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.
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 
3 An iterative method and strong convergence results
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.
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