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}{xn} defined by{yn=βnxn+(1−βn)Jrnxn,xn+1=αnu+(1−αn)yn, where JrnJrn is the resolvent of m-accretive operator A   and u∈Cu∈C is an arbitrary (but fixed) element in C   and sequences {αn}{αn} in (0,1)(0,1), {βn}{βn} in [0,1][0,1] is established. Under certain appropriate assumptions on the sequences {αn}{αn}, {βn}{βn} and {rn}{rn}, that {xn}{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.