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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 217407, 27 pages
doi:10.1155/2011/217407
Research Article
A New Hybrid Algorithm for a System of
Mixed Equilibrium Problems, Fixed Point Problems
for Nonexpansive Semigroup, and Variational
Inclusion Problem
Thanyarat Jitpeera and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of
Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam,
Received 14 December 2010; Accepted 15 January 2011
Academic Editor: Jen Chih Yao
Copyright q 2011 T. Jitpeera and P. Kumam. 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.
The purpose of this paper is to consider a shrinking projection method for finding the common
element of the set of common fixed points f or nonexpansive semigroups, the set of common fixed
points for an infinite family of a ξ-strict pseudocontraction, the set of solutions of a system of
mixed equilibrium problems, and the set of solutions of the variational inclusion problem. Strong
convergence of the sequences generated by the proposed iterative scheme is obtained. The results
presented in this paper extend and improve some well-known results in the literature.
1. Introduction
Throughout this paper, we assume that H be a real Hilbert space with inner product ·, · and
norm ·,andletC be a nonempty closed convex subset of H. We denote weak convergence
and strong convergence by notations and → , respectively. Let I {F
k
}
k∈Γ
be a countable
family of bifunctions from C × C to R, where R is the set of real numbers and Γ is an arbitrary
index set. Let ϕ : C →R∪{∞} be a proper extended real-valued function. The system of
mixed equilibrium problems is to find x ∈ C such that
F
k
x, y
ϕ
y
≥ ϕ
x
, ∀k ∈ Γ, ∀y ∈ C. 1.1
The set of solutions of 1.1 is denoted by SMEPF
k
,ϕ,thatis,
SMEP
F
k
,ϕ
x ∈ C : F
k
x, y
ϕ
y
≥ ϕ
x
, ∀k ∈ Γ, ∀y ∈ C
. 1.2
2 Fixed Point Theory and Applications
If Γ is a singleton, the problem 1.1 reduces to find the following mixed equilibrium
problem see also the work of Flores-Baz
´
an in 1. For finding x ∈ C such that,
F
x, y
ϕ
y
≥ ϕ
x
, ∀y ∈ C, 1.3
the set of solutions of 1.3 is denoted by MEPF, ϕ. Combettes and Hirstoaga 2 introduced
the following system of equilibrium problems. For finding x ∈ C such that,
F
k
x, y
≥ 0, ∀k ∈ Γ, ∀y ∈ C, 1.4
the set of solutions of 1.4 is denoted by SEPI ,thatis,
SEP
I
x ∈ C : F
k
x, y
≥ 0, ∀k ∈ Γ, ∀y ∈ C
. 1.5
If Γ is a singleton, the problem 1.4 becomes the following equilibrium problem. For finding
x ∈ C such that
F
x, y
≥ 0, ∀y ∈ C. 1.6
The set of solution of 1.6 is denoted by EPF.
The equilibrium problem include fixed point problems, optimization problems, varia-
tional inequalities problems, Nash equilibrium problems, noncooperative games, economics
and the mixed equilibrium problems as special cases see, e.g., 3–8. Some methods have
been proposed to solve the equilibrium problem, see, for instance, 9–17.
Recall that, a mapping T : C → C is said to be nonexpansive if
Tx − Ty
≤
x − y
, ∀x, y ∈ C. 1.7
We denote the set of fixed points of T by FT,thatisFT{x ∈ C : x Tx}.
Definition 1.1. A family S {Ss :0≤ s ≤∞}of mappings of C into itself is called a
nonexpansive semigroup on C if it satisfies the following conditions:
1 S0x x, for all x ∈ C;
2 Ss tSsSt, for all s, t ≥ 0;
3 Ssx − Ss
y≤x − y, for all x, y ∈ C and s ≥ 0;
4 for all x ∈ C, s → Ssx is continuous.
We denoted by FS the set of all common fixed points of S {Ss : s ≥ 0},thatis,
FS
s≥0
FSs. It is know that FS is closed and convex.
Let B : H → H be a single-valued nonlinear mapping and M : H → 2
H
be a set-
valued mapping. The variational inclusion problem is to find x ∈ H such that
θ ∈ B
x
M
x
, 1.8
Fixed Point Theory and Applications 3
where θ is the zero vecter in H. The set of solutions of problem 1.8 is denoted by IB, M.
A set-valued mapping M : H → 2
H
is called monotone if for all x, y ∈ H, f ∈ Mx and
g ∈ My imply x − y, f − g≥0. A monotone mapping M is maximal if its graph GM :
{f, x ∈ H × H : f ∈ Mx} of M is not properly contained in the graph of any other
monotone mapping. It is known that a monotone mapping M is maximal if and only if for
x, f ∈ H × H, x − y, f − g≥0 for all y, g ∈ GM imply f ∈ Mx.
Definition 1.2. A mapping B : C → H is said to be a k-Lipschitz continous
if there exists a
constant k>0 such that
Bx − By
≤ k
x − y
, ∀x, y ∈ C. 1.9
Definition 1.3. A mapping B : C → H is said to be a β-inverse-strongly monotone if there exists
a constant β>0 with the property
Bx − By, x − y
≥ β
Bx − By
2
, ∀x, y ∈ C.
1.10
Remark 1.4. It is obvious that any β-inverse-strongly monotone mappings B is monotone
and 1/β-Lipschitz continuous. It is easy to see that for any λ constant is in 0, 2β, then the
mapping I − λB is nonexpansive, where I is the identity mapping on H.
Definition 1.5. Let η : C × C → H is called Lipschitz continuous, if there exists a constant
L>0 such that
η
x, y
≤ L
x − y
, ∀x, y ∈ C. 1.11
Let K : C →Rbe a differentiable functional on a convex set C, which is called:
1 η-convex 18 if
K
y
−K
x
≥
K
x
,η
y, x
, ∀x, y ∈ C, 1.12
where K
x is the Fr
´
echet derivative of K at x;
2 η-strongly convex 19 if there exists a constant σ>0 such that
K
y
−K
x
−
K
x
,η
y, x
≥
σ
2
x − y
2
, ∀x, y ∈ C.
1.13
In particular, if ηx, yx − y for all x, y ∈ C, then K is said to be strongly convex.
Definition 1.6. Let M : H → 2
H
be a set-valued maximal monotone mapping, then the single-
valued mapping J
M,λ
: H → H defined by
J
M,λ
x
I λM
−1
x
, x ∈ H
1.14
is called the resolvent operator associated with M, where λ is any positive number and I is the
identity mapping. The following characterizes the resolvent operator.
4 Fixed Point Theory and Applications
R1 The resolvent operator J
M,λ
is single-valued and nonexpansive for all λ>0, that is,
J
M,λ
x
− J
M,λ
y
≤
x − y
, ∀x, y ∈ H, ∀λ>0. 1.15
R2 The resolvent operator J
M,λ
is 1-inverse-strongly monotone; see 20,thatis,
J
M,λ
x − J
M,λ
y
2
≤
x − y, J
M,λ
x
− J
M,λ
y
, ∀x, y ∈ H.
1.16
R3 The solution of problem 1.8 is a fixed point of the operator J
M,λ
I − λB for all
λ>0; see also 21,thatis,
I
B, M
F
J
M,λ
I − λB
, ∀λ>0. 1.17
R4 If 0 <λ≤ 2β, then the mapping J
M,λ
I − λB : H → H is nonexpansive.
R5 IB,M is closed and convex.
