# The generalized cubic functional equation and the stability of cubic Jordan $$*$$ ∗ -derivations

The generalized cubic functional equation and the stability of cubic Jordan $$*$$ ∗ -derivations 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} f ( x + m y ) + f ( x - m y ) = 2 2 cos m π 2 + m 2 - 1 f ( x ) - 1 2 cos m π 2 + m 2 - 1 f ( 2 x ) + m 2 { f ( x + y ) + f ( x - y ) } for an integer $$m \ge 1$$ m ≥ 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 $$C^*$$ C ∗ -algebras and $$JC^*$$ J C ∗ -algebras. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ANNALI DELL'UNIVERSITA' DI FERRARA Springer Journals

# The generalized cubic functional equation and the stability of cubic Jordan $$*$$ ∗ -derivations

, Volume 59 (2) – Jul 13, 2013
16 pages      /lp/springer-journals/the-generalized-cubic-functional-equation-and-the-stability-of-cubic-bMkVUJZREe
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Analysis; Geometry; History of Mathematical Sciences; Numerical Analysis; Algebraic Geometry
ISSN
0430-3202
eISSN
1827-1510
DOI
10.1007/s11565-013-0185-9
Publisher site
See Article on Publisher Site

### 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} f ( x + m y ) + f ( x - m y ) = 2 2 cos m π 2 + m 2 - 1 f ( x ) - 1 2 cos m π 2 + m 2 - 1 f ( 2 x ) + m 2 { f ( x + y ) + f ( x - y ) } for an integer $$m \ge 1$$ m ≥ 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 $$C^*$$ C ∗ -algebras and $$JC^*$$ J C ∗ -algebras.

### Journal

ANNALI DELL'UNIVERSITA' DI FERRARASpringer Journals

Published: Jul 13, 2013

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