Báo cáo hóa học: " Research Article The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Strictly Pseudocontractive Mappings" ppt
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (590.16 KB, 25 trang )
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,
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.
Let S : E → E be a mapping. Then S is called
1 nonexpansive if
Sx − Sy
≤
x − y
, ∀x, y ∈ E, 1.1
2 Journal of Inequalities and Applications
2 strictly pseudocontractive with the coefficient k ∈ 0, 1 if
Sx − Sy
2
≤
x − y
2
k
I − S
x −
I − S
y
2
, ∀x, y ∈ E,
1.2
3 pseudocontractive if
Sx − Sy
2
≤
x − y
2
I − S
x −
I − S
y
2
, ∀x, y ∈ E.
1.3
The class of strictly pseudocontractive mappings falls into the one between classes of
nonexpansive mappings and pseudocontractive mappings. Within the past several decades,
many authors have been devoted to the studies on the existence and convergence of fixed
points for strictly pseudocontractive mappings. In 2008, Zhou 1 considered a convex
combination method to study strictly pseudocontractive mappings. More precisely, take
k ∈ 0, 1, and define a mapping S
k
by
S
k
x kx
1 − k
Sx, ∀x ∈ E, 1.4
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
/
∅,where is the set of real numbers.
There exists the generalized mixed equilibrium problem for finding x ∈ E such that
F
x, y
Ax, y − x
ϕ
y
− ϕ
x
≥ 0, ∀y ∈ E. 1.7
The set of solutions of 1.7 is denoted by GMEPF, ϕ, A,thatis,
GMEP
F, ϕ, A
x ∈ E : F
x, y
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
x
≥ 0, ∀y ∈ E. 1.9
The set of solutions of 1.9 is denoted by MEPF, ϕ.
If A ≡ 0andϕ ≡ 0, then the problem 1.7 is reduced into the equilibrium problem 3
for finding x ∈ E such that
F
x, y
≥ 0, ∀y ∈ E. 1.10
The set of solutions of 1.10 is denoted by EPF. This pro blem contains fixed point problems
and includes as special cases numerous problems in physics, optimization, and economics.
Some methods have been proposed to solve the equilibrium problem; please consult 4, 5.
If F ≡ 0andϕ ≡ 0, then the problem 1.7 is reduced into the Hartmann-Stampacchia
variational inequality 6 for finding x ∈ E such that
Ax, y − x
≥ 0, ∀y ∈ E. 1.11
The set of solutions of 1.11 is denoted by VIE, A. The variational inequality has been
extensively studied in the literature. See, for example, 7–10 and the references therein.
Many authors solved the problems GMEPF, ϕ, A,MEPF, ϕ,andEPF based on
iterative methods; see, for instance, 4, 5, 11–25 and reference therein.
In 2007, Tada and Takahashi 26 intr oduced a hybrid method for finding the common
element of the set of fixed point of nonexpansive mapping and the set of s olutions of
equilibrium problems in Hilbert spaces. Let {x
n
} and {u
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
n
x
n
α
n
Su
n
,
E
n
{
z ∈ H :
w
n
− z
≤
x
n
− z
}
,
D
n
{
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
finding a common element of the set of fixed point of strictly pseudocontractive mappings
4 Journal of Inequalities and Applications
and the set of solutions of variational inequalities with inverse-strongly monotone mappings
in Hilbert spaces:
x
1
∈ E,
z
n
P
E
x
n
− μ
n
Cx
n
,
y
n
P
E
x
n
− λ
n
Bx
n
,
x
n1
n
f
x
n
β
n
x
n
γ
n
α
1
n
S
k
x
n
α
2
n
y
n
α
3
n
z
n
, ∀n ≥ 1.
1.13
Then, they proved that, under certain appropriate conditions imposed on {
n
}, {β
n
}, {γ
n
},
{α
1
n
}, {α
2
n
},and{α
3
n
},thesequence{x
n
} generated by 1.13 converges strongly to q ∈
FS ∩ VIE, B ∩ VIE, C,whereq P
FS∩VIE,B∩VIE,C
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
solutions of generalized mixed equilibrium problems, and the set of common solutions of the
variational inequalities for inverse-strongly monotone mappings in Hilbert spaces. Finally,
we apply our results to Convex Feasibility Problems CFP. The results obtained in this paper
improve and extend the corresponding results announced by the previously known results.
2. Preliminaries
Let H be a real Hilbert space, and let E be a nonempty closed convex subset of H.Inareal
Hilbert space H,itiswellknownthat
λx
1 − λ
y
2
λ
x
2
1 − λ
y
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
The mapping P
E
is called the metric projection of H onto E.
It is well known that P
E
is a firmly nonexpansive mapping of H onto E,thatis,
x − y, P
E
x − P
E
y
≥
P
E
x − P
E
y
2
, ∀x, y ∈ H.
2.3
Journal of Inequalities and Applications 5
Moreover, P
E
x is characterized by the following properties: P
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
2
2.4
for all x ∈ H, y ∈ E.
Lemma 2.1. Let E be a nonempty closed convex subset of a real Hilbert space H.Givenx ∈ H and
z ∈ E,then,
z P
E
x ⇐⇒
x − z, y − z
≤ 0, ∀y ∈ E. 2.5
Lemma 2.2. Let H be a Hilbert space, let E be a nonempty closed convex subset of H,andletB be a
mapping of E into H.Letu ∈ E.Then,forλ>0,
u ∈ VI
E, B
⇐⇒ u P
E
u − λBu
, 2.6
where P
E
is the metric projection of H onto E.
