A New Modified Block Iterative Algorithm for a System of
Equilibrium Problems and a Fixed Point Set of Uniformly

φ-Asymptotically Nonexpansive Mappings∗

Siwaporn Saewan and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi

(KMUTT),Bangmod, Bangkok 10140, Thailand
Email: 52501406@st.kmutt.ac.th, poom.kum@kmutt.ac.th

Abstract In this paper, we construct a new modified block hybrid projection algorithm for
finding a common element of the set of common fixed points of an infinite family of closed and
uniformly quasi -φ- asymptotically nonexpansive mappings, the set of the variational inequal-
ity for an α-inverse-strongly monotone operator, the set of solutions of a system of equilibrium
problems. Moreover, we obtain a strong convergence theorem for the sequences generated by
this process in the framework Banach spaces. The results presented in this paper improve and
generalize some well-known results in the literature.

Keywords Modified block iterative algorithm Inverse-strongly monotone operator Varia-
tional inequality A system of equilibrium problem Uniformly quasi-φ-asympto
tically nonexpansive mapping.

1. Introduction

Let C be a nonempty closed convex subset of a real Banach space E with ‖ · ‖ and E∗ the
dual space of E and A : C → E∗ be an operator. The classical variational inequality problem
for an operator A is to find x∗ ∈ C such that

〈Ax∗, y − x∗〉 ≥ 0, ∀y ∈ C. (1)

The set of solution of (1) is denote by V I(A,C). 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) α−inverse-strongly monotone if there exists a constant α > 0 such that

〈Ax−Ay, x− y〉 ≥ α‖x− y‖2, ∀x, y ∈ C.

Such a problem is connected with the convex minimization problem, the complementary prob-
lem, the problem of finding a point x∗ ∈ E satisfying Ax∗ = 0.
∗This work was completed with the support by the National Research University Project of Thailand’s Office of

the Higher Education Commission (under NRU-CSEC project no.54000267).

ISSN 1078-6236 International Institute for General Systems Studies, Inc.

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 123-148

Quasi-



Let {fi}i∈Γ : C × C −→ R be a bifunction, {ϕi}i∈Γ : C −→ R be a real-valued function,
where Γ is an arbitrary index set. The system of equilibrium problems, is to find x ∈ C such
that

fi(x, y) ≥ 0, i ∈ Γ, ∀y ∈ C. (2)

If Γ is a singleton, then problem (2) reduces to the equilibrium problem, is to find x ∈ C such
that

f(x, y) ≥ 0, ∀y ∈ C. (3)

The above formulation (3) was shown in [5] to cover monotone inclusion problems, sad-
dle point problems, variational inequality problems, minimization problems, optimization prob-
lems, variational inequality problems, vector equilibrium problems, Nash equilibria in noncoop-
erative 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
[5, 13] and references therein. Some solution methods have been proposed to solve the EP (f);
see, for example, [5, 13, 15, 16, 21, 24, 30, 29, 28, 35, 46] and references therein.

For each p > 1, the generalized duality mapping Jp : E → 2E
∗

is defined by

Jp(x) = {x∗ ∈ E∗ : 〈x, x∗〉 = ‖x‖p, ‖x∗‖ = ‖x‖p−1}

for all x ∈ E. In particular, J = J2 is called the normalized duality mapping. 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〉+ ‖y‖2, ∀x, y ∈ E. (4)

As well know 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 φ that

(‖x‖ − ‖y‖)2 ≤ φ(x, y) ≤ (‖x‖+ ‖y‖)2, ∀x, y ∈ E. (5)

If E is a Hilbert space, then φ(x, y) = ‖x − y‖2, for all x, y ∈ E. On the author hand, the
generalized projection (Alber [2]) ΠC : E → C is a map that assigns to an arbitrary point
x ∈ E the minimum point of the functional φ(x, y), that is, ΠCx = x̄, where x̄ is the solution
to the minimization problem

φ(x̄, x) = inf
y∈C

φ(y, x), (6)

existence and uniqueness of the operator ΠC follows from the properties of the functional
φ(x, y) and strict monotonicity of the mapping J (see, for example, [1, 2, 12, 17, 37]).

Remark 1.1. If E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E,
φ(x, y) = 0 if and only if x = y. It is sufficient to show that if φ(x, y) = 0 then x = y. From
( 4 ), 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 [12, 37] for more details.

124 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



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 ∈ C and a mapping T is said to be nonexpansive
if ‖Tx − Ty‖ ≤ ‖x − y‖, ∀x, y ∈ C. A point x ∈ 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 ∈ C : Tx = x}. Recall that
a point p in C is said to be an asymptotic fixed point of T [31] 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 F̃ (T ).

A mapping T from C into itself is said to be relatively nonexpansive [25, 36, 45] if F̃ (T ) =
F (T ) and φ(p, Tx) ≤ φ(p, x) for all x ∈ C and p ∈ F (T ). The asymptotic behavior of
a relatively nonexpansive mapping was studied in [6, 7, 8]. T is said to be φ-nonexpansive, if
φ(Tx, Ty) ≤ φ(x, y) for x, y ∈ C. T is said to be relatively quasi-nonexpansive if F (T ) 6= ∅
and φ(p, Tx) ≤ φ(p, x) for all x ∈ C and p ∈ F (T ). T is said to be quasi-φ-asymptotically
nonexpansive if F (T ) 6= ∅ and there exists a real sequence {kn} ⊂ [1,∞) with kn → 1 such
that φ(p, Tnx) ≤ knφ(p, x) for all n ≥ 1 x ∈ C and p ∈ F (T ).

We note that the class of relatively quasi-nonexpansive mappings is more general than the
class of relatively nonexpansive mappings [6, 7, 8, 23, 33] which requires the strong restriction:
F (T ) = F̃ (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. ([9]) (1) Let {Ti}∞i=1 : C → C be a sequence of mapping. {Ti}∞i=1 is said to be
a family of uniformly quasi-φ-asymptotically nonexpansive mappings, if ∩∞i=1F (Ti) 6= ∅, and
there exists a sequence {kn} ⊂ [1,∞) with kn → 1 such that for each i ≥ 1

φ(p, Tni x) ≤ knφ(p, x), ∀p ∈ ∩∞i=1F (Ti), x ∈ C, ∀n ≥ 1. (7)

(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. (8)

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

continuous.

In 2004, Matsushita and Takahashi [22] introduced the following iteration: a sequence {xn}
defined by

xn+1 = ΠCJ
−1(αnJxn + (1− αn)JTxn), (9)

where the initial guess element x0 ∈ C is arbitrary, {αn} 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 [23] proposed the following hybrid iteration method (it
is also called the CQ method) with generalized projection for relatively nonexpansive mapping

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 125



T in a Banach space E:
x0 ∈ C chosen arbitrarily,
yn = J−1(αnJxn + (1− αn)JTxn),
Cn = {z ∈ C : φ(z, yn) ≤ φ(z, xn)},
Qn = {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0},
xn+1 = ΠCn∩Qnx0.

