Real Normed Space Research Articles

Nonlinear strict cone separation theorems in real normed spaces

Nonlinear strict cone separation theorems in real normed spaces

Further remarks on local K-boundedness of K-subadditive set-valued maps

AbstractLet X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map $$F:X\rightarrow \mathcal{C}\mathcal{C}(Y)$$ F : X → C C ( Y ) , for which the functionals $$f_{y^*}(x)=\inf y^*(F(x))$$ f y ∗ ( x ) = inf y ∗ ( F ( x ) ) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional $$y^*$$ y ∗ , has to be locally K-bounded on X. Our results refer to the papers Banakh and Jabłońska (Israel J Math 230:361–386, 2019), Jabłońska and Nikodem (Math Inequal Appl 22:1081–1089, 2019) and Nikodem (Aequationes Math 62:175–183, 2001).

Ordering stability of Nash equilibria for a class of differential games

Abstract This study is concerned with the stability of Nash equilibria for a class of n n -person noncooperative differential games. More precisely, due to a preorder induced by a convex cone on a real linear normed space, we define a new concept called ordering stability of equilibria against the perturbation of the right-hand side functions of state equations for the differential game. Moreover, using the set-valued analysis theory, we present the sufficient conditions of the ordering stability for such differential games.

Study on Orthogonal Sets for Birkhoff Orthogonality

We introduce the notion of orthogonal sets for Birkhoff orthogonality, which we will call Birkhoff orthogonal sets in this paper. As a generalization of orthogonal sets in Hilbert spaces, Birkhoff orthogonal sets are not necessarily linearly independent sets in finite-dimensional real normed spaces. We prove that the Birkhoff orthogonal set A={x1,x2,…,xn}(n≥3) containing n−3 right symmetric points is linearly independent in smooth normed spaces. In particular, we obtain similar results in strictly convex normed spaces when n=3 and in both smooth and strictly convex normed spaces when n=4. These obtained results can be applied to the mutually Birkhoff orthogonal sets studied in recently.

On k-rotundity and k-uniform rotundity in direct sums of normed spaces

A real normed linear space X is k-rotund if and only if for any linearly independent x1,…,xk+1∈X, ‖x1+⋯+xk+1‖<‖x1‖+⋯+‖xk+1‖. We show that X is k-uniformly rotund if and only if the set {∑i=1k+1‖xi‖−‖∑i=1k+1xi‖:x1,…,xk+1∈BX,V(x1,…,xk+1,0)≥ϵ} is bounded below by a positive number for every positive ϵ. Using these characterizations of k-rotundity and k-uniform rotundity, we obtain necessary and sufficient conditions for a space in a class of direct sums of finitely many normed spaces, to be n-rotund and n-uniformly rotund respectively for some n∈N. Moreover, we prove that, given n∈N, the 1-direct sum of normed spaces X1,…,Xd is n-uniformly rotund if and only if there exist ki∈N such that Xi is ki-uniformly rotund for i∈{1,…,d} and n=k1+⋯+kd.

Convex functions and approximate Birkhoff–James orthogonality

AbstractFor the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions.

Componentwise Dinkelbach algorithm for nonlinear fractional optimization problems

The paper deals with fractional optimization problems where the objective function (ratio of two functions) is defined on a Cartesian product of two real normed spaces X and Y. Within this framework, we are interested to determine the so-called partial minimizers, i.e. points in X × Y with the property that any of its variables minimizes the objective function, restricted to this variable, with respect to the other one. While any global minimizer is obviously a partial minimizer, the reverse implication holds true only under additional assumptions (e.g. separability properties of the involved functions). By exploiting the particularities of the objective function, we deliver a Dinkelbach type algorithm for computing partial minimizers of fractional optimization problems. Further assumptions on the involved spaces and functions, such as Lipschitz-type continuity, partial Fréchet differentiability, and coercivity, enable us to establish the convergence of our algorithm to a partial minimizer.

