Topological Vector Spaces Research Articles

Remarks on generalized weak KKM multimaps

The concept of generalized KKM maps was initiated by Kassay-Kolumbán in 1990 and Chang-Zhang in 1991. Recently, Balaj and his colleagues extended generalized KKM maps w.r.t. a multimap to weak KKM maps and generalized weak KKM maps w.r.t. a multimap and applied them to various problems in the KKM theory. However, their results are mainly concerned within the realm of topological vector spaces. Our aim in this article is to extend some of them to abstract convex spaces. Some related facts are also discussed.

Lineability and K-linear discontinuous functions

It is well known that there are Q-linear discontinuous functions from R to R. Moreover, such functions have dense graph in the real plane. In this paper, we study the existence of linear spaces contained in the family of linear discontinuous functions in the generalized context of topological vector spaces and fields endowed with an absolute value. Furthermore, in the case of fields endowed with an absolute value, we study the algebraic genericity of the families of (1) linear discontinuous functions that are not everywhere surjective and (2) linear discontinuous functions that are strongly everywhere surjective. This extends a number of previous results of lineability theory.Generalizations of classical results of Q-linear discontinuous functions are also analyzed.

Fenchel–Rockafellar theorem in infinite dimensions via generalized relative interiors

In this paper we provide further studies of Fenchel duality theory in the general framework of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong duality under some qualification conditions via generalized relative interiors imposed on the epigraphs and the domains of the functions involved. Our results directly generalize the classical Fenchel–Rockafellar theorem on strong duality from finite dimensions to LCTV spaces.

Well-posedness for set-valued equilibrium problems

"In this paper we extend a concept of well-posedness for vector equilibrium problems to the more general framework of set-valued equilibrium problems in topological vector spaces using an appropriate reformulation of the concept of minimality for sets. Su cient conditions for well-posedness are given in the generalized convex settings and we are able to single out classes of well-posed set-valued equilibrium problems. On the other hand, in order to relax some conditions, we introduce a concept of minimizing sequences for a set-valued problem, in the set criterion sense, and further we will have a concept of well-posedness for the set-valued equilibrium problem we are interested in. Suficient results are also given for this well-posedness concept."

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.

Distributions in [formula omitted] with a thick curve

A theory of distributions called “thick distributions” was developed to incorporate a point singularity in the test function space. In this present article we consider a more general situation where test functions are singular on a curve in R3. We construct a topological vector space of such test functions and, by duality, the space D⁎,C′(R3) of distributions that are thick on this curve. We study several operations, including partial differentiation. We introduce the notion of line thick delta functions which is a lifting of line delta functions to D⁎,C′(R3). These new distributions, in particular thick line delta functions, may have applications in providing more accurate models to some problems from physics, biology and engineering where a “line source” or a “tube source” is present. As an example of such applications, we propose a more refined model of a growth factor's reaction and diffusion of a very thin blood capillary in a bulk tumor, and give a solution of the corresponding PDE.

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.

Almost all positive continuous linear functionals can be extended

Let F be an ordered topological vector space (over mathbb {R}) whose positive cone F_+ is weakly closed, and let E subseteq F be a subspace. We prove that the set of positive continuous linear functionals on E that can be extended (positively and continuously) to F is weak-* dense in the topological dual wedge E_+'. Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces.

From the Fan-KKM principle to extended real-valued equilibria and to variational-hemivariational inequalities with application to nonmonotone contact problems

This paper starts off by the celebrated Knaster–Kuratowski–Mazurkiewicz principle in the formulation by Ky Fan. We provide a novel variant of this principle and build an existence theory for extended real-valued equilibrium problems with general, then monotone and pseudomonotone bifunctions. We develop our existence theory first in general topological vector spaces, then in reflexive Banach spaces, where we investigate the issue of coerciveness for existence on unbounded sets. Thereafter we use the Clarke generalized differential calculus for locally Lipschitz functions and derive existence results for nonlinear variational-hemivariational inequalities and hemivariational quasivariational inequalities. As application, we treat a unilateral contact problem in solid mechanics with nonmonotone friction.

On equicontinuity and tightness of bi-continuous semigroups

In contrast to classical strongly continuous semigroups, the study of bi-continuous semigroups comes with some freedom in the properties of the associated locally convex topology. This paper aims to give minimal assumptions in order to recover typical features like tightness and equicontinuity with respect to the mixed topology as well as to carefully clarify on mutual relations between previously studied variants of these notions. The abstract results—exploiting techniques from topological vector spaces—are thoroughly discussed by means of several example classes, such as semigroups on spaces of bounded continuous functions.

