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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 567147, 20 pages
doi:10.1155/2009/567147
Research Article
A Viscosity Approximation Method for Finding
Common Solutions of Variational Inclusions,
Equilibrium Problems, and Fixed Point Problems in
Hilbert Spaces
Somyot Plubtieng
1, 2
and Wanna Sriprad
1, 2
1
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
2
PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University, Bangkok
10400, Thailand
Correspondence should be addressed to Somyot Plubtieng,
Received 12 February 2009; Accepted 18 May 2009
Recommended by William A. Kirk
We introduce an iterative method for finding a common element of the set of common fixed points
of a countable family of nonexpansive mappings, the set of solutions of a variational inclusion with
set-valued maximal monotone mapping, and inverse strongly monotone mappings and the set of
solutions of an equilibrium problem in Hilbert spaces. Under suitable conditions, some strong
convergence theorems for approximating this common elements are proved. The results presented
in the paper improve and extend the main results of J. W. Peng et al. 2008 and many others.
Copyright q 2009 S. Plubtieng and W. Sriprad. 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.
1. Introduction
Let H be a real Hilbert space whose inner product and norm are denoted by ·, · and ·,
respectively. Let C be a nonempty closed convex subset of H,andletF be a bifunction of
C × C into R, where R is the set of real numbers. The equilibrium problem for F : C × C → R
is to find x ∈ C such that
F
x, y
≥ 0, ∀y ∈ C. 1.1
The set of solutions of 1.1 is denoted by EPF. Recently, Combettes and Hirstoaga 1
introduced an iterative scheme of finding the best approximation to t he initial data when
EPF is nonempty and proved a strong convergence theorem. Let A : C → H be a nonlinear
2 Fixed Point Theory and Applications
map. The classical variational inequality which is denoted by VIA, C is to find u ∈ C such
that
Au, v − u≥0, ∀v ∈ C. 1.2
The variational inequality has been extensively studied in literature. See, for example, 2, 3
and the references therein. Recall that a mapping T of C into itself is called nonexpansive if
Su − Sv≤u − v, ∀u, v ∈ C. 1.3
A mapping f : C → C is called contractive if there exists a constant β ∈ 0, 1 such that
fu− fv≤βu − v, ∀u, v ∈ C. 1.4
We denote by FS the set of fixed points of S.
Some methods have been proposed to solve the equilibrium problem and fixed
point problem of nonexpansive mapping; see, for instance, 3–
6 and the references therein.
Recently, Plubtieng and Punpaeng 6 introduced the following iterative scheme. Let x
1
∈ H
and let {x
n
},and{u
n
} be sequences generated by
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ H,
x
n1
α
n
γf
x
n
I − α
n
A
Su
n
, ∀n ∈ N.
1.5
They proved that if the sequences {α
n
} and {r
n
} of parameters satisfy appropriate conditions,
then the sequences {x
n
} and {u
n
} both converge strongly to the unique solution of the
variational inequality
A − γf
z, z − x
≥ 0, ∀x ∈ F
S
∩ EP
F
, 1.6
which is the optimality condition for the minimization problem
min
x∈FS∩EP F
1
2
Ax, x−h
x
, 1.7
where h is a potential function for γf.
Let A : H → H be a single-valued nonlinear mapping, and let M : H → 2
H
be a
set-valued mapping. We consider the following variational inclusion, which is to find a point
u ∈ H such that
θ ∈ A
u
M
u
, 1.8
where θ is the zero vector in H. The set of solutions of problem 1.8 is denoted by IA, M.
If A 0, then problem 1.8 becomes the inclusion problem introduced by Rockafellar 7.
Fixed Point Theory and Applications 3
If M ∂δ
C
, where C is a nonempty closed convex subset of H and δ
C
: H → 0, ∞ is the
indicator function of C, that is,
δ
C
x
⎧
⎨
⎩
0,x∈ C,
∞,x
/
∈ C,
1.9
then the variational inclusion problem 1.8 is equivalent to variational inequality problem
1.2. It is known that 1.8 provides a convenient framework for the unified study of optimal
solutions in many optimization related areas including mathematical programming, com-
plementarity, variational inequalities, optimal control, mathematical economics, equilibria,
game theory. Also various types of variational inclusions problems have been extended and
generalized see 8 and the references therein.
Very recently, Peng et al. 9 introduced the following iterative scheme for finding
a common element of the set of solutions to the problem 1.8, the set of solutions of an
equilibrium problem, and the set of fixed points of a nonexpansive mapping in Hilbert space.
Starting with x
1
∈ H, define sequence, {x
n
}, {y
n
},and{u
n
} by
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ H,
x
n1
α
n
γf
x
n
1 − α
n
Sy
n
,
y
n
J
M,λ
u
n
− λAu
n
, ∀n ≥ 0,
1.10
for all n ∈ N, where λ ∈ 0, 2α, {α
n
}⊂0, 1 and {r
n
}⊂0, ∞. They proved that
under certain appropriate conditions imposed on {α
n
} and {r
n
}, the sequences {x
n
}, {y
n
},
and {u
n
} generated by 1.10 converge strongly to z ∈ FT ∩ IA, M ∩ EPF, where
z P
FS∩IA,M∩EPF
fz.
Motivated and inspired by Plubtieng and Punpaeng 6, Peng et al. 9 and Aoyama et
al. 10, we introduce an iterative scheme for finding a common element of the set of solutions
of the variational inclusion problem 1.8 with multi-valued maximal monotone mapping
and inverse-strongly monotone mappings, the set of solutions of an equilibrium problem and
the set of fixed points of a nonexpansive mapping in Hilbert space. Starting with an arbitrary
x
1
∈ H, define sequences {x
n
}, {y
n
} and {u
n
} by
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ H,
x
n1
α
n
γf
x
n
I − α
n
B
S
n
y
n
,
y
n
J
M,λ
u
n
− λAu
n
, ∀n ≥ 0,
1.11
for all n ∈ N, where λ ∈ 0, 2α, {α
n
}⊂0, 1,andlet{r
n
}⊂0, ∞; B be a strongly bounded
linear operator on H,and{S
n
} is a sequence of nonexpansive mappings on H. Under suitable
conditions, some strong convergence t heorems for approximating to this common elements
are proved. Our results extend and improve some corresponding results in 3, 9 and the
references therein.
