A fixed point approach to almost double derivations and Lie \(*\)-double derivations
The stability problem of functional equations originated from a question of Ulam in 1940, concerning the stability of group homomorphisms. In 1941, D. H. Hyers gave a first affirmative answer to the question for Banach spaces. Let \(A\) be a subalgebra of an algebra \(B\), \(X\) be a \(B\)-module and \(\sigma:A\to B\) be a linear mapping. A linear mapping \(f :A\to B\) is called \(\sigma\)-derivation if \(f(ab)=f(a)\sigma(b)+\sigma(a)f(b)\) for all \(a,b\in A\). Let \(A\) be an algebra and \(\delta,\varepsilon:A\to A\) be linear mappings. \textit{M. Mirzavaziri} and \textit{E. O. Tehrani} [Bull. Iran. Math. Soc. 35, No. 1, 147--154 (2009; Zbl 1189.46058)] said that a linear mapping \(f:A\to A\) is a \((g,h)\)-double derivation if it satisfies \(f(ab)=f(a)b+g(a)h(b)+h(a)g(b)\). In the present paper, the authors by using the fixed point method prove the Hyers-Ulam stability of double derivations associated with the following mapping: \[ \begin{multlined} \sum_{k=2}^n\left(\sum_{i_1=2}^k\sum_{i_2=i_1+1}^{k+1}\dots \sum_{i_{n-k+1}=i_{n-k}+1}^n\right)\\ \times f\left(\sum_{i=1,i\neq i_1,\dots,i_{n-k+1}}^n x_i -\sum_{r=1}^{n-k+1}x_{i_r}\right)+f\left(\sum_{i=1}^n x_i\right)=2^{n-1}f(x_1)\end{multlined} \] for a fixed positive integer \(n\) with \(n\geq 2\). Also they investigate this problem in a Lie \(C^*\)-algebra.
scientific article; zbMATH DE number 6135255
scientific article