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A General Iterative Method of Fixed Points for Mixed Equilibrium Problems and Variational Inclusion Problems
Journal of Inequalities and Applications volume 2010, Article number: 370197 (2010)
Abstract
The purpose of this paper is to investigate the problem of finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings and inverse-strongly monotone mappings, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. We propose a new iterative scheme for finding the common element of the above three sets. Our results improve and extend the corresponding results of the works by Zhang et al. (2008), Peng et al. (2008), Peng and Yao (2009), as well as Plubtieng and Sriprad (2009) and some well-known results in the literature.
1. Introduction
Throughout this paper, we assume that 
is a real Hilbert space with inner product and norm being denoted by 
and 
, respectively, 
denoting the family of all subsets of 
and leting 
be a closed convex subset of 
. 
mapping 
is called nonexpansive if 
for all 
. We use 
to denote the set of fixed points of 
, that is, 
. It is assumed throughout the paper that 
is a nonexpansive mapping such that 
. Recall that a self-mapping 
is contraction on 
if there exists a constant 
and 
such that 
. Let 
be a strongly positive bounded linear operator on 
: that is, there is a constant 
with property
Let 
be a single-valued nonlinear mapping and let 
be a set-valued mapping. We consider the following variational inclusion problem, which is to find a point 
such that
where 
is the zero vector in 
. The set of solutions of problem (1.2) is denoted by 
.
If 
, where 
is a proper convex lower semi-continuous function and 
is the subdifferential of 
, then the variational inclusion problem (1.2) is equivalent to find 
such that
which is called the mixed quasivariational inequality (see [1]).
If 
, where 
is a nonempty closed convex subset of 
and 
is the indicator function of 
, that is,
then the variational inclusion problem (1.2) is equivalent to find 
such that
This problem is called Hartman-Stampacchia variational problem (see [2–4]).
A set-valued mapping 
is called monotone if, for all 
, 
and 
imply that 
. A monotone mapping 
is maximal if the graph of 
of 
is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 
is maximal if and only if, for 
, 
for every 
implies that 
.
Let the set-valued mapping 
be a maximal monotone. We define the resolvent operator
that is associate with 
and 
as follows:
where 
is a positive number. It is worth mentioning that the resolvent operator 
is single-valued, nonexpansive, and 1-inverse-strongly monotone (see [5, 6]).
Let 
be a proper extended real-valued function and let 
be a bifunction of 
into 
, where 
is the set of real numbers. Ceng and Yao [7] considered the following mixed equilibrium problem for finding 
such that
The set of solutions of (1.7) is denoted by 
. We see that x is a solution of problem (1.7) implying that 
. If 
, then the mixed equilibrium problem (1.7) becomes the following equilibrium problem that is to find 
such that
The set of solutions of (1.8) is denoted by EP(F). Given a mapping 
, let 
for all 
. Then 
if and only if 
for all 
that is, 
is a solution of the variational inequality. The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.8). Some methods have been proposed to solve the equilibrium problem (see [8–21]).
In 2008, Zhang et al. [6] introduced an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with a multivalued maximal monotone mapping and an inverse-strongly monotone mapping and the set of fixed points of nonexpansive mapping in Hilbert spaces. The iterative scheme is 
and:
for all 
. They proved the strong convergence theorem under some mind conditions. In the same year, Peng et al. [22] introduced an iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse-strongly monotone mapping, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in Hilbert spaces. The sequence 
is generated as follows:
They proved that if 
and 
satisfy appropriate conditions, then 
converges strongly to 
, where 
.
In 2009, Plubtieng and Sriprad [23] introduced an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mapping, 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. Starting with an arbitrary 
, define the sequences 
, 
and 
by
where 
is an inverse-strongly monotone mapping and 
is a bounded linear operator on 
. They proved that if the sequences 
and 
of parameters satisfy appropriate conditions, then 
is generated by (1.11) converging strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem
where 
is a potential function for 
, that is, 
, for all 
.
Let 
, where 
, be a family of finitely nonexpansive mappings. Let the mapping 
be defined by
where 
. Such a mapping 
is called the W-mapping generated by 
and 
. Nonexpansivity of each 
ensures the nonexpansivity of 
. Moreover, in [24, Lemma 
], it is shown that 
.
