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Consider L^0, the F-space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin's classical result on the factorization of sublinear continuous operators from a Banach space to L^0, we prove a theorem that characterizes those maps from any quasi-metric space into L^0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces. [ABSTRACT FROM AUTHOR]

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The goal of this research article is to study some fixed point results in dislocated quasi metric spaces. These results are proved for certain generalized contraction conditions involving linear and rational expressions. Our results extend and generalize some of the well-known fixed point results of the literature available in fixed point theory. Suitable examples of the established results are also given. [ABSTRACT FROM AUTHOR]

In this paper we prove new fixed point results in partially ordered bicomplete quasimetric spaces. Our results extends/generalized celebrated results, on the one hand, by Nieto and Rodríguez-López for contraction mappings in partially ordered complete metric spaces and, on the other hand, by Schellekens for contraction mappings in bicomplete quasi-metric spaces. Moreover, it is also shown that neither our assumptions can be weakened nor our results can be deduced from the celebrated Kleene's fixed point theorem. Finally, an application of our results to the asymptotic analysis of recurrence equations is given. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the analysis of the Hepatitis B model, specifically the S E I A I C R model, within the scope of the Caputo–Fabrizio fractional operator. A new state variable the number of vaccinated individuals is also added to the model. This addition enriches the scope of the hepatitis B model inviting a deeper exploration of the subject matter. Our study proves the existence of a disease-free fixed point within the proposed compartmental model. To ensure the existence and uniqueness of the fixed point, we employ a fixed-point result in the b-complete b-dislocated quasi-metric space, utilizing a Geraghty-type contraction mapping. This approach establishes the fixed point of a disease-free state within the model. Furthermore, we employ a two-step Adams–Bashforth numerical scheme, serving as a validation of both the significance of fractional-order derivatives and the validity of our obtained theoretical results. Together our research presents an innovative perspective on the S E I A I C R hepatitis B model pushing the boundaries of understanding and shedding light on the dynamics of disease transmission with the impact of vaccine. [ABSTRACT FROM AUTHOR]

The survey of the available literature shows that a lot of important invariant point problems of Banach and Heilpern types have been examined in both metric and quasimetric spaces. However, a handful of the existing results employed the recent approaches of interpolative contractions. Therefore, based on the new idea of interpolation techniques in fixed point theory, this article studies new notions of L-fuzzy contractions and investigates conditions for the existence of L-fuzzy fixed points for such mappings. On the fact that fixed points of point-to-point mappings satisfying interpolative-type contraction are not always unique, whence making the concepts more fitted for invariant point results of crisp set-valued maps, new multi-valued analogues of the key findings put forward in this work are derived. Comparative illustrations, which indicate the preeminence of the results presented herein, are constructed. From application viewpoint, one of the theorems so obtained is employed to introduce new solvability conditions of Fredholm-type integral inclusions. [ABSTRACT FROM AUTHOR]

In this paper, we present some new results on the existence of fixed points for multivalued mappings endowed with Q-function in the setting of quasi-metric space. Examples are also provided in support of our main results. We conclude that our results in fact either improve or generalize some classical metric fixed point results as well. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki's theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki's theorem, which was applied to discuss types of difference equations and recurrence equations. [ABSTRACT FROM AUTHOR]

We first give a pre-order principle whose form is very general. Combining the pre-order principle and generalized Gerstewitz functions, we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a subset of the ordering cone multiplied by the quasi-metric. From this, we obtain a number of new results which essentially improve the related results. Particularly, the earlier lower boundedness condition has been weakened. Finally, we apply the new EVPs to Psychology. [ABSTRACT FROM AUTHOR]

By implying α-admissible mapping, this study expands and investigates generalized contraction mappings in quasi-metric spaces, aiming to establish the existence of fixed points. Moreover, we show that the main outcomes of the paper encompass several previously reported results in the literature. [ABSTRACT FROM AUTHOR]

Roughly speaking, Ekeland's Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. 1, 1–21) and Tomonari Suzuki (J. Math. Anal. Appl. 320 (2006), no. 2, 787–794 and Nonlinear Anal. 72 (2010), no. 5, 2204–2209)) proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Please note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key role played by reflexivity. In recent years, an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, i.e., to quasi-metric spaces (see references). Applications to optimization, behavioral sciences, and others were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev's and Suzuki's versions, to the quasi-pseudometric case. At the end, we ask for the possibility of extending it to asymmetric normed spaces (i.e., the extension of Suzuki's results). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We introduce the new idea of (α − θ σ) -contraction in quasi-metric spaces in this paper. For these kinds of mappings, we then prove new fixed-point theorems on left K, left M, and left S m y t h -complete quasi-metric spaces. We also apply our results to infer the existence of a solution to a second-order boundary-value problem. [ABSTRACT FROM AUTHOR]

Embeddings in the framework of grand variable exponent function spaces are studied. In particular, embeddings from grand variable exponent Hajłasz-Sobolev-Morrey spaces to variable exponent H ¨older spaces are established. The regularity of a fractional integral operator defined with respect to a non-doubling measure is also investigated. In particular, mapping properties of this operator from a grand variable exponent Morrey space to a grand variable parameter H ¨older space are studied. The results are proved under the log-H ¨older continuity condition on the exponents. The spaces are defined, generally speaking, on quasi-metric measure spaces, however, the results are new even for Euclidean spaces. [ABSTRACT FROM AUTHOR]

We continue the study of the concept of one-local retract in the settings of modular metrics. This concept has been studied in metric spaces and quasi-metric spaces by different authors with different motivations. In this article, we extend the well-known results on one-local retract in metric point of view to the framework of modular metrics. In particular, we show that any self-map ψ: Xw → Xw satisfying the property w(λ, ψ(x), ψ(y)) ≤ w(λ, x, y) for all x, y ∈ X and λ > 0, has at least one fixed point whenever the collection of all qw-admissible subsets of Xw is both compact and normal. [ABSTRACT FROM AUTHOR]

We present a systematic approach to the problem whether a topologically infinite-dimensional space can be made homogeneous in the Coifman–Weiss sense. The answer to the question is negative, as expected. Our leading representative of spaces with this property is $\mathbb {T}^\omega = \mathbb {T} \times \mathbb {T} \times \cdots $ with the natural product topology. [ABSTRACT FROM AUTHOR]

In this paper we work in the category of approach spaces with contractions [14], the objects of which are sets endowed with a numerical distance between sets and points. Approach spaces are to be considered a simultaneous generalization of both quasi-metric and topological spaces. Especially the fundamental notion of distance is reminiscent of the closure operator in a topological space and of the point-to-set distance in a quasi-metric space. The embedding of the category of topological spaces with continuous maps and of quasi-metric spaces with non-expansive maps is extremely nice. Every approach space has both a quasi-metric coreflection as well as a topological coreflection. Different approach spaces though can have the same topological as well as the same quasi-metric coreflection, in other words, in general these coreflections do not determine the approach space. In this paper we investigate approach spaces for which these coreflections, do determine the approach space. We will call such spaces supported. We prove that in the setting of compact approach spaces many examples of supported approach spaces can be found. Thus, compact spaces that are base-regular, which is a weakening of regularity, are always supported. An important feature of supported approach spaces is the behaviour of contractions. On a supported domain contractivity is characterized by the combination of continuity for the topological coreflection and non-expansiveness for the quasi-metric coreflection. This result implies that a supported approach space actually is the infimum of its quasi-metric and its topological coreflection. In the course of our study we also give several more examples of both supported and non-supported approach spaces. [ABSTRACT FROM AUTHOR]

In order to generalize classical Banach contraction principle in the setup of quasi-metric spaces, we introduce Suzuki-type contractions of quasi-metric spaces and prove some fixed point results. Further, we suggest a correction in the definition of another class of quasi-metrics known as Δ -symmetric quasi-metrics satisfying a weighted symmetry property. We discuss equivalence of various types of completeness of Δ -symmetric quasi-metric spaces. At the end, we consider the existence of fixed points of generalized Suzuki-type contractions of Δ -symmetric quasi-metric spaces. Some examples have been furnished to make sure that generalizations we obtain are the proper ones. [ABSTRACT FROM AUTHOR]

