Abstract
In this paper, we introduce a new iterative algorithm which is constructed by using
the hybrid projection method for finding a common solution of a system of equilibrium
problems of bifunctions satisfying certain conditions and a common solution of fixed
point problems of a family of uniformly Lipschitz continuous and asymptotically
MSC: 47H05, 47H09, 47H10.
Keywords:
asymptotically strict pseudocontraction in the intermediate sense; hybrid projection method; system of equilibrium problems; fixed point problems1 Introduction
Let C be a closed and convex subset of a real Hilbert space H with the inner product
The set of solutions of (1.1) is denoted by
If Γ is a singleton, then the problem (1.1) is reduced to the equilibrium problem of finding
The set of solutions of (1.3) is denoted by
Recall the following definitions.
A mapping
A mapping A is called αinversestrongly monotone[1,2], if there exists a positive real number α such that
Clearly, if A is αinversestrongly monotone, then A is monotone.
A mapping A is called βstrongly monotone if there exists a positive real number β such that
A mapping A is called LLipschitz continuous if there exists a positive real number L such that
It is easy to see that if A is an αinversestrongly monotone mapping from C into H, then A is
In 2009, Qin et al.[3] introduced the following algorithm for a finite family of asymptotically
Let
It is called the explicit iterative sequence of a finite family of asymptotically
Next, Sahu et al.[4] introduced new iterative schemes for asymptotically strictly pseudocontractive mappings in the intermediate sense. To be more precise, they proved the following theorem.
Theorem (SXY)LetCbe a nonempty closed and convex subset of a real Hilbert spaceHand
where
In 2010, Hu and Cai [5] considered the asymptotically strictly pseudocontractive mappings in the intermediate sense concerning the equilibrium problem. They obtained the following result in a real Hilbert space. Next, Ceng et al.[6] introduced the viscosity approximation method for a modified Mann iteration process for asymptotically strict pseudocontractive mappings in the intermediate sense and they proved the strong convergence of a general CQalgorithm and extended the concept of asymptotically strictly pseudocontractive mappings in the intermediate sense to the Banach space setting called nearly asymptotically strictly pseudocontractive mappings in the intermediate sense. Finally, they established a weak convergence theorem for a fixed point of nearly asymptotically strictly pseudocontractive mappings in the intermediate sense which are not necessarily Lipschitz continuous mappings.
Theorem (HC)LetCbe a nonempty closed and convex subset of a real Hilbert spaceHand
Let
where
In 2011, Duan and Zhao [7] introduced new iterative schemes for finding a common solution set of a system of equilibrium problems and a solution of a fixed point set of asymptotically strict pseudocontractions in the intermediate sense and they proved these schemes converge strongly.
In 2012, Shui Ge [8] introduced a new hybrid algorithm with variable coefficients for a fixed point problem of a uniformly Lipschitz continuous mapping and asymptotically pseudocontractive mapping in the intermediate sense on unbounded domains and he proved strong convergence in a real Hilbert space.
Theorem (Ge)LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH,
where
Assume that the positive real number
Then
In this paper, motivated and inspired by the previously mentioned above results, we
introduce a new iterative algorithm by the hybrid projection method for finding a
common solution of a system of equilibrium problems of bifunctions satisfying certain
conditions and a common solution of fixed point problems of a family of uniformly
Lipschitz continuous and asymptotically
2 Preliminaries
Let H be a real Hilbert space with the inner product
We know that
and
We will adopt the following notations:
(1) → for strong convergence and ⇀ for weak convergence.
(2)
(3) A nonlinear mapping S :
Definition 2.1 Let S be a mapping from C to C. Then
(1) S is said to be nonexpansive if
(2) S is said to be uniformly Lipschitz continuous if there exists a constant
(3) S is said to be asymptotically nonexpansive if there exists a sequence
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk
(see [9]) in 1972. It is known that if C is a nonempty, bounded, closed, and convex subset of a real Hilbert space H, then every asymptotically nonexpansive selfmapping has a fixed point. Further,
the set
(4) S is said to be asymptotically nonexpansive in the intermediate sense [10,11] if it is continuous and the following inequality holds:
Putting
The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk and Bruck et al. (see [10,11]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that if C is a nonempty, bounded, closed, and convex subset of a real Hilbert space H, then every asymptotically nonexpansive selfmapping in the intermediate sense has a fixed point (see [12]).
