WEAK CONVERGENCE THEOREMS FOR GENERALIZED HYBRID MAPPINGS IN BANACH SPACES

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.