Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 238053, 5 pages http://dx.doi.org/10.1155/2014/238053
Research Article Hybrid Implicit Iteration Process for a Finite Family of Non-Self-Nonexpansive Mappings in Uniformly Convex Banach Spaces Qiaohong Jiang, Jinghai Wang, and Jianhua Huang Institute of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, China Correspondence should be addressed to Qiaohong Jiang;
[email protected] Received 30 April 2014; Accepted 29 June 2014; Published 10 July 2014 Academic Editor: Luigi Muglia Copyright © 2014 Qiaohong Jiang 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. Weak and strong convergence theorems are established for hybrid implicit iteration for a finite family of non-self-nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper extend and improve some recent results.
1. Introduction The convergence problem of an implicit (or nonimplicit) iterative process to a common fixed point for a finite family of nonexpansive mappings (or asymptotically nonexpansive mappings) in Hilbert spaces or uniformly convex Banach spaces has been considered by many authors (see [1–9]). In 2001, Xu and Ori [1] introduced the following implicit iteration scheme for common fixed points of a finite family of nonexpansive mappings {𝑇𝑖 }𝑁 𝑖=1 in Hilbert spaces: 𝑥𝑛 = 𝛼𝑛 𝑥𝑛−1 + (1 − 𝛼𝑛 ) 𝑇𝑛 𝑥𝑛 ,
𝑛 ≥ 1,
(1)
where 𝑇𝑛 = 𝑇𝑛(mod 𝑁) , and they proved the weak convergence theorem. In 2005, Zeng and Yao [2] introduced the following implicit iteration process with a perturbed mapping 𝐹 in Hilbert space 𝐻. For an arbitrary initial point 𝑥0 ∈ 𝐻, the sequence {𝑥𝑛 }∞ 𝑛=1 is generated as follows: 𝑥𝑛 = 𝛼𝑛 𝑥𝑛−1 + (1 − 𝛼𝑛 ) [𝑇𝑛 𝑥𝑛 − 𝜆 𝑛 𝜇𝐹 (𝑇𝑛 𝑥𝑛 )] ,
𝑛 ≥ 1, (2)
where 𝑇𝑛 = 𝑇𝑛(mod 𝑁) . Using this iteration process, they proved the following weak and strong convergence theorems for a family of nonexpansive mappings in Hilbert spaces.
Theorem 1 (see [2]). Let 𝐻 be a real Hilbert space and let 𝐹 : 𝐻 → 𝐻 be a mapping such that, for some constants 𝑘, 𝜂 > 0, 𝐹 is 𝑘-Lipschitzian and 𝜂-strongly monotone. Let {𝑇𝑛 }𝑁 𝑛=1 be 𝑁 nonexpansive self-mappings of 𝐻 such that ⋂𝑁 𝐹𝑖𝑥(𝑇 𝑛 ) ≠ 0. 𝑛=1 ∞ 2 Let 𝜇 ∈ (0, 2𝜂/𝑘 ) and 𝑥0 ∈ 𝐻. Let {𝛼𝑛 }𝑛=1 ⊂ (0, 1) and ∞ {𝜆 𝑛 }∞ 𝑛=1 ⊂ [0, 1) satisfying conditions ∑𝑛=1 𝜆 𝑛 < ∞ and 𝛼 ≤ 𝛼𝑛 ≤ 𝛽, 𝑛 ≥ 1, for some 𝛼, 𝛽 ∈ (0, 1). Then the sequence {𝑥𝑛 }∞ 𝑛=1 defined by (2) converges weakly to a common fixed point of the mappings {𝑇𝑛 }𝑁 𝑛=1 . Theorem 2 (see [2]). Let 𝐻 be a real Hilbert space and let 𝐹 : 𝐻 → 𝐻 be a mapping such that, for some constants 𝑘, 𝜂 > 0, 𝐹 is 𝑘-Lipschitzian and 𝜂-strongly monotone. Let {𝑇𝑛 }𝑁 𝑛=1 be 𝑁 nonexpansive self-mappings of 𝐻 such that ⋂𝑁 𝐹𝑖𝑥(𝑇 𝑖 ) ≠ 0. 𝑖=1 ⊂ (0, 1) Let 𝜇 ∈ (0, 2𝜂/𝑘2 ) and 𝑥0 ∈ 𝐻. Let {𝛼𝑛 }∞ 𝑛=1 ∞ ⊂ [0, 1) satisfying conditions 𝜆 < ∞ and {𝜆 𝑛 }∞ ∑ 𝑛=1 𝑛=1 𝑛 and 𝛼 ≤ 𝛼𝑛 ≤ 𝛽, 𝑛 ≥ 1, for some 𝛼, 𝛽 ∈ (0, 1). Then the sequence {𝑥𝑛 }∞ 𝑛=1 defined by (2) converges strongly to a common fixed point of the mappings {𝑇𝑛 }𝑁 𝑛=1 if and only if lim inf 𝑛 → ∞ 𝑑(𝑥𝑛 , ⋂𝑁 𝐹𝑖𝑥(𝑇 )) = 0. 𝑛 𝑛=1 The purpose of this paper is to extend Theorems 1 and 2 from Hilbert spaces to uniformly convex Banach spaces and from self-mappings to non-self-mappings. Our results are more general and applicable than the results of Zeng and Yao [2] because the strong monotonicity condition imposed on 𝐹 by them is not required in this paper.
