Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 840319, 25 pages
doi:10.1155/2011/840319
Research Article
The Shrinking Projection Method for Common
Solutions of Generalized Mixed Equilibrium
Problems and Fixed Point Problems for Strictly
Pseudocontractive Mappings
Thanyarat Jitpeera and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Bangmod, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam, [email protected]
Received 21 September 2010; Revised 14 December 2010; Accepted 20 January 2011
Academic Editor: Jewgeni Dshalalow
Copyright q 2011 T. Jitpeera and P. Kumam. 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 introduce the shrinking hybrid projection method for finding a common element of the set of
fixed points of strictly pseudocontractive mappings, the set of common solutions of the variational
inequalities with inverse-strongly monotone mappings, and the set of common solutions of
generalized mixed equilibrium problems in Hilbert spaces. Furthermore, we prove strong
convergence theorems for a new shrinking hybrid projection method under some mild conditions.
Finally, we apply our results to Convex Feasibility Problems CFP. The results obtained in this
paper improve and extend the corresponding results announced by Kim et al. 2010 and the
previously known results.
1. Introduction
Let H be a real Hilbert space with inner product ·, · and norm ·,andletE be a nonempty
closed convex subset of H.LetT : E → E be a mapping. In the sequel, we will use FT
to denote the set of fixed points of T,thatis,FT{x ∈ E : Tx x}.Wedenoteweak
convergence and strong convergence by notations and → , respectively.
I − S
x −
I − S
y
2
, ∀x, y ∈ E,
1.2
3 pseudocontractive if
Sx − Sy
2
≤
x − y
2
where S is strictly pseudocontractive mappings. Under appropriate restrictions on k,itis
proved that the mapping S
k
is nonexpansive. Therefore, the techniques of studying nonex-
pansive mappings can be applied to study more general strictly pseudocontractive mappings.
Recall that letting A : E → H be a mapping, then A is ca lled
1 monotone if
Ax − Ay, x − y
≥ 0, ∀x, y ∈ E, 1.5
2 β-inverse-strongly monotone if there exists a constant β>0suchthat
Ax − Ay, x − y
≥ β
Ax − Ay
2
, ∀x, y ∈ E.
1.6
The domain of the function ϕ : E →
∪{∞} is the set dom ϕ {x ∈ E : ϕx < ∞}.
Let ϕ : E →
∪{∞} be a proper extended real-valued function and let F be a bifunction of
E × E into
such that E ∩ dom ϕ
Ax, y − x
ϕ
y
− ϕ
x
≥ 0, ∀y ∈ E
. 1.8
Journal of Inequalities and Applications 3
We see that x is a solution of a problem 1.7 which implies that x ∈ dom ϕ {x ∈ E : ϕx <
∞}.
In particular, if A ≡ 0, then the problem 1.7 is reduced into the m ixed equilibrium
problem 2 for finding x ∈ E such that
F
x, y
ϕ
y
− ϕ
n
} be sequences generated by the
following iterative algorithm:
x
1
x ∈ H,
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ E,
w
n
1 − α
z ∈ H :
x
n
− z, x − x
n
≥ 0
}
,
x
n1
P
E
n
∩D
n
x, ∀n ≥ 1.
1.12
Then, they proved that, under certain appropriate conditions imposed on {α
n
} and {r
n
},the
sequence {x
n
} generated by 1.12 converges strongly to P
FS∩EPF
x.
In 2009, Qin and Kang 27 introduced an explicit viscosity approximation method for
n
,
x
n1
n
f
x
n
β
n
x
n
γ
n
α
1
n
S
k
x
n
α
2
n
y
fq.
In 2010, Kumam and Jaiboon 28 introduced a new method for finding a common
element of the set of fixed point of strictly pseudocontractive mappings, the set of common
solutions of variational inequalities with inverse-strongly monotone mappings, and the set of
common solutions of a system of generalized mixed equilibrium problems in Hilbert spaces.
Then, they proved that, under certain a ppropriate conditions imposed on {
n
}, {β
n
},and
{α
i
n
},wherei 1, 2, 3, 4, 5. The sequence {x
n
} converges strongly to q ∈ Θ : FS∩VIE, B∩
VIE, C ∩ GMEPF
1
,ϕ,A
1
∩ GMEPF
2
,ϕ,A
2
,whereq P
Θ
I − A γfq.
In this paper, motivate, by Tada and Takahashi 26, Qin and Kang 27,andKumam
and Jaiboon 28, we introduce a new shrinking projection method for finding a common
element of the set of fixed points of strictly pseudocontractive mappings, the set of common
2
− λ
1 − λ
x − y
2
,
2.1
for all x, y ∈ H and λ ∈ 0, 1.
For any x ∈ H,thereexistsaunique nearest point in E, denoted by P
E
x,suchthat
x − P
E
x
≤
x − y
, ∀y ∈ E. 2.2
E
x ∈ E and
x − P
E
x, y − P
E
x
≤ 0,
x − y
2
≥
x − P
E
x
2
y − P
E
x
convex. Define S
k
: E → E by S
k
kx 1 − kSx for each x ∈ E.ThenS
k
is nonexpansive such
that FS
k
FS.
Lemma 2.4 see 29. Let E be a closed convex subset of a real Hilbert space H,andletS : E → E
be a nonexpansive m apping. Then I − S is demiclosed at zero; that is,
x
n
x, x
n
− Sx
n
−→ 0 2.7
implies x Sx.
Lemma 2.5 see 30. Each Hilbert space H satisfies the Kadec-Klee property, for any sequence {x
n
}
with x
n
xand x
n
→x together implying x
n
− x→0.
x
n
− u
≤
u − q
2.8
for all n,thenx
n
→ q.
6 Journal of Inequalities and Applications
Lemma 2.8 see 33. Let E be a nonempty closed convex subset of a strictly convex Banach space
X.Let{T
n
: n ∈ } be a sequence of nonexpansive mappings on E. Suppose
∞
n1
FT
n
is nonempty.
Let δ
n
be a sequence of positive number with
∞
B1 for each x ∈ H and r>0, there exists a bounded subset D
x
⊆ E and y
x
∈ E such
that, for any z ∈ E \ D
x
,
F
z, y
x
ϕ
y
x
− ϕ
z
1
r
y
x
− z, z − x
− ϕ
z
1
r
y − z, z − x
≥ 0, ∀y ∈ E
, 2.11
for all z ∈ H. Then, the following hold:
1 for each x ∈ H, T
F
r
x
/
∅,
2 T
F
r
is single valued,
3 T
F
r
is firmly nonexpansive, that is, for any x, y ∈ H,
Lemma 2.10. Let H be a Hilbert space, let E be a nonempty closed convex subset of H,andlet
A : E → H be ρ-inverse-strongly monotone. If 0 <r≤ 2ρ,thenI − ρA is a nonexpansive mapping
in H.
Proof. For all x, y ∈ E and 0 <r≤ 2ρ,wehave
I − rA
x −
I − rA
y
2
x − y
− r
Ax − Ay
2
r
2
Ax − Ay
2
x − y
2
r
r − 2ρ
Ax − Ay
2
≤
Θ : F
S
∩ GMEP
F
1
,ϕ,A
1
∩ GMEP
F
2
,ϕ,A
2
∩ VI
E, B
∩ VI
E, C
/
∅. 3.1
Let {x
n
u
n
A
1
x
n
,u− u
n
1
r
n
u − u
n
,u
n
− x
n
≥ 0, ∀u ∈ E,
F
2
v
n
n
≥ 0, ∀v ∈ E,
y
n
P
E
x
n
− λ
n
Bx
n
,z
n
P
E
x
n
− μ
n
Cx
n
,
8 Journal of Inequalities and Applications
t
n1
{
w ∈ E
n
:
t
n
− w
≤
x
n
− w
}
,
x
n1
P
E
n1
x
0
, ∀n ≥ 0,
3.2
where {α
i
n
≤ 2ω,wherea, b are two positive constants,
C4 c ≤ λ
n
≤ 2β and d ≤ μ
n
≤ 2ξ,wherec, d are two positive constants,
C5 lim
n →∞
|λ
n1
− λ
n
| lim
n →∞
|μ
n1
− μ
n
| 0.
Then, {x
n
} converges strongly to P
Θ
x
0
.
Proof. Letting p ∈ Θ and by Lemma 2.9,weobtain
p P
E
A
2
p.
3.3
Note that u
n
T
F
1
r
n
I − r
n
A
1
x
n
∈ dom ϕ and v
n
T
F
2
s
n
I − s
n
A
2
x
A
1
p≤x
n
− p,
v
n
− p
T
F
2
s
n
I − s
n
A
2
x
n
is closed and convex for any n ≥ 1.
From the assumption, we see that E
1
E is closed and convex. Suppose that E
k
is
closed and convex for some k ≥ 1. Next, we show that E
k1
is closed and convex for some k.
For any p ∈ E
k
,weobtain
t
k
− p
≤
x
k
− p
3.5
is equivalent to
for each n ≥ 1. From the assumption, we see that Θ ⊂ E E
1
.
Suppose Θ ⊂ E
k
for some k ≥ 1. For any p ∈ Θ ⊂ E
k
,sincey
n
P
E
x
n
− λ
n
Bx
n
and
z
n
P
E
x
n
− μ
n
Cx
n
,foreachλ
n
p − λ
n
Bp
≤
x
n
− λ
n
Bx
n
−
p − λ
n
Bp
P
E
x
n
− μ
n
Cx
n
− P
E
p − μ
n
Cp
≤
x
n
− μ
≤
x
n
− p
.
3.7
From Lemma 2.3,wehaveS
k
is nonexpansive with FS
k
FS. It follows that
t
n
− p
α
1
n
n
S
k
x
n
− p
α
2
n
y
n
− p
α
3
n
z
n
− p
α
2
n
x
n
− p
α
3
n
x
n
− p
α
4
n
x
n
− p
n
0 and lim
n →∞
x
n
− t
n
0.
From x
n
P
E
n
x
0
,weget
x
0
− x
n
,x
n
− y
≥ 0 3.9
for each y ∈ E
n
.UsingΘ ⊂ E
n
n
− x
0
x
0
− p
−
x
0
− x
n
,x
0
− x
n
x
0
− x
n
,x
0
− p
≤−
x
0
− p
, ∀p ∈ Θ,n∈ . 3.12
From x
n
P
E
n
x
0
and x
n1
P
E
n1
x
0
∈ E
n1
⊂ E
n
,wehave
x
0
0
x
0
− x
n1
−
x
0
− x
n
,x
0
− x
n
x
0
− x
n
,x
0
− x
n1
≤−
n
x
0
− x
n1
,
3.14
and then
x
0
− x
n
≤
x
0
− x
n1
, ∀n ∈
. 3.15
Thus, the sequence {x
n
−x
0
} is a bounded and nondecreasing sequence, so lim
− x
0
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n1
x
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n
2
x
n
− x
− x
n1
x
0
− x
n1
2
≤−
x
n
− x
0
2
x
0
− x
n1
2
.
3.17
By 3.16,weobtain
x
n
− x
n1
x
n1
− t
n
≤ 2
x
n
− x
n1
. 3.19
By 3.18,weobtain
lim
n →∞
x
n
− t
n
0.
For p ∈ Θ,wenotethat
z
n
− p
2
P
E
x
n
− μ
n
Cx
n
− P
E
p − μ
n
Cp
− μ
n
Cx
n
− Cp
2
≤
x
n
− p
2
− 2μ
n
x
n
− p, Cx
n
− Cp
μ
2
2
x
n
− p
2
− μ
n
2ξ − μ
n
Cx
n
− Cp
2
.
3.21
12 Journal of Inequalities and Applications
Similarly, we also have
2
.
3.22
We note that
u
n
− p
2
T
F
1
r
n
I − r
n
A
1
x
n
− T
A
1
p
2
x
n
− p
− r
n
A
1
x
n
− A
1
p
2
− A
1
p
2
≤
x
n
− p
2
− 2r
n
ρ
A
1
x
n
− A
1
p
2
A
1
x
n
− A
1
p
2
x
n
− p
2
− r
n
2ρ − r
n
− s
n
2ω − s
n
A
2
x
n
− A
2
p
2
.
3.24
Observing that
t
n
− p
2
≤ α
− p
2
α
4
n
u
n
− p
2
α
5
n
v
n
− p
2
≤ α
1
n
4
n
u
n
− p
2
α
5
n
v
n
− p
2
.
3.25
Substituting 3.21, 3.22, 3.23,and3.24 into 3.25,weobtain
t
n
− p
Bx
n
− Bp
2
α
3
n
x
n
− p
2
− μ
n
2ξ − μ
n
n
− A
1
p
2
α
5
n
x
n
− p
2
− s
n
2ω − s
n
A
2
n
− Bp
2
− α
3
n
μ
n
2ξ − μ
n
Cx
n
− Cp
2
− α
4
n
r
n
2ρ − r
n
p
2
.
3.26
Journal of Inequalities and Applications 13
It follows that
α
3
n
μ
n
2ξ − μ
n
Cx
n
− Cp
2
≤
x
n
− α
4
n
r
n
2ρ − r
n
A
1
x
n
− A
1
p
2
− α
5
n
s
n
2ω − s
n
x
n
− t
n
.
3.27
From C2, C4,and3.20,wehave
lim
n →∞
Cx
n
− Cp
0.
3.28
Since s
n
∈ 0, 2ω,wealsohave
α
5
n
s
n
2ω − s
2
− α
2
n
λ
n
2β − λ
n
Bx
n
− Bp
2
− α
3
n
μ
n
2ξ − μ
n
x
n
− p
t
n
− p
x
n
− t
n
.
3.29
From C2, C3,and3.20,weobtain
lim
n →∞
A
2
0.
3.31
14 Journal of Inequalities and Applications
On the other hand, letting p ∈ Θ for each n ≥ 1, we get p T
F
1
r
n
I − r
n
A
1
p.SinceT
F
1
r
n
is firmly
nonexpansive, we have
u
n
− p
2
2
≤
I − r
n
A
1
x
n
−
I − r
n
A
1
p, u
n
− p
1
2
I − r
n
A
1
x
n
−
I − r
n
A
1
p −
u
n
− p
2
≤
1
2
x
n
− A
1
p
2
≤
1
2
x
n
− p
2
u
n
− p
n
A
1
x
n
− A
1
p
2
.
3.32
So, we obtain
u
n
− p
2
≤
x
n
.
3.33
Observe that
y
n
− p
2
P
E
x
n
− λ
n
Bx
n
− P
E
p − λ
n
I − λ
n
B
x
n
−
I − λ
n
B
p
2
y
n
− p
2
−
− p
2
y
n
− p
2
−
x
n
− y
n
− λ
n
Bx
n
− Bp
n
2
− λ
2
n
Bx
n
− Bp
2
2λ
n
x
n
− y
n
,Bx
n
− Bp
,
3.34
and hence
x
n
− y
n
Bx
n
− Bp
.
3.35
Journal of Inequalities and Applications 15
By using the same argument in 3.33 and 3.35,wecanget
v
n
− p
2
≤
x
n
,
z
n
− p
2
≤
x
n
− p
2
−
x
n
− z
n
2
2μ
n
− p
2
α
2
n
y
n
− p
2
α
3
n
z
n
− p
2
α
4
n
x
n
− p
2
−
x
n
− y
n
2
2λ
n
x
n
− y
n
Bx
− z
n
Cx
n
− Cp
α
4
n
x
n
− p
2
−
x
n
− u
n
2
−
x
n
− v
n
2
2s
n
x
n
− v
n
A
2
x
n
− A
2
p
n
− y
n
Bx
n
− Bp
− α
3
n
x
n
− z
n
2
2μ
n
α
3
n
x
n
A
1
x
n
− A
1
p
− α
5
n
x
n
− v
n
2
2s
n
α
5
n
x
n
x
n
− p
2
−
t
n
− p
2
− α
2
n
x
n
− y
n
2
2λ
n
α
3
n
x
n
− z
n
Cx
n
− Cp
2r
n
α
4
n
x
n
− u
n
A
1
A
2
x
n
− A
2
p
≤
x
n
− p
t
n
− p
x
n
n
− z
n
Cx
n
− Cp
2r
n
α
4
n
x
n
− u
n
A
1
x
n
− A
1
n →∞
x
n
− u
n
0.
3.39
By using the same argument, we can prove that
lim
n →∞
x
n
− y
n
lim
n →∞
x
n
− z
n
lim
n →∞
t
n
− z
n
lim
n →∞
t
n
− v
n
0.
3.41
Step 5. We show that
z ∈ F
S
∩ GMEP
F
1
,ϕ,A
1
∩ GMEP
F
x α
3
P
E
1 − μC
x α
4
T
F
1
r
I − rA
1
x
α
5
T
F
2
s
I − sA
2
x, ∀x ∈ E,
3.43
1 − λB
∩ F
P
E
1 − μC
∩ F
T
F
1
r
I − rA
1
∩ F
T
F
2
s
I − sA
2
Journal of Inequalities and Applications 17
We note that
Px
n
− x
n
≤
Px
n
− t
n
t
n
− x
n
α
1
S
x
n
α
5
T
F
2
s
I − sA
2
x
n
−
α
1
n
S
k
x
n
α
2
n
P
E
x
n
α
5
n
T
F
2
s
I − sA
2
x
n
t
n
− x
n
≤
x
n
α
2
− α
2
n
P
E
I − λ
n
B
x
n
α
3
P
E
I − μ
n
C
x
n
α
4
− α
4
n
T
s
I − sA
2
x
n
t
n
− x
n
≤
α
1
− α
1
n
I − λ
n
B
x
n
α
3
μ
n
− μ
Cx
n
α
3
− α
3
n
T
F
1
r
I − rA
1
x
n
α
5
− α
5
n
T
− α
i
n
|
λ
n
− λ
|
μ
n
− μ
t
n
− x
n
,
3.45
T
F
2
s
I − sA
2
x
n
, sup
n≥1
P
E
I − λ
n
B
x
n
,
sup
n≥1
n
.
