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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
set of equilibrium problems and in the fixed point set of a strictly pseudocontractive mapping.
Strong convergence of the purposed hybrid projection algorithm is obtained in Hilbert spaces.
1. Introduction and Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm ·.LetC be a nonempty
closed convex subset of H and S : C → C a nonlinear mapping. In this paper, we use FS
to denote the fixed point set of S. Recall that the mapping S is said to be nonexpansive if
Sx − Sy
≤
x − y
, ∀x, y ∈ C. 1.1
S is said to be k-strictly pseudocontractive if there exists a constant k ∈ 0, 1 such that
Sx − Sy
2
≤
x − y
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
x − Sx
−
y − Sy
2
, ∀x, y ∈ C. 1.3
2 Journal of Inequalities and Applications
The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn
1 in 1967. It is easy to see that the class of strictly pseudocontractive mappings falls into the
class of nonexpansive mappings and the class of pseudocontractions.
Let A : C → H be a mapping. Recall that A is said to be monotone if
Ax − Ay, x − y
≥ 0, ∀x, y ∈ C. 1.4
A is said to be inverse-strongly monotone if there exists a constant α>0 such that
Ax − Ay, x − y
≥ α
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
Ax, y − x
≥ 0, ∀y ∈ C
. 1.7
To study the generalized equilibrium problems 1.6, we may assume that F satisfies
the following conditions:
A1 Fx, x0 for all x ∈ C;
A2 F is monotone, that is, Fx, yFy,x ≤ 0 for all x, y ∈ C;
A3 for each x, y, z ∈ C,
lim sup
t↓0
F
tz
1 − t
x, y
≤ F
x, y
; 1.8
A4 for each x ∈ C, y → Fx, y is convex and weakly lower semicontinuous.
Next, we give two special cases of the problem 1.6.
I If A ≡ 0, then the generalized equilibrium problem 1.6 is reduced to the following
equilibrium problem:
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.
Recently, many authors studied the problems 1.6 and 1.9 based on iterative
methods; see, for example, 2–18.
In 2007, Tada and Takahashi 17 considered the problem 1.9 and proved the
following result.
Theorem TT. Let C be a nonempty closed convex subset of H.LetF be a bifunction from C × C to R
satisfying A1–A4 and let S be a nonexpansive mapping of C into H such that FS ∩ EPF
/
∅.
Let {x
n
} and {u
n
} be sequences generated by x
1
x ∈ H and let
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
w
n
1 − α
n
x
n
α
n
Su
n
,
C
n
{
z ∈ H :
w
n
− z
≤
x
n
− z
}
,
D
n
{
z ∈ H :
x
n
− z, x − x
n
≥ 0
}
,
x
n1
P
C
n
∩D
n
x,
1.12
for every n ≥ 1,where{α
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
n
→ 0,thenx ∈ FT.
The following lemma can be found in 2, 3.
4 Journal of Inequalities and Applications
Lemma 1.2. Let C be a nonempty closed convex subset of H and let F : C × C → R be a bifunction
satisfying A1–A4. Then, for any r>0 and x ∈ H,thereexistsz ∈ C such that
F
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C. 1.13
Further, define
T
r
x
z ∈ C : F
z, y
1
r
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
; 1.15
c FT
r
EPF;
d EPF is closed and convex.
Lemma 1.3 see 1. Let C be a nonempty closed convex subset of a real Hilbert space H and S :
C → C a k-strict pseudocontraction with a fixed point. Define S : C → C by S
a
x ax 1 − aSx
for each x ∈ C.Ifa ∈ k, 1,thenS
a
is nonexpansive with FS
a
FS.
2. Main Results
Theorem 2.1. 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.LetA : C → H be an α-inverse-
strongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, and S : C → C
Journal of Inequalities and Applications 5
a k-strict pseudocontraction. 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
Ax
n
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ 0, ∀u ∈ C,
F
2
v
n
,v
Bx
n
,v− v
n
1
s
n
v − v
n
,v
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
β
n
z
n
1 − β
n
Sz
n
,
C
n1
w ∈ C
n
:
y
n
− w
≤
x
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
}
satisfy the following restrictions:
a 0 ≤ α
n
≤ a<1;
b 0 ≤ k ≤ β
n
<b<1;
c 0 ≤ c ≤ γ
n
≤ d<1;
d 0 <e≤ r
n
≤ f<2α and 0 <e
≤ s
n
≤ f
< 2β.
Then the sequence {x
n
} generated in Υ converges strongly to some point x,wherex P
F
x
1
.