In 2007, Takahashi et al. 22 proved the following strong convergence theorem
for a nonexpansive mapping by using the shrinking projection method in mathematical
programming. For C
1
C and x
1
P
C
1
x
0
, they define a sequence {x
n
} as follows:
y
n
α
n
x
n
1 − α
n
Tx
n
,
C
n1
z ∈ C
n
:
y
n
− z
≤
x
n
− z
,
x
n1
P
C
n1
x
0
, ∀n ≥ 0,
1.18
where 0 ≤ α
n
<a<1. They proved that the sequence {x
n
} generated by 1.18 converges
weakly to z ∈ FT, where z P
FT
x
0
.
In 2008, S. Takahashi and W. Takahashi 23 introduced the following iterative scheme
for finding a common element of the set of solution of generalized equilibrium problem
and the set of fixed points of a nonexpansive mapping in a Hilbert space. They proved the
strong convergence theorems under certain appropriate conditions imposed on parameters.
Next, Zhang et al. 24 introduced the following new iterative scheme for finding a common
element of the set of solution to the problem 1.8 and the set of fixed points of a nonexpansive
mapping in a real Hilbert space. Starting with an arbitrary x
1
x ∈ H, define a sequence {x
n
}
by
y
n
J
M,λ
x
n
− λBx
n
,
x
n1
α
n
x
1 − α
n
Ty
n
, ∀n ≥ 1,
1.19
where J
M,λ
I λM
−1
is the resolvent operator associated with M and a positive number
λ and {α
n
} is a sequence in the interval 0, 1. Peng et al. 25 introduced the iterative scheme
by the viscosity approximation method for finding a common element of the set of solutions
Fixed Point Theory and Applications 5
to the problem 1.8, the set of solutions of an equilibrium problem, and the set of fixed points
of a nonexpansive mapping in a Hilbert space.
In 2009, Saeidi 26 introduced a more general iterative algorithm for finding a
common element of the set of solution for a system of equilibrium problems and the set
of common fixed points for a finite family of nonexpansive mappings and a nonexpansive
semigroup. In 2010, Katchang and Kumam 27 obtained a strong convergence theorem for
finding a common element of the set of fixed points of a family of finitely nonexpansive
mappings, the set of solutions of a mixed equilibrium problem and the set of solutions
of a variational inclusion problem for an inverse-strongly monotone mapping. Let W
n
be
W-mapping defined by 2.8, f be a contraction mapping and A, B be inverse-strongly
monotone mappings. Let J
M,λ
I λM
−1
be the resolvent operator associated with M
and a positive number λ. Starting with arbitrary initial x
1
∈ H, defined a sequence {x
n
} by
F
u
n
,y
ϕ
y
− ϕ
u
n
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
y
n
J
M,λ
u
n
− λAu
n
,
v
n
J
M,λ
y
n
− λAy
n
,
x
n1
α
n
γf
x
n
β
n
x
n
1 − β
n
I − α
n
B
W
n
v
n
, ∀n ≥ 1.
1.20
They proved that under certain appropriate conditions imposed on {α
n
}, {β
n
},and{r
n
},the
sequence {x
n
} generated by 1.20 converges strongly to p ∈ Ω :
∞
i1
FS
i
∩ IA, M ∩
MEPF, ϕ, where p P
Ω
I −Bγfp. Later, Kumam et al. 28 proved a strongly convergence
theorem of the iterative sequence generated by the shrinking projection method for finding a
common element of the set of solutions of generalized mixed equilibrium problems, the set
of fixed points of a finite family of quasinonexpansive mappings, and the set of solutions of
variational inclusion problems.
Liu et al. 29 introduced a hybrid iterative scheme for finding a common element
of the set of solutions of mixed equilibrium problems, the set of common fixed points
for nonexpansive semigroup and the set of solution of quasivariational inclusions with
multivalued maximal monotone mappings and inverse-strongly monotone mappings.
Recently, Jitpeera and Kumam 30 considered a shrinking projection method of finding
the common element of the set of common fixed points for a finite family of a ξ-strict
pseudocontraction, the set of solutions of a systems of equilibrium problems and the set
of solutions of variational inclusions. Then, they proved strong convergence theorems of
the iterative sequence generated by the shrinking projection method under some suitable
conditions in a real Hilbert space. Very recently, Hao 18 introduced a general iterative
method for finding a common element of solution set of quasi variational inclusion problems
and of the common fixed point set of an infinite family of nonexpansive mappings.
In this paper, motivated and inspired by the previously mentioned results, we
introduce an iterative scheme by the shrinking projection method for finding the common
element of the set of common fixed points for nonexpansive semigroups, the set of common
fixed points for an infinite family of a ξ-strict pseudocontraction, the set of solutions of a
systems of mixed equilibrium problems and the set of solutions of the variational inclusions
problem. Then, we prove a strong convergence theorem of the iterative sequence generated
by the shrinking projection method under some suitable conditions. The results obtained in
this paper extend and improve several recent results in this area.
6 Fixed Point Theory and Applications
2. Preliminaries
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Recall that
the nearest point projection P
C
from H onto C assigns to each x ∈ H the unique point in
P
C
x ∈ C satisfying the property x − P
C
x min
y∈C
x − y.
The following characterizes the projection P
C
. We recall some lemmas which will be
needed in the rest of this paper.
Lemma 2.1. For a given z ∈ H, u ∈ C, u P
C
z ⇔u − z, v − u≥0, for all v ∈ C.
It is well known that P
C
is a firmly nonexpansive mapping of H onto C and satisfies
P
C
x − P
C
y
2
≤
P
C
x − P
C
y, x − y
, ∀x, y ∈ H.
2.1
Moreover, P
C
x is characterized by the following properties: P
C
x ∈ C and for all x ∈ H, y ∈ C,
x − P
C
x, y − P
C
x
≤ 0. 2.2
Lemma 2.2 see 20. Let M : H → 2
H
be a maximal monotone mapping and let B : H → H
be a Lipshitz continuous mapping. Then the mapping L M B : H → 2
H
is a maximal monotone
mapping.
Lemma 2.3 see 31. Let C be a closed convex subset of H.Let{x
n
} be a bounded sequence in H.
Assume that
1 the weak ω-limit set ω
w
x
n
⊂ C,
2 for each z ∈ C, lim
n →∞
x
n
− z exists.
Then {x
n
} is weakly convergent to a point in C.
Lemma 2.4 see 32. Each Hilbert space H satisfies Opial’s condition, that is, for any sequence
{x
n
}⊂H with x
n
x, the inequality lim inf
n →∞
x
n
− x < lim inf
n →∞
x
n
− y, hold for each
y ∈ H with y
/
x.
Lemma 2.5 see 33. Each Hilbert space H, satisfies the Kadec-Klee property, that is, for any
sequence {x
n
} with x
n
xand x
n
→x together imply x
n
− x→0.
For solving the system of mixed equilibrium problem, let us assume that function F
k
:
C × C →R, k 1, 2, ,N satisfies the following conditions:
H1 F
k
is monotone, that is, F
k
x, yF
k
y, x ≤ 0, for all x, y ∈ C;
H2 for each fixed y ∈ C, x → F
k
x, y is convex and upper semicontinuous;
H3 for each fixed x ∈ C, y → F
k
x, y is convex.
