báo cáo hóa học:" Research Article Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces" pptx
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (575.34 KB, 24 trang )
Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 392741, 24 pages
doi:10.1155/2011/392741
Research Article
Strong Convergence of a New Iterative Method
for Infinite Family of Generalized Equilibrium and
Fixed-Point Problems of Nonexpansive Mappings
in Hilbert Spaces
Shenghua Wang
1, 2
and Baohua Guo
1, 2
1
National Engineering Laboratory for Biomass Power Generation Equipment,
North China Electric Power University, Baoding 071003, China
2
Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
Correspondence should be addressed to Shenghua Wang,
Received 15 October 2010; Accepted 18 November 2010
Academic Editor: Qamrul Hasan Ansari
Copyright q 2011 S. Wang and B. Guo. 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 an iterative algorithm for finding a common element of the set of solutions of an
infinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansive
mappings in a Hilbert space. We prove some strong convergence theorems for the proposed
iterative scheme to a fixed point of the family of nonexpansive mappings, which is the unique
solution of a variational inequality. As an application, we use the result of this paper to solve a
multiobjective optimization problem. Our result extends and improves the ones of Colao et al.
2008 and some others.
1. Introduction
Let H be a real Hilbert space and T be a mapping of H into itself. T is said to be nonexpansive
if
Tx − Ty
≤
x − y
, ∀x, y ∈ H. 1.1
If there exists a point u ∈ H such that Tu u, then the point u is called a fixed point of T.The
set of fixed points of T is denoted by FT. It is well known that FT is closed convex and
also nonempty if T has a bounded trajectory see 1.
2 Fixed Point Theory and Applications
Let f : H → H be a mapping. If there exists a constant 0 ≤ κ<1 such that
fx − fy
≤ κ
x − y
, ∀x, y ∈ H, 1.2
then f is called a contraction with the constant κ. Recall that an operator A : H → H is called
to be strongly positive with coefficient
γ>0if
Ax, x
≥
γx
2
, ∀x ∈ H.
1.3
Let u ∈ H be a fixed point, A be a strongly positive linear bounded operator on
H and {T}
N
n1
be a finite family of nonexpansive mappings of H into itself such that F
N
n1
FT
n
/
∅.
In 2003, Xu 2 introduced the following iterative scheme:
x
n1
I −
n1
A
T
n1
x
n
n1
u, ∀n ≥ 1, 1.4
where I is the identical mapping on H and T
n
T
n mod N
, and proved some strong
convergence theorems for the iterative scheme to the solution of the quadratic minimization
problem
min
x∈F
1
2
Ax, x
−
x, u
1.5
under suitable hypotheses on
n
and the additional hypothesis:
F F
T
1
T
2
···T
N
F
T
N
T
1
···T
N−1
··· F
T
2
T
3
···T
N
T
1
. 1.6
Recently, Marino and Xu 3 introduced a new iterative scheme from an arbitrary point
x
0
∈ H by the viscosity approximation method as follows:
x
n1
n
γf
x
n
I −
n
A
Tx
n
, ∀n ≥ 1, 1.7
and prove that the scheme strongly converges to the unique solution x
∗
of the variational
inequality:
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ F
T
, 1.8
which is the optimality condition for the minimization problem:
min
x∈F
1
2
Ax, x
− h
x
,
1.9
where h is a potential function for γf i.e., h
xγfx for all x ∈ H.
Fixed Point Theory and Applications 3
Let {T
n
}
N
n1
be a finite family of nonexpansive mappings of H into itself. In 2007, Yao
4 defined the mappings
U
n,1
λ
n,1
T
1
1 − λ
n,1
I,
U
n,2
λ
n,2
T
2
U
n,1
1 − λ
n,2
I,
.
.
.
U
n,N−1
λ
n,N−1
T
N−1
U
n,N−2
1 − λ
n,N−1
I,
W
n
≡ U
n,N
λ
n,N
T
N
U
n,N−1
1 − λ
n,N
I
1.10
and, by extending 1.10 , proposed the iterative scheme:
x
n1
n
γf
x
n
βx
n
1 − β
I −
n
A
W
n
x
n
, ∀n ≥ 1. 1.11
Then he proved that the iterative scheme 1.10 strongly converges to the unique solution x
∗
of the variational inequality:
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ F, 1.12
where F
N
n1
FT
n
, which is the optimality condition for the minimization problem:
min
x∈F
1
2
Ax, x−h
x
,
1.13
where h is a potential function for γf However, Colao et al. pointed out in 5 that there is a
gap in Yao’s proof.
Let C be a nonempty closed convex subset of H and G : C × C → R be a bifunction.
The equilibrium problem for the function G is to determine the equilibrium points, that is,
the set
EP
G
x ∈ C : G
x, y
≥ 0, ∀y ∈ C
. 1.14
Let A : C → H be a nonlinear mapping. Let EPG, A denote the set of all solutions to the
following equilibrium problem:
EP
cG, A
x ∈ C : G
x, y
Az, y − z
≥ 0, ∀y ∈ C
. 1.15
In the case of A ≡ 0, EPG, A is deduced to EP.InthecaseofG ≡ 0, EPG, A is also denoted
by VIC, A.
4 Fixed Point Theory and Applications
In 2007, S. Takahashi and W. Takahashi 6 introduced a viscosity approximation
method for finding a common element of EPG and FT from an arbitrary initial element
x
1
∈ H
G
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
x
n1
n
f
x
n
1 −
n
Tu
n
, ∀n ≥ 1,
1.16
and proved that, under certain appropriate conditions over
n
and r
n
, the sequences {x
n
} and
{u
n
} both converge strongly to z P
FT∩EPG
fz.
By combing the schemes 1.7 and 1.16, Plubtieg and Punpaeng 7 proposed the
following algorithm:
G
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
x
n1
n
γf
x
n
I −
n
A
Tu
n
, ∀n ≥ 1,
1.17
and proved that the iterative schemes {x
n
} and {u
n
} converge strongly to the unique solution
z of the variational inequality:
A − γf
z, x − z
≥ 0, ∀x ∈ F
T
∩ EP
G
, 1.18
which is the optimality condition for the minimization problem:
min
x∈FT∩EP
G
1
2
Ax, x
− h
x
,
1.19
where h is a potential function for γf.
Very recently, for finding a common element of the set of a finite family of
nonexpansive mappings and the set of solutions of an equilibrium problem, by combining
the schemes 1.11 and 1.17, Colao et al. 5 proposed the following explicit scheme:
G
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
x
n1
n
γf
x
n
βx
n
1 − β
I −
n
A
W
n
u
n
, ∀n ≥ 1,
1.20
and proved under some certain hypotheses that both sequences {x
n
} and {u
n
} converge
strongly to a point x
∗
∈ F which is an equilibrium point for G and is the unique solution
of the variational inequality:
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ F ∩ EP
G
, 1.21
where F
N
n1
FT
n
.
