Abstract
Define on the set \(G:=\mathbb R^+\times \mathbb R\) the operation \((t,a)*(s,b)=(ts,tb+a)\). \((G,*)\) is a non-commutative group with the neutral element (1, 0). We consider a non-commutative translation equation \(F(\eta ,F(\xi ,x))=F(\eta *\xi ,x)\), \(\eta , \xi \in G\), \(x\in I\), \(F(1,0)=\mathrm{id}\), where I is an open interval and \(F:G\times I\rightarrow I\) is a continuous mapping. This equation can be written in the form: \(F((t,a),F((s,b),x))=F((ts,tb+a),x)\), \( t,s \in \mathbb R^+\), \(x\in I\). For \(t=1\) the family \(\{F(t,a)\}\) defines an additive iteration group, however for \(a=0\) it defines a multiplicative iteration group. We show that if F(t, 0) for some \(t\ne 1\) has exactly one fixed point \(x_t\), \((F(t,0)-\mathrm{id})(x_t-\mathrm{id})\ge 0\) and for an \(a>0 \)\(F(1,a)>\mathrm {id}\), then there exists a unique homeomorphism \(\varphi :I\rightarrow \mathbb R\) such that \(F((s,b),x)=\varphi ^{-1}(s\varphi (x)+b)\) for \(s\in \mathbb R^+\) and \(b\in \mathbb R\).
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Dedicated to Professor Karol Baron on the occasion of his 70th birthday
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Zdun, M.C. On multiplicatively-additive iteration groups. Aequat. Math. 93, 205–215 (2019). https://doi.org/10.1007/s00010-018-0622-z
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DOI: https://doi.org/10.1007/s00010-018-0622-z