4 Fixed Point Theory and Applications
2. Preliminaries
This section collects some lemmas which will be used in the proofs for the main results in the
next section.
Let H be a real Hilbert space with inner product ·, · and norm ·, respectively.
It is wellknown that for all x, y ∈ H and λ ∈ 0, 1, there holds
λx 1 − λy
2
λ
x
2
1 − λ
y
2
− λ
1 − λ
x − y
2
. 2.1
Let C be a nonempty closed convex subset of H. T hen, for any x ∈ H, there exists a unique
nearest point of C, denoted by P
C
x, such that x − P
C
x≤x − y for all y ∈ C. Such a P
C
is
called the metric projection from H into C. We know that P
C
is nonexpansive. It is also known
that, P
C
x ∈ C and
x − P
C
x, P
C
x − z≥0, ∀x ∈ H and z ∈ C. 2.2
It is easy to see that 2.2 is equivalent to
x − z
2
≥
x − P
C
x
2
z − P
C
x
2
, ∀x ∈ H, z ∈ C. 2.3
For solving the equilibrium problem for a bifunction F : C × C → R, let us assume
that F satisfies the following conditions:
A1 Fx, x0 for all x ∈ C;
A2 F is monotone, that is, Fx, yFy, x ≤ 0 for all x, y ∈ C;
A3 for each x, y, z ∈ C,
lim
t → 0
F
tz
1 − t
x, y
≤ F
x, y
; 2.4
A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous.
The following lemma appears implicitly in 11 and 1.
Lemma 2.1 See 1, 11. Let C be a nonempty closed convex subset of H and let F be a bifunction
of C × C in to R 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. 2.5
Define a mapping T
r
: H → C as follows:
T
r
x
z ∈ C : F
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C
, 2.6
Fixed Point Theory and Applications 5
for all z ∈ H. Then, the following hold:
1 T
r
is single-valued;
2 T
r
is firmly nonexpansive, that is, for any x, y ∈ H,
T
r
x − T
r
y
2
≤
T
r
x − T
r
y, x − y
; 2.7
3 FT
r
EPF;
4 EPF is closed and convex.
We also need the following lemmas for proving our main result.
Lemma 2.2 See 12. Let H be a Hilbert space, C a nonempty closed convex subset of H, f : H →
H a contraction with coefficient 0 <α<1, and B a strongly positive linear bounded operator with
coefficient
γ>0.Then:
1 if 0 <γ<
γ/α,thenx − y, B − γfx − B − γfy≥γ − γαx − y
2
,x,y ∈ H.
2 if 0 <ρ<B
−1
,thenI − ρB≤1 − ργ.
Lemma 2.3 See 13. Assume {a
n
} is a sequence of nonnegative real numbers such that
a
n1
≤
1 − γ
n
a
n
δ
n
,n≥ 0, 2.8
where {γ
n
} is a sequence in 0, 1 and {δ
n
} is a sequence in R such that
1
∞
n1
γ
n
∞;
2 lim sup
n →∞
δ
n
/γ
n
≤ 0 or
∞
n1
|δ
n
| < ∞.
Then lim
n →∞
a
n
0.
Recall that a mapping A : H → H is called α-inverse-strongly monotone, if there
exists a positive number α such that
Au − Av, u − v≥α
Au − Av
2
, ∀u, v ∈ H. 2.9
Let I be the identity mapping on H. It is well known that if A : H → H is α-inverse-
strongly monotone, then A is 1/α-Lipschitz continuous and monotone mapping. In addition,
if 0 <λ≤ 2α, then I − λA is a nonexpansive mapping.
A set-valued 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 : H → 2
H
is maximal if its graph
GM : {x, f ∈ 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 every y, g ∈ GM implies f ∈ Mx.
Let the set-valued mapping M : H → 2
H
be maximal monotone. We define the
resolvent operator J
M,λ
associated with M and λ as follows:
J
M,λ
u
I λM
−1
u
, ∀u ∈ H, 2.10
6 Fixed Point Theory and Applications
where λ is a positive number. It is worth mentioning that the resolvent operator J
M,λ
is
single-valued, nonexpansive and 1-inverse-strongly monotone, see for example 14 and that
a solution of problem 1.8 is a fixed point of the operator J
M,λ
I − λA for all λ>0, see for
instance 15.
Lemma 2.4 See 14. Let M : H → 2
H
be a maximal monotone mapping and A : H → H be
a Lipschitz-continuous mapping. Then the mapping S M A : H → 2
H
is a maximal monotone
mapping.
Remark 2.5 See 9. Lemma 2.4 implies that IA, M is closed and convex if M : H → 2
H
is
a maximal monotone mapping and A : H → H be an inverse strongly monotone mapping.
Lemma 2.6 See 10. Let C be a nonempty closed subset of a Banach space and let {S
n
} a sequence
of mappings of C into itself. Suppose that
∞
n1
sup{S
n1
z − S
n
z : z ∈ C} < ∞. Then, for each
x ∈ C, {S
n
x} converges strongly to some point of C. Moreover, let S be a mapping from C into itself
defined by
Sx lim
n →∞
S
n
x, ∀x ∈ C. 2.11
Then lim
n →∞
sup{Sz − S
n
z : z ∈ C} 0.
3. Main Results
We begin this section by proving a strong convergence theorem of an implicit iterative
sequence {x
n
} obtained by the viscosity approximation method for finding a common
element of the set of solutions of the variational inclusion, the set of solutions of an
equilibrium problem and the set of fixed points of a nonexpansive mapping.
Throughout the rest of this paper, we always assume that f is a contraction of H into
itself with coefficient β ∈ 0, 1,andB is a strongly positive bounded linear operator with
coefficient
γ and 0 <γ<γ/β.LetS be a nonexpansive mapping of H into H.LetA : H → H
be an α-inverse-strongly monotone mapping, M : H → 2
H
be a maximal monotone mapping
and let J
M,λ
be defined as in 2.10.Let{T
r
n
} be a sequence of mappings defined as Lemma 2.1.