Fixed Point Theory and Applications 5
The equilibrium problems have been considered by many authors; see, for example,
6, 8–19 and the reference therein. But, in these references, the authors only considered at
most finite family of equilibrium problems and few of authors investigate the infinite family
of equilibrium problems in a Hilbert space or Banach space. In this paper, we consider a new
iterative scheme for obtaining a common element in the solution set of an infinite family
of generalized equilibrium problems and in the common fixed-point set of a finite family
of nonexpansive mappings in a Hilbert space. Let {T
n
}
N
n1
N ≥ 1 be a finite family of
nonexpansive mappings of H into itself, be {G
n
} : C × C → R be an infinite family of
bifunctions, and be {A
n
} : C → H be an infinite f amily of k
n
-inverse-strongly monotone
mappings. Let {r
n
} be a sequence such that r
n
⊂ r, 2k
n
with r>0 for each n ≥ 1. Define the
mapping T
r
i
: H → C by
T
r
i
x
z ∈ C : G
i
z, y
1
r
i
y − z, z − x
≥ 0, ∀y ∈ C
,x∈ H, i ≥ 1. 1.22
Assume that Ω
N
i1
FT
i
∞
i1
EPG
i
,A
i
/
∅. For an arbitrary initial point x
1
∈ H,we
define the iterative scheme {x
n
} by
z
n
α
n
x
n
n
i1
α
i−1
− α
i
T
r
i
I − r
i
A
i
x
n
,
x
n1
n
γf
x
n
δ
n
Bx
n
I − δ
n
B −
n
A
W
n
z
n
, ∀n ≥ 1,
1.23
where α
0
1, {α
n
}, {
n
} and {δ
n
} are three sequences in 0,1, A and B are both strongly
positive linear bounded operators on H, W
n
is defined by 1.10, and prove that, under
some certain appropriate hypotheses on the control sequences, the sequence {x
n
} strongly
converges to a point x
∗
∈ Ω, which is the unique solution of the variational inequality:
A − γf
x
∗
,x− x
∗
≥ 0, ∀x ∈ Ω. 1.24
If A
i
≡ A
0
, G
i
≡ G and r
i
≡ r, then 1.23 is reduced to the iterative scheme:
z
n
α
n
x
n
1 − α
n
T
r
I − rA
0
x
n
,
x
n1
n
γf
x
n
δ
n
Bx
n
I − δ
n
B −
n
A
W
n
z
n
, ∀n ≥ 1.
1.25
The proof method of the main result of this paper is different with ones of others in the
literatures and our result extends and improves the ones of Colao et al. 5 and some others.
6 Fixed Point Theory and Applications
2. Preliminaries
Let C be a closed convex subset of a Hilbert space H. For any point x ∈ H, there exists a
unique nearest point in C, denoted by P
C
x, such that
x − P
C
x
≤
x − y
, ∀y ∈ C. 2.1
Then P
C
is called the metric projection of H onto C. It is well known that P
C
is a nonexpansive
mapping of H onto C and satisfies the following:
x − y, P
C
x − P
C
y
≥
P
C
x − P
C
y
2
, ∀x, y ∈ H.
2.2
Let A be a mapping from C into H, then A is called monotone if
x − y, Ax − Ay
≥ 0 2.3
for all x, y ∈ C. However, A is called an α-inverse-strongly monotone mapping if there exists
a positive real number α such that
x − y, Ax − Ay
≥ α
Ax − Ay
2
2.4
for all x, y ∈ C.LetI denote the identity mapping of H, then for all x, y ∈ C and λ>0, one
has 20
I − λA
x −
I − λA
y
2
≤
x − y
2
λ
λ − 2α
Ax − Ay
2
.
2.5
Hence, if λ ∈ 0, 2α, then I − λA is a nonexpansive mapping of C into H.
If there exists u ∈ C such that
v − u, Au
≥ 0 2.6
for all v ∈ C, then u is called the solution of this variational inequality. The set of all solutions
of the variational inequality is denoted by VIC, A.
In this paper, we need the following lemmas.
Fixed Point Theory and Applications 7
Lemma 2.1 see 21. Given x ∈ H and y ∈ C.ThenP
C
x y if and only if there holds the
inequality
x − y, y − z
≥ 0, ∀z ∈ C. 2.7
Lemma 2.2 see 22. Let {s
n
} be a sequence of nonnegative real numbers satisfying
s
n1
≤
1 − η
n
s
n
η
n
τ
n
ξ
n
, ∀n ≥ 0, 2.8
where {η
n
}, {τ
n
}, and {ξ
n
} satisfy the conditions:
1 {η
n
}⊂0, 1,
∞
n1
η
n
∞ or, equivalently,
∞
n0
1 − η
n
0;
2 lim sup
n →∞
τ
n
≤ 0;
3 ξ
n
≥ 0 n ≥ 0,
∞
n0
ξ
n
< ∞.
Then lim
n →∞
s
n
0.
Let H be a Hilbert space. For all x, y ∈ H, the following equality holds:
x y
2
x
2
2
y, x y
−
y
2
.
2.9
Therefore, the following lemma naturally holds.
Lemma 2.3. Let H be a real Hilbert space. The following identity holds:
x y
2
≤
x
2
2
y, x y
, ∀x, y ∈ H.
2.10
Lemma 2.4 see 3. Assume that A is a strongly positive linear bounded operator on a Hilbert
space H with coefficient
γ>0 and 0 <ρ≤A
−1
.ThenI − ρA≤1 − ργ.
Lemma 2.5 see 2. Assume that {a
n
} is a sequence of nonnegative numbers such that
a
n1
≤
1 − γ
n
a
n
δ
n
, ∀n ≥ 0, 2.11
where {γ
n
} is a sequence in 0, 1 and δ
n
is a sequence in R such that
1
∞
n1
γ
n
∞;
2 lim sup
n →∞
δ
n
/γ
n
≤ 0 or
∞
n1
|δ
n
| < ∞.
Then lim
n →∞
a
n
0.
8 Fixed Point Theory and Applications
Lemma 2.6 see 23. Let C be a nonempty closed convex subset of a Hilbert space H and let G :
C × C → R be a bifunction which satisfies the following:
A1 Gx, x0 for all x ∈ C;
A2 G is monotone, that is, Gx, yGy, x ≤ 0 for all x, y ∈ C;
A3 For each x, y, z ∈ C,
lim
t↓0
G
tz
1 − t
x, y
≤ G
x, y
;
2.12
A4 For each x ∈ C, y → Gx, y is convex and lower semicontinuous.
