Báo cáo hóa học: " A new modified block iterative algorithm for uniformly quasi-j-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems" ppt

Saewan and Kumam Fixed Point Theory and Applications 2011, 2011:35
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A new modified block iterative algorithm for
uniformly quasi-j-asymptotically nonexpansive
mappings and a system of generalized mixed
equilibrium problems
Siwaporn Saewan and Poom Kumam*
* Correspondence: poom.

Department of Mathematics,
Faculty of Science, King Mongkut’s
University of Technology Thonburi
(KMUTT), Bangmod, Bangkok
10140, Thailand

Abstract
In this paper, we introduce a new modified block iterative algorithm for finding a
common element of the set of common fixed points of an infinite family of closed
and uniformly quasi-j-asymptotically nonexpansive mappings, the set of the
variational inequality for an a-inverse-strongly monotone operator, and the set of
solutions of a system of generalized mixed equilibrium problems. We obtain a strong
convergence theorem for the sequences generated by this process in a 2-uniformly
convex and uniformly smooth Banach space. Our results extend and improve ones
from several earlier works.
2000 MSC: 47H05; 47H09; 47H10.
Keywords: modified block iterative algorithm, inverse-strongly monotone operator,
variational inequality, a system of generalized mixed equilibrium problem, uniformly

quasi-j-asymptotically nonexpansive mapping

1 Introduction
Let C be a nonempty closed convex subset of a real Banach space E with ||·|| and let
E* be the dual space of E. Let {fi}iẻ : C ì C đ be a bifunction, {i}iẻ : C đ be a
real-valued function, and {Bi}iẻ : C đ E* be a monotone mapping, where Γ is an arbitrary index set. The system of generalized mixed equilibrium problems is to find x Ỵ C
such that
fi (x, y) + Bi x, y − x + ϕi (y) − ϕi (x) ≥ 0,

i ∈ , ∀y ∈ C.

(1:1)

If Γ is a singleton, then problem (1.1) reduces to the generalized mixed equilibrium
problem, which is to find x Ỵ C such that
f (x, y) + Bx, y − x + ϕ(y) − ϕ(x) ≥ 0,

∀y ∈ C.

(1:2)

The set of solutions to (1.2) is denoted by GMEP(f, B,  ), i.e.,
GMEP(f , B, ϕ) = {x ∈ C : f (x, y) + Bx, y − x + ϕ(y) − ϕ(x) ≥ 0, ∀y ∈ C}.

(1:3)

© 2011 Saewan and Kumam; 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.

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If B ≡ 0, the problem (1.2) reduces into the mixed equilibrium problem for f, denoted
by MEP (f, ), which is to find x Ỵ C such that
f (x, y) + ϕ(y) − ϕ(x) ≥ 0,

∀y ∈ C.

(1:4)

If f ≡ 0, the problem (1.2) reduces into the mixed variational inequality of Browder
type, denoted by VI(C, B, ), which is to find x Ỵ C such that
Bx, y − x + ϕ(y) − ϕ(x) ≥ 0,

∀y ∈ C.

(1:5)

If B ≡ 0 and  ≡ 0 the problem (1.2) reduces into the equilibrium problem for f,
denoted by EP(f ), which is to find x Ỵ C such that
f (x, y) ≥ 0,

∀y ∈ C.

(1:6)

If f ≡ 0, the problem (1.4) reduces into the minimize problem, denoted by Argmin(),

which is to find x Ỵ C such that
ϕ(y) − ϕ(x) ≥ 0,

∀y ∈ C.

(1:7)

The above formulation (1.5) was shown in [1] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems,
optimization problems, vector equilibrium problems, Nash equilibria in noncooperative
games. In addition, there are several other problems, for example, the complementarity
problem, fixed point problem and optimization problem, which can also be written in
the form of an EP(f). In other words, the EP(f) is an unifying model for several
problems arising in physics, engineering, science, optimization, economics, etc. In the
last two decades, many papers have appeared in the literature on the existence of solutions of EP(f); see, for example [1,2] and references therein. Some solution methods
have been proposed to solve the EP(f); see, for example, [1-15] and references therein.
The normalized duality mapping J : E ® 2E* is defined by
J(x) = {x∗ ∈ E∗ : x, x∗ =

x 2 , x∗ =

x }

for all x Ỵ E. If E is a Hilbert space, then J = I, where I is the identity mapping.
Consider the functional defined by
φ(x, y) =

x

2

− 2 x, Jy +

∀x, y ∈ E.

y 2,

(1:8)

As well known that if C is a nonempty closed convex subset of a Hilbert space H
and PC : H ® C is the metric projection of H onto C, then PC is nonexpansive. This
fact actually characterizes Hilbert spaces and consequently, it is not available in more
general Banach spaces. It is obvious from the definition of function j that
( x

−

y )2 ≤ φ(x, y) ≤ ( x

+

y )2 ,

∀x, y ∈ E.

(1:9)

If E is a Hilbert space, then j(x, y) = ||x - y||2, for all x, y Ỵ E. On the other hand,
the generalized projection [16] ΠC : E ® C is a map that assigns to an arbitrary point x
¯
Ỵ E the minimum point of the functional j(x, y), that is, C x = x, where x is the

¯
solution to the minimization problem
φ(¯ , x) = inf φ(y, x),
x
y∈C

(1:10)


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existence and uniqueness of the operator ΠC follows from the properties of the functional j(x, y) and strict monotonicity of the mapping J (see, for example, [16-20]).
Remark 1.1. If E is a reflexive, strictly convex, and smooth Banach space, then for x,
y Ỵ E, j(x, y) = 0 if and only if x = y. It is sufficient to show that if j(x, y) = 0 then x
= y. From (1.8), we have ||x|| = ||y||. This implies that 〈x, Jy〉 = ||x||2 = ||Jy||2. From
the definition of J, one has Jx = Jy. Therefore, we have x = y; see [18,20] for more
details.
Let C be a closed convex subset of E, a mapping T : C ® C is said to be L-Lipschitz
continuous if ||Tx - Ty|| ≤ L||x - y||, ∀x, y Ỵ C and a mapping T is said to be nonexpansive if ||Tx - Ty|| ≤ ||x - y||, ∀x, y Ỵ C. A point x Ỵ C is a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T; that is, F(T) = {x Ỵ C : Tx =
x}. Recall that a point p in C is said to be an asymptotic fixed point of T [21] if C contains a sequence {xn} which converges weakly to p such that limn®∞ ||xn - Txn|| = 0.
The set of asymptotic fixed points of T will be denoted by F(T).
A mapping T from C into itself is said to be relatively nonexpansive [22-24] if
F(T) = F(T) and j(p, Tx) ≤ j(p, x) for all x Ỵ C and p Ỵ F(T). T is said to be relatively
quasi-nonexpansive if F(T) ≠ ∅ and j(p, Tx) ≤ j(p, x) for all x Ỵ C and p Ỵ F(T). T is
said to be j-nonexpansive, if j(Tx, Ty) ≤ j(x, y) for x, y Ỵ C. T is said to be quasi-jasymptotically nonexpansive if F(T) ≠ ∅ and there exists a real sequence {kn} ⊂ [1, ∞)
with kn ® 1 such that j(p, Tnx) ≤ knj(p, x) for all n ≥ 1 x Ỵ C and p Ỵ F(T). The
asymptotic behavior of a relatively nonexpansive mapping was studied in [25-27].
We note that the class of relatively quasi-nonexpansive mappings is more general

than the class of relatively nonexpansive mappings [25-29] which requires the strong

restriction: F(T) = F(T). A mapping T is said to be closed if for any sequence {xn} ⊂ C
with xn ® x and Txn ® y, then Tx = y. It is easy to know that each relatively nonexpansive mapping is closed.
Definition 1.2. (Chang et al. [30]) (1) Let {Ti }∞ : C → C be a sequence of mapping.
i=1
{Ti }∞ is said to be a family of uniformly quasi-j-asymptotically nonexpansive mapi=1
pings, if F := ∩∞ F(Ti ) = ∅, and there exists a sequence {kn} ⊂ [1, ∞) with kn ® 1 such
i=1
that for each i ≥ 1
φ(p, Tin x) ≤ kn φ(p, x),

∀p ∈ F , x ∈ C, ∀n ≥ 1.

(1:11)

(2) A mapping T : C ® C is said to be uniformly L-Lipschitz continuous, if there
exists a constant L >0 such that
Tnx − Tny

≤L

x−y ,

∀x, y ∈ C.

(1:12)

Recall that let A : C ® E* be a mapping. Then A is called
(i) monotone if

Ax − Ay, x − y ≥ 0,

∀x, y ∈ C,

(ii) a-inverse-strongly monotone if there exists a constant a >0 such that
Ax − Ay, x − y ≥ α

Ax − Ay 2 ,

∀x, y ∈ C.

Remark 1.3. It is easy to see that an a-inverse-strongly monotone is monotone and
1
α -Lipschitz

continuous.


