Fan-KKM Theorem Research Articles

On the existence and Lipschitz-like property of the solution map to parametric equilibrium problems governed by trifunction with applications

In this paper, we study a class of parametric equilibrium problems governed by a trifunction. We apply the Fan-KKM theorem under new type of monotonicity and coercivity conditions for trifunctions to obtain the existence result of a solution for the equilibrium problem. For the solution map of parametric equilibrium problems, we obtain the Hölder-type Lipschitz-like property relative to a closed and convex set, which generalizes some existing results in the literature. We illustrate the application of our abstract results in the study of a new model of a fluid flow problem with slip and leak boundary conditions of frictional type.

Applications of Fell Topology to Closed Set-Valued Optimizations in Partially Ordered First Countable Topological Vector Spaces

Let (X, ) be a topological vector space equipped with a partial order which is induced by a nonempty pointed closed and convex cone K in X. Let (X) = 2 X be the power set of X and (X) = 2 X \\{}. In this paper, we consider the so called upward preordering on (X), which has been used by many authors (see [47, 9, 1114, 1819]). We first prove some properties of this ordering relation Let (X) denote the collection of all -closed subsets of X and (X) = (X)\\{}, which is equipped with the Fell topology Let C be a nonempty subset of X and let F: C (X) be a closed set-valued mapping. By applying the Fell topology on (X) and the Fan-KKM Theorem, we prove some existence theorems for some -minimization and -maximization problems with respect to F subject to the subset C for first countable topological vector spaces. These results will be applied to solve some closed ball-valued optimization problems in partially ordered Hilbert spaces. Some examples will be provided in

On F-implicit Minimal Vector Variational Inequalities

In this paper, by introducing some new concepts in minimal spaces, we prove a generalized form of the Fan-KKM theorem in minimal vector spaces. A new class of minimal generalized vector F-implicit variational inequality problems and, as an application of Fan-KKM theorem is investigated. Moreover, an existence theorem for this kind of problems under some suitable assumptions in minimal vector spaces is given.

Solvability and algorithms of generalized nonlinear variational-like inequalities in reflexive Banach spaces

This paper deals with solvability and algorithms for a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By employing the Banach’s fixed point theorem, Schauder’s fixed point theorem, and FanKKM theorem, we obtain a sufficient condition which guarantees the existence of solutions for the generalized nonlinear variational-like inequality. We introduce also an auxiliary variational-like inequality and, by utilizing the minimax inequality, get the existence and uniqueness of solutions for the auxiliary variational-like inequality, which is used to suggest an iterative algorithm for solving the generalized nonlinear variational-like inequality. Under certain conditions, by means of the auxiliary principle technique, we both establish the existence and uniqueness of solutions for the generalized nonlinear variational-like inequality and discuss the convergence of iterative sequences generated by the iterative algorithm. Our results extend, improve, and unify several known results in the literature.

An extension of Tychonoff’s fixed point theorem to pseudonorm adjoint topological vector spaces

In this paper, we introduce a type of pseudonorms and quasi-pseudonorms on real vector spaces. By these concepts, we introduce the concept of pseudonorm adjoint topological vector spaces, which include locally convex topological vector spaces as special cases. By the Fan-KKM theorem, we prove a fixed point theorem in pseudonorm adjoint topological vector spaces, that is a proper extension of Tychonoff’s fixed point theorem on locally convex topological vector spaces. Then we provide an example to show that this extension is indeed a proper extension.

Applications of Optimization Theory to Social Benefit Maximizations in Macroeconomics with Uncertainty

In this paper, we use the upward power preorder relation on the power sets of ordered sets to construct a mathematical social benefit maximization model in a fiscal year with respect to the initially planned fiscal budgetary policies and the practically implementation of the fiscal budgetary policies. We only consider the uncertainty case for the social benefit functions. By applying the Fan-KKM Theorem, we prove the existence of a generalized optimal complete feasible fiscal budget policy and the existence of a generalized Pareto optimal complete feasible fiscal policy, respectively.

Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces

The aim of this article is to study new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. We establish some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some invexity assumptions. The celebrated Fan-KKM theorem is used to obtain the existence of solution of generalized nonsmooth exponential-type vector variational like inequality problems. In support of our main result, some examples are given. Our results presented in this article improve, extend, and generalize some known results offer in the literature.

Split equilibrium problems for related games and applications to economic theory

ABSTRACTIn this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan-KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As an application of this existence to economics, an example is provided that studies the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual (playing twice) extended Bertrand duopoly model of price competition.

Split Equilibrium Problems for Related Games and Applications to Economic Theory

In this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan–KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As an application of this existence to economics, an example is provided that studies the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual (playing twice) extended Bertrand duopoly model of price competition.

On vector variational-like inequalities involving right upper-Dini-derivative functions

In this paper, we introduce (weak) Stampacchia and Minty arcwise connected vector variational-like inequalities in the term of right upper-Dini-derivative and establish not only the relations of introduced inequalities with vector optimization problems but also the existence results, by using KKM-Fan theorem and Brouwer fixed point theorem. Examples are provided to illustrate the derived results.

