WEAK CONVERGENCE THEOREMS FOR GENERALIZED HYBRID MAPPINGS IN BANACH SPACES

WATARU  TAKAHASHI; JEN-CHIH  YAO

WEAK CONVERGENCE THEOREMS FOR GENERALIZED HYBRID MAPPINGS IN BANACH SPACES

Authors

WATARU TAKAHASHI Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Japan and Applied Mathematics, National Sun Yaysen University, Taiwan
JEN-CHIH YAO Department of Applied Mathematics, National Sun Yatsen University, Kaohsiung 80424, Taiwan

Keywords:

Banach space, Nonexpansive mapping, Nonspreading mapping, Hybrid mapping, Fixed point, Weak convergence

Abstract

Let E be a real Banach space and let C be a nonempty subset of E. A mapping T is called generalized hybrid if there are \alpha, \beta \in R such that \alpha\|Tx-Ty\|^2 + (1-\alpha)\|x-Tx\|^2 \leq \beta\|Tx-y\|^2 + (1-\beta)\|x-y\|^2 for all x,y \in C. In this paper, we first deal with some properties for generalized hybrid mappings in a Banach space. Then, we prove weak convergence theorems of Mann's type for such mappings in a Banach space satisfying Opial's condition.