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In 2004, Matsushita and Takahashi [31] introduced the following iteration: a
sequence {xn} defined by
xn+1 =

CJ

−1

(αn Jxn + (1 − αn )JTxn ),

(1:13)

where the initial guess element x0 Ỵ C is arbitrary, {an} is a real sequence in [0, 1], T
is a relatively nonexpansive mapping and ΠC denotes the generalized projection from E
onto a closed convex subset C of E. They proved that the sequence {xn} converges
weakly to a fixed point of T.
In 2005, Matsushita and Takahashi [28] proposed the following hybrid iteration
method (it is also called the CQ method) with generalized projection for relatively
nonexpansive mapping T in a Banach space E:
⎧
⎪ x0 ∈ C chosen arbitrarily,
⎪
⎪
⎪ yn = J−1 (αn Jxn + (1 − αn )JTxn ),
⎨
Cn = {z ∈ C : φ(z, yn ) ≤ φ(z, xn )},
(1:14)
⎪
⎪ Qn = {z ∈ C : xn − z, Jx0 − Jxn ≥ 0},
⎪
⎪
⎩
xn+1 = Cn ∩Qn x0 .
They proved that {xn} converges strongly to ΠF(T)x0, where ΠF(T) is the generalized
projection from C onto F(T).
In 2008, Iiduka and Takahashi [32] introduced the following iterative scheme for
finding a solution of the variational inequality problem for an inverse-strongly monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E : x1 =
x Ỵ C and

xn+1 =

CJ

−1

(Jxn − λn Axn ),

(1:15)

for every n = 1, 2, 3, ..., where ΠC is the generalized metric projection from E onto C,
J is the duality mapping from E into E* and {ln} is a sequence of positive real numbers.
They proved that the sequence {xn} generated by (1.15) converges weakly to some element of VI(A, C). Takahashi and Zembayashi [33,34] studied the problem of finding a
common element of the set of fixed points of a nonexpansive mapping and the set of
solutions of an equilibrium problem in the framework of Banach spaces.
In 2009, Wattanawitoon and Kumam [14] using the idea of Takahashi and Zembayashi [33] extended the notion from relatively nonexpansive mappings or j-nonexpansive
mappings to two relatively quasi-nonexpansive mappings and also proved some strong
convergence theorems to approximate a common fixed point of relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework
of Banach spaces. Cholamjiak [35] studied the following iterative algorithm:
⎧
⎪ zn = C J−1 (Jxn − λn Axn ),
⎪
⎪
⎪ yn = J−1 (αn Jxn + βn JTxn + γn JSzn ),
⎪
⎪
⎨
1
un ∈ C such that f (un , y) +
y − un , Jun − Jyn ≥ 0, ∀y ∈ C,

(1:16)
⎪
rn
⎪
⎪C
⎪
⎪ n+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn )},
⎪
⎩x
=
x ,
n+1

Cn+1 0

where J is the duality mapping on E. Assume that {an}, {bn} and {gn} are sequences in
[0, 1]. Then, he proved that {xn} converges strongly to q = ΠFx0, where F := F (T ) ∩ F
(S) ∩ EP(f ) ∩ VI(A, C).


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In 2010, Saewan et al. [29] introduced a new hybrid projection iterative scheme
which is difference from the algorithm (1.16) of Cholamjiak in [[35], Theorem 3.1] for
two relatively quasi-nonexpansive mappings in a Banach space. Motivated by the
results of Takahashi and Zembayashi [34], Cholumjiak and Suantai [36] proved the following strong convergence theorem by the hybrid iterative scheme for approximation
of common fixed point of countable families of relatively quasi-nonexpansive mappings
in a uniformly convex and uniformly smooth Banach space: x0 Ỵ E, x1 = C1 x0, C1 = C

⎧
−1
⎪ yn,i = J (αn Jxn + (1 − αn )JTxn ),
⎪
⎨
fm
fm−1
f1
un,i = Trm,n Trm−1,n · · · Tr1,n yn,i ,
(1:17)
⎪ Cn+1 = {z ∈ Cn : supi>1 φ(z, Jun,i ) ≤ φ(w, Jxn )},
⎪
⎩
xn+1 = Cn+1 x0 , n ≥ 1.
Then, they proved that under certain appropriate conditions imposed on {an}, and
{rn, i}, the sequence {xn} converges strongly to ΠF(T)∩EP(f)x0.
We note that the block iterative method is a method which often used by many
authors to solve the convex feasibility problem (see, [37,38], etc.). In 2008, Plubtieng
and Ungchittrakool [39] established strong convergence theorems of block iterative
methods for a finite family of relatively nonexpansive mappings in a Banach space
by using the hybrid method in mathematical programming. Chang et al. [30] proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-j-asymptotically
nonexpansive mapping, and they obtained the strong convergence theorems in a
Banach space.
In 2010, Saewan and Kumam [40] obtained the following result for the set of solutions of the generalized equilibrium problems and the set of common fixed points of
an infinite family of closed and uniformly quasi-j-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with Kadec-Klee
property.
Theorem SK Let C be a nonempty closed and convex subset of a uniformly smooth
and strictly convex Banach space E with the Kadec-Klee property. Let f be a bifunction
from C × C to ℝ satisfying (A1)-(A4). Let B be a continuous monotone mapping of C
into E*. Let {Si }∞ : C → C be an infinite family of closed uniformly Li-Lipschitz coni=1

tinuous and uniformly quasi-j-asymptotically nonexpansive mappings with a sequence
{kn} ⊂ [1, ∞), kn ® 1 such that F := ∩∞ F(Si ) ∩ GEP(f , B)is a nonempty and bounded
i=1
subset in C. For an initial point x 0 Ỵ E with x1 = C1 x0 and C1 = C, we define the
sequence {xn} as follows:
⎧
⎪ yn = J−1 (βn Jxn + (1 − βn )Jzn ),
⎪
⎪
⎪ zn = J−1 (αn,0 Jxn + ∞ αn,i JSn xn ),
⎪
⎪
i=1
i
⎨
1
un ∈ C such that f (un , y) + Byn , y − un +
y − un , Jun − Jyn ≥ 0,
⎪
rn
⎪
⎪C
⎪
⎪ n+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn ) + θn },
⎪
⎩x
=
x , ∀n ≥ 0,
n+1

∀y ∈ C, (1:18)

Cn+1 0

where J is the duality mapping on E, θn = supqỴF (k n - 1)j(q, xn), {an, i}, {bn} are
sequences in [0, 1] and {r n } ⊂ [a, ∞) for some a >0. If ∞ αn,i = 1for all n ≥ 0
i=0
and lim infn ® ∞ an, 0an, i > 0 for all i ≥ 1, then {xn} converges strongly to p Ỵ F ,
where p = ΠFx0.


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Quite recently, Qin et al. [9] purposed the problem of approximating a common
fixed point of two asymptotically quasi-j-nonexpansive mappings based on hybrid
projection methods. Strong convergence theorems are established in a real Banach
space. Zegeye et al. [15] introduced an iterative process which converges strongly to a
common element of set of common fixed points of countably infinite family of closed
relatively quasi- nonexpansive mappings, the solution set of generalized
equilibrium problem and the solution set of the variational inequality problem for an
a-inverse-strongly monotone mapping in Banach spaces.
Motivated and inspired by the work of Chang et al. [30], Qin et al. [7], Takahashi
and Zembayashi [33], Wattanawitoon and Kumam [14], Zegeye [41] and Saewan and
Kumam [40], we introduce a new modified block hybrid projection algorithm for finding a common element of the set of the variational inequality for an a-inverse-strongly
monotone operator, the set of solutions of the system of generalized mixed equilibrium
problems and the set of common fixed points of an infinite family of closed and uniformly quasi-j-asymptotically nonexpansive mappings in the framework Banach
spaces. The results presented in this paper improve and generalize some well-known
results in the literature.

2 Preliminaries

x+y
A Banach space E is said to be strictly convex if 2 < 1 for all x, y Ỵ E with ||x|| =
||y|| = 1 and x ≠ y. Let U = {x Ỵ E : ||x|| = 1} be the unit sphere of E. Then a Banach
space E is said to be smooth if the limit
lim
t→0

−

x + ty

x

t

exists for each x, y Ỵ U. It is also said to be uniformly smooth if the limit is attained
uniformly for x, y Ỵ U. Let E be a Banach space. The modulus of convexity of E is the
function δ : [0, 2] ® [0, 1] defined by
δ(ε) = inf 1−

x+y
: x, y ∈ E, x = y = 1, x − y ≥ ε .
2

A Banach space E is uniformly convex if and only if δ(ε) >0 for all ε Î (0, 2]. Let p be
a fixed real number with p ≥ 2. A Banach space E is said to be p-uniformly convex if
there exists a constant c >0 such that δ(ε) ≥ cεp for all ε Ỵ [0, 2]; see [42,43] for more
details. Observe that every p-uniformly convex is uniformly convex. One should note
that no a Banach space is p-uniformly convex for 1 < p <2. It is well known that a
Hilbert space is 2-uniformly convex, uniformly smooth. It is also known that if E is

uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded
subset of E.
Remark 2.1. The following basic properties can be found in Cioranescu [18].
(i) If E is a uniformly smooth Banach space, then J is uniformly continuous on each
bounded subset of E.
(ii) If E is a smooth, strictly convex, and reflexive Banach space, then the normalized
duality mapping J : E ® 2E* is single-valued, one-to-one, and onto.
(iii) A Banach space E is uniformly smooth if and only if E* is uniformly convex.
(iv) Each uniformly convex Banach space E has the Kadec-Klee property, that is, for
any sequence {xn} ⊂ E, if xn ⇀ x Ỵ E and ||xn|| ® ||x||, then xn ® x.

Page 6 of 24


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We also need the following lemmas for the proof of our main results.
Lemma 2.2. (Beauzamy [44] and Xu [45]). If E be a 2-uniformly convex Banach
space, then for all x, y Ỵ E we have
x−y ≤

2
c2

Jx − Jy ,

where J is the normalized duality mapping of E and 0 < c ≤ 1.
The best constant

1
c

in lemma is called the p-uniformly convex constant of E.

