In this chapter we present various extension results for Lipschitz functions obtained by Kirszbra... more In this chapter we present various extension results for Lipschitz functions obtained by Kirszbraun, McShane, Valentine and Flett—the analogs of Hahn-Banach and Tietze extension theorems. A discussion on the corresponding property for semi-Lipschitz functions defined on quasi-metric spaces and for Lipschitz functions with values in a quasi-normed space is included as well.
In this chapter we study the problem of the uniform approximation of some classes of functions (e... more In this chapter we study the problem of the uniform approximation of some classes of functions (e.g. uniformly continuous) by Lipschitz functions, based on the existence of Lipschitz partitions of unity or on some extension results for Lipschitz functions. A result due to Baire on the approximation of semi-continuous functions by continuous ones, based on McShane’s extension method, is also included. The chapter ends with a study of homotopy of Lipschitz functions and a brief presentation of Lipschitz manifolds.
The paper is concerned with b-metric and generalized b-metric spaces. One proves the existence of... more 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.
The paper is concerned with b-metric and generalized b-metric spaces. One proves the existence of... more 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. MSC2010: 54E25 54E35 47H09 47H10 46A16 26A16
The aim of this paper is to discuss the relations between various notions of sequential completen... more 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.
We prove versions of Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaces,... more 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. 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.
The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not ... more 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.
We introduce the notion of generalized b-metric, as a b-metric which can take infinite values, an... more We introduce the notion of generalized b-metric, as a b-metric which can take infinite values, and prove the existence and uniqueness of the completion of some particular b-metric spaces (called generalized strong b-metric spaces). Some fixed point results in b-metric spaces and their counterparts in generalized b-metric spaces are proved.
Let (X, d) be a metric space. A function f: X-t R is called Lipschitz if there exists a number M ... more Let (X, d) be a metric space. A function f: X-t R is called Lipschitz if there exists a number M 2 0 such that ,f(.Y)-.f(y)l sz ML&Y, y) (1) for all X, y t X. The smallest constant M verifying (I) is called the norm of,f and is denoted by /f' ,r. We have 'Ifs ,y-= sup{/ f(x)-f(y)l/d(x, y) : X, y E x, x z yj. (2) Denote by Lip X the linear space of all Lipschitz functions on X. Actually, il. j X is not a norm on the space Lip X, since I& =: 0 iffis constant. Now let Y be a nonvoid subset of X. A norm-preserving extension of a function fg Lip Y to X is a function F t Lip X such that F iy =: f and lIfl~r-: j' Fll,. By a result of Banach [I] (see also Czipser and Geher [2]) every f~ Lip Y has a norm-preserving extension F in Lip I'. Two of these extensions are given by and F,(x) = sup{f(y)-s,fl,Y d(.u, ~1) : .V E Y) (3) F,(.r) = inf{f(-r)-C ~ j-y d(x, y) : y E Y). (4) Every norm-preserving extension F offsatisfies F,(x) 5; F(x)-(-F?(x) (5) for all x t X (see [7]). Now, let X be a normed linear space and let Y be a nonvoid convex subset of X. Concerning the convex norm-preserving extension to X of the convex functions in Lip Y, we can prove the following theorem: 236 0021-9045/78/0243~236~02.00/0
The aim of this paper is to study the basic properties of the Thompson metric dT in the general c... more The aim of this paper is to study the basic properties of the Thompson metric dT in the general case of a linear spaces X ordered by a cone K. We show that dT has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of dT , and some results concerning the topology of dT , including a brief study of the dT-convergence of monotone sequences. It is shown most results are true without any assumption of Archimedean-type property for K. One considers various completeness properties and one studies the relations between them. Since dT is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. Thompson metric dT and order-unit (semi)norms | • |u are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although dT and | • |u are only topological (and not metrical) equivalent on Ku, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.
We prove a version of Ekeland Variational Principle (EkVP) in a weighted graph $G$ and its equiva... more We prove a version of Ekeland Variational Principle (EkVP) in a weighted graph $G$ and its equivalence to Caristi fixed point theorem and to Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in terms of the graph $G$. The main tool used in the proof is the OSC property for sequences in a graph. Converse results, meaning the completeness of graphs for which one of these principles holds is also considered.
The paper deals with divergence phenomena for various approxima-
tion processes of analysis such... more The paper deals with divergence phenomena for various approxima-
tion processes of analysis such as Fourier series, Lagrange interpolation,
Walsh-Fourier series. We prove the existence of superdense (meaning
residual, dense and uncountable) families of functions in appropriate
function spaces over an interval T ⊂ R. One proves that for each function
in the family, the corresponding approximation process is unboundedly
divergent on a superdense subset of T of full measure.
The aim of the present paper is to show that many Phelps type duality result, relating the extens... more The aim of the present paper is to show that many Phelps type duality result, relating the extension properties of various classes of functions (continuous, linear continuous, bounded bilinear, Holder-Lipschitz) with the approximation properties of some annihilating spaces, can be derived in a unitary and simple way from a formula for the distance to the kernel of a linear operator, extending the well-known distance formula to hyperplanes in normed spaces. The case of spaces \(c_0\) and \(l^\infty\) is treated in details.
In this chapter we present various extension results for Lipschitz functions obtained by Kirszbra... more In this chapter we present various extension results for Lipschitz functions obtained by Kirszbraun, McShane, Valentine and Flett—the analogs of Hahn-Banach and Tietze extension theorems. A discussion on the corresponding property for semi-Lipschitz functions defined on quasi-metric spaces and for Lipschitz functions with values in a quasi-normed space is included as well.
