Dirichlet summability and strong nonlinear ergodic theorems in Hilbert spaces

Abstract

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 { 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 non-expansive. Let μ={ μ n } be a ( D, μ) method which means a sequence of real numbers satisfying the following conditions: (D1) μ 0⩾0 and inf n⩾0 {μ n+1−μ n}=τ for some τ>0 and (D2) sup s>0 (1/g(s)) ∑ n=0 ∞ n{ e −μ ns − e −μ n+1s }<∞ , where g(s)= ∑ n=0 ∞ e −μ ns which converges for any s>0. Such a sequence μ={ μ 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 Abel summation method. Given a mapping T : C→C , we define a μ(T;x)= lim sup n→∞ log||∑ k=0 n T kx|| μ n if lim sup n→∞ ∑ k=0 n T kx >0, −∞ if lim sup n→∞ ∑ k=0 n T 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 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 T be a nonlinear self-mapping of a non-empty closed convex subset C of H and let μ={ μ n } be a ( D, μ) method. Then the following statements hold: (1) If for x∈ C, ∑ n=0 ∞ e −μ ns T nx converges in H for any s>0, then a μ ( T; x)⩽0. (2) If a μ ( T; x)<∞ for x∈ C, then ∑ n=0 ∞ e −μ ns T nx converges in H for any real s with s> max (0,a μ(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∈ C and let σ x(y)= lim sup n→∞ ||T nx−y|| 2 for any y∈ C. Then σ 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 μ={ μ n } be a ( D, μ) method, fix an element x∈ C and suppose that for each m, 〈 T j x, T j+ m x〉 converges as j→∞, the convergence being uniform for m⩾0. Then the Dirichlet mean D s ( μ) [ T] x converges strongly as s→0+ to the asymptotic center of the sequence { T n x}.