Asymptotic behaviour of unbounded nonexpansive sequences in Banach spaces

Abstract

Let X be a real Banach space, (x0 )n>O a nonexpansive sequence in X (i.e., llxil xj+1 11 0), and C the closed convex hull of the sequence (x,+i x0),>o We prove that lim0+OO llxn/nll = infn>l II(xn xo)/nll = infZEC lizil and deduce a simple short proof for the following result. (i) If X is reflexive and strictly convex, then x,,/n converges weakly in X to the element of minimum norm PCO in C with IIPCOII = inf Xn -XO = lim Xn n>l fl n-'+0O fl (ii) If X* has Frechet differentiable norm, then X/nl converges strongly to PCO. 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.