Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 312602, 18 pages
doi:10.1155/2010/312602
Research Article
Hybrid Projection Algorithms for
Generalized Equilibrium Problems and
Strictly Pseudocontractive Mappings
Jong Kyu Kim,
1
Sun Young Cho,
2
and Xiaolong Qin
3
1
Department of Mathematics Education, Kyungnam University, Masan 631-701, Republic of Korea
2
Department of Mathematics, Gyeongsang National University, Chinju 660-701, Republic of Korea
3
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
Correspondence should be addressed to Jong Kyu Kim,
Received 12 October 2009; Accepted 19 July 2010
Academic Editor: Andr
´
as Ront
´
o
Copyright q 2010 Jong Kyu Kim 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.
The purpose of this paper is to consider the problem of finding a common element in the solution
2
k
x − Sx
−
y − Sy
2
, ∀x, y ∈ C. 1.2
S is said to be pseudocontractive if
Sx − Sy
2
≤
x − y
2
Ax − Ay
2
, ∀x, y ∈ C. 1.5
Let F be a bifunction of C × C into R, where R denotes the set of real numbers and
A : C → H an inverse-strongly monotone mapping. In this paper, we consider the following
generalized equilibrium problem.
Find x ∈ C such that F
x, y
Ax, y − x
≥ 0, ∀y ∈ C. 1.6
In this paper, the set of such an x ∈ C is denoted by EPF, A,thatis,
EP
F, A
x ∈ C : F
x, y
Find x ∈ C such that F
x, y
≥ 0, ∀y ∈ C. 1.9
In this paper, the set of such an x ∈ C is denoted by EPF,thatis,
EP
F
x ∈ C : F
x, y
≥ 0, ∀y ∈ C
. 1.10
Journal of Inequalities and Applications 3
II If F ≡ 0, then the problem 1.6 is reduced to the following classical variational
inequality. Find x ∈ C such that
Ax, y − x
≥ 0, ∀y ∈ C. 1.11
It is known t hat x ∈ C is a solution to 1.11 if and only if x is a fixed point of the
mapping P
C
I − ρA, where ρ>0 is a constant and I is the identity mapping.
− x
n
≥ 0, ∀y ∈ C,
w
n
1 − α
n
x
n
α
n
Su
n
,
C
n
{
z ∈ H :
w
n
− z
≤
n
}⊂a, 1 for some a ∈ 0, 1 and {r
n
}⊂0, ∞ satisfies lim inf
n →∞
r
n
>
0. Then, {x
n
} converges strongly to P
FS∩EPF
x.
In this paper, we consider the generalized equilibrium problem 1.6 and a strictly
pseudocontractive mapping based on the shrinking projection algorithm which was first
introduced by Takahashi et al. 18. A strong convergence of common elements of the
fixed point sets of the strictly pseudocontractive mapping and of the solution sets of the
generalized equilibrium problem is established in the framework of Hilbert spaces. The
results presented in this paper improve and extend the corresponding results announced
by Tada and Takahashi 17.
In order to prove our main results, we also need the following definitions and lemmas.
Lemma 1.1 see 19. Let C be a nonempty closed convex subset of a Hilbert space H and T : C →
C a k-strict pseudocontraction. Then T is 1 k/1 − k-Lipschitz and I − T is demiclosed, this is, if
{x
n
} is a sequence in C with x
n
xand x
n
− Tx
y − z, z − x
≥ 0, ∀y ∈ C
1.14
for all r>0 and x ∈ H. Then, the following hold:
a T
r
is single-valued;
b T
r
is firmly nonexpansive, that is, for any x, y ∈ H,
T
r
x − T
r
y
2
≤
T
r
x − T
r
y, x − y
2
,B∩FS is not empty. Let {x
n
} be a sequence generated in the following manner:
x
1
∈ C,
C
1
C,
F
1
u
n
,u
Ax
n
,u− u
n
1
r
n
u − u
− x
n
≥ 0, ∀v ∈ C,
z
n
γ
n
u
n
1 − γ
n
v
n
,
y
n
α
n
x
n
1 − α
n
n
− w
,
x
n1
P
C
n1
x
1
,n≥ 1,
Υ
where {α
n
}, {β
n
}, and {γ
n
} are sequences in 0, 1. Assume that {α
n
}, {β
n
}, {γ
n
}, {r
n
}, and {s
n
u
n
T
r
n
x
n
− r
n
Ax
n
, ∀n ≥ 1 2.1
and v
n
can be rewritten as
v
n
T
s
n
x
n
− s
n
Bx
n
x −
I − r
n
A
y
2
x − y
− r
n
Ax − Ay
2
x − y
Ax − Ay
2
≤
x − y
2
.
