Stefan Cobzas

We prove a version of the Ekeland Variational Principle (EkVP) in a weighted graph G and its equivalence to Caristi fixed point theorem and to the 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 prope...

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...

In recent years Lipschitz-type functions have been studied in an extensive manner. Particularly, we have locally Lipschitz functions (both the ball radius and the local Lipschitz constant depend on the point), uniformly locally Lipschitz functions (the ball radius is independent of the point) and Lipschitz in the small functions (both the ball radi...

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. Th...

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 analysi...

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.~Cobza\c{s}, Topology Appl. \textbf{265} (2019), 106831, 22, for quasi-metri...

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 (2020) [24]. 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 pro...

The paper is concerned with compact bilinear operators on asymmetric normed spaces. One proves a Schauder type theorem on the compactness of the conjugate of a compact bilinear operator and one studies the ideal properties of spaces of compact bilinear operators. These extend some results of Ramanujan and Schock, Linear and Multilinear Algebra (198...

The present paper is concerned with Ekeland Variational Principle (EkVP) and its equivalents (Caristi-Kirk fixed point theorem, Takahashi minimization principle, Oettli-Th\'era equilibrium version of EkVP) in quasi-uniform spaces. These extend some results proved by Hamel, Nonlinear Anal. \textbf{62} (2005), 913--924, in uniform spaces, as well as...

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 the present paper is to present various circumstances in which fixed point results imply completeness. For metric spaces this is the cas...

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 sequen...

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 seque...

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...

Geodesic metric spaces are a natural generalization of Riemannian manifolds and provide a suitable setting for the study of problems from various areas of mathematics with important applications. In this chapter we review selected properties of Lipschitz mappings in geodesic metric spaces focusing mainly on certain extension theorems which generali...

In this chapter we introduce several Banach spaces of Lipschitz functions (Lipschitz functions vanishing at a fixed point, bounded Lipschitz functions, little Lipschitz functions) on a metric space and present some of their properties. A detailed study of free Lipschitz spaces is carried out, including several ways to introduce them and duality res...

The main results in this chapter are Aharoni’s theorem (Theorem 7.3.3) on the bi-Lipschitz embeddability of separable metric spaces in the Banach space c0 and a result of Väisälä (Theorem 7.4.6) on the characterization of the completeness of a normed space X by the non-existence of bi-Lipschitz surjections of X onto X ∖{0}. Other results are discus...

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, b...

For the reader’s convenience we collect in this chapter some notions and results used throughout the book. In this part we give references only to some appropriate books where the mentioned results can be found along with references to the original papers were they were first proved.

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 s...

In this chapter we prove the existence of some Lipschitz functions (the analog of Urysohn’s lemma for Lipschitz functions) and of Lipschitz partitions of unity. We also study algebraic operations with Lipschitz functions, sequences of Lipschitz functions, Lipschitz properties for differentiable functions (including a characterization in terms of Di...

We have seen (see Sect. 10.1007/978-3-030-16489-8_1#Sec17) that any Lipschitz function is uniformly continuous. In this chapter we shall present Lipschitz properties of convex functions and convex operators, and equi-Lipschitz properties for families of convex vector-functions. In the vector case, meaning convex functions defined on a locally conve...

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 speci...

In this paper, driven by Behavioral applications to human dynamics, we consider the characterization of completeness in pseudo-quasimetric spaces in term of a generalization of Ekeland’s variational principle in such spaces, and provide examples illustrating significant improvements to some previously obtained results, even in complete metric space...

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 present paper is concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex subset $\Omega$ 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 conti...

Some versions of the fundamental principles of the functional analysis in asymmetric normed spaces – the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Principle – are proved. The proofs are based on an asymmetric version of a lemma of Zabreijko (1969) [12] on the continuity of the countably subadditive functionals. At t...

Let X be a metrizable space. Let FP(Y) and AP(X) be the free paratopological group over X and the free Abelian paratopological group over X, respectively. Firstly, we use asymmetric locally convex spaces to prove that if Y is a subspace of X then AP(Y) is topological subgroup of AP(X). Then, we mainly prove that: (a) if the tightness of AP(X) is co...

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $...

Introduction.- 1. Quasi-metric and Quasi-uniform Spaces. 1.1. Topological properties of quasi-metric and quasi-uniform spaces.- 1.2. Completeness and compactness in quasi-metric and quasi-uniform spaces.- 2. Asymmetric Functional Analysis.- 2.1. Continuous linear operators between asymmetric normed spaces.- 2.2. Hahn-Banach type theorems and the se...

An asymmetric seminorm is a positive sublinear functional p on a real vector space X. If p(x) = p(−x) = 0 implies x = 0, then p is called an asymmetric norm. The conjugate asymmetric seminorm is given by p(x) = p(−x), x ∈ X, and ps = p ∨ p is a seminorm, respectively a norm on X if p is an asymmetric norm. An important example is the asymmetric nor...

The first chapter of the book is concerned with the topological properties of quasimetric, quasi-uniform, asymmetric normed and asymmetric locally convex spaces. A quasi-metric on a set X is a positive function ρ on X × X satisfying all the axioms of a metric excepting symmetry: it is possible that ρ(y, x) ≠ ρ(x, y) for some x, y ∈ X. Similarly, a...

In this paper we prove two versions of Ekeland Variational Principle in asymmetric locally convex spaces. The first one is based on a version of Ekeland Variational Principle in asymmetric normed spaces proved in S. Cobzaş, Topology Appl. 158 (8) (2011) 1073–1084. For the proof we need to study the completeness with respect to the asymmetric norm p...

In this paper we prove a quasi-metric version of Ekeland Variational Principle and study its connections with the completeness properties of the underlying quasi-metric space. The equivalence with Caristi–Kirkʼs fixed point theorem and a proof of Clarkeʼs fixed point theorem for directional contractions within this framework are also considered.

