Hybrid iteration method for common fixed points of an infinite family of nonexpansive mappings in Banach spaces
Wei-Qi Deng · Fixed Point Theory and Applications · 2012
Let E be a real uniformly convex Banach space, and let K be a nonempty closed convex subset of E. Let 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 x1 ∈ K, the modified hybrid iteration scheme {x n } is defined as follows: , where A: K → K is an L-Lipschitzian mapping, 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 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.