Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 165098, 18 pages
doi:10.1155/2010/165098
Research Article
A New Iterative Method for Solving
Equilibrium Problems and Fixed Point Problems for
Infinite Family of Nonexpansive Mappings
Shenghua Wang,1 Yeol Je Cho,2 and Xiaolong Qin3
1
School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
Department of Mathematics Education and the RINS, Gyeongsang National University,
Chinju 660-701, Republic of Korea
3
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
2
Correspondence should be addressed to Yeol Je Cho, yjcho@gnu.ac.kr
Received 7 January 2010; Revised 21 May 2010; Accepted 11 July 2010
Academic Editor: Simeon Reich
Copyright q 2010 Shenghua Wang et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We introduce a new iterative scheme for finding a common element of the solutions sets of a
finite family of equilibrium problems and fixed points sets of an infinite family of nonexpansive
mappings in a Hilbert space. As an application, we solve a multiobjective optimization problem
using the result of this paper.
1. Introduction
Let H be a Hilbert space and C be a nonempty, closed, and convex subset of H. Let Φ be a
bifunction of C × C into R, where R is the set of real numbers. The equilibrium problem for
the bifunction Φ : C × C → R is to find x ∈ C such that
Φ x, y ≥ 0,
∀y ∈ C.
1.1
The set of solutions of the above inequality is denoted by EPΦ. Many problems arising
from physics, optimization, and economics can reduce to finding a solution of an equilibrium
problem.
In 2007, S. Takahashi and W. Takahashi 1 first introduced an iterative scheme by
the viscosity approximation method for finding a common element of the solutions set of
equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space
2
Fixed Point Theory and Applications
H and proved a strong convergence theorem which is based on Combettes and Hirstoaga’s
result 2 and Wittmann’s result 3. More precisely, they obtained the following theorem.
Theorem 1.1 see 1. Let C be a nonempty closed and convex subset of H. Let Φ : C × C → R be
a bifunction which satisfies the following conditions:
A1 Φx, x 0 for all x ∈ C;
A2 Φ is monotone, that is, Φx, y Φy, x ≤ 0 for all x, y ∈ C;
A3 For all x, y, z ∈ C,
limΦ tz 1 − tx, y ≤ Φ x, y ;
t↓0
1.2
A4 For each x ∈ C, y → Φx, y is convex and lower semicontinuous.
Let S : C → H be a nonexpansive mapping with FixS ∩ EPΦ /
∅, where FixS denotes
the set of fixed points of the mapping S, and let f : H → H be a contraction, if there exists a constant
λ ∈ 0, 1 such that fx − fy ≤ λx − y for all x, y ∈ H. Let {xn } and {un } be the sequences
generated by x1 ∈ H and
1
Φ un , y y − un , un − xn ≥ 0,
rn
xn1 αn fxn 1 − αn Sun ,
∀y ∈ C,
1.3
∀n ≥ 1,
where {αn } ⊂ 0, 1 and {rn } ⊂ 0, ∞ satisfy the following conditions:
lim αn 0,
n→∞
∞
αn ∞,
∞
n1
n1
lim inf rn > 0,
n→∞
∞
|αn1 − αn | < ∞,
1.4
|rn1 − rn | < ∞.
n1
Then the sequences {xn } and {un } converge strongly to a point z ∈ FixS ∩ EPΦ, where
z PFixS∩EPΦ fz
1.5
P is the metric projection of H onto C and PFixS∩EPΦ fz denotes nearest point in FixS ∩ EPΦ
from fz.
Recently, many results on equilibrium problems and fixed points problems in the
context of the Hilbert space and Banach space are introduced see, e.g., 4–8.
Fixed Point Theory and Applications
3
Let F : H → H be a nonlinear mapping. The variational inequality problem
corresponding to the mapping F is to find a point x∗ ∈ C such that
Fx∗ , x − x∗ ≥ 0,
∀x ∈ C.
1.6
The variational inequality problem is denoted by VIF, C 9.
The mapping F is called κ-Lipschitzian and η-strongly monotone if there exist
constants κ, η > 0 such that
Fx − Fy ≤ κx − y, ∀x, y ∈ H,
2
Fx − Fy, x − y ≥ ηx − y , ∀x, y ∈ H,
1.7
1.8
respectively. It is well known that if F is strongly monotone and Lipschitzian on C, then
VIF, C has a unique solution. An important problem is how to find a solution of VIF, C.
Recently, there are many results to solve the VIF, C see, e.g., 10–14.
Let C be a nonempty closed and convex subset of a Hilbert space H, {Tn }∞
n1 : H → H
:
C
×
C
→
R
be
m
bifunctions
be a countable family of nonexpansive mappings, and {Φi }m
i1
FixT
∩
EPΦ
∩
·
·
·
∩
EPΦ
satisfying conditions A1–A4 such that Ω ∞
/ ∅. Let
n
1
m n1
r1 , . . . , rm ∈ 0, ∞. For each i 1, . . . , m, define the mapping Tri : H → C by
Tri x 1
z ∈ C : Φi z, y y − z, z − x ≥ 0, ∀y ∈ C ,
ri
∀x ∈ H.
1.9
Lemma 2.5 see below shows that, for each 1 ≤ i ≤ m, Tri is firmly nonexpansive and
hence nonexpansive and FixTri EPΦi . Suppose that F : H → H is a κ-Lipschitzian and
∅.
η-strong monotone operator and let μ ∈ 0, 2η/κ2 . Assume that VIΦi F, Ω /
In this paper, motivated and inspired by the above research results, we introduce the
following iterative process for finding an element in Ω: for an arbitrary initial point x1 ∈ H,
zn γ1 Tr1 xn γ2 Tr2 xn · · · γm Trm xn ,
xn1 αn xn n
αi−1 − αi σn Ti xn 1 − αn 1 − σn T λn zn ,
∀n ≥ 1,
1.10
i1
where T λn zn zn − λn μFzn , α0 1, {αn }∞
is a strictly decreasing sequence in 0, α with
n1 m
∞
m
0 < α < 1, {λn }∞
n1 ⊂ 0, 1, {γi }i1 ⊂ 0, 1 with
i1 γi 1, and {σn }n1 ⊂ a, b with 0 < a, b <
1. Then we prove that the iterative process {xn } defined by 1.10 strongly converge to an
element x∗ ∈ Ω, which is the unique solution of the variational inequality
Fx∗ , x − x∗ ≥ 0,
∀x ∈ Ω.
As an application of our main result, we solve a multiobjective optimization problem.
1.11
4
Fixed Point Theory and Applications
2. Preliminaries
Let H be a Hilbert space and T a nonexpansive mapping of H into itself such that FixT / ∅.
For all x ∈ FixT and x ∈ H, we have
2 ≥ T x − T x
2 T x − x
2 T x − x x − x
2
x − x
T x − x2 x − x
2 2T x − x, x − x
2.1
and hence
T x − x2 ≤ 2x − T x, x − x,
∀x ∈ FixT , x ∈ H.
2.2
It is well known that, for all x, y ∈ H and t ∈ 0, 1,
tx 1 − ty2 ≤ tx2 1 − ty2 ,
2.3
2
n
n
ti xi 2
ti xi ≤
i1
i1
2.4
which implies that
for all {xi }ni1 ⊂ H and {ti }ni1 ⊂ 0, 1 with ni1 ti 1.
Let C be a nonempty closed and convex subset of H and, for any x ∈ H, there exists
unique nearest point in C, denoted by PC x, such that
PC x − x ≤ y − x,
∀y ∈ C.
2.5
Moreover, we have the following:
z PC x ⇐⇒ x − z, z − y ≥ 0,
∀y ∈ C.
2.6
Let I denote the identity operator of H and let {xn } be a sequence in a Hilbert space
H and x ∈ H. Throughout this paper, xn → x denotes that {xn } strongly converges to x and
xn x denotes that {xn } weakly converges to x.
We need the following lemmas for our main results.
Lemma 2.1 see 15. Let C be a nonempty closed and convex subset of a Hilbert space H and T a
nonexpansive mapping from C into itself. Then I − T is demiclosed at zero, that is,
xn x,
xn − T xn −→ 0
implies x T x.
2.7
Fixed Point Theory and Applications
5
Lemma 2.2 see 10, Lemma 3.1b. Let H be a Hilbert space and T : H → H be a nonexpansive
mapping. Let F : H → H be a mapping which is κ-Lipschitzian and η-strong monotone on T H.
