Abstract

Recently, two retractions (projections) which are different from the metric projection and the sunny nonexpansive retraction in a Banach space were introduced. In this paper, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for a commutative family of positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions. In the proof, we use the theory of invariant means essentially. We apply our nonlinear ergodic theorem to get some nonlinear ergodic theorems in Banach spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

Wataru Takahashi

Dept. of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
and: Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

wataru@is.titech.ac.jp

Ngai-Ching Wong

Dept. of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
and: Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan

wong@math.nsysu.edu.tw

Jen-Chih Yao

Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan
and: Dept. of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

yaojc@kmu.edu.tw

Keywords

  • Banach space
  • fixed point
  • invariant mean
  • mean convergence
  • nonexpansive mapping
  • positively homogeneous mapping
  • semitopological semigroup

Mathematics Subject Classification

47H10, 47H25

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Suggested citation

W. Takahashi, N.-C. Wong, J.-C. Yao. “Nonlinear Ergodic Theorem for Commutative Families of Positively Homogeneous Nonexpansive Mappings in Banach Spaces and Applications.” Journal of Convex Analysis 21 (2014), No. 2, 535–552. https://doi.org/10.68381/jca21028

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