(10)

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 [14] 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 ∈ C and

xn+1 = ΠCJ
−1(Jxn − λnAxn), (11)

for every n = 1, 2, 3, . . ., where ΠC is the generalized metric projection fromE ontoC, J is the
duality mapping from E into E∗ and {λn} is a sequence of positive real numbers. They proved
that the sequence {xn} generated by (11) converges weakly to some element of V I(A,C).
Takahashi and Zembayashi [39, 40], 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 [41] using the idea of Takahashi and Zembayashi [39]

extend the notion from relatively nonexpansive mappings or φ-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 problen in the framework of Banach spaces. Cholamjiak [10], proved
the following iteration:

zn = ΠCJ
−1(Jxn − λnAxn),

yn = J−1(αnJxn + βnJTxn + γnJSzn),
un ∈ C such that f(un, y) + 1

rn
〈y − un, Jun − Jyn〉 ≥ 0, ∀y ∈ C,

Cn+1 = {z ∈ Cn : φ(z, un) ≤ φ(z, xn),
xn+1 = ΠCn+1x0,

(12)

where J is the duality mapping on E. Assume that αn, βn and γn are sequence in [0, 1]. Then
{xn} converges strongly to q = ΠFx0, where F := F (T ) ∩ F (S) ∩ EP (f) ∩ V I(A,C).

In 2010, Saewan et al. [33] introduced a new hybrid projection iterative scheme which is
difference from the algorithm (12) of Cholamjiak in [10, Theorem 3.1] for two relatively quasi-
nonexpansive mappings in a Banach space. Motivated by the results of Takahashi and Zem-
bayashi [40], Cholumjiak and Suantai [11] 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 ∈ E, x1 = ΠC1x0, C1 = C

yn,i = J−1(αnJxn + (1− αn)JTxn, )

un,i = T fmrm,nT
fm−1
rm−1,n · · ·T

f1
r1,nyn,i

Cn+1 = {z ∈ Cn : supi>1 φ(z, Jun,i) ≤ φ(w, Jxn)},
xn+1 = ΠCn+1x0, n ≥ 1.

(13)

126 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



Then, they proved that under certain appropriate conditions imposed on {αn}, and {rn,i}, the
sequence {xn} converges strongly to ΠCn+1x0.

We note that the block iterative method is a method which often used by many authors to
solve the convex feasibility problem (see, [18, 20], etc.). In 2008, Plubtieng and Ungchittrakool
[27] 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 mathe-
matical programming. Chang et al. [9] proposed the modified block iterative algorithm for
solving the convex feasibility problems for an infinite family of closed and uniformly quasi-φ-
asymptotically nonexpansive mapping, they obtain the strong convergence theorems in a Ba-
nach space. In 2010, Saewan and Kumam [34] obtain 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-φ-asymptotically nonexpansive mappings in a uniformly smooth
and strictly convex Banach space E with Kadec-Klee property.

Very recently, Qin, Cho and Kang [28] purposed the problem of approximating a common
fixed point of two asymptotically quasi-φ-nonexpansive mappings based on hybrid projection
methods. Strong convergence theorems are established in a real Banach space. H. Zegeye, E.
U. Ofoedu and N. Shahzad [46] 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 a α-inverse strongly monotone mapping in
Banach spaces.

Motivated and inspired by the work of Chang et al. [9], Qin et al. [30], Takahashi and
Zembayashi [39], Wattanawitoon and Kumam [41], Zegeye [44] and Saewan and Kumam [34],
we introduce a new modified block hybrid projection algorithm for finding a common element
of the set of the variational inequality for an α-inverse-strongly monotone operator, the set
of solutions of the system of equilibrium problems and the set of common fixed points of an
infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings in a
2-uniformly convex and uniformly smooth Banach space. The results presented in this paper
improve and generalize some well-known results in the literature.

2. Preliminaries

A Banach space E is said to be strictly convex if ‖x+y
2 ‖ < 1 for all x, y ∈ E with

‖x‖ = ‖y‖ = 1 and x 6= y. Let U = {x ∈ 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 ∈ U. It is also said to be uniformly smooth if the limit is attained uniformly
for x, y ∈ 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

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

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 127



A Banach space E is uniformly convex if and only if δ(ε) > 0 for all ε ∈ (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 ∈ [0, 2]; see [3, 38] 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 [12].
(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 reflexive and strictly convex Banach space, then J−1 is norm-weak∗-continuous.
(iii) 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.

(iv) A Banach space E is uniformly smooth if and only if E∗ is uniformly convex.
(v) Each uniformly convex Banach space E has the Kadec-Klee property, that is, for any se-
quence {xn} ⊂ E, if xn ⇀ x ∈ E and ‖xn‖ → ‖x‖, then xn → x.

We also need the following lemmas for the proof of our main results.

Lemma 2.2. (Beauzamy [4] and Xu[42]). If E be a 2-uniformly convex Banach space. Then for
all x, y ∈ 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 [4] and Zalinescu [43]). IfE be a p-uniformly convex Banach space and
let p be a given real number with p ≥ 2. Then for all x, y ∈ E, jx ∈ Jp(x) and jy ∈ Jp(y)

〈x− y, jx − jy〉 ≥
cp

2p−2p
‖x− y‖p,

where Jp is the generalized duality mapping of E and 1
c is the p-uniformly convexity constant

of E.

Lemma 2.4. (Kamimura and Takahashi [17]). Let E be a uniformly convex and smooth Banach
space and let {xn} and {yn} be two sequences of E. If φ(xn, yn)→ 0 and either {xn} or {yn}
is bounded, then ‖xn − yn‖ → 0.

Lemma 2.5. (Alber [2]). Let C be a nonempty closed convex subset of a smooth Banach space
E and x ∈ E. Then x0 = ΠCx if and only if

〈x0 − y, Jx− Jx0〉 ≥ 0, ∀y ∈ C.

Lemma 2.6. (Alber [2, Lemma 2.4]). Let E be a reflexive, strictly convex and smooth Banach
space, let C be a nonempty closed convex subset of E and let x ∈ E. Then

φ(y,ΠCx) + φ(ΠCx, x) ≤ φ(y, x), ∀y ∈ C.

128 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



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 [2]

V (x, x∗) = ‖x‖2 − 2〈x, x∗〉+ ‖x∗‖2, (14)

for all x ∈ E and x∗ ∈ E∗, that is, V (x, x∗) = φ(x, J−1(x∗)).

Lemma 2.7. (Alber [2]). Let E be a reflexive, strictly convex smooth Banach space and let V
be as in (14) . Then

V (x, x∗) + 2〈J−1(x∗)− x, y∗〉 ≤ V (x, x∗ + y∗),

for all x ∈ E and x∗, y∗ ∈ E∗.

A set valued mapping B : E ⇒ E∗ with graph G(B) = {(x, x∗) : x∗ ∈ Bx}, domain
D(B) = {x ∈ E : Bx 6= ∅}, and rang R(B) = ∪{Bx : x ∈ D(B)}. B is said to be monotone
if 〈x − y, x∗ − y∗〉 ≥ 0 whenever (x, x∗) ∈ G(B), (y, y∗) ∈ G(B). We denote a set valued
operator B form E to E∗ by B ⊂ E × E∗. A monotone B is said to be maximal if its graph is
not property contained in the graph of any other monotone operator. IfB is maximal monotone,
then the solution set B−10 is closed and convex.

