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 Mat. Zametki, 2002, Volume 71, Issue 2, Pages 168–181 (Mi mz337)

On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions

K. O. Besov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We obtain sufficient conditions for the continuity of the general nonlinear superposition operator (generalized Nemytskii operator) acting from the space $C^m(\overline \Omega)$ of differentiable functions on a bounded domain $\Omega$ to the Lebesgue space $L_p(\Omega)$. The values of operators on a function $u\in C^m(\overline \Omega)$ are locally determined by the values of both the function $u$ itself and all of its partial derivatives up to order $m$ inclusive. In certain particular cases, the sufficient conditions obtained are proved to be necessary as well. The results are illustrated by several examples, and an application to the theory of Sobolev spaces is also given.

DOI: https://doi.org/10.4213/mzm337

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English version:
Mathematical Notes, 2002, 71:2, 154–165

Bibliographic databases:

UDC: 517.988.5

Citation: K. O. Besov, “On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions”, Mat. Zametki, 71:2 (2002), 168–181; Math. Notes, 71:2 (2002), 154–165

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz337
• https://doi.org/10.4213/mzm337
• http://mi.mathnet.ru/eng/mz/v71/i2/p168

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This publication is cited in the following articles:
1. Besov, KO, “Eigenfunctions of some nonlinear nonlocal operators”, Differential Equations, 38:4 (2002), 510
2. Walczak, J, “Simplified models of nonlinear multipoles in frequency domain”, Przeglad Elektrotechniczny, 85:4 (2009), 227
3. Gulgowski J., “Approximation of Solutions to Second Order Nonlinear Picard Problems with Caratheodory Right-Hand Side”, Cent. Eur. J. Math., 12:1 (2014), 155–166
4. Beldzinski M., Galewski M., Steglinski R., “Solvability of Abstract Semilinear Equations By a Global Diffeomorphism Theorem”, Results Math., 73:3 (2018), UNSP 122
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