Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 572838, 8 pages
doi:10.1155/2010/572838
Research Article
New Hybrid Iterative Schemes for
an Infinite Family of Nonexpansive Mappings in
Hilbert Spaces
Baohua Guo and Shenghua Wang
School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
Correspondence should be addressed to Shenghua Wang, sheng-huawang@hotmail.com
Received 20 November 2009; Accepted 4 February 2010
Academic Editor: Anthony To Ming Lau
Copyright q 2010 B. Guo and S. Wang. 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.
We propose some new iterative schemes for finding common fixed point of an infinite family of
nonexpansive mappings in a Hilbert space and prove the strong convergence of the proposed
schemes. Our results extend and improve ones of Nakajo and Takahashi 2003.
1. Introduction and Preliminaries
Let H be a Hilbert space and C a nonempty closed convex subset of H. Let T be a nonlinear
mapping of C into itself. We use FT and PC to denote the set of fixed points of T and the
metric projection from H onto C, respectively.
Recall that T is said to be nonexpansive if
T x − T y ≤ x − y
1.1
for all x, y ∈ C.
For approximating the fixed point of a nonexpansive mapping in a Hilbert space,
Mann 1 in 1953 introduced a famous iterative scheme as follows:
∀x1 ∈ C, xn1 1 − αn xn αn T xn ,
∀n ≥ 1,
1.2
where T is a nonexpansive mapping of C into itself and {αn } is a sequence in 0, 1. It is well
known that {xn } defined in 1.2 converges weakly to a fixed point of T .
2
Fixed Point Theory and Applications
Attempts to modify the normal Mann iteration method 1.2 for nonexpansive
mappings so that strong convergence is guaranteed have recently been made; see, for
example, 2–9.
Nakajo and Takahashi 5 proposed the following modification of Mann iteration
method 1.2 for a single nonexpansive mapping T in a Hilbert space H:
x0 ∈ C chosen arbitrarily,
yn αn xn 1 − αn T xn ,
Cn z ∈ C : yn − z ≤ xn − z ,
1.3
Qn {z ∈ C : xn − z, x0 − xn ≥ 0},
xn1 PCn ∩Qn x0 ,
where PK denotes the metric projection from H onto a closed convex subset K of H.
They proved that if the sequence {αn } is bounded above from one, then the sequence {xn }
generated by 1.3 converges strongly to PF T x0 .
In this paper, we introduce some new iterative schemes for infinite family of
nonexpansive mappings in a Hilbert space and prove the strong convergence of the
algorithms. Our results extend and improve the corresponding one of Nakajo and Takahashi
5.
The following two lemmas will be used for the main results of this paper.
Lemma 1.1. Let C be a closed convex subset of a real Hilbert space H and let PC be the metric
projection from H onto C (i.e., for x ∈ H, PC x is the only point in C such that x − PC x inf{x − z : z ∈ C}. Given x ∈ H and z ∈ C, then z PC x if and only if there holds the following
relation:
x − z, y − z ≤ 0,
∀y ∈ C.
1.4
Lemma 1.2 see 10. Let H be a real Hilbert space. Then the following equation holds:
tx 1 − ty2 tx2 1 − ty2 − t1 − tx − y2 ,
∀x ∈ C, ∀t ∈ 0, 1.
1.5
2. Main Results
Theorem 2.1. Let C be a nonempty closed convex subset of a Hilbert space H. Let {Ti }∞
i1 : C → C
∞
∅. Let {xn } be a sequence
be an infinite family of nonexpansive mappings such that F i1 FTi /
generated by the following manner:
x1 x ∈ C chosen arbitrarily,
yi,n 1 − αn xn αn Ti xn , i 1, 2, . . . ,
∞
2
2
Cn v ∈ C :
βi yi,n − v ≤ xn − v ,
i1
Fixed Point Theory and Applications
3
Dn n
j1
xn1 PDn x,
Cj ,
n ≥ 1,
2.1
where {αn } is a sequence in 0, 1 satisfying lim infn → ∞ αn > 0, and {βn } is a sequence in 0, 1
satisfying ∞
n1 βn 1. Then {xn } defined by 2.1 converges strongly to PF x.
