Smooth variational principles in Radon-Nikodým spaces

E-ISSN： 1755-1633|60|1|109-118

ISSN： 0004-9727

Source： Bulletin of the Australian Mathematical Society, Vol.60, Iss.1, 1999-08, pp. : 109-118

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Abstract

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.