2
Journal of Applied Mathematics
2. Preliminaries Throughout this paper, we assume that 𝐸 is a real Banach space. 𝑇 : 𝐷(𝑇) ⊆ 𝐸 → 𝐸 is a mapping, where 𝐷(𝑇) is the domain of 𝑇. 𝐹(𝑇) denotes the set of fixed points of a mapping 𝑇. Recall that 𝐸 is said to satisfy Opial’s condition [10], if, for each sequence {𝑥𝑛 } in 𝐸, the condition that the sequence 𝑥𝑛 → 𝑥 weakly implies that lim sup 𝑥𝑛 − 𝑥 < lim sup 𝑥𝑛 − 𝑦 , 𝑛→∞
𝑛→∞
(3)
for all 𝑦 ∈ 𝐸 with 𝑦 ≠ 𝑥. Definition 3. Let 𝐾 be a closed subset of 𝐸 and let 𝑇 : 𝐾 → 𝐸, 𝑓 : 𝐸 → 𝐸 be two mappings.
In order to prove our main results of this paper, we need the following lemmas. Lemma 6 (see [4]). Let {𝑎𝑛 }, {𝑏𝑛 }, and {𝛿𝑛 } be three nonnegative sequences satisfying 𝑎𝑛+1 ≤ (1 + 𝛿𝑛 ) 𝑎𝑛 + 𝑏𝑛 ,
(3) 𝑇 is said to be nonexpansive, if ‖𝑇𝑥 − 𝑇𝑦‖ ≤ ‖𝑥 − 𝑦‖ for any 𝑥, 𝑦 ∈ 𝐸. (4) 𝑓 is said to be 𝐿-Lipschitzian if there exists constant 𝐿 > 0 such that ‖𝑓𝑥 − 𝑓𝑦‖ ≤ 𝐿‖𝑥 − 𝑦‖ for any 𝑥, 𝑦 ∈ 𝐸. Definition 4. A nonempty subset 𝐾 of 𝐸 is said to be a retract of 𝐸, if there exists a continuous mapping 𝑟 : 𝐸 → 𝐾 such that 𝑟𝑥 = 𝑥, for any 𝑥 ∈ 𝐾. And 𝑟 is called the retraction of 𝐸 onto 𝐾. Remark 5 (see [3]). It is known that every nonempty closed convex subset 𝐾 of a uniformly convex Banach space 𝐸 is a retract of 𝐸 and the retraction 𝑟 is a nonexpansive mapping. Suppose that 𝐾 is a nonempty closed convex subset of 𝐸, which is also a retract of 𝐸. Let 𝑥0 ∈ 𝐾 be any given point. Let {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} : 𝐾 → 𝐸 be 𝑁 nonexpansive mappings with 𝑇𝑛 = 𝑇𝑛(mod 𝑁) . Let 𝑓 : 𝐸 → 𝐸 be an 𝐿-Lipschitzian mapping. Assume that {𝛼𝑛 } is a sequence in (0,1) and {𝜆 𝑛 } ⊂ [0, 1), given 𝜇 > 0. Then the sequence {𝑥𝑛 } defined by 𝑥𝑛 = 𝛼𝑛 𝑥𝑛−1 + (1 − 𝛼𝑛 ) 𝑟𝑇𝑛𝜆 𝑛 𝑥𝑛 := 𝛼𝑛 𝑥𝑛−1 + (1 − 𝛼𝑛 ) 𝑟 [𝑇𝑛 𝑥𝑛 − 𝜆 𝑛 𝜇𝑓 (𝑇𝑛 𝑥𝑛 )] ,
𝑛 ≥ 1, (4)
is called hybrid implicit iteration for a finite family of nonexpansive mappings {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} in Banach spaces, where 𝑛 𝑇𝑛𝑛 = 𝑇𝑛(mod 𝑁) and 𝜇 is a fixed constant. The purpose of this paper is to study weak and strong convergence of hybrid implicit iteration {𝑥𝑛 } defined by (4) to a common fixed point of {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} : 𝐾 → 𝐸 in Banach spaces. The results we obtained in this paper extend and improve the corresponding results of Xu and Ori [1], Zeng and Yao [2], and others.
