Abstract
In this paper, we suggest and analyze a modified projection method for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. We prove the strong convergence of the sequence generated by the proposed method to a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem. Several special cases are also discussed. The results presented in this paper extend and improve some well-known results in the literature.
MSC: 49J30, 47H09, 47J20.
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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. Recently, Ceng et al. [1] considered the general system of variational inequalities, which involves finding such that
where is a nonlinear mapping for each . The solution set of (1.1) is denoted by . As pointed out in [2] the system of variational inequalities is used as a tool to study the Nash equilibrium problem, see, for example, [3–6] and the references therein. We believe that the problem (1.1) could be used to study the Nash equilibrium problem for a two players game. The theory of variational inequalities is well established and it has a wide range of applications in science, engineering, management, and social sciences, see, for example, [4–7] and the references therein.
Ceng et al. [1] transformed problem (1.1) into a fixed-point problem (see Lemma 2.2) and introduced an iterative method for finding the common element of the set . Based on the one-step iterative method [8] and the multi-step iterative method [9], Latif et al. [10] proposed a multi-step hybrid viscosity method that generates a sequence via an explicit iterative algorithm to compute the approximate solutions of a system of variational inequalities defined over the intersection of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings, and the solution set of a nonexpansive mapping. Under very mild conditions, they proved that the sequence converges strongly to a unique solution of system of variational inequalities defined over the set consisting of the set of solutions of an equilibrium problem, the set of common fixed points of nonexpansive mappings, and the set of fixed points of a mapping, and to a unique solution of the triple hierarchical variational inequality problem.
On the other hand, by combining the regularization method, Korpelevich’s extragradient method, the hybrid steepest-descent method, and the viscosity approximation method, Ceng et al. [2] introduced and analyzed implicit and explicit iterative schemes for computing a common element of the solution set of system of variational inequalities and a set of zeros of an accretive operator in Banach space. Under suitable assumptions, they proved the strong convergence of the sequences generated by the proposed schemes.
If , then the problem (1.1) reduces to finding such that
which has been introduced and studied by Verma [11, 12].
If and , then the problem (1.2) collapses to the classical variational inequality: finding , such that
For the recent applications, numerical techniques, and physical formulation, see [1–45].
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)
a 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 ,
The fixed-point problem for the mapping T is to find such that
We denote by the set of solutions of (1.5). It is well known that is closed and convex, and is well defined.
The equilibrium problem, denoted by EP, 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 [25, 38]. In 1994, Censor and Elfving [19] introduced and studied the following split feasibility problem.
Let C and Q be nonempty closed convex subsets of the infinite-dimensional real Hilbert spaces and , respectively, and let , where denotes the collection of all bounded linear operators from to . The problem is to find such that
Very recently, Ceng et al. [22] introduced and analyzed an extragradient method with regularization for finding a common element of the solution set of the split feasibility problem and the set of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. By combining Mann’s iterative method and the extragradient method, Ceng et al. [21] proposed three different kinds of Mann type iterative methods for finding a common element of the solution set of the split feasibility problem and the set of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces.
Recently, Censor et al. [23] introduced a new variational inequality problem which we call the split variational inequality problem (SVIP). Let and be two real Hilbert spaces. Given the operators and , a bounded linear operator , and the nonempty, closed, and convex subsets and , the SVIP is formulated as follows: find a point such that
and such that
In [36], Moudafi introduced an iterative method which can be regarded as an extension of the method given by Censor et al. [23] for the following split monotone variational inclusions:
and such that
where is a set-valued mapping for . Later Byrne et al. [18] generalized and extended the work of Censor et al. [23] and Moudafi [36].
