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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 941090, 20 pages
doi:10.1155/2011/941090
Research Article
Convergence Analysis for a System of
Generalized Equilibrium Problems and a Countable
Family of Strict Pseudocontractions
Prasit Cholamjiak
1
and Suthep Suantai
1, 2
1
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Suthep Suantai,
Received 18 October 2010; Accepted 27 December 2010
Academic Editor: Jen Chih Yao
Copyright q 2011 P. Cholamjiak and S. Suantai. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We introduce a new iterative algorithm for a system of generalized equilibrium problems and
a countable family of strict pseudocontractions in Hilbert spaces. We then prove that the sequence
generated by the proposed algorithm converges strongly to a common element in the solutions set
of a system of generalized equilibrium problems and the common fixed points set of an infinitely
countable family of strict pseudocontractions.
1. Introduction
Let H be a real Hilbert space with the inner product ·, · and inducted norm ·.LetC
be a nonempty, closed, and convex subset of H.Let{f
k
}
k∈Λ
: C × C → be a family of
bifunctions, and let {A
k
}
k∈Λ
: C → H be a family of nonlinear mappings, where Λ is an
arbitrary index set. The system of generalized equilibrium problems is to find x ∈ C such that
f
k
x, y
A
k
x, y − x
≥ 0, ∀y ∈ C, k ∈ Λ. 1.1
If Λ is a singleton, then 1.1 reduces to find x ∈ C such that
f
x, y
Ax, y − x
≥ 0, ∀y ∈ C. 1.2
The solutions set of 1.2 is denoted by GEPf, A.Iff ≡ 0, then the solutions set of 1.2
is denoted by VIC, A,andifA ≡ 0, then the solutions set of 1.2 is denoted by EPf.
2 Fixed Point Theory and Applications
The problem 1.2 is very general in the sense that it includes, as special cases, optimization
problems, variational inequalities, minimax problems, and the Nash equilibrium problem
in noncooperative games; see also 1, 2. Some methods have been constructed to solve the
system of equilibrium problems see, e.g., 3–7. R ecall that a mapping A : C → H is
said to be
1 monotone if
Ax − Ay, x − y
≥ 0, ∀x, y ∈ C, 1.3
2 α-inverse-strongly monotone if there exists a constant α>0suchthat
Ax − Ay, x − y
≥ α
Ax − Ay
2
, ∀x, y ∈ C.
1.4
It is easy to see that if A is α-inverse-strongly monotone, then A is monotone and
1/α-Lipschitz.
For solving the equilibrium problem, let us assume that f satisfies the following
conditions:
A1 fx, x0forallx ∈ C,
A2 f is monotone, that is, fx, yfy, x ≤ 0forallx, y ∈ C,
A3 for each x, y, z ∈ C, lim
t → 0
ftz 1 − tx, y ≤ fx, y,
A4 for each x ∈ C, y → fx, y is convex and lower semicontinuous.
Throughout this paper, we denote the fixed points set of a nonlinear mapping
T : C → C by FT{x ∈ C : Tx x}.RecallthatT is said to be a κ-strict pseudocontraction if
there exists a constant 0 ≤ κ<1suchthat
Tx − Ty
2
≤
x − y
2
κ
I − T
x −
I − T
y
2
.
1.5
It is well known that 1.5 is equivalent to
Tx − Ty,x − y
≤
x − y
2
−
1 − κ
2
I − T
x −
I − T
y
2
.
1.6
It is worth mentioning that the class of strict pseudocontractions includes properly the
class of nonexpansive mappings. It is also known that every κ-strict pseudocontraction is
1 κ/1 − κ-Lipschitz; see 8.
In 1953, Mann 9 introduced the iteration as follows: a sequence {x
n
} defined by
x
0
∈ C and
x
n1
α
n
x
n
1 − α
n
Sx
n
,n≥ 0, 1.7
where {α
n
}
∞
n0
⊂ 0, 1.IfS is a nonexpansive mapping with a fixed point and the control
sequence {α
n
}
∞
n0
is chosen so that
∞
n0
α
n
1 − α
n
∞, then the sequence {x
n
} defined
Fixed Point Theory and Applications 3
by 1.7 converges weakly to a fixed point of S this is also valid in a uniformly convex
Banach space with the Fr
´
echet differentiable norm 10.
In 1967, Browder and Petryshyn 11 introduced the class of strict pseudocontractions
and proved existence and weak convergence theorems in a real Hilbert setting by using Mann
iterative algorithm 1.7 with a constant sequence α
n
α for all n ≥ 0. Recently, Marino
and Xu 8 and Zhou 12 extended the results of Browder and Petryshyn 11 to Mann’s
iteration process 1.7. Since 1967, the construction of fixed points for pseudocontractions via
the iterative process has been extensively investigated by many authors see, e.g., 13–22.
Let C be a nonempty, closed, and convex subset of a real Hilbert space H.Let
S : C → C be a nonexpansive mapping, f : C × C →
a bifunction, and let A : C → H be
an inverse-strongly monotone mapping.
In 2008, Moudafi 23 introduced an iterative m ethod for approximating a common
element of the fixed points set of a nonexpansive mapping S and the solutions set of a
generalized equilibrium problem GEPf, A as follows: a sequence {x
n
} defined by x
0
∈ C
and
f
y
n
,y
Ax
n
,y− y
n
1
r
n
y − y
n
,y
n
− x
n
≥ 0, ∀y ∈ C,
x
n1
α
n
x
n
1 − α
n
Sy
n
,n≥ 1,
1.8
where {α
n
}
∞
n0
⊂ 0, 1 and {r
n
}
∞
n0
⊂ 0, ∞. He proved that the sequence {x
n
} generated by
1.8 converges weakly to an element in GEPf, A ∩ FS under suitable conditions.
