On the method of the steepest, descent

Let H be a Hilbert space over the reals and let f: H → R¹ be a function of class C¹.  We have shown in |1|  that the differential equation

du/dt= -T (u(t) ),u(0)= u_o                                                                                                                                               (1)

(T = grad f) has global solutions if

 i) ≥ c||x-y ||² , c> 0,

ii) f is bounded from below;

iii) T is locally Lipschitzian.

To be precise, in [1;Th.3] we heve assumed f to be of class C² and f" to be locally bounded. However, the hypothesis  f"  is locally bounded implies that  f'´ = T is locally Lipschitzian, and this is what matters to show existence and uniqueness.