Strong convergence theorems by the hybrid method for families of mappings in Banach spaces

Let CC be a nonempty closed convex subset of a uniformly convex Banach space EE whose norm is Gâteaux differentiable and let {Tn}{Tn} be a family of mappings of CC into itself such that the set of all common fixed points of {Tn}{Tn} is nonempty. We consider a sequence {xn}{xn} generated by the hybrid method in mathematical programming. And we give the conditions of {Tn}{Tn} under which {xn}{xn} converges strongly to a common fixed point of {Tn}{Tn} and generalize the results of [K. Nakajo, K. Shimoji, W. Takahashi, Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces, Taiwanese J. Math. 10 (2006) 339–360; S. Ohsawa, W. Takahashi, Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces, Arch. Math. 81 (2003) 439–445].