For x ∈ H and r>0, define a mapping T
r
: H → C by
T
r
x
z ∈ C : G
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C
. 2.13
Then T
r
is well defined and the following hold:
1 T
r
is single-valued;
2 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
;
2.14
3 FT
r
EPG;
4 EPG is closed and convex.
It is easy to see that if there exists some point v ∈ C such that v T
r
I − rAv, where
A : C → H is an α-inverse strongly monotone mapping, then v ∈ EPG, A. In fact, since
v T
r
I − rAv, one has
G
v, y
1
r
y − v, v −
I − rA
v
≥ 0, ∀y ∈ C,
2.15
that is,
G
v, y
y − v, Av≥0, ∀y ∈ C. 2.16
Hence, v ∈ EPG, A.
Let C be a nonempty convex subset of a Banach space. Let {T
i
}
N
i1
be a finite family of
nonexpansive mappings of C into itself and λ
1
,λ
2
, ,λ
N
be real numbers such that 0 ≤ λ
i
≤ 1
Fixed Point Theory and Applications 9
for each i 1, 2, ,N. Define a mapping W of C into itself as follows:
U
1
λ
1
T
1
1 − λ
1
I,
U
2
λ
2
T
2
U
1
1 − λ
2
I,
.
.
.
U
N−1
λ
N−1
T
N−1
U
N−2
1 − λ
N−1
I,
W U
N
λ
N
T
N
U
N−1
1 − λ
N
I.
2.17
Such a mapping W is called the W-mapping generated by T
1
,T
2
, ,T
N
and λ
1
,λ
2
, ,λ
N
see 5, 24, 25.
Lemma 2.7 see 26. Let C be a nonempty closed convex subset of a Banach space. Let
T
1
,T
2
, ,T
N
be nonexpansive mappings of C into itself such that
N
i1
FT
i
/
∅ and let
λ
1
,λ
2
, ,λ
N
be real numbers such that 0 <λ
i
< 1 for each i 1, 2, ,N − 1 and 0 <λ
N
≤ 1.Let
W be the W-mapping of C generated by T
1
,T
2
, ,T
N
and λ
1
,λ
2
, ,λ
N
.ThenFW
N
i1
FT
i
.
Lemma 2.8 see 5. Let C be a nonempty convex subset of a Banach space. Let {T
i
}
N
i1
be a
finite family of nonexpansive mappings of C into itself and let {λ
n,i
}
N
i1
be sequences in 0, 1 such
that λ
n,i
→ λ
i
for each i 1, 2, ,N. Moreover, for each n ∈ N,letW and W
n
be the W-
mappings generated by T
1
,T
2
, ,T
N
and λ
1
,λ
2
, ,λ
N
and T
1
,T
2
, ,T
N
and λ
n,1
,λ
n,2
, ,λ
n,N
,
respectively. Then, for all x ∈ C, it follows that
lim
n →∞
W
n
x − Wx
0.
2.18
3. Main Results
Now, we give our main results in this paper.
Theorem 3.1. Let H be a Hilbert space and C be a nonempty closed convex subset of H.Letf : H →
H be a contraction with coefficient 0 <κ<1, A, B : H → H be strongly positive linear bounded
self-adjoint operators with coefficients
γ>0 and β>2B B
2
, respectively, {T
n
}
N
n1
: H →
H N ≥ 1 be a finite family of nonexpansive mappings, {G
n
} : C × C → R be an infinite family
of bifunctions satisfying A1–A4, and {A
n
} : C → H be an infinite family of inverse-strongly
monotone mappings with constants {k
n
} such that Ω
N
i1
FT
i
∩
∞
i1
EPG
i
,A
i
/
∅.Let
{
n
} and {δ
n
} be two sequences in 0, 1, {λ
n,i
}
N
i1
be asequence in a, b with 0 <a≤ b<1, {r
n
}
be a sequence in r, 2k
n
with r>0 and {α
n
} be a strictly decreasing sequence 0, 1. Set α
0
1.
Take a fixed number γ>0 with 0 <
γ − γκ < 1. Assume that
E1 lim
n →∞
n
0and
∞
n1
n
∞;
E2 lim
n →∞
|λ
n1,i
− λ
n,i
| 0 for each i 1, 2, ,N;
E3 0 ≤ δ
n
n
≤ 1 for all n ≥ 1;
E4 {δ
n
}⊂0, min{c, 1/β, 2BB
2
−β
β −B
2
− 2B
2
8βB/4βB} with
c<1;
E5
∞
n1
|
n1
−
n
| < ∞,
∞
n1
|α
n1
− α
n
| < ∞,
∞
n1
|δ
n1
− δ
n
| < ∞.
10 Fixed Point Theory and Applications
Then the sequence {x
n
} defined by 1.23 converges strongly to x
∗
∈ Ω, which is the unique solution
of the variational inequality: 1.24, that is,
x
∗
P
Ω
I −
A − γf
x
∗
. 3.1
Proof. Since
n
→ 0asn →∞by the condition E1, we may assume, without loss of
generality, that
n
< 1 − δ
n
BA
−1
for all n ≥ 1. Noting that A and B are both the linear
bounded self-adjoint operators, one has
A sup
{|
Ax, x
|
: x ∈ H, x 1
}
,
B sup
{|
Bx,x
|
: x ∈ H, x 1
}
.
3.2
Observing that
I − δ
n
B −
n
A
x, x
1 − δ
n
Bx,x−
n
Ax, x
≥ 1 − δ
n
B−
n
A
≥ 0,
3.3
we obtain that I − δ
n
B −
n
A is positive for all n ≥ 1. It follows that
I − δ
n
B −
n
A
sup
{
I − δ
n
B −
n
A
x, x
: x ∈ H, x 1
}
sup
{
1 −
δ
n
B
n
A
x, x
: x ∈ H, x 1
}
≤ 1 − δ
n
β −
n
γ.
3.4
For each n ≥ 1, define a quadratic function fδ
n
in δ
n
as follows:
f
δ
n
2
βBδ
2
n
β −B
2
− 2B
δ
n
. 3.5
Note that
f
0
f
2
B
−
β
B
2
2β
B
0, 3.6
f
⎛
⎜
⎜
⎝
2
B
B
2
− β
β −
B
2
− 2
B
2
8β
B
4β
B
⎞
⎟
⎟
⎠
1.
3.7
Hence, for each δ
n
satisfying the condition E4, one has
0 < 2
β
B
δ
2
n
β −
B
2
− 2
B
δ
n
< 1. 3.8
Fixed Point Theory and Applications 11
Moreover, it follows from 3.7, f1/B > 1andE4 that
δ
n
<
1
B
, ∀n ≥ 1.
3.9
Next, we proceed the proof with following steps.
Step 1. {x
n
} is bounded.
