Hybrid Iteration Method for Fixed Points of Nonexpansive Mappings in Arbitrary Banach Spaces
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Research Article Hybrid Iteration Method for Fixed Points of Nonexpansive Mappings in Arbitrary Banach Spaces M. O. Osilike, F. O. Isiogugu, and P. U. Nwokoro Received 20 June 2007; Accepted 23 November 2007 Recommended by Nanjing Huang
We prove that recent results of Wang (2007) concerning the iterative approximation of fixed points of nonexpansive mappings using a hybrid iteration method in Hilbert spaces can be extended to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator. Copyright © 2007 M. O. Osilike et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let E be a real Banach space. A mapping T : E→E is said to be L-Lipschitzian if there exists L > 0 such that Tx − T y ≤ Lx − y ,
∀x, y ∈ E.
(1.1)
T is said to be nonexpansive if L = 1 in (1.1). Several authors have studied various methods for the iterative approximation of fixed points of nonexpansive mappings. Recently, Wang [1] studied the following iteration method in Hilbert spaces. The hybrid iteration method. Let H be a Hilbert space, T : H →H a nonexpansive mapping with F(T) = {x ∈ H : Tx = x}=∅, and F : H →H an L-Lipschitzian mapping which is also η-strongly monotone, where T is η-strongly monotone if there exists η > 0 such that
Tx − T y,x − y ≥ ηx − y 2 ,
∀x, y ∈ H.
(1.2)
2
Fixed Point Theory and Applications
∞ ∞ Let {αn }∞ n=1 and {λn }n=1 be real sequences in [0,1), and μ > 0, then the sequence {xn }n=1 is generated from an arbitrary x1 ∈ H by
xn+1 = αn xn + 1 − αn T λn+1 xn ,
n ≥ 1,
(1.3)
where T λn+1 xn := Txn − λn+1 μF(Txn ), μ > 0. Wang’s work was motivated by earlier results of Xu and Kim [2] and Yamada [3], in addition to several other related results. Using this iteration method, Wang proved the following main results. Lemma 1.1 (see [1, page 3]). Let H be a Hilbert space, T : H →H a nonexpansive mapping with F(T) = {x ∈ H : Tx = x} = ∅, and F : H →H an η-strongly monotone and LLipschitzian mapping. Let {xn }∞ n=1 be the sequence generated from an arbitrary x1 ∈ H by
xn+1 = αn xn + 1 − αn T λn+1 xn ,
n ≥ 1,
(1.4)
∞ where T λn+1 xn := Txn − λn+1 μF(Txn ), μ > 0, and let {αn }∞ n=1 and {λn }n=1 be real sequences in [0,1) satisfying the following conditions: (i) 0 < α ≤ αn ≤ β < 1, for some α,β ∈ (0,1), (ii) ∞ n =1 λ n < ∞ , (iii) 0 < μ < 2η/L2 . Then, (a) lim n→∞ xn − x∗ exists for each x∗ ∈ F(T), (b) lim n→∞ xn − Txn = 0. ∞ ∞ ∞ Theorem 1.2 (see [1, page 5]). Let H, T, F(T), F, {T λn+1 }∞ n=1 , {xn }n=1 , {αn }n=1 , {λn }n=1 , μ,α, and β be as in Lemma 1.1. Let {xn }∞ n=1 be the sequence generated from an arbitrary x1 ∈ H by
xn+1 = αn xn + 1 − αn T λn+1 xn ,
n ≥ 1.
(1.5)
Then, (a) {xn }∞ n=1 converges weakly to a fixed point of T, (b) {xn }∞ n=1 converges strongly to a fixed point of T if and only if lim inf n→∞ d(xn ,F(T)) = 0, where d(x,F(T)) := inf
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