WEAK CONVERGENCE THEOREMS FOR GENERALIZED HYBRID MAPPINGS IN BANACH SPACES - Takahashi, W.; YAO, J. (Journal of Nonlinear Analysis and Optimization: Theory & Applications (JNAO), 2011)
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 α, β R such that α\|Tx-Ty\|^2 + (1-α)\|x-Tx\|^2 ≤ β\|Tx-y\|^2 + (1-β)\|x-y\|^2 for all x,y 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.