The concept of 
-mappings was introduced in [25, 26]. It is now one of the main tools in studying convergence of iterative methods for approaching common fixed points of nonlinear mappings; more recent progresses can be found in [24, 27, 28] and the references cited therein.
Following from 
-mappings, Peng and Yao [29] introduced iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of a finite family of nonexpansive mappings, and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. The sequence 
is generated by
where 
is monotone and Lipschitz continuous mapping and 
is an inverse-strongly monotone mapping. They proved the weak convergence theorem if the sequences 
and 
of parameters satisfy appropriate conditions.
In this paper, motivated by the above results and the iterative schemes considered by Zhang et al. in [6], Peng et al. in [22], Peng and Yao in [29], and Plubtieng and Sriprad in [23], we present a new general iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mapping and inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Zhang et al. [6], Peng et al. [22], Peng and Yao [29], Plubtieng and Sriprad [23], and some authors.
2. Preliminaries
Let 
be a real Hilbert space with norm 
and inner product 
and let 
be a closed convex subset of 
. When 
is a sequence in 
, 
means that 
converges weakly to 
and 
means the strong convergence. In a real Hilbert space 
, we have
for all 
and 
. For every point 
, there exists a unique nearest point in 
, denoted by 
, such that
is called the metric projection of 
onto 
It is well known that 
is a nonexpansive mapping of 
onto 
and satisfies
for every 
Moreover, 
is characterized by the following properties:
for all 
.
Recall that a mapping 
of 
into itself is called 
-inverse-strongly monotone if there exists a positive real number 
such that
for all 
It is obvious that any 
-inverse-strongly monotone mapping 
is 
-Lipschitz monotone and continuous mapping. We also have that, for all 
and 
So, if 
then 
is a nonexpansive mapping from 
into itself.
It is also known that H satisfies the Opial condition [30], that is, for any sequence 
with 
, the inequality
holds for every 
with 
.
For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction F, 
, and the set C.
(A1)
for all 
(A2)
is monotone, that is, 
for all 
(A3)For each 
(A4)For each 
is convex and lower semicontinuous.
(A5)For each 
is weakly upper semicontinuous.
(B1)For each 
and 
, there exist a bounded subset 
and 
such that for any 
,
(B2)C is a bounded set.
We need the following lemmas for proving our main result.
Lemma 2.1 (Peng and Yao [31]).
Let C be a nonempty closed convex subset of H. Let 
be a bifunction satisfying (A1)–(A5) and let 
be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For 
and 
, define a mapping 
as follows:
for all 
. Then, the following hold.
(1)For each 
.
(2)
is single valued.
(3)
is firmly nonexpansive, that is, for any 
(4)
(5)
is closed and convex.
Lemma 2.2 (Xu [32]).
Assume that 
is a sequence of nonnegative real numbers such that
where 
is a sequence in 
and 
is a sequence in 
such that
(1)
(2)
or 
Then 
Lemma 2.3 (Osilike and Igbokwe [33]).
Let 
be an inner product space. Then for all 
and 
with 
one has
Lemma 2.4 (Colao et al. [28]).
Let 
be a nonempty convex subset of a Banach space. Let 
be a family of finitely nonexpansive mappings of 
into itself and let 
be sequences in 
such that 
. Moreover for every integer 
, let 
and 
be the 
-mappings generated by 
and 
and 
and 
, respectively. Then for every 
, it follows that
Lemma 2.5 (Suzuki [34]).
Let 
and 
be bounded sequences in a Banach space 
and let 
be a sequence in 
with 
Suppose that 
for all integers 
and 
Then, 
Lemma 2.6 (Marino and Xu [35]).
Assume that 
is a strongly positive linear bounded operator on a Hilbert space 
with coefficient 
and 
. Then 
.
Lemma 2.7 (Brézis [5]).
Let
be a maximal monotone mapping and 
be a Lipschitz continuous mapping. Then the mapping 
is a maximal monotone mapping.
Remark 2.8.
Lemma 2.7 implies that 
is closed and convex if 
is a maximal monotone mapping and 
is a Lipschitz continuous mapping.
Lemma 2.9 (Zhang et al. [6]).
is a solution of variational inclusion (1.2) if and only if 
for all 
that is,
3. Main Result
In this section, we prove 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 in a real Hilbert space.
Theorem 3.1.
