# Article

 Title: Notes on approximation in the Musielak-Orlicz sequence spaces of multifunctions  (English) Author: Kasperski, Andrzej Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 36 Issue: 1 Year: 1995 Pages: 19-24 . Category: math . Summary: We introduced the notion of $({\bold X},\operatorname{dist},{\Cal V})$-boundedness of a filtered family of operators in the Musielak-Orlicz sequence space $X_{\varphi }$ of multifunctions. This notion is used to get the convergence theorems for the families of ${\bold X}$-linear operators, ${\bold X}$-dist-sublinear operators and ${\bold X}$-dist-convex operators. Also, we prove that $X_{\varphi }$ is complete. Keyword: Musielak-Orlicz space Keyword: multifunction Keyword: modular space of multifunctions Keyword: approximation Keyword: singular kernel MSC: 28B20 MSC: 41A65 MSC: 46E30 MSC: 54C60 idZBL: Zbl 0832.54021 idMR: MR1334410 . Date available: 2009-01-08T18:15:43Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118728 . Reference: [1] Kasperski A.: Modular approximation by a filtered family of sublinear operators.Commentationes Math. XXVII (1987), 109-114. Zbl 0636.46025, MR 0910964 Reference: [2] Kasperski A.: Modular approximation in $\utilde{X}_{\varphi }$ by a filtered family of $\utilde{X}_{\varphi }$-linear operators.Functiones et Approximatio XX (1992), 183-187. MR 1201727 Reference: [3] Kasperski A.: Modular approximation in $\utilde{X}_{\varphi }$ by a filtered family of dist-sublinear operators and dist-convex operators.Mathematica Japonica 38 (1993), 119-125. MR 1204190 Reference: [4] Kasperski A.: Approximation of elements of the spaces $X^{1}_{\varphi }$ and $X_{\varphi }$ by nonlinear singular kernels.Annales Math. Silesianae, Vol. 6, Katowice, 1992, pp. 21-29. Zbl 0821.41021, MR 1217338 Reference: [5] Kasperski A.: Notes on approximation in the Musielak-Orlicz space of multifunctions.Commentationes Math., in print. Reference: [6] Musielak J.: Modular approximation by a filtered family of linear operators.Functional Analysis and Approximation, Proc. Conf. Oberwolfach, August 9-16, 1980'', BirkhäuserVerlag, Basel 1981, pp. 99-110. Zbl 0471.46017, MR 0650267 Reference: [7] Musielak J.: Orlicz Spaces and Modular Spaces.Lecture Notes in Mathematics, Vol. 1034, Springer-Verlag, Berlin, 1983. Zbl 0557.46020, MR 0724434 .

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