A Mean Ergodic Theorem for Affine Nonexpansive Mappings in Nonpositive Curvature Metric Spaces

Khatibzadeh, Hadi; Pouladi, Hadi

doi:10.2478/auom-2021-0008

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.

References

[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.
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[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
Search in Google Scholar Back to article
[4] G. Birkhoff, The mean ergodic theorem, Duke Math. J. 5 (1939), no. 1, 19–20.
Search in Google Scholar Back to article
[5] M. R. Bridson and A. Hafliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2011.
Search in Google Scholar Back to article
[6] P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl. 320 (2006), no. 2, 983 – 987.
Search in Google Scholar Back to article
[7] H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977), no. 5, 509–541.
Search in Google Scholar Back to article
[8] H. Khatibzadeh and S. Ranjbar, On the Halpern iteration in CAT(0) spaces, Ann. Funct. Anal. 6 (2015), no. 3, 155–165.
Search in Google Scholar Back to article
[9] W. A. Kirk, Geodesic geometry and fixed point theory II, Proceedings of the International Conference in Fixed Point Theory and Applications (Valencia, Spain), 2003, pp. 113–142.
Search in Google Scholar Back to article
[10] M. K. Kuo, Tauberian conditions for almost convergence, Positivity 13 (2009), no. 4, 611–619.
Search in Google Scholar Back to article
[11] T. Liimatainen, Optimal Riemannian metric for a volumorphism and a mean ergodic theorem in complete global Alexandrov nonpositively curved spaces, Analysis, geometry and quantum field theory, Contemp. Math., vol. 584, Amer. Math. Soc., Providence, RI, 2012, pp. 163–178.10.1090/conm/584/11593
Search in Google Scholar Back to article
[12] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167–190.10.1007/BF02393648
Search in Google Scholar Back to article
[13] A. Papadopoulos, Metric spaces, convexity and nonpositive curvature, IRMA lectures in mathematics and theoretical physics, European Mathematical Society, 2005.10.4171/010
Search in Google Scholar Back to article
[14] A. Pazy, On the asymptotic behavior of iterates of nonexpansive mappings in Hilbert space, Israel J. Math. 26 (1977), no. 2, 197–204.
Search in Google Scholar Back to article
[15] S. Saejung, Halpern’s iteration in CAT(0) spaces, Fixed Point Theory Appl. (2010), Art. ID 471781, 13.
Search in Google Scholar Back to article
[16] K. T. Sturm, Probability measures on metric spaces of nonpositive curvature, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 357–390.10.1090/conm/338/06080
Search in Google Scholar Back to article
[17] J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Natl. Acad. Sci. USA 18 (1932), no. 1, 70–82.
Search in Google Scholar Back to article

A Mean Ergodic Theorem for Affine Nonexpansive Mappings in Nonpositive Curvature Metric Spaces

Abstract

Paradigm

My account