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The superposition operator in Musielak-Orlicz spaces of vector-valfued functions. (English) Zbl 0637.47036

Abstract analysis, Proc. 14th Winter Sch., Srní/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 411-417 (1987).
[For the entire collection see Zbl 0627.00012.]
The author surveys some properties (boundedness, continuity etc.) of the nonlinear superposition operator \(Fx(s)=f(s,x(s))\) in Musielak-Orlicz spaces which generalize the classical Orlicz spaces [see e.g. M. A. Krasnosel’skij, Ya. B. Rutitskij, “Convex functions and Orlicz spaces”, Fizmatgiz Moscow (1958; Zbl 0084.101)] and are examples of so- called modular spaces [see e.g. J. Musielak, “Orlicz spaces and modular spaces”, Lect. Notes Math. 1034 (1983; Zbl 0557.46020)]. Details may be found in two previous papers of the same author [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 25, 321-337 (1985; Zbl 0608.47068), and Bull. Acad. Polon. Sci. Sér. Math. 33, 531-540 (1985; Zbl 0587.46027)]. An application to Hammerstein integral equations is discussed in the last section.
Reviewer: J.Appell

MSC:

47J05 Equations involving nonlinear operators (general)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47H99 Nonlinear operators and their properties
45G10 Other nonlinear integral equations
46E40 Spaces of vector- and operator-valued functions
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