Lemma 2.3 see 1. Let E be a nonempty closed convex subset of a real H ilbert space H,andlet
S : E → E be a k-strictly pseudocontractive mapping with a fixed point. Then FS is closed a nd
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.
Lemma 2.6 see 31. Let E be a closed convex subset of H.Let{x
n
} be a bounded sequence in H.
Assume that
1 the weak ω-limit set ω
w
x
n
⊂ E,
2 for each z ∈ E, lim
n →∞
x
n
− z exists.
Then {x
n
} is weakly convergent to a point in E.
Lemma 2.7 see 32. Let E be a closed convex subset of H.Let{x
n
} be a sequence in H and u ∈ H.
Let q P
E
u.If{x
n
} is ω
w
x
n
⊂ E and satisfies the condition
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
∞
n1
δ
n
1. Then a mapping S on E defined by
Sx
∞
n1
δ
n
T
n
x
2.9
for x ∈ E is well defined, nonexpansive, and FS
∞
n1
FT
n
holds.
For solving the mixed equilibrium problem, let us give the following assumptions for
the bifunction F, the function A,andthesetE:
A1 Fx, x0forallx ∈ E
A2 F is monotone, that is, Fx, yFy, x ≤ 0forallx, y ∈ E
A3 for each x, y, z ∈ E, lim
t →0
Ftz 1 − tx, y ≤ Fx, y
A4 for each x ∈ E, y → Fx, y is convex and lower semicontinuous
A5 for each y ∈ E, x → Fx, y is weakly upper semicontinuous
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
< 0,
2.10
B2 E is a bounded set.
By similar argument as i n the proof of Lemma 2.9 in 34, we have the following lemma
appearing.
Lemma 2.9. Let E be a nonempty closed convex subset of H.LetF : E × E →
be a bifunction
that satisfies (A1)–(A5), and let ϕ: E →
∪{∞} be a proper lower semicontinuous and convex
function. Assume that either (B1) or (B2) holds. For r>0 and x ∈ H, define a mapping T
F
r
: H → E
as follows:
T
F
r
x
z ∈ E : F
z, y
ϕ
y
− 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,
T
F
r
x − T
F
r
y
2
≤
T
F
r
x − T
F
r
y, x − y
,
2.12
4 FT
F
r
MEPF, ϕ,
5 MEPF, ϕ is closed and convex.
Journal of Inequalities and Applications 7
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
x − y
2
− 2r
x − y, Ax − Ay
r
2
Ax − Ay
2
≤
x − y
2
− 2rρ
Ax − Ay
r
2
Ax − Ay
2
x − y
2
r
r − 2ρ
Ax − Ay
2
≤
x − y
2
.
2.13
So, I − ρA is a nonexpansive mapping of E into H.
3. Main Results
In this section, we prove a strong convergence theorem of the new shrinking p rojection
method for finding a common element of the set of fixed points of strictly pseudocontractive
mappings, the set of common solutions of generalized mixed equilibrium problems and
the set of common solutions of the variational inequalities with inverse-strongly monotone
mappings in Hilbert spaces.
Theorem 3.1. Let E be a nonempty closed convex subset of a real Hilbert space H.LetF
1
and F
2
be two bifunctions from E × E to satisfying (A1)–(A5), and let ϕ : E → ∪{∞} be a pr oper
lower semicontinuous and convex function with either (B1) or (B2). Let A
1
, A
2
, B, C be four ρ, ω,
β, ξ-inverse-strongly monotone mappings of E into H, respectively. Let S : 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
∩ GMEP
F
1
,ϕ,A
1
∩ GMEP
F
2
,ϕ,A
2
∩ VI
E, B
∩ VI
E, C
/
∅. 3.1
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
,u
n
∈ E, v
n
∈ E,
F
1
u
n
,u
ϕ
u
− ϕ
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
,v
ϕ
v
− ϕ
v
n
A
2
x
n
,v− v
n
1
s
n
v − v
n
,v
n
− x
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
n
α
1
n
S
k
x
n
α
2
n
y
n
α
3
n
z
n
α
4
n
u
n
α
5
n
v
n
,
E
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
} are sequences in 0, 1,wherei 1, 2, 3, 4, 5, r
n
∈ 0, 2ρ, s
n
∈ 0, 2ω,and{λ
n
}, {μ
n
}
are positive sequences. Assume that the control sequences satisfy the following restrictions:
C1
5
i1
α
i
n
1,
C2 lim
n →∞
α
i
n
α
i
∈ 0, 1,wherei 1, 2, 3, 4, 5,
C3 a ≤ r
n
≤ 2ρ and b ≤ s
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
p − λ
n
Bp
P
E
p − μ
n
Cp
T
F
1
r
n
I − r
n
A
1
p T
F
2
s
n
I − s
n
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
n
∈ dom ϕ,thenwehave
u
n
− p
T
F
1
r
n
I − r
n
A
1
x
n
− T
F
1
r
n
I − r
n
A
1
p≤x
n
− p,
v
n
− p
T
F
2
s
n
I − s
n
A
2
x
n
− T
F
2
s
n
I − s
n
A
2
p
≤
x
n
− p
.
3.4
Next, we will divide the proof into six steps.
