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, [email protected]
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
≥ 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
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 −
,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
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
}⊂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
∞
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
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
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
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
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
,
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
}
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
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
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
∞
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
}
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
}
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
> 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
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
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.
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∈
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
x
t
− p
Su
t
− Sp
≤
u
t
− p
≤
y
t
− 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
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
− 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
≤
2
≤
Θ
k−1
y
t
− p
2
r
k
r
k
− 2α
k
A
k
Θ
− p
2
≤
y
t
− p
2
M
i1
r
i
r
i
− 2α
i
A
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
t
− A
i
p
2
≤
x
y
t
− A
i
p
2
≤
x
t
− p
2
tM
1
M
i1
r
i
r
i
− 2α
2
}.So,wehave
x
t
− p
2
≤
u
t
− p
2
≤
x
t
− p
2
tM
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
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
A
k
Θ
k−1
y
t
−
p − r
k
A
k
p
, Θ
k
y
t
− p
1
2
Θ
k−1
y
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
−
Θ
k−1
y
t
− Θ
k
y
t
− r
k
A
k
Θ
k−1
Θ
k
y
t
− p
2
−
Θ
k−1
y
t
− Θ
k
y
t
2
2r
k
This implies that
Θ
k
y
t
− p
2
≤
Θ
k−1
y
t
− p
2
−
A
k
Θ
k−1
y
t
− A
k
p
≤
Θ
k−1
y
t
− p
2
−
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
i
y
t
2
M
2
M
i1
A
i
Θ
i−1
y
t
− A
i
p
≤
i1
A
i
Θ
i−1
y
t
− A
i
p
.
3.13
Hence,
x
t
− p
2
≤
i
y
t
2
M
2
M
i1
A
i
Θ
i−1
y
t
− A
i
p
.
3.14
From 3.10,weobtain
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
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
−→ 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
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
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
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
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
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
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
,z
t
≤ tf
k
z
t
,y
1 − t
f
k
z
t
,x
∗
≤ tf
k
z
t
,y
.
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
∗
∗
2
Su
t
− x
∗
2
≤
u
t
− x
∗
2
≤
y
t
− x
∗
2
≤
t
2
x
t
− x
∗
2
− 2t
x
t
− x
∗
,x
t
− x
∗
− 2t
x
∗
,x
t
− x
∗
x
t
2
.
3.31
In particular,
x
n
− x
∗
2
≤
x
∗
,x
∗
− x
n
t
n
2
x
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
}
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 →∞
x − γy −
1 − γ
T
n
y
2
γ
x − y
1 − γ
T
n
x − T
n
y
2
x −
I − T
n
y
2
≤ γ
x − y
2
1 − γ
x − y
2
1 − γ
n
y
2
x − y
2
1 − γ
κ − γ
I − T
n
x −
I − T
n
,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
S
n
u
n
− p
≤ β
n
x
n
− p
1 − β
n
u
n
− p
x
n
− p
1 − β
n
y
n
− p
≤ β
n
x
n
− p
1 − β
n
x
n
− p
α
n
1 − β
n
p
≤ max
x
n
− p
,
p
Θ
M
y
n1
,
u
n1
− u
n
Θ
M
y
n1
− Θ
M
y
n
≤
y
n
≤
S
n1
u
n1
− S
n1
u
n
S
n1
u
n
− S
n
u
n
≤
u
n1
− u
n
n
≤
1 − α
n1
x
n1
−
1 − α
n
x
n
sup
z∈
{
u
n
}
S
n1
z − S
n
z
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
0.
4.8
Observe that
y
n
− x
n
P
C
1 − α
n
x
n
− P
C
x
n
≤ α
n
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
n
− A
i
p
2
,
4.10
u
− Θ
i
y
n
2
M
2
M
i1
A
i
Θ
i−1
y
n
− A
i
p
,
n
S
n
u
n
− p
2
≤ β
n
x
n
− p
2
1 − β
n
u
α
n
M
1
M
i1
r
i
r
i
− 2α
i
A
i
Θ
i−1
y
n
− A
i
p
i
− 2α
i
A
i
Θ
i−1
y
n
− A
i
p
2
,
4.12
which implies that
1 − β
n
M
i1
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
2
1 − β
n
S
n
u
n
− p
2
≤ β
n
x
n
− p
2
1 − β
n
− p
2
α
n
M
1
−
M
i1
Θ
i−1
y
n
− Θ
i
y
n
2
M
α
n
M
1
−
1 − β
n
M
i1
Θ
i−1
y
n
− Θ
i
y
n
2
M
Θ
i−1
y
n
− Θ
i
y
n
2
≤
x
n
− p
2
−
x
n1
− p
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
y
n
− Θ
M−1
y
n
Θ
M−1
y
n
− Θ
M−2
y
n
···
Θ
1
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
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 →∞
2
1 − β
n
S
n
u
n
− x
∗
2
≤ β
n
x
n
− x
∗
2
1 − β
n
n
x
n
− x
∗
2
1 − β
n
1 − α
n
x
n
− x
∗
− α
n
x
∗
2
≤ β
n
x
∗
,x
n
− x
∗
α
2
n
x
∗
2
1 − α
n
1 − β
n
x
n
− x
∗
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
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
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
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
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