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Stability of bi-\(\theta \)-derivations on JB\(^{\ast}\)-triples. (English) Zbl 1259.39021

The stability problem of functional equations originated from a question of S. M. Ulam [A collection of mathematical problems. New York and London: Interscience Publishers (1960; Zbl 0086.24101)] concerning the stability of group homomorphisms in metric groups. D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; Zbl 0061.26403)] gave a first affirmative answer to the question of Ulam for Banach spaces. T. M. Rassias [Proc. Am. Math. Soc. 72, 297–300 (1978; Zbl 0398.47040)] provided a generalization of the Hyers’ theorem by considering the stability problem with unbounded Cauchy difference \[ \|f(x+y) - f(x) - f(y)\| \leq \varepsilon (\|x\|^p + \|y\|^p) \] for all \(x,y\in X\), where \(f: X\to Y\) is a mapping from a normed space \(X\) into a Banach space \(Y\), \(\varepsilon >0\) and \(p\in [0, 1)\).
Let \(J\) be a complex \(JB^*\)-algebra, \(\theta : J \to J\) a \(\mathbb{C}\)-linear mapping and let \(D: J\times J \to J\) be a \(\mathbb{C}\)-bilinear mapping. The authors investigate a bi-\(\theta\)-derivation \(D\) on \(J\) defined by \[ \begin{aligned} D(\{x,y,z\}, w ) &= \{ D(x, w), \theta(y), \theta(z)\} + \{\theta(x), D(y, w), \theta(z)\} + \{\theta(x), \theta(y), D(z,w)\}, \\ D(x, \{y,z,w\}) &= \{D(x,y), \theta(z), \theta(w)\} + \{\theta(y), D(x, z), \theta(w)\}+ \{\theta(y), \theta(z), D(x, w)\} \end{aligned} \] for all \(x,y,z,w\in J\).
The authors prove the Hyers-Ulam stability of bi-\(\theta\)-derivations on \(JB^*\)-triples by using the bi-additive functional equation \[ f(x+y, z-w)+f(x-y, z+w) = 2f(x,z) - 2 f(y, w), \] which was introduced by J.-H. Bae and W.-G. Park [Bull. Korean Math. Soc. 47, No. 1, 195–209 (2010; Zbl 1188.39026)].

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
47B48 Linear operators on Banach algebras
17Cxx Jordan algebras (algebras, triples and pairs)
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