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An implicit iteration process for common fixed points of two infinite families of asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1266.47090
Summary: Let K be a nonempty, closed, and convex subset of a real uniformly convex Banach space E. Let {T λ } λΛ and {S λ } λΛ be two infinite families of asymptotically nonexpansive mappings from K to itself with F:={xK:T λ x=x=S λ x, λΛ}. For an arbitrary initial point x 0 K, {x n } is defined as follows: x n =α n x n-1 +β n (T n-1 * ) m n-1 x n-1 +γ n (T n * ) m n y n , y n =α n ' x n +β n ' (S n-1 * ) m n-1 x n-1 +γ n ' (S n * ) m n x n , n=1,2,3,, where T n * =T λ i n and S n * =S λ i n with i n and m n satisfying the positive integer equation: n=i+(m-1)m/2, mi; {T λ i } i=1 and {S λ i } i=1 are two countable subsets of {T λ } λΛ and {S λ } λΛ , respectively; {α n }, {β n }, {γ n }, {α n ' }, {β n ' }, and {γ n ' } are sequences in [δ,1-δ] for some δ(0,1), satisfying α n +β n +γ n =1 and α n ' +β n ' +γ n ' =1. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings {T λ } λΛ and {S λ } λΛ is obtained. The results extend those of the authors whose related works are restricted to the situation of finite families of asymptotically nonexpansive mappings.
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties