Common fixed points of a finite family of asymptotically pseudocontractive maps
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Convergence theorems for approximation of common fixed points of a finite family of asymptotically pseudocontractive mappings are proved in Banach spaces using an averaging implicit iteration process. 1. Introduction Let E be a real Banach space and let J denote the normalized duality mapping from E into ∗ 2E given by J(x) = { f ∈ E∗ : x, f = x2 = f 2 }, where E∗ denotes the dual space of E and ·, · denotes the generalized duality pairing. If E∗ is strictly convex, then J is single-valued. In the sequel, we will denote the single-valued duality mapping by j. Let K be a nonempty subset of E. A mapping T : K → K is said to be asymptotically pseudocontractive (see, e.g., [3]) if there exists a sequence {an }∞ n=1 ⊆ [1, ∞) such that limn→∞ an = 1 and
T n x − T n y, j(x − y) ≤ an x − y 2 ,
∀n ≥ 1,
(1.1)
for all x, y ∈ K, j(x − y) ∈ J(x − y). In Hilbert spaces H, a self-mapping T of a nonempty subset K of H is asymptotically pseudocontractive if it satisfies the simpler inequality n T x − T n y 2 ≤ an x − y 2 + x − y − T n x − T n y 2 ,
∀n ≥ 1
(1.2)
for all x, y ∈ K and for some sequence {an }∞ n=1 ⊆ [1, ∞) such that limn→∞ an = 1. The class of asymptotically pseudocontractive mappings contains the important class of asymptotically nonexpansive mappings (i.e., mappings T : K → K such that n T x − T n y ≤ an x − y ,
∀n ≥ 1, ∀x, y ∈ K,
(1.3)
and for some sequence {an }∞ n=1 ⊆ [1, ∞) such that limn→∞ an = 1). T is called asymptotically quasi-nonexpansive if F(T) = {x ∈ K : Tx = x} = ∅ and (1.3) is satisfied for all x ∈ K and for all y ∈ F(T). If there exists L > 0 such that T n x − T n y ≤ Lx − y for Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:2 (2004) 81–88 2000 Mathematics Subject Classification: 47H09, 47H10, 47J05, 65J15 URL: http://dx.doi.org/10.1155/S1687182004312027
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Asymptotically pseudocontractive maps
all n ≥ 1 and for all x, y ∈ K, then T is said to be uniformly L-Lipschitzian. A mapping T : K → K is said to be semicompact (see, e.g., [4]) if for any sequence {xn }∞ n=1 in K ∞ such that limn→∞ xn − Txn = 0, there exists a subsequence {xn j }∞ of { x } n n=1 such j =1 ∞ that {xn j } j =1 converges strongly to some x∗ ∈ K. In [5], Xu and Ori introduced an implicit iteration process and proved weak convergence theorem for approximation of common fixed points of a finite family of nonexpansive mappings (i.e., a subclass of asymptotically nonexpansive mappings for which Tx − T y ≤ x − y ∀x, y ∈ K). In [4], Sun modified the implicit iteration process of Xu and Ori and applied the modified averaging iteration process for the approximation of fixed points of asymptotically quasi-nonexpansive maps. If K is a nonempty closed convex subset of E, and {Ti }Ni=1 is N asymptotically quasi-nonexpansive self-maps of K, then for x0 ∈ K and {αn }∞ n=1 ⊆ (0,1), the iteration process is generated as follows:
x1 = α1 x0 + 1 − α1 T1 x1 , x2 = α2 x1 + 1 − α2 T2 x2 , .. .
xN = αN xN −1 + 1 − αN TN xN
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