- Research Article
- Open Access
An iterative method for common solutions of equilibrium problems and hierarchical fixed point problems
© Bnouhachem et al.; licensee Springer. 2014
- Received: 30 May 2014
- Accepted: 5 September 2014
- Published: 24 September 2014
The purpose of this paper is to investigate the problem of finding an approximate point of the common set of solutions of an equilibrium problem and a hierarchical fixed point problem in the setting of real Hilbert spaces. We establish the strong convergence of the proposed method under some mild conditions. Several special cases are also discussed. Numerical examples are presented to illustrate the proposed method and convergence result. The results presented in this paper extend and improve some well-known results in the literature.
MSC:49J30, 47H09, 47J20, 49J40.
- equilibrium problems
- variational inequalities
- hierarchical fixed point problems
- fixed point problems
- projection method
The solution set of EP (1.1) is denoted by .
The equilibrium problem provides a unified, natural, innovative and general framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity and optimization. The theory of equilibrium problems has witnessed an explosive growth in theoretical advances and applications across all disciplines of pure and applied sciences; see [1–10] and the references therein.
It is a well-known classical variational inequality problem. We now have a variety of techniques to suggest and analyze various iterative algorithms for solving variational inequalities and the related optimization problems; see [1–27] and the references therein.
We denote by the set of solutions of (1.3). It is well known that is closed and convex, and is well defined.
In this paper, motivated by the work of Ceng et al. [18, 20], Yao et al. , Bnouhachem  and others, we propose an iterative method for finding an approximate element of the common set of solutions of EP (1.1) and HFPP (1.4) in the setting of real Hilbert spaces. We establish a strong convergence theorem for the sequence generated by the proposed method. The proposed method is quite general and flexible and includes several known methods for solving of variational inequality problems, equilibrium problems, and hierarchical fixed point problems; see, for example, [12, 13, 15, 18, 19, 21] and the references therein.
We present some definitions which will be used in the sequel.
If , then T is called nonexpansive.
If , then T is called contraction.
- (a)monotone if
- (b)strongly monotone if there exists an such that
- (c)α-inverse strongly monotone if there exists an such that
The following lemma provides some basic properties of the projection onto C.
, , ;
, , .
Assumption 2.1 
Let be a bifunction satisfying the following assumptions:
(A1) , ;
(A2) is monotone, that is, , ;
(A3) For each , ;
(A4) For each , is convex and lower semicontinuous.
Lemma 2.2 
is nonempty single-valued;
- (ii)is firmly nonexpansive, that is,
is closed and convex.
Lemma 2.3 
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, that is, if is a sequence in C converges weakly to x and converges strongly to 0, then .
Lemma 2.4 
Lemma 2.5 
Lemma 2.6 
the weak w-limit set , where
for each , exists.
Then the sequence is weakly convergent to a point in C.
Lemma 2.7 
In this section, we propose and analyze an iterative method for finding the common solutions of EP (1.1) and HFPP (1.4).
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction that satisfy conditions (A1)-(A4), and let be nonexpansive mappings such that . Let be a k-Lipschitzian mapping and η-strongly monotone, and let be a τ-Lipschitzian mapping.
, and ,
Remark 3.1 Algorithm 3.1 can be viewed as an extension and improvement for some well-known methods.
If , , , and , then we obtain an extension and improvement of a method considered in .
Lemma 3.1 Let . Then , , and are bounded.
Proof It follows from Lemma 2.2 that . Let , then . Define .
for and . Hence, is bounded, and consequently, we deduce that , , , , , and are bounded. □
The weak w-limit set .
Since is bounded, without loss of generality, we can assume that . It follows from Lemma 2.3 that . Therefore, . □
therefore, from Lemma 2.4, the operator is strongly monotone, and we get the uniqueness of the solution of variational inequality (3.13), and denote it by .
Thus, all the conditions of Lemma 2.7 are satisfied. Hence, we deduce that . This completes the proof. □
Putting in Algorithm 3.1, we obtain the following result, which can be viewed as an extension and improvement of the method studied in .
Putting , , , and , we obtain an extension and improvement of the method considered in .
To illustrate Algorithm 3.1 and the convergence result, we consider the following examples.
Then the sequence satisfies condition (e).
In all the tests we take , , and for Algorithm 3.1. In this example , , . It is easy to show that the parameters satisfy , , where .