In this paper, we consider the following pair of equilibrium problems, called split equilibrium problems: 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 (1.9)-(1.10) 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 (1.11) links with some monotone variational inequalities and convex programming problems; see [45]. Various methods have been proposed to solve the hierarchical fixed-point problem; see Mainge and Moudafi [34] and Cianciaruso et al. [26]. In 2010, Yao et al. [45] introduced the following strong convergence iterative algorithm to solve the problem (1.11):
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 (1.12) converges strongly to , which is the unique solution of the following variational inequality:
In 2011, Ceng et al. [20] 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 (1.14) converges strongly to the unique solution of the variational inequality
In this paper, motivated by the work of Censor et al. [23], Moudafi [36], Byrne et al. [18], Yao et al. [45], Ceng et al. [20], Bnouhachem [15–17] 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), (1.9)-(1.10), and (1.11) in 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 a system of variational inequality problems, split equilibrium problems, and hierarchical fixed-point problems, see, e.g. [20, 26, 34, 40, 45] 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 2.1 Let denote the projection of H onto C. Then we have the following inequalities:
Lemma 2.2 [1]
For any , is a solution of (1.1) if and only if is a fixed point of the mapping defined by
where , and is for the -inverse strongly monotone mappings for each .
Assumption 2.1 [14]
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)
for fixed and , there exists a bounded subset K of and such that
Lemma 2.3 [28]
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, and for and , define a mapping as follows:
Then satisfies conditions (i)-(iv) of Lemma 2.3. , where is the solution set of the following equilibrium problem:
Lemma 2.4 [27]
Assume that satisfies Assumption 2.1, and let be defined as in Lemma 2.3. Let and . Then
Lemma 2.5 [31]
Let C be a nonempty closed convex subset of a real Hilbert space H. If is a nonexpansive mapping with , then the mapping is demiclosed at 0, i.e., if is a sequence in C weakly converges to x and if converges strongly to 0, then .
Lemma 2.6 [20]
Let be a τ-Lipschitzian mapping and let be a k-Lipschitzian and η-strongly monotone mapping, then for , is -strongly monotone i.e.,
Lemma 2.7 [39]
Suppose that and . Let be a k-Lipschitzian and η-strongly monotone operator. In association with a nonexpansive mapping , define the mapping by
Then is a contraction provided , that is,
where .
Lemma 2.8 [43]
Assume 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 2.9 [13]
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.
3 The proposed method and some properties
In this section, we suggest and analyze our method for finding the common solutions of the system of the variational inequality problem (1.1), the split equilibrium problem (1.9)-(1.10), and the hierarchical fixed-point problem (1.11).
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. Assume that and are the bifunctions satisfying Assumption 2.1 and is upper semicontinuous in the first argument. Let be a -inverse strongly monotone mapping for each and 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 for each , and , L is the spectral radius of the operator , and is the adjoint of A. Suppose the parameters satisfy , , where . Also, and are sequences in satisfying the following conditions:
-
(a)
and ,
-
(b)
,
-
(c)
and ,
-
(d)
and .
Remark 3.1 Our method can be viewed as an extension and improvement for some well-known results, for example the following.
-
The proposed method is an extension and improvement of the method of Wang and Xu [42] for finding the approximate element of the common set of solutions of a split 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.[45] for finding the approximate element of the common set of solutions of a split equilibrium problem and a hierarchical fixed-point problem in a real Hilbert space.
-
The contractive mapping f with a coefficient in other papers [39, 40, 45] 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 3.1 Let . Then , , , and are bounded.
Proof Let ; we have and . Then
From the definition of L it follows that
It follows from (1.4) that
Applying (3.4) and (3.3) to (3.2) and from the definition of γ, we get
Let ; we have
where
We set . Since is a -inverse strongly monotone mapping, it follows that
Since is -inverse strongly monotone mappings, for each , we get
We denote . 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 2.7.
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 Since and . It follows from Lemma 2.4 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
Next, we estimate
It follows from (3.8) and (3.9) that
From (3.1) and the above inequality, we get
Next, we estimate
where the second inequality follows from Lemma 2.7. From (3.10) and (3.11), we have
Here
It follows by conditions (a)-(d) of Algorithm 3.1 and Lemma 2.8 that
Next, we show that . Since by using (3.2), (3.5), and (3.7), we obtain
which implies that
Then from the above inequality, we get
Since , , , , , and , we obtain
and
Since is firmly nonexpansive, we have
Hence, we get
From (3.13), (3.7), and the above inequality, we have
which implies that
Hence
Since , , , and , we obtain
From (2.2), we get
Hence
where the last inequality follows from (3.15). On the other hand, from (3.1) and (2.2), we obtain
Hence
where the last inequality follows from (3.17). From (3.13) and the above inequality, we have
which implies that
Hence
Since , , , and , , , we obtain
Since
we get
It follows from (3.16) and (3.18) 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 2.5 that . Therefore . □
Theorem 3.1 The sequence generated by Algorithm 3.1 converges strongly to z, 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 monotonicity of that
and
Since , , and , it is easy to observe that . It follows by Assumption 2.1(iv) that , .