A convergence theorem for ap−Henstock-Kurzweil integral and its relation to topology

In this paper we discuss about the ap?Henstock-Kurzweil integrable functions on a topological vector spaces. Basic results of ap?Henstock-Kurzweil integrable functions are discussed here. We discuss the equivalence of the ap?Henstock-Kurzweil integral on a topological vector spaces and the vector valued ap?Henstock-Kurzweil integral. Finally, several convergence theorems are studied.

Compactness Principles for Topological Vector Spaces

Abstract We show that the many of the canonical quantifications of interrelated concepts, centered around compactness in the setting of metric spaces, can be easily generalized to the setting of topological linear spaces. Among other things, we obtain a generalization of the Hausdorff Total Boundedness Principle, of the Grothendieck Compactness Principle, as well as of the Convex Compactness Principle for topological vector spaces.

Rosenthal L∞-theorem revisited

A remarkable Rosenthal L∞-theorem is extended to operators T : L∞(Γ, E) → F , where Γ is an infinite set, E a locally bounded (for instance, normed or p-normed) space, and F any topological vector space.

Open discrete shrinkings in function spaces

Given a sequence U={Un:n∈ω} of non-empty open subsets of a space X, a family {Vn:n∈ω} is a shrinking of U if Vn is a non-empty open set and Vn⊂Un for every n∈ω. A space X has discrete shrinking property if every sequence of non-empty open subsets of X has a discrete shrinking. We show that if L is a non-metrizable locally convex topological vector space and Lw is the set L with the weak topology of the space L, then Lw has the discrete shrinking property. In particular, a space X is uncountable if and only if Cp(X) has the discrete shrinking property. The same statement is not true for Cp(X,[0,1]) so we study when the discrete shrinking property holds in Cp(X,[0,1]). We prove, among other things, that Cp(X,[0,1]) features discrete shrinking property if X is an essentially uncountable space.

Denjoy-type Integrals in Locally Convex Topological Vector Space

In this paper, we introduce AC* and ACG*-type properties and then, using theseconditions along with other concepts, define two Denjoy-type integrals of a function with values in a locally convex topological vector space (LCTVS). We show, among others, that these newly defined integrals are included in the SH integral, a stronger version of the Henstock integral for LCTVS-valued functions.

A universal coregular countable second-countable space

A Hausdorff topological space X is called superconnected (resp. coregular) if for any nonempty open sets U1,…Un⊆X, the intersection of their closures U‾1∩…∩U‾n is not empty (resp. the complement X∖(U‾1∩…∩U‾n) is a regular topological space). A canonical example of a coregular superconnected space is the projective space QP∞ of the topological vector space Q<ω={(xn)n∈ω∈Qω:|{n∈ω:xn≠0}|<ω} over the field of rationals Q. The space QP∞ is the quotient space of Q<ω∖{0}ω by the equivalence relation x∼y iff Q⋅x=Q⋅y.We prove that every countable second-countable coregular space is homeomorphic to a subspace of QP∞, and a topological space X is homeomorphic to QP∞ if and only if X is countable, second-countable, and admits a decreasing sequence of closed sets (Xn)n∈ω such that (i) X0=X, ⋂n∈ωXn=∅, (ii) for every n∈ω and a nonempty relatively open set U⊆Xn the closure U‾ contains some set Xm, and (iii) for every n∈ω the complement X∖Xn is a regular topological space. Using this topological characterization of QP∞ we find topological copies of the space QP∞ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.

ON THE SL-INTEGRAL OF LCTVS-VALUED FUNCTIONS

A function F:[a,b]→X is said to be an SL function if it satisfies the Strong Lusin (SL) condition given as follows: for every θ-nbd U and a set E⊂[a,b] of measure zero, there exists a gauge δ such that for every δ-fine partial partition D={([xi-1,xi],ti):1≤i≤n} of [a,b] with ti∈E, there exist θ-nbds U1,U2,…,Un such that ∑i=1nUi⊆V and F(xi)-F(xi-1)∈Ui for each i=1,2,…,n. In this paper, we introduce the SL integral of a function taking values on a locally convex topological vector space (LCTVS). Further, we show that this integral is equivalent to a stronger version of the Henstock integral.

On abstract economies with an arbitrary set of players and action sets in locally-convex topological vector spaces

In this paper we show the existence of equilibrium in an abstract economy with an arbitrary set of players, each of whose compact convex action sets lie in different locally convex spaces, and whose preferences obey weaker continuity postulates that have been used in the original treatments of Arrow and Debreu in the early fifties, and of Shafer and Sonnenschein in the mid-seventies. Our theorem synthesizes results around the cardinality of the set of agents, and incorporates in particular recent results of Carmona–Podczeck and He–Yannelis. The proof of the theorem is based on nonstandard analysis and its novelty has independent methodological interest.

Chaos on Fuzzy Dynamical Systems

Given a continuous map f:X→X on a metric space, it induces the maps f¯:K(X)→K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)→F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X→[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).