(5)
∞ If ∑∞ 𝑛=1 𝛿𝑛 < ∞ and ∑𝑛=1 𝑏𝑛 < ∞, then lim𝑛 → ∞ 𝑎𝑛 exists.
Lemma 7 (see [5]). Let 𝐸 be a uniformly convex Banach space. Let 𝑏, 𝑐 be two constants with 0 < 𝑏 < 𝑐 < 1. Suppose that {𝑡𝑛 } is a sequence in [b, c] and {𝑥𝑛 }, {𝑦𝑛 } are two sequences in 𝐸. Then the conditions
lim 𝑡 𝑥 𝑛→∞ 𝑛 𝑛
+ (1 − 𝑡𝑛 ) 𝑦𝑛 = 𝑑,
lim sup 𝑥𝑛 ≤ 𝑑,
(1) 𝑇 is said to be demiclosed at the origin, if, for each sequence {𝑥𝑛 } in 𝐾, the condition 𝑥𝑛 → 𝑥0 weakly and 𝑇𝑥𝑛 → 0 strongly implies 𝑇𝑥0 = 0. (2) 𝑇 is said to be semicompact, if, for any bounded sequence {𝑥𝑛 } in 𝐾, such that ‖𝑥𝑛 − 𝑇𝑥𝑛 ‖ → 0 (𝑛 → ∞), there exists a subsequence {𝑥𝑛𝑖 } ⊂ {𝑥𝑛 } converging to some 𝑥∗ in 𝐾.
∀𝑛 = 1, 2, . . . .
𝑛→∞
(6)
lim sup 𝑦𝑛 ≤ 𝑑 𝑛→∞
imply that lim𝑛 → ∞ ‖𝑥𝑛 −𝑦𝑛 ‖ = 0, where 𝑑 ≥ 0 is some constant. Lemma 8 (see [6]). Let 𝐾 be a nonempty closed convex subset of real Banach space 𝐸 and 𝑇 : 𝐾 → 𝐸 a nonexpansive mapping. If 𝑇 has a fixed point, then 𝐼 − 𝑇 is demiclosed at zero, where 𝐼 is the identity mapping of 𝐸.
3. Main Results Theorem 9. Suppose that 𝐸 is a real uniformly convex Banach space satisfying Opial’s condition and 𝐾 is a nonempty closed convex subset of 𝐸 with a nonexpansive retraction 𝑟 : 𝐸 → 𝐾. Let {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} : 𝐾 → 𝐸 be 𝑁 nonexpansive mappings with 𝐹 = ⋂𝑁 𝑛=1 𝐹(𝑇𝑛 ) ≠ 0 and let 𝑓 : 𝐸 → 𝐸 be an 𝐿Lipschitzian mapping. Assume that {𝛼𝑛 } is a sequence in (0, 1) and {𝜆 𝑛 } ⊂ [0, 1) satisfying the following conditions: (i) ∑∞ 𝑛=1 𝜆 𝑛 < ∞; (ii) there exist constants 𝜏1 , 𝜏2 ∈ (0, 1) such that 𝜏1 ≤ (1 − 𝛼𝑛 ) ≤ 𝜏2 ,
∀𝑛 ≥ 1.