We provide an overview of the known problem on the existence of a bi-Lipschitz equivalent metric with respect to a given quasi-ultrametric, revisiting known results and counterexamples in an unified approach based on the existence of a relaxed polygonal inequality. We present new proofs and characterizations of classical results using different techniques. [ABSTRACT FROM AUTHOR]

Our aim in this article is to show that all metric fixed point theorems hold for quasi-metric spaces (X, δ) (without symmetry). In fact, we show some well-known theorems on metric spaces hold for quasi-metric spaces from the beginning. We check this fact for the Banach contraction principle, the Covitz-Nadler fixed point theorem, the Rus-Hicks-Rhoades fixed point theorem, and others. In these theorems the concepts of continuity and completeness can be replaced by orbital continuity and T-orbital completeness for a selfmap T, respectively. Consequently, we improve and generalize the basic known theorems in the metric fixed point theory. [ABSTRACT FROM AUTHOR]

Necessary and sufficient condition governing the boundedness of the multilinear fractional integral operator T γ , μ (f) (x) = ∫ X m f 1 (y 1) ⋯ f (y m) (ρ (x , y 1) + ⋯ + ρ (x , y m) ) m - γ d μ (y) defined with respect to a measure μ from grand product space ∏ j = 1 m L p j) , θ (X , μ) to grand Lebesgue space L q) , θ q / p (X , μ) is established, where (X , ρ , μ) is a quasi-metric measure space, 1 p = ∑ j = 1 m 1 p j , p < q and θ > 0 . In particular, we show that for that boundedness, the necessary and sufficient condition on μ is that it is upper Ahlfors β m —regular for certain number β m . To establish this result, first we derive the boundedness of T γ , μ between more general grand spaces: T γ , μ : ∏ j = 1 m L p j) , Ψ (·) (X , μ) ↦ L q) , Φ (·) (X , μ) . As a corollary, we have a complete characterization of the multilinear Sobolev inequality (i.e., when q is the Sobolev exponent of p) in these spaces. In this case criterion on μ is that it is upper Ahlfors 1– regular. Some weighted inequalities for the bilinear fractional integral operator B γ (f , g) (x) = ∫ R n f (x - t) g (x + t) d t | t | n - γ , 0 < γ < n , in the classical Lebesgue spaces are also discussed. [ABSTRACT FROM AUTHOR]

Weighted extrapolation for pairs of functions in mixed‐norm Banach function spaces defined on the product of quasi‐metric measure spaces (X,d,μ)$(X, d, \mu)$ and (Y,ρ,ν)$(Y, \rho , \nu)$ are derived. As special cases, we have appropriate results for mixed‐norm Lebesgue, Lorentz, and Orlicz spaces. Some of the derived results are applied to get weighted extrapolation in mixed‐norm grand Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper, some properties of locally antisymmetrically connected spaces which are the localized version of the antisymmetrically connected T0-quasi-metric spaces constructed as the natural counterparts of connected complementary graphs, are presented in terms of asymmetric norms. According to that, we investigated some different aspects and examples of local antisymmetric connectedness in the framework of asymmetrically normed real vector spaces. Specifically, it is proved that the structures of antisymmetric connectedness and local antisymmetric connectedness coincide for the T0-quasi-metrics induced by the asymmetric norms which associate the theory of quasi-metrics with functional analysis. [ABSTRACT FROM AUTHOR]