Lemma 2.3. (Beauzamy [44] and Zalinescu [46]). If E be a p-uniformly convex
Banach space and let p be a given real number with p ≥ 2, then for all x, y Ỵ E, jx Ỵ Jp
(x) and jy Ỵ Jp(y)
x − y, jx − jy ≥

cp
2p−2 p

x − y p,

where Jp is the generalized duality mapping of E and 1is the p-uniformly convexity
c
constant of E.
Lemma 2.4. (Kamimura and Takahashi [19]). Let E be a uniformly convex and
smooth Banach space and let {xn} and {yn} be two sequences of E. If j(xn, yn) ® 0 and
either {xn} or {yn} is bounded, then ||xn -yn|| ® 0.
Lemma 2.5. (Alber [16]). Let C be a nonempty closed convex subset of a smooth
Banach space and x Ỵ E. Then x0 = ΠCx if and only if
x0 − y, Jx − Jx0 ≥ 0,

∀y ∈ C.

Lemma 2.6. (Alber [[16], Lemma 2.4]). Let E be a reflexive, strictly convex and
smooth Banach space, and let C be a nonempty closed convex subset of E and let x Ỵ

E. Then
φ(y,

C x)

+ φ(

C x, x)

≤ φ(y, x),

∀y ∈ C.

Let E be a reflexive, strictly convex, smooth Banach space and J is the duality mapping from E into E*. Then J-1 is also single value, one-to-one, surjective, and it is the
duality mapping from E* into E. We make use of the following mapping V studied in
Alber [16]:
V(x, x∗ ) =

x

2

− 2 x, x∗ +

x∗

2

,

(2:1)

for all x Ỵ E and x* Ỵ E*, that is, V (x, x*) = j(x, J-1(x*)).
Lemma 2.7. (Alber [16]). Let E be a reflexive, strictly convex smooth Banach space
and let V be as in (2.1). Then
V(x, x∗ ) + 2 J−1 (x∗ ) − x, y∗ ≤ V(x, x∗ + y∗ ),

for all x Ỵ E and x*, y* Ỵ E*.
A set valued mapping U : E ⇉ E* with graph G(U) = (x, x*) : x* Ỵ Ux}, domain D(U)
= {x Ỵ E : Ux ≠ ∅}, and range R(U) = ∪{Ux : x Ỵ D(U)}. U is said to be monotone if 〈x
- y, x* -y*〉 ≥ 0 whenever (x, x*) Ỵ G(U), (y, y*) Ỵ G(U). We denote a set valued operator U from E to E* by U ⊂ E × E*. A monotone U is said to be maximal if its graph is
not property contained in the graph of any other monotone operator. If U is maximal
monotone, then the solution set U-10 is closed and convex. Let E be a reflexive, strictly


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convex and smooth Banach space, and it is known that U is a maximal monotone if
and only if R(J + rU) = E* for all r >0. Define the resolvent of U by Jrx = xr. In other
words, Jr = (J + rU)-1 for all r >0. Jr is a single-valued mapping from E to D(U). Also,
U-1(0) = F(Jr) for all r >0, where F(Jr) is the set of all fixed points of Jr. Define, for r
>0, the Yosida approximation of U by Trx = (Jx - JJrx)/r for all x Ỵ C: We know that
Trx Ỵ U (Jrx) for all r >0 and x Ỵ E.
Let A be an inverse-strongly monotone mapping of C into E* which is said to be
hemicontinuous if for all x, y Ỵ C, and the mapping F of [0, 1] into E*, defined by F(t)
= A(tx + (1 - t)y), is continuous with respect to the weak* topology of E*. We define
by NC(v) the normal cone for C at a point v Î C, that is,
NC (v) = {x∗ ∈ E∗ : v − y, x∗ ≥ 0, ∀y ∈ C}.

(2:2)

Lemma 2.8. (Rockafellar [47]). Let C be a nonempty, closed convex subset of a
Banach space E, and A is a monotone, hemicontinuous operator of C into E*. Let U ⊂
E × E* be an operator defined as follows:
Uv =

Av + NC (v), v ∈ C;
∅
otherwise.

(2:3)

Then U is maximal monotone and U -10 = VI(A, C).
Lemma 2.9. (Chang et al. [30]). Let E be a uniformly convex Banach space, r >0 be a
positive number and B r (0) be a closed ball of E. Then, for any given sequence
{xi }∞ ⊂ Br (0)and for any given sequence {λi }∞ of positive number with ∞ λn = 1,
i=1
i=1
n=1
there exists a continuous, strictly increasing, and convex function g : [0, 2r) ® [0, ∞)
with g(0) = 0 such that, for any positive integer i, j with i < j,
2

∞

λn xn
n=1

∞

≤

λn

xn

2

− λi λj g( xi − xj ).

(2:4)

n=1

Lemma 2.10. (Chang et al. [30]). Let E be a real uniformly smooth and strictly convex Banach space, and C be a nonempty closed convex subset of E. Let T : C ® C be a
closed and quasi-j-asymptotically nonexpansive mapping with a sequence {kn} ⊂ [1, ∞),
kn ® 1. Then F (T ) is a closed convex subset of C:
For solving the equilibrium problem for a bifunction f : C ì C đ , let us assume
that f satisfies the following conditions:
(A1) f(x, x) = 0 for all x Î C;
(A2) f is monotone, i.e., f(x, y) + f(y, x) ≤ 0 for all x, y Ỵ C;
(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 a f(x, y) is convex and lower semicontinuous.
For example, let A be a continuous and monotone operator of C into E* and define
f (x, y) = Ax, y − x ,

∀x, y ∈ C.

Then, f satisfies (A1)-(A4). The following result is in Blum and Oettli [1].
Motivated by Combettes and Hirstoaga [2] in a Hilbert space and Takahashi and
Zembayashi [33] in a Banach space, Zhang [48] obtained the following lemma.


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Lemma 2.11. (Zhang [[48], Lemma 1.5]). Let C be a closed convex subset of a
smooth, strictly convex and reflexive Banach space E. Assume that f be a bifunction
from C × C to ℝ satisfying (A1)-(A4), A : C ® E* be a continuous and monotone mapping and  : C ® ℝ be a semicontinuous and convex functional. For r >0 and let x Ỵ
E. Then, there exists z Ỵ C such that
Q(z, y) +

1
y − z, Jz − Jx ≥ 0,
r

∀y ∈ C,

where Q(z, y) = f(z, y) + 〈Bz, y - z〉 + (y) (z), x, y Ỵ C. Furthermore, define a mapping Tr : E ® C as follows:
Tr x = z ∈ C : Q(z, y) +

1
y − z, Jz − Jx ≥ 0, ∀y ∈ C .
r

Then the following hold:
1. Tr is single-valued;
2. Tr is firmly nonexpansive, i.e., for all x, y Ỵ E, 〈Trx - Try, JTrx - JTry〉 ≤ 〈Trx Try, Jx -Jy〉;
3. F(Tr ) = F(Tr ) = GMEP(f , B, ϕ);
4. GMEP(f, B, ) is closed and convex;
5. j(p, Trz) + j(Trz, z) ≤ j(p, z), ∀p Ỵ F(Tr) and z Ỵ E.

3 Main results
In this section, we prove the new convergence theorems for finding the set of solutions
of system of generalized mixed equilibrium problems, the common fixed point set of a
family of closed and uniformly quasi-j-asymptotically nonexpansive mappings, and the
solution set of variational inequalities for an a-inverse strongly monotone mapping in
a 2-uniformly convex and uniformly smooth Banach space.
Theorem 3.1. Let C be a nonempty closed and convex subset of a 2-uniformly convex
and uniformly smooth Banach space E. For each j = 1, 2, ..., m let fj be a bifunction from C
× C to ℝ which satisfies conditions (A1)-(A4), Bj : C ® E* be a continuous and monotone
mapping and j : C i® ℝ be a lower semicontinuous and convex function. Let A be an ainverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y Ỵ C and
u Ỵ VI(A, C) ≠ ∅. Let {Si }∞ : C → Cbe an infinite family of closed uniformly Li-Lipschitz
i=1
continuous and uniformly quasi-j-asymptotically nonexpansive mappings with a sequence
{kn} ⊂ [1, ∞), kn ® 1 such that F := (∩∞ F(Si )) ∩ (∩m GMEP(fj , Bj , ϕj ))(∩VI(A, C)) is a
i=1
j=1
nonempty and bounded subset in C. For an initial point x0 Ỵ E with x1 = C1 x0and C1 =
C, we define the sequence {xn} as follows:
⎧
−1
⎪ vn = C J (Jxn − λn Axn ),
⎪

⎪ z = J−1 (α Jx + ∞ α JSn v ),
⎪ n
n,0 n
⎪
i=1 n,i i n
⎪
⎨ y = J−1 (β Jx + (1 − β )Jz ),
n
n n
n
n
Qm Qm−1
Q2 Q1
⎪ un = Trm,n Trm−1,n · · · Tr2,n Tr1,n yn ,
⎪
⎪
⎪C
⎪ n+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn ) + θn },
⎪
⎩
xn+1 = Cn+1 x0 , ∀n ≥ 1,

(3:1)

where θn = supqỴF(kn -1)j(q, xn), for each i ≥ 0, {an,i} and {bn} are sequences in [0, 1],
{rj, n} ⊂ [d, ∞) for some d >0 and {ln} ⊂ [a, b] for some a, b with 0 < a < b < c2a/2,


