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Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings
Fixed Point Theory and Applications volume 2014, Article number: 200 (2014)
We give new hybrid variants of extragradient methods for finding a common solution of an equilibrium problem and a family of nonexpansive mappings. We present a scheme that combines the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this scheme is modified by projecting on a suitable convex set to get a better convergence property under certain assumptions in a real Hilbert space.
MSC:65K10, 65K15, 90C25, 90C33.
In this paper, we always assume that ℋ is a real Hilbert space with the inner product and the induced norm . Let C be a nonempty closed convex subset of ℋ and the bifunction . Then f is called strongly monotone on C with iff
monotone on C iff
pseudomonotone on C iff
Lipschitz-type continuous on C in the sense of Mastroeni  iff there exist positive constants , such that
An equilibrium problem, shortly , is to find a point in
Let a mapping T of C into itself. Then T is called contractive with constant iff
The mapping T is called strictly pseudocontractive iff there exists a constant such that
In the case , the mapping T is called nonexpansive on C. We denote by the set of fixed points of T.
Let , , be a family of nonexpansive mappings where Γ stands for an index set. In this paper, we are interested in the problem of finding a common element of the solution set of problem and the set of fixed points , namely:
where the function f and the mappings , , satisfy the following conditions:
(A1) for all and f is pseudomonotone on C,
(A2) f is Lipschitz-type continuous on C with constants and ,
(A3) f is upper semicontinuous on C,
(A4) For each , is convex and subdifferentiable on C,
Under these assumptions, for each and , there exists a unique element such that
Problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, equilibrium equilibriums, fixed point problems (see, e.g., [2–7]). Recently, it has become an attractive field for many researchers in both theory and its solution methods (see, e.g., [3, 4, 8–12] and the references therein). Most of these algorithms are based on inequality (1.2) for solving the underlying equilibrium problem when . Motivated by this idea for finding a common point of and the fixed point set of a nonexpansive mapping T, Takahashi and Takahashi  first introduced an iterative scheme by the viscosity approximation method. The sequence is defined by
where is contractive. Under certain conditions over the parameters and , they showed that the sequences and strongly converge to , where denotes the projection on C. At each iteration n in all of these algorithms, it requires to solve approximation auxiliary equilibrium problems for finding a common solution of an equilibrium problem and a fixed point problem. In order to avoid this requirement, Anh  recently proposed a hybrid extragradient algorithm for finding a common point of the set . Starting with an arbitrary initial point , iteration sequences are defined by
Under certain conditions onto parameters and , he showed that the sequences , and weakly converge to the point in a real Hilbert space. At each main iteration n of the scheme, he only solved strongly convex problems on C, but the proof of convergence was still done under the assumptions that .
For finding a common point of a family of nonexpansive mappings (), as a corollary of Theorem 2.1 in , Zhou proposed the following iteration scheme:
Under the restrictions of the control sequences , he showed that the sequence defined by (1.4) strongly converges to in a real Hilbert space ℋ, where .
In this paper, motivated by Ceng et al. [16, 17], Wang and Guo , Zhou , Nadezhkina and Takahashi , Cho et al. , Takahashi and Takahashi , Anh [6, 12] and Anh et al. [20, 21], we introduce several modified hybrid extragradient schemes to modify the iteration schemes (1.3) and (1.4) to obtain new strong convergence theorems for a family of nonexpansive mappings and the equilibrium problem in the framework of a real Hilbert space ℋ.
To investigate the convergence of this scheme, we recall the following technical lemmas which will be used in the sequel.
Lemma 1.1 (, Lemma 3.1)
Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Let be a pseudomonotone and Lipschitz-type continuous bifunction. For each , let be convex and subdifferentiable on C. Suppose that the sequences , , are generated by scheme (1.3) and . Then
Lemma 1.2 Let C be a closed convex subset of a real Hilbert space ℋ, and let be the metric projection from ℋ on to C (i.e., for , is the only point in C such that ). Given and . Then if only if there holds the relation for all .
Lemma 1.3 Let ℋ be a real Hilbert space. Then the following equations hold:
for all .
for all and .
2 Convergence theorems
Now, we prove the main convergence theorem.
Theorem 2.1 Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Suppose that assumptions (A1)-(A5) are satisfied and is a family of nonexpansive mappings from C into itself and a nonempty common fixed points set F. Let be a sequence generated by the following scheme:
Then the sequences , and strongly converge to the same point .
Proof The proof of this theorem is divided into several steps.
Step 1. Claim that and are closed and convex for all .
We have to show that for any fixed point but arbitrary , is closed and convex for every . This can be proved by induction on n. It is obvious that is closed and convex. Assume that is closed and convex for some . We have that the set
is closed and convex, and , hence is closed and convex. Then is closed and convex for all . We can write under the form
Then is closed and convex. Thus, is closed and convex.
Step 2. Claim that for all .
First, we show that by induction on n. It suffices to show that .
We have is obvious. Suppose for some . We have to show that . Indeed, let , by inductive hypothesis, we have and
On the other hand, for all and , we have
Combining this with (2.1), we obtain
This follows that
By the definition of , we have , and so for all , which deduces that . This shows that for all .
Next, we will prove by induction on . It suffices to show that . Indeed, so . Suppose that . Let , then . Using Lemma 1.1, we get
Then we have and hence . This shows that , which yields that for all .
Step 3. Claim that the sequence is bounded and there exists the limit .
From , it follows that
Then, using Step 2, we have and
Combining this and assumption (A5), the projection is well defined and there exits a unique point p such that . So, we have
Then the sequence is bounded. So, the sequences , , , also are bounded. Since and (2.3), we have
and hence . This together with the boundedness of implies that the limit exists.
Step 4. We claim that .
Since , for any positive integer and (2.3), we have
Passing the limit in (2.5) as , we get . Hence, is a Cauchy sequence in a real Hilbert space ℋ and so .
Step 5. We claim that , where .
First we show that . Since , we have . Then and
which yields that
Combining this and for all , we get
For each , by (2.2) we have
Using this, the boundedness of sequences , and (2.6), we obtain
By a similar way, we also have . Then it follows from the inequality
On the other hand, we have
Combining this, (2.6) and (2.8), we obtain . By the definition of the sequence , we have
which yields that
It follows from Step 4 that . Hence .
Now we show that . By Step 5, we have as .
Since is the unique solution of the strongly convex problem
From this it follows that
where and . By the definition of the normal cone , we have
On the other hand, since is subdifferentiable on C, by the well-known Moreau-Rockafellar theorem, there exists such that
Combining this with (2.9), we have
Then, using , (2.7), , as and the upper semicontinuity of f, we have
This means that . By taking the limit in (2.4), we have
which implies that . Thus, the subsequences , , strongly converge to the same point . This completes the proof. □
Now, notice that
From (2.10) and using the methods in Theorem 2.1, we can get the following convergence result.
Theorem 2.2 Let C be a nonempty closed convex subset of a real Hilbert space ℋ. Suppose that assumptions (A1)-(A5) are satisfied and is a family of nonexpansive mappings from C into itself and a nonempty common fixed points set F. Let be a sequence generated by the following scheme:
Then the sequences , and converge strongly to the same point .
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The author declares that he has no competing interests.
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Cite this article
Thanh, D.D. Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings. Fixed Point Theory Appl 2014, 200 (2014). https://doi.org/10.1186/1687-1812-2014-200
- equilibrium problems
- fixed point
- Lipschitz-type continuous
- extragradient method
- nonexpansive mappings