In this manuscript, we introduce the notions of fuzzy strong controlled metric spaces, fuzzy strong controlled quasi-metric spaces, and non-Archimedean fuzzy strong controlled quasi-metric spaces and generalize the famous Banach contraction principle. We prove several fixed point results in the context of non-Archimedean fuzzy strong controlled quasi-metric space. Furthermore, we use our main result to obtain the existence of a solution for a recurrence problem linked with the study of Quicksort algorithms. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the Ekeland Variational Principle (EkVP) and its equivalents (Caristi–Kirk fixed point theorem, Takahashi minimization principle, Oettli-Théra equilibrium version of EkVP) in quasi-uniform spaces. These extend some results proved by Hamel and Löhne [A minimal point theorem in uniform spaces. In: Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vols 1, 2. Dordrecht: Kluwer Academic Publishers; 2003. p. 577–593] and Hamel [Equivalents to Ekeland's variational principle in uniform spaces. Nonlinear Anal. 2005;62:913–924] in uniform spaces, as well as those proved in quasi-metric spaces by various authors. The case of F-quasi-gauge spaces, a non-symmetric version of F-gauge spaces introduced by Fang [The variational principle and fixed point theorems in certain topological spaces. J Math Anal Appl. 1996;202:398–412], is also considered. The paper ends with the quasi-uniform versions of some minimization principles proved by Arutyunov and Gel'man [The minimum of a functional in a metric space, and fixed points. Zh Vychisl Mat Mat Fiz. 2009;49:1167–1174] and Arutyunov [Caristi's condition and existence of a minimum of a lower bounded function in a metric space. Applications to the theory of coincidence points. Proc Steklov Inst Math. 2015;291(1):24–37] in complete metric spaces. [ABSTRACT FROM AUTHOR]

In this note, grand variable exponent Hajłasz-Morrey spaces are introduced and embeddings from these spaces to Hölder spaces are established under the log-Hölder continuity condition on exponents. The boundedness of the fractional integral operator from a grand variable exponent Morrey space to a grand variable parameter Hölder space is also proved. In general, the function spaces are defined on quasi-metric measure spaces, however, the results are new even for the Euclidean spaces. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a new generalized type of metric spaces called the right quasi-metric spaces. We state and prove convergence theorems to a fixed point for any map in these spaces. Finally, we give applications of our results. These results generalize the corresponding results in M. A. Ahmed and F. M. Zeyada (On convergence of a sequence in complete metric spaces and its applications to some iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl., 274(1), 458-465, 2002). [ABSTRACT FROM AUTHOR]

In this paper, T-orbitally complete dislocated quasi-metric spaces are utilized to validate fixed point results for freshly developed Geraghty quasi-contraction type mappings. The Geraghty quasi-contraction type mappings generalize Ciric’s quasi-contraction mappings and other Geraghty quasicontractive type mappings in the literature. Without establishing a continuity condition on the mapping, fixed point results are obtained, generalizing some relevant work in the literature. [ABSTRACT FROM AUTHOR]

It is known that if any function acting from precompact metric space to itself increases the distance between some pair of points then it must decrease distance between some other pair of points. We show that this is not the case for quasi-metric spaces. After that, we present some sufficient conditions under which the previous property holds true for hereditarily precompact quasi-metric spaces. [ABSTRACT FROM AUTHOR]

We prove versions of Ekeland, Takahashi and Caristi principles in pre-ordered quasi-metric spaces, the equivalence between these principles, as well as their equivalence to some completeness results for the underlying quasi-metric space. These extend the results proved in S. Cobzaş, Topology Appl.265 (2019), 106831, 22, for quasi-metric spaces. The key tools are Picard sequences for some special set-valued mappings on a pre-ordered quasi-metric space X, defined in terms of the preorder and of a function φ on X. [ABSTRACT FROM AUTHOR]

Grand Lebesgue spaces, well-known on sets of finite measure, are studied on general measure spaces (X , μ) with admission of μ (X) = ∞ . In this case, the grand space is defined via the so-called grandizer. It is shown that the classical Lebesgue space is embedded into the grand Lebesgue space if and only if the grandizer is in L 1 (X) . Embedding between grand Lebesgue spaces with different grandizers is also considered. The main results concern the boundedness of maximal, singular, and fractional operators in grand Lebesgue spaces over quasi-metric measure spaces (X , d , μ) . Application to such operators over S n - 1 and homogeneous groups on R n is also given. [ABSTRACT FROM AUTHOR]