Assume that λ ∈ 0, 1 and μ ∈ 0, 2η/κ2 . Define a mapping T λ : H → H by
T λ x T x − λμFT x,
2.8
∀x ∈ H.
Then T λ x − T λ y ≤ 1 − λτx − y for all x, y ∈ H, where τ 1 −
1 − μ2η − μκ2 ∈ 0, 1.
If T I, Lemma 2.2 still holds.
Lemma 2.3 see 16. Let {sn }, {cn } be the sequences of nonnegative real numbers and {an } ⊂
0, 1. Suppose that {bn } is a real number sequence such that
sn1 ≤ 1 − an sn bn cn ,
∀n ≥ 0.
2.9
Assume that ∞
n0 cn < ∞. Then the following results hold.
(1) If bn ≤ βan for all n ≥ 0, where β ≥ 0, then {sn } is a bounded sequence.
(2) If
∞
an ∞,
n0
lim sup
n→∞
bn
≤ 0,
an
2.10
then limn → ∞ sn 0.
Lemma 2.4 see 17. Let C be a nonempty closed and convex subset of a Hilbert space H and
Φ : C × C → R be a bifunction which satisfies the conditions (A1)–(A4). Let r > 0 and x ∈ H. Then
there exists z ∈ C such that
1
Φ z, y y − z, z − x ≥ 0,
r
∀y ∈ C.
2.11
Lemma 2.5 see 2. Let H be a Hilbert space and C be a nonempty closed and convex subset of H.
Assume that Φ : C × C → R satisfies the conditions (A1)–(A4). For all r > 0 and x ∈ H, define a
mapping Tr : H → C as follows:
Tr x 1
z ∈ C : Φ z, y y − z, z − x ≥ 0, ∀y ∈ C ,
r
∀x ∈ H.
2.12
Then the following holds:
1 Tr is single-valued;
2 Tr is firmly nonexpansive, that is, for any x, y ∈ H,
Tr x − Tr y2 ≤ Tr x − Tr y, x − y ;
3 FixTr EPΦ;
4 EPΦ is closed and convex.
2.13
6
Fixed Point Theory and Applications
The following lemma is an immediate consequence of an inner product.
Lemma 2.6. Let H be a real Hilbert space. Then the following identity holds:
x y2 ≤ x2 2 y, x y ,
∀x, y ∈ H.
2.14
3. Main Results
First, we prove some lemmas as follows.
Lemma 3.1. The sequence {xn } generated by 1.10 is bounded.
Proof. Let uin Tri xn for each i 1, 2, . . . , m. Lemma 2.5 shows that each Tri is firmlynonexpansive and hence nonexpansive. Hence, for each 1 ≤ i ≤ m and p ∈ Ω, we have
uin − p Tr xn − Tr p ≤ xn − p,
i
i
∀n ≥ 1,
3.1
m zn − p ≤
γi uin − p ≤ xn − p,
∀n ≥ 1.
3.2
∀x, y ∈ H,
3.3
i1
By Lemma 2.2, we have
λn
T x − T λn y ≤ 1 − λn τx − y,
where τ 1 −
1 − μ2η − μκ2 ∈ 0, 1. Therefore, by 3.2 and 3.3, we obtain note that
{αn } is strictly decreasing and T λn p − p −λn μFp
n
λn
xn1 − p αn xn − p αi−1 − αi σn Ti xn − p 1 − αn 1 − σn T zn − p i1
n
≤ αn xn − p αi−1 − αi σn Ti xn − p 1 − αn 1 − σn T λn zn − p
i1
n
≤ αn xn − p αi−1 − αi σn xn − p
i1
1 − αn 1 − σn T λn zn − T λn p T λn p − p
n
≤ αn xn − p αi−1 − αi σn xn − p
i1
1 − αn 1 − σn 1 − λn τzn − p λn μF p Fixed Point Theory and Applications
7
n
≤ αn xn − p αi−1 − αi σn xn − p
i1
1 − αn 1 − σn 1 − λn τxn − p λn μF p 1 − 1 − αn 1 − σn λn τxn − p 1 − αn 1 − σn λn μF p .
