Setting Of Quasi-metric Spaces Research Articles

Completeness in Quasi-Pseudometric Spaces—A Survey

The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. 17 (1967), 226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., 49 (1984), 36. A discussion on nets defined over ordered or pre-ordered directed sets is also included.

On Hybrid Contractions in the Context of Quasi-Metric Spaces

In this manuscript, we will investigate the existence of fixed points for mappings that satisfy some hybrid type contraction conditions in the setting of quasi-metric spaces. We provide examples to assure the validity of the given results. The results of this paper generalize several known theorems in the recent literature.

Existence of fixed points in quasi metric spaces

In this paper, we obtain some new fixed point theorems for two pairs of weakly compatible mappings in the setting of quasi metric spaces using $C$- class functions. Several interesting corollaries are also deduced. The results obtained extend various well known results of the literature in the setting of quasi metric space. We also construct an example to demonstrate the usability of the proved results.

Takahashi’s minimization theorem and some related results in quasi-metric spaces

In this paper, we establish Takahashi’s minimization theorem in the setting of quasi-metric spaces and provide its equivalence with Ekeland’s variational principle given in Cobzas (Topol Appl 158:1073–1084, 2011). We present an equilibrium version of Ekeland’s variational principle and extended Takahashi’s minimization theorem in the setting of quasi-metric spaces but without using the triangle inequality of the involved bifunction. We establish an equivalent chain of theorems containing Takahashi’s minimization theorem, Ekeland’s variational principle, the equilibrium version of Ekeland’s variational principle and Caristi–Kirk’s fixed point theorem for set-valued maps in the setting of quasi-metric spaces. As applications, we give an error bound for the solution set of the equilibrium problems and provide sufficient conditions for the existence of weak sharp solutions of equilibrium problems.

Abstract approach to non homogeneous Harnack inequality in doubling quasi metric spaces

We develop an abstract theory to obtain Harnack inequality for non homogeneous PDEs in the setting of quasi metric spaces. The main idea is to adapt the notion of double ball and critical density property given by Di Fazio, Gutiérrez, Lanconelli, taking into account the right hand side of the equation. Then we apply the abstract procedure to the case of subelliptic equations in non divergence form involving Grushin vector fields and to the case of X-elliptic operators in divergence form.

Equilibrium versions of variational principles in quasi-metric spaces and the robust trap problem

Using a pre-order principle in [Qiu JH. A pre-order principle and set-valued Ekeland variational principle. J Math Anal Appl. 2014;419:904–937], we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective bimap is defined on the product of left-complete quasi-metric spaces and taking values in a quasi-order linear space, and the perturbation consists of the quasi-metric and a positive vector . Here, the ordering is only to be -closed, which is strictly weaker than to be topologically closed. From the general equilibrium version, we deduce a number of particular equilibrium versions of EVP with set-valued bimaps or with vector-valued bimap. As applications of the equilibrium versions of EVP, we present several interesting results on equilibrium problems, vector optimization and fixed point theory in the setting of quasi-metric spaces. These results extend and improve the related known results. Using the obtained EVPs, we further study the existence and the robustness of traps in Behavioural Sciences.

Matkowski theorems in the context of quasi-metric spaces and consequences on G-metric spaces

Abstract In this paper, we prove the characterization of a Matkowski's theorem in the setting of quasi-metric spaces. As a result, we observe that some recent fixed point results in the context of G-metric spaces are consequences of our main result.

Coincidence point theorems in quasi-metric spaces without assuming the mixed monotone property and consequences in G-metric spaces

Abstract In this paper, we present some coincidence point theorems in the setting of quasi-metric spaces that can be applied to operators which not necessarily have the mixed monotone property. As a consequence, we particularize our results to the field of metric spaces, partially ordered metric spaces and G-metric spaces, obtaining some very recent results. Finally, we show how to use our main theorems to obtain coupled, tripled, quadrupled and multidimensional coincidence point results.

Generalized α-ψ contractive mappings in quasi-metric spaces and related fixed-point theorems

Abstract In this paper, we characterize α-ψ contractive mappings in the setting of quasi-metric spaces and investigate the existence and uniqueness of a fixed point of such mappings. We notice that by using our result some fixed-point theorems in the context of G-metric space can be deduced. MSC:46T99, 47H10, 54H25, 46J10.

Fixed Point Results in Quasimetric Spaces

AbstractIn the setting of quasimetric spaces, we prove some new results on the existence of fixed points for contractive type maps with respect to "Equation missing"-function. Our results either improve or generalize many known results in the literature.

Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory

In this paper, we introduce the concept of a Q -function defined on a quasi-metric space which generalizes the notion of a τ -function and a w -distance. We establish Ekeland-type variational principles in the setting of quasi-metric spaces with a Q -function. We also present an equilibrium version of the Ekeland-type variational principle in the setting of quasi-metric spaces with a Q -function. We prove some equivalences of our variational principles with Caristi–Kirk type fixed point theorems for multivalued maps, the Takahashi minimization theorem and some other related results. As applications of our results, we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps. We also extend the Nadler’s fixed point theorem for multivalued maps to a Q -function and in the setting of complete quasi-metric spaces. As a consequence, we prove the Banach contraction theorem for a Q -function and in the setting of complete quasi-metric spaces. The results of this paper extend and generalize many results appearing recently in the literature.

The relationship between the Vietoris topology and the Hausdorff quasi-uniformity

We investigate the problem of characterizing those quasi-uniform spaces for which the Vietoris topology is compatible with the Hausdorff quasi-uniformity. In particular, we prove that the Vietoris topology of a quasi-uniform space (X,u) is compatible with the Hausdorff quasi-uniformity on the family K0(X) of nonempty compact subsets of X, if and only if for each K ∈ K0(X), u-1|K is precompact. We show that for a T1 quasi-uniform space (X,u), the Vietoris topology is compatible with the Hausdorff quasi-uniformity on the family of nonempty closed subsets of X, if and only if u is equinormal and u-1 is hereditarily precompact. We also discuss the problem in the setting of quasi-metric spaces and show, among other results, that the Vietoris topology of a quasi-metric space (X,d) is compatible with the Hausdorff extended quasi-pseudo-metric on the family of nonempty subsets of X, if and only if the quasi-uniformity induced by d coincides with the Pervin quasi-uniformity of X.