A digraph D = (V, A) is semi-complete if for each pair of distinct vertices x and y in V, either xy or yx belong to A. A subset ℓ of vertices is a line of D if there are two distinct vertices x and y such that for any vertex z ε V, z ε ℓ if and only if a directed shortest path exists containing x , y and z . A classic result proved by Erdös says that any set of n points in the Euclidean plane endowed with the Euclidean distance defines a metric space with at least n different lines unless there is a line containing the n points. Chen and Chvátal in 2008 conjectured that the same results is true for any metric spaces where lines are defined in a manner similar to above. In this paper we prove that in any semi-complete digraphs with n vertices the number of lines defined by vertices connected by an arc is at least n. Then, the quasi-metric spaces defined by semi-complete digraphs fulfill Chen and Chvátal conjecture in a stronger manner as, on the one hand, they always have at least n lines, and on the other hand, these n lines are defined by vertices at distance one. [ABSTRACT FROM AUTHOR]

It is a classic result that a set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chvátal conjectured in 2008 that the same results is true in metric spaces for an adequate definition of line. More recently, this conjecture was studied in the context of quasi-metric spaces. One way to study lines in an space is though its betweenness. Given a quasi-metric space (V,ρ), its induced quasi-metric be-tweenness is the set of triples (x, y, z) ϵ V3 such that ρ(x, z) = ρ(x, y) +ρ(y, z). In this work, we prove the existence of a quasi-metric space on four points a, b, c and d whose quasi-metric betweenness is ẞ = {(c, a, b), (a, b, c), (d, b, a), (b, a, d)}. This space has only three lines, none of which has four points. Moreover, we show that the betweenness of any quasi-metric space on four points with this property is isomorphic to B. Since B is not metric, we conclude that Chen and Chvatal's conjecture is valid for any metric space on four points. [ABSTRACT FROM AUTHOR]

One of the well-studied generalizations of a metric space is known as a partial metric space. The partial metric space was further generalized to the so-called M-metric space. In this paper, we introduce the Double-Controlled Quasi M-metric space as a new generalization of the M-metric space. In our new generalization of the M-metric space, the symmetry condition is not necessarily satisfied and the triangle inequality is controlled by two binary functions. We establish some fixed point results, along with the examples and applications to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we explore the existence and uniqueness of fixed points for the new constructed contraction mapping on dislocated quasi-metric spaces by using Geraghty contraction and F-contraction. Moreover, we support our results by a couple of non-trivial examples. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the concepts of d- and d-1 -proximal Ćirić contraction mappings. Also, we present new definitions and notations by taking into account the lack of symmetry property of quasi-metric spaces. Moreover, we give some examples to support our definitions and notations. Then, we prove some right and left best proximity point results for these mappings on best orbitally complete quasi-metric spaces. Hence, we obtain some generalizations of famous results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of generalized contractive mappings to establish a fixed point theorem for this class of mappings in complete quasimetric spaces. In fact, at first we present the notion of ordered cyclic weakly (ψ, ϕ, A, B)- contraction and then we establish a fixed point theorem for such mappings in complete ordered s-quasimetric spaces. This can be considered as an improvement of some old fixed point theorems in the literature. Finally, we provide an example to show that our result is genuine generalizations of some fixed point theorems. [ABSTRACT FROM AUTHOR]

Roughly speaking, Ekeland’s Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324–353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space. Later, Pando Georgiev (J. Math. Anal. Appl. 131 (1988), no. 1, 1–21) and Tomonari Suzuki (J. Math. Anal. Appl. 320 (2006), no. 2, 787–794 and Nonlinear Anal. 72 (2010), no. 5, 2204–2209)) proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Please note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key role played by reflexivity. In recent years, an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, i.e., to quasi-metric spaces (see references). Applications to optimization, behavioral sciences, and others were obtained. The aim of the present paper is to extend the strong Ekeland principle, both Georgiev’s and Suzuki’s versions, to the quasi-pseudometric case. At the end, we ask for the possibility of extending it to asymmetric normed spaces (i.e., the extension of Suzuki’s results).