Due to the weak convergence, recently, S. Takahashi and W. Takahashi 24 introduced
another modification iterative method of 1.8 for finding a common element of the fixed
points set of a nonexpansive mapping and the solutions set of a generalized equilibrium
problem in the framework of a real Hilbert space. To be more precise, they proved the
following theorem.
Theorem 1.1 see 24. Let C be a closed convex subset of a real Hilbert space H,andlet
f : C × C →
be a bifunction satisfying (A1)–(A4). Let A be an α-inverse-strongly monotone
mapping of C into H,andletS be a nonexpansive mapping of C into itself such that FS ∩
GEPf, A
/
∅.Letu ∈ C and x
1
∈ C,andlet{y
n
}⊂C and {x
n
}⊂C be sequences generated by
f
y
n
,y
Ax
n
,y− y
n
1
r
n
y − y
n
,y
n
− x
n
≥ 0, ∀y ∈ C,
x
n1
β
n
x
n
1 − β
n
S
α
n
u
1 − α
n
y
n
,n≥ 1,
1.9
where {α
n
}
∞
n1
⊂ 0, 1, {β
n
}
∞
n1
⊂ 0, 1 and {r
n
}
∞
n1
⊂ 0, 2α satisfy
i lim
n →∞
α
n
0 and
∞
n1
α
n
∞,
ii 0 <c≤ β
n
≤ d<1,
iii 0 <a≤ r
n
≤ b<2α,
iv lim
n →∞
r
n
− r
n1
0.
Then, {x
n
} converges strongly to z P
FS ∩ GEPf,A
u.
4 Fixed Point Theory and Applications
Recently, Yao et al. 25 introduced a new modified Mann iterative algorithm which is
different from those in the literature for a nonexpansive mapping in a real Hilbert space. To
be more precise, they proved the following theorem.
Theorem 1.2 see 25. Let C be a nonempty, closed, and convex subset of a real Hilbert space H.
Let S : C → C be a nonexpansive mapping such that FS
/
∅.Let{α
n
}
∞
n0
,andlet{β
n
}
∞
n0
be
two real sequences in 0, 1. For given x
0
∈ C arbitrarily, let the sequence {x
n
}, n ≥ 0, be generated
iteratively by
y
n
P
C
1 − α
n
x
n
,
x
n1
1 − β
n
x
n
β
n
Sy
n
.
1.10
Suppose that the following conditions are satisfied:
i lim
n →∞
α
n
0 and
∞
n0
α
n
∞,
ii 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1,
then, the sequence {x
n
} generated by 1.10 strongly converges to a fixed point of S.
We know the following crucial lemmas concerning the equilibrium problem in Hilbert
spaces.
Lemma 1.3 see 1. Let C be a nonempty, closed, and convex subset of a real Hilbert space H,letf
be a bifunction from C × C to
satisfying (A1)–(A4). Let r>0 and x ∈ H. Then, there exists z ∈ C
such that
f
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C.
1.11
Lemma 1.4 see 26. Let C be a nonempty , closed, and convex subset of a real Hilbert space H.Let
f be a bifunction from C × C to
satisfying (A1)–(A4). For x ∈ H and r>0, define the mapping
T
f
r
: H → 2
C
as follows:
T
f
r
x
z ∈ C : f
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C
. 1.12
Then, the following statements hold:
1 T
f
r
is single-valued,
2 T
f
r
is firmly nonexpansive, that is, for any x, y ∈ H,
T
f
r
x − T
f
r
y
2
≤
T
f
r
x − T
f
r
y, x − y
,
1.13
3 FT
f
r
EPf,
4 EPf is closed and convex.
Fixed Point Theory and Applications 5
Let C be a nonempty, closed, a nd convex subset of a real Hilbert space H.Letr
k
> 0
for each k ∈{1, 2, ,M}.Let{f
k
}
M
k1
: C × C → be a family of bifunctions, let {A
k
}
M
k1
:
C → H be a family of α
k
-inverse-strongly monotone mappings, and let {T
n
}
∞
n1
: C → C
be a countable f amily of κ-strict pseudocontractions. For each k ∈{1, 2, ,M},denotethe
mapping T
f
k
,A
k
r
k
: C → C by T
f
k
,A
k
r
k
: T
f
k
r
k
I − r
k
A
k
,whereT
f
k
r
k
: H → C is the mapping
defined as in Lemma 1.4.
Motivated and inspired by Marino and Xu 8,Moudafi23, S. Takahashi and W.
Takahashi 24,andYaoetal.25, we consider the following iteration: x
1
∈ C and
y
n
P
C
1 − α
n
x
n
,
u
n
T
f
M
,A
M
r
M
T
f
M−1
,A
M−1
r
M−1
···T
f
2
,A
2
r
2
T
f
1
,A
1
r
1
y
n
,
x
n1
β
n
x
n
1 − β
n
γu
n
1 − γ
T
n
u
n
,n≥ 1,
1.14
where {α
n
}
∞
n1
⊂ 0, 1, {β
n
}
∞
n1
⊂ 0, 1 and γ ∈ 0, 1.
In this paper, we first prove a path convergence result for a nonexpansive mapping
and a system of generalized equilibrium problems. Then, we prove a strong convergence
theorem of the iteration process 1.14 for a system of generalized equilibrium problems and
a countable family of strict pseudocontractions in a real Hilbert space. Our results extend the
main results obtained by Yao et al. 25 in several aspects.