For any and , let , and 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 . Now we set . It follows from (3.14) that and . Therefore from the definition of , we have
Since is upper semicontinuous in the first argument, taking lim sup in the above inequality as and using Assumption 2.1(iv), we obtain
which implies that and hence .
Next, we show that . Since and there exists a subsequence of such that , it is easy to observe that . For any , using (2.5), we have
This implies that is nonexpansive. On the other hand
Since (see (3.18)), we have . It follows from Lemma 2.5 that , which implies from Lemma 2.2 that .
Thus we have
Observe that the constants satisfy and
therefore, from Lemma 2.6, the operator is strongly monotone, and we get the uniqueness of the solution of the variational inequality (3.20) 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 2.8 are satisfied. Hence we deduce that . This completes the proof. □
Remark 3.2 In hierarchical fixed-point problem (1.11), if , then we can get the variational inequality (3.20). In (3.20), if then we get the variational inequality , , which just is the variational inequality studied by Suzuki [39] extending the common set of solutions of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem.
4 Conclusions
In this paper, we suggest and analyze an iterative method for finding the approximate element of the common set of solutions of (1.1), (1.9)-(1.10), and (1.11) in real Hilbert space, which can be viewed as a refinement and improvement of some existing methods for solving a system of variational inequality problem, a split equilibrium problem, and a hierarchical fixed-point problem. Some existing methods (e.g. [20, 26, 34, 40, 45]) can be viewed as special cases of Algorithm 3.1. Therefore, the new algorithm is expected to be widely applicable.
References
Ceng LC, Wang CY, Yao JC: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 2008, 67: 375–390. 10.1007/s00186-007-0207-4
Ceng LC, Al-Mezel SA, Anasri QH: Implicit and explicit iterative methods for systems of variational inequalities and zeros of accretive operators. Abstr. Appl. Anal. 2013., 2013: Article ID 631382
Ansari QH, Yao JC: Systems of generalized variational inequalities and their applications. Appl. Anal. 2000, 76: 203–217. 10.1080/00036810008840877
Aubin JP: Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam; 1979.
Facchinei F, Pang JS I. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York; 2003.
Facchinei F, Pang JS II. In Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York; 2003.
Ansari QH, Lalitha CS, Mehta M: Generalized Convexity, Nonsmooth Variational Inequalities and Nonsmooth Optimization. CRC Press, Boca Raton; 2013.
Ceng LC, Anasri QH, Yao JC: Iterative methods for triple hierarchical variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 2011, 151: 489–512. 10.1007/s10957-011-9882-7
Marino G, Muglia L, Yao Y: Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points. Nonlinear Anal. 2012, 75: 1787–1798. 10.1016/j.na.2011.09.019
Latif A, Ceng LC, Ansari QH: Multi-step hybrid viscosity method for systems of variational inequalities defined over sets of solutions of equilibrium problem and fixed point problems. Fixed Point Theory Appl. 2012., 2012: Article ID 186
Verma RU: Projection methods, algorithms, and a new system of nonlinear variational inequalities. Comput. Math. Appl. 2001, 41: 1025–1031. 10.1016/S0898-1221(00)00336-9
Verma RU: General convergence analysis for two-step projection methods and applications to variational problems. Appl. Math. Lett. 2005, 18: 1286–1292. 10.1016/j.aml.2005.02.026
Acedo GL, Xu HK: Iterative methods for strictly pseudo-contractions in Hilbert space. Nonlinear Anal. 2007, 67: 2258–2271. 10.1016/j.na.2006.08.036
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Bnouhachem A, Noor MA: An iterative method for approximating the common solutions of a variational inequality, a mixed equilibrium problem and a hierarchical fixed point problem. J. Inequal. Appl. 2013., 2013: Article ID 490
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., 2013: Article ID 278
Bnouhachem, A: Strong convergence algorithm for split equilibrium problems and hierarchical fixed point problems. Sci. World J. (in press)
