Boundedness results for a certain third order nonlinear differential equation (Q984335)

The authors consider a class of third order non-linear differential equations of the form \[ \dddot{x} + f(\ddot {x}) + g(\dot {x}) + h(x) = p(t,x,\dot {x},\ddot {x}), \tag{1} \] where \(f, g, h \in C(\mathbb R ,\mathbb R )\), \(p \in C(\mathbb R ^ + \times \mathbb R ^3,\mathbb R )\), \( \mathbb R ^ + = [0,\infty )\). The authors give some new sufficient conditions which guarantee the boundedness and uniform ultimate boundedness of the solutions (1). By defining an appropriate Lyapunov function, they prove their main results.