Abstract
In this paper, we suggest and analyze an iterative scheme for finding an approximate element of the common set of solutions of a split equilibrium problem, a variational inequality problem and a hierarchical fixed point problem in a real Hilbert space. We also consider the strong convergence of the proposed method under some conditions. Results proved in this paper may be viewed as an improvement and refinement of the previously known results.
MSC:49J30, 47H09, 47J20.
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Dedication
Dedicated to Professor Bingsheng He on the occasion of his sixty-fifth birthday
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 D 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
This alternative formulation has played a significant part in developing various projection-type methods for solving variational inequalities. We now have a variety of techniques to suggest and analyze various iterative algorithms for solving variational inequalities and the related optimization problems; see [1–29].
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 such that
-
(c)
α-inverse strongly monotone if there exists 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. It is well known that every nonexpansive operator satisfies, for all , the inequality
and therefore we get, for all ,
see, e.g., [9], Theorem 1 and [10], Theorem 3.
A mapping is called a 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 (5). It is well known that the class of strict pseudo-contractions strictly includes the class of nonexpansive mappings, then is closed and convex and is well defined (see [29]).
The equilibrium problem denoted by EP is to find such that
The solution set of (6) is denoted by . Numerous problems in physics, optimization and economics reduce to finding a solution of (6); see [7, 12, 23, 24]. In 1997, Combettes and Hirstoaga [8] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty. Recently Plubtieng and Punpaeng [23] introduced an iterative method for finding the common element of the set .
Recently, Censor et al. [4] introduced a new variational inequality problem which we call the split variational inequality problem (SVIP). Let and be two real Hilbert spaces. Given operators and , a bounded linear operator , and nonempty, closed and convex subsets and , the SVIP is formulated as follows: Find a point such that
and such that
In [22], Moudafi introduced an iterative method which can be regarded as an extension of the method given by Censor et al. [4] for the following split monotone variational inclusions:
and such that
where is a set-valued mapping for . Later Byrne et al. [3] generalized and extended the work of Censor et al. [4] and Moudafi [22].
Very recently, Kazmi and Rivzi [13] studied the following pair of equilibrium problems called a split equilibrium problem: Let and be nonlinear bifunctions and be a bounded linear operator, then the split equilibrium problem (SEP) is to find such that
and such that
The solution set of SEP (9)-(10) is denoted by .
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 (11) links with some monotone variational inequalities and convex programming problems; see [11, 27]. Various methods have been proposed to solve the hierarchical fixed point problem; see Moudafi [21], Mainge and Moudafi in [15], Marino and Xu in [17] and Cianciaruso et al. [5]. In 2010, Yao et al. [27] introduced the following strong convergence iterative algorithm to solve problem (11):
where is a contraction mapping and and are two sequences in . Under some certain restrictions on parameters, Yao et al. proved that the sequence generated by (12) converges strongly to , which is the unique solution of the following variational inequality:
By changing the restrictions on parameters, the authors obtained another result on the iterative scheme (12), the sequence generated by (12) converges strongly to a point , which is the unique solution of the following variational inequality:
Let be a nonexpansive mapping and be a countable family of nonexpansive mappings. In 2011, Gu et al. [11] introduced the following iterative algorithm:
where , is a strictly decreasing sequence in and is a sequence in . Under some certain conditions on parameters, Gu et al. proved that the sequence generated by (15) converges strongly to , which is the unique solution of one of variational inequalities (13) and (14).
In this paper, motivated by the work of Censor et al. [4], Moudafi [22], Byrne et al. [3] Kazmi and Rivzi [13], Yao et al. [27] and Gu et al. [11] and by the recent work going on in this direction, we give an iterative method for finding an approximate element of the common set of solutions of (1), (9)-(10) and (11) for a strictly pseudo-contraction mapping in a real Hilbert space. We establish a strong convergence theorem based on this method. The presented method improves and generalizes many known results for solving equilibrium problems, variational inequality problems and hierarchical fixed point problems; see, e.g., [5, 11, 15, 27] 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 the projection of H onto C.
Lemma 2.1 Let denote the projection of H onto C. Then we have the following inequalities,
Assumption 2.1 [2]
Let be a bifunction satisfying the following assumptions:
-
(i)
, ;
-
(ii)
F is monotone, i.e., , ;
-
(iii)
For each , ;
-
(iv)
For each , is convex and lower semicontinuous;
-
(v)
Fixed and , there exists a bounded subset K of and such that
Lemma 2.2 [8]
Assume that satisfies Assumption 2.1. For and , define a mapping as follows:
Then the following hold:
-
(i)
is nonempty and single-valued;
-
(ii)
is firmly nonexpansive, i.e.,
-
(iii)
;
-
(iv)
is closed and convex.
Assume that satisfies Assumption 2.1. For and , define a mapping as follows:
Then satisfies conditions (i)-(iv) of Lemma 2.2. , where is the solution set of the following equilibrium problem:
Lemma 2.3 [6]
Assume that satisfies Assumption 2.1, and let be defined as in Lemma 2.2. Let and . Then
Lemma 2.4 [28]
Let C be a nonempty closed convex subset of a real Hilbert space H. If is a k-strict pseudo-contraction, then:
-
(i)
The mapping is demiclosed at 0, i.e., if is a sequence in C weakly converging to x and if converges strongly to 0, then ;
-
(ii)
The set of T is closed and convex so that the projection is well defined.