2. Preliminaries
Let C be a nonempty, closed, and convex subset of a real Hilbert space H.Foreachx ∈ H,
there exists a unique nearest point in C, denoted by P
C
x,suchthatx−P
C
x min
y∈C
x −y.
P
C
is called the metric projection of H onto C.Itisalsoknownthatforx ∈ H and z ∈ C,
z P
C
x is equivalent to x − z, y − z≤0forally ∈ C.Furthermore,
y − P
C
x
2
x − P
C
x
2
≤
x − y
2
,
2.1
for all x ∈ H, y ∈ C. In a real Hilbert space, we also know that
λx
1 − λ
y
2
λ
x
2
1 − λ
y
2
− λ
1 − λ
x − y
2
,
2.2
for all x, y ∈ H and λ ∈ 0, 1.
In the sequel, we need the following lemmas.
Lemma 2.1 see 27, 28. Let E be a real uniformly convex Banach space, and let C be a nonempty,
closed, and convex subset of E,andletS : C → C be a nonexpansive mapping such that FS
/
∅,
then I − S is demiclosed at zero.
Lemma 2.2 see 29. Let {x
n
} and {z
n
} be two sequences in a Banach space E such that
x
n1
β
n
x
n
1 − β
n
z
n
,n≥ 1, 2.3
6 Fixed Point Theory and Applications
where {β
n
}
∞
n1
satisfies conditions: 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1.Iflim sup
n →∞
z
n1
−
z
n
−x
n1
− x
n
≤ 0,thenx
n
− z
n
→0 as n →∞.
Lemma 2.3 see 30. Assume that {a
n
}
∞
n1
is a sequence of nonnegative real numbers such that
a
n1
≤
1 − γ
n
a
n
γ
n
δ
n
,n≥ 1, 2.4
where {γ
n
}
∞
n1
is a sequence in 0, 1 and {δ
n
}
∞
n1
is a sequence in such that
a
∞
n1
γ
n
∞; b lim sup
n →∞
δ
n
≤ 0 or
∞
n1
|γ
n
δ
n
| < ∞.
Then, lim
n →∞
a
n
0.
Lemma 2.4 see 31. Let C be a nonempty , closed, and convex subset of a real Hilbert space H.Let
the mapping A : C → H be α-inverse-strongly monotone, and let r>0 be a constant. Then, we have
I − rA
x −
I − rA
y
2
≤
x − y
2
r
r − 2α
Ax − Ay
2
,
2.5
for all x, y ∈ C. In particular, if 0 ≤ r ≤ 2α,thenI − rA is nonexpansive.
To deal with a family of mappings, the following conditions are introduced: let C
be a subset of a real Hilbert space H,andlet{T
n
}
∞
n1
be a family of mappings of C such
that
∞
n1
FT
n
/
∅. Then, {T
n
} is said to satisfy the AKTT-condition 32 if for each bounded
subset B of C,
∞
n1
sup
{
T
n1
z − T
n
z
: z ∈ B
}
< ∞.
2.6
Lemma 2.5 see 32. Let C be a nonempty and closed subset of a Hilbert space H,andlet{T
n
} be
a family of mappings of C into itself which satisfies the AKTT-condition. Then, for each x ∈ C, {T
n
x}
converges strongly to a point in C. Moreover, let the mapping T be defined by
Tx lim
n →∞
T
n
x, ∀x ∈ C.
2.7
Then, for each bounded subset B of C,
lim sup
n →∞
{
Tz − T
n
z
: z ∈ B
}
0.
2.8
The following results can be found in 33, 34.
Lemma 2.6 see 33, 34. Let C be a closed, and convex subset o f a Hilbert space H. Suppose that
{T
n
}
∞
n1
is a family of κ-strictly pseudocontractive mappings from C into H with
∞
n1
FT
n
/
∅ and
{μ
n
}
∞
n1
is a real sequence in 0, 1 such that
∞
n1
μ
n
1. Then, the following conclusions hold:
1 G :
∞
n1
μ
n
T
n
: C → H is a κ-strictly pseudocontractive mapping,
2 FG
∞
n1
FT
n
.
Fixed Point Theory and Applications 7
Lemma 2.7 see 34. Let C be a closed and convex subset of a Hilbert space H. Suppose that {S
i
}
∞
i1
is a countable family of κ-strictly pseudocontractive mappings of C into itself with
∞
i1
FS
i
/
∅.
For each n ∈
,defineT
n
: C → C by
T
n
x
n
i1
μ
i
n
S
i
x, x ∈ C,
2.9
where {μ
i
n
} is a family of nonnegative numbers satisfying
i
n
i1
μ
i
n
1 for all n ∈ ,
ii μ
i
: lim
n →∞
μ
i
n
> 0 for all i ∈ ,
iii
∞
n1
n
i1
|μ
i
n1
− μ
i
n
| < ∞.
Then,
1 Each T
n
is a κ-strictly pseudocontractive mapping.
2 {T
n
} satisfies AKTT-condition.
3 If T : C → C is defined by
Tx
∞
i1
μ
i
S
i
x, x ∈ C,
2.10
then Tx lim
n →∞
T
n
x and FT
∞
n1
FT
n
∞
i1
FS
i
.
In the sequel, we will write {T
n
},T satisfies the AKTT-condition if {T
n
} satisfies the
AKTT-condition and T is defined by Lemma 2.5 with FT
∞
n1
FT
n
.