Step 1. We show that {x
n
} is well defined and E
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
t
k
− p
2
2
t
k
− x
k
,x
k
− p
≤ 0.
3.6
Thus, E
k1
is closed and convex. Then, E
n
is closed and convex for any n ≥ 1. This implies
that {x
n
} is well defined.
Journal of Inequalities and Applications 9
Step 2. We show that Θ ⊂ E
n
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
≤ 2β and μ
n
≤ 2ξ by Lemma 2.10,wehaveI − λ
n
B and
I − μ
n
C are nonexpansive. Thus, we obtain
y
n
− p
P
E
x
n
− λ
n
Bx
n
− P
E
p − λ
n
Bp
≤
x
n
− λ
n
Bx
n
−
p − λ
n
Bp
I − λ
n
B
x
n
−
I − λ
n
B
p
≤
x
n
− p
,
z
n
− p
P
E
x
n
− μ
n
Cx
n
− P
E
p − μ
n
Cp
≤
x
n
− μ
n
Cx
n
−
p − μ
n
Cp
I − μ
n
C
x
n
−
I − μ
n
C
p
≤
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
S
k
x
n
α
2
n
y
n
α
3
n
z
n
α
4
n
u
n
α
5
n
v
n
− p
≤ α
1
n
S
k
x
n
− p
α
2
n
y
n
− p
α
3
n
z
n
− p
α
4
n
u
n
− p
α
5
n
v
n
− p
≤ α
1
n
x
n
− p
α
2
n
x
n
− p
α
3
n
x
n
− p
α
4
n
x
n
− p
α
5
n
x
n
− p
x
n
− p
.
3.8
It follows that p ∈ E
k1
. This implies that Θ ⊂ E
n
for each n ≥ 1.
Step 3. We claim that lim
n →∞
x
n1
− x
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
,wehave
x
0
− x
n
,x
n
− p
≥ 0foreachp ∈ Θ,n∈ . 3.10
10 Journal of Inequalities and Applications
Hence, for p ∈ Θ,weobtain
0 ≤
x
0
− x
n
,x
n
− p
x
0
− x
n
,x
n
− x
0
x
0
− p
−
x
0
− x
n
,x
0
− x
n
x
0
− x
n
,x
0
− p
≤−
x
0
− x
n
2
x
0
− x
n
x
0
− p
.
3.11
It follows that
x
0
− x
n
≤
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
− x
n
,x
n
− x
n1
≥ 0. 3.13
For n ∈
,wecompute
0 ≤
x
0
− x
n
,x
n
− x
n1
x
0
− x
n
,x
n
− x
0
x
0
− x
n1
−
x
0
− x
n
,x
0
− x
n
x
0
− x
n
,x
0
− x
n1
≤−
x
0
− x
n
2
x
0
− x
n
,x
0
− x
n1
≤−
x
0
− x
n
2
x
0
− x
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
n →∞
x
n
−x
0
exists; that is, there exists m such that
m lim
n →∞
x
n
− x
0
.
3.16
Journal of Inequalities and Applications 11
From 3.13,weget
x
n
− x
n1
2
x
n
− x
0
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n1
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n
x
n
− x
n1
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n
2
x
n
− x
0
,x
n
− x
n1
x
0
− x
n1
2
−
x
n
− x
0
2
2
x
n
− x
0
,x
n
− x
n1
x
0
− x
n1
2
≤−
x
n
− x
0
2
x
0
− x
n1
2
.
3.17
By 3.16,weobtain
lim
n →∞
x
n
− x
n1
0.
3.18
Since x
n1
P
E
n1
x
0
∈ E
n1
⊂ E
n
,wehave
x
n
− t
n
≤
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.
3.20
Step 4. We claim that the following statements hold:
S1 lim
n →∞
x
n
− u
n
0,
S2 lim
n →∞
x
n
− y
n
0,
S3 lim
n →∞
x
n
− z
n
0,
S4 lim
n →∞
x
n
− v
n
0.
For p ∈ Θ,wenotethat
z
n
− p
2
P
E
x
n
− μ
n
Cx
n
− P
E
p − μ
n
Cp
2
≤
x
n
− μ
n
Cx
n
−
p − μ
n
Cp
2
x
n
− p
− μ
n
Cx
n
− Cp
2
≤
x
n
− p
2
− 2μ
n
x
n
− p, Cx
n
− Cp
μ
2
n
Cx
n
− Cp
2
≤
x
n
− p
2
μ
n
μ
n
− 2ξ
Cx
n
− Cp
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
y
n
− p
2
≤
x
n
− p
2
− λ
n
2β − λ
n
Bx
n
− Bp
2
.
3.22
We note that
u
n
− p
2
T
F
1
r
n
I − r
n
A
1
x
n
− T
F
1
r
n
I − r
n
A
1
p
2
≤
I − r
n
A
1
x
n
−
I − r
n
A
1
p
2
x
n
− p
− r
n
A
1
x
n
− A
1
p
2
x
n
− p
2
− 2r
n
x
n
− p, A
1
x
n
− A
1
p
r
2
n
A
1
x
n
− A
1
p
2
≤
x
n
− p
2
− 2r
n
ρ
A
1
x
n
− A
1
p
2
r
2
n
A
1
x
n
− A
1
p
2
x
n
− p
2
r
n
r
n
− 2ρ
A
1
x
n
− A
1
p
2
x
n
− p
2
− r
n
2ρ − r
n
A
1
x
n
− A
1
p
2
.