3.46
From C2, C5,and3.20,weobtain
lim
n →∞
x
n
−Px
n
0.
3.47
18 Journal of Inequalities and Applications
Since {x
n
i
} is bounded, there exists a subsequence {x
n
i
} of {x
n
} which converges weakly to
z. Without loss of generality, we may assume that {x
n
i
} z. It follows from 3.47,that
∈ Θ ⊂ E
n
and x
n
P
E
n
x
0
,wehave
x
0
− x
n
x
0
− P
E
n
x
0
≤
≤
x
0
− z
.
3.50
Since z ∈ ω
w
x
n
⊂ Θ,weobtain
x
0
− z
x
0
− P
x
0
− z
≤ lim inf
i →∞
x
0
− x
n
≤ lim sup
i →∞
x
0
− x
n
≤
x
0
− z
.
− x
n
x
0
− z.UsingLemma 2.5,weobtainthat
x
n
− z
x
n
− x
0
−
z − x
0
−→ 0 3.54
as n →∞and hence x
n
→ z in norm. This completes the proof.
If the mapping S is nonexpansive, then S
k
S
0
,ϕ,A
2
∩ VI
E, B
∩ VI
E, C
/
∅. 3.55
Let {x
n
} be a sequence generated by the following iterative algorithm 3.1,where{α
i
n
} ar e sequences
in 0, 1,wherei 1, 2, 3, 4, 5, r
n
∈ 0, 2ρ, s
n
∈ 0, 2ω,and{λ
n
}, {μ
n
} are p ositive sequences.
Assume that the control sequences satisfy (C1)–(C5) in Theorem 3.1.Then,{x
n
∩ EP
F
2
∩ VI
E, B
∩ VI
E, C
/
∅. 3.56
Let {x
n
} be a sequence generated by the following iterative algorithm:
x
0
∈ H, E
1
E, x
1
P
E
1
x
0
,v
1
s
n
v − v
n
,v
n
− x
n
≥ 0, ∀v ∈ E,
z
n
P
E
x
n
− μ
n
Cx
n
,
y
n
n
u
n
α
5
n
v
n
,
E
n1
{
w ∈ E
n
:
t
n
− w
≤
x
n
− w
}
,
x
n1
n
,uF
1
v
n
,v0inCorollary 3.3,thenP
E
I and we get
u
n
y
n
x
n
and v
n
z
n
x
n
; hence, we can obtain the following result immediately.
Corollary 3.4. Let E be a nonempty closed convex subset of a real Hilbert space H.LetS : E → E be
a k-strictly p seudocontractive mapping with a fixed point. Define a mapping S
k
: E → E by S
k
x
kx1−kSx, for all x ∈ E. Suppose that FS
/
∅.Let{x
n
,
E
n1
{
w ∈ E
n
:
t
n
− w
≤
x
n
− w
}
,
x
n1
P
E
n1
x
0
, ∀n ≥ 1,
3.58
tomography 37, and radiation therapy treatment planning 38.
The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.
Theorem 4.1. Let E be a nonempty closed convex subset of a real Hilbert space H.Let{F
j
}
M
j1
be a
family of bifunction from E × E to
satisfying (A1)–(A5), and let ϕ : E → ∪{∞} be a proper
lower semicontinuous and convex function with either (B1) or (B2). Let A
j
: E → H be ρ
j
-inverse-
strongly monotone mapping for each j ∈{1, 2 , 3, ,M}.LetB
i
: E → H be β
i
-inverse-strongly
monotone mapping for each i ∈{1, 2, 3, ,N}.LetS : E → E be a k-strictly pseudocontractive
mapping with a fixed point. Define a mapping S
k
: E → E by S
k
x kx 1 − kSx, for all x ∈ E.