3. Path Convergence Results
Let C be a nonempty, closed, and convex subset of a real Hilbert space H.LetS : C → C be
a nonexpansive mapping. Let {f
k
}
M
k1
: C × C → be a family of bifunctions, let {A
k
}
M
k1
:
C → H be a family of α
k
-inverse-strongly monotone mappings, and let r
k
∈ 0, 2α
k
.Foreach
k ∈{1, 2, ,M}, we denote the mapping T
f
k
,A
k
r
k
: C → C by
T
f
k
,A
k
r
k
: T
f
k
r
k
I − r
k
A
k
,
3.1
where T
f
k
r
k
is the mapping defined as in Lemma 1.4.Foreacht ∈ 0, 1,wedefinethemapping
S
t
: C → C as follows:
S
t
x ST
f
M
,A
M
r
M
T
f
M−1
,A
M−1
r
M−1
···T
f
1
,A
1
r
1
P
C
1 − t
x
, ∀x ∈ C.
3.2
By Lemmas 1.42 and 2.4, we know that T
f
k
r
k
and I − r
k
A
k
are nonexpansive for each
k ∈{1, 2 , ,M}. So, the mapping T
f
k
,A
k
r
k
is also nonexpansive for each k ∈{1, 2, ,M}.
8 Fixed Point Theory and Applications
Moreover, we can check easily that S
t
is a contraction. Then, the Banach contraction principle
ensures that there exists a unique fixed point x
t
of S
t
in C,thatis,
x
t
ST
f
M
,A
M
r
M
T
f
M−1
,A
M−1
r
M−1
···T
f
1
,A
1
r
1
P
C
1 − t
x
t
,t∈
0, 1
.
3.3
Theorem 3.1. Let C be a nonempty , closed, and convex subset of a real Hilbert space H.Let
S : C → C be a nonexpansive mapping. Let {f
k
}
M
k1
: C × C → be a family of bifunctions,
let {A
k
}
M
k1
: C → H be a family of α
k
-inverse-strongly monotone mappings, and let r
k
∈
0, 2α
k
.Foreachk ∈{1, 2, ,M}, let the mapping T
f
k
,A
k
r
k
be defined by 3.1. Assume that
F :
M
k1
GEPf
k
,A
k
∩
∞
n1
FT
n
/
∅.Foreacht ∈ 0, 1, let the net {x
t
} be g enerated b y
3.3.Then,ast → 0,thenet{x
t
} converges strongly to an element in F.
Proof. First, we show that {x
t
} is bounded. For each t ∈ 0, 1,lety
t
P
C
1 − tx
t
and
u
t
T
f
M
,A
M
r
M
T
f
M−1
,A
M−1
r
M−1
···T
f
1
,A
1
r
1
y
t
.From3.3,wehaveforeachp ∈ F that
x
t
− p
Su
t
− Sp
≤
u
t
− p
≤
y
t
− p
≤
1 − t
x
t
− p
t
p
. 3.4
It follows that
x
t
− p
≤
p
. 3.5
Hence, {x
t
} is bounded and so are {y
t
} and {u
t
}. Observe that
y
t
− x
t
≤ t
x
t
−→ 0, 3.6
as t → 0since{x
t
} is bounded.
Next, we show that u
t
− x
t
→0ast → 0. Denote Θ
k
T
f
k
,A
k
r
k
T
f
k−1
,A
k−1
r
k−1
···T
f
1
,A
1
r
1
for
any k ∈{1, 2, ,M} and Θ
0
I.Wenotethatu
t
Θ
M
y
t
for each t ∈ 0, 1.FromLemma 2.4,
we have for each k ∈{1, 2, ,M} and p ∈ F that
Θ
k
y
t
− p
2
T
f
k
,A
k
r
k
Θ
k−1
y
t
− T
f
k
,A
k
r
k
Θ
k−1
p
2
T
f
k
r
k
Θ
k−1
y
t
− r
k
A
k
Θ
k−1
y
t
− T
f
k
r
k
Θ
k−1
p − r
k
A
k
Θ
k−1
p
2
≤
Θ
k−1
y
t
− r
k
A
k
Θ
k−1
y
t
−
Θ
k−1
p − r
k
A
k
Θ
k−1
p
2
≤
Θ
k−1
y
t
− p
2
r
k
r
k
− 2α
k
A
k
Θ
k−1
y
t
− A
k
p
2
.