3.23
Similarly, we also have
v
n
− p
2
≤
x
n
− p
2
− s
n
2ω − s
n
A
2
x
n
− A
2
p
2
.
3.24
Observing that
t
n
− p
2
≤ α
1
n
S
k
x
n
− p
2
α
2
n
y
n
− p
2
α
3
n
z
n
− p
2
α
4
n
u
n
− p
2
α
5
n
v
n
− p
2
≤ α
1
n
x
n
− p
2
α
2
n
y
n
− p
2
α
3
n
z
n
− p
2
α
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
2
≤ α
1
n
x
n
− p
2
α
2
n
x
n
− p
2
− λ
n
2β − λ
n
Bx
n
− Bp
2
α
3
n
x
n
− p
2
− μ
n
2ξ − μ
n
Cx
n
− Cp
2
α
4
n
x
n
− p
2
− r
n
2ρ − r
n
A
1
x
n
− A
1
p
2
α
5
n
x
n
− p
2
− s
n
2ω − s
n
A
2
x
n
− A
2
p
2
x
n
− p
2
− α
2
n
λ
n
2β − λ
n
Bx
n
− Bp
2
− α
3
n
μ
n
2ξ − μ
n
Cx
n
− Cp
2
− α
4
n
r
n
2ρ − r
n
A
1
x
n
− A
1
p
2
− α
5
n
s
n
2ω − s
n
A
2
x
n
− A
2
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
− p
2
−
t
n
− p
2
− α
2
n
λ
n
2β − λ
n
Bx
n
− Bp
2
− α
4
n
r
n
2ρ − r
n
A
1
x
n
− A
1
p
2
− α
5
n
s
n
2ω − s
n
A
2
x
n
− A
2
p
2
≤
x
n
− p
t
n
− p
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
n
A
2
x
n
− A
2
p
2
≤
x
n
− p
2
−
t
n
− p
2
− α
2
n
λ
n
2β − λ
n
Bx
n
− Bp
2
− α
3
n
μ
n
2ξ − μ
n
Cx
n
− Cp
2
− α
4
n
r
n
2ρ − r
n
A
1
x
n
− A
1
p
2
≤
x
n
− p
t
n
− p
x
n
− t
n
.
3.29
From C2, C3,and3.20,weobtain
lim
n →∞
A
2
x
n
− A
2
p
0.
3.30
Similarly, by 3.28 and 3.30,wecanprovethat
lim
n →∞
Bx
n
− Bp
lim
n →∞
A
1
x
n
− A
1
p
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
T
F
1
r
n
I − r
n
A
1
x
n
− T
F
1
r
n
I − r
n
A
1
p
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
2
u
n
− p
2
−
I − r
n
A
1
x
n
−
I − r
n
A
1
p −
u
n
− p
2
≤
1
2
x
n
− p
2
u
n
− p
2
−
x
n
− u
n
− r
n
A
1
x
n
− A
1
p
2
≤
1
2
x
n
− p
2
u
n
− p
2
−
x
n
− u
n
2
2r
n
x
n
− u
n
A
1
x
n
− A
1
p
− r
2
n
A
1
x
n
− A
1
p
2
.
3.32
So, we obtain
u
n
− p
2
≤
x
n
− p
2
−
x
n
− u
n
2
2r
n
x
n
− u
n
A
1
x
n
− A
1
p
.
3.33
Observe that
y
n
− p
2
P
E
x
n
− λ
n
Bx
n
− P
E
p − λ
n
Bp
2
≤
I − λ
n
B
x
n
−
I − λ
n
B
p, y
n
− p
1
2
I − λ
n
B
x
n
−
I − λ
n
B
p
2
y
n
− p
2
−
I − λ
n
B
x
n
−
I − λ
n
B
p −
y
n
− p
2
≤
1
2
x
n
− p
2
y
n
− p
2
−
x
n
− y
n
− λ
n
Bx
n
− Bp
2
≤
1
2
x
n
− p
2
y
n
− p
2
−
x
n
− y
n
2
− λ
2
n
Bx
n
− Bp
2
2λ
n
x
n
− y
n
,Bx
n
− Bp
,
3.34
and hence
y
n
− p
2
≤
x
n
− p
2
−
x
n
− y
n
2
2λ
n
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
− p
2
−
x
n
− v
n
2
2s
n
x
n
− v
n
A
2
x
n
− A
2
p
,
z
n
− p
2
≤
x
n
− p
2
−
x
n
− z
n
2
2μ
n
x
n
− z
n
Cx
n
− Cp
.
3.36
Substituting 3.33, 3.35,and3.36 into 3.25,weobtain
t
n
− p
2
≤ α
1
n
x
n
− p
2
α
2
n
y
n
− p
2
α
3
n
z
n
− p
2
α
4
n
u
n
− p
2
α
5
n
v
n
− p
2
≤ α
1
n
x
n
− p
2
α
2
n
x
n
− p
2
−
x
n
− y
n
2
2λ
n
x
n
− y
n
Bx
n
− Bp
α
3
n
x
n
− p
2
−
x
n
− z
n
2
2μ
n
x
n
− z
n
Cx
n
− Cp
α
4
n
x
n
− p
2
−
x
n
− u
n
2
2r
n
x
n
− u
n
A
1
x
n
− A
1
p
α
5
n
x
n
− p
2
−
x
n
− v
n
2
2s
n
x
n
− v
n
A
2
x
n
− A
2
p
x
n
− p
2
− α
2
n
x
n
− y
n
2
2λ
n
α
2
n
x
n
− y
n
Bx
n
− Bp
− α
3
n
x
n
− z
n
2
2μ
n
α
3
n
x
n
− z
n
Cx
n
− Cp
− α
4
n
x
n
− u
n
2
2r
n
α
4
n
x
n
− u
n
A
1
x
n
− A
1
p
− α
5
n
x
n
− v
n
2
2s
n
α
5
n
x
n
− v
n
A
2
x
n
− A
2
p
.