3.4
By induction, we obtain xn1 ≤ max{x1 − p, μ/τFp}. Hence {xn } is bounded and so
are {zn } and {uin } for each i 1, 2, . . . , m. Since F is κ-Lipschitzian, we have
Fzn ≤ Fzn − F p F p 3.5
≤ κzn − p F p ≤ κzn κp F p ,
which shows that {Fzn } is bounded. This completes the proof.
Lemma 3.2. If the following conditions hold:
∞
λn ∞,
∞
n1
n1
|λn − λn1 | < ∞,
∞
|σn − σn1 | < ∞,
3.6
n1
then limn → ∞ xn1 − xn 0.
Proof. For each i 1, 2, . . . , m, since each Tri is nonexpansive, we have
uin−1 − uin Tri xn−1 − Tri xn ≤ xn−1 − xn ,
∀n ≥ 1.
3.7
By 3.7, we have
zn − zn−1 γ1 u1n − u1n−1 γ2 u2n − u2n−1 · · · γm umn − umn−1 ≤
m
m
γi uin − uin−1 ≤
γi xn − xn−1 i1
xn−1 − xn ,
i1
∀n ≥ 1.
By the definition of the iterative sequence 1.10, we have
xn1 − xn αn xn − xn−1 αn xn−1 n
αi−1 − αi σn Ti xn − Ti xn−1 i1
n
αi−1 − αi σn Ti xn−1 1 − αn 1 − σn T λn zn − T λn zn−1
i1
1 − αn 1 − σn T λn zn−1 − αn−1 xn−1 −
n−1
αi−1 − αi σn−1 Ti xn−1
i1
− 1 − αn−1 1 − σn−1 T λn−1 zn−1
3.8
8
Fixed Point Theory and Applications
αn xn − xn−1 αn − αn−1 xn−1 n
αi−1 − αi σn Ti xn − Ti xn−1 i1
n
1 − αn 1 − σn T λn zn − T λn zn−1 αi−1 − αi σn Ti xn−1
i1
−
n−1
αi−1 − αi σn−1 Ti xn−1 1 − αn 1 − σn T λn zn−1
i1
− 1 − αn−1 1 − σn−1 T λn−1 zn−1
αn xn − xn−1 αn − αn−1 xn−1 n
αi−1 − αi σn Ti xn − Ti xn−1 i1
n−1
1 − αn 1 − σn T λn zn − T λn zn−1 αi−1 − αi σn − σn−1 Ti xn−1
i1
αn−1 − αn σn Tn xn−1 αn−1 − αn 1 − σn σn−1 − σn 1 − αn−1 zn−1
{1 − αn−1 1 − σn−1 λn−1 − λn −αn−1 − αn 1 − σn σn−1 − σn 1 − αn−1 λn }μFzn−1 ,
3.9
and hence
xn1 − xn ≤ αn xn − xn−1 αn−1 − αn xn−1 n
αi−1 − αi σn xn − xn−1 i1
1 − αn 1 − σn 1 − λn τzn − zn−1 n−1
αi−1 − αi |σn − σn−1 |Ti xn−1 i1
αn−1 − αn Tn xn−1 αn−1 − αn |σn−1 − σn |zn−1 |λn−1 − λn | αn−1 − αn |σn−1 − σn |μFzn−1 αn xn − xn−1 n
αi−1 − αi σn xn − xn−1 i1
1 − αn 1 − σn 1 − λn τzn − zn−1 αn−1 − αn xn−1 Tn xn−1 zn−1 μFzn−1 n−1
αi−1 − αi |σn − σn−1 |Ti xn−1 |σn−1 − σn | zn−1 μFzn−1 i1
|λn−1 − λn |μFzn−1 .
3.10
Fixed Point Theory and Applications
9
It follows from 3.8 and 3.10 that
xn1 − xn ≤ αn xn − xn−1 n
αi−1 − αi σn xn − xn−1 i1
1 − αn 1 − σn 1 − λn τxn−1 − xn αn−1 − αn xn−1 Tn xn−1 zn−1 μFzn−1 n−1
αi−1 − αi |σn − σn−1 |Ti xn−1 |σn−1 − σn | zn−1 μFzn−1 i1
3.11
|λn−1 − λn |μFzn−1 ≤ 1 − 1 − αn 1 − σn λn τxn − xn−1 αn−1 − αn 3 μ M
|σn − σn−1 | 2 μ M |λn−1 − λn |μM
≤ 1 − 1 − α1 − bλn τxn − xn−1 αn−1 − αn 3 μ M
|σn − σn−1 | 2 μ M |λn−1 − λn |μM,
where M max{supn≥1 xn , supn≥1 zn , supi≥1,n≥1 Ti xn , supn≥1 Fzn }. Since {αn } is
strictly decreasing, we have ∞
n2 αn−1 − αn α1 < ∞. Further, from the assumptions, it
follows that
∞
αn−1 − αn 3 μ M |σn − σn−1 | 2 μ M |λn−1 − λn |μM < ∞.