Saewan and Kumam Fixed Point Theory and Applications 2011, 2011:35
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Page 10 of 24

where 1is the 2-uniformly convexity constant of E. If
c

∞
i=0

αn,i = 1for all n ≥ 0, lim infn

(1 - bn) > 0 and lim infn®∞ an,0an, i > 0 for all i ≥ 1, then {xn} converges strongly
to p Î F, where p = ΠF x0.
Proof. We first show that Cn+1 is closed and convex for each n ≥ 0. Clearly, C1 = C is
closed and convex. Suppose that Cn is closed and convex for each n Ỵ N. Since for any
z Ỵ Cn, we know j(z, un) ≤ j(z, xn) + θn is equivalent to 2〈z, Jxn - Jun〉 ≤ ||xn||2 - ||
un||2 + θn. So, Cn+1 is closed and convex.
Next, we show that F ⊂ C n for all n ≥ 0. Since un = m yn, when
n
® ∞

j
n

Q Qj−1
Q2 Q1
= Trj,nj Trj−1,n · · · Tr2,n Tr1,n, j = 1, 2, 3, ..., m,

0
n

= I, by the convexity of ||·||2, property of

j, Lemma 2.9 and by uniformly quasi-j-asymptotically nonexpansive of Sn for each q
Ỵ F ⊂ Cn, we have
φ(q, un ) = φ(q, m yn )
n
≤ φ(q, yn )
= φ(q, J−1 (βn Jxn + (1 − βn )Jzn )
= q 2 − 2 q, βn Jxn + (1 − βn )Jzn + βn Jxn + (1 − βn )Jzn 2
≤ q 2 − 2βn q, Jxn − 2(1 − βn ) q, Jzn + βn xn 2 + (1 − βn )
= βn φ(q, xn ) + (1 − βn )φ(q, zn )

(3:2)
zn

2

and
φ(q, zn ) = φ(q, J−1 (αn,0 Jxn + ∞ αn,i JSn vn ))
i=1
i
= q 2 − 2 q, αn,0 Jxn + ∞ αn,i JSn vn + αn,0 Jxn + ∞ αn,i JSn vn 2
i=1
i=1
i
i
∞
2
= q − 2αn,0 q, Jxn − 2 i=1 αn,i q, JSn vn + αn,0 Jxn + ∞ αn,i JSn vn 2
i=1

i
i
≤ q 2 − 2αn,0 q, Jxn − 2 ∞ αn,i q, JSn vn + αn,0 Jxn 2 + ∞ αn,i JSn vn
i=1
i=1
i
i
n
−αn,0 αn,j g Jvn − JSj vn
= q 2 − 2αn,0 q, Jxn + αn,0 Jxn 2 − 2 ∞ αn,i q, JSn vn
i=1
i
+ ∞ αn,i JSn vn 2 − αn,0 αn,j g Jvn − JSn vn
i=1
i
j
= αn,0 φ(q, xn ) + ∞ αn,i φ(q, Sn vn ) − αn,0 αn,j g Jvn − JSn vn
i=1
i
j
≤ αn,0 φ(q, xn ) + ∞ αn,i kn φ(q, vn ) − αn,0 αn,j g Jvn − JSn vn .
i=1
j

2

(3:3)

It follows from Lemma 2.7 that
φ(q, vn ) = φ(q, C J−1 (Jxn − λn Axn ))

≤ φ(q, J−1 (Jxn − λn Axn ))
= V(q, Jxn − λn Axn )
≤ V(q, (Jxn − λn Axn ) + λn Axn ) − 2 J−1 (Jxn − λn Axn ) − q, λn Axn
= V(q, Jxn ) − 2λn J−1 (Jxn − λn Axn ) − q, Axn
= φ(q, xn ) − 2λn xn − q, Axn + 2 J−1 (Jxn − λn Axn ) − xn , −λn Axn .

(3:4)

Since q ỴVI(A, C) and A is an a-inverse-strongly monotone mapping, we have
−2λn xn − q, Axn = −2λn xn − q, Axn − Aq − 2λn xn − q, Aq
≤ −2λn xn − q, Axn − Aq
≤ −2αλn Axn − Aq 2 .

(3:5)

From Lemma 2.2 and ||Axn|| ≤ ||Axn - Aq||, ∀q Ỵ VI(A, C), we also have
2 J−1 (Jxn − λn Axn ) − xn , −λn Axn = 2 J−1 (Jxn − λn Axn ) − J−1 (Jxn ), −λn Axn
≤ 2 J−1 (Jxn − λn Axn ) − J−1 (Jxn )
λn Axn
4
−1 (Jx − λ Ax ) − JJ−1 (Jx )
≤ 2 JJ
λn Axn
n
n
n
n
c
4
= 2 Jxn − λn Axn − Jxn

λn Axn
c
4
= 2 λn Axn 2
c
4
= 2 λ2 Axn 2
c n
4
≤ 2 λ2 Axn − Aq 2 .
c n

(3:6)


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Substituting (3.5) and (3.6) into (3.4), we obtain
φ(q, vn ) ≤ φ(q, xn ) − 2αλn Axn − Aq 2 + c4 λ2 Axn − Aq
2 n
2
= φ(q, xn ) + 2λn ( 2 λn − α) Axn − Aq 2
c
≤ φ(q, xn ).

2

Substituting (3.7) into (3.3), we also have

φ(q, zn ) ≤ αn,0 φ(q, xn ) + ∞ αn,i kn φ(q, xn ) − αn,0 αn,j g Jvn − JSn vn
i=1
j
≤ αn,0 kn φ(q, xn ) + ∞ αn,i kn φ(q, xn ) − αn,0 αn,j g Jvn − JSn vn
i=1
j
= kn φ(q, xn ) − αn,0 αn,j g Jvn − JSn vn
j
≤ φ(q, xn ) + supq∈F (kn − 1)φ(q, xn ) − αn,0 αn,j g Jvn − JSn vn
j
= φ(q, xn ) + θn − αn,0 αn,j g Jvn − JSn vn
j
≤ φ(q, xn ) + θn .

(3:8)

and substituting (3.8) into (3.2), we also have
φ(q, un ) ≤ φ(q, xn ) + θn .

(3:9)

This shows that q Ỵ Cn+1 implies that F ⊂ Cn+1 and hence, F ⊂ Cn for all n ≥ 0. This
implies that the sequence {xn} is well defined. From definition of Cn+1 that xn = Cn x0
and xn+1 = Cn+1 x0 , ∈ Cn+1 ⊂ Cn, we have
φ(xn , x0 ) ≤ φ(xn+1 , x0 ),

∀n ≥ 0.

(3:10)

By Lemma 2.6, we get
φ(xn , x0 ) = φ( Cn x0 , x0 )
≤ φ(q, x0 ) − φ(q, xn )
≤ φ(q, x0 ), ∀q ∈ F.

(3:11)

From (3.10) and (3.11), then {j(x n , x 0 )} are nondecreasing and bounded. So, we
lim
obtain that n→∞ φ(xn , x0 ) exists. In particular, by (1.9), the sequence {(||xn|| - ||x0||)2 is
bounded. This implies {xn} is also bounded. Denote
M = sup{ xn } < ∞.

(3:12)

n≥0

Moreover, by the definition of θn and (3.12), it follows that
θn → 0

as n → ∞.

Next, we show that {xn} is a Cauchy sequence in C. Since xm =
m > n, by Lemma 2.6, we have

(3:13)
C m x0

∈ Cm ⊂ Cn, for

φ(xm , xn ) = φ(xm , Cn x0 )
≤ φ(xm , x0 ) − φ( Cn x0 , x0 )
= φ(xm , x0 ) − φ(xn , x0 ).

Since limn®∞ j(xn, x0) exists and we take m, n ® ∞, we get j(xm, xn) ® 0. From
Lemma 2.4, we have limn®∞ ||xm - xn|| = 0. Thus, {xn} is a Cauchy sequence, and by
the completeness of E, there exists a point p ẻ C such that xn đ p as n ® ∞.
Now, we claim that ||Jun - Jxn|| ® 0, as n ® ∞. By definition of xn = Cn x0, we have
φ(xn+1 , xn ) = φ(xn+1 , Cn x0 )
≤ φ(xn+1 , x0 ) − φ( Cn x0 , x0 )
= φ(xn+1 , x0 ) − φ(xn , x0 ).


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Page 12 of 24

Since limn®∞ j(xn, x0) exists, we also have
lim φ(xn+1 , xn ) = 0.

(3:14)

n→∞

Again from Lemma 2.4 that
lim

n→∞

xn+1 − xn = 0.

(3:15)

Since J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain
lim

n→∞

Jxn+1 − Jxn = 0.

Since xn+1 =

Cn+1 x0

(3:16)

∈ Cn+1 ⊂ Cn and the definition of Cn+1, we have

φ(xn+1 , un ) ≤ φ(xn+1 , xn ) + θn .

By (3.13) and (3.14) that
lim φ(xn+1 , un ) = 0.

(3:17)

n→∞

Again applying Lemma 2.4, we have
lim

n→∞

xn+1 − un = 0.

(3:18)

Since
un − x n

=
≤

un − xn+1 + xn+1 − xn
un − xn+1 + xn+1 − xn

.

It follows from (3.15) and (3.18) that
lim

n→∞

un − xn = 0.

(3:19)

Since J is uniformly norm-to-norm continuous on bounded subsets of E, we also
have
lim

n→∞

Jun − Jxn = 0.

