Approximating Fixed Points of Nonexpansive Mappings in Hyperspaces
نویسندگان
چکیده
Let X be a nonempty compact subset of a Banach space (E,‖·‖), and let C(X) and CC(X) denote the families of all nonempty compact and all nonempty compact convex subsets of X , respectively. It is well known that (C(X),H) is compact, where H is the Hausdorff metric induced by ‖·‖. For A,B ∈ CC(X) and t ∈ R = (−∞,+∞), let A + B = {a + b : a∈ A, b ∈ B}, and let tA= {ta : a∈ A}. In the sequel, we assume that X is a nonempty compact convex subset of E. Hu and Huang [1] proved that (CC(X),H) is a compact subset of (C(X),H). It is clear that tA+ (1− t)B ∈ CC(X) for all A,B ∈ CC(X) and t ∈ [0,1]. That is, CC(X) has convexity structure. Let I be a nonempty subset of CC(X). A mapping T : (I,H)→(I,H) is said to be nonexpansive if H(TA,TB) ≤H(A,B) for all A,B ∈ I. Within the past 20 years or so, a few researchers have applied the Mann iteration method and the Ishikawa iteration method to approximate fixed points of nonexpansive mappings in several classes of subsets of Banach spaces. For details we refer to [2–11]. Recently, Hu and Huang [1] established the following result.
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