Affiliation:
1. Department of Mathematics , University of Zanjan , 45195-313, Zanjan, Iran .
Abstract
Abstract
In this paper, we consider the orbits of an affine nonexpansive mapping in Hadamard (nonpositive curvature metric) spaces and prove an ergodic theorem for the inductive mean, which extends the von Neumann linear ergodic theorem. The main result shows that the sequence given by the inductive means of iterations of an affine nonexpansive mapping with a nonempty fixed point set converges strongly to a fixed point of the mapping. A Tauberian theorem is also proved in order to ensure convergence of the iterations.
Reference17 articles.
1. [1] B. Ahmadi Kakavandi and M. Amini, Non-linear ergodic theorems in complete non-positive curvature metric spaces, Bull. Iranian Math. Soc. 37 (2011), no. 3, 11–20.
2. [2] K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), no. 8, 2350 – 2360.
3. [3] M. Bacak, Convex analysis and optimization in Hadamard spaces, De Gruyter Series in Nonlinear Analysis and Applications, 22. De Gruyter, 2014.10.1515/9783110361629
4. [4] G. Birkhoff, The mean ergodic theorem, Duke Math. J. 5 (1939), no. 1, 19–20.
5. [5] M. R. Bridson and A. Hafliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2011.