Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 396080, 14 pages
doi:10.1155/2010/396080
Research Article
Convergence Theorems Concerning Hybrid
Methods for Strict Pseudocontractions and
Systems of Equilibrium Problems
Peichao Duan
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Peichao Duan,
Received 23 May 2010; Revised 23 August 2010; Accepted 26 August 2010
Academic Editor: S. Reich
Copyright q 2010 Peichao Duan. 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.
Let {S
i
}
N
i1
be N strict pseudo-contractions defined on a closed and convex subset C of a real
Hilbert space H. We consider the problem of finding a common element of fixed point set
of these mappings and the solution set of a system of equilibrium problems by parallel and
cyclic algorithms. In this paper, new iterative schemes are proposed for solving this problem.
Furthermore, we prove that these schemes converge strongly by hybrid methods. The results
presented in this paper improve and extend some well-known results in the literature.
1. Introduction
Let H be a real Hilbert space with inner product ·, · and norm ·.LetC be a nonempty,
closed, and convex subset of H.
Let {F
x − y
2
κ
I − S
x −
I − S
y
2
1.3
for all x, y ∈ C;see2. We denote the fixed point set of S by FS,thatis,FS{x ∈
C : Sx x}.
Note that the class of strict pseudocontractions properly includes the class of
nonexpansive mappings which are mapping S on C such that
Sx − Sy
≤
n
}.
2. Preliminaries
We will use the facts and tools in a real Hilbert space H which are listed below.
Lemma 2.1. Let H be a real Hilbert space. Then the following identities hold:
i x − y
2
x
2
−y
2
− 2x − y, y, for all x,y ∈ H,
ii tx1−ty
2
tx
2
1−ty
2
−t1−tx−y
2
, for all t ∈ 0, 1, for all x, y ∈ H.
Lemma 2.2 see 6. Let H be a real Hilbert space. Given a nonempty, closed, and convex subset
C ⊂ H, points x, y, z ∈ H, and a real number a ∈ R, then the set
v ∈ C :
y − v
2.2
for all y ∈ C. Such a P
C
is called the metric or the nearest point projection of H onto C.As
we all know y P
C
x if and only if there holds the relation
x − y, y − z
≥ 0 ∀z ∈ C. 2.3
Lemma 2.3 see 13. Let C be a nonempty, closed, and convex subset of H.Let{x
n
} be a sequence
in H and u ∈ H.Letq P
C
u. Suppose that {x
n
} is such that ω
w
x
n
⊂ C and satisfies the following
condition:
x
n
− u
≤
n
xand I − Tx
n
→ 0,thenI −Tx 0.
iii If T : C → C is a κ-strict pseudocontraction, then the fixed point set FT of T is closed
and convex. Therefore the projection P
FT
is well defined.
iv Given an integer N ≥ 1, assume that, for each 1 ≤ i ≤ N, T
i
: C → C is a κ
i
-strict
pseudocontraction for some 0 ≤ κ
i
< 1. Assume that {λ
i
}
N
i1
is a positive sequence such that
N
i1
λ
i
1.Then
N
i1
λ
i
T
i
N
i1
F
T
i
.
2.6
Lemma 2.5 see 2. Let S : C → H be a κ-strict pseudocontraction. Define T : C → H by
Tx λx 1 − λSx for any x ∈ C. Then, for any λ ∈ κ, 1, T is a nonexpansive mapping with
FTFS.
4 Journal of Inequalities and Applications
For solving the equilibrium problem, let one assume that the bifunction F satisfies the following
conditions:
A1 Fx, x0 for all x ∈ C;
A2 F is monotone, that is, Fx, yFy,x ≤ 0 for any x, y ∈ C;
A3 for each x, y, z ∈ C, lim sup
t → 0
Ftz 1 − tx, y ≤ Fx, y;
A4 Fx, · is convex and lower semicontinuous for each x ∈ C.
Lemma 2.6 see 3. Let C be a nonempty, closed, and convex subset of H,letF be bifunction from
C × C to R which satisfies conditions (A1)–(A4), and let r>0 and x ∈ H. Then there exists z ∈ C
y − z, z − x
≥ 0, ∀y ∈ C
2.8
for all x ∈ H. Then, the following statements hold:
i T
r
is single valued;
ii T
r
is firmly nonexpansive, that is, for any x, y ∈ H,
T
r
x − T
r
y
2
≤
T
r
x − T
r
∩ ∩
M
k1
EPF
k
/
∅. Assume also that {η
n
i
}
N
i1
is
a finite sequence of positive numbers such that
N
i1
η
n
i
1 for all n ∈ N and inf
n≥1
η
n
i
> 0 for all
1 ≤ i ≤ N. Let the mapping A
n
be defined by
T
F
M−1
r
M−1,n
···T
F
2
r
2,n
T
F
1
r
1,n
x
n
,
A
λ
n
n
λ
n
I
1 − λ
n
A
≤
x
n
− z
,
Q
n
{
z ∈ C :
x
n
− z, x
1
− x
n
≥ 0
}
,
x
n1
P
C
k
r
k,n
···T
F
2
r
2,n
T
F
1
r
1,n
for every k ∈{1, 2, ,M} and Θ
0
n
I for all n ∈ N.
