Publ. Math. Debrecen 47 / 1-2 (1995), 15–27 On Nemytskii operator in the space of set-valued functions of bounded p-variation in the sense of Riesz By N. MERENTES (Caracas) and S. RIVAS (Caracas) Abstract. We sonsider the Nemytskii operator, i.e. the composition operator defined by (Nu)(t)= H(t, u(t)), where H is a given set-valued function. It is shown that if the operator N maps the space of set-valued functions of bounded p-variation in the sense of Riesz into the space of set-valued functions of bounded q-variation in the sense of Riesz, there is 1 ≤ q ≤ p< ∞, and if it is globally Lipschitzian, then it has to be of the form (Nu)(t)= A(t)u(t)+ B(t), where A(t) are linear continuous set-valued and B is a set-valued function of bounded q-variation in the sense of Riesz. This generalizes results of G. [8], A. and W. [7], N. and K. [3]. Introduction In [7] A. Smajdor and W. Smajdor proved that every Nemytskii operator N , i.e. (Nu)(t)= H (t, u(t)) mapping the space Lip([a, b], cc(Y )) into itself and globally Lipschitzian has to be of the form (Nu)(t)= A(t)u(t)+ B(t), u ∈ Lip([a, b], cc(Y )), t ∈ [a, b], where A(t) are linear continuous set-valued functions and B is a set- valued function belonging to the space Lip([a, b]), cc(Y )). For the first time a theorem of such a type for single-valued functions was proved by J. Matkowski [1] in the space of Lipschitz functions. Similar charac- terizations of the Nemytskii operator have been also obtained by G. Za- wadzka (see [8]) in the space of set-valued functions of bounded variation in the classical Jordan sense. For single-valued functions it was proved by Mathematics Subject Classification : 47H99, 54C60, 39B70. Key words and phrases : Nemytskii operator, Composition operator, p-variation in the sense of Riesz, set-valued function.