Normed Linear Spaces Research Articles

Krasnoselskii N-Tupled Fixed Point Theorem with Applications to Fractional Nonlinear Dynamical System

In this paper, N-tupled fixed point theorems for two monotone nondecreasing mappings in complete normed linear space are established. The extension of Krasnoseskii fixed point theorem for a version of N-tupled fixed point is given. Our theoretical results are applied to prove the existence of a mild solution of the system of N-nonlinear fractional evolution equations. Finally, an example of a nonlinear fractional dynamical system is given to illustrate the results.

Rough $I$-statistical convergence of double sequences in normed linear spaces

Rough $I$-statistical convergence of double sequences in normed linear spaces

On some geometric properties of operator spaces

In this article, we study some geometric properties like parallelism, orthogonality, and semirotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces X and Y , assuming X to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces X and Y . We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff–James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semirotund points (semirotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semirotund operators and prove that B ( X , Y ) is a semirotund space which is not strictly convex if X , Y are finite-dimensional Banach spaces and Y is strictly convex.

The Additive Operator Preserving Birkhoff Orthogonal

The Birkhoff orthogonal plays an important role in the geometric study of Banach spaces. It has been con rmed that a Birkhoff orthogonality preserving linear operator between two normed linear spaces must necessarily be a scalar multiple of a linear isometry. In this paper, the author gives a new result that a Birkhoff orthogonality preserving additive operator between two-dimensional normed linear spaces is necessarily a scalar multiple of a linear isometry.

Convexity of Suns in Tangent Directions

A direction $$d$$ is called a tangent direction to the unit sphere $$S$$ if the conditions that $$s \in S$$ and $${\text{lin}}(s + d)$$ is a supporting line to $$S$$ at the point s imply that $${\text{lin}}(s + d)$$ is a semitangent line to S, i.e., is the limit of secants at s. A set M is called convex in a direction $$d$$ if $$x,y \in M$$ and $$(y - x)\parallel d$$ imply that $$[x,y] \subset M$$ . In an arbitrary normed linear space, an arbitrary sun (in particular, a boundedly compact Chebyshev set) is proved to be convex in any tangent direction of the unit sphere.

Множество регулярности многозначного отображения в пространстве с векторнозначной метрикой

We consider multivalued mappings acting in spaces with a vector-valued metric. A vector-valued metric is understood as a mapping satisfying the axioms “of an ordinary metric” with values in the cone of a linear normed space. The concept of the regularity set of a multivalued mapping is defined. A set of regularity is used in the study of inclusions in spaces with a vector-valued metric.

New Fixed Point Theorems via Contraction Mappings in Complete Intuitionistic Fuzzy Normed Linear Space

Contraction mappings provide us with one of the major sources of fixed point theorems. In many mathematical models, the existence of a solution may often be described by the existence of a fixed point for a suitable map. Therefore, study of such mappings and fixed point results becomes well motivated in the setting of intuitionistic fuzzy normed linear spaces (IFNLSs) as well. In this paper, we define some new contraction mappings and establish fixed point theorems in a complete IFNLS. Our results unify and generalize several classical results existing in the literature.

Співвідношення двоїстості та критерії екстремальності елемента для задачі відшукання відстані між двома опуклими множинами лінійного нормованого простору

P. L. Chebyshov started the conception of the best approximation of a continuous function on a segment using algebraic polynomials of some order at the middle of the XIX century. Later, the notion of the best approximation was considered for the case of general linear normed spaces. Over time, it became clear that a many tasks of best approximation are partial consequence of the problem of the best approximation of an element of a linear normed space by a convex set. This task is called the problem of finding the distance from the given element of the linear normed space to the convex plurality of this space as well. Important questions of the study of this problem are the establishment of the relation of duality and the criterion of the extremality of its element, the specification of this relation and the criterion for some partial cases and their application.M. P. Korniichuk and V. M. Tikhomirov established the general relation of duality and criterion of the extremality of an element for the problem of the best approximation of an element of a linear normed space by a convex set based on the dual interrelation.An important problem, the partial case of which is the problem of the best approximation of an element of a linear normed space by a convex set of this space, is the problem of finding the distance between two convex sets of linear normed space, which is considered in this paper. In the article establishes the relation of duality for this problem, which reduces this problem to the problem of evaluationing the upper bound in the conjugate space. The obtained relation is used to obtain the criterion of the extremality of an element. Thes results are used to find the distance between two bullets and between the bullet and the hyperplane of the linear normed space.

Continuity of Bounded Linear Operators on Normed Linear Spaces

Summary In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized. In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization.

Rough {I}_{2}-lacunary statistical convergence of double sequences

In this paper, we introduce and study the notion of rough mathcal {I}_{2}-lacunary statistical convergence of double sequences in normed linear spaces. We also introduce the notion of rough mathcal{I}_{2}-lacunary statistical limit set of a double sequence and discuss some properties of this set.

A generalized orthogonality relation via norm derivatives in real normed linear spaces

We introduce a generalized orthogonality relation in real normed linear spaces via norm derivatives. The relation between this concept and other types of orthogonalities such as Birkhoff–James orthogonality and orthogonality relations connected with norm derivatives is investigated. As an application, some new characterizations of smooth real normed linear spaces are obtained. Moreover, some mapping theorems with respect to this new type of orthogonality are presented.

Jensen\u2019s and Hermite\u2013Hadamard\u2019s Type Inequalities for Lower and Strongly Convex Functions on Normed Spaces

In this paper, we obtain some Jensen’s and Hermite–Hadamard’s type inequalities for lower, upper, and strongly convex functions defined on convex subsets in normed linear spaces. The case of inner product space is of interest since in these case the concepts of lower convexity and strong convexity coincides. Applications for univariate functions of real variable and the connections with earlier Hermite–Hadamard’s type inequalities are also provided.

Uniqueness of completions and related topics

A bounded subset of a normed linear space is said to be (diametrically) complete if it cannot be enlarged without increasing the diameter. A complete super set of a bounded set K having the same diameter as K is called a completion of K. In general, a bounded set may have different completions. We study normed linear spaces having the property that there exists a nontrivial segment with a unique completion. It turns out that this property is strictly weaker than the property that each complete set is a ball, and it is strictly stronger than the property that each set of constant width is a ball. Extensions of this property are also discussed.

Locally Convex Limit Spaces of Measurable Functions with Order Units and Its Duals

We consider linear normed spaces of measurable functions dominated by positive measurable function powered by real positive parameter. Also, we consider its dual and predual, and we propose a method for constructing a limit spaces of these functional spaces taken by power parameter. We prove that these limit spaces are (LF)-spaces and also prove that the limit spaces presume the relation of duality, i.e., the limit space of predual spaces is predual for the limit space of dominated functions, and the limit space of duals is dual for it. Also, the limit space of predual spaces is embedded into the limit space of dual spaces.

On asphericity of convex bodies in linear normed spaces

In 1960, Dvoretzky proved that in any infinite dimensional Banach space X and for any epsilon> 0 there exists a subspace L of X of arbitrary large dimension ϵ-iometric to Euclidean space. A main tool in proving this deep result was some results concerning asphericity of convex bodies. In this work, we introduce a simple technique and rigorous formulas to facilitate calculating the asphericity for each set that has a nonempty boundary set with respect to the flat space generated by it. We also give a formula to determine the center and the radius of the smallest ball containing a nonempty nonsingleton set K in a linear normed space, and the center and the radius of the largest ball contained in it provided that K has a nonempty boundary set with respect to the flat space generated by it. As an application we give lower and upper estimations for the asphericity of infinite and finite cross products of these sets in certain spaces, respectively.

Weak Sharp Minima for Convex Infinite Optimization Problems in Normed Linear Spaces

In the present paper, we study the characterization issue of the weak sharp minima properties for convex infinite optimization problems in normed linear spaces. We develop a new approach to establish several complete geometric characterizations for the global/bounded/local weak sharp minima property, which extend/improve the corresponding ones in this direction by removing/relaxing the key topological assumptions made on the index set. As by-products, some complete characterizations of the global/bounded/local weak sharp minima are obtained for a subset of the level set of a given convex function (not necessarily the level set itself) in terms of the normal cones and the subdifferentials of the involved convex subset and convex function. These characterization results are of independent interest in extending/improving the existing ones on characterizing the weak sharp minima for convex optimization problems.

KKM-type theorems for best proximal points in normed linear space

KKM-type theorems for best proximal points in normed linear space

О МНОЖЕСТВАХ МЕТРИЧЕСКОЙ РЕГУЛЯРНОСТИ ОТОБРАЖЕНИЙ В ПРОСТРАНСТВАХ С ВЕКТОРНОЗНАЧНОЙ МЕТРИКОЙ

Spaces with vector-valued metric are considered. The values of a vectorvalued metric are elements of a cone in some linear normed space. The concept of the set of metric regularity for mapping in spaces with vector-valued metric is formulated. A statement on the stability of the set of metric regularity of a given mapping for its Lipschitz perturbations in spaces with vector-valued metric is obtained.

Riesz Lemma in Normed Quasilinear Spaces and Its an Application

Aseev (Proc Steklov Inst Math 2:23–52, 1986) started a new field in functional analysis by introducing the concept of normed quasilinear spaces which is a generalization of classical normed linear spaces. Then, we introduced the normed proper quasilinear spaces in addition to the notions of regular and singular dimension of a quasilinear space, Cakan and Yilmaz (J Nonlinear Sci Appl 8:816–836, 2015). In this study, we classify the normed proper quasilinear spaces as “solid-floored” and “non solid-floored”. Thus, some properties of normed proper quasilinear spaces become more comprehensible. Also we present the counterpart of classical Riesz lemma in normed quasilinear spaces.

A Note on Topological and Strict Transitivity

Let \(X\) be a normed linear space and \(L(X)\) be the algebra of continuous linear operators on \(X\). We give a necessary condition for topological transitivity of subsets of \(L(X)\) which gives a necessary condition for hypercyclicity and supercyclicity of a single operator on \(X\). Also, we prove the strict transitivity of some particular families of operators on locally convex spaces and Hilbert spaces.