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.


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