Proof. Note that u
n
can be rewritten as
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
, ∀n ≥ 1. 2.2
Fix p ∈F. It follows that
p Sp T
r
n
p − r
n
Ap
T
s
n
p − s
n
Bp
, ∀n ≥ 1. 2.3
6 Journal of Inequalities and Applications
Note that I − r
n
A is nonexpansive for each n ≥ 1. Indeed, for any x,y ∈ C,weseefromthe
restriction d that
I − r
n
A
x −
I − r
n
A
y
2
x − y
− r
n
Ax − Ay
2
x − y
2
− 2r
n
x − y, Ax − Ay
r
2
n
Ax − Ay
2
≤
x − y
2
− r
n
2α − r
n
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
≤
x
n
− p
,
u
n
− p
≤
x
n
− p
. 2.5
This implies that
z
n
− p
≤ γ
n
u
n
− p
1 − γ
n
v
n
− p
≤
x
n
− p
. 2.6
Now, we are in a position to show that C
n
is closed and convex for each n ≥ 1. From the
assumption, we see that C
1
C is closed and convex. Suppose that C
m
is closed and convex
for some m ≥ 1. We show that C
m1
is closed and convex for the same m. Indeed, for any
w ∈ C
m
,weseethat
y
m
− w
≤
x
m
− w
2.7
is equivalent to
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
. Suppose that F⊂C
m
for some m ≥ 1. Putting
S
n
β
n
I
1 − β
n
S, ∀n ≥ 1, 2.9
we see from Lemma 1.3 that S
n
is a nonexpansive mapping for each n ≥ 1. For any w ∈F⊂
C
m
,weseefrom2.6 that
y
m
− w
α
m
x
m
1 − α
m
S
m
z
m
− w
≤ α
m
x
m
− w
1 − α
m
z
m
− w
≤
x
m
− w
.
2.10
Journal of Inequalities and Applications 7
This shows that w ∈ C
m1
. This proves that F⊂C
n
for each n ≥ 1. Note x
n
P
C
n
x
1
. For each
w ∈F⊂C
n
, we have
x
1
− x
n
≤
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
n
x
1
and x
n1
P
C
n1
x
1
∈ C
n1
⊂ C
n
, we have
0 ≤
x
1
− x
n
,x
n
− x
n1
x
1
− x
n
,x
n
− x
1
x
1
− x
n1
≤−
x
1
− x
n
2
x
1
− x
n
x
1
− x
n1
.
2.13
It follows that
x
n
− x
1
≤
x
n1
− x
1
. 2.14
This proves that lim
n →∞
x
n
− x
1
exists. Notice that
x
n
− x
n1
2
x
n
− x
1
x
1
− x
n1
2
x
n
− x
1
2
2
x
n
− x
1
,x
1
− x
n1
x
1
− x
n1
2
x
n
− x
1
2
2
x
n
− x
1
,x
1
− x
n
x
n
− x
n1
x
1
− x
n1
2
x
n
− x
1
2
− 2
x
n
− x
1
2
2
x
n
− x
1
,x
n
− x
n1
x
1
− x
n1
2
≤
x
1
− x
n1
2
−
x
n
− x
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
n1
≤
x
n
− x
n1
. 2.17
This implies that
y
n
− x
n
≤
y
n
− x
n1
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
− y
n
x
n
− α
n
x
n
−
1 − α
n
S
n
z
n
1 − α
n
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
2
T
r
n
I − r
n
Ax
n
− T
r
n
I − r
n
Ap
2
x
n
− p
− r
n
Ax
n
− Ap
2
x
n
− p
2
− 2r
n
x
n
− p, Ax
n
− Ap
r
2
n
Ax
n
− Ap
2
≤
x
n
− p
2
− r
n
2α − r
n
Ax
n
− Ap
2
.
2.22
In a similar way, we also have
v
n
− p
2
≤
x
n
− p
2
− s
n
2β − s
n
Bx
n
− Bp
2
. 2.23
Note that
y
n
− p
2
α
n
x
n
1 − α
n
S
n
z
n
− p
2
≤ α
n
x
n
− p
2
1 − α
n
S
n
z
n
− p
2
≤ α
n
x
n
− p
2
1 − α
n
z
n
− p
2
≤ α
n
x
n
− p
2
1 − α
n
γ
n
u
n
− p
2
1 − α
n
1 − γ
n
v
n
− p
2
.
2.24
Substituting 2.22 and 2.23 into 2.24, we arrive at
y
n
− p
2
≤
x
n
− p
2
−
1 − α
n
γ
n
r
n
2α − r
n
Ax
n
− Ap
2
−
1 − α
n
1 − γ
n
s
n
2β − s
n
Bx
n
− Bp
2
.
