An Implicit Iteration Process for Common Fixed Points of Two Infinite Families of Asymptotically Nonexpansive Mappings in Banach Spaces

Let K be a nonempty, closed, and convex subset of a real uniformly convex Banach space... Let {T-lambda}(lambda is an element of Lambda) and {S-lambda}(lambda is an element of Lambda) be two infinite families of asymptotically nonexpansive mappings from.. to itself with.. := {x is an element of K : T(lambda)x = x = S(lambda)x, lambda is an element of Lambda} not equal theta. For an arbitrary initial point x(0) is an element of K, {x(n)} is defined as follows: x(n) = alpha(n)x(n-1) + beta(n) (Tn-1*)(mn-1) gamma(n)(T-n*)(mn) y(n), y(n) = alpha(n)'x(n) + beta(n)' (Sn-1*)(mn-1)n =, 2, 3, ..., where T-n* = T-lambda in and S-n* = S-lambda in. with i(n) and m(n) satisfying the positive integer equation: n = i + (m - 1)m/2, m >= i; {T-lambda}(i=1)(infinity) and {S-lambda i}(i=1)(infinity) are two countable subsets of {alpha(n)} and {beta(n)} respectively; {alpha(n) }, {beta(n) }, {gamma(n)}, {alpha(n)'}, {beta(n)'}, and {gamma(n)'} are sequences in [delta, 1-delta] for some (0, 1), satisfying alpha(n) + beta(n) + gamma(n) = 1 and alpha(n)' + beta(n)' + gamma(n)' = 1. Under some suitable conditions, a strong convergence theorem for common fixed points of the mappings {T-lambda}(lambda is an element of Lambda) and {S-lambda}(lambda is an element of Lambda) is obtained. The results extend those of the authors whose related researches are restricted to the situation of finite families of asymptotically nonexpansive mappings.