báo cáo hóa học: " Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings" potx

Peng Fixed Point Theory and Applications 2011, 2011:12
/>
RESEARCH

Open Access

Some extragradient methods for common
solutions of generalized equilibrium problems
and fixed points of nonexpansive mappings
Jian-Wen Peng
Correspondence: jwpeng6@yahoo.
com.cn
School of Mathematics, Chongqing
Normal University, Chongqing
400047, PR China

Abstract
In this article, we introduce some new iterative schemes based on the extragradient
method (and the hybrid method) for finding a common element of the set of
solutions of a generalized equilibrium problem, and the set of fixed points of a
family of infinitely nonexpansive mappings and the set of solutions of the variational
inequality for a monotone, Lipschitz-continuous mapping in Hilbert spaces. We
obtain some strong convergence theorems and weak convergence theorems. The
results in this article generalize, improve, and unify some well-known convergence
theorems in the literature.
Keywords: Generalized equilibrium problem, Extragradient method, Hybrid method,
Nonex-pansive mapping, Strong convergence, Weak convergence

1. Introduction
Let H be a real Hilbert space with inner product 〈.,.〉 and induced norm ||·||. Let C be
a nonempty closed convex subset of H. Let F be a bifunction from C × C to R and let

B : C ® H be a nonlinear mapping, where R is the set of real numbers. Moudafi [1],
Moudafi and Thera [2], Peng and Yao [3,4], Takahashi and Takahashi [5] considered
the following generalized equilibrium problem:
Find x ∈ C Such that F(x, y) + Bx, y − x ≥ 0, ∀y ∈ C.

(1:1)

The set of solutions of (1.1) is denoted by GEP(F, B). If B = 0, the generalized equilibrium problem (1.1) becomes the equilibrium problem for F : C × C đ R, which is to
find x ẻ C such that
F(x, y) ≥ 0

for all y ∈ C.

(1:2)

The set of solutions of (1.2) is denoted by EP(F).
The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see for instance [1-7].
Recall that a mapping S : C ® C is nonexpansive if there holds that
||Sx − Sy|| ≤ ||x − y||

for all x, y ∈ C.

We denote the set of fixed points of S by Fix(S).
© 2011 Peng; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
License ( which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.


Peng Fixed Point Theory and Applications 2011, 2011:12
/>

Let the mapping A : C ® H be monotone and k-Lipschitz-continuous. The variational inequality problem is to find x Ỵ C such that
Ax, y − x ≥ 0

for all y Ỵ C. The set of solutions of the variational inequality problem is denoted by
V I(C, A).
Several algorithms have been proposed for finding the solution of problem (1.1). Moudafi [1] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert
space, and proved a weak convergence theorem. Moudafi and Thera [2] introduced an
auxiliary scheme for finding a solution of problem (1.1) in a Hilbert space and obtained a
weak convergence theorem. Peng and Yao [3,4] introduced some iterative schemes for
finding a common element of the set of solutions of problem (1.1), the set of fixed points
of a nonexpansive mapping and the set of solutions of the variational inequality for a
monotone, Lipschitz-continuous mapping and obtain both strong convergence theorems,
and weak convergence theorems for the sequences generated by the corresponding processes in Hilbert spaces. Takahashi and Takahashi [5] introduced an iterative scheme for
finding a common element of the set of solutions of problem (1.1) and the set of fixed
points of a nonexpansive mapping in a Hilbert space, and proved a strong convergence
theorem.
Some methods also have been proposed to solve the problem (1.2); see, for instance,
[8-19] and the references therein. Takahashi and Takahashi [9] introduced an iterative
scheme by the viscosity approximation method for finding a common element of the
set of solutions of problem (1.2) and the set of fixed points of a non-expansive mapping, and proved a strong convergence theorem in a Hilbert space. Su et al. [10] introduced and researched an iterative scheme by the viscosity approximation method for
finding a common element of the set of solutions of problem (1.2) and the set of fixed
points of a nonexpansive mapping and the set of solutions of the variational inequality
problem for an a-inverse-strongly monotone mapping in a Hilbert space. Tada and
Takahashi [11] introduced two iterative schemes for finding a common element of the
set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping
in a Hilbert space, and obtained both strong convergence and weak convergence theorems. Plubtieng and Punpaeng [12] introduced an iterative processes based on the
extragradient method for finding the common element of the set of fixed points of a
nonexpansive mapping, the set of solutions of an equilibrium problem and the set of
solutions of variational inequality problem for an a-inverse-strongly monotone mapping. Chang et al. [13] introduced an iterative processes based on the extragradient
method for finding the common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings

and the set of solutions of variational inequality problem for an a-inverse-strongly
monotone mapping. Yao et al. [14] and Ceng and Yao [15] introduced some iterative
viscosity approximation schemes for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a family of infinitely nonexpansive
mappings in a Hilbert space. Colao et al. [16] introduced an iterative viscosity approximation scheme for finding a common element of the set of solutions of problem (1.2)

Page 2 of 19


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 3 of 19

and the set of fixed points of a family of finitely nonexpansive mappings in a Hilbert
space. We observe that the algorithms in [13-16] involves the W-mapping generated
by a family of infinitely (finitely) nonexpansive mappings which is an effective tool in
nonlinear analysis (see [20,21]). However, the W-mapping generated by a family of infinitely (finitely) nonexpansive mappings is too completed to use for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a
family of infinitely (finitely) nonexpansive mappings. It is natural to raise and to give
an answer to the following question: Can one construct algorithms for finding a common element of the set of solutions of a generalized equilibrium problem (an equilibrium problem), the common set of fixed points of a family of infinitely nonexpansive
mappings and the set of solutions of a variational inequality without the W-mapping
generated by a family of infinitely (finitely) nonexpansive mappings? In this article, we
will give a positive answer to this question.
Recently, OHaraa et al. [22] introduced and researched an iterative approach for
finding a nearest point of infinitely many nonexpansive mappings in a Hilbert spaces
without using the W-mapping generated by a family of infinitely (finitely) nonexpansive
mappings. Inspired by the ideas in [1-6,8-16,22] and the references therein, we introduce some new iterative schemes based on the extragradient method (and the hybrid
method) for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings,
and the set of solutions of the variational inequality for a monotone, Lipschitz–continuous mapping without using the W-mapping generated by a family of infinitely
(finitely) nonexpansive mappings. We obtain both strong convergence theorems and
weak convergence theorems for the sequences generated by the corresponding processes. The results in this article generalize, improve, and unify some well-known convergence theorems in the literature.

2. Preliminaries
Let H be a real Hilbert space with inner product 〈·,·〉 and norm ||·||. Let C be a nonempty closed convex subset of H. Let symbols ® and ⇀ denote strong and weak convergences, respectively. In a real Hilbert space H, it is well known that
λx + (1 − λ)y

2

=λ x

2

+ (1 − λ) y

2

− λ(1 − λ) x − y

2

for all x, y Ỵ H and l Ỵ [0, 1].
For any x Î H, there exists the unique nearest point in C, denoted by PC(x), such
that ||x - PC(x)|| ≤ ||x - y|| for all y Ỵ C. The mapping PC is called the metric projection of H onto C. We know that PC is a nonexpansive mapping from H onto C. It is
also known that PCx Ỵ C and
x − PC (x), PC (x) − y ≥ 0

(2:1)

for all x Ỵ H and y Ỵ C.
It is easy to see that (2.1) is equivalent to
x−y

2

≥ x − PC (x)

for all x Î H and y Î C.

2

+ y − PC (x)

2

(2:2)


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
A mapping A of C into H is called monotone if
Ax − Ay, x − y ≥ 0

for all x, y Ỵ C. A mapping A of C into H is called a-inverse-strongly monotone if
there exists a positive real number a such that
x − y, Ax − Ay ≥ α Ax − Ay

2

for all x, y Ỵ C. A mapping A : C ® H is called k-Lipschitz-continuous if there exists
a positive real number k such that
Ax − Ay ≤ k x − y

for all x, y Ỵ C. It is easy to see that if A is a-inverse-strongly monotone, then A is
monotone and Lipschitz-continuous. The converse is not true in general. The class of
a-inverse-strongly monotone mappings does not contain some important classes of
mappings even in a finite-dimensional case. For example, if the matrix in the corresponding linear complementarity problem is positively semidefinite, but not positively
definite, then the mapping A will be monotone and Lipschitz-continuous, but not ainverse-strongly monotone (see [23]).
Let A be a monotone mapping of C into H. In the context of the variational inequality problem, the characterization of projection (2.1) implies the following:
u ∈ VI(C, A) ⇒ u = PC (u − λAu), λ > 0.

and
u = PC (u − λAu) for some λ > 0 ⇒ u ∈ VI(C, A).

It is also known that H satisfies the Opial’s condition [24], i.e., for any sequence {xn}
⊂ H with xn ⇀ x, the inequality
lim inf xn − x < lim inf xn − y
n→∞

n→∞

holds for every y Ỵ H with x ≠ y.
A set-valued mapping T : H ® 2H is called monotone if for all x, y Ỵ H, f Î Tx and g Î
Ty imply 〈x - y, f - g〉 ≥ 0. A monotone mapping T : H ® 2H is maximal if its graph G(T)
of T is not properly contained in the graph of any other monotone mapping. It is known
that a monotone mapping T is maximal if and only if for (x, f) ẻ H ì H, 〈x - y, f - g〉 ≥ 0
for every (y, g) Ỵ G(T) implies f Ỵ Tx. Let A be a monotone, k-Lipschitz-continuous
mapping of C into H and NCv be normal cone to C at v Ỵ C, i.e., NCv = {w Ỵ H : 〈v - u,
w〉 ≥ 0, ∀u Ỵ C}. Define
Tv =

Av + NC v if v ∈ C,
∅

if v ∈ C.
/

Then, T is maximal monotone and 0 Ỵ Tv if and only if v Ỵ V I(C, A) (see [25]).
For solving the equilibrium problem, let us assume that the bifunction F satisfies the
following condition:
(A1) F(x, x) = 0 for all x Ỵ C;
(A2) F is monotone, i.e., F(x, y) + F(y, x) ≤ 0 for any x, y Ỵ C;

Page 4 of 19


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 5 of 19

(A3) for each x, y, z Ỵ C,
lim F(tz + (1 − t)x, y) ≤ F(x, y);
t↓0

(A4) for each x Î C, y ↦ F(x, y) is convex and lower semicontinuous.
We recall some lemmas which will be needed in the rest of this article.
Lemma 2.1.[7] Let C be a nonempty closed convex subset of H, let F be a bifunction from
C × C to R satisfying (A1)-(A4). Let r >0 and x Ỵ H. Then, there exists z Ỵ C such that
F(z, y) +

1
y − z, z − x ≥ 0,
r

for all y ∈ C.

Lemma 2.2.[8] Let C be a nonempty closed convex subset of H, let F be a bi-function from C × C to R satisfying (A1)-(A4). For r >0 and x Ỵ H, define a mapping Tr :
H ® C as follows:
Tr (x) = {z ∈ C : F(z, y) +

1
y − z, z − x ≥ 0, ∀y ∈ C}
r

for all x Ỵ H. Then, the following statements hold:
(1) Tr is single-valued;
(2) Tr is firmly nonexpansive, i.e., for any x, y Ỵ H,
Tr (x) − Tr (y)

2

≤ Tr (x) − Tr (y), x − y ;

(3) F(Tr) = EP (F);
(4) EP(F) is closed and convex.

3. The main results
We first show a strong convergence of an iterative algorithm based on extragradient
and hybrid methods which solves the problem of finding a common element of the set
of solutions of a generalized equilibrium problem, the set of fixed points of a family of
infinitely nonexpansive mappings, and the set of solutions of the variational inequality
for a monotone, Lipschitz-continuous mapping in a Hilbert space.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone and

k-Lipschitz-continuous mapping of C into H and B be an a-inverse-strongly monotone
mapping of C into H. Let S1, S2,... be a family of infinitely nonexpansive mappings of C
into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ GEP(F, B) = ∅. Assume that for all i Ỵ
i=1
{1, 2,...} and for any bounded subset K of C, thenthere holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )

Let {xn}, {un}, {yn} and {zn} be sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪
⎪
⎪
⎪ F(u , y) + Bx , y − u + 1 y − u , u − x ≥ 0, ∀y ∈ C,
⎪
⎪
n
n
n
n n
n
⎪
r
⎪
⎪
⎨ y = (1 − γ )u + γ P (u n− λ Au ),

n
n n
n C n
n
n
⎪ zn = (1 − αn − βn )xn + αn yn + βn Sn PC (un − λn Ayn ),
⎪
⎪ C = {z ∈ C : ||z − z||2 ≤ ||x − z||2 + (3 − 3γ + α )b2 ||Au ||2 },
⎪ n
⎪
n
n
n
n
n
⎪
⎪
⎪ Qn = {z ∈ C : xn − z, x − xn ≥ 0},
⎪
⎪
⎩
xn+1 = PCn Qn x

(3:1)


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 6 of 19

1
), {rn} ⊂ [d, e] for some
4k
d, e Ỵ (0, 2a), and {an}, {bn}, {gn} are three sequences in [0, 1] satisfying the conditions:

for every n = 1, 2,... where {ln} ⊂ [a, b] for some a, b ∈ (0,

(i) an + bn ≤ 1 for all n Ỵ N;
lim
(ii) n→∞ αn = 0;
(iii) lim inf βn > 0;
n→∞
3
lim
(iv) n→∞ γn = 1 and γn > for all n Ỵ N;
4

Then, {xn}, {un}, {yn} and {zn} converge strongly to w = PΩ(x).
Proof. It is obvious that Cn is closed, and Qn is closed and convex for every n = 1,
2,.... Since
Cn = {z ∈ H : zn − xn

+ 2 zn − xn , xn − z ≤ (3 − 3γn + αn )b2 Aun 2 },

2

we also have that Cn is convex for every n = 1, 2,.... It is easy to see that 〈xn - z, x xn〉 ≥ 0 for all z Ỵ Qn and by (2.1), xn = PQ n x. Let tn = PC(un - lnAyn) for every n = 1,
2,.... Let u Ỵ Ω and let {Trn } >be a sequence of mappings defined as in Lemma 2.2.
Then u = PC (u − λn Au) = Trn (u − rn Bu). From un = Trn (xn − rn Bxn ) ∈ C and the ainverse strongly monotonicity of B, we have
un − u

= Trn (xn − rn Bxn ) − Trn (u − rn Bu)

2

≤ xn − rn Bxn − (u − rn Bu)
≤ xn − u

2

2

2

2
− 2rn xn − u, Bxn − Bu + rn Bxn − Bu

≤ xn − u

2

− 2rn α Bxn − Bu

= xn − u

2

2

2

+ rn Bxn − Bu

+ rn (rn − 2α) Bxn − Bu

2

(3:2)

2

2

≤ xn − u 2 .