3.37
It follows that
α
4
n
x
n
− u
n
2
≤
x
n
− p
2
−
t
n
− p
2
− α
2
n
x
n
− y
n
2
2λ
n
α
2
n
x
n
− y
n
Bx
n
− Bp
− α
3
n
x
n
− z
n
2
2μ
n
α
3
n
x
n
− z
n
Cx
n
− Cp
2r
n
α
4
n
x
n
− u
n
A
1
x
n
− A
1
p
− α
5
n
x
n
− v
n
2
2s
n
α
5
n
x
n
− v
n
A
2
x
n
− A
2
p
≤
x
n
− p
t
n
− p
x
n
− t
n
2λ
n
α
2
n
x
n
− y
n
Bx
n
− Bp
2μ
n
α
3
n
x
n
− z
n
Cx
n
− Cp
2r
n
α
4
n
x
n
− u
n
A
1
x
n
− A
1
p
2s
n
α
5
n
x
n
− v
n
A
2
x
n
− A
2
p
.
3.38
16 Journal of Inequalities and Applications
From C2, 3.20, 3.28, 3.30,and3.31,wehave
lim
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 →∞
x
n
− v
n
0.
3.40
Applying 3.20, 3.39,and3.40,wecanobtain
lim
n →∞
t
n
− u
n
lim
n →∞
t
n
− y
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
2
,ϕ,A
2
∩ VI
E, B
∩ VI
E, C
. 3.42
Assume that λ
n
→ λ ∈ c, 2β and μ
n
→ μ ∈ d, 2ξ.
Define a mapping P : E → E by
Px α
1
S
k
x α
2
P
E
1 − λB
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
where lim
n →∞
α
i
n
α
i
∈ 0, 1,wheni 1, 2, 3, 4, 5. By C1,thenwehave
5
i1
α
i
n
1.
From Lemma 2.8,wehaveP is nonexpansive and
F
P
F
S
k
∩ F
P
E
1 − λB
∩ F
P
E
1 − μC
∩ F
T
F
1
r
I − rA
1
∩ F
T
F
2
s
I − sA
2
F
S
k
∩ GMEP
F
1
,ϕ,A
1
∩ GMEP
F
2
,ϕ,A
2
∩ VI
E, B
∩ VI
E, C
.
3.44
Journal of Inequalities and Applications 17
We note that
Px
n
− x
n
≤
Px
n
− t
n
t
n
− x
n
α
1
S
k
x
n
α
2
P
E
1 − λB
x
n
α
3
P
E
1 − μC
x
n
α
4
T
F
1
r
I − rA
1
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
1 − λ
n
B
x
n
α
3
n
P
E
1 − μ
n
C
x
n
α
4
n
T
F
1
r
I − rA
1
x
n
α
5
n
T
F
2
s
I − sA
2
x
n
t
n
− x
n
≤
α
1
− α
1
n
S
k
x
n
α
2
P
E
I − λB
x
n
− P
E
I − λ
n
B
x
n
α
2
− α
2
n
P
E
I − λ
n
B
x
n
α
3
P
E
I − μC
x
n
− P
E
I − μ
n
C
x
n
α
3
− α
3
n
P
E
I − μ
n
C
x
n
α
4
− α
4
n
T
F
1
r
I − rA
1
x
n
α
5
− α
5
n
T
F
2
s
I − sA
2
x
n
t
n
− x
n
≤
α
1
− α
1
n
S
k
x
n
α
2
|
λ
n
− λ
|
Bx
n
α
2
− α
2
n
P
E
I − λ
n
B
x
n
α
3
μ
n
− μ
Cx
n
α
3
− α
3
n
P
E
I − μ
n
C
x
n
α
4
− α
4
n
T
F
1
r
I − rA
1
x
n
α
5
− α
5
n
T
F
2
s
I − sA
2
x
n
t
n
− x
n
≤ K
1
5
i1
α
i
− α
i
n
|
λ
n
− λ
|
μ
n
− μ
t
n
− x
n
,
3.45
where K
1
is an appropriate constant such that
K
1
max
sup
n≥1
T
F
1
r
I − rA
1
x
n
, sup
n≥1
T
F
2
s
I − sA
2
x
n
, sup
n≥1
P
E
I − λ
n
B
x
n
,
sup
n≥1
P
E
I − μ
n
C
x
n
, sup
n≥1
Bx
n
, sup
n≥1
Cx
n
, sup
n≥1
S
k
x
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
lim
n →∞
x
n
i
−Px
n
i
0.
3.48
It follows from Lemma 2.4 that z ∈ FP.By3.44,wehavez ∈ Θ.
Step 6. Finally, we show that x
n
→ z,wherez P
Θ
x
0
.
Since Θ is nonempty closed convex subset of H, there exists a unique z
∈ Θ such that
z
P
Θ
x
0
.Sincez
∈ 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
3.49
for all n ≥ 1. From 3.49, {x
n
} is bounded, so ω
w
x
n
/
∅. By the weak lower semicontinuity
of the norm, we have
x
0
− z
≤ lim inf
i →∞
x
0
− x
n
i
≤
x
0
− z
.
3.50
Since z ∈ ω
w
x
n
⊂ Θ,weobtain
x
0
− z
x
0
− P
Θ
x
0
≤
x
0
− z
. 3.51
Using 3.49 and 3.50,weobtainz
z.Thus,ω
w
x
n
{z} and x
n
z.Sowehave
x
0
− z
≤
x
0
− z
≤ lim inf
i →∞
x
0
− x
n
≤ lim sup
i →∞
x
0
− x
n
≤
x
0
− z
.
3.52
Thus,
x
0
− z
lim
i →∞
x
0
− x
n
x
0
− z
.
3.53
From x
n
z,weobtainx
0
− 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
S. We can obtain the following result
from Theorem 3.1 immediately.
Corollary 3.2. Let E be a nonempty closed convex subset of a real Hilbert space H.LetF
1
and F
2
be two bifunctions 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
1
, A
2
, B, C be four ρ, ω, β,
Journal of Inequalities and Applications 19
ξ-inverse-strongly monotone mappings of E into H, respectively. Let S : E → E be a nonexpansive
mapping with a fixed point. Suppose that
Θ : F
S
∩ GMEP
F
1
,ϕ,A
1
∩ GMEP
F
2
,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
} converges strongly
to P
Θ
x
0
.
If ϕ 0andA
1
A
2
0inTheorem 3 .1, then we can obtain the following result
immediately.
Corollary 3.3. Let E be a nonempty closed convex subset of a real Hilbert space H.LetF
1
and F
2
be
two bifunctions 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 B,C be two β, ξ-inverse-strongly
monotone mappings of E into H, respectively. Let S : E → E be a nonexpansive mapping with a fixed
point. Suppose that
Θ : F
S
∩ EP
F
1
∩ 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
,u
n
∈ E, v
n
∈ E,
F
1
u
n
,u
1
r
n
u − u
n
,u
n
− x
n
≥ 0, ∀u ∈ E,
F
2
v
n
,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
P
E
x
n
− λ
n
Bx
n
,
t
n
α
1
n
Sx
n
α
2
n
y
n
α
3
n
z
n
α
4
n
u
n
α
5
n
v
n
,
E
n1
{
w ∈ E
n
:
t
n
− w
≤
x
n
− w
}
,
x
n1
P
E
n1
x
0
, ∀n ≥ 1,
3.57
where {α
i
n
} are sequences in (0,1), where i 1, 2, 3, 4, 5, r
n
∈ 0, ∞, s
n
∈ 0, ∞ and {λ
n
}, {μ
n
} are
positive sequences. Assume that the control sequences satisfy the condition (C1)–(C5) in Theorem 3.1.
Then, {x
n
} converges strongly to P
Θ
x
0
.
If B 0, C 0, and F
1
u
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
} be a sequence generated by the following
20 Journal of Inequalities and Applications
iterative algorithm:
x
0
∈ H, E
1
E, x
1
P
E
1
x
0
,
t
n
α
n
S
k
x
n
1 − α
n
x
n
,
E
n1
{
w ∈ E
n
:
t
n
− w
≤
x
n
− w
}
,
x
n1
P
E
n1
x
0
, ∀n ≥ 1,
3.58
where {α
n
} are sequences in 0, 1. Assume that the control sequences sa tisfy the condition
lim
n →∞
α
n
α ∈ 0, 1 in Theorem 3.1.Then,{x
n
} converges strongly to a point P
FS
x
0
.
4. Convex Feasibility Problem
Finally, we consider the following Convex Feasibility Problem CFP:findinganx ∈
M
j1
C
j
,
where M ≥ 1 is an integer and each C
i
is assumed to be the solutions of equilibrium problem
with the bifunction F
j
, j 1, 2, 3, ,M and the solution set of the variational inequality
problem. There is a considerable investigation on CFP in the setting of Hilbert spaces which
captures applications in various disciplines such as image restoration 35, 36,computer
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
∩
⎛
⎝
M
j1
GMEP
F
j
,ϕ,A
j
⎞
⎠
∩
N
i1
VI
E, B
i
/
∅. 4.1
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
ϕ
v
1
− ϕ
v
n,1
A
1
x
n
,v
1
− v
n,1
1
r
1
v
1
− v
n,1
,v
n,1
− 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
,v
2
− v
n,2
1
r
2
v
2
− v
n,2
,v
n,2
− x
n
≥ 0, ∀v
2
∈ E,
.
.
.
F
M
v
n,M
,v
M
ϕ
v
M
− ϕ
v
n,M
A
M
x
n
,v
M
− v
n,M
1
r
M
v
M
− v
n,M
,v
n,M
− x
n
≥ 0, ∀v
M
∈ E,
y
n,1
P
E
x
n
− λ
n,1
B
1
x
n
,
y
n,2
P
E
x
n
− λ
n,2
B
2
x
n
,
.
.
.
y
n,N
P
E
x
n
− λ
n,N
B
N
x
n
,
t
n
α
n,0
S
k
x
n
N
i1
α
n,i
y
n,i
M
j1
α
n,j
v
n,j
,
E
n1
{
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
, ,α
n,N
and α
n,1
,α
n,2
, ,α
n,M
∈ 0, 1 such that
N
i0
α
n,i
M
j1
α
n,j
1,
{λ
n,i
} are positive sequences in 0, 1. Assume that the control sequences satisfy the following
restrictions:
C1 lim
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
,wherec
i
is some positive constants for each 1 ≤ i ≤ N,
C5 lim
n →∞
|λ
n1,i
− λ
n,i
| 0,foreach1 ≤ i ≤ N.
Then, {x
n
} converges strongly to P
Θ
x
0
.
If A
j
0, for each 1 ≤ j ≤ M and F
i
v
n,i
,v
i
0, for each 1 ≤ i ≤ N in Theorem 4.1,
then v
n,i
x
n
; 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
/
∅.
4.3
22 Journal of Inequalities and Applications
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
,
y
n,1
P
E
x
n
− λ
n,1
B
1
x
n
,
y
n,2
P
E
x
n
− λ
n,2
B
2
x
n
,
.
.
.
y
n,N
P
E
x
n
− λ
n,N
B
N
x
n
,
t
n
α
n,0
S
k
x
n
N
i1
α
n,i
y
n,i
,
E
n1
{
w ∈ E
n
:
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
∈ 0, 1 such that
N
i0
α
n,i
1, {λ
n,i
} are positive sequences in 0, 1.
Assume that the control sequences satisfy the following restrictions:
C1 lim
n →∞
α
i
n
α
i
∈ 0, 1,foreach0 ≤ i ≤ N,
C2 c
i
≤ λ
n,i
≤ 2β
i
,wherec
i
is some positive constants for each 1 ≤ i ≤ N,
C3 lim
n →∞
|λ
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
j
-inverse-strongly
monotone mapping for each j ∈{1, 2, 3, ,M}.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
∩
⎛
⎝
M
j1
GMEP
F
j
,ϕ,A
j
⎞
⎠
/
∅. 4.5
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
ϕ
v
1
− ϕ
v
n,1
A
1
x
n
,v
1
− v
n,1
1
r
1
v
1
− v
n,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
x
n
,v
2
− v
n,2
1
r
2
v
2
− v
n,2
,v
n,2
− x
n
≥ 0, ∀v
2
∈ E,
.
.
.
F
M
v
n,M
,v
M
ϕ
v
M
− ϕ
v
n,M
A
M
x
n
,v
M
− v
n,M
1
r
M
v
M
− v
n,M
,v
n,M
− x
n
≥ 0, ∀v
M
∈ E,
t
n
α
n,0
S
k
x
n
M
j1
α
n,j
v
n,j
,
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,0
and α
n,1
,α
n,2
, ,α
n,M
∈ 0, 1 such that α
n,0
M
j1
α
n,j
1. Assume that the control
sequences satisfy the following restrictions:
C1 lim
n →∞
α
0
n
α
0
∈ 0, 1,
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.
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.
References
1 H. Zhou, “Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces,”
Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 2, pp. 456–462, 2008.
2 L C. Ceng and J C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point
problems,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.
3 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The
Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
24 Journal of Inequalities and Applications
4 S. D. Fl
˚
am and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathemat-
ical Programming, vol. 78, pp. 29–41, 1997.
5 S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and
fixed point problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 331, no.
1, pp. 506–515, 2007.
6 P. Hartman and G. Stampacchia, “On some non-linear elliptic differential-functional equations,” Acta
Mathematica, vol. 115, pp. 271–310, 1966.
7 J.C. Yao and O. Chadli, “Pseudomonotone complementarity problems and variational inequalities,”
in Handbook of Generalized Convexity and Generalized Monotonicity, J. P. Crouzeix, N. Haddjissas, and S.
Schaible, Eds., vol. 76, pp. 501–558, Springer, New York, NY, USA, 2005.
8 H. Zegeye, E. U. Ofoedu, and N. Shahzad, “Convergence theorems for equilibrium problem,
variational inequality problem and countably infinite relatively quasi-nonexpansive mappings,”
Applied Mathematics and Computation, vol. 216, no. 12, pp. 3439–3449, 2010.
9 H. Zegeye and N. Shahzad, “A hybrid approximation method for equilibrium, variational inequality
and fixed point problems,” Nonlinear Analysis. Hybrid Systems, vol. 4, no. 4, pp. 619–630, 2010.
10 L. C. Zeng, S. Schaible, and J. C. Yao, “Iterative algorithm for generalized set-valued strongly
nonlinear mixed variational-like inequalities,” Journal of Optimization Theory and Applications, vol. 124,
no. 3, pp. 725–738, 2005.
11 X. Qin, Y. J. Cho, and S. M. Kang, “Viscosity approximation methods for generalized equilibrium
problems and fixed point problems with applications,” Nonlinear Analysis. Theory, Methods &
Applications, vol. 72, no. 1, pp. 99–112, 2010.
12 X. Gao and Y. Guo, “Strong convergence of a modified iterative algorithm for mixed-equilibrium
problems in Hilbert spaces,” Journal of Inequalities and Applications, vol. 2008, Article ID 454181, 23
pages, 2008.
13 C. Jaiboon and P. Kumam, “A hybrid extragradient viscosity approximation method for solving
equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Fixed
Point Theory and Applications, vol. 2009, Article ID 374815, 32 pages, 2009.