2.25
Journal of Inequalities and Applications 9
It follows that
1 − α
n
γ
n
r
n
2α − r
n
Ax
n
− Ap
2
≤
x
n
− p
2
−
y
n
− p
2
≤
x
n
− p
y
n
− p
x
n
− y
n
.
2.26
In view of the restrictions a–d and 2.19,weobtainthat
lim
n →∞
Ax
n
− Ap
0. 2.27
It also follows from 2.25 that
1 − α
n
1 − γ
n
s
n
2β − s
n
Bx
n
− Bp
2
≤
x
n
− p
2
−
y
n
− p
2
≤
x
n
− p
y
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
T
r
n
I − r
n
A
x
n
− T
r
n
I − r
n
A
p
2
≤
I − r
n
A
x
n
−
I − r
n
A
p, u
n
− p
1
2
I−r
n
Ax
n
− I −r
n
Ap
2
u
n
−p
2
−
I−r
n
A
x
n
−
I−r
n
A
p−
u
n
−p
2
≤
1
2
x
n
− p
2
u
n
− p
2
−
x
n
− u
n
− r
n
Ax
n
− Ap
2
1
2
x
n
−p
2
u
n
−p
2
−
x
n
−u
n
2
−2r
n
x
n
−u
n
,Ax
n
−Ap
r
2
n
Ax
n
−Ap
2
.
2.30
This implies that
u
n
− p
2
≤
x
n
− p
2
−
x
n
− u
n
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
n
− p
2
−
x
n
− v
n
2
2s
n
x
n
− v
n
Bx
n
− Bp
. 2.32
10 Journal of Inequalities and Applications
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
2
2r
n
1 − α
n
γ
n
x
n
− u
n
Ax
n
− Ap
−
1 − α
n
1 − γ
n
x
n
− v
n
2
2s
n
1 − α
n
1 − γ
n
x
n
− v
n
Bx
n
− Bp
≤
x
n
− p
2
−
1 − α
n
γ
n
x
n
− u
n
2
2r
n
x
n
− u
n
Ax
n
− Ap
−
1 − α
n
1 − γ
n
x
n
− v
n
2
2s
n
x
n
− v
n
Bx
n
− Bp
.
2.33
It follows that
1 − α
n
γ
n
x
n
− u
n
2
≤
x
n
− p
2
−
y
n
− p
2
2r
n
x
n
− u
n
Ax
n
− Ap
2s
n
x
n
− v
n
Bx
n
− Bp
≤
x
n
− p
y
n
− p
x
n
− y
n
2r
n
x
n
− u
n
Ax
n
− Ap
2s
n
x
n
− v
n
Bx
n
− Bp
.
2.34
In view of the restrictions a and c,weobtainfrom2.27 and 2.29 that
lim
n →∞
x
n
− u
n
0. 2.35
It also follows from 2.33 that
1 − α
n
1 − γ
n
x
n
− v
n
2
≤
x
n
− p
2
−
y
n
− p
2
2r
n
x
n
− u
n
Ax
n
− Ap
2s
n
x
n
− v
n
Bx
n
− Bp
≤
x
n
− p
y
n
− p
x
n
− y
n
2r
n
x
n
− u
n
Ax
n
− Ap
2s
n
x
n
− v
n
Bx
n
− Bp
.
2.36
Thanks to the restrictions a and c,weobtainfrom2.27 and 2.29 that
lim
n →∞
x
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
. 2.38
From 2.35 and 2.37,weseethat
lim
n →∞
x
n
− z
n
0. 2.39
Journal of Inequalities and Applications 11
On the other hand, we see from 2.21 that
β
n
z
n
− x
n
1 − β
n
Sz
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
Sz
n
− x
n
≤
1 k
1 − k
x
n
− z
n
Sz
n
− x
n
. 2.42
It follows from 2.39 and 2.41 that
lim
n →∞
x
n
− Sx
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
1
u
n
,u
Ax
n
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ 0. 2.44
From the condition A2,weseethat
Ax
n
,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
n
i
u − u
n
i
,
u
n
i
− x
n
i
r
n
i
≥ F
1
u, u
n
i
. 2.46
For any t with 0 <t≤ 1andu ∈ C, let u
t
tu 1 − tξ. Since u ∈ C and ξ ∈ C, we have 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
n
i
−
u
t
− u
n
i
,
u
n
i
− x
n
i
r
n
i
F
1
u
t
,u
n
i
u
t
−u
n
i
,Au
t
−Au
n
i
u
t
−u
n
i
,Au
n
i
−Ax
n
i
−
u
t
−u
n
i
,
u
n
i
−x
n
i
r
n
i
F
1
u
t
,u
n
i
.
2.47
12 Journal of Inequalities and Applications
Since A is Lipschitz continuous, we obtain from 2.35 that Au
n
i
− 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
,ξ
. 2.49
From A1, A4,and2.49,weseethat
0 F
1
u
t
,u
t
≤ tF
1
u
t
,u
1 − t
F
1
u
t
,ξ
≤ tF
1
u
t
,u
1 − t
u
t
− ξ, Au
t
tF
1
u
t
,u
1 − t
t
u − ξ, Au
t
,
2.50
which yields that
F
1
u
t
,u
1 − t
u − ξ, Au
t
≥ 0. 2.51
Letting t → 0 in the above inequality, we arrive at
F
1
ξ, u
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
x
1
− x
n1
≤
x
1
− x
. 2.53
On the other hand, we have
x
1
− x
≤
x
1
− ξ
≤ lim inf
i →∞
x
1
− x
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 →∞
x
1
− x
n
i
x
1
− x
. 2.55
This implies x
n
i
→ ξ x. Since {x
n
i
} is an arbitrary subsequence of {x
n
},weobtainthat
x
n
→ x as n →∞. This completes the proof.
If S is nonexpansive, then we have from Theorem 2.1 the following result immediately.
Corollary 2.2. 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.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
Ax
n
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ 0, ∀u ∈ C,
F
2
v
n
,v
Bx
n
,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
n
u
n
1 − γ
n
v
n
,
C
n1
w ∈ C
n
:
y
n
− w
≤
x
n
− 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;
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
.
As applications of Theorem 2.1, we consider the problems 1.9 and 1.11.
14 Journal of Inequalities and Applications
Theorem 2.3. 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 k-strict pseudocontraction. Let {r
n
} and {s
n
} be two positive real sequences. Assume
that F : VIC, A∩VIC, B∩FS is not empty. Let {x
n
} 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
n
B
x
n
,
y
n
α
n
x
n
1 − α
n
β
n
z
n
1 − β
n
Sz
n
,
C
n1
w ∈ C
n
:
y
n
− w
≤
x
n
− w
,
x
n1
P
C
n1
x
1
,n≥ 1,
2.57
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
}
satisfy the following restrictions:
a 0 ≤ α
n
≤ a<1;
b 0 ≤ k ≤ β
n
<b<1;
c 0 ≤ c ≤ γ
n
≤ d<1;
d 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
.
Proof. Putting F
1
F
2
≡ 0, we see that
Ax
n
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ 0, ∀u ∈ C, 2.58
is equivalent to
x
n
− r
n
Ax
n
− u
n
,u
n
− u
≥ 0, ∀u ∈ C. 2.59
This implies that u
n
P
C
x
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
∈ C,
C
1
C,
z
n
γ
n
P
C
I − r
n
A
x
n
1 − γ
n
P
C
I − s
n
B
x
n
,
y
n
α
n
x
n
1 − α
n
Sz
n
,
C
n1
w ∈ C
n
:
y
n
− w
≤
x
n
− w
,
x
n1
P
C
n1
x
1
,n≥ 1,
2.60
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;
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
} and {s
n
} be two positive real sequences. Assume that F :
EPF
1
∩ FPF
2
∩ 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
1
r
n
u − u
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
u
n
1 − γ
n
v
n
,
y
n
α
n
x
n
1 − α
n
β
n
z
n
1 − β
n
Sz
n
,
C
n1
w ∈ C
n
:
y
n
− w
≤
x
n
− w
,
x
n1
P
C
n1
x
1
,n≥ 1,
2.61
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
}
satisfy the following restrictions:
a 0 ≤ α
n
≤ a<1;
b 0 ≤ k ≤ β
n
<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
F
2
be two bifunctions from C × C to R which satisfies A1–A4.LetT
A
: C → C be a
k
α
-strict pseudocontraction, B : C → C a k
β
-strict pseudocontraction, and S : C → C a
k-strict pseudocontraction. Let {r
n
} and {s
n
} be two positive real sequences. Assume that F :
EPF
1
,I−T
A
∩FPF
2
,I−T
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
I − T
A
x
n
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ 0, ∀u ∈ C,
F
2
v
n
,v
I − T
B
x
n
,v− v
n
1
s
n
v − v
n
,v
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
β
n
z
n
1 − β
n
Sz
n
,
C
n1
w ∈ C
n
:
y
n
− w
≤
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
n
}
satisfy the following restrictions:
a 0 ≤ α
n
≤ a<1;
b 0 ≤ k ≤ β
n
<b<1;
c 0 ≤ c ≤ γ
n
≤ d<1;
d 0 <e≤ r
n
≤ f<1 − k
α
and 0 <e
≤ s
n
≤ f
< 1 − k
β
.
Then the sequence {x
n
} converges strongly to some point x,wherex P
F
x
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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