Existence results of the system of generalized variational inequalities with multi-valued mappings

In this article, we study a generalized variational inequality problem with multi-valued mappings over product sets and the system of generalized variational inequalities with multi-valued mappings which are equivalent problems. By developing the idea of generalized densely relatively pseudomonotone mappings, and by using well-known Fan-KKM theorem and fixed point theorem, we prove existence results of our problem. We construct an example of generalized Nash equilibrium problem in the context of our problem, and as an application of our results, we establish an existence of coincident point result.

Study of equilibrium problem in Hadamard manifolds by extending some concepts of nonlinear analysis

In this paper, we attempt to define a new KKM map for nonself maps in the setting of Hadamard manifolds, which is then utilized to state the finite intersection property in this framework. Subsequently, this property is used to develop the Fan-KKM theorem for nonself maps on Hadamard manifolds. Moreover, a new definition of upper semicontinuouty and a generalization of the closedness of a set are also proposed. Inspired by this extension, we establish a new existence theorem of a solution to the equilibrium problem for nonself maps on Hadamard Manifolds. As an application of our KKM theorem, we obtain an existence result of maximal elements for nonself set valued mappings in Hadamard manifold frameworks. Finally, for the sake of clarity, a number of examples are presented throughout the paper and our results are compared with the results of some other papers.

A class of generalized invex functions and vector variational-like inequalities

In this paper, a class of generalized invex functions, called (alpha,rho,eta)-invex functions, is introduced, and some examples are presented to illustrate their existence. Then we consider the relationships of solutions between two types of vector variational-like inequalities and multi-objective programming problem. Finally, the existence results for the discussed variational-like inequalities are proposed by using the KKM-Fan theorem.

Existence theorems for generalized vector equilibria with variable ordering relation

In this paper we study the solvability of the generalized vector equilibrium problem (for short, GVEP) with a variable ordering relation in reflexive Banach spaces. The existence results of strong solutions of GVEPs for monotone multifunctions are established with the use of the KKM-Fan theorem. We also investigate the GVEPs without monotonicity assumptions and obtain the corresponding results of weak solutions by applying the Brouwer fixed point theorem. These results are also the extension and improvement of some recent results in the literature.

Existence and Connectedness of Solutions for Generalized Vector Quasi-Equilibrium Problems

In this paper, we establish three existence theorems for strongly efficient solutions, weakly efficient solutions and efficient solutions of generalized vector quasi-equilibrium problems by using the Fan-KKM theorem and the scalarization method. Moreover, we investigate the connectedness of the sets of weakly efficient solutions and efficient solutions for generalized vector quasi-equilibrium problems and make a new attempt to establish the path connectedness of the set of weakly efficient solutions for generalized vector quasi-equilibrium problems.

Existence and stability of solutions for a class of generalized vector equilibrium problems

In this paper, two existence theorems concerning the strong efficient solutions and the weakly efficient solutions of generalized vector equilibrium problems are derived by using the Fan-KKM Theorem and an existence theorem for the efficient solutions of generalized vector equilibrium problems is established by using the scalarization method. Moreover, the lower semicontinuity of the strong efficient solution mapping and the weakly efficient solution mapping to parametric generalized vector equilibrium problems are showed under suitable conditions with neither monotonicity nor any information of the solution mappings. Finally, some applications to the vector optimization problems and the Stackelberg equilibrium problem are also given.

Extended f-Vector Equilibrium Problem

We introduce and study extended f-vector equilibrium problem. By using KKM-Fan Theorem as basic tool, we prove existence theorem in the setting of Hausdorff topological vector space and reflexive Banach space. Some examples are also given.

Exponential type vector variational-like inequalities and nonsmooth vector optimization problems

This paper is devoted to study a new class of exponential form of vector variational-like inequality problems comprised of locally Lipschitz functions having exponential type invexities. In the setting of Banach space, we investigate the relationship of nonsmooth exponential type vector variational-like inequality problems with vector optimization problems involving nonsmooth \((p,r)\)-invex functions. Also, we explore the conditions for solvability of the aforesaid nonsmooth exponential type vector variational-like inequality problems using Fan-KKM Theorem. Moreover, we provide examples to elucidate our results.

Generalized strongly vector equilibrium problem for set-valued mappings

In this paper, we introduce and study a generalized strongly vector equilibrium problem for set-valued mappings in real Banach spaces. Using Minty type lemma and KKM-Fan theorem as basic tools, we prove existence theorems for generalized strongly vector equilibrium problem with and without monotonicity, respectively. Some examples are given.

Existence Results for Strong Mixed Vector Equilibrium Problem for Multivalued Mappings

We consider a strong mixed vector equilibrium problem in topological vector spaces. Using generalized Fan-Browder fixed point theorem (Takahashi 1976) and generalized pseudomonotonicity for multivalued mappings, we provide some existence results for strong mixed vector equilibrium problem without using KKM-Fan theorem. The results in this paper generalize, improve, extend, and unify some existence results in the literature. Some special cases are discussed and an example is constructed.