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
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
≥ 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,
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
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
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
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
≤
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
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
− 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
},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
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
}⊂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
x
,
F
y, z
x
ϕ
z
x
− ϕ
y
1
r
K
y
−K
x
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
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
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
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
,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
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
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
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}
∞
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
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
1 − α
n
1
t
n
t
n
0
S
s
W
n
v
n
ds,
C
n1
⎧
⎨
⎩
z ∈ C
n
:
t
n
0
S
s
W
n
v
n
ds
2
⎫
⎬
⎭
,
x
n1
P
C
n1
x
0
: 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
,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
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
≤
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
k
y
k
− λ
k
Ay
k
− J
M
1
,λ
k
p − λ
k
Ap
≤
y
k
− λ
k
Ay
y
k
− p
,
3.5
y
k
− p
J
M
2
,δ
k
u
k
− δ
k
Bu
≤
u
k
− p
≤
x
k
− p
,
3.6
which yields
z
k
− p
2
k
ds − p
2
≤ α
k
v
k
− p
2
1 − α
k
1
−
1
t
k
t
k
0
SsW
k
v
k
ds
2
≤ α
k
v
k
− p
2
t
k
0
SsW
k
v
k
ds
2
≤
v
k
− p
2
− α
k
1 − α
k
z
k
− p
2
≤
x
k
− p
2
− α
k
1 − α
k
v
k
−
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
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
− p
−x
0
− x
n
,x
0
− x
n
x
0
− x
n
,x
0
− p
≤−
x
0
− x
n
2
, ∀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
− 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
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
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
n
− x
0
2
x
n
− x
0
,x
n
− x
n1
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
. 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
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
n
− p
1
2
I
k
n
x
n
− p
2
I
k−1
n
x
n
− p
k
n
x
n
− I
k
n
p
2
≤
I
k−1
n
x
n
− p
2
−
k
n
x
n
− I
k
n
p
2
≤
I
k−1
n
x
n
− p
2
−
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
k−1
n
x
n
2
.
3.28
Consequently, we have
I
k
n
x
n
− I
k−1
n
x
n
2
≤
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
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
y
n
− λ
n
Ay
n
− p − λ
n
Ap
2
y
n
− p − λ
n
Ay
n
− Ap
2
≤
y
n
− 2λ
n
γ
Ay
n
− Ap
2
λ
2
n
Ay
n
− Ap
2
≤
x
n
− p
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
W
n
v
n
ds − p
2
≤ α
n
v
n
− p
2
1 − α
n
v
n
−
1
t
n
t
n
0
SsW
n
v
n
ds
2
≤ α
n
v
n
− p
≤ 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
λ
n
− 2γ
Ay
n
− Ap
2
x
n
− p
2
1 − α
n
λ
n
2
≤
1 − α
n
λ
n
2γ − λ
n
Ay
n
− Ap
2
≤
x
n
− p
− 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
I − λ
n
A
y
n
− J
M
1
,λ
n
I − λ
n
A
p
y
n
−
I − λ
n
A
p
2
v
n
− p
2
−
I − λ
n
Ay
n
− I − λ
y
n
− v
n
− λ
n
Ay
n
− Ap
2
≤
1
2
x
n
− p
2
2λ
n
y
n
− v
n
,Ay
n
− Ap
,
3.37
which yields
v
n
− p
2
≤
x
n
− p
Similarly, we can obtain
y
n
− p
2
≤
x
n
− p
2
−
u
n
− y
n
2
2δ
n
x
n
− p
2
1 − α
n
v
n
− p
2
≤ α
n
x
n
− p
2
n
Ay
n
− Ap
x
n
− p
2
−
1 − α
n
y
n
− v
1 − α
n
y
n
− v
n
2
≤
x
n
− p
2
−
z
n
− p
2
x
n
− p
z
n
− p
2
1 − α
n
λ
n
y
n
− v
n
v
n
−
1
t
n
t
n
0
SsW
n
v
n
ds
2
≤
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
lim
n →∞
1
t
n
t
n
0
S
s
W
n
v
n
ds − S
h
1
t
n
n
≤
v
n
−K
n
W
n
v
n
K
n
W
n
v
n
− S
s
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
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
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
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
I
k
n
x
n
1
r
k
K
I
k
n
x
n
−K
I
k−1
n
x
n
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
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
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
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 ∈
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
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
,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
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
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
− z
≤ lim inf
n
i
→∞
x
0
− x
n
i
≤
x
0
− z
.
3.60
However, since z ∈ ω
w
x
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
− 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
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
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,
y
n
P
C
u
n
− δ
n
Bu
n
,
v
n
P
C
y
n
− λ
n
Ay
n
,
z
n
α
n
n
:
z
n
− z
2
≤
x
n
− z
2
− α
n
1 − α
n
v
n
−
1
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;
y, z
x
ϕ
z
x
− ϕ
y
1
r
k
K
y
−K
x
,η
∈ 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
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
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
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
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
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
,
v
n
1 − λ
n
y
n
λ
n
S
A
y
n
,
z
n
α
n
v
n
1 − α
n
1
≤
x
n
− z
2
− α
n
1 − α
n
v
n
−
1
t
n
t
n
0
S
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
x
− ϕ
y
1
r
k
K
y
−K
x
,η
z
x
,y
< 0;
3.78
C4 {α
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
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
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