Lemma 2.6 see 34. Let C be a nonempty closed convex subset of a real Hilbert space H and let ϕ
be a lower semicontinuous and convex functional from C to R.LetF be a bifunction from C × C to R
Fixed Point Theory and Applications 7
satisfying (H1)–(H3). Assume that
i η : C × C → H is k Lipschitz continuous with constant k>0 such that;
a ηx, yηy,x0, for all x, y ∈ C,
b η·, · is affine in the first variable,
c for each fixed x ∈ C, y → ηx, y is sequentially continuous from the weak topology
to the weak topology,
ii K : C →Ris η-strongly convex with constant σ>0 and its derivative K
is sequentially
continuous from the weak topology to the strong topology;
iii for each x ∈ C, there exist a bounded subset D
x
⊂ C and z
x
∈ C such that for any
y ∈ C \ D
x
,
F
y, z
x
ϕ
z
x
− ϕ
y
1
r
K
y
−K
x
,η
z
x
,y
< 0.
2.3
For given r>0,LetK
F
r
: C → C be the mapping defined by:
K
F
r
x
y ∈ C : F
y, z
ϕ
z
− ϕ
y
1
r
K
y
−K
x
,η
z, y
≥ 0, ∀z ∈ C
2.4
for all x ∈ C. Then the following hold
1 K
F
r
is single-valued;
2 K
F
r
is nonexpansive if K
is Lipschitz continuous with constant ν>0 such that σ ≥ kν;
3 FK
F
r
MEPF, ϕ;
4 MEPF, ϕ is closed and convex.
Lemma 2.7 see 35. Let V : C → H be a ξ-strict pseudocontraction, then
1 the fixed point set FV of V is closed convex so that the projection P
FV
is well defined;
2 define a mapping T : C → H by
Tx tx
1 − t
Vx, ∀x ∈ C. 2.5
If t ∈ ξ, 1,thenT is a nonexpansive mapping such that FV FT.
A family of mappings {V
i
: C → H}
∞
i1
is called a family of uniformly ξ-strict
pseudocontractions, if there exists a constant ξ ∈ 0, 1 such that
V
i
x − V
i
y
2
≤
x − y
2
ξ
I − V
i
x − I − V
i
y
2
, ∀x, y ∈ C, ∀i ≥ 1.
2.6
8 Fixed Point Theory and Applications
Let {V
i
: C → C}
∞
i1
be a countable family of uniformly ξ-strict pseudocontractions. Let
{T
i
: C → C}
∞
i1
be the sequence of nonexpansive mappings defined by 2.5,thatis,
T
i
x tx
1 − t
V
i
x, ∀x ∈ C, ∀i ≥ 1,t∈
ξ, 1
. 2.7
Let {T
i
} be a sequence of nonexpansive mappings of C into itself defined by 2.7 and
let {μ
i
} be a sequence of nonnegative numbers in 0, 1. For each n ≥ 1, define a mapping W
n
of C into itself as follows:
U
n,n1
I,
U
n,n
μ
n
T
n
U
n,n1
1 − μ
n
I,
U
n,n−1
μ
n−1
T
n−1
U
n,n
1 − μ
n−1
I,
.
.
.
U
n,k
μ
k
T
k
U
n,k1
1 − μ
k
I,
U
n,k−1
μ
k−1
T
k−1
U
n,k
1 − μ
k−1
I,
.
.
.
U
n,2
μ
2
T
2
U
n,3
1 − μ
2
I,
W
n
U
n,1
μ
1
T
1
U
n,2
1 − μ
1
I.
2.8
Such a mapping W
n
is nonexpansive from C to C and it is called the W-mapping generated
by T
1
,T
2
, ,T
n
and μ
1
,μ
2
, ,μ
n
. For each n, k ∈ N, let the mapping U
n,k
be defined by 2.8.
Then we can have the following crucial conclusions concerning W
n
.
Lemma 2.8 see 36. Let C be a nonempty closed convex subset of a real Hilbert space H.Let
T
1
,T
2
, be nonexpansive mappings of C into itself such that
∞
i1
FT
i
is nonempty, let μ
1
,μ
2
,
be real numbers such that 0 ≤ μ
i
≤ b<1 for every i ≥ 1. Then, for every x ∈ C and k ∈ N,
lim
n →∞
U
n,k
x exists.
Using this lemma, one can define a mapping U
∞,k
and W : C → C as follows U
∞,k
x
lim
n →∞
U
n,k
x and
Wx : lim
n →∞
W
n
x lim
n →∞
U
n,1
x, ∀x ∈ C.
2.9
Such a mapping W is called the W-mapping. Since W
n
is nonexpansive and FW
∞
i1
FT
i
,
W : C → C is also nonexpansive. Indeed, observe that for each x, y ∈ C such that
Wx − Wy
lim
n →∞
W
n
x − W
n
y
≤
x − y
.
2.10
Fixed Point Theory and Applications 9
Lemma 2.9 see 36. Let C be a nonempty closed convex subset of a Hilbert space H, {T
i
: C → C}
be a countable family of nonexpansive mappings with
∞
i1
FT
i
/
∅, {μ
i
} be a real sequence such that
0 <μ
i
≤ b<1, for all i ≥ 1.ThenFW
∞
i1
FT
i
.
Lemma 2.10 see 37. Let C be a nonempty closed convex subset of a Hilbert space H, {T
i
: C →
C} be a countable family of nonexpansive mappings with
∞
i1
FT
i
/
∅, {μ
i
} be a real sequence such
that 0 <μ
i
≤ b<1, for all i ≥ 1.IfD is any bounded subset of C,then
lim
n →∞
sup
x∈D
Wx − W
n
x
0.
2.11
Lemma 2.11 see 38. Let C be a nonempty bounded closed convex subset of a Hilbert space H and
let S {Ss :0≤ s<∞} be a nonexpansive semigroup on C, then for any h ≥ 0,
lim
t →∞
sup
x∈C
1
t
t
0
T
s
xds − T
h
1
t
t
0
T
s
xds
0. 2.12
Lemma 2.12 see 39. Let C be a nonempty bounded closed convex subset of H, {x
n
} be a sequence
in C and S {Ss :0≤ s<∞} be a nonexpansive semigroup on C. If the following conditions are
satisfied:
1 x
n
z;
2 lim sup
s →∞
lim sup
n →∞
Ssx
n
− x
n
0,thenz ∈ FS.
3. Main Results
In this section, we will introduce an iterative scheme by using a shrinking projection method
for finding the common element of the set of common fixed points for nonexpansive semi-
groups, the set of common fixed points for an infinite family of ξ-strict pseudocontraction,
the set of solutions of a systems of mixed equilibrium problems and the set of solutions of the
variational inclusions problem in a real Hilbert space.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H,let{F
k
: C ×
C →R, k 1, 2, ,N} be a finite family of mixed equilibrium functions satisfying conditions
(H1)–(H3). Let S {Ss :0≤ s<∞} be a nonexpansive semigroup on C and let {t
n
} be a
positive real divergent sequence. Let {V
i
: C → C}
∞
i1
be a countable family of uniformly ξ-strict
pseudocontractions, {T
i
: C → C}
∞
i1
be the countable family of nonexpansive mappings defined by
T
i
x tx 1 − tV
i
x, for all x ∈ C, for all i ≥ 1, t ∈ ξ, 1, W
n
be the W-mapping defined by 2.8
and W be a mapping defined by 2.9 with FW
/
∅.LetA, B : C → H be γ,β-inverse-strongly
monotone mappings and M
1
,M
2
: H → 2
H
be maximal monotone mappings such that
Θ : F
S
∩ F
W
∩
N
k1
SMEP
F
k
∩ I
A, M
1
∩ I
B, M
2
/
∅.