3.7
Fixed Point Theory and Applications 9
It follows that
u
t
− p
2
Θ
M
y
t
− p
2
≤
y
t
− p
2
M
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
t
− A
i
p
2
P
C
1 − t
x
t
− p
2
M
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
t
− A
i
p
2
≤
x
t
− p
t
x
t
2
M
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
t
− A
i
p
2
≤
x
t
− p
2
tM
1
M
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
t
− A
i
p
2
,
3.8
where M
1
sup
0<t<1
{2x
t
− px
t
tx
t
2
}.So,wehave
x
t
− p
2
≤
u
t
− p
2
≤
x
t
− p
2
tM
1
M
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
t
− A
i
p
2
,
3.9
which implies that
lim
t → 0
A
k
Θ
k−1
y
t
− A
k
p
0,
3.10
for each k ∈{1, 2, ,M}.SinceT
f
k
r
k
is firmly nonexpansive for each k ∈{1, 2, ,M},we
have for each p ∈ F and k ∈{1, 2, ,M} that
Θ
k
y
t
− p
2
T
f
k
,A
k
r
k
Θ
k−1
y
t
− T
f
k
,A
k
r
k
Θ
k−1
p
2
T
f
k
r
k
Θ
k−1
y
t
− r
k
A
k
Θ
k−1
y
t
− T
f
k
r
k
Θ
k−1
p − r
k
A
k
Θ
k−1
p
2
≤
Θ
k−1
y
t
− r
k
A
k
Θ
k−1
y
t
−
p − r
k
A
k
p
, Θ
k
y
t
− p
1
2
Θ
k−1
y
t
− r
k
A
k
Θ
k−1
y
t
−
p − r
k
A
k
p
2
Θ
k
y
t
− p
2
−
Θ
k−1
y
t
− r
k
A
k
Θ
k−1
y
t
−
p − r
k
A
k
p
−
Θ
k
y
t
− p
2
10 Fixed Point Theory and Applications
≤
1
2
Θ
k−1
y
t
− p
2
Θ
k
y
t
− p
2
−
Θ
k−1
y
t
− Θ
k
y
t
− r
k
A
k
Θ
k−1
y
t
− A
k
p
2
≤
1
2
Θ
k−1
y
t
− p
2
Θ
k
y
t
− p
2
−
Θ
k−1
y
t
− Θ
k
y
t
2
2r
k
Θ
k−1
y
t
− Θ
k
y
t
A
k
Θ
k−1
y
t
− A
k
p
.
3.11
This implies that
Θ
k
y
t
− p
2
≤
Θ
k−1
y
t
− p
2
−
Θ
k−1
y
t
− Θ
k
y
t
2
2r
k
Θ
k−1
y
t
− Θ
k
y
t
A
k
Θ
k−1
y
t
− A
k
p
≤
Θ
k−1
y
t
− p
2
−
Θ
k−1
y
t
− Θ
k
y
t
2
M
2
A
k
Θ
k−1
y
t
− A
k
p
,
3.12
where M
2
max
1≤k≤M
sup
0<t<1
{2r
k
Θ
k−1
y
t
− Θ
k
y
t
}. This shows that
u
t
− p
2
Θ
M
y
t
− p
2
≤
y
t
− p
2
−
M
i1
Θ
i−1
y
t
− Θ
i
y
t
2
M
2
M
i1
A
i
Θ
i−1
y
t
− A
i
p
≤
x
t
− p
2
tM
1
−
M
i1
Θ
i−1
y
t
− Θ
i
y
t
2
M
2
M
i1
A
i
Θ
i−1
y
t
− A
i
p
.
3.13
Hence,
x
t
− p
2
≤
u
t
− p
2
≤
x
t
− p
2
tM
1
−
M
i1
Θ
i−1
y
t
− Θ
i
y
t
2
M
2
M
i1
A
i
Θ
i−1
y
t
− A
i
p
.
3.14
From 3.10,weobtain
M
i1
Θ
i−1
y
t
− Θ
i
y
t
−→ 0,
3.15
as t → 0. So, we can conclude that
lim
t → 0
Θ
k−1
y
t
− Θ
k
y
t
0,
3.16
Fixed Point Theory and Applications 11
for each k ∈{1, 2, ,M}. Observing
u
n
− y
t
Θ
M
y
t
− y
t
≤
Θ
M
y
t
− Θ
M−1
y
t
Θ
M−1
y
t
− Θ
M−2
y
t
···
Θ
1
y
t
− y
t
,
3.17
it follows by 3.16 that
lim
t → 0
u
t
− y
t
0.
3.18
From 3.6 and 3.18,wehave
lim
t → 0
u
t
− x
t
0.
3.19
Hence,
x
t
− Sx
t
Su
t
− Sx
t
≤
u
t
− x
t
−→ 0, 3.20
as t → 0.
Next, we show that {x
t
} is relatively norm compact as t → 0. Let {t
n
}⊂0, 1 be
a sequence such that t
n
→ 0asn →∞.Putx
n
: x
t
n
.From3.20,weobtain
lim
n →∞
x
n
− Sx
n
0.
3.21
Since {x
n
} is bounded, without loss of generality, we may assume that {x
n
} converges weakly
to x
∗
∈ C. Applying Lemma 2.1 to 3.21, we can conclude that x
∗
∈ FS.
Next, we show that x
∗
∈
M
k1
GEPf
k
,A
k
.NotethatΘ
k
y
n
T
f
k
,A
k
r
k
Θ
k−1
y
n
T
f
k
r
k
Θ
k−1
y
n
− r
k
A
k
Θ
k−1
y
n
for each k ∈{1, 2, ,M}.Hence,foreachy ∈ C and k ∈
{1, 2, ,M},weobtain
f
k
Θ
k
y
n
,y
1
r
k
y − Θ
k
y
n
, Θ
k
y
n
−
Θ
k−1
y
n
− r
k
A
k
Θ
k−1
y
n
≥ 0.
3.22
From A2,wehave
1
r
k
y − Θ
k
y
n
, Θ
k
y
n
−
Θ
k−1
y
n
− r
k
A
k
Θ
k−1
y
n
≥ f
k
y, Θ
k
y
n
, ∀y ∈ C.
3.23
Therefore,
y − Θ
k
y
n
j
,
Θ
k
y
n
j
− Θ
k−1
y
n
j
r
k
A
k
Θ
k−1
y
n
j
≥ f
k
y, Θ
k
y
n
j
, ∀y ∈ C.
