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The generalized cubic functional equation and the stability of cubic Jordan \(*\)-derivations

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Abstract

In the current work, we obtain the general solution of the following generalized cubic functional equation

$$\begin{aligned}&f(x+my)+f(x-my)\\&\quad =2\left( 2\cos \left( \frac{m\pi }{2}\right) +m^2-1\right) f(x)-\frac{1}{2}\left( \cos \left( \frac{m\pi }{2}\right) +m^2-1\right) f(2x)\\&\qquad +m^2\{f(x+y)+f(x-y)\} \end{aligned}$$

for an integer \(m \ge 1\). We prove the Hyers–Ulam stability and the superstability for this cubic functional equation by the directed method and a fixed point approach. We also employ the mentioned functional equation to establish the stability of cubic Jordan \(*\)-derivations on \(C^*\)-algebras and \(JC^*\)-algebras.

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Acknowledgments

The authors express their sincere thanks to the reviewers for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially.

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Correspondence to Abasalt Bodaghi.

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Bodaghi, A., Moosavi, S.M. & Rahimi, H. The generalized cubic functional equation and the stability of cubic Jordan \(*\)-derivations. Ann Univ Ferrara 59, 235–250 (2013). https://doi.org/10.1007/s11565-013-0185-9

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Keywords

  • Banach algebra
  • Cubic derivation
  • Cubic functional equation
  • Hyers–Ulam stability
  • Superstability

Mathematics Subject Classification (2010)

  • 39B52
  • 39B72
  • 46L05
  • 47B47