Consider a sequence of mappings {S
n
} on H defined by
S
n
x α
n
γf
x
I − α
n
B
SJ
M,λ
I − λA
T
r
n
x, x ∈ H, n ≥ 1, 3.1
where {α
n
}⊂0, B
−1
. By Lemma 2.2,wenotethatS
n
is a contraction. Therefore, by the
Banach contraction principle, S
n
has a unique fixed point x
n
∈ H such that
x
n
α
n
γf
x
n
I − α
n
B
SJ
M,λ
I − λA
T
r
n
x
n
. 3.2
Theorem 3.1. Let H be a real Hilbert space, let F be a bifunction from H × H → R satisfying
(A1)–(A4) and let S be a nonexpansive mapping on H.LetA : H → H be an α-inverse-strongly
monotone mapping, M : H → 2
H
be a maximal monotone mapping such that Ω : FS ∩ EPF ∩
IA, M
/
∅. Let f be a contraction of H into itself with a constant β ∈ 0, 1 and let B be a strongly
Fixed Point Theory and Applications 7
bounded linear operator on H with coefficient
γ>0 and 0 <γ<γ/β.Let{x
n
}, {y
n
} and {u
n
} be
sequences generated by x
1
∈ H and
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥0, ∀y ∈ H
x
n
α
n
γf
x
n
I − α
n
B
Sy
n
,
y
n
J
M,λ
u
n
− λAu
n
∀n ≥ 0,
3.3
where λ ∈ 0, 2α, {r
n
}⊂0, ∞ and {α
n
}⊂0, 1 satisfy lim
n →∞
α
n
0 and lim inf
n →∞
r
n
>
0. Then, {x
n
}, {y
n
} and {u
n
} converges strongly to a point z in Ω which solves the variational
inequality:
B − γf
z, z − x≤0,x∈ Ω. 3.4
Equivalently, we have z P
Ω
I − B γfz.
Proof. First, we assume that α
n
∈ 0, B
−1
.ByLemma 2.2,weobtainI − α
n
B≤1 − α
n
γ.Let
v ∈ Ω. Since u
n
T
r
n
x
n
, we have
u
n
− v T
r
n
x
n
− T
r
n
v≤x
n
− v∀n ∈ N. 3.5
We note from v ∈ Ω that v J
M,λ
v − λAv.AsI − λA is nonexpansive, we have
y
n
− v J
M,λ
u
n
− λAu
n
− J
M,λ
v − λAv
≤
u
n
− λAu
n
−
v − λAv
≤u
n
− v≤x
n
− v
3.6
for all n ∈ N. Thus, we have
x
n
− v α
n
γf
x
n
I − α
n
B
Sy
n
− v
≤ α
n
γf
x
n
− Bv I − α
n
By
n
− v
≤ α
n
γf
x
n
− Bv
1 − α
n
γ
x
n
− v
≤ α
n
γ
f
x
n
− f
v
γf
v
− Bv
1 − α
n
γ
x
n
− v
≤ α
n
γβx
n
− v α
n
γf
v
− Bv
1 − α
n
γ
x
n
− v
1 − α
n
γ − γβ
x
n
− v α
n
γf
v
− Bv.
3.7
It follows that x
n
− v≤γfv − Bv/γ − γβ, ∀n ≥ 1. Hence {x
n
} is bounded and we
also obtain that {u
n
},{y
n
}, {fx
n
},{Sy
n
} and {Au
n
} are bounded. Next, we show that y
n
−
Sy
n
→0. Since α
n
→ 0, we note that
x
n
− Sy
n
α
n
γf
x
n
− BSy
n
−→0asn −→ ∞ . 3.8
8 Fixed Point Theory and Applications
Moreover, it follows from Lemma 2.1 that
u
n
− v
2
T
r
n
x
n
− T
r
n
v
2
≤T
r
n
x
n
− T
r
n
v, x
n
− v u
n
− v, x
n
− v
1
2
u
n
− v
2
x
n
− v
2
−
x
n
− u
n
2
,
3.9
and hence u
n
− v
2
≤x
n
− v
2
−x
n
− u
n
2
. Therefore, we have
x
n
− v
2
α
n
γfx
n
I − α
n
BSy
n
− v
2
I − α
n
BSy
n
− vα
n
γfx
n
− Bv
2
≤
1 − α
n
γ
2
Sy
n
− v
2
2α
n
γf
x
n
− Bv, x
n
− v
≤
1 − α
n
γ
2
y
n
− v
2
2α
n
γf
x
n
− f
v
,x
n
− v 2α
n
γf
v
− Bv, x
n
− v
≤
1 − α
n
γ
2
u
n
− v
2
2α
n
γf
x
n
− f
v
,x
n
− v 2α
n
γf
v
− Bv, x
n
− v
≤
1 − α
n
γ
2
x
n
− v
2
−
x
n
− u
n
2
2α
n
γβ
x
n
− v
2
2α
n
γf
v
− Bvx
n
− v
1 − 2α
n
γ − γβ
α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
x
n
− u
n
2
2α
n
γf
v
− Bvx
n
− v
≤
x
n
− v
2
α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
x
n
− u
n
2
2α
n
γf
v
− Bvx
n
− v,
3.10
and hence
1 − α
n
γ
2
x
n
− u
n
2
≤ α
n
γ
2
x
n
− v
2
2α
n
γf
v
− Bvx
n
− v. 3.11
Since {x
n
} is bounded and α
n
→ 0, it follows that x
n
− u
n
→0asn →∞.