3.24
12 Fixed Point Theory and Applications
For each t ∈ 0, 1 and y ∈ C,putz
t
ty 1 − tx
∗
. Then, we have z
t
∈ C.From3.24,
we get that
z
t
− Θ
k
y
n
j
,A
k
z
t
≥
z
t
− Θ
k
y
n
j
,A
k
z
t
−
z
t
− Θ
k
y
n
j
,
Θ
k
y
n
j
− Θ
k−1
y
n
j
r
k
A
k
Θ
k−1
y
n
j
f
k
z
t
, Θ
k
y
n
j
z
t
− Θ
k
y
n
j
,A
k
z
t
− A
k
Θ
k
y
n
j
z
t
− Θ
k
y
n
j
,A
k
Θ
k
y
n
j
− A
k
Θ
k−1
y
n
j
−
z
t
− Θ
k
y
n
j
,
Θ
k
y
n
j
− Θ
k−1
y
n
j
r
k
f
k
z
t
, Θ
k
y
n
j
.
3.25
We note that A
k
Θ
k
y
n
j
− A
k
Θ
k−1
y
n
j
≤1/α
k
Θ
k
y
n
j
− Θ
k−1
y
n
j
→0, Θ
k
y
n
j
x
∗
as j →∞,
and {A
k
}
M
k1
is a family of monotone mappings. It follows from 3.25 that
z
t
− x
∗
,A
k
z
t
≥ f
k
z
t
,x
∗
. 3.26
So, by A1, A4 and 3.26,wehaveforeachy ∈ C and k ∈{1, 2, ,M} that
0 f
k
z
t
,z
t
≤ tf
k
z
t
,y
1 − t
f
k
z
t
,x
∗
≤ tf
k
z
t
,y
1 − t
z
t
− x
∗
,A
k
z
t
tf
k
z
t
,y
t
1 − t
y − x
∗
,A
k
z
t
.
3.27
This implies that
f
k
z
t
,y
1 − t
y − x
∗
,A
k
z
t
≥ 0, ∀y ∈ C. 3.28
Letting t → 0in3.28, it follows from A3 that
f
k
x
∗
,y
y − x
∗
,A
k
x
∗
≥ 0, ∀y ∈ C. 3.29
Fixed Point Theory and Applications 13
Hence x
∗
∈
M
k1
GEPf
k
,A
k
;consequently,x
∗
∈ F.Further,weseethat
x
t
− x
∗
2
Su
t
− x
∗
2
≤
u
t
− x
∗
2
≤
y
t
− x
∗
2
≤
x
t
− x
∗
− tx
t
2
x
t
− x
∗
2
− 2t
x
t
,x
t
− x
∗
t
2
x
t
2
x
t
− x
∗
2
− 2t
x
t
− x
∗
,x
t
− x
∗
− 2t
x
∗
,x
t
− x
∗
t
2
x
t
2
.
3.30
So, we have
x
t
− x
∗
2
≤
x
∗
,x
∗
− x
t
t
2
x
t
2
.
3.31
In particular,
x
n
− x
∗
2
≤
x
∗
,x
∗
− x
n
t
n
2
x
n
2
.
3.32
Since x
n
x
∗
,wehavex
n
→ x
∗
as n →∞. By using the same argument as in the proof of
Theorem 3.1 of 25, we can show that x
t
→ x
∗
∈ F as t → 0. This completes the proof.
4. Strong Convergence Results
Theorem 4.1. Let C be a nonempty, closed and convex subset of a real Hilbert space H.Let{f
k
}
M
k1
:
C × C →
be a family of bifunctions, let {A
k
}
M
k1
: C → H be a family of α
k
-inverse-strongly
monotone mappings and let {T
n
}
∞
n1
: C → C be a countable family of κ-strict pseudocontractions for
some 0 <κ<1 such that F :
M
k1
GEPf
k
,A
k
∩
∞
n1
FT
n
/
∅. Assume that {α
n
}
∞
n1
⊂ 0, 1,
{β
n
}
∞
n1
⊂ 0, 1, γ ∈ κ, 1 and r
k
∈ 0, 2α
k
for each k ∈{1, 2, ,M} satisfy the following
conditions:
i lim
n →∞
α
n
0 and
∞
n1
α
n
∞,
ii 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1.
Suppose that {T
n
},T satisfies the AKTT-condition. Then, {x
n
} generated by 1.14
converges strongly to an element in F.
Proof. For e ach n ∈
,defineS
n
: C → C by S
n
x γx 1 − γT
n
x, x ∈ C. Then, FS
n
FT
n
FT,sinceγ ∈ 0, 1. M oreover, we know that {S
n
} satisfies the AKTT-condition,
since {T
n
} satisfies the AKTT-condition. By Lemma 2.5,wecandefinethemappingS : C →
C by Sx lim
n →∞
S
n
x for x ∈ C.Hence,Sx γx 1 − γTx, x ∈ C,sinceT
n
x → Tx for
14 Fixed Point Theory and Applications
x ∈ C. Further, we know that S
n
is nonexpansive for each n ∈ . Indeed, for each x,y ∈ C
and n ∈
,wehave
S
n
x − S
n
y
2
γx
1 − γ
T
n
x − γy −
1 − γ
T
n
y
2
γ
x − y
1 − γ
T
n
x − T
n
y
2
γ
x − y
2
1 − γ
T
n
x − T
n
y
2
− γ
1 − γ
I − T
n
x −
I − T
n
y
2
≤ γ
x − y
2
1 − γ
x − y
2
1 − γ
κ
I − T
n
x −
I − T
n
y
2
− γ
1 − γ
I − T
n
x −
I − T
n
y
2
x − y
2
1 − γ
κ − γ
I − T
n
x −
I − T
n
y
2
≤
x − y
2
.