From (2.2), the monotonicity of A, and u Ỵ V I(C, A), we have
tn − u

2

2

≤ un − λn Ayn − u

− un − λn Ayn − tn

2

= un − u

2

− un − tn

2

+ 2λn Ayn , u − tn

= un − u

2

− un − tn

2

+ 2λn ( Ayn − Au, u − yn + Au, u − yn + Ayn , yn − tn )

≤ un − u
≤ un − u

2

= un − u

2

2

− un − tn

− un − yn

2

− un − yn

2

2

+ 2λn Ayn , yn − tn

− 2 un − yn , yn − tn − yn − tn
− y n − tn

2

2

+ 2λn Ayn , yn − tn

+ 2 un − λn Ayn − yn , tn − yn .

Further, Since yn = (1 - gn)un + gnPC(un - lnAun) and A is k-Lipschitz-continuous, we
have
un − λn Ayn − yn , tn − yn
= un − λn Aun − yn , tn − yn + λn Aun − λn Ayn , tn − yn
≤ un − λn Aun − (1 − γn )un − γn PC (un − λn Aun ), tn − yn + λn Aun − Ayn

tn − y n

≤ γn un − λn Aun − PC (un − λn Aun ), tn − yn − (1 − γn )λn Aun , tn − yn + λn k un − yn

tn − yn .


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 7 of 19

In addition, from the definition of PC, we have
un − λn Aun − PC (un − λn Aun ), tn − yn
= un − λn Aun − PC (un − λn Aun ), tn − (1 − γn )un − γn PC (un − λn Aun )
= (1 − γn ) un − λn Aun − PC (un − λn Aun ), tn − un
+γn un − λn Aun − PC (un − λn Aun ), tn − PC (un − λn Aun )
≤ (1 − γn ) un − λn Aun − PC (un − λn Aun )

tn − un

≤ (1 − γn )λn un − Aun − un ( tn − yn + yn − un )
≤ (1 − γn )λn Aun ( tn − yn + yn − un ).

It follows from b <
tn − u

2

1
4k,

≤ un − u

2

3
and (3.2) that
4

γn >

2

− un − yn

+2(1 − γn )b Aun
≤ un − u

2

2

− un − yn

+(1 − γn )(b Aun
= un − u

2

2

+ 2γn (1 − γn )b Aun ( tn − yn + yn − un )

tn − yn + 2bk un − yn

− yn − tn

2

2

− yn − tn
2

+ (1 − γn )(2b2 Aun

+ tn − y n
2

− (γn − bk) un − yn

2

tn − y n
2

+ tn − yn

2

) + bk( un − yn

+ tn − y n
2

+ (1 − 2γn + bk) tn − yn

≤ un − u

2

+ 3(1 − γn )b2 Aun

≤ xn − u

2

2

+ yn − un

2

2

)

(3:3)

)

+ 3(1 − γn )b Aun
2

2

+ 3(1 − γn )b2 Aun 2 .

2

In addition, from u Î V I(C, A) and (3.2), we have
2

yn − u

= (1 − γn )(un − u) + γn (PC (un − λn Aun ) − u)

≤ (1 − γn ) un − u

2

+ γn PC (un − λn Aun ) − PC (u)

≤ (1 − γn ) un − u
≤ (1 − γn ) un − u

2

2

+ γn un − λn Aun − u

+ γn [ un − u

2

2

2

2

− 2λn Aun , un − u + λ2 Aun 2 ]
n

≤ un − u

2

+ b2 Aun

≤ xn − u

2

(3:4)

+ b2 Aun 2 .

2

Therefore, from (3.2) to (3.4) and zn = (1 - an - bn)xn + anyn + bnSntn and u = Snu,
we have
zn − u

2

= (1 − αn − βn )xn + αn yn + βn Sn tn − u

≤ (1 − αn − βn ) xn − u

2

+ αn yn − u

2

+ βn Sn tn − u
2

≤ (1 − αn − βn ) xn − u

2

+ αn yn − u

≤ (1 − αn − βn ) xn − u

2

+ αn [ un − u

+βn [ un − u
≤ xn − u

2

2

2

+ β n tn − u
2

2
2

+ b2 Aun 2 ]

(3:5)

+ 3(1 − γn )b2 Aun 2 ]

+ (3 − 3γn + αn )b2 Aun 2 ],

for every n = 1, 2,... and hence u Ỵ Cn. So, Ω ⊂ Cn for every n = 1, 2,.... Next, let us
show by mathematical induction that xn is well defined and Ω ⊂ Cn ∩ Qn for every n
= 1, 2,.... For n = 1 we have x1 = x Ỵ C and Q1 = C. Hence, we obtain Ω ⊂C1 ∩ Q1.
Suppose that xk is given and Ω ⊂ Ck ∩ Qk for some k Ỵ N. Since Ω is nonempty, Ck ∩
Qk is a nonempty closed convex subset of H. Hence, there exists a unique element xk+1
Ỵ Ck ∩ Qk such that xk+1 = PCk ∩Qk x. It is also obvious that there holds 〈xk+1 - z, x - xk

+1〉 ≥ 0 for every z Ỵ Ck ∩ Qk. Since Ω ⊂ Ck ∩ Qk, we have 〈xk+1 - z, x - xk+1〉 ≥ 0 for
every z Ỵ Ω and hence Ω ⊂ Qk+1. Therefore, we obtain Ω ⊂ Ck+1 ∩ Qk+1.


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 8 of 19

Let l0 = PΩx. From xn+1 = PCn ∩Qn x and l0 v Ω ⊂ Cn ∩ Qn, we have
xn+1 − x ≤ l0 − x

(3:6)

for every n = 1, 2,.... Therefore, {xn} is bounded. From (3.2) to (3.5) and the lipschitz
continuity of A, we also obtain that {un}, {yn}, {Aun}, {tn} and {zn} are bounded. Since
xn+1 Ỵ Cn ∩ Qn ⊂ Cn and xn = PQ n x, we have
xn − x ≤ xn+1 − x

for every n = 1, 2,.... It follows from (3.6) that limn®∞ ||xn - x|| exists.
Since xn = PQ n x and xn+1 Ỵ Qn, using (2.2), we have
xn+1 − xn

2

≤ xn+1 − x

2

− xn − x

2

for every n = 1, 2,.... This implies that
lim xn+1 − xn = 0.

n→∞

Since xn+1 Ỵ Cn, we have ||zn - xn+1||2 ≤ ||xn - xn+1||2 + (3 - 3gn + an)b2||Aun||2 and
hence it follows from limn®∞ gn = 1 and limn®∞ an = 0 that limn®∞ ||zn - xn+1|| = 0. Since
||xn − zn || ≤ ||xn − xn+1 || + ||xn+1 − zn ||

for every n = 1, 2,..., we have ||xn - zn|| ® 0.
For u Ỵ Ω, from (3.5), we obtain
||zn − u||2 − ||xn − u||2
≤ (−αn − βn )||xn − u||2 + αn ||yn − u||2 + βn ||Sn tn − u||2
≤ (3 − 3γn + αn )b2 ||Aun ||2 .

