Abstract
In this paper, we first get that the set of common fixed points of a family of new generalized nonexpansive mappings defined by Takahashi and Yao [30] is a sunny generalized nonexpansive retract. We next consider a method which unifies the hybrid method introduced by Solodov and Svaiter [24] and the shrinking projection method introduced by Takahashi, Takeuchi and Kubota [29] and then prove a strong convergence result by this method for finding a common fixed point of the family of new generalized nonexpansive mappings in a Banach space. Using this result, we prove two strong convergence theorems by the hybrid method and the shrinking projection method. Using these two results, we obtain well-known and new strong convergence theorems in a Hilbert space and a Banach space.