4.1
Hence, S
n
is nonexpansive for each n ∈ and so is S.
Next, we show that {x
n
} is bounded. Denote Θ
k
T
f
k
,A
k
r
k
T
f
k−1
,A
k−1
r
k−1
···T
f
1
,A
1
r
1
for any
k ∈{1, 2, ,M} and Θ
0
I.Wenotethatu
n
Θ
M
y
n
.From1.14,wehaveforeach
p ∈ F that
x
n1
− p
β
n
x
n
1 − β
n
S
n
u
n
≤ β
n
x
n
− p
1 − β
n
S
n
u
n
− p
≤ β
n
x
n
− p
1 − β
n
u
n
− p
β
n
x
n
− p
1 − β
n
Θ
M
y
n
− p
≤ β
n
x
n
− p
1 − β
n
y
n
− p
≤ β
n
x
n
− p
1 − β
n
1 − α
n
x
n
− p
α
n
p
1 − α
n
1 − β
n
x
n
− p
α
n
1 − β
n
p
≤ max
x
n
− p
,
p
.
4.2
Hence, by induction, {x
n
} is bounded and so are {y
n
} and {u
n
}.
Next, we show that
lim
n →∞
x
n1
− x
n
0.
4.3
Since u
n
Θ
M
y
n
and u
n1
Θ
M
y
n1
,
u
n1
− u
n
Θ
M
y
n1
− Θ
M
y
n
≤
y
n1
− y
n
.
4.4
Fixed Point Theory and Applications 15
Set z
n
S
n
u
n
, n ∈ . So, we have from 1.14 and 4.4 that
z
n1
− z
n
S
n1
u
n1
− S
n
u
n
≤
S
n1
u
n1
− S
n1
u
n
S
n1
u
n
− S
n
u
n
≤
u
n1
− u
n
S
n1
u
n
− S
n
u
n
≤
y
n1
− y
n
S
n1
u
n
− S
n
u
n
≤
1 − α
n1
x
n1
−
1 − α
n
x
n
sup
z∈
{
u
n
}
S
n1
z − S
n
z
≤
x
n1
− x
n
α
n1
x
n1
α
n
x
n
sup
z∈
{
u
n
}
S
n1
z − S
n
z
.
4.5
Since {S
n
} satisfies the AKTT-condition and lim
n →∞
α
n
0, it follows that
lim sup
n →∞
z
n1
− z
n
−
x
n1
− x
n
≤ 0.
4.6
So, by Lemma 2.2 and ii,weobtain
lim
n →∞
z
n
− x
n
0.
4.7
Hence,
lim
n →∞
x
n1
− x
n
lim
n →∞
1 − β
n
z
n
− x
n
0.
4.8
Observe that
y
n
− x
n
P
C
1 − α
n
x
n
− P
C
x
n
≤ α
n
x
n
−→ 0, 4.9
as n →∞. Similar to the proof of Theorem 3.1,weobtainforeachp ∈ F that
u
n
− p
2
≤
x
n
− p
2
α
n
M
1
M
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
n
− A
i
p
2
,
4.10
u
n
− p
2
≤
x
n
− p
2
α
n
M
1
−
M
i1
Θ
i−1
y
n
− Θ
i
y
n
2
M
2
M
i1
A
i
Θ
i−1
y
n
− A
i
p
,
4.11
16 Fixed Point Theory and Applications
for some M
1
> 0andM
2
> 0. Then, from 4.10,wehave
x
n1
− p
2
≤ β
n
x
n
− p
2
1 − β
n
S
n
u
n
− p
2
≤ β
n
x
n
− p
2
1 − β
n
u
n
− p
2
≤ β
n
x
n
− p
2
1 − β
n
×
x
n
− p
2
α
n
M
1
M
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
n
− A
i
p
2
≤
x
n
− p
2
α
n
M
1
1 − β
n
M
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
n
− A
i
p
2
,
4.12
which implies that
1 − β
n
M
i1
r
i
2α
i
− r
i
A
i
Θ
i−1
y
n
− A
i
p
2
≤
x
n1
− p
2
−
x
n
− p
2
α
n
M
1
.
4.13
So, from 4.8, i, ii and 0 <r
k
< 2α
k
for each k 1, 2, ,M,wehave
lim
n →∞
A
k
Θ
k−1
y
n
− A
k
p
0,
4.14
for each k ∈{1, 2, ,M}. Similarly, from 4.11,wehave
x
n1
− p
2
≤ β
n
x
n
− p
2
1 − β
n
S
n
u
n
− p
2
≤ β
n
x
n
− p
2
1 − β
n
u
n
− p
2
≤ β
n
x
n
− p
2
1 − β
n
×
x
n
− p
2
α
n
M
1
−
M
i1
Θ
i−1
y
n
− Θ
i
y
n
2
M
2
M
i1
A
i
Θ
i−1
y
n
− A
i
p
≤
x
n
− p
2
α
n
M
1
−
1 − β
n
M
i1
Θ
i−1
y
n
− Θ
i
y
n
2
M
2
M
i1
A
i
Θ
i−1
y
n
− A
i
p
.
4.15
This implies that
1 − β
n
M
i1
Θ
i−1
y
n
− Θ
i
y
n
2
≤
x
n
− p
2
−
x
n1
− p
2
α
n
M
1
M
2
M
i1
A
i
Θ
i−1
y
n
− A
i
p
.
