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Almost contractive retractions in Orlicz spaces

Published online by Cambridge University Press:  17 April 2009

Grzegorz Lewicki  and
Giulio Trombetta
Grzegorz Lewicki
Affiliation:
Department of Mathematics, Jagiellonian University, 30-059 Kraków, Reymonta 4, Poland, e-mail: [email protected]
Giulio Trombetta
Affiliation:
Department of Mathematics, University of Calabria, 87036 Arcavacata di Rende, Cosenza, Italy, e-mail: [email protected]
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Abstract

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Let Bk denote the Euclidean unit ball in ℝκ equipped with the k-dimensional Lebesgue measure and let φ: ℝ+ → ℝ+ be a convex function satisfying φ(0) = 0, φ(t) > 0 for some t > 0. Denote by Eφ = Eφ(Bk) the Orlicz space of finite elements (see (1.6)) generated by φ. The aim of this paper is to show that there exists a retraction of the closed unit ball in Eφ onto the unit sphere in Eφ being a (2 + ɛ)γφ;-set contraction (Theorem 3.6), which generalises [9, Corollary 6] proved for the case of Lp[−1, 1], 1 ≤ p < ∞. Here γφ, denote the Hausdorff measure of noncompactness. This theorem is proved both for the Amemiya and the Luxemburg norms. Also some related results concerning the case of s-convex (0 < s ≤ 1) functions are presented.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 68 , Issue 3 , December 2003 , pp. 353 - 369
Copyright
Copyright © Australian Mathematical Society 2003

References

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