Let E be a reflexive, strictly convex and smooth Banach space, it is knows that B is a
maximal monotone if and only if R(J + rB) = E∗ for all r > 0. Define the resolvent of B by
Jrx = xr. In other words, Jr = (J + rB)−1J for all r > 0. Jr is a single-valued mapping
from E to D(B). Also, B−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 B by Trx = (Jx − JJrx)/r for
all x ∈ C. We know that Trx ∈ B(Jrx) for all r > 0 and x ∈ E.

Let A be an inverse-strongly monotone mapping of C into E∗ which is said to be hemicon-
tinuous if for all x, y ∈ C, the mapping F of [0, 1] intoE∗, 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 ∈ C, that is,

NC(v) = {x∗ ∈ E∗ : 〈v − y, x∗〉 ≥ 0, ∀y ∈ C}. (15)

Lemma 2.8. (Rockafellar [32]). LetC be a nonempty, closed convex subset of a Banach spaceE
and A is a monotone, hemicontinuous operator of C into E∗. Let B ⊂ E × E∗ be an operator
defined as follows:

Bv =

{
Av +NC(v), v ∈ C;
∅, otherwise. (16)

Then B is maximal monotone and B−10 = V I(A,C).

Lemma 2.9. (Chang et al.[9]) . Let E be a uniformly convex Banach space, r > 0 be a positive
number andBr(0) be a closed ball ofE. Then, for any given sequence {xi}∞i=1 ⊂ Br(0) and for
any given sequence {λi}∞i=1 of positive number with

∑∞
n=1 λn = 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,

‖
∞∑
n=1

λnxn‖2 ≤
∞∑
n=1

λn‖xn‖2 − λiλjg(‖xi − xj‖). (17)

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 129



Lemma 2.10. (Chang et al.[9]) . 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-φ-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 × C → R, let us assume that f
satisfies the following conditions:

(A1) f(x, x) = 0 for all x ∈ C;

(A2) f is monotone, i.e., f(x, y) + f(y, x) ≤ 0 for all x, y ∈ C;

(A3) for each x, y, z ∈ C,
lim
t↓0

f(tz + (1− t)x, y) ≤ f(x, y);

(A4) for each x ∈ C, y 7→ f(x, y) is convex and lower semi-continuous.

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 [5].

Lemma 2.11. (Blum and Oettli [5]). LetC be a closed convex subset of a smooth, strictly convex
and reflexive Banach space E, let f be a bifunction from C×C to R satisfying (A1)-(A4), and
let r > 0 and x ∈ E. Then, there exists z ∈ C such that

f(z, y) +
1

r
〈y − z, Jz − Jx〉 ≥ 0, ∀y ∈ C.

Lemma 2.12. (Takahashi and Zembayashi [39]). Let C be a closed convex subset of a uniformly
smooth, strictly convex and reflexive Banach space E and let f be a bifunction from C × C to
R satisfying conditions (A1)-(A4). For all r > 0 and x ∈ E, define a mapping T fr : E → C
as follows:

T fr x = {z ∈ C : f(z, y) +
1

r
〈y − z, Jz − Jx〉 ≥ 0, ∀y ∈ C}.

Then the following hold:

(1) T fr is single-valued;

(2) T fr is a firmly nonexpansive-type mapping [19], that is, for all x, y ∈ E,

〈T fr x− T fr y, JT fr x− JT fr y〉 ≤ 〈T fr x− T fr y, Jx− Jy〉;

(3) F (T fr ) = EP (f);

(4) EP (f) is closed and convex.

Lemma 2.13. (Takahashi and Zembayashi [39]). Let C be a closed convex subset of a smooth,
strictly convex, and reflexive Banach space E, let f be a bifunction from C ×C to R satisfying
(A1)-(A4) and let r > 0. Then, for x ∈ E and q ∈ F (T fr ),

φ(q, T fr x) + φ(T fr x, x) ≤ φ(q, x).

130 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



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-φ-asymptotically nonexpansive mappings, and the solution set of
variational inequalities for an α-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). Let A be an α-inverse-strongly monotone mapping
of C into E∗ satisfying ‖Ay‖ ≤ ‖Ay −Au‖, ∀y ∈ C and u ∈ V I(A,C) 6= ∅. Let {Si}∞i=1 :
C → C be an infinite family of closed uniformly Li-Lipschitz continuous and uniformly quasi-
φ-asymptotically nonexpansive mappings with a sequence {kn} ⊂ [1,∞), kn → 1 such that
F := (∩∞i=1F (Si))∩ (∩mj=1EP (fj))(∩V I(A,C)) is a nonempty and bounded subset in C. For
an initial point x0 ∈ E with x1 = ΠC1x0 and C1 = C, we define the sequence {xn} as follows:



vn = ΠCJ
−1(Jxn − λnAxn),

zn = J−1(αn,0Jxn +
∑∞

i=1 αn,iJS
n
i vn),

yn = J−1(βnJxn + (1− βn)Jzn),

un = T fmrm,nT
fm−1
rm−1,n ...T

f2
r2,nT

f1
r1,nyn,

Cn+1 = {z ∈ Cn : φ(z, un) ≤ φ(z, xn) + θn},
xn+1 = ΠCn+1x0, ∀n ≥ 1,

(18)

where J is the duality mapping on E, θn = supq∈F (kn − 1)φ(q, xn), for each i ≥ 0, {αn,i}
and {βn} are sequences in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for
some a, b with 0 < a < b < c2α/2, where 1

c is the 2-uniformly convexity constant of E. If∑∞
i=0 αn,i = 1 for all n ≥ 0, lim infn−→∞ βn(1− βn) > 0 and lim infn−→∞ αn,0αn,i > 0 for

all i ≥ 1, then {xn} converges strongly to p ∈ F , where p = ΠFx0.

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 ∈ N. Since for any z ∈ Cn, we
known that

φ(z, un) ≤ φ(z, xn) + θn

is equivalent to

2〈z, Jxn − Jun〉 ≤ ‖xn‖2 − ‖un‖2 + θn.

Hence, Cn+1 is closed and convex.

