Uniform boundedness of solutions to functional differential equations with infinite delay (Q1919010)

The author deals with solutions to functional-differential equations with infinite delay (1) \(x'=f(t,x_t)\) which are uniformly bounded and uniformly ultimate bounded with respect to spaces \(C_g\) of unbounded initial functions. The author establishes conditions which improve slightly \textit{R. H. Hering}'s one [J. Math. Anal. Appl. 163, No. 2, 521-535 (1992; Zbl 0755.34064)]. Neither a reference on the existence of solutions to (1) is explicit nor a proof is presented.