Let p ∈ Ω. Lemma 2.6 shows that every T
r
i
is firmly nonexpansive and hence
nonexpansive. Since r<r
i
< 2k
i
, I −r
i
A
i
is nonexpansive for each i ≥ 1. Therefore, T
r
i
I −r
i
A
i
is nonexpansive for each i ≥ 1. Noting that {α
n
} is strictly decreasing, α
0
1, we have
z
n
− p
α
n
x
n
− p
n
i1
α
i−1
− α
i
T
r
i
I − r
i
A
i
x
n
− T
r
i
I − r
i
A
i
p
≤ α
n
x
n
− p
n
i1
α
i−1
− α
i
T
r
i
I − r
i
A
i
x
n
− T
r
i
I − r
i
A
i
p
≤ α
n
x
n
− p
n
i1
α
i−1
− α
i
x
n
− p
x
n
− p
3.10
and hence
W
n
z
n
− p
W
n
z
n
− W
n
p
≤
z
n
− p
≤
x
n
− p
. 3.11
Then, from 3.4 and 3.11, it follows that noting that B is linear and
β>2B B
2
⇒ β>
B
x
n1
− p
n
γf
x
n
− Ap
δ
n
Bx
n
− Bp
I − δ
n
B −
n
A
W
n
z
n
− p
≤
n
γf
x
n
− Ap
δ
n
B
x
n
− p
I − δ
n
B −
n
A
W
n
z
n
− p
≤
n
γκ
x
n
− p
n
γf
p
− Ap
δ
n
B
x
n
− p
1 − δ
n
β −
n
γ
W
n
z
n
− p
≤
n
γκ
x
n
− p
n
γf
p
− Ap
δ
n
B
x
n
− p
1 − δ
n
β −
n
γ
x
n
− p
≤
1 −
n
γ − γκ
x
n
− p
n
γf
p
− Ap
.
3.12
12 Fixed Point Theory and Applications
It follows from
n
∈ 0, 1 and 0 < γ − γκ < 1that0<
n
γ − γκ < 1. Therefore, by the simple
induction, we have
x
n
− p
≤ max
x
1
− p
,
γf
p
− Ap
γ − γκ
, ∀n ≥ 1, 3.13
which shows that {x
n
} is bounded, so is {z
n
}.
Step 2. x
n1
− x
n
→0asn →∞.
First, we prove
lim
n →∞
W
n1
z
n
− W
n
z
n
0.
3.14
Let i ∈{0, 1, ,N − 2} and set
M
1
sup
n
z
n
T
1
z
n
N
i2
T
i
U
n,i−1
z
n
< ∞.
3.15
It follows from the definition of W
n
that
U
n1,N−i
z
n
− U
n,N−i
z
n
λ
n1,N−i
T
N−i
U
n1,N−i−1
z
n
1 − λ
n1,N−i
z
n
− λ
n,N−i
T
N−i
U
n,N−i−1
z
n
−
1 − λ
n,N−i
z
n
≤ λ
n1,N−i
T
N−i
U
n1,N−i−1
z
n
− T
N−i
U
n,N−i−1
z
n
|
λ
n1,N−i
− λ
n,N−i
|
T
N−i
U
n,N−i−1
z
n
|
λ
n1,N−i
− λ
n,N−i
|
z
n
≤
U
n1,N−i−1
z
n
− U
n,N−i−1
z
n
z
n
T
N−i
U
n,N−i−1
z
n
|
λ
n1,N−i
− λ
n,N−i
|
≤
U
n1,N−i−1
z
n
− U
n,N−i−1
z
n
M
1
|
λ
n1,N−i
− λ
n,N−i
|
3.16
for each i ∈{0, 1, ,N− 2}. Thus, using the above recursive inequalities repeatedly, we have
W
n1
z
n
− W
n
z
n
U
n1,N
z
n
− U
n,N
z
n
≤ M
1
N
i2
|
λ
n1,i
− λ
n,i
|
|
λ
n1,1
− λ
n,1
|
z
n
T
1
z
n
≤ M
1
N
i1
|
λ
n1,i
− λ
n,i
|
.
3.17
Fixed Point Theory and Applications 13
Also, we have
z
n
− z
n−1
α
n
x
n
n
i1
α
i−1
− α
i
T
r
i
I − r
i
A
i
x
n
− α
n−1
x
n−1
−
n
i1
α
i−1
− α
i
T
r
i
I − r
i
A
i
x
n−1
α
n−1
− α
n
T
r
n
I − r
n
A
n
x
n−1
≤ α
n
x
n
− x
n−1
|
α
n
− α
n−1
|
x
n−1
n
i1
α
i−1
− α
i
x
n
− x
n−1
|
α
n−1
− α
n
|
T
r
n
I − r
n
A
n
x
n−1
x
n
− x
n−1
|
α
n
− α
n−1
|
x
n−1
|
α
n−1
− α
n
|
T
r
n
I − r
n
A
n
x
n−1
≤
x
n
− x
n−1
|
α
n
− α
n−1
|
L,
3.18
where L sup{x
n−1
T
r
n
I − r
n
A
n
x
n−1
}.
Next, we prove lim
n →∞
x
n1
− x
n
0. Observe noting that B is linear that
x
n1
− x
n
n
γ
f
x
n
− f
x
n−1
n
γf
x
n−1
δ
n
B
x
n
− x
n−1
δ
n
Bx
n−1
I − δ
n
B −
n
A
W
n
z
n
− W
n
z
n−1
I − δ
n
B −
n
A
W
n
z
n−1
−
n−1
γf
x
n−1
− δ
n−1
Bx
n−1
−
I − δ
n−1
B −
n−1
A
W
n−1
z
n−1
n
γ
f
x
n
− f
x
n−1
δ
n
B
x
n
− x
n−1
I − δ
n
B −
n
A
W
n
z
n
− W
n
z
n−1
n
−
n−1
γf
x
n−1
δ
n
− δ
n−1
Bx
n−1
W
n
z
n−1
− W
n−1
z
n−1
δ
n−1
BW
n−1
z
n−1
− δ
n
BW
n
z
n−1
n−1
AW
n−1
z
n−1
−
n
AW
n
z
n−1
n
γ
f
x
n
− f
x
n−1
δ
n
B
x
n
− x
n−1
I − δ
n
B −
n
A
W
n
z
n
− W
n
z
n−1
n
−
n−1
γf
x
n−1
δ
n
− δ
n−1
Bx
n−1
W
n
z
n−1
− W
n−1
z
n−1
δ
n−1
− δ
n
BW
n−1
z
n−1
δ
n
B
W
n−1
z
n−1
− W
n
z
n−1
n−1
−
n
AW
n−1
z
n−1
n
A
W
n−1
z
n−1
− W
n
z
n−1
.