Next, we show that F ⊂ Cn for all n ≥ 0. Since by the convexity of ‖ · ‖2, property of φ,
Lemma 2.9 and by uniformly quasi-φ-asymptotically nonexpansive of Sn for each q ∈ F ⊂ Cn,

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 131



we have

φ(q, un) = φ(q, T fmrm,n
T fm−1
rm−1,n

...T f2r2,nT
f1
r1,nyn)

≤ φ(q, yn)

= φ(q, J−1(βnJxn + (1− βn)Jzn)

= ‖q‖2 − 2〈q, βnJxn + (1− βn)Jzn〉+ ‖βnJxn + (1− βn)Jzn‖2

≤ ‖q‖2 − 2βn〈q, Jxn〉 − 2(1− βn)〈q, Jzn〉+ βn‖xn‖2 + (1− βn)‖zn‖2

= βnφ(q, xn) + (1− βn)φ(q, zn), (19)

and

φ(q, zn) = φ(q, J−1(αn,0Jxn +
∑∞

i=1 αn,iJS
n
i vn))

= ‖q‖2 − 2〈q, αn,0Jxn +
∑∞

i=1 αn,iJS
n
i vn〉+ ‖αn,0Jxn +

∑∞
i=1 αn,iJS

n
i vn‖2

= ‖q‖2 − 2αn,0〈q, Jxn〉 − 2
∑∞

i=1 αn,i〈q, JSni vn〉+ ‖αn,0Jxn +
∑∞

i=1 αn,iJS
n
i vn‖2

≤ ‖q‖2 − 2αn,0〈q, Jxn〉 − 2
∑∞

i=1 αn,i〈q, JSni vn〉+ αn,0‖Jxn‖2 +
∑∞

i=1 αn,i‖JSni vn‖2
−αn,0αn,jg‖Jvn − JSnj vn‖

= ‖q‖2 − 2αn,0〈q, Jxn〉+ αn,0‖Jxn‖2 − 2
∑∞

i=1 αn,i〈q, JSni vn〉
+
∑∞

i=1 αn,i‖JSni vn‖2 − αn,0αn,jg‖Jvn − JSnj vn‖
= αn,0φ(q, xn) +

∑∞
i=1 αn,iφ(q, Sni vn)− αn,0αn,jg‖Jvn − JSnj vn‖

≤ αn,0φ(q, xn) +
∑∞

i=1 αn,iknφ(q, vn)− αn,0αn,jg‖Jvn − JSnj vn‖.
(20)

It follows from Lemma 2.7, that

φ(q, vn) = φ(q,ΠCJ
−1(Jxn − λnAxn))

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

(21)
Since q ∈ V I(A,C) and A is an α-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.

(22)

By Lemma 2.2 and ‖Axn‖ ≤ ‖Axn −Aq‖, ∀q ∈ V I(A,C), we also have

2〈J−1(Jxn − λnAxn)− xn,−λnAxn〉 = 2〈J−1(Jxn − λnAxn)− J−1(Jxn),−λnAxn〉
≤ 2‖J−1(Jxn − λnAxn)− J−1(Jxn)‖‖λnAxn‖
≤ 4

c2
‖JJ−1(Jxn − λnAxn)− JJ−1(Jxn)‖‖λnAxn‖

= 4
c2
‖Jxn − λnAxn − Jxn‖‖λnAxn‖

= 4
c2
‖λnAxn‖2

= 4
c2
λ2
n‖Axn‖2

≤ 4
c2
λ2
n‖Axn −Aq‖2.

(23)

132 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



Substituting (22) and (23) into (21), we have

φ(q, vn) ≤ φ(q, xn)− 2αλn‖Axn −Aq‖2 + 4
c2
λ2
n‖Axn −Aq‖2

= φ(q, xn) + 2λn( 2
c2
λn − α)‖Axn −Aq‖2

≤ φ(q, xn).
(24)

Substituting (24) into (20), we also have

φ(q, zn) ≤ αn,0φ(q, xn) +
∑∞

i=1 αn,iknφ(q, xn)− αn,0αn,jg‖Jvn − JSnj vn‖
≤ αn,0knφ(q, xn) +

∑∞
i=1 αn,iknφ(q, xn)− αn,0αn,jg‖Jvn − JSnj vn‖

= knφ(q, xn)− αn,0αn,jg‖Jvn − JSnj vn‖
≤ φ(q, xn) + supq∈F (kn − 1)φ(q, xn)− αn,0αn,jg‖Jvn − JSnj vn‖
= φ(q, xn) + θn − αn,0αn,jg‖Jvn − JSnj vn‖
≤ φ(q, xn) + θn.

(25)

and substituting (25) into (19), we obtain

φ(q, un) ≤ φ(q, xn) + θn. (26)

Thus, this show that q ∈ 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 = ΠCnx0

and xn+1 = ΠCn+1x0,∈ Cn+1 ⊂ Cn we have

φ(xn, x0) ≤ φ(xn+1, x0), ∀n ≥ 0. (27)

Form Lemma 2.6, it follows that

φ(xn, x0) = φ(ΠCnx0, x0)
≤ φ(q, x0)− φ(q, xn)
≤ φ(q, x0), ∀q ∈ F.

(28)

By (27) and (28), then {φ(xn, x0)} are nondecreasing and bounded. So, we obtain that
lim
n→∞

φ(xn, x0) exists. In particular, by (5), the sequence {(‖xn‖− ‖x0‖)2} is bounded. This

implies {xn} is also bounded. We denote

M := sup
n≥0
{‖xn‖} <∞. (29)

Moreover, by the definition of θn and (29), it follows that

θn −→ 0 as n −→∞. (30)

Next, we show that {xn} is a Cauchy sequence in C. Since xm = ΠCmx0 ∈ Cm ⊂ Cn, for
m > n, by Lemma 2.6, we have

φ(xm, xn) = φ(xm,ΠCnx0)
≤ φ(xm, x0)− φ(ΠCnx0, x0)
= φ(xm, x0)− φ(xn, x0).

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 133



Since limn−→∞ φ(xn, x0) exists and we taking m,n→∞ then, we get φ(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 and there exist a point p ∈ C such that xn → p as n→∞.

Now, we claim that ‖Jun−Jxn‖ → 0, as n→∞. By definition of xn = ΠCnx0, we have

φ(xn+1, xn) = φ(xn+1,ΠCnx0)
≤ φ(xn+1, x0)− φ(ΠCnx0, x0)
= φ(xn+1, x0)− φ(xn, x0).

Since lim
n→∞

φ(xn, x0) exists, we also have

lim
n→∞

φ(xn+1, xn) = 0. (31)

Again form Lemma 2.4, that
lim
n→∞

‖xn+1 − xn‖ = 0. (32)

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

lim
n→∞

‖Jxn+1 − Jxn‖ = 0. (33)

Since xn+1 = ΠCn+1x0 ∈ Cn+1 ⊂ Cn and the definition of Cn+1, we have

φ(xn+1, un) ≤ φ(xn+1, xn) + θn.

By (30) and (31), that
lim
n→∞

φ(xn+1, un) = 0. (34)

Applying Lemma 2.4, we have

lim
n→∞

‖xn+1 − un‖ = 0. (35)

Since
‖un − xn‖ = ‖un − xn+1 + xn+1 − xn‖

≤ ‖un − xn+1‖+ ‖xn+1 − xn‖
It follows from (32) and (35), that

lim
n→∞

‖un − xn‖ = 0. (36)

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

lim
n→∞

‖Jun − Jxn‖ = 0. (37)

Next, we will show that xn → p ∈ F := ∩mj=1EP (fj) ∩ (∩∞i=1F (Si)) ∩ V I(A,C).