2.4
This shows that I − r
n
A is nonexpansive for each n ≥ 1. In a similar way, we can obtain that
I − s
n
B is nonexpansive for each n ≥ 1. It follows that
u
n
− p
≤
≤ γ
n
u
n
− p
1 − γ
n
v
n
− p
≤
x
n
− p
. 2.6
y
m
2
−
x
m
2
− 2
w, y
m
− x
m
≥ 0. 2.8
Thus C
m1
is closed and convex. This shows that C
n
is closed and convex for each n ≥ 1.
Next, we show that F⊂C
n
for each n ≥ 1. From the assumption, we see that F⊂C
C
1
x
m
1 − α
m
S
m
z
m
− w
≤ α
m
x
m
− w
1 − α
m
z
m
− w
≤
x
1
− w
. 2.11
In particular, we have
x
1
− x
n
≤
x
1
− P
F
x
1
. 2.12
This implies that {x
n
} is bounded. Since x
n
P
C
− x
n
,x
n
− x
1
x
1
− x
n1
≤−
x
1
− x
n
2
x
1
− x
n
x
1
− x
n1
2
x
n
− x
1
x
1
− x
n1
2
x
n
− x
1
2
2
x
n
− x
1
,x
1
n
− x
n1
x
1
− x
n1
2
x
n
− x
1
2
− 2
x
n
− x
1
2
2
1
2
.
2.15
It follows that
lim
n →∞
x
n
− x
n1
0. 2.16
Since x
n1
P
C
n1
x
1
∈ C
n1
,weseethat
y
n
− x
x
n
− x
n1
≤ 2
x
n
− x
n1
. 2.18
8 Journal of Inequalities and Applications
From 2.16,weobtainthat
lim
n →∞
x
n
− y
n
0. 2.19
On the other hand, we have
x
n
− S
n
z
n
. 2.20
From the assumption 0 ≤ α
n
≤ a<1and2.19, we have
lim
n →∞
x
n
− S
n
z
n
0. 2.21
For any p ∈F, we have
u
n
− p
Ax
n
− Ap
2
x
n
− p
2
− 2r
n
x
n
− p, Ax
n
− Ap
r
2
n
.
2.22
In a similar way, we also have
v
n
− p
2
≤
x
n
− p
2
− s
n
2β − s
n
Bx
n
− p
2
≤ α
n
x
n
− p
2
1 − α
n
S
n
z
n
− p
2
≤ α
2
1 − α
n
γ
n
u
n
− p
2
1 − α
n
1 − γ
n
v
n
n
r
n
2α − r
n
Ax
n
− Ap
2
−
1 − α
n
1 − γ
n
s
n
2β − s
n
2
≤
x
n
− p
2
−
y
n
− p
2
≤
x
n
− p
1 − α
n
1 − γ
n
s
n
2β − s
n
Bx
n
− Bp
2
≤
x
n
− p
x
n
− y
n
.