3.19
14 Fixed Point Theory and Applications
Hence, by 3.4 and 3.18,weget
x
n1
− x
n
≤
n
γκ
x
n
− x
n−1
δ
n
B
x
n
− x
n−1
1 − δ
n
β −
n
γ
z
n
− z
n−1
|
n
−
n−1
|
γ
f
x
n−1
|
δ
n
− δ
n−1
|
Bx
n−1
W
n
z
n−1
− W
n−1
z
n−1
|
δ
n−1
− δ
n
|
BW
n−1
z
n−1
δ
n
B
W
n−1
z
n−1
− W
n
z
n−1
|
n−1
−
n
|
AW
n−1
z
n−1
n
A
W
n−1
z
n−1
− W
n
z
n−1
≤
n
γκ
x
n
− x
n−1
δ
n
B
x
n
− x
n−1
1 − δ
n
β −
n
γ
x
n
− x
n−1
|
α
n
− α
n−1
|
L
|
n
−
n−1
|
γ
f
x
n−1
|
δ
n
− δ
n−1
|
Bx
n−1
W
n
z
n−1
− W
n−1
z
n−1
|
δ
n−1
− δ
n
|
BW
n−1
z
n−1
δ
n
B
W
n−1
z
n−1
− W
n
z
n−1
|
n−1
−
n
|
AW
n−1
z
n−1
n
A
W
n−1
z
n−1
− W
n
z
n−1
≤
1 −
δ
n
β −
B
n
γ − γκ
x
n
− x
n−1
L
|
α
n
− α
n−1
|
2
|
n−1
−
n
|
M
2
2
|
δ
n−1
− δ
n
|
M
2
1 δ
n
B
n
A
W
n−1
z
n−1
− W
n
z
n−1
,
3.20
where M
2
sup
n
{γfx
n−1
Bx
n−1
BW
n−1
z
n−1
AW
n−1
z
n−1
}.
Set M
3
min{β −B, γ − γκ}. It follows from 0 ≤ γ − γκ < 1andβ>B due to
β>2B B
2
that 0 ≤ M
2
< 1. Thus we have
x
n1
− x
n
≤
1 −
δ
n
n
M
3
x
n
− x
n−1
δ
n
n
M
3
×
1
δ
n
n
M
3
δ
n
B
δ
n
n
M
3
n
A
δ
n
n
M
3
×
W
n−1
z
n−1
− W
n
z
n−1
L
|
α
n
− α
n−1
|
2
|
n−1
−
n
|
M
2
2
|
δ
n−1
− δ
n
|
M
2
.
3.21
Set
η
n
δ
n
n
M
3
,
τ
n
1
δ
n
n
M
3
δ
n
B
δ
n
n
M
3
n
A
δ
n
n
M
3
W
n−1
z
n−1
− W
n
z
n−1
,
ξ
n
L
|
α
n
− α
n−1
|
2
|
n−1
−
n
|
M
2
2
|
δ
n−1
− δ
n
|
M
2
.
3.22
Fixed Point Theory and Applications 15
Then it follows from 3.21 that
x
n1
− x
n
≤
1 − η
n
x
n
− x
n−1
η
n
τ
n
ξ
n
. 3.23
It follows from the assumption condition E1, E3, E5,and3.14 that
η
n
⊂
0, 1
,
∞
n1
η
n
∞, lim
n →∞
τ
n
0,
∞
n1
ξ
n
< ∞.
3.24
By applying Lemma 2.2 to 3.23,weobtainx
n1
− x
n
→0asn →∞.
Step 3. x
n
− W
n
z
n
→ 0asn →∞.
For all n ≥ 1, we have
x
n
− W
n
z
n
≤
x
n
− x
n1
x
n1
− W
n
z
n
x
n
− x
n1
n
γf
x
n
δ
n
Bx
n
I − δ
n
B −
n
A
W
n
z
n
− W
n
z
n
≤
x
n
− x
n1
n
γf
x
n
− AW
n
z
n
δ
n
Bx
n
− BW
n
z
n
≤
x
n
− x
n1
n
γf
x
n
− AW
n
z
n
δ
n
B
x
n
− W
n
z
n
3.25
and hence noting 3.9
x
n
− W
n
z
n
≤
1
1 − δ
n
B
x
n
− x
n1
n
1 − δ
n
γ
f
x
n
AW
n
z
n
.
3.26
It follows from the assumption conditions E1, E2,andStep 2 that
x
n
− W
n
z
n
−→ 0
n −→ ∞
. 3.27
Step 4. x
n
− z
n
→ 0asn →∞.
16 Fixed Point Theory and Applications
Notice that, for any x ∈ Ω,
z
n
− x
2
α
n
x
n
− x
n
i1
α
i−1
− α
i
T
r
i
I − r
i
A
i
x
n
− T
r
i
I − r
i
A
i
x
2
≤ α
n
x
n
− x
2
n
i1
α
i−1
− α
i
I − r
i
A
i
x
n
−
I − r
i
A
i
x
2
≤ α
n
x
n
− x
2
n
i1
α
i−1
− α
i
x
n
− x
2
r
i
r
i
− 2k
i
A
i
x
n
− A
i
x
2
x
n
− x
2
n
i1
α
i−1
− α
i
r
i
r
i
− 2k
i
A
i
x
n
− A
i
x
2
.
3.28
Let y
n
γfx
n
−AW
n
z
n
and λ sup{γfx
n
−AW
n
z
n
: n ≥ 1}.Byusing3.8, 3.9, 3.28,
Lemmas 2.3, and 2.4, we have noting that δ
n
< 1/β
x
n1
− x
2
I − δ
n
B
W
n
z
n
− x
δ
n
Bx
n
− Bx
n
γf
x
n
− AW
n
z
n
2
≤
I − δ
n
B
W
n
z
n
− x
δ
n
Bx
n
− Bx
2
2
n
y
n
,x
n1
− x
I − δ
n
B
W
n
z
n
− W
n
x
δ
n
B
x
n
− x
2
2
n
y
n
,x
n1
− x
≤
1 − δ
n
β
z
n
− x
2
δ
n
B
2
x
n
− x
2
2δ
n
1 − δ
n
β
B
x
n
− x
2
2λ
2
n
≤
1 − δ
n
β
x
n
− x
2
n
i1
α
i−1
− α
i
r
i
r
i
− 2k
i
A
i
x
n
− A
i
x
2
δ
n
B
2
x
n
− x
2
2δ
n
1 − δ
n
β
B
x
n
− x
2
2λ
2
n
1 −
δ
n
β − δ
n
B
2
− 2δ
n
1 − δ
n
β
B
x
n
− x
2
1 − δ
n
β
n
i1
α
i−1
− α
i
r
i
r
i
− 2k
i
A
i
x
n
− A
i
x
2
2λ
2
n
≤
x
n
− x
2
1 − δ
n
β
n
i1
α
i−1
− α
i
r
i
r
i
− 2k
i
A
i
x
n
− A
i
x
2
2λ
2
n
.
