Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces

Let CC be a nonempty closed convex subset of a Banach space EE and let {Sn}{Sn} be a family of nonexpansive mappings of CC into itself such that the set of common fixed points of {Sn}{Sn} is nonempty. We first introduce a sequence {xn}{xn} of CC defined by x1=x∈Cx1=x∈C and xn+1=αnf(xn)+(1−αn)Snxnfor alln∈N, where {αn}⊂(0,1){αn}⊂(0,1) and ff is a contraction of CC into itself. Further, we give the conditions of {αn}{αn} and {Sn}{Sn} under which {xn}{xn} converges strongly to a common fixed point of {Sn}{Sn}. This result generalizes the strong convergence theorem for nonexpansive mappings by Suzuki [T. Suzuki, A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 135 (2007) 99–106] and the strong convergence theorem for accretive operators by Kamimura and Takahashi [S. Kamimura, W. Takahashi, Weak and strong convergence of solutions to accretive operator inclusions and applications, Set-Valued Anal. 8 (2000) 361–374], simultaneously. Using this result, we improve and extend the two above-mentioned results.