In this chapter we study the problem of the uniform approximation of some classes of functions (e... more In this chapter we study the problem of the uniform approximation of some classes of functions (e.g. uniformly continuous) by Lipschitz functions, based on the existence of Lipschitz partitions of unity or on some extension results for Lipschitz functions. A result due to Baire on the approximation of semi-continuous functions by continuous ones, based on McShane’s extension method, is also included. The chapter ends with a study of homotopy of Lipschitz functions and a brief presentation of Lipschitz manifolds.
The paper is concerned with b-metric and generalized b-metric spaces. One proves the existence of... more 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.
The paper is concerned with b-metric and generalized b-metric spaces. One proves the existence of... more 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. MSC2010: 54E25 54E35 47H09 47H10 46A16 26A16
The aim of this paper is to discuss the relations between various notions of sequential completen... more 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.
We prove versions of Ekeland, Takahashi and Caristi principles in preordered quasi-metric spaces,... more 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. 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.
The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not ... more 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.
We introduce the notion of generalized b-metric, as a b-metric which can take infinite values, an... more We introduce the notion of generalized b-metric, as a b-metric which can take infinite values, and prove the existence and uniqueness of the completion of some particular b-metric spaces (called generalized strong b-metric spaces). Some fixed point results in b-metric spaces and their counterparts in generalized b-metric spaces are proved.
Let (X, d) be a metric space. A function f: X-t R is called Lipschitz if there exists a number M ... more Let (X, d) be a metric space. A function f: X-t R is called Lipschitz if there exists a number M 2 0 such that ,f(.Y)-.f(y)l sz ML&Y, y) (1) for all X, y t X. The smallest constant M verifying (I) is called the norm of,f and is denoted by /f' ,r. We have 'Ifs ,y-= sup{/ f(x)-f(y)l/d(x, y) : X, y E x, x z yj. (2) Denote by Lip X the linear space of all Lipschitz functions on X. Actually, il. j X is not a norm on the space Lip X, since I& =: 0 iffis constant. Now let Y be a nonvoid subset of X. A norm-preserving extension of a function fg Lip Y to X is a function F t Lip X such that F iy =: f and lIfl~r-: j' Fll,. By a result of Banach [I] (see also Czipser and Geher [2]) every f~ Lip Y has a norm-preserving extension F in Lip I'. Two of these extensions are given by and F,(x) = sup{f(y)-s,fl,Y d(.u, ~1) : .V E Y) (3) F,(.r) = inf{f(-r)-C ~ j-y d(x, y) : y E Y). (4) Every norm-preserving extension F offsatisfies F,(x) 5; F(x)-(-F?(x) (5) for all x t X (see [7]). Now, let X be a normed linear space and let Y be a nonvoid convex subset of X. Concerning the convex norm-preserving extension to X of the convex functions in Lip Y, we can prove the following theorem: 236 0021-9045/78/0243~236~02.00/0
The aim of this paper is to study the basic properties of the Thompson metric dT in the general c... more The aim of this paper is to study the basic properties of the Thompson metric dT in the general case of a linear spaces X ordered by a cone K. We show that dT has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of dT , and some results concerning the topology of dT , including a brief study of the dT-convergence of monotone sequences. It is shown most results are true without any assumption of Archimedean-type property for K. One considers various completeness properties and one studies the relations between them. Since dT is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. Thompson metric dT and order-unit (semi)norms | • |u are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although dT and | • |u are only topological (and not metrical) equivalent on Ku, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.
We prove a version of Ekeland Variational Principle (EkVP) in a weighted graph $G$ and its equiva... more We prove a version of Ekeland Variational Principle (EkVP) in a weighted graph $G$ and its equivalence to Caristi fixed point theorem and to Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in terms of the graph $G$. The main tool used in the proof is the OSC property for sequences in a graph. Converse results, meaning the completeness of graphs for which one of these principles holds is also considered.
The paper deals with divergence phenomena for various approxima-
tion processes of analysis such... more The paper deals with divergence phenomena for various approxima-
tion processes of analysis such as Fourier series, Lagrange interpolation,
Walsh-Fourier series. We prove the existence of superdense (meaning
residual, dense and uncountable) families of functions in appropriate
function spaces over an interval T ⊂ R. One proves that for each function
in the family, the corresponding approximation process is unboundedly
divergent on a superdense subset of T of full measure.
The aim of the present paper is to show that many Phelps type duality result, relating the extens... more The aim of the present paper is to show that many Phelps type duality result, relating the extension properties of various classes of functions (continuous, linear continuous, bounded bilinear, Holder-Lipschitz) with the approximation properties of some annihilating spaces, can be derived in a unitary and simple way from a formula for the distance to the kernel of a linear operator, extending the well-known distance formula to hyperplanes in normed spaces. The case of spaces \(c_0\) and \(l^\infty\) is treated in details.
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Papers by Stefan Cobzas
tion processes of analysis such as Fourier series, Lagrange interpolation,
Walsh-Fourier series. We prove the existence of superdense (meaning
residual, dense and uncountable) families of functions in appropriate
function spaces over an interval T ⊂ R. One proves that for each function
in the family, the corresponding approximation process is unboundedly
divergent on a superdense subset of T of full measure.
tion processes of analysis such as Fourier series, Lagrange interpolation,
Walsh-Fourier series. We prove the existence of superdense (meaning
residual, dense and uncountable) families of functions in appropriate
function spaces over an interval T ⊂ R. One proves that for each function
in the family, the corresponding approximation process is unboundedly
divergent on a superdense subset of T of full measure.