Proof. We first show that Dn is closed and convex. By Lemma 1.2, one observes that
∞
2
βi yi,n − v ≤ xn − v2
i1
2.2
is equivalent to
∞
∞
2
βi yi,n − xn 2 ≤ 2
βi yi,n − xn , v
i1
2.3
i1
for all n ≥ 1. So, Cn is closed and convex for all n ≥ 1 and hence Dn nj1 Cj is also closed
and convex for all n ≥ 1. This implies that PDn x is well defined.
Next, we show that F ⊂ Dn for all n ≥ 1. To end this, we need to prove that F ⊂ Cn for
all n ≥ 1. Indeed, for each p ∈ F, we have
∞
∞
2 2
2 2
βi yi,n − p ≤
βi αn xn − p 1 − αn Ti xn − p ≤ xn − p .
i1
i1
2.4
This implies that
p ∈ Cn ,
∀n ≥ 1.
2.5
Therefore, F ⊂ Cn and Cn is nonempty for all n ≥ 1. On the other hand, from the definition of
Dn , we see that F ⊂ Dn ni1 Cj for all n ≥ 1.
From xn1 PDn x, we have
xn1 − x ≤ v − x,
∀v ∈ Dn , n ≥ 1.
2.6
Since z PF x ∈ F ⊂ Dn for all n ≥ 1, one has
xn1 − x ≤ z − x.
2.7
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Fixed Point Theory and Applications
This implies that {xn } is bounded. For each fixed i ≥ 1, by 2.1 we have noting that z PF x ∈ F ∞
i1 FTi yi,n ≤ yi,n − z z ≤ yi,n − z z
≤ 1 − αn xn − z αn Ti xn − z z
≤ 1 − αn xn − z αn xn − z z
2.8
xn − z z
≤ xn 2z
for all n ≥ 1. Since {xn } is bounded, {yi,n } is bounded for each i ≥ 1.
On the other hand, observing that Dn1 ⊂ Dn for all n ≥ 1, we have
xn2 PDn1 x ∈ Dn1 ⊂ Dn
2.9
for all n ≥ 1. Since xn1 PDn x, we have
xn1 − x ≤ xn2 − x
2.10
for all n ≥ 1. It follows from 2.7 and 2.10 that the limit of {xn − x} exists.
Since Dm ⊂ Dn and xm1 PDm x ∈ Dm ⊂ Dn for all m ≥ n and xn1 PDn x, by
Lemma 1.1 one has
xn1 − x, xm1 − xn1 ≥ 0.
2.11
It follows from 2.11 that
xm1 − xn1 2 xm1 − x − xn1 − x2
xm1 − x2 xn1 − x2 − 2xn1 − x, xm1 − x
xm1 − x2 xn1 − x2 − 2xn1 − x, xm1 − xn1 xn1 − x
2.12
xm1 − x2 − xn1 − x2 − 2xn1 − x, xm1 − xn1 ≤ xm1 − x2 − xn1 − x2 .
Since the limit of xn1 − x exists, we get
lim xm − xn 0.
m,n → ∞
2.13
It follows that {xn } is a Cauchy sequence. Since H is a Hilbert space and C is closed and
convex, one can assume that
xn −→ q ∈ C,
as n −→ ∞.
2.14
Fixed Point Theory and Applications
5
By taking m n 1 in 2.12, one arrives that
lim xn2 − xn1 0,
2.15
lim xn1 − xn 0.
2.16
n→∞
that is,
n→∞
Noticing that xn1 PDn x ∈ Dn ⊂ Cn , we get
∞
2
βi yi,n − xn1 ≤ xn − xn1 2 .
i1
This implies that limn → ∞
∞
i1
2.17
βi yi,n − xn1 2 0. Since each βi ∈ 0, 1, we conclude that
yi,n − xn1 −→ 0,
as n −→ ∞, i 1, 2, . . . .
2.18
From 2.16 and 2.18, we get
yi,n − xn ≤ yi,n − xn1 xn1 − xn −→ 0,
as n −→ ∞, i 1, 2, . . . .
2.19
By Ti xn − xn 1/αn yi − xn and lim infn → ∞ αn > 0, we have
lim Ti xn − xn 0,
i 1, 2, . . . .
n→∞
2.20
This implies that
q∈F
∞
i1
FTi .
2.21
Finally, we prove that q z PF x. From xn1 PDn x and F ⊂ Dn , one gets
x − xn1 , xn1 − v ≥ 0,
∀v ∈ F.
2.22
Taking the limit in 2.22 and noting that xn → q as n → ∞, we get that
x − q, q − v ≥ 0,
∀v ∈ F.
