On Nemytskij operator in the space of absolutely continuous set-valued functions

We consider the Nemytskij operator, defined by (Nφ)(x):=G(x, φ(x), where G is a given set-valued function. It is shown that if N maps AC (I, C), the space of all absolutely continuous functions on the interval I:= [0,1] with values in a cone C in a reflexive Banach space, into AC(I, K), the space of all absolutely continuous set-valued functions on I with values in the set K, consisting of all compact intervals (including degenerate ones) on the real line ℝ, and N is uniformly continuous, then the generator G is of the form G(x, y) = A(x)(y) + B(x), where the function A(x) is additive and uniformly continuous for every x ∈ I and, moreover, the functions x → A(x)(y) and B are absolutely continuous. Moreover, a condition, under which the Nemytskij operator maps the space AC(I, C) into AC(I, K) and is Lipschitzian, is given.