2.28
By virtue of the restrictions a–d and 2.19,wegetthat
lim
n →∞
Bx
n
− Bp
0. 2.29
On the other hand, we have from Lemma 1.1 that
u
n
− p
2
x
n
−
I − r
n
A
p, u
n
− p
1
2
I−r
n
Ax
n
− I −r
n
Ap
2
2
≤
1
2
x
n
− p
2
u
n
− p
2
−
x
n
− u
2
−
x
n
−u
n
2
−2r
n
x
n
−u
n
,Ax
n
−Ap
r
2
n
Ax
n
−Ap
2
2r
n
x
n
− u
n
Ax
n
− Ap
. 2.31
In a similar way, we can also obtain that
v
n
− p
2
≤
x
Substituting 2.31 and 2.32 into 2.24,weobtainthat
y
n
− p
2
≤
x
n
− p
2
−
1 − α
n
γ
n
x
n
− u
n
n
x
n
− v
n
2
2s
n
1 − α
n
1 − γ
n
x
n
− v
n
Bx
n
− Bp
− u
n
Ax
n
− Ap
−
1 − α
n
1 − γ
n
x
n
− v
n
2
2s
n
x
x
n
− p
2
−
y
n
− p
2
2r
n
x
n
− u
n
Ax
n
− Ap
x
n
− y
n
2r
n
x
n
− u
n
Ax
n
− Ap
2s
n
x
n
x
n
− v
n
2
≤
x
n
− p
2
−
y
n
− p
2
2r
n
x
n
− p
y
n
− p
x
n
− y
n
2r
n
x
n
− u
n
n
− v
n
0. 2.37
Note that
z
n
− x
n
≤ γ
n
u
n
− x
n
1 − γ
n
v
n
− x
n
− x
n
−→ 0 2.40
as n →∞. In view of 2.39 and the restriction b,weobtainthat
lim
n →∞
x
n
− Sz
n
0. 2.41
Note that
Sx
n
− x
n
≤
Sx
n
− Sz
n
0. 2.43
Since {x
n
} is bounded, we assume that a subsequence {x
n
i
} of {x
n
} converges weakly to ξ.
Next, we show that ξ ∈ FS ∩ EPF
1
,A ∩ EPF
2
,B. First, we prove that ξ ∈ EPF
1
,A.
Since u
n
T
r
n
x
n
− r
n
Ax
n
for any u ∈ C, we have
F
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ F
1
u, u
n
. 2.45
Replacing n by n
i
, we arrive at
Ax
n
i
,u− u
t
∈ C.
It follows from 2.46 that
u
t
− u
n
i
,Au
t
≥
u
t
− u
n
i
,Au
t
−
Ax
n
i
,u
t
− u
u
t
−u
n
i
,Au
t
−Au
n
i
u
t
−u
n
i
,Au
n
i
−Ax
n
i
−
u
− Ax
n
i
→ 0asi →∞.On
the other hand, we get from the monotonicity of A that
u
t
− u
n
i
,Au
t
− Au
n
i
≥ 0. 2.48
It follows from A4 and 2.47 that
u
t
− ξ, Au
t
≥ F
1
u
t
≤ tF
1
u
t
,u
1 − t
u
t
− ξ, Au
t
tF
1
u
t
,u
1 − t
t
u − ξ, Aξ
≥ 0. 2.52
This shows that ξ ∈ EPF
1
,A. In a similar way, we can obtain that ξ ∈ EPF
2
,B.
Next, we show that ξ ∈ FS. We can conclude from Lemma 1.1 the desired conclusion
easily. This proves that ξ ∈F. Put
x P
F
x
1
. Since x P
F
x
1
⊂ C
n1
and x
n1
P
C
n1
x
1
,we
have
n
i
≤ lim sup
i →∞
x
1
− x
n
i
≤
x
1
− x
.
2.54
Journal of Inequalities and Applications 13
We, t herefore, obtain that
x
1
− ξ
lim
i →∞
2
be two bifunctions from C × C to R which satisfies A1–A4.LetA : C → H be an α-
inverse-strongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, and
S : C → C a nonexpansive mapping. Let {r
n
} and {s
n
} be two positive real sequences. Assume that
F : EPF
1
,A ∩ FPF
2
,B ∩ FS is not empty. Let {x
n
} be a sequence generated in the following
manner:
x
1
∈ C,
C
1
C,
F
1
u
n
,u
,v− v
n
1
s
n
v − v
n
,v
n
− x
n
≥ 0, ∀v ∈ C,
y
n
α
n
x
n
1 − α
n
S
γ
− w
,
x
n1
P
C
n1
x
1
,n≥ 1,
2.56
where {α
n
} and {γ
n
} are sequences in 0, 1. Assume that {α
n
}, {γ
n
}, {r
n
}, and {s
n
} satisfy the
following restrictions:
a 0 ≤ α
n
≤ a<1;
} be a sequence generated in the following
manner:
x
1
∈ C,
C
1
C,
z
n
γ
n
P
C
I − r
n
A
x
n
1 − γ
n
P
C
I − s
,
C
n1
w ∈ C
n
:
y
n
− w
≤
x
n
− w
,
x
n1
P
C
n1
x
1
n
≤ f<2α and 0 <e
≤ s
n
≤ f
< 2β.
