Existence and uniqueness of heteroclinic orbits for the equation'λu”+u’=f(u)

If f is a continuous even function which is decreasing on (0, cc) and such that ±α are its only zeros and are simple, then in three-dimensional phase space the unstable manifold of the equilibrium u = -α and the stable manifold of u = α are both two dimensional. If A < 0 it is shown that there is a unique bounded orbit of the equation λu'” + u’ =f(u), and that this is a heteroclinic orbit joining these two equilibria. Other results on the existence and uniqueness of heteroclinic orbits are also established when f is not even and when f is not monotone on (0,∞).