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Zbl 1087.39027
Chang, Ick-Soon; Jun, Kil-Woung; Jung, Yong-Soo
The modified Hyers-Ulam-Rassias stability of a cubic type functional equation.
(English)
[J] Math. Inequal. Appl. 8, No. 4, 675-683 (2005). ISSN 1331-4343

The functional equation $$\multline f(x+y+2z)+f(x+y-2z)+f(2x)+f(2y)+7f(x)+7f(-x)\\ =2\bigl(f(x+y)+2f(x+z)+2f(x-z)+2f(y+z)+2f(y-z)\bigr) \endmultline\tag{1}$$ of a cubic type (fulfilled e.g. by $f(x)=ax^3+b$) is considered for functions mapping a~real vector space~$X$ into a~Banach space~$Y$. Its general solution is given and the stability in the sense of Hyers, Ulam, Rassias and Găvruta is proved. Instead of the classical method of the Hyers sequence" the so-called fixed point alternative is used in the proof. The desired cubic function near the approximate solution of~(1) is the fixed point of some operator acting on functions $g:X\to Y$ such that $g(0)=0$.
[Szymon Wasowicz (Bielsko-Biała)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains

Keywords: fixed point alternative; cubic functional equation; Banach space

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