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

Correspondence: dwq1273@126. com College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P. R. China Abstract Let E be a real uniformly convex Banach space, and let K be a nonempty closed convex subset of E. Let {Ti}i=1 be a sequence of nonexpansive mappings from K to itself with F :={x Î K :Tix = x, ∀i ≥ 1}≠ ∅. For an arbitrary initial point x1 Î K, the modified hybrid iteration scheme {xn} is defined as follows: xn+1 = αnxn + (1 − αn) ( T∗ nxn − λn+1μA(T∗ nxn) ) , n ≥ 1 , where A: K ® K is an LLipschitzian mapping, T∗ n = Ti with i satisfying: n = [(k-i+1)(i+k)/2]+[1+(i-1)(i+2)/2],k ≥ i-1(i = 1,2,...),{ln} ⊂ [0,1), and {an} is a sequence in [a, 1 a] for some a Î (0,1). Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of the nonexpansive mappings {Ti}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.