(3:20)

Next, we will show that p ∈ F := ∩m GMEP(fj , Bj , ϕj ) ∩ (∩∞ F(Si )) ∩ VI(A, C).
i=1
j=1
(a) We show that p ∈ ∩∞ F(Si ). Since xn+1 =
i=1
(3.8), we have

Cn+1 x0

∈ Cn+1 ⊂ Cn, it follow from

φ(xn+1 , zn ) ≤ φ(xn+1 , xn ) + θn ,

by (3.13) and (3.14), we get
lim φ(xn+1 , zn ) = 0

n→∞

(3:21)

again from Lemma 2.4 that
lim

n→∞

xn+1 − zn = 0.

(3:22)

Since J is uniformly norm-to-norm continuous, we obtain
lim

n→∞

Jxn+1 − Jzn = 0.

(3:23)


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From (3.50), we note that
Jxn+1 − Jzn

∞

= Jxn+1 − (αn,0 Jxn +

αn,i JSn vn )
i

i=1

∞

= αn,0 Jxn+1 − αn,0 Jxn +
∞

= αn,0 (Jxn+1 − Jxn ) +
∞

=

i=1
∞

≥

i=1

i=1

∞

αn,i Jxn+1 −

i=1

αn,i JSn vn
i

αn,i (Jxn+1 − JSn vn )

i

i=1

αn,i (Jxn+1 − JSn vn ) − αn,0 (Jxn − Jxn+1 )
i

αn,i

Jxn+1 − JSn vn
i

−αn,0

Jxn − Jxn+1 ,

and hence
1

Jxn+1 − JSn vn ≤
i

∞
i=1

αn,i

( Jxn+1 − Jzn

From (3.16), (3.23) and lim inf n→∞

∞
i=1

+αn,0

Jxn − Jxn+1 ).

(3:24)

αn,i > 0, we obtain that

lim ||Jxn+1 − JSn vn || = 0.
i

(3:25)

n→∞

Since J-1 is uniformly norm-to-norm continuous on bounded sets, we have
lim

n→∞

xn+1 − Sn vn = 0.
i

(3:26)

Using the triangle inequality that

xn − Sn vn
i

xn − xn+1 + xn+1 − Sn vn
i
xn − xn+1 + xn+1 − Sn vn
i

=
≤

.

From (3.15) and (3.26), we have
lim

n→∞

xn − Sn vn = 0.
i

(3:27)

On the other hand, we note that
φ(q, xn ) − φ(q, un ) + θn =
≤

xn 2 − un 2 − 2 q, Jxn − Jun + θn
xn − un ( xn + un ) + 2 q

Jxn − Jun

+ θn .

It follows from θn ® 0, ||xn - un|| ® 0 and ||Jxn - Jun|| ® 0, that
φ(q, xn ) − φ(q, un ) + θn → 0

as n → ∞.

(3:28)

From (3.2), (3.3) and (3.7) that
φ(q, un ) ≤ φ(q, yn )
≤ βn φ(q, xn ) + (1 − βn )φ(q, zn )
≤ βn φ(q, xn ) + (1 − βn )[αn,0 φ(q, xn ) +
− αn,0 αn,j g

Jvn −

JSn vn
j

∞
i=1

αn,i kn φ(q, vn )

]
∞

= βn φ(q, xn ) + (1 − βn )αn,0 φ(q, xn ) + (1 − βn )
− (1 − βn )αn,0 αn,j g

∞

≤ βn φ(q, xn ) + (1 − βn )αn,0 φ(q, xn ) + (1 − βn )

≤ βn φ(q, xn ) + (1 − βn )αn,0 kn φ(q, xn ) + (1 − βn )
− (1 − βn )

i=1

αn,i kn 2λn (α −

2
c2 λn )

= kn φ(q, xn ) − (1 − βn )

i=1

αn,i kn 2λn (α −

i=1

i=1

αn,i kn 2λn (α −

i=1

2
c2 λn )

αn,i kn φ(q, xn )

αn,i kn 2λn (α −

2
c2 λn )

Axn − Aq

Axn − Aq ]
2

αn,i kn 2λn (α −

2
c2 λn )

Axn − Aq 2 ]

2

∞

2
c2 λn )
∞

q∈F
∞

i=1

Axn − Aq

≤ φ(q, xn ) + sup(kn − 1)φ(q, xn ) − (1 − βn )
= φ(q, xn ) + θn − (1 − βn )

αn,i kn [φ(q, xn ) − 2λn (α −

i=1
∞

≤ βn kn φ(q, xn ) + (1 − βn )kn φ(q, xn ) − (1 − βn )
∞

αn,i kn φ(q, vn )

i=1
∞

≤ βn φ(q, xn ) + (1 − βn )αn,0 φ(q, xn ) + (1 − βn )
∞

αn,i kn φ(q, vn )

i=1

Jvn − JSn vn
j

2
c2 λn )

Axn − Aq 2 ,

Axn − Aq

2

2


Saewan and Kumam Fixed Point Theory and Applications 2011, 2011:35
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Page 14 of 24

and hence
2a(α −

2b
c2 )

Axn − Aq

2

≤ 2λn (α −
≤

2
c2 λn )

Axn − Aq

1
(1 − βn )

∞
i=1

αn,i kn

2

(φ(q, xn ) − φ(q, un ) + θn ).

(3:29)

From (3.28), {ln} ⊂ [a, b] for some a, b with 0 < a < b < c2 a/2, lim infn ®∞(1 - bn)
>0 and lim infn ® ∞ an,0an, i > 0, for i ≥ 0 and kn ® 1 as n ® ∞, we obtain that
lim

n→∞

Axn − Aq

= 0.

(3:30)

From Lemmas 2.6, 2.7 and (3.6), we compute
φ(xn , vn ) = φ(xn , C J−1 (Jxn − λn Axn ))
≤ φ(xn , J−1 (Jxn − λn Axn ))
= V(xn , Jxn − λn Axn )
≤ V(xn , (Jxn − λn Axn ) + λn Axn ) − 2 J−1 (Jxn − λn Axn ) − xn , λn Axn
= φ(xn , xn ) + 2 J−1 (Jxn − λn Axn ) − xn , −λn Axn
= 2 J−1 (Jxn − λn Axn ) − xn , −λn Axn
4λ2
≤ c2n Axn − Aq 2
2
≤ 4b
Axn − Aq 2 .
c2

Applying Lemma 2.4 and (3.30) that
lim

n→∞

xn − vn = 0

(3:31)

and we also obtain
lim ||Jxn − Jvn || = 0.

(3:32)

n→∞

Since Sn is continuous, for any i ≥ 1
i
lim

n→∞

Sn xn − Sn vn = 0.
i
i

(3:33)

Again by the triangle inequality, we get
xn − Sn xn
i

≤

xn − Sn vn
i

Sn vn − Sn xn
i
i

+

.

From (3.27) and (3.33), we have
lim

n→∞

xn − Sn xn = 0,
i

∀i ≥ 1.

(3:34)

By using triangle inequality, we get
Sn xn − p ≤ Sn xn − xn
i
i

+

xn − p ,

∀i ≥ 1.

We know that xn ® p as n ® ∞ and from (3.34)
Sn xn → p
i

for each i ≥ 1.

Moreover, by the assumption that ∀i ≥ 1, Si is uniformly Li-Lipschitz continuous, and
hence we have.
Sn+1 xn − Sn xn
i
i

≤ Sn+1 xn − Sn+1 xn+1
i
i
≤ (Li + 1) xn+1 − xn

By (3.15) and (3.34), it yields that

+ Sn+1 xn+1 − xn+1
i
+ Sn+1 xn+1 − xn+1
i

+ xn+1 − xn
+ xn − Sn xn
i

+
.

xn − Sn xn
i

(3:35)

Sn+1 xn − Sn xn → 0. From Sn xn → p, we have
i
i
i

Sn+1 xn → p, that is Si Sn xn → p. In view of closeness of Si, we have Sip = p, for all i ≥ 1.
i
i
p ∈ ∩∞ F(Si ).
This implies that
i=1


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Page 15 of 24

(b) We show that p ∈ ∩m GMEP(fj , Bj , ϕj ).
j=1
my
n n,

Let un =

j
n

when

φ(q, un ) = φ(q,
≤ φ(q,
≤ φ(q,
.
.
.

Q Qj−1
Q2 Q1
= Trj,nj Trj−1,n · · · Tr2,n Tr1,n, j = 1,2,3, ..., m and

0
n

my )
n n
m−1 y )
n
n
m−2
yn )
n

= I, we obtain

(3:36)

j
n yn ).

≤ φ(q,

By Lemma (2.11)(5), we have for j = 1, 2, 3, ..., m
j
n yn , yn )

φ(

j

+ θn ≤ φ(q, yn ) − φ(q, n yn ) + θn
j
≤ φ(q, xn ) − φ(q, n yn ) + θn
≤ φ(q, xn ) − φ(q, un ) + θn .
j
n yn , yn )

From (3.13) and (3.28), we get φ(
Lemma 2.4 implies that
j
n yn

lim

n→∞

Since xn+1 =

(3:37)

→ 0as n ® ∞, for j = 1, 2, 3, ..., m and

− yn = 0, ∀j = 1, 2, 3, . . . , m.

Cn+1 x0

(3:38)

∈ Cn+1 ⊂ Cn, it follows from (3.2) and (3.8) that

φ(xn+1 , yn ) ≤ φ(xn+1 , xn ) + θn .

