On T. Kato's problem regarding bounded solutions of differential- functional equations (Q2640764)

A complete analysis of the equation \(y'(t)=ay(\alpha t)+by(t)\) has been given by \textit{T. Kato} and \textit{J. B. McLeod} [Bull. Amer. Math. Soc. 77, 891-937 (1971; Zbl 0236.34064)]. Kato has formulated the problem of the investigation of equations that contain terms with both ``contractions'' and ``extensions'' of the argument. The purpose of this paper is to give an answer to T. Kato's question for the equation \[ y''(t)=\sum^{\ell}_{j=-\ell,j\neq 0}a_ jy(q^ jt)+\lambda y(t), \] where \(q>1\), \(a_ j\), \(\lambda\in {\mathbb{R}}\). In the above equation, the arguments of terms \(y(q^{-j}t)\) \((j=1,...,\ell)\) are contractions and the arguments of \(y(q^ jt)\) \((j=1,...,\ell)\) are extensions.