Mean value theorems for correspondences (Q696140)

Starting from some properties satisfied by the normal cone \(N(S,x)\) to a subset \(S\) of a real Banach space \(X\) and \(x\in X\), the author defines axiomatically the notion of normal sprout \(\hat N^*(S,x)\). A typical example of a normal sprout is \(\hat N^*(S,x) = \partial d_S(x), \) where \(d_S\) denotes the distance function to the set \(S\). He defines also the notion of coderivative sprout of a multimapping \(F:X \rightrightarrows Y, X,Y \) Banach spaces, as a natural extension of that of coderivative, and proves some mean value results expressing, in essence, that the multimapping \(F\) satisfies locally a Lipschitz type condition with respect to the Pompeiu-Hausdorff metric. The paper ends with some open problems.