By (3.13) and (3.14), we have
lim φ(xn+1 , yn ) = 0.

n→∞

Applying Lemma 2.4 that
lim

n→∞

xn+1 − yn = 0.

(3:39)

Using the triangle inequality, we obtain
xn − yn

≤

xn − xn+1

+

xn+1 − yn

.

From (3.15) and (3.39), we get
lim

n→∞

xn − yn = 0.

Since xn ® p and ||xn - yn|| ® 0, we have yn ® p as n ® ∞.
Again by using the triangle inequality, we have for j = 1, 2, 3, ..., m
p−

j
n yn

≤

p − yn

+

j
n yn

yn −

.

From (3.38) and yn ® p as n ® ∞, we get
lim

n→∞

j
n yn

p−

= 0, ∀j = 1, 2, 3, . . . , m.

(3:41)

By using the triangle inequality, we obtain
j
n yn

−

j−1
n yn

≤

j
n yn

−p

+

p−

j−1
n yn

.

From (3.41), we have
lim

n→∞

j
n yn

−

j−1
n yn

= 0, ∀j = 1, 2, 3, . . . , m.

(3:42)


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Page 16 of 24

Since {rj, n} ⊂ [d, ∞) and J is uniformly continuous on any bounded subset of E,
J

lim

j
n yn

n→∞

−J
rj,n

j−1
n yn

= 0, ∀j = 1, 2, 3, . . . , m.

(3:43)

From Lemma 2.11, we get for j = 1, 2, 3, ..., m
Qj (

j
n yn , y)

+

1
y−
rj,n

j
n yn , J

j
n yn

−J

j−1
n yn

≥ 0,

∀y ∈ C.

From (A2),
1
y−
rj,n

j
n yn , J

j
n yn

−J

j−1
n yn

≥ Qj (y,

j
n yn ),

∀y ∈ C, ∀j = 1, 2, 3, . . . , m.

From (3.41) and (3.43), we have
0 ≥ Qj (y, p),

∀y ∈ C, ∀j = 1, 2, 3, . . . , m.

(3:44)

For t with 0 < t ≤ 1 and y Ỵ C; let yt = ty + (1 - t)p. Then, we get that yt Ỵ C. From
(3.44), and it follows that
Qj (yt , p) ≤ 0,

∀y ∈ C, ∀j = 1, 2, 3, . . . , m.

(3:45)

By the conditions (A1) and (A4), we have for j = 1, 2, 3, ..., m
0 = Qj (yt , yt )
≤ tQj (yt , y) + (1 − t)Qj (yt , p)
≤ tQj (yt , y)
= Qj (yt , y).

(3:46)

From (A3) and letting t đ 0, This implies that p ẻ GMEP(fj, Bj, j), ∀j = 1, 2, 3, ...,
m. Therefore p ∈ ∩m GMEP(fj , Bj , ϕj )
j=1
(c) We show that p ẻ VI(A, C). Indeed, define U E ì E* by
Uv =

Av + NC (v), v ∈ C;
∅,
v ∈ C.
/

(3:47)

By Lemma 2.8, U is maximal monotone and U -10 = VI(A, C). Let (v, w) Ỵ G(U).
Since w Ỵ Uv = Av + NC(v), we get w - Av Î NC(v).
From vn Î C, we have
v − vn , w − Av ≥ 0.

On the other hand, since vn =

(3:48)
CJ

−1 (Jx

n

− λn Axn ). Then, by Lemma 2.5, we have

v − vn , Jvn − (Jxn − λn Axn ) ≥ 0,

and thus
v − vn ,

Jxn − Jvn
− Axn ≤ 0.
λn

(3:49)


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Page 17 of 24

1
It follows from (3.48), (3.49) and A is monotone and α -Lipschitz continuous that

v − vn , w ≥ v − vn , Av

Jxn − Jvn
− Axn
λn
Jxn − Jvn
+ v − zvn ,
λn

≥ v − vn , Av + v − vn ,
= v − vn , Av − Axn

= v − vn , Av − Avn + v − vn , Avn − Axn + v − vn ,
≥−

v − vn
vn − xn
≥ −H
α

vn − xn
− v − vn
α
Jxn − Jvn
,
+
a

Jxn − Jvn
a

Jxn − Jvn
λn

where H = supn≥1 ||v - vn||. Take the limit as n i® ∞, (3.31) and (3.32), we obtain 〈v
- p, w〉 ≥ 0. By the maximality of B we have p Ỵ B-10, that is p Ỵ VI(A, C). Hence,
from (a), (b) and (c), we obtain p Ỵ F.
Finally, we show that p = ΠFx0. From xn = Cn x0, we have 〈Jx0 - Jxn, xn - z〉 ≥ 0, ∀z Ỵ
Cn. Since F ⊂ Cn, we also have
Jx0 − Jxn , xn − y ≥ 0,

∀y ∈ F.

Taking limit n ® ∞, we obtain
Jx0 − Jp, p − y ≥ 0,

∀y ∈ F.

By Lemma 2.5, we can conclude that p = ΠFx0 and xn ® p as n ® ∞. This completes the proof. □
If Si = S for each i Î N, then Theorem 3.1 is reduced to the following corollary.
Corollary 3.2. Let C be a nonempty closed and convex subset of a 2-uniformly convex
and uniformly smooth Banach space E. For each j = 1, 2, ..., m let fj be a bifunction
from C × C to ℝ which satisfies conditions (A1)-(A4), Bj : C ® E* be a continuous and
monotone mapping and j : C ® ℝ be a lower semicontinuous and convex function.
Let A be an a-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay Au||, ∀y Ỵ C and u Ỵ VI(A, C) ≠ ∅. Let S : C ® C be a closed L-Lipschitz continuous
and quasi-j-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1, ∞), kn
® 1 such that F := F(S) ∩ (∩m GMEP(fj , Bj , ϕj )) ∩ VI(A, C)is a nonempty and bounded
j=1
subset in C. For an initial point x 0 Ỵ E with x1 =
sequence {xn} as follows:
⎧

−1
⎪ vn = C J (Jxn − λn Axn ),
⎪
⎪
⎪ zn = J−1 (αn Jxn + (1 − αn )JSn vn ),
⎪
⎪
⎨ y = J−1 (β Jx + (1 − β )Jz ),
n

n

n

n

n

Qm Qm−1
Q2 Q1
⎪ un = Trm,n Trm−1,n · · · Tr2,n Tr1,n yn ,
⎪
⎪
⎪C
⎪ n+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn ) + θn },
⎪
⎩
xn+1 = Cn+1 x0 , ∀n ≥ 1,

C1 x0and

C 1 = C, we define the

(3:50)

where θn = supqỴF (kn - 1)j(q, xn), {an}, {bn} are sequences in [0, 1], {rj, n} ⊂ [d, ∞) for
some d >0 and {ln} ⊂ [a, b] for some a, b with 0 < a < b < c2a/2, where 1c is the 2c
uniformly convexity constant of E. If lim infn®∞(1 - bn) >0 and lim infnđ(1 - an) >0,
then {xn} converges strongly to p ẻ F, where p = ΠF x0.


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Page 18 of 24

For a special case that i = 1, 2, we can obtain the following results on a pair of quasi_-asymptotically nonexpansive mappings immediately from Theorem 3.1.
Corollary 3.3. Let C be a nonempty closed and convex subset of a 2-uniformly convex
and uniformly smooth Banach space E. For each j = 1, 2, ..., m let fj be a bifunction
from C × C to ℝ which satisfies conditions (A1)-(A4), Bj : C ® E* be a continuous and
monotone mapping and j : C ® ℝ be a lower semicontinuous and convex function.
Let A be an a-inversestrongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay Au||, ∀y Ỵ C and u Ỵ VI(A, C) ≠ ∅. Let S, T : C ® C be two closed quasi-j-asymptotically nonexpansive mappings and L S , L T -Lipschitz continuous, respectively with a
sequence
{k n }
⊂
[1,
∞),
kn
®
1
such

that
m
F := F(S) ∩ F(T) ∩ (∩j=1 GMEP(fj , Bj , ϕj )) ∩ VI(A, C) is a nonempty and bounded subset
in C. For an initial point x0 Î E with x1 =
{xn} as follows:
⎧
−1
⎪ vn = C J (Jxn − λn Axn ),
⎪
⎪ z = J−1 (α Jx + β JSn v + γ JT n v ),
⎪ n
n n
n
n
n
n
⎪
⎪
⎨ y = J−1 (δ Jx + (1 − δ )Jz ),
n

n

n

n

C1 x0and

C1 = C, we define the sequence

n

Qm Qm−1
Q2 Q1
⎪ un = Trm,n Trm−1,n · · · Tr2,n Tr1,n yn ,
⎪
⎪
⎪C
⎪ n+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn ) + θn },
⎪
⎩
xn+1 = Cn+1 x0 , ∀n ≥ 0,

