On Nemytskij operator of substitution in the C¹ space of set-valued functions

We consider the Nemytskij operator, i.e., the operator of substitution, defined by (Nφ)(x):= G(x, φ(x)), where G is a given multifunction. It is shown that N maps C¹(7, C), the space of all continuously differentiable functions on the interval I with values in a cone C in a Banach space, into C¹(7, cc(Z)), the space of all continuously differentiable set-functions on I with compact and convex values in a Banach space Z and N fulfils the Lipschitz condition if and only if the generator G is of the form (Equation presented) where A(x,) is continuous, linear function, A(.,y) and B are continuously differentiable and the function x → A(x,.) is Lipschitzian.