3.29
This shows that
1 − δ
n
β
n
i1
α
i−1
− α
i
r
i
2k
i
− r
i
A
i
x
n
− A
i
x
2
≤
x
n
− x
2
−
x
n1
− x
2
2λ
2
n
≤
x
n
− x
x
n1
− x
x
n
− x
n1
2λ
2
n
3.30
Fixed Point Theory and Applications 17
and hence, for each i ≥ 1,
1 − δ
n
β
α
i−1
− α
i
r
i
2k
i
− r
i
A
i
x
n
− A
i
x
2
≤
x
n
− x
2
−
x
n1
− x
2
2λ
2
n
≤
x
n
− x
x
n1
− x
x
n
− x
n1
2λ
2
n
.
3.31
Since δ
n
→ 0, x
n
− x
n1
→0andα
i−1
− α
i
> 0, we have
lim
n →∞
A
i
x
n
− A
i
x
0,i≥ 1.
3.32
Now, for x ∈ Ω, we have, f rom Lemma 2.2,
T
r
i
I − r
i
A
i
x
n
− x
2
T
r
i
I − r
i
A
i
x
n
− T
r
i
I − r
i
A
i
x
2
≤
T
r
i
I − r
i
A
i
x
n
− T
r
i
I − r
i
A
i
x,
I − r
i
A
i
x
n
−
I − r
i
A
i
x
T
r
i
I − r
i
A
i
x
n
− x, x
n
− x
r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
1
2
T
r
i
I − r
i
A
i
x
n
− x
2
x
n
− x
2
−
x
n
− T
r
i
I − r
i
A
i
x
n
2
r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
3.33
and hence
T
r
i
I − r
i
A
i
x
n
− x
2
≤
x
n
− x
2
−
x
n
− T
r
i
I − r
i
A
i
x
n
2
2r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
.
3.34
Therefore,
z
n
− x
2
≤ α
n
x
n
− x
2
n
i1
α
i−1
− α
i
T
r
i
I − r
i
A
i
x
n
− x
2
≤ α
n
x
n
− x
2
n
i1
α
i−1
− α
i
x
n
− x
2
−
x
n
− T
r
i
I − r
i
A
i
x
n
2
2r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
x
n
− x
2
−
n
i1
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x
n
2
2
n
i1
α
i−1
− α
i
r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
.
3.35
18 Fixed Point Theory and Applications
By using 3.8, 3.9, 3.35, Lemmas 2.3 and 2.4, we have noting that δ
n
< 1/β
x
n1
− x
2
I − δ
n
B
W
n
z
n
− x
δ
n
Bx
n
− Bx
n
γf
x
n
− AW
n
z
n
2
≤
I − δ
n
B
W
n
z
n
− x
δ
n
Bx
n
− Bx
2
2
n
y
n
,x
n1
− x
I − δ
n
B
W
n
z
n
− W
n
x
δ
n
B
x
n
− x
2
2
n
y
n
,x
n1
− x
≤
1 − δ
n
β
z
n
− x
2
δ
n
B
2
x
n
− x
2
2δ
n
1 − δ
n
β
B
x
n
− x
2
2λ
2
n
≤
1 − δ
n
β
x
n
− x
2
−
n
i1
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x
n
2
2
n
i1
α
i−1
− α
i
r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
δ
n
B
2
x
n
− x
2
2δ
n
1 − δ
n
β
B
x
n
− x
2
2λ
2
n
1 −
δ
n
β − δ
n
B
2
− 2δ
n
1 − δ
n
β
B
x
n
− x
2
−
1 − δ
n
β
n
i1
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x
n
2
2
1 − δ
n
β
n
i1
α
i−1
− α
i
r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
2λ
2
n
≤
x
n
− x
2
−
1 − δ
n
β
n
i1
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x
n
2
2
1 − δ
n
β
n
i1
α
i−1
− α
i
r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
2λ
2
n
3.36
and hence
1 − δ
n
β
n
i1
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x
n
2
≤
x
n
− x
2
−
x
n1
− p
2
2
1 − δ
n
β
×
n
i1
α
i−1
− α
i
r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
2λ
2
n
≤
x
n
− x
x
n1
− x
x
n
− x
n1
2
1 − δ
n
β
n
i1
α
i−1
− α
i
r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
2λ
2
n
.
3.37
Fixed Point Theory and Applications 19
This shows that for, each i ≥ 1,
1 − δ
n
β
α
i−1
− α
i
x
n
− T
r
i
I − r
i
A
i
x
n
2
≤
x
n
− x
x
n1
− x
x
n
− x
n1
2
1 − δ
n
β
n
i1
α
i−1
− α
i
r
i
T
r
i
I − r
i
A
i
x
n
− x, A
i
x − A
i
x
n
2λ
2
n
.
3.38
Since {α
n
} is strictly decreasing, δ
n
→ 0,
n
→ 0, A
i
x
n
− A
i
x → 0andx
n
− x
n1
→0, we
have, for each i ≥ 1,
x
n
− T
r
i
I − r
i
A
i
x
n
−→ 0,n−→ ∞ . 3.39
Now, from z
n
− x
n
n
i1
α
i−1
− α
i
T
r
i
x
n
− x
n
we get
z
n
− x
n
≤
n
i1
α
i−1
− α
i
T
r
i
x
n
− x
n
.
3.40
Since x
n
− T
r
i
x
n
→0and0<α
i−1
− α
i
for each i ≥ 1, one has
z
n
− x
n
−→ 0, as n −→ ∞ . 3.41
Step 5. lim sup
n →∞
γf − Ax
∗
,x
n
− x
∗
≤0.
To prove this, we pick a subsequence {x
n
j
} of {x
n
} such that
lim sup
n →∞
γf − A
x
∗
,x
n
− x
∗
lim
j →∞
γf − A
x
∗
,x
n
j
− x
∗
.
3.42
Without loss of generality, we may further assume that x
n
j
x. Obviously, to prove Step 5,
we only need to prove that x ∈ Ω.
Indeed, for each i ≥ 1, since x
n
− T
r
i
I − r
i
A
i
x
n
→ 0, x
n
j
→ x and T
r
i
I − r
i
A
i
is
nonexpansive, by demiclosed principle of nonexpansive mapping we have
x ∈ F
T
r
i
I − r
i
A
i
EP
G
i
,A
i
,i≥ 1. 3.43
Assume that λ
n
m
,k
→ λ
k
∈ 0, 1 for each k 1, 2, ,N.LetW be the W-mapping
generated by T
1
, ,T
N
and λ
1
, ,λ
N
. Then, by Lemma 2.8, we have
W
n
m
x −→ Wx, ∀x ∈ H. 3.44
20 Fixed Point Theory and Applications
Moreover, it follows from Lemma 2.7 that
N
n1
FT
i
FW. Assume that x
/
∈ FW.
