## Approximation of elements of the spaces $$X_ \varphi^ 1$$ and $$X_ \varphi$$ by nonlinear, singular kernels.(English)Zbl 0821.41021

Summary: Let $$l^ \varphi$$ be a Musielak-Orlicz sequence space. Let $$X^ 1_ \varphi$$ and $$X_ \varphi$$ be the modular spaces of multifunctions generated by $$l^ \varphi$$. Let $$K_{w,j} : \mathbb{R} \to \mathbb{R}$$ for $$j = 0,1,2, \dots, w \in \mathbb{W}$$, where $$\mathbb{W}$$ is an abstract set of indices. Assuming certain singularity assumption on the nonlinear kernel $$K_{w,j}$$ and setting $$T_ w (F) = (T_ w(F)(i))^ \infty_{i = 0}$$ with $$T_ w (F)) (i) = \{\sum^ i_{j = 0} K_{w,j} (f(j)) : f(j) \in F(j)\}$$, convergence theorems $$T_ w (F) @>>\varphi, {\mathcal W}>F$$ in $$X^ 1_ \varphi$$ and $$T_ w (F)@>>d, \varphi, {\mathcal W}>F$$ in $$X_ \varphi$$ are obtained.

### MSC:

 41A35 Approximation by operators (in particular, by integral operators) 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 54C60 Set-valued maps in general topology