Then the sequence {x
n
} converges strongly to some point x,wherex P
F
x
1
.
Proof. Putting F
1
F
2
≡ 0, we see that
Ax
n
,u− u
n
1
r
n
− r
n
Ax
n
. We also have v
n
P
C
x
n
− s
n
Bx
n
. We can obtain from
Theorem 2.1 the desired results immediately.
Corollary 2.4. Let C be a nonempty closed convex subset of a real Hilbert space H.LetA : C → H
be an α-inverse-strongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping,
and S : C → C a nonexpansive mapping. Let {r
n
} and {s
n
} be two positive real sequences. Assume
Journal of Inequalities and Applications 15
that F : VIC, A∩VIC, B∩FS is not empty. Let {x
n
} be a sequence generated in the following
manner:
x
1
n
,
y
n
α
n
x
n
1 − α
n
Sz
n
,
C
n1
w ∈ C
n
:
y
n
− w
following restrictions:
a 0 ≤ α
n
≤ a<1;
b 0 ≤ c ≤ γ
n
≤ d<1;
c 0 <e≤ r
n
≤ f<2α and 0 <e
≤ s
n
≤ f
< 2β.
Then the sequence {x
n
} converges strongly to some point x,wherex P
F
x
1
.
Theorem 2.5. Let C be a nonempty closed convex subset of a real Hilbert space H.LetF
1
and
F
2
be two bifunctions from C × C to R which satisfies A1–A4.LetS : C → C be a k-
strict pseudocontraction. Let {r
n
,u
n
− x
n
≥ 0, ∀u ∈ C,
F
2
v
n
,v
1
s
n
v − v
n
,v
n
− x
n
≥ 0, ∀v ∈ C,
z
n
γ
n
Sz
n
,
C
n1
w ∈ C
n
:
y
n
− w
≤
x
n
− w
,
x
n1
<b<1;
16 Journal of Inequalities and Applications
c 0 ≤ c ≤ γ
n
≤ d<1;
d 0 <e≤ r
n
≤ f<∞ and 0 <e
≤ s
n
≤ f
< ∞.
Then the sequence {x
n
} converges strongly to some point x,wherex P
F
x
1
.
Proof. Putting A B 0, we can obtain from Theorem 2.1 the desired conclusion
immediately.
Remark 2.6. Theorem 2.5 is generalization of Theorem TT. To be more precise, we consider a
pair of bifunctions and a strictly pseudocontractive mapping.
Let T : C → C be a k-strict pseudocontraction. It is known that I − T is a 1 − k/2-
inverse-strongly monotone mapping. The following results are not hard to derive.
Theorem 2.7. Let C be a nonempty closed convex subset of a real Hilbert space H.LetF
1
and
C
1
C,
F
1
u
n
,u
I − T
A
x
n
,u− u
n
1
r
n
u − u
n
,u
n
n
− x
n
≥ 0, ∀v ∈ C,
z
n
γ
n
u
n
1 − γ
n
v
n
,
y
n
α
n
x
n
1 − α
n
x
n
− w
,
x
n1
P
C
n1
x
1
,n≥ 1,
2.62
where {α
n
}, {β
n
}, and {γ
n
} are sequences in 0, 1. Assume that {α
n
}, {β
n
}, {γ
n
}, {r
n
}, and {s
1
.
Journal of Inequalities and Applications 17
Acknowledgment
This work was supported by a National Research Foundation of Korea Grant funded by the
Korean Government 2009-0076898.
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