(i) We show that xn → p ∈ ∩∞i=1F (Si). Since xn+1 = ΠCn+1x0 ∈ Cn+1 ⊂ Cn, it follow
from (25), we have

φ(xn+1, zn) ≤ φ(xn+1, xn) + θn,

by (30) and (31), we get
lim
n→∞

φ(xn+1, zn) = 0 (38)

134 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



again form Lemma 2.4, that
lim
n→∞

‖xn+1 − zn‖ = 0. (39)

Since
‖zn − xn‖ ≤ ‖zn − xn+1‖+ ‖xn+1 − xn‖

from (32) and (39), we have

lim
n→∞

‖zn − xn‖ = 0. (40)

By using the triangle inequality, we obtain

‖xn+1 − zn‖ ≤ ‖xn+1 − xn‖+ ‖xn − zn‖. (41)

By (32) and (40), we get
lim
n→∞

‖xn+1 − zn‖ = 0. (42)

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

lim
n→∞

‖Jxn+1 − Jzn‖ = 0. (43)

From (67), we note that

‖Jxn+1 − Jzn‖ = ‖Jxn+1 − (αn,0Jxn +
∑∞

i=1 αn,iJS
n
i vn)‖

= ‖αn,0Jxn+1 − αn,0Jxn +
∑∞

i=1 αn,iJxn+1 −
∑∞

i=1 αn,iJS
n
i vn‖

= ‖αn,0(Jxn+1 − Jxn) +
∑∞

i=1 αn,i(Jxn+1 − JSni vn)‖
= ‖

∑∞
i=1 αn,i(Jxn+1 − JSni vn)− αn,0(Jxn − Jxn+1)‖

≥
∑∞

i=1 αn,i‖Jxn+1 − JSni vn‖ − αn,0‖Jxn − Jxn+1‖,

and hence

‖Jxn+1 − JSni vn‖ ≤ 1∑∞
i=1 αn,i

(‖Jxn+1 − Jzn‖+ αn,0‖Jxn − Jxn+1‖). (44)

From (33), (43) and lim inf
n→∞

∑∞
i=1 αn,i > 0, we get

lim
n→∞

‖Jxn+1 − JSni vn‖ = 0. (45)

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

lim
n→∞

‖xn+1 − Sni vn‖ = 0. (46)

Using the triangle inequality, that

‖xn − Sni vn‖ = ‖xn − xn+1 + xn+1 − Sni vn‖
≤ ‖xn − xn+1‖+ ‖xn+1 − Sni vn‖.

From (32) and (46), we have

lim
n→∞

‖xn − Sni vn‖ = 0. (47)

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 135



On the other hand, we observe that

φ(q, xn)− φ(q, un) + θn = ‖xn‖2 − ‖un‖2 − 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 −→∞. (48)

From (19), (20) and (24), we compute

φ(q, un) ≤ φ(q, yn)
≤ βnφ(q, xn) + (1− βn)φ(q, zn)
≤ βnφ(q, xn) + (1− βn)[αn,0φ(q, xn) +

∑∞
i=1 αn,iknφ(q, vn)

−αn,0αn,jg‖Jvn − JSnj vn‖]
= βnφ(q, xn) + (1− βn)αn,0φ(q, xn) + (1− βn)

∑∞
i=1 αn,iknφ(q, vn)

−(1− βn)αn,0αn,jg‖Jvn − JSnj vn‖
≤ βnφ(q, xn) + (1− βn)αn,0φ(q, xn) + (1− βn)

∑∞
i=1 αn,iknφ(q, vn)

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

i=1 αn,ikn[φ(q, xn)−
2λn(α− 2

c2
λn)‖Axn −Aq‖2]

≤ βnφ(q, xn) + (1− βn)αn,0knφ(q, xn) + (1− βn)
∑∞

i=1 αn,iknφ(q, xn)
−(1− βn)

∑∞
i=1 αn,ikn2λn(α− 2

c2
λn)‖Axn −Aq‖2

= βnknφ(q, xn) + (1− βn)knφ(q, xn)− (1− βn)
∑∞

i=1 αn,ikn2λn(α−
2
c2
λn)‖Axn −Aq‖2

≤ knφ(q, xn)− (1− βn)
∑∞

i=1 αn,ikn2λn(α− 2
c2
λn)‖Axn −Aq‖2]

≤ φ(q, xn) + supq∈F (kn − 1)φ(q, xn)−
(1− βn)

∑∞
i=1 αn,ikn2λn(α− 2

c2
λn)‖Axn −Aq‖2

≤ φ(q, xn) + θn − (1− βn)
∑∞

i=1 αn,ikn2λn(α− 2
c2
λn)‖Axn −Aq‖2

and hence

2a(α− 2b
c2

)‖Axn −Aq‖2 ≤ 2λn(α− 2
c2
λn)‖Axn −Aq‖2

≤ 1
(1−βn)

∑∞
i=1 αn,ikn

(φ(q, xn)− φ(q, un) + θn).
(49)

From (48), {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, lim infn−→∞(1− βn) > 0
and lim infn−→∞ αn,0αn,i > 0, for i ≥ 0 and kn −→ 1 as n −→∞, we obtain that

lim
n→∞

‖Axn −Aq‖ = 0. (50)

From Lemma 2.6, Lemma 2.7 and (23), we compute

φ(xn, vn) = φ(xn,ΠCJ
−1(Jxn − λnAxn))

≤ φ(xn, J
−1(Jxn − λnAxn))

= V (xn, Jxn − λnAxn)
≤ V (xn, (Jxn − λnAxn) + λnAxn)− 2〈J−1(Jxn − λnAxn)− xn, λnAxn〉
= φ(xn, xn) + 2〈J−1(Jxn − λnAxn)− xn,−λnAxn〉
= 2〈J−1(Jxn − λnAxn)− xn,−λnAxn〉
≤ 4λ2n

c2
‖Axn −Aq‖2

≤ 4b2

c2
‖Axn −Aq‖2.

136 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



Applying Lemma 2.4 and (50) that

lim
n→∞

‖xn − vn‖ = 0 (51)

and we also obtain
lim
n→∞

‖Jxn − Jvn‖ = 0 (52)

From Sni is continuous, for any i ≥ 1

lim
n→∞

‖Sni xn − Sni vn‖ = 0. (53)

Again by the triangle inequality, we get

‖xn − Sni xn‖ ≤ ‖xn − Sni vn‖+ ‖Sni vn − Sni xn‖.

From (47) and (53), we have

lim
n→∞

‖xn − Sni xn‖ = 0, ∀i ≥ 1. (54)

Since J is uniformly continuous on any bounded subset of E, we obtain

limn→∞ ‖Jxn − JSni xn‖ = 0, ∀i ≥ 1. (55)

Since xn −→ p and J is uniformly continuous, it yields Jxn −→ Jp.
Hence, from (55), we get

JSni xn −→ Jp, ∀i ≥ 1. (56)

Since J−1 : E∗ −→ E is norm-weake*-continuous, we have

Sni xn ⇀ p, for each i ≥ 1. (57)

On the other hand, for each i ≥ 1, we have

|‖Sni xn‖ − ‖p‖| = |‖J(Sni xn)‖ − ‖Jp‖| ≤ ||J(Sni xn)− Jp||.

