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A note on Picard iterates of nonexpansive mappings

Eun Suk Kim, W. A. Kirk (2001)

Annales Polonici Mathematici

Let X be a Banach space, C a closed subset of X, and T:C → C a nonexpansive mapping. It has recently been shown that if X is reflexive and locally uniformly convex and if the fixed point set F(T) of T has nonempty interior then the Picard iterates of the mapping T always converge to a point of F(T). In this paper it is shown that if T is assumed to be asymptotically regular, this condition can be weakened much further. Finally, some observations are made about the geometric conditions imposed.

A Note on Preserved Smoothness

Tang, Wee-Kee (1996)

Serdica Mathematical Journal

* Supported by NSERC (Canada)Let X be a Banach space equipped with norm || · ||. We say that || · || is Gâteaux differentiable at x if for every h ∈ SX(|| · ||), (∗) lim t→0 (||x + th|| − ||x||) / t exists. We say that the norm || · || is Gâteaux differentiable if || · || is Gâteaux differentiable at all x ∈ SX(|| · ||).

A note on the Hyers-Ulam problem

Yunbai Dong (2015)

Colloquium Mathematicae

Let X,Y be real Banach spaces and ε > 0. Suppose that f:X → Y is a surjective map satisfying | ∥f(x)-f(y)∥ - ∥x-y∥ | ≤ ε for all x,y ∈ X. Hyers and Ulam asked whether there exists an isometry U and a constant K such that ∥f(x) - Ux∥ ≤ Kε for all x ∈ X. It is well-known that the answer to the Hyers-Ulam problem is positive and K = 2 is the best possible solution with assumption f(0) = U0 = 0. In this paper, using the idea of Figiel's theorem on nonsurjective isometries, we give a new proof of...

A note on the structure of WUR Banach spaces

Spiros A. Argyros, Sophocles Mercourakis (2005)

Commentationes Mathematicae Universitatis Carolinae

We present an example of a Banach space E admitting an equivalent weakly uniformly rotund norm and such that there is no Φ : E → c 0 ( Γ ) , for any set Γ , linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, Hájek and Zizler. The space E is actually the dual space Y * of a space Y which is a subspace of a WCG space.

A note on Tsirelson type ideals

Boban Veličković (1999)

Fundamenta Mathematicae

Using Tsirelson’s well-known example of a Banach space which does not contain a copy of c 0 or l p , for p ≥ 1, we construct a simple Borel ideal I T such that the Borel cardinalities of the quotient spaces P ( ℕ ) / I T and P ( ℕ ) / I 0 are incomparable, where I 0 is the summable ideal of all sets A ⊆ ℕ such that ∑ n ∈ A 1 / ( n + 1 ) < ∞ . This disproves a “trichotomy” conjecture for Borel ideals proposed by Kechris and Mazur.

A note on weakly Lindelöf determined Banach spaces

A. González, Vicente Montesinos (2009)

Czechoslovak Mathematical Journal

We prove that weakly Lindelöf determined Banach spaces are characterized by the existence of a ``full'' projectional generator. Some other results pertaining to this class of Banach spaces are given.

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