scholarly journals On the Generalized Riesz Derivative

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1089
Author(s):  
Chenkuan Li ◽  
Joshua Beaudin
Keyword(s):  
Banach Space ◽  
Integral Representation ◽  
Unit Sphere ◽  
Normed Space ◽  
Continuous Mapping ◽  
End Points ◽  
Riesz Derivative ◽  
One To One ◽  
Surface Integrals

The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one and linearly continuous mapping from the normed space W k + 1 ( R ) to the Banach space C ( R ) . In addition, we show that R Z D x 2 s u ( x ) is continuous at the end points and well defined for s = 1 2 + k . Furthermore, we extend the generalized Riesz derivative R Z D x 2 s u ( x ) to the space C k ( R n ) , where k is an n-tuple of nonnegative integers, based on the normalization of distribution and surface integrals over the unit sphere. Finally, several examples are presented to demonstrate computations for obtaining the generalized Riesz derivatives.

scholarly journals The conjugate of a smooth Banach space

1970 ◽  
Vol 2 (3) ◽  
pp. 415-425 ◽  
Author(s):  
D. G. Tacon
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Unit Sphere ◽  
Strong Duality ◽  
Smooth Banach Space ◽  
Supporting Hyperplane ◽  
Linear Map ◽  
Strictly Convex ◽  
One To One ◽  
Smooth Space

A Banach space X is smooth if at every point of the unit sphere there is only one supporting hyperplane of the unit ball; and strictly convex, or rotund, if the unit sphere contains no line segment.Although there is a strong duality between these notions, Klee has produced a smooth space whose conjugate is not rotund. However there is no known example of a smooth space with conjugate not isomorphic to a rotund space.The main purpose of this note is to show that if X is a smooth space with a certain property, X* is isomorphic to a rotund space. This will follow from a mapping theorem which implies the existence of a set Γ and a continuous one-to-one linear map T of X* into co(Γ).

scholarly journals Towards the fixed point property for superreflexive spaces

2001 ◽  
Vol 64 (3) ◽  
pp. 435-444 ◽  
Author(s):  
Andrzej Wiśnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unit Sphere ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometrical Condition

A Banach space X is said to have property (Sm) if every metrically convex set A ⊂ X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

Perturbation of the fixed point problem for continuous mappings

2020 ◽  
pp. 241-249
Author(s):  
Zukhra T. Zhukovskaya ◽  
Sergey E. Zhukovskiy
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Continuous Mapping ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Bounded Subset ◽  
Completely Continuous ◽  
Continuous Map ◽  
Continuous Mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

CLASSES OF SEQUENTIALLY LIMITED OPERATORS

2015 ◽  
Vol 58 (3) ◽  
pp. 573-586
Author(s):  
JAN H. FOURIE ◽  
ELROY D. ZEEKOEI
Keyword(s):  
Banach Space ◽  
Vector Space ◽  
Normed Space ◽  
Operator Ideal ◽  
Operator Ideals ◽  
Ideal Norm ◽  
Relevant Operator

AbstractThe purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals.

Stability of an n-Dimensional Functional Equation in Banach Space and Fuzzy Normed Space

2019 ◽  
pp. 159-181
Author(s):  
Sandra Pinelas ◽  
V. Govindan ◽  
K. Tamilvanan
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Normed Space ◽  
Fuzzy Normed Space

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

An integral representation for selfdecomposable banach space valued random variables

1983 ◽  
Vol 62 (2) ◽  
pp. 247-262 ◽  
Author(s):  
Zbigniew J. Jurek ◽  
Wim Vervaat
Keyword(s):  
Banach Space ◽  
Integral Representation ◽  
Random Variables

scholarly journals A Note on the Topological Group c0

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 77
Author(s):  
Michael Megrelishvili
Keyword(s):  
Banach Space ◽  
Topological Group ◽  
Unit Sphere ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Semigroup Compactification ◽  
Weakly Almost Periodic ◽  
Open Question

A well-known result of Ferri and Galindo asserts that the topological group c 0 is not reflexively representable and the algebra WAP ( c 0 ) of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame ( c 0 ) of tame functions. Respectively, it is an open question if c 0 is representable on a Rosenthal Banach space. In the present work we show that Tame ( c 0 ) is small in a sense that the unit sphere S and 2 S cannot be separated by a tame function f ∈ Tame ( c 0 ) . As an application we show that the Gromov’s compactification of c 0 is not a semigroup compactification. We discuss some questions.

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

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