By (56), we obtain ‖Sni xn‖ −→ ‖p‖ for each i ≥ 1. Since E is uniformly convex Banach
spaces then E has the Kadec-Klee property, we get

Sni xn −→ p for each i ≥ 1.

By the assumption that ∀i ≥ 1, Si is uniformly Li-Lipschitz continuous, hence we have.

‖Sn+1
i xn − Sni xn‖ ≤ ‖Sn+1

i xn − Sn+1
i xn+1‖+ ‖Sn+1

i xn+1 − xn+1‖+
‖xn+1 − xn‖+ ‖xn − Sni xn‖

≤ (Li + 1)‖xn+1 − xn‖+ ‖Sn+1
i xn+1 − xn+1‖+ ‖xn − Sni xn‖.

(58)
By (32) and (54), it follows that ‖Sn+1

i xn − Sni xn‖ → 0. From Sni xn −→ p, we have
Sn+1
i xn → p, that is SiSni xn → p. In view of closeness of Si, we have Sip = p, for all i ≥ 1.

This imply that p ∈ ∩∞i=1F (Si).

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 137



(ii) We show that xn → p ∈ ∩mj=1EP (fj).
Applying (19) and (25), we get φ(p, yn) ≤ φ(p, xn)+θn. From Lemma 2.13 and un = Ωm

n yn,
when Ωj

n = T
Qj
rj,nT

Qj−1
rj−1,n ...T

Q2
r2,nT

Q1
r1,n , j = 1, 2, 3, ...,m, Ω0

n = I , for p ∈ F , we observe that

φ(p, un) = φ(p,Ωm
n yn)

≤ φ(p,Ωm−1
n yn)

...
≤ φ(p,Ωj

nyn)
...

≤ φ(p, yn)
≤ φ(p, xn) + θn ∀j = 1, 2, 3, ...,m.

(59)

Follows from Lemma 2.13, that

φ(un,Ω
j
nyn) ≤ φ(p,Ωj

nyn)− φ(p, un)
≤ φ(p, xn)− φ(p, un) + θn
= ‖p‖2 − 2〈p, Jxn〉+ ‖xn‖2 − (‖p‖2 − 2〈p, Jun〉+ ‖un‖2) + θn
= ‖xn‖2 − ‖un‖2 − 2〈p, Jxn − Jun〉+ θn
≤ ‖xn − un‖(‖xn + un‖) + 2‖p‖‖Jxn − Jun‖+ θn.

(60)
From (36), (37), θn → 0 as n→∞ and Lemma 2.4, we get

lim
n→∞

‖un − Ωj
nyn‖ = 0∀j = 1, 2, 3, ...,m. (61)

By using triangle inequality, we have

‖xn − Ωj
nyn‖ ≤ ‖xn − un‖+ ‖un − Ωj

nyn‖.

From (36) and (61), we have

lim
n→∞

‖xn − Ωj
nyn‖ = 0 ∀j = 1, 2, 3, ...,m. (62)

Again by using triangle inequality, we have

‖Ωj
nyn − Ωj−1

n yn‖ ≤ ‖Ωj
nyn − xn‖+ ‖xn − Ωj−1

n yn‖.

From (62),we also have

lim
n→∞

‖Ωj
nyn − Ωj−1

n yn‖ = 0 ∀j = 1, 2, 3, ...,m. (63)

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

lim
n→∞

‖JΩj
nyn − JΩj−1

n yn‖ = 0 ∀j = 1, 2, 3, ...,m.

From rj,n > 0 we have ‖JΩj
nyn−JΩj−1

n yn‖
rj,n

→ 0 as n→∞ ∀j = 1, 2, 3, ...,m, and

fj(Ω
j
nyn, y) + 1

rj,n
〈y − Ωj

nyn, JΩj
nyn − JΩj−1

n yn〉 ≥ 0, ∀y ∈ C.

138 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



By (A2), that

‖y − Ωj
nyn‖‖JΩj

nyn−JΩj−1
n yn‖

rn
≥ 1

rj,n
〈y − Ωj

nyn, JΩj
nyn − JΩj−1

n yn〉
≥ −fj(Ωj

nyn, y)

≥ fj(y,Ω
j
nyn), ∀y ∈ C,

and Ωj
nyn → p we get f(y, p) ≤ 0 for all y ∈ C. For 0 < t < 1, define yt = ty + (1 − t)p.

Then yt ∈ C which imply that fj(yt, p) ≤ 0. From (A1), we obtain that

0 = fj(yt, yt) ≤ tfj(yt, y) + (1− t)fj(yt, p) ≤ tfj(yt, y).

Thus fj(yt, y) ≥ 0. From (A3), we have fj(p, y) ≥ 0 for all y ∈ C and j = 1, 2, 3, ...,m.
Hence p ∈ EP (fj) ∀j = 1, 2, 3, ...,m. This imply that p ∈ ∩mj=1EP (fj).

(iii) We show that xn → p ∈ V I(A,C). Indeed, define B ⊂ E × E∗ by

Bv =

{
Av +NC(v), v ∈ C;
∅, v /∈ C. (64)

By Lemma 2.8, B is maximal monotone and B−10 = V I(A,C). Let (v, w) ∈ G(B). Since
w ∈ Bv = Av +NC(v), we get w −Av ∈ NC(v).
From vn ∈ C, we have

〈v − vn, w −Av〉 ≥ 0. (65)

On the other hand, since vn = ΠCJ
−1(Jxn − λnAxn). Then by Lemma 2.5, we have

〈v − vn, Jvn − (Jxn − λnAxn)〉 ≥ 0,

and thus
〈v − vn, Jxn−Jvnλn

−Axn〉 ≤ 0. (66)

It follows from (65), (66) and A is monotone and 1
α -Lipschitz continuous, that

〈v − vn, w〉 ≥ 〈v − vn, Av〉
≥ 〈v − vn, Av〉+ 〈v − vn, Jxn−Jvnλn

−Axn〉
= 〈v − vn, Av −Axn〉+ 〈v − zvn, Jxn−Jvnλn

〉
= 〈v − vn, Av −Avn〉+ 〈v − vn, Avn −Axn〉+ 〈v − vn, Jxn−Jvnλn

〉
≥ −‖v − vn‖‖vn−xn‖α − ‖v − vn‖‖Jxn−Jvn‖a

≥ −H(‖vn−xn‖α + ‖Jxn−Jvn‖
a ),

where H = supn≥1 ‖v − vn‖. Take the limit as n → ∞, (51) and (52), we obtain 〈v −
p, w〉 ≥ 0. By the maximality of B we have p ∈ B−10, that is p ∈ V I(A,C).

Finally, we show that p = ΠFx0. From xn = ΠCnx0, we have 〈Jx0 − Jxn, xn − z〉 ≥ 0,
∀z ∈ Cn. Since F ⊂ Cn, we also have

〈Jx0 − Jxn, xn − y〉 ≥ 0, ∀y ∈ F.

