Convergence Theorems for Countable Family Lipschitzian Mappings in Uniformly Convex Banach Spaces

The purpose of this paper is to prove a convergence theorem for a countable family Lipschitzian mappings in uniformly convex Banach spaces. Let E be a real uniformly convex Banach space and satisfy Opial's condition, K be a nonempty closed convex subset of E. Let {Tn} be a sequence of Ln-Lipschitzian mappings from K into itself with Σ∞n=1(Ln-1) < ∞ and let ∩∞n=1 F(Tn) be nonempty. Let {xn} be a sequence in K defined by x1 ∈ K and xn+1 = αnxn + (1 - αn)Tnxn, for all n ∈ N, where {αn} is a sequence in [0,1) with Σ∞n=1 an(1-an)=∞. Let Σ∞n=1 sup{Tn+1z - Tnz : z ∈ B} < ∞ for any bounded subset B of K and T be a mapping of K into itself defined by Tz = limn→∞ Tnz for all z ∈ K and suppose that F(T) = ∩∞n=1 F(Tn), then {xn} converges weakly to w ∈ F(T).