Some generalizations of locally and weakly locally uniformly convex space

2011 ◽  
Vol 74 (12) ◽  
pp. 3896-3902 ◽  
Author(s):  
Z.H. Zhang ◽  
C.Y. Liu
Keyword(s):  
Convex Space ◽  
Uniformly Convex ◽  
Uniformly Convex Space

On nearly uniformly convex and k-uniformly convex spaces

1984 ◽  
Vol 95 (2) ◽  
pp. 325-327 ◽  
Author(s):  
V. I. Istrăt‚escu ◽  
J. R. Partington
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Normal Structure ◽  
Uniformly Convex ◽  
Uniformly Convex Space ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

AbstractIn this note we prove that every nearly uniformly convex space has normal structure and that K-uniformly convex spaces are super-reflexive.We recall [1] that a Banach space is said to be Kadec–Klee if whenever xn → x weakly and ∥n∥ = ∥x∥ = 1 for all n then ∥xn −x∥ → 0. The stronger notions of nearly uniformly convex spaces and uniformly Kadec–Klee spaces were introduced by R. Huff in [1]. For the reader's convenience we recall them here.

On Nonreflexive Banach Spaces Which Contain No c0 or lp

1980 ◽  
Vol 32 (6) ◽  
pp. 1382-1389 ◽  
Author(s):  
P. G. Casazza ◽  
Bor-Luh Lin ◽  
R. H. Lohman
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Reflexive Banach Space ◽  
Analytical Description ◽  
Basic Sequence ◽  
Isomorphic Copy ◽  
Uniformly Convex ◽  
Infinite Dimensional ◽  
Symmetric Basis ◽  
Uniformly Convex Space

The first infinite-dimensional reflexive Banach space X such that no subspace of X is isomorphic to c0 or lp, 1 ≦ p < ∞, was constructed by Tsirelson [8]. In fact, he showed that there exists a Banach space with an unconditional basis which contains no subsymmetric basic sequence and which contains no superreflexive subspace. Subsequently, Figiel and Johnson [4] gave an analytical description of the conjugate space T of Tsirelson's example and showed that there exists a reflexive Banach space with a symmetric basis which contains no superreflexive subspace; a uniformly convex space with a symmetric basis which contains no isomorphic copy of lp, 1 < p < ∞; and a uniformly convex space which contains no subsymmetric basic sequence and hence contains no isomorphic copy of lp, 1 < p < ∞. Recently, Altshuler [2] showed that there is a reflexive Banach space with a symmetric basis which has a unique symmetric basic sequence up to equivalence and which contains no isomorphic copy of lp, 1 < p < ∞.

scholarly journals Denseness of operators which attain their numerical radius

1984 ◽  
Vol 36 (1) ◽  
pp. 130-133 ◽  
Author(s):  
I. D. Berg ◽  
Brailey Sims
Keyword(s):  
Linear Operator ◽  
Compact Operator ◽  
Convex Space ◽  
Bounded Linear Operator ◽  
Uniformly Convex ◽  
Numerical Radius ◽  
Bounded Linear ◽  
Uniformly Convex Space ◽  
Small Norm

AbstractWe show that a bounded linear operator on a uniformly convex space may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius.

Uniformly Lipschitzian Families of Transformations in Banach Spaces

1974 ◽  
Vol 26 (5) ◽  
pp. 1245-1256 ◽  
Author(s):  
K. Goebel ◽  
W. A. Kirk ◽  
R. L. Thele
Keyword(s):  
Fixed Point ◽  
Convex Space ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Real Numbers ◽  
Fixed Sequence ◽  
Uniformly Convex Space ◽  
Asymptotically Nonexpansive ◽  
The Fixed Point Theorem ◽  
Image Position

The observations of this paper evolved from the concept of 'asymptotic nonexpansiveness' introduced by two of the writers in a previous paper [10]. Let X be a Banach space and K ⊆ X. A mapping T : K → K is called asymptotically nonexpansive if for each x, y ∊ Kwhere {ki} is a fixed sequence of real numbers such that ki→1 as i → ∞ . It is proved in [10] that if K is a bounded closed and convex subset of a uniformly convex space X then every asymptotically nonexpansive mapping T : K → K has a fixed point. This theorem generalizes the fixed point theorem of Browder-Göhde-Kirk [2 ; 12 ; 16] for nonexpansive mappings (mappings T for which ||T(x) — T(y)|| ≦ ||x — y||, x, y ∊ K) in a uniformly convex space. (A generalization along similar lines also has been obtained by Edelstein [4].)

Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Preeyalak Chuadchawna ◽  
Ali Farajzadeh ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Uniformly Convex Hyperbolic Space

Abstract In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

scholarly journals Improving KKM Theory on General Metric Type Spaces

2021 ◽  
Vol 9 (1) ◽  
pp. 1-12
Author(s):  
Sehie Park
Keyword(s):  
Convex Space ◽  
Metric Spaces ◽  
Space Theory ◽  
Type Space ◽  
Generalized Metric ◽  
Abstract Convex Space ◽  
Type Spaces

Abstract A generalized metric type space is a generic name for various spaces similar to hyperconvex metric spaces or extensions of them. The purpose of this article is to introduce some KKM theoretic works on generalized metric type spaces and to show that they can be improved according to our abstract convex space theory. Most of these works are chosen on the basis that they can be improved by following our theory. Actually, we introduce abstracts of each work or some contents, and add some comments showing how to improve them.

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