Since limn®∞ gn = 1 and limn®∞ an = 0, {xn}, {yn}, {Aun}, and {zn} are bounded, we
have
lim βn (||Sn tn − u||2 − ||xn − u||2 ) = 0.

n→∞

By lim inf

n®∞

bn > 0, we get

lim ||Sn tn − u||2 − ||xn − u||2 = 0.

n→∞

From (3.3) and u = Snu, we have
lim ||Sn tn − u||2 − ||xn − u||2 ≤ lim ||tn − u||2 − ||xn − u||2

n→∞

n→∞

≤ lim 3(1 − γn )b2 ||Aun ||2 = 0.
n→∞

Thus, limn®∞ ||tn - u||2 - ||xn - u||2 = 0.
From (3.3) and (3.2), we have
(γn − bk)||un − yn ||2 + (2γn − 1 − bk)||tn − yn ||2
≤ ||xn − u||2 − ||tn − u||2 + 3(1 − γn )b2 ||Aun ||2 .

It follows that
lim (γn − bk)||un − yn ||2 + (2γn − 1 − bk)||tn − yn ||2 = 0.

n→∞


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 9 of 19

1
1

and 2γn − 1 − bk > . Conse2
4
quently, limn®∞ ||un - yn|| = limn®∞ ||tn - yn|| = 0. Since A is Lipschitz-continuous,
we have limn®∞ ||Atn - Ayn|| = 0. It follows from ||un - tn|| ≤ ||un - yn|| + ||tn - yn||
that limn®∞ ||un - tn|| = 0.
We rewrite the definition of zn as

The assumptions on gn and ln imply that γn − bk >

zn − xn = αn (yn − xn ) + βn (Sn tn − xn ).

From limn®∞ ||zn - xn|| = 0, limn®∞ an = 0, the boundedness of {xn}, {yn} and lim
infn®∞ bn > 0 we infer that limn®∞ ||Sntn - xn|| = 0.
By (3.2)-(3.5), we have
||zn − u||2 ≤ (1 − αn − βn )||xn − u||2 + αn [||un − u||2 + b2 ||Aun ||2 ] + βn [||un − u||2 + 3(1 − γn )b2 ||Aun ||2 ]
≤ (1 − αn − βn )||xn − u||2 + αn [||xn − u||2 + rn (rn − 2α)||Bxn − Bu||2 + b2 ||Aun ||2 ]
+ βn [||xn − u||2 + rn (rn − 2α)||Bxn − Bu||2 + 3(1 − γn )b2 ||Aun ||2 ]

(3:7)

≤ ||xn − u||2 + (αn + βn )rn (rn − 2α)||Bxn − Bu||2 + (3βn − 3βn γn + αn )b2 ||Aun ||2 ].

Hence, we have
(αn + βn )d(2α − e)||Bxn − Bu||2
≤ (αn + βn )rn (2α − rn )||Bxn − Bu||2
≤ ||xn − u||2 − ||zn − u||2 + (3βn − 3βn γn + αn )b2 ||Aun ||2
≤ (||xn − u|| + ||zn − u||)||xn − zn || + (3βn − 3βn γn + αn )b2 ||Aun ||2 .

Since lim n®∞ a n = 1, lim infn®∞ b n > 0, lim n®∞ g n = 1, ||x n - z n || ® 0 and the
sequences {xn} and {zn} are bounded, we obtain ||Bxn - Bu|| đ 0.

For u ẻ , we have, from Lemma 2.2,
||un − u||2 = ||Trn (xn − rn Bxn ) − Trn (u − rn Bu)||2
≤ Trn (xn − rn Bxn ) − Trn (u − rn Bu), xn − rn Bxn − (u − rn Bu)
1
= {||un − u||2 + ||xn − rn Bxn − (u − rn Bu)||2 − ||xn − rn Bxn − (u − rn Bu) − (un − u)||2 }
2
1
≤ {||un − u||2 + ||xn − u||2 − ||xn − rn Bxn − (u − rn Bu) − (un − u)||2 }
2
1
2
= {||un − u||2 + ||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un − rn ||Bxn − Bu||2 }.
2

Hence,
2
||un − u||2 ≤ ||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un − rn ||Bxn − Bu||2

≤ ||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un .

Then, by (3.5), we have
||zn − u||2 ≤ (1 − αn − βn )||xn − u||2 + αn [||un − u||2 + b2 ||Aun ||2 ] + βn [||un − u||2 + 3(1 − γn )b2 ||Aun ||2 ]
≤ (1 − αn − βn )||xn − u||2 + αn [(||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un ) + b2 ||Aun ||2 ]
+ βn [(||xn − u||2 − ||xn − un ||2 + 2rn Bxn − Bu, xn − un ) + 3(1 − γn )b2 ||Aun ||2 ]
≤ ||xn − u||2 + (−αn − βn )||xn − un ||2 + 2rn (αn + βn )||Bxn − Bu|| ||xn − un || + (3βn − 3βn γn + αn )b2 ||Aun ||2

Hence,
(αn + βn )||xn − un ||2 ≤ ||xn − u||2 − ||zn − u||2 + 2rn (αn + βn )||Bxn − Bu|| ||xn − un || + (3βn − 3βn γn + αn )b2 ||Aun ||2
≤ (||xn − u|| + ||zn − u||)||xn − zn || + 2rn (αn + βn )||Bxn − Bu|| ||xn − un || + (3βn − 3βn γn + αn )b2 ||Aun ||2 .

Since limn®∞ an = 0, lim infn®∞ bn > 0, limn®∞ gn = 1, ||xn - zn|| ® 0, ||Bxn - Bu|| ® 0
and the sequences {xn}, {un} and {zn} are bounded, we obtain ||xn - un|| ® 0. From ||zn tn|| ≤ ||zn - xn||+||xn - un||+||un - tn||, we have ||zn - tn|| ® 0.


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 10 of 19

From ||tn - xn|| ≤ ||tn - un|| + ||xn - un||, we also have ||tn - xn|| ® 0.
Since zn = (1 - an - bn)xn + anyn + bnSntn, we have bn(Sntn - tn) = (1 - an - bn)(tn xn) + an(tn - yn) + (zn - tn). Then
βn ||Sn tn − tn || ≤ (1 − αn − βn )||tn − xn || + αn ||tn − yn || + ||zn − tn ||

and hence ||Sntn - tn|| ® 0. At the same time, observe that for all i Ỵ {1, 2,...},
||Si tn − tn || ≤ ||Si tn − Si (Sn tn )|| + ||Si (Sn tn ) − Sn tn || || + ||Sn tn − tn ||.
≤ 2||Sn tn − tn || + sup ||Si (Sn x) − Sn x||.
x∈K

It follows from (3.8) and the condition (*) that for all i Ỵ {1, 2,...},
lim ||Si tn − tn || = 0.

(3:9)

n→∞

As {xn} is bounded, there exists a subsequence {xni } of {xn} such that xni ⇀ w. From ||
xn - un|| ® 0, we obtain that uni ⇀ w. From ||un - tn|| ® 0, we also obtain that tni ⇀
w. Since {uni} ⊂ C and C is closed and convex, we obtain w Ỵ C.
First, we show w Ỵ GEP(F, B). By un = Trn (xn − rn Bxn ) ∈ C, we know that
F(un , y) + Bxn , y − un +

1
y − un , un − xn ≥ 0, ∀y ∈ C.
rn

It follows from (A2) that
Bxn , y − un +

1
y − un , un − xn ≥ F(y, un ), ∀y ∈ C.
rn

Hence,
Bxni , y − uni + y − uni ,

uni − xni
≥ F(y, uni ), ∀y ∈ C.
rni

(3:10)

For t with 0 < t ≤ 1 and y Ỵ C, let yt = ty + (1 - t)w. Since y Ỵ C and w Ỵ C, we
obtain yt Ỵ C. So, from (3.10) we have
yt − uni , Byt ≥ yt − uni , Byt − yt − uni , Bxni
un − xni
− yt − uni , i
+ F(yt , uni )
rni
= yt − uni , Byt − Buni + yt − uni , Buni − Bxni
un − xni
+ F(yt , uni ).

