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Strong convergence algorithm for approximating the common solutions of a variational inequality, a mixed equilibrium problem and a hierarchical fixed-point problem
Journal of Inequalities and Applications volume 2014, Article number: 154 (2014)
Abstract
This paper investigates the common set of solutions of a variational inequality, a mixed equilibrium problem, and a hierarchical fixed-point problem in a Hilbert space. A numerical method is proposed to find the approximate element of this common set. The strong convergence of this method is proved under some conditions. The proposed method is shown to be an improvement and extension of some known results.
MSC: 49J30, 47H09, 47J20.
1 Introduction
Let H be a real Hilbert space, whose inner product and norm are denoted by and . Let C be a nonempty closed convex subset of H and A be a mapping from C into H. A classical variational inequality problem, denoted by , is to find a vector such that
The solution of is denoted by . It is easy to observe that
We now have a variety of techniques to suggest and analyze various iterative algorithms for solving variational inequalities and the related optimization problems; see [1–25]. The fixed-point theory has played an important role in the development of various algorithms for solving variational inequalities. Using the projection operator technique, one usually establishes an equivalence between the variational inequalities and the fixed-point problem. This alternative equivalent formulation was used by Lions and Stampacchia [1] to study the existence of a solution of the variational inequalities.
We introduce the following definitions, which are useful in the following analysis.
Definition 1.1 The mapping is said to be
-
(a)
monotone if
-
(b)
strongly monotone if there exists an such that
-
(c)
α-inverse strongly monotone if there exists an such that
-
(d)
nonexpansive if
-
(e)
k-Lipschitz continuous if there exists a constant such that
-
(f)
contraction on C if there exists a constant such that
It is easy to observe that every α-inverse strongly monotone T is monotone and Lipschitz continuous. A mapping is called k-strict pseudo-contraction if there exists a constant such that
The fixed-point problem for the mapping T is to find such that
We denote by the set of solutions of (1.3). It is well known that the class of strict pseudo-contractions includes the class of Lipschitzian mappings, then is closed and convex and is well defined (see [2]).
The mixed equilibrium problem, denoted by , is to find such that
where is a bifunction, and is a nonlinear mapping. This problem was introduced and studied by Moudafi and Théra [3] and Moudafi [4]. The set of solutions of (1.4) is denoted by
If , then it is reduced to the equilibrium problem, which is to find such that
The solution set of (1.6) is denoted by . Numerous problems in physics, optimization, and economics reduce to finding a solution of (1.6); see [5–9]. In 1997, Combettes and Hirstoaga [10] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty. Recently Plubtieng and Punpaeng [7] introduced an iterative method for finding the common element of the set .
Let be a nonexpansive mapping. The following problem is called a hierarchical fixed point problem: Find such that
It is known that the hierarchical fixed-point problem (1.7) links with some monotone variational inequalities and convex programming problems; see [11, 12, 26]. Various methods have been proposed to solve the hierarchical fixed-point problem; see Moudafi [13], Mainge and Moudafi in [14], Marino and Xu in [15] and Cianciaruso et al. [16]. Very recently, Yao et al. [12] introduced the following strong convergence iterative algorithm to solve problem (1.7):
where is a contraction mapping, and and are two sequences in . Under some certain restrictions on the parameters, Yao et al. proved that the sequence generated by (1.8) converges strongly to , which is the unique solution of the following variational inequality:
In 2011, Ceng et al. [17] investigated the following iterative method:
where U is a Lipschitzian mapping, and F is a Lipschitzian and strongly monotone mapping. They proved that under some approximate assumptions on the operators and parameters, the sequence generated by (1.10) converges strongly to the unique solution of the variational inequality
In this paper, motivated by the work of Yao et al. [12], Ceng et al. [17], Bnouhachem [18, 19] and by the recent work going in this direction, we give an iterative method for finding the approximate element of the common set of solutions of (1.1), (1.4), and (1.7) in a real Hilbert space. We establish a strong convergence theorem based on this method. We would like to mention that our proposed method is quite general and flexible and includes many known results for solving equilibrium problems, variational inequality problems, and hierarchical fixed-point problems; see, e.g., [11, 12, 14–17, 24, 25] and relevant references cited therein.
2 Preliminaries
In this section, we list some fundamental lemmas that are useful in the consequent analysis. The first lemma provides some basic properties of projection onto C.
Lemma 2.1Letdenote the projection ofHontoC. Then we have the following inequalities:
Lemma 2.2 [20]
Letbe a bifunction satisfying the following assumptions:
-
(i)
, ;
-
(ii)
is monotone, i.e., , ;
-
(iii)
for each, ;
-
(iv)
for each, is convex and lower semicontinuous.
