ITERATIVE APPROXIMATION TO M-ACCRETIVE OPERATOR EQUATIONS IN BANACH SPACES
Jong An Park, Yang Seob Park · 1996
Abstract. In 1994 Z.Liang constructed an iterative method for the solution of nonlinear equations involving m-accretive operators in uniformly smooth Banach spaces. In this paper we apply the slight variants of Liang’s iterative methods and generalize the results of Z.Liang. Moreover our proof is more simple than Liang’s proof. 1. Preliminaries Let (X, ‖ · ‖) be a Banach space. A Banach space (X, ‖ · ‖) is called smooth if the norm of X is Gâteaux differentiable on X − {0}. The normalized duality mapping J is defined by J(x) = {x ∗ ∈ X∗|(x, x∗) = ‖x‖2, ‖x∗ ‖ = ‖x‖}, where X ∗ is the dual of X and ( , ) is the dual pairing. In a smooth Banach space J is single-valued. A Banach space (X, ‖ · ‖) is called uniformly smooth if X ∗ (the dual of X) is uniformly convex. In a uniformly smooth Banach space the duality mapping J is uniformly continuous on any bounded subset of X. Denote the closed ball {x ∈ X: ‖x − y ‖ ≤ r} by B(x, r). And the domain and range of a operator A is denoted by D(A), R(A) respec-tively. An operator A: D(A) ⊂ X − → X is said to be accretive if for any x, y in D(A) there exists j ∈ J(x − y) such that (Ax−Ay, j) ≥ 0. An accretive operator A is called m-accretive if R(A+ λI) = X for all λ> 0, where I denotes the identity operator. If A: D(A) ⊂ X − → X