Suppose that
Θ : F
S
n
} be a sequence generated by the following iterative algorithm:
x
0
∈ H, E
1
E, x
1
P
E
1
x
0
,v
1
,v
2
, ,v
M
∈ E,
F
1
v
n,1
,v
1
ϕ
− x
n
≥ 0, ∀v
1
∈ E,
Journal of Inequalities and Applications 21
F
2
v
n,2
,v
2
ϕ
v
2
− ϕ
v
n,2
A
2
x
n,M
,v
M
ϕ
v
M
− ϕ
v
n,M
A
M
x
n
,v
M
− v
n,M
1
r
M
E
x
n
− λ
n,2
B
2
x
n
,
.
.
.
y
n,N
P
E
x
n
− λ
n,N
B
N
x
n
,
w ∈ E
n
:
t
n
− w
≤
x
n
− w
}
,
x
n1
P
E
n1
x
0
, ∀n ≥ 1,
4.2
where α
n,0
,α
n,1
,α
n,2
n →∞
α
i
n
α
i
∈ 0, 1,foreach0 ≤ i ≤ N,
C2 lim
n →∞
α
j
n
α
j
∈ 0, 1,foreach1 ≤ j ≤ M,
C3 a
j
≤ r
j
≤ 2ρ
j
,wherea
j
is some positive constants for each 1 ≤ j ≤ M,
C4 c
i
≤ λ
n,i
≤ 2β
i
; hence, we can obtain the following result immediately.
Theorem 4.2. Let E be a nonempty closed convex subset of a real Hilbert space H.Letϕ : E →
∪
{∞} be a proper lower semicontinuous and convex function with either (B1) or (B2). Let B
i
: E → H
be β
i
-inverse-strongly monotone mapping for each i ∈{1, 2, 3, ,N}.LetS : E → E be a k-strictly
pseudocontractive mapping with a fixed point. Define a mapping S
k
: E → E by S
k
x kx1−kSx,
for all x ∈ E. Suppose that
Θ : F
S
∩
N
i1
VI
E, B
i
x
n
,
y
n,2
P
E
x
n
− λ
n,2
B
2
x
n
,
.
.
.
y
n,N
P
E
x
n
− λ
:
t
n
− w
≤
x
n
− w
}
,
x
n1
P
E
n1
x
0
, ∀n ≥ 1,
4.4
where α
n,0
,α
n,1
,α
n,2
, ,α
n,N
n1,i
− λ
n,i
| 0,foreach1 ≤ i ≤ N.
Then, {x
n
} converges strongly to P
Θ
x
0
.
If B
i
0, for each 1 ≤ i ≤ N in Theorem 4.1,thenwegety
n,i
x
n
.Hence,wecanobtain
the following result immediately.
Theorem 4.3. Let E be a nonempty closed convex subset of a real Hilbert space H.Letbea{F
j
}
M
j1
be a
family of bifunction from E×E to
satisfying (A1)–(A5), and let ϕ : E → ∪{∞} be a proper lower
semicontinuous and convex function with either (B1) or (B2). Let A
j
: E → H be ρ
Journal of Inequalities and Applications 23
Let {x
n
} be a sequence generated by the following iterative algorithm:
x
0
∈ H, E
1
E, x
1
P
E
1
x
0
,v
1
,v
2
, ,v
M
∈ E,
F
1
v
n,1
,v
1
,v
n,1
− x
n
≥ 0, ∀v
1
∈ E,
F
2
v
n,2
,v
2
ϕ
v
2
− ϕ
v
n,2
A
2
v
n,M
,v
M
ϕ
v
M
− ϕ
v
n,M
A
M
x
n
,v
M
− v
n,M
1
r
M
,
E
n1
{
w ∈ E
n
:
t
n
− w
≤
x
n
− w
}
,
x
n1
P
E
n1
x
0
, ∀n ≥ 1,
4.6
where α
n →∞
α
j
n
α
j
∈ 0, 1,foreach1 ≤ j ≤ M,
C3 a
j
≤ r
j
≤ 2ρ
j
,wherea
j
is some positive constants for each 1 ≤ j ≤ M.
Then, {x
n
} converges strongly to P
Θ
x
0
.
Acknowledgments
The a uthors would like to than k the anonymous referees for helpful comments to improve
this paper, and the second author was supported by the Commission on Higher Education
and the Thailand Research Fund under Grant MRG5380044. Moreover, they also would like to
thank the National Research University Project of Thailand’s Office of the Higher Education
Commission for financial support under NRU-CSEC Project no. 54000267.
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