(7)
Then, the implicit iterative process {𝑥𝑛 } defined by (4) converges weakly to a common fixed point of {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} in 𝐸. Proof. Since 𝐹 = ⋂𝑁 𝑛=1 𝐹(𝑇𝑖 ) ≠ 0, for each 𝑞 ∈ 𝐹, we have 𝜆 𝑥𝑛 − 𝑞 = 𝛼𝑛 (𝑥𝑛−1 − 𝑞) + (1 − 𝛼𝑛 ) (𝑟𝑇𝑛 𝑛 𝑥𝑛 − 𝑞) = 𝛼𝑛 (𝑥𝑛−1 − 𝑞) + (1 − 𝛼𝑛 ) (𝑟𝑇𝑛𝜆 𝑛 𝑥𝑛 − 𝑟𝑞) ≤ 𝛼𝑛 𝑥𝑛−1 − 𝑞 + (1 − 𝛼𝑛 ) 𝑇𝑛𝜆 𝑛 𝑥𝑛 − 𝑞 ≤ 𝛼𝑛 𝑥𝑛−1 − 𝑞 + (1 − 𝛼𝑛 ) 𝑇𝑛 𝑥𝑛 − 𝑞 + (1 − 𝛼𝑛 ) 𝜆 𝑛 𝜇 𝑓 (𝑇𝑛 𝑥𝑛 )
Journal of Applied Mathematics
3
≤ 𝛼𝑛 𝑥𝑛−1 − 𝑞 + (1 − 𝛼𝑛 ) 𝑥𝑛 − 𝑞 + (1 − 𝛼𝑛 ) 𝜆 𝑛 𝜇 𝑓 (𝑇𝑛 𝑥𝑛 ) − 𝑓 (𝑞) + (1 − 𝛼𝑛 ) 𝜆 𝑛 𝜇 𝑓 (𝑞) ≤ 𝛼𝑛 𝑥𝑛−1 − 𝑞 + (1 − 𝛼𝑛 ) 𝑥𝑛 − 𝑞 + 𝜆 𝑛 𝜇𝐿 𝑥𝑛 − 𝑞 + 𝜆 𝑛 𝜇 𝑓 (𝑞) .
by condition (i) and (8) and (15), that lim sup 𝑟𝑇𝑛𝜆 𝑛 𝑥𝑛 − 𝑞 = lim sup 𝑟𝑇𝜆𝑛 𝑛 𝑥𝑛 − 𝑟𝑞 𝑛→∞
𝑛→∞
≤ lim sup 𝑇𝑛𝜆 𝑛 𝑥𝑛 − 𝑞 𝑛→∞
≤ lim sup 𝑇𝑛 𝑥𝑛 − 𝑇𝑛 𝑞 𝑛→∞
(8)
+ lim sup𝜆 𝑛 𝜇 𝑓 (𝑇𝑛 𝑥𝑛 ) − 𝑓 (𝑇𝑛 𝑞)
Simplifying we have
𝑛→∞
+ lim sup𝜆 𝑛 𝜇 𝑓 (𝑇𝑛 𝑞)
𝜆 𝜇𝐿 𝜆 𝜇 𝑥𝑛 − 𝑞 ≤ 𝑥𝑛−1 − 𝑞 + 𝑛 𝑥𝑛 − 𝑞 + 𝑛 𝑓 (𝑞) . (9) 𝛼𝑛 𝛼𝑛
𝑛→∞
≤ lim sup 𝑥𝑛 − 𝑞
By condition (ii), 1 − 𝜏2 ≤ 𝛼𝑛 ; hence from (9) we have 𝜆 𝜇𝐿 𝑥𝑛 − 𝑞 ≤ 𝑥𝑛−1 − 𝑞 + 𝑛 1 − 𝜏2
𝜆 𝜇 𝑥𝑛 − 𝑞 + 𝑛 1 − 𝜏2
𝑛→∞
1 − 𝜏2 𝑥 − 𝑞 𝑥𝑛 − 𝑞 ≤ 1 − 𝜏2 − 𝜆 𝑛 𝜇𝐿 𝑛−1
+ lim sup𝜆 𝑛 𝜇 𝑓 (𝑞) ≤ 𝑑. 𝑛→∞
(17) Since 𝐸 is a uniformly convex Banach space, from (15)–(17) and Lemma 7 we know that
lim 𝑥𝑛−1 𝑛→∞
𝜆𝑛𝜇 + 𝑓 (𝑞) 1 − 𝜏2 − 𝜆 𝑛 𝜇𝐿
+
𝑛→∞
(10)
By condition (i), we know that 𝜆 𝑛 → 0 and 𝜆 𝑛 𝜇𝐿 → 0 as 𝑛 → ∞; therefore there exists a positive integer 𝑛0 such that 𝜆 𝑛 𝜇𝐿 ≤ (1 − 𝜏2 )/2, for all 𝑛 ≥ 𝑛0 ; then we have
= (1 +
+ lim sup𝜆 𝑛 𝜇𝐿 𝑥𝑛 − 𝑞
𝑓 (𝑞) .
