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Smooth variational principles in Radon-Nikodým spaces

Published online by Cambridge University Press:  17 April 2009

Robert Deville  and
Abdelhakim Maaden
Robert Deville
Affiliation:
Université Bordeaux 1Faculté des SciencesLaboratoire de Mathématiques pures351 cours de la Libération33400 TalenceFrance
Abdelhakim Maaden
Affiliation:
Université Cadi-AyyadFaculté des Sciences et TechniquesDepartement de MathematiquesB.P. 523 Béni-MellalMaroc
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Abstract

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We prove that if f is a real valued lower semicontinuous function on a Banach space X, for which there exist a > 0 and b ∈ ℝ such that f(x) ≥ 2a∥x∥ + b, x ∈ X, and if X has the Radon-Nikody´m property, then for every Ε > 0 there exists a real function φ X such that φ is Fréchet differentiable, ∥φ∥∞ < Ε, ∥φ′∥∞ < Ε, φ′ is weakly continuous and f + φ attains a minimum on X. In addition, if we assume that the norm in X is β-smooth, we can take the function φ = g1 + g2 where g1 is radial and β-smooth, g2 is Fréchet differentiable, ∥g1∥∞ < Ε, ∥g2∥∞ < Ε, ∥g′1∥∞ < Ε, ∥g′1∥∞ < Ε, g′2 is weakly continuous and f + g1 + g2 attains a minimum on X.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 1 , August 1999 , pp. 109 - 118
Copyright
Copyright © Australian Mathematical Society 1999

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