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An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems
Fixed Point Theory and Applications volume 2009, Article number: 632819 (2009)
Abstract
The purpose of this paper is to investigate the problem of approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. First, we propose an extragradient method for solving the mixed equilibrium problems and the fixed point problems. Subsequently, we prove the strong convergence of the proposed algorithm under some mild assumptions.
1. Introduction
Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be a real-valued function and be an equilibrium bifunction, that is, for each . We consider the following mixed equilibrium problem (MEP) which is to find such that
In particular, if , this problem reduces to the equilibrium problem (EP), which is to find such that
Denote the set of solutions of (MEP) by and the set of solutions of (EP) by . The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases; see, for example, [1–5]. Some methods have been proposed to solve the equilibrium problems, see, for example, [5–21].
In 1997, Flåm and Antipin [15] introduced an iterative algorithm of finding the best approximation to the initial data when and proved a strong convergence theorem. Recently by using the viscosity approximation method S. Takahashi and W. Takahashi [8] introduced another iterative algorithm for finding a common element of the set of solutions of (EP) and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let be a nonexpansive mapping and be a contraction. Starting with arbitrary initial , define the sequences and recursively by
-
S.
Takahashi and W. Takahashi proved that the sequences and defined by (TT) converge strongly to with the following restrictions on algorithm parameters and :
(i) and ;
(ii);
(iii)(A1): ; and (R1): .
Subsequently, some iterative algorithms for equilibrium problems and fixed point problems have further developed by some authors. In particular, Zeng and Yao [16] introduced a new hybrid iterative algorithm for mixed equilibrium problems and fixed point problems and Mainge and Moudafi [22] introduced an iterative algorithm for equilibrium problems and fixed point problems.
On the other hand, for solving the equilibrium problem (EP), Moudafi [23] presented a new iterative algorithm and proved a weak convergence theorem. Ceng et al. [24] introduced another iterative algorithm for finding an element of . Let be a -strict pseudocontraction for some such that . For given , let the sequences and be generated iteratively by
where the parameters and satisfy the following conditions:
(i) for some ;
(ii) and .
Then, the sequences and generated by (CAY) converge weakly to an element of .
At this point, we should point out that all of the above results are interesting and valuable. At the same time, these results also bring us the following conjectures.
Questions
(1)Could we weaken or remove the control condition (iii) on algorithm parameters in S. Takahashi and W. Takahashi [8]?
(2)Could we construct an iterative algorithm for -strict pseudocontractions such that the strong convergence of the presented algorithm is guaranteed?
(3)Could we give some proof methods which are different from those in [8, 12, 16, 24].
It is our purpose in this paper that we introduce a general iterative algorithm for approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. Subsequently, we prove the strong convergence of the proposed algorithm under some mild assumptions. Our results give positive answers to the above questions.
2. Preliminaries
Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of .
Let be a mapping. We use to denote the set of the fixed points of . Recall what follows.
(i) is called demicontractive if there exists a constant such that
for all and , which is equivalent to
For such case, we also say that is a -demicontractive mapping.
(ii) is called nonexpansive if
for all .
(iii) is called quasi-nonexpansive if
for all and .
(iv) is called strictly pseudocontractive if there exists a constant such that
for all .
It is worth noting that the class of demicontractive mappings includes the class of the nonexpansive mappings, the quasi-nonexpansive mappings and the strictly pseudo-contractive mappings as special cases.
Let us also recall that is called demiclosed if for any sequence and , we have
It is well-known that the nonexpansive mappings, strictly pseudo-contractive mappings are all demiclosed. See, for example, [25–27].
An operator is said to be -strongly monotone if there exists a positive constant such that
for all .
Now we concern the following problem: find such that
In this paper, for solving problem (2.8) with an equilibrium bifunction , we assume that satisfies the following conditions:
(H1) is monotone, that is, for all ;
(H2) for each fixed , is concave and upper semicontinuous;
(H3) for each , is convex.
A mapping is called Lipschitz continuous, if there exists a constant such that
A differentiable function on a convex set is called
(i)-convex if
where is the Frechet derivative of at ;
(ii)-strongly convex if there exists a constant such that
Let be a nonempty closed convex subset of a real Hilbert space , be real-valued function and be an equilibrium bifunction. Let be a positive number. For a given point , the auxiliary problem for (MEP) consists of finding such that
Let be the mapping such that for each , is the solution set of the auxiliary problem, that is, ,
We need the following important and interesting result for proving our main results.
Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Assume what follows.
(i) is Lipschitz continuous with constant such that
(a),
(b) is affine in the first variable,
(c)for each fixed is sequentially continuous from the weak topology to the weak topology.
(ii) is -strongly convex with constant and its derivative is sequentially continuous from the weak topology to the strong topology.
(iii)For each , there exist a bounded subset and such that for any ,
Then there hold the following:
(i) is single-valued;
(ii) is nonexpansive if is Lipschitz continuous with constant such that and
where for ;
(iii);
(iv) is closed and convex.
3. Main Results
Let be a real Hilbert space, be a lower semicontinuous and convex real-valued function, be an equilibrium bifunction. Let be a mapping and be a mapping. In this section, we first introduce the following new iterative algorithm.
