Mircea Balaj

University of Oradea · Department of Mathematics and Computer Science

The aim of this paper is to investigate a variational relation problem of Stampacchia type. The existence results presented here are different by the ones obtained in other works which deal with this subject. As applications, we establish existence theorems for two types of variational inequality problems

If X is a convex subset of a topological vector space and f is a real bifunction defined on X×X, the problem of finding a point x0∈X such that f(x0,y)≥0 for all y∈X, is called an equilibrium problem. When the bifunction f is defined on the cartesian product of two distinct sets X and Y we will call it a generalized equilibrium problem. In this pape...

Given a nonempty convex subset X of a topological vector space and a real bifunction f defined on $$X \times X$$ X × X , the associated equilibrium problem consists in finding a point $$x_0 \in X$$ x 0 ∈ X such that $$f(x_0, y) \ge 0$$ f ( x 0 , y ) ≥ 0 , for all $$y \in X$$ y ∈ X . A standard condition in equilibrium problems is that the values of...

Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed point and saddle point problems, and noncooperative games as particular cases. In this paper sufficient conditions for the existence of solutions of an equilibrium problem are given by weakening the assumption of quasiconvexity of the...

In this paper, existence results for scalar and vector equilibrium problems involving two bifunctions are established. To this aim, a new concept of generalized pseudomonotonicity for a pair of bifunctions is introduced. It leads to existence criteria different from the ones encountered in the literature. The given applications refer to minimax ine...

In this article, we introduce the notion of bifunction with the property of negative transitivity and, using the Berge-Klee intersection theorem, we establish existence theorems of the solutions for the equilibrium problems, when the involved bifunction has this property. Then, using standard scalarization techniques, we obtain existence criteria o...

In this paper, we introduce the concept of generalized weak KKM mapping that is more general than many others encountered in the KKM theory. Then, two previous intersection theorems of the author are extended from weak KKM mappings to generalized weak KKM mappings. Applications of these results to set-valued equilibrium problems and minimax inequal...

By a quasi-equilibrium problem is understood an equilibrium problem with a constraint set depending on the current point. In this paper, we establish existence results for scalar and vector quasi-equilibrium problems, when the standard equilibrium condition is replaced by a weaker one.

In this paper, the concept of weak KKM set-valued mapping is extended from topological vector spaces to hyperconvex metric spaces. For these mappings we obtain several intersection theorems that prove to be useful in establishing existence criteria for weak and strong solutions of the general variational inequality problem and minimax inequalities.

Let X be a convex set in a vector space, Y be a nonempty set and S,T:X⇉Y two set-valued mappings. S is said to be a weak KKM mapping w.r.t. T if for each nonempty finite subset A of X and any x∈conv A, T(x)∩S(A)≠∅. Recently, the authors obtained two intersection theorems for a pair of such mappings, when X is a compact convex subset of a topologica...

In this paper, we obtain three intersection theorems that can be considered versions of Theorem 3.1 from the paper [Agarwal, R. P., Balaj, M. and O’Regan, D., Intersection theorems with applications in optimization, J. Optim. Theory Appl., 179 (2018), 761–777]. As will be seen, there are two major differences between the hypotheses of the above men...

In this paper, we establish two intersection theorems which are useful in considering some optimization problems (complementarity problems, variational inequalities, minimax inequalities, saddle point problems).

Let Φ be the class of all real functions φ:[0,∞[×[0,∞[→[0,∞[ that satisfy the following condition: there exists α∈]0,1[such thatφ((1-α)r,αr)<r,for allr>0. In this paper, we show that if X is a nonempty compact convex subset of a real normed vector space, any two closed set-valued mappings T,S:X⇉X, with nonempty and convex values, have a common fixe...

In this paper, the concept of weak convex set-valued mapping is introduced and various conditions for a set-valued mapping to be weak convex are given. Then, existence theorems for the Stampacchia variational inequality problem are established, when the involved mapping is weak convex.

In this paper, we establish several common fixed point theorems for families of set-valued mappings defined in hyperconvex metric spaces. Then we give several applications of our results.

Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are special cases of a variational relation problem proposed in a recent paper by Agarwal et al. (J. Optim. Theory Appl. 2012;155:417–429). The purpose of this paper is to establish new existence results for the solutions of this problem. The main ingredie...

Our purpose in this paper is to present two methods for obtaining common fixed point theorems in topological vector spaces. Both methods combine an intersection theorem and a fixed point theorem, but the order in which they are applied differs.

