ASYMPTOTIC-BEHAVIOR OF UNBOUNDED NONEXPANSIVE SEQUENCES IN BANACH-SPACES

Let X be a real Banach space, (x(n))n greater-than-or-equal-to 0 a nonexpansive sequence in X (i.e., \\x(i+1) - x(j+1)\\ less-than-or-equal-to \\x(i) - x(j)\\ for all i, j greater-than-or-equal-to 0), and C the closed convex hull of the sequence (x(n+1) - x(n))n greater-than-or-equal-to 0. We prove that lim(n-->+infinity) \\x(n)/n\\ = inf(n greater-than-or-equal-to 1) \\(x(n) - x0)/n\\ = inf(z is-an-element-of C) \\z\\ and deduce a simple short proof for the following result. (i) If X is reflexive and strictly convex, then x(n)/n converges weakly in X to the element of minimum norm P(C)0 in C with [GRAPHICS] (ii) If X* has Frechet differentiable norm, then x(n)/n converges strongly to P(C)0. This result contains previous results by Pazy, Kohlberg and Neyman, Plant and Reich, and Reich and is also optimal since the assumptions made on X in (i) or (ii) are also necessary for the respective conclusion to hold.