On the Proximal Point Algorithm

Let A be a maximal monotone operator in a real Hilbert space H and let {un} be the sequence in H given by the proximal point algorithm, defined by un=(I+cnA)−1(un−1−fn), ∀n≥1, with u0=z, where cn>0 and fn∈H. We show, among other things, that under suitable conditions, un converges weakly or strongly to a zero of A if and only if lim inf n→+∞|wn|<+∞, where wn=(∑k=1nck)−1∑k=1nckuk. Our results extend previous results by several authors who obtained similar results by assuming A−1(0)≠φ.