The purpose of this paper is to present a unifying approach for several variational relation problems which can be regarded as general models for various vector equilibrium problems encountered in the recent literature. Existence criteria for the solutions of these problems will be established in locally convex spaces by using the Kakutani-Fan-Glic...

In this paper, we consider variational relation problems involving a binary relation. The framework presented is more general than that in [J. Optim. Theory Appl. 138 (2008) , 65–76] and in other recent papers which deal with this subject.

In [29], Luc investigated the existence of solution for an abstract problem in variational analysis, called by him, variational relation problem. In this paper, we study Luc's problem with certain lower semicontinuous set-valued constraints, �first, when one of the variable is missing, and then in its general form.

The aim of this paper is to establish new existence results for a certain type of variational relation problem studied by Agarwal et al. in a very recent paper. This problem is a general model for several quasivariational inclusion and quasiequilibrium problems investigated in many recent papers. The main tools in proving the existence theorems are...

In this paper, we use fixed point techniques to establish existence criteria of the solution for a system of two variational relations with lower semicontinuous set-valued mappings.

Variational relation problems, a recent concept introduced by Luc, are general models for a large class of problems in optimization and nonlinear analysis. In this paper we establish an existence theorem for the solution of the following variational relation problem: Find $x^*\in X$ such that $(x^*, y)\in R$ for every $y\in Y$, where $X$ is a nonem...

In the last decade, more and more authors have attempted to treat in a unified manner, by means of some binary or ternary relations, various equilibrium or quasi-equilibrium problems encountered in literature. Several types of variational relation problems have been investigated in many recent papers in which, as for other mathematical models, the...

In this paper, we establish a common fixed point theorem for a family of self set-valued mappings on a compact and convex set in a locally convex topological vector space. As applications, we obtain an existence theorem of solutions for a variational inequality of Stampacchia type and some Ky Fan-type minimax inequalities.

Given a nonempty convex set $X$ in a locally convex Hausdorff topological vector space, a nonempty set $Y$ and two set-valued mappings $T X\rightrightarrows X$, $S Y\rightrightarrows X$ we prove that under suitable conditions one can find an $x\in X$ which is simultaneously a fixed point for $T$ and a common point for the family of values of $S$. A...

The variational relation problems have been introduced in 2008, by Dinh The Luc, as general models for a large class of problems in nonlinear analysis and applied mathematics. Since this manner of approach proved to be a powerful tool for studying a wide class of problems in nonlinear analysis and applied mathematics, several types of variational r...

Variational relation problems were introduced by Luc ] as a general model for a large class of problems in nonlinear analysis and applied mathematics. Since this manner of approach provides unified results for several mathematical problems it has been used in many recent papers. In this paper we investigate the existence of solutions for three type...

In this paper we study the sequence of successive approximations, the fixed points and the Ishikawa iterates for the Bernstein max-product operator.

The purpose of this paper is to present a unified approach to study the existence of solutions for two types of variational relation problems, which encompass several generalized equilibrium problems, variational inequalities and variational inclusions investigated in the recent literature. By using two well-known fixed point theorems, we estab...

In this paper, we establish two matching theorems involving three maps. As applications, KKM type theorems, intersections theorems, analytic alternatives and min-imax inequalities are obtained.

In this paper we give two matching theorems of Ky Fan type concerning open or closed coverings of nonempty convex sets in a topological vector space. One of them will permit us to put in evidence, when X and Y are convex sets in topological vector spaces, a new subclass of KKM (X, Y) different by any admissible class Ac (X, Y). For this class of...

In this paper, we first obtain an existence theorem of the solutions for a variational relation problem. An existence theorem for a variational inclusion problem, a KKM theorem and an extension of the well know Ky Fan inequality will be established, as particular cases. Some applications concerning a saddle point problem with constraints, existence...

In this paper we establish two alternative principles of the following type: If X and Y are convex subsets of two locally convex Hausdorff topological vector spaces and ${F,S:X \multimap Y}$ are two set-valued mappings satisfying certain conditions, then either there exists ${x_0 \in X}$ such that ${F(x_0) = \emptyset}$ or ${\bigcap_{x \in...

Let A, B and Y be nonempty sets, S 1 :A⇉A, S 2 :A⇉B, T:A×B⇉Y be set-valued mappings with nonempty values and R(a,b,y) be a relation linking elements a∈A, b∈B and y∈Y. In [J. Optim. Theory Appl. 138, No. 1, 65–76 (2008; Zbl 1148.49009)], D. T. Luc established existence theorems for solutions of the following problem: find a ¯∈A such that a ¯ is a fi...

In this paper, using the Kakutani–Fan–Glicksberg fixed point theorem, we obtain an existence theorem for a generalized vector quasi-equilibrium problem of the following type: for a suitable choice of the sets X, Z and V and of the mappings T:X ⊸ X, R:X ⊸ X, Q:X ⊸ Z, F:X× X×Z ⊸ V, C:X ⊸ V, find [(x)\tilde]\widetilde{x} ∈X such that [(x)\tilde]\wid...

In this paper, using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain an existence theorem of a point which is simultaneously fixed point for a given mapping and equilibrium point for a very general vector equilibrium problem. Finally some particular cases are discussed and three applications are given.

In this paper we exploit the method of variational relations to establish existence of solutions to a general inclusion problem. The result is applied to variational relation problems in which several relations are simultaneously considered. Particular cases of variational inclusion and intersection of set-valued maps are also discussed.

In this paper we obtain a very general theorem of ρ-compatibility for three multivalued mappings, one of them from the class ж. More exactly, we show that given a G- convex space Y, two topological spaces X and Z, a (binary) relation ρ on 2Z and three mappings P: XZ, Q: YZ and T∈ ж (Y, X) satisfying a set of conditions we can find (x, y)∈ X× Y s...

In this paper, we study various types of variational relation problems. We establish the existence of solutions for these types of problems and point out some important particular cases and their applications. We also show that some existence theorems of solution for these types of problems and some existence theorems of variational inclusion probl...

In this paper, using the Kakutani-Fan-Glicksberg fixed point theorem, we obtain an existence theorem of a point which is simultaneously fixed point for a given mapping and equilibrium point for a very general vector equilibrium problem. Finally some particular cases are discussed and three applications are given.

In this paper, we introduce a new KKM property for pairs of families of maps. Two pairs of classes of maps having this property are given.

In this article, we introduce the concept of a family of set-valued mappings generalized Knaster–Kuratowski–Mazurkiewicz (KKM) w.r.t. other family of set-valued mappings. We then prove that if X is a nonempty compact convex subset of a locally convex Hausdorff topological vector space and 𝒯 and 𝒮 are two families of self set-valued mappings of X su...

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 b...

In this paper we obtain first a very general coincidence theorem. From this we derive a new coincidence theorem and two alternative theorems concerning existence of maximal elements. Applications of these results to generalized equilibrium problems and minimax inequalities are given in the last sections.

Using a fixed point theorem by Kuo, Jeng and Huang, we obtain in G-convex spaces a very general intersection theorem concerning the values of three maps. From this result we derive successively alternative theorems concerning maximal elements, analytic alternatives and minimax inequalities.

In this paper we investigate two new types of generalized equilibrium problems. The theory is self contained and relies only on Brouwer’s fixed point theorem. Keywords G-convex space, Fixed point, Weakly G–KKM-mapping, Generalized equilibrium problem, Weakly equilibrium problem Mathematics Subject Classification (2000) 54H25, 54C60, 91B50

Deguire and Lassonde [P. Deguire, M. Lassonde, Familles sélectantes, Topol. Methods Nonlinear Anal. 5 (1995) 261–269] extend the concept of continuous selection and introduce the notion of selecting family for a family of set-valued mappings. In this paper, we first establish a new existence theorem of selecting families. The existence of the selec...

In this paper we establish several minimax inequalities closely related to the von Neumann-Sion minimax principle.

In this paper we obtain a very general intersection theorem for the values of a map. From this we derive existence theorems for two types of vectorial equilibrium problems, an analytic alternative and a minimax inequality involving three real functions.

New fixed point theorems for permissible maps between $Fr{\acute{e}}chet$ spaces are presented. The proof relies on index theory developed by Dzedzej and on viewing a $Fr{\acute{e}}chet$ space as the projective limit of a sequence of Banach spaces.

In this paper we obtain a very general intersection theorem for the values of a map. From this we derive an analytic alternative, a new type of minimax inequality and an alternative theorem concerning three abstract equilibrium problems.

In this paper, we obtain three alternative theorems in GG-convex spaces, each of them involving three set-valued mappings. From these results, we derive new alternative theorems and two very general minimax inequalities.

