Moreau-Rockafellar type theorems for nonconvex and non-locally lipschitz integral functional on L p(T,X)

For a separable real Banach space X with separable dual X ∗, we consider the integral functional on over a positive finite measure space (T,Г,μ) where E is a closed subspace. This paper characterizes Moreau-Rockefellar type theorems by the generalized subgradient operator acting on the integral functional F(x). We prove that under appropriate conditions, and show that equality holds if the generalized regularity holds for ft (.) at some pointz(t) ∊ X. This result extends thc locally Lipschitzian case of Clarke’s result and implies several Moreau-Rockafellar type theorems