On the stability of solutions to a certain fourth-order delay differential equation (Q840349)

Consider the delay differential equation \[ x^{(4)}(t)+ \varphi(\ddot x(t)) x^{(3)}(t)+ h(\ddot x(t- r))+ \Phi(\dot x(t- r))+ f(x(t-r))= 0,\tag{\(*\)} \] where \(\varphi\) and \(h\) are continuous, \(\Phi\) and \(f\) are continuously differentiable, additionally it holds \(h(0)= \Phi(0)= f(0)= 0\). The author derives conditions such that the trivial solution of \((*)\) is stable. The proof is based on the construction of a suitable Lyapunov functional.