Generalized Convex Spaces Research Articles

Fuzzy generalized convex spaces

On completely distributive lattice, the notion of fuzzy generalized convex space is introduced. It can be characterized by many means including fuzzy generalized hull space, fuzzy generalized restricted hull space, fuzzy generalized convexly enclosed relation space and fuzzy generalized derived hull space.

Generalized convexity and applications to fixed points and equilibria

In this paper, we give a uniform approach for generalized convexity by using the concept of L-convexity defined by Ben El-Mechaiekh et al. (J Math Anal Appl 222:138–150, 1998). We prove that the generalized notion of L-space contains well-known generalized convex spaces defined in the literature in topological vector spaces as well as several generalized convexity structures defined on metric spaces. In this context, we give a generalized version of the Fan–Knaster–Kuratowski–Mazurkiewicz Principle (FKKM Principle) in L-spaces and a Browder-Fan type theorem about the existence of fixed points for open lower section set-valued maps defined in an L-space. As an application, we prove the existence of equilibria for an abstract economy with an infinite number of agents.

The robustness of generalized abstract fuzzy economies in generalized convex spaces

In this paper, we study the economic model proposed by Anderlini and Canning, a parameterized class of generalized abstract fuzzy economies together with an associated abstract rationality function, and show that the structural stability of this model implies its robustness to ε -equilibria.

Fixed point theorems in minimal generalized convex spaces

This paper deals with coincidence and fixed point theorems in minimal generalized convex spaces. By establishing a kind of KKM Principle in minimal generalized convex space, we obtain some results on coincidence point and fixed point theorems. Generalized versions of Ky Fan?s lemma, Fan-Browder fixed point theorem, Nash equilibrium theorem and some Urai?s type fixed point theorems in minimal generalized convex spaces are given.

Generalized saddle points theorems for set-valued mappings in locally generalized convex spaces

In the paper, the notions of generalized loose (weak) saddle points for set-valued mappings are introduced. By applying the fixed point theorem and the scalarization method, some new existence theorems for generalized loose (weak) saddle points are established in locally G -convex space. As applications, some vector minimax theorems are given also.

Fixed points, coincidence points and maximal elements with applications to generalized equilibrium problems and minimax theory

In this paper, using a generalization of the Fan–Browder fixed point theorem, we obtain a new fixed point theorem for multivalued maps in generalized convex spaces from which we derive several coincidence theorems and existence theorems for maximal elements. Applications of these results to generalized equilibrium problems and minimax theory will be given in the last sections of the paper.

A Unified Fixed Point Theory in Generalized Convex Spaces

Let \({\mathfrak {B}}\)be the class of ‘better’ admissible multimaps due to the author. We introduce new concepts of admissibility (in the sense of Klee) and of Klee approximability for subsets of G-convex uniform spaces and show that any compact closed multimap in\({\mathfrak {B}}\)from a G-convex space into itself with the Klee approximable range has a fixed point. This new theorem contains a large number of known results on topological vector spaces or on various subclasses of the class of admissible G-convex spaces. Such subclasses are those of Φ-spaces, sets of the Zima–Hadžic type, locally G-convex spaces, and LG-spaces. Mutual relations among those subclasses and some related results are added.

The Knaster–Kuratowski–Mazurkiewicz theorem and abstract convexities

The Knaster–Kuratowski–Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called “intersection” theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from R n to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. Math. 11 (1959) 556–575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173–187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285–302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.

GENERALIZED KKM MAPS, MAXIMAL ELEMENTS AND ALMOST FIXED POINTS

In the framework of generalized convex spaces, we show that generalized KKM maps can be regarded as ordinary KKM maps, and obtain some applications to equilibrium result, maximal element theorems, and almost fixed point theorems on multimaps of the Zima type.

On scalar equilibrium problem in generalized convex spaces

In this paper we prove the existence of a solution to the scalar equilibrium problem in generalized convex space. As applications, we derive results on the best approximations and simultaneous approximations. The results of this paper generalize some known results in the literature.

Another form of KKM type theorem and its applications on generalized convex spaces

In this paper, another form of KKM type theorem on generalized convex spaces is obtained and the problems of von Neumann-Fan type sup inf sup inequalities and variational inequalities are discussed for their applications. The main results improve and generalize the corresponding results in previous papers.

Fixed point theorems for compact multimaps on almost [formula omitted]-convex sets in generalized convex spaces

In this paper, we introduce the concept of the strict Kakutani multimap (KKM) property and investigate the fixed point problem for multimaps having this property on almost Γ -convex subsets of locally G -convex uniform spaces. Our new fixed point theorem generalizes the well-known Fan–Glicksberg fixed point theorem and partially extends the Himmelberg fixed point theorem. Its application to minimax theorem is also given.

Remarks on Best Approximations in Generalized Convex Spaces

In this paper, we prove a best approximation theorem in generalized convex spaces. As an application, we derive a result on the existence of a maximal element and a coincidence point theorem in generalized convex spaces. The results of this paper generalize some known results in the literature.

On almost coincidence points in generalized convex spaces

We prove an almost coincidence point theorem in generalized convex spaces. As an application, we derive a result on the existence of a maximal element and an almost coincidence point theorem in hyperconvex spaces. The results of this paper generalize some known results in the literature.

Coincidence and fixed point theorems for functions in S-KKM class on generalized convex spaces

We establish a coincidence theorem in -KKM class by means of the basic defining property for multifunctions in -KKM. Based on this coincidence theorem, we deduce some useful corollaries and investigate the fixed point problem on uniform spaces.

Fixed points theorems and quasi-variational inequalities in G-convex spaces

We obtain a generalized continuous selection theorem and a coincidence theorem for generalized convex spaces. Some new Himmelberg type theorems and Eilenberg-Montgomery and Gorniéwicz type fixed point theorems for mappings with KKM property are established in noncompact LG-spaces. Moreover, applications to these fixed point theorems for existence of equilibria are given.

Comments on collectively fixed points in generalized convex spaces

Based on a Fan–Browder type fixed point theorem due to the author, we deduce new general collectively fixed point theorems for a family of Browder type multimaps defined on a product of generalized convex spaces.

Generalized Vector Equilibrium Problems in Generalized Convex Spaces

In this paper, we introduce and study a class of abstract generalized vector equilibrium problems (AGVEP) in generalized convex spaces which includes most vector equilibrium problems, vector variational inequality problems, generalized vector equilibrium problems, and generalized vector variational inequality problems as special cases. By using the generalized GKKM and generalized SKKM type theorems due to the first author, some new existence results of equilibrium points for the AGVEP are established in noncompact generalized convex spaces. As consequences, some recent results in the literature are obtained under much weaker assumptions.

Comments on some fixed point theorems in hyperconvex metric spaces

In this paper, we show that a number of known fixed point theorems for the Fan–Browder type maps or acyclic maps defined on (subsets of) hyperconvex metric spaces are simple consequences of the previously known theorems for corresponding maps defined on generalized convex spaces.

Maximal elements forG B-majorized mappings in productG-convex spaces and applications (II)

A new family of set-valued mappings from a topological space into generalized convex spaces was introduced and studied. By using the continuous partition of unity theorem and Brouwer fixed point theorem, several existence theorems of maximal elements for the family of set-valued mappings were proved under noncompact setting of product generalized convex spaces. These theorems improve, unify and generalize many important results in recent literature.