Adv. Math. Econ. 15, 67–88 (2011)

Fixed point theorems and ergodic theorems for nonlinear mappings in Banach spaces

Pavel Kocourek^{1}, Wataru Takahashi^{2 1}, and Jen-Chih Yao^{1} ^{3}

1 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (e-mail: pakocica@gmail.com; yaojc@math.nsysu.edu.tw)

2 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan (e-mail: wataru@is.titech.ac.jp)

Received: September 29, 2010 Revised: October 25, 2010

JEL classification: C62, C68

Mathematical Subject Classification (2010): 47H09, 47H10, 47H25

Abstract. In this paper, we first introduce a class of nonlinear mappings called generalized nonspreading which contains the class of nonspreading mappings in a Banach space and then prove a fixed point theorem, a nonlinear mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for such nonlinear mappings in a Banach space. Using these theorems, we obtain some fixed point theorems, nonlinear mean convergence theorems and weak convergence theorems in a Banach space.

Key words: Banach space, fixed point, mean convergence, weak convergence, nonexpansive mapping, nonspreading mapping

3The second author and the third author were partially supported by Grant-in-Aid for Scientific Research No. 19540167 from Japan Society for the Promotion of Science and by the grant NSC, respectively.