On the functional equation \(1/p\cdot \{f(x/p)+\cdots +f((x+p- 1)/p)\}=\lambda f(\mu x)\) (Q1070127)

Periodic with period one solutions \(f: {\mathbb{R}}\to {\mathbb{R}}\) of the equation from the title are considered, where \(p\geq 2\) is an integer and \(\lambda\), \(\mu\) are positive numbers. It is proved that every locally integrable solution equals zero a.e. in each of the following three cases: 1. \(\mu\) is irrational and \(\lambda\) \(\neq 1\); 2. \(\mu =s/r\), \((s,r)=1\), \(p\not\equiv 0 (mod r)\) or \(p=r\), and \(\lambda\) \(\neq 1\); 3. \(\mu\) is an integer and \(\lambda >1\). Assuming additionally that \(\mu\) is an integer, the following is also proved. 4. If \(\lambda\in (0,1)\) then there exist non-trivial continuous solutions. 5. If \(\lambda\) \(\neq 1\) and \(\lambda \mu p>1\), then the zero function is the only solution of bounded variation over [0,1]. 6. If \(\lambda \mu p<1\), then there exist non- trivial continuously differentiable solutions. 7. If \(\lambda \mu p=1\), then the zero function is the only absolutely continuous solution and there exist non-trivial continuous solutions which are also of bounded variation.