Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2008, Article ID 417089, 15 pages
doi:10.1155/2008/417089
Research Article
A New Hybrid Iterative Algorithm for
Fixed-Point Problems, Variational Inequality
Problems, and Mixed Equilibrium Problems
Yonghong Yao,1 Yeong-Cheng Liou,2 and Jen-Chih Yao3
1
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
3
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
2
Correspondence should be addressed to Jen-Chih Yao, yaojc@math.nsysu.edu.tw
Received 29 August 2007; Accepted 6 February 2008
Recommended by Tomonari Suzuki
We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed
points of an infinite family of nonexpansive mappings, the set of solutions of the variational
inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. This
study, proves a strong convergence theorem by the proposed hybrid iterative algorithm which
solves fixed-point problems, variational inequality problems, and mixed equilibrium problems.
Copyright q 2008 Yonghong Yao 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.
1. Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping
f : C → C is called contractive if there exists a constant α ∈ 0, 1 such that fx − fy ≤
αx − y for all x, y ∈ C. A mapping T : C → C is said to be nonexpansive if T x − T y ≤ x − y
for all x, y ∈ C. Denote the set of fixed points of T by FT .
Let ϕ : C → R be a real-valued function and Θ : C × C → R be an equilibrium bifunction,
that is, Θu, u 0 for each u ∈ C. The mixed equilibrium problem for short, MEP is to find
x∗ ∈ C such that
MEP: Θ x∗ , y ϕy − ϕ x∗ ≥ 0 ∀y ∈ C.
1.1
In particular, if ϕ ≡ 0, this problem reduces to the equilibrium problem for short, EP, which
is to find x∗ ∈ C such that
1.2
EP: Θ x∗ , y ≥ 0 ∀y ∈ C.
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Fixed Point Theory and Applications
Denote the set of solutions of MEP by Ω. The mixed equilibrium problems include fixedpoint problems, optimization problems, variational inequality problems, Nash equilibrium
problems, and the equilibrium problems as special cases see, e.g., 1–5. Some methods
have been proposed to solve the MEP and EP see, e.g., 5–14. In 1997, Combettes and
Hirstoaga 13 introduced an iterative method of finding the best approximation to the initial
data and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi
8 introduced another iterative scheme for finding a common element of the set of solutions
of EP and the set of fixed-point points of a nonexpansive mapping. Yao et al. 12 considered
an iterative scheme for finding a common element of the set of solutions of EP and the set
of common fixed points of an infinite nonexpansive mappings. Very recently, Zeng and Yao
14 considered a new iterative scheme for finding a common element of the set of solutions
of MEP and the set of common fixed points of finitely many nonexpansive mappings. Their
results extend and improve many results in the literature.
Let A of C into H be a nonlinear mapping. It is well known that the variational inequality
problem is to find u ∈ C such that
Au, v − u ≥ 0 ∀v ∈ C.
1.3
The set of solutions of the variational inequality problem is denoted by V IC, A. A mapping
A : C → H is called β-inverse-strongly monotone if there exists a positive real number β such
that
Au − Av, u − v ≥ βAu − Av2
∀u, v ∈ C.
1.4
Recently, some authors have proposed new iterative algorithms to approximate a common
element of the set of fixed points of a nonxpansive mapping and the set of solutions of the
variational inequality. For the details, see 15, 16 and the references therein.
Motivated by the recent works, in this paper we introduce a new hybrid iterative
algorithm for finding a common element of the set of fixed points of an infinite family of
nonexpansive mappings, the set of solutions of the variational inequality of a monotone
mapping, and the set of solutions of a mixed equilibrium problem. We prove a strong
convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point
problems, variational inequality problems, and mixed equilibrium problems.
2. Preliminaries
Let H be a real Hilbert space with inner product ·, · and norm ·. Let C be a nonempty closed
convex subset of H. Then for any x ∈ H, there exists a unique nearest point in C, denoted by
PC x such that
x − PC x ≤ x − y ∀y ∈ C.
2.1
Such a PC is called the metric projection of H onto C. It is well known that PC is a nonexpansive
mapping and satisfies
2
x − y, PC x − PC y ≥ PC x − PC y ∀x, y ∈ H.
2.2
Yonghong Yao et al.
3
Moreover, PC is characterized by the following properties:
x − PC x, y − PC x ≤ 0,
2 2
x − y2 ≥ x − PC x y − PC x ∀x ∈ H, y ∈ C.
2.3
It is clear that
u ∈ V IC, A ⇐⇒ u PC u − λAu
∀λ > 0.
2.4
In this paper, for solving the mixed equilibrium problems for an equilibrium bifunction
Θ : C × C → R, we assume that Θ satisfies the following conditions:
H1 Θ is monotone, that is, Θx, y Θy, x ≤ 0 for all x, y ∈ C;
H2 for each fixed y ∈ C, x → Θx, y is concave and upper semicontinuous;
H3 for each x ∈ C, y → Θx, y is convex.
A mapping η : C × C → H is called Lipschitz continuous if there exists a constant λ > 0
such that
ηx, y ≤ λx − y ∀x, y ∈ C.
2.5
A differentiable function K : C → R on a convex set C is called:
i η-convex if
Ky − Kx ≥ K x, ηy, x
∀x, y ∈ C,
2.6
where K is the Fréchet derivative of K at x;
ii η-strongly convex if there exists a constant σ > 0 such that
σ
x − y2
Ky − Kx − K x, ηy, x ≥
2
∀x, y ∈ C.
2.7
Let C be a nonempty closed convex subset of a real Hilbert space H, ϕ : C → R be a realvalued function, and Θ : C × C → R be an equilibrium bifunction. Let r be a positive number.
For a given point x ∈ C, the auxiliary problem for MEP consists of finding y ∈ C such that
Θy, z ϕz − ϕy 1 K y − K x, ηz, y ≥ 0 ∀z ∈ C.
r
2.8
Let Sr : C → C be the mapping such that for each x ∈ C, Sr x is the solution set of the auxiliary
problem MEP, that is,
Sr x y ∈ C : Θy, z ϕz − ϕy 1 K y − K x, ηz, y ≥ 0 ∀z ∈ C
r
We first need the following important and interesting result.
∀x ∈ C.
2.9
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Fixed Point Theory and Applications
Lemma 2.1 see 14. Let C be a nonempty closed convex subset of a real Hilbert space H and let
ϕ : C → R be a lower semicontinuous and convex functional. Let Θ : C × C → R be an equilibrium
bifunction satisfying conditions (H1)–(H3). Assume that
i η : C × C → H is Lipschitz continuous with constant λ > 0 such that
a ηx, y ηy, x 0 for all x, y ∈ C,
b η·, · is affine in the first variable,
c for each fixed y ∈ C, x → ηy, x is sequentially continuous from the weak topology to
the weak topology;
ii K : C → R is η-strongly convex with constant σ > 0 and its derivative K is sequentially
continuous from the weak topology to the strong topology;
iii for each x ∈ C, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈ C \ Dx ,
1
Θ y, zx ϕzx − ϕy K y − K x, η zx , y < 0.
r
2.10
Then there hold the following:
i Sr is single-valued;
ii Sr is nonexpansive if K is Lipschitz continuous with constant ν > 0 such that σ ≥ λν and
∀ x1 , x2 ∈ C × C,
K x1 − K x2 , η u1 , u2 ≥ K u1 − K u2 , η u1 , u2
2.11
where ui Sr xi for i 1, 2;
iii FSr Ω;
vi Ω is closed and convex.
We also need the following lemmas for proving our main results.
Lemma 2.2 see 17. Let {xn } and {zn } be bounded sequences in a Banach space X and let {βn } be
a sequence in 0, 1 with 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1. Suppose xn1 1 − βn zn βn xn
for all integers n ≥ 0 and lim supn→∞ zn1 − zn − xn1 − xn ≤ 0. Then limn→∞ zn − xn 0.