Jacobson’s theorem on derivations of primitive rings with nonzero socle: a proof and applications

We provide a proof of Jacobson’s theorem on derivations of primitive rings with nonzero socle. Both Jacobson’s theorem and its formulation (in terms of the so-called differential operators on left vector spaces over a division ring) underlie our paper. We apply Jacobson’s theorem to describe derivations of standard operator rings on real, complex, or quaternionic left normed spaces. Indeed, when the space is infinite-dimensional, every derivation of such a standard operator ring is of the form Arightarrow AB-BA for some continuous linear operator B on the space. Our quaternionic approach allows us to generalize Rickart’s theorem on representation of primitive complete normed associative complex algebras with nonzero socle to the case of primitive real or complex associative normed Q-algebras with nonzero socle. We prove that additive derivations of the Jordan algebra of a continuous nondegenerate symmetric bilinear form on any infinite-dimensional real or complex Banach space are in a one-to-one natural correspondence with those continuous linear operators on the space which are skew-adjoint relative to the form. Finally we prove that additive derivations of a real or complex (possibly non-associative) H^*-algebra with no nonzero finite-dimensional direct summand are linear and continuous.

Location problem and inner product spaces

In this work we solve a problem that has been open for more than 110 years (see [21]). We prove that a real normed space (X,‖⋅‖) of dimension greater than or equal to three is an inner product space if and only if, for every three points a1,a2,a3∈X, the set of points at which the function x∈X→γ(‖x−a1‖,‖x−a2‖,‖x−a3‖) attains its minimum, intersects the convex hull of these three points, where γ is a symmetric monotone norm on R3.

MIN-PHASE-ISOMETRIES IN STRICTLY CONVEX NORMED SPACES

AbstractSuppose that X and Y are two real normed spaces. A map $f:X\rightarrow Y$ is called a min-phase-isometry if it satisfies $$ \begin{align*} \min\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\min\{\|x+y\|,\|x-y\|\} \quad (x,y\in X). \end{align*} $$ We present properties of min-phase-isometries in the case that Y is strictly convex and show that if a min-phase-isometry f (not necessarily surjective) fixes the origin, then it is phase-equivalent to a linear isometry, that is, $f(x)=\varepsilon (x)g(x)$ for $x\in X$ , where $g:X\rightarrow Y$ is a linear isometry and $\varepsilon $ is a map from X to $\{-1,1\}$ .

Generalized orthogonality equations in finite-dimensional normed spaces

Let X, Y be real normed spaces and let rho '_+, rho '_- be norm derivatives. In this work, we solve a system of functional equations ρ+′(f(x),f(y))=g(x)ρ+′(x,y),ρ-′(f(x),f(y))=g(x)ρ-′(x,y),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll}\\rho '_+(f(x),f(y))=g(x)\\rho '_+(x,y),\\\\ \\rho '_-(f(x),f(y))=g(x)\\rho '_-(x,y), \\end{array}\\right. } \\end{aligned}$$\\end{document}with unknown functions f:X!rightarrow !Y, g:Xrightarrow mathbb {R}. Moreover, we give partial answer to open problem posed in 2010.

Strong m-Convexity of Set-Valued Functions

Abstract In this research we introduce the concept of strong m-convexity for set-valued functions defined on m-convex subsets of real linear normed spaces, a variety of properties and examples of these functions are shown, an inclusion of Jensen type is also exhibited.

A solution to the multidimensional additive homological equation

We prove that, for a finite-dimensional real normed space $V$, every bounded mean zero function $f\in L_\infty([0,1];V)$ can be written in the form $f=g\circ T-g$ for some $g\in L_\infty([0,1];V)$ and some ergodic invertible measure preserving transformation $T$ of $[0,1]$. Our method moreover allows us to choose $g$, for any given $\varepsilon&gt;0$, to be such that $\|g\|_\infty\leq (S_V+\varepsilon)\|f\|_\infty$, where $S_V$ is the Steinitz constant corresponding to $V$.

