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The modified Hyers-Ulam-Rassias stability of a cubic type functional equation. (English) Zbl 1087.39027

The functional equation

f(x+y+2z)+f(x+y-2z)+f(2x)+f(2y)+7f(x)+7f(-x)=2f ( x + y ) + 2 f ( x + z ) + 2 f ( x - z ) + 2 f ( y + z ) + 2 f ( y - z )(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:XY such that g(0)=0.

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges