Weak convergence of an iterative sequence for accretive operators in Banach spaces

Abstract Let "Equation missing" be a nonempty closed convex subset of a smooth Banach space "Equation missing" and let "Equation missing" be an accretive operator of "Equation missing" into "Equation missing". We first introduce the problem of finding a point "Equation missing" such that "Equation missing""Equation missing" where "Equation missing" is the duality mapping of "Equation missing". Next we study a weak convergence theorem for accretive operators in Banach spaces. This theorem extends the result by Gol'shteĭn and Tret'yakov in the Euclidean space to a Banach space. And using our theorem, we consider the problem of finding a fixed point of a strictly pseudocontractive mapping in a Banach space and so on.