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Continuity of hysteresis operators in Sobolev spaces. (English) Zbl 0705.47054

The classical Prandtl, Ishlinskij and Preisach hysteresis oprators are continuous in the spaces C and \(L_ p\) [see e.g., M. A. Krasnosel’skij and A. V. Pokrovskij: Systems with hysteresis, Nauka, Moscow (1983; Zbl 0665.47038)]. In this interesting paper, the authors show that these operators are continuous in \(W^ 1_ p\) for \(1\leq p<\infty\), but not for \(p=\infty\).
Reviewer: J.Appell

MSC:

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58C07 Continuity properties of mappings on manifolds
74H99 Dynamical problems in solid mechanics

Citations:

Zbl 0665.47038
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References:

[1] M. A. Krasnoselskii A. V. Pokrovskii: Systems with hysteresis. (Russian) Moscow, Nauka, 1983.
[2] A. V. Pokrovskii: On the theory of hysteresis nonlinearities. (Russian) Dokl. Akad. Nauk SSSR 210 (1973), no. 6, 1284-1287.
[3] P. Krejčí: On Maxwell equations with the Preisach hysteresis operator: the one-dimensional time-periodic case. Apl. Mat. 34 (1989), 364-374. · Zbl 0701.35098
[4] A. Visintin: On the Preisach model for hysteresis. Nonlinear Anal. T. M. A. 8 (1984), 977-996. · Zbl 0563.35007
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