Algorithm 3.1.
Let be a positive parameter. Let be a sequence in and be a sequence in . Define the sequences , and by the following manner:
Now we give a strong convergence result concerning Algorithm 3.1 as follows.
Theorem 3.2.
Let be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Let be an -Lipschitz continuous and -strongly monotone mapping and be a demiclosed and -demicontractive mapping such that . Assume what follows.
(i) is Lipschitz continuous with constant such that
(a),
(b) is affine in the first variable,
(c)for each fixed , is sequentially continuous from the weak topology to the weak topology.
(ii) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant such that .
(iii)For each ; there exist a bounded subset and such that, for any ,
(iv) for some , and .
Then the sequences , , and generated by (3.1) converge strongly to which solves the problem (2.8) provided is firmly nonexpansive.
Proof.
First, we prove that , , and are all bounded. Without loss of generality, we may assume that . Given and , we have
that is,
Take . From (3.1), we have
Therefore,
where .
Note that and are firmly nonexpansive. Hence, we have
which implies that
From (2.2) and (3.1), we have
From (3.6)–(3.9), we have
This implies that is bounded, so are and .
From (3.1), we can write . Thus, from (3.9), we have
Since , . Therefore, from (3.8) and (3.11), we obtain
We note that and are bounded. So there exists a constant such that
Consequently, we get
Now we divide two cases to prove that converges strongly to .
Case 1.
Assume that the sequence is a monotone sequence. Then is convergent. Setting .
(i)If , then the desired conclusion is obtained.
(ii)Assume that . Clearly, we have
this together with and (3.14) implies that
that is to say
Let be a weak limit point of . Then there exists a subsequence of , still denoted by which weakly converges to . Noting that , we also have
Combining (3.1) and (3.17), we have
Since is demiclosed, then we obtain .
Next we show that . Since , we derive
From the monotonicity of , we have
and hence
Since and weakly, from the weak lower semicontinuity of and in the second variable , we have
for all . For and , let . Since and , we have and hence . From the convexity of equilibrium bifunction in the second variable , we have
and hence . Then, we have
for all and hence .
Therefore, we have
Thus, if is a solution of problem (2.8), we have
Suppose that there exists another subsequence which weakly converges to . It is easily checked that and
Therefor, we have
Since is -strongly monotone, we have
By (3.17)–(3.30), we get
From (3.12), for , we deduce that there exists a positive integer number large enough, when ,
This implies that
Since and is bounded, hence the last inequality is a contraction. Therefore, , that is to say, .
Case 2.
Assume that is not a monotone sequence. Set and let be a mapping for all by
Clearly, is a nondecreasing sequence such that as and for . From (3.14), we have
thus
Therefore,
Since , for all , from (3.12), we get
which implies that
Since is bounded, there exists a subsequence of , still denoted by which converges weakly to . It is easily checked that . Furthermore, we observe that
Hence, for all ,
Therefore
which implies that
Thus,
It is immediate that
Furthermore, for , it is easily observed that if (i.e., ), because for . As a consequence, we obtain for all ,
Hence , that is, converges strongly to . Consequently, it easy to prove that and converge strongly to . This completes the proof.
Remark 3.3.
The advantages of these results in this paper are that less restrictions on the parameters are imposed.
As direct consequence of Theorem 3.2, we obtain the following.
Corollary 3.4.
Let be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Let be an -Lipschitz continuous and -strongly monotone mapping and be a nonexpansive mapping such that . Assume what follows.
(i) is Lipschitz continuous with constant such that;
(a),
(b) is affine in the first variable,
(c)for each fixed , is sequentially continuous from the weak topology to the weak topology.
(ii) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant such that .
(iii)For each ; there exist a bounded subset and such that, for any ,
(iv) for some , and .
Then the sequences , , and generated by (3.1) converge strongly to which solves the problem (2.8) provided is firmly nonexpansive.
Corollary 3.5.
Let be a real Hilbert space. Let be a lower semicontinuous and convex functional. Let be an equilibrium bifunction satisfying conditions (H1)–(H3). Let be an -Lipschitz continuous and -strongly monotone mapping and be a strictly pseudo-contractive mapping such that . Assume what follows.
(i) is Lipschitz continuous with constant such that
(a),
(b) is affine in the first variable,
(c)for each fixed , is sequentially continuous from the weak topology to the weak topology.
(ii) is -strongly convex with constant and its derivative is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant such that .
(iii)For each ; there exist a bounded subset and such that, for any ,
(iv) for some , and .
Then the sequences , and generated by (3.1) converge strongly to which solves the problem (2.8) provided is firmly nonexpansive.
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Acknowledgment
The authors are extremely grateful to the anonymous referee for his/her useful comments and suggestions. The first author was partially supposed by National Natural Science Foundation of China Grant 10771050. The second author was partially supposed by the Grant NSC 97-2221-E-230-017.
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Yao, Y., Liou, YC. & Wu, YJ. An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems. Fixed Point Theory Appl 2009, 632819 (2009). https://doi.org/10.1155/2009/632819
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DOI: https://doi.org/10.1155/2009/632819