Scalarization of the Normal Fréchet Regularity of Set–valued Mappings

Let M be a set–valued mapping defined between two Banach spaces E and F . Several important aspects of behavior of M can be characterized in terms of the distance function to images ∆M defined by ∆M (x, y) := d ( y, M(x) ) for all (x, y) ∈ E × F . In this paper, we use this function to scalarize the Fréchet normal regularity of set–valued mappings. The Fréchet subdifferential regularity of ∆M is also investigated for points outside of the graph. An application to the metric regularity for set–valued mappings is given.