Abstract
Let E be a real uniformly convex Banach space, and let{Ti : i ∈ I} be N nonexpansive mappings from E into itself with F = {x ∈ E : Tix = x, i ∈ I} ≠ ϕ, where I = {1, 2, …, N}. From an arbitrary initial point x1 ∈ E, hybrid iteration scheme {xn} is defined as follows: xn+1 = αnxn + (1 − αn)(Tnxn − λn+1μA(Tnxn)), n ≥ 1, where A : E → E is an L‐Lipschitzian mapping, Tn = Ti, i = n(mod N), 1 ≤ i ≤ N, μ > 0, {λn}⊂[0, 1), and {αn}⊂[a, b] for some a, b ∈ (0, 1). Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of the mappings {Ti : i ∈ I} are obtained. The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007).
Highlights
Let E be a Banach space endowed with the norm ·
A mapping T : E → E is said to be nonexpansive if T x − T y ≤ x − y for any x, y ∈ E
The interest and importance of construction of fixed points of nonexpansive mappings stem mainly from the fact that it may be applied in many areas, such as imagine recovery and Mathematical Problems in Engineering signal processing see, e.g., 1–3
Summary
Let E be a Banach space endowed with the norm ·. Let {xn} be the sequence generated from an arbitrary x1 ∈ E by xn 1 αnxn 1 − αn T λn 1 xn, n ≥ 1, 1.4 where T λn 1 xn T xn −λn 1μA T xn , μ > 0, and {αn} ⊂ 0, 1 and {λn} ⊂ 0, 1 satisfy the following conditions: 1 0 < α ≤ αn ≤ 1 for all n ≥ 1 and some α ∈ 0, 1 ; 2
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