− 𝑟𝑇𝑛𝜆 𝑛 𝑥𝑛 = 0.
(18)
By (18), we have that
𝜆 𝑛 𝜇𝐿 ) 𝑥 − 𝑞 1 − 𝜏2 − 𝜆 𝑛 𝜇𝐿 𝑛−1
(11)
𝜆𝑛𝜇 𝑓 (𝑞) . 1 − 𝜏2 − 𝜆 𝑛 𝜇𝐿
It follows from (11) that
𝜆 𝑥𝑛 − 𝑥𝑛−1 = (𝛼𝑛 − 1) 𝑥𝑛−1 + (1 − 𝛼𝑛 ) 𝑟𝑇𝑛 𝑛 𝑥𝑛 ≤ (1 − 𝛼𝑛 ) 𝑥𝑛−1 − 𝑟𝑇𝑛𝜆 𝑛 𝑥𝑛 → 0, (𝑛 → ∞) . (19) It follows from (18) and (19) that
2𝜆 𝜇𝐿 2𝜆 𝑛 𝜇 𝑥𝑛 − 𝑞 ≤ (1 + 𝑛 ) 𝑥𝑛−1 − 𝑞 + 𝑓 (𝑞) , 1 − 𝜏2 1 − 𝜏2 (12)
lim 𝑥𝑛 𝑛→∞
∀𝑛 ≥ 𝑛0 . Taking 𝑎𝑛 = ‖𝑥𝑛 −𝑞‖, 𝛿𝑛 = 2𝜆 𝑛 𝜇𝐿/(1−𝜏2 ), and 𝑏𝑛 = (2𝜆 𝑛 𝜇/(1− 𝜏2 ))‖𝑓(𝑞)‖ and by using ∑∞ 𝑛=1 𝜆 𝑛 < ∞, it is easy to see that ∞
∑ 𝛿𝑛 < ∞,
𝑛=1
lim 𝑥𝑛 𝑛→∞
∞
∑ 𝑏𝑛 < ∞.
(13)
𝑛=1
It follows from Lemma 6 that lim𝑛 → ∞ ‖𝑥𝑛 − 𝑞‖ exists for each 𝑞 ∈ 𝐹. Hence, there exists 𝑀 > 0, such that (14) 𝑥𝑛 − 𝑞 ≤ 𝑀, 𝑛 ≥ 0.
− 𝑟𝑇𝑛𝜆 𝑛 𝑥𝑛 ≤ lim (𝑥𝑛 − 𝑥𝑛−1 𝑛→∞ + 𝑥𝑛−1 − 𝑟𝑇𝑛𝜆 𝑛 𝑥𝑛 ) = 0, (20) − 𝑥𝑛+𝑗 = 0,
∀𝑗 = 1, 2, . . . , 𝑁.
(21)
By (14), we know 𝑓 (𝑇𝑛 𝑥𝑛 ) ≤ 𝑓 (𝑇𝑛 𝑥𝑛 ) − 𝑓 (𝑞) + 𝑓 (𝑞)
≤ 𝐿 𝑥𝑛 − 𝑞 + 𝑓 (𝑞) ≤ 𝐿𝑀 + 𝑓 (𝑞) .