(3:51)

where θn = supqỴF (kn - 1)j(q, xn), {an}, {bn}, {gn} and {δn} are sequences in [0, 1], {rj,
} ⊂ [d, ∞) for some d >0 and {l n } ⊂ [a, b] for some a, b with 0 < a < b < c2 a/2,
n
where 1is the 2-uniformly convexity constant of E. If an + bn + gn = 1 for all n ≥ 0 and
c
lim infn ®∞ anbn >0, lim infn ®∞ angn >0, lim infn ®∞ bngn >0 and lim infn ®∞(1 - δn)
>0, then {xn} converges strongly to p Ỵ F, where p = ΠFx0.
Corollary 3.4. Let C be a nonempty closed and convex subset of a 2-uniformly convex
and uniformly smooth Banach space E. For each j = 1, 2, ..., m let fj be a bifunction
from C × C to ℝ which satisfies conditions (A1)-(A4), Bj : C ® E* be a continuous and
monotone mapping and j : C ® ℝ be a lower semicontinuous and convex function.
Let A be an a-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay Au||, ∀y Ỵ C and u Ỵ VI(A, C) ≠ ∅. Let {Si }∞ : C → Cbe an infinite family of closed
i=1
quasi-jnonexpansive

mappings
such
that
∞ F(S ) ∩ (∩m GMEP(f , B , ϕ )) ∩ VI(A, C) = ∅.
F := ∩i=1
For an initial point x 0 Ỵ E with
i
j j j
j=1
x1 =

C1 x0and

C1 = C, we define the sequence {xn} as follows:
⎧
−1
⎪ vn = C J (Jxn − λn Axn ),
⎪
⎪
⎪ zn = J−1 (αn,0 Jxn + ∞ αn,i JSi vn ),
⎪
i=1
⎪
⎨ y = J−1 (β Jx + (1 − β )Jz ),
n
n n
n
n
Qm Qm−1
Q2 Q1

⎪ un = Trm,n Trm−1,n · · · Tr2,n Tr1,n yn ,
⎪
⎪
⎪C
⎪ n+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn ),
⎪
⎩
xn+1 = Cn+1 x0 , ∀n ≥ 0,

(3:52)

where {an, i} and {bn} are sequences in [0, 1], {rj, n} ⊂ [d, ∞) for some d >0 and {ln} ⊂
[a, b] for some a, b with 0 < a < b < c2a/2, where 1is the 2-uniformly convexity conc

stant of E. If ∞ αn,i = 1for all n ≥ 0, lim infn ®∞(1 -bn) >0 and lim infn ®∞ an, 0an, i
i=0
> 0 for all i ≥ 1, then {xn} converges strongly to p Ỵ F, where p = ΠFx0.
Proof. Since {Si }∞ : C → C is an infinite family of closed quasi-j-nonexpansive mapi=1
pings, it is an infinite family of closed and uniformly quasi-j-asymptotically


Saewan and Kumam Fixed Point Theory and Applications 2011, 2011:35
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Page 19 of 24

nonexpansive mappings with sequence kn = 1. Hence, the conditions appearing in Theorem 3.1 F is a bounded subset in C and for each i ≥ 1, Si is uniformly Li-Lipschitz
continuous are of no use here. By virtue of the closeness of mapping Si for each i ≥ 1,
it yields that p Ỵ F (Si) for each i ≥ 1, that is, p ∈ ∩∞ F(Si ). Therefore, all conditions in
i=1
Theorem 3.1 are satisfied. The conclusion of Corollary 3.4 is obtained from Theorem

3.1 immediately. □

4 Deduced theorems
Corollary 4.1. [[41], Theorem 3.2] Let C be a nonempty closed and convex subset of a
2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C
× C to ℝ satisfying (A1)-(A4) and  : C ® ℝ is convex and lower semicontinuous. Let
A be an a-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay Au||, ∀y Î C and u Î VI(A, C) ≠ ∅. Let {Si }N : C → Cbe a finite family of closed
i=1
quasi-jnonexpansive
mappings
such
that
F := ∩N F(Si ) ∩ GMEP(f , B, ϕ) ∩ VI(A, C) = ∅. For an initial point x0 Î E with 1and C1
i=1
c
= C, we define the sequence {xn} as follows:
⎧
⎪ zn = C J−1 (Jxn − λn Axn ),
⎪
⎪
⎪
⎪ yn = J−1 (α0 Jxn + N αi JSi zn ),
⎪
i=1
⎨
1
f (u , y) + Bun , y − un + ϕ(y) − ϕ(un ) +
y − un , Jun − Jyn ≥ 0,
⎪ n
rn

⎪
⎪
⎪ Cn+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn ),
⎪
⎪
⎩
xn+1 = Cn+1 x0 , ∀n ≥ 0,

∀y ∈ C,(4:1)

where {ai} is sequence in [0, 1], {rn} ⊂ [d, ∞) for some d >0 and {ln} ⊂ [a, b] for some
a, b with 0 < a < b < c2a/2, where 1c is the 2-uniformly convexity constant of E. If ai Î
c
αi = 1then {xn} converges strongly to p Î F, where p = ΠF x0.
Remark 4.2. Theorems 3.1, Corollaries 3.4 and 4.1 improve and extend the corresponding results of Wattanawitoon and Kumam [14] and Zegeye [41] in the following
senses:

(0, 1) such that

N
i=0

• from a solution of the classical equilibrium problem to the generalized mixed
equilibrium problem with an infinite family of quasi-j-asymptotically mappings;
• for the mappings, we extend the mappings from nonexpansive mappings, relatively quasi-nonexpansive mappings or quasi-j-nonexpansive mappings and a finite
family of closed relatively quasi-nonexpansive mappings to an infinite family of
quasi-j-asymptotically nonex-pansive mappings.
Corollary 4.3. Let C be a nonempty closed and convex subset of a uniformly convex
and uniformly smooth Banach space E. Let f be a bifunction from C × C to ℝ satisfying
(A1)-(A4) and  : C ® ℝ is convex and lower semicontinuous. Let B be a continuous

monotone mapping of C into E*. Let {Si }∞ : C → Cbe an infinite family of closed and
i=1
uniformly quasi-j-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1, ∞),
kn ® 1 and uniformly Li-Lipschitz continuous such that F := ∩∞ F(Si ) ∩ GMEP(f , B, ϕ)
i=1
is a nonempty and bounded subset in C. For an initial point x 0 Ỵ E with
x1 = C1 x0and C1 = C, we define the sequence {xn} as follows:


Saewan and Kumam Fixed Point Theory and Applications 2011, 2011:35
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⎧
⎪ yn = J−1 (αn,0 Jxn + ∞ αn,i JSn xn ),
i=1
⎪
i
⎪
⎪
1
⎨
f (un , y) + Bun , y − un + ϕ(y) − ϕ(un ) +
y − un , Jun − Jyn ≥ 0,
rn
⎪C
⎪ n+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn ) + θn },
⎪
⎪
⎩x
∀n ≥ 0,
n+1 =

Cn+1 x0 ,

Page 20 of 24

∀y ∈ C,
(4:2)

where θn = supqỴF (kn - 1)j(q, xn), {an,i} is sequence in [0, 1], {rn} ⊂ [a, ∞) for some a
>0. If ∞ αn,i = 1for all n ≥ 0 and lim infn®∞ an, 0 an, i >0 for all i ≥ 1, then {xn} coni=0
verges strongly to p Ỵ F, where p = ΠF x0.
Proof. Put A ≡ 0 in Theorem 3.1 Then, we get that zn = xn. Thus, the method of
proof of Theorem 3.1 gives the required assertion without the requirement that E be
2-uniformly convex. □
If setting B ≡ 0 and  ≡ 0 in Corollary 4.3, then we have the following corollary.
Corollary 4.4. Let C be a nonempty closed and convex subset of a uniformly convex
and uniformly smooth Banach space E. Let f be a bifunction from C × C to ℝ satisfying
(A1)-(A4) and  : C ® ℝ is convex and lower semicontinuous. Let {Si }∞ : C → Cbe an
i=1
infinite family of closed and uniformly quasi-j-asymptotically nonexpansive mappings
with a sequence {kn} ⊂ [1, ∞), kn ® 1 and uniformly Li-Lipschitz continuous such that
F := ∩∞ F(Si ) ∩ EP(f ) is a nonempty and bounded subset in C. For an initial point x0
i=1
Ỵ E with x1 = C1 x0and C1 = C, we define the sequence {xn} as follows:
⎧
⎪ yn = J−1 (αn,0 Jxn + ∞ αn,i JSn xn ),
i=1
⎪
i
⎪
⎪

1
⎨
f (un , y) +
y − un , Jun − Jyn ≥ 0, ∀y ∈ C,
(4:3)
rn
⎪C
⎪
⎪ n+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn ) + θn },
⎪
⎩x
=
x , ∀n ≥ 0,
n+1

Cn+1 0

where θn = supqỴF (kn - 1)j(q, xn), {an, i} is sequence in [0, 1], {rn} ⊂ [a, ∞) for some a
>0. If ∞ αn,i = 1for all n ≥ 0 and lim infn®∞ an, 0 an, i >0 for all i ≥ 1, then {xn} coni=0
verges strongly to p Î F, where p = ΠF x0.
Remark 4.5. Corollaries 4.3 and 4.4 improve and extend the corresponding results of
Zegeye [41] and Wattanawitoon and Kumam [14] in the sense from a finite family of
closed relatively quasi-nonexpansive mappings and closed relatively quasi-nonexpansive
mappings to more general than an infinite family of closed and uniformly quasi-jasymptotically nonexpansive mappings.
Remark 4.6. Moreover, Our theorems improve, generalize, unify and extend Qin et
al. [9], Zeg-eye et al. [15], Zegeye [41] and Wattanawitoon and Kumam [14,49] and
several results recently announced.

5 Applications
5.1 Application to complementarity problems

Let K be a nonempty, closed convex cone in E. We define the polar K* of K as follows:
K ∗ = {y∗ ∈ E∗ : x, y∗ ≥ 0, ∀x ∈ K}.

(5:1)

If A : K ® E* is an operator, then an element u Ỵ K is called a solution of the complementarity problem [20] if
Au ∈ K ∗

and

u, Au = 0.

The set of solutions of the complementarity problem is denoted by CP(A, K).

(5:2)


Saewan and Kumam Fixed Point Theory and Applications 2011, 2011:35
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Theorem 5.1. Let K be a nonempty closed and convex subset of a 2-uniformly convex
and uniformly smooth Banach space E. For each j = 1, 2, ..., m let fj be a bifunction
from C × C to ℝ which satisfies conditions (A1)-(A4), Bj : C ® E* be a continuous and
monotone mapping and j : C ® ℝ be a lower semicontinuous and convex function.
Let A be an a-inverse-strongly monotone mapping of K into E* satisfying ||Ay|| ≤ ||Ay Au||, ∀y Î K and u Î CP(A, K) ≠ ∅. Let {Si }∞ : K → K be an infinite family of closed
i=1
uniformly Li-Lipschitz continuous and uniformly quasi-j-asymptotically nonexpansive
mappings with a sequence {k n } ⊂ [1, ∞), k n ® 1 such that
F := ∩∞ F(Si ) ∩ (∩m GMEP(fj , Bj , ϕj )) ∩ CP(A, K) is a nonempty and bounded subset in
i=1

j=1
K. For an initial point x0 Î E with x1 = C1 x0and K1 = K, we define the sequence {xn}
as follows:
⎧
−1
⎪ vn = K J (Jxn − λn Axn ),
⎪
⎪ z = J−1 (α Jx + ∞ α JSn v ),
⎪ n
n,0 n
⎪
i=1 n,i i n
⎪
⎨ y = J−1 (β Jx + (1 − β )Jz ),
n
n n
n
n
(5:3)
Qm Qm−1
Q2 Q1
⎪ un = Trm,n Trm−1,n · · · Tr2,n Tr1,n yn ,
⎪
⎪
⎪K
⎪ n+1 = {z ∈ Kn : φ(z, un ) ≤ φ(z, xn ) + θn },
⎪
⎩
xn+1 = Kn+1 x0 , ∀n ≥ 0,
where J is the duality mapping on E, θn = supqỴF (kn - 1)j(q, xn), for each i ≥ 0, {an,

} and {bn} are sequences in [0, 1], {rj, n} ⊂ [d, ∞) for some d >0 and {ln} ⊂ [a, b] for
i
some a, b with 0 < a < b < c2a/2, where 1is the 2-uniformly convexity constant of E. If
c
∞
i=0

αn,i = 1for all n ≥ 0, lim infn®∞(1 - bn) >0 and lim infn ® ∞ an, 0an, i > 0 for all i
≥ 1, then {xn} converges strongly to p Ỵ F, where p = ΠF x0.
Proof. As in the proof of Takahashi in [[20], Lemma 7.11], we get that VI(A, K) = CP
(A, K). So, we obtain the result. □
5.2 Application to zero points

Next, we consider the problem of finding a zero point of an inverse-strongly monotone
operator of E into E*. Assume that A satisfies the conditions:
(C1) A is a-inverse-strongly monotone,
(C2) A -10 = {u Ỵ E : Au = 0} ≠ ∅.
Theorem 5.2. Let C be a nonempty closed and convex subset of a 2-uniformly convex
and uniformly smooth Banach space E. For each j = 1, 2, ..., m let fj be a bifunction
from C × C to R which satisfies conditions (A1)-(A4), Bj : C ® E* be a continuous and
monotone mapping and j : C ® ℝ be a lower semicontinuous and convex function.
Let A be an operator of E into E* satisfying (C1) and (C2). Let {Si }∞ : C → Cbe an infii=1
nite family of closed uniformly Li- Lipschitz continuous and uniformly quasi-j-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1, ∞), kn ® 1 such that
F := ∩∞ F(Si ) ∩ (∩m GMEP(fj , Bj , ϕj )) ∩ A−1 0
j=1
i=1

is a nonempty and bounded subset in C: For an initial point x 0 Ỵ E with
x1 = C1 x0and C1 = C, we define the sequence {xn} as follows:
⎧

−1
⎪ vn = C J (Jxn − λn Axn ),
⎪
⎪ z = J−1 (α Jx + ∞ α JSn v ),
⎪ n
n,0 n
⎪
i=1 n,i i n
⎪
⎨ y = J−1 (β Jx + (1 − β )Jz ),
n
n n
n
n
(5:4)
Qm Qm−1
Q2 Q1
⎪
⎪ un = Trm,n Trm−1,n · · · Tr2,n Tr1,n yn ,
⎪
⎪C
⎪ n+1 = {z ∈ Cn : φ(z, un ) ≤ φ(z, xn ) + θn },
⎪
⎩
xn+1 = Cn+1 x0 , ∀n ≥ 0,

Page 21 of 24


Saewan and Kumam Fixed Point Theory and Applications 2011, 2011:35

/>
i}

where J is the duality mapping on E, θn = supqỴF (kn - 1)j(q, xn), for each i ≥ 0, {an,
and {bn} are sequences in [0, 1], {rj, n] ⊂ [d, ∞) for some d >0 and {ln} ⊂ [a, b] for

some a, b with 0 < a < b < c2a/2, where 1is the 2-uniformly convexity constant of E. If
c
∞
i=0

αn,i = 1for all n ≥ 0, lim infn®∞(1 - bn) >0 and lim infn ®∞ an, 0 an, i > 0 for all i
≥ 1, then {xn} converges strongly to p Ỵ F, where p = ΠF x0.
Proof. Setting C = E in Corollary 3.4, we also get ΠE = I. We also have VI(A, C) = VI
(A, E) {x Î E : Ax = 0} ≠ ∅ and then the condition ||Ay|| ≤ ||Ay - Au|| holds for all y
Ỵ E and u Ỵ A- 10. So, we obtain the result. □
5.3 Application to Hilbert spaces

If E = H, a Hilbert space, then E is 2-uniformly convex (we can choose c = 1) and uniformly smooth real Banach space and closed relatively quasi-nonexpansive map
reduces to closed quasi-nonexpansive map. Moreover, J = I, identity operator on H
and ΠC = PC, projection mapping from H into C: Thus, the following corollaries hold.
Theorem 5.3. Let C be a nonempty closed and convex subset of a Hilbert space H.
For each j = 1, 2, ..., m let fj be a bifunction from C × C to ℝ which satisfies conditions
(A1)-(A4), Bj : C ® E* be a continuous and monotone mapping and j : C ® ℝ be a
lower semicontinuous and convex function. Let A be an a-inverse-strongly monotone
mapping of C into H satisfying ||Ay|| ≤ ||Ay - Au||, ∀y Ỵ C and u Ỵ VI(A, C) ≠ ∅. Let
{Si }∞ : C → Cbe an infinite family of closed and uniformly quasi-j-asymptotically
i=1
nonexpansive mappings with a sequence {k n } ⊂ [1, ∞), k n ® 1 and uniformly L i Lipschitz continuous such that F := ∩∞ F(Si ) ∩ (∩m GMEP(fj , Bj , ϕj )) ∩ VI(A, C) is a
i=1

j=1
nonempty and bounded subset in C. For an initial point x0 Î H with x1 = PC1 x0 and
C1 = C, we define the sequence {xn} as follows:
⎧
⎪ zn = PC (xn − λn Axn ),
⎪
⎪
⎪ yn = αn,0 xn + ∞ αn,i Sn zn ,
⎨
i=1
i
Qm Qm−1
Q2 Q1
(5:5)
un = Trm,n Trm−1,n · · · Tr2,n Tr1,n yn ,
⎪
⎪ Cn+1 = {z ∈ Cn : z − un ≤ z − xn +θn },
⎪
⎪
⎩
xn+1 = PCn+1 x0 , ∀n ≥ 0,
where θn = supqỴF (kn - 1)|||q -xn||, {an, i} is sequence in [0, 1], {rj, n} ⊂ [a, ∞) for some
a >0 and {ln} ⊂ [a, b] for some a; b with 0 < a < b < a/2. If ∞ αn,i = 1for all n ≥ 0
i=0
and lim infn ®∞an,0an, i > 0 for all i ≥ 1, then {xn} converges strongly to p Ỵ F, where p
= ΠF x0.
Remark 5.4. Theorem 5.3 improves and extends the Corollary 3.7 in Zegeye [41] in
the aspect for the mappings, and we extend the mappings from a finite family of closed
relatively quasi-nonexpansive mappings to a more general infinite family of closed and
uniformly quasi-j-asymptotically nonexpansive mappings.

6 Competing interests
The authors declare that they have no competing interests.
7 Authors’ contributions
All authors contribute equally and significantly in this research work. All authors read
and approved the final manuscript.

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Saewan and Kumam Fixed Point Theory and Applications 2011, 2011:35
/>
Acknowledgements
This research was supported by grant from under the program Strategic Scholarships for Frontier Research Network
for the Join Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand. Also,
Siwaporn Saewan was supported by the King Mongkuts Diamond scholarship for Ph.D. program at King Mongkuts
University of Technology Thonburi (KMUTT) (under project NRU-CSEC No.54000267). Moreover, this work was
supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of
the Higher Education Commission and Poom Kumam was supported by the Higher Education Commission, the
Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (Grant No.MRG5380044).
Finally, the authors are very grateful to the referees for their careful reading, comments and suggestions, which
improved the presentation of this article.
Received: 21 March 2011 Accepted: 15 August 2011 Published: 15 August 2011
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