Paraconvex analysis on -manifolds†

Abstract

Let be real Banach spaces. Let f(·) be a real-valued locally uniformly approximate convex function defined on an open subset . Let be an open subset. Let σ (·) be a differentiable mapping of ΩX into ΩY such that the differentials of are locally uniformly continuous function of x. Then f(σ (·)) is also a locally uniformly approximate convex function. Therefore the function f(σ (·)) is Fréchet differentiable on a dense G δ-set, provided X is an Asplund space, and Gateaux differentiable on a dense G δ-set, provided X is separable. As a consequence, we obtain that a locally uniformly approximate convex function defined on a -manifold is Fréchet differentiable on a dense G δ-set, provided is an Asplund space, and Gateaux differentiable on a dense G δ-set, provided E is separable. †Dedicated to Professor Diethard Pallaschke on the occasion of his 65th birthday.