Therefore u
n
Θ
M
n
x
n
. The proof is divided into six steps.
Step 1. The sequence {x
n
} is well defined.
It is obvious that C
n
Θ
M
n
x
n
− Θ
M
n
p
≤
x
n
− p
3.3
for all n ∈ N. From Proposition 2.4, Lemma 2.5,and3.3,weget
x
n
− p
1 − α
n
A
λ
n
n
u
n
− p
≤
x
n
− z, x
1
− x
n1
≥ 0, ∀z ∈ C
n
∩ Q
n
. 3.5
As Ω ⊂ C
n
∩ Q
n
by induction assumption, the inequality holds, in particular, for all z ∈ Ω.
This together with the definition of Q
n1
implies that Ω ⊂ Q
n1
. Hence Ω ⊂ Q
n
holds for all
n ≥ 1. Thus Ω ⊂ C
n
∩ Q
n
, and therefore the sequence {x
n
} is well defined.
Step 2.
. This together with the fact that
Ω ⊂ Q
n
further implies that
x
n
− x
1
≤
x
1
− p
∀p ∈ Ω. 3.7
Then {x
n
} is bounded and 3.6 holds. From 3.3, 3.4, and Proposition 2.4 i, we also obtain
that {u
n
}, {y
n
},and {S
i
x
n
1
≥0. This together with Lemma 2.1
i implies that
x
n1
− x
n
2
x
n1
− x
1
−
x
n
− x
1
2
x
n1
− x
−
x
n
− x
1
2
.
3.9
Then x
n
−x
1
≤x
n1
−x
1
, that is, the sequence {x
n
−x
1
} is nondecreasing. Since {x
n
−x
1
}
is bounded, lim
n →∞
x
≤
x
n1
− x
n
y
n
− x
n1
≤ 2
x
n1
− x
n
. 3.11
By 3.6,weobtain
lim
n →∞
y
F
k
r
k,n
that for each k ∈
{1, 2, ,M}, we have
Θ
k
n
x
n
− p
2
T
F
k
r
k,n
Θ
k−1
Θ
k
n
x
n
− p
2
Θ
k−1
n
x
n
− p
2
−
2
≤
Θ
k−1
n
x
n
− p
2
−
Θ
k
n
x
n
− Θ
k−1
n
x
− p
2
−
Θ
k
n
x
n
− Θ
k−1
n
x
n
2
−
Θ
k−1
n
Θ
1
n
x
n
− Θ
0
n
x
n
2
≤
x
n
− p
2
−
x
n
− p
2
1 − α
n
A
λ
n
n
u
n
− p
2
≤ α
n
1 − α
n
Θ
k
n
x
n
− p
2
≤ α
n
x
n
− p
2
1 − α
x
n
− p
2
−
1 − α
n
Θ
k
n
x
n
− Θ
k−1
n
x
n
− p
2
−
y
n
− p
2
≤
x
n
− y
n
x
n
− p
n
Θ
M−1
n
x
n
− Θ
M−2
n
x
n
···
Θ
1
n
x
n
1 − α
n
A
λ
n
n
u
n
, we observe that
1 − α
n
A
λ
n
n
u
n
− u
n
− u
n
≤
y
n
− u
n
α
n
x
n
− u
n
.
3.20
From {α
n
}⊂0,a, 3.19,andy
n
− u
n
n
n
x
n
− A
λ
n
n
u
n
A
λ
n
n
u
n
− u
n
A
λ
n
n
x
n
− x
n
λ
n
x
n
1 − λ
n
A
n
x
n
− x
n
⊂ Ω. 3.23
We first show that ω
w
x
n
⊂∩
N
i1
FS
i
. To this end, we take ω∈
w
x
n
and assume that x
n
j
ω
as j →∞for some subsequence {x
n
j
} of x
n
.Without loss of generality, we may assume that
η
n
j
η
i
S
i
. Note that, by Proposition 2.4, A is a κ-strict pseudocontraction and
FA∩
N
i1
FS
i
. Since
Ax
n
j
− x
n
j
≤
A
n
j
η
n
j
i
− η
i
S
i
x
n
j
A
n
j
N
i1
FS
i
,
and hence ω
w
x
n
⊂∩
N
i1
FS
i
.
Next we will show that ω ∈∩
M
k1
EPF
k
. Indeed, by Lemma 2.6, we have that, for each
k 1, 2, ,M,
F
k
Θ
k
n
x
n
y − Θ
k
n
x
n
, Θ
k
n
x
n
− Θ
k−1
n
x
n
≥ F
k
y, Θ
k
n
x
n
, ∀y ∈ C.
3.29
Hence,
y − Θ
n
j
x
n
j
, ∀y ∈ C.
