On strong convergence by the hybrid method for families of mappings in Hilbert spaces

Let CC be a nonempty closed convex subset of a real Hilbert space 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. We give the new conditions of {Tn}{Tn} under which {xn}{xn} converges strongly to a common fixed point of {Tn}{Tn} and generalize the unified result for families of nonexpansive mappings [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] and the results for asymptotically nonexpansive mappings and semigroups [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152; S. Plubtieng and K. Ungchittrakool, Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 67 (2007) 2306–2315].