Let C be a nonempty closed convex subset of a smooth Banach space E and let A be an accretive operator of C into E . We first introduce the problem of finding a point u ∈ C such that 〈 A u , J ( v − u ) 〉 ≥ 0  for all  v ∈ C , where J is the duality mapping of E . 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.