Smooth variational principles in Radon-Nikodým spaces

Robert Deville, Abdelhakim Maaden

Research output: Contribution to journalArticlepeer-review


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 (cursive Greek chi) ≥ 2a∥cursive Greek chi∥ + b, cursive Greek chi ∈ X, and if X has the Radon-Nikodým property, then for every ε > 0 there exists a real function ρ on X such that ρ is Fréchet differentiable, ∥ρ∥ < ε, ∥ρ′∥ < ε, ρ′ is weakly continuous and f + ρ attains a minimum on X. In addition, if we assume that the norm in X is β-smooth, we can take the function ρ = g 1 + g 2 where g 1 is radial and β-smooth, g 2 is Fréchet differentiable, ∥g 1 < ε, ∥g 2 < ε, ∥g′ 1∞ < ε, ∥g′ 2∥∞ < ε, g′ 2 is weakly continuous and f + g 1 + g 2 attains a minimum on X.

Original languageEnglish (US)
Pages (from-to)109-118
Number of pages10
JournalBulletin of the Australian Mathematical Society
Issue number1
StatePublished - Aug 1999
Externally publishedYes


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