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 similaritie...

The aim of the present paper is to study precompactness and compactness within the framework of asymmetric locally convex spaces, defined and studied by the author in [S. Cobzaş, Asymmetric locally convex spaces, Int. J. Math. Math. Sci. 2005 (16) (2005) 2585–2608]. The obtained results extend some results on compactness in asymmetric normed spaces...

A classical theorem of S. Mazur and S. Ulam asserts that any surjective isometry between two normed spaces is an affine mapping. D. Mushtari proved in 1968 the same result in the case of random normed spaces in the sense of A. Sherstnev. The aim of the present paper is to show that the result holds also for the probabilistic normed spaces as define...

The aim of the present paper is to define compact operators on asymmetric normed spaces and to study some of their properties. The dual of a bounded linear operator is defined and a Schauder type theorem is proved within this framework. The paper contains also a short discussion on various completeness notions for quasi-metric and for quasi-uniform...

In the appendix to the book by F. F. Bonsal, Lectures on Some Fixed Point Theorems of Functional Analysis (Tata Institute, Bombay, 1962) a proof by Singbal of the Schauder-Tychonoff fixed point theorem, based on a locally convex variant of Schauder mapping method, is included. The aim of this note is to show that this method can be adapted to yield...

The aim of the present paper is to prove that the family of all closed nonempty subsets of a complete probabilistic metric space L is complete with respect to the probabilistic Pompeiu-Hausdorff metric H. The same is true for the families of all closed bounded, respectively compact, nonempty subsets of L. If L is a complete random normed space in t...

The aim of the present paper is to introduce the asymmetric locally convex spaces and to prove some basic properties. Among these I do mention the analogs of the Eidelheit-Tuckey separation theorems, of the Alaoglu-Bourbaki theorem on the weak compactness of the polar of a neighborhood of 0, and a Krein-Milman-type theorem. These results extend tho...

The aim of this paper is to present some generic existence results for nearest and farthest points in connection with some geometric properties of Banach spaces.

The aim of the present paper is to show that some results in Banach spaces have their analogs in spaces with asymmetric seminorms. A space with asymmetric seminorm is a pair (X, p),where X is a real vector space and p a positive sublinear functional onX. Due to the asymmetry of p (it is possible that p(- x) ≠ p(x) for somex), there are differences...

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The present paper is concerned with the characterization of the elements of best approximation in a subspace Y of a space with asymmetric norm, in terms of some linear functionals vanishing on Y. The approach is based on some extension results, proved in Section 3, for bounded linear functionals on such spaces. Also, the well known formula for the...

The aim of this paper is to show that the Lipschitz adjoint of a Lipschitz mapping F, defined by I. Sawashima [Lect. Notes Econ. Math. Syst. 419, 247–259 (1995; Zbl 0942.47052)], corresponds in a canonical way to the adjoint of a linear operator associated to F.

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, Hölder-Lipschitz) with the approximation properties of some annihilating spaces, can be derived in a unitary and simple way from a formula for the distance...

The aim of the present paper is to present a short survey of the basic properties of Šerstnev random normed spaces, with emphasis on best approximation problems in such spaces.

A closed nonvoid subset Z of a Banach space X is called antiproximinal if no point outside Z has a nearest point in Z. The aim of the present paper is to prove that, for a compact Hausdorff space T and a real Banach space E, the Banach space C(T, E), of all continuous functions defined on T and with values in E, contains an antiproximinal bounded c...

The aim of this paper is to prove a compactness criterion in spaces of Lipschitz and Fréchet differentiable mappings.

Let X be a real Banach space, Z a closed nonvoid subset of X, and J: Z --> R a lower semicontinuous function bounded from below. If X is reflexive and has the Kadets property then the set of all x is an element of X for which there exists z(0) is an element of Z such that J(z(0)) + parallel to x - z(0)parallel to = inf{J(z) + parallel to x - z para...

The paper deals with divergence phenomena for various approximation 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 $T\subset \mathbb R.$ One proves that for...

The aim of the present paper is to survey the results known by the author on so-called antiproximinal sets in Banach spaces.

The paper investigates the relationship between the extension properties of bounded bilinear functionals and the approximation properties in 2-normed spaces.

The aim of this note is to relax a necessary condition under which a set Y is convex; namely to show that if it has an algebraically interior point, is closed and totally supported, then it is convex. A new proof of a I. Muntean’s result is given.

Given a family of linear continuous mappings of a topological vector space X into another topological vector space Y, the set A of singularities for is defined as the set of all x in X for which A is an unbounded set in Y. The following general principle of condensation of singularities for nonequicontinuous families is obtained: If either X is Hau...

We present some results on the generic existence of solutions for some optimization problems in Banach spaces – the problem of best approximation by elements of closed sets, the problem of farthest point in closed bounded subsets, and their perturbed analogues.

Let f(x)=∑ i,j=1 n a i,j x i x j +2∑ i=1 n a i,n+1 x i +a n+1,n+1 be a second degree polynomial function of n real variables, where a i,j =a j,i , for all i,j, 1≤i,j≤n. Denote by Δ the determinant Δ=det[a i,j ] 1≤i,j≤n+1 · Expanding Δ with respect to the last row, we obtain Δ=∑ i=1 n+1 (-1) n+1+i a n+1,i δ i , where δ i is the determinant of order...

The aim of the present note is to give a new proof of the fact that a pointwisely bounded family of continuous convex mappings defined on an open convex subset Ω of a barrelled locally convex space and with values in a locally convex space, ordered by a normal cone, is locally equi-Lipschitz on Ω and equi-Lipschitz on every compact subset of Ω.