4.16
Fixed Point Theory and Applications 17
From i, ii, 4.8,and4.14, it follows that
lim
n →∞
Θ
k−1
y
n
− Θ
k
y
n
0,
4.17
for each k ∈{1, 2, ,M}.
Next, we show that
lim
n →∞
x
n
− Sx
n
0.
4.18
Observing
u
n
− y
n
Θ
M
y
n
− y
n
≤
Θ
M
y
n
− Θ
M−1
y
n
Θ
M−1
y
n
− Θ
M−2
y
n
···
Θ
1
y
n
− y
n
,
4.19
it follows, by 4.17,that
lim
n →∞
u
n
− y
n
0.
4.20
From 4.9 and 4.20,wehave
lim
n →∞
u
n
− x
n
0.
4.21
We see that
x
n
− Sx
n
≤
x
n
− S
n
u
n
S
n
u
n
− S
n
x
n
S
n
x
n
− Sx
n
≤
x
n
− S
n
u
n
u
n
− x
n
sup
z∈
{
x
n
}
S
n
z − Sz
.
4.22
So, by 4.7, 4.21,andLemma 2.5,wehave
lim
n →∞
x
n
− Sx
n
0.
4.23
Let the net {x
t
} be defined by 3.3.ByTheorem 3.1,wehavex
t
→ x
∗
∈ F as t → 0.
Moreover, by proving in the same manner as in Theorem 3.2 of 25, we can show that
lim sup
n →∞
x
∗
,x
∗
− x
n
≤ 0.
4.24
18 Fixed Point Theory and Applications
Finally, we show that x
n
→ x
∗
∈ F as n →∞.From1.14,wehave
x
n1
− x
∗
2
≤ β
n
x
n
− x
∗
2
1 − β
n
S
n
u
n
− x
∗
2
≤ β
n
x
n
− x
∗
2
1 − β
n
u
n
− x
∗
2
≤ β
n
x
n
− x
∗
2
1 − β
n
y
n
− x
∗
2
≤ β
n
x
n
− x
∗
2
1 − β
n
1 − α
n
x
n
− x
∗
− α
n
x
∗
2
≤ β
n
x
n
− x
∗
2
1 − β
n
×
1 − α
n
x
n
− x
∗
2
− 2α
n
1 − α
n
x
∗
,x
n
− x
∗
α
2
n
x
∗
2
1 − α
n
1 − β
n
x
n
− x
∗
2
α
n
1 − β
n
2
1 − α
n
x
∗
,x
∗
− x
n
α
n
x
∗
2
.
4.25
By i and 4.24, it follows from Lemma 2.3 that x
n
→ x
∗
∈ F. This completes the proof.
As a direct consequence of Lemmas 2.6 and 2.7 and Theorem 4.1,weobtainthe
following result.
Theorem 4.2. Let C be a nonempty, closed, and convex subset of a real Hilbert space H.Let{f
k
}
M
k1
:
C × C →
be a family of bifunctions, let {A
k
}
M
k1
: C → H be a family of α
k
-inverse-strongly
monotone mappings, and let {S
i
}
∞
i1
be a sequence of κ
i
-strict pseudocontractions of C into itself such
that F :
M
k1
GEPf
k
,A
k
∩
∞
i1
FS
i
/
∅ and sup{κ
i
: i ∈ } κ>0. Assume that
γ ∈ κ, 1 and r
k
∈ 0, 2α
k
for each k ∈{1, 2, ,M}. Define the sequence {x
n
} by x
1
∈ C and
y
n
P
C
1 − α
n
x
n
,
u
n
T
f
M
,A
M
r
M
T
f
M−1
,A
M−1
r
M−1
···T
f
2
,A
2
r
2
T
f
1
,A
1
r
1
y
n
,
x
n1
β
n
x
n
1 − β
n
γu
n
1 − γ
n
i1
μ
i
n
S
i
u
n
,n≥ 1,
4.26
where {α
n
}
∞
n1
and {β
n
}
∞
n1
are real sequences in 0, 1 which satisfy (i)-(ii) of Theorem 4.1 and {μ
i
n
}
is a real sequence which satisfies (i)–(iii) of Lemma 2.7.Then,{x
n
} converges strongly to an element
in F.
Remark 4.3. Theorems 4.1 and 4.2 extend the main results in 25 from a nonexpansive map-
ping to an infinite family of strict pseudocontractions and a system of generalized equilibrium
problems.
Remark 4.4. If we take A
k
≡ 0andf
k
≡ 0foreachk 1, 2, ,M, then Theorems 3.1, 4.1,
and 4.2 can be applied to a system of equilibrium problems and to a system of variational
inequality problems, respectively.
Fixed Point Theory and Applications 19
Remark 4.5. Let S
1
,S
2
, be an infinite family of nonexpansive mappings of C into itself,
and let ξ
1
,ξ
2
, be real numbers such that 0 <ξ
i
< 1foralli ∈ .Moreover,letW
n
and
W be the W-mappings 35 generated by S
1
,S
2
, ,S
n
and ξ
1
,ξ
2
, ,ξ
n
and S
1
,S
2
, and
ξ
1
,ξ
2
, Then, we know from 7, 35 that {W
n
},W satisfies the AKTT-condition. Therefore,
in Theorem 4.1, the mapping T
n
can be also replaced by W
n
.
Acknowledgments
The authors would like to thank the referees for valuable suggestions. This research is
supported by the Centre of Excellence in Mathematics, the Commission on Higher Education,
and the Thailand Research Fund. The first author is supported by the Royal Golden Jubilee
Grant no. PHD/0261/2551 and by the Graduate School, Chiang Mai University, Thailand.
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