Periodic behavior of a nonlinear third order vibrating system

A theorem is proved to show that the third order differential equation x‴+f(t,x,x′,x″)=0 has nontrivial solutions characterized by x′(0)=x′(τ)=0 when x,x′,x″ and f(t,x,x′, x″) are bounded. A second condition is introduced to prove the existence of periodic solution for this equation. It is shown that the equation has a τ-periodic solution if f(t,x,x′,x″) is an even function with respect to x′. The existence and periodicity conditions would be applied to third order systems such as viscoelastic mechanical vibration isolator system. The concepts of Green's function and the Schauder's fixed-point theorem have been used for proving the third-order-existence theorem.