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 strict pseudocontractive mappings in the intermediate sense. We prove the strong convergence theorem for a new iterative algorithm under some mild conditions in Hilbert spaces. Finally, we also give a numerical example which supports our results.
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 and the norm . Let be a family of bifunctions from into ℝ, where ℝ is the set of real numbers and Γ is an arbitrary index set. The system of equilibrium problems is to find such that
The set of solutions of (1.1) is denoted by , where , that is,
If Γ is a singleton, then the problem (1.1) is reduced to the equilibrium problem of finding such that
The set of solutions of (1.3) is denoted by .
Recall the following definitions.
A mapping is called monotone if
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 Lipschitz continuous.
In 2009, Qin et al.[3] introduced the following algorithm for a finite family of asymptotically strictly pseudocontractions.
Let and be a sequence in . The sequence is as follows:
It is called the explicit iterative sequence of a finite family of asymptotically strictly pseudocontractions . Since for each , it can be written as , where , is a positive integer and , as , we can rewrite the above table in the following compact form:
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 spaceHandbe a uniformly continuous asymptoticallyκstrictly pseudocontractive mapping in the intermediate sense with a sequencesuch thatis nonempty and bounded. Letbe a sequence insuch thatfor all. Letbe a sequence generated by the following (CQ) algorithm:
whereand. Thenconverges strongly to, whereis a metric projection fromHinto.
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 spaceHandbe an integer, be a bifunction satisfying (A1)(A4), andbe anαinversestrongly monotone mapping. Let for each, be a uniformly continuousstrictly asymptotically pseudocontractive mapping in the intermediate sense for somewith sequencessuch thatandsuch that. Let, , and. Assume thatis nonempty and bounded. Letandbe sequences insuch that, for all, and.
Letandbe sequences generated by the following algorithm:
where, as, and. Thenconverges strongly to.
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, be a uniformlyLLipschitz continuous mapping and asymptotically pseudocontractive mapping in the intermediate sense with sequencesand. Letfor each. Letbe the sequence generated by the following hybrid algorithm with variable coefficients:
where
Assume that the positive real numberis chosen so thatand thatandare sequences insuch thatfor someand for some.
Thenconverges strongly to a fixed point ofT.
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 strict pseudocontractive mappings in the intermediate sense in a real Hilbert space. Then, we prove a strong convergence theorem of the iterative algorithm generated by this conditions. Finally, we also give a numerical example which supports our results. The results obtained in this paper extend and improve several recent results in this area.
2 Preliminaries
Let H be a real Hilbert space with the inner product and the norm . Let C be a closed and convex subset of H. For any point , there exists a unique nearest point in C, denoted by , such that
is called the metric projection of H onto C defined by the following:
We know that is a nonexpansive mapping H onto C. It is also known that satisfies
and
We will adopt the following notations:
(1) → for strong convergence and ⇀ for weak convergence.
(2) denotes the weak wlimit set of .
(3) A nonlinear mapping S : is a selfmapping in C. We denote the set of fixed points of S by (i.e., ). Recall the following definitions.
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 such that
(3) S is said to be asymptotically nonexpansive if there exists a sequence with as such that
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 of fixed points of S is closed and convex.
(4) S is said to be asymptotically nonexpansive in the intermediate sense [10,11] if it is continuous and the following inequality holds:
Putting , we see that as . Then (2.4) is reduced to
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 such that
(6) S is said to be a λstrict pseudocontraction if there exists a coefficient such that
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 . We also remark that if , then S is called a pseudocontractive mapping.
(7) S is said to be an asymptoticallyλstrict pseudocontraction with the sequence (see also [13]) if there exists a sequence with as and a constant such that
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 . We also remark that if , then S is said to be an asymptotically pseudocontractive mapping which was introduced by Schu [15] in 1991.
(8) S is said to be an asymptoticallyλstrict pseudocontraction in the intermediate sense with the sequence [4,5] if there exists a sequence with as and a constant such that
Putting , we see that as . Then (2.8) is reduced to
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:
(A2) F is monotone, i.e., , for all ;
(A4) for each , is convex and lower semicontinuous.
Lemma 2.2 ([16])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. For anyand given also a real number, the set
is closed and convex.
Lemma 2.3 ([17])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letsatisfy (A1)(A4), and letand. Then there existssuch that
Lemma 2.4 ([18])
Assume thatsatisfies (A1)(A4). Forand, define a mappingas follows:
Then the following hold:
(2) is firmly nonexpansive, i.e., for any,
LetHbe a real Hilbert space. Then the following identities hold:
Lemma 2.6 ([4])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH, andbe a uniformlyLLipschitz continuous and asymptoticallyλstrict pseudocontraction in the intermediate sense. Thenis closed and convex.
Lemma 2.7 ([4])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceHandbe a uniformlyLLipschitz continuous and asymptoticallyλstrict pseudocontraction in the intermediate sense. Then the mappingis demiclosed at zero, that is, if the sequenceinCis such thatand, then.