3.30
From 3.13,weobtainthatΘ
k
n
j
x
n
j
ωas j →∞for each k 1, 2, ,M especially,
u
n
j
Θ
M
n
j
x
n
j
. Together with 3.13 and A4 we have, for each k 1, 2, ,M,that
0 ≥ F
k
1 − t
F
k
y
t
,ω
≤ tF
k
y
t
,y
. 3.32
Dividing by t, we get, for each k 1, 2, ,M,that
F
k
y
t
,y
≥ 0, ∀y ∈ C. 3.33
Letting t → 0andfromA3,weget
F
k
1 − α
n
A
n
x
n
,
C
n
z ∈ C :
y
n
− z
2
≤
x
n
− z
2
− x
n
≥ 0
}
,
x
n1
P
C
n
∩Q
n
x
0
.
3.35
In this paper, we first turn the strict pseudocontraction A
n
into nonexpansive mapping
A
λ
n
n
then replace C
n
with a more simple form in the iterative algorithm.
Remark 3.3. If F
k
x, y0, N 1, and λ
i1
FS
i
∩ ∩
M
k1
EPF
k
/
∅. Assume also that {η
n
i
}
N
i1
is a finite
sequence of positive numbers such that
N
i1
η
n
i
1 for all n and inf
n≥1
η
n
i
> 0 for all 1 ≤ i ≤ N.
F
M
r
M,n
T
F
M−1
r
M−1,n
···T
F
2
r
2,n
T
F
1
r
1,n
x
n
,
A
λ
n
n
λ
n
I
: y
n
− z≤x
n
− z
,
x
n1
P
C
n1
x
1
,
3.37
where {α
n
}⊂0,a for some a ∈ 0, 1, {λ
n
}⊂κ, b for some b ∈ κ, 1, and, {r
k,n
}⊂0, ∞
satisfies lim inf
n →∞
r
k,n
> 0 for all k ∈{1, 2, ,M}. Then, {x
n
} converge strongly to P
1
.
Step 3. x
n1
− x
n
→0asn →∞.
Step 4. A
n
x
n
− x
n
→0asn →∞.
Step 5. ω
w
x
n
⊂ Ω.
Step 6. x
n
→ q.
The proof of Step 2–Step 6 is similar to that of Theorem 3.1.
Remark 3.6. If M 1, we can obtain the two corresponding theorems in 10.
4. Cyclic Algorithm
Let C be a closed, and convex subset of a Hilbert space H,andlet{S
i
}
N−1
i0
1
α
0
x
0
1 − α
0
S
0
x
0
,
x
2
α
1
x
1
1 − α
1
S
1
x
1
0
x
N
,
···.
4.2
In general, x
n1
is defined by
x
n1
α
n
x
n
1 − α
n
S
n
x
n
, 4.3
where S
n
S
i
,withi n mod N, 0 ≤ i ≤ N − 1.
n
, u
n
, and y
n
be sequences which are generated by the following algorithm:
u
n
T
F
M
r
M,n
T
F
M−1
r
M−1,n
···T
F
2
r
2,n
T
F
1
r
1,n
x
n
n
u
n
,
C
n
z ∈ C :
y
n
− z
≤
x
n
− z
,
Q
n
{
z ∈ C :
n →∞
r
k,n
> 0 for all k ∈{1, 2, ,M}. Then, {x
n
} converge strongly to P
F
x
0
.
Proof. The proof of this theorem is similar to that of Theorem 3.1. The main points are the
following.
Step 1. The sequence {x
n
} is well defined.
Step 2. x
n
− x
0
≤q − x
0
for all n, where q P
Ω
x
0
.
Step 3. x
n1
− x
n
}. We may assume that l n
m
mod N for all m. Since, by x
n1
− x
n
→0, we also
have x
n
m
j
ωfor all j ≥ 0, we deduce that
x
n
m
j
− S
lj
x
n
m
j
x
converges strongly to q.
The strong convergence to q of {x
n
} is a consequence of Step 2,Step5, and Lemma 2.3.
Theorem 4.2. Let C be a nonempty, closed, and convex subset of a real Hilbert space H, and let
F
k
,k ∈{1, 2, ,M}, be bifunctions from C × C to R which satisfies conditions (A1)–(A4). Let,
for each 0 ≤ i ≤ N − 1, S
i
: C → C be a κ
i
-strict pseudocontraction for some 0 ≤ κ
i
< 1.Let
Journal of Inequalities and Applications 13
κ max{κ
i
:0≤ i ≤ N − 1}. Assume that Ω∩
N−1
i0
FS
i
∩ ∩
M
k1
EPF
k
/
F
1
r
1,n
x
n
,
S
λ
n
n
λ
n
I
1 − λ
n
S
n
,
y
n
α
n
x
n
n
− z
,
x
n1
P
C
n1
x
0
,
4.6
where {α
n
}⊂0,a for some a ∈ 0, 1, {λ
n
}⊂κ, b for some b ∈ κ, 1, and {r
k,n
}⊂0, ∞
satisfies lim inf
n →∞
r
k,n
> 0 for all k ∈{1, 2, }. Then, {x
n
} converge strongly to P
Ω
x
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