An iterative algorithm for system of generalized equilibrium problems and fixed point problem
© Bnouhachem; licensee Springer. 2014
Received: 1 September 2014
Accepted: 12 November 2014
Published: 1 December 2014
In this paper, we propose an iterative algorithm for finding a common solution of a system of generalized equilibrium problems and a fixed point problem of strictly pseudo-contractive mapping 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 generalized equilibrium problems and a hierarchical fixed point problem. Preliminary numerical experiments are included to verify the theoretical assertions of the proposed method. The iterative algorithm and results presented in this paper generalize, unify, and improve the previously known results of this area.
MSC:49J30, 47H09, 47J20.
where is two bifunctions and is a nonlinear mapping for each . The solution set of (1.1) is denoted by Ω.
which was studied by Takahashi and Takahashi . Inspired by the work of Takahashi and Takahashi , and Ceng et al. , Ceng et al.  introduced and analyzed an iterative scheme for finding the approximate solutions of the generalized equilibrium problem (1.2), a system of general generalized equilibrium problems (1.1) and a fixed point problem of a nonexpansive mapping in a Hilbert space. Under appropriate conditions, they proved that the sequence converges strongly to a common solution of these three problems. Recently, Ansari  studied the existence of solutions of equilibrium problems in the setting of metric spaces. Inspired by the method in , Latif et al.  introduced and analyzed an iterative algorithm by the hybrid iterative method for finding a solution of the system of generalized equilibrium problems with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inclusions, and the common fixed point problem of an asymptotically strict pseudo-contractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. Under mild conditions, they proved the weak convergence of this iterative algorithm.
this problem was considered and investigated by Ceng et al. . As pointed out in  that the system of variational inequalities is used as a tool to study the Nash equilibrium problem; see, for example, [9–11] and the references therein.
The theory of variational inequalities emerged as a rapidly growing area of research because of its applications in nonlinear analysis, optimization, economics, game theory; see for example [14–17]. For recent applications, numerical techniques, and physical formulation, see [1–50].
We denote by the set of solutions of (1.5). It is well known that is closed and convex, and is well defined (see ).
where is a nonexpansive mapping. They proved that under some approximate assumptions on the operators and parameters, the sequence generated by (1.11) converges strongly to the unique solution of the variational inequality (1.10). In 2014, Ansari et al.  presented a hybrid iterative algorithm for computing a fixed point of a pseudo-contractive mapping and for finding a solution of triple hierarchical variational inequality in the setting of real Hilbert space. Under very appropriate conditions, they proved that the sequence generated by the proposed algorithm converges strongly to a fixed point which is also a solution of this triple hierarchical variational inequality.
In this paper, motivated by the work of Ceng et al. , Yao et al. , Bnouhachem [33, 34] 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) and (1.6) in real Hilbert space. We establish a strong convergence theorem based on this method. In order to verify the theoretical assertions and to compare the numerical results between the system of generalized equilibrium problems and the generalized equilibrium problems, an example is given. Our results can be viewed as significant extensions of the previously known results.
We present some definitions which will be used in the sequel.
If , then T is called nonexpansive.
If , then T is called a contraction.
- (a)strongly monotone if there exists an such that
- (b)α-inverse strongly monotone if there exists an such that
- (c)a k-strict pseudo-contraction, if there exists a constant such that
Assumption 2.1 
Let be a bifunction satisfying the following assumptions:
(A1) , ;
(A2) F is monotone, i.e., , ;
(A3) for each , ;
(A4) for each , is convex and lower semicontinuous.
We list some fundamental lemmas that are useful in the consequent analysis.
Lemma 2.1 
is nonempty and single-valued;
- (ii)is firmly nonexpansive, i.e.,
is closed and convex.
Lemma 2.2 
where , , and is a -inverse strongly monotone mapping for each .
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, i.e., if is a sequence in C that weakly converges to x, and if converges strongly to 0, then .
Lemma 2.4 
Lemma 2.5 
Let C be a nonempty closed convex subset of a real Hilbert space H, and be a k-strict pseudo-contraction mapping. Define by for all . Then as , B is a nonexpansive mapping such that .
Lemma 2.6 
Lemma 2.7 
Let , be bounded sequences in a Banach space E and be a sequence in with .
Suppose , and . Then .
Lemma 2.8 
Lemma 2.9 
the weak w-limit set where ;
for each , exists.
Then is weakly convergent to a point in C.
Lemma 2.10 
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 generalized equilibrium problem (1.1) and the hierarchical fixed point problem (1.6). Let C be a nonempty closed convex subset of a real Hilbert space H. Let be two bifunctions satisfying (A1)-(A4). Let be a -inverse strongly monotone mapping for each , and let be a σ-strict pseudo-contraction mapping such that . Let be a k-Lipschitzian mapping and be η-strongly monotone, and let be a τ-Lipschitzian mapping.
If , , and , then Algorithm 3.1 reduces to Algorithm 3.2 for finding the common solutions of the generalized equilibrium problem (1.2) and the hierarchical fixed point problem (1.6).
Remark 3.1 If , , and , we obtain an extension and improvement of the method of Yao et al.  and Wang and Xu  for finding the approximate element of the common set of solutions of a system of generalized equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.
Lemma 3.1 Let . Then , , and are bounded.
where the third inequality follows from Lemma 2.6 and the fourth inequality follows from (3.4). By induction on n, we obtain , for and . Hence is bounded, and consequently we deduce that , , , , , and are bounded. □
The weak w-limit set ().
Since is bounded and without loss of generality we can assume that , from (3.11), it is easy to observe that . It follows from Lemma 2.3 that . Therefore . □
Since , from Lemma 2.4, the operator is μη-ρτ-strongly monotone, and we get the uniqueness of the solution of the variational inequality (3.13) and denote it by .
Let and .
Thus all the conditions of Lemma 2.8 are satisfied. Hence we deduce that . This completes the proof. □
To verify the theoretical assertions, we consider the following example.
Example 4.1 Let , , and .
It is easy to show that the sequence satisfies condition (a).
The sequence satisfies condition (b).
The values of , , and with initial value
Algorithm 3.2 with
Algorithm 3.2 with
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
Algorithm 3.2 with
Algorithm 3.2 with
n = 1
n = 2
n = 3
n = 4
n = 5
n = 6
n = 7
n = 8
n = 9
n = 10
In this paper, we suggest and analyze an iterative method for finding the approximate element of the common set of solutions of (1.1) and (1.6) in real Hilbert space, which can be viewed as a refinement and improvement of some existing methods for solving equilibrium problem, and a hierarchical fixed point problem. Strong convergence of the proposed method is proved under mild assumptions. Furthermore, some preliminary numerical results are reported to verify the theoretical assertions of the proposed method and show that our algorithm for the system of generalized equilibrium problems is more attractive in practice than our algorithm for the generalized equilibrium problems.
The author would like to thank Prof. Xindan Li, Dean of School of Management and Engineering of Nanjing University, for providing excellent research facilities.
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