Convex Linear Topological Space Research Articles

On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces

ABSTRACT This article discusses differential equations in infinite-dimensional spaces. We introduce a general version of the Lipschitz-type continuity criterion without the notion of a distance and prove an existence and uniqueness result for ordinary differential equations in locally convex topological linear spaces. It is shown that remarkable techniques of classical analysis, such as the method of successive approximations, are still adaptable tools in the study of the abstract models.

Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result

The existence of \begin{document}$ L^{2} $\end{document} -nonnegative solutions for nonlinear quadratic integral equations on a bounded closed interval is investigated. Two existence results for different classes of functions are shown. As a consequence an existence theorem for the Chandrasekhar integral quadratic equation, well-known in theory of radiative transfer, is obtained. The aim is achieved by means of a new fixed point theorem for multimaps in locally convex linear topological spaces.

Variational methods for strongly indefinite problems

This paper reviews some recent development in studying variational methods for strongly indefinite problems under the support of National Natural Science Foundation of China. We firstly describe the ideas for establishing the variational setting, then state a deformation theory in locally convex topological linear spaces and several critical point theorems for dealing with strongly indefinite problems based on the deformation. Some applications of the theory are given including mainly: Existence and multiplicity of solutions to non-autonomous stationary Dirac system, in particular, the existence, concentration phenomena and the exponential decay of semi-classical states; existence and multiplicity of global solutions to the nonlinear (non-autonomous, unbounded Hamilton-type) reaction-diffusion system, especially the existence, concentration phenomena and decay of its ground states of the system with singular perturbation; depth study of homoclinic orbits Hamiltonian system, global Schrodinger equation; other initial work, such as bifurcation of Dirac equation on spin manifolds.

Weighted Lacunary Statistical Convergence

In this paper, we extend the concept of weighted lacunary statistical convergence to Hausdorff locally convex topological linear spaces and present some inclusion theorems.

New nonlinear separation theorems and applications in vector optimization

By means of the Minkowski-type nonlinear scalarization functional, in this paper, we establish some nonlinear separation theorems via relative algebraic interior and vector closure in a general real linear space, via relative topological interior and topological closure in a real topological linear space and via quasi relative interior and topological closure in a real separated locally convex topological linear space, respectively. These new separation theorems can be applied to study those vector optimization problems with the ordering cones having possibly empty topological interior and even relative topological interior or relative algebraic interior. As their applications, we give some nonlinear scalarization characterizations of the corresponding weak efficient solutions of vector optimization problems. Moreover, we also present some concrete examples to illustrate the main results in some infinite dimensional spaces.

Formula omitted]-spaces

A regular topological space X is defined to be a P0-space if it has countable Pytkeev network. A network N for X is called a Pytkeev network if for any point x∈X, neighborhood Ox⊂X of x and subset A⊂X accumulating at a x there is a set N∈N such that N⊂Ox and N∩A is infinite. The class of P0-spaces contains all metrizable separable spaces and is (properly) contained in the Michael's class of ℵ0-spaces. It is closed under many topological operations: taking subspaces, countable Tychonoff products, small countable box-products, countable direct limits, hyperspaces of compact subsets. For an ℵ0-space X and a P0-space Y the function space Ck(X,Y) endowed with the compact-open topology is a P0-space. For any sequential ℵ0-space X the free abelian topological group A(X) and the free locally convex linear topological space L(X) both are P0-spaces. A sequential space is a P0-space if and only if it is an ℵ0-space. A topological space is metrizable and separable if and only if it is a P0-space with countable fan tightness.

