Opuscula Math. 27, no. 1 (2007), 13-24

Opuscula Mathematica

# On Lipschitzian operators of substitution generated by set-valued functions

Jakub Jan Ludew

Abstract. We consider the Nemytskii operator, i.e., the operator of substitution, defined by $$(N \phi)(x):=G(x,\phi(x))$$, where $$G$$ is a given multifunction. It is shown that if $$N$$ maps a Hölder space $$H_{\alpha}$$ into $$H_{\beta}$$ and $$N$$ fulfils the Lipschitz condition then $G(x,y)=A(x,y)+B(x),\tag{1}$ where $$A(x,\cdot)$$ is linear and $$A(\cdot ,y),\, B \in H_{\beta}$$. Moreover, some conditions are given under which the Nemytskii operator generated by $$(1)$$ maps $$H_{\alpha}$$ into $$H_{\beta}$$ and is Lipschitzian.

Keywords: Nemytskii operator, Hölder functions, set-valued functions, Jensen equation.

Mathematics Subject Classification: 39B99, 47H04, 47H30, 54C60.

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• Jakub Jan Ludew
• Silesian University of Technology, Institute of Mathematics, ul. Kaszubska 23, 44-100 Gliwice, Poland