3.1
10 Fixed Point Theory and Applications
Let r
k
> 0, k 1, 2, ,N, which are constants. Let {x
n
}, {y
n
}, {v
n
}, {z
n
}, and {u
n
} be sequences
generated by x
0
∈ C, C
1
C, x
1
P
C
1
x
0
, u
n
∈ C and
x
0
x ∈ C chosen arbitrarily,
u
n
K
F
N
r
N,n
K
F
N−1
r
N−1,n
K
F
N−2
r
N−2,n
···K
F
2
r
2,n
K
F
1
r
1,n
x
n
,
y
n
J
M
2
,δ
n
u
n
− δ
n
Bu
n
,
v
n
J
M
1
,λ
n
y
n
− λ
n
Ay
n
,
z
n
α
n
v
n
1 − α
n
1
t
n
t
n
0
S
s
W
n
v
n
ds,
C
n1
⎧
⎨
⎩
z ∈ C
n
:
z
n
− z
2
≤
x
n
− z
2
− α
n
1 − α
n
v
n
−
1
t
n
t
n
0
S
s
W
n
v
n
ds
2
⎫
⎬
⎭
,
x
n1
P
C
n1
x
0
,n∈ N,
3.2
where K
F
k
r
k
: C → C, k 1, 2, ,Nis the mapping defined by 2.4 and {α
n
} be a sequence in 0, 1
for all n ∈ N. Assume the following conditions are satisfied:
C1 η
k
: C × C → H is L
k
-Lipschitz continuous with constant k 1, 2, ,Nsuch that
a η
k
x, yη
k
y, x0, for all x, y ∈ C,
b x → η
k
x, y is affine,
c for each fixed y ∈ C, y → η
k
x, y is sequentially continuous from the weak topology
to the weak topology;
C2 K
k
: C →Ris η
k
-strongly convex with constant σ
k
> 0 and its derivative K
k
is not only
sequentially continuous from the weak topology to the strong topology but also Lipschitz
continuous with a Lipschitz constant ν
k
> 0 such that σ
k
>L
k
ν
k
;
C3 for each k ∈{1, 2, ,N} and for all x ∈ C, there exist a bounded subset D
x
⊂ C and
z
x
∈ C such that for any y ∈ C \ D
x
,
F
k
y, z
x
ϕ
z
x
− ϕ
y
1
r
k
K
y
−K
x
,η
z
x
,y
< 0,
3.3
C4 {α
n
}⊂c, d,forsomec, d ∈ ξ, 1;
C5 {λ
n
}⊂a
1
,b
1
,forsomea
1
,b
1
∈ 0, 2γ;
C6 {δ
n
}⊂a
2
,b
2
,forsomea
2
,b
2
∈ 0, 2β;
C7 lim inf
n →∞
r
k,n
> 0, for each k ∈ 1, 2, 3, ,N.
Then, {x
n
} and {u
n
} converge strongly to z P
Θ
x
0
.
Fixed Point Theory and Applications 11
Proof. Pick any p ∈ Θ. Taking I
k
n
K
F
k
r
k,n
K
F
k−1
r
k−1,n
K
F
k−2
r
k−2,n
···K
F
2
r
2,n
K
F
1
r
1,n
for k ∈{1, 2, 3, ,N} and
I
0
n
I for all n ∈ N. From the definition of K
F
k
r
k,n
is nonexpansive for each k 1, 2, 3, ,N,
then I
k
n
also and p I
F
k
r
k,n
p,wenotethatu
n
I
N
n
x
n
. If follows that
u
n
− p
I
N
n
x
n
− I
N
n
p
≤
x
n
− p
. 3.4
Next, we will divide the proof into eight steps.
Step 1. We first show by induction that Θ ⊂ C
n
for each n ≥ 1.
Taking p ∈ Θ,wegetthatp J
M
1
,λ
k
p − λ
k
ApJ
M
2
,δ
k
p − δ
k
Bp. Since J
M
1
,λ
k
,J
M
2
,δ
k
are nonexpansive. From the assumption, we see that Θ ⊂ C C
1
.SupposeΘ ⊂ C
k
for some
k ≥ 1. For any p ∈ ΘC
k
, we have
v
k
− p
J
M
1
,λ
k
y
k
− λ
k
Ay
k
− J
M
1
,λ
k
p − λ
k
Ap
≤
y
k
− λ
k
Ay
k
−
p − λ
k
Ap
≤
I − λ
k
A
y
k
−
I − λ
k
A
p
≤
y
k
− p
,
3.5
y
k
− p
J
M
2
,δ
k
u
k
− δ
k
Bu
k
− J
M
2
,δ
k
p − δ
k
Bp
≤
u
k
− δ
k
Bu
k
−
p − δ
k
Bp
≤
u
k
− p
≤
x
k
− p
,
3.6
which yields
z
k
− p
2
α
k
v
k
− p
1 − α
k
1
t
k
t
k
0
S
s
W
k
v
k
ds − p
2
≤ α
k
v
k
− p
2
1 − α
k
1
t
k
t
k
0
SsW
k
v
k
ds − p
2
− α
k
1 − α
k
v
k
−
1
t
k
t
k
0
SsW
k
v
k
ds
2
≤ α
k
v
k
− p
2
1 − α
k
v
k
− p
2
− α
k
1 − α
k
v
k
−
1
t
k
t
k
0
SsW
k
v
k
ds
2
≤
v
k
− p
2
− α
k
1 − α
k
v
k
−
1
t
k
t
k
0
SsW
k
v
k
ds
2
.
3.7
12 Fixed Point Theory and Applications
Applying 3.5 and 3.6,weget
z
k
− p
2
≤
x
k
− p
2
− α
k
1 − α
k
v
k
−
1
t
k
t
k
0
SsW
k
v
k
ds
2
.
3.8
Hence p ∈ C
k1
. This implies that Θ ⊂ C
n
for each n ≥ 1.
Step 2. Next, we show that {x
n
} is well defined and C
n
is closed and convex for any n ∈ N.
It is obvious that C
1
C is closed and convex. Suppose that C
k
is closed and convex
for some k ≥ 1. Now, we show that C
k1
is closed and convex for some k. For any p ∈ C
k
,we
obtain
z
k
− p
2
≤
x
k
− p
2
3.9
is equivalent to
z
k
− x
k
2
2
z
k
− x
k
,x
k
− p
≤ 0.
3.10
Thus C
k1
is closed and convex. Then, C
n
is closed and convex for any n ∈ N. This implies
that {x
n
} is well-defined.
Step 3. Next, we show that {x
n
} is bounded and lim
n →∞
x
n
− x
0
exists. From x
n
P
C
n
x
0
,we
have
x
0
− x
n
,x
n
− y
≥ 0, 3.11
for each y ∈ C
n
.UsingΘ ⊂ C
n
, we also have
x
0
− x
n
,x
n
− p
≥ 0, ∀p ∈ Θ,n∈ N. 3.12
So, for p ∈ Θ, we observe that
0 ≤
x
0
− x
n
,x
n
− p
x
0
− x
n
,x
n
− x
0
x
0
− p
−x
0
− x
n
,x
0
− x
n
x
0
− x
n
,x
0
− p
≤−
x
0
− x
n
2
x
0
− x
n
x
0
− p
.
3.13
This implies that
x
0
− x
n
≤
x
0
− p
, ∀p ∈ Θ,n∈ N. 3.14
Fixed Point Theory and Applications 13
Hence, we get {x
n
} is bounded. It follows by 3.5–3.7,that{v
n
}, {y
n
},and{W
n
v
n
} are also
bounded. From x
n
P
C
n
x
0
,andx
n1
P
C
n1
x
0
∈ C
n1
⊂ C
n
,weobtain
x
0
− x
n
,x
n
− x
n1
≥ 0. 3.15
It follows that, we have for each n ∈ N
0 ≤
x
0
− x
n
,x
n
− x
n1
x
0
− x
n
,x
n
− x
0
x
0
− x
n1
−x
0
− x
n
,x
0
− x
n
x
0
− x
n
,x
0
− x
n1
≤−
x
0
− x
n
2
x
0
− x
n
x
0
− x
n1
.
