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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