14 C. Jaiboon and P. Kumam, “Strong convergence for generalized equilibrium problems, fixed point
problems and relaxed cocoercive variational inequalities,” Journal of Inequalities and Applications,vol.
2010, Article ID 728028, 43 pages, 2010.
15 C. Jaiboon, P. Kumam, and U. W. Humphries, “Weak convergence theorem by an extragradient
method for variational inequality, equilibrium and fixed point problems,”
Bulletin of the Malaysian
Mathematical Sciences Society, vol. 32, no. 2, pp. 173–185, 2009.
16 J. S. Jung, “Strong convergence of composite iterative methods for equilibrium problems and fixed
point problems,” Applied Mathematics and Computation, vol. 213, no. 2 , pp. 498–505, 2009.
17 J. K. Kim, S. Y. Cho, and X. Qin, “Hybrid projection algorithms for generalized equilibrium problems
and strictly pseudocontractive mappings,” Journal of Inequalities and Applications, vol. 2010, Article ID
312602, 18 pages, 2010.
18 P. Kumam and C. Jaiboon, “A new hybrid iterative method for mixed equilibrium problems and
variational inequality problem for relaxed cocoercive mappings with application to optimization
problems,” Nonlinear Analysis. Hybrid Systems, vol. 3, no. 4, pp. 510–530, 2009.
19 S Y. Matsushita, K. Nakajo, and W. Takahashi, “Strong convergence theorems obtained by a
generalized projections hybrid method for families of mappings in Banach spaces,” Nonlinear
Analysis. Theory, Methods & Applications, vol. 73, no. 6, pp. 1466–1480, 2010.
20 J W. Peng and J C. Yao, “Strong convergence theorems of iterative scheme based on the extragradient
method for mixed equilibrium problems and fixed point problems,” Mathematical and Computer
Modelling, vol. 49, no. 9-10, pp. 1816–1828, 2009.
21 P. Katchang and P. Kumam, “A general iterative method of fixed points for mixed equilibrium
problems and variational inclusion problems,” Journal of Inequalities and Applications, vol. 2010, Article
ID 370197, 25 pages, 2010.
22 W. Kumam, C. Jaiboon, P. Kumam, and A. Singta, “A shrinking projection method for gener-
alized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-
nonexpansive mappings,” Journal of Inequalities and Applications, vol. 2010, Article ID 458247, 25 pages,
2010.
23 Y. Yao, Y C. Liou, and J C. Yao, “A new hybrid iterative algorithm for fixed-point problems, vari-
ational inequality problems, and mixed equilibrium problems,” Fixed Point Theory and Applications,
vol. 2008, Article ID 417089, 15 pages, 2008.
Journal of Inequalities and Applications 25
24 H. Zegeye and N. Shahzad, “A hybrid scheme for finite families of equilibrium, variational inequality
and fixed point problems,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 1, pp. 263–
272, 2011.
25 H. Zegeye and N. Shahzad, “Approximating common solution of variationalinequality problems for
two monotone mappings in Banach spaces,” Optimization Letters. In press.
26 A. Tada and W. Takahashi, “Weak and strong c onvergence theorems for a nonexpansive mapping and
an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370,
2007.
27 X. Qin and S. M. Kang, “Convergence theorems on an iterative method for variational inequality
problems and fixed point problems,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 33, no.
1, pp. 155–167, 2010.
28 P. Kumam and C. Jaiboon, “A system of generalized mixed equilibrium problems and fixed point
problems for pseudocontractive mappings in Hilbert spaces,” Fixed P oint Theory and Applications,vol.
2010, Article ID 361512, 33 pages, 2010.
29 F. E. Browder, “Nonlinear operators and nonlinear equations of evolution in Banach spaces,”
Proceedings of Symposia in Pure Mathematics, vol. 18, pp. 78–81, 1976.
30 W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
31 G. L. Acedo and H K. Xu, “Iterative methods for strict pseudo-contractions in Hilbert spaces,”
Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 7, pp. 2258–2271, 2007.
32 C. Martinez-Yanes and H K. Xu, “Strong convergence of the CQ method for fixed point iteration
processes,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006.
33 R. E. Bruck Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,”
Transactions of the American Mathematical Society, vol. 179, pp. 251–262, 1973.
34 J W. Peng and J C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium
problems, fixed point problems and variational inequality problems,” Taiwanese Journal of Mathemat-
ics, vol. 12, no. 6, pp. 1401–1432, 2008.
35 P. L. Combettes, “The convex feasibility problem: in image recovery,” in Advances in Imaging and
Electron Physics, P. H awkes, Ed., pp. 155–270, Academic Press, Orlando, Fla, USA, 1996.
36 T. Kotzer, N. Cohen, and J. Shamir, “Images to ration by a novel method of parallel projection onto
constraint sets,” Optics Letters, vol. 20, pp. 1172–1174, 1995.
37 M. I. Sezan and H. Stark, “Application of convex projection theory to image recovery in tomograph
and related areas,” in Image Recovery: Theory and Application, H. Stark, Ed., pp. 155–270, Academic
Press, Orlando, Fla, USA, 1987.
38 Y. Cen sor and S . A. Zeni os, Parallel Optimization, Numerical Mathematics and Scientific Computation,
Oxford University Press, New York, NY, USA, 1997.