3.16
It follows that
x
0
− x
n
≤
x
0
− x
n1
. 3.17
Thus, since the sequence {x
n
− x
0
} is a bounded and nondecreasing sequence, so
lim
n →∞
x
n
− x
0
exists, that is
m lim
n →∞
x
n
− x
0
.
3.18
Step 4. Next, we show that lim
n →∞
x
n1
− x
n
0 and lim
n →∞
x
n
− z
n
0.
Applying 3.15,weget
x
n
− x
n1
2
x
n
− x
0
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n1
x
0
− x
n1
2
x
n
− x
0
2
2
x
n
− x
0
,x
0
− x
n
x
n
− x
n1
x
0
− x
n1
2
x
n
− x
0
2
− 2
x
n
− x
0
,x
n
− x
0
2
x
n
− x
0
,x
n
− x
n1
x
0
− x
n1
2
−
x
n
− x
0
2
2
x
n
− x
0
,x
n
− x
n1
x
0
− x
n1
2
≤−
x
n
− x
0
2
x
0
− x
n1
2
.
3.19
Thus, by 3.18,weobtain
lim
n →∞
x
n
− x
n1
0.
3.20
14 Fixed Point Theory and Applications
On the other hand, from x
n1
P
C
n1
x
0
∈ C
n1
⊂ C
n
, which implies that
x
n1
− z
n
≤
x
n1
− x
n
. 3.21
It follows by 3.21, we also have
z
n
− x
n
≤
z
n
− x
n1
x
n1
− x
n
≤ 2
x
n
− x
n1
. 3.22
By 3.20,weobtain
lim
n →∞
x
n
− z
n
0.
3.23
Step 5. Next, we show that
lim
n →∞
I
k
n
x
n
− I
k−1
n
x
n
0
3.24
for every k ∈{1, 2, 3, ,N}. Indeed, for p ∈ Θ,notethatK
F
k
r
k,n
is the firmly nonexpansie, so
we have
I
k
n
x
n
− I
k
n
p
2
K
F
k
r
k,n
I
k−1
n
x
n
− K
F
k
r
k,n
p
2
≤
I
k
n
x
n
− p, I
k−1
n
x
n
− p
1
2
I
k
n
x
n
− p
2
I
k−1
n
x
n
− p
2
−
I
k
n
x
n
− I
k−1
n
x
n
2
.
3.25
Thus, we get
I
k
n
x
n
− I
k
n
p
2
≤
I
k−1
n
x
n
− p
2
−
I
k
n
x
n
− I
k−1
n
x
n
2
.
3.26
It follows that
u
n
− p
2
≤
I
k
n
x
n
− I
k
n
p
2
≤
I
k−1
n
x
n
− p
2
−
I
k
n
x
n
− I
k−1
n
x
n
2
≤
x
n
− p
2
−
I
k
n
x
n
− I
k−1
n
x
n
2
.
3.27
Fixed Point Theory and Applications 15
By 3.5, 3.6, 3.7,and3.27, we have for each k ∈{1, 2, 3, ,N}
z
n
− p
2
≤
v
n
− p
2
≤
u
n
− p
2
≤
x
n
− p
2
−
I
k
n
x
n
− I
k−1
n
x
n
2
.
3.28
Consequently, we have
I
k
n
x
n
− I
k−1
n
x
n
2
≤
x
n
− p
2
−
z
n
− p
2
≤
x
n
− z
n
x
n
− p
z
n
− p
.
3.29
Equation 3.23 implies that for every k ∈{1, 2, 3, ,N}
lim
n →∞
I
k
n
x
n
− I
k−1
n
x
n
0.
3.30
Step 6. Next, we show that lim
n →∞
y
n
− v
n
0 and lim
n →∞
K
n
W
n
v
n
− v
n
0, where
K
n
1/t
n
t
n
0
Ssds.
For any given p ∈ Θ, λ
n
∈ 0, 2γ, δ
n
∈ 0, 2β and p J
M
1
,λ
n
p − λ
n
ApJ
M
2
,δ
n
p −
δ
n
Bp. Since I − λ
n
A and I − δ
n
B are nonexpansive, we have
v
n
− p
2
J
M
1
,λ
n
y
n
− λ
n
Ay
n
− J
M
1
,λ
n
p − λ
n
Ap
2
≤
y
n
− λ
n
Ay
n
− p − λ
n
Ap
2
y
n
− p − λ
n
Ay
n
− Ap
2
≤
y
n
− p
2
− 2λ
n
y
n
− p, Ay
n
− Ap λ
2
n
Ay
n
− Ap
2
≤
x
n
− p
2
− 2λ
n
γ
Ay
n
− Ap
2
λ
2
n
Ay
n
− Ap
2
≤
x
n
− p
2
λ
n
λ
n
− 2γ
Ay
n
− Ap
2
.
3.31
Similarly, we can show that
y
n
− p
2
≤
x
n
− p
2
δ
n
δ
n
− 2β
Bu
n
− Bp
2
.
3.32
16 Fixed Point Theory and Applications
Observe that
z
n
− p
2
α
n
v
n
− p
1 − α
n
1
t
n
t
n
0
S
s
W
n
v
n
ds − p
2
≤ α
n
v
n
− p
2
1 − α
n
1
t
n
t
n
0
SsW
n
v
n
ds − p
2
− α
n
1 − α
n
v
n
−
1
t
n
t
n
0
SsW
n
v
n
ds
2
≤ α
n
v
n
− p
2
1 − α
n
1
t
n
t
n
0
SsW
n
v
n
ds − p
2
≤ α
n
x
n
− p
2
1 − α
n
v
n
− p
2
.
3.33
Substituting 3.31 into 3.33 and using conditions C4 and C5, we have
z
n
− p
2
≤ α
n
x
n
− p
2
1 − α
n
x
n
− p
2
λ
n
λ
n
− 2γ
Ay
n
− Ap
2
x
n
− p
2
1 − α
n
λ
n
λ
n
− 2γ
Ay
n
− Ap
2
.
3.34
It follows that
1 − d
a
1
2γ − b
1
Ay
n
− Ap
2
≤
1 − α
n
λ
n
2γ − λ
n
Ay
n
− Ap
2
≤
x
n
− p
2
−
z
n
− p
2
≤
x
n
− z
n
x
n
− p
z
n
− p
.
3.35
By 3.23,weobtain
lim
n →∞
Ay
n
− Ap
0.
3.36
Fixed Point Theory and Applications 17
Since the resolvent operator J
M
1
,λ
n
is 1-inverse-strongly monotone, we obtain
v
n
− p
2
J
M
1
,λ
n
y
n
− λ
n
Ay
n
− J
M
1
,λ
n
p − λ
n
Ap
2
J
M
1
,λ
n
I − λ
n
A
y
n
− J
M
1
,λ
n
I − λ
n
A
p
2
≤
I − λ
n
A
y
n
−
I − λ
n
A
p, v
n
− p
1
2
I − λ
n
A
y
n
−
I − λ
n
A
p
2
v
n
− p
2
−
I − λ
n
Ay
n
− I − λ
n
Ap − v
n
− p
2
≤
1
2
y
n
− p
2
v
n
− p
2
−
y
n
− v
n
− λ
n
Ay
n
− Ap
2
≤
1
2
x
n
− p
2
v
n
− p
2
−
y
n
− v
n
2
−λ
2
n
Ay
n
− Ap
2
2λ
n
y
n
− v
n
,Ay
n
− Ap
,
3.37
which yields
v
n
− p
2
≤
x
n
− p
2
−
y
n
− v
n
2
2λ
n
y
n
− v
n
Ay
n
− Ap
.