Put M sup
n≥1
{γfv − Bvx
n
− v}.From3.10, it follows by the nonexpansive of
J
M,λ
and the inverse strongly monotonicity of A that
x
n
− v
2
≤
1 − α
n
γ
2
y
n
− v
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1 − α
n
γ
2
u
n
− λAu
n
− v − λAv
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1 − α
n
γ
2
u
n
− v
2
λ
λ − 2α
Au
n
− Av
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1−α
n
γ
2
x
n
−v
2
1−α
n
γ
2
λ
λ−2α
Au
n
−Av
2
2α
n
γβ
x
n
−v
2
2α
n
M
1−2α
n
γ −γβ
α
n
γ
2
x
n
−v
2
1−α
n
γ
2
λ
λ−2α
Au
n
−Av
2
2α
n
M
≤
x
n
− v
2
α
n
γ
2
x
n
− v
2
1 − α
n
γ
2
λ
λ − 2α
Au
n
− Av
2
2α
n
M,
3.12
Fixed Point Theory and Applications 9
which implies that
1 − α
n
γ
2
λ
2α − λ
Au
n
− Av
2
≤ α
n
γ
2
x
n
− v
2
2α
n
M. 3.13
Since α
n
→ 0, we have Au
n
−Av→0asn →∞. Since J
M,λ
is 1–inverse-strongly monotone
and I − λA is nonexpansive, we have
y
n
− v
2
J
M,λ
u
n
− λAu
n
− J
M,λ
v − λAv
2
≤
u
n
− λAu
n
−
v − λAv
,y
n
− v
1
2
u
n
−λAu
n
−v−λAv
2
y
n
−v
2
−
u
n
−λAu
n
−v−λAv−y
n
−v
2
≤
1
2
u
n
− v
2
y
n
− v
2
−
u
n
− y
n
− λAu
n
− Av
2
1
2
u
n
−v
2
y
n
−v
2
−
u
n
−y
n
2
2λ
u
n
−y
n
,Au
n
−Av
−λ
2
Au
n
− Av
2
.
3.14
Thus, we have
y
n
− v
2
≤
u
n
− v
2
−
u
n
− y
n
2
2λ
u
n
− y
n
,Au
n
− Av
− λ
2
Au
n
− Av
2
. 3.15
From 3.5, 3.10,and3.15, we have
x
n
− v
2
≤
1 − α
n
γ
2
y
n
− v
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1 − α
n
γ
2
u
n
− v
2
−
u
n
− y
n
2
2λ
u
n
− y
n
,Au
n
− Av
− λ
2
Au
n
− Av
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1 − α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
u
n
− y
n
2
2
1 − α
n
γ
2
λ
u
n
− y
n
,Au
n
− Av
−
1 − α
n
γ
2
λ
2
Au
n
− Av
2
2α
n
γβ
x
n
− v
2
2α
n
M
1 − 2α
n
γ − γβ
α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
u
n
− y
n
2
2
1 − α
n
γ
2
λ
u
n
− y
n
,Au
n
− Av
−
1 − α
n
γ
2
λ
2
Au
n
− Av
2
2α
n
M
≤
x
n
− v
2
α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
u
n
− y
n
2
2
1 − α
n
γ
2
λ
u
n
− y
n
,Au
n
− Av
−
1 − α
n
γ
2
λ
2
Au
n
− Av
2
2α
n
M.
3.16
Thus, we get
1 − α
n
γ
2
u
n
− y
n
2
≤ α
n
γ
2
x
n
− v
2
2
1 − α
n
γ
2
λ
u
n
− y
n
,Au
n
− Av
−
1 − α
n
γ
2
λ
2
Au
n
− Av
2
2α
n
M.
3.17
10 Fixed Point Theory and Applications
Since α
n
→ 0, Au
n
− Av→0asn →∞, we have u
n
− y
n
→0asn →∞. It follows
from the inequality y
n
− Sy
n
≤y
n
− u
n
u
n
− x
n
x
n
− Sy
n
that y
n
− Sy
n
→0as
n →∞. Moreover, we have x
n
− y
n
≤x
n
− u
n
u
n
− y
n
→0asn →∞.
Put U ≡ SJ
M,λ
I − λA. Since both S and J
M,λ
I − λA are nonexpansive, we have U
is a nonexpansive mapping on H and t hen we have x
n
α
n
γfx
n
I − α
n
BUT
r
n
x
n
for all
n ∈ N. It follows by Theorem 3.1 of Plubtieng and Punpaeng 6 that {x
n
} converges strongly
to z ∈ FU ∩ EPF, where z P
FU∩EP F
γf I − Bz and B − γfz, u − z≥0, for all
u ∈ FU ∩ EPF. We will show that z ∈ FS ∩ IA, M. Since {x
n
} converges strongly to
z, we also have x
n
z. Let us show z ∈ FS. Assume z
/
∈ FS. Since x
n
− y
n
→0and
x
n
z, we have y
n
zSince z
/
Sz, it follows by the Opial’s condition that
lim inf
n →∞
y
n
− z < lim inf
n →∞
y
n
− Sz≤lim inf
n →∞
y
n
− Sy
n
Sy
n
− Sz
≤ lim inf
n →∞
y
n
− z.
3.18
This is a contradiction. Hence z ∈ FS. We now show that z ∈ IA, M. In fact, since A
is α–inverse-strongly monotone, A is an 1/α-Lipschitz continuous monotone mapping and
DAH. It follows from Lemma 2.4 that MA is maximal monotone. Let p, g ∈ GMA,
that is, g − Ap ∈ Mp. Again since y
n
J
M,λ
u
n
− λAu
n
, we have u
n
− λAu
n
∈ I λMy
n
,
that is,
1
λ
u
n
− y
n
− λAu
n
∈ M
y
n
. 3.19
By the maximal monotonicity of M A, we have
p − y
n
,g− Ap −
1
λ
u
n
− y
n
− λAu
n
≥ 0, 3.20
and so
p − y
n
,g
≥
p − y
n
,Ap
1
λ
u
n
− y
n
− λAu
n
p − y
n
,Ap− Ay
n
Ay
n
− Au
n
1
λ
u
n
− y
n
≥ 0
p − y
n
,Ay
n
− Au
n
p − y
n
,
1
λ
u
n
− y
n
.