Then x
/
W x. Since x ∈ FT
r
i
I − r
i
A
i
for each i ≥ 1, by Step 3, 3.44 and Opial’s property
of the Hilbert space H, we have
lim inf
n →∞
x
n
m
− x
< lim inf
n →∞
x
n
m
− W x
≤ lim inf
n →∞
x
n
m
− W
n
m
z
n
m
W
n
m
z
n
m
− W
n
m
x
W
n
m
x − W x
≤ lim inf
n →∞
x
n
m
− W
n
m
z
n
m
z
n
m
− x
W
n
m
x − W x
≤ lim inf
n →∞
x
n
m
− W
n
m
z
n
m
z
n
m
− x
n
m
x
n
m
− T
r
i
I − r
i
A
i
x
n
m
x − T
r
i
I − r
i
A
i
x
n
m
W
n
m
x − W x
≤ lim inf
n →∞
x
n
m
− W
n
m
z
n
m
z
n
m
− x
n
m
x
n
m
− T
r
i
I − r
i
A
i
x
n
m
T
r
i
I − r
i
A
i
x − T
r
i
I − r
i
A
i
x
n
m
W
n
m
x − W x
≤ lim inf
n →∞
x
n
m
− x
,
3.45
which is a contradiction. Therefore, x ∈ FW. Hence, x ∈ Ω
N
i1
FT
i
∩
∞
i1
EPG
i
,A
i
.
Step 6. The sequence {x
n
} strongly converges to some point x
∗
∈ H.
By using Lemmas 2.3 and 2.4, we have
x
n1
− x
∗
2
I − δ
n
B −
n
A
W
n
z
n
− x
∗
δ
n
Bx
n
− Bx
∗
n
γf
x
n
− Ax
∗
2
≤
I − δ
n
B −
n
A
W
n
z
n
− x
∗
δ
n
Bx
n
− Bx
∗
2
2
n
γf
x
n
− Ax
∗
,x
n1
− x
∗
≤
1 − δ
n
B
−
n
γ
z
n
− x
∗
δ
n
B
x
n
− x
∗
2
2
n
γf
x
n
− Ax
∗
,x
n1
− x
∗
≤
1 − δ
n
B
−
n
γ
x
n
− x
∗
δ
n
B
x
n
− x
∗
2
2
n
γf
x
n
− Ax
∗
,x
n1
− x
∗
1 −
n
γ
x
n
− x
∗
2
2
n
γf
x
n
− Ax
∗
,x
n1
− x
∗
≤
1 −
n
γ
2
x
n
− x
∗
2
2
n
γκ
x
n
− x
∗
x
n1
− x
∗
2
n
γf
x
∗
− Ax
∗
,x
n1
− x
∗
≤
1 −
n
γ
2
x
n
− x
∗
2
n
γκ
x
n
− x
∗
2
x
n1
− x
∗
2
2
n
γf
x
∗
− Ax
∗
,x
n1
− x
∗
,
3.46
Fixed Point Theory and Applications 21
which implies that
x
n1
− x
∗
2
≤
1 −
n
γ
2
n
γκ
1 −
n
γκ
x
n
− x
∗
2
2
n
1 −
n
γκ
γf
x
∗
− Ax
∗
,x
n1
− x
∗
1 − 2
n
γ
n
γκ
1 −
n
γκ
x
n
− x
∗
2
2
n
γ
2
1 −
n
γκ
x
n
− x
∗
2
2
n
1 −
n
γκ
γf
x
∗
− Ax
∗
,x
n1
− x
∗
≤
1 −
2
n
γ − κγ
1 −
n
γκ
x
n
− x
∗
2
2
n
γ − κγ
1 −
n
γκ
1
γ − κγ
γf
x
∗
− Ax
∗
,x
n1
− x
∗
n
γ
2
2
γ − κγ
M
,
3.47
where M
is an appropriate constant such that M
sup
n≥1
{x
n
− x
∗
}.Put
s
n
2
n
γ − κγ
1 −
n
κγ
,
t
n
1
γ − κγ
γf
x
∗
− Ax
∗
,x
n1
− x
∗
n
γ
2
2
γ − κγ
M
.
3.48
Then we have
x
n1
− x
∗
2
≤
1 − s
n
x
n
− x
∗
s
n
t
n
.
3.49
It follows from the assumption condition E1 and 3.42 that
lim
n →∞
s
n
0,
∞
n1
s
n
∞, lim sup
n →∞
t
n
≤ 0.
3.50
Thus, applying Lemma 2.5 to 3.49, it follows that x
n
→ x
∗
as n →∞. This completes the
proof.
By Theorem 3.1, we have the following direct corollaries.
Corollary 3.2. Let H be a Hilbert space and C be a nonempty closed convex subset of H.Letf :
H → H be a contraction with coefficient 0 <κ<1, A : H → H be strongly positive linear
bounded self-adjoint operator with coefficient
γ>0, {T
n
}
N
n1
: H → H N ≥ 1 be a finite family of
nonexpansive mappings, G : C × C → R be a bifunction satisfying (A1)–(A4), and A
0
: C → H
be an α-inverse strongly monotone mapping such that Ω
N
i1
FT
i
∩ EPG, A
/
∅.Let{ε
n
} and
{δ
n
} be two sequences in 0, 1, {λ
n,i
}
N
i1
be a sequence in a, b with 0 <a≤ b<1, r be a number in
0, 2α, and {α
n
} be a sequence 0, 1. Take a fixed number γ>0 with 0 < γ − γκ < 1. Assume that
22 Fixed Point Theory and Applications
E1 lim
n →∞
n
0 and
∞
n1
ε
n
∞;
E2 lim
n →∞
|λ
n1,i
− λ
n,i
| 0 for each i 1, 2, ,N;
E3 0 ≤ δ
n
n
≤ 1 for all n ≥ 1;
E4 {δ
n
}⊂0, min{c, 1/β, 2BB
2
−β
β −B
2
− 2B
2
8βB/4βB} with
c<1;
E5
∞
n1
|
n1
−
n
| < ∞,
∞
n1
|α
n1
− α
n
| < ∞,
∞
n1
|δ
n1
− δ
n
| < ∞.
Then the sequence {x
n
} defined by 1.25 converges strongly to x
∗
∈ Ω, which is the unique solution
of the variational inequality:
x
∗
P
Ω
I −
A − γf
x
∗
. 3.51
Remark 3.3. In the proof process of Theorem 3.1, we do not use Suzuki’s Lemma see 27,
which was used by many others for obtaining x
n1
− x
n
→0asn →∞see 4, 5, 28.The
proof method of x ∈
∞
i1
EPG
i
,A
i
is simple and different with ones of others.