Let 
be a nonempty closed convex subset of a real Hilbert Space 
. Let 
be a bifunction of 
into real numbers 
satisfying (A1)–(A5) and let 
be a proper lower semicontinuous and convex function. Let 
be a contraction of 
into itself with coefficient 
. Let 
be an 
-inverse-strongly monotone mapping of 
into itself, 
be a maximal monotone mapping, and let 
be a strongly bounded linear operator on 
with coefficient 
and 
. Let 
be a family of finitely nonexpansive mappings of 
into 
such that 
and let 
be the 
-mapping generated by 
and 
. Assume that either (B1) or (B2) holds. Let 
be a sequence generated by 
and
for every 
, where 
and 
satisfy the following conditions:
(i)
and 
,
(ii)
,
(iii)
,
(iv)
for all 
.
Then 
converges strongly to 
, where 
, which is the unique solution of the variational inequality
Proof.
Since 
, we may assume, without loss of generality, that 
for all 
. We assume that 
. Since 
is linear bounded self-adjoint operator on 
, we have
Observe that
this shows that 
is positive. It follows that
Let 
, let 
be a sequence of mappings defined as in Lemma 2.1, and let 
. For any 
, we have
Since 
, we have 
. From 
and 
being nonexpansive, then we have
for all 
. It follows that
for every 
. It follows by mathematical induction that
Therefore, 
is bounded. We also obtain that 
and 
, are all bounded.
Next, we show that 
Observing that 
and 
, we get
Take 
in (3.10) and 
in (3.11); by using condition (A2), we obtain
Thus 
. Without loss of generality, let us assume that there exists a real number 
such that 
for all 
. Then, we have
and hence
where 
.
On the other hand, again since 
and 
are nonexpansive, we obtain
It follows from (3.14) and (3.15) that
Define the sequence 
by 
for all
. Then, observe that
It follows that
From the definition of 
, since 
and 
are nonexpansive, we have
where 
is an approximate constant such that 
, 
.
Since 
for all 
and 
, we compute
It follows that
Substituting (3.21) into (3.19) yields that
and hence
Combining (3.16) and (3.23), we obtain
So
Conditions (i)–(iv) imply that
Hence, by Lemma 2.5, we have
Consequently,
From (ii), (3.14), (3.16), and (3.28), we also have 
and 
as 
We note that
It follows that
From (i), (iii), and (3.28), we obtain
Next, we shall show that 
. For any 
, since 
is firmly nonexpansive, we have
It follows that
Therefore, we have
Then, we obtain
By (i), (iii), and (3.28) imply that
Since 
we have
We note that, by (3.34), nonexpansiveness of 
and the inverse-strong monotonicity of 
imply that
It follows from (i), (iii), and (3.28) that
which implies that
On the other hand, since 
is firmly nonexpansive, we have
which yields that
From (3.34) and (3.42), we obtain
Hence, we get
By (i), (iii), (3.28), and (3.40), we have
Similarly, we can prove that
By the same idea in (3.42), (3.45) and using (3.46), then we obtain that
From
hence
and also
Observe that 
is a contraction of 
into itself. Indeed, for all 
, we have
Since 
is complete, there exists a unique fixed point 
such that 
Next, we show that
Indeed, we can choose a subsequence 
of 
such that
Since 
is bounded, there exists a subsequence 
of 
which converges weakly to 
. Without loss of generality, we can assume that 
From 
we obtain 
. Let us show that 
Since 
, we have
From (A2), we also have
and hence
From 
and 
we get 
. Since 
it follows by (A4) and the weakly lower semicontinuity of 
that
For 
with 
and 
let 
Since 
and 
we have 
and hence 
So, from (A1), (A4), and the convexity of 
, we have
Dividing by t, we get 
. From (A3) and the weakly lower semicontinuity of 
, we have 
for all 
and hence 
Next, we show that 
. Assume that 
. Since 
and 
, from Opial's condition, we have
which is a contradiction. Thus, we obtain 
.
Next, we show that 
. In fact, having 
as 
-inverse-strongly monotone, implies that 
is 
-Lipschitz continuous monotone mapping and that domain of 
is equal to 
. It follows from Lemma 2.7 that 
is a maximal monotone. Let 
, that is, 
. Since 
, we have 
, that is,
With 
being a maximal monotone, we have
and so
It follows from 
, 
, and 
that
It follows from the maximal monotonicity of 
that 
, that is, 
. This implies that 
.
