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$C^{1}$ -smoothness of Nemytskii operators on Sobolev-type spaces of periodic functions

Irina Kmit (2011)

Commentationes Mathematicae Universitatis Carolinae

We consider a class of Nemytskii superposition operators that covers the nonlinear part of traveling wave models from laser dynamics, population dynamics, and chemical kinetics. Our main result is the $C^{1}$ -continuity property of these operators over Sobolev-type spaces of periodic functions.

Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz

N. Merentes, J. L. Sánchez Hernández (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Let (X,∥·∥) and (Y,∥·∥) be two normed spaces and K be a convex cone in X. Let CC(Y) be the family of all non-empty convex compact subsets of Y. We consider the Nemytskiĭ operators, i.e. the composition operators defined by (Nu)(t) = H(t,u(t)), where H is a given set-valued function. It is shown that if the operator N maps the space $R V_{φ ₁} ([a, b]; K)$ into $R W_{φ ₂} ([a, b]; C C (Y))$ (both are spaces of functions of bounded φ-variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t,u(t)) = A(t)u(t)...

Characterization of globally Lipschitzian composition operators in the Banach space ${BV}_{p}^{2} [a, b]$

Janusz Matkowski, Nelson Merentes (1992)

Archivum Mathematicum

We give a characterization of the globally Lipschitzian composition operators acting in the space $B V_{p}^{2} [a, b]$

Common fixed points of biased maps of type $(A)$ and applications.

Pathak, H.K., Cho, Y.J., Kang, S.M. (1998)

International Journal of Mathematics and Mathematical Sciences

Commutators in real interpolation with quasi-power parameters.

Fan, Ming (2002)

Abstract and Applied Analysis

Compactness and existence results for ordinary differential equations in Banach spaces.

Appell, J., Väth, M., Vignoli, A. (1999)

Zeitschrift für Analysis und ihre Anwendungen

Composing infinitely differentiable functions.

J. Aööell, V.I. Nazarov, P.P. Zabrejko (1991)

Mathematische Zeitschrift

Continuity and Fréchet-Differentiability of Nemytskij Operators in Hölder Spaces.

Manfred Goebel (1992)

Monatshefte für Mathematik

Continuity of hysteresis operators in Sobolev spaces

Pavel Krejčí, Vladimír Lovicar (1990)

Aplikace matematiky

We prove that the classical Prandtl, Ishlinskii and Preisach hysteresis operators are continuous in Sobolev spaces $W^{1, p} (0, T)$ for $1 \leq p < + \infty$ , (localy) Lipschitz continuous in $W^{1, 1} (0, T)$ and discontinuous in $W^{1, \infty} (0, T)$ for arbitrary $T > 0$ . Examples show that this result is optimal.

Currently displaying 1 – 12 of 12