(22)
From (20), (22), and condition (i) we have
We can assume that
lim 𝑥 𝑛→∞ 𝑛
− 𝑞 = 𝑑,
(15)
where 𝑑 ≥ 0 is some number. Since {‖𝑥𝑛 − 𝑞‖} is a convergent sequence, {𝑥𝑛 } is a bounded sequence in 𝐾. Since 𝜆 𝑥𝑛 − 𝑞 = 𝛼𝑛 (𝑥𝑛−1 − 𝑞) + (1 − 𝛼𝑛 ) (𝑟𝑇𝑛 𝑛 𝑥𝑛 − 𝑞) , (16)
lim 𝑥 𝑛→∞ 𝑛
− 𝑇𝑛 𝑥𝑛
≤ lim (𝑥𝑛 − 𝑇𝑛𝜆 𝑛 𝑥𝑛 + 𝑇𝑛𝜆 𝑛 𝑥𝑛 − 𝑇𝑛 𝑥𝑛 ) (23) 𝑛→∞ ≤ lim (𝑥𝑛 − 𝑇𝑛𝜆 𝑛 𝑥𝑛 + 𝜆 𝑛 𝜇 (𝐿𝑀 + 𝑓 (𝑞))) = 0. 𝑛→∞
4
Journal of Applied Mathematics
Consequently, for any 𝑗 = 1, 2, . . . , 𝑁, from (21) and (23) we have 𝑥 − 𝑇 𝑥 ≤ 𝑥 − 𝑥 + 𝑥 − 𝑇 𝑥 𝑛 𝑛+𝑗 𝑛 𝑛+𝑗 𝑛+𝑗 𝑛+𝑗 𝑛 𝑛+𝑗 + 𝑇𝑛+𝑗 𝑥𝑛+𝑗 − 𝑇𝑛+𝑗 𝑥𝑛 ≤ 2 𝑥𝑛 − 𝑥𝑛+𝑗 + 𝑥𝑛+𝑗 − 𝑇𝑛+𝑗 𝑥𝑛+𝑗 → 0, (𝑛 → ∞) . (24) This implies that the sequence 𝑁
∞ ⋃{𝑥𝑛 − 𝑇𝑛+𝑗 𝑥𝑛 }𝑛=1 → 0
(𝑛 → ∞) .
(25)
𝑗=1
Since, for each 𝑙 = 1, 2, . . . , 𝑁, {‖𝑥𝑛 − 𝑇𝑙 𝑥𝑛 ‖}∞ 𝑛=1 is a subse∞ quence of ⋃𝑁 {‖𝑥 − 𝑇 𝑥 ‖} , therefore we have 𝑛 𝑛+𝑗 𝑛 𝑛=1 𝑗=1
lim 𝑥 𝑛→∞ 𝑛
− 𝑇𝑙 𝑥𝑛 = 0,
∀𝑙 = 1, 2, . . . 𝑁.
(26)
Since 𝐸 is uniformly convex, every bounded subset of 𝐸 is weakly compact. Since {𝑥𝑛 } is a bounded sequence in 𝐸, there exists a subsequence {𝑥𝑛𝑗 } ⊂ {𝑥𝑛 } such that {𝑥𝑛𝑗 } converges weakly to 𝑢 ∈ 𝐸. From (26) we have lim 𝑥𝑛𝑗 − 𝑇𝑙 𝑥𝑛𝑗 = 0, 𝑛→∞
∀𝑙 = 1, 2, . . . 𝑁.
(27)
By Lemma 8, we know that 𝑢 ∈ 𝐹(𝑇𝑙 ). By the arbitrariness of 𝑙 ∈ {1, 2, . . . 𝑁}, we have that 𝑢 ∈ 𝐹 = ⋂𝑁 𝑙=1 𝐹(𝑇𝑖 ). Suppose that there exists some subsequence {𝑥𝑛𝑘 } ⊂ {𝑥𝑛 } such that 𝑥𝑛𝑘 → V ∈ 𝐸 weakly and V ≠ 𝑢. From Lemma 8, V ∈ 𝐹. By (12) we know that lim𝑛 → ∞ ‖𝑥𝑛 −𝑢‖ and lim𝑛 → ∞ ‖𝑥𝑛 −V‖ exist. Since 𝐸 satisfies Opial’s condition, we have lim 𝑥𝑛 − 𝑢 = lim 𝑥𝑛𝑗 − 𝑢 < lim 𝑥𝑛𝑗 − V 𝑛→∞ 𝑗→∞ 𝑗→∞ = lim 𝑥𝑛 − V = lim 𝑥𝑛𝑘 − V 𝑛→∞ 𝑘→∞
