On the stability of solutions for non-autonomous delay differential equations of third-order (Q710432)

Consider the delay-differential equation \[ \dddot x(t)+ a(t) \varphi(x(t),\dot x(t))\ddot x(t)+ b(t)\psi(x(t),\dot x(t))+c(t) h(x(t-r))=0,\tag{\(*\)} \] where \(a\), \(b\), \(c\), \(\varphi\), \(\psi\) and \(h\) are sufficiently smooth functions, \(a\), \(b\) and \(c\) are positive, \(\psi(x,0)\equiv 0\), \(h(0)= 0\), \(r\) is a positive parameter. By using a Lyapunov functional, the author derives sufficient conditions for the uniform asymptotic stability of the zero solution of \((*)\).