponding results in the existing literature.

Approximation algorithm for fixed points of nonlinear operators and solutions of mixed equilibrium problems and variational inclusion problems with applications

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of fixed point of nonexpansive mappings, set of a mixed equilibrium problem and the set of variational inclusions in a real Hilbert space. We prove that the sequence \(x_n\) which is generated by the proposed iterative algorithm converges strongly to a common element of four sets above. Furthermore, we give an application to optimization and some numerical examples which support our main theorem in the last part. Our result extended and improve the existing result of Yao et al. [19] and references therein.

A Viscosity Approximation Scheme for Finding Common Solutions of Mixed Equilibrium Problems, a Finite Family of Variational Inclusions, and Fixed Point Problems in Hilbert Spaces

We introduce an iterative method for finding a common element of set of fixed points of nonexpansive mappings, the set of solutions of a finite family of variational inclusion with set‐valued maximal monotone mappings and inverse strongly monotone mappings, and the set of solutions of a mixed equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of Plubtemg and Sripard and many others.

Existence and algorithm of solutions for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces

Some classes of mixed equilibrium problems and bilevel mixed equilibrium problems are introduced and studied in reflexive Banach spaces. First, by using a minimax inequality, some new existence results of solutions and the behavior of solution set for the mixed equilibrium problems and the bilevel mixed equilibrium problems are proved under suitable assumptions without the coercive conditions. Next, by using auxiliary principle technique, some new iterative algorithms for solving the mixed equilibrium problems and the bilevel mixed equilibrium problems are suggested and analyzed. The strong convergence of the iterative sequences generated by the proposed algorithms is proved under suitable assumptions without the coercive conditions. These results are new and generalize some recent results in this field.

A Viscosity of Cesàro Mean Approximation Methods for a Mixed Equilibrium, Variational Inequalities, and Fixed Point Problems

We introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a -inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Cesaro mean approximation method. We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang (2009), Peng and Yao (2009), Shimizu and Takahashi (1997), and some authors.

A General Iterative Method of Fixed Points for Mixed Equilibrium Problems and Variational Inclusion Problems

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings and inverse-strongly monotone mappings, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. We propose a new iterative scheme for finding the common element of the above three sets. Our results improve and extend the corresponding results of the works by Zhang et al. (2008), Peng et al. (2008), Peng and Yao (2009), as well as Plubtieng and Sriprad (2009) and some well-known results in the literature.

A Hybrid Projection Algorithm for Finding Solutions of Mixed Equilibrium Problem and Variational Inequality Problem

AbstractWe propose a modified hybrid projection algorithm to approximate a common fixed point of a "Equation missing"-strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.