3.21
It follows from u
n
− y
n
→0, Au
n
− Ay
n
→0andy
n
zthat
lim
n →∞
p − y
n
,g
p − z, g
≥ 0. 3.22
Since A M is maximal monotone, this implies that θ ∈ M Az, that is, z ∈ IA, M.
Hence, z ∈ Ω : FS∩EPF∩IA, M. Since FS∩IA, MFS∩FJ
M,λ
I−λA ⊂ FU,
we have Ω ⊂ FU∩EPF. It implies that z is the unique solution of the variational inequality
3.4.
Fixed Point Theory and Applications 11
Now we prove the following theorem which is the main result of this paper.
Theorem 3.2. Let H be a real Hilbert space, let F be a bifunction from H × H → R satisfying
(A1)–(A4) and let {S
n
} be a sequence of nonexpansive mappings on H.LetA : H → H be an
α-inverse-strongly monotone mapping, M : H → 2
H
be a maximal monotone mapping such that
Ω :
∞
n1
FS
n
∩ EPF ∩ IA, M
/
∅. Let f be a contraction of H into itself with a constant
β ∈ 0, 1 and let B be a strongly bounded linear operator on H with coefficient
γ>0 and 0 <γ<γ/β.
Let {x
n
}, {y
n
} and {u
n
} be sequences generated by x
1
∈ H and
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥0, ∀y ∈ H
x
n1
α
n
γf
x
n
I − α
n
B
S
n
y
n
,
y
n
J
M,λ
u
n
− λAu
n
∀n ≥ 0,
3.23
for all n ∈ N,whereλ ∈ 0, 2α, {α
n
}⊂0, 1 and {r
n
}⊂0, ∞ satisfy
lim
n →∞
α
n
0,
∞
n1
α
n
∞,
∞
n1
|
α
n1
− α
n
|
< ∞,
lim inf
n →∞
r
n
> 0,
∞
n1
|
r
n1
− r
n
|
< ∞.
3.24
Suppose that
∞
n1
sup{S
n1
z − S
n
z : z ∈ K} < ∞ for any bounded subset K of H.LetS be a
mapping of H into itself defined by Sx lim
n →∞
S
n
x, for all x ∈ H and suppose that FS
∞
n1
FS
n
. Then, {x
n
}, {y
n
} and {u
n
} converges strongly to z,wherez P
Ω
I − B γfz is a
unique solution of the variational inequalities 3.4.
Proof. Since α
n
→ 0, we may assume that α
n
≤B
−1
for all n. First we will prove that {x
n
} is
bonded. Let v ∈ Ω. Then, we have
x
n1
− v α
n
γf
x
n
I − α
n
B
S
n
y
n
− v
≤ α
n
γf
x
n
− Bv I − α
n
By
n
− v
≤ α
n
γf
x
n
− Bv
1 − α
n
γ
x
n
− v
≤ α
n
γ
f
x
n
− f
v
γf
v
− Bv
1 − α
n
γ
x
n
− v
≤ α
n
γβx
n
− v α
n
γf
v
− Bv
1 − α
n
γ
x
n
− v
1 − α
n
γ − γβ
x
n
− v α
n
γf
v
− Bv
1 − α
n
γ − γα
x
n
− v α
n
γ − γα
γf
v
− Bv
γ − γα
.
3.25
It follows from 3.25 and induction that
x
n
− v≤max
x
1
− v,
1
γ − γα
γf
p
− B
p
,n≥ 0. 3.26
12 Fixed Point Theory and Applications
Hence {x
n
} is bounded and therefore {u
n
}, {y
n
}, {fx
n
}, {S
n
y
n
} and {Au
n
} are also
bounded. Next, we show that x
n1
− x
n
→0. Since I − λA is nonexpansive, it follows
that
y
n1
− y
n
J
M,λ
u
n1
− λAu
n1
− J
M,λ
u
n
− λAu
n
≤
u
n1
− λAu
n1
−
u
n
− λAu
n
≤u
n1
− u
n
.
3.27
Then, we have
x
n2
− x
n1
α
n1
γf
x
n1
I − α
n1
B
S
n1
y
n1
− α
n
γf
x
n
−
I − α
n
B
S
n
y
n
α
n1
γf
x
n1
I − α
n1
B
S
n1
y
n1
− α
n
γf
x
n
−
I − α
n
B
S
n
y
n
−
I − α
n1
B
S
n1
y
n
I − α
n1
B
S
n1
y
n
−
I − α
n
B
S
n1
y
n
I − α
n
B
S
n1
y
n
I − α
n1
B
S
n1
y
n1
− S
n1
y
n
α
n
− α
n1
BS
n1
y
n
I − α
n
B
S
n1
y
n
−S
n
y
n
α
n1
−α
n
γf
x
n
α
n1
γf
x
n1
−f
x
n
≤
1−α
n1
γ
y
n1
−y
n
|
α
n
−α
n1
|
BS
n1
y
n
1−α
n
γ
S
n1
y
n
−S
n
y
n
|
α
n
− α
n1
|
γf
x
n
α
n1
γβx
n1
− x
n
≤
1 − α
n1
γ
y
n1
− y
n
α
n1
γβx
n1
− x
n
|
α
n
− α
n1
|
BS
n1
y
n
γf
x
n
S
n1
y
n
− S
n
y
n
≤
1 − α
n1
γ
u
n1
− u
n
α
n1
γβ
x
n1
− x
n
|
α
n
− α
n1
|
M
sup
S
n1
z − S
n
z
: z ∈
y
n
,
3.28
where M : sup{max{BS
n1
y
n
, γfx
n
} : n ≥ 0} < ∞. On the other hand, we note that
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ H
, 3.29
F
u
n1
,y
1
r
n1
y − u
n1
,u
n1
− x
n1
≥ 0, ∀y ∈ H.