Lemma 2.5 [16]
Let H be a real Hilbert space. Then the following inequality holds:
Lemma 2.6 [26]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(1)
;
-
(2)
or .
Then .
Lemma 2.7 [1]
Let C be a closed convex subset of H. Let be a bounded sequence in H. Assume that
-
(i)
the weak w-limit set , where ;
-
(ii)
for each , exists.
Then is weakly convergent to a point in C.
Lemma 2.8 [29]
Let H be a Hilbert space, C be a closed and convex subset of H, and be a k-strict pseudo-contraction mapping. Define a mapping by , . Then, as , V is a nonexpansive mapping such that .
Lemma 2.9 [11]
Let H be a Hilbert space, C be a closed and convex subset of H, and be a nonexpansive mapping such that . Then
3 The proposed method and some properties
In this section, we suggest and analyze our method for finding common solutions of the variational inequality (1), the split equilibrium problem (9)-(10) and the hierarchical fixed point problem (11).
Let and be two real Hilbert spaces and and be nonempty closed convex subsets of Hilbert spaces and , respectively. Let be a bounded linear operator. Let be an α-inverse strongly monotone mapping. Assume that and are the bifunctions satisfying Assumption 2.1 and is upper semicontinuous in the first argument. Let be a nonexpansive mapping and be a countable family of -strict pseudo-contraction mappings such that , where . Let f be a ρ-contraction mapping.
Algorithm 3.1 For a given arbitrarily, let the iterative sequences , , and be generated by
where , , , and , L is the spectral radius of the operator and is the adjoint of A and , is a strictly decreasing sequence in and is a sequence in satisfying the following conditions:
-
(a)
and ,
-
(b)
,
-
(c)
and ,
-
(d)
and ,
-
(e)
and .
Lemma 3.1 Let . Then , , and are bounded.
Proof First, we show that the mapping is nonexpansive. For any ,
Let , we have and . Then
From the definition of L, it follows that
It follows from (3) that
Applying (23) and (22) to (21) and from the definition of γ, we get
Since the mapping D is α-inverse strongly monotone, we have
Next, we prove that the sequence is bounded, without loss of generality, we can assume that for all . From Lemma 2.8, we have is a nonexpansive mapping and . Since , we get
By induction on n, we obtain , for and . Hence is bounded and consequently, we deduce that , and are bounded. □
Lemma 3.2 Let and be the sequence generated by Algorithm 3.1. Then we have
-
(a)
;
-
(b)
The weak w-limit set ().
Proof From the nonexpansivity of the mapping and , we have
Next, we estimate
It follows from (27) and (28) that
On the other hand, and . It follows from Lemma 2.3 that
where and . 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 (29) and (30) that
Next, we estimate
From (31) and (32), we have
where
Since , , and are bounded, we deduce that , , , and are bounded. We can conclude that , , , , and .
It follows by conditions (a)-(e) of Algorithm 3.1 and Lemma 2.6 that
Next, we show that . Since and , by using (24) and (25), we obtain
Then, from the above inequality, we get
Since , , , and , we obtain
and
Since is firmly nonexpansive, we have
where the last inequality follows from (21) and (24). Hence, we get
From (34), (25) and the above inequality, we have
Hence
Since , , and , we obtain
From (17), we get
Hence
From (34) and the above inequality, we have
Hence
Since , , and , we obtain
It follows from (36) and (37) that
Now, let , since for each , and , we have , and
It follows that
From Lemma 2.9 and the above inequality, we get
Since , , and , we obtain
Since and is strictly decreasing, we have
Hence, we obtain
Since is bounded, without loss of generality, we can assume that . It follows from Lemma 2.4 that . Therefore . □
Theorem 3.1 The sequence generated by Algorithm 3.1 converges strongly to , which is the unique solution of the variational inequality
which is the optimality condition for a minimization problem
where h is a potential function for f (i.e., for ) and .
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 . It follows by Assumption 2.1(iv) that , .
For any and , let , we have . Then, from Assumption 2.1(i) and (iv), we have
Therefore . From Assumption 2.1(iii), we have , which implies that .
Next, we show that . Since is bounded and , there exists a subsequence of such that and since A is a bounded linear operator so that . Now set . It follows from (35) that and . Therefore from the definition of , we have
Since is upper semicontinuous in the first argument, taking lim sup to the above inequality as and using Assumption 2.1(iv), we obtain
which implies that and hence .
Furthermore, we show that . Let
where is the normal cone to C at . Then T is maximal monotone and if and only if (see [25]). Let denote the graph of T and let . Since and , we have
On the other hand, it follows from and that
and
Therefore, from (41) and inverse strong monotonicity of D, 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
Since , Λ and are convex, then is convex. Next, we claim that , where .
Since is bounded, there exists a subsequence of such that
Next, we show that . From (16), we get
which implies that
Let and .
