On a functional equation by Baak, Boo and Rassias

Abstract

In Baak et al. (J Math Anal Appl 314(1):150–161, 2006) the authors considered the functional equation

$$\begin{aligned} r f\left( \frac{1}{r}\,\sum _{j=1}^{d}x_j\right)+ & {} \sum _{i(j)\in \{0,1\} \atop \sum _{1\le j\le d} i(j)=\ell }r f\left( \frac{1}{r}\,\sum _{j=1}^d (-1)^{i(j)}x_j\right) \\= & {} \left( {d-1\atopwithdelims ()\ell }-{d-1\atopwithdelims ()\ell -1} +1\right) \sum _{j=1}^{d} f(x_j) \end{aligned}$$

where $d,\ell \in \mathbb {N}$, $1<\ell <d/2$ and $r\in \mathbb {Q}{\setminus }\{0\}$. The authors determined all odd solutions $f:X\rightarrow Y$ for vector spaces X, Y over $\mathbb {Q}$. In Oubbi (Can Math Bull 60:173–183, 2017) the author considered the same equation but now for arbitrary real $r\not =0$ and real vector spaces X, Y. Generalizing similar results from (J Math Anal Appl 314(1):150–161, 2006) he additionally investigates certain stability questions for the equation above, but as for that equation itself for odd approximate solutions only. The present paper deals with the general solution of the equation and the corresponding stability inequality. In particular it is shown that under certain circumstances non-odd solutions of the equation exist.