3.12
n2
Therefore, by Lemma 2.3, we have limn → ∞ xn1 − xn 0. This completes the proof.
Lemma 3.3. If the following conditions hold:
lim λn 0,
n→∞
∞
λn ∞,
∞
n1
n1
|λn − λn1 | < ∞,
∞
|σn − σn1 | < ∞,
3.13
n1
then lim xn − uin 0 for each i 1, 2, . . . , m.
n→∞
Proof. For any p ∈ Ω and i 1, 2, . . . , m, it follows from Lemma 2.52 that
uin − p2 Tr xn − Tr p2 ≤ Tr xn − Tr p, xn − p uin − p, xn − p
i
i
i
i
1 uin − p2 xn − p2 − uin − xn 2 ,
2
3.14
10
Fixed Point Theory and Applications
and hence uin − p2 ≤ xn − p2 − uin − xn 2 . Further, we have
2
m
m 2
zn − p2 γi uin − p
γi uin − p ≤
i1
i1
≤
m 2
γi xn − p − uin − xn 2
3.15
i1
m
2 xn − p − γi uin − xn 2 ,
∀n ≥ 1.
i1
Therefore, from 2.4 and 3.3, we have
2
n
2 λ
n
xn1 − p αn xn − p αi−1 − αi σn Ti xn − p 1 − αn 1 − σn T zn − p
i1
n
2
2 2
≤ αn xn − p αi−1 − αi σn Ti xn − p 1 − αn 1 − σn T λn zn − p
i1
2
2
≤ αn xn − p 1 − αn σn xn − p
2
1 − αn 1 − σn T λn zn − T λn p T λn p − p
2
2
≤ αn xn − p 1 − αn σn xn − p
2
1 − αn 1 − σn 1 − λn τzn − p λn μFp
2
2
≤ αn xn − p 1 − αn σn xn − p 1 − αn 1 − σn 2
2 × 1 − λn τzn − p 2λn 1 − λn τμzn − pF p λn μ2 F p 2
2
≤ αn xn − p 1 − αn σn xn − p 1 − αn 1 − σn m
2 2
xn − p − γi uin − xn × 1 − λn τ
i1
2
2λn 1 − λn τμzn − pF p λn μ2 F p m
2
1 − 1 − αn 1 − σn λn τxn − p − 1 − αn 1 − σn 1 − λn τ γi uin − xn 2
i1
2
2λn μ1 − αn 1 − σn 1 − λn τzn − pF p 1 − αn 1 − σn λn μ2 Fp
Fixed Point Theory and Applications
11
m
2
≤ xn − p − 1 − αn 1 − σn 1 − λn τ γi uin − xn 2
i1
2
1 − αn 1 − σn λn μ2 Fp
2λn μ1 − αn 1 − σn 1 − λn τzn − pF p .
3.16
It follows that
γi 1 − αn 1 − σn 1 − λn τuin − xn 2
2
≤ xn − p xn1 − p xn − xn1 λn μ2 Fp 2μzn − pF p 3.17
for each i 1, 2, . . . , m. Note that 0 < γi < 1 for i 1, 2, . . . , m. From the assumptions,
Lemma 3.2, and the previous inequality, we conclude that uin − xn → 0 as n → ∞ for
each i 1, 2, . . . , m. Further, we have
zn − xn ≤
m
γi uin − xn −→ 0
n −→ ∞.
3.18
i1
This completes the proof.
Lemma 3.4. If the following conditions hold:
lim λn 0,
n→∞
∞
λn ∞,
∞
n1
n1
|λn − λn1 | < ∞,
∞
|σn − σn1 | < ∞,
3.19
n1
then limn → ∞ xn − Ti xn 0 for all i ≥ 1.
