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Cocycles of continuous iteration semigroups. (English) Zbl 1044.39017
Among the results offered by the authors is the following and its converse. For every solution \(\Delta:\; ]0,\infty[\times]p,q[\to Y\) of \(\Delta(s+t,u)=\Delta(s,u)\Delta(t,u+s)\) and for every \(u_0\in(p,q),\; c_0\in Y\) (where \(-\infty\leq p<q\leq \infty\) and \(Y\) is a commutative group under multiplication), there exists a unique \(\delta:(p,\infty)\to Y\) such that \(\delta(u_0)=c_0\) and \(\Delta(t,u)=\delta(u+t)/\delta(u)\), \(t\in]0,\infty[\), \(u\in]p,q[.\) As the authors mention, the functional equation for \(\Delta\) reduces to Sincov’s equation \(f(s,u)=f(s,t)f(t,u)\) for \(f(s,t)=\Delta(t,s-t).\)

MSC:
39B12 Iteration theory, iterative and composite equations
39B52 Functional equations for functions with more general domains and/or ranges
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