In view of Lemma 1.1, one sees that q z PF x. This completes the proof.
2.23
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Fixed Point Theory and Applications
Corollary 2.2. Let C be a nonempty closed convex subset of a Hilbert space H. Let T : C → C be a
nonexpansive mapping such that FT /
∅. Let {xn } be a sequence generated by the following manner:
x1 x ∈ C chosen arbitrarily,
yn 1 − αn xn αn T xn ,
Cn v ∈ C : yn − v ≤ xn − v ,
Dn n
j1
xn1 PDn x,
2.24
Cj ,
n ≥ 1,
where {αn } is a sequence in 0, 1 satisfying that lim infn → ∞ αn > 0. Then {xn } defined by 2.24
converges strongly to PF x.
Proof. Set Tn T for all n ≥ 1, β1 1 and βn 0 for all n ≥ 2 in Theorem 2.1. By Theorem 2.1,
we obtain the desired result.
Theorem 2.3. Let C be a nonempty closed convex subset of a Hilbert space H. Let {Ti }∞
i1 : C → C
∞
∅. Let {xn } be a sequence
be an infinite family of nonexpansive mappings such that F i1 FTi /
generated by the following manner:
x1 x ∈ C chosen arbitrarily,
yn αn xn n
αi−1 − αi Ti xn ,
i1
Cn v ∈ C : yn − v ≤ xn − v ,
Dn n
j1
xn1 PDn x,
2.25
Cj ,
n ≥ 1,
where {αn }∞
n1 is a strictly decreasing sequence in 0, 1 and set α0 1. Then {xn } defined by 2.25
converges strongly to PF x.
Proof. Obviously, Cn is closed and convex for all n ≥ 1 and hence Dn nj1 Cj is also closed
and convex for all n ≥ 1. Next, we prove that F ⊂ Dn for all n ≥ 1. For any p ∈ F, we have
n
yn − p αn xn − p αi−1 − αi Ti xn − p i1
n
≤ αn xn − p αi−1 − αi Ti xn − p
i1
Fixed Point Theory and Applications
7
n
≤ αn xn − p αi−1 − αi xn − p
xn − p.
i1
2.26
This shows that p ∈ Cn for all n ≥ 1. Therefore, p ∈ Dn nj1 Cj for all n ≥ 1. It follows that
F ⊂ Dn for all n ≥ 1.
By using the method of Theorem 2.1, we can conclude that {xn } is bounded, xn → p,
xn − xn1 → 0, and yn − xn1 → 0 as n → ∞. This implies that xn − yn → 0 as n → ∞.
Next, we show that p ∈ F. To end this, we see a fact. For p and xn , we have
xn − p2 ≥ Ti xn − Ti p2 Ti xn − p2 Ti xn − xn xn − p2
2
Ti xn − xn 2 xn − p 2Ti xn − xn , xn − p
2.27
Ti xn − xn 2 ≤ 2xn − Ti xn , xn − p
2.28
and hence
for each i 1, 2, . . . .
Observe that yn n
i1 αi−1
n
− αi xn − Ti xn − 1 − αn xn αn xn , that is,
αi−1 − αi xn − Ti xn xn − yn .
i1
2.29
It follows from 2.28 and 2.29 that
n
n
αi−1 − αi xn − Ti xn 2 ≤ 2 αi−1 − αi xn − Ti xn , xn − p
i1
i1
2 xn − yn , xn − p
≤ 2xn − yn xn − p.
2.30
Since {αn } is strictly decreasing, xn − yn → 0, and xn → p as n → ∞, we get
xn − Ti xn −→ 0 as n −→ ∞
2.31
for each i 1, 2, . . . . Since each Ti is nonexpansive, one has p ∈ FTi and hence
p∈F
∞
i1
FTi .
2.32
Finally, by using the method of Theorem 2.1, we can conclude that p PF x. This
completes the proof.
8
Fixed Point Theory and Applications
Remark 2.4. In this paper, we extend result of Nakajo and Takahashi 5 from a single
nonexpansive mapping to an infinite family of nonexpansive mappings.
Remark 2.5. The iterative schemes introduced in this paper are new and of independent
interest.
Remark 2.6. It is of interest to extend the algorithm 2.25 to certain Banach spaces.
Acknowledgment
The work was supported by Youth Foundation of North China Electric Power University.
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