On the stability of the translation equation and dynamical systems (Q412651)

The author proves the following result on the stability of the translation equation. Let \(H: \mathbb{R}\times I \to I\) (\(I\subset\mathbb{R}\) is an interval) be a continuous, with respect to each variable, \(\delta\)-approximate (\(\delta\geq 0\)) solution of the translation equation, i.e., NEWLINE\[NEWLINE |H(s,H(t,x))-H(t+s,x)|\leq\delta,\qquad x\in I,\;t,s\in\mathbb{R}. NEWLINE\]NEWLINE Then, there exists a continuous mapping \(F: \mathbb{R}\times I \to I\) satisfying the translation equation: NEWLINE\[NEWLINE H(s,H(t,x))=H(t+s,x)|,\qquad x\in I,\;t,s\in\mathbb{R} NEWLINE\]NEWLINE and such that NEWLINE\[NEWLINE |F(t,x)-H(t,x)|\leq 10\delta,\qquad x\in I,\;t\in\mathbb{R}. NEWLINE\]NEWLINE In a laborious proof of this theorem, an earlier result by \textit{J. Chudziak} [Appl. Math. Letters 25, No. 3, 532--537 (2012; Zbl 1244.39019)] is used.NEWLINENEWLINEMoreover, the stability of dynamical systems (solutions of the translation equation satisfying additionally the identity condition \(F(0,x)=x,\;x\in I\)) is considered. It is shown that generally the stability does not hold true.