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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.
MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties