On the method of the steepest, descent

Abstract

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 equationdu/dt= -T (u(t) ),u(0)= uo (1)(T = grad f) has global solutions if i) ≥ c||x-y ||2 , c and gt; 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.