Proof. By the definition of the iterative sequence 1.10, we have
xn1 n
αi−1 − αi σn xn − Ti xn − 1 − αn σn xn αn xn 1 − αn 1 − σn T λn zn ,
i1
3.20
that is,
n
αi−1 − αi σn xn − Ti xn xn − xn1 − xn αn xn 1 − αn σn xn 1 − αn 1 − σn T λn zn
i1
xn − xn1 1 − αn σn − 1xn 1 − αn 1 − σn T λn zn
xn − xn1 1 − αn 1 − σn T λn zn − xn .
3.21
12
Fixed Point Theory and Applications
Hence, for any p ∈ Ω, we get
n
αi−1 − αi σn xn − Ti xn , xn − p 1 − αn 1 − σn T λn zn − xn , xn − p xn − xn1 , xn − p.
i1
3.22
Since each Ti is nonexpansive, by 2.2, we have
Ti xn − xn 2 ≤ 2xn − Ti xn , xn − p.
3.23
Hence, combining this inequality with 3.22, we get
n
1
αi−1 − αi σn Ti xn − xn 2 ≤ 1 − αn 1 − σn T λn zn − xn , xn − p xn − xn1 , xn − p,
2 i1
3.24
which implies that note that {αn } is a strictly decreasing sequence
21 − αn 1 − σn λn
2
T zn − xn , xn − p xn − xn1 , xn − p
αi−1 − αi σn
αi−1 − αi σn
2
21 − αn 1 − σn λn
≤
xn − xn1 xn − p.
T zn − xn xn − p αi−1 − αi σn
αi−1 − αi σn
3.25
Ti xn − xn 2 ≤
From Lemma 3.3, limn → ∞ λn 0, and the inequality
λn
T zn − xn ≤ zn − xn λn μFzn ,
3.26
lim T λn zn − xn 0.
3.27
we obtain
n→∞
Therefore, from Lemma 3.2, 3.25, and 3.27, it follows that
lim Ti xn − xn 0,
n→∞
This completes the proof.
∀i ≥ 1.
3.28
Fixed Point Theory and Applications
13
Next we prove the main results of this paper.
Theorem 3.5. Assume that the following conditions hold:
∞
lim λn 0,
n→∞
∞
|λn − λn1 | < ∞,
λn ∞,
n1
n1
∞ ∞
n1
n1
γn − γn1 < ∞,
3.29
|σn − σn1 | < ∞.
Then the sequence {xn } generated by 1.10 converges strongly to an element in Ω, which is the unique
solution of the variational inequality VIF, Ω.
Proof. Since VIF, Ω /
∅, we can select an element x∗ ∈ VIF, Ω, which implies that
Fx∗ , x∗ − x ≥ 0,
∀x ∈ Ω.
3.30
lim sup−Fx∗ , xn1 − x∗ ≤ 0.
3.31
First, we prove that
n→∞
Since {xn } is bounded, there exists a subsequence {xnj } of {xn } such that
lim sup−Fx∗ , xn − x∗ lim −Fx∗ , xnj − x∗ .
j →∞
n→∞
3.32
Without loss of generality, we may further assume that xnj x for some x ∈ H. From
Lemmas 3.4 and 2.1, we get x ∈ FixTn for all n ≥ 1. Hence we have x ∈ ∞
n1 FixTn .
It follows from Lemma 2.5 that each Tri is firmly nonexpansive and hence nonexpansive.
Lemma 3.3 shows that Tri xn − xn → 0 as n → ∞. Therefore, from Lemma 2.1, it follows
FixTri . Lemma 2.5 shows
that x ∈ FixTri for each i 1, . . . , m, which shows that x ∈ m
m i1
that FixTri EPΦi for each i 1, . . . , m. Hence x ∈ i1 EPΦi . By using the above
argument, we conclude that
x ∈ Ω ∞
FixTn ∩ EPΦ1 ∩ · · · ∩ EPΦm .
n1
3.33
Noting that x∗ is a solution of the VIF, Ω, we obtain
lim sup−Fx∗ , xn − x∗ −Fx∗ , x − x∗ ≤ 0.