Lemma 2.3 see 18. Assume {an } is a sequence of nonnegative real numbers such that an1 ≤
1 − γn an δn , where {γn } is a sequence in 0, 1 and {δn } is a sequence such that
1 ∞
n1 γn ∞;
2 lim supn→∞ δn /γn ≤ 0 or ∞
n1 |δn | < ∞.
Then limn→∞ an 0.
3. Iterative algorithm and strong convergence theorems
In this section, we first introduce a new iterative algorithm. Consequently, we will establish a
strong convergence theorem for this iteration algorithm. To be more specific, let T1 , T2 , . . . be
Yonghong Yao et al.
5
infinite mappings of C into itself and let ξ1 , ξ2 , . . . be real numbers such that 0 ≤ ξi ≤ 1 for every
i ∈ N. For any n ∈ N, define a mapping Wn of C into itself as follows:
Un,n1 I,
Un,n ξn Tn Un,n1 1 − ξn I,
Un,n−1 ξn−1 Tn−1 Un,n 1 − ξn−1 I,
..
.
Un,k ξk Tk Un,k1 1 − ξk I,
3.1
Un,k−1 ξk−1 Tk−1 Un,k 1 − ξk−1 I,
..
.
Un,2 ξ2 T2 Un,3 1 − ξ2 I,
Wn Un,1 ξ1 T1 Un,2 1 − ξ1 I.
Such Wn is called the W-mapping generated by Tn , Tn−1 , . . . , T2 , T1 and ξn , ξn−1 , . . . , ξ2 , ξ1 . For the
iterative algorithm for a finite family of nonexpansive mappings, we refer the reader to 19.
We have the following crucial Lemmas 3.1 and 3.2 concerning Wn which can be found in
20. Now we only need the following similar version in Hilbert spaces.
Lemma 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , . . . be
nonexpansive mappings of C into itself such that ∞
n1 FTn is nonempty, and let ξ1 , ξ2 , . . . be real
numbers such that 0 < ξi ≤ b < 1 for any i ∈ N. Then for every x ∈ C and k ∈ N, the limit
limn→∞ Un,k x exists.
Lemma 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , . . . be
nonexpansive mappings of C into itself such that ∞
is nonempty, and let ξ1 , ξ2 , . . . be real
n1 FTn numbers such that 0 < ξi ≤ b < 1 for any i ∈ N. Then FW ∞
n1 FTn .
The following remark 12 is important to prove our main results.
Remark 3.3. Using Lemma 3.1, one can define a mapping W of C into itself as Wx limn→∞ Wn x limn→∞ Un,1 x for every x ∈ C. If {xn } is a bounded sequence in C, then we
have
lim Wxn − Wn xn 0.
n→∞
3.2
Throughout this paper, we will assume that 0 < ξi ≤ b < 1 for every i ∈ N.
Now we introduce the following iteration algorithm.
Algorithm 3.4. Let r > 0 be a constant. Let ϕ : C → R be a lower semicontinuous and convex
functional and let Θ : C × C → R be an equilibrium bifunction. Let A : C → H be a βinverse-strongly monotone mapping and Wn be the W-mapping defined by 3.1. Let f be a
6
Fixed Point Theory and Applications
contraction of C into itself with coefficient α ∈ 0, 1 and given x0 ∈ C arbitrarily. Suppose that
the sequences {xn } and {yn } are generated iteratively by
1 Θ zn , x ϕx − ϕ zn K zn − K xn , η x, zn ≥ 0 ∀x ∈ C,
r
yn PC zn − λn Azn ,
xn1 αn f Wn xn βn xn γn Wn PC yn − λn Ayn
∀n ≥ 0,
3.3
where {αn }, {βn }, and {γn } are three sequences in 0, 1, and {λn } is a sequence in 0, 2β.
Now we study the strong convergence of the hybrid iterative algorithm 3.3.
Theorem 3.5. Let C be a nonempty closed convex subset of a real Hilbert space H and let ϕ : C → R
be a lower semicontinuous and convex functional. Let Θ : C × C → R be an equilibrium bifunction
satisfying conditions (H1)–(H3) and let T1 , T2 , . . . be an infinite family of nonexpansive mappings of C
into itself. Let A : C → H be a β-inverse-strongly monotone mapping such that ∩∞
n1 FTn ∩V IA, C∩
Ω
∅
.
Suppose
{α
},
{β
},
and
{γ
}
are
three
sequences
in
0,
1
with
α
β
/
n
n
n
n
n γn 1, n ≥ 0. Assume
that
i η : C × C → H is Lipschitz continuous with constant λ > 0 such that
a ηx, y ηy, x 0 for all x, y ∈ C,
b η·, · is affine in the first variable,
c for each fixed y ∈ C, x → ηy, x is sequentially continuous from the weak topology to
the weak topology;
ii K : C → R is η-strongly convex with constant σ > 0 and its derivative K is not only
sequentially continuous from the weak topology to the strong topology but also Lipschitz
continuous with constant ν > 0 such that σ ≥ λν;
iii for each x ∈ C; there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y ∈ C \ Dx ,
1
Θ y, zx ϕ zx − ϕy K y − K x, η zx , y < 0;
3.4
r
iv limn→∞ αn 0, ∞
n0 αn ∞, 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1, λn ∈ a, b ⊂
0, 2β, and limn→∞ λn1 − λn 0.
Let f be a contraction of C into itself and given x0 ∈ C arbitrarily. Then the sequence {xn } generated
by 3.3 converges strongly to x∗ PΓ fx∗ , where Γ ∩∞
n1 FTn ∩ V IA, C ∩ Ω provided that Sr is
firmly nonexpansive.
Proof. We first note that f is a contraction with coefficient α ∈ 0, 1. Then PΓ fx − PΓ fy ≤
fx − fy ≤ αx − y for all x, y ∈ C. Therefore PΓ f is a contraction of C into itself which
implies that there exists a unique element x∗ ∈ C such that x∗ PΓ fx∗ .
Next we divide the following proofs into several steps.
Step 1 {xn }, {yn }, and {zn } are bounded. Let x∗ ∈ Γ. From the definition of Sr , we know that
zn Sr xn . It follows that
zn − x∗ Sr xn − Sr x∗ ≤ xn − x∗ .
3.5
Yonghong Yao et al.
7
For all x, y ∈ C and λn ∈ 0, 2β, we note that
I − λn A x − I − λn A y 2 x − y − λn Ax − Ay2
x − y2 − 2λn Ax − Ay, x − y λ2n Ax − Ay2
≤ x − y2 λn λn − 2β Ax − Ay2 ,
3.6
which implies that I − λn A is nonexpansive.
Set un PC yn − λn Ayn for all n ≥ 0. From 2.4, we have that x∗ PC x∗ − λn Ax∗ . It
follows from 3.6 that
yn − x∗ PC zn − λn Azn − PC x∗ − λn Ax∗ ≤ zn − λn Azn − x∗ − λn Ax∗ ≤ zn − x∗ ≤ xn − x∗ ,
un − x∗ PC yn − λn Ayn − PC x∗ − λn Ax∗ ≤ yn − λn Ayn − x∗ − λn Ax∗ ≤ yn − x∗ ≤ xn − x∗ .
Hence we obtain that
xn1 − x∗ αn f Wn xn − x∗ βn xn − x∗ γn Wn un − x∗ ≤ αn f Wn xn − x∗ βn xn − x∗ γn Wn un − x∗ ≤ αn f Wn xn − f x∗ αn f x∗ − x∗ βn xn − x∗ γn un − x∗ ≤ αn βxn − x∗ αn f x∗ − x∗ βn xn − x∗ γn xn − x∗ ∗
f x − x∗ 1 − βαn
1 − 1 − βαn xn − x∗ 1−β
f x∗ − x∗ f x∗ − x∗ ≤ max xn − x∗ ,
≤ max x0 − x∗ ,
.
1−β
1−β
3.7
3.8
Therefore {xn } is bounded, so are {yn } and {zn }.
