Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded $ \varphi $-Variation in the Sense of Riesz
Volume 52 / 2004
Abstract
Let $ (X,\|\cdot\| )$ and $ (Y,\|\cdot\| )$ 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 Nemytskiĭ operators, i.e. the composition operators 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 $RV_{\varphi_1}([a,b];K)$ into $ RW_{\varphi_2}([a,b];CC(Y))$ (both are spaces of functions of bounded $\varphi$-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 $\varphi_2$-variation in the sense of Riesz. This generalizes results of G. Zawadzka \cite{GZ}, A. Smajdor and W. Smajdor \cite{ASW}, N. Merentes and K. Nikodem \cite{MN}, and N. Merentes and S. Rivas \cite{MS}.
