Smooth variational principles in Radon-Nikodým spaces

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.