Hausdorff Topological Vector Spaces Research Articles

Coercive and Noncoercive Mixed Generalized Complementarity Problems

Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the recession analysis, to establish the existence of solutions for mixed generalized complementarity problems without coercivity assumptions in Banach spaces.

CHARACTERISATION OF EQUIVALENT NORMS ON A LINEAR SPACE USING EXPONENTIAL VECTOR SPACE

In this paper we have found a necessary and sufficient condition for equivalence of two norms on a linear space using the theory of exponential vector space. Exponential vector space (‘evs’ in short) is an ordered algebraic structure which can be considered as an algebraic ordered extension of vector space. This structure is axiomatised on the basis of the intrinsic properties of the hyperspace C(X) comprising all nonempty compact subsets of a Hausdorff topological vector space X. Exponential vector space is a conglomeration of a semigroup structure, a scalar multiplication and a compatible partial order. We have shown that the collection of all norms defined on a linear space, together with the constant function zero, forms a topological exponential vector space. Then using the concept of comparing function (a concept defined on a topological exponential vector space) we have proved the aforesaid necessary and sufficient condition; also we have proved using comparing function that in an infinite dimensional linear space there are uncountably many non-equivalent norms.

A Note on Equivalence of G-Cone Metric Spaces and G-Metric Spaces

This paper contains the equivalence between tvs-G cone metric and G-metric using a scalarization function $\zeta_p$, defined over a locally convex Hausdorff topological vector space. This function ensures that most studies on the existence and uniqueness of fixed-point theorems on G-metric space and tvs-G cone metric spaces are equivalent. We prove the equivalence between the vector-valued version and scalar-valued version of the fixed-point theorems of those spaces. Moreover, we present that if a real Banach space is considered instead of a locally convex Hausdorff space, then the theorems of this article extend some results of G-cone metric spaces and ensure the correspondence between any G-cone metric space and the G-metric space.

Browder’s Theorem through Brouwer’s Fixed Point Theorem

A parametric version of Brouwer’s fixed point theorem, called Browder’s theorem, states that for every continuous mapping , where X is a nonempty, compact, and convex set in a Euclidean space, the set of fixed points of f, namely, the set , has a connected component whose projection onto the first coordinate is . Browder’s original proof relies on the theory of the fixed point index. We provide an alternative proof that uses Brouwer’s fixed point theorem and is valid whenever X is a nonempty, compact, and convex subset of a Hausdorff topological vector space.

THE STUDY OF EQUILIBRIA FOR GENERALIZED GAMES IN HAUSDORFF TOPOLOGICAL VECTOR SPACES

Recent fixed point theory of the author is used to establish equilibria to multiperson games for majorized coercive or compact type maps defined on Hausdorff topological vector spaces.

On the cohomological equation of a linear contraction

In this paper, we study the discrete cohomological equation of a contracting linear automorphism A of the Euclidean space Rd. More precisely, if δ is the cobord operator defined on the Fréchet space E = Cl (Rd) (0 ≤ l ≤ ∞) by: δ(h) = h − h ◦ A, we show that: • If E = C0(Rd), the range δ (E) of δ has infinite codimension and its closure is the hyperplane E0 consisting of the elements of E vanishing at 0. Consequently, H1 (A, E) is infinite dimensional non Hausdorff topological vector space and then the automorphism A is not cohomologically C0-stable. • If E = Cl (Rd), with 1 ≤ l ≤ ∞, the space δ (E) coincides with the closed hyperplane E0. Consequently, the cohomology space H1 (A, E) is of dimension 1 and the automorphism A is cohomologically Cl-stable.

A Perturbation Approach to Vector Optimization Problems: Lagrange and Fenchel–Lagrange Duality

