On Nemytskij Operator of Substitution in the C1 Space of Set-Valued Functions

Abstract

Abstract. We consider the Nemytskij operator, i.e., the operator of substitution, defined by (Νφ)(χ) := G (χ, φ(χ)), where G is a given multifunction. It is shown that Ν maps C 1 (7, C), the space of all continuously differentiable functions on the interval I with values in a cone C in a Banach space, into C 1 (I, cc(Z)), the space of all continuously differentiable set-functions on I with compact and convex values in a Banach space Ζ and Ν fulfils the Lipschitz condition if and only if the generator G is of the form