Letand. Then there existssuch that
Lemma 2.3 [10]
Assume thatsatisfies assumptions (i)-(iv) of Lemma 2.2, and forand, define a mappingas follows:
Then the following hold:
-
(i)
is single-valued;
-
(ii)
is firmly nonexpansive, i.e.,
-
(iii)
;
-
(iv)
is closed and convex.
Lemma 2.4 [21]
LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Ifis a nonexpansive mapping with, then the mappingis demiclosed at 0, i.e., ifis a sequence inCweakly converging toxand ifconverges strongly to 0, then.
Lemma 2.5 [17]
Letbe aτ-Lipschitzian mapping, and letbe ak-Lipschitzian andη-strongly monotone mapping, then for, is-strongly monotone, i.e.,
Lemma 2.6 [22]
Suppose thatand. Letbe ak-Lipschitzian andη-strongly monotone operator. In association with a nonexpansive mapping, define the mappingby
Thenis a contraction provided, that is,
where.
Lemma 2.7 [23]
Assume that is a sequence of nonnegative real numbers such that
whereis a sequence in, andis a sequence such that
-
(1)
;
-
(2)
or.
Then.
Lemma 2.8 [27]
LetCbe a closed convex subset ofH. Letbe a bounded sequence in H. Assume that
-
(i)
the weakw-limit set, where;
-
(ii)
for each, exists.
Thenis weakly convergent to a point inC.
3 The proposed method and some properties
In this section, we suggest and analyze our method for finding the common solutions of the variational inequality (1.1), the mixed equilibrium problem (1.4), and the hierarchical fixed-point problem (1.7).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be θ, α-inverse strongly monotone mappings, respectively. Let be a bifunction satisfying assumptions (i)-(iv) of Lemma 2.2 and be a nonexpansive mappings such that . Let be a k-Lipschitzian mapping and be η-strongly monotone, and let be a τ-Lipschitzian mapping.
Algorithm 3.1 For an arbitrary given , let the iterative sequences , , , and be generated by
where , . Suppose that the parameters satisfy , , where . Also, and are sequences in satisfying the following conditions:
-
(a)
and ,
-
(b)
,
-
(c)
and ,
-
(d)
and ,
-
(e)
and .
Remark 3.1 Our method can be viewed as an extension and improvement for some well-known results, for example, the following.
-
If , we obtain an extension and improvement of the method of Wang and Xu [24] for finding the approximate element of the common set of solutions of a mixed equilibrium problem and a hierarchical fixed-point problem in a real Hilbert space.
-
If we have the Lipschitzian mapping , , , and , we obtain an extension and improvement of the method of Yao et al.[12] for finding the approximate element of the common set of solutions of a mixed equilibrium problem and a hierarchical fixed-point problem in a real Hilbert space.
-
The contractive mapping f with a coefficient in other papers [12, 15, 22, 25] is extended to the cases of the Lipschitzian mapping U with a coefficient constant .
This shows that Algorithm 3.1 is quite general and unifying.
Lemma 3.1Let. Then, , , andare bounded.
Proof First, we show that the mapping is nonexpansive. For any ,
Similarly, we can show that the mapping is nonexpansive. It follows from Lemma 2.3 that . Let ; we have .
Since the mapping A is α-inverse strongly monotone, we have
We define . Next, we prove that the sequence is bounded, and without loss of generality we can assume that for all . From (3.1), we have
where the third inequality follows from Lemma 2.6.
By induction on n, we obtain for and . Hence, is bounded and, consequently, we deduce that , , , , , , , and are bounded. □
Lemma 3.2Letandbe the sequence generated by Algorithm 3.1. Then we have:
-
(a)
.
-
(b)
The weakw-limit set, ().
Proof From the nonexpansivity of the mapping and , we have
Next, we estimate that
It follows from (3.4) and (3.5) that
On the other hand, and , we have
and
Take in (3.7) and in (3.8), we get
and
Adding (3.9) and (3.10) and using the monotonicity of , we have
which implies that
and then
Without loss of generality, let us assume that there exists a real number μ such that for all positive integers n. Then we get
It follows from (3.6) and (3.11) that
Next, we estimate that
where the second inequality follows from Lemma 2.6. From (3.12) and (3.13), we have
Here
It follows by conditions (a)-(e) of Algorithm 3.1 and Lemma 2.7 that
Next, we show that . Since , by using (3.2) and (3.3), we obtain
which implies that
Then, from the inequality above, we get
Since , , , , and , we obtain and .