New characterizations of minimal cusco maps

We give new characterizations of minimal cusco maps in the class of all set-valued maps extending results from \cite{BZ1, GM}. Let $X$ be a topological space and $Y$ a Hausdorff locally convex linear topological space. Let $F: X \to Y$ be a set-valued map. The following are equivalent: (1)~$F$ is minimal cusco; (2)~$F$ has nonempty compact values, there is a quasicontinuous, subcontinuous selection $f$ of $F$ such that $F(x) = \overline{co\,}\overline f(x)$ for every $x \in X$; (3)~$F$ has nonempty compact values, there is a densely defined subcontinuous, quasicontinuous selection $f$ of $F$ such that $F(x) = \overline{co}\,\overline f(x)$ for every $x \in X$; (4)~$F$ has nonempty compact convex values, $F$ has a closed graph, every extreme function of $F$ is quasicontinuous, subcontinuous and any two extreme functions of $F$ have the same closures of their graphs. Some applications to known results are given.

Kakutani-type fixed point theorems: A survey

A Kakutani-type fixed point theorem refers to a theorem of the following kind: Given a group or semigroup S of continuous affine transformations s : Q → Q, where Q is a nonempty compact convex subset of a Hausdorff locally convex linear topological space, then under suitable conditions S has a common fixed point in Q, i.e., a point $${a \in Q}$$ such that s(a) = a for each $${s \in S}$$ . In 1938, Kakutani gave two conditions under each of which a common fixed point of S in Q exists. They are (1) the condition that S be a commutative semigroup, and (2) the condition that S be an equicontinuous group. The present survey discusses subsequent generalizations of Kakutani’s two theorems above.

Fixed Point Properties Related to Multivalued Mappings

AbstractWe discuss fixed point properties of convex subsets of locally convex linear topological spaces. We derive equivalence among fixed point properties concerning several types of multivalued mappings.

On super efficiency in set-valued optimization

The set-valued optimization problem with constraints is considered in the sense of super efficiency in locally convex linear topological spaces. Under the assumption of iccone-convexlikeness, by applying the seperation theorem, Kuhn-Tucker’s, Lagrange’s and saddle points optimality conditions, the necessary conditions are obtained for the set-valued optimization problem to attain its super efficient solutions. Also, the sufficient conditions for Kuhn-Tucker’s, Lagrange’s and saddle points optimality conditions are derived.

Strong Unicity of Restricted p-Centers

Given a continuous seminorm p on a separated locally convex linear topological space X, we study in this paper strong uniqueness of the so-called restricted p-centers of sets. First we explore finite extremal characterizations of strongly unique restricted p-centers of subsets of X from its finite dimensional subspaces. Our main goal here is to investigate strong uniqueness of restricted p-centers of certain sets from the so-called p-RS sets, which are defined as closed convex sets that are obtained by imposing convex side constraints arising from boundedness of coefficients on translates of certain subspaces of X.

Narrow Convergence in Spaces of Set-Valued Measures

Summary. We prove an analogue of Topsoe's criterion for relative compactness of a family of probability measures which are regular with respect to a family sets. We consider measures whose values are compact convex sets in a locally convex linear topological space. Introduction. Let T be an abstract set, K a family of subsets of T, and (E;F) a dual pair of real vector spaces, with E endowed with the weak topology (E;F). Let cc(E;F) be the set of all convex compact non-empty subsets of E, and f M+(T;K;cc(E;F)) the set of K-inner regular positive set-valued measures defined on a -fieldB of subsets of T and with values in cc(E;F). We denote by M+(T;K) the set ofK-inner regular non-negative measures defined on B provided with the topology of weak convergence. Prokhorov (11) has proved that if T is a Polish space andB the set of Borel subsets, then the relatively compact subsets of M+(T;K) are precisely the tight ones. But this result is not valid for all topological space (see e.g. (5), (10), (18)). In (16) Topsoe has characterized the relatively compact subsets of M+(T;K) in general situations. Before and after Topsoe's paper there were others (e.g. (1), (3), (18), (5)-(10)). In this paper we generalize to the space f

Cardinal Invariants of the Topology of Uniform Convergence on Compact Sets on the Space of Minimal USCO Maps

For a Baire space X the set of all minimal USCO real-valued maps on X coincides with the space D∗(X) of locally bounded densely continuous real-valued forms on X. When X is a locally compact space, the space D∗ k(X) of locally bounded densely continuous real-valued forms on X, under the topology of uniform convergence on compact sets, is a locally convex linear topological space. This paper gives characterizations and bounds for the cardinal function properties on D∗ k(X) of character, pseudocharacter, density, weight, netweight and cellularity. Examples are given to show how these properties can be the same or different. We answer also some questions posed in [17].