3.38
Similarly, we can obtain
y
n
− p
2
≤
x
n
− p
2
−
u
n
− y
n
2
2δ
n
u
n
− y
n
Bu
n
− Bp
.
3.39
Substituting 3.38 into 3.33, and using condition C4 and C5, we have
z
n
− p
2
≤ α
n
x
n
− p
2
1 − α
n
v
n
− p
2
≤ α
n
x
n
− p
2
1 − α
n
x
n
− p
2
−
y
n
− v
n
2
2λ
n
y
n
− v
n
Ay
n
− Ap
x
n
− p
2
−
1 − α
n
y
n
− v
n
2
2
1 − α
n
λ
n
y
n
− v
n
Ay
n
− Ap
.
3.40
It follows that
1 − α
n
y
n
− v
n
2
≤
x
n
− p
2
−
z
n
− p
2
2
1 − α
n
λ
n
y
n
− v
n
Ay
n
− Ap
≤
x
n
− z
n
x
n
− p
z
n
− p
2
1 − α
n
λ
n
y
n
− v
n
Ay
n
− Ap
.
3.41
By 3.23 and 3.36,weget
lim
n →∞
y
n
− v
n
0.
3.42
18 Fixed Point Theory and Applications
From 3.8 and C4, we also have
α
n
1 − α
n
v
n
−
1
t
n
t
n
0
SsW
n
v
n
ds
2
≤
x
n
− p
2
−
z
n
− p
2
≤
x
n
− z
n
x
n
− p
z
n
− p
.
3.43
Since K
n
1/t
n
t
n
0
Ssds,weobtain3.23, we have
lim
n →∞
K
n
W
n
v
n
− v
n
0.
3.44
Since {W
n
v
n
} is a bounded sequence in C,fromLemma 2.11 for all h ≥ 0, we have
lim
n →∞
K
n
W
n
v
n
− S
h
K
n
W
n
v
n
lim
n →∞
1
t
n
t
n
0
S
s
W
n
v
n
ds − S
h
1
t
n
t
n
0
S
s
W
n
v
n
ds
0.
3.45
From 3.44 and 3.45,weget
v
n
− S
s
v
n
≤
v
n
−K
n
W
n
v
n
K
n
W
n
v
n
− S
s
K
n
W
n
v
n
S
s
K
n
W
n
v
n
− S
s
v
n
≤ 2
v
n
−K
n
W
n
v
n
K
n
W
n
v
n
− S
s
K
n
W
n
v
n
.
3.46
So, we have
lim
n →∞
v
n
− S
s
v
n
0.
3.47
Step 7. Next, we show that q ∈ Θ : FS∩FW∩
N
k1
SMEPF
k
∩IA, M
1
∩IB, M
2
/
∅.
Since {v
n
i
} is bounded, there exists a subsequence {v
n
i
j
} of {v
n
i
} which converges
weakly to q ∈ C. Without loss of generality, we can assume that v
n
i
q.
1 First, we prove that q ∈ FS. Indeed, from Lemma 2.12 and 3.47,wegetq ∈
FS,thatis,q Ssq, for all s ≥ 0.
2 We show that q ∈ FW
∞
n1
FW
n
, where FW
n
∞
i1
FT
i
, for all n ≥ 1
and FW
n1
⊂ FW
n
. Assume that q/∈ FW, then there exists a positive integer m such
that q/∈ FT
m
and so q/∈
m
i1
FT
i
. Hence for any n ≥ m, q/∈
n
i1
FT
i
FW
n
,thatis,
q
/
W
n
q. T his together with q Ssq, for all s ≥ 0 shows q Ssq
/
SsW
n
q, for all s ≥ 0,
Fixed Point Theory and Applications 19
therefore we have q
/
K
n
W
n
q, for all n ≥ m. It follows from the Opial’s condition and 3.44
that
lim inf
i →∞
v
n
i
− q
< lim inf
i →∞
v
n
i
−K
n
i
W
n
i
q
≤ lim inf
i →∞
v
n
i
−K
n
i
W
n
i
v
n
i
K
n
i
W
n
i
v
n
i
−K
n
i
W
n
i
q
≤ lim inf
i →∞
v
n
i
− q
,
3.48
which is a contradiction. Thus, we get q ∈ FW.
3 We prove that q ∈
N
k1
SMEPF
k
,ϕ. Since I
k
n
K
F
k
r
k
, k 1, 2, ,Nand u
k
n
I
k
n
x
n
,
we have
F
k
I
k
n
x
n
,x
ϕ
x
− ϕ
I
k
n
x
n
1
r
k
K
I
k
n
x
n
−K
I
k−1
n
x
n
,η
x, I
k
n
x
n
≥ 0, ∀x ∈ C.
3.49
It follows that
1
r
k
K
I
k
n
i
x
n
i
−K
I
k−1
n
i
x
n
i
,η
x, I
k
n
i
x
n
i
≥−F
k
I
k
n
i
x
n
i
,x
− ϕ
x
ϕ
I
k
n
i
x
n
i
3.50
for all x ∈ C.From3.30 and by conditions C1c and C2,weget
lim
n
i
→∞
1
r
k
K
I
k
n
i
x
n
i
−K
I
k−1
n
i
x
n
i
,η
x, I
k
n
i
x
n
i
0.
3.51
By the assumption and by condition H1, we know that the function ϕ and the mapping
x → −F
k
x, y both are convex and lower semicontinuous, hence they are weakly lower
semicontinuous.
These together with K
I
k
n
i
x
n
i
−K
I
k−1
n
i
x
n
i
/r
k
→ 0andI
k
n
i
x
n
i
q, we have
0 lim inf
n
i
→∞
K
I
k
n
i
x
n
i
−K
I
k−1
n
i
x
n
i
r
k
,η
x, I
k
n
i
x
n
i
≥ lim inf
n
i
→∞
−F
k
I
k
n
i
x
n
i
,x
− ϕ
x
ϕ
I
k
n
i
x
n
i
.
3.52
Then, we obtain
F
k
q, x
ϕ
x
− ϕ
q
≥ 0, ∀x ∈ C, ∀k , 1, 2, ,N. 3.53
Therefore q ∈
N
k1
SMEPF
k
,ϕ.
4 Lastly, we prove that q ∈ IA, M
1
∩ IB, M
2
.
20 Fixed Point Theory and Applications
We observe that A is an 1/γ-Lipschitz monotone mapping and DAH.From
Lemma 2.2, we know that M
1
A is maximal monotone. Let v, g ∈ GM
1
A that is,
g − Av ∈ M
1
v. Since v
n
i
J
M
1
,λ
n
i
y
n
i
− λ
n
i
Ay
n
i
, we have
y
n
i
− λ
n
i
Ay
n
i
∈
I λ
n
i
M
1
v
n
i
, 3.54
that is,
1
λ
n
i
y
n
i
− v
n
i
− λ
n
i
Ay
n
i
∈ M
1
v
n
i
.
