## 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 (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 language | English (US) |
---|---|

Pages (from-to) | 109-118 |

Number of pages | 10 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 60 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1999 |

Externally published | Yes |