Differentiability of strongly paraconvex vector-valued functions

Abstract

In the paper the notion of strongly $\alpha(\cdot)$-$K$-paraconvex functions is introduced. It is shown that a strongly $\alpha(\cdot)$-$K$-paraconvex function defined on a convex set contained in a Banach space $X$ with values in $\mathbb{R}^n$ is: (a) Fréchet differentiable on a dense $G_{\delta}$-set provided $X$ is an Asplund space, (b) Gateaux differentiable on a dense $G_{\delta}$-set provided $X$ is separable.