naa a22 4500 510811523 CHVBK 20180411083444.0 cr unu---uuuuu 180411e20130401xx s 000 0 eng 10.1134/S0081543813010136 doi (NATIONALLICENCE)springer-10.1134/S0081543813010136 Rigidity and stability of the Leibniz and the chain rule [Elektronische Daten] [Hermann König, Vitali Milman] We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T 1, T 2,A: C k (ℝ) → C(ℝ) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ C k (ℝ), namely, V (f · g) = (T 1 f) · g + f · (T 2 g) for k = 1, V (f · g) = (T 1 f) · g + f · (T 2 g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T 1 f) ○ g · (T 2 g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T 1 f) · (Ag) + (Af) · (T 2 g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T 1 and T 2 must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · Tg + B(f, g), and show under suitable conditions on B in the first case that B = 0 and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation. Pleiades Publishing, Ltd., 2013 König Hermann Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, D-24118, Kiel, Germany aut Milman Vitali School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel aut Proceedings of the Steklov Institute of Mathematics SP MAIK Nauka/Interperiodica 280/1(2013-04-01), 191-207 0081-5438 280:1<191 2013 280 11501 https://doi.org/10.1134/S0081543813010136 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1134/S0081543813010136 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- König Hermann Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, D-24118, Kiel, Germany aut NATIONALLICENCE 700 1- Milman Vitali School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel aut NATIONALLICENCE 773 0- Proceedings of the Steklov Institute of Mathematics SP MAIK Nauka/Interperiodica 280/1(2013-04-01), 191-207 0081-5438 280:1<191 2013 280 11501 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer