Asymptotic behaviour of solutions of second order nonlinear autonomous differential equations (Q1364604)

The paper studies asymptotic properties, as \(t\rightarrow + \infty\), of solutions of the differential equation \(u''=f(u,u')\). The main result are sufficient conditions for every solution to possess the property \(\lim_{t\rightarrow +\infty} u'(t)=a\) and \(\lim_{\rightarrow +\infty} u=\pm\infty\). The investigation is done in the phase plane and the phase portrait of to equivalent system is obtained by using the zero-isocline function. The obtained result is applied to the Liénard equation to which the conditions become also necessary. Four examples are examined.