Quasi-pseudometric Spaces Research Articles

Expansive homeomorphisms on quasi-metric spaces

The investigation of expansive homeomorphisms in metric spaces began with Utz in 1950. Thereafter, several authors have extensively studied this concept for different motivations. In this current article, we study expansive homeomorphism in the context of quasi-pseudometric spaces. This is motivated by the fact that any expansive homeomorphism on quasi-pseudometric space is again expansive homeomorphism on its induced pseudometric space but the converse is not true in general. Moreover, the study of orbit structures has been taken to consideration in this article. For instance, we investigate the denseness of orbits in the context of quasi-metric spaces.

Completeness of fuzzy quasi-pseudometric spaces

The purpose of this paper is to present the relations among the completeness of sequences, of filters and of nets in the framework of fuzzy quasi-pseudometric spaces. In particular, we show that right completeness of filters and of sequences are equivalent under special conditions of fuzzy quasi-pseudometrics. By introducing a kind of more general right K-Cauchy nets in fuzzy quasi-pseudometric spaces, the equivalence between the completeness of the nets and the sequential completeness is established.

W-convexity on the Isbell-convex hull of an asymmetrically normed real vector space

K?nzi and Yilzid introduced the concept of convexity structures in the sense of Takahashi in quasi-pseudometric spaces [7]. In this article, we continue the study of this theory, introducing the concept of W-convexity for real-valued pair of functions defined on an asymmetrically normed real vector space. Moreover, we show that all minimal pairs of functions defined on an asymmetrically normed real vector space equipped with a convex structure which is W-convex whenever W is translation-invariant.

Ekeland, Takahashi and Caristi principles in quasi-pseudometric spaces

We prove versions of Ekeland, Takahashi and Caristi principles in sequentially right K-complete quasi-pseudometric spaces (meaning asymmetric pseudometric spaces), the equivalence between these principles, as well as their equivalence to the completeness of the underlying quasi-pseudometric space.The key tools are Picard sequences for some special set-valued mappings corresponding to a function φ on a quasi-pseudometric space, allowing a unitary treatment of all these principles.

Some advances in the theory of quasi-pseudometric type spaces

Abstract In this paper, we extend most of the results proved in [4]. In particular, we give some topological properties of the quasi-pseudometric type spaces. Moreover, some fixed point and common fixed point theorems are obtained in the setting of quasi-pseudometric spaces, introduced some months ago by Kazeem et al in [4].

Best Proximity Point Theorem in Quasi-Pseudometric Spaces

In quasi-pseudometric spaces (not necessarily sequentially complete), we continue the research on the quasi-generalized pseudodistances. We introduce the concepts of semiquasiclosed map and contraction of Nadler type with respect to generalized pseudodistances. Next, inspired by Abkar and Gabeleh we proved new best proximity point theorem in a quasi-pseudometric space. A best proximity point theorem furnishes sufficient conditions that ascertain the existence of an optimal solution to the problem of globally minimizing the errorinf⁡{d(x,y):y∈T(x)}, and hence the existence of a consummate approximate solution to the equationT(X)=x.

New Results in the Startpoint Theory for Quasipseudometric Spaces

We give two generalizations of Theorem 35 proved by Gaba (2014). More precisely, we change the structure of the contractive condition; namely, we introduce a function Φ instead of a simple constant c.

Asymmetric structures, discontinuous contractions and iterative approximation of fixed and periodic points

In quasi-pseudometric spaces (X,p) (not necessarily Hausdorff), the concepts of the left quasi-closed maps (generalizing continuous maps) and generalized quasi-pseudodistances J:X×X→[0,∞) (generalizing in metric spaces: metrics, Tataru distances, w-distances of Kada et al., τ-distances of Suzuki and τ-functions of Lin and Du) are introduced, the asymmetric structures on X determined by J (generalizing the asymmetric structure on X determined by quasi-pseudometric p) are described and the contractions T:X→X with respect to J (generalizing Banach and Rus contractions) are defined. Moreover, if (X,p) are left sequentially complete (in the sense of Reilly, Subrahmanyam and Vamanamurthy), then, for these contractions T:X→X such that T [ q ] is left quasi-closed for some q∈N, the global minimum of the map x→J(x, T [ q ] (x)) is studied and theorems concerning the existence of global optimal approximate solutions of the equation T [ q ] (x)=x are established. The results are new in quasi-pseudometric and quasi-metric spaces and even in metric spaces. Examples showing the difference between our results and the well-known ones are provided. In the literature the fixed and periodic points in not Hausdorff spaces were not studied.MSC:41A65, 47H09, 47H10, 37C25, 54E15.

A Solution to the Completion Problem for Quasi-Pseudometric Spaces

The different notions of Cauchy sequence and completeness proposed in the literature for quasi-pseudometric spaces do not provide a satisfactory theory of completeness and completion for all quasi-pseudometric spaces. In this paper, we introduce a notion of completeness which is classical in the sense that it is made up of equivalence classes of Cauchy sequences and constructs a completion for any given quasi-pseudometric space. This new completion theory extends the existing completion theory for metric spaces and satisfies the requirements posed by Doitchinov for a nice theory of completeness.

Q-Hyperconvexity in Quasipseudometric Spaces and Fixed Point Theorems

In a previous work, we started investigating the concept of hyperconvexity in quasipseudometric spaces which we calledq-hyperconvexity or Isbell-convexity. In this paper, we continue our studies of this concept, generalizing further known results about hyperconvexity from the metric setting to our theory. In particular, in the present paper, we consider subspaces ofq-hyperconvex spaces and also present some fixed point theorems for nonexpansive self-maps on a boundedq-hyperconvex quasipseudometric space. In analogy with a metric result, we show among other things that a set-valued mappingT∗on aq-hyperconvexT0-quasimetric space (X, d) which takes values in the space of nonempty externallyq-hyperconvex subsets of (X, d) always has a single-valued selectionTwhich satisfiesd(T(x),T(y))≤dH(T∗(x),T∗(y))wheneverx,y∈X. (Here,dHdenotes the usual (extended) Hausdorff quasipseudometric determined bydon the set𝒫0(X)of nonempty subsets ofX.)

The B-completion of a [formula omitted]-quasi-metric space

We continue our study of the conjugate invariant method for completing an arbitrary T 0 -quasi-metric space which we have introduced in an earlier article under the name of the B-completion. We present examples that show that B-completeness of quasi-pseudometric spaces is not preserved under quasi-uniform isomorphisms. This leads to investigations of how some well-known operations applied to quasi-pseudometrics affect balancedness of their Cauchy filter pairs. We also observe that the B-completion of a totally bounded T 0 -quasi-metric space is totally bounded, but can be strictly larger than the bicompletion.