Nonlinear ergodic theorems of Dirichlet's type in Hilbert spaces

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. A mapping T: C --> C is called asymptotically nonexpansive with Lipschitz constants {alpha (n)} if parallel toT(n)x - T(n)y parallel to less than or equal to (1 +alpha (n))parallel tox-y parallel to for all n greater than or equal to 0 and all x, y is an element of C, where alpha (n) greater than or equal to 0 for all n greater than or equal to 0 and alpha (n)-->0 as n --> infinity, In particular, if alpha (n) = 0 for all n greater than or equal to 0, then T is called nonexpansive. We introduce a new summation method which will be called a (D, mu)-method, extending the Abel method of summability. Let mu = {mu (n)} be a sequence of real numbers satisfying the following conditions: (D1) mu (0) greater than or equal to 0 and inf (n greater than or equal too) {mu (n+1) - mu (n)} = delta for some delta > 0 and (D2) sup(s >0) (1/g(s))(-1) Sigma (infinity)(n=0), n{e(-mu nS) -e(-mun+1S)} < infinity, where g(s) = Sigma (infinity)(n=0) e(-mu ns) which converges for s > 0. Such a sequence mu={mu (n)} determines a strongly regular method of summability (Dirichlet summability) and is called a (D,mu)-method. Given a mapping T: C --> C, we define [GRAPHICS] for x is an element of C. Then we can define the so-called Dirichlet means D-s((mu)) [T]x of the sequence {T(n)x} by the formula D-s((mu))[T]x = (1/g(s)) Sigma (infinity)(n=0)e(-mu ns)T(n)x, s > 0, whenever a(mu)(Tx) less than or equal to 0. In particular, when mu (n) = n + 1, we get the Abel means (1 - r) Sigma (infinity)(n=0) r(n)T(n)x, 0 < r < 1. In the above setting, our results are stated as follows: Theorem 1. Let C be a nonempty bounded closed convex subset of H and let T be an asymptotically nonexpansive nonlinear mapping of C into itself. Let mu = {mu (n)} be a (D,mu)-method. Then for each x is an element of C, D-s((mu))[T]x converges weakly as s -->0+ to the asymptotic center of {T(n)x}. We say that a (D,mu)-method mu = {mu (n)} is proper if for each {beta (n)} is an element of l(infinity) for which(1/g(s))(-1) Sigma (infinity)(n=0)e(-mu ns)beta (n) converges to some delta as s -->0+, we have lim(s -->0+) (1/g(s))(2) Sigma (infinity)(n=0) Sigma (infinity)(k=0) e(-(mun+muk)s)beta(\n - k \) = delta. Theorem 2. Let C be a nonempty bounded closed convex subset of H and let T be a nonexpansive nonlinear mapping of C into itself. Let mu = {mu (n)} be a (D,mu)-method and suppose that (i) 0 is an element of C and T(0) = 0, (ii) for some c > 0, T satisfies for all u, v is an element of C the inequality \ < Tu,Tv > - <u,v > \ less than or equal to c{parallel tou parallel to (2) - parallel to Tu parallel to (2) + parallel tov parallel to (2) - parallel to Tv parallel to (2)}, and (iii) there is an t(infinity)-element {beta (n)} such that for any x is an element of C, \ (T(p)x,T(q)x) - beta(\p - q \)\ less than or equal to gamma (min(p,q)), where gamma (min(p,q))-->0 as min(p,q)--> infinity. Then for each x is an element of C, D-s((mu))[T]x converges strongly as s -->0+ to the asymptotic center of {T(n)x}. (C) 2001 Elsevier Science Ltd. All rights reserved.