Approximation of a zero point of accretive operator in Banach spaces

This paper introduces a composite iteration scheme for approximating a zero point of accretive operator in the framework of uniformly smooth Banach spaces and the reflexive Banach space which has a weak continuous duality map, respectively. Strong convergence of the composite iteration scheme {xn} defined by yn = βnxn +(1−βn)Jrnxn, xn+1 = αnu+ (1− αn)yn, where Jrn is the resolvent of m-accretive operator A and u ∈ C is an arbitrary (but fixed) element in C and sequences {αn} in (0, 1), {βn} in [0, 1] is established. Under certain appropriate assumptions on the sequences {αn}, {βn} and {rn}, that {xn} defined by the above iteration scheme converges to a zero point of A is proved. The results improve and extend results of T.H. Kim, H.K. Xu and some others. © 2006 Elsevier Inc. All rights reserved.