Some remarks concerning on a fuzzy real normed space

In mathematics, computer science, statistics, and other fields, the notion of fuzzy metric and fuzzy normed is crucial. Finding an acceptable fuzzy metric or norm for making some measurements can solve a lot of difficulties. In the sense of Mohammedali, fuzzy real normed spaces (FRNS) are the issue of this paper. Real normed and fuzzy real normed are advanced together with a ground-breaking analysis of their interactions. The structure of FRNS in terms of families of sublinear functional is established. After investigating some properties of "sublinear functional" that corresponding to an FRN, the concept of the family of star sublinear funtional based on the popular t-conorm (a parameter some families) is introduced with proved that a descending and separating family of star sublinear functional, denoted by produces a FRNS. In addition, the concept of generating space of quasinorm and quasi-norm family (GSQNF) has been presented. Furthermore, the decomposition theorem of a FRN into a family of quasinorm is formulated. The correlation between FRNS and the quasinorm family's generating space is explored and demonstrated.

On semi and quasi γ ρρ-orthogonality in real normed space

This article aims to study some other properties of - orthogonality. We explain the relation between - orthogonality and semi_inner_product_ space ( s_ i_ p_ s) in real norm space. Also we explain the concept of -quasi-inner product space ( - q- i. p. s) and consequences relative to a new notion are studied.

New form of orthogonality in the real normed spaces

This paper aims to introduce a new form of orthogonality in the real normed space. This new orthogonality depends on a norm derivative concept. Some of its substantial characteristics are present, also we use this relation to define -angular property.

Regularly abstract convex functions with respect to the set of Lipschitz continuous concave functions

Given a set of functions defined on a set X, á function is called abstract -convex if it is the upper envelope of its -minorants, i.e. such its minorants which belong to the set ; and f is called regularly abstract -convex if it is the upper envelope of its maximal (with respect to the pointwise ordering) -minorants. In the paper we first present the basic notions of (regular) -convexity for the case when is an abstract set of functions. For this abstract case a general sufficient condition based on Zorn's lemma for a -convex function to be regularly -convex is formulated. The goal of the paper is to study the particular class of regularly -convex functions, when is the set of real-valued Lipschitz continuous classically concave functions defined on a real normed space X. For an extended-real-valued function to be -convex it is necessary and sufficient that f be lower semicontinuous and bounded from below by a Lipschitz continuous function; moreover, each -convex function is regularly -convex as well. We focus on -subdifferentiability of functions at a given point. We prove that the set of points at which an -convex function is -subdifferentiable is dense in its effective domain. This result extends the well-known classical Brøndsted-Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using the subset of the set consisting of such Lipschitz continuous concave functions that vanish at the origin we introduce the notions of -subgradient and -subdifferential of a function at a point which generalize the corresponding notions of the classical convex analysis. Some properties and simple calculus rules for -subdifferentials as well as -subdifferential conditions for global extremum points are established. Symmetric notions of abstract -concavity and -superdifferentiability of functions where is the set of Lipschitz continuous convex functions are also considered.

On Several Bounds for Types of Angular Distances

In this study, we introduce the expression dλ(x,y):=λ∥x∥+(1−λ)∥y∥−∥λx+(1−λ)y∥ on the real normed space X(X,∥·∥), where x,y∈X and λ∈R. We characterize this expression and find various estimates of it. We also obtain a generalization and some refinements of Maligranda’s inequality. Finally, we give some relations between dλ(x,y) and several types of angular distances between two nonzero vectors in a real normed space.

Some Moduli of Angles in Banach Spaces

In this paper, we mainly discuss the angle modulus of convexity δXa(ϵ) and the angle modulus of smoothness ρXa(ϵ) in a real normed linear space X, which are closely related to the classical modulus of convexity δX(ϵ) and the modulus of smoothness ρX(ϵ). Some geometric properties of the two moduli were investigated. In particular, we obtained a characterization of uniform non-squareness in terms of ρXa(1). Meanwhile, we studied the relationships between δXa(ϵ), ρXa(ϵ) and other geometric constants of real normed linear spaces through some equalities and inequalities. Moreover, these two coefficients were computed for some concrete spaces.

Approximate solutions for set optimization with an order cone that has nonempty quasirelative interiors

In a real normed linear space, when the quasirelative interior is not empty, a class of order relation is introduced with Minkowski difference. Two classes of nonlinear functions are introduced, and their properties are discussed. A class of approximately efficient solutions and approximate weakly efficient solutions are introduced for set optimization. With nonlinear functions, optimality conditions are established for approximate solutions. Some examples are given to illustrate our main results.