APPROXIMATIONS OF THE ITERATIVE SEQUENCES FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN BANACH SPACES
Shih-sen Chang, YeolJe Cho, Haiyun Zhou · 2008
Abstract. In this paper, we first introduce some iterative sequences of Halpern type for asymptotically nonexpansive mappings and nonex-pansive mappings in Banach spaces and then we discuss strong conver-gence for the iterative processes. The results presented in this paper extend, supplement and improve the correspoding main results of Re-ich [11], Shimizu and Takahashi [13], Shioji and Takahashi [15], [16] and Wittmann [18]. Throughout this paper, we assume that E is a real Banach space, E ¤ is the dual space of E, D is a nonempty subset of E and J: E → 2E ¤ is the normalized duality mapping defined by (1) J(x) = {f ∈ E¤, ⟨x, f ⟩ = ∥x∥∥f∥, ∥f ∥ = ∥x∥}, ∀x ∈ E. Definition 1. Let T: D → D be a mapping. (1) The mapping T is said to be asymptotically nonexpansive ([7]) if there exists a sequence {kn} ⊂ [1,∞) with limn!1 kn = 1 such that (2) ∥Tnx − Tny ∥ ≤ kn∥x − y∥ for all x, y ∈ D and n ∈ N. (2) The mapping T is said to be nonexpansive if the sequence {kn} appeared in (2) is a constant sequence {1}, i.e., ∥Tx − Ty ∥ ≤ ∥x − y∥, ∀x, y ∈ D. Definition 2. Let U = {x ∈ E: ∥x ∥ = 1}. The norm of E is said to be uniformly Gâteaux differentiable if, for each y ∈ U, the limit lim t!0 ∥x+ ty ∥ − ∥x∥ t exits uniformly for all x ∈ U. It is well-known that the following proposition is true: