On parametric implicit vector variational inequality problems

A class of nonlinear variational inequalities involving strongly G-monotone mappings in a locally convex Hausdorff topological vector space setting is presented. The obtained result generalizes the results on nonlinear variational inequalities on monotone and strongly monotone mappings.

In this article, we obtain some stability results for parametric weak vector equilibrium problem with set-valued mappings. By using a scalarization method, we establish sufficient conditions for the semicontinuity of the approximate solution mappings to parametric set-valued weak vector equilibrium problem under weak assumptions. These results extend and improve some known results in the literature.

In this paper, we first introduce a new class of bilevel weak vector variational inequality problems in locally convex Hausdorff topological vector spaces. Then, using the Kakutani-Fan-Glicksberg fixed-point theorem, we establish some existence conditions of the solution for this problem.

We consider the parametric weak vector equilibrium problem. By using a weaker assumption of Peng and Chang (2014), the sufficient conditions for continuity of the solution mappings to a parametric weak vector equilibrium problem are established. Examples are provided to illustrate the essentialness of imposed assumptions. As advantages of the results, we derive the continuity of solution mappings for vector optimization problems.

Altman [1] showed that Riesz-Schauder theory remains valid for a completely continuous linear operator on a locally convex Hausdorflf topological vector space over the complex field. In a later paper [2], he proved an analogue of the Aronszajn-Smith result; specifically, he showed that such an operator possesses a proper closed invariant subspace. The purpose of this paper is to show that Ringrose's theory of superdiagonal forms for compact linear operators [3] can be generalized to the case of a completely continuous linear operator on a locally convex Hausdorff topological vector space over the complex field. However, the proof given in [3] requires considerable modification.

On the solution semicontinuity to a parametric generalized vector quasivariational inequality

Let X be a completely regular Hausdorff space, let V be a system of weights on X and let T be a locally convex Hausdorff topological vector space. Then CVb(X, T) is a locally convex space of vector-valued continuous functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper we characterize multiplication operators on the space CVb(X, T) induced by operator-valued mappings and then obtain a (linear) dynamical system on this weighted function space.

In this paper, we consider vector variational inequality and vector F-complementarity problems in the setting of topological vector spaces. We extend the concept of upper sign continuity for vector-valued functions and provide some existence results for solutions of vector variational inequalities and vector F-complementarity problems. Moreover, the nonemptyness and compactness of solution sets of these problems are investigated under suitable assumptions. We use a version of Fan-KKM theorem and Dobrowolski's fixed point theorem to establish our results. The results of this paper generalize and improve several results recently appeared in the literature.

In [1] it is shown that barrelledness and quasi-barrelledness are merely the two extreme examples of a property, called 2-symmetry, which may be possessed by a locally convex Hausdorff topological vector space. The object of this note is to show how recent characterisations [2; 3] of barrelled and quasi-barrelled spaces may be subsumed under characterisations of I-symmetric spaces, and to exhibit some properties of these spaces. First we need some definitions and simple results.

Convex sets in topological vector spaces

Tropical convexity in Riesz spaces

A Survey of Some Results Closely Related to the Knaster–Kuratowski–Mazurkiewicz Theorem

In this paper, we first introduce the new notions of $$\overline{{\mathbb {R}}}_+$$-local-intersection property, $$\overline{{\mathbb {R}}}_+$$-local-inclusion property, $$\overline{{\mathbb {R}}}_+$$-strongly transfer lower semicontinuity, $${\mathbb {R}}_-$$-weakly transfer lower semicontinuity and $${\mathbb {R}}_-$$-strongly transfer lower semicontinuity for a set-valued mapping $$\Phi $$ in topological vector spaces. Then, using these notions, we characterize the existence of set-valued equilibrium without assuming any form of convexity of function and/or convexity and compactness of sets. Furthermore, we apply the basic results obtained in the paper to Browder variational inclusion, with weekend conditions on the involved set-valued operators and provide an affirmative answer to some open question posed by Alleche and Rǎdulescu in their final remark of (Optim Lett, 2018. https://doi.org/10.1007/s11590-1233-2). Some application of our results to characterize the existence of set-valued pure strategy Nash equilibrium in games with discontinuous and non-convex payoff functions and nonconvex and/or noncompact strategy spaces is presented. Finally, we characterize the existence of single-valued equilibrium problem without assuming any form of convexity of function and/or convexity and compactness of sets in the setting of topological vector spaces. Our results improve and generalize many known results in the current literature.

Asymptotic closure condition and Fenchel duality for DC optimization problems in locally convex spaces

In this paper, the authors prove some existence results of solutions for a new class of generalized bi-quasi-variational inequalities (GBQVI) for quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators defined on compact sets in locally convex Hausdorff topological vector spaces. In obtaining these results on GBQVI for quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators, we shall use Chowdhury and Tan’s generalized version [3] of Ky Fan’s minimax inequality [7] as the main tool.