Strong convergence theorems for equilibrium problems involving a family of nonexpansive mappings
© Thanh; licensee Springer. 2014
Received: 7 May 2014
Accepted: 10 September 2014
Published: 25 September 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 the case , the mapping T is called nonexpansive on C. We denote by the set of fixed points of T.
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 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 .
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)
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 .
for all .
for all and .
2 Convergence theorems
Now, we prove the main convergence theorem.
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 .
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 .
By the definition of , we have , and so for all , which deduces that . This shows that for all .
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 .
and hence . This together with the boundedness of implies that the limit exists.
Step 4. We claim that .
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 .
It follows from Step 4 that . Hence .
Now we show that . By Step 5, we have as .
which implies that . Thus, the subsequences , , strongly converge to the same point . This completes the proof. □
From (2.10) and using the methods in Theorem 2.1, we can get the following convergence result.
Then the sequences , and converge strongly to the same point .
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