### abstract

- We solve T. Kato’s problem on the existence of bounded (almost-periodic) solutions for the functional-differential equation y '' (t)=∑ j=-l,j≠0 l a j y(q j t)+λy(t), where q>1. Theorem: There exists a number K>0 such that (1) for λ<-K the equation has a nontrivial bounded solution on the entire axis; (2) for λ<-K each bounded solution is almost-periodic; (3) for λ>K the equation does not have nontrivial bounded solutions on the entire axis.