On the proximal point algorithm

Let A be a maximal monotone operator in a real Hilbert space H and let {u(n)} be the sequence in H given by the proximal point algorithm, defined by u (n) =(I+c(n) A)(-1)(u(n-1)-f(n) ), for all n >= 1, with u(0) = z, where c(n) > 0 and f(n) is an element of H. We show, among other things, that under suitable conditions, u(n) converges weakly or strongly to a zero of A if and only if lim inf(n ->+infinity) vertical bar w(n)vertical bar +infinity, where w(n) = (Sigma(n)(k=1) c(k))(-1) Sigma(n)(k=1) c(k)u(k). Our results extend previous results by several authors who obtained similar results by assuming A(-1)(0) not equal phi.