Stability of the entropy equation
The authors prove the stability result of a Pexider-type functional equation stemming from the entropy function. The main result reads as follows. Let \(X\) be a Banach space, \(f:[0,1]\to \mathbb R_{+}\), \(g:[0,1]\to X\) continuous functions satisfying \(f(0)=f(1)=0\), \(g(0)=g(1)=0\), \(\lim_{n\to\infty}nf(1/n)=\infty\) and let \(L:\mathbb R_{+}\to X\) be a continuous mapping. If, for \(\varepsilon>0\), \[ \left\| L\left(\sum_{j=1}^{3}k_jf(p_j)\right)-\sum_{j=1}^{3}k_jg(p_j)\right\| \leq\varepsilon \] for \(0\leq p_j\leq 1\), \(k_j\in\mathbb N_{0}\), \(\sum_{j=1}^{3}k_jp_j=1\), then there exists a unique \(x_0\in X\) such that \[ \| L(r)-rx_0\| \leq 3\varepsilon,\quad r\in\mathbb R_{+} \] and \[ \| g(p)-f(p)x_0\| \leq 12\varepsilon p,\quad p\in [0,1]. \] The result generalizes the one of \textit{Z.\ Dudek} [Demonstr. Math. 34, 641--650 (2001; Zbl 1007.39017)]. A reverse, in some way, result is also proved.
scientific article; zbMATH DE number 2162490