Gregori and Romaguera introduced, in 2004, the notion of a KM-fuzzy quasi-metric space as a natural asymmetric generalization of the concept of fuzzy metric space in the sense of Kramosil and Michalek. Ever since, various authors have discussed several aspects of such spaces, including their topological and (quasi-)metric properties as well as their connections with domain theory and their relationship with other fuzzy structures. In particular, the development of the fixed point theory for these spaces and other related ones, such as fuzzy partial metric spaces, has received remarkable attention in the last 15 years. Continuing this line of research, we here establish general fixed point theorems for left and right complete Hausdorff KM-fuzzy quasi-metric spaces, which are applied to deduce characterizations of these distinguished kinds of fuzzy quasi-metric completeness. Our approach, which mixes conditions of Suzuki-type with contractions of α − ϕ -type in the well-known proposal of Samet et al., allows us to extend and improve some recent theorems on complete fuzzy metric spaces. The obtained results are accompanied by illustrative and clarifying examples. [ABSTRACT FROM AUTHOR]

The current paper deals with some new classes of Finsler metrics with reversible geodesics. We construct weighted quasi-metrics associated with these metrics. Further, we investigate some important geometric properties of weighted quasi-metric space. Finally, we discuss the embedding of quasi-metric spaces with generalized weight. [ABSTRACT FROM AUTHOR]

In this paper, we study quasi-metric spaces using domain theory. Given a quasi-metric space (X , d), we use $({\bf B}(X,d),\leq^{d^{+}}\!)$ to denote the poset of formal balls of the associated quasi-metric space (X , d). We introduce the notion of local Yoneda-complete quasi-metric spaces in terms of domain-theoretic properties of $({\bf B}(X,d),\leq^{d^{+}}\!)$. The manner in which this definition is obtained is inspired by Romaguera–Valero theorem and Kostanek–Waszkiewicz theorem. Furthermore, we obtain characterizations of local Yoneda-complete quasi-metric spaces via local nets in quasi-metric spaces. More precisely, we prove that a quasi-metric space is local Yoneda-complete if and only if every local net has a d -limit. Finally, we prove that every quasi-metric space has a local Yoneda completion. [ABSTRACT FROM AUTHOR]

The ambition of this work is to introduce the notion of left (right) K-sequentially complete ordered dislocated fuzzy quasimetric spaces and to define a relevant Hausdorff metric on compact sets. A new approach, given in (Shoaib et al. in Filomat 34(2):323–338, 2020) has been used to obtain fixed-point results for multivalued mappings fulfilling generalized contraction in the latest framework. For the authenticity of our result, an example is formulated. [ABSTRACT FROM AUTHOR]

In 1994, S.G. Matthews introduced the notions of partial metric and weighted quasi-metric and, in addition, he showed that both are dual. In fact, he developed a method for generating weightable quasi-metrics from partial metrics and conversely. Recently, J.J. Miñana and O. Valero introduced a general method with the aim of inducing quasi-metrics from partial metrics via real-valued functions in such a way that the original Matthews method can be retrieved as a special case when an appropriate function is chosen. However, they showed that the aforementioned method does not induce in general weighted quasi-metrics and, thus, they posed the question about what properties must be fulfilled by this type of functions in order to guarantee that the generated quasi-metric is weighted. In this paper, we answer such a question providing a characterization of such functions through the use of what we have called weight difference function and a new type of quadrangular triplets. Finally, a new characterization of weightable quasi-metric spaces is also given and a few properties of weightable quasi-metric spaces are discussed. [ABSTRACT FROM AUTHOR]

In 2023 Haihambo and Olela Otafudu introduced and studied the notion of quasi-uniform entropy h Q U (ψ) for a uniformly continuous self-map ψ of a quasi-metric or a quasi-uniform space X. In this paper, we discuss the connection between the topological entropy functions h , h f and the quasi-uniform entropy function h Q U on a quasi-uniform space X , where h and h f are the topological entropy functions defined using compact sets and finite open covers, respectively. In particular, we have shown that for a uniformly continuous self-map ψ of a T 0 -quasi-uniform space (X , U) we have h (ψ) ≤ h Q U (ψ) when X is compact and h Q U (ψ) ≤ h f (ψ) with equality if X is a compact T 2 space. [ABSTRACT FROM AUTHOR]

