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• W1987609093 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 {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 infn⩾0{μn+1−μn}=τ for some τ>0 and (D2) sups>0(1/g(s))∑n=0∞n{e−μns−e−μn+1s}<∞, where g(s)=∑n=0∞e−μns 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 T:C→C, we defineaμ(T;x)=limsupn→∞log||∑k=0nTkx||μniflimsupn→∞∑k=0nTkx>0,−∞iflimsupn→∞∑k=0nTkx=0for x∈C. Then we can define the so-called Dirichlet means Ds(μ)[T]x of the sequence {Tnx} byDs(μ)[T]x=1g(s)∑n=0∞e−μnsTnx,s>0,whenever aμ(T,x)⩽0. In particular, when μn=n+1, we get the Abel means (1−r)∑n=0∞rnTnx,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={μn} be a (D,μ) method. Then the following statements hold: If for x∈C, ∑n=0∞e−μnsTnx converges in H for any s>0, then aμ(T;x)⩽0. If aμ(T;x)<∞ for x∈C, then ∑n=0∞e−μnsTnx converges in H for any real s with s>max(0,aμ(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 letσx(y)=limsupn→∞||Tnx−y||2for 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. 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={μn} be a (D,μ) method, fix an element x∈C and suppose that for each m, 〈Tjx,Tj+mx〉 converges as j→∞, the convergence being uniform for m⩾0. Then the Dirichlet mean Ds(μ)[T]x converges strongly as s→0+ to the asymptotic center of the sequence {Tnx}." @default.
• W1987609093 created "2016-06-24" @default.
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• W1987609093 date "2003-07-01" @default.
• W1987609093 modified "2023-09-27" @default.
• W1987609093 title "Dirichlet summability and strong nonlinear ergodic theorems in Hilbert spaces" @default.
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• W1987609093 doi "https://doi.org/10.1016/s0362-546x(03)00049-x" @default.
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