Then, the implicit iterative process {𝑥𝑛 } defined by (4) converges strongly to a common fixed point of {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} if and only if lim inf 𝑛 → ∞ 𝑑(𝑥𝑛 , 𝐹(𝑇𝑙 )) = 0 (for all 𝑙 = 1, 2, . . . , 𝑁). Proof. From (12) and (14) in the proof of Theorem 9, we have 𝑥𝑛 − 𝑞 ≤ (1 + 𝛿𝑛 ) 𝑥𝑛−1 − 𝑞 + 𝑏𝑛 ≤ 𝑥𝑛−1 − 𝑞 + 𝑀𝛿𝑛 + 𝑏𝑛 = 𝑥𝑛−1 − 𝑞 + 𝛽𝑛 ,
(30)
where 𝛿𝑛 = 2𝜆 𝑛 𝜇𝐿/(1 − 𝜏2 ), 𝑏𝑛 = (2𝜆 𝑛 𝜇/(1 − 𝜏2 ))‖𝑓(𝑞)‖, and 𝛽𝑛 = 𝑀𝛿𝑛 + 𝑏𝑛 . Hence, 𝑑(𝑥𝑛 , 𝐹) ≤ 𝑑(𝑥𝑛−1 , 𝐹) + 𝛽𝑛 . Since ∑∞ 𝑛=1 𝛽𝑛 < ∞, it follows from Lemma 6 that lim𝑛 → ∞ 𝑑(𝑥𝑛 , 𝐹) exists. If {𝑥𝑛 }∞ 𝑛=1 converges strongly to a common fixed point 𝑝 of {𝑇1 , 𝑇2 , . . . , 𝑇𝑁}, then lim𝑛 → ∞ ‖𝑥𝑛 − 𝑝‖ = 0. Since 0 ≤ 𝑑 (𝑥𝑛 , 𝐹) ≤ 𝑥𝑛 − 𝑝 , (31) we know that lim inf 𝑛 → ∞ 𝑑(𝑥𝑛 , 𝐹) = 0. Conversely, suppose lim inf 𝑛 → ∞ 𝑑(𝑥𝑛 , 𝐹) = 0; then lim𝑛 → ∞ 𝑑(𝑥𝑛 , 𝐹) = 0. Moreover, we have ∑∞ 𝑛=1 𝛽𝑛 < ∞; thus for arbitrary 𝜖 > 0, there exists a positive integer 𝑁 such that 𝑑(𝑥𝑛 , 𝐹) < 𝜖/4 and ∑∞ 𝑗=𝑛 𝛽𝑗 < 𝜖/4 for all 𝑛 ≥ 𝑁. It follows from (30) that, for all 𝑛, 𝑚 ≥ 𝑁 and for all 𝑝 ∈ 𝐹, we have 𝑥𝑛 − 𝑥𝑚 ≤ 𝑥𝑛 − 𝑝 + 𝑥𝑚 − 𝑝 𝑛
𝑚
𝑗=𝑁+1
𝑗=𝑁+1
≤ 𝑥𝑁 − 𝑝 + ∑ 𝛽𝑗 + 𝑥𝑁 − 𝑝 + ∑ 𝛽𝑗 ∞
≤ 2 𝑥𝑁 − 𝑝 + 2 ∑ 𝛽𝑗 . 𝑗=𝑁
(32) Taking infimum over all 𝑝 ∈ 𝐹, we obtain ∞
𝑥𝑛 − 𝑥𝑚 ≤ 2𝑑 (𝑥𝑁, 𝐹) + 2 ∑ 𝛽𝑗 < 𝜖, 𝑗=𝑁
∀𝑛, 𝑚 ≥ 𝑁. (33)
(28)
Thus, {𝑥𝑛 }∞ 𝑛=1 is a Cauchy sequence. Letting lim𝑛 → ∞ 𝑥𝑛 = 𝑢, then, from Lemma 8, we have 𝑢 ∈ 𝐹. This completes the proof of the theorem.
which is a contradiction. Hence 𝑢 = V. This implies that {𝑥𝑛 } converges weakly to a common fixed point of {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} in 𝐸.
