New regularity conditions and Fenchel dualities for DC optimization problems involving composite functions

We consider the following DC composite optimization problem (P): inf(x is an element of X) {f(1)(x) - f(2)(x) + g(1)(h(x)) - g(2)(h(x))}, where f(1), f(2) and g(1), g(2) are proper convex functionals defined on locally convex Hausdorff topological vector spaces X and Y respectively, and h is a proper K-convex mapping from X to Y. By using the properties of the epigraph of the conjugate functions, we introduce some new regularity conditions and obtain complete characterizations for weak / strong / stable Fenchel dualities and for zero / stable zero duality gap properties of problem (P).