On the existence of a limiting regime in the sense of Demidovic for a certain fourth-order nonlinear differential equation (Q1108438)

A solution X(t) of the ordinary fourth order DE \[ (*)\quad x+a\dot{\ddot x}+b\ddot x+c\dot x+h(x)=p(t,x,\dot x,\ddot x\dot{\ddot x}), \] with a,b,c positive constants, h and p continuous, is said to be a Demidovic limit regime if \((X^ 2+\dot X^ 2+\ddot X^ 2+\dot{\ddot X}^ 2)^{1/2}\leq m\) for a finite m and all \(t\in {\mathbb{R}}\), and if every other solution converges to X as \(t\to \infty\). The author gives some sufficient conditions on h and p under which a limit regime exists and is periodic or almost periodic when p, as a function of t, has those properties.