Multipliers and operators on the temperedultradistribution spaces of Roumieu type for thedistributional Hankel‐type transformation spaces

The tempered ultradistribution space of Roumieu type for the space Hμ,ν is defined, which is a subspace of the Hausdörff locally convex topological linear space. Further, results are obtained for the multipliers and operators on the tempered ultradistribution spaces for the distributional Hankel‐type transformation spaces.

Duality and saddle-points for convex-like vector optimization problems on real linear spaces

Usually, finite dimensional linear spaces, locally convex topological linear spaces or normed spaces are the framework for vector and multiojective optimization problems. Likewise, several generalizations of convexity are used in order to obtain new results. In this paper we show several Lagrangian type duality theorems and saddle-points theorems. From these, we obtain some characterizations of several efficient solutions of vector optimization problems (VOP), such as weak and proper efficient solutions in Benson’s sense. These theorems are generalizations of preceding results in two ways. Firstly, because we consider real linear spaces without any particular topology, and secondly because we work with a recently appeared convexlike type of convexity. This new type, designated GVCL in this paper, is based on a new algebraic closure which we named vector closure.

On super efficiency in set-valued optimisation in locally convex spaces

The set-valued optimisation problem with constraints is considered in the sense of super efficiency in locally convex linear topological spaces. Under the assumption of nearly cone-subconvexlikeness, by applying the separation theorem for convex sets, Kuhn-Tucker and Lagrange necessary conditions for the set-valued optimisation problem to attain its super efficient solutions are obtained. Also, Kuhn-Tucker and Lagrange sufficient conditions are derived. Finally two kinds of unconstrained programs equivalent to set-valued optimisation problems are established.

Deformation in locally convex topological linear spaces

Deformation in locally convex topological linear spaces

Fundamental Theorems of Asset Pricing for Good Deal Bounds

We prove fundamental theorems of asset pricing for good deal bounds in incomplete markets. These theorems relate arbitrage‐freedom and uniqueness of prices for over‐the‐counter derivatives to existence and uniqueness of a pricing kernel that is consistent with market prices and the acceptance set of good deals. They are proved using duality of convex optimization in locally convex linear topological spaces. The concepts investigated are closely related to convex and coherent risk measures, exact functionals, and coherent lower previsions in the theory of imprecise probabilities.

Topological degree theory and fixed point theorems in fuzzy normed space

In this paper, the Leray–Schauder topological degree theory is developed in a fuzzy normed space. Since the linear topology on this fuzzy normed space is not necessarily locally convex, and since each Menger probabilistic normed space can be considered as a special fuzzy normed space, the degree theory in this paper is different from the degree theory in locally convex linear topological space presented by Nagumo (Amer. J. Math. 73 (1951) 497–511), and it also is an extension of the degree theory in Menger probabilistic normed space studied by Zhang and Chen (Appl. Math. Mech. 10(6) (1989) 477–486). Applying this degree theory, some fixed point theorems for operators are given in fuzzy normed spaces, and some former corresponding results are extended and improved.

Deformation in locally convex topological linear spaces

We are concerned with a deformation theory in locally convex topological linear spaces. Aspecial “nice” partition of unity is given. This enables us to construct certain vector fields which are locally Lipschitz continuous with respect to the locally convex topology. The existence, uniqueness and continuous dependence of flows associated to the vector fields are established. Deformations related to strongly indefinite functionals are then obtained. Finally, as applications, we prove some abstract critical point theorems.