Existence Theorems Of Maximal Elements Research Articles

On Maximal Elements with Applications to Abstract Economies, Fixed Point Theory and Eigenvector Problems

Two existence theorems of maximal elements in H-spaces are obtained without compactness. More accurately, we deal with the correspondence to be of L -majorized mappings in the setting of noncompact strategy sets but merely requiring a milder coercive condition. As applications, we obtain an equilibrium existence theorem for general abstract economies, a new fixed point theorem, and give a sufficient condition for the existence of solutions of the eigenvector problem (EIVP).

NONEMPTY INTERSECTION THEOREMS AND SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS IN FC-SPACES

By using some existence theorems of maximal elements for a family of set-valued mappings involving a better admissible set-valued mapping under noncompact setting of FC-spaces, we present some non-empty intersection theorems for a family in product FC-spaces. Then, as applications, some new existence theorems of equilibrium for a system of generalized vector equilibrium problems are proved in product FC-spaces. Our results improve and generalize some recent results.

Fixed Points, Maximal Elements and Equilibria of Generalized Games in Abstract Convex Spaces

We firstly prove some new fixed point theorems for set‐valued mappings in noncompact abstract convex space. Next, two existence theorems of maximal elements for class of 𝒜C,θ mapping and 𝒜C,θ‐majorized mapping are obtained. As in applications, we establish new equilibria existence theorems for qualitative games and generalized games. Our theorems improve and generalize the most known results in recent literature.

Fixed Points and Existence Theorems of Maximal Elements with Applications in FC-Spaces

We present some fixed point theorems and existence theorems of maximal elements in FC-space from which we derive several coincidence theorems. Applications of these results to generalized equilibrium problems and minimax theory will be given. Our results improve and generalize some recent results.

New systems of generalized quasi-variational inclusions in FC-spaces and applications

In this paper, we study some new systems of generalized quasi-variational inclusion problems in FC-spaces without convexity structure. By applying an existence theorem of maximal elements of set-valued mappings due to the author, some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems are proved in noncompact FC-spaces. As applications, some existence results of solutions for the system of quasi-optimization problems and mathematical programs with the systems of generalized quasi-variational inclusion constraints are obtained in FC-spaces.

Systems of generalized quasi-variational inclusion (disclusion) problems in FC-spaces

By applying an existence theorem of maximal elements of set-valued mappings in FC-spaces proposed by the author, some new existence theorems of solutions for systems of generalized quasi-variational inclusion (disclusion) problems are proved in FC-spaces without convexity structures. These results improve and generalize some results in recent publications from closed convex subsets of topological vector spaces to FC-spaces under weaker conditions.

Fixed point theorems and existence of equilibrium points of noncompact abstract economies for [formula omitted]-majorized mappings in [formula omitted]-spaces

In this paper, we first establish two fixed point theorems in noncompact F C -spaces. Next two existence theorems of maximal elements for L F ∗ -correspondence and L F ∗ -majorized correspondence are obtained. Finally, by applying the existence theorems of maximal elements, some new equilibrium existence theorems for one person game, qualitative games and noncompact abstract economies with L F ∗ -majorized correspondences are obtained in F C -spaces. These results generalize some known results in recent literature.

Pareto equilibria for generalized constrained multiobjective games in FC-spaces without local convexity structure

In this paper, we further study a class of generalized constrained multiobjective games where the number of players may be finite or infinite, the strategy sets may be general FC-spaces without local convexity structure, and all payoff functions get their values in infinite-dimensional topological vector spaces. By using an existence theorem of maximal elements for a family of set-valued mappings in FC-spaces due to the author, an existence theorem of solutions for a system of generalized vector quasivariational inclusions is first proved in general FC-spaces. By applying the existence result of solutions of the system of generalized vector quasivariational inclusions, some existence theorems of (weak) Pareto equilibria for the generalized constrained multiobjective games are established in noncompact product FC-spaces. Some special cases of our results are also discussed. Our results are new and different from the corresponding known results in the literature.

SYSTEMS OF GENERALIZED VECTOR QUASI-VARIATIONAL INCLUSION PROBLEMS AND APPLICATION TO MATHEMATICAL

In this paper, we introduce and study some new systems of generalized vector quasi-variational inclusion problems involving condensing mappings in locally $FC$-uniform spaces. These systems contain many known systems of generalized vector quasi-variational inclusion problems, systems of generalized vector quasi-equilibrium problems and systems of vector quasi-optimization problems as special cases. By applying an existence theorem of maximal elements of a family of set-valued mappings involving condensing mapping due to author, we prove some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems. As applications, some existence results of solutions of the mathematical programs with systems of generalized vector quasi-variational inclusion constraints are established in noncompact locally $FC$-uniform

Some Maximal Elements' Theorems in -Spaces

Let be a finite or infinite index set, let be a topological space, and let be a family of -spaces. For each , let be a set-valued mapping. Some new existence theorems of maximal elements for a set-valued mapping and a family of set-valued mappings involving a better admissible set-valued mapping are established under noncompact setting of -spaces. Our results improve and generalize some recent results.

Equilibria of nonparacompact generalized games with [formula omitted]-majorized correspondences in FC-spaces

The new concept of FC-hull for a nonempty subset in FC-spaces is introduced, which includes the convex hull in topological vector spaces, the H -convex hull in H -spaces and G -convex hull in G -convex spaces as special cases. On this basis, new classes of ℒ c -correspondences and ℒ c -majorized correspondences without open lower sections are introduced in FC-spaces. Some existence theorems of maximal elements for ℒ c -correspondences and ℒ c -majorized correspondences are proved in FC-spaces. As applications, some equilibrium existence theorems for qualitative games and generalized games with infinite set of players and ℒ c -majorized correspondences are established in FC-spaces. These results unify, improve and generalize many corresponding results in recent literature.

