arXiv subjects

S. Cobzaş

Publications and source records attributed to S. Cobzaş.

10 recordsLinked to original sources

arXiv2024

Hahn-Banach type extension results for linear operators on asymmetric normed spaces

We present some results related to Hahn-Banach extension theorem for linear operators on asymmetric normed spaces. L. Nachbin, Trans. Amer. Math. Soc. 68 (1950), proved that a Banach space has the extension property for linear operators (a property also called injectivity) if and only if it has the Binary Intersection Property (BIP), meaning that every family of mutually intersecting closed balls has nonempty intersection. Its analog for quasi-metric spaces, called mixed BIP, was considered by Kemajou et al. Topology Appl. 159 (2012). The equivalence of mixed BIP to the injectivity of an asymmetric normed space was proved by Conradie et al., Topology Appl. 231 (2017), derived from some properties of the injective hull of a quasi-metric space. The aim of the present paper is to give a direct proof of this result by adapting Nachbin's ideas to the asymmetric case. Keywords: quasi-metric space, asymmetric normed space, injectivity, binary intersection property, hyperconvexity, Isbell-completeness, Isbell-convexity

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arXiv2024

The strong Ekeland variational principle in quasi-pseudometric spaces

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. \textbf{131} (1988), no.~1, 1--21) and Tomonari Suzuki (J. Math. Anal. Appl. \textbf{320} (2006), no.~2, 787--794 and Nonlinear Anal. \textbf{72} (2010), no.~5, 2204--2209)), proved a Strong Ekeland Variational Principle, meaning the existence of strong minima for such perturbations. Note that Suzuki also considered the case of functions defined on Banach spaces, emphasizing the key-role played by reflexivity. In the last years an increasing interest was manifested by many researchers to extend EkVP to the asymmetric case, that is, to quasi-metric spaces (see the 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 and Suzuki versions, to the quasi-pseudometric case. At the end we ask for the possibility to extend it to asymmetric normed spaces (i.e., the extension of Suzuki's results).

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arXiv2022

Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaces

We prove versions of Ekeland, Takahashi and Caristi principles in preordered 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. \textbf{265} (2019), 106831, 22, for quasi-metric spaces. The key tools are Picard sequences for some special set-valued mappings on a preordered quasi-metric space $X$, defined in terms of the preorder and of a function $φ$ on $X$. Key words: preordered quasi-metric space; completeness in quasi-metric spaces; variational principles; Ekeland variational principle; Takahashi minimization principle; fixed point; Caristi fixed point theorem.

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arXiv2021

Compact bilinear operators on asymmetric normed spaces

The paper is concerned with compact bilinear operators on asymmetric normed spaces. The study of multilinear operators on asymmetric normed spaces was initiated by Latreche and Dahia, Colloq. Math. (2020). We go further in this direction and prove a Schauder type theorem on the compactness of the adjoint of a compact bilinear operator and study the ideal properties of spaces of compact bilinear operators. These extend some results of Ramanujan and Schock, Linear and Multilinear Algebra (1985), and Ruch, ibid. (1989), on compact bilinear operators on Banach spaces. On the space of bilinear forms one introduces the analog of the weak$^*$-topology, called the $w^2$-topology, and one proves an Alaoglu-Bourbaki type theorem -- the $w^2$-compactness of the closed unit ball.

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arXiv2020

Completeness in quasi-pseudometric spaces

The aim of this paper is to discus 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. \textbf{17} (1967), 226--240, and V.~Gregori and J.~Ferrer, Proc. Lond. Math. Soc., III Ser., \textbf{49} (1984), 36.

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arXiv2019

Fixed points and completeness in metric and in generalized metric spaces

The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not a necessary condition -- there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present various circumstances in which fixed point results imply completeness. For metric spaces this is the case of Ekeland variational principle and of its equivalent - Caristi fixed point theorem. Other fixed point results having this property will be also presented in metric spaces, in quasi-metric spaces and in partial metric spaces. A discussion on topology and order and on fixed points in ordered structures and their completeness properties is included as well.

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arXiv2019

B-metric spaces, fixed points and Lipschitz functions

The paper is concerned with b-metric and generalized b-metric spaces. One proves the existence of the completion of a generalized b-metric space and some fixed point results. The behavior of Lipschitz functions on b-metric spaces of homogeneous type, as well as of Lipschitz functions defined on, or with values in quasi-Banach spaces, is studied.

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arXiv2019

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.

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arXiv2017

Lipschitz properties of convex mappings

The present paper is concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex subset $Ω$ of a locally convex space $X$ and taking values in a locally convex space $Y$ ordered by a normal cone. One proves also equi-Lipschitz properties for pointwise bounded families of continuous convex mappings, provided the source space $X$ is barrelled. Some results on Lipschitz properties of continuous convex functions defined on metrizable topological vector spaces are included as well. The paper has a methodological character - its aim is to show that some geometric properties (monotonicity of the slope, the normality of the seminorms) allow to extend the proofs from the scalar case to the vector one. In this way the proofs become more transparent and natural.

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arXiv2010

Functional analysis in asymmetric normed spaces

The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous linear functionals, duality, geometry of asymmetric normed spaces, compact operators) emphasizing similarities as well as differences with respect to the classical theory. The main difference comes form the fact that the dual of an asymmetric normed space $X$ is not a linear space, but merely a convex cone in the space of all linear functionals on $X.$ Due to this fact, a careful treatment of the duality problems (e.g. reflexivity) and of other results as, for instance, the extension of fundamental principles of functional analysis -the open mapping theorem and the closed graph theorem - to this setting, is needed.

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