Convex Hausdorff Topological Vector Spaces Research Articles

Vector variational-like inequalities with \(\eta\)-generally convex mappings

Because of applications in optimization problems, mathematical programming, equilibrium problem and operations research, considerable progresses have been achieved in both theory and applications of vector variational-like inequalities. In this work, we consider vector variational-like inequalities with \(\eta\)-generally convex mappings and prove some existence results for our inequalities in the setting of Hausdorff topological vector space. The results presented in this article are more general and can be used to solve many known problems related to vector variational inequalities, variational-like inequalities and vector variational-like inequalities. References G. Y. Chen and X.Q. Yang, Vector complementarity problems and its equivalences with weak minimal element in ordered spaces, J. Math. Anal. Appl. 153(1990), 136--158. F. Giannessi, Theorems of Alternative, Quadratic Programs and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi and J. L. Lions, John Wiley and Sons, Chichester, 151--186, 1980. P. Hartman and G. Stampacchia, On some nonlinear elliptic differential functional equations, Acta Math. 115(1966), 271--310. I. V. Konnov and J. C. Yao, On the generalized vector variational inequality problem, J. Math. Anal. Appl. 206(1997), 42--58. B. S. Lee and G. M. Lee, A vector version of Minty's lemma and application, Appl. Math. Lett. 12(5)(1999), 43--50. G. M. Lee, D. S. Kim, B. S. Lee and N. D. Yen, Vector variational inequality as a tool for studying vector optimization problems, Nonlinear Analysis 34(1998), 745--765. G. M. Lee, B. S. Lee and S. S. Chang, On vector quasi-variational inequalities, J. Math. Anal. Appl. 203(1996), 626--638. B. S. Lee, G. M. Lee and D. S. Kim, Generalized vector variational-like inequalities on locally convex Hausdorff topological vector spaces, Indian J. Pure Appl. Math. 28(1997), 33--41. S. Park, B. S. Lee and G. M. Lee, A general vector-valued variational inequality and its fuzzy extension, Internat. J. Math. and Math. Sci. 21(4)(1998), 637--642. S. K. Mishra and M. A. Noor, On vector variational-like inequality problems, J. Math. Anal. Applics. 311(1)(2005), 69--75. Q. H. Ansari, On generalized vector variational-like inequalities, Ann. Sci. Math. Quebec 19(1995), 131--137. A. H. Siddiqi, Q. H. Ansari and R. Ahmad, On vector variational-like inequalities, Indian J. Pure Appl. Math. 28(8)(1997), 1009--1016. A. H. Siddiqi, Q. H. Ansari and A. Khaliq, On vector variational inequalities, J. Optim. Theory Appl. 84(1)(1995), 171--180. X. Q. Yang, Vector variational inequality and its duality, Nonlinear Analysis, T. M. A. 21(1993), 869--877. S. J. Yu and J. C. Yao, On vector variational inequalities, J. Optim. Theory Appl. 89(1996), 749--769. X. M. Yang and X. Q. Yang, Vector variational-like inequality with pseudoinvexity, Optimization 55(1-2)(2006), 157--170. Yali Zhao and Zunquan Xia, On the existence of solutions to generalized vector variational-like inequalities, Nonlinear Analysis 64(9)(2006), 2075--2083. G. Y. Chen, Existence of solution for a vector variational inequality; An extension of the Hartman-Stampacchia theorem, J. Optim. Theory Appl. 74(1992), 445--456. G. Y. Chen and G.H. Cheng, Vector Variational Inequality and Vector Optimization Problem, In Lecture Notes in Economics and Mathematical Systems, volume 258, Springer--Verlag, (1987). G. Y. Chen and B.D. Craven, A vector variational inequality and optimization over an efficient set, Zeitscrift fur Operations Research 3(1990), 1--12.

Existence of solutions on weak vector equilibrium problems

Weak vector equilibrium problems with bi-variable mappings from product space of two bounded complete locally convex Hausdorff topological vector spaces to another topological vector space are studied. The existence theorems of solutions are proved by the FKKM fixed point theorem. Viscosity principle of vector equilibrium problems is dealt with. The relations between solutions of the vector equilibrium problem and those of its perturbation problem are presented.

A noncompact generalized quasi-variational inequality

By applying a fixed point theorem of Yannelis and Prabhakar, a generalized quasi-variational inequality is proved on a noncompact convex subset of a locally convex Hausdorff topological vector space.

On the Geometry of Locally Nonconical Convex Sets

A convex subset Q of a Hausdorff topological vector space is called locally nonconical (LNC) if for every two points x,y∈Q there is a relative neighborhood U of x in Q such that U+\(\tfrac{1}{2}\)(y-x) ⊂Q. A geometric characterization (Theorem 2.2) of closed LNC sets with nonempty interior in a Hilbert space is supplied. It states that any proper line segment ]x,y[ contained in bd(Q), the topological boundary of Q, lies inside a relative neighborhood in bd(Q) composed of parallel line segments. It is shown that one half of this characterization, at least, generalizes to the setting of a locally convex Hausdorff topological vector space (LCHTVS). This leads to the observation that the set ext(Q) of extreme points of any LNC set Q in an LCHTVS is closed. Finally, it is proven that, in the same setting, all LNC sets are uniformly stable and, hence, stable.

A class of nonlinear variational inequalities involvingpseudomonotone operators

We present the solvability of a class of nonlinear variational inequalities involving pseudomonotone operators in a locally convex Hausdorff topological vector spaces setting. The obtained result generalizes similar variational inequality problems on monotone operators.