Step 2 xn1 − xn → 0. Setting xn1 βn xn 1 − βn Vn for all n ≥ 0. It follows that
xn2 − βn1 xn1 xn1 − βn xn
−
1 − βn1
1 − βn
αn1 f Wn1 xn1 γn1 Wn1 un1 αn f Wn xn γn Wn un
−
1 − βn1
1 − βn
αn1 f Wn1 xn1
αn f Wn xn
γn1 −
Wn1 un1 − Wn1 un
1 − βn1
1 − βn
1 − βn1
γn1
γn
γn Wn1 un −
Wn1 un − Wn un ,
1 − βn1 1 − βn
1 − βn
Vn1 − Vn 3.9
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Fixed Point Theory and Applications
which implies that
Vn1 − Vn ≤
αn1 f Wn1 xn1 Wn1 un 1 − βn1
αn f Wn xn Wn1 un 1 − βn
γn1 un1 − un γn Wn1 un − Wn un .
1 − βn1
1 − βn
3.10
Now we estimate un1 − un and Wn1 un − Wn un .
From 3.1, since Ti and Un,i are nonexpansive, we have
Wn1 un − Wn un ξ1 T1 Un1,2 un − ξ1 T1 Un,2 un ≤ ξ1 Un1,2 un − Un,2 un ξ1 ξ2 T2 Un1,3 un − ξ2 T2 Un,3 un ≤ ξ1 ξ2 Un1,3 un − Un,3 un 3.11
≤ ···
≤ ξ1 ξ2 · · · ξn Un1,n1 un − Un,n1 un ≤M
n
ξi ,
i1
where M is a constant such that sup{Un1,n1 un − Un,n1 un , n ≥ 0} ≤ M.
At the same time, we observe that
yn1 − yn PC zn1 − λn1 Azn1 − PC zn − λn Azn ≤ zn1 − λn1 Azn1 − zn − λn Azn zn1 − λn1 Azn1 − zn − λn1 Azn λn − λn1 Azn ≤ zn1 − λn1 Azn1 − zn − λn1 Azn λn − λn1 Azn ≤ zn1 − zn λn − λn1 Azn ,
un1 − un PC yn1 − λn1 Ayn1 − PC yn − λn Ayn ≤ yn1 − λn1 Ayn1 − yn − λn Ayn yn1 − λn1 Ayn1 − yn − λn1 Ayn λn − λn1 Ayn ≤ yn1 − yn λn − λn1 Ayn ≤ zn1 − zn λn − λn1 Ayn Azn .
3.12
Yonghong Yao et al.
9
Since zn Sr xn and zn1 Sr xn1 , from the nonexpansivity of Sr , we get
zn1 − zn ≤ xn1 − xn .
Substituting 3.11–3.13 into 3.10, we have
Vn1 − Vn − xn1 − xn ≤ αn1 f Wn1 xn1 Wn1 un 1 − βn1
n
αn f Wn xn Wn1 un M ξi
1 − βn
i1
λn − λn1 Ayn Azn .
Since αn → 0, λn1 − λn → 0, and ξi ∈ a, b, we have
lim sup Vn1 − Vn − xn1 − xn ≤ 0.
n→∞
Hence by Lemma 2.2, we have
3.14
3.15
lim Vn − xn 0.
3.16
lim xn1 − xn 0.
3.17
n→∞
Consequently,
3.13
n→∞
Step 3 un − Wun → 0. Note that xn1 − xn αn fWn xn − xn γn Wn un − xn . Then we have
xn − Wn un ≤ 1 xn − xn1 αn f Wn xn − xn −→ 0.
γn
3.18
For x∗ ∈ Γ, noting that Sr is firmly nonexpansive, we have
zn − x∗ 2 Sr xn − Sr x∗ 2
≤ S r x n − S r x ∗ , xn − x ∗
z n − x ∗ , xn − x ∗
and hence
3.19
1 zn − x∗ 2 xn − x∗ 2 − xn − zn 2 ,
2
zn − x∗ 2 ≤ xn − x∗ 2 − xn − zn 2 .
3.20
So, we have
xn1 − x∗ 2 ≤ αn f Wn xn − x∗ 2 βn xn − x∗ 2 γn Wn un − x∗ 2
2
2
2
≤ αn f Wn xn − x∗ βn xn − x∗ γn un − x∗ 2
2
2
≤ αn f Wn xn − x∗ βn xn − x∗ γn zn − x∗ 2
2
2 2 ≤ αn f Wn xn − x∗ βn xn − x∗ γn xn − x∗ − xn − zn 2 2
2
≤ αn f Wn xn − x∗ xn − x∗ − γn xn − zn ,
3.21
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Fixed Point Theory and Applications
that is,
xn − zn 2 ≤ 1 αn f Wn xn − x∗ 2 xn − x∗ 2 − xn1 − x∗ 2
γn
≤
2 1 αn f Wn xn − x∗ xn1 − xn xn − x∗ xn1 − x∗ γn
3.22
−→ 0.
From 3.6, we obtain that
xn1 − x∗ 2 ≤ αn f Wn xn − x∗ 2 βn xn − x∗ 2 γn yn − x∗ 2
2
2
2 ≤ αn f Wn xn − x∗ βn xn − x∗ γn zn − λn Azn − x∗ − λn Ax∗ 2
2
2
2 ≤ αn f Wn xn −x∗ βn xn −x∗ γn zn −x∗ λn λn − 2β Azn − Ax∗ 2 2
2
≤ αn f Wn xn − x∗ xn − x∗ γn ab − 2βAzn − Ax∗ .
3.23
Then we have
2
2 2 2
−γn ab − 2βAzn − Ax∗ ≤ αn f Wn xn − x∗ xn − x∗ − xn1 − x∗ −→ 0,
3.24
which implies that
lim Azn − Ax∗ 0.
n→∞
3.25
We note that
yn − x∗ 2 PC zn − λn Azn − PC x∗ − λn Ax∗ 2
≤
zn − λn Azn − x∗ − λn Ax∗ , yn − x∗
1 zn − λn Azn − x∗ − λn Ax∗ 2 yn − x∗ 2
2
2 − zn − λn Azn − x∗ − λn Ax∗ − yn − x∗ 1 zn − x∗ 2 yn − x∗ 2 − zn − yn − λn Azn − Ax∗ 2
2
1 zn −x∗ 2 yn −x∗ 2 − zn −yn 2 2λn Azn −Ax∗ , zn −yn −λ2n Azn −Ax∗ 2 .
2
3.26
≤
Then we derive
yn − x∗ 2 ≤ zn − x∗ 2 − zn − yn 2 2λn Azn − Ax∗ , zn − yn − λ2n Azn − Ax∗ 2
2 2
≤ xn − x∗ − zn − yn 2λn Azn − Ax∗ , zn − yn .
3.27
Yonghong Yao et al.
11
Hence
xn1 − x∗ 2
2
2
2 2
≤ αn f Wn xn −x∗ βn xn −x∗ γn xn −x∗ − zn −yn 2λn Azn −Ax∗ , zn −yn ,
3.28
which implies that
zn −yn ≤
2 2 2
1 αn f Wn xn −x∗ xn −x∗ − xn1 −x∗ 2γn λn Azn −Ax∗ zn −yn −→ 0.
γn
3.29
Since yn − un PC zn − λn Azn − PC yn − λn Ayn ≤ zn − yn , we have
Wun − un ≤ Wun − Wn un Wn un − un ≤ Wun − Wn un Wn un − xn xn − zn zn − yn yn − un ≤ Wun − Wn un Wn un − xn xn − zn 2zn − yn .
Combining the above inequality, 3.18–3.29, and Remark 3.3, we have
lim Wun − un 0.
n→∞
3.30
3.31
Step 4 limn→∞ fx∗ − x∗ , xn − x∗ , where x∗ PΓ fx∗ . To show the above inequality, we can
choose a subsequence {uni } of {un } such that
lim f x∗ − x∗ , unj − x∗ lim sup f x∗ − x∗ , un − x∗ .
3.32
j→∞
n→∞
Since {unj } is bounded, there exists a subsequence {unji } of {unj } which converges weakly to w. Without
loss of generality, we can assume that unj w. From Wun − un → 0, we obtain Wunj w.