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 139



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 ∈ 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). Let A be an α-inverse-strongly monotone mapping of
C into E∗ satisfying ‖Ay‖ ≤ ‖Ay − Au‖, ∀y ∈ C and u ∈ V I(A,C) 6= ∅. Let S : C → C
be a closed L-Lipschitz continuous and quasi-φ-asymptotically nonexpansive mappings with a
sequence {kn} ⊂ [1,∞), kn → 1 such that F := (F (S)) ∩ (∩mj=1EP (fj)) ∩ (V I(A,C)) is a
nonempty and bounded subset in C. For an initial point x0 ∈ E with x1 = ΠC1x0 and C1 = C,
we define the sequence {xn} as follows:

vn = ΠCJ
−1(Jxn − λnAxn),

zn = J−1(αnJxn + (1− αn)JSnvn),
yn = J−1(βnJxn + (1− βn)Jzn),

un = T fmrm,nT
fm−1
rm−1,n ...T

f2
r2,nT

f1
r1,nyn,

Cn+1 = {z ∈ Cn : φ(z, un) ≤ φ(z, xn) + θn},
xn+1 = ΠCn+1x0, ∀n ≥ 1,

(67)

where J is the duality mapping onE, θn = supq∈F (kn−1)φ(q, xn), {αn}, {βn} are sequences
in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b <
c2α/2, where 1

c is the 2-uniformly convexity constant of E. If lim infn−→∞(1 − βn) > 0 and
lim infn−→∞ αn(1− αn) > 0, then {xn} converges strongly to p ∈ F , where p = ΠFx0.

For a special case that i = 1, 2, we can obtain the following results on a pair of quasi-φ-
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). Let A be an α-inverse-strongly monotone mapping
of C into E∗ satisfying ‖Ay‖ ≤ ‖Ay − Au‖, ∀y ∈ C and u ∈ V I(A,C) 6= ∅. Let S, T :
C → C be two closed quasi-φ-asymptotically nonexpansive mappings and LS , LT -Lipschitz
continuous, respectively with a sequence {kn} ⊂ [1,∞), kn → 1 such that F := F (S) ∩
F (T ) ∩ (∩mj=1EP (fj)) ∩ V I(A,C) is a nonempty and bounded subset in C. For an initial
point x0 ∈ E with x1 = ΠC1x0 and C1 = C, we define the sequence {xn} as follows:

vn = ΠCJ
−1(Jxn − λnAxn),

zn = J−1(αnJxn + βnJS
nvn + γnJT

nvn),
yn = J−1(δnJxn + (1− δn)Jzn),

un = T fmrm,nT
fm−1
rm−1,n ...T

f2
r2,nT

f1
r1,nyn,

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

(68)

140 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



where J is the duality mapping on E, θn = supq∈F (kn − 1)φ(q, xn), {αn}, {βn}, {γn} and
{δn} are sequences in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for some a, b
with 0 < a < b < c2α/2, where 1

c is the 2-uniformly convexity constant ofE. If αn+βn+γn =
1 for all n ≥ 0 and lim infn−→∞ αnβn > 0, lim infn−→∞ αnγn > 0, lim infn−→∞ βnγn > 0
and lim infn−→∞ δn(1− δn) > 0, then {xn} converges strongly to p ∈ 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 spaceE. For each j = 1, 2, ...,m let fj be a bifunction fromC×C to
R which satisfies conditions (A1)-(A4). Let A be an α-inverse-strongly monotone mapping of C
into E∗ satisfying ‖Ay‖ ≤ ‖Ay−Au‖, ∀y ∈ C and u ∈ V I(A,C) 6= ∅. Let {Si}∞i=1 : C →
C be an infinite family of closed quasi-φ- nonexpansive mappings such that F := ∩∞i=1F (Si)∩
(∩mj=1EP (fj)) ∩ V I(A,C) 6= ∅. For an initial point x0 ∈ E with x1 = ΠC1x0 and C1 = C,
we define the sequence {xn} as follows:

vn = ΠCJ
−1(Jxn − λnAxn),

zn = J−1(αn,0Jxn +
∑∞

i=1 αn,iJSivn),
yn = J−1(βnJxn + (1− βn)Jzn),

un = T fmrm,nT
fm−1
rm−1,n ...T

f2
r2,nT

f1
r1,nyn,

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

(69)

where J is the duality mapping on E, {αn,i} and {βn} is sequence in [0, 1], {rj,n} ⊂ [d,∞)
for some d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where 1

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

∑∞
i=0 αn,i = 1 for all n ≥ 0, lim infn−→∞(1− βn) > 0

and lim infn−→∞ αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly to p ∈ F , where
p = ΠFx0.

Proof Since {Si}∞i=1 : C −→ C is an infinite family of closed quasi-φ-nonexpansive
mappings, it is an infinite family of closed and uniformly quasi-φ-asymptotically nonexpan-
sive 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 ∈ F (Si) for
each i ≥ 1, that is, p ∈ ∩∞i=1F (Si). Therefore all conditions in Theorem 3.1 are satisfied. The
conclusion of Corollary 3.4 is obtained from Theorem 3.1 immediately.

Corollary 3.5. [44, Theorem 3.2] Let C be a nonempty closed and convex subset of a 2-uniformly
convex and uniformly smooth Banach spaceE. Let f be a bifunction fromC×C to R satisfying
(A1)-(A4). LetA be an α-inverse-strongly monotone mapping of C into E∗ satisfying ‖Ay‖ ≤
‖Ay − Au‖, ∀y ∈ C and u ∈ V I(A,C) 6= ∅. Let {Si}Ni=1 : C → C be a finite family of
closed quasi-φ- nonexpansive mappings such that F := ∩Ni=1F (Si)∩EP (f)∩V I(A,C) 6= ∅.
For an initial point x0 ∈ E with x1 = ΠC1x0 and C1 = C, we define the sequence {xn} as
follows: 

zn = ΠCJ
−1(Jxn − λnAxn),

yn = J−1(α0Jxn +
∑N

i=1 αiJSizn),
f(un, y) + 1

rn
〈y − un, Jun − Jyn〉 ≥ 0, ∀y ∈ C,

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

(70)

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 141



where J is the duality mapping on E, {αn,i} is sequence in [0, 1], {rn} ⊂ [d,∞) for some
d > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where 1

c is the 2-uniformly
convexity constant of E. If αi ∈ (0, 1) such that

∑N
i=0 αi = 1, then {xn} converges strongly to

p ∈ F , where p = ΠFx0.

Corollary 3.6. Let C be a nonempty closed and convex subset of a uniformly convex and uni-
formly smooth Banach space E. Let f be a bifunction from C × C to R satisfying (A1)-(A4).
Let {Si}∞i=1 : C → C be an infinite family of closed and uniformly quasi-φ-asymptotically
nonexpansive mappings with a sequence {kn} ⊂ [1,∞), kn → 1 and uniformly Li-Lipschitz
continuous such that F := ∩∞i=1F (Si) ∩ EP (f) is a nonempty and bounded subset in C. For
an initial point x0 ∈ E with x1 = ΠC1x0 and C1 = C, we define the sequence {xn} as follows:

yn = J−1(αn,0Jxn +
∑∞

i=1 αn,iJS
n
i xn),

f(un, y) + 1
rn
〈y − un, Jun − Jyn〉 ≥ 0, ∀y ∈ C,

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

(71)

where J is the duality mapping onE, θn = supq∈F (kn−1)φ(q, xn), {αn,i} is sequence in [0, 1],
{rn} ⊂ [a,∞) for some a > 0. If

∑∞
i=0 αn,i = 1 for all n ≥ 0 and lim infn−→∞ αn,0αn,i > 0

for all i ≥ 1, then {xn} converges strongly to p ∈ F , where p = ΠFx0.

