Journal of Convex Analysis 21 (2014), No. 2, 535--552
Copyright Heldermann Verlag 2014
Nonlinear Ergodic Theorem for Commutative Families of Positively Homogeneous Nonexpansive Mappings in Banach Spaces and Applications
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
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.
Keywords: Banach space, fixed point, invariant mean, mean convergence, nonexpansive mapping, positively homogeneous mapping, semitopological semigroup.
MSC: 47H10, 47H25
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