## Stability problem for Jensen-type functional equations of cubic mappings.(English)Zbl 1118.39013

The authors introduce the following Jensen type functional equations of cubic mappings between real linear spaces: \begin{aligned} 4f\biggl(\frac{3x+y}{4}\biggr)&+ 4f\biggl(\frac{x+3y}{4}\biggr) =6f\biggl(\frac{x+y}{2}\biggr)+f(x)+f(y),\\ 9f\biggl(\frac{2x+y}{3}\biggr)&+ 9f\biggl(\frac{x+2y}{3}\biggr) =16f\biggl(\frac{x+y}{2}\biggr)+ f(x)+f(y). \end{aligned} The general solution of both functional equations is $$f(x) = C(x)+Q(x)+A(x)+f(0)$$, where $$C$$ is cubic, $$Q$$ is quadratic and $$A$$ is additive, see K.-W. Jun and H.-M. Kim [Math. Inequal. Appl. 6, No. 2, 289–302 (2003; Zbl 1032.39015)]. The authors also prove the generalized Hyers-Ulam-Rassias stability of both equations.

### MSC:

 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges

Zbl 1032.39015
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