Strong Convergence of Modified Mann Iterations for Nonexpansive Mappings

Abstract. In this paper, we introduce a modified Mann iterative process as following ⎧⎪⎨ x0 ∈ C chosen arbitrarily, yn = αnxn + (1 − αn)Txn, xn+1 = βnf(xn) + (1 − βn)yn, where f is a contraction and sequences {αn}∞n=0 and {βn}∞n=0 in (0, 1). Under certain appropriate assumptions on the sequences {αn} and {βn}, we prove the sequence {xn} defined by the above iteration scheme converges strongly to a fixed point of T which solves some variational inequality. Our results improve and extend the results of Kim, Xu and some others