4. Deduced 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 4.1. 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 ϕj : C → R be a lower
semicontinuous and convex function. Let A be an α-inverse-strongly monotone mapping of C
into H satisfying ‖Ay‖ ≤ ‖Ay −Au‖, ∀y ∈ C and u ∈ V I(A,C) 6= ∅. Let {Si}∞i=1 : C →
C be an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings
with a sequence {kn} ⊂ [1,∞), kn → 1 and uniformly Li-Lipschitz continuous such that
F := ∩∞i=1F (Si)∩ (∩mj=1GMEP (fj , Bj , ϕj))∩ V I(A,C) is a nonempty and bounded subset
in C. For an initial point x0 ∈ H with x1 = PC1x0 and C1 = C, we define the sequence {xn}
as follows: 

zn = PC(xn − λnAxn),
yn = αn,0xn +

∑∞
i=1 αn,iS

n
i zn,

un = T fmrm,nT
fm−1
rm−1,n ...T

f2
r2,nT

f1
r1,nyn,

Cn+1 = {z ∈ Cn : ‖z − un‖ ≤ ‖z − xn‖+ θn},
xn+1 = PCn+1x0, ∀n ≥ 0,

(72)

where θn = supq∈F (kn − 1)‖q − xn‖, {αn,i} is sequence in [0, 1], {rj,n} ⊂ [a,∞) for some
a > 0 and {λn} ⊂ [a, b] for some a, b with 0 < a < b < α/2. If

∑∞
i=0 αn,i = 1 for all n ≥ 0

142 Saewan:A New Modified Block Iterative Algorithm for a System of. . . . . .



and lim infn−→∞ αn,0αn,i > 0 for all i ≥ 1, then {xn} converges strongly to p ∈ F , where
p = ΠFx0.

Remark 4.2. Theorem 4.1 improve and extend the Corollary 3.7 in Zegeye [44] in the aspect
for the mappings, we extend the mappings from a finite family of closed relatively quasi-
nonexpansive mappings to more general an infinite family of closed and uniformly quasi-φ-
asymptotically nonexpansive mappings.

5. Applications

5.1 Zero Points of an Inverse-strongly Monotone Operator

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 α-inverse-strongly monotone,

(C2) A−10 = {u ∈ E : Au = 0} 6= ∅.

Theorem 5.1. 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 ϕj : C → R be a lower semicontinuous and convex function. Let A be an
operator of E into E∗ satisfying (C1) and (C2). Let {Si}∞i=1 : C → C be an infinite family of
closed uniformly Li-Lipschitz continuous and uniformly quasi-φ-asymptotically nonexpansive
mappings with a sequence {kn} ⊂ [1,∞), kn → 1 such that

F := ∩∞i=1F (Si) ∩ (∩mj=1GMEP (fj , Bj , ϕj)) ∩A−10

is a nonempty and bounded subset in C. For an initial point x0 ∈ E with x1 = ΠC1x0 and
C1 = C, we define the sequence {xn} as follows:

zn = J−1(αn,0Jxn +
∑∞

i=1 αn,iJS
n
i vn),

yn = J−1(βnJxn + (1− βn)Jzn),

un = T fmrm,nT
fm−1
rm−1,n ...T

f2
r2,nT

f1
r1,nyn,

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

(73)

where J is the duality mapping on E, θn = supq∈F (kn − 1)φ(q, xn), for each i ≥ 0, {αn,i}
and {βn} are sequences in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for
some a, b with 0 < a < b < c2α/2, where 1

c is the 2-uniformly convexity constant of E. If∑∞
i=0 αn,i = 1 for all n ≥ 0, lim infn−→∞(1− βn) > 0 and lim infn−→∞ αn,0αn,i > 0 for all

i ≥ 1, then {xn} converges strongly to p ∈ F , where p = ΠFx0.

Proof. Setting C = E in Corollary 3.4, we also get ΠE = I. We also have V I(A,C) =
V I(A,E) = {x ∈ E : Ax = 0} 6= ∅ and then the condition ‖Ay‖ ≤ ‖Ay − Au‖ holds for all
y ∈ E and u ∈ A−10. So, we obtain the result.

Advances in Systems Science and Applications (2011), Vol. 11, No. 1-2 143



5.2 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}. (74)

If A : K −→ E∗ is an operator, then an element u ∈ K is called a solution of the complemen-
tarity problem ([37]) if

Au ∈ K∗, and 〈u,Au〉 = 0. (75)

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

Theorem 5.2. 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 ϕj : C → R be a lower semicontinuous and convex function. Let A be an α-
inverse-strongly monotone mapping of K into E∗ satisfying ‖Ay‖ ≤ ‖Ay − Au‖, ∀y ∈ K
and u ∈ CP (A,K) 6= ∅. Let {Si}∞i=1 : K → K be an infinite family of closed uniformly
Li-Lipschitz continuous and uniformly quasi-φ-asymptotically nonexpansive mappings with a
sequence {kn} ⊂ [1,∞), kn → 1 such that F := ∩∞i=1F (Si) ∩ (∩mj=1GMEP (fj , Bj , ϕj)) ∩
CP (A,K) is a nonempty and bounded subset in K. For an initial point x0 ∈ E with x1 =
ΠK1x0 and K1 = K, we define the sequence {xn} as follows:

vn = ΠKJ
−1(Jxn − λnAxn),

zn = J−1(αn,0Jxn +
∑∞

i=1 αn,iJS
n
i vn),

yn = J−1(βnJxn + (1− βn)Jzn),

un = T fmrm,nT
fm−1
rm−1,n ...T

f2
r2,nT

f1
r1,nyn,

Kn+1 = {z ∈ Kn : φ(z, un) ≤ φ(z, xn) + θn},
xn+1 = ΠKn+1x0, ∀n ≥ 0,

(76)

where J is the duality mapping on E, θn = supq∈F (kn − 1)φ(q, xn), for each i ≥ 0, {αn,i}
and {βn} are sequences in [0, 1], {rj,n} ⊂ [d,∞) for some d > 0 and {λn} ⊂ [a, b] for
some a, b with 0 < a < b < c2α/2, where 1

c is the 2-uniformly convexity constant of E. If∑∞
i=0 αn,i = 1 for all n ≥ 0, lim infn−→∞(1− βn) > 0 and lim infn−→∞ αn,0αn,i > 0 for all

i ≥ 1, then {xn} converges strongly to p ∈ F , where p = ΠFx0.

Proof . As in the proof of Takahashi in [37, Lemma 7.11], we get that V I(A,K) = CP (A,K). So,
we obtain the result.

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