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

On Lipschitzian operators of substitution generated by set-valued functions

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.