In 1961, using his own generalization of the Knaster-Kuratowski-Mazurkiewicz (simply, KKM) theorem, Fan [2] established an elementary but very basic “geometric” lemma for multimaps and gave several applications. In 1968, Browder [1] obtained a fixed point theorem which is the more convenient form of Fan’s lemma. With this result alone, Browder carried through a complete treatment of a wide rang...

We introduce a class of generalized vector quasivariational-like inequality problems in Banach spaces. We derive some new existence results by using KKM-Fan theorem and an equivalent fixed point theorem. As an application of our results, we have obtained as special cases the existence results for vector quasi-equilibrium problems, generalized vector quasivariational inequality and vector quasi-...

In this paper, we introduce a class of generalized vector equilibrium problems for set-valued mappings which include a number of vector equilibrium problems, generalized vector variational-like inequality problems and generalized vector variational inequality problems as special cases. By using the KKM-Fan theorem and Nadler’s lemma, we prove the existence of solution to this class of generaliz...

In this paper we prove a new combinatorial theorem for labellings of trees, and show that it is equivalent to a KKM-type theorem for finite covers of trees and to discrete and continuous fixed point theorems on finite trees. This is in analogy with the equivalence of the classical Sperner’s lemma, KKM lemma, and the Brouwer fixed point theorem on simplices. Furthermore, we use these ideas to de...

We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmoot...

* Correspondence: [email protected]. th Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Abstract In this article, we consider the solutions of the system of generalized variational inequality problems in Banach spaces. By employing the generalized projection operator, the well-known Fan’s KKM theorem and Kakutani-Fan-Glicksberg fixed point theorem, we...

In 1929, Knaster et al. 1 had proved the well-known KKM theorem on n-simplex. Besides, in 1961, Fan 2 had generalized the KKM theorem to an infinite dimensional topological vector space. Later, Amini et al. 3 had introduced the class of KKM-typemappings onmetric spaces and established some fixed point theorems for this class. In this paper, we define a weaker Meir-Keeler type function and estab...

In this work, we study the following Generalized Variational Inequality Problem (for short, GVIP): Given a closed convex set K in a reflexive Banach space E with the dual E, a multifunction T : K → 2E∗ , and a vector b ∈ E, find x̄ ∈ K such that there exists ū ∈ T(x̄) satisfying 〈ū− b, y− x̄〉 ≥ 0, for all y ∈ K. By using generalized projection and the well-known Fan–KKM Theorem, we prove existence...

(1) x0 ∈ K such that c1 ≤ f(x0, y) ≤ c2, for all y ∈ K, where K is a given set, f : K × K → R is a given function and c1 and c2 two real numbers such that c1 ≤ c2. If K is a nonempty closed subset in a locally convex semi-reflexive topological vector space, then the problem (1) is problem by Isac, Sehgal and Singh [16] (Open Problem 2). In [20], Li gave the answer of this problem by introducing...

In this paper, we introduce two classes of generalized variational-like inequalities with compositely monotone multifunctions in Banach spaces. Using the KKM-Fan lemma and the Nadler’s result, we prove the existence of solutions for generalized variational-like inequalities with compositely relaxed η − α monotone multifunctions in reflexive Banach spaces. On the other hand we also derive the so...