n→∞
3.34
14
Fixed Point Theory and Applications
It follows from Lemma 2.6 that
xn1 − x∗ 2
n
∗
∗
λn
λn ∗
αn xn − x αi−1 − αn σn Ti xn − x 1 − αn 1 − σn T zn − T x
i1
2
1 − αn 1 − σn T x − x λn ∗
∗
2
n
∗
∗
λn
λn ∗ ≤ αn xn − x αi−1 − αn σn Ti xn − x 1 − αn 1 − σn T zn − T x i1
21 − αn 1 − σn T λn x∗ − x∗ , xn1 − x∗ ≤ αn xn − x∗ 2 n
2
αi−1 − αn σn Ti xn − x∗ 2 1 − αn 1 − σn T λn zn − T λn x∗ i1
∗
3.35
∗
21 − αn 1 − σn λn μ−Fx , xn1 − x ≤ αn xn − x∗ 2 n
αi−1 − αn σn xn − x∗ 2 1 − αn 1 − σn 1 − λn τzn − x∗ 2
i1
21 − αn 1 − σn λn μ−Fx∗ , xn1 − x∗ αn xn − x∗ 2 1 − αn σn xn − x∗ 2 1 − αn 1 − σn 1 − λn τzn − x∗ 2
21 − αn 1 − σn λn μ−Fx∗ , xn1 − x∗ ≤ αn xn − x∗ 2 1 − αn σn xn − x∗ 2 1 − αn 1 − σn 1 − λn τxn − x∗ 2
21 − αn 1 − σn λn μ−Fx∗ , xn1 − x∗ 1 − 1 − αn 1 − σn λn τxn − x∗ 21 − αn 1 − σn λn μ−Fx∗ , xn1 − x∗ .
Let an 1 − αn 1 − σn λn τ and bn 21 − αn 1 − σn λn μ−Fx∗ , xn1 − x∗ for all n ≥ 1.
Then, from the assumptions and 3.31, we have
0 < an < 1,
∞
n1
an ∞,
lim sup
n→∞
bn
0.
an
3.36
Therefore, by applying Lemma 2.3 to 3.35, we conclude that the sequence {xn } strongly
converges to a point x∗ .
In order to prove the uniqueness of solution of the VIF, Ω, we assume that u∗ is
another solution of VIF, Ω. Similarly, we can conclude that {xn } converges strongly to a
point u∗ . Hence x∗ u∗ , that is, x∗ is the unique solution of VIF, Ω. This completes the
proof.
Fixed Point Theory and Applications
15
As direct consequences of Theorem 3.5, we obtain the following corollaries.
Corollary 3.6. Let C be a nonempty closed and convex subset of a Hilbert space H. For each i 1, 2, . . . , m let Φi : C × C → R be m bifunctions which satisfy conditions (A1)–(A4) such that
m
∅. Let μ ∈ 0, 2, and let {αn }∞
⊂ 0, α be a strictly decreasing sequence with 0 < α <
i1 EPΦi /
n1
∞
n
m
⊂
0,
1
with
γ
1, {λn }n1 ⊂ 0, 1, {γi }m
i1
i1 i 1, r1 , r2 , . . . , rm ∈ 0, ∞, and {σn }n1 ⊂ a, b
with 0 < a, b < 1. For an arbitrary initial x1 ∈ H, define the iterative sequence {xn } by
xn1
zn γ1 Tr1 xn γ2 Tr2 xn · · · γm Trm xn ,
αn 1 − αn σn xn 1 − αn 1 − σn 1 − λn μ zn ,
∀n ≥ 1.
3.37
If the following conditions hold:
lim λn 0,
n→∞
∞
λn ∞,
∞
n1
n1
∞
|λn − λn1 | < ∞,
|σn − σn1 | < ∞,
3.38
n1
then the sequence {xn } converges strongly to an element x∗ ∈
m
i1
EPΦi .
Proof. Put F I and Ti I for each i ≥ 1 in Theorem 3.5. Then we know that F is 1
Lipschitzian and 1-strongly monotone, ni1 αi−1 −αi Ti xn 1−αn xn and T λn zn 1−λn μzn .
Therefore, by Theorem 3.5, we conclude the desired result.