3.30
Putting y u
n1
in 3.29 and y u
n
in 3.30, Fu
n
,u
n1
1/r
n
u
n1
− u
n
,u
n
− x
n
≥0
and Fu
n1
,u
n
1/r
n1
u
n
− u
n1
,u
n1
− x
n1
≥0. By A2, we have
u
n1
− u
n
,
u
n
− x
n
r
n
−
u
n1
− x
n1
r
n1
≥ 0 3.31
and hence
u
n1
− u
n
,u
n
− u
n1
u
n1
− x
n
−
r
n
r
n1
u
n1
− x
n1
≥0. 3.32
Fixed Point Theory and Applications 13
Since lim inf
n →∞
r
n
> 0, we assume that there exists a real number b such that r
n
>b>0for
all n ∈ N. Thus, we have
u
n1
− u
n
2
≤
u
n1
− u
n
,x
n1
− x
n
1 −
r
n
r
n1
u
n1
− x
n1
≤u
n1
− u
n
x
n1
− x
n
1 −
r
n
r
n1
u
n1
− x
n1
,
3.33
and hence
u
n1
− u
n
≤
x
n1
− x
n
1
r
n1
|
r
n1
− r
n
|
u
n1
− x
n1
≤
x
n1
− x
n
1
b
|
r
n1
− r
n
|
L,
3.34
where L sup{u
n
− x
n
: n ∈ N}.From3.28, we have
x
n2
− x
n1
≤
1 − α
n1
γ
x
n1
− x
n
1
b
|
r
n1
− r
n
|
L
α
n1
γβx
n1
− x
n
|
α
n
− α
n1
|
M sup
S
n1
z − S
n
z : z ∈
y
n
1 − α
n1
γ α
n1
γβ
x
n1
− x
n
1 − α
n1
γ
b
|
r
n1
− r
n
|
L
|
α
n
− α
n1
|
M sup
S
n1
z − S
n
z : z ∈
y
n
≤
1 − α
n1
γ − γβ
x
n1
− x
n
L
b
|
r
n1
− r
n
|
|
α
n
− α
n1
|
M
sup
S
n1
z − S
n
z
: z ∈
y
n
.
3.35
Since {y
n
} is bounded, it follows that
∞
n1
sup{S
n1
z − S
n
z : z ∈{y
n
}} < ∞. Hence, by
Lemma 2.3, we have x
n1
− x
n
→∞as n →∞.From3.34 and |r
n1
− r
n
|→0, we have
lim
n →∞
u
n1
− u
n
0. Moreover, we have from 3.27 that lim
n →∞
y
n1
− y
n
0.
We note from 3.23 that x
n
α
n−1
γfx
n−1
1 − α
n−1
BS
n−1
y
n−1
. Then, we have
x
n
− S
n
y
n
≤x
n
− S
n−1
y
n−1
S
n−1
y
n−1
− S
n−1
y
n
S
n−1
y
n
− S
n
y
n
≤ α
n−1
γf
x
n−1
− BS
n−1
y
n−1
y
n−1
− y
n
sup
S
n−1
z − S
n
z : z ∈
y
n
.
3.36
Since α
n
→ 0, y
n−1
− y
n
→0andsup{S
n−1
z−S
n
z : z ∈{y
n
}} → 0, we get x
n
− S
n
y
n
→
0. From the proof of Theorem 3.1, we have
u
n
− v
2
≤
x
n
− v
2
−
x
n
− u
n
2
, 3.37
14 Fixed Point Theory and Applications
for all v ∈ Ω. Therefore, we have
x
n1
− v
2
α
n
γfx
n
I − α
n
BS
n
y
n
− v
2
I − α
n
BS
n
y
n
− vα
n
γfx
n
− Bv
2
≤
1 − α
n
γ
2
S
n
y
n
− v
2
2α
n
γf
x
n
− Bv, x
n
− v
≤
1 − α
n
γ
2
y
n
− v
2
2α
n
γf
x
n
− f
v
,x
n
− v 2α
n
γf
v
− Bv, x
n
− v
≤
1 − α
n
γ
2
u
n
− v
2
2α
n
γf
x
n
− f
v
,x
n
− v 2α
n
γf
v
− Bv, x
n
− v
≤
1 − α
n
γ
2
x
n
− v
2
−
x
n
− u
n
2
2α
n
γβ
x
n
− v
2
2α
n
γf
v
− Bv
x
n
− v
1 − 2α
n
γ − γβ
α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
x
n
− u
n
2
2α
n
γf
v
− Bv
x
n
− v
≤
x
n
− v
2
α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
x
n
− u
n
2
2α
n
γf
v
− Bvx
n
− v
3.38
and hence
1 − α
n
γ
2
x
n
− u
n
2
≤ α
n
γ
2
x
n
− v
2
2α
n
γf
v
− Bvx
n
− v
x
n
− v
2
−
x
n1
− v
2
≤ α
n
γ
2
x
n
− v
2
2α
n
γf
v
− Bvx
n
− v
x
n
− x
n1
x
n
− v x
n1
− v
.
3.39
Since {x
n
} is bounded, α
n
→ 0andx
n
− x
n1
→0, it follows that x
n
− u
n
→0asn →∞.
Put M sup
n≥1
{γfv − Bvx
n
− v}. It follows from 3.38, the nonexpansive of
J
M,λ
and the inverse strongly monotonicity of A that
x
n1
− v
2
≤
1 − α
n
γ
2
y
n
− v
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1 − α
n
γ
2
u
n
− λAu
n
− v − λAv
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1 − α
n
γ
2
u
n
− v
2
λ
λ − 2α
Au
n
− Av
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1 − α
n
γ
2
x
n
− v
2
1 − α
n
γ
2
λ
λ − 2α
Au
n
− Av
2
2α
n
γβ
x
n
− v
2
2α
n
M
1−2α
n
γ −γβ
α
n
γ
2
x
n
−v
2
1−α
n
γ
2
λ
λ−2α
Au
n
−Av
2
2α
n
M
≤
x
n
− v
2
α
n
γ
2
x
n
− v
2
1 − α
n
γ
2
λ
λ − 2α
Au
n
− Av
2
2α
n
M.
3.40
Fixed Point Theory and Applications 15
This implies that
1 − α
n
γ
2
λ
2α − λ
Au
n
− Av
2
≤ α
n
γ
2
x
n
− v
2
2α
n
M
x
n
− v
2
−
x
n1
− v
2
≤ α
n
γ
2
x
n
−v
2
2α
n
Mx
n
−x
n1
x
n
−vx
n1
−v
.
