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On the functional equations \(\phi (x)=\alpha \phi (\alpha x)+(1- \alpha)\phi (1-(1-\alpha)x)\). (English) Zbl 0608.39003
The authors deal with the functional equation \(\phi (x)=\alpha \phi (\alpha x)+(1-\alpha)\phi (1-(1-\alpha)x)\), \(x\in [0,1]\), where \(\alpha\in (0,1)\) is a fixed constant and \(\phi\) :[0,1]\(\to {\mathbb{R}}\). They construct the general solution of the equation starting from an arbitrary function \(\phi_ 0\) defined on the interval [0,\(\alpha\) ] and such that \(\phi_ 0(0)=\phi_ 0(\alpha)\). Moreover it is proved that if a solution \(\phi\) is either Riemann integrable or bounded and continuous at least at one point, then it is constant.
Reviewer: G.L.Forti
MSC:
39B99 Functional equations and inequalities
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