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Reflexivity and approximate fixed points

Eva Matoušková, Simeon Reich (2003)

Studia Mathematica

A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence x n k for which s u p f ∈ S X * l i m s u p k → ∞ f ( x n k ) is attained at some f in the dual unit sphere S X * . A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every f ∈ S X * , there exists g ∈ S X * such that l i m s u p n → ∞ f ( x ₙ ) < l i m i n f n → ∞ g ( x ₙ ) . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded closed convex...

Remarks on an article of J.P. King

Heiner Gonska, Paula Piţul (2005)

Commentationes Mathematicae Universitatis Carolinae

The present note discusses an interesting positive linear operator which was recently introduced by J.P. King. New estimates in terms of the first and second modulus of continuity are given, and iterates of the operators are considered as well. For general King operators the second moments are minimized.

Remarks on fixed points of rotative Lipschitzian mappings

Jarosław Górnicki (1999)

Commentationes Mathematicae Universitatis Carolinae

Let C be a nonempty closed convex subset of a Banach space E and T : C → C a k -Lipschitzian rotative mapping, i.eṡuch that ∥ T x - T y ∥ ≤ k · ∥ x - y ∥ and ∥ T n x - x ∥ ≤ a · ∥ x - T x ∥ for some real k , a and an integer n > a . The paper concerns the existence of a fixed point of T in p -uniformly convex Banach spaces, depending on k , a and n = 2 , 3 .

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