(5) S is said to be contractive if there exists a coefficient
(6) S is said to be a λstrict pseudocontraction if there exists a coefficient
The class of strict pseudocontractions was introduced by Brower and Petryshyn (see
[1]) in 1967. Clearly, if S is a nonexpansive mapping, then S is a strict pseudocontraction with
(7) S is said to be an asymptoticallyλstrict pseudocontraction with the sequence
The class of asymptotically strict pseudocontractions was introduced by Qihou [14] in 1996. Clearly, if S is an asymptotically nonexpansive mapping, then S is an asymptotically strict pseudocontraction with
(8) S is said to be an asymptoticallyλstrict pseudocontraction in the intermediate sense with the sequence
Putting
The class of asymptotically strict pseudocontractions in the intermediate sense was introduced by Sahu, Xu, and Yao [4] as a generalization of a class of asymptotically strict pseudocontractions.
For solving the equilibrium problem, let us give the following assumptions for the bifunction F and the set C:
(A1)
(A2) F is monotone, i.e.,
(A3) for each
(A4) for each
Lemma 2.2 ([16])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. For any
is closed and convex.
Lemma 2.3 ([17])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let
Lemma 2.4 ([18])
Assume that
Then the following hold:
(1)
(2)
(3)
(4)
LetHbe a real Hilbert space. Then the following identities hold:
(i)
(ii)
(iii)
Lemma 2.6 ([4])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH, and
Lemma 2.7 ([4])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceHand
Lemma 2.8 ([20])
LetCbe a nonempty closed and convex subset of a real Hilbert space H. Let
Then
Lemma 2.9 ([4])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let
for all
3 Main results
In this section, we prove a strong convergence theorem which solves the problem of finding a common solution of a system of equilibrium problems and a common solution of fixed point problems in Hilbert spaces.
Theorem 3.1LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let
Let
where
Proof The proof is split into seven steps.
Step 1. We will show that
From Lemma 2.4, we get
Therefore,
Step 2. We will show that
By the assumption of
Note that
is equivalent to
Taking
and
Combining (3.3) with (3.4), we obtain that
That is,
In view of the convexity of
Step 3. We will show that
Put
Next, we show that
We observe that
By virtue of convexity of
Substituting (3.5) and (3.6) into (3.7), we obtain
Therefore,
Step 4. We will show that
Since Ω is a nonempty closed and convex subset of H, there exists a unique
This implies that
Step 5. We will show that
Since
Therefore,
Thus, the sequence
The fact that
It is easy to see that
Since
It follows that
Since
For each
Thus, we get
This implies that for each
Therefore, by the convexity of
It follows that
From (3.15) and (3.17), we obtain
Then we have
Therefore,
From (3.13) and (3.19), we get
It follows that
Since for any positive integer
From the conditions
From (3.15) and (3.19), we obtain
It is obvious that the relations
Therefore, we compute
Applying Lemma 2.9 and (3.21), we get
From (3.22) and (3.24), it follows that
Since
for any
Moreover, for each
This implies that
Step 6. We will show that
(6.1) We will show that
We take
Note that
It follows from Lemma 2.7 that
(6.2) We will show that
By Lemma 2.3, for each
From (A2), we get
Taking
From (3.18), we obtain that
For any
Dividing by t, for each
Letting
Therefore,
Step 7. We will show that
Set
Since Ω is a nonempty closed and convex subset of H, there exists a unique
4 Deduced theorems
If we take
Theorem 4.1LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let
Let
where
Remark 4.2 Theorem 4.1 improves and extends the theorem of Tada and Takahashi [21] and the corollary of Duan and Zhao [7].
If we set
Theorem 4.3LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Let
Let
where
Remark 4.4 Theorem 4.1 improves and extends the theorem of Sahu, Xu, and Yao [4], the theorem of Qin, Cho, Kang, and Shang [3] and the corollary of Duan and Zhao [7].
5 Numerical examples
In this section, in order to demonstrate the effectiveness, realization and convergence of algorithm of Theorem 3.1, we consider the following simple example that was presented in reference [4].
Example 5.1 Let
where
It is easy to see that
Set
For each
For each
For each
It follows that
for all
Therefore, S is an asymptotically kstrict pseudocontractive mapping in the intermediate sense.
In Theorem 3.1, we set
Under the above assumption in Theorem 3.1 is simplified as follows:
In fact, in onedimensional case,
The numerical results for an initial guess
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in this research. All authors read and approved the final manuscript.
Acknowledgements
This research was supported by the Faculty of Science, KMUTT Research Fund 25532554.
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