Dirichlet summability and strong nonlinear ergodic theorems in Hilbert spaces

Let C be a non-empty closed convex subset of a real Hilbert space H. Following Goebel and Kirk, a mapping T: C --> C is called asymptotically non-expansive with Lipschitz constants {alpha(n)} if parallel toT(n)x - T(n)yparallel to less than or equal to (1 + alpha(n))parallel tox - yparallel 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 non-expansive. Let mu = {mu(n)} be a (D, mu) method which means 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 to 0 {mu(n+1) - mu(n)} = tau for some tau > 0 and (D2) sup(s>0) (1/g(s)) E-n=0(infinity) n{e(-muns) - e(-mun+1s)} < infinity, where g(s) = Sigma(n=0)(infinity) e-(muns) which converges for any s > 0. Such a sequence mu = {mu(n)} is easily seen to determine a strongly regular method of summability which is called the Dirichlet method of summability as a natural extension of the Abet summation 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 D-s((mu))[T]x = 1/g(s)Sigma(n=0)(infinity)e(-muns)T(n)x, s>0, whenever a(mu)(T,x) less than or equal to 0. In particular, when mu(n) = n + 1, we get the Abel means (1 - r) Sigma(n=0)(infinity) r(n)T(n)x, 0 < r < 1. In the above setting, our results are stated as follows. Theorem 1. Let T be a nonlinear self-mapping of a non-empty closed convex subset C of H and let mu = {mu(n)} be a (D, mu) method. Then the following statements hold: (1) If for x is an element of C, Sigma(n=0)(infinity) e(-muns)T(n) x converges in H for any s > 0, then a(mu)(T;x) less than or equal to 0. (2) If amu(T;x) < infinity for x is an element of C, then Sigman=0infinity e(-muns)T(n) x converges in H for any real s with s > max (0, a(mu)(T; x)). Let T be an asymptotically non-expansive self-mapping of a non-empty bounded closed convex subset C of H. Fix an element x E C and let sigmax*(y) = lim sup(n-->infinity) \\T-n x - y \\(2) for any gamma is an element of C. Then sigma(x)(y) has a unique minimizer, the point which we call the asymptotic center of the sequence {T(n)x} in the sense of Edelstein. Theorem 2. Let C be a non-empty bounded closed convex subset of H and let T be an asymptotically non-expansive self-mapping of C. Let mu = {mu(n)}be a (D,mu) method, fix an element x is an element of C and suppose that for each m, [T(j)x, T(j+m)x] converges as j --> infinity, the convergence being uniform for m greater than or equal to 0. Then the Dirichlet mean D-s((mu))[T]x converges strongly as s --> 0+ to the asymptotic center of the sequence {T-n x}. (C) 2003 Elsevier Science Ltd. All rights reserved.