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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.

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.

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 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.

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.

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 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.

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.

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.

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.