Stability of the entropy equation

Summary.

Let X be a Banach space. We prove the stability of the functional equation

$$ L\left( {\sum\limits_{j = 1}^3 {k_j f(p_j )} } \right) = \sum\limits_{j = 1}^3 {k_j g(p_j )} $$

for $0 \leq p_j \leq 1,\,k_j \in \mathbb{N}_0 = \mathbb{N} \cup \{ 0\} ,\sum\nolimits_{j = 1}^3 {k_j p_j = 1,}$ where $f:[0,1] \to \mathbb{R}_ + ,g:[0,1] \to X$ and $L:\mathbb{R}_ + \to X$ are unknown continuous functions satisfying some additional conditions.

As a corollary we obtain a generalization of the results of Z. Dudek.