Corollary 3.7. Let C be a nonempty closed and convex subset of a Hilbert space H. Let {Ti }∞
i1 be
FixT
and
F
:
H
→
H
a countable family of nonexpansive mappings of H such that C ∞
i
i1
2
an operator which is κ-Lipschitzian and η-strong monotone on H. Let μ ∈ 0, 2η/κ . Assume that
∞
VIF, C /
∅. Let {αn }∞
n1 ⊂ 0, α with 0 < α < 1 be a strictly decreasing sequence, {λn }n1 ⊂ 0, 1
and {σn }∞
n1 ⊂ a, b with 0 < a, b < 1. For an arbitrary initial x1 ∈ H, define the iterative sequence
{xn } by
xn1 αn xn n
αi−1 − αi σn Ti xn 1 − αn 1 − σn PC xn − λn μFPC xn ,
∀n ≥ 1,
3.39
i1
where α0 1. If the following conditions hold:
lim λn 0,
n→∞
∞
λn ∞,
∞
n1
n1
|λn − λn1 | < ∞,
∞
|σn − σn1 | < ∞,
3.40
n1
then the sequence {xn } strongly converges to an element x∗ ∈ C, which is the unique solution of the
variational inequality
Fx∗ , x − x∗ ≥ 0,
∀x ∈ C.
3.41
Proof. Put Φi x, y 0 for each i 1, 2, . . . , m and x, y ∈ C. Set r1 r2 · · · rm 1
in Theorem 3.5. Then, by 2.6, we have Tr1 xn Tr2 xn · · · Trm xn PC xn . Therefore, by
Theorem 3.5, we conclude the desired result.
16
Fixed Point Theory and Applications
Remark 3.8. 1 Recently, many authors have studied the iteration sequences for infinite
family of nonexpansive mappings. But our iterative sequence 1.10 is very different from
others because we do not use W-mapping generated by the infinite family of nonexpansive
mappings and we have no any restriction with the infinite family of nonlinear mappings.
2 We do not use Suzuki’s lemma 18 for obtaining the result that limn → ∞ xn1 −
xn 0. However, many authors have used Suzuki’s lemma 18 for obtaining the result that
limn → ∞ xn1 − xn 0 in the process of studying the similar algorithms. For example, see
5, 19, 20 and so on.
4. Application
In this section, we study a kind of multiobjective optimization problem based on the result of
this paper. That is, we give an iterative sequence which solves the following multiobjective
optimization problem with nonempty set of solutions:
min h1 x,
4.1
min h2 x,
x ∈ C,
where h1 x and h2 x are both convex and lower semicontinuous functions defined on a
nonempty closed and convex subset of C of a Hilbert space H. We denote by A the set of
solutions of 4.1 and assume that A /
∅.
We denote the sets of solutions of the following two optimization problems by A1 and
A2 , respectively,
min h1 x
x ∈ C,
4.2
min h2 x
x ∈ C.
Obviously, if we find a solution x ∈ A1 ∩ A2 , then one must have x ∈ A.
Now, let Φ1 and Φ2 be two bifunctions from C × C to R defined by Φ1 x, y h1 y − h1 x and Φ2 x, y h2 y − h2 x, respectively. It is easy to see that EPΦ1 A1
and EPΦ2 A2 , where EPΦi denotes the set of solutions of the equilibrium problem:
Φi x, y ≥ 0, ∀y ∈ C, i 1, 2,
4.3
respectively. In addition, it is easy to see that Φ1 and Φ2 satisfy the conditions A1–A4.
Therefore, by setting m 2 in Corollary 3.6, we know that, for any initial guess x1 ∈ H,
1
h1 y − h1 u1n y − u1n , u1n − xn ≥ 0,
r1n
xn1
∀y ∈ C,
1
h2 y − h2 u2n y − u2n , u2n − xn ≥ 0, ∀y ∈ C,
r2n
zn γ1 u1n 1 − γ1 u2n ,
αn 1 − αn σn xn 1 − αn 1 − σn 1 − λn μ zn , ∀n ≥ 1.
4.4
Fixed Point Theory and Applications
17
By Corollary 3.6, we know that the sequence {xn } converges strongly to a solution x∗ ∈
EPΦ1 ∩ EPΦ2 A1 ∩ A2 , which is a solution of the multiobjective optimization problem
4.1.
Acknowledgment
This work was supported by the Korea Research Foundation Grant funded by the Korean
Government KRF-2008-313-C00050.
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