Maximal element with applications to Nash equilibrium problems in Hadamard manifolds

ABSTRACT

The purpose of this paper is to study the existence of maximal elements with applications to Nash equilibrium problems for generalized games in Hadamard manifolds. By employing a KKM lemma, we establish a new maximal element theorem in Hadamard manifolds. As applications, some existence results of Nash equilibria for generalized games are derived. The results in this paper unify, improve and extend some known results from the literature.     

An extragradient method for equilibrium problems on Hadamard manifolds

In this paper, we propose an extragradient algorithm for solving equilibrium problems on Hadamard manifolds to the case where the equilibrium bifunction is not necessarily pseudomonotone. Under mild assumptions, we establish global convergence results. We show that the multiobjective optimization problem satisfies all the hypotheses of our result of convergence, when formulated as an equilibrium problem.     

Vector variational inequalities and vector optimization problems on Hadamard manifolds

In this paper, we introduce a Minty type vector variational inequality, a Stampacchia type vector variational inequality, and the weak forms of them, which are all defined by means of subdifferentials on Hadamard manifolds. We also study the equivalent relations between the vector variational inequalities and nonsmooth convex vector optimization problems. By using the equivalent relations and an analogous to KKM lemma, we give some existence theorems for weakly efficient solutions of convex vector optimization problems under relaxed compact assumptions.     

Nonmonotone equilibrium problems: coercivity conditions and weak regularization

We consider a general equilibrium problem in a finite-dimensional space setting and propose a new coercivity condition for existence of solutions. We also show that it enables us to create a broad family of regularization methods with preserving well-definiteness and convergence of the iteration sequence without additional monotonicity assumptions. Some examples of applications are also given.     

A concentration theorem of expanders on Hadamard manifolds

In this note, we prove a concentration theorem of expanders. As a simple corollary, one can prove that expanding sequences of graphs do not admit coarse embeddings into Hadamard manifolds with bounded sectional curvatures.     

Korpelevich’s method for variational inequality problems on Hadamard manifolds

The concept of pseudomonotone vector field on Hadamard manifold is introduced. A variant of Korpelevich??s method for solving the variational inequality problem is extended from Euclidean spaces to constant curvature Hadamard manifolds. Under a pseudomonotone assumption on the underlying vector field, we prove that the sequence generated by the method converges to a solution of variational inequality, whenever it exists. Moreover, we give an example to show the effectiveness of our method.     

The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions

We extend the Tikhonov regularization method widely used in optimization and monotone variational inequality studies to equilibrium problems. It is shown that the convergence results obtained from the monotone variational inequality remain valid for the monotone equilibrium problem. For pseudomonotone equilibrium problems, the Tikhonov regularized subproblems have a unique solution only in the limit, but any Tikhonov trajectory tends to the solution of the original problem, which is the unique solution of the strongly monotone equilibrium problem defined on the basis of the regularization bifunction.     

Existence results for a class of hemivariational inequality problems on Hadamard manifolds

In this paper, a class of hemivariational inequality problems are introduced and studied on Hadamard manifolds. Using the properties of Clarke’s generalized directional derivative and Fan-KKM lemma, an existence theorem of solution in connection with the hemivariational inequality problem is obtained when the constraint set is bounded. By employing some coercivity conditions and the properties of Clarke’s generalized directional derivative, an existence result and the boundedness of the set of solutions for the underlying problem are investigated when the constraint set is unbounded. Moreover, a sufficient and necessary condition for ensuring the nonemptiness of the set of solutions concerned with the hemivariational inequality problem is also given.     

Dynamical systems for solving variational inclusion and fixed point problems on Hadamard manifolds

In this paper, we investigate a new class of dynamical systems for solving variational inclusion and fixed point problems on Hadamard manifolds. Then we prove that the dynamical system has a unique solution under some suitable assumptions. Moreover, the global exponential stability and invariance property of the dynamical systems are also established. Our main results in this work are new and extend the existing ones in the literature.     

A common fixed point theorem with applications to vector equilibrium problems

In this paper, using the Brouwer fixed point theorem, we establish a common fixed point theorem for a family of set-valued mappings. As applications of this result we obtain existence theorems for the solutions of two types of vector equilibrium problems, a Ky Fan-type minimax inequality and a generalization of a known result due to Iohvidov.     

Coincidence theorems and its applications to equilibrium problems

Given two maps h : X ×K ? \mathbbR{h : X \times K \rightarrow \mathbb{R}} and g : X → K such that, for all x ? X, h(x, g(x)) = 0{x \in X, h(x, g(x)) = 0} , we consider the equilibrium problem of finding [(x)\tilde] ? X{\tilde{x} \in X} such that h([(x)\tilde], g(x)) 3 0{h(\tilde{x}, g(x)) \geq 0} for every x ? X{x \in X} . This question is related to a coincidence problem.     

Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications

In this paper, maximal element theorem on Hadamard manifolds is established. First, we prove the existence of solutions for maximal element theorem on Hadamard manifolds. Further, we prove that most of problems in maximal element theorem on Hadamard manifolds (in the sense of Baire category) are essential and that, for any problem in maximal element theorem on Hadamard manifolds, there exists at least one essential component of its solution set. As applications, we study existence and stability of solutions for variational relation problems on Hadamard manifolds, and existence and stability of weakly Pareto-Nash equilibrium points for n-person multi-objective games on Hadamard manifolds.     

Generalized Browder-type fixed point theorem with strongly geodesic convexity on Hadamard manifolds with applications

In this paper, a generalized Browder-type fixed point theorem on Hadamard manifolds is introduced, which can be regarded as a generalization of the Browder-type fixed point theorem for the set-valued mapping on an Euclidean space to a Hadamard manifold. As applications, a maximal element theorem, a section theorem, a Ky Fan-type Minimax Inequality and an existence theorem of Nash equilibrium for non-cooperative games on Hadamard manifolds are established.     

On regularization of plurisubharmonic functions on manifolds

We study the question of when a -plurisubharmonic function on a complex manifold, where is a fixed -form, can be approximated by a decreasing sequence of smooth -plurisubharmonic functions. We show in particular that it is always possible in the compact Kähler case.

     

On maximal monotonicity of bifunctions on Hadamard manifolds

We study some conditions for a monotone bifunction to be maximally monotone by using a corresponding vector field associated to the bifunction and vice versa. This approach allows us to establish existence of solutions to equilibrium problems in Hadamard manifolds obtained by perturbing the equilibrium bifunction.     

Existence results for equilibrium problems with applications to evolution equations

In this paper, we consider evolution equations with time-dependent pseudomonotone and quasimonotone operators by a new approach based on equilibrium problems theory. We establish new existence results for equilibrium problems associated to pseudomonotone and quasimonotone bifunctions in the topological sense. The results obtained are then applied to derive existence of solutions for evolution equations. The approach is new and leads to improve and unify most of the results obtained in this direction.     

Tikhonov regularization method for a system of equilibrium problems in Banach spaces

The purpose of this paper is to investigate the Tikhonov regularization method for solving a system of ill-posed equilibrium problems in Banach spaces with a posteriori regularization-parameter choice. An application to convex minimization problems with coupled constraints is also given.