Approximation of a zero point of accretive operator in Banach spaces

Abstract

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 { x n } defined by { y n = β n x n + ( 1 − β n ) J r n x n , x n + 1 = α n u + ( 1 − α n ) y n , where J r n is the resolvent of m-accretive operator A and u ∈ C is an arbitrary (but fixed) element in C and sequences { α n } in ( 0 , 1 ) , { β n } in [ 0 , 1 ] is established. Under certain appropriate assumptions on the sequences { α n } , { β n } and { r n } , that { x n } 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.