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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 Au,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'shteȋn 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.

Let C be a nonempty, closed convex subset of a Banach space E. In this paper, motivated by Alber [Ya.I. Alber, Metric and generalized projection operators in 15-50], we introduce the following iterative scheme for finding a solution of the variational inequality problem for an inversestrongly-monotone operator A in a Banach space: x 1 = x ∈ C and x n+1 = Π C J −1 (J x n − λ n Ax n ) for every n = 1, 2, . . . , where Π C is the generalized projection from E onto C, J is the duality mapping from E into E * and {λ n } is a sequence of positive real numbers. Then we show a weak convergence theorem (Theorem 3.1). Finally, using this result, we consider the convex minimization problem, the complementarity problem, and the problem of finding a point u ∈ E satisfying 0 = Au.

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