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Stability of a cubic functional equation on groups. (English) Zbl 1059.39023
The functional equation \[ f(3x+y)+f(3x-y)=3f(x+y)+3f(x-y)+48f(x)\tag{1} \] is considered for functions mapping an abelian group \(G\) into a Banach space \(X\). Since \(f(x)=cx^3\) fulfils (1), the authors call it a {cubic functional equation} and any its solution the {cubic function}. The general solution of (1) is given. It is of the form \(f(x)=F(x,x)\), where \(F:G\times G\to X\) is such a function that \(F(\cdot,y)\) is additive for any \(y\in G\) and \(F(x,\cdot)\) is quadratic for all \(x\in G\). The generalized Hyers-Ulam-Rassias stability of the equation (1) is established. Namely, it is proved that if \[ \bigl\| f(3x+y)+f(3x-y)-3f(x+y)-3f(x-y)-48f(x)\bigr\| \leq \phi(x,y),\quad x,y\in G, \] where \(\phi:G\times G\to[0,\infty)\) fulfils the conditions \[ \sum_{i=0}^{\infty}\frac{\phi(3^ix,0)}{27^i}<\infty\quad\text{and} \quad\lim\limits_{n\to\infty}\frac{\phi(3^nx,3^ny)}{27^n}=0, \] then the function \[ C(x)=\lim\limits_{n\to\infty}\frac{f(3^nx)}{27^n},\quad x\in G \] is the unique cubic function close to \(f\), i.e. satisfying the inequality \[ \bigl\| f(x)-C(x)\bigr\| \leq \frac{1}{54}\sum_{i=0}^{\infty}\frac{\phi(3^ix,0)}{27^i}, \quad x\in G. \]

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