In this paper we establish two intersection theorems for a map whose dual is upper or lower semicontinuous. As applications, analytic alternatives, and minimax inequalities are obtained.

In this work we obtain two minimax inequalities in G-convex spaces which extend and improve a large number of generalizations of the Ky Fan minimax inequality and of the von Neumann-Sion minimax principle.

In this paper we establish two minimax theorems of Sion-type in G-convex spaces. As applications we obtain generalisations of some theorems concerning compatibility of some systems of inequalities.(Received October 04 2004)

In this paper fixed point theorems for maps with nonempty convex values and having the local intersection property are given. As applications several minimax inequalities are obtained.

In this paper we extend to G-convex spaces a characterization theorem of general- ized KKM mappings obtained by Pathak and Khan. As applications we establish coincidence, fixed point and matching theorems.

Applying a KKM-type theorem due to Park we obtain an acyclic version of a minimax inequality established by Ky Fan. The result is applied to formulate acyclic versions of other minimax results and of a theorem of Ky Fan concerning compatibility of some systems of inequalities.

We obtain several intersection theorems for the values of a concave-convex or only concave set-valued mapping. From each of these theorems we derive a Sion type minimax theorem.

Using Fan-Glicksberg fixed point theorem we obtain in this paper a fixed point theorem for the composition of two Kakutani maps.As application of this we get a new fixed point theorem, section properties and minimax inequalities. In order to give a simple proof for von Neuman minimax theorem, Kakutani (11) extended the well-known Brower's fixed poi...

We introduce a new concept of generalized Knaster–Kuratowski–Mazurkiewicz (KKM) family of sets and related to this we obtain fixed point theorems and sections results in homotopically trivial spaces.

In this paper necessary and sufficient conditions for the compatibility of some systems of quasi-convex or convex inequalities are established. Finally a new proof for a theorem of Shioji and Takahashi is given.

In this paper we give a new proof for a fixed point theorem due to ˇCiriˇc. Our method permits the localization of the fixed point into a certain closed ball.

Using a collectively fixed point theorem we obtain section theorems, minimax inequalities and fixed point theorems for families of composed maps.

In this paper we give a new proof for a fixed point theorem due toČiričtoˇtoČirič.Our method permits the localization of the fixed point into a certain closed ball. 2000 Mathematics Subject Classification: 54H25, 54E50.

In [11] Kim obtained fixed point theorems for maps defined on some "locally G-convex" subsets of a generalized convex space. Theorem 2 in Kim's article determines us to introduce, in this paper, the notion of K-G-convex space. In this framework we obtain fixed point theorems, section properties and minimax inequalities.

A unified generalization of two Halpern's fixed point theorems is given. It is then used to obtain a coincidence theorem and a minimax inequality.

We obtain generalizations of the Fan's matching theorem for an open (or closed) covering related to an admissible map. Each of these is restated as a KKM theorem. Finally, applications concerning coincidence theorems and section results are given.

In this paper we obtain a Fan’s matching theorem in H-spaces. Applications concerning intersection results, fixed point theorems, minimax inequalities are given.

Using the ‘multiplied’ version of Helly's theorem given by Bárány (Discrete Math. 40 (1982) 141–152) we generalize some selection and separation results obtained in Behrends and Nikodem (Studia Math. 116 (1) (1995) 43–48), Nikodem and Wa̧sowicz (Aequationes Math. 49 (1995) 160–164) and Wa̧sowicz (J. Appl. Anal. 1 (2) (1995) 173–179). In particular,...

In this paper we obtain sets of conditions under which the convex hull of a family of convex cones is an acute cone. Some intersectional results for families of convex sets there are also given. Finally, using two combinatorial results concerning families of convex cones, a lower bound for the Ramsey numbers R2n (m.3n) is established.

In this paper the main result in (1), concerning (n + 1)-families of sets in general position in Rn, is generalized. Finally we prove the following theorem: If {A1,A2,...,An+1} is a family of compact convexly connected sets in general position in R n , then for each proper subset I of {1,2,...,n + 1} the set of hyperplanes separating ∪{Ai : i ∈ I}...

. A family of sets in R n is said to be in general position if any m--flat, 0 m n Gamma 1; intersects at most m+ 1 members of the family. Using the Fan-- Glicksberg--Kakutani fixed point theorem, we prove that if fA 1 ; A 2 ; : : : ; A n+1 g is a family of compact convexly connected sets then for any proper subset I of f1; 2; : : : ; n+1g; there ex...