Existence theorems of generalized quasi-variational inequalities with upper hemi-continuous and demi operators on non-compact sets

Suppose that E is a topological vector space and X is a non-empty subset of E. Let S: X --> 2(X) and T:X --> 2(E)* be two maps. Then the generalized quasi-variational inequality problem (GQVI) is to find a point (y) over cap is an element of S((y) over cap) and a point (w) over cap is an element of T((y) over cap) such that Re[(w) over cap, (y) over cap - x] less than or equal to 0 far all x is an element of S((y) over cap). We shall use Chowdhury and Tan's generalized version [4] of Ky Fan's minimax inequality [7] as a tool to obtain some general theorems on solutions of the GQVI in locally convex Hausdorff topological vector spaces. We obtain the existence theorems of GQVI on paracompact sets X where the set-valued operators T are demi operators [3] and are upper hemi-continuous [5] along line segments in X to the weak * -topology on E*.

Variational inequalities in locally convex Hausdorff topological vector spaces

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.

Generalized nonlinear variational inequalities

In this paper, we investigate the existence results of solutions for a class of generalized nonlinear variational inequalities (GNVI) with nonmonotone set-valued operators defined on noncompact sets in locally convex Hausdorff topological vector spaces. Results that generalize the corresponding recent results of Chang and Zhang are derived.

On the Generalized Nonlinear Variational Inequalities and Quasi-Variational Inequalities

The purpose of this paper is to investigate the existence of solutions for a class of generalized nonlinear inequalities (GNVI) and generalized nonlinear quasi-variational inequalities (GNQVI) with non-monotone set-valued operators defined on non-compact sets in locally convex Hausdorff topological vector spaces. Results that generalize and unify corresponding recent results of Ding and Tarafdar are derived.

Best approximation theorems for composites of upper semicontinuous maps

Let (E, τ) be a Hausdorff topological vector space and (X, ω) a weakly compact convex subset of E with the relative weak topology ω. Recently, there have appeared best approximation and fixed point theorems for convex-valued upper semicontinuous maps F: (X, ω) → 2(E, τ) whenever (E, τ) is locally convex. In this paper, these results are extended to a very broad class of multifunctions containing composites of acyclic maps in a topological vector space having sufficiently many linear functionals. Moreover, we also obtain best approximation theorems for classes of multifunctions defined on approximatively compact convex subsets of locally convex Hausdorff topological vector spaces or closed convex subsets of Banach spaces with the Oshman property.

Generalized nonlinear variational inequalities with nonmonotone setvalued mappings

In this paper, we study a new class of generalized nonlinear variational inequality problem GNVIP(F, g, b, K) in locally convex Hausdorff topological vector spaces. Several existence theorems of solutions for GNVIP(F, g, b, K) with nonmonotone type set-valued mappings F are proved.

Some fixed point theorems for composites of acyclic maps

We obtain fixed point theorems for a new class of multifunctions containing compact composites of acyclic maps defined on a convex subset of a locally convex Hausdorff topological vector space. Our new results are applied to approximatively compact, convex sets or to Banach spaces with the Oshman property.

Maximal elements and equilibria of condensing multimaps

In this paper, we shall prove a maximal element existence theorem for condensing multimaps and next prove an equilibrium existence theorem for generalized games in locally convex Hausdorff topological vector spaces.

Multiplication Operators and Dynamical Systems

AbstractLet 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.

Extending Edgar's ordering to locally convex spaces

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

On the continuity of closed convex hull

It is shown that the closed convex hull operator h is continuous on the space K of nonempty compact subsets of a real locally convex Hausdorff topological vector space, so long as h( K )⊂ K .

Optimality for set functions with values in ordered vector spaces

Let (X, Г, μ) be a finite atomless measure space,L a convex subfamily of Φ, andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:L→Y beC-convex andG:L→Z beD-convex set functions. Consider the following optimization problem (P): minimizeF(ω), subject to ω∈L andG(ω)≤ D θ. The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.

On variational inequalities for fuzzy mappings

This paper presents some results of variational inequalities for fuzzy mappings in locally convex Hausdorff topological vector spaces. As an example of applications we utilize the results to study the existence of fixed points for some kind of fuzzy mappings. The results given in this paper generalize some recent results.

Remark on a generalized quasivariational inequality

In this note, we prove a generalization of a generalized quasi-variational inequality of Shih and Tan in locally convex Hausdorff topological vector spaces.

Generalized quasi-variational inequalities in locally convex topological vector spaces

Let E be a Hausdorff topological vector space and X ⊂ E an arbitrary nonempty set. Denote by E′ the dual space of E and the pairing between E′ and E by 〈 w, x〉 for w ϵ E′ and x ϵ E. Given a point-to-set map S: X → 2 X and a point-to-set map T: X → 2 E′ , the generalized quasi-variational inequality problem (GQVI) is to find a point y ̂ ϵ S( y ̂ ) and a point u ̂ ϵ T( y ̂ ) such that Re〈 u ̂ , y ̂ − x〉 ⩽ 0 for all x ϵ S( y ̂ ) . By using the Ky Fan minimax principle or its generalized version as a tool, some general theorems on solutions of the GQVI in locally convex Hausdorff topological vector spaces are obtained which include a fixed point theorem due to Ky Fan and I. L. Glicksberg, and two different multivalued versions of the Hartman-Stampacchia variational inequality.