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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 .

Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions

Luděk Zajíček (2014)

Commentationes Mathematicae Universitatis Carolinae

We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on Γ -almost everywhere Fréchet differentiability of Lipschitz functions on c 0 (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at Γ -almost every x at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are Γ -almost everywhere Fréchet differentiable....

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