The values of , , and with initial values and
n = 1
n = 2
n = 3
n = 4
n = 5
n = 6
n = 7
n = 8
n = 9
n = 10
The values of , , and with initial value
n = 1
n = 2
n = 3
n = 4
n = 5
n = 6
n = 7
n = 8
n = 9
n = 10
Example 4.2 In this example we take the same mappings and parameters as in Example 4.1 except T and .
In this paper, we suggested and analyzed an iterative method for finding an element of the common set of solutions of (1.1) and (1.4) in real Hilbert spaces. This method can be viewed as a refinement and improvement of some existing methods for solving variational inequality problem, equilibrium problem and a hierarchical fixed point problem. Some existing methods, for example, [12, 13, 15, 18, 19, 21], can be viewed as special cases of Algorithm 3.1. Therefore, Algorithm 3.1 is expected to be widely applicable. In the hierarchical fixed point problem (1.4), if , then we can get the variational inequality (3.13). In (3.13), if then we get the variational inequality , , which just is a variational inequality studied by Suzuki .
In this research, the second and third author were financially supported by King Fahd University of Petroleum & Minerals (KFUPM), Dhahran, Saudi Arabia. It was partially done during the visit of third author to KFUPM, Dhahran, Saudi Arabia.
- Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetMATHGoogle Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming using proximal like algorithms. Math. Program. 1997, 78: 29–41.View ArticleMathSciNetGoogle Scholar
- Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert space. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetMATHGoogle Scholar
- Ceng LC, Yao JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 2008, 214: 186–201. 10.1016/j.cam.2007.02.022View ArticleMathSciNetMATHGoogle Scholar
- Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008, 69: 1025–1033. 10.1016/j.na.2008.02.042View ArticleMathSciNetMATHGoogle Scholar
- Reich S, Sabach S: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp. Math. 568. Optimization Theory and Related Topics 2012, 225–240.View ArticleGoogle Scholar
- Kassay G, Reich S, Sabach S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 2011, 21: 1319–1344. 10.1137/110820002View ArticleMathSciNetMATHGoogle Scholar
- Chang SS, Joseph Lee HW, 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.2008.04.035View ArticleMathSciNetMATHGoogle Scholar
- 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 186Google Scholar
- Marino C, 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.2011.09.019View ArticleMathSciNetMATHGoogle Scholar
- Suwannaut S, Kangtunyakarn A: The combination of the set of solutions of equilibrium problem for convergence theorem of the set of fixed points of strictly pseudo-contractive mappings and variational inequalities problem. Fixed Point Theory Appl. 2013., 2013: Article ID 291Google Scholar
- 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.2010.06.038View ArticleMathSciNetMATHGoogle Scholar
- Mainge PE, Moudafi A: Strong convergence of an iterative method for hierarchical fixed-point problems. Pac. J. Optim. 2007, 3(3):529–538.MathSciNetMATHGoogle Scholar
- Moudafi A: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 2007, 23(4):1635–1640. 10.1088/0266-5611/23/4/015View ArticleMathSciNetMATHGoogle Scholar
- 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 208692Google Scholar
- Marino G, Xu HK: Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 2011, 149(1):61–78. 10.1007/s10957-010-9775-1View ArticleMathSciNetMATHGoogle Scholar
- Crombez G: A hierarchical presentation of operators with fixed points on Hilbert spaces. Numer. Funct. Anal. Optim. 2006, 27: 259–277. 10.1080/01630560600569957View ArticleMathSciNetMATHGoogle Scholar
- 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.2011.05.005View ArticleMathSciNetMATHGoogle Scholar
- 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., 2014: Article ID 22Google Scholar
- 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/s10957-011-9882-7View ArticleMathSciNetMATHGoogle Scholar
- Tian M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 2010, 73: 689–694. 10.1016/j.na.2010.03.058View ArticleMathSciNetMATHGoogle Scholar
- Geobel K, Kirk WA Stud. Adv. Math. 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.View ArticleGoogle Scholar
- Suzuki N: Moudafi’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl. 2007, 325: 342–352. 10.1016/j.jmaa.2006.01.080View ArticleMathSciNetMATHGoogle Scholar
- Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240–256. 10.1112/S0024610702003332View ArticleMathSciNetMATHGoogle Scholar
- Acedo GL, Xu HK: Iterative methods for strictly pseudo-contractions in Hilbert space. Nonlinear Anal. 2007, 67: 2258–2271. 10.1016/j.na.2006.08.036View ArticleMathSciNetMATHGoogle Scholar
- 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 121Google Scholar
- 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 383740Google Scholar
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