Uniform stability and boundedness of solutions of nonlinear delay differential equations of the third order (Q2839633)

The authors consider uniform stability and boundedness of solutions of the equation NEWLINE\[NEWLINE\dddot{x}+f(x,\dot{x},\ddot{x})\ddot{x}+g(x(t-r(t)),\dot{x}(t-r(t)))+h(x(t-r(t))=p(t,x,\dot{x},\ddot{x})\tag{1}NEWLINE\]NEWLINE where \(0\leq r(t)\leq \gamma\) (\(\gamma>0\) is a constant) and \(f, g, h\) and \(p\) are continuous functions in their respective arguments.NEWLINENEWLINEBy reducing equation (1) to an equivalent system, the authors construct a complete Lyapunov functional and use this to obtain new results on the uniform asymptotic stability of the zero solution (\(p=0\)). Furthermore, they also consider the situation when \(p\not=0\) and obtain sufficient conditions for the existence of solutions that are uniformly bounded and uniformly ultimately bounded.