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Paper   IPM / M / 436
School of Mathematics
  Title:   An ergodic theorem for sequences in a Hilbert space
  Author(s):  B. Djafari Rouhani
  Status:   Published
  Journal: Nonlinear Anal. Forum
  Vol.:  4
  Year:  1999
  Pages:   33-48
  Supported by:  IPM
By suitable modifying our methods in [10], we prove the following nonlinear ergodic theorem, extending H. Brezis and F.E. Browder [4, Theorem 2] and R. Wittmann's mean ergodic theorem [15, Theorem 2.3].
For any sequence (xn)n ≥ 0 in a real Hilbert space H satisfying: (xj|xj+l) ≤ (xk|xk+l)+ϵ(k,l,jk) for all k,l ≥ 0 and jk with ϵ bounded and limk,l,m→∞ϵ(k,l,m)=0, and any strongly regular summation mehtod {an,j}, the sequence yn=∑j=0an,j xj converges strongly to the same limit. Some identifications of the limit are also given.

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