− yt − uni , i
rni

Since ||uni − xni || → 0, we have ||Buni − Bxni || → 0. Further, from the inverse-strongly
monotonicity of B, we have yt − uni , Byt − Buni ≥ 0. Hence, from (A4),
and uni

uni −xni
rni

→0

w, we have
yt − w, Byt ≥ F(yt , w),

as i ® ∞. From (A1), (A4) and (3.11), we also have
0 =F(yt , yt ) ≤ tF(yt , y) + (1 − t)F(yt , w)
≤ tF(yt , y) + (1 − t) yt − w, Byt
= tF(yt , y) + (1 − t)t y − w, Byt .

(3:11)


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 11 of 19

and hence
0 ≤ F(yt , y) + (1 − t) y − w, Byt .

Letting t ® 0, we have, for each y Ỵ C,
F(w, y) + y − w, Bw ≥ 0.

This implies that w Ỵ GEP(F, B).
/ i=1
We next show that w ∈ ∩∞ Fix(Si ). Assume w ∈ ∩∞ Fix(Si ). Since tni
i=1
w = Si0 w for some i0 Ỵ {1, 2,...} from the Opial condition, we have
lim inf ||tni − w||
i→∞

w and

< lim inf ||tni − Si0 w||
i→∞

≤ lim inf{||tni − Si0 tni || + ||Si0 tni − Si0 w||}
i→∞

≤ lim inf ||tni − w||.
i→∞

This is a contradiction. Hence, we get w ∈ ∩∞ Fix(Si ).
i=1
Finally we show w Ỵ V I(C, A). Let
Tv =

Av + NC v if v ∈ C,
∅
if v ∈ C.

where NCv is the normal cone to C at v Ỵ C. We have already mentioned that in this
case the mapping T is maximal monotone, and 0 Ỵ Tv if and only if v Ỵ V I(C, A). Let
(v, g) Ỵ G(T). Then Tv = Av + NCv and hence g - Av Ỵ NCv.
Hence, we have 〈v - t, g - Av〉 ≥ 0 for all t Ỵ C. On the other hand, from tn = PC(un lnAyn) and v Ỵ C, we have
un − λn Ayn − tn , tn − v ≥ 0

and hence
v − tn ,

tn − un
+ Ayn ≥ 0.
λn

Therefore, we have
v − tni , g

≥ v − tni , Av
tni − uni
+ Ayni
λni
tn − uni
= v − tni , Av − Ayni − i
λni
tn − uni
= v − tni , Av − Atni + Atni − Ayni − i
λni

≥ v − tni , Av − v − tni ,

= v − tni , Av − Atni + v − tni , Atni − Ayni − v − tni ,
≥ v − tni , Atni − Ayni − v − tni ,

tni − uni
λni

tni − uni
λni

Hence, we obtain 〈v - w, g〉 ≥ 0 as i ® ∞. Since T is maximal monotone, we have w
Ỵ T-10 and hence w Ỵ V I(C, A). This implies that w Ỵ Ω.


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 12 of 19

From l0 = PΩx, w Ỵ Ω and (3.6), we have
||l0 − x|| ≤ ||w − x|| ≤ lim inf ||xni − x|| ≤ lim sup ||xni − x|| ≤ ||l0 − x||.
i→∞

i→∞

Hence, we obtain
lim ||xni − x|| = ||w − x||.

i→∞

w − x, we have xni − x → w − x, and hence xni → w. Since xn = PQ n x
From xni − x

and l0 Ỵ Ω ⊂ Cn ∩ Qn ⊂ Qn, we have
−||l0 − xni ||2 ≤ l0 − xni , xni − x + l0 − xni , x − l0 ≥ l0 − xni , x − l0 .

As i ® ∞, we obtain - ||l0 - w||2 ≥ 〈l0 - w, x - l0〉 ≥ 0 by l0 = PΩx and w Î Ω. Hence,
we have w = l0. This implies that xn ® l0. It is easy to see un ® l0, yn ® l0 and zn ®
l0. The proof is now complete.
By combining the arguments in the proof of Theorem 3.1 and those in the proof of
Theorem 3.1 in [3], we can easily obtain the following weak convergence theorem for
an iterative algorithm based on the extragradient method which solves the problem of
finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings and the
set of solutions of the variational inequality for a monotone, Lipschitz-continuous
mapping in a Hilbert space.
Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone, and
k-Lipschitz-continuous mapping of C into H and B be an a-inverse-strongly monotone
mapping of C into H. Let S1, S2,... be a family of infinitely nonexpansive mappings of C
into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ GEP(F, B) = ∅. Assume that for all i Ỵ
i=1
{1, 2,...} and for any bounded subset K of C, thenthere holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )

Let {xn}, {un} and {yn} be the sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪
⎪

⎪
1
⎨
y − un , un − xn ≥ 0,
F(un , y) + Bxn , y − un +
rn
⎪
⎪ yn = PC (un − λn Aun ),
⎪
⎪
⎩x
n+1 = βn xn + (1 − βn )Sn PC (un − λn Ayn )

∀y ∈ C,

(3:12)

for every n = 1, 2,.... If {ln} ⊂ [a, b] for some a, b ∈ (0, 1 ), {bn} ⊂ [δ, ε] for some δ, ε
k
Ỵ (0, 1) and {rn} ⊂ [d, e] for some d, e Ỵ (0, 2a). Then, {xn}, {un} and {yn} converge
weakly to w Ỵ Ω, where w = limn®∞ PΩxn.

4. Applications
By Theorems 3.1 and 3.2, we can obtain many new and interesting convergence theorems in a real Hilbert space. We give some examples as follows:
Let A = 0, by Theorems 3.1 and 3.2, respectively, we obtain the following results.
Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let B be an a-inversestrongly monotone mapping of C into H. Let S 1 , S 2 ,... be a family of infinitely


Peng Fixed Point Theory and Applications 2011, 2011:12

/>
Page 13 of 19

nonexpansive mappings of C into itself such that

= ∩∞ Fix(Si ) ∩ GEP(F, B) = ∅.
i=1

Assume that for all i Î {1, 2,...} and for any bounded subset K of C, thenthere holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )

Let {xn}, {un} {yn}, and {zn} be the sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪
⎪
⎪
⎪ F(u , y) + Bx , y − u + 1 y − u , u − x ≥ 0, ∀y ∈ C,
⎪
⎪
n
n
n
n n
n
⎪

⎨
rn
zn = (1 − αn − βn )xn + αn un + βn Sn un ,
⎪ C = {z ∈ C : ||z − z||2 ≤ ||x − z||2 },
⎪ n
⎪
n
n
⎪
⎪ Q = {z ∈ C : x − z, x − x ≥ 0},
⎪ n
⎪
n
n
⎪
⎩
xn+1 = PCn ∩Qn x
for every n = 1, 2,.... where {rn} ⊂ [d, e] for some d, e Ỵ (0, 2a), and {an}, {bn} are
sequences in [0, 1] satisfying the conditions:
(i) an + bn ≤ 1 for all n Ỵ N;
(ii) lim αn = 0;
n→∞

(iii) lim inf βn > 0 for all n Ỵ N;
n→∞
Then, {xn}, {un}, and {zn} converge strongly to w = P∑(x).
Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let B be an a-inversestrongly monotone mapping of C into H. Let S1, S2,... be a family of infinitely nonexpansive mappings of C into itself such that

= ∩∞ Fix(Si ) ∩ GEP(F, B) = ∅. Assume

i=1

that for all i Ỵ {1, 2,...} and for any bounded subset K of C, thenthere holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )

Let {xn} and {un} be sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪
⎨
1
y − un , un − xn ≥ 0,
F(un , y) + Bxn , y − un +
⎪
rn
⎪
⎩x
= β x + (1 − β )S u
n+1

n n

n

∀y ∈ C,

n n

for every n = 1, 2,.... If {bn} ⊂ [δ, ε] for some δ, ε Ỵ (0, 1) and {rn} ⊂ [d, e] for some
d, e Ỵ (0, 2a). Then, {xn} and {un} converge weakly to w Ỵ ∑, where w = limn®∞ P∑xn.
Theorem 4.3. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let A be a monotone and kLipschitz-continuous mapping of C into H and B be an a-inverse-strongly monotone
mapping of C into H. Let S1, S2,... be a family of infinitely nonexpansive mappings of C
into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ GEP(F, B) = ∅. Assume that for all i Ỵ
i=1
{1, 2,...} and for any bounded subset K of C, thenthere holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 14 of 19

Let {xn}, {un}, {yn}, and {zn} be sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪
⎪ F(u , y) + Bx , y − u + 1 y − u , u − x ≥ 0,
⎪
n
n
n

n n
n
⎪
rn
⎪
⎪
⎪ yn = PC (un − λn Aun ),
⎨
z = (1 − βn )xn + βn Sn PC (un − λn Ayn ),
⎪ n
⎪ Cn = {z ∈ C : ||zn − z||2 ≤ ||xn − z||2 ,
⎪
⎪
⎪
⎪ Q = {z ∈ C : x − z, x − x ≥ 0},
⎪ n
n
n
⎪
⎩
xn+1 = PCn ∩Qn x

∀y ∈ C,

1
), {rn} ⊂ [d, e] for some
4k
d, e Ỵ (0, 2a), and {bn} is a sequence in [0, 1] satisfying lim inf βn > 0. Then, {x n},
n→∞

for every n = 1, 2,... where {ln} ⊂ [a, b] for some a, b ∈ (0,

{un}, {yn}, and {zn} converge strongly to w = PΩ(x).
Proof. Putting gn = 1 and an = 0, by Theorem 3.1, we obtain the desired result.
Let B = 0, by Theorems 3.1, 3.2, and 4.3, we obtain the following results.
Theorem 4.4. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let A be a monotone and kLipschitz-continuous mapping of C into H. Let S1, S2,... be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ EP(F) = ∅.
i=1
Assume that for all i Ỵ {1, 2,...}, and for any bounded subset K of C, there holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )

Let {xn}, {un}, {yn}, and {zn} be the sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪
⎪
⎪ F(un , y) + r1 y − un , un − xn ≥ 0, ∀y ∈ C,
⎪
⎪
n
⎪
⎪ yn = (1 − γn )un + γn PC (un − λn Aun ),
⎨
z = (1 − αn − βn )xn + αn yn + βn Sn PC (un − λn Ayn ),
⎪ n
⎪ Cn = {z ∈ C : ||zn − z||2 ≤ ||xn − z||2 + (3 − 3γn + αn )b2 ||Aun ||2 },

⎪
⎪
⎪
⎪ Q = {z ∈ C : x − z, x − x ≥ 0},
⎪ n
n
n
⎪
⎩
xn+1 = PCn ∩Qn x
1
), {rn} ⊂ [d, +∞) for
4k
some d >0, and {an}, {bn}, {gn} are three sequences in [0, 1] satisfying the following
conditions:

for every n = 1, 2,.... where {ln} ⊂ [a, b] for some a, b ∈ (0,

(i) an + bn ≤ 1 for all n Î N;
lim
(ii) n→∞ αn = 0;
(iii) lim inf βn > 0;
n→∞
3
lim
(iv) n→∞ γn = 1 and γn > for all n Ỵ N;
4

Then, {xn}, {un}, {yn} and {zn} converge strongly to w = PΛ(x).
Theorem 4.5. Let C be a nonempty closed convex subset of a real Hilbert space H.

Let F be a bifunction from C×C to R satisfying (A1)-(A4). Let A be a monotone and kLipschitz-continuous mapping of C into H. Let S1, S2,... be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ EP(F) = ∅.
i=1
Assume that for all i Ỵ {1, 2,...} and for any bounded subset K of C, thenthere holds


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
lim sup ||Sn x − Si (Sn x)|| = 0.

Page 15 of 19

( )

n→∞ x∈K

Let {xn}, {un}, and {yn} be the sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪
⎪
⎪
1
⎨
y − un , un − xn ≥ 0, ∀y ∈ C,
F(un , y) +
rn
⎪
⎪ yn = PC (un − λn Aun ),
⎪
⎪

⎩x
= β x + (1 − β )S P (u − λ Ay )
n+1

n n

n

n C

n

n

n

for every n = 1, 2,.... If {ln} ⊂ [a, b] for some a, b ∈ (0, 1 ),{bn} ⊂ [δ, ε], for some δ, ε
k
Ỵ (0, 1) and {rn} ⊂ [d, +∞] for some d > 0, then {xn}, {un} and {yn} converge weakly to
w Ỵ Λ, where w = limn®∞ PΛxn.
Theorem 4.6. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone and
k-Lipschitz-continuous mapping of C into H. Let S1, S2,... be a family of infinitely nonexpansive mappings of C into itself such that = ∩∞ Fix(Si ) ∩ VI(C, A) ∩ EP(F) = ∅.
i=1
Assume that for all i Ỵ {1, 2,...} and for any bounded subset K of C, thenthere holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )

Let {xn}, {un} {yn}, and {zn} be the sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪
⎪
⎪
⎪ F(u , y) + 1 y − u , u − x ≥ 0, ∀y ∈ C,
⎪
⎪
n
n n
n
⎪
rn
⎪
⎪
⎨ y = P (u − λ Au ),
n
C n
n
n
⎪ zn = (1 − βn )xn + βn Sn PC (un − λn Ayn ),
⎪
⎪ C = {z ∈ C : ||z − z||2 ≤ ||x − z||2 ,
⎪ n
⎪
n
n
⎪

⎪
⎪ Qn = {z ∈ C : xn − z, x − xn ≥ 0},
⎪
⎪
⎩
xn+1 = PCn ∩Qn x
1
), {rn} ⊂ [d, +∞) and
4k
for some d >0, and {bn} is a sequence in [0, 1] satisfying lim inf βn > 0. Then, {xn},
n→∞

for every n = 1, 2,.... where {ln} ⊂ [a, b] for some a, b ∈ (0,

{un}, {yn}, and {zn} converge strongly to w = PΛ(x).
Let B = 0 and F(x, y) = 0 for x, y Ỵ C, by Theorems 3.1 and 4.3, we obtain the following results.
Theorem 4.7. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let A be a monotone and k-Lipschitz-continuous mapping of C into H. Let S1, S2,... be
a family of infinitely nonexpansive mappings of C into itself such that
= ∩∞ Fix(Si ) ∩ VI(C, A) = ∅. Assume that for all i Ỵ {1, 2,...} and for any bounded
i=1
subset K of C, thenthere holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )

Let {xn}, {yn}, and {zn} be the sequences generated by
⎧

⎪ x1 = x ∈ C,
⎪
⎪
⎪ yn = (1 − γn )xn + γn PC (xn − λn Axn ),
⎪
⎪
⎨
zn = (1 − αn − βn )xn + αn yn + βn Sn PC (xn − λn Ayn ),
2
2
2
2
⎪
⎪ Cn = {z ∈ C : ||zn − z|| ≤ ||xn − z|| + (3 − 3γn + αn )b ||Axn || },
⎪
⎪ Qn = {z ∈ C : xn − z, x − xn ≥ 0},
⎪
⎪
⎩
xn+1 = PCn ∩Qn x


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 16 of 19

for every n = 1, 2,.... where {ln} ⊂ [a, b] for some a, b ∈ (0,

1
), and {an}, {bn}, {gn}are

4k

three sequences in [0, 1] satisfying the following conditions:
(i) an + bn ≤ 1 for all n Ỵ N;
lim
(ii) n→∞ αn = 0;
(iii) lim inf βn > 0;
n→∞
3
lim
(iv) n→∞ γn = 1 and γn > for all n Ỵ N;
4

Then, {xn}, {yn}, and {zn} converge strongly to w = PΓ(x).
Theorem 4.8. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let A be a monotone and k-Lipschitz-continuous mapping of C into H. Let S1, S2,... be
a family of infinitely nonexpansive mappings of C into itself such that
= ∩∞ Fix(Si ) ∩ VI(C, A) = ∅. Assume that for all i Ỵ {1, 2,...} and for any bounded
i=1
subset K of C, thenthere holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )

Let {xn}, {yn}, and {zn} be the sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪

⎪
⎪ yn = PC (xn − λn Axn ),
⎪
⎪
⎨
zn = (1 − βn )xn + βn Sn PC (xn − λn Ayn ),
Cn = {z ∈ C : ||zn − z||2 ≤ ||xn − z||2 ,
⎪
⎪
⎪
⎪ Qn = {z ∈ C : xn − z, x − xn ≥ 0},
⎪
⎪
⎩
xn+1 = PCn ∩Qn x
1
), and {b n } is a
4k
sequence in [0, 1] satisfying lim inf βn > 0. Then, {xn}, {yn}, and {zn} converge strongly
n→∞

for every n = 1, 2,.... where {l n } ⊂ [a, b] for some a, b ∈ (0,

to w = PΓ(x).
Let F(x, y) = 0 for x, y Ỵ C, then by Theorem 3.2 and the proof of Theorem 4.7 in
[3], we obtain the following result.
Theorem 4.9. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let A be a monotone and k-Lipschitz-continuous mapping of C into H and B be an ainverse-strongly monotone mapping of C into H. Let S1, S2,... be a family of infinitely
nonexpansive
mappings

of
C
into
itself
such
that
∞
= ∩i=1 Fix(Si ) ∩ VI(C, A) ∩ VI(C, B) = ∅. Assume that for all i Ỵ {1, 2,...} and for any
bounded subset K of C, thenthere holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

( )

Let {xn}, {un}, and {yn} be the sequences generated by
⎧
⎪ x1 = x ∈ C,
⎪
⎨
un = PC (xn − rn Bxn ),
⎪ yn = PC (un − λn Aun ),
⎪
⎩
xn+1 = αn xn + (1 − αn )Sn PC (un − λn Ayn )


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
Page 17 of 19

for every n = 1, 2,.... if {ln} ⊂ [a, b] for some a, b ∈ (0, 1 ), {bn} ⊂ [δ, ε] for some δ, ε
k
Ỵ (0, 1) and {rn} ⊂ [d, e] for some d, e Ỵ (0, 2a). Then, {xn} and {un} converge weakly
to w Ỵ Ξ, where w = limn®∞ PΞxn.
Remark 4.1.
(i) For all n ≥ 1, let Sn = S be a nonexpansive mapping, by Theorems 3.2, 4.2, 4.7,
4.8, and 4.9 we recover Theorem 3.1 in [5], Theorem 3.1 in [1], Theorem 5 in [26],
Theorem 3.1 in [23], and Theorem 4.7 in [3]. In addition, let A = 0, by Theorems
4.6 and 4.5, respectively, we recover Theorems 3.1 and 4.1 in [11].
(ii) For all n ≥ 1, let Sn = S be a nonexpansive mapping, by Theorems 3.1, 4.3, and
4.4, respectively, we recover Theorems 4.3, 4.4, and 4.7 in [4] with some modified
conditions on F.
(iii) Theorems 3.1, 3.2, 4.3-4.7 also improve the main results in [10,12,13] because
the inverse strongly monotonicity of A has been replaced by the monotonicity and
Lipschitz continuity of A.
The following result illustrates that there are the nonexpansive mappings S1, S2 ,...
satisfying the condition (*).
Lemma 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let
T be a nonexpansive mapping of C into itself such that Fix(T) ≠ ∅. If we define
n−1 j
1
Sn (x) =
T x for n Î {1, 2,...}, and x Î C, then the following results hold:
j=0
n
(a) For any bounded subset K of C, there holds
lim sup ||Sn x − T(Sn x)|| = 0.

n→∞ x∈K

(b) ∩∞ Fix(Si ) = Fix(T).
i=1
(c) for all i Ỵ {1, 2,...} and for any bounded subset K of C, there holds
lim sup ||Sn x − Si (Sn x)|| = 0.

n→∞ x∈K

Proof.
(a) It is due to Bruck [27,28] (please also see Lemma 3.1 in [22]).
(b) It follows from (a) that ∩∞ Fix(Si ) ⊆ Fix(T).
i=1
Moreover, it is obvious that ∩∞ Fix(Si ) ⊇ Fix(T). Hence, ∩∞ Fix(Si ) = Fix(T).
i=1
i=1
(c) It can be proved by mathematical induction. In fact, it is clear that this conclusion holds for i = 1. Assume that the conclusion holds for i = m, that is, for any
bounded subset K of C, there holds
lim sup ||Sn x − Sm (Sn x)|| = 0.

(4:1)

n→∞ x∈K

We now prove that the conclusion also holds for i = m + 1. In fact, we observe that
lim sup ||Sn x − Sm+1 (Sn x)|| ≤ lim sup ||Sn x − Sm (Sn x)|| + lim sup ||Sm (Sn x) − Sm+1 (Sn x)||
n→∞ x∈K
n→∞ x∈K
⎡
⎤
m−1

(4:2)
1
j
⎣ 1 ||T m (Sn x)|| +
⎦.
≤ lim sup ||Sn x − Sm (Sn x)|| + lim sup
||T (Sn x)||
n→∞ x∈K
n→∞ x∈K
m+1
m(m + 1)
n→∞ x∈K

j=0


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
It is easy to verify that S1, S2,... are nonexpansive mappings. It follows from (4.1) and
(4.2) that for any bounded subset K of C, there holds
lim sup ||Sn x − Sm+1 (Sn x)|| = 0.

n→∞ x∈K

From Lemma 4.1, we know that by Theorems 3.1 and 3.2, respectively, we can obtain
the following results.
Theorem 4.10. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone and
k-Lipschitz-continuous mapping of C into H and B be an a-inverse-strongly monotone
mapping of C into H. Let T be a nonexpansive mapping of C into itself such that Θ =

1
Fix(T)∩VI(C, A)∩GEP(F, B) ≠ ∅. Let {ln} ⊂ [a, b] for some a, b ∈ (0, ), {rn} ⊂ [d, e]
4k
and for some d, e Ỵ (0, 2a), and {an}, {bn}, and {gn} be three sequences in [0, 1] satisfying the following conditions:
(i) an + bn ≤ 1 for all n Ỵ N;
(ii) lim αn = 0;
n→∞

(iii) lim inf βn > 0;
n→∞
n−1 j
1
3
lim
(iv) n→∞ γn = 1 and γn > for all n Ỵ N; If we define Sn (x) =
T x for n Ỵ
j=0
n
4
{1, 2,...}, and x Ỵ C, then the sequences {xn}, {un}, {yn}, and {zn} generated by algorithm (3.1) converge strongly to w = PΘ(x).

Theorem 4.11. Let C be a nonempty closed convex subset of a real Hilbert space H.
Let F be a bifunction from C × C to R satisfying (A1)-(A4). Let A be a monotone and
k-Lipschitz-continuous mapping of C into H and B be an a-inverse-strongly monotone
mapping of C into H, and T be a nonexpansive mapping of C into itself such that Θ =
Fix(T)∩VI(C, A)∩GEP(F, B) ≠ ∅. Assume that {ln} ⊂ [a, b] for some a, b ∈ (0, 1 ) {bn}
k
⊂ [δ, ε] for some δ, ε Ỵ (0, 1), and {r n } ⊂ [d, e] some d, e Ỵ (0, 2a). If we define
n−1 j
1

Sn (x) =
T x for n Î {1, 2,...} and x Î C, then the sequences {xn}, {un}, and {yn}
j=0
n
generated by algorithm (3.12) converge weakly to w ẻ , where w = limnđ Pxn.

5. Competing interests
The authors declare that they have no competing interests.
Acknowledgements
This research was supported by the National Natural Science Foundation of China, the Natural Science Foundation of
Chongqing (Grant No. CSTC, 2009BB8240), and the Special Fund of Chongqing Key Laboratory (CSTC). The author is
grateful to the referees for their detailed comments and helpful suggestions, which have improved the presentation
of this article.
Received: 30 November 2010 Accepted: 29 June 2011 Published: 29 June 2011
References
1. Moudafi, A: Weak convergence theorems for nonexpansive mappings and equilibrium Problems. J Nonlinear Convex
Anal. 9, 37–43 (2008)
2. Moudafi, A, Thera, M: Proximal and dynamical approaches to equilibrium problems. Lecture Notes in Econom and Math
System, Springer-Verlag, Berlin. 477, 187–201 (1999)
3. Peng, JW, Yao, JC: Weak convergence of an iterative scheme for generalized equilibrium problems. Bull Aust Math Soc.
79, 437–453 (2009). doi:10.1017/S0004972708001378

Page 18 of 19


Peng Fixed Point Theory and Applications 2011, 2011:12
/>
4.
5.
6.

7.
8.
9.
10.
11.
12.
13.

14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.

27.
28.

Peng, JW, Yao, JC: A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point
problems and variational inequality problems. Taiwan J Math. 12(6):1401–1432 (2008)
Takahashi, S, Takahashi, W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive
mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008). doi:10.1016/j.na.2008.02.042
Flam, SD, Antipin, AS: Equilibrium programming using proximal-like algorithms. Math Program. 78, 29–41 (1997)

Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math Stud. 63, 123–145
(1994)
Combettes, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal. 6, 117–136 (2005)
Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and fixed point problems in
Hilbert spaces. J Math Anal Appl. 331, 506–515 (2006)
Su, Y, Shang, M, Qin, X: An iterative method of solutions for equilibrium and optimization problems. Nonlinear Anal. 69,
2709–2719 (2008). doi:10.1016/j.na.2007.08.045
Tada, A, Takahashi, W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium
problem. J Optim Theory Appl. 133, 359–370 (2007). doi:10.1007/s10957-007-9187-z
Plubtieng, S, Punpaeng, R: A new iterative method for equilibrium problems and fixed point problems of nonexpansive
mappings and monotone mappings. Appl Math Comput. 197, 548–558 (2008). doi:10.1016/j.amc.2007.07.075
Chang, SS, Joseph Lee, HW, Chan, CK: A new method for solving equilibrium problem, fixed point problem and
variational inequality problem with application to optimization. Nonlinear Anal. 70, 3307–3319 (2009). doi:10.1016/j.
na.2008.04.035
Yao, YH, Liou, YC, Yao, JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family
of nonexpansive mappings. Fixed Point Theory Appl 12 (2007). Article ID 64363
Ceng, LC, Yao, JC: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of
infinitely many nonexpansive mappings. Appl Math Comput. 198, 729–741 (2007)
Colao, V, Marino, G, Xu, H-K: An iterative method for finding common solutions of equilibrium and fixed point
problems. J Math Anal Appl. 344, 340–352 (2008). doi:10.1016/j.jmaa.2008.02.041
Iusem, AN, Sosa, W: Iterative algorithms for equilibrium problems. Optimization. 52, 301–316 (2003). doi:10.1080/
0233193031000120039
Nguyen, TTV, Strodiot, JJ, Nguyen, VH: A bundle method for solving equilibrium problems. Math Program Ser B. 116,
529–552 (2009). doi:10.1007/s10107-007-0112-x
Ceng, LC, AI-Homidan, S, Ansari, QH, Yao, JC: An iterative scheme for equilibrium problems and fixed point problems of
strict pseudo-contraction mappings. J Comput Appl Math. 223, 967–974 (2008)
Takahashi, W, Shimoji, K: Convergence theorems for nonexpansive mappings and feasibility problems. Math Comput
Model. 32, 1463–1471 (2000). doi:10.1016/S0895-7177(00)00218-1
Shimoji, K, Takahashi, W: Strong convergence to common fixed points of infinite nonexpansive mappings and
applications. Taiwanese J Math. 5(2):387–404 (2001)

OHaraa, JG, Pillay, P, Xu, HK: Iterative approaches to finding nearest common fixed points of nonexpansive mappings in
Hilbert spaces. Nonlinear Anal. 54, 1417–1426 (2003). doi:10.1016/S0362-546X(03)00193-7
Nadezhkina, N, Takahashi, W: Strong convergence theorem by a hybrid method for non-expansive mappings and
Lipschitz-continuous monotone mappings. SIAM J Optim. 16(4):1230–1241 (2006). doi:10.1137/050624315
Opial, Z: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull Am Math
Soc. 73, 561–597 (1967)
Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc. 149, 75–88 (1970).
doi:10.1090/S0002-9947-1970-0282272-5
Ceng, LC, Hadjisavvas, N, Wong, NC: Strong convergence theorem by a hybrid extragradient-like approximation method
for variational inequalities and fixed point problems. J Global Optim. 46, 635–646 (2010) (2010). doi:10.1007/s10898-0099454-7
Bruck, RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J Math. 32,
107–116 (1979). doi:10.1007/BF02764907
Bruck, RE: On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach
spaces. Israel J Math. 38, 304–314 (1981). doi:10.1007/BF02762776

doi:10.1186/1687-1812-2011-12
Cite this article as: Peng: Some extragradient methods for common solutions of generalized equilibrium
problems and fixed points of nonexpansive mappings. Fixed Point Theory and Applications 2011 2011:12.

Submit your manuscript to a
journal and benefit from:
7 Convenient online submission
7 Rigorous peer review
7 Immediate publication on acceptance
7 Open access: articles freely available online
7 High visibility within the field
7 Retaining the copyright to your article

Submit your next manuscript at 7 springeropen.com

Page 19 of 19