Boundedness of solutions for a second order differential equation with causal operators

Abstract

This paper deals with the asymptotic behavior of solutions to the second order functional differential equation \begin{equation}\label{e1}\ddot x+(Lx)\dot x+{\rm grad}\,F(x)=f(t),\end{equation}on the half-axis $R_+=[0,\9)$, with special concern on the boundedness of its solutions. Condition are provided on the data, such that all the solutions of (1) are defined on the whole $R_+$, together with their first derivatives. The basic local existence result is available in [4].