Displaying similar documents to “On Nemytskii Lipschitzian operator”

Point derivations for Lipschitz functions andClarke's generalized derivative

Vladimir Demyanov, Diethard Pallaschke (1997)

Applicationes Mathematicae

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Clarke’s generalized derivative f 0 ( x , v ) is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed x ∈ X and fixed v ∈ E the function f 0 ( x , v ) is continuous and sublinear in f ∈ L i p ( X , d ) . It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz’s product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given. ...

On local convexity of nonlinear mappings between Banach spaces

Iryna Banakh, Taras Banakh, Anatolij Plichko, Anatoliy Prykarpatsky (2012)

Open Mathematics

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We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.

Averages of uniformly continuous retractions

A. Jiménez-Vargas, J. Mena-Jurado, R. Nahum, J. Navarro-Pascual (1999)

Studia Mathematica

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Let X be an infinite-dimensional complex normed space, and let B and S be its closed unit ball and unit sphere, respectively. We prove that the identity map on B can be expressed as an average of three uniformly retractions of B onto S. Moreover, for every 0≤ r < 1, the three retractions are Lipschitz on rB. We also show that a stronger version where the retractions are required to be Lipschitz does not hold.