Hybrid iteration method for common fixed points of an infinite family of nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space, and let K be a nonempty closed convex subset of E . Let { T i } i = 1 ∞ be a sequence of nonexpansive mappings from K to itself with F :={ x ∈ K : T i x = x , ∀ i ≥ 1}≠ ∅. For an arbitrary initial point x 1 ∈ K , the modified hybrid iteration scheme { x n } is defined as follows: x n + 1 = α n x n + ( 1 - α n ) T n * x n - λ n + 1 μ A ( T n * x n ) , n ≥ 1 , where A : K → K is an L -Lipschitzian mapping, T n * = T i with i satisfying: n = [( k - i +1)( i + k )/2]+[1+( i -1)(i+2)/2], k ≥ i -1(i = 1,2,.),{ λ n } ⊂ [0,1), and { α n } is a sequence in [ a , 1 - a ] for some a ∈ (0,1). Under some suitable conditions, the strong and weak convergence theorems of { x n } to a common fixed point of the nonexpansive mappings { T i } i = 1 ∞ are obtained. The results in this article extend those of the authors whose related researches are restricted to the situation of finite families of nonexpansive mappings. Mathematics Subject Classifications 2000: 47H09; 47J25.