Since
it follows that
Thus all the conditions of Lemma 2.6 are satisfied. Hence we deduce that .
is a contraction, there exists a unique such that . From (16), it follows that z is the unique solution of problem (39). This completes the proof. □
Theorem 3.2 Let and be two real Hilbert spaces and and be nonempty closed convex subsets of Hilbert spaces and , respectively. Let be a bounded linear operator. Let be an α-inverse strongly monotone mapping. Assume that and are the bifunctions satisfying Assumption 2.1 and is upper semicontinuous in the first argument. Let be a nonexpansive mapping and be a countable family of -strict pseudo-contraction mappings such that , where . Let f be a ρ-contraction mapping. For a given arbitrarily, let the iterative sequences , , and be generated by
where , , , and , L is the spectral radius of the operator and is the adjoint of A and , is a strictly decreasing sequence in and is a sequence in satisfying the following conditions:
-
(a)
and ,
-
(b)
,
-
(c)
and ,
-
(d)
,
-
(e)
there exists a constant such that ,
-
(f)
and ,
-
(g)
and .
Then the sequence generated by Algorithm (42) converges strongly to , which is the unique solution of the variational inequality
Proof From , without loss of generality, we can assume that for all . Hence . By a similar argument as that in Lemmas 3.1 and 3.2, we can deduce that is bounded, , (see (38)) and . Then we have
It follows that for all ,
From (44) and (45), we have
Set . From (32) and (33), we obtain
Let and . From conditions (a) and (d), we have
By Lemma 2.6, we obtain
From (42), we have
Hence it follows that
and hence
Let . For any , we have
Since S is a nonexpansive mapping, f is a ρ-contraction mapping and is a -strict pseudo-contraction mapping. Then and are monotone and f is strongly monotone with a coefficient . We can deduce
From (16), we get
Then, from (46)-(49), we have
Then we obtain
By condition (e) of Theorem 3.2, there exists a constant such that . Since , , and as , then every weak cluster point of is also a strong cluster point. Since is bounded, by Lemma 3.2 there exists a subsequence of converging to a point , and by some similar arguments in Theorem 3.1, we can show that .
From (46)-(49), it follows that for any ,
Since , , and , letting in (50), we obtain
i.e.,
In the following, we show that (43) has a unique solution. Assume that is another solution. Then we have
Adding (51) and (52), we get
Then . Since (43) has a unique solution, it follows that . Since every weak cluster point of is also a strong cluster point, we conclude that . This completes the proof. □
4 Applications
In this section, we obtain the following results by using a special case of the proposed method. The first result can be viewed as an extension and improvement of the method of Gu et al. [11] for finding an approximate element of the common set of solutions of a split equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.
Corollary 4.1 Let and be two real Hilbert spaces and and be nonempty closed convex subsets of Hilbert spaces and , respectively. Let be a bounded linear operator. Let be an α-inverse strongly monotone mapping. Assume that and are the bifunctions satisfying Assumption 2.1 and is upper semicontinuous in the first argument. Let be a nonexpansive mapping and be a countable family of -strict pseudo-contraction mappings such that , where . Let f be a ρ-contraction mapping. For a given arbitrarily, let the iterative sequences , , and be generated by
where and , L is the spectral radius of the operator and is the adjoint of A and , is a strictly decreasing sequence in and is a sequence in satisfying the following conditions:
-
(a)
and ,
-
(b)
,
-
(c)
and ,
-
(d)
,
-
(e)
there exists a constant such that ,
-
(f)
and .
Then the sequence generated by Algorithm (53) converges strongly to , which is the unique solution of the variational inequality
Proof Put and , in Theorem 3.2. Then conclusion of Corollary 4.1 is obtained. □
The following result can be viewed as an extension and improvement of the method of Yao et al. [27] for finding an approximate element of the common set of solutions of a split equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.
Corollary 4.2 Let and be two real Hilbert spaces and and be nonempty closed convex subsets of Hilbert spaces and , respectively. Let be a bounded linear operator. Let be an α-inverse strongly monotone mapping. Assume that and are the bifunctions satisfying Assumption 2.1 and is upper semicontinuous in the first argument. Let be a nonexpansive mapping and be a k-strict pseudo-contraction mapping such that . Let f be a ρ-contraction mapping. For a given arbitrarily, let the iterative sequences , , and be generated by
where and , L is the spectral radius of the operator and is the adjoint of A and , is a strictly decreasing sequence in and is a sequence in satisfying the following conditions:
-
(a)
and ,
-
(b)
,
-
(c)
and ,
-
(d)
,
-
(e)
there exists a constant such that ,
-
(f)
and .
Then the sequence generated by Algorithm (54) converges strongly to , which is the unique solution of the variational inequality
Proof Put , and , in Theorem 3.2. Then conclusion of Corollary 4.2 is obtained. □
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Bnouhachem, A. Algorithms of common solutions for a variational inequality, a split equilibrium problem and a hierarchical fixed point problem. Fixed Point Theory Appl 2013, 278 (2013). https://doi.org/10.1186/1687-1812-2013-278
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DOI: https://doi.org/10.1186/1687-1812-2013-278