First, we show w ∈ FW ∩∞
∈ FW. Since unj w and
n1 FTn . Assume that ω /
w/
Ww, by Opial’s condition, we have
lim infun − w < lim infun − Ww
j→∞
j
j→∞
j
≤ lim inf unj − Wunj Wunj − Ww
j→∞
3.33
≤ lim infunj − w,
j→∞
which is a contradiction. Hence we get w ∈ FW. By the same argument as that in the proof of [21,
Theorem 3.1], we can prove that w ∈ V IA, C; and by the same argument as that in the proof of [14,
Theorem 4.1], we also can prove that w ∈ Ω. Hence w ∈ Γ.
Since x∗ PΓ fx∗ ∈ Γ and un − xn → 0, we have
lim sup f x∗ − x∗ , xn − x∗ lim f x∗ − x∗ , xnj − x∗
n→∞
j→∞
lim f x∗ − x∗ , unj − x∗ f x∗ − x∗ , w − x∗ ≤ 0.
j→∞
3.34
12
Fixed Point Theory and Applications
Step 5 xn → x∗ , where x∗ PΓ fx∗ . From 3.3, we have
xn1 − x∗ 2 ≤ βn xn − x∗ γn Wn un − x∗ 2 2αn f Wn xn − x∗ , xn1 − x∗
2
≤ βn xn − x∗ γn un − x∗ 2αn f Wn xn − f x∗ , xn1 − x∗
2αn f x∗ − x∗ , xn1 − x∗
2
2 ≤ 1 − αn xn − x∗ 2αn f Wn xn − f x∗ xn1 − x∗ 2αn f x∗ − x∗ , xn1 − x∗
2
2 ≤ 1 − αn xn − x∗ 2ααn xn − x∗ xn1 − x∗ 2αn f x∗ − x∗ , xn1 − x∗
2
2 2 2 ≤ 1−αn xn −x∗ ααn xn −x∗ xn1 −x∗ 2αn f x∗ −x∗ , xn1 −x∗ ,
3.35
that is,
αn xn1 −x∗ 2 ≤ 1− 21−α αn xn −x∗2 21−α αn
xn −x∗2 1 f x∗ −x∗ , xn1 −x∗ .
1−ααn
1−ααn
21−α
1−α
3.36
It is easy to see that
∞
n0 21
lim sup
n→∞
− α/1 − ααn αn ∞ and
αn xn − x∗ 2 1 f x∗ − x∗ , xn1 − x∗
21 − α
1−α
≤ 0.
3.37
Applying Lemma 2.3 and 3.34 to 3.36, we conclude that xn → x∗ as n → ∞. This completes the
proof.
Concerning Sr , we give the following remark.
Remark 3.6. For each x1 , x2 ∈ C, we denote u1 Sr x1 and u2 Sr x2 . Then for all y ∈ C, we
have
r Θ u1 , y ϕy − ϕ u1 K u1 − K x1 , η y, u1 ≥ 0,
3.38
r Θ u2 , y ϕy − ϕ u2 K u2 − K x2 , η y, u2 ≥ 0.
3.39
Taking y u2 in 3.38 and y u1 in 3.39, and adding up these two inequalities, we obtain
r Θ u1 , u2 ϕ u2 − ϕ u1 K u1 − K x1 , η u2 , u1
r Θ u2 , u1 ϕ u1 − ϕ u2 K u2 − K x2 , η u1 , u2 ≥ 0.
3.40
Note that ηu1 , u2 ηu2 , u1 0 and Θu1 , u2 Θu2 , u1 ≤ 0. Hence from 3.40, we deduce
K x1 − K u1 , η u1 , u2 K u2 − K x2 , η u1 , u2 ≥ 0,
3.41
Yonghong Yao et al.
13
which implies that
K x1 − K x2 , η u1 , u2 ≥ K u1 − K u2 , η u1 , u2 .
3.42
Since K : C → H is η-strongly monotone with constant μ > 0, then from 3.42, we conclude
that
2
K x1 − K x2 , η u1 , u2 ≥ μu1 − u2 .