3.41
Since α
n
→ 0andx
n
− x
n1
→0, we have Au
n
− Av→0asn →∞.From 3.5, 3.15
and 3.38, we have
x
n1
− v
2
≤
1 − α
n
γ
2
y
n
− v
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1 − α
n
γ
2
u
n
− v
2
−
u
n
− y
n
2
2λ
u
n
− y
n
,Au
n
− Av
− λ
2
Au
n
− Av
2
2α
n
γβ
x
n
− v
2
2α
n
M
≤
1 − α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
u
n
− y
n
2
2
1 − α
n
γ
2
λ
u
n
− y
n
,Au
n
− Av
−
1 − α
n
γ
2
λ
2
Au
n
− Av
2
2α
n
γβ
x
n
− v
2
2α
n
M
1 − 2α
n
γ − γβ
α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
u
n
− y
n
2
2
1 − α
n
γ
2
λ
u
n
− y
n
,Au
n
− Av
−
1 − α
n
γ
2
λ
2
Au
n
− Av
2
2α
n
M
≤
x
n
− v
2
α
n
γ
2
x
n
− v
2
−
1 − α
n
γ
2
u
n
− y
n
2
2
1 − α
n
γ
2
λ
u
n
− y
n
,Au
n
− Av
−
1 − α
n
γ
2
λ
2
Au
n
− Av
2
2α
n
M.
3.42
Thus, we obtain
1 − α
n
γ
2
u
n
− y
n
2
≤ α
n
γ
2
x
n
− v
2
2
1 − α
n
γ
2
λ
u
n
− y
n
,Au
n
− Av
−
1 − α
n
γ
2
λ
2
Au
n
− Av
2
2α
n
M
x
n
− v
2
−
x
n1
− v
2
≤ α
n
γ
2
x
n
− v
2
2
1 − α
n
γ
2
λ
u
n
− y
n
,Au
n
− Av
−
1−α
n
γ
2
λ
2
Au
n
−Av
2
2α
n
Mx
n
−x
n1
x
n
−vx
n1
−v
3.43
Since α
n
→ 0, Au
n
− Av→0andx
n
− x
n1
→0, we have u
n
− y
n
→0asn →∞.It
follows from the inequality y
n
−S
n
y
n
≤y
n
−u
n
u
n
−x
n
x
n
−S
n
y
n
that y
n
−S
n
y
n
→0
as n →∞. Moreover, we note that x
n
− y
n
≤x
n
− u
n
u
n
− y
n
→0asn →∞. Since
Sy
n
− y
n
≤Sy
n
− S
n
y
n
S
n
y
n
− y
n
≤ sup
Sz − S
n
z : z ∈
y
n
S
n
y
n
− y
n
3.44
16 Fixed Point Theory and Applications
for all n ∈ N, it follows that lim
n →∞
Sy
n
− y
n
0. Next, we show that
lim sup
n →∞
B − γf
z, z − x
n
≤ 0, 3.45
where z P
Ω
I − B γfz is a unique solution of the variational inequality 3.4.Toshow
this inequality, we choose a subsequence {x
n
i
} of {x
n
} such that
lim
i →∞
B − γf
z, z − x
n
i
lim sup
n →∞
B − γf
z, z − x
n
. 3.46
Since {u
n
i
} is bounded, there exists a subsequence {u
n
i
j
} of {u
n
i
} which converges weakly to
w. Without loss of generality, we can assume that u
n
i
w.From u
n
− y
n
→0, we obtain
y
n
i
w. Let us show w ∈ EPF. It follows by 3.23 and A2 that
1
r
n
y − u
n
,u
n
− x
n
≥ F
y, u
n
3.47
and hence
y − u
n
i
,
u
n
i
− x
n
i
r
n
i
≥ F
y, u
n
i
. 3.48
Since u
n
i
− x
n
i
/r
n
i
→ 0andu
n
i
w,it follows by A4 that 0 ≥ Fy,w for all y ∈ H. For t
with 0 <t≤ 1andy ∈ H, let y
t
ty 1 − tw. Since y ∈ H and w ∈ H, we have y
t
∈ H and
hence Fy
t
,w ≤ 0. So, from A1 and A4 we have
0 F
y
t
,y
t
≤ tF
y
t
,y
1 − t
F
y
t
,w
≤ tF
y
t
,y
, 3.49
and hence 0 ≤ Fy
t
,y.FromA3, we have 0 ≤ Fw, y for all y ∈ H and hence w ∈ EPF.
By the same argument as in proof of Theorem 3.1, we have w ∈ FS ∩ IA, M and hence
w ∈ Ω. This implies that
lim sup
n →∞
B − γf
z, z − x
n
lim
i →∞
B − γf
z, z − x
n
i
B − γf
z, z − w≤0. 3.50
Fixed Point Theory and Applications 17
Finally we prove that x
n
→ z.From3.23, we have
x
n1
− z
2
α
n
γfx
n
I − α
n
BS
n
y
n
− z
2
α
n
γfx
n
− BzI − α
n
BS
n
y
n
− z
2
≤
I − α
n
BS
n
y
n
− z
2
2α
n
γf
x
n
− Bz,x
n1
− z
≤
1−α
n
γ
2
y
n
−z
2
2α
n
γf
x
n
−f
z
,x
n1
−z2α
n
γf
z
−Bz,x
n1
−z
≤
1−α
n
γ
2
x
n
−z
2
2α
n
γβx
n
−zx
n1
−z2α
n
γf
z
−Bz,x
n1
−z
≤
1−α
n
γ
2
x
n
−z
2
α
n
γβ
x
n
−z
2
x
n1
−z
2
2α
n
γf
z
−Bz,x
n1
−z.
