The author considers the following functional equation
\[ \begin{split} rf\bigg(\frac{\sum_{j=1}^d x_j}{r}\biggr)+ \sum_{s(j)=0,1; \sum_{j=1}^d s(j)=l} rf \biggl(\frac{\sum_{j=1}^d (-1)^{s(j)}x_j}{r}\biggr)=\\ \biggl[\binom{l}{d-1}-\binom{l-1}d -1+1\biggr] \sum_{j=1}^d f(x_j),\end{split}\tag{*} \]
where \(f:X \to Y\) (\(X\), \(Y\) vector spaces) is an odd mapping, \(r\) is \(1\) or \(3\), and \(d\) and \(l\) are integers with \(1<l<d/2\). After proving that the solutions of equation (*) are the additive functions, some results of stability, in the sense of Ulam-Hyers, are proved. These are then used for proving that some odd bijective mappings from a unital \(C^{*}\)-algebra A into itself satisfying certain inequalities are automorphisms.