Nonlinear ergodic theorems of Dirichletʹs type in Hilbert spaces Original Research Article

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 {αn} if ∥Tnx−Tny∥≤(1+αn)∥x−y∥ for all n≥0 and all x,y∈C, where αn≥0 for all n≥0 and αn→0 as n→∞. In particular, if αn=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={μn} be a sequence of real numbers satisfying the following conditions: View Within Article Such a sequence μ={μn} determines a strongly regular method of summability (Dirichlet summability) and is called a (D,μ)-method. Given a mapping T:C→C, we define View the MathML source for x∈C. Then we can define the so-called Dirichlet means Ds(μ)[T]x of the sequence {Tnx} by the formula View the MathML source whenever aμ(T,x)≤0. In particular, when μn=n+1, we get the Abel means View the MathML source. In the above setting, our results are stated as follows: Theorem 1.LetCbe a nonempty bounded closed convex subset ofHand letTbe an asymptotically nonexpansive nonlinear mapping ofCinto itself. Letμ={μn} be a (D,μ)-method. Then for eachView the MathML sourceconverges weakly ass→0+ to the asymptotic center of {Tnx}. We say that a (D,μ)-method μ={μn} is proper if for each {β(n)}∈ℓ∞ for which (1/g(s))−1∑n=0∞e−μnsβ(n) converges to some δ as s→0+, we have View the MathML source Theorem 2.LetCbe a nonempty bounded closed convex subset ofHand letTbe a nonexpansive nonlinear mapping ofCinto itself. Letμ={μn} be a (D,μ)-method and suppose that (i) 0∈CandT(0)=0, (ii) for somec>0,Tsatisfies for allu,v∈Cthe 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 anyView the MathML sourcewhereγmin(p,q)→0 as min(p,q)→∞. Then for eachView the MathML sourceconverges strongly ass→0+ to the asymptotic center of {Tnx}.