3.51
This implies that
x
n1
− z
2
≤
1 − 2α
n
γ
α
n
γ
2
α
n
γβ
1 − α
n
γβ
x
n
− z
2
2α
n
1 − α
n
γβ
γf
z
− Az, x
n1
− z
1 −
2
γ − γβ
α
n
1 − α
n
γβ
x
n
− z
2
α
n
γ
2
1 − α
n
γβ
x
n
− z
2
2α
n
1 − α
n
γβ
γf
z
− Az, x
n1
− z
≤
1 − γ
n
x
n
− z
2
δ
n
,
3.52
where γ
n
:2α
n
γ − γβ/1 − α
n
γβ and δ
n
: α
n
/1 − α
n
γβ{α
n
γ
2
x
n
− z
2
2γfz −
Az, x
n1
− z}. It easily verified that γ
n
→ 0,
∞
n1
γ
n
∞ and lim sup
n →∞
δn/γ
n
≤ 0. Hence,
by Lemma 2.1, the sequence {x
n
} converges strongly to z.
As in 10, Theorem 4.1, we can generate a sequence {S
n
} of nonexpansive mappings
satisfying condition
∞
n1
sup{S
n1
z − S
n
z : z ∈ K} < ∞. for any bounded K of H by using
convex combination of a general sequence {T
k
} of nonexpansive mappings with a common
fixed point.
Corollary 3.3. Let H be a real Hilbert space, let F be a bifunction from H × H → R satisfying
(A1)–(A4) and let A : H → H be an α-inverse-strongly monotone mapping, M : H → 2
H
be a
maximal monotone mapping. Let f be a contraction of H into itself with a constant β ∈ 0, 1 and let
B be a strongly bounded linear operator on H with coefficient
γ>0 and 0 <γ<γ/β.Let{β
k
n
} be a
family of nonnegative numbers with indices n, k ∈ N with k ≤ n such that
i
k1
β
k
n
1, for all n ∈ N;
ii lim
n →∞
β
k
n
> 0, for every k ∈ N;
iii
∞
n1
n
k1
|β
k
n1
− β
k
n
| < ∞.
18 Fixed Point Theory and Applications
Let {T
k
} be a sequence of nonexpansive mappings on H with Ω : F
∞
k1
FT
k
∩ EPF ∩
IA, M
/
∅ and let {x
n
}, {y
n
} and {u
n
} be sequences generated by x
1
∈ H and
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥0, ∀y ∈ H
x
n1
α
n
γf
x
n
I − α
n
B
n
k1
β
k
n
T
k
y
n
,
y
n
J
M,λ
u
n
− λAu
n
∀n ≥ 0,
3.53
for all n ∈ N,whereλ ∈ 0, 2α, {α
n
}⊂0, 1 and {r
n
}⊂0, ∞ satisfy
lim
n →∞
α
n
0,
∞
n1
α
n
∞,
∞
n1
|
α
n1
− α
n
|
< ∞,
lim inf
n →∞
r
n
> 0,
∞
n1
|
r
n1
− r
n
|
< ∞.
3.54
Then, {x
n
}, {y
n
} and {u
n
} converges strongly to z in Ω which solves the variational inequality:
B − γf
z, z − x≥0,x∈ Ω. 3.55
If S
n
S, B ≡ I and γ 1 in Theorem 3.2, we obtain the following corollary.
Corollary 3.4 see Peng et al. 9. Let H be a real Hilbert space, let F be a bifunction from C×C →
R satisfying (A1)–(A4) and let S be a nonexpansive mapping on H.LetA : H → H be an α-
inverse-strongly monotone mapping, M : H → 2
H
be a maximal monotone mapping such that
Ω : FS ∩ EPF ∩ IA, M
/
∅. Let f be a contraction of H into itself with a constant β ∈ 0, 1.
Let {x
n
}, {y
n
} and {u
n
} be sequences generated by x
1
∈ H and
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥0, ∀y ∈ H
x
n1
α
n
f
x
n
I − α
n
Sy
n
,
y
n
J
M,λ
u
n
− λAu
n
∀n ≥ 0,
3.56
for all n ∈ N,whereλ ∈ 0, 2α, {α
n
}⊂0, 1 and {r
n
}⊂0, ∞ satisfy
lim
n →∞
α
n
0,
∞
n1
α
n
∞,
∞
n1
|
α
n1
− α
n
|
< ∞,
lim inf
n →∞
r
n
> 0,
∞
n1
|
r
n1
− r
n
|
< ∞.
3.57
Then, {x
n
}, {y
n
} and {u
n
} converges strongly to z ∈ Ω,wherez P
Ω
fz.
Fixed Point Theory and Applications 19
If S
n
S, A ≡ 0andM ≡ 0inTheorem 3.2, we obtain the following corollary.
Corollary 3.5 see S. Plubtieng and R. Punpaeng 6. Let H be a real Hilbert space, let F be a
bifunction from H × H → R satisfying (A1)–(A4) and let S be a nonexpansive mapping on H such
that Ω : FS ∩ EPF
/
∅. Let f be a contraction of H into itself with a constant β ∈ 0, 1 and let
B be a strongly bounded linear operator on H with coefficient
γ>0 and 0 <γ<γ/β.Let{x
n
}, {u
n
}
and be sequences generated by x
1
∈ H and
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥0, ∀y ∈ H
x
n1
α
n
γf
x
n
I − α
n
B
Su
n
,
3.58
for all n ∈ N,where{α
n
}⊂0, 1 and {r
n
}⊂0, ∞ satisfy
lim
n →∞
α
n
0,
∞
n1
α
n
∞,
∞
n1
|
α
n1
− α
n
|
< ∞,
lim inf
n →∞
r
n
> 0,
∞
n1
|
r
n1
− r
n
|
< ∞.
3.59
Then, {x
n
} and {u
n
} converges strongly to a point z in Ω which solves the variational inequality:
B − γf
z, z − x≥0,x∈ Ω. 3.60
Acknowledgments
The first author thank the National Research Council of Thailand to Naresuan University,
2009 for the financial support. Moreover, the second author would like t o thank the “National
Centre of Excellence in Mathematics”, PERDO, under the Commission on Higher Education,
Ministry of Education, Thailand.
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