In this paper, we develop some connections between pointwise quasi-metric spaces and Scott spaces in domain theory. The main results include (i) the category of Scott quasi-metrics with S-morphisms is equivalent to that of pointwise quasi-metrics in the sense of Shi; (ii) a topological space (X , T) is quasi-metrizable if and only if the topologically generated space (I X , ω I (T)) (where ω I (T) denotes the family of all lower semi-continuous mappings from X to the unit interval I) can be induced by a pointwise quasi-metric with a property M; (iii) the notion of Scott quasi-uniformity is presented, and it is shown that d -spaces of domain theory are exactly the Scott quasi-uniformizable spaces; (iv) the relationship between Scott quasi-metrics (introduced by the first and second authors) and Scott quasi-uniformities is established. In specific, the Scott quasi-metrics are exactly the Scott quasi-uniformities that has a countable base. [ABSTRACT FROM AUTHOR]

In order to generalize classical Banach contraction principle in the setup of quasi-metric spaces, we introduce Suzuki-type contractions of quasi-metric spaces and prove some fixed point results. Further, we suggest a correction in the definition of another class of quasi-metrics known as Δ-symmetric quasi-metrics satisfying a weighted symmetry property. We discuss equivalence of various types of completeness of Δ-symmetric quasi-metric spaces. At the end, we consider the existence of fixed points of generalized Suzuki-type contractions of Δ-symmetric quasi-metric spaces. Some examples have been furnished to make sure that generalizations we obtain are the proper ones.

We study the fractional maximal commutators Mb,η$$ {M}_{b,\eta } $$ and the commutators [b,Mη]$$ \left[b,{M}_{\eta}\right] $$ of the fractional maximal operator with b∈BMO(X)$$ b\in BMO(X) $$ in the variable Lebesgue spaces Lp(·)(X)$$ {L}&amp;#x0005E;{p\left(\cdotp \right)}(X) $$ over bounded quasi‐metric measure spaces. We give necessary and sufficient conditions for the boundedness of the operators Mb,η$$ {M}_{b,\eta } $$ and [b,Mη]$$ \left[b,{M}_{\eta}\right] $$ on the spaces Lp(·)(X)$$ {L}&amp;#x0005E;{p\left(\cdotp \right)}(X) $$ when b∈BMO(X)$$ b\in BMO(X) $$. Furthermore, we obtain some new characterizations for certain subspaces of BMO(X)$$ BMO(X) $$. [ABSTRACT FROM AUTHOR]

This paper deals with the question of achieving a suitable extension of the notion of Suzuki-type contraction to the framework of quasi-metric spaces, which allows us to obtain reasonable fixed point theorems in the quasi-metric context. This question has no an easy answer; in fact, we here present an example of a self map of Smyth complete quasi-metric space (a very strong kind of quasi-metric completeness) that fulfills a simple and natural contraction of Suzuki-type but does not have fixed points. Despite it, we implement an approach to obtain two fixed point results, whose validity is supported with several examples. Finally, we present a general method to construct non- T 1 quasi-metric spaces in such a way that it is possible to systematically generate non-Banach contractions which are of Suzuki-type. Thus, we can apply our results to deduce the existence and uniqueness of solution for some kinds of functional equations which is exemplified with a distinguished case. [ABSTRACT FROM AUTHOR]

We establish general versions of the Ekeland variational principle (EVP), where we include two perturbation bifunctions to discuss and obtain better perturbations for obtaining three improved versions of the principle. Here, unlike the usual studies and applications of the EVP, which aim at exact minimizers via a limiting process, our versions provide good-enough approximate minimizers aiming at applications in particular situations. For the presentation of applications chosen in this paper, the underlying space is a partial quasi-metric one. To prove the aforementioned versions, we need a new proof technique. The novelties of the results are in both theoretical and application aspects. In particular, for applications, using our versions of the EVP together with new concepts of Ekeland points and stop and go dynamics, we study in detail human dynamics in terms of a psychological traveler problem, a typical model in behavioral sciences. [ABSTRACT FROM AUTHOR]