arXiv: 1108.5953
Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692
Ceng LC, Anasri QH, Yao JC: Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Anal. 2011, 74: 5286–5302. 10.1016/j.na.2011.05.005
Ceng LC, Anasri QH, Yao JC: Mann type iterative methods for finding a common solution of split feasibility and fixed point problems. Positivity 2012, 16(3):471–495. 10.1007/s11117-012-0174-8
Ceng LC, Anasri QH, Yao JC: An extragradient method for split feasibility and fixed point problems. Comput. Math. Appl. 2012, 64: 633–642. 10.1016/j.camwa.2011.12.074
Censor Y, Gibali A, Reich S: Algorithms for the split variational inequality problem. Numer. Algorithms 2012, 59(2):301–323. 10.1007/s11075-011-9490-5
Chang SS, Lee HWJ, Chan CK: Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces. Appl. Math. Lett. 2007, 20: 329–334. 10.1016/j.aml.2006.04.017
Chang SS, Lee HWJ, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 2009, 70: 3307–3319. 10.1016/j.na.2008.04.035
Cianciaruso F, Marino G, Muglia L, Yao Y: On a two-steps algorithm for hierarchical fixed point problems and variational inequalities. J. Inequal. Appl. 2009., 2009: Article ID 208692
Cianciaruso F, Marino G, Muglia L, Yao Y: A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl. 2010., 2010: Article ID 383740
Combettes PL, Hirstoaga SA: Equilibrium programming using proximal like algorithms. Math. Program. 1997, 78: 29–41.
Crombez G: A geometrical look at iterative methods for operators with fixed points. Numer. Funct. Anal. Optim. 2005, 26: 157–175. 10.1081/NFA-200063882
Crombez G: A hierarchical presentation of operators with fixed points on Hilbert spaces. Numer. Funct. Anal. Optim. 2006, 27: 259–277. 10.1080/01630560600569957
Geobel K, Kirk WA Cambridge Stud. Adv. Math. 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.
Gu G, Wang S, Cho YJ: Strong convergence algorithms for hierarchical fixed points problems and variational inequalities. J. Appl. Math. 2011., 2011: Article ID 164978
Korpelevic GM: An extragradient method for finding saddle points and for other problems. Èkon. Mat. Metody 1976, 12(4):747–756.
Mainge PE, Moudafi A: Strong convergence of an iterative method for hierarchical fixed-point problems. Pac. J. Optim. 2007, 3(3):529–538.
Marino G, Xu HK: Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 2011, 149(1):61–78. 10.1007/s10957-010-9775-1
Moudafi A: Split monotone variational inclusions. J. Optim. Theory Appl. 2011, 50: 275–283.
Moudafi A: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 2007, 23(4):1635–1640. 10.1088/0266-5611/23/4/015
Qin X, Shang M, Su Y: A general iterative method for equilibrium problem and fixed point problem in Hilbert spaces. Nonlinear Anal. 2008, 69: 3897–3909. 10.1016/j.na.2007.10.025
Suzuki N: Moudafi’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl. 2007, 325: 342–352. 10.1016/j.jmaa.2006.01.080
Tian M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 2010, 73: 689–694. 10.1016/j.na.2010.03.058
Verma RU: Generalized system for relaxed cocoercive variational inequalities and projection methods. J. Optim. Theory Appl. 2004, 121(1):203–210.
Wang Y, Xu W: Strong convergence of a modified iterative algorithm for hierarchical fixed point problems and variational inequalities. Fixed Point Theory Appl. 2013., 2013: Article ID 121
Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332
Yang H, Zhou L, Li Q: A parallel projection method for a system of nonlinear variational inequalities. Appl. Math. Comput. 2010, 217: 1971–1975. 10.1016/j.amc.2010.06.053
Yao Y, Cho YJ, Liou YC: Iterative algorithms for hierarchical fixed points problems and variational inequalities. Math. Comput. Model. 2010, 52(9–10):1697–1705. 10.1016/j.mcm.2010.06.038
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Bnouhachem, A. A modified projection method for a common solution of a system of variational inequalities, a split equilibrium problem and a hierarchical fixed-point problem. Fixed Point Theory Appl 2014, 22 (2014). https://doi.org/10.1186/1687-1812-2014-22
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DOI: https://doi.org/10.1186/1687-1812-2014-22