3.55
By virtue of the maximal monotonicity of M
1
A, we have
v − v
n
i
,g− Av −
1
λ
n
i
y
n
i
− v
n
i
− λ
n
i
Ay
n
i
≥ 0,
3.56
and so
v − v
n
i
,g≥
v − v
n
i
,Av
1
λ
n
i
y
n
i
− v
n
i
− λ
n
i
Ay
n
i
v − v
n
i
,Av− Av
n
i
Av
n
i
− Ay
n
i
1
λ
n
i
y
n
i
− v
n
i
≥ 0
v − v
n
i
,Av
n
i
− Ay
n
i
v − v
n
i
,
1
λ
n
i
y
n
i
− v
n
i
.
3.57
By 3.42, v
n
i
qand A is inverse-strongly monotone, we obtain that lim
n →∞
Ay
n
−Av
n
0
and it follows that
lim
n
i
→∞
v − v
n
i
,g
v − q, g
≥ 0.
3.58
It follows from the maximal monotonicity of M
1
A that θ ∈ M
1
Aq,thatis,q ∈ IA, M
1
.
Since {y
n
i
} is bounded, there exists a subsequence {y
n
i
j
} of {y
n
i
} which converges weakly to
q ∈ C. Without loss of generality, we can assume that y
n
i
q. In similar way, we can obtain
q ∈ IB, M
2
, hence q ∈ IA, M
1
∩ IB, M
2
.
Step 8. Finally, we show that x
n
→ z and u
n
→ z, where z P
Θ
x
0
.
Since Θ is nonempty closed convex subset of H, there exists a unique z
∈ Θ such that
z
P
Θ
x
0
. Since z
∈ Θ ⊂ C
n
and x
n
P
C
n
x
0
, we have
x
0
− x
n
≤
x
0
− P
C
n
x
0
≤
x
0
− z
3.59
for all n ∈ N.From3.59 and {x
n
} is bounded, so ω
w
x
n
/
∅.
Fixed Point Theory and Applications 21
By the weakly lower semicontinuous of the norm, we have
x
0
− z
≤ lim inf
n
i
→∞
x
0
− x
n
i
≤
x
0
− z
.
3.60
However, since z ∈ ω
w
x
n
⊂ Θ, we have
x
0
− z
≤
x
0
− P
C
n
x
0
≤
x
0
− z
. 3.61
Using 3.59 and 3.60,weobtainz
z.Thusω
w
x
n
{z} and x
n
z. So, we have
x
0
− z
≤
x
0
− z
≤ lim inf
n →∞
x
0
− x
n
≤ lim sup
n →∞
x
0
− x
n
≤
x
0
− z
.
3.62
Thus, we obtain that
x
0
− z
lim
n →∞
x
0
− x
n
x
0
− z
.
3.63
From x
n
z,weobtainx
0
− x
n
x
0
− z. Using the Kadec-Klee property, we obtain that
x
n
− z
x
n
− x
0
−
z − x
0
−→ 0asn −→ ∞ 3.64
and hence x
n
→ z in norm. Finally, noticing u
n
− z I
N
n
x
n
− I
N
n
z≤x
n
− z.Wealso
conclude that u
n
→ z in norm. This completes the proof.
Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H,let{F
k
: C ×
C →R,k 1, 2, ,N} be a finite family of mixed equilibrium functions satisfying conditions
(H1)–(H3). Let S {Ss :0≤ s<∞} be a nonexpansive semigroup on C and let {t
n
} be a
positive real divergent sequence. Let {V
i
: C → C}
∞
i1
be a countable family of uniformly ξ-strict
pseudocontractions, {T
i
: C → C}
∞
i1
be the countable family of nonexpansive mappings defined by
T
i
x tx 1 − tV
i
x, for all x ∈ C, for all i ≥ 1,t ∈ ξ,1, W
n
be the W-mapping defined by 2.8
and W be a mapping defined by 2.9 with FW
/
∅.LetA, B : C → H be γ,β-inverse-strongly
monotone mapping. Such that
Θ : F
S
∩ F
W
∩
N
k1
SMEP
F
k
∩ VI
C, A
∩ VI
C, B
/
∅.
3.65
22 Fixed Point Theory and Applications
Let r
k
> 0, k 1, 2, ,N, which are constants. Let {x
n
}, {y
n
}, {v
n
}, {z
n
}, and {u
n
} be sequences
generated by x
0
∈ C, C
1
C, x
1
P
C
1
x
0
, u
n
∈ C and
x
0
x ∈ C chosen arbitrarily,
u
n
K
F
N
r
N,n
K
F
N−1
r
N−1,n
K
F
N−2
r
N−2,n
···K
F
2
r
2,n
K
F
1
r
1,n
x
n
,
y
n
P
C
u
n
− δ
n
Bu
n
,
v
n
P
C
y
n
− λ
n
Ay
n
,
z
n
α
n
v
n
1 − α
n
1
t
n
t
n
0
S
s
W
n
v
n
ds,
C
n1
⎧
⎨
⎩
z ∈ C
n
:
z
n
− z
2
≤
x
n
− z
2
− α
n
1 − α
n
v
n
−
1
t
n
t
n
0
S
s
W
n
v
n
ds
2
⎫
⎬
⎭
,
x
n1
P
C
n1
x
0
,n∈ N,
3.66
where K
F
k
r
k
: C → C, k 1, 2, ,N is the mapping defined by 2.4 and {α
n
} be a sequence in
0, 1 for all n ∈ N. Assume the following conditions are satisfied:
C1 η
k
: C × C → H is L
k
-Lipschitz continuous with constant k 1, 2, ,Nsuch that
a η
k
x, yη
k
y, x0, for all x, y ∈ C,
b x → η
k
x, y is affine,
c for each fixed y ∈ C, y → η
k
x, y is sequentially continuous from the weak topology
to the weak topology;
C2 K
k
: C →Ris η
k
-strongly convex with constant σ
k
> 0 and its derivative K
k
is not only
sequentially continuous from the weak topology to the strong topology but also Lipschitz
continuous with a Lipschitz constant ν
k
> 0 such that σ
k
>L
k
ν
k
;
C3 for each k ∈{1, 2, ,N} and for all x ∈ C, there exist a bounded subset D
x
⊂ C and
z
x
∈ C such that for any y ∈ C \ D
x
,
F
k
y, z
x
ϕ
z
x
− ϕ
y
1
r
k
K
y
−K
x
,η
z
x
,y
< 0,
3.67
C4 {α
n
}⊂c, d,forsomec, d ∈ ξ, 1;
C5 {λ
n
}⊂a
1
,b
1
,forsomea
1
,b
1
∈ 0, 2γ;
C6 {δ
n
}⊂a
2
,b
2
,forsomea
2
,b
2
∈ 0, 2β;
C7 lim inf
n →∞
r
k,n
> 0, for each k ∈ 1, 2, 3, ,N.
Then, {x
n
} and {u
n
} converge strongly to z P
Θ
x
0
.
Fixed Point Theory and Applications 23
Proof. In Theorem 3.1, take M
i
iC
: H → 2
H
, where
iC
:0 → 0, ∞ is the indicator
function of C,thatis,
iC
x
⎧
⎨
⎩
0,x∈ C,
∞,x/∈ C,
3.68
for i 1, 2. Then 1.8 is equivalent to variational inequality problem, that is, to find x ∈ C
such that
Ax,y − x
≥ 0, ∀y ∈ C. 3.69
Again, since M
i
iC
,fori 1, 2, then
J
M
1
,λ
n
P
C
J
M
2
,δ
n
. 3.70
So, we have
v
n
P
C
y
n
− λ
n
Ay
n
J
M
1
,λ
n
y
n
− λ
n
Ay
n
,
y
n
P
C
u
n
− δ
n
Bu
n
J
M
2
,δ
n
u
n
− δ
n
Bu
n
.