3.43
Take Kx x2 /2, ηy, x y − x, and μ 1. Then from 3.43, we have
2
x1 − x2 , u1 − u2 ≥ u1 − u2 .
3.44
This indicates that Sr is firmly nonexpansive.
Corollary 3.7. Let C be a nonempty closed convex subset of a real Hilbert space H and let ϕ : C → R
be a lower semicontinuous and convex functional. Let Θ : C × C → R be an equilibrium bifunction
satisfying conditions (H1)–(H3). Let A : C → H be a β-inverse-strongly monotone mapping such that
V IA, C ∩ Ω /
∅. Suppose {αn }, {βn }, and {γn } are three sequences in 0, 1 with αn βn γn 1, n ≥ 0. Assume that
i η : C × C → H is Lipschitz continuous with constant λ > 0 such that
a ηx, y ηy, x 0 for all x, y ∈ C,
b η·, · is affine in the first variable,
c for each fixed y ∈ C, x → ηy, x is sequentially continuous from the weak topology to
the weak topology;
ii K : C → R is η-strongly convex with constant σ > 0 and its derivative K is not only
sequentially continuous from the weak topology to the strong topology but also Lipschitz
continuous with constant ν > 0 such that σ ≥ λν;
iii for each x ∈ C, there exist a bounded subset Dx ⊂ C and zx ∈ C such that, for any y ∈ C\Dx ,
1
Θ y, zx ϕ zx − ϕy K y − K x, η zx , y < 0;
r
3.45
14
Fixed Point Theory and Applications
∞
iv limn→∞ αn 0, n0 αn ∞, 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1, λn ∈ a, b ⊂
0, 2β, and limn→∞ λn1 − λn 0.
Let f be a contraction of C into itself and given x0 ∈ C arbitrarily. Let the sequences {xn }, {yn }, and
{zn } be generated iteratively by
1 Θ zn , x ϕx − ϕ zn K zn − K xn , η x, zn ≥ 0
r
yn PC zn − λn Azn ,
xn1 αn f xn βn xn γn PC yn − λn Ayn
∀x ∈ C,
3.46
∀n ≥ 0.
Then the sequence {xn } generated by 3.46 converges strongly to x∗ PΓ fx∗ , where Γ V IA, C∩
Ω provided that Sr is firmly nonexpansive.
Proof. Take Tn x x for all n 1, 2, . . . , and for all x ∈ C in 3.1. Then Wn x x for all x ∈ C.
The conclusion follows immediately from Theorem 3.5. This completes the proof.
Corollary 3.8. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1 , T2 , . . . be
an infinite family of nonexpansive mappings of C into itself. Let A : C → H be a β-inverse-strongly
monotone mapping such that ∩∞
/ ∅. Suppose {αn }, {βn }, and {γn } are three
n1 FTn ∩ V IA, C sequences in 0, 1 with αn βn γn 1, n ≥ 0. Assume that
i limn→∞ αn 0 and ∞
n0 αn ∞;
ii 0 < lim infn→∞ βn ≤ lim supn→∞ βn < 1;
iii λn ∈ a, b ⊂ 0, 2β and limn→∞ λn1 − λn 0.
Let f be a contraction of C into itself and given x0 ∈ C arbitrarily. Then the sequence
{xn }, generated iteratively by
yn PC xn − λn Axn ,
xn1 αn f Wn xn βn xn γn Wn PC yn − λn Ayn
∀n ≥ 0,
3.47
converges strongly to x∗ PΓ fx∗ , where Γ ∩∞
n1 FTn ∩ V IA, C.
Proof. Set ϕx 0 and Θx, y 0 for all x, y ∈ C and put r 1. Take Kx x2 /2 and
ηy, x y − x for all x, y ∈ C. Then we have zn PC xn xn . Hence the conclusion follows.
This completes the proof.
Acknowledgments
The authors are extremely grateful to the anonymous referees and Professor T. Suzuki for
their useful comments and suggestions. The first author was partially supported by National
Natural Science Foundation of China, Grant no. 10771050. The second author was partially
supported by the Grant no. NSC 96-2221-E-230-003.
Yonghong Yao et al.
15
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