4. Applications for Multiobjective Optimization Problem
In this section, we study a kind of multiobjective optimization problem by using the result
of this paper. That is, we will give an iterative algorithm of solution for the following
multiobjective optimization problem with the nonempty set of solutions:
min h
1
x
,
min h
2
x
,
x ∈ C, 4.1
where h
1
x and h
2
x are both the convex and lower semicontinuous functions defined on a
closed convex subset of C of a Hilbert space H.
We denote by A the set of solutions of the problem 4.1 and assume that A
/
∅.Also,
we denote the sets of solutions of the following two optimization problems by A
1
and A
2
,
respectively,
min h
1
x
,x∈ C, 4.2
and
min h
2
x
,x∈ C. 4.3
Note that, if we find a solution x ∈ A
1
∩ A
2
, then one must have x ∈ A obviously.
Fixed Point Theory and Applications 23
Now, let G
1
and G
2
be two bifunctions from C × C to R defined by
G
1
x, y
h
1
y
− h
1
x
,G
2
x, y
h
2
y
− h
2
x
, ∀
x, y
∈ C × C, 4.4
respectively. It is easy to see that EPG
1
A
1
and EPG
2
A
2
, where EPG
i
denotes the
set of solutions of the equilibrium problem:
G
i
x, y
≥ 0, ∀y ∈ C, i 1, 2, 4.5
respectively. In addition, it is easy to see that G
1
and G
2
satisfy the conditions A1–A4.Let
{α
n
} be a sequence in 0,1 and r
1
,r
2
∈ 0, 1. Define a sequence {x
n
} by
x
1
∈ H,
z
n
α
n
T
r
1
x
n
1 − α
n
T
r
2
x
n
,
x
n1
n
γf
x
n
1 −
n
z
n
, ∀n ≥ 1.
4.6
By Theorem 3.1 with A I, N 1, T
1
I and δ
n
0 for all n ≥ 1, the sequence {x
n
}
converges strongly to a solution x
∗
P
A
1
∩A
2
γfx
∗
, which is a solution of the multiobjective
optimization problem 4.1.
References
1 K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced
Mathematics, Cambridge University Press, Cambridge, UK, 1990.
2 H K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical
Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
3 G. Marino and H K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”
Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
4 Y. Yao, “A general iterative method for a finite family of nonexpansive mappings,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 66, no. 12, pp. 2676–2687, 2007.
5 V. Colao, G. Marino, and H K. Xu, “An iterative method for finding common solutions of equilibrium
and fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340–
352, 2008.
6 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.
7 S. Plubtieng and R. Punpaeng, “A general iterative method for equilibrium problems and fixed point
problems in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 455–
469, 2007.
8 L C. Ceng and J C. Yao, “Hybrid viscosity approximation schemes for equilibrium problems
and fixed point problems of infinitely many nonexpansive mappings,” Applied Mathematics and
Computation, vol. 198, no. 2, pp. 729–741, 2008.
9 L C. Ceng, S. Al-Homidan, Q. H. Ansari, and J C. Yao, “An iterative scheme for equilibrium
problems and fixed point problems of strict pseudo-contraction mappings,” Journal of Computational
and Applied Mathematics, vol. 223, no. 2, pp. 967–974, 2009.
10 L. C. Ceng, A. Petrus¸el, and J. C. Yao, “Iterative approaches to solving equilibrium problems and
fixed point problems of infinitely many nonexpansive mappings,” Journal of Optimization Theory and
Applications, vol. 143, no. 1, pp. 37–58, 2009.
11 S S. Chang, Y. J. Cho, and J. K. Kim, “Approximation methods of solutions for equilibrium problem
in Hilbert spaces,” Dynamic Systems and Applications, vol. 17, no. 3-4, pp. 503–513, 2008.
24 Fixed Point Theory and Applications
12 S. Chang, H. W. Joseph Lee, and C. K. Chan, “A new method for solving equilibrium problem
fixed point problem and variational inequality problem with application to optimization,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3307–3319, 2009.
13 P. Kumam and P. Katchang, “A viscosity of extragradient approximation method for finding equi-
librium problems, variational inequalities and fixed point problems for nonexpansive mappings,”
Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 475–486, 2009.
14 A. Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems,”
Journal of Nonlinear and Convex Analysis, vol. 9, no. 1, pp. 37–43, 2008.
15 S. Plubtieng and R. Punpaeng, “A new iterative method for equilibrium problems and fixed
point problems of nonexpansive mappings and monotone mappings,” Applied Mathematics and
Computation, vol. 197, no. 2, pp. 548–558, 2008.
16 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.
17 X. Qin, S. Y. Cho, and S. M. Kang, “Strong convergence of shrinking projection methods for quasi-φ-
nonexpansive mappings and equilibrium problems,” Journal of Computational and Applied Mathematics,
vol. 234, no. 3, pp. 750–760, 2010.
18 S. H. Wang, G. Marino, and F. H. Wang, “Strong convergence theorems for a generalized equilibrium
problem with a relaxed monotone mapping and a countable family of nonexpansive mappings in a
hilbert space,” Fixed Point Theory and Applications, vol. 2010, Article ID 230304, p. 22, 2010.
19 S. H. Wang, Y. J. Cho, and X. L. Qin, “A new iterative method for solving equilibrium problems
and fixed point problems for infinite family of nonexpansive mappings,” Fixed Point Theory and
Applications, vol. 2010, Article ID 165098, p. 18, 2010.
20 H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-
strongly monotone mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 3, pp.
341–350, 2005.
21 W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Application, Yokohama
Publishers, Yokohama, Japan, 2000.
22 H K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society,
vol. 66, no. 1, pp. 240–256, 2002.
23 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of
Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.
24 W. Takahashi, “Weak and strong convergence theorems for families of nonexpansive mappings and
their applications,” Annales Universitatis Mariae Curie-Skłodowska, vol. 51, no. 2, pp. 277–292, 1997.
25 W. Takahashi and K. Shimoji, “Convergence theorems for nonexpansive mappings and feasibility
problems,” Mathematical and Computer Modelling, vol. 32, no. 11–13, pp. 1463–1471, 2000.
26 S. Atsushiba and W. Takahashi, “Strong convergence theorems for a finite family of nonexpansive
mappings and applications,” Indian Journal of Mathematics, vol. 41, no. 3, pp. 435–453, 1999.
27 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter non-
expansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications,
vol. 305, no. 1, pp. 227–239, 2005.
28 Y. J. Cho and X. Qin, “Convergence of a general iterative method for nonexpansive mappings in
Hilbert spaces,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 458–465, 2009.
