Convolution of Robust Functions

Recently Zheng [1,2], in the setting of global optimization, introduced the concepts of robust set and robust function as a generalization of open set and upper semicontinuous (u.s.c) function, respectively. The aims of this paper are to study the structure of robust sets defined on a normed space $X$ as well as to extend some multivalued convergence results obtained by the author in [3,4] and Greco et al. [6] for semicontinuous functions to the class of robust functions. More precisely, we introduce the concepts of level-convergence and epigraphic convergence on $\mathcal{R}(X)$ the space of nonnegative robust functions on a normed space $X$ and, on one hand, we study its properties and relationships, and on the other, we present some results on level-approximation and epi-approximation of functions by using convolution of robust functions.