Cauchy's functional equation in the mean (Q794217)

The following theorem is proved. If \(\alpha \geq 1\) and \(f\in L^{\alpha}(0,z)\) for every \(z>0\) and if \[ \lim_{z\to \infty}(z^{- 2}\int^{z}_{0}\int^{z}_{0}| f(x+y)-f(x)- f(y)|^{\alpha}dxdy)=0, \] then there exists an A such that \(\lim_{z\to \infty}(z^{-1}\int^{z}_{0}| f(x)- Ax|^{\alpha}dx)=0.\) Similar theorems and connections to \textit{D. H. Hyers'} [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.264)] and \textit{E. Wirsing's} [Symp. Math. 4, 45-57 (1970; Zbl 0223.10036)] results are mentioned.