Nonlinear ergodic theorems of Dirichlet's type in Hilbert spaces

Abstract

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 { α n } if ∥ T n x− T n y∥≤(1+ α n )∥ x− y∥ for all n≥0 and all x, y∈ C, where α n ≥0 for all n≥0 and α n →0 as n→∞. In particular, if α n =0 for all n≥0, then T is called nonexpansive. We introduce a new summation method which will be called a ( D, μ)-method, extending the Abel method of summability. Let μ={ μ n } be a sequence of real numbers satisfying the following conditions: ( D1) μ 0≥0 and inf n≥0 {μ n+1−μ n}=δ for some δ>0 and ( D2) sup s>0 (1/g(s)) −1∑ n=0 ∞n{ e −μ ns − e −μ n+1s }<∞, where g(s)=∑ n=0 ∞ e −μ ns which converges for s>0. Such a sequence μ={ μ n } determines a strongly regular method of summability (Dirichlet summability) and is called a ( D, μ)-method. Given a mapping T: C→ C, we define a μ(T,x)= lim sup n→∞ log∥∑ k=0 nT kx∥ μ n if lim sup n→∞ ∑ k=0 nT kx >0, −∞ if lim sup n→∞ ∑ k=0 nT kx =0 for x∈ C. Then we can define the so-called Dirichlet means D s ( μ) [ T] x of the sequence { T n x} by the formula D s (μ)[T]x=(1/g(s))∑ n=0 ∞ e −μ ns T nx, s>0, whenever a μ ( T, x)≤0. In particular, when μ n = n+1, we get the Abel means (1−r)∑ n=0 ∞r nT nx, 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 μ={ μ n } be a ( D, μ)- method. Then for each x∈C, D s (μ)[T]x converges weakly as s→0+ to the asymptotic center of { T n x}. We say that a ( D, μ)-method μ={ μ n } is proper if for each { β( n)}∈ℓ ∞ for which (1/ g( s)) −1∑ n=0 ∞e − μ n s β( n) converges to some δ as s→0+, we have lim s→0+ 1 g(s) 2 ∑ n=0 ∞ ∑ k=0 ∞ e −(μ n+μ k)s β(|n−k|)=δ. 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 μ={ μ n } be a ( D, μ)- method and suppose that (i) 0∈ C and T(0)=0, (ii) for some c>0, T satisfies for all u, v∈ C the inequality |〈 Tu, Tv〉−〈 u, v〉|≤ c{∥ u∥ 2−∥ Tu∥ 2+∥ v∥ 2−∥ Tv∥ 2}, and (iii) there is an ℓ ∞- element { β( n)} such that for any x∈C, |(T px,T qx)−β(|p−q|)|≤γ min(p,q) , where γ min( p, q) →0 as min( p, q)→∞. Then for each x∈C, D s (μ)[T]x converges strongly as s→0+ to the asymptotic center of { T n x}.