Since is firmly nonexpansive, we have
Hence,
From (3.15), (3.3), and the inequality above, we have
which implies that
Hence,
Since , , , and , we obtain
From (2.2), we get
Hence,
From (3.15) and the inequality above, we have
which implies that
Hence,
Since , , , and , we obtain
It follows from (3.16) and (3.17) that
Since , we have
Since , , , and and are bounded, and , we obtain
Since is bounded, without loss of generality we can assume that . It follows from Lemma 2.4 that . Therefore, . □
Theorem 3.1The sequencegenerated by Algorithm 3.1 converges strongly toz, which is the unique solution of the variational inequality
Proof Since is bounded and from Lemma 3.2, we have . Next, we show that . Since , we have
It follows from the monotonicity of that
and
Since , and , it is easy to observe that . For any and , let , and we have . Then from (3.20), we obtain
Since D is Lipschitz continuous and , we obtain . From the monotonicity of D and , it follows from (3.21) that
Hence, from assumptions (i)-(iv) of Lemma 2.2 and (3.22), we have
which implies that . Letting , we have
which implies that .
Furthermore, we show that . Let
where is the normal cone to C at . Then T is maximal monotone and if and only if (see [28]). Let denote the graph of T, and let ; since and , we have
On the other hand, it follows from and that
and
Therefore, from (3.24) and the inverse strong monotonicity of A, we have
Since and , it is easy to observe that . Hence, we obtain . Since T is maximal monotone, we have , and hence . Thus we have
Observe that the constants satisfy and
therefore, from Lemma 2.5, the operator is strongly monotone, and we get the uniqueness of the solution of the variational inequality (3.19) and denote it by .
Next, we claim that . Since is bounded, there exists a subsequence of such that
Next, we show that . We have
which implies that
Let and .
We have
and
It follows that
Thus all the conditions of Lemma 2.7 are satisfied. Hence we deduce that . This completes the proof. □
4 Applications
In this section, we obtain the following results by using a special case of the proposed method for example.
Putting in Algorithm 3.1, we obtain the following result which can be viewed as an extension and improvement of the method of Wang and Xu [24] for finding the approximate element of the common set of solutions of a mixed equilibrium problem and a hierarchical fixed-point problem in a real Hilbert space.
Corollary 4.1LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letbeθ-inverse strongly monotone mappings. Letbe a bifunction satisfying assumptions (i)-(iv) of Lemma 2.2 andbe a nonexpansive mappings such that. Letbe ak-Lipschitzian mapping and beη-strongly monotone, and letbe aτ-Lipschitzian mapping. For an arbitrary given, let the iterative sequences, , , andbe generated by
where, , . Suppose that the parameters satisfy, , where. Also, , , andare sequences satisfying conditions (a)-(d) of Algorithm 3.1. The sequenceconverges strongly toz, which is the unique solution of the variational inequality
Putting, , , and, we obtain an extension and improvement of the method of Yao et al. [12]for finding the approximate element of the common set of solutions of a mixed equilibrium problem and a hierarchical fixed-point problem in a real Hilbert space.
Corollary 4.2LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letbeθ-inverse strongly monotone mappings. Letbe a bifunction satisfying assumptions (i)-(iv) of Lemma 2.2 andbe a nonexpansive mappings such that. Letbe aτ-Lipschitzian mapping. For an arbitrary given, let the iterative sequences, , , andbe generated by
where, , are sequences insatisfying conditions (a)-(d) of Algorithm 3.1. The sequenceconverges strongly toz, which is the unique solution of the variational inequality
Remark 4.1 Some existing methods (e.g., [12, 14, 16, 17, 25]) can be viewed as special cases of Algorithm 3.1. Therefore, the new algorithm is expected to be widely applicable.
To verify the theoretical assertions, we consider the following example.
Example 4.1 Let , , , and .
We have
and
The sequence satisfies condition (a).
Condition (b) is satisfied. We compute
It is easy to show . Similarly, we can show . The sequences and satisfy condition (c). We have
and
Then, the sequence satisfies condition (d). We compute
Then, the sequence satisfies condition (e).
Let ℝ be the set of real numbers, , and let the mapping be defined by
let the mapping be defined by
let the mapping be defined by
let the mapping be defined by
let the mapping be defined by
and let the mapping be defined by
It is easy to show that A is a 1-inverse strongly monotone mapping, T and S are nonexpansive mappings, F is a 1-Lipschitzian mapping and -strongly monotone and U is -Lipschitzian. It is clear that
By the definition of , we have
Then
Let . is a quadratic function of y with coefficient , , . We determine the discriminant Δ of B as follows:
We have , . If it has at most one solution in ℝ, then , we obtain
For every , from (4.1), we rewrite (3.1) as follows:
In all the tests we take and . In our example, , , . It is easy to show that the parameters satisfy , , where . All codes were written in Matlab, the values of , , , and with different n are reported in Table 1.
Remark 4.2 Table 1 and Figure 1 show that the sequences , , , and converge to 0, where .
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Bnouhachem, A. Strong convergence algorithm for approximating the common solutions of a variational inequality, a mixed equilibrium problem and a hierarchical fixed-point problem. J Inequal Appl 2014, 154 (2014). https://doi.org/10.1186/1029-242X-2014-154
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DOI: https://doi.org/10.1186/1029-242X-2014-154