Maximal elements and generalized games involving condensing mappings in locally FC-uniform spaces and applications (I)

First,the notions of the measure of noncompactnees and condensing set- valued mappings are introduced in locally FC-uniform spaces without convexity struc- ture.A new existence theorem of maximal elements of a family of set-valued mappings involving condensing mappings is proved in locally FC-uniform spaces.As applications, some new equilibrium existence theorems of generalized game involving condensing map- pings are established in locally FC-uniform spaces.These results improve and generalize some known results in literature to locally FC-uniform spaces.Some further applications of our results to the systems of generalized vector quasi-equilibrium problems will be given in a follow-up paper.

Maximal elements and generalized games involving condensing mappings in locally FC-uniform spaces and applications (II)

First, the notions of the measure of noncompactness and condensing set-valued mappings are introduced in locally FC-uniform spaces without convexity structure. A new existence theorem of maximal elements of a family of set-valued mappings involving condensing mappings is proved in locally FC-uniform spaces. As applications, some new equilibrium existence theorems of generalized game involving condensing mappings are established in locally FC-uniform spaces. These results improve and generalize some known results in literature to locally FC-uniform spaces. Some further applications of our results to the systems of generalized vector quasi-equilibrium problems will be given in a follow-up paper.

Fixed points, minimax inequalities and equilibria of noncompact abstract economies in FC -spaces

Several new fixed point theorems are first proved in noncompact F C -spaces. By applying these fixed point theorems, some new minimax inequalities and new existence theorems of maximal elements of L F -correspondences and L F -majorized correspondences are established in noncompact F C -spaces. Finally, by applying the existence theorems of maximal elements, some new equilibrium existence theorems for one person games, qualitative games and noncompact abstract economies with L F -majorized correspondences are obtained in F C -spaces. Our theorems improve and generalize most known results due to Borglin and Keiding, Ding, Kim and Tan, Ding and Tan, Tan and Yuan, Tarafdar, Toussaint, Tulcea, Yannelis and Yannelis and Prabhakar etc.

An existence theorem for maximal elements and its applications

An existence theorem of maximal elements for an \(\mathcal{L}*\)-majorized correspondence defined on a non-paracompact H-space is established. As applications of the result, an equilibrium existence theorem for a non-paracompact generalized game in H-spaces with infinitely many players and with \(\mathcal{L}*\)-majorized correspondences is given.

Maximal elements of a family of G B-majorized mappings in product FC-spaces and applications

A new family of G B-majorized mappings from a topological space into a finite continuous topological spaces (in short, FC-space) involving a better admissible set-valued mapping is introduced. Some existence theorems of maximal elements for the family of G B-majorized mappings are proved under noncompact setting of product FC-spaces. Some applications to fixed point and system of minimax inequalities are given in product FC-spaces. These theorems improve, unify and generalize many important results in recent literature.

Maximal elements of [formula omitted]-majorized mappings in product [formula omitted]-spaces and applications (I)

Let I be any index set. Some new families of G KKM -mappings and G KKM -majorized mappings from a topological space X into finite continuous topological spaces ( Y i , φ N i ) (in short, FC -spaces) involving a set-valued mapping T ∈ KKM ( Y , X ) with KKM property are introduced where Y = ∏ i ∈ I Y i . Some existence theorems of maximal elements for the families of G KKM -majorized mappings and G KKM -mappings are proved under noncompact setting of product FC -spaces. As applications, some new equilibrium existence theorems of generalized games are established in noncompact product FC -spaces. These theorems improve and generalize some recent results in the literature. Some further applications to the system of generalized vector quasi-equilibrium problems will be given in a follow-up paper.

Equilibrium existence theorems of generalized games for generalized [formula omitted]-majorized mapping in topological space

In this paper, we first introduce generalized L θ , F c -correspondence and generalized L θ , F c -majorized mappings without open lower sections in the nonconvexity setting of topological space. Some existence theorems of maximal elements for generalized L θ , F c -correspondence and generalized L θ , F c -majorized mappings are obtained in topological spaces without convexity. As applications, we establish new equilibrium existence theorems for qualitative games and generalized games with infinite sets of players and generalized L θ , F c -majorized preference correspondences in topological spaces without convexity. Our results generalize and improve the corresponding results in the recent literature.

Some fixed-point theorems and minimax inequalities in FC-space

By applying the existence theorem of maximal elements, some new collectively fixed-point theorems for a family of set-valued mappings defined on the product space of noncompact FC-space are proved and some new theorems about minimax inequality involving two functions are given to show the relations of fixed-point theorem and minimax inequality in FC-spaces. These results improve and generalize many important results in the recent literature.

Nonempty Intersection Theorems and Generalized Multi-objective Games in Product FC-Spaces

A new class of generalized multi-objective games is introduced and studied in FC-spaces where the number of players may be finite or infinite, and all payoff are all set-valued mappings and get their values in a topological space. By using an existence theorems of maximal elements for a family of set-valued mappings in product FC-spaces due to author, some new nonempty intersection theorems for a family of set-valued mappings are first proved in FC-spaces. As applications, some existence theorems of weak Pareto equilibria for the generalized multi-objective games are established in noncompact FC-spaces. These theorems improve, unify and generalize the corresponding results in recent literatures.