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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 topological vector space. In this paper, we obtain open versions of the above mentioned theorems when X is a compact convex set in R n . As applications, we establish several minimax inequalities and existence criteria for the solutions of three types of set-valued equilibrium problems.

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 mentioned theorem and those of our results. Applications of the main results are considered in the last two sections of the paper.

Let $$\Phi $$ be the class of all real functions $$\varphi : [0, \infty [ \times [0, \infty [ \rightarrow [0, \infty [$$ that satisfy the following condition: there exists $$\alpha \in ]0, 1[ \text { such that } \varphi ((1 - \alpha ) r, \alpha r) 0$$ . 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 $$T, S: X \rightrightarrows X$$ , with nonempty and convex values, have a common fixed point whenver there exists a function $$\varphi \in \Phi $$ such that $$\begin{aligned} \Vert y - u\Vert \le \varphi (\Vert y - x\Vert , \Vert u - x\Vert ), \; \text { for all } x\in X, y\in T(x), u\in S(x). \end{aligned}$$ Next, we prove that the same conclusion holds when at least one of the set-valued mappings is lower semicontinuous with nonempty closed and convex values. Our common fixed point theorems turn out to be useful for a unitary treatment of several problems from optimization and nonlinear analysis (quasi-equilibrium problems, quasi-optimization problems, constrained fixed point problems, quasi-variational inequalities).

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).

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.

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 ingredients in the proofs are some continuous selection and fixed point theorems, and an interesting section result. In the last section, we prove that, applied for some concrete relations, our results are different from those obtained by other authors.

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 aim of this article is to establish new existence results for a certain type of variational relation problem studied by Agarwal et al. in a very recent article. This problem is a general model for several quasivariational inclusions and quasiequilibrium problems investigated in many recent articles. The main tools in proving the existence theorems are two well-known fixed point theorems (Eilenberg-Montgomery and Kakutany-Fan Glicksberg, respectively). In the last section, we prove that, applied in special cases, our results are different or stronger than several earlier results.

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.

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 main focus has been on sufficient conditions for the existence of solution. The existence of solution has been also studied for several types of systems of two variational relation problems called either mixed variational relation problems or simultaneous variational relation problems. In this paper, we deal with the existence of solutions for two types of mixed variational relation problems in which the involved relations (one binary and the other ternary) are linked by a certain condition of compatibility.

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 a double dagger parts per thousand X, S: Y a double dagger parts per thousand X we prove that under suitable conditions one can find an x a X which is simultaneously a fixed point for T and a common point for the family of values of S. Applying our intersection theorem, we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems.

In this article, we study the existence of solutions for a quasivariational relation problem and then give applications to the existence of solutions for set-valued Ekeland's principle, generalized vector Ekeland's variational principle and generalized equilibrium problems. Our results and techniques of proof are different from any existence result in the literature.

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 relation problems or systems of variational relation problems have been investigated in many recent papers. The present paper fits into this interesting group of works, establishing existence criteria for the solutions of two very general types of variational relation problems.

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 establish several existence criteria for the solutions of these problems.

In this paper we give two matching theorems of Ky Fan type concerning open or closed coverings of nonempty convex sets in a topo- logical 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 set-valued mappings we establish a KKM-type theorem which will be then used for obtaining existence theorems for the solutions of two types of simultaneous relation problems.

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 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 $${\bigcap_{x \in X}S(x) \neq \emptyset}$$ . As first applications of the alternative principles we obtain two matching theorems of Ky Fan type. Next, are given several analytic alternatives and minimax inequalities. Finally we establish two very general alternative theorems concerning existence of solutions of a vector equilibrium problem.

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 of a common fixed point for two mappings and an optimization problem with constraints, will be given in the last section of the paper.

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 problems are equivalent to a generalized KKM theorem. Applying our results we obtain existence theorems of common fixed point, generalized maximal element theorems, a generalized coincidence theorems and a section theorem.

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 such that 𝒮 is generalized KKM w.r.t. 𝒯, under some natural conditions, the set-valued mappings S ∈ 𝒮 have a fixed point. Other common fixed point theorems and minimax inequalities of Ky Fan type are obtained as applications.

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.

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 selecting families will be then used in order to obtain several fixed component theorems of Fan–Browder type, an intersection result, a maximal element theorem for a family of set-valued mappings and a minimax inequality.

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 G -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 minimax theorems of Sion-type inG-convex spaces. As applications we obtain generalisations of some theorems concerning compatibility of some systems of inequalities.

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 (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.

Using the ‘multiplied’ version of Helly's theorem given by Barany (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 Wasowicz (Aequationes Math. 49 (1995) 160–164) and Wasowicz (J. Appl. Anal. 1 (2) (1995) 173–179). In particular, it is shown that if three set-valued functions Φ 1 ,Φ 2 ,Φ 3 :I→ cc (R) satisfy the condition Φ i (tx+(1−t)y)∩[tΦ j (x)+(1−t)Φ k (y)]≠∅ for all x,y∈I,t∈[0,1] and every permutation (i,j,k) of the set {1,2,3} then at least one of them has an affine selection.

Variational relation problems were introduced by Luc in [1] as a general model fora large class of problems in nonlinear analysis and applied mathematics. Since this manner ofapproach provides unied results for several mathematical problems it has been used in manyrecent papers (see [2-9]). In this paper we investigate the existence of solutions for three typesof variational relation problems which encompass several generalized equilibrium problems, variational inequalities and variational inclusions studied in a long list of papers in the eld.

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

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 obtain a very general theorem of rho-compatibility for three multivalued mappings, one of them from the class B. More exactly, we show that given a G-convex space Y, two topological spaces X and Z, a (binary) relation rho on 2(Z) and three mappings P : X (sic) Z, Q : Y (sic) Z and T is an element of B(Y, X) satisfying a set of conditions we can find ((x) over tilde, (y) over tilde) is an element of X x Y such that (x) over tilde is an element of T((y) over tilde) and P((x) over tilde)rho Q((y) over tilde). Two particular cases of this general result will be then used to establish existence theorems for the solutions of some general equilibrium problems.

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 \(\widetilde{x}\) ∈X such that \(\widetilde{x}\) ∈T(\(\widetilde{x}\)) and (∀)y∈R(\(\widetilde{x}\)), (α)z∈Q(\(\widetilde{x}\)), ρ(F((\(\widetilde{x}\),y,z), C(\(\widetilde{x}\))), where ρ is a given binary relation on 2V and α is any of the quantifiers ∈, ∃. Finally, several particular cases are discussed and some applications are given.

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.