On the stability of solutions of certain fourth order delay differential equations. (Q1417026)

The author considers the fourth-order nonlinear differential equations \[ x^{(4)} + \alpha_1 x^{(3)} + \alpha_2 \ddot x + \alpha_3 \dot x + f(x(t-r))=0, \] and \[ x^{(4)} + \alpha_1 x^{(3)} + \alpha_2 \ddot x + \phi( \dot x(t-r)) + f(x)=0, \] where \(r, \alpha_1,\alpha_2, \alpha_3\) are positive constants, \(\phi(x), f(x)\) are continuous, \(\phi(0)=f(0)=0\). By constructing Lyapunov functionals, the author derives conditions for the asymptotic stability of the zero-solution to the equations above.