Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz

Abstract

Summary. Let (X, · ) and (Y, · ) be two normed spaces and K be a convex cone in X. Let CC(Y ) be the family of all non-empty convex compact subsets of Y . We consider the Nemytskiȋ operators, i.e. the composition operators defined by (N u )(t) = H(t, u(t)),where H is a given set-valued function. It is shown that if the operator N maps the space b]; CC(Y )) (both are spaces of functions of bounded ϕ-variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t, u(t)) = A(t)u(t)+B(t), where A(t)is a linear continuous set-valued function and B is a set-valued function of bounded ϕ 2 -variation in the sense of Riesz. This generalizes results of G. Zawadzka [12], A. Smajdor and W. Smajdor [11], N. Merentes and K. Nikodem [5], and N. Merentes and S. Rivas [7].

Nemytskii Operator in the Space of Set-Valued Functions of Bounded <i>φ</i>-Variation

“…N While in [8], we generalize article [6] by introducing a weight function. Now, we intend to generalize [7] in a similar form we did in [8], i.e., the propose of this paper is proving an analogous result in which the Nemytskii operator maps the space N    , ;  . The first such theorem for singlevalued functions was proved in [2] on the space of Lipschitz functions.…”

Funciones regulares en el plano y operadores de Nemytskii

“…In [6] Josephy proved that the autonomous Nemytskii operator generated by h : R → R is a self mapping of BV ([a, b]; R) if and only if h is locally Lipschitz on R. Subsequently, several authors proved this result for many other functions spaces (see [1,8,9,10,11,12], for example).…”

SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BV2α(I)

“…Subsequently, parallel results have been proved in various spaces of functions of bounded variation (cf. [15,16,17]). …”