Dirichlet summability and strong nonlinear ergodic theorems in Hilbert spaces Original Research Article

Let C be a non-empty closed convex subset of a real Hilbert space H. Following Goebel and Kirk, a mapping View the MathML source is called asymptotically non-expansive 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 non-expansive. Let μ={μn} be a (D,μ) method which means a sequence of real numbers satisfying the following conditions: (D1) μ0⩾0 and View the MathML source for some τ>0 and (D2) View the MathML source, where View the MathML source which converges for any s>0. Such a sequence μ={μn} 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 View the MathML source, 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 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.LetTbe a nonlinear self-mapping of a non-empty closed convex subsetCofHand letμ={μn} be a (D,μ) method. Then the following statements hold: (1) If forx∈C, View the MathML sourceconverges inHfor anys>0, thenaμ(T;x)⩽0. (2) Ifaμ(T;x)<∞ forx∈C, thenView the MathML sourceconverges inHfor any realswithView the MathML source. 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 View the MathML source for any y∈C. Then σx(y) has a unique minimizer, the point which we call the asymptotic center of the sequence {Tnx} in the sense of Edelstein. Theorem 2.LetCbe a non-empty bounded closed convex subset ofHand letTbe an asymptotically non-expansive self-mapping ofC. Letμ={μn} be a (D,μ) method, fix an elementx∈Cand suppose that for eachm, 〈Tjx,Tj+mx〉 converges asj→∞, the convergence being uniform form⩾0. Then the Dirichlet meanDs(μ)[T]xconverges strongly ass→0+ to the asymptotic center of the sequence {Tnx}.