In this paper we study the general minimization vector problem (P), concerning a perturbation mapping, defined in locally convex Hausdorff topological vector spaces where the "WInf" stands for the weak infimum with respect to an ordering generated by a convex cone $K$. Several representations of the epigraph of the conjugate mapping of the perturbation mapping are established. From these, variants vector Farkas lemmas are then proved. Armed with these basic tools, the {\it dual} and the so-called {\it loose dual problem} of (P) are defined, and then stable strong duality results between these pairs of primal-dual problems are established. The results just obtained are then applied to a general class (CCCV) of composed vector optimization problems with cone-constrained. For this classes of problems, four perturbation mappings are suggested. Each of these mappings yields several forms of vector Farkas lemmas and two forms of dual problems for (CCVP). Concretely, one of the suggested perturbation mapping give rises to well-known {\it Lagrange} and {\it loose Lagrange dual problems} for (CCVP) while each of the three others, yields two kinds of Fenchel-Lagrange dual problems for (CCVP). Stable strong duality for these pairs of primal-dual problems are proved. Several special cases of (CCVP) are also considered at the end of the paper, including: vector composite problems (without constraints), cone-constrained vector problems, and scalar composed problems. The results obtained in this papers when specified to the two concrete mentioned vector problems go some Lagrange duality results appeared recently, and also lead to new results on stable strong Fenchel-Lagrange duality results, which, to the best knowledge of the authors, appear for the first time in the literature.

New representations of epigraphs of conjugate mappings and Lagrange, Fenchel–Lagrange duality for vector optimization problems

In this paper, we concern the vector problem: where X, Y, Z are locally convex Hausdorff topological vector spaces, and are proper mappings, C is a nonempty convex subset of X, and S is a non-empty closed, convex cone in Z. Several new presentations of epigraphs of composite conjugate mappings associated to (VP) are established under variant qualifying conditions. The significance of these representations is twofold: Firstly, they play a key role in establishing new kinds of vector Farkas lemmas which serve as tools in the study of vector optimization problems; secondly, they pay the way to define Lagrange and two new kinds of Fenchel–Lagrange dual problems for (VP). Strong and stable strong duality results corresponding to these mentioned dual problems of (VP) are established using the new Farkas-type results just obtained. It is shown that in the special case where , the Lagrange and Fenchel–Lagrange dual problems for (VP), go back to Lagrange, and Fenchel–Lagrange dual problems for scalar problems, and the resulting duality results cover, and in some cases, extend the corresponding ones for scalar problems in the literature.

Approximate Fixed Point Theory for Countably Condensing Maps and Multimaps and Applications

In this paper, we establish an approximate fixed point theorem for Γ-τ-sequentially continuous countably -condensing mappings. As application, we obtain an approximate fixed point result for demicontinuous mappings and use this to prove new results in asymptotic fixed point theory. Moreover, we obtain new fixed point results for some countably -condensing mappings and multivalued mappings and fixed point results of Krasnoselskii-Daher type for the sum of two τ-sequentially continuous mappings in non-separable Hausdorff topological vector spaces. We also present a multivalued version of an approximation result of Ky Fan for τ-Γ-s.l.sc multivalued mappings. Apart from that, we show the applicability of our results to the theory of Volterra integral equations in Banach spaces and we prove the existence of limiting-weak solutions for differential equations in Banach spaces not necessarily reflexive. Our results extend the results of Banaś and Ben Amar, Barroso and Seda.

Fuzzy Kakutani-Fan-Glicksberg fixed point theorem and existence of Nash equilibria for fuzzy games

In this paper, we give fuzzy generalizations to the well-known Brouwer fixed point theorem and Kakutani-Fan-Glicksberg fixed point theorem. As applications, we apply the fuzzy Kakutani-Fan-Glicksberg fixed point theorem to derive an existence theorem for Nash equilibria in generalized fuzzy games with locally convex Hausdorff topological vector spaces for strategy spaces and/or discontinuous payoff functions which generalizes existence theorems for Nash equilibria in generalized games.

A weak [formula omitted]-core existence theorem of games with nonordered preferences and a continuum of agents

Inspired by Kajii (1992) and Askoura (2011, 2017), we introduce the notion of the weak α-core for games with nonordered preferences and a continuum of agents. First, we extend the work of Kajii (1992) to games with spaces of strategies defined on Hausdorff topological vector spaces. Furthermore, we prove the nonemptiness of the weak α-core. Finally, we establish the relations between normal-form games, games with nonordered preferences and games with pseudo-utilities.

A new class of bilevel weak vector variational inequality problems

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.

A new approach to zero duality gap of vector optimization problems using characterizing sets

In this paper we propose results on zero duality gap in vector optimization problems posed in a real locally convex Hausdorff topological vector space with a vector-valued objective function to be minimized under a set and a convex cone constraint. These results are then applied to linear programming.

