Convergence Theorems for Countable Family Lipschitzian Mappings in Uniformly Convex Banach Spaces
Jing Sun, Yanrong Yu, Rudong Chen · 2011
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 Σn=1∞(Ln-1)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 Σn=1∞an(1-an)=∞. Let Σn=1∞sup{∥Tn+1z - Tnz∥ : z ∈ B}z= limn→∞ Tnz for all z ∈ K and suppose that F(T) = ∩n=1∞F(Tn), then {xn} converges weakly to w ∈ F(T).