scholarly journals A uniformly convex Banach space with a Schauder basis which is subsymmetric but not symmetric

1976 ◽  
Vol 54 (1) ◽  
pp. 207-207 ◽  
Author(s):  
Lakhpat R. Pujara
Keyword(s):  
Banach Space ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Schauder Basis ◽  
Uniformly Convex

scholarly journals Fixed points of mappings satisfying semicontractivity conditions

1992 ◽  
Vol 53 (1) ◽  
pp. 25-38
Author(s):  
Jürgen Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Opial’S Condition ◽  
Asymptotically Nonexpansive

AbstractLet A be a subset of a Banach space E. A mapping T: A →A is called asymptoically semicontractive if there exists a mapping S: A×A→A and a sequence (kn) in [1, ∞] such that Tx=S(x, x) for all x ∈A while for each fixed x ∈A, S(., x) is asymptotically nonexpansive with sequence (kn) and S(x,.) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mpping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided s has a certain asymptoticregurity property.

scholarly journals Approximating fixed points by ishikawa iterates

1989 ◽  
Vol 40 (1) ◽  
pp. 113-117 ◽  
Author(s):  
M. Maiti ◽  
M.K. Ghosh
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex

In a uniformly convex Banach space the convergence of Ishikawa iterates to a fixed point is discussed for nonexpansive and generalised nonexpansive mappings.

Asymptotic centres of filters on uniformly rotund spaces

1976 ◽  
Vol 15 (1) ◽  
pp. 87-96
Author(s):  
John Staples
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Convex Banach Space ◽  
Bounded Sequence ◽  
Uniform Convexity ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Filter Base

The notion of asymptotic centre of a bounded sequence of points in a uniformly convex Banach space was introduced by Edelstein in order to prove, in a quasi-constructive way, fixed point theorems for nonexpansive and similar maps.Similar theorems have also been proved by, for example, adding a compactness hypothesis to the restrictions on the domain of the maps. In such proofs, which are generally less constructive, it may be possible to weaken the uniform convexity hypothesis.In this paper Edelstein's technique is extended by defining a notion of asymptotic centre for an arbitrary set of nonempty bounded subsets of a metric space. It is shown that when the metric space is uniformly rotund and complete, and when the set of bounded subsets is a filter base, this filter base has a unique asymptotic centre. This fact is used to derive, in a uniform way, several fixed point theorems for nonexpansive and similar maps, both single-valued and many-valued.Though related to known results, each of the fixed point theorems proved is either stronger than the corresponding known result, or has a compactness hypothesis replaced by the assumption of uniform convexity.

The Banach-Saks Theorem in C(S)

1974 ◽  
Vol 26 (1) ◽  
pp. 91-97 ◽  
Author(s):  
Nicholas R. Farnum
Keyword(s):  
Banach Space ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex

A Banach space X has the Banach-Saks property if every sequence (xn) in X converging weakly to x has a subsequence (xnk) with (1/p)Σk=1xnk converging in norm to x. Originally, Banach and Saks [2] proved that the spaces Lp (p > 1) have this property. Kakutani [4] generalized their result by proving this for every uniformly convex Banach space, and in [9] Szlenk proved that the space L1 also has this property.

scholarly journals Coarse infinite-dimensionality of hyperspaces of finite subsets

2021 ◽  
Author(s):  
Thomas Weighill ◽  
Takamitsu Yamauchi ◽  
Nicolò Zava
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Dimensional Properties ◽  
Bounded Geometry ◽  
Infinite Dimensional

AbstractWe consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

scholarly journals Best proximity point theorems for weak cyclic Kannan contractions

Filomat ◽  
2011 ◽  
Vol 25 (1) ◽  
pp. 145-154 ◽  
Author(s):  
Ancuţa Petric
Keyword(s):  
Banach Space ◽  
Best Proximity Point ◽  
Convex Banach Space ◽  
Existence Results ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Best Proximity Points ◽  
Kannan Contraction

In this paper we introduce the notion of weak cyclic Kannan contraction. We give some convergence and existence results for best proximity points for weak cyclic Kannan contractions in the setting of a uniformly convex Banach space.

scholarly journals Weak Convergence of the Projection Type Ishikawa Iteration Scheme for Two Asymptotically Nonexpansive Nonself-Mappings

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Tanakit Thianwan
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Iteration Scheme ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Ishikawa Iteration ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Fréchet Differentiable

We study weak convergence of the projection type Ishikawa iteration scheme for two asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach spaceEwhich has a Fréchet differentiable norm or its dualE*has the Kadec-Klee property. Moreover, weak convergence of projection type Ishikawa iterates of two asymptotically nonexpansive nonself-mappings without any condition on the rate of convergence associated with the two maps in a uniformly convex Banach space is established. Weak convergence theorem without making use of any of the Opial's condition, Kadec-Klee property, or Fréchet differentiable norm is proved. Some results have been obtained which generalize and unify many important known results in recent literature.

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