Existence of solutions for extended generalized complementarity problems

In this paper we introduce extended generalized complementarity problems in Hausdorff topological vector spaces in duality and study the existence of their solutions. We use a different method than those in literature on the existence of solutions of complementarity problems, which are usually based on arguments from generalized monotonicity. This leads us to obtain new results and improve many existing results in literature. We also prove some existence results for extended generalized complementarity problems in reflexive Banach spaces by means of a Tikhonov regularization procedure under a copositivity assumption and arguments from the recession analysis.

Local dense solutions for equilibrium problems with applications to noncooperative games

In this work, we establish several results concerning the existence of solutions for set-valued and single-valued equilibrium problems in real Hausdorff topological vector spaces. Firstly we introduce some generalizations of convexity and continuity conditions to set-valued mappings and then apply them to special dense subsets of the domain to obtain the existence of local dense solutions of equilibrium problems. Then the existence of the global solutions follows from a condition that is weaker than semistrict quasiconvexity. Specifically, we give an existence theorem for noncooperative n-person games, under assumptions imposed on a locally segment-dense subset of the strategy set of each player.

Existence conditions for solutions of bilevel vector equilibrium problems with application to traffic network problems with equilibrium constraints

In this paper, we introduce some strong and weak bilevel vector equilibrium problems in locally convex Hausdorff topological vector spaces and present some conditions for the existence of solutions to these problems by using the Kakutani–Fan–Glicksberg fixed-point theorem. Furthermore, as a real-world application, we obtain the existence of solutions to traffic network problems with equilibrium constraints.

Continuous orthosymmetric multilinear maps and homogeneous polynomials on Riesz spaces

We show that any continuous orthosymmetric multilinear map from an Archimedean Riesz space into a Hausdorff topological vector space is symmetric. Then, we establish a linear representation for continuous orthogonally additive homogeneous polynomials. This representation will be used to introduce and describe a new class of homogeneous polynomials, namely that of polyorthomorphisms. In particular, we prove that, for a Riesz space E and a natural number $$n\ge 2$$ , the space $${{\mathcal{P}}}_{orth}(^nE)$$ of all polyorthomorphisms of degree n is a Riesz space.

Approximate Benson efficient solutions for set-valued equilibrium problems

In locally convex Hausdorff topological vector spaces, the approximate Benson efficient solution is proposed for set-valued equilibrium problems and its relationship to the Benson efficient solution is discussed. Under the assumption of generalized convexity, by using a separation theorem for convex sets, Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued equilibrium problems are established, respectively.

A new approach to strong duality for composite vector optimization problems

This paper generalizes and unifies different recent results as well as provides a new methodology concerning vector optimization problems involving composite mappings in locally convex Hausdorff topological vector spaces. The Lagrangian and weak Lagrangian dual problems are proposed. Characterizations of strong duality results are proved at the same time with characterizations of Farkas lemmas for composite vector mappings in a general setting (i.e. without any assumptions on convexity or continuity of the mappings involved). Corresponding results in the convex setting are also proposed by establishing as main tools some variants of representations of epigraphs of conjugate mappings in our composite vector framework. As by-products, several Farkas-type results for composite vector functions are proposed, which extend and cover several known ones recently appeared in the literature. Lastly, the results are applied to get duality results for a class of convex semi-vector bilevel optimization problems.

Real-valued non compactness measures in topological vector spaces and a pplications

A non compactness measure with values in the lattice of extended non negative real numbers $$[0, +\infty ]$$ is introduced in the general setting of a Hausdorff topological vector space E. This generalizes the classical Kuratowski and Hausdorff non compactness measures. In order to achieve this, we introduce the notions of basic and sufficient collections of zero neighborhoods. We then show that our measure satisfies most of the properties of the classical non compactness measures. We particularly show that if E is locally p-convex for some $$0 < p \le 1$$ , our measure is stable by the transition to the closed p-convex hull. This allows us to obtain, as applications, generalizations of the well-known three fixed point theorems, namely Schauder, Darbo, and Sadovskii’s ones in the setting of locally p-convex spaces. As another application, we establish a quantification of Ascoli theorem in the space C(X, E) of vector-valued continuous functions on a Hausdorff completely regular space with values in a topological vector space E, giving an alternative of Ambrosetti theorem initially stated in the metric spaces setting.