Lemma 2.8 ([20])
LetCbe a nonempty closed and convex subset of a real Hilbert space H. Letbe a sequence inHand, and let. Suppose thatis such thatand satisfies the condition
Lemma 2.9 ([4])
LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe an asymptoticallyλstrict pseudocontractive mapping in the intermediate sense with the sequence. Then
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. Letbe a positive integer. Letbe a bifunction satisfying (A1)(A4). Letbe a uniformly Lipschitz continuous and asymptoticallystrict pseudocontractive mapping in the intermediate sense for somewith the sequencessuch thatandsuch that. Let, and. Assume thatis nonempty and bounded. Let, be sequences insuch that, , , andbe a sequence insuch that.
Letbe a sequence generated by the following algorithm:
where, asandand, where. Thenconverges strongly to some point, where.
Proof The proof is split into seven steps.
Step 1. We will show that is well defined.
From Lemma 2.4, we get is closed and convex. From the assumption of and Lemma 2.6, it follows that is closed and convex.
Therefore, is closed and convex. Hence, is well defined.
Step 2. We will show that is closed and convex for each .
By the assumption of , it is easy to see that is closed for each . We only show that is convex for each .
Note that is convex. Suppose that is convex for some . Next, we show that is convex for the same k. For each , we see that
is equivalent to
Taking and in and putting , it follows that , and so
and
Combining (3.3) with (3.4), we obtain that
That is,
In view of the convexity of , we see that . This implies that . Therefore, is convex. Hence, is closed and convex for each .
Step 3. We will show that for each .
Put for every and for all . Therefore, . It is obvious that . Suppose that for some .
Next, we show that for the same k. Taking and for each , we see that is nonexpansive and . We note that
We observe that
By virtue of convexity of , one has
Substituting (3.5) and (3.6) into (3.7), we obtain
Therefore, , and so for each . Hence, is well defined.
Step 4. We will show that is bounded.
Since Ω is a nonempty closed and convex subset of H, there exists a unique such that . By the assumption, we have for any . Then
This implies that is bounded. Therefore, , , and are also bounded.
Step 5. We will show that and as , .
Thus, the sequence is nondecreasing. Since is bounded, exists. On the other hand, from (3.10), we have
The fact that exists implies that
It is easy to see that
It follows that
Since as and from (3.13), we obtain
For each , it follows from the firmly nonexpansive that for each , we have
Thus, we get
Therefore, by the convexity of and (3.8) and the nonexpansivity of , we get
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 , we can write , where , note that
From the conditions and , we get
From (3.15) and (3.19), we obtain
It is obvious that the relations and hold.
Therefore, we compute
Applying Lemma 2.9 and (3.21), we get
From (3.22) and (3.24), it follows that
Since
Moreover, for each , we obtain
This implies that
We take and assume that for some subsequence of .
Note that is uniformly Lipschitz continuous and (3.27), we obtain
It follows from Lemma 2.7 that
By Lemma 2.3, for each , we have
From (A2), we get
From (3.18), we obtain that as for each (especially ). Considering this together with (3.18) and (A4), we have for each that
For any and , we let . Since and , we obtain that , and so . It follows that
Dividing by t, for each , we get
Step 7. We will show that converges strongly to .
Since Ω is a nonempty closed and convex subset of H, there exists a unique such that . It follows from Lemma 2.8 that , where . This completes proof. □
4 Deduced theorems
If we take in Theorem 3.1, then we obtain the following result.
Theorem 4.1LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a positive integer. Letbe a bifunction satisfying (A1)(A4). Letbe a uniformly Lipschitz continuous and asymptoticallystrict pseudocontractive mapping in the intermediate sense for somewith the sequencessuch thatandsuch that. Let, and. Assume thatis nonempty and bounded. Let, be sequences insuch that, , , , be a sequence insuch that.
Letbe a sequence generated by the following algorithm:
where, asandand, where. Thenconverges strongly to some point, 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 and for all in Theorem 3.1, then we obtain the following result.
Theorem 4.3LetCbe a nonempty closed and convex subset of a real Hilbert spaceH. Letbe a positive integer. Letbe a uniformly Lipschitz continuous and asymptoticallystrict pseudocontractive mapping in the intermediate sense for somewith the sequencessuch thatandsuch that. Let, and. Assume thatis nonempty and bounded. Let, be sequences insuch that, , , , be a sequence insuch that.
Letbe a sequence generated by the following algorithm:
where, asandand, where. Thenconverges strongly to some point, 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 and . For each , we define
It is easy to see that is discontinuous at and S is not Lipschitz continuous.
It follows that
Therefore, S is an asymptotically kstrict pseudocontractive mapping in the intermediate sense.
In Theorem 3.1, we set , , , . We apply it to find the fixed point of S of Example 5.1.
Under the above assumption in Theorem 3.1 is simplified as follows:
In fact, in onedimensional case, is a closed interval. If we set , then the projection point of onto can be expressed as
The numerical results for an initial guess are shown in Table 1. From the table, we see that the iterations converge to 0 which is the unique fixed point of S. The convergence of each iteration is also shown in Figure 1 for comparison.
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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