Hybrid Iteration Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings in Banach Spaces

Abstract EN

Let E be a real uniformly convex Banach space, and let{Ti:i∈I} be N nonexpansive mappings from E into itself with F={x∈E:Tix=x, i∈I}≠ϕ, where I={1,2,…,N}.

From an arbitrary initial point x1∈E, hybrid iteration scheme {xn} is defined as follows: xn+1=αnxn+(1−αn)(Tnxn−λn+1μA(Tnxn)), n≥1, where A:E→E is an L-Lipschitzian mapping, Tn=Ti, i=n(mod N), 1≤i≤N, μ>0, {λn}⊂[0,1), and {αn}⊂[a,b] for some a,b∈(0,1).

Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of the mappings {Ti:i∈I} are obtained.

The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007).