3.71
Hence, we can obtain the desired conclusion from Theorem 3.1 immediately.
Next, we consider another class of important mappings.
Definition 3.3. A mapping S : C → C is called strictly pseudocontraction if there exists a
constant 0 ≤ κ<1 such that
Sx − Sy
2
≤
x − y
2
κ
I − Sx − I − Sy
2
, ∀x, y ∈ C.
3.72
If κ 0, then S is nonexpansive. In this case, we say that S : C → C is a κ-strictly
pseudocontraction. Putting B I − S. Then, we have
I − Bx − I − By
2
≤
x − y
2
κ
Bx − By
2
, ∀x, y ∈ C.
3.73
Observe that
I − Bx − I − By
2
x − y
2
Bx − By
2
− 2
x − y, Bx − By
, ∀x, y ∈ C.
3.74
Hence, we obtain
x − y, Bx − By
≥
1 − κ
2
Bx − By
2
, ∀x, y ∈ C.
3.75
Then, B is 1 − κ/2-inverse-strongly monotone mapping.
24 Fixed Point Theory and Applications
Now, we obtain the following result.
Theorem 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H,let{F
k
: C ×
C →R, k 1, 2, ,N} be a finite family of mixed equilibrium functions satisfying conditions
(H1)–(H3). Let S {Ss :0≤ s<∞} be a nonexpansive semigroup on C and let {t
n
} be a
positive real divergent sequence. Let {V
i
: C → C}
∞
i1
be a countable family of uniformly ξ-strict
pseudocontractions, {T
i
: C → C}
∞
i1
be the countable family of nonexpansive mappings defined by
T
i
x tx 1 − tV
i
x, for all x ∈ C, for all i ≥ 1, t ∈ ξ, 1, W
n
be the W-mapping defined by 2.8
and W be a mapping defined by 2.9 with FW
/
∅.LetA, B : C → H be γ,β-inverse-strongly
monotone mapping and S
A
,S
B
be κ
γ
,κ
β
-strictly pseudocontraction mapping of C into C for some
0 ≤ κ
γ
< 1, 0 ≤ κ
β
< 1 such that
Θ : F
S
∩ F
W
∩
N
k1
SMEP
F
k
∩ F
S
A
∩ F
S
B
/
∅.
3.76
Let r
k
> 0, k 1, 2, ,N, which are constants. Let {x
n
}, {y
n
}, {v
n
}, {z
n
}, and {u
n
} be sequences
generated by x
0
∈ C, C
1
C, x
1
P
C
1
x
0
, u
n
∈ C and
x
0
x ∈ C chosen arbitrarily,
u
n
K
F
N
r
N,n
K
F
N−1
r
N−1,n
K
F
N−2
r
N−2,n
···K
F
2
r
2,n
K
F
1
r
1,n
x
n
,
y
n
1 − δ
n
u
n
δ
n
S
B
u
n
,
v
n
1 − λ
n
y
n
λ
n
S
A
y
n
,
z
n
α
n
v
n
1 − α
n
1
t
n
t
n
0
S
s
W
n
v
n
ds,
C
n1
⎧
⎨
⎩
z ∈ C
n
:
z
n
− z
2
≤
x
n
− z
2
− α
n
1 − α
n
v
n
−
1
t
n
t
n
0
S
s
W
n
v
n
ds
2
⎫
⎬
⎭
,
x
n1
P
C
n1
x
0
,n∈ N,
3.77
where K
F
k
r
k
: C → C, k 1, 2, ,Nis the mapping defined by 2.4 and {α
n
} be a sequence in 0, 1
for all n ∈ N. Assume the following conditions are satisfied:
C1 η
k
: C × C → H is L
k
-Lipschitz continuous with constant k 1, 2, ,Nsuch that
a η
k
x, yη
k
y, x0, for all x, y ∈ C,
b x → η
k
x, y is affine,
c for each fixed y ∈ C, y → η
k
x, y is sequentially continuous from the weak topology
to the weak topology;
C2 K
k
: C →Ris η
k
-strongly convex with constant σ
k
> 0 and its derivative K
k
is not only
sequentially continuous from the weak topology to the strong topology but also Lipschitz
continuous with a Lipschitz constant ν
k
> 0 such that σ
k
>L
k
ν
k
;
Fixed Point Theory and Applications 25
C3 for each k ∈{1, 2, ,N} and for all x ∈ C, there exist a bounded subset D
x
⊂ C and
z
x
∈ C such that for any y ∈ C \ D
x
,
F
k
y, z
x
ϕ
z
x
− ϕ
y
1
r
k
K
y
−K
x
,η
z
x
,y
< 0;
3.78
C4 {α
n
}⊂c, d,forsomec, d ∈ ξ, 1;
C5 {λ
n
}⊂a
1
,b
1
,forsomea
1
,b
1
∈ 0, 2γ;
C6 {δ
n
}⊂a
2
,b
2
,forsomea
2
,b
2
∈ 0, 2β;
C7 lim inf
n →∞
r
k,n
> 0, for each k ∈ 1, 2, 3, ,N
Then, {x
n
} and {u
n
} converge strongly to z P
Θ
x
0
.
Proof. Taking A ≡ I −S
A
and B ≡ I −S
B
, then we see that A, B is 1 −κ
γ
/2, 1−κ
β
/2-inverse-
strongly monotone mapping, respectively. We have FS
A
VIC, A and FS
B
VIC, B.
So, we have
y
n
P
C
u
n
− δ
n
Bu
n
P
C
1 − δ
n
u
n
δ
n
S
B
u
n
1 − δ
n
u
n
δ
n
S
B
u
n
∈ C,
v
n
P
C
y
n
− λ
n
Ay
n
P
C
1 − λ
n
y
n
λ
n
S
A
y
n
1 − λ
n
y
n
λ
n
S
A
y
n
∈ C.
3.79
By using Theorem 3.2, it is easy to obtain the desired conclusion.
Acknowledgments
The authors would like to thank the Faculty of Science, King Monkut’s University of Tech-
nology Thonburi for its financial support. Moreover, P. Kumam was supported by the Com-
mission on Higher Education and the Thailand Research Fund under Grant MRG5380044.
References
1 F. Flores-Baz
´
an, “Existence theorems for generalized noncoercive equilibrium problems: the quasi-
convex case,” SIAM Journal on Optimization, vol. 11, no. 3, pp. 675–690, 2000.
2 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of
Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.
3 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The
Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
4 O. Chadli, N. C. Wong, and J. C. Yao, “Equilibrium problems with applications to eigenvalue
problems,” Journal of Optimization Theory and Applications, vol. 117, no. 2, pp. 245–266, 2003.
5 O. Chadli, S. Schaible, and J. C. Yao, “Regularized equilibrium problems with application to
noncoercive hemivariational inequalities,” Journal of Optimization Theory and Applications, vol. 121,
no. 3, pp. 571–596, 2004.
6 I. V. Konnov, S. Schaible, and J. C. Yao, “Combined relaxation method for mixed equilibrium
problems,” Journal of Optimization Theory and Applications, vol. 126, no. 2, pp. 309–322, 2005.
7 A. Moudafi and M. Th
´
era, “Proximal and dynamical approaches to equilibrium problems,” in Lecture
Notes in Econcmics and Mathematical Systems, vol. 477, pp. 187–201, Springer, Berlin, Germany, 1999.
8 L C. Zeng, S Y. Wu, and J C. Yao, “Generalized KKM theorem with applications to generalized
minimax inequalities and generalized equilibrium problems,” Taiwanese Journal of Mathematics, vol.
10, no. 6, pp. 1497–1514, 2006.
