Browsing by Subject "multifunkcja lipschitzowska"

Now showing 1 - 1 of 1

Access status: Open Access ,
O operatorach złożenia generowanych przez multifunkcje
(Data obrony: 2009) Ludew, Jakub Jan
Wydział Matematyki Stosowanej
For a given function $h : I \times \mathbb{R} \rightarrow \mathbb{R}, I=[0,1]$ the mapping $H : \mathbb{R} \rightarrow \mathbb{R}$, defined by $(H\varphi(x) = h(x,\varphi (x)), \varphi \in \mathbb{R}^{I}$ is called a composition operator of a generator $h$. In 1982 J. Matkowski proved that a composition operator mapping the Banach space Lip(I) of Lipschitzian functions $\varphi : I \rightarrow \mathbb{R}$ into itself is globally Lipschitzian if and only if there exist functions $a, b \in Lip(I)$ such that postać $h(x,y) = a(x)y + b(x), x \in I, y \in \mathbb{R}$. Then this result has been extended to some other function Banach spaces in papers by J. Matkowski and his students. Composition operators generated by multifunctions in the spaces of Lipschitzian and bounded variation functions and multifunctions have been studied by A. Smajdor, W. Smajdor, G. Zawadzka and others.The main goal of the doctoral thesis is to examine composition operators generated by set-valued functions. In my paper I prove that Lipschitzian composition operators acting in the function spaces of: functions satisfying the Hölder condition, functions of $C^{1}$, absolutely continuous and continuous functions of bounded variation with values in corresponding multifunction spaces have to be generated by a set-valued function of the form $H(x,y)=A(x,y) + B(x)$, where $A(x, \cdot)$ is continuous linear multifunction, and $B, A(\cdot ,y)$ belong to the corresponding multifunction spaces mentioned above. The doctoral thesis also contains opposite theorems. Two of them strengthen respective results in the papers of A. Smajdor, W. Smajdor and G. Zawadzka. The basic tool is Rådström's embedding theorem.