Since 
, it follows that
By (3.49), (3.50), and the last inequality, we have
Finally, we show that 
converges strongly to 
. Indeed, from (3.1), we have
where 
. By (3.65), we get 
. Hence by Lemma 2.2 to (3.66), we conclude that 
. This completes the proof.
Using Theorem 3.1, we obtain the following corollaries.
Corollary 3.2.
Let 
be a nonempty closed convex subset of a real Hilbert Space 
. Let 
be a bifunction of 
into real numbers 
satisfying (A1)–(A5) and let 
be a contraction of 
into itself with coefficient 
. Let 
be an 
-inverse-strongly monotone mapping of 
into itself and let 
be a maximal monotone mapping such that 
. Let 
be a sequence generated by 
and
for every 
, where 
and 
satisfy the conditions (i)–(iii) in Theorem 3.1. Then 
converges strongly to 
Proof.
Taking 
for 
, 
and 
, in Theorem 3.1, we can conclude the desired conclusion easily. This completes the proof.
Corollary 3.3.
Let 
be a nonempty closed convex subset of a real Hilbert Space 
. Let 
be a bifunction of 
into real numbers 
satisfying (A1)–(A5) and let 
be a proper lower semicontinuous and convex function. Let 
be a contraction of 
into itself with coefficient 
. Let 
be an 
-inverse-strongly monotone mapping of 
into 
and let 
be a strongly bounded linear operator on 
with coefficient 
and 
. Let 
be a family of finitely nonexpansive mappings of 
into 
such that 
and let 
be the 
-mapping generated by 
and 
. Assume that either (B1) or (B2) holds. Let 
be a sequence generated by 
and
For every 
. where 
and 
satisfy the condition (i)–(iv) in Theorem 3.1. Then 
converges strongly to 
which is the unique solution of the variational inequality
Equivalently, one has 
Proof.
From Theorem 3.1 put 
; then 
. So we have 
and 
. The conclusion of Corollary 3.3 can be obtained from Theorem 3.1 immediately.
4. Application
In this section, we study a kind of optimization problem by using the result of this paper. We will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:
where 
is a convex and lower semicontinuous functional defined convex subset 
of a Hilbert space 
. We denote by 
the set of solutions of (4.1). Let 
be a bifunction defined by 
. We consider the equilibrium problem (1.8); it is obvious that 
. Therefore, from Theorem 3.1, we give the following corollary.
Corollary 4.1.
Let 
be a nonempty closed convex subset of a real Hilbert Space 
. Let 
be a bifunction of 
into real numbers 
satisfying (A1)–(A5) and let 
be a lower semicontinuous and convex function. Let 
be a contraction of 
into itself with coefficient 
. Let 
be an 
-inverse-strongly monotone mapping of 
into itself, let 
be a maximal monotone mapping, and let 
be a strongly bounded linear operator on 
with coefficient 
and 
. Let 
be a family of finitely nonexpansive mappings of 
into 
such that 
and let 
be the 
-mapping generated by 
and 
. Let 
be a sequence generated by 
and
for every 
, where 
and 
satisfy the following conditions:
(i)
and 
.
(ii)
.
(iii)
.
(iv)
for all 
.
Then 
converges strongly to 
, where 
, which is the unique solution of the variational inequality
Proof.
From Theorem 3.1 put 
and 
. The conclusion of Corollary 4.1 can be obtained from Theorem 3.1 immediately.
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Acknowledgments
The authors would like to thank the Centre of Excellence in Mathematics, under the Commission on Higher Education, Ministry of Education, Thailand. Mr. Phayap Katchang was supported by King Mongkut's Diamond scholarship for fostering special academic skills by KMUTT for Ph.D. Program at KMUTT. Moreover, the authors are also very grateful to Professor Yeol Je Cho and Professor Jong Kyu Kim for the hospitality.
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Katchang, P., Kumam, P. A General Iterative Method of Fixed Points for Mixed Equilibrium Problems and Variational Inclusion Problems. J Inequal Appl 2010, 370197 (2010). https://doi.org/10.1155/2010/370197
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DOI: https://doi.org/10.1155/2010/370197
Keywords
- Variational Inequality
- Equilibrium Problem
- Nonexpansive Mapping
- Maximal Monotone
- Real Hilbert Space