Theorem 11. Suppose that 𝐸 is a real uniformly convex Banach space and 𝐾 is a nonempty closed convex nonexpansive retract of 𝐸 with 𝑟 : 𝐸 → 𝐾 as a nonexpansive retraction. Let {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} : 𝐾 → 𝐸 be 𝑁 nonexpansive mappings with 𝐹 = ⋂𝑁 𝑛=1 𝐹(𝑇𝑛 ) ≠ 0 and at least there exists a 𝑇𝑙 , 1 ≤ 𝑙 ≤ 𝑁, which is semicompact. Let 𝑓 : 𝐸 → 𝐸 be 𝐿-Lipschitzian mapping. Assume that {𝛼𝑛 } is a sequence in (0, 1) and {𝜆 𝑛 } ⊂ [0, 1) satisfying the following conditions:
< lim 𝑥𝑛𝑘 − 𝑢 = lim 𝑥𝑛 − 𝑢 , 𝑛→∞ 𝑘→∞
Theorem 10. Suppose that 𝐸 is a real uniformly convex Banach space and 𝐾 is a nonempty closed convex nonexpansive retract of 𝐸 with 𝑟 : 𝐸 → 𝐾 as a nonexpansive retraction. Let {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} : 𝐾 → 𝐸 be 𝑁 nonexpansive mappings with 𝐹 = ⋂𝑁 𝑛=1 𝐹(𝑇𝑛 ) ≠ 0 and let 𝑓 : 𝐸 → 𝐸 be an 𝐿Lipschitzian mapping. Assume that {𝛼𝑛 } is a sequence in (0, 1) and {𝜆 𝑛 } ⊂ [0, 1) satisfying the following conditions: (i)
∑∞ 𝑛=1
𝜆 𝑛 < ∞; ∀𝑛 ≥ 1.
(ii) there exist constants 𝜏1 , 𝜏2 ∈ (0, 1) such that 𝜏1 ≤ (1 − 𝛼𝑛 ) ≤ 𝜏2 ,
(ii) there exist constants 𝜏1 , 𝜏2 ∈ (0, 1) such that 𝜏1 ≤ (1 − 𝛼𝑛 ) ≤ 𝜏2 ,
(i) ∑∞ 𝑛=1 𝜆 𝑛 < ∞;
(29)
∀𝑛 ≥ 1.
(34)
Then, the implicit iterative process {𝑥𝑛 } defined by (4) converges strongly to a common fixed point of {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} in 𝐸.
Journal of Applied Mathematics
5
Proof. From the proof of Theorem 9, {𝑥𝑛 } is bounded, and lim𝑛 → ∞ ‖𝑥𝑛 − 𝑇𝑙 𝑥𝑛 ‖ = 0, for all 𝑙 = 1, 2, . . . 𝑁. We especially have lim 𝑥 − 𝑇1 𝑥𝑛 = 0. (35) 𝑛→∞ 𝑛 By the assumption of Theorem 11, we may assume that 𝑇1 is semicompact, without loss of generality. Then, it follows from (35) that there exists a subsequence {𝑥𝑛𝑘 } of {𝑥𝑛 } such that {𝑥𝑛𝑘 } converges strongly to 𝑝 ∈ 𝐾. Thus from (26) we have 𝑝 − 𝑇𝑙 𝑝 = lim 𝑥𝑛𝑘 − 𝑇𝑙 𝑥𝑛𝑘 = 𝑛lim 𝑥 − 𝑇𝑙 𝑥𝑛 = 0, →∞ 𝑛 𝑘→∞ ∀𝑙 = 1, 2, . . . 𝑁. (36) This implies that 𝑝 ∈ 𝐹. In addition, since lim𝑛 → ∞ ‖𝑥𝑛 − 𝑝‖ exists, therefore lim𝑛 → ∞ ‖𝑥𝑛 − 𝑝‖ = 0; that is, {𝑥𝑛 } converges strongly to a fixed point of {𝑇1 , 𝑇2 , . . . , 𝑇𝑁} in 𝐸. The proof is completed.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Authors’ Contribution The main idea of this paper was proposed by Qiaohong Jiang. All authors contributed equally to the writing of this paper. All authors read and approved the final paper.
Acknowledgment The research was supported by the Fujian Nature Science Foundation under Grant no. 2014J01008.
References [1] H. Xu and R. G. Ori, “An implicit iteration process for nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 22, no. 5-6, pp. 767–773, 2001. [2] L.-C. Zeng and J.-C. Yao, “Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2507–2515, 2006. [3] R. P. Agarwal and M. Meehan, Fixed Point Theory and Applications, Cambridge Tracts in Mathematics, Cambridge University Press, 2005. [4] K. K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993. [5] J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991. [6] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, Mass, USA, 1990.
[7] S. S. Chang, K. K. Tan, H. W. J. Lee, and C. K. Chan, “On the convergence of implicit iteration process with error for a finite family of asymptotically nonex pansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 313, pp. 273–283, 2006. [8] H. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. [9] Y. Yao and J. Yao, “On modified iterative method for nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 186, no. 2, pp. 1551–1558, 2007. [10] Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
Advances in
Operations Research Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at http://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
